{"category": "Math", "title": "A determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata", "abstract": "We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the determinant."}
{"category": "Math", "title": "From dyadic $\\Lambda_{\\alpha}$ to $\\Lambda_{\\alpha}$", "abstract": "In this paper we show how to compute the $\\Lambda_{\\alpha}$ norm, $\\alpha\\ge 0$, using the dyadic grid. This result is a consequence of the description of the Hardy spaces $H^p(R^N)$ in terms of dyadic and special atoms."}
{"category": "Math", "title": "Partial cubes: structures, characterizations, and constructions", "abstract": "Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djokovi\\'{c}'s and Winkler's relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given. The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated."}
{"category": "Math", "title": "Computing genus 2 Hilbert-Siegel modular forms over $\\Q(\\sqrt{5})$ via the Jacquet-Langlands correspondence", "abstract": "In this paper we present an algorithm for computing Hecke eigensystems of Hilbert-Siegel cusp forms over real quadratic fields of narrow class number one. We give some illustrative examples using the quadratic field $\\Q(\\sqrt{5})$. In those examples, we identify Hilbert-Siegel eigenforms that are possible lifts from Hilbert eigenforms."}
{"category": "Math", "title": "Distribution of integral Fourier Coefficients of a Modular Form of Half Integral Weight Modulo Primes", "abstract": "Recently, Bruinier and Ono classified cusp forms $f(z) := \\sum_{n=0}^{\\infty} a_f(n)q ^n \\in S_{\\lambda+1/2}(\\Gamma_0(N),\\chi)\\cap \\mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes $p \\geq 5$. As applications of our main theorem we derive distribution properties, for modulo primes $p\\geq5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newman's conjecture for overpartitions."}
{"category": "Math", "title": "$p$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral Weight", "abstract": "Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(\\mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on $\\Gamma_{0}(4N)$ for $N=1,2,4$. A proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications we obtain congruences of Borcherds exponents, congruences of quotient of Eisentein series and congruences of values of $L$-functions at a certain point are also studied. Furthermore, the congruences of the Fourier coefficients of Siegel modular forms on Maass Space are obtained using Ikeda lifting."}
{"category": "Math", "title": "Iterated integral and the loop product", "abstract": "In this article we discuss a relation between the string topology and differential forms based on the theory of Chen's iterated integrals and the cyclic bar complex."}
{"category": "Math", "title": "Approximation for extinction probability of the contact process based on the Gr\\\"obner basis", "abstract": "In this note we give a new method for getting a series of approximations for the extinction probability of the one-dimensional contact process by using the Gr\\\"obner basis."}
{"category": "Math", "title": "Stochastic Lie group integrators", "abstract": "We present Lie group integrators for nonlinear stochastic differential equations with non-commutative vector fields whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding Lie algebra via the exponential map. We construct an approximation to the stochastic flow in the Lie algebra via closed operations and then push back to the Lie group and then to the manifold, thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell--Gaines methods. These involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential Lie series. They become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying ordinary differential integrator. Further, we show that some Castell--Gaines methods are uniformly more accurate than the corresponding stochastic Taylor schemes. Lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes."}
{"category": "Math", "title": "Placeholder Substructures II: Meta-Fractals, Made of Box-Kites, Fill Infinite-Dimensional Skies", "abstract": "Zero-divisors (ZDs) derived by Cayley-Dickson Process (CDP) from N-dimensional hypercomplex numbers (N a power of 2, at least 4) can represent singularities and, as N approaches infinite, fractals -- and thereby,scale-free networks. Any integer greater than 8 and not a power of 2 generates a meta-fractal or \"Sky\" when it is interpreted as the \"strut constant\" (S) of an ensemble of octahedral vertex figures called \"Box-Kites\" (the fundamental building blocks of ZDs). Remarkably simple bit-manipulation rules or \"recipes\" provide tools for transforming one fractal genus into others within the context of Wolfram's Class 4 complexity."}
{"category": "Math", "title": "Pfaffians, hafnians and products of real linear functionals", "abstract": "We prove pfaffian and hafnian versions of Lieb's inequalities on determinants and permanents of positive semi-definite matrices. We use the hafnian inequality to improve the lower bound of R\\'ev\\'esz and Sarantopoulos on the norm of a product of linear functionals on a real Euclidean space (this subject is sometimes called the `real linear polarization constant' problem)."}
{"category": "Math", "title": "Multilinear function series in conditionally free probability with amalgamation", "abstract": "As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity results, a version of the central limit theorem and an analogue of the conditionally free R-transform constructed by means of multilinear function series."}
{"category": "Math", "title": "An algorithm for the classification of smooth Fano polytopes", "abstract": "We present an algorithm that produces the classification list of smooth Fano d-polytopes for any given d. The input of the algorithm is a single number, namely the positive integer d. The algorithm has been used to classify smooth Fano d-polytopes for d<=7. There are 7622 isomorphism classes of smooth Fano 6-polytopes and 72256 isomorphism classes of smooth Fano 7-polytopes."}
{"category": "Math", "title": "The Hardy-Lorentz Spaces $H^{p,q}(R^n)$", "abstract": "In this paper we consider the Hardy-Lorentz spaces $H^{p,q}(R^n)$, with $0<p\\le 1$, $0<q\\le \\infty$. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them."}
{"category": "Math", "title": "Intersection Bodies and Generalized Cosine Transforms", "abstract": "Intersection bodies represent a remarkable class of geometric objects associated with sections of star bodies and invoking Radon transforms, generalized cosine transforms, and the relevant Fourier analysis. The main focus of this article is interrelation between generalized cosine transforms of different kinds in the context of their application to investigation of a certain family of intersection bodies, which we call $\\lam$-intersection bodies. The latter include $k$-intersection bodies (in the sense of A. Koldobsky) and unit balls of finite-dimensional subspaces of $L_p$-spaces. In particular, we show that restrictions onto lower dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms preserve their integral-geometric structure. We apply this result to the study of sections of $\\lam$-intersection bodies. New characterizations of this class of bodies are obtained and examples are given. We also review some known facts and give them new proofs."}
{"category": "Math", "title": "Littlewood-Richardson polynomials", "abstract": "We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are polynomials in the parameters which we call the Littlewood-Richardson polynomials. We give a combinatorial rule for their calculation by modifying an earlier result of B. Sagan and the author. The new rule provides a formula for these polynomials which is manifestly positive in the sense of W. Graham. We apply this formula for the calculation of the product of equivariant Schubert classes on Grassmannians which implies a stability property of the structure coefficients. The first manifestly positive formula for such an expansion was given by A. Knutson and T. Tao by using combinatorics of puzzles while the stability property was not apparent from that formula. We also use the Littlewood-Richardson polynomials to describe the multiplication rule in the algebra of the Casimir elements for the general linear Lie algebra in the basis of the quantum immanants constructed by A. Okounkov and G. Olshanski."}
{"category": "Math", "title": "A Note About the {Ki(z)} Functions", "abstract": "In the article [Petojevic 2006], A. Petojevi\\' c verified useful properties of the $K_{i}(z)$ functions which generalize Kurepa's [Kurepa 1971] left factorial function. In this note, we present simplified proofs of two of these results and we answer the open question stated in [Petojevic 2006]. Finally, we discuss the differential transcendency of the $K_{i}(z)$ functions."}
{"category": "Math", "title": "Dynamical Objects for Cohomologically Expanding Maps", "abstract": "The goal of this paper is to construct invariant dynamical objects for a (not necessarily invertible) smooth self map of a compact manifold. We prove a result that takes advantage of differences in rates of expansion in the terms of a sheaf cohomological long exact sequence to create unique lifts of finite dimensional invariant subspaces of one term of the sequence to invariant subspaces of the preceding term. This allows us to take invariant cohomological classes and under the right circumstances construct unique currents of a given type, including unique measures of a given type, that represent those classes and are invariant under pullback. A dynamically interesting self map may have a plethora of invariant measures, so the uniquess of the constructed currents is important. It means that if local growth is not too big compared to the growth rate of the cohomological class then the expanding cohomological class gives sufficient \"marching orders\" to the system to prohibit the formation of any other such invariant current of the same type (say from some local dynamical subsystem). Because we use subsheaves of the sheaf of currents we give conditions under which a subsheaf will have the same cohomology as the sheaf containing it. Using a smoothing argument this allows us to show that the sheaf cohomology of the currents under consideration can be canonically identified with the deRham cohomology groups. Our main theorem can be applied in both the smooth and holomorphic setting."}
{"category": "Math", "title": "A transcendental approach to Koll\\'ar's injectivity theorem", "abstract": "We treat Koll\\'ar's injectivity theorem from the analytic (or differential geometric) viewpoint. More precisely, we give a curvature condition which implies Koll\\'ar type cohomology injectivity theorems. Our main theorem is formulated for a compact K\\\"ahler manifold, but the proof uses the space of harmonic forms on a Zariski open set with a suitable complete K\\\"ahler metric. We need neither covering tricks, desingularizations, nor Leray's spectral sequence."}
{"category": "Math", "title": "Injective Morita contexts (revisited)", "abstract": "This paper is an exposition of the so-called injective Morita contexts (in which the connecting bimodule morphisms are injective) and Morita $\\alpha$contexts (in which the connecting bimodules enjoy some local projectivity in the sense of Zimmermann-Huisgen). Motivated by situations in which only one trace ideal is in action, or the compatibility between the bimodule morphisms is not needed, we introduce the notions of Morita semi-contexts and Morita data, and investigate them. Injective Morita data will be used (with the help of static and adstatic modules) to establish equivalences between some intersecting subcategories related to subcategories of modules that are localized or colocalized by trace ideals of a Morita datum. We end up with applications of Morita $\\alpha$-contexts to $\\ast$-modules and injective right wide Morita contexts."}
{"category": "Math", "title": "Operator algebras associated with unitary commutation relations", "abstract": "We define nonselfadjoint operator algebras with generators $L_{e_1},..., L_{e_n}, L_{f_1},...,L_{f_m}$ subject to the unitary commutation relations of the form \\[ L_{e_i}L_{f_j} = \\sum_{k,l} u_{i,j,k,l} L_{f_l}L_{e_k}\\] where $u= (u_{i,j,k,l})$ is an $nm \\times nm$ unitary matrix. These algebras, which generalise the analytic Toeplitz algebras of rank 2 graphs with a single vertex, are classified up to isometric isomorphism in terms of the matrix $u$."}
{"category": "Math", "title": "Clustering in a stochastic model of one-dimensional gas", "abstract": "We give a quantitative analysis of clustering in a stochastic model of one-dimensional gas. At time zero, the gas consists of $n$ identical particles that are randomly distributed on the real line and have zero initial speeds. Particles begin to move under the forces of mutual attraction. When particles collide, they stick together forming a new particle, called cluster, whose mass and speed are defined by the laws of conservation. We are interested in the asymptotic behavior of $K_n(t)$ as $n\\to \\infty$, where $K_n(t)$ denotes the number of clusters at time $t$ in the system with $n$ initial particles. Our main result is a functional limit theorem for $K_n(t)$. Its proof is based on the discovered localization property of the aggregation process, which states that the behavior of each particle is essentially defined by the motion of neighbor particles."}
{"category": "Math", "title": "Approximate solutions to the Dirichlet problem for harmonic maps between hyperbolic spaces", "abstract": "Our main result in this paper is the following: Given $H^m, H^n$ hyperbolic spaces of dimensional $m$ and $n$ corresponding, and given a Holder function $f=(s^1,...,f^{n-1}):\\partial H^m\\to \\partial H^n$ between geometric boundaries of $H^m$ and $H^n$. Then for each $\\epsilon >0$ there exists a harmonic map $u:H^m\\to H^n$ which is continuous up to the boundary (in the sense of Euclidean) and $u|_{\\partial H^m}=(f^1,...,f^{n-1},\\epsilon)$."}
{"category": "Math", "title": "Groups with finitely many conjugacy classes and their automorphisms", "abstract": "We combine classical methods of combinatorial group theory with the theory of small cancellations over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate elements. Moreover, we present several results concerning embeddings into such groups. As another application of these techniques, we prove that every countable group $C$ can be realized as a group of outer automorphisms of a group $N$, where $N$ is a finitely generated group having Kazhdan's property (T) and containing exactly two conjugacy classes."}
{"category": "Math", "title": "Geometry of Locally Compact Groups of Polynomial Growth and Shape of Large Balls", "abstract": "We get asymptotics for the volume of large balls in an arbitrary locally compact group G with polynomial growth. This is done via a study of the geometry of G and a generalization of P. Pansu's thesis. In particular, we show that any such G is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We also show that large balls in G have an asymptotic shape, i.e. after a suitable renormalization, they converge to a limiting compact set which can be interpreted geometrically. We then discuss the speed of convergence, treat some examples and give an application to ergodic theory. We also answer a question of Burago about left invariant metrics and recover some results of Stoll on the irrationality of growth series of nilpotent groups."}
{"category": "Math", "title": "On Ando's inequalities for convex and concave functions", "abstract": "For positive semidefinite matrices $A$ and $B$, Ando and Zhan proved the inequalities $||| f(A)+f(B) ||| \\ge ||| f(A+B) |||$ and $||| g(A)+g(B) ||| \\le ||| g(A+B) |||$, for any unitarily invariant norm, and for any non-negative operator monotone $f$ on $[0,\\infty)$ with inverse function $g$. These inequalities have very recently been generalised to non-negative concave functions $f$ and non-negative convex functions $g$, by Bourin and Uchiyama, and Kosem, respectively. In this paper we consider the related question whether the inequalities $||| f(A)-f(B) ||| \\le ||| f(|A-B|) |||$, and $||| g(A)-g(B) ||| \\ge ||| g(|A-B|) |||$, obtained by Ando, for operator monotone $f$ with inverse $g$, also have a similar generalisation to non-negative concave $f$ and convex $g$. We answer exactly this question, in the negative for general matrices, and affirmatively in the special case when $A\\ge ||B||$. In the course of this work, we introduce the novel notion of $Y$-dominated majorisation between the spectra of two Hermitian matrices, where $Y$ is itself a Hermitian matrix, and prove a certain property of this relation that allows to strengthen the results of Bourin-Uchiyama and Kosem, mentioned above."}
{"category": "Math", "title": "Duality and Tameness", "abstract": "We prove a duality theorem for certain graded algebras and show by various examples different kinds of failure of tameness of local cohomology."}
{"category": "Math", "title": "A geometric realization of sl(6,C)", "abstract": "Given an orientable weakly self-dual manifold X of rank two, we build a geometric realization of the Lie algebra sl(6,C) as a naturally defined algebra L of endomorphisms of the space of differential forms of X. We provide an explicit description of Serre generators in terms of natural generators of L. This construction gives a bundle on X which is related to the search for a natural Gauge theory on X. We consider this paper as a first step in the study of a rich and interesting algebraic structure."}
{"category": "Math", "title": "Rigid subsets of symplectic manifolds", "abstract": "We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P.Albers and P.Biran-O.Cornea), as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states."}
{"category": "Math", "title": "On Equivariant Embedding of Hilbert C^* modules", "abstract": "We prove that an arbitrary (not necessarily countably generated) Hilbert $G$-$\\cla$ module on a G-C^* algebra $\\cla$ admits an equivariant embedding into a trivial $G-\\cla$ module, provided G is a compact Lie group and its action on $\\cla$ is ergodic."}
{"category": "Math", "title": "Invariance and the twisted Chern character : a case study", "abstract": "We give details of the proof of the remark made in \\cite{G2} that the Chern characters of the canonical generators on the K homology of the quantum group $SU_q(2)$ are not invariant under the natural $SU_q(2)$ coaction. Furthermore, the conjecture made in \\cite{G2} about the nontriviality of the twisted Chern character coming from an odd equivariant spectral triple on $SU_q(2)$ is settled in the affirmative."}
{"category": "Math", "title": "Placeholder Substructures III: A Bit-String-Driven ''Recipe Theory'' for Infinite-Dimensional Zero-Divisor Spaces", "abstract": "Zero-divisors (ZDs) derived by Cayley-Dickson Process (CDP) from N-dimensional hypercomplex numbers (N a power of 2, at least 4) can represent singularities and, as N approaches infinite, fractals -- and thereby,scale-free networks. Any integer greater than 8 and not a power of 2 generates a meta-fractal or \"Sky\" when it is interpreted as the \"strut constant\" (S) of an ensemble of octahedral vertex figures called \"Box-Kites\" (the fundamental building blocks of ZDs). Remarkably simple bit-manipulation rules or \"recipes\" provide tools for transforming one fractal genus into others within the context of Wolfram's Class 4 complexity."}
{"category": "Math", "title": "Smooth maps with singularities of bounded K-codimensions", "abstract": "We will prove the relative homotopy principle for smooth maps with singularities of a given {\\cal K}-invariant class with a mild condition. We next study a filtration of the group of homotopy self-equivalences of a given manifold P by considering singularities of non-negative {\\cal K}-codimensions."}
{"category": "Math", "title": "Proper J-holomorphic discs in Stein domains of dimension 2", "abstract": "We prove the existence of global Bishop discs in a strictly pseudoconvex Stein domain in an almost complex manifold of complex dimension 2."}
{"category": "Math", "title": "Anisotropic thermo-elasticity in 2D -- Part I: A unified approach", "abstract": "In this note we develop tools and techniques for the treatment of anisotropic thermo-elasticity in two space dimensions. We use a diagonalisation technique to obtain properties of the characteristic roots of the full symbol of the system in order to prove $L^p$--$L^q$ decay rates for its solutions."}
{"category": "Math", "title": "New simple modular Lie superalgebras as generalized prolongs", "abstract": "Over algebraically closed fields of characteristic p>2, prolongations of the simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. Several new simple Lie superalgebras are discovered, serial and exceptional, including superBrown and superMelikyan superalgebras. Simple Lie superalgebras with Cartan matrix of rank 2 are classified."}
{"category": "Math", "title": "On smooth foliations with Morse singularities", "abstract": "Let $M$ be a smooth manifold and let $\\F$ be a codimension one, $C^\\infty$ foliation on $M$, with isolated singularities of Morse type. The study and classification of pairs $(M,\\F)$ is a challenging (and difficult) problem. In this setting, a classical result due to Reeb \\cite{Reeb} states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper \\cite{Ku-Ee} classify manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices). In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, $Sing(\\F)$ of the foliation $\\F$, we consider {\\em{weakly stable}} components, that we define as those components admitting a neighborhood where all leaves are compact. If $Sing(\\F)$ admits only weakly stable components, given by smoothly embedded curves diffeomorphic to $S^1$, we are able to extend Haefliger's theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result."}
{"category": "Math", "title": "Frobenius-Schur indicators for semisimple Lie algebras", "abstract": "Let g be a finite dimensional complex semisimple Lie algebra, and let V be a finite dimensional represenation of g. We give a closed formula for the mth Frobenius-Schur indicator, m>1, of V in representation-theoretic terms. We deduce that the indicators take integer values, and that for a large enough m, the mth indicator of V equals the dimension of the zero weight space of V. For the classical Lie algebras sl(n), so(2n), so(2n+1) and sp(2n), this is the case for m greater or equal to 2n-1, 4n-5, 4n-3 and 2n+1, respectively."}
{"category": "Math", "title": "Monoid generalizations of the Richard Thompson groups", "abstract": "The groups G_{k,1} of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call M_{k,1}, and to inverse monoids, called Inv_{k,1}; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids M_{k,1} have connections with circuit complexity (studied in another paper). Here we prove that M_{k,1} and Inv_{k,1} are congruence-simple for all k. Their Green relations J and D are characterized: M_{k,1} and Inv_{k,1} are J-0-simple, and they have k-1 non-zero D-classes. They are submonoids of the multiplicative part of the Cuntz algebra O_k. They are finitely generated, and their word problem over any finite generating set is in P. Their word problem is coNP-complete over certain infinite generating sets. Changes in this version: Section 4 has been thoroughly revised, and errors have been corrected; however, the main results of Section 4 do not change. Sections 1, 2, and 3 are unchanged, except for the proof of Theorem 2.3, which was incomplete; a complete proof was published in the Appendix of reference [6], and is also given here."}
{"category": "Math", "title": "Group-theoretical properties of nilpotent modular categories", "abstract": "We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects of C have integral Frobenius-Perron dimensions then C is group-theoretical. As a consequence, we obtain that semisimple quasi-Hopf algebras of prime power dimension are group-theoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasi-Lie bialgebras in terms of Manin pairs)."}
{"category": "Math", "title": "Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions", "abstract": "Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,...,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=\\si_1\\si_2 cdots\\si_d$ of a Coxeter element $c$ of $W$, such that $\\si_i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,...,d$, and such that the factorisation is \"minimal\" in the sense that the sum of the ranks of the $T_i$'s, $i=1,2,...,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author. The type $A_n$ decomposition numbers have been computed by Goulden and Jackson, albeit using a somewhat different language. We explain how to extract the type $B_n$ decomposition numbers from results of B\\'ona, Bousquet, Labelle and Leroux on map enumeration. Our formula for the type $D_n$ decomposition numbers is new. These results are then used to determine, for a fixed positive integer $l$ and fixed integers $r_1\\le r_2\\le ...\\le r_l$, the number of multi-chains $\\pi_1\\le \\pi_2\\le ...\\le \\pi_l$ in Armstrong's generalised non-crossing partitions poset, where the poset rank of $\\pi_i$ equals $r_i$, and where the \"block structure\" of $\\pi_1$ is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type $D_n$ generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong's $F=M$ Conjecture in type $D_n$."}
{"category": "Math", "title": "Hecke-Clifford algebras and spin Hecke algebras I: the classical affine type", "abstract": "Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the centers. We further develop connections of these algebras with the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras."}
{"category": "Math", "title": "Some non-braided fusion categories of rank 3", "abstract": "We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary re-proof of the fact that the quadruple dual functor is naturally isomorphic to the identity."}
{"category": "Math", "title": "Classification of superpotentials", "abstract": "We extend our previous classification of superpotentials of ``scalar curvature type\" for the cohomogeneity one Ricci-flat equations. We now consider the case not covered in our previous paper, i.e., when some weight vector of the superpotential lies outside (a scaled translate of) the convex hull of the weight vectors associated with the scalar curvature function of the principal orbit. In this situation we show that either the isotropy representation has at most 3 irreducible summands or the first order subsystem associated to the superpotential is of the same form as the Calabi-Yau condition for submersion type metrics on complex line bundles over a Fano K\\\"ahler-Einstein product."}
{"category": "Math", "title": "Linkedness and ordered cycles in digraphs", "abstract": "The minimum semi-degree of a digraph D is the minimum of its minimum outdegree and its minimum indegree. We show that every sufficiently large digraph D with minimum semi-degree at least n/2 +k-1 is k-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis from 1990. We also determine the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every ordered sequence of k distinct vertices of D there is a directed cycle which encounters these vertices in this order."}
{"category": "Math", "title": "The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts", "abstract": "In this note we consider the time of the collision $\\tau$ for $n$ independent Brownian motions $X^1_t,...,X_t^n$ with drifts $a_1,...,a_n$, each starting from $x=(x_1,...,x_n)$, where $x_1<...<x_n$. We show the exact asymptotics of $P_x(\\tau>t) = C h(x)t^{-\\alpha}e^{-\\gamma t}(1 + o(1))$ as $t\\to\\infty$ and identify $C,h(x),\\alpha,\\gamma$ in terms of the drifts."}
{"category": "Math", "title": "Interpolating and sampling sequences in finite Riemann surfaces", "abstract": "We provide a description of the interpolating and sampling sequences on a space of holomorphic functions with a uniform growth restriction defined on finite Riemann surfaces."}
{"category": "Math", "title": "Curvature flows in semi-Riemannian manifolds", "abstract": "We prove that the limit hypersurfaces of converging curvature flows are stable, if the initial velocity has a weak sign, and give a survey of the existence and regularity results."}
{"category": "Math", "title": "Orbifold cohomology of abelian symplectic reductions and the case of weighted projective spaces", "abstract": "These notes accompany a lecture about the topology of symplectic (and other) quotients. The aim is two-fold: first to advertise the ease of computation in the symplectic category; and second to give an account of some new computations for weighted projective spaces. We start with a brief exposition of how orbifolds arise in the symplectic category, and discuss the techniques used to understand their topology. We then show how these results can be used to compute the Chen-Ruan orbifold cohomology ring of abelian symplectic reductions. We conclude by comparing the several rings associated to a weighted projective space. We make these computations directly, avoiding any mention of a stacky fan or of a labeled moment polytope."}
{"category": "Math", "title": "On Existence of Boundary Values of Polyharmonic Functions", "abstract": "In trigonometric series terms all polyharmonic functions inside the unit disk are described. For such functions it is proved the existence of their boundary values on the unit circle in the space of hyperfunctions. The necessary and sufficient conditions are presented for the boundary value to belong to certain subspaces of the space of hyperfunctions."}
{"category": "Math", "title": "Mapping radii of metric spaces", "abstract": "It is known that every closed curve of length \\leq 4 in R^n (n>0) can be surrounded by a sphere of radius 1, and that this is the best bound. Letting S denote the circle of circumference 4, with the arc-length metric, we here express this fact by saying that the \"mapping radius\" of S in R^n is 1. Tools are developed for estimating the mapping radius of a metric space X in a metric space Y. In particular, it is shown that for X a bounded metric space, the supremum of the mapping radii of X in all convex subsets of normed metric spaces is equal to the infimum of the sup norms of all convex linear combinations of the functions d(x,-): X --> R (x\\in X). Several explicit mapping radii are calculated, and open questions noted."}
{"category": "Math", "title": "Leray numbers of projections and a topological Helly type theorem", "abstract": "Let X be a simplicial complex on the vertex set V. The rational Leray number L(X) of X is the minimal d such that the rational reduced homology of any induced subcomplex of X vanishes in dimensions d and above. Let \\pi be a simplicial map from X to a simplex Y, such that the cardinality of the preimage of any point in |Y| is at most r. It is shown that L(\\pi(X)) \\leq r L(X)+r-1. One consequence is a topological extension of a Helly type result of Amenta."}
{"category": "Math", "title": "On the Markov trace for Temperley--Lieb algebras of type $E_n$", "abstract": "We show that there is a unique Markov trace on the tower of Temperley--Lieb type quotients of Hecke algebras of Coxeter type $E_n$ (for all $n \\geq 6$). We explain in detail how this trace may be computed easily using tom Dieck's calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan--Lusztig polynomials."}
{"category": "Math", "title": "On the number of topological types occurring in a parametrized family of arrangements", "abstract": "Let ${\\mathcal S}(\\R)$ be an o-minimal structure over $\\R$, $T \\subset \\R^{k_1+k_2+\\ell}$ a closed definable set, and $$ \\displaylines{\\pi_1: \\R^{k_1+k_2+\\ell}\\to \\R^{k_1 + k_2}, \\pi_2: \\R^{k_1+k_2+\\ell}\\to \\R^{\\ell}, \\ \\pi_3: \\R^{k_1 + k_2} \\to \\R^{k_2}} $$ the projection maps. For any collection ${\\mathcal A} = \\{A_1,...,A_n\\}$ of subsets of $\\R^{k_1+k_2}$, and $\\z \\in \\R^{k_2}$, let $\\A_\\z$ denote the collection of subsets of $\\R^{k_1}$, $\\{A_{1,\\z},..., A_{n,\\z}\\}$, where $A_{i,\\z} = A_i \\cap \\pi_3^{-1}(\\z), 1 \\leq i \\leq n$. We prove that there exists a constant $C = C(T) > 0,$ such that for any family ${\\mathcal A} = \\{A_1,...,A_n\\}$ of definable sets, where each $A_i = \\pi_1(T \\cap \\pi_2^{-1}(\\y_i))$, for some $\\y_i \\in \\R^{\\ell}$, the number of distinct stable homotopy types of $\\A_\\z, \\z \\in \\R^{k_2}$, is bounded by $ \\displaystyle{C \\cdot n^{(k_1+1)k_2},} $ while the number of distinct homotopy types is bounded by $ \\displaystyle{C \\cdot n^{(k_1+3)k_2}.} $ This generalizes to the general o-minimal setting, bounds of the same type proved in \\cite{BV} for semi-algebraic and semi-Pfaffian families. One main technical tool used in the proof of the above results, is a topological comparison theorem which might be of independent interest in the study of arrangements."}
{"category": "Math", "title": "Direct Theorems in the Theory of Approximation of the Banach Space Vectors by Entire Vectors of Exponential Type", "abstract": "For an arbitrary operator A on a Banach space X which is a generator of C_0-group with certain growth condition at the infinity, the direct theorems on connection between the smoothness degree of a vector $x\\in X$ with respect to the operator A, the order of convergence to zero of the best approximation of x by exponential type entire vectors for the operator A, and the k-module of continuity are given. Obtained results allows to acquire Jackson-type inequalities in many classic spaces of periodic functions and weighted $L_p$ spaces."}
{"category": "Math", "title": "Spline Single-Index Prediction Model", "abstract": "For the past two decades, single-index model, a special case of projection pursuit regression, has proven to be an efficient way of coping with the high dimensional problem in nonparametric regression. In this paper, based on weakly dependent sample, we investigate the single-index prediction (SIP) model which is robust against deviation from the single-index model. The single-index is identified by the best approximation to the multivariate prediction function of the response variable, regardless of whether the prediction function is a genuine single-index function. A polynomial spline estimator is proposed for the single-index prediction coefficients, and is shown to be root-n consistent and asymptotically normal. An iterative optimization routine is used which is sufficiently fast for the user to analyze large data of high dimension within seconds. Simulation experiments have provided strong evidence that corroborates with the asymptotic theory. Application of the proposed procedure to the rive flow data of Iceland has yielded superior out-of-sample rolling forecasts."}
{"category": "Math", "title": "The small deviations of many-dimensional diffusion processes and rarefaction by boundaries", "abstract": "We lead the algorithm of expansion of sojourn probability of many-dimensional diffusion processes in small domain. The principal member of this expansion defines normalizing coefficient for special limit theorems."}
{"category": "Math", "title": "Complete Shrinking Ricci Solitons have Finite Fundamental Group", "abstract": "We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of Garcia-Rio and Fernandez-Lopez in the compact case."}
{"category": "Math", "title": "On the pseudospectrum of elliptic quadratic differential operators", "abstract": "We study the pseudospectrum of a class of non-selfadjoint differential operators. Our work consists in a detailed study of the microlocal properties, which rule the spectral stability or instability phenomena appearing under small perturbations for elliptic quadratic differential operators. The class of elliptic quadratic differential operators stands for the class of operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this paper a simple necessary and sufficient condition on the Weyl symbol of these operators, which ensures the stability of their spectra. When this condition is violated, we prove that it occurs some strong spectral instabilities for the high energies of these operators, in some regions which can be far away from their spectra. We give a precise geometrical description of them, which explains the results obtained for these operators in some numerical simulations giving the computation of false eigenvalues far from their spectra by algorithms for eigenvalues computing."}
{"category": "Math", "title": "Solutions of fractional reaction-diffusion equations in terms of the H-function", "abstract": "This paper deals with the investigation of the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and Riesz-Feller fractional derivative as the space-derivative. The solution is derived by the application of the Laplace and Fourier transforms in closed form in terms of the H-function. The results derived are of general nature and include the results investigated earlier by many authors, notably by Mainardi et al. (2001, 2005) for the fundamental solution of the space-time fractional diffusion equation, and Saxena et al. (2006a, b) for fractional reaction- diffusion equations. The advantage of using Riesz-Feller derivative lies in the fact that the solution of the fractional reaction-diffusion equation containing this derivative includes the fundamental solution for space-time fractional diffusion, which itself is a generalization of neutral fractional diffusion, space-fractional diffusion, and time-fractional diffusion. These specialized types of diffusion can be interpreted as spatial probability density functions evolving in time and are expressible in terms of the H-functions in compact form."}
{"category": "Math", "title": "Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains", "abstract": "A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the Euler equations of incompressible fluids. The fast singular oscillating limits of the Euler equations are studied for parametrically resonant cylinders. Resonances of fast swirling Beltrami waves deplete the Euler nonlinearity. The resonant Euler equations are systems of three-dimensional rigid body equations, coupled or not. Some cases of these resonant systems have homoclinic cycles, and orbits in the vicinity of these homoclinic cycles lead to bursts of the Euler solution measured in Sobolev norms of order higher than that corresponding to the enstrophy."}
{"category": "Math", "title": "Cofibrations in the Category of Frolicher Spaces. Part I", "abstract": "Cofibrations are defined in the category of Fr\\\"olicher spaces by weakening the analog of the classical definition to enable smooth homotopy extensions to be more easily constructed, using flattened unit intervals. We later relate smooth cofibrations to smooth neighborhood deformation retracts. The notion of smooth neighborhood deformation retract gives rise to an analogous result that a closed Fr\\\"olicher subspace $A$ of the Fr\\\"olicher space $X$ is a smooth neighborhood deformation retract of $X$ if and only if the inclusion $i: A\\hookrightarrow X$ comes from a certain subclass of cofibrations. As an application we construct the right Puppe sequence."}
{"category": "Math", "title": "Resolvent estimates related with a class of dispersive equations", "abstract": "We present a simple proof of the resolvent estimates of elliptic Fourier multipliers on the Euclidean space, and apply them to the analysis of time-global and spatially-local smoothing estimates of a class of dispersive equations. For this purpose we study in detail the properties of the restriction of Fourier transform on the unit cotangent sphere associated with the symbols of multipliers."}
{"category": "Math", "title": "The Colin de Verdi\\`ere number and graphs of polytopes", "abstract": "The Colin de Verdi\\`ere number $\\mu(G)$ of a graph $G$ is the maximum corank of a Colin de Verdi\\`ere matrix for $G$ (that is, of a Schr\\\"odinger operator on $G$ with a single negative eigenvalue). In 2001, Lov\\'asz gave a construction that associated to every convex 3-polytope a Colin de Verdi\\`ere matrix of corank 3 for its 1-skeleton. We generalize the Lov\\'asz construction to higher dimensions by interpreting it as minus the Hessian matrix of the volume of the polar dual. As a corollary, $\\mu(G) \\ge d$ if $G$ is the 1-skeleton of a convex $d$-polytope. Determination of the signature of the Hessian of the volume is based on the second Minkowski inequality for mixed volumes and on Bol's condition for equality."}
{"category": "Math", "title": "Some properties of the complex Monge-Ampere operator in Cegrell's classes and applications", "abstract": "In this article we will first prove a result about convergence in capacity. Using the achieved result we will obtain a general decompositon theorem for complex Monge-Ampere measues which will be used to prove a comparison principle for the complex Monge-Ampere operator."}
{"category": "Math", "title": "An equilibrium problem for the limiting eigenvalue distribution of banded Toeplitz matrices", "abstract": "We study the limiting eigenvalue distribution of $n\\times n$ banded Toeplitz matrices as $n\\to \\infty$. From classical results of Schmidt-Spitzer and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as $n\\to\\infty$. In this paper, we characterize the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimes an energy functional defined on admissible vectors of measures. In addition, we show that each of the other components is the limiting measure of the normalized counting measure on certain generalized eigenvalues."}
{"category": "Math", "title": "Exponential growth rates in a typed branching diffusion", "abstract": "We study the high temperature phase of a family of typed branching diffusions initially studied in [Ast\\'{e}risque 236 (1996) 133--154] and [Lecture Notes in Math. 1729 (2000) 239--256 Springer, Berlin]. The primary aim is to establish some almost-sure limit results for the long-term behavior of this particle system, namely the speed at which the population of particles colonizes both space and type dimensions, as well as the rate at which the population grows within this asymptotic shape. Our approach will include identification of an explicit two-phase mechanism by which particles can build up in sufficient numbers with spatial positions near $-\\gamma t$ and type positions near $\\kappa \\sqrt{t}$ at large times $t$. The proofs involve the application of a variety of martingale techniques--most importantly a ``spine'' construction involving a change of measure with an additive martingale. In addition to the model's intrinsic interest, the methodologies presented contain ideas that will adapt to other branching settings. We also briefly discuss applications to traveling wave solutions of an associated reaction--diffusion equation."}
{"category": "Math", "title": "On Some Subgroup Chains Related to Kneser's Theorem", "abstract": "A recent result of Balandraud shows that for every subset S of an abelian group G, there exists a non trivial subgroup H such that |TS| <= |T|+|S|-2 holds only if the stabilizer of TS contains H. Notice that Kneser's Theorem says only that the stabilizer of TS must be a non-zero subgroup. This strong form of Kneser's theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud's results in the abelian case."}
{"category": "Math", "title": "Dual billiards, Fagnano orbits and regular polygons", "abstract": "We study the notion of Fagnano orbits for dual polygonal billiards. We used them to characterize regular polygons and we study the iteration of the developing map."}
{"category": "Math", "title": "Renewals for exponentially increasing lifetimes, with an application to digital search trees", "abstract": "We show that the number of renewals up to time $t$ exhibits distributional fluctuations as $t\\to\\infty$ if the underlying lifetimes increase at an exponential rate in a distributional sense. This provides a probabilistic explanation for the asymptotics of insertion depth in random trees generated by a bit-comparison strategy from uniform input; we also obtain a representation for the resulting family of limit laws along subsequences. Our approach can also be used to obtain rates of convergence."}
{"category": "Math", "title": "Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem", "abstract": "In this paper we study the shape of least-energy solutions to a singularly perturbed quasilinear problem with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that at limit, the global maximum point of least-energy solutions goes to a point on the boundary faster than the linear rate and this point on the boundary approaches to a point where the mean curvature of the boundary achieves its maximum. We also give a complete proof of exponential decay of least-energy solutions."}
{"category": "Math", "title": "An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries", "abstract": "Semimartingale reflecting Brownian motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of interest in applied probability because of their role as heavy traffic approximations for some stochastic networks. In this paper, assuming certain conditions on the domains and directions of reflection, a perturbation result, or invariance principle, for SRBMs is proved. This provides sufficient conditions for a process that satisfies the definition of an SRBM, except for small random perturbations in the defining conditions, to be close in distribution to an SRBM. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. We use the invariance principle to show weak existence of SRBMs under mild conditions. We also use the invariance principle, in conjunction with known uniqueness results for SRBMs, to give some sufficient conditions for validating approximations involving (i) SRBMs in convex polyhedrons with a constant reflection vector field on each face of the polyhedron, and (ii) SRBMs in bounded domains with piecewise smooth boundaries and possibly nonconstant reflection vector fields on the boundary surfaces."}
{"category": "Math", "title": "Unit groups of integral finite group rings with no noncyclic abelian finite subgroups", "abstract": "It is shown that in the units of augmentation one of an integral group ring $\\mathbb{Z} G$ of a finite group $G$, a noncyclic subgroup of order $p^{2}$, for some odd prime $p$, exists only if such a subgroup exists in $G$. The corresponding statement for $p=2$ holds by the Brauer--Suzuki theorem, as recently observed by W. Kimmerle."}
{"category": "Math", "title": "Origamis with non congruence Veech groups", "abstract": "As main result we show that for each g > 1 there is some translation surface of genus g whose Veech group is a non congruence subgroup of SL(2,Z). We use origamis/square-tiled surfaces to produce our examples. The article is divided into two parts: In the first part we introduce translation surfaces, origamis, Veech groups and Teichmueller curves and show for two origamis in genus 2 that their Veech groups are non congruence groups; in the second part we provide a technique that produces sequences of origamis whose Veech groups are decreasing. This is used to prove the main result."}
{"category": "Math", "title": "Reduced phase space and toric variety coordinatizations of Delzant spaces", "abstract": "In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex algebraic geometry view-point), and we give an explicit formula for the toric variety coordinates in terms of the reduced phase space coordinates. We use considerations in the maximal coordinate neighborhoods to give simple proofs of some of the basic facts about the Delzant space, as a reduced phase space, and as a toric variety. These can be viewed as a first application of the coordinatizations, and serve to make the presentation more self-contained."}
{"category": "Math", "title": "A Survey of Huebschmann and Stasheff's Paper: Formal Solution of the Master Equation via HPT and Deformation Theory", "abstract": "These notes, based on the paper \"Formal Solution of the Master Equation via HPT and Deformation Theory\" by Huebschmann and Stasheff, were prepared for a series of talks at Illinois State University with the intention of applying Homological Perturbation Theory to the derived bracket constructions of Kosmann-Schwarzbach and T. Voronov, and eventually writing Part II of the paper \"Higher Derived Brackets and Deformation Theory I\" by the present authors."}
{"category": "Math", "title": "The classification of surfaces with p_g=q=1 isogenous to a product of curves", "abstract": "A projective surface S is said to be isogenous to a product if there exist two smooth curves C, F and a finite group G acting freely on C \\times F so that S=(C \\times F)/G. In this paper we classify all surfaces with p_g=q=1 which are isogenous to a product."}
{"category": "Math", "title": "On iterated image size for point-symmetric relations", "abstract": "Let $\\Gamma =(V,E)$ be a point-symmetric reflexive relation and let $v\\in V$ such that $|\\Gamma (v)|$ is finite (and hence $|\\Gamma (x)|$ is finite for all $x$, by the transitive action of the group of automorphisms). Let $j\\in \\N$ be an integer such that $\\Gamma ^j(v)\\cap \\Gamma ^{-}(v)=\\{v\\}$. Our main result states that $$ |\\Gamma ^{j} (v)|\\ge | \\Gamma ^{j-1} (v)| + |\\Gamma (v)|-1.$$ As an application we have $ |\\Gamma ^{j} (v)| \\ge 1+(|\\Gamma (v)|-1)j.$ The last result confirms a recent conjecture of Seymour in the case of vertex-symmetric graphs. Also it gives a short proof for the validity of the Caccetta-H\\\"aggkvist conjecture for vertex-symmetric graphs and generalizes an additive result of Shepherdson."}
{"category": "Math", "title": "On line arrangements with applications to 3-nets", "abstract": "We show a one-to-one correspondence between arrangements of d lines in the projective plane, and lines in P^{d-2}. We apply this correspondence to classify (3,q)-nets over the complex numbers for all q<=6. When q=6, we have twelve possible combinatorial cases, but we prove that only nine of them are realizable. This new case shows several new properties for 3-nets: different dimensions for moduli, strict realization over certain fields, etc. We also construct a three dimensional family of (3,8)-nets corresponding to the Quaternion group."}
{"category": "Math", "title": "Contrasting Two Transformation-Based Methods for Obtaining Absolute Extrema", "abstract": "In this note we contrast two transformation-based methods to deduce absolute extrema and the corresponding extremizers. Unlike variation-based methods, the transformation-based ones of Carlson and Leitmann and the recent one of Silva and Torres are direct in that they permit obtaining solutions by inspection."}
{"category": "Math", "title": "The affine part of the Picard scheme", "abstract": "We describe the maximal torus and maximal unipotent subgroup of the Picard variety of a proper scheme over a perfect field."}
{"category": "Math", "title": "Optimal Shape Design for Stokes Flow Via Minimax Differentiability", "abstract": "This paper is concerned with a shape sensitivity analysis of a viscous incompressible fluid driven by Stokes equations with nonhomogeneous boundary condition. The structure of shape gradient with respect to the shape of the variable domain for a given cost function is established by using the differentiability of a minimax formulation involving a Lagrangian functional combining with function space parametrization technique or function space embedding technique. We apply an gradient type algorithm to our problem. Numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible."}
{"category": "Math", "title": "Penalization approach for mixed hyperbolic systems with constant coefficients satisfying a Uniform Lopatinski Condition", "abstract": "In this paper, we describe a new, systematic and explicit way of approximating solutions of mixed hyperbolic systems with constant coefficients satisfying a Uniform Lopatinski Condition via different Penalization approaches."}
{"category": "Math", "title": "On the polynomial automorphisms of a group", "abstract": "We prove that if a group is nilpotent (resp. metabelian), then so is the subgroup of its automorphism group generated by all polynomial automorphisms."}
{"category": "Math", "title": "Generic representations of orthogonal groups: projective functors in the category Fquad", "abstract": "In this paper, we continue the study of the category of functors Fquad, associated to F_2-vector spaces equipped with a nondegenerate quadratic form, initiated in two previous papers of the author. We define a filtration of the standard projective objects in Fquad; this refines to give a decomposition into indecomposable factors of the two first standard projective objects in Fquad. As an application of these two decompositions, we give a complete description of the polynomial functors of the category Fquad."}
{"category": "Math", "title": "Manifolds admitting a $\\tilde G_2$-structure", "abstract": "We find a necessary and sufficient condition for a compact 7-manifold to admit a $\\tilde G_2$-structure. As a result we find a sufficient condition for an open 7-manifold to admit a closed 3-form of $\\tilde G_2$-type."}
{"category": "Math", "title": "Invariance principle for additive functionals of Markov chains", "abstract": "We consider a sequence of additive functionals {\\phi_n}, set on a sequence of Markov chains {X_n} that weakly converges to a Markov process X. We give sufficient condition for such a sequence to converge in distribution, formulated in terms of the characteristics of the additive functionals, and related to the Dynkin's theorem on the convergence of W-functionals. As an application of the main theorem, the general sufficient condition for convergence of additive functionals in terms of transition probabilities of the chains X_n is proved."}
{"category": "Math", "title": "Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity", "abstract": "In this paper we study a class of backward stochastic differential equations (BSDEs) of the form dY(t)= -AY(t)dt -f_0(t,Y(t))dt -f_1(t,Y(t),Z(t))dt + Z(t)dW(t) on the interval [0,T], with given final condition at time T, in an infinite dimensional Hilbert space H. The unbounded operator A is sectorial and dissipative and the nonlinearity f_0(t,y) is dissipative and defined for y only taking values in a subspace of H. A typical example is provided by the so-called polynomial nonlinearities. Applications are given to stochastic partial differential equations and spin systems."}
{"category": "Math", "title": "Complexity Considerations, cSAT Lower Bound", "abstract": "This article discusses completeness of Boolean Algebra as First Order Theory in Goedel's meaning. If Theory is complete then any possible transformation is equivalent to some transformation using axioms, predicates etc. defined for this theory. If formula is to be proved (or disproved) then it has to be reduced to axioms. If every transformation is deducible then also optimal transformation is deducible. If every transformation is exponential then optimal one is too, what allows to define lower bound for discussed problem to be exponential (outside P). Then we show algorithm for NDTM solving the same problem in O(n^c) (so problem is in NP), what proves that P \\neq NP. Article proves also that result of relativisation of P=NP question and oracle shown by Baker-Gill-Solovay distinguish between deterministic and non-deterministic calculation models. If there exists oracle A for which P^A=NP^A then A consists of infinite number of algorithms, DTMs, axioms and predicates, or like NDTM infinite number of simultaneous states."}
{"category": "Math", "title": "Using decomposed household food acquisitions as inputs of a Kinetic Dietary Exposure Model", "abstract": "Foods naturally contain a number of contaminants that may have different and long term toxic effects. This paper introduces a novel approach for the assessment of such chronic food risk that integrates the pharmacokinetic properties of a given contaminant. The estimation of such a Kinetic Dietary Exposure Model (KDEM) should be based on long term consumption data which, for the moment, can only be provided by Household Budget Surveys such as the SECODIP panel in France. A semi parametric model is proposed to decompose a series of household quantities into individual quantities which are then used as inputs of the KDEM. As an illustration, the risk assessment related to the presence of methyl mercury in seafood is revisited using this novel approach."}
{"category": "Math", "title": "Optimal control of stochastic differential equations with dynamical boundary conditions", "abstract": "In this paper we investigate the optimal control problem for a class of stochastic Cauchy evolution problem with non standard boundary dynamic and control. The model is composed by an infinite dimensional dynamical system coupled with a finite dimensional dynamics, which describes the boundary conditions of the internal system. In other terms, we are concerned with non standard boundary conditions, as the value at the boundary is governed by a different stochastic differential equation."}
{"category": "Math", "title": "Linearisation of finite abelian subgroups of the Cremona group of the plane", "abstract": "This article gives the proof of results announced in [J. Blanc, Finite Abelian subgroups of the Cremona group of the plane, C.R. Acad. Sci. Paris, S\\'er. I 344 (2007), 21-26.] and some description of automorphisms of rational surfaces. Given a finite Abelian subgroup of the Cremona group of the plane, we provide a way to decide whether it is birationally conjugate to a group of automorphisms of a minimal surface. In particular, we prove that a finite cyclic group of birational transformations of the plane is linearisable if and only if none of its non-trivial elements fix a curve of positive genus. For finite Abelian groups, there exists only one surprising exception, a group isomorphic to Z/2ZxZ/4Z, whose non-trivial elements do not fix a curve of positive genus but which is not conjugate to a group of automorphisms of a minimal rational surface. We also give some descriptions of automorphisms (not necessarily of finite order) of del Pezzo surfaces and conic bundles."}
{"category": "Math", "title": "Integral representations for convolutions of non-central multivariate gamma distributions", "abstract": "Three types of integral representations for the cumulative distribution functions of convolutions of non-central p-variate gamma distributions are given by integration of elementary complex functions over the p-cube Cp = (-pi,pi]x...x(-pi,pi]. In particular, the joint distribution of the diagonal elements of a generalized quadratic form XAX' with n independent normally distributed column vectors in X is obtained. For a single p-variate gamma distribution function (p-1)-variate integrals over Cp-1 are derived. The integrals are numerically more favourable than integrals obtained from the Fourier or laplace inversion formula."}
{"category": "Math", "title": "On complete subsets of the cyclic group", "abstract": "A subset $X$ of an abelian $G$ is said to be {\\em complete} if every element of the subgroup generated by $X$ can be expressed as a nonempty sum of distinct elements from $X$. Let $A\\subset \\Z_n$ be such that all the elements of $A$ are coprime with $n$. Solving a conjecture of Erd\\H{o}s and Heilbronn, Olson proved that $A$ is complete if $n$ is a prime and if $|A|>2\\sqrt{n}.$ Recently Vu proved that there is an absolute constant $c$, such that for an arbitrary large $n$, $A$ is complete if $|A|\\ge c\\sqrt{n},$ and conjectured that 2 is essentially the right value of $c$. We show that $A$ is complete if $|A|> 1+2\\sqrt{n-4}$, thus proving the last conjecture."}
{"category": "Math", "title": "Hilbert functions of points on Schubert varieties in Orthogonal Grassmannians", "abstract": "A solution is given to the following problem: how to compute the multiplicity, or more generally the Hilbert function, at a point on a Schubert variety in an orthogonal Grassmannian. Standard monomial theory is applied to translate the problem from geometry to combinatorics. The solution of the resulting combinatorial problem forms the bulk of the paper. This approach has been followed earlier to solve the same problem for the Grassmannian and the symplectic Grassmannian. As an application, we present an interpretation of the multiplicity as the number of non-intersecting lattice paths of a certain kind. Taking the Schubert variety to be of a special kind and the point to be the \"identity coset,\" our problem specializes to a problem about Pfaffian ideals treatments of which by different methods exist in the literature. Also available in the literature is a geometric solution when the point is a \"generic singularity.\""}
{"category": "Math", "title": "The Graham conjecture implies the Erdos-Turan conjecture", "abstract": "Erd\\\"{o}s and Tur\\'{a}n once conjectured that any set $A\\subset\\mathbb{N}$ with $\\sum_{a\\in A}{1}/{a}=\\infty$ should contain infinitely many progressions of arbitrary length $k\\geq3$. For the two-dimensional case Graham conjectured that if $B\\subset \\mathbb{N}\\times\\mathbb{N}$ satisfies $$\\sum\\limits_{(x,y)\\in B}\\frac{1}{x^2+y^2}=\\infty,$$ then for any $s\\geq2$, $B$ contains an $s\\times s$ axes-parallel grid. In this paper it is shown that if the Graham conjecture is true for some $s\\geq2$, then the Erd\\\"{o}s-Tur\\'{a}n conjecture is true for $k=2s-1$."}
{"category": "Math", "title": "M-regularity of the Fano surface", "abstract": "Let $(A,\\Theta)$ be a principally polarised abelian variety, and let Y be a subvariety. Pareschi and Popa conjectured that Y has minimal cohomology class if and only if the structure sheaf of Y satisfies a property that they call M-regularity. Let now X be a smooth cubic threefold. By a classical result due to Clemens and Griffiths, its intermediate Jacobian J(X) is a principally polarised abelian variety; furthermore the Fano surface of lines on X can be embedded in J(X) and has minimal cohomology class. In this short note we show that its structure sheaf is M-regular."}
{"category": "Math", "title": "The Lifshitz-Slyozov-Wagner equation for reaction-controlled kinetics", "abstract": "We rigorously derive a weak form of the Lifshitz-Slyozov-Wagner equation as the homogenization limit of a Stefan-type problem describing reaction-controlled coarsening of a large number of small spherical particles. Moreover, we deduce that the effective mean-field description holds true in the particular limit of vanishing surface-area density of particles."}
{"category": "Math", "title": "Canonical singular hermitian metrics on relative canonical bundles", "abstract": "We introduce a new class of canonical AZD's (called the supercanonical AZD's) on the canonical bundles of smooth projective varieties with pseudoeffective canonical classes. We study the variation of the supercanonical AZD $\\hat{h}_{can}$ under projective deformations and give a new proof of the invariance of plurigenera."}
{"category": "Math", "title": "Counting characters in linear group actions", "abstract": "Let $G$ be a finite group and $V$ be a finite $G$--module. We present upper bounds for the cardinalities of certain subsets of $\\Irr(GV)$, such as the set of those $\\chi\\in\\Irr(GV)$ such that, for a fixed $v\\in V$, the restriction of $\\chi$ to $<v>$ is not a multiple of the regular character of $<v>$. These results might be useful in attacking the non--coprime $k(GV)$--problem."}
{"category": "Math", "title": "On the KK-theory of strongly self-absorbing C*-algebras", "abstract": "Let $\\Dh$ and $A$ be unital and separable $C^{*}$-algebras; let $\\Dh$ be strongly self-absorbing. It is known that any two unital $^*$-homomorphisms from $\\Dh$ to $A \\otimes \\Dh$ are approximately unitarily equivalent. We show that, if $\\Dh$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\\Dh$ is asymptotically inner. Moreover, the space of automorphisms of $\\Dh$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,\\Aut(\\Dh)]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\\Dh$. As an application, we give a description of the Kasparov group $KK(\\Dh, A\\ot \\Dh)$ in terms of $^*$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\\Dh, A\\ot \\Dh)$ is isomorphic to $K_0(A\\ot \\Dh)$."}
{"category": "Math", "title": "A new approach to mutual information", "abstract": "A new expression as a certain asymptotic limit via \"discrete micro-states\" of permutations is provided to the mutual information of both continuous and discrete random variables."}
{"category": "Math", "title": "The local structure of conformally symmetric manifolds", "abstract": "This is a final step in a local classification of pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric."}
{"category": "Math", "title": "Solvability of linear equations within weak mixing sets", "abstract": "We introduce a new class of \"random\" subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of systems of linear equations within every WM set and within every normal set. We also show that partition-regular system of linear equations with integer coefficients is solvable in any WM set."}
{"category": "Math", "title": "On the homology of two-dimensional elimination", "abstract": "We study birational maps with empty base locus defined by almost complete intersection ideals. Birationality is shown to be expressed by the equality of two Chern numbers. We provide a relatively effective method of their calculation in terms of certain Hilbert coefficients. In dimension two the structure of the irreducible ideals leads naturally to the calculation of Sylvester determinants via a computer-assisted method. For degree at most 5 we produce the full set of defining equations of the base ideal. The results answer affirmatively some questions raised by D. Cox."}
{"category": "Math", "title": "Proper holomorphic mappings of the spectral unit ball", "abstract": "We prove an Alexander type theorem for the spectral unit ball $\\Omega_n$ showing that there are no non-trivial proper holomorphic mappings in $\\Omega_n$, $n\\geq 2$."}
{"category": "Math", "title": "Number of moduli of irreducible families of plane curves with nodes and cusps", "abstract": "Consider the family S of irreducible plane curves of degree n with d nodes and k cusps as singularities. Let W be an irreducible component of S. We consider the natural rational map from W to the moduli space of curves of genus g=(n-1)(n-2)/2-d-k. We define the \"number of moduli of W\" as the dimension of the image of W with respect to this map. If W has the expected dimension equal to 3n+g-1-k, then the number of moduli of W is at most equal to the min(3g-3, 3g-3+\\rho-k), dove \\rho is the Brill-Neother number of the linear series of degree n and dimension 2 on a smooth curve of genus g. We say that W has the expected number of moduli if the equality holds. In this paper we construct examples of families of irreducible plane curves with nodes and cusps as singularities having expected number of moduli and with non-positive Brill-Noether number."}
{"category": "Math", "title": "Uniqueness theorems for Cauchy integrals", "abstract": "If $\\mu$ is a finite complex measure in the complex plane $\\C$ we denote by $C^\\mu$ its Cauchy integral defined in the sense of principal value. The measure $\\mu$ is called reflectionless if it is continuous (has no atoms) and $C^\\mu=0$ at $\\mu$-almost every point. We show that if $\\mu$ is reflectionless and its Cauchy maximal function $C^\\mu_*$ is summable with respect to $|\\mu|$ then $\\mu$ is trivial. An example of a reflectionless measure whose maximal function belongs to the \"weak\" $L^1$ is also constructed, proving that the above result is sharp in its scale. We also give a partial geometric description of the set of reflectionless measures on the line and discuss connections of our results with the notion of sets of finite perimeter in the sense of De Giorgi."}
{"category": "Math", "title": "On the number of moduli of plane sextics with six cusps", "abstract": "Let S be the variety of irreducible sextics with six cusps as singularities. Let W be one of irreducible components of W. Denoting by M_4 the space of moduli of smooth curves of genus 4, the moduli map of W is the rational map from W to M_4 sending the general point of W, corresponding to a plane curve D, to the point of M_4 parametrizing the normalization curve of D. The number of moduli of W is, by definition the dimension of the image of W with respect to the moduli map. We know that this number is at most equal to seven. In this paper we prove that both irreducible components of S have number of moduli exactly equal to seven."}
{"category": "Math", "title": "Skew-Hadamard matrices of orders 188 and 388 exist", "abstract": "We construct several difference families on cyclic groups of orders 47 and 97, and use them to construct skew-Hadamard matrices of orders 188 and 388. Such difference families and matrices are constructed here for the first time. The matrices are constructed by using the Goethals-Seidel array."}
{"category": "Math", "title": "On the HOMFLY and Tutte polynomials", "abstract": "A celebrated result of F. Jaeger states that the Tutte polynomial of a planar graph is determined by the HOMFLY polynomial of an associated link. Here we are interested in the converse of this result. We consider the question `to what extent does the Tutte polynomial determine the HOMFLY polynomial of any knot?' We show that the HOMFLY polynomial of a knot is determined by Tutte polynomials of plane graphs associated to the knot."}
{"category": "Math", "title": "Quivers with potentials and their representations I: Mutations", "abstract": "We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras."}
{"category": "Math", "title": "Necessary optimality conditions for the calculus of variations on time scales", "abstract": "We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems of the calculus of variations with delta-differential side conditions (Lagrange problem of the calculus of variations on time scales)."}
{"category": "Math", "title": "Ample subvarieties and rationally connected fibrations", "abstract": "Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration."}
{"category": "Math", "title": "Bounds for Multiplicities of Unitary Representations of Cohomological Type in Spaces of Cusp Forms", "abstract": "Let $\\Goo$ be a semisimple real Lie group with unitary dual $\\Ghat$. The goal of this note is to produce new upper bounds for the multiplicities with which representations $\\pi \\in \\Ghat$ of cohomological type appear in certain spaces of cusp forms on $\\Goo$."}
{"category": "Math", "title": "Topological Free Entropy Dimension of in Unital C^*-algebras", "abstract": "The notion of topological free entropy dimension of $n-$tuples of elements in a unital C$^*$ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological orbit dimension of one self-adjoint element in a unital C$^*$ algebra. Moreover, we calculate the values of topological free entropy dimensions of families of generators of some unital C$^*$ algebras (for example: irrational rotation C$^*$ algebras or minimal tensor product of two reduced C$^*$ algebras of free groups)."}
{"category": "Math", "title": "Finite branch solutions to Painleve VI around a fixed singular point", "abstract": "Every finite branch solutions to the sixth Painleve equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The proof of this result relies on algebraic geometry of Painleve VI, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, resolutions of Kleinian singularities, and power geometry of algebraic differential equations. In the course of the proof we are also able to classify all finite branch solutions up to Backlund transformations."}
{"category": "Math", "title": "A generalization of Chebyshev polynomials and non rooted posets", "abstract": "In this paper we give a generalization of Chebyshev polynomials and using this we describe the M\\\"obius function of the generalized subword order from a poset {a1,...as,c |ai<c}, which contains an affirmative answer for the conjecture by Bj\\\"orner, Sagan, Vatter.[5,10]"}
{"category": "Math", "title": "Entropic Measure and Wasserstein Diffusion", "abstract": "We construct a new random probability measure on the sphere and on the unit interval which in both cases has a Gibbs structure with the relative entropy functional as Hamiltonian. It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval respectively. The associated integration by parts formula is used to construct two classes of diffusion processes on probability measures (on the sphere or the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin's Brownian motion on the homeomorphism group. The second one is a probability valued stochastic perturbation of the heat flow, whose intrinsic metric is the quadratic Wasserstein distance. It may be regarded as the canonical diffusion process on the Wasserstein space."}
{"category": "Math", "title": "On second order shape optimization methods for electrical impedance tomography", "abstract": "This paper is devoted to the analysis of a second order method for recovering the \\emph{a priori} unknown shape of an inclusion $\\omega$ inside a body $\\Omega$ from boundary measurement. This inverse problem - known as electrical impedance tomography - has many important practical applications and hence has focussed much attention during the last years. However, to our best knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: we investigate the existence of second order derivative of the state $u$ with respect to perturbations of the shape of the interface $\\partial\\omega$, then we choose a cost function in order to recover the geometry of $\\partial \\omega$ and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer."}
{"category": "Math", "title": "Maximum solutions of normalized Ricci flows on 4-manifolds", "abstract": "We consider maximum solution $g(t)$, $t\\in [0, +\\infty)$, to the normalized Ricci flow. Among other things, we prove that, if $(M, \\omega) $ is a smooth compact symplectic 4-manifold such that $b_2^+(M)>1$ and let $g(t),t\\in[0,\\infty)$, be a solution to (1.3) on $M$ whose Ricci curvature satisfies that $|\\text{Ric}(g(t))|\\leq 3$ and additionally $\\chi(M)=3 \\tau (M)>0$, then there exists an $m\\in \\mathbb{N}$, and a sequence of points $\\{x_{j,k}\\in M\\}$, $j=1, ..., m$, satisfying that, by passing to a subsequence, $$(M, g(t_{k}+t), x_{1,k},..., x_{m,k}) \\stackrel{d_{GH}}\\longrightarrow (\\coprod_{j=1}^m N_j, g_{\\infty}, x_{1,\\infty}, ...,, x_{m,\\infty}),$$ $t\\in [0, \\infty)$, in the $m$-pointed Gromov-Hausdorff sense for any sequence $t_{k}\\longrightarrow \\infty$, where $(N_{j}, g_{\\infty})$, $j=1,..., m$, are complete complex hyperbolic orbifolds of complex dimension 2 with at most finitely many isolated orbifold points. Moreover, the convergence is $C^{\\infty}$ in the non-singular part of $\\coprod_1^m N_{j}$ and $\\text{Vol}_{g_{0}}(M)=\\sum_{j=1}^{m}\\text{Vol}_{g_{\\infty}}(N_{j})$, where $\\chi(M)$ (resp. $\\tau(M)$) is the Euler characteristic (resp. signature) of $M$."}
{"category": "Math", "title": "$C^r$-Lohner algorithm", "abstract": "We present a Lohner type algorithm for the computation of rigorous bounds for solutions of ordinary differential equations and its derivatives with respect to initial conditions up to arbitrary order. As an application we prove the existence of multiple invariant tori around some elliptic periodic orbits for the pendulum equation with periodic forcing and for Michelson system."}
{"category": "Math", "title": "Computation of Power Loss in Likelihood Ratio Tests for Probability Densities Extended by Lehmann Alternatives", "abstract": "We compute the loss of power in likelihood ratio tests when we test the original parameter of a probability density extended by the first Lehmann alternative."}
{"category": "Math", "title": "A note on higher-order differential operations", "abstract": "In this paper we consider successive iterations of the first-order differential operations in space ${\\bf R}^3.$"}
{"category": "Math", "title": "Some combinatorial aspects of differential operation compositions on space $R^n$", "abstract": "In this paper we present a recurrent relation for counting meaningful compositions of the higher-order differential operations on the space $R^{n}$ (n=3,4,...) and extract the non-trivial compositions of order higher than two."}
{"category": "Math", "title": "Hyperbolicity in unbounded convex domains", "abstract": "We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory."}
{"category": "Math", "title": "A procedure for finding the k-th power of a matrix", "abstract": "We give a new procedure in Maple for finding the k-th power of a martix. The algorithm is based on the article [1]."}
{"category": "Math", "title": "Energy conservation and Onsager's conjecture for the Euler equations", "abstract": "Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in 3D conserve energy only if they have a certain minimal smoothness, (of order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space $B^{1/3}_{3,c(\\NN)}$. We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood-Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to $B^{2/3}_{3,c(\\NN)}$ conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range."}
{"category": "Math", "title": "Stability of a colocated finite volume scheme for the incompressible Navier-Stokes equations", "abstract": "We introduce a finite volume scheme for the two-dimensional incompressible Navier-Stokes equations. We use a triangular mesh. The unknowns for the velocity and pressure are both piecewise constant (colocated scheme). We use a projection (fractional-step) method to deal with the incompressibility constraint. We prove that the differential operators in the Navier-Stokes equations and their discrete counterparts share similar properties. In particular, we state an inf-sup (Babuska-Brezzi) condition. We infer from it the stability of the scheme."}
{"category": "Math", "title": "K_0-theory of n-potents in rings and algebras", "abstract": "Let $n \\geq 2$ be an integer. An \\emph{$n$-potent} is an element $e$ of a ring $R$ such that $e^n = e$. In this paper, we study $n$-potents in matrices over $R$ and use them to construct an abelian group $K_0^n(R)$. If $A$ is a complex algebra, there is a group isomorphism $K_0^n(A) \\cong \\bigl(K_0(A)\\bigr)^{n-1}$ for all $n \\geq 2$. However, for algebras over cyclotomic fields, this is not true in general. We consider $K_0^n$ as a covariant functor, and show that it is also functorial for a generalization of homomorphism called an \\emph{$n$-homomorphism}."}
{"category": "Math", "title": "Frobenius splitting and geometry of $G$-Schubert varieties", "abstract": "Let $X$ be an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic. Let $B$ denote a Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the form $\\diag(G) \\cdot V$, where $V$ is a $B \\times B$-orbit closure in $X$. In the case where $X$ is the wonderful compactification of a group of adjoint type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable pieces. We prove that $X$ admits a Frobenius splitting which is compatible with all $G$-Schubert varieties. Moreover, when $X$ is smooth, projective and toroidal, then any $G$-Schubert variety in $X$ admits a stable Frobenius splitting along an ample divisors. Although this indicates that $G$-Schubert varieties have nice singularities we present an example of a non-normal $G$-Schubert variety in the wonderful compactification of a group of type $G_2$. Finally we also extend the Frobenius splitting results to the more general class of $\\mathcal R$-Schubert varieties."}
{"category": "Math", "title": "Stability of a finite volume scheme for the incompressible fluids", "abstract": "We introduce a finite volume scheme for the two-dimensional incompressible Navier-Stokes equations. We use a triangular mesh. The unknowns for the velocity and pressure are respectively piecewise constant and affine. We use a projection method to deal with the incompressibility constraint. We show that the differential operators in the Navier-Stokes equations and their discrete counterparts share similar properties. In particular we state an inf-sup (Babuska-Brezzi) condition. Using these properties we infer the stability of the scheme."}
{"category": "Math", "title": "Convergence of a finite volume scheme for the incompressible fluids", "abstract": "We consider a finite volume scheme for the two-dimensional incompressible Navier-Stokes equations. We use a triangular mesh. The unknowns for the velocity and pressure are respectively piecewise constant and affine. We use a projection method to deal with the incompressibility constraint. In a former paper, the stability of the scheme has been proven. We infer from it its convergence."}
{"category": "Math", "title": "Nonimmersions of RP^n implied by tmf, revisited", "abstract": "In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RP^n with n as small as 48, and yet it was stated there that the first new result occurred when n=1536. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f^*(X)=X_1+X_2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in the original paper. Finally, we present a complete determination of tmf^{8*}(RP^\\infty\\times RP^\\infty) and tmf^*(CP^\\infty\\times CP^\\infty) in positive dimensions."}
{"category": "Math", "title": "Fundamental solutions for a class of non-elliptic homogeneous differential operators", "abstract": "We compute temperate fundamental solutions of homogeneous differential operators with real-principal type symbols. Via analytic continuation of meromorphic distributions, fundamental solutions for these non-elliptic operators can be constructed in terms of radial averages and invariant distributions on the unit sphere."}
{"category": "Math", "title": "On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields", "abstract": "Let $ v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform \\operatorname H_{v, \\epsilon}f(x) := \\text{p.v.}\\int_{-\\epsilon}^ \\epsilon f(x-yv(x)) \\frac{dy}y where $ \\epsilon $ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.\\thinspace M.\\thinspace Stein, that if $ v$ is Lipschitz, there is a positive $ \\epsilon $ for which the transform above is bounded on $ L ^{2}$. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to $ v$. This sufficient condition is that this new maximal function be bounded on some $ L ^{p}$, for some $ 1<p<2$. We show that the maximal function is bounded from $ L ^{2}$ to weak $ L ^{2}$ for all Lipschitz maximal function. The relationship between our results and other known sufficient conditions is explored."}
{"category": "Math", "title": "An S_3-symmetric Littlewood-Richardson rule", "abstract": "The classical Littlewood-Richardson coefficients C(lambda,mu,nu) carry a natural $S_3$ symmetry via permutation of the indices. Our \"carton rule\" for computing these numbers transparently and uniformly explains these six symmetries; previously formulated Littlewood-Richardson rules manifest at most three of the six."}
{"category": "Math", "title": "On the (3,N) Maurer-Cartan equation", "abstract": "Deformations of the 3-differential of 3-differential graded algebras are controlled by the (3,N) Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of N-differential graded algebras, and use these results to study N Lie algebroids."}
{"category": "Math", "title": "Local well-posedness of nonlinear dispersive equations on modulation spaces", "abstract": "By using tools of time-frequency analysis, we obtain some improved local well-posedness results for the NLS, NLW and NLKG equations with Cauchy data in modulation spaces $M{p, 1}_{0,s}$."}
{"category": "Math", "title": "A matroid-friendly basis for the quasisymmetric functions", "abstract": "A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the space of quasisymmetric functions associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and Reiner. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F is injective on the set of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer is given to the Hilbert basis question raised by Billera, Jia, and Reiner."}
{"category": "Math", "title": "Moduli spaces of rational tropical curves", "abstract": "This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. We define the Deligne-Mumford compactification of this space and tropical $\\psi$-class divisors."}
{"category": "Math", "title": "PI degree parity in q-skew polynomial rings", "abstract": "For k a field of arbitrary characteristic, and R a k-algebra, we show that the PI degree of an iterated skew polynomial ring R[x_1;\\tau_1,\\delta_1]...b[x_n;\\tau_n,\\delta_n] agrees with the PI degree of R[x_1;\\tau_1]...b[x_n;\\tau_n] when each (\\tau_i,\\delta_i) satisfies a q_i-skew relation for q_i \\in k^{\\times} and extends to a higher q_i-skew \\tau_i-derivation. We confirm the quantum Gel'fand-Kirillov conjecture for various quantized coordinate rings, and calculate their PI degrees. We extend these results to completely prime factor algebras."}
{"category": "Math", "title": "Counting on rectangular areas", "abstract": "In the first section of this paper we prove a theorem for the number of columns of a rectangular area that are identical to the given one. In the next section we apply this theorem to derive several combinatorial identities by counting specified subsets of a finite set."}
{"category": "Math", "title": "Normalized Ricci flow on nonparabolic surfaces", "abstract": "This paper studies normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically -1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature -1. A relative estimate of Green's function is proved as a tool."}
{"category": "Math", "title": "Transfinite diameter, Chebyshev constant and energy on locally compact spaces", "abstract": "We study the relationship between transfinite diameter, Chebyshev constant and Wiener energy in the abstract linear potential analytic setting pioneered by Choquet, Fuglede and Ohtsuka. It turns out that, whenever the potential theoretic kernel has the maximum principle, then all these quantities are equal for all compact sets. For continuous kernels even the converse statement is true: if the Chebyshev constant of any compact set coincides with its transfinite diameter, the kernel must satisfy the maximum principle. An abundance of examples is provided to show the sharpness of the results."}
{"category": "Math", "title": "A priori estimates for weak solutions of complex Monge-Amp\\`ere equations", "abstract": "Let $X$ be a compact K\\\"ahler manifold and $\\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\\mathcal E}_{\\chi}(X,\\om)$ of $\\om$-plurisubharmonic functions of finite weighted Monge-Amp\\`ere energy. When the weight $\\chi$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Amp\\`ere capacity, then it belongs to the range of the Monge-Amp\\`ere operator on some class ${\\mathcal E}_{\\chi}(X,\\om)$. This is done by establishing a priori estimates on the capacity of sublevel sets of the solutions. Our result extends U.Cegrell's and S.Kolodziej's results and puts them into a unifying frame. It also gives a simple proof of S.T.Yau's celebrated a priori ${\\mathcal C}^0$-estimate."}
{"category": "Math", "title": "Spectral perturbation bounds for selfadjoint operators", "abstract": "We give general spectral and eigenvalue perturbation bounds for a selfadjoint operator perturbed in the sense of the pseudo-Friedrichs extension. We also give several generalisations of the aforementioned extension. The spectral bounds for finite eigenvalues are obtained by using analyticity and monotonicity properties (rather than variational principles) and they are general enough to include eigenvalues in gaps of the essential spectrum."}
{"category": "Math", "title": "Higher ramification and varieties of secant divisors on the generic curve", "abstract": "For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to completely clarify this problem in the case of the generic curve C of given genus. Using degeneration techniques and a few facts about the birational geometry of moduli spaces of stable pointed curves we determine precisely under which conditions the cycle of e-secant k-planes in non-empty and we compute its dimension. We also precisely determine the dimension of the variety of linear series on C carrying e-secant k-planes. In a different direction, in the last part of the paper we study the distribution of ramification points of the powers of a line bundle on C having prescribed ramification at a given point."}
{"category": "Math", "title": "The concrete theory of numbers: initial numbers and wonderful properties of numbers repunit", "abstract": "In this work initial numbers and repunit numbers have been studied. All numbers have been considered in a decimal notation. The problem of simplicity of initial numbers has been studied. Interesting properties of numbers repunit are proved: $gcd(R_a, R_b) = R_{gcd(a,b)}$; $R_{ab}/(R_aR_b)$ is an integer only if $gcd(a,b) = 1$, where $a\\geq1$, $b\\geq1$ are integers. Dividers of numbers repunit, are researched by a degree of prime number."}
{"category": "Math", "title": "Non-monotone convergence in the quadratic Wasserstein distance", "abstract": "We give an easy counter-example to Problem 7.20 from C. Villani's book on mass transport: in general, the quadratic Wasserstein distance between $n$-fold normalized convolutions of two given measures fails to decrease monotonically."}
{"category": "Math", "title": "Extension theorems of Sakai type for separately holomorphic and meromorphic functions", "abstract": "We first exhibit counterexamples to some open questions related to a theorem of Sakai. Then we establish an extension theorem of Sakai type for separately holomorphic/meromorphic functions."}
{"category": "Math", "title": "Uniform measures and countably additive measures", "abstract": "Uniform measures are defined as the functionals on the space of bounded uniformly continuous functions that are continuous on bounded uniformly equicontinuous sets. If every cardinal has measure zero then every countably additive measure is a uniform measure. The functionals sequentially continuous on bounded uniformly equicontinuous sets are exactly uniform measures on the separable modification of the underlying uniform space."}
{"category": "Math", "title": "Gorenstein locus of minuscule Schubert varieties", "abstract": "In this article, we describe explicitely the Gorenstein locus of all minuscule Schubert varieties. This proves a special case of a conjecture of A. Woo and A. Yong (see math.AG/0603273) on the Gorenstein locus of Schubert varieties."}
{"category": "Math", "title": "A unified approach to the theory of separately holomorphic mappings", "abstract": "We extend the theory of separately holomorphic mappings between complex analytic spaces. Our method is based on Poletsky theory of discs, Rosay Theorem on holomorphic discs and our recent joint-work with Pflug on cross theorems in dimension 1. It also relies on our new technique of conformal mappings and a generalization of Siciak's relative extremal function. Our approach illustrates the unified character: ``From local informations to global extensions\". Moreover, it avoids systematically the use of the classical method of doubly orthogonal bases of Bergman type."}
{"category": "Math", "title": "Metropolis algorithm and equienergy sampling for two mean field spin systems", "abstract": "In this paper we study the Metropolis algorithm in connection with two mean--field spin systems, the so called mean--field Ising model and the Blume--Emery--Griffiths model. In both this examples the naive choice of proposal chain gives rise, for some parameters, to a slowly mixing Metropolis chain, that is a chain whose spectral gap decreases exponentially fast (in the dimension $N$ of the problem). Here we show how a slight variant in the proposal chain can avoid this problem, keeping the mean computational cost similar to the cost of the usual Metropolis. More precisely we prove that, with a suitable variant in the proposal, the Metropolis chain has a spectral gap which decreases polynomially in 1/N. Using some symmetry structure of the energy, the method rests on allowing appropriate jumps within the energy level of the starting state."}
{"category": "Math", "title": "L^2 rho form for normal coverings of fibre bundles", "abstract": "We define the secondary invariants L^2- eta and -rho forms for families of generalized Dirac operators on normal coverings of fibre bundles. On the covering family we assume transversally smooth spectral projections, and Novikov--Shubin invariants bigger than 3(dim B+1) to treat the large time asymptotic for general operators. In the particular case of a bundle of spin manifolds, we study the L^2- rho class in relation to the space of positive scalar curvature vertical metrics."}
{"category": "Math", "title": "On the Nonexistence of Nontrivial Involutive n-Homomorphisms of C*-algebras", "abstract": "An n-homomorphism between algebras is a linear map $\\phi : A \\to B$ such that $\\phi(a_1 ... a_n) = \\phi(a_1)... \\phi(a_n)$ for all elements $a_1, >..., a_n \\in A.$ Every homomorphism is an n-homomorphism, for all n >= 2, but the converse is false, in general. Hejazian et al. [7] ask: Is every *-preserving n-homomorphism between C*-algebras continuous? We answer their question in the affirmative, but the even and odd n arguments are surprisingly disjoint. We then use these results to prove stronger ones: If n >2 is even, then $\\phi$ is just an ordinary *-homomorphism. If n >= 3 is odd, then $\\phi$ is a difference of two orthogonal *-homomorphisms. Thus, there are no nontrivial *-linear n-homomorphisms between C*-algebras."}
{"category": "Math", "title": "Algebraic geometry of Gaussian Bayesian networks", "abstract": "Conditional independence models in the Gaussian case are algebraic varieties in the cone of positive definite covariance matrices. We study these varieties in the case of Bayesian networks, with a view towards generalizing the recursive factorization theorem to situations with hidden variables. In the case when the underlying graph is a tree, we show that the vanishing ideal of the model is generated by the conditional independence statements implied by graph. We also show that the ideal of any Bayesian network is homogeneous with respect to a multigrading induced by a collection of upstream random variables. This has a number of important consequences for hidden variable models. Finally, we relate the ideals of Bayesian networks to a number of classical constructions in algebraic geometry including toric degenerations of the Grassmannian, matrix Schubert varieties, and secant varieties."}
{"category": "Math", "title": "A variation of Gronwall's lemma", "abstract": "We prove a variation of Gronwall's lemma."}
{"category": "Math", "title": "When the Cramer-Rao Inequality provides no information", "abstract": "We investigate a one-parameter family of probability densities (related to the Pareto distribution, which describes many natural phenomena) where the Cramer-Rao inequality provides no information."}
{"category": "Math", "title": "Lower order terms in the 1-level density for families of holomorphic cuspidal newforms", "abstract": "The Katz-Sarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of L-functions agree with the N -> oo scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for many families by studying the n-level densities; for suitably restricted test functions the main terms agree with random matrix theory. In particular, all one-parameter families of elliptic curves with rank r over Q(T) and the same distribution of signs of functional equations have the same limiting behavior. We break this universality and find family dependent lower order correction terms in many cases; these lower order terms have applications ranging from excess rank to modeling the behavior of zeros near the central point, and depend on the arithmetic of the family. We derive an alternate form of the explicit formula for GL(2) L-functions which simplifies comparisons, replacing sums over powers of Satake parameters by sums of the moments of the Fourier coefficients lambda_f(p). Our formula highlights the differences that we expect to exist from families whose Fourier coefficients obey different laws (for example, we expect Sato-Tate to hold only for non-CM families of elliptic curves). Further, by the work of Rosen and Silverman we expect lower order biases to the Fourier coefficients in families of elliptic curves with rank over Q(T); these biases can be seen in our expansions. We analyze several families of elliptic curves and see different lower order corrections, depending on whether or not the family has complex multiplication, a forced torsion point, or non-zero rank over Q(T)."}
{"category": "Math", "title": "A Contraction Theory Approach to Stochastic Incremental Stability", "abstract": "We investigate the incremental stability properties of It\\^o stochastic dynamical systems. Specifically, we derive a stochastic version of nonlinear contraction theory that provides a bound on the mean square distance between any two trajectories of a stochastically contracting system. This bound can be expressed as a function of the noise intensity and the contraction rate of the noise-free system. We illustrate these results in the contexts of stochastic nonlinear observers design and stochastic synchronization."}
{"category": "Math", "title": "A Symplectic Test of the L-Functions Ratios Conjecture", "abstract": "Recently Conrey, Farmer and Zirnbauer conjectured formulas for the averages over a family of ratios of products of shifted L-functions. Their L-functions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from n-level correlations and densities to mollifiers and moments to vanishing at the central point. There are now many results showing agreement between the main terms of number theory and random matrix theory; however, there are very few families where the lower order terms are known. These terms often depend on subtle arithmetic properties of the family, and provide a way to break the universality of behavior. The L-functions Ratios Conjecture provides a powerful and tractable way to predict these terms. We test a specific case here, that of the 1-level density for the symplectic family of quadratic Dirichlet characters arising from even fundamental discriminants d \\le X. For test functions supported in (-1/3, 1/3) we calculate all the lower order terms up to size O(X^{-1/2+epsilon}) and observe perfect agreement with the conjecture (for test functions supported in (-1, 1) we show agreement up to errors of size O(X^{-epsilon}) for any epsilon). Thus for this family and suitably restricted test functions, we completely verify the Ratios Conjecture's prediction for the 1-level density."}
{"category": "Math", "title": "Axiom A polynomial skew products of C^2 and their postcritical sets", "abstract": "A polynomial skew product of C^2 is a map of the form f(z,w) = (p(z), q(z,w)), where p and q are polynomials, such that f is regular of degree d >= 2. For polynomial maps of C, hyperbolicity is equivalent to the condition that the closure of the postcritical set is disjoint from the Julia set; further, critical points either iterate to an attracting cycle or infinity. For polynomial skew products, Jonsson (Math. Ann., 1999) established that f is Axiom A if and only if the closure of the postcritical set is disjoint from the right analog of the Julia set. Here we present the analogous conclusion: critical orbits either escape to infinity or accumulate on an attracting set. In addition, we construct new examples of Axiom A maps demonstrating various postcritical behaviors."}
{"category": "Math", "title": "Gibbs fragmentation trees", "abstract": "We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous' beta-splitting model, which has an extended parameter range $\\beta>-2$ with respect to the ${\\rm beta}(\\beta+1,\\beta+1)$ probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter Poisson--Dirichlet models for exchangeable random partitions of $\\mathbb {N}$, with an extended parameter range $0\\le\\alpha\\le1$, $\\theta\\ge-2\\alpha$ and $\\alpha<0$, $\\theta =-m\\alpha$, $m\\in \\mathbb {N}$."}
{"category": "Math", "title": "Conservation laws for invariant functionals containing compositions", "abstract": "The study of problems of the calculus of variations with compositions is a quite recent subject with origin in dynamical systems governed by chaotic maps. Available results are reduced to a generalized Euler-Lagrange equation that contains a new term involving inverse images of the minimizing trajectories. In this work we prove a generalization of the necessary optimality condition of DuBois-Reymond for variational problems with compositions. With the help of the new obtained condition, a Noether-type theorem is proved. An application of our main result is given to a problem appearing in the chaotic setting when one consider maps that are ergodic."}
{"category": "Math", "title": "One-dimensional Brownian particle systems with rank dependent drifts", "abstract": "We study interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. Our main objective has been to study the long range behavior of the spacings between the Brownian motions arranged in increasing order. For finitely many Brownian motions interacting in this manner, we characterize drifts for which the family of laws of the vector of spacings is tight, and show its convergence to a unique stationary joint distribution given by independent exponential distributions with varying means. We also study one particular countably infinite system, where only the minimum Brownian particle gets a constant upward drift, and prove that independent and identically distributed exponential spacings remain stationary under the dynamics of such a process. Some related conjectures in this direction have also been discussed."}
{"category": "Math", "title": "Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part II", "abstract": "We study the Dirichlet problem associated to the equation for self-similar surfaces for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are small enough and satisfy some symmetries. This is the second step towards the construction of new examples of complete embedded self similar surfaces under mean curvature flow."}
{"category": "Math", "title": "Enumerating limit groups", "abstract": "We prove that the set of limit groups is recursive, answering a question of Delzant. One ingredient of the proof is the observation that a finitely presented group with local retractions (a la Long and Reid) is coherent and, furthermore, there exists an algorithm that computes presentations for finitely generated subgroups. The other main ingredient is the ability to algorithmically calculate centralizers in relatively hyperbolic groups. Applications include the existence of recognition algorithms for limit groups and free groups."}
{"category": "Math", "title": "A Direct Method for Solving Optimal Switching Problems of One-Dimensional Diffusions", "abstract": "In this paper, we propose a direct solution method for optimal switching problems of one-dimensional diffusions. This method is free from conjectures about the form of the value function and switching strategies, or does not require the proof of optimality through quasi-variational inequalities. The direct method uses a general theory of optimal stopping problems for one-dimensional diffusions and characterizes the value function as sets of the smallest linear majorants in their respective transformed spaces."}
{"category": "Math", "title": "Mediatic graphs", "abstract": "Any medium can be represented as an isometric subgraph of the hypercube, with each token of the medium represented by a particular equivalence class of arcs of the subgraph. Such a representation, although useful, is not especially revealing of the structure of a particular medium. We propose an axiomatic definition of the concept of a `mediatic graph'. We prove that the graph of any medium is a mediatic graph. We also show that, for any non-necessarily finite set S, there exists a bijection from the collection M of all the media on a given set S (of states) onto the collection G of all the mediatic graphs on S."}
{"category": "Math", "title": "Stable algebras of entire functions", "abstract": "Suppose that $h$ and $g$ belong to the algebra $\\B$ generated by the rational functions and an entire function $f$ of finite order on ${\\Bbb C}^n$ and that $h/g$ has algebraic polar variety. We show that either $h/g\\in\\B$ or $f=q_1e^p+q_2$, where $p$ is a polynomial and $q_1,q_2$ are rational functions. In the latter case, $h/g$ belongs to the algebra generated by the rational functions, $e^p$ and $e^{-p}$."}
{"category": "Math", "title": "Test vectors for trilinear forms, when two representations are unramified and one is special", "abstract": "Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three admissible, irreducible, finite dimensional representations of G, the space of G-invariant linear forms has dimension at most one. When a non zero linear form exists, one wants to find an element of V which is not in its kernel: this is a test vector. Gross and Prasad found explicit test vectors for some triple of representations. In this paper, others are found, and they almost complete the case when the conductor of each representation is at most 1."}
{"category": "Math", "title": "Generic character sheaves on disconnected groups and character values", "abstract": "We relate a generic character sheaf on a disconnected reductive group with a character of a representation of the rational points of the group over a finite field extending a result known in the connected case."}
{"category": "Math", "title": "Tautological relations in Hodge field theory", "abstract": "We propose a Hodge field theory construction that captures algebraic properties of the reduction of Zwiebach invariants to Gromov-Witten invariants. It generalizes the Barannikov-Kontsevich construction to the case of higher genera correlators with gravitational descendants. We prove the main theorem stating that algebraically defined Hodge field theory correlators satisfy all tautological relations. From this perspective the statement that Barannikov-Kontsevich construction provides a solution of the WDVV equation looks as the simplest particular case of our theorem. Also it generalizes the particular cases of other low-genera tautological relations proven in our earlier works; we replace the old technical proofs by a novel conceptual proof."}
{"category": "Math", "title": "Constructions of Kahler-Einstein metrics with negative scalar curvature", "abstract": "We show that on Kahler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by H. Tsuji converges uniformly to the Kahler-Einstein metric. For algebraic surfaces of general type and orbifolds with isolated singularities, we prove a convergence result for a modified version of Tsuji's iterative construction."}
{"category": "Math", "title": "A Denjoy Theorem for commuting circle diffeomorphisms with mixed Holder derivatives", "abstract": "We prove that if d is an integer number bigger than 1 and f_1,...,f_d are commuting circle diffeomorphisms respectively of class C^(1+\\tau_k), where \\tau_1 + ... + \\tau_k > 1, then these maps are simultaneously conjugate to rotations provided that their rotation numbers are independent over the rationals."}
{"category": "Math", "title": "Explicit HRS-Tilting", "abstract": "For an abelian category $A$ equipped with a torsion pair, we give an explicit description for the abelian category $B$ introduced by Happel-Reiten-Smalo, and also for the category of chain complexes $Ch(B)$ and the derived category $D(B)$ of $B$. We also describe the DG structure on $Ch(B)$. As a consequence, we find new proofs of certain results of Happel-Reiten-Smalo. The main ingredient is the category of {\\em decorated} complexes."}
{"category": "Math", "title": "Lectures on derived and triangulated categories", "abstract": "These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included."}
{"category": "Math", "title": "Group actions on algebraic stacks via butterflies", "abstract": "We introduce an explicit method for studying actions of a group stack G on an algebraic stack X. As an example, we study in detail the case where X=P(n_0,...,n_r) is a weighted projective stack over an arbitrary base S. To this end, we give an explicit description of the group stack of automorphisms of, the weighted projective general linear 2-group PGL(n_0,...,n_r). As an application, we use a result of Colliot-Thelene to show that for every linear algebraic group G over an arbitrary base field k (assumed to be reductive if char(k)>0) such that Pic}(G)=0, every action of G on P(n_0,...,n_r) lifts to a linear action of G on A^{r+1}."}
{"category": "Math", "title": "Flops connect minimal models", "abstract": "A remark on a paper by Birkar-Cascini-Hacon-McKernan."}
{"category": "Math", "title": "A product formula for volumes of varieties", "abstract": "A simple application of the semipositivity."}
{"category": "Math", "title": "Cyclic cohomology of certain nuclear Fr\\'echet and DF algebras", "abstract": "We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain topological algebras. To this end we show that, for a continuous morphism $\\phi: \\X\\to \\Y$ of complexes of complete nuclear $DF$-spaces, the isomorphism of cohomology groups $H^n(\\phi): H^n(\\X) \\to H^n(\\Y)$ is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective $\\hat{\\otimes}$-algebras: the tensor algebra $E \\hat{\\otimes} F$ generated by the duality $(E, F, < \\cdot, \\cdot >)$ for nuclear Fr\\'echet spaces $E$ and $F$ or for nuclear $DF$-spaces $E$ and $F$; nuclear biprojective K\\\"{o}the algebras $\\lambda(P)$ which are Fr\\'echet spaces or $DF$-spaces; the algebra of distributions $\\mathcal{E}^*(G)$ on a compact Lie group $G$."}
{"category": "Math", "title": "Curvature estimates for Weingarten hypersurfaces in Riemannian manifolds", "abstract": "We prove curvature estimates for general curvature functions. As an application we show the existence of closed, strictly convex hypersurfaces with prescribed curvature $F$, where the defining cone of $F$ is $\\C_+$. $F$ is only assumed to be monotone, symmetric, homogeneous of degree 1, concave and of class $C^{m,\\al}$, $m\\ge4$."}
{"category": "Math", "title": "Almost sure functional central limit theorem for non-nestling random walk in random environment", "abstract": "We consider a non-nestling random walk in a product random environment. We assume an exponential moment for the step of the walk, uniformly in the environment. We prove an invariance principle (functional central limit theorem) under almost every environment for the centered and diffusively scaled walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively."}
{"category": "Math", "title": "Theta constants identities for Jacobians of cyclic 3-sheeted covers of the sphere and representations of the symmetric group", "abstract": "We find identities between theta constants with rational characteristics evaluated at period matrix of $R,$ a cyclic 3 sheeted cover of the sphere with $3k$ branch points $\\lambda_1...\\lambda_{3k}.$ These identities follow from Thomae formula \\cite{BR}. This formula expresses powers of theta constants as polynomials in $\\lambda_1...\\lambda_{3k}.$ We apply the representation of the symmetric group to find relations between the polynomials and hence between the associated theta constants."}
{"category": "Math", "title": "Topology of spaces of equivariant symplectic embeddings", "abstract": "We compute the homotopy type of the space of T^n-equivariant symplectic embeddings from the standard 2n-dimensional ball of some fixed radius into a 2n-dimensional symplectic-toric manifold M, and use this computation to define a Z-valued step function on the positive real line which is an invariant of the symplectic-toric type of M. We conclude with a discussion of the partially equivariant case of this result."}
{"category": "Math", "title": "Toric symplectic ball packing", "abstract": "We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric-combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory. Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section."}
{"category": "Math", "title": "Maximal ball packings of symplectic-toric manifolds", "abstract": "Let M be a symplectic-toric manifold of dimension at least four. This paper investigates the so called symplectic ball packing problem in the toral equivariant setting. We show that the set of toric symplectic ball packings of M admits the structure of a convex polytope. Previous work of the first author shows that up to equivalence, only CP^1 x CP^1 and CP^2 admit density one packings when n=2 and only CP^n admits density one packings when n>2. In contrast, we show that for a fixed n>=2 and each r in (0, 1), there are uncountably many inequivalent 2n-dimensional symplectic-toric manifolds with a maximal toric packing of density r. This result follows from a general analysis of how the densities of maximal packings change while varying a given symplectic-toric manifold through a family of symplectic-toric manifolds that are equivariantly diffeomorphic but not equivariantly symplectomorphic."}
{"category": "Math", "title": "Symplectic homology, autonomous Hamiltonians, and Morse-Bott moduli spaces", "abstract": "We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between linearized contact homology of a fillable contact manifold and symplectic homology of its filling."}
{"category": "Math", "title": "Higher dimensional conundra", "abstract": "We study asymptotics of various Euclidean geometric phenomena as the dimension tend to infinity."}
{"category": "Math", "title": "Heights and metrics with logarithmic singularities", "abstract": "We prove lower bound and finiteness properties for arakelovian heights with respect to pre-log-log hermitian ample line bundles. These heights were introduced by Burgos, Kramer and K\\\"uhn, in their extension of the arithmetic intersection theory of Gillet and Soul\\'e, aimed to deal with hermitian vector bundles equipped with metrics admitting suitable logarithmic singularities. Our results generalize the corresponding properties for the heights of Bost-Gillet-Soul\\'e, as well as the properties established by Faltings for heights of points attached to hermitian line bundles whose metrics have logarithmic singularities. We also discuss various geometric constructions where such pre-log-log hermitian ample line bundles naturally arise."}
{"category": "Math", "title": "Working with 2s and 3s", "abstract": "We establish an equivalent condition to the validity of the Collatz conjecture, using elementary methods. We derive some conclusions and show several examples of our results. We also offer a variety of exercises, problems and conjectures."}
{"category": "Math", "title": "Decartes' Perfect Lens", "abstract": "We give a new, elementary, purely analytical development of \\textsc{Descartes}' theorem that a smooth connected surface is a perfect focusing lens if and only if it is a connected subset of the ovoid obtained by revolving a cartesian oval around its axis of symmetry."}
{"category": "Math", "title": "Observations on degenerate saddle point problems", "abstract": "We investigate degenerate saddle point problems, which can be viewed as limit cases of standard mixed formulations of symmetric problems with large jumps in coefficients. We prove that they are well-posed in a standard norm despite the degeneracy. By wellposedness we mean a stable dependence of the solution on the right-hand side. A known approach of splitting the saddle point problem into separate equations for the primary unknown and for the Lagrange multiplier is used. We revisit the traditional Ladygenskaya--Babu\\v{s}ka--Brezzi (LBB) or inf--sup condition as well as the standard coercivity condition, and analyze how they are affected by the degeneracy of the corresponding bilinear forms. We suggest and discuss generalized conditions that cover the degenerate case. The LBB or inf--sup condition is necessary and sufficient for wellposedness of the problem with respect to the Lagrange multiplier under some assumptions. The generalized coercivity condition is necessary and sufficient for wellposedness of the problem with respect to the primary unknown under some other assumptions. We connect the generalized coercivity condition to the positiveness of the minimum gap of relevant subspaces, and propose several equivalent expressions for the minimum gap. Our results provide a foundation for research on uniform wellposedness of mixed formulations of symmetric problems with large jumps in coefficients in a standard norm, independent of the jumps. Such problems appear, e.g., in numerical simulations of composite materials made of components with contrasting properties."}
{"category": "Math", "title": "Markov basis and Groebner basis of Segre-Veronese configuration for testing independence in group-wise selections", "abstract": "We consider testing independence in group-wise selections with some restrictions on combinations of choices. We present models for frequency data of selections for which it is easy to perform conditional tests by Markov chain Monte Carlo (MCMC) methods. When the restrictions on the combinations can be described in terms of a Segre-Veronese configuration, an explicit form of a Gr\\\"obner basis consisting of moves of degree two is readily available for performing a Markov chain. We illustrate our setting with the National Center Test for university entrance examinations in Japan. We also apply our method to testing independence hypotheses involving genotypes at more than one locus or haplotypes of alleles on the same chromosome."}
{"category": "Math", "title": "Microlocal Asymptotic Analysis in Algebras of Generalized Functions", "abstract": "We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators."}
{"category": "Math", "title": "B-pairs and (phi,Gamma)-modules", "abstract": "Main change from v1 : theorem C has been modified, see remark 3.1.7 (2). We study the category of B-pairs (W_e,W_dR^+) where W_e is a free B_cris^{phi=1}-module with a semilinear and continuous action of G_K and where W_dR^+ is a G_K-stable B_dR^+ -lattice in B_dR \\otimes W_e. This category contains the category of p-adic representations and is naturally equivalent to the category of all (phi,Gamma)-modules over the Robba ring."}
{"category": "Math", "title": "On the weight structure of cyclic codes over $GF(q)$, $q>2$", "abstract": "The interrelation between the cyclic structure of an ideal, i.e., a cyclic code over Galois field $GF(q)$, $q>2$, and its classes of proportional elements is considered. This relation is used in order to define the code's weight structure. The equidistance conditions of irreducible nonprimitive codes over GF(q) are given. Besides that, the minimum distance for some class of nonprimitive cyclic codes is found."}
{"category": "Math", "title": "Orbits of tori extended by finite groups and their polynomial hulls: the case of connected complex orbits", "abstract": "Let $V$ be a complex linear space, $G\\subset\\GL(V)$ be a compact group. We consider the problem of description of polynomial hulls $\\wh{Gv}$ for orbits $Gv$, $v\\in V$, assuming that the identity component of $G$ is a torus $T$. The paper contains a universal construction for orbits which satisfy the inclusion $Gv\\subset T^\\bbC v$ and a characterization of pairs $(G,V)$ such that it is true for a generic $v\\in V$. The hull of a finite union of $T$-orbits in $T^\\bbC v$ can be distinguished in $\\clos T^\\bbC v$ by a finite collection of inequalities of the type $\\abs{z_1}^{s_1}...\\abs{z_n}^{s_n}\\leq c$. In particular, this is true for $Gv$. If powers in the monomials are independent of $v$, $Gv\\subset T^\\bbC v$ for a generic $v$, and either the center of $G$ is finite or $T^\\bbC$ has an open orbit, then the space $V$ and the group $G$ are products of standard ones; the latter means that $G=S_nT$, where $S_n$ is the group of all permutations of coordinates and $T$ is either $\\bbT^n$ or $\\SU(n)\\cap\\bbT^n$, where $\\bbT^n$ is the torus of all diagonal matrices in $\\rU(n)$. The paper also contains a description of polynomial hulls for orbits of isotropy groups of bounded symmetric domains. This result is already known, but we formulate it in a different form and supply with a shorter proof."}
{"category": "Math", "title": "Transitive powers of Young-Jucys-Murphy elements are central", "abstract": "Although powers of the Young-Jucys-Murphya elements X_i = (1 i) + ... +(i-1 i), i = 1, ..., n, in the symmetric group S_n acting on {1, ...,n} do not lie in the centre of the group algebra of S_n, we show that transitive powers, namely the sum of the contributions from elements that act transitively on {1, >...,n}, are central. We determine the coefficients, which we call star factorization numbers, that occur in the resolution of transitive powers with respect to the class basis of the centre of S_n, and show that they have a polynomiality property. These centrality and polynomiality properties have seemingly unrelated consequences. First, they answer a question raised by Pak about reduced decompositions; second, they explain and extend the beautiful symmetry result discovered by Irving and Rattan; and thirdly, we relate the polynomiality to an existing polynomiality result for a class of double Hurwitz numbers associated with branched covers of the sphere, which therefore suggests that there may be an ELSV-type formula associated with the star factorization numbers."}
{"category": "Math", "title": "On the S_n-module structure of the noncommutative harmonics", "abstract": "Using a noncommutative analog of Chevalley's decomposition of polynomials into symmetric polynomials times coinvariants due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius series for their two sets of noncommutative harmonics with respect to the left action of the symmetric group (acting on variables). We use these results to derive the Frobenius series for the enveloping algebra of the derived free Lie algebra in n variables."}
{"category": "Math", "title": "Classification of Noncommuting Quadrilaterals of Factors", "abstract": "A quadrilateral of factors is an irreducible inclusion of factors $N \\subset M$ with intermediate subfactors $P$ and $Q$ such that $P$ and $Q$ generate $M$ and the intersection of $P$ and $Q$ is $N$. We investigate the structure of a non-commuting quadrilateral of factors with all the elementary inclusions $P\\subset M$, $Q\\subset M$, $N\\subset P$, and $N\\subset Q$ 2-supertransitive. In particular we classify such quadrilaterals with the indices of the elementary subfactors less than or equal to 4. We also compute the angles between $P$ and $Q$ for quadrilaterals coming from $\\alpha$-induction and asymptotic inclusions."}
{"category": "Math", "title": "Subfactors and Hadamard Matrices", "abstract": "To any complex Hadamard matrix H one associates a spin model commuting square, and therefore a hyperfinite subfactor. The standard invariant of this subfactor captures certain \"group-like\" symmetries of H. To gain some insight, we compute the first few relative commutants of such subfactors for Hadamard matrices of small dimensions. Also, we show that subfactors arising from Dita type matrices have intermediate subfactors, and thus their standard invariants have some extra structure besides the Jones projections."}
{"category": "Math", "title": "High-dimensional variable selection", "abstract": "This paper explores the following question: what kind of statistical guarantees can be given when doing variable selection in high-dimensional models? In particular, we look at the error rates and power of some multi-stage regression methods. In the first stage we fit a set of candidate models. In the second stage we select one model by cross-validation. In the third stage we use hypothesis testing to eliminate some variables. We refer to the first two stages as \"screening\" and the last stage as \"cleaning.\" We consider three screening methods: the lasso, marginal regression, and forward stepwise regression. Our method gives consistent variable selection under certain conditions."}
{"category": "Math", "title": "Sharp Asymptotics for KPP Pulsating Front Speed-up and Diffusion Enhancement by Flows", "abstract": "We study KPP pulsating front speed-up and effective diffusivity enhancement by general periodic incompressible flows. We prove the existence of and determine the limits $c^*(A)/A$ and $D(A)/A^2$ as $A\\to\\infty$, where $c^*(A)$ is the minimal front speed and $D(A)$ the effective diffusivity."}
{"category": "Math", "title": "Pulsating Front Speed-up and Quenching of Reaction by Fast Advection", "abstract": "We consider reaction-diffusion equations with combustion-type non-linearities in two dimensions and study speed-up of their pulsating fronts by general periodic incompressible flows with a cellular structure. We show that the occurence of front speed-up in the sense $\\lim_{A\\to\\infty} c_*(A)=\\infty$, with $A$ the amplitude of the flow and $c_*(A)$ the (minimal) front speed, only depends on the geometry of the flow and not on the reaction function. In particular, front speed-up happens for KPP reactions if and only if it does for ignition reactions. We also show that the flows which achieve this speed-up are precisely those which, when scaled properly, are able to quench any ignition reaction."}
{"category": "Math", "title": "Deaconstructing Functions on Quadratic Surfaces into Multipoles", "abstract": "Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\\lambda + \\sum_{k = 1}^d [\\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are linear forms in $x, y$ and $z$ and $\\lambda$ is a real number. The coefficients of these linear forms, viewed as 3D vectors, are called \\emph{multipole} vectors of $P$. In this paper we consider similar multipole representations of polynomial and analytic functions on other quadratic surfaces $Q(x, y, z) = c$, real and complex. Over the complex numbers, the above representation is not unique, although the ambiguity is essentially finite. We investigate the combinatorics that depicts this ambiguity. We link these results with some classical theorems of harmonic analysis, theorems that describe decompositions of functions into sums of spherical harmonics. We extend these classical theorems (which rely on our understanding of the Laplace operator $\\Delta_{S^2}$) to more general differential operators $\\Delta_Q$ that are constructed with the help of the quadratic form $Q(x, y, z)$. Then we introduce modular spaces of multipoles. We study their intricate geometry and topology using methods of algebraic geometry and singularity theory. The multipole spaces are ramified over vector or projective spaces, and the compliments to the ramification sets give rise to a rich family of $K(\\pi, 1)$-spaces, where $\\pi$ runs over a variety of modified braid groups."}
{"category": "Math", "title": "Arithmetic homology and an integral version of Katos conjecture", "abstract": "We define an integral Borel-Moore homology theory over finite fields, called arithmetic homology, and an integral version of Kato homology. Both types of groups are expected to be finitely generated, and sit in a long exact sequence with higher Chow groups of zero-cycles."}
{"category": "Math", "title": "On the Kaehler rank of compact complex surfaces", "abstract": "Harvey and Lawson introduced the Kaehler rank and computed it in connection to the cone of positive exact currents of bidimension (1,1) for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions."}
{"category": "Math", "title": "Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations", "abstract": "The large time behavior of solutions to Cauchy problem for viscous Hamilton-Jacobi equation is classified. The large time asymptotics are given by very singular self-similar solutions on one hand and by self-similar viscosity solutions on the other hand"}
{"category": "Math", "title": "Asymptotic profiles of solutions to convection-diffusion equations", "abstract": "The large time behavior of zero mass solutions to the Cauchy problem for a convection-diffusion equation. We provide conditions on the size and shape of the initial datum such that the large time asymptotics of solutions is given either by the derivative of the Guass-Weierstrass kernel or by a self-similar solution or by a hyperbolic N-wave"}
{"category": "Math", "title": "The Manin conjecture in dimension 2", "abstract": "These lecture notes describe the current state of affairs for Manin's conjecture in the context of del Pezzo surfaces."}
{"category": "Math", "title": "Dynamics of the Tippe Top via Routhian Reduction", "abstract": "We consider a tippe top modeled as an eccentric sphere, spinning on a horizontal table and subject to a sliding friction. Ignoring translational effects, we show that the system is reducible using a Routhian reduction technique. The reduced system is a two dimensional system of second order differential equations, that allows an elegant and compact way to retrieve the classification of tippe tops in six groups as proposed in [1] according to the existence and stability type of the steady states."}
{"category": "Math", "title": "Quadratic BSDEs with random terminal time and elliptic PDEs in infinite dimension", "abstract": "In this paper we study one dimensional backward stochastic differential equations (BSDEs) with random terminal time not necessarily bounded or finite when the generator F(t,Y,Z) has a quadratic growth in Z. We provide existence and uniqueness of a bounded solution of such BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained results are then applied to prove existence and uniqueness of a mild solution to elliptic partial differential equations in Hilbert spaces."}
{"category": "Math", "title": "Pseudodifferential operators and weighted normed symbol spaces", "abstract": "In this work we study some general classes of pseudodifferential operators whose symbols are defined in terms of phase space estimates."}
{"category": "Math", "title": "Entwining Structures in Monoidal Catrgories", "abstract": "Interpreting entwining structures as special instances of J. Beck's distributive law, the concept of entwining module can be generalized for the setting of arbitrary monoidal category. In this paper, we use the distributive law formalism to extend in this setting basic properties of entwining modules."}
{"category": "Math", "title": "Sur les repr\\'esentations du groupe fondamental d'une vari\\'et\\'e priv\\'ee d'un diviseur \\`a croisements normaux simples", "abstract": "Given a projective variety X over an algebraically closed field of characteristic zero, we show that finite parabolic bundles along a fixed simple normal crossings divisor D are in one to one correspondence with representations of the \\'etale fundamental group of X-D."}
{"category": "Math", "title": "Tannakian Categories attached to abelian Varieties", "abstract": "Starting from certain perverse sheaves on an abelian variety, including the intersection cohomology sheaves of curves and smooth ample divisors, we construct a semisimple super-Tannakian category."}
{"category": "Math", "title": "Cluster tilting for one-dimensional hypersurface singularities", "abstract": "In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfy $\\tau^2=\\id$. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities."}
{"category": "Math", "title": "Complexity of Janet basis of a D-module", "abstract": "We prove a double-exponential upper bound on the degree and on the complexity of constructing a Janet basis of a $D$-module. This generalizes a well known bound on the complexity of a Gr\\\"obner basis of a module over the algebra of polynomials. We would like to emphasize that the obtained bound can not be immediately deduced from the commutative case."}
{"category": "Math", "title": "Intersection local time for two independent fractional Brownian motions", "abstract": "We prove the existence of the intersection local time for two independent, d -dimensional fractional Brownian motions with the same Hurst parameter H. Assume d greater or equal to 2, then the intersection local time exists if and only if Hd<2."}
{"category": "Math", "title": "Dynamics of shear homeomorphisms of tori and the Bestvina-Handel algorithm", "abstract": "Sharkovskii proved that the existence of a periodic orbit in a one-dimensional dynamical system implies existence of infinitely many periodic orbits. We obtain an analog of Sharkovskii's theorem for periodic orbits of shear homeomorphisms of the torus. This is done by obtaining a dynamical order relation on the set of simple orbits and simple pairs. We then use this order relation for a global analysis for a quantum chaotic physical system called the kicked accelerated particle."}
{"category": "Math", "title": "Coniveau over $p$-adic fields and points over finite fields", "abstract": "If the $\\ell$-adic cohomology of a projective smooth variety, defined over a $\\frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $\\ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This slightly improves our earlier result math/0405318: we needed there the model to be regular (but then our result was more general: we obtained a congruence for the number of points, and $K$ could be local of characteristic $p>0$)."}
{"category": "Math", "title": "A geometric proof that $e$ is irrational and a new measure of its irrationality", "abstract": "We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q > 1$, then $|e - p/q| > 1/(S(q)+1)!$, where $S(q)$ is the smallest positive integer such that $S(q)!$ is a multiple of $q$. We relate this measure for $e$ to a known one and to the greatest prime factor of an integer. We make two conjectures and recall a theorem of Cantor that can be proved by a similar construction."}
{"category": "Math", "title": "Center Manifold and Lie Symmetry Calculations on a Quasi-chemical Model for Growth-death Kinetics in Food", "abstract": "Food scientists at the U.S. Army's Natick Solider Center have developed a model for the lifecyle of the bacteria Staphylococcus aureus in intermediate moisture bread. In this article, we study this model using dynamical systems and Lie symmetry methods. We calculate center manifolds and Lie symmetries for different cases of parameter values and compare our results to those of the food scientists."}
{"category": "Math", "title": "Study of a finite volume - finite element scheme for a nuclear transport model", "abstract": "We consider a problem of nuclear waste contamination. It takes into account the thermal effects. The temperature and the contaminant's concentration fulfill convection-diffusion-reaction equations. The velocity and the pressure in the flow satisfy the Darcy equation, with a viscosity depending on both concentration and temperature. The equations are nonlinear and strongly coupled. Using both finite volume and nonconforming finite element methods, we introduce a scheme adapted to this problem. We prove the stability and convergence of this scheme and give some error estimates."}
{"category": "Math", "title": "Two characterizations of crooked functions", "abstract": "We give two characterizations of crooked functions: one based on the minimum distance of a Preparata-like code, and the other based on the distance-regularity of a crooked graph."}
{"category": "Math", "title": "Thistlethwaite's theorem for virtual links", "abstract": "The celebrated Thistlethwaite theorem relates the Jones polynomial of a link with the Tutte polynomial of the corresponding planar graph. We give a generalization of this theorem to virtual links. In this case, the graph will be embedded into a (higher genus) surface. For such graphs we use the generalization of the Tutte polynomial discovered by B.Bollobas and O.Riordan."}
{"category": "Math", "title": "Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise", "abstract": "We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t,x), and upper bounds of Gaussian-type on the two-point density of (u(s,y),u(t,x)). In particular, this estimate quantifies how this density degenerates as (s,y) converges to (t,x). From these results, we deduce upper and lower bounds on hitting probabilities of the process {u(t,x)}_{t \\in \\mathbb{R}_+, x \\in [0,1]}, in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d >6 and are not polar when d<6. We also show that the Hausdorff dimension of the range of the process is 6 when d>6, and give analogous results for the processes t \\mapsto u(t,x) and x \\mapsto u(t,x). Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes."}
{"category": "Math", "title": "Mutant knots and intersection graphs", "abstract": "We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. The converse statement is easy and well known. We discuss relationship between our results and certain Lie algebra weight systems."}
{"category": "Math", "title": "On the largest prime factor of the Mersenne numbers", "abstract": "Let P(k) be the largest prime factor of the positive integer k. In this paper, we prove that the series $\\sum_{n\\ge 1}\\frac{(\\log n)^a}{P(2^n-1)}$ is convergent for each constant a<1/2, which gives a more precise form of a result of C. L. Stewart from 1977."}
{"category": "Math", "title": "On the classification of Floer-type theories", "abstract": "In this paper we outline a program for the classification of Floer-type theories, (or defining invariants of finite type for families). We consider Khovanov complexes as a local system on the space of knots introduced by V. Vassiliev and construct the wall-crossing morphism. We extend this system to the singular locus by the cone of this morphism and introduce the definition of the local system of finite type. This program can be further generalized to the manifolds of dimension 3 and 4."}
{"category": "Math", "title": "Siegel's theorem for Drinfeld modules", "abstract": "We prove an analog of Siegel's theorem for integral points in the context of Drinfeld modules. The result holds for finitely generated submodules of the additive group over a function field of transcendence dimension 1."}
{"category": "Math", "title": "A dynamical version of the Mordell-Lang conjecture for the additive group", "abstract": "We prove a dynamical version of the Mordell-Lang conjecture in the context of Drinfeld modules. We use analytic methods similar to the ones employed by Skolem, Chabauty, and Coleman for studying diophantine equations."}
{"category": "Math", "title": "On the reductive Borel-Serre compactification: $L^p$-cohomology of arithmetic groups (for large $p$)", "abstract": "The $L^2$-cohomology of a locally symmetric variety is known to have the topological interpretation as the intersection homology of its Baily-Borel Satake compactification. In this article, we observe that even without the Hermitian hypothesis, the $L^p$-cohomology of an arithmetic quotient, for $p$ finite and sufficiently large, is isomorphic to the ordinary cohomology of its reductive Borel-Serre compactification. We use this to generalize a theorem of Mumford concerning homogeneous vector bundles, their invariant Chern forms and the canonical extensions of the bundles; here, though, we are referring to canonical extensions to the reductive Borel-Serre compactification of any arithmetic quotient. To achieve that, we give a systematic discussion of vector bundles and Chern classes on stratified"}
{"category": "Math", "title": "Tautological classes on moduli spaces of curves with linear series and a push-forward formula when $\\rho=0$", "abstract": "We define tautological Chow classes on the moduli space of curves with linear series. In the case where the forgetful morphism to the moduli space of curves has relative dimension zero, we describe the images of these classes in the Chow group of Mgbar. As an application, we compute the (virtual) slopes of several different classes of divisors on Mgbar."}
{"category": "Math", "title": "Equivariant symmetric bilinear torsions", "abstract": "We extend the main result in the previous paper of Zhang and the author relating the Milnor-Turaev torsion with the complex valued analytic torsion to the equivariant case."}
{"category": "Math", "title": "Hardy and Rellich type inequalities with remainders for Baouendi-Grushin vector fields", "abstract": "In this paper we study Hardy and Rellich type inequalities for Baouendi-Grushin vector fields : $\\nabla_{\\gamma}=(\\nabla_x, |x|^{2\\gamma}\\nabla_y)$ where $\\gamma>0$, $\\nabla_x$ and $\\nabla_y$ are usual gradient operators in the variables $x\\in \\mathbb{R}^m$ and $y\\in\\mathbb{R}^k$, respectively. In the first part of the paper, we prove some weighted Hardy type inequalities with remainder terms. In the second part, we prove two versions of weighted Rellich type inequality on the whole space. We find sharp constants for these inequalities. We also obtain their improved versions for bounded domains."}
{"category": "Math", "title": "Comparaison entre cohomologie cristalline et cohomologie \\'etale $p$-adique sur certaines vari\\'et\\'es de Shimura", "abstract": "Let $X$ be an integral model at a prime $p$ of a Shimura variety of PEL type having good reduction, associated to a reductive group $G$. To $\\mathbb{Z}_p$ reprsententations of the group $G$ can be associated two kinds of sheaves : crystals on the special fiber of $X$, and locally constant \\'etale sheaves on the generic fiber. We establish a comparison between the cohomology of these two kinds of sheaves."}
{"category": "Math", "title": "Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients", "abstract": "This work is devoted to the strong unique continuation problem for second order parabolic equations with nonsmooth coefficients. Introduction and bibliography have been revised."}
{"category": "Math", "title": "The Green function estimates for strongly elliptic systems of second order", "abstract": "We establish existence and pointwise estimates of fundamental solutions and Green's matrices for divergence form, second order strongly elliptic systems in a domain $\\Omega \\subseteq \\mathbb{R}^n$, $n \\geq 3$, under the assumption that solutions of the system satisfy De Giorgi-Nash type local H\\\"{o}lder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation."}
{"category": "Math", "title": "A Dynamic Algorithm for Blind Separation of Convolutive Sound Mixtures", "abstract": "We study an efficient dynamic blind source separation algorithm of convolutive sound mixtures based on updating statistical information in the frequency domain, andminimizing the support of time domain demixing filters by a weighted least square method. The permutation and scaling indeterminacies of separation, and concatenations of signals in adjacent time frames are resolved with optimization of $l^1 \\times l^\\infty$ norm on cross-correlation coefficients at multiple time lags. The algorithm is a direct method without iterations, and is adaptive to the environment. Computations on recorded and synthetic mixtures of speech and music signals show excellent performance."}
{"category": "Math", "title": "On families of rational curves in the Hilbert square of a surface (with an Appendix by Edoardo Sernesi)", "abstract": "Under natural hypotheses we give an upper bound on the dimension of families of singular curves with hyperelliptic normalizations on a surface S with p_g(S) >0 via the study of the associated families of rational curves in Hilb^2(S). We use this result to prove the existence of nodal curves of geometric genus 3 with hyperelliptic normalizations, on a general K3 surface, thus obtaining specific 2-dimensional families of rational curves in its Hilbert square. We describe two infinite series of examples of general, primitively polarized K3's such that their Hilbert squares contain a IP^2 or a threefold birational to a IP^1-bundle over a K3. We discuss some consequences on the Mori cone of the Hilbert square of a general K3."}
{"category": "Math", "title": "Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients", "abstract": "Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on $\\mathbb{R}^n$. In particular, in the case when $n=2$ they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded, measurable, time-independent coefficients, and extend their results to the systems of parabolic equations."}
{"category": "Math", "title": "Triangulated categories without models", "abstract": "We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories."}
{"category": "Math", "title": "U-max-Statistics", "abstract": "In 1948, W. Hoeffding introduced a large class of unbiased estimators called U-statistics, defined as the average value of a real-valued k-variate function h calculated at all possible sets of k points from a random sample. In the present paper we investigate the corresponding extreme value analogue, which we shall call U-max-statistics. We are concerned with the behavior of the largest value of such function h instead of its average. Examples of U-max-statistics are the diameter or the largest scalar product within a random sample. U-max-statistics of higher degrees are given by triameters and other metric invariants."}
{"category": "Math", "title": "Generalizing circles over algebraic extensions", "abstract": "This paper deals with a family of spatial rational curves that were introduced by Andradas, Recio and Sendra, under the name of hypercircles, as an algorithmic cornerstone tool in the context of improving the rational parametrization (simplifying the coefficients of the rational functions, when possible) of algebraic varieties. A real circle can be defined as the image of the real axis under a Moebius transformation in the complex field. Likewise, and roughly speaking, a hypercircle can be defined as the image of a line (\"the ${\\mathbb{K}}$-axis\") in a $n$-degree finite algebraic extension $\\mathbb{K}(\\alpha)\\thickapprox\\mathbb{K}^n$ under the transformation $\\frac{at+b}{ct+d}:\\mathbb{K}(\\alpha)\\to\\mathbb{K}(\\alpha)$. The aim of this article is to extend, to the case of hypercircles, some of the specific properties of circles. We show that hypercircles are precisely, via $\\mathbb{K}$-projective transformations, the rational normal curve of a suitable degree. We also obtain a complete description of the points at infinity of these curves (generalizing the cyclic structure at infinity of circles). We characterize hypercircles as those curves of degree equal to the dimension of the ambient affine space and with infinitely many ${\\mathbb{K}}$-rational points, passing through these points at infinity. Moreover, we give explicit formulae for the parametrization and implicitation of hypercircles. Besides the intrinsic interest of this very special family of curves, the understanding of its properties has a direct application to the simplification of parametrizations problem, as shown in the last section."}
{"category": "Math", "title": "Decreasing families of dynamically determined intervals in the power-law family", "abstract": "We study the rate of growth of ratios of intervals delimited by the post-critical orbit of a map in the quasi-quadratic family $x\\mapsto -|x|^\\alpha +a.$ The critical order $\\alpha$ is an arbitrary real number $\\alpha>1.$ The range of the parameter $a$ is confined to an interval $(1,a_{\\alpha})$ of length depending on the critical order. We prove that in every power-law family there is a unique parameter $p_{\\alpha}$ corresponding to the kneading sequence $RLRRRLRC.$ Subsequently, we obtain monotonicity results concerning ratios of all intervals labeled by infinite post-critical orbit in the case of the kneading sequence $RLRL...$ This extends the results from \\cite{P}, via refinement of the tools based on special properties of power-law mappings in non-euclidean metric."}
{"category": "Math", "title": "The p-adic generalized twisted (h,q)-euler-l-function and its applications", "abstract": "The purpose of this paper is to construct the p-adic twisted (h,q)-Euler-l-function, which interpolates the twisted generalized twisted Euler numbers attached to chi at a negative integer."}
{"category": "Math", "title": "La formule de Lie-Trotter pour les semi-groupes fortement continus", "abstract": "In this research project we presents the general properties, the spectral properties and the representation formulas for $C_0$-semigroups of linear operators in Banach spaces"}
{"category": "Math", "title": "Equivalences of Higher Derived Brackets", "abstract": "This note elaborates on Th. Voronov's construction [math/0304038,math/0412202] of $L_\\infty$-structures via higher derived brackets with a Maurer-Cartan element. It is shown that gauge equivalent Maurer-Cartan elements induce $L_\\infty$-isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are discussed."}
{"category": "Math", "title": "Sobolev solution for semilinear PDE with obstacle under monotonicity condition", "abstract": "We prove the existence and uniqueness of the solution of a semilinear PDE's and also PDE's with obstacle under monotonicity condition. Moreover we give the probabilistic interpretation of the Sobolev's solutions in term of Backward SDE and reflected Backward SDE respectively."}
{"category": "Math", "title": "Exact distribution of the sample variance from a gamma parent distribution", "abstract": "Several representations of the exact cdf of the sum of squares of n independent gamma-distributed random variables Xi are given, in particular by a series of gamma distribution functions. Using a characterization of the gamma distribution by Laha, an expansion of the exact distribution of the sample variance is derived by a Taylor series approach with the former distribution as its leading term. In particular for integer orders alpha some further series are provided, including a convex combination of gamma distributions for alpha = 1 and nearly of this type for alpha > 1. Furthermore, some representations of the distribution of the angle Phi between (X1,...,Xn) and (1,...,1) are given by orthogonal series. All these series are based on the same sequence of easily computed moments of cos(Phi)."}
{"category": "Math", "title": "Asymptotic stability at infinity for bidimensional Hurwitz vector fields", "abstract": "Let $X:U-->R^2$ be a differentiable vector field. Set $Spc(X)={eigenvalues of DX(z) : z\\in U}$. This $X$ is called Hurwitz if $Spc(X)\\subset{z\\in C:\\Re(z)<0}$. Suppose that $X$ is Hurwitz and $U\\subset R^2$ is the complement of a compact set. Then by adding to $X$ a constant $v$ one obtains that the infinity is either an attractor or a repellor for $X+v.$"}
{"category": "Math", "title": "Generalized characteristic polynomials of graph bundles", "abstract": "In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic polynomial of the graph. Since the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, consequently, the Bartholdi zeta function of a graph bundle can be computed by using our computational formulae."}
{"category": "Math", "title": "Some invariants of pretzel links", "abstract": "We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same."}
{"category": "Math", "title": "Curvature in Synthetic Differential Geometry of Groupoids", "abstract": "We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one."}
{"category": "Math", "title": "The classification ofseparable simple C*-algebras which are inductive limits of continuous-trace C*-algebraswith spectrum homeomorphic to the closed interval [0,1]", "abstract": "A classification is given of certain separable nuclear C*-algebras not necessarily of real rank zero, namely, the class of separable simple C*-algebras which are inductive limits of continuous-trace C*-algebras whose building blocks have spectrum homeomorphic to the closed interval [0,1], or to a disjoint union of copies of this space. Also, the range of the invariant is calculated."}
{"category": "Math", "title": "Even infinite dimensional real Banach spaces", "abstract": "This article is a continuation of a paper of the first author \\cite{F} about complex structures on real Banach spaces. We define a notion of even infinite dimensional real Banach space, and prove that there exist even spaces, including HI or unconditional examples from \\cite{F} and $C(K)$ examples due to Plebanek \\cite{P}. We extend results of \\cite{F} relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space, and give characterizations of even spaces in terms of this group. We also generalize results of \\cite{F} about totally incomparable complex structures to essentially incomparable complex structures, while showing that the complex version of a space defined by S. Argyros and A. Manoussakis \\cite{AM} provide examples of essentially incomparable complex structures which are not totally incomparable."}
{"category": "Math", "title": "Large deviations of Poisson cluster processes", "abstract": "In this paper we prove scalar and sample path large deviation principles for a large class of Poisson cluster processes. As a consequence, we provide a large deviation principle for ergodic Hawkes point processes."}
{"category": "Math", "title": "Sparse Estimators and the Oracle Property, or the Return of Hodges' Estimator", "abstract": "We point out some pitfalls related to the concept of an oracle property as used in Fan and Li (2001, 2002, 2004) which are reminiscent of the well-known pitfalls related to Hodges' estimator. The oracle property is often a consequence of sparsity of an estimator. We show that any estimator satisfying a sparsity property has maximal risk that converges to the supremum of the loss function; in particular, the maximal risk diverges to infinity whenever the loss function is unbounded. For ease of presentation the result is set in the framework of a linear regression model, but generalizes far beyond that setting. In a Monte Carlo study we also assess the extent of the problem in finite samples for the smoothly clipped absolute deviation (SCAD) estimator introduced in Fan and Li (2001). We find that this estimator can perform rather poorly in finite samples and that its worst-case performance relative to maximum likelihood deteriorates with increasing sample size when the estimator is tuned to sparsity."}
{"category": "Math", "title": "Williams' decomposition of the L\\'evy continuous random tree and simultaneous extinction probability for populations with neutral mutations", "abstract": "We consider an initial Eve-population and a population of neutral mutants, such that the total population dies out in finite time. We describe the evolution of the Eve-population and the total population with continuous state branching processes, and the neutral mutation procedure can be seen as an immigration process with intensity proportional to the size of the population. First we establish a Williams' decomposition of the genealogy of the total population given by a continuous random tree, according to the ancestral lineage of the last individual alive. This allows us give a closed formula for the probability of simultaneous extinction of the Eve-population and the total population."}
{"category": "Math", "title": "Wavelet frames, Bergman spaces and Fourier transforms of Laguerre functions", "abstract": "The Fourier transforms of Laguerre functions play the same canonical role in wavelet analysis as do the Hermite functions in Gabor analysis. We will use them as analyzing wavelets in a similar way the Hermite functions were recently by K. Groechenig and Y. Lyubarskii in \"Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007)\". Building on the work of K. Seip, \"Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39 (1993)\", concerning sampling sequences on weighted Bergman spaces, we find a sufficient density condition for constructing frames by translations and dilations of the Fourier transform of the nth Laguerre function. As in Groechenig-Lyubarskii theorem, the density increases with n, and in the special case of the hyperbolic lattice in the upper half plane it is given by b\\log a<\\frac{4\\pi}{2n+\\alpha}, where alpha is the parameter of the Laguerre function."}
{"category": "Math", "title": "Where the monotone pattern (mostly) rules", "abstract": "We consider pattern containment and avoidance with a very tight definition that was used first by Riordan more than 60 years ago. Using this definition, we prove the monotone pattern is easier to avoid than almost any other pattern of the same length. We also show that with this definition, almost all patterns of length $k$ are avoided by the same number of permutations of length $n$. The corresponding statements are not known to be true for more relaxed definitions of pattern containment. This is the first time we know of that expectations are used to compare numbers of permutations avoiding certain patterns."}
{"category": "Math", "title": "On a {K_4,K_{2,2,2}}-ultrahomogeneous graph", "abstract": "The existence of a connected 12-regular $\\{K_4,K_{2,2,2}\\}$-ultrahomogeneous graph $G$ is established, (i.e. each isomorphism between two copies of $K_4$ or $K_{2,2,2}$ in $G$ extends to an automorphism of $G$), with the 42 ordered lines of the Fano plane taken as vertices. This graph $G$ can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of $K_4$ and as the edge-disjoint union of 21 induced copies of $K_{2,2,2}$, with no more copies of $K_4$ or $K_{2,2,2}$ existing in $G$. Moreover, each edge of $G$ is shared by exactly one copy of $K_4$ and one of $K_{2,2,2}$. While the line graphs of $d$-cubes, ($3\\le d\\in\\ZZ$), are $\\{K_d, K_{2,2}\\}$-ultrahomogeneous, $G$ is not even line-graphical. In addition, the chordless 6-cycles of $G$ are seen to play an interesting role and some self-dual configurations associated to $G$ with 2-arc-transitive, arc-transitive and semisymmetric Levi graphs are considered."}
{"category": "Math", "title": "A Diagrammatic Category for the Representation Theory of U_q(sl_n)", "abstract": "This thesis provides a partial answer to a question posed by Greg Kuperberg in q-alg/9712003 and again by Justin Roberts as problem 12.18 in \"Problems on invariants of knots and 3-manifolds\", math.GT/0406190, essentially: \"Can one describe the category of representations of the quantum group U_q(sl_n) (thought of as a spherical category) via generators and relations?\" For each n \\geq 0, I define a certain tensor category of trivalent graphs, modulo isotopy, and construct a functor from this category onto (a full subcategory of) the category of representations of the quantum group U_q(sl_n). One would like to describe completely the kernel of this functor, by providing generators. The resulting quotient of the diagrammatic category would then be a category equivalent to the representation category of U_q(sl_n). I make significant progress towards this, describing certain generators of the kernel, and some obstructions to further elements. It remains a conjecture that these relations generate the kernel. My results extend those of q-alg/9712003, MR1659228, math.QA/0310143 and math.GT/0506403. The argument is essentially by constructing a diagrammatic version of the forgetful functor coming from the inclusion of U_q(sl_{n-1}) in U_q(sl_n}. We know this functor is faithful, so a diagram is in the kernel for n exactly if its image under the diagrammatic forgetful functor is in the kernel for n-1. This allows us to perform inductive calculations, both establishing families of elements of the kernel, and finding obstructions."}
{"category": "Math", "title": "Concrete Classification and Centralizers of Certain $\\mathbb{Z}^2 \\rtimes {\\rm SL}(2,\\mathbb{Z})$-actions", "abstract": "We introduce a new class of actions of the group $\\G$ on finite von Neumann algebras and call them twisted Bernoulli shift actions. We classify these actions up to conjugacy and give an explicit description of their centralizers. We also distinguish many of those actions on the AFD $\\mathrm{II}_1$ factor in view of outer conjugacy."}
{"category": "Math", "title": "Estimates for singular integrals and extrapolation", "abstract": "We prove a sharp Lp estimate for a singular Radon transform according to a size condition of its kernel, which is useful for extrapolation."}
{"category": "Math", "title": "Riemannian and Lorentzian structures on the non symmetric space SO(2m)/Sp(m)", "abstract": "In this work, we are interested in a non symmetric homogeneous space, namely $SO(2m)/Sp(m)$. We show that this space admits a structure of $Z_2^2$-symmetric space. We describe all the non degenerated metrics and classify the Riemannian and Lorentzian ones."}
{"category": "Math", "title": "Discrete Nonholonomic Lagrangian Systems on Lie Groupoids", "abstract": "This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of the evolution operator and the discrete nonholonomic momentum equation are also considered. Finally, we illustrate with several classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile robot)."}
{"category": "Math", "title": "When the orbit algebra of group is an integral domain? Proof of a conjecture of P.J. Cameron", "abstract": "P.J.Cameron introduced the orbit algebra of a permutation group and conjectured that this algebra is an integral domain if and only if the group has no finite orbit. We prove that this conjecture holds and in fact that the age algebra of a relational structure $R$ is an integral domain if and only if $R$ is age-inexhaustible. We deduce these results from a combinatorial lemma asserting that if a product of two non-zero elements of a set algebra is zero then there is a finite common tranversal of their supports. The proof is built on Ramsey theorem and the integrity of a shuffle algebra."}
{"category": "Math", "title": "Saturated actions by finite dimensional Hopf *-algebras on C*-algebras", "abstract": "If a finite group action $\\alpha$ on a unital $C^*$-algebra $M$ is saturated, the canonical conditional expectation $E:M\\to M^\\alpha$ onto the fixed point algebra is known to be of index finite type with $Index(E)=|G|$ in the sense of Watatani. More generally if a finite dimensional Hopf $*$-algebra $A$ acts on $M$ and the action is saturated, the same is true with $Index (E)=\\dim(A)$. In this paper we prove that the converse is true. Especially in case $M$ is a commutative $C^*$-algebra $C(X)$ and $\\alpha$ is a finite group action, we give an equivalent condition in order that the expectation $E:C(X)\\to C(X)^\\alpha$ is of index finite type, from which we obtain that $\\alpha$ is saturated if and only if $G$ acts freely on $X$. Actions by compact groups are also considered to show that the gauge action $\\gamma$ on a graph $C^*$-algebra $C^*(E)$ associated with a locally finite directed graph $E$ is saturated."}
{"category": "Math", "title": "Matrix Ordered Operator Algebras", "abstract": "We study the question when for a given *-algebra $\\mathcal{A}$ a sequence of cones $C_n\\in M_n(\\mathcal{A})$ can be realized as cones of positive operators in a faithful *-representation of $\\mathcal{A}$ on a Hilbert space. A characterization of operator algebras which are completely boundedly isomorphic to $C\\sp*$-algebras is presented."}
{"category": "Math", "title": "Properly infinite C(X)-algebras and K_1-injectivity", "abstract": "We investigate if a unital C(X)-algebra is properly infinite when all its fibres are properly infinite. We show that this question can be rephrased in several different ways, including the question if every unital properly infinite C*-algebra is K_1-injective. We provide partial answers to these questions, and we show that the general question on proper infiniteness of C(X)-algebras can be reduced to establishing proper infiniteness of a specific C([0,1])-algebra with properly infinite fibres."}
{"category": "Math", "title": "A separable deformation of the quaternion group algebra", "abstract": "The Donald-Flanigan conjecture asserts that for any finite group and for any field, the corresponding group algebra can be deformed to a separable algebra. The minimal unsolved instance, namely the quaternion group over a field of characteristic 2 was considered as a counterexample. We present here a separable deformation of the quaternion group algebra. In a sense, the conjecture for any finite group is open again."}
{"category": "Math", "title": "On the residue fields of Henselian valued stable fields, II", "abstract": "Let $E$ be a primarily quasilocal field, $M/E$ a finite Galois extension and $D$ a central division $E$-algebra of index divisible by $[M\\colon E]$. In addition to the main result of Part I, this part of the paper shows that if the Galois group $G(M/E)$ is not nilpotent, then $M$ does not necessarily embed in $D$ as an $E$-subalgebra. When $E$ is quasilocal, we find the structure of the character group of its absolute Galois group; this enables us to prove that if $E$ is strictly quasilocal and almost perfect, then the divisible part of the multiplicative group $E ^{\\ast}$ equals the intersection of the norm groups of finite Galois extensions of $E$."}
{"category": "Math", "title": "A parachute for the degree of a polynomial in algebraically independent ones", "abstract": "We give a simpler proof as well as a generalization of the main result of an article of Shestakov and Umirbaev. This latter article being the first of two that solve a long-standing conjecture about the non-tameness, or \"wildness\", of Nagata's automorphism. As corollaries we get interesting informations about the leading terms of polynomials forming an automorphism in any dimension and reprove the tameness of automorphisms in dimension two."}
{"category": "Math", "title": "One-way permutations, computational asymmetry and distortion", "abstract": "Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of one-way transformations. We introduce a computational asymmetry function that measures the amount of one-wayness of permutations. We also introduce the word-length asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to word-length. Moreover, boolean circuits have a representation in terms of elements of a Thompson group, in such a way that circuit size is polynomially equivalent to word-length. We show that circuits built with gates that are not constrained to have fixed-length inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixed-length inputs and outputs). Finally, we show that the computational asymmetry function is closely related to certain distortion functions: The computational asymmetry function is polynomially equivalent to the distortion of the path length in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of certain Thompson monoids. We also show that the results of Razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions."}
{"category": "Math", "title": "On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups", "abstract": "Let T be a random field invariant under the action of a compact group G We give conditions ensuring that independence of the random Fourier coefficients is equivalent to Gaussianity. As a consequence, in general it is not possible to simulate a non-Gaussian invariant random field through its Fourier expansion using independent coefficients."}
{"category": "Math", "title": "L^2-Betti numbers of coamenable quantum groups", "abstract": "We prove that a compact quantum group is coamenable if and only if its corepresentation ring is amenable. We further propose a Foelner condition for compact quantum groups and prove it to be equivalent to coamenability. Using this Foelner condition, we prove that for a coamenable compact quantum group with tracial Haar state, the enveloping von Neumann algebra is dimension flat over the Hopf algebra of matrix coefficients. This generalizes a theorem of Lueck from the group case to the quantum group case, and provides examples of compact quantum groups with vanishing L^2-Betti numbers."}
{"category": "Math", "title": "Can One Estimate The Unconditional Distribution of Post-Model-Selection Estimators?", "abstract": "We consider the problem of estimating the unconditional distribution of a post-model-selection estimator. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model selection criterion like AIC or by a hypothesis testing procedure) and then estimating the parameters in the selected model (e.g., by least-squares or maximum likelihood), all based on the same data set. We show that it is impossible to estimate the unconditional distribution with reasonable accuracy even asymptotically. In particular, we show that no estimator for this distribution can be uniformly consistent (not even locally). This follows as a corollary to (local) minimax lower bounds on the performance of estimators for the distribution; performance is here measured by the probability that the estimation error exceeds a given threshold. These lower bounds are shown to approach 1/2 or even 1 in large samples, depending on the situation considered. Similar impossibility results are also obtained for the distribution of linear functions (e.g., predictors) of the post-model-selection estimator."}
{"category": "Math", "title": "On algebraic automorphisms and their rational invariants", "abstract": "Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism F, we denote by k(X)^F its field of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let n(F) be the transcendence degree of k(X)^F over k. In this paper, we study the class of automorphisms F of X for which n(F)= dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form F=A_g, where A is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(F)=1."}
{"category": "Math", "title": "An adaptive numerical method for the Vlasov equation based on a multiresolution analysis", "abstract": "In this paper, we present very first results for the adaptive solution on a grid of the phase space of the Vlasov equation arising in particles accelarator and plasma physics. The numerical algorithm is based on a semi-Lagrangian method while adaptivity is obtained using multiresolution analysis."}
{"category": "Math", "title": "Critical points for surface maps and the Benedicks-Carleson theorem", "abstract": "We give an alternative proof of the Benedicks-Carleson theorem on the existence of strange attractors in H\\'enon-like families in the plane. To bypass a huge inductive argument, we introduce an induction-free explicit definition of dynamically critical points. The argument is sufficiently general and in particular applies to the case of non-invertible maps as well. It naturally raises the question of an intrinsic characterization of dynamically critical points for dissipative surface maps."}
{"category": "Math", "title": "Complete Segal spaces arising from simplicial categories", "abstract": "In this paper, we compare several functors which take simplicial categories or model categories to complete Segal spaces, which are particularly nice simplicial spaces which, like simplicial categories, can be considered to be models for homotopy theories. We then give a characterization, up to weak equivalence, of complete Segal spaces arising from these functors."}
{"category": "Math", "title": "A Systematic Scan for 7-colourings of the Grid", "abstract": "We study the mixing time of a systematic scan Markov chain for sampling from the uniform distribution on proper 7-colourings of a finite rectangular sub-grid of the infinite square lattice, the grid. A systematic scan Markov chain cycles through finite-size subsets of vertices in a deterministic order and updates the colours assigned to the vertices of each subset. The systematic scan Markov chain that we present cycles through subsets consisting of 2x2 sub-grids and updates the colours assigned to the vertices using a procedure known as heat-bath. We give a computer-assisted proof that this systematic scan Markov chain mixes in O(log n) scans, where n is the size of the rectangular sub-grid. We make use of a heuristic to compute required couplings of colourings of 2x2 sub-grids. This is the first time the mixing time of a systematic scan Markov chain on the grid has been shown to mix for less than 8 colours. We also give partial results that underline the challenges of proving rapid mixing of a systematic scan Markov chain for sampling 6-colourings of the grid by considering 2x3 and 3x3 sub-grids."}
{"category": "Math", "title": "Hilbert Spaces with Generic Predicates", "abstract": "We study the model theory of expansions of Hilbert spaces by generic predicates. We first prove the existence of model companions for generic expansions of Hilbert spaces in the form first of a distance function to a random substructure, then a distance to a random subset. The theory obtained with the random substructure is {\\omega}-stable, while the one obtained with the distance to a random subset is $TP_2$ and $NSOP_1$. That example is the first continuous structure in that class."}
{"category": "Math", "title": "Weak Amenability of Hyperbolic Groups", "abstract": "We prove that hyperbolic groups are weakly amenable. This partially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable. We take a combinatorial approach in the spirit of Haagerup and prove that for the word length metric d on a hyperbolic group, the Schur multipliers associated with r^d have uniformly bounded norms for 0<r<1. We then combine this with a Bozejko-Picardello type inequality to obtain weak amenability."}
{"category": "Math", "title": "Bergman kernels and equilibrium measures for ample line bundles", "abstract": "Let L be an ample holomorphic line bundle over a compact complex Hermitian manifold X. Any fixed smooth Hermitian metric on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k:th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels. The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric associated to the fixed metric, as well as in terms of the Monge-Ampere measure of the fixed metric itself on a certain support set. It is also shown that the equilibrium metric has Lipschitz continuous first derivatives. These results can be seen as generalizations of well-known results concerning the case when the curvature of the fixed metric is positive (the corresponding equilibrium metric is then simply the fixed metric itself)."}
{"category": "Math", "title": "The LIL for $U$-statistics in Hilbert spaces", "abstract": "We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for $U$-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued $U$-statistics of arbitrary order, which are of independent interest."}
{"category": "Math", "title": "On the support genus of a contact structure", "abstract": "The algorithm given by Akbulut-Ozbagci constructs an explicit open book decomposition on a contact three-manifold described by a contact surgery on a link in the three-sphere. In this article, we will improve this algorithm by using Giroux's contact cell decomposition process. Our algorithm is more economical on choosing the supporting genus of the open book; in particular it gives a good upper bound for the recently defined ``minimal supporting genus invariant'' of contact structures."}
{"category": "Math", "title": "Holographic formula for Q-curvature", "abstract": "This paper derives an explicit formula for Branson's Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincare metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses Q-curvature as a multiple of the Pfaffian and the divergence of a natural one-form. The paper also outlines the relation between holographic formulae for Q-curvature and a new theory of conformally covariant families of differential operators due to the second author."}
{"category": "Math", "title": "Spectral averaging for trace compatible operators", "abstract": "In this note the notions of trace compatible operators and infinitesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Krein's formula is established. Some examples of trace compatible affine spaces of operators are given."}
{"category": "Math", "title": "Some Properties of and Open Problems on Hessian Nilpotent Polynomials", "abstract": "In the recent progress [BE1], [M], [Z1] and [Z2], the well-known Jacobian conjecture ([BCW], [E]) has been reduced to a problem on HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent) and their (deformed) inversion pairs. In this paper, we prove several results on HN polynomials, their (deformed) inversion pairs as well as the associated symmetric polynomial or formal maps. We also propose some open problems for further study of these objects."}
{"category": "Math", "title": "Two Results on Homogeneous Hessian Nilpotent Polynomials", "abstract": "Let $z=(z_1, ..., z_n)$ and $\\Delta=\\sum_{i=1}^n \\frac {\\partial^2}{\\partial z^2_i}$ the Laplace operator. A formal power series $P(z)$ is said to be {\\it Hessian Nilpotent}(HN) if its Hessian matrix $\\Hes P(z)=(\\frac {\\partial^2 P}{\\partial z_i\\partial z_j})$ is nilpotent. In recent developments in [BE1], [M] and [Z], the Jacobian conjecture has been reduced to the following so-called {\\it vanishing conjecture}(VC) of HN polynomials: {\\it for any homogeneous HN polynomial $P(z)$ $($of degree $d=4$$)$, we have $\\Delta^m P^{m+1}(z)=0$ for any $m>>0$.} In this paper, we first show that, the VC holds for any homogeneous HN polynomial $P(z)$ provided that the projective subvarieties ${\\mathcal Z}_P$ and ${\\mathcal Z}_{\\sigma_2}$ of $\\mathbb C P^{n-1}$ determined by the principal ideals generated by $P(z)$ and $\\sigma_2(z):=\\sum_{i=1}^n z_i^2$, respectively, intersect only at regular points of ${\\mathcal Z}_P$. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps $F=z-\\nabla P$ with $P(z)$ HN if $F$ has no non-zero fixed point $w\\in \\mathbb C^n$ with $\\sum_{i=1}^n w_i^2=0$. Secondly, we show that the VC holds for a HN formal power series $P(z)$ if and only if, for any polynomial $f(z)$, $\\Delta^m (f(z)P(z)^m)=0$ when $m>>0$."}
{"category": "Math", "title": "A Vanishing Conjecture on Differential Operators with Constant Coefficients", "abstract": "In the recent progress [BE1], [Me] and [Z2], the well-known JC (Jacobian conjecture) ([BCW], [E]) has been reduced to a VC (vanishing conjecture) on the Laplace operators and HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent). In this paper, we first show that the vanishing conjecture above, hence also the JC, is equivalent to a vanishing conjecture for all 2nd order homogeneous differential operators $\\Lambda$ and $\\Lambda$-nilpotent polynomials $P$ (the polynomials $P(z)$ satisfying $\\Lambda^m P^m=0$ for all $m\\ge 1$). We then transform some results in the literature on the JC, HN polynomials and the VC of the Laplace operators to certain results on $\\Lambda$-nilpotent polynomials and the associated VC for 2nd order homogeneous differential operators $\\Lambda$. This part of the paper can also be read as a short survey on HN polynomials and the associated VC in the more general setting. Finally, we discuss a still-to-be-understood connection of $\\Lambda$-nilpotent polynomials in general with the classical orthogonal polynomials in one or more variables. This connection provides a conceptual understanding for the isotropic properties of homogeneous $\\Lambda$-nilpotent polynomials for the 2nd order homogeneous full rank differential operators $\\Lambda$ with constant coefficients."}
{"category": "Math", "title": "Theoretical Aspects of the SOM Algorithm", "abstract": "The SOM algorithm is very astonishing. On the one hand, it is very simple to write down and to simulate, its practical properties are clear and easy to observe. But, on the other hand, its theoretical properties still remain without proof in the general case, despite the great efforts of several authors. In this paper, we pass in review the last results and provide some conjectures for the future work."}
{"category": "Math", "title": "Retract rationality and Noether's problem", "abstract": "Let K be any field and G be a finite group. We will prove that, if K is any field, p an odd prime number, and G is a non-abelian group of exponent p with |G|=p^3 or p^4 satisfying [K(\\zeta_p):K] <= 2, then K(G) is rational over K. We will also show that K(G) is retract rational if G belongs to a much larger class of p-groups. In particular, generic G-polynomials of G-Galois extensions exist for these groups."}
{"category": "Math", "title": "Noether's problem for some p-groups", "abstract": "Let K be any field and G be a finite group. Noether's problem asks whether the fixed field is rational (=purely transcendental) over K. We will prove that if G is a non-abelian p-group of order p^n containing a cyclic subgroup of index p and K is any field containing a primitive p^{n-2}-th root of unity, then K(G) is rational over K."}
{"category": "Math", "title": "The canonical volume of threefolds of general type with $\\chi<1$", "abstract": "We prove that the canonical volume $K^3\\geq {1/30}$ for all projective 3-folds of general type with $\\chi(\\mathcal{O})\\leq 0$. This bound is sharp."}
{"category": "Math", "title": "Dynamical Equilibrium, trajectories study in an economical system. The case of the labor market", "abstract": "The paper deals with the study of labor market dynamics, and aims to characterize its equilibriums and possible trajectories. The theoretical background is the theory of the segmented labor market. The main idea is that this theory is well adapted to interpret the observed trajectories, due to the heterogeneity of the work situations."}
{"category": "Math", "title": "One-parameter families of functions in the Pick class", "abstract": "In the one-parameter family of power-law maps of the form $f_a(x)=-|x|^{\\alpha}+a,$ $\\alpha >1,$ we give examples of mutually related dynamically determined quantities, depending on the parameter $a$, such that one is a Pick function of the following one. These Pick functions are extendable by reflection through the $(1,+\\infty)$ half-axis and have completely monotone derivatives there."}
{"category": "Math", "title": "The homology of the Steinberg variety and Weyl group coinvariants", "abstract": "Let G be a complex, connected, reductive algebraic group with Weyl group W and Steinberg variety Z. We show that the graded Borel-Moore homology of Z is isomorphic to the smash product of the coinvariant algebra of W and the group algebra of W."}
{"category": "Math", "title": "Latin bitrades derived from groups", "abstract": "A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. Dr\\'apal (\\cite{Dr9}) showed that a latin bitrade is equivalent to three derangements whose product is the identity and whose cycles pairwise have at most one point in common. By letting a group act on itself by right translation, we show how some latin bitrades may be derived from groups without specifying an independent group action. Properties of latin trades such as homogeneousness, minimality (via thinness) and orthogonality may also be encoded succinctly within the group structure. We apply the construction to some well-known groups, constructing previously unknown latin bitrades. In particular, we show the existence of minimal, $k$-homogeneous latin trades for each odd $k\\geq 3$. In some cases these are the smallest known such examples."}
{"category": "Math", "title": "Integral representations for solutions of exponential Gauss-Manin systems", "abstract": "Let f,g be two algebraically independent regular functions from the smooth affine complex variety U to the affine line. The associated exponential Gauss-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system $\\mathcal{O}_U e^g $ with respect to f. We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition."}
{"category": "Math", "title": "Solving algebraic equations in roots of unity", "abstract": "This paper is devoted to finding solutions of polynomial equations in roots of unity. It was conjectured by S. Lang and proved by M. Laurent that all such solutions can be described in terms of a finite number of parametric families called maximal torsion cosets. We obtain new explicit upper bounds for the number of maximal torsion cosets on an algebraic subvariety of the complex algebraic $n$-torus ${\\mathbb G}_{\\rm m}^n$. In contrast to earlier works that give the bounds of polynomial growth in the maximum total degree of defining polynomials, the proofs of our results are constructive. This allows us to obtain a new algorithm for determining maximal torsion cosets on an algebraic subvariety of ${\\mathbb G}_{\\rm m}^n$."}
{"category": "Math", "title": "Quantum random walks and vanishing of the second Hochschild cohomology", "abstract": "Given a conditionally completely positive map $\\mathcal L$ on a unital $\\ast$-algebra $\\A$, we find an interesting connection between the second Hochschild cohomology of $\\A$ with coefficients in the bimodule $E_{\\mathcal L}=\\B^a(\\A \\oplus M)$ of adjointable maps, where $M$ is the GNS bimodule of $\\mathcal L$, and the possibility of constructing a quantum random walk (in the sense of \\cite{AP,LP,L,KBS}) corresponding to $\\mathcal L$."}
{"category": "Math", "title": "Another Riemann-Farey Computation", "abstract": "Another approach to constructing an upper bound for the Riemann-Farey sum is described."}
{"category": "Math", "title": "Quenched limits for transient, zero speed one-dimensional random walk in random environment", "abstract": "We consider a nearest-neighbor, one dimensional random walk $\\{X_n\\}_{n\\geq0}$ in a random i.i.d. environment, in the regime where the walk is transient but with zero speed, so that $X_n$ is of order $n^s$ for some $s<1$. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible: There exist sequences $\\{n_k\\}$ and $\\{x_k\\}$ depending on the environment only, such that $X_{n_k}-x_k=o(\\log n_k)^2$ (a localized regime). On the other hand, there exist sequences $\\{t_m\\}$ and $\\{s_m\\}$ depending on the environment only, such that $\\log s_m/\\log t_m\\to s<1$ and $P_{\\omega}(X_{t_m}/s_m\\leq x)\\to1/2$ for all $x>0$ and $\\to0$ for $x\\leq0$ (a spread out regime)."}
{"category": "Math", "title": "Asymptotics of the Euler number of bipartite graphs", "abstract": "We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing the Euler number of certain subgraphs of the Cartesian product of a graph $G$ with the path $P_m$ in terms of self adjoint operators. The asymptotic expansion of the Euler number is given in terms of the eigenvalues of the associated operator. For two classes of graphs, the comb graphs and the Cartesian product $P_2 \\Box P_m$, we numerically solve the eigenvalue problem."}
{"category": "Math", "title": "Generalized Smirnov statistics and the distribution of prime factors", "abstract": "We apply recent bounds of the author (math.PR/0609224) for generalized Smirnov statistics to the distribution of integers whose prime factors satisfy certain systems of inequalities."}
{"category": "Math", "title": "Le module dendriforme sur le groupe cyclique", "abstract": "The structure of anticyclic operad on the Dendriform operad defines in particular a matrix of finite order acting on the vector space spanned by planar binary trees. We compute its characteristic polynomial and propose a (compatible) conjecture for the characteristic polynomial of the Coxeter transformation for the Tamari lattice, which is mostly a square root of this matrix."}
{"category": "Math", "title": "Representation Theorems for Quadratic ${\\cal F}$-Consistent Nonlinear Expectations", "abstract": "In this paper we extend the notion of ``filtration-consistent nonlinear expectation\" (or \"${\\cal F}$-consistent nonlinear expectation\") to the case when it is allowed to be dominated by a $g$-expectation that may have a quadratic growth. We show that for such a nonlinear expectation many fundamental properties of a martingale can still make sense, including the Doob-Meyer type decomposition theorem and the optional sampling theorem. More importantly, we show that any quadratic ${\\cal F}$-consistent nonlinear expectation with a certain domination property must be a quadratic $g$-expectation. The main contribution of this paper is the finding of the domination condition to replace the one used in all the previous works, which is no longer valid in the quadratic case. We also show that the representation generator must be deterministic, continuous, and actually must be of the simple form."}
{"category": "Math", "title": "Regularity properties in the classification program for separable amenable C*-algebras", "abstract": "We report on recent progress in the program to classify separable amenable C*-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program's successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup."}
{"category": "Math", "title": "Polar actions on compact Euclidean hypersurfaces", "abstract": "Given an isometric immersion $f\\colon M^n\\to \\R^{n+1}$ of a compact Riemannian manifold of dimension $n\\geq 3$ into Euclidean space of dimension $n+1$, we prove that the identity component $Iso^0(M^n)$ of the isometry group $Iso(M^n)$ of $M^n$ admits an orthogonal representation $\\Phi\\colon Iso^0(M^n)\\to SO(n+1)$ such that $f\\circ g=\\Phi(g)\\circ f$ for every $g\\in Iso^0(M^n)$. If $G$ is a closed connected subgroup of $Iso(M^n)$ acting locally polarly on $M^n$, we prove that $\\Phi(G)$ acts polarly on $\\R^{n+1}$, and we obtain that $f(M^n)$ is given as $\\Phi(G)(L)$, where $L$ is a hypersurface of a section which is invariant under the Weyl group of the $\\Phi(G)$-action. We also find several sufficient conditions for such an $f$ to be a rotation hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of dimension $n\\geq 3$ are characterized by their underlying warped product structure."}
{"category": "Math", "title": "Energy Functionals for the Parabolic Monge-Ampere Equation", "abstract": "We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere equation."}
{"category": "Math", "title": "Compatible Actions and Cohomology of Crystallographic Groups", "abstract": "We compute the cohomology of crystallographic groups with holonomy of prime order. As an application we compute the group of gerbes associated to many six--dimensional toroidal orbifolds arising in string theory."}
{"category": "Math", "title": "Stochastic Heat Equation Driven by Fractional Noise and Local Time", "abstract": "The aim of this paper is to study the $d$-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and it has the covariance of a fractional Brownian motion with Hurst parameter $% H\\in (0,1)$ in time. Two types of equations are considered. First we consider the equation in the It\\^{o}-Skorohod sense, and later in the Stratonovich sense. An explicit chaos development for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion."}
{"category": "Math", "title": "Specht modules and Kazhdan--Lusztig cells in type $B_n$", "abstract": "Dipper, James and Murphy generalized the classical Specht module theory to Hecke algebras of type $B_n$. On the other hand, for any choice of a monomial order on the parameters in type $B_n$, we obtain corresponding Kazhdan--Lusztig cell modules. In this paper, we show that the Specht modules are naturally equivalent to the Kazhdan--Lusztig cell modules {\\em if} we choose the dominance order on the parameters, as in the ``asymptotic case'' studied by Bonnaf\\'e and the second named author. We also give examples which show that such an equivalence does not hold for other choices of monomial orders."}
{"category": "Math", "title": "Spectrum of the Laplacian on Quaternionic Kahler Manifolds", "abstract": "Let $M^{4n}$ be a complete quaternionic K\\\"ahler manifold with scalar curvature bounded below by $-16n(n+2)$. We get a sharp estimate for the first eigenvalue $\\lambda_1(M)$ of the Laplacian which is $\\lambda_1(M)\\le (2n+1)^2$. If the equality holds, then either $M$ has only one end, or $M$ is diffeomorphic to $\\mathbb{R}\\times N$ with N given by a compact manifold. Moreover, if $M$ is of bounded curvature, $M$ is covered by the quaterionic hyperbolic space $\\mathbb{QH}^n$ and $N$ is a compact quotient of the generalized Heisenberg group. When $\\lambda_1(M)\\ge \\frac{8(n+2)}3$, we also prove that $M$ must have only one end with infinite volume."}
{"category": "Math", "title": "Weak type radial convolution operators on free group", "abstract": "Radial convolution operators on free groups with nonnegative kernel of weak type $(2,2)$ and of restricted weak type $(2,2)$ are characterized. Estimates of weak type $(p,p)$ are obtained as well for $1<p<2.$"}
{"category": "Math", "title": "Analycity and smoothing effect for the coupled system of equations of Korteweg - de Vries type with a single point singularity", "abstract": "We study that a solution of the initial value problem associated for the coupled system of equations of Korteweg - de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has analyticity in time and smoothing effect up to real analyticity if the initial data only has a single point singularity at $x=0.$"}
{"category": "Math", "title": "Smoothing properties for the higher order nonlinear Schr\\\"{o}dinger equation with constant coefficients", "abstract": "We study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear Schr\\\"odinger equation with constant coefficients which appears as a model for propagation of pulse in optical fiber."}
{"category": "Math", "title": "On $L^{1}$-Convergence of Fourier Series Under $MVBV$ Condition", "abstract": "Let $f\\in L_{2\\pi}$ be a real-valued even function with its Fourier series $ \\frac{a_{0}}{2}+\\sum_{n=1}^{\\infty}a_{n}\\cos nx,$ and let $S_{n}(f,x), n\\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence $\\{a_{n}\\}$ is decreasing and $\\lim\\limits_{n\\to \\infty}a_{n}=0$, then $$ \\lim\\limits_{n\\to \\infty}\\Vert f-S_{n}(f)\\Vert_{L}=0 {if and only if} \\lim\\limits_{n\\to \\infty}a_{n}\\log n=0. $$ We weaken the monotone condition in this classical result to the so-called mean value bounded variation ($MVBV$) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper which gives the $L^{1}$% -convergence of a function $f\\in L_{2\\pi}$ in complex space. We also give results on $L^{1}$-approximation of a function $f\\in L_{2\\pi}$ under the $% MVBV$ condition."}
{"category": "Math", "title": "The Weil representation and Hecke operators for vector valued modular forms", "abstract": "We define Hecke operators on vector valued modular forms transforming with the Weil representation associated to a discriminant form. We describe the properties of the corresponding algebra of Hecke operators and study the action on modular forms."}
{"category": "Math", "title": "New versions of Schur-Weyl duality", "abstract": "After reviewing classical Schur-Weyl duality, we present some other contexts which enjoy similar features, relating to Brauer algebras and classical groups."}
{"category": "Math", "title": "Lower bounds in some power sum problems", "abstract": "We study the power sum problem max_{v=1,...,m} | sum_{k=1}^n z_k^v | and by using features of Fejer kernels we give new lower bounds in the case of unimodular complex numbers z_k and m cn^2 for constants c>1."}
{"category": "Math", "title": "Coloring ordinals by reals", "abstract": "We study combinatorial principles we call Homogeneity Principle HP(\\kappa) and Injectivity Principle IP(\\kappa,\\lambda) for regular \\kappa>\\aleph_1 and \\lambda\\leq\\kappa which are formulated in terms of coloring the ordinals <\\kappa by reals."}
{"category": "Math", "title": "N-homogeneous superalgebras", "abstract": "We develop the theory of N-homogeneous algebras in a super setting, with particular emphasis on the Koszul property. To any Hecke operator on a vector superspace, we associate certain superalgebras and generalizing the ordinary symmetric and Grassmann algebra, respectively. We prove that these algebras are N-Koszul. For the special case where the Hecke operator is the ordinary supersymmetry, we derive an $N$-generalized super-version of MacMahon's classical \"master theorem\"."}
{"category": "Math", "title": "Axioms for a local Reidemeister trace in fixed point and coincidence theory on differentiable manifolds", "abstract": "We give axioms which characterize the local Reidemeister trace for orientable differentiable manifolds. The local Reidemeister trace in fixed point theory is already known, and we provide both uniqueness and existence results for the local Reidemeister trace in coincidence theory."}
{"category": "Math", "title": "On (n+2)-dimensional n-Lie algebras", "abstract": "I show that an (n+2)-dimensional n-Lie algebra over an algebraically closed field must have a subalgeba of codimension 1."}
{"category": "Math", "title": "Finite Representations of the braid group commutator subgroup", "abstract": "We study the representations of the commutator subgroup K_{n} of the braid group B_{n} into a finite group . This is done through a symbolic dynamical system. Some experimental results enable us to compute the number of subgroups of K_{n} of a given (finite) index, and, as a by-product, to recover the well known fact that every representation of K_{n} into S_{r}, with n > r, must be trivial."}
{"category": "Math", "title": "Solutions of certain fractional kinetic equations and a fractional diffusion equation", "abstract": "In view of the usefulness and importance of the kinetic equation in certain physical problems, the authors derive the explicit solution of a fractional kinetic equation of general character, that unifies and extends earlier results. Further, an alternative shorter method based on a result developed by the authors is given to derive the solution of a fractional diffusion equation."}
{"category": "Math", "title": "Relative Rigidity, Quasiconvexity and C-Complexes", "abstract": "We introduce and study the notion of relative rigidity for pairs $(X,\\JJ)$ where 1) $X$ is a hyperbolic metric space and $\\JJ$ a collection of quasiconvex sets 2) $X$ is a relatively hyperbolic group and $\\JJ$ the collection of parabolics 3) $X$ is a higher rank symmetric space and $\\JJ$ an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such $\\JJ$'s to a quasi-isometry between the corresponding $X$'s. A related notion is that of a $C$-complex which is the adaptation of a Tits complex to this context. We prove the relative rigidity of the collection of pairs $(X, \\JJ)$ as above. This generalises a result of Schwarz for symmetric patterns of geodesics in hyperbolic space. We show that a uniformly proper map induces an isomorphism of the corresponding $C$-complexes. We also give a couple of characterizations of quasiconvexity. of subgroups of hyperbolic groups on the way."}
{"category": "Math", "title": "The Loewner driving function of trajectory arcs of quadratic differentials", "abstract": "We obtain a first order differential equation for the driving function of the chordal Loewner differential equation in the case where the domain is slit by a curve which is a trajectory arc of certain quadratic differentials. In particular this includes the case when the curve is a path on the square, triangle or hexagonal lattice in the upper halfplane or, indeed, in any domain with boundary on the lattice. We also demonstrate how we use this to calculate the driving function numerically. Equivalent results for other variants of the Loewner differential equation are also obtained: Multiple slits in the chordal Loewner differential equation and the radial Loewner differential equation. The method also works for other versions of the Loewner differential equation. The proof of our formula uses a generalization of Schwarz-Christoffel mapping to domains bounded by trajectory arcs of rotations of a given quadratic differential that is of interest in its own right."}
{"category": "Math", "title": "Identities for number series and their reciprocals: Dirac delta function approach", "abstract": "Dirac delta function (delta-distribution) approach can be used as efficient method to derive identities for number series and their reciprocals. Applying this method, a simple proof for identity relating prime counting function (pi-function) and logarithmic integral (Li-function) can be obtained."}
{"category": "Math", "title": "The Chow ring of the moduli space and its related homogeneous space of bundles on P^2 with charge 1", "abstract": "For an algebraically closed field K with ch K \\not = 2, we determine the Chow ring of the moduli space of holomorphic bundles on a projective plane with the structure group SO(n,K) and half the first Pontryagin index being equal to 1, each of which is trivial on a fixed line and has a fixed holomorphic trivialization there."}
{"category": "Math", "title": "Additional Explanatory Notes on the Analytic Proof of the Finite Generation of the Canonical Ring", "abstract": "This set of notes provides some additional explanatory material on the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type."}
{"category": "Math", "title": "Tait's conjectures and odd crossing number amphicheiral knots", "abstract": "We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last remaining?) problem of Tait, with the construction of amphicheiral knots of almost all odd crossing numbers. An application to the non-triviality problem for the Jones polynomial is also outlined."}
{"category": "Math", "title": "The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes", "abstract": "We compute the complete Fadell-Husseini index of the 8 element dihedral group D_8 acting on S^d \\times S^d, both for F_2 and for integer coefficients. This establishes the complete goup cohomology lower bounds for the two hyperplane case of Gr\"unbaum's 1960 mass partition problem: For which d and j can any j arbitrary measures be cut into four equal parts each by two suitably-chosen hyperplanes in R^d? In both cases, we find that the ideal bounds are not stronger than previously established bounds based on one of the maximal abelian subgroups of D_8."}
{"category": "Math", "title": "R-matrices in Rime", "abstract": "We replace the ice Ansatz on matrix solutions of the Yang-Baxter equation by a weaker condition which we call \"rime\". Rime solutions include the standard Drinfeld-Jimbo R-matrix. Solutions of the Yang--Baxter equation within the rime Ansatz which are maximally different from the standard one we call \"strict rime\". A strict rime non-unitary solution is parameterized by a projective vector. We show that this solution transforms to the Cremmer-Gervais R-matrix by a change of basis with a matrix containing symmetric functions in the components of the parameterizing vector. A strict unitary solution (the rime Ansatz is well adapted for taking a unitary limit) is shown to be equivalent to a quantization of a classical \"boundary\" r-matrix of Gerstenhaber and Giaquinto. We analyze the structure of the elementary rime blocks and find, as a by-product, that all non-standard R-matrices of GL(1|1)-type can be uniformly described in a rime form. We discuss then connections of the classical rime solutions with the Bezout operators. The Bezout operators satisfy the (non-)homogeneous associative classical Yang--Baxter equation which is related to the Rota-Baxter operators. We classify the rime Poisson brackets: they form a 3-dimensional pencil. A normal form of each individual member of the pencil depends on the discriminant of a certain quadratic polynomial. We also classify orderable quadratic rime associative algebras. For the standard Drinfeld-Jimbo solution, there is a choice of the multiparameters, for which it can be non-trivially rimed. However, not every Belavin-Drinfeld triple admits a choice of the multiparameters for which it can be rimed. We give a minimal example."}
{"category": "Math", "title": "Quadratic centers defining elliptic surfaces", "abstract": "Let $X$ be a quadratic vector field with a center whose generic orbits are algebraic curves of genus one. To each $X$ we associate an elliptic surface (a smooth complex compact surface which is a genus one fibration). We give the list of all such vector fields and determine the corresponding elliptic surfaces."}
{"category": "Math", "title": "Zeta function and cryptographic exponent of supersingular curves of genus 2", "abstract": "We compute in a direct (not algorithmic) way the zeta function of all supersingular curves of genus 2 over a finite field k, with many geometric automorphisms. We display these computations in an appendix where we select a family of representatives of all these curves up to geometric isomorphism and we exhibit equations and the zeta function of all their twists. As an application we obtain a direct computation of the cryptographic exponent of the Jacobians of these curves."}
{"category": "Math", "title": "The Allen-Cahn Action functional in higher dimensions", "abstract": "The Allen-Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen-Cahn equation. Formal calculations suggest a reduced action functional in the sharp interface limit. We prove in two and three space dimensions the corresponding lower bound. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the limit of diffuse surface area measures and introduce a generalized velocity and generalized reduced action functional in a class of evolving measures. As a corollary we obtain the Gamma convergence of the action functional in a class of regularly evolving hypersurfaces."}
{"category": "Math", "title": "Metrical characterization of super-reflexivity and linear type of Banach spaces", "abstract": "We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain's result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniforms embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with a suitable metric."}
{"category": "Math", "title": "Some new observations on interpolation in the spectral unit ball", "abstract": "We present several results associated to a holomorphic-interpolation problem for the spectral unit ball \\Omega_n, n\\geq 2. We begin by showing that a known necessary condition for the existence of a $\\mathcal{O}(D;\\Omega_n)$-interpolant (D here being the unit disc in the complex plane), given that the matricial data are non-derogatory, is not sufficient. We provide next a new necessary condition for the solvability of the two-point interpolation problem -- one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of \\Omega_n, n\\geq 2."}
{"category": "Math", "title": "On the Young-Fibonacci insertion algorithm", "abstract": "This work is concerned with some properties of the Young-Fibonacci insertion algorithm and its relation with Fomin's growth diagrams. It also investigates a relation between the combinatorics of Young-Fibonacci tableaux and the study of Okada's algebra associated to the Young-Fibonacci lattice. The original algorithm was introduced by Roby and we redefine it in such a way that both the insertion and recording tableaux of any permutation are \\emph{conveniently} interpreted as chains in the Young-Fibonacci lattice. A property of Killpatrick's evacuation is given a simpler proof, but this evacuation is no longer needed in making Roby's and Fomin's constructions coincide. We provide the set of Young-Fibonacci tableaux of size $n$ with a structure of graded poset, induced by the weak order on permutations of the symmetric group, and realized by transitive closure of elementary transformations on tableaux. We show that this poset gives a combinatorial interpretation of the coefficients in the transition matrix from the analogue of complete symmetric functions to analogue of the Schur functions in Okada's algebra. We end with a quite similar observation for four posets on Young-tableaux studied by Taskin."}
{"category": "Math", "title": "Growth rates for geometric complexities and counting functions in polygonal billiards", "abstract": "We introduce a new method for estimating the growth of various quantities arising in dynamical systems. We apply our method to polygonal billiards on surfaces of constant curvature. For instance, we obtain power bounds of degree two plus epsilon in length for the number of billiard orbits between almost all pairs of points in a planar polygon."}
{"category": "Math", "title": "The Jumping Phenomenon of Hodge Numbers", "abstract": "Let $X$ be a compact complex manifold, consider a small deformation $\\phi: \\mathcal{X} \\to B$ of $X$, the dimension of the Dolbeault cohomology groups $H^q(X_t,\\Omega_{X_t}^p)$ may vary under this defromation. This paper will study such phenomenons by studying the obstructions to deform a class in $H^q(X,\\Omega_X^p)$ with the parameter $t$ and get the formula for the obstructions."}
{"category": "Math", "title": "V-cycle optimal convergence for DCT-III matrices", "abstract": "The paper analyzes a two-grid and a multigrid method for matrices belonging to the DCT-III algebra and generated by a polynomial symbol. The aim is to prove that the convergence rate of the considered multigrid method (V-cycle) is constant independent of the size of the given matrix. Numerical examples from differential and integral equations are considered to illustrate the claimed convergence properties."}
{"category": "Math", "title": "Some notes on analytic torsion of the Rumin complex on contact manifolds", "abstract": "We propose a definition for analytic torsion of the Rumin complex on contact manifolds. This is given by the derivative at zero of a well-chosen combination of zeta functions of a fourth-order modified Rumin Laplacian. The regular value at zero (before differentiation) of this well-chosen combination of zeta functions is shown to be a contact invariant. The variation of our analytic torsion is given as the integral of local terms, together with a global term coming from the null-space of the Laplacian."}
{"category": "Math", "title": "Characterization of Closed Vector Fields in Finsler Geometry", "abstract": "The $\\pi$-exterior derivative ${\\o}d$, which is the Finslerian generalization of the (usual) exterior derivative $d$ of Riemannian geometry, is defined. The notion of a ${\\o}d$-closed vector field is introduced and investigated. Various characterizations of ${\\o}d$-closed vector fields are established. Some results concerning ${\\o}d$-closed vector fields in relation to certain special Finsler spaces are obtained."}
{"category": "Math", "title": "Pure inductive limit state and Kolmogorov's property", "abstract": "Let $(\\clb,\\lambda_t,\\psi)$ be a $C^*$-dynamical system where $(\\lambda_t: t \\in \\IT_+)$ be a semigroup of injective endomorphism and $\\psi$ be an $(\\lambda_t)$ invariant state on the $C^*$ subalgebra $\\clb$ and $\\IT_+$ is either non-negative integers or real numbers. The central aim of this exposition is to find a useful criteria for the inductive limit state $\\clb \\raro^{\\lambda_t} \\clb$ canonically associated with $\\psi$ to be pure. We achieve this by exploring the minimal weak forward and backward Markov processes associated with the Markov semigroup on the corner von-Neumann algebra of the support projection of the state $\\psi$ to prove that Kolmogorov's property [Mo2] of the Markov semigroup is a sufficient condition for the inductive state to be pure. As an application of this criteria we find a sufficient condition for a translation invariant factor state on a one dimensional quantum spin chain to be pure. This criteria in a sense complements criteria obtained in [BJKW,Mo2] as we could go beyond lattice symmetric states."}
{"category": "Math", "title": "Jones index of a quantum dynamical semigroup", "abstract": "In this paper we consider a semigroup of completely positive maps $\\tau=(\\tau_t,t \\ge 0)$ with a faithful normal invariant state $\\phi$ on a type-$II_1$ factor $\\cla_0$ and propose an index theory. We :achieve this via a more general Kolmogorov's type of construction for stationary Markov processes which naturally associate a nested isomorphic von-Neumann algebras. In particular this construction generalizes well known Jones construction associated with a sub-factor of type-II$_1$ factor."}
{"category": "Math", "title": "On the formal cohomology of local rings", "abstract": "Let $\\mathfrak a$ denote an ideal of a local ring $(R, \\mathfrak m).$ Let $M$ be a finitely generated $R$-module. There is a systematic study of the formal cohomology modules $\\varprojlim \\HH^i(M/\\mathfrak a^nM), i \\in \\mathbb Z.$ We analyze their $R$-module structure, the upper and lower vanishing and non-vanishing in terms of intrinsic data of $M,$ and its functorial behavior. These cohomology modules occur in relation to the formal completion of the punctured spectrum $\\Spec R \\setminus V(\\mathfrak m).$ As a new cohomological data there is a description on the formal grade $\\fgrade(\\mathfrak a, M)$ defined as the minimal non-vanishing of the formal cohomology modules. There are various exact sequences concerning the formal cohomology modules. Among them a Mayer-Vietoris sequence for two ideals. It applies to new connectedness results. There are also relations to local cohomological dimensions."}
{"category": "Math", "title": "On Lyubeznik's invariants and endomorphisms of local cohomology modules", "abstract": "Let $(R, \\mathfrak m)$ denote an $n$-dimensional Gorenstein ring. For an ideal $I \\subset R$ of height $c$ we are interested in the endomorphism ring $B = \\Hom_R(H^c_I(R), H^c_I(R)).$ It turns out that $B$ is a commutative ring. In the case of $(R,\\mathfrak m)$ a regular local ring containing a field $B$ is a Cohen-Macaulay ring. Its properties are related to the highest Lyubeznik number $l = \\dim_k \\Ext_R^d(k,H^c_I(R)).$ In particular $R \\simeq B$ if and only if $l = 1.$ Moreover, we show that the natural homomorphism $\\Ext_R^d(k, H^c_I(R)) \\to k$ is non-zero."}
{"category": "Math", "title": "On Virasoro Constraints for Orbifold Gromov-Witten Theory", "abstract": "Virasoro constraints for orbifold Gromov-Witten theory are described. These constraints are applied to the degree zreo, genus zero orbifold Gromov-Witten potentials of the weighted projective stacks $\\mathbb{P}(1,N)$, $\\mathbb{P}(1,1,N)$ and $\\mathbb{P}(1,1,1,N)$ to obtain formulas of descendant cyclic Hurwitz-Hodge integrals."}
{"category": "Math", "title": "On a new version of the Ito's formula for the stochastic heat equation", "abstract": "We derive an It\\^o's-type formula for the one dimensional stochastic heat equation driven by a space-time white noise. The proof is based on elementary properties of the $\\mathcal{S}$-transform and on the explicit representation of the solution process. We also discuss the relationship with other versions of this It\\^o's-type formula existing in literature."}
{"category": "Math", "title": "Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents", "abstract": "When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the values of irreducible characters of the extended groups on these elements. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way."}
{"category": "Math", "title": "A Note on Sums of Powers", "abstract": "We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers."}
{"category": "Math", "title": "The Hopf Algebra Structure of the Character Rings of Classical Groups", "abstract": "The character ring \\CGL of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra \\Sym$ of symmetric functions. Here we study the character rings \\CO and \\CSp of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that \\CO and \\CSp also admit natural Hopf algebra structures that are isomorphic to that of \\CGL, and hence to \\Sym. The isomorphisms are determined explicitly, along with the specification of standard bases for \\CO and \\CSp analogous to those used for \\Sym. A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur-Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the \\CGL case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras \\CO and \\CSp are not self-dual. The dual Hopf algebras \\CO^* and \\CSp^* are identified. Finally, the Hopf algebra of the universal rational character ring \\CGLrat of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified."}
{"category": "Math", "title": "New Approach to Arakelov Geometry", "abstract": "This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of generalized rings and schemes, which include classical rings and schemes together with \"exotic\" objects such as F_1 (\"field with one element\"), Z_\\infty (\"real integers\"), T (tropical numbers) etc., thus providing a systematic way of studying such objects. This theory of generalized rings and schemes is developed up to construction of algebraic K-theory, intersection theory and Chern classes. Then existence of Arakelov models of algebraic varieties over Q is shown, and our general results are applied to such models."}
{"category": "Math", "title": "Hyperbolic Balance Laws with a Non Local Source", "abstract": "This paper is devoted to hyperbolic systems of balance laws with non local source terms. The existence, uniqueness and Lipschitz dependence proved here comprise previous results in the literature and can be applied to physical models, such as Euler system for a radiating gas and Rosenau regularization of the Chapman-Enskog expansion."}
{"category": "Math", "title": "A Bishop surface with a vanishing Bishop invariant", "abstract": "We derive a complete set of invariants for a formal Bishop surface near a point of complex tangent with a vanishing Bishop invariant under the action of formal transformations. We prove that the modular space of Bishop surfaces with a vanishing Bishop invariant and with a fixed Moser invariant $s<\\infty$ is of infinite dimension. We also prove that the equivalence class of the germ of a generic real analytic Bishop surface near a complex tangent with a vanishing Bishop invariant can not be determined by a finite part of the Taylor expansion of its defining equation. This answers, in the negative, a problem raised by J. Moser in 1985 after his joint work with Webster in 1983 and his own work in 1985."}
{"category": "Math", "title": "Bi-Lipschitz geometry of weighted homogeneous surface singularities", "abstract": "We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent."}
{"category": "Math", "title": "Combinatorial structure of Kirillov-Reshetikhin crystals of type D_n(1), B_n(1), A_{2n-1}(2)", "abstract": "We provide the explicit combinatorial structure of the Kirillov-Reshetikhin crystals B^{r,s} of type D_n(1), B_n(1), and A_{2n-1}(2). This is achieved by constructing the crystal analogue sigma of the automorphism of the D_n(1) (resp. B_n(1) or A_{2n-1}(2)) Dynkin diagram that interchanges the 0 and 1 node. The involution sigma is defined in terms of new plus-minus diagrams that govern the D_n to D_{n-1} (resp. B_n to B_{n-1}, or C_n to C_{n-1}) branching. It is also shown that the crystal B^{r,s} is perfect. These crystals have been implemented in MuPAD-Combinat; the implementation is discussed in terms of many examples."}
{"category": "Math", "title": "Combinatorial Gray codes for classes of pattern avoiding permutations", "abstract": "The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, Schr\\\"oder, Pell, even index Fibonacci numbers and the central binomial coefficients. Consequently, this provides Gray codes for $\\s_n(\\tau)$ for all $\\tau\\in \\s_3$ and the obtained Gray codes have distances 4 and 5."}
{"category": "Math", "title": "A duality approach to representations of Baumslag-Solitar groups", "abstract": "We give an operator theoretic approach to the constructions of multiresolutions as they are used in a number of basis constructions with wavelets, and in Hilbert spaces on fractals. Our approach starts with the following version of the classical Baumslag-Solitar relations $u t = t^2 u$ where $t$ is a unitary operator in a Hilbert space $\\mathcal H$ and $u$ is an isometry in $\\mathcal H$. There are isometric dilations of this system into a bigger Hilbert space, relevant for wavelets. For a variety of carefully selected dilations, the ``bigger'' Hilbert space may be $L^2(\\br)$, and the dilated operators may be the unitary operators which define a dyadic wavelet multiresolutions of $L^2(\\br)$ with the subspace $\\mathcal H$ serving as the corresponding resolution subspace. That is, the initialized resolution which is generated by the wavelet scaling function(s). In the dilated Hilbert space, the Baumslag-Solitar relations then take the more familiar form $u t u^{-1} = t^2$. We offer an operator theoretic framework including the standard construction; and we show how the representations of certain discrete semidirect group products serve to classify the possibilities. For this we analyze and compare several types of unitary representations of these semidirect products: the induced representations in Mackey's theory, the wavelet representations on $L^2(\\br)$, the irreducible representation on the dual, the finite dimensional representations, and the the regular representation."}
{"category": "Math", "title": "A biased view of symplectic cohomology", "abstract": "These are lecture notes from my talks at the \"Current Developments in Mathematics\" conference (Harvard, 2006). They cover a variety of topics involving symplectic cohomology. In particular, a discussion of (algorithmic) classification issues in symplectic and contact topology is included."}
{"category": "Math", "title": "Three-manifolds of positive Ricci curvature and convex weakly umbilic boundary", "abstract": "In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is convex (nonnegative Second Fundamental form), its metric can be deformed via the Ricci flow to a metric of constant curvature and totally geodesic boundary."}
{"category": "Math", "title": "Cocycles and Ma\\~{n}e sequences with an application to ideal fluids", "abstract": "Exponential dichotomy of a strongly continuous cocycle $\\bFi$ is proved to be equivalent to existence of a Ma\\~{n}e sequence either for $\\bFi$ or for its adjoint. As a consequence we extend some of the classical results to general Banach bundles. The dynamical spectrum of a product of two cocycles, one of which is scalar, is investigated and applied to describe the essential spectrum of the Euler equation in an arbitrary spacial dimension."}
{"category": "Math", "title": "Prewavelet Solution to Poisson Equations", "abstract": "Finite element method is one of powerful numerical methods to solve PDE. Usually, if a finite element solution to a Poisson equation based on a triangulation of the underlying domain is not accurate enough, one will discard the solution and then refine the triangulation uniformly and compute a new finite element solution over the refined triangulation. It is wasteful to discard the original finite element solution. We propose a prewavelet method to save the original solution by adding a prewavelet subsolution to obtain the refined level finite element solution. To increase the accuracy of numerical solution to Poisson equations, we can keep adding prewavelet subsolutions. Our prewavelets are orthogonal in the $H^1$ norm and they are compactly supported except for one globally supported basis function in a rectangular domain. We have implemented these prewavelet basis functions in MATLAB and used them for numerical solution of Poisson equation with Dirichlet boundary conditions. Numerical simulation demonstrates that our prewavelet solution is much more efficient than the standard finite element method."}
{"category": "Math", "title": "A Remark on Compact Minimal Surfaces in $S^5$ with Non-Negative Gaussian Curvature", "abstract": "In this paper we classify compact minimal surfaces in $S^5$ with non-negative Gaussian curvature using the notion of a contact angle."}
{"category": "Math", "title": "On parametrization of linear pseudo-differential boundary value control systems", "abstract": "The paper considers pseudo-differential boundary value control systems. The underlying operators form an algebra D with the help of which we are able to formulate typical boundary value control problems. The symbolic calculus gives tools to form e.g. compositions, formal adjoints, generalized right or left inverses and compatibility conditions. By a parametrizability we mean that for a given control system Au=0 one finds an operator S such that Au=0 if and only if u=Sf. The computation rules of D (or its appropriate subalgebra D') guarantee that in many applications S can be refinely analyzed or even explicitly calculated. We outline some methods of homological algebra for the study of parametrization S. Especially the projectivity of a certain factor module (defined by the system equations) implies the parametrizability. We give some examples to illustrate our computational methods."}
{"category": "Math", "title": "Associated Graded Algebras and Coalgebras", "abstract": "We investigate the notion of associated graded coalgebra (algebra) of a bialgebra with respect to a subbialgebra (quotient bialgebra) and characterize those which are bialgebras of type one in the framework of abelian braided monoidal categories."}
{"category": "Math", "title": "Poincare Duality Pairs in Dimension Three", "abstract": "We extend Hendriks' classification theorem and Turaev's realisation and splitting theorems for Poincare duality complexes in dimension three to the relative case of Poincare duality pairs. The results for Poincare duality complexes are recovered by restricting the results to the case of Poincare duality pairs with empty boundary. Up to oriented homotopy equivalence, three-dimensional Poincare duality pairs are classified by their fundamental triple consisting of the fundamental group system, the orientation character and the image of the fundamental class under the classifying map. Using the derived module category we provide necessary and sufficient conditions for a given triple to be realised by a three-dimensional Poincare duality pair. The results on classification and realisation yield splitting or decomposition theorems for three-dimensional Poincare duality pairs, that is, conditions under which a given three-dimensional Poincare duality pair decomposes as interior or boundary connected sum of two three-dimensional Poincare duality pairs."}
{"category": "Math", "title": "The Jumping Phenomenon of the Dimensions of Cohomology Groups of Tangent Sheaf", "abstract": "Let $X$ be a compact complex manifold, consider a small deformation $\\phi: \\mathcal{X} \\to B$ of $X$, the dimensions of the cohomology groups of tangent sheaf $H^q(X_t,\\mathcal{T}_{X_t})$ may vary under this deformation. This paper will study such phenomenons by studying the obstructions to deform a class in $H^q(X,\\mathcal{T}_X)$ with the parameter $t$ and get the formula for the obstructions."}
{"category": "Math", "title": "Conformal Structures in Noncommutative Geometry", "abstract": "It is well-known that a compact Riemannian spin manifold can be reconstructed from its canonical spectral triple which consists of the algebra of smooth functions, the Hilbert space of square integrable spinors and the Dirac operator. It seems to be a folklore fact that the metric can be reconstructed up to conformal equivalence if one replaces the Dirac operator D by sign(D). We give a precise formulation and proof of this fact."}
{"category": "Math", "title": "Geometric Invariant Theory and Generalized Eigenvalue Problem", "abstract": "Let $H$ be a connected reductive subgroup of a complex connected reductive group $G$. Fix maximal tori and Borel subgroups of $H$ and $G$. Consider the pairs $(V,V')$ of irreducible representations of $H$ and $G$ such that $V$ is a submodule of $V'$. We are interested in the cone $LR(G,H)$ generated by the pairs of dominant weights of such a pair of representations. Our main result gives a minimal set of inequalities describing $LR(G,H)$ as a part of the dominant chamber. In way, we obtain results about the faces of the Dolgachev-Hu's $G$-ample cone and variations of this cone."}
{"category": "Math", "title": "L-theory of groups with unstable derived series", "abstract": "In this short note we prove that the Farrell-Jones Fibered Isomorphism Conjecture in L-theory, after inverting 2, is true for a group whose some derived subgroup is free."}
{"category": "Math", "title": "An algebraic proof of Gabrielov's theorem about analytic homomorphisms in any characteristic", "abstract": "The proof of proposition 3.6 is not correct"}
{"category": "Math", "title": "Homogeneous edge-disjoint $K_{2s}$ and $T_{st,t}$ unions", "abstract": "Let $r>2$ and $\\sigma\\in(0,r-1)$ be integers. We require $t<2s$, where $t=2^{\\sigma+1}-1$ and $s=2^{r-\\sigma-1}$. Generalizing a known $\\{K_4,T_{6,3}\\}$-ultrahomogenous graph $G_3^1$, we find that a finite, connected, undirected, arc-transitive graph $G_r^\\sigma$ exists each of whose edges is shared by just two maximal subgraphs, namely a clique $X_0=K_{2s}$ and a $t$-partite regular-Tur\\'an graph $X_1=T_{st,t}$ on $s$ vertices per part. Each copy $Y$ of $X_i$ ($i=0,1$) in $G_r^\\sigma$ shares each edge with just one copy of $X_{1-i}$ and all such copies of $X_{1-i}$ are pairwise distinct. Moreover, $G_r^\\sigma$ is an edge-disjoint union of copies of $X_i$, for $i=0,1$. We prove that $G_r^\\sigma$ is $\\{K_{2s},T_{st,t}\\}$-homogeneous if $t<2s$, and just $\\{T_{st,t}\\}$-homogeneous otherwise, meaning that there is an automorphism of $G_r^\\sigma$ between any two such copies of $X_i$ relating two preselected arcs."}
{"category": "Math", "title": "(2+1)-Einstein spacetimes of finite type", "abstract": "The aim of this survey is to give an overview on the geometry of Einstein maximal globally hyperbolic 2+1 spacetimes of arbitrary curvature, conatining a complete Cauchy surface of finite type. In particular a specialization to the finite type case of the canonicla Wick rotation-rescaling theory, previously developed by the authors, is provided. This includes, for arbitrary curvatures, parameterizations in terms of suitable measured geodesic laminations on open hyperbolic surfaces of finite type. The same geometric objects also parameterize complex projective structures on the surfaces. The coincidence of such parameter space is explained by means of geometric correlations between spacetimes of different curvatures and projective surfaces realized via canonical WR-rescaling along the cosmological times. We also specialize on AdS case mostly referring to recent results achieved by other authors. In particular we describe maximal causal extensions of AdS globally hyperbolic spacetimes and an AdS approach to the theory of earthquakes for hyperbolic surfaces of finite type. A general earthquake theorem is proved for the so called enhanced Teichmuller space. The case of spacetimes with conical timelike singularities is also treated."}
{"category": "Math", "title": "Free pre-Lie algebras are free as Lie algebras", "abstract": "We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad."}
{"category": "Math", "title": "Localization of injective modules over arithmetical rings", "abstract": "It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in addition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Pr\\\"ufer domain of finite character, localizations of injective $R$-modules are injective."}
{"category": "Math", "title": "On the Computational Complexity of MCMC-based Estimators in Large Samples", "abstract": "In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasi-Bayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the Laplace-Bernstein-Von Mises central limit theorem, which states that in large samples the posterior or quasi-posterior approaches a normal density. Using the conditions required for the central limit theorem to hold, we establish polynomial bounds on the computational complexity of general Metropolis random walks methods in large samples. Our analysis covers cases where the underlying log-likelihood or extremum criterion function is possibly non-concave, discontinuous, and with increasing parameter dimension. However, the central limit theorem restricts the deviations from continuity and log-concavity of the log-likelihood or extremum criterion function in a very specific manner. Under minimal assumptions required for the central limit theorem to hold under the increasing parameter dimension, we show that the Metropolis algorithm is theoretically efficient even for the canonical Gaussian walk which is studied in detail. Specifically, we show that the running time of the algorithm in large samples is bounded in probability by a polynomial in the parameter dimension $d$, and, in particular, is of stochastic order $d^2$ in the leading cases after the burn-in period. We then give applications to exponential families, curved exponential families, and Z-estimation of increasing dimension."}
{"category": "Math", "title": "An exact sequence for contact- and symplectic homology", "abstract": "A symplectic manifold $W$ with contact type boundary $M = \\partial W$ induces a linearization of the contact homology of $M$ with corresponding linearized contact homology $HC(M)$. We establish a Gysin-type exact sequence in which the symplectic homology $SH(W)$ of $W$ maps to $HC(M)$, which in turn maps to $HC(M)$, by a map of degree -2, which then maps to $SH(W)$. Furthermore, we give a description of the degree -2 map in terms of rational holomorphic curves with constrained asymptotic markers, in the symplectization of $M$."}
{"category": "Math", "title": "Stability in random Boolean cellular automata on the integer lattice", "abstract": "We consider random boolean cellular automata on the integer lattice, i.e., the cells are identified with the integers from 1 to $N$. The behaviour of the automaton is mainly determined by the support of the random variable that selects one of the sixteen possible Boolean rules, independently for each cell. A cell is said to stabilize if it will not change its state anymore after some time. We classify the random boolean automata according to the positivity of their probability of stabilization. Here is an example of a consequence of our results: if the support contains at least 5 rules, then asymptotically as $N$ tends to infinity the probability of stabilization is positive, whereas there exist random boolean cellular automata with 4 rules in their support for which this probability tends to 0."}
{"category": "Math", "title": "Equilibrium states for interval maps: the potential $-t\\log |Df|$", "abstract": "Let $f:I \\to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential $\\phi_t:x\\mapsto -t\\log|Df(x)|$ for $t$ close to 1, and also that the pressure function $t \\mapsto P(\\phi_t)$ is analytic on an appropriate interval near $t = 1$."}
{"category": "Math", "title": "On Abelian Difference Sets with Parameters of 3-dimensional Projective Geometries", "abstract": "A difference set is said to have classical parameters if $ (v,k, \\lambda) = (\\frac{q^d-1}{q-1}, \\frac{q^{d-1}-1}{q-1}, \\frac{q^{d-2}-1}{q-1}).$ The case $d=3$ corresponds to planar difference sets. We focus here on the family of abelian difference sets with $d=4$. The only known examples of such difference sets correspond to the projective geometries $PG(3,q)$. We consider an arbitrary difference set with the parameters of $PG(3,q)$ in an abelian group and establish constraints on its structure. In particular, we discern embedded substructures."}
{"category": "Math", "title": "Motives for perfect PAC fields with pro-cyclic Galois group", "abstract": "Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective."}
{"category": "Math", "title": "Adjoint Functors and Heteromorphisms", "abstract": "Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades, the notion of adjoint functors has moved to center-stage as category theory's primary tool to characterize what is important in mathematics. Our focus here is to present a theory of adjoint functors. The basis for the theory is laid by first showing that the object-to-object \"heteromorphisms\" between the objects of different categories (e.g., insertion of generators as a set to group map) can be rigorously treated within category theory. The heteromorphic theory shows that all adjunctions arise from the birepresentations of the heteromorphisms between the objects of different categories."}
{"category": "Math", "title": "Factor Analysis and Alternating Minimization", "abstract": "In this paper we make a first attempt at understanding how to build an optimal approximate normal factor analysis model. The criterion we have chosen to evaluate the distance between different models is the I-divergence between the corresponding normal laws. The algorithm that we propose for the construction of the best approximation is of an the alternating minimization kind."}
{"category": "Math", "title": "Minimax State Observation in Linear One Dimensional 2-Point Boundary Value Problems", "abstract": "In this paper we study observation problem for linear 2-point BVP Dx=Bf assuming that information about system input f and random noise \\eta in system state observation model y=Hx+\\eta$ is incomplete (f and M\\eta\\eta' are some arbitrary elements of given sets). A criterion of guaranteed (minimax) estimation error finiteness is proposed. Representations of minimax estimations are obtained in terms of 2-point BVP solutions. It is proved that in general case we can only estimate a projection of system state onto some linear manifold $F$. In particular, $F=L_2$ if $dim N(D H) = 0$. Also we propose a procedure which decides if given linear functional belongs to $F$."}
{"category": "Math", "title": "The Picard group of $M_{1,1}$", "abstract": "We compute the Picard group of the moduli stack of elliptic curves and its canonical compactification over general base schemes."}
{"category": "Math", "title": "Maximally Sparse Polynomials have Solid Amoebas", "abstract": "Let $f$ be an ordinary polynomial in $\\mathbb{C}[z_1,..., z_n]$ with no negative exponents and with no factor of the form $z_1^{\\alpha_1}... z_n^{\\alpha_n}$ where $\\alpha_i$ are non zero natural integer. If we assume in addicting that $f$ is maximally sparse polynomial (that its support is equal to the set of vertices of its Newton polytope), then a complement component of the amoeba $\\mathscr{A}_f$ in $\\mathbb{R}^n$ of the algebraic hypersurface $V_f\\subset (\\mathbb{C}^*)^n$ defined by $f$, has order lying in the support of $f$, which means that $\\mathscr{A}_f$ is solid. This gives an affirmative answer to Passare and Rullg\\aa rd question in [PR2-01]."}
{"category": "Math", "title": "Hydrodynamic Limit for a Particle System with degenerate rates", "abstract": "We study the hydrodynamic limit for some conservative particle systems with degenerate rates, namely with nearest neighbor exchange rates which vanish for certain configurations. These models belong to the class of {\\sl kinetically constrained lattice gases} (KCLG) which have been introduced and intensively studied in physics literature as simple models for the liquid/glass transition. Due to the degeneracy of rates for KCLG there exists {\\sl blocked configurations} which do not evolve under the dynamics and in general the hyperplanes of configurations with a fixed number of particles can be decomposed into different irreducible sets. As a consequence, both the Entropy and Relative Entropy method cannot be straightforwardly applied to prove the hydrodynamic limit. In particular, some care should be put when proving the One and Two block Lemmas which guarantee local convergence to equilibrium. We show that, for initial profiles smooth enough and bounded away from zero and one, the macroscopic density profile for our KCLG evolves under the diffusive time scaling according to the porous medium equation. Then we prove the same result for more general profiles for a slightly perturbed dynamics obtained by adding jumps of the Symmetric Simple Exclusion. The role of the latter is to remove the degeneracy of rates and at the same time they are properly slowed down in order not to change the macroscopic behavior. The equilibrium fluctuations and the magnitude of the spectral gap for this perturbed model are also obtained."}
{"category": "Math", "title": "Hyperbolicity of Semigroup Algebras", "abstract": "Let $A$ be a finite dimensional $Q-$algebra and $\\Gamma subset A$ a $Z-$order. We classify those $A$ with the property that $Z^2$ does not embed in $\\mathcal{U}(\\Gamma)$. We call this last property the hyperbolic property. We apply this in the case that $A = KS$ a semigroup algebra with $K = Q$ or $K = Q(\\sqrt{-d})$. In particular, when $KS$ is semi-simple and has no nilpotent elements, we prove that $S$ is an inverse semigroup which is the disjoint union of Higman groups and at most one cyclic group $C_n$ with $n \\in \\{5,8,12\\}$."}
{"category": "Math", "title": "Grothendieck rings of basic classical Lie superalgebras", "abstract": "The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalised root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called Weyl groupoids."}
{"category": "Math", "title": "Representations of Lie algebras arising from polytopes", "abstract": "We present an extremely elementary construction of the simple Lie algebras over the complex numbers in all of their minuscule representations, using the vertices of various polytopes. The construction itself requires no complicated combinatorics and essentially no Lie theory other than the definition of a Lie algebra; in fact, the Lie algebras themselves appear as by-products of the construction."}
{"category": "Math", "title": "Irreducible representations and Artin L-functions of quasi-cyclotomic fields", "abstract": "We determine all irreducible representations of primary quasi-cyclotomic fields in this paper. The methods can be applied to determine the irreducible representations of any quasi-cyclotomic field. We also compute the Artin L-functions for a class of quasi-cyclotomic fields."}
{"category": "Math", "title": "Graphs with chromatic roots in the interval (1,2)", "abstract": "We present an infinite family of 3-connected non-bipartite graphs with chromatic roots in the interval (1,2) thus resolving a conjecture of Jackson's in the negative. In addition, we briefly consider other graph classes that are conjectured to have no chromatic roots in (1,2)."}
{"category": "Math", "title": "Gersten's conjecture", "abstract": "The purpose of this article is to prove that Gersten's conjecture for a commutative regular local ring is true. As its applications, we will prove the vanishing conjecture for certain Chow groups, generator conjecture for certain $K$-groups and Bloch's formula for absolute case."}
{"category": "Math", "title": "Inference on Eigenvalues of Wishart Distribution Using Asymptotics with respect to the Dispersion of Population Eigenvalues", "abstract": "In this paper we derive some new and practical results on testing and interval estimation problems for the population eigenvalues of a Wishart matrix based on the asymptotic theory for block-wise infinite dispersion of the population eigenvalues. This new type of asymptotic theory has been developed by the present authors in Takemura and Sheena (2005) and Sheena and Takemura (2007a,b) and in these papers it was applied to point estimation problem of population covariance matrix in a decision theoretic framework. In this paper we apply it to some testing and interval estimation problems. We show that the approximation based on this type of asymptotics is generally much better than the traditional large-sample asymptotics for the problems."}
{"category": "Math", "title": "Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas", "abstract": "This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz.\\spacefactor =1000 Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wright's constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area. The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which appears in slightly different ways in formulas for all seven random variables. We further want to give explicit relations between the many different similar notations and definitions that have been used by various authors. There are also some new results, mainly to fill in gaps left in the literature. Some short proofs are given, but most proofs are omitted and the reader is instead referred to the original sources."}
{"category": "Math", "title": "Symmetries in the system of type $A_5^{(2)}$", "abstract": "In this paper, we propose a 3-parameter family of coupled Painlev\\'e III systems in dimension four with affine Weyl group symmetry of type $A_5^{(2)}$. We also propose its symmetric form in which the $A_5^{(2)}$-symmetries become clearly visible."}
{"category": "Math", "title": "Cutting surfaces and applications to periodic points and chaotic-like dynamics", "abstract": "In this paper we propose an elementary topological approach which unifies and extends various different results concerning fixed points and periodic points for maps defined on sets homeomorphic to rectangles embedded in euclidean spaces. We also investigate the associated discrete semidynamical systems in view of detecting the presence of chaotic-like dynamics."}
{"category": "Math", "title": "Symmetries in the system of type $D_4^{(1)}$", "abstract": "In this paper, we propose a 4-parameter family of coupled Painlev\\'e III systems in dimension four with affine Weyl group symmetry of type $D_4^{(1)}$. We also propose its symmetric form in which the $D_4^{(1)}$-symmetries become clearly visible."}
{"category": "Math", "title": "Existence of graphs with sub exponential transitions probability decay and applications", "abstract": "In this paper, we present a complete proof of the construction of graphs with bounded valency such that the simple random walk has a return probability at time $n$ at the origin of order $exp(-n^{\\alpha}),$ for fixed $\\alpha \\in [0,1[$ and with Folner function $exp(n^{\\frac{2\\alpha}{1-\\alpha}})$. We begin by giving a more detailled proof of this result contained in (see \\cite{ershdur}). In the second part, we give an application of the existence of such graphs. We obtain bounds of the correct order for some functional of the local time of a simple random walk on an infinite cluster on the percolation model."}
{"category": "Math", "title": "A new class of rank one transformations with singular spectrum", "abstract": "We introduce a new tool to study the spectral type of rank one transformations using the method of central limit theorem for trigonometric sums. We get some new applications."}
{"category": "Math", "title": "Navier-Stokes equations with periodic boundary conditions and pressure loss", "abstract": "We present in this note the existence and uniqueness results for the Stokes and Navier-Stokes equations which model the laminar flow of an incompressible fluid inside a two-dimensional channel of periodic sections. The data of the pressure loss coefficient enables us to establish a relation on the pressure and to thus formulate an equivalent problem."}
{"category": "Math", "title": "Reproductive strong solutions of Navier-Stokes equations with non homogeneous boundary conditions", "abstract": "The object of the present paper is to show the existence and the uniqueness of a reproductive strong solution of the Navier-Stokes equations, i.e. the solution $\\boldsymbol{u} $ belongs to $\\text{}\\mathbf{L}% ^{\\infty}(0,T;V) \\cap \\mathbf{L}^{2}(0,T;\\mathbf{H}% ^{2}(\\Omega))$ and satisfies the property $\\boldsymbol{u}% (\\boldsymbol{x,}T) =\\boldsymbol{u}% (\\boldsymbol{x,}0) =\\boldsymbol{u}_{0}(\\boldsymbol{x})$. One considers the case of an incompressible fluid in two dimensions with nonhomogeneous boundary conditions, and external forces are neglected."}
{"category": "Math", "title": "Solutions fortes de l'\\'equation de l'\\'energie", "abstract": "In this paper, we give some existence results of stong solutions for the energy equation associated to the Navier-Stokes equations with nonhomogeneous boundary conditions in two dimension."}
{"category": "Math", "title": "On Brownian flights", "abstract": "Let K be a compact subset of ${\\mathbb R}^n$. We choose at random with uniform law a point at distance $\\epsilon$ of K and start a Brownian motion (BM) from this point. We study the probability that this BM hits K for the first time at a distance $\\geq r$ from the starting point."}
{"category": "Math", "title": "Energy solutions for polymer aqueous solutions in two dimension", "abstract": "The aim of this article is to study a nonlinear system modeling a Non-Newtonian fluid of polymer aqueous solutions. We are interested here in the existence of weak solutions for the stationary problem in a bounded plane domain or in two-dimensional exterior domain. Due to the third order of derivatives in the non-linear term, it's difficult to obtain solution satisfying energy inequality. But with a good choice of boundary conditions, an adapt special basis and the use of the good properties of the trilinear form associated to the non-linear term, we obtain energy solutions. The problem in bounded domain is treated and the more difficult problem on non bounded domain too."}
{"category": "Math", "title": "Coupled Painlev\\'e VI systems in dimension four with affine Weyl group symmetry of type $D_6^{(1)}$, II", "abstract": "We give a reformulation of a six-parameter family of coupled Painlev\\'e VI systems with affine Weyl group symmetry of type $D_6^{(1)}$ from the viewpoint of its symmetry and holomorphy properties."}
{"category": "Math", "title": "Asymptotics of the fast diffusion equation via entropy estimates", "abstract": "We consider non-negative solutions of the fast diffusion equation $u_t=\\Delta u^m$ with $m \\in (0,1)$, in the Euclidean space R^d, d?3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to $t\\to\\infty$ for $m\\ge m_c=(d-2)/d$, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if $m\\ge m_c$, or close enough to the extinction time if m < mc. Such results are new in the range $m\\le m_c$ where previous approaches fail. In the range mc < m < 1 we improve on known results."}
{"category": "Math", "title": "Catalan Traffic and Integrals on the Grassmannians of Lines", "abstract": "We prove that certain numbers occurring in a problem of paths enumeration, studied by Niederhausen (Catlan Traffic at the Beach, The Electronic Journal of Combinatorics, 9, (2002), 1--17), are top intersection numbers in the cohomology ring of the grassmannians of the lines in the complex projective (n+1)-space."}
{"category": "Math", "title": "A note on Seshadri constants on general $K3$ surfaces", "abstract": "We prove a lower bound on the Seshadri constant $\\epsilon (L)$ on a $K3$ surface $S$ with $\\Pic S \\simeq \\ZZ[L]$. In particular, we obtain that $\\epsilon (L)=\\alpha$ if $L^2=\\alpha^2$ for an integer $\\alpha$."}
{"category": "Math", "title": "Extended centres of finitely generated prime algebras", "abstract": "Let $K$ be a field and let $A$ be a finitely generated prime $K$-algebra. We generalize a result of Smith and Zhang, showing that if $A$ is not PI and does not have a locally nilpotent ideal, then the extended centre of $A$ has transcendence degree at most ${\\rm GKdim}(A)-2$ over $K$. As a consequence, we are able to show that if $A$ is a prime $K$-algebra of quadratic growth, then either the extended centre is a finite extension of K or $A$ is PI. Finally, we give an example of a finitely generated non-PI prime $K$-algebra of GK dimension 2 with a locally nilpotent ideal such that the extended centre has infinite transcendence degree over $K$."}
{"category": "Math", "title": "Local well-posedness of Musiela's SPDE with L\\'evy noise", "abstract": "We determine sufficient conditions on the volatility coefficient of Musiela's stochastic partial differential equation driven by an infinite dimensional L{\\'e}vy process so that it admits a unique local mild solution in spaces of functions whose first derivative is square integrable with respect to a weight."}
{"category": "Math", "title": "The prime spectrum of algebras of quadratic growth", "abstract": "We study prime algebras of quadratic growth. Our first result is that if $A$ is a prime monomial algebra of quadratic growth then $A$ has finitely many prime ideals $P$ such that $A/P$ has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the nonzero prime ideals $P$ such that $A/P$ has GK dimension 2 is non-empty, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has nonzero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra $A$ of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that $A/P$ has GK dimension 1."}
{"category": "Math", "title": "Fusion algebras with negative structure constants", "abstract": "We introduce fusion algebras with not necessarily positive structure constants and without identity element. We prove that they are semisimple when tensored with $\\mathbb{C}$ and that their characters satisfy orthogonality relations. Then we define the proper notion of subrings and factor rings for such algebras. For certain algebras $R$ we prove the existence of a ring $R'$ with nonnegative structure constants such that $R$ is a factor ring of $R'$. We give some examples of interesting factor rings of the representation ring of the quantum double of a finite group. Then, we investigate the algebras associated to Hadamard matrices. For an $n\\times n$-matrix the corresponding algebra is a factor ring of a subalgebra of $\\mathbb{Z}[{(\\mathbb{Z}/2\\mathbb{Z})}^{n-2}]$."}
{"category": "Math", "title": "L'indice de Maslov dans les $JB^*$-triples", "abstract": "We construct a homotopy invariant index for pathes in the set of invertible tripotents in a JB*-triple that satisfy a Fredholm type condition with respect to a fixed invertible tripotent. That index generalizes the Maslov index in the Fredholm-Lagrangian of a symplectic Hilbert space."}
{"category": "Math", "title": "Ordinary differential systems in dimension three with affine Weyl group symmetry of types $D_4^{(1)},B_3^{(1)},G_2^{(1)},D_3^{(2)}$ and $A_2^{(2)}$", "abstract": "We present a four-parameter family of ordinary differential systems in dimension three with affine Weyl group symmetry of type $D_4^{(1)}$. By obtaining its first integral, we can reduce this system to the second-order non-linear ordinary differential equations of Painlev\\'e type. We also study this system restricted its parameters. Each system can be obtained by connecting some invariant divisors in the system of type $D_4^{(1)}$. Each system admits affine Weyl group symmetry of types $B_3^{(1)},G_2^{(1)},D_3^{(2)}$ and $A_2^{(2)}$, respectively. These symmetries, holomorphy conditions and invariant divisors are new."}
{"category": "Math", "title": "Symmetry in the Painlev\\'e systems and their extensions to four-dimensional systems", "abstract": "We give a new approach to the symmetries of the Painlev\\'e equations $P_{V},P_{IV},P_{III}$ and $P_{II}$, respectively. Moreover, we make natural extensions to fourth-order analogues for each of the Painlev\\'e equations $P_{V}$ and $P_{III}$, respectively, which are natural in the sense that they preserve the symmetries."}
{"category": "Math", "title": "Multivariate Wavelet Frames", "abstract": "We proved that for any matrix dilation and for any positive integer $n$, there exists a compactly supported tight wavelet frame with approximation order $n$. Explicit methods for construction of dual and tight wavelet frames with a given number of vanishing moments are suggested."}
{"category": "Math", "title": "Indices of the iterates of $R^3$-homeomorphisms at Lyapunov stable fixed points", "abstract": "Given any positive sequence (\\{c_n\\}_{n \\in {\\Bbb N}}), we construct orientation preserving homeomorphisms (f:{\\Bbb R}^3 \\to {\\Bbb R}^3) such that (Fix(f)=Per(f)=\\{0\\}), (0) is Lyapunov stable and (\\limsup \\frac{|i(f^m, 0)|}{c_m}= \\infty). We will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms."}
{"category": "Math", "title": "The Invariant Ring of Triples of 3x3 Matrices over a Field of Arbitrary Characteristic", "abstract": "Let R_{n,d} be the ring of invariants of d-tuples of n x n matrices under the simultaneous conjugation action of the general linear group. A minimal generating system and a homogeneous system of parameters for R_{3,3} are determined. Homogeneous systems of parameters for R_{3,2}, R_{4,2} are also pointed out."}
{"category": "Math", "title": "Indecomposable invariants of quivers for dimension (2,...,2) and maximal paths", "abstract": "An upper bound on degrees of elements of a minimal generating system for invariants of quivers of dimension (2,...,2) is established over a field of arbitrary characteristic and its precision is estimated. The proof is based on the reduction to the problem of description of maximal paths satisfying certain condition."}
{"category": "Math", "title": "Hopf algebras of dimension pq, II", "abstract": "Let H be a Hopf algebra of dimension pq over an algebraically closed field of characteristic zero, where p, q are odd primes with p < q < 4p+12. We prove that H is semisimple and thus isomorphic to a group algebra, or the dual of a group algebra."}
{"category": "Math", "title": "New $_5F_4$ hypergeometric transformations, three-variable Mahler measures, and formulas for $1/\\pi$", "abstract": "New relations are established between families of three-variable Mahler measures. Those identities are then expressed as transformations for the $_5F_4$ hypergeometric function. We use these results to obtain two explicit $_5F_4$ evaluations, and several new formulas for $1/\\pi$."}
{"category": "Math", "title": "Visible Points on Curves over Finite Fields", "abstract": "For a prime $p$ and an absolutely irreducible modulo $p$ polynomial $f(U,V) \\in \\Z[U,V]$ we obtain an asymptotic formulas for the number of solutions to the congruence $f(x,y) \\equiv a \\pmod p$ in positive integers $x \\le X$, $y \\le Y$, with the additional condition $\\gcd(x,y)=1$. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over $a$ for a fixed prime $p$, and also on average over $p$ for a fixed integer $a$."}
{"category": "Math", "title": "An Abstract Regularity Lemma", "abstract": "We extend Szemeredi's Regularity Lemma (SRL) to abstract measure spaces. Our main aim is to find general conditions under which the original proof of Szemeredi still works. To illustrate that our approach has some merit, we outline several applications. Some of these applications seem to be tailored to our approach: in particular, we are not aware of any alternative proofs."}
{"category": "Math", "title": "On Carmichael's Conjecture", "abstract": "In this article we prove that equation $\\phi(x)=n$, for a fixed $n$, admits a finite number of solutions, we find the general form of these solutions, and we show that: if $x_0$ is a unique solution of this equation then $x_0$ is a product of a very large number of primes (we conjecture that the number of such primes is infinite)."}
{"category": "Math", "title": "Existence and Stability for Fokker-Planck equations with log-concave reference measure", "abstract": "We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker-Planck equation as the steepest descent flow of the relative Entropy functional in the space of probability measures, endowed with the Wasserstein distance. Applications include stochastic partial differential equations and convergence of equilibrium fluctuations for a class of random interfaces."}
{"category": "Math", "title": "Coupled Painlev\\'e III systems with affine Weyl group symmetry of types $B_5^{(1)},D_5^{(1)}$ and $D_6^{(2)}$", "abstract": "We find and study four kinds of five-parameter family of six-dimensional coupled Painlev\\'e III systems with affine Weyl group symmetry of types $D_5^{(1)},B_5^{(1)}$ and $D_6^{(2)}$. We show that each system is equivalent by an explicit birational and symplectic transformation, respectively. We also show that we characterize each system from the viewpoint of holomorphy."}
{"category": "Math", "title": "Discontinuity of the Lempert function and the Kobayashi--Royden metric of the spectral ball", "abstract": "Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given."}
{"category": "Math", "title": "Coupled Painlev\\'e III systems with affine Weyl group symmetry of types $B_4^{(1)}$, $D_4^{(1)}$ and $D_5^{(2)}$", "abstract": "We find and study four kinds of a 4-parameter family of four-dimensional coupled Painlev\\'e III systems with affine Weyl group symmetry of types $B_4^{(1)}$, $D_4^{(1)}$ and $D_5^{(2)}$. We also show that these systems are equivalent by an explicit birational and symplectic transformation, respectively."}
{"category": "Math", "title": "Coupled Painlev\\'e VI systems in dimension four with affine Weyl group symmetry of types $B_6^{(1)}$, $D_6^{(1)}$ and $D_7^{(2)}$", "abstract": "We find four kinds of six-parameter family of coupled Painlev\\'e VI systems in dimension four with affine Weyl group symmetry of types $B_6^{(1)}$, $D_6^{(1)}$ and $D_7^{(2)}$. Each system is the first example which gave higher-order Painlev\\'e equations of types $B_l^{(1)},D_l^{(1)}$ and $D_l^{(2)}$, respectively. Each system can be expressed as a polynomial Hamiltonian system. We show that these systems are equivalent by an explicit birational and symplectic transformation, respectively. By giving each holomorphy condition, we can recover each system. These symmetries, holomorphy conditions and invariant divisors are new. We also give an explicit description of a confluence process from the system of type $D_6^{(1)}$ to the system of type $A_5^{(1)}$ by taking the coupling confluence process from the Painlev\\'e VI system to the Painlev\\'e V system."}
{"category": "Math", "title": "Lineare Rekurrenzen, Potenzreihen und ihre erzeugenden Funktionen", "abstract": "Diese kurze Einfuehrung in Theorie und Berechnung linearer Rekurrenzen versucht, eine Luecke in der Literatur zu fuellen. Zu diesem Zweck sind viele ausfuehrliche Beispiele angegeben. This short introduction to theory and usage of linear recurrences tries to fill a gap in the literature by giving many extensive examples."}
{"category": "Math", "title": "Un module inversible associ\\'e au ruban de M\\\"obius, et quelques autres", "abstract": "After attaching explicitly to the M\\\"obius strip an invertible module over the ring of real polynomial functions on the real circle, we expound as directly as possible the many faces and the main algebraic properties of invertible modules. The goal is to make this algebraic concept accessible to a wide mathematical audience."}
{"category": "Math", "title": "On the Stability Functional for Conservation Laws", "abstract": "This note is devoted to the explicit construction of a functional defined on all pairs of $\\L1$ functions with small total variation, which is equivalent to the $\\L1$ distance and non increasing along the trajectories of a given system of conservation laws. Two different constructions are provided, yielding an extension of the original stability functional by Bressan, Liu and Yang."}
{"category": "Math", "title": "Structural adaptation via $L_p$-norm oracle inequalities", "abstract": "In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary $\\rL_p$--losses. These results are applied for adaptive estimation in the additive multi--index model."}
{"category": "Math", "title": "A universal procedure for aggregating estimators", "abstract": "In this paper we study the aggregation problem that can be formulated as follows. Assume that we have a family of estimators $\\mathcal{F}$ built on the basis of available observations. The goal is to construct a new estimator whose risk is as close as possible to that of the best estimator in the family. We propose a general aggregation scheme that is universal in the following sense: it applies for families of arbitrary estimators and a wide variety of models and global risk measures. The procedure is based on comparison of empirical estimates of certain linear functionals with estimates induced by the family $\\mathcal{F}$. We derive oracle inequalities and show that they are unimprovable in some sense. Numerical results demonstrate good practical behavior of the procedure."}
{"category": "Math", "title": "Homotopy coherent nerve in Deformation theory", "abstract": "In this note we explain that homotopy coherent simplicial nerve has to used intead of the standard definition in the author's papers on formal deformation theory. A convenient version of the notion of fibered category is presented which is useful once one works with simplicial categories."}
{"category": "Math", "title": "Postnikov-Stability for Complexes", "abstract": "We present a novel notion of stable objects in the derived category of coherent sheaves on a smooth projective variety. As one application we compactify a moduli space of stable bundles using genuine complexes."}
{"category": "Math", "title": "On groups of central type, non-degenerate and bijective cohomology classes", "abstract": "A finite group $G$ is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class $[c]\\in H^2(G,\\C^*)$ ($G$ acts trivially on $\\C^*$). Groups of central type play a fundamental role in the classification of semisimple triangular complex Hopf algebras and can be determined by their representation theoretical properties. Suppose that a finite group $Q$ acts on an abelian group $A$ so that there exists a bijective 1-cocycle $\\pi\\in Z^1(Q,\\ach)$, where $\\ach=\\rm{Hom}(A,\\C^*)$ is endowed with the diagonal $Q$-action. Under this assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate 2-cocycle in $Z^2(G,\\C^*)$, where $G:=A\\rtimes Q$. Hence, the semidirect product $G$ is of central type. In this paper we present a more general correspondence between bijective and non-degenerate cohomology classes. In particular, given a bijective class $[\\pi]\\in H^1(Q,\\ach)$ as above, we construct non-degenerate classes $[c_{\\pi}]\\in H^2(G,\\C^*)$ for certain extensions $1\\to A\\to G\\to Q\\to 1$ which are not necessarily split. We thus strictly extend the above family of central type groups."}
{"category": "Math", "title": "Combinatorics Of RNA Structures With Pseudoknots", "abstract": "In this paper we derive the generating function of RNA structures with pseudoknots. We enumerate all $k$-noncrossing RNA pseudoknot structures categorized by their maximal sets of mutually intersecting arcs. In addition we enumerate pseudoknot structures over circular RNA. For 3-noncrossing RNA structures and RNA secondary structures we present a novel 4-term recursion formula and a 2-term recursion, respectively. Furthermore we enumerate for arbitrary $k$ all $k$-noncrossing, restricted RNA structures i.e. $k$-noncrossing RNA structures without 2-arcs i.e. arcs of the form $(i,i+2)$, for $1\\le i\\le n-2$."}
{"category": "Math", "title": "Substitution tilings with statistical circular symmetry", "abstract": "Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for all substitution tilings showing tiles in infinitely many orientations."}
{"category": "Math", "title": "Quelques plats pour la m\\'etrique de Hofer", "abstract": "We show, by an elementary and explicit construction, that the group of Hamiltonian diffeomorphisms of certain symplectic manifolds, endowed with Hofer's metric, contains subgroups quasi-isometric to Euclidean spaces of arbitrary dimension."}
{"category": "Math", "title": "Self-dual tilings with respect to star-duality", "abstract": "The concept of star-duality is described for self-similar cut-and-project tilings in arbitrary dimensions. This generalises Thurston's concept of a Galois-dual tiling. The dual tilings of the Penrose tilings as well as the Ammann-Beenker tilings are calculated. Conditions for a tiling to be self-dual are obtained."}
{"category": "Math", "title": "Integration on moduli spaces of stable curves through localization", "abstract": "We introduce a new method of calculating intersections on \\bar{M}_{g,n}, using localization of equivariant cohomology. As an application, we give a proof of Mirzakhani's recursion relation for calculating intersections of mixed psi and kappa_1 classes."}
{"category": "Math", "title": "Periodic solutions for the Schroedinger equation with nonlocal smoothing nonlinearities in higher dimension", "abstract": "We consider the nonlinear Schroedinger equation in higher dimension with Dirichlet boundary conditions and with a non-local smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in the sense that they continue linear solutions with an arbitrarily large number of resonant modes. The main difficulty in the proof consists in solving a \"small divisor problem\" which we do by using a renormalisation group approach."}
{"category": "Math", "title": "Characteristic forms of complex Cartan geometries", "abstract": "We calculate relations on characteristic classes which are obstructions preventing closed K\\\"ahler manifolds from carrying holomorphic Cartan geometries. We apply these relations to give global constraints on the phase spaces of complex analytic determined and underdetermined systems of differential equations."}
{"category": "Math", "title": "Families of varieties of general type over compact bases", "abstract": "Let f: X -> Y be a smooth family of canonically polarized complex varieties over a smooth base. Generalizing the classical Shafarevich hyperbolicity conjecture, Viehweg conjectured that Y is necessarily of log general type if the family has maximal variation. A somewhat stronger and more precise version of Viehweg's conjecture was shown by the authors in arXiv:math/0511378 in the case where Y is a quasi-projective surface. Assuming that the minimal model program holds, this very short paper proves the same result for projective base manifolds Y of arbitrary dimension."}
{"category": "Math", "title": "Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices", "abstract": "Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M-function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M-function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M-function corresponds to the Dirichlet to Neumann map."}
{"category": "Math", "title": "Homomorphic images of Branch groups, and Serre's property (FA)", "abstract": "It is shown that a finitely generated branch group has Serre's property (FA) if and only if it does not surject onto the infinite cyclic group or the infinite dihedral group. An example of a finitely generated self-similar branch group surjecting onto the infinite cyclic group is constructed."}
{"category": "Math", "title": "L-stable functors", "abstract": "We generalize and greatly simplify the approach of Lydakis and Dundas-R\\\"ondigs-{\\O}stv{\\ae}r to construct an L-stable model structure for small functors from a closed symmetric monoidal model category V to a V-model category M, where L is a small cofibrant object of V. For the special case V=M=S_* pointed simplicial sets and L=S^1 this is the classical case of linear functors and has been described as the first stage of the Goodwillie tower of a homotopy functor. We show, that our various model structures are compatible with a closed symmetric monoidal product on small functors. We compare them with other L-stabilizations described by Hovey, Jardine and others. This gives a particularly easy construction of the classical and the motivic stable homotopy category with the correct smash product. We establish the monoid axiom under certain conditions."}
{"category": "Math", "title": "Formal completions of N\\'eron models for algebraic tori", "abstract": "We calculate the formal group law which represents the completion of the N\\'eron model of an algebraic torus over the rationals that splits in a tamely ramified abelian extension. As a tools in the proof, we define and give criterions to compute the Weil restriction of a formal group law and the analog of the fixed part of a formal group law with respect to the action of a (finite) group."}
{"category": "Math", "title": "Distributions of Roots of Reduced Cubic Equations with Random Coefficients", "abstract": "If the coefficients of polynomials are selected by some random process, the zeros of the resulting polynomials are in some sense random. In this paper the author rephrases the above in more precise language, and calculates the joint conditional densities of a random vector whose values determine almost surely the zeros of a \"random\" reduced cubic."}
{"category": "Math", "title": "Uniform convergence in the mapping class group", "abstract": "We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact."}
{"category": "Math", "title": "Computing invariants of algebraic group actions in arbitrary characteristic", "abstract": "Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring K[X]^G in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes K[X]^G in terms of a so-called colon-operation. From this, generators of K[X]^G can be obtained in finite time if it is finitely generated. Under the additional hypothesis that K[X] is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is K[X]^G. Along the way, we develop some techniques for dealing with non-finitely generated algebras. In particular, we introduce the finite generation locus ideal."}
{"category": "Math", "title": "On the cycling operation in braid groups", "abstract": "The cycling operation is a special kind of conjugation that can be applied to elements in Artin's braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In their seminal paper on braid-cryptography, Ko, Lee et al. proposed the {\\it cycling problem} as a hard problem in braid groups that could be interesting for cryptography. In this paper we give a polynomial solution to that problem, mainly by showing that cycling is surjective, and using a result by Maffre which shows that pre-images under cycling can be computed fast. This result also holds in every Artin-Tits group of spherical type. On the other hand, the conjugacy search problem in braid groups is usually solved by computing some finite sets called (left) ultra summit sets (left-USS), using left normal forms of braids. But one can equally use right normal forms and compute right-USS's. Hard instances of the conjugacy search problem correspond to elements having big (left and right) USS's. One may think that even if some element has a big left-USS, it could possibly have a small right-USS. We show that this is not the case in the important particular case of rigid braids. More precisely, we show that the left-USS and the right-USS of a given rigid braid determine isomorphic graphs, with the arrows reversed, the isomorphism being defined using iterated cycling. We conjecture that the same is true for every element, not necessarily rigid, in braid groups and Artin-Tits groups of spherical type."}
{"category": "Math", "title": "Cohomotopy invariants and the universal cohomotopy invariant jump formula", "abstract": "Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of $S^1$-equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which have clear functorial properties with respect to diffeomorphisms of 4-manifolds. Our invariants and the Bauer-Furuta classes are directly comparable for 4-manifolds with $b_1=0$; they are equivalent when $b_1=0$ and $b_+>1$, but are finer in the case $b_1=0$, $b_+=1$ (they detect the wall-crossing phenomena). We study fundamental properties of the new invariants in a very general framework. In particular we prove a universal cohomotopy invariant jump formula and a multiplicative property. The formalism applies to other gauge theoretical problems, e.g. to the theory of gauge theoretical (Hamiltonian) Gromov-Witten invariants."}
{"category": "Math", "title": "Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds", "abstract": "We prove an existence theorem for gauge invariant $L^2$-normal neighborhoods of the reduction loci in the space ${\\cal A}_a(E)$ of oriented connections on a fixed Hermitian 2-bundle $E$. We use this to obtain results on the topology of the moduli space ${\\cal B}_a(E)$ of (non-necessarily irreducible) oriented connections, and to study the Donaldson $\\mu$-classes globally around the reduction loci. In this part of the article we use essentially the concept of harmonic section in a sphere bundle with respect to an Euclidean connection. Second, we concentrate on moduli spaces of instantons on definite 4-manifolds with arbitrary first Betti number. We prove strong generic regularity results which imply (for bundles with \"odd\" first Chern class) the existence of a connected, dense open set of \"good\" metrics for which all the reductions in the Uhlenbeck compactification of the moduli space are simultaneously regular. These results can be used to define new Donaldson type invariants for definite 4-manifolds. The idea behind this construction is to notice that, for a good metric $g$, the geometry of the instanton moduli spaces around the reduction loci is always the same, independently of the choice of $g$. The connectedness of the space of good metrics is important, in order to prove that no wall-crossing phenomena (jumps of invariants) occur. Moreover, we notice that, for low instanton numbers, the corresponding moduli spaces are a priori compact and contain no reductions at all so, in these cases, the existence of well-defined Donaldson type invariants is obvious. The natural question is to decide whether these new Donaldson type invariants yield essentially new differential topological information on the base manifold have, or have a purely topological nature."}
{"category": "Math", "title": "Families of holomorphic bundles", "abstract": "The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds: For instance we prove that stability and semi-stability are Zariski open properties in families when the Gauduchon degree map is a topological invariant, or when the parameter manifold is compact. Second we show that, for a generically stable family of bundles over a K\\\"ahler manifold, the Petersson-Weil form extends as a closed positive current on the whole parameter space of the family. This extension theorem uses classical tools from Yang-Mills theory developed by Donaldson (e.g. the Donaldson functional and the heat equation for Hermitian metrics on a holomorphic bundle). We apply these results to study families of bundles over a K\\\"ahlerian manifold $Y$ parameterized by a non-K\\\"ahlerian surface $X$, proving that such families must satisfy very restrictive conditions. These results play an important role in our program to prove existence of curves on class VII surfaces."}
{"category": "Math", "title": "The E-theoretic descent functor for groupoids", "abstract": "The paper establishes, for a wide class of locally compact groupoids $\\Gamma$, the E-theoretic descent functor at the $C^{*}$-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. The second section shows that $\\Gamma$-actions on a $C_{0}(X)$-algebra $B$, where $X$ is the unit space of $\\Gamma$, can be usefully formulated in terms of an action on the associated bundle $B^{\\sharp}$. The third section shows that the functor $B\\to C^{*}(\\Gamma,B)$ is continuous and exact, and uses the disintegration theory of J. Renault. The last section establishes the existence of the descent functor under a very mild condition on $\\Gamma$, the main technical difficulty involved being that of finding a $\\Gamma$-algebra that plays the role of C_{b}(T,B)^{cont}$ in the group case."}
{"category": "Math", "title": "Instantons and curves on class VII surfaces", "abstract": "We develop a general strategy, based on gauge theoretical methods, to prove existence of curves on class VII surfaces. We prove that, for $b_2=2$, every minimal class VII surface has a cycle of rational curves hence, by a result of Nakamura, is a global deformation of a one parameter family of blown up primary Hopf surfaces. The case $b_2=1$ has been solved in a previous article. The fundamental object intervening in our strategy is the moduli space ${\\mathcal M}^{\\pst}(0,{\\mathcal K})$ of polystable bundles ${\\mathcal E}$ with $c_2({\\mathcal E})=0$, $\\det({\\mathcal E})={\\mathcal K}$. For large $b_2$ the geometry of this moduli space becomes very complicated. The case $b_2=2$ treated here in detail requires new ideas and difficult techniques of both complex geometric and gauge theoretical nature."}
{"category": "Math", "title": "Donaldson theory on non-K\\\"ahlerian surfaces and class $VII$ surfaces with $b_2=1$", "abstract": "We prove that any class $VII$ surface with $b_2=1$ has curves. This implies the \"Global Spherical Shell conjecture\" in the case $b_2=1$: Any minimal class $VII$ surface with $b_2=1$ admits a global spherical shell, hence it is isomorphic to one of the surfaces in the known list. The main idea of the proof is to show that a certain moduli space of PU(2)-instantons on a surface $X$ with no curves (if such a surface existed) would contain a closed Riemann surface $Y$ whose general points correspond to non-filtrable holomorphic bundles on $X$. Then we pass from a family of bundles on $X$ parameterized by $Y$ to a family of bundles on $Y$ parameterized by $X$, and we use the algebraicity of $Y$ to obtain a contradiction. The proof uses essentially techniques from Donaldson theory: compactness theorems for moduli spaces of PU(2)-instantons and the Kobayashi-Hitchin correspondence on surfaces."}
{"category": "Math", "title": "Isolated fixed points and moment maps of symplectic manifolds", "abstract": "Withdrawn due to an incompleteness of the main results."}
{"category": "Math", "title": "On Multiplier Hermitian Structures on Compact Kahler Manifolds", "abstract": "Mabuchi introduced multiplier Hermitian structures on compact Kahler manifolds and defined metrics similar to Kahler-Einstein metrics under these structures. In this note we generalize the inequality of Moser-Trudinger type on Kahler-Einstein manifolds to this case."}
{"category": "Math", "title": "Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs", "abstract": "An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2,4,6,8}. We provide sufficient conditions for the existence of such a subgraph."}
{"category": "Math", "title": "Application of the equal dissipation rate principle to automatic generation of strut-and-tie models", "abstract": "This work presents an extended formulation of maximal stiffness design, within the framework of the topology optimization. The mathematical formulation of the optimization problem is based on the postulated principle of equal dissipation rate during inelastic deformation. This principle leads to the enforcement of stress constraints in domains where inelastic deformation would occur. During the transition from the continuous structure to the truss-like one (strut-and-tie model) the dissipation rate is kept constant using the projected gradient method. The equal dissipation rate in the resulting truss and in the original continuous structure can be regarded as an equivalence statement and suggests an alternative understanding of physical motivation behind the strut-and-tie modeling. The performance of the proposed formulation is demonstrated with the help of two examples."}
{"category": "Math", "title": "First integrals for non linear hyperbolic equations", "abstract": "Given a solution of a nonlinear wave equation on the flat space-time (with a real analytic nonlinearity), we relate its Cauchy data at two different times by nonlinear representation formulas in terms of asymptotic series. We first show how to construct formally these series by mean of generating functions based on an algebraic framework inspired by the construction of Fock spaces in quantum field theory. Then we build an analytic setting in which all these constructions really make sense and give rise to convergent series."}
{"category": "Math", "title": "Classification of line-transitive point-imprimitive linear spaces with line size at most 12", "abstract": "In this paper we complete a classification of finite linear spaces $\\cS$ with line size at most 12 admitting a line-transitive point-imprimitive subgroup of automorphisms. The examples are the Desarguesian projective planes of orders $4,7, 9$ and 11, two designs on 91 points with line size 6, and 467 designs on 729 points with line size 8."}
{"category": "Math", "title": "Determining factors behind the PageRank log-log plot", "abstract": "We study the relation between PageRank and other parameters of information networks such as in-degree, out-degree, and the fraction of dangling nodes. We model this relation through a stochastic equation inspired by the original definition of PageRank. Further, we use the theory of regular variation to prove that PageRank and in-degree follow power laws with the same exponent. The difference between these two power laws is in a multiple coefficient, which depends mainly on the fraction of dangling nodes, average in-degree, the power law exponent, and damping factor. The out-degree distribution has a minor effect, which we explicitly quantify. Our theoretical predictions show a good agreement with experimental data on three different samples of the Web."}
{"category": "Math", "title": "Computation of Atomic Fibers of Z-Linear Maps", "abstract": "For given matrix $A\\in\\Z^{d\\times n}$, the set $P_{b}=\\{z:Az=b,z\\in\\Z^n_+\\}$ describes the preimage or fiber of $b\\in\\Z^d$ under the $\\Z$-linear map $f_A:\\Z^n_+\\to\\Z^d$, $x\\mapsto Ax$. The fiber $P_{b}$ is called atomic, if $P_{b}=P_{b_1}+P_{b_2}$ implies $b=b_1$ or $b=b_2$. In this paper we present a novel algorithm to compute such atomic fibers. An algorithmic solution to appearing subproblems, computational examples and applications are included as well."}
{"category": "Math", "title": "Dynamics of meromorphic functions with direct or logarithmic singularities", "abstract": "We show that if a meromorphic function has a direct singularity over infinity, then the escaping set has an unbounded component and the intersection of the escaping set with the Julia set contains continua. This intersection has an unbounded component if and only if the function has no Baker wandering domains. We also give estimates of the Hausdorff dimension and the upper box dimension of the Julia set of a meromorphic function with a logarithmic singularity over infinity. The above theorems are deduced from more general results concerning functions which have \"direct or logarithmic tracts\", but which need not be meromorphic in the plane. These results are obtained by using a generalization of Wiman-Valiron theory. The method is also applied to complex differential equations."}
{"category": "Math", "title": "Tverberg's theorem with constraints", "abstract": "The topological Tverberg theorem claims that for any continuous map of the (q-1)(d+1)-simplex to R^d there are q disjoint faces such that their images have a non-empty intersection. This has been proved for affine maps, and if $q$ is a prime power, but not in general. We extend the topological Tverberg theorem in the following way: Pairs of vertices are forced to end up in different faces. This leads to the concept of constraint graphs. In Tverberg's theorem with constraints, we come up with a list of constraints graphs for the topological Tverberg theorem. The proof is based on connectivity results of chessboard-type complexes. Moreover, Tverberg's theorem with constraints implies new lower bounds for the number of Tverberg partitions. As a consequence, we prove Sierksma's conjecture for $d=2$, and $q=3$."}
{"category": "Math", "title": "Multidimensional SDE with anticipating initial process and reflection", "abstract": "In this paper, the strong solutions $ (X, L)$ of multidimensional stochastic differential equations with reflecting boundary and possible anticipating initial random variables is established. The key is to obtain some substitution formula for Stratonovich integrals via a uniform convergence of the corresponding Riemann sums and to prove continuity of functionals of $ (X, L)$."}
{"category": "Math", "title": "Constructing a quadrilateral inside another one", "abstract": "Connect each vertex of a convex quadrilateral Q to the midpoint of the next (proceeding counterclockwise) side. The four connecting lines create an interior quadrilateral I. We study the ratio area(I)/area(Q). We also determine what happens to area(I)/area(Q) when the four midpoints are replaced by points which divide the sides in the ratio of rho to (1-rho) proceeding clockwise. Here rho is any fixed number satisfying 0 < rho < 1."}
{"category": "Math", "title": "Two Generator Subalgebras of Lie Algebras", "abstract": "J. G. Thompson showed that a finite group G is solvable if and only if every two -generated subgroup is solvable. Recently, Grunevald, Kunyavskii, Nikolova, and Plotkin have shown that the analogue holds for finite-dimensional Lie algebras over infinite fields of characteristic greater than 5. It is a natural question to ask to what extent the two-generated subalgebras determine the structure of the algebra. It is to this question that this paper is addressed. Here, we consider the classes of strongly-solvable and of supersolvable Lie algebras, and the property of triangulability."}
{"category": "Math", "title": "Complex quotients by nonclosed groups and their stratifications", "abstract": "We define the notion of complex stratification by quasifolds and show that such spaces occur as complex quotients by certain nonclosed subgroups of tori associated to convex polytopes. The spaces thus obtained provide a natural generalization to the nonrational case of the notion of toric variety associated with a rational convex polytope."}
{"category": "Math", "title": "Brauer Algebras of Simply Laced Type", "abstract": "The diagram algebra introduced by Brauer that describes the centralizer algebra on tensor products of the natural representation of an orthogonal group has a presentation by generators and relations that only depends on the graph of type An on n nodes. Here we describe an algebra depending on an arbitrary graph of type M. We study its structure when the type is An, Dn, E6, E7, E8. We determine the representations and find the dimensions. The algebra is generically semisimple and contains the group algebra of the Coxeter type M as a subalgebra. It is a ring homomorphism of the Birman-Murakami-Wenzl algebra of these types. This fact will be used in later work determining the structure of Birman-Murakami-Wenzl algebras of simply laced spherical type."}
{"category": "Math", "title": "The order of the decay of the hole probability for Gaussian random SU(m+1) polynomials", "abstract": "We show that for Gaussian random SU(m+1) polynomials of a large degree N the probability that there are no zeros in the disk of radius r is less than $e^{-c_{1,r} N^{m+1}}$, and is also greater than $e^{-c_{2,r} N^{m+1}}$. Enroute to this result, we also derive a more general result: probability estimates for the event where the volume of the zero set of a random polynomial of high degree deviates significantly from its mean."}
{"category": "Math", "title": "A Cohen-Macaulay algebra has only finitely many semidualizing modules", "abstract": "We prove the result stated in the title, which answers the equicharacteristic case of a question of Vasconcelos."}
{"category": "Math", "title": "The Birman-Murakami-Algebras Algebras of Type Dn", "abstract": "The Birman-Murakami-Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free of rank (2^n+1)n!!-(2^(n-1)+1)n! over a specified commutative ring R, where n!! is the product of the first n odd integers. We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type Dn is the image af an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the polynomial ring Z with delta and its inverse adjoined. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley-Lieb algebra of type Dn is a subalgebra of the BMW algebra of the same type."}
{"category": "Math", "title": "Nahm transform and parabolic minimal Laplace transform", "abstract": "We prove that Nahm transform for integrable connections with a finite number of regular singularities and an irregular singularity of rank 1 on the Riemann sphere is equivalent -- up to considering integrable connections as holonomic $\\D$-modules -- to minimal Laplace transform. We assume semi-simplicity and resonance-freeness conditions, and we work in the framework of objects with a parabolic structure. In particular, we describe the definition of the parabolic version of Laplace transform due to C. Sabbah. The proof of the main result relies on the study of a twisted de Rham complex."}
{"category": "Math", "title": "Remarks to Glazek's results on n-ary groups", "abstract": "It is a survey of the results obtained by K. Glazek's and his co-workers. We restrict our attention to the problems of axiomatizations of n-ary groups, classes of n-ary groups, properties of skew elements and homomorphisms induced by skew elements, constructions of covering groups, classifications and representations of n-ary groups. Some new results are added too."}
{"category": "Math", "title": "Tangle and Brauer Diagram Algebras of Type Dn", "abstract": "A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole or order 2. The algebra is shown to be isomorphic to the Birman-Murakami-Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which in our set-up, occurs when the Coxeter type is of type A with index n-1. The proof involves a diagrammatic version of the Brauer algebra of type Dn in which the Temperley-Lieb algebra of type Dn is a subalgebra."}
{"category": "Math", "title": "Parabolic surfaces in hyperbolic space with constant curvature", "abstract": "We study parabolic linear Weingarten surfaces in hyperbolic space $\\rlopezh^3$. In particular, we classify two family of parabolic surfaces: surfaces with constant Gaussian curvature and surfaces that satisfy the relation $a\\kappa_1+b\\kappa_2=c$, where $\\kappa_i$ are the principal curvatures, and $a,b$ and $c$ are constant."}
{"category": "Math", "title": "Equivariant Bundles and Isotropy Representations", "abstract": "We introduce a new construction, the isotropy groupoid, to organize the orbit data for split $\\Gamma$-spaces. We show that equivariant principal $G$-bundles over split $\\Gamma$-CW complexes $X$ can be effectively classified by means of representations of their isotropy groupoids. For instance, if the quotient complex $A=\\Gamma\\backslash X$ is a graph, with all edge stabilizers toral subgroups of $\\Gamma$, we obtain a purely combinatorial classification of bundles with structural group $G$ a compact connected Lie group. If $G$ is abelian, our approach gives combinatorial and geometric descriptions of some results of Lashof-May-Segal and Goresky-Kottwitz-MacPherson."}
{"category": "Math", "title": "Heat Equations and the Weighted $\\bar\\partial$-Problem", "abstract": "The purpose of this article is to establish regularity and pointwise upper bounds for the (relative) fundamental solution of the heat equation associated to the weighted dbar-operator in $L^2(C^n)$ for a certain class of weights. The weights depend on a parameter, and we find pointwise bounds for heat kernel, as well as its derivatives in time, space, and the parameter. We also prove cancellation conditions for the heat semigroup. We reduce the $n$-dimensional case to the one-dimensional case, and the estimates in one-dimensional case are achieved by Duhamel's principle and commutator properties of the operators. As an application, we recover estimates of heat kernels on polynomial models in $C^2$."}
{"category": "Math", "title": "Bialgebra cohomology, pointed Hopf algebras, and deformations", "abstract": "We give explicit formulas for maps in a long exact sequence connecting bialgebra cohomology to Hochschild cohomology. We give a sufficient condition for the connecting homomorphism to be surjective. We apply these results to compute all bialgebra two-cocycles of certain Radford biproducts (bosonizations). These two-cocycles are precisely those associated to the finite dimensional pointed Hopf algebras in the recent classification of Andruskiewitsch and Schneider, in an interpretation of these Hopf algebras as graded bialgebra deformations of Radford biproducts."}
{"category": "Math", "title": "Results for a turbulent system with unbounded viscosities: weak formulations, existence of solutions, boundedness, smoothness'", "abstract": "We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solutions are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solution"}
{"category": "Math", "title": "Holonomy representations which are a diagonal direct sum of two faithful representations", "abstract": "We study holonomy representations admitting a pair of supplementary faithful sub-representations. In particular the cases where the sub-representations are isomorphic respectively dual to each other are treated. In each case we have a closer look at the classification in small dimension."}
{"category": "Math", "title": "Representations admitting two pairs of supplementary invariant spaces", "abstract": "We examine the lattice generated by two pairs of supplementary vector subspaces of a finite-dimensional vector space by intersection and sum, with the aim of applying the results to the study of representations admitting two pairs of supplementary invariant spaces, or one pair and a reflexive form. We show that such a representation is a direct sum of three canonical sub-representations which we characterize. We then focus on holonomy representations with the same property."}
{"category": "Math", "title": "Exploring Continuous Tensegrities", "abstract": "A discrete tensegrity framework can be thought of as a graph in Euclidean n-space where each edge is of one of three types: an edge with a fixed length (bar) or an edge with an upper (cable) or lower (strut) bound on its length. Roth and Whiteley, in their 1981 paper \"Tensegrity Frameworks\", showed that in certain cases, the struts and cables can be replaced with bars when analyzing the framework for infinitesimal rigidity. In that case we call the tensegrity \"bar equivalent\". In specific, they showed that if there exists a set of positive weights, called a positive \"stress\", on the edges such that the weighted sum of the edge vectors is zero at every vertex, then the tensegrity is bar equivalent. In this paper we consider an extended version of the tensegrity framework in which the vertex set is a (possibly infinite) set of points in Euclidean n-space and the edgeset is a compact set of unordered pairs of vertices. These are called \"continuous tensegrities\". We show that if a continuous tensegrity has a strictly positive stress, it is bar equivalent and that it has a semipositive stress if and only if it is partially bar equivalent. We also show that if a tensegrity is minimally bar equivalent (it is bar equivalent but removing any open set of edges makes it no longer so), then it has a strictly positive stress. In particular, we examine the case where the vertices form a rectifiable curve and the possible motions of the curve are limited to local isometries of it. Our methods provide an attractive proof of the following result: There is no locally arclength preserving motion of a circle that increases any antipodal distance without decreasing some other one."}
{"category": "Math", "title": "Pseudocontinuation and cyclicity for random power series", "abstract": "We prove that a random function in the Hardy space $H^2$ is a non-cyclic vector for the backward shift operator almost surely. The question of existence of a local pseudocontinuation for a random analytic function is also studied."}
{"category": "Math", "title": "Group amenability properties for von Neumann algebras", "abstract": "In his study of amenable unitary representations, M. E. B. Bekka asked if there is an analogue for such representations of the remarkable fixed-point property for amenable groups. In this paper, we prove such a fixed-point theorem in the more general context of a $G$-amenable von Neumann algebra $M$, where $G$ is a locally compact group acting on $M$. The F{\\o}lner conditions of Connes and Bekka are extended to the case where $M$ is semifinite and admits a faithful, semifinite, normal trace which is invariant under the action of $G$."}
{"category": "Math", "title": "The equivariant analytic index for proper groupoid actions", "abstract": "The paper constructs the analytic index for an elliptic pseudodifferential family of $L^{m}_{\\rho,\\de}$-operators invariant under the proper action of a continuous family groupoid on a $G$-compact, $C^{\\infty,0}$ $G$-space."}
{"category": "Math", "title": "The analytic index for proper, Lie groupoid actions", "abstract": "Many index theorems (both classical and in noncommutative geometry) can be interpreted in terms of a Lie groupoid acting properly on a manifold and leaving an elliptic family of pseudodifferential operators invariant. Alain Connes in his book raised the question of an index theorem in this general context. In this paper, an analytic index for many such situations is constructed. The approach is inspired by the classical families theorem of Atiyah and Singer, and the proof generalizes, to the case of proper Lie groupoid actions, some of the results proved for proper locally compact group actions by N. C. Phillips."}
{"category": "Math", "title": "The Fourier algebra for locally compact groupoids", "abstract": "We introduce and investigate using Hilbert modules the properties of the {\\em Fourier algebra} $A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard."}
{"category": "Math", "title": "Continuous family groupoids", "abstract": "In this paper, we define and investigate the properties of continuous family groupoids. This class of groupoids is necessary for investigating the groupoid index theory arising from the equivariant Atiyah-Singer index theorem for families, and is also required in noncommutative geometry. The class includes that of Lie groupoids, and the paper shows that, like Lie groupoids, continuous family groupoids always admit (an essentially unique) continuous left Haar system of smooth measures. We also show that the action of a continuous family groupoid $G$ on a continuous family $G$-space fibered over another continuous family $G$-space $Y$ can always be regarded as an action of the continuous family groupoid $G*Y$ on an ordinary $G*Y$-space."}
{"category": "Math", "title": "Tychonoff's theorem for locally compact space and an elementary approach to the topology of path spaces", "abstract": "The path spaces of a directed graph play an important role in the study of graph $\\css$. These are topological spaces that were originally constructed using groupoid and inverse semigroup techniques. In this paper, we develop a simple, purely topological, approach to this construction, based on Tychonoff's theorem. In fact, the approach is shown to work even for higher dimensional graphs satisfying the finitely aligned condition, and we construct the groupoid of the graph. Motivated by these path space results, we prove a Tychonoff theorem for an infinite, countable product of locally compact spaces. The main idea is to include certain finite products of the spaces along with the infinite product. We show that the topology is, in a reasonable sense, a pointwise topology."}
{"category": "Math", "title": "Approximating reals by sums of rationals", "abstract": "We consider the question of approximating any real number $\\alpha$ by sums of $n$ rational numbers $\\frac{a_1}{q_1} + \\frac{a_2}{q_2} + ... + \\frac{a_n}{q_n}$ with denominators $1 \\leq q_1, q_2, ..., q_n \\leq N$. This leads to an inquiry on approximating a real number by rational numbers with a prescribed number of prime factors in the denominator."}
{"category": "Math", "title": "Behavior of corank one singular points on wave fronts", "abstract": "Let $M^2$ be an oriented 2-manifold and $f:M^2\\to R^3$ a $C^\\infty$-map. A point $p\\in M^2$ is called a singular point if $f$ is not an immersion at $p$. The map $f$ is called a front (or wave front), if there exists a unit $C^\\infty$-vector field $\\nu$ such that the image of each tangent vector $df(X)$ $(X\\in TM^2)$ is perpendicular to $\\nu$, and the pair $(f,\\nu)$ gives an immersion into $R^3\\times S^2$. In our previous paper, we gave an intrinsic formulation of wave fronts in $R^3$. In this paper, we shall investigate the behavior of cuspidal edges near corank one singular points and establish Gauss-Bonnet-type formulas under the intrinsic formulation."}
{"category": "Math", "title": "Motzkin numbers of higher rank: Generating function and explicit expression", "abstract": "The generating function and an explicit expression is derived for the (colored) Motzkin numbers of higher rank introduced recently. Considering the special case of rank one yields the corresponding results for the conventional colored Motzkin numbers for which in addition a recursion relation is given."}
{"category": "Math", "title": "Symmetric Crystals for $\\gl_\\infty$", "abstract": "In the preceding paper, we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for $\\gl_\\infty$. In the present paper, we prove the existence of the symmetric crystal and the global basis for $\\gl_\\infty$."}
{"category": "Math", "title": "Sums of squares over totally real fields are rational sums of squares", "abstract": "Let $K$ be a totally real number field with Galois closure $L$. We prove that if $f \\in \\mathbb Q[x_1,...,x_n]$ is a sum of $m$ squares in $K[x_1,...,x_n]$, then $f$ is a sum of \\[4m \\cdot 2^{[L: \\mathbb Q]+1} {[L: \\mathbb Q] +1 \\choose 2}\\] squares in $\\mathbb Q[x_1,...,x_n]$. Moreover, our argument is constructive and generalizes to the case of commutative $K$-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programing problems."}
{"category": "Math", "title": "Analytic crossing probabilities for certain barriers by Brownian motion", "abstract": "We calculate crossing probabilities and one-sided last exit time densities for a class of moving barriers on an interval $[0,T]$ via Schwartz distributions. We derive crossing probabilities and first hitting time densities for another class of barriers on $[0,T]$ by proving a Schwartz distribution version of the method of images. Analytic expressions for crossing probabilities and related densities are given for new explicit and semi-explicit barriers."}
{"category": "Math", "title": "Sharp Global Bounds for the Hessian on Pseudo-Hermitian Manifolds", "abstract": "We find sharp bounds for the norm inequality on a Pseudo-hermitian manifold, where the L^2 norm of all second derivatives of the function involving horizontal derivatives is controlled by the L^2 norm of the sub-Laplacian. Perturbation allows us to get a-priori bounds for solutions to sub-elliptic PDE in non-divergence form with bounded measurable coefficients. The method of proof is through a Bochner technique. The Heisenberg group is seen to be en extremal manifold for our inequality in the class of manifolds whose Ricci curvature is non-negative."}
{"category": "Math", "title": "An analogue of Gutzmer's formula for Hermite expansions", "abstract": "We prove an analogue of Gutzmer's formula for Hermite expansions. As a consequence we obtain a new proof of a characterisation of the image of $ L^2(\\R^n) $ under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions."}
{"category": "Math", "title": "Kirillov-Reshetikhin conjecture : the general case", "abstract": "We prove the Kirillov-Reshetikhin (KR) conjecture in the general case : for all twisted quantum affine algebras we prove that the characters of KR modules solve the twisted Q-system and we get explicit formulas for the character of their tensor products (the untwisted simply-laced case was treated by Najakima, and the untwisted case by the author). The proof is uniform and provides several new developments for the representation theory of twisted quantum affine algebras, including twisted Frenkel-Reshetikhin q-characters (expected by Frenkel-Reshetikhin and Frenkel-Mukhin). We also prove the twisted T-system. As an application we get explicit formulas for the twisted q-characters of fundamental representations for all types, including the formulas for types D_4^{(3)}, E_6^{(2)} conjectured by Reshetikhin. We prove the formulas for KR modules in types A_n^{(2)} and D_4^{(3)} conjectured by Kuniba-Suzuki. Eventually our results imply the conjectural branching rules [HKOTT] to the quantum subalgebra of finite type."}
{"category": "Math", "title": "Quadrature formulas for the Laplace and Mellin transforms", "abstract": "A discrete Laplace transform and its inversion formula are obtained by using a quadrature of the continuous Fourier transform which is given in terms of Hermite polynomials and its zeros. This approach yields a convergent discrete formula for the two-sided Laplace transform if the function to be transformed falls off rapidly to zero and satisfy certain conditions of integrability, achieving convergence also for singular functions. The inversion formula becomes a quadrature formula for the Bromwich integral. This procedure also yields a quadrature formula for the Mellin transform and its corresponding inversion formula that can be generalized straightforwardly for functions of several variables."}
{"category": "Math", "title": "Kahane-Khinchin type Averages", "abstract": "We prove a Kahane-Khinchin type result with a few random vectors, which are distributed independently with respect to an arbitrary log-concave probability measure on $\\R^n$. This is an application of small ball estimate and Chernoff's method, that has been recently used in the context of Asymptotic Geometric Analysis in [1], [2]."}
{"category": "Math", "title": "Primitive flag-transitive generalized hexagons and octagons", "abstract": "Suppose that an automorphism group $G$ acts flag-transitively on a finite generalized hexagon or octagon $\\cS$, and suppose that the action on both the point and line set is primitive. We show that $G$ is an almost simple group of Lie type, that is, the socle of $G$ is a simple Chevalley group."}
{"category": "Math", "title": "Gaussian conditional independence relations have no finite complete characterization", "abstract": "We show that there can be no finite list of conditional independence relations which can be used to deduce all conditional independence implications among Gaussian random variables. To do this, we construct, for each $n> 3$ a family of $n$ conditional independence statements on $n$ random variables which together imply that $X_1 \\ind X_2$, and such that no subset have this same implication. The proof relies on binomial primary decomposition."}
{"category": "Math", "title": "Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. I", "abstract": "In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers ${\\cal C}^{[\\bullet]}$ and the relative Jacobian ${\\cal J}$ of a family of curves ${\\cal C}/S$. As one of the applications, we show that in the case of a single curve $C$ this action induces a integral form of a Lefschetz $\\operatorname{sl}_2$-action on the Chow groups of $C^{[N]}$. Another application gives a new grading on the ring of 0-cycles on the Jacobian $J$ of $C$ (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes in $CH^*({\\cal C}^{[\\bullet]})$ and in $CH^*({\\cal J})$ and prove for them analogs of the properties established in the case of the Jacobian of a single curve by Beauville in math.AG/0204188. We also show that the our algebras of operators preserve the subrings of tautological cycles and act on them via some explicit differential operators."}
{"category": "Math", "title": "On the Complement of the Projective Hull in C^n", "abstract": "We prove that if $K$ is a compact subset of an affine variety O = P^n - D (where D is a projective hypersuface), and if K is a compact subset of a closed analytic subvariety V \\subset O, then the projective hull K^ of K has the property that K^ \\cap O is contained in V. If V is smooth and 1-dimensional, then K^ \\cap O is also closed in O. The result has applications to graphs in C^2 of functions in the disk algebra."}
{"category": "Math", "title": "Double covering of the Painlev\\'e I equation and its singular analysis", "abstract": "In this note, we will do analysis of accessible singular points for a polynomial Hamiltonian system obtained by taking a double covering of the Painlev\\'e I equation. We will show that this system passes the Painlev\\'e $\\alpha$-test for all accessible singular points $P_i \\ (i=1,2,3)$. We note its holomorphy condition of the first Painlev\\'e system."}
{"category": "Math", "title": "A sequence of blowing-ups connecting moduli of sheaves and the Donaldson polynomial under change of polarization", "abstract": "Let $H$ and $H'$ be two ample line bundles over a nonsingular projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r=2,c_1,c_2)$. In a moduli-theoretic way that comes from elementary transforms, we connect $M(H)$ and $M(H')$ by a sequence of blowing-ups when walls separating $H$ and $H'$ are not necessarily good. As an application, we also consider the polarization change problem of Donaldson polynomials."}
{"category": "Math", "title": "Large components in random induced subgraphs of n-cubes", "abstract": "In this paper we study random induced subgraphs of the binary $n$-cube, $Q_2^n$. This random graph is obtained by selecting each $Q_2^n$-vertex with independent probability $\\lambda_n$. Using a novel construction of subcomponents we study the largest component for $\\lambda_n=\\frac{1+\\chi_n}{n}$, where $\\epsilon\\ge \\chi_n\\ge n^{-{1/3}+ \\delta}$, $\\delta>0$. We prove that there exists a.s. a unique largest component $C_n^{(1)}$. We furthermore show that $\\chi_n=\\epsilon$, $| C_n^{(1)}|\\sim \\alpha(\\epsilon) \\frac{1+\\chi_n}{n} 2^n$ and for $o(1)=\\chi_n\\ge n^{-{1/3}+\\delta}$, $| C_n^{(1)}| \\sim 2 \\chi_n \\frac{1+\\chi_n}{n} 2^n$ holds. This improves the result of \\cite{Bollobas:91} where constant $\\chi_n=\\chi$ is considered. In particular, in case of $\\lambda_n=\\frac{1+\\epsilon} {n}$, our analysis implies that a.s. a unique giant component exists."}
{"category": "Math", "title": "Studies on the Garnier system in two variables", "abstract": "We study some Hamiltonian structures of the Garnier system in two variables from the viewpoints of its symmetry and holomorphy properties. We also give a generalization of {\\it Okamoto transformation \\it}of the sixth Painlev\\'e system."}
{"category": "Math", "title": "Blowing-ups describing the polarization change of moduli schemes of semistable sheaves of general rank", "abstract": "Let $H$ and $H'$ be two ample line bundles over a smooth projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r,c_1,c_2)$. We construct a sequence of blowing-ups which describes how $M(H)$ differs from $M(H')$ not only when $r=2$ but also when $r$ is arbitrary. Means we here utilize are elementary transforms and the notion of a sheaf with flag."}
{"category": "Math", "title": "Complex dimensions of real manifolds, attached analytic discs and parametric argument principle", "abstract": "Let $\\Omega$ be a smooth real analytic submanifold of a complex manifold $X$. We establish and study the link between the following 3 subjects: 1) topological properties of smooth families of attached analytic discs, the manifold $\\Omega$ admits, 2) lower bounds for dimensions of complex tangent spaces of $\\Omega$, 3) a generalization of the argument principle for smooth families of holomorphic mappings from the standard complex disc to $X$. In particular, we obtain characterization of complex manifolds and their boundaries in terms of attached analytic discs. The special case when $\\Omega$ is the graph, leads to new characterizations of holomorphic and $CR$ functions, and in particular, to solutions of some open problems about such functions."}
{"category": "Math", "title": "Coupled Painlev\\'e III system with affine Weyl group symmetry of type $D_6^{(1)}$", "abstract": "We find and study a six-parameter family of coupled Painlev\\'e III systems in dimension six with affine Weyl group symmetry of type $D_6^{(1)}$. We also find and study its degenerate systems with affine Weyl group symmetry of types $B_5^{(1)}$ and $D_5^{(2)}$."}
{"category": "Math", "title": "Helicity-type integral invariants for Hamiltonian systems", "abstract": "In this note, we consider generalizations of the asymptotic Hopf invariant, or helicity, for Hamiltonian systems with one-and-a-half degrees of freedom and symplectic diffeomorphisms of a two-disk to itself."}
{"category": "Math", "title": "Convex comparison of service disciplines in real time queues", "abstract": "We present a comparison of the service disciplines in real-time queueing systems (the customers have a deadline before which they should enter the service booth). We state that giving priority to customers having an early deadline minimizes the average stationary lateness. We show this result by comparing adequate random vectors with the Schur-Convex majorization ordering."}
{"category": "Math", "title": "Control of mechanical systems on Lie groups and ideal hydrodynamics", "abstract": "In contrast to the Euler-Poincar{\\'e} reduction of geodesic flows of left- or right-invariant metrics on Lie groups to the corresponding Lie algebra (or its dual), one can consider the reduction of the geodesic flows to the group itself. The reduced vector field has a remarkable hydrodynamic interpretation: it is a velocity field for a stationary flow of an ideal fluid. Right- or left-invariant symmetry fields of the reduced field define vortex manifolds for such flows. Consider now a mechanical system, whose configuration space is a Lie group and whose Lagrangian is invariant to left translations on that group, and assume that the mass geometry of the system may change under the action of internal control forces. Such system can also be reduced to the Lie group. With no controls, this mechanical system describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and thus its reduced flow is a stationary ideal fluid flow on the Lie group. The standard control problem for such system is to find the conditions, under which the system can be brought from any initial position in the configuration space to another preassigned position by changing its mass geometry. We show that under these conditions, by changing the mass geometry, one can also bring one vortex manifold to any other preassigned vortex manifold."}
{"category": "Math", "title": "Schwartz functions on Nash manifolds", "abstract": "In this paper we extend the notions of Schwartz functions, tempered functions and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for this case classically known properties of Schwartz functions on $R^n$ and build some additional tools which are important in representation theory."}
{"category": "Math", "title": "A 2-generated 2-related group with no non-trivial finite factors", "abstract": "We construct a 2-generated 2-related group without non-trivial finite factors. That answers a question of J. Button."}
{"category": "Math", "title": "Some group theory problems", "abstract": "This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For each problem, I try to mention the author. If the author is not given, the problem, to the best of my knowledge, was formulated by me first."}
{"category": "Math", "title": "On weakly convex star-shaped polyhedra", "abstract": "Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps."}
{"category": "Math", "title": "The spectral laws of Hermitian block-matrices with large random blocks", "abstract": "We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix with $k-$weakly dependent entries need not to be the semicircle law in the limit."}
{"category": "Math", "title": "Uniqueness thresholds on trees versus graphs", "abstract": "Counter to the general notion that the regular tree is the worst case for decay of correlation between sets and nodes, we produce an example of a multi-spin interacting system which has uniqueness on the $d$-regular tree but does not have uniqueness on some infinite $d$-regular graphs."}
{"category": "Math", "title": "Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations", "abstract": "In this paper we show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n)."}
{"category": "Math", "title": "Excedance numbers for permutations in complex reflection groups", "abstract": "Recently, Bagno, Garber and Mansour studied a kind of excedance number on the complex reflection groups and computed its multidistribution with the number of fixed points on the set of involutions in these groups. In this note, we consider the similar problems in more general cases and make a correction of one result obtained by them."}
{"category": "Math", "title": "Some smooth Finsler deformations of hyperbolic surfaces", "abstract": "Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectrum rigidity and boundary rigidity all fail to extend to the Finsler category."}
{"category": "Math", "title": "La controverse de 1874 entre Camille Jordan et Leopold Kronecker", "abstract": "During the whole of 1874, Camille Jordan and Leopold Kronecker quar- relled vigorously over the organisation of the theory of bilinear forms. That theory promised a \"general\" and \"homogeneous\" treatment of numerous questions arising in various 19th-century theoretical contexts, and it hinged on two theorems, stated independently by Jordan and Weierstrass, that would today be considered equivalent. It was, however, the perceived difference between those two theorems that sparked the 1874 controversy. Focusing on this quarrel allows us to explore the algebraic identity of the polynomial practices of the manipulations of forms in use before the advent of structural approaches to linear algebra. The latter approaches identified these practices with methods for the classification of similar matrices. We show that the prac- tices -- Jordan's canonical reduction and Kronecker's invariant computation -- reflect identities inseparable from the social context of the time. Moreover, these practices reveal not only tacit knowledge, local ways of thinking, but also -- in light of a long history tracing back to the work of Lagrange, Laplace, Cau- chy, and Hermite -- two internal philosophies regarding the significance of generality which are inseparable from two disciplinary ideals opposing algebra and arithmetic. By interrogating the cultural identities of such practices, this study aims at a deeper understanding of the history of linear algebra without focusing on issues related to the origins of theories or structures."}
{"category": "Math", "title": "L'identit\\'e alg\\'ebrique d'une pratique port\\'ee par la discussion sur l'\\'equation \\`a l'aide de laquelle on d\\'etermine les in\\'egalit\\'es s\\'eculaires des plan\\`etes (1766-1874)", "abstract": "What did \"algebra\" mean before the development of the algebraic theories of the 20th century ? This paper stresses the identities taken by the algebraic practices developped during the century long discussion around the equation around the equation of secular inequalities (1766- 1874). In 1874, a strong controversy on the theory of bilinear and quadratic forms opposed Camille Jordan and Leopold Kronecker. The arithmetical ideal of Kronecker faced Jordan's claim for the simplicity of his algebraic canonical form. As the controversy combined mathematical and historical arguments, it gave rise to the writing of a history of the methods used by Lagrange, Laplace and Weierstrass in a century long mathematical discussion around the \"equation of secular inequalities\"."}
{"category": "Math", "title": "Tables of graphs of binary and ternary sequences differentiation", "abstract": "Let $x$ be a cyclic sequence of $n$ elements of the finite field $\\mathbb{F}_q$ (the first element immediately follows the $n$-th one). Let us define the operation $\\Delta$ as the transition from $x$ to the sequence of differences of the neighbouring elements from $x$. The aim of this work is to give graphs of the dynamic system $\\Delta$ for $q=2$, $n\\le 300$ and $q=3$, $n\\le 150$. These results enable us to define more precisely the Arnold hypotheses and to prove them."}
{"category": "Math", "title": "The pseudo-effective cone of a non-K\\\"ahlerian surface and applications", "abstract": "We describe the positive cone and the pseudo-effective cone of a non-K\\\"ahlerian surface. We use these results for two types of applications: - Describe the set $\\sigma(X)$ of possible total Ricci scalars associated with Gauduchon metrics of fixed volume 1 on a fixed non-K\\\"ahhlerian surface, and decide whether the assignment $X\\mapsto\\sigma(X)$ is a deformation invariant. - Study the stability of the canonical extension $$0\\to {\\cal K}_X\\to {\\cal A}\\to{\\cal O}_X\\to 0$$ of a class VII surface $X$ with positive $b_2$. This extension plays an important role in our strategy to prove the GSS conjecture using gauge theoretical methods."}
{"category": "Math", "title": "Pseudo-localization of singular integrals and noncommutative Calderon-Zygmund theory", "abstract": "In this paper we obtain the weak type (1,1) boundedness of Calderon-Zygmund operators acting over operator-valued functions. Our main tools for its solution are a noncommutative form of Calderon-Zygmund decomposition in conjunction with a pseudo-localization principle for singular integrals, which is new even in the classical setting and of independent interest. Perhaps because of the hidden role of pseudo-localization and almost orthogonality, this problem has remained open for quite some time. We also consider Calderon-Zygmund operators associated to certain operator-valued kernels."}
{"category": "Math", "title": "A link polynomial via a vertex-edge-face state model", "abstract": "We construct a 2-variable link polynomial, called $W_L$, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two 1-variable polynomials, one of which is the Alexander polynomial. We refine $W_L$ to an ordered set of 3-variable polynomials for those links in 3-space which contain a Hopf link as a sublink."}
{"category": "Math", "title": "Group-theoretic Description of Riemannian Spaces", "abstract": "It is shown that a locally geometrical structure of arbitrarily curved Riemannian space is defined by a deformed group of its diffeomorphisms"}
{"category": "Math", "title": "On the alpha-Amenability of Hypergroups", "abstract": "Let $UC(K)$ denote the Banach space of all bounded uniformly continuous functions on a hypergroup $K$. The main results of this article concern on the $\\alpha$-amenability of $UC(K)$ and quotients and products of hypergroups. It is also shown that a Sturm-Liouville hypergroup with a positive index is $\\alpha$-amenable if and only if $\\alpha=1$."}
{"category": "Math", "title": "On Loops in the Hyperbolic Locus of the Complex H\\'enon Map and Their Monodromies", "abstract": "We prove John Hubbard's conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex H\\'enon map. Indeed, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Our main tool is a rigorous computational algorithm for verifying the uniform hyperbolicity of chain recurrent sets. In addition, we show that the dynamics of the real H\\'enon map is completely determined by the monodromy of a certain loop, providing the parameter of the map is contained in the hyperbolic horseshoe locus of the complex H\\'enon map."}
{"category": "Math", "title": "Canonical Deformed Groups of Diffeomorphisms and Finite Parallel Transports in Riemannian Spaces", "abstract": "We show that finite parallel transports of vectors in Riemannian spaces, determined by the multiplication law in the deformed groups of diffeomorphisms, and sequences of infinitesimal parallel transports of vectors along geodesics are equivalent."}
{"category": "Math", "title": "Globally stable quasistatic evolution in plasticity with softening", "abstract": "We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each time interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response."}
{"category": "Math", "title": "On the geometric quantization of twisted Poisson manifolds", "abstract": "We study the geometric quantization process for twisted Poisson manifolds. First, we introduce the notion of Lichnerowicz-twisted Poisson cohomology for twisted Poisson manifolds and we use it in order to characterize their prequantization bundles and to establish their prequantization condition. Next, we introduce a polarization and we discuss the quantization problem. In each step, several examples are presented."}
{"category": "Math", "title": "Finite determination of regular (a,b)-modules", "abstract": "The concept of (a,b)-module comes from the study the Gauss-Manin lattices of an isolated singularity of a germ of an holomorphic function. It is a very simple ''abstract algebraic structure'', but very rich, whose prototype is the formal completion of the Brieskorn-module of an isolated singularity. The aim of this article is to prove a very basic theorem on regular (a,b)-modules showing that a given regular (a,b)-module is completely characterized by some ''finite order jet'' of its structure. Moreover a very simple bound for such a sufficient order is given in term of the rank and of two very simple invariants : the regularity order which count the number of times you need to apply \\ $b^{-1}.a \\simeq \\partial_z.z$ in order to reach a simple pole (a,b)-module. The second invariant is the ''width'' which corresponds, in the simple pole case, to the maximal integral difference between to eigenvalues of $b^{-1}.a$ (the logarithm of the monodromy). In the computation of examples this theorem is quite helpfull because it tells you at which power of $b$ in the expansions you may stop without loosing any information."}
{"category": "Math", "title": "Low regularity local well-posedness of the Derivative Nonlinear Schr\\\"odinger Equation with periodic initial data", "abstract": "The Cauchy problem for the derivative nonlinear Schr\\\"odinger equation with periodic boundary condition is considered. Local well-posedness for periodic initial data u_0 in the space ^H^s_r, defined by the norms ||u_0||_{^H^s_r}=||<xi>^s ^u_0||_{l^r'} is shown in the parameter range s>= 1/2, 2>r>4/3. The proof is based on an adaptation of the gauge transform to the periodic setting and an appropriate variant of the Fourier restriction norm method."}
{"category": "Math", "title": "On the inverse braid monoid", "abstract": "Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial $n$ may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely we prove an inclusion into a monoid of partial monomorphisms of a free group. This gives a solution of the word problem. Another solution is obtained by an approach similar to that of Garside. We give also the analogues of Artin presentation with two generators and Sergiescu graph-presentations."}
{"category": "Math", "title": "Deformation quantization modules on complex symplectic manifolds", "abstract": "We study modules over the algebroid stack $\\W[\\stx]$ of deformation quantization on a complex symplectic manifold $\\stx$ and recall some results: construction of an algebra for $\\star$-products, existence of (twisted) simple modules along smooth Lagrangian submanifolds, perversity of the complex of solutions for regular holonomic $\\W[\\stx]$-modules, finiteness and duality for the composition of ``good'' kernels. As a corollary, we get that the derived category of good $\\W[\\stx]$-modules with compact support is a Calabi-Yau category. We also give a conjectural Riemann-Roch type formula in this framework."}
{"category": "Math", "title": "On the Conditions to Extend Ricci Flow", "abstract": "Along a Ricci flow solution on a closed manifold, we show that if Ricci curvature is uniformly bounded from below, then a scalar curvature integral bound is enough to extend flow. Moreover, this integral bound condition is optimal in some sense."}
{"category": "Math", "title": "Order preserving transformations of the Hilbert grassmannian: complex case", "abstract": "Let $H$ be a separable complex Hilbert space. Denote by ${\\mathcal G}_{\\infty}(H)$ the Grassmannian consisting of closed linear subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every continuous order preserving bijective transformation of ${\\mathcal G}_{\\infty}(H)$ is induced by an invertible bounded semi-linear operator."}
{"category": "Math", "title": "Generic dynamics of 4-dimensional C2 Hamiltonian systems", "abstract": "We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C2-residual set of Hamiltonians for which every regular energy surface is either Anosov or it is in the closure of energy surfaces with zero Lyapunov exponents a.e. This is in the spirit of the Bochi-Mane dichotomy for area-preserving diffeomorphisms on compact surfaces and its continuous-time version for 3-dimensional volume-preserving flows."}
{"category": "Math", "title": "Involutory quasi-Hopf algebras", "abstract": "We introduce and investigate the basic properties of an involutory (dual) quasi-Hopf algebra. We also study the representations of an involutory quasi-Hopf algebra and prove that an involutory dual quasi-Hopf algebra with non-zero integral is cosemisimple."}
{"category": "Math", "title": "Representation Of Level Paths Of An Analytic Function", "abstract": "We find an arc-parameterization of the contour on which an given analytic function has constant modulus. This contour is seen to satisfy a differential equation which we explicitly give."}
{"category": "Math", "title": "On bounds and algorithms for frequency synchronization for collaborative communication systems", "abstract": "Cooperative diversity systems are wireless communication systems designed to exploit cooperation among users to mitigate the effects of multipath fading. In fairly general conditions, it has been shown that these systems can achieve the diversity order of an equivalent MISO channel and, if the node geometry permits, virtually the same outage probability can be achieved as that of the equivalent MISO channel for a wide range of applicable SNR. However, much of the prior analysis has been performed under the assumption of perfect timing and frequency offset synchronization. In this paper, we derive the estimation bounds and associated maximum likelihood estimators for frequency offset estimation in a cooperative communication system. We show the benefit of adaptively tuning the frequency of the relay node in order to reduce estimation error at the destination. We also derive an efficient estimation algorithm, based on the correlation sequence of the data, which has mean squared error close to the Cramer-Rao Bound."}
{"category": "Math", "title": "Some integer sequences based on derangements", "abstract": "Sequences whose terms are equal to the number of functions with specified properties are considered. Properties are based on the notion of derangements in a more general sense. Several sequences which generalize the standard notion of derangements are thus obtained. These sequences generate a number of integer sequences from the well-known Sloane's encyclopedia."}
{"category": "Math", "title": "Bruhat order for two subspaces and a flag", "abstract": "The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a finite-dimensional vector space V; or, equivalently, the closure of an orbit of the group GL(V) acting on the direct product of two full flag varieties. We obtain a similar result for triples consisting of two subspaces and a partial flag in V; this is equivalent to describing the closure of a GL(V)-orbit in the product of two Grassmannians and one flag variety. We give a rank criterion to check whether such a triple can be degenerated to another one, and we classify the minimal degenerations. Our methods involve only elementary linear algebra and combinatorics of graphs (originating in Auslander-Reiten quivers)."}
{"category": "Math", "title": "Kazhdan--Lusztig polynomials for maximally-clustered hexagon-avoiding permutations", "abstract": "We provide a non-recursive description for the bounded admissible sets of masks used by Deodhar's algorithm to calculate the Kazhdan--Lusztig polynomials $P_{x,w}(q)$ of type $A$, in the case when $w$ is hexagon avoiding and maximally clustered. This yields a combinatorial description of the Kazhdan--Lusztig basis elements of the Hecke algebra associated to such permutations $w$. The maximally-clustered hexagon-avoiding elements are characterized by avoiding the seven classical permutation patterns $\\{3421, 4312, 4321, 46718235, 46781235, 56718234, 56781234\\}$. We also briefly discuss the application of heaps to permutation pattern characterization."}
{"category": "Math", "title": "Hausdorff Dimension of Exponential Parameter Rays and Their Endpoints", "abstract": "We investigate the set $I$ of parameters $\\kappa$ for which the singular value of $z\\mapsto e^z+\\kappa$ converges to $\\infty$. The set $I$ consists of uncountably many parameter rays, plus landing points of some of these rays. We show that the parameter rays have Hausdorff dimension 1, while the ray endpoints in $I$ alone have dimension 2. Analogous results were known for dynamical planes of exponential maps; our result shows that this also holds in parameter space."}
{"category": "Math", "title": "Pure Virtual Braids Homotopic to the Identity Braid", "abstract": "Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid."}
{"category": "Math", "title": "Modules over operads and functors", "abstract": "In the theory of operads we consider functors of generalized symmetric powers defined by sums of coinvariant modules under actions of symmetric groups. One observes classically that the construction of symmetric functors provides an isomorphism from the category of symmetric modules to a subcategory of the category of functors on the base category. The purpose of this book is to obtain a similar relationship for functors on a category of algebras over an operad. We observe that right modules over operads, symmetric modules equipped with a right operad action, give rise to functors on categories of algebras and we prove that this construction yields an embedding of categories. Then we check that right modules over operads form a model category. In addition we prove that weak-equivalences of right modules correspond to pointwise weak-equivalences at the functor level. As a conclusion, we obtain that right modules over operads supply good models for the homotopy of associated functors on algebras over operads."}
{"category": "Math", "title": "Triacontagonal coordinates for the E(8) root system", "abstract": "This note gives an explicit formula for the elements of the E(8) root system. The formula is triacontagonally symmetric in that one may clearly see an action by the cyclic group with 30 elements. The existence of such a formula is due to the fact that the Coxeter number of E(8) is 30."}
{"category": "Math", "title": "Extensions of operator algebras I", "abstract": "We transcribe a portion of the theory of extensions of C*-algebras to general operator algebras. We also include several new general facts about approximately unital ideals in operator algebras and the C*-algebras which they generate."}
{"category": "Math", "title": "Whitehead double and Milnor invariants", "abstract": "We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length < k are all zero into a link with vanishing Milnor invariants of length < 2k, and we provide formulas for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self Delta-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link L is link-homotopic to the unlink if and only if a link L with a single component Whitehed doubled is self Delta-equivalent to the unlink."}
{"category": "Math", "title": "Ordered involutive operator spaces", "abstract": "This is a companion to recent papers of the authors; here we construct the `noncommutative Shilov boundary' of a (possibly nonunital) selfadjoint ordered space of Hilbert space operators. The morphisms in the universal property of the boundary preserve order. As an application, we consider `maximal' and `minimal' unitizations of such ordered operator spaces."}
{"category": "Math", "title": "The contour of splitting trees is a L\\'evy process", "abstract": "Splitting trees are those random trees where individuals give birth at constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump--Mode--Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential. Here, we allow the birth rate to be infinite, that is, pairs of birth times and lifespans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure. A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point $(v,\\tau)$ of some individual $v$ (vertex) in a discrete tree, where $\\tau$ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping $\\varphi$ from the tree into the real line which preserves this order. The inverse of $\\varphi$ is called the exploration process, and the projection of this inverse on chronological levels the contour process. For splitting trees truncated up to level $\\tau$, we prove that thus defined contour process is a L\\'evy process reflected below $\\tau$ and killed upon hitting 0. This allows to derive properties of (i) splitting trees: conceptual proof of Le Gall--Le Jan's theorem in the finite variation case, exceptional points, coalescent point process, age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems, asymptotic numbers of individuals with infinite vs finite descendances."}
{"category": "Math", "title": "Elliptic hypergeometric functions", "abstract": "This is a brief overview of the status of the theory of elliptic hypergeometric functions to the end of 2012 written as a complementary chapter to the Russian edition of the book by G.E. Andrews, R. Askey, and R. Roy, Special Functions, Encycl. of Math. Appl. 71, Cambridge Univ. Press, 1999."}
{"category": "Math", "title": "Self similar expanding solutions of the planar network flow", "abstract": "We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schn\\\"urer and Schulze which treats the case of three half-lines. There are multiple solutions, and these are parametrized by combinatorial objects, namely Steiner trees with respect to a complete negatively curved metric on the unit ball which span $k$ specified points on the boundary at infinity. We also provide a sharp formulation of the regularity of these solutions at $t=0$."}
{"category": "Math", "title": "Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coalescence processes", "abstract": "We show that for $0<\\alpha<1$ and $\\theta>-\\alpha$, the Poisson-Dirichlet distribution with parameter $(\\alpha, \\theta)$ is the unique reversible distribution of a rather natural fragmentation-coalescence process. This completes earlier results in the literature for certain split and merge transformations and the parameter $\\alpha =0$."}
{"category": "Math", "title": "A second order minimality condition for the Mumford-Shah functional", "abstract": "A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\\Gamma)$, $\\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given."}
{"category": "Math", "title": "The existence of superinvolutions", "abstract": "Superinvolutions on graded associative algebras constitute a source of Lie and Jordan superalgebras. Graded versions of the classical Albert and Albert-Riehm Theorems on the existence of superinvolutions are proven. Surprisingly, the existence of superinvolutions of the first kind is a rare phenomenon, as nontrivial central division superalgebras are never endowed with this kind of superinvolutions."}
{"category": "Math", "title": "Surgery formula for Seiberg--Witten invariants of negative definite plumbed 3-manifolds", "abstract": "We derive a cut-and-paste surgery formula of Seiberg--Witten invariants for negative definite plumbed rational homology 3-spheres. It is similar to (and motivated by) Okuma's recursion formula [arXiv:math.AG/0610464, 4.5] targeting analytic invariants of splice quotient singularities. The two formulas combined provide automatically a proof of the equivariant version [arXiv:math.AG/0310084, 5.2(b)] of the `Seiberg--Witten invariant conjecture' [arXiv:math.AG/0111298] for these singularities."}
{"category": "Math", "title": "Fold cobordisms and stable homotopy groups", "abstract": "We show that the cobordism groups of negative codimensional folds maps contain direct sums of stable homotopy groups of Thom spaces of vector bundles like the circle and the infinite dimensional projective space. We give geometrical invariants which detect these direct summands."}
{"category": "Math", "title": "Uniformly continuous maps between ends of R-trees", "abstract": "There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps between the end spaces are translated to some classes of coarse maps (or even classes of metrically proper lipschitz maps) between the trees."}
{"category": "Math", "title": "Severi varieties and self rational maps of K3 surfaces", "abstract": "Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parametrizing nodal curves of given genus and degree lying on some K3 surface. We also establish a number of numerical constraints satisfied by such non trivial rational maps, that is of topological degree >1."}
{"category": "Math", "title": "Grafting hyperbolic metrics and Eisenstein series", "abstract": "The family hyperbolic metric for the plumbing variety $\\{zw=t\\}$ and the non holomorphic Eisenstein series $E(\\zeta;2)$ are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of Riemann surfaces. Applications include an asymptotic expansion for the Weil-Petersson metric and a local form of symplectic reduction."}
{"category": "Math", "title": "L^2 harmonics forms on non compact manifolds", "abstract": "The source of these notes is a series of lectures given at the CIMPA's summer school \"Recent Topics in Geometric Analysis\"."}
{"category": "Math", "title": "Dynamic rays of bounded-type entire functions", "abstract": "We construct an entire function in the Eremenko-Lyubich class $\\B$ whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in $\\B$, in particular those of finite order, every escaping point can be connected to $\\infty$ by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926."}
{"category": "Math", "title": "The Hartogs extension theorem on (n-1)-complete complex spaces", "abstract": "Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete normal complex space X of pure dimension n >= 2 and for every compact set K in D such that D - K is connected, holomorphic or meromorphic functions in D - K extend holomorphically or meromorphically to D. Normality is an unvavoidable assumption for holomorphic extension, but we show that meromorphic extension holds on a reduced globally irreducible (not necessarily normal) X of pure dimension n >=2 provided that the regular part of D - K is connected."}
{"category": "Math", "title": "On limit cycles appearing by polynomial perturbation of Darbouxian integrable systems", "abstract": "We prove an existential finiteness Varchenko-Khovanskii type result for integrals of rational 1-forms over the level curves of Darbouxian integrals."}
{"category": "Math", "title": "Absolute Galois acts faithfully on the components of the moduli space of surfaces: A Belyi-type theorem in higher dimension", "abstract": "Given an object over the algebraic closure Qbar of Q, there is often no reason for invariants of the corresponding holomorphic object to be preserved by the absolute Galois group Gal(Qbar/Q), and in general this is not true, although it is sometimes surprising to observe in practice. The case of covers of the projective line branched only over the points 0, 1, and infinity, through Belyi's theorem, leads to Grothendieck's dessins d'enfants program for understanding the absolute Galois group through its faithful action on such covers. This note is motivated by Catanese's question about a higher-dimensional analogue: does the absolute Galois group act faithfully on the deformation equivalence classes of smooth surfaces? (These equivalence classes are of course by definition the strongest deformation invariants.) We give a short proof of a weaker result: the absolute Galois group acts faithfully on the irreducible components of the moduli space of smooth surfaces (of general type, canonically polarized). Bauer, Catanese, and Grunewald have recently answered Catanese's original question using a different construction."}
{"category": "Math", "title": "Characterization of polynomials", "abstract": "In 1954 it was proved if f is infinitely differentiable in the interval I and some derivative (of order depending on x) vanishes at each x, then f is a polynomial. Later it was generalized for multi-variable case. In this paper we give an extension for distributions."}
{"category": "Math", "title": "On C$^2$-smooth Surfaces of Constant Width", "abstract": "A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along its normal lines. We also give a characterization of surfaces of constant width that have rational support function. Our techniques, which are complex differential geometric in nature, allow us to construct explicit smooth surfaces of constant width in ${\\mathbb{E}}^3$, and their focal sets. They also allow for easy construction of tetrahedrally symmetric surfaces of constant width."}
{"category": "Math", "title": "Equifocality of a singular riemannian foliation", "abstract": "A singular foliation on a complete riemannian manifold M is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. We prove that the regular leaves are equifocal, i.e., the end point map of a normal foliated vector field has constant rank. This implies that we can reconstruct the singular foliation by taking all parallel submanifolds of a regular leaf with trivial holonomy. In addition, the end point map of a normal foliated vector field on a leaf with trivial holonomy is a covering map. These results generalize previous results of the authors on singular riemannian foliations with sections."}
{"category": "Math", "title": "Symmetrical invariants of some modular Lie algebras of Cartan type", "abstract": "Let $L$ be one of the finite dimensional Lie algebras $W_n({\\bf m}),$ $S_n({\\bf m}),$ $ H_n({\\bf m})$ of Cartan type over an algebraically closed field of prime characteristic $p>0.$ For an elements $F$ of the symmetrical algebra $S(L)$ we found necessary and sufficient condition in order to the element $ad(\\partial_1)^{p^{m_1}-1} ad(\\partial_2)^{p^{m_2}-1}... ad(\\partial_n)^{p^{m_n}-1}(F)$ belongs to the symmetrical invariants algebra $S(L)^L.$ Also, for $p=3,5$ the algebra of symmetrical invariants $S(H_2)^{H_2}$ is calculated in explicit way."}
{"category": "Math", "title": "The moduli space of cubic fourfolds", "abstract": "We describe the GIT compactification of the moduli space of cubic fourfolds, with a special emphasis on the role played by singularities. Our main result is that a cubic fourfold with only isolated simple (A-D-E) singularities is GIT stable. Conversely, with some minor exceptions, the stability for cubic fourfolds is characterized by this condition."}
{"category": "Math", "title": "Zeroth Hochschild homology of preprojective algebras over the integers", "abstract": "We determine the Z-module structure of the preprojective algebra and its zeroth Hochschild homology, for any non-Dynkin quiver (and hence the structure working over any base commutative ring, of any characteristic). This answers (and generalizes) a conjecture of Hesselholt and Rains, producing new $p$-torsion classes in degrees 2p^l, l >= 1, We relate these classes by p-th power maps and interpret them in terms of the kernel of Verschiebung maps from noncommutative Witt theory. An important tool is a generalization of the Diamond Lemma to modules over commutative rings, which we give in the appendix. In the previous version, additional results are included, such as: the Poisson center of $\\text{Sym } HH_0(\\Pi)$ for all quivers, the BV algebra structure on Hochschild cohomology, including how the Lie algebra structure $HH_0(\\Pi_Q)$ naturally arises from it, and the cyclic homology groups of $\\Pi_Q$."}
{"category": "Math", "title": "Local structure of the moduli space of K3 surfaces over finite characteristic", "abstract": "Let k be a perfect field of characteristic p > 2. In this note, we show that the local moduli space of a non-supersingular K3 surface over k with trivial deformation of the associated enlarged formal Brauer group admits a natural p-divisible formal group structure."}
{"category": "Math", "title": "Reduced bias nonparametric lifetime density and hazard estimation", "abstract": "Kernel-based nonparametric hazard rate estimation is considered with a special class of infinite-order kernels that achieves favorable bias and mean square error properties. A fully automatic and adaptive implementation of a density and hazard rate estimator is proposed for randomly right censored data. Careful selection of the bandwidth in the proposed estimators yields estimates that are more efficient in terms of overall mean squared error performance, and in some cases achieves a nearly parametric convergence rate. Additionally, rapidly converging bandwidth estimates are presented for use in second-order kernels to supplement such kernel-based methods in hazard rate estimation. Simulations illustrate the improved accuracy of the proposed estimator against other nonparametric estimators of the density and hazard function. A real data application is also presented on survival data from 13,166 breast carcinoma patients."}
{"category": "Math", "title": "Glicci simplicial complexes", "abstract": "One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this question for Stanley-Reisner ideals defined by simplicial complexes that are weakly vertex-decomposable. This class of complexes includes matroid, shifted and Gorenstein complexes respectively. Moreover, we construct a simplicial complex which shows that the property of being glicci depends on the characteristic of the base field. As an application of our methods we establish new evidence for two conjectures of Stanley on partitionable complexes and on Stanley decompositions."}
{"category": "Math", "title": "Milnor Invariants for Spatial Graphs", "abstract": "Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the classical case. Unlike previous attempts at generalizing link homotopy to spatial graphs, our new relation allows analogues of some standard link homotopy results and invariants. In particular we can define a type of Milnor group for a spatial graph under component homotopy, and this group determines whether or not the spatial graph is splittable. More surprisingly, we will also show that whether the spatial graph is splittable up to component homotopy depends only on the link homotopy class of the links contained within it. Numerical invariants of the relation will also be produced."}
{"category": "Math", "title": "On the Marginal Distributions of Stationary AR(1) Sequences", "abstract": "In this note we correct an omission in our paper (Satheesh and Sandhya, 2005) in defining semi-selfdecomposable laws and also show with examples that the marginal distributions of a stationary AR(1) process need not even be infinitely divisible."}
{"category": "Math", "title": "Singularities of Schr\\\"oder maps and unhyperbolicity of rational functions", "abstract": "We study transcendental singularities of a Schr\\\"oder map arising from a rational function $f$, using results from complex dynamics and Nevanlinna theory. These maps are transcendental meromorphic functions of finite order in the complex plane. We show that their transcendental singularities lie over the set where $f$ is not semihyperbolic (unhyperbolic). In addition, if they are direct, then they lie over only attracting periodic points of $f$, and moreover, if $f$ is a polynomial, then both direct and indirect singularities lie over attracting, parabolic and Cremer periodic points of $f$. We also obtain concrete examples of both kinds of transcendental singularities of Schr\\\"oder maps as well as a new proof of the Pommerenke-Levin-Yoccoz inequality and a new formulation of the Fatou conjecture."}
{"category": "Math", "title": "Geometric structure of sumsets", "abstract": "Given a finite set of lattice points, we compare its sumsets and lattice points in its dilated convex hulls. Both of these are known to grow as polynomials. Generally, the former are subsets of the latter. In this paper, we will see that sumsets occupy all the central lattice points in convex hulls, giving us a kind of approximation to lattice points in polytopes."}
{"category": "Math", "title": "Jet schemes, arc spaces and the Nash problem", "abstract": "This paper is an introduction to the jet schemes and the arc space of an algebraic variety. We also introduce the Nash problem on arc families."}
{"category": "Math", "title": "Groups of diffeomorphisms and geometric loops of manifolds over ultra-normed fields", "abstract": "The article is devoted to the investigation of groups of diffeomorphisms and loops of manifolds over ultra-metric fields of zero and positive characteristics. Different types of topologies are considered on groups of loops and diffeomorphisms relative to which they are generalized Lie groups or topological groups. Among such topologies pairwise incomparable are found as well. Topological perfectness of the diffeomorphism group relative to certain topologies is studied. There are proved theorems about projective limit decompositions of these groups and their compactifications for compact manifolds. Moreover, an existence of one-parameter local subgroups of diffeomorphism groups is investigated."}
{"category": "Math", "title": "Generators of Jacobians of Hyperelliptic Curves", "abstract": "This paper provides a probabilistic algorithm to determine generators of the m-torsion subgroup of the Jacobian of a hyperelliptic curve of genus two."}
{"category": "Math", "title": "Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting", "abstract": "We extend classical results of Kostant and al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras."}
{"category": "Math", "title": "J-class weighted shifts on the space of bounded sequences of complex numbers", "abstract": "We provide a characterization of $J$-class and $J^{mix}$-class unilateral weighted shifts on $l^{\\infty}(\\mathbb{N})$ in terms of their weight sequences. In contrast to the previously mentioned result we show that a bilateral weighted shift on $l^{\\infty}(\\mathbb{Z})$ cannot be a $J$-class operator."}
{"category": "Math", "title": "J-class operators and hypercyclicity", "abstract": "The purpose of the present work is to treat a new notion related to linear dynamics, which can be viewed as a \"localization\" of the notion of hypercyclicity. In particular, let $T$ be a bounded linear operator acting on a Banach space $X$ and let $x$ be a non-zero vector in $X$ such that for every open neighborhood $U\\subset X$ of $x$ and every non-empty open set $V\\subset X$ there exists a positive integer $n$ such that $T^{n}U\\cap V\\neq\\emptyset$. In this case $T$ will be called a $J$-class operator. We investigate the class of operators satisfying the above property and provide various examples. It is worthwhile to mention that many results from the theory of hypercyclic operators have their analogues in this setting. For example we establish results related to the Bourdon-Feldman theorem and we characterize the $J$-class weighted shifts. We would also like to stress that even non-separable Banach spaces which do not support topologically transitive operators, as for example $l^{\\infty}(\\mathbb{N})$, do admit $J$-class operators."}
{"category": "Math", "title": "Wreath products in modular group algebras of some finite 2-groups", "abstract": "Let $K$ be field of characteristic 2 and let $G$ be a finite non-abelian 2-group with the cyclic derived subgroup $G'$, and there exists a central element $z$ of order 2 in $Z(G) \\backslash G'$. We prove that the unit group of the group algebra $KG$ possesses a section isomorphic to the wreath product of a group of order 2 with the derived subgroup of the group $G$, giving for such groups a positive answer to the question of A. Shalev."}
{"category": "Math", "title": "A Class of pairwise-independent Joinings", "abstract": "We introduce a special class of pairwise-independent self-joinings for a stationary process: Those for which one coordinate is a continuous function of the two others. We investigate which properties on the process the existence of such a joining entails. In particular, we prove that if the process is aperiodic, then it has positive entropy. Our other results suggest that such pairwise independent, non-independent self-joinings exist only in very specific situations: Essentially when the process is a subshift of finite type topologically conjugate to a full-shift. This provides an argument in favor of the conjecture that 2-fold mixing implies 3-fold-mixing."}
{"category": "Math", "title": "Topology and Factorization of Polynomials", "abstract": "For any polynomial $P \\in \\mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$ is the number of irreducible factors of $P$. Moreover, the knowledge of $F(P)$ gives a complete factorization of the polynomial $P$ by taking gcd's. This generalizes previous results by Ruppert and Gao in the case $n=2$."}
{"category": "Math", "title": "Natural boundary of Dirichlet series", "abstract": "We prove some conditions on the existence of natural boundaries of Dirichlet series. We show that generically the presumed boundary is the natural one. We also give an application of natural boundaries in determining asymptotic results."}
{"category": "Math", "title": "Some remarks on generalized roundness", "abstract": "By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank $\\geq 2$ equals 1. This answers a question of J-F. Lafont and S. Prassidis."}
{"category": "Math", "title": "Extension of symmetries on Einstein manifolds with boundary", "abstract": "We investigate the validity of the isometry extension property for (Riemannian) Einstein metrics on manifolds with boundary. Given a metric on the boundary, this is the issue of whether any Killing field of the boundary metric extends to a Killing field of any bulk or filling Einstein metric inducing the given data on the boundary. Under a mild condition on the fundamental group, this is proved to be the case at least when the Killing field preserves the mean curvature of the boundary."}
{"category": "Math", "title": "$A^\\nabla$-tensors on lightlike hypersurfaces", "abstract": "This paper introduces $\\anabla$-tensors on lightlike hypersurfaces $M^{n+1}$ of signature $(0,n)$, $(n\\geq 1)$ and investigates on their properties in connection with the null geometry of $M$. In particular, we show that there is an interplay between existence of $\\anabla$-tensors of certain type and lightlike warped product structures."}
{"category": "Math", "title": "Determinant Formulas Relating to Tableaux of Bounded Height", "abstract": "Chen et al. recently established bijections for $(d+1)$-noncrossing/ nonnesting matchings, oscillating tableaux of bounded height $d$, and oscillating lattice walks in the $d$-dimensional Weyl chamber. Stanley asked what is the total number of such tableaux of length $n$ and of any shape. We find a determinant formula for the exponential generating function. The same idea applies to prove Gessel's remarkable determinant formula for permutations with bounded length of increasing subsequences. We also give short algebraic derivations for some results of the reflection principle."}
{"category": "Math", "title": "A note on conformal connections on lightlike hypersurfaces", "abstract": "Degenerate submanifolds of pseudo-Riemannian manifolds are quite difficult to study because there is no prefered connection when the submanifold is not totally geodesic. For the particular case of degenerate totally umbilical hypersurfaces, we show that there are \"Weyl\" connections adapted to the induced structure on the hypersurface. We begin the study of these with their holonomy."}
{"category": "Math", "title": "Einstein-Weyl structures on lightike hypersurfaces", "abstract": "We study Weyl structures on lightlikes hypersurfaces endowed with a conformal structure of certain type and specific screen distribution: the Weyl screen structures. We investigate various differential geometric properties of Einstein-Weyl screen structures on lightlike hypersurfaces and show that, for ambiant Lorentzian space $\\mathbb{R}^{n+2}_{1}$ and a totally umbilical screen foliation, there is a strong interplay with the induced (Riemannian) Weyl-structure on the leaves. Finally, we establish necessary and sufficient conditions for a Weyl structure defined by the $1-$form of an almost contact structure given by an additional complex structure in case of an ambiant Kaehler manifold to be closed."}
{"category": "Math", "title": "Four-dimensional Painlev\\'e systems of types $D_5^{(1)}$ and $B_4^{(1)}$", "abstract": "We find and study a five-parameter family of four-dimensional coupled Painlev\\'e V systems with affine Weyl group symmetry of type $D_5^{(1)}$. We then give an explicit description of a confluence from those systems to a four-parameter family of four-dimensional coupled Painlev\\'e III systems with affine Weyl group symmetry of type $B_4^{(1)}$."}
{"category": "Math", "title": "Noncommutative Berezin transforms and model theory", "abstract": "We initiate the study of a class of noncommutative domains of n-tuples of bounded linear operators on a Hilbert space, which is generated by certain positivity conditions on polynomials in n noncommutative indeterminates. We obtain Fatou type results, functional calculi, and a model theory for n-tuples of operators in these domains. The main role in this study is played by a class of noncommutative Berezin transforms which is introduced in this paper. Our results extend to noncommutative varieties."}
{"category": "Math", "title": "L^2-Betti numbers of plane algebraic curves", "abstract": "In [DJL07] it was shown that if A is an affine hyperplane arrangement in C^n, then at most one of the L^2-Betti numbers of its complement is non--zero. We will prove an analogous statement for complements of any algebraic curve in C^2. Furthermore we also recast and extend results of [LM06] in terms of L^2-Betti numbers."}
{"category": "Math", "title": "Realizations of Seifert matrices by hyperbolic knots", "abstract": "Recently Kearton showed that any Seifert matrix of a knot is S--equivalent to the Seifert matrix of a prime knot. We show in this note that such a matrix is in fact S--equivalent to the Seifert matrix of a hyperbolic knot. This result follows from reinterpreting this problem in terms of Blanchfield pairings and by applying results of Kawauchi."}
{"category": "Math", "title": "Almost Product Evaluation of Hankel Determinants", "abstract": "An extensive literature exists describing various techniques for the evaluation of Hankel determinants. The prevailing methods such as Dodgson condensation, continued fraction expansion, LU decomposition, all produce product formulas when they are applicable. We mention the classic case of the Hankel determinants with binomial entries ${3k+2 \\choose k}$ and those with entries ${3k \\choose k}$; both of these classes of Hankel determinants have product form evaluations. The intermediate case, ${3k+1 \\choose k}$ has not been evaluated. There is a good reason for this: these latter determinants do not have product form evaluations. In this paper we evaluate the Hankel determinant of ${3k+1 \\choose k}$. The evaluation is a sum of a small number of products, an almost product. The method actually provides more, and as applications, we present the salient points for the evaluation of a number of other Hankel determinants with polynomial entries, along with product and almost product form evaluations at special points."}
{"category": "Math", "title": "Smoothed Wigner transforms in the numerical simulation of semiclassical (high-frequency) wave propagation", "abstract": "The numerical simulation of wave propagation in semiclassical (high-frequency) problems is well known to pose a formidable challenge. In this work, a new phase-space approach for the numerical simulation of semiclassical wave propagation, making use of the smoothed Wigner Transform (SWT), is proposed. There are numerous works which use the Wigner Transform (WT) in the study of a variety of wave propagation problems including high-frequency limits for linear, nonlinear and/or random waves. The WT however is well known to present significant difficulties in the formulation of numerical schemes. Working with concrete examples for the semiclassical linear Schrodinger equation it is seen that the SWT approach is indeed significantly faster (in a well-defined sense) to work with than the WT and than full numerical solutions of the original equation in the semiclassical regime. Comparisons with exact and numerical solutions are used to keep track of numerical errors."}
{"category": "Math", "title": "On the $p$-adic meromorphy of the function field height zeta function", "abstract": "In this brief note, we will investigate the number of points of bounded (twisted) height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the $p$-adic analytic properties of the height zeta function. In particular, we will show that for a large class of projective varieties this function is $p$-adic meromorphic."}
{"category": "Math", "title": "Twisted conjugacy classes in R. Thompson's group F", "abstract": "In this short article, we prove that any automorphism of the R. Thompson's group $F$ has infinitely many twisted conjugacy classes. The result follows from the work of Matthew Brin, together with a standard facts on R. Thompson's group $F$, and elementary properties of the Reidemeister numbers."}
{"category": "Math", "title": "Complexity of Villamayor's algorithm in the non exceptional monomial case", "abstract": "We study monomial ideals, always locally given by a monomial, like a reasonable first step to estimate in general the number of monoidal transformations of Villamayor's algorithm of resolution of singularities. The resolution of a monomial ideal $<X_1^{a_1}\\cdot ... \\cdot X_n^{a_n}>$ is interesting due to its identification with the particular toric problem $<Z^c- X_1^{a_1}\\cdot ... \\cdot X_n^{a_n}>$. In the special case, when all the exponents $a_i$ are greater than or equal to the critical value $c$, we construct the largest branch of the resolution tree which provides an upper bound involving partial sums of Catalan numbers. This case will be called ``minimal codimensional case''. Partial sums of Catalan numbers (starting $1,2,5,...$) are $1,3,8,22,...$ These partial sums are well known in Combinatorics and count the number of paths starting from the root in all ordered trees with $n+1$ edges. Catalan numbers appear in many combinatorial problems, counting the number of ways to insert $n$ pairs of parenthesis in a word of $n+1$ letters, plane trees with $n+1$ vertices, $... $, etc. The non minimal case, when there exists some exponent $a_{i_0}$ smaller than $c$, will be called ``case of higher codimension''. In this case, still unresolved, we give an example to state the foremost troubles. Computation of examples has been helpful in both cases to study the behaviour of the resolution invariant. Computations have been made in Singular (see \\cite{sing}) using the \\emph{desing} package by G. Bodn\\'ar and J. Schicho, see \\cite{lib}."}
{"category": "Math", "title": "Cohomology of the minimal nilpotent orbit", "abstract": "We compute the integral cohomology of the minimal non-trivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the sub-root system generated by the long simple roots. The modulo $\\ell$ reduction of the Springer correspondent representation involves the sign representation exactly when $\\ell$ divides the order of this cohomology group. The primes dividing the torsion of the rest of the cohomology are bad primes."}
{"category": "Math", "title": "Finite Gorenstein representation type implies simple singularity", "abstract": "Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive R-module is free."}
{"category": "Math", "title": "Orthogonal complex structures on domains in R^4", "abstract": "An orthogonal complex structure on a domain in R^4 is a complex structure which is integrable and is compatible with the Euclidean metric. This gives rise to a first order system of partial differential equations which is conformally invariant. We prove two Liouville-type uniqueness theorems for solutions of this system, and use these to give an alternative proof of the classification of compact locally conformally flat Hermitian surfaces first proved by Pontecorvo. We also give a classification of non-degenerate quadrics in CP^3 under the action of the conformal group. Using this classification, we show that generic quadrics give rise to orthogonal complex structures defined on the complement of unknotted solid tori which are smoothly embedded in R^4."}
{"category": "Math", "title": "A New Proof of Pappus's Theorem", "abstract": "Any stretching of Ringel's non-Pappus pseudoline arrangement when projected into the Euclidean plane, implicitly contains a particular arrangement of nine triangles. This arrangement has a complex constraint involving the sines of its angles. These constraints cannot be satisfied by any projection of the initial arrangement. This is sufficient to prove Pappus's theorem. The derivation of the constraint is via systems of inequalities arising from the polar coordinates of the lines. These systems are linear in r for any given theta, and their solubility can be analysed in terms of the signs of determinants. The evaluation of the determinants is via a normal form for sums of products of sines, giving a powerful system of trigonometric identities. The particular result is generalized to arrangements derived from three edge connected totally cyclic directed graphs, conjectured to be sufficient for a complete analysis of angle constraining arrangements of lines, and thus a full response to Ringel's slope conjecture. These methods are generally applicable to the realizability problem for rank 3 oriented matroids."}
{"category": "Math", "title": "Five-parameter family of partial differential systems in two variables", "abstract": "We find a five-parameter family of partial differential systems in two variables with two polynomial Hamiltonians. We give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new."}
{"category": "Math", "title": "Corestrictions of algebras and splitting fields", "abstract": "Given a field $F$, an \\'etale extension $L/F$ and an Azumaya algebra $A/L$, one knows that there are extensions $E/F$ such that $A \\otimes_F E$ is a split algebra over $L \\otimes_F E$. In this paper we bound the degree of a minimal splitting field of this type from above and show that our bound is sharp in certain situations, even in the case where $L/F$ is a split extension. This gives in particular a number of generalizations of the classical fact that when the tensor product of two quaternion algebras is not a division algebra, the two quaternion algebras must share a common quadratic splitting field. In another direction, our constructions combined with results of Karpenko also show that for any odd prime number $p$, the generic algebra of index $p^n$, and exponent $p$ cannot be expressed nontrivially as the corestriction of an algebra over any extension field if $n < p^2$."}
{"category": "Math", "title": "Finite Euler products and the Riemann Hypothesis", "abstract": "We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the approximation by products is good in this region, the zeta-function has at most finitely many zeros in it. We then construct a parameterized family of non-analytic functions with this same property. With the possible exception of a finite number of zeros off the critical line, every function in the family satisfies a Riemann Hypothesis. Moreover, when the parameter is not too large, they have about the same number of zeros as the zeta-function, their zeros are all simple, and they \"repel\". The structure of these functions makes the reason for the simplicity and repulsion of their zeros apparent and suggests a mechanism that might be responsible for the corresponding properties of the zeta-function's zeros. Computer evidence suggests that the zeros of functions in the family are remarkably close to those of the zeta-function (even for small values of the parameter), and we show that they indeed converge to them as the parameter increases. Furthermore, between zeros of the zeta-function, the moduli of functions in the family tend to twice the modulus of the zeta-function. Both assertions assume the Riemann Hypothesis. We end by discussing analogues for other L-functions and show how they give insight into the study of the distribution of zeros of linear combinations of L-functions."}
{"category": "Math", "title": "The center of the generic algebra of degree p", "abstract": "Let $F$ be an algebraically closed field of characteristic zero, and let $p$ be an odd prime. We show that the center of the generic division algebra of degree $p$ is stably rational over $F$. Equivalently, if we let $V=M_p(F) \\oplus M_p(F)$ and $PGL_p$ act on $V$ by simultaneous conjugation, then we show that the function field of the quotient variety $V/PGL_p$ is stably rational over $F$."}
{"category": "Math", "title": "Spectral methods for orthogonal rational functions", "abstract": "An operator theoretic approach to orthogonal rational functions on the unit circle with poles in its exterior is presented in this paper. This approach is based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two different alternatives are discussed, depending whether we use for the matrix representation the standard basis of orthogonal rational functions, or a new one with poles alternatively located in the exterior and the interior of the unit circle. The corresponding representations are linear fractional transformations with matrix coefficients acting respectively on Hessenberg and five-diagonal unitary matrices. In consequence, the orthogonality measure can be recovered from the spectral measure of an infinite unitary matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg and five-diagonal matrices. As an application of this operator approach, we obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation, generalizing to the rational case known results for orthogonal polynomials on the unit circle. Finally, we extend these results to orthogonal polynomials on the real line with poles in the lower half plane."}
{"category": "Math", "title": "Uniqueness of solutions of Ricci flow on complete noncompact manifolds", "abstract": "We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures."}
{"category": "Math", "title": "The enumeration of maximally clustered permutations", "abstract": "The maximally clustered permutations are characterized by avoiding the classical permutation patterns 3421, 4312, and 4321. This class contains the freely-braided permutations and the fully-commutative permutations. In this work, we show that the generating functions for certain fully-commutative pattern classes can be transformed to give generating functions for the corresponding freely-braided and maximally clustered pattern classes. Moreover, this transformation of generating functions is rational. As a result, we obtain enumerative formulas for the pattern classes mentioned above as well as the corresponding hexagon-avoiding pattern classes where the hexagon-avoiding permutations are characterized by avoiding 46718235, 46781235, 56718234, and 56781234."}
{"category": "Math", "title": "Elimination Theory for Tropical Varieties", "abstract": "Tropical algebraic geometry offers new tools for elimination theory and implicitization. We determine the tropicalization of the image of a subvariety of an algebraic torus under any homomorphism from that torus to another torus."}
{"category": "Math", "title": "Missing Data: A Comparison of Neural Network and Expectation Maximisation Techniques", "abstract": "The estimation of missing input vector elements in real time processing applications requires a system that possesses the knowledge of certain characteristics such as correlations between variables, which are inherent in the input space. Computational intelligence techniques and maximum likelihood techniques do possess such characteristics and as a result are important for imputation of missing data. This paper compares two approaches to the problem of missing data estimation. The first technique is based on the current state of the art approach to this problem, that being the use of Maximum Likelihood (ML) and Expectation Maximisation (EM. The second approach is the use of a system based on auto-associative neural networks and the Genetic Algorithm as discussed by Adbella and Marwala3. The estimation ability of both of these techniques is compared, based on three datasets and conclusions are made."}
{"category": "Math", "title": "Cherednik algebras for algebraic curves", "abstract": "For any smooth algebraic curve C, Pavel Etingof introduced a `global' Cherednik algebra as a natural deformation of the cross product of the algebra of differential operators on C^n and the symmetric group. We provide a construction of the global Cherednik algebra in terms of quantumn Hamiltonian reduction. We study a category of character D-modules on a representation scheme associated to C and define a Hamiltonian reduction functor from that category to category O for the global Cherednik algebra. In the special case where the curve C is the multiplicative group, the global Cherednik algebra reduces to the trigonometric Cherednik algebra of type A, and our character D-modules become holonomic D-modules on GL_n \\times C^n. The corresponding perverse sheaves are reminiscent of (and include as special cases) Lusztig's character sheaves."}
{"category": "Math", "title": "Well displacing representations and orbit maps", "abstract": "We discuss in this article a property of action of groups by isometries called \"well displacing\". An action is said to be well displacing, if the displacement function is equivalent to the the displacement function for the action on the Cayley graph. We relate this property with the fact that orbit maps are quasi-isometric embeddings. We first describe countrexamples that shows this two notions are unrelated in general. On the other hand we explain that for a certain class of groups -- in particular hyperbolic groups -- these two properties are equivalent. In the course of our discussion, we introduce an intrinsic property of the group -- that we called the U-property -- which says quantitatively how the norm an element is controlled by the translation length of finitely many related conjugacy classes. This property play a central role in our discussion."}
{"category": "Math", "title": "Construction of type ${\\rm II_1}$ factors with prescribed countable fundamental group", "abstract": "In the context of Free Probability Theory, we study two different constructions that provide new examples of factors of type ${\\rm II_1}$ with prescribed fundamental group. First we investigate state-preserving group actions on the almost periodic free Araki-Woods factors satisfying both a condition of mixing and a condition of free malleability in the sense of Popa. Typical examples are given by the free Bogoliubov shifts. Take an ICC $w$-rigid group $G$ such that $\\mathcal{F}(L(G)) = \\{1\\}$ (e.g. $G = \\Z^2 \\rtimes \\SL(2, \\Z)$). For any countable subgroup $S \\subset \\R^*_+$, we show that there exists an action of $G$ on $L(\\F_\\infty)$ such that $L(\\F_\\infty) \\rtimes G$ is a type ${\\rm II_1}$ factor and its fundamental group is $S$. The second construction is based on a free product. Take $(B(H), \\psi)$ any factor of type ${\\rm I}$ endowed with a faithful normal state and denote by $\\Gamma \\subset \\R^*_+$ the subgroup generated by the point spectrum of $\\psi$. We show that the centralizer $(L(G) \\ast B(H))^{\\tau \\ast \\psi}$ is a type ${\\rm II_1}$ factor and its fundamental group is $\\Gamma$. Our proofs rely on Popa's deformation/rigidity strategy using his intertwining-by-bimodules technique."}
{"category": "Math", "title": "The descent statistic on involutions is not log-concave", "abstract": "We establish a combinatorial connection between the sequence $(i_{n,k})$ counting the involutions on $n$ letters with $k$ descents and the sequence $(a_{n,k})$ enumerating the semistandard Young tableaux on $n$ cells with $k$ symbols. This allows us to show that the sequences $(i_{n,k})$ are not log-concave for some values of $n$, hence answering a conjecture due to F. Brenti."}
{"category": "Math", "title": "Transverse LS-Category for Riemannian Foliations", "abstract": "We study the transverse Lusternik-Schnirelmann category of a Riemannian foliation on a compact manifold. We obtain a necessary and sufficient condition when the transverse LS category is finite. We also introduce a variation on the concept of transverse LS category, the essential transverse category, and show that this is finite for every Riemannian foliation and coincides with the transverse category if the latter is finite. Moreover we prove that the essential transverse category is a lower bound for the number of critical leaf closures of a basic differentiable function on M."}
{"category": "Math", "title": "The Equivariant LS-Category of Polar Actions", "abstract": "We will provide a lower bound for the equivariant Lusternik-Schnirelmann category of an arbitrary proper action in terms of the stratification by orbit types, and an upper bound for proper polar actions in terms of the equivariant Lusternik-Schnirelmann category of its generalized Weyl group. As an application we reprove a theorem of Singhof that determines the classical Lusternik-Schnirelmann category for U(n) and SU(n)."}
{"category": "Math", "title": "Chung's law for homogeneous Brownian functionals", "abstract": "Consider the first exit time $T_{a,b}$ from a finite interval $[-a,b]$ for an homogeneous fluctuating functional $X$ of a linear Brownian motion. We show the existence of a finite positive constant $\\k$ such that $$\\lim_{t\\to\\infty}t^{-1}\\log \\p[ T_{ab} > t] = -\\k.$$ Following Chung's original approach, we deduce a \"liminf\" law of the iterated logarithm for the two-sided supremum of $X$. This extends and gives a new point of view on a result of Khoshnevisan and Shi."}
{"category": "Math", "title": "Immersions of spheres and algebraically constructible functions", "abstract": "Let L be an algebraic set and let g : R^(n+1) \\times L --> R^(2n) (n is even) be a polynomial mapping such that for each l in L there is r(l)>0 such that the mapping g_l = g(.,l) restricted to the sphere S^n(r) is an immersion for every 0<r<(l), so that the intersection number I(g_l|S^n(r)) is defined. Then the function which maps l in L to I(g_l|S^n(r)) is algebraically constructible."}
{"category": "Math", "title": "Hypercontractivity, Nash inequalities, and subordination for classes of nonlinear semigroups", "abstract": "A suitable notion of hypercontractivity for a nonlinear semigroup $\\{T_t\\}$ is shown to imply Gagliardo--Nirenberg inequalities for its generator $H$, provided a subhomogeneity property holds for the energy functional $(u,Hu)$. We use this fact to prove that, for semigroups generated by operators of $p$--Laplacian--type, hypercontractivity implies ultracontractivity. We then introduce the notion of subordinated nonlinear semigroups when the corresponding Bernstein function is $f(x)=x^\\alpha$, and write down an explicit formula for the associated generator. It is shown that, in certain cases, hypercontractivity still holds for the subordinated semigroup and, hence, that Nash--type inequalities holds as well for the subordinated generator."}
{"category": "Math", "title": "Invariant forms, associated bundles and Calabi-Yau metrics", "abstract": "We develop a method, initially due to Salamon, to compute the space of ``invariant'' forms on an associated bundle X=P\\times_G V, with a suitable notion of invariance. We determine sufficient conditions for this space to be d-closed. We apply our method to the construction of Calabi-Yau metrics on TCP^1 and TCP^2."}
{"category": "Math", "title": "Low frequency dispersive estimates for the wave equation in higher dimensions", "abstract": "We prove dispersive estimates at low frequency in dimensions n greater or equal to 4 for the wave equation for a very large class of real-valued potentials, provided the zero is neither an eigenvalue nor a resonance."}
{"category": "Math", "title": "Weakly Compact \"Matrices\", Fubini-Like Property and Extension of Densely Defined Semigroups of Operators", "abstract": "Taking matrix as a synonym for a numerical function on the Cartesian product of two (in general, infinite) sets, a simple purely algebraic \"reciprocity property\" says that the set of rows spans a finite-dim space iff the set of columns does so. Similar topological reciprocity properties serve to define strongly compact and weakly compact matrices, featured in the well-known basic facts about almost periodic functions and about compact operators. Some properties, especially for the weak compact case, are investigated, such as the connection with the matrix having a Fubini-like property for general means. These are applied to prove possibility of extension to the entire semigroup of bounded densely defined semigroups of operators in a Banach space with weak continuity properties."}
{"category": "Math", "title": "Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic", "abstract": "We show that algebraic analogues of universal group covers, surjective group homomorphisms from a $\\mathbb{Q}$-vector space to $F^{\\times}$ with \"standard kernel\", are determined up to isomorphism of the algebraic structure by the characteristic and transcendence degree of $F$ and, in positive characteristic, the restriction of the cover to finite fields. This extends the main result of \"Covers of the Multiplicative Group of an Algebraically Closed Field of Characteristic Zero\" (B. Zilber, JLMS 2007), and our proof fills a hole in the proof given there."}
{"category": "Math", "title": "Enumerating permutations avoiding more than three Babson - Steingr\\'\\i msson patterns", "abstract": "Not long ago, Claesson and Mansour proposed some conjectures about the enumeration of the permutations avoiding more than three Babson - Steingr\\'\\i msson patterns (generalized patterns of type $(1,2)$ or $(2,1)$). The avoidance of one, two or three patterns has already been considered. Here, the cases of four and five forbidden patterns are solved and the exact enumeration of the permutations avoiding them is given, confirming the conjectures of Claesson and Mansour. The approach we use can be easily extended to the cases of more than five forbidden patterns."}
{"category": "Math", "title": "The Special McKay correspondence as an equivalence of derived categories", "abstract": "We give a new moduli construction of the minimal resolution of the singularity of type 1/r(1,a) by introducing the Special McKay quiver. To demonstrate that our construction trumps that of the G-Hilbert scheme, we show that the induced tautological line bundles freely generate the bounded derived category of coherent sheaves on X by establishing a suitable derived equivalence. This gives a moduli construction of the Special McKay correspondence for abelian subgroups of GL(2)."}
{"category": "Math", "title": "Cohomology of Fiber Products of Local Rings", "abstract": "Let $S$ and $T$ be local rings with common residue field $k$, let $R$ be the fiber product $S \\times_k T$, and let $M$ be an $S$-module. The Poincar\\'e series $P^R_M$ of $M$ has been expressed in terms of $P^S_M$, $P^S_k$ and $P^T_k$ by Kostrikin and Shafarevich, and by Dress and Kr\\\"amer. Here, an explicit minimal resolution, as well as theorems on the structure of $\\Ext_R(k,k)$ and $\\Ext_R(M,k)$ are given that illuminate these equalities. Structure theorems for the cohomology modules of fiber products of modules are also given. As an application of these results, we compute the depth of cohomology modules over a fiber product."}
{"category": "Math", "title": "Triangulations of projective modules", "abstract": "We show that the category of projective modules over a graded commutative ring admits a triangulation with respect to module suspension if and only if the ring is a finite product of graded fields and exterior algebras on one generator over a graded field (with a unit in the appropriate degree). We also classify the ungraded commutative rings for which the category of projective modules admits a triangulation with respect to the identity suspension. Applications to two analogues of the generating hypothesis in algebraic topology are given, and we translate our results into the setting of modules over a symmetric ring spectrum or S-algebra."}
{"category": "Math", "title": "A characterization of round spheres in terms of blocking light", "abstract": "A closed Riemannian manifold is said to have cross blocking if whenever distinct points p and q are at distance less than the diameter, all light rays from p can be shaded away from q with at most two point shades. Similarly, a closed Riemannian manifold is said to have sphere blocking if for each point p, all the light rays from p are shaded away from p by a single point shade. We prove that Riemannian manifolds with cross and sphere blocking are isometric to round spheres."}
{"category": "Math", "title": "Cancellation for inclusions of C*-algebras of finite depth", "abstract": "Let B be a unital C*-algebra, let A be a unital subalgebra, and let E be a conditional expectation from B to A with index-finite type and a quasi-basis of n elements. Then the topological stable rank satisfies \\tsr (B) \\leq \\tsr (A) + n - 1. As an application, we show that if a unital inclusion A \\subset B of C*-algebras has index-finite type and finite depth, and A is simple with stable rank one and Property (SP), then B has cancellation. In particular, if A is a simple unital C*-algebra with stable rank one and Property (SP), and a finite group G acts on A, then the crossed product has cancellation. Separately, if the group is the integers, we obtain cancellation under the additional hypotheses that the group action is outer and is trivial on K_0 (A)."}
{"category": "Math", "title": "Bernstein-Szego Polynomials Associated with Root Systems", "abstract": "We introduce multivariate generalizations of the Bernstein-Szego polynomials, which are associated to the root systems of the complex simple Lie algebras. The multivariate polynomials in question generalize Macdonald's Hall-Littlewood polynomials associated with root systems. For the root system of type A1 (corresponding to the Lie algebra SL (2;C)) the classic Bernstein-Szego polynomials are recovered."}
{"category": "Math", "title": "Crossed products by finite group actions with the Rokhlin property", "abstract": "We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (1) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (2) Simple unital AH algebras with slow dimension growth and real rank zero. (3) C*-algebras with real rank zero or stable rank one. (4) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective."}
{"category": "Math", "title": "Recursive boson system in the Cuntz algebra ${\\cal O}_{\\infty}$", "abstract": "Bosons and fermions are often written by elements of other algebras. M. Abe gave a recursive realization of the boson by formal infinite sums of the canonical generators of the Cuntz algebra ${\\cal O}_{\\infty}$. We show that such formal infinite sum always makes sense on a certain dense subspace of any permutative representation of ${\\cal O}_{\\infty}$. In this meaning, we can regard as if the algebra ${\\cal B}$ of bosons was a unital $*$-subalgebra of ${\\cal O}_{\\infty}$ on a given permutative representation by keeping their unboundedness. By this relation, we compute branching laws arising from restrictions of representations of ${\\cal O}_{\\infty}$ on ${\\cal B}$. For example, it is shown that the Fock representation of ${\\cal B}$ is given as the restriction of the standard representation of ${\\cal O}_{\\infty}$ on ${\\cal B}$."}
{"category": "Math", "title": "Unified quantum invariants and their refinements for homology 3-spheres with 2-torsion", "abstract": "For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and at any even root of unity the SU(2) quantum invariant. Moreover, this unified invariant splits into a sum of the refined unified invariants dominating spin and cohomological refinements of quantum SU(2) invariants. New results on the Ohtsuki series and the integrality of quantum invariants are the main applications of our construction."}
{"category": "Math", "title": "Hamiltonian Graphs and the Traveling Salesman Problem", "abstract": "A new characterisation of Hamiltonian graphs using f-cutset matrix is proposed. A new exact polynomial time algorithm for the travelling salesman problem (TSP) based on this new characterisation is developed. We then define so called ordered weighted adjacency list for given weighted complete graph and proceed to the main result of the paper, namely, the exact algorithm based on utilisation of ordered weighted adjacency list and the simple properties that any path or circuit must satisfy. This algorithm performs checking of sub-lists, containing (p-1) entries (edge pairs) for paths and p entries (edge pairs) for circuits, chosen from ordered adjacency list in a well defined sequence to determine exactly the shortest Hamiltonian path and shortest Hamiltonian circuit in a weighted complete graph of p vertices. The procedure has intrinsic advantage of landing on the desired solution in quickest possible time and even in worst case in polynomial time. A new characterisation of shortest Hamiltonian tour for a weighted complete graph satisfying triangle inequality (i.e. for tours passing through every city on a realistic map of cities where cities can be taken as points on a Euclidean plane) is also proposed. Finally, we propose a classical algorithm for unstructured search and also three new quantum algorithms for unstructured search which exponentially speed up the searching ability in the unstructured database and discuss its effect on the NP-Complete problems."}
{"category": "Math", "title": "Characterizing group $C^\\ast$-algebras through their unitary groups: the Abelian case", "abstract": "We study to what extent group $C^\\ast$-algebras are characterized by their unitary groups. A complete characterization of which Abelian group $C^\\ast$-algebras have isomorphic unitary groups is obtained. We compare these results with other unitary-related invariants of $C^\\ast(\\Gamma)$, such as the $K$-theoretic $K_1(C^\\ast(\\Gamma))$ and find that $C^\\ast$-algebras of nonisomorphic torsion-free Abelian groups may have isomorphic $K_1$-groups, in sharp contrast with the well-known fact that $C^\\ast(\\Gamma)$ (even $\\Gamma$) is characterized by the topological group structure of its unitary group when $\\Gamma $ is torsion-free and Abelian."}
{"category": "Math", "title": "Mittag-Leffler conditions on modules", "abstract": "We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions. In particular, this implies that tilting modules satisfy a useful finiteness condition over their endomorphism ring. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture."}
{"category": "Math", "title": "Reconstruction Algebras of Type A", "abstract": "We introduce a new class of algebras, called reconstruction algebras, and present some of their basic properties. These non-commutative rings dictate in every way the process of resolving the Cohen-Macaulay singularities C^2/G where G is a finite small cyclic subgroup of GL(2,C)."}
{"category": "Math", "title": "A weighted graph problem from commutative algebra", "abstract": "We give an especially simple proof of a theorem in graph theory that forms the key part of the solution to a problem in commutative algebra, on how to characterize the integral closure of a polynomial ring generated by quadratic monomials."}
{"category": "Math", "title": "Demazure embeddings are smooth", "abstract": "We prove Brion's conjecture stating that the closure of the orbit of a self-normalizing spherical subalgebra in the corresponding Grassmanian is smooth"}
{"category": "Math", "title": "Maximal C*-algebras of quotients and injective envelopes of C*-algebras", "abstract": "A new C*-enlargement of a C*-algebra $A$ nested between the local multiplier algebra $M_{\\text{loc}}(A)$ of $A$ and its injective envelope $I(A)$ is introduced. Various aspects of this maximal C*-algebra of quotients, $Q_{\\text{max}}(A)$, are studied, notably in the setting of AW*-algebras. As a by-product we obtain a new example of a type I C*-algebra $A$ such that $M_{\\text{loc}}(M_{\\text{loc}}(A))\\ne M_{\\text{loc}}(A)$."}
{"category": "Math", "title": "Existence and Number of Solutions of Diophantine Quadratic Equations with Two Unknowns in $Z$ and $N$", "abstract": "In this short note we study the existence and number of solutions in the set of integers ($Z$) and in the set of natural numbers ($N$) of Diopahntine Equations of second degree with two variables of the general form $ax^2-by^2=c$."}
{"category": "Math", "title": "Multiparameter Riesz Commutators", "abstract": "It is shown that product BMO of Chang and Fefferman, defined on the product of Euclidean spaces can be characterized by the multiparameter commutators of Riesz transforms. This extends a classical one-parameter result of Coifman, Rochberg, and Weiss, and at the same time extends the work of Lacey and Ferguson and Lacey and Terwilleger on multiparameter commutators with Hilbert transforms. The method of proof requires the real-variable methods throughout, which is new in the multi-parameter context."}
{"category": "Math", "title": "Codazzi spinors and globally hyperbolic manifolds with special holonomy", "abstract": "In this paper we examine the structure of Riemannian manifolds with a special kind of Codazzi tensors. We use them to construct globally hyperbolic Lorentzian manifolds with complete Cauchy hypersurfaces for any weakly irreducible holonomy representation with parallel spinors, i.e. with a holonomy group which is a semidirect product between $\\R^{n-2}$ and one of $\\1, SU(k), Sp(1), G_2$ and $Spin(7)$."}
{"category": "Math", "title": "Self-similar and self-affine sets; measure of the intersection of two copies", "abstract": "Let K be a self-similar or self-affine set in R^d, let \\mu be a self-similar or self-affine measure on it, and let G be the group of affine maps, similitudes, isometries or translations of R^d. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpinski sponge) we prove theorems of the following types, which are closely related to each other; Non-stability: There exists a constant c<1 such that for every g\\in G we have either \\mu(K\\cap g(K)) <c \\mu(K) or K\\subset g(K). Measure and topology: For every g\\in G we have \\mu(K\\cap g(K)) > 0 \\iff int_K (K\\cap g(K)) is nonempty (where int_K is interior relative to K). Extension: The measure \\mu has a G-invariant extension to R^d. Moreover, in many situations we characterize those g's for which \\mu(K\\cap g(K) > 0, and we also get results about those $g$'s for which $g(K)\\su K$ or $g(K)\\supset K$ holds."}
{"category": "Math", "title": "Catalan's intervals and realizers of triangulations", "abstract": "The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable orders defined on the set of Catalan objects of a given size. These lattices are ordered by inclusion: the Stanley lattice is an extension of the Tamari lattice which is an extension of the Kreweras lattice. The Stanley order can be defined on the set of Dyck paths of size $n$ as the relation of \\emph{being above}. Hence, intervals in the Stanley lattice are pairs of non-crossing Dyck paths. In a former article, the second author defined a bijection $\\Phi$ between pairs of non-crossing Dyck paths and the realizers of triangulations (or Schnyder woods). We give a simpler description of the bijection $\\Phi$. Then, we study the restriction of $\\Phi$ to Tamari's and Kreweras' intervals. We prove that $\\Phi$ induces a bijection between Tamari intervals and minimal realizers. This gives a bijection between Tamari intervals and triangulations. We also prove that $\\Phi$ induces a bijection between Kreweras intervals and the (unique) realizers of stack triangulations. Thus, $\\Phi$ induces a bijection between Kreweras intervals and stack triangulations which are known to be in bijection with ternary trees."}
{"category": "Math", "title": "Torsion units in integral group ring of the Mathieu simple group M22", "abstract": "We investigate the possible character values of torsion units of the normalized unit group of the integral group ring of Mathieu sporadic group $M_{22}$. We confirm the Kimmerle conjecture on prime graphs for this group and specify the partial augmentations for possible counterexamples to the stronger Zassenhaus conjecture."}
{"category": "Math", "title": "Contractible Lie groups over local fields", "abstract": "Let G be a Lie group over a local field of positive characteristic which admits a contractive automorphism f (i.e., the forward iterates f^n(x) of each group element x converge to the neutral element 1). We show that then G is a torsion group of finite exponent and nilpotent. We also obtain results concerning the interplay between contractive automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon. Some of the results even extend to Lie groups over arbitrary complete ultrametric fields."}
{"category": "Math", "title": "The General Form Of Cyclic Orthonormal Generators In R^N", "abstract": "In this paper we give a definition of cyclic orthonormal generators (cogs) in R^N. We give a general canonical form for their expression. Further, we give an explicit formula for computing the canonical form of any given cog."}
{"category": "Math", "title": "Kazhdan and Haagerup properties from the median viewpoint", "abstract": "We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affine isometric actions on $L^p$-spaces. It also allows us to answer an open problem on a dynamical characterization of property (T), generalizing results of Robertson-Steger."}
{"category": "Math", "title": "Generalizing the notion of Koszul algebra", "abstract": "We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class."}
{"category": "Math", "title": "Large Deviations for Partition Functions of Directed Polymers and Some Other Models in an IID Field", "abstract": "Consider the partition function of a directed polymer in an IID field. We assume that both tails of the negative and the positive part of the field are at least as light as exponential. It is a well-known fact that the free energy of the polymer is equal to a deterministic constant for almost every realization of the field and that the upper tail of the large deviations is exponential. The lower tail of the large deviations is typically lighter than exponential. In this paper we provide a method to obtain estimates on the rate of decay of the lower tail of the large deviations, which are sharp up to multiplicative constants. As a consequence, we show that the lower tail of the large deviations exhibits three regimes, determined according to the tail of the negative part of the field. Our method is simple to apply and can be used to cover other oriented and non-oriented models including first/last-passage percolation and the parabolic Anderson model"}
{"category": "Math", "title": "Gorenstein algebras and Hochschild cohomology", "abstract": "For homomorphism K-->S of commutative rings, where K is Gorenstein and S is essentially of finite type and flat as a K-module, the property that all non-trivial fiber rings of K-->S are Gorenstein is characterized in terms of properties of the cohomology modules Ext_n^{S\\otimes_KS}S{S\\otimes_KS}."}
{"category": "Math", "title": "A new proof of the Beez-Cartan theorem", "abstract": "This paper has been withdrawn by the author."}
{"category": "Math", "title": "Dynamic programming principle for one kind of stochastic recursive optimal control problem and Hamilton-Jacobi-Bellman equations", "abstract": "In this paper, we study one kind of stochastic recursive optimal control problem with the obstacle constraints for the cost function where the cost function is described by the solution of one reflected backward stochastic differential equations. We will give the dynamic programming principle for this kind of optimal control problem and show that the value function is the unique viscosity solution of the obstacle problem for the corresponding Hamilton-Jacobi-Bellman equations."}
{"category": "Math", "title": "On the Theory of Colorful Graphs", "abstract": "The theory of colorful graphs can be developed by working in Galois field modulo (p), p > 2 and a prime number. The paper proposes a program of possible conversion of graph theory into a pleasant colorful appearance. We propose to paint the usual black (indicating presence of an edge) and white (indicating absence of an edge) edges of graphs using multitude of colors and study their properties. All colorful graphs considered here are simple, i.e. not having any multiple edges or self-loops. This paper is an invitation to the program of generalizing usual graph theory in this direction."}
{"category": "Math", "title": "Congruent numbers, elliptic curves, and the passage from the local to the global", "abstract": "The ancient unsolved problem of congruent numbers has been reduced to one of the major questions of contemporary arithmetic: the finiteness of the number of curves over $\\bf Q$ which become isomorphic at every place to a given curve. We give an elementary introduction to congruent numbers and their conjectural characterisation, discuss local-to-global issues leading to the finiteness problem, and list a few results and conjectures in the arithmetic theory of elliptic curves."}
{"category": "Math", "title": "On the smoothness of H\\\"older-doubling measures", "abstract": "In this paper we consider the question of whether the doubling character of a measure supported on a subset of $\\RR^m$ determines the regularity of its support (in a classical sense). This problem was studied by David, Kenig and Toro for codimension 1 sets under the assumption that the support be flat. Here we study the higher codimension case and remove the flatness hypothesis."}
{"category": "Math", "title": "Microfractured media with a scale and Mumford-Shah energies", "abstract": "We want to understand he concentration of damage in microfractured elastic media. Due to the different scallings of the volume and area (or area and length in two dimensions) the traditional method of homogenization using periodic arrays of cells seems to fail when applied to the Mumford-Shah functional and to periodically fractured domains. In the present paper we are departing from traditional homogenization. The main result implies the use of Mumford-Shah energies and leads to an explanation of the observed concentration of damage in microfractured elastic bodies."}
{"category": "Math", "title": "Relative Cuntz-Pimsner Algebras, Partial Isometric Crossed Products and Reduction of Relations", "abstract": "The article discusses the interrelation between relative Cuntz-Pimsner algebras and partial isometric crossed products, and presents a procedure that reduces any given Hilbert bimodule to the \"smallest\" Hilbert bimodule yielding the same relative Cuntz-Pimsner algebra as the initial one. In the context of crossed products this reduction procedure corresponds to reduction of C*-dynamical systems."}
{"category": "Math", "title": "Weyl modules for the twisted loop algebras", "abstract": "The notion of a Weyl module, previously defined for the untwisted affine algebras, is extended here to the twisted affine algebras. We describe an identification of the Weyl modules for the twisted affine algebras with suitably chosen Weyl modules for the untwisted affine algebras. This identification allows us to use known results in the untwisted case to compute the dimensions and characters of the Weyl modules for the twisted algebras."}
{"category": "Math", "title": "An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age structured equations", "abstract": "For monotone linear differential systems with periodic coefficients, the (first) Floquet eigenvalue measures the growth rate of the system. We define an appropriate arithmetico-geometric time average of the coefficients for which we can prove that the Perron eigenvalue is smaller than the Floquet eigenvalue. We apply this method to Partial Differential Equations, and we use it for an age-structured systems of equations for the cell cycle. This opposition between Floquet and Perron eigenvalues models the loss of circadian rhythms by cancer cells."}
{"category": "Math", "title": "Compositions of Graphs Revisited", "abstract": "The idea of graph compositions, which was introduced by A. Knopfmacher and M. E. Mays, generalizes both ordinary compositions of positive integers and partitions of finite sets. In their original paper they developed formulas, generating functions, and recurrence relations for composition counting functions for several families of graphs. Here we show that some of the results involving compositions of bipartite graphs can be derived more easily using exponential generating functions."}
{"category": "Math", "title": "Recovery of edges from spectral data with noise -- a new perspective", "abstract": "We consider the problem of detecting edges in piecewise smooth functions from their N-degree spectral content, which is assumed to be corrupted by noise. There are three scales involved: the \"smoothness\" scale of order 1/N, the noise scale of order $\\eta$ and the O(1) scale of the jump discontinuities. We use concentration factors which are adjusted to the noise variance, $\\eta$ >> 1/N, in order to detect the underlying O(1)-edges, which are separated from the noise scale, $\\eta$ << 1."}
{"category": "Math", "title": "Word length in surface groups with characteristic generating sets", "abstract": "A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length of any element with nonzero crossing number is positive, and bounded below by a constant depending only (and explicitly) on a bound on the crossing numbers of generating elements. This answers a question of B. Farb."}
{"category": "Math", "title": "Produit d'entrelacement et action triangulaire d'alg\\`ebres de Lie", "abstract": "Formal actions of Lie algebras over vector spaces are introduced in a purely algebraic way, as a mimic of infinitesimal operations of Banach Lie algebras over Banach analytic manifolds. In analogy with the case of abstract groups, complete wreath products and triangular actions are then defined for Lie algebras acting \"en cascade\" over vector spaces. Finally, a Kaloujnine-Krasner type theorem for Lie algebra extensions is proved. ----- En mimant les lois d'op\\'erations infinit\\'esimales des alg\\`ebres de Lie sur les vari\\'et\\'e s analytiques banachiques, on introduit de mani\\`ere purement alg\\`ebrique la notion d'action formelle d'une alg\\`ebre de Lie sur un espace vectoriel. Ensuite, par analogie avec le cas des groupes abstraits, et en faisant op\\'erer les alg\\`ebres de Lie \"en cascade\", on d\\'efinit produit d'entrelacement (\"wreath product\") et action triangulaire pour les alg\\`ebres de Lie. On d\\'emontre enfin un th\\'eor\\`eme du type Kaloujnine-Krasner pour les extensions d'alg\\`ebres de Lie."}
{"category": "Math", "title": "Characterizing Sparse Graphs by Map Decompositions", "abstract": "A {\\bf map} is a graph that admits an orientation of its edges so that each vertex has out-degree exactly 1. We characterize graphs which admit a decomposition into $k$ edge-disjoint maps after: (1) the addition of {\\it any} $\\ell$ edges; (2) the addition of {\\it some} $\\ell$ edges. These graphs are identified with classes of {\\it sparse} graphs; the results are also given in matroidal terms."}
{"category": "Math", "title": "Unstable structures definable in o-minimal theories", "abstract": "Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the structure induced from N is o-minimal. In particular X is linearly ordered. As part of the proof we show: Theorem 1: If the M-dimenson of N is 1 then any 1-N-type is either strongly stable or finite by o-minimal. Theorem 2: If N is N-minimal then it is 1-M-dimensional."}
{"category": "Math", "title": "C_{0}-Hilbert Modules", "abstract": "We provide the definition and fundamental properties of algebraic elements with respect to an operator satisfying hypothesis (h). Furthermore, we analyze Hilbert modules using C_0-operators relative to a bounded finitely connected region Omega in the complex plane."}
{"category": "Math", "title": "An Integrated Human-Computer System for Controlling Interstate Disputes", "abstract": "In this paper we develop a scientific approach to control inter-country conflict. This system makes use of a neural network and a feedback control approach. It was found that by controlling the four controllable inputs: Democracy, Dependency, Allies and Capability simultaneously, all the predicted dispute outcomes could be avoided. Furthermore, it was observed that controlling a single input Dependency or Capability also avoids all the predicted conflicts. When the influence of each input variable on conflict is assessed, Dependency, Capability, and Democracy emerge as key variables that influence conflict."}
{"category": "Math", "title": "A converse to the Second Whitehead Lemma", "abstract": "We show that finite-dimensional Lie algebras over a field of characteristic zero such that the second cohomology group in every finite-dimensional module vanishes, are, essentially, semisimple."}
{"category": "Math", "title": "Sharp $L^1$ estimates for singular transport equations", "abstract": "We provide $L^1$ estimates for a class of transport equations containing singular integral operators. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates is of a more general interest."}
{"category": "Math", "title": "The integrals in Gradshteyn and Rhyzik. Part 1: A family of logarithmic integrals", "abstract": "We present the evaluation of a family of logarithmic integrals. This provides a unified proof of several formulas in the classical table of integrals by I. S. Gradshteyn and I. M. Rhyzik."}
{"category": "Math", "title": "The integrals in Gradshteyn and Rhyzik. Part 2: Elementary logarithmic integrals", "abstract": "We describe methods to evaluate elementary logarithmic integrals. The integrand is the product of a rational function and a linear polynomial in ln x."}
{"category": "Math", "title": "Controlled Lagrangians and Stabilization of Discrete Mechanical Systems I", "abstract": "Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria and equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. In particular, to make the discrete theory effective, one can make an appropriate selection of momentum levels or, alternatively, introduce a new parameter into the controlled Lagrangian to complete the kinetic matching procedure. Specifically, new terms in the controlled shape equation that are necessary for potential matching in the discrete setting are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline. The paper also discusses digital and model predictive controllers."}
{"category": "Math", "title": "Groups generated by 3-state automata over a 2-letter alphabet, II", "abstract": "Classification of groups generated by 3-state automata over a 2-letter alphabet started in the first paper (see http://www.arxiv.org/abs/math/0612178) is continued."}
{"category": "Math", "title": "On Classification of Finite Dimensional Complex Filiform Leibniz Algebras (Part 2)", "abstract": "The paper is devoted to classification problem of finite dimensional complex none Lie filiform Leibniz algebras. Actually, the observations show there are two resources to get classification of filiform Leibniz algebras. The first of them is naturally graded none Lie filiform Leibniz algebras and the another one is naturally graded filiform Lie algebras. Using the first resource we get two disjoint classes of filiform Leibniz algebras. The present paper deals with the second of the above two classes, the first class has been considered in our previous paper. The algebraic classification here means to specify the representatives of the orbits, whereas the geometric classification is the problem of finding generic structural constants in the sense of algebraic geometry. Our main effort in this paper is the algebraic classification. We suggest here an algebraic method based on invariants. Utilizing this method for any given low dimensional case all filiform Leibniz algebras can be classified. Moreover, the results can be used for geometric classification of orbits of such algebras."}
{"category": "Math", "title": "Manifolds admitting stable forms", "abstract": "In this note we give a direct method to classify all stable forms on $\\R^n$ as well as to determine their automorphism groups. We show that in dimension 6,7,8 stable forms coincide with non-degnerate forms. We present necessary conditions and sufficient conditions for a manifold to admit a stable form. We also discuss rich properties of the geometry of such manifolds."}
{"category": "Math", "title": "Parameterized Gromov-Witten invariants and topology of symplectomorphism groups", "abstract": "In this note we introduce parameterized Gromov-Witten invariants for symplectic fiber bundles and study the topology of the symplectomorphism group. We also give sample applications showing the non-triviality of certain homotopy groups of some symplectomorphism groups."}
{"category": "Math", "title": "Existence and symmetry of minimizers for nonconvex radially symmetric variational problems", "abstract": "Nonconvex functionals with spherical symmetry are studied. Existence of one and radial symmetry of all global minimizers is shown with an approach based on convex relaxation."}
{"category": "Math", "title": "On the number of collisions in $\\Lambda$-coalescents", "abstract": "We examine the total number of collisions $C_n$ in the $\\Lambda$-coalescent process which starts with $n$ particles. A linear growth and a stable limit law for $C_n$ are shown under the assumption of a power-like behaviour of the measure $\\Lambda$ near 0 with exponent $0<\\alpha<1$."}
{"category": "Math", "title": "An abundance of invariant polynomials satisfying the Riemann hypothesis", "abstract": "In 1999, Iwan Duursma defined the zeta function for a linear code as a generating function of its Hamming weight enumerator. It can also be defined for other homogeneous polynomials not corresponding to existing codes. If the homogeneous polynomial is invariant under the MacWilliams transform, then its zeta function satisfies a functional equation and we can formulate an analogue of the Riemann hypothesis. As far as existing codes are concerned, the Riemann hypothesis is believed to be closely related to the extremal property. In this article, we show there are abundant polynomials invariant by the MacWilliams transform which satisfy the Riemann hypothesis. The proof is carried out by explicit construction of such polynomials. To prove the Riemann hypothesis for a certain class of invariant polynomials, we establish an analogue of the Enestr\"om-Kakeya theorem."}
{"category": "Math", "title": "Distal actions and ergodic actions on compact groups", "abstract": "Let $K$ be a compact metrizable group and $\\Ga$ be a group of automorphisms of $K$. We first show that each $\\ap \\in \\Ga$ is distal on $K$ implies $\\Ga$ itself is distal on $K$, a local to global correspondence provided $\\Ga$ is a generalized $\\ov{FC}$-group or $K$ is a connected finite-dimensional group. We show that $\\Ga$ contains an ergodic automorphism when $\\Ga$ is nilpotent and ergodic on a connected finite-dimensional compact abelian group $K$."}
{"category": "Math", "title": "Quasi elementary contractions of Fano manifolds", "abstract": "Let X be a smooth complex Fano variety. We define and study 'quasi elementary' contractions of fiber type f: X -> Y. These have the property that rho(X) is at most rho(Y)+rho(F), where rho is the Picard number and F is a general fiber of f. In particular any elementary extremal contraction of fiber type is quasi elementary. We show that if Y has dimension at most 3 and Picard number at least 4, then Y is smooth and Fano; if moreover rho(Y) is at least 6, then X is a product. This yields sharp bounds on rho(X) when dim(X)=4 and X has a quasi elementary contraction, and other applications in higher dimensions."}
{"category": "Math", "title": "The Gehring Lemma in Metric Spaces", "abstract": "We present a proof for the Gehring lemma in a metric measure space endowed with a doubling measure. As an application we show the self improving property of Muckenhoupt weights."}
{"category": "Math", "title": "Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrodinger system", "abstract": "In this paper, we establish two new types of invariant sets for the coupled nonlinear Schrodinger system on $\\mathbb{R}^n$, and derive two sharp thresholds of blow-up and global existence for its solution. Some analogous results for the nonlinear Schrodinger system posed on the hyperbolic space $\\mathbb{H}^n$ and on the standard 2-sphere $\\mathbb{S}^2$ are also presented. Our arguments and constructions are improvements of some previous works on this direction. At the end, we give some heuristic analysis about the strong instability of the solitary waves."}
{"category": "Math", "title": "Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump", "abstract": "We consider a one-dimensional jumping Markov process $\\{X^x_t\\}_{t \\geq 0}$, solving a Poisson-driven stochastic differential equation. We prove that the law of $X^x_t$ admits a smooth density for $t>0$, under some regularity and non-degeneracy assumptions on the coefficients of the S.D.E. To our knowledge, our result is the first one including the important case of a non-constant rate of jump. The main difficulty is that in such a case, the map $x \\mapsto X^x_t$ is not smooth. This seems to make impossible the use of Malliavin calculus techniques. To overcome this problem, we introduce a new method, in which the propagation of the smoothness of the density is obtained by analytic arguments."}
{"category": "Math", "title": "Subgroups of direct products of limit groups", "abstract": "If $G_1,...,G_n$ are limit groups and $S\\subset G_1\\times...\\times G_n$ is of type $\\FP_n(\\mathbb Q)$ then $S$ contains a subgroup of finite index that is itself a direct product of at most $n$ limit groups. This settles a question of Sela."}
{"category": "Math", "title": "Multifractal Analysis of inhomogeneous Bernoulli products", "abstract": "We are interested to the multifractal analysis of inhomogeneous Bernoulli products which are also known as coin tossing measures. We give conditions ensuring the validity of the multifractal formalism for such measures. On another hand, we show that these measures can have a dense set of phase transitions."}
{"category": "Math", "title": "An obstruction to a knot being deform-spun via Alexander polynomials", "abstract": "We show that if a co-dimension two knot is deform-spun from a lower-dimensional co-dimension 2 knot, there are constraints on the Alexander polynomials. In particular this shows, for all n, that not all co-dimension 2 knots in S^n are deform-spun from knots in S^{n-1}."}
{"category": "Math", "title": "Excision for K-theory of connective ring spectra", "abstract": "We extend Geisser and Hesselholt's result on ``bi-relative K-theory'' from discrete rings to connective ring spectra. That is, if $\\mathcal A$ is a homotopy cartesian $n$-cube of ring spectra (satisfying connectivity hypotheses), then the $(n+1)$-cube induced by the cyclotomic trace $$K(\\mathcal A)\\to TC(\\mathcal A)$$ is homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision."}
{"category": "Math", "title": "On the necessity of new ramification breaks", "abstract": "Ramification invariants are necessary, but not in general sufficient, to determine the Galois module structure of ideals in local number field extensions. This insufficiency is associated with elementary abelian extensions, where one can define a refined ramification filtration -- one with more ramification breaks [JNTB 17 (2005)]. The first refined break number comes from the usual ramification filtration and is therefore necessary. Here we study the second refined break number."}
{"category": "Math", "title": "Complex asymptotics of Poincar\\'e functions and properties of Julia sets", "abstract": "The asymptotic behaviour of the solutions of Poincar\\'e's functional equation $f(\\lambda z)=p(f(z))$ ($\\lambda>1$) for $p$ a real polynomial of degree $\\geq2$ is studied in angular regions of the complex plain. The constancy of an occurring periodic function is characterised in terms of geometric properties of the Julia set of $p$. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of $f$ is related to the harmonic measure on the Julia set of $p$."}
{"category": "Math", "title": "Un th\\'eor\\`eme de Beilinson-Bernstein pour les D-modules arithm\\'etiques", "abstract": "One proves a Beilinson-Bernstein theorem in the context of arithmetic D-modules introduced by Berthelot, for flag varieties. This generalizes in the arithmetic context previous results of Brylinski-Kashiwara and Beilinson-Bernstein in the complex case."}
{"category": "Math", "title": "Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm", "abstract": "Suppose $\\Gamma$ is a Carleson Jordan curve with logarithmic whirl points, $\\varrho$ is a Khvedelidze weight, $p:\\Gamma\\to(1,\\infty)$ is a continuous function satisfying $|p(\\tau)-p(t)|\\le -\\mathrm{const}/\\log|\\tau-t|$ for $|\\tau-t|\\le 1/2$, and $L^{p(\\cdot)}(\\Gamma,\\varrho)$ is a weighted generalized Lebesgue space with variable exponent. We prove that all semi-Fredholm operators in the algebra of singular integral operators with $N\\times N$ matrix piecewise continuous coefficients are Fredholm on $L_N^{p(\\cdot)}(\\Gamma,\\varrho)$."}
{"category": "Math", "title": "Lawvere completeness in Topology", "abstract": "It is known since 1973 that Lawvere's notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for $(\\mathbb{T},\\mathsf{V})$-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that $\\mathsf{V}$ has a canonical $(\\mathbb{T},\\mathsf{V})$-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of $(\\mathbb{T},\\mathsf{V})$-categories."}
{"category": "Math", "title": "A construction of generalized Harish-Chandra modules for locally reductive Lie algebras", "abstract": "We study cohomological induction for a pair $(\\frak g,\\frak k)$, $\\frak g$ being an infinite dimensional locally reductive Lie algebra and $\\frak k \\subset\\frak g$ being of the form $\\frak k_0 + C_\\gg(\\frak k_0)$, where $\\frak k_0\\subset\\frak g$ is a finite dimensional reductive in $\\frak g$ subalgebra and $C_{\\gg} (\\frak k_0)$ is the centralizer of $\\frak k_0$ in $\\frak g$. We prove a general non-vanishing and $\\frak k$-finiteness theorem for the output. This yields in particular simple $(\\frak g,\\frak k)$-modules of finite type over $\\frak k$ which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in \\cite{PZ1} and \\cite{PZ2}. We study explicit versions of the construction when $\\frak g$ is a root-reductive or diagonal locally simple Lie algebra."}
{"category": "Math", "title": "Three Applications of the Cuntz Semigroup", "abstract": "Building on work of Elliott and coworkers, we present three applications of the Cuntz semigroup: (i) for many simple C$^*$-algebras, the Thomsen semigroup is recovered functorially from the Elliott invariant, and this yields a new proof of Elliott's classification theorem for simple, unital AI algebras; (ii) for the algebras in (i), classification of their Hilbert modules is similar to the von Neumann algebra context; (iii) for the algebras in (i), approximate unitary equivalence of self-adjoint operators is characterised in terms of the Elliott invariant."}
{"category": "Math", "title": "Instability of an equilibrium of a partial differential equation", "abstract": "A nonlinear parabolic differential equation with a quadratic nonlinearity is presented which has at least one equilibrium. The linearization about this equilibrium is asymptotically stable, but by using a technique inspired by H. Fujita, we show that the equilibrium is unstable in the nonlinear setting. The perturbations used have the property that they are small in every $L^p$ norm, yet they result in solutions which fail to be global."}
{"category": "Math", "title": "Metric Properties of Conflict Sets", "abstract": "In this paper we show that the tangent cone of a conflict set in $R^n$ is a linear affine cone over a conflict set of smaller dimension and has dimension $n-1$. Moreover we give an example where the conflict sets is not normally embedded and not locally bi-Lipschitz equivalent to the corresponding tangent cone."}
{"category": "Math", "title": "Covers of Elliptic Curves and the Lower Bound for Slopes of Effective Divisors on $\\bar{\\mathcal M}_{g}$", "abstract": "Consider genus $g$ curves that admit degree $d$ covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family $Y$ that naturally maps into the moduli space of stable genus $g$ curves $\\bar{\\mathcal M}_{g}$. We study the geometry of $Y$, and produce a combinatorial method by which to investigate its slope, irreducible components, genus and orbifold points. As a by-product of our approach, we find some equalities from classical number theory. Moreover, a correspondence between our method and the viewpoint of square-tiled surfaces is established. We also use our results to study the lower bound for slopes of effective divisors on $\\bar{\\mathcal M}_{g}$."}
{"category": "Math", "title": "Polynomial cocycles of Alexander quandles and applications", "abstract": "Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions."}
{"category": "Math", "title": "Long Borel Hierarchies", "abstract": "We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length $\\omega_2$. This implies that $\\omega_1$ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has length any given limit ordinal less than $\\omega_2$, e.g., $\\omega$ or $\\omega_1+\\omega_1$. Latex2e: 24 pages plus 8 page appendix Latest version at: www.math.wisc.edu/~miller"}
{"category": "Math", "title": "The Steenrod problem of realizing polynomial cohomology rings", "abstract": "In this paper we completely classify which graded polynomial R-algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental case R = Z, our result states that the only polynomial cohomology rings over Z which can occur, are tensor products of copies of H^*(CP^\\infty;Z) = Z[x_2], H^*(BSU(n);Z) = Z[x_4,x_6,...,x_{2n}], and H^*(BSp(n):Z) = Z[x_4,x_8,...,x_{4n}] confirming an old conjecture. Our classification extends Notbohm's solution for R = F_p, p odd. Odd degree generators, excluded above, only occur if R is an F_2-algebra and in that case the recent classification of 2-compact groups by the authors can be used instead of the present paper. Our proofs are short and rely on the general theory of p-compact groups, but not on classification results for these."}
{"category": "Math", "title": "Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)", "abstract": "This paper is dedicated to triangulated categories endowed with weight structures (a new notion; D. Pauksztello has independently introduced them as co-t-structures). This axiomatizes the properties of stupid truncations of complexes in $K(B)$. We also construct weight structures for Voevodsky's categories of motives and for various categories of spectra. A weight structure $w$ defines Postnikov towers of objects; these towers are canonical and functorial 'up to morphisms that are zero on cohomology'. For $Hw$ being the heart of $w$ (in $DM_{gm}$ we have $Hw=Chow$) we define a canonical conservative 'weakly exact' functor $t$ from our $C$ to a certain weak category of complexes $K_w(Hw)$. For any (co)homological functor $H:C\\to A$ for an abelian $A$ we construct a weight spectral sequence $T:H(X^i[j])\\implies H(X[i+j])$ where $(X^i)=t(X)$; it is canonical and functorial starting from $E_2$. This spectral sequences specializes to the 'usual' (Deligne's) weight spectral sequences for 'classical' realizations of motives and to Atiyah-Hirzebruch spectral sequences for spectra. Under certain restrictions, we prove that $K_0(C)\\cong K_0(Hw)$ and $K_0(End C)\\cong K_0(End Hw)$. The definition of a weight structure is almost dual to those of a t-structure; yet several properties differ. One can often construct a certain $t$-structure which is 'adjacent' to $w$ and vice versa. This is the case for the Voevodsky's $DM^{eff}_-$ (one obtains certain new Chow weight and t-structures for it; the heart of the latter is 'dual' to $Chow^{eff}$) and for the stable homotopy category. The Chow t-structure is closely related to unramified cohomology."}
{"category": "Math", "title": "When does a satellite knot fiber?", "abstract": "Necessary and sufficient conditions are given for a satellite knot to be fibered. Any knot $\\tilde k$ embeds in an unknotted solid torus $\\tilde V$ with arbitrary winding number in such a way that no satellite knot with pattern $(\\tilde V, \\tilde k)$ is fibered. In particular, there exist nonfibered satellite knots with fibered pattern and companion knots and nonzero winding number."}
{"category": "Math", "title": "Swan conductors for p-adic differential modules, II: Global variation", "abstract": "Using a local construction from a previous paper, we exhibit a numerical invariant, the differential Swan conductor, for an isocrystal on a variety over a perfect field of positive characteristic overconvergent along a boundary divisor; this leads to an analogous construction for certain p-adic and l-adic representations of the etale fundamental group of a variety. We then demonstrate some variational properties of this definition for overconvergent isocrystals, paying special attention to the case of surfaces."}
{"category": "Math", "title": "Ergodic Theory: Recurrence", "abstract": "We survey the impact of the Poincar\\'e recurrence principle in ergodic theory, especially as pertains to the field of ergodic Ramsey theory."}
{"category": "Math", "title": "Three remarks on one dimensional bi-Lipschitz conjugacies", "abstract": "We show that bi-Lipschitz conjugacies between non singular one dimensional systems are forced to be smooth, at least in the minimal (and ergodic) case. This is however far from being true in the non minimal case. These results clarify a classical work by Ghys and Tsuboi."}
{"category": "Math", "title": "Prime Graphs and Exponential Composition of Species", "abstract": "In this paper, we enumerate prime graphs with respect to the Cartesian multiplication of graphs. We use the unique factorization of a connected graph into the product of prime graphs given by Sabidussi to find explicit formulas for labeled and unlabeled prime graphs. In the case of species, we construct the exponential composition of species based on the arithmetic product of species of Maia and M\\'endez and the quotient species, and express the species of connected graphs as the exponential composition of the species of prime graphs."}
{"category": "Math", "title": "On the decay properties of solutions to a class of Schr\\\"odinger equations", "abstract": "We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrodinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate involving the projections P+ and P- onto the positive and negative frequencies."}
{"category": "Math", "title": "Enumeration of Point-Determining Graphs", "abstract": "Point-determining graphs are graphs in which no two vertices have the same neighborhoods, co-point-determining graphs are those whose complements are point-determining, and bi-point-determining graphs are those both point-determining and co-point-determining. Bicolored point-determining graphs are point-determining graphs whose vertices are properly colored with white and black. We use the combinatorial theory of species to enumerate these graphs as well as the connected cases."}
{"category": "Math", "title": "Brundan-Kazhdan-Lusztig and super duality conjectures", "abstract": "We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was initiated by Brundan. We show that the Brundan-Kazhdan-Lusztig (BKL) polynomials for Lie superalgebra gl(m|n) in our parabolic setup can be identified with the usual parabolic Kazhdan-Lusztig polynomials. We establish some special cases of the BKL conjecture on the parabolic category O of gl(m|n)-modules and additional results which support the BKL conjecture and super duality conjecture."}
{"category": "Math", "title": "A general homological Kleiman-Bertini theorem", "abstract": "Let G be a smooth algebraic group acting on a variety X. Let F and E be coherent sheaves on X. We show that if all the higher Tor sheaves of F against G-orbits vanish, then for generic g in G, the sheaf Tor^X_j(gF, E) vanishes for all j >0. This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical Kleiman-Bertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0."}
{"category": "Math", "title": "Double solid twistor spaces: the case of arbitrary signature", "abstract": "In a recent paper (math.DG/0701278) we constructed a series of new Moishezon twistor spaces which is a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on nCP^2 for arbitrary n>2, which can be regarded as a generalization of the twistor spaces of a 'double solid type' on 3CP^2 studied by Kreussler, Kurke, Poon and the author. Similarly to the twistor spaces of 'double solid type' on 3CP^2, projective models of present twistor spaces have a natural structure of double covering of a CP^2-bundle over CP^1. We explicitly give a defining polynomial of the branch divisor of the double covering whose restriction to fibers are degree four. If n>3 these are new twistor spaces, to the best of the author's knowledge. We also compute the dimension of the moduli space of these twistor spaces. Differently from math.DG/0701278, the present investigation is based on analysis of pluri-(half-)anticanonical systems of the twistor spaces."}
{"category": "Math", "title": "Arithmetic progressions of primes in short intervals", "abstract": "Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any given length."}
{"category": "Math", "title": "Semiparametric efficiency in GMM models with auxiliary data", "abstract": "We study semiparametric efficiency bounds and efficient estimation of parameters defined through general moment restrictions with missing data. Identification relies on auxiliary data containing information about the distribution of the missing variables conditional on proxy variables that are observed in both the primary and the auxiliary database, when such distribution is common to the two data sets. The auxiliary sample can be independent of the primary sample, or can be a subset of it. For both cases, we derive bounds when the probability of missing data given the proxy variables is unknown, or known, or belongs to a correctly specified parametric family. We find that the conditional probability is not ancillary when the two samples are independent. For all cases, we discuss efficient semiparametric estimators. An estimator based on a conditional expectation projection is shown to require milder regularity conditions than one based on inverse probability weighting."}
{"category": "Math", "title": "The Cox Ring of $\\bar{M}_{0,6}$", "abstract": "We prove that the Cox ring of $\\bar{M}_{0,6}$, the moduli space of stable, rational curves with 6 marked points, is finitely generated by sections corresponding to the boundary divisors and divisors which are pull-backs of the hyperelliptic locus in $\\bar{M}_3$, the moduli space of stable, genus 3 curves, via morphisms that send a 6-pointed rational curve to a curve with 3 nodes by identifying 3 pairs of points. In particular, this gives a self-contained proof of Hassett and Tschinkel's result about the effective cone of $\\bar{M}_{0,6}$ being generated by the above mentioned divisors."}
{"category": "Math", "title": "Link concordance, homology cobordism, and Hirzebruch-type defects from iterated p-covers", "abstract": "We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion which is invisible via signature invariants. Applications illustrating these features include the following: (1) There are infinitely many homology equivalent rational 3-spheres which are indistinguishable via multisignatures, eta-invariants, and L2-signatures but have distinct homology cobordism types. (2) There is an infinite family of 2-torsion (amphichiral) knots, including the figure eight knot, with non-slice iterated Bing doubles; as a special case, we give the first proof of the conjecture that the Bing double of the figure eight knot is not slice. (3) There exist infinitely many torsion elements at any depth of the Cochran-Orr-Teichner filtration of link concordance."}
{"category": "Math", "title": "Lens space surgeries on A'Campo's divide knots", "abstract": "It is proved that every knot in the major subfamilies of J. Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) plane curve as a \"divide knot\" defined by N. A'Campo in the context of singularity theory of complex curves. For each knot given by Berge's parameters, the corresponding plane curve is constructed. The surgery coefficients are also considered. Such presentations support us to study each knot itself, and the relationship among the knots in the set of lens space surgeries."}
{"category": "Math", "title": "Gauss map on the theta divisor and Green's functions", "abstract": "In an earlier paper we constructed a Cartier divisor on the theta divisor of a principally polarised abelian variety whose support is precisely the ramification locus of the Gauss map. In this note we discuss a Green's function associated to this locus. For jacobians we relate this Green's function to the canonical Green's function of the corresponding Riemann surface."}
{"category": "Math", "title": "On Hadwiger Conjecture", "abstract": "We propose an algorithm to reduce a k-chromatic graph to a complete graph of largest possible order through a well defined sequence of contractions. We introduce a new matrix called transparency matrix and state its properties. We then define correct contraction procedure to be executed to get largest possible complete graph from given connected graph. We finally give a characterization for k-chromatic graphs and use it to settle Hadwigers conjecture."}
{"category": "Math", "title": "Compact Corigid Objects in Triangulated Categories and Co-t-structures", "abstract": "In the work of Hoshino, Kato and Miyachi, the authors look at t-structures induced by a compact object, C, of a triangulated category, T, which is rigid in the sense of Iyama and Yoshino. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on T whose heart es equivalent to Mod(End(C)^op). Rigid objects in a triangulated category can be thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, S, of a triangulated category, T, induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End(S)^op), and hence an abelian subcategory of T."}
{"category": "Math", "title": "The Mathematics", "abstract": "This is an essay that considering the knowledge structure and language of a different nature, attempts to build on an explanation of the object of study and characteristics of the mathematical science. We end up with a learning cycle of mathematics and a paradigm for education, namely Learn to structure."}
{"category": "Math", "title": "Homological Epimorphisms of Differential Graded Algebras", "abstract": "Let R and S be differential graded algebras. In this paper we give a characterisation of when a differential graded R-S-bimodule M induces a full embedding of derived categories M\\otimes - :D(S)--> D(R). In particular, this characterisation generalises the theory of Geigle and Lenzing's homological epimorphisms of rings. Furthermore, there is an application of the main result to Dwyer and Greenlees's Morita theory."}
{"category": "Math", "title": "The Complexity of Orbits of Computably Enumerable Sets", "abstract": "The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, $\\E$, such that the question of membership in this orbit is $\\Sigma^1_1$-complete. This result and proof have a number of nice corollaries: the Scott rank of $\\E$ is $\\wock +1$; not all orbits are elementarily definable; there is no arithmetic description of all orbits of $\\E$; for all finite $\\alpha \\geq 9$, there is a properly $\\Delta^0_\\alpha$ orbit (from the proof). A few small corrections made in this version"}
{"category": "Math", "title": "Packing-Dimension Profiles and Fractional Brownian Motion", "abstract": "In order to compute the packing dimension of orthogonal projections Falconer and Howroyd (1997) introduced a family of packing dimension profiles ${\\rm Dim}_s$ that are parametrized by real numbers $s>0$. Subsequently, Howroyd (2001) introduced alternate $s$-dimensional packing dimension profiles $\\hbox{${\\rm P}$-$\\dim$}_s$ and proved, among many other things, that $\\hbox{${\\rm P}$-$\\dim$}_s E={\\rm Dim}_s E$ for all integers $s>0$ and all analytic sets $E\\subseteq\\R^N$. The goal of this article is to prove that $\\hbox{${\\rm P}$-$\\dim$}_s E={\\rm Dim}_s E$ for all real numbers $s>0$ and analytic sets $E\\subseteq\\R^N$. This answers a question of Howroyd (2001, p. 159). Our proof hinges on a new property of fractional Brownian motion."}
{"category": "Math", "title": "Dynamical percolation on general trees", "abstract": "H\\\"aggstr\\\"om, Peres, and Steif (1997) have introduced a dynamical version of percolation on a graph $G$. When $G$ is a tree they derived a necessary and sufficient condition for percolation to exist at some time $t$. In the case that $G$ is a spherically symmetric tree, H\\\"aggstr\\\"om, Peres, and Steif (1997) derived a necessary and sufficient condition for percolation to exist at some time $t$ in a given target set $D$. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time $t\\in D$, in the case that the underlying tree is not necessary spherically symmetric. This answers a question of Yuval Peres (personal communication). We present also a formula for the Hausdorff dimension of the set of exceptional times of percolation."}
{"category": "Math", "title": "Formality of function spaces", "abstract": "Let $X$ be a nilpotent space such that there exists $p\\geq 1$ with $H^p(X,\\mathbb Q) \\ne 0$ and $H^n(X,\\mathbb Q)=0$ if $n>p$. Let $Y$ be a m-connected space with $m\\geq p+1$ and $H^*(Y,\\mathbb Q)$ is finitely generated as algebra. We assume that $X$ is formal and there exists $p$ odd such that $H^p(X,\\mathbb Q) \\ne 0$. We prove that if the space $\\mathcal F(X,Y)$ of continuous maps from $X$ to $Y$ is formal, then $Y$ has the rational homotopy type of a product of Eilenberg Mac Lane spaces. At the opposite, we exhibit an example of a formal space $\\mathcal F(S^2,Y)$ where $Y$ is not rationally equivalent to a product of Eilenberg Mac Lane spaces."}
{"category": "Math", "title": "A duality between pairs of split decompositions for a $Q$-polynomial distance-regular graph", "abstract": "Let $\\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D \\geq 3$ and standard module $V$. Recently Ito and Terwilliger introduced four direct sum decompositions of $V$; we call these the $(\\mu,\\nu)$--{\\it split decompositions} of $V$, where $\\mu, \\nu \\in \\lbrace \\downarrow, \\uparrow \\rbrace$. In this paper we show that the ($\\downarrow,\\downarrow$)--split decomposition and the ($\\uparrow,\\uparrow$)--split decomposition are dual with respect to the standard Hermitian form on $V$. We also show that the ($\\downarrow,\\uparrow$)--split decomposition and the ($\\uparrow,\\downarrow$)--split decomposition are dual with respect to the standard Hermitian form on $V$."}
{"category": "Math", "title": "Brownian subordinators and fractional Cauchy problems", "abstract": "A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involve subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes, and consequently, an equivalence between these two families of partial differential equations."}
{"category": "Math", "title": "The spine which was no spine", "abstract": "Let T_n be the Teichmueller space of flat metrics on the n-dimensional torus and identify SL(n,Z) with the corresponding mapping class group. We prove that the subset Y consisting of those points at which the systoles generate the fundamental group of the torus is, for n > 4, not contractible. In particular, Y is not an SL(n,Z)-equivariant deformation retract of T_n."}
{"category": "Math", "title": "The integrals in Gradshteyn and Ryzhik. Part 3: Combinations of Logarithms and Exponentials", "abstract": "We present the evaluation of a family of exponential-logarithmic integrals. These have integrands of the form P(exp(x),ln(x)) where P is a polynomial. The examples presented here appear in sections 4.33, 4.34 and 4.35 in the classical table of integrals by I. Gradshteyn and I. Ryzhik."}
{"category": "Math", "title": "The integrals in Gradshteyn and Ryzhik. Part 4: The Gamma function", "abstract": "We present a systematic derivation of some definite integrals in the classical table of Gradshteyn and Ryzhik that can be reduced to the gamma function."}
{"category": "Math", "title": "Normalizers of Irreducible Subfactors", "abstract": "We consider normalizers of an irreducible inclusion $N\\subseteq M$ of $\\mathrm{II}_1$ factors. In the infinite index setting an inclusion $uNu^*\\subseteq N$ can be strict, forcing us to also investigate the semigroup of one-sided normalizers. We relate these normalizers of $N$ in $M$ to projections in the basic construction and show that every trace one projection in the relative commutant $N'\\cap < M,e_N>$ is of the form $u^*e_Nu$ for some unitary $u\\in M$ with $uNu^*\\subseteq N$. This enables us to identify the normalizers and the algebras they generate in several situations. In particular each normalizer of a tensor product of irreducible subfactors is a tensor product of normalizers modulo a unitary. We also examine normalizers of irreducible subfactors arising from subgroup--group inclusions $H\\subseteq G$. Here the normalizers are the normalizing group elements modulo a unitary from $L(H)$. We are also able to identify the finite trace $L(H)$-bimodules in $\\ell^2(G)$ as double cosets which are also finite unions of left cosets."}
{"category": "Math", "title": "Geodesics on an ellipsoid in Minkowski space", "abstract": "We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the time- and space-like geodesics and invariant 1-form on the space of null geodesics. We prove a Poncelet-type theorem for null geodesics on the ellipsoid: if such a geodesic close up after several oscillations in the \"pseudo-Riemannian belt\", so do all other null geodesics on this ellipsoid."}
{"category": "Math", "title": "Cohomology of line bundles on compactified Jacobians", "abstract": "Let C be an integral projective curve with surficial singularities. We prove that topologically trivial line bundles on the compactified Jacobian of C are in one-to-one correspondence with line bundles on C (the autoduality conjecture), and compute the cohomology of the line bundles. We also show that the natural Fourier-Mukai functor between the derived categories of quasi-coherent sheaves on the Jacobian and on the compactified Jacobian is fully faithful."}
{"category": "Math", "title": "On Solving General Linear Equations in the Set of Natural Numbers", "abstract": "In this paper one shows if the number of natural solutions of a general linear equation is limited or not. Also, it is presented a method of solving the Diophantine equation $ax-by=c$ in the set of natural numbers, and an example of solving in $N$ a Diophantine equation with three variables."}
{"category": "Math", "title": "Asymptotics for eigenvalues of a non-linear integral system", "abstract": "We show the asymptotic behavior of the eigenvalues of the non-linear integral system related to the (p,q)-Laplacian."}
{"category": "Math", "title": "1-Factorizations of Cayley graphs", "abstract": "In this note we prove that all connected Cayley graphs of every finite group $Q \\times H$ are 1-factorizable, where $Q$ is any non-trivial group of 2-power order and $H$ is any group of odd order."}
{"category": "Math", "title": "On the automorphism group of a possible symmetric $(81,16,3)$ design", "abstract": "In this paper we study the automorphism group of a possible symmetric $(81,16,3)$ design."}
{"category": "Math", "title": "On the Motion of Vortex Sheets with Surface Tension in the 3D Euler Equations with Vorticity", "abstract": "We prove well-posedness of vortex sheets with surface tension in the 3D incompressible Euler equations with vorticity."}
{"category": "Math", "title": "Chiral Equivariant Cohomology III", "abstract": "This is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra, and contains and generalizes the classical equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In this paper, we compute this cohomology for spheres and show that for any simple connected group G, there is a sphere with infinitely many actions of G which have distinct chiral equivariant cohomology, but identical classical equivariant cohomology. Unlike the classical case, the description of the chiral equivariant cohomology of spheres requires a substantial amount of new structural theory, which we fully develop in this paper. This includes a quasi-conformal structure, equivariant homotopy invariance, and the values of this cohomology on homogeneous spaces. These results rely on crucial features of the underlying vertex algebra valued complex that have no classical analogues."}
{"category": "Math", "title": "Support vector machine for functional data classification", "abstract": "In many applications, input data are sampled functions taking their values in infinite dimensional spaces rather than standard vectors. This fact has complex consequences on data analysis algorithms that motivate modifications of them. In fact most of the traditional data analysis tools for regression, classification and clustering have been adapted to functional inputs under the general name of functional Data Analysis (FDA). In this paper, we investigate the use of Support Vector Machines (SVMs) for functional data analysis and we focus on the problem of curves discrimination. SVMs are large margin classifier tools based on implicit non linear mappings of the considered data into high dimensional spaces thanks to kernels. We show how to define simple kernels that take into account the unctional nature of the data and lead to consistent classification. Experiments conducted on real world data emphasize the benefit of taking into account some functional aspects of the problems."}
{"category": "Math", "title": "Un r\\'esultat de consistance pour des SVM fonctionnels par interpolation spline", "abstract": "This Note proposes a new methodology for function classification with Support Vector Machine (SVM). Rather than relying on projection on a truncated Hilbert basis as in our previous work, we use an implicit spline interpolation that allows us to compute SVM on the derivatives of the studied functions. To that end, we propose a kernel defined directly on the discretizations of the observed functions. We show that this method is universally consistent."}
{"category": "Math", "title": "Multilayer Perceptron with Functional Inputs: an Inverse Regression Approach", "abstract": "Functional data analysis is a growing research field as more and more practical applications involve functional data. In this paper, we focus on the problem of regression and classification with functional predictors: the model suggested combines an efficient dimension reduction procedure [functional sliced inverse regression, first introduced by Ferr\\'e & Yao (Statistics, 37, 2003, 475)], for which we give a regularized version, with the accuracy of a neural network. Some consistency results are given and the method is successfully confronted to real-life data."}
{"category": "Math", "title": "Weakly null sequences with upper estimates", "abstract": "We prove that if $(v_i)$ is a normalized basic sequence and X is a Banach space such that every normalized weakly null sequence in X has a subsequence that is dominated by $(v_i)$, then there exists a uniform constant $C\\geq1$ such that every normalized weakly null sequence in X has a subsequence that is C-dominated by $(v_i)$. This extends a result of Knaust and Odell, who proved this for the cases in which $(v_i)$ is the standard basis for $\\ell_p$ or $c_0$."}
{"category": "Math", "title": "The centralizer of a C1 generic diffeomorphism is trivial", "abstract": "In this announcement, we describe the solution in the C1 topology to a question asked by S. Smale on the genericity of trivial centralizers: the set of diffeomorphisms of a compact connected manifold with trivial centralizer residual in Diff^1 but does not contain an open and dense subset."}
{"category": "Math", "title": "Hermitian manifolds of pointwise constant antiholomorphic sectional curvatures", "abstract": "In dimension greater than four, we prove that if a Hermitian non-Kaehler manifold is of pointwise constant antiholomorphic sectional curvatures, then it is of constant sectional curvatures."}
{"category": "Math", "title": "Concavity, Abel-transform and the Abel-inverse theorem in smooth complete toric varieties", "abstract": "We extend the usual projective Abel-Radon transform to the larger context of a smooth complete toric variety X. We define and study toric concavity attached to an algebraic splitting vector bundle on X and we prove a toric version of the Abel-inverse theorem."}
{"category": "Math", "title": "Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems", "abstract": "We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in $L_2$ for small complex $L_\\infty$ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for $k$-forms are well posed for small perturbations of block matrices."}
{"category": "Math", "title": "Arithmetic of curves over two dimensional local field", "abstract": "We study the class field theory of curve defined over two dimensional local field. The approch used here is a combination of the work of Kato-Saito, and Yoshida where the base field is one dimensional"}
{"category": "Math", "title": "A local Paley-Wiener theorem for compact symmetric spaces", "abstract": "The Fourier coefficients of a smooth $K$-invariant function on a compact symmetric space $M=U/K$ are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients"}
{"category": "Math", "title": "Forward stagewise regression and the monotone lasso", "abstract": "We consider the least angle regression and forward stagewise algorithms for solving penalized least squares regression problems. In Efron, Hastie, Johnstone & Tibshirani (2004) it is proved that the least angle regression algorithm, with a small modification, solves the lasso regression problem. Here we give an analogous result for incremental forward stagewise regression, showing that it solves a version of the lasso problem that enforces monotonicity. One consequence of this is as follows: while lasso makes optimal progress in terms of reducing the residual sum-of-squares per unit increase in $L_1$-norm of the coefficient $\\beta$, forward stage-wise is optimal per unit $L_1$ arc-length traveled along the coefficient path. We also study a condition under which the coefficient paths of the lasso are monotone, and hence the different algorithms coincide. Finally, we compare the lasso and forward stagewise procedures in a simulation study involving a large number of correlated predictors."}
{"category": "Math", "title": "Needlet algorithms for estimation in inverse problems", "abstract": "We provide a new algorithm for the treatment of inverse problems which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. Our goal is to devise an inversion procedure which has the advantages of localization and multiscale analysis of wavelet representations without losing the stability and computability of the SVD decompositions. To this end we utilize the construction of localized frames (termed \"needlets\") built upon the SVD bases. We consider two different situations: the \"wavelet\" scenario, where the needlets are assumed to behave similarly to true wavelets, and the \"Jacobi-type\" scenario, where we assume that the properties of the frame truly depend on the SVD basis at hand (hence on the operator). To illustrate each situation, we apply the estimation algorithm respectively to the deconvolution problem and to the Wicksell problem. In the latter case, where the SVD basis is a Jacobi polynomial basis, we show that our scheme is capable of achieving rates of convergence which are optimal in the $L_2$ case, we obtain interesting rates of convergence for other $L_p$ norms which are new (to the best of our knowledge) in the literature, and we also give a simulation study showing that the NEED-D estimator outperforms other standard algorithms in almost all situations."}
{"category": "Math", "title": "An analytic KAM-Theorem", "abstract": "We prove an analytic KAM-Theorem, which is used in [1], where the differential part of KAM-theory is discussed. Related theorems on analytic KAM-theory exist in the literature (e. g., among many others, [7], [8], [13]). The aim of the theorem presented here is to provide exactly the estimates needed in [1]."}
{"category": "Math", "title": "Simulation of Laser Propagation in a Plasma with a Frequency Wave Equation", "abstract": "The aim of this work is to perform numerical simulations of the propagation of a laser in a plasma. At each time step, one has to solve a Helmholtz equation in a domain which consists in some hundreds of millions of cells. To solve this huge linear system, one uses a iterative Krylov method with a preconditioning by a separable matrix. The corresponding linear system is solved with a block cyclic reduction method. Some enlightments on the parallel implementation are also given. Lastly, numerical results are presented including some features concerning the scalability of the numerical method on a parallel architecture."}
{"category": "Math", "title": "A new Cement to Glue non-conforming Grids with Robin interface conditions: the finite element case", "abstract": "We design and analyze a new non-conforming domain decomposition method based on Schwarz type approaches that allows for the use of Robin interface conditions on non-conforming grids. The method is proven to be well posed, and the iterative solver to converge. The error analysis is performed in 2D piecewise polynomials of low and high order and extended in 3D for $P_1$ elements. Numerical results in 2D illustrate the new method."}
{"category": "Math", "title": "Properties of B\\\"or\\\"oczki tilings in high dimensional hyperbolic spaces", "abstract": "We consider families of B\\\"or\\\"oczky tilings in hyperbolic space in arbitrary dimension, study some basic properties and classify all possible symmetries. In particular, it is shown that these tilings are non-crystallographic, and that there are uncountably many tilings with a fixed prototile."}
{"category": "Math", "title": "The Euler characteristic of local systems on the moduli of curves and abelian varieties of genus three", "abstract": "We show how to calculate the Euler characteristic of a local system associated to an irreducible representation of the symplectic group of genus 3 on the moduli space of curves of genus 3 and the moduli space of principally polarized abelian varieties of dimension 3."}
{"category": "Math", "title": "Higher Order Asymptotic Formulas for Traces of Toeplitz Matrices with Symbols in H\\\"older-Zygmund Spaces", "abstract": "We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to H\\\"older-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refine previous asymptotic trace formulas by Szeg\\\"o and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to B\\\"ottcher and Silbermann."}
{"category": "Math", "title": "Measure of the Julia Set of the Feigenbaum map with infinite criticality", "abstract": "We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets go to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments."}
{"category": "Math", "title": "On two conjectures for curves on $K3$ surfaces", "abstract": "We prove that the gonality among the smooth curves in a complete linear system on a $K3$ surface is constant except for the Donagi-Morrison example. This was proved by Ciliberto and Pareschi under the additional condition that the linear system is ample. As a consequence we prove that exceptional curves on $K3$ surfaces satisfy the Eisenbud-Lange-Martens-Schreyer conjecture and explicitly describe such curves. They turn out to be natural extensions of the Eisenbud-Lange-Martens-Schreyer examples of exceptional curves on $K3$ surfaces."}
{"category": "Math", "title": "Mod\\'elisations prospectives de l'occupation du sol. Le cas d'une montagne m\\'editerran\\'eenne", "abstract": "The authors apply three methods of prospective modelling to high resolution georeferenced land cover data in a Mediterranean mountain area: GIS approach, non linear parametric model and neuronal network. Land cover prediction to the latest known date is used to validate the models. In the frame of spatial-temporal dynamics in open systems results are encouraging and comparable. Correct prediction scores are about 73 %. The results analysis focuses on geographic location, land cover categories and parametric distance to reality of the residues. Crossing the three models show the high degree of convergence and a relative similitude of the results obtained by the two statistic approaches compared to the GIS supervised model. Steps under work are the application of the models to other test areas and the identification of respective advantages to develop an integrated model."}
{"category": "Math", "title": "Decomposition of spaces of distributions induced by Hermite expansions", "abstract": "Decomposition systems with rapidly decaying elements (needlets) based on Hermite functions are introduced and explored. It is proved that the Triebel-Lizorkin and Besov spaces on $\\R^d$ induced by Hermite expansions can be characterized in terms of the needlet coefficients. It is also shown that the Hermite Triebel-Lizorkin and Besov spaces are, in general, different from the respective classical spaces."}
{"category": "Math", "title": "The order of the largest complete minor in a random graph", "abstract": "Let ccl(G) denote the order of the largest complete minor in a graph G (also called the contraction clique number) and let G(n,p) denote a random graph on n vertices with edge probability p. Bollobas, Catlin and Erdos asymptotically determined ccl(G (n,p)) when p is a constant. Luczak, Pittel and Wierman gave bounds on ccl(G(n,p)) when p is very close to 1/n, i.e. inside the phase transition. Extending the results of Bollobas, Catlin and Erdos, we determine ccl(G(n,p)) quite tightly, for p>C/n where C is a large constant. If p=C/n, for an arbitrary constant C>1, then we show that asymptotically almost surely ccl(G (n,p)) is of order square-root of n. This answers a question of Krivelevich and Sudakov."}
{"category": "Math", "title": "A generalization of Vassiliev's h-principle", "abstract": "This thesis consists of two parts which share only a slight overlap. The first part is concerned with the study of ideals in the ring $C^\\infty(M,R)$ of smooth functions on a compact smooth manifold M or more generally submodules of a finitely generated $C^\\infty(M,R)$-module V. We define a topology on the space of all submodules of V of a fixed finite codimension d. Its main property is that it is compact Hausdorff and, in the case of ideals in the ring itself, it contains as a subspace the configuration space of d distinct unordered points in M and therefore gives a \"compactification\" of this configuration space. We present a concrete description of this space for low codimensions. The main focus is then put on the second part which is concerned with a generalization of Vassiliev's h-principle. This principle in its simplest form asserts that the jet prolongation map $j^r:C^\\infty(M,E)\\to\\Gamma(J^r(M,E))$, defined on the space of smooth maps from a compact manifold M to a Euclidean space E and with target the space of smooth sections of the jet bundle $J^r(M,E)$, is a cohomology isomorphism when restricted to certain \"nonsingular\" subsets (these are defined in terms of a certain subset $R\\subseteq J^r(M,E)$). Our generalization then puts this theorem in a more general setting of topological $C^\\infty(M,R)$-modules. As a reward we get a strengthening of this result asserting that all the homotopy fibres have zero homology."}
{"category": "Math", "title": "Affine surfaces with trivial Makar-Limanov invariant", "abstract": "We study the class of 2-dimensional affine k-domains R satisfying ML(R) = k, where k is an arbitrary field of characteristic zero. In particular, we obtain the following result: Let R be a localization of a polynomial ring in finitely many variables over a field of characteristic zero. If ML(R) = K for some field K included in R and such that R has transcendence degree 2 over K, then R is K-isomorphic to K[X,Y,Z]/(XY-P(Z)) for some nonconstant polynomial P(Z) in K[Z]."}
{"category": "Math", "title": "On the interpolation constant for subadditive operators in Orlicz spaces", "abstract": "Let $1\\le p<q\\le\\infty$ and let $T$ be a subadditive operator acting on $L^p$ and $L^q$. We prove that $T$ is bounded on the Orlicz space $L^\\phi$, where $\\phi^{-1}(u)=u^{1/p}\\rho(u^{1/q-1/p})$ for some concave function $\\rho$ and \\[ \\|T\\|_{L^\\phi\\to L^\\phi}\\le C\\max\\{\\|T\\|_{L^p\\to L^p},\\|T\\|_{L^q\\to L^q}\\}. \\] The interpolation constant $C$, in general, is less than 4 and, in many cases, we can give much better estimates for $C$. In particular, if $p=1$ and $q=\\infty$, then the classical Orlicz interpolation theorem holds for subadditive operators with the interpolation constant C=1. These results generalize our results for linear operators obtained in \\cite{KM01}."}
{"category": "Math", "title": "The Cuntz semigroup as an invariant for C*-algebras", "abstract": "A category is described to which the Cuntz semigroup belongs and as a functor into which it preserves inductive limits."}
{"category": "Math", "title": "Unifying derived deformation theories", "abstract": "We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy categories of DGLAs and SHLAs (L infinity algebras) considered by Kontsevich, Hinich and Manetti are equivalent, and are compatible with the derived stacks of Toen--Vezzosi and Lurie. Another application is that the cohomology groups associated to any classical deformation problem (in any characteristic) admit the same operations as Andre--Quillen cohomology."}
{"category": "Math", "title": "Merging of opinions in game-theoretic probability", "abstract": "This paper gives game-theoretic versions of several results on \"merging of opinions\" obtained in measure-theoretic probability and algorithmic randomness theory. An advantage of the game-theoretic versions over the measure-theoretic results is that they are pointwise, their advantage over the algorithmic randomness results is that they are non-asymptotic, but the most important advantage over both is that they are very constructive, giving explicit and efficient strategies for players in a game of prediction."}
{"category": "Math", "title": "The symplectic ideal and a double centraliser theorem", "abstract": "We interpret a result of S. Oehms as a statement about the symplectic ideal. We use this result to prove a double centraliser theorem for the symplectic group acting on \\bigoplus_{r=0}^s\\otimes^rV, where V is the natural module for the symplectic group. This result was obtained in characteristic zero by H. Weyl. Furthermore we use this to extend to arbitrary connected reductive groups G with simply connected derived group the earlier result of the author that the algebra K[G]^g of infinitesimal invariants in the algebra of regular functions on G is a unique factorisation domain."}
{"category": "Math", "title": "On logical characterization of henselianity", "abstract": "We give some sufficient conditions under which any valued field that admits quantifier elimination in the Macintyre language is henselian. Then, without extra assumptions, we prove that if a valued field of characteristic $(0,0)$ has a $\\Z$-group as its value group and admits quantifier elimination in the main sort of the Denef-Pas style language $\\mathcal{L}_{RRP}$ then it is henselian. In fact the proof of this suggests that a quite large class of Denef-Pas style languages is natural with respect to henselianity."}
{"category": "Math", "title": "Large deviations for multidimensional SDEs with reflection", "abstract": "The large deviations principles are established for a class of multidimensional degenerate stochastic differential equations with reflecting boundary conditions. The results include two cases where the initial conditions are adapted and anticipated."}
{"category": "Math", "title": "Moduli of toric vector bundles", "abstract": "We give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy's Law, in the sense of Vakil. The preliminary sections of the paper give a self-contained introduction to Klyachko's classification of toric vector bundles."}
{"category": "Math", "title": "Enhanced negative type for finite metric trees", "abstract": "Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities that quantify the extent of the \"strictness\" of the 1-negative type inequalities for finite metric trees. These inequalities of \"enhanced 1-negative type\" are sufficiently strong to imply that any given finite metric tree must have strict p-negative type for all values of p in an open interval that contains the number 1. Moreover, these open intervals can be characterized purely in terms of the unordered distribution of edge weights that determine the path metric on the particular tree, and are therefore largely independent of the tree's internal geometry. From these calculations we are able to extract a new non linear technique for improving lower bounds on the maximal p-negative type of certain finite metric spaces. Some pathological examples are also considered in order to stress certain technical points."}
{"category": "Math", "title": "Invariants via word for curves and fronts", "abstract": "We construct the infinite sequence of invariants for curves in surfaces by using word theory that V. Turaev introduced. For plane closed curves, we add some extra terms, e.g. the rotation number. From these modified invariants, we get the Arnold's basic invariants and some other invariants. We also express how these invariants classify plane closed curves. In addition, we consider other classes of plane curves: long curves and fronts."}
{"category": "Math", "title": "Sur l'holonomie de D-modules arithm\\'etiques associ\\'es \\`a des F-isocristaux surconvergents sur des courbes lisses", "abstract": "We show that the arithmetic D-module associated to an overconvergent F-isocrystal over a smooth curve is holonomic. We first prove that unipotent F-isocrystals are holonomic D-module by using the fact that such F-isocrystals come from logarithmic F-isocrystals. We deduce the general case from the semi-stable theorem for F-isocrystals over curves of Matsuda-Trihan which relies on the p-adic monodromy theorem independently proved by Andr\\'e, Kedlaya and Mebkhout. The main result has already been proved by D. Caro."}
{"category": "Math", "title": "Various Approaches for Predicting Land Cover in Mountain Areas", "abstract": "Using former maps, geographers intend to study the evolution of the land cover in order to have a prospective approach on the future landscape; predictions of the future land cover, by the use of older maps and environmental variables, are usually done through the GIS (Geographic Information System). We propose here to confront this classical geographical approach with statistical approaches: a linear parametric model (polychotomous regression modeling) and a nonparametric one (multilayer perceptron). These methodologies have been tested on two real areas on which the land cover is known at various dates; this allows us to emphasize the benefit of these two statistical approaches compared to GIS and to discuss the way GIS could be improved by the use of statistical models."}
{"category": "Math", "title": "Almost bi-Lipschitz embeddings and almost homogeneous sets", "abstract": "This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be embedded into a Hilbert space using an almost bi-Lipschitz mapping (bi-Lipschitz to within logarithmic corrections). The image of this set is no longer homogeneous, but `almost homogeneous'. We therefore study the problem of embedding an almost homogeneous subset $X$ of a Hilbert space $H$ into a finite-dimensional Euclidean space. In fact we show that if $X$ is a compact subset of a Banach space and $X-X$ is almost homogeneous then, for $N$ sufficiently large, a prevalent set of linear maps from $X$ into $\\Re^N$ are almost bi-Lipschitz between $X$ and its image. We are then able to use the Kuratowski embedding of $(X,d)$ into $L^\\infty(X)$ to prove a similar result for compact metric spaces."}
{"category": "Math", "title": "Reflection subgroups of Coxeter groups", "abstract": "We use geometry of Davis complex of a Coxeter group to prove the following result: if G is an infinite indecomposable Coxeter group and $H\\subset G$ is a finite index reflection subgroup then the rank of H is not less than the rank of G. This generalizes results of math/0305093. We also describe some properties of the nerves of the group and the subgroup in the case of equal ranks."}
{"category": "Math", "title": "Higher order asymptotic formulas for Toeplitz matrices with symbols in generalized H\\\"older spaces", "abstract": "We prove higher order asymptotic formulas for determinants and traces of finite block Toeplitz matrices generated by matrix functions belonging to generalized H\\\"older spaces with characteristic functions from the Bari-Stechkin class. We follow the approach of B\\\"ottcher and Silbermann and generalize their results for symbols in standard H\\\"older spaces."}
{"category": "Math", "title": "Optimal relocation strategies for spatially mobile consumers", "abstract": "We develop a model of the behaviour of a dynamically optimizing economic agent who makes consumption-saving and spatial relocation decisions. We formulate an existence result for the model, derive the necessary conditions for optimality and study the behaviour of the economic agent, focusing on the case of a wage distribution with a single maximum."}
{"category": "Math", "title": "Existence and uniqueness of optimal maps on Alexandrov spaces", "abstract": "The purpose of this paper is to show that in a finite dimensional metric space with Alexandrov's curvature bounded below, Monge's transport problem for the quadratic cost admits a unique solution."}
{"category": "Math", "title": "On a two-dimensional analog of Szemeredi's Theorem in Abelian groups", "abstract": "Let G be a finite Abelian group and A be a subset G\\times G of cardinality at least |G|^2/(log log |G|)^c, where c>0 is an absolute constant. We prove that A contains a triple {(k,m), (k+d,m), (k,m+d)}, where d does not equal 0. This theorem is a two-dimensional generalization of Szemeredi's theorem on arithmetic progressions."}
{"category": "Math", "title": "Parallel Transport and Functors", "abstract": "Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to substitute categories of functors for categories of smooth fibre bundles with connection. We indicate that this concept can be generalized to connections in categorified bundles, and how this generalization improves the understanding of higher dimensional parallel transport."}
{"category": "Math", "title": "The Dagum family of isotropic correlation functions", "abstract": "A function $\\rho:[0,\\infty)\\to(0,1]$ is a completely monotonic function if and only if $\\rho(\\Vert\\mathbf{x}\\Vert^2)$ is positive definite on $\\mathbb{R}^d$ for all $d$ and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function \\[\\rho(\\beta ,\\gamma)(x)=1-\\biggl(\\frac{x^{\\beta}}{1+x^{\\beta}}\\biggr )^{\\gamma},\\qquad x\\ge 0, \\beta,\\gamma>0,\\] called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any $d$-dimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions."}
{"category": "Math", "title": "The level 1 case of Serre's conjecture revisited", "abstract": "We prove existence of conjugate Galois representations, and we use it to derive a simple method of weight reduction. As a consequence, an alternative proof of the level 1 case of Serre's conjecture follows."}
{"category": "Math", "title": "The accessory parameter problem in positive characteristic", "abstract": "We study the existence of Fuchsian differential equations in positive characteristic with nilpotent p-curvature, and given local invariants. In the case of differential equations with logarithmic local mononodromy, we determine the minimal possible degree of a polynomial solution."}
{"category": "Math", "title": "When are Swing options bang-bang and how to use it", "abstract": "In this paper we investigate a class of swing options with firm constraints in view of the modeling of supply agreements. We show, for a fully general payoff process, that the premium, solution to a stochastic control problem, is concave and piecewise affine as a function of the global constraints of the contract. The existence of bang-bang optimal controls is established for a set of constraints which generates by affinity the whole premium function. When the payoff process is driven by an underlying Markov process, we propose a quantization based recursive backward procedure to price these contracts. A priori error bounds are established, uniformly with respect to the global constraints."}
{"category": "Math", "title": "Many Random Walks Are Faster Than One", "abstract": "We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time - the expected time required to visit every node in a graph at least once - and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds."}
{"category": "Math", "title": "Simple connectedness of quasitilted algebras", "abstract": "Let A be a basic connected finite dimensional algebra over an algebraically closed field. Assuming that A is quasitilted, we prove that A is simply connected if and only if its first Hochschild cohomology group HH^1(A) vanishes. This generalises a result of I. Assem, F.U. Coelho and S. Trepode and which proves the same equivalence for tame quasitilted algebras."}
{"category": "Math", "title": "Automorphism groups and anti-pluricanonical curves", "abstract": "We show the existence of an anti-pluricanonical curve on every smooth projective rational surface X which has an infinite group G of automorphisms of either null entropy or of type Z . Z (semi-direct product), provided that the pair (X, G) is minimal. This was conjectured by Curtis T. McMullen (2005) and further traced back to Marat Gizatullin and Brian Harbourne (1987). We also prove (perhaps) the strongest form of the famous Tits alternative theorem."}
{"category": "Math", "title": "Adjoints of composition operators with rational symbol", "abstract": "Building on techniques developed by Cowen and Gallardo-Guti\\'{e}rrez, we find a concrete formula for the adjoint of a composition operator with rational symbol acting on the Hardy space $H^{2}$. We consider some specific examples, comparing our formula with several results that were previously known."}
{"category": "Math", "title": "The Extended Bloch Group and the Cheeger-Chern-Simons Class", "abstract": "We present a formula for the full Cheeger-Chern-Simons class of the tautological flat complex vector bundle of rank two over BSL(2,\\C^\\delta). Our formula improves the formula by Dupont and Zickert, where the class is only computed modulo 2-torsion."}
{"category": "Math", "title": "Rothberger's property in finite powers", "abstract": "We show that several classical Ramseyan statements, and a forcing statement, are each equivalent to having Rothberger's property in all finite powers."}
{"category": "Math", "title": "Space-time percolation", "abstract": "The contact model for the spread of disease may be viewed as a directed percolation model on $\\ZZ \\times \\RR$ in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The corresponding process when the time-axis is unoriented is an undirected percolation model to which now standard techniques may be applied. One may construct in similar vein a random-cluster model on $\\ZZ \\times \\RR$, with associated continuum Ising and Potts models. These models are of independent interest, in addition to providing a path-integral representation of the quantum Ising model with transverse field. This representation may be used to obtain a bound on the entanglement of a finite set of spins in the quantum Ising model on $\\ZZ$, where this entanglement is measured via the entropy of the reduced density matrix. The mean-field version of the quantum Ising model gives rise to a random-cluster model on $K_n \\times \\RR$, thereby extending the Erdos-Renyi random graph on the complete graph $K_n$."}
{"category": "Math", "title": "Strong Stein neighborhood bases", "abstract": "Let D be a smooth bounded pseudoconvex domain in C^n. We give several characterizations for the closure of D to have a strong Stein neighborhood basis in the sense that D has a defining function r such that {z\\in C^n:r(z)<a} is pseudoconvex for sufficiently small a>0. We also show that this condition is invariant under proper holomorphic maps that extend smoothly to the boundary."}
{"category": "Math", "title": "Hodge Spaces for Real Toric Varieties", "abstract": "We define the Z/2Z Hodge spaces H_{pq}(\\Sigma) of a fan \\Sigma. If \\Sigma is the normal fan of a reflexive polytope \\Delta then we use polyhedral duality to compute the Z/2Z Hodge Spaces of \\Sigma. In particular, if the cones of dimension at most e in the face fan \\Sigma^* of \\Delta are smooth then we compute H_{pq}(\\Sigma) for p<e-1. If \\Sigma^* is a smooth fan then we completely determine the spaces H_{pq}(\\Sigma) and we show the toric variety X associated to \\Sigma is maximal, meaning that the sum of the Z/2Z Betti numbers of X(R) is equal to the sum of the Z/2Z Betti numbers of X(C)."}
{"category": "Math", "title": "Leonard triples and hypercubes", "abstract": "Let $V$ denote a vector space over C with finite positive dimension. By a {\\em Leonard triple} on $V$ we mean an ordered triple of linear operators on $V$ such that for each of these operators there exists a basis of $V$ with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let $D$ denote a positive integer and let $Q_D$ denote the graph of the $D$-dimensional hypercube. Let $X$ denote the vertex set of $Q_D$ and let $A$ denote the adjacency matrix of $Q_D$. Fix $x \\in X$ and let $A^*$ denote the corresponding dual adjacency matrix. Let $T$ denote the subalgebra of $Mat_X(C)$ generated by $A, A^*$. We refer to $T$ as the {\\em Terwilliger algebra of} $Q_D$ {\\em with respect to} $x$. The matrices $A$ and $A^*$ are related by the fact that $2 \\im A = A^* A^e - A^e A^*$ and $2 \\im A^* = A^e A - A A^e$, where $2 \\im A^e = A A^* - A^* A$ and $\\im^2=-1$. We show that the triple $A$, $A^*$, $A^e$ acts on each irreducible $T$-module as a Leonard triple. We give a detailed description of these Leonard triples."}
{"category": "Math", "title": "Bringing errors into focus", "abstract": "This lecture presents recent advances in the theory of errors propagation. We first explain in which cases the propagation of errors may be performed with a first order differential calculus or needs a second order differential calculus. Then we point out the link between error propagation and the concept of second order vector in differential geometry, emphasizing the existence of a slight ambiguity concerning the bias operator. The third part in devoted to the powerful framework of Dirichlet forms whose main feature is to apply easily to infinite dimensional models including the Wiener space (giving an interpretation of Malliavin calculus in terms of errors), the Poisson space and the Monte Carlo space. In the fourth part we show how an error in the usual mathematical sense, i.e. an approximate quantity, may yield a Dirichlet form and we introduce the four bias operators. Eventually we connect the Dirichlet form with statistics by identifying the square of field operator with the inverse of the Fisher information matrix."}
{"category": "Math", "title": "Bijectiveness of the Nash Map for Quasi-Ordinary Hypersurface Singularities", "abstract": "In this paper we give a positive answer to a question of Nash concerning the arc space of a singularity, for the class of quasi-ordinary hypersurface singularities, extending to this case previous results and techniques of Shihoko Ishii."}
{"category": "Math", "title": "Birationality of \\'etale morphisms via surgery", "abstract": "We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\\Bbb C^n$, then any local diffeomorphism $F:X \\to \\Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover away from $D$ has degree $d=1$ or $d=\\infty$ (however all degrees $d > 1$ are possible if $F$ fails to be a local diffeomorphism at even a single point). In particular, any \\'etale morphism $F:X \\to \\Bbb C^n$ of algebraic varieties which covers away from such a hypersurface $D$ must be birational."}
{"category": "Math", "title": "Subrings of the asymptotic Hecke algebra of type $H_4$", "abstract": "The structure of subring $J^{\\Gamma \\cap \\Gamma^{-1}}$ of the asymptotic Hecke algebra is described for $\\Gamma$ a left cell of the Coxeter group of type $H_4$. A small set of generators is produced. The subalgebras spanned by a subset of the basis ${t_x}_{x\\in \\Gamma\\cap\\Gamma^{-1}}$ are determined."}
{"category": "Math", "title": "Displacement energy of coisotropic submanifolds and Hofer's geometry", "abstract": "We prove that the displacement energy of a stable coisotropic submanifold is bounded away from zero if the ambient symplectic manifold is closed, rational and satisfies a mild topological condition."}
{"category": "Math", "title": "Strichartz and Smoothing Estimates for Schr\\\"odinger Operators with Almost Critical Magnetic Potentials in Three and Higher Dimensions", "abstract": "In this paper we consider magnetic Schr\\\"odinger operators in R^n, n \\ge 3. Under almost optimal conditions on the potentials in terms of decay and regularity we prove smoothing and Strichartz estimates, as well as a limiting absorption principle. For large gradient perturbations the latter is not a corollary of the free case as the differentiated free resolvent does not have small operator norm on any weighted L^2 spaces. We instead show that the spectral radius of such operators decreases to zero, hence their perturbation of the identity is still invertible. The key estimates are based on an angular decomposition of the free resolvent, or rather a bound that holds uniformly for all possible angular decompositions. The proof avoids the Fourier transform and instead uses H\\\"ormander's variable coefficient Plancherel theorem for oscillatory integrals."}
{"category": "Math", "title": "The momentum map in Poisson geometry", "abstract": "Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction."}
{"category": "Math", "title": "Bivariate linear mixed models using SAS proc MIXED", "abstract": "Bivariate linear mixed models are useful when analyzing longitudinal data of two associated markers. In this paper, we present a bivariate linear mixed model including random effects or first-order auto-regressive process and independent measurement error for both markers. Codes and tricks to fit these models using SAS Proc MIXED are provided. Limitations of this program are discussed and an example in the field of HIV infection is shown. Despite some limitations, SAS Proc MIXED is a useful tool that may be easily extendable to multivariate response in longitudinal studies."}
{"category": "Math", "title": "Mixed models for longitudinal left-censored repeated measures", "abstract": "Longitudinal studies could be complicated by left-censored repeated measures. For example, in Human Immunodeficiency Virus infection, there is a detection limit of the assay used to quantify the plasma viral load. Simple imputation of the limit of the detection or of half of this limit for left-censored measures biases estimations and their standard errors. In this paper, we review two likelihood-based methods proposed to handle left-censoring of the outcome in linear mixed model. We show how to fit these models using SAS Proc NLMIXED and we compare this tool with other programs. Indications and limitations of the programs are discussed and an example in the field of HIV infection is shown."}
{"category": "Math", "title": "Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion", "abstract": "The present article is devoted to a fine study of the convergence of renormalized weighted quadratic and cubic variations of a fractional Brownian motion $B$ with Hurst index $H$. In the quadratic (resp. cubic) case, when $H<1/4$ (resp. $H<1/6$), we show by means of Malliavin calculus that the convergence holds in $L^2$ toward an explicit limit which only depends on $B$. This result is somewhat surprising when compared with the celebrated Breuer and Major theorem."}
{"category": "Math", "title": "Multidimensional continued fractions and a Minkowski function", "abstract": "The Minkowski Question Mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only homeomorphism of an n-simplex that conjugates the piecewise-fractional map associated to the Monkemeyer continued fraction algorithm with an appropriate tent map."}
{"category": "Math", "title": "Complete surfaces with positive extrinsic curvature in product spaces", "abstract": "We prove that every complete connected immersed surface with positive extrinsic curvature $K$ in $H^2\\times R$ must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study the behavior of the end. Then, we focus our attention on surfaces with positive constant extrinsic curvature ($K-$surfaces). We establish that the only complete $K-$surfaces in $S^2\\times R$ and $H^2\\times R$ are rotational spheres. Here are the key steps to achieve this. First height estimates for compact $K-$surfaces in a general ambient space $M^2\\times R$ with boundary in a slice are obtained. Then distance estimates for compact $K-$surfaces (and H-$surfaces) in $H^2\\times R$ with boundary on a vertical plane are obtained. Finally we construct a quadratic form with isolated zeroes of negative index."}
{"category": "Math", "title": "Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue", "abstract": "This article surveys many standard results about the braid group with emphasis on simplifying the usual algebraic proofs. We use van der Waerden's trick to illuminate the Artin-Magnus proof of the classic presentation of the algebraic mapping-class group of a punctured disc. We give a simple, new proof of the Dehornoy-Larue braid-group trichotomy, and, hence, recover the Dehornoy right-ordering of the braid group. We then turn to the Birman-Hilden theorem concerning braid-group actions on free products of cyclic groups, and the consequences derived by Perron-Vannier, and the connections with the Wada representations. We recall the very simple Crisp-Paris proof of the Birman-Hilden theorem that uses the Larue-Shpilrain technique. Studying ends of free groups permits a deeper understanding of the braid group; this gives us a generalization of the Birman-Hilden theorem. Studying Jordan curves in the punctured disc permits a still deeper understanding of the braid group; this gave Larue, in his PhD thesis, correspondingly deeper results, and, in an appendix, we recall the essence of Larue's thesis, giving simpler combinatorial proofs."}
{"category": "Math", "title": "Iteration of closed geodesics in stationary Lorentzian manifolds", "abstract": "Following the lines of a celebrated result by R. Bott (Comm. Pure Appl. Math. 9, 1956) we study the Morse index of the iterated of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic $\\gamma$, we prove the existence of a locally constant integer valued map $\\Lambda_\\gamma$ on the unit circle with the property that the Morse index of the iterated $\\gamma^N$ is equal, up to a correction term $\\epsilon_\\gamma\\in\\{0,1\\}$, to the sum of the values of $\\Lambda_\\gamma$ at the $N$-th roots of unity. The discontinuities of $\\Lambda_\\gamma$ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincar\\'e map of $\\gamma$. We discuss some applications of the theory."}
{"category": "Math", "title": "The Oka principle for sections of stratified fiber bundles", "abstract": "A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n to Y. If X is a reduced Stein space and Z is a stratified holomorphic fiber bundle over X all of whose fibers satisfy CAP, then sections of Z over X enjoy the Oka property with (jet) interpolation and approximation. Previously this has been proved by the author in the case when X is a Stein manifold without singularities (Ann. Math., 163 (2006), 689-707, math.CV/0402278; Ann. Inst. Fourier, 55 (2005), 733-751, math.CV/0411048). We also give existence results for holomorphic sections under certain connectivity hypothesis on the fibers. In the final part of the paper we obtain the Oka property for sections of submersions with stratified sprays over Stein spaces."}
{"category": "Math", "title": "Quasi Ordinary Singularities, Essential Divisors and Poincare Series", "abstract": "We define Poincar\\'e series associated to a toric or analytically irreducible quasi-ordinary hypersurface singularity, (S,0), by a finite sequence of monomial valuations, such that at least one of them is centered at the origin 0. This involves the definition of a multi-graded ring associated to the analytic algebra of the singularity by the sequence of valuations. We prove that the Poincar\\'e series is a rational function with integer coefficients, which can be defined also as an integral with respect of the Euler characteristic, over the projectivization of the analytic algebra of the singularity, of a function defined by the valuations. In particular, the Poincar\\'e series associated to the set of divisorial valuations associated to the essential divisors, considered both over the singular locus and over the point 0, is an analytic invariant of the singularity. In the quasi-ordinary hypersurface case we prove that this Poincar\\'e series determines and it is determined by the normalized sequence of characteristic monomials. These monomials in the analytic case define a complete invariant of the embedded topological type of the hypersurface singularity."}
{"category": "Math", "title": "A new proof of Vazsonyi's conjecture", "abstract": "We present a self-contained proof that the number of diameter pairs among n points in Euclidean 3-space is at most 2n-2. The proof avoids the ball polytopes used in the original proofs by Grunbaum, Heppes and Straszewicz. As a corollary we obtain that any three-dimensional diameter graph can be embedded in the projective plane."}
{"category": "Math", "title": "Maxwell strata in Euler's elastic problem", "abstract": "The classical Euler's problem on stationary configurations of elastic rod with fixed endpoints and tangents at the endpoints is considered as a left-invariant optimal control problem on the group of motions of a two-dimensional plane $\\E(2)$. The attainable set is described, existence and boundedness of optimal controls are proved. Extremals are parametrized by Jacobi's elliptic functions of natural coordinates induced by the flow of the mathematical pendulum on fibers of the cotangent bundle of $\\E(2)$. The group of discrete symmetries of Euler's problem generated by reflections in the phase space of the pendulum is studied. The corresponding Maxwell points are completely described via the study of fixed points of this group. As a consequence, an upper bound on cut points in Euler's problem is obtained."}
{"category": "Math", "title": "N^p Spaces", "abstract": "We introduce a new norm, called $N^{p}$-norm $(1\\leq{p}<\\infty)$ on a space $N^{p}(V,W)$ where $V$ and $W$ are abstract operator spaces. By proving some fundamental properties of the space $N^{p}(V,W)$, we also obtain that if $W$ is complete, then the space $N^{p}(V,W)$ is also a Banach space with respect to this norm for $1\\leq{p}<\\infty$."}
{"category": "Math", "title": "Exponential sums with coefficients 0 or 1 and concentrated L^{p} norms", "abstract": "Let f be a sum of exponentials of the form exp(2 pi i N x), where the N are distinct integers. We call f an idempotent trigonometric polynomial (because the convolution of f with itself is f) or, simply, an idempotent. We show that for every p > 1, and every set E of the torus T = R/Z with |E| > 0, there are idempotents concentrated on E in the Lp sense. More precisely, for each p > 1, there is an explicitly calculated constant Cp > 0 so that for each E with |E| > 0 and epsilon > 0 one can find an idempotent f such that the pth root of the ratio of the integral over E of the pth power of |f| to the integral over T of the pth power of |f| is greater than Cp - epsilon. This is in fact a lower bound result and, though not optimal, it is close to the best that our method gives. We also give both heuristic and computational evidence for the still open problem of whether the Lp concentration phenomenon fails to occur when p = 1."}
{"category": "Math", "title": "Additive preserving rank one maps on Hilbert $C^\\ast$-modules", "abstract": "In this paper, we characterize a class of additive maps on Hilbert $C^\\ast$-modules which maps a \"rank one\" adjointable operators to another rank one operators."}
{"category": "Math", "title": "The Kalman--Yakubovich--Popov inequality for passive discrete time-invariant systems", "abstract": "We consider the Kalman - Yakubovich - Popov (KYP) inequality \\[ \\begin{pmatrix} X-A^* XA-C^*C & -A^*X B- C^*D\\cr -B^*X A-D^* C & I- B^*X B-D^*D \\end{pmatrix} \\ge 0 \\] for contractive operator matrices $ \\begin{pmatrix} A&B\\cr C &D \\end{pmatrix}:\\begin{pmatrix}\\mathfrak{H}\\cr\\mathfrak{M} \\end{pmatrix}\\to\\begin{pmatrix}\\mathfrak{H}\\cr\\mathfrak{N} \\end{pmatrix}, $ where $\\mathfrak{H},$ $\\mathfrak{M}$, and $\\mathfrak{N}$ are separable Hilbert spaces. We restrict ourselves to the solutions $X$ from the operator interval $[0, I_\\mathfrak{H}]$. Several equivalent forms of KYP are obtained. Using the parametrization of the blocks of contractive operator matrices, the Kre\\u{\\i}n shorted operator, and the M\\\"obius representation of the Schur class operator-valued function we find several equivalent forms of the KYP inequality. Properties of solutions are established and it is proved that the minimal solution of the KYP inequality satisfies the corresponding algebraic Riccati equation and can be obtained by the iterative procedure with the special choice of the initial point. In terms of the Kre\\u{\\i}n shorted operators a necessary condition and some sufficient conditions for uniqueness of the solution are established."}
{"category": "Math", "title": "Central Limit Theorem for the Excited Random Walk in dimension $d \\geq 2$", "abstract": "We prove that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension $d \\geq 2$."}
{"category": "Math", "title": "Linear systems on a class of anticanonical rational threefolds", "abstract": "Let X be the blow-up of the three dimensional complex projective space along r general points of a smooth elliptic quartic curve B of P^3 and let L be any line bundle of X. The aim of this paper is to provide an explicit algorithm for determining the dimension of H^0(X,L)."}
{"category": "Math", "title": "Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups", "abstract": "We classify Lagrangian subcategories of the representation category of a twisted quantum double of a finite group. In view of results of 0704.0195v2 this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of the representation category of a twisted quantum double of a finite group G and module categories over the category of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld's characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories."}
{"category": "Math", "title": "A Simplification of Combinatorial Link Floer Homology", "abstract": "We define a new combinatorial complex computing the hat version of link Floer homology over Z/2Z, which turns out to be significantly smaller than the Manolescu-Ozsvath-Sarkar one."}
{"category": "Math", "title": "Surjunctivity for cellular automata in Besicovitch spaces", "abstract": "The Besicovitch pseudodistance measures the relative size of the set of points where two functions take different values; the quotient space modulo the induced equivalence relation is endowed with a natural metric. We study the behavior of cellular automata in the new topology and show that, under suitable additional hypotheses, they retain certain properties possessed in the usual product topology; in particular, that injectivity still implies surjectivity."}
{"category": "Math", "title": "On the near-equality case of the Positive Mass Theorem", "abstract": "The Positive Mass Conjecture states that any complete asymptotically flat manifold of nonnnegative scalar curvature has nonnegative mass. Moreover, the equality case of the Positive Mass Conjecture states that in the above situation, if the mass is zero, then the Riemannian manifold must be Euclidean space. The Positive Mass Conjecture was proved by R. Schoen and S.-T. Yau for all manifolds of dimension less than 8, and it was proved by E. Witten for all spin manifolds. In this paper, we consider complete asymptotically flat manifolds of nonnegative scalar curvature that are also harmonically flat in an end. We show that, whenever the Positive Mass Theorem holds, any appropriately normalized sequence of such manifolds whose masses converge to zero must have metrics that are uniformly converging to Euclidean metrics outside a compact region. This result is an ingredient in a forthcoming proof, co-authored with H. Bray, of the Riemannian Penrose inequality in dimensions less than 8."}
{"category": "Math", "title": "Modules-at-infinity for quantum vertex algebras", "abstract": "This is a sequel to \\cite{li-qva1} and \\cite{li-qva2} in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian $DY_{\\hbar}(sl_{2})$, denoted by $DY_{q}(sl_{2})$ and $DY_{q}^{\\infty}(sl_{2})$ with $q$ a nonzero complex number. For each nonzero complex number $q$, we construct a quantum vertex algebra $V_{q}$ and prove that every $DY_{q}(sl_{2})$-module is naturally a $V_{q}$-module. We also show that $DY_{q}^{\\infty}(sl_{2})$-modules are what we call $V_{q}$-modules-at-infinity. To achieve this goal, we study what we call $\\S$-local subsets and quasi-local subsets of $\\Hom (W,W((x^{-1})))$ for any vector space $W$, and we prove that any $\\S$-local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with $W$ as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity."}
{"category": "Math", "title": "On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres", "abstract": "The $n$-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the $(n+1)$-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2- planes and is a compact Hermitian symmetric space of rank 2. In this paper we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique. Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as Gauss images of isoparametric hypersurfaces in spheres with $g(=1,2,3)$ distinct principal curvatures."}
{"category": "Math", "title": "Signed q-Analogs of Tornheim's Double Series", "abstract": "We introduce signed q-analogs of Tornheim's double series, and evaluate them in terms of double q-Euler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series, and correct some mistakes in the literature."}
{"category": "Math", "title": "Inflated Beta Distributions", "abstract": "This paper considers the issue of modeling fractional data observed in the interval [0,1), (0,1] or [0,1]. Mixed continuous-discrete distributions are proposed. The beta distribution is used to describe the continuous component of the model since its density can have quite diferent shapes depending on the values of the two parameters that index the distribution. Properties of the proposed distributions are examined. Also, maximum likelihood and method of moments estimation is discussed. Finally, practical applications that employ real data are presented."}
{"category": "Math", "title": "Polar Cremona Transformations and Monodromy of Polynomials", "abstract": "Consider the gradient map associated to any non-constant homogeneous polynomial $f\\in \\C[x_0,...,x_n]$ of degree $d$, defined by \\[\\phi_f=grad(f): D(f)\\to \\CP^n, (x_0:...:x_n)\\to (f_0(x):...:f_n(x))\\] where $D(f)=\\{x\\in \\CP^n; f(x)\\neq 0\\}$ is the principal open set associated to $f$ and $f_i=\\frac{\\partial f}{\\partial x_i}$. This map corresponds to polar Cremona transformations. In Proposition \\ref{p1} we give a new lower bound for the degree $d(f)$ of $\\phi_f$ under the assumption that the projective hypersurface $V:f=0 $ has only isolated singularities. When $d(f)=1$, Theorem \\ref{t4} yields very strong conditions on the singularities of $V$."}
{"category": "Math", "title": "Sum-product estimates via directed expanders", "abstract": "Let $\\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\\F_q[x_1, x_2]$. For a set $A \\subset \\F_q$, define $P(A):=\\{P(x_1, x_2) | x_i \\in A \\}$. Using certain constructions of expanders, we characterize all polynomials $P$ for which the following holds \\vskip2mm \\centerline{\\it If $|A+A|$ is small, then $|P(A)|$ is large.} \\vskip2mm The case $P=x_1x_2$ corresponds to the well-known sum-product problem."}
{"category": "Math", "title": "Strange duality and the Hitchin/WZW connection", "abstract": "For a compact Riemann surface X of positive genus, the space of sections of certain theta bundle on moduli of bundles of rank r and level k admits a natural map to (the dual of) a similar space of sections of rank k and level r (the strange duality isomorphism). Both sides of the isomorphism carry projective connections as X varies in a family. We prove that this map is (projectively) flat."}
{"category": "Math", "title": "Limit cycles in the presence of convection, a travelling wave analysis", "abstract": "We consider a diffusion model with limit cycle reaction functions, in the presence of convection. We select a set of functions derived from a realistic reaction model: the Schnakenberg equations. This resultant form is unsymmetrical. We find a transformation which maps the irregular equations into model form. Next we transform the dependent variables into polar form. From here, a travelling wave analysis is performed on the radial variable. Results are complex, but we make some simple estimates. We carry out numerical experiments to test our analysis. An initial `knock' starts the propagation of pattern. The speed of the travelling wave is not quite as expected. We investigate further. The system demonstrates distinctly different behaviour to the left and the right. We explain how this phenomenon occurs by examining the underlying behaviour."}
{"category": "Math", "title": "Partial sums of the M{\\\"o}bius function", "abstract": "Assuming the Riemann Hypothesis we establish an upper bound for the sum of the M{\\\" o}bius function up to $x$. Our method is based on estimating the frequency with which intervals of a given length can contain an unusual number of ordinates of zeros of the Riemann zeta-function."}
{"category": "Math", "title": "Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant", "abstract": "Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analogs of $T$. This leads to a relation between certain multiple polylogarithm values and multiple zeta values."}
{"category": "Math", "title": "On monoidal equivalences and Ann-equivalences", "abstract": "In this paper, we show another proof of the problem by constructing a strict monoidal category M(C) consisting of M-functors and M-morphisms of a category C and we prove C is equivalent to it. The proof is based on a basic character of monoidal equivalences. Ideas and techniques of these proofs can been used to prove the equivalence between an Ann-category and an almost strict Ann-category."}
{"category": "Math", "title": "Orbifoldes speciales et classification bimeromorphe des varietes kaehleriennes compactes", "abstract": "This is a sequel to [Ca01]=math.AG/0110051. We define the bimeromorphic {\\it category} of geometric orbifolds. These interpolate between (compact K\\\" ahler) manifolds and such manifolds with logarithmic structure. These geometric orbifolds are considered from the point of view of their geometry, and thus equipped with the usual invariants of varieties: morphisms and bimeromorphic maps, differential forms, fundamental groups and universal covers, fields of definition and rational points. The most elementary properties, directly adapted from the case of varieties without orbifold structure, are established here. The arguments of [Ca01] can then be directly adapted to extend the main structure results to this orbifold category. We hope to come back to deeper aspects later. The motivation is that the natural frame for the theory of classification of compact K\\\" ahler (and complex projective) manifolds includes at least the category of orbifolds, as shown in [Ca01] by the fonctorial decomposition of {\\it special} manifolds as tower of orbifolds with either $\\kappa_+=-\\infty$ or $\\kappa=0$, and also, seemingly, by the minimal model program, in which most proofs work only after the adjunction of a \"boundary\". Also, fibrations enjoy in the bimeromorphic category of geometric orbifolds extension properties not satisfied in the category of varieties without orbifold structure, permitting to express invariants of the total space from those of the generic fibre and of the base. For example, the natural sequence of fundamental groups is exact there; also the total space is special if so are the generic fibre and the base. This makes this category suitable to lift properties from orbifolds having either $\\kappa_+=-\\infty$ or $\\kappa=0$ to those which are special."}
{"category": "Math", "title": "On the torsion of Brieskorn modules of homogeneous polynomials", "abstract": "Let $f\\in \\mathbb{C}[X_1,..., X_n]$ be a homogeneous polynomial and B(f) be the corresponding Brieskorn module. We describe the torsion of the Brieskorn module B(f) for n=2 and show that any torsion element has order 1. For n>2, we find some examples in which the torsion order is strictly greater than 1."}
{"category": "Math", "title": "Stallings' Foldings and Subgroups of Amalgams of Finite Groups", "abstract": "In the 1980's Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for ``sewing'' on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems."}
{"category": "Math", "title": "Algorithmic Problems in Amalgams of Finite Groups", "abstract": "Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. It turns out that Stallings' methods can be effectively generalized for the class of amalgams of finite groups. In the present paper we employ subgroup graphs constructed by the generalized Stallings' folding algorithm to solve various algorithmic problems in amalgams of finite groups."}
{"category": "Math", "title": "The bounded isometry conjecture for the Kodaira-Thurston manifold and 4-Torus", "abstract": "The purpose of this note is to study the bounded isometry conjecture proposed by Lalonde and Polterovich. In particular, we show that the conjecture holds for the Kodaira-Thurston manifold with the standard symplectic form and for the 4-torus with all linear symplectic forms."}
{"category": "Math", "title": "Symmetries in Differential Geometry: A Computational Approach to Prolongations", "abstract": "The aim of this work is to develop a systematic manner to close overdetermined systems arising from conformal Killing tensors (CKT). The research performs this action for 1-tensor and 2-tensors. This research makes it possible to develop a new general method for any rank of CKT. This method can also be applied to other types of Killing equations, as well as to overdetermined systems constrained by some other conditions. The major methodological apparatus of the research is a decomposition of the section bundles where the covariant derivatives of the CKT land via generalized gradients. This decomposition generates a tree in which each row represents a higher derivative. After using the conformal Killing equation, just a few components (branches) survive, which means that most of them can be expressed in terms of lower order terms. This results in a finite number of independent jets. Thus, any higher covariant derivative can be written in terms of these jets. The findings of this work are significant methodologically and, more specifically, in the potential for the discovery of symmetries. First, this work has uncovered a new method that could be used to close overdetermined systems arising from conformal Killing tensors (CKT). Second, through an application of this method, this research finds higher symmetry operators of first and second degree, which are known by other means, for the Laplace operator. The findings also reveal the first order symmetry operators for the Yamabe case. Moreover, the research leads to conjectures about the second order symmetries of the Yamabe operator."}
{"category": "Math", "title": "Manifolds with 1/4-pinched Curvature are Space Forms", "abstract": "Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking."}
{"category": "Math", "title": "A succinct method for investigating the sums of infinite series through differential formulae", "abstract": "Translation of \"Methodus succincta summas serierum infinitarum per formulas differentiales investigandi\" (1780). Euler wants to represent some given series of functions S(x)=X(x)+X(x+1)+X(x+2)+etc. in a different way. He writes S as a series in derivatives of X with unknown coefficients. He makes a generating function V(z) out of these coefficients, which is the same as a generating function that involves the Bernoulli numbers."}
{"category": "Math", "title": "Fourier-stable subrings in the Chow rings of abelian varieties", "abstract": "We study subrings in the Chow ring $\\CH^*(A)_{{\\Bbb Q}}$ of an abelian variety $A$, stable under the Fourier transform with respect to an arbitrary polarization. We prove that by taking Pontryagin products of classes of dimension $\\leq 1$ one gets such a subring. We also show how to construct finite-dimensional Fourier-stable subrings in $\\CH^*(A)_{{\\Bbb Q}}$. Another result concerns the relation between the Pontryagin product and the usual product on the $\\CH^*(A)_{{\\Bbb Q}}$. We prove that the operator of the usual product with a cycle is a differential operator with respect to the Pontryagin product and compute its order in terms of the Beauville's decomposition of $\\CH^*(A)_{{\\Bbb Q}}$."}
{"category": "Math", "title": "Relatively computably enumerable reals", "abstract": "A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and Y does not compute X. A real X is relatively simple and above if there is a real Y <_T X such that X is c.e.(Y) and there is no infinite subset Z of the complement of X such that Z is c.e.(Y). We prove that every nonempty Pi^0_1 class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above."}
{"category": "Math", "title": "Simply connected minimal symplectic 4-manifolds with signature less than --1", "abstract": "For each pair $(e,\\sigma)$ of integers satisfying $2e+3\\sigma\\ge 0$, $\\sigma\\leq -2$, and $e+\\sigma\\equiv 0\\pmod{4}$, with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic $e$ and signature $\\sigma$. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature -1) with Euler characteristic $4k$ (resp. $4k+1$) for all $k\\ge 46$ (resp. $k\\ge 49$)."}
{"category": "Math", "title": "On the structure of positive maps between matrix algebras", "abstract": "A partial description of the structure of positive unital maps $\\phi: M_2(\\bC) \\to M_{n+1}(\\bC)$ ($n\\geq 2$) is given."}
{"category": "Math", "title": "Interval Conjectures for level Hilbert functions", "abstract": "We conjecture that the set of all Hilbert functions of (artinian) level algebras enjoys a very natural form of regularity, which we call the {\\em Interval Conjecture} (IC): If, for some positive integer $\\alpha $, $(1,h_1,...,h_i,...,h_e)$ and $(1,h_1,...,h_i+\\alpha ,...,h_e)$ are both level $h$-vectors, then $(1,h_1,...,h_i+\\beta ,...,h_e)$ is also level for each integer $\\beta =0,1,..., \\alpha .$ In the Gorenstein case, i.e. when $h_e=1$, we also supply the {\\em Gorenstein Interval Conjecture} (GIC), which naturally generalizes the IC, and basically states that the same property simultaneously holds for any two symmetric entries, say $h_i$ and $h_{e-i}$, of a Gorenstein $h$-vector. These conjectures are inspired by the research performed in this area over the last few years. A series of recent results seems to indicate that it will be nearly impossible to characterize explicitly the sets of all Gorenstein or of all level Hilbert functions. Therefore, our conjectures would at least provide the existence of a very strong - and natural - form of order in the structure of such important and complicated sets. We are still far from proving the conjectures at this point. However, we will already solve a few interesting cases, especially when it comes to the IC, in this paper. Among them, that of Gorenstein $h$-vectors of socle degree 4, that of level $h$-vectors of socle degree 2, and that of non-unimodal level $h$-vectors of socle degree 3 and any given codimension."}
{"category": "Math", "title": "Uniqueness and non-uniqueness of chains on half lines", "abstract": "We establish a one-to-one correspondence between one-sided and two-sided regular systems of conditional probabilities on the half-line that preserves the associated chains and Gibbs measures. As an application, we determine uniqueness and non-uniqueness regimes in one-sided versions of ferromagnetic Ising models with long range interactions. Our study shows that the interplay between chain and Gibbsian theories yields more information than that contained within the known theory of each separate framework. In particular: (i) A Gibbsian construction due to Dyson yields a new family of chains with phase transitions; (ii) these transitions show that a square summability uniqueness condition of chains is false in the general non-shift-invariant setting, and (iii) an uniqueness criterion for chains shows that a Gibbsian conjecture due to Kac and Thompson is false in this half-line setting."}
{"category": "Math", "title": "Real singular Del Pezzo surfaces and threefolds fibred by rational curves, I", "abstract": "Let W -> X be a real smooth projective threefold fibred by rational curves. Koll\\'ar proved that if W(R) is orientable a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Let k : = k(N) be the integer defined as follows: If g : N -> F is a Seifert fibration, one defines k : = k(N) as the number of multiple fibres of g, while, if N is a connected sum of lens spaces, k is defined as the number of lens spaces different from P^3(R). Our Main Theorem says: If X is a geometrically rational surface, then k <= 4. Moreover we show that if F is diffeomorphic to S^1xS^1, then W(R) is connected and k = 0. These results answer in the affirmative two questions of Koll\\'ar who proved in 1999 that k <= 6 and suggested that 4 would be the sharp bound. We derive the Theorem from a careful study of real singular Del Pezzo surfaces with only Du Val singularities."}
{"category": "Math", "title": "Restricted infinitesimal deformations of restricted simple Lie algebras", "abstract": "We compute the restricted infinitesimal deformations of the restricted simple Lie algebras over an algebraically closed field of characteristic different from 2 and 3."}
{"category": "Math", "title": "A Berry-Esseen type inequality for convex bodies with an unconditional basis", "abstract": "We provide a sharp rate of convergence in the central limit theorem for random vectors with an unconditional, log-concave density. The argument relies on analysis of the Neumann laplacian on convex domains and on the theory of optimal transportation of measures."}
{"category": "Math", "title": "Universal deformation rings and dihedral 2-groups", "abstract": "Let $k$ be an algebraically closed field of characteristic 2, and let $W$ be the ring of infinite Witt vectors over $k$. Suppose $D$ is a dihedral 2-group. We prove that the universal deformation ring $R(D,V)$ of an endo-trivial $kD$-module $V$ is always isomorphic to $W[\\mathbb{Z}/2\\times\\mathbb{Z}/2]$. As a consequence we obtain a similar result for modules $V$ with stable endomorphism ring $k$ belonging to an arbitrary nilpotent block with defect group $D$. This confirms for such $V$ conjectures on the ring structure of the universal deformation ring of $V$ which had previously been shown for $V$ belonging to cyclic blocks or to blocks with Klein four defect groups."}
{"category": "Math", "title": "Analyticity of layer potentials and $L^{2}$ solvability of boundary value problems for divergence form elliptic equations with complex $L^{\\infty}$ coefficients", "abstract": "We consider divergence form elliptic operators of the form $L=-\\dv A(x)\\nabla$, defined in $R^{n+1} = \\{(x,t)\\in R^n \\times R \\}$, $n \\geq 2$, where the $L^{\\infty}$ coefficient matrix $A$ is $(n+1)\\times(n+1)$, uniformly elliptic, complex and $t$-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on $L^2(\\mathbb{R}^{n})=L^2(\\partial\\mathbb{R}_{+}^{n+1})$, is stable under complex, $L^{\\infty}$ perturbations of the coefficient matrix. Using a variant of the $Tb$ Theorem, we also prove that the layer potentials are bounded and invertible on $L^2(\\mathbb{R}^n)$ whenever $A(x)$ is real and symmetric (and thus, by our stability result, also when $A$ is complex, $\\Vert A-A^0\\Vert_{\\infty}$ is small enough and $A^0$ is real, symmetric, $L^{\\infty}$ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with $L^2$ (resp. $\\dot{L}^2_1)$ data, for small complex perturbations of a real symmetric matrix. Previously, $L^2$ solvability results for complex (or even real but non-symmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients $A_{j,n+1}=0=A_{n+1,j}$, $1\\leq j\\leq n$, which corresponds to the Kato square root problem."}
{"category": "Math", "title": "Dahlberg's bilinear estimate for solutions of divergence form complex elliptic equations", "abstract": "We consider divergence form elliptic operators $L=-\\dv A(x)\\nabla$, defined in $\\mathbb{R}^{n+1}=\\{(x,t)\\in\\mathbb{R}^{n}\\times\\mathbb{R}\\}, n \\geq 2$, where the $L^{\\infty}$ coefficient matrix $A$ is $(n+1)\\times(n+1)$, uniformly elliptic, complex and $t$-independent. Using recently obtained results concerning the boundedness and invertibility of layer potentials associated to such operators, we show that if $Lu=0$ in $\\mathbb{R}^{n+1}_+$, then for any vector-valued ${\\bf v} \\in W^{1,2}_{loc},$ we have the bilinear estimate $$|\\iint_{\\mathbb{R}^{n+1}_+} \\nabla u \\cdot \\bar{{\\bf v}} dx dt |\\leq C\\sup_{t>0} \\|u(\\cdot,t)\\|_{L^2(\\mathbb{R}^n)}(\\||t \\nabla {\\bf v}\\|| + \\|N_*{\\bf v}\\|_{L^2(\\mathbb{R}^n)}),$$ where $\\||F\\|| \\equiv (\\iint_{\\mathbb{R}^{n+1}_+} |F(x,t)|^2 t^{-1} dx dt)^{1/2},$ and where $N_*$ is the usual non-tangential maximal operator. The result is new even in the case of real symmetric coefficients, and generalizes the analogous result of Dahlberg for harmonic functions on Lipschitz graph domains."}
{"category": "Math", "title": "A proof of the local $Tb$ Theorem for standard Calder\\'{o}n-Zygmund operators", "abstract": "We give a proof of a so-called \"local $Tb$\" Theorem for singular integrals whose kernels satisfy the standard Calder\\'on-Zygmund conditions. The present theorem, which extends an earlier result of M. Christ \\cite{Ch}, was proved in \\cite{AHMTT} for \"perfect dyadic\" Calder\\'on-Zygmund operators. The proof in \\cite{AHMTT} essentially carries over to the case considered here, with some technical adjustments."}
{"category": "Math", "title": "Algebraic K-theory of hyperbolic 3-simplex reflection groups", "abstract": "A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in the isometry group of hyperbolic 3-space, with fundamental domain a geodesic simplex (possibly with some ideal vertices). The classification of these groups is known, and there are exactly 9 cocompact examples, and 23 non-cocompact examples. We provide a complete computation of the lower algebraic K-theory of the integral group ring of all the hyperbolic 3-simplex reflection groups."}
{"category": "Math", "title": "Une nouvelle condition d'independance pour le theoreme de la limite centrale", "abstract": "We prove a central limit theorem with aassumptions which are many weak than classical conditions"}
{"category": "Math", "title": "A real convexity theorem for quasi-hamiltonian actions", "abstract": "The main result of this paper is a quasi-hamiltonian analogue of a special case of the O'Shea-Sjamaar convexity theorem for usual momentum maps. We denote by U a simply connected compact connected Lie group and we fix an involutive automorphism of maximal rank on this Lie group (such an automorphism always exists). We then denote by M a quasi-hamiltonian U-space and we prove that the image under the momentum map of the fixed-point set of a form-reversing compatible involution of M is a convex polytope, which is in fact equal to the full momentum polytope. This theorem was announced in arXiv:math/0609517v1. As an application, we obtain an example of lagrangian subspace in representation spaces of surface groups."}
{"category": "Math", "title": "Poisson approximation for non-backtracking random walks", "abstract": "Random walks on expander graphs were thoroughly studied, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. Alon, Benjamini, Lubetzky and Sodin studied non-backtracking random walks on regular graphs, and showed that their mixing rate may be up to twice as fast as that of the simple random walk. As an application, they showed that the maximal number of visits to a vertex, made by a non-backtracking random walk of length $n$ on a high-girth $n$-vertex regular expander, is typically $(1+o(1))\\frac{\\log n}{\\log\\log n}$, as in the case of the balls and bins experiment. They further asked whether one can establish the precise distribution of the visits such a walk makes. In this work, we answer the above question by combining a generalized form of Brun's sieve with some extensions of the ideas in Alon et al. Let $N_t$ denote the number of vertices visited precisely $t$ times by a non-backtracking random walk of length $n$ on a regular $n$-vertex expander of fixed degree and girth $g$. We prove that if $g=\\omega(1)$, then for any fixed $t$, $N_t/n$ is typically $\\frac{1}{\\mathrm{e}t!}+o(1)$. Furthermore, if $g=\\Omega(\\log\\log n)$, then $N_t/n$ is typically $\\frac{1+o(1)}{\\mathrm{e}t!}$ uniformly on all $t \\leq (1-o(1))\\frac{\\log n}{\\log\\log n}$ and 0 for all $t \\geq (1+o(1))\\frac{\\log n}{\\log\\log n}$. In particular, we obtain the above result on the typical maximal number of visits to a single vertex, with an improved threshold window. The essence of the proof lies in showing that variables counting the number of visits to a set of sufficiently distant vertices are asymptotically independent Poisson variables."}
{"category": "Math", "title": "The Witten-Reshetikhin-Turaev Invariants of Lens Spaces", "abstract": "We derive an explicit formula for the Witten-Reshetikhin-Turaev SO(3)-invariants of lens spaces. We use the representation of the mapping class group of the torus corresponding to the Witten-Reshetikhin-Turaev SO(3)-TQFT to give such formula."}
{"category": "Math", "title": "Realisation de Hodge du polylogarithme d'un schema abelien", "abstract": "The main result of this article is the fact that the currents defined by Levin give a description of the polylogarithm of an abelian scheme at the topological level. This result was a conjecture of Levin. This provides a method to explicit the Eisenstein classes of an abelian scheme at the topological level. These classes are of special interest since they have a motivic origin by a theorem of Kings. In a forthcoming article, we use the main result of this paper to prove that the Eisenstein classes of the universal abelian scheme over an Hilbert-Blumenthal variety degenerate at the boundary of the Baily-Borel compactification of the base in a special value of an $L$-function associated to the underlying totally real number field. As a corollary, we get a non vanishing result for some of these Eisenstein classes in this geometric situation."}
{"category": "Math", "title": "How to release Frege's system from Russell's antinomy", "abstract": "The conditions for proper definitions in mathematics are given, in terms of the theory of definition, on the basis of the criterions of eliminability and non-creativity. As a definition, Russell's antinomy is a violation of the criterion of eliminability (Behmann, 1931; Bochvar, 1943). Following the path of the criterion of non-creativity, this paper develops a new analysis of Comprehension schema and, as a consequence, proof that Russell's antinomy argumentation, despite the words of Frege himself, does not hold in Grundgesetze der Arithmetik. According to Basic Law (III), the class of classes not belonging to themselves is a class defined by a function which can not take as argument its own course of value. In other words, the class of classes not belonging to themselves is a class whose classes are not identical to the class itself."}
{"category": "Math", "title": "Remarks on families of singular curves with hyperelliptic normalizations", "abstract": "We give restrictions on the existence of families of curves on smooth projective surfaces $S$ of nonnegative Kodaira dimension all having constant geometric genus $g \\geq 2$ and hyperelliptic normalizations. In particular, we prove a Reider-like result whose proof is ``vector bundle-free'' and relies on deformation theory and bending-and-breaking of rational curves in $\\Sym^2(S)$. We also give examples of families of such curves."}
{"category": "Math", "title": "A note on equicontinuity of families of operators and automorphisms", "abstract": "This note concerns uniform equicontinuity of families of operators on a separable Hilbert space H, and of families of maps on B(H). It is shown that a one parameter group of automorphisms is uniformly equicontinuous if and only if the group of unitaries which implements it is so. A \"geometrical\" necessary and sufficient condition is given for a family of operators to be uniformly equicontinuous."}
{"category": "Math", "title": "Composite lacunary polynomials and the proof of a conjecture of Schinzel", "abstract": "Let $g(x)$ be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if $g(h(x))$ has boundedly many terms, then $h(x)\\in \\C[x]$ must also have boundedly many terms. Solving an older conjecture raised by R\\'enyi and by Erd\\\"os, Schinzel had proved this in the special cases $g(x)=x^d$; however that method does not extend to the general case. Here we prove the full Schinzel's conjecture (actually in sharper form) by a completely different method. Simultaneously we establish an \"algorithmic\" parametric description of the general decomposition $f(x)=g(h(x))$, where $f$ is a polynomial with a given number of terms and $g,h$ are arbitrary polynomials. As a corollary, this implies for instance that a polynomial with $l$ terms and given coefficients is non-trivially decomposable if and only if the degree-vector lies in the union of certain finitely many subgroups of $\\Z^l$."}
{"category": "Math", "title": "Two examples of surfaces with normal crossing singularities", "abstract": "This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in [EGA,III.2.6.2] and raised again by Viehweg."}
{"category": "Math", "title": "The \"fundamental theorem\" for the algebraic K-theory of spaces. III. the nil-term", "abstract": "In this paper we identify the ``nil-terms'' for Waldhausen's algebraic K-theory of spaces functor as the reduced K-theory of a category of equivariant spaces equipped with a homotopically nilpotent endomorphism."}
{"category": "Math", "title": "Extremal Graph Theory for Metric Dimension and Diameter", "abstract": "A set of vertices $S$ \\emph{resolves} a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \\emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of $G$. Let $\\mathcal{G}_{\\beta,D}$ be the set of graphs with metric dimension $\\beta$ and diameter $D$. It is well-known that the minimum order of a graph in $\\mathcal{G}_{\\beta,D}$ is exactly $\\beta+D$. The first contribution of this paper is to characterise the graphs in $\\mathcal{G}_{\\beta,D}$ with order $\\beta+D$ for all values of $\\beta$ and $D$. Such a characterisation was previously only known for $D\\leq2$ or $\\beta\\leq1$. The second contribution is to determine the maximum order of a graph in $\\mathcal{G}_{\\beta,D}$ for all values of $D$ and $\\beta$. Only a weak upper bound was previously known."}
{"category": "Math", "title": "Domestic canonical algebras and simple Lie algebras", "abstract": "For each simply-laced Dynkin graph $\\Delta$ we realize the simple complex Lie algebra of type $\\Delta$ as a quotient algebra of the complex degenerate composition Lie algebra $L(A)_{1}^{\\mathbb{C}}$ of a domestic canonical algebra $A$ of type $\\Delta$ by some ideal $I$ of $L(A)_{1}^{\\mathbb{C}}$ that is defined via the Hall algebra of $A$, and give an explicit form of $I$. Moreover, we show that each root space of $L(A)_{1}^{\\mathbb{C}}/I$ has a basis given by the coset of an indecomposable $A$-module $M$ with root easily computed by the dimension vector of $M$."}
{"category": "Math", "title": "Universal derived equivalences of posets", "abstract": "By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for any abelian category A, a functor between the categories of complexes of diagrams over X and Y with values in A. This functor induces a triangulated functor between the corresponding derived categories. This allows us to prove, for pairs X, Y of posets sharing certain common underlying combinatorial structure, that for any abelian category A, regardless of its nature, the categories of diagrams over X and Y with values in A are derived equivalent."}
{"category": "Math", "title": "The moduli space of cubic fourfolds via the period map", "abstract": "We characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space. It follows then that the GIT compactification of the moduli space of cubic fourfolds is isomorphic to the Looijenga's compactification associated to this arrangement. This work builds on and is a natural continuation of our previous paper on the GIT compactification of the moduli space of cubic fourfolds."}
{"category": "Math", "title": "Contraction semigroups of elliptic quadratic differential operators", "abstract": "We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the non-selfadjoint operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this paper that under the assumption of ellipticity, as soon as the real part of their Weyl symbols is a non-zero non-positive quadratic form, the norm of contraction semigroups generated by these operators decays exponentially in time."}
{"category": "Math", "title": "The period map for cubic fourfolds", "abstract": "The period map for cubic fourfolds takes values in a locally symmetric variety of orthogonal type of dimension 20. We determine the image of this period map (thus confirming a conjecture of Hassett) and give at the same time a new proof of the theorem of Voisin that asserts that this period map is an open embedding. An algebraic version of our main result is an identification of the algebra of SL(6)-invariant polynomials on the space of cubic forms in 6 complex variables with a certain algebra of meromorphic automorphic forms on a symmetric domain of orthogonal type of dimension 20. We also describe the stratification of the moduli space of semistable cubic fourfolds in terms of a Dynkin-Vinberg diagram."}
{"category": "Math", "title": "Ultrametric and tree potential", "abstract": "We study infinite tree and ultrametric matrices, and their action on the boundary of the tree. For each tree matrix we show the existence of a symmetric random walk associated to it and we study its Green potential. We provide a representation theorem for harmonic functions that includes simple expressions for any increasing harmonic function and the Martin kernel. In the boundary, we construct the Markov kernel whose Green function is the extension of the matrix and we simulate it by using a cascade of killing independent exponential random variables and conditionally independent uniform variables. For ultrametric matrices we supply probabilistic conditions to study its potential properties when immersed in its minimal tree matrix extension."}
{"category": "Math", "title": "On a Class of Ideals of the Toeplitz Algebra on the Bergman Space of the Unit Ball", "abstract": "Let $\\mathfrak{T}$ denote the full Toeplitz algebra on the Bergman space of the unit ball $\\mathbb{B}_n.$ For each subset $G$ of $L^{\\infty},$ let $\\mathfrak{CI}(G)$ denote the closed two-sided ideal of $\\mathfrak{T}$ generated by all $T_fT_g-T_gT_f$ with $f,g\\in G.$ It is known that $\\mathfrak{CI}(C(\\bar{\\mathbb{B}}_n))=\\mathcal{K}$ - the ideal of compact operators and $\\mathfrak{CI}(C(\\mathbb{B}_n))=\\mathfrak{T}.$ Despite these ``extremal cases'', $\\mathfrak{T}$ does contain other non-trivial ideals. This paper gives a construction of a class of subsets $G$ of $L^{\\infty}$ so that $\\mathcal{K}\\subsetneq\\mathfrak{CI}(G)\\subsetneq\\mathfrak{T}.$"}
{"category": "Math", "title": "An Inequality for Mixed Monge-Amp\\`ere measures", "abstract": "We generalize an inequality for mixed Monge-Amp\\`ere measures. We also give an example that shows that our assumptions are sharp. The corresponding result in the setting of compact K\\\"ahler manifold is also discussed."}
{"category": "Math", "title": "Some remarks on sinc integrals and their connection with combinatorics, geometry and probability", "abstract": "We give an alternative, combinatorial/geometrical evaluation of a class of improper sinc integrals studied by the Borweins. A probabilistic interpretation is also noted and used to shed light on a related combinatorial identity."}
{"category": "Math", "title": "Vicious walkers and random contraction matrices", "abstract": "The ensemble $\\CUE^{(q)}$ of truncated random unitary matrices is a deformation of the usual Circular Unitary Ensemble depending on a discrete non-negative parameter $q.$ $\\CUE^{(q)}$ is an exactly solved model of random contraction matrices originally introduced in the context of scattering theory. In this article, we exhibit a connection between $\\CUE^{(q)}$ and Fisher's random-turns vicious walker model from statistical mechanics. In particular, we show that the moment generating function of the trace of a random matrix from $\\CUE^{(q)}$ is a generating series for the partition function of Fisher's model, when the walkers are assumed to represent mutually attracting particles."}
{"category": "Math", "title": "A remark on left invariant metrics on compact Lie groups", "abstract": "We show that a left invariant metric on a compact Lie group $G$ which is obtained by stretching a biinvariant metric in the direction of a subalgebra $\\h$ of $\\g$ always has some negative sectional curvature, unless the semi-simple part of $\\h$ is an ideal of $\\g$."}
{"category": "Math", "title": "Classically normal pure states", "abstract": "A pure state f of a von Neumann algebra M is called classically normal if f is normal on any von Neumann subalgebra of M on which f is multiplicative. Assuming the continuum hypothesis, a separably represented von Neumann algebra M has classically normal, singular pure states iff there is a central projection p in M such that Mp is a factor of type I_\\infty, II, or III."}
{"category": "Math", "title": "The alternating sign matrix polytope", "abstract": "We define the alternating sign matrix polytope as the convex hull of nxn alternating sign matrices and prove its equivalent description in terms of inequalities. This is analogous to the well known result of Birkhoff and von Neumann that the convex hull of the permutation matrices equals the set of all nonnegative doubly stochastic matrices. We count the facets and vertices of the alternating sign matrix polytope and describe its projection to the permutohedron as well as give a complete characterization of its face lattice in terms of modified square ice configurations. Furthermore we prove that the dimension of any face can be easily determined from this characterization."}
{"category": "Math", "title": "Conjugate points in Euler's elastic problem", "abstract": "For the classical Euler's elastic problem, conjugate points are described. Inflectional elasticae admit the first conjugate point between the first and the third inflection points. All the rest elasticae do not have conjugate points."}
{"category": "Math", "title": "Secondary Characteristic Classes on Loop Spaces", "abstract": "A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. The connection and curvature forms of these metrics take values in pseudodifferential operators. We develop a theory of Wodzicki-Chern-Simons classes using the s=0, 1 connections and the Wodzicki residue. These classes distinguish the smooth homotopy type of some circle actions on M = S^2 x S^3, and imply that the fundamental group of Diff(M) is infinite."}
{"category": "Math", "title": "On the Chow ring of the stack of rational nodal curves", "abstract": "The goal of this paper is to compute the rational Chow ring of the stack consisting of nodal curves of genus 0 with at most 3 nodes: it is a Q-algebra with 10 generators and 11 relations."}
{"category": "Math", "title": "Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods", "abstract": "We define angles from-to and between infinite dimensional subspaces of a Hilbert space, inspired by the work of E. J. Hannan, 1961/1962 for general canonical correlations of stochastic processes. The spectral theory of selfadjoint operators is used to investigate the properties of the angles, e.g., to establish connections between the angles corresponding to orthogonal complements. The classical gaps and angles of Dixmier and Friedrichs are characterized in terms of the angles. We introduce principal invariant subspaces and prove that they are connected by an isometry that appears in the polar decomposition of the product of corresponding orthogonal projectors. Point angles are defined by analogy with the point operator spectrum. We bound the Hausdorff distance between the sets of the squared cosines of the angles corresponding to the original subspaces and their perturbations. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh-Ritz method, where the former subspace serves as a trial subspace and the orthogonal projector of the latter subspace serves as an operator in the Rayleigh-Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the trial subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. Finally, we consider the classical alternating projectors method and propose its ultimate acceleration, using the conjugate gradient approach. The corresponding convergence rate estimate is obtained in terms of the angles. We illustrate a possible acceleration for the domain decomposition method with a small overlap for the 1D diffusion equation."}
{"category": "Math", "title": "Reflected backward SDEs with two barriers under monotonicity and general increasing conditions", "abstract": "In this paper, we prove the existence and uniqueness result of the reflected BSDE with two continuous barriers under monotonicity and general increasing condition on $y$, with Lipschitz condition on $z$."}
{"category": "Math", "title": "Small Chvatal rank", "abstract": "We propose a variant of the Chvatal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra {xx : Ax <= b} as b varies. The number of steps needed is called the small Chvatal rank (SCR) of A. We characterize matrices for which SCR is zero via the notion of supernormality which generalizes unimodularity. SCR is studied in the context of the stable set problem in a graph, and we show that many of the well-known facet normals of the stable set polytope appear in at most two rounds of our procedure. Our results reveal a uniform hypercyclic structure behind the normals of many complicated facet inequalities in the literature for the stable set polytope. Lower bounds for SCR are derived both in general and for polytopes in the unit cube."}
{"category": "Math", "title": "Equivariant Chow ring and Chern classes of wonderful symmetric varieties of minimal rank", "abstract": "We describe the equivariant Chow ring of the wonderful compactification $X$ of a symmetric space of minimal rank, via restriction to the associated toric variety $Y$. Also, we show that the restrictions to $Y$ of the tangent bundle $T_X$ and its logarithmic analogue $S_X$ decompose into a direct sum of line bundles. This yields closed formulae for the equivariant Chern classes of $T_X$ and $S_X$, and, in turn, for the Chern classes of reductive groups considered by Kiritchenko."}
{"category": "Math", "title": "Geometry of limit sets for expansive Markov systems", "abstract": "We describe the geometric and dynamical properties of expansive Markov systems."}
{"category": "Math", "title": "Non-positive curvature and the Ptolemy inequality", "abstract": "We provide examples of non-locally compact geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolemy."}
{"category": "Math", "title": "Dirichlet series for finite combinatorial rank dynamics", "abstract": "We introduce a class of group endomorphisms -- those of finite combinatorial rank -- exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to have a closed rational form. Analytic properties of the Dirichlet series are related to orbit-growth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbit-growth asymptotics are found, all of which are polynomially bounded."}
{"category": "Math", "title": "Drinfeld second realization of the quantum affine superalgebras of $D^{(1)}(2,1;x)$ via the Weyl groupoid", "abstract": "We obtain Drinfeld second realization of the quantum affine superalgebras associated with the affine Lie superalgebra $D^{(1)}(2,1;x)$. Our results are analogous to those obtained by Beck for the quantum affine algebras. Beck's analysis uses heavily the (extended) affine Weyl groups of the affine Lie algebras. In our approach the structures are based on a Weyl groupoid."}
{"category": "Math", "title": "Geometric representation of interval exchange maps over algebraic number fields", "abstract": "We consider the restriction of interval exchange transformations to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically, and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable interval exchange maps with zero and non-zero drift vector, and carry out some investigations of their properties. In particular, we look for evidence of the finite decomposition property: each lattice is the union of finitely many orbits."}
{"category": "Math", "title": "Noncommutative tori and the Riemann-Hilbert correspondence", "abstract": "We study the interplay between noncommutative tori and noncommutative elliptic curves through a category of equivariant differential modules on $\\mathbb{C}^*$. We functorially relate this category to the category of holomorphic vector bundles on noncommutative tori as introduced by Polishchuk and Schwarz and study the induced map between the corresponding K-theories. In addition, there is a forgetful functor to the category of noncommutative elliptic curves of Soibelman and Vologodsky, as well as a forgetful functor to the category of vector bundles on $\\mathbb{C}^*$ with regular singular connections. The category that we consider has the nice property of being a Tannakian category, hence it is equivalent to the category of representations of an affine group scheme. Via an equivariant version of the Riemann-Hilbert correspondence we determine this group scheme to be (the algebraic hull of) $\\mathbb{Z}^2$. We also obtain a full subcategory of the category of holomorphic bundles of the noncommutative torus, which is equivalent to the category of representations of $\\mathbb{Z}$. This group is the proposed topological fundamental group of the noncommutative torus (understood as a degenerate elliptic curve) and we study Nori's notion of \\'etale fundamental group in this context."}
{"category": "Math", "title": "Lattice polytopes having h^*-polynomials with given degree and linear coefficient", "abstract": "The h^*-polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let P be a lattice polytope with h^*-polynomial of degree d and with linear coefficient h^*_1. We show that P has to be a lattice pyramid over a lower-dimensional lattice polytope, if the dimension of P is greater or equal to h^*_1 (2d+1) + 4d-1. This result has a purely combinatorial proof and generalizes a recent theorem of Batyrev."}
{"category": "Math", "title": "The hook fusion procedure for Hecke algebras", "abstract": "We derive a new expression for the q-analogue of the Young symmetrizer which generate irreducible representations of the Hecke algebra. We obtain this new expression using Cherednik's fusion procedure. However, instead of splitting Young diagrams into their rows or columns, we consider their principal hooks. This minimises the number of auxiliary parameters needed in the fusion procedure."}
{"category": "Math", "title": "Monge-Amp\\`ere Measures for Convex Bodies and Bernstein-Markov Type Inequalities", "abstract": "We use geometric methods to calculate a formula for the complex Monge-Amp\\`ere measure $(dd^cV_K)^n$, for $K \\Subset \\RR^n \\subset \\CC^n$ a convex body and $V_K$ its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies $K$. We apply this to show that two methods for deriving Bernstein-Markov-type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same results for all convex bodies. A key role is played by the geometric result that the extremal inscribed ellipses appearing in approximation theory are the maximal area ellipses determining the complex Monge-Amp\\`ere solution $V_K$."}
{"category": "Math", "title": "Weil-Petersson Metric Geometry Quick Overview", "abstract": "A quick overview is provided on the current development of the WP metric geometry."}
{"category": "Math", "title": "On compact manifolds admitting indefinite metrics with parallel Weyl tensor", "abstract": "Compact pseudo-Riemannian manifolds that have parallel Weyl tensor without being conformally flat or locally symmetric are known to exist in infinitely many dimensions greater than 4. We prove some general topological properties of such manifolds, namely, vanishing of the Euler characteristic and real Pontryagin classes, and infiniteness of the fundamental group. We also show that, in the Lorentzian case, each of them is at least 5-dimensional and admits a two-fold cover which is a bundle over the circle."}
{"category": "Math", "title": "Cyclicity of period annuli and principalization of Bautin ideals", "abstract": "We prove that the maximal number of limit cycles which bifurcate from an open period annulus under a given multi-parameter analytic deformation of a given analytic vector field is the same as in an appropriate one-parameter analytic deformation of the field, provided that this cyclicity is finite. Along the same lines we give also a bound of the cyclicity of homoclinic saddle loops."}
{"category": "Math", "title": "Quotients of cluster categories", "abstract": "Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories are actually just quotients of cluster categories. The other half can be obtained as quotients of 2-cluster categories, the \"lowest\" type of higher cluster categories. Hence, in Dynkin types A and D, all higher cluster phenomena are implicit in cluster categories and 2-cluster categories. In contrast, the same is not true in Dynkin type E."}
{"category": "Math", "title": "Exotic rational elliptic surfaces without 1-handles", "abstract": "Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface $E(1)_{2,3}$ requires both 1- and 3-handles. In this article, we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as $E(1)_{2,3}$ and admits neither 1- nor 3-handles, by using rational blow-downs and Kirby calculus. Our manifold gives the first example of either a counterexample to the Harer-Kas-Kirby conjecture or a homeomorphic but non-diffeomorphic pair of simply connected closed smooth 4-manifolds with the same non-vanishing Seiberg-Witten invariants."}
{"category": "Math", "title": "The cyclic homology of monogenic extensions in the noncommutative setting", "abstract": "We study the Hochschild and cyclic homologies of noncommutative monogenic extensions. As an aplication we compute the Hochschild and cyclic homologies of the rank~1 Hopf algebras introduced by L. Krop and D. Radford in [Finite dimensional Hopf algebras of rank 1 in characteristic 0, Journal of Algebra 302, no. 1, 214-230} (2006)]."}
{"category": "Math", "title": "Cusp Algebras", "abstract": "We consider simple cusp algebras, that is certain subalgebras of the algebra of holomorphic functions on a disk that are annihilated by some distributions living on a singleton. We determine when these algebras can be holized in two dimensions, and when these holizations are globally biholomorphic."}
{"category": "Math", "title": "Bases invariantes de friabilit\\'e", "abstract": "Given a finite residue field $k$, one looks for a smoothness basis that is invariant under the automorphism group of $k$. We construct models for some finite fields that admit such a basis. This work aims at accelerating algorithms for computing discrete logarithms in some finite residue fields."}
{"category": "Math", "title": "The Group of Hamiltonian Homeomorphisms in the L^\\infty-norm", "abstract": "The group Hameo (M,\\omega) of Hamiltonian homeomorphisms of a connected symplectic manifold (M,\\omega) was defined and studied in [7] and further in [6]. In these papers, the authors consistently used the L^{(1,\\infty)}-Hofer norm (and not the L^\\infty-Hofer norm) on the space of Hamiltonian paths (see below for the definitions). A justification for this choice was given in [7]. In this article we study the L^\\infty-case. In view of the fact that the Hofer norm on the group Ham (M,\\omega) of Hamiltonian diffeomorphisms does not depend on the choice of the L^{(1,\\infty)}-norm vs. the L^\\infty-norm [9], Y.-G. Oh and D. McDuff (private communications) asked whether the two notions of Hamiltonian homeomorphisms arising from the different norms coincide. We will give an affirmative answer to this question in this paper."}
{"category": "Math", "title": "Pure states on free group C*-algebras", "abstract": "We prove that all of the pure states of the reduced C*-algebra of the free goup on $\\aleph_1$ generators are *-automorphism equivalent and extract some consequences of that fact."}
{"category": "Math", "title": "Puzzles, Tableaux and Mosaics", "abstract": "We define mosaics, which are naturally in bijection with Knutson-Tao puzzles. We define an operation on mosaics, which shows they are also in bijection with Littlewood-Richardson skew-tableaux. Another consequence of this construction is that we obtain bijective proofs of commutativity and associativity for the ring structures defined either of these objects. In particular, we obtain a new, easy proof of the Littlewood-Richardson rule. Finally we discuss how our operation is related to other known constructions, particularly jeu de taquin."}
{"category": "Math", "title": "A transcendental approach to Koll\\'ar's injectivity theorem II", "abstract": "We treat a relative version of the main theorem in my previous paper: A transcendental approach to Koll\\'ar's injectivity theorem. More explicitly, we give a curvature condition that implies Koll\\'ar type cohomology injectivity theorems in the relative setting. To carry out this generalization, we use the Ohsawa-Takegoshi twisted version of Nakano's identity."}
{"category": "Math", "title": "F-thresholds of hypersurfaces", "abstract": "We continue our study of F-thresholds begun in math/0607660 by an in depth analysis of the hypersurface case. We use the D--module theoretic description of generalized test ideals which allows us to show that in any F--finite regular ring the F-thresholds of hypersurfaces are discrete and rational (in math/0607660 the finite type over a field case was shown for arbitrary ideals). Furthermore we show that any limit of F-pure thresholds of principal ideals in bouneded dimension is again an F-pure-threshold, hence in particular the limit is rational. The study of the set of F-pure-thresholds leads to natural analogs of conjectures of Shokurov and Koll\\'{a}r (for log canonical thresholds) in the case of F-pure-thresholds."}
{"category": "Math", "title": "Sch\\'emas en groupes et poids de Diamond-Serre", "abstract": "This note is a correction of (statement and proof of) proposition 3.3.1 of Toby Gee's preprint intitled *On the weights of mod p Hilbert modular forms*. The aim is to compare Galois representations arising from extensions of some group schemes (over the ring of integers of a p-adic field) endowed with a descent data, and extensions of some crystalline representations with given Hodge-Tate weights. The main tool of the proof is the theory of Breuil."}
{"category": "Math", "title": "A simple solution to Ulam's liar game with one lie", "abstract": "Ulam asked for the maximum number of questions required to determine an integer between one and one million by asking questions whose answer is `Yes' or `No' and where one untruthful answer is allowed. Pelc showed that the number of questions required is 25. Here we give a simple proof of this result."}
{"category": "Math", "title": "Fluctuations of eigenvalues and second order Poincar\\'e inequalities", "abstract": "Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of `second order Poincar\\'e inequalities': just as ordinary Poincar\\'e inequalities give variance bounds, second order Poincar\\'e inequalities give central limit theorems. The proof of the main result employs Stein's method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices."}
{"category": "Math", "title": "Statistical minimax approach of the Hausdorff moment problem", "abstract": "The purpose of this paper is to study the problem of estimating a compactly supported density of probability from noisy observations of its moments. In fact, we provide a statistical approach to the famous Hausdorff classical moment problem. We prove an upper bound and a lower bound on the rate of convergence of the mean squared error showing that the considered estimator attains minimax rate over the corresponding smoothness classes."}
{"category": "Math", "title": "Crown theory for the upper halfplane", "abstract": "I exemplify part of my recent work on the upper halfplane."}
{"category": "Math", "title": "Microlocalization of rational Cherednik algebras", "abstract": "We construct a microlocalization of the rational Cherednik algebras $H$ of type $S_n$. This is achieved by a quantization of the Hilbert scheme $\\Hilb^n\\C^2$ of $n$ points in $\\C^2$. We then prove the equivalence of the category of $H$-modules and the one of modules over its microlocalization under certain conditions on the parameter."}
{"category": "Math", "title": "Diagonalizing the Frobenius", "abstract": "Over a Noetherian, local ring R of prime characteristic p, the Frobenius functor F induces a diagonalizable map on certain quotients of rational Grothendieck groups. This leads to an explicit formula for the Dutta multiplicity, and it is shown that a weaker version of Serre's vanishing conjecture holds if only chi(F(X)) = p^{dim R}chi(X) for all bounded complexes X of finitely generated, projective modules with finite length homology."}
{"category": "Math", "title": "Dualities and intersection multiplicities", "abstract": "Let R be a commutative, noetherian, local ring. Topological Q-vector spaces modelled on full subcategories of the derived category of R are constructed in order to study intersection multiplicities."}
{"category": "Math", "title": "Extremal first Dirichlet eigenvalue of doubly connected plane domains and dihedral symmetry", "abstract": "We deal with the following eigenvalue optimization problem: Given a bounded domain $D\\subset \\R^2$, how to place an obstacle $B$ of fixed shape within $D$ so as to maximize or minimize the fundamental eigenvalue $\\lambda_1$ of the Dirichlet Laplacian on $D\\setminus B$. This means that we want to extremize the function $\\rho\\mapsto \\lambda_1(D\\setminus \\rho (B))$, where $\\rho$ runs over the set of rigid motions such that $\\rho (B)\\subset D$. We answer this problem in the case where both $D$ and $B$ are invariant under the action of a dihedral group $\\mathbb{D}_n$, $n\\ge2$, and where the distance from the origin to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of $B$ coincide with those of $D$."}
{"category": "Math", "title": "Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold", "abstract": "For any bounded regular domain $\\Omega$ of a real analytic Riemannian manifold $M$, we denote by $\\lambda_{k}(\\Omega)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $\\Omega$. In this paper, we consider $\\lambda_k$ and as a functional upon the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $\\lambda_k$. These results rely on Hadamard type variational formulae that we establish in this general setting."}
{"category": "Math", "title": "Diffusion covariation and co-jumps in bidimensional asset price processes with stochastic volatility and infinite activity Levy jumps", "abstract": "In this paper we consider two processes driven by diffusions and jumps. The jump components are Levy processes and they can both have finite activity and infinite activity. Given discrete observations we estimate the covariation between the two diffusion parts and the co-jumps. The detection of the co-jumps allows to gain insight in the dependence structure of the jump components and has important applications in finance. Our estimators are based on a threshold principle allowing to isolate the jumps. This work follows Gobbi and Mancini (2006) where the asymptotic normality for the estimator of the covariation, with convergence speed given by the squared root of h, was obtained when the jump components have finite activity. Here we show that the speed is the squared root of h only when the activity of the jump components is moderate."}
{"category": "Math", "title": "Causal inference in longitudinal studies with history-restricted marginal structural models", "abstract": "A new class of Marginal Structural Models (MSMs), History-Restricted MSMs (HRMSMs), was recently introduced for longitudinal data for the purpose of defining causal parameters which may often be better suited for public health research or at least more practicable than MSMs \\citejoffe,feldman. HRMSMs allow investigators to analyze the causal effect of a treatment on an outcome based on a fixed, shorter and user-specified history of exposure compared to MSMs. By default, the latter represent the treatment causal effect of interest based on a treatment history defined by the treatments assigned between the study's start and outcome collection. We lay out in this article the formal statistical framework behind HRMSMs. Beyond allowing a more flexible causal analysis, HRMSMs improve computational tractability and mitigate statistical power concerns when designing longitudinal studies. We also develop three consistent estimators of HRMSM parameters under sufficient model assumptions: the Inverse Probability of Treatment Weighted (IPTW), G-computation and Double Robust (DR) estimators. In addition, we show that the assumptions commonly adopted for identification and consistent estimation of MSM parameters (existence of counterfactuals, consistency, time-ordering and sequential randomization assumptions) also lead to identification and consistent estimation of HRMSM parameters."}
{"category": "Math", "title": "On regularity of SLE_8 curves", "abstract": "This paper has been withdrawn by the author due to essential mistakes in some previous versions."}
{"category": "Math", "title": "A novel characterization of the Iwasawa decomposition of a simple Lie group", "abstract": "It is about the uniqueness of the Iwasawa decomposition."}
{"category": "Math", "title": "On Power Stable Ideals", "abstract": "We define the notion of a power stable ideal in a polynomial ring $ R[X]$ over an integral domain $ R $. It is proved that a maximal ideal $\\chi$ $ M $ in $ R[X]$ is power stable if and only if $ P^t $ is $ P$- primary for all $ t\\geq 1 $ for the prime ideal $ P = M \\cap R $. Using this we prove that for a Hilbert domain $R$ any radical ideal in $R[X]$ which is a finite intersection G-ideals is power stable. Further, we prove that if $ R $ is a Noetherian integral domain of dimension 1 then any radical ideal in $ R[X] $ is power stable. Finally, it is proved that if every ideal in $ R[X]$ is power stable then $ R $ is a field."}
{"category": "Math", "title": "Uniform random sampling of planar graphs in linear time", "abstract": "This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Gim\\'enez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost. Then, the expected time complexity of generation is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with $n$ vertices, which was a little over $O(n^7)$."}
{"category": "Math", "title": "Some result on K-algebras", "abstract": "We give a new proof of the classical result due to Rodney Y. Sharp and Peter Vamos on the dimension of tensor product of a finite number of field extensions of a given field."}
{"category": "Math", "title": "Domino Tiling Congruence Modulo 4", "abstract": "The number of domino tilings of a region with reflective symmetry across a line is combinatorially shown to depend on the number of domino tilings of particular subregions, modulo 4. This expands upon previous congruency results for domino tilings, modulo 2, and leads to a variety of corollaries, including that the number of domino tilings of a k x 2k rectangle is congruent to 1 mod 4."}
{"category": "Math", "title": "A General Fredholm Theory II: Implicit Function Theorems", "abstract": "This is the revised version of the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory. Some proofs have been improved and a glossary added."}
{"category": "Math", "title": "Novikov algebras and Novikov structures on Lie algebras", "abstract": "We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures."}
{"category": "Math", "title": "Mutant knots with symmetry", "abstract": "Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m>2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directed m-string satellites of mutants share the same Homfly polynomial for m<6 in general, and for all choices of m when the satellite is based on a cable knot pattern. We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantum sl(3) invariants."}
{"category": "Math", "title": "Third Order Newton's Method for Zernike Polynomial Zeros", "abstract": "The Zernike radial polynomials are a system of orthogonal polynomials over the unit interval with weight x. They are used as basis functions in optics to expand fields over the cross section of circular pupils. To calculate the roots of Zernike polynomials, we optimize the generic iterative numerical Newton's Method that iterates on zeros of functions with third order convergence. The technique is based on rewriting the polynomials as Gauss hypergeometric functions, reduction of second order derivatives to first order derivatives, and evaluation of some ratios of derivatives by terminating continued fractions. A PARI program and a short table of zeros complete up to polynomials of 20th order are included."}
{"category": "Math", "title": "Global Solutions to the Ultra-Relativistic Euler Equations", "abstract": "We prove a global existence theorem for the $3\\times 3$ system of relativistic Euler equations in one spacial dimension. It is shown that in the ultra-relativistic limit, there is a family of equations of state that satisfy the second law of thermodynamics for which solutions exist globally. With this limit and equation of state, which includes equations of state for both an ideal gas and one dominated by radiation, the relativistic Euler equations can be analyzed by a Nishida-type method leading to a large data existence theorem, including the entropy and particle number evolution, using a Glimm scheme. Our analysis uses the fact that the equations of state are of the form $p=p(n,S)$, but whose form simplifies to $p=a^{2}\\rho$ when viewed as a function of $\\rho$ alone."}
{"category": "Math", "title": "Invertibility of the Gabor frame operator on the Wiener amalgam space", "abstract": "We use a generalization of Wiener's $1/f$ theorem to prove that for a Gabor frame with the generator in the Wiener amalgam space $W(L^{\\infty}, \\ell^{1}_{\\nu})(\\mathbb{R}^{d})$, the corresponding frame operator is invertible on this space. Therefore, for such a Gabor frame, the generator of the canonical dual belongs also to $W(L^{\\infty}, \\ell^{1}_{\\nu})(\\mathbb{R}^{d}) $"}
{"category": "Math", "title": "On the approximate normality of eigenfunctions of the Laplacian", "abstract": "The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an eigenfunction of the Laplacian with $L^2$-norm one and eigenvalue $-\\mu$, then $$d_{TV}(f(X),Z)\\le\\frac{2}{\\mu}\\E\\big|\\|\\nabla f(X)\\|^2-\\E\\|\\nabla f(X) \\|^2\\big|.$$ This result is applied to construct specific examples of spherical harmonics of arbitrary (odd) degree which are close to Gaussian in distribution. A second application is given to random linear combinations of eigenfunctions on flat tori."}
{"category": "Math", "title": "Semisimple Algebraic Groups in Characteristic Zero", "abstract": "It is shown that the classification theorems for semisimple algebraic groups in characteristic zero can be derived quite simply and naturally from the corresponding theorems for Lie algebras by using a little of the theory of tensor categories. This article is extracted from Milne 2007."}
{"category": "Math", "title": "Thom polynomials and Schur functions: the singularities I_{2,2}(-)", "abstract": "We give the Thom polynomials for the singularities $I_{2,2}$ associated with maps $({\\bf C}^{\\bullet},0) \\to ({\\bf C}^{\\bullet+k},0)$ with parameter $k\\ge 0$. Our computations combine the characterization of Thom polynomials via the ``method of restriction equations'' of Rimanyi et al. with the techniques of Schur functions."}
{"category": "Math", "title": "On Two of Erd\\\"os's Open Problems", "abstract": "In this short note we present some remarks and conjectures on two of Erd\\\"os's open problems in number theory."}
{"category": "Math", "title": "Quantum cohomology of G/P and homology of affine Grassmannian", "abstract": "Let G be a simple and simply-connected complex algebraic group, P \\subset G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH^*(G/P) of a flag variety is, up to localization, a quotient of the homology H_*(Gr_G) of the affine Grassmannian \\Gr_G of G. As a consequence, all three-point genus zero Gromov-Witten invariants of $G/P$ are identified with homology Schubert structure constants of H_*(Gr_G), establishing the equivalence of the quantum and homology affine Schubert calculi. For the case G = B, we use the Mihalcea's equivariant quantum Chevalley formula for QH^*(G/B), together with relationships between the quantum Bruhat graph of Brenti, Fomin and Postnikov and the Bruhat order on the affine Weyl group. As byproducts we obtain formulae for affine Schubert homology classes in terms of quantum Schubert polynomials. We give some applications in quantum cohomology. Our main results extend to the torus-equivariant setting."}
{"category": "Math", "title": "The spectral shift function and spectral flow", "abstract": "This paper extends Krein's spectral shift function theory to the setting of semifinite spectral triples. We define the spectral shift function under these hypotheses via Birman-Solomyak spectral averaging formula and show that it computes spectral flow."}
{"category": "Math", "title": "Every compact group arises as the outer automorphism group of a II_1 factor", "abstract": "We show that any compact group can be realized as the outer automorphism group of a factor of type II_1. This has been proved in the abelian case by Ioana, Peterson and Popa applying Popa's deformation/rigidity techniques to amalgamated free product von Neumann algebras. Our methods are a generalization of theirs."}
{"category": "Math", "title": "Counting hyperelliptic curves that admit a Koblitz model", "abstract": "Let k be a finite field of odd characteristic. We find a closed formula for the number of k-isomorphism classes of pointed, and non-pointed, hyperelliptic curves of genus g over k, admitting a Koblitz model. These numbers are expressed as a polynomial in the cardinality q of k, with integer coefficients (for pointed curves) and rational coefficients (for non-pointed curves). The coefficients depend on g and the set of divisors of q-1 and q+1. These formulas show that the number of hyperelliptic curves of genus g suitable (in principle) of cryptographic applications is asymptotically (1-e^{-1})2q^{2g-1}, and not 2q^{2g-1} as it was believed. The curves of genus g=2 and g=3 are more resistant to the attacks to the DLP; for these values of g the number of curves is respectively (91/72)q^3+O(q^2) and (3641/2880)q^5+O(q^4)."}
{"category": "Math", "title": "Fonction Z\\^eta de Hurwitz p-adique et irrationalit\\'e", "abstract": "The knowledge on irrationality of p-adic zeta values has recently progressed. The irrationality of zeta_2(2), \\zeta_2(3) and of a few other p-adic series of Dirichlet was obtained by F. Calegari. F. Beukers gave a more elementary proof of this result. In parallel, T. Rivoal has just obtained, in the complex case, some Pade approximants of Lerch functions. It is this work which, transposed to C_p, enables us to obtain results of irrationality and linear independence."}
{"category": "Math", "title": "On b-function, spectrum and multiplier ideals", "abstract": "We give a survey on b-function, spectrum, and multiplier ideals together with certain interesting relations among them including the case of arbitrary subvarieties."}
{"category": "Math", "title": "Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space", "abstract": "We classify the entire minimal vertical graphs in the 3 dimensional Heisenberg group Nil endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in Nil, is given in terms of the Abresch-Rosenberg holomorphic differential for minimal surfaces in Nil."}
{"category": "Math", "title": "Contractible groups and linear dilatation structures", "abstract": "A dilatation structure on a metric space, arXiv:math/0608536v4, is a notion in between a group and a differential structure, accounting for the approximate self-similarity of the metric space. The basic objects of a dilatation structure are dilatations (or contractions). The axioms of a dilatation structure set the rules of interaction between different dilatations. Linearity is also a property which can be explained with the help of a dilatation structure. In this paper we show that we can speak about two kinds of linearity: the linearity of a function between two dilatation structures and the linearity of the dilatation structure itself. Our main result here is a characterization of contractible groups in terms of dilatation structures. To a normed conical group (normed contractible group) we can naturally associate a linear dilatation structure. Conversely, any linear and strong dilatation structure comes from the dilatation structure of a normed contractible group."}
{"category": "Math", "title": "Embedding Degree of Hyperelliptic Curves with Complex Multiplication", "abstract": "Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication. In this paper we show that if the l-Sylow subgroup of the Jacobian is not cyclic, then the embedding degree of the Jacobian with respect to l is one."}
{"category": "Math", "title": "Rational homotopy type of subspace arrangements with a geometric lattice", "abstract": "Let A be a subspace arrangement with a geometric lattice such that codim(x) > 1 for every x in A. Using rational homotopy theory, we prove that the complement M(A) is rationally elliptic if and only if the sum of the orthogonal subspaces is a direct sum. The homotopy type of M(A) is also given: it is a product of odd dimensional spheres. Finally, some other equivalent conditions are given, such as Poincare duality. Those results give a complete description of arrangements (with geometric lattice and with the codimension condition on the subspaces) such that M(A) is rationally elliptic, and show that most arrangements have an hyperbolic complement."}
{"category": "Math", "title": "Homotopy Lie algebra of the complements of subspace arrangements with geometric lattices", "abstract": "Let A be a geometric arrangement such that codim(x) > 1 for every x in A. We prove that, if the complement space M(A) is rationally hyperbolic, then there exists an injective from a free Lie algebra L(u,v) to the homotopy Lie algebra of M(A)."}
{"category": "Math", "title": "A simple uniform approach to complexes arising from forests", "abstract": "In this paper we present a unifying approach to study the homotopy type of several complexes arising from forests. We show that this method applies uniformly to many complexes that have been extensively studied."}
{"category": "Math", "title": "Bispectrality of multivariable Racah-Wilson polynomials", "abstract": "We construct a commutative algebra A_x of difference operators in R^p, depending on p+3 real parameters which is diagonalized by the multivariable Racah polynomials R_p(n;x) considered by Tratnik [27]. It is shown that for specific values of the variables x=(x_1,x_2,...,x_p) there is a hidden duality between n and x. Analytic continuation allows us to construct another commutative algebra A_n in the variables n=(n_1,n_2,...,n_p) which is also diagonalized by R_p(n;x). Thus R_p(n;x) solve a multivariable discrete bispectral problem in the sense of Duistermaat and Grunbaum [8]. Since a change of the variables and the parameters in the Racah polynomials gives the multivariable Wilson polynomials [26], this change of variables and parameters in A_x and A_n leads to bispectral commutative algebras for the multivariable Wilson polynomials."}
{"category": "Math", "title": "Busemann points of Artin groups of dihedral type", "abstract": "We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is the limits of geodesic rays. In the case of the dual generators, it turns out that all boundary points are Busemann points, but this is not true for the Artin generators. We also characterise the geodesics with respect to the dual generators, which allows us to calculate the associated geodesic growth series."}
{"category": "Math", "title": "Nonorientable 3-manifolds admitting coloured triangulations with at most 30 tetrahedra", "abstract": "We present the census of all non-orientable, closed, connected 3-manifolds admitting a rigid crystallization with at most 30 vertices. In order to obtain the above result, we generate, manipulate and compare, by suitable computer procedures, all rigid non-bipartite crystallizations up to 30 vertices."}
{"category": "Math", "title": "Multiple Solutions for a Henon-Like Equation on the Annulus", "abstract": "For the equation (-\\Delta u = | |x|-2 |^\\alpha u^{p-1}), (1 < |x| < 3), we prove the existence of two solutions for (\\alpha) large, and of two additional solutions when (p) is close to the critical Sobolev exponent (2^*=2N/(N-2)). A symmetry--breaking phenomenon appears, showing that the least--energy solutions cannot be radial functions."}
{"category": "Math", "title": "Trivialization of C(X)-algebras with strongly self-absorbing fibres", "abstract": "Suppose $A$ is a separable unital $C(X)$-algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_{1}$-injective $C^{*}$-algebra $D$. We show that $A$ and $C(X) \\otimes D$ are isomorphic as $C(X)$-algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case."}
{"category": "Math", "title": "Small clones and the projection property", "abstract": "In 1986, the second author classified the minimal clones on a finite universe into five types. We extend this classification to infinite universes and to multiclones. We show that every non-trivial clone contains a \"small\" clone of one of the five types. From it we deduce, in part, an earlier result, namely that if $\\mathcal C$ is a clone on a universe $A$ with at least two elements, that contains all constant operations, then all binary idempotent operations are projections and some $m$-ary idempotent operation is not a projection some $m\\geq 3$ if and only if there is a Boolean group $G$ on $A$ for which $\\mathcal C$ is the set of all operations $f(x_1,..., x_n)$ of the form $a+\\sum_{i\\in I}x_i$ for $a\\in A$ and $I\\subseteq \\{1,..., n\\}$."}
{"category": "Math", "title": "Differentiable and deformation type of algebraic surfaces, real and symplectic structures", "abstract": "Lecture 1: Projective and K\\\"ahler Manifolds, the Enriques classification, construction techniques. Lecture 2: Surfaces of general type and their Canonical models. Deformation equivalence and singularities. Lecture 3: Deformation and diffeomorphism, canonical symplectic structure for surfaces of general type. Lecture 4: Irrational pencils, orbifold fundamental groups, and surfaces isogenous to a product. Lecture 5: Lefschetz pencils, braid and mapping class groups, and diffeomorphism of ABC-surfaces. Epilogue: Deformation, diffeomorphism and symplectomorphism type of surfaces of general type."}
{"category": "Math", "title": "On the distance between separatrices for the discretized logistic differential equation", "abstract": "We consider the discretization y(t+\\epsilon)=y(t-\\epsilon)+2\\epsilon\\big(1-y(t)^{2}\\big), $\\epsilon>0$ a small parameter, of the logistic differential equation $y'=2\\big(1-y^{2}\\big)$, which can also be seen as a discretization of the system {y'=2\\big(1-v^{2}\\big), v'= 2\\big(1-y^{2}\\big). This system has two saddle points at $A=(1,1)$, $B=(-1, -1)$ and there exist stable and unstable manifolds. We will show that the stable manifold $W_{s}^{+}$ of the point $A=(1,1)$ and the unstable manifold $W_{i}^{-}$ of the point $B=(-1, -1)$ for the discretization do not coincide. The vertical distance between these two manifolds is exponentially small but not zero, in particular we give an asymptotic estimate of this distance. For this purpose we will use a method adapted from the paper of Sch\\\"afke-Volkmer \\cite{SV} using formal series and accurate estimates of the coefficients."}
{"category": "Math", "title": "The Rotor-Router Model on Regular Trees", "abstract": "The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We show that the set of occupied sites for this model on an infinite regular tree is a perfect ball whenever it can be, provided the initial rotor configuration is acyclic (that is, no two neighboring vertices have rotors pointing to one another). This is proved by defining the rotor-router group of a graph, which we show is isomorphic to the sandpile group. We also address the question of recurrence and transience: We give two rotor configurations on the infinite ternary tree, one for which chips exactly alternate escaping to infinity with returning to the origin, and one for which every chip returns to the origin. Further, we characterize the possible \"escape sequences\" for the ternary tree, that is, binary words a_1 ... a_n for which there exists a rotor configuration so that the k-th chip escapes to infinity if and only if a_k=1."}
{"category": "Math", "title": "On the square root of the centre of the Hecke algebra of type A", "abstract": "In this paper we investigate non-central elements of the Iwahori-Hecke algebra of the symmetric group whose squares are central. In particular, we describe a commutative subalgebra generated by certain non-central square roots of central elements, and the generic existence of a rank-three submodule of the Hecke algebra contained in the square root of the centre but not in the centre. The generators for this commutative subalgebra include the longest word and elements related to trivial and sign characters of the Hecke algebra. We find elegant expressions for the squares of such generators in terms of both the minimal basis of the centre and the elementary symmetric functions of Murphy elements."}
{"category": "Math", "title": "On Shalika Periods and a Theorem of Jacquet-Martin", "abstract": "Let \\pi be a cuspidal automorphic representation of GL_4 with central character \\mu^2. It is known that \\pi has Shalika period with respect to \\mu if and only if the L-function L^S(s, \\pi, \\bigwedge^2\\otimes\\mu^{-1}) has a pole at s=1. Recentlt, Jacquet and Martin considered the analogous question for cuspidal representations \\pi_D of the inner form GL_2(D)(\\A), and obtained a partial result via the relative trace formula. In this paper, we provide a complete solution to this problem via the method of theta correspondence, and give necessary and sufficient conditions for the existence of Shalika period for \\pi_D. We also resolve the analogous question in the local setting."}
{"category": "Math", "title": "Axiomatic theory of divergent series and cohomological equations", "abstract": "A general theory of summation of divergent series based on the Hardy-Kolmogorov axioms is developed. A class of functional series is investigated by means of ergodic theory. The results are formulated in terms of solvability of some cohomological equations, all solutions to which are nonmeasurable. In particular, this realizes a construction of a nonmeasurable function as first conjectured by Kolmogorov."}
{"category": "Math", "title": "A new integral basis for the centre of the Hecke algebra of type A", "abstract": "We describe a recursive algorithm that produces an integral basis for the centre of the Hecke algebra of type A consisting of linear combinations of monomial symmetric polynomials of Jucys--Murphy elements. We also discuss the existence of integral bases for the centre of the Hecke algebra that consist solely of monomial symmetric polynomials of Jucys--Murphy elements. Finally, for n=3, we show that only one such basis exists for the centre of the Hecke algebra, by proving that there are exactly four bases for the centre of the corresponding symmetric group algebra which consist solely of monomial symmetric polynomials of Jucys--Murphy elements."}
{"category": "Math", "title": "Bauer-Furuta invariants under Z_2-actions", "abstract": "S.Bauer and M.Furuta defined a stable cohomotopy refinement of the Seiberg-Witten invariants. In this paper, we prove a vanishing theorem of Bauer-Furuta invariants for 4-manifolds with smooth Z/2-actions. As an application, we give a constraint on smooth Z/2-actions on homotopy K3#K3, and construct a nonsmoothable locally linear Z/2-action on K3#K3. We also construct a nonsmoothable locally linear Z/2-action on $K3$."}
{"category": "Math", "title": "Weight 2 blocks of general linear groups and modular Alvis-Curtis duality", "abstract": "We obtain the structure of weight 2 blocks and [2:1]-pairs of q-Schur algebras, and compute explicitly the modular Alvis-Curtis duality for weight 2 blocks of finite general linear groups in non-defining characteristic."}
{"category": "Math", "title": "Application of Girsanov Theorem to Particle Filtering of Discretely Observed Continuous-Time Non-Linear Systems", "abstract": "This article considers the application of particle filtering to continuous-discrete optimal filtering problems, where the system model is a stochastic differential equation, and noisy measurements of the system are obtained at discrete instances of time. It is shown how the Girsanov theorem can be used for evaluating the likelihood ratios needed in importance sampling. It is also shown how the methodology can be applied to a class of models, where the driving noise process is lower in the dimensionality than the state and thus the laws of state and noise are not absolutely continuous. Rao-Blackwellization of conditionally Gaussian models and unknown static parameter models is also considered."}
{"category": "Math", "title": "Proof of the Double Bubble Conjecture in R^n", "abstract": "The least-area hypersurface enclosing and separating two given volumes in R^n is the standard double bubble."}
{"category": "Math", "title": "Perturbations of quadratic centers of genus one", "abstract": "We propose a program for finding the cyclicity of period annuli of quadratic systems with centers of genus one. As a first step, we classify all such systems and determine the essential one-parameter quadratic perturbations which produce the maximal number of limit cycles. We compute the associated Poincare-Pontryagin-Melnikov functions whose zeros control the number of limit cycles. To illustrate our approach, we determine the cyclicity of the annuli of two particular reversible systems."}
{"category": "Math", "title": "Estimates of the topological entropy from below for continuous self-maps on some compact manifolds", "abstract": "Extending our results in \"Entropy conjecture for continuous maps of nilmanifolds\", to appear in Israel Jour. of Math., we confirm that Entropy Conjecture holds for every continuous self-map of a compact $K(\\pi,1)$ manifold with the fundamental group $\\pi$ torsion free and virtually nilpotent, in particular for every continuous map of an infra-nilmanifold. In fact we prove a stronger version, a lower estimate of the topological entropy of a map by logarithm of the spectral radius of an associated \"linearization matrix\" with integer entries. From this, referring to known estimates of Mahler measure of polynomials, we deduce some absolute lower bounds for the entropy."}
{"category": "Math", "title": "Determining full conditional independence by low-order conditioning", "abstract": "A concentration graph associated with a random vector is an undirected graph where each vertex corresponds to one random variable in the vector. The absence of an edge between any pair of vertices (or variables) is equivalent to full conditional independence between these two variables given all the other variables. In the multivariate Gaussian case, the absence of an edge corresponds to a zero coefficient in the precision matrix, which is the inverse of the covariance matrix. It is well known that this concentration graph represents some of the conditional independencies in the distribution of the associated random vector. These conditional independencies correspond to the \"separations\" or absence of edges in that graph. In this paper we assume that there are no other independencies present in the probability distribution than those represented by the graph. This property is called the perfect Markovianity of the probability distribution with respect to the associated concentration graph. We prove in this paper that this particular concentration graph, the one associated with a perfect Markov distribution, can be determined by only conditioning on a limited number of variables. We demonstrate that this number is equal to the maximum size of the minimal separators in the concentration graph."}
{"category": "Math", "title": "Boundary Harnack inequalities for regional fractional Laplacian", "abstract": "We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stable-like processes on G taking \\kappa(x,y)/|x-y|^{n+\\alpha}dxdy, x,y\\in G as the jumping measure. When G is a C^{1,\\beta-1} open set, 1<\\alpha<\\beta\\leq 2 and \\kappa\\in C^{1}(\\overline{G}\\times \\overline{G}) bounded between two positive numbers, we prove a boundary Harnack inequality giving dist(x,\\partial G)^{\\alpha-1} order decay for harmonic functions near the boundary. For a C^{1,\\beta-1} open set D\\subset \\overline{D}\\subset G, 0<\\alpha\\leq (1\\vee\\alpha)<\\beta\\leq 2, we prove a boundary Harnack inequality giving dist(x,\\partial D)^{\\alpha/2} order decay for harmonic functions near the boundary. These inequalities are generalizations of the known results for the homogeneous case on C^{1,1} open sets. We also prove the boundary Harnack inequality for regional fractional Laplacian on Lipschitz domain."}
{"category": "Math", "title": "Finite Just Non-Dedekind Groups", "abstract": "A group is just non-Dedekind (JND) if it is not a Dedekind group but all of whose proper homomorphic images are Dedekind groups. The aim of the paper is to classify finite JND-groups."}
{"category": "Math", "title": "Deligne-Lusztig varieties and period domains over finite fields", "abstract": "We prove that the Drinfeld halfspace is essentially the only Deligne-Lusztig variety which is at the same time a period domain over a finite field. This is done by comparing a cohomology vanishing theorem for DL-varieties, due to Digne, Michel, and Rouquier, with a non-vanishing theorem for PD, due to the first author. We also discuss an affineness criterion for DL-varieties."}
{"category": "Math", "title": "Chen's double sieve, Goldbach's conjecture and the twin prime problem", "abstract": "We give a more comrehensive treatment of Chen's double sieve and improve related constants in Goldbach's conjecture and the twin prime problem."}
{"category": "Math", "title": "Gromov-Witten theory and Noether-Lefschetz theory", "abstract": "Noether-Lefschetz divisors in the moduli of K3 surfaces are the loci corresponding to Picard rank at least 2. We relate the degrees of the Noether-Lefschetz divisors in 1-parameter families of K3 surfaces to the Gromov-Witten theory of the 3-fold total space. The reduced K3 theory and the Yau-Zaslow formula play an important role. We use results of Borcherds and Kudla-Millson for O(2,19) lattices to determine the Noether-Lefschetz degrees in classical families of K3 surfaces of degrees 2, 4, 6 and 8. For the quartic K3 surfaces, the Noether-Lefschetz degrees are proven to be the Fourier coefficients of an explicitly computed modular form of weight 21/2 and level 8. The interplay with mirror symmetry is discussed. We close with a conjecture on the Picard ranks of moduli spaces of K3 surfaces."}
{"category": "Math", "title": "Existence of extremal Beltrami coefficients with non-constant modulus", "abstract": "Suppose $[\\mu]_{T(\\Delta)}$ is a point of the universal Teichm\\\"uller space $T(\\Delta)$. In 1998, it was shown by Bo\\v{z}in et al. that there exists $\\mu$ such that $\\mu$ has non-constant modulus and is uniquely extremal in $[\\mu]_{T(\\Delta)}$. It is a natural problem whether there is always an extremal Beltrmai coefficient of constant modulus in $[\\mu]_{T(\\Delta)}$ if $[\\mu]_{T(\\Delta)}$ admits more than one extremal Beltrami coefficient. The purpose of this paper is to show that the answer is negative. An infinitesimal version is also obtained. Extremal sets of extremal Beltrami coefficients are considered and an open problem is proposed."}
{"category": "Math", "title": "Existence results for mean field equations with turbulence", "abstract": "In this paper we consider the following form of the so-called Mean field equation arising from the statistical mechanics description of two dimensional turbulence \\begin{equation}\\label{eq:study} - \\D_g u = \\rho_1 (\\frac{e^{u}}{\\int_\\Sig e^{u} dV_g}-1)-\\rho_2 (\\frac{e^{-u}}{\\int_\\Sig e^{-u} dV_g} - 1) \\end{equation} on a given closed orientable Riemannian surface ($\\Sigma, g$) with volume 1, where $\\rho_1, \\rho_2$ are real parameters. Exploiting the variational structure of the problem and running a min-max scheme introduced by Djadli and Malchiodi, we prove that if $k$ is a positive integer, $\\rho_1$ and $\\rho_2$ two real numbers such that $\\rho_1\\in (8k\\pi, 8(k+1)\\pi)$ and $\\rho_2<4\\pi$ then $\\eqref{eq:study}$ is solvable."}
{"category": "Math", "title": "Generalized Brjuno functions associated to $\\alpha$-continued fractions", "abstract": "For \\alpha in the interval [0,1], we consider the one-parameter family of \\alpha-continued fraction maps, which include the Gauss map (\\alpha=1) and the nearest integer (\\alpha=1/2) and by-excess (\\alpha=0) continued fraction maps. To each of these expansions, and to each choice of a positive function u on the interval I_\\alpha=(0,max(\\alpha,1-\\alpha)) we associate a generalized Brjuno function B_(\\alpha,u)(x). For \\alpha=1/2 or \\alpha=1, and u(x)=-\\log(x), these functions were introduced by Yoccoz in his work on the linearization of holomorphic maps. Their regularity properties, including BMO regularity and their extension to the complex plane, have been thoroughly investigated. We compare the functions obtained with different values of \\alpha and we prove that the set of (\\alpha,u)-Brjuno numbers does not depend on the choice of \\alpha provided that \\alpha>0. We then consider the case \\alpha=0, u(x)=-\\log(x) and we prove that x is a Brjuno number (for \\alpha> 0) if and only if both x and -x are Brjuno numbers for \\alpha=0."}
{"category": "Math", "title": "Subelliptic Spin_C Dirac operators, I", "abstract": "We consider modifications of the classical dbar-Neumann conditions that define Fredholm problems for the Spin_C Dirac operator. In part II, we use boundary layer methods to obtain subelliptic estimates for these boundary value problems. Using these results, we obtain an expression for the finite part of the holomorphic Euler characteristic of a strictly pseudoconvex manifold as the index of a Spin_C-Dirac operator with a subelliptic boundary condition. We also prove an analogue of the Agranovich-Dynin formula expressing the change in the index in terms of a relative index on the boundary. If X is a complex manifold partitioned by a strictly pseudoconvex hypersurface, then we obtain formulae for the holomorphic Euler characteristic of X as sums of indices of Spin_C-Dirac operators on the components. This is a subelliptic analogue of Bojarski's formula in the elliptic case."}
{"category": "Math", "title": "Hives and the fibres of the convolution morphism", "abstract": "By the geometric Satake correspondence, the number of components of certain fibres of the affine Grassmannian convolution morphism equals the tensor product multiplicity for representations of the Langlands dual group. On the other hand, in the case of GL_n, combinatorial objects called hives also count tensor product multiplicities. The purpose of this paper is to give a simple bijection between hives and the components of these fibres. In particular, we give a description of the individual components. We also describe a conjectural generalization involving the octahedron recurrence."}
{"category": "Math", "title": "Subelliptic Spin_C Dirac operators, II Basic Estimates", "abstract": "We assume that the manifold with boundary, X, has a Spin_C-structure with spinor bundle S. Along the boundary, this structure agrees with the structure defined by an infinite order integrable almost complex structure and the metric is Kahler. The induced CR-structure on bX is integrable and either strictly pseudoconvex or strictly pseudoconcave. We assume that E->X is a complex vector bundle, which has an infinite order integrable complex structure along bX, compatible with that defined along bX. In this paper use boundary layer methods to prove subelliptic estimates for the twisted Spin_C- Dirac operator acting on sections on S\\otimes E. We use boundary conditions that are modifications of the classical dbar-Neumann condition. These results are proved by using the extended Heisenberg calculus."}
{"category": "Math", "title": "Universality results for largest eigenvalues of some sample covariance matrix ensembles", "abstract": "For sample covariance matrices with iid entries with sub-Gaussian tails, when both the number of samples and the number of variables become large and the ratio approaches to one, it is a well-known result of A. Soshnikov that the limiting distribution of the largest eigenvalue is same as the of Gaussian samples. In this paper, we extend this result to two cases. The first case is when the ratio approaches to an arbitrary finite value. The second case is when the ratio becomes infinity or arbitrarily small."}
{"category": "Math", "title": "Cobordism, Relative Indices and Stein Fillings", "abstract": "In this paper we build on the framework developed in \"Subelliptic Boundary Value Problems for the Spin_C Dirac Operator, I, II, III\" to obtain a more complete understanding of the gluing properties for indices of boundary value problems for the SpinC-Dirac operator with sub-elliptic boundary conditions. We extend our analytic results for sub-elliptic boundary value problems for the SpinC-Dirac operator, and gluing results for the indices of these boundary problems to SpinC-manifolds with several pseudoconvex (pseudoconcave) boundary components. These results are applied to study Stein fillability for compact, 3-dimensional, contact manifolds."}
{"category": "Math", "title": "Slicing, skinning, and grafting", "abstract": "We prove that a Bers slice is never algebraic, meaning that its Zariski closure in the character variety has strictly larger dimension. A corollary is that skinning maps are never constant. The proof uses grafting and the theory of complex projective structures."}
{"category": "Math", "title": "A homotopy method for finding eigenvalues and eigenvectors", "abstract": "Suppose we want to find the eigenvalues and eigenvectors for the linear operator L, and suppose that we have solved this problem for some other \"nearby\" operator K. In this paper we show how to represent the eigenvalues and eigenvectors of L in terms of the corresponding properties of K."}
{"category": "Math", "title": "Differential operators on toric varieties and Fourier transform", "abstract": "We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans are related to each other by reflections of one-dimensional cones. The simplest class of examples is provided by the toric varieties related by such reflections to projective spaces. It includes the blow-up at a point in affine space and resolution of singularities of varieties appearing in the study of the minimal orbit of sl(n+1)."}
{"category": "Math", "title": "Selfsimilar Equivalence of Porous Medium and p-Laplacian Flows", "abstract": "We demonstrate the equivalence between the two popular models of nonlinear diffusion, the porous medium equation and the p-Laplacian equation. The equivalence is shown at the level of selfsimilar solutions."}
{"category": "Math", "title": "Holomorphic fiber bundle with Stein base and Stein fibers", "abstract": "In this article, we prove that if $\\Pi: X\\to \\Omega$ is a surjective holomorphic map, with $\\Omega$ a Stein space and $X$ a complex manifold of dimension $n\\geq 3,$ and if, for every $x\\in \\Omega$ there exists an open neighborhood $U$ such that $\\Pi^{-1}(U)$ is Stein, then $X$ is Stein"}
{"category": "Math", "title": "On Vojta's $1+\\epsilon$ Conjecture", "abstract": "I gave a geometric proof of Vojta's 1 + epsilon conjecture. Some gaps in the published paper were spotted and kindly pointed out to me by Paul Vojta. These were addressed in \"Erratum\"."}
{"category": "Math", "title": "Fibers of tropicalization", "abstract": "We use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the Zariski topology. For subvarieties of tori over fields of generalized power series, points in each tropical fiber are obtained \"constructively\" using Kedlaya's transfinite version of Newton's method."}
{"category": "Math", "title": "The Large Sieve Inequality for Quadratic Polynomial Amplitudes", "abstract": "We provide here a modest improvement upon a large sieve inequality for quadratic polynomial amplitudes orginally due to Liangyi Zhao."}
{"category": "Math", "title": "Nonlinarity of Boolean functions and hyperelliptic curves", "abstract": "We study the nonlinearity of functions defined on a finite field with 2^m elements which are the trace of a polynomial of degree 7 or more general polynomials. We show that for m odd such functions have rather good nonlinearity properties. We use for that recent results of Maisner and Nart about zeta functions of supersingular curves of genus 2. We give some criterion for a vectorial function not to be almost perfect nonlinear."}
{"category": "Math", "title": "Recursive Parameter Estimation: Convergence", "abstract": "We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We propose a wide class of recursive estimation procedures for the general statistical model and study convergence."}
{"category": "Math", "title": "Rate of Convergence in Recursive Parameter Estimation procedures", "abstract": "We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We study rate of convergence of recursive estimation procedures for the general statistical model."}
{"category": "Math", "title": "The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion", "abstract": "The paper focuses on discrete-type approximations of solutions to non-homogeneous stochastic differential equations (SDEs) involving fractional Brownian motion (fBm). We prove that the rate of convergence for Euler approximations of solutions of pathwise SDEs driven by fBm with Hurst index $H>1/2$ can be estimated by $O(\\delta^{2H-1})$ ($\\delta$ is the diameter of partition). For discrete-time approximations of Skorohod-type quasilinear equation driven by fBm we prove that the rate of convergence is $O(\\delta^H)$."}
{"category": "Math", "title": "Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian", "abstract": "We investigate integrable second order equations of the form F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0. Familiar examples include the Boyer-Finley equation, the potential form of the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota equation, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit."}
{"category": "Math", "title": "Recursive Parameter Estimation: Asymptotic expansion", "abstract": "We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. The model considered in the paper is very general as we do not impose any preliminary restrictions on the probabilistic nature of the observation process and cover a wide class of nonlinear recursive procedures. In this paper we study asymptotic behaviour of the recursive estimators. The results of the paper can be used to determine the form of a recursive procedure which is expected to have the same asymptotic properties as the corresponding non-recursive one defined as a solution of the corresponding estimating equation."}
{"category": "Math", "title": "Riemann surfaces, ribbon graphs and combinatorial classes", "abstract": "This survey paper begins with the description of the duality between arc systems and ribbon graphs embedded in a punctured surface. Then we explain how to cellularize the moduli space of curves in two different ways: using Jenkins-Strebel differentials and using hyperbolic geometry. We also briefly discuss how these two methods are related. Next, we recall the definition of Witten cycles and we illustrate their connection with tautological classes and Weil-Petersson geometry. Finally, we exhibit a simple direct argument to prove that Witten classes are stable."}
{"category": "Math", "title": "Semimartingale Stochastic Approximation Procedures and Recursive Estimation", "abstract": "The semimartingale stochastic approximation procedure, namely, the Robbins-Monro type SDE is introduced which naturally includes both generalized stochastic approximation algorithms with martingale noises and recursive parameter estimation procedures for statistical models associated with semimartingales. General results concerning the asymptotic behaviour of the solution are presented. In particular, the conditions ensuring the convergence, rate of convergence and asymptotic expansion are established. The results concerning the Polyak weighted averaging procedure are also presented."}
{"category": "Math", "title": "Subjective Questions and Answers for a Mathematics Instructor of Higher Education", "abstract": "This article of mathematical education reflects author's experience with job applications and teaching methods and procedures to employ in the American Higher Education. It is organized as a standard questionnaire."}
{"category": "Math", "title": "Transversals in trees", "abstract": "A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n nodes by saying that a tree T succeeds a tree T' if c(T,k) is at least c(T',k) for all k and strictly greater than c(T',k) for at least one k. We prove that, for every choice of positive integers d and n, the set of all rooted trees on n nodes where each node has at most d children has a unique minimal element with respect to this partial order and we describe this tree."}
{"category": "Math", "title": "Computing the core of ideals in arbitrary characteristic", "abstract": "Let $R$ be a local Gorenstein ring with infinite residue field of arbitrary characteristic. Let $I$ be an $R$--ideal with $g=\\height I >0$, analytic spread $\\ell$, and let $J$ be a minimal reduction of $I$. We further assume that $I$ satisfies $G_{\\ell}$ and ${\\depth} R/I^j \\geq \\dim R/I -j+1$ for $1 \\leq j \\leq \\ell-g$. The question we are interested in is whether $\\core{I}=J^{n+1}:\\ds \\sum_{b \\in I} (J,b)^n$ for $n \\gg 0$. In the case of analytic spread one Polini and Ulrich show that this is true with even weaker assumptions (\\cite[Theorem 3.4]{PU}). We give a negative answer to this question for higher analytic spreads and suggest a formula for the core of such ideals."}
{"category": "Math", "title": "Index theory for linear self-adjoint operator equations and nontrivial solutions for asymptotically linear operator equations", "abstract": "We will first establish an index theory for linear self-adjoint operator equations. And then with the help of this index theory we will discuss existence and multiplicity of solutions for asymptotically linear operator equations by making use of the dual variational methods and Morse theory. Finally, some interesting examples concerning second order Hamiltonian systems, first order Hamiltonian systems and elliptical partial differential equations will be presented to illustrate our results."}
{"category": "Math", "title": "The Cauchy Operator for Basic Hypergeometric Series", "abstract": "We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's ${}_2\\phi_1$ transformation formula and Sears' ${}_3\\phi_2$ transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator $T(bD_q)$. Using this operator, we obtain extensions of the Askey-Wilson integral, the Askey-Roy integral, Sears' two-term summation formula, as well as the $q$-analogues of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers-Szeg\\\"o polynomials, or the continuous big $q$-Hermite polynomials."}
{"category": "Math", "title": "Periodic Orbits of Twisted Geodesic Flows and The Weinstein-Moser Theorem", "abstract": "In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more general theorem concerning periodic orbits of autonomous Hamiltonian flows near Morse-Bott non-degenerate, symplectic extrema. Namely, we show that all energy levels near such extrema carry periodic orbits, provided that the ambient manifold meets certain topological requirements. This result is a partial generalization of the Weinstein-Moser theorem. The proof of the generalized Weinstein-Moser theorem is a combination of a Sturm-theoretic argument and a Floer homology calculation."}
{"category": "Math", "title": "Anticipated backward stochastic differential equations", "abstract": "In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations."}
{"category": "Math", "title": "Free Boolean algebras over unions of two well orderings", "abstract": "Given a partially ordered set $P$ there exists the most general Boolean algebra $F(P)$ which contains $P$ as a generating set, called the {\\it free Boolean algebra} over $P$. We study free Boolean algebras over posets of the form $P=P_0\\cup P_1$, where $P_0,P_1$ are well orderings. We call them {\\it nearly ordinal algebras}. Answering a question of Maurice Pouzet, we show that for every uncountable cardinal $\\kappa$ there are $2^\\kappa$ pairwise non-isomorphic nearly ordinal algebras of cardinality $\\kappa$. Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product $(\\omega_1+1)\\times(\\omega_1+1)$, thus showing that there are only $\\aleph_1$ many of them. In contrast with the last result, we show that there are $2^{\\aleph_1}$ topological types of closed subsets of the Tikhonov plank $(\\omega_1+1)\\times(\\omega+1)$."}
{"category": "Math", "title": "Radon transform on real symmetric varieties: kernel and cokernel", "abstract": "We define and study the (minimal) Radon transform on a real symmetric variety."}
{"category": "Math", "title": "Some Blow-Up Problems for a Semilinear Parabolic Equation with a Potential", "abstract": "The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=\\Delta u+V(x) |u|^{p-1}u$ in $\\Omega \\times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data $u(x,0)=M\\vf (x)$ as $M$ goes to infinity, which have been found in \\cite{cer}, are improved under some reasonable and weaker conditions compared with \\cite{cer}."}
{"category": "Math", "title": "Remark on the Garnier system in two variables", "abstract": "We remark on the Garnier system in two variables."}
{"category": "Math", "title": "Decomposition and Enumeration of Triangulated Surfaces", "abstract": "We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genus-surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most a fixed number of vertices. We specialize the theory for the case where the number of vertices is at most 11 and we get theoretical restrictions on genus-surfaces allowing us to get the list of triangulations of closed surfaces with at most 11 vertices."}
{"category": "Math", "title": "A rigidity theorem for the mapping class group action on the space of unmeasured foliations on a surface", "abstract": "Let $S$ be a surface of finite type which is not a sphere with at most four punctures, a torus with at most two punctures, or a closed surface of genus two. Let $\\mathcal{MF}$ be the space of equivalence classes of measured foliations of compact support on $S$ and let $\\mathcal{UMF}$ be the quotient space of $\\mathcal{MF}$ obtained by identifying two equivalence classes whenever they can be represented by topologically equivalent foliations, that is, forgetting the transverse measure. The extended mapping class group $\\Gamma^*$ of $S$ acts as by homeomorphisms of $\\mathcal{UMF}$. We show that the restriction of the action of the whole homeomorphism group of $\\mathcal{UMF}$ on some dense subset of $\\mathcal{UMF}$ coincides with the action of $\\Gamma^*$ on that subset. More precisely, let $\\mathcal{D}$ be the natural image in $\\mathcal{UMF}$ of the set of homotopy classes of not necessarily connected essential disjoint and pairwise nonhomotopic simple closed curves on $S$. The set $\\mathcal{D}$ is dense in $\\mathcal{UMF}$, it is invariant by the action of $\\Gamma^*$ on $\\mathcal{UMF}$ and the restriction of the action of $\\Gamma^*$ on $\\mathcal{D}$ is faithful. We prove that the restriction of the action on $\\mathcal{D}$ of the group $\\mathrm{Homeo}(\\mathcal{UMF})$ coincides with the action of $\\Gamma^*(S)$ on that subspace."}
{"category": "Math", "title": "Cohen-Macaulay multigraded modules", "abstract": "Let S be a standard N^r-graded algebra over a local ring A, and let M be a finitely generated Z^r-graded S-module. We characterize the Cohen-Macaulayness of M in terms of the vanishing of certain sheaf cohomology modules. As a consequence, we apply our result to study the Cohen-Macaulayness of multi-Rees modules (also called Rees modification). Our work extends previous studies on the Cohen-Macaulayness of multi-Rees algebras."}
{"category": "Math", "title": "When does elementary bi-embeddability imply isomorphism?", "abstract": "A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if a countable theory T has the Schroder-Bernstein property then it is classifiable (it is superstable and has NDOP and NOTOP) and satisfies a slightly stronger condition than nonmultidimensionality, namely: there cannot be a model M of T, a type p over M, and an automorphism f of M such that for every two distinct natural numbers i and j, f^i(p) is orthogonal to f^j(p). We also make some conjectures about how the class of theories with the Schroder-Bernstein property can be characterized."}
{"category": "Math", "title": "The Schroder-Bernstein property for theories of abelian groups", "abstract": "A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if G is an abelian group, then the follwing are equivalent: 1. Th(G, +) has the Schroder-Bernstein property; 2. Th(G, +) is omega-stable; 3. G is the direct sum of a divisible group and a torsion group of bounded exponent; 4. Th(G, +) is superstable, and if (H, +) is a saturated elementary extension of (G,+), every map in Aut(H/H^0) is unipotent."}
{"category": "Math", "title": "The Riemann Mapping Theorem for semianalytic domains and o-minimality", "abstract": "We consider the Riemann Mapping Theorem in the case of a bounded simply connected and semianalytic domain. We show that the germ at 0 of the Riemann map (i.e. biholomorphic map) from the upper half plane to such a domain can be realized in a certain quasianalytic class if the angle of the boundary at the point to which 0 is mapped, is greater than 0. This quasianalytic class was introduced and used by Ilyashenko in his work on Hilbert's 16th problem. With this result we can prove that the Riemann map from a bounded simply connected semianalytic domain onto the unit ball is definable in an o-minimal structure, provided that at singular boundary points the angles of the boundary are irrational multiples of $\\pi$."}
{"category": "Math", "title": "Revesibility of chordal SLE", "abstract": "We prove that the chordal SLE$_\\kappa$ trace is reversible for $\\kappa\\in(0,4]$."}
{"category": "Math", "title": "2-Frame flow dynamics and hyperbolic rank rigidity in nonpositive curvature", "abstract": "This paper presents hyperbolic rank rigidity results for rank 1, nonpositively curved spaces. Let $M$ be a compact, rank 1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd dimensional, or if $M$ is even dimensional and has sectional curvature pinched as follows: $-\\Lambda^2 < K < -\\lambda^2$ where $\\lambda/\\Lambda > >.93$. When $-a^2$ is the upper curvature bound this gives a shorter proof of the hyperbolic rank rigidity theorem of Hamenst\\\"{a}dt, subject to the pinching condition in even dimension; in all other cases it is a new result. We also present a rigidity result using only an assumption on maximal Lyapunov exponents in direct analogy with work done by Connell. The proof of the main theorem is simplified considerably by assuming strict negative curvature; in fact, in all dimensions but 7 and 8 it is then an immediate consequence of ergodicity of the $(dim(M)-1)$-frame flow. In these exceptional dimensions, recourse to the dynamics of the 2-frame flow must be made and the scheme of proof developed there can be generalized to deal with rank 1, nonpositively curved spaces."}
{"category": "Math", "title": "On the Riemann zeta-function, Part I: Outline", "abstract": "Results of a multipart work are outlined. Use is made therein of the conjunction of the Riemann hypothesis, RH, and hypotheses advanced by the author. Let z(n) be the nth nonreal zero of the Riemann zeta-function with positive imaginary part in order of magnitude thereof. A relation is obtained, of the pair z(n) and the first derivative thereat of the zeta-function, to the preceding such pairs and the values of zeta at the points one-half plus a nonnegative multiple of four. That relation is derived from two forms of the density of the Laplace representation, on a certain vertical strip, of a meromorphic function constructed from zeta. Specific functions which play a central role therein are proven to have analytic extensions to the entire complex plane. It is established that the Laplace density is positive. That positivity implies RH and that each nonreal zero of zeta is simple. A metric geometry expression of the positivity of the density is derived. An analogous context is delineated relative to Dirichlet L-functions and the Ramanujan tau Dirichlet function."}
{"category": "Math", "title": "On the Green's matrices of strongly parabolic systems of second order", "abstract": "We establish existence and various estimates of fundamental matrices and Green's matrices for divergence form, second order strongly parabolic systems in arbitrary cylindrical domains under the assumption that solutions of the systems satisfy an interior H\\\"{o}lder continuity estimate. We present a unified approach valid for both the scalar and the vectorial cases."}
{"category": "Math", "title": "On level crossings for a general class of piecewise-deterministic Markov processes", "abstract": "We consider a piecewise-deterministic Markov process governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. We study the point process of upcrossings of a level $b$ by the Markov process. Our main result shows that, under a suitable scaling $\\nu(b)$, the point process converges, as $b$ tends to infinity, weakly to a geometrically compound Poisson process. We also prove a version of Rice's formula relating the stationary density of the process to level crossing intensities. This formula provides an interpretation of the scaling factor $\\nu(b)$. While our proof of the limit theorem requires additional assumptions, Rice's formula holds whenever the (stationary) overall intensity of jumps is finite."}
{"category": "Math", "title": "On the degree of Polar Transformations -- An approach through Logarithmic Foliations", "abstract": "We investigate the degree of the polar transformations associated to a certain class of multi-valued homogeneous functions. In particular we prove that the degree of the pre-image of generic linear spaces by a polar transformation associated to a homogeneous polynomial $F$ is determined by the zero locus of $F$. For zero dimensional-dimensional linear spaces this was conjecture by Dolgachev and proved by Dimca-Papadima using topological arguments. Our methods are algebro-geometric and rely on the study of the Gauss map of naturally associated logarithmic foliations."}
{"category": "Math", "title": "A criterion for transience of multidimensional branching random walk in random environment", "abstract": "We develop a criterion for transience for a general model of branching Markov chains. In the case of multi-dimensional branching random walk in random environment (BRWRE) this criterion becomes explicit. In particular, we show that \\emph{Condition L} of Comets and Popov is necessary and sufficient for transience as conjectured. Furthermore, the criterion applies to two important classes of branching random walks and implies that the critical branching random walk is transient resp. dies out locally."}
{"category": "Math", "title": "On Mordell-Weil groups of elliptic curves induced by Diophantine triples", "abstract": "We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square."}
{"category": "Math", "title": "Set theoretic solutions of the Yang-Baxter equation, graphs and computations", "abstract": "We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of solutions and use our graphical methods for the computation of solutions of finite order and their automorphisms. Results include a detailed study of solutions of multipermutation level 2."}
{"category": "Math", "title": "Self-duality of Selmer groups", "abstract": "The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K."}
{"category": "Math", "title": "On the invertibility of \"rectangular\" bi-infinite matrices and applications in time--frequency analysis", "abstract": "Finite dimensional matrices having more columns than rows have no left inverses while those having more rows than columns have no right inverses. We give generalizations of these simple facts to bi--infinite matrices and use those to obtain density results for $p$--frames of time--frequency molecules in modulation spaces and identifiability results for operators with bandlimited Kohn--Nirenberg symbols."}
{"category": "Math", "title": "Necessary Conditions for Geometric Realizability of Simplicial Complexes", "abstract": "We associate with any simplicial complex $\\K$ and any integer $m$ a system of linear equations and inequalities. If $\\K$ has a simplicial embedding in $\\R^m$ then the system has an integer solution. This result extends the work of I. Novik (2000)."}
{"category": "Math", "title": "Measurement of time--varying Multiple--Input Multiple--Output Channels", "abstract": "We derive a criterion on the measurability / identifiability of Multiple--Input Multiple--Output (MIMO) channels based on the size of the so-called spreading support of its subchannels. Novel MIMO transmission techniques provide high-capacity communication channels in time-varying environments and exact knowledge of the transmission channel operator is of key importance when trying to transmit information at a rate close to channel capacity."}
{"category": "Math", "title": "The Dirichlet problem in the plane with semianalytic raw data, quasianalyticity and o-minimal structures", "abstract": "We investigate the Dirichlet solution for a semianalytic continuous function on the boundary of a semianalytic bounded domain in the plane. We show that the germ of the Dirichlet solution at a boundary point with angle greater than 0 lies in a certain quasianalytic class used by Ilyashenko in his work on Hilbert's 16th problem. With this result we can prove that the Dirichlet solution is definable in an o-minimal structure if the angle at a singular boundary point of the domain is an irrational multiple of $\\pi$."}
{"category": "Math", "title": "Linear Prediction of Long-Memory Processes: Asymptotic Results on Mean-squared Errors", "abstract": "We present two approaches for linear prediction of long-memory time series. The first approach consists in truncating the Wiener-Kolmogorov predictor by restricting the observations to the last $k$ terms, which are the only available values in practice. We derive the asymptotic behaviour of the mean-squared error as $k$ tends to $ + \\infty$. By contrast, the second approach is non-parametric. An AR($k$) model is fitted to the long-memory time series and we study the error that arises in this misspecified model."}
{"category": "Math", "title": "Ideals of varieties parameterized by certain symmetric tensors", "abstract": "The ideal of a Segre variety is generated by the 2-minors of a generic hypermatrix of indeterminates. We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of weak generic hypermatrix which allows us to treat also the case of projection of Veronese surfaces from a set of generic points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2."}
{"category": "Math", "title": "Riesz and Szeg\\\"o type factorizations for noncommutative Hardy spaces", "abstract": "Let $\\A$ be a finite subdiagonal algebra in Arveson's sense. Let $H^p(\\A)$ be the associated noncommutative Hardy spaces, $0<p\\le\\8$. We extend to the case of all positive indices most recent results about these spaces, which include notably the Riesz, Szeg\\\"o and inner-outer type factorizations. One new tool of the paper is the contractivity of the underlying conditional expectation on $H^p(\\A)$ for $p<1$."}
{"category": "Math", "title": "BMO functions and Carleson measures with values in uniformly convex spaces", "abstract": "This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\\mathbb T$, respectively. For $1< q<\\infty$ and a Banach space $B$ we prove that there exists a positive constant $c$ such that $$\\sup_{z_0\\in D}\\int_{D}(1-|z|)^{q-1}\\|\\nabla f(z)\\|^q P_{z_0}(z) dA(z) \\le c^q\\sup_{z_0\\in D}\\int_{\\T}\\|f(z)-f(z_0)\\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ iff $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\\frac{1-|z_0|^2}{|1-\\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm."}
{"category": "Math", "title": "Sums of lens spaces bounding rational balls", "abstract": "We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3-spheres up to rational homology cobordisms, and to determine the concordance order of each 2-bridge knot."}
{"category": "Math", "title": "Noncommutative Burkholder/Rosenthal inequalities II: applications", "abstract": "We show norm estimates for the sum of independent random variables in noncommutative $L_p$-spaces for $1<p<\\infty$ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications, we derive an equivalence for the $p$-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative $L_p$ for $2<p<\\infty$."}
{"category": "Math", "title": "Intersections of polynomial orbits, and a dynamical Mordell-Lang conjecture", "abstract": "We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang conjecture."}
{"category": "Math", "title": "$C^*$-alg\\`ebres gradu\\'ees par un semi-treillis", "abstract": "Graded $C^*$-algebras by a semi lattice were introduced and studied by Anne Boutet de Monvel, Vladimir Georgescu and their collaborators in relation with the quantum N body problem. This thesis is devoted to a systematic study of these algebras and their properties. In particular, we show that they are completely described by their homogenous subalgebras and that they are invariant under several operations such as tensor products, crossed products (by actions of locally compact groups that respect the graduation). We give the $K$-theory of graded $C^*$-algebras and finally, we study some examples of such algebras."}
{"category": "Math", "title": "Realizing Kasparov's KK-theory groups as the homotopy classes of maps of a Quillen model category", "abstract": "In this article we build a Quillen model category structure on the category of sequentially complete l.m.c.-C*-algebras such that the corresponding homotopy classes of maps Ho(A,B) for separable C*-algebras A and B coincide with the Kasparov groups KK(A,B). This answers an open question posed by Mark Hovey about the possibility of describing KK-theory for C*-algebras using the language of Quillen model categories."}
{"category": "Math", "title": "Amenable groups and Hadamard spaces with a totally disconnected isometry group", "abstract": "Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected group acting continuously, properly and cocompactly on $X$. We show that a closed subgroup of $G$ is amenable if and only if it is (topologically locally finite)-by-(virtually abelian). We are led to consider a set $\\bdfine X$ which is a refinement of the visual boundary $\\bd X$. For each $x \\in \\bdfine X$, the stabilizer $G_x$ is amenable."}
{"category": "Math", "title": "Integral group ring of the Mathieu simple group M24", "abstract": "We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group $M_{24}$. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs."}
{"category": "Math", "title": "Energy of zeros of random sections on Riemann Surface", "abstract": "The purpose of this paper is to determine the asymptotic of the average energy of a configuration of N zeros of system of random polynomials of degree N as N tends to infinity and more generally the zeros of random holomorphic sections of a line bundle L over any Riemann surface M. And we compare our results to the well-known minimum of energies."}
{"category": "Math", "title": "Intrinsically Linked Graphs with Knotted Components", "abstract": "We construct a graph G such that any embedding of G into R^{3} contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nontrivial knots. We then turn our attention to complete graphs and show that for any given n, every embedding of a large enough complete graph contains a two component link whose linking number is a nonzero multiple of n."}
{"category": "Math", "title": "Multivariable generalizations of the Schur class: positive kernel characterization and transfer function realization", "abstract": "The operator-valued Schur-class is defined to be the set of holomorphic functions $S$ mapping the unit disk into the space of contraction operators between two Hilbert spaces. There are a number of alternate characterizations: the operator of multiplication by $S$ defines a contraction operator between two Hardy Hilbert spaces, $S$ satisfies a von Neumann inequality, a certain operator-valued kernel associated with $S$ is positive-definite, and $S$ can be realized as the transfer function of a dissipative (or even conservative) discrete-time linear input/state/output linear system. Various multivariable generalizations of this class have appeared recently,one of the most encompassing being that of Muhly and Solel where the unit disk is replaced by the strict unit ball of the elements of a dual correspondence $E^{\\sigma}$ associated with a $W^{*}$-correspondence $E$ over a $W^{*}$-algebra $\\cA$ together with a $*$-representation $\\sigma$ of $\\cA$. The main new point which we add here is the introduction of the notion of reproducing kernel Hilbert correspondence and identification of the Muhly-Solel Hardy spaces as reproducing kernel Hilbert correspondences associated with a completely positive analogue of the classical Szeg\\\"o kernel. In this way we are able to make the analogy between the Muhly-Solel Schur class and the classical Schur class more complete. We also illustrate the theory by specializing it to some well-studied special cases; in some instances there result new kinds of realization theorems."}
{"category": "Math", "title": "Noetherian types of homogeneous compacta and dyadic compacta", "abstract": "The Noetherian type of a space is the least $\\kappa$ such that it has a base that is $\\kappa$-like with respect to containment. Just as all known homogeneous compacta have cellularity at most $2^\\omega$, they satisfy similar upper bounds in terms of Noetherian type and related cardinal functions. We prove these and many other results about these cardinal functions. For example, every homogeneous dyadic compactum has Noetherian type $\\omega$. Assuming GCH, every point in a homogeneous compactum $X$ has a local base that is $c(X)$-like with respect to containment. If every point in a compactum has a well-quasiordered local base, then some point has a countable local $\\pi$-base."}
{"category": "Math", "title": "The Maximal Integral Domain Generated By A Commutative Ring", "abstract": "In this paper, we exhibit the creation of the maximal integral domain mid(R) generated by a nonzero commutative ring R."}
{"category": "Math", "title": "Drinfeld-Hecke algebras over cocommutative algebras", "abstract": "If A is a cocommutative algebra with coproduct, then so is the smash product algebra of a symmetric algebra Sym(V) with A, where V is an A-module. Such smash product algebras, with A a group ring or a Lie algebra, have families of deformations that have been studied widely in the literature; examples include symplectic reflection algebras and infinitesimal Hecke algebras. We introduce a family of deformations of these smash product algebras for general A, and characterize the PBW property. We then characterize the Jacobi identity for \"grouplike\" algebras (that include group rings and the nilCoxeter algebra), and precisely identify the PBW deformations in the example where A is the nilCoxeter algebra. We end with the more prominent case - where A is a Hopf algebra. We show the equivalence of several versions of the \"deformed\" relations in the smash product, and identify the PBW deformations which are Hopf algebras as well."}
{"category": "Math", "title": "Tensor Product of the Fundamental Representations for the Quantum Loop Algebras of Type A at Roots of Unity", "abstract": "In this paper, we consider the necessary and sufficient conditions for the tensor product of the fundamental representations for the restricted quantum loop algebras of type A at roots of unity to be irreducible."}
{"category": "Math", "title": "Gluing pseudoholomorphic curves along branched covered cylinders II", "abstract": "This paper and its prequel (\"Part I\") prove a generalization of the usual gluing theorem for two index 1 pseudoholomorphic curves U_+ and U_- in the symplectization of a contact 3-manifold. We assume that for each embedded Reeb orbit gamma, the total multiplicity of the negative ends of U_+ at covers of gamma agrees with the total multiplicity of the positive ends of U_- at covers of gamma. However, unlike in the usual gluing story, here the individual multiplicities are allowed to differ. In this situation, one can often glue U_+ and U_- to an index 2 curve by inserting genus zero branched covers of R-invariant cylinders between them. This paper shows that the signed count of such gluings equals a signed count of zeroes of a certain section of an obstruction bundle over the moduli space of branched covers of the cylinder. Part I obtained a combinatorial formula for the latter count and, assuming the result of the present paper, deduced that the differential d in embedded contact homology satisfies d^2=0. The present paper completes all of the analysis that was needed in Part I. The gluing technique explained here is in principle applicable to more gluing problems. We also prove some lemmas concerning the generic behavior of pseudoholomorphic curves in symplectizations, which may be of independent interest."}
{"category": "Math", "title": "Vanishing and injectivity theorems for LMMP", "abstract": "This preprint has been withdrawn. It is because I will never publish this preprint since everything has been contained in my new preprint: arXiv:0907.1506. Please refer to arXiv:0907.1506. Please do not cite this preprint any more."}
{"category": "Math", "title": "Notes on the log minimal model program", "abstract": "This preprint has been withdrawn. It is because I will never publish this preprint since everything has been contained in my new preprint: arXiv:0907.1506. Please refer to arXiv:0907.1506. Please do not cite this preprint any more."}
{"category": "Math", "title": "The abelianization of a symmetric mapping class group", "abstract": "We determine the abelianization of the symmetric mapping class group of a double unbranched cover using the Riemann theta constant, Schottky theta constant, and the theta multiplier. We also give lower bounds of the abelianizations of some finite index subgroups of the mapping class group."}
{"category": "Math", "title": "Mirzakharni's recursion formula is equivalent to the Witten-Kontsevich theorem", "abstract": "In this paper, we give a proof of Mirzakhani's recursion formula of Weil-Petersson volumes of moduli spaces of curves using the Witten-Kontsevich theorem. We also describe properties of intersections numbers involving higher degree $\\kappa$ classes."}
{"category": "Math", "title": "A Blichfeldt-type inequality for the surface area", "abstract": "In 1921 Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that $#(K\\cap\\Z^n)\\leq n! \\vol(K)+n$, whenever $K\\subset\\R^n$ is a convex body containing $n+1$ affinely independent integral points. Here we prove an analogous inequality with respect to the surface area $\\F(K)$, namely $ #(K\\cap\\Z^n) < \\vol(K) + ((\\sqrt{n}+1)/2) (n-1)! \\F(K)$. The proof is based on a slight improvement of Blichfeldt's bound in the case when $K$ is a non-lattice translate of a lattice polytope, i.e., $K=t+P$, where $t\\in\\R^n\\setminus\\Z^n$ and $P$ is an $n$-dimensional polytope with integral vertices. Then we have $#((t+P)\\cap\\Z^n)\\leq n! \\vol(P)$. Moreover, in the 3-dimensional case we prove a stronger inequality, namely $#(K\\cap\\Z^n) < \\vol(K) + 2 \\F(K)$."}
{"category": "Math", "title": "Another property of minimal surfaces in Euclidean space", "abstract": "The new property of minimal surfaces is obtained in this article."}
{"category": "Math", "title": "Smooth multiparameter perturbation of polynomials and operators", "abstract": "This paper has been withdrawn by the authors due to a gap in the proof of the main result (in 5.3)."}
{"category": "Math", "title": "Pr\\\"ufer's Ideal Numbers as Gelfand's maximal Ideals", "abstract": "Polyadic arithmetics is a branch of mathematics related to $p$--adic theory. The aim of the present paper is to show that there are very close relations between polyadic arithmetics and the classic theory of commutative Banach algebras. Namely, let $\\ms A$ be the algebra consisting of all complex periodic functions on $\\Z$ with the uniform norm. Then the polyadic topological ring can be defined as the ring of all characters $\\ms A\\to\\C$ with convolution operations and the Gelfand topology."}
{"category": "Math", "title": "Casimir operators, abelian subspaces and u-cohomology", "abstract": "This survey paper is an exposition of old and recent results of Kostant and al. on the relationships between the exterior algebra of a simple Lie algebra and the action of the Casimir operator on it. Our exposition relies on u-cohomology and it is basically self-contained."}
{"category": "Math", "title": "The General Definition of the Complex Monge-Amp\\`ere Operator on Compact K\\\"ahler Manifolds", "abstract": "We introduce a wide subclass ${\\cal F}(X,\\omega)$ of quasi-plurisubharmonic functions in a compact K\\\"ahler manifold, on which the complex Monge-Amp\\`ere operator is well-defined and the convergence theorem is valid. We also prove that ${\\cal F}(X,\\omega)$ is a convex cone and includes all quasi-plurisubharmonic functions which are in the Cegrell class."}
{"category": "Math", "title": "Determination of the body force of a two-dimensional isotropic elastic body", "abstract": "Let $\\Omega$ represent a two$-$dimensional isotropic elastic body. We consider the problem of determining the body force $F$ whose form $\\phi(t)(f_1(x),f_2(x))$ with $\\phi$ be given inexactly. The problem is nonlinear and ill-posed. Using the Fourier transform, the methods of Tikhonov's regularization and truncated integration, we construct a regularized solution from the data given inexactly and derive the explicitly error estimate. Numerical part is given"}
{"category": "Math", "title": "Discontinuity and Involutions on Countable Sets", "abstract": "For any infinite subset $X$ of the rationals and a subset $F \\subseteq X$ which has no isolated points in $X$ we construct a function $f: X \\to X$ such that $f(f(x))=x$ for each $x\\in X$ and $F $ is the set of discontinuity points of $f$."}
{"category": "Math", "title": "TYZ expansion for the Kepler manifold", "abstract": "The main goal of the paper is to address the issue of the existence of Kempf's distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for the Kepler manifold - an important example of non compact manfold. Motivated by the recent results for compact manifolds we construct Kempf's distortion function and derive a precise TYZ asymptotic expansion for the Kepler manifold. We get an exact formula: finite asymptotic expansion of $n-1$ terms and exponentially small error terms uniformly with respect to the discrete quantization parameter $m\\to \\infty $ and $\\rho \\to \\infty$, $\\rho$ being the polar radius in $\\C^n$. Moreover, the coefficents are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the Kepler manifold. We also prove and derive an asymptotic expansion of the obtstruction term with the coefficients being defined by geometrical quantities. We show that our estimates are sharp by analyzing the nonharmonic behaviour of $T_m$ and the error term of the approximation of the Fubini--Study metric by $m\\omega$ for $m\\to +\\infty$. The arguments of the proofs combine geometrical methods, quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic-Gevrey spaces."}
{"category": "Math", "title": "Extremal metrics on Hartogs domains", "abstract": "An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural \\K metric $g_F$. In this paper we prove that if $g_F$ is an extremal \\K metric then $(D_F, g_F)$ is biholomorphically isometric to the $n$-dimensional complex hyperbolic space."}
{"category": "Math", "title": "On the complexity of solving ordinary differential equations in terms of Puiseux series", "abstract": "We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algorithm is based on a differential version of the Newton-Puiseux procedure for algebraic equations."}
{"category": "Math", "title": "A tree approach to $p$-variation and to integration", "abstract": "We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of $p$-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths theory, are written as integrals on the tree. Examples include some stochastic paths such as martingales, L\\'evy processes and fractional Brownian motions (for which an estimator of the Hurst parameter is given)."}
{"category": "Math", "title": "Predictability, entropy and information of infinite transformations", "abstract": "We show that a certain type of quasi finite, conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic, measure preserving transformation which is not quasi finite; and consider distribution asymptotics of information showing that e.g. for Boole's transformation, information is asymptotically mod-normal with square root normalization. Lastly we see that certain ergodic, probability preserving transformations with zero entropy have analogous properties and consequently entropy dimension of at most 1/2."}
{"category": "Math", "title": "Distributions vectorielles homog\\`enes sur une alg\\`ebre de Jordan", "abstract": "We study distributions on a Euclidean Jordan algebra V with values in a finite dimensional representation space for the identity component G of the structure group of V and homogeneous equivariance condition. We show that such distributions exist if and only if the representation is spherical, and that then the dimension of the space of these distributions is r+1 (where r is the rank of V). We give also construction of these distributions and of those that are invariant under the semi-simple part of G."}
{"category": "Math", "title": "Gerby Localization, Z_3-Hodge Integrals and the GW Theory of C^3/Z_3", "abstract": "We exhibit a set of recursive relations that completely determine all equivariant Gromov-Witten invariants of the quotient orbifold C^3/Z_3. We interpret such invariants as G-Hodge Integrals, and produce relations among them via Atiyah-Bott localization on moduli spaces of twisted stable maps to gerbes over the projective line."}
{"category": "Math", "title": "Quantum cohomology of [C^N/\\mu_r]", "abstract": "We give a construction of the moduli space of stable maps to the classifying stack B\\mu_r of a cyclic group by a sequence of r-th root constructions on M_{0, n}. We prove a closed formula for the total Chern class of \\mu_r-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus zero Gromov-Witten theory of stacks of the form [C^N/\\mu_r]. We deduce linear recursions for all genus-zero Gromov-Witten invariants."}
{"category": "Math", "title": "Local dynamics for fibered holomorphic transformations", "abstract": "Fibered holomorphic dynamics are skew-product transformations over an irrational rotation, whose fibers are holomorphic functions. In this paper we study such a dynamics on a neighborhood of an invariant curve. We obtain some results analogous to the results in the non fibered case."}
{"category": "Math", "title": "Multiplicity one Conjectures", "abstract": "In the first part, in the local non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We conjecture that such distributions are invariant by transposition. This would imply multiplicity at most one for restrictions from GL(n+1) to GL(n). We reduce ourselves to distributions with \"singular\" support and then finish the proof for n< 9. In the second part we show that similar Theorems for orthogonal or unitary groups follow from the case of GL(n)"}
{"category": "Math", "title": "Limits of Hypergraphs, Removal and Regularity Lemmas. A Non-standard Approach", "abstract": "We study the integral and measure theory of the ultraproduct of finite sets. As a main application we construct limit objects for hypergraph sequences. We give a new proof for the Hypergraph Removal Lemma and the Hypergraph Regularity Lemma."}
{"category": "Math", "title": "Rational functions with linear relations", "abstract": "We find all polynomials f,g,h over a field K such that g and h are linear and f(g(x))=h(f(x)). We also solve the same problem for rational functions f,g,h, in case the field K is algebraically closed."}
{"category": "Math", "title": "A remarkable moduli space of rank 6 vector bundles related to cubic surfaces", "abstract": "We study the moduli space $\\fM^s(6;3,6,4)$ of simple rank 6 vector bundles $\\E$ on $\\PP^3$ with Chern polynomial $1+3t+6t^2+4t^3$ and properties of these bundles, especially we prove some partial results concerning their stability. We first recall how these bundles are related to the construction of sextic nodal surfaces in $\\PP^3$ having an even set of 56 nodes (cf. \\cite{CaTo}). We prove that there is an open set, corresponding to the simple bundles with minimal cohomology, which is irreducible of dimension 19 and bimeromorphic to an open set $\\fA^0$ of the G.I.T. quotient space of the projective space $\\fB:=\\{B\\in \\PP(U^\\vee\\otimes W\\otimes V^\\vee)\\}$ of triple tensors of type $(3,3,4)$ by the natural action of $SL(W)\\times SL(U)$. We give several constructions for these bundles, which relate them to cubic surfaces in 3-space $\\PP^3$ and to cubic surfaces in the dual space $(\\PP^3)^{\\vee}$. One of these constructions, suggested by Igor Dolgachev, generalizes to other types of tensors. Moreover, we relate the socalled {\\em cross-product involution} for $(3,3,4)$-tensors, introduced in \\cite{CaTo}, with the Schur quadric associated to a cubic surface in $\\PP^3$ and study further properties of this involution."}
{"category": "Math", "title": "The Gorenstein Colength of an Artinian Local Ring", "abstract": "In this paper, we make the notion of approximating an Artinian local ring by a Gorenstein Artin local ring precise using the concept of Gorenstein colength. We also answer the question as to when the Gorenstein colength is at most two."}
{"category": "Math", "title": "Multiresolution wavelet analysis of integer scale Bessel functions", "abstract": "We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived from geometric Hilbert-space considerations, the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of Hilbert spaces. We study the theory of representations of the $C^{\\ast}$-algebra $% O_{\\nu +1}$ arising from this multiresolution analysis. A connection with Markov chains and representations of $O_{\\nu +1}$ is found. Projection valued measures arising from the multiresolution analysis give rise to a Markov trace for quantum groups $SO_q$."}
{"category": "Math", "title": "Combinatorial Hopf algebras and K-homology of Grassmanians", "abstract": "Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be thought of as K-theoretic analogues of the by now classical ``square'' of Hopf algebras consisting of symmetric functions, quasisymmetric functions, noncommutative symmetric functions and the Malvenuto-Reutenauer Hopf algebra of permutations. In addition, we develop a theory of set-valued P-partitions and study three new families of symmetric functions which are weight generating functions of reverse plane partitions, weak set-valued tableaux and valued-set tableaux."}
{"category": "Math", "title": "Bagging multiple comparisons from microarray data", "abstract": "The problem of large-scale simultaneous hypothesis testing is re-visited. Bagging and subagging procedures are put forth with the purpose of improving the discovery power of the tests. The procedures are implemented in both simulated and real data. It is shown that bagging and subagging significantly improve power at the cost of a small increase in false discovery rate with the proposed `maximum contrast' subagging having an edge over bagging, i.e., yielding similar power but significantly smaller false discovery rates."}
{"category": "Math", "title": "Real interpolation of Sobolev spaces", "abstract": "We prove that $W^{1}_{p}$ is an interpolation space between $W^{1}_{p_{1}}$ and $W^{1}_{p_{2}}$ for $p>q_{0}$ and $1\\leq p_{1}<p<p_{2}\\leq \\infty$ on some classes of manifolds and general metric spaces, where $q_{0}$ depends on our hypotheses."}
{"category": "Math", "title": "Dye's theorem in the almost continuous category", "abstract": "We prove an almost continuous version of Dye's theorem: any two non-atomic probability measure preserving homeomorphisms of Polish spaces are almost continuously orbit equivalent. More precisely they are orbit equivalent by a map which is defined and continuous on a Polish subset of full measure with an inverse satisfying the same conditions. This result includes all of the recent results on almost continuous orbit equivalence. We also deal with the case of infinite invariant measures."}
{"category": "Math", "title": "On the convergence to equilibrium of Kac's random walk on matrices", "abstract": "We consider Kac's random walk on $n$-dimensional rotation matrices, where each step is a random rotation in the plane generated by two randomly picked coordinates. We show that this process converges to the Haar measure on $\\mathit{SO}(n)$ in the $L^2$ transportation cost (Wasserstein) metric in $O(n^2\\ln n)$ steps. We also prove that our bound is at most a $O(\\ln n)$ factor away from optimal. Previous bounds, due to Diaconis/Saloff-Coste and Pak/Sidenko, had extra powers of $n$ and held only for $L^1$ transportation cost. Our proof method includes a general result of independent interest, akin to the path coupling method of Bubley and Dyer. Suppose that $P$ is a Markov chain on a Polish length space $(M,d)$ and that for all $x,y\\in M$ with $d(x,y)\\ll1$ there is a coupling $(X,Y)$ of one step of $P$ from $x$ and $y$ (resp.) that contracts distances by a $(\\xi+o(1))$ factor on average. Then the map $\\mu\\mapsto\\mu P$ is $\\xi$-contracting in the transportation cost metric."}
{"category": "Math", "title": "Variance reduction for particle filters of systems with time-scale separation", "abstract": "We present a particle filter construction for a system that exhibits time-scale separation. The separation of time-scales allows two simplifications that we exploit: i) The use of the averaging principle for the dimensional reduction of the system needed to solve for each particle and ii) the factorization of the transition probability which allows the Rao-Blackwellization of the filtering step. Both simplifications can be implemented using the coarse projective integration framework. The resulting particle filter is faster and has smaller variance than the particle filter based on the original system. The method is tested on a multiscale stochastic differential equation and on a multiscale pure jump diffusion motivated by chemical reactions."}
{"category": "Math", "title": "A note on De Concini and Procesi's curious identity", "abstract": "We give a short, case-free and combinatorial proof of de Concini and Procesi's formula for the volume of the simplicial cone spanned by the simple roots of any finite root system. The argument presented here also extends their formula to include the non-crystallographic root systems."}
{"category": "Math", "title": "Double Shuffle Relations of Euler Sums", "abstract": "In this paper we shall develop a theory of (extended) double shuffle relations of Euler sums which generalizes that of multiple zeta values (see Ihara, Kaneko and Zagier, \\emph{Derivation and double shuffle relations for multiple zeta values}. Compos. Math. \\textbf{142} (2)(2006), 307--338). After setting up the general framework we provide some numerical evidence for our two main conjectures. At the end we shall prove the following long standing conjecture: for every positive integer n $$\\zeta(\\{3\\}^n)=8^n\\zeta(\\{\\ol2,1\\}^n).$$ The main idea is to use the double shuffle relations and the distribution relation. This particular distribution relation doesn't follow from the double shuffle relation in general. But we believe it does follow from the extended double shuffle relations."}
{"category": "Math", "title": "Real interpoaltion of Sobolev spaces associated to a weight", "abstract": "We study the interpolation property of Sobolev spaces of order 1 denoted by $W^{1}_{p,V}$, arising from Schr\\\"{o}dinger operators with positive potential. We show that for $1\\leq p_1<p<p_2<q_{0}$ with $p>s_0$, $W^{1}_{p,V}$ is a real interpolation space between $W_{p_1,V}^{1}$ and $W_{p_2,V}^{1}$ on some classes of manifolds and Lie groups. The constants $s_{0}, q_{0}$ depend on our hypotheses."}
{"category": "Math", "title": "The weighted complexity and the determinant functions of graphs", "abstract": "The complexity of a graph can be obtained as a derivative of a variation of the zeta function or a partial derivative of its generalized characteristic polynomial evaluated at a point [\\textit{J. Combin. Theory Ser. B}, 74 (1998), pp. 408--410]. A similar result for the weighted complexity of weighted graphs was found using a determinant function [\\textit{J. Combin. Theory Ser. B}, 89 (2003), pp. 17--26]. In this paper, we consider the determinant function of two variables and discover a condition that the weighted complexity of a weighted graph is a partial derivative of the determinant function evaluated at a point. Consequently, we simply obtain the previous results and disclose a new formula for the Bartholdi zeta function. We also consider a new weighted complexity, for which the weights of spanning trees are taken as the sum of weights of edges in the tree, and find a similar formula for this new weighted complexity. As an application, we compute the weighted complexities of the product of the complete graphs."}
{"category": "Math", "title": "Remarks on regularity conditions of the Navier-Stokes equations", "abstract": "Let $v$ and $\\o$ be the velocity and the vorticity of the a suitable weak solution of the 3D Navier-Stokes equations in a space-time domain containing $z_0 =(x_0, t_0)$, and $Q_{z_0, r} =B_{x_0, r}\\times (t_0-r^2, t_0)$ be a parabolic cylinder in the domain. We show that if $v\\times \\frac{\\o}{|\\o|}\\in L^{\\gamma, \\alpha}_{x,t} (Q_{z_0, r})$ or ${\\o}\\times \\frac{v}{|v|}\\in L^{\\gamma, \\alpha}_{x,t} (Q_{z_0, r})$, where $L^{\\gamma, \\alpha}_{x,t}$ denotes the Serrin type of class, then $z_0$ is a regular point for $v$. This refines previous local regularity criteria for the suitable weak solutions."}
{"category": "Math", "title": "On Uniqueness of Boundary Blow-up Solutions of a Class of Nonlinear Elliptic Equations", "abstract": "We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained."}
{"category": "Math", "title": "Construction of maximal unramified p-extensions with prescribed Galois groups", "abstract": "In the present paper, we shall show that for any prime number p, every finite p-group occurs as the Galois Group of the maximal unramified p-extension over a certain number field of finite degree. We shall also show that for any given pro-p-group G with countably many generators, there exists a number field (not necessary of finite degree) whose maximal unramified p-extension has Galois group isomorphic to G. This means that the set of the isomorphism classes of the Galois groups of the maximal unramified p-extensions over the number fields (including of infinite degree) is precisely equal to that of all the pro-p-groups with countably many generators."}
{"category": "Math", "title": "Local Sentences and Mahlo Cardinals", "abstract": "Local sentences were introduced by J.-P. Ressayre who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models. Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact (consistency) strength was left open in [O. Finkel and J.-P. Ressayre, Stretchings, Journal of Symbolic Logic, Volume 61 (2), 1996, p. 563-585 ]. Here, we solve this problem, using a combinatorial result of J. H. Schmerl. In fact, we show that the stretching principles are equivalent to the existence of n-Mahlo cardinals for appropriate integers n. This is done by proving first that for each integer n, there is a local sentence phi_n which has well ordered models of order type alpha, for every infinite ordinal alpha > omega which is not an n-Mahlo cardinal."}
{"category": "Math", "title": "On randomized stopping", "abstract": "A general result on the method of randomized stopping is proved. It is applied to optimal stopping of controlled diffusion processes with unbounded coefficients to reduce it to an optimal control problem without stopping. This is motivated by recent results of Krylov on numerical solutions to the Bellman equation."}
{"category": "Math", "title": "Stability of associated primes of monomial ideals", "abstract": "Let $I$ be a monomial ideal of a polynomial ring $R$. In this paper we determine a number $B$ such that $\\Ass (I^n/I^{n+1}) = \\Ass (I^{B}/I^{B+1})$ for all $n\\geq B$."}
{"category": "Math", "title": "The orbifold transform and its applications", "abstract": "We discuss the notion of the orbifold transform, and illustrate it on simple examples. The basic properties of the transform are presented, including transitivity and the exponential formula for symmetric products. The connection with the theory of permutation orbifolds is addressed, and the general results illustrated on the example of torus partition functions."}
{"category": "Math", "title": "Extremal Presentations for Classical Lie Algebras", "abstract": "The long-root elements in Lie algebras of Chevalley type have been well studied and can be characterized as extremal elements, that is, elements $x$ such that the image of $(\\ad x)^2$ lies in the subspace spanned by $x$. In this paper, assuming an algebraically closed base field of characteristic not 2, we find presentations of the Lie algebras of classical Chevalley type by means of minimal sets of extremal generators. The relations are described by simple graphs on the sets. For example, for $C_n$ the graph is a path of length $2n$, and for $A_n$ the graph is the triangle connected to a path of length $n-3$."}
{"category": "Math", "title": "An elementary proof of the convergence of Ricci flow on compact surfaces", "abstract": "This paper has been withdrawn by the author for further modification."}
{"category": "Math", "title": "On the motive of certain subvarieties of fixed flags", "abstract": "We compute de Chow motive of certain subvarieties of the flags manifold and show that it is an Artin motive."}
{"category": "Math", "title": "Uniqueness of polynomial canonical representations", "abstract": "Let P(z) and Q(y) be polynomials of the same degree k>=1 in the complex variables z and y, respectively. In this extended abstract we study the non-linear functional equation P(z)=Q(y(z)), where y(z) is restricted to be analytic in a neighborhood of z=0. We provide sufficient conditions to ensure that all the roots of Q(y) are contained within the range of y(z) as well as to have y(z)=z as the unique analytic solution of the non-linear equation. Our results are motivated from uniqueness considerations of polynomial canonical representations of the phase or amplitude terms of oscillatory integrals encountered in the asymptotic analysis of the coefficients of mixed powers and multivariable generating functions via saddle-point methods. Uniqueness shall prove important for developing algorithms to determine the Taylor coefficients of the terms appearing in these representations. The uniqueness of Levinson's polynomial canonical representations of analytic functions in several variables follows as a corollary of our one-complex variables results."}
{"category": "Math", "title": "Lasso type classifiers with a reject option", "abstract": "We consider the problem of binary classification where one can, for a particular cost, choose not to classify an observation. We present a simple proof for the oracle inequality for the excess risk of structural risk minimizers using a lasso type penalty."}
{"category": "Math", "title": "Normalization of twisted Alexander invariants", "abstract": "Twisted Alexander invariants of knots are well-defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, and refine the duality theorem. We further obtain necessary conditions on the invariants for a knot to be fibered, and study behavior of the highest degrees of the invariants."}
{"category": "Math", "title": "Singular link Floer homology", "abstract": "We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that this homology is acyclic under some conditions which naturally make its Euler characteristic vanish."}
{"category": "Math", "title": "The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals", "abstract": "We present evalauations and provide proofs of definite integrals involving the function x^p cos^n x. These formulae are generalizations of 3.761.11 and 3.822.1, among others, in the classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik."}
{"category": "Math", "title": "Generalised Einstein condition and cone construction for parabolic geometries", "abstract": "This paper attempts to define a generalisation of the standard Einstein condition (in conformal/metric geometry) to any parabolic geometry. To do so, it shows that any preserved involution $\\sigma$ of the adjoint bundle $\\mc{A}$ gives rise, given certain algebraic conditions, to a unique preferred affine connection $\\nabla$ with covariantly constant rho-tensor $\\mathsf{P}$, compatible with the algebraic bracket on $\\mc{A}$. These conditions can reasonably be considered the generalisations of the Einstein condition, and recreate the standard Einstein condition in conformal geometry. The existence of such an involution is implies by some simpler structures: preserved metrics when the overall algebra $\\mf{g}$ is $\\mf{sl}(m,\\mbb{F})$, preserved complex structures anti-commuting with the skew-form for $\\mf{g}=\\mf{sp}(2m,\\mbb{F})$, and preserved subundles of the tangent bundle, of a certain rank, for all the other non-exceptional simple Lie algebras. Examples of Einstein involutions are constructed or referenced for several geometries. The existence of cone constructions for certain Einstein involutions is then demonstrated."}
{"category": "Math", "title": "Torsion units in integral group ring of Higman-Sims simple group", "abstract": "Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Higman-Sims simple sporadic group HS. As a consequence, we confirm the Kimmerle's conjecture on prime graphs for this sporadic group."}
{"category": "Math", "title": "The Reduced Genus-One Gromov-Witten Invariants of Calabi-Yau Hypersurfaces", "abstract": "We compute the reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces. As a consequence, we confirm the 1993 Bershadsky-Cecotti Ooguri-Vafa (BCOV) prediction for the standard genus 1 GW-invariants of a quintic threefold. We combine constructions from a series of previous papers with the classical localization theorem to relate the reduced genus 1 invariants of a CY-hypersurface to previously computed integrals on moduli spaces of stable genus 0 maps into projective space. The resulting, rather unwieldy, expressions for a genus 1 equivariant generating function simplify drastically, using a regularity property of a genus 0 equivariant generating function in half of the cases. Finally, by disregarding terms that cannot effect the non-equivariant part of the former, we relate the answer to an explicit hypergeometric series in a simple way. The approach described in this paper is systematic. It is directly applicable to computing reduced genus 1 GW-invariants of other complete intersections and should apply to higher-genus localization computations."}
{"category": "Math", "title": "Misere quotients for impartial games: Supplementary material", "abstract": "We provide supplementary appendices to the paper Misere quotients for impartial games. These include detailed solutions to many of the octal games discussed in the paper, and descriptions of the algorithms used to compute most of our solutions."}
{"category": "Math", "title": "Nonuniform Thickness and Weighted Distance", "abstract": "Nonuniform tubular neighborhoods of curves in Euclidean n-space are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtained for nonuniform thickness. All singularities within almost injectivity radius are classified by the Horizontal Collapsing Property. Examples are provided to show the distinction between the different types of injectivity radii, as well as showing that the standard differentiable injectivity radius fails to be upper semicontinuous on a singular set of weight functions."}
{"category": "Math", "title": "Exploded Fibrations", "abstract": "Initiated by Gromov, the study of holomorphic curves in symplectic manifolds has been a powerfull tool in symplectic topology, however the moduli space of holomorphic curves is often very difficult to find. A common technique is to study the limiting behavior of holomorphic curves in a degenerating family of complex structures which corresponds to a kind of adiabatic limit. The category of exploded fibrations is an extension of the smooth category in which some of these degenerations can be described as smooth families. The first part of this paper is devoted to defining exploded fibrations and a slightly more specialized category of exploded torus fibrations. Later sections contain the transverse interesction theory for exploded fibrations and some examples of holomorphic curves in exploded torus fibrations, including a brief discussion of the relationship between tropical geometry and exploded torus fibrations. In the final section, the perturbation theory of holomorphic curves in exploded torus fibrations is sketched."}
{"category": "Math", "title": "The Nahm transform for calorons", "abstract": "In this paper, we complete the proof of an equivalence given by Nye and Singer of the equivalence between calorons (instantons on $S^1\\times R^3$) and solutions to Nahm's equations over the circle, both satisfying appropriate boundary conditions. Many of the key ingredients are provided by a third way of encoding the same data which involves twistors and complex geometry."}
{"category": "Math", "title": "The asymptotic volume of the Birkhoff polytope", "abstract": "Let m,n be positive integers. Define T(m,n) to be the transportation polytope consisting of the m x n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case B(n)=T(n,n) is the much-studied Birkhoff-von Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determine the asymptotic volume of T(m,n) as n goes to infinity, with m=m(n) such that m/n neither decreases nor increases too quickly. In particular, we give an asymptotic formula for the volume of B(n)."}
{"category": "Math", "title": "Standard Bases in K[[t_1,...,t_m]][x_1,...,x_n]^s", "abstract": "In this paper we study standard bases for submodules of K[[t_1,...,t_m]][x_1,...,x_n]^s respectively of their localisation with respect to a t-local monomial ordering. The main step is to prove the existence of a division with remainder generalising and combining the division theorems of Grauert and Mora. Everything else then translates naturally. Setting either m=0 or n=0 we get standard bases for polynomial rings respectively for power series rings as a special case. We then apply this technique to show that the t-initial ideal of an ideal over the Puiseux series field can be read of from a standard basis of its generators. This is an important step in the constructive proof that each point in the tropical variety of such an ideal admits a lifting."}
{"category": "Math", "title": "A tight bound on the collection of edges in MSTs of induced subgraphs", "abstract": "Let $G=(V,E)$ be a complete $n$-vertex graph with distinct positive edge weights. We prove that for $k\\in\\{1,2,...,n-1\\}$, the set consisting of the edges of all minimum spanning trees (MSTs) over induced subgraphs of $G$ with $n-k+1$ vertices has at most $nk-\\binom{k+1}{2}$ elements. This proves a conjecture of Goemans and Vondrak \\cite{GV2005}. We also show that the result is a generalization of Mader's Theorem, which bounds the number of edges in any edge-minimal $k$-connected graph."}
{"category": "Math", "title": "An algorithm for lifting points in a tropical variety", "abstract": "The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseux-valued ``lift'' of this point in the algebraic variety. This theorem is so fundamental because it justifies why a tropical variety (defined combinatorially using initial ideals) carries information about algebraic varieties: it is the image of an algebraic variety over the Puiseux series under the valuation map. We have implemented the ``lifting algorithm'' using Singular and Gfan if the base field are the rational numbers. As a byproduct we get an algorithm to compute the Puiseux expansion of a space curve singularity in (K^{n+1},0)."}
{"category": "Math", "title": "Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity", "abstract": "In this short note, we give a link between the regularity of the solution $u$ to the 3D Navier-Stokes equation, and the behavior of the direction of the velocity $u/|u|$. It is shown that the control of $\\Div (u/|u|)$ in a suitable $L_t^p(L_x^q)$ norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. But in this case the condition is not on the vorticity, but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines."}
{"category": "Math", "title": "Packing dimension of mean porous measures", "abstract": "We prove that the packing dimension of any mean porous Radon measure on $\\mathbb R^d$ may be estimated from above by a function which depends on mean porosity. The upper bound tends to $d-1$ as mean porosity tends to its maximum value. This result was stated in \\cite{BS}, and in a weaker form in \\cite{JJ1}, but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure $\\mu$ on $\\mathbb R$ such that $\\mu(A)=0$ for all mean porous sets $A\\subset\\mathbb R$."}
{"category": "Math", "title": "Cyclotomic factors of the descent set polynomial", "abstract": "We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group S_n only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics."}
{"category": "Math", "title": "Ces\\`aro means of orthogonal expansions in several variables", "abstract": "Ces\\`aro $(C,\\delta)$ means are studied for orthogonal expansions with respect to the weight function $\\prod_{i=1}^{d}|x_i|^{2\\k_i}$ on the unit sphere, and for the corresponding weight functions on the unit ball and the Jacobi weight on the simplex. A sharp pointwise estimate is established for the $(C,\\d)$ kernel with $\\d > -1$ and for the kernel of the projection operator, which allows us to derive the exact order for the norm of the Ces\\`aro means and the projection operator on these domains."}
{"category": "Math", "title": "Recursive calculation of effective resistances in distance-regular networks based on Bose-Mesner algebra and Christoffel-Darboux identity", "abstract": "Recently in \\cite{jss1}, the authors have given a method for calculation of the effective resistance (resistance distance) on distance-regular networks, where the calculation was based on stratification introduced in \\cite{js} and Stieltjes transform of the spectral distribution (Stieltjes function) associated with the network. Also, in Ref. \\cite{jss1} it has been shown that the resistance distances between a node $\\alpha$ and all nodes $\\beta$ belonging to the same stratum with respect to the $\\alpha$ ($R_{\\alpha\\beta^{(i)}}$, $\\beta$ belonging to the $i$-th stratum with respect to the $\\alpha$) are the same. In this work, an algorithm for recursive calculation of the resistance distances in an arbitrary distance-regular resistor network is provided, where the derivation of the algorithm is based on the Bose-Mesner algebra, stratification of the network, spectral techniques and Christoffel-Darboux identity. It is shown that the effective resistance on a distance-regular network is an strictly increasing function of the shortest path distance defined on the network. In the other words, the two-point resistance $R_{\\alpha\\beta^{(m+1)}}$ is strictly larger than $R_{\\alpha\\beta^{(m)}}$. The link between the resistance distance and random walks on distance-regular networks is discussed, where the average commute time (CT) and its square root (called Euclidean commute time (ECT)) as a distance are related to the effective resistance. Finally, for some important examples of finite distance- regular networks, the resistance distances are calculated. {\\bf Keywords: resistance distance, association scheme, stratification, distance-regular networks, Christoffel-Darboux identity} {\\bf PACs Index: 01.55.+b, 02.10.Yn}"}
{"category": "Math", "title": "A Spectral Sequence for the K-theory of Tiling Spaces", "abstract": "Let $\\Tt$ be an aperiodic and repetitive tiling of $\\RM^d$ with finite local complexity. We present a spectral sequence that converges to the $K$-theory of $\\Tt$ with $E_2$-page given by a new cohomology that will be called PV in reference to the Pimsner-Voiculescu exact sequence. It is a generalization of the Serre spectral sequence. The PV cohomology of $\\Tt$ generalizes the cohomology of the base space of a fibration with local coefficients in the $K$-theory of its fiber. We prove that it is isomorphic to the \\v{C}ech cohomology of the hull of $\\Tt$ (a compactification of the family of its translates)."}
{"category": "Math", "title": "Characterization of rank two locally nilpotent derivations in dimension three", "abstract": "In this paper we give an algorithmic characterization of rank two locally nilpotent derivations in dimension three. Together with an algorithm for computing the plinth ideal, this gives a method for computing the rank of a locally nilpotent derivation in dimension three."}
{"category": "Math", "title": "Triangulable locally nilpotent derivations in dimension three", "abstract": "In this paper we give an algorithm to recognize triangulable locally nilpotent derivations in dimension three. In case the given derivation is triangulable, our method produces a coordinate system in which it exhibits a triangular form."}
{"category": "Math", "title": "Deformations of generalized complex and generalized Kahler structures", "abstract": "In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which admits deformations to generalized complex manifolds and obtain non-trivial generalized Kahler structures on Fano surfaces and toric Kahler manifolds. In particular, we show that holomorphic Poisson structures on a Kahler manifold induce deformations of generalized Kahler structures."}
{"category": "Math", "title": "Finite Element Model Updating Using Bayesian Approach", "abstract": "This paper compares the Maximum-likelihood method and Bayesian method for finite element model updating. The Maximum-likelihood method was implemented using genetic algorithm while the Bayesian method was implemented using the Markov Chain Monte Carlo. These methods were tested on a simple beam and an unsymmetrical H-shaped structure. The results show that the Bayesian method gave updated finite element models that predicted more accurate modal properties than the updated finite element models obtained through the use of the Maximum-likelihood method. Furthermore, both these methods were found to require the same levels of computational loads."}
{"category": "Math", "title": "Forms of higher degree permitting composition", "abstract": "Nondegenerate forms N of degree d on a unital nonassociative algebra A over a ring R which permit composition, i.e., satisfy N(1)=1 and N(xy)=N(x)N(y) for all x,y in A, are studied. These forms were first classified by Schafer over fields of characteristic 0 or >d. We investigate cubic and quartic nondegenerate forms which permit composition over certain rings and curves. Classes of highly degenerate cubic forms N over fields which permit composition are constructed."}
{"category": "Math", "title": "Convex-transitive characterizations of Hilbert spaces", "abstract": "In this paper we investigate real convex-transitive Banach spaces X, which admit a 1-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. The results obtained can be regarded as partial answers to the well-known Banach-Mazur rotation problem, as well as to a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces."}
{"category": "Math", "title": "From multiplicative unitaries to quantum groups II", "abstract": "It is shown that all important features of a $\\mathrm{C}^*$-algebraic quantum group $(A,\\Delta)$ defined by a modular multiplicative $W$ depend only on the pair $(A,\\Delta)$ rather than the multiplicative unitary operator $W$. The proof is based on thorough study of representations of quantum groups. As an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary - without assuming existence of Haar weights."}
{"category": "Math", "title": "Reflexivity in Derived Categories", "abstract": "An adjoint pair of contravariant functors between abelian categories can be extended to the adjoint pair of their derived functors in the associated derived categories. We describe the reflexive complexes and interpret the achieved results in terms of objects of the initial abelian categories. In particular we prove that, for functors of any finite cohomological dimension, the objects of the initial abelian categories which are reflexive as stalk complexes form the largest class where a Cotilting Theorem in the sense of Colby and Fuller works."}
{"category": "Math", "title": "A Bayesian approach to the estimation of maps between riemannian manifolds", "abstract": "Let \\Theta be a smooth compact oriented manifold without boundary, embedded in a euclidean space and let \\gamma be a smooth map \\Theta into a riemannian manifold \\Lambda. An unknown state \\theta \\in \\Theta is observed via X=\\theta+\\epsilon \\xi where \\epsilon>0 is a small parameter and \\xi is a white Gaussian noise. For a given smooth prior on \\Theta and smooth estimator g of the map \\gamma we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces \\Theta and \\Lambda, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of \\gamma is found based on the modern theory of harmonic maps and hypo-elliptic differential operators."}
{"category": "Math", "title": "On asymptotic dimension of amalgamated products and right-angled Coxeter groups", "abstract": "We prove the inequality $$ \\as A\\ast_CB\\le\\max\\{\\as A,\\as B,\\as C+1\\} $$ and we apply this inequality to show that the asymptotic dimension of any right-angled Coxeter group does not exceed the dimension of its Davis' complex."}
{"category": "Math", "title": "Some remarks on spherical harmonics", "abstract": "The article contains several observations on spherical harmonics and their nodal sets: a construction for harmonics with prescribed zeroes; a kind of canonical representation of this type for harmonics on $\\bbS^2$; upper and lower bounds for nodal length and inner radius (the upper bounds are sharp); precise upper bound for the number of common zeroes of two spherical harmonics on $\\bbS^2$; the mean Hausdorff measure on the intersection of $k$ nodal sets for harmonics of different degrees on $\\bbS^m$, where $k\\leq m$ (in particular, the mean number of common zeroes of $m$ harmonics)."}
{"category": "Math", "title": "The strong Novikov conjecture for low degree cohomology", "abstract": "We show that for each discrete group G, the rational assembly map K_*(BG) \\otimes Q \\to K_*(C*_{max} G) \\otimes \\Q is injective on classes dual to the subring generated by cohomology classes of degree at most 2 (identifying rational K-homology and homology via the Chern character). Our result implies homotopy invariance of higher signatures associated to these cohomology classes. This consequence was first established by Connes-Gromov-Moscovici and Mathai. Our approach is based on the construction of flat twisting bundles out of sequences of almost flat bundles as first described in our previous work. In contrast to the argument of Mathai, our approach is independent of (and indeed gives a new proof of) the result of Hilsum-Skandalis on the homotopy invariance of the index of the signature operator twisted with bundles of small curvature."}
{"category": "Math", "title": "Adelic resolution for homology sheaves", "abstract": "A generalization of the usual ideles group is proposed, namely, we construct certain adelic complexes for sheaves of $K$-groups on schemes. More generally, such complexes are defined for any abelian sheaf on a scheme. We focus on the case when the sheaf is associated to the presheaf of a homology theory with certain natural axioms, satisfied by $K$-theory. In this case it is proven that the adelic complex provides a flasque resolution for the above sheaf and that the natural morphism to the Gersten complex is a quasiisomorphism. The main advantage of the new adelic resolution is that it is contravariant and multiplicative in contrast to the Gersten resolution. In particular, this allows to reprove that the intersection in Chow groups coincides up to sign with the natural product in the corresponding $K$-cohomology groups. Also, we show that the Weil pairing can be expressed as a Massey triple product in $K$-cohomology groups with certain indices."}
{"category": "Math", "title": "Sample eigenvalue based detection of high dimensional signals in white noise using relatively few samples", "abstract": "We present a mathematically justifiable, computationally simple, sample eigenvalue based procedure for estimating the number of high-dimensional signals in white noise using relatively few samples. The main motivation for considering a sample eigenvalue based scheme is the computational simplicity and the robustness to eigenvector modelling errors which are can adversely impact the performance of estimators that exploit information in the sample eigenvectors. There is, however, a price we pay by discarding the information in the sample eigenvectors; we highlight a fundamental asymptotic limit of sample eigenvalue based detection of weak/closely spaced high-dimensional signals from a limited sample size. This motivates our heuristic definition of the effective number of identifiable signals which is equal to the number of \"signal\" eigenvalues of the population covariance matrix which exceed the noise variance by a factor strictly greater than 1+sqrt(Dimensionality of the system/Sample size). The fundamental asymptotic limit brings into sharp focus why, when there are too few samples available so that the effective number of signals is less than the actual number of signals, underestimation of the model order is unavoidable (in an asymptotic sense) when using any sample eigenvalue based detection scheme, including the one proposed herein. The analysis reveals why adding more sensors can only exacerbate the situation. Numerical simulations are used to demonstrate that the proposed estimator consistently estimates the true number of signals in the dimension fixed, large sample size limit and the effective number of identifiable signals in the large dimension, large sample size limit."}
{"category": "Math", "title": "On the Selmer groups of abelian varieties over function fields of characteristic p>0", "abstract": "In this paper, we study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We propose also a conjecture which might be considered as a counterpart of the principal conjecture in the case over a number field. \\par This is a preprint which is distributed since 2005 which is still in the process of submision. Following a recent modification of some technical mistakes in the previous version of the paper as well as an amelioration of the presentation of the paper, we decide wider distribution via the archive."}
{"category": "Math", "title": "Homotopy groups of Hom complexes of graphs", "abstract": "The notion of $\\times$-homotopy from \\cite{DocHom} is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space $\\Hom_*(G,H)$ with the homotopy groups of $\\Hom_*(G,H^I)$. Here $\\Hom_*(G,H)$ is a space which parametrizes pointed graph maps from $G$ to $H$ (a pointed version of the usual $\\Hom$ complex), and $H^I$ is the graph of based paths in $H$. As a corollary it is shown that $\\pi_i \\big(\\Hom_*(G,H) \\big) \\cong [G,\\Omega^i H]_{\\times}$, where $\\Omega H$ is the graph of based closed paths in $H$ and $[G,K]_{\\times}$ is the set of $\\times$-homotopy classes of pointed graph maps from $G$ to $K$. This is similar in spirit to the results of \\cite{BBLL}, where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed."}
{"category": "Math", "title": "Densely ordered braid subgroups", "abstract": "Dehornoy showed that the Artin braid groups $B_n$ are left-orderable. This ordering is discrete, but we show that, for $n >2$ the Dehornoy ordering, when restricted to certain natural subgroups, becomes a dense ordering. Among subgroups which arise are the commutator subgroup and the kernel of the Burau representation (for those $n$ for which the kernel is nontrivial). These results follow from a characterization of least positive elements of any normal subgroup of $B_n$ which is discretely ordered by the Dehornoy ordering."}
{"category": "Math", "title": "Boundary Regularity for Conformally Compact Einstein Metrics in Even Dimensions", "abstract": "We study boundary regularity for conformally compact Einstein metrics in even dimensions by generalizing the ideas of Michael Anderson. Our method of approach is to view the vanishing of the Ambient Obstruction tensor as an nth order system of equations for the components of a compactification of the given metric. This, together with boundary conditions that the compactification is shown to satisfy provide enough information to apply classical boundary regularity results. These results then provide local and global versions of finite boundary regularity for the components of the compactification."}
{"category": "Math", "title": "Resonance between Cantor sets", "abstract": "Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\\log b/\\log a$ is irrational, then \\[ \\dim(C_a+C_b) = \\min(\\dim(C_a) + \\dim(C_b),1), \\] where $\\dim$ is Hausdorff dimension. More generally, given two self-similar sets $K,K'$ in $\\RR$ and a scaling parameter $s>0$, if the dimension of the arithmetic sum $K+sK'$ is strictly smaller than $\\dim(K)+\\dim(K') \\le 1$ (``geometric resonance''), then there exists $r<1$ such that all contraction ratios of the similitudes defining $K$ and $K'$ are powers of $r$ (``algebraic resonance''). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation."}
{"category": "Math", "title": "A geometric categorification of tensor products of $U_q(sl_2)$-modules", "abstract": "We give a purely geometric categorification of tensor products of finite-dimensional simple $U_q(sl_2)$-modules and $R$-matrices on them. The work is developed in the framework of category of perverse sheaves and the categorification theorems are understood as consequences of Deligne's theory of weights."}
{"category": "Math", "title": "New identities in dendriform algebras", "abstract": "Dendriform structures arise naturally in algebraic combinatorics (where they allow, for example, the splitting of the shuffle product into two pieces) and through Rota-Baxter algebra structures (the latter appear, among others, in differential systems and in the renormalization process of pQFT). We prove new combinatorial identities in dendriform dialgebras that appear to be strongly related to classical phenomena, such as the combinatorics of Lyndon words, rewriting rules in Lie algebras, or the fine structure of the Malvenuto-Reutenauer algebra. One of these identities is an abstract noncommutative, dendriform, generalization of the Bohnenblust-Spitzer identity and of an identity involving iterated Chen integrals due to C.S. Lam."}
{"category": "Math", "title": "Normalization of bundle holomorphic contractions and applications to dynamics", "abstract": "We establish a Poincar\\'e-Dulac theorem for sequences (G_n)_n of holomorphic contractions whose differentials d_0 G_n split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps. Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of CP(k). In this context, our normalization result allows to precisely estimate the distortions of ellipsoids along typical orbits. As an application, we show how the Lyapunov exponents of the equilibrium measure are approximated in terms of the multipliers of the repulsive cycles."}
{"category": "Math", "title": "Strong peak points and denseness of strong peak functions", "abstract": "Let $C_b(K)$ be the set of all bounded continuous (real or complex) functions on a complete metric space $K$ and $A$ a closed subspace of $C_b(K)$. Using the variational method, it is shown that the set of all strong peak functions in $A$ is dense if and only if the set of all strong peak points is a norming subset of $A$. As a corollary we show that if $X$ is a locally uniformly convex, complex Banach space, then the set of all strong peak functions in $\\mathcal{A}(B_X)$ is a dense $G_\\delta$ subset. Moreover if $X$ is separable, smooth and locally uniformly convex, then the set of all norm and numerical strong peak functions in $\\mathcal{A}_u(B_X:X)$ is a dense $G_\\delta$ subset. In case that a set of uniformly strongly exposed points of a (real or complex) Banach space $X$ is a norming subset of $\\mathcal{P}({}^n X)$ for some $n\\ge 1$, then the set of all strongly norm attaining elements in $\\mathcal{P}({}^n X)$ is dense, in particular, the set of all points at which the norm of $\\mathcal{P}({}^n X)$ is Fr\\'echet differentiable is a dense $G_\\delta$ subset."}
{"category": "Math", "title": "Floer homology and singular knots", "abstract": "We define Floer homology theories for oriented, singular knots in S^3 and show that one of these theories can be defined combinatorially for planar singular knots."}
{"category": "Math", "title": "A duality theorem for generalized local cohomology", "abstract": "We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality."}
{"category": "Math", "title": "On Asymptotic Proximity of Distributions", "abstract": "We consider some general facts concerning convergence P_{n}-Q_{n}\\to 0 as n\\to \\infty, where P_{n} and Q_{n} are probability measures in a complete separable metric space. The main point is that the sequences {P_{n}} and {Q_{n}} are not assumed to be tight. We compare different possible definitions of the above convergence, and establish some general properties."}
{"category": "Math", "title": "Nilpotent bicone and characteristic submodule of a reductive Lie algebra", "abstract": "The nilpotent bicone of a finite dimensional complex reductive Lie algebra g is the subset of elements in g x g whose subspace generated by the components is contained in the nilpotent cone of g. The main result of this note is that the nilpotent bicone is a complete intersection. This affirmatively answers a conjecture of Kraft-Wallach concerning the nullcone. In addition, we introduce and study the characteristic submodule of g. The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. In order to study the nilpotent bicone, we introduce another subvariety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed in http://arxiv.org/abs/math/0008002v5 ."}
{"category": "Math", "title": "Rational torus-equivariant homotopy I: calculating groups of stable maps", "abstract": "We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show it is of finite injective dimension. It can be used as a model for rational $G$-spectra in the sense that there is a homology theory \\piA_*: G-spectra/Q --> A(G) on rational G-spectra with values in A(G), and the associated Adams spectral sequence converges for all rational $G$-spectra and collapses at a finite stage."}
{"category": "Math", "title": "Circle-equivariant classifying spaces and the rational equivariant sigma genus", "abstract": "The circle-equivariant spectrum MString_C is the equivariant analogue of the cobordism spectrum MU<6> of stably almost complex manifolds with c_1=c_2=0. Given a rational elliptic curve C, the second author has defined a ring T-spectrum EC representing the associated T-equivariant elliptic cohomology. The core of the present paper is the construction, when C is a complex elliptic curve, of a map of ring T-spectra MString_C --> EC which is the rational equivariant analogue of the sigma orientation of Ando-Hopkins-Strickland. We support this by a theory of characteristic classes for calculation, and a conceptual description in terms of algebraic geometry. In particular, we prove a conjecture of the first author."}
{"category": "Math", "title": "Binary Search Tree insertion, the Hypoplactic insertion, and Dual Graded Graphs", "abstract": "Fomin (1994) introduced a notion of duality between two graded graphs on the same set of vertices. He also introduced a generalization to dual graded graphs of the classical Robinson-Schensted-Knuth algorithm. We show how Fomin's approach applies to the binary search tree insertion algorithm also known as sylvester insertion, and to the hypoplactic insertion algorithm."}
{"category": "Math", "title": "Finite dimensional representations of DAHA and affine Springers fibers : the spherical case", "abstract": "We classify finite dimensional simple spherical representations of rational double affine Hecke algebras, and we study a remarkable family of finite dimensional simple spherical representations of double affine Hecke algebras."}
{"category": "Math", "title": "On the Riemann zeta-function, Part II", "abstract": "An odd meromorphic function f(s) is constructed from the Riemann zeta-function evaluated at one-half plus s. We determine the two-sided Laplace transform representation of f(s) on open vertical strips, V'(4w), disjoint from the (translated) critical strip. V'(4w) consists of all s with real part, Re(s), of absolute value greater than one-half and Re(s) between successive poles 4w, 4(w + 1) of f(s), with w an integer. The corresponding Laplace density is related to confluent hypergeometric functions. That density is shown to be positive for nonzero w other than -1. Those results are obtained without relying on any unproven hypothesis. They are used together with the Riemann hypothesis and hypotheses advanced by the author to obtain conditional results concerning the zeta-function. Those results are presented in Part I. Their proofs are derived in Parts III-V. A metric geometry expression of the positivity of the Laplace densities arising is established in Part VI."}
{"category": "Math", "title": "Asymptotics for Duration-Driven Long Range Dependent Processes", "abstract": "We consider processes with second order long range dependence resulting from heavy tailed durations. We refer to this phenomenon as duration-driven long range dependence (DDLRD), as opposed to the more widely studied linear long range dependence based on fractional differencing of an $iid$ process. We consider in detail two specific processes having DDLRD, originally presented in Taqqu and Levy (1986), and Parke (1999). For these processes, we obtain the limiting distribution of suitably standardized discrete Fourier transforms (DFTs) and sample autocovariances. At low frequencies, the standardized DFTs converge to a stable law, as do the standardized sample autocovariances at fixed lags. Finite collections of standardized sample autocovariances at a fixed set of lags converge to a degenerate distribution. The standardized DFTs at high frequencies converge to a Gaussian law. Our asymptotic results are strikingly similar for the two DDLRD processes studied. We calibrate our asymptotic results with a simulation study which also investigates the properties of the semiparametric log periodogram regression estimator of the memory parameter."}
{"category": "Math", "title": "A conic manifold perspective of elliptic operators on graphs", "abstract": "We give a simple, explicit, sufficient condition for the existence of a sector of minimal growth for second order regular singular differential operators on graphs. We specifically consider operators with a singular potential of Coulomb type and base our analysis on the theory of elliptic cone operators."}
{"category": "Math", "title": "Log Fano varieties over function fields of curves", "abstract": "Consider a smooth log Fano variety over the function field of a curve. Suppose that the boundary has positive normal bundle. Choose an integral model over the curve. Then integral points are Zariski dense, after removing an explicit finite set of points on the base curve."}
{"category": "Math", "title": "Differential systems associated with tableaux over Lie algebras", "abstract": "We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [Comm. Anal. Geom 14 (2006), 475-496; math.DG/0412169]. The definition of a tableau over a Lie algebra is revisited and extended in the light of the formalism of the Spencer cohomology; the question of involutiveness for the associated systems and their prolongations is addressed; examples are discussed."}
{"category": "Math", "title": "Genus-Zero Two-Point Hyperplane Integrals in the Gromov-Witten Theory", "abstract": "In this paper we compute certain two-point integrals over a moduli space of stable maps into projective space. Computation of one-point analogues of these integrals constitutes a proof of mirror symmetry for genus-zero one-point Gromov-Witten invariants of projective hypersurfaces. The integrals computed in this paper constitute a significant portion in the proof of mirror symmetry for genus-one GW-invariants completed in a separate paper. These integrals also provide explicit mirror formulas for genus-zero two-point GW-invariants of projective hypersurfaces. The approach described in this paper leads to a reconstruction algorithm for all genus-zero GW-invariants of projective hypersurfaces."}
{"category": "Math", "title": "Fold maps and immersions from the viewpoint of cobordism", "abstract": "We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. By using geometric invariants defined in the author's earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group by natural homomorphisms between cobordism groups of immersions. We also compute the ranks of the oriented (right-left) bordism groups of simple fold maps."}
{"category": "Math", "title": "Real Zeros and Partitions without singleton blocks", "abstract": "We prove that the generating polynomials of partitions of an $n$-element set into non-singleton blocks, counted by the number of blocks, have real roots only and we study the asymptotic behavior of the leftmost roots. We apply this information to find the most likely number of blocks."}
{"category": "Math", "title": "Rigidity of pseudo-Anosov flows transverse to R-covered foliations", "abstract": "A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two, then the foliation is weakly conjugate to the the stable foliation of an R-covered Anosov flow. The proof uses the universal circle to R-covered foliations."}
{"category": "Math", "title": "Generalized functions as sequence spaces with ultranorms", "abstract": "We review our recent formulation of Colombeau type algebras as Hausdorff sequence spaces with ultranorms, defined by sequences of exponential weights. We extend previous results and give new perspectives related to echelon type spaces, possible generalisations, asymptotic algebras, concepts of association, and applications thereof."}
{"category": "Math", "title": "Measures related to (e,n)-complexity functions", "abstract": "The (e,n)-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance $\\epsilon$ during the time interval n. Behavior of the (e, n)-complexity functions as n goes to infinity is reflected in the properties of special measures. These measures are constructed as limits of atomic measures supported at points of (e,n)-separated sets. We study such measures. In particular, we prove that they are invariant if the (e,n)-complexity function grows subexponentially. Keywords: Topological entropy, complexity functions, separability."}
{"category": "Math", "title": "Un probl\\`eme de type Yamabe sur les vari\\'et\\'es compactes spinorielles compactes", "abstract": "Let $(M,g,\\si)$ be a compact spin manifold of dimension $n \\geq 2$. Let $\\lambda_1^+(\\tilde{g})$ be the smallest positive eigenvalue of the Dirac operator in the metric $\\tilde{g} \\in [g]$ conformal to $g$. We then define $\\lamin(M,[g],\\si) = \\inf_{\\tilde{g} \\in [g]} \\lambda_1^+(\\tilde{g}) \\Vol(M,\\tilde{g})^{1/n} $. We show that $0< \\lamin(M,[g],\\si) \\leq \\lamin(\\mS^n)$. %=\\frac{n}{2} \\om_n^{{1 \\over n}}$ . We find sufficient conditions for which we obtain strict inequality $\\lamin(M,[g],\\si) < \\lamin(\\mS^n)$. This strict inequality has applications to conformal spin geometry. ----- Soit $(M,g,\\si)$ une vari\\'et\\'e spinorielle compacte de dimension $n \\geq 2$. %Si $\\tilde{g} \\in [g]$ est une m\\'etrique conforme \\`a $g$, On note $\\lambda_1^+(\\tilde{g})$ la plus petite valeur propre $>0$ de l'op\\'erateur de Dirac dans la m\\'etrique $\\tilde{g} \\in [g]$ conforme \\`a $g$. On d\\'efinit $\\lamin(M,[g],\\si) = \\inf_{\\tilde{g} \\in [g]} \\lambda_1^+(\\tilde{g}) \\Vol(M,\\tilde{g})^{1/n} $. On montre que $0< \\lamin(M,[g],\\si) \\leq \\lamin(\\mS^n)$. %= \\frac{n}{2} \\om_n^{{1 \\over n}}$ On trouve des conditions suffisantes pour lesquelles on obtient l'in\\'egalit\\'e stricte $\\lamin(M,[g],\\si) < \\lamin(\\mS^n)$. Cette in\\'egalit\\'e stricte a des applications en g\\'eom\\'etrie spinorielle conforme."}
{"category": "Math", "title": "Abstract Convexity and Cone-Vexing Abstractions", "abstract": "This talk is a write-up on some origins of abstract convexity and afew vexing limitations on the range of abstraction in convexity."}
{"category": "Math", "title": "Wild ramification and the characteristic cycle of an l-adic sheaf", "abstract": "We propose a geometric method to measure the wild ramification of a smooth etale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of positive characteristic with arbitrary residue field. We also define the characteristic cycle of an l-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent bundle and prove that the intersection with the 0-section computes the characteristic class, and hence the Euler number. Definition 2.1.1 is corrected in v2."}
{"category": "Math", "title": "Maximal hypoellipticity and Dolbeault cohomology representations for U(p,q)", "abstract": "Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormander condition. We prove here that this operator is not maximal hypoelliptic when G=U(p,q)."}
{"category": "Math", "title": "Global Schauder estimates for a class of degenerate Kolmogorov equations", "abstract": "We consider a class of possibly degenerate second order elliptic operators $\\cal A$ on $\\R^n$. This class includes hypoelliptic Ornstein-Uhlenbeck type operators having an additional first order term with unbounded coefficients. We establish global Schauder estimates in H\\\"older spaces both for elliptic equations and for parabolic Cauchy problems involving ${\\cal A}$. The H\\\"older function spaces are defined with respect to a non-euclidean metric related to the operator $\\cal A$."}
{"category": "Math", "title": "Decoding of scroll codes", "abstract": "We define and study a class of codes obtained from scrolls over curves of any genus over finite fields. These codes generalize Goppa codes in a natural way, and the orthogonal complements of these codes belong to the same class. We show how syndromes of error vectors correspond to certain vector bundle extensions, and how decoding is associated to finding destabilizing subbundles."}
{"category": "Math", "title": "Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions", "abstract": "Consider a homogenized spectral pencil of exactly solvable linear differential operators $T_{\\la}=\\sum_{i=0}^k Q_{i}(z)\\la^{k-i}\\frac {d^i}{dz^i}$, where each $Q_{i}(z)$ is a polynomial of degree at most $i$ and $\\la$ is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers $n$ there exist exactly $k$ distinct values $\\la_{n,j}$, $1\\le j\\le k$, of the spectral parameter $\\la$ such that the operator $T_{\\la}$ has a polynomial eigenfunction $p_{n,j}(z)$ of degree $n$. These eigenfunctions split into $k$ different families according to the asymptotic behavior of their eigenvalues. We conjecture and prove sequential versions of three fundamental properties: the limits $\\Psi_{j}(z)=\\lim_{n\\to\\infty} \\frac{p_{n,j}'(z)}{\\la_{n,j}p_{n,j}(z)}$ exist, are analytic and satisfy the algebraic equation $\\sum_{i=0}^k Q_{i}(z) \\Psi_{j}^i(z)=0$ almost everywhere in $\\bCP$. As a consequence we obtain a class of algebraic functions possessing a branch near $\\infty\\in \\bCP$ which is representable as the Cauchy transform of a compactly supported probability measure."}
{"category": "Math", "title": "Cohomology of affine Artin groups and applications", "abstract": "The result of this paper is the determination of the cohomology of Artin groups of type A_n, B_n and \\tilde{A}_{n} with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B_n with coefficients over the module \\Q[q^{\\pm 1},t^{\\pm 1}]. Here the first (n-1) standard generators of the group act by (-q)-multiplication, while the last one acts by (-t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type \\tilde{A}_{n} as well as the cohomology of the classical braid group {Br}_{n} with coefficients in the n-dimensional representation presented in \\cite{tong}. The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be K(\\pi,1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived."}
{"category": "Math", "title": "Sinc Approximation of the Heat Distribution on the Boundary of a Two-Dimensional Finite Slab", "abstract": "We consider the two-dimensional problem of recovering globally in time the heat distribution on the surface of a layer inside of a heat conducting body from two interior temperature measurements. The problem is ill-posed. The approximation function is represented by a two-dimensional Sinc series and the error estimate is given."}
{"category": "Math", "title": "Laguerre polynomials and the inverse Laplace transform using discrete data", "abstract": "We consider the problem of finding a function defined on $(0,\\infty)$ from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution."}
{"category": "Math", "title": "The contact invariant in sutured Floer homology", "abstract": "We describe an invariant of a contact 3-manifold with convex boundary as an element of Juh\\'asz's sutured Floer homology. Our invariant generalizes the contact invariant in Heegaard Floer homology in the closed case, due to Ozsv\\'ath and Szab\\'o. This version has some clarifications and new figures."}
{"category": "Math", "title": "The K(\\pi, 1) problem for the affine Artin group of type \\widetilde{B}_n and its cohomology", "abstract": "In this paper we prove that the complement to the affine complex arrangement of type \\widetilde{B}_n is a K(\\pi, 1) space. We also compute the cohomology of the affine Artin group G of type \\widetilde{B}_n with coefficients over several interesting local systems. In particular, we consider the module Q[q^{\\pm 1}, t^{\\pm 1}], where the first n-standard generators of G act by (-q)-multiplication while the last generator acts by (-t)-multiplication. Such representation generalizes the analog 1-parameter representation related to the bundle structure over the complement to the discriminant hypersurface, endowed with the monodromy action of the associated Milnor fibre. The cohomology of G with trivial coefficients is derived from the previous one."}
{"category": "Math", "title": "Local Existence for Nonlinear Wave Equation with Radial Data in 2+1 Dimensions", "abstract": "We get a local existence result in $H^s$ with $s>3/2$ for second order quasilinear wave equation with radial initial data in 2+1 dimensions, based on an improvement of Strichartz estimate in the radial case. Moreover, we get the corresponding local well-posed result for semilinear wave equation. The required index of regularity here is 1/4 less than the index 7/4, which is essentially sharp in general."}
{"category": "Math", "title": "Augmented Teichmuller Spaces and Orbifolds", "abstract": "We study complex-analytic properties of the augmented Teichmuller spaces ATS introduced by Lipman Bers. These spaces are obtained by adding to the classical Teichmuller space TS the points corresponding to nodal Riemann surfaces. Unlike TS, the space ATS is not a complex manifold (it is not even locally compact). We prove however that the quotient of ATS by any finite index subgroup of the Teichmuller modular group has a canonical structure of a complex orbifold. Using this structure we construct natural maps from ATS to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest."}
{"category": "Math", "title": "On the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions", "abstract": "We present decompositions of various positive kernels as integrals or sums of positive kernels. Within this framework we study the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. As a tool, we define a new function of two complex variables, which is a natural generalization of the classical Gamma function for the setting we consider"}
{"category": "Math", "title": "Continuous selections and sigma-spaces", "abstract": "Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections."}
{"category": "Math", "title": "Combinatorial Morse theory and minimality of hyperplane arrangements", "abstract": "We find an explicit combinatorial gradient vector field on the well known complex S (Salvetti complex) which models the complement to an arrangement of complexified hyperplanes. The argument uses a total ordering on the facets of the stratification of R^n associated to the arrangement, which is induced by a generic system of polar coordinates. We give a combinatorial description of the singular facets, finding also an algebraic complex which computes local homology. We also give a precise construction in the case of the braid arrangement."}
{"category": "Math", "title": "Asymmetric potentials and motor effect: a large deviation approach", "abstract": "We provide a mathematical analysis of appearance of the concentrations (as Dirac masses) of the solution to a Fokker-Planck system with asymmetric potentials. This problem has been proposed as a model to describe motor proteins moving along molecular filaments. The components of the system describe the densities of the different conformations of the proteins. Our results are based on the study of a Hamilton-Jacobi equation arising, at the zero diffusion limit, after an exponential transformation change of the phase function that rises a Hamilton-Jacobi equation. We consider different classes of conformation transitions coefficients (bounded, unbounded and locally vanishing)."}
{"category": "Math", "title": "Bernstein polynomials, Bergman kernels and toric K\\\"ahler varieties", "abstract": "It does not seem to have been observed previously that the classical Bernstein polynomials $B_N(f)(x)$ are closely related to the Bergman-Szego kernels $\\Pi_N$ for the Fubini-Study metric on $\\CP^1$: $B_N(f)(x)$ is the Berezin symbol of the Toeplitz operator $\\Pi_N f(N^{-1} D_{\\theta})$. The relation suggests a generalization of Bernstein polynomials to any toric Kahler variety and Delzant polytope $P$. When $f$ is smooth, $B_N(f)(x)$ admits a complete asymptotic expansion. Integrating it over $P$ gives a complete asymptotic expansion for Dedekind-Riemann sums of smooth functions over lattice points in $N P$ related to Euler-MacLaurin sum formulae of Guillemin-Sternberg and others."}
{"category": "Math", "title": "Symplectic Energy and Lagrangian Intersection Under Legendrian Deformations", "abstract": "Let M be a compact symplectic manifold, and L be a closed Lagrangian submanifold which can be lifted to a Legendrian submanifold in the contactization of M. For any Legendrian deformation of L satisfying some given conditions, we get a new Lagrangian submanifold L'. We prove that the number of intersection of L and L' can be estimated from below by the sum of $Z_2$-Betti numbers of L, provided they intersect transversally."}
{"category": "Math", "title": "New Integral Representations of Whittaker Functions for Classical Lie Groups", "abstract": "We propose integral representations of the Whittaker functions for the classical Lie algebras sp(2l), so(2l) and so(2l+1). These integral representations generalize the integral representation of gl(l+1)-Whittaker functions first introduced by Givental. One of the salient features of the Givental representation is its recursive structure with respect to the rank of the Lie algebra gl(l+1). The proposed generalization of the Givental representation to the classical Lie algebras retains this property. It was shown elsewhere that the integral recursion operator for gl(l+1)-Whittaker function in the Givental representation coincides with a degeneration of the Baxter Q-operator for $\\hat{gl(l+1)}$-Toda chains. We construct Q-operator for affine Lie algebras $\\hat{so(2l)}$, $\\hat{so(2l+1)}$ and a twisted form of $\\hat{gl(2l)}$. We demonstrate that the relation between recursion integral operators of the generalized Givental representation and degenerate Q-operators remains valid for all classical Lie algebras."}
{"category": "Math", "title": "Simple formulas for lattice paths avoiding certain periodic staircase boundaries", "abstract": "There is a strikingly simple classical formula for the number of lattice paths avoiding the line x = ky when k is a positive integer. We show that the natural generalization of this simple formula continues to hold when the line x = ky is replaced by certain periodic staircase boundaries--but only under special conditions. The simple formula fails in general, and it remains an open question to what extent our results can be further generalized."}
{"category": "Math", "title": "Weakly commensurable arithmetic groups, lengths of closed geodesics and isospectral locally symmetric spaces", "abstract": "We introduce the notion of weak commensurabilty of arithmetic subgroups and relate it to the length equivalence and isospectrality of locally symmetric spaces. We prove many strong consequences of weak commensurabilty and derive from these many interesting results about isolength and isospectral locally symmetric spaces."}
{"category": "Math", "title": "A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus", "abstract": "We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of [Sch\\\"{u}tzenberger '77] for standard Young tableaux. We apply this to give a new combinatorial rule for the K-theory Schubert calculus of Grassmannians via K-theoretic jeu de taquin, providing an alternative to the rules of [Buch '02] and others. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety G/P, extending [Thomas-Yong '06]. We also present analogues of results of Fomin, Haiman, Schensted and Sch\\\"{u}tzenberger."}
{"category": "Math", "title": "Functional analytic background for a theory of infinite-dimensional reductive Lie groups", "abstract": "Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability."}
{"category": "Math", "title": "On arithmetic progressions on genus two curves", "abstract": "We study arithmetic progression in the $x$-coordinate of rational points on genus two curves. As we know, there are two models for the curve $C$ of genus two: $C: y^2=f_{5}(x)$ or $C: y^2=f_{6}(x)$, where $f_{5}, f_{6}\\in\\Q[x]$, $\\operatorname{deg}f_{5}=5, \\operatorname{deg}f_{6}=6$ and the polynomials $f_{5}, f_{6}$ do not have multiple roots. First we prove that there exists an infinite family of curves of the form $y^2=f(x)$, where $f\\in\\Q[x]$ and $\\operatorname{deg}f=5$ each containing 11 points in arithmetic progression. We also present an example of $F\\in\\Q[x]$ with $\\operatorname{deg}F=5$ such that on the curve $y^2=F(x)$ twelve points lie in arithmetic progression. Next, we show that there exist infinitely many curves of the form $y^2=g(x)$ where $g\\in\\Q[x]$ and $\\operatorname{deg}g=6$, each containing 16 points in arithmetic progression. Moreover, we present two examples of curves in this form with 18 points in arithmetic progression."}
{"category": "Math", "title": "Stably isomorphic dual operator algebras", "abstract": "We prove that two unital dual operator algebras A, B are stably isomorphic if and only if they are Delta-equivalent, if and only if they have completely isometric normal representations a, b on Hilbert spaces H, K respectively and there exists a ternary ring of operators M \\subset B(H,K) such that a(A)=[M* b(B) M]^{-w^*} and b(B)=[M a(A) M*]^{-w^*}."}
{"category": "Math", "title": "Inductive characterizations of hyperquadrics", "abstract": "We give two characterizations of hyperquadrics: one as non-degenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point; the other as $LQEL$-manifolds with large secant defects."}
{"category": "Math", "title": "Compound basis for the space of symmetric functions", "abstract": "The aim of this note is to introduce a compound basis for the space of symmetric functions. Our basis consists of products of Schur functions and $Q$-functions. The basis elements are indexed by the partitions. It is well known that the Schur functions form an orthonormal basis for our space. A natural question arises. How are these two bases connected? In this note we present some numerical results of the transition matrix for these bases. In particular we will see that the determinant of the transition matrix is a power of 2. This is not a surprising fact. However the explicit formula involves an interesting combinatorial feature. Our compound basis comes from the twisted homogeneous realization of the basic representation of the affine Lie algebras. This note is not written in a standard style of mathematical articles. It is more like a draft of a talk. In particular proofs are not given here. Details and proofs will be published elsewhere."}
{"category": "Math", "title": "Codage arithmetique pour la description d'une distribution", "abstract": "Using predictive adaptive arithmetic coding and the Minimum Description Length principle, we derive an efficient tool for model selection problems : the RIC information criterion. We then present an extension of these coding techniques to non-parametrical estimation of a distribution and illustrate it on the gray scales histogram of an image. Key-words : Information criteria, MDL, model selection, non-parametrical estimation, histograms."}
{"category": "Math", "title": "Rational points on certain elliptic surfaces", "abstract": "Let $\\mathcal{E}_{f}:y^2=x^3+f(t)x$, where $f\\in\\Q[t]\\setminus\\Q$, and let us assume that $\\op{deg}f\\leq 4$. In this paper we prove that if $\\op{deg}f\\leq 3$, then there exists a rational base change $t\\mapsto\\phi(t)$ such that on the surface $\\cal{E}_{f\\circ\\phi}$ there is a non-torsion section. A similar theorem is valid in case when $\\op{deg}f=4$ and there exists $t_{0}\\in\\Q$ such that infinitely many rational points lie on the curve $E_{t_{0}}:y^2=x^3+f(t_{0})x$. In particular, we prove that if $\\op{deg}f=4$ and $f$ is not an even polynomial, then there is a rational point on $\\cal{E}_{f}$. Next, we consider a surface $\\cal{E}^{g}:y^2=x^3+g(t)$, where $g\\in\\Q[t]$ is a monic polynomial of degree six. We prove that if the polynomial $g$ is not even, there is a rational base change $t\\mapsto\\psi(t)$ such that on the surface $\\cal{E}^{g\\circ\\psi}$ there is a non-torsion section. Furthermore, if there exists $t_{0}\\in\\Q$ such that on the curve $E^{t_{0}}:y^2=x^3+g(t_{0})$ there are infinitely many rational points, then the set of these $t_{0}$ is infinite. We also present some results concerning diophantine equation of the form $x^2-y^3-g(z)=t$, where $t$ is a variable."}
{"category": "Math", "title": "Quantisation commutes with reduction at discrete series representations of semisimple groups", "abstract": "Using the analytic assembly map that appears in the Baum-Connes conjecture in noncommutative geometry, we generalise the $\\Spin^c$-version of the Guillemin-Sternberg conjecture that `quantisation commutes with reduction' to (discrete series representations of) semisimple groups $G$ with maximal compact subgroups $K$ acting cocompactly on symplectic manifolds. We prove this statement in cases where the image of the momentum map in question lies in the set of strongly elliptic elements, the set of elements of $\\g^*$ with compact stabilisers. This assumption on the image of the momentum map is equivalent to the assumption that $M = G \\times_K N$, for a compact Hamiltonian $K$-manifold $N$. The proof comes down to a reduction to the compact case. This reduction is based on a `quantisation commutes with induction'-principle, and involves a notion of induction of Hamiltonian group actions. This principle, in turn, is based on a version of the naturality of the assembly map for the inclusion of $K$ into $G$."}
{"category": "Math", "title": "Computation of RS-pullback transformations for algebraic Painleve VI solutions", "abstract": "Algebraic solutions of the sixth Painleve equation can be computed using pullback transformations of hypergeometric equations with respect to specially ramified rational coverings. In particular, as was noticed by the second author and Doran, some algebraic solutions can be constructed from a rational covering alone, without computation of the pullbacked Fuchsian equation. But the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painleve VI solutions. This paper presents computations of explicit RS-pullback transformations, and derivation of algebraic Painleve VI solutions from them."}
{"category": "Math", "title": "On the Definitions of Difference Galois Groups", "abstract": "We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of Picard-Vessiot extensions over fields with not necessarily algebraically closed subfields of constants."}
{"category": "Math", "title": "Optimal Stopping with Rank-Dependent Loss", "abstract": "For $\\tau$ a stopping rule adapted to a sequence of $n$ iid observations, we define the loss to be $\\ex [ q(R_\\tau)]$, where $R_j$ is the rank of the $j$th observation, and $q$ is a nondecreasing function of the rank. This setting covers both the best choice problem with $q(r)={\\bf 1}(r>1)$, and Robbins' problem with $q(r)=r$. As $n\\to\\infty$ the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit, thus answering a question asked by Bruss in the context of Robbins' problem."}
{"category": "Math", "title": "On the Riemann zeta-function, Part III", "abstract": "An odd meromorphic function f(s) is constructed from the Riemann zeta-function evaluated at one-half plus s. The partial fraction expansion, p(s), of f(s) is obtained using the conjunction of the Riemann hypothesis and hypotheses advanced by the author. That compound hypothesis and the expansion p(s) are employed in Part IV to derive the two-sided Laplace transform representation of f(s) on the open vertical strip of all s with real part between zero and four."}
{"category": "Math", "title": "An action of the cactus group", "abstract": "We construct an action of the big cactus group (the fundamental group of the Deligne-Mumford compactification of the moduli space of real curves of genus zero with n undistinguished marked points) on Fock-Goncharov's SL_m analog of the decorated Teichmuller space of ideal n-gons."}
{"category": "Math", "title": "Integral group ring of the McLaughlin simple group", "abstract": "We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the McLaughlin sporadic group McL. As a consequence, we confirm for this group the Kimmerle's conjecture on prime graphs."}
{"category": "Math", "title": "Discrete Tomography of Icosahedral Model Sets", "abstract": "The discrete tomography of B-type and F-type icosahedral model sets is investigated, with an emphasis on reconstruction and uniqueness problems. These are motivated by the request of materials science for the unique reconstruction of quasicrystalline structures from a small number of images produced by quantitative high resolution transmission electron microscopy."}
{"category": "Math", "title": "Integral group ring of Rudvalis simple group", "abstract": "Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group Ru. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs."}
{"category": "Math", "title": "Curve shortening and the topology of closed geodesics on surfaces", "abstract": "We study \"flat knot types\" of geodesics on compact surfaces M^2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M^2. We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial."}
{"category": "Math", "title": "Necessary and sufficient conditions for solvability of the Hartman-Wintner problem for difference equations", "abstract": "For homogeneous difference equation of the second order we study the analogy of Hartman-Wintner problem on asymptotic integration of fundamental system of solutions as argument tends to infinity."}
{"category": "Math", "title": "A conditional 0-1 law for the symmetric sigma-field", "abstract": "Let (\\Omega,\\mathcal{B},P) be a probability space, \\mathcal{A} a sub-sigma-field of \\mathcal{B}, and \\mu a regular conditional distribution for P given \\mathcal{A}. For various, classically interesting, choices of \\mathcal{A} (including tail and symmetric) the following 0-1 law is proved: There is a set A_0 in \\mathcal{A} such that P(A_0)=1 and \\mu(\\omega)(A) is 0 or 1 for all A in \\mathcal{A} and \\omega in A_0. Provided \\mathcal{B} is countably generated (and certain regular conditional distributions exist), the result applies whatever P is."}
{"category": "Math", "title": "Singularly perturbed periodic and semiperiodic differential operators", "abstract": "Qualitative and spectral properties of the form-sums S_{\\pm}(V):=D_{\\pm}^{2m}\\dotplus V(x),\\quad m\\in \\mathbb{N}, in the Hilbert space $L_{2}(0,1)$ are studied. Here the periodic $(D_{+})$ and the semiperiodic $(D_{-})$ differential operators are $D_{\\pm}: u\\mapsto -i u'$, and $V(x)$ is a 1-periodic complex-valued distribution in the Sobolev spaces $H_{per}^{-m\\alpha}$, $\\alpha\\in [0,1]$."}
{"category": "Math", "title": "Strong q-convexity in uniform neighborhoods of subvarieties in coverings of complex spaces", "abstract": "The main result is that, for any projective compact analytic subset A of dimension q>0 in a reduced complex space X, there is a neighborhood U of A such that, for any covering space Z of X in which the lifting B of A has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a smooth exhaustion function on Z which is strongly q-convex on the lifting of U outside a uniform neighborhood of the q-dimensional compact irreducible components B."}
{"category": "Math", "title": "Euler Coefficients and Restricted Dyck Paths", "abstract": "We consider the problem of enumerating Dyck paths staying weakly above the x-axis with a limit to the number of consecutive up steps, or a limit to the number of consecutive down steps. We use Finite Operator Calculus to obtain formulas for the number of all such paths reaching a given point in the first quadrant. All our results are based on the Eulerian coefficients."}
{"category": "Math", "title": "Posets of annular non-crossing partitions of types B and D", "abstract": "We study the set $\\sncb (p,q)$ of annular non-crossing permutations of type B, and we introduce a corresponding set $\\ncb (p,q)$ of annular non-crossing partitions of type B, where $p$ and $q$ are two positive integers. We prove that the natural bijection between $\\sncb (p,q)$ and $\\ncb (p,q)$ is a poset isomorphism, where the partial order on $\\sncb (p,q)$ is induced from the hyperoctahedral group $B_{p+q}$, while $\\ncb (p,q)$ is partially ordered by reverse refinement. In the case when $q=1$, we prove that $\\ncb (p,1)$ is a lattice with respect to reverse refinement order. We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between $\\sncd (p,q)$ and $\\ncd (p,q)$. For $q=1$, the poset $\\ncd (p,1)$ coincides with a poset ``$NC^{(D)} (p+1)$'' constructed in a paper by Athanasiadis and Reiner in 2004, and is a lattice by the results of that paper."}
{"category": "Math", "title": "U-Invariants for forms of higher degree", "abstract": "Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. Both old and new results on these invariants are collected."}
{"category": "Math", "title": "All automorphisms of the Calkin algebra are inner", "abstract": "We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a 2006 Phillips--Weaver construction of an outer automorphism using the Continuum Hypothesis, this gives a complete solution to a 1977 problem of Brown-Douglas-Fillmore. We also give a simpler and self-contained proof of the Phillips--Weaver result."}
{"category": "Math", "title": "Some classes of multiplicative forms of higher degree", "abstract": "Several notions of multiplicativity are introduced for forms of degree $d\\geq 3$ over a field of characteristic 0 or greater than d. Examples of multiplicative and strongly multiplicative forms of higher degree are given. Conditions restricting the structure of a strongly multiplicative form are found."}
{"category": "Math", "title": "Minimal Surfaces in $S^3$ with Constant Contact Angle", "abstract": "We provide a characterization of the Clifford Torus in S3 via moving frames and contact structure equations. More precisely, we prove that minimal surfaces in S3 with constant contact angle must be the Clifford Torus. Some applications of this result are then given, and some examples are discussed."}
{"category": "Math", "title": "Antisymmetric elements in group rings with an orientation morphism", "abstract": "Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $\\phi_{\\sigma} : RG\\to RG$ denote the involution defined by $\\phi_{\\sigma} (\\sum r_{g}g) = \\sum r_{g} \\sigma (g) g^{-1}$, where $\\sigma:G\\to \\{\\pm 1\\}$ is a group homomorphism (called an orientation morphism). An element $x$ in $RG$ is said to be antisymmetric if $\\phi_{\\sigma} (x) =-x$. We give a full characterization of the groups $G$ and its orientations for which the antisymmetric elements of $RG$ commute."}
{"category": "Math", "title": "Shelling-type orderings of regular CW-complexes and acyclic matchings of the Salvetti complex", "abstract": "Motivated by the work of Salvetti and Settepanella we introduce certain total orderings of the faces of any shellable regular CW-complex (called `shelling-type orderings') that can be used to explicitly construct maximum acyclic matchings of the poset of cells of the given complex. Building on an application of this method to the classical zonotope shellings we describe a class of maximum acyclic matchings for the Salvetti complex of a linear complexified arrangement. To do this, we introduce and study a new combinatorial stratification of the Salvetti complex. For the obtained acyclic matchings we give an explicit description of the critical cells that depends only on the chosen linear extension of the poset of regions. It is always possible to choose the linear extension so that the critical cells can be explicitly constructed from the chambers of the arrangement via the bijection to no-broken-circuit sets defined by Jewell and Orlik. Our method can be generalized to arbitraty oriented matroids."}
{"category": "Math", "title": "Symplectic Group Actions and Covering Spaces", "abstract": "For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic cover. At the same time we give a classification of all Hamiltonian covers of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied."}
{"category": "Math", "title": "Differentiable perturbations of Ornstein-Uhlenbeck operators", "abstract": "We prove an extension theorem for a small perturbation of the Ornstein-Uhlenbeck operator $(L,D(L))$ in the space of all uniformly continuous and bounded functions $f:H\\to \\Rset$, where $H$ is a separable Hilbert space. We consider a perturbation of the form $N_0\\phi=L\\phi+< D\\phi,F>$ where $F:H\\to H$ is bounded and Fr\\'echet differentiable with uniformly continuous and bounded differential. Hence, we prove that $N_0$ is $m$-dissipative and its closure in $C_b(H)$ coincides with the infinitesimal generator of a diffusion semigroup associated to a stochastic differential equation in $H$."}
{"category": "Math", "title": "Computation of highly ramified coverings", "abstract": "An almost Belyi covering is an algebraic covering of the projective line, such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line. In general, there are 1-dimensional families of these coverings with a fixed ramification pattern. (That is, Hurwitz spaces for these coverings are curves.) In this paper, three almost Belyi coverings of degrees 11, 12, and 20 are explicitly constructed. We demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth Painleve equation."}
{"category": "Math", "title": "Second order polynomial Hamiltonian systems with ${\\tilde W}(E_6^{(1)}),{\\tilde W}(E_7^{(1)})$ and $W(E_8^{(1)})$-symmetry", "abstract": "We find and study a six (resp. seven, eight)-parameter family of polynomial Hamiltonian systems of second order, respectively. This system admits the affine Weyl group symmetry of type $E_6^{(1)}$ (resp. $E_7^{(1)}, E_8^{(1)}$) as the group of its B{\\\"a}cklund transformations. Each system is the first example which gave second-order polynomial Hamiltonian system with ${\\tilde W}(E_6^{(1)})$ (resp. ${\\tilde W}(E_7^{(1)}), W(E_8^{(1)})$)-symmetry. We also show that its space of initial conditions $S$ is obtained by gluing eight (resp. nine, ten) copies of ${\\Bbb C}^2$ via the birational and symplectic transformations."}
{"category": "Math", "title": "Small time Edgeworth-type expansions for weakly convergent nonhomogeneous Markov chains", "abstract": "We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points."}
{"category": "Math", "title": "Configurations of saddle connections of quadratic differentials on CP1 and on hyperelliptic Riemann surfaces", "abstract": "Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done in Masur and Zorich, although in that case the connected components were not distinguished. We classify the configurations for quadratic differentials on the Riemann sphere and on hyperelliptic connected components of the moduli space of quadratic differentials. We show that, in genera greater than five, any configuration that appears in the hyperelliptic connected component of a stratum also appears in the non-hyperelliptic one."}
{"category": "Math", "title": "Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations", "abstract": "We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. Additionally, we use the local tangent-normal decomposition, available when the symmetry group is proper, to construct local skew-product splittings in a neighborhood of any point in the open and dense principal orbit type. The general methods introduced in the first part of the paper are then adapted to the Hamiltonian case, which is studied with special care and illustrated with several examples. The Hamiltonian category deserves a separate study since in that situation the presence of symmetries implies in most cases the existence of conservation laws, mathematically described via momentum maps, that should be taken into account in the analysis."}
{"category": "Math", "title": "A 5-quantifier (\\in,=)-expression ZF-equivalent to the Axiom of Choice", "abstract": "In this paper I present an (\\in, =)-sentence, AC**, with only 5 quantifiers, that logically implies the axiom of choice, AC. Furthermore, using a weak fragment of ZF set theory, I prove that AC implies AC**. Up to now 6 quantifiers were the minimum and 3 quantifiers don't suffice since all 3-quantifier (\\in, =)-sentences are decided in a weak fragment of ZF set theory. Thus the gap is reduced to the undecided case of a 4 quantifier sentence ZF-equivalent to AC."}
{"category": "Math", "title": "Dirichlet-like space and capacity in complex analysis in several variables", "abstract": "For a Kahler manifold X, we study a space of test functions W* which is a complex version of H1. We prove for W* the classical results of the theory of Dirichlet spaces: the functions in W* are defined up to a pluripolar set and the functional capacity associated to W* tests the pluripolar sets. This functional capacity is a Choquet capacity. The space W* is not reflexive and the smooth functions are not dense in it for the strong topology. So the classical tools of potential theory do not apply here. We use instead pluripotential theory and Dirichlet spaces associated to a current."}
{"category": "Math", "title": "A few brief comments on the results of manuscript arXiv:math/0405153v3", "abstract": "The results in the recently posted manuscript arXiv:math/0405153v3 are incorrect. The correct version of the aimed results is not original. The preprint contains material from references that are not properly quoted."}
{"category": "Math", "title": "Inner Metric Geometry of Complex Algebraic Surfaces with Isolated Singularities", "abstract": "We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to the inner metric, to cones. The technique used to prove the nonexistence of the metric conic structure is related to a development of Metric Homology. The class of the examples is rather large and it includes some surfaces of Brieskorn."}
{"category": "Math", "title": "(Co)cyclic (co)homology of bialgebroids: An approach via (co)monads", "abstract": "For a (co)monad T_l on a category M, an object X in M, and a functor \\Pi: M \\to C, there is a (co)simplex Z^*:=\\Pi T_l^{* +1} X in C. Our aim is to find criteria for para-(co)cyclicity of Z^*. Construction is built on a distributive law of T_l with a second (co)monad T_r on M, a natural transformation i:\\Pi T_l \\to \\Pi T_r, and a morphism w: T_r X \\to T_l X in M. The relations i and w need to satisfy are categorical versions of Kaygun's axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads T_l=T \\otimes_R (-) and T_r = (-)\\otimes_R T on the category of R-bimodules. The functor \\Pi can be chosen such that Z^n= T\\hat{\\otimes}_R... \\hat{\\otimes}_R T \\hat{\\otimes}_R X is the cyclic R-module tensor product. A natural transformation i:T \\hat{\\otimes}_R (-) \\to (-) \\hat{\\otimes}_R T is given by the flip map and a morphism w: X \\otimes_R T \\to T\\otimes_R X is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. Stable anti Yetter-Drinfel'd modules over certain bialgebroids, so called x_R-Hopf algebras, are introduced. In the particular example when T is a module coring of a x_R-Hopf algebra B and X is a stable anti Yetter-Drinfel'd B-module, the para-cyclic object Z_* is shown to project to a cyclic structure on T^{\\otimes_R *+1} \\otimes_B X. For a B-Galois extension S \\to T, a stable anti Yetter-Drinfel'd B-module T_S is constructed, such that the cyclic objects B^{\\otimes_R *+1} \\otimes_B T_S and T^ {\\hat{\\otimes}_S *+1} are isomorphic. As an application, we compute Hochschild and cyclic homology of a groupoid with coefficients, by tracing it back to the group case. In particular, we obtain explicit expressions for ordinary Hochschild and cyclic homology of a groupoid."}
{"category": "Math", "title": "A mirror symmetric solution to the quantum Toda lattice", "abstract": "We use representation theory to construct integral formulas for solutions to the quantum Toda lattice in general type. This result generalizes work of Givental for SL(n)/B in a uniform way to arbitrary type and can be interpreted as a kind of mirror theorem for the full flag variety G/B. We also prove the existence of a totally positive critical point of the 'superpotential' in every mirror fiber."}
{"category": "Math", "title": "Cohomology of the adjoint of Hopf algebras", "abstract": "A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the YBE are given and quandle cocycles are constructed from groupoid cocycles."}
{"category": "Math", "title": "On the Collection of Integers that Index the Fixed Points of Maps on the Space of Rational Functions", "abstract": "Given integers s and t, define a function phi_{s,t} on the space of all formal complex series expansions by phi_{s,t} (sum a_n x^n) = sum a_{sn+t} x^n. We define an integer r to be distinguished with respect to (s,t) if r and s are relatively prime and and r divides t (1 + s + ... s^{ord_r(s)-1}). The vector space consisting of all rational functions whose Taylor expansions at zero are fixed by phi_{s,t} was previously classified by constructing a basis that is partially indexed by integers that are distinguished with respect to the pair (s,t). In this paper, we study the properties of the set of distinguished integers with respect to (s,t). In particular, we demonstrate that the set of distinguished integers with respect to (s,t) can be written as a union of infinitely many arithmetic progressions. In addition, we construct another generating set for the collection of rational functions that are fixed by phi_{s,t} and discuss the relationship between this generating set and the basis that was generated previously."}
{"category": "Math", "title": "On the icosahedron: from two to three dimensions", "abstract": "In his famous book, Felix Klein describes a complex variable for the quotients of the ordinary sphere by the finite groups of rotations and in particular for the most complex situation of the quotient by the symmetry group of the icosahedron. The purpose of this work and its sequels is to obtain similar results for the quotients of the three--dimensional sphere. Various properties of the group $SU(2)$ and of its representations are used to obtain explicit expressions for coordinates and the relations they satisfy."}
{"category": "Math", "title": "Translation Groupoids and Orbifold Bredon Cohomology", "abstract": "We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps. We use this result to define an orbifold version of Bredon cohomology."}
{"category": "Math", "title": "Yangian of the Strange Lie Superalgebra of $\\boldsymbol{Q_{n-1}}$ Type, Drinfel'd Approach", "abstract": "The Yangian of the strange Lie superalgebras in Drinfel'd realization is defined. The current system generators and defining relations are described."}
{"category": "Math", "title": "Evaluating Throwing Ability in Baseball", "abstract": "We present a quantitative analysis of throwing ability for major league outfielders and catchers. We use detailed game event data to tabulate success and failure events in outfielder and catcher throwing opportunities. We attribute a run contribution to each success or failure which are tabulated for each player in each season. We use four seasons of data to estimate the overall throwing ability of each player using a Bayesian hierarchical model. This model allows us to shrink individual player estimates towards an overall population mean depending on the number of opportunities for each player. We use the posterior distribution of player abilities from this model to identify players with significant positive and negative throwing contributions."}
{"category": "Math", "title": "The absorption theorem for affable equivalence relations", "abstract": "We prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being `small' in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case--when Y is a finite set--this result is highly non-trivial. The result itself--called the absorption theorem--is a powerful and crucial tool for the study of the orbit structure of minimal Z^n-actions on the Cantor set [GMPS]. The absorption theorem is a significant generalization of the main theorem proved in [GPS2]. However, we shall need a few key results from [GPS2] in order to prove the absorption theorem."}
{"category": "Math", "title": "Billiards in L-shaped tables with barriers", "abstract": "We compute the volumes of the eigenform loci in the moduli space of genus two Abelian differentials. From this, we obtain asymptotic formulas for counting closed billiards paths in certain L-shaped polygons with barriers."}
{"category": "Math", "title": "Birkhoff billiards are insecure", "abstract": "We prove that every compact plane billiard, bounded by a smooth curve, is insecure: there exist pairs of points $A,B$ such that no finite set of points can block all billiard trajectories from $A$ to $B$."}
{"category": "Math", "title": "Quantum doubles of certain rank two pointed Hopf algebras", "abstract": "A certain class of rank two pointed Hopf algebras is considered. The simple modules of their Drinfel'd double is described using Radford's method \\cite{rad}. The socle of the tensor product of two such modules is computed and a formula similar to the one in \\cite{one} is obtained in some conditions. Cases when such a tensor product is completely irreducible are also given in the last section."}
{"category": "Math", "title": "Generalized CCR Flows", "abstract": "We introduce a new construction of $E_0$-semigroups, called generalized CCR flows, with two kinds of descriptions: those arising from sum systems and those arising from pairs of $C_0$-semigroups. We get a new necessary and sufficient condition for them to be of type III, when the associated sum system is of finite index. Using this criterion, we construct examples of type III $E_0$-semigroups, which can not be distinguished from $E_0$-semigroups of type I by the invariants introduced by Boris Tsirelson. Finally, by considering the local von Neumann algebras, and by associating a type III factor to a given type III $E_0$-semigroup, we show that there exist uncountably many type III $E_0$-semigroups in this family, which are mutually non-cocycle conjugate."}
{"category": "Math", "title": "Infinitesimal spectral flow and scattering matrix", "abstract": "In this note the notion of infinitesimal scattering matrix is introduced. It is shown that under certain assumption, the scattering operator of a pair of trace compatible operators is equal to the chronological exponential of the infinitesimal scattering matrix and that the trace of the infinitesimal scattering matrix is equal to the absolutely continuous part of the infinitesimal spectral flow. As a corollary, a variant of the Birman-Krein formula is derived. An interpretation of Pushnitski's $\\mu$-invariant is given."}
{"category": "Math", "title": "Actions of symbolic dynamical systems on $C^*$-algebras II. Simplicity of $C^*$-symbolic crossed products and some examples", "abstract": "We have introduced a notion of $C^*$-symbolic dynamical system in [K. Matsumoto: Actions of symbolic dynamical systems on $C^*$-algebras, to appear in J. Reine Angew. Math.], that is a finite family of endomorphisms of a $C^*$-algebra with some conditions. The endomorphisms are indexed by symbols and yield both a subshift and a $C^*$-algebra of a Hilbert $C^*$-bimodule. The associated $C^*$-algebra with the $C^*$-symbolic dynamical system is regarded as a crossed product by the subshift. We will study a simplicity condition of the $C^*$-algebras of the $C^*$-symbolic dynamical systems. Some examples such as irrational rotation Cuntz-Krieger algebras will be studied."}
{"category": "Math", "title": "Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions", "abstract": "In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or -- equivalently -- a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem {from} being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains $\\Omega \\subset\\mathbb{R}^n$, the negative part of the corresponding Green's function is \"small\" when compared with its singular positive part, provided $n\\ge 3$. Moreover, the biharmonic Green's function in balls $B\\subset\\mathbb{R}^n$ under Dirichlet (i.e. clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if $n=2$. In the present paper, such a stability result is proved for $n\\ge 3$. Keywords: Biharmonic Green's functions, positivity, almost positivity, blow-up procedure."}
{"category": "Math", "title": "Diffusion constants and martingales for senile random walks", "abstract": "We derive diffusion constants and martingales for senile random walks with the help of a time-change. We provide direct computations of the diffusion constants for the time-changed walks. Alternatively, the values of these constants can be derived from martingales associated with the time-changed walks. Using an inverse time-change, the diffusion constants for senile random walks are then obtained via these martingales. When the walks are diffusive, weak convergence to Brownian motion can be shown using a martingale functional limit theorem."}
{"category": "Math", "title": "Sparsity oracle inequalities for the Lasso", "abstract": "This paper studies oracle properties of $\\ell_1$-penalized least squares in nonparametric regression setting with random design. We show that the penalized least squares estimator satisfies sparsity oracle inequalities, i.e., bounds in terms of the number of non-zero components of the oracle vector. The results are valid even when the dimension of the model is (much) larger than the sample size and the regression matrix is not positive definite. They can be applied to high-dimensional linear regression, to nonparametric adaptive regression estimation and to the problem of aggregation of arbitrary estimators."}
{"category": "Math", "title": "Probabilita` e Paradosso (Probability and Paradox)", "abstract": "In this paper we present three simple applications of probability and highlight and discuss their paradoxical flavour."}
{"category": "Math", "title": "Invariant measures for a stochastic Kuramoto-Sivashinky equation", "abstract": "For the 1-dimensional Kuramoto-Sivashinsky equation with random forcing term, existence and uniqueness of solutions is proved. Then, the Markovian semigroup is well defined; its properties are analyzed, in order to provide sufficient conditions for existence and uniqueness of invariant measures for this stochastic equation. Finally, regularity results are presented."}
{"category": "Math", "title": "Ascending number of knots and links", "abstract": "We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number."}
{"category": "Math", "title": "A functional limit theorem for a 2D-random walk with dependent marginals", "abstract": "We prove a non-standard functional limit theorem for a two dimensional simple random walk on some randomly oriented lattices. This random walk, already known to be transient, has different horizontal and vertical fluctuations leading to different normalizations in the functional limit theorem, with a non-Gaussian horizontal behavior. We also prove that the horizontal and vertical components are not asymptotically independent."}
{"category": "Math", "title": "On critical normal sections for two-dimensional immersions in R^n and a Riemann-Hilbert problem", "abstract": "For orthonormal normal sections of two-dimensional immersions in R^4 we define torsion coefficients and a functional for the total torsion. We discuss normal sections which are critical for this functional. In particular, a global estimate for the torsion coefficients of a critical normal section in terms of the curvature of the normal bundle is provided."}
{"category": "Math", "title": "Moduli Spaces of PU(2)-Instantons on Minimal Class VII Surfaces with b_2=1", "abstract": "We describe explicitly the moduli spaces $M^{pst}_g(S,E)$ of polystable holomorphic structures $E$ with $\\det E\\cong K$ on a rank 2 vector bundle $E$ with $c_1(E)=c_1(K)$ and $c_2(E)=0$ for all minimal class VII surfaces $S$ with $b_2(S)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are non-elliptic and non-Kaehler complex surfaces and have recently been completely classified. When $S$ is a half or parabolic Inoue surface, $M^{pst}_g(S,E)$ is always a compact one-dimensional complex disc. When $S$ is an Enoki surface, one obtains a complex disc with finitely many transverse self-intersections whose number becomes arbitrarily large when $g$ varies in the space of Gauduchon metrics. $M^{pst}_g(S,E)$ can be identified with a moduli space of PU(2)-instantons. The moduli spaces of simple bundles of the above type leads to interesting examples of non-Hausdorff singular one-dimensional complex spaces."}
{"category": "Math", "title": "The Skolem-Bang Theorems in Ordered Fields with an $IP$", "abstract": "This paper is concerned with the extent to which the Skolem-Bang theorems in Diophantine approximations generalise from the standard setting of $<R,Z>$ to structures of the form $<F,I>$, where $F$ is an ordered field and $I$ is an integer part of $F$. We show that some of these theorems are hold unconditionally in general case (ordered fields with an integer part). The remainder results are based on Dirichlet's and Kronecker's theorems. Finally we extend Dirichlet's theorem to ordered fields with $IE_1$ integer part."}
{"category": "Math", "title": "A nonlinearly ill-posed problem of reconstructing the temperature from interior data", "abstract": "We consider the problem of reconstructing, from the interior data $u(x,1)$, a function $u$ satisfying a nonlinear elliptic equation $$ \\Delta u = f(x,y,u(x,y)), x \\in \\RR, y > 0. $$"}
{"category": "Math", "title": "Convergence to Equilibrium for the Cahn-Hilliard Equation with Wentzell Boundary Condition", "abstract": "In this paper we consider the Cahn-Hilliard equation endowed with Wentzell boundary condition which is a model of phase separation in a binary mixture contained in a bounded domain with permeable wall. Under the assumption that the nonlinearity is analytic with respect to unknown dependent function, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable \\L ojasiewicz-Simon type inequality with boundary term. Estimates of convergence rate are also provided."}
{"category": "Math", "title": "Mexican Hat Wavelet on the Heisenberg Group", "abstract": "In this article wavelets (admissible vectors) on the Heisenberg group are studied from the point of view of Calderon's formula. Further we shall show that for the class of Schwartz functions the Calderon admissibility condition is equivalent to the usual admissibility property which will be introduced in this work. Furthermore motivated by a well-known example on the real line, the Mexican-Hat wavelet, we demonstrate the existence and construction of an analogous wavelet on the Heisenberg Lie group with 2 vanishing moments, which together with all of its derivatives has Gaussian decay."}
{"category": "Math", "title": "On Some Properties of Linear Mapping Induced by Linear Descriptor Differential Equation", "abstract": "In this paper we introduce linear mapping D from WnF\\subset Ln into Lm\\times Rm, induced by linear differential equation d/dt Fx(t)-C(t)x(t)=f(t),Fx(t_0)=f_0. We prove that D is closed dense defined mapping for any m\\times n-matrix F. Also adjoint mapping D* is constructed and its domain WmF is described. Some kind of so-called \"integration by parts\" formula for vectors from WnF, WmF is suggested. We obtain a necessary and sufficient condition for existence of generalized solution of equation Dx=(f,f_0). Also we find a sufficient criterion for closureness of the R(D) in Lm\\times Rm which is formulated in terms of transparent conditions for blocks of matrix C(t). Some examples are supplied to illustrate obtained results."}
{"category": "Math", "title": "Notes on planar semimodular lattices. I. Construction", "abstract": "We construct all planar semimodular lattices in three simple steps from the direct product of two chains."}
{"category": "Math", "title": "Adaptive classification of temporal signals in fixed-weights recurrent neural networks: an existence proof", "abstract": "We address the important theoretical question why a recurrent neural network with fixed weights can adaptively classify time-varied signals in the presence of additive noise and parametric perturbations. We provide a mathematical proof assuming that unknown parameters are allowed to enter the signal nonlinearly and the noise amplitude is sufficiently small."}
{"category": "Math", "title": "Rings of integers of type $K(\\pi,1)$", "abstract": "We investigate the Galois group $G_S(p)$ of the maximal $p$-extension unramified outside a finite $S$ of primes of a number field in the (tame) case, when no prime dividing $p$ is in $S$. We show that the cohomology of $G_S(p)$ is 'often' isomorphic to the etale cohomology of the scheme $\\Spec(\\O_k \\sm S)$, in particular, $G_S(p)$ is of cohomological dimension~2 then."}
{"category": "Math", "title": "On the cohomology of vector fields on parallelizable manifolds", "abstract": "In the present paper we determine for each parallelizable smooth compact manifold $M$ the cohomology spaces $H^2(V_M,\\bar\\Omega^p_M)$ of the Lie algebra $V_M$ of smooth vector fields on $M$ with values in the module $\\bar\\Omega^p_M = \\Omega^p_M/d\\Omega^{p-1}_M$. The case of $p=1$ is of particular interest since the gauge algebra $C^\\infty (M,k)$ has the universal central extension with center $\\bar\\Omega^1_M$, generalizing affine Kac-Moody algebras. The second cohomology $H^2(V_M, \\bar\\Omega^1_M)$ classifies twists of the semidirect product of $V_M$ with the universal central extension $C^\\infty (M,k) \\oplus \\bar\\Omega^1_M$."}
{"category": "Math", "title": "Linear response formula for piecewise expanding unimodal maps", "abstract": "The average R(t) of a smooth function with respect to the SRB measure of a smooth one-parameter family f_t of piecewise expanding interval maps is not always Lipschitz. We prove that if f_t is tangent to the topological class of f_0, then R(t) is differentiable at zero, and the derivative coincides with the resummation previously proposed by the first named author of the (a priori divergent) series given by Ruelle's conjecture."}
{"category": "Math", "title": "Isometries of CAT(0) cube complexes are semi-simple", "abstract": "We show that an automorphism of an arbitrary CAT(0) cube complex either has a fixed point or preserves some combinatorial axis. It follows that when a group contains a distorted cyclic subgroup, it admits no proper action on a discrete space with walls."}
{"category": "Math", "title": "Product preserving bundle functors on multifibered and multifoliate manifolds", "abstract": "We show that the set of the equivalence classes of multifoliate structures is in one-to-one correspondence with the set of equivalence classes of finite complete projective systems of vector space epimorphisms. After that we give the complete description of all product preserving bundle functors on the categories of multifibered and multifoliate manifolds."}
{"category": "Math", "title": "Beyond the PI Controllers in First-Order Time-Delay Systems", "abstract": "In this paper the following three control systems for first-order time-delay plants are studied and compared: the feedback proportional-integral controller (PI), the Smith Predictor (SP) and a proposed variable structure consisting of two blocks. This structure acts as an open-loop proportional controller, after a setpoint change, and as a closed-loop integrating controller, when the error enters in a preset band. A chart, provided with the borderlines of the stability zone and with the curves of two design parameters, is implemented for each controller. The first parameter is the overshoot of the controlled variable, evaluated during a step change of the setpoint and made equal to a preset value. The second parameter, only for the PI and SP controllers, is the integral of the squared error (ISE), which must have the minimum allowable value. The ISE is also assumed as comparison index and the proposed controller appears as the best."}
{"category": "Math", "title": "The variety of exterior powers of linear maps", "abstract": "Let $K$ be a field and $V$ and $W$ be $K$-vector spaces of dimension $m$ and $n$. Let $\\phi$ be the canonical map from $Hom(V,W)$ to $Hom(\\wedge^t V,\\wedge^t W)$. We investigate the Zariski closure $X_t$ of the image $Y_t$ of $\\phi$. In the case $t=\\min(m,n)$, $Y_t=X_t$ is the cone over a Grassmannian, but $X_t$ is larger than $Y_t$ for $1<t<\\min(m,n)$. We analyze the $G=\\GL(V)\\times\\GL(W)$-orbits in $X_t$ via the corresponding $G$-stable prime ideals. It turns out that they are classified by two numerical invariants, one of which is the rank and the other a related invariant that we call small rank. Surprisingly, the orbits in $X_t\\setminus Y_t$ arise from the images $Y_u$ for $u<t$ and simple algebraic operations. In the last section we determine the singular locus of $X_t$. Apart from well-understood exceptional cases, it is formed by the elements of rank $\\le 1$ in $Y_t$."}
{"category": "Math", "title": "Dimension and enumeration of primitive ideals in quantum algebras", "abstract": "In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of $2\\times n$ quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the ``variety of $2\\times n$ quantum matrices''."}
{"category": "Math", "title": "Annotations to a certain passage of Descartes for finding the quadrature of the circle", "abstract": "Translation from the Latin of \"Annotationes in locum quendam Cartesii ad circuli quadraturam spectantem\" (1763). The passage Euler is referring to is the \"Excerpta\" in part 6, p. 6 of Descartes' 1701 \"Opuscula posthuma\". Before reading this paper I had not heard of the \"quadratrix\" before, and I recommend learning a bit about it before reading this. I found Thomas Heath, \"A history of Greek mathematics\", vol. I, chapter VII to be helpful, in particular pp. 226-230. The quadratrix is a \"mechanical curve\" that can be used to rectify the circle. The usual problem of squaring the circle is to construct a square with the same area (or perimeter) as a given circle, in a finite number of steps using compass and straightedge. Descartes worked in the reverse direction: from a given square he constructed the radius of a circle with the same perimeter, but in an infinite number of steps. In this paper Euler reconstructs Descartes' argument and develops some consequences of it. Euler finds that \\[ \\sum_{n=0}^\\infty \\frac{1}{2^n} \\tan \\frac{1}{2^n}\\phi = \\frac{1}{\\phi} - 2\\cot 2\\phi. \\] Integrating this yields \\[ \\prod_{n=1}^\\infty \\sec \\frac{1}{2^n} \\phi = \\frac{2\\phi}{\\sin 2\\phi}. \\] I'd like to thank Davide Crippa from the University of Paris 7 for some helpful back and forth about this paper. One of the only citations to this paper that I have found is in Pietro Ferroni, De calculo integralium exercitatio mathematica, Allegrini, Florence, 1792, pp. xxi--xxiii. The full text of it is available on Google Books."}
{"category": "Math", "title": "Combinatorial independence in measurable dynamics", "abstract": "We develop a fine-scale local analysis of measure entropy and measure sequence entropy based on combinatorial independence. The concepts of measure IE-tuples and measure IN-tuples are introduced and studied in analogy with their counterparts in topological dynamics. Local characterizations of the Pinsker von Neumann algebra and its sequence entropy analogue are given in terms of combinatorial independence, l_1 geometry, and Voiculescu's completely positive approximation entropy. Among the novel features of our local study is the treatment of general discrete acting groups, with the structural assumption of amenability in the case of entropy."}
{"category": "Math", "title": "O-minimal cohomology: finiteness and invariance results", "abstract": "We prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language. We also study the cohomology of the intersection of a definable decreas-ing family of definably compact sets, under the additional assumption that the o-minimal structure expands a field."}
{"category": "Math", "title": "Reiter's properties (P_1) and (P_2) for locally compact quantum groups", "abstract": "A locally compact group $G$ is amenable if and only if it has Reiter's property $(P_p)$ for $p=1$ or, equivalently, all $p \\in [1,\\infty)$, i.e., there is a net $(m_\\alpha)_\\alpha$ of non-negative norm one functions in $L^p(G)$ such that $\\lim_\\alpha \\sup_{x \\in K} \\| L_{x^{-1}} m_\\alpha - m_\\alpha \\|_p = 0$ for each compact subset $K \\subset G$ ($L_{x^{-1}} m_\\alpha$ stands for the left translate of $m_\\alpha$ by $x^{-1}$). We extend the definitions of properties $(P_1)$ and $(P_2)$ from locally compact groups to locally compact quantum groups in the sense of J. Kustermans and S. Vaes. We show that a locally compact quantum group has $(P_1)$ if and only if it is amenable and that it has $(P_2)$ if and only if its dual quantum group is co-amenable. As a consequence, $(P_2)$ implies $(P_1)$."}
{"category": "Math", "title": "Why prove things?", "abstract": "We illustrate the concept of mathematical proof."}
{"category": "Math", "title": "Admissible orders of Jordan loops", "abstract": "A commutative loop is Jordan if it satisfies the identity $x^2 (y x) = (x^2 y) x$. Using an amalgam construction and its generalizations, we prove that a nonassociative Jordan loop of order $n$ exists if and only if $n\\geq 6$ and $n\\neq 9$. We also consider whether powers of elements in Jordan loops are well-defined, and we construct an infinite family of finite simple nonassociative Jordan loops."}
{"category": "Math", "title": "The fundamental theorem of complex multiplication", "abstract": "The goal of this expository article is to present a proof that is as direct and elementary as possible of the fundamental theorem of complex multiplication (Shimura, Taniyama, Langlands, Tate, Deligne et al.). The article is a revision of part of my manuscript, Complex Multiplication, April 7, 2006."}
{"category": "Math", "title": "The symplectic geometry of cotangent bundles from a categorical viewpoint", "abstract": "We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow (math/0604379, math/0612399) and the authors (math/0701783), before discussing a new approach using family Floer cohomology and the ``wrapped Fukaya category''. The latter, inspired by Viterbo's symplectic homology, emphasises the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply-connected case."}
{"category": "Math", "title": "Graphs on surfaces and Khovanov homology", "abstract": "Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram $L$, there is an associated ribbon graph whose quasi-trees correspond bijectively to spanning trees of the graph obtained by checkerboard coloring $L$. This correspondence preserves the bigrading used for the spanning tree model of Khovanov homology, whose Euler characteristic is the Jones polynomial of $L$. Thus, Khovanov homology can be expressed in terms of ribbon graphs, with generators given by ordered chord diagrams."}
{"category": "Math", "title": "The spectral radius of subgraphs of regular graphs", "abstract": "We give a bound on the spectral radius of subgraphs of regular graphs with given order and diameter. We give a lower bound on the smallest eigenvalue of a nonbipartite regular graph of given order and diameter."}
{"category": "Math", "title": "Quasi-tree expansion for the Bollob\\'as-Riordan-Tutte polynomial", "abstract": "Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollob\\'as-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We generalize the spanning tree expansion of the Tutte polynomial to a quasi-tree expansion of the Bollob\\'as-Riordan-Tutte polynomial."}
{"category": "Math", "title": "Deconvolution with unknown error distribution", "abstract": "We consider the problem of estimating a density $f_X$ using a sample $Y_1,...,Y_n$ from $f_Y=f_X\\star f_{\\epsilon}$, where $f_{\\epsilon}$ is an unknown density. We assume that an additional sample $\\epsilon_1,...,\\epsilon_m$ from $f_{\\epsilon}$ is observed. Estimators of $f_X$ and its derivatives are constructed by using nonparametric estimators of $f_Y$ and $f_{\\epsilon}$ and by applying a spectral cut-off in the Fourier domain. We derive the rate of convergence of the estimators in case of a known and unknown error density $f_{\\epsilon}$, where it is assumed that $f_X$ satisfies a polynomial, logarithmic or general source condition. It is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density, if the density $f_X$ belongs to a Sobolev space $H_{\\mathbh p}$ and $f_{\\epsilon}$ is ordinary smooth or supersmooth."}
{"category": "Math", "title": "Biclosed bicategories: localisation of convolution", "abstract": "We give a summary (without proofs) of the main results in the author's thesis entitled ``Construction of biclosed categories'' (University of New South Wales, Australia, 1970). This summary is reprinted directly from Report 81-0030 of the School of Mathematics and Physics, Macquarie University, April 1981. In particular, it gives sufficient conditions for existence of an extension of a (pro)monoidal category structure along a given dense functor to a cocomplete category. The two basic procedures used in the proof turn out to be special cases of the final result, the two respective dense functors then being the Yoneda embedding followed by a localisation. The final result has a standard universal property based on left Kan extension of (pro)monoidal functors along the given dense functor, however this property is not stated explicitly here."}
{"category": "Math", "title": "The Dixmier-Moeglin equivalence and a Gel'fand-Kirillov problem for Poisson polynomial algebras", "abstract": "The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, such as semiclassical limits of quantum matrices, quantum symplectic and euclidean spaces, quantum symmetric and antisymmetric matrices. For a similarly large class of Poisson polynomial rings, it is proved that the quotient field of the algebra (respectively, of any Poisson prime factor ring) is a rational function field $F(x_1,...,x_n)$ over the base field (respectively, over an extension field of the base field) with $\\{x_i,x_j\\}= \\lambda_{ij} x_ix_j$ for suitable scalars $\\lambda_{ij}$, thus establishing a quadratic Poisson version of the Gel'fand-Kirillov problem. Finally, partial solutions to the isomorphism problem for Poisson fields of the type just mentioned are obtained."}
{"category": "Math", "title": "The two-parameter Poisson--Dirichlet point process", "abstract": "The two-parameter Poisson--Dirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (that is, the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Using this, we apply the theory of point processes to reveal the mathematical structure of the two-parameter Poisson--Dirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we are able to extend several results previously known for the one-parameter case. The Markov--Krein identity for the generalized Dirichlet process is discussed from the point of view of functional analysis based on the two-parameter Poisson--Dirichlet distribution."}
{"category": "Math", "title": "Multiscale Resolution of Shortwave-Longwave Interaction", "abstract": "In the study of time-dependent waves, it is computationally expensive to solve a problem in which high frequencies (shortwaves, with wavenumber k = kmax) and low frequencies (longwaves, near k=kmin) mix. Consider a problem in which low frequencies scatter off a sharp impurity. The impurity generates high frequencies which propagate and spread throughout the computational domain, while the domain must be large enough to contain several longwaves. Conventional spectral methods have computational cost proportional to O(kmax/kmin \\log (kmax/kmin)). We present here a multiscale algorithm (implemented for the Schrodinger equation, but generally applicable) which solves the problem with cost (in space and time) O(kmax L log(kmax / kmin) \\log(kmax L)). Here, L is the width of the region in which the algorithm resolves all frequencies, and is independent of kmin."}
{"category": "Math", "title": "A Morita context and Galois extensions for Quasi-Hopf algebras", "abstract": "If H is a finite dimensional quasi-Hopf algebra and A is a left H-module algebra, we prove that there is a Morita context connecting the smash product A#H and the subalgebra of invariants A^{H}. We define also Galois extensions and prove the connection with this Morita context, as in the Hopf case."}
{"category": "Math", "title": "On a generalized Sturm theorem", "abstract": "Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. What we propose here is a Sturm oscillation theorem for systems of even order having strongly indefinite leading coefficient."}
{"category": "Math", "title": "Small Bialgebras with a Projection: Applications", "abstract": "In this paper we continue the investigation started in [A.M.St.-Small], dealing with bialgebras $A$ with an $H$-bilinear coalgebra projection over an arbitrary subbialgebra $H$ with antipode. These bialgebras can be described as deformed bosonizations $R#_{\\xi} H$ of a pre-bialgebra $R$ by $H$ with a cocycle $\\xi$. Here we describe the behavior of $\\xi$ in the case when $R$ is f.d. and thin i.e. it is connected with one dimensional space of primitive elements. This is used to analyze the arithmetic properties of $A$. Meaningful results are obtained when $H$ is cosemisimple. By means of Ore extension construction, we provide some examples of atypical situations (e.g. the multiplication of $R$ is not $H$-colinear or $\\xi$ is non-trivial)."}
{"category": "Math", "title": "Integral Chow rings of toric stacks", "abstract": "The purpose of this paper is to prove that integral Chow rings of toric stacks are naturally isomorphic to Stanley-Reisner rings."}
{"category": "Math", "title": "Linear balls and the multiplicity conjecture", "abstract": "A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley--Reisner ring has a linear resolution. It turns out that the Stanley--Reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture. A class of shellable spheres arising naturally from commutative algebra whose Stanley--Reisner rings satisfy the multiplicity conjecture will be presented."}
{"category": "Math", "title": "p-torsion of Genus Two Curves Over Prime Fields of Characteristic p", "abstract": "Consider the Jacobian of a hyperelliptic genus two curve defined over a prime field of characteristic p and with complex multiplication. In this paper we show that the p-Sylow subgroup of the Jacobian is either trivial or of order p."}
{"category": "Math", "title": "New Generalization of Perturbed Ostrowski Type Inequalities and Applications", "abstract": "Generalizations of Ostrowski type inequality for functions of Lipschitzian type are established. Applications in numerical integration and cumulative distribution functions are also given."}
{"category": "Math", "title": "Large Scale Properties of the IIIC for 2D Percolation", "abstract": "We reinvestigate the 2D problem of the inhomogeneous incipient infinite cluster where, in an independent percolation model, the density decays to p_c with an inverse power, \\lambda, of the distance to the origin. Assuming the existence of critical exponents (as is known in the case of the triangular site lattice) if the power is less than 1/\\nu, with \\nu the correlation length exponent, we demonstrate an infinite cluster with scale dimension given by D_H=2-\\beta\\lambda. Further, we investigate the critical case \\lambda_c=1/\\nu and show that iterated logarithmic corrections will tip the balance between the possibility and impossibility of an infinite cluster."}
{"category": "Math", "title": "On PAC Extensions and Scaled Trace Forms", "abstract": "Any non-degenerate quadratic form over a Hilbertian field (e.g., a number field) is isomorphic to a scaled trace form. In this work we extend this result to more general fields. In particular, prosolvable and prime-to-p extensions of a Hilbertian field. The proofs are based on the theory of PAC extensions."}
{"category": "Math", "title": "Convergence of excursion point processes and its applications to functional limit theorems of Markov processes on a half-line", "abstract": "Invariance principles are obtained for a Markov process on a half-line with continuous paths on the interior. The domains of attraction of the two different types of self-similar processes are investigated. Our approach is to establish convergence of excursion point processes, which is based on It\\^{o}'s excursion theory and a recent result on convergence of excursion measures by Fitzsimmons and the present author."}
{"category": "Math", "title": "Orthogonal arrays from Hermitian varieties", "abstract": "An orthogonal array OA(q^{2n-1},q^{2n-2}, q,2) is constructed from the action of a subset of PGL(n+1,q^2) on some non--degenerate Hermitian varieties in PG(n,q^2). It is also shown that the rows of this orthogonal array correspond to some blocks of an affine design, which for q> 2 is a non--classical model of the affine space AG(2n-1,q)."}
{"category": "Math", "title": "A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields", "abstract": "We give a complete list of normal forms for the 2-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie."}
{"category": "Math", "title": "Pseudoprocesses governed by higher-order fractional differential equations", "abstract": "We study here a heat-type differential equation of order n greater than two, in the case where the time-derivative is supposed to be fractional. The corresponding solution can be described as the transition function of a pseudoprocess (coinciding with the one governed by the standard, non-fractional, equation) with a time argument T which is itself random. The distribution of T is presented together with some features of the solution (such as analytic expressions for its moments)."}
{"category": "Math", "title": "Integer symmetric matrices having all their eigenvalues in the interval [-2,2]", "abstract": "We completely describe all integer symmetric matrices that have all their eigenvalues in the interval [-2,2]. Along the way we classify all signed graphs, and then all charged signed graphs, having all their eigenvalues in this same interval. We then classify subsets of the above for which the integer symmetric matrices, signed graphs and charged signed graphs have all their eigenvalues in the open interval (-2,2)."}
{"category": "Math", "title": "Spinal partitions and invariance under re-rooting of continuum random trees", "abstract": "We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson--Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform re-rooting."}
{"category": "Math", "title": "Measure of full dimension for some nonconformal repellers", "abstract": "We prove the existence of an ergodic measure with full Hausdorff dimension for a class of nonlinear nonconformal skew-product transformations. In order to do so we establish a variational principle for the topological pressure of certain noncompact sets."}
{"category": "Math", "title": "Four Drafts of The Representation Theory of the Group of Infinite Matrices over Finite Fields", "abstract": "Preface (A.Vershik) - about these texts (3.); I.Interpolation between inductive and projective limits of finite groups with applicatons to linear groups over finite fields; II.The characters of the groups of almost triangle matrices over finite filed; III.A Law of Large Numbers for the characters of GL_n(k) over finite field k; IV.An outline of construction of factor representations of the group GLB(F_q)."}
{"category": "Math", "title": "Bounding slopes of $p$-adic modular forms", "abstract": "Let $p$ be prime, $N$ be a positive integer prime to $p$, and $k$ be an integer. Let $P_k(t)$ be the characteristic series for Atkin's $U$ operator as an endomorphism of $p$-adic overconvergent modular forms of tame level $N$ and weight $k$. Motivated by conjectures of Gouvea and Mazur, we strengthen Wan's congruence between coefficients of $P_k$ and $P_{k'}$ for $k'$ close $p$-adically to $k$. For $p-1 | 12$, $N = 1$, $k = 0$, we compute a matrix for $U$ whose entries are coefficients in the power series of a rational function of two variables. We apply this computation to show for $p = 3$ a parabola below the Newton polygon $N_0$ of $P_0$, which coincides with $N_0$ infinitely often. As a consequence, we find a polygonal curve above $N_0$. This tightest bound on $N_0$ yields the strongest congruences between coefficients of $P_0$ and $P_k$ for $k$ of large 3-adic valuation."}
{"category": "Math", "title": "Arithmetic properties related to the shuffle-product", "abstract": "Properties of the shuffle product suggest the definition of a quadratic form with domain and values in formal power series over a field of characteristic 2. This quadratic form preserves rational (respectively algebraic) power series and its restriction to the affine subspace of series with constant term 1 is bijective. Conjecturally, this bijection restricts to a bijection of rational (respectively algebraic) formal power series."}
{"category": "Math", "title": "Burkholder's submartingales from a stochastic calculus perspective", "abstract": "We provide a simple proof, as well as several generalizations, of a recent result by Davis and Suh, characterizing a class of continuous submartingales and supermartingales that can be expressed in terms of a squared Brownian motion and of some appropriate powers of its maximum. Our techniques involve elementary stochastic calculus, as well as the Doob-Meyer decomposition of continuous submartingales. These results can be used to obtain an explicit expression of the constants appearing in the Burkholder-Davis-Gundy inequalities. A connection with some balayage formulae is also established."}
{"category": "Math", "title": "Computing parametric rational generating functions with a primal Barvinok algorithm", "abstract": "Computations with Barvinok's short rational generating functions are traditionally being performed in the dual space, to avoid the combinatorial complexity of inclusion--exclusion formulas for the intersecting proper faces of cones. We prove that, on the level of indicator functions of polyhedra, there is no need for using inclusion--exclusion formulas to account for boundary effects: All linear identities in the space of indicator functions can be purely expressed using half-open variants of the full-dimensional polyhedra in the identity. This gives rise to a practically efficient, parametric Barvinok algorithm in the primal space."}
{"category": "Math", "title": "Picard-Fuchs Differential Equations for Families of K3 Surfaces", "abstract": "This thesis studies some examples of families of K3 surfaces with Picard lattices of maximal rank. These families occur as invariants of finite automorphism groups. The Picard-Fuchs differential equations describing the variation of Hodge structure in these families are considered. Techniques are developed to find the corresponding monodromy groups as arithmetic Fuchsian groups acting on the families' period spaces."}
{"category": "Math", "title": "Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations", "abstract": "This article is devoted to a Log improvement of Prodi-Serrin criterion for global regularity to solutions to Navier-Stokes equations in dimension 3. It is shown that the global regualrity holds under the condition that |u|^5/ log (1+|u|) is integrable in space time variables."}
{"category": "Math", "title": "A LLT-like test for proving the primality of Fermat numbers", "abstract": "This paper provides a proof of a LLT-like test for Fermat numbers, based on the properties of Lucas Sequences and on the method of Lehmer."}
{"category": "Math", "title": "Classification of singularities in the complete conformally flat Yamabe flow", "abstract": "We show that an eternal solution to a complete, locally conformally flat Yamabe flow, $\\frac{\\partial}{\\partial t} g = -Rg$, with uniformly bounded scalar curvature and positive Ricci curvature at $t = 0$, where the scalar curvature assumes its maximum is a gradient steady soliton. As an application of that, we study the blow up behavior of $g(t)$ at the maximal time of existence, $T < \\infty$. We assume that $(M,g(\\cdot, t))$ satisfies (i) the injectivity radius bound {\\bf or} (ii) the Schouten tensor is positive at time $t = 0$ and the scalar curvature bounded at each time-slice. We show that the singularity the flow develops at time $T$ is always of type I."}
{"category": "Math", "title": "Existence of Infinitely Many Solutions for a Quasilinear Elliptic Problem on Time Scales", "abstract": "We study a boundary-value quasilinear elliptic problem on a generic time scale. Making use of the fixed-point index theory, sufficient conditions are given to obtain existence, multiplicity, and infinite solvability of positive solutions."}
{"category": "Math", "title": "The geometry of systems of third order differential equations induced by second order Lagrangians", "abstract": "A dynamical system on the total space of the fibre bundle of second order accelerations, $T^2M$, is defined as a third order vector field $S$ on $T^2M$, called semispray, which is mapped by the second order tangent structure into one of the Liouville vector field. For a regular Lagrangian of second order we prove that this semispray is uniquely determined by two associated Cartan-Poincar\\'e one-forms. To study the geometry of this semispray we construct a nonlinear connection, which is a Lagrangian subbundle for the presymplectic structure. Using this semispray and the associated nonlinear connection we define covariant derivatives of first and second order. With respect to this, the second order dynamical derivative of the Lagrangian metric tensor vanishes."}
{"category": "Math", "title": "Logarithmic knot invariants arising from restricted quantum groups", "abstract": "We construct knot invariants from the radical part of projective modules of restricted quantum groups. We also show a relation between these invariants and the colored Alexander invariants."}
{"category": "Math", "title": "Algebraic Polymorphisms", "abstract": "In this paper we consider a special class of polymorphisms with invariant measure, - (cf.[1])- the algebraic polymorphisms of compact groups. A general polymorphism is -- by definition -- a many-valued map with invariant measure, and the conjugate operator of a polymorphism is a Markov operator (i.e., a positive operator on $L^2$ of norm 1 which preserves the constants). In the algebraic case a polymorphism is a correspondence in the sense of algebraic geometry, but here we investigate it from a dynamical point of view. The most important examples are the algebraic polymorphisms of torus, where we introduce a parametrization of the semigroup of toral polymorphisms in terms of rational matrices and describe the spectra of the corresponding Markov operators."}
{"category": "Math", "title": "Stable functions and common stabilizations of Heegaard splittings", "abstract": "We present a new proof of Reidemeister and Singer's Theorem that any two Heegaard splittings of the same 3-manifold have a common stabilization. The proof leads to an upper bound on the minimal genus of a common stabilization in terms of the number of negative slope inflection points and type-two cusps in a Rubinstein-Scharlemann graphic for the two splittings."}
{"category": "Math", "title": "Hyperbolic knots with three toroidal Dehn surgeries", "abstract": "It is conjectured that a hyperbolic knot admits at most three Dehn surgeries which yield closed three manifolds containing incompressible tori. We show that there exist infinitely many hyperbolic knots which attain the conjectural maximum number. Interestingly, those surgeries correspond to consecutive integers."}
{"category": "Math", "title": "An analogue of the space of conformal blocks in (4k+2)-dimensions", "abstract": "Based on projective representations of smooth Deligne cohomology groups, we introduce an analogue of the space of conformal blocks to compact oriented (4k+2)-dimensional Riemannian manifolds with boundary. For the standard (4k+2)-dimensional disk, we compute the space concretely to prove that its dimension is finite."}
{"category": "Math", "title": "Symplectic quasi-states and semi-simplicity of quantum homology", "abstract": "We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D.McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds."}
{"category": "Math", "title": "Constant Angle Surfaces in $\\H^2\\times \\R$", "abstract": "In this paper we classify constant angle surfaces in $\\H^2\\times\\R$, where $\\H^2$ is the hyperbolic plane."}
{"category": "Math", "title": "Difference sets and shifted primes", "abstract": "We show that if A is a subset of {1, ..., n} such that it has no pairs of elements whose difference is equal to p-1 with p a prime number, then the size of A is O(n(loglog n)^(-clogloglogloglog n)) for some positive constant c."}
{"category": "Math", "title": "Inducing stability conditions", "abstract": "We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application we examine stability conditions on Kummer and Enriques surfaces and we improve the derived version of the Torelli Theorem for the latter surfaces already present in the litterature. We also study the relationship between stability conditions on projective spaces and those on their canonical bundles."}
{"category": "Math", "title": "On the Lenstra constant associated to the Rosen continued fractions", "abstract": "The purpose of this paper is to describe the relation between the Legendre and the Lenstra constants. Indeed we show that they are equal whenever the Legendre constant exists; in particular, this holds for both Rosen continued fractions and $\\alpha$-continued fractions. We also give the explicit value of the entropy of the Rosen map with respect to the absolutely continuous invariant probability measure."}
{"category": "Math", "title": "Contracted ideals and the Groebner fan of the rational normal curve", "abstract": "The paper has two goals: the study the associated graded ring of contracted homogeneous ideals in $K[x,y]$ and the study of the Groebner fan of the ideal $P$ of the rational normal curve in ${\\bf P}^d$. These two problems are, quite surprisingly, very tightly related. We completely classify the contracted ideals with a Cohen-Macaulay associated graded rings in terms of the numerical invariants arising from Zariski's factorization. We determine explicitly all the initial ideals (monomial or not) of $P$ that are Cohen-Macaulay."}
{"category": "Math", "title": "On the spectrum of the normalized graph Laplacian", "abstract": "The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this spectrum under local and global operations like motif doubling, graph joining or splitting. The eigenvalue 1 plays a particular role, and we therefore emphasize those constructions that change its multiplicity in a controlled manner, like the iterated duplication of nodes."}
{"category": "Math", "title": "Circular law, Extreme Singular values and Potential theory", "abstract": "Consider the empirical spectral distribution of complex random $n\\times n$ matrix whose entries are independent and identically distributed random variables with mean zero and variance $1/n$. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements."}
{"category": "Math", "title": "Blow-up in the Parabolic Scalar Curvature Equation", "abstract": "The \\textit{parabolic scalar curvature equation} is a reaction-diffusion type equation on an $(n-1)$-manifold $\\Sigma$, the time variable of which shall be denoted by $r$. Given a function $R$ on $[r_0,r_1)\\times\\Sigma$ and a family of metrics $\\gamma(r)$ on $\\Sigma$, when the coefficients of this equation are appropriately defined in terms of $\\gamma$ and $R$, positive solutions give metrics of prescribed scalar curvature $R$ on $[r_0,r_1)\\times\\Sigma$ in the form \\[ g=u^2dr^2+r^2\\gamma.\\] If the area element of $r^2\\gamma$ is expanding for increasing $r$, then the equation is parabolic, and the basic existence problem is to take positive initial data at some $r=r_0$ and solve for $u$ on the maximal interval of existence, which above was implicitly assumed to be $I=[r_0,r_1)$; one often hopes that $r_1=\\infty$. However, the case of greatest physical interest, $R>0$, often leads to blow-up in finite time so that $r_1<\\infty$. It is the purpose of the present work to investigate the situation in which the blow-up nonetheless occurs in such a way that $g$ is continuously extendible to $\\bar M=[r_0,r_1]\\times\\Sigma$ as a manifold with totally geodesic outer boundary at $r=r_1$."}
{"category": "Math", "title": "Polarized Variation of Hodge Structures of Calabi-Yau Type and Characteristic Subvarieties Over Bounded Symmetric Domains", "abstract": "In this paper we extend the construction of the canonical polarized variation of Hodge structures over tube domain considered by B. Gross in \\cite{G} to bounded symmetric domain and introduce a series of invariants of infinitesimal variation of Hodge structures, which we call characteristic subvarieties. We prove that the characteristic subvariety of the canonical polarized variations of Hodge structures over irreducible bounded symmetric domains are identified with the characteristic bundles defined by N. Mok in \\cite{M}. We verified the generating property of B. Gross for all irreducible bounded symmetric domains, which was predicted in \\cite{G}."}
{"category": "Math", "title": "Poisson approximation for large clusters in the supercritical FK model", "abstract": "Using the Chen-Stein method, we show that the spatial distribution of large finite clusters in the supercritical FK model approximates a Poisson process when the ratio weak mixing property holds."}
{"category": "Math", "title": "On measure solutions of backward stochastic differential equations", "abstract": "We consider backward stochastic differential equations (BSDE) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. In case the terminal condition is bounded and the generator fulfills the usual continuity and boundedness conditions, we show that measure solutions with equivalent measures just reinterpret classical ones. In case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in case of non-uniqueness classical solutions that fail to be measure solutions can coexists with different measure solutions."}
{"category": "Math", "title": "Stability conditions on curves", "abstract": "We study some examples of Bridgeland-Douglas stability conditions on triangulated categories. From one side we give a complete description of the stability manifolds for smooth projective curves of positive genus. From the other side we study stability conditions on triangulated categories generated by an exceptional collection. In the case of the projective line this leads to the connectedness and simply-connectedness of the stability manifold."}
{"category": "Math", "title": "On implications in sectionally pseudocomplemented posets", "abstract": "A sectionally pseudocomplemented poset P is one which has the top element and in which every principal order filter is a pseudocomplemented poset. The sectional pseudocomplements give rise to an implication-like operation on P which coincides with the relative pseudocomplementation if P is relatively psudocomplemented. We characterise this operation and study some elementary properties of upper semilattices, lower semilattices and lattices equipped with this kind of implication. We deal also with a few weaker versions of implication. Sectionally pseudocomplemented lattices have already been studied in the literature."}
{"category": "Math", "title": "Parabolic equations with measurable coefficients in $L_p$-spaces with mixed norms", "abstract": "The unique solvability of parabolic equations in Sobolev spaces with mixed norms is presented. The second order coefficients (except $a^{11}$) are assumed to be only measurable in time and one spatial variable, and VMO in the other spatial variables. The coefficient $a^{11}$ is measurable in one spatial variable and VMO in the other variables."}
{"category": "Math", "title": "Crossed products of locally C*-algebras and Morita equivalence", "abstract": "We introduce the notion of strong Morita equivalence for group actions on locally C*-algebras and prove that the crossed products associated with two strongly Morita equivalent continuous inverse limit actions of a locally compact group G on the locally C*-algebras A and B are strongly Morita equivalent. This generalizes a result of F. Combes, Proc. London Math. Soc. 49(1984) and R. E. Curto, P.S. Muhly, D. P. Williams, Proc. Amer. Soc. 90(1984)."}
{"category": "Math", "title": "Dynamics of the Teichmueller flow on compact invariant sets", "abstract": "Let Q(S) be the moduli space of area one holomorphic quadratic differentials for an oriented surface S of genus g with m punctures and 3g-3+m>1. We show that the supremum over all compact subsets K of Q(S) of the asymptotic growth rate of the number of periodic orbits of the Teichmueller flow which are contained in K equals h=6g-6+2m. Moreover, h is also the supremum of the topological entropies of the restriction of the Teichmueller flow to compact invariant subsets of Q(S)."}
{"category": "Math", "title": "On Morita equivalence of group actions on locally C*-algebras", "abstract": "We extend to the context of locally C*-algebras a result of F. Combes [Proc. London Math. Soc. 49(1984), 289-306]."}
{"category": "Math", "title": "On pseudo-harmonic maps in conformal geometry", "abstract": "We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We obtain an extension of Siu's rigidity to Kahler-Weyl geometry and apply the latter to Vaisman's conjecture. Other applications include topological obstructions to the existence of Kahler-Weyl structures. For example, we show that no co-compact lattice in SO(1,n), n>2, can be the fundamental group of a compact Kahler-Weyl manifold of certain type."}
{"category": "Math", "title": "The Cut-off Covering Spectrum", "abstract": "We introduce the $R$ cut-off covering spectrum and the cut-off covering spectrum of a complete length space or Riemannian manifold. The spectra measure the sizes of localized holes in the space and are defined using covering spaces called $\\delta$ covers and $R$ cut-off $\\delta$ covers. They are investigated using $\\delta$ homotopies which are homotopies via grids whose squares are mapped into balls of radius $\\delta$. On locally compact spaces, we prove that these new spectra are subsets of the closure of the length spectrum. We prove the $R$ cut-off covering spectrum is almost continuous with respect to the pointed Gromov-Hausdorff convergence of spaces and that the cut-off covering spectrum is also relatively well behaved. This is not true of the covering spectrum defined in our earlier work which was shown to be well behaved on compact spaces. We close by analyzing these spectra on Riemannian manifolds with lower bounds on their sectional and Ricci curvature and their limit spaces."}
{"category": "Math", "title": "A note on toric Deligne-Mumford stacks", "abstract": "We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks."}
{"category": "Math", "title": "Notes on Schubert classes of a loop group", "abstract": "In these notes, we survey the homology of the loop group Omega(K) of a compact group K, also known as the affine Grassmannian of a complex loop group. Using the Bott picture of H_*(Omega(K)), the homology algebra or Pontryagin ring, we obtain two new results: A. Factorization of affine Schubert homology classes. B. Definition of affine Schubert polynomials representing the affine Schubert homology classes in all types, in terms similar to ordinary Schubert polynomials."}
{"category": "Math", "title": "Width and mean curvature flow", "abstract": "Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show: Each curve in the tightened sweepout whose length is close to the length of the longest curve in the sweepout must itself be close to a closed geodesic. In particular, there are curves in the sweepout that are close to closed geodesics. Finding closed geodesics on the 2-sphere by using sweepouts goes back to Birkhoff in 1917. As an application, we bound from above, by a negative constant, the rate of change of the width for a one-parameter family of convex hypersurfaces that flows by mean curvature. The width is loosely speaking up to a constant the square of the length of the shortest closed curve needed to ``pull over'' $M$. This estimate is sharp and leads to a sharp estimate for the extinction time; cf. [CM1], [CM2] where a similar bound for the rate of change for the two dimensional width is shown for homotopy 3-spheres evolving by the Ricci flow (see also Perelman)."}
{"category": "Math", "title": "The incidence class and the hierarchy of orbits", "abstract": "R. Rim\\'anyi defined the incidence class of two singularities X and Y as $[X]|_Y$, the restriction of the Thom polynomial of X to Y. He conjectured that (under mild conditions) the incidence is not zero if and only if Y is in the closure of X. Generalizing this notion we define the incidence class of two orbits X and Y of a representation. We give a sufficient condition (positivity) for Y to have the property that the incidence class $[X]|_Y$ is not zero if and only if Y is in the closure of X for any other orbit X. We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits. In other words in these cases the incidence classes completely determine the hierarchy of the orbits. We also study the case of singularities where positivity doesn't hold for all orbits."}
{"category": "Math", "title": "Some Extensions of Witt's Theorem", "abstract": "We extend Witt's theorem to several kinds of simultaneous isometries of subspaces. We determine sufficient and necessary conditions for the extension of an isometry of subspaces $\\phi:E\\to E'$ to an isometry $\\phi_V:V\\to V'$ that also sends a given subspace to another, or a given self-dual flag to another, or a Witt's decomposition to another and a special self-dual flag to another. We also determine sufficient and necessary conditions for the isometry of generic flags or the simultaneous isometry of (subspace, self-dual flag) pairs."}
{"category": "Math", "title": "A survey of total positivity", "abstract": "We survey the history of totally positive matrices and the generalization to Lie groups. We describe a reduction of a bilinear form to a canonical form (generalizing the case of symplectic nondegenerate forms) using ideas from total positivity; we also place this in a Lie group context. We give a short exposition of results of Fock and Goncharov on the study of homomorphisms of the fundamental group of a closed surface into a Lie group."}
{"category": "Math", "title": "Fast computation by block permanents of cumulative distribution functions of order statistics from several populations", "abstract": "The joint cumulative distribution function for order statistics arising from several different populations is given in terms of the distribution function of the populations. The computational cost of the formula in the case of two populations is still exponential in the worst case, but it is a dramatic improvement compared to the general formula by Bapat and Beg. In the case when only the joint distribution function of a subset of the order statistics of fixed size is needed, the complexity is polynomial, for the case of two populations."}
{"category": "Math", "title": "A cube of resolutions for knot Floer homology", "abstract": "We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot."}
{"category": "Math", "title": "Extremal metrics for spectral functions of Dirac operators in even and odd dimensions", "abstract": "Let (M^n, g) be a closed smooth Riemannian spin manifold and denote by D its Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for the zeta function and functional determinant of D^2, and prove finiteness of the Morse index at stationary metrics, and local extremality at such metrics under general, i.e. not only conformal, change of metrics. In even dimensions, which is also a new case for the conformal Laplacian, the relevant stability operator is of log-polyhomogeneous pseudodifferential type, and we prove new results of independent interest, on the spectrum for such operators. We use this to prove local extremality under variation of the Riemannian metric, which in the important example when (M^n, g) is the round n-sphere, gives a partial verification of Branson's conjecture on the pattern of extremals. Thus det(D^2) has a local (max, max, min, min) when the dimension is (4k, 4k + 1, 4k + 2, 4k + 3), respectively."}
{"category": "Math", "title": "Indecomposable p-algebras and Galois subfields in generic abelian crossed products", "abstract": "Let F be a Henselian valued field with char(F) = p and D a semi-ramified, \"not strongly degenerate\" p-algebra. We show that all Galois subfields of D are inertial. Using this as a tool we study generic abelian crossed product p-algebras, proving among other things that the noncyclic generic abelian crossed product p-algebras defined by non-degenerate matrices are indecomposable p-algebras. To construct examples of these indecomposable p-algebras with exponent p and large index we study the relationship between degeneracy in matrices defining abelian crossed products and torsion in CH^2 of Severi-Brauer varieties."}
{"category": "Math", "title": "Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average", "abstract": "We prove that the set of Farey fractions of order $T$, that is, the set $\\{\\alpha/\\beta \\in \\Q : \\gcd(\\alpha, \\beta) = 1, 1 \\le \\alpha, \\beta \\le T\\}$, is uniformly distributed in residue classes modulo a prime $p$ provided $T \\ge p^{1/2 +\\eps}$ for any fixed $\\eps>0$. We apply this to obtain upper bounds for the Lang--Trotter conjectures on Frobenius traces and Frobenius fields ``on average'' over a one-parametric family of elliptic curves."}
{"category": "Math", "title": "Discrete Control Systems", "abstract": "Discrete control systems, as considered here, refer to the control theory of discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are based on a discrete variational principle, and are part of the broader field of geometric integration. Geometric integrators are numerical integration methods that preserve geometric properties of continuous systems, such as conservation of the symplectic form, momentum, and energy. They also guarantee that the discrete flow remains on the manifold on which the continuous system evolves, an important property in the case of rigid-body dynamics. In nonlinear control, one typically relies on differential geometric and dynamical systems techniques to prove properties such as stability, controllability, and optimality. More generally, the geometric structure of such systems plays a critical role in the nonlinear analysis of the corresponding control problems. Despite the critical role of geometry and mechanics in the analysis of nonlinear control systems, nonlinear control algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. The field of discrete control system aims to address this deficiency by restricting the approximation to choice of a discrete-time model, and developing an associated control theory that does not introduce any additional approximation. In particular, this involves the construction of a control theory for discrete-time models based on geometric integrators that yields numerical implementations of nonlinear and geometric control algorithms that preserve the crucial underlying geometric structure."}
{"category": "Math", "title": "On the closure of the diagonal of a $T_1$-space", "abstract": "Let X be a topological space. The closure of \\Delta = {(x, x) : x \\in X} in X \\times X is a symmetric relation on X. We characterise those equivalence relations on an infinite set that arise as the closure of the diagonal with respect to a T_1-topology."}
{"category": "Math", "title": "Distributive lattice orderings and Priestley duality", "abstract": "The ordering relation of a bounded distributive lattice L is a (distributive) (0, 1)-sublattice of L \\times L. This construction gives rise to a functor \\Phi from the category of bounded distributive lattices to itself. We examine the interaction of \\Phi with Priestley duality and characterise those bounded distributive lattices L such that there is a bounded distributive lattice K such that \\Phi(K) is (isomorphic to) L."}
{"category": "Math", "title": "The Self-injective Cluster Tilted Algebras", "abstract": "We are going to determine all the self-injective cluster tilted algebras. All are of finite representation type and special biserial."}
{"category": "Math", "title": "The Ladder Construction of Pruefer Modules", "abstract": "Let R be a ring (associative, with 1). A non-zero module M is said to be a Pruefer module provided there exists a surjective, locally nilpotent endomorphism with kernel of finite length. The aim of this note is construct Pruefer modules starting from a pair of module homomorphisms w,v: U_0 -> U_1, where w is injective and its cokernel is of finite length."}
{"category": "Math", "title": "A Note on Equimultiple Deformations", "abstract": "While the tangent space to an equisingular family of curves can be discribed by the sections of a twisted ideal sheaf, this is no longer true if we only prescribe the multiplicity which a singular point should have. However, it is still possible to compute the dimension of the tangent space with the aid of the equimulitplicity ideal. In this note we consider families L_m={(C,p) | mult_p(C)=m} with C in some linear system |L| on a smooth projective surface S and for a fixed positive integer m, and we compute the dimension of the tangent space to L_m at a point (C,p) depending on whether p is a unitangential singular point of C or not. We deduce that the expected dimension of L_m at (C,p) in any case is just dim|L|+2-m*(m+1)/2. The result is used in the study of triple-point defective surfaces in some joint papers with Luca Chiantini."}
{"category": "Math", "title": "Triple-Point Defective Regular Surfaces", "abstract": "In this paper we study the linear series |L-3p| of hyperplane sections with a triple point p of a surface S embedded via a very ample line bundle L for a general point p. If this linear series does not have the expected dimension we call (S,L) triple-point defective. We show that on a triple-point defective regular surface through a general point every hyperplane section has either a triple component or the surface is rationally ruled and the hyperplane section contains twice a fibre of the ruling."}
{"category": "Math", "title": "Triple-Point Defective Ruled Surfaces", "abstract": "In arXive:0705.3912 we studied triple-point defective very ample linear systems on regular surfaces, and we showed that they can only exist if the surface is ruled. In the present paper we show that we can drop the regularity assumption, and we classify the triple-point defective very ample linear systems on ruled surfaces."}
{"category": "Math", "title": "Modular equations of order $p$ and Theta functions", "abstract": "An algorithm to obtain equations between theta functions with integral characteristics evaluated at $\\tau$ and $p\\tau$ for $g>1$ is presented."}
{"category": "Math", "title": "Transition maps between the 24 bases for a Leonard pair", "abstract": "Let $V$ denote a vector space with finite positive dimension. We consider a pair of linear transformations $A : V \\to V$ and $A^* : V \\to V$ that satisfy (i) and (ii) below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. In an earlier paper we described 24 special bases for $V$. One feature of these bases is that with respect to each of them the matrices that represent $A$ and $A^*$ are (i) diagonal and irreducible tridiagonal or (ii) irreducible tridiagonal and diagonal or (iii) lower bidiagonal and upper bidiagonal or (iv) upper bidiagonal and lower bidiagonal. For each ordered pair of bases among the 24, there exists a unique linear transformation from $V$ to $V$ that sends the first basis to the second basis; we call this the transition map. In this paper we find each transition map explicitly as a polynomial in $A,A^*$."}
{"category": "Math", "title": "On the Structure of Some Reduced Amalgamated Free Product C*-Algebras", "abstract": "We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K_0 group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the later result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag-Solitar groups BS(m,n)."}
{"category": "Math", "title": "A convexity theorem for real projective structures", "abstract": "Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes sharing a common facet is convex. We prove that the real projective structure on M is (1) convex if P contains no triangular polytope, and (2) properly convex if, in addition, P contains a polytope whose dual polytope is thick. Triangular polytopes and polytopes with thick duals are defined as analogues of triangles and polygons with at least five edges, respectively."}
{"category": "Math", "title": "Speculations on some characteristic properties of numbers", "abstract": "Translation of the Latin original \"Speculationes circa quasdam insignes proprietates numerorum\" (1784). E564 in the Enestrom index. In this paper Euler talks about Farey sequences and proves some results about the phi function, the number of positive integers less than and relatively prime to an integer. Euler uses the notation pi instead of phi."}
{"category": "Math", "title": "Row Ideals and Fibers of Morphisms", "abstract": "We study the fibers of a projective morphism and some related algebraic problems. We characterize the analytic spread of a homogeneous ideal through properties of its syzygy matrix. Powers of linearly presented ideals need not be linearly presented, but we identify a weaker linearity property that is preserved under taking powers."}
{"category": "Math", "title": "p^k-torsion of genus two curves over F_{p^m}", "abstract": "We determine the isogeny classes of abelian surfaces over F_q whose group of F_q-rational points has order divisible by q^2. We also solve the same problem for Jacobians of genus-2 curves."}
{"category": "Math", "title": "Generalized CRF-structures", "abstract": "A generalized F-structure is a complex, isotropic subbundle $E$ of $T_cM\\oplus T^*_cM$ ($T_cM=TM\\otimes_{\\mathds{R}}\\mathds{C}$ and the metric is defined by pairing) such that $E\\cap\\bar E^{\\perp}=0$. If $E$ is also closed by the Courant bracket, $E$ is a generalized CRF-structure. We show that a generalized F-structure is equivalent with a skew-symmetric endomorphism $\\Phi$ of $TM\\oplus T^*M$ that satisfies the condition $\\Phi^3+\\Phi=0$ and we express the CRF-condition by means of the Courant-Nijenhuis torsion of $\\Phi$. The structures that we consider are generalizations of the F-structures defined by Yano and of the CR (Cauchy-Riemann) structures. We construct generalized CRF-structures from: a classical F-structure, a pair $(\\mathcal{V},\\sigma)$ where $\\mathcal{V}$ is an integrable subbundle of $TM$ and $\\sigma$ is a 2-form on $M$, a generalized, normal, almost contact structure of codimension $h$. We show that a generalized complex structure on a manifold $\\tilde M$ induces generalized CRF-structures into some submanifolds $M\\subseteq\\tilde M$. Finally, we consider compatible, generalized, Riemannian metrics and we define generalized CRFK-structures that extend the generalized K\\\"ahler structures and are equivalent with quadruples $(\\gamma,F_+,F_-,\\psi)$, where $(\\gamma,F_\\pm)$ are classical, metric CRF-structures, $\\psi$ is a 2-form and some conditions expressible in terms of the exterior differential $d\\psi$ and the $\\gamma$-Levi-Civita covariant derivative $\\nabla F_\\pm$ hold. If $d\\psi=0$, the conditions reduce to the existence of two partially K\\\"ahler reductions of the metric $\\gamma$. The paper ends by an Appendix where we define and characterize generalized Sasakian structures."}
{"category": "Math", "title": "On the principal ideal theorem in arithmetic topology", "abstract": "In this paper we state and prove the analogous of the principal ideal theorem of algebraic number theory for the case of 3-manifolds from the point of view of arithmetic topology."}
{"category": "Math", "title": "Symmetric Crystals and LLTA Type Conjectures for the Affine Hecke Algebras of Type B", "abstract": "In the previous paper \"Symmetric Crystals and Affine Hecke Algebras of Type B\", we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for $\\gl_\\infty$. In the first half of this paper (sections 2 and 3), we give a survey of the LLTA type theorem of the affine Hecke algebra of type $A$. In the latter half (sections 4, 5 and 6), we review the construction of the symmetric crystals and the LLTA type conjectures for the affine Hecke algebra of type $B$."}
{"category": "Math", "title": "Degeneration of A-infinity modules", "abstract": "In this paper we use A-infinity modules to study the derived category of a finite dimensional algebra over an algebraically closed field. We study varieties parameterising A-infinity modules. These varieties carry an action of an algebraic group such that orbits correspond to quasi-isomorphism classes of complexes in the derived category. We describe orbit closures in these varieties, generalising a result of Zwara and Riedtmann for modules."}
{"category": "Math", "title": "On the Weinstein conjecture in higher dimensions", "abstract": "The existence of a \"Plastikstufe\" for a contact structure implies the Weinstein conjecture for all supporting contact forms."}
{"category": "Math", "title": "The Cones associated to some Transversal Polymatroids", "abstract": "In this paper we describe the facets cone associated to transversal polymatroid presented by $\\mathcal{A} = \\{\\{1,2\\},\\{2,3\\},...,\\{n-1,n\\},\\{n,1\\}\\}.$ Using the Danilov-Stanley theorem to characterize the canonicale module, we deduce that the base ring associated to this polymatroid is Gorenstein ring. Also, starting from this polymatroid we describe the transversal polymatroids with Gorenstein base ring in dimension 3 and with the help $\\it Normaliz$ in dimension 4."}
{"category": "Math", "title": "Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem with a Mixed Nonhomogeneous Condition", "abstract": "We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\\eta u(1,t)=g(t), u(x,0)=\\^u_{0}(x), u_{t}(x,0)={\\^u}_{1}(x), where \\eta \\geq 0, q\\geq 2 are given constants {\\^u}_{0}, {\\^u}_{1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under more restrictive conditions it is proved that the solution u(t) and its derivative u_{x}(t) decay exponentially to 0 as t tends to infinity."}
{"category": "Math", "title": "New Examples of Biharmonic Submanifolds in $CP^n$ and $S^{2n+1}$", "abstract": "We construct biharmonic real hypersurfaces and Lagrangian submanifolds of Clifford torus type in $CP^n$ via the Hopf fibration; and get new examples of biharmonic submanifolds in $S^{2n+1}$ as byproducts ."}
{"category": "Math", "title": "Classification of manifolds with weakly 1/4-pinched curvatures", "abstract": "We show that a compact Riemannian manifold with weakly 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form."}
{"category": "Math", "title": "Knot concordance and Blanchfield duality", "abstract": "We introduce a new technique for showing classical knots and links are not slice. As one application we resolve a long-standing question as to whether certain natural families of knots contain topologically slice knots. We also present a simpler proof of the result of Cochran-Teichner that the successive quotients of the integral terms of the Cochran-Orr-Teichner filtration of the knot concordance group have rank 1. For links we have similar results. We show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. Our main examples are actually boundary links but cannot be detected in the algebraic boundary link concordance group, nor by any $\\rho$ invariants associated to solvable representations into finite unitary groups."}
{"category": "Math", "title": "Identities by Generalized $L-$Summing Method", "abstract": "In this paper, we introduce 3-dimensional $L-$summing method, which is a rearrangement of the summation $\\sum A_{abc}$ with $1\\leq a,b,c\\leq n$. Applying this method on some special arrays, we obtain some identities on the Riemann zeta function and digamma function. Also, we give a Maple program for this method to obtain identities with input various arrays and out put identities concerning some elementary functions and hypergeometric functions. Finally, we introduce a further generalization of $L-$summing method in higher dimension spaces."}
{"category": "Math", "title": "About Brezis-Merle Problem with Lipschitz condition", "abstract": "We give blow-up analysis for a Brezis-Merle's problem on the boundary. Also we give a proof of a compactness result with Lipschitz condition and weaker assumption on the regularity of the domain (smooth domain or $ C^{2,\\alpha} $ domain)."}
{"category": "Math", "title": "Mathematical Model for the Evaporation of a Liquid Fuel Droplet, Subject to Nonlinear Constraints", "abstract": "We study the mathematical evolution of a liquid fuel droplet inside a vessel. In particular, we analyze the evolution of the droplet radius on a finite time interval. The model problem involves an hyperbolic system coupled with the pressure and velocity of the surrounding gas. Existence of bounded solutions for the mass fraction of the liquid, submitted to nonlinear constraints, is shown. Numerical simulations are given, in agreement with known physical experiments."}
{"category": "Math", "title": "Linear dilatation structures and inverse semigroups", "abstract": "Here we prove that for dilatation structures linearity (see arXiv:0705.1440v1) is equivalent to a statement about the inverse semigroup generated by the family of dilatations of the space. The result is new for Carnot groups and the proof seems to be new even for vector spaces."}
{"category": "Math", "title": "Strict Partitions of Maximal Projective Degree", "abstract": "The projective degrees of strict partitions of n were computed for all n < 101 and the partitions with maximal projective degree were found for each n. It was observed that maximizing partitions for successive values of n \"lie close to each other\" in a certain sense. Conjecturing that this holds for larger values of n, the partitions of maximal degree were computed for all n < 221. The results are consistent with a recent conjecture on the limiting shape of the strict partition of maximal projective degree."}
{"category": "Math", "title": "p-adic logarithms for polynomial dynamics", "abstract": "We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of the affine space A^g over a p-adic field, endowed with polynomial actions on each coordinate of A^g. We use analytic methods similar to the ones employed by Skolem, Chabauty, and Coleman for studying diophantine equations."}
{"category": "Math", "title": "The K\\\"ahler-Ricci flow and the $\\bar\\partial$ operator on vector fields", "abstract": "The limiting behavior of the normalized K\\\"ahler-Ricci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi K-energy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi K-energy is bounded from below and if the lowest positive eigenvalue of the $\\bar\\partial^\\dagger \\bar\\partial$ operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in $C^\\infty$ to a K\\\"ahler-Einstein metric."}
{"category": "Math", "title": "Distortion in Groups of Circle and Surface Diffeomorphisms", "abstract": "In these lectures we consider how algebraic properties of discrete subgroups of Lie groups restrict the possible actions of those groups on surfaces. The results show a strong parallel between the possible actions of such a group on the circle $S^1$ and the measure preserving actions on surfaces. Our aim is the study of the (non)-existence of actions of lattices in a large class of non-compact Lie groups on surfaces. A definitive analysis of the analogous question for actions on $S^1$ was carried out by \\'E. Ghys. Our approach is topological and insofar as possible we try to isolate properties of a group which provide the tools necessary for our analysis. The two key properties we consider are almost simplicity and the existence of a distortion element. Both will be defined and described in the lectures. Our techniques are almost all from low dimensional dynamics. But we are interested in how algebraic properties of a group -- commutativity, nilpotence, etc. affect the possible kinds of dynamics which can occur. For most of the results we will consider groups of diffeomorphisms which preserve a Borel probability measure."}
{"category": "Math", "title": "On the Spectrum of the Dirichlet Laplacian in a Narrow Strip", "abstract": "We derive a two-terms asymptotics for eigenvalues of the Dirichlet Laplacian in a narrow strip of variable width. The asymptotics is taken with respect to a small paprameter that characterizes the width of the strip."}
{"category": "Math", "title": "C*- Algebras and Thermodynamic Formalism", "abstract": "We present a detailed exposition (for a Dynamical System audience) of the content of the paper: R. Exel and A. Lopes, $C^*$ Algebras, approximately proper equivalence relations and Thermodynamic Formalism, {\\it Erg. Theo. and Dyn. Syst.}, Vol 24, pp 1051-1082 (2004). We show only the uniqueness of the \\beta-KMS (in a certain C*-Algebra obtained from the operators acting in $L^2$ of a Gibbs invariant probability $\\mu$) and its relation with the eigen-probability $\\nu_\\beta$ for the dual of a certain Ruele operator. We consider an example for a case of Hofbauer type where there exist a Phase transition for the Gibbs state. There is no Phase transition for the KMS state."}
{"category": "Math", "title": "Semidefinite Representation of Convex Sets", "abstract": "Let $S =\\{x\\in \\re^n: g_1(x)\\geq 0, ..., g_m(x)\\geq 0\\}$ be a semialgebraic set defined by multivariate polynomials $g_i(x)$. Assume $S$ is convex, compact and has nonempty interior. Let $S_i =\\{x\\in \\re^n: g_i(x)\\geq 0\\}$, and $\\bdS$ (resp. $\\bdS_i$) be the boundary of $S$ (resp. $S_i$). This paper discusses whether $S$ can be represented as the projection of some LMI representable set. Such $S$ is called semidefinite representable or SDP representable. The contributions of this paper: {\\bf (i)} Assume $g_i(x)$ are all concave on $S$. If the positive definite Lagrange Hessian (PDLH) condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function $\\ell^Tx$ on $S$ is positive definite at the minimizer, then $S$ is SDP representable. {\\bf (ii)} If each $g_i(x)$ is either sos-concave ($-\\nabla^2g_i(x)=W(x)^TW(x)$ for some possibly nonsquare matrix polynomial $W(x)$) or strictly quasi-concave on $S$, then $S$ is SDP representable. {\\bf (iii)} If each $S_i$ is either sos-convex or poscurv-convex ($S_i$ is compact convex, whose boundary has positive curvature and is nonsingular, i.e. $\\nabla g_i(x) \\not = 0$ on $\\bdS_i \\cap S$), then $S$ is SDP representable. This also holds for $S_i$ for which $\\bdS_i \\cap S$ extends smoothly to the boundary of a poscurv-convex set containing $S$. {\\bf (iv)} We give the complexity of Schm\\\"{u}dgen and Putinar's matrix Positivstellensatz, which are critical to the proofs of (i)-(iii)."}
{"category": "Math", "title": "Solving ill-conditioned linear algebraic systems by the dynamical systems method (DSM)", "abstract": "An iterative scheme for the Dynamical Systems Method (DSM) is given such that one does not have to solve the Cauchy problem occuring in the application of the DSM for solving ill-conditioned linear algebraic systems. The novelty of the algorithm is that the algorithm does not have to find the regularization parameter $a$ by solving a nonlinear equation. Numerical experiments show that DSM competes favorably with the Variational Regularization."}
{"category": "Math", "title": "On canonical bases and internality criteria", "abstract": "A criterion is given for a type in a finite rank stable theory to be (almost) internal to a given nonmodular minimal type. The motivation comes from results of Campana which give criteria for compact complex analytic spaces to be algebraic (namely Moishezon), in terms of the existence of \"generating\" families of algebraic subvarieties. A model-theoretic anologue/generalisation of Campana's results is given under the hypothesis that the theory has the \"canonical base property\" (CBP), a property that is conjectured to hold in all stable finite rank theories and which states that the type of the canonical base over a realisation is almost internal to the minimal types of the theory."}
{"category": "Math", "title": "Aperiodic substitutional systems and their Bratteli diagrams", "abstract": "In the paper we study aperiodic substitutional dynamical systems arisen from non-primitive substitutions. We prove that the Vershik homeomorphism $\\phi$ of a stationary ordered Bratteli diagram is homeomorphic to an aperiodic substitutional system if and only if no restriction of $\\phi$ to a minimal component is homeomorphic to an odometer. We also show that every aperiodic substitutional system generated by a substitution with nesting property is homeomorphic to the Vershik map of a stationary ordered Bratteli diagram. It is proved that every aperiodic substitutional system is recognizable. The classes of $m$-primitive substitutions and associated to them derivative substitutions are studied. We discuss also the notion of expansiveness for Cantor dynamical systems of finite rank."}
{"category": "Math", "title": "Examples of Free Actions on Products of Spheres", "abstract": "We construct a non-abelian extension $\\Gamma$ of $S^1$ by $\\cy 3 \\times \\cy 3$, and prove that $\\Gamma$ acts freely and smoothly on $S^{5} \\times S^{5}$. This gives new actions on $S^{5} \\times S^{5}$ for an infinite family $\\cP$ of finite 3-groups. We also show that any finite odd order subgroup of the exceptional Lie group $G_2$ admits a free smooth action on $S^{11}\\times S^{11}$. This gives new actions on $S^{11}\\times S^{11}$ for an infinite family $\\cE $ of finite groups. We explain the significance of these families $\\cP $, $\\cE $ for the general existence problem, and correct some mistakes in the literature."}
{"category": "Math", "title": "Remarks on the existence of bilaterally symmetric extremal K\\\"ahler metrics on $\\mathbb{CP}^2\\sharp 2\\bar{\\mathbb{CP}^2}$", "abstract": "In this short note we show that the existence of bilaterally symmetric extremal K\\\"ahler metrics on $\\mathbb{CP}^2\\sharp 2\\bar{\\mathbb{CP}^2}$."}
{"category": "Math", "title": "Bethe algebra and algebra of functions on the space of differential operators of order two with polynomial solutions", "abstract": "We show that the following two algebras are isomorphic. The first is the algebra $A_P$ of functions on the scheme of monic linear second-order differential operators on $\\C$ with prescribed regular singular points at $z_1,..., z_n, \\infty$, prescribed exponents $\\La^{(1)}, ..., \\La^{(n)}, \\La^{(\\infty)}$ at the singular points, and having the kernel consisting of polynomials only. The second is the Bethe algebra of commuting linear operators, acting on the vector space $\\Sing L_{\\La^{(1)}} \\otimes ... \\otimes L_{\\La^{(n)}}[\\La^{(\\infty)}]$ of singular vectors of weight $\\La^{(\\infty)}$ in the tensor product of finite dimensional polynomial $gl_2$-modules with highest weights $\\La^{(1)},..., \\La^{(n)}$."}
{"category": "Math", "title": "Almost sure functional central limit theorem for ballistic random walk in random environment", "abstract": "We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central limit theorem, under almost every environment for the diffusively scaled centered walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively."}
{"category": "Math", "title": "Finding Minimal Permutation Representations of Finite Groups", "abstract": "A minimal permutation representation of a finite group G is a faithful G-set with the smallest possible size. We study the structure of such representations and show that for certain groups they may be obtained by a greedy construction. In these situations (except when central involutions intervene) all minimal permutation representations have the same set of orbit sizes. Using the same ideas we also show that if the size d(G) of a minimal faithful G-set is at least c|G| for some c>0 then d(G) = |G|/m + O(1) for an integer m, with the implied constant depending on c."}
{"category": "Math", "title": "The Call of Mathematics", "abstract": "A few remarks on how mathematics quests for freedom."}
{"category": "Math", "title": "Interaction of Order and Convexity", "abstract": "This is an overview of merging the techniques of Riesz space theory and convex geometry."}
{"category": "Math", "title": "EF equivalent not isomorphic models", "abstract": "We construct non-isomorphic models M, N, e.g. of cardinality aleph_1 such that in the Ehrenfeucht-Fraisse game of length zeta < omega_1 the isomorphism player wins"}
{"category": "Math", "title": "The Automorphism Group of Toric Deligne-Mumford Stacks", "abstract": "We prove that the automorphism group of a toric Deligne-Mumford stack is isomorphic to the $2$-group associated to the stacky fan."}
{"category": "Math", "title": "On long increasing chains modulo flat ideals", "abstract": "We prove that e.g. there is no omega_4-sequence in (omega_3)^{omega_3} increasing modulo the ideal of countable sets."}
{"category": "Math", "title": "No limit model in inaccessible", "abstract": "Our aim is to improve the negative results i.e. non-existence of limit models, and the failure of the generic pair property from math.LO/0609636 to inaccessible lambda as promised there. The motivation is that in [Sh:F756] the positive results are for lambda measurable hence inaccessible, whereas in math.LO/0609636 in the negative results obtained only on non-strong limit cardinals."}
{"category": "Math", "title": "Noetherian ring with free additive groups", "abstract": "There are Noetherian rings (in fact domains) with a free additive group, in every infinite cardinality. (This is an expanded version of [SgSh:217] which appears in the Abstracts of the American Mathematical Society 7 (1986): 369.)"}
{"category": "Math", "title": "Abstract elementary classes near aleph_1", "abstract": "We prove in ZFC, no psi in L_{omega_1,omega}[Q] have unique model of uncountable cardinality, this confirms theBaldwin conjecture. But we analyze this in more general terms. We introduce and investigate a.e.c. and also versions of limit models, and prove some basic properties like representation by PC class, for any a.e.c. For PC_{aleph_0}-representable a.e.c. we investigate the conclusion of having not too many non-isomorphic models in aleph_1 and aleph_2, but have to assume 2^{aleph_0}<2^{aleph_1} and even 2^{aleph_1}<2^{aleph_2}."}
{"category": "Math", "title": "Specht modules and semisimplicity criteria for Brauer and Birman--Murakami--Wenzl Algebras", "abstract": "A construction of bases for cell modules of the Birman--Murakami--Wenzl (or B--M--W) algebra $B_n(q,r)$ by lifting bases for cell modules of $B_{n-1}(q,r)$ is given. By iterating this procedure, we produce cellular bases for B--M--W algebras on which a large abelian subalgebra, generated by elements which generalise the Jucys--Murphy elements from the representation theory of the Iwahori--Hecke algebra of the symmetric group, acts triangularly. The triangular action of this abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters $q$ and $r$, for B--M--W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori--Hecke algebra of the symmetric group."}
{"category": "Math", "title": "Picone identities for half-linear differential equations of fourth order", "abstract": "Picone-type identities are established for half-linear ODEs of fourth order (one-dimensional p-biLaplacian). It is shown that in the linear case they reduce to the known identities for fourth order linear ODEs. Picone-type identity known for two half-linear second-order equations is also generalised to set of equations greater than two."}
{"category": "Math", "title": "About construction of orthogonal wavelets with compact support and with scaling coefficient N", "abstract": "In this paper a simple method of construction of scaling function $\\phi (x)$ and orthogonal wavelets with the compact support for any natural coefficient of scaling $N\\ge 2$ is given. Examples of construction of wavelets for coefficients of scaling N=2 and N=3 are produced."}
{"category": "Math", "title": "A preferential attachment model with random initial degrees", "abstract": "In this paper, a random graph process ${G(t)}_{t\\geq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{t\\geq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+\\delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $\\tau=\\min\\{\\tau_{W}, \\tau_{P}\\}$, where $\\tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{t\\geq 1}$ and $\\tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed."}
{"category": "Math", "title": "Diameters in preferential attachment models", "abstract": "In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent \\tau>2. We prove that the diameter of the PA-model is bounded above by a constant times \\log{t}, where t is the size of the graph. When the power-law exponent \\tau exceeds 3, then we prove that \\log{t} is the right order, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for \\tau>3, distances are of the order \\log{t}. For \\tau\\in (2,3), we improve the upper bound to a constant times \\log\\log{t}, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order \\log\\log{t}. These bounds partially prove predictions by physicists that the typical distance in PA-graphs are similar to the ones in other scale-free random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order \\log\\log{t} when \\tau\\in (2,3), and of order \\log{t} when \\tau>3."}
{"category": "Math", "title": "On $\\pi - \\pi$ theorem for manifold pairs with boundaries", "abstract": "Surgery obstruction of a normal map to a simple Poincare pair $(X,Y)$ lies in the relative surgery obstruction group $L_*(\\pi_1(Y)\\to\\pi_1(X))$. A well known result of Wall, the so called $\\pi$-$\\pi$ theorem, states that in higher dimensions a normal map of a manifold with boundary to a simple Poincare pair with $\\pi_1(X)\\cong\\pi_1(Y)$ is normally bordant to a simple homotopy equivalence of pairs. In order to study normal maps to a manifold with a submanifold, Wall introduced surgery obstruction group for manifold pairs $LP_*$ and splitting obstruction groups $LS_*$. In the present paper we formulate and prove for manifold pairs with boundaries the results which are similar to the $\\pi$-$\\pi$ theorem. We give direct geometric proofs, which are based on the original statements of Wall's results and apply obtained results to investigate surgery on filtered manifolds."}
{"category": "Math", "title": "Riesz bases of root vectors of indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions. II", "abstract": "We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions affinely dependent on the eigenparameter. We give sufficient conditions under which the root vectors of this Sturm-Liouville problem can be selected to form a Riesz basis of a corresponding weighted Hilbert space."}
{"category": "Math", "title": "On the fundamental group of $\\mathbb R^3$ modulo the Case-Chamberlin continuum", "abstract": "It has been known for a long time that the fundamental group of the quotient of $\\RR ^3$ by the Case-Chamberlin continuum is nontrivial. In the present paper we prove that this group is in fact, uncountable."}
{"category": "Math", "title": "Riesz Bases of Root Vectors of Indefinite Sturm-Liouville problems with eigenparameter dependent boundary conditions, I", "abstract": "We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigenparameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space."}
{"category": "Math", "title": "Classification of framed links in 3-manifolds", "abstract": "We present a short proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in details: {\\bf Theorem.} Let $M$ be a connected oriented closed smooth 3-manifold. Let $L_1(M)$ be the set of framed links in $M$ up to a framed cobordism. Let $\\deg:L_1(M)\\to H_1(M;\\Z)$ be the map taking a framed link to its homology class. Then for each $\\alpha\\in H_1(M;\\Z)$ there is a 1-1 correspondence between the set $\\deg\\nolimits^{-1}\\alpha$ and the group $\\Bbb Z_{2d(\\alpha)}$, where $d(\\alpha)$ is the divisibility of the projection of $\\alpha$ to the free part of $H_1(M;\\Bbb Z)$."}
{"category": "Math", "title": "Quantum Lie algebras via modified Reflection Equation Algebra", "abstract": "We discuss the consistency of the axioms which the definition of quantum Lie algebras is usually based on."}
{"category": "Math", "title": "An Elliptic Type Gradient Estimate For the Schr\\\"{o}dinger Equation", "abstract": "In this paper, the author discusses the elliptic type gradient estimate for the solution of the time-dependent Schr\\\"{o}dinger equations on noncompact manifolds. As its application, the dimension-free Harnack inequality and the Liouville type theorem for the Schr\\\"{o}dinger equation are proved."}
{"category": "Math", "title": "Quantized anti de Sitter spaces and non-formal deformation quantizations of symplectic symmetric spaces", "abstract": "We realize quantized anti de Sitter space black holes, building Connes spectral triples, similar to those used for quantized spheres but based on Universal Deformation Quantization Formulas (UDF) obtained from an oscillatory integral kernel on an appropriate symplectic symmetric space. More precisely we first obtain a UDF for Lie subgroups acting on a symplectic symmetric space M in a locally simply transitive manner. Then, observing that a curvature contraction canonically relates anti de Sitter geometry to the geometry of symplectic symmetric spaces, we use that UDF to define what we call Dirac-isospectral noncommutative deformations of the spectral triples of locally anti de Sitter black holes. The study is motivated by physical and cosmological considerations."}
{"category": "Math", "title": "Closed geodesics on positively curved Finsler spheres", "abstract": "In this paper, we prove that for every Finsler $n$-sphere $(S^n, F)$ for $n\\ge 3$ with reversibility $\\lambda$ and flag curvature $K$ satisfying $(\\frac{\\lambda}{\\lambda+1})^2<K\\le 1$, either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincar\\'e map has at least one eigenvalue which is of the form $\\exp(\\pi i \\mu)$ with an irrational $\\mu$. Furthermore, there always exist three prime closed geodesics on any $(S^3, F)$ satisfying the above pinching condition."}
{"category": "Math", "title": "Rational BV-algebra in String Topology", "abstract": "Let $M$ be a 1-connected closed manifold and $LM$ be the space of free loops on $M$. In \\cite{C-S} M. Chas and D. Sullivan defined a structure of BV-algebra on the singular homology of $LM$, $H_\\ast(LM; \\bk)$. When the field of coefficients is of characteristic zero, we prove that there exists a BV-algebra structure on $\\hH^\\ast(C^\\ast (M); C^\\ast (M))$ which carries the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between $\\hH^\\ast (C^\\ast (M); C^\\ast (M)) $ and the shifted $ H_{\\ast+m} (LM; {\\bk})$. We also prove that the Chas-Sullivan product and the BV-operator behave well with the Hodge decomposition of $H_\\ast (LM) $."}
{"category": "Math", "title": "Spectrum and multiplier ideals of arbitrary subvarieties", "abstract": "We introduce a spectrum for arbitrary varieties. This generalizes the definition by Steenbrink for hypersurfaces. In the isolated complete intersection singularity case, it coincides with the one given by Ebeling and Steenbrink except for the coefficients of integral exponents. We show a relation to the graded pieces of the multiplier ideals by using a relation to the filtration $V$ of Kashiwara and Malgrange. This implies a partial generalization of a theorem of Budur in the hypersurface case. The point is to consider the direct sum of the graded pieces of the multiplier ideals as a module over the algebra defining the normal cone of the subvariety. We also give a combinatorial description in the case of monomial ideals."}
{"category": "Math", "title": "A mean value theorem for systems of integrals", "abstract": "More than a century ago, G. Kowalewski stated that for each n continuous functions on a compact interval [a,b], there exists an n-point quadrature rule (with respect to Lebesgue measure on [a,b]), which is exact for given functions. Here we generalize this result to continuous functions with an arbitrary positive and finite measure on an arbitrary interval. The proof relies on a version of Caratheodory's convex hull theorem for a continuous curve, that we also prove in the paper. As applications, we give a representation of the covariance for two continuous functions of a random variable, and a most general version of Gruess' inequality."}
{"category": "Math", "title": "On residual properties of pure braid groups of closed surfaces", "abstract": "We prove that pure braid groups of closed surface are almost-direct products of residually torsion free nilpotent groups and hence residually torsion free nilpotent. As a Corollary, we prove also that braid groups on 2 strands of closed surfaces are residually nilpotent."}
{"category": "Math", "title": "Dynamical Diophantine Approximation", "abstract": "Let $\\mu$ be a Gibbs measure of the doubling map $T$ of the circle. For a $\\mu$-generic point $x$ and a given sequence $\\{r_n\\} \\subset \\R^+$, consider the intervals $(T^nx - r_n \\pmod 1, T^nx + r_n \\pmod 1)$. In analogy to the classical Dvoretzky covering of the circle we study the covering properties of this sequence of intervals. This study is closely related to the local entropy function of the Gibbs measure and to hitting times for moving targets. A mass transference principle is obtained for Gibbs measures which are multifractal. Such a principle was shown by Beresnevich and Velani \\cite{BV} only for monofractal measures. In the symbolic language we completely describe the combinatorial structure of a typical relatively short sequence, in particular we can describe the occurrence of ''atypical'' relatively long words. Our results have a direct and deep number-theoretical interpretation via inhomogeneous diadic diophantine approximation by numbers belonging to a given (diadic) diophantine class."}
{"category": "Math", "title": "Ratliff-Rush Closure of Ideals in Integral Domains", "abstract": "This paper studies the Ratliff-Rush closure of ideals in integral domains. By definition, the Ratliff-Rush closure of an ideal $I$ of a domain $R$ is the ideal given by $\\tilde{I}:=\\bigcup(I^{n+1}:_{R}I^{n})$ and an ideal $I$ is said to be a Ratliff-Rush ideal if $\\tilde{I}=I$. We completely characterize integrally closed domains in which every ideal is a Ratliff-Rush ideal and we give a complete description of the Ratliff-Rush closure of an ideal in a valuation domain."}
{"category": "Math", "title": "Geometric Weil representation: local field case", "abstract": "Let k be an algebraically closed field of characteristic >2, F=k((t)) and Mp(F) denote the metaplectic extension of Sp_{2d}(F). In this paper we propose a geometric analog of the Weil representation of Mp(F). This is a category of certain perverse sheaves on some stack, on which Mp(F) acts by functors. This construction will be used in math.RT/0701170 (and subsequent publications) for a proof of the geometric Langlands functoriality for some dual reductive pairs."}
{"category": "Math", "title": "An Isoperimetric Function for Bestvina-Brady Groups", "abstract": "Given a right-angled Artin group A, the associated Bestvina-Brady group is defined to be the kernel of the homomorphism A \\to \\mathbb{Z} that maps each generator in the standard presentation of A to a fixed generator of \\mathbb{Z}. We prove that the Dehn function of an arbitrary finitely presented Bestvina-Brady group is bounded above by n^4. This is the best possible universal upper bound."}
{"category": "Math", "title": "Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain", "abstract": "In this article we study a controllability problem for a parabolic and a hyperbolic partial differential equations in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the solution into an open subdomain and at a given time, when the right hand side source term is known. The mapping that associates this trace to the shape of the domain is nonlinear. We show (i) an approximate controllability property for the linearized parabolic problem and (ii) an exact local controllability property for the linearized and the nonlinear equations in the hyperbolic case. We then address the same questions in the context of a finite difference spatial semi-discretization in both the parabolic and hyperbolic problems. In this discretized case again we prove a local controllability result for the parabolic problem, and an exact controllability for the hyperbolic case, applying a local surjectivity theorem together with a unique continuation property of the underlying adjoint discrete system."}
{"category": "Math", "title": "Efficient independent component analysis", "abstract": "Independent component analysis (ICA) has been widely used for blind source separation in many fields such as brain imaging analysis, signal processing and telecommunication. Many statistical techniques based on M-estimates have been proposed for estimating the mixing matrix. Recently, several nonparametric methods have been developed, but in-depth analysis of asymptotic efficiency has not been available. We analyze ICA using semiparametric theories and propose a straightforward estimate based on the efficient score function by using B-spline approximations. The estimate is asymptotically efficient under moderate conditions and exhibits better performance than standard ICA methods in a variety of simulations."}
{"category": "Math", "title": "A survey of hypertoric geometry and topology", "abstract": "Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkahler geometry. The aim of this survey is to give clear definitions and statements of known results, serving both as a reference and as a point of entry to this beautiful subject."}
{"category": "Math", "title": "Stability of viscous shocks in isentropic gas dynamics", "abstract": "In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier--Stokes equations, or $p$-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength. By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there is no unstable spectrum in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number $M\\approx 3000$ for $1 \\le \\gamma \\leq 3$. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant $\\gamma$. We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected."}
{"category": "Math", "title": "Global well-posedness of the KP-I initial-value problem in the energy space", "abstract": "We prove that the KP-I initial value problem is globally well-posed in the natural energy space of the equation."}
{"category": "Math", "title": "Stability of isentropic viscous shock profiles in the high-Mach number limit", "abstract": "By a combination of asymptotic ODE estimates and numerical Evans function calculations, we establish stability of viscous shock solutions of the isentropic compressible Navier--Stokes equations with $\\gamma$-law pressure (i) in the limit as Mach number $M$ goes to infinity, for any $\\gamma\\ge 1$ (proved analytically), and (ii) for $M\\ge 2,500$, $\\gamma\\in [1,2.5]$ (demonstrated numerically). This builds on and completes earlier studies by Matsumura--Nishihara and Barker--Humpherys--Rudd--Zumbrun establishing stability for low and intermediate Mach numbers, respectively, indicating unconditional stability, independent of shock amplitude, of viscous shock waves for $\\gamma$-law gas dynamics in the range $\\gamma \\in [1,2.5]$. Other $\\gamma$-values may be treated similarly, but have not been checked numerically. The main idea is to establish convergence of the Evans function in the high-Mach number limit to that of a pressureless, or ``infinitely compressible'', gas with additional upstream boundary condition determined by a boundary-layer analysis. Recall that low-Mach number behavior is incompressible."}
{"category": "Math", "title": "The ODE method for some self-interacting diffusions on non-compact spaces", "abstract": "Self-interacting diffusions are solutions to SDEs with a drift term depending on the process and its normalized occupation measure $\\mu_t$ (via an interaction potential and a confinement potential). We establish a relation between the asymptotic behavior of $\\mu_t$ and the asymptotic behavior of a deterministic dynamical flow (defined on the space of the Borel probability measures). We extend previous results on $\\mathbb{R}^d$ or more generally a smooth complete connected Riemannian manifold without boundary. We will also give some sufficient conditions for the convergence of $\\mu_t$. Finally, we will illustrate our study with an example on $\\mathbb{R}^2$."}
{"category": "Math", "title": "The equation w(x,y)=u over free groups", "abstract": "Using the theory developed by Olga Kharlampovich, Alexei Miasnikov, and, independently, by Zlil Sela to describe the set of homomorphisms of a f.g. group G into a free group F, we describe the solutions to equations with coefficients from F and unknowns x,y of the form w(x,y) = u, where u lies in F and w(x,y) is a word in {x,y}^\\{pm 1}. We also give an example of a single equation whose solutions cannot be described with only one \"level\" of automorphisms."}
{"category": "Math", "title": "Convergence of the Min-Sum Algorithm for Convex Optimization", "abstract": "We establish that the min-sum message-passing algorithm and its asynchronous variants converge for a large class of unconstrained convex optimization problems."}
{"category": "Math", "title": "Sums and products in finite fields: an integral geometric viewpoint", "abstract": "We prove that if $A \\subset {\\Bbb F}_q$ is such that $$|A|>q^{{1/2}+\\frac{1}{2d}},$$ then $${\\Bbb F}_q^{*} \\subset dA^2=A^2+...+A^2 d \\text{times},$$ where $$A^2=\\{a \\cdot a': a,a' \\in A\\},$$ and where ${\\Bbb F}_q^{*}$ denotes the multiplicative group of the finite field ${\\Bbb F}_q$. In particular, we cover ${\\Bbb F}_q^{*}$ by $A^2+A^2$ if $|A|>q^{{3/4}}$. Furthermore, we prove that if $$|A| \\ge C_{size}^{\\frac{1}{d}}q^{{1/2}+\\frac{1}{2(2d-1)}},$$ then $$|dA^2| \\ge q \\cdot \\frac{C^2_{size}}{C^2_{size}+1}.$$ Thus $dA^2$ contains a positive proportion of the elements of ${\\Bbb F}_q$ under a considerably weaker size assumption.We use the geometry of ${\\Bbb F}_q^d$, averages over hyper-planes and orthogonality properties of character sums. In particular, we see that using operators that are smoothing on $L^2$ in the Euclidean setting leads to non-trivial arithmetic consequences in the context of finite fields."}
{"category": "Math", "title": "Brauer-Siegel theorem for elliptic surfaces", "abstract": "We consider higher-dimensional analogues of the classical Brauer-Siegel theorem focusing on the case of abelian varieties over global function fields. We prove such an analogue in the case of constant families of elliptic curves and abelian varieties."}
{"category": "Math", "title": "Representable posets and their order components", "abstract": "A partially ordered set P is representable if there is a bounded distributive lattice such that its ordered set of prime ideals is order-isomorphic to P. We show that if the order components of a poset P are representable, then so is P. Moreover, we provide an example disproving the converse."}
{"category": "Math", "title": "Entiers al\\'eatoires, ensembles de Sidon, densit\\'e dans le groupe de Bohr et ensembles d'analyticit\\'e", "abstract": "We study properties of a sequence $\\Lambda$ obtained by a randomselection of integers $n$, where $n\\in\\Lambda$ with probability $\\varpi_{n}$, independently of the other choices. We distinguish two cases : if $\\limsup_{n\\to\\infty}n\\varpi_{n}<\\infty$, $\\Lambda$ is a.s. a Sidon set, non-dense in the Bohr group ; if $\\lim_{n\\to\\infty}n\\varpi_{n}=\\infty$, then $\\Lambda$ is a.s. a set of analyticity and is dense in the Bohr group."}
{"category": "Math", "title": "A Small Polyhedral Z-Acyclic 2-Complex in R4", "abstract": "We present a small 4-dimensional polyhedral realization of a 2-dimensional Z-acyclic but non-contractible simplicial complex."}
{"category": "Math", "title": "Higher exponential maps and explicit reciprocity laws I", "abstract": "This paper has been withdrawn, as it is superseded by arXiv:0806.2122 (Bloch-Kato exponential maps for local fields with imperfect residue fields), which is a more recent version of the same paper."}
{"category": "Math", "title": "The higher Hilbert pairing via (phi,G)-modules", "abstract": "We prove the Tate duality for higher dimensional local fields of mixed characteristic (0,p), when p is an odd prime, using the theory of higher fields of norms. Assuming that p is not ramified in the basefield, we then use this construction to define the higher Hilbert pairing. In particular, we show that the Hilbert pairing is non-degenerate, and we re-discover the formulae of Brueckner and Vostokov."}
{"category": "Math", "title": "Order convergence and compactness", "abstract": "Let $(P,\\leq)$ be a partially ordered set and let $\\tau$ be a compact topology on $P$ that is finer than the interval topology. Then $\\tau$ is contained in the order (convergence) topology on $(P,\\tau)$. So any Priestley topology is contained in the order topology."}
{"category": "Math", "title": "Optimal control of Goursat-Volterra systems", "abstract": "We analyze an optimal control problem for systems of integral equations of Volterra type with two independent variables. These systems generalize both, the hyperbolic control problems for systems of Goursat-Darboux type, and the optimal control of ordinary (i.e. with one independent variable) Volterra integral equations. We prove extremal principles akin to Pontryagin's maximum principle."}
{"category": "Math", "title": "Convolutions on compact groups and Fourier algebras of coset spaces", "abstract": "In this note we study two related questions. (1) For a compact group G, what are the ranges of the convolution maps on A(GXG) given for u,v in A(G) by $u X v |-> u*v' ($v'(s)=v(s^{-1})$) and $u X v |-> u*v$? (2) For a locally compact group G and a compact subgroup K, what are the amenability properties of the Fourier algebra of the coset space A(G/K)? The algebra A(G/K) was defined and studied by the first named author. In answering the first question, we obtain for compact groups which do not admit an abelian subgroup of finite index, some new subalgebras of A(G). Using those algebras we can find many instances in which A(G/K) fails the most rudimentary amenability property: operator weak amenability. However, using different techniques, we show that if the connected component of the identity of G is abelian, then A(G/K) always satisfies the stronger property that it is hyper-Tauberian, which is a concept developed by the second named author. We also establish a criterion which characterises operator amenability of A(G/K) for a class of groups which includes the maximally almost periodic groups."}
{"category": "Math", "title": "Amenability constants for semilattice algebras", "abstract": "For any finite unital commutative idempotent semigroup S, a unital semilattice, we show how to compute the amenability constant of its semigroup algebra l^1(S), which is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We show that there is no commutative semilattice with amenability constant between 5 and 9."}
{"category": "Math", "title": "Approximation orders for interpolation by surface splines to rough functions", "abstract": "In this paper we consider the approximation of functions by radial basis function interpolants. There is a plethora of results about the asymptotic behaviour of the error between appropriately smooth functions and their interpolants, as the interpolation points fill out a bounded domain in R^d. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function - the native space. In many cases, the native space contains functions possessing a certain amount of smoothness. We address the question of what can be said about these error estimates when the function being interpolated fails to have the required smoothness. These are the rough functions of the title. We limit our discussion to surface splines, as an exemplar of a wider class of radial basis functions, because we feel our techniques are most easily seen and understood in this setting."}
{"category": "Math", "title": "Remarks on affine complete distributive lattices", "abstract": "We characterise the Priestley spaces corresponding to affine complete bounded distributive lattices. Moreover we prove that the class of affine complete bounded distributive lattices is closed under products and free products. We show that every (not necessarily bounded) distributive lattice can be embedded in an affine complete one and that $\\mathbb{Q} \\cap [0,1]$ is initial in the class of affine complete lattices."}
{"category": "Math", "title": "Splitting families and the Noetherian type of $\\beta\\omega-\\omega$", "abstract": "Extending some results of Malykhin, we prove several independence results about base properties of $\\beta\\omega-\\omega$ and its powers, especially the Noetherian type $Nt(\\beta\\omega-\\omega)$, the least $\\kappa$ for which $\\beta\\omega-\\omega$ has a base that is $\\kappa$-like with respect to containment. For example, $Nt(\\beta\\omega-\\omega)$ is never less than the splitting number, but can consistently be that $\\omega_1$, $2^\\omega$, $(2^\\omega)^+$, or strictly between $\\omega_1$ and $2^\\omega$. $Nt(\\beta\\omega-\\omega)$ is also consistently less than the additivity of the meager ideal. $Nt(\\beta\\omega-\\omega)$ is closely related to the existence of special kinds of splitting families."}
{"category": "Math", "title": "Abstract factorials", "abstract": "A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X of the positive integers we construct a \"factorial set\" with which one may define a multitude of abstract factorials on X. We study the possible equality of consecutive factorials, a dichotomy involving the limit superior of the ratios of consecutive factorials and we provide many examples outlining the applications of the ensuing theory; examples dealing with prime numbers, Fibonacci numbers, and highly composite numbers among other sets of integers. One of our results states that given any abstract factorial the series of reciprocals of its factorials always converges to an irrational number. Thus, for example, for any positive integer k the series of the reciprocals of the k-th powers of the cumulative product of the divisors of the numbers from 1 to n is irrational."}
{"category": "Math", "title": "Error estimates for interpolation of rough data using the scattered shifts of a radial basis function", "abstract": "The error between appropriately smooth functions and their radial basis function interpolants, as the interpolation points fill out a bounded domain in R^d, is a well studied artifact. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function -- the native space. The native space contains functions possessing a certain amount of smoothness. This paper establishes error estimates when the function being interpolated is conspicuously rough."}
{"category": "Math", "title": "Slanted matrices, Banach frames, and sampling", "abstract": "In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted matrices boundedness below of the corresponding operator in $\\ell^p$ for some $p$ implies boundedness below in $\\ell^p$ for all $p$. We use the established resultto enrich our understanding of Banach frames and obtain new results for irregular sampling problems. We also present a version of a non-commutative Wiener's lemma for slanted matrices."}
{"category": "Math", "title": "On the Landau-Siegel Zeros Conjecture", "abstract": "We provide a proof of a variant of the Landau-Siegel Zeros conjecture."}
{"category": "Math", "title": "On Stability of Sampling-Reconstruction Models", "abstract": "A useful sampling-reconstruction model should be stable with respect to different kind of small perturbations, regardless whether they result from jitter, measurement errors, or simply from a small change in the model assumptions. In this paper we prove this result for a large class of sampling models. We define different classes of perturbations and quantify the robustness of a model with respect to them. We also use the theory of localized frames to study the frame algorithm for recovering the original signal from its samples."}
{"category": "Math", "title": "Learning about a Categorical Latent Variable under Prior Near-Ignorance", "abstract": "It is well known that complete prior ignorance is not compatible with learning, at least in a coherent theory of (epistemic) uncertainty. What is less widely known, is that there is a state similar to full ignorance, that Walley calls near-ignorance, that permits learning to take place. In this paper we provide new and substantial evidence that also near-ignorance cannot be really regarded as a way out of the problem of starting statistical inference in conditions of very weak beliefs. The key to this result is focusing on a setting characterized by a variable of interest that is latent. We argue that such a setting is by far the most common case in practice, and we show, for the case of categorical latent variables (and general manifest variables) that there is a sufficient condition that, if satisfied, prevents learning to take place under prior near-ignorance. This condition is shown to be easily satisfied in the most common statistical problems."}
{"category": "Math", "title": "Rings of Algebraic Numbers in Infinite Extensions of $\\Q$ and Elliptic Curves Retaining Their Rank", "abstract": "We show that elliptic curves whose Mordell-Weil groups are finitely generated over some infinite extensions of $\\Q$, can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers."}
{"category": "Math", "title": "Minimum volume cusped hyperbolic three-manifolds", "abstract": "This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic 3-manifold with volume <= 2.848 can be obtained by a Dehn filling on one of 21 cusped hyperbolic 3-manifolds. We also show how this result can be used to construct a complete list of all one-cusped hyperbolic three-manifolds with volume <= 2.848 and all closed hyperbolic three-manifolds with volume <= 0.943. In particular, the Weeks manifold is the unique smallest volume closed orientable hyperbolic 3-manifold."}
{"category": "Math", "title": "Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres", "abstract": "We prove that for every $\\Q$-homological Finsler 3-sphere $(M,F)$ with a bumpy and irreversible metric $F$, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics."}
{"category": "Math", "title": "Multiple closed geodesics on bumpy Finsler $n$-spheres", "abstract": "In this paper we prove that for every bumpy Finsler metric $F$ on every rationally homological $n$-dimensional sphere $S^n$ with $n\\ge 2$, there exist always at least two distinct prime closed geodesics."}
{"category": "Math", "title": "Almost-minimal nonuniform lattices of higher rank", "abstract": "If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be interpreted as a statement about the existence of totally geodesic subspaces of finite-volume, noncompact, locally symmetric spaces of higher rank.) Another formulation of the result states that if G is any isotropic, almost simple algebraic group over Q (the rational numbers), such that the real rank of G is greater than 1, then G contains an isotropic, almost simple Q-subgroup H, such that H is quasisplit, and the real rank of H is greater than 1."}
{"category": "Math", "title": "On local properties of Hochschild cohomology of a C$^*$- algebra", "abstract": "Let $A$ be a C$^*$-algebra, and let $X$ be a Banach $A$-bimodule. B. E. Johnson showed that local derivations from $A$ into $X$ are derivations. We extend this concept of locality to the higher cohomology of a $C^*$-algebra %for $n$-cocycles from $A^{(n)}$ into $X$ and show that, for every $n\\in \\N$, bounded local $n$-cocycles from $A^{(n)}$ into $X$ are $n$-cocycles."}
{"category": "Math", "title": "Coherence without unique normal forms", "abstract": "Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a feature that is inherent to the coherence problem itself. This is demonstrated by the theory of iterated monoidal categories, which model iterated loop spaces and have a coherence theorem but fail to be confluent. We develop a framework for expressing coherence problems in terms of term rewriting systems equipped with a two dimensional congruence. Within this framework we provide general solutions to two related coherence theorems: Determining whether there is a decision procedure for the commutativity of diagrams in the resulting structure and determining sufficient conditions ensuring that ``all diagrams commute''. The resulting coherence theorems rely on neither the termination nor the confluence of the underlying rewriting system. We apply the theory to iterated monoidal categories and obtain a new, conceptual proof of their coherence theorem."}
{"category": "Math", "title": "A Projective C*-Algebra Related to K-Theory", "abstract": "The C*-algebra qC is the smallest of the C*-algebras qA introduced by Cuntz in the context of KK-theory. An important property of qC is the natural isomorphism of K0 of D with classes of homomorphism from qC to matrix algebras over D. Our main result concerns the exponential (boundary) map from K0 of a quotient B to K1 of an ideal I. We show if a K0 element is realized as a homomorphism from qC to B then its boundary is realized as a unitary in the unitization of I. The picture we obtain of the exponential map is based on a projective C*-algebra P that is universal for a set of relations slightly weaker than the relations that define qC. A new, shorter proof of the semiprojectivity of qC is described. Smoothing questions related the relations for qC are addressed."}
{"category": "Math", "title": "Edge number of knots and links", "abstract": "We introduce a new numerical invariant of knots and links made from the partitioned diagrams. It measures the complexity of knots and links."}
{"category": "Math", "title": "An inverse spectral theory for finite CMV matrices", "abstract": "For finite dimensional CMV matrices the classical inverse spectral problems are considered. We solve the inverse problem of reconstructing a CMV matrix by its Weyl's function, the problem of reconstructing the matrix by two spectra of CMV matrices with different \"boundary conditions\", and the problem of reconstructing the CMV matrix by its spectrum and the spectrum of the CMV matrix obtained from it by truncation."}
{"category": "Math", "title": "Extending the range of error estimates for radial approximation in Euclidean space and on spheres", "abstract": "We adapt Schaback's error doubling trick [R. Schaback. Improved error bounds for scattered data interpolation by radial basis functions. Math. Comp., 68(225):201--216, 1999.] to give error estimates for radial interpolation of functions with smoothness lying (in some sense) between that of the usual native space and the subspace with double the smoothness. We do this for both bounded subsets of R^d and spheres. As a step on the way to our ultimate goal we also show convergence of pseudoderivatives of the interpolation error."}
{"category": "Math", "title": "Enhancing SPH using moving least-squares and radial basis functions", "abstract": "In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using: moving least-squares approximation (MLS); radial basis functions (RBF). Using MLS approximation is appealing because polynomial consistency of the particle approximation can be enforced. RBFs further appeal as they allow one to dispense with the smoothing-length -- the parameter in the SPH method which governs the number of particles within the support of the shape function. Currently, only ad hoc methods for choosing the smoothing-length exist. We ensure that any enhancement retains the conservative and meshfree nature of SPH. In doing so, we derive a new set of variationally-consistent hydrodynamic equations. Finally, we demonstrate the performance of the new equations on the Sod shock tube problem."}
{"category": "Math", "title": "On simplicial toric varieties of codimension 2", "abstract": "We describe classes of toric varieties of codimension 2 which are either minimally defined by 3 binomial equations over any algebraically closed field, or are set-theoretic complete intersections in exactly one positive characteristic."}
{"category": "Math", "title": "Infinitesimal cubical structure, and higher connections", "abstract": "In the context of Synthetic Differential Geometry, we describe a notion of higher connection with values in a cubical groupoid. We do this by exploiting a certain structure of cubical complex derived from the first neighbourhood of the diagonal of a manifold. This cubical complex consists of infinitesimal parallelelpipeda."}
{"category": "Math", "title": "On Shimura curves in the Schottky locus", "abstract": "We show that a given rational Shimura curve Y with strictly maximal Higgs field in the moduli space of g-dimensional abelian varieties does not generically intersect the Schottky locus for large g. We achieve this by using a result of Viehweg and Zuo which says that if Y parameterizes a family of curves of genus g, then the corresponding family of Jacobians is isogenous over Y to the g-fold product of a modular family of elliptic curves. After reducing the situation from the field of complex numbers to a finite field, we will see, combining the Weil and Sato-Tate conjectures, that this is impossible for large genus g."}
{"category": "Math", "title": "Maximal lattice free bodies, test sets and the Frobenius problem", "abstract": "Maximal lattice free bodies are maximal polytopes without interior integral points. Scarf initiated the study of maximal lattice free bodies relative to the facet normals in a fixed matrix. In this paper we give an efficient algorithm for computing the maximal lattice free bodies of an integral matrix A. An important ingredient is a test set for a certain integer program associated with A. This test set may be computed using algebraic methods. As an application we generalize the Scarf-Shallcross algorithm for the three-dimensional Frobenius problem to arbitrary dimension. In this context our method is inspired by the novel algorithm by Einstein, Lichtblau, Strzebonski and Wagon and the Groebner basis approach by Roune."}
{"category": "Math", "title": "Moduli spaces of critical Riemannian metrics with L^{n/2} norm curvature bounds", "abstract": "We consider the moduli space of the extremal K\\\"ahler metrics on compact manifolds. We show that under the conditions of two-sided total volume bounds, $L^{n\\over2}$-norm bounds on $\\Riem$, and Sobolev constant bounds, this Moduli space can be compactified by including (reduced) orbifolds with finitely many singularities. Most of our results go through for certain other classes of critical Riemannian metrics."}
{"category": "Math", "title": "Deformations of asymptotically cylindrical G_2 manifolds", "abstract": "We prove that for a 7-dimensional manifold M with cylindrical ends the moduli space of exponentially asymptotically cylindrical torsion-free G_2 structures is a smooth manifold (if non-empty), and study some of its local properties. We also show that the holonomy of the induced metric of an exponentially asymptotically cylindrical G_2 manifold M is exactly G_2 if and only if its fundamental group is finite and neither M nor any double cover of M is homeomorphic to a cylinder."}
{"category": "Math", "title": "On partial polynomial interpolation", "abstract": "The Alexander-Hirschowitz theorem says that a general collection of $k$ double points in ${\\bf P}^n$ imposes independent conditions on homogeneous polynomials of degree $d$ with a well known list of exceptions. We generalize this theorem to arbitrary zero-dimensional schemes contained in a general union of double points. We work in the polynomial interpolation setting. In this framework our main result says that the affine space of polynomials of degree $\\le d$ in $n$ variables, with assigned values of any number of general linear combinations of first partial derivatives, has the expected dimension if $d\\neq 2$ with only five exceptional cases. If $d=2$ the exceptional cases are fully described."}
{"category": "Math", "title": "Network tomography based on 1-D projections", "abstract": "Network tomography has been regarded as one of the most promising methodologies for performance evaluation and diagnosis of the massive and decentralized Internet. This paper proposes a new estimation approach for solving a class of inverse problems in network tomography, based on marginal distributions of a sequence of one-dimensional linear projections of the observed data. We give a general identifiability result for the proposed method and study the design issue of these one dimensional projections in terms of statistical efficiency. We show that for a simple Gaussian tomography model, there is an optimal set of one-dimensional projections such that the estimator obtained from these projections is asymptotically as efficient as the maximum likelihood estimator based on the joint distribution of the observed data. For practical applications, we carry out simulation studies of the proposed method for two instances of network tomography. The first is for traffic demand tomography using a Gaussian Origin-Destination traffic model with a power relation between its mean and variance, and the second is for network delay tomography where the link delays are to be estimated from the end-to-end path delays. We compare estimators obtained from our method and that obtained from using the joint distribution and other lower dimensional projections, and show that in both cases, the proposed method yields satisfactory results."}
{"category": "Math", "title": "Singularities of generic projection hypersurfaces", "abstract": "Linearly projecting smooth projective varieties provides a method of obtaining hypersurfaces birational to the original varieties. We show that in low dimension, the resulting hypersurfaces only have Du Bois singularities. Moreover, we conclude that these Du Bois singularities are in fact semi log canonical. However, we demonstrate the existence of counterexamples in high dimension -- the generic linear projection of certain varieties of dimension 30 or higher is neither semi log canonical nor Du Bois."}
{"category": "Math", "title": "The Label Algorithm For Irreducible Decomposition of Monomial Ideals", "abstract": "The paper that was here is a preprint that was never turned into a proper paper. In particular it does not have enough citations to the literature. The paper \"The Slice Algorithm For Irreducible Decomposition of Monomial Ideals\" contains a much better description of the Label algorithm than this preprint did. If you still wish to read the original preprint then access the arXiv's version 1 of this paper, instead of version 2 which is what you are reading now."}
{"category": "Math", "title": "Explicit bounds for the approximation error in Benford's law", "abstract": "Benford's law states that for many random variables X > 0 its leading digit D = D(X) satisfies approximately the equation P(D = d) = log_{10}(1 + 1/d) for d = 1,2,...,9. This phenomenon follows from another, maybe more intuitive fact, applied to Y := log_{10}(X): For many real random variables Y, the remainder U := Y - floor(Y) is approximately uniformly distributed on [0,1). The present paper provides new explicit bounds for the latter approximation in terms of the total variation of the density of Y or some derivative of it. These bounds are an interesting alternative to traditional Fourier methods which yield mostly qualitative results. As a by-product we obtain explicit bounds for the approximation error in Benford's law."}
{"category": "Math", "title": "Dilation Theory for Rank 2 Graph Algebras", "abstract": "An analysis is given of $*$-representations of rank 2 single vertex graphs. We develop dilation theory for the non-selfadjoint algebras $\\A_\\theta$ and $\\A_u$ which are associated with the commutation relation permutation $\\theta$ of a 2 graph and, more generally, with commutation relations determined by a unitary matrix $u$ in $M_m(\\bC) \\otimes M_n(\\bC)$. We show that a defect free row contractive representation has a unique minimal dilation to a $*$-representation and we provide a new simpler proof of Solel's row isometric dilation of two $u$-commuting row contractions. Furthermore it is shown that the C*-envelope of $\\A_u$ is the generalised Cuntz algebra $\\O_{X_u}$ for the product system $X_u$ of $u$; that for $m\\geq 2 $ and $n \\geq 2 $ contractive representations of $\\Ath$ need not be completely contractive; and that the universal tensor algebra $\\T_+(X_u)$ need not be isometrically isomorphic to $\\A_u$."}
{"category": "Math", "title": "Atomic Representations of Rank 2 Graph Algebras", "abstract": "We provide a detailed analysis of atomic *-representations of rank 2 graphs on a single vertex. They are completely classified up to unitary equivalence, and decomposed into a direct sum or direct integral of irreducible atomic representations. The building blocks are described as the minimal *-dilations of defect free representations modelled on finite groups of rank 2."}
{"category": "Math", "title": "Periodicity in Rank 2 Graph Algebras", "abstract": "Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of $\\ca(\\Fth)$. The periodic C*-algebras are characterized, and it is shown that $\\ca(\\Fth) \\simeq \\rC(\\bT) \\otimes \\fA$ where $\\fA$ is a simple C*-algebra."}
{"category": "Math", "title": "Adiabatic limit of the eta invariant over cofinite quotient of PSL(2,R)", "abstract": "We study the adiabatic limit of the eta invariant of the Dirac operator over cofinite quotient of PSL(2,R), which is a noncompact manifold with a nonexact fibred-cusp metric near the ends."}
{"category": "Math", "title": "Pointwise convergence for semigroups in vector-valued $L^p$ spaces", "abstract": "Suppose that T_t is a symmetric diffusion semigroup on L^2(X) and consider its tensor product extension to the Bochner space L^p(X,B), where B belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf--Dunford--Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of the semigroup's extension to L^p(X,B). As an application, we show that such continuations exhibit pointwise convergence."}
{"category": "Math", "title": "Stable real algebraic vector bundles over a Klein bottle", "abstract": "Let X be a geometrically connected smooth projective curve of genus one, defined over the field of real numbers, such that X does not have any real points. We classify the isomorphism classes of all stable real algebraic vector bundles over X."}
{"category": "Math", "title": "Multiple solutions to the likelihood equations in the Behrens-Fisher problem", "abstract": "The Behrens-Fisher problem concerns testing the equality of the means of two normal populations with possibly different variances. The null hypothesis in this problem induces a statistical model for which the likelihood function may have more than one local maximum. We show that such multimodality contradicts the null hypothesis in the sense that if this hypothesis is true then the probability of multimodality converges to zero when both sample sizes tend to infinity. Additional results include a finite-sample bound on the probability of multimodality under the null and asymptotics for the probability of multimodality under the alternative."}
{"category": "Math", "title": "Patience Sorting and Its Generalizations", "abstract": "This dissertation collects together results on Patience Sorting and its generalizations. It incorporates the results of math.CO/0506358, math.CO/0507031, and math.CO/0512122, as well as previously unpublished results."}
{"category": "Math", "title": "Domains of definition of Monge-Amp\\`ere operators on compact K\\\"ahler manifolds", "abstract": "Let $(X,\\omega)$ be a compact K\\\"ahler manifold. We introduce and study the largest set $DMA(X,\\omega)$ of $\\omega$-plurisubharmonic (psh) functions on which the complex Monge-Amp\\`ere operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set $PSH(X,\\om)$ of all $\\om$-psh functions. We prove that certain twisted Monge-Amp\\`ere operators are well defined for all $\\omega$-psh functions. As a consequence, any $\\om$-psh function with slightly attenuated singularities has finite weighted Monge-Amp\\`ere energy."}
{"category": "Math", "title": "The Length of a Shortest Geodesic Loop", "abstract": "We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in their recent paper [HP] to give a geometric characterization of dynamically convex Finsler metrics on the 2-sphere."}
{"category": "Math", "title": "Attractors for gradient flows of non convex functionals and applications", "abstract": "This paper addresses the long-time behavior of gradient flows of non convex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of non convex evolution problems. In particular, we discuss the long-time behavior of solutions of quasi-stationary phase field models and prove the existence of a global attractor."}
{"category": "Math", "title": "L-infinity Algebras and Deformations of Holomorphic Maps", "abstract": "We construct the deformation functor associated with a pair of morphisms of differential graded Lie algebras, and use it to study infinitesimal deformations of holomorphic maps of compact complex manifolds. In particular, using L-infinity structures, we give an explicit description of the differential graded Lie algebra that controls this problem."}
{"category": "Math", "title": "M_2-rank differences for partitions without repeated odd parts", "abstract": "We prove formulas for the generating functions for M_2-rank differences for partitions without repeated odd parts. These formulas are in terms of modular forms and generalized Lambert series."}
{"category": "Math", "title": "Refined bound for sum-free sets in groups of prime order", "abstract": "Improving upon earlier results of Freiman and the present authors, we show that if $p$ is a sufficiently large prime and $A$ is a sum-free subset of the group of order $p$, such that $n:=|A|>0.318p$, then $A$ is contained in a dilation of the interval $[n,p-n]\\pmod p$."}
{"category": "Math", "title": "Classification of quasifinite representations with nonzero central charges for type $A_1$ EALA with coordinates in quantum torus", "abstract": "In this paper, we first construct a Lie algebra $L$ from rank 3 quantum torus, and show that it is isomorphic to the core of EALAs of type $A_1$ with coordinates in rank 2 quantum torus. Then we construct two classes of irreducible ${\\bf Z}$-graded highest weight representations, and give the necessary and sufficient conditions for these representations to be quasifinite. Next, we prove that they exhaust all the generalized highest weight irreducible ${\\bf Z}$-graded quasifinite representations. As a consequence, we determine all the irreducible ${\\bf Z}$-graded quasifinite representations with nonzero central charges. Finally, we construct two classes of highest weight ${\\bf Z}^2$-graded quasifinite representations by using these ${\\bf Z}$-graded modules."}
{"category": "Math", "title": "Coupled Painlev\\'e systems with affine Weyl group symmetry of types $D_3^{(2)}$ and $D_5^{(2)}$", "abstract": "In this paper, we find a two-parameter family of coupled Painlev\\'e systems in dimension four with affine Weyl group symmetry of type $D_3^{(2)}$. We also find a four-parameter family of 2-coupled $D_3^{(2)}$-systems in dimension eight with affine Weyl group symmetry of type $D_5^{(2)}$. We show that for each system, we give its symmetry and holomorphy conditions, respectively. These symmetries, holomorphy conditions and invariant divisors are new."}
{"category": "Math", "title": "The diffeomorphism group of a K3 surface and Nielsen realization", "abstract": "The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed oriented 4-manifold which contains a K3 surface as a connected summand then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying in the rational cohomology of B(pi_0 Diff(M)) are nonzero. We detect these classes by showing that they are nonzero when pulled back to the moduli space of Einstein metrics on a K3 surface."}
{"category": "Math", "title": "Skew Divided Difference Operators and Schubert Polynomials", "abstract": "We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with positive integer coefficients."}
{"category": "Math", "title": "On the Vanishing and the Finiteness of Supports of Generalized Local Cohomology Modules", "abstract": "Let $(R,\\fr m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M, N$ two finitely generated $R$-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that $H^j_I(M,N)=0$ for all $j>\\dim(R)$, provided $M$ is of finite projective dimension. Next, we study and give characterizations for the least and the last integer $r$ such that $\\Supp(H^r_I(M,N))$ is infinite."}
{"category": "Math", "title": "Almost-free finite covers", "abstract": "Finite covers are a technique for building new structures from simpler ones. The original motivation to study finite covers is in the Ladder theorem of Zilber which describes how totally categorical structures are built from strictly minimal sets by a sequence of covers. Let W be a first-order structure and r be an Aut(W)-congruence on W. In this paper we define the almost-free finite covers of W with respect to r, and we show how to construct them. These are a generalization of free finite covers. A consequence of a result of Evans and Hrushovski in the paper \"On the automorphism groups of finite covers\" is that any finite cover of W with binding groups all equal to a simple non-abelian permutation group is almost-free with respect to some r on W. Our main result gives a description (up to isomorphism) in terms of the Aut(W)-congruences on W of the kernels of principal finite covers of W with bindings groups equal at any point to a simple non-abelian regular permutation group G. Then we analyze almost-free finite covers of the set of ordered n-tuples of distinct elements from a countable set Omega, regarded as a structure with automorphism group equal to the Sym(Omega) and we show a result of biinterpretability."}
{"category": "Math", "title": "The Pseudospectrum of Systems of Semiclassical Operators", "abstract": "The pseudospectra (or spectral instability) of non-selfadjoint operators is a topic of current interest in applied mathematics. In fact, for non-selfadjoint operators the resolvent could be very large outside the spectrum, making the numerical computation of the complex eigenvalues very hard. This has importance, for example, in quantum mechanics, random matrix theory and fluid dynamics. The occurence of pseudospectra for non-selfadjoint semiclassical differential operators is due to the existence of quasimodes, i.e., approximate local solutions to the eigenvalue problem. For scalar operators, the quasimodes appear since the bracket condition is not satisfied for topological reasons, see the paper by Dencker, Sjostrand and Zworski in Comm. Pure Appl. Math. 57 (2004), 384-415. In this paper we shall investigate how these result can be generalized to square systems of semiclassical differential operators of principal type. These are the systems whose principal symbol vanishes of first order on its kernel. We show that the resolvent blows up as in the scalar case, except in a nowhere dense set of degenerate values. We also define quasi-symmetrizable systems and systems of subelliptic type for which we prove estimates on the resolvent."}
{"category": "Math", "title": "Twisted Whittaker model and factorizable sheaves", "abstract": "Let G be a reductive group. The geometric Satake equivalence realized the category of representations of the Langlands dual group ^LG in terms of spherical perverse sheaves (or D-modules) on the affine Grassmannian Gr_G=G((t))/G[[t]] of the original group G. In the present paper we perform a first step in realizing the category of representations of the quantum group corresponding to ^LG in terms of the geometry of Gr_G. The idea of the construction belongs to Jacob Lurie."}
{"category": "Math", "title": "Topological pressure for one-dimensional holomorphic dynamical systems", "abstract": "For a class of one-dimensional holomorphic maps f of the Riemann sphere we prove that for a wide class of potentials h the topological pressure is entirely determined by the values of h on the repelling periodic points of f. This is a version of a classical result of Bowen for hyperbolic diffeomorphisms in the holomorphic non-uniformly hyperbolic setting."}
{"category": "Math", "title": "Bounds on exponential sums over small multiplicative subgroups", "abstract": "We show that there is significant cancellation in certain exponential sums over small multiplicative subgroups of finite fields, giving an exposition of the arguments by Bourgain and Chang."}
{"category": "Math", "title": "Equivalence of symmetric union diagrams", "abstract": "Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is easy to see that every symmetric union represents a ribbon knot, but the converse is still an open problem. Besides existence it is natural to consider the question of uniqueness. In order to attack this question we extend the usual Reidemeister moves to a family of moves respecting the symmetry, and consider the symmetric equivalence thus generated. This notion being in place, we discuss several situations in which a knot can have essentially distinct symmetric union representations. We exhibit an infinite family of ribbon two-bridge knots each of which allows two different symmetric union representations."}
{"category": "Math", "title": "Variable Selection Incorporating Prior Constraint Information into Lasso", "abstract": "We propose the variable selection procedure incorporating prior constraint information into lasso. The proposed procedure combines the sample and prior information, and selects significant variables for responses in a narrower region where the true parameters lie. It increases the efficiency to choose the true model correctly. The proposed procedure can be executed by many constrained quadratic programming methods and the initial estimator can be found by least square or Monte Carlo method. The proposed procedure also enjoys good theoretical properties. Moreover, the proposed procedure is not only used for linear models but also can be used for generalized linear models({\\sl GLM}), Cox models, quantile regression models and many others with the help of Wang and Leng (2007)'s LSA, which changes these models as the approximation of linear models. The idea of combining sample and prior constraint information can be also used for other modified lasso procedures. Some examples are used for illustration of the idea of incorporating prior constraint information in variable selection procedures."}
{"category": "Math", "title": "Energy identity for approximations of harmonic maps from surfaces", "abstract": "We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two approximations and on estimates for the concentration radius of bubbles."}
{"category": "Math", "title": "An Algebraic Analysis of Conchoids to Algebraic Curves", "abstract": "We study conchoids to algebraic curve from the perspective of algebraic geometry, analyzing their main algebraic properties. We introduce the formal definition of conchoid of an algebraic curve by means of incidence diagrams. We prove that, with the exception of a circle centered at the focus and taking $d$ as its radius, the conchoid is an algebraic curve having at most two irreducible components. In addition, we introduce the notions of special and simple components of a conchoid. Moreover we state that, with the exception of lines passing through the focus, the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to the initial curve, and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve."}
{"category": "Math", "title": "On the Riemann zeta-function, Parts IV-V", "abstract": "In Part I an odd meromorphic function f(s) has been constructed from the Riemann zeta-function evaluated at one-half plus s. The conjunction of the Riemann hypothesis and hypotheses advanced by the author in Part I is assumed. In Part IV we derive the two-sided Laplace transform representation of f(s) on the open vertical strip V of all s with real part between zero and four. An additional hypothesis is used to prove that the Laplace density of f(s) on the strip V is positive. Let z(n) be the nth critical zero of the Riemann zeta-function of positive imaginary part in order of magnitude thereof. In Part V an expression is derived for z(1). A relation is obtained of the pair z(n) and the first derivative thereat of the zeta-function to the preceding such pairs."}
{"category": "Math", "title": "Endomorphisms of projective varieties", "abstract": "We study complex projective manifolds X that admit surjective endomorphisms f:X->X of degree at least two. In case f is etale, we prove structure theorems that describe X. In particular, a rather detailed description is given if X is a uniruled threefold. As to the ramified case, we first prove a general theorem stating that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild conditions. This is applied to the study of ramified endomorphisms of Fano manifolds with second Betti number one. It is conjectured that the projective space is the only Fano manifold admitting admitting an endomorphism of degree d>1, and we prove that in several cases. A part of the argumentation is based on a new characterization of the projective space as the only manifold that admits an ample subsheaf in its tangent bundle."}
{"category": "Math", "title": "Nested quantum Dyck paths and nabla(s_lambda)", "abstract": "We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable \"diagonal inversion\" statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the inverse Kostka matrix leads to an elementary proof of our proposed formula when q=1. We also outline a possible approach for proving all the extant nabla conjectures that reduces everything to the construction of sign-reversing involutions on explicit collections of signed, weighted objects."}
{"category": "Math", "title": "On the Small Ball Inequality in All Dimensions", "abstract": "Let h_R denote an L ^{\\infty} normalized Haar function adapted to a dyadic rectangle R contained in the unit cube in dimension d. We establish a non-trivial lower bound on the L^{\\infty} norm of the `hyperbolic' sums $$ \\sum _{|R|=2 ^{-n}} \\alpha(R) h_R (x) $$ The lower bound is non-trivial in that we improve the average case bound by n^{\\eta} for some positive \\eta, a function of dimension d. As far as the authors know, this is the first result of this type in dimension 4 and higher. This question is related to Conjectures in (1) Irregularity of Distributions, (2) Approximation Theory and (3) Probability Theory. The method of proof of this paper gives new results on these conjectures in all dimensions 4 and higher. This paper builds upon prior work of Jozef Beck, from 1989, and first two authors from 2006. These results were of the same nature, but only in dimension 3."}
{"category": "Math", "title": "New Enhanced Chaotic Number Generators", "abstract": "We introduce new families of enhanced chaotic number generators in order to compute very fast long series of pseudorandom numbers. The key feature of these generators being the use of chaotic numbers themselves for sampling chaotic subsequence of chaotic numbers in order to hide the generating function. We explore numerically the properties of these new families and underline their very high qualities and usefulness as CPRNG when series are computed up to 10 trillions iterations."}
{"category": "Math", "title": "New Invariants of Long Virtual Knots", "abstract": "This paper extends the construction of invariants for virtual knots to virtual long knots and introduces two new invariant modules of virtual long knots. Several interesting features are described that distinguish virtual long knots from their classical counterparts with respect to their symmetries and the concatenation product."}
{"category": "Math", "title": "On the values of integrals with the variable taken from $x=0$ to $x=\\infty$", "abstract": "This is a translation from the Latin original, \"De valoribus integralium a termino variabilis x=0 usque ad x=infinity extensorum\" (1781). This is E675 in the Enestrom index. Euler wants to find the location of the end point of a clothoid, a type of spiral. He proves some general results about the gamma function."}
{"category": "Math", "title": "Boundary cross theorem in dimension 1 with singularities", "abstract": "Let $D$ and $G$ be copies of the open unit disc in $\\C,$ let $A$ (resp. $B$) be a measurable subset of $\\partial D$ (resp. $\\partial G$), let $W$ be the 2-fold cross $\\big((D\\cup A)\\times B\\big)\\cup \\big(A\\times(B\\cup G)\\big),$ and let $M$ be a relatively closed subset of $W.$ Suppose in addition that $A$ and $B$ are of positive one-dimensional Lebesgue measure and that $M$ is fiberwise polar (resp. fiberwise discrete) and that $M\\cap (A\\times B)=\\varnothing.$ We determine the \"envelope of holomorphy\" $\\hat{W\\setminus M}$ of $W\\setminus M$ in the sense that any function locally bounded on $W\\setminus M,$ measurable on $A\\times B,$ and separately holomorphic on $\\big((A\\times G) \\cup (D\\times B)\\big)\\setminus M$ \"extends\" to a function holomorphic on $\\hat{W\\setminus M}.$"}
{"category": "Math", "title": "Algebraic Cycles and Mumford-Griffiths Invariants", "abstract": "Let $X$ be a projective algebraic manifold and let $CH^r(X)$ be the Chow group of algebraic cycles of codimension $r$ on $X$, modulo rational equivalence. Working with a candidate Bloch-Beilinson filtration $\\{F^{\\nu}\\}_{\\nu\\geq 0}$ on $CH^r(X)\\otimes {\\Bbb Q}$ due to the second author, we construct a space of arithmetic Hodge theoretic invariants $\\nabla J^{r,\\nu}(X)$ and corresponding map $\\phi_{X}^{r,\\nu} : Gr_{F}^{\\nu}CH^r(X)\\otimes {\\Bbb Q} \\to \\nabla J^{r,\\nu}(X)$, and determine conditions on $X$ for which the kernel and image of $\\phi_{X}^{r,\\nu}$ are ``uncountably large''."}
{"category": "Math", "title": "The Euclidean distortion of the lamplighter group", "abstract": "We show that the cyclic lamplighter group $C_2 \\bwr C_n$ embeds into Hilbert space with distortion ${\\rm O}(\\sqrt{\\log n})$. This matches the lower bound proved by Lee, Naor and Peres in \\cite{LeeNaoPer}, answering a question posed in that paper. Thus the Euclidean distortion of $C_2 \\bwr C_n$ is $\\Theta(\\sqrt{\\log n})$. Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni, Maurey and Mityagin \\cite{AhaMauMit} and by Gromov (see \\cite{deCTesVal}), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups."}
{"category": "Math", "title": "Stability of Solutions to Damped Equations with Negative Stiffness", "abstract": "This article concerns the stability of a model for mass-spring systems with positive damping and negative stiness. It is well known that when the coefficients are frozen in time the system is unstable. Here we find conditions on the variable cofficients to prove stability. In particular, we disprove the believe that if the eigenvalues of the system change slowly in time the system remains unstable. We extend some of our results for nonlinear systems."}
{"category": "Math", "title": "On group-theoretic models of randomness and genericity", "abstract": "We compare the random group model of Gromov and the model of generic groups of Arzhantseva and Ol'shanskii."}
{"category": "Math", "title": "Exponents of 2-multiarrangements and multiplicity lattices", "abstract": "We introduce a concept of multiplicity lattices of 2-multiarrangements, determine the combinatorics and geometry of that lattice, and give a criterion and method to construct a basis for derivation modules effectively."}
{"category": "Math", "title": "Py-Calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus", "abstract": "We show that Py-Calabi quasi-morphism on the group of Hamiltonian diffeomorphisms of surfaces of higher genus gives rise to a quasi-state."}
{"category": "Math", "title": "Weak amenability and 2-weak amenability of Beurling algebras", "abstract": "Let $L^1_\\om(G)$ be a Beurling algebra on a locally compact abelian group $G$. We look for general conditions on the weight which allows the vanishing of continuous derivations of $L^1_\\om(G)$. This leads us to introducing vector-valued Beurling algebras and considering the translation of operators on them. This is then used to connect the augmentation ideal to the behavior of derivation space. We apply these results to give examples of various classes of Beurling algebras which are weakly amenable, 2-weakly amenable or fail to be even 2-weakly amenable."}
{"category": "Math", "title": "Braided Sweedler cohomology", "abstract": "We introduced a braided Sweedler cohomology, which is adequate to work with the H-braided cleft extensions studied in [J. A. Guccione and J. J. Guccione, Theory of braided Hopf crossed products, Journal of Algebra, Vol 261 (2003) 54-101]"}
{"category": "Math", "title": "A filtered version of the bipolar theorem of Brannath and Schachermayer", "abstract": "We extend the Bipolar Theorem of Brannath and Schachermayer (1999) to the space of nonnegative cadlag supermartingales on a filtered probability space. We formulate the notion of fork-convexity as an analogue to convexity in this setting. As an intermediate step in the proof of our main result we establish a conditional version of the Bipolar theorem. In an application to mathematical finance we describe the structure of the set of dual processes of the utility maximization problem of Kramkov and Schachermayer (1999) and give a budget-constraint characterization of admissible consumption processes in an incomplete semimartingale market."}
{"category": "Math", "title": "On Cuspidal Representations of General Linear Groups over Discrete Valuation Rings", "abstract": "We define a new notion of cuspidality for representations of $\\GL_n$ over a finite quotient $\\Oh_k$ of the ring of integers $\\Oh$ of a non-Archimedean local field $F$ using geometric and infinitesimal induction functors, which involve automorphism groups $G_\\lambda$ of torsion $\\Oh$\\nobreakdash-modules. When $n$ is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of $\\GL_n(F)$. We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of $\\GL_n(\\Oh_k)$ for $k\\geq 2$ for all $n$ is equivalent to the construction of the representations of all the groups $G_\\lambda$. A functional equation for zeta functions for representations of $\\GL_n(\\Oh_k)$ is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for $\\GL_4(\\Oh_2)$ are constructed. Not all these representations are strongly cuspidal."}
{"category": "Math", "title": "The $(g,K)$-module structures of principal series representations of $Sp(3,R)$", "abstract": "We describe explicitly the whole structures of the $(g,K)$-modules of principal series representations of $Sp(3,R)$. We apply this result to determine the holonomic system characterizing those Whittaker functions."}
{"category": "Math", "title": "Modeling Hourly Ozone Concentration Fields", "abstract": "This paper presents a dynamic linear model for modeling hourly ozone concentrations over the eastern United States. That model, which is developed within an Bayesian hierarchical framework, inherits the important feature of such models that its coefficients, treated as states of the process, can change with time. Thus the model includes a time--varying site invariant mean field as well as time varying coefficients for 24 and 12 diurnal cycle components. This cost of this model's great flexibility comes at the cost of computational complexity, forcing us to use an MCMC approach and to restrict application of our model domain to a small number of monitoring sites. We critically assess this model and discover some of its weaknesses in this type of application."}
{"category": "Math", "title": "Enumeration of curves via floor diagrams", "abstract": "In this note we compute some enumerative invariants of real and complex projective spaces by means of some enriched graphs called floor diagrams."}
{"category": "Math", "title": "The Jones and Alexander polynomials for singular links", "abstract": "We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibility of long singular knots."}
{"category": "Math", "title": "Spherical harmonics and the icosahedron", "abstract": "We define a sextic invariant J on the seven-dimensional space of degree three spherical harmonics and show that J is positive if and only if the nodal set of the spherical harmonic contains the vertices of exactly two regular icosahedra. The proof uses the geometry of the Clebsch diagonal cubic surface, Atiyah's classification of vector bundles on an elliptic curve and a Fano threefold introduced by Mukai."}
{"category": "Math", "title": "Sign refinement for combinatorial link Floer homology", "abstract": "Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer coefficients thanks to a sign refinement. In this paper, thanks to the spin extension of the permutation group we give an alternative construction of the combinatorial link Floer chain complex associated to a grid diagram with integer coefficients. We prove that the filtered homology of this complex is an invariant for the link and that it gives the previous sign refinement by means of a 2-cohomological class corresponding to the spin extension of the permutation group."}
{"category": "Math", "title": "Total singular value decomposition. Robust SVD, regression and location-scale", "abstract": "Singular Value Decomposition (SVD) is the basic body of many statistical algorithms and few users question whether SVD is properly handling its job. SVD aims at evaluating the decomposition that best approximates a data matrix, given some rank restriction. However often we are interested in the best components of the decomposition rather than in the best approximation . This conflict of objectives leads us to introduce {\\em Total SVD}, where the word \"Total\" is taken as in \"Total\" least squares. SVD is a least squares method and, therefore, is very sensitive to gross errors in the data matrix. We make SVD robust by imposing a weight to each of the matrix entries. Breakdown properties are excellent. Algorithmic aspects are handled; they rely on high dimension fixed point computations."}
{"category": "Math", "title": "Reading Off Kurosh Decompositions", "abstract": "Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used by many authors to solve a wide collection of decision problems for free groups and their subgroups. In the present paper we employ our generalized Stallings' folding method to introduce a procedure, which given a subgroup H of a free product of finite groups reads off its Kurosh decomposition from the subgroup graph of H."}
{"category": "Math", "title": "Flat modules over valuation rings", "abstract": "Let $R$ be a valuation ring and let $Q$ be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if $Q$ is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of $R$ is a zero-divisor and that each singly projective module is locally projective if and only if $R$ is self injective. Moreover, $R$ is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or $\\pi$-coherent. A complete characterization of semihereditary commutative rings which are $\\pi$-coherent is given. When $R$ is a commutative ring with a self FP-injective quotient ring $Q$, it is proved that each flat $R$-module is finitely projective if and only if $Q$ is perfect."}
{"category": "Math", "title": "Harmonic G-structures", "abstract": "For closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold M, where G-structures are considered as sections of the quotient bundle O(M)/G. Then, we deduce the corresponding first and second variation formulae and the characterising conditions for critical points by means of tools closely related with the study of G-structures. In this direction, we show the role in the energy functional played by the intrinsic torsion of the G-structure. Moreover, we analyse the particular case G=U(n) for even-dimensional manifolds. This leads to the study of harmonic almost Hermitian manifolds and harmonic maps from M into O(M)/U(n)."}
{"category": "Math", "title": "Number of points of Prym varieties over finite fields", "abstract": "We establish some upper and lower bounds for the number of rational points of Prym varieties over finite fields."}
{"category": "Math", "title": "The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations", "abstract": "In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices $(M_N)_N$ defined by $M_N=W_N/\\sqrt{N}+A_N$ where $W_N$ is an $N\\times N$ Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincar\\'{e} inequality. The matrix $A_N$ is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of $A_N$ are sufficiently far from zero, the corresponding eigenvalues of $M_N$ almost surely exit the limiting semicircle compact support as the size $N$ becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of $W_N$. On the other hand, when $A_N$ is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of $W_N$."}
{"category": "Math", "title": "Resurgent functions and splitting problems", "abstract": "The present text is an introduction to \\'Ecalle's theory of resurgent functions and alien calculus, in connection with problems of exponentially small separatrix splitting. An outline of the resurgent treatment of Abel's equation for resonant dynamics in one complex variable is included. The emphasis is on examples of nonlinear difference equations, as a simple and natural way of introducing the concepts."}
{"category": "Math", "title": "A quasianalyticity property for monogenic solutions of small divisor problems", "abstract": "We discuss the quasianalytic properties of various spaces of functions suitable for one-dimensional small divisor problems. These spaces are formed of functions C^1-holomorphic on certain compact sets K_j of the Riemann sphere (in the Whitney sense), as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of K_j with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the K_j's."}
{"category": "Math", "title": "Existence and non-existence results for a logistic-type equation on manifolds", "abstract": "We study the steady state solutions of a generalized logistic type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the manifold, which is mostly taken into account by means of conditions on the volume growth of geodesic balls."}
{"category": "Math", "title": "The M-estimator in a multi-phase random nonlinear model", "abstract": "This paper considers M-estimation of a nonlinear regression model with multiple change-points occuring at unknown times. The multi-phase random design regression model, discontinuous in each change-point, have an arbitrary error $\\epsilon$. In the case when the number of jumps is known, the M-estimator of locations of breaks and of regression parameters are studied. These estimators are consistent and the distribution of the regression parameter estimators is Gaussian. The estimator of each change-point converges, with the rate $n^{-1}$, to the smallest minimizer of the independent compound Poisson processes. The results are valid for a large class of error distributions."}
{"category": "Math", "title": "Number of irreducible polynomials in several variables over finite fields", "abstract": "We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field."}
{"category": "Math", "title": "Dijkgraaf-Witten invariants of surfaces and projective representations of groups", "abstract": "We compute the Dijkgraaf-Witten invariants of surfaces in terms of projective representations of groups. As an application we prove that the complex Dijkgraaf-Witten invariants of surfaces of positive genus are positive integers."}
{"category": "Math", "title": "A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile", "abstract": "Consider a $N\\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $$ Y_{ij}^{n}=\\frac{\\sigma_{ij}(n)}{\\sqrt{n}} X_{ij}^{n} $$ the $X_{ij}^{n}$ being centered, independent and identically distributed random variables with unit variance and $(\\sigma_{ij}(n); 1\\le i\\le N, 1\\le j\\le n)$ being an array of numbers we shall refer to as a variance profile. We study in this article the fluctuations of the random variable $$ \\log\\det(Y_n Y_n^* + \\rho I_N) $$ where $Y^*$ is the Hermitian adjoint of $Y$ and $\\rho > 0$ is an additional parameter. We prove that when centered and properly rescaled, this random variable satisfies a Central Limit Theorem (CLT) and has a Gaussian limit whose parameters are identified. A complete description of the scaling parameter is given; in particular it is shown that an additional term appears in this parameter in the case where the 4$^\\textrm{th}$ moment of the $X_{ij}$'s differs from the 4$^{\\textrm{th}}$ moment of a Gaussian random variable. Such a CLT is of interest in the field of wireless communications."}
{"category": "Math", "title": "Extremal functions for the sharp $L^2-$ Nash inequality", "abstract": "We give geometrical conditions under which there exist extremal functions for the sharp $L^2$-Nash inequality."}
{"category": "Math", "title": "Monge-Ampere measures on pluripolar sets", "abstract": "In this article we solve the complex Monge-Ampere equation for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Kolodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge-Ampere measure, then it is a complex Monge-Ampere measure."}
{"category": "Math", "title": "The geometrical quantity in damped wave equations on a square", "abstract": "The energy in a square membrane $\\Omega$ subject to constant viscous damping on a subset $\\omega\\subset \\Omega$ decays exponentially in time as soon as $\\omega$ satisfies a geometrical condition known as the \"Bardos-Lebeau-Rauch\" condition. The rate $\\tau(\\omega)$ of this decay satisfies $\\tau(\\omega)= 2 \\min(-\\mu(\\omega), g(\\omega))$ (see Lebeau [Math. Phys. Stud. 19 (1996) 73-109]). Here $\\mu(\\omega)$ denotes the spectral abscissa of the damped wave equation operator and $g(\\omega)$ is a number called the geometrical quantity of $\\omega$ and defined as follows. A ray in $\\Omega$ is the trajectory generated by the free motion of a mass-point in $\\Omega$ subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity $g(\\omega)$ is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g(\\omega)$ when $\\omega$ is a finite union of squares."}
{"category": "Math", "title": "The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field", "abstract": "Let R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of the rationals in R, and let st: O^n \\to \\mathbb{R}^n be the standard part map. For X \\subseteq R^n define st(X):=st(X \\cap O^n). We let \\mathbb{R}_{\\ind} be the structure with underlying set \\mathbb{R} and expanded by all sets of the form st(X), where X \\subseteq R^{n} is definable in R and n=1,2,.... We show that the subsets of \\mathbb{R}^n that are definable in \\mathbb{R}_{\\ind} are exactly the finite unions of sets of the form st(X) \\setminus st(Y), where X,Y \\subseteq R^n are definable in R. A consequence of the proof is a partial answer to a question by Hrushovski, Peterzil and Pillay about the existence of measures with certain invariance properties on the lattice of bounded definable sets in R^n."}
{"category": "Math", "title": "On factorizations of smooth nonnegative matrix-values functions and on smooth functions with values in polyhedra", "abstract": "We discuss the possibility to represent smooth nonnegative matrix-valued functions as finite linear combinations of fixed matrices with positive real-valued coefficients whose square roots are Lipschitz continuous. This issue is reduced to a similar problem for smooth functions with values in a polyhedron."}
{"category": "Math", "title": "Equivalence of operations with respect to discriminator clones", "abstract": "For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of the other using operations from C. In this paper we prove that if C is a discriminator clone on a finite set, then there are only finitely many C-equivalence classes. Moreover, we show that the smallest discriminator clone is minimal with respect to this finiteness property. For discriminator clones of Boolean functions we explicitly describe the associated equivalence relations."}
{"category": "Math", "title": "A New Proof of the New Intersection Theorem", "abstract": "In 1987 Roberts completed the proof of the New Intersection Theorem (NIT) by settling the mixed characteristic case using local Chern characters, as developed by Fulton and also by Roberts. His proof has been the only one recorded of the NIT in mixed characteristic. This paper gives a new proof of this theorem, one which mostly parallels Roberts' original proof, but avoids the use of local Chern characters. Instead, the proof here uses Adams operations on K-theory with supports as developed by Gillet-Soule."}
{"category": "Math", "title": "A non commutative sewing lemma", "abstract": "In a preceding paper [E.J.ofProb.34,860-892,(2006)], we proved a sewing lemma which was a key result for the study of Holder continuous functions. In this paper we give a non-commutative version of this lemma with some applications."}
{"category": "Math", "title": "Asymptotic results on the length of coalescent trees", "abstract": "We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations in the partial tree. This is a first step to study the asymptotic distribution of a natural estimator of DNA mutation rate for species with large families."}
{"category": "Math", "title": "On a class number formula for real quadratic number fields", "abstract": "For an even Dirichlet character psi, we obtain a formula for L(1,psi) in terms of a sum of Dirichlet L-series evaluated at s=2 and s=3 and a rapidly convergent numerical series involving the central binomial coefficients. We then derive a class number formula for real quadratic number fields by taking L(s,psi) to be the quadratic L-series associated with these fields."}
{"category": "Math", "title": "Random spatial growth with paralyzing obstacles", "abstract": "We study models of spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (indicated by red). The green sources expand and may merge with others (there is no `inter-green' competition). The red substance remains passive as long as it is isolated. However, when a green cluster comes in touch with the red substance, it is immediately invaded by the latter, stops growing and starts to act as red substance itself. In our main model space is represented by a graph, of which initially each vertex is randomly green, red or white (vacant), and the growth of the green clusters is similar to that in first-passage percolation. The main issues we investigate are whether the model is well-defined on an infinite graph (e.g. the $d$-dimensional cubic lattice), and what can be said about the distribution of the size of a green cluster just before it is paralyzed. We show that, if the initial density of red vertices is positive, and that of white vertices is sufficiently small, the model is indeed well-defined and the above distribution has an exponential tail. In fact, we believe this to be true whenever the initial density of red is positive. This research also led to a relation between invasion percolation and critical Bernoulli percolation which seems to be of independent interest."}
{"category": "Math", "title": "Two Erdos problems on lacunary sequences: Chromatic number and Diophantine approximation", "abstract": "Let ${n_k}$ be an increasing lacunary sequence, i.e., $n_{k+1}/n_k>1+r$ for some $r>0$. In 1987, P. Erdos asked for the chromatic number of a graph $G$ on the integers, where two integers $a,b$ are connected by an edge iff their difference $|a-b|$ is in the sequence ${n_k}$. Y. Katznelson found a connection to a Diophantine approximation problem (also due to Erdos): the existence of $x$ in $(0,1)$ such that all the multiples $n_j x$ are at least distance $\\delta(x)>0$ from the set of integers. Katznelson bounded the chromatic number of $G$ by $Cr^{-2}|\\log r|$. We apply the Lov\\'asz local lemma to establish that $\\delta(x)>cr|\\log r|^{-1}$ for some $x$, which implies that the chromatic number of $G$ is at most $Cr^{-1} |\\log r|$. This is sharp up to the logarithmic factor."}
{"category": "Math", "title": "A p-adic quasi-quadratic point counting algorithm", "abstract": "In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality $q$ with time complexity $O(n^{2+o(1)})$ and space complexity $O(n^2)$, where $n=\\log(q)$. In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of theta constants. The theta null values are computed with respect to a semi-canonical theta structure of level $2^\\nu p$ where $\\nu >0$ is an integer and $p=\\mathrm{char}(\\F_q)>2$. The results of this paper suggest a global positive answer to the question whether there exists a quasi-quadratic time algorithm for the computation of the number of rational points on a generic ordinary abelian variety defined over a finite field."}
{"category": "Math", "title": "Selbstduale Vertexoperatorsuperalgebren und das Babymonster (Self-dual Vertex Operator Super Algebras and the Baby Monster)", "abstract": "We investigate self-dual vertex operator algebras (VOAs) and super algebras (SVOAs). Using the genus one correlation functions, it is shown that self-dual SVOAs exist only for half-integral central charges. It is described how self-dual SVOAs can be constructed from self-dual VOAs of larger central charge. The analogy with integral lattices and binary codes is emphasized. One main result is the construction of the shorter Moonshine module, a self-dual SVOA of central charge 23.5 on which the Baby monster - the second largest sporadic simple group - acts by automorphisms. The shorter Moonshine module has the character q^(-47/48)*(1+ 4371q^(3/2)+ 96256q^2+ 1143745q^(5/2) +...) and is the \"shorter cousin\" of the Moonshine module. Its lattice and code analog are the shorter Leech lattice and shorter Golay code. We conjecture that the shorter Moonshine module is the unique SVOA with this character. The final chapter introduces the notion of extremal VOAs and SVOAs. These are self-dual (S)VOAs with character having the same first few coefficients as the vacuum representation of the Virasoro algebra of the same central charge. We show that extremal VOAs exist at least for the central charges 8, 16, 24, 32, 40 and that extremal SVOAs exist only for the central charges c=0.5, 1, ..., 7.5, 8, 12, 14, 15, 15.5, 23.5 and 24. Examples for c=24 (resp. 23.5) are the (shorter) Moonshine module. Again, our results are similar to results known for codes and lattices."}
{"category": "Math", "title": "Braided doubles", "abstract": "Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. They are a class of algebras with triangular decomposition, arising from a deformation problem, the solutions to which are called quasi-Yetter-Drinfeld modules. A basic family of quasi-YD modules is provided by braidings (matrices satisfying the quantum Yang-Baxter equation); these give rise to quantum versions of the Weyl algebra, where the role of polynomial rings is played by Nichols-Woronowicz algebras. Rational Cherednik algebras for t = 0 emerge as subalgebras in doubles of Nichols-Woronowicz algebras. For nonzero t, the Nichols-Woronowicz algebra is replaced with an algebra associated to the classical Yang-Baxter equation."}
{"category": "Math", "title": "Approximately bisimilar symbolic models for nonlinear control systems", "abstract": "Control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs. Although these models are very powerful when dealing with physical phenomena, they are less suitable to describe software and hardware interfacing the physical world. For this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics, where each \"symbol\" corresponds to an \"aggregate\" of states in the continuous model. Since these symbolic models are of the same nature of the models used in computer science to describe software and hardware, they provide a unified language to study problems of control in which software and hardware interact with the physical world. Furthermore the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes. In this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent (bisimilar) to a symbolic model. The approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired. Furthermore if the state space of the control system is bounded the obtained symbolic model is finite. For digital control systems, and under the stronger assumption of incremental input-to-state stability, symbolic models can be constructed through a suitable quantization of the inputs."}
{"category": "Math", "title": "Aperiodic Pointlikes and Beyond", "abstract": "We prove that if $\\pi$ is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are $\\pi$-groups. In particular, when $\\pi$ is the empty set, we obtain Henckell's decidability of aperiodic pointlikes. Our proof, restricted to the case of aperiodic semigroups, is simpler than the original proof."}
{"category": "Math", "title": "The Compositions of the Differential Operations and Gateaux Directional Derivative", "abstract": "In this paper we determine the number of the meaningful compositions of higher order of the differential operations and Gateaux directional derivative."}
{"category": "Math", "title": "Giga-Periodic Orbits for Weakly Coupled Tent and Logistic Discretized Maps", "abstract": "We introduce new models of very weakly coupled logistic and tent maps for which orbits of very long period are found. The length of these periods is far greater than one billion. The property of these models relatively to the distribution of the iterated points (invariant measure) is described."}
{"category": "Math", "title": "Self-equivalence 3rd order ODEs by time-fixed transformations", "abstract": "Let y''' = f(x, y, y', y'') be a 3rd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates."}
{"category": "Math", "title": "Proof of generalized Riemann hypothesis for Dedekind zetas and Dirichlet L-functions", "abstract": "A short proof of the generalized Riemann hypothesis (gRH in short) for zeta functions $\\zeta_{k}$ of algebraic number fields $k$ - based on the Hecke's proof of the functional equation for $\\zeta_{k}$ and the method of the proof of the Riemann hypothesis derived in [$M_{A}$] (algebraic proof of the Riemann hypothesis) is given. The generalized Riemann hypothesis for Dirichlet L-functions is an immediately consequence of (gRH) for $\\zeta_{k}$ and suitable product formula which connects the Dedekind zetas with L-functions."}
{"category": "Math", "title": "A Geometric Zero-One Law", "abstract": "Each relational structure X has an associated Gaifman graph, which endows X with the properties of a graph. Suppose that X is infinite, connected and of bounded degree. A first-order sentence in the language of X is almost surely true (resp. a.s. false) for finite substructures of X if for every element x in X, the fraction of substructures of the ball of radius n around x which satisfy the sentence approaches 1 (resp. 0) as n approaches infinity. Suppose further that, for every finite substructure, X has a disjoint isomorphic substructure. Then every sentence is a.s. true or a.s. false for finite substructures of X. This is one form of the geometric zero-one law. We formulate it also in a form that does not mention the ambient infinite structure. In addition, we investigate various questions related to the geometric zero-one law."}
{"category": "Math", "title": "Linearizing torsion classes in the Picard group of algebraic curves over finite fields", "abstract": "We address the problem of computing in the group of $\\ell^k$-torsion rational points of the jacobian variety of algebraic curves over finite fields, with a view toward computing modular representations."}
{"category": "Math", "title": "Baer and Mittag-Leffler modules over tame hereditary algebras", "abstract": "We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones."}
{"category": "Math", "title": "On the antipode of a co-Frobenius (co)quasitriangular Hopf algebra", "abstract": "We extend to the co-Frobenius case a result of Drinfeld and Radford related to the fourth power of the antipode of a finite dimensional (co)quasitriangular Hopf algebra."}
{"category": "Math", "title": "Parametrization of Pythagorean triples by a single triple of polynomials", "abstract": "It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials."}
{"category": "Math", "title": "Remarks on polynomial parametrization of sets of integer points", "abstract": "If, for a subset S of Z^k, we compare the conditions of being parametrizable (a) by a single k-tuple of polynomials with integer coefficients, (b) by a single k-tuple of integer-valued polynomials and, (c) by finitely many k-tuples of polynomials with integer coefficients (variables ranging through the integers in each case) then (a) implies (b) (obviously), (b) implies (c), and neither converse holds. Condition (b) is equivalent to the set S being the set of integer values taken by some k-tuple of polynomials with rational coefficients as the variables range through the integers. We also show that every co-finite subset of Z^k is parametrizable a single k-tuple of polynomials with integer coefficients."}
{"category": "Math", "title": "The rectifiability of singular sets for geometric flows (I)--Yang-Mills flow", "abstract": "We prove that monotonicity of density and energy inequality imply the rectifiability of the singular sets for Yang-Mills flow."}
{"category": "Math", "title": "On the decycling of powers and products of cycles", "abstract": "We calculate exact values of the decycling numbers of $C_{m} \\times C_{n}$ for $m=3,4$, of $C_{n}^{2}$, and of $C_{n}^{3}$."}
{"category": "Math", "title": "Ring extension problem, Shukla cohomology and Ann-category theory", "abstract": "Every ring extension of $A$ by $R$ induces a pair of group homomorphisms $\\mathcal{L}^{*}:R\\to End_\\Z(A)/L(A);\\mathcal{R}^{*}:R\\to End_\\Z(A)/R(A),$ preserving multiplication, satisfying some certain conditions. A such 4-tuple $(R,A,\\mathcal{L}^{*},\\mathcal{R}^{*})$ is called a ring pre-extension. Each ring pre-extension induces a $R$-bimodule structure on bicenter $K_A$ of ring $A,$ and induces an obstruction $k,$ which is a 3-cocycle of $\\Z$-algebra $R,$ with coefficients in $R$-bimodule $K_A$ in the sense of Shukla. Each obstruction $k$ in this sense induces a structure of a regular Ann-category of type $(R,K_A).$ This result gives us the first application of Ann-category in extension problems of algebraic structures, as well as in cohomology theories."}
{"category": "Math", "title": "Even more simple cardinal invariants", "abstract": "Using GCH, we force the following: There are continuum many simple cardinal characteristics with pairwise different values."}
{"category": "Math", "title": "Loop Spaces and Langlands Parameters", "abstract": "We apply the technique of S^1-equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of geometric representation theory. Namely, we categorify the well known relationship between free loop spaces, cyclic homology and de Rham cohomology to recover the category of D-modules on a smooth stack X as a localization of the category of S^1-equivariant coherent sheaves on its loop space LX. The main observation is that this procedure connects categories of equivariant D-modules on flag varieties with categories of equivariant coherent sheaves on the Steinberg variety and its relatives. This provides a direct connection between the geometry of finite and affine Hecke algebras and braid groups, and a uniform geometric construction of all of the categorical parameters for representations of real and complex reductive groups. This paper forms the first step in a project to apply the geometric Langlands program to the complex and real local Langlands programs, which we describe."}
{"category": "Math", "title": "Parabolic equations with partially VMO coefficients and boundary value problems in Sobolev spaces with mixed norms", "abstract": "Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except $a^{11}$) are measurable in both time and one spatial variable, and VMO in the other spatial variables. The coefficient $a^{11}$ is measurable in time and VMO in the spatial variables. The unique solvability of equations in the whole space is applied to solving Dirichlet and oblique derivative problems for parabolic equations defined in a half-space."}
{"category": "Math", "title": "Positive Complex Sectional Curvature, Ricci Flow and the Differential Sphere Theorem", "abstract": "The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of B\\\"ohm-Wilking, that the normalized Ricci flow deforms such a metric to a metric of constant positive curvature. Using earlier work of Yau and Zheng it can be shown that a metric with strictly (pointwise) 1/4-pinched sectional curvature has positive complex sectional curvature. This gives a direct proof of Brendle-Schoen's recent differential sphere theorem, bypassing any discussion of positive isotropic curvature."}
{"category": "Math", "title": "Product Formulae for Ozsvath-Szabo 4-manifold Invariants", "abstract": "We give formulae for the Ozsvath-Szabo invariants of 4-manifolds X obtained by fiber sum of two manifolds M_1, M_2 along surfaces S_1, S_2 having trivial normal bundle and genus g>0. The formulae follow from a general theorem on the Ozsvath-Szabo invariants of the result of gluing two 4-manifolds along a common boundary, which is phrased in terms of relative invariants of the pieces. These relative invariants take values in a version of Heegaard Floer homology with coefficients in modules over certain Novikov rings; the fiber sum formula follows from the theorem that this \"perturbed\" version of Heegaard Floer theory recovers the usual Ozsvath-Szabo invariants, when the 4-manifold in question has b^+>1. The construction allows an extension of the definition of the Ozsvath-Szabo invariants to 4-manifolds having b^+ = 1 depending on certain choices, in close analogy with Seiberg-Witten theory. The product formulae lead quickly to calculations of the Ozsvath-Szabo invariants of various 4-manifolds; in all cases the results are in accord with the conjectured equivalence between the Ozsvath-Szabo and Seiberg-Witten invariants."}
{"category": "Math", "title": "Complex Ratios of Cubic Polynomials", "abstract": "Let $p(w)=(w-w_{1})(w-w_{2})(w-w_{3}),$with $\\func{Re}w_{1}<\\func{Re}w_{2}<\\func{Re}w_{3}$. Assume that if the critical points of $p$ are not identical, then they cannot have equal real parts. Define the ratios $\\sigma_{1}=\\dfrac{z_{1}-w_{1}}{w_{2}-w_{1}}$ and $\\sigma _{2}=\\dfrac{z_{2}-w_{2}}{w_{3}-w_{2}}$. $(\\sigma_{1},\\sigma_{2})$ is called the \\QTR{it}{ratio vector} of $p$. This extends the definition of ratio vectors given in earlier papers for polynomials of degree $n$ with all real roots. We then derive bounds on the real part, imaginary part, and modulus of the ratios and also some relations between the ratios. In particular, we prove that $\\func{Re}\\sigma_{1}\\leq \\func{Re}\\sigma_{2}$. We also show that the ratios are real if and only if the roots of $p$ are collinear."}
{"category": "Math", "title": "Modular representations of the ortho-symplectic supergroups", "abstract": "A Chevalley type integral basis for the ortho-symplectic Lie superalgebra is constructed. The simple modules of the ortho-symplectic supergroup over an algebraically closed field of prime characteristic not equal to 2 are classified, where a key combinatorial ingredient comes from the Mullineux conjecture on modular representations of the symmetric group. A Steinberg tensor product theorem for the ortho-symplectic supergroup is also obtained."}
{"category": "Math", "title": "R-Matrix Poisson Algebras and Their Deformations", "abstract": "We classify in this paper Poisson structures on modules over semisimple Lie algebras arising from classical r-matrices. We then study their quantizations and the relation to classical invariant theory."}
{"category": "Math", "title": "A class of series acceleration formulae for Catalan's constant", "abstract": "In this note, we develop transformation formulae and expansions for the log tangent integral, which are then used to derive series acceleration formulae for certain values of Dirichlet L-functions, such as Catalan's constant. The formulae are characterized by the presence of an infinite series whose general term consists of a linear recurrence damped by the central binomial coefficient and a certain quadratic polynomial. Typically, the series can be expressed in closed form as a rational linear combination of Catalan's constant and pi times the logarithm of an algebraic unit."}
{"category": "Math", "title": "A certain continuity property of the residues of the poles of $\\sum_{n \\geq 1} \\Lambda(n) e^{-2 \\pi i q n } n^{-s}$ with respect to $q \\in \\mathbb{Q} \\cap (0, 1)$ and the Riemann hypothesis", "abstract": "The purpose of this article is to present some result which may characterize nontrivial zeros of the Riemann zeta-function off the critical line $\\text{Re}(s) = 1/2$, if any exists. In brief, it concerns the residues of the poles of the function $M(s, q) \\equiv \\sum_{n \\geq 1} \\Lambda(n) e^{- 2\\pi i q n } n^{-s}$, where $\\Lambda$ is the arithmetical Mangoldt $\\Lambda$-function. Suppose that $M(s, 1/2)$ has a pole for some complex number $\\rho_{*}$, with $\\text{Re}(\\rho_{*}) > 1/2$. Then we discuss a certain continuity property of the residues of the poles of $M(\\rho_{*}, q)$ with respect to the variable $q \\in \\mathbb{Q} \\cap (1/2, 1)$."}
{"category": "Math", "title": "Ends in Uniform Spanning Forests", "abstract": "It has hitherto been known that in a transitive unimodular graph, each tree in the wired spanning forest has only one end a.s. We dispense with the assumptions of transitivity and unimodularity, replacing them with a much broader condition on the isoperimetric profile that requires just slightly more than uniform transience."}
{"category": "Math", "title": "Coverings of skew-products and crossed products by coactions", "abstract": "Consider a projective limit G of finite groups G_n. Fix a compatible family \\delta^n of coactions of the G_n on a C*-algebra A. From this data we obtain a coaction \\delta of G on A. We show that the coaction crossed product of A by \\delta is isomorphic to a direct limit of the coaction crossed products of A by the \\delta^n. If A = C*(\\Lambda) for some k-graph \\Lambda, and if the coactions \\delta^n correspond to skew-products of \\Lambda, then we can say more. We prove that the coaction crossed-product of C*(\\Lambda) by \\delta may be realised as a full corner of the C*-algebra of a (k+1)-graph. We then explore connections with Yeend's topological higher-rank graphs and their C*-algebras."}
{"category": "Math", "title": "A theorem on circle configurations", "abstract": "A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an analytic solution to the Apollonian problem. The general theorem for n-spheres is also considered."}
{"category": "Math", "title": "On the apparition of singularities of vector fields transported by volume preserving diffeomorphisms", "abstract": "We consider possible generation of singularities of a vector field transported by diffeomorphisms with derivatives of uniformly bounded determinants. A particular case of volume preserving diffeomrphism is the most important, since it has direct applications to the incompressible, inviscid hydrodynamics. We find relations between the directions of the vector field and the eigenvectors of the derivative of the back-to-label map near the singularity. We also find an invariant when we follow the motion of the integral curves of the vector field. For the 3D Euler equations these results have immediate implications about the directions of the vortex stretching and the material stretching near the possible singularities. We also have similar applications to the other inviscid, incompressible fluid equations such as the 2D quasi-geostrophic equation and the 3D magnetohydrodynamics equations."}
{"category": "Math", "title": "$\\Lambda$-adic Kolyvagin systems", "abstract": "In this paper, we study the deformations of Kolyvagin systems that are known to exist in a wide variety of cases, by the work of B. Howard, B. Mazur, and K. Rubin for the residual Galois representations, along the cyclotomic Iwasawa algebra. We prove, under certain technical hypotheses, that a cyclotomic deformation of a Kolyvagin system exists. We also briefly discuss how our techniques could be extended to prove that one could deform Kolyvagin systems for other deformations as well. We discuss several applications of this result, particularly relation of these $\\Lambda$-adic Kolyvagin systems to p-adic L-functions (in view of the conjectures of Perrin-Riou on p-adic L-functions) and applications to main conjectures; also applications to the study of Iwasawa theory of Rubin-Stark units."}
{"category": "Math", "title": "Contact 5-manifolds with SU(2)-structure", "abstract": "We consider 5-manifolds with a contact form arising from a hypo structure, which we call \\emph{hypo-contact}. We provide conditions which imply that there exists such a structure on an oriented hypersurface of a 6-manifold with a half-flat SU(3)-structure. For half-flat manifolds with a Killing vector field $X$ preserving the SU(3)-structure we study the geometry of the orbits space. Moreover, we describe the solvable Lie algebras admitting a \\emph{hypo-contact} structure. This allows us exhibit examples of Sasakian $\\eta$-Einstein manifolds, as well as to prove that such structures give rise to new metrics with holonomy SU(3) and to new metrics with holonomy $G_2$."}
{"category": "Math", "title": "Bounds for Hilbert coefficients", "abstract": "We compute the Hilbert coefficients of a graded module with pure resolution and discuss lower and upper bounds for these coefficients for arbitrary graded modules."}
{"category": "Math", "title": "Asymptotic Behavior of Total Times For Jobs That Must Start Over If a Failure Occurs", "abstract": "Many processes must complete in the presence of failures. Different systems respond to task failure in different ways. The system may resume a failed task from the failure point (or a saved checkpoint shortly before the failure point), it may give up on the task and select a replacement task from the ready queue, or it may restart the task. The behavior of systems under the first two scenarios is well documented, but the third ({\\em RESTART}) has resisted detailed analysis. In this paper we derive tight asymptotic relations between the distribution of {\\em task times} without failures to the {\\em total time} when including failures, for any failure distribution. In particular, we show that if the task time distribution has an unbounded support then the total time distribution $H$ is always heavy-tailed. Asymptotic expressions are given for the tail of $H$ in various scenarios. The key ingredients of the analysis are the Cram\\'er--Lundberg asymptotics for geometric sums and integral asymptotics, that in some cases are obtained via Tauberian theorems and in some cases by bare-hand calculations."}
{"category": "Math", "title": "Automorphisms fixing every normal subgroup of a nilpotent-by-abelian group", "abstract": "Among other things, we prove that the group of automorphisms fixing every normal subgroup of a nilpotent-by-abelian group is nilpotent-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most 3. An example shows that this bound cannot be improved."}
{"category": "Math", "title": "Regularity of harmonic functions for anisotropic fractional Laplacian", "abstract": "We prove that bounded harmonic functions of anisotropic fractional Laplacians are H\\\"older continuous under mild regularity assumptions on the corresponding L\\'evy measure. Under some stronger assumptions the Green function, Poisson kernel and the harmonic functions are even differentiable of order up to three."}
{"category": "Math", "title": "On the kernel of the norm in some unramified number fields extensions", "abstract": "We determine the -1 Tate cohomology group of the units for principal abelian extensions of type (p^a, p^b) of a number field."}
{"category": "Math", "title": "Stability of Euclidean space under Ricci flow", "abstract": "We study the Ricci flow for initial metrics which are C^0 small perturbations of the Euclidean metric on R^n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In proving this stability result, we introduce a monotone integral quantity which measures the deviation of the evolving metric from the Euclidean metric. We also investigate the convergence of the diffeomorphisms relating Ricci harmonic map heat flow to Ricci flow."}
{"category": "Math", "title": "Finite strain viscoplasticity with nonlinear kinematic hardening: phenomenological modeling and time integration", "abstract": "This article deals with a viscoplastic material model of overstress type. The model is based on a multiplicative decomposition of the deformation gradient into elastic and inelastic part. An additional multiplicative decomposition of inelastic part is used to describe a nonlinear kinematic hardening of Armstrong-Frederick type. Two implicit time-stepping methods are adopted for numerical integration of evolution equations, such that the plastic incompressibility constraint is exactly satisfied. The first method is based on the tensor exponential. The second method is a modified Euler-Backward method. Special numerical tests show that both approaches yield similar results even for finite inelastic increments. The basic features of the material response, predicted by the material model, are illustrated with a series of numerical simulations."}
{"category": "Math", "title": "Numerical Clifford Analysis for Nonlinear Schrodinger Problem", "abstract": "The aim of this work is to study the numerical solution of the nonlinear Schrodinger problem using a combination between Witt basis and finite difference approximations. We construct a discrete fundamental solution for the non-stationary Schrodinger operator and we show the convergence of the numerical scheme. Numerical examples are given at the end of the paper."}
{"category": "Math", "title": "Carleson Measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on Complex Balls", "abstract": "We characterize the Carleson measures for the Drury-Arveson Hardy space and other Hilbert spaces of analytic functions of several complex variables. This provides sharp estimates for Drury's generalization of Von Neumann's inequality. The characterization is in terms of a geometric condition, the \"split tree condition\", which reflects the nonisotropic geometry underlying the Drury-Arveson Hardy space."}
{"category": "Math", "title": "Representations of Graded Multi-Loop Lie Algebras", "abstract": "Let g_A (respectively, g_A(\\mu)) be the graded multi-loop Lie algebra (respectively graded twisted multi-loop Lie algebra)\" associated with the simple finite dimensional Lie algebra g over the complex field C. In this paper, we prove that irreducible integrable g_A(\\mu)-modules with finite dimensional weight spaces are either highest weight modules or their duals and classify the isomorphism classes of irreducible integrable g_A-modules and g_A(\\mu)-modules with finite dimensional weight spaces."}
{"category": "Math", "title": "Fractal analysis for sets of non-differentiability of Minkowski's question mark function", "abstract": "In this paper we study various fractal geometric aspects of the Minkowski question mark function $Q.$ We show that the unit interval can be written as the union of the three sets $\\Lambda_{0}:=\\{x:Q'(x)=0\\}$, $\\Lambda_{\\infty}:=\\{x:Q'(x)=\\infty\\}$, and $\\Lambda_{\\sim}:=\\{x:Q'(x)$ does not exist and $Q'(x)\\not=\\infty\\}.$ The main result is that the Hausdorff dimensions of these sets are related in the following way. $\\dim_{H}(\\nu_{F})<\\dim_{H}(\\Lambda_{\\sim})= \\dim_{H} (\\Lambda_{\\infty}) = \\dim_{H} (\\mathcal{L}(h_{\\mathrm{top}}))<\\dim_{H}(\\Lambda_{0})=1.$ Here, $\\mathcal{L}(h_{\\mathrm{top}})$ refers to the level set of the Stern-Brocot multifractal decomposition at the topological entropy $h_{\\mathrm{top}}=\\log2$ of the Farey map $F,$ and $\\dim_{H}(\\nu_{F})$ denotes the Hausdorff dimension of the measure of maximal entropy of the dynamical system associated with $F.$ The proofs rely partially on the multifractal formalism for Stern-Brocot intervals and give non-trivial applications of this formalism."}
{"category": "Math", "title": "Braided enveloping algebras associated to quantum parabolic subalgebras", "abstract": "Associated to each subset $J$ of the nodes $I$ of a Dynkin diagram is a triangular decomposition of the corresponding Lie algebra $\\mathfrak{g}$ into three subalgebras $\\widetilde{\\mathfrak{g}_{J}}$ (generated by $e_{j}$, $f_{j}$ for $j\\in J$ and $h_{i}$ for $i\\in I$), $\\mathfrak{n}^{-}_{D}$ (generated by $f_{d}$, $d\\in D=I\\setminus J$) and its dual $\\mathfrak{n}_{D}^{+}$. We demonstrate a quantum counterpart, generalising work of Majid and Rosso, by exhibiting analogous triangular decompositions of $U_{q}(\\mathfrak{g})$ and identifying a graded braided Hopf algebra that quantizes $\\mathfrak{n}_{D}^{-}$. This algebra has many similar properties to $U_{q}^{-}(\\mathfrak{g})$, in many cases being a Nichols algebra and therefore completely determined by its associated braiding."}
{"category": "Math", "title": "Permutation actions on equivariant cohomology", "abstract": "This survey paper describes two geometric representations of the permutation group using the tools of toric topology. These actions are extremely useful for computational problems in Schubert calculus. The (torus) equivariant cohomology of the flag variety is constructed using the combinatorial description of Goresky-Kottwitz-MacPherson, discussed in detail. Two permutation representations on equivariant and ordinary cohomology are identified in terms of irreducible representations of the permutation group. We show how to use the permutation actions to construct divided difference operators and to give formulas for some localizations of certain equivariant classes. This paper includes several new results, in particular a new proof of the Chevalley-Monk formula and a proof that one of the natural permutation representations on the equivariant cohomology of the flag variety is the regular representation. Many examples, exercises, and open questions are provided."}
{"category": "Math", "title": "Jordan Triple Elementary Maps on Rings", "abstract": "We prove that Jordan triple elementary surjective maps on unital rings containing a nontrivial idempotent are automatically additive."}
{"category": "Math", "title": "Twisted Fermat curves over totally real fields", "abstract": "Let p be a prime number, F a totally real field such that [F(mu_p): F]=2 and [F:Q] is odd. For delta \\in F^times, let [delta] denote its class in F^times/F^{times p}. In this paper, we show Main Theorem. There are infinitely many classes [delta]\\in F^times/F^{times p} such that the twisted affine Fermat curves W_delta: X^p+Y^p=delta have no F-rational points."}
{"category": "Math", "title": "Superrosy dependent groups having finitely satisfiable generics", "abstract": "We study a model theoretic context (finite thorn rank, NIP, with finitely satisfiable generics) which is a common generalization of groups of finite Morley rank and definably compact groups in o-minimal structures. We show that assuming thorn rank 1, the group is abelian-by-finite, and assuming thorn rank 2 the group is solvable by finite. Also a field is algebraically closed."}
{"category": "Math", "title": "Additivity of Jordan Elementary Maps on Rings", "abstract": "We prove that Jordan elementary surjective maps on rings are automatically additive."}
{"category": "Math", "title": "The derived category of quasi-coherent sheaves and axiomatic stable homotopy", "abstract": "We prove in this paper that for a quasi-compact and semi-separated (non necessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(A_qc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category of a usual scheme."}
{"category": "Math", "title": "Lectures on flips and minimal models", "abstract": "This document contains notes from the lectures of Corti, Koll\\'ar, Lazarsfeld, and Musta\\c{t}\\u{a} at the workshop ``Minimal and canonical models in algebraic geometry\" at MSRI, Berkeley, April 2007. The lectures give an overview of the recent advances on canonical and minimal models of algebraic varieties obtained by Hacon--McKernan and Birkar--Cascini--Hacon--McKernan."}
{"category": "Math", "title": "The Order of the Giant Component of Random Hypergraphs", "abstract": "We establish central and local limit theorems for the number of vertices in the largest component of a random $d$-uniform hypergraph $\\hnp$ with edge probability $p=c/\\binnd$, where $(d-1)^{-1}+\\eps<c<\\infty$. The proof relies on a new, purely probabilistic approach, and is based on Stein's method as well as exposing the edges of $H_d(n,p)$ in several rounds."}
{"category": "Math", "title": "Local Limit Theorems and Number of Connected Hypergraphs", "abstract": "Let $H_d(n,p)$ signify a random $d$-uniform hypergraph with $n$ vertices in which each of the ${n}\\choose{d}$ possible edges is present with probability $p=p(n)$ independently, and let $H_d(n,m)$ denote a uniformly distributed with $n$ vertices and $m$ edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of $H_d(n,p)$ and $H_d(n,m)$ for the regime ${{n-1}\\choose{d-1}} p,dm/n >(d-1)^{-1}+\\epsilon$. As an application, we obtain an asymptotic formula for the probability that $H_d(n,p)$ or $H_d(n,m)$ is connected. In addition, we infer a local limit theorem for the conditional distribution of the number of edges in $H_d(n,p)$ given connectivity. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach."}
{"category": "Math", "title": "False discovery rate control with multivariate $p$-values", "abstract": "Multivariate statistics are often available as well as necessary in hypothesis tests. We study how to use such statistics to control not only false discovery rate (FDR) but also positive FDR (pFDR) with good power. We show that FDR can be controlled through nested regions of multivariate $p$-values of test statistics. If the distributions of the test statistics are known, then the regions can be constructed explicitly to achieve FDR control with maximum power among procedures satisfying certain conditions. On the other hand, our focus is where the distributions are only partially known. Under certain conditions, a type of nested regions are proposed and shown to attain (p)FDR control with asymptotically maximum power as the pFDR control level approaches its attainable limit. The procedure based on the nested regions is compared with those based on other nested regions that are easier to construct as well as those based on more straightforward combinations of the test statistics."}
{"category": "Math", "title": "Classifying Compactly generated t-structures on the derived category of a Noetherian ring", "abstract": "We classify complactly generated t-structures on the derived category of modules over a commutative Noetherian ring R in terms of decreasing filtrations by supports on Spec(R). A decreasing filtration by supports \\phi : Z -> Spec(R) satisfies the weak Cousin condition if for any integer i \\in Z, the set \\phi(i) contains all the inmediate generalizations of each point in \\phi(i+1). Every t-structure on D^b_fg(R) (equivalently, on D^-_fg(R)) is induced by complactly generated t-structures on D(R) whose associated filtrations by supports satisfy the weak Cousin condition. If the ring R has dualizing complex we prove that these are exactly the t-structures on D^b_fg(R). More generally, if R has a pointwise dualizing complex we classify all compactly generated t-structures on D_fg(R)."}
{"category": "Math", "title": "A note on the $K$-stability on toric manifolds", "abstract": "In this note, we prove that on polarized toric manifolds the relative $K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and also we show that the modified $K$-energy is proper in the space of $G_0$-invariant K\\\"ahler metrics for the case of toric surfaces which admit the extremal metrics."}
{"category": "Math", "title": "Ring completion of rig categories", "abstract": "We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category R we construct a natural additive group completion R' that retains the multiplicative structure, hence has become a ring category. If we start with a commutative rig category R (also known as a symmetric bimonoidal category), the additive group completion R' will be a commutative ring category. In an accompanying paper we show how this can be used to prove the conjecture from [BDR] that the algebraic K-theory of the connective topological K-theory spectrum ku is equivalent to the algebraic K-theory of the rig category V of complex vector spaces."}
{"category": "Math", "title": "Nontrivial elements of Sha explained through K3 surfaces", "abstract": "In this paper we present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group."}
{"category": "Math", "title": "Strichartz estimates and local smoothing estimates for asymptotically flat Schr\\\"odinger equations", "abstract": "In this article we study global-in-time Strichartz estimates for the Schr\\\"odinger evolution corresponding to long-range perturbations of the Euclidean Laplacian. This is a natural continuation of a recent article of the third author, where it is proved that local smoothing estimates imply Strichartz estimates. In the aforementioned paper, the third author proved the local smoothing estimates for small perturbations of the Laplacian. Here we consider the case of large perturbations in three increasingly favorable scenarios: (i) without non-trapping assumptions we prove estimates outside a compact set modulo a lower order spatially localized error term, (ii) with non-trapping assumptions we prove global estimates modulo a lower order spatially localized error term, and (iii) for time independent operators with no resonance or eigenvalue at the bottom of the spectrum we prove global estimates for the projection onto the continuous spectrum."}
{"category": "Math", "title": "Trees and Markov convexity", "abstract": "We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors."}
{"category": "Math", "title": "More spectral bounds on the clique and independence numbers", "abstract": "We give some new bounds for the clique and independence numbers of a graph in terms of its eigenvalues."}
{"category": "Math", "title": "A primer on computational group homology and cohomology", "abstract": "These are expanded lecture notes of a series of expository talks surveying basic aspects of group cohomology and homology. They were written for someone who has had a first course in graduate algebra but no background in cohomology. You should know the definition of a (left) module over a (non-commutative) ring, what $\\zzz[G]$ is (where $G$ is a group written multiplicatively and $\\zzz$ denotes the integers), and some ring theory and group theory. However, an attempt has been made to (a) keep the presentation as simple as possible, (b) either provide an explicit reference of proof of everything. Several computer algebra packages are used to illustrate the computations, though for various reasons we have focused on the free, open source packages such as GAP and SAGE."}
{"category": "Math", "title": "Operator space Lp embedding theory I", "abstract": "Given any $1 < q \\le 2$, we use new free probability techniques to construct a completely isomorphic embedding of $\\ell_q$ (equipped with its natural operator space structure) into the predual of a sufficiently large QWEP von Neumann algebra."}
{"category": "Math", "title": "A generating function for non-standard orthogonal polynomials involving differences: the Meixner case", "abstract": "In this paper we deal with a family of non--standard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so--called $\\Delta$--Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the $\\Delta$--Meixner--Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre--Sobolev orthogonal polynomials."}
{"category": "Math", "title": "Embeddability of multiple cones", "abstract": "The main result of this paper is that if $X$ is a Peano continuum such that its $n$-th cone $C^n(X)$ embeds into $\\RR^{n+2}$ then $X$ embeds into $S^2$. This solves a problem proposed by W. Rosicki."}
{"category": "Math", "title": "Queues with heterogeneous servers and uninformed customers: who works the most?", "abstract": "In this paper, we consider systems that can be modelled by $M \\mid M \\mid n$ queues with heterogeneous servers and non informed customers. Considering any two servers: we show that the probability that the fastest server is busy is smaller than the probability that the slowest server is busy. Moreover, we show that the effective rate of service done by the fastest server is larger than effective rate of service done by the slowest server."}
{"category": "Math", "title": "An extension of Perelman's soul theorem for singular spaces", "abstract": "In this paper, we study open complete metric spaces with non-negative curvature. Among other things, we establish an extension of Perelman's soul theorem for possibly singular spaces: \"Let X be a complete, non-compact, finite dimensional Alexandrov space with non-negative curvature. Suppose that X has no boundary and has positive curvature on a non-empty open subset. Then X must be a contractible space\". The proof of this result uses the detailed analysis of concavity of distance functions and Busemann functions on singular spaces with non-negative curvature. We will introduce a family of angular excess functions to measure convexity and extrinsic curvature of convex hypersurfaces in singular spaces. We also derive a new comparison for trapezoids in non-negatively curved spaces, which led to desired convexity estimates for the proof of our new soul theorem."}
{"category": "Math", "title": "Relations among modular points on elliptic curves", "abstract": "Given a correspondence between a modular curve and an elliptic curve A we study the group of relations among the CM points of A. In particular we prove that the intersection of any finite rank subgroup of A with the set of CM points of A is finite. We also prove a local version of this global result with an effective bound valid also for certain infinite rank subgroups. We deduce the local result from a ``reciprocity'' theorem for CL (canonical lift) points on A. Furthermore we prove similar global and local results for intersections between subgroups of A and isogeny classes in A. Finally we prove Shimura curve analogues and, in some cases, higher-dimensional versions of these results."}
{"category": "Math", "title": "Subrepresentation Theorem for p-adic Symmetric Spaces", "abstract": "The notion of relative cuspidality for distinguished representations attached to $p$-adic symmetric spaces is introduced. A characterization of relative cuspidality in terms of Jacquet modules is given and a generalization of Jacquet's subrepresentation theorem to the relative case (symmetric space case) is established."}
{"category": "Math", "title": "Weighted lattice polynomials", "abstract": "We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a median based decomposition formula."}
{"category": "Math", "title": "Norm and Numerical Peak Holomorphic Functions on Banach Spaces", "abstract": "We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let $A_b(B_X:X)$ be the Banach space of all bounded continuous functions $f$ on the unit ball $B_X$ of a Banach space $X$ and their restrictions $f|_{B_X^\\circ}$ to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense $G_\\delta$ subset of $A_b(B_X:X)$. We also prove that if $X$ is a smooth Banach space with the Radon-Nikod\\'ym property, then the set of all numerical strong peak functions is dense in $A_b(B_X:X)$. In particular, when $X=L_p(\\mu)$ $(1<p<\\infty)$ or $X=\\ell_1$, it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense $G_\\delta$ subset of $A_b(B_X:X)$. In the meanwhile, we study the properties of the numerical radius of an holomorphic function and the numerical index of subspaces of $A_b(B_X:X)$. As an application, the existence and properties of numerical boundary of $A_b(B_X:X)$ are studied. Finally, the numerical peak function in $A_b(B_X:X)$ is characterized when $X=\\ell_\\infty^n$ and some negative results on the denseness of numerical (strong) peak holomorphic functions are given."}
{"category": "Math", "title": "Expressing Combinatorial Optimization Problems by Systems of Polynomial Equations and the Nullstellensatz", "abstract": "Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. In the first part of this paper, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colorable subgraph. For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows. We show that the minimum-degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3- colorability, we found only graphs with Nullstellensatz certificates of degree four."}
{"category": "Math", "title": "Affine Structures on a Ringed Space and Schemes", "abstract": "In this paper we will first introduce the notion of affine structures on a ringed space and then obtain several properties. Affine structures on a ringed space, arising mainly from complex analytical spaces of algebraic schemes over number fields, behave like differential structures on a smooth manifold. As one does for differential manifolds, we will use pseudogroups of affine transformations to define affine atlases on a ringed space. An atlas on a space is said to be an affine structure if it is maximal. An affine structure is admissible if there is a sheaf on the underlying space such that they are coincide on all affine charts, which are in deed affine open sets of a scheme. In a rigour manner, a scheme is defined to be a ringed space with a specified affine structure if the affine structures are in action in some special cases such as analytical spaces of algebraic schemes. Particularly, by the whole of affine structures on a space, we will obtain respectively necessary and sufficient conditions that two spaces are homeomorphic and that two schemes are isomorphic, which are the two main theorems of the paper. It follows that the whole of affine structures on a space and a scheme, as local data, encode and reflect the global properties of the space and the scheme, respectively."}
{"category": "Math", "title": "Stabilizing the monodromy of an open book decomposition", "abstract": "We prove that any mapping class on a compact oriented surface with nonempty boundary can be made pseudo-Anosov and right-veering after a sequence of positive stabilizations."}
{"category": "Math", "title": "On products of T-ideals in free algebras and free group algebras", "abstract": "Let F be a field and A a free associative F-algebra or a group algebra of a free group with an infinite set X of generators. We find a necessary and sufficient condition for the inclusion I' into I, where I=I_1...I_k and I'=I'_1...I'_l are any products of T-ideals in A. A canonical reformulation in terms of products of group representation varieties answers a question posed in 1986"}
{"category": "Math", "title": "Hodge polynomials of the moduli spaces of rank 3 pairs", "abstract": "Let $X$ be a smooth projective curve of genus $g\\geq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,\\phi)$ on $X$ consists of two holomorphic vector bundles $E_1$ and $E_2$ over $X$ and a holomorphic map $\\phi:E_2 \\to E_1$. There is a concept of stability for triples which depends on a real parameter $\\sigma$. In this paper, we determine the Hodge polynomials of the moduli spaces of $\\sigma$-stable triples with $\\rk(E_1)=3$, $\\rk(E_2)=1$, using the theory of mixed Hodge structures. This gives in particular the Poincar\\'e polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles."}
{"category": "Math", "title": "High distance Heegaard splittings via fat train tracks", "abstract": "We define \"fat\" train tracks and use them to give a combinatorial criterion for the Hempel distance of Heegaard splittings for closed orientable 3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on knots in the three sphere."}
{"category": "Math", "title": "Finite Schur filtration dimension for modules over an algebra with Schur filtration", "abstract": "Let G be GL_N or SL_N as reductive linear algebraic group over a field k of positive characteristic p. We prove several results that were previously established only when N < 6 or p > 2^N. Let G act rationally on a finitely generated commutative k-algebra A. Assume that A as a G-module has a good filtration or a Schur filtration. Let M be a noetherian A-module with compatible G action. Then M has finite good/Schur filtration dimension, so that there are at most finitely many nonzero H^i(G,M). Moreover these H^i(G,M) are noetherian modules over the ring of invariants A^G. Our main tool is a resolution involving Schur functors of the ideal of the diagonal in a product of Grassmannians."}
{"category": "Math", "title": "On the geometry of generalized Gaussian distributions", "abstract": "In this paper we consider the space of those probability distributions which maximize the $q$-R\\'enyi entropy. These distributions have the same parameter space for every $q$, and in the $q=1$ case these are the normal distributions. Some methods to endow this parameter space with Riemannian metric is presented: the second derivative of the $q$-R\\'enyi entropy, Tsallis-entropy and the relative entropy give rise to a Riemannian metric, the Fisher-information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovri\\'c, Min-Oo and Ruh. These metrics are different therefore our differential geometrical calculations based on a unified metric, which covers all the above mentioned metrics among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors and scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter $q$ modulates the statistical distinguishability of close points. We show that some frequently used metric in quantum information geometry can be easily recovered from classical metrics."}
{"category": "Math", "title": "An extension of the inductive approach to the lace expansion", "abstract": "We extend the inductive approach to the lace expansion, previously developed to study models with critical dimension 4, to be applicable more generally. In particular, the result of this note has recently been used to prove Gaussian asymptotic behaviour for the Fourier transform of the two-point function for sufficiently spread-out lattice trees in dimensions d>8, and it is potentially also applicable to percolation in dimensions d>6."}
{"category": "Math", "title": "Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition", "abstract": "We consider asymptotic behavior of the following fourth order equation \\[ \\Delta^2 u= \\rho \\frac{e^{u}}{\\int_\\Om e^{u} dx} {in} \\Om, u= \\partial_\\nu u=0 {on} \\partial \\Omega \\] where $\\Om$ is a smooth oriented bounded domain in $\\R^4$. Assuming that $0<\\rho \\leq C$, we completely characterize the asymptotic behavior of the unbounded solutions."}
{"category": "Math", "title": "On compositions of d.c. functions and mappings", "abstract": "A d.c. (delta-convex) function on a normed linear space is a function representable as a difference of two continuous convex functions. We show that an infinite dimensional analogue of Hartman's theorem on stability of d.c. functions under compositions does not hold in general. However, we prove that it holds in some interesting particular cases. Our main results about compositions are proved in the more general context of d.c. mappings between normed linear spaces."}
{"category": "Math", "title": "Covering Homology", "abstract": "We introduce the notion of \"covering homology\" of a commutative ring spectrum with respect to certain families of coverings of topological spaces. The construction of covering homology is extracted from Bokstedt, Hsiang and Madsen's topological cyclic homology. In fact covering homology with respect to the family of orientation preserving isogenies of the circle is equal to topological cyclic homology. Our basic tool for the analysis of covering homology is a cofibration sequence involving homotopy orbits and a restriction map similar to the restriction map used in Bokstedt, Hsiang and Madsen's construction of topological cyclic homology. Covering homology with respect to families of isogenies of a torus is constructed from iterated topological Hochschild homology. It receives a trace map from iterated algebraic K-theory and the hope is that the rich structure, and the calculability of covering homology will make covering homology useful in the exploration of J. Rognes' ``red shift conjecture''."}
{"category": "Math", "title": "Tight estimates for convergence of some non-stationary consensus algorithms", "abstract": "The present paper is devoted to estimating the speed of convergence towards consensus for a general class of discrete-time multi-agent systems. In the systems considered here, both the topology of the interconnection graph and the weight of the arcs are allowed to vary as a function of time. Under the hypothesis that some spanning tree structure is preserved along time, and that some nonzero minimal weight of the information transfer along this tree is guaranteed, an estimate of the contraction rate is given. The latter is expressed explicitly as the spectral radius of some matrix depending upon the tree depth and the lower bounds on the weights."}
{"category": "Math", "title": "Classification of multiplicity free Hamiltonian actions of complex tori on Stein manifolds", "abstract": "A Hamiltonian action of a complex torus on a symplectic complex manifold is said to be {\\it multiplicity free} if a general orbit is a lagrangian submanifold. To any multiplicity free Hamiltonian action of a complex torus $T\\cong (\\C^\\times)^n$ on a Stein manifold $X$ we assign a certain 5-tuple consisting of a Stein manifold $Y$, an \\'{e}tale map $Y\\to \\t^*$, a set of divisors on $Y$ and elements of $H^2(Y,\\Z)^{\\oplus n}, H^2(Y,\\C)$. We show that $X$ is uniquely determined by this invariants. Furthermore, we describe all 5-tuples arising in this way."}
{"category": "Math", "title": "Quotients of continuous convex functions on nonreflexive Banach spaces", "abstract": "On each nonreflexive Banach space X there exists a positive continuous convex function f such that 1/f is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that X is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction gives also a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals."}
{"category": "Math", "title": "Some additive applications of the isopermetric approach", "abstract": "Let $G$ be a group and let $X$ be a finite subset. The isoperimetric method investigates the objective function $|(XB)\\setminus X|$, defined on the subsets $X$ with $|X|\\ge k$ and $|G\\setminus (XB)|\\ge k$. A subset with minimal where this objective function attains its minimal value is called a $k$--fragment. In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and obtaingeneralizations and short proofs for several known results. We also give some new applications. Some of the results obtained here will be used in coming papers to improve Kempermann structure Theory."}
{"category": "Math", "title": "Non-Abelian Hopf Cohomology II -- the General Case --", "abstract": "We introduce and study non-abelian cohomology sets of Hopf algebras with coefficients in Hopf comodule algebras. We prove that these sets generalize as well Serre's non-abelian group cohomology theory as the cohomological theory constructed by the authors in a previous article. We establish their functoriality and compute explicit examples. Further we classify Hopf torsors."}
{"category": "Math", "title": "A new infinite game in Banach spaces with applications", "abstract": "We consider the following two-player game played on a separable, infinite-dimensional Banach space X. Player S chooses a positive integer k_1 and a finite-codimensional subspace X_1 of X. Then player P chooses x_1 in the unit sphere of X_1. Moves alternate thusly, forever. We study this game in the following setting. Certain normalized, 1-unconditional sequences (u_i) and (v_i) are fixed so that S has a winning strategy to force P to select x_i's so that if the moves are (k_1,X_1,x_1,k_2,X_2,x_2,...), then (x_i) is dominated by (u_{k_i}) and/or (x_i) dominates (v_{k_i}). In particular, we show that for suitable (u_i) and (v_i) if X is reflexive and S can win both of the games above, then X embeds into a reflexive space Z with an FDD which also satisfies analogous block upper (u_i) and lower (v_i) estimates. Certain universal space consequences ensue."}
{"category": "Math", "title": "Banach spaces of bounded Szlenk index", "abstract": "For a countable ordinal a we denote by C_a the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by a. We show that each C_a admits a separable, reflexive universal space. We also show that spaces in the class C_{omega^{a*omega}} embed into spaces of the same class with a basis. As a consequence we deduce that each C_a is analytic in the Effros-Borel structure of subspaces of C[0,1]."}
{"category": "Math", "title": "Rigidity of Graded Regular Algebras", "abstract": "We prove a graded version of Alev-Polo's rigidity theorem: the homogenization of the universal enveloping algebra of a semisimple Lie algebra and the Rees ring of the Weyl algebras $A_n(k)$ cannot be isomorphic to their fixed subring under any finite group action. We also show the same result for other classes of graded regular algebras including the Sklyanin algebras."}
{"category": "Math", "title": "Les espaces de Berkovich sont excellents", "abstract": "In this paper, we first study the local rings of a Berkovich analytic space from the point of view of commutative algebra. We show that those rings are excellent ; we introduce the notion of a an analytically separable extension of non-archimedean complete fields (it includes the case of the finite separable extensions, and also the case of any complete extension of a perfect complete non-archimedean field) and show that the usual commutative algebra properties (Rm, Sm, Gorenstein, Cohen-Macaulay, Complete Intersection) are stable under analytically separable ground field extensions; we also establish a GAGA principle with respect to those properties for any finitely generated scheme over an affinoid algebra. A second part of the paper deals with more global geometric notions : we define, show the existence and establish basic properties of the irreducible components of analytic space ; we define, show the existence and establish basic properties of its normalization ; and we study the behaviour of connectedness and irreducibility with respect to base change."}
{"category": "Math", "title": "Sur la p-dimension des corps", "abstract": "Let A be an excellent integral henselian local noetherian ring, k its residue field of characteristic p>0 and K its fraction field. Using an algebraization technique introduced by the first named author, and the one-dimension case already proved by Kazuya KATO, we prove the following formula: cd_p(K) = dim(A) + p-rank(k), if k is separably closed and K of characteristic zero. A similar statement is valid without those assumptions on k and K."}
{"category": "Math", "title": "The Thurston norm via Normal Surfaces", "abstract": "Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by non-negative integer weights, 14 for each 3-simplex, that describe how many copies of each oriented normal disc type there are. The Euler characteristic and homology class are both linear functions of the weights. There is a convex polytope in the space of weights, defined by linear equations given by the combinatorics of the triangulation, whose image under the homology map is the unit ball, B, of the Thurston norm. Applications of this approach include (1) an algorithm to compute B and hence the Thurston norm of any homology class, (2) an explicit exponential bound on the number of vertices of B in terms of the number of simplices in the triangulation, (3) an algorithm to determine the fibred faces of B and hence an algorithm to decide whether a 3-manifold fibres over the circle."}
{"category": "Math", "title": "Hamiltonian S^1 manifolds are uniruled", "abstract": "The main result of this note is that every closed Hamiltonian S^1 manifold is uniruled, i.e. it has a nonzero Gromov--Witten invariant one of whose constraints is a point. The proof uses the Seidel representation of \\pi_1 of the Hamiltonian group in the small quantum homology of M as well as the blow up technique recently introduced by Hu, Li and Ruan. It applies more generally to manifolds that have a loop of Hamiltonian symplectomorphisms with a nondegenerate fixed maximum. Some consequences for Hofer geometry are explored. An appendix discusses the structure of the quantum homology ring of uniruled manifolds."}
{"category": "Math", "title": "$\\R$-trees and laminations for free groups III: Currents and dual $\\R$-tree metrics", "abstract": "This is the third of a series of three articles where we introduce laminations for the free-groups. We explore here the link between currents and laminations and prove that the situation is more complicated than in the surface case of real tree dual to a measured geodesic lamination."}
{"category": "Math", "title": "A characterization of the Riesz distribution", "abstract": "Bobecka and Wesolowski (2002) have shown that, in the Olkin and Rubin characterization of the Wishart distribution (See Casalis and Letac (1996)), when we use the division algorithm defined by the quadratic representation and replace the property of invariance by the existence of twice differentiable densities, we still have a characterization of the Wishart distribution. In the present work, we show that, when we use the division algorithm defined by the Cholesky decomposition, we get a characterization of the Riesz distribution."}
{"category": "Math", "title": "Crystal isomorphisms for irreducible highest weight U_{v}{\\hat{sl}}_{e})-modules of higher level", "abstract": "We study the crystal graphs of irreducible $U_{v}(\\hat{sl}}_{e})$-modules of higher level l. Generalizing works of the first author, we obtain a simple description of the bijections between the classes of multipartitions which naturally label these graphs: the Uglov multipartitions. This is achieved by expliciting an embedding of the $U_{v}(\\hat{sl}}_{e})$-crystals of level l into $U_{v}(\\hat{sl}_{\\infty})$-crystals associated to highest weight modules."}
{"category": "Math", "title": "Explicit enumeration of triangulations with multiple boundaries", "abstract": "We enumerate rooted triangulations of a sphere with multiple holes by the total number of edges and the length of each boundary component. The proof relies on a combinatorial identity due to W.T. Tutte."}
{"category": "Math", "title": "Bethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum", "abstract": "We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dimensional $gl_2$-modules. As a byproduct we show that there exist exactly $\\binom {n}{l}-\\binom{n}{l-1}$ two-dimensional vector subspaces $V \\subset \\C[u]$ with a basis $f,g\\in V$ such that $\\deg f = l, \\deg g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$."}
{"category": "Math", "title": "Maximal slope of tensor product of Hermitian vector bundles", "abstract": "We give an upper bound for the maximal slope of the tensor product of several non-zero Hermitian vector bundles on the spectrum of an algebraic integer ring. By Minkowski's theorem, we need to estimate the Arakelov degree of an arbitrary Hermitian line subbundle $\\bar M$ of the tensor product. In the case where the generic fiber of $M$ is semistable in the sense of geometric invariant theory, the estimation is established by constructing, through the classical invariant theory, a special polynomial which does not vanish on the generic fibre of $M$. Otherwise we use an explicte version of a result of Ramanan and Ramanathan to reduce the general case to the former one."}
{"category": "Math", "title": "Culminating paths", "abstract": "Let a and b be two positive integers. A culminating path is a path of Z^2 that starts from (0,0), consists of steps (1,a) and (1,-b), stays above the x-axis and ends at the highest ordinate it ever reaches. These paths were first encountered in bioinformatics, in the analysis of similarity search algorithms. They are also related to certain models of Lorentzian gravity in theoretical physics. We first show that the language on a two letter alphabet that naturally encodes culminating paths is not context-free. Then, we focus on the enumeration of culminating paths. A step by step approach, combined with the kernel method, provides a closed form expression for the generating fucntion of culminating paths ending at a (generic) height k. In the case a=b, we derive from this expression the asymptotic behaviour of the number of culminating paths of length n. When a>b, we obtain the asymptotic behaviour by a simpler argument. When a<b, we only determine the exponential growth of the number of culminating paths. Finally, we study the uniform random generation of culminating paths via various methods. The rejection approach, coupled with a symmetry argument, gives an algorithm that is linear when a>= b, with no precomputation stage nor non-linear storage required. The choice of the best algorithm is not as clear when a<b. An elementary recursive approach yields a linear algorithm after a precomputation stage involving O(n^3) arithmetic operations, but we also present some alternatives that may be more efficient in practise."}
{"category": "Math", "title": "A Liouville type theorem for special Lagrangian Equations with constraints", "abstract": "We derive a Liouville type result for special Lagrangian equations with certain \"convexity\" and restricted linear growth assumptions on the solutions."}
{"category": "Math", "title": "The sum-product estimate for large subsets of prime fields", "abstract": "Let $\\mathbb{F}_p$ be the field of a prime order $p.$ It is known that for any integer $N\\in [1,p]$ one can construct a subset $A\\subset\\mathbb{F}_p$ with $|A|= N$ such that $$ \\max\\{|A+A|, |AA|\\}\\ll p^{1/2}|A|^{1/2}. $$ In the present paper we prove that if $A\\subset \\mathbb{F}_p$ with $|A|>p^{2/3},$ then $$ \\max\\{|A+A|, |AA|\\}\\gg p^{1/2}|A|^{1/2}. $$"}
{"category": "Math", "title": "Some naturally ocurring examples of A-infinity bialgebras", "abstract": "Let p be an odd prime. When n>2, we show that each tensor factor of form E \\otimes \\Gamma in H(Z,n;Z_p) is an A-infinity bialgebra with non-trivial structure. We give explicit formulas for the structure maps and the quadratic relations among them. Thus E \\otimes \\Gamma is a naturally occurring example of an A-infinity bialgebra whose internal structure is well-understood."}
{"category": "Math", "title": "Contact metric $(\\kappa,\\mu)$-spaces as bi-Legendrian manifolds", "abstract": "We regard a contact metric manifold whose Reeb vector field belongs to the $(\\kappa,\\mu)$-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric $(\\kappa,\\mu)$-spaces in terms of a canonical connection which can be naturally defined on them."}
{"category": "Math", "title": "The topology of the spectrum for Gelfand pairs on Lie groups", "abstract": "Given a Gelfand pair of Lie groups, we identify the spectrum with a suitable subset of C^n and we prove the equivalence between Gelfand topology and euclidean topology."}
{"category": "Math", "title": "Standard vs. Reduced Genus-One Gromov-Witten Invariants", "abstract": "We give an explicit formula for the difference between the standard and reduced genus-one Gromov-Witten invariants. Combined with previous work on geometric properties of the latter, this paper makes it possible to compute the standard genus-one GW-invariants of complete intersections. In particular, we obtain a closed formula for the genus-one GW-invariants of a Calabi-Yau projective hypersurface and verify a recent mirror symmetry prediction for a sextic fourfold as a special case."}
{"category": "Math", "title": "The Jacobian algebras", "abstract": "The Jacobian algebras are introduced and their various properties are studied."}
{"category": "Math", "title": "An ansatz for the asymptotics of hypergeometric multisums", "abstract": "Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the asymptotic expansion of such sequences is to find a recurrence satisfied by them, convert it into a differential equation satisfied by their generating series, and analyze the singulatiries in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set, for which we conjecture (and in some cases prove) that it contains all the singularities of the generating series. Our construction of this finite set is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. The finite set can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive $K$-theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples."}
{"category": "Math", "title": "On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles", "abstract": "We give some new advances in the research of the maximum number of triangles that we may obtain in a simple arrangements of n lines or pseudo-lines."}
{"category": "Math", "title": "Factorization of quadratic polynomials in the ring of formal power series over Z", "abstract": "We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\\ge 1$ and $p$ prime, we show that $p^n+p^m\\beta x+\\alpha x^2$ is reducible in $Z[[x]]$ if and only if it is reducible in $Z_p[x]$, the ring of polynomials over the $p$-adic integers."}
{"category": "Math", "title": "On knot Floer homology in double branched covers", "abstract": "Let L be a link in an thickened annulus. We specify the embedding of this annulus in the three sphere, and consider its complement thought of as the axis to L. In the right circumstances this axis lifts to a null-homologous knot in the double branched cover of the three sphere, branched over the embedded copy of L. This paper shows that the knot Floer homology of this lift, with mod 2 coefficients, can be computed from a spectral sequence starting at a type of Khovanov homology already described by Asaeda, Przytycki, and Sikora. We extend the known results about this type of Khovanov homology, and use it to provide a very simple explanation of the case when L is alternating for the obvious projection."}
{"category": "Math", "title": "On knot Floer homology for some fibered knots", "abstract": "A companion paper to \"On knot Floer homology in branched double covers\" applied to braided branched loci. We reprove the main result of that paper concerning alternating branched loci when projected to an annulus, without using Khovanov homology. This provides two advantages: 1) the results hold for integer coefficients and 2) the spin^c structures are more readily discernable. We apply this result to a branch locus which is a braid, and use the braid structure to find information about a fibered knot in the branched double cover. In some cases this provides all the information about the knot Floer homology and can be used to derive information about the Heegaard-Floer homology of associated fibered three manifolds. Results for certain positive braids are also included, establishing results similar to E. Eftekhary's in the Heegaard-Floer setting."}
{"category": "Math", "title": "The zero-one law for planar random walks in i.i.d. random environments revisited", "abstract": "In this note we present a simplified proof of the zero-one law by Merkl and Zerner (2001) for directional transience of random walks in i.i.d. random environments (RWRE) on the square lattice. Also, we indicate how to construct a two-dimensional counterexample in a non-uniformly elliptic and stationary environment which has better ergodic properties than the example given by Merkl and Zerner."}
{"category": "Math", "title": "Fast Adaptive Algorithms in the Non-Standard Form for Multidimensional Problems", "abstract": "We present a fast, adaptive multiresolution algorithm for applying integral operators with a wide class of radially symmetric kernels in dimensions one, two and three. This algorithm is made efficient by the use of separated representations of the kernel. We discuss operators of the class $(-\\Delta+\\mu^{2}I)^{-\\alpha}$, where $\\mu\\geq0$ and $0<\\alpha<3/2$, and illustrate the algorithm for the Poisson and Schr\\\"{o}dinger equations in dimension three. The same algorithm may be used for all operators with radially symmetric kernels approximated as a weighted sum of Gaussians, making it applicable across multiple fields by reusing a single implementation. This fast algorithm provides controllable accuracy at a reasonable cost, comparable to that of the Fast Multipole Method (FMM). It differs from the FMM by the type of approximation used to represent kernels and has an advantage of being easily extendable to higher dimensions."}
{"category": "Math", "title": "On the lower bound of the spectral norm of symmetric random matrices with independent entries", "abstract": "We show that the spectral radius of an $N\\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by $ 2 \\*\\sigma - o(N^{-6/11+\\epsilon}), $ where $\\sigma^2 $ is the variance of the matrix entries and $\\epsilon $ is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any $\\epsilon >0, $ one has $$ \\|A_N\\| =2 \\*\\sigma + o(N^{-6/11+\\epsilon}) $$ with probability going to 1 as $N \\to \\infty. $"}
{"category": "Math", "title": "An analogue of Szego's limit theorem in free probability theory", "abstract": "In the paper, we discuss orthogonal polynomials in free probability theory. Especially, we prove an analogue of of Szego's limit theorem in free probability theory."}
{"category": "Math", "title": "Log canonical thresholds of certain Fano hypersurfaces", "abstract": "We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an application, we show that they have a Kaehler-Einstein metric if they are general."}
{"category": "Math", "title": "Boundedness of projection operators and Ces\\`aro means in weighted $L^p$ space on the unit sphere", "abstract": "For the weight function $\\prod_{i=1}^{d+1}|x_i|^{2\\k_i}$ on the unit sphere, sharp local estimates of the orthogonal projection operators are obtained and used to prove the convergence of the Ces\\`aro $(C,\\delta)$ means in the weighted $L^p$ space for $\\delta$ below the critical index. Similar results are also proved for corresponding weight functions on the unit ball and on the simplex."}
{"category": "Math", "title": "A Conjecture on Primes and a Step towards Justification", "abstract": "We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification."}
{"category": "Math", "title": "Construction of Bayesian Deformable Models via Stochastic Approximation Algorithm: A Convergence Study", "abstract": "The problem of the definition and the estimation of generative models based on deformable templates from raw data is of particular importance for modelling non aligned data affected by various types of geometrical variability. This is especially true in shape modelling in the computer vision community or in probabilistic atlas building for Computational Anatomy (CA). A first coherent statistical framework modelling the geometrical variability as hidden variables has been given by Allassonni\\`ere, Amit and Trouv\\'e (JRSS 2006). Setting the problem in a Bayesian context they proved the consistency of the MAP estimator and provided a simple iterative deterministic algorithm with an EM flavour leading to some reasonable approximations of the MAP estimator under low noise conditions. In this paper we present a stochastic algorithm for approximating the MAP estimator in the spirit of the SAEM algorithm. We prove its convergence to a critical point of the observed likelihood with an illustration on images of handwritten digits."}
{"category": "Math", "title": "Polynomial maps that are roots of power series", "abstract": "We introduce a class of polynomial maps that we call polynomial roots of powerseries, and show that automorphisms with this property generate the automorphism group in any dimension. In particular we determine generically which polynomial maps that preserve the origin are roots of powerseries. We study the one-dimensional case in greater depth."}
{"category": "Math", "title": "Limit laws for k-coverage of paths by a Markov-Poisson-Boolean model", "abstract": "Let P := {X_i,i >= 1} be a stationary Poisson point process in R^d, {C_i,i >= 1} be a sequence of i.i.d. random sets in R^d, and {Y_i^t; t \\geq 0, i >= 1} be i.i.d. {0,1}-valued continuous time stationary Markov chains. We define the Markov-Poisson-Boolean model C_t := {Y_i^t(X_i + C_i), i >= 1}. C_t represents the coverage process at time t. We first obtain limit laws for k-coverage of an area at an arbitrary instant. We then obtain the limit laws for the k-coverage seen by a particle as it moves along a one-dimensional path."}
{"category": "Math", "title": "Free Actions of Finite Groups on $S^n \\times S^n$", "abstract": "Let $p$ be an odd prime. We construct a non-abelian extension $\\Gamma$ of $S^1$ by $Z/p \\times Z/p$, and prove that any finite subgroup of $\\Gamma$ acts freely and smoothly on $S^{2p-1} \\times S^{2p-1}$. In particular, for each odd prime $p$ we obtain free smooth actions of infinitely many non-metacyclic rank two $p$-groups on $S^{2p-1} \\times S^{2p-1}$. These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres."}
{"category": "Math", "title": "Stringy E-functions of hypersurfaces and of Brieskorn singularities", "abstract": "We show that for a hypersurface Batyrev's stringy E-function can be seen as a residue of the Hodge zeta function, a specialization of the motivic zeta function of Denef and Loeser. This is a nice application of inversion of adjunction. If an affine hypersurface is given by a polynomial that is non-degenerate with respect to its Newton polyhedron, then the motivic zeta function and thus the stringy E-function can be computed from this Newton polyhedron (by work of Artal, Cassou-Nogues, Luengo and Melle based on an algorithm of Denef and Hoornaert). We use this procedure to obtain an easy way to compute the contribution of a Brieskorn singularity to the stringy E-function. As a corollary, we prove that stringy Hodge numbers of varieties with a certain class of strictly canonical Brieskorn singularities are nonnegative. We conclude by computing an interesting 6-dimensional example. It shows that a result, implying nonnegativity of stringy Hodge numbers in lower dimensional cases, obtained in our previous paper, is not true in higher dimension."}
{"category": "Math", "title": "Upper Bounds on the Number of Vertices of Weight <=k in Particular Arrangements of Pseudocircles", "abstract": "In arrangements of pseudocircles (Jordan curves) the weight of a vertex (intersection point) is the number of pseudocircles that contain the vertex in its interior. We give improved upper bounds on the number of vertices of weight <=k in certain arrangements of pseudocircles in the plane. In particular, forbidding certain subarrangements we improve the known bound of 6n-12 (cf. Kedem et al., 1986) for vertices of weight 0 in arrangements of n pseudocircles to 4n-6. In complete arrangements (i.e. arrangements with each two pseudocircles intersecting) we identify two subarrangements of three and four pseudocircles, respectively, whose absence gives improved bounds for vertices of weight 0 and more generally for vertices of weight <=k."}
{"category": "Math", "title": "Measure-valued stochastic recurrences and the stability of queues", "abstract": "In this paper we present a stability criterion for finite measure-valued stochastic recursions, generalizing Loynes's Theorem to spaces of measures. This result provides conditions for the reach of a \"total stationary state\" for the queue with an infinity of servers and the single-server SRPT queue. Indeed, we give in both cases a condition of existence of a stationary measure-valued recursive sequence characterizing the queueing system exhaustively."}
{"category": "Math", "title": "On the Dirac delta as initial condition for nonlinear Schr\\\"odinger equations", "abstract": "In this article we will study the initial value problem for some Schr\\\"odinger equations with Diraclike initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to conclude a stability result in the defocusing setting. These problems are related to the existence of a singular dynamics for Schr\\\"odinger maps through the so called Hasimoto transformation."}
{"category": "Math", "title": "Mutations for quivers with potentials: Oberwolfach talk, April 2007", "abstract": "This is an extended abstract of my talk at the Oberwolfach Workshop \"Algebraic Groups\" (April 22 - 28, 2007). It is based on a joint work with H.Derksen and J.Weyman (arXiv:0704.0649v2 [math.RA])."}
{"category": "Math", "title": "Central limit theorems in linear structural error-in-variables models with explanatory variables in the domain of attraction of the normal law", "abstract": "Linear structural error-in-variables models with univariate observations are revisited for studying modified least squares estimators of the slope and intercept. New marginal central limit theorems (CLT's) are established for these estimators, assuming the existence of four moments for the measurement errors and that the explanatory variables are in the domain of attraction of the normal law. The latter condition for the explanatory variables is used the first time, and is so far the most general in this context. It is also optimal, or nearly optimal, for our CLT's. Moreover, due to the obtained CLT's being in Studentized and self-normalized forms to begin with, they are a priori nearly, or completely, data-based, and free of unknown parameters of the joint distribution of the error and explanatory variables. Consequently, they lead to a variety of readily available, or easily derivable, large-sample approximate confidence intervals (CI's) for the slope and intercept. In contrast, in related CLT's in the literature so far, the variances of the limiting normal distributions, in general, are complicated and depend on various, typically unknown, moments of the error and explanatory variables. Thus, the corresponding CI's for the slope and intercept in the literature, unlike those of the present paper, are available only under some additional model assumptions."}
{"category": "Math", "title": "Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions", "abstract": "We study the almost sure convergence of randomly truncated stochastic algorithms. We present a new convergence theorem which extends the already known results by making vanish the classical condition on the noise terms. The aim of this work is to prove an almost sure convergence result of randomly truncated stochastic algorithms under easily verifiable conditions"}
{"category": "Math", "title": "Maximal probabilities of convolution powers of discrete uniform distributions", "abstract": "We prove optimal constant over root $n$ upper bounds for the maximal probabilities of $n$th convolution powers of discrete uniform distributions."}
{"category": "Math", "title": "Two multivariate central limit theorems", "abstract": "In this paper, explicit error bounds are derived in the approximation of rank $k$ projections of certain $n$-dimensional random vectors by standard $k$-dimensional Gaussian random vectors. The bounds are given in terms of $k$, $n$, and a basis of the $k$-dimensional space onto which we project. The random vectors considered are two generalizations of the case of a vector with independent, identically distributed components. In the first case, the random vector has components which are independent but need not have the same distribution. The second case deals with finite exchangeable sequences of random variables."}
{"category": "Math", "title": "Holomorphic Extension of CR Functions from Quadratic Cones", "abstract": "It is proved that CR functions on a quadratic cone M in $\\C^n$, n>1, admit one-sided holomorphic extension if and only if M does not have two-sided support, a geometric condition on M which generalizes minimality in the sense of Tumanov. A biholomorphic classification of quadratic cones in $\\C^2$ is also given."}
{"category": "Math", "title": "On the zero set of the Kobayashi--Royden pseudometric of the spectral unit ball", "abstract": "Given $A\\in\\Omega_n,$ the $n^2$-dimensional spectral unit ball, we show that $B$ is a \"generalized\" tangent vector at $A$ to an entire curve in $\\Omega_n$ if and only if $B$ is in the tangent cone $C_A$ to the isospectral variety at $A.$ In the case of $\\Omega_3,$ the zero set of this metric is completely described."}
{"category": "Math", "title": "Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves", "abstract": "Let ${\\cal M}_{g,[n]}$, for $2g-2+n>0$, be the D-M moduli stack of smooth curves of genus $g$ labeled by $n$ unordered distinct points. The main result of the paper is that a finite, connected \\'etale cover ${\\cal M}^\\l$ of ${\\cal M}_{g,[n]}$, defined over a sub-$p$-adic field $k$, is \"almost\" anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces. The precise result is the following. Let $\\pi_1({\\cal M}^\\l_{\\ol{k}})$ be the geometric algebraic fundamental group of ${\\cal M}^\\l$ and let ${Out}^*(\\pi_1({\\cal M}^\\l_{\\ol{k}}))$ be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of ${\\cal M}^\\l$ (this is the \"$\\ast$-condition\" motivating the \"almost\" above). Let us denote by ${Out}^*_{G_k}(\\pi_1({\\cal M}^\\l_{\\ol{k}}))$ the subgroup consisting of elements which commute with the natural action of the absolute Galois group $G_k$ of $k$. Let us assume, moreover, that the generic point of the D-M stack ${\\cal M}^\\l$ has a trivial automorphisms group. Then, there is a natural isomorphism: $${Aut}_k({\\cal M}^\\l)\\cong{Out}^*_{G_k}(\\pi_1({\\cal M}^\\l_{\\ol{k}})).$$ This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-$p$-adic fields."}
{"category": "Math", "title": "Shape derivative of the first eigenvalue of the 1-Laplacian", "abstract": "We compute the shape derivative of the first eigenvalue of the 1-Laplacian. As an application, we find that a ball is critical among all volume-preserving deformations."}
{"category": "Math", "title": "Schrodinger equations and Hamiltonian systems of PDEs with selfdual boundary conditions", "abstract": "Selfdual variational calculus is further refined and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, Hamiltonian systems of PDEs, as well as certain nonlinear Schrodinger evolutions. This allows for the resolution of such equations under general time boundary conditions which include the more traditional ones such as initial value problems, periodic and anti-periodic orbits, but also yield new ones such as \"periodic orbits up to an isometry\" for evolution equations that may not have periodic solutions. In the process, we introduce a method for perturbing selfdual functionals in order to induce coercivity and compactness, while keeping the system selfdual."}
{"category": "Math", "title": "On the spectrum of the twisted Dolbeault Laplacian over K\\\"ahler manifolds", "abstract": "We use Dirac operator techniques to a establish sharp lower bound for the first eigenvalue of the Dolbeault Laplacian twisted by Hermitian-Einstein connections on vector bundles of negative degree over compact K\\\"ahler manifolds."}
{"category": "Math", "title": "On the optimality of Stein factors", "abstract": "The application of Stein's method for distributional approximation often involves so called Stein factors (also called 'magic factors') in the bound of the solutions to Stein equations. However, in some cases these factors contain additional (undesirable) logarithmic terms. It has been shown for many Stein factors that the known bounds are sharp and thus that these additional logarithmic terms cannot be avoided in general. However, no probabilistic examples have appeared in the literature that would show that these terms in the Stein factors are not just unavoidable artefacts, but that they are there for a good reason. In this article we close this gap by constructing such examples. This also leads to a new interpretation of the solutions to Stein equations."}
{"category": "Math", "title": "Adaptive Optimal Nonparametric Regression and Density Estimation Based on Fourier-Legendre Expansion", "abstract": "Motivated by finance and technical applications, the objective of this paper is to consider adaptive estimation of regression and density distribution based on Fourier-Legendre expansion, and construction of confidence intervals - also adaptive. The estimators are asymptotically optimal and adaptive in the sense that they can adapt to unknown smoothness."}
{"category": "Math", "title": "Survey on eigenvalues of the Dirac operator and geometric structures", "abstract": "We give a survey of results relating the restricted holonomy of a Riemannian spin manifold with lower bounds on the spectrum of its Dirac operator, giving a new proof of a result originally due to Kirchberg."}
{"category": "Math", "title": "Some remarks on the generalized Tanaka-Webster connection of a contact metric manifold", "abstract": "We find necessary and sufficient conditions for the bi-Legendrian connection $\\nabla$ associated to a bi-Legendrian structure $(\\cal F,\\cal G)$ on a contact metric manifold $(M,\\phi,\\xi,\\eta,g)$ being a metric connection and then we give conditions ensuring that $\\nabla$ coincides with the (generalized) Tanaka-Webster connection of $(M,\\phi,\\xi,\\eta,g)$. Using these results, we give some interpretations of the Tanaka-Webster connection and we study the interactions between the Tanaka-Webster, the bi-Legendrian and the Levi Civita connection in a Sasakian manifold."}
{"category": "Math", "title": "Expected gaps between prime numbers", "abstract": "We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations. Using this recursion we can estimate the numbers of a gap or of a constellation that occur between a prime and its square. This recursion also has explicit implications for open questions about gaps between prime numbers, including three questions posed by Erd\\\"os and Tur\\'an."}
{"category": "Math", "title": "Latin Square Thue-Morse Sequences are Overlap-Free", "abstract": "We define a morphism based upon a Latin square that generalizes the Thue-Morse morphism. We prove that fixed points of this morphism are overlap-free sequences generalizing results of Allouche - Shallit and Frid."}
{"category": "Math", "title": "Entropy and Variational principles for holonomic probabilities of IFS", "abstract": "Associated to a IFS one can consider a continuous map $\\hat{\\sigma} : [0,1]\\times \\Sigma \\to [0,1]\\times \\Sigma$, defined by $\\hat{\\sigma}(x,w)=(\\tau_{X_{1}(w)}(x), \\sigma(w))$ were $\\Sigma=\\{0,1, ..., d-1\\}^{\\mathbb{N}}$, $\\sigma: \\Sigma \\to \\Sigma$ is given by$\\sigma(w_{1},w_{2},w_{3},...)=(w_{2},w_{3},w_{4}...)$ and $X_{k} : \\Sigma \\to \\{0,1, ..., n-1\\}$ is the projection on the coordinate $k$. A $\\rho$-weighted system, $\\rho \\geq 0$, is a weighted system $([0,1], \\tau_{i}, u_{i})$ such that there exists a positive bounded function $h : [0,1] \\to \\mathbb{R}$ and probability $\\nu $ on $[0,1]$ satisfying $ P_{u}(h)=\\rho h, \\quad P_{u}^{*}(\\nu)=\\rho\\nu$. A probability $\\hat{\\nu}$ on $[0,1]\\times \\Sigma$ is called holonomic for $\\hat{\\sigma}$ if $ \\int g \\circ \\hat{\\sigma} d\\hat{\\nu}= \\int g d\\hat{\\nu}, \\forall g \\in C([0,1])$. We denote the set of holonomic probabilities by ${\\cal H}$. Via disintegration, holonomic probabilities $\\hat{\\nu}$ on $[0,1]\\times \\Sigma$ are naturally associated to a $\\rho$-weighted system. More precisely, there exist a probability $\\nu$ on $[0,1]$ and $u_i, i\\in\\{0, 1,2,..,d-1\\}$ on $[0,1]$, such that is $P_{u}^*(\\nu)=\\nu$. We consider holonomic ergodic probabilities. For a holonomic probability we define entropy. Finally, we analyze the problem: given $\\phi \\in \\mathbb{B}^{+}$, find the solution of the maximization pressure problem $$p(\\phi)=$$"}
{"category": "Math", "title": "Universal inequalities for the eigenvalues of Laplace and Schr\\\"odinger operators on submanifolds,", "abstract": "We establish inequalities for the eigenvalues of Schr\\\"{o}dinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, P\\'olya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schr\\\"{o}dinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group. Among the consequences of this analysis are an extension of Reilly's inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces."}
{"category": "Math", "title": "On the fixed point property in direct sums of Banach spaces with strictly monotone norms", "abstract": "It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis."}
{"category": "Math", "title": "Principal bundles on $p$-adic curves and parallel transport", "abstract": "We define functorial isomorphisms of parallel transport along \\'etale paths for a class of principal $G$-bundles on a $p$-adic curve. Here $G$ is a connected reductive algebraic group of finite presentation and the considered principal bundles are just those with potentially strongly semistable reduction of degree zero. The constructed isomorphisms yield continous functors from the \\'etale fundamental groupoid of the given curve to the category of topological spaces with a simply transitive continous right $G(\\mathbb{C}_{p})$-action. This generalizes a construction in the case of vector bundles on a $p$-adic curve by Deninger and Werner. It may be viewed as a partial $p$-adic analogue of the classical theory by Ramanathan of principal bundles on compact Riemann surfaces, which generalizes the classical Narasimhan--Seshadri theory of vector bundles on compact Riemann surfaces."}
{"category": "Math", "title": "Controller synthesis for bisimulation equivalence", "abstract": "The objective of this paper is to solve the controller synthesis problem for bisimulation equivalence in a wide variety of scenarios including discrete-event systems, nonlinear control systems, behavioral systems, hybrid systems and many others. This will be accomplished by showing that the arguments underlying proofs of existence and methods for the construction of controllers are extraneous to the particular class of systems being considered and thus can be presented in greater generality."}
{"category": "Math", "title": "Irrationality of motivic series of Chow varieties", "abstract": "The Euler characteristic of all the Chow varieties, of a fixed projective variety, can be collected in a formal power series called the Euler-Chow series. This series coincides with the Hilbert series when the Picard group is a finite generated free abelian group. It is an interesting open problem to find for which varieties this series is rational. A few cases have been computed, and it is suspected that the series is not rational for the blow up of P^2 at nine points in general position. It is very natural to extend this series to Chow motives and ask the question if the series is rational or to find a counterexample. In this short paper we generalized the series and show by an example that the series is not rational. This opens the question of what is the geometrical meaning of the Euler-Chow series."}
{"category": "Math", "title": "Infinite Product Decomposition of Orbifold Mapping Spaces", "abstract": "Physicists showed that the generating function of orbifold elliptic genera of symmetric orbifolds can be written as an infinite product. We show that there exists a geometric factorization on space level behind this infinite product formula in much more general framework, where factors in the infinite product correspond to isomorphism classes of connected finite covering spaces of manifolds involved. From this formula, a concept of geometric Hecke operators for functors emerges. We show that these Hecke operators indeed satisfy the usual identity of Hecke operators for the case of 2-dimensional tori."}
{"category": "Math", "title": "Cap Products in String Topology", "abstract": "Chas and Sullivan showed that the homology of the free loop space LM of an oriented closed smooth finite dimensional manifold M admits the structure of a Batalin-Vilkovisky (BV) algebra equipped with an associative product called the loop product and a Lie bracket called the loop bracket. We show that the cap product is compatible with the above two products in the loop homology. Namely, the cap product with cohomology classes coming from M via the circle action acts as derivations on loop products as well as on loop brackets. We show that Poisson identities and Jacobi identities hold for the cap product action, extending the BV structure in the loop homology to the one including the cohomology of M. Finally, we describe the cap product in terms of the BV algebra structure in the loop homology."}
{"category": "Math", "title": "Coarse differentiation of quasi-isometries II: Rigidity for Sol and Lamplighter groups", "abstract": "In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1]."}
{"category": "Math", "title": "On sums of primes from Beatty sequences", "abstract": "Let $k \\ge 2$ and $\\alpha_1, \\beta_1, ..., \\alpha_k, \\beta_k$ be reals such that the $\\alpha_i$'s are irrational and greater than 1. Suppose further that some ratio $\\alpha_i/\\alpha_j$ is irrational. We study the representations of an integer $n$ in the form $$ p_1 + p_2 + ... + p_k = n, $$ where $p_i$ is a prime from the Beatty sequence $$ \\mathcal B_i = \\left\\{n \\in \\mathbb N : n = [ \\alpha_i m + \\beta_i ] \\text{for some} m \\in \\mathbb Z \\right\\}. $$"}
{"category": "Math", "title": "Flat Moebius strips of given isotopy types in R^3 whose centerlines are geodesics or lines of curvature", "abstract": "We construct real analytic flat Moebius strips of arbitrary isotopy types, whose centerlines are geodesics or lines of curvature."}
{"category": "Math", "title": "The Local Langlands Conjecture for GSp(4)", "abstract": "We prove the local Langlands conjecture for $GSp_4(F)$ where $F$ is a non-archimedean local field of characteristic zero."}
{"category": "Math", "title": "Growth and mixing", "abstract": "Given a bi-Lipschitz measure-preserving homeomorphism of a compact metric measure space of finite dimension, consider the sequence formed by the Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of this sequence assuming that our homeomorphism mixes a Lipschitz function. In particular, we get a universal lower bound which depends on the dimension of the space but not on the rate of mixing. Furthermore, we get a lower bound on the growth rate in the case of rapid mixing. The latter turns out to be sharp: the corresponding example is given by a symbolic dynamical system associated to the Rudin-Shapiro sequence."}
{"category": "Math", "title": "Winding numbers and SU(2)-representations of knot groups", "abstract": "Given an abelian group $A$ and a Lie group $G$, we construct a bilinear pairing from $A\\times\\pi_1({\\mathcal R})$ to $\\pi_1(G)$, where $\\mathcal R$ is a subvariety of the variety of representations $A\\to G$. In the case where $A$ is the peripheral subgroup of a torus or two-bridge knot group, $G=S^1$ and $\\mathcal R$ is a certain variety of representations arising from suitable SU(2)-representations of the knot group, we show that this pairing is not identically zero. We discuss the consequences of this result for the SU(2)-representations of fundamental groups of manifolds obtained by Dehn surgery on such knots."}
{"category": "Math", "title": "An Optimal Algorithm to Generate Pointed Trivalent Diagrams and Pointed Triangular Maps", "abstract": "A trivalent diagram is a connected, two-colored bipartite graph (parallel edges allowed but not loops) such that every black vertex is of degree 1 or 3 and every white vertex is of degree 1 or 2, with a cyclic order imposed on every set of edges incident to to a same vertex. A rooted trivalent diagram is a trivalent diagram with a distinguished edge, its root. We shall describe and analyze an algorithm giving an exhaustive list of rooted trivalent diagrams of a given size (number of edges), the list being non-redundant in that no two diagrams of the list are isomorphic. The algorithm will be shown to have optimal performance in that the time necessary to generate a diagram will be seen to be bounded in the amortized sense, the bound being independent of the size of the diagrams. That's what we call the CAT property. One objective of the paper is to provide a reusable theoretical framework for algorithms generating exhaustive lists of complex combinatorial structures with attention paid to the case of unlabeled structures and to those generators having the CAT property."}
{"category": "Math", "title": "A Counterexample to the Quantizability of Modules", "abstract": "Let a Poisson structure on a manifold M be given. If it vanishes at a point m, the evaluation at m defines a one dimensional representation of the Poisson algebra of functions on M. We show that this representation can, in general, not be quantized. Precisely, we give a counterexample for M=R^n, such that: (i) The evaluation map at 0 can not be quantized to a representation of the algebra of functions with product the Kontsevich product associated to the Poisson structure. (ii) For any formal Poisson structure extending the given one and vanishing at zero up to second order in epsilon, (i) still holds. We do not know whether the second claim remains true if one allows the higher order terms in epsilon to attain nonzero values at zero."}
{"category": "Math", "title": "A method to find ideal points from ideal triangulations", "abstract": "We give a simple method to find ideal points of the character variety of a 3-manifold from an ideal triangulation."}
{"category": "Math", "title": "Congruence for rational points over finite fields and coniveau over local fields", "abstract": "If the $\\ell$-adic cohomology of a projective smooth variety, defined over a local field $K$ with finite residue field $k$, is supported in codimension $\\ge 1$, then every model over the ring of integers of $K$ has a $k$-rational point. For $K$ a $p$-adic field, this is math/0405318, Theorem 1.1. If the model $\\sX$ is regular, one has a congruence $|\\sX(k)|\\equiv 1 $ modulo $|k|$ for the number of $k$-rational points 0704.1273, Theorem 1.1. The congruence is violated if one drops the regularity assumption."}
{"category": "Math", "title": "Spacelike mean curvature one surfaces in de Sitter 3-space", "abstract": "The first author studied spacelike constant mean curvature one (CMC-1) surfaces in de Sitter 3-space when the surfaces have no singularities except within some compact subset and are of finite total curvature on the complement of this compact subset. However, there are many CMC-1 surfaces whose singular sets are not compact. In fact, such examples have already appeared in the construction of trinoids given by Lee and the last author via hypergeometric functions. In this paper, we improve the Osserman-type inequality given by the first author. Moreover, we shall develop a fundamental framework that allows the singular set to be non-compact, and then will use it to investigate the global behavior of CMC-1 surfaces."}
{"category": "Math", "title": "Riemann-Roch theorems and elliptic genus for virtually smooth Schemes", "abstract": "For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable sheaves."}
{"category": "Math", "title": "v1-periodic homotopy groups of the Dwyer-Wilkerson space", "abstract": "The Dwyer-Wilkerson space DI(4) is the only exotic 2-compact group. We compute its v1-periodic homotopy groups."}
{"category": "Math", "title": "A proof of Culter's theorem on the existence of periodic orbits in polygonal outer billiards", "abstract": "We discuss a recent result by C. Culter: every polygonal outer billiard has a periodic trajectory."}
{"category": "Math", "title": "On adapted coordinate systems", "abstract": "The notion of an adapted coordinate system, introduced by V.I.Arnol'd, plays an important role in the study of asymptotic expansions of oscillatory integrals. In two dimensions, A.N.Varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for a class of analytic functions without multiple components. Varchenko's proof is based on Hironaka's theorem on the resolution of singularities. In this article, we present a new, elementary and concrete approach to these results, which is based on the Puiseux series expansion of roots of the given function. Our method applies to arbitrary real analytic functions, and even extends to arbitrary smooth functions of finite type. Moreover, by avoiding Hironaka's theorem, we can give necessary and sufficient conditions for the adaptedness of a given coordinate system in the smooth, finite type setting."}
{"category": "Math", "title": "Sharp $L^p$-estimates for maximal operators associated to hypersurfaces in $\\bR^3$ for $p>2.$", "abstract": "We study the boundedness problem for maximal operators $\\M$ associated to smooth hypersurfaces $S$ in 3-dimensional Euclidean space. For $p>2,$ we prove that if no affine tangent plane to $S$ passes through the origin and $S$ is analytic, then the associated maximal operator is bounded on $L^p(\\RR^3)$ if and only if $p>h(S),$ where $h(S)$ denotes the so-called height of the surface $S.$ For non-analytic finite type $S$ we obtain the same statement with the exception of the exponent $p=h(S).$ Our notion of height $h(S)$ is closely related to A. N. Varchenko's notion of height $h(\\phi)$ for functions $\\phi$ such that $S$ can be locally represented as the graph of $\\phi$ after a rotation of coordinates. Several consequences of this result are discussed. In particular we verify a conjecture by E.M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on $S$ and the $L^p$-boundedness of the associated maximal operator $\\M$, and a conjecture by Iosevich and Sawyer which relates the $L^p$-boundedness of $\\M$ to an integrability condition on $S$ for the distance function to tangential hyperplanes, in dimension three. In particular, we also give ess. sharp uniform estimates for the Fourier transform of the surface measure on $S,$ thus extending a result by V.N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V.I. Arnol'd in our context."}
{"category": "Math", "title": "A Family of $q$-Dyson Style Constant Term Identities", "abstract": "By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud $q$-Dyson Theorem, we establish a family of $q$-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of the $q$-Dyson product, including three conjectures of Sills' as special cases and generalizing Stembridge's first layer formulas for characters of $SL(n,\\mathbb{C})$."}
{"category": "Math", "title": "A natural Lie superalgebra bundle on rank three WSD manifolds", "abstract": "We determine the structure of the $*$-Lie superalgebra generated by a set of carefully chosen natural operators of an orientable WSD manifold of rank three. This Lie superalgebra is formed by global sections of a natural Lie superalgebra bundle, and turns out to be a product of $\\mathbf{sl}(4,\\C)$ with the full special linear superalgebras of some graded vector spaces isotypical with respect to a natural action of $\\mathbf{so}(3,\\R)$. We provide an explicit description of one of the real forms of this superalgebra, which is geometrically natural being made of $\\mathbf{so}(3,\\R)$-invariant operators which preserve the Poincar\\'e (odd Hermitean) inner product on the bundle of forms."}
{"category": "Math", "title": "Equivariant Lefschetz number of differential operators", "abstract": "Let $G$ be a compact Lie group acting on a compact complex manifold $M$. We prove a trace density formula for the $G$-Lefschetz number of a differential operator on $M$. We generalize Engeli and Felder's recent results to orbifolds."}
{"category": "Math", "title": "Simultaneous approximation of a real number by all conjugates of an algebraic number", "abstract": "Using a method of H. Davenport and W. M. Schmidt, we show that, for each positive integer n, the ratio 2/n is the optimal exponent of simultaneous approximation to real irrational numbers 1) by all conjugates of algebraic numbers of degree n, and 2) by all but one conjugates of algebraic integers of degree n+1."}
{"category": "Math", "title": "Differential Equations on Complex Projective Hypersurfaces of Low Dimension", "abstract": "Let $n=2,3,4,5$ and let $X$ be a smooth complex projective hypersurface of $\\mathbb P^{n+1}$. In this paper we find an effective lower bound for the degree of $X$, such that every holomorphic entire curve in $X$ must satisfy an algebraic differential equation of order $k=n=\\dim X$, and also similar bounds for order $k>n$. Moreover, for every integer $n\\ge 2$, we show that there are no such algebraic differential equations of order $k<n$ for a smooth hypersurface in $\\mathbb P^{n+1}$."}
{"category": "Math", "title": "Polynomial functors and opetopes", "abstract": "We give an elementary and direct combinatorial definition of opetopes in terms of trees, well-suited for graphical manipulation and explicit computation. To relate our definition to the classical definition, we recast the Baez-Dolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the Baez-Dolan construction, starting with the trivial monad. We show that our notion of opetope agrees with Leinster's. Next we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the Baez-Dolan construction. A final section is devoted to example computations, and indicates also how the calculus of opetopes is well-suited for machine implementation."}
{"category": "Math", "title": "Infinite-dimensional diffusions as limits of random walks on partitions", "abstract": "The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtained from certain Markov chains on partitions of natural numbers n in a scaling limit as n goes to infinity. These Markov chains arise in a natural way, due to the approximation of the infinite symmetric group by the increasing chain of the finite symmetric groups. Each z-measure P_z serves as a unique invariant distribution for the corresponding diffusion process, and the process is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so that the process is reversible. We describe the spectrum of its generator and compute the associated (pre)Dirichlet form."}
{"category": "Math", "title": "Local Structure of Ideal Shapes of Knots, II, Constant Curvature Case", "abstract": "The thickness, NIR(K) of a knot or link K is defined to be the radius of the largest solid tube one can put around the curve without any self intersections, which is also known as the normal injectivity radius of K. For C^{1,1} curves K, NIR(K)=min{(1/2)DCSC(K),(1/(supkappa(K))))}, where kappa(K) is the generalized curvature, and the double critical self distance DCSD(K) is the shortest length of the segments perpendicular to K at both end points. The knots and links in ideal shapes (or tight knots or links) belong to the minima of ropelength = length/thickness within a fixed isotopy class. In this article, we prove that NIR(K)=(1/2)DCSC(K), for every relative minimum K of ropelength in R^n for certain dimensions n, including n=3."}
{"category": "Math", "title": "Group gradings on simple Lie algebras of type A in positive characteristic", "abstract": "In this paper we consider gradings by a finite abelian group $G$ on the Lie algebra $\\mathfrak{sl}_n(F)$ over an algebraically closed field $F$ of characteristic different from 2 and not dividing $n$."}
{"category": "Math", "title": "Estimates of the best Sobolev constant of the embedding of $BV(\\Omega)$ into $L^1(\\partial\\Omega)$ and related shape optimization problems", "abstract": "In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality $\\lambda_1(\\Omega)\\|u\\|_{L^1(\\partial\\Omega)} \\le \\|u\\|_{W^{1,1}(\\Omega)}$ that are independent of $\\Omega$. This estimates generalize those of \\cite{BS} concerning the $p$-Laplacian to the case $p=1$. We apply our results to prove existence of an extremal for this embedding. We then study an optimal design problem related to $\\lambda_1$, and eventually compute the shape derivative of the functional $\\Omega\\to\\lambda_1(\\Omega)$. As a consequence, we obtain that a ball of $\\R^n$ of radius $n$ is critical for volume-preserving deformations."}
{"category": "Math", "title": "Tridiagonal pairs of Krawtchouk type", "abstract": "Let $K$ denote an algebraically closed field with characteristic 0 and let $V$ denote a vector space over $K$ with finite positive dimension. Let $A,A^*$ denote a tridiagonal pair on $V$ with diameter $d$. We say that $A,A^*$ has Krawtchouk type whenever the sequence $\\lbrace d-2i\\rbrace_{i=0}^d$ is a standard ordering of the eigenvalues of $A$ and a standard ordering of the eigenvalues of $A^*$. Assume $A,A^*$ has Krawtchouk type. We show that there exists a nondegenerate symmetric bilinear form $< , >$ on $V$ such that $<Au,v>= < u,Av>$ and $<A^*u,v >= < u,A^*v>$ for $u,v\\in V$. We show that the following tridiagonal pairs are isomorphic: (i) $A,A^*$; (ii) $-A,-A^*$; (iii) $A^*,A$; (iv) $-A^*,-A$. We give a number of related results and conjectures."}
{"category": "Math", "title": "Rota-Baxter Categories", "abstract": "We introduce Rota-Baxter categories and construct examples of such structures."}
{"category": "Math", "title": "Index reduction for Brauer classes via stable sheaves (with an appendix by Bhargav Bhatt)", "abstract": "We use twisted sheaves to study the problem of index reduction for Brauer classes. In general terms, this problem may be phrased as follows: given a field $k$, a $k$-variety $X$, and a class $\\alpha \\in \\Br(k)$, compute the index of the class $\\alpha_{k(X)} \\in \\Br(X)$ obtained from $\\alpha$ by extension of scalars to $k(X)$. We give a general method for computing index reduction which refines classical results of Schofield and van den Bergh. When $X$ is a curve of genus 1, we use Atiyah's theorem on the structure of stable vector bundles with integral slope to show that our formula simplifies dramatically, giving a complete solution to the index reduction problem in this case. Using the twisted Fourier-Mukai transform, we show that a similarly simple formula describes homogeneous index reduction on torsors under higher-dimensional abelian varieties."}
{"category": "Math", "title": "Regularity of Dirichlet nearly minimizing multiple-valued functions", "abstract": "In this paper, we extend the related notions of Dirichlet quasiminimizer, $\\omega-$minimizer and almost minimizer to the framework of multiple-valued functions in the sense of Almgren and prove Holder regularity results. We also give examples of those minimizers with various branch sets."}
{"category": "Math", "title": "Some extremal problems related to Bell-type inequalities", "abstract": "The best approximation by bounded product functions is calculated for some very simple two-valued functions of two variables."}
{"category": "Math", "title": "Generalizations of Khovanskii's theorems on growth of sumsets in abelian semigroups", "abstract": "We show that if $P$ is a lattice polytope in the nonnegative orthant of $\\R^k$ and $\\chi$ is a coloring of the lattice points in the orthant such that the color $\\chi(a+b)$ depends only on the colors $\\chi(a)$ and $\\chi(b)$, then the number of colors of the lattice points in the dilation $nP$ of $P$ is for large $n$ given by a polynomial (or, for rational $P$, by a quasipolynomial). This unifies a classical result of Ehrhart and Macdonald on lattice points in polytopes and a result of Khovanski\\u\\i{} on sumsets in semigroups. We also prove a strengthening of multivariate generalizations of Khovanski\\u\\i's theorem. Another result of Khovanski\\u\\i{} states that the size of the image of a finite set after $n$ applications of mappings from a finite family of mutually commuting mappings is for large $n$ a polynomial. We give a combinatorial proof of a multivariate generalization of this theorem."}
{"category": "Math", "title": "Computational techniques for proving identities in symmetric compositions", "abstract": "We present in this work a complete session in a Mathematica notebook. The aim of this notebook is to check identities in symmetric compositions. This notebook is a complement of our work [1] and it has all the explicit computations. We refer the reader to that paper which can be seen in http://www.uibk.ac.at/mathematik/loos/jordan/index.html. First of all we will present a few number of comands in order to simplify identities by extracting scalars, SOut. The rest of the strategy holds on the powerfull of using patterns and rules."}
{"category": "Math", "title": "Characterization of the matrix whose norm is determined by its action on decreasing sequences", "abstract": "Let $A=(a_{j,k})_{j,k \\ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\\|A\\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences."}
{"category": "Math", "title": "A slight improvement to Korenblum's constant", "abstract": "Let $A^2(D)$ be the Bergman space over the open unit disk $D$ in the complex plane. Korenblum conjectured that there is an absolute constant $c \\in (0,1)$ such that whenever $|f(z)|\\le |g(z)|$ in the annulus $c<|z|<1$ then $||f(z)|| \\le ||g(z)||$.In 2004 C.Wang gave an upper bound on $c$,that is, $c < 0.67795$, and in 2006 A.Schuster gave a lower bound ,$c > 0.21 $ .In this paper we slightly improve the upper bound for $c$."}
{"category": "Math", "title": "On the threshold for k-regular subgraphs of random graphs", "abstract": "The $k$-core of a graph is the largest subgraph of minimum degree at least $k$. We show that for $k$ sufficiently large, the $(k + 2)$-core of a random graph $\\G(n,p)$ asymptotically almost surely has a spanning $k$-regular subgraph. Thus the threshold for the appearance of a $k$-regular subgraph of a random graph is at most the threshold for the $(k+2)$-core. In particular, this pins down the point of appearance of a $k$-regular subgraph in $\\G(n,p)$ to a window for $p$ of width roughly $2/n$ for large $n$ and moderately large $k$."}
{"category": "Math", "title": "Symmetric linear functions of the restricted quantum group $\\bar{U}_qsl_2(\\mathbb{C})$", "abstract": "We determine a set of primitive idempotents and the basic algebra of the restricted quantum group $\\bar{U}_qsl_2(\\mathbb{C})$. As a result, we can show the dimension of the space of symmetric linear functions of $\\bar{U}_qsl_2(\\mathbb{C})$ is $3p-1$"}
{"category": "Math", "title": "Comparison Geometry for the Bakry-Emery Ricci Tensor", "abstract": "For Riemannian manifolds with a measure $(M,g, e^{-f} dvol_g)$ we prove mean curvature and volume comparison results when the $\\infty$-Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or $\\partial_r f$ is bounded from below, generalizing the classical ones (i.e. when $f$ is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when $f$ is bounded. Simple examples show the bound on $f$ is necessary for these results."}
{"category": "Math", "title": "Thurston obstructions and Ahlfors regular conformal dimension", "abstract": "Let $f: S^2 \\to S^2$ be an expanding branched covering map of the sphere to itself with finite postcritical set $P_f$. Associated to $f$ is a canonical quasisymmetry class $\\GGG(f)$ of Ahlfors regular metrics on the sphere in which the dynamics is (non-classically) conformal. We show \\[ \\inf_{X \\in \\GGG(f)} \\hdim(X) \\geq Q(f)=\\inf_\\Gamma \\{Q \\geq 2: \\lambda(f_{\\Gamma,Q}) \\geq 1\\}.\\] The infimum is over all multicurves $\\Gamma \\subset S^2-P_f$. The map $f_{\\Gamma,Q}: \\R^\\Gamma \\to \\R^\\Gamma$ is defined by \\[ f_{\\Gamma, Q}(\\gamma) =\\sum_{[\\gamma']\\in\\Gamma} \\sum_{\\delta \\sim \\gamma'} \\deg(f:\\delta \\to \\gamma)^{1-Q}[\\gamma'],\\] where the second sum is over all preimages $\\delta$ of $\\gamma$ freely homotopic to $\\gamma'$ in $S^2-P_f$, and $ \\lambda(f_{\\Gamma,Q})$ is its Perron-Frobenius leading eigenvalue. This generalizes Thurston's observation that if $Q(f)>2$, then there is no $f$-invariant classical conformal structure."}
{"category": "Math", "title": "Test ideals vs. multiplier ideals", "abstract": "The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via vanishing theorems. In this note we give several examples to emphasize the different behavior of test ideals and multiplier ideals. Our main result is that every ideal in an F-finite regular local ring can be written as a generalized test ideal. We also prove the semicontinuity of $F$-pure thresholds (though the analogue of the Generic Restriction Theorem for multiplier ideals does not hold)."}
{"category": "Math", "title": "Some Comments around The Examples against The Generalized Jacobian Conjecture", "abstract": "We have studied a faded problem, the Jacobian Conjecture ~: \\noindent {\\sf The Jacobian Conjecture $(JC_n)$}~: If $f_1, \\cdots, f_n$ are elements in a polynomial ring $k[X_1, \\cdots, X_n]$ over a field $k$ of characteristic $0$ such that the Jacobian $\\det(\\partial f_i/ \\partial X_j) $ is a nonzero constant, then $k[f_1, \\cdots, f_n] = k[X_1, \\cdots, X_n]$. For this purpose, we generalize it to the following form~: \\noindent {\\sf The Generalized Jacobian Conjecture $(GJC)$}~: {\\it Let $\\varphi : S \\rightarrow T$ be an unramified homomorphism of Noetherian domains with $T^\\times = \\varphi(S^\\times)$. Assume that $T$ is a factorial domain and that $S$ is a simply connected normal domain. Then $\\varphi$ is an isomorphism. } For the consistency of our discussion, we raise some serious (or idiot) questions and some comments concerning the examples appeared in the papers published by the certain excellent mathematicians (though we are unwilling to deal with them). Since the existence of such examples would be against our original target Conjecture$(GJC)$, we have to dispute their arguments about the existence of their respective (so called) counter-examples. Our conclusion is that they are not perfect counter-examples as are shown explicitly."}
{"category": "Math", "title": "Additive Regression Model for Continuous Time Processes", "abstract": "In the setting of additive regression model for continuous time process, we establish the optimal uniform convergence rates and optimal asymptotic quadratic error of additive regression. To build our estimate, we use the marginal integration method."}
{"category": "Math", "title": "A one dimensional analysis of turbulence and its intermittence for the d-dimensional stochastic Burgers equation", "abstract": "The inviscid limit of the stochastic Burgers equation is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one dimensional) action function. We demonstrate that the geometry of the caustic, level surfaces and Maxwell set can change infinitely rapidly causing turbulent behaviour which is stochastic in nature. The intermittence of this turbulence is demonstrated in terms of the recurrence of two processes."}
{"category": "Math", "title": "Some Uniform Limit Results in Additive Regression Model", "abstract": "We establish some uniform limit results in the setting of additive regression model estimation. Our results allow to give an asymptotic 100% confidence bands for these components. These results are stated in the framework of i.i.d random vectors when the marginal integration estimation method is used."}
{"category": "Math", "title": "Stein's method and Poisson process approximation for a class of Wasserstein metrics", "abstract": "Based on Stein's method, we derive upper bounds for Poisson process approximation in the $L_1$-Wasserstein metric $d_2^{(p)}$, which is based on a slightly adapted $L_p$-Wasserstein metric between point measures. For the case $p=1$, this construction yields the metric $d_2$ introduced in [Barbour and Brown Stochastic Process. Appl. 43 (1992) 9--31], for which Poisson process approximation is well studied in the literature. We demonstrate the usefulness of the extension to general $p$ by showing that $d_2^{(p)}$-bounds control differences between expectations of certain $p$th order average statistics of point processes. To illustrate the bounds obtained for Poisson process approximation, we consider the structure of 2-runs and the hard core model as concrete examples."}
{"category": "Math", "title": "A one dimensional analysis of singularities and turbulence for the stochastic Burgers equation in d-dimensions", "abstract": "The inviscid limit of the stochastic Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimising Hamilton-Jacobi function, the classical mechanical caustic and the Maxwell set and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one dimensional) action function. We give an explicit expression for an algebraic surface containing the Maxwell set and caustic in the polynomial case. Those parts of the caustic and Maxwell set which are singular are characterised. We demonstrate how the geometry of the caustic, level surfaces and Maxwell set can change infinitely rapidly causing turbulent behaviour which is stochastic in nature, and we determine its intermittence in terms of the recurrent behaviour of two processes."}
{"category": "Math", "title": "Asymptotic stability of solitons of the gKdV equations with general nonlinearity", "abstract": "We consider the generalized Korteweg-de Vries equation \\partial_t u + \\partial_x (\\partial_x^2 u + f(u))=0, \\quad (t,x)\\in [0,T)\\times \\mathbb{R}, (1) with general $C^3$ nonlinearity $f$. Under an explicit condition on $f$ and $c>0$, there exists a solution in the energy space $H^1$ of (1) of the type $u(t,x)=Q_c(x-x_0-ct)$, called soliton. In this paper, under general assumptions on $f$ and $Q_c$, we prove that the family of soliton solutions around $Q_c$ is asymptotically stable in some local sense in $H^1$, i.e. if $u(t)$ is close to $Q_{c}$ (for all $t\\geq 0$), then $u(t)$ locally converges in the energy space to some $Q_{c_+}$ as $t\\to +\\infty$. Note in particular that we do not assume the stability of $Q_{c}$. This result is based on a rigidity property of equation (1) around $Q_{c}$ in the energy space whose proof relies on the introduction of a dual problem. These results extend the main results in previous works devoted to the pure power case."}
{"category": "Math", "title": "Two-sided optimal bounds for Green function of half-spaces for relativistic $\\alpha$-stable process", "abstract": "The purpose of this paper is to find optimal estimates for the Green function of a half-space of {\\it the relativistic $\\alpha$-stable process} with parameter $m$ on $\\Rd$ space. This process has an infinitesimal generator of the form $mI-(m^{2/\\alpha}I-\\Delta)^{\\alpha/2},$ where $0<\\alpha<2$, $m>0$, and reduces to the isotropic $\\alpha$-stable process for $m=0$. Its potential theory for open bounded sets has been well developed throughout the recent years however almost nothing was known about the behaviour of the process on unbounded sets. The present paper is intended to fill this gap and we provide two-sided sharp estimates for the Green function for a half-space. As a byproduct we obtain some improvements of the estimates known for bounded sets specially for balls. The advantage of these estimates is a clarification of the relationship between the diameter of the ball and the parameter $m$ of the process. The main result states that the Green function is comparable with the Green function for the Brownian motion if the points are away from the boundary of a half-space and their distance is greater than one. On the other hand for the remaining points the Green function is somehow related the Green function for the isotropic $\\alpha$-stable process. For example, for $d\\ge3$, it is comparable with the Green function for the isotropic $\\alpha$-stable process, provided that the points are close enough."}
{"category": "Math", "title": "Refined asymptotics around solitons for gKdV equations", "abstract": "We consider the generalized Korteweg-de Vries equation $$ \\partial_t u + \\partial_x (\\partial_x^2 u + f(u))=0, \\quad (t,x)\\in [0,T)\\times \\mathbb{R}$$ with general $C^2$ nonlinearity $f$. Under an explicit condition on $f$ and $c>0$, there exists a solution in the energy space $H^1$ of the type $u(t,x)=Q_c(x-x_0-ct)$, called soliton. Stability theory for $Q_c$ is well-known. In previous works, we have proved that for $f(u)=u^p$, $p=2,3,4$, the family of solitons is asymptotically stable in some local sense in $H^1$, i.e. if $u(t)$ is close to $Q_{c}$ (for all $t\\geq 0$), then $u(t,.+\\rho(t))$ locally converges in the energy space to some $Q_{c_+}$ as $t\\to +\\infty$, for some $c^+\\sim c$. Then, the asymptotic stability result could be extended to the case of general assumptions on $f$ and $Q_c$. The objective of this paper is twofold. The main objective is to prove that in the case $f(u)=u^p$, $p=2,3,4$, $\\rho(t)-c_+ t$ has limit as $t\\to +\\infty$ under the additional assumption $x_+ u\\in L^2$. The second objective of this paper is to provide large time stability and asymptotic stability results for two soliton solutions for the case of general nonlinearity $f(u)$, when the ratio of the speeds of the solitons is small. The motivation is to accompany forthcoming works devoted to the collision of two solitons in the nonintegrable case. The arguments are refinements of previous works specialized to the case $u(t)\\sim Q_{c_1}+Q_{c_2}$, for $0< c_2 \\ll c_1$."}
{"category": "Math", "title": "All creatures great and small", "abstract": "Let kappa be an uncountable regular cardinal. Assuming 2^kappa=kappa^+, we show that the clone lattice on a set of size kappa is not dually atomic."}
{"category": "Math", "title": "A Computational Approach to Essential and Nonessential Objective Functions in Linear Multicriteria Optimization", "abstract": "The question of obtaining well-defined criteria for multiple criteria decision making problems is well-known. One of the approaches dealing with this question is the concept of nonessential objective function. A certain objective function is called nonessential if the set of efficient solutions is the same both with or without that objective function. In this paper we put together two methods for determining nonessential objective functions. A computational implementation is done using a computer algebra system."}
{"category": "Math", "title": "Depth three towers and Jacobson-Bourbaki correspondence", "abstract": "We introduce a notion of depth three tower of three rings C < B < A as a useful generalization of depth two ring extension. If A = End B_C and B | C is a Frobenius extension, this also captures the notion of depth three for a Frobenius extension in math.RA/0107064 and math.RA/0108067 such that if B | C is depth three, then A | C is depth two (cf. math.QA/0001020). If A, B and C correspond to a tower of subgroups G > H > K via the group algebra over a fixed base ring, the depth three condition is the condition that subgroup K has normal closure K^G contained in H. For a depth three tower of rings, there is a pre-Galois theory for the ring End {}_BA_C and coring (A \\otimes_B A)^C involving Morita context bimodules and left coideal subrings. This is applied in the last two sections to a specialization of a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings."}
{"category": "Math", "title": "Constant Weight Codes: A Geometric Approach Based on Dissections", "abstract": "We present a novel technique for encoding and decoding constant weight binary codes that uses a geometric interpretation of the codebook. Our technique is based on embedding the codebook in a Euclidean space of dimension equal to the weight of the code. The encoder and decoder mappings are then interpreted as a bijection between a certain hyper-rectangle and a polytope in this Euclidean space. An inductive dissection algorithm is developed for constructing such a bijection. We prove that the algorithm is correct and then analyze its complexity. The complexity depends on the weight of the code, rather than on the block length as in other algorithms. This approach is advantageous when the weight is smaller than the square root of the block length."}
{"category": "Math", "title": "A general convergence result for the Ricci flow in higher dimensions", "abstract": "Let (M,g_0) be a compact Riemannian manifold of dimension n \\geq 4. We show that the normalized Ricci flow deforms g_0 to a constant curvature metric provided that (M,g_0) x R has positive isotropic curvature. This condition is stronger than 2-positive flag curvature but weaker than 2-positive curvature operator."}
{"category": "Math", "title": "On the cost-subdifferentials of cost-convex functions", "abstract": "We are interested in the cost-convex potentials in optimal mass transport theory, and we show by direct and geometric arguments the equivalence between cost-subdifferentials and ordinary subdifferentials of cost-convex functions, under the assumptions A0, A1, A2, and A3W on cost functions introduced by Ma, Trudinger, and Wang. The connectivity of contact sets of optimal transport maps follows as a direct corollary. Our approach is quite different from the previous result of Loeper which he obtained as the first step toward his Hoelder regularity theory of potential functions, and which was based upon approximation using the regularity theory of Ma, Trudinger, and Wang. The result in this paper improves his result, by relaxing certain geometrical assumptions on the domains and targets; it also completes his Hoelder regularity theory of potential functions on the round sphere, by making it self-contained."}
{"category": "Math", "title": "Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: $\\lambda$-Aluthge transform", "abstract": "Let $\\lambda \\in (0,1)$ and let $T$ be a $r\\times r$ complex matrix with polar decomposition $T=U|T|$. Then, the $\\la$- Aluthge transform is defined by $$ \\Delta_\\lambda (T )= |T|^{\\lambda} U |T |^{1-\\lambda}. $$ Let $\\Delta_\\lambda^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, $n\\in\\mathbb{N}$. We prove that the sequence $\\{\\Delta_\\lambda^{n}(T)\\}_{n\\in\\mathbb{N}}$ converges for every $r\\times r$ {\\bf diagonalizable} matrix $T$. We show regularity results for the two parameter map $(\\la, T) \\mapsto \\alulit{\\infty}{T}$, and we study for which matrices the map $(0,1)\\ni \\lambda \\mapsto \\Delta_\\lambda^{\\infty}(T)$ is constant."}
{"category": "Math", "title": "On The Universality Of Central Loops", "abstract": "LC-loops, RC-loops and C-loops are collectively called central loops. It is shown that an LC(RC)-loop is a left(right) universal loop. But an LC(RC)-loop is a universal loop if and only if it is a right(left) universal loop. It is observed that not all RC-loops or LC-loops or C-loops are universal loops. But if an RC-loop(LC-loop, C-loop) is universal, then it is a right Bol loop(left Bol loop, Moufang loop) respectively. If a loop and its right or left isotope are commutative then the loop is a C-loop if and only if its right or left isotope is a C-loop. If a C-loop is central square and its right or left isotope is an alternative central square loop, then the latter is a C-loop. Necessary and sufficient condition for an LC-loop(RC-loop) to be a left(right)G-loop is established. Consequently, necessary and sufficient conditions for an LC-loop, and an RC-loop to be a G-loop are established. A necessary and sufficient condition for a C-loop to be a G-loop is established."}
{"category": "Math", "title": "Remark on the rank of elliptic curves", "abstract": "A covariant functor on the elliptic curves with complex multiplication is constructed. The functor takes values in the noncommutative tori with real multiplication. A conjecture on the rank of an elliptic curve is formulated."}
{"category": "Math", "title": "Loop coproducts in string topology and triviality of higher genus TQFT operations", "abstract": "Cohen and Godin constructed positive boundary topological quantum field theory (TQFT) structure on the homology of free loop spaces of oriented closed smooth manifolds by associating a certain operations called string operations to orientable surfaces with parametrized boundaries. We show that all TQFT string operations associated to surfaces of genus at least one vanish identically. This is a simple consequence of properties of the loop coproduct which will be discussed in detail. One interesting property is that the loop coproduct is nontrivial only on the degree $d$ homology group of the connected component of $LM$ consisting of contractible loops, where $d=\\dim M$, with values in the degree 0 homology group of constant loops. Thus the loop coproduct behaves in a dramatically simpler way than the loop product."}
{"category": "Math", "title": "A Homotopy Theoretic Proof of the BV Identity in Loop Homology", "abstract": "Chas and Sullivan proved the existence of a Batalin-Vilkovisky algebra structure in the homology of free loop spaces on closed finite dimensional smooth manifolds using chains and chain homotopies. This algebraic structure involves an associative product called the loop product, a Lie bracket called the loop bracket, and a square 0 operator called the BV operator. Cohen and Jones gave a homotopy theoretic description of the loop product in terms of spectra. In this paper, we give an explicit homotopy theoretic description of the loop bracket and, using this description, we give a homological proof of the BV identity connecting the loop product, the loop bracket, and the BV operator. The proof is based on an observation that the loop bracket and the BV derivation are given by the same cycle in the free loop space, except that they differ by parametrization of loops."}
{"category": "Math", "title": "Framed bicategories and monoidal fibrations", "abstract": "In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the `morphisms between 0-cells', such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1-cells. We avoid complicated coherence problems by describing base change `nonalgebraically', using categorical fibrations. The resulting `framed bicategories' assemble into 2-categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a `monoidal fibration', meaning a parametrized family of monoidal categories, and produces an analogue of the framed bicategory of spans. Combining the two, we obtain a construction which includes both enriched and internal categories as special cases."}
{"category": "Math", "title": "Bayesian Covariance Matrix Estimation using a Mixture of Decomposable Graphical Models", "abstract": "A Bayesian approach is used to estimate the covariance matrix of Gaussian data. Ideas from Gaussian graphical models and model selection are used to construct a prior for the covariance matrix that is a mixture over all decomposable graphs. For this prior the probability of each graph size is specified by the user and graphs of equal size are assigned equal probability. Most previous approaches assume that all graphs are equally probable. We show empirically that the prior that assigns equal probability over graph sizes outperforms the prior that assigns equal probability over all graphs, both in identifying the correct decomposable graph and in more efficiently estimating the covariance matrix."}
{"category": "Math", "title": "Limits of PGL(3)-translates of plane curves, I", "abstract": "We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map, determined by the curve, from the $P^8$ of 3x3 matrices to the $P^N$ of plane curves of degree $d$. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, performed in our earlier paper \"Linear orbits of arbitrary plane curves\"."}
{"category": "Math", "title": "On reconstruction formulas and algorithms for the thermoacoustic tomography", "abstract": "The paper surveys recent progress in establishing uniqueness and developing inversion formulas and algorithms for the thermoacoustic tomography. In mathematical terms, one deals with a rather special inverse problem for the wave equation. In the case of constant sound speed, it can also be interpreted as a problem concerning the spherical mean transform."}
{"category": "Math", "title": "Limits of PGL(3)-translates of plane curves, II", "abstract": "Every complex plane curve C determines a subscheme S of the $P^8$ of 3x3 matrices, whose projective normal cone (PNC) captures subtle invariants of C. In \"Limits of PGL(3)-translates of plane curves, I\" we obtain a set-theoretic description of the PNC and thereby we determine all possible limits of families of plane curves whose general element is isomorphic to C. The main result of this article is the determination of the PNC as a cycle; this is an essential ingredient in our computation in \"Linear orbits of arbitrary plane curves\" of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, an invariant of natural enumerative significance."}
{"category": "Math", "title": "Compactified moduli of projective bundles", "abstract": "We present a method for compactifying stacks of $\\PGL_n$-torsors (Azumaya algebras) on algebraic spaces. In particular, when the ambient space is a smooth projective surface we use our methods to show that various moduli spaces are irreducible and carry natural virtual fundamental classes. We also prove a version of the Skolem-Noether theorem for certain algebra objects in the derived category, which allows us to give an explicit description of the boundary points in our compactified moduli problem."}
{"category": "Math", "title": "Entrelacement d'alg\\`ebres de Lie [Wreath products for Lie algebras]", "abstract": "Full details are given for the definition and construction of the wreath product of two arbitrary Lie algebras, in the hope that it can lead to the definition of a suitable Lie group to be the wreath product of two given Lie groups. In the process, quite a few new notions are needed, and introduced. Such are, for example : Formal series with variables in a vector space and coefficients in some other vector space. Derivation of a formal series relative to another formal series. The Lie algebra of a vector space. Formal actions of Lie algebras over vector spaces. The basic formal action of a Lie algebra over itself (as a formal version of the analytic aspect of the infinitesimal operation law of a Lie groupuscule). More generally, the wreath product of two Lie algebras is defined, relative to a formal action of the second onto an arbitrary vector space. Main features are : A description of the triangular actions of wreath products over product vector spaces, and a Kaloujnine-Krasner type theorem : In essence, it says that all Lie extensions of a given Lie algebra by another Lie algebra are, indeed, subalgebras of their wreath product."}
{"category": "Math", "title": "Non-unique ergodicity, observers' topology and the dual algebraic lamination for $\\R$-trees", "abstract": "We continue in this article the study of laminations dual to very small actions of a free group F on R-trees. We prove that this lamination determines completely the combinatorial structure of the R-tree (the so-called observers' topology). On the contrary the metric is not determined by the lamination, and an R-tree may be equipped with different metrics which have the same observers' topology."}
{"category": "Math", "title": "The oscillation stability problem for the Urysohn sphere: A combinatorial approach", "abstract": "We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for $\\ell_2$ in the context of the Urysohn space $\\Ur$. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances."}
{"category": "Math", "title": "Hopf Algebras of Heap Ordered Trees and Permutations", "abstract": "It is known that there is a Hopf algebra structure on the vector space with basis all heap-ordered trees. We give a new bialgebra structure on the space with basis all permutations and show that there is a direct bialgebra isomorphism between the Hopf algebra of heap-ordered trees and the bialgebra of permutations."}
{"category": "Math", "title": "On a Smale Theorem and Nonhomogeneous Equilibria in Cooperative Systems", "abstract": "A standard result by Smale states that n dimensional strongly cooperative dynamical systems can have arbitrary dynamics when restricted to unordered invariant hyperspaces. In this paper this result is extended to the case when all solutions of the strongly cooperative system are bounded and converge towards one of only two equilibria outside of the hyperplane. An application is given in the context of strongly cooperative systems of reaction diffusion equations. It is shown that such a system can have a continuum of spatially inhomogeneous steady states, even when all solutions of the underlying reaction system converge to one of only three equilibria."}
{"category": "Math", "title": "An explicit construction of the Quillen homotopical category of dg Lie algebras", "abstract": "Let $\\g_1$ and $\\g_2$ be two dg Lie algebras, then it is well-known that the $L_\\infty$ morphisms from $\\g_1$ to $\\g_2$ are in 1-1 correspondence to the solutions of the Maurer-Cartan equation in some dg Lie algebra $\\Bbbk(\\g_1,\\g_2)$. Then the gauge action by exponents of the zero degree component $\\Bbbk(\\g_1,\\g_2)^0$ on $MC\\subset\\Bbbk(\\g_1,\\g_2)^1$ gives an explicit \"homotopy relation\" between two $L_\\infty$ morphisms. We prove that the quotient category by this relation (that is, the category whose objects are $L_\\infty$ algebras and morphisms are $L_\\infty$ morphisms modulo the gauge relation) is well-defined, and is a localization of the category of dg Lie algebras and dg Lie maps by quasi-isomorphisms. As localization is unique up to an equivalence, it is equivalent to the Quillen-Hinich homotopical category of dg Lie algebras [Q1,2], [H1,2]. Moreover, we prove that the Quillen's concept of a homotopy coincides with ours. The last result was conjectured by V.Dolgushev [D]."}
{"category": "Math", "title": "A note on Poisson homogeneous spaces", "abstract": "We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H, and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra associated to G/H. We also construct a Poisson groupoid over G/H which is symplectic near the identity section. This note serves as preparation for forthcoming papers, in which we will compute explicitly the Poisson cohomology and study their symplectic groupoids for certain examples of Poisson homogeneous spaces related to semi-simple Lie groups."}
{"category": "Math", "title": "Verification theorem and construction of $\\epsilon$-optimal controls for control of abstract evolution equations", "abstract": "We study several aspects of the dynamic programming approach to optimal control of abstract evolution equations, including a class of semilinear partial differential equations. We introduce and prove a verification theorem which provides a sufficient condition for optimality. Moreover we prove sub- and superoptimality principles of dynamic programming and give an explicit construction of $\\epsilon$-optimal controls."}
{"category": "Math", "title": "The rational homotopy type of a blow-up in the stable case", "abstract": "Suppose that f:V->W is an embedding of closed oriented manifolds whose normal bundle has the structure of a complex vector bundle. It is well known in both complex and symplectic geometry that one can then construct a manifold W' which is the blow-up of W along V. Assume that dim(W)>2.dim(V)+2 and that H^1(f) is injective. We construct an algebraic model of the rational homotopy type of the blow-up W' from an algebraic model of the embedding and the Chern classes of the normal bundle. This implies that if the space W is simply connected then the rational homotopy type of W' depends only on the rational homotopy class of f and on the Chern classes of the normal bundle."}
{"category": "Math", "title": "On Zermelo'-like problems: a Gauss-Bonnet inequality and a E. Hopf theorem", "abstract": "The goal of this paper is to describe Zermelo's navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to change the Riemannian metric on the manifold the control curvature of Zermelo's problem has a simple to handle expression which naturally leads to a generalization of the classical Gauss-Bonnet formula in an inequality. This Gauss-Bonnet inequality enables to generalize for Zermelo's problems the E. Hopf theorem on flatness of Riemannian tori without conjugate points."}
{"category": "Math", "title": "On the generalization of the Costas property in the continuum", "abstract": "We extend the definition of the Costas property to functions in the continuum, namely on intervals of the reals or the rationals, and argue that such functions can be used in the same applications as discrete Costas arrays. We construct Costas bijections in the real continuum within the class of piecewise continuously differentiable functions, but our attempts to construct a fractal-like Costas bijection there are successful only under slight but necessary deviations from the usual arithmetic laws. Furthermore, we are able, contingent on the validity of Artin's conjecture, to set up a limiting process according to which sequences of Welch Costas arrays converge to smooth Costas bijections over the reals. The situation over the rationals is different: there, we propose an algorithm of great generality and flexibility for the construction of a Costas fractal bijection. Its success, though, relies heavily on the enumerability of the rationals, and therefore it cannot be generalized over the reals in an obvious way."}
{"category": "Math", "title": "A geometric invariant theory construction of moduli spaces of stable maps", "abstract": "We construct the moduli spaces of stable maps, \\bar M_g,n(P^r,d), via geometric invariant theory (GIT). This construction is only valid over Spec C, but a special case is a GIT presentation of the moduli space of stable curves of genus g with n marked points, \\bar M_g,n; this is valid over Spec Z. Our method follows that used in the case n=0 by Gieseker to construct \\bar M_g, though our proof that the semistable set is nonempty is entirely different."}
{"category": "Math", "title": "Fixed point theorem for discontinuous mappings on PN spaces", "abstract": "We present a study on strong t-continuity and measure of discontinuity on PN spaces. As an application, we prove a fixed point theorem for a self mapping on PN spaces by means of measure of discontinuity."}
{"category": "Math", "title": "On (g, \\phi)-contraction in fuzzy metric spaces", "abstract": "In this paper, we give a generalization of Hicks type contractions and Golet type contractions on fuzzy metric spaces. We prove some fixed point theorems for this new type contractions mappings on fuzzy metric spaces."}
{"category": "Math", "title": "Wall-Crossing Morphisms in Khovanov-Rozansky Homology", "abstract": "We define a wall-crossing morphism for Khovanov-Rozansky homology; that is, a map between the KR homology of knots related by a crossing change. Using this map, we extend KR homology to an invariant of singular knots categorifying the Vasilliev derivative of the HOMFLY polynomial, and of $\\mathfrak{sl}_n$ quantum invariants."}
{"category": "Math", "title": "A universal property of the monoidal 2-category of cospans of finite linear orders and surjections", "abstract": "We prove that the monoidal 2-category of cospans of finite linear orders and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2-dimensional separable algebra condition."}
{"category": "Math", "title": "A universal enveloping for L-infinity algebras", "abstract": "For any L-infinity algebra L, we construct an A-infinity structure on the space of symmetric tensors Sym*(L), which generalizes the classical universal enveloping for Lie algebras. Our construction is based on an invariant homotopy on a cobar construction of the symmetric coalgebra, which is obtained through its relation with permutahedra and Young tableaux."}
{"category": "Math", "title": "BGG correspondence for toric complete intersections", "abstract": "We generalize the classical Bernstein-Gelfand-Gelfand correspondence to complete intersections in toric varieties."}
{"category": "Math", "title": "Controlling for individual heterogeneity in longitudinal models, with applications to student achievement", "abstract": "Longitudinal data tracking repeated measurements on individuals are highly valued for research because they offer controls for unmeasured individual heterogeneity that might otherwise bias results. Random effects or mixed models approaches, which treat individual heterogeneity as part of the model error term and use generalized least squares to estimate model parameters, are often criticized because correlation between unobserved individual effects and other model variables can lead to biased and inconsistent parameter estimates. Starting with an examination of the relationship between random effects and fixed effects estimators in the standard unobserved effects model, this article demonstrates through analysis and simulation that the mixed model approach has a ``bias compression'' property under a general model for individual heterogeneity that can mitigate bias due to uncontrolled differences among individuals. The general model is motivated by the complexities of longitudinal student achievement measures, but the results have broad applicability to longitudinal modeling."}
{"category": "Math", "title": "Rate of Convergence of Space Time Approximations for stochastic evolution equations", "abstract": "Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators driven by a finite dimensional Brownian motion are considered. Under some regularity condition assumed for the solution, the rate of convergence of various numerical approximations are estimated under strong monotonicity and Lipschitz conditions. The abstract setting involves general consistency conditions and is then applied to a class of quasilinear stochastic PDEs of parabolic type."}
{"category": "Math", "title": "The absolute order on the symmetric group, constructible partially ordered sets and Cohen-Macaulay complexes", "abstract": "The absolute order is a natural partial order on a Coxeter group W. It can be viewed as an analogue of the weak order on W in which the role of the generating set of simple reflections in W is played by the set of all reflections in W. By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen-Macaulay. This answers in part a question raised by V. Reiner and the first author. The Euler characteristic of the order complex of the proper part of the absolute order on the symmetric group is also computed."}
{"category": "Math", "title": "Jordan structures and non-associative geometry", "abstract": "In this survey paper we give an overview over constructions of geometries associated to Jordan structures (algebras, triple systems and pairs), featuring analogs of these constructions with the Lie functor on the one hand and with the approach of non-commutative geometry on the other hand."}
{"category": "Math", "title": "Absolute continuity of the measures of the Dunkl intetwining operator and its dualand applications", "abstract": "In this paper we consider the representing measures $\\mu_{x},x\\in \\QTR{Bbb}{R}^{d}$, and $\\nu_{y},y\\in \\QTR{Bbb}{R}^{d}$, of the Dunkl intertwining operator and of its dual. When the multiplicity function is positive, we prove that for all $x\\in \\QTR{Bbb}{R}_{\\QTO{mbox}{reg}}^{d}$ we have $d\\mu_{x}(y)=\\QTR{cal}{K}(x,y)dy$ and for almost all $y\\in \\QTR{Bbb}{R}^{d}$ we have $d\\nu_{y}(x)=\\QTR{cal}{K}(x,y)\\omega_{k}(x)dx,$ where $\\QTR{cal}{K}(x,.)$ is a positive integrable function on $\\QTR{Bbb}{R}^{d}$ with support in $\\{y\\in \\QTR{Bbb}{R}^{d}/\\Vert y\\Vert \\leq \\Vert x\\Vert \\}$ and the function $\\QTR{cal}{K}(.,y)$ is locally integrable on $\\QTR{Bbb}{R}^{d}$ with respect to the measure $\\omega_{k}(x)dx$ and with support in $\\{x\\in \\QTR{Bbb}{R}^{d}/\\Vert x\\Vert \\geq \\Vert y\\Vert \\}$. Next we present some applications of this result."}
{"category": "Math", "title": "Sensitivity of principal Hessian direction analysis", "abstract": "We provide sensitivity comparisons for two competing versions of the dimension reduction method principal Hessian directions (pHd). These comparisons consider the effects of small perturbations on the estimation of the dimension reduction subspace via the influence function. We show that the two versions of pHd can behave completely differently in the presence of certain observational types. Our results also provide evidence that outliers in the traditional sense may or may not be highly influential in practice. Since influential observations may lurk within otherwise typical data, we consider the influence function in the empirical setting for the efficient detection of influential observations in practice."}
{"category": "Math", "title": "Generalized Banach contraction in probabilistic metric/normed spaces", "abstract": "In this paper, we present the generalization of B-contraction and C-contraction due to Sehgal and Hicks respectively. We also study some properties of C-contraction in probabilistic metric space."}
{"category": "Math", "title": "Regularization by free additive convolution, square and rectangular cases", "abstract": "The free convolution is the binary operation on the set of probability measures on the real line which allows to deduce, from the individual spectral distributions, the spectral distribution of a sum of independent unitarily invariant square random matrices or of a sum of free operators in a non commutative probability space. In the same way, the rectangular free convolution allows to deduce, from the individual singular distributions, the singular distribution of a sum of independent unitarily invariant rectangular random matrices. In this paper, we consider the regularization properties of these free convolutions on the whole real line. More specifically, we try to find continuous semigroups $(\\mu_t)$ of probability measures such that $\\mu_0$ is the Dirac mass at zero and such that for all positive $t$ and all probability measure $\\nu$, the free convolution of $\\mu_t$ with $\\nu$ (or, in the rectangular context, the rectangular free convolution of $\\mu_t$ with $\\nu$) is absolutely continuous with respect to the Lebesgue measure, with a positive analytic density on the whole real line. In the square case, we prove that in semigroups satisfying this property, no measure can have a finite second moment, and we give a sufficient condition on semigroups to satisfy this property, with examples. In the rectangular case, we prove that in most cases, for $\\mu$ in a continuous rectangular-convolution-semigroup, the rectangular convolution of $\\mu$ with $\\nu$ either has an atom at the origin or doesn't put any mass in a neighborhood of the origin, thus the expected property does not hold. However, we give sufficient conditions for analyticity of the density of the rectangular convolution of $\\mu$ with $\\nu$ except on a negligible set of points, as well as existence and continuity of a density everywhere."}
{"category": "Math", "title": "Inverse Conductivity Problem for a Parabolic Equation using a Carlemen Estimate with one Observation", "abstract": "For the heat equation in a bounded domain we give a stability result for a smooth diffusion coefficient. The key ingredients are a global Carleman-type estimate, a Poincar\\'e-type estimate and an energy estimate with a single observation acting on a part of the boundary."}
{"category": "Math", "title": "3-quasi-Sasakian manifolds", "abstract": "In the present paper we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy."}
{"category": "Math", "title": "Representations of tame quivers and affine canonical bases", "abstract": "An integral PBW-basis of type $A_1^{(1)}$ has been constructed by Zhang [Z] and Chen [C] using the Auslander-Reiten quiver of the Kronecker quiver. We associate a geometric order to elements in this basis following an idea of Lusztig [L1] in the case of finite type. This leads to an algebraic realization of a bar-invariant basis of $\\uq2$. For any affine symmetric type, we obtain an integral PBW-basis of the generic composition algebra, by using an algebraic construction of the integral basis for a tube in [DDX], an embedding of the module category of the Kronecker quiver into the module category of the tame quiver, and a list of the root vectors of indecomposable modules according to the preprojective, regular, and preinjective components of the Auslander-Reiten quiver of the tame quiver. When the basis elements are ordered to be compatible with the geometric order given by the dimensions of the orbit varieties and the extension varieties, we can show that the transition matrix between the PBW-basis and a monomial basis is triangular with diagonal entries equal to 1. Therefore we obtain a bar-invariant basis. By a orthogonalization for the PBW-basis with the inner product, we finally give an algebraic way to realize the canonical bases of the quantized enveloping algebras of all symmetric affine Kac-Moody Lie algebras."}
{"category": "Math", "title": "Rational points on certain hyperelliptic curves over finite fields", "abstract": "Let $K$ be a field, $a, b\\in K$ and $ab\\neq 0$. Let us consider the polynomials $g_{1}(x)=x^n+ax+b, g_{2}(x)=x^n+ax^2+bx$, where $n$ is a fixed positive integer. In this paper we show that for each $k\\geq 2$ the hypersurface given by the equation \\begin{equation*} S_{k}^{i}: u^2=\\prod_{j=1}^{k}g_{i}(x_{j}),\\quad i=1, 2. \\end{equation*} contains a rational curve. Using the above and Woestijne's recent results \\cite{Woe} we show how one can construct a rational point different from the point at infinity on the curves $C_{i}:y^2=g_{i}(x), (i=1, 2)$ defined over a finite field, in polynomial time."}
{"category": "Math", "title": "The absolute Galois group acts faithfully on the connected components of the moduli space of surfaces of general type", "abstract": "We show that the Galois group $Gal(\\bar{\\Q} /\\Q)$ operates faithfully on the set of connected components of the moduli spaces of surfaces of general type, and also that for each element $\\sigma \\in Gal(\\bar{\\Q} /\\Q)$ different from the identity and from complex conjugation, there is a surface of general type such that $X$ and the Galois conjugate variety $X^{\\sigma}$ have nonisomorphic fundamental groups. The result was announced by the second author at the Alghero Conference 'Topology of algebraic varieties' in september 2006. Before the present paper was actually written, we received a very interesting preprint by Robert Easton and Ravi Vakil (\\cite{e-v}), where it is proven, with a completely different type of examples, that the Galois group $Gal(\\bar{\\Q} /\\Q)$ operates faithfully on the set of irreducible components of the moduli spaces of surfaces of general type. We also give other simpler examples of surfaces with nonisomorphic fundamental groups which are Galois conjugate, hence have isomorphic algebraic fundamental groups."}
{"category": "Math", "title": "Singular unitarity in \"quantization commutes with reduction\"", "abstract": "Let $M$ be a connected compact quantizable K\\\"ahler manifold equipped with a Hamiltonian action of a connected compact Lie group $G$. Let $M//G=\\phi^{-1}(0)/G=M_0$ be the symplectic quotient at value 0 of the moment map $\\phi$. The space $M_0$ may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over $M_0$ and the $G$-invariant subspace of the quantum Hilbert space over $M$. In this paper, without any regularity assumption on the quotient $M_0$, we discuss the relation between the inner products of these two quantum Hilbert spaces under the above natural isomorphism; we establish asymptotic unitarity to leading order in Planck's constant of a modified map of the above isomorphism under a ``metaplectic correction'' of the two quantum Hilbert spaces."}
{"category": "Math", "title": "On Lesieur's Measured Quantum Groupoids", "abstract": "In his thesis ([L1]), which is published in an expended and revised version ([L2]), Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras, using intensively the notion of pseudo-multiplicative unitary, which had been introduced in a previous article of the author, in collaboration with Jean-Michel Vallin [EV]. In [L2], the axioms given are very complicated and are here simplified."}
{"category": "Math", "title": "Jacobi-Nijenhuis algebroids and their modular classes", "abstract": "Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids."}
{"category": "Math", "title": "Some necessary and sufficient conditions for parastrophic invariance of the associative law in quasigroups", "abstract": "Every quasigroup $(S,\\cdot)$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(S,\\pi_i)$, $i\\in \\{1,2,3,4,5,6\\}$. It is shown that isotopy-isomorphy is a necessary and sufficient condition for any two distinct quasigroups $(S,\\pi_i)$ and $(S,\\pi_j)$, $i,j\\in \\{1,2,3,4,5,6\\}$ to be parastrophic invariant relative to the associative law. In addition, a necessary and sufficient condition for any two distinct quasigroups $(S,\\pi_i)$ and $(S,\\pi_j)$, $i,j\\in \\{1,2,3,4,5,6\\}$ to be parastrophic invariance under the associative law is either if the $\\pi_i$-parastrophe of $H$ is equivalent to the $\\pi_i$-parastrophe of the holomorph of the $\\pi_i$-parastrophe of $S$ or if the $\\pi_i$-parastrophe of $H$ is equivalent to the $\\pi_k$-parastrophe of the $\\pi_i$-parastrophe of the holomorph of the $\\pi_i$-parastrophe of $S$, for a particular $k\\in \\{1,2,3,4,5,6\\}$."}
{"category": "Math", "title": "Weak Inverse Property Loops and Some Isotopy-Isomorphy Properties", "abstract": "Two distinct isotopy-isomorphy conditions, different from those of J. M. Osborn and Wilson's condition, for a weak inverse property loop(WIPL) are shown. Only one of them characterizes isotopy-isomorphy in WIPLs while the other is just a sufficient condition for isotopy-isomorphy. Under the sufficient condition called the ${\\cal T}$ condition, Artzy's result that isotopic cross inverse property loops(CIPL) are isomorphic is proved for WIP loops."}
{"category": "Math", "title": "On a Pair of Universal Weak Inverse Property Loops", "abstract": "A new condition called ${\\cal T}$ condition is introduced for the first time and used to study a pair of isotopic loops. Under this condition, a loop in the pair is a WIPL if and only if the other loop is a WIPL. Furthermore, such WIPLs are isomorphic. The translation elements $f$ and $g$ of a CIPL with the ${\\cal T}$ condition(such that its $f,g$-isotope is an automorphic inverse property loop) are found to be alternative, flexible, centrum and equal elements. A necessary and sufficient condition for a pair WIPLs with a weak ${\\cal T}$ condition to be isomorphic is shown. A CIPL and an isomorph are observed to have this weak ${\\cal T}$ condition."}
{"category": "Math", "title": "Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids", "abstract": "For points in $d$ real dimensions, we introduce a geometry for general digit sets. We introduce a positional number system where the basis for our representation is a fixed $d$ by $d$ matrix over $\\bz$. Our starting point is a given pair $(A, \\mathcal D)$ with the matrix $A$ assumed expansive, and $\\mathcal D$ a chosen complete digit set, i.e., in bijective correspondence with the points in $\\bz^d/A^T\\bz^d$. We give an explicit geometric representation and encoding with infinite words in letters from $\\mathcal D$. We show that the attractor $X(A^T,\\mathcal D)$ for an affine Iterated Function System (IFS) based on $(A,\\mathcal D)$ is a set of fractions for our digital representation of points in $\\br^d$. Moreover our positional \"number representation\" is spelled out in the form of an explicit IFS-encoding of a compact solenoid $\\sa$ associated with the pair $(A,\\mathcal D)$. The intricate part (Theorem \\ref{thenccycl}) is played by the cycles in $\\bz^d$ for the initial $(A,\\mathcal D)$-IFS. Using these cycles we are able to write down formulas for the two maps which do the encoding as well as the decoding in our positional $\\mathcal D$-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces."}
{"category": "Math", "title": "Frames of subspaces and operators", "abstract": "We study the relationship between operators, orthonormal basis of subspaces and frames of subspaces (also called fusion frames) for a separable Hilbert space $\\mathcal{H}$. We get sufficient conditions on an orthonormal basis of subspaces $\\mathcal{E} = \\{E_i \\}_{i\\in I}$ of a Hilbert space $\\mathcal{K}$ and a surjective $T\\in L(\\mathcal{K}, \\mathcal{H})$ in order that $\\{T(E_i)\\}_{i\\in I}$ is a frame of subspaces with respect to a computable sequence of weights. We also obtain generalizations of results in [J. A. Antezana, G. Corach, M. Ruiz and D. Stojanoff, Oblique projections and frames. Proc. Amer. Math. Soc. 134 (2006), 1031-1037], which related frames of subspaces (including the computation of their weights) and oblique projections. The notion of refinament of a fusion frame is defined and used to obtain results about the excess of such frames. We study the set of admissible weights for a generating sequence of subspaces. Several examples are given."}
{"category": "Math", "title": "Bootstrapping confidence intervals for the change-point of time series", "abstract": "We study an AMOC time series model with an abrupt change in the mean and dependent errors that fulfill certain mixing conditions. We obtain confidence intervals for the unknown change-point via bootstrapping methods. Precisely we use a block bootstrap of the estimated centered error sequence. Then we reconstruct a sequence with a change in the mean using the same estimators as before. The difference between the change-point estimator of the resampled sequence and the one for the original sequence can be use as an approximation of the difference between the real change-point and its estimator. This enables us to construct confidence intervals using the empirical distribution of the resampled time series. A simulation study shows that the resampled confidence intervals are usually closer to their target levels and at the same time smaller than the asymptotic intervals."}
{"category": "Math", "title": "Power-free values, repulsion between points, differing beliefs and the existence of error", "abstract": "Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is square-free."}
{"category": "Math", "title": "On the derived category of 1-motives, I", "abstract": "We consider the category of Deligne 1-motives over a perfect field k of exponential characteristic p and its derived category for a suitable exact structure after inverting p. As a first result, we provide a fully faithful embedding into an etale version of Voevodsky's triangulated category of geometric motives. Our second main result is that this full embedding \"almost\" has a left adjoint, that we call \\LAlb. Applied to the motive of a variety we thus get a bounded complex of 1-motives, that we compute fully for smooth varieties and partly for singular varieties. As an application we give motivic proofs of Roitman type theorems (in characteristic 0)."}
{"category": "Math", "title": "Elementary Maps on Triangular Algebras", "abstract": "In this note we prove that elementary maps on triangular algebras are automically additive."}
{"category": "Math", "title": "Additivity of Maps on Triangular Algebras", "abstract": "We prove that every multiplicative bijective map, Jordan bijective map, and Jordan triple bijective map from a triangular algebra onto any ring is automatically additive."}
{"category": "Math", "title": "The law of the supremum of a stable L\\'{e}vy process with no negative jumps", "abstract": "Let $X=(X_t)_{t\\ge0}$ be a stable L\\'{e}vy process of index $\\alpha \\in(1,2)$ with no negative jumps and let $S_t=\\sup_{0\\le s\\le t}X_s$ denote its running supremum for $t>0$. We show that the density function $f_t$ of $S_t$ can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann--Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for $f_t$. Recalling the familiar relation between $S_t$ and the first entry time $\\tau_x$ of $X$ into $[x,\\infty)$, this further translates into an explicit series representation for the density function of $\\tau_x$."}
{"category": "Math", "title": "Local stability of ergodic averages", "abstract": "The mean ergodic theorem is equivalent to the assertion that for every function K and every epsilon, there is an n with the property that the ergodic averages A_m f are stable to within epsilon on the interval [n,K(n)]. We show that even though it is not generally possible to compute a bound on the rate of convergence of a sequence of ergodic averages, one can give explicit bounds on n in terms of K and || f || / epsilon. This tells us how far one has to search to find an n so that the ergodic averages are \"locally stable\" on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general proof-theoretic methods falling under the heading of \"proof mining.\""}
{"category": "Math", "title": "On dynamical bit sequences", "abstract": "Let X^{(k)}(t) = (X_1(t), ..., X_k(t)) denote a k-vector of i.i.d. random variables, each taking the values 1 or 0 with respective probabilities p and 1-p. As a process indexed by non-negative t, $X^{(k)}(t)$ is constructed--following Benjamini, Haggstrom, Peres, and Steif (2003)--so that it is strong Markov with invariant measure ((1-p)\\delta_0+p\\delta_1)^k. We derive sharp estimates for the probability that ``X_1(t)+...+X_k(t)=k-\\ell for some t in F,'' where F \\subset [0,1] is nonrandom and compact. We do this in two very different settings: (i) Where \\ell is a constant; and (ii) Where \\ell=k/2, k is even, and p=q=1/2. We prove that the probability is described by the Kolmogorov capacitance of F for case (i) and Howroyd's 1/2-dimensional box-dimension profiles for case (ii). We also present sample-path consequences, and a connection to capacities that answers a question of Benjamini et. al. (2003)"}
{"category": "Math", "title": "Thom polynomials for maps of curves with isolated singularities", "abstract": "Thom (residual) polynomials in characteristic classes are used in the analysis of geometry of functional spaces. They serve as a tool in description of classes Poincar\\'e dual to subvarieties of functions of prescribed types. We give explicit universal expressions for residual polynomials in spaces of functions on complex curves having isolated singularities and multisingularities, in terms of few characteristic classes. These expressions lead to a partial explicit description of a stratification of Hurwitz spaces."}
{"category": "Math", "title": "Helix, shadow boundary and minimal submanifolds", "abstract": "Inspired by a Blaschke's work about analytic convex surfaces, we study {\\em shadow boundaries} of Riemannian submanifolds $M$, which are defined by a parallel vector field along $M$. Since a shadow boundary is just a closed subset of $M$, first, we will give a condition that guarantee its smoothness. It depends on the second fundamental form of the submanifold. It is natural to search for what kind of properties might have such submanifolds of $M$? Could they be totally geodesic or minimal? Answers to these and related questions are given in this work."}
{"category": "Math", "title": "On Colorings of Squares of Outerplanar Graphs", "abstract": "We study vertex colorings of the square $G^2$ of an outerplanar graph $G$. We find the optimal bound of the inductiveness, chromatic number and the clique number of $G^2$ as a function of the maximum degree $\\Delta$ of $G$ for all $\\Delta\\in \\nats$. As a bonus, we obtain the optimal bound of the choosability (or the list-chromatic number) of $G^2$ when $\\Delta \\geq 7$. In the case of chordal outerplanar graphs, we classify exactly which graphs have parameters exceeding the absolute minimum."}
{"category": "Math", "title": "On multipartite posets", "abstract": "A poset $\\mathbf{P} = (X,\\preceq)$ is {\\em $m$-partite} if $X$ has a partition $X = X_1 \\cup ... \\cup X_m$ such that (1) each $X_i$ forms an antichain in $\\mathbf{P}$, and (2) $x\\prec y$ implies $x\\in X_i$ and $y\\in X_j$ where $i<j$. In this article we derive a tight asymptotic upper bound on the order dimension of $m$-partite posets in terms of $m$ and their bipartite sub-posets in a constructive and elementary way."}
{"category": "Math", "title": "Randomly coloring planar graphs with fewer colors than the maximum degree", "abstract": "We study Markov chains for randomly sampling $k$-colorings of a graph with maximum degree $\\Delta$. Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar graphs when $k=\\Omega(\\Delta/\\log{\\Delta})$. Our results can be partially extended to the more general case where the maximum eigenvalue of the adjacency matrix of the graph is at most $\\Delta^{1-\\eps}$, for fixed $\\eps > 0$. The main challenge when $k \\le \\Delta + 1$ is the possibility of \"frozen\" vertices, that is, vertices for which only one color is possible, conditioned on the colors of its neighbors. Indeed, when $\\Delta = O(1)$, even a typical coloring can have a constant fraction of the vertices frozen. Our proofs rely on recent advances in techniques for bounding mixing time using \"local uniformity\" properties."}
{"category": "Math", "title": "Vertex coloring acyclic digraphs and their corresponding hypergraphs", "abstract": "We consider vertex coloring of an acyclic digraph $\\Gdag$ in such a way that two vertices which have a common ancestor in $\\Gdag$ receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding {\\em down-chromatic number} and derive an upper bound as a function of $D(\\Gdag)$, the maximum number of descendants of a given vertex, and the degeneracy of the corresponding hypergraph. Finally we determine an asymptotically tight upper bound of the down-chromatic number in terms of the number of vertices of $\\Gdag$ and $D(\\Gdag)$."}
{"category": "Math", "title": "On the dynamics of codimension one holomorphic foliations with ample normal bundle", "abstract": "We investigate the accumulation to singular points of leaves of codimension one foliations whose normal bundle is ample, with emphasis on the nonexistence of Levi-flat hypersurfaces."}
{"category": "Math", "title": "Murre's conjectures and explicit Chow--Kuenneth projectors for some varieties", "abstract": "In this paper, we investigate Murre's conjectures on the structure of rational Chow groups and exhibit explicit Chow--Kuenneth projectors for some examples. More precisely, the examples we study are the varieties which have a nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle explicit Chow--Kuenneth projectors are obtained which satisfy Murre's conjectures and the motivic Hard Lefschetz theorem is verified."}
{"category": "Math", "title": "Equivalence of sparse circulants: the bipartite \\'Ad\\'am problem", "abstract": "We consider n-by-n circulant matrices having entries 0 and 1. Such matrices can be identified with sets of residues mod n, corresponding to the columns in which the top row contains an entry 1. Let A and B be two such matrices, and suppose that the corresponding residue sets S_A and S_B have size at most 3. We prove that the following are equivalent: (1) there are integers u,v mod n, with u a unit, such that S_A = uS_B + v; (2) there are permutation matrices P,Q such that A=PBQ. Our proof relies on some new results about vanishing sums of roots of unity. We give examples showing this result is not always true for denser circulants, as well as results showing it continues to hold in some situations. We also explain how our problem relates to the Adam problem on isomorphisms of circulant directed graphs."}
{"category": "Math", "title": "Integrable systems and complex geometry", "abstract": "In this paper, we discuss an interaction between complex geometry and integrable systems. Section 1 reviews the classical results on integrable systems. New examples of integrable systems, which have been discovered, are based on the Lax representation of the equations of motion. These systems can be realized as straight line motions on a Jacobi variety of a so-called spectral curve. In section 2, we study a Lie algebra theoretical method leading to integrable systems and we apply the method to several problems. In section 3, we discuss the concept of the algebraic complete integrability (a.c.i.) of hamiltonian systems. Algebraic integrability means that the system is completely integrable in the sens of the phase space being folited by tori, which in addition are real parts of a complex algebraic tori (abelian varieties). The method is devoted to illustrate how to decide about the a.c.i. of hamiltonian systems and is applied to some examples. Finally, in section 4 we study an a.c.i. in the generalized sense which appears as covering of a.c.i. system. The manifold invariant by the complex flow is covering of abelian variety."}
{"category": "Math", "title": "Examples of Non-Rigid CAT(0) Groups from the Category of Knot Groups", "abstract": "C Croke and B Kleiner have constructed an example of a CAT(0) group with more than one visual boundary. J Wilson has proven that this same group has uncountably many distinct boundaries. In this article we prove that the knot group of any connected sum of two non-trivial torus knots also has uncountably many distinct CAT(0) boundaries."}
{"category": "Math", "title": "A method for the resolution of the Jacobi equation Y''+RY = 0 on the manifold Sp(2)/SU(2)", "abstract": "In this paper a method for the resolution of the differential equation of the Jacobi vector fields in the manifold V1 = Sp(2)/SU(2) is exposed. These results are applied to determine areas and volumes of geodesic spheres and balls."}
{"category": "Math", "title": "Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems", "abstract": "In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point)"}
{"category": "Math", "title": "A uniform Sobolev inequality under Ricci flow", "abstract": "Let ${\\bf M}$ be a compact Riemannian manifold and the metrics $g=g(t)$ evolve by the Ricci flow. We prove the following result. The Sobolev imbedding by Aubin or Hebey, perturbed by a scalar curvature term and modulo sharpness of constants, holds uniformly for $({\\bf M}, g(t))$ for all time if the Ricci flow exists for all time; and if the Ricci flow develops a singularity in finite time, then the same Sobolev imbedding holds uniformly after a standard normalization. As a consequence, long time non-collapsing results are derived, which improve Perelman's local non-collapsing results. An application to 3-d Ricci flow with surgery is also presented."}
{"category": "Math", "title": "The vanishing of the contact invariant in the presence of torsion", "abstract": "We prove that the Ozsvath-Szabo contact invariant of a closed contact 3-manifold with positive Giroux torsion vanishes."}
{"category": "Math", "title": "Partitions with independent iterates in random dynamical systems", "abstract": "Consider an invertible measure-preserving transformation of a probability space. A finite partition of the space is called weakly independent if there are infinitely many images of this partition under powers of the transformation that are jointly independent. Krengel proved that a transformation is weakly mixing if and only if weakly independent partitions of the underlying space are dense among all finite partitions. Using the tools developed in the later papers of del Junco-Reinhold-Weiss and del Junco-Begun we obtain Krengel- type results for weakly mixing random dynamical systems (or equivalently, skew products that are relatively weakly mixing)."}
{"category": "Math", "title": "Effective equidistribution of S-integral points on symmetric varieties", "abstract": "Let K be a global field of characteristic not 2. Let Z be a symmetric variety defined over K and S a finite set of places of K. We obtain counting and equidistribution results for the S-integral points of Z. Our results are effective when K is a number field."}
{"category": "Math", "title": "Small values of Lusternik-Schnirelmann and systolic categories for manifolds", "abstract": "We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We examine its ramifications in systolic topology, and provide a sufficient condition for ensuring a lower bound of 3 for systolic category."}
{"category": "Math", "title": "Critical Delays and Polynomial Eigenvalue Problems", "abstract": "In this work we present a new method to compute the delays of delay differential equations (DDEs), such that the DDE has a purely imaginary eigenvalue. For delay differential equations with multiple delays, the critical curves or critical surfaces in delay space (that is, the set of delays where the DDE has a purely imaginary eigenvalue) are parameterized. We show how the method is related to other works in the field by treating the case where the delays are integer multiples of some delay value, i.e., commensurate delays. The parametrization is done by solving a {\\em quadratic eigenvalue problem} which is constructed from the vectorization of a matrix equation and hence typically of large size. For commensurate delay differential equations, the corresponding equation is a polynomial eigenvalue problem. As a special case of the proposed method, we find a closed form for a parameterization of the critical surface for the scalar case. We provide several examples with visualizations where the computation is done with some exploitation of the structure of eigenvalue problems."}
{"category": "Math", "title": "t-Wise Independence with Local Dependencies", "abstract": "In this note we prove a large deviation bound on the sum of random variables with the following dependency structure: there is a dependency graph $G$ with a bounded chromatic number, in which each vertex represents a random variable. Variables that are represented by neighboring vertices may be arbitrarily dependent, but collections of variables that form an independent set in $G$ are $t$-wise independent."}
{"category": "Math", "title": "Right and Left Joint System Representation of a Rational Matrix Function in General Position (System Representation Theory for Dummies)", "abstract": "For a rational matrix function R of one variable in general position, the matrix functions R(x)/R(y) and R(y)\\R(x) of two variables are considered. For these matrix functions of two variables, representations which are analogous to the system representations (or realizations) of a rational matrix function of one variable are constructed. This representation is called the joint right [the joint left] system representation."}
{"category": "Math", "title": "Pascal's triangle and word bases for blob algebra ideals", "abstract": "We give an elementary proof of isomorphism of the blob (diagram) algebra and the corresponding extended Temperley-Lieb algebra (defined by presentation)."}
{"category": "Math", "title": "Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies", "abstract": "Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction."}
{"category": "Math", "title": "On fundamental groups related to the Hirzebruch surface F_1", "abstract": "Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in C^2 or in CP^2. In this article, we show that these groups, for the Hirzebruch surface F_{1,(a,b)}, are almost-solvable. That is - they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces."}
{"category": "Math", "title": "A classification of CO spaces which are continuous images of compact ordered spaces", "abstract": "A compact Hausdorff space X is called a CO space, if every closed subset of X is homeomorphic to an open subset of X. Every successor ordinal with its order topology is a CO space. We find an explicit characterization of the class K of CO spaces which are a continuous image of a Dedkind complete totally ordered set. (The topology of a totally ordered set is taken to be its order topology). We show that every member of K can be described as a finite disjoint sum of very simple spaces. Every summand has either form: (1) mu + 1 + nu^*, where mu and nu are cardinals, and nu^* is the reverse order of nu; or (2) the summand is the 1-point-compactification of a discrete space with cardinality aleph_1."}
{"category": "Math", "title": "Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections", "abstract": "We propose new methods for the numerical continuation of point-to-cycle connecting orbits in 3-dimensional autonomous ODE's using projection boundary conditions. In our approach, the projection boundary conditions near the cycle are formulated using an eigenfunction of the associated adjoint variational equation, avoiding costly and numerically unstable computations of the monodromy matrix. The equations for the eigenfunction are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find connecting orbits are discussed in general and illustrated with several examples, including the Lorenz equations. Complete AUTO demos, which can be easily adapted to any autonomous 3-dimensional ODE system, are freely available."}
{"category": "Math", "title": "The Explicit Chaotic Representation of the powers of increments of Levy Processes", "abstract": "An explicit formula for the chaotic representation of the powers of increments, (X_{t+t_0}-X_{t_0})^n, of a Levy process is presented. There are two different chaos expansions of a square integrable functional of a Levy process: one with respect to the compensated Poisson random measure and the other with respect to the orthogonal compensated powers of the jumps of the Levy process. Computationally explicit formulae for both of these chaos expansions of (X_{t+t_0}-X_{t_0})^n are given in this paper. Simulation results verify that the representation is satisfactory. The CRP of a number of financial derivatives can be found by expressing them in terms of (X_{t+t_0}-X_{t_0})^n using Taylor's expansion."}
{"category": "Math", "title": "Constraints, MMSNP and expander relational structures", "abstract": "We give a poly-time construction for a combinatorial classic known as Sparse Incomparability Lemma, studied by Erdos, Lovasz, Nesetril, Rodl and others: We show that every Constraint Satisfaction Problem is poly-time equivalent to its restriction to structures with large girth. This implies that the complexity classes CSP and Monotone Monadic Strict NP introduced by Feder and Vardi are computationally equivalent. The technical novelty of the paper is a concept of expander relations and a new type of product for relational structures: a generalization of the zig-zag product, the twisted product."}
{"category": "Math", "title": "Forbidden lists (NP and CSP for combinatorialists)", "abstract": "We present a definition of the class NP in combinatorial context as the set of languages of structures defined by finitely many forbidden lifted substructures. We apply this to special syntactically defined subclasses and show how they correspond to naturally defined (and intensively studied) combinatorial problems. We show that some types of combinatorial problems like edge colorings and graph decompositions express the full computational power of the class NP. We then characterize Constraint Satisfaction Problems (i.e. H-coloring problems) which are expressible by finitely many forbidden lifted substructures. This greatly simplifies and generalizes the earlier attempts to characterize this problem. As a corollary of this approach we perhaps find a proper setting of Feder and Vardi analysis of CSP languages within the class MMSNP."}
{"category": "Math", "title": "Correction of paper published in J. Combinatorial Theory 21, 1976: On the Existence of Hadamard Matrices", "abstract": "In the paper On the Existence of Hadamard Matrices in J. Combinatorial Theory 21, 1976, it is shown that for a natural number q > 3, we can construct an Hadamard Matrix of order 2^s q for s \\geq t where t = [2 log_2(q-3)]. I will show that this bound is not a consequence of the proof given in the paper and explain the error in the argumentation."}
{"category": "Math", "title": "The two possible values of the chromatic number of a random graph", "abstract": "Given d \\in (0,infty) let k_d be the smallest integer k such that d < 2k\\log k. We prove that the chromatic number of a random graph G(n,d/n) is either k_d or k_d+1 almost surely."}
{"category": "Math", "title": "Lambda Mu Calculus and Duality: Call-by-Name and Call-by-Value", "abstract": "Under the extension of Curry-Howard's correspondence to classical logic, Gentzen's NK and LK systems can be seen as syntax-directed systems of simple types respectively for Parigot's Lambda Mu Calculus and Curien-Herbelin's Lambda Bar Mu Mu Tidle Calculus. We aim at showing their computational equivalence. We define translations between these calculi. We prove simulation theorems for an undirected evaluation as well as for call-by-name and call-by-value evaluations."}
{"category": "Math", "title": "Global well-posedness for dissipative Korteweg-de Vries equations", "abstract": "This paper is devoted to the well-posedness for dissipative KdV equations $u_t+u_{xxx}+|D_x|^{2\\alpha}u+uu_x=0$, $0<\\alpha\\leq 1$. An optimal bilinear estimate is obtained in Bourgain's type spaces, which provides global well-posedness in $H^s(\\R)$, $s>-3/4$ for $\\alpha\\leq1/2$ and $s>-3/(5-2\\alpha)$ for $\\alpha>1/2$."}
{"category": "Math", "title": "Permutations with Extremal number of Fixed Points", "abstract": "We extend Stanley's work on alternating permutations with extremal number of fixed points in two directions: first, alternating permutations are replaced by permutations with a prescribed descent set; second, instead of simply counting permutations we study their generating polynomials by number of excedances. Several techniques are used: Desarmenien's desarrangement combinatorics, Gessel's hook-factorization and the analytical properties of two new permutation statistics \"DEZ\" and \"lec\". Explicit formulas for the maximal case are derived by using symmetric function tools."}
{"category": "Math", "title": "On path factors of (3,4)-biregular bigraphs", "abstract": "A (3,4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor."}
{"category": "Math", "title": "On a complex differential Riccati equation", "abstract": "We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical \"one-dimensional\" results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral theorem."}
{"category": "Math", "title": "Noether Symmetries and Conservations Laws For Non-critical Kohn-Laplace Equations on Three-Dimensional Heisenberg Group", "abstract": "We show which Lie point symmetries of non-critical semilinear Kohn-Laplace equations on the Heisenberg group $H^1$ are Noether symmetries and we establish their respectives conservations laws."}
{"category": "Math", "title": "On the Linearization of the Painleve' III-VI Equations and Reductions of the Three-Wave Resonant System", "abstract": "We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3x3 matrix Fuchs--Garnier pairs for the third and fifth Painleve' equations, together with the previously known Fuchs--Garnier pair for the fourth and sixth Painleve' equations. These Fuchs--Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by M. Jimbo and T. Miwa. As an application of the 3x3 matrix pairs, we found an integral auto-transformation for the standard Fuchs--Garnier pair for the fifth Painleve' equation. It generates an Okamoto-like B\\\"acklund transformation for the fifth Painleve' equation. Another application is an integral transformation relating two different 2x2 matrix Fuchs--Garnier pairs for the third Painleve' equation."}
{"category": "Math", "title": "Partial Inertial Manifolds for infinite-dimensional dynamical systems: Example for P.D.E.s with a state-dependent delay", "abstract": "We propose a new notion of Partial Inertial Manifold to study the long-time asymptotic behavior of dissipative differential equations. As shown on an example, such manifolds may exist in the cases when the classical Inertial manifold does not exist (or not known to exist)."}
{"category": "Math", "title": "Concentration of the Spectral Measure for Large Random Matrices with Stable Entries", "abstract": "We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value."}
{"category": "Math", "title": "A discrete mean value of the derivative of the Riemann zeta function", "abstract": "In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of $\\zeta'(\\rho)$ where $\\zeta(s)$ is the Riemann zeta function and $\\rho$ is a non-trivial zero of the Riemann zeta function."}
{"category": "Math", "title": "Extreme values of zeta prime rho", "abstract": "In this article we exhibit small and large values of $\\zeta'(\\rho)$ by applying Soundararajan's resonance method. Our results assume the Riemann hypothesis."}
{"category": "Math", "title": "The fundamental progroupoid of a general topos", "abstract": "It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the \\emph{fundamental progroupoid}, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and can not be replaced by a localic groupoid. The classifying topos in not any more a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver July 2004."}
{"category": "Math", "title": "The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation", "abstract": "In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide."}
{"category": "Math", "title": "Coherence and phase synchronization: generalization to pairs of multivariate time series, and removal of zero-lag contributions", "abstract": "Coherence and phase synchronization between time series corresponding to different spatial locations are usually interpreted as indicators of the connectivity between locations. In neurophysiology, time series of electric neuronal activity are essential for studying brain interconnectivity. Such signals can either be invasively measured from depth electrodes, or computed from very high time resolution, non-invasive, extracranial recordings of scalp electric potential differences (EEG: electroencephalogram) and magnetic fields (MEG: magnetoencephalogram) by means of a tomography such as sLORETA (standardized low resolution brain electromagnetic tomography). There are two problems in this case. First, in the usual situation of unknown cortical geometry, the estimated signal at each brain location is a vector with three components (i.e. a current density vector), which means that coherence and phase synchronization must be generalized to pairs of multivariate time series. Second, the inherent low spatial resolution of the EEG/MEG tomography introduces artificially high zero-lag coherence and phase synchronization. In this report, solutions to both problems are presented. Two additional generalizations are briefly mentioned: (1) conditional coherence and phase synchronization; and (2) non-stationary time-frequency analysis. Finally, a non-parametric randomization method for connectivity significance testing is outlined. The new connectivity measures proposed here can be applied to pairs of univariate EEG/MEG signals, as is traditional in the published literature. However, these calculations cannot be interpreted as connectivity, since it is in general incorrect to associate an extracranial electrode or sensor to the underlying cortex."}
{"category": "Math", "title": "Surfaces with K^2<3\\chi and finite fundamental group", "abstract": "In this paper we continue the study of algebraic fundamentale group of minimal surfaces of general type S satisfying K_S^2<3\\chi(S). We show that, if K_S^2= 3\\chi(S)-1 and the algebraic fundamental group of S has order 8, then S is a Campedelli surface. In view of the results of math.AG/0512483 and math.AG/0605733, this implies that the fundamental group of a surface with K^2<3\\chi that has no irregular etale cover has order at most 9, and if it has order 8 or 9, then S is a Campedelli surface. To obtain this result we establish some classification results for minimal surfaces of general type such that K^2=3p_g-5 and such that the canonical map is a birational morphism. We also study rational surfaces with a Z_2^3-action."}
{"category": "Math", "title": "On existence of log minimal models", "abstract": "In this paper, we prove that the log minimal model program in dimension $d-1$ implies the existence of log minimal models for effective lc pairs (eg of nonnegative Kodaira dimension) in dimension $d$. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension $d-1$. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and the Shokurov reduction theorem. Other applications appear in a paper of Birkar-Paun."}
{"category": "Math", "title": "Birational geometry", "abstract": "This article contains the notes of a graduate course on birational geometry focusing on the minimal model program. Topics covered include singularities, vanishing, nonvanishing, cone and contraction, base point freeness, finite generation, flips, termination, minimal models and Mori fibre spaces."}
{"category": "Math", "title": "Necessary Conditions for Schur-Positivity", "abstract": "In recent years, there has been considerable interest in showing that certain conditions on skew shapes A and B are sufficient for the difference s_A - s_B of their skew Schur functions to be Schur-positive. We determine necessary conditions for the difference to be Schur-positive. Our conditions are motivated by those of Reiner, Shaw and van Willigenburg that are necessary for s_A = s_B, and we deduce a strengthening of their result as a special case."}
{"category": "Math", "title": "On the asymptotic behavior of Faber polynomials for domains with piecewise analytic boundary", "abstract": "For a function g(w) analytic and univalent in {w:1<|w|<\\infty} with a simple pole at \\infty and a continuous extension to {w:|w|\\geq 1}, we consider the Faber polynomials F_n(z), n=0,1,2,..., associated to g(w) via their generating function g'(w)/(g(w)-z)=\\sum_{n=0}^\\infty F_n(z)w^{-(n+1)}. Assuming that g(w) maps the unit circle T onto a piecewise analytic curve L whose exterior domain has no outward-pointing cusps, and under an additional assumption concerning the \"Lehman expansion\" of g(w) about those points of T mapped onto corners of L, we obtain asymptotic formulas for F_n(z) that yield fine results on the location, limiting distribution and accumulation points of the zeros of the Faber polynomials. The asymptotic formulas are shown to hold uniformly and the exact rate of decay of the error terms involved is provided."}
{"category": "Math", "title": "Generalised knot groups distinguish the square and granny knots (with an appendix by David Savitt)", "abstract": "Given a knot K we may construct a group G_n(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These \"generalised knot groups\" were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well known example of a pair of distinct knots with isomorphic fundamental groups. We show that G_n(SK) and G_n(GK) are non-isomorphic for all n>1. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of G_n(K), n>1, carries more information about K than the isomorphism type of the fundamental group. An appendix by David Savitt contains some results on representations of the trefoil group in PSL(2,p) that are needed for the proof."}
{"category": "Math", "title": "Simultaneous packing and covering in the two-dimensional Euclidean plane II", "abstract": "This paper determines the optimal upper bound for the simultaneous packing and covering constants of the two-dimensional centrally symmetric convex domains. It solved a problem opening for more than thirty years."}
{"category": "Math", "title": "A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra", "abstract": "A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal of this paper is to show the abundance of elements of N_p. Our main result shows that any divisor n of q-1, where q is a power of p, such that $n\\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its special case for p=2 which was proved in a previous paper by a different method."}
{"category": "Math", "title": "Characters of representations of affine Kac-Moody Lie algebras at the critical level", "abstract": "We present an explicit character formula for the irreducible highest weight representations of the non-twisted affine Kac-Moody Lie algebra at the critical level which are integrable over the corresponding finite-dimensional simple Lie algebra."}
{"category": "Math", "title": "An elementary proof of global existence for nonlinear wave equations in an exterior domain", "abstract": "The aim of this article is to present an elementary proof of a global existence result for nonlinear wave equations satifying the null condition in an exterior domain. The novelty of our proof is to avoid completely the scaling operator which would make the argument complicated in the mixed problem, by using a new weighted pointwise estimates of a tangential derivative to the light cone."}
{"category": "Math", "title": "General concepts of graphs", "abstract": "A little general abstract combinatorial nonsense delivered in this note is a presentation of some old and basic concepts, central to discrete mathematics, in terms of new words. The treatment is from a structural and systematic point of view. This note consists essentially of definitions and summaries."}
{"category": "Math", "title": "Subgeometric ergodicity of Markov chains", "abstract": "The goal of this paper is to give a short and self contained proof of general bounds for subgeometric rates of convergence, under practical conditions. The main result whose proof, based on coupling, provides an intuitive understanding of the results of Nummelin and Tuominen (1983) and Tuominen and Tweedie (1994). To obtain practical rates, a very general drift condition, recently introduced in Douc et al (2004) is used."}
{"category": "Math", "title": "On the K\\\"ahler classes of constant scalar curvature metrics on blow ups", "abstract": "In this note we clarify the structure of the moduli space of constant scalar curvature Kaehler metrics as one approaches the boundary of the Kaehler cone on cscK manifolds blown up at finite set of points, in the spirit of the previous work arXiv:math/0504115. Results about which Kaehler classes can be reached and about the position of the blown up points are given."}
{"category": "Math", "title": "Extreme-Value Analysis of Standardized Gaussian Increments", "abstract": "Let $\\{X_i,i=1,2,...\\}$ be i.i.d. standard gaussian variables. Let $S_n=X_1+...+X_n$ be the sequence of partial sums and $$ L_n=\\max_{0\\leq i<j\\leq n}\\frac{S_j-S_i}{\\sqrt{j-i}}. $$ We show that the distribution of $L_n$, appropriately normalized, converges as $n\\to\\infty$ to the Gumbel distribution. In some sense, the the random variable $L_n$, being the maximum of $n(n+1)/2$ dependent standard gaussian variables, behaves like the maximum of $Hn \\log n$ independent standard gaussian variables. Here, $H\\in (0,\\infty)$ is some constant. We also prove a version of the above result for the Brownian motion."}
{"category": "Math", "title": "Optimal H2 order-one reduction by solving eigenproblems for polynomial equations", "abstract": "A method is given for solving an optimal H2 approximation problem for SISO linear time-invariant stable systems. The method, based on constructive algebra, guarantees that the global optimum is found; it does not involve any gradient-based search, and hence avoids the usual problems of local minima. We examine mostly the case when the model order is reduced by one, and when the original system has distinct poles. This case exhibits special structure which allows us to provide a complete solution. The problem is converted into linear algebra by exhibiting a finite-dimensional basis for a certain space, and can then be solved by eigenvalue calculations, following the methods developed by Stetter and Moeller. The use of Buchberger's algorithm is avoided by writing the first-order optimality conditions in a special form, from which a Groebner basis is immediately available. Compared with our previous work the method presented here has much smaller time and memory requirements, and can therefore be applied to systems of significantly higher McMillan degree. In addition, some hypotheses which were required in the previous work have been removed. Some examples are included."}
{"category": "Math", "title": "Contributions of Issai Schur to Analysis", "abstract": "The name Schur is associated with many terms and concepts that are widely used in a number of diverse fields of mathematics and engineering. This survey article focuses on Schur's work in analysis. Here too, Schur's name is commonplace: The Schur test and Schur-Hadamard multipliers (in the study of estimates for Hermitian forms), Schur convexity, Schur complements, Schur's results in summation theory for sequences (in particular, the fundamental Kojima-Schur theorem), the Schur-Cohn test, the Schur algorithm, Schur parameters and the Schur interpolation problem for functions that are holomorphic and bounded by one in the unit disk. In this survey, we discuss all of the above mentioned topics and then some, as well as some of the generalizations that they inspired. There are nine sections of text, each of which is devoted to a separate theme based on Schur's work. Each of these sections has an independent bibliography. There is very little overlap. A tenth section presents a list of the papers of Schur that focus on topics that are commonly considered to be analysis. We begin with a review of Schur's less familiar papers on the theory of commuting differential operators."}
{"category": "Math", "title": "Quelques math\\'ematiques de la cryptographie \\`a cl\\'es publiques", "abstract": "I present examples of mathematical objects that are of interest for public key cryptography. Text for the Journ\\'ee Annuelle 2007 of the SMF."}
{"category": "Math", "title": "A mathematical model for a copolymer in an emulsion", "abstract": "In this paper we review some recent results, obtained jointly with Stu Whittington, for a mathematical model describing a copolymer in an emulsion. The copolymer consists of hydrophobic and hydrophilic monomers, concatenated randomly with equal density. The emulsion consists of large blocks of oil and water, arranged in a percolation-type fashion. To make the model mathematically tractable, the copolymer is allowed to enter and exit a neighboring pair of blocks only at diagonally opposite corners. The energy of the copolymer in the emulsion is minus $\\alpha$ times the number of hydrophobic monomers in oil minus $\\beta$ times the number of hydrophilic monomers in water. Without loss of generality we may assume that the interaction parameters are restricted to the cone $\\{(\\alpha,\\beta)\\in \\mathbb{R}^2\\colon |\\beta|\\leq\\alpha\\}$. We show that the phase diagram has two regimes: (1) in the supercritical regime where the oil blocks percolate, there is a single critical curve in the cone separating a localized and a delocalized phase; (2) in the subcritical regime where the oil blocks do not percolate, there are three critical curves in the cone separating two localized phases and two delocalized phases, and meeting at two tricritical points. The different phases are characterized by different behavior of the copolymer inside the four neighboring pairs of blocks."}
{"category": "Math", "title": "Homotopy classes of total foliations and bi-contact structures on three-manifolds", "abstract": "On every compact and orientable three-manifold, we construct total foliations (three codimension 1 foliations that are transverse at every point). This construction can be performed on any homotopy class of plane fields with vanishing Euler class. As a corollary we obtain similar results on bi-contact structures."}
{"category": "Math", "title": "On Transformation of Potapov's Fundamental Matrix Inequality", "abstract": "According to V.P.Potapov, a classical interpolation problem can be reformulated in terms of a so-called Fundamental Matrix Inequality (FMI). To show that every solution of the FMI satisfies the interpolation problem, we usualy have to transform the FMI in some special way. In this paper the number of of transformations of the FMI which come into play are motivated and demonstrated by simple, but typical examples."}
{"category": "Math", "title": "An Abstract Interpolation Problem and the Extension Theory of Hermitian Operators", "abstract": "The algebraic structure of V.P. Potapov's Fundamental Matrix Inequality (FMI) is discussed and its interpolation meaning is analyzed. Functional model spaces are involved. A general Abstract Interpolation Problem is formulated which seems to cover all the classical and recent problems in the field and the solution set of this problem is described using the Arov--Grossman formula. The extension theory of isometric operators is the proper language for treating interpolation problems of this type."}
{"category": "Math", "title": "On the Theory of Matrix Valued Functions Belonging to the Smirnov Class", "abstract": "A theory of matrix-valued functions from the matricial Smirnov class ${\\goth N}_n^+({\\Bbb D})$ is systematically developed. In particular, the maximum principle of V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman's theorem are presented for matrix-valued functions from the Smirnov class ${\\goth N}_n^+({\\Bbb D})$. We also consider a family $F_{\\lambda} =F-\\lambda I$ of functions belonging to the matricial Smirnov class which is indexed by a complex parameter $\\lambda$. We show that with the exception of a ''very small'' set of such $\\lambda$ the corresponding inner factor in the inner-outer factorization of the function $F_{\\lambda}$ is a Blaschke-Potapov product. The main goal of this paper is to provide users of analytic matrix-function theory with a standard source for references related to the matricial Smirnov class."}
{"category": "Math", "title": "Survival Probabilities for N-ary Subtrees on a Galton-Watson Family Tree", "abstract": "The family tree of a Galton-Watson branching process may contain N-ary subtrees, i.e. subtrees whose vertices have at least N>0 children. For family trees without infinite N-ary subtrees, we study how fast N-ary subtrees of height t disappear as t goes to infinity."}
{"category": "Math", "title": "Nagata's embedding theorem", "abstract": "In 1962-63, M. Nagata showed that an abstract variety could be embedded into a complete variety. Later, P. Deligne translated Nagata's proof into the language of schemes, but did not publish his notes. This paper, which is to appear as an appendix in a forthcoming book, gives an elaboration of Deligne's notes. It also contains some complementary results on extending divisors and vector sheaves to suitable completions."}
{"category": "Math", "title": "How to disentangle two braided Hopf algebras", "abstract": "We show how to define the tensor product of two braided Hopf algebras."}
{"category": "Math", "title": "Affine Buildings and Tropical Convexity", "abstract": "The notion of convexity in tropical geometry is closely related to notions of convexity in the theory of affine buildings. We explore this relationship from a combinatorial and computational perspective. Our results include a convex hull algorithm for the Bruhat--Tits building of SL$_d(K)$ and techniques for computing with apartments and membranes. While the original inspiration was the work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel and Tevelev in algebraic geometry, our tropical algorithms will also be applicable to problems in other fields of mathematics."}
{"category": "Math", "title": "On a problem of Duke-Erdos-Rodl on cycle-connected subgraphs", "abstract": "In this short note, we prove that for \\beta < 1/5 every graph G with n vertices and n^{2-\\beta} edges contains a subgraph G' with at least cn^{2-2\\beta} edges such that every pair of edges in G' lie together on a cycle of length at most 8. Moreover edges in G' which share a vertex lie together on a cycle of length at most 6. This result is best possible up to the constant factor and settles a conjecture of Duke, Erdos, and Rodl."}
{"category": "Math", "title": "The Heegaard structure of Dehn filled manifolds", "abstract": "We expect manifolds obtained by Dehn filling to inherit properties from the knot manifold. To what extent does that hold true for the Heegaard structure? We study four changes to the Heegaard structure that may occur after filling: (1) Heegaard genus decreases, (2) a new Heegaard surface is created, (3) a non-stabilized Heegaard surface destabilizes, and (4) two or more non-isotopic Heegaard surfaces become isotopic. We survey general results that give quite satisfactory restrictions to phenomena (1) and (2) and, in a parallel thread, give a complete classification of when all four phenomena occur when filling most torus knot exteriors. This latter thread yields sufficient (and perhaps necessary) conditions for the occurrence of phenomena (3) and (4)."}
{"category": "Math", "title": "Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics", "abstract": "Functional limit theorem for continuous-time random walks (CTRW) are found in general case of dependent waiting times and jump sizes that are also position dependent. The limiting anomalous diffusion is described in terms of fractional dynamics. Probabilistic interpretation of generalized fractional evolution is given in terms of the random time change (subordination) by means of hitting times processes."}
{"category": "Math", "title": "Proof of Riemann Hypothesis", "abstract": "In this article, we will prove Riemann Hypothesis by using the mean value theorem of integrals. The function $ \\xi(s) $ is introduced by Riemann, which zeros are identical equal to non-trivial zeros of zeta function.The function $ \\xi(s) $ is an entire function, and its real part and imaginary part can be represented as infinite integral form. In the special condition, the mean value theorem of integrals is established for infinite integral. Using the mean value theorem of integrals and the isolation of zeros of analytic function, we determined that all zeros of the function $ \\xi(s) $ have real part equal to$\\frac{1}{2}$, namely, all non-trivial zeros of zeta function lies on the critical line. Riemann Hypothesis is true."}
{"category": "Math", "title": "The characterization of the Carleson measures for analytic Besov spaces: a simple proof", "abstract": "We give a simple proof of the characterization of the Carleson measures for the weighted analytic Besov spaces. Such characterization provides some information on the radial variation of an analytic Besov function."}
{"category": "Math", "title": "On Jordan Derivations of Triangular Algebras", "abstract": "In this short note we prove that every Jordan derivation of triangular algebras is a derivation."}
{"category": "Math", "title": "The wreath product of Z with Z has Hilbert compression exponent 2/3", "abstract": "Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all $\\alpha\\ge 0$ such that there exists a Lipschitz mapping $f:G\\to L_2$ and a constant $c>0$ such that for all $x,y\\in G$ we have $\\|f(x)-f(y)\\|_2\\ge cd(x,y)^\\alpha.$ In \\cite{AGS06} it was shown that the Hilbert compression exponent of the wreath product $\\Z\\bwr \\Z $ is at most $\\frac34$, and in \\cite{NP07} was proved that this exponent is at least $\\frac23$. Here we show that $\\frac23$ is the correct value. Our proof is based on an application of K. Ball's notion of Markov type."}
{"category": "Math", "title": "Canonical representatives for residue classes of a polynomial ideal and orthogonality", "abstract": "The aim of this paper is to unveil an unexpected relationship between the normal form of a polynomial with respect to a polynomial ideal and the more geometric concept of orthogonality. We present a new way to calculate the normal form of a polynomial with respect to a polynomial ideal I in the ring of multivariate polynomials over a field K, provided the field K is finite and the ideal I is a vanishing ideal. In order to use the concept of orthogonality, we introduce a symmetric bilinear form on a vector space over a finite field."}
{"category": "Math", "title": "Generators of II_1 Factors", "abstract": "In 2005, Shen introduced a new invariant, $\\mathcal G(N)$, of a diffuse von Neumann algebra $N$ with a fixed faithful trace, and he used this invariant to give a unified approach to showing that large classes of ${\\mathrm{II}}_1$ factors $M$ are singly generated. This paper focuses on properties of this invariant. We relate $\\mathcal G(M)$ to the number of self-adjoint generators of a ${\\mathrm{II}}_1$ factor $M$: if $\\mathcal G(M)<n/2$, then $M$ is generated by $n+1$ self-adjoint operators, whereas if $M$ is generated by $n+1$ self-adjoint operators, then $\\mathcal G(M)\\leq n/2$. The invariant $\\mathcal G(\\cdot)$ is well-behaved under amplification, satisfying $\\mathcal G(M_t)=t^{-2}\\mathcal G(M)$ for all $t>0$. In particular, if $\\mathcal G(\\mathcal L\\mathbb F_r)>0$ for any particular $r>1$, then the free group factors are pairwise non-isomorphic and are not singly generated for sufficiently large values of $r$. Estimates are given for forming free products and passing to finite index subfactors and the basic construction. We also examine a version of the invariant $\\mathcal G_{\\text{sa}}(M)$ defined only using self-adjoint operators; this is proved to satisfy $\\mathcal G_{\\text{sa}}(M)=2\\mathcal G(M)$. Finally we give inequalities relating a quantity involved in the calculation of $\\mathcal G(M)$ to the free-entropy dimension $\\delta_0$ of a collection of generators for $M$."}
{"category": "Math", "title": "SPM Bulletin 21", "abstract": "Contents: 1. Editor's note; 2. Personal impressions from the SPM07 meeting; 3. Research announcements; 3.1. Coloring ordinals by reals; 3.2. Long Borel Hierarchies; 3.3. Rothberger's property in finite powers; 3.4. Special subsets of the reals and tree forcing notions; 3.5. All automorphisms of the Calkin algebra are inner; 3.6. Continuous selections and sigma-spaces; 3.7. On the closure of the diagonal of a T1-space; 3.8. Splitting families and Noetherian type; 3.9. Even more simple cardinal invariants; 3.10. A classification of CO spaces which are continuous images of compact ordered spaces; 4. Problem of the Issue"}
{"category": "Math", "title": "Spherical Continuous Wavelet Transforms arising from sections of the Lorentz group", "abstract": "We consider the conformal group of the unit sphere $S^{n-1},$ the so-called proper Lorentz group Spin$^+(1,n),$ for the study of spherical continuous wavelet transforms (CWT). Our approach is based on the method for construction of general coherent states associated to square integrable group representations over homogeneous spaces. The underlying homogeneous space is an extension to the whole of the group Spin$^+(1,n)$ of the factorization of the gyrogroup of the unit ball by an appropriate gyro-subgroup. Sections on this homogeneous space are constituted by rotations of the subgroup Spin$(n)$ and M\\\"{o}bius transformations of the type $\\phi_a(x)=(x-a)(1+ax)^{-1},$ where $a$ belongs to a given section on a homogeneous space of the unit ball. This extends in a natural way the work of Antoine and Vandergheynst to anisotropic conformal dilations."}
{"category": "Math", "title": "Fano threefolds with noncyclic torsion in the divisor class group", "abstract": "In this note we study Fano threefolds with noncyclic torsion in the divisor class group. Since they can all be obtained as quotients of Fano threefolds, we get also all examples that can be obtained as quotients of low codimension Fanos in the weighted projective space."}
{"category": "Math", "title": "Kleinian groups with ubiquitous surface subgroups", "abstract": "We show that every finitely-generated free subgroup of a right-angled, co-compact Kleinian reflection group is contained in a surface subgroup."}
{"category": "Math", "title": "The Riemann hypothesis - an elementary analytic approach based on complex Laplace transform", "abstract": "An elementary analytic proof of the famous Riemann hypothesis is given. The main \"accent\" of the proof is a both using of the 2-dimensional double real and complex Laplace integral representations of the Green function $\\mid z \\mid^{-2}$."}
{"category": "Math", "title": "Formation of singularities for a transport equation with nonlocal velocity", "abstract": "We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time."}
{"category": "Math", "title": "Skew-Hadamard matrices of orders 436, 580 and 988 exist", "abstract": "We construct two difference families on each of the cyclic groups of order 109, 145 and 247, and use them to construct skew-Hadamard matrices of orders 436, 580 and 988. Such difference families and matrices are constructed here for the first time. The matrices are constructed by using the Goethals-Seidel array."}
{"category": "Math", "title": "On pathwise uniqueness for reflecting Brownian motion in $C^{1+\\gamma}$ domains", "abstract": "Pathwise uniqueness holds for the Skorokhod stochastic differential equation in $C^{1+\\gamma}$ domains in $\\mathbb{R}^d$ for $\\gamma >1/2$ and $d\\geq3$."}
{"category": "Math", "title": "A transient Markov chain with finitely many cutpoints", "abstract": "We give an example of a transient reversible Markov chain that almost surely has only a finite number of cutpoints. We explain how this is relevant to a conjecture of Diaconis and Freedman and a question of Kaimanovich. We also answer Kaimanovich's question when the Markov chain is a nearest-neighbor random walk on a tree."}
{"category": "Math", "title": "Twisted Alexander polynomial of links in the projective space", "abstract": "We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial."}
{"category": "Math", "title": "Quantum Teichm\\\"uller spaces and Kashaev's 6j-symbols", "abstract": "The Kashaev invariants of 3-manifolds are based on $6j$-symbols from the representation theory of the Weyl algebra, a Hopf algebra corresponding to the Borel subalgebra of $U_q(sl(2,\\C))$. In this paper, we show that Kashaev's $6j$-symbols are intertwining operators of local representations of quantum Teichm\\\"uller spaces. This relates Kashaev's work with the theory of quantum Teichm\\\"uller space, which was developed by Chekhov-Fock, Kashaev and continued by Bonahon-Liu."}
{"category": "Math", "title": "Eigenvalues and lambda constants on Riemannian submersions", "abstract": "Given a Riemannian submersion, we study the relation between lambda constants introduced by G.Perelman on the base manifold and the total space of a Riemannian submersion. We also discuss the relationship between the first eigenvalues of Laplacians on the base manifold and that of the total space. The quantities on warped products are discussed in detail."}
{"category": "Math", "title": "Heegaard Floer homology and fibred 3--manifolds", "abstract": "Given a closed 3--manifold $Y$, we show that the Heegaard Floer homology determines whether $Y$ fibres over the circle with a fibre of negative Euler characteristic. This is an analogue of an earlier result about knots proved by Ghiggini and the author."}
{"category": "Math", "title": "Properties of positive solutions of an Elliptic Equation with negative exponents", "abstract": "In this paper, we study the existence and non-existence result of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics. We can get some useful local gradient estimate and L1 lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given."}
{"category": "Math", "title": "Lines of minima are uniformly quasi-geodesic", "abstract": "We continue the comparison between lines of minima and Teichmueller geodesics begun in [CRS1]. We show that in the Teichmueller space of a surface S, lines of minima are quasi-geodesic with respect to the Teichmueller metric. The quasi-geodesic constants depend only on the topological type of S."}
{"category": "Math", "title": "Geometry of curves with exceptional secant planes: linear series along the general curve", "abstract": "We study linear series on a general curve of genus $g$, whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill-Noether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formula along judiciously-chosen test families. As an application, we compute the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. We pay special attention to the extremal case of $d$-secant $(d-2)$-planes to $(2d-1)$-dimensional series, which appears in the study of Hilbert schemes of points on surfaces. In that case, our formula may be rewritten in terms of hypergeometric series, which allows us both to prove that it is nonzero and to deduce its asymptotics in $d$."}
{"category": "Math", "title": "Collapse of unit horizontal bundles equipped with a metric of Cheeger-Gromoll type", "abstract": "We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse theorem for unit horizontal bundles associated with a sequence of warped Riemannian submersions."}
{"category": "Math", "title": "Marcus-Lushnikov processes, Smoluchowski's and Flory's models", "abstract": "The Marcus-Lushnikov process is a finite stochastic particle system in which each particle is entirely characterized by its mass. Each pair of particles with masses $x$ and $y$ merges into a single particle at a given rate $K(x,y)$. We consider a {\\it strongly gelling} kernel behaving as $K(x,y)=x^\\alpha y + x y^\\alpha$ for some $\\alpha\\in (0,1]$. In such a case, it is well-known that {\\it gelation} occurs, that is, giant particles emerge. Then two possible models for hydrodynamic limits of the Marcus-Lushnikov process arise: the Smoluchowski equation, in which the giant particles are inert, and the Flory equation, in which the giant particles interact with finite ones. We show that, when using a suitable cut-off coagulation kernel in the Marcus-Lushnikov process and letting the number of particles increase to infinity, the possible limits solve either the Smoluchowski equation or the Flory equation. We also study the asymptotic behaviour of the largest particle in the Marcus-Lushnikov process without cut-off and show that there is only one giant particle. This single giant particle represents, asymptotically, the lost mass of the solution to the Flory equation."}
{"category": "Math", "title": "Weak collapsing and geometrisation of aspherical 3-manifolds", "abstract": "Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred or contains an incompressible torus. This result gives an alternative approach for the last step in Perelman's proof of the Geometrisation Conjecture for aspherical 3-manifolds."}
{"category": "Math", "title": "Preserving zeros of a polynomial", "abstract": "We study non-linear surjective mappings on subsets of ${\\cal M}_n(F)$, which preserve the zeros of some fixed polynomials in noncommuting variables. Keywords: Matrix algebra, Multilinear polynomials, Preservers."}
{"category": "Math", "title": "Homogenization of nonlinear scalar conservation laws", "abstract": "We study the limit as $\\e\\to 0$ of the entropy solutions of the equation $\\p_t \\ue + \\dv_x[A(\\frac{x}{\\e},\\ue)] =0$. We prove that the sequence $\\ue$ two-scale converges towards a function $u(t,x,y)$, and $u$ is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence result in $L^1_{\\text{loc}}$."}
{"category": "Math", "title": "The size of the largest component below phase transition in inhomogeneous random graphs", "abstract": "We study the \"rank 1 case\" of the inhomogeneous random graph model. In the subcritical case we derive an exact formula for the asymptotic size of the largest connected component scaled to log n. This result is new, it completes the corresponding known result in the supercritical case. We provide some examples of application of a new formula."}
{"category": "Math", "title": "Constrained Ramsey Numbers", "abstract": "For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdos-Rado Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <= O(st^2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f(S, P_t) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t."}
{"category": "Math", "title": "On the strong chromatic number of random graphs", "abstract": "Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \\cup ... \\cup V_r, all of size exactly k, there exists a proper vertex k-coloring of G with each color appearing exactly once in each V_i. In the case when k does not divide n, G is defined to be strongly k-colorable if the graph obtained by adding k \\lceil n/k \\rceil - n isolated vertices is strongly k-colorable. The strong chromatic number of G is the minimum k for which G is strongly k-colorable. In this paper, we study the behavior of this parameter for the random graph G(n, p). In the dense case when p >> n^{-1/3}, we prove that the strong chromatic number is a.s. concentrated on one value \\Delta+1, where \\Delta is the maximum degree of the graph. We also obtain several weaker results for sparse random graphs."}
{"category": "Math", "title": "Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm", "abstract": "Let $\\F_q$ ($q=p^r$) be a finite field. In this paper the number of irreducible polynomials of degree $m$ in $\\F_q[x]$ with prescribed trace and norm coefficients is calculated in certain special cases and a general bound for that number is obtained improving the bound by Wan if $m$ is small compared to $q$. As a corollary, sharp bounds are obtained for the number of elements in $\\F_{q^3}$ with prescribed trace and norm over $\\F_q$ improving the estimates by Katz in this special case. Moreover, a characterization of Kloosterman sums over $\\F_{2^r}$ divisible by three is given generalizing the earlier result by Charpin, Helleseth, and Zinoviev obtained only in the case $r$ odd. Finally, a new simple proof for the value distribution of a Kloosterman sum over the field $\\F_{3^r}$, first proved by Katz and Livne, is given."}
{"category": "Math", "title": "A family of acyclic functors", "abstract": "We determine a family of functors from a poset to abelian groups such that the higher direct limits vanish on them. This is done by first characterizing the projective functors. Then a spectral sequence arising from the grading of the poset is used. Also the dual version for injective functors and higher inverse limits is included. Graded posets include simplicial complexes, subdivision categories and simplex-like posets."}
{"category": "Math", "title": "Linear precision for parametric patches", "abstract": "We give a precise mathematical formulation for the notions of a parametric patch and linear precision, and establish their elementary properties. We relate linear precision to the geometry of a particular linear projection, giving necessary (and quite restrictive) conditions for a patch to possess linear precision. A main focus is on linear precision for Krasauskas' toric patches, which we show is equivalent to a certain rational map on CP^d being a birational isomorphism. Lastly, we establish the connection between linear presision for toric surface patches and maximum likelihood degree for discrete exponential families in algebraic statistics, and show how iterative proportional fitting may be used to compute toric patches."}
{"category": "Math", "title": "A method for integral cohomology of posets", "abstract": "We present a method to compute integral cohomology of posets. This toolbox is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes and simplex-like posets. The method is based on homological algebra arguments in the category of functors and on a spectral sequence built upon the poset. We show its relation to discrete Morse theory. As application we give an alternative proof of Webb's conjecture for saturated fusion systems and we compute the cohomology of Coxeter complexes for finite and infinite Coxeter groups."}
{"category": "Math", "title": "Diagonalization and representation results for nonpositive sesquilinear form measures", "abstract": "We study decompositions of operator measures and more general sesquilinear form measures $E$ into linear combinations of positive parts, and their diagonal vector expansions. The underlying philosophy is to represent $E$ as a trace class valued measure of bounded variation on a new Hilbert space related to $E$. The choice of the auxiliary Hilbert space fixes a unique decomposition with certain properties, but this choice itself is not canonical. We present relations to Naimark type dilations and direct integrals."}
{"category": "Math", "title": "Independent transversals in locally sparse graphs", "abstract": "Let G be a graph with maximum degree \\Delta whose vertex set is partitioned into parts V(G) = V_1 \\cup ... \\cup V_r. A transversal is a subset of V(G) containing exactly one vertex from each part V_i. If it is also an independent set, then we call it an independent transversal. The local degree of G is the maximum number of neighbors of a vertex v in a part V_i, taken over all choices of V_i and v \\not \\in V_i. We prove that for every fixed \\epsilon > 0, if all part sizes |V_i| >= (1+\\epsilon)\\Delta and the local degree of G is o(\\Delta), then G has an independent transversal for sufficiently large \\Delta. This extends several previous results and settles (in a stronger form) a conjecture of Aharoni and Holzman. We then generalize this result to transversals that induce no cliques of size s. (Note that independent transversals correspond to s=2.) In that context, we prove that parts of size |V_i| >= (1+\\epsilon)[\\Delta/(s-1)] and local degree o(\\Delta) guarantee the existence of such a transversal, and we provide a construction that shows this is asymptotically tight."}
{"category": "Math", "title": "The role of the cotangent bundle in resolving ideals of fat points in the plane", "abstract": "We study the connection between the generation of a fat point scheme supported at general points in the plane and the behaviour of the cotangent bundle with respect to some rational curves particularly relevant for the scheme. We put forward two conjectures, giving examples and partial results in support of them."}
{"category": "Math", "title": "On elliptic Dunkl operators", "abstract": "We attach elliptic Dunkl operators to an abelian variety with a finite group action. This generalizes elliptic Dunkl operators for Weyl groups, defined by Buchstaber, Felder, and Veselov in 1994. We show that these operators commute, and use them to define representations from category O of elliptic Cherednik algebras. We also consider the monodromy representations of differential equations defined by elliptic Dunkl operators, and show that they yield finite dimensional rrepresentations of generalized double affine Hecke algebras."}
{"category": "Math", "title": "Vector invariants of a class of pseudo-reflection groups and multisymmetric syzygies", "abstract": "First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field. Special case of the result is a finite presentation of the algebra of multisymmetric polynomials. Reducedness of the invariant commuting scheme is proved as a by-product. The algebra of multisymmetric polynomials over an arbitrary base ring is revisited."}
{"category": "Math", "title": "Minimality in topological groups and Heisenberg type groups", "abstract": "We study relatively minimal subgroups in topological groups. We find, in particular, some natural relatively minimal subgroups in unipotent groups which are defined over \"good\" rings. By \"good\" rings we mean archimedean absolute valued (not necessarily associative) division rings. Some of the classical rings which we consider besides the field of reals are the ring of quaternions and the ring of octonions. This way we generalize in part a previous result which was obtained by Dikranjan and Megrelishvili and involved the Heisenberg group."}
{"category": "Math", "title": "Canonical heights and the arithmetic complexity of morphisms on projective space", "abstract": "Let F and G be morphisms of degree at least 2 from P^N to P^N that are defined over the algebraic closure of Q. We define the arithmetic distance d(F,G) between F and G to be the supremum over all algebraic points P of |h_F(P)-h_G(P)|, where h_F and h_G are the canonical heights associated to the morphisms F and G, respectively. We prove comparison theorems relating d(F,G) to more naive height functions and show that for a fixed G, the set of F satisfying d(F,G) < B is a set of bounded height. In particular, there are only finitely many such F defined over any given number field."}
{"category": "Math", "title": "Nonarchimedean Green functions and dynamics on projective space", "abstract": "Let F: P^N_K --> P^N_K be a morphism of degree d > 1 defined over a field K that is algebraically closed and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function G_F associated to F is Holder continuous on P^N(K) and that the Fatou set F is equal to the set of points at which G_F is locally constant. Further, G_F vanishes precisely on the set of points P such that F has good reduction at every point in the forward orbit of P. We also prove that the iterates of F are locally uniformly Lipschitz on the Fatou set of F."}
{"category": "Math", "title": "A resolution of the K(2)-local sphere at the prime 3", "abstract": "We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L_{K(2)S^0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E_2^hF where F is a finite subgroup of the Morava stabilizer group and E_2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n=2 at p=3 represents the edge of our current knowledge: n=1 is classical and at n=2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic."}
{"category": "Math", "title": "Mapping Class Groups do not have Kazhdan's Property (T)", "abstract": "We prove that the mapping class group of a closed oriented surface of genus at least two does not have Kazhdan's property (T)."}
{"category": "Math", "title": "Operads of compatible structures and weighted partitions", "abstract": "In this paper we describe operads encoding two different kinds of compatibility of algebraic structures. We show that there exist decompositions of these in terms of black and white products and we prove that they are Koszul for a large class of algebraic structures by using the poset method of B. Vallette. In particular we show that this is true for the operads of compatible Lie, associative and pre-Lie algebras."}
{"category": "Math", "title": "Gorenstein Multiple Structures on Smooth Algebraic Varieties", "abstract": "We characterize the Gorenstein nilpotent scheme structures on a smooth algebraic variety as support, in terms of a duality property of the graded objects associated to two canonical filtrations."}
{"category": "Math", "title": "Non-abelian Reidemeister torsion for twist knots", "abstract": "This paper gives an explicit formula for the SL_2(C)-non-abelian Reidemeister torsion as defined in [Dub06] in the case of twist knots. For hyperbolic twist knots, we also prove that the non-abelian Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot."}
{"category": "Math", "title": "The analog of the Schauder inequality for closed surfaces in Euclidean spaces", "abstract": "The analog of the Schauder inequality for closed surfaces in Euclidean spaces is obtained in this article."}
{"category": "Math", "title": "On the derivative of the Minkowski question mark function $?(x)$", "abstract": "Let $ x = [0;a_1,a_2,...]$ be the decomposition of the irrational number $x \\in [0,1]$ into regular continued fraction. Then for the derivative of the Minkowski function $?(x)$ we prove that $?'(x) = +\\infty$ provided $ \\limsup_{t\\to \\infty}\\frac{a_1+...+a_t}{t} <\\kappa_1 =\\frac{2\\log \\lambda_1}{\\log 2} = 1.388^+$, and $?'(x) = 0$ provided $ \\liminf_{t\\to \\infty}\\frac{a_1+...+a_t}{t} >\\kappa_2 = \\frac{4L_5-5L_4}{L_5-L_4}= 4.401^+$ (here $ L_j = \\log (\\frac{j+\\sqrt{j^2+4}}{2}) - j\\cdot\\frac{\\log 2}{2}$). Constants $\\kappa_1,\\kappa_2$ are the best possible. Also we prove that $?'(x) = +\\infty$ holds for all $x$ with partial quotients bounded by 4."}
{"category": "Math", "title": "Towards the Distribution of the Size of a Largest Planar Matching and Largest Planar Subgraph in Random Bipartite Graphs", "abstract": "We address the following question: When a randomly chosen regular bipartite multi--graph is drawn in the plane in the ``standard way'', what is the distribution of its maximum size planar matching (set of non--crossing disjoint edges) and maximum size planar subgraph (set of non--crossing edges which may share endpoints)? The problem is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam's problem). We present combinatorial identities which relate the number of r-regular bipartite multi--graphs with maximum planar matching (maximum planar subgraph) of at most d edges to a signed sum of restricted lattice walks in Z^d, and to the number of pairs of standard Young tableaux of the same shape and with a ``descend--type'' property. Our results are obtained via generalizations of two combinatorial proofs through which Gessel's identity can be obtained (an identity that is crucial in the derivation of a bivariate generating function associated to the distribution of LISs, and key to the analytic attack on Ulam's problem). We also initiate the study of pattern avoidance in bipartite multigraphs and derive a generalized Gessel identity for the number of bipartite 2-regular multigraphs avoiding a specific (monotone) pattern."}
{"category": "Math", "title": "Existence of Kirillov-Reshetikhin crystals for nonexceptional types", "abstract": "Using the methods of Kang et al. and recent results on the characters of Kirillov-Reshetikhin modules by Nakajima and Hernandez, the existence of Kirillov-Reshetikhin crystals B^{r,s} is established for all nonexceptional affine types. We also prove that the crystals B^{r,s} of type B_n^{(1)}, D_n^{(1)}, and A_{2n-1}^{(2)} are isomorphic to recently constructed combinatorial crystals for r not a spin node."}
{"category": "Math", "title": "Periodic harmonic functions on lattices and points count in positive characteristic", "abstract": "This survey addresses pluri-periodic harmonic functions on lattices with values in a positive characteristic field. We mention, as a motivation, the game \"Lights Out\" following the work of Sutner, Goldwasser-Klostermeyer-Ware, Barua-Ramakrishnan-Sarkar, Hunzikel-Machiavello-Park e.a.; see also 2 previous author's preprints for a more detailed account. Our approach explores harmonic analysis and algebraic geometry over a positive characteristic field. The Fourier transform allows us to interpret pluri-periods of harmonic functions on lattices as torsion multi-orders of points on the corresponding affine algebraic variety."}
{"category": "Math", "title": "AF-embedding of the crossed products of AH-algebras by finitely generated abelian groups", "abstract": "Let $X$ be a compact metric space and let $\\Lambda$ be a $\\Z^k$ ($k\\ge 1$) action on $X.$ We give a solution to a version of Voiculescu's problem of AF-embedding: The crossed product $C(X)\\rtimes_{\\Lambda}\\Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $\\Lambda$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $\\Lambda: G\\to Aut(C)$ be a monomorphism. We show that $C\\rtimes_{\\Lambda} G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $\\Lambda$-invariant tracial state."}
{"category": "Math", "title": "Characterization of intrinsically harmonic forms", "abstract": "Let $M$ be a closed oriented manifold of dimension $n$ and $\\omega$ a closed 1-form on it. We discuss the question whether there exists a Riemannian metric for which $\\omega$ is co-closed. For closed 1-forms with nondegenerate zeros the question was answered completely by Calabi in 1969. The goal of this paper is to give an answer in the general case, i.e. not making any assumptions on the zero set of $\\omega$."}
{"category": "Math", "title": "Infinitesimal Castelnuovo Theory in Abelian Varieties", "abstract": "The purpose of this article is to show that the Castelnuovo theory for abelian varieties, developed by G. Pareschi and M. Popa, can be infinitesimalized. More precisely, we prove that an irreducible principally polarized abelian variety has a finite scheme in extremal position, in the sense of Castelnuovo theory for abelian varieties, if, and only if, it is a Jacobian and the scheme is contained in a unique Abel-Jacobi curve."}
{"category": "Math", "title": "A refinement of Khovanov-Rozansky link homology", "abstract": "This paper has been withdrawn by the author due to an error in the proof of Theorem 1."}
{"category": "Math", "title": "Sign lemma for dimension shifting", "abstract": "There is a surprising occurrence of some minus signs in the isomorphisms produced in the well-known technique of dimension shifting in calculating derived functors in homological algebra. We explicitly determine these signs. Getting these signs right is important in order to avoid basic contradictions. We illustrate the lemma by some de Rham cohomology and Chern class considerations for compact Riemann surfaces."}
{"category": "Math", "title": "Algebraic K-theory and cubical descent", "abstract": "In this note we apply Guillen-Navarro descent theorem, \\cite{GN02}, to define a descent variant of the algebraic $K$-theory of varieties over a field of characteristic zero, $\\mathcal{KD}(X)$, which coincides with $\\mathcal{K}(X)$ for smooth varieties. After a result of Haesemeyer, this new theory is equivalent to the homotopy algebraic $K$-theory introduced by Weibel. We also prove that there is a natural weight filtration on the groups $KH_\\ast(X)$."}
{"category": "Math", "title": "Uniqueness of $\\bf C^*$- and $\\bf C_+$-actions on Gizatullin surfaces", "abstract": "A Gizatullin surface is a normal affine surface $V$ over $\\bf C$, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $\\bf C^*$-actions and $\\bf A^1$-fibrations on such a surface $V$ up to automorphisms. The latter fibrations are in one to one correspondence with $\\bf C_+$-actions on $V$ considered up to a \"speed change\". Non-Gizatullin surfaces are known to admit at most one $\\bf A^1$-fibration $V\\to S$ up to an isomorphism of the base $S$. Moreover an effective $\\bf C^{*}$-action on them, if it does exist, is unique up to conjugation and inversion $t\\mapsto t^{-1}$ of $\\bf C^*$. Obviously uniqueness of $\\bf C^*$-actions fails for affine toric surfaces; however we show in this case that there are at most two conjugacy classes of $\\bf A^1$-fibrations. There is a further interesting family of non-toric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $\\bf C^*$-actions and $\\bf A^1$-fibrations. In the present paper we obtain a criterion as to when $\\bf A^1$-fibrations of Gizatullin surfaces are conjugate up to an automorphism of $V$ and the base $S$. We exhibit as well a large subclasses of Gizatullin $\\bf C^{*}$-surfaces for which a $\\bf C^*$-action is essentially unique and for which there are at most two conjugacy classes of $\\bf A^1$-fibrations over $\\bf A^1$."}
{"category": "Math", "title": "Expanders and the Affine Building of ${\\rm Sp}_n$", "abstract": "For $n \\geq 2$ and a local field $K$, let $\\Delta_n$ denote the affine building naturally associated to the symplectic group ${\\rm Sp}_n(K)$. We compute the spectral radius of the subgraph $Y_n$ of $\\Delta_n$ induced by the special vertices in $\\Delta_n$, from which it follows that $Y_n$ is an analogue of a family of expanders and is non-amenable."}
{"category": "Math", "title": "Partition Identities and the Coin Exchange Problem", "abstract": "The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases."}
{"category": "Math", "title": "Existence theorem and blow-up criterion of the strong solutions to the Magneto-micropolar fluid equations", "abstract": "In this paper we study the magneto-micropolar fluid equations in $\\R^3$, prove the existence of the strong solution with initial data in $H^s(\\R^3)$ for $s> {3/2}$, and set up its blow-up criterion. The tool we mainly use is Littlewood-Paley decomposition, by which we obtain a Beale-Kato-Majda type blow-up criterion for smooth solution $(u,\\omega,b)$ which relies on the vorticity of velocity $\\nabla\\times u$ only."}
{"category": "Math", "title": "The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem", "abstract": "Let f(x) be a continuous function from a compact real interval into itself with a periodic orbit of minimal period m, where m is not an integral power of 2. Then, by Sharkovsky's theorem, for every positive integer n with m \\prec n in the Sharkovsky's ordering defined below, a lower bound on the number of periodic orbits of f(x) with minimal period n is 1. Could we improve this lower bound from 1 to some larger number? In this paper, we give a complete answer to this question."}
{"category": "Math", "title": "The periodic table of $n$-categories for low dimensions II: degenerate tricategories", "abstract": "We continue the project begun in ``The periodic table of $n$-categories for low dimensions I'' by examining degenerate tricategories and comparing them with the structures predicted by the Periodic table. For triply degenerate tricategories we exhibit a triequivalence with the partially discrete tricategory of commutative monoids. For the doubly degenerate case we explain how to construct a braided monoidal category from a given doubly degenerate category, but show that this does not induce a straightforward comparison between \\bfseries{BrMonCat} and \\bfseries{Tricat}. We show how to alter the natural structure of \\bfseries{Tricat} in two different ways to provide a comparison, but show that only the more brutal alteration yields an equivalence. Finally we study degenerate tricategories in order to give the first fully algebraic definition of monoidal bicategories and the full tricategory structure \\bfseries{MonBicat}."}
{"category": "Math", "title": "Cohomology of diffeological spaces and foliations", "abstract": "Let (M,F) be a foliated manifold. We study the relationship between the basic cohomology Hb(M,F) of the foliation and the De Rham cohomology H(DF) of the space of leaves M/F as a quotient diffeological space. We prove that for an arbitrary foliation there is a morphism from H(DF) to Hb(M,F). It is an isomorphism when F is a Q-foliation."}
{"category": "Math", "title": "Hypersurfaces in H^{n+1} and conformally invariant equations: the generalized Christoffel and Nirenberg problems", "abstract": "Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with the famous problem of prescribing scalar curvature on $\\S^n$ for conformal metrics, posed by Nirenberg and Kazdan-Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory interpretation of such an intrinsic problem from conformal geometry. Our third objective is to place the above result into a more general framework. Specifically, we will show how the problem of prescribing the hyperbolic Gauss map and a given function of the hyperbolic curvature radii in $H^{n+1}$ is strongly related to some important problems on conformally invariant PDEs in terms of the Schouten tensor. This provides a bridge between the theory of conformal metrics on $\\S^n$ and the theory of hypersurfaces with prescribed hyperbolic Gauss map in $\\H^{n+1}$. The fourth objective is to use the above correspondence to prove that for a wide family of Weingarten functionals $W(\\k_1,..., \\k_n)$, the only compact immersed hypersurfaces in $H^{n+1}$ on which $W$ is constant are round spheres."}
{"category": "Math", "title": "Stable Border Bases for Ideals of Points", "abstract": "Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ which exhibits structural stability, that is, if $\\widetilde X$ is any set of points differing only slightly from $X$, there exists a polynomial set $\\widetilde B$ structurally similar to $B$, which is a basis of the perturbed ideal $ I(\\widetilde X)$."}
{"category": "Math", "title": "On large automorphism groups of algebraic curves in positive characteristic", "abstract": "In his investigation on large $K$-automorphism groups of an algebraic curve, Stichtenoth obtained an upper bound on the order of the first ramification group of an algebraic curve $\\cX$ defined over an algebraically closed field of characteristic $p$. Stichtenoth's bound has raised the problem of classifying all $\\K$-automorphism groups $G$ of $\\cX$ with the following property: There is a point $P\\in \\cX$ for which \\begin{equation} |G_P^{(1)}|>\\frac{p}{p-1}g. \\end{equation} Such a classification is obtained here by proving Theorem 1.3"}
{"category": "Math", "title": "Lower bounds for moments of zeta prime rho", "abstract": "Assuming the Riemann Hypothesis, we establish lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\\zeta(s)$. Our proof is based upon a recent method of Rudnick and Soundararajan that provides analogous bounds for moments of $L$-functions at the central point, averaged over families."}
{"category": "Math", "title": "Special isogenies and tensor product multiplicities", "abstract": "We show that any bijection between two root systems that preserves angles (but not necessarily lengths) gives rise to inequalities relating tensor product multiplicities for the corresponding complex semisimple Lie groups (or Lie algebras). We explain the inequalities in two ways: combinatorially, using Littelmann's Path Model, and geometrically, using isogenies between algebaric groups defined over an algebraically closed field of positive characteristic."}
{"category": "Math", "title": "Hodge-Stickelberger polygons for L-functions of exponential sums of P(x^s)", "abstract": "Let P(x) be a one-variable Laurent polynomial of degree (d_1,d_2) over a finite field of characteristic p. For any fixed positive integer s not divisible by p, we prove that the (normalized) p-adic Newton polygon of the L-functions of exponential sums of P(x^s) has a tight lower bound which we call `Hodge-Stickelberger polygon', depending only on d_1,d_2,s, and (p mod s). This Hodge-Stickelberger polygon is a weighted convolution of a `Hodge polygon' for L-function of exponential sum of P(x) and the `Newton polygon' for L-function of exponential sum of x^s (given by the classical Stickelberger theory). We prove an analogous Hodge-Stickelberger lower bound for multivariable Laurent polynomials as well. We prove this Hodge-Stickelberger polygon is the limit of generic Newton polygons of P(x^s) in a sense that was made explicit in the paper."}
{"category": "Math", "title": "Nonlinear perturbations of Fuchsian systems: corrections and linearization, normal forms", "abstract": "Nonlinear perturbation of Fuchsian systems are studied in a region including two singularities. It is proved that such systems are generally not analytically equivalent to their linear part (they are not linearizable) and the obstructions are found constructively, as a countable set of numbers. Furthermore, assuming a polynomial character of the nonlinear part, it is shown that there exists a unique formal \"correction\" of the nonlinear part so that the \"corrected\" system is formally linearizable. Normal forms of these systems are found, providing also their classification."}
{"category": "Math", "title": "On the cyclicity of weight-homogeneous centers", "abstract": "Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this result to a particular family of planar polynomial systems having a nilpotent center without meromorphic first integral."}
{"category": "Math", "title": "Analytic linearization of nonlinear perturbations of Fuchsian systems", "abstract": "Nonlinear perturbation of Fuchsian systems are studied in regions including two singularities. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable). Nevertheless, it is shown that in the case when the linear part has commuting monodromy, and the eigenvalues have positive real parts, there exists a unique correction function of the nonlinear part so that the corrected system becomes analytically linearizable."}
{"category": "Math", "title": "Kummer subfields of tame division algebras over Henselian valued fields", "abstract": "By generalizing the method used by Tignol and Amitsur in [TA85], we determine necessary and sufficient conditions for an arbitrary tame central division algebra D over a Henselian valued field E to have Kummer subfields [Corollary 2.11 and Corollary 2.12]. We prove also that if D is a tame semiramified division algebra of prime power degree p^n over E such that p\\neq char(\\bar E) and rk(\\Gamma_D/\\Gamma_E)\\geq 3 [resp., such that p\\neq char(\\bar E) and p^3 divides exp(\\Gamma_D/\\Gamma_E)], then D is non-cyclic [Proposition 3.1] [resp., D is not an elementary abelian crossed product [Proposition 3.2]]."}
{"category": "Math", "title": "FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension", "abstract": "We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed."}
{"category": "Math", "title": "A Note on Carleman's Inequality", "abstract": "We study a weighted version of Carleman's inequality via Carleman's original approach. As an application of our result, we prove a conjecture of Bennett."}
{"category": "Math", "title": "Koszul duality in deformation quantization, I", "abstract": "Let $\\alpha$ be a polynomial Poisson bivector on a finite-dimensional vector space $V$ over $\\mathbb{C}$. Then Kontsevich [K97] gives a formula for a quantization $f\\star g$ of the algebra $S(V)^*$. We give a construction of an algebra with the PBW property defined from $\\alpha$ by generators and relations. Namely, we define an algebra as the quotient of the free tensor algebra $T(V^*)$ by relations $x_i\\otimes x_j-x_j\\otimes x_i=R_{ij}(\\hbar)$ where $R_{ij}(\\hbar)\\in T(V^*)\\otimes\\hbar \\mathbb{C}[[\\hbar]]$, $R_{ij}=\\hbar \\Sym(\\alpha_{ij})+\\mathcal{O}(\\hbar^2)$, with one relation for each pair of $i,j=1...\\dim V$. We prove that the constructed algebra obeys the PBW property, and this is a generalization of the Poincar\\'{e}-Birkhoff-Witt theorem. In the case of a linear Poisson structure we get the PBW theorem itself, and for a quadratic Poisson structure we get an object closely related to a quantum $R$-matrix on $V$. At the same time we get a free resolution of the deformed algebra (for an arbitrary $\\alpha$). The construction of this PBW algebra is rather simple, as well as the proof of the PBW property. The major efforts should be undertaken to prove the conjecture that in this way we get an algebra isomorphic to the Kontsevich star-algebra."}
{"category": "Math", "title": "Galois theory of iterated endomorphisms", "abstract": "Given an abelian algebraic group $A$ over a global field $F$, $\\alpha \\in A(F)$, and a prime $\\ell$, the set of all preimages of $\\alpha$ under some iterate of $[\\ell]$ generates an extension of $F$ that contains all $\\ell$-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of $A$ we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes $\\p$ in the ring of integers of $F$ such that the order of $(\\alpha \\bmod{\\p})$ is prime to $\\ell$. We compute this density in the general case for several classes of $A$, including elliptic curves and one-dimensional tori. For example, if $F$ is a number field, $A/F$ is an elliptic curve with surjective 2-adic representation and $\\alpha \\in A(F)$ with $\\alpha \\not\\in 2A(F(A[4]))$, then the density of $\\mathfrak{p}$ with ($\\alpha \\bmod{\\p}$) having odd order is 11/21."}
{"category": "Math", "title": "Stochastic Parabolic Equations of Full Second Order", "abstract": "A procedure is described for defining a generalized solution for stochastic differential equations using the Cameron-Martin version of the Wiener Chaos expansion. Existence and uniqueness of this Wiener Chaos solution is established for parabolic stochastic PDEs such that both the drift and the diffusion operators are of the second order."}
{"category": "Math", "title": "From Random Processes to Generalized Fields: A Unified Approach to Stochastic Integration", "abstract": "The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of chaos space of the random field leads to two such extensions, corresponding to the \\Ito-Skorokhod and the Stratononovich integrals, and provides an efficient tool to study these integrals, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields."}
{"category": "Math", "title": "Inversely Unstable Solutions of Two-Dimensional Systems on Genus-p Surfaces and the Topology of Knotted Attractors", "abstract": "In this paper, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behaviour. This is achieved by studying the appearance of inversely unstable solutions within each invariant set."}
{"category": "Math", "title": "Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions", "abstract": "Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the non-linearities."}
{"category": "Math", "title": "Commuting linear operators and algebraic decompositions", "abstract": "For commuting linear operators $P_0,P_1,..., P_\\ell$ we describe a range of conditions which are weaker than invertibility. When any of these conditions hold we may study the composition $P=P_0P_1... P_\\ell$ in terms of the component operators or combinations thereof. In particular the general inhomogeneous problem $Pu=f$ reduces to a system of simpler problems. These problems capture the structure of the solution and range spaces and, if the operators involved are differential, then this gives an effective way of lowering the differential order of the problem to be studied. Suitable systems of operators may be treated analogously. For a class of decompositions the higher symmetries of a composition $P$ may be derived from generalised symmmetries of the component operators $P_i$ in the system."}
{"category": "Math", "title": "The calculus structure of the Hochschild homology/cohomology of preprojective algebras of Dynkin quivers", "abstract": "The Hochschild homology/cohomology an associative algebra, together with the Connes differential, the contraction map and the Lie derivative, forms the structure of calculus. In this paper we compute explicitely the calculus structure of preprojective algebras of Dynkin quivers over a field of characteristic zero. This also completes the work in math.AG/0502301, where the Batalin-Vilkovisky structure of the Hochschild cohomology of preprojective algebras of non-Dynkin quivers are computed and the calculus can be easily computed from that."}
{"category": "Math", "title": "Higher K-theory via universal invariants", "abstract": "Using the formalism of Grothendieck's derivators, we construct `the universal localizing invariant of dg categories'. By this, we mean a morphism U_l from the pointed derivator associated with the Morita homotopy theory of dg categories to a triangulated strong derivator M^loc such that U_l commutes with filtered homotopy colimits, preserves the point, sends each exact sequence of dg categories to a triangle and is universal for these properties. Similary, we construct the `the universal additive invariant of dg categories', i.e. the universal morphism of derivators U_a to a strong triangulated derivator M^add which satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen K-theory appears as a mapping space in the target of the universal additive invariant. This is the first conceptual characterization of Quillen-Waldhausen's K-theory since its definition in the early 70's. As an application we obtain for free the higher Chern characters from K-theory to cyclic homology."}
{"category": "Math", "title": "A Simple Method Which Generates Infinitely Many Congruence Identities", "abstract": "A simple method called symbolic representation for piecewise linear functions on the real line is introduced and used to compute the numbers of periodic points of all periods for some such functions. Since, for every positive integer m, the number of periodic points of minimal period m must be divisible by m, we obtain infinitely many congruence identities."}
{"category": "Math", "title": "On the Cordial Deficiency of Complete Multipartite Graphs", "abstract": "We calculate the cordial edge deficiencies of the complete multipartite graphs and find an upper bound for their cordial vertex deficiencies. We also give conditions under which the tensor product of two cordial graphs is cordial."}
{"category": "Math", "title": "A system for constructing relatively small polyhedra from Sonob\\'e modules", "abstract": "We develop a quite elementary graph theoretic system for designing small-size augmented origami polyhedra out of Sonob\\'e modules beginning with a (convex or not) deltahedron."}
{"category": "Math", "title": "Analysis of the expected number of bit comparisons required by Quickselect", "abstract": "When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard sorting and searching algorithms have been analyzed with respect to key comparisons but not with respect to bit comparisons. In this paper, we investigate the expected number of bit comparisons required by Quickselect (also known as Find). We develop exact and asymptotic formulae for the expected number of bit comparisons required to find the smallest or largest key by Quickselect and show that the expectation is asymptotically linear with respect to the number of keys. Similar results are obtained for the average case. For finding keys of arbitrary rank, we derive an exact formula for the expected number of bit comparisons that (using rational arithmetic) requires only finite summation (rather than such operations as numerical integration) and use it to compute the expectation for each target rank."}
{"category": "Math", "title": "Adelic amoebas disjoint from open halfspaces", "abstract": "We show that a conjecture of Einsiedler, Kapranov, and Lind on adelic amoebas of subvarieties of tori and their intersections with open halfspaces of complementary dimension is false for subvarieties of codimension greater than one that have degenerate projections to smaller dimensional tori. We prove a suitably modified version of the conjecture using algebraic methods, functoriality of tropicalization, and a theorem of Zhang on torsion points in subvarieties of tori."}
{"category": "Math", "title": "Minimal intersection of curves on surfaces", "abstract": "In the eighties Goldman discovered a Lie algebra structure on the vector space generated by the free homotopy classes of oriented curves on an oriented surface. The Lie bracket [a,b] is defined as the signed sum over the intersection points of a and b of the loop product of at the intersection points. If one of the classes has a simple representative we give a combinatorial group theory description of the terms of the Lie bracket and prove that this bracket has as many terms, counted with multiplicity, as the minimal number of intersection points of a and b. In other words the bracket with a simple element has no cancellation and determines minimal intersection numbers. We show that analogous results hold for the Lie bracket (also discovered by Goldman) of unoriented curves. We give three applications: a factorization of Thurston's map defining the boundary of Teichmuller space, various decompositions of the underlying vector space of conjugacy classes into ad invariant subspaces and a connection between bijections of the set of conjugacy classes of curves on a surface preserving the Goldman bracket and the mapping class group."}
{"category": "Math", "title": "Reducible Families of Curves with Ordinary Multiple Points on Surfaces in Projective Three-Space", "abstract": "In math.AG/0108089, math.AG/0212090 and math.AG/0308247 we gave numerical conditions which ensure that an equisingular family is irreducible respectively T-smooth. Combining results by Greuel, Lossen and Shustin and an idea from math.AG/9802009 we give in the present paper series of examples of families of irreducible curves on surfaces in projective three-space with only ordinary multiple points which are reducible and where at least one component does not have the expected dimension. The examples show that for families of curves with ordinary multiple points the conditions for T-smoothness in math.AG/0308247 have the right asymptotics."}
{"category": "Math", "title": "Some obstructed equisingular families of curves on surfaces in projective three-space", "abstract": "Very few examples of obstructed equsingular families of curves on surfaces other than the projective plane are known. Combining results from Westenberger and Hirano with an idea from math.AG/9802009 we give in the present paper series of examples of families of irreducible curves with simple singularities on surfaces in projective three-space which are not T--smooth, i.e. do not have the expected dimension, and we compare this with conditions (showing the same asymptotics) which ensure the existence of a T--smooth component."}
{"category": "Math", "title": "On the topological stable rank of non-selfadjoint operator algebras", "abstract": "We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu's non-commutative disc algebras and to free semigroup algebras as well."}
{"category": "Math", "title": "Transitive spaces of operators", "abstract": "We investigate algebraic and topological transitivity and, more generally, k-transitivity for linear spaces of operators. In finite dimensions, we determine minimal dimensions of k-transitive spaces for every k, and find relations between the degree of transitivity of a product or tensor product on the one hand and those of the factors on the other. We present counterexamples to some natural conjectures. Some infinite dimensional analogues are discussed. A simple proof is given of Arveson's result on the weak-operator density of transitive spaces that are masa bimodules."}
{"category": "Math", "title": "Transitive decompositions of graphs and their links with geometry and origami", "abstract": "A transitive decomposition of a graph is a partition of the edge or arc set giving a set of subgraphs which are preserved and permuted transitively by a group of automorphisms of the graph. In this paper we give some background to the study of transitive decompositions and highlight a connection with partial linear spaces. We then describe a simple method for constructing transitive decompositions using graph quotients, and we show how this may be used in an application to modular origami."}
{"category": "Math", "title": "Les classes d'Eisenstein des varietes de Hilbert-Blumenthal", "abstract": "This article deals with the Eisenstein classes of Hilbert-Blumenthal families of abelian varieties. We first give a coordinate expression of these one at the topological level, using currents defined by Levin. Then we study the degeneration of these Eisenstein classes at a cusp of the Baily-Borel compactification of the Hilbert-Blumenthal variety. We show, using the explicit description of the Eisenstein classes obtained previously, that these classes degenerate in special values of an $L$-function associated to the underlying totally real number field. We deduce then both a geometric proof the Klingen-Siegel Theorem and a non vanishing result for some of these Eisenstein classes ."}
{"category": "Math", "title": "An extension of a Bourgain--Lindenstrauss--Milman inequality", "abstract": "Let || . || be a norm on R^n. Averaging || (\\eps_1 x_1, ..., \\eps_n x_n) || over all the 2^n choices of \\eps = (\\eps_1, ..., \\eps_n) in {-1, +1}^n, we obtain an expression ||| . ||| which is an unconditional norm on R^n. Bourgain, Lindenstrauss and Milman showed that, for a certain (large) constant \\eta > 1, one may average over (\\eta n) (random) choices of \\eps and obtain a norm that is isomorphic to ||| . |||. We show that this is the case for any \\eta > 1."}
{"category": "Math", "title": "Congruence Identities Arising From Dynamical Systems", "abstract": "By counting the numbers of periodic points of all periods for some interval maps, we obtain infinitely many new congruence identities in number theory."}
{"category": "Math", "title": "The Intrinsic Fundamental Group of a Linear Category", "abstract": "We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering. The grading deduced from a Galois covering enables us to describe the canonical monomorphism from its automorphism group to the first Hochschild-Mitchell cohomology vector space."}
{"category": "Math", "title": "Robot motion planning, weights of cohomology classes, and cohomology operations", "abstract": "The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC(X) of the configuration space X of the system. Previously known lower bounds for TC(X) use the structure of the cohomology algebra of X. In this paper we show how cohomology operations can be used to sharpen these lower bounds for TC(X). As an application of this technique we calculate explicitly the topological complexity of various lens spaces. The results of the paper were inspired by the work of E. Fadell and S. Husseini on weights of cohomology classes appearing in the classical lower bounds for the Lusternik - Schnirelmann category. In the appendix to this paper we give a very short proof of a generalized version of their result."}
{"category": "Math", "title": "Alexander polynomials: Essential variables and multiplicities", "abstract": "We explore the codimension one strata in the degree-one cohomology jumping loci of a finitely generated group, through the prism of the multivariable Alexander polynomial. As an application, we give new criteria that must be satisfied by fundamental groups of smooth, quasi-projective complex varieties. These criteria establish precisely which fundamental groups of boundary manifolds of complex line arrangements are quasi-projective. We also give sharp upper bounds for the twisted Betti ranks of a group, in terms of multiplicities constructed from the Alexander polynomial. For Seifert links in homology 3-spheres, these bounds become equalities, and our formula shows explicitly how the Alexander polynomial determines all the characteristic varieties."}
{"category": "Math", "title": "Matching polytopes, toric geometry, and the non-negative part of the Grassmannian", "abstract": "In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian (Gr_{kn})_{\\geq 0}. This is a cell complex whose cells Delta_G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Delta_G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We then use the data of P(G) to define an associated toric variety X_G. We use our technology to prove that the cell decomposition of (Gr_{kn})_{\\geq 0} is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr_{kn})_{\\geq 0} is 1."}
{"category": "Math", "title": "Varieties with very little transcendental cohomology", "abstract": "Given a complex algebraic variety X, we define a natural number called the motivic dimension which measures the amount of transcendental (co)homology of X. It is zero precisely when all the (co)homolgy is spanned by algebraic cycles. Most of this paper is concerned with giving estimates on this number, along with examples where it is small. As an application, we check or recheck the Hodge conjectue in a number of examples: uniruled fourfolds, rationally connected fivefolds, fourfolds fibred by surfaces with p_g=0, Hilbert schemes of a small number points on surfaces with p_g=0, and generic hypersurfaces."}
{"category": "Math", "title": "Logarithmic comparison theorem versus Gauss-Manin system for isolated singularities", "abstract": "For quasihomogeneous isolated hypersurface singularities, the logarithmic comparison theorem has been characterized explicitly by Holland and Mond. In the non quasihomogeneous case, we give a necessary condition for the logarithmic comparison theorem in terms of the Gauss-Manin system of the singularity. It shows in particular that the logarithmic comparison theorem can hold for a non quasihomogeneous singularity only if 1 is an eigenvalue of the monodromy."}
{"category": "Math", "title": "The Harmonic Series and the nth Term Test for Divergence", "abstract": "The divergence of the harmonic series is proved by direct comparison with a series whose nth partial sum telescopes to the natural logarithm of n. The key idea is to apply the classical inequality x>=log(1+x) (valid for x>-1) with x=1/k and sum over k, 1<=k<=n-1."}
{"category": "Math", "title": "Big-Pieces-of-Lipschitz-Images Implies a Sufficient Carleson Estimate in a Metric Space", "abstract": "This note is intended to be a supplement to the bi-Lipschitz decomposition of Lipschitz maps shown in [Sch]. We show that in the case of 1-Ahlfors-regular sets, the condition of having `Big Pieces of bi-Lipschitz Images' (BPBI) is equivalent to a Carleson condition."}
{"category": "Math", "title": "Newton-Hodge Filtration for Self-Dual F-Crystals", "abstract": "In this paper we study F-crystals with self-dual structure over base schemes of characteristic p. We generalize Katz's Newton-Hodge Filtration Theorem to F-crystals with self-dual structure."}
{"category": "Math", "title": "The noncommutative residue and canonical trace in the light of Stokes' and continuity properties", "abstract": "We show that the noncommutative residue density, resp. the cut-off regularised integral are the only closed linear, resp. continuous closed linear forms on certain classes of symbols. This leads to alternative proofs of the uniqueness of the noncommutative residue, resp. the canonical trace as linear, resp. continuous linear forms on certain classes of classical pseudodifferential operators which vanish on brackets. The uniqueness of the canonical trace actually holds on classes of classical pseudodifferential with vanishing residue density which include non integer order operators in all dimensions and odd-class (resp. even-class) operators in odd (resp. even) dimensions. The description of the canonical trace for non integer order operators as an integrated global density on the manifold is extended to odd-class (resp. even-class) operators in odd (resp. even) dimensions on the grounds of defect formulae for regularised traces of classical pseudodifferential operators."}
{"category": "Math", "title": "Chern-Weil calculus extended to a class of infinite dimensional manifolds", "abstract": "We discuss possible extensions of the classical Chern-Weil formalism to an infinite dimensional setup. This is based on joint work with Steven Rosenberg, joint work with Simon Scott and joint work with Jouko Mickelsson."}
{"category": "Math", "title": "Gray Curvature Identities for Almost Contact Metric Manifolds", "abstract": "The aim of this research is the study of Gray curvature identities, introduced by Alfred Gray in \\cite{kn:Gra76} for the class of almost hermitian manifolds. As known till now, there is no equivalent for the class of almost contact manifolds. For this purpose we use the Boohby-Wang fibration and the warped manifolds construction in order to establish which identities could be satisfied by an almost contact manifold."}
{"category": "Math", "title": "A Groshev Theorem for Small Linear Forms", "abstract": "In this paper the absolute value or distance from the origin analogue of the classical Khintchine-Groshev theorem is established for a single linear form with a `slowly decreasing' error function."}
{"category": "Math", "title": "Prescribing the behaviour of geodesics in negative curvature", "abstract": "Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon described by Hall in special arithmetic situations and by Schmidt-Sheingorn for hyperbolic surfaces is in fact only a negative curvature property."}
{"category": "Math", "title": "Planar maps whose second iterate has a unique fixed point", "abstract": "Let a>0, F: R^2 -> R^2 be a differentiable (not necessarily C^1) map and Spec(F) be the set of (complex) eigenvalues of the derivative F'(p) when p varies in R^2. (a) If Spec(F) is disjoint of the interval [1,1+a[, then Fix(F) has at most one element, where Fix(F) denotes the set of fixed points of F. (b) If Spec(F) is disjoint of the real line R, then Fix(F^2) has at most one element. (c) If F is a C^1 map and, for all p belonging to R^2, the derivative F'(p) is neither a homothety nor has simple real eigenvalues, then Fix(F^2) has at most one element, provided that Spec(F) is disjoint of either (c1) the union of the number 0 with the intervals ]-\\infty, -1] and [1,\\infty[, or (c2) the interval [-1-a, 1+a]. Conditions under which Fix(F^n), with n>1, is at most unitary are considered."}
{"category": "Math", "title": "Betti numbers for fat point ideals in the plane: a geometric approach", "abstract": "We consider the open problem of determining the graded Betti numbers for fat point subschemes supported at general points of the projective plane. We relate this problem to the open geometric problem of determining the splitting type of the pullback of the cotangent bundle on the plane to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of the plane. We also prove our Betti number conjecture in a broad range of cases. An appendix discusses many more cases in which our conjecture has been verified computationally and provides a new and more efficient computational approach for computing graded Betti numbers in certain degrees. It also demonstrates how to derive explicit conjectural values for the Betti numbers and how to compute splitting types."}
{"category": "Math", "title": "The Kashiwara-Vergne conjecture", "abstract": "This is a Bourbaki's seminar text. We introduce the combinatorial Kashiwara-Vergne conjecture on the Baker-Campbell-Hausdorff serie. After recalling previous results and consequences, we explain the Alekseev-Meinrenken's proof [math.QA/0506499]"}
{"category": "Math", "title": "On the genealogy on conditioned stable L\\'evy forest", "abstract": "We give a realization of the stable L\\'evy forest of a given size conditioned by its mass from the path of the unconditioned forest. Then, we prove an invariance principle for this conditioned forest by considering $k$ independent Galton-Watson trees whose offspring distribution is in the domain of attraction of any stable law conditioned on their total progeny to be equal to $n$. We prove that when $n$ and $k$ tend towards $+\\infty$, under suitable rescaling, the associated coding random walk, the contour and height processes converge in law on the Skorokhod space respectively towards the \"first passage bridge\" of a stable L\\'evy process with no negative jumps and its height process."}
{"category": "Math", "title": "A refinement of multi-dimensional persistence", "abstract": "We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these modules, and present several examples. We also provide a geometric interpretation for the higher tor-modules of homology modules of multi-filtered simplicial complexes."}
{"category": "Math", "title": "Minimal data rate stabilization of nonlinear systems over networks with large delays", "abstract": "Control systems over networks with a finite data rate can be conveniently modeled as hybrid (impulsive) systems. For the class of nonlinear systems in feedfoward form, we design a hybrid controller which guarantees stability, in spite of the measurement noise due to the quantization, and of an arbitrarily large delay which affects the communication channel. The rate at which feedback packets are transmitted from the sensors to the actuators is shown to be arbitrarily close to the infimal one."}
{"category": "Math", "title": "A Dynamical Systems Approach to the Kadison-Singer Problem", "abstract": "In these notes we develop a link between the Kadison-Singer problem and questions about certain dynamical systems. We conjecture that whether or not a given state has a unique extension is related to certain dynamical properties of the state. We prove that if any state corresponding to a minimal idempotent point extends uniquely to the von Neumann algebra of the group, then every state extends uniquely to the von Neumann algebra of the group. We prove that if any state arising in the Kadsion-Singer problem has a unique extension, then the injective envelope of a C*-crossed product algebra associated with the state necessarily contains the full von Neumann algebra of the group. We prove that this latter property holds for states arising from rare ultrafilters and $\\delta$-stable ultrafilters, independent, of the group action and also for states corresponding to non-recurrent points in the corona of the group."}
{"category": "Math", "title": "Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion", "abstract": "We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/2$. For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved by any approximation method using an equidistant discretization of the driving fractional Brownian motion. We find that there are mainly two cases: either the solution can be approximated perfectly or the best possible rate of convergence is $n^{-H-1/2},$ where $n$ denotes the number of evaluations of the fractional Brownian motion. In addition, we present an implementable approximation scheme that obtains the optimal rate of convergence in the latter case."}
{"category": "Math", "title": "Some applications of the Mellin transform to branching processes", "abstract": "We introduce a Mellin transform of functions which live on all of $\\bR$ and discuss its applications to the limiting theory of Bellman-Harris processes, and specifically Luria-Delbr\\\"uck processes. More precisely, we calculate the life-time distribution of particles in a Bellman-Harris process from their first-generation offspring and limiting distributions, and prove a formula for the Laplace transform of the distribution of types in a Luria-Delbr\\\"uck process in the Mittag-Leffler limit. As a by-product, we show how to easily derive the (classical) Mellin transforms of certain stable probability distributions from their Fourier transform."}
{"category": "Math", "title": "The twisted Mellin transform", "abstract": "The \"twisted Mellin transform\" is a slightly modified version of the usual classical Mellin transform on $L^2(\\mathbb R)$. In this short note we investigate some of its basic properties. From the point of views of combinatorics one of its most important interesting properties is that it intertwines the differential operator, $df/dx$, with its finite difference analogue, $\\nabla f= f(x)-f(x-1)$. From the point of view of analysis one of its most important properties is that it describes the asymptotics of one dimensional quantum states in Bargmann quantization."}
{"category": "Math", "title": "Some new equivalences of Anderson's paving conjectures", "abstract": "Anderson's paving conjectures are known to be equivalent to the Kadison-Singer problem. We prove some new equivalences of Anderson's conjectures that require the paving of smaller sets of matrices. We prove that if the strictly upper triangular operatorss are pavable, then every 0 diagonal operator is pavable. This result follows from a new paving condition for positive operators. In addition, we prove that if the upper triangular Toeplitz operators are paveable, then all Toeplitz operators are paveable."}
{"category": "Math", "title": "Compactness of the Complex Green Operator", "abstract": "Let $\\Omega\\subset\\C^n$ be a bounded smooth pseudoconvex domain. We show that compactness of the complex Green operator $G_{q}$ on $(0,q)$-forms on $b\\Omega$ implies compactness of the $\\bar{\\partial}$-Neumann operator $N_{q}$ on $\\Omega$. We prove that if $1 \\leq q \\leq n-2$ and $b\\Omega$ satisfies $(P_q)$ and $(P_{n-q-1})$, then $G_{q}$ is a compact operator (and so is $G_{n-1-q}$). Our method relies on a jump type formula to represent forms on the boundary, and we prove an auxiliary compactness result for an `annulus' between two pseudoconvex domains. Our results, combined with the known characterization of compactness in the $\\bar{\\partial}$-Neumann problem on locally convexifiable domains, yield the corresponding characterization of compactness of the complex Green operator(s) on these domains."}
{"category": "Math", "title": "A quantitative version of the Besicovitch projection theorem via multiscale analysis", "abstract": "By using a multiscale analysis, we establish quantitative versions of the Besicovitch projection theorem (almost every projection of a purely unrectifiable set in the plane of finite length has measure zero) and a standard companion result, namely that any planar set with at least two projections of measure zero is purely unrectifiable. We illustrate these results by providing an explicit (but weak) upper bound on the average projection of the $n^{th}$ generation of a product Cantor set."}
{"category": "Math", "title": "A note for Gromov's distance functions on the space of mm-spaces", "abstract": "This is just a note for \\cite[Chapter$3{1/2}_+$]{gromov}. Maybe this note is obvious for a reader who knows metric geometry. I wish that someone study further in this direction."}
{"category": "Math", "title": "Harder-Narasimhan categories", "abstract": "We propose a generalization of Quillen's exact category -- arithmetic exact category and we discuss conditions on such categories under which one can establish the notion of Harder-Narasimhan filtrations and Harder-Narsimhan polygons. Furthermore, we show the functoriality of Harder-Narasimhan filtrations (indexed by $\\mathbb R$), which can not be stated in the classical setting of Harder and Narasimhan's formalism."}
{"category": "Math", "title": "Convergence of Harder-Narasimhan polygons", "abstract": "We establish in this article convergence results of normalized Harder-Narasimhan polygons both in geometric and in arithmetic frameworks by introducing the Harder-Narasimhan filtration indexed by $\\mathbb R$ and the associated Borel probability measure."}
{"category": "Math", "title": "On the connected component of compact composition operators on the Hardy space", "abstract": "We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space $H^2$ on the unit disc. This answers a question posed by Shapiro and Sundberg in 1990. We also establish an improved version of a theorem of MacCluer, giving a lower bound for the essential norm of a difference of composition operators in terms of the angular derivatives of their symbols. As a main tool we use Aleksandrov-Clark measures."}
{"category": "Math", "title": "On singular cubic surfaces", "abstract": "We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kahler-Einstein metric on two singular cubic surfaces."}
{"category": "Math", "title": "Le th\\'eor\\`eme de Riemann-Roch et ses applications", "abstract": "The Riemann-Roch theorem is of utmost importance in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of this article is to present a simple direct proof of this theorem and explore some of its numerous consequences. We also give an analytic proof of the Riemann-Hurwitz formula. As an application, we compute the genus of some interesting algebraic curves."}
{"category": "Math", "title": "Bounds on the concentration function in terms of Diophantine approximation", "abstract": "We demonstrate a simple analytic argument that may be used to bound the Levy concentration function of a sum of independent random variables. The main application is a version of a recent inequality due to Rudelson and Vershynin, and its multidimensional generalisation."}
{"category": "Math", "title": "Singular Loci of Hibi toric varieties", "abstract": "We first construct explicit bases for the cotangent spaces at singular points on Hibi toric varieties, i.e., toric varieties associated to distributive lattices. We then determine the singular loci of these toric varieties."}
{"category": "Math", "title": "Universal flattening of Frobenius", "abstract": "For a variety $X$ of positive characteristic and a non-negative integer $e$, we define its $e$-th F-blowup to be the universal flattening of the $e$-iterated Frobenius of $X$. Thus we have the sequence (a set labeled by non-negative integers) of blowups of $X$. Under some condition, the sequence stabilizes and leads to a nice (for instance, minimal or crepant) resolution. For tame quotient singularities, the sequence leads to the $G$-Hilbert scheme."}
{"category": "Math", "title": "Holonomic rank of A-hypergeometric differential-difference equations", "abstract": "We introduce A-hypergeometric differential-difference equation and prove that its holonomic rank is equal to the normalized volume of A with giving a set of convergent series solutions."}
{"category": "Math", "title": "The p-modular Descent Algebra of the Symmetric Group", "abstract": "The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the radical, and its nilpotency index. It also allows the irreducible representations of the descent algebra to be described."}
{"category": "Math", "title": "On the Descent Algebra of Type $D$", "abstract": "Here we give a combinatorial interpretation of Solomon's rule for multiplication in the descent algebra of Weyl groups of type $D$, $\\Sigma D_n$. From here we show that $\\Sigma D_n$ is a homomorphic image of the descent algebra of the hyperoctahedral group, $\\Sigma B_{n-2}$."}
{"category": "Math", "title": "Contraction groups in complete Kac-Moody groups", "abstract": "Let $G$ be an abstract Kac-Moody group over a finite field and $\\bar{G}$ be the closure of the image of $G$ in the automorphism group of its positive building. We show that if the Dynkin diagram associated to $G$ is irreducible and neither of spherical nor of affine type, then the contraction groups of elements in $\\bar{G}$ which are not topologically periodic are not closed. (In those groups there always exist elements which are not topologically periodic.)"}
{"category": "Math", "title": "A Proof of Solomon's Rule", "abstract": "We put forward a proof of Solomon's rule, in terms of matrices, for multiplication in the descent algebra of the symmetric group. Our proof exploits the graphs that we can obtain from all the subsets of the set of transpositions, $\\{(i,i+1)\\}_{i=1}^{n-1}$."}
{"category": "Math", "title": "Universally defined representations of Lie conformal superalgebras", "abstract": "We distinguish a class of irreducible finite representations of conformal Lie (super)algebras. These representations (called universally defined) are the simplest ones from the computational point of view: a universally defined representation of a conformal Lie (super)algebra $L$ is completely determined by commutation relations of $L$ and by the requirement of associative locality of generators. We describe such representations for conformal superalgebras $W_n$, $n\\ge 0$, with respect to a natural set of generators. We also consider the problem for superalgebras $K_n$. In particular, we find a universally defined representation for the Neveu--Schwartz conformal superalgebra $K_1$ and show that the analogues of this representation for $n\\ge 2$ are not universally defined."}
{"category": "Math", "title": "Small Deviation Probability via Chaining", "abstract": "We obtain several extensions of Talagrand's lower bound for the small deviation probability using metric entropy. For Gaussian processes, our investigations are focused on processes with sub-polynomial and, respectively, exponential behaviour of covering numbers. The corresponding results are also proved for non-Gaussian symmetric stable processes, both for the cases of critically small and critically large entropy. The results extensively use the classical chaining technique; at the same time they are meant to explore the limits of this method."}
{"category": "Math", "title": "Associative algebras related to conformal algebras", "abstract": "In this note, we introduce a class of algebras that are in some sense related to conformal algebras. This class (called TC-algebras) includes Weyl algebras and some of their (associative and Lie) subalgebras. By a conformal algebra we generally mean what is known as $H$-pseudo-algebra over the polynomial Hopf algebra $H=\\Bbbk[T_1,..., T_n]$. Some recent results in structure theory of conformal algebras are applied to get a description of TC-algebras."}
{"category": "Math", "title": "On the behavior of Castelnuovo-Mumford regularity with respect to some functors", "abstract": "We investigate the behavior of Castelnuovo-Mumford regularity with respect to some classical functors : Tor, the Frobenius functor in positive characteristic, taking a power or a product (on ideals). These generalizes and refines previous results on these issues by several authors. As an application we provide results on the regularity of an intersection of subschemes of a projective scheme, under appropriate geometric hypotheses. Results on the rigidity of multiple Tor modules and on the characterization of their vanishing are given, motivated by geometric applications."}
{"category": "Math", "title": "Sums of $L$-functions over the rational function field", "abstract": "Friedberg, Hoffstein and Lieman have constructed two related multiple Dirichlet series from quadratic and higher-order $L$-functions and Gauss sums. We compute these multiple Dirichlet series explicitly in the case of the rational function field. This is done by utilizing the functional equation of the $L$-functions and the functional equation relating the two multiple Dirichlet series. We also point out a very simple correspondence between these series and their $p$-parts."}
{"category": "Math", "title": "Mapping Class Groups and Interpolating Complexes: Rank", "abstract": "A family of interpolating graphs $\\calC (S, \\xi)$ of complexity $\\xi$ is constructed for a surface $S$ and $-2 \\leq \\xi \\leq \\xi (S)$. For $\\xi = -2, -1, \\xi (S) -1$ these specialise to graphs quasi-isometric to the marking graph, the pants graph and the curve graph respectively. We generalise Theorems of Brock-Farb and Behrstock-Minsky to show that the rank of $\\calC (S, \\xi)$ is $r_\\xi$, the largest number of disjoint copies of subsurfaces of complexity greater than $\\xi $ that may be embedded in $S$. The interpolating graphs $\\calC (S, \\xi)$ interpolate between the pants graph and the curve graph."}
{"category": "Math", "title": "Obtaining New Dividing Formulas n|Q(n) From the Known Ones", "abstract": "In this note, we present a few methods (Theorems 1, 2, and 3) from discrete dynamical systems theory of obtaining new functions Q(n) from the known ones so that the dividing formulas n|Q(n) hold."}
{"category": "Math", "title": "On certain bounds for first-crossing-time probabilities of a jump-diffusion process", "abstract": "We consider the first-crossing-time problem through a constant boundary for a Wiener process perturbed by random jumps driven by a counting process. On the base of a sample-path analysis of the jump-diffusion process we obtain explicit lower bounds for the first-crossing-time density and for the first-crossing-time distribution function. In the case of the distribution function, the bound is improved by use of processes comparison based on the usual stochastic order. The special case of constant jumps driven by a Poisson process is thoroughly discussed."}
{"category": "Math", "title": "Scaled entropy of filtrations of $\\sigma$-fields", "abstract": "We study the notion of the scaled entropy of a filtration of $\\sigma$-fields (= decreasing sequence of $\\sigma$-fields) introduced by the first author ({V4}). We suggest a method for computing this entropy for the sequence of $\\sigma$-fields of pasts of a Markov process determined by a random walk over the trajectories of a Bernoulli action of a commutative or nilpotent countable group (Theorems~5,~6). Since the scaled entropy is a metric invariant of the filtration, it follows that the sequences of $\\sigma$-fields of pasts of random walks over the trajectories of Bernoulli actions of lattices (groups ${\\Bbb Z}^d$) are metrically nonisomorphic for different dimensions $d$, and for the same $d$ but different values of the entropy of the Bernoulli scheme. We give a brief survey of the metric theory of filtrations, in particular, formulate the standardness criterion and describe its connections with the scaled entropy and the notion of a tower of measures."}
{"category": "Math", "title": "Embedding Theorems and Boundary-value Problems for cusp domains", "abstract": "We study the Robin boundary-value problem for bounded domains with isolated singularities. Because for such domains trace spaces of space $H^1(D)$ on its boundaries are weighted Sobolev spaces $L^{2, \\xi}(\\partial D)$ existence and uniqueness of corresponding Robin boundary-value problems depends on properties of embedding operators $I_1: H^{1}(D)\\to L^{2}(D)$ and $I_{2}:H^{1}(D)\\to L^{2,\\xi}(\\partial D)$ i.e. on type of singularities. We obtain an exact description of the weights $\\xi$ for bounded domains with 'outside peaks' on its boundaries. This result allows us to formulate correctly the corresponding Robin boundary-value problems for elliptic operators."}
{"category": "Math", "title": "Near-homeomorphisms of Nobeling manifolds", "abstract": "We characterize maps between $n$-dimensional N\\\"obeling manifolds that can be approximated by homeomorphisms."}
{"category": "Math", "title": "The Ricci iteration and its applications", "abstract": "In this Note we introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics as discretizations of certain geometric flows. We pose a conjecture on their convergence towards canonical Kahler metrics and study the case where the first Chern class is negative, zero or positive. This construction has several applications in Kahler geometry, among them an answer to a question of Nadel and a construction of multiplier ideal sheaves."}
{"category": "Math", "title": "Chains in the noncrossing partition lattice", "abstract": "We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification of finite Coxeter groups) using basic facts about noncrossing partitions. We solve these recursions for each finite Coxeter group in the classification. Among other results, we obtain a simpler proof of a known uniform formula for the number of maximal chains of noncrossing partitions and a new uniform formula for the number of edges in the noncrossing partition lattice. All of our results extend to the m-divisible noncrossing partition lattice."}
{"category": "Math", "title": "Extremal subsets of {1,...,n} avoiding solutions to linear equations in three variables", "abstract": "We refine previous results to provide examples, and in some cases precise classifications, of extremal subsets of {1,...,n} containing no solutions to a wide class of non-invariant, homogeneous linear equations in three variables, i.e.: equations of the form ax+by=cz with a+b \\neq c."}
{"category": "Math", "title": "Filling inequalities do not depend on topology", "abstract": "Gromov's universal filling inequalities relate the filling radius and the filling volume of a Riemannian manifold to its volume. The main result of the present article is that in dimensions at least three the optimal constants in the filling inequalities depend only on dimension and orientability, not on the manifold itself. This contrasts with the analogous situation for the optimal systolic inequality, which does depend on the manifold."}
{"category": "Math", "title": "A Characterization of the Morse Minimal Set up to Topological Conjugacy", "abstract": "We establish necessary and sufficient conditions for a dynamical system to be topologically conjugate to the Morse minimal set, the shift orbit closure of the Morse sequence, and conditions for topological conjugacy to the closely related Teoplitz minimal set."}
{"category": "Math", "title": "On the constant in the Mertens product for arithmetic progressions. I. Identities", "abstract": "The aim of the paper is the proof of new identities for the constant in the Mertens product for arithmetic progressions. We deal with the problem of the numerical computation of these constants in another paper."}
{"category": "Math", "title": "On the tautological ring of a Jacobian modulo rational equivalence", "abstract": "We consider the Chow ring with rational coefficients of the Jacobian of a curve. Assume D is a divisor in a base point free g^r_d of the curve such that the canonical divisor K is a multiple of the divisor D. We find relations between tautological cycles. We give applications for curves having a degree d covering of P^1 whose ramification points are all of order d, and then for hyperelliptic curves."}
{"category": "Math", "title": "The angel wins", "abstract": "The angel-devil game is played on an infinite two-dimensional ``chessboard''. The squares of the board are all white at the beginning. The players called angel and devil take turns in their steps. When it is the devil's turn, he can turn a square black. The angel always stays on a white square, and when it is her turn she can fly at a distance of at most J steps (each of which can be horizontal, vertical or diagonal) to a new white square. Here J is a constant. The devil wins if the angel does not find any more white squares to land on. The result of the paper is that if J is sufficiently large then the angel has a strategy such that the devil will never capture her. This deceptively easy-sounding result has been a conjecture, surprisingly, for about thirty years. Several other independent solutions have appeared simultaneously, some of them prove that J=2 is sufficient (see the Wikipedia on the angel problem). Still, it is hoped that the hierarchical solution presented here may prove useful for some generalizations."}
{"category": "Math", "title": "Approximate calculation of operator semigroups by perturbation of generators", "abstract": "Let $\\Omega$ be an operator semigroup with generator $A$ in a sequentially complete locally convex topological vector space $E$. For a semigroup with generator $A+D$, where $D$ is a bounded linear operator on $E$, two integral equations are derived. A theorem on continuous dependence of a semigroup on its generator is proved. An application to random walk on $\\mathbb{Z}$ is given."}
{"category": "Math", "title": "Higher Tits indices of algebraic groups", "abstract": "Let G be a semisimple algebraic group over a field k. We introduce the higher Tits indices of G as the set of all Tits indices of G over all field extensions K/k. In the context of quadratic forms this notion coincides with the notion of the higher Witt indices introduced by M. Knebusch and classified by N. Karpenko and A. Vishik. We classify the higher Tits indices for exceptional algebraic groups. Our main tools involve the Chow groups and the Chow motives of projective homogeneous varieties, Steenrod operations, and the notion of the J-invariant of algebraic groups."}
{"category": "Math", "title": "Adelic Maass spaces on U(2,2)", "abstract": "Generalizing the results of Kojima, Gritsenko and Krieg, we define an adelic version of the Maass space for hermitian modular forms of weight k regarded as functions on adelic points of the quasi-split unitary group U(2,2) associated with an imaginary quadratic extension F/Q of discriminant D_F. When the class number h_F of F is odd, we show that the Maass space is invariant under the action of the local Hecke algebras of U(2,2)(Q_p) for all p not dividing D_F. As a consequence we obtain a Hecke-equivariant injective map from the Maass space to the h_F-fold direct product of the space of elliptic modular forms of weight k-1 and level D_F."}
{"category": "Math", "title": "A Simple Counterexample to Havil's \"Reformulation\" of the Riemann Hypothesis", "abstract": "This is an elementary note. It corrects a mistake in the reformulation of the Riemann Hypothesis in J. Havil's book Gamma: Exploring Euler's Constant."}
{"category": "Math", "title": "Counting closed geodesics on rank one manifolds", "abstract": "We establish a precise asymptotic formula for the number of homotopy classes of periodic orbits for the geodesic flow on rank one manifolds of nonpositive curvature. This extends a celebrated result of G. A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper. We also establish some useful properties of the measure of maximal entropy."}
{"category": "Math", "title": "Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields", "abstract": "We study the weak convergence (in the high-frequency limit) of the frequency components associated with Gaussian-subordinated, spherical and isotropic random fields. In particular, we provide conditions for asymptotic Gaussianity and we establish a new connection with random walks on the the dual of SO(3), which mirrors analogous results previously established for fields defined on Abelian groups (see Marinucci and Peccati (2007)). Our work is motivated by applications to cosmological data analysis, and specifically by the probabilistic modelling and the statistical analysis of the Cosmic Microwave Background radiation, which is currently at the frontier of physical research. To obtain our main results, we prove several fine estimates involving convolutions of the so-called Clebsch-Gordan coefficients (which are elements of unitary matrices connecting reducible representations of SO(3)); this allows to intepret most of our asymptotic conditions in terms of coupling of angular momenta in a quantum mechanical system. Part of the proofs are based on recently established criteria for the weak convergence of multiple Wiener-It\\^o integrals."}
{"category": "Math", "title": "The K\\\"ahler-Ricci flow with positive bisectional curvature", "abstract": "We show that the K\\\"ahler-Ricci flow on a manifold with positive first Chern class converges to a K\\\"ahler-Einstein metric assuming positive bisectional curvature and certain stability conditions."}
{"category": "Math", "title": "A Polynomial Invariant Of Twisted Graph Diagrams", "abstract": "Twisted graph diagrams are virtual graph diagrams with bars on edges. A bijection between abstract graph diagrams and twisted graph diagrams is constructed. Then a polynomial invariant of Yamada-type is developed which provides a lower bound for the virtual crossing number of virtual graph diagrams."}
{"category": "Math", "title": "Smarandache Type Function Obtained by Duality", "abstract": "In this paper we extend the Smarandache function from the set $N*$ of positive integers to the set $Q$ pf rational numbers. Using the inverse formula, this function is also regarded as a generating function. We put in evidence a procedure to construct a (numerical) function starting from a given function in two particular cases. Also, connections between this function and Euler totient function as well as with Riemann zeta function are established."}
{"category": "Math", "title": "Comparing composites of left and right derived functors", "abstract": "We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category whose vertical and horizontal arrows are left and right Quillen functors, respectively, and that passage to derived functors is functorial at the level of this double category. The theory of conjunctions and mates in double categories, which generalizes the theory of adjunctions and mates in 2-categories, then gives us canonical ways to compare composites of left and right derived functors. We give a number of sample applications, most of which are improvements of existing proofs in the literature."}
{"category": "Math", "title": "Parametrized spaces model locally constant homotopy sheaves", "abstract": "We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of the loop space of X. This gives a homotopy-theoretic version of the correspondence between covering spaces over X and sets with an action of the fundamental group of X. We then use these two equivalences to study base change functors for parametrized spaces."}
{"category": "Math", "title": "On the Nonexistence of Quadratic Lyapunov Functions for Consensus Algorithms", "abstract": "We provide an example proving that there exists no quadratic Lyapunov function for a certain class of linear agreement/consensus algorithms, a fact that had been numerically verified in [5]. We also briefly discuss sufficient conditions for the existence of such a Lyapunov function."}
{"category": "Math", "title": "A general method for investigating the roots of all equations by approximation", "abstract": "Translation of the Latin original, \"Methodus generalis investigandi radices omnium aequationum per approximationem\" (1776). E643 in the Enestrom index. Euler gives a series to find powers of roots of polynomials."}
{"category": "Math", "title": "Dimensional reduction as a tool for mesh refinement and tracking singularities of PDEs", "abstract": "We present a collection of algorithms which utilize dimensional reduction to perform mesh refinement and study possibly singular solutions of time-dependent partial differential equations. The algorithms are inspired by constructions used in statistical mechanics to evaluate the properties of a system near a critical point. The first algorithm allows the accurate determination of the time of occurrence of a possible singularity. The second algorithm is an adaptive mesh refinement scheme which can be used to approach efficiently the possible singularity. Finally, the third algorithm uses the second algorithm until the available resolution is exhausted (as we approach the possible singularity) and then switches to a dimensionally reduced model which, when accurate, can follow faithfully the solution beyond the time of occurrence of the purported singularity. An accurate dimensionally reduced model should dissipate energy at the right rate. We construct two variants of each algorithm. The first variant assumes that we have actual knowledge of the reduced model. The second variant assumes that we know the form of the reduced model, i.e. the terms appearing in the reduced model, but not necessarily their coefficients. In this case, we also provide a way of determining the coefficients. We present numerical results for the Burgers equation with zero and nonzero viscosity to illustrate the use of the algorithms."}
{"category": "Math", "title": "Hecke operators in equivariant elliptic cohomology and generalized moonshine", "abstract": "This paper studies connections between generalized moonshine and elliptic cohomology with a focus on the action of the Hecke correspondence and its implications for the notion of replicability."}
{"category": "Math", "title": "Disconnected synchronized regions of complex dynamical networks", "abstract": "This paper addresses the synchronized region problem, which is reduced to a matrix stability problem, for complex dynamical networks. For any natural number $n$, the existence of a network which has $n$ disconnected synchronized regions is theoretically demonstrated. This shows the complexity in network synchronization. Convexity characteristic of stability for matrix pencils is further discussed. Smooth and generalized smooth Chua's circuit networks are finally discussed as examples for illustration."}
{"category": "Math", "title": "Analysis and control of network synchronizability", "abstract": "In this paper, the investigation is first motivated by showing two examples of simple regular symmetrical graphs, which have the same structural parameters, such as average distance, degree distribution and node betweenness centrality, but have very different synchronizabilities. This demonstrates the complexity of the network synchronizability problem. For a given network with identical node dynamics, it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix. Several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability. In searching for conditions under which the network synchronizability may be increased by adding edges, it is found that for networks with disconnected complementary graphs, adding edges never decreases their synchronizability. This implies that better understanding and careful manipulation of the complementary graphs are important and useful for enhancing the network synchronizability. Moreover, it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region. Therefore, to effectively enhance the network synchronizability, a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region, for the case where the synchronous state is an equilibrium point of the network."}
{"category": "Math", "title": "Complexity in linearly coupled dynamical networks: Some unusual phenomena in energy accumulation", "abstract": "This paper addresses the energy accumulation problem, in terms of the $H_2$ norm, of linearly coupled dynamical networks. An interesting outer-coupling relationship is constructed, under which the $H_2$ norm of the newly constructed network with column-input and row-output shaped matrices increases exponentially fast with the node number $N$: it increases generally much faster than $2^N$ when $N$ is large while the $H_2$ norm of each node is 1. However, the $H_2$ norm of the network with a diffusive coupling is equal to $\\gamma_2 N$, i.e., increasing linearly, when the network is stable, where $\\gamma_2$ is the $H_2$ norm of a single node. And the $H_2$ norm of the network with antisymmetrical coupling also increases, but rather slowly, with the node number $N$. Other networks with block-diagonal-input and block-diagonal-output matrices behave similarly. It demonstrates that the changes of $H_2$ norms in different networks are very complicated, despite the fact that the networks are linear. Finally, the influence of the $H_2$ norm of the locally linearized network on the output of a network with Lur'e nodes is discussed."}
{"category": "Math", "title": "The $p$-modular descent algebras", "abstract": "The concept of descent algebras over a field of characteristic zero is extended to define descent algebras over a field of prime characteristic. Some basic algebraic structure of the latter, including its radical and irreducible modules, is then determined. The decomposition matrix of the descent algebras of Coxeter group types $A$, $B$, and $D$ are calculated, and used to derive a description of the decomposition matrix of an arbitrary descent algebra. The Cartan matrix of a variety of descent algebras over a finite field is then obtained."}
{"category": "Math", "title": "Properties of the descent algebras of type $D$", "abstract": "We establish simple combinatorial descriptions of the radical and irreducible representations specifically for the descent algebra of a Coxeter group of type $D$ over any field."}
{"category": "Math", "title": "Statistical testing procedure for the interaction effects of several controllable factors in two-valued input-output systems", "abstract": "Suppose several two-valued input-output systems are designed by setting the levels of several controllable factors. For this situation, Taguchi method has proposed to assign the controllable factors to the orthogonal array and use ANOVA model for the standardized SN ratio, which is a natural measure for evaluating the performance of each input-output system. Though this procedure is simple and useful in application indeed, the result can be unreliable when the estimated standard errors of the standardized SN ratios are unbalanced. In this paper, we treat the data arising from the full factorial or fractional factorial designs of several controllable factors as the frequencies of high-dimensional contingency tables, and propose a general testing procedure for the main effects or the interaction effects of the controllable factors."}
{"category": "Math", "title": "Enumerative properties of Ferrers graphs", "abstract": "We define a class of bipartite graphs that correspond naturally with Ferrers diagrams. We give expressions for the number of spanning trees, the number of Hamiltonian paths when applicable, the chromatic polynomial, and the chromatic symmetric function. We show that the linear coefficient of the chromatic polynomial is given by the excedance set statistic."}
{"category": "Math", "title": "Tropical hyperplane arrangements and oriented matroids", "abstract": "We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection."}
{"category": "Math", "title": "Mackey functors on compact closed categories", "abstract": "We develop and extend the theory of Mackey functors as an application of enriched category theory. We define Mackey functors on a lextensive category $\\E$ and investigate the properties of the category of Mackey functors on $\\E$. We show that it is a monoidal category and the monoids are Green functors. Mackey functors are seen as providing a setting in which mere numerical equations occurring in the theory of groups can be given a structural foundation. We obtain an explicit description of the objects of the Cauchy completion of a monoidal functor and apply this to examine Morita equivalence of Green functors."}
{"category": "Math", "title": "On the stabilization of the elasticity system by the boundary", "abstract": "We obtain free of resonances regions for the elasticity system in the exterior of a strictly convex body with dissipative boundary conditions under some natural assumptions on the behaviour of the geodesics on the boundary."}
{"category": "Math", "title": "Thue equations and torsion groups of elliptic curves", "abstract": "A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations."}
{"category": "Math", "title": "On the shadow boundary of a centrally symmetric convex body", "abstract": "We discuss the concept of the shadow boundary of a centrally symmetric convex ball $K$ (actually being the unit ball of a Minkowski normed space) with respect to a direction ${\\bf x}$ of the Euclidean n-space $R^n$. We introduce the concept of general parameter spheres of $K$ corresponding to this direction and prove that the shadow boundary is a topological manifold if all of the non-degenerated general parameter spheres are, too. In this case, using the approximation theorem of cell-like maps we get that they are homeomorphic to the $(n-2)$-dimensional sphere $S^{(n-2)}$. We also prove that the bisector (equidistant set of the corresponding normed space) in the direction ${\\bf x}$ is homeomorphic to $R^{(n-1)}$ iff all of the non-degenerated general parameter spheres are $(n-2)$-manifolds implying that if the bisector is a homeomorphic copy of $R^{(n-1)}$ then the corresponding shadow boundary is a topological $(n-2)$-sphere."}
{"category": "Math", "title": "On Scattering for CMV Matrices", "abstract": "Adamjan-Arov (Lax--Phillips) model space is considered as a scattering representation space for a CMV matrix in context of an extended Marchenko--Faddeev scattering theory. That is, there exists a basis in which the multiplication by independent variable is a CMV matrix. This basis as well as Verblunski coefficients are computed explicitly in terms of Nehari interpolation. Asymptotically the Verblynski coefficients go to zero. Moreover, relations between the basis and wandering subspaces are established. Transformation from scattering representation to spectral representation is given."}
{"category": "Math", "title": "Rational formality of function spaces", "abstract": "Let $X$ be a nilpotent space such that there exists $N\\geq 1$ with $H^N(X,\\mathbb Q) \\ne 0$ and $H^n(X,\\mathbb Q)=0$ if $n>N$. Let $Y$ be a m-connected space with $m\\geq N+1$ and $H^*(Y,\\mathbb Q)$ is finitely generated as algebra. We assume that the odd part of the rational Hurewicz homomorphism: $\\pi_{odd}(X)\\otimes \\mathbb Q\\to H_{odd}(X,\\mathbb Q)$ is non-zero. We prove that if the space $\\mathcal F(X,Y)$ of continuous maps from $X$ to $Y$ is rationally formal, then $Y$ has the rational homotopy type of a finite product of Eilenberg Mac Lane spaces. At the opposite, we exhibit an example of a rationally formal space $\\mathcal F(S^2,Y)$ where $Y$ is not rationally equivalent to a product of Eilenberg Mac Lane spaces."}
{"category": "Math", "title": "Moduli space of Brody curves, energy and mean dimension", "abstract": "We study the mean dimension of the moduli space of Brody curves. We introduce the notion of \"mean energy\" and show that this can be used to estimate the mean dimension."}
{"category": "Math", "title": "Explicit birational geometry of threefolds of general type", "abstract": "Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V)>0$ and $P_{24}(V)>1$ (which answers an open problem of J. Kollar and S. Mori). We also prove that the canonical volume has an universal lower bound $\\text{Vol}(V) \\geq 1/2660$ and that the pluri-canonical map $\\Phi_m$ is birational onto its image for all $m\\geq 77$. As an application of our method, we prove Fletcher's conjecture on weighted hyper-surface 3-folds with terminal quotient singularities. Another featured result is the optimal lower bound $\\text{Vol}(V)\\geq {1/420}$ among all those 3-folds $V$ with $\\chi({\\mathcal O}_V)\\leq 1$."}
{"category": "Math", "title": "Injectivity and Projectivity in Analysis and Topology", "abstract": "We give new proofs for many injectivity results in analysis that make more careful use of the duality between unital abelian C*-algebras and compact Hausdorff spaces. We then extend many of these results to incorporate group actions. Our approach uses only elementary topological constructions and eliminates the need for results from the theory of Boolean algebras and AW*-algebras that were used in earlier proofs."}
{"category": "Math", "title": "The forgotten monoid", "abstract": "We study properties of the forgotten monoid which appeared in work of Lascoux and Schutzenberger and recently resurfaced in the construction of dual equivalence graphs by Assaf. In particular, we provide an explicit characterization of the forgotten classes in terms of inversion numbers and show that there are n^2-3n+4 forgotten classes in the symmetric group S_n. Each forgotten class contains a canonical element that can be characterized by pattern avoidance. We also show that the sum of Gessel's quasi-symmetric functions over a forgotten class is a 0-1 sum of ribbon-Schur functions."}
{"category": "Math", "title": "Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula", "abstract": "Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and an addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas. Applications to the classical polynomials are given."}
{"category": "Math", "title": "Recollement of Deformed Preprojective Algebras and the Calogero-Moser Correspondence", "abstract": "The aim of this paper is to clarify the relation between the following objects: $ (a) $ rank 1 projective modules (ideals) over the first Weyl algebra $ A_1(\\C)$; $ (b) $ simple modules over deformed preprojective algebras $ \\Pi_{\\lambda}(Q) $ introduced by Crawley-Boevey and Holland; and $ (c) $ simple modules over the rational Cherednik algebras $ H_{0,c}(S_n) $ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on $\\A$-modules over $ A_1 $ to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities $ \\C^2/\\Gamma $, where $ \\Gamma $ is a finite cyclic subgroup of $ \\SL(2, \\C) $."}
{"category": "Math", "title": "Quasi-invariance properties of a class of subordinators", "abstract": "We study absolute-continuity properties of a class of stochastic processes, including the gamma and the Dirichlet processes. We prove that the laws of a general class of non-linear transformations of such processes are locally equivalent to the law of the original process and we compute explicitly the associated Radon-Nikodym densities. This work unifies and generalizes to random non-linear transformations several previous results on quasi-invariance of gamma and Dirichlet processes."}
{"category": "Math", "title": "Independence property and hyperbolic groups", "abstract": "We prove that existentially closed $CSA$-groups have the independence property. This is done by showing that there exist words having the independence property relatively to the class of torsion-free hyperbolic groups."}
{"category": "Math", "title": "On a generalization of Dehn's algorithm", "abstract": "Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms."}
{"category": "Math", "title": "On the discreteness and rationality of F-jumping coefficients", "abstract": "This paper studies the jumping coefficients of principal ideals of regular local rings. Recently M. Blickle, M. Mustata and K. Smith showed that, when $R$ is of essentially finite type over a field and $F$-finite, bounded intervals contain finitely many jumping coefficients and that those are rational. In a later paper they extended these results to principal ideals of $F$-finite complete regular local rings. The aim of this paper is to extend these results on the discreteness and rationality of jumping coefficients to principal ideals of arbitrary (i.e. not necessarily $F$-finite) excellent regular local rings containing fields of positive characteristic. Our proof uses a very different method: we do not use $D$-modules and instead we analyze the modules of nilpotents elements in the injective hull or $R$ under some non-standard Frobenius actions. This new method undoubtedly holds a potential for more applications."}
{"category": "Math", "title": "Using integral transforms to estimate higher order derivatives", "abstract": "Integral transformations are used to estimate high order derivatives of various special functions. Applications are given to numerical integration, where estimates of high order derivatives of the integrand are needed to achieve bounds on the error. The main idea is to find a suitable integral representation of the function whose derivatives are to be estimated, differentiate repeatedly under the integral sign, and estimate the resulting integral."}
{"category": "Math", "title": "Duality of antidiagonals and pipe dreams", "abstract": "Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A. Knutson and E. Miller, Gr\\\"obner geometry of Schubert polynomials, Ann. Math. 161, 1245-1318]. The original proof of the duality was roundabout, relying on the algebra of certain monomial ideals and a recursive characterization of reduced pipe dreams. This paper provides a direct combinatorial proof."}
{"category": "Math", "title": "Implicit function density computation", "abstract": "If two random variables X and A are functionally related via f(X)=A for some strictly monotone continuously differentiable function f:R->R, the distribution of X may easily be computed from the distribution of A."}
{"category": "Math", "title": "On McQuillan's \"tautological inequality\" and the Weyl-Ahlfors theory of associated curves", "abstract": "In 1941, L. Ahlfors gave another proof of a 1933 theorem of H. Cartan on approximation to hyperplanes of holomorphic curves in P^n. Ahlfors' proof built on earlier work of H. and J. Weyl (1938), and proved Cartan's theorem by studying the associated curves of the holomorphic curve. This work has subsequently been reworked by H.-H. Wu in 1970, using differential geometry, M. Cowen and P. A. Griffiths in 1976, further emphasizing curvature, and by Y.-T. Siu in 1987 and 1990, emphasizing meromorphic connections. This paper gives another variation of the proof, motivated by successive minima as in the proof of Schmidt's Subspace Theorem, and using McQuillan's \"tautological inequality.\" In this proof, essentially all of the analysis is encapsulated within a modified McQuillan-like inequality, so that most of the proof primarily uses methods of algebraic geometry, in particular flag varieties. A diophantine conjecture based on McQuillan's inequality is also posed."}
{"category": "Math", "title": "A primer on the (2+1) Einstein universe", "abstract": "The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension $2 + 1$, in which there is a rich interplay with symplectic geometry."}
{"category": "Math", "title": "Extreme values for Benedicks-Carleson quadratic maps", "abstract": "We consider the quadratic family of maps given by $f_{a}(x)=1-a x^2$ with $x\\in [-1,1]$, where $a$ is a Benedicks-Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes $X_0,X_1,...$, given by $X_{n}=f_a^n$, for every integer $n\\geq0$, where each random variable $X_n$ is distributed according to the unique absolutely continuous, invariant probability of $f_a$. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of $M_n=\\max\\{X_0,...,X_{n-1}\\}$ is the same as that which would apply if the sequence $X_0,X_1,...$ was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of $M_n$ is of Type III (Weibull)."}
{"category": "Math", "title": "Strong Uniqueness of the Ricci Flow", "abstract": "In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let $g(t)$ be a smooth complete solution to the Ricci flow on $\\mathbb{R}^{3}$, with the canonical Euclidean metric $E$ as initial data, then $g(t)$ is trivial, i.e. $g(t)\\equiv E$."}
{"category": "Math", "title": "Configurations of Extremal Even Unimodular Lattices", "abstract": "We extend the results of Ozeki on the configurations of extremal even unimodular lattices. Specifically, we show that if L is such a lattice of rank 56, 72, or 96, then L is generated by its minimal-norm vectors."}
{"category": "Math", "title": "Estimates of Gromov's box distance", "abstract": "In 1999, M. Gromov introduced the box distance function $\\sikaku$ on the space of all mm-spaces. In this paper, by using the method of T. H. Colding (cf. \\cite[Lemma 5.10]{Colding}), we estimate $\\sikaku(\\mathbb{S}^n,\\mathbb{S}^m)$ and $\\sikaku (\\mathbb{C}P^n, \\mathbb{C}P^m)$, where $\\mathbb{S}^n$ is the $n$-dimensional unit sphere in $\\mathbb{R}^{n+1}$ and $\\mathbb{C}P^n$ is the $n$-dimensional complex projective space equipped with the Fubini-Study metric. In paticular, we give the complete answer to an Exercise of Gromov's Green book (cf. \\cite[Section $3{1/2}.18$]{gromov}). We also estimate $\\sikaku \\big(SO(n), SO(m)\\big)$ from below, where SO(n) is the special orthogonal group."}
{"category": "Math", "title": "Stark units and main conjectures for totally real fields", "abstract": "Main theorem of [Buyukboduk, arXiv:0706.0377v1] suggests that it should be possible to lift the Kolyvagin systems of Stark units constructed in [Buyukboduk, arXiv:math/0703426v1] to a Kolyvagin system over the cyclotomic Iwasawa algebra. This is what we prove in this paper. This construction gives the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement of local Iwasawa theory. This statement, however, turns out to be interesting in its own right as it suggests a relation between solutions to $p$-adic and complex Stark conjectures."}
{"category": "Math", "title": "On the well-posedness of the Cauchy problem for the generalized Korteweg-de Vries-Burgers equation", "abstract": "Considered is the generalized Korteweg-de Vries-Burgers equation $$ u_{t}+u_{xxx}+uu_{x}+|D_{x}|^{2\\alpha}u=0,\\quad t\\in \\mathbb{R}^{+}, x\\in \\mathbb{R}, $$ with $0\\leq \\alpha\\le 1$. We prove a sharp results on the associated Cauchy problem in the Sobolev space $ {H}^s(\\mathbb{R})$. For $s>-\\min\\{\\frac {3+2\\alpha}4, 1\\}$ we give the well-posedness of solutions of the Cauchy problem, while for $\\frac 12\\le\\alpha\\le 1$ and for $s<-\\min\\{\\frac {3+2\\alpha}4, 1\\}$ we show some ill-posedness issues."}
{"category": "Math", "title": "On $(2k)$-Minimal Submanifolds", "abstract": "Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total $(2k)$-th Gauss-Bonnet curvature function, called $(2k)$-minimal submanifolds. We prove that they are characterized by the vanishing of a higher mean curvature, namely the $(2k+1)$-Gauss-Bonnet curvature. Furthermore, we show that several properties of usual minimal submanifolds can be naturally generalized to $(2k)$-minimal submanifolds."}
{"category": "Math", "title": "Spinorial Characterization of Surfaces into 3-dimensional homogeneous Manifolds", "abstract": "We give a spinorial characterization of isometrically immersed surfaces into 3-dimensional homogeneous manifolds with 4-dimensional isometry group in terms of the existence of a particular spinor, called generalized Killing spinor. This generalizes results by T. Friedrich for $\\R^3$ and B. Morel for $\\Ss^3$ and $\\HH^3$. The main argument is the interpretation of the energy-momentum tensor of a genralized Killing spinor as the second fondamental form up to a tensor depending on the structure of the ambient space"}
{"category": "Math", "title": "Polynomial rate convergence to an invariant measure for the continuum time limit of the Minority Game", "abstract": "In this paper we show that the continuum time version of the Minority Game satisfies the criteria for the application of a theorem on the existence of an invariant measure. We consider the special case of a game with \"sufficiently\" asymmetric initial condition where the number of possible choices for each individual is S=2 and $\\Gamma<+\\infty$. An upper bound for the asymptotic behavior, as the number of agents grows to infinity, of the waiting time for reaching the stationary state is then obtained."}
{"category": "Math", "title": "Finite jet determination of CR mappings", "abstract": "We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M of C^N, N >= 2, which is essentially finite and of finite type at each of its points, for every point p on M there exists an integer l(p), depending upper-semicontinuously on p, such that for every smooth generic submanifold M' of C^N of the same dimension as M, if h_1 and h_2: (M,p)->M' are two germs of smooth finite CR mappings with the same l(p) jet at p, then necessarily their k-jets agree for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in C^N of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Omega and Omega' are two bounded domains in C^N with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary of Omega, such that if H_1 and H_2: Omega -> Omega' are two proper holomorphic mappings extending smoothly up to the boundary of Omega near some point boundary point p and agreeing up to order k at p, then necessarily H_1=H_2."}
{"category": "Math", "title": "Multi-triangulations as complexes of star polygons", "abstract": "Maximal $(k+1)$-crossing-free graphs on a planar point set in convex position, that is, $k$-triangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at $k$-triangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of $k$-triangulations, as well as some new results. This interpretation also opens-up new avenues of research, that we briefly explore in the last section."}
{"category": "Math", "title": "A nilpotent quotient algorithm for L-presented groups", "abstract": "The main part of this paper contains a description of a nilpotent quotient algorithm for L-presented groups and a report on applications of its implementation in the computer algebra system GAP. The appendix introduces two new infinite series of L-presented groups. Apart from being of interest in their own right, these new L-presented groups serve as examples for applications of the nilpotent quotient algorithm."}
{"category": "Math", "title": "Higher-Order Calculus of Variations on Time Scales", "abstract": "We prove a version of the Euler-Lagrange equations for certain problems of the calculus of variations on time scales with higher-order delta derivatives."}
{"category": "Math", "title": "Remarks on the calculus of variations on time scales", "abstract": "The calculus of variations is a classical subject which has gain throughout the last three hundred years a level of rigor and elegance that only time can give. In this note we show that, contrary to the classical field, available formulations and results on the recent calculus of variations on time scales are still at the heuristic level."}
{"category": "Math", "title": "Connections between Romanovski and other polynomials", "abstract": "A connection between Romanovski polynomials and those polynomials that solve the one-dimensional Schr\\\"odinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established. The map is constructed by reworking the Rodrigues formula in an elementary and natural way. The generating function is summed in closed form from which recursion relations and addition theorems follow. Relations to some classical polynomials are also given."}
{"category": "Math", "title": "On contact tops and integrable tops", "abstract": "In this paper, we introduce a geometric structure called top, which is a trivialized bundle of plane pencils over a Riemannian 3-manifold, defined as the set of kernels of a circle of 1-forms (e.g. of contact and integrable forms) with particular properties with respect to the metric. We classify the manifolds which admit tops and we describe the associated metrics."}
{"category": "Math", "title": "Remarks on the density of the law of the occupation time for Bessel bridges and stable excursions", "abstract": "Smoothness and asymptotic behaviors are studied for the densities of the law of the occupation time on the positive line for Bessel bridges and the normalized excursion of strictly stable processes. The key role is played by these properties for functions defined by Riemann--Liouville fractional integrals."}
{"category": "Math", "title": "Verhulst's logistic curve", "abstract": "We observe that the elementary logistic differential equation dP/dt=(1-P/M)kP may be solved by first changing the variable to R=(M-P)/P. This reduces the logistic differential equation to the simple linear differential equation dR/dt=-kR, which can be solved without using the customary but slightly more elaborate methods applied to the original logistic DE. The resulting solution in terms of R can be converted by simple algebra to the familiar sigmoid expression involving P. A biological argument is given for introducing logistic growth via the simpler DE for R. It is also shown that the sigmoid P may be written in terms of the hyperbolic tangent by a simple translation that is also motivated by a biological argument."}
{"category": "Math", "title": "Representing a product system representation as a contractive semigroup and applications to regular isometric dilations", "abstract": "In this paper we propose a new technical tool for analyzing representations of Hilbert $C^*$-product systems. Using this tool, we give a new proof that every doubly commuting representation over $\\mathbb{N}^k$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\\mathbb{R}_+^k$."}
{"category": "Math", "title": "Mirror symmetry and T-duality in the complement of an anticanonical divisor", "abstract": "We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context."}
{"category": "Math", "title": "Convolution estimates and model surfaces of low codimension", "abstract": "We give examples of measures on certain k-surfaces in R^d. These measures satisfy convolution estimates which are nearly optimal."}
{"category": "Math", "title": "Computing word length in alternate presentations of Thompson's group F", "abstract": "We introduce a new method for computing the word length of an element of Thompson's group F with respect to a \"consecutive\" generating set of the form X_n={x_0,x_1,...,x_n}, which is a subset of the standard infinite generating set for F. We use this method to show that (F,X_n) is not almost convex, and has pockets of increasing, though bounded, depth dependent on n."}
{"category": "Math", "title": "Curvature line parametrization from circle patterns", "abstract": "We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete nets on the smooth surface one can obtain second-order approximation globally. Also a simple geometric construction for approximate determination of principal directions of smooth surfaces is given."}
{"category": "Math", "title": "Smooth maps of a foliated manifold in a symplectic manifold", "abstract": "The immersions of a smooth manifold $M$ in a symplectic manifold $(N,\\sigma)$ inducing a given closed form $\\omega$ on $M$ satisfy the $C^0$-dense $h$-principle in the space of all continuous maps which pull back the deRham cohomology class of $\\sigma$ onto that of $\\omega$. In this paper we prove a foliated version of this result due to Gromov."}
{"category": "Math", "title": "Convex Hull Realizations of the Multiplihedra", "abstract": "We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A_infinity-categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph associahedra."}
{"category": "Math", "title": "Stringy product on twisted orbifold K-theory for abelian quotients", "abstract": "In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, and we explicitely calculate the stringy product for a weighted projective orbifold. In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is $(\\integer/2)^3$."}
{"category": "Math", "title": "On finite groups whose derived subgroup has bounded rank", "abstract": "Let $G$ be a finite group with derived subgroup of rank $r$. We prove that $\\gzz\\leq |G'|^{2r}$. Motivated by the results of I. M. Isaacs in \\cite{isa} we show that if $G$ is capable then $\\gz\\leq |G'|^{4r}$. This answers a question of L. Pyber. We prove that if $G$ is a capable $p$-group then the rank of $G/\\mathbf{Z}(G)$ is bounded above in terms of the rank of $G'$."}
{"category": "Math", "title": "Multiplicity free expansions of Schur $P$-functions", "abstract": "After deriving inequalities on coefficients arising in the expansion of a Schur $P$-function in terms of Schur functions we give criteria for when such expansions are multiplicity free. From here we study the multiplicity of an irreducible spin character of the twisted symmetric group in the product of a basic spin character with an irreducible character of the symmetric group, and determine when it is multiplicity free."}
{"category": "Math", "title": "Compact symmetric solutions to the postage stamp problem", "abstract": "We derive lower and upper bounds on possible growth rates of certain sets of positive integers $A_k=\\{1= a_1 < a_2 < ... < a_{k}\\}$ such that all integers $n\\in \\{0, 1, 2, ..., ka_{k}\\}$ can be represented as a sum of no more than $k$ elements of $A_k$, with repetition."}
{"category": "Math", "title": "On tensor products of polynomial representations", "abstract": "We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $GL(n,\\mathbb{C})$ is isomorphic to another. As a consequence we discover families of Littlewood-Richardson coefficients that are non-zero, and a condition on Schur non-negativity."}
{"category": "Math", "title": "Schur positivity of skew Schur function differences and applications to ribbons and Schubert classes", "abstract": "Some new relations on skew Schur function differences are established both combinatorially using Sch\\\"utzenberger's jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain differences of skew Schur functions are Schur positive. Applying these results to a basis of symmetric functions involving ribbon Schur functions confirms the validity of a Schur positivity conjecture due to McNamara. A further application reveals that certain differences of products of Schubert classes are Schubert positive."}
{"category": "Math", "title": "Zeta functions and topological entropy of the Markov-Dyck shifts", "abstract": "The Markov-Dyck shifts arise from finite directed graphs. An expression for the zeta function of a Markov-Dyck shift is given. The derivation of this expression is based on a formula in Keller (G. Keller, {\\it Circular codes, loop counting, and zeta-functions}, J. Combinatorial Theory {\\bf 56} (1991), pp. 75--83). For a class of examples that includes the Fibonacci-Dyck shift the zeta functions and topological entropy ae determined."}
{"category": "Math", "title": "A Blass-Sagan bijection on Eulerian equivalence classes", "abstract": "Following the treatment of Blass and Sagan, we present an algorithmic bijection between the Eulerian equivalence classes of totally cyclic orientations and the spanning trees without internal activity edges for a given graph."}
{"category": "Math", "title": "Orientations, lattice polytopes, and group arrangements I: Chromatic and tension polynomials of graphs", "abstract": "This is the first one of a series of papers on association of orientations, lattice polytopes, and abelian group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the Tutte polynomial $T(G;x,y)$ of a graph $G$ at $(1,0)$ as the number of cut-equivalence classes of acyclic orientations on $G$."}
{"category": "Math", "title": "Braided Hom-Lie bialgebras", "abstract": "We introduce the new concept of braided Hom-Lie bialgebras which is a generalization of Sommerh\\\"{a}user-Majid's braided Lie bialgebras and Yau's Hom-Lie bialgebras. Using this concept we give the unified product construction for Hom-Lie bialgebras which can be seen as a Hom-Lie version of Bespalov-Drabant's cocycle cross product bialgebras. Some special cases of unified products such as crossed product and matched pair of braided Hom-Lie bialgebras are investigated. As an application, we solve the Agore-Militaru extending problem for Hom-Lie bialgebras by using some non-abelian cohomology theory. Furthermore, one dimensional flag extending structures for Hom-Lie bialgebras are also investigated."}
{"category": "Math", "title": "The role of string topology in symplectic field theory", "abstract": "We outline a program for incorporating holomorphic curves with Lagrangian boundary conditions into symplectic field theory, with an emphasis on ideas, geometric intuition, and a description of the resulting algebraic structures."}
{"category": "Math", "title": "On geodesics of Berger tangent sphere bundle of Hermitian locally symmetric manifold", "abstract": "We propose a special deformation of the Sasaki metric on tangent and unit tangent bundle of a Hermitian locally symmetric manifold. Geodesics of this deformed metric have different projections on a base manifold for tangent or unit tangent bundle cases in contrast to usual Sasaki metric. Nevertheless, the projections of geodesics of the unit tangent bundle still preserve the property to have all geodesic curvatures constant."}
{"category": "Math", "title": "Relaxation of the flow of triods by Curve Shortening Flow via the vector-valued parabolic Allen-Cahn equation", "abstract": "In this paper we find solutions $u_\\epsilon$ to a certain class of vector-valued parabolic Allen-Cahn equation that as $\\epsilon \\to 0$ develops as interface a given triod evolving under curve shortening flow."}
{"category": "Math", "title": "Parity properties of Costas arrays defined via finite fields", "abstract": "A Costas array of order $n$ is an arrangement of dots and blanks into $n$ rows and $n$ columns, with exactly one dot in each row and each column, the arrangement satisfying certain specified conditions. A dot occurring in such an array is even/even if it occurs in the $i$-th row and $j$-th column, where $i$ and $j$ are both even integers, and there are similar definitions of odd/odd, even/odd and odd/even dots. Two types of Costas arrays, known as Golomb-Costas and Welch-Costas arrays, can be defined using finite fields. When $q$ is a power of an odd prime, we enumerate the number of even/even odd/odd, even/odd and odd/even dots in a Golomb-Costas array. We show that three of these numbers are equal and they differ by $\\pm 1$ from the fourth. For a Welch-Costas array of order $p-1$, where $p$ is an odd prime, the four numbers above are all equal to $(p-1)/4$ when $p\\equiv 1\\pmod{4}$, but when $p\\equiv 3\\pmod{4}$, we show that the four numbers are defined in terms of the class number of the imaginary quadratic field $\\mathbb{Q}(\\sqrt{-p})$, and thus behave in a much less predictable manner."}
{"category": "Math", "title": "Invariance of Gamma-dimension for projective families", "abstract": "This article is withdrawn because of a mistake in the main result of the paper."}
{"category": "Math", "title": "Radius and profile of random planar maps with faces of arbitrary degrees", "abstract": "We prove some asymptotic results for the radius and the profile of large random rooted planar maps with faces of arbitrary degrees. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted planar maps and certain four-type trees with positive labels, we derive our results from a conditional limit theorem for four-type spatial Galton-Watson trees."}
{"category": "Math", "title": "Filtering and estimation in stochastic volatility models with rationally distributed disturbances", "abstract": "This paper deals with the filtering problem for a class of discrete time stochastic volatility models in which the disturbances have rational probability density functions. This includes the Cauchy distributions and Student t-distributions with odd number of degrees of freedom. Using state space realizations to represent the rational probability density functions we are able to solve the filtering problem exactly. However the size of the involved state space matrices grows exponentially with each time step of the filter. Therefore we use stochastically balanced truncation techniques to approximate the high order rational functions involved. In a simulation study we show the applicability of this approach. In addition a simple method of moments estimator is derived."}
{"category": "Math", "title": "Corps de nombres peu ramifies et formes automorphes autoduales", "abstract": "Let S be a finite set of primes, p in S, and Q_S a maximal algebraic extension of Q unramified outside S and infinity. Assume that |S|>=2. We show that the natural maps Gal(Q_p^bar/Q_p) --> Gal(Q_S/Q) are injective. Much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of GL(2n,A_Q) with prescribed properties at all places, that we study via the twisted trace formula of J. Arthur. The techniques we develop shed also some lights on the orthogonal/symplectic alternative for selfdual representations of GL(2n)."}
{"category": "Math", "title": "On the residual finiteness and other properties of (relative) one-relator groups", "abstract": "A relative one-relator presentation has the form P = < X,H ; R > where X is a set, H is a group, and R is a group word on X and H. We show that if the group word on X obtained from R by deleting all the terms from H has what we call the unique max-min property, then the group defined by P is residually finite if and only if H is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form P (Theorem 6)."}
{"category": "Math", "title": "Flocking in noisy environments", "abstract": "We consider a perturbed version of the dynamics of a flock introduced by Cucker and Smale (\"Emergent behaviour in flocks\") and prove, under similar conditions, that nearly-alignment (a concept that is precised in the text) is achieved with a certain probability, bounded from below."}
{"category": "Math", "title": "Cyclic projectors and separation theorems in idempotent convex geometry", "abstract": "Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem."}
{"category": "Math", "title": "Probabilistic Representations of Solutions of the Forward Equations", "abstract": "In this paper we prove a stochastic representation for solutions of the evolution equation $ \\partial_t \\psi_t = {1/2}L^*\\psi_t $ where $ L^* $ is the formal adjoint of an elliptic second order differential operator with smooth coefficients corresponding to the infinitesimal generator of a finite dimensional diffusion $ (X_t).$ Given $ \\psi_0 = \\psi $, a distribution with compact support, this representation has the form $ \\psi_t = E(Y_t(\\psi))$ where the process $ (Y_t(\\psi))$ is the solution of a stochastic partial differential equation connected with the stochastic differential equation for $ (X_t) $ via Ito's formula."}
{"category": "Math", "title": "Automorphisms of Generalized Down-Up Algebras", "abstract": "A generalization of down-up algebras was introduced by Cassidy and Shelton (J. Algebra 279 (2004), no. 1), the so-called generalized down-up algebras. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f,r,s,\\gamma) such that r is not a root of unity, listing explicitly the elements of the group. In the last section we apply these results to Noetherian down-up algebras, thus obtaining a characterization of the automorphism group of Noetherian down-up algebras A(\\alpha, \\beta, \\gamma) for which the roots of the polynomial X^2-\\alpha X-\\beta are not both roots of unity."}
{"category": "Math", "title": "Three experimental pearls in Costas arrays", "abstract": "The results of 3 experiments in Costas arrays are presented, for which theoretical explanation is still not available: the number of dots on the main diagonal of exponential Welch arrays, the parity populations of Golomb arrays generated in fields of characteristic 2, and the maximal cross-correlation between pairs of Welch or Golomb arrays generated in fields of size equal to a Sophie Germain prime."}
{"category": "Math", "title": "Integral degree of a ring and reduction numbers", "abstract": "The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains bounds for the Castelnuovo-Mumford regularity of the Rees algebra and for the Artin-Rees numbers."}
{"category": "Math", "title": "Characterization of the Radon-Nikodym Property in terms of inverse limits", "abstract": "We clarify the relation between inverse systems, the Radon-Nikodym property, the Asymptotic Norming Property of James-Ho, and the GFDA spaces introduced in our earlier paper on differentiability of Lipschitz maps into Banach spaces."}
{"category": "Math", "title": "Solving SPDEs driven by colored noise: a chaos approach", "abstract": "An Ito-Skorokhod bi-linear equation driven by infinitely many independent colored noises is considered in a normal triple of Hilbert spaces. The special feature of the equation is the appearance of the Wick product in the definition of the Ito-Skorokhod integral, requiring innovative approaches to computing the solution. A chaos expansion of the solution is derived and several truncations of this expansion are studied. A recursive approximation of the solution is suggested and the corresponding approximation error bound is computed."}
{"category": "Math", "title": "The slice-ribbon conjecture for 3-stranded pretzel knots", "abstract": "We determine the smooth concordance order of the 3-stranded pretzel knots P(p,q,r) with p,q,r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As corollaries we give new proofs of results first obtained by Fintushel-Stern and Casson-Gordon."}
{"category": "Math", "title": "On Point Coverings of Boxes in $\\mathbb R^d$", "abstract": "Families of boxes in $\\mathbb R^d$ are considered. In the paper an upper bound on the size of a minimum transversal in terms of the space dimension and the independence number of the given family was improved."}
{"category": "Math", "title": "Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers", "abstract": "Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or \"shock-like\", boundary layers of the isentropic compressible Navier-Stokes equations with gamma-law pressure by a combination of asymptotic ODE estimates and numerical Evans function computations. Our results indicate stability for gamma in the interval [1, 3] for all compressive boundary-layers, independent of amplitude, save for inflow layers in the characteristic limit (not treated). Expansive inflow boundary-layers have been shown to be stable for all amplitudes by Matsumura and Nishihara using energy estimates. Besides the parameter of amplitude appearing in the shock case, the boundary-layer case features an additional parameter measuring displacement of the background profile, which greatly complicates the resulting case structure. Moreover, inflow boundary layers turn out to have quite delicate stability in both large-displacement and large-amplitude limits, necessitating the additional use of a mod-two stability index studied earlier by Serre and Zumbrun in order to decide stability."}
{"category": "Math", "title": "Twisted conjugacy classes in nilpotent groups", "abstract": "A group is said to have the $R_\\infty$ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether $G$ has the $R_\\infty$ property when $G$ is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer $n\\ge 5$, there is a compact nilmanifold of dimension $n$ on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the $R_\\infty$ property. The $R_{\\infty}$ property for virtually abelian and for $\\mathcal C$-nilpotent groups are also discussed."}
{"category": "Math", "title": "Separating populations with wide data: A spectral analysis", "abstract": "In this paper, we consider the problem of partitioning a small data sample drawn from a mixture of $k$ product distributions. We are interested in the case that individual features are of low average quality $\\gamma$, and we want to use as few of them as possible to correctly partition the sample. We analyze a spectral technique that is able to approximately optimize the total data size--the product of number of data points $n$ and the number of features $K$--needed to correctly perform this partitioning as a function of $1/\\gamma$ for $K>n$. Our goal is motivated by an application in clustering individuals according to their population of origin using markers, when the divergence between any two of the populations is small."}
{"category": "Math", "title": "Undercomplete Blind Subspace Deconvolution via Linear Prediction", "abstract": "We present a novel solution technique for the blind subspace deconvolution (BSSD) problem, where temporal convolution of multidimensional hidden independent components is observed and the task is to uncover the hidden components using the observation only. We carry out this task for the undercomplete case (uBSSD): we reduce the original uBSSD task via linear prediction to independent subspace analysis (ISA), which we can solve. As it has been shown recently, applying temporal concatenation can also reduce uBSSD to ISA, but the associated ISA problem can easily become `high dimensional' [1]. The new reduction method circumvents this dimensionality problem. We perform detailed studies on the efficiency of the proposed technique by means of numerical simulations. We have found several advantages: our method can achieve high quality estimations for smaller number of samples and it can cope with deeper temporal convolutions."}
{"category": "Math", "title": "Non-archimedean analytification of algebraic spaces", "abstract": "It is now a classical result that an algebraic space locally of finite type over $\\mathbf{C}$ is analytifiable if and only if it is locally separated. In this paper we study non-archimedean analytifications of algebraic spaces. We construct a quotient for any etale non-archimedean analytic equivalence relation whose diagonal is a closed immersion, and deduce that any separated algebraic space locally of finite type over any non-archimedean field $k$ is analytifiable in both the category of rigid spaces and the category of analytic spaces over $k$. Also, though local separatedness remains a necessary condition for analytifiability in either of these categories, we present many surprising examples of non-analytifiable locally separated smooth algebraic spaces over $k$ that can even be defined over the prime field."}
{"category": "Math", "title": "The SSM Toolbox for Matlab", "abstract": "State Space Models (SSM) is a MATLAB 7.0 software toolbox for doing time series analysis by state space methods. The software features fully interactive construction and combination of models, with support for univariate and multivariate models, complex time-varying (dynamic) models, non-Gaussian models, and various standard models such as ARIMA and structural time-series models. The software includes standard functions for Kalman filtering and smoothing, simulation smoothing, likelihood evaluation, parameter estimation, signal extraction and forecasting, with incorporation of exact initialization for filters and smoothers, and support for missing observations and multiple time series input with common analysis structure. The software also includes implementations of TRAMO model selection and Hillmer-Tiao decomposition for ARIMA models. The software will provide a general toolbox for doing time series analysis on the MATLAB platform, allowing users to take advantage of its readily available graph plotting and general matrix computation capabilities."}
{"category": "Math", "title": "Products of Brauer Severi surfaces", "abstract": "Let $\\{P_i\\}_{1 \\leq i \\leq r}$ and $\\{Q_i\\}_{1 \\leq i \\leq r}$ be two collections of Brauer Severi surfaces (resp. conics) over a field $k$. We show that the subgroup generated by the $P_i's$ in $Br(k)$ is the same as the subgroup generated by the $Q_i's$ \\iff $\\Pi P_i $ is birational to $\\Pi Q_i$. Moreover in this case $\\Pi P_i$ and $\\Pi Q_i$ represent the same class in $M(k)$, the Grothendieck ring of $k$-varieties. The converse holds if $char(k)=0$. Some of the above implications also hold over a general noetherian base scheme."}
{"category": "Math", "title": "Complexity test modules", "abstract": "A method is provided for computing an upper bound of the complexity of a module over a local ring, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which all modules have finite complexity."}
{"category": "Math", "title": "Stability of Hodge bundles and a numerical characterization of Shimura varieties", "abstract": "Consider a family f:A --> U of g-dimensional abelian varieties over a quasiprojective manifold U. Suppose that the induced map from U to the moduli scheme of polarized abelian varieties is generically finite and that there is a projective manifold Y, containing U as the complement of a normal crossing divisor S, such that the sheaf of logarithmic one forms is nef and that its determinant is ample with respect to U. We characterize whether $U$ is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map from U to the moduli scheme or by the existence of CM points. More precisely, we show that U is a Shimura variety, if and only if two conditions hold. First, each irreducible local subsystem V of the complex weight one variation of Hodge structures is either unitary or satisfies the Arakelov equality. Secondly, for each factor M in the universal cover of U whose tangent bundle behaves like the one of a complex ball, an iterated Kodaira-Spencer map associated with V has minimal possible length in the direction of M."}
{"category": "Math", "title": "Classification of hyperfinite factors up to completely bounded isomorphism of their preduals", "abstract": "In this paper we consider the following problem: When are the preduals of two hyperfinite (=injective) factors $\\M$ and $\\N$ (on separable Hilbert spaces) cb-isomorphic (i.e., isomorphic as operator spaces)? We show that if $\\M$ is semifinite and $\\N$ is type III, then their preduals are not cb-isomorphic. Moreover, we construct a one-parameter family of hyperfinite type III$_0$-factors with mutually non cb-isomorphic preduals, and we give a characterization of those hyperfinite factors $\\M$ whose preduals are cb-isomorphic to the predual of the unique hyperfinite type III$_1$-factor. In contrast, Christensen and Sinclair proved in 1989 that all infinite dimensional hyperfinite factors with separable preduals are cb-isomorphic. More recently Rosenthal, Sukochev and the first-named author proved that all hyperfinite type III$_\\lambda$-factors, where $0< \\lambda\\leq 1$, have cb-isomorphic preduals."}
{"category": "Math", "title": "Stationarity and Self-similarity Characterization of the Set-indexed Fractional Brownian Motion", "abstract": "The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to statisfy a strenghtened definition of increment stationarity. This new definition for stationarity property allows to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dimensional fractional Brownian motion, the limitation of its definition for a self-similarity parameter 0<H<1/2 is studied, as illustrated by some examples. When the indexing collection is totally ordered, the sifBm can be defined for 0<H<1."}
{"category": "Math", "title": "An exact geometric mass formula", "abstract": "We show an exact geometric mass formula for superspecial points in the reduction of any quaternionic Shimura variety modulo at a good prime $p$."}
{"category": "Math", "title": "Baxter operator and Archimedean Hecke algebra", "abstract": "In this paper we introduce Baxter integral Q-operators for finite-dimensional Lie algebras gl(n+1) and so(2n+1). Whittaker functions corresponding to these algebras are eigenfunctions of the Q-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G=GL(n+1) proved earlier by Stade. We also identify eigenvalues of the Baxter Q-operator acting on Whittaker functions with local Archimedean L-factors. The Baxter Q-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra H(G(R),K), K being a maximal compact subgroup of G. Finally we stress an analogy between Q-operators and certain elements of the non-Archimedean Hecke algebra H(G(Q_p),G(Z_p))."}
{"category": "Math", "title": "Relations between tautological cycles on Jacobians", "abstract": "We study tautological cycle classes on the Jacobian of a curve. We prove a new result about the ring of tautological classes on a general curve that allows, among other things, easy dimension calculations and leads to some general results about the structure of this ring. Next we obtain a vanishing result for some of the generating classes p_i; this gives an improvement of an earlier result of Herbaut. Finally we lift a result of Herbaut and van der Geer-Kouvidakis to the Chow ring (as opposed to its quotient modulo algebraic equivalence) and we give a method to obtain further explicit cycle relations. As an ingredient for this we prove a theorem about how Polishchuk's operator D lifts to the tautological subalgebra of Chow(J)."}
{"category": "Math", "title": "The size of isoperimetric surfaces in 3-manifolds and a rigidity result for the upper hemisphere", "abstract": "We characterize the standard $\\mathbb{S}^3$ as the closed Ricci-positive 3-manifold with scalar curvature at least 6 having isoperimetric surfaces of largest area: $4\\pi$. As a corollary we answer in the affirmative an interesting special case of a conjecture of Min-Oo's on the scalar curvature rigidity of the upper hemisphere.."}
{"category": "Math", "title": "Peak Quasisymmetric Functions and Eulerian Enumeration", "abstract": "Via duality of Hopf algebras, there is a direct association between peak quasisymmetric functions and enumeration of chains in Eulerian posets. We study this association explicitly, showing that the notion of $\\cd$-index, long studied in the context of convex polytopes and Eulerian posets, arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by Stembridge. Thus Eulerian posets having a nonnegative $\\cd$-index (for example, face lattices of convex polytopes) correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis. We diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions, and study the $g$-polynomial for Eulerian posets as an algebra homomorphism."}
{"category": "Math", "title": "Metric Embedding for Nearest Neighbor Classification", "abstract": "The distance metric plays an important role in nearest neighbor (NN) classification. Usually the Euclidean distance metric is assumed or a Mahalanobis distance metric is optimized to improve the NN performance. In this paper, we study the problem of embedding arbitrary metric spaces into a Euclidean space with the goal to improve the accuracy of the NN classifier. We propose a solution by appealing to the framework of regularization in a reproducing kernel Hilbert space and prove a representer-like theorem for NN classification. The embedding function is then determined by solving a semidefinite program which has an interesting connection to the soft-margin linear binary support vector machine classifier. Although the main focus of this paper is to present a general, theoretical framework for metric embedding in a NN setting, we demonstrate the performance of the proposed method on some benchmark datasets and show that it performs better than the Mahalanobis metric learning algorithm in terms of leave-one-out and generalization errors."}
{"category": "Math", "title": "A symmetric Finsler space with Chern connection", "abstract": "We define a symmetry for a Finsler space with Chern connection and investigate its implementation and properties and find a relation between them and flag curvature."}
{"category": "Math", "title": "On elliptic differential operators with shifts", "abstract": "We give an index formula for elliptic differential operators whose coefficients include shifts forming an infinite group."}
{"category": "Math", "title": "Homogeneous geodesics in homogeneous Finsler spaces", "abstract": "In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on compact semi-simple Lie group is established. We introduce the notion of naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous geodesics. In particular, we prove that the S-curvature vanishes along the homogeneous geodesics."}
{"category": "Math", "title": "Wedderburn polynomials over division rings, II", "abstract": "A polynomial $f(t)$ in an Ore extension $K[t;S,D]$ over a division ring $K$ is a Wedderburn polynomial if $f(t)$ is monic and is the minimal polynomial of an algebraic subset of $K$. These polynomials have been studied in \"Wedderburn polynomials over division rings,I (Journal of Pure and Applied Algebra, Vol. 186, (2004), 43-76). In this paper, we continue this study and give some applications to triangulation, diagonalization and eigenvalues of matrices over a division ring in the general setting of $(S,D)$-pseudo-linear transformations. In the last section we introduce and study the notion of $G$-algebraic sets which, in particular, permits generalization of Wedderburn's theorem relative to factorization of central polynomials."}
{"category": "Math", "title": "On probabilities for separating sets of order statistics", "abstract": "Consider a set of order statistics that arise from sorting samples from two different populations, each with their own, possibly different distribution function. The probability that these order statistics fall in disjoint, ordered intervals, and that of the smallest statistics, a certain number come from the first populations, are given in terms of the two distribution functions. The result is applied to computing the joint probability of the number of rejections and the number of false rejections for the Benjamini-Hochberg false discovery rate procedure."}
{"category": "Math", "title": "Avoidable Sets in The Bicyclic Inverse Semigroup", "abstract": "A subset $U$ of a set $S$ with a binary operation is called {\\it avoidable} if $S$ can be partitioned into two subsets $A$ and $B$ such that no element of $U$ can be written as a product of two distinct elements of $A$ or as the product of two distinct elements of $B$. The avoidable sets of the bicyclic inverse semigroup are classified."}
{"category": "Math", "title": "The diameter of random Cayley digraphs of given degree", "abstract": "We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\\to\\infty$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \\sqrt{n \\log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the probability converges exponentially fast to 1."}
{"category": "Math", "title": "Cohen-Macaulayness and computation of Newton graded toric rings", "abstract": "Let $H$ be a positive semigroup in $\\mathbb{Z}^d$ generated by $A$, and let $K[H]$ be the associated semigroup ring over a field $K$. We investigate heredity of the Cohen-Macaulay property from $K[H]$ to both its $A$-Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen-Macaulay property. On the positive side we show that for every $H$ there exist generating sets $A$ for which the Newton graduation preserves Cohen-Macaulayness. This gives an elementary proof for an important vanishing result on $A$-hypergeometric Euler-Koszul homology. As a tool for our investigations we develop an algorithm to compute algorithmically the Newton filtration on a toric ring."}
{"category": "Math", "title": "Crossed products of k-graph C*-algebras by Z^l", "abstract": "An action of Z^l by automorphisms of a k-graph induces an action of Z^l by automorphisms of the corresponding k-graph C*-algebra. We show how to construct a (k+l)-graph whose C*-algebra coincides with the crossed product of the original k-graph algebra by Z^l. We then investigate the structure of the crossed-product C*-algebra."}
{"category": "Math", "title": "The mean curvature flow for isoparametric submanifolds", "abstract": "A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametric condition, develops singularities in finite time, and converges in finite time to a smooth submanifold of lower dimension. We also give a precise description of the collapsing."}
{"category": "Math", "title": "A formula for the hypergeometric function of type $BC_n$", "abstract": "Formulae of Berezin and Karpelevic for the radial parts of invariant differential operators and the spherical function on a complex Grassmann manifold are generalized to the hypergeometric functions associated with root system of type $BC_n$ under condition that the multiplicity of the middle roots is zero or one."}
{"category": "Math", "title": "A phase transition behavior for Brownian motions interacting through their ranks", "abstract": "Consider a time-varying collection of n points on the positive real axis, modeled as exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as $n$ tends to infinity. Under a certain `continuity at the edge' condition on the drifts, we show that one of the following must happen: either (i) all points converge to zero, or (ii) the maximum goes to one and the rest go to zero, or (iii) the processes converge in law to a non-trivial Poisson-Dirichlet distribution. The proof employs, among other things, techniques from Talagrand's analysis of the low temperature phase of Derrida's Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the Poisson-Dirichlet law."}
{"category": "Math", "title": "Reparametrizations of Continuous Paths", "abstract": "A reparametrization (of a continuous path) is given by a surjective weakly increasing self-map of the unit interval. We show that the monoid of reparametrizations (with respect to compositions) can be understood via ``stop-maps'' that allow to investigate compositions and factorizations, and we compare it to the distributive lattice of countable subsets of the unit interval. The results obtained are used to analyse the space of traces in a topological space, i.e., the space of continuous paths up to reparametrization equivalence. This space is shown to be homeomorphic to the space of regular paths (without stops) up to increasing reparametrizations. Directed versions of the results are important in directed homotopy theory."}
{"category": "Math", "title": "Algebraic definition of Holonomy on Poisson Manifold", "abstract": "We give an algebraic construction of connection on the symplectic leaves of Poisson manifold, introduced in \\cite{Ginzburg}. This construction is suitable for the definition of the linearized holonomy on a regular symplectic foliation."}
{"category": "Math", "title": "An explicit stationary phase formula for the local formal Fourier-Laplace transform", "abstract": "We give an explicit formula (i.e., a formal stationary phase formula) for the local Fourier-Laplace transform of a formal germ of meromorphic connection of one complex variable with a possibly irregular singularity. This is a complex analogue of the formulas in the preprint math/0702436v1."}
{"category": "Math", "title": "Cusps of Hilbert modular varieties", "abstract": "Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3-manifold is diffeomorphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3-manifolds that cannot arise as a cusp cross-section of a 1-cusped nonsingular Hilbert modular surface."}
{"category": "Math", "title": "Bohr and Rogosinski abscissas for ordinary Dirichlet series", "abstract": "We prove that the abscissas of Bohr and Rogosinski for ordinary Dirichlet series, mapping the right half-plane into the bounded convex domain $G\\subset \\mathbb{C} $ are independent of the domain $G$. Furthermore, we obtain new estimates about these abscissas."}
{"category": "Math", "title": "An introduction to Seiberg-Witten theory on closed 3-manifolds", "abstract": "This is a version of the author's diploma thesis written at the University of Cologne in 2002/03. The topic is the construction of Seiberg-Witten invariants of closed 3-manifolds. In analogy to the four dimensional case, the structure of the moduli space is investigated. The Seiberg-Witten invariants are defined and their behaviour under deformation of the Riemannian metric is analyzed. Since it is essentially an exposition of results which were already known during the time of writing, the thesis has not been published. In particular, the author does not claim any originality concerning the results. Moreover, new developments of the theory are not included. However, the detailed account--together with the appendices on the required functional analytic and geometric background--might be of interest for people starting to work in the area of gauge field theory."}
{"category": "Math", "title": "Projective structures and projective bundles over compact Riemann surfaces", "abstract": "A projective structure on a compact Riemann surface X of genus g is given by an atlas with transition functions in PGL(2,C). Equivalently, a projective structure is given by a projective sl(2,C)-bundle over X equipped with a section s and a foliation F which is both transversal to the fibers and the section s. From this latter geometric bundle picture, we survey on classical problems and results on projective structures. We will give a complete description of projective (actually affine) structures on the torus with an explicit versal family of foliated bundle picture."}
{"category": "Math", "title": "Low Regularity local well-posedness for the 1+3 dimensional Dirac-Klein-Gordon system", "abstract": "We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1+3 dimensions is locally well-posed in a range of Sobolev spaces for the Dirac spinor and the meson field. The result contains and extends the earlier known results for the same problem. Our proof relies on the null structure in the system, and bilinear spacetime estimates of Klainerman-Machedon type."}
{"category": "Math", "title": "Almost Everywhere Convergence of Inverse Dunkl Transform on the Real Line", "abstract": "In this paper, we will first show that the maximal operator $S_*^\\alpha$ of spherical partial sums $S_R^\\alpha$, associated to Dunkl transform on $\\mathbb{R}$ is bounded on $L^p(\\mathbb{R}, |x|^{2\\alpha+1} dx)$ functions when $\\frac{4(\\alpha+1)}{2\\alpha+3}<p<\\frac{4(\\alpha+1)}{2\\alpha+1}$, and it implies that, for every $L^p(\\mathbb{R}, |x|^{2\\alpha+1} dx)$ function $f(x)$, $S_R^\\alpha f(x)$ converges to $f(x)$ almost everywhere as $R\\to \\infty$. On the other hand we obtain a sharp version by showing that $S_*^\\alpha$ is bounded from the Lorentz space $L^{p_i,1}(\\mathbb{R}, |x|^{2\\alpha+1})$ into $L^{p_i,\\infty}(\\mathbb{R}, |x|^{2\\alpha+1}),\\quad i=0,1$ where $p_0=\\frac{4(\\alpha+1)}{2\\alpha+3}$ and $p_1=\\frac{4(\\alpha+1)}{2\\alpha+1}$."}
{"category": "Math", "title": "Hypergroups with Unique Alpha-Means", "abstract": "Let $K$ be a commutative hypergroup and $\\alpha\\in \\hat{K}$. We show that $K$ is $\\alpha$-amenable with the unique $\\alpha$-mean $m_\\alpha$ if and only if $m_\\alpha\\in L^1(K)\\cap L^2(K)$ and $\\alpha$ is isolated in $\\hat{K}$. In contrast to the case of amenable noncompact locally compact groups, examples of polynomial hypergroups with unique $\\alpha$-means ($\\alpha\\not=1$) are given. Further examples emphasize that the $\\alpha$-amenability of hypergroups depends heavily on the asymptotic behavior of Haar measures and characters."}
{"category": "Math", "title": "On a class of $\\mathrm{II}_1$ factors with at most one Cartan subalgebra", "abstract": "We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\\Bbb F_r)$ generates an amenable von Neumann subalgebra. Moreover, any II$_1$ factor of the form $Q \\vt L(\\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group $\\Bbb F_r$, $2\\leq r \\leq \\infty$, on a probability space $(X,\\mu)$ is profinite then the group measure space factor $L^\\infty(X)\\rtimes \\Bbb F_r$ has unique Cartan subalgebra, up to unitary conjugacy."}
{"category": "Math", "title": "Familles fuchsiennes d'\\'equations aux (q-)diff\\'erences et confluence", "abstract": "In a first part, we give a method for solving a family of fuchsian systems of operators of pseudo-derivations associated to a family of homographies with two parameters which unify and generalize the differential, the difference and the $q$-difference cases. In a second part, we study the problems of confluence related to these families."}
{"category": "Math", "title": "Orbit-counting for nilpotent group shifts", "abstract": "We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens' theorem for the full $G$-shift for a finitely-generated torsion-free nilpotent group $G$. Using bounds for the M{\\\"o}bius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \\[ \\sum_{|\\tau|\\le N}\\frac{1}{e^{h|\\tau|}}\\sim CN^{\\alpha}(\\log N)^{\\beta} \\] where $|\\tau|$ is the cardinality of the finite orbit $\\tau$. For the usual orbit-counting function we find upper and lower bounds together with numerical evidence to suggest that for actions of non-cyclic groups there is no single asymptotic in terms of elementary functions."}
{"category": "Math", "title": "Cohomology of Congruence Subgroups of SL(4,\\Z) II", "abstract": "In a previous paper [3] we computed cohomology groups H^5 (Gamma_0 (N), \\C), where Gamma_0 (N) is a certain congruence subgroup of SL (4, \\Z), for a range of levels N. In this note we update this earlier work by extending the range of levels and describe cuspidal cohomology classes and additional boundary phenomena found since the publication of [3]. The cuspidal cohomology classes in this paper are the first cuspforms for GL(4) concretely constructed in terms of Betti cohomology."}
{"category": "Math", "title": "Comparison of relative cohomology theories with respect to semidualizing modules", "abstract": "We compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules. We prove a general balance result for relative cohomology over a Cohen-Macaulay ring with a dualizing module, and we demonstrate the failure of the naive version of balance one might expect for these functors. We prove that the natural comparison morphisms between relative cohomology modules are isomorphisms in several cases, and we provide a Yoneda-type description of the first relative Ext functor. Finally, we show by example that each distinct relative cohomology construction does in fact result in a different functor."}
{"category": "Math", "title": "Singularity categories, Schur Functors and Triangular Matrix Rings", "abstract": "We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz-Happel's theorem, we can describe singularity categories of certain non-Gorenstein rings via the stable category of maximal Cohen-Macaulay modules. Three concrete examples of finite-dimensional algebras with the same singularity category are discussed."}
{"category": "Math", "title": "Nearly Tight Frames and Space-Frequency Analysis on Compact Manifolds", "abstract": "Let $\\bf M$ be a smooth compact oriented Riemannian manifold, and let $\\Delta$ be the Laplace-Beltrami operator on ${\\bf M}$. Say $0 \\neq f \\in \\mathcal{S}(\\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 \\Delta)$. Suppose $f$ satisfies Daubechies' criterion, and $b > 0$. For each $j$, write ${\\bf M}$ as a disjoint union of measurable sets $E_{j,k}$ with diameter at most $ba^j$, and comparable to $ba^j$ if $ba^j$ is sufficiently small. Take $x_{j,k} \\in E_{j,k}$. We then show that the functions $\\phi_{j,k}(x)=[\\mu(E_{j,k})]^{1/2} \\bar{K_{a^j}}(x_{j,k},x)$ form a frame for $(I-P)L^2({\\bf M})$, for $b$ sufficiently small (here $P$ is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for $b$ sufficiently small). Moreover, based upon how well-localized a function $F \\in (I-P)L^2$ is in space and in frequency, we can describe which terms in the summation $F \\sim SF = \\sum_j \\sum_k < F,\\phi_{j,k} > \\phi_{j,k}$ are so small that they can be neglected. If $n=2$ and $\\bf M$ is the torus or the sphere, and $f(s)=se^{-s}$ (the \"Mexican hat\" situation), we obtain two explicit approximate formulas for the $\\phi_{j,k}$, one to be used when $t$ is large, and one to be used when $t$ is small. Finally we explain in what sense the kernel $K_t(x,y)$ should itself be regarded as a continuous wavelet on ${\\bf M}$, and characterize the H\\\"older continuous functions on ${\\bf M}$ by the size of their continuous wavelet transforms, for H\\\"older exponents strictly between 0 and 1."}
{"category": "Math", "title": "Dilatation structures with the Radon-Nikodym property", "abstract": "In this paper I explain what is a pair of dilatation structures, one looking down to another. Such a pair of dilatation structures leads to the intrinsic definition of a distribution as a field of topological filters. To any pair of dilatation structures there is an associated notion of differentiability which generalizes the Pansu differentiability. This allows the introduction of the Radon-Nikodym property for dilatation structures, which is the straightforward generalization of the Radon-Nikodym property for Banach spaces. After an introducting section about length metric spaces and metric derivatives, is proved that for a dilatation structure with the Radon-Nikodym property the length of absolutely continuous curves expresses as an integral of the norms of the tangents to the curve, as in Riemannian geometry. Further it is shown that Radon-Nikodym property transfers from any \"upper\" dilatation structure looking down to a \"lower\" dilatation structure, theorem \\ref{ttransfer}. Im my opinion this result explains intrinsically the fact that absolutely continuous curves in regular sub-Riemannian manifolds are derivable almost everywhere, as proved by Margulis-Mostow, Pansu (for Carnot groups) or Vodopyanov."}
{"category": "Math", "title": "Betti numbers of Springer fibers in type A", "abstract": "We determine the Betti numbers of the Springer fibers in type A. To do this, we construct a cell decomposition of the Springer fibers. The codimension of the cells is given by an analogue of the Coxeter length. This makes our cell decomposition well suited for the calculation of Betti numbers."}
{"category": "Math", "title": "Coloring complexes and arrangements", "abstract": "Steingrimsson's coloring complex and Jonsson's unipolar complex are interpreted in terms of hyperplane arrangements. This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type B_n arrangements."}
{"category": "Math", "title": "The wave equation on asymptotically de Sitter-like spaces", "abstract": "In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds $(X^\\circ,g)$ which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y+ and Y-, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to plus infinity, and to the other manifold as the parameter goes to minus infinity, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y-."}
{"category": "Math", "title": "Construction of eigenvarieties in small cohomological dimensions for semi-simple, simply connected groups", "abstract": "We study low order terms of Emerton's spectral sequence for simply connected, simple groups. As a result, for real rank 1 groups, we show that Emerton's method for constructing eigenvarieties is successful in cohomological dimension 1. For real rank 2 groups, we show that a slight modification of Emerton's method allows one to construct eigenvarieties in cohomological dimension 2."}
{"category": "Math", "title": "Conjugate varieties with distinct real cohomology algebras", "abstract": "Using constructions of Voisin, we exhibit a smooth projective variety defined over a number field k and two complex embeddings of k, such that the two complex manifolds induced by these embeddings have non isomorphic cohomology algebras with real coefficients. This contrasts with the fact that the cohomology algebras with l-adic coefficients are canonically isomorphic for any prime number l, and answers a question of Grothendieck."}
{"category": "Math", "title": "The Universal sl_2 Link Homology Theory", "abstract": "We explore the complex associated to a link in the geometric formalism of Khovanov's (n=2) link homology theory, determine its exact underlying algebraic structure and find its precise universality properties for link homology functors. We present new methods of extracting all known link homology theories directly from this universal complex, and determine its relative strength as a link invariant by specifying the amount of information held within the complex. We achieve these goals by finding a complex isomorphism which reduces the complex into one in a simpler category. We introduce few tools and methods, including surface classification modulo the 4TU/S/T relations and genus generating operators, and use them to explore the relation between the geometric complex and its underlying algebraic structure. We identify the universal topological quantum field theory (TQFT) that can be used to create link homology and find that it is ``smaller'' than what was previously reported by Khovanov. We find new homology theories that hold a controlled amount of information relative to the known ones. The universal complex is computable efficiently using our reduction theorem. This allows us to explore the phenomenological aspects of link homology theory through the eyes of the universal complex in order to explain and unify various phenomena (such as torsion and thickness). The universal theory also enables us to state results regarding specific link homology theories derived from it. The methods developed in this thesis can be combined with other known techniques (such as link homology spectral sequences) or used in the various extensions of Khovanov link homology (such as sl_3 link homology)."}
{"category": "Math", "title": "The Mellin transform and spectral properties of toric varieties", "abstract": "In this article we apply results of \\cite{W} on the twisted Mellin transform to problems in toric geometry. In particular we use these results to describe the asymptotics of probability densities associated with the monomial eigenstates, $z^k$, $k \\in \\ZZ^d$, in Bargmann space and prove an \"upstairs\" version of the spectral density theorem of \\cite{BGU}. We also obtain for the $z^k$'s, \"upstairs\" versions of the results of \\cite{STZ} on distribution laws for eigenstates on toric varieties."}
{"category": "Math", "title": "Comparing powers and symbolic powers of ideals", "abstract": "We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension."}
{"category": "Math", "title": "The Bergman kernel and projection on non-smooth worm domains", "abstract": "This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel."}
{"category": "Math", "title": "Point counting on reductions of CM elliptic curves", "abstract": "We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\\mathbf{Q}$-curves in certain cases. This generalizes earlier results of Gross, Stark, and others."}
{"category": "Math", "title": "Weyl modules and opers without monodromy", "abstract": "We prove that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. This result may be used to test the local geometric Langlands correspondence proposed in our earlier work arXiv:math/0508382."}
{"category": "Math", "title": "Combinatorial and hybrid principles for sigma-directed families of countable sets modulo finite", "abstract": "We consider strong combinatorial principles for sigma-directed families of countable sets in the ordering by inclusion modulo finite, e.g. P-ideals of countable sets. We try for principles as strong as possible while remaining compatible with CH, and we also consider principles compatible with the existence of nonspecial Aronszajn trees. The main thrust is towards abstract principles with game theoretic formulations. Some of these principles are purely combinatorial, while the ultimate principles are primarily combinatorial but also have aspects of forcing axioms."}
{"category": "Math", "title": "Higher Rank TQFT Representations of SL(2,Z) are Reducible", "abstract": "In this article we give examples which show that the TQFT representations of the mapping class groups derived from quantum SU(N) for N>2 are generically decomposable. One general decomposition of the representations is induced by the symmetry which exchanges SU(N) representation labels by their conjugates. The respective summands of a given parity are typically still reducible into many further components. Specifically, we give an explicit basis for an irreducible direct summand in the SL(2,Z) representation obtained from quantum PSU(3) when the order of the root of unity is a prime r=2 mod 3. We show that this summand is isomorphic to the respective PSU(2) representation."}
{"category": "Math", "title": "Equivariant Littlewood-Richardson Skew Tableaux", "abstract": "We give a positive equivariant Littlewood-Richardson rule also discovered independently by Molev. Our proof generalizes a proof by Stembridge of the ordinary Littlewood-Richardson rule. We describe a weight-preserving bijection between our indexing tableaux and the Knutson-Tao puzzles."}
{"category": "Math", "title": "Gorenstein cohomology in abelian categories", "abstract": "We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context. Our main theorem is a balance result for relative cohomology that simultaneously recovers theorems of Holm and the current authors as special cases."}
{"category": "Math", "title": "Randomized series and Geometry of Banach spaces", "abstract": "We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\\ge 2$ and $1<p<\\infty$, it is shown that $\\ell_\\infty^n$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a K\\\"othe function space $E$ is upper locally uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is midpoint locally uniformly convex."}
{"category": "Math", "title": "The Bloch-Okounkov correlation functions of negative levels", "abstract": "Bloch and Okounkov introduced an $n$-point correlation function on the fermionic Fock space and found a closed formula in terms of theta functions. This function affords several distinguished interpretations and in particular can be formulated as correlation functions on irreducible $\\hat{gl}_\\infty$-modules of level one. These correlation functions have been generalized for irreducible integrable modules of $\\hat{gl}_\\infty$ and its classical Lie subalgebras of positive levels by the authors. In this paper we extend further these results and compute the correlation functions as well as the $q$-dimensions for modules of $\\hat{gl}_\\infty$ and its classical subalgebras at negative levels."}
{"category": "Math", "title": "Gale duality for complete intersections", "abstract": "We show that every complete intersection of Laurent polynomials in an algebraic torus is isomorphic to a complete intersection of master functions in the complement of a hyperplane arrangement, and vice versa. We call this association Gale duality because the exponents of the monomials in the polynomials annihilate the weights of the master functions. We use Gale duality to give a Kouchnirenko theorem for the number of solutions to a system of master functions and to compute some topological invariants of generic master function complete intersections."}
{"category": "Math", "title": "On Monge-Ampere equations with homogeneous right hand side", "abstract": "We study the regularity and behavior at the origin of solutions to the two-dimensional degenerate Monge-Ampere equation with homogeneous right hand side of degree alpha, alpha>-2. We show that when alpha > 0, solutions admit only two possible behaviors near the origin, radial and non-radial. We also show that the radial behavior is unstable. For alpha<0 we prove that solutions admit only the radial behavior near the origin."}
{"category": "Math", "title": "Sharp linear and bilinear restriction estimates for paraboloids in the cylindrically symmetric case", "abstract": "For cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and bilinear adjoint restriction estimates. As corollaries, we first extend the ranges of exponents for the classical \\textit{linear or bilinear adjoint restriction conjectures} for such functions and verify the \\textit{linear adjoint restriction conjecture} for the paraboloid. We also interpret the restriction estimates in terms of solutions to the Schr\\\"odinger equation and establish the analogous results when the paraboloid is replaced by the lower third of the sphere."}
{"category": "Math", "title": "Uppers to zero and semistar operations in polynomial rings", "abstract": "Given a stable semistar operation of finite type $\\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $[\\star]$ on the polynomial ring $D[X]$, such that $D$ is a $\\star$-quasi-Pr\\\"ufer domain if and only if each upper to zero in $D[X]$ is a quasi-$[\\star]$-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott \\cite[Section 2]{hmm} in the star operation setting. Moreover, we show that $D$ is a Pr\\\"ufer $\\star$-multiplication (resp., a $\\star$-Noetherian; a $\\star$-Dedekind) domain if and only if $D[X]$ is a Pr\\\"ufer $[\\star]$-multiplication (resp., a $[\\star]$-Noetherian; a $[\\star]$-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel-Popescu localizing systems of finite type on an integral domain $D$ (Problem 45 of \\cite{cg}), in terms of multiplicatively closed sets of the polynomial ring $D[X]$."}
{"category": "Math", "title": "Valdivia compact Abelian groups", "abstract": "Let R denote the smallest class of compact spaces containing all metric compacta and closed under limits of continuous inverse sequences of retractions. Class R is striclty larger than the class of Valdivia compact spaces. We show that every compact connected Abelian group which is a topological retract of a space from class R is necessarily isomorphic to a product of metric groups. This completes the result of V. Uspenskij and the author, where a compact connected Abelian group outside class R has been described."}
{"category": "Math", "title": "Order from Randomness", "abstract": "We consider an elementary discrete process which starts from purely random configuration and leads to well-ordered and stable state. Complete analytical solution to this problem is presented."}
{"category": "Math", "title": "On systematic scan for sampling H-colourings of the path", "abstract": "This paper is concerned with sampling from the uniform distribution on H-colourings of the n-vertex path using systematic scan Markov chains. An H-colouring of the n-vertex path is a homomorphism from the n-vertex path to some fixed graph H. We show that systematic scan for H-colourings of the n-vertex path mixes in O(log n) scans for any fixed H. This is a significant improvement over the previous bound on the mixing time which was O(n^5) scans. Furthermore we show that for a slightly more restricted family of H (where any two vertices are connected by a 2-edge path) systematic scan also mixes in O(log n) scans for any scan order. Finally, for completeness, we show that a random update Markov chain mixes in O(n log n) updates for any fixed H, improving the previous bound on the mixing time from O(n^5) updates."}
{"category": "Math", "title": "Inverse Systems and I-Favorable Spaces", "abstract": "A compact space X is I-favorable if, and only if X can be representing as a limit of $\\sigma$-complete inverse system of compact metrizable spaces with skeletal bonding maps."}
{"category": "Math", "title": "Geometric generalizations in Kresin-Maz'ya Sharp Real-Part Theorems", "abstract": "In the present article we give geometric generalizations of the estimates from Chapters 5,6,7 from \\cite{krem:gnus}, while extending their sharpness to new cases."}
{"category": "Math", "title": "Random Sampling of Entire Functions of Exponential Type in Several Variables", "abstract": "We consider the problem of random sampling for band-limited functions. When can a band-limited function $f$ be recovered from randomly chosen samples $f(x_j), j\\in \\mathbb{N}$? We estimate the probability that a sampling inequality of the form A\\|f\\|_2^2 \\leq \\sum_{j\\in \\mathbb{N}} |f(x_j)|^2 \\leq B \\|f\\|_2^2 hold uniformly all functions $f\\in L^2(\\mathbb{R}^d)$ with supp $\\hat{f} \\subseteq [-1/2,1/2]^d$ or some subset of \\bdl functions. In contrast to discrete models, the space of band-limited functions is infinite-dimensional and its functions \"live\" on the unbounded set $\\mathbb{R}^d$. This fact raises new problems and leads to both negative and positive results. (a) With probability one, the sampling inequality fails for any reasonable definition of a random set on $\\mathbb{R}^d$, e.g., for spatial Poisson processes or uniform distribution over disjoint cubes. (b) With overwhelming probability, the sampling inequality holds for certain compact subsets of the space of band-limited functions and for sufficiently large sampling size."}
{"category": "Math", "title": "Invariant deformations of orbit closures in $\\mathfrak{sl}_n$", "abstract": "We study deformations of orbit closures for the action of a connected semisimple group $G$ on its Lie algebra $\\mathfrak{g}$, especially when $G$ is the special linear group. The tools we use are on the one hand the invariant Hilbert scheme and on the other hand the sheets of $\\mathfrak{g}$. We show that when $G$ is the special linear group, the connected components of the invariant Hilbert schemes we get are the geometric quotients of the sheets of $\\mathfrak{g}$. These quotients were constructed by Katsylo for a general semisimple Lie algebra $\\mathfrak{g}$; in our case, they happen to be affine spaces."}
{"category": "Math", "title": "Mok-Siu-Yeung type formulas on contact locally sub-symmetric spaces", "abstract": "We derive Mok-Siu-Yeung type formulas for horizontal maps from compact contact locally sub-symmetric spaces into strictly pseudoconvex CR manifolds and we obtain some rigidity theorems for the horizontal pseudoharmonic maps."}
{"category": "Math", "title": "Arithmetic lattices and weak spectral geometry", "abstract": "This note is an expansion of three lectures given at the workshop \"Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces\" held at Kyoto University in December of 2006 and will appear in the proceedings for this workshop."}
{"category": "Math", "title": "D-modules over rings with finite F-representation type", "abstract": "Smith and Van den Bergh introduced the notion of finite F-representation type as a characteristic $p$ analogue of the notion of finite representation type. In this paper, we prove two finiteness properties of rings with finite F-representation type. The first property states that if $R=\\bigoplus_{n \\ge 0}R_n$ is a Noetherian graded ring with finite (graded) F-representation type, then for every non-zerodivisor $x \\in R$, $R_x$ is generated by $1/x$ as a $D_{R}$-module. The second one states that if $R$ is a Gorenstein ring with finite F-representation type, then $H_I^n(R)$ has only finitely many associated primes for any ideal $I$ of $R$ and any integer $n$. We also include a result on the discreteness of F-jumping exponents of ideals of rings with finite (graded) F-representation type as an appendix."}
{"category": "Math", "title": "Graphical Representation of some Duality Relations in Stochastic Population Models", "abstract": "We derive a unified stochastic picture for the duality of a resampling-selection model with a branching-coalescing particle process (cf. http://www.ams.org/mathscinet-getitem?mr=MR2123250) and for the self-duality of Feller's branching diffusion with logistic growth (cf. math/0509612). The two dual processes are approximated by particle processes which are forward and backward processes in a graphical representation. We identify duality relations between the basic building blocks of the particle processes which lead to the two dualities mentioned above."}
{"category": "Math", "title": "Countable groups of isometries on Banach spaces", "abstract": "A group G is representable in a Banach space X if G is isomorphic to the group of isometries on X in some equivalent norm. We prove that a countable group G is representable in a separable real Banach space X in several general cases, including when $G=\\{-1,1\\} \\times H$, H finite and $\\dim X \\geq |H|$, or when G contains a normal subgroup with two elements and X is of the form c_0(Y) or $\\ell_p(Y)$, $1 \\leq p <+\\infty$. This is a consequence of a result inspired by methods of S. Bellenot and stating that under rather general conditions on a separable real Banach space X and a countable bounded group G of isomorphisms on X containing -Id, there exists an equivalent norm on X for which G is equal to the group of isometries on X. We also extend methods of K. Jarosz to prove that any complex Banach space of dimension at least 2 may be renormed to admit only trivial real isometries, and that any real Banach space which is a cartesian square may be renormed to admit only trivial and conjugation real isometries. It follows that every real space of dimension at least 4 and with a complex structure up to isomorphism may be renormed to admit exactly two complex structures up to isometry, and that every real cartesian square may be renormed to admit a unique complex structure up to isometry."}
{"category": "Math", "title": "Understanding preservation theorems: Chapter VI of Proper and Improper Forcing", "abstract": "This expository paper covers the first two sections of chapter VI of Shelah's book \"Proper and Improper Forcing,\" including the preservationn (using CS iterations of proper forcings) of omega-omega bounding, Sacks property, Lavewr property, P-point property, and others."}
{"category": "Math", "title": "The classification question for Leavitt path algebras", "abstract": "We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between ${\\mathbb Z}$-graded algebras. As our main application of this theorem, we obtain isomorphisms between the Leavitt path algebras of specified graphs. From these isomorphisms we are able to achieve two ends. First, we show that the $K_0$ groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in $K_0$, classifies the algebras in these sets up to isomorphism. Second, we show that the isomorphism between matrix rings over the classical Leavitt algebras, established previously using number-theoretic methods, can be reobtained via appropriate isomorphisms between Leavitt path algebras."}
{"category": "Math", "title": "Simplicial volume of Hilbert modular varieties", "abstract": "The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact locally symmetric spaces of finite volume, to so-called Hilbert modular varieties. The key idea is to reduce the problem to the compact case by first relating the simplicial volume of these manifolds to the Lipschitz simplicial volume and then taking advantage of a proportionality principle for the Lipschitz simplicial volume. Moreover, using computations of Bucher-Karlsson for the simplicial volume of products of closed surfaces, we obtain the exact value of the simplicial volume of Hilbert modular surfaces."}
{"category": "Math", "title": "The Implicit Function Theorem for continuous functions", "abstract": "In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of these classic theorems are proved when we consider differenciable (not necessarily C^1) maps."}
{"category": "Math", "title": "On the rank of a Coxeter group", "abstract": "Let W be a Coxeter group with Coxeter generators S. The rank of the Coxeter system (W,S) is the cardinality |S| of S. The Coxeter system (W,S) has finite rank if and only if W is finitely generated. If (W,S) has infinite rank, then |S| = |W|, since every element of W is represented by a finite product of elements of S. Thus if W is not finitely generated, the rank of (W,S) is uniquely determined by W. If W is finitely generated, then W may have sets of Coxeter generators S and S' of different ranks. In this paper, we determine the set of all possible ranks for an arbitrary finitely generated Coxeter group W."}
{"category": "Math", "title": "Triviality of vector bundles on sufficiently twisted ind-Grassmannians", "abstract": "Twisted ind-Grassmannians are ind-varieties $\\GG$ obtained as direct limits of Grassmannians $G(r_m,V^{r_m})$, for $m\\in\\ZZ_{>0}$, under embeddings $\\phi_m:G(r_m,V^{r_m})\\to G(r_{m+1}, V^{r_{m+1}})$ of degree greater than one. It has been conjectured in \\cite{PT} and \\cite{DP} that any vector bundle of finite rank on a twisted ind-Grassmannian is trivial. We prove this conjecture under the assumption that the ind-Grassmannian $\\GG$ is sufficiently twisted, i.e. that $\\lim_{m\\to\\infty}\\frac{r_m}{\\deg \\phi_1...\\deg\\phi_m}=0$."}
{"category": "Math", "title": "Holomorphic curves in Exploded Torus Fibrations: Compactness", "abstract": "The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the limits considered in tropical geometry appear smooth, also degenerations corresponding to an algebraic family with normal crossing singularities are smooth.) In this paper we prove a compactness theorem for (pseudo)-holomorphic curves in exploded torus fibrations. In the case of smooth manifolds, this is just a version of Gromov's compactness theorem in a topology strong enough for gluing analysis."}
{"category": "Math", "title": "A Linearization of Connes' Embedding Problem", "abstract": "We show that Connes' embedding problem for II_1-factors is equivalent to a statement about distributions of sums of self-adjoint operators with matrix coefficients. This is an application of a linearization result for finite von Neumann algebras, which is proved using asymptotic second order freeness of Gaussian random matrices."}
{"category": "Math", "title": "Nonparametric estimation for dependent data with an application to panel time series", "abstract": "In this paper we consider nonparametric estimation for dependent data, where the observations do not necessarily come from a linear process. We study density estimation and also discuss associated problems in nonparametric regression using the 2-mixing dependence measure. We compare the results under 2-mixing with those derived under the assumption that the process is linear. In the context of panel time series where one observes data from several individuals, it is often too strong to assume the joint linearity of processes. Instead the methods developed in this paper enable us to quantify the dependence through 2-mixing which allows for nonlinearity. We propose an estimator of the panel mean function and obtain its rate of convergence. We show that under certain conditions the rate of convergence can be improved by allowing the number of individuals in the panel to increase with time."}
{"category": "Math", "title": "Rips complexes and covers in the uniform category", "abstract": "James \\cite{Jam} introduced uniform covering maps as an analog of covering maps in the topological category. Subsequently Berestovskii and Plaut \\cite{BP3} introduced a theory of covers for uniform spaces generalizing their results for topological groups \\cite{BP1}-\\cite{BP2}. Their main concepts are discrete actions and pro-discrete actions, respectively. In case of pro-discrete actions Berestovskii and Plaut provided an analog of the universal covering space and their theory works well for the so-called coverable spaces. As will be seen in Section \\ref{SECTION-Comparison}, \\cite{BP3} generalizes only regular covering maps in topology and pro-discrete actions may not be preserved by compositions. In this paper we redefine the uniform covering maps and we generalize pro-discrete actions using Rips complexes and the chain lifting property. We expand the concept of generalized paths of Krasinkiewicz and Minc \\cite{KraMin}."}
{"category": "Math", "title": "Thorn independence in the field of real numbers with a small multiplicative group", "abstract": "We characterize thorn-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets."}
{"category": "Math", "title": "Maps between moduli spaces of vector bundles and the base locus of the theta divisor", "abstract": "Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections, the kernel of the evaluation map of sections is again a vector bundle. Also, new vector bundles can be produced by taking elementary transformations centered at a fixed point. Under suitable conditions on degree and rank, these constructions can be carried out globally. While all this processes seem quite elementary, very little is known about the resulting maps. The purpose of this paper is to fill in this gap."}
{"category": "Math", "title": "Notes on $\\pi_1$ of Smooth Loci of Log Del Pezzo Surfaces", "abstract": "It is known that the fundamental groups of smooth loci of Log del Pezzo Surfaces are finite groups. The aim of this note is to study these finite groups. A short table containing these groups is given. And lots of groups on the table are proved to be fundamental groups."}
{"category": "Math", "title": "Coxeter polytopes with a unique pair of non-intersecting facets", "abstract": "We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results of Lann\\'er, Kaplinskaja, Esselmann, and the second author, this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8."}
{"category": "Math", "title": "Multiscale inference about a density", "abstract": "We introduce a multiscale test statistic based on local order statistics and spacings that provides simultaneous confidence statements for the existence and location of local increases and decreases of a density or a failure rate. The procedure provides guaranteed finite-sample significance levels, is easy to implement and possesses certain asymptotic optimality and adaptivity properties."}
{"category": "Math", "title": "Raynaud vector bundles", "abstract": "We construct vector bundles $R^r_\\mu$ on a smooth projective curve $X$ having the property that for all sheaves $E$ of slope $\\mu$ and rank $r$ on $X$ we have an equivalence: $E$ is a semistable vector bundle $\\iff$ $Hom(R^r_\\mu,E)=0$. As a byproduct of our construction we obtain effective bounds on $r$ such that the linear system $|R \\cdot \\Theta|$ has base points on the moduli space $U_X(r,r(g-1))$."}
{"category": "Math", "title": "On the L^p-distorsion of finite quotients of amenable groups", "abstract": "We study the L^p-distortion of finite quotients of amenable groups. In particular, for every number p larger or equal than 2, we prove that the l^p-distortion of the finite lamplighter group grows like (\\log n)^{1/p}. We also give the asymptotic behavior of the l^p-distortion of finite quotients of certain metabelian polycyclic groups and of the solvable Baumslag-Solitar groups BS(m,1). The proofs are short and elementary."}
{"category": "Math", "title": "Whitehead-Torsion und Faserungen", "abstract": "This work treats on the question whether a given map f: M -> B of smooth closed manifolds is homotopic to a smooth fiber bundle. We define a first obstruction in H^1(B;Wh(\\pi_1(E))) and, provided that this obstruction vanishes and one additional condition is verified, a second obstruction in Wh(\\pi_1(E)) >. Both elements vanish if the answer to the above question is positive. In the case where B is the 1-sphere and the dimension of M exceeds five, we show that the converse is also true, using a relationship with two obstructions defined by Farrell in this particular situation."}
{"category": "Math", "title": "On the Greenfield-Wallach and Katok conjectures", "abstract": "We survey recent progress on the Greenfield-Wallach and Katok conjectures on globally hypoelliptic and cohomology free vector fields and derive a proof of the conjectures in dimension three. The argument is primarily based on recent work of F. and J. Rodriguez Hertz which allows us to reduce the question to the case of a Reeb flow for a contact form. The contact case is settled by invoking the Weinstein conjecture (which has been recently announced by C. Taubes)."}
{"category": "Math", "title": "Distributions associated with general runs and patterns in hidden Markov models", "abstract": "This paper gives a method for computing distributions associated with patterns in the state sequence of a hidden Markov model, conditional on observing all or part of the observation sequence. Probabilities are computed for very general classes of patterns (competing patterns and generalized later patterns), and thus, the theory includes as special cases results for a large class of problems that have wide application. The unobserved state sequence is assumed to be Markovian with a general order of dependence. An auxiliary Markov chain is associated with the state sequence and is used to simplify the computations. Two examples are given to illustrate the use of the methodology. Whereas the first application is more to illustrate the basic steps in applying the theory, the second is a more detailed application to DNA sequences, and shows that the methods can be adapted to include restrictions related to biological knowledge."}
{"category": "Math", "title": "The order completion method for systems of nonlinear PDEs: Pseudo-topological perspectives", "abstract": "By setting up appropriate uniform convergence structures, we are able to reformulate the Order Completion Method of Oberguggenberger and Rosinger in a setting that more closely resembles the usual topological constructions for solving PDEs. As an application, we obtain existence and uniqueness results for the solutions of arbitrary continuous, nonlinear PDEs."}
{"category": "Math", "title": "Local results for flows whose speed or height satisfies a bound of the form $\\frac c t$", "abstract": "In this paper we prove local results for solutions to the Ricci flow (heat flow) whose speed (height) is bounded by $\\frac c t$ for some time interval $ t \\in (0,T)$. These results are contained in chapter 7 of the author's habilitation thesis, University of Freiburg, Germany, 2006."}
{"category": "Math", "title": "Stochastic control problems for systems driven by normal martingales", "abstract": "In this paper we study a class of stochastic control problems in which the control of the jump size is essential. Such a model is a generalized version for various applied problems ranging from optimal reinsurance selections for general insurance models to queueing theory. The main novel point of such a control problem is that by changing the jump size of the system, one essentially changes the type of the driving martingale. Such a feature does not seem to have been investigated in any existing stochastic control literature. We shall first provide a rigorous theoretical foundation for the control problem by establishing an existence result for the multidimensional structure equation on a Wiener--Poisson space, given an arbitrary bounded jump size control process; and by providing an auxiliary counterexample showing the nonuniqueness for such solutions. Based on these theoretical results, we then formulate the control problem and prove the Bellman principle, and derive the corresponding Hamilton--Jacobi--Bellman (HJB) equation, which in this case is a mixed second-order partial differential/difference equation. Finally, we prove a uniqueness result for the viscosity solution of such an HJB equation."}
{"category": "Math", "title": "Induction and computation of Bass Nil Groups for finite groups", "abstract": "Let G be a finite group. We show that the Bass Nil-groups $NK_n(RG)$, $n \\in Z$, are generated from the p-subgroups of G by induction maps, certain twisting maps depending on elements in the centralizers of the p-subgroups, and the Verschiebung homomorphisms. As a consequence, the groups $NK_n(RG)$ are generated by induction from elementary subgroups. For $NK_0(ZG)$ we get an improved estimate of the torsion exponent."}
{"category": "Math", "title": "How much information about a dynamical system do its recurrences contain?", "abstract": "We show that, under suitable assumptions, Poincare recurrences of a dynamical system determine its topology in phase space. Therefore, dynamical systems with the same recurrences are topologically equivalent."}
{"category": "Math", "title": "Stopped diffusion processes: boundary corrections and overshoot", "abstract": "For a stopped diffusion process in a multidimensional time-dependent domain $\\D$, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size $\\Delta$ and stopping it at discrete times $(i\\Delta)_{i\\in\\N^*}$ in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal $n(t,x)$ at any point $(t,x)$ on the parabolic boundary of $\\D$, and its amplitude is equal to $0.5826 (...) |n^*\\sigma|(t,x)\\sqrt \\Delta$ where $\\sigma$ stands for the diffusion coefficient of the process. The procedure is thus extremely easy to use. In addition, we prove that the rate of convergence w.r.t. $\\Delta$ for the associated weak error is higher than without shifting, generalizin g previous results by \\cite{broa:glas:kou:97} obtained for the one dimensional Brownian motion. For this, we establish in full generality the asymptotics of the triplet exit time/exit position/overshoot for the discretely stopped Euler scheme. Here, the overshoot means the distance to the boundary of the process when it exits the domain. Numerical experiments support these results."}
{"category": "Math", "title": "Continuous spectrum for a class of nonhomogeneous differential operators", "abstract": "We study the boundary value problem $-{\\rm div}((|\\nabla u|^{p_1(x)-2}+|\\nabla u|^{p_2(x)-2})\\nabla u)=\\lambda|u|^{q(x)-2}u$ in $\\Omega$, $u=0$ on $\\partial\\Omega$, where $\\Omega$ is a bounded domain in $\\RR^N$ with smooth boundary, $\\lambda$ is a positive real number, and the continuous functions $p_1$, $p_2$, and $q$ satisfy $1<p_2(x)<q(x)<p_1(x)<N$ and $\\max_{y\\in\\bar\\Omega}q(y)<\\frac{N p_2(x)}{N-p_2(x)}$ for any $x\\in\\bar\\Omega$. The main result of this paper establishes the existence of two positive constants $\\lambda_0$ and $\\lambda_1$ with $\\lambda_0\\leq\\lambda_1$ such that any $\\lambda\\in[\\lambda_1,\\infty)$ is an eigenvalue, while any $\\lambda\\in(0,\\lambda_0)$ is not an eigenvalue of the above problem."}
{"category": "Math", "title": "Toward the classification of cohomology-free vector fields", "abstract": "In 1984, Anatole Katok conjectured that the only closed orientable manifolds that support cohomology-free vector fields are tori and these vector fields are smoothly conjugated to Diophantine (constant) ones. In this work we present a proof of Katok conjecture for 3-manifolds."}
{"category": "Math", "title": "The pentagon relation for the quantum dilogarithm and quantized M_{0,5}", "abstract": "We introduce and study a Schwarz space S in the space of functions on the real line. It is a module over the algebra L of regular functions on the (modular double of the) non-commutative q-deformation of the moduli space of configurations of 5 cyclically ordered points on the projective line. The algebra L has an order five automorphism corresponding to the cyclic shift of the points. The quantum dilogarithm gives rise to an automorphism of the space Schwarz S intertwining the automorphism of L. This easily implies the pentagon relation for the quantum dilogarithm function. The triple (L, S, the automorphism) is the quantized moduli space of configurations of 5 points on the projective line. It is the simplest example of a quantized cluster X-variety."}
{"category": "Math", "title": "Weighted Low-Regularity Solutions of the KP-I Initial Value Problem", "abstract": "In this paper we establish local well-posedness of the KP-I problem, with initial data small in the intersection of the natural energy space with the space of functions which are square integrable when multiplied by the weight y. The result is proved by the contraction mapping principle. A similar (but slightly weaker) result was the main Theorem in the paper \" Low regularity solutions for the Kadomstev-Petviashvili I equation \" by Colliander, Kenig and Staffilani (GAFA 13 (2003),737-794 and math.AP/0204244). Ionescu found a counterexample (included in the present paper) to the main estimate used in the GAFA paper, which renders incorrect the proof there. The present paper thus provides a correct proof of a strengthened version of the main result in the GAFA paper."}
{"category": "Math", "title": "Minimal $\\gamma$--sheaves", "abstract": "In this note we show that finitely generated unit $O_X[\\sigma]$--modules for $X$ regular and $F$--finite have a minimal root (in the sense of [Lyubeznik, F-modules] Definition~3.6). This problem was posed by Lyubeznik and answered by himself in the case that $X=\\Spec R$ is a complete local ring. One immediate consequence of this result is that the parameter test module of tight closure theory commutes with localization. As a further application of the methods in this paper we give new proofs of the results on discreteness and rationality of $F$--thresholds [arXiv:0705.1210] and on $D$-module generation [arXiv:math/0502405v1]. The new proofs are valid in a slightly more general setting such that they also party cover the generalizations recently obtained in [arXiv:0706.3028]."}
{"category": "Math", "title": "Equisingularite reelle : invariants locaux et conditions de regularite", "abstract": "For germs of subanalytic sets, we define two finite sequences of new numerical invariants. The first one is obtained by localizing the classical Lipschitz-Killing curvatures, the second one is the real analogue of the evanescent characteristics introduced by M. Kashiwara. We show that each invariant of one sequence is a linear combination of the invariants of the other sequence. We then connect our invariants to the geometry of the discriminants of all dimension. Finally we prove that these invariants are continuous along Verdier strata of a closed subanalytic set."}
{"category": "Math", "title": "Universal abelian covers of rational surface singularities and multi-index filtrations", "abstract": "In previous papers, there were computed the Poincare series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincare series were written as the integer parts of certain fractional power series, an interpretation of whom was not given. Here we show that, up to a simple change of variables, these fractional power series are specializations of the equivariant Poincare series for filtrations on the ring of germs of functions on the universal abelian cover of the surface singularity. We compute these equivariant Poincare series."}
{"category": "Math", "title": "An effective criterion and a new example for ballistic diffusions in random environment", "abstract": "In the setting of multidimensional diffusions in random environment, we carry on the investigation of condition $(T')$, introduced by Sznitman [Ann. Probab. 29 (2001) 723--764] and by Schmitz [Ann. Inst. H. Poincar\\'{e} Probab. Statist. 42 (2006) 683--714] respectively in the discrete and continuous setting, and which implies a law of large numbers with nonvanishing limiting velocity (ballistic behavior) as well as a central limit theorem. Specifically, we show that when $d\\geq2$, $(T')$ is equivalent to an effective condition that can be checked by local inspection of the environment. When $d=1$, we prove that condition $(T')$ is merely equivalent to almost sure transience. As an application of the effective criterion, we show that when $d\\geq4$ a perturbation of Brownian motion by a random drift of size at most $\\epsilon>0$ whose projection on some direction has expectation bigger than $\\epsilon^{2-\\eta},\\eta>0$, satisfies condition $(T')$ when $\\epsilon$ is small and hence exhibits ballistic behavior. This class of diffusions contains new examples of ballistic behavior which in particular do not fulfill the condition in [Ann. Inst. H. Poincar\\'{e} Probab. Statist. 42 (2006) 683--714], (5.4) therein, related to Kalikow's condition."}
{"category": "Math", "title": "Chernoff's theorem for evolution families", "abstract": "A generalized version of Chernoff's theorem has been obtained. Namely, the version of Chernoff's theorem for semigroups obtained in a paper by Smolyanov, Weizsaecker, and Wittich is generalized for a time-inhomogeneous case. The main theorem obtained in the current paper, Chernoff's theorem for evolution families, deals with a family of time-dependent generators of semigroups $A_t$ on a Banach space, a two-parameter family of operators $Q_{t,t+\\Delta t}$ satisfying the relation: $\\frac{\\partial}{\\partial \\Delta t}Q_{t,t+\\Delta t}|_{\\Delta t = 0}=A_t$, whose products $Q_{t_i,t_{i+1}}... Q_{t_{k-1},t_k}$ are uniformly bounded for all subpartitions $s = t_0 < t_1 < >... < t_n = t$. The theorem states that $Q_{t_0,t_1}... Q_{t_{n-1},t_n}$ converges to an evolution family $U(s,t)$ solving a non-autonomous Cauchy problem. Furthermore, the theorem is formulated for a particular case when the generators $A_t$ are time dependent second order differential operators. Finally, an example of application of this theorem to a construction of time-inhomogeneous diffusions on a compact Riemannian manifold is given. Keywords: Chernoff's theorem, evolution family, strongly continuous semigroup, evolution families generated by manifold valued stochastic processes."}
{"category": "Math", "title": "The classification of torsion endo-trivial modules", "abstract": "This paper is a major step in the classification of endotrivial modules over p-groups. Let G be a finite p-group and k be a field of characteristic p. A kG-module M is an endo-trivial module if {\\End_k(M)\\cong k\\oplus F} as kG-modules, where F is a free module. The classification of endo-trivial modules is the crucial step for understanding the more general class of endo-permutation modules. The endo-permutation modules play an important role in module theory, in particular as source modules, and in block theory where they appear in the description of source algebras. Endo-trivial modules are also important in the study of both derived equivalences and stable equivalences of group algebras and block algebras. The collection of isomorphism classes of endo-trivial modules modulo projectives is an abelian group under tensor product. The main result of this paper is that this group is torsion free except in the case that G is cyclic, quaternion or semi-dihedral. Hence for any p-group which is not cyclic, quaternion or semi-dihedral and any finitely generated kG-module M, if M \\otimes_k M \\otimes_k ... \\otimes_k M \\cong k \\oplus P for some projective module P and some finite number of tensor products, then M \\cong k \\oplus Q for some projective module Q. The proof uses a reduction to the cases in which G is an extraspecial or almost extraspecial p-group, proved in a previous paper of the authors, and makes extensive use of the theory of support varieties for modules."}
{"category": "Math", "title": "Inf-sup estimates for the Stokes problem in a periodic channel", "abstract": "We derive estimates of the Babu\\u{s}ka-Brezzi inf-sup constant $\\beta$ for two-dimensional incompressible flow in a periodic channel with one flat boundary and the other given by a periodic, Lipschitz continuous function $h$. If $h$ is a constant function (so the domain is rectangular), we show that periodicity in one direction but not the other leads to an interesting connection between $\\beta$ and the unitary operator mapping the Fourier sine coefficients of a function to its Fourier cosine coefficients. We exploit this connection to determine the dependence of $\\beta$ on the aspect ratio of the rectangle. We then show how to transfer this result to the case that $h$ is $C^{1,1}$ or even $C^{0,1}$ by a change of variables. We avoid non-constructive theorems of functional analysis in order to explicitly exhibit the dependence of $\\beta$ on features of the geometry such as the aspect ratio, the maximum slope, and the minimum gap thickness (if $h$ passes near the substrate). We give an example to show that our estimates are optimal in their dependence on the minimum gap thickness in the $C^{1,1}$ case, and nearly optimal in the Lipschitz case."}
{"category": "Math", "title": "Equivariant path fields on topological manifolds", "abstract": "A classical theorem of H. Hopf asserts that a closed connected smooth manifold admits a nowhere vanishing vector field if and only if its Euler characteristic is zero. R. Brown generalized Hopf's result to topological manifolds, replacing vector fields with path fields. In this note, we give an equivariant analog of Brown's theorem for locally smooth $G$-manifolds where $G$ is a finite group."}
{"category": "Math", "title": "On graphs with subgraphs of large independence numbers", "abstract": "Let G be a graph on n vertices in which every induced subgraph on s=\\log^3 n vertices has an independent set of size at least t=\\log n. What is the largest q=q(n) so that every such G must contain an independent set of size at least q ? This is one of several related questions raised by Erdos and Hajnal. We show that q(n)=\\Theta(\\log^2 n/\\log \\log n), investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey-type problem."}
{"category": "Math", "title": "Embedding nearly-spanning bounded degree trees", "abstract": "We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-\\epsilon)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\\geq 2 and 0<\\epsilon<1, there exists a constant c=c(d,\\epsilon) such that a random graph G(n,c/n) contains almost surely a copy of every tree T on (1-\\epsilon)n vertices with maximum degree at most d. We also prove that if an (n,D,\\lambda)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most \\lambda in their absolute values) has large enough spectral gap D/\\lambda as a function of d and \\epsilon, then G has a copy of every tree T as above."}
{"category": "Math", "title": "Making a K_4-free graph bipartite", "abstract": "We show that every K_4-free graph G with n vertices can be made bipartite by deleting at most n^2/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdos."}
{"category": "Math", "title": "Ramsey numbers and the size of graphs", "abstract": "For two graph H and G, the Ramsey number r(H, G) is the smallest positive integer n such that every red-blue edge coloring of the complete graph K_n on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions of Erdos and Harary, in this note we study how the Ramsey number r(K_s, G) depends on the size of the graph G. For s \\geq 3, we prove that for every G with m edges, r(K_s,G) \\geq c (m/\\log m)^{\\frac{s+1}{s+3}} for some positive constant c depending only on s. This lower bound improves an earlier result of Erdos, Faudree, Rousseau, and Schelp, and is tight up to a polylogarithmic factor when s=3. We also study the maximum value of r(K_s,G) as a function of m."}
{"category": "Math", "title": "Local resilience of graphs", "abstract": "In this paper, we initiate a systematic study of graph resilience. The (local) resilience of a graph G with respect to a property P measures how much one has to change G (locally) in order to destroy P. Estimating the resilience leads to many new and challenging problems. Here we focus on random and pseudo-random graphs and prove several sharp results."}
{"category": "Math", "title": "Induced Ramsey-type theorems", "abstract": "We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers."}
{"category": "Math", "title": "Effective Termination of Kohn's Algorithm for Subelliptic Multipliers", "abstract": "This note discusses the problem of the effective termination of Kohn's algorithm for subelliptic multipliers for bounded smooth weakly pseudoconvex domains of finite type. We give a complete proof for the case of special domains of finite type and indicate briefly how this method is to be extended to the case of general bounded smooth weakly pseudoconvex domains of finite type."}
{"category": "Math", "title": "On some power sum problems of Montgomery and Turan", "abstract": "We use an estimate for character sums over finite fields of Katz to solve open problems of Montgomery and Turan. Let h=>2 be an integer. We prove that inf_{|z_k| => 1} max_{v=1,...,n^h} |sum_{k=1}^n z_k^v| <= (h-1+o(1)) sqrt n. This gives the right order of magnitude for the quantity and improves on a bound of Erdos-Renyi by a factor of the order sqrt log n."}
{"category": "Math", "title": "Bounds on the number of real solutions to polynomial equations", "abstract": "We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large."}
{"category": "Math", "title": "Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization", "abstract": "The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization."}
{"category": "Math", "title": "The Borel conjecture for manifolds with virtually solvable fundamental groups", "abstract": "The article has been withdrawn by the author. Wolfgang Lueck and Peter Linnell pointed out that the proof of Lemma 3.8 does not apply to the unrestricted case of wreath product. It is not clear at this stage how to complete the proof of Theorem 3.1 using the present version of Lemma 3.8. The valid results originating from this article will be added in a later paper."}
{"category": "Math", "title": "Adaptive dynamics in logistic branching populations", "abstract": "We consider a trait-structured population subject to mutation, birth and competition of logistic type, where the number of coexisting types may fluctuate. Applying a limit of rare mutations to this population while keeping the population size finite leads to a jump process, the so-called `trait substitution sequence', where evolution proceeds by successive invasions and fixations of mutant types. The probability of fixation of a mutant is interpreted as a fitness landscape that depends on the current state of the population. It was in adaptive dynamics that this kind of model was first invented and studied, under the additional assumption of large population. Assuming also small mutation steps, adaptive dynamics' theory provides a deterministic ODE approximating the evolutionary dynamics of the dominant trait of the population, called `canonical equation of adaptive dynamics'. In this work, we want to include genetic drift in this models by keeping the population finite. Rescaling mutation steps (weak selection) yields in this case a diffusion on the trait space that we call `canonical diffusion of adaptive dynamics', in which genetic drift (diffusive term) is combined with directional selection (deterministic term) driven by the fitness gradient. Finally, in order to compute the coefficients of this diffusion, we seek explicit first-order formulae for the probability of fixation of a nearly neutral mutant appearing in a resident population. These formulae are expressed in terms of `invasibility coefficients' associated with fertility, defense, aggressiveness and isolation, which measure the robustness (stability w.r.t. selective strengths) of the resident type. Some numerical results on the canonical diffusion are also given."}
{"category": "Math", "title": "Decay estimates of a tangential derivative to the light cone for the wave equation and their application", "abstract": "We consider wave equations in three space dimensions, and obtain new weighted $L^\\infty$-$L^\\infty$ estimates for a tangential derivative to the light cone. As an application, we give a new proof of the global existence theorem, which was originally proved by Klainerman and Christodoulou, for systems of nonlinear wave equations under the null condition. Our new proof has the advantage of using neither the scaling nor the pseudo-rotation operators."}
{"category": "Math", "title": "Explicit formulas for biharmonic submanifolds in Sasakian space forms", "abstract": "We classify the biharmonic Legendre curves in a Sasakian space form, and obtain their explicit parametric equations in the $(2n+1)$-dimensional unit sphere endowed with the canonical and deformed Sasakian structures defined by Tanno. Then, composing with the flow of the Reeb vector field, we transform a biharmonic integral submanifold into a biharmonic anti-invariant submanifold. Using this method we obtain new examples of biharmonic submanifolds in spheres and, in particular, in $\\mathbb{S}^{7}$."}
{"category": "Math", "title": "Harmonic analysis of additive Levy processes", "abstract": "Let $X_1,...,X_N$ denote $N$ independent $d$-dimensional L\\'evy processes, and consider the $N$-parameter random field \\[\\X(\\bm{t}):= X_1(t_1)+...+X_N(t_N).\\] First we demonstrate that for all nonrandom Borel sets $F\\subseteq\\R^d$, the Minkowski sum $\\X(\\R^N_+)\\oplus F$, of the range $\\X(\\R^N_+)$ of $\\X$ with $F$, can have positive $d$-dimensional Lebesgue measure if and only if a certain capacity of $F$ is positive. This improves our earlier joint effort with Yuquan Zhong \\ycite{KXZ:03} by removing a symmetry-type condition there. Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical [non-probabilistic] harmonic analysis that might be of independent interest. As was shown in \\fullocite{KXZ:03}, the potential theory of the type studied here has a large number of consequences in the theory of L\\'evy processes. We present a few new consequences here."}
{"category": "Math", "title": "\\C-flows A^z of linear maps A expressed in terms of A^{-1},A^{-2},...,A^{-n} and analytic functions of z", "abstract": "Suppose A\\in GL_n(\\C) has a relation A^p=c_{p-1}A^{p-1}+.... + c_1 A+ c_0I where the c_i in \\C. This article describes how to construct analytic functions c_i(z) such that A^z=c_{p-1}(z)A^{p-1}+... + c_1(z) A+ c_0(z)I . One of the theorems gives a possible description of the c_i(z): c_i(z)=C^z\\alpha where C\\in Mat_p(\\C) is (similar to) the companion matrix of X^p-c_{p-1}X^{p-1}-... -c_1X-c_0I, and \\alpha:= (c_{p-1},...,c_1,c_0)^t."}
{"category": "Math", "title": "Lattice polytopes of degree 2", "abstract": "A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimensions. In particular, there is only a finite number of quadratic polynomials with fixed leading coefficient being the $h^*$-polynomial of a lattice polytope."}
{"category": "Math", "title": "A characterization property on field equivalent to algebraicity on Banach spaces", "abstract": "In his article \"A discrete form of the theorem that each field endomorphism of $\\mathbb{R}$ ($\\mathbb{Q}_p$) is the identity\", Tyszka introduce a logical property which is equivalent to algebraicity in $\\mathbb{R}$ and in $\\mathbb{Q}_p$. Amazingly, the property is no longer equivalent to algebraicity in $\\mathbb{C}$. This article present a similirar property which is equivalent to algebraicity in any field of characteristic zero which is also a Banach space, and prove a weaker equivalency for fields of positive charcteristic (which are also Banach spaces)."}
{"category": "Math", "title": "Infinitely generated Derksen and Makar-Limanov invariant", "abstract": "In this paper, we give an example of a finitely generated 3-dimensional C-algebra which has infinitely generated Derksen invariant as well as Makar-Limaonv invariant."}
{"category": "Math", "title": "Constructing (almost) rigid rings and a UFD having infinitely generated Derksen and Makar-Limanov invariant", "abstract": "An example is given of a UFD which has infinitely generated Derksen invariant. The ring is \\textquotedblleft almost rigid\\textquotedblright\\ meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem."}
{"category": "Math", "title": "Zero-Bidimension and Various Classes of Bitopological Spaces", "abstract": "The sum theorem and its corollaries are proved for a countable family of zero-dimensional (in the sense of small and large inductive bidimensions) p-closed sets, using a new notion of relative normality whose topological correspondent is also new. The notion of almost $n$-dimensionality is considered from the bitopological point of view. Bitopological spaces in which every subset is i-open in its $j$-closure (i.e.,(i,j)-submaximal spaces) are introduced and their properties are studied. Based on the investigations begun in [5] and [14], sufficient conditions are found for bitopological spaces to be(1,2)-Baire in the class of p-normal spaces. Furthermore, (i,j)-I-spaces are introduced and both the relations between(i,j)-submaximal, (i,j)-nodec and (i,j)-I-spaces, and their properties are studied when two topologies on a set are either independent of each other or interconnected by the inclusion, S-, C- and N-relations or by their combinations. The final part of the paper deals with the questions of preservation of $(i,j)$-submaximal and $(2,1)\\dd I$-spaces to an image, of $D$-spaces to an image and an inverse image for both the topological and the bitopological cases. Two theorems are formulated containing, on the one hand, topological conditions and, on the other hand, bitopological ones, under which a topological space is a $D$-space."}
{"category": "Math", "title": "The Automorphism Group of Certain Factorial Threefolds and a Cancellation Problem", "abstract": "The automorphism groups of certain factorial complex affine threefolds admitting locally trivial actions of the additive group are determined. As a consequence new counterexamples to a generalized cancellation problem are obtained."}
{"category": "Math", "title": "SiZer for time series: A new approach to the analysis of trends", "abstract": "Smoothing methods and SiZer are a useful statistical tool for discovering statistically significant structure in data. Based on scale space ideas originally developed in the computer vision literature, SiZer (SIgnificant ZERo crossing of the derivatives) is a graphical device to assess which observed features are `really there' and which are just spurious sampling artifacts. In this paper, we develop SiZer like ideas in time series analysis to address the important issue of significance of trends. This is not a straightforward extension, since one data set does not contain the information needed to distinguish `trend' from `dependence'. A new visualization is proposed, which shows the statistician the range of trade-offs that are available. Simulation and real data results illustrate the effectiveness of the method."}
{"category": "Math", "title": "Transportation-information inequalities for Markov processes", "abstract": "In this paper, one investigates the following type of transportation-information $T_cI$ inequalities: $\\alpha(T_c(\\nu,\\mu))\\le I(\\nu|\\mu)$ for all probability measures $\\nu$ on some metric space $(\\XX, d)$, where $\\mu$ is a given probability measure, $T_c(\\nu,\\mu)$ is the transportation cost from $\\nu$ to $\\mu$ with respect to some cost function $c(x,y)$ on $\\XX^2$, $I(\\nu|\\mu)$ is the Fisher-Donsker-Varadhan information of $\\nu$ with respect to $\\mu$ and $\\alpha: [0,\\infty)\\to [0,\\infty]$ is some left continuous increasing function. Using large deviation techniques, it is shown that $T_cI$ is equivalent to some concentration inequality for the occupation measure of a $\\mu$-reversible ergodic Markov process related to $I(\\cdot|\\mu)$, a counterpart of the characterizations of transportation-entropy inequalities, recently obtained by Gozlan and L\\'eonard in the i.i.d. case . Tensorization properties of $T_cI$ are also derived."}
{"category": "Math", "title": "On the Theory of Relative Bitopological and Topological Properties", "abstract": "In the first part of the work (Sections 2-6) a special attention is given to relative separation axioms and relative connectedness, in particular, many relative versions of p-T_0, p-T_1, p-T_2, (i,j)- and p-regularities, (i,j)- and p-complete regularities, p-real normality and p-normality are discussed. Moreover, relative properties of (i,j)- and p-compactness types, including relative versions of (i,j)- and p-paracompactness, (i,j)- and p-Lindeofness, (i,j)- and p-pseudocompactness are also introduced and investigated. The second part (Sections 7-12) is devoted, on the one hand, to relative bitopological inductive and covering dimension functions and, on the other hand, to relative versions of Baire spaces for both the topological and the bitopological case. At the end, note that relative (bi)topological properties play a special role not only in the development of respective theories, but also in the strengthening of the previously known results."}
{"category": "Math", "title": "Optimal curves of low genus over finite fields", "abstract": "The Hasse-Weil-Serre bound is improved for curves of low genera over finite fields with discriminant in {-3,-4,-7,-8,-11,-19} by studying optimal curves."}
{"category": "Math", "title": "The extended Burnside ring and module categories", "abstract": "In this note an `extended Burnside ring' is defined, generated by classes of semisimple module categories over Rep(G) with quasifibre functors. Here G is a finite group and representations are taken over an algebraically closed field of characteristic 0. It is shown that this is equivalent to a ring generated by centrally extended G-sets and hence the name. Ring homomorphisms into the multiplicative group of the field are computed with an explicit formula and tables of these homomorphisms are given for the groups S_4 and S_5 which are of particular interest in the context of reductive algebraic groups. ----- L'anneau de Burnside \\'etendu et cat\\'egories de modules. Dans cette note un `Anneau de Burnside \\'etendu' est d\\'efini, gener\\'e par des classes de cat\\'egories de modules semisimples sur Rep(G) avec des foncteurs quasifibres. Ici G est un groupe fini, et des repr\\'esentations sont prises sur un corps alg\\'ebriquement clos de caract\\'eristique nulle. Il est demontr\\'e que ceci \\'equivaut \\`a un anneau gener\\'e par des G-ensembles centralement \\'etendus, d'o\\`u le nom. Des homomorphismes d'anneau dans le groupe multiplicatif du corps sont comput\\'ees avec une formule explicite et des tableaux de ces homomorphismes sont fournis pour les groupes S_4 et S_5 qui sont d'un int\\'er\\^et particulier dans le contexte de groupes alg\\'ebriques r\\'eductifs."}
{"category": "Math", "title": "Automorphic Forms and Reeb-Like Foliations on Three-Manifolds", "abstract": "In this paper, we consider different ways of generating dynamical systems on 3-manifolds. We first derive explicit differential equations for dynamical systems defined on generic hyperbolic 3-manifolds by using automorphic function theory to uniformize the upper half-space model. It is achieved via the modification of the standard Poincare theta series to generate systems invariant within each individual fundamental region such that the solution trajectories match up on the appropriate sides after the identifications which generate a hyperbolic 3-manifold. Then we consider the gluing pattern in the conformal ball model. At the end we shall study the construction of dynamical systems by using the Reeb foliation."}
{"category": "Math", "title": "Dynamical Systems On Three Manifolds Part II: 3-Manifolds,Heegaard Splittings and Three-Dimensional Systems", "abstract": "The global behaviour of nonlinear systems is extremely important in control and systems theory since the usual local theories will only give information about a system in some neighbourhood of an operating point. Away from that point, the system may have totally different behaviour and so the theory developed for the local system will be useless for the global one. In this paper we shall consider the analytical and topological structure of systems on 2- and 3- manifolds and show that it is possible to obtain systems with 'arbitrarily strange' behaviour, i.e., arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We shall do this by considering Heegaard Splittings of these manifolds and the resulting systems defined on the boundaries."}
{"category": "Math", "title": "Topology of the regular part for infinitely renormalizable quadratic polynomials", "abstract": "In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable quadratic polynomials and prove that when they satisfy a-priori bounds, the topology is rigid modulo its combinatorics."}
{"category": "Math", "title": "On the methods to construct UFD counterexamples to a cancellation problem", "abstract": "In a previous paper, the author together with prof. dr. Finston constructed a class of UFDs A_{n,m} where n,m\\in \\N^*. These rings are all stably equivalent (A_{n,m}[T]\\cong A_{p,q}[T] for all n,m,p,q) but are only isomorphic themselves if (n,m)=(p,q). These examples are the first UFD examples over a characteristically closed field satisfying this behavior. In this paper, we describe the methods used in this article, and show that they are very general, enabling the reader to construct many more such examples, based on the same principles."}
{"category": "Math", "title": "On the set of complex points of a 2-sphere", "abstract": "Let $G$ be a strictly pseudoconvex domain in $\\mathbb{C}^2$ with $C^\\infty$-smooth boundary $\\partial G$. Let $S$ be a 2-dimensional sphere embedded into $\\partial G$. Denote by $\\mathcal{E}$ the set of all complex points on $S$. We study how the structure of the set $\\mathcal{E}$ depends on the smoothness of $S$"}
{"category": "Math", "title": "Symmetry in semidefinite programs", "abstract": "This paper is a tutorial in a general and explicit procedure to simplify semidefinite programs which are invariant under the action of a symmetry group. The procedure is based on basic notions of representation theory of finite groups. As an example we derive the block diagonalization of the Terwilliger algebra of the binary Hamming scheme in this framework. Here its connection to the orthogonal Hahn and Krawtchouk polynomials becomes visible."}
{"category": "Math", "title": "Immediate Calculation of some Poisson Type Integrals Using Supermathematics Circular Ex-Centric Functions", "abstract": "This article presents two methods, in parallel, of solving more complex integrals, among which is the Poisson's integral, in order to emphasize the obvious advantages of a new method of integration, which uses the supermathematics circular ex-centric functions."}
{"category": "Math", "title": "Quantum Unique Ergodicity for Eisenstein Series on the Hilbert Modular Group over a Totally Real Field", "abstract": "W. Luo and P. Sarnak have proved the quantum unique ergodicity property for Eisenstein series on $\\rm{PSL}(2,\\mathbb{Z}) \\backslash H$. We extend their result to Eisenstein series on $\\rm{PSL}(2,O) \\backslash H^n$, where $O$ is the ring of integers in a totally real field of degree $n$ over $Q$ with narrow class number one, using the Eisenstein series considered by I. Efrat. We also give an expository treatment of the theory of Hecke operators on non-holomorphic Hilbert modular forms."}
{"category": "Math", "title": "Subgroup separability in residually free groups", "abstract": "We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type $\\mathrm{FP}_\\infty$ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups."}
{"category": "Math", "title": "Isoperimetric profile and random walks on locally compact solvable groups", "abstract": "We study a large class of amenable locally compact groups containing all solvable algebraic groups over a local field and their discrete subgroups. We show that the isoperimetric profile of these groups is in some sense optimal among amenable groups. We use this fact to compute the probability of return of symmetric random walks, and to derive various other geometric properties which are likely to be only satisfied by these groups."}
{"category": "Math", "title": "A mixed problem for the infinity laplacian via Tug-of-War games", "abstract": "In this paper we prove that a function $ u\\in\\mathcal{C}(\\bar{\\Omega})$ is the continuous value of the Tug-of-War game described in \\cite{PSSW} if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions {-\\Delta_{\\infty}u(x)=0\\quad & \\text{in} \\Omega, \\frac{\\partial u}{\\partial n}(x)=0\\quad & \\text{on} \\Gamma_N, u(x)=F(x)\\quad & \\text{on} \\Gamma_D. By using the results in \\cite{PSSW}, it follows that this viscous PDE problem has a unique solution, which is the unique {\\it absolutely minimizing Lipschitz extension} to the whole $\\bar{\\Omega}$ (in the sense of \\cite{Aronsson} and \\cite{PSSW}) of the boundary data $ F:\\Gamma_D\\to\\R $."}
{"category": "Math", "title": "Theory of the Siegel Modular Variety", "abstract": "In this paper, we discuss the theory of the Siegel modular variety in the aspects of arithmetic and geometry. This article covers the theory of Siegel modular forms, the Hecke theory, a lifting of elliptic cusp forms, geometric properties of the Siegel modular variety, (hypothetical) motives attached to Siegel modular forms and a cohomology of the Siegel modular variety."}
{"category": "Math", "title": "Diagonal fibrations are pointwise fibrations", "abstract": "On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, the diagonal fibrations. In this note, we prove that every diagonal fibration is a pointwise fibration."}
{"category": "Math", "title": "Riesz integral representation theory", "abstract": "We present a Riesz integral representation theory in which functions, operators and measures take values in uniform commutative monoids (a commutative monoid with a uniformity making the binary operation of the monoid uniformly continuous). It describes the operators to which the theory can be applied and the finitely-additive measures they generate. Operators satisfying the conditions will be called ``Riesz integrals''. Given an underlying ``Riesz system'', it is shown that every Riesz integral generates a certain kind of finitely additive measure called here a ``Riesz measure''. The correspondence between Riesz integrals and Riesz measures is a bijection. A straightforward calculation shows that if an operator has such a representation, then it must have the Hammerstein property. For topological vector spaces, the theory yields necessary and sufficient conditions for operators with the Hammerstein property to be Riesz integrals. We note that uniform commutative monoids arise naturally when considering set-valued functions, and that the axioms of a Riesz system rule out certain spaces of infinitely differentiable functions."}
{"category": "Math", "title": "On the Ring of Integer-valued Quasi-polynomials", "abstract": "The paper studies some properties of the ring of integer-valued quasi-polynomials. On this ring, theory of generalized Euclidean division and generalized GCD are presented. Applications to finite simple continued fraction expansion and Smith normal form of integral matrices with integer parameters are also given."}
{"category": "Math", "title": "Schwarzian Derivatives and Uniform Local Univalence", "abstract": "Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings."}
{"category": "Math", "title": "Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints", "abstract": "Regularization of ill-posed linear inverse problems via $\\ell_1$ penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an $\\ell_1$ penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to $\\ell_1$-constraints, using a gradient method, with projection on $\\ell_1$-balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration."}
{"category": "Math", "title": "A Metric on Shape Space with Explicit Geodesics", "abstract": "This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc...) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided"}
{"category": "Math", "title": "Moments from their very truncations", "abstract": "It is known that positive definiteness is not enough for the multidimensional moment problem to be solved. We would like throw in to the garden of existing in this matter so far results one more, a result which takes into considerations the utmost possible truncations."}
{"category": "Math", "title": "Smooth models of quiver moduli", "abstract": "For any moduli space of stable representations of quivers, certain smooth varieties, compactifying projective space fibrations over the moduli space, are constructed. The boundary of this compactification is analyzed. Explicit formulas for the Betti numbers of the smooth models are derived. In the case of moduli of simple representations, explicit cell decompositions of the smooth models are constructed."}
{"category": "Math", "title": "Reducibility of the polynomial representation of the degenerate double affine Hecke algebra", "abstract": "In this note we determine the values of parameters c for which the polynomial representation of the degenerate double affine Hecke algebra (DAHA), i.e. the trigonometric Cherednik algebra, is reducible. Namely, we show that c is a reducibility point for the polynomial representation of the trigonometric Cherednik algebra for a root system R if and only if it is a reducibility point for the rational Cherednik algebra for the Weyl group of some root subsystem R' of R of the same rank; such subsystems for any R are given by the well known Borel-de Siebenthal algorithm. This result has been proved by Cherednik using a case-by-case method."}
{"category": "Math", "title": "Cutoff Resolvent Estimates and the Semilinear Schr\\\"odinger Equation", "abstract": "This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schr\\\"odinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in regularity in the local smoothing estimate. As an application, we apply well-known techniques to obtain well-posedness results for the semi-linear Schr\\\"odinger equation."}
{"category": "Math", "title": "The age grading and the Chen-Ruan cup product", "abstract": "We prove that the obstruction bundle used to define the cup-product in Chen-Ruan cohomology is determined by the so-called `age grading' or `degree-shifting numbers'. Indeed, the obstruction bundle can be directly computed using the age grading. We obtain a Kunneth Theorem for Chen-Ruan cohomology as a direct consequence of an elementary property of the age grading, and explain how several other results - including associativity of the cup-product - can be proved in a similar way."}
{"category": "Math", "title": "Power Loss for Inhomogeneous Poisson Processes", "abstract": "In this work, based on a realization of an inhomogeneous Poisson process whose intensity function depends on a real unknown parameter, we consider a simple hypothesis against a sequence of close (contiguous) alternatives. Under certain regularity conditions we obtain the power loss of the score test with respect to the Neyman-Pearson test. The power loss measures the performance of a second order efficient test by the help of third order asymptotic properties of the problem under consideration."}
{"category": "Math", "title": "Convergence properties of Donaldson's $T$-iterations on the Riemann sphere", "abstract": "In a recent paper Donaldson defines three operators on a space of Hermitian metrics on a complex projective manifold: $T, T_{\\nu}, T_K.$ Iterations of these operators converge to balanced metrics, and these themselves approximate constant scalar curvature metrics. In this paper we investigate the convergence properties of these iterations by examining the case of the Riemann sphere as well as higher dimensional $\\mathbb{CP}^n$."}
{"category": "Math", "title": "A note on p-adic q-integrals associated with q-Euler numbers", "abstract": "In this we give a detailed proof of fermionic p-adic q-measures on Z_p and we will treat some interesting formulae related q-extension of Euler numbers and polynomials."}
{"category": "Math", "title": "Hausdorff Dimension and Hausdorff Measure for Non-integer based Cantor-type Sets", "abstract": "We consider digits-deleted sets or Cantor-type sets with $\\beta$-expansions. We calculate the Hausdorff dimension $d$ of these sets and show that $d$ is continuous with respect to $\\beta$. The $d$-dimentional Hausdorff measure of these sets is finite and positive."}
{"category": "Math", "title": "2-Selmer Groups and the Birch-Swinnerton-Dyer Conjecture for the Congruent Number Curve", "abstract": "We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the ``independence'' of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors."}
{"category": "Math", "title": "A fast direct solver for network matrices", "abstract": "A fast direct inversion scheme for the large sparse systems of linear equations resulting from the discretization of elliptic partial differential equations in two dimensions is given. The scheme is described for the particular case of a discretization on a uniform square grid, but can be generalized to more general geometries. For a grid containing $N$ points, the scheme requires $O(N \\log^{2}N)$ arithmetic operations and $O(N \\log N)$ storage to compute an approximate inverse. If only a single solve is required, then the scheme requires only $O(\\sqrt{N} \\log N)$ storage; the same storage is sufficient for computing the Dirichlet-to-Neumann operator as well as other boundary-to-boundary operators. The scheme is illustrated with several numerical examples. For instance, a matrix of size $10^6 \\times 10^6$ is inverted to seven digits accuracy in four minutes on a 2.8GHz P4 desktop PC."}
{"category": "Math", "title": "Geometric Invariant Theory via Cox Rings", "abstract": "We consider actions of reductive groups on a varieties with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description of the chamber structure of the linearized ample cone and several Gelfand-MacPherson type correspondences relating quotients of reductive groups to quotients of torus actions. Moreover, our approach provides information on the geometry of many of the resulting quotient spaces."}
{"category": "Math", "title": "A counterexample to generalizations of the Milnor-Bloch-Kato conjecture", "abstract": "We construct an example of a torus $T$ over a field $K$ for which the Galois symbol $K(K; T,T)/n K(K; T,T) \\to H^2(K, T[n]\\otimes T[n])$ is not injective for some $n$. Here $K(K; T,T)$ is the Milnor $K$-group attached to $T$ introduced by Somekawa. We show also that the motive $M(T\\times T)$ gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson)."}
{"category": "Math", "title": "On Wahl's proof of $\\mu(6)=65$", "abstract": "D. Jaffe and D. Ruberman proved in 1997 that a sextic hypersurface in $\\mathbb{P}^3$ has at most 65 nodes (the bound is sharp by Barth's construction). Almost at the same time, J. Wahl proposed a much shorter proof of the same result, by proving that a linear code $V\\subset \\F^{66}$ with weights in $\\{24,32,40\\}$ has dimension $\\dim(V)\\leq12$. He claimed that Jaffe-Ruberman's theorem follows as a corollary since the code associated to a sextic with n nodes has dimension at least $n-53$ and an incorrect result stated by Casnati and Catanese asserted that the possible cardinalities of an even set of nodes on a sextic were only 24, 32 and 40. Recently Catanese and Tonoli showed that the possible cardinalities of an even set of nodes on a sextic are exactly 24, 32, 40, 56. According to the above cardinalities, the theorem of Jaffe and Ruberman reduces to the following: Let $V\\subset \\F^{66}$ be a code with weights in $\\{24,32,40,56\\}$. Then $\\dim(V)\\leq12$. In this short note we give an elementary proof of this theorem using and integrating Wahl's ideas."}
{"category": "Math", "title": "Multi-dimensional BSDE with Oblique Reflection and Optimal Switching", "abstract": "In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimations. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As applications, we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type, and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities."}
{"category": "Math", "title": "Combinatorial and geometric methods in topology", "abstract": "Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central role played by hyperbolic geometry within three-dimensional topology. We will also point out the striking difference with the two-dimensional case, and we will review some of the results of the combinatorial and computational approach to three-manifolds developed by different mathematicians over the last several years."}
{"category": "Math", "title": "Quartic equations and 2-division on elliptic curves", "abstract": "Let K be a field of characteristic different from 2 and C an elliptic curve over K given by a Weierstrass equation. To divide an element of the group C by 2, one must solve a certain quartic equation. We characterise the quartics arising from this procedure and find how far the quartic determines the curve and the point. We find the quartics coming from 2-division of 2- and 3-torsion points, and generalise this correspondence to singular plane cubics. We use these results to study the question of which degree 4 maps of curves can be realised as duplication of a multisection on an elliptic surface."}
{"category": "Math", "title": "On Hamiltonian stationary Lagrangian spheres in non-Einstein Kaehler surfaces", "abstract": "Hamiltonian stationary Lagrangian spheres in Kaehler-Einstein surfaces are minimal. We prove that in the family of non-Einstein Kaehler surfaces given by the product $\\Sigma_1\\times\\Sigma_2$ of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example is defined when the surfaces $\\Sigma_1$ and $ \\Sigma_2$ are spheres."}
{"category": "Math", "title": "Improvements on removing non-optimal support points in D-optimum design algorithms", "abstract": "We improve the inequality used in Pronzato [2003. Removing non-optimal support points in D-optimum design algorithms. Statist. Probab. Lett. 63, 223-228] to remove points from the design space during the search for a $D$-optimum design. Let $\\xi$ be any design on a compact space $\\mathcal{X} \\subset \\mathbb{R}^m$ with a nonsingular information matrix, and let $m+\\epsilon$ be the maximum of the variance function $d(\\xi,\\mathbf{x})$ over all $\\mathbf{x} \\in \\mathcal{X}$. We prove that any support point $\\mathbf{x}_{*}$ of a $D$-optimum design on $\\mathcal{X}$ must satisfy the inequality $d(\\xi,\\mathbf{x}_{*}) \\geq m(1+\\epsilon/2-\\sqrt{\\epsilon(4+\\epsilon-4/m)}/2)$. We show that this new lower bound on $d(\\xi,\\mathbf{x}_{*})$ is, in a sense, the best possible, and how it can be used to accelerate algorithms for $D$-optimum design."}
{"category": "Math", "title": "Tightness of voter model interfaces", "abstract": "Consider a long-range, one-dimensional voter model started with all zeroes on the negative integers and all ones on the positive integers. If the process obtained by identifying states that are translations of each other is positively recurrent, then it is said that the voter model exhibits interface tightness. In 1995, Cox and Durrett proved that one-dimensional voter models exhibit interface tightness if their infection rates have a finite third moment. Recently, Belhaouari, Mountford, and Valle have improved this by showing that a finite second moment suffices. The present paper gives a new short proof of this fact. We also prove interface tightness for a long range swapping voter model, which has a mixture of long range voter model and exclusion process dynamics."}
{"category": "Math", "title": "Relaxation Enhancement by Time-Periodic Flows", "abstract": "We study enhancement of diffusive mixing by fast incompressible time-periodic flows. The class of relaxation-enhancing flows that are especially efficient in speeding up mixing has been introduced in [2]. The relaxation-enhancing property of a flow has been shown to be intimately related to the properties of the dynamical system it generates. In particular, time-independent flows $u$ such that the operator $u \\cdot \\nabla$ has sufficiently smooth eigenfunctions are not relaxation-enhancing. Here we extend results of [2] to time-periodic flows $u(x,t)$ and in particular show that there exist flows such that for each fixed time the flow is Hamiltonian, but the resulting time-dependent flow is relaxation-enhancing. Thus we confirm the physical intuition that time dependence of a flow may aid mixing. We also provide an extension of our results to the case of a nonlinear diffusion model. The proofs are based on a general criterion for the decay of a semigroup generated by an operator of the form $\\Gamma+iAL(t)$ with a negative unbounded self-adjoint operator $\\Gamma$, a time-periodic self-adjoint operator-valued function $L(t)$, and a parameter $A>>1$."}
{"category": "Math", "title": "Some Two Color, Four Variable Rado Numbers", "abstract": "There exists a minimum integer $N$ such that any 2-coloring of $\\{1,2,...,N\\}$ admits a monochromatic solution to $x+y+kz =\\ell w$ for $k,\\ell \\in \\mathbb{Z}^+$, where $N$ depends on $k$ and $\\ell$. We determine $N$ when $\\ell-k \\in \\{0,1,2,3,4,5\\}$, for all $k,\\ell$ for which ${1/2}((\\ell-k)^2-2)(\\ell-k+1)\\leq k \\leq \\ell-4$, as well as for arbitrary $k$ when $\\ell=2$."}
{"category": "Math", "title": "Bounds on Van der Waerden Numbers and Some Related Functions", "abstract": "For positive integers $s$ and $k_1, k_2, ..., k_s$, let $w(k_1,k_2,...,k_s)$ be the minimum integer $n$ such that any $s$-coloring $\\{1,2,...,n\\} \\to \\{1,2,...,s\\}$ admits a $k_i$-term arithmetic progression of color $i$ for some $i$, $1 \\leq i \\leq s$. In the case when $k_1=k_2=...=k_s=k$ we simply write $w(k;s)$. That such a minimum integer exists follows from van der Waerden's theorem on arithmetic progressions. In the present paper we give a lower bound for $w(k,m)$ for each fixed $m$. We include a table with values of $w(k,3)$ which match this lower bound closely for $5 \\leq k \\leq 16$. We also give an upper bound for $w(k,4)$, an upper bound for $w(4;s)$, and a lower bound for $w(k;s)$ for an arbitrary fixed $k$. We discuss a number of other functions that are closely related to the van der Waerden function."}
{"category": "Math", "title": "A presentation for Hilden's subgroup of the braid group", "abstract": "Consider the unit ball, B = D x [0,1], containing n unknotted arcs a_1, ... a_n such that the boundary of each a_i lies in D x {0}. We give a finite presentation for the mapping class group of B fixing the arcs {a_1, ..., a_n} setwise and fixing D x {1} pointwise. This presentation is calculated using the action of this group on a simply-connected complex."}
{"category": "Math", "title": "Free $n$-distributions: holonomy, sub-Riemannian structures, Fefferman constructions and dual distributions", "abstract": "This paper analyses the parabolic geometries generated by a free $n$-distribution in the tangent space of a manifold. It shows that certain holonomy reductions of the associated normal Tractor connections, imply preferred connections with special properties, along with Riemannian or sub-Riemannian structures on the manifold. It constructs examples of these holonomy reductions in the simplest cases. The main results, however, lie in the free 3-distributions. In these cases, there are normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively. There is also a fascinating construction of a `dual' distribution when the holonomy reduces to $G_2'$."}
{"category": "Math", "title": "An optimization problem with volume constrain in Orlicz spaces", "abstract": "We consider the optimization problem of minimizing $\\int_{\\Omega}G(|\\nabla u|) dx$ in the class of functions $W^{1,G}(\\Omega)$, with a constrain on the volume of $\\{u>0\\}$. The conditions on the function $G$ allow for a different behavior at 0 and at $\\infty$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution $u$ is locally Lipschitz continuous and that the free boundary, $\\partial\\{u>0\\}\\cap \\Omega$, is smooth."}
{"category": "Math", "title": "Pure motives, mixed motives and extensions of motives associated to singular surfaces", "abstract": "We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface and study its fundamental properties. Using Voevodsky's category of effective geometrical motives, we then study the motive of the exceptional divisor in a non-singular blow-up. If all geometric irreducible components of the divisor are of genus zero, then Voevodsky's formalism allows us to construct certain one-extensions of Chow motives, as canonical sub-quotients of the motive with compact support of the smooth part of the surface. Specializing to Hilbert--Blumenthal surfaces, we recover a motivic interpretation of a recent construction of A. Caspar."}
{"category": "Math", "title": "Cheeger constants of surfaces and isoperimetric inequalities", "abstract": "We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\\sqrt t$, then it grows at least as fast as a linear function. This generalizes a result of Gromov for simply connected surfaces. We study the isoperimetric problem in dimension 3. We show that if the filling volume function in dimension 2 is Euclidean, while in dimension 3 is sub-Euclidean and there is a $g$ such that minimizers in dimension 3 have genus at most $g$, then the filling function in dimension 3 is `almost' linear."}
{"category": "Math", "title": "Quadratic optimal functional quantization of stochastic processes and numerical applications", "abstract": "In this paper, we present an overview of the recent developments of functional quantization of stochastic processes, with an emphasis on the quadratic case. Functional quantization is a way to approximate a process, viewed as a Hilbert-valued random variable, using a nearest neighbour projection on a finite codebook. A special emphasis is made on the computational aspects and the numerical applications, in particular the pricing of some path-dependent European options."}
{"category": "Math", "title": "Synchronization in networks of networks: the onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators", "abstract": "The onset of synchronization in networks of networks is investigated. Specifically, we consider networks of interacting phase oscillators in which the set of oscillators is composed of several distinct populations. The oscillators in a given population are heterogeneous in that their natural frequencies are drawn from a given distribution, and each population has its own such distribution. The coupling among the oscillators is global, however, we permit the coupling strengths between the members of different populations to be separately specified. We determine the critical condition for the onset of coherent collective behavior, and develop the illustrative case in which the oscillator frequencies are drawn from a set of (possibly different) Cauchy-Lorentz distributions. One motivation is drawn from neurobiology, in which the collective dynamics of several interacting populations of oscillators (such as excitatory and inhibitory neurons and glia) are of interest."}
{"category": "Math", "title": "The Popescu-Gabriel theorem for triangulated categories", "abstract": "The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topological triangulated categories has recently been obtained by A. Heider."}
{"category": "Math", "title": "A note on the stability for Kawahara-KdV type equations", "abstract": "In this paper we establish the nonlinear stability of solitary traveling-wave solutions for the Kawahara-KdV equation $$u_t+uu_x+u_{xxx}-\\gamma_1 u_{xxxxx}=0,$$ and the modified Kawahara-KdV equation $$u_t+3u^2u_x+u_{xxx}-\\gamma_2 u_{xxxxx}=0,$$ where $\\gamma_i\\in\\mathbb{R}$ is a positive number when $i=1,2$. The main approach used to determine the stability of solitary traveling-waves will be the theory developed by Albert"}
{"category": "Math", "title": "On a Teichmueller functor between the categories of complex tori and the Effros-Shen algebras", "abstract": "A covariant functor from the category of the complex tori to the category of the Effros-Shen algebras is constructed. The functor maps isomorphic complex tori to the stably isomorphic Effros-Shen algebras. Our construction is based on the Teichmueller theory of the Riemann surfaces."}
{"category": "Math", "title": "An Algorithm to Construct A Basis for the Module of Logarithmic Vector Fields", "abstract": "We consider logarithmic vector fields parametrized by finite collections of weighted hyperplanes. For a finite collection of weighted hyperplanes in a two-dimensional vector space, it is known that the set of such vector fields is a free module of rank two whose basis elements are homogeneous. We give an algorithm to construct a homogeneous basis for the module."}
{"category": "Math", "title": "Invariant weighted algebras $L_p^w(G)$", "abstract": "The paper deals with weighted spaces $L_p^w(G)$ on a locally compact group G. If w is a positive measurable function on G then we define the space $L_p^w(G)$, $p\\ge1$, as $L_p^w(G)=\\{f:fw\\in L_p(G)\\}$. We consider weights such that these weighted spaces are algebras with respect to usual convolution. It is shown that for p>1 such weights exists on any sigma-compact group. We prove also a criterion known earlier in special cases: $L_1^w(G)$ is an algebra if and only if w is submultiplicative. It is proved that invariant algebras $L_p^w(G)$, $p>1$, have approximate units of standard form, but this may not be true for a non-invariant algebra."}
{"category": "Math", "title": "A Morse type uniqueness theorem for non-parametric minimizing hypersurfaces", "abstract": "A classical result about minimal geodesics on R^2 with Z^2 periodic metric that goes back to H.M. Morse asserts that a minimal geodesic that is asymptotic to a periodic minimal geodesic cannot intersect any periodic minimal geodesic of the same period. This paper treats a similar theorem for nonparametric minimizing hypersurfaces without selfintersections -- as were studied by J. Moser, V. Bangert, P.H. Rabinowitz, E. Stredulinsky and others."}
{"category": "Math", "title": "Lagrangian Mechanics and Variational Integrators on Two-Spheres", "abstract": "Euler-Lagrange equations and variational integrators are developed for Lagrangian mechanical systems evolving on a product of two-spheres. The geometric structure of a product of two-spheres is carefully considered in order to obtain global equations of motion. Both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints. We derive global expressions for the Euler-Lagrange equations on two-spheres which are more compact than existing equations written in terms of angles. Since the variational integrators are derived from Hamilton's principle, they preserve the geometric features of the dynamics such as symplecticity, momentum maps, or total energy, as well as the structure of the configuration manifold. Computational properties of the variational integrators are illustrated for several mechanical systems."}
{"category": "Math", "title": "The Boltzmann-Hamel Equations for Optimal Control", "abstract": "We extend the Boltzmann-Hamel equations to the optimal control setting, producing a set of equations for both kinematic and dynamic nonholonomic optimal control problems. In particular, we will show the dynamic optimal control problem can be written as a minimal set of 4n-2m first order differential equations of motion."}
{"category": "Math", "title": "Explicit Heegner Points: Kolyvagin's Conjecture and Non-trivial Elements in the Shafarevich-Tate Group", "abstract": "Kolyvagin used Heegner points to associate a system of cohomology classes to an elliptic curve over $\\Q$ and conjectured that the system contains a non-trivial class. His conjecture has profound implications on the structure of Selmer groups. We provide new computational and theoretical evidence for Kolyvagin's conjecture. More precisely, we explicitly compute Heegner points over ring class fields and use these points to verify the conjecture for specific elliptic curves of rank two. We explain how Kolyvagin's conjecture implies that if the analytic rank of an elliptic curve is at least two then the $\\Z_p$-corank of the corresponding Selmer group is at least two as well. We also use explicitly computed Heegner points to produce non-trivial classes in the Shafarevich-Tate group."}
{"category": "Math", "title": "The Existence of Type II Singularities for the Ricci Flow on $S^{n+1}$", "abstract": "In this paper we prove the existence of Type II singularities for the Ricci flow on $S^{n+1}$ for all $n\\geq 2$."}
{"category": "Math", "title": "Manifolds with Pointwise Ricci Pinched Curvature", "abstract": "In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor."}
{"category": "Math", "title": "A Simple Proof for the Generalized Frankel Conjecture", "abstract": "In this short paper, we will give a simple and transcendental proof for Mok's theorem of the generalized Frankel conjecture. This work is based on the maximum principle in \\cite{BS2} proposed by Brendle and Schoen."}
{"category": "Math", "title": "Cluster-tilted algebras and slices", "abstract": "We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the Auslander-Reiten quiver of the cluster-tilted algebra, which we call local slice."}
{"category": "Math", "title": "Modified A-hypergeometric Systems", "abstract": "We will introduce a modified system of A-hypergeometric system (GKZ system) by applying a change of variables for Groebner deformations and study its Groebner basis and the indicial polynomials along the \"exceptional hypersurface\"."}
{"category": "Math", "title": "The linear and non linear Rayleigh-Taylor instability for the quasi isobaric profile", "abstract": "We study the stability of the system of the Euler equation in the neighborhood of a stationary profile associated with the quasi isobaric model in a gravity field. This stationary profile is not bounded below, hence the operator is not coercive. We use this linear result to deduce a nonlinear result"}
{"category": "Math", "title": "Smooth K-Theory", "abstract": "We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We construct a multiplicative lift of the Chern character from smooth K-theory to smooth rational cohomology and verify that the cohomological version of the Atiyah-Singer index theorem for families lifts to smooth cohomology."}
{"category": "Math", "title": "Asymptotic Expansion of the One-Loop Approximation of the Chern-Simons Integral in an Abstract Wiener Space Setting", "abstract": "In an abstract Wiener space setting, we constract a rigorous mathematical model of the one-loop approximation of the perturbative Chern-Simons integral, and derive its explicit asymptotic expansion for stochastic Wilson lines."}
{"category": "Math", "title": "Quantum subgroups of a simple quantum group at roots of 1", "abstract": "Let G be a connected, simply connected, simple complex algebraic group and let e be a primitive l-th root of 1, with l odd and 3 does not divide l if G is of type G_{2}. We determine all Hopf algebra quotients of the quantized coordinate algebra of G at e."}
{"category": "Math", "title": "Herman's Theory Revisited", "abstract": "We prove that a $C^{2+\\alpha}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_\\delta$, $0<\\delta<\\alpha\\le1$, is $C^{1+\\alpha-\\delta}$-smoothly conjugate to a rigid rotation. We also derive the most precise version of Denjoy's inequality for such diffeomorphisms."}
{"category": "Math", "title": "Finite Sections of Weighted Carleman's Inequality", "abstract": "We study finite sections of weighted Carleman's inequality following the approach of De Bruijn. Similar to the unweighted case, we obtain an asymptotic expression for the optimal constant."}
{"category": "Math", "title": "Herman's Theory Revisited (Extension)", "abstract": "We prove that a $C^{3+\\beta}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_\\delta$, $0<\\beta<\\delta<1$, is $C^{2+\\beta-\\delta}$-smoothly conjugate to a rigid rotation."}
{"category": "Math", "title": "Minimax Robust Function Reconstruction in Reproducing Kernel Hilbert Spaces", "abstract": "In this paper, we present a unified approach to function approximation in reproducing kernel Hilbert spaces (RKHS) that establishes a previously unrecognized optimality property for several well-known function approximation techniques, such as minimum-norm interpolation, smoothing splines, and pseudo-inverses. We consider the problem of approximating a function belonging to an arbitrary real-valued RKHS on R^d based on approximate observations of the function. The observations are approximate in the sense that the actual observations (i.e., the true function values) are known only to belong to a convex set of admissible observations. We seek a minimax optimal approximation for the function that minimizes the supremum of the RKHS norm on the error between the true function and the chosen approximation subject only to the conditions that the true function belongs to a uniformly bounded uncertainty set of functions that satisfy the constraints on the observations and that the approximation is a member of the RKHS. We refer to such a solution as a minimax robust reconstruction. We characterize the solution to the minimax robust reconstruction problem and show that it is equivalent to solving a straightforward convex optimization problem. We demonstrate that a minimax robust reconstruction will generally be more stable than an approximation based on interpolation through a nominal set of observations and that, subject to some mild regularity conditions on the convex set of admissible observations, the minimax robust reconstruction is unconditionally stable. We motivate our results by characterizing the minimax robust reconstruction for several specific convex observational models and discuss relationships with other approaches to function approximation."}
{"category": "Math", "title": "Gallai Multigraphs", "abstract": "A complete edge-colored graph or multigraph is called Gallai if it lacks rainbow triangles. We give a construction of all finite Gallai multigraphs."}
{"category": "Math", "title": "Orthogonal Systems in Finite Graphs", "abstract": "Given a finite graph G there is a corresponding group given by the presentation with generators the vertices of G and a relation [x,y]=1 for generators x and y precisely when (x,y) is an edge of G. Such groups are known as partially commutative groups (or right-angled Artin groups). In this paper we construct orthogonality theory for graphs with the study of partially commutative groups in mind. The theory developed here provides tools for the study of the structure of the centraliser lattice of partially commutative groups and for their automorphism groups."}
{"category": "Math", "title": "Perverse coherent sheaves and the geometry of special pieces in the unipotent variety", "abstract": "Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent middle extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construction (called \"S2-extension\") in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical \"S2-ification\" of appropriate X. The construction also has applications to the \"Macaulayfication\" problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown in the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S2-extension to give a uniform construction of the desired variety."}
{"category": "Math", "title": "Calculation of local Fourier transforms for formal connections", "abstract": "We calculate the local Fourier transforms for formal connections. In particular, we verify an analogous conjecture suggested in Laumon's paper: \"Transformation de Fourier, constantes d'equations fonctionnelles et conjecture de Weil, 2.6.3\"."}
{"category": "Math", "title": "Study of the linear ablation growth rate for the quasi isobaric model of Euler equations with thermal conductivity", "abstract": "In this paper, we study a linear system related to the 2d system of Euler equations with thermal conduction in the quasi-isobaric approximation of Kull-Anisimov [14]. This model is used for the study of the ablation front instability, which appears in the problem of inertial confinement fusion. This physical system contains a mixing region, in which the density of the gaz varies quickly, and one denotes by L0 an associated characteristic length. The system of equations is linearized around a stationary solution, and each perturbed quantity is written using the normal modes method. The resulting linear system is not self-adjoint, of order 5, with coefficients depending on x and on physical parameters $\\alpha, \\beta$. We calculate Evans function associated with this linear system, using rigorous constructions of decreasing at $\\pm \\infty$ solutions of systems of ODE. We prove that for $\\alpha$ small, there is no bounded solution of the linearized system."}
{"category": "Math", "title": "On the One-Dimensional Optimal Switching Problem", "abstract": "We explicitly solve the optimal switching problem for one-dimensional diffusions by directly employing the dynamic programming principle and the excessive characterization of the value function. The shape of the value function and the smooth fit principle then can be proved using the properties of concave functions."}
{"category": "Math", "title": "Markov type of Alexandrov spaces of nonnegative curvature", "abstract": "We prove that Alexandrov spaces $X$ of nonnegative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of $X$ into a 2-uniformly convex Banach space is extended as a Lipschitz continuous map on the entire space $X$."}
{"category": "Math", "title": "Width and finite extinction time of Ricci flow", "abstract": "This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2-spheres. For instance, when $M$ is a homotopy 3-sphere, the width is loosely speaking the area of the smallest 2-sphere needed to ``pull over'' $M$. Second, we use this to conclude that Hamilton's Ricci flow becomes extinct in finite time on any homotopy 3-sphere. We have chosen to write this since the results and ideas given here are quite useful and seem to be of interest to a wide audience."}
{"category": "Math", "title": "Canonical volume forms on compact K\\\"{a}hler manifolds", "abstract": "We construct a canonical singular hermitian metric with semipositive curvature current on the canonical line bundle of a compact K\\\"{a}hler manifold with pseudoeffective canonical bundle."}
{"category": "Math", "title": "Holomorphy conditions of Fuji-Suzuki coupled Painlev\\'e VI system", "abstract": "In this note, we give some holomorphy conditions of Fuji-Suzuki coupled Painlev\\'e VI system. We also give two translation operators acting on the constant parameter $\\eta$. We note a confluence process from the Fuji-Suzuki system to the Noumi-Yamada system of type $A_5^{(1)}$."}
{"category": "Math", "title": "Higher derivatives and the inverse derivative of a tensor-valued function of a tensor", "abstract": "The n-th derivative of a tensor valued function of a tensor is defined by a finite number of coefficients each with closed form expression."}
{"category": "Math", "title": "Generalized Gevrey ultradistributions", "abstract": "We first introduce new algebras of generalized functions containing Gevrey ultradistributions and then develop a Gevrey microlocal analysis suitable for these algebras. Finally, we give an application through an extension of the well-known H\\\"{o}rmander's theorem on the wave front of the product of two distributions."}
{"category": "Math", "title": "Hodge-theoretic Atiyah-Meyer formulae and the stratified multiplicative property", "abstract": "In this note we survey Hodge-theoretic formulae of Atiyah-Meyer type for genera and characteristic classes of complex algebraic varieties, and derive some new and interesting applications. We also present various extensions to the singular setting of the Chern-Hirzebruch-Serre signature formula."}
{"category": "Math", "title": "Incompressible, quasi-rigid deformations of 2-dimensional domains", "abstract": "his paper proposes a sensible definition of a deformation metric between 2-dimensional surfaces obtained from each other by an area preserving (incompressible) mapping, and an algorithm for obtaining this metric, as well as the optimal deformation."}
{"category": "Math", "title": "Serial coalgebras and their valued Gabriel quivers", "abstract": "We study serial coalgebras by means of their valued Gabriel quivers. In particular, Hom-computable and representation-directed coalgebras are characterized. The Auslander-Reiten quiver of a serial coalgebra is described. Finally, a version of Eisenbud-Griffith theorem is proved, namely, every subcoalgebra of a prime, hereditary and strictly quasi-finite coalgebra is serial."}
{"category": "Math", "title": "Minors in expanding graphs", "abstract": "Extending several previous results we obtained nearly tight estimates on the maximum size of a clique-minor in various classes of expanding graphs. These results can be used to show that graphs without short cycles and other H-free graphs contain large clique-minors, resolving some open questions in this area."}
{"category": "Math", "title": "Additive approximation for edge-deletion problems", "abstract": "A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E_P(G). Our first result states that for any monotone graph property P, any \\epsilon >0 and n-vertex input graph G one can approximate E_P(G) up to an additive error of \\epsilon n^2 Our second main result shows that such approximation is essentially best possible and for most properties, it is NP-hard to approximate E_P(G) up to an additive error of n^{2-\\delta}, for any fixed positive \\delta. The proof requires several new combinatorial ideas and involves tools from Extremal Graph Theory together with spectral techniques. Interestingly, prior to this work it was not even known that computing E_P(G) precisely for dense monotone properties is NP-hard. We thus answer (in a strong form) a question of Yannakakis raised in 1981."}
{"category": "Math", "title": "Model theoretic connected components of groups", "abstract": "We give a general exposition of model theoretic connected components of groups. We show that if a group G has NIP, then there exists the smallest invariant (over some small set) subgroup of G with bounded index (Theorem 5.3). This result extends theorem of Shelah. We consider also in this context the multiplicative and the additive groups of some rings (including infite fields)."}
{"category": "Math", "title": "On semiparametric regression with O'Sullivan penalised splines", "abstract": "This is an expos\\'e on the use of O'Sullivan penalised splines in contemporary semiparametric regression, including mixed model and Bayesian formulations. O'Sullivan penalised splines are similar to P-splines, but have an advantage of being a direct generalisation of smoothing splines. Exact expressions for the O'Sullivan penalty matrix are obtained. Comparisons between the two reveals that O'Sullivan penalised splines more closely mimic the natural boundary behaviour of smoothing splines. Implementation in modern computing environments such as Matlab, R and BUGS is discussed."}
{"category": "Math", "title": "Dimension Data, Local and Global Conjugacy in Reductive Groups", "abstract": "Let G be a connected reductive group (over $\\mathbb{C}$) and H a connected semisimple subgroup. The dimension data of H (realative to its given embedding in G) is the collection of the numbers $\\{{\\rm dim} V^{H}\\}$, where V runs over all the finite dimensional representations of G. By a Theorem of Larsen-Pink ([L-P90]), the dimension data determines H up to isomorphism, and if G = GL (n) even up to conjugacy. Professor Langlands raised the question as to whether the strong (conjugacy) result holds for arbitrary G. In this paper We provided the following (negative) answer: If H is simple of type A_{4 n}, $B_{2 n} (n \\geq 2)$, $C_{2 n} (n \\geq 2)$, E_{6}, E_{8}, F_{4} and G_{2}, then there exist (for suitable $N$) pairs of embeddings i and i' of H into $G = SO (2 N)$ such that there image i (H) and i' (H) have the same dimension data but are not conjugate. In fact we have shown that i (H) and i' (H) are \\emph{locally conjugate}, i.e., that i (h) and i' (h) are conjugate in G for all semisimple $h \\in H$. If one assumes functoriality, this result will furnish the failure of multiplicity one for automorphic forms on such G over global fields. Such things are known in the disconnected cases, especially when H is finite, as in the works of Blasius [Blasius94] for $SL (n) (n \\geq 3)$ and Gan-Gurevich-Jiang2002 ([Gan]) for G_{2}."}
{"category": "Math", "title": "Tsirelson like operator spaces", "abstract": "We construct nontrivial examples of weak-$C_p$ ($1\\leq p \\leq \\infty$) operator spaces with the local operator space structure very close to $C_p = [R, C]_{\\frac{1}{p}}$. These examples are non-homogeneous Hilbertian operator spaces, and their constructions are similar to that of 2-convexified Tsirelson's space by W. B. Johnson."}
{"category": "Math", "title": "A Maurey type result for operator spaces", "abstract": "The little Grothendieck theorem for Banach spaces says that every bounded linear operator between $C(K)$ and $\\ell_2$ is 2-summing. However, it is shown in \\cite{J05} that the operator space analogue fails. Not every cb-map $v : \\K \\to OH$ is completely 2-summing. In this paper, we show an operator space analogue of Maurey's theorem : Every cb-map $v : \\K \\to OH$ is $(q,cb)$-summing for any $q>2$ and hence admits a factorization $\\|v(x)\\| \\leq c(q) \\|v\\|_{cb} \\|axb\\|_q$ with $a,b$ in the unit ball of the Schatten class $S_{2q}$."}
{"category": "Math", "title": "Non-degeneracy of Wiener functionals arising from rough differential equations", "abstract": "Malliavin Calculus is about Sobolev-type regularity of functionals on Wiener space, the main example being the Ito map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of solution to (possibly stochastic) differential equations. We combine arguments of both theories and discuss existence of a density for solutions to stochastic differential equations driven by a general class of non-degenerate Gaussian processes, including processes with sample path regularity worse than Brownian motion."}
{"category": "Math", "title": "Balanced Cayley graphs and balanced planar graphs", "abstract": "A balanced graph is a bipartite graph with no induced circuit of length 2 mod 4. These graphs arise in linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for balanced regular graphs are also presented."}
{"category": "Math", "title": "Nodal curves with general moduli on K3 surfaces", "abstract": "We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a d-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p-2, for p any integer between 3 and 11 and g = p - d between 2 and p. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S,X) to the moduli space of curves of genus g that associates to X the isomorphism class [C] of its normalization."}
{"category": "Math", "title": "Algorithmic Problems in Amalgams of Finite Groups: Conjugacy and Intersection Properties", "abstract": "Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. In the present paper we employ the generalized Stallings' methods, developed by the author, to solve various algorithmic problems concerning finitely generated subgroups of amalgams of finite groups."}
{"category": "Math", "title": "The random Tukey depth", "abstract": "The computation of the Tukey depth, also called halfspace depth, is very demanding, even in low dimensional spaces, because it requires the consideration of all possible one-dimensional projections. In this paper we propose a random depth which approximates the Tukey depth. It only takes into account a finite number of one-dimensional projections which are chosen at random. Thus, this random depth requires a very small computation time even in high dimensional spaces. Moreover, it is easily extended to cover the functional framework. We present some simulations indicating how many projections should be considered depending on the sample size and on the dimension of the sample space. We also compare this depth with some others proposed in the literature. It is noteworthy that the random depth, based on a very low number of projections, obtains results very similar to those obtained with other depths."}
{"category": "Math", "title": "Ore extensions satisfying a polynomial identity", "abstract": "Necessary and sufficient conditions for an Ore extension $S=R[x;\\si,\\de]$ to be a {\\rm PI} ring are given in the case $\\si$ is an injective endomorphism of a semiprime ring $R$ satisfying the {\\rm ACC} on annihilators. Also, for an arbitrary endomorphism $\\tau$ of $R$, a characterization of Ore extensions $R[x;\\tau]$ which are {\\rm PI} rings is given, provided the coefficient ring $R$ is noetherian."}
{"category": "Math", "title": "Bernstein Type Results for Lagrangian Graphs with Partially Harmonic Gauss Map", "abstract": "We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized."}
{"category": "Math", "title": "Infimum of the exponential volume growth and the bottom of the essential spectrum of the Laplacian", "abstract": "The purpose of this paper is to point out that `supremum' in two inequalities of Brooks should be replaced with `infimum'. The results of this paper are already known by Professor Higuci. Hence, I want to delete this paper from this preprint server."}
{"category": "Math", "title": "Energy-Momentum tensor on foliations", "abstract": "In this paper, we give a new lower bound for the eigenvalues of the Dirac operator on a compact spin manifold. This estimate is motivated by the fact that in its limiting case a skew-symmetric tensor (see Equation \\eqref{eq:16}) appears that can be identified geometrically with the O'Neill tensor of a Riemannian flow, carrying a transversal parallel spinor. The Heisenberg group which is a fibration over the torus is an example of this case. Sasakian manifolds are also considered as particular examples of Riemannian flows. Finally, we characterize the 3-dimensional case by a solution of the Dirac equation"}
{"category": "Math", "title": "Category of Noncommutative CW Complexes", "abstract": "We expose the notion of noncommutative CW (NCCW) complexes, define noncommutative (NC) mapping cylinder and NC mapping cone, and prove the noncommutative Approximation Theorem. The long exact homotopy sequences associated with arbitrary morphisms are also deduced."}
{"category": "Math", "title": "On the non-existence of L-space surgery structure", "abstract": "We exhibit homology spheres which never yield lens spaces by any integral Dehn surgery by using Ozsvath Szabo's contact invariant."}
{"category": "Math", "title": "On the divisibility of characteristic classes of non-oriented surface bundles", "abstract": "In this note we introduce a construction which assigns to an arbitrary manifold bundle its fiberwise orientation covering. This is used to show that the zeta classes of unoriented surface bundles are not divisible in the stable range."}
{"category": "Math", "title": "Eigenvalues of the transversal Dirac operator on Kahler foliations", "abstract": "In this paper, we prove Kirchberg inequalities for any kahler spin foliations. Their limiting cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal kahlerian twistor operators."}
{"category": "Math", "title": "Eigenvalues of the basic Dirac operator on quaternion-Kahler foliations", "abstract": "In this paper, we give an optimal lower bound for the eigenvalues of the basic Dirac operator on a quaternion-Kahler foliation. The limiting case is characterized by the existence of quaternion-Kahler Killing spinors. We end this paper by giving some examples."}
{"category": "Math", "title": "Unit distances and diameters in Euclidean spaces", "abstract": "We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d >= 4 and n sufficiently large, depending on d. As a corollary we determine the exact maximum number of unit distances for all even d >= 6, and the exact maximum number of diameters for all d >= 4, for all $n$ sufficiently large, depending on d."}
{"category": "Math", "title": "Exponentially Stable Nonlinear Systems have Polynomial Lyapunov Functions on Bounded Regions", "abstract": "This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of sufficiently smooth nonlinear ordinary differential equations on bounded sets. The main result states that if there exists an n-times continuously differentiable Lyapunov function which proves exponential stability on a bounded subset of R^n, then there exists a polynomial Lyapunov function which proves exponential stability on the same region. Such a continuous Lyapunov function will exist if, for example, the right-hand side of the differential equation is polynomial or at least n-times continuously differentiable. The proof is based on a generalization of the Weierstrass approximation theorem to differentiable functions in several variables. Specifically, we show how to use polynomials to approximate a differentiable function in the Sobolev norm W^{1,\\infty} to any desired accuracy. We combine this approximation result with the second-order Taylor series expansion to find that polynomial Lyapunov functions can approximate continuous Lyapunov functions arbitrarily well on bounded sets. Our investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions."}
{"category": "Math", "title": "Convex and star-shaped sets associated with stable distributions", "abstract": "It is known that each symmetric stable distribution in $R^d$ is related to a norm on $R^d$ that makes $R^d$ embeddable in $L_p([0,1])$. In case of a multivariate Cauchy distribution the unit ball in this norm corresponds is the polar set to a convex set in $R^d$ called a zonoid. This work interprets general stable laws using convex or star-shaped sets and exploits recent advances in convex geometry in order to come up with new probabilistic results for multivariate stable distributions. In particular, it provides expressions for moments of the Euclidean norm of a stable vector, mixed moments and various integrals of the density function. It is shown how to use geometric inequalities in order to bound important parameters of stable laws. It is shown that each symmetric stable laws appears as the limit for the sum of sub-Gaussian laws and an estimate for the probability distance to a sub-Gaussian law is given. Operations with convex sets induce the well-known and new operations with stable vectors. Furthermore, covariation, regression and orthogonality concepts for stable laws acquire geometric interpretations. A similar collection of results is presented for one-sided stable laws."}
{"category": "Math", "title": "Uniform Star-factors of Graphs with Girth Three", "abstract": "A {\\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each component of which is a star. Recently, Hartnell and Rall studied a family $\\mathscr{U}$ of graphs satisfying the property that every star-factor of a member graph has the same number of edges. They determined the family $\\mathscr{U}$ when the girth is at least five. In this paper, we investigate the family of graphs with girth three and determine all members of this family."}
{"category": "Math", "title": "Star-uniform Graphs", "abstract": "A {\\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each of its component is a star. Clearly, every graph without isolated vertices has a star factor. A graph $G$ is called {\\it star-uniform} if all star-factors of $G$ have the same number of components. To characterize star-uniform graphs was an open problem posed by Hartnell and Rall, which is motivated by the minimum cost spanning tree and the optimal assignment problems. We use the concepts of factor-criticality and domination number to characterize all star-uniform graphs with the minimum degree at least two. Our proof is heavily relied on Gallai-Edmonds Matching Structure Theorem."}
{"category": "Math", "title": "Uniformly Weighted Star-Factors of Graphs", "abstract": "A {\\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each component of which is a star. An {\\it edge-weighting} of $G$ is a function $w: E(G)\\longrightarrow \\mathbb{N}^+$, where $\\mathbb{N}^+$ is the set of positive integers. Let $\\Omega$ be the family of all graphs $G$ such that every star-factor of $G$ has the same weights under a fixed edge-weighting $w$. In this paper, we present a simple structural characterization of the graphs in $\\Omega$ that have girth at least five."}
{"category": "Math", "title": "Publication history of von Staudt's Geometrie der Lage", "abstract": "From a census of forty copies, we can distinguish three different editions of von Staudt's Geometrie der Lage: the first of 1847 and two undated ones from the 1870's."}
{"category": "Math", "title": "Positive Forms and Stability of Linear Time-Delay Systems", "abstract": "We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parametrization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program."}
{"category": "Math", "title": "The number of imaginary quadratic fields with a given class number", "abstract": "We investigate the number ${\\Cal F}(h)$ of imaginary quadratic fields with class number $h$. We establish an asymptotic formula for the average value of ${\\Cal F}(h)$. We also establish a modest non-trivial upper bound for ${\\Cal F}(h)$ and give an application to a question of Rosen and Silverman on the odd part of the class number. Finally, we speculate on the asymptotic nature of ${\\Cal F}(h)$."}
{"category": "Math", "title": "Hamiltonian Stationary Shrinkers and Expanders for Lagrangian Mean Curvature Flows", "abstract": "We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of special Lagrangians or Lagrangian minimal surfaces in the variational approach. It is known that these cone singularities cannot be resolved by any smooth oriented Lagrangian submanifolds. The shrinkers and expanders that we found can be glued together to yield solutions of the Brakke motion-a weak formulation of the mean curvature flow. For any coprime pair $(p,q)$ other than $(2,1)$, we construct such a solution that resolves any single Schoen-Wolfson cone $C_{p,q}$. This thus provides an evidence to Schoen-Wolfson's conjecture that the $(2,1)$ cone is the only area-minimizing cone. Higher dimensional generalizations are also obtained."}
{"category": "Math", "title": "Singular points of real quartic curves via computer algebra", "abstract": "There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D.A. Gudkov. There are nine types of singular points for irreducible complex quartic curves and ten types of singular points for reducible complex quartic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, the proof consists of a sequence of large symbolic computations that can be done nicely using Maple."}
{"category": "Math", "title": "A construction of numerical Campedelli Surfaces with \\Z/6 torsion group", "abstract": "We produce a family of numerical Campedelli surfaces with \\Z/6 torsion by constructing the (Gorenstein codimension 5) canonical ring of the \\'{e}tale six to one cover using serial unprojection. In Section 2 we develop the necessary algebraic machinery. Section 3 contains the numerical Campedelli surface construction, while Section 4 contains remarks and open questions."}
{"category": "Math", "title": "A new graphical tool of outliers detection in regression models based on recursive estimation", "abstract": "We present in this paper a new tool for outliers detection in the context of multiple regression models. This graphical tool is based on recursive estimation of the parameters. Simulations were carried out to illustrate the performance of this graphical procedure. As a conclusion, this tool is applied to real data containing outliers according to the classical available tools."}
{"category": "Math", "title": "The refined transfer, bundle structures and algebraic K-theory", "abstract": "We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits a reduction to a fiber bundle with compact topological manifold fibers. The criteria lead to a new and unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjective splitting of the assembly map for Waldhausen's functor A(X). We also give concrete examples of fibrations having a reduction to a fiber bundle with compact topological manifold fibers but which fail to admit a compact fiber smoothing. The examples are detected by algebraic K-theory invariants. We consider a refinement of the Becker-Gottlieb transfer. We show that a version of the axioms described by Becker and Schultz uniquely determines the refined transfer for the class of fibrations admitting a reduction to a fiber bundle with compact topological manifold fibers. In an appendix, we sketch a theory of characteristic classes for fibrations. The classes are primary obstructions to finding a compact fiber smoothing."}
{"category": "Math", "title": "Orthogonal functions generalizing Jack polynomials", "abstract": "The rational Cherednik algebra $\\HH$ is a certain algebra of differential-reflection operators attached to a complex reflection group $W$. Each irreducible representation $S^\\lambda$ of $W$ corresponds to a standard module $M(\\lambda)$ for $\\HH$. This paper deals with the infinite family $G(r,1,n)$ of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra $\\ttt$ of $\\HH$ discovered by Dunkl and Opdam. In this case, the irreducible $W$-modules are indexed by certain sequences $\\lambda$ of partitions. We first show that $\\ttt$ acts in an upper triangular fashion on each standard module $M(\\lambda)$, with eigenvalues determined by the combinatorics of the set of standard tableaux on $\\lambda$. As a consequence, we construct a basis for $M(\\lambda)$ consisting of orthogonal functions on $\\CC^n$ with values in the representation $S^\\lambda$. For $G(1,1,n)$ with $\\lambda=(n)$ these functions are the non-symmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of $M(\\lambda)$ in the case in which the orthogonal functions are all well-defined."}
{"category": "Math", "title": "LAMN property for hidden processes: the case of integrated diffusions", "abstract": "In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process $X$. Our data are given by $ \\int_0^1 X_{\\frac{s+i}{n}} \\dd \\mu (s)$ for $i=0,...,n-1$ and the unknown parameter appears in the diffusion coefficient of the process $X$ only. Although the data are nor Markovian neither Gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic expansion. We actually find that the asymptotic information of this model is the same one as for a usual discrete sampling of $X$."}
{"category": "Math", "title": "Yang-Mills Connections On Orientable and Nonorientable Surfaces", "abstract": "In math.SG/0605587, we studied Yang-Mills functional on the space of connections on a principal G_R-bundle over a closed, connected, nonorientable surface, where G_R is any compact connected Lie group. In this sequel, we generalize the discussion in \"The Yang-Mills equations over Riemann surfaces\" by Atiyah and Bott, and math.SG/0605587. We obtain explicit descriptions (as representation varieties) of Morse strata of Yang-Mills functional on orientable and nonorientable surfaces for non-unitary classical groups SO(n) and Sp(n). It turns out to be quite different from the unitary case. we use Laumon and Rapoport's method in \"The Langlands lemma and the Betti numbers of stacks of G-bundles on a curve\" to invert the Atiyah-Bott recursion relation, and write down explicit formulas of rational equivariant Poincar\\'{e} series of the semistable stratum of the space of holomorphic structures on a principal $SO(n,\\bC)$-bundle or a principal $Sp(n,\\bC)$-bundle."}
{"category": "Math", "title": "On the affineness of Deligne-Lusztig varieties", "abstract": "We prove that the Deligne-Lusztig variety associated to minimal length elements in any $\\d$-conjugacy class of the Weyl group is affine, which was conjectured by Orlik and Rapoport in \\cite{OR}."}
{"category": "Math", "title": "Maximum Likelihood Estimator for Hidden Markov Models in continuous time", "abstract": "The paper studies large sample asymptotic properties of the Maximum Likelihood Estimator (MLE) for the parameter of a continuous time Markov chain, observed in white noise. Using the method of weak convergence of likelihoods due to I.Ibragimov and R.Khasminskii, consistency, asymptotic normality and convergence of moments are established for MLE under certain strong ergodicity conditions of the chain."}
{"category": "Math", "title": "On Fox quotients of arbitrary group algebras", "abstract": "For a group $G$, N-series $\\cal G$ of $G$ and commutative ring $R$ let $I^n_{R,\\cal G}(G)$, $n\\ge 0$, denote the filtration of the group algebra $R(G)$ induced by $\\cal G$, and $I_R(G)$ its augmentation ideal. For subgroups $H$ of $G$, left ideals $J$ of $R(H)$ and right $H$-submodules $M$ of $I_Z(G)$ the quotients $I_R(G)J/MJ$ are studied by homological methods, notably for $M= I_Z(G)I_Z(H)$, $I_Z(H)I_Z(G) + I_Z([H,G])Z(G)$ and $Z(G)I_Z(N) +I^n_{Z,\\cal G}(G)$ with $N \\lhd G$ where the group $I_R(G)J/MJ$ is completely determined for $n=2$. The groups $I^{n-1}_{Z,\\cal G}(G)I_Z(H)/I^n_{Z,\\cal G}(G)I_Z(H)$ are studied and explicitly computed for $n\\le 3$ in terms of enveloping rings of certain graded Lie rings and of torsion products of abelian groups."}
{"category": "Math", "title": "A statistical theory for the measurement and estimation of Rayleigh fading channel", "abstract": "In this paper, we propose a statistical theory on measurement and estimation of Rayleigh fading channels in wireless communications and provide complete solutions to the fundamental problems: What is the optimum estimator for the statistical parameters associated with the Rayleigh fading channel, and how many measurements are sufficient to estimate these parameters with the prescribed margin of error and confidence level? Our proposed statistical theory suggests that two testing signals of different strength be used. The maximum likelihood (ML) estimator is obtained for estimation of the statistical parameters of the Rayleigh fading channel that is both sufficient and complete statistic. Moreover, the ML estimator is the minimum variance (MV) estimator that in fact achieves the Cramer-Rao lower bound."}
{"category": "Math", "title": "The relative second Fox and third dimension subgroup of arbitrary groups", "abstract": "Let $I_R(G)$ denote the augmentation ideal of the group algebra $R(G)$ of a group $G$ with coefficients in a commutative ring $R$. We give a complete description of the third relative dimension subgroup $G\\cap(1+I_R(K)I_R(G)+I^3_R(G))$ and the second relative Fox subgroup $G\\cap(1+I_R(K)I_R(H)+I^2_R(G)I_R(H))$ for any subgroups $K$ and $H$ of $G$."}
{"category": "Math", "title": "On the second cohomology of semidirect products", "abstract": "Let $G$ be a group which is the semidirect product of a normal subgroup $N$ and a subgroup $T$, and let $M$ be a $G$-module with not necessarily trivial $G$-action. Then we embed the simultaneous restriction map $res=(res^G_N,res^G_T)^t : H^2(G,M) \\to H^2(N,M)^T \\times H^2(T,M)$ into a natural five term exact sequence consisting of one and two-dimensional cohomology groups of the factors $N$ and $T$. The elements of $H^2(G,M)$ are represented in terms of group extensions of $G$ by $M$ constructed from extensions of $N$ and $T$."}
{"category": "Math", "title": "The distribution of smooth numbers in arithmetic progressions", "abstract": "For a wide range of $x$ and $y$ we show that ${\\Cal S}(x,y)$, the set of integers below $x$ composed only of prime factors below $y$, is equidistributed in the reduced residue classes $\\pmod q$ for all $q<y^{4\\sqrt{e}-\\epsilon}$. This improves earlier work of Granville; any improvement of this range of $q$ would have interesting consequences for Vinogradov's conjecture on the least quadratic non-residue. For larger ranges of $q$ we prove the existence of a large subgroup of the group of reduced residues such that ${\\Cal S}(x,y)$ is equidistributed within cosets of that subgroup."}
{"category": "Math", "title": "Categorical aspects of toric topology", "abstract": "We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces X_K, such as moment-angle complexes Z_K, quasitoric manifolds M, and Davis-Januszkiewicz spaces DJ(K). We first exhibit X_K as the homotopy colimit of a diagram of spaces over the small category cat(K), whose objects are the faces of a finite simplicial complex K and morphisms their inclusions. Then we study the corresponding cat(K)-diagrams in various algebraic Quillen model categories, and interpret their homotopy colimits as algebraic models for X_K. Such models encode many standard algebraic invariants, and their existence is assured by the Quillen structure. We provide several illustrative calculations, often over the rationals, including proofs that quasitoric manifolds (and various generalisations) are rationally formal; that the rational Pontrjagin ring of the loop space \\Omega DJ(K) is isomorphic to the quadratic dual of the Stanley-Reisner algebra Q[K] for flag complexes K; and that DJ(K) is coformal precisely when K is flag. We conclude by describing algebraic models for the loop space \\Omega DJ(K) for any complex K, which mimic our previous description as a homotopy colimit of topological monoids."}
{"category": "Math", "title": "Learning from dependent observations", "abstract": "In most papers establishing consistency for learning algorithms it is assumed that the observations used for training are realizations of an i.i.d. process. In this paper we go far beyond this classical framework by showing that support vector machines (SVMs) essentially only require that the data-generating process satisfies a certain law of large numbers. We then consider the learnability of SVMs for $\\a$-mixing (not necessarily stationary) processes for both classification and regression, where for the latter we explicitly allow unbounded noise."}
{"category": "Math", "title": "The Teichm\\\"uller distance between finite index subgroups of $PSL_2(\\mathbb{Z})$", "abstract": "For a given $\\epsilon >0$, we show that there exist two finite index subgroups of $PSL_2(\\mathbb{Z})$ which are $(1+\\epsilon)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $\\epsilon>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Modular torus) and a $(1+\\epsilon)$-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichm\\\"uller space $T(\\S)$ of the punctured solenoid $\\S$ under the action of the corresponding Modular group (which is the mapping class group of $\\S$ \\cite{NS}, \\cite{Odd}) has the closure in $T(\\S)$ strictly larger than the orbit and that the closure is necessarily uncountable."}
{"category": "Math", "title": "On the maximum size of an anti-chain of linearly separable sets and convex pseudo-discs", "abstract": "We show that the maximum cardinality of an anti-chain composed of intersections of a given set of n points in the plane with half-planes is close to quadratic in n. We approach this problem by establishing the equivalence with the problem of the maximum monotone path in an arrangement of n lines. For a related problem on antichains in families of convex pseudo-discs we can establish the precise asymptotic bound: it is quadratic in n. The sets in such a family are characterized as intersections of a given set of n points with convex sets, such that the difference between the convex hulls of any two sets is nonempty and connected."}
{"category": "Math", "title": "Differential Equations Driven by Gaussian Signals I", "abstract": "We consider multi-dimensional Gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of stochastic area(s). Gaussian rough paths are constructed with a variety of weak and strong approximation results. Together with a new RKHS embedding, we obtain a powerful - yet conceptually simple - framework in which to analysize differential equations driven by Gaussian signals in the rough paths sense."}
{"category": "Math", "title": "How many random edges make a dense hypergraph non-2-colorable?", "abstract": "We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. We obtain a tight bound on the number of random edges required to ensure non-2-colorability. We prove that for any k-uniform hypergraph with Omega(n^{k-epsilon}) edges, adding omega(n^{k epsilon/2}) random edges makes the hypergraph almost surely non-2-colorable. This is essentially tight, since there is a 2-colorable hypergraph with Omega(n^{k-\\epsilon}) edges which almost surely remains 2-colorable even after adding o(n^{k \\epsilon / 2}) random edges."}
{"category": "Math", "title": "Occupation time fluctuations of Poisson and equilibrium branching systems in critical and large dimensions", "abstract": "Limit theorems are presented for the rescaled occupation time fluctuation process of a critical finite variance branching particle system in $\\mathbb{R}^{d}$ with symmetric $\\alpha$-stable motion starting off from either a standard Poisson random field or the equilibrium distribution for critical $d=2\\alpha$ and large $d>2\\alpha$ dimensions. The limit processes are generalised Wiener processes. The obtained convergence is in space-time, finite-dimensional distributions sense. With the addtional assumption on the branching law we obtain functional convergence."}
{"category": "Math", "title": "Consistency of support vector machines for forecasting the evolution of an unknown ergodic dynamical system from observations with unknown noise", "abstract": "We consider the problem of forecasting the next (observable) state of an unknown ergodic dynamical system from a noisy observation of the present state. Our main result shows, for example, that support vector machines (SVMs) using Gaussian RBF kernels can learn the best forecaster from a sequence of noisy observations if (a) the unknown observational noise process is bounded and has a summable $\\alpha$-mixing rate and (b) the unknown ergodic dynamical system is defined by a Lipschitz continuous function on some compact subset of $\\mathbb{R}^d$ and has a summable decay of correlations for Lipschitz continuous functions. In order to prove this result we first establish a general consistency result for SVMs and all stochastic processes that satisfy a mixing notion that is substantially weaker than $\\alpha$-mixing."}
{"category": "Math", "title": "Label-setting methods for Multimode Stochastic Shortest Path problems on graphs", "abstract": "Stochastic shortest path (SSP) problems arise in a variety of discrete stochastic control contexts. An optimal solutions to such a problem is typically computed using the value function, which can be found by solving the corresponding dynamic programming equations. In the deterministic case, these equations can be often solved by the highly efficient label-setting methods (such as Dijkstra's and Dial's algorithms). In this paper we define and study a class of Multimode Stochastic Shortest Path problems and develop sufficient conditions for the applicability of label-setting methods. We illustrate our approach on a number of discrete stochastic control examples. We also discuss the relationship of SSPs with discretizations of static Hamilton-Jacobi equations and provide an alternative derivation for several fast (non-iterative) numerical methods for these PDEs."}
{"category": "Math", "title": "Integral pinched 3-manifolds are space forms", "abstract": "In this paper we prove that, under an explicit integral pinching assumption between the $L^2$-norm of the Ricci curvature and the $L^2$-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits an Einstein metric with positive curvature. In particular this implies that the manifold is diffeomorphic to a quotient of ${\\Bbb S}^3$."}
{"category": "Math", "title": "Asymptotic enumeration of sparse nonnegative integer matrices with specified row and column sums", "abstract": "Let \\svec = (s_1,...,s_m) and \\tvec = (t_1,...,t_n) be vectors of nonnegative integer-valued functions of m,n with equal sum S = sum_{i=1}^m s_i = sum_{j=1}^n t_j. Let M(\\svec,\\tvec) be the number of m*n matrices with nonnegative integer entries such that the i-th row has row sum s_i and the j-th column has column sum t_j for all i,j. Such matrices occur in many different settings, an important example being the contingency tables (also called frequency tables) important in statistics. Define s=max_i s_i and t=max_j t_j. Previous work has established the asymptotic value of M(\\svec,\\tvec) as m,n\\to\\infty with s and t bounded (various authors independently, 1971-1974), and when \\svec,\\tvec are constant vectors with m/n,n/m,s/n >= c/log n for sufficiently large (Canfield and McKay, 2007). In this paper we extend the sparse range to the case st=o(S^(2/3)). The proof in part follows a previous asymptotic enumeration of 0-1 matrices under the same conditions (Greenhill, McKay and Wang, 2006). We also generalise the enumeration to matrices over any subset of the nonnegative integers that includes 0 and 1."}
{"category": "Math", "title": "Large deviations for symmetrised empirical measures", "abstract": "In this paper we prove a Large Deviation Principle for the sequence of symmetrised empirical measures $\\frac{1}{n} \\sum_{i=1}^{n} \\delta_{(X^n_i,X^n_{\\sigma_n(i)})}$ where $\\sigma_n$ is a random permutation and $((X_i^n)_{1 \\leq i \\leq n})_{n \\geq 1}$ is a triangular array of random variables with suitable properties. As an application we show how this result allows to improve the Large Deviation Principles for symmetrised initial-terminal conditions bridge processes recently established by Adams, Dorlas and K\\\"{o}nig."}
{"category": "Math", "title": "On Fox and augmentation quotients of semidirect products", "abstract": "Let $G$ be a group which is the semidirect product of a normal subgroup $N$ and some subgroup $T$. Let $I^n(G)$, $n\\ge 1$, denote the powers of the augmentation ideal $I(G)$ of the group ring $\\Z(G)$. Using homological methods the groups $Q_n(G,H) = I^{n-1}(G)I(H)/I^{n}(G)I(H)$, $H=G,N,T$, are functorially expressed in terms of enveloping algebras of certain Lie rings associated with $N$ and $T$, in the following cases: for $n\\le 4$ and arbitrary $G,N,T$ (except from one direct summand of $Q_4(G,N)$), and for all $n\\ge 2$ if certain filtration quotients of $N$ and $T$ are torsionfree."}
{"category": "Math", "title": "Polarization types of isogenous Prym-Tyurin varieties", "abstract": "Let p:C-->Y be a covering of smooth, projective curves which is a composition of \\pi:C-->C' of degree 2 and g:C'-->Y of degree n. Let f:X-->Y be the covering of degree 2^n, where the curve X parametrizes the liftings in C^{(n)} of the fibers of g:C'-->Y. Let P(X,\\delta) be the associated Prym-Tyurin variety, known to be isogenous to the Prym variety P(C,C'). Most of the results in the paper focus on calculating the polarization type of the restriction of the canonical polarization of JX on P(X,\\delta). We obtain the polarization type when n=3. When Y=P^1 we conjecture that P(X,\\delta) is isomorphic to the dual of the Prym variety P(C,C'). This was known when n=2, we prove it when n=3, and for arbitrary n if \\pi:C-->C' is \\'{e}tale. Similar results are obtained for some other types of coverings."}
{"category": "Math", "title": "Willmore Legendrian surfaces in pseudoconformal 5-sphere", "abstract": "Let $ X: M \\hook S^5$ be a compact Legendrian surface in pseudoconformal(CR) 5-sphere. We introduce a pseudoconformally invariant Willmore type second order functional $ \\W(X)$, and study its critical points called Willmore Legendrian surfaces. The fifth order structure equations show that Willmore dual can be defined for a class of Willmore Legendrian surfaces. Moreover when this dual is constant, Willmore Legendrian surface admits a Weierstra\\ss type representation in terms of immersed meromorphic curve in $ \\C^2$ satisfying an appropriate real period condition via pseudoconformal stereographic projection. We show that every compact Riemann surface admits a generally one to one, conformal, Willmore Legendrian immersion in $ S^5$ with constant Willmore dual. As a corollary, every compact Riemann surface can be conformally immersed in $ \\C^2$ as an exact, algebraic Lagrangian surface."}
{"category": "Math", "title": "Radial Dunkl Processes : Existence and uniqueness, Hitting time, Beta Processes and Random Matrices", "abstract": "We begin with the study of some properties of the radial Dunkl process associated to a reduced root system $R$. It is shown that this diffusion is the unique strong solution for all $t \\geq 0$ of a SDE with singular drift. Then, we study $T_0$, the first hitting time of the positive Weyl chamber : we prove, via stochastic calculus, a result already obtained by Chybiryakov on the finiteness of $T_0$. The second and new part deals with the law of $T_0$ for which we compute the tail distribution, as well as some insight via stochastic calculus on how root systems are connected with eigenvalues of standard matrix-valued processes. This gives rise to the so-called $\\beta$-processes. The ultraspherical $\\beta$-Jacobi case still involves a reduced root system while the general case is closely connected to a non reduced one. This process lives in a convex bounded domain known as principal Weyl alcove and the strong uniqueness result remains valid. The last part deals with the first hitting time of the alcove's boundary and the semi group density which enables us to answer some open questions."}
{"category": "Math", "title": "Quadratic maps between groups", "abstract": "The notion of quadratic maps between arbitrary groups appeared at several places in the literature on quadratic algebra. Here a unified extensive treatment of their properties is given; the relation with a relative version of Passi's polynomial maps and groups of degree 2 is established and used to study the structure of the latter."}
{"category": "Math", "title": "Regulators of canonical extensions are torsion: the smooth divisor case", "abstract": "In this paper, we prove a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees $>1$) are torsion, of a flat bundle on a smooth complex projective variety. We consider the case of a smooth quasi--projective variety with an irreducible smooth divisor at infinity. We define the Chern-Simons classes of Deligne's canonical extension of a flat vector bundle with unipotent monodromy at infinity, which lift the Deligne Chern classes and prove that these classes are torsion."}
{"category": "Math", "title": "Self Improving Sobolev-Poincare Inequalities, Truncation and Symmetrization", "abstract": "In a previous paper we developed a new method to obtain symmetrization inequalities of Sobolev type for functions in $W_{0}^{1,1}(\\Omega)$. In this paper we extend our method to Sobolev functions that do not vanish at the boundary."}
{"category": "Math", "title": "A remark on fractional integrals on modulation spaces", "abstract": "In this paper, we give the necessary and sufficient conditions for the boundedness of fractional integral operators on the modulation spaces."}
{"category": "Math", "title": "On the $L^2$-boundedness of pseudo-differential operators and their commutators with symbols in $\\alpha$-modulation spaces", "abstract": "In this paper, we consider the $L^2$-boundedness of pseudo-differential operators with symbols in $\\alpha$-modulation spaces."}
{"category": "Math", "title": "Singular Hecke algebras, Markov traces, and HOMFLY-type invariants", "abstract": "We define the singular Hecke algebra ${\\mathcal H} (SB_n)$ as the quotient of the singular braid monoid algebra ${\\mathbb C} (q) [SB_n]$ by the Hecke relations $\\sigma_k^2 = (q-1) \\sigma_k +q$, $1 \\le k\\le n-1$, and define the Markov traces on the sequence $\\{{\\mathcal H}(SB_n)\\}_{n=1}^{+\\infty}$ in the same way as for the Markov traces on the tower of (non-singular) Hecke algebras of the symmetric groups. We prove that the Markov traces are in one-to-one correspondance with the invariants that satisfies some skein relation, and compute an explicit classification of the Markov traces. Thanks to this classification, we define some universal HOMFLY-type invariant which has the property that it distinguishes all the pairs of singular links that can be distinguished by an invariant which satisfies the required skein relation."}
{"category": "Math", "title": "A computation of Poisson kernels for some standard weighted biharmonic operators in the unit disc", "abstract": "We compute Poisson kernels for integer weight parameter standard weighted biharmonic operators in the unit disc with Dirichlet boundary conditions. The computations performed extend the supply of explicit examples of such kernels and suggest similar formulas for these Poisson kernels to hold true in more generality. Computations have been carried out using the open source computer algebra package Maxima."}
{"category": "Math", "title": "Random Normal Matrices and Polynomial Curves", "abstract": "We show that in the large matrix limit, the eigenvalues of the normal matrix model for matrices with spectrum inside a compact domain with a special class of potentials homogeneously fill the interior of a polynomial curve uniquely defined by the area of its interior domain and its exterior harmonic moments which are all given as parameters of the potential. Then we consider the orthogonal polynomials corresponding to this matrix model and show that, under certain assumptions, the density of the zeros of the highest relevant orthogonal polynomial in the large matrix limit is (up to some constant factor) given by the discontinuity of the Schwarz function of this polynomial curve."}
{"category": "Math", "title": "Complete isometries between subspaces of noncommutative Lp-spaces", "abstract": "We prove some noncommutative analogues of a theorem by Plotkin and Rudin about isometries between subspaces of Lp-spaces. Let 0<p<\\infty, p not an even integer. The main result of this paper states that in the category of unital subspaces of noncommutative probability Lp-spaces, under some boundedness condition, the unital completely isometric maps come from *-isomorphisms of the underlying von Neumann algebras. Some applications are given, including to non commutative H^p spaces."}
{"category": "Math", "title": "An algebraic proof of the commutativity of the intersection with divisors", "abstract": "We present a purely algebraic proof of the commutativity of the operation defined by intersection with divisors on the Chow group of a local Noetherian domain."}
{"category": "Math", "title": "Another generalization of Mason's ABC-theorem", "abstract": "We show a generalization of Mason's ABC-theorem, with the only conditions that the greatest common divisor has been divided out and no proper subsum of the (possibly multivariate) polynomial sum f_1 + f_2 + ... + f_n = 0 vanishes. As a result, we show that the generalized Fermat-Catalan equation for polynomials: g_1^{d_1} + g_1^{d_2} + ... + g_n^{d_n} = 0 has no non-constant solutions if the greatest common divisor of the terms equals one, no proper subsum vanishes and the hyperbolic sum 1/d_1 + 1/d_2 + ... + 1/d_n is at most 1/(n-2). Furthermore, we show that the generalized Fermat-equation for polynomials g_1^d + g_1^d + ... + g_n^d = 0 has no 'interesting' solutions if d >= n(n-2)."}
{"category": "Math", "title": "Modular Abelian Variety of Odd Modular Degree", "abstract": "We will study modular Abelian varieties with odd congruence numbers, by studying the cuspidal subgroup of $J_0(N)$. We show the conductor of such Abelian varieties must be of a special type, for example if $N$ is odd then $N=p^\\alpha$ or $N=pq$ for some prime $p$ and $q$. We then focus our attention to modular elliptic curves, and using result of Agashe, Ribet, and Stein, we try to classify all elliptic curves of odd modular degree. Our studies prove many cases of the Stein and Watkins's conjecture on elliptic curves with odd modular degree."}
{"category": "Math", "title": "Rubinstein distance on configurations spaces", "abstract": "By a method inspired of the Stein's method, we derive an upper-bound of the Rubinstein distance between two absolutely continuous probability measures on configurations space. As an application, we show that the best way to approximate a Modulated Poisson Process (see below for the definition) by a Poisson process is to equate their intensity."}
{"category": "Math", "title": "Subrings which are closed with respect to taking the inverse", "abstract": "Let S be a subring of the ring R. We investigate the question of whether S intersected by U(R) is equal to U(S) holds for the units. In many situations our answer is positive. There is a special emphasis on the case when R is a full matrix ring and S is a structural subring of R defined by a reflexive and transitive relation."}
{"category": "Math", "title": "Blocking: New examples and properties of products", "abstract": "We say that a pair of points x and y is secure if there exist a finite set of blocking points such that any geodesic between x and y passes through one of the blocking points. The main point of this paper is to exhibit new examples of blocking phenomena both in the manifold and the billiard table setting. As an approach to this, we study if the product of secure configurations (or manifolds) is also secure. We introduce the concept of midpoint security that imposes that the geodesic reaches a blocking point exactly at its midpoint. We prove that products of midpoint secure configurations are midpoint secure. On the other hand, we give an example of a compact C^1 surface that contains secure configurations that are not midpoint secure. This surface provides the first example of an insecure product of secure configurations, as well as billiard table examples."}
{"category": "Math", "title": "M-estimation of Boolean models for particle flow experiments", "abstract": "Probability models are proposed for passage time data collected in experiments with a device designed to measure particle flow during aerial application of fertilizer. Maximum likelihood estimation of flow intensity is reviewed for the simple linear Boolean model, which arises with the assumption that each particle requires the same known passage time. M-estimation is developed for a generalization of the model in which passage times behave as a random sample from a distribution with a known mean. The generalized model improves fit in these experiments. An estimator of total particle flow is constructed by conditioning on lengths of multi-particle clumps."}
{"category": "Math", "title": "The game chromatic number of random graphs", "abstract": "Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number \\chi_g(G) is the minimum k for which the first player has a winning strategy. In this paper we analyze the asymptotic behavior of this parameter for a random graph G_{n,p}. We show that with high probability the game chromatic number of G_{n,p} is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs."}
{"category": "Math", "title": "Treelets--An adaptive multi-scale basis for sparse unordered data", "abstract": "In many modern applications, including analysis of gene expression and text documents, the data are noisy, high-dimensional, and unordered--with no particular meaning to the given order of the variables. Yet, successful learning is often possible due to sparsity: the fact that the data are typically redundant with underlying structures that can be represented by only a few features. In this paper we present treelets--a novel construction of multi-scale bases that extends wavelets to nonsmooth signals. The method is fully adaptive, as it returns a hierarchical tree and an orthonormal basis which both reflect the internal structure of the data. Treelets are especially well-suited as a dimensionality reduction and feature selection tool prior to regression and classification, in situations where sample sizes are small and the data are sparse with unknown groupings of correlated or collinear variables. The method is also simple to implement and analyze theoretically. Here we describe a variety of situations where treelets perform better than principal component analysis, as well as some common variable selection and cluster averaging schemes. We illustrate treelets on a blocked covariance model and on several data sets (hyperspectral image data, DNA microarray data, and internet advertisements) with highly complex dependencies between variables."}
{"category": "Math", "title": "$z$-Classes of Isometries of The Hyperbolic Space", "abstract": "Let $G$ be a group. Two elements $x, y$ are said to be {\\it $z$-equivalent} if their centralizers are conjugate in $G$. The class equation of $G$ is the partition of $G$ into conjugacy classes. Further decomposition of conjugacy classes into $z$-classes provides an important information about the internal structure of the group. Let $I(\\H^n)$ denote the group of isometries of the hyperbolic $n$-space. We show that the number of $z$-classes in $I(\\H^n)$ is finite. We actually compute their number, cf. theorem 1.3. We interpret the finiteness of $z$-classes as accounting for the finiteness of \"dynamical types\" in $I(\\H^n)$. Along the way we also parametrize conjugacy classes. We mainly use the linear model for the hyperbolic space for this purpose. This description of parametrizing conjugacy classes appears to be new."}
{"category": "Math", "title": "L.V.Kantorovich and Linear Programming", "abstract": "I want to write about what I know and remember about the activities of Leonid Vital'evich Kantorovich, an outstanding scientist of the 20th century; about his dramatic struggle for recognition of his mathematical economic theories; about the initial stage of the history of linear programming; about beautuful Kantorovich metric, about the creation of a new area of mathematical activity related to economic applications, which is called sometimes operation research, sometimes mathematical economics, sometimes linear and convex programming, or economic cybernetics, etc.; about its place in the modern mathematical landscape; and, finally, about several personal impressions of this distinguished scientist. The notes in no way pretend to exhaust these topics."}
{"category": "Math", "title": "Generalized handlebody sets and non-Haken 3-manifolds", "abstract": "In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance between the boundary sets of the handlebodies is zero if and only if the ambient manifold contains a non-separating, two sided incompressible surface. We show that the boundary set is 2-dense in the curve complex, i.e. every vertex is within two edges of a point in the boundary set."}
{"category": "Math", "title": "Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-\\'{E}mery Ricci curvature", "abstract": "In this paper, we prove two generalized versions of the Cheeger-Gromoll splitting theorem via the non-negativity of the Bakry-\\'Emery Ricci curavture on complete Riemannian manifolds."}
{"category": "Math", "title": "A general Schwarz Lemma for almost-Hermitian manifolds", "abstract": "We prove a version of Yau's Schwarz Lemma for general almost-complex manifolds equipped with Hermitian metrics. This requires an extension to this setting of the Laplacian comparison theorem. As an application we show that the product of two almost-complex manifolds does not admit any complete Hermitian metric with bisectional curvature bounded between two negative constants that satisfies some additional mild assumptions."}
{"category": "Math", "title": "Adaptive FDR control under independence and dependence", "abstract": "In the context of multiple hypotheses testing, the proportion $\\pi_0$ of true null hypotheses in the pool of hypotheses to test often plays a crucial role, although it is generally unknown a priori. A testing procedure using an implicit or explicit estimate of this quantity in order to improve its efficency is called adaptive. In this paper, we focus on the issue of False Discovery Rate (FDR) control and we present new adaptive multiple testing procedures with control of the FDR. First, in the context of assuming independent $p$-values, we present two new procedures and give a unified review of other existing adaptive procedures that have provably controlled FDR. We report extensive simulation results comparing these procedures and testing their robustness when the independence assumption is violated. The new proposed procedures appear competitive with existing ones. The overall best, though, is reported to be Storey's estimator, but for a parameter setting that does not appear to have been considered before. Second, we propose adaptive versions of step-up procedures that have provably controlled FDR under positive dependences and unspecified dependences of the $p$-values, respectively. While simulations only show an improvement over non-adaptive procedures in limited situations, these are to our knowledge among the first theoretically founded adaptive multiple testing procedures that control the FDR when the $p$-values are not independent."}
{"category": "Math", "title": "Filtering the Wright-Fisher diffusion", "abstract": "We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t1 < t2 < . . ., the observations y(ti) are such that, given the process (x(t)), the random variables (y(ti)) are independent and the conditional distribution of y(ti) only depends on x(ti). When this conditional distribution has a specific form, we prove that the model ((x(ti), y(ti)), i 1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations."}
{"category": "Math", "title": "Transformations of infinitely divisible distributions via improper stochastic integrals", "abstract": "Let $X^{(\\mu)}(ds)$ be an $\\mathbb{R}^d$-valued homogeneous independently scattered random measure over $\\mathbb{R}$ having $\\mu$ as the distribution of $X^{(\\mu)}((t,t+1])$. Let $f(s)$ be a nonrandom measurable function on an open interval $(a,b)$ where $-\\infty\\leqslant a<b\\leqslant\\infty$. The improper stochastic integral $\\int_{a+}^{b-} f(s)X^{(\\mu)}(ds)$ is studied. Its distribution $\\Phi_f(\\mu)$ defines a mapping from $\\mu$ to an infinitely divisible distribution on $\\mathbb{R}^d$. Three modifications (compensated, essential, and symmetrized) and absolute definability are considered. After their domains are characterized, necessary and sufficient conditions for the domains to be very large (or very small) in various senses are given. The concept of the dual in the class of purely non-Gaussian infinitely divisible distributions on $\\mathbb{R}^d$ is introduced and employed in studying some examples. The $\\tau$-measure $\\tau$ of function $f$ is introduced and whether $\\tau$ determines $\\Phi_f$ is discussed. Related transformations of L\\'evy measures are also studied."}
{"category": "Math", "title": "A local ring such that the map between Grothendieck groups with rational coefficient induced by completion is not injective", "abstract": "In this paper, we construct a local ring $A$ such that the kernel of the map $G_0(A)\\subq \\to G_0(\\hat{A})\\subq$ is not zero, where $\\hat{A}$ is the comletion of $A$ with respect to the maximal ideal, and $G_0()\\subq$ is the Grothendieck group of finitely generated modules with rational coefficient. In our example, $A$ is a two-dimensional local ring which is essentially of finite type over ${\\Bbb C}$, but it is not normal."}
{"category": "Math", "title": "Linear automorphism groups of relatively free groups", "abstract": "Let G be a free group in a variety of groups, but G is not absolutely free. We prove that the group of automorphisms Aut(G) is linear iff G is a virtually nilpotent group."}
{"category": "Math", "title": "Detecting spatial patterns with the cumulant function. Part I: The theory", "abstract": "In climate studies, detecting spatial patterns that largely deviate from the sample mean still remains a statistical challenge. Although a Principal Component Analysis (PCA), or equivalently a Empirical Orthogonal Functions (EOF) decomposition, is often applied on this purpose, it can only provide meaningful results if the underlying multivariate distribution is Gaussian. Indeed, PCA is based on optimizing second order moments quantities and the covariance matrix can only capture the full dependence structure for multivariate Gaussian vectors. Whenever the application at hand can not satisfy this normality hypothesis (e.g. precipitation data), alternatives and/or improvements to PCA have to be developed and studied. To go beyond this second order statistics constraint that limits the applicability of the PCA, we take advantage of the cumulant function that can produce higher order moments information. This cumulant function, well-known in the statistical literature, allows us to propose a new, simple and fast procedure to identify spatial patterns for non-Gaussian data. Our algorithm consists in maximizing the cumulant function. To illustrate our approach, its implementation for which explicit computations are obtained is performed on three family of of multivariate random vectors. In addition, we show that our algorithm corresponds to selecting the directions along which projected data display the largest spread over the marginal probability density tails."}
{"category": "Math", "title": "Compact Manifolds Covered by a Torus", "abstract": "Let $X$ be a connected compact complex manifold admitting a finite surjective map $A \\to X$ from a complex torus $A.$ We prove that up to finite \\'etale cover, $X$ is a product of projective spaces and a torus."}
{"category": "Math", "title": "Triangles with two given integral sides", "abstract": "We study some Diophantine problems related to triangles with two given integral sides. We solve two problems posed by Zolt\\'an Bertalan and we also provide some generalization."}
{"category": "Math", "title": "Arithmetic progressions of squares, cubes and $n$-th powers", "abstract": "In this paper we continue the investigations about unlike powers in arithmetic progression. We provide sharp upper bounds for the length of primitive non-constant arithmetic progressions consisting of squares/cubes and $n$-th powers."}
{"category": "Math", "title": "Note on a paper \"An Extension of a Theorem of Euler\" by Hirata-Kohno et al", "abstract": "In this paper we extend a result of Hirata-Kohno, Laishram, Shorey and Tijdeman on the Diophantine equation $n(n+d)...(n+(k-1)d)=by^2,$ where $n,d,k\\geq 2$ and $y$ are positive integers such that $\\gcd(n,d)=1.$"}
{"category": "Math", "title": "Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds", "abstract": "To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M\\times (-1,0)$. We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M\\times (-1,0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice of contact form $\\theta$), i.e., is an invariant of the contact structure $H$. The approximately Einstein ACH metric $g$ is a generalisation of, and exhibits similar asymptotic boundary behaviour to, Fefferman's approximately Einstein complete K\\\"ahler metric $g_+$ on strictly pseudoconvex domains. The present work demonstrates that the CR-invariant log term coefficient in the asymptotic volume expansion of $g_+$ is in fact a contact invariant. We discuss some implications this may have for CR $Q$-curvature. The formal power series method of finding $g$ is obstructed at finite order. We show that part of this obstruction is given as a one-form on $H^*$. This is a new result peculiar to the partially integrable setting."}
{"category": "Math", "title": "Does heterosexual transmission drive the HIV/AIDS epidemic in Sub-Saharan Africa (or elsewhere)?", "abstract": "A two-sex Basic Reproduction Number (BRN) is used to investigate the conditions under which the Human Immunodeficiency Virus (HIV) may spread through heterosexual contacts in Sub-Saharan Africa. (The BRN is the expected number of new infections generated by one infected individual; the disease spreads if the BRN is larger than 1). A simple analytical expression for the BRN is derived on the basis of recent data on survival rates, transmission probabilities, and levels of sexual activity. Baseline results show that in the population at large (characterized by equal numbers of men and women) the BRN is larger than 1 if every year each person has 82 sexual contacts with different partners. the BRN is also larger than 1 for commercial sex workers (CSWs) and their clients (two populations of different sizes) if each CSW has about 256 clients per year and each client visits one CSW every two weeks. A sensitivity analysis explores the effect on the BRN of a doubling (or a halving) of the transmission probabilities. Implications and extensions are discussed."}
{"category": "Math", "title": "Infinite Horizon and Ergodic Optimal Quadratic Control for an Affine Equation with Stochastic Coefficients", "abstract": "We study quadratic optimal stochastic control problems with control dependent noise state equation perturbed by an affine term and with stochastic coefficients. Both infinite horizon case and ergodic case are treated. To this purpose we introduce a Backward Stochastic Riccati Equation and a dual backward stochastic equation, both considered in the whole time line. Besides some stabilizability conditions we prove existence of a solution for the two previous equations defined as limit of suitable finite horizon approximating problems. This allows to perform the synthesis of the optimal control."}
{"category": "Math", "title": "Spectral Analysis of a Family of Second-Order Elliptic Operators with Nonlocal Boundary Condition Indexed by a Probabilty Measure", "abstract": "Let $D\\subset R^d$ be a bounded domain and let \\[ L=\\frac12\\nabla\\cdot a\\nabla +b\\cdot\\nabla \\] %\\[ %L=\\frac12\\sum_{i,j=1}^da_{i,j}\\frac{\\partial^2}{\\partial x_i\\partial x_j}+\\sum_{i=1}^db_i\\frac{\\partial}{\\partial x_i}, %\\] be a second order elliptic operator on $D$. Let $\\nu$ be a probability measure on $D$. Denote by ${\\mathcal L}$ the differential operator whose domain is specified by the following non-local boundary condition: $$ {\\mathcal D_{{\\mathcal L}}}=\\{f\\in C^2(\\ol{D}): \\int_D f d\\nu = f|_{\\partial D}\\}, $$ and which coincides with $L$ on its domain. It is known that $\\mathcal L$ possesses an infinite sequence of eigenvalues, and that with the exception of the zero eigenvalue, all eigenvalues have negative real part. Define the spectral gap of $\\mathcal {L}$, indexed by $\\nu$, by \\gamma_1(\\nu)\\equiv\\sup\\{\\re \\lambda:0\\neq \\lambda is an eigenvalue for {\\mathcal L}\\}. In this paper we investigate the eigenvalues of $\\mathcal L$ in general and the spectral gap $\\gamma_1(\\nu)$ in particular. The operator $\\mathcal L$ is the generator of a diffusion process with random jumps from the boundary, and $\\gamma_1(\\nu)$ measures the exponential rate of convergence of this process to its invariant measure."}
{"category": "Math", "title": "The bitwisted Cartesian model for the free loop fibration", "abstract": "Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal truncating twisting function, the (co)chain complex of the corresponding bitwisted Cartesian product agrees with the standard Cartier (Hochschild) chain complex of the simplicial (co)chains. The modelling polytopes $F_n$ are constructed. An explicit diagonal on $F_n$ is defined and a multiplicative model for the free loop fibration $\\Omega Y\\to \\Lambda Y\\to Y$ is obtained. As an application we establish an algebra isomorphism $H^*(\\Lambda Y;\\mathbb{Z}) \\approx S(U)\\otimes \\Lambda(s^{_{-1}}U)$ for the polynomial cohomology algebra $H^*(Y;\\mathbb{Z})=S(U).$"}
{"category": "Math", "title": "Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus", "abstract": "We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique."}
{"category": "Math", "title": "Strong confidence intervals for autoregression", "abstract": "In this short note I apply the methodology of game-theoretic probability to calculating non-asymptotic confidence intervals for the coefficient of a simple first order scalar autoregressive model. The most distinctive feature of the proposed procedure is that with high probability it produces confidence intervals that always cover the true parameter value when applied sequentially."}
{"category": "Math", "title": "Asymptotic enumeration of 2-covers and line graphs", "abstract": "In this paper we find asymptotic enumerations for the number of line graphs on $n$-labelled vertices and for different types of related combinatorial objects called 2-covers. We find that the number of 2-covers, $s_n$, and proper 2-covers, $t_n$, on $[n]$ both have asymptotic growth $$ s_n\\sim t_n\\sim B_{2n}2^{-n}\\exp(-\\frac12\\log(2n/\\log n))= B_{2n}2^{-n}\\sqrt{\\frac{\\log n}{2n}}, $$ where $B_{2n}$ is the $2n$th Bell number, while the number of restricted 2-covers, $u_n$, restricted, proper 2-covers on $[n]$, $v_n$, and line graphs $l_n$, all have growth $$ u_n\\sim v_n\\sim l_n\\sim B_{2n}2^{-n}n^{-1/2}\\exp(-[\\frac12\\log(2n/\\log n)]^2). $$ In our proofs we use probabilistic arguments for the unrestricted types of 2-covers and and generating function methods for the restricted types of 2-covers and line graphs."}
{"category": "Math", "title": "Characterization of geodesic flows on T^2 with and without positive topological entropy", "abstract": "In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus $T^2$ for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering $\\Br^2$. Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on $T^2$."}
{"category": "Math", "title": "The Large Sieve Inequality for Integer Polynomial Amplitudes", "abstract": "We obtain a close to the best possible version of the large sieve inequality with amplitudes given by the values of a polynomial with integer coefficients of degree $\\geq 2$."}
{"category": "Math", "title": "Minimal geodesics and topological entropy on T^2", "abstract": "Let (T^2, g) be a two-dimensional Riemannian torus. In this paper we prove that the topological entropy of the geodesic flow restricted to the set of initial conditions of minimal geodesics vanishes, independent of the choice of the Riemannian metric."}
{"category": "Math", "title": "A geometrical link between the circle and sexagesimal system", "abstract": "This paper presents a simple geometrical fact which could relate to the history of mathematics and astronomy. This fact shows a natural link between the circle and the multiples of 6 and it makes it possible to obtain a simple representation of the 12 months of the year, the 24 hours of the day, the 30 days (average number) of the month and the 360 days (approximate number) of the year, which brings us closer to the sexagesimal division of time. Moreover this representation reminds one of the movement of the planets around a centre. Using this fact one will be able also to find geometrically the principal divisor of number 60, to represent numbers in base 60 with a kind of abacus or calculation table and to make a division of the circle into 6 and 12 equal parts. Afterwards one will be able to obtain a division in 360 unequal parts but relatively close to one another, and the goal isn't precisely to obtain an optimal division of the circle in 360 equal parts but to prove that the idea to divide the circle in 360 equal parts can subsequently be suggested by these geometrical facts that have been showed. In this article the author will not answer the following questions: a) What is the origin of the sexagesimal system? b) By which way could one manage to adopt the sexagesimal system starting from the knowledge of the facts exposed in this article and starting from the knowledge of the astronomical data? These questions could be treated, using information of this article, by the readers or later on by the author."}
{"category": "Math", "title": "Generic initial ideals and fibre products", "abstract": "We study the behavior of generic initial ideals with respect to fibre products. In our main result we determine the generic initial ideal of the fibre product with respect to the reverse lexicographic order. As an application we compute the symmetric algebraic shifted complex of two disjoint simplicial complexes as was conjectured by Kalai. This result is the symmetric analogue of a theorem of Nevo who determined the exterior algebraic shifted complex of two disjoint simplicial complexes as predicted by Kalai."}
{"category": "Math", "title": "Normal forms for parabolic Monge-Ampere equations", "abstract": "We find normal forms for parabolic Monge-Ampere equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the analytic case, this equation is shown to have solutions. The other normal forms exhaust the different classes of parabolic Monge-Ampere equations with symmetry properties, namely, the existence of classical or nonholonomic intermediate integrals. Our approach is based on the equivalence between parabolic Monge-Ampere equations and particular distributions on a contact manifold, and involves a classification of vector fields lying in the contact structure. These are divided into three types and described in terms of the simplest ones (characteristic fields of first order PDE's)."}
{"category": "Math", "title": "Ideals in the ring of Colombeau generalized numbers", "abstract": "In this paper, the structure of the ideals in the ring of Colombeau generalized numbers is investigated. Connections with the theories of exchange rings, Gelfand rings and lattice-ordered rings are given. Characterizations for prime, projective, pure and topologically closed ideals are given, answering in particular the questions about prime ideals in [Aragona-Juriaans]. Also z-ideals in the sense of [Mason] are characterized. The quotient rings modulo maximal ideals are shown to be canonically isomorphic with nonstandard fields of asymptotic numbers. Finally, a detailed study of the ideals allows us to prove that (under some set-theoretic assumption) the Hahn-Banach extension property does not hold for a large class of topological modules over the ring of Colombeau generalized numbers."}
{"category": "Math", "title": "A double demonstration of a theorem of Newton, which gives a relation between the coefficient of an algebraic equation and the sums of the powers of its roots", "abstract": "Translation from the Latin original, \"Demonstratio gemina theorematis Neutoniani, quo traditur relatio inter coefficientes cuiusvis aequationis algebraicae et summas potestatum radicum eiusdem\" (1747). E153 in the Enestrom index. In this paper Euler gives two proofs of Newton's identities, which express the sums of powers of the roots of a polynomial in terms of its coefficients. The first proof takes the derivative of a logarithm. The second proof uses induction and the fact that in a polynomial of degree $n$, the coefficient of $x^{n-k}$ is equal to the sum of the products of $k$ roots, times $(-1)^k$."}
{"category": "Math", "title": "The Ring of Integers in the Canonical Structures of the Planes", "abstract": "The \\emph{canonical structures of the plane} are those that result, up to isomorphism, from the rings that have the form $\\mathds{R}[x]/(ax^2+bx+c)$ with $a\\neq 0$.That ring is isomorphic to $\\mathds{R}[\\theta]$, where $\\theta$ is the equivalence class of x, which satisfies $\\theta^2 = (-\\dfrac{c}{a}) + \\theta (-\\dfrac{b}{a})$. On the other hand, it is known that, up to isomorphism, there are only three canonical structures: the corresponding to $\\theta^2 = -1$ (the complex numbers), $\\theta^2 = 1$ (the perplex or hyperbolic numbers) and $\\theta^2 = 0$ (the parabolic numbers). This article copes with the algebraic structure of the rings of integers $\\mathds{Z}[\\theta]$ in the perplex and parabolic cases by \\emph{analogy} to the complex cases: the ring of Gaussian integers. For those rings a \\emph{division algorithm} is proved and it is obtained, as a consequence, the characterization of the prime and irreducible elements."}
{"category": "Math", "title": "Two results from Morita theory of stable model categories", "abstract": "We prove two results from Morita theory of stable model categories. Both can be regarded as topological versions of recent algebraic theorems. One is on recollements of triangulated categories, which have been studied in the algebraic case by J{\\o}rgensen. We give a criterion which answers the following question: When is there a recollement for the derived category of a given symmetric ring spectrum in terms of two other symmetric ring spectra? The other result is on well generated triangulated categories in the sense of Neeman. Porta characterizes the algebraic well generated categories as localizations of derived categories of DG categories. We prove a topological analogon: a topological triangulated category is well generated if and only if it is triangulated equivalent to a localization of the derived category of a symmetric ring spectrum with several objects. Here `topological' means triangulated equivalent to the homotopy category of a spectral model category. Moreover, we show that every well generated spectral model category is Quillen equivalent to a Bousfield localization of a category of modules via a single Quillen functor."}
{"category": "Math", "title": "Jacobi Forms of Critical Weight and Weil Representations", "abstract": "Jacobi forms can be considered as vector valued modular forms, and Jacobi forms of critical weight correspond to vector valued modular forms of weight $\\frac12$. Since the only modular forms of weight $\\frac12$ on congruence subgroups of $\\SL$ are theta series the theory of Jacobi forms of critical weight is intimately related to the theory of Weil representations of finite quadratic modules. This article explains this relation in detail, gives an account of various facts about Weil representations which are useful in this context, and it gives some applications of the theory developed herein by proving various vanishing theorems and by proving a conjecture on Jacobi forms of weight one on $\\SL$ with character."}
{"category": "Math", "title": "Beltrami operators, non--symmetric elliptic equations and quantitative Jacobian bounds", "abstract": "In recent studies on the G-convergence of Beltrami operators, a number of issues arouse concerning injectivity properties of families of quasiconformal mappings. Bojarski, D'Onofrio, Iwaniec and Sbordone formulated a conjecture based on the existence of a so-called primary pair. Very recently, Bojarski proved the existence of one such pair. We provide a general, constructive, procedure for obtaining a new rich class of such primary pairs. This proof is obtained as a slight adaptation of previous work by the authors concerning the nonvanishing of the Jacobian of pairs of solutions of elliptic equations in divergence form in the plane. It is proven here that the results previously obtained when the coefficient matrix is symmetric also extend to the non-symmetric case. We also prove a much stronger result giving a quantitative bound for the Jacobian determinant of the so-called \\emph{periodic} $\\sigma$-harmonic sense preserving homeomorphisms of $\\mathbb C$ onto itself."}
{"category": "Math", "title": "Transient NN random walk on the line", "abstract": "We prove strong theorems for the local time at infinity of a nearest neighbor transient random walk. First, laws of the iterated logarithm are given for the large values of the local time. Then we investigate the length of intervals over which the walk runs through (always from left to right) without ever returning."}
{"category": "Math", "title": "Completely periodic directions and orbit closures of many pseudo-Anosov Teichmueller discs in Q(1,1,1,1)", "abstract": "In this paper, we investigate the closure of a large class of Teichm\\\"uller discs in the stratum Q(1,1,1,1) or equivalently, in a GL^+_2(R)-invariant locus L of translation surfaces of genus three. We describe a systematic way to prove that the GL^+_2(R)-orbit closure of a translation surface in L is the whole of L. The strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of Ratner's theorem to unipotent subgroups acting on an ``adequate'' splitting. This analysis applies for example to all Teichmueller discs stabilized obtained by Thurston's construction with a trace field of degree three which moreover ``obviously not Veech''. We produce an infinite series of such examples and show moreover that the favourable splitting situation does not arise everywhere on L, contrary to the situation in genus two. We also study completely periodic directions on translation surfaces in L. For instance, we prove that completely periodic directions are dense on surfaces obtained by Thurston's construction."}
{"category": "Math", "title": "Nonlinear PDEs and Scale Dependence", "abstract": "The properties of nonlinear PDEs that generate filtered solutions are explored with particular attention given to the constraints on the residual term. The analysis is carried out for nonlinear PDEs with an emphasis on evolution problems recast on space-time-scale. We examine the role of approximation that allow for the generation of solutions on isolated scale slices."}
{"category": "Math", "title": "Local properties of J-complex curves in Lipschitz-continuous structures", "abstract": "We prove the existence of primitive curves and positivity of intersections of $J$-complex curves for Lipschitz-continuous almost complex structures. These results are deduced from the Comparison Theorem for $J$-holomorphic maps in Lipschitz structures, previously known for $J$ of class $C^{1, Lip}$. We also give the optimal regularity of curves in Lipschitz structures. It occurs to be $C^{1,LnLip}$, i.e. the first derivatives of a $J$-complex curve for Lipschitz $J$ are Log-Lipschitz-continuous. A simple example that nothing better can be achieved is given. Further we prove the Genus Formula for $J$-complex curves and determine their principal Puisieux exponents (all this for Lipschitz-continuous $J$-s)."}
{"category": "Math", "title": "Regulators of rank one quadratic twists", "abstract": "We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of an odd quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions."}
{"category": "Math", "title": "Caratheodory-Fejer interpolation and related topics in locally convex spaces", "abstract": "We study Caratheodory-Herglotz functions whose values are continuous operators from a locally convex topological space which admits the factorization property into its conjugate dual space. We show how this case can be reduced to the case of functions whose values are bounded operators from a Hilbert space into itself."}
{"category": "Math", "title": "Note sur les invariants du groupe affine", "abstract": "In the paper, it is proved that any $C^{1}$-function on GL(n) which is locally $P$-invariant (here $P$ is the affine (sub)group of GL(n)) is locally $G$-invairant. There is also a statement for distributions (a very weak form of Baruch's results)."}
{"category": "Math", "title": "Non-commutative Zariski geometries and their classical limit", "abstract": "We undertake a case study of two series of nonclassical Zariski geometries. We show that these geometries can be realised as representations of certain noncommutative $C^*$-algebras and introduce a natural limit construction which for each of the two series produces a classical U(1)-gauge field over a 2-dimensional Riemann surface."}
{"category": "Math", "title": "Notes sur la notion d'invariant caract\\'eristique", "abstract": "Let $G$ be a Lie group acting on a vector space $V$. Given a set of $G$-invariants, one can ask the question : does this set of invariants characterize the group $G$ ? We recall here some known results, ask questions and state some conjectures for different choices of invariants : polynomial functions, orbits, distributions, and different types of groups."}
{"category": "Math", "title": "Log-canonical threshold for curves on a smooth surface", "abstract": "It is shown that the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram."}
{"category": "Math", "title": "Stability of a functional equation of Deeba on semigroups", "abstract": "Let $S$ be a semigroup and $X$ a Banach space. The functional equation $\\phi (xyz)+ \\phi (x) + \\phi (y) + \\phi (z) = \\phi (xy) + \\phi (yz) + \\phi (xz)$ is said to be stable for the pair $(X, S)$ if and only if $f: S\\to X$ satisfying $\\| f(xyz)+f(x) + f(y) + f(z) - f(xy)- f(yz)-f(xz)\\| \\leq \\delta $ for some positive real number $\\delta$ and all $x, y, z \\in S$, there is a solution $\\phi : S \\to X$ such that $f-\\phi$ is bounded. In this paper, among others, we prove the following results: 1) this functional equation, in general, is not stable on an arbitrary semigroup; 2) this equation is stable on periodic semigroups; 3) this equation is stable on abelian semigroups; 4) any semigroup with left (or right) law of reduction can be embedded into a semigroup with left (or right) law of reduction where this equation is stable."}
{"category": "Math", "title": "Representation theory, Radon transform and the heat equation on a Riemannian symmetric space", "abstract": "Let X=G/K be a Riemannian symmetric space of the noncompact type. We give a short exposition of the representation theory related to X, and discuss its holomorphic extension to the complex crown, a G-invariant subdomain in the complexified symmetric space X_\\C=G_\\C/K_\\C. Applications to the heat transform and the Radon transform for X are given."}
{"category": "Math", "title": "Holomorphic geometric models for representations of $C^*$-algebras", "abstract": "Representations of $C^*$-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. Applications of this technique to dilation theory of completely positive maps are explored and the critical role of complexified homogeneous spaces in connection with the Stinespring dilations is pointed out. The general results are further illustrated by a discussion of several specific topics, including similarity orbits of representations of amenable Banach algebras, similarity orbits of conditional expectations, geometric models of representations of Cuntz algebras, the relationship to endomorphisms of ${\\mathcal B}({\\mathcal H})$, and non-commutative stochastic analysis."}
{"category": "Math", "title": "Large Deviations Principle for Self-Intersection Local Times for random walk in dimension d>4", "abstract": "We obtain a large deviations principle for the self-intersection local times for a symmetric random walk in dimension d>4. As an application, we obtain moderate deviations for random walk in random sceneries in some region of parameters."}
{"category": "Math", "title": "The number of open paths in an oriented $\\rho$-percolation model", "abstract": "We study the asymptotic properties of the number of open paths of length $n$ in an oriented $\\rho$-percolation model. We show that this number is $e^{n\\alpha(\\rho)(1+o(1))}$ as $n \\to \\infty$. The exponent $\\alpha$ is deterministic, it can be expressed in terms of the free energy of a polymer model, and it can be explicitely computed in some range of the parameters. Moreover, in a restricted range of the parameters, we even show that the number of such paths is $n^{-1/2} W e^{n\\alpha(\\rho)}(1+o(1))$ for some nondegenerate random variable $W$. We build on connections with the model of directed polymers in random environment, and we use techniques and results developed in this context."}
{"category": "Math", "title": "The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth", "abstract": "We prove that the mapping torus group $\\FN \\rtimes_{\\alpha} \\Z$ of any automorphism $\\alpha$ of a free group $\\FN$ of finite rank $n \\geq 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\\mathcal H(\\alpha)$ of subgroups of $\\FN$ which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of $\\alpha$. Furthermore, we show that $\\FN \\rtimes_{\\alpha} \\Z$ is strongly hyperbolic relative to the mapping torus of the family $\\mathcal H(\\alpha)$. As an application, we use a result of Drutu-Sapir to deduce that $\\FN \\rtimes_{\\alpha} \\Z$ has Rapic Decay."}
{"category": "Math", "title": "A Statistical Theory for the Analysis of Uncertain Systems", "abstract": "This paper addresses the issues of conservativeness and computational complexity of probabilistic robustness analysis. We solve both issues by defining a new sampling strategy and robustness measure. The new measure is shown to be much less conservative than the existing one. The new sampling strategy enables the definition of efficient hierarchical sample reuse algorithms that reduce significantly the computational complexity and make it independent of the dimension of the uncertainty space. Moreover, we show that there exists a one to one correspondence between the new and the existing robustness measures and provide a computationally simple algorithm to derive one from the other."}
{"category": "Math", "title": "Probabilistic Robustness Analysis -- Risks, Complexity and Algorithms", "abstract": "It is becoming increasingly apparent that probabilistic approaches can overcome conservatism and computational complexity of the classical worst-case deterministic framework and may lead to designs that are actually safer. In this paper we argue that a comprehensive probabilistic robustness analysis requires a detailed evaluation of the robustness function and we show that such evaluation can be performed with essentially any desired accuracy and confidence using algorithms with complexity linear in the dimension of the uncertainty space. Moreover, we show that the average memory requirements of such algorithms are absolutely bounded and well within the capabilities of today's computers. In addition to efficiency, our approach permits control over statistical sampling error and the error due to discretization of the uncertainty radius. For a specific level of tolerance of the discretization error, our techniques provide an efficiency improvement upon conventional methods which is inversely proportional to the accuracy level; i.e., our algorithms get better as the demands for accuracy increase."}
{"category": "Math", "title": "Mesures limites pour l'equation de Helmholtz dans le cas non captif", "abstract": "Cet article est consacre a l'etude des mesures limites associees a la solution de l'equation de Helmholtz avec un terme source se concentrant en un point. Le potentiel est suppose regulier et l'operateur non-captif. La solution de l'equation de Schrodinger semi-classique s'ecrit alors micro-localement comme somme finie de distributions lagrangiennes. Sous une hypothese geometrique, qui generalise l'hypothese du viriel, on en deduit que la mesure limite existe et qu'elle verifie des proprietes standard. Enfin, on donne un exemple d'operateur qui ne verifie pas l'hypothese geometrique et pour lequel la mesure limite n'est pas unique. Le cas de deux termes sources est aussi traite."}
{"category": "Math", "title": "Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group", "abstract": "We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil_3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil_3 is entire. Also, the sister surface of an entire minimal graph in Nil_3 is an entire constant mean curvature 1/2 graph in H^2 x R, and conversely. This gives a classification of all entire constant mean curvature 1/2 graphs in H^2 x R. Finally we construct properly embedded constant mean curvature 1/2 annuli in H^2 x R."}
{"category": "Math", "title": "The Euler characteristic of a category as the sum of a divergent series", "abstract": "The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible when the complex is the nerve of a finite category. This provides an alternative definition of the Euler characteristic of a category, which is in many cases equivalent to the original one (math.CT/0610260)."}
{"category": "Math", "title": "Repr\\'esentations de Springer pour les groupes de r\\'eflexions complexes imprimitifs", "abstract": "To a spetsial complex reflection group, equipped with a root lattice in the sense of Nebe, we attach a certain finite set playing a role which is analogous to the role of the set of unipotent classes of an algebraic group. In the case of imprimitive groups, we give a combinatoric parametrization of it in terms of Malle-Shoji generalized symbols. This result provides a link between the works of Shoji on Green functions for complex reflection groups and of Broue, Kim, Malle, Rouquier, et. al. on the cyclotomic Hecke algebras and their families of characters. ----- A un groupe de reflexions complexe spetsial, muni d'un reseau radiciel au sens de Nebe, nous associons un certain ensemble fini qui doit jouer un role analogue a celui de l'ensemble des classes unipotentes d'un groupe algebrique. Dans le cas des groupes imprimitifs, nous en donnons un parametrage combinatoire en termes des symboles generalises de Malle et Shoji. Ce resultat fournit un lien entre les travaux de Shoji sur les fonctions de Green pour les groupes de reflexions complexes et ceux de Broue, Kim, Malle, Rouquier, et al. sur les algebres de Hecke cyclotomiques et leurs familles de caracteres."}
{"category": "Math", "title": "Explicit Formula for Constructing Binomial Confidence Interval with Guaranteed Coverage Probability", "abstract": "In this paper, we derive an explicit formula for constructing the confidence interval of binomial parameter with guaranteed coverage probability. The formula overcomes the limitation of normal approximation which is asymptotic in nature and thus inevitably introduce unknown errors in applications. Moreover, the formula is very tight in comparison with classic Clopper-Pearson's approach from the perspective of interval width. Based on the rigorous formula, we also obtain approximate formulas with excellent performance of coverage probability."}
{"category": "Math", "title": "Fredholm Modules on P.C.F. Self-Similar Fractals and their Conformal Geometry", "abstract": "The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct Fredholm modules on post critically finite fractals by regular harmonic structures. The modules are d-summable, the summability exponent d coinciding with the spectral dimension of the generalized laplacian operator associated with the regular harmonic structures. The characteristic tools of the noncommutative infinitesimal calculus allow to define a d-energy functional which is shown to be a self-similar conformal invariant."}
{"category": "Math", "title": "Logarithmic limit sets of real semi-algebraic sets", "abstract": "This paper is about the logarithmic limit sets of real semi-algebraic sets, and, more generally, about the logarithmic limit sets of sets definable in an o-minimal, polynomially bounded structure. We prove that most of the properties of the logarithmic limit sets of complex algebraic sets hold in the real case. This include the polyhedral structure and the relation with the theory of non-archimedean fields, tropical geometry and Maslov dequantization."}
{"category": "Math", "title": "Regular and Completely Regular Differential Operators", "abstract": "We define the concept of completely regular ordinary differential operators and give various criteria for operators to belong to this class. We give also criteria for Birkhof regularity of ordinary differential operators in terms of the growth of the Green function and basis property."}
{"category": "Math", "title": "Spectral isolation of naturally reductive metrics on simple Lie groups", "abstract": "We show that within the class of left-invariant naturally reductive metrics $\\mathcal{M}_{\\operatorname{Nat}}(G)$ on a compact simple Lie group $G$, every metric is spectrally isolated. We also observe that any collection of isospectral compact symmetric spaces is finite; this follows from a somewhat stronger statement involving only a finite part of the spectrum."}
{"category": "Math", "title": "Data-driven efficient score tests for deconvolution problems", "abstract": "We consider testing statistical hypotheses about densities of signals in deconvolution models. A new approach to this problem is proposed. We constructed score tests for the deconvolution with the known noise density and efficient score tests for the case of unknown density. The tests are incorporated with model selection rules to choose reasonable model dimensions automatically by the data. Consistency of the tests is proved."}
{"category": "Math", "title": "Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x", "abstract": "We consider a singular Sturm-Liouville expression with the indefinite weight sgn x. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of $\\infty$. Moreover, in this situation, the point $\\infty$ is a regular critical point. We construct an operator $A=(\\sgn x)(-d^2/dx^2+q)$ with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators."}
{"category": "Math", "title": "Deformation theory of representations of prop(erad)s", "abstract": "We study the deformation theory of morphisms of properads and props thereby extending to a non-linear framework Quillen's deformation theory for commutative rings. The associated chain complex is endowed with a Lie algebra up to homotopy structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with a canonical Lie algebra up to homotopy structure in general and a Lie algebra structure only in the Koszul case. In particular, we explicit the deformation complex of morphisms from the properad of associative bialgebras. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of a Lie algebra up to homotopy structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras."}
{"category": "Math", "title": "Brown representability for space-valued functors", "abstract": "In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every small contravariant functor from spaces to spaces which takes coproducts to products up to homotopy and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie's classification of linear functors."}
{"category": "Math", "title": "A spectral sequence to compute L2-Betti numbers of groups and groupoids", "abstract": "We construct a spectral sequence for L2-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L2-Betti numbers of groups and manifolds and obstructions to the existence of normal subrelations in measured equivalence relations."}
{"category": "Math", "title": "The concrete theory of numbers : Problem of simplicity of Fermat number-twins", "abstract": "The problem of simplicity of Fermat number-twins $f_{n}^{\\pm}=2^{2^n}\\pm3$ is studied. The question for what $n$ numbers $f_{n}^{\\pm}$ are composite is investigated. The factor-identities for numbers of a kind $x^2 \\pm k $ are found."}
{"category": "Math", "title": "A vanishing theorem for sheaves of small differential operators in positive characteristic", "abstract": "Let $X$ be a smooth variety over an algebraically closed field $k$ of positive characteristic, ${\\rm D}_X$ the sheaf of PD-differential operators, and ${\\bar D}_X$ its central reduction, the sheaf of small differential operators. In this paper we show that if $X$ is a line-hyperplane incidence variety (a partial flag variety of type $(1,n,n+1)$) or a quadric of arbitrary dimension (in this case the characteristic is supposed to be odd) then ${\\rm H}^{i}(X,{\\bar D}_X)=0$ for $i>0$. Using this vanishing result and the derived localization theorem for crystalline differential operators (\\cite{BMR}) we show that the Frobenius pushforward of the structure sheaf is a tilting bundle on these varieties, provided that $p>h$, the Coxeter number of the corresponding group."}
{"category": "Math", "title": "D-affinity and Frobenius morphism on quadrics", "abstract": "We compute decomposition of Frobenius push-forwards of line bundles on quadrics into a direct sum of line bundles and spinor bundles. As an application we show when the Frobenius push-forward gives a tilting bundle and we apply it to study D-modules on quadrics."}
{"category": "Math", "title": "Toroidal embeddings and polyhedral divisors", "abstract": "Given an effective action of an (n-1)-dimensional torus on an n-dimensional normal affine variety, Mumford constructs a toroidal embedding, while Altmann and Hausen give a description in terms of a polyhedral divisor on a curve. We compare the fan of the toroidal embedding with this polyhedral divisor."}
{"category": "Math", "title": "The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence", "abstract": "We study the pure braid groups $P_n(RP^2)$ of the real projective plane $RP^2$, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence $1 \\to P_m(RP^2 {x_1,...,x_n} \\to P_{n+m}(RP^2) \\stackrel{p_{\\ast}}{\\to} P_n(RP^2) \\to 1$, where $n\\geq 2$ and $m\\geq 1$, and $p_{\\ast}$ is the homomorphism which corresponds geometrically to forgetting the last $m$ strings. This problem is equivalent to that of the existence of a section for the associated fibration $p: F_{n+m}(RP^2) \\to F_n(RP^2)$ of configuration spaces. Van Buskirk proved in 1966 that $p$ and $p_{\\ast}$ admit a section if $n=2$ and $m=1$. Our main result in this paper is to prove that there is no section if $n\\geq 3$. As a corollary, it follows that $n=2$ and $m=1$ are the only values for which a section exists. As part of the proof, we derive a presentation of $P_n(RP^2)$: this appears to be the first time that such a presentation has been given in the literature."}
{"category": "Math", "title": "Extension of automorphisms to C*-crossed products with non-trivial centre", "abstract": "Given a quasi-special endomorphism $\\rho$ of a C*-algebra A with nontrivial center, we study an extension problem for automorphisms of A to a minimal cross-product B of A by $\\rho$. Exploiting some aspects of the underlying generalized Doplicher-Roberts duality theory based on Pimsner algebras, an obstruction to the existence of such extensions is found and described in terms of sections of a suitable group bundle."}
{"category": "Math", "title": "The intrinsic torsion of almost quaternion-Hermitian manifolds", "abstract": "We study the intrinsic torsion of almost quaternion-Hermitian manifolds via the exterior algebra. In particular, we show how it is determined by particular three-forms formed from simple combinations of the exterior derivatives of the local Kaehler forms. This gives a practical method to compute the intrinsic torsion and is applied in a number of examples. In addition we find simple characterisations of HKT and QKT geometries entirely in the exterior algebra and compute how the intrinsic torsion changes under a twist construction."}
{"category": "Math", "title": "Sobolev regularity of solutions of the cohomological equation", "abstract": "We refine the theory of the cohomological equation for translation flows on higher genus surfaces with the goal of proving optimal results on the Sobolev regularity of solutions and of distributional obstructions. For typical translation surfaces our results are sharp and we find the expected relation between the regularity of the distributional obstructions and the Lyapunov exponents of the Kontsevich-Zorich renormalization cocycle. As a consequence we exactly determine the dimension of the space of obstructions in each Sobolev regularity class in terms of the Kontsevich-Zorich exponents. For a fixed arbitrary translation surface and a typical direction, our results are probably not optimal but are the best which can be achieved with the available harmonic analysis techniques we have introduced in an earlier paper."}
{"category": "Math", "title": "Weighted lattice polynomials of independent random variables", "abstract": "We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems."}
{"category": "Math", "title": "Indecomposable representations of quivers on infinite-dimensional Hilbert spaces", "abstract": "We study indecomposable representations of quivers on separable infinite-dimensional Hilbert spaces by bounded operators. We consider a complement of Gabriel's theorem for these representations. Let $\\Gamma$ be a finite, connected quiver. If its underlying undirected graph contains one of extended Dynkin diagrams $\\tilde{A_n} (n \\geq 0)$, $\\tilde{D_n} (n \\geq 4)$, $\\tilde{E_6}$,$\\tilde{E_7}$ and $\\tilde{E_8}$, then there exists an indecomposable representation of $\\Gamma$ on separable infinite-dimensional Hilbert spaces."}
{"category": "Math", "title": "Non-abelian free groups admit non-essentially free actions on rooted trees", "abstract": "We show that every countable non-abelian free group $\\Gamma $ admits a spherically transitive action on a rooted tree $T$ such that the action of $\\Gamma $ on the boundary of $T$ is not essentially free. This reproves a result of Bergeron and Gaboriau. The existence of such an action answers a question of Grigorchuk, Nekrashevich and Sushchanskii."}
{"category": "Math", "title": "Scalar extension of bicoalgebroids", "abstract": "After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter--Drinfel'd modules over a bicoalgebroid. It is proved that the Yetter--Drinfel'd category is monoidal and pre--braided just as in the case of bialgebroids, and is embedded into the one--sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of Brzezinski and Militaru [2] and Balint and Slachanyi [1], originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given. Identifying the comodule category over a bicoalgebroid with the category of coalgebras of the associated comonad, we obtain a comonadic (weakened) version of Schauenburg's theorem. Finally, we take a look at the scalar extension and braided cocommutative coalgebras from a (co--)monadic point of view."}
{"category": "Math", "title": "Multiplicity of solutions of a zero mass nonlinear equation on a Riemannian manifold", "abstract": "The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory."}
{"category": "Math", "title": "Analog of the Skewes number for twin primes", "abstract": "The results of the computer investigation of the sign changes of the difference between the number of twin primes $\\pi_2(x)$ and the Hardy--Littlewood conjecture $c_2\\Li_2(x)$ are reported. It turns out that $\\pi_2(x) - c_2\\Li_2(x)$ changes the sign at unexpectedly low values of $x$ and for $x<2^{42}$ there are over 90000 sign changes of this difference. It is conjectured that the number of sign changes of $\\pi_2(x) - c_2\\Li_2(x)$ for $x\\in (1, T)$ is given by $\\sqrt T/\\log(T)$."}
{"category": "Math", "title": "Moduli spaces of coherent systems of small slope on algebraic curves", "abstract": "Let $C$ be an algebraic curve of genus $g\\ge2$. A coherent system on $C$ consists of a pair $(E,V)$, where $E$ is an algebraic vector bundle over $C$ of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of the space of sections of $E$. The stability of the coherent system depends on a parameter $\\alpha$. We study the geometry of the moduli space of coherent systems for $0<d\\le2n$. We show that these spaces are irreducible whenever they are non-empty and obtain necessary and sufficient conditions for non-emptiness."}
{"category": "Math", "title": "The integral of the supremum process of Brownian motion", "abstract": "In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) A(T), covered by the process in the time interval [0,T]. The Laplace transform of A(T) follows as a consequence. The main proof involves a double Laplace transform of A(T) and is based on excursion theory and local time for Brownian motion."}
{"category": "Math", "title": "Tail estimates for the Brownian excursion area and other Brownian areas", "abstract": "Several Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of the distribution function and for the moments. Our main tool is the two-dimensional saddle point method."}
{"category": "Math", "title": "Internal sets and internal functions in Colombeau theory", "abstract": "Inspired by nonstandard analysis, we define and study internal subsets and internal functions in algebras of Colombeau generalized functions. We prove a saturation principle for internal sets and provide applications to Colombeau algebras."}
{"category": "Math", "title": "A vanishing theorem for a class of logarithmic D-modules", "abstract": "Let $O_X$ (resp. $D_X$) be the sheaf of holomorphic functions (resp. the sheaf of linear differential operators with holomorphic coefficients) on $X$ (=the complex affine n-space). Let $Y$ be a locally weakly quasi-homogeneous free divisor defined by a polynomial $f$. In this paper we prove that, locally, the annihilating ideal of $1/f^k$ over $D_X$ is generated by linear differential operators of order 1 (for $k$ big enough). For this purpose we prove a vanishing theorem for the extension groups of a certain logarithmic $D_X$--module with $O_X$. The logarithmic $D_X$--module is naturally associated with $Y$. This result is related to the so called Logarithmic Comparison Theorem."}
{"category": "Math", "title": "The low-dimensional homotopy of the stable mapping class group", "abstract": "Due to the deep work of Tillmann, Madsen, Weiss and Galatius, the cohomology of the stable mapping class group $\\gaminf$ is known with rational or finite field coefficients. Little is known about the integral cohomology. In this paper, we study the first four cohomology groups. Also, we compute the first few steps of the Postnikov tower of $B \\gaminf^+$, the Quillen plus construction applied to $B \\gaminf$. Our method relies on the Madsen-Weiss theorem, a few known computations of stable homotopy groups of spheres and projective spaces and on a certain action of the binary icosahedral group on a surface. Using the latter, we can also describe an explicit geometric generator of the third homotopy group $\\pi_3 (B \\gaminf)$."}
{"category": "Math", "title": "On the location and classification of all prime numbers", "abstract": "We will describe an algorithm to arrange all the positive and negative integer numbers. This array of numbers permits grouping them in six different Classes, $\\alpha$, $\\beta$, $\\gamma$, $\\delta$, $\\epsilon$, and $\\zeta$. Particularly, numbers belong to Class $\\alpha$ are defined as $\\alpha=1+6 n$, and those of Class $\\beta$, as $\\beta=5+6n$, where $n=0,\\pm1,\\pm2,\\pm3,\\pm4,...$ These two Classes $\\alpha$ and $\\beta$,contain: i) all prime numbers, except + 2, -2 and $\\pm$3, which belong to $\\epsilon$, $\\delta$, and $\\gamma$ Classes, respectively, and ii) all the other odd numbers, except those that are multiple of $\\pm$3, according to the sequence $\\pm$9, $\\pm$15, $\\pm$21, $\\pm$27, ... Besides, products between numbers of the Class $\\alpha$, and also those between numbers of the Class $\\beta$, generates numbers belonging to the Class $\\alpha$. On the other side, products between numbers of Class $\\alpha$ with numbers of Class $\\beta$, result in numbers of Class $\\beta$. Then, both Classes $\\alpha$ and $\\beta$ include: i) all the prime numbers except $\\pm$2 and $\\pm$3, and ii) all the products between $\\alpha$ numbers, as $\\alpha\\cdot\\alpha^{\\prime}$; all the products between $\\beta$ numbers, as $\\beta\\cdot\\beta^{\\prime}$; and also all the products between numbers of Classes $\\alpha$ and $\\beta$, as $\\alpha\\cdot\\beta$, which necessarily are composite numbers, whose factorization is completely determined."}
{"category": "Math", "title": "Self-dual polygons and self-dual curves", "abstract": "We study projectively self-dual polygons and curves in the projective plane. Our results provide a partial answer to problem No 1994-17 in the book of Arnold's problems."}
{"category": "Math", "title": "Compositions inside a rectangle and unimodality", "abstract": "Let c^{k,l}(n) be the number of compositions (ordered partitions) of the integer n whose Ferrers diagram fits inside a k-by-l rectangle. The purpose of this note is to give a simple, algebraic proof of a conjecture of Vatter that the sequence c^{k,l}(0), c^{k,l}(1), ..., c^{k,l}(kl) is unimodal. The problem of giving a combinatorial proof of this fact is discussed, but is still open."}
{"category": "Math", "title": "Hyperbolic geometry and moduli of real cubic surfaces", "abstract": "Let M_0^R be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H^4 and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in PO(4,1). We also derive several new and several old results on the topology of M_0^R. Let M_s^R be the moduli space of real cubic surfaces that are stable in the sense of geometric invariant theory. We show that this space carries a hyperbolic structure whose restriction to M_0^R is that just mentioned. The corresponding lattice in PO(4,1), for which we find an explicit fundamental domain, is nonarithmetic."}
{"category": "Math", "title": "A note on Reeb dynamics on the tight 3-sphere", "abstract": "We show that a nondegenerate tight contact form on the 3-sphere has exactly two simple closed Reeb orbits if and only if the differential in linearized contact homology vanishes. Moreover, in this case the Floquet multipliers and Conley-Zehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4-space."}
{"category": "Math", "title": "Coloring and The Lonely Graph", "abstract": "We improve upper bounds on the chromatic number proven independently in \\cite{reedNote} and \\cite{ingo}. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. Using this, we prove that if a graph has an optimal coloring with more than $\\frac{\\omega}{2}$ singleton color classes, then it satisfies $\\chi \\leq \\frac{\\omega + \\Delta + 1}{2}$. It follows that a graph satisfying $n - \\Delta < \\alpha + \\frac{\\omega - 1}{2}$ must also satisfy $\\chi \\leq \\frac{\\omega + \\Delta + 1}{2}$, improving the bounds in \\cite{reedNote} and \\cite{ingo}. We then give a simple argument showing that if a graph satisfies $\\chi > \\frac{n + 3 - \\alpha}{2}$, then it also satisfies $\\chi(G) \\leq \\left\\lceil\\frac{\\omega(G) + \\Delta(G) + 1}{2}\\right\\rceil$. From this it follows that a graph satisfying $n - \\Delta < \\alpha + \\omega$ also satisfies $\\chi(G) \\leq \\left\\lceil\\frac{\\omega(G) + \\Delta(G) + 1}{2}\\right\\rceil$ improving the bounds in \\cite{reedNote} and \\cite{ingo} even further at the cost of a ceiling. In the next sections, we generalize our main lemma to constrained colorings (e.g. r-bounded colorings). We present a generalization of Reed's conjecture to r-bounded colorings and prove the conjecture for graphs with maximal degree close to their order. Finally, we outline some applications (in \\cite{BorodinKostochka} and \\cite{ColoringWithDoublyCriticalEdge}) of the theory presented here to the Borodin-Kostochka conjecture and coloring graphs containing a doubly critical edge."}
{"category": "Math", "title": "A few remarks concerning the geometry of the moduli spaces of semistable sheaves supported on plane curves of multiplicity four", "abstract": "This paper has been withdrawn by the author due to a crucial error in the proofs. The error has been corrected and the paper has been expanded in arXiv:0910.5327"}
{"category": "Math", "title": "On the variety of two dimensional real associative algebras", "abstract": "This paper consists of a description of the variety of two dimensional associative algebras within the framework of Nonstandard Analysis. By decomposing each algebra in A^2 as sum of a Jordan algebra and a Lie algebra, we calculate the isomorphism classes of two dimensional real associative algebras over the field of real numbers and determine the components and the contractions of the variety."}
{"category": "Math", "title": "Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature", "abstract": "Existence and uniqueness in ${\\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\\overrightarrow{Om}$ pointing in the direction of $\\overrightarrow{OM}$, divided by the square of the norm of $\\overrightarrow{OM}$ (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from $O$ (the hyperbolic space); radial upper and lower solutions are constructed which, relying on a previous result in the Cartesian setting, imply their existence."}
{"category": "Math", "title": "On the Calabi-Yau problem for maximal surfaces in L^3", "abstract": "In this paper we construct an example of a weakly complete maximal surface in the Lorentz-Minkowski space L^3, which is bounded by a hyperboloid. Moreover, all the singularities of our example are of lightlike type."}
{"category": "Math", "title": "Nonexistence of permutation binomials of certain shapes", "abstract": "Suppose x^m + c*x^n is a permutation polynomial over GF(p), where p>5 is prime, m>n>0, and c is in GF(p)^*. We prove that gcd(m-n,p-1) is not 2 or 4. In the special case that either (p-1)/2 or (p-1)/4 is prime, this was conjectured in a recent paper by Masuda, Panario and Wang."}
{"category": "Math", "title": "Hilbert $\\widetilde{\\C}$-modules: structural properties and applications to variational problems", "abstract": "We develop a theory of Hilbert $\\widetilde{\\C}$-modules by investigating their structural and functional analytic properties. Particular attention is given to finitely generated submodules, projection operators, representation theorems for $\\widetilde{\\C}$-linear functionals and $\\widetilde{\\C}$-sesquilinear forms. By making use of a generalized Lax-Milgram theorem, we provide some existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form."}
{"category": "Math", "title": "Permutation binomials over finite fields", "abstract": "We prove that if x^m + c*x^n permutes the prime field GF(p), where m>n>0 and c is in GF(p)^*, then gcd(m-n,p-1) > sqrt{p} - 1. Conversely, we prove that if q>=4 and m>n>0 are fixed and satisfy gcd(m-n,q-1) > 2q*(log log q)/(log q), then there exist permutation binomials over GF(q) of the form x^m + c*x^n if and only if gcd(m,n,q-1) = 1."}
{"category": "Math", "title": "Boundary of the braid groups and Markov--Ivanovsky normal form", "abstract": "We describe random walk boundaries (in particular, the Poisson--Furstenberg, or PF-boundary) for a vast family of groups in terms of the hyperbolic boundary of a special free subgroup. We prove that almost all trajectories of the random walk (with respect to an arbitrary nondegenerate measure on the group) converge to points of that boundary. This implies the stability (in the sense of \\cite{Ver}) of the so-called Markov--Ivanovsky normal form for braids."}
{"category": "Math", "title": "On some permutation polynomials over F_q of the form x^r*h(x^((q-1)/d)))", "abstract": "In a recent paper, Akbary and Wang gave a sufficient condition for x^u + x^r to permute GF(q), in terms of the period of a certain sequence involving sums of cosines. As an application they gave necessary and sufficient conditions in case u,r,q satisfy certain special properties. We show that the Akbary-Wang sufficient condition follows from a more general sufficient condition which does not involve sums of cosines. This leads to vastly simpler proofs of the Akbary-Wang results, as well as generalizations to polynomials of the form x^r*h(x^{(q-1)/d})."}
{"category": "Math", "title": "Some families of permutation polynomials over finite fields", "abstract": "We give necessary and sufficient conditions for a polynomial of the form x^r*(1+x^v+x^(2v)+...+x^(kv))^t to permute the elements of the finite field GF(q). Our results yield especially simple criteria in case (q-1)/gcd(q-1,v) is a small prime."}
{"category": "Math", "title": "Norm convergence of multiple ergodic averages for commuting transformations", "abstract": "Let $T_1, ..., T_l: X \\to X$ be commuting measure-preserving transformations on a probability space $(X, \\X, \\mu)$. We show that the multiple ergodic averages $\\frac{1}{N} \\sum_{n=0}^{N-1} f_1(T_1^n x) ... f_l(T_l^n x)$ are convergent in $L^2(X,\\X,\\mu)$ as $N \\to \\infty$ for all $f_1,...,f_l \\in L^\\infty(X,\\X,\\mu)$; this was previously established for $l=2$ by Conze and Lesigne and for general $l$ assuming some additional ergodicity hypotheses on the maps $T_i$ and $T_i T_j^{-1}$ by Frantzikinakis and Kra (with the $l=3$ case of this result established earlier by Zhang). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the $l=2$ case of our arguments are a finitary analogue of those of Conze and Lesigne."}
{"category": "Math", "title": "Weyl closure of hypergeometric systems", "abstract": "We show that A-hypergeometric systems and Horn hypergeometric systems are Weyl closed for very generic parameters."}
{"category": "Math", "title": "Khovanov-Rozansky homology and the braid index of a knot", "abstract": "We prove the existence of a knot whose braid index the Morton-Franks-Williams inequality fails to detect but a related inequality (KR-MFW inequality), which uses new information of Khovanov-Rozansky homology, detects. We also prove, by examples, that there exists infinitely many knots for which the KR-MFW inequality fails to detect the braid indices."}
{"category": "Math", "title": "Stochastic Differential Games with Reflection and Related Obstacle Problems for Isaacs Equations", "abstract": "In this paper we first investigate zero-sum two-player stochastic differential games with reflection with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straight-forward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions of the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs heavily from those used for control problems with reflection, it has its own techniques and its own interest. On the other hand, we also prove a new estimate for RBSDEs being sharper than that in El Karoui, Kapoudjian, Pardoux, Peng and Quenez [7], which turns out to be very useful because it allows to estimate the $L^p$-distance of the solutions of two different RBSDEs by the $p$-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution of the approximating Isaacs equation which is constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle."}
{"category": "Math", "title": "Linearity Defect and Regularity over a Koszul Algebra", "abstract": "Let A be a Koszul algebra, and $mod A$ the category of finitely generated graded left A-modules. The \"linearity defect\" ld_A(M) of $M \\in mod A$ is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which is the Koszul dual S^! of a polynomial ring S. Eisenbud et al. showed that $ld_E(M) < \\infty$ for all $M \\in mod E$. Improving their result, we show the following (and many other facts): (*) If A is a Koszul complete intersection, then $reg_{A^!} (M) < \\infty$ and $ld_{A^!} (M) < \\infty$ for all $M \\in mod A^!$. (**) There is a uniform bound of $ld(J)$, where J is a graded ideal of E."}
{"category": "Math", "title": "On multipliers of Hilbert modules over locally C*-algebras", "abstract": "In this paper, we investigate the structure of the multiplier module of a Hilbert module over a locally C*-algebra and the relationship between the set of all adjointable operators from a Hilbert A-module E to a Hilbert A-module F and the set of all adjointable operators from the multiplier module M(E)of E to the multiplier module M(F) of F."}
{"category": "Math", "title": "Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p", "abstract": "Let $F$ be a function field of characteristic $p>0$, $\\F/F$ a Galois extension with $Gal(\\F/F)\\simeq \\Z_l^d$ (for some prime $l\\neq p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Sel_E(L)_r$ ($r$ any prime) as $L$ varies through the subextensions of $\\F$ via appropriate versions of Mazur's Control Theorem. As a consequence we prove that $Sel_E(\\F)_r$ is a cofinitely generated (in some cases cotorsion) $\\Z_r[[Gal(\\F/F)]]$-module."}
{"category": "Math", "title": "Surface subgroups of right-angled Artin groups", "abstract": "We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\\geq 5$. We construct another eight \"forbidden\" graphs and show that every graph $K$ on $\\le 8$ vertices either contains one of our examples, or contains a hole of length $\\ge 5$, or has the property that $A(K)$ does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a \\RAAG to contain no hyperbolic surface subgroups. We prove that for one of these \"forbidden\" subgraphs $P_2(6)$, the right angled Artin group $A(P_2(6))$ is a subgroup of a (right angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir)."}
{"category": "Math", "title": "Realisability and Localisation", "abstract": "Let $A$ be a differential graded algebra with cohomology ring $H^*A$. A graded module over $H^*A$ is called \\emph{realisable} if it is (up to direct summands) of the form $H^*M$ for some differential graded $A$-module $M$. Benson, Krause and Schwede have stated a local and a global obstruction for realisability. The global obstruction is given by the Hochschild class determined by the secondary multiplication of the $A_{\\infty}$-algebra structure of $H^*A$. In this thesis we mainly consider differential graded algebras $A$ with graded-commutative cohomology ring. We show that a finitely presented graded $H^*A$-module $X$ is realisable if and only if its $\\mathfrak{p}$-localisation $X_{\\mathfrak{p}}$ is realisable for all graded prime ideals $\\mathfrak{p}$ of $H^*A$. In order to obtain such a local-global principle also for the global obstruction, we define the \\emph{localisation of a differential graded algebra $A$ at a graded prime $\\mathfrak{p}$ of $H^*A$}, denoted by $A_{\\mathfrak{p}}$, and show the existence of a morphism of differential graded algebras inducing the canonical map $H^*A \\to (H^*A)_{\\mathfrak{p}}$ in cohomology. The latter result actually holds in a much more general setting: we prove that every smashing localisation on the derived category of a differential graded algebra is induced by a morphism of differential graded algebras. Finally we discuss the relation between realisability of modules over the group cohomology ring and the Tate cohomology ring."}
{"category": "Math", "title": "Natural Moves for Knots and Links", "abstract": "We propose some natural generalizations of Reidemeister moves that do not increase the number of crossings in the generated diagrams. Experimentations make us conjecture that this class of monotonic moves is complete for computing canonical forms and then deciding isotopy."}
{"category": "Math", "title": "On the skein exact squence for knot Floer homology", "abstract": "The aim of this paper is to study the skein exact sequence for knot Floer homology. We prove precise graded version of this sequence, and also one using $\\HFm$. Moreover, a complete argument is also given purely within the realm of grid diagrams."}
{"category": "Math", "title": "Banach-like metrics and metrics of compact sets", "abstract": "We present and study a family of metrics on the space of compact subsets of $R^N$ (that we call ``shapes''). These metrics are ``geometric'', that is, they are independent of rotation and translation; and these metrics enjoy many interesting properties, as, for example, the existence of minimal geodesics. We view our space of shapes as a subset of Banach (or Hilbert) manifolds: so we can define a ``tangent manifold'' to shapes, and (in a very weak form) talk of a ``Riemannian Geometry'' of shapes. Some of the metrics that we propose are topologically equivalent to the Hausdorff metric; but at the same time, they are more ``regular'', since we can hope for a local uniqueness of minimal geodesics. We also study properties of the metrics obtained by isometrically identifying a generic metric space with a subset of a Banach space to obtain a rigidity result."}
{"category": "Math", "title": "Green's formula with $\\bbc^{*}$-action and Caldero-Keller's formula for cluster algebras", "abstract": "It is known that Green's formula over finite fields gives rise to the comultiplications of Ringel-Hall algebras and quantum groups (see\\cite{Green}, also see \\cite{Lusztig}). In this paper, we deduce the projective version of Green's formula in a geometric way. Then following the method of Hubery in \\cite{Hubery2005}, we apply this formula to proving Caldero-Keller's multiplication formula for acyclic cluster algebras of arbitrary type."}
{"category": "Math", "title": "Unit distance graphs with ambiguous chromatic number", "abstract": "First Laszlo Szekely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite graphs whose chromatic numbers depend on the axioms chosen for set theory. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this paper we construct a new class of graphs with ambiguous chromatic number. They are unit distance graphs with vertex set R^n, and hence may be seen as further evidence that the chromatic number of the plane might depend on set theory."}
{"category": "Math", "title": "Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability", "abstract": "In his celebrated paper on area distortion for quasiconformal mappings, Astala showed optimal area distortion bounds and dimension distortion estimates for planar quasiconformal mappings. He asked (Question 4.4) whether a finer result held, namely absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings. This was proved in one particular case relevant for removability questions, in joint work of Astala, Clop, Mateu, Orobitg and the author (\"Distortion of Hausdorff measures and improved Painlev'e removability for bounded quasiregular mappings\", Duke Math J., to appear [ACMOU]) (Theorem 1.1), the other cases remaining open. A related question that we left open in [ACMOU] (Question 4.2) (which was asked by Astala to the author before [ACMOU] in an equivalent form in a personal communication) is whether BMO removability for K-quasiregular mappings and ($L^{\\infty}$) removability for K-quasiregular mappings are indeed different problems. In this paper we give a series of examples answering in the positive Question 4.2 in [ACMOU], at the same time proving sharpness in two different senses of Theorem 1.1 in [ACMOU], and also giving examples that would yield sharpness in those two different senses as well for the absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings, were it to be proved."}
{"category": "Math", "title": "Global parametrices and dispersive estimates for variable coefficient wave equations", "abstract": "In this article we consider variable coefficient, time-dependent wave equations. Using phase space methods we construct outgoing parametrices and prove Strichartz-type estimates globally in time. This is done in the context of C^2 metrics which satisfy a weak aymptotic flatness condition at infinity."}
{"category": "Math", "title": "Nonlinear Dynamics of the 3D Pendulum", "abstract": "A 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom. The pendulum is acted on by a gravitational force. Symmetry assumptions are shown to lead to the planar 1D pendulum and to the spherical 2D pendulum models as special cases. The case where the rigid body is asymmetric and the center of mass is distinct from the pivot location leads to the 3D pendulum. Full and reduced 3D pendulum models are introduced and used to study important features of the nonlinear dynamics: conserved quantities, equilibria, invariant manifolds, local dynamics near equilibria and invariant manifolds, and the presence of chaotic motions. These results demonstrate the rich and complex dynamics of the 3D pendulum."}
{"category": "Math", "title": "Eigenfunctions of transfer operators and cohomology", "abstract": "The eigenfunctions with eigenvalues 1 or -1 of the transfer operator of Mayer are in bijective correspondence with the eigenfunctions with eigenvalue 1 of a transfer operator connected to the nearest integer continued fraction algorithm. This is shown by relating these eigenspaces of these operators to cohomology groups for the modular group with coefficients in certain principal series representations."}
{"category": "Math", "title": "Fa\\`a di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations", "abstract": "We consider the combinatorial Dyson-Schwinger equation X=B^+(P(X)) in the non-commutative Connes-KreimerHopf algebra of planar rooted trees H, where B^+ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra A_P of\\H. We describe all the formal series P such that A_P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of H, organized into three isomorphism classes: a first one, restricted to a olynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Fa\\`a di Bruno Hopf algebra. By taking the quotient, the last classe gives an infinite set of embeddings of the Fa\\`a di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Fa\\`a di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, togetherwith a non commutative version of this embedding."}
{"category": "Math", "title": "Gaussian Approximations of Multiple Integrals", "abstract": "Fix an integer k, and let I(l), l=1,2,..., be a sequence of k-dimensional vectors of multiple Wiener-It\\^o integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as l diverges, the law of I(l) is asymptotically close (for example, in the sense of Prokhorov's distance) to the law of a k-dimensional Gaussian vector having the same covariance matrix as I(l). The main feature of our results is that they require minimal assumptions (basically, boundedness of variances) on the asymptotic behaviour of the variances and covariances of the elements of I(l). In particular, we will not assume that the covariance matrix of I(l) is convergent. This generalizes the results proved in Nualart and Peccati (2005), Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As shown in Marinucci and Peccati (2007b), the criteria established in this paper are crucial in the study of the high-frequency behaviour of stationary fields defined on homogeneous spaces."}
{"category": "Math", "title": "An Infinite Family of Quadratic Quadrinomial APN Functions", "abstract": "We present an infinite family of quadratic APN functions on a finite field of dimension over GF(2) divisible by 3."}
{"category": "Math", "title": "Ubiquity and a general logarithm law for geodesics", "abstract": "There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that they are both in fact a simple consequence of the notion of `local ubiquity'. The local ubiquity framework introduced here is a much simplified and more transparent version of that in \\cite{memoirs}. Furthermore, it leads to a single local ubiquity theorem that unifies the Lebesgue and Hausdorff theories. As an application of our framework we consider the theory of metric Diophantine approximation on limit sets of Kleinian groups. In particular, we obtain a general Hausdorff measure version of Sullivan's logarithm law for geodesics -- an aspect overlooked in \\cite{memoirs}."}
{"category": "Math", "title": "Euler Scheme and Tempered Distributuions", "abstract": "Given a smooth R^d-valued diffusion, we study how fast the Euler scheme with time step 1/n converges in law. To be precise, we look for which class of test functions f the approximate expectation E[f(X^{n,x}_1)] converges with speed 1/n to E[f(X^x_1)]. If X is uniformly elliptic, we show that this class contains all tempered distributions, and all measurable functions with exponential growth. We give applications to option pricing and hedging, proving numerical convergence rates for prices, deltas and gammas."}
{"category": "Math", "title": "The geometry of the critical set of nonlinear periodic Sturm-Liouville operators", "abstract": "We study the critical set C of the nonlinear differential operator F(u) = -u\" + f(u) defined on a Sobolev space of periodic functions H^p(S^1), p >= 1. Let R^2_{xy} \\subset R^3 be the plane z = 0 and, for n > 0, let cone_n be the cone x^2 + y^2 = tan^2 z, |z - 2 pi n| < pi/2; also set Sigma = R^2_{xy} U U_{n > 0} cone_n. For a generic smooth nonlinearity f: R -> R with surjective derivative, we show that there is a diffeomorphism between the pairs (H^p(S^1), C) and (R^3, Sigma) x H where H is a real separable infinite dimensional Hilbert space."}
{"category": "Math", "title": "Affine systems: asymptotics at infinity for fractal measures", "abstract": "We study measures on $\\mathbb{R}^d$ which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions. The construction includes measures arising from affine and contractive iterated function systems with and without overlap (IFSs), i.e., limit measures $\\mu$ induced by a finite family of affine mappings in $\\mathbb{R}^d$ (the focus of our paper), as well as equilibrium measures in complex dynamics. By a systematic analysis of the Fourier transform of the measure $\\mu$ at hand (frequency domain), we identify asymptotic laws, spectral types, dichotomy, and chaos laws. In particular we show that the cases when $\\mu$ is singular carry a gradation, ranging from Cantor-like fractal measures to measures exhibiting chaos, i.e., a situation when small changes in the initial data produce large fluctuations in the outcome, or rather, the iteration limit (in this case the measures). Our method depends on asymptotic estimates on the Fourier transform of $\\mu$ for paths at infinity in $\\mathbb{R}^d$. We show how properties of $\\mu$ depend on perturbations of the initial data, e.g., variations in a prescribed finite set of affine mappings in $\\mathbb{R}^d$, in parameters of a rational function in one complex variable (Julia sets and equilibrium measures), or in the entries of a given infinite positive definite matrix."}
{"category": "Math", "title": "Some intersection numbers of divisors on toroidal compactifications of A_g", "abstract": "We study the top intersection numbers of the boundary and Hodge class divisors on toroidal compactifications of the moduli space $A_g$ of principally polarized abelian varieties and compute those numbers that live away from the stratum which lies over the closure of $A_{g-3}$ in the Satake compactification."}
{"category": "Math", "title": "Knotted Polyhedral Tori", "abstract": "For every knot K with stick number k there is a knotted polyhedral torus of knot type K with 3k vertices. We prove that at least 3k-2 vertices are necessary."}
{"category": "Math", "title": "Monge-Ampere equations and moduli spaces of manifolds of circular type", "abstract": "A (bounded) manifold of circular type is a complex manifold M of dimension n admitting a (bounded) exhaustive real function u, defined on M minus a point x_o, so that: a) it is a smooth solution on $M\\setminus {x_o}$ to the Monge-Amp\\`ere equation $(d d^c u)^n = 0$; b) x_o is a singular point for u of logarithmic type and e^u extends smoothly on the blow up of M at x_o; c) $d d^c (e^u) >0$ at any point of $M\\setminus {x_o}$. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of $\\bC^n$. The moduli spaces of bounded manifolds of circular type are studied. In particular, for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form. It is also shown that the class of normalizing maps for an n-dimensional manifold M is a new holomorphic invariant with the following property: it is parameterized by the points of a finite dimensional real manifold of dimension n^2 when M is a (non-convex) circular domain while it is of dimension $n^2 + 2 n$ when M is a strictly convex domain. New characterizations of the circular domains and of the unit ball are also obtained."}
{"category": "Math", "title": "Conformal quaternionic contact curvature and the local sphere theorem", "abstract": "A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser tensor in CR geometry. It is shown that a quaternionic contact manifold is locally quaternionic contact conformal to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-sasakian structure on the sphere iff the quaternionic contact conformal curvature vanishes."}
{"category": "Math", "title": "Smoothness Theorem for Differential BV Algebras", "abstract": "Associated to a differential BV algebra are two differential graded Lie algebras: we call one classical and the other, which contains a formal h-bar parameter, quantum. The classical dgLa is always smooth formal. In this paper, we give necessary and sufficient conditions for the quantum dgLa to be smooth formal. These conditions are equivalent to the degeneration of a version of the noncommutative Hodge to de Rham spectral sequence. References added."}
{"category": "Math", "title": "Poisson transform for higher-rank graph algebras and its applications", "abstract": "Higher-rank graph generalisations of the Popescu-Poisson transform are constructed, allowing us to develop a dilation theory for higher rank operator tuples. These dilations are joint dilations of the families of operators satisfying relations encoded by the graph structure which we call $\\Lambda$-contractions or $\\Lambda$-coisometries. Besides commutant lifting results and characterisations of pure states on higher rank graph algebras several applications to the structure theory of non-selfadjoint graph operator algebras are presented generalising recent results in special cases."}
{"category": "Math", "title": "Harmonic morphisms and hyperelliptic graphs", "abstract": "We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2-edge-connected graph G which is not a cycle, there is at most one involution $\\iota$ on G for which the quotient $G/\\iota$ is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a non-constant harmonic morphism to the graph B_2 consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points."}
{"category": "Math", "title": "Multiplicity Bounds for Quadratic Monomial Ideals", "abstract": "We prove the multiplicity bounds conjectured by Herzog-Huneke-Srinivasan and Herzog-Srinivasan in the following cases: the strong conjecture for edge ideals of bipartite graphs, and the weaker Taylor bound conjecture for all quadratic monomial ideals. We attach a directed graph to a bipartite graph with perfect matching, and describe operations on the directed graph that would reduce the problem to a Cohen-Macaulay bipartite graph. We determine when equality holds in the conjectured bound for edge ideals of bipartite graphs, and verify that when equality holds, the resolution is pure. We characterize bipartite graphs that have Cohen-Macaulay edge ideals and quasi-pure resolutions."}
{"category": "Math", "title": "A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension", "abstract": "We study the extension of the Kechris-Solecki-Todorcevic dichotomy on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the finite dimension. We first prove that the natural extension does not work in the case of the infinite dimension, for the notion of continuous homomorphism used in the original theorem. Then we solve the problem in the case of the infinite dimension. Finally, we prove that the natural extension works in the case of the infinite dimension, but for the notion of Baire-measurable homomorphism."}
{"category": "Math", "title": "Surfaces from Circles", "abstract": "In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation groups. We discuss discretizations based on M\\\"obius invariant building blocks such as circles and spheres. Concrete problems considered in these lectures include the Willmore energy as well as conformal and curvature line parametrizations of surfaces. In particular we discuss geometric properties of a recently found discrete Willmore energy. The convergence to the smooth Willmore functional is shown for special refinements of triangulations originating from a curvature line parametrization of a surface. Further we treat special classes of discrete surfaces such as isothermic and minimal. The construction of these surfaces is based on the theory of circle patterns, in particular on their variational description."}
{"category": "Math", "title": "Cartan and Berwald connections in the pullback formalism", "abstract": "Adopting the pullback approach to global Finsler geometry, the aim of the present paper is to provide new intrinsic (coordinate-free) proofs of intrinsic versions of the existence and uniqueness theorems for the Cartan and Berwald connections on a Finsler manifold. To accomplish this, the notions of semispray and nonlinear connection associated with a given regular connection, in the pullback bundle, is introduced and investigated. Moreover, it is shown that for the Cartan and Berwald connections, the associated semispray coincides with the canonical spray and the associated nonlinear connection coincides with the Barthel connection. An explicit intrinsic expression relating both connections is deduced."}
{"category": "Math", "title": "Theory of dimension for large discrete sets and applications", "abstract": "We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections between the Erdos and Falconer distance problems in geometric combinatorics and geometric measure theory, respectively."}
{"category": "Math", "title": "The Norm Index Theorem (An Analytic Proof)", "abstract": "We give an analytic proof of the norm index theorem $[I_:K^* N(I_L)] =[L:K]$ for cyclic extensions of number fields using spectral theory of the idele class group."}
{"category": "Math", "title": "New Generalizations of Poisson Algebras", "abstract": "We introduce many new generalizations of Poisson algebras which can be constructed inside the associative algebra of linear transformations over a vector space."}
{"category": "Math", "title": "Proving the triviality of rational points on Atkin-Lehner quotients of Shimura curves", "abstract": "In this paper we give a method for studying global rational points on certain quotients of Shimura curves by Atkin-Lehner involutions. We obtain explicit conditions on such quotients for rational points to be ``trivial'' (coming from CM points only) and exhibit an explicit infinite family of such quotients satisfying these conditions."}
{"category": "Math", "title": "Functoriality and special values of L-functions", "abstract": "This is a semi-expository article concerning Langlands functoriality and Deligne's conjecture on the special values of $L$-functions. The emphasis is on symmetric power $L$-functions associated to a holomorphic cusp form, while appealing to a recent work of Mahnkopf on the special values of automorphic $L$-functions."}
{"category": "Math", "title": "l-Adic representations associated to modular forms over imaginary quadratic fields", "abstract": "Let pi be a regular algebraic cuspidal automorphic representation of GL(2) over an imaginary quadratic number field K such that the central character of pi is invariant under the non-trivial automorphism of K. We show that pi is associated with an l-adic Galois representation rho over K such that at each prime of K outside an explicit finite set the Frobenius polynomial of rho agrees with the Hecke polynomial of pi."}
{"category": "Math", "title": "Piecewise principal comodule algebras", "abstract": "A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra B. We prove that principality is a piecewise property: given N comodule-algebra surjections P->Pi whose kernels intersect to zero, P is principal if and only if all Pi's are principal. Furthermore, assuming the principality of P, we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with B. Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N-families of surjections P->Pi and such that the comodule algebra of global sections is P."}
{"category": "Math", "title": "Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations", "abstract": "Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact K\\\"ahler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions on the possible $f$-vectors of convex polytopes. While the statements of these theorems depend on the choice of a K\\\"ahler class, or its analog, there is usually a cone of possible K\\\"ahler classes. It is then natural to ask whether the HLT and HRR remain true in a mixed context. In this note we present a unified approach to proving the mixed HLT and HRR, generalizing the previously known results, and proving it in new cases such as the intersection cohomology of non-rational polytopes."}
{"category": "Math", "title": "Poincare duality and Periodicity", "abstract": "We construct periodic families of Poincare complexes, partially solving a question of Hodgson that was posed in the proceedings of the 1982 Northwestern homotopy theory conference. We also construct infinite families of Poincare complexes whose top cell falls off after one suspension but which fail to embed in a sphere of codimension one. We give a homotopy theoretic description of the four-fold periodicity in knot cobordism."}
{"category": "Math", "title": "Graphs of functions and vanishing free entropy", "abstract": "Suppose X is an n-tuple of selfadjoint elements in a tracial von Neumann algebra M. If z is a selfadjoint element in M and for some selfadjoint element y in the von Neumann algebra generated by X $\\delta_0(y, z) < \\delta_0(y) + \\delta_0(z)$, then $\\chi(X \\cup \\{z\\}) = -\\infty$ (here $\\chi$ and $\\delta_0$ denote the microstates free entropy and free entropy dimension, respectively). In particular, if z lies in the von Neumann algebra generated by X, then $\\chi(X \\cup \\{z\\}) = -\\infty$. The statement and its proof are motivated by geometric-measure-theoretic results on graphs of functions. A similar statement for the nonmicrostates free entropy is obtained under the much stronger hypothesis that z lies in the algebra generated by X."}
{"category": "Math", "title": "A generalization of the Shestakov-Umirbaev inequality", "abstract": "We give a generalization of the Shestakov-Umirbaev inequality which plays an important role in their solution of the tame generators conjecture."}
{"category": "Math", "title": "Pareto Optima of Multicriteria Integer Linear Programs", "abstract": "We settle the computational complexity of fundamental questions related to multicriteria integer linear programs, when the dimensions of the strategy space and of the outcome space are considered fixed constants. In particular we construct: 1. polynomial-time algorithms to exactly determine the number of Pareto optima and Pareto strategies; 2. a polynomial-space polynomial-delay prescribed-order enumeration algorithm for arbitrary projections of the Pareto set; 3. an algorithm to minimize the distance of a Pareto optimum from a prescribed comparison point with respect to arbitrary polyhedral norms; 4. a fully polynomial-time approximation scheme for the problem of minimizing the distance of a Pareto optimum from a prescribed comparison point with respect to the Euclidean norm."}
{"category": "Math", "title": "Conditions for Generic Initial Ideals to be Almost Reverse Lexicographic", "abstract": "Let $I$ be a homogeneous Artinian ideal in a polynomial ring $R=k[x_1,...,x_n]$ over a field $k$ of characteristic 0. We study an equivalent condition for the generic initial ideal $\\gin(I)$ with respect to reverse lexicographic order to be almost reverse lexicographic. As a result, we show that Moreno-Socias conjecture implies Fr\\\"{o}berg conjecture. And for the case $\\Codim I \\le 3$, we show that $R/I$ has the strong Lefschetz property if and only if $\\gin(I)$ is almost reverse lexicographic. Finally for a monomial complete intersection Artinian ideal $I=(x_1^{d_1},...,x_n^{d_n})$, we prove that $\\gin(I)$ is almost reverse lexicographic if $d_i > \\sum_{j=1}^{i-1} d_j - i + 1$ for each $i \\ge 4$. Using this, we give a positive partial answer to Moreno-Socias conjecture, and to Fr\\\"{o}berg conjecture."}
{"category": "Math", "title": "Generalized Bounded Variation and Inserting point masses", "abstract": "Let $d\\mu$ be a probability measure on the unit circle and $d\\nu$ be the measure formed by adding a pure point to $d\\mu$. We give a simple formula for the Verblunsky coefficients of $d\\nu$ based on a result of Simon. Then we consider $d\\mu_0$, a probability measure on the unit circle with $\\ell^2$ Verblunsky coefficients $(\\alpha_n (d\\mu_0))_{n=0}^{\\infty}$ of bounded variation. We insert $m$ pure points to $d\\mu$, rescale, and form the probability measure $d\\mu_m$. We use the formula above to prove that the Verblunsky coefficients of $d\\mu_m$ are in the form $\\alpha_n(d\\mu_0) + \\sum_{j=1}^m \\frac{\\ol{z_j}^{n} c_j}{n} + E_n$, where the $c_j$'s are constants of norm 1 independent of the weights of the pure points and independent of $n$; the error term $E_n$ is in the order of $o(1/n)$. Furthermore, we prove that $d\\mu_m$ is of $(m+1)$-generalized bounded variation - a notion that we shall introduce in the paper. Then we use this fact to prove that $\\lim_{n \\to \\infty} \\vp_n^*(z, d\\mu_m)$ is continuous and is equal to $D(z, d\\mu_m)^{-1}$ away from the pure points."}
{"category": "Math", "title": "Szego kernels, Toeplitz operators, and equivariant fixed point formulae", "abstract": "Let $\\gamma$ be an automorphism of a polarized complex projective manifold $(M,L)$. Then $\\gamma$ induces an automorphism $\\gamma_k$ of the space of global holomorphic sections of the $k$-th tensor power of $L$, for every $k=1,2,...$; for $k\\gg 0$, the Lefschetz fixed point formula expresses the trace of $\\gamma_k$ in terms of fixed point data. More generally, one may consider the composition of $\\gamma_k$ with the Toeplitz operator associated to some smooth function on $M$. Still more generally, in the presence of the compatible action of a compact and connected Lie group preserving $(M,L,\\gamma)$, one may consider induced linear maps on the equivariant summands associated to the irreducible representations of $G$. In this paper, under familiar assumptions in the theory of symplectic reductions, we show that the traces of these maps admit an asymptotic expansion as $k\\to +\\infty$, and compute its leading term."}
{"category": "Math", "title": "The characteristic quasi-polynomials of the arrangements of root systems and mid-hyperplane arrangements", "abstract": "Let $q$ be a positive integer. In our recent paper, we proved that the cardinality of the complement of an integral arrangement, after the modulo $q$ reduction, is a quasi-polynomial of $q$, which we call the characteristic quasi-polynomial. In this paper, we study general properties of the characteristic quasi-polynomial as well as discuss two important examples: the arrangements of reflecting hyperplanes arising from irreducible root systems and the mid-hyperplane arrangements. In the root system case, we present a beautiful formula for the generating function of the characteristic quasi-polynomial which has been essentially obtained by Ch. Athanasiadis and by A. Blass and B. Sagan. On the other hand, it is hard to find the generating function of the characteristic quasi-polynomial in the mid-hyperplane arrangement case. We determine them when the dimension is less than six."}
{"category": "Math", "title": "Asymptotically Optimal Estimator of the Parameter of Semi-Linear Autoregression", "abstract": "The difference equations $\\xi_{k}=af(\\xi_{k-1})+\\epsilon_{k}$, where $(\\epsilon_k)$ is a square integrable difference martingale, and the differential equation ${\\rm d}\\xi=-af(\\xi){\\rm d}t+{\\rm d}\\eta$, where $\\eta$ is a square integrable martingale, are considered. A family of estimators depending, besides the sample size $n$ (or the observation period, if time is continuous) on some random Lipschitz functions is constructed. Asymptotic optimality of this estimators is investigated."}
{"category": "Math", "title": "Equisingularity classes of birational projections of normal singularities to a plane", "abstract": "Given a birational normal extension S of a two-dimensional local regular ring R, we describe all the equisingularity types of the complete ideals J in R whose blowing-up has some point at which the local ring is analytically isomorphic to S. The problem of classifying the germs of such normal surface singularities was already posed by Spivakovsky (Ann. of Math. 1990). This problem has two parts: discrete and continous. The continous part is to some extent equivalent to the problem of the moduli of plane curve singularities, while the main result of this paper solves completely the discrete part."}
{"category": "Math", "title": "The AS-Cohen-Macaulay property for quantum flag manifolds of minuscule weight", "abstract": "It is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights, their Schubert varieties, big cells, and determinantal varieties are AS-Cohen-Macaulay. The main ingredient in the proof is the notion of a quantum graded algebra with a straightening law, introduced by T.H. Lenagan and L. Rigal [J. Algebra 301 (2006), 670-702]. Using Stanley's Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS-Gorenstein."}
{"category": "Math", "title": "Probability Measures and Effective Randomness", "abstract": "We study the question, ``For which reals $x$ does there exist a measure $\\mu$ such that $x$ is random relative to $\\mu$?'' We show that for every nonrecursive $x$, there is a measure which makes $x$ random without concentrating on $x$. We give several conditions on $x$ equivalent to there being continuous measure which makes $x$ random. We show that for all but countably many reals $x$ these conditions apply, so there is a continuous measure which makes $x$ random. There is a meta-mathematical aspect of this investigation. As one requires higher arithmetic levels in the degree of randomness, one must make use of more iterates of the power set of the continuum to show that for all but countably many $x$'s there is a continuous $\\mu$ which makes $x$ random to that degree."}
{"category": "Math", "title": "Continuous first-passage percolation and continuous greedy paths model: linear growth", "abstract": "We study a random growth model on $\\R^d$ introduced by Deijfen. This is a continuous first-passage percolation model. The growth occurs by means of spherical outbursts with random radii in the infected region. We aim at finding conditions on the distribution of the random radii to determine whether the growth of the process is linear or not. To do so, we compare this model with a continuous analogue of the greedy lattice paths model and transpose results in the lattice setting to the continuous setting."}
{"category": "Math", "title": "Lower bounds for sup + inf and sup * inf and an Extension of Chen-Lin result in dimension 3", "abstract": "We give two results about Harnack type inequalities. First, on compact smooth Riemannian surface without boundary, we have an estimate of the type $\\sup +\\inf$. The second result concerns the solutions of prescribed scalar curvature equation on the unit ball of ${\\mathbb R}^n$ with Dirichlet condition. Next, we give an inequality of the type $(\\sup_K u)^{2s-1} \\times \\inf_{\\Omega} u \\leq c$ for positive solutions of $\\Delta u=Vu^5$ on $\\Omega \\subset {\\mathbb R}^3$, where $K$ is a compact set of $\\Omega$ and $V$ is $s-$ h\\\"olderian, $s\\in ]-1/2,1]$. For the case $s=1/2$, we prove that if $\\min_{\\Omega} u>m>0$ and the h\\\"olderian constant $A$ of $V$ is small enough (in certain meaning), we have the uniform boundedness of the supremum of the solutions of the previous equation on any compact set of $\\Omega$. ----- Nous donnons quelques estimations des solutions d'equations elliptiques sur les surfaces de Riemann et sur des ouverts en dimension n> 2. Nous traitons le cas holderien pour l'equation de la courbure scalaire prescrite en dimension 3."}
{"category": "Math", "title": "Corings with exact rational functors and injective objects", "abstract": "We describe how some aspects of abstract localization on module categories have applications to the study of injective comodules over some special types of corings. We specialize the general results to the case of Doi-Koppinen modules, generalizing previous results in this setting."}
{"category": "Math", "title": "Uniqueness at infinity in time for the Maxwell-Schr\"odinger system with arbitrarily large asymptotic data", "abstract": "We prove the uniqueness of solutions of the Maxwell-Schr\"odinger system with given asymptotic behaviour at infinity in time. The assumptions include suitable restrictions on the growth of solutions for large time and on the accuracy of their asymptotics, but no restriction on their size. The result applies to the solutions with prescribed asymptotics constructed in a previous paper."}
{"category": "Math", "title": "Module Shifts and Measure Rigidity in Linear Cellular Automata", "abstract": "Suppose R is a finite commutative ring of prime characteristic, A is a finite R-module, M:=Z^D x N^E, and F is an R-linear cellular automaton on A^M. If mu is an F-invariant measure which is multiply shift-mixing in a certain way, then we show that mu must be the Haar measure on a coset of some submodule shift of A^M. Under certain conditions, this means mu must be the uniform Bernoulli measure on A^M."}
{"category": "Math", "title": "Constructing Simplicial Branched Covers", "abstract": "Branched covers are applied frequently in topology - most prominently in the construction of closed oriented PL d-manifolds. In particular, strong bounds for the number of sheets and the topology of the branching set are known for dimension d<=4. On the other hand, Izmestiev and Joswig described how to obtain a simplicial covering space (the partial unfolding) of a given simplicial complex, thus obtaining a simplicial branched cover [Adv. Geom. 3(2):191-255, 2003]. We present a large class of branched covers which can be constructed via the partial unfolding. In particular, for d<=4 every closed oriented PL d-manifold is the partial unfolding of some polytopal d-sphere."}
{"category": "Math", "title": "On natural and conformally equivariant quantizations", "abstract": "The concept of conformally equivariant quantizations was introduced by Duval, Lecomte and Ovsienko in \\cite{DLO} for manifolds endowed with flat conformal structures. They obtained results of existence and uniqueness (up to normalization) of such a quantization procedure. A natural generalization of this concept is to seek for a quantization procedure, over a manifold $M$, that depends on a pseudo-Riemannian metric, is natural and is invariant with respect to a conformal change of the metric. The existence of such a procedure was conjectured by P. Lecomte in \\cite{Leconj} and proved by C. Duval and V. Ovsienko in \\cite{DO1} for symbols of degree at most 2 and by S. Loubon Djounga in \\cite{Loubon} for symbols of degree 3. In two recent papers \\cite{MR,MR1}, we investigated the question of existence of projectively equivariant quantizations using the framework of Cartan connections. Here we will show how the formalism developed in these works adapts in order to deal with the conformally equivariant quantization for symbols of degree at most 3. This will allow us to easily recover the results of \\cite{DO1} and \\cite{Loubon}. We will then show how it can be modified in order to prove the existence of conformally equivariant quantizations for symbols of degree 4."}
{"category": "Math", "title": "Constructing Combinatorial 4-Manifolds", "abstract": "Every closed oriented PL 4-manifold is a branched cover of the 4-sphere branched over a PL-surface with finitely many singularities by Piergallini [Topology 34(3):497-508, 1995]. This generalizes a long standing result by Hilden and Montesinos to dimension four. Izmestiev and Joswig [Adv. Geom. 3(2):191-225, 2003] gave a combinatorial equivalent of the Hilden and Montesinos result, constructing closed oriented combinatorial 3-manifolds as simplicial branched covers of combinatorial 3-spheres. The construction of Izmestiev and Joswig is generalized and applied to the result of Piergallini, obtaining closed oriented combinatorial 4-manifolds as simplicial branched covers of simplicial 4-spheres."}
{"category": "Math", "title": "Characteristic Functions of Liftings", "abstract": "We introduce characteristic functions for certain contractive liftings of row contractions. These are multi-analytic operators which classify the liftings up to unitary equivalence and provide a kind of functional model. The most important cases are subisometric and coisometric liftings. We also identify the most general setting which we call reduced liftings. We derive properties of these new characteristic functions and discuss the relation to Popescu's definition for completely non-coisometric row contractions. Finally we apply our theory to completely positive maps and prove a one-to-one correspondence between the fixed point sets of completely positive maps related to each other by a subisometric lifting."}
{"category": "Math", "title": "On the transformation semitopological semigroup", "abstract": "In this paper we introduce the notion of weighted (weakly) almost periodic compactifcation of a semitopological semigroup and generalize this notion to corresponding notion for transformation semigroup.The inclusion relation and equality of some well known function spaces on a weighted transformation semigroup is also investigated."}
{"category": "Math", "title": "Parastrophic invariance of Smarandache quasigroups", "abstract": "Every quasigroup $(L,\\cdot)$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(L,\\pi_i)$, $i\\in \\{1,2,3,4,5,6\\}$. It is shown that $(L,\\pi_i)$ is a Smarandache quasigroup with associative subquasigroup $(S,\\pi_i) \\forall i\\in \\{1,2,3,4,5,6\\}$ if and only if for any of some four $j\\in \\{1,2,3,4,5,6\\}$, $(S,\\pi_j)$ is an isotope of $(S,\\pi_i)$ or $(S,\\pi_k)$ for one $k\\in \\{1,2,3,4,5,6\\}$ such that $i\\ne j\\ne k$. Hence, $(L,\\pi_i)$ is a Smarandache quasigroup with associative subquasigroup $(S,\\pi_i) \\forall i\\in \\{1,2,3,4,5,6\\}$ if and only if any of the six Khalil conditions is true for any of some four of $(S,\\pi_i)$."}
{"category": "Math", "title": "Smarandache Isotopy Theory Of Smarandache: Quasigroups And Loops", "abstract": "The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration includes: Smarandache; isotopy and isomorphy classes, Smarandache $f,g$ principal isotopes and G-Smarandache loops."}
{"category": "Math", "title": "Tangential projections and secant defective varieties", "abstract": "Going one step further in Zak's classification of Scorza varieties with secant defect equal to one, we characterize the Veronese embedding of $\\P^n$ given by the complete linear system of quadrics and its smooth projections from a point as the only smooth irreducible complex and non-degenerate projective subvarieties of $\\P^N$ that can be projected isomorphically into $\\P^{2n}$ when $N\\geq\\binom{n+2}{2}-2$."}
{"category": "Math", "title": "On the Derivatives of Central Loops", "abstract": "The right(left) derivative, $a^{-1},e-$ and $e,a^{-1}-$ isotopes of a C-loop are shown to be C-loops. Furthermore, for a central loop $(L,F)$, it is shown that $\\big\\{F,F^{a^{-1}},F_{a^{-1},e}\\big\\}$ and $\\big\\{F,F_{a^{-1}},F_{e,a^{-1}}\\big\\}$ are systems of isotopic C-loops that obey a form of generalized distributive law. Quasigroup isotopes $(L,\\otimes)$ and $(L,\\ominus)$ of a loop $(L,\\theta)$ and its parastrophe $(L,\\theta ^*)$ respectively are proved to be isotopic if either $(L,\\otimes)$ or $(L,\\ominus )$ is commutative. If $(L,\\theta)$ is a C-loop, then it is shown that $\\big\\{(L,\\theta),(L,\\theta ^*),(L,\\otimes),(L,\\oplus)\\big\\}$ is a system of isotopic C-quasigroup under the above mentioned condition. It is shown that C-loops are isotopic to some finite indecomposable groups of the classes ${\\cal D}_i,i=1,2,3,4,5$ and that the center of such C-loops have a rank of 1,2 or 3."}
{"category": "Math", "title": "On Isotopic Characterization of Central Loops", "abstract": "The representation sets of central loops are investigated and the results obtained are used to construct a finite C-loop. It is shown that for certain types of isotopisms, the central identities are isotopic invariant."}
{"category": "Math", "title": "On Some Autotopisms Of Non-Steiner Central Loops", "abstract": "An algebraic process for the construction of an autotopism for a non-Steiner C-loop is described and this is demonstrated with an example using a known finite C-loop. In every C-loop, two of its parastrophes are not equivalent(equal) it, if and only if both the first and second components of the constructed autotopism and its inverse autotopism are not equal to the identity map. Hence, none of the other three parastrophes is equivalent(equal) to the C-loop. It is proved that the set of autotopisms that prevent a C-loop from being a Steiner loop forms a Steiner triple system."}
{"category": "Math", "title": "On central loops and the central square property", "abstract": "The representation sets of a central square C-loop are investigated. Isotopes of central square C-loops of exponent 4 are shown to be both C-loops and A-loops."}
{"category": "Math", "title": "Algebraic properties of some varieties of central loops", "abstract": "Isotopes of C-loops with unique non-identity squares are shown to be both C-loops and A-loops. The relationship between C-loops and Steiner loops is further studied. Central loops with the weak and cross inverse properties are also investigated. C-loops are found to be Osborn loops if every element in them are squares."}
{"category": "Math", "title": "Invariant measure for a three dimensional nonlinear wave equation", "abstract": "We study the long time behavior of the subcritical (subcubic) defocussing nonlinear wave equation on the three dimensional ball, for random data of low regularity. We prove that for a large set of radial initial data in $\\cap_{s<1/2} H^s(B(0,1))$ the equation is (globally in time) well posed and we construct an invariant measure."}
{"category": "Math", "title": "Random data Cauchy theory for supercritical wave equations I: Local theory", "abstract": "We study the local existence of strong solutions for the cubic nonlinear wave equation with data in $H^s(M)$, $s<1/2$, where $M$ is a three dimensional compact riemannian manifold. This problem is supercritical and can be shown to be strongly ill-posed (in the Hadamard sense). However, after a suitable randomization, we are able to construct local strong solution for a large set of initial data in $H^s(M)$, where $s\\geq 1/4$ in the case of a boundary less manifold and $s\\geq 8/21$ in the case of a manifold with boundary."}
{"category": "Math", "title": "Random data Cauchy theory for supercritical wave equations II : A global existence result", "abstract": "We prove that the subquartic wave equation on the three dimensional ball $\\Theta$, with Dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in $\\cap_{s<1/2} H^s(\\Theta)$. We obtain this result as a consequence of a general random data Cauchy theory for supercritical wave equations developed in our previous work \\cite{BT2} and invariant measure considerations which allow us to obtain also precise large time dynamical informations on our solutions."}
{"category": "Math", "title": "Abstract homotopical methods for theoretical computer science", "abstract": "The purpose of this paper is to collect the homotopical methods used in the development of the theory of flows initialized by author's paper ``A model category for the homotopy theory of concurrency''. It is presented generalizations of the classical Whitehead theorem inverting weak homotopy equivalences between CW-complexes using weak factorization systems. It is also presented methods of calculation of homotopy limits and homotopy colimits using Quillen adjunctions and Reedy categories."}
{"category": "Math", "title": "Explicit computations of all finite index bimodules for a family of II_1 factors", "abstract": "We study II_1 factors M and N associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result: every finite index M-N-bimodule (in particular, every isomorphism between M and N) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index M-M-bimodules is identified with an extended Hecke fusion algebra, providing the first explicit computations of the fusion algebra of a II_1 factor. We obtain in particular explicit examples of II_1 factors with trivial fusion algebra, i.e. only having trivial finite index subfactors."}
{"category": "Math", "title": "Standard Relations of Multiple Polylogarithm Values at Roots of Unity", "abstract": "Let $N$ be a positive integer. In this paper we shall study the special values of multiple polylogarithms at $N$th roots of unity, called multiple polylogarithm values (MPVs) of level $N$. These objects are generalizations of multiple zeta values and alternating Euler sums, which was studied by Euler, and more recently, many mathematicians and theoretical physicists.. Our primary goal in this paper is to investigate the relations among the MPVs of the same weight and level by using the regularized double shuffle relations, regularized distribution relations, lifted versions of such relations from lower weights, and seeded relations which are produced by relations of weight one MPVs. We call relations from the above four families \\emph{standard}. Let $d(w,N)$ be the $\\Q$-dimension of $\\Q$-span of all MPVs of weight $w$ and level $N$. Then we obtain upper bound for $d(w,N)$ by the standard relations which in general are no worse or no better than the one given by Deligne and Goncharov depending on whether $N$ is a prime-power or not, respectively, except for 2- and 3-powers, in which case standard relations seem to be often incomplete whereas Deligne shows that their bound should be sharp by a variant of Grothedieck's period conjecture. This suggests that in general there should be other linear relations among MPVs besides the standard relations, some of which are written down in this paper explicitly with good numerical verification. We also provide a few conjectures which are supported by our computational evidence."}
{"category": "Math", "title": "Conditional large and moderate deviations for sums of discrete random variables. Combinatoric applications", "abstract": "We prove large and moderate deviation principles for the distribution of an empirical mean conditioned by the value of the sum of discrete i.i.d. random variables. Some applications for combinatoric problems are discussed."}
{"category": "Math", "title": "Coherent systems of genus 0, III: Computation of flips for k=1", "abstract": "In this paper we continue the investigation of coherent systems of type $(n,d,k)$ on the projective line which are stable with respect to some value of a parameter $\\alpha$. We consider the case $k=1$ and study the variation of the moduli spaces with $\\alpha$. We determine inductively the first and last moduli spaces and the flip loci, and give an explicit description for ranks 2 and 3. We also determine the Hodge polynomials explicitly for ranks 2 and 3 and in certain cases for arbitrary rank."}
{"category": "Math", "title": "Stable curves and screens on fatgraphs", "abstract": "The mapping class group invariant ideal cell decomposition of the Teichmueller space of a punctured surface times an open simplex has been used in a number of computations. This paper answers a question about the asymptotics of this decomposition, namely, in a given cell of the decomposition, which curves can be short? Screens are a new combinatorial structure which provide an answer to this question. The heart of the calculation here involves Ptolemy transformations and the triangle inequalities on lambda lengths."}
{"category": "Math", "title": "Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity", "abstract": "We analyze the coarse geometry of the Weil-Petersson metric on Teichm\\\"uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the \"pants graph.\" We show that in dimension~3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no non-trivial collection of subsets with respect to which it strongly relatively hyperbolic; this extends a theorem of [BDM] in dimension 6 and higher into the intermediate range (it is hyperbolic if and only if the dimension is 1 or 2 [BF]). Stability and relative stability of quasi-geodesics in dimensions up through 3 provide for a strong understanding of the behavior of geodesics and a complete description of the CAT(0)-boundary of the Weil-Petersson metric via curve-hierarchies and their associated \"boundary laminations.\""}
{"category": "Math", "title": "On strict inclusions in hierarchies of convex bodies", "abstract": "Let $\\mathcal I_k$ be the class of convex $k$-intersection bodies in $\\mathbb{R}^n$ (in the sense of Koldobsky) and $\\mathcal I_k^m$ be the class of convex origin-symmetric bodies all of whose $m$-dimensional central sections are $k$-intersection bodies. We show that 1) $\\mathcal I_k^m\\not\\subset \\mathcal I_k^{m+1}$, $k+3\\le m<n$, and 2) $\\mathcal I_l \\not\\subset \\mathcal I_k$, $1\\le k<l < n-3$."}
{"category": "Math", "title": "On l^p norms of weighted mean matrices", "abstract": "We study $l^{p}$ operator norms of weighted mean matrices using the approaches of Kaluza-Szeg\\\"o and Redheffer. As an application, we prove a conjecture of Bennett."}
{"category": "Math", "title": "Integrable operators and squares of Hankel Matrices", "abstract": "In this note, we find sufficient conditions for an operator with kernel of the form $A(x)B(y)-A(x)B(y)/(x-y)$ (which we call a Tracy-Widom type operator) to be the square of a Hankel operator. We consider two contexts: infinite matrices on $\\ell^2$, and integral operators on the Hardy space $H^2(\\mathbb{T})$. The results can be applied to the discrete Bessel kernel, which is significant in random matrix theory."}
{"category": "Math", "title": "Modified Shephard's problem on projections of convex bodies", "abstract": "We disprove a conjecture of A. Koldobsky asking whether it is enough to compare $(n-2)$-derivatives of the projection functions of two symmetric convex bodies in the Shephard problem in order to get a positive answer in all dimensions."}
{"category": "Math", "title": "Descent on Elliptic Curves and Hilbert's Tenth Problem", "abstract": "Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert's Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers."}
{"category": "Math", "title": "Heisenberg Uncertainty Principle for the q-Bessel Fourier transform", "abstract": "In this paper we uses an I.I. Hirschman-W. Beckner entropy argument to give an uncertainty inequality for the $q$-Bessel Fourier transform: $$ \\mathcal{F}_{q,v}f(x)=c_{q,v}\\int_{0}^{\\infty}f(t)j_{v}(xt,q^{2})t^{2v +1}d_{q}t, $$ where $j_v(x,q)$ is the normalized Hahn-Exton $q$-Bessel function."}
{"category": "Math", "title": "Subsets of F_p^n without three term arithmetic progressions have several large Fourier coefficients", "abstract": "Suppose that f : F_p^n -> [0,1] has expected value t in [p^(-n/9),1] (so, the density t can be quite low!). Furthermore, suppose that support(f) has no three-term arithmetic progressions. Then, we develop non-trivial lower bounds for f_j, which is the jth largest Fourier coefficient of f. This result is similar in spirit to that appearing in an earlier paper [1] by the author; however, in that paper the focus was on the ``small'' Fourier coefficients, whereas here the focus is on the ``large'' Fourier coefficients. Furthermore, the proof in the present paper requires much more sophisticated arguments than those of that other paper."}
{"category": "Math", "title": "Quasi-isometries Between Tubular Groups", "abstract": "We give a method of constructing maps between tubular groups inductively according to a set of strategies. This map will be a quasi-isometry exactly when the set of strategies is consistent. Conversely, if there exists a quasi-isometry between tubular groups, then there is a consistent set of strategies for them. There is an algorithm that will in finite time either produce a consistent set of strategies or decide that such a set does not exist. Consequently, this algorithm decides whether or not the groups are quasi-isometric."}
{"category": "Math", "title": "Variation of Periods Modulo p in Arithmetic Dynamics", "abstract": "Let F : V --> V be a self-morphism of a quasiprojective variety defined over a number field K and let P be a point in V(K) with infinite orbit under iteration of F. For each prime ideal p of good reduction, let m_p(F,P) be the size of the F-orbit of the reduction of P modulo p. Fix any e > 0. We show that for almost all primes p, in the sense of analytic density, the orbit size m_p(F,P) is larger than (log(N(p)))^(1-e), where N(p) is the norm of the ideal p."}
{"category": "Math", "title": "The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring", "abstract": "Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well known patch or constructible topology. The proof is accomplished by use of a von Neumann regular ring canonically associated with $R$."}
{"category": "Math", "title": "Conjugacy classes and invariant subrings of R-automorphisms of R[x]", "abstract": "We consider the group G of R-automorphisms of the polynomial ring R[x] especially in the case where R is the ring of integers modulo n. We describe conjugacy classes in G, and in the case where n = 4, we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G."}
{"category": "Math", "title": "Noncommutative geometry through monoidal categories I", "abstract": "After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way."}
{"category": "Math", "title": "Regulatory Dynamics on Random Networks: Asymptotic Periodicity and Modularity", "abstract": "We study the dynamics of discrete-time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long-term dynamics of the system. We prove that, in a random regulatory network, initial conditions converge almost surely to a periodic attractor. We study the subnetworks, which we call modules, where the periodic asymptotic oscillations are concentrated. We proof that those modules are dynamically equivalent to independent regulatory networks."}
{"category": "Math", "title": "Polynomials with a common composite", "abstract": "Let f and g be nonconstant polynomials over an arbitrary field K. In this paper we study the intersection of the polynomial rings K[f] and K[g], and in particular we ask whether this intersection is larger than K. We completely resolve this question when K has characteristic zero, and in positive characteristic we present various results, examples, and algorithms."}
{"category": "Math", "title": "An accurate finite element method for elliptic interface problems", "abstract": "A finite element method for elliptic problems with discontinuous coefficients is presented. The discontinuity is assumed to take place along a closed smooth curve. The proposed method allows to deal with meshes that are not adapted to the discontinuity line. The (nonconforming) finite element space is enriched with local basis functions. We prove an optimal convergence rate in the $H^1$--norm. Numerical tests confirm the theoretical results."}
{"category": "Math", "title": "K theory of smooth complete toric varieties and related spaces", "abstract": "The K-rings of non-singular complex pro jective varieties as well as quasi- toric manifolds were described in terms of generators and relations in an earlier work of the author with V. Uma. In this paper we obtain a similar description for complete non-singular toric varieties. Indeed, our approach enables us to obtain such a description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope."}
{"category": "Math", "title": "About the isotropy constant of random convex sets", "abstract": "Let K be the symmetric convex hull of m independent random vectors uniformly distributed on the unit sphere of R^n. We prove that, for every $\\delta>0$, the isotropy constant of K is bounded by a constant $c(\\delta)$ with high probability, provided that $m\\geq (1+\\delta)n$."}
{"category": "Math", "title": "k-Ordered Hamilton cycles in digraphs", "abstract": "Given a digraph D, the minimum semi-degree of D is the minimum of its minimum indegree and its minimum outdegree. D is k-ordered Hamiltonian if for every ordered sequence of k distinct vertices there is a directed Hamilton cycle which encounters these vertices in this order. Our main result is that every digraph D of sufficiently large order n with minimum semi-degree at least (n+k)/2 -1 is k-ordered Hamiltonian. The bound on the minimum semi-degree is best possible. An undirected version of this result was proved earlier by Kierstead, S\\'ark\\\"ozy and Selkow."}
{"category": "Math", "title": "Involutions on 3-Manifolds and Self-dual, Binary Codes", "abstract": "We study a correspondence between orientation reversing involutions on compact 3-manifolds with only isolated fixed points and binary, self-dual codes. We show in particular that every such code can be obtained from such an involution. We further relate doubly even codes to Pin^- -structures and Spin-manifolds."}
{"category": "Math", "title": "Parameter Estimation in Manneville-Pomeau Processes", "abstract": "In this work we study a class of stochastic processes $\\{X_t\\}_{t\\in\\N}$, where $X_t = (\\phi \\circ T_s^t)(X_0)$ is obtained from the iterations of the transformation T_s, invariant for an ergodic probability \\mu_s on [0,1] and a continuous by part function $\\phi:[0,1] \\to \\R$. We consider here $T_s:[0,1]\\to [0,1]$ the Manneville-Pomeau transformation. The autocorrelation function of the resulting process decays hyperbolically (or polynomially) and we obtain efficient methods to estimate the parameter s from a finite time series. As a consequence we also estimate the rate of convergence of the autocorrelation decay of these processes. We compare different estimation methods based on the periodogram function, on the smoothed periodogram function, on the variance of the partial sum and on the wavelet theory."}
{"category": "Math", "title": "Secant dimensions of low-dimensional homogeneous varieties", "abstract": "We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all Segre-Veronese embeddings of P^1 * P^1, P^1 * P^1 * P^1, and P^2 * P^1, as well as for the variety F of incident point-line pairs in P^2. For P^2 * P^1 and F the results are new, while the proofs for the other two varieties are more compact than existing proofs. Our main tool is the second author's tropical approach to secant dimensions."}
{"category": "Math", "title": "Exchangeable partitions derived from Markovian coalescents with simultaneous multiple collisions", "abstract": "Kingman derived the Ewens sampling formula for random partitions from the genealogy model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process. M\\\"ohle described the recursion which determines the generalization of the Ewens sampling formula when the lines of descent are governed by a coalescent with multiple collisions. In a recent work by Dong, Gnedin and Pitman, authors exploit an analogy with the theory of regenerative composition and partition structures, and provide various characterizations of the associated exchangeable random partitions. This paper gives parallel results for the further generalized model with lines of descent following a coalescent with simultaneous multiple collisions."}
{"category": "Math", "title": "Equivariant monads and equivariant lifts versus a 2-category of distributive laws", "abstract": "Fix a monoidal category C. The 2-category of monads in the 2-category of C-actegories, colax C-equivarant functors, and C-equivariant natural transformations of colax functors, may be recast in terms of pairs consisting of a usual monad and a distributive law between the monad and the action of C, morphisms of monads respecting the distributive law, and transformations of monads satisfying some compatibility with the actions and distributive laws involved. The monads in this picture may be generalized to actions of monoidal categories, and actions of PRO-s in particular. If C is a PRO as well, then in special cases one gets various distributive laws of a given classical type, for example between a comonad and an endofunctor or between a monad and a comonad. The usual pentagons are in general replaced by multigons, and there are also ``mixed'' multigons involving two distinct distributive laws. Beck's bijection between the distributive laws and lifts of one monad to the Eilenberg-Moore category of another monad is here extended to an isomorphism of 2-categories. The lifts of maps of above mentioned pairs are colax C-equivariant. We finish with a short treatment of relative distributive laws between two pseudoalgebra structures which are relative with respect to the distributivity of two pseudomonads involved, what gives a hint toward the generalizations."}
{"category": "Math", "title": "Analysis of the accuracy and convergence of equation-free projection to a slow manifold", "abstract": "In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis, and A. Zagaris, Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes, SIAM J. Appl. Dyn. Syst. 4 (2005) 711-732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the m-th member of the class of algorithms (m = 0, 1, ...) finds iteratively an approximation of the appropriate zero of the (m+1)-st time derivative of the remaining variables and uses this root to approximate the location of the point on the slow manifold corresponding to these values of the observables. This article is the first of two articles in which the accuracy and convergence of the iterative algorithms are analyzed. Here, we work directly with explicit fast--slow systems, in which there is an explicit small parameter, epsilon, measuring the separation of time scales. We show that, for each m = 0, 1, ..., the fixed point of the iterative algorithm approximates the slow manifold up to and including terms of O(epsilon^m). Moreover, for each m, we identify explicitly the conditions under which the m-th iterative algorithm converges to this fixed point. Finally, we show that when the iteration is unstable (or converges slowly) it may be stabilized (or its convergence may be accelerated) by application of the Recursive Projection Method. Alternatively, the Newton-Krylov Generalized Minimal Residual Method may be used. In the subsequent article, we will consider the accuracy and convergence of the iterative algorithms for a broader class of systems-in which there need not be an explicit small parameter-to which the algorithms also apply."}
{"category": "Math", "title": "Computation of the cover of Shimura curves $X_0(2) \\to X(1)$ for the cyclic cubic field of discriminant 13^2", "abstract": "We compute the canonical model of the cover of Shimura curves $X_0(2) \\to X(1)$ for the cubic field of discriminant 13^2 described at the end of Elkies' paper \"Shimura curves for level 3 subgroups of the (2,3,7) triangle group\". Last, we list the coordinates of some rational CM points on X(1)."}
{"category": "Math", "title": "Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces", "abstract": "This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the Krasnoselski-Mann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from prima-facie ineffective proofs in the fixed point theory of such mappings."}
{"category": "Math", "title": "On the solutions of a boundary value problem arising in free convection with prescribed heat flux", "abstract": "For given $a\\in\\R$, c<0, we are concerned with the solution $f^{}_b$ of the differential equation $f^{\\prime\\prime\\prime}+ff^{\\prime\\prime}+\\g(f^{\\prime})=0$, satisfying the initial conditions $f(0)=a$, $f'(0)=b$, $f''(0)=c< 0$, where g is some nonnegative subquadratic locally Lipschitz function. It is proven that there exists $b_*>0$ such that $f^{}_b$ exists on $[0,+\\infty)$ and is such that $f'_b(t)\\to 0$ as $t\\to+\\infty$, if and only if $b\\geq b_*$. This allows to answer questions about existence, uniqueness and boundedness of solutions to a boundary value problem arising in fluid mechanics, and especially in boundary layer theory."}
{"category": "Math", "title": "A partial $A_\\infty$-structure on the cohomology of $C_n\\times C_m$", "abstract": "Suppose k is a field of characteristic 2, and $n,m\\geq 4$ powers of 2. Then the $A_\\infty$-structure of the group cohomology algebras $H^*(C_n,k)$ and $(H^*(C_m,k)$ are well known. We give results characterizing an $A_\\infty$-structure on $H^*(C_n\\times C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations."}
{"category": "Math", "title": "Asymptotic regimes for the occupancy scheme of multiplicative cascades", "abstract": "In the classical occupancy scheme, one considers a fixed discrete probability measure ${\\bf p}=(p_i: {i\\in{\\cal I}})$ and throws balls independently at random in boxes labeled by ${\\cal I}$, such that $p_i$ is the probability that a given ball falls into the box $i$. In this work, we are interested in asymptotic regimes of this scheme in the situation induced by a refining sequence $({\\bf p}(k) : k\\in\\N)$ of random probability measures which arise from some multiplicative cascade. Our motivation comes from the study of the asymptotic behavior of certain fragmentation chains"}
{"category": "Math", "title": "Birational Calabi-Yau 3-folds and BPS state counting", "abstract": "This paper contains some applications of Bridgeland-Douglas stability conditions on triangulated categories, and Joyce's work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of motivic Gopakumar-Vafa invariants as counting invariants of D2-branes, and show that they are invariant under birational transformations between Calabi-Yau 3-folds. The result is similar to the fact that birational Calabi-Yau 3-folds have the same betti numbers or Hodge numbers."}
{"category": "Math", "title": "Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I", "abstract": "We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic $(0,p)$ of integral canonical models of projective Shimura varieties of Hodge type with respect to h--hyperspecial subgroups as pro-\\'etale covers of N\\'eron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle."}
{"category": "Math", "title": "On certain period relations for cusp forms on GL_n", "abstract": "Let $\\pi$ be a regular algebraic cuspidal automorphic representation of ${\\rm GL}_n({\\mathbb A}_F)$ for a number field $F$. We consider certain periods attached to $\\pi$. These periods were originally defined by Harder when $n=2$, and later by Mahnkopf when $F = {\\mathbb Q}$. In the first part of the paper we analyze the behaviour of these periods upon twisting $\\pi$ by algebraic Hecke characters. In the latter part of the paper we consider Shimura's periods associated to a modular form. If $\\phi_{\\chi}$ is the cusp form associated to a character $\\chi$ of a quadratic extension, then we relate the periods of $\\phi_{\\chi^n}$ to those of $\\phi_{\\chi}$, and as a consequence give another proof of Deligne's conjecture on the critical values of symmetric power $L$-functions associated to dihedral modular forms. Finally, we make some remarks on the symmetric fourth power $L$-functions."}
{"category": "Math", "title": "Iterating the Pimsner construction", "abstract": "For $A$ a $C^*$-algebra, $E_1, E_2$ two Hilbert bimodules over $A$, and a fixed isomorphism $\\chi : E_1\\otimes_AE_2\\to E_2\\otimes_AE_1$, we consider the problem of computing the $K$-theory of the Cuntz-Pimsner algebra ${\\mathcal O}_{E_2\\otimes_A{\\mathcal O}_{E_1}}$ obtained by extending the scalars and by iterating the Pimsner construction. The motivating examples are a commutative diagram of Douglas and Howe for the Toeplitz operators on the quarter plane, and the Toeplitz extensions associated by Pimsner and Voiculescu to compute the $K$-theory of a crossed product. The applications are for Hilbert bimodules arising from rank two graphs and from commuting endomorphisms of abelian $C^*$-algebras."}
{"category": "Math", "title": "On the explicit construction of higher deformations of partition statistics", "abstract": "The modularity of the partition generating function has many important consequences, for example asymptotics and congruences for $p(n)$. In a series of papers the author and Ono \\cite{BO1,BO2} connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms and which were first considered in \\cite{BF}. Here we do a further step towards understanding how weak Maass forms arise from interesting partition statistics by placing certain 2-marked Durfee symbols introduced by Andrews \\cite{An1} into the framework of weak Maass forms. To do this we construct a new class of functions which we call quasiweak Maass forms because they have quasimodular forms as components. As an application we prove two conjectures of Andrews. It seems that this new class of functions will play an important role in better understanding weak Maass forms of higher weight themselves, and also their derivatives. As a side product we introduce a new method which enables us to prove transformation laws for generating functions over incomplete lattices."}
{"category": "Math", "title": "On the subdivision of small categories", "abstract": "We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of spaces and small categories, by using partially ordered sets. This yields a new conceptual proof to the well-known fact that these two homotopy categories are equivalent."}
{"category": "Math", "title": "Cyclotomic $q$-Schur algebras associated to the Ariki-Koike algebra", "abstract": "Let $S$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $H_{n,r}$ of rank $n$, introduced by Dipper-James-Mathas. For each $p = (r_1, ..., r_g)$ such that $r_1 + ... + r_g = r$, we define a subalgebra $S^p$ of $S$ and its quotient algebra $\\bar S^p$. It is shown that $S^p$ is a standardly based algebra and $\\bar S^p$ is a cellular algebra. By making use of these algebras, we show that certain decomposition numbers for $S$ can be expressed as a product of decomposition numbers for cyclotomic $q$-Schur algebras associated to smaller Ariki_koike algebras $H_{n_k,r_k}$."}
{"category": "Math", "title": "Fields of invariants of finite linear groups", "abstract": "The survey is devoted to the rationality question of finite linear groups. We concentrate on lower-dimensional cases, especially on the case of dimension four."}
{"category": "Math", "title": "Relative log convergent cohomology and relative rigid cohomology I", "abstract": "In this paper, we develop the theory of relative log convergent cohomology. We prove the coherence of relative log convergent cohomology in certain case by using the comparison theorem between relative log convergent cohomlogy and relative log crystalline cohomology, and we relates relative log convergent cohomology to relative rigid cohomology to show the validity of Berthelot's conjecture on the coherence and the overconvergence of relative rigid cohomology for proper smooth families when they admit nice proper log smooth compactification to which the coefficient extends logarithmically."}
{"category": "Math", "title": "Relative log convergent cohomology and relative rigid cohomology II", "abstract": "In this paper, we develop the theory of relative log convergent cohomology of radius $\\lambda$ ($0 < \\lambda \\leq 1$), which is a generalization of the notion of relative log convergent cohomology in the previous paper. By comparing this cohomology with relative log crystalline cohomology, relative rigid cohomology and its variants and by using some technique of hypercovering, we prove a version of Berthelot's conjecture on the overconvergence of relative rigid cohomology for proper smooth families."}
{"category": "Math", "title": "Statistical properties of a generalized threshold network model", "abstract": "The threshold network model is a type of finite random graphs. In this paper, we introduce a generalized threshold network model. A pair of vertices with random weights is connected by an edge when real-valued functions of the pair of weights belong to given Borel sets. We extend several known limit theorems for the number of prescribed subgraphs to show that the strong law of large numbers can be uniform convergence. We also prove two limit theorems for the local and global clustering coefficients."}
{"category": "Math", "title": "On the Graf's addition theorem for Hahn Exton q-Bessel function", "abstract": "In this paper we study the positivity of the generalized $q$-translation associated with the $q$-Bessel Hahn Exton function which is deduced by a new formulation of the Graf's addition formula related to this function."}
{"category": "Math", "title": "Random environment on coloured trees", "abstract": "In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this model and also show how our model generalizes many other probabilistic models, including random walk in random environment on trees, recursive distributional equations and multi-type branching random walk on $\\mathbb{R}$."}
{"category": "Math", "title": "Algebraic Connections vs. Algebraic {$\\cD$}-modules: inverse and direct images", "abstract": "In the dictionary between the language of (algebraic integrable) connections and that of (algebraic) $\\cD$-modules, to compare the definitions of inverse images for connections and $\\cD$-modules is easy. But the comparison between direct images for connections (the classical construction of the Gauss-Manin connection for smooth morphisms) and for $\\cD$-modules, although known to specialists, has been explicitly proved only recently in a paper of Dimca, Maaref, Sabbah and Saito in 2000, where the authors' main technical tool was M. Saito's equivalence between the derived category of $\\cD$-modules and a localized category of differential complexes. The aim of this short paper is to give a simplified summary of the [DMSS] argument, and to propose an alternative proof of this comparison which is simpler, in the sense that it does not use Saito equivalence. Moreover, our alternative strategy of comparison works in a context which is a precursor to the Gauss-Manin connection (at the level of $f^{-1}\\cD_Y$-modules, for a morphism $f:X\\to Y$), and may be of some intrinsic interest."}
{"category": "Math", "title": "On decomposition numbers with Jantzen filtration of cyclotomic $q$-Schur algebras", "abstract": "Let $\\Sc(\\vL)$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $\\He_{n,r}$, introduced by Dipper-James-Mathas. In this paper, we consider $v$-decomposition numbers of $\\Sc(\\vL)$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji-Wada, a product formula for $v$-decomposition numbers of $\\Sc(\\vL)$, which asserts that certain $v$-decomposition numbers are expressed as a product of $v$-decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike algebras $\\He_{n_i,r_i}$ of smaller rank. Moreover we prove a similar formula for $v$-decomposition numbers of $\\He_{n,r}$ by using a Schur functor."}
{"category": "Math", "title": "On the divisor function and the Riemann zeta-function in short intervals", "abstract": "We obtain, for $T^\\epsilon \\le U=U(T)\\le T^{1/2-\\epsilon}$, asymptotic formulas for $$ \\int_T^{2T}(E(t+U) - E(t))^2 dt,\\quad \\int_T^{2T}(\\Delta(t+U) - \\Delta(t))^2 dt, $$ where $\\Delta(x)$ is the error term in the classical divisor problem, and $E(T)$ is the error term in the mean square formula for $|\\zeta(1/2+it)|$. Upper bounds of the form $O_\\epsilon(T^{1+\\epsilon}U^2)$ for the above integrals with biquadrates instead of square are shown to hold for $T^{3/8} \\le U =U(T) \\ll T^{1/2}$. The connection between the moments of $E(t+U) - E(t)$ and $|\\zeta(1/2+it)|$ is also given. Generalizations to some other number-theoretic error terms are discussed."}
{"category": "Math", "title": "E_0-dilation of strongly commuting CP_0-semigroups", "abstract": "We prove that every strongly commuting pair of CP_0-semigroups has a minimal E_0-dilation. This is achieved in two major steps, interesting in themselves: 1: we show that a strongly commuting pair of CP_0-semigroups can be represented via a two parameter product system representation; 2: we prove that every fully coisometric product system representation has a fully coisometric, isometric dilation. In particular, we obtain that every commuting pair of CP_0-semigroups on B(H), H finite dimensional, has an E_0-dilation."}
{"category": "Math", "title": "On Excess of the Odious Primes", "abstract": "We give a more strong heuristic justification of our conjecture on the excess of the odious primes."}
{"category": "Math", "title": "On the discriminant of elliptic curves with non-trivial torsion", "abstract": "For those elliptic curves defined over the rational with non--trivial torsion subgroup, we find a tight relationship between the torsion subgroup itself and a Galois group naturally arising from the curve."}
{"category": "Math", "title": "Harmonic analysis on local fields and adelic spaces I", "abstract": "We develop a harmonic analysis on objects of some category $C_2$ of infinite-dimensional filtered vector spaces over a finite field. It includes two-dimensional local fields and adelic spaces of algebraic surfaces defined over a finite field. The main result is the theory of the Fourier transform on these objects and two-dimensional Poisson formulas."}
{"category": "Math", "title": "The spectral data for Hamiltonian stationary Lagrangian tori in R^4", "abstract": "This article determines the spectral data, in the integrable systems sense, for all weakly conformally immersed Hamiltonian stationary Lagrangian in $\\R^4$. This enables us to describe their moduli space and the locus of branch points of such an immersion. This is also an informative example in integrable systems geometry, since the group of ambient isometries acts non-trivially on the spectral data and the relevant energy functional (the area) need not be constant under deformations by higher flows."}
{"category": "Math", "title": "On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions", "abstract": "We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. When the interspecific competition parameter tends to infinity, the system solution converges to that of a freeboundary problem. If all stationary solutions of this limit problem are non-degenerate and if a certain linear combination of the boundary data does not identically vanish, then for sufficiently large interspecific competition, all non-negative solutions of the competition-diffusion system converge to stationary states as time tends to infinity. Such dynamics are much simpler than those found for the corresponding system with either homogeneous Neumann or homogeneous Dirichlet boundary conditions."}
{"category": "Math", "title": "A note on the Hayman-Wu theorem", "abstract": "The Hayman-Wu theorem states that the preimage of a line or circle L under a conformal mapping from the unit disc to a simply-connected domain U has total Euclidean length bounded by an absolute constant. The best possible constant is known to lie in the interval [pi^2, 4 pi), thanks to work of {\\O}yma and Rohde. Earlier, Brown Flinn showed that the total length is at most pi^2 in the special case in which U contains L. Let r be the anti-M\\\"obius map that fixes L pointwise. In this note we extend the sharp bound pi^2 to the case where each connected component of the intersection of U with r(U) is bounded by one arc of U and its image under r. We also strengthen the bounds slightly by replacing Euclidean length with the strictly larger spherical length restricted to the unit disc."}
{"category": "Math", "title": "The Poisson Kernel for Hardy Algebras", "abstract": "This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras $H^{\\infty}(E)$, which we call Hardy algebras, and which are noncommutative generalizations of classical $H^{\\infty}$, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which ``reproduces'' the values, on $\\mathbb{D}((E^{\\sigma})^*)$, of the ``functions'' coming from $H^{\\infty}(E)$. We present results that are natural generalizations of the Poisson integral formuala. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the ``model space'' for the canonical model that can be attached to a point in the disc $\\mathbb{D}((E^{\\sigma})^*)$. We also connect our Poission kernel to various \"point evaluations\" and to the idea of curvature."}
{"category": "Math", "title": "A new approach to the giant component problem", "abstract": "We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np=1+omega(n)n^{-1/3}, where omega(n) tends to infinity arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs."}
{"category": "Math", "title": "Canonical connections on paracontact manifolds", "abstract": "The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $\\mathcal{D}$-homothetic transformation is determined as a special gauge transformation. The $\\eta$-Einstein manifold are defined, it is prove that their scalar curvature is a constant and it is shown that in the paraSasakian case these spaces can be obtained from Einstein paraSasakian manifolds with a $\\mathcal{D}$-homothetic transformations. It is shown that an almost paracontact structure admits a connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor of the paracontact structure is skew-symmetric and the defining vector field is Killing."}
{"category": "Math", "title": "A note on the radial solutions for the supercritical Henon equation", "abstract": "We prove the existence of a positive radial solution for the H\\'enon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions."}
{"category": "Math", "title": "Equations for Chow and Hilbert Quotients", "abstract": "We give explicit equations for the Chow and Hilbert quotients of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety. As a consequence we provide GIT descriptions of these canonical quotients, and obtain other GIT quotients of X by variation of GIT quotient. We apply these results to find equations for the moduli space \\bar{M}_{0,n} of stable genus zero n-pointed curves as a subvariety of a smooth toric variety defined via tropical methods."}
{"category": "Math", "title": "Universal L^s -rate-optimality of L^r-optimal quantizers by dilatation and contraction", "abstract": "Let $ r, s>0 $. For a given probability measure $P$ on $\\mathbb{R}^d$, let $(\\alpha_n)_{n \\geq 1}$ be a sequence of (asymptotically) $L^r(P)$- optimal quantizers. For all $\\mu \\in \\mathbb{R}^d $ and for every $\\theta >0$, one defines the sequence $(\\alpha_n^{\\theta, \\mu})_{n \\geq 1}$ by : $\\forall n \\geq 1, \\alpha_n^{\\theta, \\mu} = \\mu + \\theta(\\alpha_n - \\mu) = \\{\\mu + \\theta(a- \\mu), a \\in \\alpha_n \\} $. In this paper, we are interested in the asymptotics of the $L^s$-quantization error induced by the sequence $(\\alpha_n^{\\theta, \\mu})_{n \\geq 1}$. We show that for a wide family of distributions, the sequence $(\\alpha_n^{\\theta, \\mu})_{n \\geq 1}$ is $L^s$-rate-optimal. For the Gaussian and the exponential distributions, one shows how to choose the parameter $\\theta$ such that $(\\alpha_n^{\\theta, \\mu})_{n \\geq 1}$ satisfies the empirical measure theorem and probably be asymptotically $L^s$-optimal."}
{"category": "Math", "title": "The Overconvergent Site II. Cohomology", "abstract": "We prove that rigid cohomology can be computed as the cohomology of a site analogous to the crystalline site. Berthelot designed rigid cohomology as a common generalization of crystalline and Monsky-Washnitzer cohomology. Unfortunately, unlike the former, the functoriality of the theory is not built-in. We defined somewhere else the \"overconvergent site\" which is functorially attached to an algebraic variety and proved that the category of modules of finite presentation on this ringed site is equivalent to the category of over- convergent isocrystals on the variety. We show here that their cohomology also coincides."}
{"category": "Math", "title": "Large continuum, oracles", "abstract": "Our main theorem is about iterated forcing for making the continuum larger than aleph_2. We present a generalization of math.LO/0303294 which is dealing with oracles for random, etc., replacing aleph_1, aleph_2 by lambda,lambda^+ (starting with lambda=lambda^{<lambda}>aleph_1). Well, instead of properness we demand absolute c.c.c. So we get, e.g. the continuum is lambda^+ but we can get cov(meagre)=lambda. We give some applications. As in math.LO/0303294, it is a \"partial\" countable support iteration but it is c.c.c."}
{"category": "Math", "title": "A solid angle polynomial with negative coefficients", "abstract": "This article has been replaced by arXiv:0906.4031"}
{"category": "Math", "title": "An extended version of additive K-theory", "abstract": "There are two infinitesimal (i.e., additive) versions of the $K$-theory of a field $F$: one was introduced by Cathelineau, which is an $F$-module, and another one introduced by Bloch-Esnault, which is an $F^*$-module. Both versions are equipped with a regulator map, when $F$ is the field of complex numbers. In our short paper we will introduce an extended version of Cathelineau's group, and a complex-valued regulator map given by the entropy. We will also give a comparison map between our extended version and Cathelineau's group. Our results were motivated by two unrelated sources: Neumann's work on the extended Bloch group (which is isomorphic to indecomposable $K_3$ of the complex numbers), and the study of singularities of generating series of hypergeometric multisums. Final version."}
{"category": "Math", "title": "On the geometry of the moduli space of spin curves", "abstract": "We determine the smooth locus and the locus of canonical singularities in the Cornalba compactification \\bar S_g of the moduli space S_g of spin curves, i.e., smooth curves of genus g with a theta characteristic. Moreover, the following lifting result for pluricanonical forms is proved: Every pluricanonical form on the smooth locus of \\bar S_g extends holomorphically to a desingularisation of \\bar S_g."}
{"category": "Math", "title": "On the girth of random Cayley graphs", "abstract": "We prove that random d-regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (log_{d-1}|G|)^{1/2}/2 and that random d-regular Cayley graphs of simple algebraic groups over F_q asymptotically almost surely have girth at least log_{d-1}|G|/dim(G). For the symmetric p-groups the girth is between log log |G| and (log|G|)^alpha with alpha<1. Several conjectures and open questions are presented."}
{"category": "Math", "title": "Polynomials with PSL(2) monodromy", "abstract": "Let K be a field of characteristic p>0, and let q be a power of p. We determine all polynomials f in K[t]\\K[t^p] of degree q(q-1)/2 such that the Galois group of f(t)-u over K(u) has a transitive normal subgroup isomorphic to PSL_2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f in K[t], of degree not a power of p, such that f is functionally indecomposable over K but f decomposes over an extension of K. Moreover, except for one ramification setup (which is treated in the companion paper arxiv:0707.1837), we describe all indecomposable polynomials f in K[t] of non-p-power degree which are exceptional, in the sense that x-y is the only absolutely irreducible factor of f(x)-f(y) which lies in K[x,y]. It is known that, when K is finite, a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of K."}
{"category": "Math", "title": "Type-II Matrices and Combinatorial Structures", "abstract": "Type-II matrices are a class of matrices used by Jones in his work on spin models. In this paper we show that type-II matrices arise naturally in connection with some interesting combinatorial and geometric structures."}
{"category": "Math", "title": "A new family of exceptional polynomials in characteristic two", "abstract": "We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this condition is equivalent to saying there are infinitely many finite extensions L/K for which the map c --> f(c) is bijective on L. Our polynomials have degree (2^e-1)*2^(e-1), where e is odd. Combined with our previous paper arxiv:0707.1835, this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic. The strategy of our proof is to identify the curves that can arise as the Galois closure of the branched cover P^1 --> P^1 induced by an exceptional polynomial f. In this case, the curves turn out to be x^(q+1)+y^(q+1)=a+T(xy), where T(z)=z^(q/2)+z^(q/4)+...+z. Our proofs rely on new properties of ramification in Galois covers of curves, as well as the computation of the automorphism groups of all curves in a certain 2-parameter family."}
{"category": "Math", "title": "Computing the complete CS decomposition", "abstract": "An algorithm is developed to compute the complete CS decomposition (CSD) of a partitioned unitary matrix. Although the existence of the CSD has been recognized since 1977, prior algorithms compute only a reduced version (the 2-by-1 CSD) that is equivalent to two simultaneous singular value decompositions. The algorithm presented here computes the complete 2-by-2 CSD, which requires the simultaneous diagonalization of all four blocks of a unitary matrix partitioned into a 2-by-2 block structure. The algorithm appears to be the only fully specified algorithm available. The computation occurs in two phases. In the first phase, the unitary matrix is reduced to bidiagonal block form, as described by Sutton and Edelman. In the second phase, the blocks are simultaneously diagonalized using techniques from bidiagonal SVD algorithms of Golub, Kahan, and Demmel. The algorithm has a number of desirable numerical features."}
{"category": "Math", "title": "Foundations of the calculus of variations in generalized function algebras", "abstract": "We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we introduce the first and second variation of a variational problem. We then derive necessary (Euler-Lagrange equations) and sufficient conditions for extremals. The concept of association is used to obtain connections to a distributional description of singular variational problems. We study variational symmetries and derive an appropriate version of N\\\"other's theorem. Finally, a number of applications to geometry, mechanics, elastostatics and elastodynamics are presented."}
{"category": "Math", "title": "On a theorem in multi-parameter potential theory", "abstract": "We prove a theorem on additive Levy processes and give applications"}
{"category": "Math", "title": "On a general theorem for additive Levy processes", "abstract": "We prove a new theorem on additive Levy processes and show that this theorem implies several proved theorems and a hard conjectured theorem."}
{"category": "Math", "title": "Jones Pairs", "abstract": "Motivated by Jones' braid group representations constructed from spin models, we define {\\sl a Jones pair} to be a pair of $\\nbyn$ matrices $(A,B)$ such that the endomorphisms $X_A$ and $\\D_B$ form a representation of a braid group. When $A$ and $B$ are type-II matrices, we call $(A,B)$ {\\sl an invertible Jones pair}. We develop the theory of Jones pairs in this thesis. Our aim is to study the connections among association schemes, spin models and four-weight spin models using the viewpoint of Jones pairs. We use Nomura's method to construct a pair of algebras from the matrices $(A,B)$, which we call the Nomura algebras of $(A,B)$. These algebras become the central tool in this thesis. We explore their properties in Chapters \\ref{Nomura} and \\ref{IINom}. In Chapter \\ref{JP}, we introduce Jones pairs. We prove the equivalence of four-weight spin models and invertible Jones pairs. We extend some existing concepts for four-weight spin models to Jones pairs. In Chapter \\ref{SpinModels}, we provide new proofs for some well-known results on the Bose-Mesner algebras associated with spin models. We document the main results of the thesis in Chapter \\ref{InvJP}. We prove that every four-weight spin model comes from a symmetric spin model (up to odd-gauge equivalence). We present four Bose-Mesner algebras associated to each four-weight spin model. We study the relations among these algebras. In particular, we provide a strategy to search for four-weight spin models. This strategy is analogous to the method given by Bannai, Bannai and Jaeger for finding spin models."}
{"category": "Math", "title": "Hausdorrf dimension for level sets and k-multiple times", "abstract": "We compute the Hausdorff dimension of the zero set of an additive Levy process."}
{"category": "Math", "title": "The topology of terminal quartic 3-folds", "abstract": "In this thesis, I determine a bound on the defect of terminal Gorenstein quartic 3-folds. More generally, I study the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 and genus at least 3. I state a geometric \"motivation\" of non Q-factoriality in the case of quartics."}
{"category": "Math", "title": "The Gauss-Bonnet-Grotemeyer Theorem in spaces of constant curvature", "abstract": "In 1963, K.P.Grotemeyer proved an interesting variant of the Gauss-Bonnet Theorem. Let M be an oriented closed surface in the Euclidean space R^3 with Euler characteristic \\chi(M), Gauss curvature G and unit normal vector field n. Grotemeyer's identity replaces the Gauss-Bonnet integrand G by the normal moment <a,n>^2G, where $a$ is a fixed unit vector. Grotemeyer showed that the total integral of this integrand is (2/3)pi times chi(M). We generalize Grotemeyer's result to oriented closed even-dimesional hypersurfaces of dimension n in an (n+1) ndimensional space form N^{n+1}(k)."}
{"category": "Math", "title": "Topologies on the space of holomorphic functions", "abstract": "We show that a fairly arbitrary Frechet space topology on the space of holomorphic functions on a domain controls the topology of uniform convergence on compact sets. In fact it turns out that the result we present can be proved more simply using the closed graph theorem. However, we believe that the techniques presented here may be used to prove a more interesting result. Details to appear later."}
{"category": "Math", "title": "Dynamical systems associated with crossed products", "abstract": "In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C^*-algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A'. Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system."}
{"category": "Math", "title": "Maltsiniotis's first conjecture for K_1", "abstract": "We show that K_1 of an exact category agrees with K_1 of the associated triangulated derivator. More generally we show that K_1 of a Waldhausen category with cylinders and a saturated class of weak equivalences coincides with K_1 of the associated right pointed derivator."}
{"category": "Math", "title": "The $\\ell^2$-homology of even Coxeter groups", "abstract": "Given a Coxeter system (W,S), there is an associated CW-complex, Sigma, on which W acts properly and cocompactly. We prove that when the nerve L of (W,S) is a flag triangulation of the 3-sphere, then the reduced $\\ell^2$-homology of Sigma vanishes in all but the middle dimension."}
{"category": "Math", "title": "Application of Extended Kalman Filter to Tactical Ballistic Missile Re-entry Problem", "abstract": "The objective is to investigate the advantages and performance of Extended Kalman Filter for the estimation of non-linear system where linearization takes place about a trajectory that was continually updated with the state estimates resulting from the measurement. Here tactile ballistic missile Re-entry problem is taken as a nonlinear system model and Extended Kalman Filter technique is used to estimate the positions and velocities at the X and Y direction at different values of ballistic coefficients. The result shows that the method gives better estimation with the increase of ballistic coefficient."}
{"category": "Math", "title": "The Elements of an Analysis of the Functions of a Set", "abstract": "The variant of calculation of functions of set and their application is offered. In particular: the new measure of system of sets generalizing classical concept of a measure is entered; the variation of set that has allowed to construct a derivative of function of set on a measure is entered, and to receive necessary conditions of optimization in terms of function of set."}
{"category": "Math", "title": "On plate decompositions of cone multipliers", "abstract": "An important inequality due to Wolff on plate decompositions of cone multipliers is known to have consequences for a variety of problems in harmonic analysis. We observe that the range in Wolff's inequality, for the conic and the spherical versions, can be improved by using bilinear restriction results. We also use this inequality to give some improved estimates on square functions associated to decompositions of cone multipliers in low dimensions. This gives a new L^4 bound for the cone multiplier operator in three dimensions."}
{"category": "Math", "title": "A reinterpretation of Emerton's $p$-adic Banach spaces", "abstract": "It is shown that the $p$-adic Banach spaces introduced by Emerton are isomorphic to the cohomology groups of the sheaf of continuous $\\Q_{p}$-valued functions on a certain space. Some applications of this result are discussed."}
{"category": "Math", "title": "The uniqueness of the helicoid in the Lorentz-Minkowski space L3", "abstract": "In this paper we deal with the uniqueness of the Lorentzian helicoid and Enneper's surface among properly embedded maximal surfaces with lightlike boundary of mirror symmetry in the Lorentz-Minkowski space L3."}
{"category": "Math", "title": "Existence of positive solutions for nonlinear systems", "abstract": "This paper deals with the existence of positive solutions for the nonlinear system q(t)\\phi(p(t)u'_{i}(t)))'+f^{i}(t,\\textbf{u})=0,\\quad 0<t<1,\\quad i=1,2,...,n. This system often arises in the study of positive radial solutions of nonlinear elliptic system. Here $\\textbf{u}=(u_{1},...,u_{n})$ and $f^{i}, i=1,2,...,n$ are continuous and nonnegative functions, $p(t), q(t)\\hbox{\\rm :} [0,1]\\to (0,\\oo)$ are continuous functions. Moreover, we characterize the eigenvalue intervals for (q(t)\\phi(p(t)u'_{i}(t)))'+\\lambda h_{i}(t)g^{i} (\\textbf{u})=0, \\quad 0<t<1,\\quad i=1,2,...,n. The proof is based on a well-known fixed point theorem in cones."}
{"category": "Math", "title": "Strong convergence of modified Ishikawa iterations for nonlinear mappings", "abstract": "In this paper, we prove a strong convergence theorem of modified Ishikawa iterations for relatively asymptotically nonexpansive mappings in Banach space. Our results extend and improve the recent results by Nakajo, Takahashi, Kim, Xu, Matsushita and some others."}
{"category": "Math", "title": "Solutions for a class of iterated singular equations", "abstract": "Some fundamental solutions of radial type for a class of iterated elliptic singular equations including the iterated Euler equation are given."}
{"category": "Math", "title": "Ergodic theory of amenable semigroup actions", "abstract": "In this paper, among other things, we state and prove the mean ergodic theorem for amenable semigroup algebras."}
{"category": "Math", "title": "Some further remarks on good sets", "abstract": "We show that in n-fold cartesian product, n >= 4, a related component need not be a full component. We also prove that when n >= 4, uniform boundedness of lengths of geodesics is not a necessary condition for boundedness of solutions of (1) for bounded function f."}
{"category": "Math", "title": "Weighted composition operators on weighted Bergman spaces of bounded symmetric domains", "abstract": "In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator $W_{\\phi,\\psi}$ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten p-class weighted composition operators."}
{"category": "Math", "title": "Infinite dimensional differential games with hybrid controls", "abstract": "A two-person zero-sum infinite dimensional differential game of infinite duration with discounted payoff involving hybrid controls is studied. The minimizing player is allowed to take continuous, switching and impulse controls whereas the maximizing player is allowed to take continuous and switching controls. By taking strategies in the sense of Elliott--Kalton, we prove the existence of value and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities."}
{"category": "Math", "title": "Borel hierarchies in infinite products of Polish spaces", "abstract": "Let H be a product of countably infinite number of copies of an uncountable Polish space X. Let $\\Sigma_\\xi$ $(\\bar {\\Sigma}_\\xi)$ be the class of Borel sets of additive class \\xi for the product of copies of the discrete topology on X (the Polish topology on X), and let ${\\cal B} = \\cup_{\\xi < \\omega_1} \\bar{\\Sigma}_\\xi$. We prove in the L\\'{e}vy--Solovay model that \\bar{\\Sigma}_\\xi =\\Sigma_{\\xi}\\cap {\\cal B} for $1 \\leq \\xi < \\omega_1$."}
{"category": "Math", "title": "Law of iterated logarithm for NA sequences with non-identical distributions", "abstract": "Based on a law of the iterated logarithm for independent random variables sequences, an iterated logarithm theorem for NA sequences with non-identical distributions is obtained. The proof is based on a Kolmogrov-type exponential inequality."}
{"category": "Math", "title": "Fractional extensions of some boundary value problems in oil strata", "abstract": "In the present paper, we solve three boundary value problems related to the temperature field in oil strata -- the fractional extensions of the incomplete lumped formulation and lumped formulation in the linear case and the fractional generalization of the incomplete lumped formulation in the radial case. By using the Caputo differintegral operator and the Laplace transform, the solutions are obtained in integral forms where the integrand is expressed in terms of the convolution of some auxiliary functions of Wright function type. A generalization of the Laplace transform convolution theorem, known as Efros' theorem is widely used."}
{"category": "Math", "title": "A variant of Davenport's constant", "abstract": "Let p be a prime number. Let G be a finite abelian p-group of exponent n (written additively) and A be a non-empty subset of $]n[:= \\{1,2,..., n\\}$ such that elements of A are incongruent modulo p and non-zero modulo p. Let $k \\geq D(G)/|A|$ be any integer where D(G) denotes the well-known Davenport's constant. In this article, we prove that for any sequence g_1, g_2, ..., g_k (not necessarily distinct) in G, one can always extract a subsequence g_{i_1}, g_{i_2}, ..., g_{i_\\ell} with $1\\leq \\ell \\leq k$ such that \\begin{equation*} \\sum_{j=1}^\\ell a_{j}g_{i_j} = 0 {in} G, \\end{equation*} where a_j \\in A for all j. We provide examples where this bound cannot be improved. Furthermore, for the cyclic groups, we prove some sharp results in this direction. In the last section, we explore the relation between this problem and a similar problem with prescribed length. The proof of Theorem~1 uses group-algebra techniques, while for the other theorems, we use elementary number theory techniques."}
{"category": "Math", "title": "Weak quantization of Poisson structures", "abstract": "In this paper we prove that any Poisson structure on a sheaf of Lie algebroids admits a weak deformation quantization, and give a sufficient condition for such a Poisson structure to admit an actual deformation quantization. We also answer the corresponding classification problems. In the complex symplectic case, we recover in particular some results of Nest-Tsygan and Polesello-Schapira. We begin the paper with a recollection of known facts about deformation theory of cosimplicial differential graded Lie algebras."}
{"category": "Math", "title": "Hidden Life of Riemann's Zeta Function 1. Arrow, Bow, and Targets", "abstract": "The Riemann Hypothesis is reformulated as statements about eigenvalues of some matrices entries of which are defined via Taylor coefficient of the zeta function. These eigenvalues demonstrate interesting visual patterns allowing one to state a number of conjectures."}
{"category": "Math", "title": "Ratio Asymptotic of Hermite-Pad\\'e Orthogonal Polynomials for Nikishin Systems. II", "abstract": "We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures ${\\mathcal{N}}(\\sigma_1,...,\\sigma_m)$ such that for each $k$, the support of $\\sigma_k$ consists of an interval $\\widetilde{\\Delta}_k$, on which $\\sigma_k^{\\prime} > 0$ almost everywhere, and a set without accumulation points in $\\mathbb{R} \\setminus \\widetilde{\\Delta}_k$."}
{"category": "Math", "title": "Coarse topology, enlargeability, and essentialness", "abstract": "Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the K-theory of the corresponding reduced C*-algebras. Our proofs do not depend on the Baum--Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture."}
{"category": "Math", "title": "A geometric model for Hochschild homology of Soergel bimodules", "abstract": "An important step in the calculation of the triply graded link homology theory of Khovanov and Rozansky is the determination of the Hochschild homology of Soergel bimodules for SL(n). We present a geometric model for this Hochschild homology for any simple group G, as equivariant intersection homology of B x B-orbit closures in G. We show that, in type A these orbit closures are equivariantly formal for the conjugation T-action. We use this fact to show that in the case where the corresponding orbit closure is smooth, this Hochschild homology is an exterior algebra over a polynomial ring on generators whose degree is explicitly determined by the geometry of the orbit closure, and describe its Hilbert series, proving a conjecture of Jacob Rasmussen."}
{"category": "Math", "title": "Functions of q-positive type", "abstract": "In this paper we characterize the subspace of $\\mathcal{L}_{q,1,v}$ of function which are the q-Bessel Fourier transform of positive functions in $\\mathcal{L}_{q,1,v}$. As application we give a q-version of the Bochner's theorem."}
{"category": "Math", "title": "Optimal Non-Linear Models for Sparsity and Sampling", "abstract": "Given a set of vectors (the data) in a Hilbert space H, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This collection of subspaces gives the best sparse representation for the given data, in a sense defined in the paper, and provides an optimal model for sampling in union of subspaces. The results are proved in a general setting and then applied to the case of low dimensional subspaces of R^N and to infinite dimensional shift-invariant spaces in L^2(R^d). We also present an iterative search algorithm for finding the solution subspaces. These results are tightly connected to the new emergent theories of compressed sensing and dictionary design, signal models for signals with finite rate of innovation, and the subspace segmentation problem."}
{"category": "Math", "title": "Exit problems associated with affine reflection groups", "abstract": "We obtain a formula for the distribution of the first exit time of Brownian motion from the alcove of an affine Weyl group. In most cases the formula is expressed compactly, in terms of Pfaffians. Expected exit times are derived in the type \\~A case. The results extend to other Markov processes. We also give formulas for the real eigenfunctions of the Dirichlet and Neumann Laplacians on alcoves, observing that the `Hot Spots' conjecture of J. Rauch is true for alcoves."}
{"category": "Math", "title": "Generalized motion of level sets by functions of their curvatures on Riemannian manifolds", "abstract": "We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M. The level sets of a function $u:M\\to\\mathbb{R}$ evolve in such a way whenever u solves an equation $u_{t}+F(Du, D^{2}u)=0$, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions that M has nonnegative curvature, F is continuous off Du=0, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is geometrically consistent, hence it allows to define a generalized evolution of level sets by very general, singular functions of their curvatures. For instance, these assumptions on F are satisfied when F is given by the evolutions of level sets by their mean curvature (even in arbitrary codimension) or by their positive Gaussian curvature. We also prove that the generalized evolution is consistent with the classical motion by the corresponding function of the curvature, whenever the latter exists. When M is not of nonnegative curvature, the same results hold if one additionally requires that F is uniformly continuous with respect to D^2 u. Finally we give some counterexamples showing that several well known properties of the evolutions in $\\mathbb{R}^{n}$ are no longer true when M has negative sectional curvature."}
{"category": "Math", "title": "Toeplitz and Hankel operators and Dixmier traces on the unit ball of \\mathbb C^n", "abstract": "We compute the Dixmier trace of pseudo-Toeplitz operators on the Fock space. As an application we find a formula for the Dixmier trace of the product of commutators of Toeplitz operators on the Hardy and weighted Bergman spaces on the unit ball of \\mathbb C^d. This generalizes an earlier work of Helton-Howe for the usual trace of the anti-symmetrization of Toeplitz operators."}
{"category": "Math", "title": "Cotorsion pairs generated by modules of bounded projective dimension", "abstract": "We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an $\\aleph_0$-noetherian ring Q of small finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and Cohen Macaulay noetherian commutative rings."}
{"category": "Math", "title": "Proper actions of lamplighter groups associated with free groups", "abstract": "Given a finite group $H$ and a free group $F_n$, we prove that the wreath product $H\\wr F_n$ admits a metrically proper, isometric action on a Hilbert space."}
{"category": "Math", "title": "A GIT Construction of Moduli Spaces of Stable Maps in Positive Characteristic", "abstract": "In a previous paper, the author and David Swinarski constructed the moduli spaces of stable maps, \\bar M_g,n(P^r,d), via geometric invariant theory (GIT). That paper required the base field to be the complex numbers, a restriction which this paper removes: here the coarse moduli spaces of stable maps are constructed via GIT over a more general base."}
{"category": "Math", "title": "The Mukai pairing, I: a categorical approach", "abstract": "We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and integral kernels as its 1-morphisms."}
{"category": "Math", "title": "The structure of surfaces mapping to the moduli stack of canonically polarized varieties", "abstract": "Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective surface that maps to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of the surface. As a result, we can describe the fibration induced by the moduli map quite explicitly. A refined affirmative answer to Viehweg's conjecture for families over surfaces follows as a corollary."}
{"category": "Math", "title": "Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems", "abstract": "We prove integral formulas for closed hypersurfaces in C^{n+1}, which furnish a relation between elementary symmetric functions in the eigenvalues of the complex Hessian matrix of the defining function and the Levi curvatures of the hypersurface. Then we follow the Reilly approach to prove an isoperimetric inequality. As an application, we obtain the ``Soap Bubble Theorem'' for star-shaped domains with positive and constant Levi curvatures bounding the classical mean curvature from above."}
{"category": "Math", "title": "Constant Mean Curvature Hypersurfaces in the (n+1)-Sphere by Gluing Spherical Building Blocks", "abstract": "The techniques developed by Butscher in arXiv:math/0703469 for constructing constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere by gluing together spherical building blocks are generalized to handle less symmetric initial configurations. The outcome is that the approximately CMC hypersurface obtained by gluing the initial configuration together can be perturbed into an exactly CMC hypersurface only when certain global geometric conditions are met. These `balancing conditions' are analogous to those that must be satisfied in the `classical' context of gluing constructions of CMC hypersurfaces in Euclidean space, although they are more restrictive in the (n+1)-sphere case. An example of an initial configuration is given which demonstrates this fact; and another example of an initial configuration is given which possesses no symmetries at all."}
{"category": "Math", "title": "Approximate zero-one laws and sharpness of the percolation transition in a class of models including 2D Ising percolation", "abstract": "One of the most well-known classical results for site percolation on the square lattice is the equation p_c + p_c^* = 1. In words, this equation means that for all values different from p_c of the parameter p the following holds: Either a.s. there is an infinite open cluster or a.s. there is an infinite closed `star' cluster. This result is closely related to the percolation transition being sharp: Below p_c the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distrubtion with an exponential tail. The analog of this result has been proved by Higuchi in 1993 for two-dimensional Ising percolation, with fixed inverse temparature beta <beta_c, and as parameter the external field h. Using sharp-threshold results (approximate zero-one laws) and a modification of an RSW-like result by Bollobas and Riordan, we show that these results hold for a large class of percolation models where the vertex values can be `nicely' represented (in a sense which will be defined precisely) by i.i.d. random variables. We point out that the ordinary percolation model belongs obviously to this class, and we show that also the above mentionedIsing model belongs to it. We hope that our results improve insight in the Ising percolation model, and will help to show that many other (not yet analyzed) weakly dependent percolation models also belong to the abovementioned class."}
{"category": "Math", "title": "Nearly optimal embeddings of trees", "abstract": "In this paper we show how to find nearly optimal embeddings of large trees in several natural classes of graphs. The size of the tree T can be as large as a constant fraction of the size of the graph G, and the maximum degree of T can be close to the minimum degree of G. For example, we prove that any graph of minimum degree d without 4-cycles contains every tree of size \\epsilon d^2 and maximum degree at most (1-2\\epsilon)d - 2. As there exist d-regular graphs without 4-cycles of size O(d^2), this result is optimal up to constant factors. We prove similar nearly tight results for graphs of given girth, graphs with no complete bipartite subgraph K_{s,t}, random and certain pseudorandom graphs. These results are obtained using a simple and very natural randomized embedding algorithm, which can be viewed as a \"self-avoiding tree-indexed random walk\"."}
{"category": "Math", "title": "Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups", "abstract": "A proof of non-existence of Lagrangian embeddings of the Klein bottle K in \\CP^2 is given. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil on \\CP^2 and study its monodromy. As the main technical tool, we develop the theory of mapping class groups, considered as quotients of special Artin braid groups, and obtain some new results about combinatorial structure of such groups."}
{"category": "Math", "title": "Ellipses of minimal area and of minimal eccentricity circumscribed about a convex quadrilateral", "abstract": "First, we fill in key gaps in Steiner's nice characterization of the most nearly circular ellipse which passes through the vertices of a convex quadrilateral, D. Steiner proved that there is only one pair of conjugate directions, M1 and M2, that belong to all ellipses of circumscription. Then he proves that if there is an ellipse, E, whose equal conjugate diameters possess the directional constants M1 and M2, then E must be an ellipse of circumscription which has minimal eccentricity. However, Steiner does not show the existence or uniqueness of such an ellipse. We prove that there is a unique ellipse of minimal eccentricity which passes through the vertices of D. We also show that there exists an ellipse which passes through the vertices of D and whose equal conjugate diameters possess the directional constants M1 and M2. We also show that there exists a unique ellipse of minimal area which passes through the vertices of D. Finally, we call a convex quadrilateral, D, bielliptic if the unique inscribed and circumscribed ellipses of minimal eccentricity have the same eccentricity. This generalizes the notion of bicentric quadrilaterals. In particular we show the existence of a bielliptic convex quadrilateral which is not bicentric."}
{"category": "Math", "title": "Conjectures on Partitions of Integers as Summations of Primes", "abstract": "In this short note we present a class of conjectures on partitions of integers as summations of primes, which are extensions of Goldbach conjecture."}
{"category": "Math", "title": "Monodromy of the p-rank strata of the moduli space of curves", "abstract": "We compute the Z/\\ell and \\ell-adic monodromy of every irreducible component of the moduli space M_g^f of curves of genus and and p-rank f. In particular, we prove that the Z/\\ell-monodromy of every component of M_g^f is the symplectic group Sp_{2g}(Z/\\ell) if g>=3 and \\ell is a prime distinct from p. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank."}
{"category": "Math", "title": "Exact Computation of Minimum Sample Size for Estimation of Binomial Parameters", "abstract": "It is a common contention that it is an ``impossible mission'' to exactly determine the minimum sample size for the estimation of a binomial parameter with prescribed margin of error and confidence level. In this paper, we investigate such a very old but also extremely important problem and demonstrate that the difficulty for obtaining the exact solution is not insurmountable. Unlike the classical approximate sample size method based on the central limit theorem, we develop a new approach for computing the minimum sample size that does not require any approximation. Moreover, our approach overcomes the conservatism of existing rigorous sample size methods derived from Bernoulli's theorem or Chernoff bounds. Our computational machinery consists of two essential ingredients. First, we prove that the minimum of coverage probability with respect to a binomial parameter bounded in an interval is attained at a discrete set of finite many values of the binomial parameter. This allows for reducing infinite many evaluations of coverage probability to finite many evaluations. Second, a recursive bounding technique is developed to further improve the efficiency of computation."}
{"category": "Math", "title": "Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras", "abstract": "We will prove that one-sided topological Markov shifts $(X_A,\\sigma_A)$ and $(X_B,\\sigma_B)$ for matrices $A$ and $B$ with entries in $\\{0,1\\}$ are topologically orbit equivalent if and only if there exists an isomorphism between the Cuntz-Krieger algebras ${\\Cal O}_A$ and ${\\Cal O}_B$ keeping their commutative $C^*$-subalgerbas $C(X_A)$ and $C(X_B)$. It is also equivalent to the condition that there exists a homeomorphism from $X_A$ to $X_B$ intertwining their topological full groups. We will also study structure of the automorphisms of ${\\Cal O}_A$ keeping the commutative $C^*$-algebra $C(X_A)$."}
{"category": "Math", "title": "Exact Computation of Minimum Sample Size for Estimating Proportion of Finite Population", "abstract": "In this paper, we develop an exact method for the determination of the minimum sample size for estimating the proportion of a finite population with prescribed margin of error and confidence level. By characterizing the behavior of the coverage probability with respect to the proportion, we show that the computational complexity can be significantly reduced and bounded regardless population size."}
{"category": "Math", "title": "Exact Computation of Minimum Sample size for Estimation of Poisson Parameters", "abstract": "In this paper, we develop an approach for the exact determination of the minimum sample size for the estimation of a Poisson parameter with prescribed margin of error and confidence level. The exact computation is made possible by reducing infinite many evaluations of coverage probability to finite many evaluations. Such reduction is based on our discovery that the minimum of coverage probability with respect to a Poisson parameter bounded in an interval is attained at a discrete set of finite many values."}
{"category": "Math", "title": "Cycle lengths in sparse graphs", "abstract": "Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdos conjectured that |C(G)| =\\Omega(d^{\\lfloor (g-1)/2\\rfloor}) for all such graphs, and we prove this conjecture. In particular, the longest cycle in a graph of average degree d and girth g has length \\Omega(d^{\\lfloor (g-1)/2\\rfloor}). The study of this problem was initiated by Ore in 1967 and our result improves all previously known lower bounds on the length of the longest cycle. Moreover, our bound cannot be improved in general, since known constructions of d-regular Moore Graphs of girth g have roughly that many vertices. We also show that \\Omega(d^{\\lfloor (g-1)/2\\rfloor}) is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g. Further results are obtained for the number of cycle lengths in H-free graphs of average degree d. In the second part of the paper, motivated by the conjecture of Erdos and Gyarfas that every graph of minimum degree at least three contains a cycle of length a power of two, we prove a general theorem which gives an upper bound on the average degree of an n-vertex graph with no cycle of even length in a prescribed infinite sequence of integers. For many sequences, including the powers of two, our theorem gives the upper bound e^{O(\\log^* n)} on the average degree of graph of order n with no cycle of length in the sequence, where \\log^* n is the number of times the binary logarithm must be applied to n to get a number which is at most"}
{"category": "Math", "title": "A formula for a quartic integral: a survey of old proofs and some new ones", "abstract": "We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations."}
{"category": "Math", "title": "The 2-adic valuation of a sequence arising from a rational integral", "abstract": "We analyze properties of the 2-adic valuations of an integer sequence that originates from an explicit evaluation of a quartic integral. We also give a combinatorial interpretation of the valuations of this sequence. Connections with the orbits arising from the Collatz (3x+1) problem are discussed."}
{"category": "Math", "title": "The integrals in Gradshteyn and Ryzhik. Part 6: the beta function", "abstract": "We present the evaluation of definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik that can be reduced to the beta function."}
{"category": "Math", "title": "The integrals in Gradshteyn and Ryzhik. Part 7: Elementary examples", "abstract": "The classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik contains some elementary integrals. We discuss their evaluations."}
{"category": "Math", "title": "The integrals in Gradshteyn and Ryzhik. Part 8: Combinations of powers, exponentials and logarithms", "abstract": "We present the evaluation of some definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik where the integrand is a combination of powers, exponentials and logarithms."}
{"category": "Math", "title": "The integrals in Gradshteyn and Ryzhik. Part9: Combinations of logarithms, rational and trigonometric functions", "abstract": "The classical table of integrals by I. S. Gradshteyn and I. M. Ryzhik contains many definite integrals where the integrand is the product of a rational function times the logarithm of another rational function. We begin the systematic evaluation of these integrals."}
{"category": "Math", "title": "The bisymplectomorphism group of a bounded symmetric domain", "abstract": "An Hermitian bounded symmetric domain in a complex vector space, given in its circled realization, is endowed with two natural symplectic forms: the flat form and the hyperbolic form. In a similar way, the ambient vector space is also endowed with two natural symplectic forms: the Fubini-Study form and the flat form. It has been shown in arXiv:math.DG/0603141 that there exists a diffeomorphism from the domain to the ambient vector space which puts in correspondence the above pair of forms. This phenomenon is called symplectic duality for Hermitian non compact symmetric spaces. In this article, we first give a different and simpler proof of this fact. Then, in order to measure the non uniqueness of this symplectic duality map, we determine the group of bisymplectomorphisms of a bounded symmetric domain, that is, the group of diffeomorphisms which preserve simultaneously the hyperbolic and the flat symplectic form. This group is the direct product of the compact Lie group of linear automorphisms with an infinite-dimensional Abelian group. This result appears as a kind of Schwarz lemma."}
{"category": "Math", "title": "Gagliardo-Nirenberg inequalities on manifolds", "abstract": "We prove Gagliardo-Nirenberg inequalities on some classes of manifolds, Lie groups and graphs."}
{"category": "Math", "title": "Semigroups on Frechet spaces and equations with infinite delays", "abstract": "In this paper, we show existence and uniqueness of a solution to a functional differential equation with infinite delay. We choose an appropriate Frechet space so as to cover a large class of functions to be used as initial functions to obtain existence and uniqueness of solutions."}
{"category": "Math", "title": "Nonconforming h-p spectral element methods for elliptic problems", "abstract": "In this paper we show that we can use a modified version of the h-p spectral element method proposed in \\cite{duttora1,duttom,duttora2,tomarth} to solve elliptic problems with general boundary conditions to exponential accuracy on polygonal domains using nonconforming spectral element functions. A geometrical mesh is used in a neighbourhood of the corners. With this mesh we seek a solution which minimizes the sum of a weighted squared norm of the residuals in the partial differential equation and the squared norm of the residuals in the boundary conditions in fractional Sobolev spaces and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in fractional Sobolev norms, to the functional being minimized. In the neighbourhood of the corners, modified polar coordinates are used and a global coordinate system elsewhere. A stability estimate is derived for the functional which is minimized based on the regularity estimate in \\cite{babguo1}. We examine how to parallelize the method and show that the set of common boundary values consists of the values of the function at the corners of the polygonal domain. The method is faster than that proposed in \\cite{duttora1,duttom,tomarth} and the h-p finite element method and stronger error estimates are obtained."}
{"category": "Math", "title": "Reducing system of parameters and the Cohen--Macaulay property", "abstract": "Let $R$ be a local ring and let ($x_1\\biss x_r$) be part of a system of parameters of a finitely generated $R$-module $M,$ where $r < \\dim_R M$. We will show that if ($y_1\\biss y_r$) is part of a reducing system of parameters of $M$ with $(y_1\\biss y_r)M=(x_1\\biss x_r)M$ then $(x_1\\biss x_r)$ is already reducing. Moreover, there is such a part of a reducing system of parameters of $M$ iff for all primes $P\\in \\supp M \\cap V_R(x_1\\biss x_r)$ with $\\dim_R R/P = \\dim_R M -r$ the localization $M_P$ of $M$ at $P$ is an $r$-dimensional \\cm\\ module over $R_P$. Furthermore, we will show that $M$ is a \\cm module iff $y_d$ is a non zero divisor on $M/(y_1\\biss y_{d-1})M$, where $(y_1\\biss y_d)$ is a reducing system of parameters of $M$ ($d := \\dim_R M$)."}
{"category": "Math", "title": "Corners of normal matrices", "abstract": "We study various conditions on matrices $B$ and $C$ under which they can be the off-diagonal blocks of a partitioned normal matrix."}
{"category": "Math", "title": "On a Result of Hardy and Ramanujan", "abstract": "In this paper, we introduce some explicit approximations for the summation $\\sum_{k\\leq n}\\Omega(k)$, where $\\Omega(k)$ is the total number of prime factors of $k$."}
{"category": "Math", "title": "Nice surjections on spaces of operators", "abstract": "A bounded linear operator is said to be nice if its adjoint preserves extreme points of the dual unit ball. Motivated by a description due to Labuschagne and Mascioni \\cite{LM} of such maps for the space of compact operators on a Hilbert space, in this article we consider a description of nice surjections on ${\\mathcal K}(X,Y)$ for Banach spaces $X,Y$. We give necessary and sufficient conditions when nice surjections are given by composition operators. Our results imply automatic continuity of these maps with respect to other topologies on spaces of operators. We also formulate the corresponding result for ${\\mathcal L}(X,Y)$ thereby proving an analogue of the result from \\cite{LM} for $L^p$ ($1 <p \\neq 2 <\\infty$) spaces. We also formulate results when nice operators are not of the canonical form, extending and correcting the results from \\cite{KS}."}
{"category": "Math", "title": "Gromov--Witten invariants and quantum cohomology", "abstract": "This article is an elaboration of a talk given at an international conference on Operator Theory, Quantum Probability, and Noncommutative Geometry held during December~20--23, 2004, at the Indian Statistical Institute, Kolkata. The lecture was meant for a general audience, and also prospective research students, the idea of the quantum cohomology based on the Gromov--Witten invariants. Of course there are many important aspects that are not discussed here."}
{"category": "Math", "title": "Representations of homogeneous quantum L\\'evy fields", "abstract": "We study homogeneous quantum L\\'{e}vy processes and fields with independent additive increments over a noncommutative *-monoid. These are described by infinitely divisible generating state functionals, invariant with respect to an endomorphic injective action of a symmetry semigroup. A strongly covariant GNS representation for the conditionally positive logarithmic functionals of these states is constructed in the complex Minkowski space in terms of canonical quadruples and isometric representations on the underlying pre-Hilbert field space. This is of much use in constructing quantum stochastic representations of homogeneous quantum L\\'{e}vy fields on It\\^{o} monoids, which is a natural algebraic way of defining dimension free, covariant quantum stochastic integration over a space-time indexing set."}
{"category": "Math", "title": "Malliavin calculus of Bismut type without probability", "abstract": "We translate in semigroup theory Bismut's way of the Malliavin calculus."}
{"category": "Math", "title": "Stochastic integral representations of quantum martingales on multiple Fock space", "abstract": "In this paper a quantum stochastic integral representation theorem is obtained for unbounded regular martingales with respect to multidimensional quantum noise. This simultaneously extends results of Parthasarathy and Sinha to unbounded martingales and those of the author to multidimensions."}
{"category": "Math", "title": "On equivariant Dirac operators for $SU_q(2)$", "abstract": "We explain the notion of minimality for an equivariant spectral triple and show that the triple for the quantum SU(2) group constructed by Chakraborty and Pal in \\cite{c-p1} is minimal. We also give a decomposition of the spectral triple constructed by Dabrowski {\\it et al} \\cite{dlssv} in terms of the minimal triple constructed in \\cite{c-p1}."}
{"category": "Math", "title": "Reducible family of height three level algebras", "abstract": "Let $R=k[x_1,..., x_r]$ be the polynomial ring in $r$ variables over an infinite field $k$, and let $M$ be the maximal ideal of $R$. Here a \\emph{level algebra} will be a graded Artinian quotient $A$ of $R$ having socle $Soc(A)=0:M$ in a single degree $j$. The Hilbert function $H(A)=(h_0,h_1,... ,h_j)$ gives the dimension $h_i=\\dim_k A_i$ of each degree-$i$ graded piece of $A$ for $0\\le i\\le j$. The embedding dimension of $A$ is $h_1$, and the \\emph{type} of $A$ is $\\dim_k \\Soc (A)$, here $h_j$. The family $\\Levalg (H)$ of level algebra quotients of $R$ having Hilbert function $H$ forms an open subscheme of the family of graded algebras or, via Macaulay duality, of a Grassmannian. We show that for each of the Hilbert functions $H=H_1=(1,3,4,4)$ and $H=H_2=(1,3,6,8,9,3)$ the family $LevAlg (H)$ parametrizing level Artinian algebras of Hilbert function $H$ has several irreducible components. We show also that these examples each lift to points. However, in the first example, an irreducible Betti stratum for Artinian algebras becomes reducible when lifted to points. These were the first examples we obtained of multiple components for $\\Levalg(H)$ in embedding dimension three. We also show that the second example is the first in an infinite sequence of examples of type three Hilbert functions $H(c)$ in which also the number of components of LevAlg(H) gets arbitrarily large. The first case where the phenomenon of multiple components can occur (i.e. the lowest embedding dimension and then the lowest type) is that of dimension three and type two. Examples of this first case have been obtained by the authors and also by J.-O. Kleppe."}
{"category": "Math", "title": "Local representations of the quantum Teichmuller space", "abstract": "We introduce a certain type of representations for the quantum Teichmuller space of a punctured surface, which we call local representations. We show that, up to finitely many choices, these purely algebraic representations are classified by classical geometric data. We also investigate the family of intertwining operators associated to such a representations. In particular, we use these intertwiners to construct a natural fiber bundle over the Teichmuller space and its quotient under the action of the mapping class group. This construction also offers a convenient framework to exhibit invariants of surface diffeomorphisms."}
{"category": "Math", "title": "On Hilbert's construction of positive polynomials", "abstract": "In 1888, Hilbert described how to find real polynomials in more than one variable which take only non-negative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert's construction and present many such polynomials."}
{"category": "Math", "title": "Variable Selection and Model Averaging in Semiparametric Overdispersed Generalized Linear Models", "abstract": "We express the mean and variance terms in a double exponential regression model as additive functions of the predictors and use Bayesian variable selection to determine which predictors enter the model, and whether they enter linearly or flexibly. When the variance term is null we obtain a generalized additive model, which becomes a generalized linear model if the predictors enter the mean linearly. The model is estimated using Markov chain Monte Carlo simulation and the methodology is illustrated using real and simulated data sets."}
{"category": "Math", "title": "Algebraic hierarchy of logics unifying fuzzy logic and quantum logic", "abstract": "In this paper, a short survey about the concepts underlying general logics is given. In particular, a novel rigorous definition of a fuzzy negation as an operation acting on a lattice to render it into a fuzzy logic is presented. According to this definition, a fuzzy negation satisfies the weak double negation condition, requiring double negation to be expansive, the antitony condition, being equivalent to the disjunctive De Morgan law and thus warranting compatibility of negation with the lattice operations, and the Boolean boundary condition stating that the universal bounds of the lattice are the negation of each other. From this perspective, the most general logics are fuzzy logics, containing as special cases paraconsistent (quantum) logics, quantum logics, intuitionistic logics, and Boolean logics, each of which given by its own algebraic restrictions. New examples of a non-contradictory logic violating the conjunctive De Morgan law, and of a typical non-orthomodular fuzzy logic along with its explicit lattice representation are given."}
{"category": "Math", "title": "Filtered Hirsch Algebras", "abstract": "Motivated by the cohomology theory of loop spaces, we consider a special class of higher order homotopy commutative differential graded algebras and construct the filtered Hirsch model for such an algebra $A$. When $x\\in H(A)$ with $\\mathbb{Z}$ coefficients and $x^{2}=0,$ the symmetric Massey products $% \\langle x\\rangle ^{n}$ with $n\\geq 3$ have a finite order (whenever defined). However, if $\\Bbbk $ is a field of characteristic zero, $\\langle x\\rangle ^{n}$ is defined and vanishes in $H(A\\otimes \\Bbbk )$ for all $n$. If $p$ is an odd prime, the Kraines formula $\\langle x\\rangle ^{p}=-\\beta \\mathcal{P}_{1}(x)$ lifts to $H^{\\ast }(A\\otimes {\\mathbb{Z}}_{p}).$ Applications of the existence of polynomial generators in the loop homology and the Hochschild cohomology with a $G$-algebra structure are given."}
{"category": "Math", "title": "An explicit formula for the action of a finite group on a commutative ring", "abstract": "Let G be a group which acts on a commutative ring k. We exhibit an induction formula which expresses an element x_G with tr_G(x_G)=1 by elements x_P with tr_P(x_P)=1, where P varies over prime order subgroups of P."}
{"category": "Math", "title": "On positive solutions of minimal growth for singular p-Laplacian with potential term", "abstract": "Let $\\Omega$ be a domain in $\\mathbb{R}^d$, $d\\geq 2$, and $1<p<\\infty$. Fix $V\\in L_{\\mathrm{loc}}^\\infty(\\Omega)$. Consider the functional $Q$ and its G\\^{a}teaux derivative $Q^\\prime$ given by Q(u):=\\frac{1}{p}\\int_\\Omega (|\\nabla u|^p+V|u|^p)\\dx, Q^\\prime (u):=-\\nabla\\cdot(|\\nabla u|^{p-2}\\nabla u)+V|u|^{p-2}u. It is assumed that $Q\\geq 0$ on $C_0^\\infty(\\Omega)$. In a previous paper we discussed relations between the absence of weak coercivity of the functional $Q$ on $C_0^\\infty(\\Omega)$ and the existence of a generalized ground state. In the present paper we study further relationships between functional-analytic properties of the functional $Q$ and properties of positive solutions of the equation $Q^\\prime (u)=0$."}
{"category": "Math", "title": "Cyclic (v;r,s;lambda) difference families with two base blocks and v <= 50", "abstract": "We construct many new cyclic (v;r,s;lambda) difference families with v less than or equal 50. In particular we construct the difference families with parameters (45;18,10;9), (45;22,22;21), (47;21,12;12), (47;19,15;12), (47;22,14;14), (48;20,10;10), (48;24,4;12), (50;25,20;20) for which the existence question was an open problem. The (45;22,22;21) difference family gives a BIBD with parameters v=45, b=90, r=44, k=22 and lambda=21, and the one with parameters (50;25,20;20) gives a pair of binary sequences of length 50 with zero periodic autocorrelation function (the periodic analog of a Golay pair). We also construct nine new D-optimal designs. A normal form for cyclic difference families is proposed and used effectively in compiling the list of known and new difference families."}
{"category": "Math", "title": "Morita equivalence of Poisson manifold via stack groupoids", "abstract": "This is a condensed exposition of the results of a future work, based on a talk of the second author at the Oberwolfach workshop \"Poisson Geometry\", April 30--4 May 2007."}
{"category": "Math", "title": "Weakly infinitely divisible measures on some locally compact Abelian groups", "abstract": "On the torus group, on the group of p-adic integers and on the p-adic solenoid, we give a construction of an arbitrary weakly infinitely divisible probability measure using a random element with values in a product of (possibly infinitely many) subgroups of the real numbers. As a special case of our results, we have a new construction of the Haar measure on the p-adic solenoid."}
{"category": "Math", "title": "Real Closed Rings and Real Closed * Rings", "abstract": "Here we try to distinguish and compare different notions of real closedness mainly one developed by N. Schwartz in his Habilitationschrift and the other developed by A. Sankaranand K. Varadarajan which we shall call real closed *. We stick to the definition of real closed rings as defined and characterized N. Schwartz and we try to determine and characterize real closed rings that are real closed *. The main result is that real closed rings have unique real closure * and that real closure of real closed * rings arent necessarily real closed *."}
{"category": "Math", "title": "Solutions of the problem of Erd\\\"os-Sierpi\\'nski: $\\sigma(n)=\\sigma(n+1)$", "abstract": "For $n\\leq 1.5 \\cdot 10^{10}$, we have found a total number of 1268 solutions to the Erd\\\"os-Sierpi\\'nski problem finding positive integer solutions of $\\sigma(n)=\\sigma(n+1)$, where $\\sigma(n)$ is the sum of the positive divisors of n. On the basis of that set of solutions the following empirical properties are enunciated: first, all the $\\sigma(n)$, $n$ being a solution, are divisible by 6; second, the repetition of solutions leads to the formulation of a new problem: \\emph{Find the natural numbers $n$ such that $\\sigma(n)=\\sigma(n+1)=\\sigma(n+k)=\\sigma(n+k+1)$ for some positive integer $k$}. A third empirical property concerns the asymptotic behavior of the function of $n$ that gives the number of solutions for $m$ less or equal to $n$, which we find to be as $n^{1/3}$. Finally some theorems related to the Erd\\\"os-Sierpi\\'nski problem are enunciated and proved."}
{"category": "Math", "title": "A generalization of Hamilton's differential Harnack inequality for the Ricci flow", "abstract": "In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M x R^2 has nonnegative isotropic curvature."}
{"category": "Math", "title": "Stabilization of the spatial oscillations of an elastic system model", "abstract": "A system of partial differential equations describing the spatial oscillations of an Euler-Bernoulli beam with a tip mass is considered. The linear system considered is actuated by two independent controls and separated into a pair of differential equations in a Hilbert space. A feedback control ensuring strong stability of the equilibrium in the sense of Lyapunov is proposed. The proof of the main result is based on the theory of strongly continuous semigroups."}
{"category": "Math", "title": "A remark on invariants for C*-algebras of stable rank one", "abstract": "It is shown that, for a C*-algebra of stable rank one (i.e., in which the invertible elements are dense), two well-known isomorphism invariants, the Cuntz semigroup and the Thomsen semigroup, contain the same information. More precisely, these two invariants, viewed appropiately, determine each other in a natural way."}
{"category": "Math", "title": "On some properties of travelling water waves with vorticity", "abstract": "We prove that for a large class of vorticity functions the crest of a corresponding travelling water wave is necessarily a point of maximal horizontal velocity. We also show that for waves with nonpositive vorticity the pressure in the flow is everywhere larger than the atmospheric pressure. A related a priori estimate for waves with nonnegative vorticity is also given."}
{"category": "Math", "title": "On the existence of extreme waves and the Stokes conjecture with vorticity", "abstract": "This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of $120^\\circ$ or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of $120^\\circ$ if the vorticity is nonnegative near the free surface."}
{"category": "Math", "title": "Fibered Correspondence", "abstract": "Base of fibered correspondence is arbitrary correspondence. Fibered correspondence is interesting when we consider relationship between different bundles. However composition of fibered correspondences may not always be defined. Reduced fibered correspondence is defined only between fibers over the same point of base. Reduced fibered correspondence in bundle is called 2-ary fibered relation. We considered fibered equivalence and isomorphism theorem in case of fibered morphisms."}
{"category": "Math", "title": "Generic initial ideals, graded Betti numbers and $k$-Lefschetz properties", "abstract": "We introduce the $k$-strong Lefschetz property ($k$-SLP) and the $k$-weak Lefschetz property ($k$-WLP) for graded Artinian $K$-algebras, which are generalizations of the Lefschetz properties. The main results obtained in this paper are as follows: 1. Let $I$ be a graded ideal of $R=K[x_1, x_2, x_3]$ whose quotient ring $R/I$ has the SLP. Then the generic initial ideal of $I$ is the unique almost revlex ideal with the same Hilbert function as $R/I$. 2. Let $I$ be a graded ideal of $R=K[x_1, x_2, ..., x_n]$ whose quotient ring $R/I$ has the $n$-SLP. Suppose that all $k$-th differences of the Hilbert function of $R/I$ are quasi-symmetric. Then the generic initial ideal of $I$ is the unique almost revlex ideal with the same Hilbert function as $R/I$. 3. We give a sharp upper bound on the graded Betti numbers of Artinian $K$-algebras with the $k$-WLP and a fixed Hilbert function."}
{"category": "Math", "title": "The crystal commutor and Drinfeld's unitarized R-matrix", "abstract": "Drinfeld defined a unitarized R-matrix for any quantum group U_q(g). This gives a commutor for the category of U_q(g) representations, making it into a coboundary category. Henriques and Kamnitzer defined another commutor which also gives U_q(g) representations the structure of a coboundary category. We show that a particular case of Henriques and Kamnitzer's construction agrees with Drinfeld's commutor. We then describe the action of Drinfeld's commutor on a tensor product of two crystal bases, and explain the relation to the crystal commutor."}
{"category": "Math", "title": "A Bayes method for a Bathtub Failure Rate via two $\\mathbf{S}$-paths", "abstract": "A class of semi-parametric hazard/failure rates with a bathtub shape is of interest. It does not only provide a great deal of flexibility over existing parametric methods in the modeling aspect but also results in a closed and tractable Bayes estimator for the bathtub-shaped failure rate (BFR). Such an estimator is derived to be a finite sum over two $\\mathbf{S}$-paths due to an explicit posterior analysis in terms of two (conditionally independent) $\\mathbf{S}$-paths. These, newly discovered, explicit results can be proved to be a Rao-Blackwellization of counterpart results in terms of partitions that are readily available by a specialization of James (2005)'s work. We develop both iterative and non-iterative computational procedures based on existing efficient Monte Carlo methods for sampling one single $\\mathbf{S}$-path. Nmerical simulations are given to demonstrate the practicality and the effectiveness of our methodology. Last but not least, two applications of the proposed method are discussed, of which one is about a Bayesian test for failure rates and the other is related to modeling with covariates."}
{"category": "Math", "title": "A spectral Erdos-Stone-Bollobas theorem", "abstract": "We give a bound on the spectral radius of a graph implying a quantitative version of the Erdos-Stone theorem."}
{"category": "Math", "title": "Irreducible Sp-representations and subgroup distortion in the mapping class group", "abstract": "We prove that various subgroups of the mapping class group $Mod(\\Sigma)$ of a surface $\\Sigma$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the \"point-pushing\" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of $H_1(\\Sigma;\\mathbb{Z})$."}
{"category": "Math", "title": "The Casson invariant and the word metric on the Torelli group", "abstract": "We bound the value of the Casson invariant of any integral homology 3-sphere $M$ by a constant times the distance-squared to the identity, measured in any word metric on the Torelli group $\\T$, of the element of $\\T$ associated to any Heegaard splitting of $M$. We construct examples which show this bound is asymptotically sharp."}
{"category": "Math", "title": "A Maslov cocycle for unitary groups", "abstract": "We introduce a 2-cocycle for symplectic and skew-hermitian hyperbolic groups over arbitrary fields and skew fields, with values in the Witt group of hermitian forms. This cocycle has good functorial properties: it is natural under extension of scalars and stable, so it can be viewed as a universal 2-dimensional characteristic class for these groups. Over R and C, it coincides with the first Chern class."}
{"category": "Math", "title": "Non-linear Symmetry-preserving Observer on Lie Groups", "abstract": "In this paper we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which reminds of the linear stationary case."}
{"category": "Math", "title": "Counting rational points on cubic hypersurfaces", "abstract": "Let X be a geometrically integral projective cubic hypersurface defined over the rationals, with dimension D and singular locus of dimension at most D-4. For any \\epsilon>0, we show that X contains O(B^{D+\\epsilon}) rational points of height at most B. The implied constant in this estimate depends upon the choice of \\epsilon and the coefficients of the cubic form defining X."}
{"category": "Math", "title": "Edge colouring models for the Tutte polynomial and related graph invariants", "abstract": "For integer q>1, we derive edge q-colouring models for (i) the Tutte polynomial of a graph G on the hyperbola H_q, (ii) the symmetric weight enumerator of the set of group-valued q-flows of G, and (iii) a more general vertex colouring model partition function that includes these polynomials and the principal specialization order q of Stanley's symmetric monochrome polynomial. In the second half of the paper we exhibit a family of non-symmetric edge q-colouring models defined on k-regular graphs, whose partition functions for q >= k each evaluate the number of proper edge k-colourings of G when G is Pfaffian."}
{"category": "Math", "title": "Parity, eulerian subgraphs and the Tutte polynomial", "abstract": "Identities obtained by elementary finite Fourier analysis are used to derive a variety of evaluations of the Tutte polynomial of a graph G at certain points (a,b) where (a-1)(b-1) equals 2 or 4. These evaluations are expressed in terms of eulerian subgraphs of G and the size of subgraphs modulo 2,3,4 or 6. In particular, a graph is found to have a nowhere-zero 4-flow if and only if there is a correlation between the event that three subgraphs A,B,C chosen uniformly at random have pairwise eulerian symmetric differences and the event that the integer part of (|A| + |B| + |C|) / 3 is even. Some further evaluations of the Tutte polynomial at points (a,b) where (a-1)(b-1) = 3 are also given that illustrate the unifying power of the methods used. The connection between results of Matiyasevich, Alon and Tarsi and Onn is highlighted by indicating how they may all be derived by the techniques adopted in this paper."}
{"category": "Math", "title": "On real log canonical thresholds", "abstract": "We introduce real log canonical threshold and real jumping numbers for real algebraic functions. A real jumping number is a root of the $b$-function up to a sign if its difference with the minimal one is less than 1. The real log canonical threshold, which is the minimal real jumping number, coincides up to a sign with the maximal pole of the distribution defined by the complex power of the absolute value of the function. However, this number may be greater than 1 if the codimension of the real zero locus of the function is greater than 1. So it does not necessarily coincide with the maximal root of the b-function up to a sign, nor with the log canonical threshold of the complexification. In fact, the real jumping numbers can be even disjoint from the non-integral jumping numbers of the complexification."}
{"category": "Math", "title": "Finite-dimensional irreducible modules for the three-point $\\mathfrak{sl}_2$ loop algebra", "abstract": "Recently Brian Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra by generators and relations. To obtain this presentation they defined a Lie algebra $\\boxtimes$ by generators and relations, and displayed an isomorphism from $\\boxtimes$ to the three-point $sl_2$ loop algebra. In this paper we describe the finite-dimensional irreducible $\\boxtimes$-modules from multiple points of view."}
{"category": "Math", "title": "Some examples of vector bundles in the base locus of the generalized theta divisor", "abstract": "This paper shows that on the moduli space of semi-stable vector bundles of fixed rank and determinant (of any degree) on a smooth curve of genus at least two, the base locus of the generalized theta divisor is large provided the rank is sufficiently large. It also shows that the base locus is large for positive multiples of the theta divisor. This work extends results already known for the case where the determinant is of degree zero."}
{"category": "Math", "title": "On points at infinity of real spectra of polynomial rings", "abstract": "Let R be a real closed field and A=R[x_1,...,x_n]. Let sper A denote the real spectrum of A. There are two kinds of points in sper A : finite points (those for which all of |x_1|,...,|x_n| are bounded above by some constant in R) and points at infinity. In this paper we study the structure of the set of points at infinity of sper A and their associated valuations. Let T be a subset of {1,...,n}. For j in {1,...,n}, let y_j=x_j if j is not in T and y_j=1/x_j if j is in T. Let B_T=R[y_1,...,y_n]. We express sper A as a disjoint union of sets of the form U_T and construct a homeomorphism of each of the sets U_T with a subspace of the space of finite points of sper B_T. For each point d at infinity in U_T, we describe the associated valuation v_{d*} of its image d* in sper B_T in terms of the valuation v_d associated to d. Among other things we show that the valuation v_{d*} is composed with v_d (in other words, the valuation ring R_d is a localization of R_{d*} at a suitable prime ideal)."}
{"category": "Math", "title": "Isometries for the Caratheodory Metric", "abstract": "Under certain hypothesises, we prove that a map which is an isometry for the Caratheodory infinitesimal metric at a point is an analytic isomorphism onto its image."}
{"category": "Math", "title": "Hilbert Function and Betti Numbers of Algebras with Lefschetz Property of Order m", "abstract": "The authors T.Harima, J.C.Migliore, U.Nagel and J.Watanabe characterized the Hilbert function of algbebras with the Lefschetz property. We extend this characterization to algebras with the Lefschetz property m times. We also give upper bounds for the Betti numbers of Artinian algebras with a given Hilbert function and with the Lefschetz property m times and describe the cases in which these bounds are reached."}
{"category": "Math", "title": "On Selberg's small eigenvalue conjecture and residual eigenvalues", "abstract": "We show that Selberg's eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic perturbation theory and number theory."}
{"category": "Math", "title": "Compatibility of quantization functors of Lie bialgebras with duality and doubling operations", "abstract": "We study the behavior of the Etingof-Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are \"compatible with duality\", i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof-Kazhdan quantization of some affine Lie superalgebras coincide with their Drinfeld-Jimbo-type quantizations."}
{"category": "Math", "title": "Hardy's theorem for the q-Bessel Fourier transform", "abstract": "In this paper we give a q-analogue of the Hardy's theorem for the $q$-Bessel Fourier transform. The celebrated theorem asserts that if a function $f$ and its Fourier transform $\\hat{f}$ satisfying $|f(x)|\\leq c.e^{-{1/2} x^2}$ and $|\\hat{f}(x)|\\leq c.e^{-{1/2} x^2}$ for all $x\\in\\mathbb{% R}$ then $f(x)=\\text{const}.e^{-{1/2} x^2}$."}
{"category": "Math", "title": "A remarkable DG-module model for configuration spaces", "abstract": "Let M be a simply-connected closed manifold and consider the (ordered) configuration space of $k$ points in M, F(M,k). In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the rational homotopy type of F(M,k). We prove that our model it is at least a Sigma_k-equivariant differential graded model. We also study Lefschetz duality at the level of cochains and describe equivariant models of the complement of a union of polyhedra in a closed manifold."}
{"category": "Math", "title": "From ballistic to diffusive behavior in periodic potentials", "abstract": "The long-time/large-scale, small-friction asymptotic for the one dimensional Langevin equation with a periodic potential is studied in this paper. It is shown that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. We show that the same result is valid for a whole one parameter family of space/time rescalings. The proofs of our main results are based on some novel estimates on the resolvent of a hypoelliptic operator."}
{"category": "Math", "title": "Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems", "abstract": "We provide a short and elementary proof for the recently proved result by G. da Prato and H. Frankowska that -- under minimal assumptions -- a closed set is invariant with respect to a stochastic control system if and only if it is invariant with respect to the (associated) deterministic control system."}
{"category": "Math", "title": "Order Independence in Asynchronous Cellular Automata", "abstract": "A sequential dynamical system, or SDS, consists of an undirected graph Y, a vertex-indexed list of local functions F_Y, and a permutation pi of the vertex set (or more generally, a word w over the vertex set) that describes the order in which these local functions are to be applied. In this article we investigate the special case where Y is a circular graph with n vertices and all of the local functions are identical. The 256 possible local functions are known as Wolfram rules and the resulting sequential dynamical systems are called finite asynchronous elementary cellular automata, or ACAs, since they resemble classical elementary cellular automata, but with the important distinction that the vertex functions are applied sequentially rather than in parallel. An ACA is said to be pi-independent if the set of periodic states does not depend on the choice of pi, and our main result is that for all n>3 exactly 104 of the 256 Wolfram rules give rise to a pi-independent ACA. In 2005 Hansson, Mortveit and Reidys classified the 11 symmetric Wolfram rules with this property. In addition to reproving and extending this earlier result, our proofs of pi-independence also provide significant insight into the dynamics of these systems."}
{"category": "Math", "title": "A proof of the multiplicity one conjecture for GL(n) in GL(n+1)", "abstract": "Let F be a non-archimedean local field of characteristic zero. We consider distributions on GL(n+1,F) which are invariant under the adjoint action of GL(n,F). We prove that any such distribution is invariant with respect to transposition. This implies that the restriction to GL(n) of any irreducible smooth representation of GL(n+1) is multiplicity free. Our paper is based on the recent work [RS] of Steve Rallis and Gerard Schiffmann where they made a remarkable progress on this problem. In [RS], they also show that our result implies multiplicity one theorem for restrictions from the orthogonal group $O(V \\oplus F)$ to $O(V)$."}
{"category": "Math", "title": "Weights of modular forms on $\\mathrm{SO}^{+}(2,l)$ and congruences between Eisenstein series and cusp forms of half-integral weight on $\\mathrm{SL}_{2}$", "abstract": "Let $E$ be a level 1, vector valued Eisenstein series of half-integral weight, normalized so that the coefficients are all in $\\mathbb{Z}$. We show that there is a level one vector valued cusp form $f$ with the same weight as $E$ and with coefficients in $\\mathbb{Z}$, which is congruent to $E$ modulo the constant term of $E$."}
{"category": "Math", "title": "Frechet topologies on hypoelliptic sheaves", "abstract": "This note is withdrawn. The result and its proof are available in the literature."}
{"category": "Math", "title": "Singular loci of Bruhat-Hibi toric varieties", "abstract": "For the toric variety X associated to the Bruhat poset of Schubert varieties in a minuscule G/P, we describe the singular locus in terms of the faces of the associated polyhedral cone. We further show that the singular locus is pure of codimension 3 in X."}
{"category": "Math", "title": "The Geometry of Genus-One Helicoids", "abstract": "We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an analogous result for periodic helicoid-like surfaces. We also give a simple condition guaranteeing that an immersed minimal surface with finite genus and bounded curvature is asymptotic to a helicoid at infinity."}
{"category": "Math", "title": "G-Compactness and Groups", "abstract": "Lascar described E_KP as a composition of E_L and the topological closure of EL. We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M_0 consisting of M and X as two sorts, where X is an affine copy of G and in M_0 we have the structure of M and the action of G on X. We prove that the Lascar group of M_0 is a semi-direct product of the Lascar group of M and G/G_L. We discuss the relationship between G-compactness of M and M_0. This example may yield new examples of non-G-compact theories."}
{"category": "Math", "title": "Regularity of Solutions to Second-Order Integral Functionals in Variational Calculus", "abstract": "We obtain regularity conditions of a new type of problems of the calculus of variations with second-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main result asserts that autonomous integral functionals of the calculus of variations with a Lagrangian having superlinearity partial derivatives with respect to the higher-order derivatives admit only minimizers with essentially bounded derivatives."}
{"category": "Math", "title": "Lectures on Poisson groupoids", "abstract": "In these lecture notes, we give a quick account of the theory of Poisson groupoids and Lie bialgebroids. In particular, we discuss the universal lifting theorem and its applications including integration of quasi-Lie bialgebroids, integration of Poisson Nijenhuis structures and Alekseev and Kosmann-Schwarzbach's theory of D/G-momentum maps."}
{"category": "Math", "title": "Pinning Complex Networks by a Single Controller", "abstract": "In this paper, without assuming symmetry, irreducibility, or linearity of the couplings, we prove that a single controller can pin a coupled complex network to a homogenous solution. Sufficient conditions are presented to guarantee the convergence of the pinning process locally and globally. An effective approach to adapt the coupling strength is proposed. Several numerical simulations are given to verify our theoretical analysis."}
{"category": "Math", "title": "Global Exponential Stability of Delayed Periodic Dynamical Systems", "abstract": "In this paper, we discuss delayed periodic dynamical systems, compare capability of criteria of global exponential stability in terms of various $L^{p}$ ($1\\le p<\\infty$) norms. A general approach to investigate global exponential stability in terms of various $L^{p}$ ($1\\le p<\\infty$) norms is given. Sufficient conditions ensuring global exponential stability are given, too. Comparisons of various stability criteria are given. More importantly, it is pointed out that sufficient conditions in terms of $L^{1}$ norm are enough and easy to implement in practice."}
{"category": "Math", "title": "Identification of a chemotactic sensitivity in a coupled system", "abstract": "Chemotaxis is the process by which cells behave in a way that follows the chemical gradient. Applications to bacteria growth, tissue inflammation, and vascular tumors provide a focus on optimization strategies. Experiments can characterize the form of possible chemotactic sensitivities. This paper addresses the recovery of the chemotactic sensitivity from these experiments while allowing for nonlinear dependence of the parameter on the state variables. The existence of solutions to the forward problem is analyzed. The identification of a chemotactic parameter is determined by inverse problem techniques. Tikhonov regularization is investigated and appropriate convergence results are obtained. Numerical results of concentration dependent chemotactic terms are explored."}
{"category": "Math", "title": "Distributive lattices defined for representations of rank two semisimple Lie algebras", "abstract": "For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems A1+A1, A2, C2, and G2. Examples appear in Figures 3.2 and 3.3. We then form the distributive lattices of order ideals of these posets. Corollary 5.4 gives elegant quotient-of-products expressions for the rank generating functions of these lattices (thereby providing answers to a 1979 question of Stanley). Also, Theorem 5.3 describes how these lattices provide a new combinatorial setting for the Weyl characters of representations of rank two semisimple Lie algebras. Most of these lattices are new; the rest of them (or related structures) have arisen in work of Stanley, Kashiwara, Nakashima, Littelmann, and Molev. In a future paper, one author shows that the posets constructed here form a Dynkin diagram-indexed answer to a combinatorially posed classification question. In a companion paper, some of these lattices are used to explicitly construct some representations of rank two semisimple Lie algebras. This implies that these lattices are strongly Sperner."}
{"category": "Math", "title": "The logarithmic Sobolev inequality along the Ricci flow", "abstract": "We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any restriction on time. One application of it is a uniform kappa-noncollapsing estimate which holds true for all time. We also obtain similar results for bounded time without assuming the eigenvalue condition. The results extend to the Ricci flow with surgeries."}
{"category": "Math", "title": "An extension of the Maskit slice for 4-dimensional Kleinian groups", "abstract": "Let $\\Gamma$ be a 3-dimensional Kleinian punctured torus group with ccidental parabolic transformations. The deformation space of $\\Gamma$ in the group of M\\\"{o}bius transformations on the 2-sphere is well-known as the Maskit slice of punctured torus groups. In this paper, we study deformations $\\Gamma'$ of $\\Gamma$ in the group of M\\\"{o}bius transformations on the 3-sphere such that $\\Gamma'$ does not contain screw parabolic transformations. We will show that the space of the deformations is realized as a domain of 3-space $\\mathbb{R}^3$, which contains the Maskit slice of punctured torus groups as a slice through a plane. Furthermore, we will show that the space also contains the Maskit slice of fourth-punctured sphere groups as a slice through another plane. Some of another slices of the space will be also studied."}
{"category": "Math", "title": "A Presentation for the Dual Symmetric Inverse Monoid", "abstract": "The dual symmetric inverse monoid $\\mathscr{I}_n^*$ is the inverse monoid of all isomorphisms between quotients of an $n$-set. We give a monoid presentation of $\\mathscr{I}_n^*$ and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements."}
{"category": "Math", "title": "Moduli spaces of quadratic complexes and their singular surfaces", "abstract": "We construct the coarse moduli space $\\cM_{qc}(\\sigma)$ of quadratic line complexes with a fixed Segre symbol $\\sigma$ as well as the moduli space $\\cM_{ss}(\\sigma)$ of the corresponding singular surfaces. We show that the map associating to a quadratic line complex its singular surface induces a morphism $\\pi: \\cM_{qc}(\\sigma) \\ra \\cM_{ss}(\\sigma)$. Finally we deduce that the varieties of cosingular quadratic line complexes are almost always curves."}
{"category": "Math", "title": "A Galois-theoretic approach to Kanev's correspondence", "abstract": "Let $G$ be a finite group, $\\Lambda$ an absolutely irreducible $\\Z[G]$-module and $w$ a weight of $\\Lambda$. To any Galois covering with group $G$ we associate two correspondences, the Schur and the Kanev correspondence. We work out their relation and compute their invariants. Using this, we give some new examples of Prym-Tyurin varieties."}
{"category": "Math", "title": "Desynchronization of pulse-coupled oscillators with delayed excitatory coupling", "abstract": "Collective behavior of pulse-coupled oscillators has been investigated widely. As an example of pulse-coupled networks, fireflies display many kinds of flashing patterns. Mirollo and Strogatz (1990) proposed a pulse-coupled oscillator model to explain the synchronization of South East Asian fireflies ({\\itshape Pteroptyx malaccae}). However, transmission delays were not considered in their model. In fact, the presence of transmission delays can lead to desychronization. In this paper, pulse-coupled oscillator networks with delayed excitatory coupling are studied. Our main result is that under reasonable assumptions, pulse-coupled oscillator networks with delayed excitatory coupling can not achieve complete synchronization, which can explain why another species of fireflies ({\\itshape Photinus pyralis}) rarely synchronizes flashing. Finally, two numerical simulations are given. In the first simulation, we illustrate that even if all the initial phases are very close to each other, there could still be big variations in the times to process the pulses in the pipeline. It implies that asymptotical synchronization typically also cannot be achieved. In the second simulation, we exhibit a phenomenon of clustering synchronization."}
{"category": "Math", "title": "The equilibrium states for semigroups of rational maps", "abstract": "We consider the dynamics of skew product maps associated with finitely generated semigroups of rational maps on the Riemann sphere. We show that under some conditions on the dynamics and the potential function \\psi, there exists a unique equilibrium state for \\psi and a unique $\\exp(\\P(\\psi)-\\psi)$-conformal measure, where P(\\psi) denotes the topological pressure of \\psi."}
{"category": "Math", "title": "Real analyticity of Hausdorff dimension for expanding rational semigroups", "abstract": "We consider the dynamics of expanding semigroups generated by finitely many rational maps on the Riemann sphere. We show that for an analytic family of such semigroups, the Bowen parameter function is real-analytic and plurisubharmonic. Combining this with a result obtained by the first author, we show that if for each semigroup of such an analytic family of expanding semigroups satisfies the open set condition, then the function of the Hausdorff dimension of the Julia set is real-analytic and plurisubharmonic. Moreover, we provide an extensive collection of classes of examples of analytic families of semigroups satisfying all the above conditions and we analyze in detail the corresponding Bowen's parameters and Hausdorff dimension function."}
{"category": "Math", "title": "A semiregularity map annihilating obstructions to deforming holomorphic maps", "abstract": "We study deformations of holomorphic maps of compact, complex, K\\\"ahler manifolds. In particular, we describe a generalization of Bloch's semiregularity map that annihilates obstructions to deform holomorphic maps with fixed codomain."}
{"category": "Math", "title": "Cyclic coverings of the $p$-adic projective line by Mumford curves", "abstract": "Exact bounds for the positions of the branch points for cyclic coverings of the $p$-adic projective line by Mumford curves are calculated in two ways. Firstly, by using Fumiharu Kato's *-trees, and secondly by giving explicit matrix representations of the Schottky groups corresponding to the Mumford curves above the projective line through combinatorial group theory."}
{"category": "Math", "title": "Applications of the q-Fourier Analysis to the Symmetric Moment Problem", "abstract": "Sufficient condition for the symmetric moment problem to be determinate is given using standards methods of $q$-Fourier analysis. This condition it cannot be a particular case of Carleman's criterion."}
{"category": "Math", "title": "$L^p$-Spectral theory of locally symmetric spaces with $Q$-rank one", "abstract": "We study the $L^p$-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces $M=\\Gamma\\backslash X$ with finite volume and arithmetic fundamental group $\\Gamma$ whose universal covering $X$ is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one."}
{"category": "Math", "title": "Primitive contractions of Calabi-Yau threefolds II", "abstract": "We construct 16 new examples of Calabi--Yau threefolds with Picard group of rank 1. Each of these examples is obtained by smoothing the image of a primitive contraction with exceptional divisor being a del Pezzo surface of degree 6, 7 or $\\mathbb{P}^1\\times \\mathbb{P}^1$."}
{"category": "Math", "title": "Landen survey", "abstract": "Landen transformations are maps on the coefficients of an integral that preserve its value. We present a brief survey of their appearnce in the literature."}
{"category": "Math", "title": "Primitive Divisors in Arithmetic Dynamics", "abstract": "Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F."}
{"category": "Math", "title": "Generic measures for hyperbolic flows on non compact spaces", "abstract": "We consider the geodesic flow on a complete connected negatively curved manifold. We show that the set of invariant borel probability measures contains a dense $G_\\delta$-subset consisting of ergodic measures fully supported on the non-wandering set. We also trat the case of non-positively curved manifolds and provide general tools to deal with hyperbolic systems defined on non-compact spaces."}
{"category": "Math", "title": "On embedding well-separable graphs", "abstract": "Call a simple graph $H$ of order $n$ well-separable, if by deleting a separator set of size $o(n)$ the leftover will have components of size at most $o(n)$. We prove, that bounded degree well-separable spanning subgraphs are easy to embed: for every $\\gamma >0$ and positive integer $\\Delta$ there exists an $n_0$ such that if $n>n_0$, $\\Delta(H) \\le \\Delta$ for a well-separable graph $H$ of order $n$ and $\\delta(G) \\ge (1-{1 \\over 2(\\chi(H)-1)} + \\gamma)n$ for a simple graph $G$ of order $n$, then $H \\subset G$. We extend our result to graphs with small band-width, too."}
{"category": "Math", "title": "Representation of finite connective spaces", "abstract": "After recalling the definition of connectivity spaces and some of their main properties, a way is proposed to represent finite connectivity spaces by directed simple graphs. Then a connectivity structure is associated to each tame link. It is showed that all spaces of a certain class (the iterated Brunnian ones) admit representations by links. Finally, I conjecture that every finite connectivity space is representable by a link. ----- Apres un rappel de la definition des espaces connectifs et de certaines de leurs principales proprietes, nous proposons une maniere de representer les espaces connectifs finis par des graphes simples orientes, puis nous associons a tout entrelacs une structure connective. Nous montrons que tout espace d'une certaine classe (les espaces brunniens iteres) admet une representation par entrelacs, et nous conjecturons finalement que tout espace connectif fini est representable par entrelacs."}
{"category": "Math", "title": "On C1-robust transitivity of volume-preserving flows", "abstract": "We prove that a divergence-free and C1-robustly transitive vector field has no singularities. Moreover, if the vector field is C4 then the linear Poincare flow associated to it admits a dominated splitting over M."}
{"category": "Math", "title": "Uniform estimates for cubic oscillatory integrals", "abstract": "This paper establishes the optimal decay rate for scalar oscillatory integrals in $n$ variables which satisfy a nondegeneracy condition on the third derivatives. The estimates proved are stable under small linear perturbations, as encountered when computing the Fourier transform of surface-carried measures. The main idea of the proof is to construct a nonisotropic family of balls which locally capture the scales and directions in which cancellation occurs."}
{"category": "Math", "title": "Stability for large forbidden subgraphs", "abstract": "We extend the classical stability theorem of Erdos and Simonovits for forbidden graphs of logarithmic order."}
{"category": "Math", "title": "Logarithmic geometry, minimal free resolutions and toric algebraic stacks", "abstract": "In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie, and Kato) and investigate their moduli. Then by applying this we define a notion of toric algebraic stacks over arbitrary schemes, which may be regarded as torus embeddings within the framework of algebraic stacks, and study some basic properties. Furthermore, we study the stack-theoretic analogue of toroidal embeddings."}
{"category": "Math", "title": "A note on the incidence coloring of outerplanar graphs", "abstract": "A proof that every outerplanar graph is \\Delta+2 colorable. This is slightly stronger then an unpublished result of Wang Shudong, Ma Fangfang, Xu Jin, and Yan Lijun proving the same for 2-connected outerplanar graphs."}
{"category": "Math", "title": "Six Conjectures which Generalize or Are Related to Andrica's Conjecture", "abstract": "Six conjectures on pairs of consecutive primes are listed in this paper, together with examples for each case."}
{"category": "Math", "title": "Derived completions in stable homotopy theory", "abstract": "We construct a notion of derived completion which applies to homomorphisms of commutative S-algebras. We study the relationship of the construction with other constructions of completions, and prove various invariance properties. The construction is expected to have applications within algebraic K-theory."}
{"category": "Math", "title": "An Elementary Proof of the Fundamental Theorem of Tropical Algebra", "abstract": "In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm for factoring tropical polynomials of a single variable. A central idea is the concept of least-coefficient polynomials as representatives for classes of functionally equivalent polynomials. This idea has importance far beyond the proof of the Fundamental Theorem of Tropical Algebra."}
{"category": "Math", "title": "The Fermat-Torricelli problem in normed planes and spaces", "abstract": "We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat-Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach."}
{"category": "Math", "title": "Hochschild cohomology, the characteristic morphism and derived deformations", "abstract": "A notion of Hochschild cohomology of an abelian category was defined by Lowen and Van den Bergh (2005) and they showed the existence of a characteristic morphism from the Hochschild cohomology into the graded centre of the (bounded) derived category. An element in the second Hochschild cohomology group corresponds to a first order deformation of the abelian category (Lowen and Van den Bergh, 2006). The problem of deforming single objects of the bounded derived category was treated by Lowen (2005). In this paper we show that the image of the Hochschild cohomology element under the characteristic morphism encodes precisely the obstructions to deforming single objects of the bounded derived category. Hence this paper provides a missing link between the above works. Finally we discuss some implications of these facts in the direction of a ``derived deformation theory''."}
{"category": "Math", "title": "The Mather measure and a Large Deviation Principle for the Entropy Penalized Method", "abstract": "We present a large deviation principle for the entropy penalized Mather problem when the Lagrangian L is generic (in this case the Mather measure $\\mu$ is unique and the support of $\\mu$ is the Aubry set). Consider, for each value of $\\epsilon $ and h, the entropy penalized Mather problem $\\min \\{\\int_{\\tn\\times\\rn} L(x,v)d\\mu(x,v)+\\epsilon S[\\mu]\\},$ where the entropy S is given by $S[\\mu]=\\int_{\\tn\\times\\rn}\\mu(x,v)\\ln\\frac{\\mu(x,v)}{\\int_{\\rn}\\mu(x,w)dw}dxdv,$ and the minimization is performed over the space of probability densities $\\mu(x,v)$ that satisfy the holonomy constraint It follows from D. Gomes and E. Valdinoci that there exists a minimizing measure $\\mu_{\\epsilon, h}$ which converges to the Mather measure $\\mu$. We show a LDP $\\lim_{\\epsilon,h\\to0} \\epsilon \\ln \\mu_{\\epsilon,h}(A),$ where $A\\subset \\mathbb{T}^N\\times\\mathbb{R}^N$. The deviation function I is given by $I(x,v)= L(x,v)+\\nabla\\phi_0(x)(v)-\\bar{H}_{0},$ where $\\phi_0$ is the unique viscosity solution for L."}
{"category": "Math", "title": "A sheaf of Hochschild complexes on quasi-compact opens", "abstract": "For a scheme X, we construct a sheaf C of complexes on X such that for every quasi-compact open subset U of X, C(U) is quasi-isomorphic to the Hochschild complex of the scheme U. Since C is moreover acyclic for taking sections on quasi-compact opens, we obtain a local to global spectral sequence for Hochschild cohomology if X is quasi-compact."}
{"category": "Math", "title": "Exceptional covers of surfaces", "abstract": "Consider a finite morphism f:X -> Y of smooth projective varieties over a finite field k. Suppose X is the vanishing locus in projective N-space of at most r forms of degree at most d. We show there is a constant C, depending only on N, r, d and deg(f) such that if #k > C, then f(k):X(k) -> Y(k) is injective if and only if it's surjective."}
{"category": "Math", "title": "Domino Waves", "abstract": "Motivated by a proposal of Daykin (Problem 71-19*, SIAM Review 13 (1971) 569), we study the wave that propagates along an infinite chain of dominoes and find the limiting speed of the wave in an extreme case."}
{"category": "Math", "title": "Higher Energies in Kahler Geometry I", "abstract": "Let $X\\hookrightarrow \\cpn $ be a smooth complex projective variety of dimension $n$. Let $\\lambda$ be an algebraic one parameter subgroup of $G:=\\gc$. Let $ 0\\leq l\\leq n+1$. We associate to the coefficients $F_{l}(\\lambda)$ of the normalized weight of $\\lambda$ on the $mth$ Hilbert point of $X$ new energies $F_{\\om,l}(\\vp)$. The (logarithmic) asymptotics of $F_{\\om,l}(\\vp)$ along the potential deduced from $\\lambda$ is the weight $F_{l}(\\lambda)$. $F_{\\om,l}(\\vp)$ reduces to the Aubin energy when $l=0$ and the K-Energy map of Mabuchi when $l=1$. When $l\\geq 2$ $F_{\\om,l}(\\vp)$ coincides (modulo lower order terms) with the functional $E_{\\om,l-1}(\\vp)$ introduced by X.X. Chen and G.Tian."}
{"category": "Math", "title": "On the linear fractional self-attracting diffusion", "abstract": "In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2<H<1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized self-intersection local time exists in L^2 if $\\frac12<H<\\frac3{4}$."}
{"category": "Math", "title": "Iterated logarithm law for anticipating stochastic differential equations", "abstract": "We prove a functional law of iterated logarithm for the following kind of anticipating stochastic differential equations $$\\xi^u_t=X_0^u+\\frac{1}{\\sqrt{\\log\\log u}}\\sum_{j=1}^k \\int_0^{t} A_j^u(\\xi^u_s)\\circ dW_{s}^j+ \\int_0^{t} A_0^u(\\xi^u_s)ds,$$ where $u>e$, $W=\\{(W_t^1,...,W_t^k), 0\\le t\\le 1\\}$ is a standard $k$-dimensional Wiener process, $A_0^u,A_1^u,..., A_k^u:\\mathbb{R}^d\\longrightarrow \\mathbb{R}^d$ are functions of class $\\mathcal{C}^2$ with bounded partial derivatives up to order 2, $X_0^u$ is a random vector not necessarily adapted and the first integral is a generalized Stratonovich integral ."}
{"category": "Math", "title": "A Non-Archimedean Wave Equation", "abstract": "Let K be a non-Archimedean local field with the normalized absolute value $|\\cdot |$. It is shown that a ``plane wave'' $f(t+\\omega_1 x_1+... +\\omega_nx_n)$, where f is a Bruhat-Schwartz complex-valued test function on K, $(t,x_1,..., x_n)\\in K^{n+1}$, $\\max\\limits_{1\\le j\\le n}|\\omega_j|=1$, satisfies, for any f, a certain homogeneous pseudo-differential equation, an analog of the classical wave equation. A theory of the Cauchy problem for this equation is developed."}
{"category": "Math", "title": "A third-order dispersive flow for closed curves into K\\\"ahler manifolds", "abstract": "This paper is devoted to studying the initial value problem for a third-order dispersive equation for closed curves into K\\\"ahler manifolds. This equation is a geometric generalization of a two-sphere valued system modeling the motion of vortex filament. We prove the local existence theorem by using geometric analysis and classical energy method."}
{"category": "Math", "title": "On the S_n-equivariant Euler characteristic of M_{2,n}", "abstract": "The Getzler's formula relates the S_n-equivariant Hodge-Deligne polynomial of the space of ordered tuples of distinct points on a given variety X with the Hodge-Deligne polynomial of X. We obtain the analogue of this formula for the case when X has a nontrivial automorphism group. Collecting together all strata of $\\mathcal{M}_2$ with different automorphism groups, we derive a formula for the S_n-equivariant Euler characteristic of $\\mathcal{M}_{2,n}$."}
{"category": "Math", "title": "A Finite Horizon Optimal Multiple Switching Problem", "abstract": "We consider the problem of optimal multiple switching in finite horizon, when the state of the system, including the switching costs, is a general adapted stochastic process. The problem is formulated as an extended impulse control problem and completely solved using probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. Finally, when the state of the system is a Markov diffusion process, we show that the vector of value functions of the optimal problem is a viscosity solution to a system of variational inequalities with inter-connected obstacles."}
{"category": "Math", "title": "A proof of the invariance of the contact angle in Electrowetting", "abstract": "We prove the invariance of the contact angle in liquid-solid wetting phenomena : an electrified droplet is spreading on a solid surface. The drop is minimizing its energy. We express the differential of this energy with respect to the shape of the drop and deduce necessary conditions for optimality . By variational method, using well-chosen test functions, we obtain the main result about the contact angle between the drop and the solid."}
{"category": "Math", "title": "The Jacobian ideal of a hyperplane arrangement", "abstract": "The Jacobian ideal of a hyperplane arrangement is an ideal in the polynomial ring whose generators are the partial derivatives of the arrangements defining polynomial. In this article, we prove that an arrangement can be reconstructed from its Jacobian ideal."}
{"category": "Math", "title": "A weak dichotomy below E_1 \\times E_3", "abstract": "If E is an equivalence relation Borel reducible to E_1 \\times E_3 then either E is Borel reducible to the equality of countable sets of reals or E_1 is Borel reducible to E. The \"either\" case admits further strengthening."}
{"category": "Math", "title": "A superadditivity and submultiplicativity property for cardinalities of sumsets", "abstract": "For finite sets of integers $A_1, A_2 ... A_n$ we study the cardinality of the $n$-fold sumset $A_1+... +A_n$ compared to those of $n-1$-fold sumsets $A_1+... +A_{i-1}+A_{i+1}+... A_n$. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets."}
{"category": "Math", "title": "A remark on global well-posedness below L^2 for the gKdV-3 equation", "abstract": "The I-method in its first version as developed by Colliander et al. is applied to prove that the Cauchy-problem for the generalised Korteweg-de Vries equation of order three (gKdV-3) is globally well-posed for large real-valued data in the Sobolev space H^s, provided s>-1/42."}
{"category": "Math", "title": "Nonlinear SDEs driven by L\\'evy processes and related PDEs", "abstract": "In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a L\\'evy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz continuous and not necessarily linear in the time-marginals of the solution as is the case in the classical McKean-Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump L\\'evy process with a smooth but unbounded L\\'evy measure, we develop a stochastic calculus of variations to prove that the time-marginals of the solutions are absolutely continuous with respect to the Lebesgue measure. In the case of a symmetric stable driving process, we deduce the existence of a function solution to a nonlinear integro-differential equation involving the fractional Laplacian."}
{"category": "Math", "title": "Prolate Spheroidal Wave Functions In q-Fourier Analysis", "abstract": "The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property [6]. They are an orthogonal basis of both $L^2(-1,1)$ and the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. We prove that there are new systems possessing this property in $q$-Fourier analysis. As application we give a new sampling formula with $q^n$ as sampling points, where 0 < q < 1."}
{"category": "Math", "title": "q-Sturm-Liouville theory and the corresponding eigenfunction expansions", "abstract": "The aim of this paper is to study the $q$-Schr\\\"{o}dinger operator $$ L= q(x)-\\Delta_q, $$ where $q(x)$ is a given function of $x$ defined over $\\mathbb{R}_{q}^{+}=\\{q^n,\\quad n\\in\\mathbb Z\\}$ and $\\Delta_q$ is the $q$-Laplace operator $$ \\Delta_{q}f(x)=\\frac{1}{x^{2}}[ f(q^{-1}x)-\\frac{1+q}{q}f(x)+\\frac{1}{q}f(qx)]. $$"}
{"category": "Math", "title": "An explicit formula for the characters of the symmetric group", "abstract": "We give an explicit expression of the normalized characters of the symmetric group in terms of the contents of the partition labelling the representation."}
{"category": "Math", "title": "Equidistribution of negative statistics and quotients of Coxeter groups of type B and D", "abstract": "We generalize some identities and q-identities previously known for the symmetric group to Coxeter groups of type B and D. The extended results include theorems of Foata and Sch\\\"utzenberger, Gessel, and Roselle on various distributions of inversion number, major index, and descent number. In order to show our results we provide caracterizations of the systems of minimal coset representatives of Coxeter groups of type B and D."}
{"category": "Math", "title": "Singular Oscillatory Integrals on R^n", "abstract": "Let Pd,n denote the space of all real polynomials of degree at most d on R^n. We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P in Pd,1. Using this estimate, we prove a sharp estimate for a singular oscillatory integral on R^n."}
{"category": "Math", "title": "Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms", "abstract": "It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4+2 leaves and that this can be improved to (n+4)/3 for cubic graphs without the diamond K_4-e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n+4)/3 leaves, and generalize this further by allowing vertices of lower degree. We show that it is necessary to exclude blossoms in order to obtain a bound of the form n/3+c. We use the new bound to obtain a simple FPT algorithm, which decides in O(m)+O^*(6.75^k) time whether a graph of size m has a spanning tree with at least k leaves. This improves the best known time complexity for MAX LEAF SPANNING TREE."}
{"category": "Math", "title": "A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves", "abstract": "We study the action of the mapping class group M(F) on the complex of curves of a non-orientable surface F. We obtain, by using a result of K. S. Brown, a presentation for M(F) defined in terms of the mapping class groups of the complementary surfaces of collections of curves, provided that F is not sporadic, i.e. the complex of curves of F is simply connected. We also compute a finite presentation for the mapping class group of each sporadic surface."}
{"category": "Math", "title": "Set partition statistics and q-Fibonacci numbers", "abstract": "We consider the set partition statistics ls and rb introduced by Wachs and White and investigate their distribution over set partitions avoiding certain patterns. In particular, we consider those set partitions avoiding the pattern 13/2, $\\Pi_n(13/2)$, and those avoiding both 13/2 and 123, $\\Pi_n(13/2,123)$. We show that the distribution over $\\Pi_n(13/2)$ enumerates certain integer partitions, and the distribution over $\\Pi_n(13/2,123)$ gives q-Fibonacci numbers. These q-Fibonacci numbers are closely related to q-Fibonacci numbers studied by Carlitz and by Cigler. We provide combinatorial proofs that these q-Fibonacci numbers satisfy q-analogues of many Fibonacci identities. Finally, we indicate how p,q-Fibonacci numbers arising from the bistatistic (ls, rb) give rise to p,q-analogues of identities."}
{"category": "Math", "title": "Random perturbations of stochastic chains with unbounded variable length memory", "abstract": "We consider binary infinite order stochastic chains perturbed by a random noise. This means that at each time step, the value assumed by the chain can be randomly and independently flipped with a small fixed probability. We show that the transition probabilities of the perturbed chain are uniformly close to the corresponding transition probabilities of the original chain. As a consequence, in the case of stochastic chains with unbounded but otherwise finite variable length memory, we show that it is possible to recover the context tree of the original chain, using a suitable version of the algorithm Context, provided that the noise is small enough."}
{"category": "Math", "title": "Coverage Probability of Random Intervals", "abstract": "In this paper, we develop a general theory on the coverage probability of random intervals defined in terms of discrete random variables with continuous parameter spaces. The theory shows that the minimum coverage probabilities of random intervals with respect to corresponding parameters are achieved at discrete finite sets and that the coverage probabilities are continuous and unimodal when parameters are varying in between interval endpoints. The theory applies to common important discrete random variables including binomial variable, Poisson variable, negative binomial variable and hypergeometrical random variable. The theory can be used to make relevant statistical inference more rigorous and less conservative."}
{"category": "Math", "title": "Regularity of solutions to higher-order integrals of the calculus of variations", "abstract": "We obtain new regularity conditions for problems of calculus of variations with higher-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main regularity result asserts that autonomous integral functionals with a Lagrangian having coercive partial derivatives with respect to the higher-order derivatives admit only minimizers with essentially bounded derivatives."}
{"category": "Math", "title": "Centralizers in the Hecke algebras of complex reflection groups", "abstract": "How far can the elementary description of centralizers of parabolic subalgebras of Hecke algebras of finite real reflection groups be generalized to the complex reflection group case? In this paper we begin to answer this question by establishing results in two directions. First, under conditions closely analogous to those existing for the real case, we give explicit relations between coefficients in an element centralizing a generator. Second, we introduce a tool for dealing with a major challenge of the complex case -- the ``instability'' of certain double cosets -- through the definition and use of a double coset graph. We use these results to find integral bases for the centralizers of generators as well as the centres of the Hecke algebras of types $G_4$ and $G(4,1,2)$. Keywords: complex reflection group; Hecke algebra; centre; centralizer; modular; double coset."}
{"category": "Math", "title": "Poincar\\'e inequality for non euclidean metrics and transportation cost inequalities on $\\mathbb{R}^d$", "abstract": "In this paper, we consider Poincar\\'e inequalities for non euclidean metrics on $\\mathbb{R}^d$. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give different equivalent functional forms of these Poincar\\'e type inequalities in terms of transportation-cost inequalities and infimum convolution inequalities. Workable sufficient conditions are given and a comparison is made with generalized Beckner-Latala-Oleszkiewicz inequalities."}
{"category": "Math", "title": "Critical percolation on random regular graphs", "abstract": "We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d \\geq 3 is fixed and n grows. We prove mean-field behavior around the critical probability p_c=1/(d-1). In particular, we show that there is a scaling window of width n^{-1/3} around p_c in which the sizes of the largest components are roughly n^{2/3} and we describe their limiting joint distribution. We also show that for the subcritical regime, i.e. p = (1-eps(n))p_c where eps(n)=o(1) but \\eps(n)n^{1/3} tends to infinity, the sizes of the largest components are concentrated around an explicit function of n and eps(n) which is of order o(n^{2/3}). In the supercritical regime, i.e. p = (1+\\eps(n))p_c where eps(n)=o(1) but eps(n)n^{1/3} tends to infinity, the size of the largest component is concentrated around the value (2d/(d-2))\\eps(n)n and a duality principle holds: other component sizes are distributed as in the subcritical regime."}
{"category": "Math", "title": "Complete sums of products of (h,q)-extension of Euler numbers and polynomials", "abstract": "In this paper we investigate some interesting of the (h,q)-extension of Euler numbers and polynomials. Finally, we will give some relations between these numbers anf polynomials"}
{"category": "Math", "title": "Geometrical relations and plethysms in the Homfly skein of the annulus", "abstract": "The oriented framed Homfly skein C of the annulus provides the natural parameter space for the Homfly satellite invariants of a knot. It contains a submodule C+ isomorphic to the algebra of the symmetric functions. We collect and expand formulae relating elements expressed in terms of symmetric functions to Turaev's geometrical basis of C+. We reformulate the formulae of Rosso and Jones for quantum sl(N) invariants of cables in terms of plethysms of symmetric functions, and use the connection between quantum sl(N) invariants and C+ to give a formula for the satellite of a cable as an element of C+. We then analyse the case where a cable is decorated by the pattern which corresponds to a power sum in the symmetric function interpretation of C+ to get direct relations between the Homfly invariants of some diagrams decorated by power sums."}
{"category": "Math", "title": "On the Orbits of Solvable Linear Groups", "abstract": "Let $G$ be a solvable linear group acting on the finite vectorpace $V$ and assume that $(|G|,|V|)=1$. In this paper we find $x,y\\in V$ such that $C_G(x)\\cap C_G(y)=1$. In particular, this answers a question of I. M. Isaacs. We complete some results of S. Dolphi, A. Seress and T. R. Wolf."}
{"category": "Math", "title": "On multidimensional item response theory -- a coordinate free approach", "abstract": "A coordinate system free definition of complex structure multidimensional item response theory (MIRT) for dichotomously scored items is presented. The point of view taken emphasizes the possibilities and subtleties of understanding MIRT as a multidimensional extension of the ``classical'' unidimensional item response theory models. The main theorem of the paper is that every monotonic MIRT model looks the same; they are all trivial extensions of univariate item response theory."}
{"category": "Math", "title": "Sylvester's Minorant Criterion, Lagrange-Beltrami Identity, and Nonnegative Definiteness", "abstract": "We consider the characterizations of positive definite as well as nonnegative definite quadratic forms in terms of the principal minors of the associated symmetric matrix. We briefly review some of the known proofs, including a classical approach via the Lagrange-Beltrami identity. For quadratic forms in up to 3 variables, we give an elementary and self-contained proof of Sylvester's Criterion for positive definiteness as well as for nonnegative definiteness. In the process, we obtain an explicit version of Lagrange-Beltrami identity for ternary quadratic forms."}
{"category": "Math", "title": "Meromorphic solutions of higher order Briot-Bouquet differential equations", "abstract": "For differential equations $P(y^{(k)},y)=0,$ where $P$ is a polynomial, we prove that all meromorphic solutions having at least one pole are elliptic functions, possibly degenerate."}
{"category": "Math", "title": "Some particular self-interacting diffusions: Ergodic behaviour and almost sure convergence", "abstract": "This paper deals with some self-interacting diffusions $(X_t,t\\geq 0)$ living on $\\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations: \\[\\mathrm{d}X_t=\\mathrm{d}B_t-g(t)\\nabla V(X_t-\\bar{\\mu}_t)\\,\\mathrm{d}t,\\] where $\\bar{\\mu}_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. We study the ergodic behaviour of $X$ and prove that it is strongly related to $g$. Actually, we show that $X$ is ergodic (in the limit quotient sense) if and only if $\\bar{\\mu}_t$ converges a.s. We also give some conditions (on $g$ and $V$) for the almost sure convergence of $X$."}
{"category": "Math", "title": "Convergence in distribution of some particular self-interacting diffusions: the simulated annealing method", "abstract": "The present paper is concerned with some self-interacting diffusions $(X_t,t\\geq 0)$ living on $\\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations: $$\\mathrm{d}X_t = \\mathrm{d}B_t - g(t)\\nabla V(X_t - \\bar{\\mu}_t) \\mathrm{d}t$$ where $\\bar{\\mu}_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. The authors have still studied the ergodic behavior of $X$ and proved that it is strongly related to $g$. We go further and give necessary and sufficient conditions (for small $g$'s) in order that $X$ converges in probability to $X_\\infty$ (which is related to the global minima of $V$)."}
{"category": "Math", "title": "Pretty clean monomial ideals and linear quotients", "abstract": "We study basic properties of monomial ideals with linear quotients. It is shown that if the monomial ideal $I$ has linear quotients, then the squarefree part of $I$ and each component of $I$ as well as $\\mm I$ have linear quotients, where $\\mm$ is the graded maximal ideal of the polynomial ring. As an analogy to the Rearrangement Lemma of Bj\\\"orner and Wachs we also show that for a monomial ideal with linear quotients the admissible order of the generators can be chosen degree increasingly. As a generalization of the facet ideal of a forest, we define monomial ideals of forest type and show that they are pretty clean. This result recovers a recent result of Tuly and Villarreal about the shellability of a clutter with the free vertex property. As another consequence of this result we show that if $I$ is a monomial ideal of forest type, then Stanley's conjecture on Stanley decomposition holds for $S/I$. We also show that a clutter is totally balanced if and only if it has the free vertex property."}
{"category": "Math", "title": "A non-homogeneous orbit of a diagonal subgroup", "abstract": "Let G=SL(n,R) with n>5. We construct examples of lattices Gamma of G, subgroup A of the diagonal group and points x in G/Gamma such that the closure of the orbit Ax is not homogeneous but does not factors through the action of a one-parameter non-unipotent group. This contradicts a conjecture of Margulis."}
{"category": "Math", "title": "Constructing simply laced Lie algebras from extremal elements", "abstract": "For any finite graph Gamma and any field K of characteristic unequal to 2 we construct an algebraic variety X over K whose K-points parameterise K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the non-edges. After that, we study the case where Gamma is a connected, simply laced Dynkin diagram of finite or affine type. We prove that X is then an affine space, and that all points in an open dense subset of X parameterise Lie algebras isomorphic to a single fixed Lie algebra. If Gamma is of affine type, then this fixed Lie algebra is the split finite-dimensional simple Lie algebra corresponding to the associated finite-type Dynkin diagram. This gives a new construction of these Lie algebras, in which they come together with interesting degenerations, corresponding to points outside the open dense subset. Our results may prove useful for recognising these Lie algebras."}
{"category": "Math", "title": "Decomposability problem on branched coverings", "abstract": "Given a branched covering of degree d between closed surfaces, it determines a collection of partitions of d, the branch data. In this work we show that any branch data are realized by an indecomposable primitive branched covering on a connected close surface N with Euler's characteristic less than or equal to 0. This shows that decomposable and indecomposable realizations may coexist. Moreover, we characterize the branch data of a decomposable primitive branched covering."}
{"category": "Math", "title": "Advanced topology on the multiscale sequence spaces S^\\nu", "abstract": "We pursue the study of the multiscale spaces $S^\\nu$ introduced by Jaffard in the context of multifractal analysis. We give the necessary and sufficient condition for $S^\\nu$ to be locally p-convex, and exhibit a sequence of $p$-norms that defines its natural topology. The strong topological dual of $S^\\nu$ is identified to another sequence space depending on $\\nu$, endowed with an inductive limit topology. As a particular case, we describe the dual of a countable intersection of Besov spaces."}
{"category": "Math", "title": "About the Characteristic Function of a Set", "abstract": "In this paper we give a method, based on the characteristic function of a set, to solve some difficult problems of set theory in undergraduate research."}
{"category": "Math", "title": "The Integral (orbifold) Chow Ring of Toric Deligne-Mumford Stacks", "abstract": "In this paper we study the integral Chow ring of toric Deligne-Mumford stacks. We prove that the integral Chow ring of a semi-projective toric Deligne-Mumford stack is isomorphic to the Stanley-Reisner ring of the associated stacky fan. The integral orbifold Chow ring is also computed. Our results are illustrated with several examples."}
{"category": "Math", "title": "Compactness results for the K\\\"ahler-Ricci flow", "abstract": "We consider the K\\\"ahler-Ricci flow $\\frac{\\partial}{\\partial t}g_{i\\bar{j}} = g_{i\\bar{j}} - R_{i\\bar{j}}$ on a compact K\\\"ahler manifold $M$ with $c_1(M) > 0$, of complex dimension $k$. We prove the $\\epsilon$-regularity lemma for the K\\\"ahler-Ricci flow, based on Moser's iteration. Assume that the Ricci curvature and $\\int_M |\\rem|^k dV_t$ are uniformly bounded along the flow. Using the $\\epsilon$-regularity lemma we derive the compactness result for the K\\\"ahler-Ricci flow. Under our assumptions, if $k \\ge 3$ in addition, using the compactness result we show that $|\\rem| \\le C$ holds uniformly along the flow. This means the flow does not develop any singularities at infinity. We use some ideas of Tian from \\cite{Ti} to prove the smoothing property in that case."}
{"category": "Math", "title": "Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves", "abstract": "Let $\\E/\\Q$ be a fixed elliptic curve over $\\Q$ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W. Duke have obtained an asymptotic formula for the number of primes $p\\le x$ such that the reduction of $\\E$ modulo p has a trivial Tate-Shafarevich group. Recent results of A. C. Cojocaru and C. David lead to a better error term. We introduce a new argument in the scheme of the proof which gives further improvement."}
{"category": "Math", "title": "Canonical lifts of the Johnson homomorphisms to the Torelli groupoid", "abstract": "We prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized Magnus expansion in the sense of Kawazumi. This Magnus expansion is used to give canonical lifts of the higher Johnson homomorphisms $\\tau_m$, for $m\\geq 1$, to the Torelli groupoid, and we provide a recursive combinatorial formula for tensor representatives of these lifts. In particular, we give an explicit 1-cocycle in the dual fatgraph complex which lifts $\\tau_2$ and thus answer affirmatively a question of Morita-Penner. To illustrate our techniques for calculating higher Johnson homomorphisms in general, we give explicit examples calculating $\\tau_m$, for $m\\leq 3$."}
{"category": "Math", "title": "Second order arithmetic means in operator ideals", "abstract": "Equality of the second order arithmetic means of two principal ideals does not imply equality of their first order arithmetic means (second order equality cancellation). We provide fairly broad sufficient conditions on one of the principal ideals for this implication to hold true. We present also sufficient conditions for second order inclusion cancellations. These conditions are formulated in terms of the growth properties of the ratio of regularity sequence associated to the sequence of s-number of a generator of the principal ideal. These results are then extended to general ideals."}
{"category": "Math", "title": "Gromov-Witten invariants of varieties with holomorphic 2-forms", "abstract": "We show that a holomorphic two-form $\\theta$ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps $\\mgn(X,\\beta)$ to the locus where $\\theta$ degenerates; it then enables us to define the localized GW-invariant, an algebro-geometric analogue of the local invariant of Lee and Parker in symplectic geometry, which coincides with the ordinary GW-invariant when X is proper. It is deformation invariant. Using this, we prove formulas for low degree GW-invariants of minimal general type surfaces with p_g>0 conjectured by Maulik and Pandharipande."}
{"category": "Math", "title": "Card shuffling and diophantine approximation", "abstract": "The ``overlapping-cycles shuffle'' mixes a deck of $n$ cards by moving either the $n$th card or the $(n-k)$th card to the top of the deck, with probability half each. We determine the spectral gap for the location of a single card, which, as a function of $k$ and $n$, has surprising behavior. For example, suppose $k$ is the closest integer to $\\alpha n$ for a fixed real $\\alpha\\in(0,1)$. Then for rational $\\alpha$ the spectral gap is $\\Theta(n^{-2})$, while for poorly approximable irrational numbers $\\alpha$, such as the reciprocal of the golden ratio, the spectral gap is $\\Theta(n^{-3/2})$."}
{"category": "Math", "title": "Triple Hilbert transforms along polynomial surfaces", "abstract": "This paper has been withdrawn since it contains some discrepancy with othe authers's recent result. We will not post this until this discrepancy is resolved."}
{"category": "Math", "title": "The asymptotic limits of zero modes of massless Dirac operators", "abstract": "Asymptotic behaviors of zero modes of the massless Dirac operator $H=\\alpha\\cdot D + Q(x)$ are discussed, where $\\alpha= (\\alpha_1, \\alpha_2, \\alpha_3)$ is the triple of $4 \\times 4$ Dirac matrices, $ D=\\frac{1}{i} \\nabla_x$, and $Q(x)=\\big(q_{jk} (x) \\big)$ is a $4\\times 4$ Hermitian matrix-valued function with $| q_{jk}(x) | \\le C < x >^{-\\rho} $, $\\rho >1$. We shall show that for every zero mode $f$, the asymptotic limit of $|x|^2f(x)$ as $|x| \\to +\\infty$ exists. The limit is expressed in terms of an integral of $Q(x)f(x)$."}
{"category": "Math", "title": "Gale duality bounds for roots of polynomials with nonnegative coefficients", "abstract": "We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most $d$. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials."}
{"category": "Math", "title": "Application of probabilistic PCR5 Fusion Rule for Multisensor Target Tracking", "abstract": "This paper defines and implements a non-Bayesian fusion rule for combining densities of probabilities estimated by local (non-linear) filters for tracking a moving target by passive sensors. This rule is the restriction to a strict probabilistic paradigm of the recent and efficient Proportional Conflict Redistribution rule no 5 (PCR5) developed in the DSmT framework for fusing basic belief assignments. A sampling method for probabilistic PCR5 (p-PCR5) is defined. It is shown that p-PCR5 is more robust to an erroneous modeling and allows to keep the modes of local densities and preserve as much as possible the whole information inherent to each densities to combine. In particular, p-PCR5 is able of maintaining multiple hypotheses/modes after fusion, when the hypotheses are too distant in regards to their deviations. This new p-PCR5 rule has been tested on a simple example of distributed non-linear filtering application to show the interest of such approach for future developments. The non-linear distributed filter is implemented through a basic particles filtering technique. The results obtained in our simulations show the ability of this p-PCR5-based filter to track the target even when the models are not well consistent in regards to the initialization and real cinematic."}
{"category": "Math", "title": "Integral Concentration of idempotent trigonometric polynomials with gaps", "abstract": "We prove that for all p>1/2 there exists a constant $\\gamma_p>0$ such that, for any symmetric measurable set of positive measure $E\\subset \\TT$ and for any $\\gamma<\\gamma_p$, there is an idempotent trigonometrical polynomial f satisfying $\\int_E |f|^p > \\gamma \\int_{\\TT} |f|^p$. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of $\\gamma_p>0$ for p>1 and conjectured that it does not exists for p=1. Furthermore, we prove that one can take $\\gamma_p=1$ when p>1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when $p\\neq 2$. This shows striking differences with the case p=2, for which the best constant is strictly smaller than 1/2, as it has been known for twenty years, and for which having arbitrarily large gaps with such concentration of the integral is not possible, according to a classical theorem of Wiener. We find sharper results for $0<p\\leq 1$ when we restrict to open sets, or when we enlarge the class of idempotent trigonometric polynomials to all positive definite ones."}
{"category": "Math", "title": "Finding Efficient Recursions for Risk Aggregation by Computer Algebra", "abstract": "We derive recursions for the probability distribution of random sums by computer algebra. Unlike the well-known Panjer-type recursions, they are of finite order and thus allow for computation in linear time. This efficiency is bought by the assumption that the probability generating function of the claim size be algebraic. The probability generating function of the claim number is supposed to be from the rather general class of D-finite functions."}
{"category": "Math", "title": "The coding complexity of L\\'evy processes", "abstract": "We investigate the high resolution coding problem for general real-valued L\\'evy processes under L^p[0,1]-norm distortion. Tight asymptotic formulas are found under mild regularity assumptions."}
{"category": "Math", "title": "Block-Toeplitz determinants, chess tableaux, and the type $\\hat{A_1}$ Geiss-Leclerc-Schroer $\\phi$-map", "abstract": "We evaluate the Geiss-Leclerc-Schroer $\\phi$-map for shape modules over the preprojective algebra $\\Lambda$ of type $\\hat{A_1}$ in terms of matrix minors arising from the block-Toeplitz representation of the loop group $\\SL_2(\\mathcal{L})$. Conjecturally these minors are among the cluster variables for coordinate rings of unipotent cells within $\\SL_2(\\mathcal{L})$. In so doing we compute the Euler characteristic of any generalized flag variety attached to a shape module by counting standard tableaux of requisite shape and parity; alternatively by counting chess tableaux of requisite shape and content."}
{"category": "Math", "title": "An sl(2) tangle homology and seamed cobordisms", "abstract": "We construct a bigraded (co)homology theory which depends on a parameter a, and whose graded Euler characteristic is the quantum sl(2) link invariant. We follow Bar-Natan's approach to tangles on one side, and Khovanov's sl(3) theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov sl(2) invariant (or Lee's modification of it) corresponds to a=0 (or a=1). In particular, our construction naturally resolves the sign ambiguity in the functoriality of Khovanov's sl(2) homology theory."}
{"category": "Math", "title": "Extremal Problems in Minkowski Space related to Minimal Networks", "abstract": "We solve the following problem of Z. F\\\"uredi, J. C. Lagarias and F. Morgan [FLM]: Is there an upper bound polynomial in $n$ for the largest cardinality of a set S of unit vectors in an n-dimensional Minkowski space (or Banach space) such that the sum of any subset has norm less than 1? We prove that |S|\\leq 2n and that equality holds iff the space is linearly isometric to \\ell^n_\\infty, the space with an n-cube as unit ball. We also remark on similar questions raised in [FLM] that arose out of the study of singularities in length-minimizing networks in Minkowski spaces."}
{"category": "Math", "title": "On universal C^*-algebras generated by n projections with scalar sum", "abstract": "We study the universal C^*-algebras generated by n projections $p_1, >..., p_n$ subject to the relation $p_1+... p_n = \\lambda 1$, $\\lambda \\in \\mathbb R$. The questions of when these C^*-algebras are type I, nuclear or exact are considered. It is proved also that among these algebras there is continuum of mutually nonisomorphic ones."}
{"category": "Math", "title": "Positively curved homogeneous metrics on spheres", "abstract": "We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the classification of all homogeneous metrics with positive curvature (apart from one special Aloff Wallach space where the set of homogeneous metrics is too large)."}
{"category": "Math", "title": "On primes in arithmetic progression having a prescribed primitive root. II", "abstract": "Let a and f be coprime positive integers. Let g be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes p such that p=a(mod f) and g is a primitive root modulo p has a natural density. In this note this density is explicitly evaluated with an Euler product as result."}
{"category": "Math", "title": "Cones over pseudo-Riemannian manifolds and their holonomy", "abstract": "By a classical theorem of Gallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone is para-K\\\"ahlerian. For Lorentzian cones we get a complete description of the possible (local) holonomy algebras in terms of the metric of the base manifold."}
{"category": "Math", "title": "Bergman metrics and geodesics in the space of K\\\"ahler metrics on toric varieties", "abstract": "Geodesics on the infinite dimensional symmetric space $\\hcal$ of K\\\"ahler metrics in a fixed K\\\"ahler class on a projective K\\\"ahler manifold X are solutions of a homogeneous complex Monge-Amp\\`ere equation in $X \\times A$, where $A \\subset \\C$ is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces $G_{\\C}/G$. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Amp\\`ere geodesics can be approximated by 1PS geodesics in the symmetric spaces of Bergman metrics. Phong-Sturm proved weak C^0 convergence of Bergman to Monge-Amp\\`ere geodesics on a general \\kahler manifold. In this article we prove convergence in $C^2(A \\times X)$ in the case of toric K\\\"ahler metrics, extending our earlier result on $\\CP^1$."}
{"category": "Math", "title": "Mixed Hodge complexes and L^2-cohomology for local systems on ball quotients", "abstract": "We study the $L^2$--cohomology of certain local systems on non-compact arithmetic ball quotients $X=\\Gamma \\backslash \\B_n$, in particular vanishing and non--vanishing results. We also give generalizations to higher dimensional ball quotients and study the mixed Hodge structure on the sheaf cohomology of a local system with the $L^2$-cohomology contributing to the lowest weight part."}
{"category": "Math", "title": "Nonnegatively and Positively curved manifolds", "abstract": "This is a survey written for a special edition of the journal of differential geometry."}
{"category": "Math", "title": "Batalin-Vilkovisky algebras and the J-homomorphism", "abstract": "Let X be a topological space. The homology of the iterated loop space $H_*\\Omega^n X$ is an algebra over the homology of the framed n-disks operad $H_*f\\mathcal{D}_n$ \\cite{Getzler:BVAlg,Salvatore-Wahl:FrameddoBVa}. We determine completely this $H_*f\\mathcal{D}_n$-algebra structure on $H_*(\\Omega^n X;\\mathbb{Q})$. We show that the action of $H_*(SO(n))$ on the iterated loop space $H_*\\Omega^n X$ is related to the J-homomorphism and that the BV-operator vanishes on spherical classes only in characteristic other than 2."}
{"category": "Math", "title": "The 2-adic valuations of Stirling numbers", "abstract": "The 2-adic valuation of the Stirling numbres is examined. We conjecture pattrens about the distributions of these valuations in residue classes modulo powers of 2."}
{"category": "Math", "title": "Quantized symplectic actions and W-algebras", "abstract": "With a nilpotent element in a semisimple Lie algebra g one associates a finitely generated associative algebra W called a W-algebra of finite type. This algebra is obtained from the universal enveloping algebra U(g) by a certain Hamiltonian reduction. We observe that W is the invariant algebra for an action of a reductive group G with Lie algebra g on a quantized symplectic affine variety and use this observation to study W. Our results include an alternative definition of W, a relation between the sets of prime ideals of W and of the corresponding universal enveloping algebra, the existence of a one-dimensional representation of W in the case of classical g and the separation of elements of W by finite dimensional representations."}
{"category": "Math", "title": "B(H) lattices, density and arithmetic mean ideals", "abstract": "This part of a multi-paper project studies the lattice properties of the arithmetic mean ideals of B(H) introduced by Dykema, Figiel, Weiss, and Wodzicki. We prove: the lattices of all principal ideals, of arithmetic mean or arithmetic mean at infinity stable principal ideals or of principal ideals with a generator that satisfies the Delta_1/2 condition, are all both upper and lower dense in the lattice of general ideals. That is, between any ideal and an ideal (nested above or below respectively) in one of these sublattices, lies another ideal in that sublattice. Among the applications: a principal ideal I is am-stable (I = I_a) if and only if any of its first order arithmetic mean ideals are am-stable if and only if the ideal satisfies the first order equality cancellation property: J_a = I_a implies J = I. We show that this cancellation property can fail even for am-stable countably generated ideals. Similar results hold for arithmetic mean at infinity ideals. Inclusion cancellations do not hold in general even for principal ideals, but for every ideal I there is a largest ideal I^ for which J_a contains I_a implies that J contains I^. When I is principal, I^ too is principal. We show that I=I^ is a strictly stronger property than am-stability. For example, for I the p < 1 power of the principal ideal J generated by diag {1/n}, I^ is the q power of J where 1/p-1/q = 1."}
{"category": "Math", "title": "The action of the mapping class group on representation varieties of PSL(2,R). Case I: The one-holed torus", "abstract": "In this paper we consider the action of the mapping class group of a surface on the space of homomorphisms from the fundamental group of a surface into PSL(2,R). Goldman conjectured that when the surface is closed and of genus bigger than one, the action on non-Teichmuller connected components of the associated moduli space (i.e. the space of homomorphisms modulo conjugation) is ergodic. One approach to this question is to use sewing techniques which requires that one considers the action on the level of homomorphisms, and for surfaces with boundary. In this paper we consider the case of the one-holed torus with boundary condition, and we determine regions where the action is ergodic. Our main result mirrors a theorem of Goldman's at the level of moduli."}
{"category": "Math", "title": "On unitarily equivalent submodules", "abstract": "The Hardy space on the unit ball in C^n provides examples of a quasi-free, finite rank Hilbert module which contains a pure submodule isometrically isomorphic to the module itself. For n=1 the submodule has finite codimension. In this note we show that this phenomenon can only occur for modules over domains in the complex plain and for finitely-connected domains only for Hardy-like spaces, the bundle shifts. Moreover, we show for essentially reductive modules that even when the codimension is infinite, the module is subnormal and again, on nice domains such as the unit ball, must be Hardy-like."}
{"category": "Math", "title": "Deformed Macdonald-Ruijsenaars operators and super Macdonald polynomials", "abstract": "It is shown that the deformed Macdonald-Ruijsenaars operators can be described as the restrictions on certain affine subvarieties of the usual Macdonald-Ruijsenaars operator in infinite number of variables. The ideals of these varieties are shown to be generated by the Macdonald polynomials related to Young diagrams with special geometry. The super Macdonald polynomials and their shifted version are introduced, the combinatorial formulas for them are given."}
{"category": "Math", "title": "ACM sets of points in multiprojective space", "abstract": "If X is a finite set of points in a multiprojective space P^n1 x ... x P^nr with r >= 2, then X may or may not be arithmetically Cohen-Macaulay (ACM). For sets of points in P^1 x P^1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space. We show that each classification for ACM points in P^1 x P^1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM."}
{"category": "Math", "title": "Separators of points in a multiprojective space", "abstract": "In this note we develop some of the properties of separators of points in a multiprojective space. In particular, we prove multigraded analogs of results of Geramita, Maroscia, and Roberts relating the Hilbert function of X and X \\{P} via the degree of a separator, and Abrescia, Bazzotti, and Marino relating the degree of a separator to shifts in the minimal multigraded free resolution of the ideal of points."}
{"category": "Math", "title": "A parameterization of the Fermat curves satisfying x^(2N)+y^(2N)=1", "abstract": "Note that the family of closed curves C_N={(x,y)\\in R^2;x^(2N)+y^(2N)=1} for N=1,2,3,... approaches the boundary of [-1,1]^2 as N \\to \\infty. In this paper we exhibit a natural parameterization of these curves and generalize to a larger class of equations."}
{"category": "Math", "title": "Stable Tameness of Two-Dimensional Polynomial Automorphisms Over a Regular Ring", "abstract": "In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. In the case R is a Dedekind Q-algebra, some stronger results are obtained. A key element in the proof is a theorem which yields the following corollary: Over an Artinian ring A all two-dimensional polynomial automorphisms having Jacobian determinant one are stably tame, and are tame if A is a Q-algebra. Another crucial ingredient, of interest in itself, is that stable tameness is a local property: If an automorphism is locally tame, then it is stably tame."}
{"category": "Math", "title": "A Random Change of Variables and Applications to the Stochastic Porous Medium Equation with Multiplicative Time Noise", "abstract": "A change of variables is introduced to reduce certain nonlinear stochastic evolution equations with multiplicative noise to the corresponding deterministic equation. The result is then used to investigate a stochastic porous medium equation."}
{"category": "Math", "title": "Intersection probabilities for a chordal SLE path and a semicircle", "abstract": "We derive a number of estimates for the probability that a chordal SLE path in the upper half plane H intersects a semicircle centred on the real line. We prove that if 0<kappa<8 and gamma:[0,infinity) to closure(H) is a chordal SLE in H from 0 to infinity, then P(gamma[0,infinity) cap C(x;rx) neq emptyset) asymp r^(4a-1) where a=2/kappa and C(x;rx) denotes the semicircle centred at x>0 of radius rx, 0<r<1/3, in the upper half plane. As an application of our results, for 0<kappa<8, we derive an estimate for the diameter of a chordal SLE path in H between two real boundary points 0 and x>0. For 4<kappa<8, we also estimate the probability that an entire semicircle on the real line is swallowed at once by a chordal SLE path in H from 0 to infinity."}
{"category": "Math", "title": "Rational Tate classes", "abstract": "In despair, as Deligne (2000) put it, of proving the Hodge and Tate conjectures, we can try to find substitutes. For abelian varieties in characteristic zero, Deligne (1982) constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of \"rational Tate classes\" on varieties over finite fields having the properties that the algebraic classes would have if the Hodge and Tate conjectures were known. v3. Submitted version."}
{"category": "Math", "title": "Traces on operator ideals and arithmetic means", "abstract": "This article - a part of a multipaper project investigating arithmetic mean ideals - investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., ``How many traces can an ideal support?\" We conjecture that the codimension can be only zero, one, or infinity. Using the arithmetic mean (am) operations on ideals introduced by Dykema, Figiel, Weiss, and Wodzicki, and the analogous am operations at infinity that we develop in this article, the conjecture is proven for all ideals not contained in the largest am-infinity stable ideal and not containing the smallest am-stable ideal. It is also proven for all soft-edged ideals (i.e., I= IK(H)) and all soft-complemented ideals (i.e., I= I/K(H)), which include many classical operator ideals. In the process, we prove that an ideal of trace class operators supports a unique trace (up to scalar multiples) if and only if it is am-infinity stable and that, for a principal ideal, am-infinity stability is equivalent to regularity at infinity of the sequence of s-numbers of the generator. Furthermore, we apply trace extension methods to two problems on elementary operators studied by V. Shulman and to Fuglede-Putnam type problems of the second author."}
{"category": "Math", "title": "Equivariant orbifold structures on the projective line and integrable hierarchies", "abstract": "Let $\\CP^1_{k,m}$ be the orbifold structure on $\\CP^1$ obtained via uniformizing the neighborhoods of 0 and $\\infty$ respectively by $z\\mapsto z^k$ and $w\\mapsto w^m.$ The diagonal action of the torus on the projective line induces naturally an orbifold action on $\\CP^1_{k,m}.$ In this paper we prove that if k and m are co-prime then Givental's prediction of the equivariant total descendent Gromov-Witten potential of $\\CP^1_{k,m}$ satisfies certain Hirota Quadratic Equations (HQE for short). We also show that after an appropriate change of the variables, similar to Getzler's change in the equivariant Gromov-Witten theory of $\\CP^1$, the HQE turn into the HQE of the 2-Toda hierarchy, i.e., the Gromov-Witten potential of $\\CP^1_{k,m}$ is a tau-function of the 2-Toda hierarchy. More precisely, we obtain a sequence of tau-functions of the 2-Toda hierarchy from the descendent potential via some translations. The later condition, that all tau-functions in the sequence are obtained from a single one via translations, imposes a serious constraint on the solution of the 2-Toda hierarchy. Our theorem leads to the discovery of a new integrable hierarchy (we suggest to be called the Equivariant Bi-graded Toda Hierarchy). We conjecture that this new hierarchy governs, i.e., uniquely determines, the equivariant Gromov-Witten invariants of $\\CP^1_{k,m}.$"}
{"category": "Math", "title": "Points on Shimura varieties over finite fields: the conjecture of Langlands and Rapoport", "abstract": "We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of PEL-type."}
{"category": "Math", "title": "A new characterization for the m-quasiinvariants of S_n and explicit basis for two row hook shapes", "abstract": "In 2002, Feigin and Veselov defined the space of m-quasiinvariants for any Coxeter group, building on earlier work of Chalykh and Veselov. While many properties of those spaces were proven from this definition, an explicit computation of a basis was only done in certain cases. In particular, Feigin and Veselov computed bases for the m-quasiinvariants of dihedral groups, including S_3, and Felder and Veselov computed the non-symmetric m-quasiinvariants of lowest degree for general S_n. In this paper, we provide a new characterization of the m-quasiinvariants of S_n, and use this to provide a basis for the isotypic component indexed by the partition [n-1,1]. This builds on a previous paper in which we computed a basis for S_3 via combinatorial methods."}
{"category": "Math", "title": "Towards a proof of the conjecture of Langlands and Rapoport", "abstract": "A conference talk discussing the conjecture of Langlands and Rapoport concerning the structure of the points on a Shimura variety modulo a prime of good reduction."}
{"category": "Math", "title": "Vanishing theorems for toric polyhedra", "abstract": "A toric polyhedron is a reduced closed subscheme of a toric variety that are partial unions of the orbits of the torus action. We prove vanishing theorems for toric polyhedra. We also give a proof of the $E_1$-degeneration of Hodge to de Rham type spectral sequence for toric polyhedra in any characteristic. Finally, we give a very powerful extension theorem for ample line bundles."}
{"category": "Math", "title": "Combinatorial Aspects of Elliptic Curves", "abstract": "Given an elliptic curve C, we study here $N_k = #C(F_{q^k})$, the number of points of C over the finite field F_{q^k}. This sequence of numbers, as k runs over positive integers, has numerous remarkable properties of a combinatorial flavor in addition to the usual number theoretical interpretations. In particular we prove that $N_k = - W_k(q, - N_1)$ where W_k(q,t) is a (q,t)-analogue of the number of spanning trees of the wheel graph. Additionally we develop a determinantal formula for N_k where the eigenvalues can be explicitly written in terms of q, N_1, and roots of unity. We also discuss here a new sequence of bivariate polynomials related to the factorization of N_k, which we refer to as elliptic cyclotomic polynomials because of their various properties."}
{"category": "Math", "title": "Random generation of finitely generated subgroups of a free group", "abstract": "We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity $\\O(n)$ in the RAM model and $\\O(n^2\\log^2n)$ in the bitcost model."}
{"category": "Math", "title": "The barnes G function and its relations with sums and products of generalized Gamma convolution variables", "abstract": "We give a probabilistic interpretation for the Barnes G-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices and the Barnes G-function are intimately related with products and sums of gamma, beta and log-gamma variables. In particular, we show that the law of the modulus of the characteristic polynomial of random unitary matrices can be expressed with the help of products of gamma or beta variables, and that the reciprocal of the Barnes G-function has a L\\'{e}vy-Khintchin type representation. These results lead us to introduce the so called generalized gamma convolution variables."}
{"category": "Math", "title": "The cubic nonlinear Schr\\\"odinger equation in two dimensions with radial data", "abstract": "We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\\\"odinger equation $iu_t + \\Delta u = \\pm |u|^2 u$ for large spherically symmetric L^2_x(\\R^2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry."}
{"category": "Math", "title": "Galilean geometry of motions", "abstract": "In this paper we show that Galilean group is a matrix Lie group and find its structure. Then provide the invariants of special Galilean geometry of motions, by Olver's method of moving coframes, we also find the corresponding $\\{e\\}-$structure."}
{"category": "Math", "title": "Index theory for actions of compact Lie groups on C*-algebras", "abstract": "We study the index theory for actions of compact Lie groups on C*-algebras with an emphasis on principal actions. Given an invariant semifinite trace on the C*-algebra we obtain semifinite spectral triples. For circle actions we consider the relation to the dual Pimsner-Voiculescu sequence. On the way we show that the notions ``saturated'' and ``principal'' are equivalent for actions by compact Lie groups."}
{"category": "Math", "title": "Applications of the change-of-rings spectral sequence to the computation of Hochschild cohomology", "abstract": "We consider the change-of-rings spectral sequence as it applies to Hochschild cohomology, obtaining a description of the differentials on the first page which relates it to the multiplicative stucture on cohomology. Using this information, we are able to completely describe the cohomology structure of monogenic algebras as well as some information on the structure of the cohomology in more general situations. We also show how to use the spectral sequence to reprove and generalize results of M. Auslander et. al. about homological epimorphisms. We derive from this a rather general version of the long exact sequence due to D. Happel for a one-point (co)-extension of a finite dimensional algebra and show how it can be put to use in concrete examples."}
{"category": "Math", "title": "Complex cobordism and algebraic topology", "abstract": "This is a historical survey, beginning where Atiyah and Sullivan leave off..."}
{"category": "Math", "title": "Soft ideals and arithmetic mean ideals", "abstract": "This article investigates the soft-interior and the soft-cover of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the arithmetic mean operations is essential for the study of the multiplicity of traces (see arXiv:0707.3169v1 [math.FA]). Many classical ideals are \"soft\", i.e., coincide with their soft interior or with their soft cover, and many ideal constructions yield soft ideals. Arithmetic mean (am) operations were proven to be intrinsic to the theory of operator ideals by the work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators and arithmetic mean operations at infinity were studied in arXiv:0707.3169v1 [math.FA]. Here we focus on the commutation relations between these operations and soft operations. In the process we characterize the am-interior and the am-infinity interior of an ideal."}
{"category": "Math", "title": "Invariants of Closed 3-Manifolds via Nullhomotopic Filling Dehn Spheres", "abstract": "We provide a calculus for the presentation of closed 3-manifolds via nullhomotopic filling Dehn spheres and we use it to define an invariant of closed 3-manifolds by applying the state-sum machinery. As a potential application of this invariant, we show how to get lower bounds for the Matveev complexity of P2-irreducible closed 3-manifolds. We also describe an easy algorithm for constructing a nullhomotopic filling Dehn sphere of each closed 3-manifold from any of its one-vertex triangulations."}
{"category": "Math", "title": "Measure-valued equations for Kolmogorov operators with unbounded coefficients", "abstract": "Given a real and separable Hilbert space H we consider the measure-valued equation \\begin{equation*} \\int_H\\phi(x)\\mu_t(dx)- \\int_H\\phi(x)\\mu(dx)= \\int_0^t(\\int_HK_0\\phi(x)\\mu_s(dx))ds, \\end{equation*} where K_0 is the Kolmogorov differential operator \\[ K_0\\phi(x)=\\frac12\\textrm{Trace}\\big[BB^*D^2\\phi(x)\\big]+< x,A^*D\\phi(x)>+< D\\phi(x),F(x)>, \\] $x\\in H$, $\\phi:H\\to \\Rset$ is a suitable smooth function, $A:D(A)\\subset H\\to H $ is linear, $F:H\\to H$ is a globally Lipschitz function and $B:H\\to H$ is linear and continuous. In order prove existence and uniqueness of a solution for the above equation, we show that $K_0$ is a core, in a suitable way, of the infinitesimal generator associated to the solution of a certain stochastic differential equation in H. We also extend the above results to a reaction-diffusion operator with polinomial nonlinearities."}
{"category": "Math", "title": "Gibbs Rapidly Samples Colorings of G(n,d/n)", "abstract": "Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd\\H{o}s-R\\'enyi random graph G(n,d/n). While the average degree in G(n,d/n) is d(1-o(1)), it contains many nodes of degree of order $\\log n / \\log \\log n$. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring or the Ising model, the mixing time of Gibbs sampling is at least $n^{1 + \\Omega(1 / \\log \\log n)}$. High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including coloring. In this work consider sampling q-colorings and show that for every $d < \\infty$ there exists $q(d) < \\infty$ such that for all $q \\geq q(d)$ the mixing time of Gibbs sampling on G(n,d/n) is polynomial in $n$ with high probability. Our results are the first polynomial time mixing results proven for the coloring model on G(n,d/n) for d > 1 where the number of colors does not depend on n. They extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. The results also generalize to the hard-core model at low fugacity and to general models of soft constraints at high temperatures."}
{"category": "Math", "title": "An exotic shuffle relation of $\\zeta(\\{2\\}^m)$ and $\\zeta(\\{3,1\\}^n)$", "abstract": "In this short note we will provide a new and shorter proof of the following exotic shuffle relation of multiple zeta values: $$\\zeta(\\{2\\}^m \\sha\\{3,1\\}^n)={2n+m\\choose m} \\frac{\\pi^{4n+2m}}{(2n+1)\\cdot (4n+2m+1)!}.$$ This was proved by Zagier when n=0, by Broadhurst when $m=0$, and by Borwein, Bradley, and Broadhurst when m=1. In general this was proved by Bowman and Bradley in \\emph{The algebra and combinatorics of shuffles and multiple zeta values}, J. of Combinatorial Theory, Series A, Vol. \\textbf{97} (1)(2002), 43--63. Our idea in the general case is to use the method of Borwein et al. to reduce the above general relation to some families of combinatorial identities which can be verified by WZ-method."}
{"category": "Math", "title": "Picard-graded Betti numbers and the defining ideals of Cox rings", "abstract": "Let X be a smooth projective variety with torsion-free Picard group. We introduce complexes of vector spaces whose homology determines the structure of the minimal free resolution of the Cox ring of X over the polynomial ring and show how the homology of these complexes can be studied by purely geometric methods. As an application of these techniques we give a simple new proof of a characterization of the Cox rings of Del Pezzo surfaces (of degree >1) conjectured by Batyrev and Popov."}
{"category": "Math", "title": "Jet Riemann-Lagrange Geometry Applied to Evolution DEs Systems from Economy", "abstract": "The aim of this paper is to construct a natural Riemann-Lagrange differential geometry on 1-jet spaces, in the sense of nonlinear connections, generalized Cartan connections, d-torsions, d-curvatures, jet electromagnetic fields and jet Yang-Mills energies, starting from some given non-linear evolution DEs systems modelling economic phenomena, like the Kaldor model of the bussines cycle or the Tobin-Benhabib-Miyao model regarding the role of money on economic growth."}
{"category": "Math", "title": "Geometrical Objects on the First Order Jet Space $J^1(T,R^5)$ Produced by the Lorenz Atmospheric DEs System", "abstract": "The aim of this paper is to construct natural geometrical objects on the 1-jet space J^1(T,R^5), where $T/subset R$, like a non-linear connection, a generalized Cartan connection, together with its d-torsions and d-curvatures, a jet electromagnetic d-field and a jet Yang-Mills energy, starting from the given Lorenz atmospheric DEs system and the pair of Euclidian metrics $/Delta = (1,/delta_{ij})$ on $T/times R^5$."}
{"category": "Math", "title": "Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces", "abstract": "We prove a closed formula expressing any multiplicative characteristic class evaluated on the tangent bundle of the Hilbert schemes of points on a non-compact simply-connected surface. As a corollary, we deduce a closed formula for the Chern character of the tangent bundles of these Hilbert schemes. We also give a closed formula for the multiplicative characteristic classes of the tautological bundles associated to a line bundle on the surface. We finally remark which implications the results here have for the Hilbert schemes of points of an arbitrary surface."}
{"category": "Math", "title": "A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem", "abstract": "The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals, and we explored their role in a multipaper project which we survey in this article. We start by presenting the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sublattice structure of ideals, arithmetic mean ideal cancellation properties of first and second order, and softness properties - a term that we introduced but a notion ubiquitous in the literature on operator ideals. Arithmetic mean closure of ideals leads us to investigate majorization for infinite sequences and this in turn leads us to an infinite Schur-Horn majorization theorem which extends theorems by A. Neumann, by Arveson and Kadison, and by Antezana, Massey, Ruiz and Stojanoff. We also list ten open questions that we encountered in the development of this material."}
{"category": "Math", "title": "Operator valued frames", "abstract": "We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity (e.g., multiframes) their dilation approach. We prove several results for operator-valued frames concerning their parametrization, duality, disjointeness, complementarity, and composition and the relationship between the two types of similarity (left and right) of such frames. We then apply these notions to prove that the collection of multiframe generators for the action of a discrete group on a Hilbert space is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. The proof is obtained by parametrizing this collection by a class of partial isometries in a larger von Neumann algebra. In the multiplicity one case this class reduces to the unitary class which is path-connected in norm, but in the infinite multiplicity case this class is path connected only in the strong operator topology and the proof depends on properties of tensor product slice maps."}
{"category": "Math", "title": "Functional Equation for Theta Series", "abstract": "In this short paper, we find the transformation formula for the theta series under the action of the Jacobi modular group on the Siegel-Jacobi space. This formula generalizes the formula (5.1) obtained by Mumford in his book[p.189, Tata Lectures on Theta I (1983), Birkhauser]."}
{"category": "Math", "title": "Origin of the numerals, Al biruni testimony", "abstract": "The origin of the numerals that we inherited from the arabo-Islamic civilization remained one enigma. The hypothesis of the Indian origin remained, with controversies, without serious rival. It was the dominant hypothesis since more of one century. Its partisans found to it and constructed a lot of arguments. The testimonies of the medieval authors have been interpreted to its advantage. The opposite opinions have been dismissed and ignored. An amalgam between the history of our modern numerals and the Indian mathematics history is made. Rational contradictions often passed under silence. A meticulous observation of the numerals permits to affirm that our numerals are in fact more or less modified Arabic letters. The \"Ghubari\" shape of the numerals shows that the symbol of a numeral corresponds to the Arabic letter whose numerical value is equal to this numeral. The numerals don't have a simple resemblance with some Arabic letters, but every number looks like the Arabic letter whose numerical value is equal to this numeral. The elements of the ''Abjadi'' calculation gives us a theoretical support, independent of the letters and numerals, witch explains our observation. Besides a re-lecture of the testimonies of the medieval authors, particularly the testimony of Al-Biruni, that is probably at the origin of all others testimonies speaking of the Indian origin of the numerals, is in agreement with the fact that our numerals are Arabic letters. We have there a second way concerning the origin of our modern numerals that is only to its beginnings. The deepened researches are necessary to understand the history of our numerals better. A rigorous re-lecture of the medieval testimonies with a new mind imposes itself."}
{"category": "Math", "title": "Operator valued frames on C*-modules", "abstract": "Frames on Hilbert C*-modules have been defined for unital C*-algebras by Frank and Larson and operator valued frames on a Hilbert space have been studied in arXiv.0707.3272v1.[math.FA]. Goal of the present paper is to introduce operator valued frames on a Hilbert C*-module for a sigma-unital C*-algebra. Theorem 1.4 reformulates the definition given by Frank and Larson in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator valued frame are limits in the strict topology of a series of elements in the multiplier algebra and hence belong to it. Theorem 3.3 shows that two operator valued frames are right similar if and only if they share the same frame projection. Theorem 3.4 establishes a one to one correspondence between Murray-von Neumann equivalence classes of projections in the multiplier algebra and right similarity equivalence classes of operator valued frames and provides a parametrization of all Parseval operator-valued frames on a given Hilbert C*-module. Left similarity is then defined and Proposition 3.9 establishes when two left unitarily equivalent frames are also right unitarily equivalent."}
{"category": "Math", "title": "Valdivia compact groups are products", "abstract": "It is shown that every Valdivia compact group is homeomorphic to a product of metrizable compacta."}
{"category": "Math", "title": "Root systems for asymmetric geometric representations of Coxeter groups", "abstract": "Results are obtained concerning the roots of asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include all Kac--Moody Weyl groups. In particular, a characterization of when a non-trivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting."}
{"category": "Math", "title": "Supercuspidal characters of reductive p-adic groups", "abstract": "We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu's construction from data satisfying a certain compactness condition. Each character is expressed in terms of a depth-zero character of a smaller group, the (linear) characters appearing in Yu's construction, Fourier transforms of orbital integrals, and certain signs and cardinalities that are described explicitly in terms of the datum associated to the representation and of the element at which the character is evaluated."}
{"category": "Math", "title": "Finite index subgroups of the modular group and their modular forms", "abstract": "Classically, congruence subgroups of the modular group, which can be described by congruence relations, play important roles in group theory and modular forms. In reality, the majority of finite index subgroups of the modular group are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences between congruence and noncongruence subgroups and modular forms will be discussed. We will mainly focus on three interesting aspects of modular forms for noncongruence subgroups: the unbounded denominator property, modularity of the Galois representation arising from noncongruence cuspforms, and Atkin and Swinnerton-Dyer congruences."}
{"category": "Math", "title": "Morita equivalences of cyclotomic Hecke algebras of type G(r,p,n)", "abstract": "We prove a Morita reduction theorem for the cyclotomic Hecke algebras H_{r,p,n}({q,Q})$ of type G(r,p,n). As a consequence, we show that computing the decomposition numbers of H_{r,p,n}(Q) reduces to computing the p'-splittable decomposition numbers of the cyclotomic Hecke algebras H_{r',p',n'}(Q'), where $1\\le r'\\le r$, $1\\le n'\\le n$, $ p'\\mid p$ and where the parameters Q' are contained in a single $(\\epsilon,q)$-orbit and $\\epsilon$ is a primitive p'th root of unity."}
{"category": "Math", "title": "On the statistics of the minimal solution of a linear Diophantine equation and uniform distribution of the real part of orbits in hyperbolic spaces", "abstract": "We study a variant of a problem considered by Dinaburg and Sinai on the statistics of the minimal solution to a linear Diophantine equation. We show that the signed ratio between the Euclidean norms of the minimal solution and the coefficient vector is uniformly distributed modulo one. We reduce the problem to an equidistribution theorem of Anton Good concerning the orbits of a point in the upper half-plane under the action of a Fuchsian group."}
{"category": "Math", "title": "Some connections between results and problems of De Giorgi, Moser and Bangert", "abstract": "Using theorems of Bangert, we prove a rigidity result which shows how a question raised by Bangert for elliptic integrands of Moser type is connected, in the case of minimal solutions without self-intersections, to a famous conjecture of De Giorgi for phase transitions."}
{"category": "Math", "title": "Non-real zeros of linear differential polynomials", "abstract": "Lower bounds are given for the number of non-real zeros of a second order linear differential polynomial with constant coefficients in a real entire function with finitely many non-real zeros."}
{"category": "Math", "title": "On the irrelevant disorder regime of pinning models", "abstract": "Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that is now rigorous. In this work, we exploit interpolation and replica coupling methods to obtain sharper results on the irrelevant disorder regime of pinning models. In particular, in this regime, we compute the first order term in the expansion of the free energy close to criticality and this term coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and quenched averaged correlation length exponents coincide, while, in general, they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals."}
{"category": "Math", "title": "On the geometry of cohomogeneity one manifolds with positive curvature", "abstract": "This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one manifolds that can admit a metric with positive curvature due to Grove-Wilking-Ziller. Two series of candidates arose in this classification for which it is not yet know if they admit positive curvature. The connection of these candidates to the Hitchin self dual Einstein orbifolds is discussed as well, including the curvature properties of the Hitchin metrics."}
{"category": "Math", "title": "Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles", "abstract": "In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold $M\\times\\mathbb{R}$, where $M$ is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure."}
{"category": "Math", "title": "On the base locus of the linear system of generalized theta functions", "abstract": "Let $\\cM_r$ denote the moduli space of semi-stable rank-$r$ vector bundles with trivial determinant over a smooth projective curve $C$ of genus $g$. In this paper we study the base locus $\\cB_r \\subset \\cM_r$ of the linear system of the determinant line bundle $\\cL$ over $\\cM_r$, i.e., the set of semi-stable rank-$r$ vector bundles without theta divisor. We construct base points in $\\cB_{g+2}$ over any curve $C$, and base points in $\\cB_4$ over any hyperelliptic curve."}
{"category": "Math", "title": "Weighted Strichartz estimates for radial Schr\\\"odinger equation on noncompact manifolds", "abstract": "We prove global weighted Strichartz estimates for radial solutions of linear Schr\\\"odinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields classical Strichartz estimates with a larger class of exponents than in the Euclidian case and improvements for the scattering theory. The manifolds, whose volume element grows polynomially or exponentially at infinity, are characterized essentially by negativity conditions on the curvature, which shows in particular that the rich algebraic structure of the Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive properties of the equation. The proofs are based on known dispersive results for the equation with potential on the Euclidean space, and on a new one, valid for C^1 potentials decaying like 1/r^2 at infinity."}
{"category": "Math", "title": "Approximation by Several Rationals", "abstract": "Following T. H. Chan, we consider the problem of approximation of a given rational fraction a/q by sums of several rational fractions a_1/q_1, ..., a_n/q_n with smaller denominators. We show that in the special cases of n=3 and n=4 and certain admissible ranges for the denominators q_1,..., q_n, one can improve a result of T. H. Chan by using a different approach."}
{"category": "Math", "title": "How Much Work Does It Take To Straighten a Plane Graph Out?", "abstract": "We prove that if one wants to make a plane graph drawing straight-line then in the worst case one has to move almost all vertices."}
{"category": "Math", "title": "Turan's theorem inverted", "abstract": "Turan's theorem implies that every graph of order n with more edges than the r-partite Turan graph contains a complete graph of order r+1. We show that the same premise implies the existence of much larger graphs. We also prove corresponding stability theorems. These results complete work started by Erdos in 1963."}
{"category": "Math", "title": "Limit laws for boolean convolutions", "abstract": "We study the distributional behavior for products, and for sums of boolean independent random variables in an infinitesimal triangular array. We show that the limit laws of boolean convolutions are determined by the limit laws of free convolutions, and vice versa. We further use these results to show several connections between the limiting distributional behavior of classical convolutions and that of boolean convolutions. The proof of our results is based on the analytical apparatus developed for free convolutions."}
{"category": "Math", "title": "On the noncommutative spin geometry of the standard Podles sphere and index computations", "abstract": "The purpose of the paper is twofold: First, known results of the noncommutative spin geometry of the standard Podles sphere are extended by discussing Poincare duality and orientability. In the discussion of orientability, Hochschild homology is replaced by a twisted version which avoids the dimension drop. The twisted Hochschild cycle representing an orientation is related to the volume form of the distinguished covariant differential calculus. Integration over the volume form defines a twisted cyclic 2-cocycle which computes the q-winding numbers of quantum line bundles. Second, a \"twisted\" Chern character from equivariant K0-theory to even twisted cyclic homology is introduced which gives rise to a Chern-Connes pairing between equivariant K0-theory and twisted cyclic cohomology. The Chern-Connes pairing between the equivariant K0-group of the standard Podles sphere and the generators of twisted cyclic cohomology relative to the modular automorphism and its inverse are computed. This includes the pairings with the twisted cyclic 2-cocycle associated to the volume form, and the one corresponding to the \"no-dimension drop\" case. From explicit index computations, it follows that the pairings with these cocycles give the q-indices of the known equivariant 0-summable Dirac operator on the standard Podles sphere."}
{"category": "Math", "title": "Kouchnirenko type formulas for local invariants of plane analytic curves", "abstract": "Let f(x,y)=0 be an equation of plane analytic curve defined in the neighborhood of the origin and let $\\pi:M\\to(\\Cn^2,0)$ be a local toric modification. We give a formula which connects a number of double points \\delta_0(f)$ with a sum $\\sum_p \\delta_p(\\tilde f)$ which runs over all intersection points of the proper preimage of f=0 with the exceptional divisor."}
{"category": "Math", "title": "On a Gibbs characterization of normalized generalized Gamma processes", "abstract": "We show that a Gibbs characterization of normalized generalized Gamma processes, recently obtained in Lijoi, Pr\\\"unster and Walker (2007), can alternatively be derived by exploiting a characterization of exponentially tilted Poisson-Kingman models stated in Pitman (2003). We also provide a completion of this result investigating the existence of normalized random measures inducing exchangeable Gibbs partitions of type $\\alpha \\in (-\\infty, 0]$."}
{"category": "Math", "title": "Fubini-Griffiths-Harris rigidity and Lie algebra cohomology", "abstract": "We prove a general extrinsic rigidity theorem for homogeneous varieties in $\\mathbb{CP}^N$. The theorem is used to show that the adjoint variety of a complex simple Lie algebra $\\mathfrak{g}$ (the unique minimal G orbit in $\\mathbb{P}\\mathfrak{g}$) is extrinsically rigid to third order. In contrast, we show that the adjoint variety of $SL_3\\mathbb{C}$, and the Segre product $\\mathit{Seg}(\\mathbb{P}^1\\times \\mathbb{P}^n)$, both varieties with osculating sequences of length two, are flexible at order two. In the $SL_3\\mathbb{C}$ example we discuss the relationship between the extrinsic projective geometry and the intrinsic path geometry. We extend machinery developed by Hwang and Yamaguchi, Se-ashi, Tanaka and others to reduce the proof of the general theorem to a Lie algebra cohomology calculation. The proofs of the flexibility statements use exterior differential systems techniques."}
{"category": "Math", "title": "When almost all sets are difference dominated", "abstract": "We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0,1,...,N}, chosen randomly according to a binomial model with parameter p(N), with N^{-1} = o(p(N)). We show that the random subset is almost surely difference dominated, as N --> oo, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O'Bryant. The proofs use recent strong concentration results. Furthermore, we exhibit a threshold phenomenon regarding the ratio of the size of the difference- to the sumset. If p(N) = o(N^{-1/2}) then almost all sums and differences in the random subset are almost surely distinct, and in particular the difference set is almost surely about twice as large as the sumset. If N^{-1/2} = o(p(N)) then both the sum and difference sets almost surely have size (2N+1) - O(p(N)^{-2}), and so the ratio in question is almost surely very close to one. If p(N) = c N^{-1/2} then as c increases from zero to infinity (i.e., as the threshold is crossed), the same ratio almost surely decreases continuously from two to one according to an explicitly given function of c. We also extend our results to the comparison of the generalized difference sets attached to an arbitrary pair of binary linear forms. For certain pairs of forms f and g, we show that there in fact exists a sharp threshold at c_{f,g} N^{-1/2}, for some computable constant c_{f,g}, such that one form almost surely dominates below the threshold, and the other almost surely above it. The heart of our approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p."}
{"category": "Math", "title": "An improved Julia-Caratheodory theorem for Schur-Agler mappings of the unit ball", "abstract": "We adapt Sarason's proof of the Julia-Caratheodory theorem to the class of Schur-Agler mappings of the unit ball, obtaining a strengthened form of this theorem. In particular those quantities which appear in the classical theorem and depend only on the component of the mapping in the complex normal direction have K-limits (not just restricted K-limits) at the boundary."}
{"category": "Math", "title": "Norms and spectral radii of linear fractional composition operators on the ball", "abstract": "We give a new proof that every linear fractional map of the unit ball induces a bounded composition operator on the standard scale of Hilbert function spaces on the ball, and obtain norm bounds analogous to the standard one-variable estimates. We also show that Cowen's one-variable spectral radius formula extends to these operators. The key observation underlying these results is that every linear fractional map of the ball belongs to the Schur-Agler class."}
{"category": "Math", "title": "Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators", "abstract": "We prove that the norm of a weighted composition operator on the Hardy space H^2 of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space associated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on H^2, and recover the standard upper bound for the norm. Similar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard functions spaces on the unit ball."}
{"category": "Math", "title": "Commensurability of geometric subgroups of mapping class groups", "abstract": "Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly disconnected). We characterise the subsurfaces which lead to virtually abelian geometric subgroups. We provide algebraic and geometric conditions under which two geometric subgroups are commensurable. We also describe the commensurator of a geometric subgroup in terms of the stabiliser of the underlying subsurface. Finally, we show some applications of our analysis to the theory of irreducible unitary representations of mapping class groups."}
{"category": "Math", "title": "Central limit theorems for multiple Skorohod integrals", "abstract": "In this paper, we prove a central limit theorem for a sequence of iterated Shorohod integrals using the techniques of Malliavin calculus. The convergence is stable, and the limit is a conditionally Gaussian random variable. Some applications to sequences of multiple stochastic integrals, and renormalized weighted Hermite variations of the fractional Brownian motion are discussed."}
{"category": "Math", "title": "Stability and intersection properties of solutions to the nonlinear biharmonic equation", "abstract": "We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation $\\Delta^2 \\phi = \\phi^p$. First, we show that there exists a critical value $p_c$, depending on the space dimension, such that the solutions are linearly unstable if $p<p_c$ and linearly stable if $p\\geq p_c$. Then, we focus on the supercritical case $p\\geq p_c$ and we show that the graphs of no two solutions intersect one another."}
{"category": "Math", "title": "Convergence of Kaehler-Ricci flow with integral curvature bound", "abstract": "Let $g(t)$, $t\\in [0, +\\infty)$, be a solution of the normalized K\\\"ahler-Ricci flow on a compact K\\\"ahler $n$-manifold $M$ with $c_{1}(M)>0$ and initial metric $g (0)\\in 2\\pi c_{1}(M)$. If there is a constant $C$ independent of $t$ such that $$ \\int_{M}|Rm(g(t))|^{n}dv_{t}\\leq C,$$ then, for any $t_{k}\\to \\infty$, a subsequence of $(M, g(t_{k}))$ converges to a compact orbifold $(X, h)$ with only finite many singular points $\\{q_{j}\\}$ in the Gromov-Hausdorff sense, where $h$ is a K\\\"ahler metric on $X\\backslash \\{q_{j}\\}$ satisfying the K\\\"ahler-Ricci soliton equation, i.e. there is a smooth function $f$ such that $$Ric(h)-h=\\nabla\\bar{\\nabla}f, {\\rm and}\\it \\nabla \\nabla f=\\bar{\\nabla} \\bar{\\nabla} f=0. $$"}
{"category": "Math", "title": "Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture", "abstract": "We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\\mathbb F}_q$, the finite field with q elements, by $A \\cdot A+... +A \\cdot A$, where A is a subset ${\\mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $\\mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields."}
{"category": "Math", "title": "Sums of products of congruence classes and of arithmetic progressions", "abstract": "Consider the congruence class R_m(a)={a+im:i\\in Z} and the infinite arithmetic progression P_m(a)={a+im:i\\in N_0}. For positive integers a,b,c,d,m the sum of products set R_m(a)R_m(b)+R_m(c)R_m(d) consists of all integers of the form (a+im)(b+jm)+(c+km)(d+\\ell m) for some i,j,k,\\ell\\in Z. It is proved that if gcd(a,b,c,d,m)=1, then R_m(a)R_m(b)+R_m(c)R_m(d) is equal to the congruence class R_m(ab+cd), and that the sum of products set P_m(a)P_m(b)+P_m(c)P_m(d) eventually coincides with the infinite arithmetic progression P_m(ab+cd)."}
{"category": "Math", "title": "McKay correspondence for canonical orders", "abstract": "Canonical orders, introduced in the minimal model program for orders, are simultaneous generalisations of Kleinian singularities and their associated skew group rings. In this paper, we construct minimal resolutions of canonical orders via non-commutative cyclic covers and skew group rings. This allows us to exhibit a derived equivalence between minimal resolutions of canonical orders and the skew group ring form of the canonical order in all but one case. The Fourier-Mukai transform used to construct this equivalence allows us to make explicit, the numerical version of the McKay correspondence for canonical orders which, relates the exceptional curves of the minimal resolution to the indecomposable reflexive modules of the canonical order."}
{"category": "Math", "title": "Loop Products and Closed Geodesics", "abstract": "We show the Chas-Sullivan product (on the homology of the free loop space of a Riemannian manifold) is related to the Morse index of its closed geodesics. We construct related products in the cohomology of the free loop space and of the based loop space, and show they are nontrivial."}
{"category": "Math", "title": "Derivatives of embedding functors I: the stable case", "abstract": "For smooth manifolds $M$ and $N$, let $\\Ebar(M, N)$ be the homotopy fiber of the map $\\Emb(M, N)\\longrightarrow \\Imm(M, N)$. Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula $V\\mapsto \\Sigma^\\infty\\Ebar(M, N\\times V)$. In this paper, we describe the Taylor polynomials of this functor, in the sense of M. Weiss' orthogonal calculus, in the case when $N$ is a nice open submanifold of a Euclidean space. This leads to a description of the derivatives of this functor when $N$ is a tame stably parallelizable manifold (we believe that the parallelizability assumption is not essential). Our construction involves a certain space of rooted forests (or, equivalently, a space of partitions) with leaves marked by points in $M$, and a certain ``homotopy bundle of spectra'' over this space of trees. The $n$-th derivative is then described as the ``spectrum of restricted sections'' of this bundle. This is the first in a series of two papers. In the second part, we will give an analogous description of the derivatives of the functor $\\Ebar(M, N\\times V)$, involving a similar construction with certain spaces of connected graphs (instead of forests) with points marked in $M$."}
{"category": "Math", "title": "Decomposition of Cartan Matrix and conjectures on Brauer character degrees", "abstract": "Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Let $J=J(F[N])$ denote the Jacboson radical of $F[N]$ and $I={\\rm Ann}(J)=\\{\\alpha \\in F[G]|J\\alpha =0\\}$. We have another algebra $F[G]/I$. We study the decomposition of Cartan matrix of $F[G]$ according to $F[G/N]$ and $F[G]/I$. This decomposition establishs some connections between Cartan invariants and chief composition factors of $G$. We find that existing zero-defect $p$-block in $N$ depends on the properties of $I$ in $F[G]$ or Cartan invariants. When we consider the Cartan invariants for a block algebra $B$ of $G$, the decomposition is related to what kind of blocks in $N$ covered by $B$. We mainly consider a block $B$ of $G$ which covers a block $b$ of $N$ with $l(b)=1$. In two cases, we prove Willems' conjecture holds for these blocks, which covers some true cases by Holm and Willems. Furthermore We give an affirmative answer to a question by Holm and Willems in our cases. Some other results about Cartan invariants are presented in our paper."}
{"category": "Math", "title": "Dynamics of symmetric holomorphic maps on projective spaces", "abstract": "We consider complex dynamics of a critically finite holomorphic map from P^k to P^k, which has symmetries associated with the symmetric group S_{k+2} acting on P^k, for each k \\ge 1. The Fatou set of each map of this family consists of attractive basins of superattracting points. Each map of this family satisfies Axiom A."}
{"category": "Math", "title": "Generating mapping class groups of nonorientable surfaces with boundary", "abstract": "We obtain simple generating sets for various mapping class groups of a nonorientable surface with punctures and/or boundary. We also compute the abelianizations of these mapping class groups."}
{"category": "Math", "title": "A comparison theorem for simplicial resolutions", "abstract": "It is well known that Barr and Beck's definition of comonadic homology makes sense also with a functor of coefficients taking values in a semi-abelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Here we focus on independence of the chosen comonad: conditions for homology to depend on the induced class of projectives only."}
{"category": "Math", "title": "Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions", "abstract": "A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process|the conditioned multitype Feller branching diffusion are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too."}
{"category": "Math", "title": "SUSY structures on deformed supermanifolds", "abstract": "We construct a geometric structure on deformed supermanifolds as a certain subalgebra of the vector fields. In the classical limit we obtain a decoupling of the infinitesimal odd and even transformations, whereas in the semiclassical limit the result is a representation of the supersymmetry algebra. In the case of mass preserving structure we describe all high energy corrections to this algebra."}
{"category": "Math", "title": "Proper holomorphic disks in the complement of varieties in \\C^2", "abstract": "For any closed analytic set X in C^2 there exists a proper holomorphic embedding of the unit disk into C^2 such that the image avoids X."}
{"category": "Math", "title": "Cohomology of Split Group Extensions and Characteristic Classes", "abstract": "There are characteristic classes that are the obstructions to the vanishing of the differentials in the Lyndon-Hochischild-Serre spectral sequence of an extension of an integral lattice L by a group G. These characteristic classes exist in a given page of the spectral sequence provided the differentials in the previous pages are all zero. When L decomposes into a sum of G-sublattices, we show that there are defining relations between the characteristic classes of L and the characteristic classes of its summands."}
{"category": "Math", "title": "Singular measures of circle homeomorphisms with two break points", "abstract": "Let $T_{f}$ be a circle homeomorphism with two break points $a_{b},c_{b}$ and irrational rotation number $\\varrho_{f}$. Suppose that the derivative $Df$ of its lift $f$ is absolutely continuous on every connected interval of the set $S^{1}\\backslash\\{a_{b},c_{b}\\}$, that $DlogDf \\in L^{1}$ and the product of the jump ratios of $ Df $ at the break points is nontrivial, i.e. $\\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\\frac{Df_{-}(c_{b})}{Df_{+}(c_{b})}\\neq1$. We prove that the unique $T_{f}$- invariant probability measure $\\mu_{f}$ is then singular with respect to Lebesgue measure $l$ on $S^{1}$."}
{"category": "Math", "title": "On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS", "abstract": "We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is ergodic. We also prove that it is ergodic iff the related skew product is."}
{"category": "Math", "title": "Degenerating families of dendrograms", "abstract": "Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist $p$-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the $p$-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are $p$-adic parameter spaces of (families of) dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram."}
{"category": "Math", "title": "Exchangeable Random Networks", "abstract": "We introduce and study a class of exchangeable random graph ensembles. They can be used as statistical null models for empirical networks, and as a tool for theoretical investigations. We provide general theorems that carachterize the degree distribution of the ensemble graphs, together with some features that are important for applications, such as subgraph distributions and kernel of the adjacency matrix. These results are used to compare to other models of simple and complex networks. A particular case of directed networks with power-law out--degree is studied in more detail, as an example of the flexibility of the model in applications."}
{"category": "Math", "title": "Parallel Tiled QR Factorization for Multicore Architectures", "abstract": "As multicore systems continue to gain ground in the High Performance Computing world, linear algebra algorithms have to be reformulated or new algorithms have to be developed in order to take advantage of the architectural features on these new processors. Fine grain parallelism becomes a major requirement and introduces the necessity of loose synchronization in the parallel execution of an operation. This paper presents an algorithm for the QR factorization where the operations can be represented as a sequence of small tasks that operate on square blocks of data. These tasks can be dynamically scheduled for execution based on the dependencies among them and on the availability of computational resources. This may result in an out of order execution of the tasks which will completely hide the presence of intrinsically sequential tasks in the factorization. Performance comparisons are presented with the LAPACK algorithm for QR factorization where parallelism can only be exploited at the level of the BLAS operations."}
{"category": "Math", "title": "Hodge metrics and the curvature of higher direct images", "abstract": "Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct images bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric."}
{"category": "Math", "title": "Renault's Equivalence Theorem for Groupoid Crossed Products", "abstract": "We provide an exposition and proof of Renault's equivalence theorem for crossed products by locally Hausdorff, locally compact groupoids. Our approach stresses the bundle approach, concrete imprimitivity bimodules and is a preamble to a detailed treatment of the Brauer semigroup for a locally Hausdorff, locally compact groupoid."}
{"category": "Math", "title": "Kuranishi homology and Kuranishi cohomology", "abstract": "A Kuranishi space is a topological space with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on moduli spaces of J-holomorphic curves in symplectic geometry. Let Y be an orbifold and R a commutative ring or Q-algebra. We define two kinds of Kuranishi homology KH_*(Y;R). The chain complex KC_*(Y;R) defining KH_*(Y;R) is spanned over R by [X,f,G], for X a compact oriented Kuranishi space with corners, f : X --> Y smooth, and G \"gauge-fixing data\" which makes Aut(X,f,G) finite. Our main result is that these are isomorphic to singular homology. We define Poincare dual Kuranishi cohomology, isomorphic to compactly-supported cohomology. We define five kinds of Kuranishi (co)bordism spanned by isomorphism classes[X,f] for X a compact oriented Kuranishi space without boundary and f : X --> Y smooth. They are new topological invariants, and we show they are very large. These theories are powerful new tools in symplectic geometry. Defining virtual cycles and chains for moduli spaces of J-holomorphic curves is trivial in Kuranishi (co)homology. There is no need to perturb moduli spaces, and no problems with transversality. This gives major simplifications in Lagrangian Floer cohomology. We define new Gromov-Witten type invariants in Kuranishi bordism, over Z not Q. We sketch how these may be used to prove the integrality conjecture for Gopakumar-Vafa invariants. This paper is surveyed in arXiv:0710.5634."}
{"category": "Math", "title": "Origin of the numerals, Zero concept", "abstract": "The partisans of the hypothesis of the Indian origin of the numerals create confusion between the history of the Indian mathematics and the history of our modern numerals. To argue the thesis of the Indian origin of the numbers they confound between: the \"intuitive zero \" of Brahmagupta, that means ''nothing'' and which is the difference of two equal numbers, the \"numeral zero\" used in the representation of the numbers and the \"mathematical zero\" defined by the modern mathematicians. \"Sifr\" designate the \"numeral zero\" and \"Shunya\" designate the \"intuitive zero\". The word \"Sifr\" is not a traduction of the word \"Shunya\" and does not derive from the Indian word \"Shunya\", since the word \"Sifr\" and its derivatives existed in Arabic long before the appearance of zero itself The facts that the \"intuitive zero\" and the \"mathematical zero\" are represented currently by the \"numerals zero\" symbol \"0\" are only consequences of the representation of the numbers by the \"Ghubari\" numerals"}
{"category": "Math", "title": "Trigonometric Series via Laplace Transforms", "abstract": "The author's method (math-ph/9804010) that uses the Laplace transform to find exact values for a large class of convergent series is extended to trigonometric series."}
{"category": "Math", "title": "Exceptional representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras", "abstract": "In this paper, we study the set of $\\Delta$-filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type $A$. Our main result is that for any fixed $\\Delta$-dimension vector, there is a unique (up to isomorphism) exceptional $\\Delta$-filtered module. We then apply this result to show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in $\\mathrm{gl}_{n}(\\field)$, thus answering positively in this $\\mathrm{gl}_{n}(\\field)$ case to a question raised independently by Michel Duflo and Dmitri Panyushev. An example of a seaweed Lie algebra in a simple Lie algebra of type $E_{8}$ not admitting an open orbit in its nilpotent radical is given."}
{"category": "Math", "title": "On the dihedral n-body problem", "abstract": "Consider n=2l>=4 point particles with equal masses in space, subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group D_l, where D_l is the group of order 2l generated by two rotations of angle pi around two secant lines in space meeting at an angle of pi/l. By adding a homogeneous gravitational (Newtonian) potential one finds a special $n$-body problem with three degrees of freedom, which is a kind of generalisation of Devaney isosceles problem, in which all orbits have zero angular momentum. In the paper we find all the central configurations and we compute the dimension of the stable/unstable manifolds."}
{"category": "Math", "title": "Sur la cat\\'egorie des bimodules de Soergel", "abstract": "The Soergel category B of a Coxeter system (W,S) is a bimodule category over a polynomial algebra on which W acts. It's a categorification of the Hecke Algebra of (W,S). In this article we give a combinatorial description of morphism spaces in B. As a corollary, we give an analogous description of the morphisms in O_0-proj, where O_0 is the principal block of the BGG category O. ----- La cat\\'egorie B de Soergel d'un syst\\`eme de Coxeter (W,S) est une cat\\'egorie de bimodules sur une alg\\`ebre de polyn\\^omes sur laquelle W agit. C'est une cat\\'egorification de l'alg\\`ebre de Hecke de (W,S). Dans cet article nous donnons une description combinatoire des espaces de morphismes dans B. En corollaire, on obtient une description analogue des morphismes dans O_0-proj, o\\`u O_0 est le bloc principal de la cat\\'egorie O de BGG."}
{"category": "Math", "title": "Algebras associated to acyclic directed graphs", "abstract": "We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings."}
{"category": "Math", "title": "An equivalent condition for a uniform space to be coverable", "abstract": "We prove that an equivalent condition for a uniform space to be coverable is that the images of the natural projections in the fundamental inverse system are uniformly open in a certain sense. As corollaries we (1) obtain a concrete way to find covering entourage, (2) correct an error in [3] and (3) show that coverable is equivalent to chain connected and uniformly joinable in the sense of arXiv:0706.3937."}
{"category": "Math", "title": "Covering R-trees", "abstract": "We show that every inner metric space X is the metric quotient of a complete R-tree via a free isometric action, which we call the covering R-tree of X. The quotient mapping is a weak submetry (hence, open) and light. In the case of compact 1-dimensional geodesic space X, the free isometric action is via a subgroup of the fundamental group of X. In particular, the Sierpin'ski gasket and carpet, and the Menger sponge all have the same covering R-tree, which is complete and has at each point valency equal to the continuum. This latter R-tree is of particular interest because it is \"universal\" in at least two senses: First, every R-tree of valency at most the continuum can be isometrically embedded in it. Second, every Peano continuum is the image of it via an open light mapping. We provide a sketch of our previous construction of the uniform universal cover in the special case of inner metric spaces, the properties of which are used in the proof."}
{"category": "Math", "title": "Two Analogs of Intrinsically Linked Graphs", "abstract": "A graph G is intrinsically S^1-linked if for every embedding of the vertices of G into S^1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S^1-linked if and only if it is not outer-planar. A graph is outer-flat if it can be embedded in the 3-ball such that all of its vertices map to the boundary of the 3-ball, all edges to the interior, and every cycle bounds a disk in the 3-ball that meets the graph only along its boundary. We show that a graph is outer-flat if and only if it is planar."}
{"category": "Math", "title": "Prime Ideals of q-Commutative Power Series Rings", "abstract": "We study the \"q-commutative\" power series ring R:=k_q[[x_1,...,x_n]], defined by the relations x_ix_j = q_{ij}x_j x_i, for multiplicatively antisymmetric scalars q_{ij} in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In particular, we prove that the prime spectrum of R is normally separated and is finitely stratified by commutative noetherian spectra. Combining this normal separation with results of Chan, Wu, Yekutieli, and Zhang, we are able to conclude that R is catenary. Following the approach of Brown and Goodearl, we also show that links between prime ideals are provided by canonical automorphisms. Moreover, for sufficiently generic q_{ij}, we find that R has only finitely many prime ideals and is a UFD (in the sense of Chatters)."}
{"category": "Math", "title": "On the type of triangle groups", "abstract": "We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups."}
{"category": "Math", "title": "Isoperimetric inequalities for eigenvalues of triangles", "abstract": "Lower bounds estimates are proved for the first eigenvalue for the Dirichlet Laplacian on arbitrary triangles using various symmetrization techniques. These results can viewed as a generalization of P\\'olya's isoperimetric bounds. It is also shown that amongst triangles, the equilateral triangle minimizes the spectral gap and (under additional assumption) the ratio of the first two eigenvalues. This last result resembles the Payne-P\\'olya-Weinberger conjecture proved by Ashbaugh and Benguria."}
{"category": "Math", "title": "Deformation bicomplex of module-algebras", "abstract": "The deformation bicomplex of a module-algebra over a bialgebra is constructed. It is then applied to study algebraic deformations in which both the module structure and the algebra structure are deformed. The cases of module-coalgebras, comodule-(co)algebras, and (co)module-bialgebras are also considered."}
{"category": "Math", "title": "GK-dimension of birationally commutative surfaces", "abstract": "Let k be an algebraically closed field, let K/k be a finitely generated field extension of transcendence degree 2 with automorphism sigma, and let A be an N-graded subalgebra of Q = K[t; sigma] with A_n finite dimensional over k for all n. Then if A is big enough in Q in an appropriate sense, we prove that GKdim A = 3,4,5 or is infinite, with the exact value depending only on the geometric properties of sigma. The proof uses techniques in the birational geometry of surfaces which are of independent interest."}
{"category": "Math", "title": "The Baum-Connes assembly map and the generalized Bass conjecture", "abstract": "We show that the image of Connes-Karoubi-Chern character, restricted to the image of the Baum-Connes assembly map in the Bott-periodized topological K-theory of the complex group algebra, lies in the elliptic summand of the (periodic) cyclic homology of the group algebra. This implies that for any (weighted) ell-1 completion of the group algebra, rational surjectivity of the Baum-Connes assembly map implies the generalized Bass conjecture for that algebra."}
{"category": "Math", "title": "Relatively hyperbolic groups, rapid decay algebras, and a generalization of the Bass conjecture", "abstract": "By deploying dense subalgebras of $\\ell^1(G)$ we generalize the Bass conjecture in terms of Connes' cyclic homology theory. In particular, we propose a stronger version of the $\\ell^1$-Bass Conjecture. We prove that hyperbolic groups relative to finitely many subgroups, each of which posses the polynomial conjugacy-bound property and nilpotent periodicity property, satisfy the $\\ell^1$-Stronger-Bass Conjecture. Moreover, we determine the conjugacy-bound for relatively hyperbolic groups and compute the cyclic cohomology of the $\\ell^1$-algebra of any discrete group."}
{"category": "Math", "title": "The Laplace transform of the digamma function: an integral due to Glasser, Manna and Oloa", "abstract": "We provide an analytic expression for the Laplace transform of the digamma fuction. This complements work of L. Glasser, D. Manna and O. Oloa on this question. The Laplace transform is continuous in the Laplace-variable a. The derivative admits a jump at a= ln 2."}
{"category": "Math", "title": "High frequency dispersive estimates in dimension two", "abstract": "We prove dispersive estimates at high frequency in dimension two for both the wave and the Schrodinger groupes for a very large class of real-valued potentials."}
{"category": "Math", "title": "p-adic Dedekind and Hardy-Berndt type sum related to Volkenborn Integral on Z_p", "abstract": "The purpose of this paper is to construct p-adic Dedekind sums and Hardy-Berndt type sums. We also construct generating function of the twisted Bernoulli polynomials and functions. Furthermore, we give some discussions on elliptic analogue of the Apostol-Dedekind sums."}
{"category": "Math", "title": "On the p-adic Beilinson conjecture for number fields", "abstract": "We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of he conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in some other cases."}
{"category": "Math", "title": "$S_t^1\\times S_s^1$-valued lightcone Gauss map of a Lorentzian surface in semi-Euclidean 4-space", "abstract": "We define the notions of $S_t^1\\times S_s^1$-valued lightcone Gauss maps, lightcone pedal surface and Lorentzian lightcone height function of Lorentzian surface in semi-Euclidean 4-space and established the relationships between singularities of these objects and geometric invariants of the surface as applications of standard techniques of singularity theory for the Lorentzian lightcone height function."}
{"category": "Math", "title": "On singular Calogero-Moser spaces", "abstract": "Using combinatorial properties of complex reflection groups, we show that the generalised Calogero-Moser space associated to the centre of the corresponding rational Cherednik algebra is singular for all values of its deformation parameter c if and only if the group is different from the wreath product $S_n\\wr C_m$ and the binary tetrahedral group. This result and a theorem of Ginzburg and Kaledin imply that there does not exist a symplectic resolution of the singular symplectic variety h+h*/W outside of these cases; conversely we show that there exists a symplectic resolution for the binary tetrahedral group (Hilbert schemes provide resolutions for the wreath product case)."}
{"category": "Math", "title": "A Cartan-Eilenberg approach to Homotopical Algebra", "abstract": "In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence between its localization with respect to weak equivalences and the localised category of cofibrant objets with respect to strong equivalences. This equivalence allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and functor categories with a triple, in the last case we find examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications, we prove the existence of filtered minimal models for \\emph{cdg} algebras over a zero-characteristic field and we formulate an acyclic models theorem for non additive functors."}
{"category": "Math", "title": "Elliptic Gauss Sums and Hecke L-values at s=1", "abstract": "The rationality of the elliptic Gauss sum coefficient is shown. The following is a specific case of our argument. Let f(u)=sl((1-i)\\varpi u), where sl() is the Gauss' lemniscatic sine and \\varpi=2.62205... is the real period of the elliptic curve y^2=x^3-x, so that f(u) is an elliptic function relative to the period lattice Z[i]. Let \\pi be a primary prime of Z[i] such that norm(\\pi)\\equiv 13\\mod 16. Let S be the quarter set mod \\pi consisting of quartic residues. Let us define G(\\pi):=\\sum_{\\nu\\in S} f(\\nu/\\pi) and \\tilde{\\pi}:=\\prod_{\\nu\\in S} f(\\nu/\\pi). The former G(\\pi) is a typical example of elliptic Gauss sum; the latter is regarded as a canonical 4-th root of -\\pi: (\\tilde{\\pi})^4=-\\pi. Then we have Theorem: G(\\pi)/(\\tilde{\\pi})^3 is a rational odd integer. G(\\pi) appears naturally in the central value of Hecke L associated to the quartic residue character mod \\pi, and our proof is based on the functional equation of L and an explicit formula of the root number. In fact, the latter is nothing but the Cassels-Matthews formula on the quartic Gauss sum."}
{"category": "Math", "title": "Exponential inequalities for self-normalized martingales with applications", "abstract": "We propose several exponential inequalities for self-normalized martingales similar to those established by De la Pe\\~{n}a. The keystone is the introduction of a new notion of random variable heavy on left or right. Applications associated with linear regressions, autoregressive and branching processes are also provided."}
{"category": "Math", "title": "Relating two Hopf algebras built from an operad", "abstract": "Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the $\\NAP$ operad, where the incidence Hopf algebra is identified with the Connes-Kreimer Hopf algebra of rooted trees."}
{"category": "Math", "title": "Scaling limits for random fields with long-range dependence", "abstract": "This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density $\\lambda$ of the sets grows to infinity and the mean volume $\\rho$ of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show that the centered and renormalized random field can have three different limits, depending on the relative speed at which $\\lambda$ and $\\rho$ are scaled. If $\\lambda$ grows much faster than $\\rho$ shrinks, the limit is Gaussian with long-range dependence, while in the opposite case, the limit is independently scattered with infinite second moments. In a special intermediate scaling regime, there exists a nontrivial limiting random field that is not stable."}
{"category": "Math", "title": "Malliavin calculus and Clark-Ocone formula for functionals of a square-integrable L\\'evy process", "abstract": "In this paper, we construct a Malliavin derivative for functionals of square-integrable L\\'evy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian motion and Poisson random measure. As an illustration, we compute the explicit martingale representation for the maximum of a L\\'evy process."}
{"category": "Math", "title": "On Categorical Theory-Building: Beyond the Formal", "abstract": "I propose a notion of theory motivated by Category theory."}
{"category": "Math", "title": "Ergodic properties of Poissonian ID processes", "abstract": "We show that a stationary IDp process (i.e., an infinitely divisible stationary process without Gaussian part) can be written as the independent sum of four stationary IDp processes, each of them belonging to a different class characterized by its L\\'{e}vy measure. The ergodic properties of each class are, respectively, nonergodicity, weak mixing, mixing of all order and Bernoullicity. To obtain these results, we use the representation of an IDp process as an integral with respect to a Poisson measure, which, more generally, has led us to study basic ergodic properties of these objects."}
{"category": "Math", "title": "p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure", "abstract": "The specializations of the motivic elliptic polylog are called motivic Eisenstein classes. For applications to special values of L-Functions, it is important to compute the realizations of these classes. In this paper, we prove that the syntomic realization of the motivic Eisenstein classes, restricted to the ordinary locus of the modular curve, may be expressed using p-adic Eisenstein-Kronecker series. These p-adic modular forms are defined using the two-variable p-adic measure with values in p-adic modular forms constructed by Katz."}
{"category": "Math", "title": "Parshin Residues via Coboundary Operators", "abstract": "The article consist of two main parts: an analog of the Leray Theory for Singular Varieties and its application to the Theory of Parshin's Residues. The first part is independent from the second. It uses the theory of Whitney stratifications. The second part is an application of the first. In particular, a geometric and very transparent proof of the Parshin's Reciprocity Law for residues is given."}
{"category": "Math", "title": "A hermitian analogue of the Broecker-Prestel theorem", "abstract": "The Broecker-Prestel local-global principle characterizes weak isotropy of quadratic forms over a formally real field in terms of weak isotropy over the henselizations and isotropy over the real closures of that field. A hermitian analogue of this principle is presented for algebras of index at most two. An improved result is also presented for algebras with a decomposable involution, algebras of pythagorean index at most two, and algebras over SAP and ED fields."}
{"category": "Math", "title": "Finite depth and Jacobson-Bourbaki correspondence", "abstract": "We introduce a notion of depth three tower of three rings C < B < A with depth two ring extension A | B recovered when B = C. If A = \\End B_C and B | C is a Frobenius extension, this captures the notion of depth three for a Frobenius extension in arXiv:math/0107064 and arXiv:math/0108067, such that if B | C is depth three, then A | C is depth two (a phenomenon of finite depth subfactors, see arXiv:math/0006057). We provide a similar definition of finite depth Frobenius extension with embedding theorem utilizing a depth three subtower of the Jones tower. If A, B and C correspond to a tower of subgroups G > H > K via the group algebra over a fixed base ring, the depth three condition is the condition that subgroup K has normal closure K^G contained in H. For a depth three tower of rings, there is a pre-Galois theory for the ring \\End {}_BA_C and coring (A \\o_B A)^C involving Morita context bimodules and left coideal subrings. This is applied in two sections to a specialization of a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings."}
{"category": "Math", "title": "On Landau--Ginzburg models for Fano varieties", "abstract": "We observe a method for finding weak Landau-Ginzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12."}
{"category": "Math", "title": "A timestepper approach for the systematic bifurcation and stability analysis of polymer extrusion dynamics", "abstract": "We discuss how matrix-free/timestepper algorithms can efficiently be used with dynamic non-Newtonian fluid mechanics simulators in performing systematic stability/bifurcation analysis. The timestepper approach to bifurcation analysis of large scale systems is applied to the plane Poiseuille flow of an Oldroyd-B fluid with non-monotonic slip at the wall, in order to further investigate a mechanism of extrusion instability based on the combination of viscoelasticity and nonmonotonic slip. Due to the nonmonotonicity of the slip equation the resulting steady-state flow curve is nonmonotonic and unstable steady-states appear in the negative-slope regime. It has been known that self-sustained oscillations of the pressure gradient are obtained when an unstable steady-state is perturbed [Fyrillas et al., Polymer Eng. Sci. 39 (1999) 2498-2504]. Treating the simulator of a distributed parameter model describing the dynamics of the above flow as an input-output black-box timestepper of the state variables, stable and unstable branches of both equilibrium and periodic oscillating solutions are computed and their stability is examined. It is shown for the first time how equilibrium solutions lose stability to oscillating ones through a subcritical Hopf bifurcation point which generates a branch of unstable limit cycles and how the stable periodic solutions lose their stability through a critical point which marks the onset of the unstable limit cycles. This implicates the coexistence of stable equilibria with stable and unstable periodic solutions in a narrow range of volumetric flow rates."}
{"category": "Math", "title": "Conformal paracontact curvature and the local flatness theorem", "abstract": "A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature if and only if the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan-Chern-Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in the complex vector space is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown."}
{"category": "Math", "title": "Nonfibered knots and representation shifts", "abstract": "The authors conjectured previously that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from symbolic dynamics."}
{"category": "Math", "title": "Binary Models for Marginal Independence", "abstract": "Log-linear models are a classical tool for the analysis of contingency tables. In particular, the subclass of graphical log-linear models provides a general framework for modelling conditional independences. However, with the exception of special structures, marginal independence hypotheses cannot be accommodated by these traditional models. Focusing on binary variables, we present a model class that provides a framework for modelling marginal independences in contingency tables. The approach taken is graphical and draws on analogies to multivariate Gaussian models for marginal independence. For the graphical model representation we use bi-directed graphs, which are in the tradition of path diagrams. We show how the models can be parameterized in a simple fashion, and how maximum likelihood estimation can be performed using a version of the Iterated Conditional Fitting algorithm. Finally we consider combining these models with symmetry restrictions."}
{"category": "Math", "title": "Equivariant Satake category and Kostant-Whittaker reduction", "abstract": "We explain (following V. Drinfeld) how the equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra equivariant with respect to the adjoint action, due to some old results of V. Ginzburg. The global cohomology functor corresponds under this identification to restricti on to the Kostant slice. We extend this description to loop rotation equivariant derived category, linking it to Harish-Chandra bimodules for the Langlands dual Lie algebra, so that the global cohomology functor corresponds to the quantum Kostant-Whittaker reduction of a Harish-Chandra bimodule. We derive a conjecture from math.AG/0306413 which identifies the loop-rotation equivariant homology of the affine Grassmannian with quantized completed Toda lattice."}
{"category": "Math", "title": "$N_{\\p}$-type quotient modules on the torus", "abstract": "Structure of the quotient modules in $\\hh$ is very complicated. A good understanding of some special examples will shed light on the general picture. This paper studies the so-call $N_{\\p}$-type quotient modules, namely, quotient modules of the form $\\hh\\ominus [z-\\p]$, where $\\p (w)$ is a function in the classical Hardy space $H^2(\\G)$ and $[z-\\p]$ is the submodule generated by $z-\\p (w)$. This type of quotient modules serve as good examples in many studies. A notable feature of the $N_{\\p}$-type quotient module is its close connections with some classical single variable operator theories."}
{"category": "Math", "title": "Approximate reduction of dynamical systems", "abstract": "The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an exact manner - as is the case with mechanical systems with symmetry--which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamical system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e., when it is behaviorally similar to a dynamical system on a lower dimensional space. These concepts are illustrated on a series of examples."}
{"category": "Math", "title": "Foliations on quaternion CR-submanifolds", "abstract": "The purpose of this paper is to study the canonical foliations of a quaternion CR-submanifold of a quaternion K\\\"{a}hler manifold."}
{"category": "Math", "title": "Occupation Statistics of Critical Branching Random Walks in Two or Higher Dimensions", "abstract": "Consider a critical nearest neighbor branching random walk on the $d$-dimensional integer lattice initiated by a single particle at the origin. Let $G_{n}$ be the event that the branching random walk survives to generation $n$. We obtain limit theorems conditional on the event $G_{n}$ for a variety of occupation statistics: (1) Let $V_{n}$ be the maximal number of particles at a single site at time $n$. If the offspring distribution has finite $\\alpha$th moment for some integer $\\alpha\\geq 2$, then in dimensions 3 and higher, $V_n=O_p(n^{1/\\alpha})$; and if the offspring distribution has an exponentially decaying tail, then $V_n=O_p(\\log n)$ in dimensions 3 and higher, and $V_n=O_p((\\log n)^2)$ in dimension 2. Furthermore, if the offspring distribution is non-degenerate then $P(V_n\\geq \\delta \\log n | G_{n})\\to 1$ for some $\\delta >0$. (2) Let $M_{n} (j)$ be the number of multiplicity-$j$ sites in the $n$th generation, that is, sites occupied by exactly $j$ particles. In dimensions 3 and higher, the random variables $M_{n} (j)/n$ converge jointly to multiples of an exponential random variable. (3) In dimension 2, the number of particles at a \"typical\" site (that is, at the location of a randomly chosen particle of the $n$th generation) is of order $O_p(\\log n)$, and the number of occupied sites is $O_p(n/\\log n)$."}
{"category": "Math", "title": "Symmetry groups of non-simply-connected four-manifolds", "abstract": "Let $M$ be a closed, connected, orientable topological four-manifold with $H_1(M)$ nontrivial and free abelian, $b_2(M)\\ne 0, 2$, and $\\chi(M)\\ne 0$. We show that if $G$ is a finite group of 2-rank $\\le 1$ which admits a homologically trivial, locally linear, effective action on $M$, then $G$ must be cyclic. With additional assumptions to ensure orientability of some components of the singular set (e.g. if $G$ acts by symplectic symmetries, or preserving a spin structure), we also rule out $C_2 \\times C_2$ actions. The proofs use equivariant cohomology, localization, and a careful study of the first cohomology groups of the (potential) singular set."}
{"category": "Math", "title": "Modules of constant Jordan type", "abstract": "We introduce the class of modules of constant Jordan type for a finite group scheme $G$ over a field $k$ of characteristic $p > 0$. This class is closed under taking direct sums, tensor products, duals, Heller shifts and direct summands, and includes endotrivial modules. It contains all modules in an Auslander-Reiten component which has at least one module in the class. Highly non-trivial examples are constructed using cohomological techniques. We offer conjectures suggesting that there are strong conditions on a partition to be the Jordan type associated to a module of constant Jordan type."}
{"category": "Math", "title": "The complex Busemann-Petty problem on sections of convex bodies", "abstract": "The complex Busemann-Petty problem asks whether origin symmetric convex bodies in $\\C^n$ with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if $n\\le 3$ and negative if $n\\ge 4.$"}
{"category": "Math", "title": "Performance Evaluation of a Multi-Agent Risk-Sensitive Tracking System", "abstract": "In this paper, we consider a simple linear exponential quadratic Gaussian (LEQG) tracking problem for a multi-agent system. We study the dynamical behaviors of the group as we vary the risk-sensitivity parameter, comparing in particular the risk averse case to the LQG case. Then we consider the evolution of the performance per agent as the number of agents in the system increases. We provide some analytical as well as simulation results. In general, more agents are beneficial only if noisy agent dynamics and/or imperfect measurements are considered. The critical value of the risk sensitivity parameter above which the cost becomes infinite increases with the number of agents. In other words, for a fixed positive value of this parameter, there is a minimum number of agents above which the cost remains finite."}
{"category": "Math", "title": "Semifinite spectral triples associated with graph C*-algebras", "abstract": "We review the recent construction of semifinite spectral triples for graph C^*-algebras. These examples have inspired many other developments and we review some of these such as the relation between the semifinite index and the Kasparov product, examples of noncommutative manifolds, and an index theorem in twisted cyclic theory using a KMS state."}
{"category": "Math", "title": "Stochastic evolution equations for nonlinear filtering of random fields in the presence of fractional Brownian sheet observation noise", "abstract": "The problem of nonlinear filtering of a random field observed in the presence of a noise, modeled by a persistent fractional Brownian sheet of Hurst index $(H_1,H_2)$ with $0.5<H_1,H_2<1$, is studied and a suitable version of the Bayes' formula for the optimal filter is obtained. Two types of spatial \"fractional\" analogues of the Duncan-Mortensen-Zakai equation are also derived: one tracks evolution of the unnormalized optimal filter along an arbitrary \"monotone increasing\" (in the sense of partial ordering in $\\mathbb{R}^2$) one-dimensional curve in the plane, while the other describes dynamics of the filter along the paths that are truly two-dimensional. Although the paper deals with the two-dimensional parameter space, the presented approach and results extend to $d$-parameter random fields with arbitrary $d\\geq 3$."}
{"category": "Math", "title": "Cha\\^{i}nes de Markov Constructives Index\\'{e}es par Z", "abstract": "Nous \\'{e}tudions les cha\\^{{\\i}}nes de Markov $(X_n)_{n\\in\\mathbf{Z}}$ gouvern\\'{e}es par une relation de r\\'{e}currence de la forme $X_{n+1}=f(X_n,V_{n+1})$, o\\`{u} $(V_n)_{n\\in\\mathbf{Z}}$ est une suite de variables al\\'{e}atoires ind\\'{e}pendantes et de m\\^{e}me loi telle pour tout $n\\in \\mathbf{Z}$, $V_{n+1}$ est ind\\'{e}pendante de la suite $((X_k,V_k))_{k\\le n}$. L'objet de l'article est de donner une condition n\\'{e}cessaire et suffisante pour que les innovations $(V_n)_{n\\in\\mathbf{Z}}$ d\\'{e}terminent compl\\`{e}tement la suite $(X_n)_{n\\in \\mathbf{Z}}$ et de d\\'{e}crire l'information manquante dans le cas contraire."}
{"category": "Math", "title": "The C*-algebras qA\\otimes K and S^2A\\otimes K are asymptotically equivalent", "abstract": "Let $A$ be a separable $C^*$-algebra. We prove that its stabilized second suspension $S^2A\\otimes \\mathcal K$ and the $C^*$-algebra $qA\\otimes \\mathcal K$ constructed by Cuntz in the framework of his picture of KK-theory are asymptotically equivalent. This means that there exist asymptotic morphisms from each to the other whose compositions are homotopic to the identity maps. This result yields an easy description of the natural transformation from KK-theory to E-theory. One more corollary is the following. T. Loring ([3]) proved that any asymptotic morphism from $\\qC$ to any $C^*$-algebra $B$ is homotopic to a $\\ast$-homomorphism. We prove that the same is true when $\\C$ is replaced by any nuclear $C^*$-algebra $A$ and when $B$ is stable."}
{"category": "Math", "title": "Rational Landen transformations on the real line", "abstract": "The rational Landen transformation is a map on the space of coefficients of a rational integrand that preserves the value of the integral. We provide a family of these transformations that apply to rational integrands on the whole line. Given an integer m, these transformations produce a numerical scheme to evaluate the integral that is of order m."}
{"category": "Math", "title": "Filtration shrinkage by level-crossings of a diffusion", "abstract": "We develop the mathematics of a filtration shrinkage model that has recently been considered in the credit risk modeling literature. Given a finite collection of points $x_1<...<x_N$ in $\\mathbb{R}$, the region indicator function $R(x)$ assumes the value $i$ if $x\\in(x_{i-1},x_i]$. We take $\\mathbb{F}$ to be the filtration generated by $(R(X_t))_{t\\geq0}$, where $X$ is a diffusion with infinitesimal generator $\\mathcal{A}$. We prove a martingale representation theorem for $\\mathbb{F}$ in terms of stochastic integrals with respect to $N$ random measures whose compensators have a simple form given in terms of certain L\\'{e}vy measures $F^{j\\pm}_i$, which are related to the differential equation $\\mathcal{A}u=\\lambda u$."}
{"category": "Math", "title": "Alternative parametrizations and reference priors for decomposable discrete graphical models", "abstract": "For a given discrete decomposable graphical model, we identify several alternative parametrizations, and construct the corresponding reference priors for suitable groupings of the parameters. Specifically, assuming that the cliques of the graph are arranged in a perfect order, the parameters we consider are conditional probabilities of clique-residuals given separators, as well as generalized log-odds-ratios. We also consider a parametrization associated to a collection of variables representing a cut for the statistical model. The reference priors we obtain do not depend on the order of the groupings, belong to a conjugate family, and are proper."}
{"category": "Math", "title": "Intrinsic tests for the equality of two correlated proportions", "abstract": "Correlated proportions arise in longitudinal (panel) studies. A typical example is the ``opinion swing'' problem: ``Has the proportion of people favoring a politician changed after his recent speech to the nation on TV?''. Since the same group of individuals is interviewed before and after the speech, the two proportions are correlated. A natural null hypothesis to be tested is whether the corresponding population proportions are equal. A standard Bayesian approach to this problem has already been considered in the literature, based on a Dirichlet prior for the cell-probabilities of the underlying two-by-two table under the alternative hypothesis, together with an induced prior under the null. In lack of specific prior information, a diffuse (e.g. uniform) distribution may be used. We claim that this approach is not satisfactory, since in a testing problem one should make sure that the prior under the alternative be adequately centered around the region specified by the null, in order to obtain a fair comparison between the two hypotheses. Following an intrinsic prior methodology, we develop two strategies for the construction of a collection of objective priors increasingly peaked around the null. We provide a simple interpretation of their structure in terms of weighted imaginary sample scenarios. We illustrate our method by means of three examples, carrying out sensitivity analysis and providing comparison with existing results."}
{"category": "Math", "title": "Some tensor products", "abstract": "We define the tensor product of 1-motives with motives of weight 0 and we construct explicitely the 1-motive underlying the quotient M_1 \\otimes M_2 / W_{-3}(M_1 \\otimes M_2)."}
{"category": "Math", "title": "The growth of additive processes", "abstract": "Let $X_t$ be any additive process in $\\mathbb{R}^d.$ There are finite indices $\\delta_i, \\beta_i, i=1,2$ and a function $u$, all of which are defined in terms of the characteristics of $X_t$, such that \\liminf_{t\\to0}u(t)^{-1/\\eta}X_t^*= \\cases{0, \\quad if $\\eta>\\delta_1$, \\cr\\infty, \\quad if $\\eta<\\delta_2$,} \\limsup_{t\\to0}u(t)^{-1/\\eta}X_t^*= \\cases{0, \\quad if $\\eta>\\beta_2$, \\cr\\infty, \\quad if $\\eta<\\beta_1$,}\\qquad {a.s.}, where $X_t^*=\\sup_{0\\le s\\le t}|X_s|.$ When $X_t$ is a L\\'{e}vy process with $X_0=0$, $\\delta_1=\\delta_2$, $\\beta_1=\\beta_2$ and $u(t)=t.$ This is a special case obtained by Pruitt. When $X_t$ is not a L\\'{e}vy process, its characteristics are complicated functions of $t$. However, there are interesting conditions under which $u$ becomes sharp to achieve $\\delta_1=\\delta_2$, $\\beta_1=\\beta_2.$"}
{"category": "Math", "title": "Maximal Arithmetic Progressions in Random Subsets", "abstract": "Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W(N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U(N)/log(N) converges almost surely to 2/log(2), while W(N)/log(N) does not converge almost surely (and in particular, limsup W(N)/log(N) is at least 3/log(2))."}
{"category": "Math", "title": "On elliptic differential operators with shifts: II. The cohomological index formula", "abstract": "This paper is a continuation of arXiv:0706.3511, where we obtained a local index formula for matrix elliptic operators with shifts. Here we establish a cohomological index formula of Atiyah-Singer type for elliptic differential operators with shifts acting between section spaces of arbitrary vector bundles. The key step is the construction of closed graded traces on certain differential algebras over the symbol algebra for this class of operators."}
{"category": "Math", "title": "Manifolds with nonnegative isotropic curvature", "abstract": "We prove that if $(M^n,g)$, $n \\ge 4$, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature, then one of the following possibilities hold: (i) $M$ admits a metric with positive isotropic curvature (ii) $(M,g)$ is isometric to a locally symmetric space (iii) $(M,g)$ is K\\\"ahler and biholomorphic to $\\C P^\\frac {n}{2}$. (iv) $(M,g)$ is quaternionic-K\\\"ahler. This is implied by the following result: Let $(M^{2n},g)$ be a compact, locally irreducible K\\\"ahler manifold with nonnegative isotropic curvature. Then either $M$ is biholomorphic to $\\C P^n$ or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. The proof is based on the recent work of S. Brendle and R. Schoen on the Ricci flow."}
{"category": "Math", "title": "Knot colouring polynomials", "abstract": "This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \\pi_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longitude pair in \\pi_K. Given a finite group G and an element x in G, we consider the set of representations \\rho from \\pi_K to G that map the meridian m_K to x, and define the colouring polynomial P(K) as the sum over all longitude images \\rho(l_K). The resulting invariant maps knots to the group ring Z[G]. It is multiplicative with respect to connected sum and equivariant with respect to symmetry operations of knots. Examples are given to show that colouring polynomials distinguish knots for which other invariants fail, in particular they can distinguish knots from their mutants, obverses, inverses, or reverses. We prove that every quandle 2-cocycle state-sum invariant of knots is a specialization of some knot colouring polynomial. This provides a complete topological interpretation of these invariants in terms of the knot group and its peripheral system. Furthermore, we show that P can be presented as a Yang-Baxter invariant, i.e. as the trace of some linear braid group representation. This entails in particular that Yang-Baxter invariants can detect non-inversible and non-reversible knots."}
{"category": "Math", "title": "A Topological Characterization Of Knots and Links Arising From Site-Specific Recombination", "abstract": "We develop a topological model of knots and links arising from a single (or multiple processive) round(s) of recombination starting with an unknot, unlink, or (2,m)-torus knot or link substrate. We show that all knotted or linked products fall into a single family, and prove that the size of this family grows linearly with the cube of the minimum number of crossings. Additionally, we prove that the only possible products of an unknot substrate are either clasp knots and links or (2,m)-torus knots and links. Finally, in the (common) case of (2,m)-torus knot or link substrates whose products have minimal crossing number m+1, we prove that the types of products are tightly prescribed, and use this to examine previously uncharacterized experimental data."}
{"category": "Math", "title": "Multivariate normal approximation in geometric probability", "abstract": "Consider a measure $\\mu_\\lambda = \\sum_x \\xi_x \\delta_x$ where the sum is over points $x$ of a Poisson point process of intensity $\\lambda$ on a bounded region in $d$-space, and $\\xi_x$ is a functional determined by the Poisson points near to $x$, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the $\\mu_\\lambda$-measures (suitably scaled and centred) of disjoint sets in $R^d$ are asymptotically independent normals as $\\lambda \\to \\infty$; here we give an $O(\\lambda^{-1/(2d + \\epsilon)})$ bound on the rate of convergence. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals."}
{"category": "Math", "title": "Multidimensional cellular automata and generalization of Fekete's lemma", "abstract": "Fekete's lemma is a well known combinatorial result on number sequences: we extend it to functions defined on $d$-tuples of integers. As an application of the new variant, we show that nonsurjective $d$-dimensional cellular automata are characterized by loss of arbitrarily much information on finite supports, at a growth rate greater than that of the support's boundary determined by the automaton's neighbourhood index."}
{"category": "Math", "title": "Landen transformations and the integration of rational functions", "abstract": "We present a rational version of the classical Landen transformation for elliptic integrals. This is employed to obtain explicit closed-form expressions for a large class of integrals of even rational functions and to develop an algorithm for the numerical integration of these functions."}
{"category": "Math", "title": "A simple example of a new class of Landen transformation", "abstract": "The rational Landen transformation is a map on the coefficients of a rational integrand that preserves the value of the integral. This is the rational analog of the classical Landen transformations for elliptic integrals that leads to the arithmetic-geometric mean of Legendre and Gauss. We present the effect of this transformation in the simplest possible case."}
{"category": "Math", "title": "Smooth representations and sheaves", "abstract": "The paper is concerned with `geometrization' of smooth (i.e. with open stabilizers) representations of the automorphism group of universal domains, and with the properties of `geometric' representations of such groups. As an application, we calculate the cohomology groups of several classes of smooth representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of an algebraically closed field."}
{"category": "Math", "title": "Covering spaces and the Kakimizu complex", "abstract": "In 1992, Osamu Kakimizu defined a complex that has become known as the Kakimizu complex of a knot. Vertices correspond to isotopy classes of minimal genus Seifert surfaces of the knot. Higher dimensional simplices correspond to collections of such classes of Seifert surfaces that admit disjoint representatives. We show that this complex is simply connected."}
{"category": "Math", "title": "Asymptotic expansions for functions of the increments of certain Gaussian processes", "abstract": "Let $G=\\{G(x),x\\ge 0\\}$ be a mean zero Gaussian process with stationary increments and set $\\sigma^2(|x-y|)= E(G(x)-G(y))^2$. Let $f$ be a function with $Ef^{2}(\\eta)<\\ff$, where $\\eta=N(0,1)$. When $\\sigma^2$ is regularly varying at zero and \\[ \\lim_{h\\to 0}{h^2\\over \\sigma^2(h)}= 0\\qquad {and}\\qquad \\lim_{h\\to 0}{\\sigma^2(h)\\over h}= 0 \\quad {but} \\quad ({d^{2}\\over ds^2}\\sigma^2(s))^{j_0} \\] is locally integrable for some integer $j_0\\ge 1$, and satisfies some additional regularity conditions, \\bea && \\int_a^bf(\\frac{G(x+h)-G(x)}{\\sigma (h)}) dx \\label{abst}\\nn &&\\qquad = \\sum_{j=0}^{j_0} (h/\\sigma(h))^{j} {E(H_{j}(\\eta) f(\\eta))\\over\\sqrt {j!}} :(G')^{j}:(I_{[a,b]}) +o({h\\over\\sigma (h)})^{j_0}\\nn \\eea in $L^2$. Here $H_j$ is the $j$-th Hermite polynomial. Also $:(G')^{j}:(I_{[a,b]})$ is a $j $-th order Wick power Gaussian chaos constructed from the Gaussian field $ G'(g) $, with covariance \\[ E(G'(g)G'(\\wt g)) = \\int \\int \\rho (x-y)g(x)\\wt g(y) dx dy\\label{3.7bqs}, \\] where $ \\rho(s)={1/2}{d^{2}\\over ds^2}\\sigma^2(s)$."}
{"category": "Math", "title": "Cohomology theories for homotopy algebras and noncommutative geometry", "abstract": "This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely $A_\\infty, C_\\infty$ and $L_\\infty$-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of $C_\\infty$-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack."}
{"category": "Math", "title": "k-Disjunctive cuts and a finite cutting plane algorithm for general mixed integer linear programs", "abstract": "In this paper we give a generalization of the well known split cuts of Cook, Kannan and Schrijver to cuts which are based on multi-term disjunctions. They will be called k-disjunctive cuts. The starting point is the question what kind of cuts is needed for a finite cutting plane algorithm for general mixed integer programs. We will deal with this question in detail and derive cutting planes based on k-disjunctions related to a given cut vector. Finally we will show how a finite cutting plane algorithm can be established using these cuts in combination with Gomory mixed integer cuts."}
{"category": "Math", "title": "A computation of invariants of a rational self-map", "abstract": "I compute the dynamical degrees in C. Voisin's example of a rational self-map of the variety of lines on a cubic fourfold."}
{"category": "Math", "title": "Potential density of rational points on the variety of lines of a cubic fourfold", "abstract": "We prove the potential density of rational points on the variety of lines of a sufficiently general cubic fourfold defined over a number field, where ``sufficiently general'' means that a condition of Terasoma type is satisfied. These varieties have trivial canonical bundle and have geometric Picard group equal to $\\mathbb{Z}$."}
{"category": "Math", "title": "Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number", "abstract": "An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number into negative powers of the nth triangular number. We also discuss the history of the Ramanujan expansion for the nth harmonic number as well as sharp estimates of its accuracy, with complete proofs, and we compare it with other approximative formulas."}
{"category": "Math", "title": "Symplectic $C_\\infty$-algebras", "abstract": "In this paper we show that a strongly homotopy commutative (or $C_\\infty$-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\\infty$-algebra (an $\\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a $\\ci$-algebra and does not generalize to algebras over other operads."}
{"category": "Math", "title": "Stable modification of relative curves", "abstract": "We generalize theorems of Deligne-Mumford and de Jong on semi-stable modifications of families of proper curves. The main result states that after a generically \\'etale alteration of the base any (not necessarily proper) family of multipointed curves with semi-stable generic fiber admits a minimal semi-stable modification. The latter can also be characterized by the property that its geometric fibers have no certain exceptional components. The main step of our proof is uniformization of one-dimensional extensions of valued fields. Riemann-Zariski spaces are then used to obtain the result over any integral base."}
{"category": "Math", "title": "On the regularity of geodesic rays associated to test configurations", "abstract": "Geodesic rays of class C^{1,1} are constructed for any test configuration of a positive line bundle L on X using resolution of singularities. The construction reduces to finding a subsolution of the corresponding Monge-Ampere equation. Geometrically, this is accomplished by the use a positive line bundle on the resolution which is trivial outside of the exceptional divisor."}
{"category": "Math", "title": "On the quantization of conjugacy classes", "abstract": "Let G be a compact, simple, simply connected Lie group. A theorem of Freed-Hopkins-Teleman identifies the level k fusion ring R_k(G) of G with the twisted equivariant K-homology at level k+h, where h is the dual Coxeter number. In this paper, we review this result using the language of Dixmier-Douady bundles. We show that the additive generators of the group R_k(G) are obtained as K-homology push-forwards of the fundamental classes of conjugacy classes in G."}
{"category": "Math", "title": "Mixed States Markov Random Fields with Symbolic Labels and Multidimensional Real Values", "abstract": "New theoretical results are presented here on the recently introduced model called mixed states MRF. Such models were introduced in the context of image motion analysis and are useful to represent information which can take both discrete values accounting for symbolic states, and real values corresponding to continuous measurements. In particular, results are given when the global energy for the Gibbs formulation expressing the mixed states model, can be decomposed into one term accounting for the discrete part of the model, and a second term related to the continuous part. This decomposition theorem permits to define conditional mixed states models in a very simple way."}
{"category": "Math", "title": "Regularly varying multivariate time series", "abstract": "A multivariate, stationary time series is said to be jointly regularly varying if all its finite-dimensional distributions are multivariate regularly varying. This property is shown to be equivalent to weak convergence of the conditional distribution of the rescaled series given that, at a fixed time instant, its distance to the origin exceeds a threshold tending to infinity. The limit object, called the tail process, admits a decomposition in independent radial and angular components. Under an appropriate mixing condition, this tail process allows for a concise and explicit description of the limit of a sequence of point processes recording both the times and the positions of the time series when it is far away from the origin. The theory is applied to multivariate moving averages of finite order with random coefficient matrices."}
{"category": "Math", "title": "Symplectic $A_\\infty$-algebras and string topology operations", "abstract": "In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincar\\'e duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading `string topology'. Our method is based on obstruction theory for $C_\\infty$-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant."}
{"category": "Math", "title": "Reflection Groups and Polytopes over Finite Fields, III", "abstract": "When the standard representation of a crystallographic Coxeter group is reduced modulo an odd prime p, one obtains a finite group G^p acting on some orthogonal space over Z_p . If the Coxeter group has a string diagram, then G^p will often be the automorphism group of a finite abstract regular polytope. In parts I and II we established the basics of this construction and enumerated the polytopes associated to groups of rank at most 4, as well as all groups of spherical or Euclidean type. Here we extend the range of our earlier criteria for the polytopality of G^p . Building on this we investigate the class of 3-infinity groups of general rank, and then complete a survey of those locally toroidal polytopes which can be described by our construction."}
{"category": "Math", "title": "Erlangen Program at Large--2: Inventing a wheel. The parabolic one", "abstract": "We discuss parabolic versions of Euler's identity e^{it}=cos t + i sin t. A purely algebraic approach based on dual numbers is known to produce a very trivial relation e^{pt} = 1+pt. Therefore we use a geometric setup of parabolic rotations to recover the corresponding non-trivial algebraic framework. Our main tool is Moebius transformations which turn out to be closely related to induced representations of the group SL(2,R). Keywords: complex numbers, dual numbers, double numbers, linear algebra, invariant, computer algebra, GiNaC"}
{"category": "Math", "title": "Tetrahedron equations and nilpotent subalgebras of U_q(sl_n)", "abstract": "A relation between q-oscillator R-matrix of the tetrahedron equation and decompositions of Poinkare-Birkhoff-Witt type bases for nilpotent subalgebras of U_q(sl_n) is observed."}
{"category": "Math", "title": "Some gradient estimates for a diffusion equation on Riemannian manifolds", "abstract": "In this note we present some gradient estimates for the diffusion equation $\\partial_t u=\\Delta u-\\nabla \\phi \\cdot \\nabla u $ on Riemannian manifolds, where $\\phi $ is a C^2 function, which generalize estimates of R. Hamilton's and Qi S. Zhang's on the heat equation."}
{"category": "Math", "title": "Multiplicative Order of Gauss Periods", "abstract": "We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski."}
{"category": "Math", "title": "The passage time distribution for a birth-and-death chain: Strong stationary duality gives a first stochastic proof", "abstract": "A well-known theorem usually attributed to Keilson states that, for an irreducible continuous-time birth-and-death chain on the nonnegative integers and any d, the passage time from state 0 to state d is distributed as a sum of d independent exponential random variables. Until now, no probabilistic proof of the theorem has been known. In this paper we use the theory of strong stationary duality to give a stochastic proof of a similar result for discrete-time birth-and-death chains and geometric random variables, and the continuous-time result (which can also be given a direct stochastic proof) then follows immediately. In both cases we link the parameters of the distributions to eigenvalue information about the chain. We also discuss how the continuous-time result leads to a proof of the Ray-Knight theorem. Intimately related to the passage-time theorem is a theorem of Fill that any fastest strong stationary time T for an ergodic birth-and-death chain on {0, >..., d} in continuous time with generator G, started in state 0, is distributed as a sum of d independent exponential random variables whose rate parameters are the nonzero eigenvalues of the negative of G. Our approach yields the first (sample-path) construction of such a T for which individual such exponentials summing to T can be explicitly identified."}
{"category": "Math", "title": "Tubular Neighborhoods of Nodal Sets and Diophantine Approximation", "abstract": "We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers."}
{"category": "Math", "title": "Clifford's theorem for coherent systems", "abstract": "Final version to appear in Archiv der Mathematik."}
{"category": "Math", "title": "A construction of noncontractible simply connected cell-like two dimensional Peano continua", "abstract": "Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts by a noncontractible n-dimensional Peano continuum for any n>0, then our construction yields a simply connected noncontractible (n + 1)-dimensional cell-like Peano continuum. In particular, starting with the circle $\\mathbb{S}^1$, one gets a noncontractible simply connected cell-like 2-dimensional Peano continuum."}
{"category": "Math", "title": "A universal property for the Jiang-Su algebra", "abstract": "We prove that the infinite tensor power of a unital separable C*-algebra absorbs the Jiang-Su algebra Z tensorially if and only if it contains, unitally, a subhomogeneous algebra without characters. This yields a succinct universal property for Z in a category so large that there are no unital separable C*-algebras without characters known to lie outside it. This category moreover contains the vast majority of our stock-in-trade separable amenable C*-algebras, and is closed under passage to separable superalgebras and quotients, and hence to unital tensor products, unital direct limits, and crossed products by countable discrete groups. One consequence of our main result is that strongly self-absorbing ASH algebras are Z-stable, and therefore satisfy the hypotheses of a recent classification theorem of W. Winter. One concludes that Z is the only projectionless strongly self-absorbing ASH algebra, completing the classification of strongly self-absorbing ASH algebras."}
{"category": "Math", "title": "Families of dendrograms", "abstract": "A conceptual framework for cluster analysis from the viewpoint of p-adic geometry is introduced by describing the space of all dendrograms for n datapoints and relating it to the moduli space of p-adic Riemannian spheres with punctures using a method recently applied by Murtagh (2004b). This method embeds a dendrogram as a subtree into the Bruhat-Tits tree associated to the p-adic numbers, and goes back to Cornelissen et al. (2001) in p-adic geometry. After explaining the definitions, the concept of classifiers is discussed in the context of moduli spaces, and upper bounds for the number of hidden vertices in dendrograms are given."}
{"category": "Math", "title": "1-join composition for $\\alpha$-critical graphs", "abstract": "Given two graphs G and H its 1-{\\it join} is the graph obtained by taking the disjoint union of G and H and adding all the edges between a nonempty subset of vertices of G and a nonempty subset of vertices of H. In general, composition operations of graphs has played a fundamental role in some structural results of graph theory and in particular the 1-join composition has played an important role in decomposition theorems of several class of graphs such as the claw-free graphs, the bull-free graphs, the perfect graphs, etc. A graph G is called {\\it $\\alpha$-critical} if $\\alpha(G\\setminus e)> \\alpha(G)$ for all the edges e of G, where $\\alpha(G)$, the {\\it stability number} of G, is equal to the maximum cardinality of a stable set of G, and a set of vertices M of G is {\\it stable} if no two vertices in M are adjacent. The study $\\alpha$-critical graphs is important, for instance a complete description of $\\alpha$-critical graphs would yield a good characterization of the stability number of G. In this paper we give necessary and sufficient conditions that G and H must satisfy in order to its 1-join will be an $\\alpha$-critical graph. Therefore we get a very useful way to construct basic $\\alpha$-critical graphs using the 1-join of graphs."}
{"category": "Math", "title": "On some properties of $\\sigma(N)$", "abstract": "We show asymptotic upper and lower bounds for the greatest common divisor of N and $\\sigma(N)$. We also show that there are infinitely many integers N with fairly large g.c.d. of N and $\\sigma(N)$."}
{"category": "Math", "title": "Rank-based inference for bivariate extreme-value copulas", "abstract": "Consider a continuous random pair $(X,Y)$ whose dependence is characterized by an extreme-value copula with Pickands dependence function $A$. When the marginal distributions of $X$ and $Y$ are known, several consistent estimators of $A$ are available. Most of them are variants of the estimators due to Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859--878] and Cap\\'{e}ra\\`{a}, Foug\\`{e}res and Genest [Biometrika 84 (1997) 567--577]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of $X$ and $Y$ are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also given."}
{"category": "Math", "title": "On the ternary Goldbach problem with primes in arithmetic progressions of a common module", "abstract": "For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k."}
{"category": "Math", "title": "A deterministic version of Pollard's p-1 algorithm", "abstract": "In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. We begin with Pollard's $p-1$ algorithm, which finds in random polynomial time the prime divisors $p$ of an integer $n$ such that $p-1$ is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the $k$-th cyclotomic method of factoring ($k\\ge 2$) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient function $\\phi$. We point out some explicit sets of integers $n$ that are completely factorable in deterministic polynomial time given $\\phi(n)$. These sets consist, roughly speaking, of products of primes $p$ satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of $p-1$. Finally, we prove that $O(\\ln n)$ oracle queries for values of $\\phi$ are sufficient to completely factor any integer $n$ in less than $\\exp\\Bigl((1+o(1))(\\ln n)^{{1/3}} (\\ln\\ln n)^{{2/3}}\\Bigr)$ deterministic time."}
{"category": "Math", "title": "Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition", "abstract": "We study reduced-order models of three-dimensional perturbations in linearized channel flow using balanced proper orthogonal decomposition (BPOD). The models are obtained from three-dimensional simulations in physical space as opposed to the traditional single-wavenumber approach, and are therefore better able to capture the effects of localized disturbances or localized actuators. In order to assess the performance of the models, we consider the impulse response and frequency response, and variation of the Reynolds number as a model parameter. We show that the BPOD procedure yields models that capture the transient growth well at a low order, whereas standard POD does not capture the growth unless a considerably larger number of modes is included, and even then can be inaccurate. In the case of a localized actuator, we show that POD modes which are not energetically significant can be very important for capturing the energy growth. In addition, a comparison of the subspaces resulting from the two methods suggests that the use of a non-orthogonal projection with adjoint modes is most likely the main reason for the superior performance of BPOD. We also demonstrate that for single-wavenumber perturbations, low-order BPOD models reproduce the dominant eigenvalues of the full system better than POD models of the same order. These features indicate that the simple, yet accurate BPOD models are a good candidate for developing model-based controllers for channel flow."}
{"category": "Math", "title": "The Hochschild cohomology of a Poincar\\'e algebra", "abstract": "In this note, we define the notion of a cactus set, and show that its geometric realization is naturally an algebra over Voronov's cactus operad, which is equivalent to the framed 2-dimensional little disks operad $\\mathcal{D}_2$. Using this, we show that the Hochschild cohomology of a Poincar\\'e algebra A is an algebra over (the chain complexes of) $\\mathcal{D}_2$."}
{"category": "Math", "title": "Characterizations of probability distributions via bivariate regression of record values", "abstract": "Bairamov et al. (Aust N Z J Stat 47:543-547, 2005) characterize the exponential distribution in terms of the regression of a function of a record value with its adjacent record values as covariates. We extend these results to the case of non-adjacent covariates. We also consider a more general setting involving monotone transformations. As special cases, we present characterizations involving weighted arithmetic, geometric, and harmonic means."}
{"category": "Math", "title": "Vertex operator algebras associated to certain admissible modules for affine Lie algebras of type A", "abstract": "Let $L(-{1/2}(l+1),0)$ be the simple vertex operator algebra associated to an affine Lie algebra of type $A_{l}^{(1)}$ with the lowest admissible half-integer level $-{1/2}(l+1)$, for even l. We study the category of weak modules for that vertex operator algebra which are in category $\\cal{O}$ as modules for the associated affine Lie algebra. We classify irreducible objects in that category and prove semisimplicity of that category."}
{"category": "Math", "title": "Splicing and the SL(2,C) Casson invariant", "abstract": "We establish a formula for the SL(2,C) Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the SL(2,C) Casson invariant vanishes for spliced sums along knots in the 3-sphere."}
{"category": "Math", "title": "A landing theorem for dynamic rays of geometrically finite entire functions", "abstract": "A transcendental entire function f is called geometrically finite if the intersection of the set of singular values with the Fatou set is compact and the intersection of the postsingular set with the Julia set is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then the Fatou set of f is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles. Let z_0 be a repelling or parabolic periodic point of such a map f. We show that, if f has finite order, then there exists an injective curve consisting of escaping points of f that connects z_0 to infinity. (This curve is called a dynamic ray.) In fact, the assumption of finite order can be weakened considerably; for example, it is sufficient to assume that f can be written as a finite composition of finite-order functions."}
{"category": "Math", "title": "Non-annulation effective et positivit\\'e locale des fibr\\'es en droites amples adjoints", "abstract": "We prove that Seshadri constants of some ample divisors are bigger than 1 on smooth threefolds whose anticanonical bundle is nef or on Fano varieties of small coindice. The main tools are (some known cases of) the Kawamata's effective non-vanishing conjecture and the adjunction theory. We prove the non-vanishing conjecture in dimension 3 in the case of line bundles of \"high\" volume using Kawamata's subadjunction formula."}
{"category": "Math", "title": "Resolution of symplectic cyclic orbifold singularities", "abstract": "In this paper we present a method to obtain resolutions of symplectic orbifolds arising from symplectic reduction of a Hamiltonian S^1-manifold at a regular value. As an application, we show that all isolated cyclic singularities of a symplectic orbifold admit a resolution and that pre-quantisations of symplectic orbifolds are symplectically fillable by a smooth manifold."}
{"category": "Math", "title": "Test configurations and Geodesic rays", "abstract": "For smooth test configurations, there always exist C^{1,1} geodesic rays in Kahler metric space parallel to the algebraic ray. The $\\yen$ invariant agrees with Futaki invariant, at least under nice assumptions. Explicit examples in Toric cases are calculated. On simple test configurations, Donaldson's correspondence between HCMA solution and holomorphic disc family is extended."}
{"category": "Math", "title": "Density theorems for bipartite graphs and related Ramsey-type results", "abstract": "In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements."}
{"category": "Math", "title": "Finite vertex algebras and nilpotence", "abstract": "I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (i.e., without nilpotent elements) finite vertex algebra is nilpotent."}
{"category": "Math", "title": "On Hadamard matrices at roots of unity", "abstract": "This paper has been withdrawn by the authors, the main result being known since Lam and Leung, J. Algebra 2000."}
{"category": "Math", "title": "Restriction of characters and products of characters", "abstract": "Let G be a finite p-group, for some prime p, and $\\psi, \\theta \\in \\Irr(G)$ be irreducible complex characters of G. It has been proved that if, in addition, $\\psi,\\theta$ are faithful characters, then the product $\\psi\\theta$ is a multiple of an irreducible or it is the nontrivial linear combination of at least $\\frac{p+1}{2}$ distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k>0, we can always find a p-group G and irreducible characters $\\Psi$ and $\\Theta$ such that $\\Psi\\Theta$ is the nontrivial combination of exactly k distinct irreducible characters. We do this by translating examples of decompositions of restrictions of characters into decompositions of products of characters."}
{"category": "Math", "title": "A short Brownian motion proof of the Riemann hypothesis", "abstract": "We give a short probabilistic (a Brownian motion) proof of the Riemann hypothesis based on some surprising, unexpected and deep algebraic conjecture (MAC in short) concerning the relation between the Riemann zeta $\\xi$ and a trivial zeta $\\zeta_{t}$. That algebraic conjecture was firstly discovered and formulated in [MA]"}
{"category": "Math", "title": "Ascent of module structures, vanishing of Ext, and extended modules", "abstract": "Let $(R,\\m)$ and $(S,\\n)$ be commutative Noetherian local rings, and let $\\phi:R\\to S$ be a flat local homomorphism such that $\\m S = \\n$ and the induced map on residue fields $R/\\m \\to S/\\n$ is an isomorphism. Given a finitely generated $R$-module $M$, we show that $M$ has an $S$-module structure compatible with the given $R$-module structure if and only if $\\Ext^i_R(S,M)=0$ for each $i\\ge 1$. We say that an $S$-module $N$ is {\\it extended} if there is a finitely generated $R$-module $M$ such that $N\\cong S\\otimes_RM$. Given a short exact sequence $0 \\to N_1\\to N \\to N_2\\to 0$ of finitely generated $S$-modules, with two of the three modules $N_1,N,N_2$ extended, we obtain conditions forcing the third module to be extended. We show that every finitely generated module over the Henselization of $R$ is a direct summand of an extended module, but that the analogous result fails for the $\\m$-adic completion."}
{"category": "Math", "title": "Noncommutative geometry and lower dimensional volumes in Riemannian geometry", "abstract": "In this paper we explain how to define \"lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes don't involve noncommutative geometry or spin structures at all."}
{"category": "Math", "title": "Uniform Uncertainty Principle and signal recovery via Regularized Orthogonal Matching Pursuit", "abstract": "This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements -- L_1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of the Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of the L_1-minimization. Our algorithm ROMP reconstructs a sparse signal in a number of iterations linear in the sparsity (in practice even logarithmic), and the reconstruction is exact provided the linear measurements satisfy the Uniform Uncertainty Principle."}
{"category": "Math", "title": "Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations", "abstract": "Symbolic models are abstract descriptions of continuous systems in which symbols represent aggregates of continuous states. In the last few years there has been a growing interest in the use of symbolic models as a tool for mitigating complexity in control design. In fact, symbolic models enable the use of well known algorithms in the context of supervisory control and algorithmic game theory, for controller synthesis. Since the 1990's many researchers faced the problem of identifying classes of dynamical and control systems that admit symbolic models. In this paper we make a further progress along this research line by focusing on control systems affected by disturbances. Our main contribution is to show that incrementally globally asymptotically stable nonlinear control systems with disturbances admit symbolic models. When specializing these results to linear systems, we show that these symbolic models can be easily constructed."}
{"category": "Math", "title": "Sampling Lissajous and Fourier knots", "abstract": "A Lissajous knot is one that can be parameterized by a single cosine function in each coordinate. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems which allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficiently sample all possible Lissajous knots with a given set of frequencies. In particular, we systematically tabulate all Lissajous knots with small frequencies and as a result substantially enlarge the tables of known Lissajous knots. A Fourier (i, j, k) knot is similar to a Lissajous knot except that each coordinate is now described by a finite sum of i, j, and k cosine functions respectively. According to Lamm, every knot is a Fourier-(1,1,k) knot for some k. By randomly searching the set of Fourier-(1,1,2) knots we find that all 2-bridge knots up to 14 crossings are either Lissajous or Fourier-(1,1,2) knots. We show that all twist knots are Fourier-(1,1,2) knots and give evidence suggesting that all torus knots are Fourier-(1,1,2) knots. As a result of our computer search, several knots with relatively small crossing numbers are identified as potential counterexamples to interesting conjectures."}
{"category": "Math", "title": "A Batalin-Vilkovisky Algebra structure on the Hochschild Cohomology of Truncated Polynomials", "abstract": "The main result of this paper is to calculate the Batalin-Vilkovisky structure of $HH^*(C^*(\\mathbf{K}P^n;R);C^*(\\mathbf{K}P^n;R))$ for $ \\mathbf{K}=\\mathbb{C}$ and $\\mathbb{H}$, and $R=\\mathbb{Z}$ and any field; and shows that in the special case when $M=\\mathbb{C}P^1=S^2$, and $R=\\mathbb{Z}$, this structure can not be identified with the BV-structure of $\\mathbb{H}_*(LS^2;\\mathbb{Z})$ computed by Luc Memichi in \\cite{menichi2}. However, the induced Gerstenhaber structures are still identified in this case. Moreover, according to a recent work of Y.Felix and J.Thomas \\cite{felix--thomas}, the main result of the present paper eventually calculates the BV-structure of the rational loop homology, $\\mathbb{H}_*(L\\mathbb{C}P^n;\\mathbb{Q})$ and $\\mathbb{H}_*(L\\mathbb{H}P^n;\\mathbb{Q})$, of projective spaces."}
{"category": "Math", "title": "Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces", "abstract": "In this paper we introduce a new kind of Backward Stochastic Differential Equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the ergodic BSDEs and the associated Hamilton-Jacobi-Bellman equation. Applications are given to ergodic control of stochastic partial differential equations."}
{"category": "Math", "title": "Orbit inequivalent actions of non-amenable groups", "abstract": "Consider two free measure preserving group actions $\\Gamma \\actson (X, \\mu), \\Delta \\actson (X, \\mu)$, and a measure preserving action $\\Delta \\actson^a (Z, \\nu)$ where $(X, \\mu), (Z, \\nu)$ are standard probability spaces. We show how to construct free measure preserving actions $\\Gamma \\actson^c (Y, m)$, $\\Delta \\actson^d (Y, m)$ on a standard probability space such that $E_{\\Delta}^d \\subset E_{\\Gamma}^c$ and $d$ has $a$ as a factor. This generalizes the standard notion of co-induction of actions of groups from actions of subgroups. We then use this construction to show that if $\\Gamma$ is a countable non-amenable group, then $\\Gamma$ admits continuum many orbit inequivalent free, measure preserving, ergodic actions on a standard probability space."}
{"category": "Math", "title": "Real embeddings, eta invariant and Chern-Simons current", "abstract": "We present an alternate proof of the Bismut-Zhang localization formula for $\\eta$-invariants without using the analytic techniques developed by Bismut-Lebeau. A Riemann-Roch property for Chern-Simons currents, which is of independent interest, is established in due course."}
{"category": "Math", "title": "Energy of harmonic functions and Gromov's proof of Stallings' theorem", "abstract": "We provide the details for Gromov's proof of Stallings' theorem on groups with infinitely many ends using harmonic functions. The main technical result of the paper is a compactness theorem for a certain family of harmonic functions."}
{"category": "Math", "title": "Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property", "abstract": "In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type ${\\rm II}\\sb 1$ factors and $M_n(\\cc)$) and symmetric gauge norms on $L^\\infty[0,1]$ and $\\cc^n$. As the first application, we obtain that the class of unitarily invariant norms on a type ${\\rm II}\\sb 1$ factor coincides with the class of symmetric gauge norms on $L^\\infty[0,1]$ and von Neumann's classical result \\cite{vN} on unitarily invariant norms on $M_n(\\cc)$. As the second application, Ky Fan's dominance theorem \\cite{Fan} is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative $L^p$-theory (e.g., non-commutative H$\\ddot{\\text{o}}$lder's inequality, duality and reflexivity of non-commutative $L^p$-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of $\\NN(\\M)$, the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor $\\M$. We obtain all extreme points of $\\NN(M_2(\\cc))$ and many extreme points of $\\NN(M_n(\\cc))$ ($n\\geq 3$). For a type ${\\rm II}\\sb 1$ factor $\\M$, we prove that if $t$ ($0\\leq t\\leq 1$) is a rational number then the Ky Fan $t$-th norm is an extreme point of $\\NN(\\M)$."}
{"category": "Math", "title": "Unitarily invariant norms related to factors", "abstract": "Let $\\M$ be a semi-finite factor and let $\\J(\\M)$ be the set of operators $T$ in $\\M$ such that $T=ETE$ for some finite projection $E$. In this paper we obtain a representation theorem for unitarily invariant norms on $\\J(\\M)$ in terms of Ky Fan norms. As an application, we prove that the class of unitarily invariant norms on $\\J(\\M)$ coincides with the class of symmetric gauge norms on a classical abelian algebra, which generalizes von Neumann's classical result \\cite{vN} on unitarily invariant norms on $M_n(\\cc)$. As another application, Ky Fan's dominance theorem \\cite{Fan} is obtained for semi-finite factors. Some classical results in non-commutative $L^p$-theory (e.g., non-commutative H$\\ddot{\\text{o}}$lder's inequality, duality and reflexivity of non-commutative $L^p$-spaces) are extended to general unitarily invariant norms related to semi-finite factors. We also prove that up to a scale the operator norm is the unique unitarily invariant norm associated to a type ${\\rm III}$ factor."}
{"category": "Math", "title": "Importance Tempering", "abstract": "Simulated tempering (ST) is an established Markov chain Monte Carlo (MCMC) method for sampling from a multimodal density $\\pi(\\theta)$. Typically, ST involves introducing an auxiliary variable $k$ taking values in a finite subset of $[0,1]$ and indexing a set of tempered distributions, say $\\pi_k(\\theta) \\propto \\pi(\\theta)^k$. In this case, small values of $k$ encourage better mixing, but samples from $\\pi$ are only obtained when the joint chain for $(\\theta,k)$ reaches $k=1$. However, the entire chain can be used to estimate expectations under $\\pi$ of functions of interest, provided that importance sampling (IS) weights are calculated. Unfortunately this method, which we call importance tempering (IT), can disappoint. This is partly because the most immediately obvious implementation is na\\\"ive and can lead to high variance estimators. We derive a new optimal method for combining multiple IS estimators and prove that the resulting estimator has a highly desirable property related to the notion of effective sample size. We briefly report on the success of the optimal combination in two modelling scenarios requiring reversible-jump MCMC, where the na\\\"ive approach fails."}
{"category": "Math", "title": "On the Brauer groups of quasilocal fields and the norm groups of their finite Galois extensions", "abstract": "This paper shows that divisible abelian torsion groups are realizable as Brauer groups of quasilocal fields. It describes the isomorphism classes of Brauer groups of primarily quasilocal fields and solves the analogous problem concerning the reduced components of the Brauer groups of two basic types of Henselian valued absolutely stable fields. For a quasilocal field E and a finite separable extension R/E, we find two sufficient conditions for validity of the norm group equality $N(R/E) = N(R_{0}/E)$, where R_{0} is the maximal abelian extension of E in R. This is used for deriving information on the arising specific relations between Galois groups and norm groups of finite Galois extensions of E."}
{"category": "Math", "title": "Jet Geometrical Objects Produced by Linear ODEs Systems and Superior Order ODEs", "abstract": "The aim of this paper is to construct a Riemann-Lagrange geometry on 1-jet spaces, in the sense of d-connections, d-torsions, d-curvatures, electromagnetic d-field and geometric electromagnetic Yang-Mills energy, starting from a given linear ODEs system or a given superior order ODE. The case of a non-homogenous linear ODE of superior order is disscused."}
{"category": "Math", "title": "Rubbling and Optimal Rubbling of Graphs", "abstract": "A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move one pebble is removed at vertices v and w adjacent to a vertex u and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number of a graph is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We determine the rubbling and optimal rubbling number of some families of graphs including cycles."}
{"category": "Math", "title": "Congruence obstructions to pseudomodularity of Fricke groups", "abstract": "A pseudomodular group is a finite coarea nonarithmetic Fuchsian group whose cusp set is exactly $\\mathbb{P}^1(\\mathbb{Q})$. Long and Reid constructed finitely many of these by considering Fricke groups, i.e., those that uniformize one-cusped tori. We prove that a zonal Fricke group with rational cusps is pseudomodular if and only if its cusp set is dense in the finite adeles of $\\mathbb{Q}$. We then deduce that infinitely many such Fricke groups are not pseudomodular."}
{"category": "Math", "title": "Traces on pseudodifferential operators and sums of commutators", "abstract": "The aim of this paper is to show that various known characterizations of traces on classical pseudodifferentials operators (PsiDOs) can actually be obtained by very elementary considerations on PsiDOs, using only basic properties of these operators. Thereby, we give a unified treatment of the determinations of the space of traces (i) on PsiDOs of noninteger orders or of regular parity-class, (ii) on integer order PsiDOs, (iii) on nonpositive order PsiDOs in dimension greather than or equal to 2, and (iv) on nonpositive order PsiDOs in dimension 1."}
{"category": "Math", "title": "An Explicit Construction of Type A Demazure Atoms", "abstract": "Demazure characters of type A, which are equivalent to key polynomials, have been decomposed by Lascoux and Sch\\\"{u}tzenberger into standard bases. We prove that the resulting polynomials, which we call Demazure atoms, can be obtained from a certain specialization of nonsymmetric Macdonald polynomials. This combinatorial interpretation for Demazure atoms accelerates the computation of the right key associated to a semi-standard Young tableau. Utilizing a related construction, we provide a new combinatorial description for the key polynomials."}
{"category": "Math", "title": "Structure and randomness in combinatorics", "abstract": "Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has proven profitable to decompose such objects into a \\emph{structured} component, a \\emph{pseudo-random} component, and a \\emph{small} component (i.e. an error term); in many cases it is the structured component which then dominates. We illustrate this philosophy in a number of model cases."}
{"category": "Math", "title": "A convexity property of expectations under exponential weights", "abstract": "Take a random variable X with some finite exponential moments. Define an exponentially weighted expectation by E^t(f) = E(e^{tX}f)/E(e^{tX}) for admissible values of the parameter t. Denote the weighted expectation of X itself by r(t) = E^t(X), with inverse function t(r). We prove that for a convex function f the expectation E^{t(r)}(f) is a convex function of the parameter r. Along the way we develop correlation inequalities for convex functions. Motivation for this result comes from equilibrium investigations of some stochastic interacting systems with stationary product distributions. In particular, convexity of the hydrodynamic flux function follows in some cases."}
{"category": "Math", "title": "Sums of the error term function in the mean square for $\\zeta(s)$", "abstract": "Sums of the form $\\sum_{n\\le x}E^k(n) (k\\in{\\bf N}$ fixed) are investigated, where $$ E(T) = \\int_0^T|\\zeta(1/2+it)|^2 dt - T\\Bigl(\\log {T\\over2\\pi} + 2\\gamma -1\\Bigr)$$ is the error term in the mean square formula for $|\\zeta(1/2+it)|$. The emphasis is on the case k=1, which is more difficult than the corresponding sum for the divisor problem. The analysis requires bounds for the irrationality measure of ${\\rm e}^{2\\pi m}$ and for the partial quotients in its continued fraction expansion."}
{"category": "Math", "title": "Effective Iitaka fibrations", "abstract": "We show that the M-canonical map of an n-dimensional complex projective manifold X of Kodaira dimension two is birational to an Iitaka fibration for a computable positive integer M. M depends on the index b of a general fibre F of the Iitaka fibration and on the Betti number of the canonical covering of F, In particular, M is a universal constant if the dimension n is smaller than or equal to 4."}
{"category": "Math", "title": "On deformations of maps and curve singularities", "abstract": "We study several deformation functors associated to the normalization of a reduced curve singularity $(X,0) \\subset (\\c^n,0)$. The main new results are explicit formulas, in terms of classical invariants of (X,0), for the cotangent cohomology groups $T^i, i = 0,1,2,$ of these functors. Thus we obtain precise statements about smoothness and dimension of the corresponding local moduli spaces. We apply the results to obtain explicit formulas resp. estimates for the $\\hoa{A}_e$-codimension of a parametrized curve singularity, where $\\hoa{A}_e$ denotes the Mather-Wall group of left-right equivalence."}
{"category": "Math", "title": "Neumann Heat kernel monotonicity", "abstract": "We prove that the diagonal of the transition probabilities for the d-dimensional Bessel processes on (0, 1], reflected at 1, which we denote by $p_R^N(t, r,r)$, is an increasing function of r for d>2 and that this is false for d=2."}
{"category": "Math", "title": "Volume and homology of one-cusped hyperbolic 3-manifolds", "abstract": "Let M be a complete, finite-volume, orientable hyperbolic manifold having exactly one cusp. If we assume that pi_1(M) has no subgroup isomorphic to a genus-2 surface group, and that either (a) H_1(M;Z_p) has dimension at least 5 for some prime p, or (b) H_1(M;Z_2) has dimension at least 4, and the subspace of H^2(M;Z_2) spanned by the image of the cup product has dimension at most 1, then vol M > 5.06 If we assume that H_1(M;Z_2) has dimension at least 7, and that the compact core of M does not contain a genus-2 closed incompressible surface, then vol M > 5.06."}
{"category": "Math", "title": "On elements of prime order in the plane Cremona group over a perfect field", "abstract": "We show that the plane Cremona group over a perfect field $k$ of characteristic $p \\ge 0$ contains an element of prime order $\\ell\\ge 7$ not equal to $p$ if and only if there exists a 2-dimensional algebraic torus $T$ over $k$ such that $T(k)$ contains an element of order $\\ell$. If $p = 0$ and $k$ does not contain a primitive $\\ell$-th root of unity, we show that there are no elements of prime order $\\ell > 7$ in $\\Cr_2(k)$ and all elements of order 7 are conjugate."}
{"category": "Math", "title": "Cohomological characterizations of projective spaces and hyperquadrics", "abstract": "We confirm Beauville's conjecture that claims that if the p-th exterior power of the tangent bundle of a smooth projective variety contains the p-th power of an ample line bundle, then the variety is either the projective space or the p-dimensional quadric hypersurface."}
{"category": "Math", "title": "Trace Estimates for Stable Processes", "abstract": "In this paper we study the behaviour in time of the trace (the partition function) of the heat semigroup associated with symmetric stable processes in domains of $\\Rd$. In particular, we show that for domains with the so called {\\it{$R$-smoothness}} property the second terms in the asymptotic as $t\\to 0$ involves the surface area of the domain, just as in the case of Brownian motion."}
{"category": "Math", "title": "All CAT(0) Boundaries of a Group of the Form HxK are CE Equivalent", "abstract": "M. Bestvina has shown that for any given torsion-free CAT(0) group G, all of its boundaries are shape equivalent. He then posed the question of whether they satisfy the stronger condition of being cell-like equivalent. In this article we prove that the answer is \"Yes\" in the situation where the group in question splits as a direct product with infinite factors. We accomplish this by proving an interesting theorem in shape theory."}
{"category": "Math", "title": "Optimalit\\'e, congruences et calculs d'invariants des vari\\'et\\'es symplectiques r\\'eelles de dimension quatre", "abstract": "This paper follows a previous one in which were introduced deformation invariants $\\chi^d_r$, $d \\in H_2 (X ; \\Z)$, $r \\in \\N$, of closed real symplectic four-manifolds $(X, \\omega, c_X)$, invariants which produced lower bounds in real enumerative geometry. We prove here using methods of symplectic field theory that the lower bounds are sharp when $r \\leq 1$ and the real locus of the manifold contains a sphere, torus or real projective plane (under stronger assumptions in this last case). We also prove that a big power of two divides $\\chi^d_r$ as soon as r is not too big and when the real locus contains a sphere or real projective plane (under the same stronger assumptions in this last case). We finally present some explicit computations in the case of the projective plane or quadric ellipsoid surface as well as the general formulas used to get them, formulas which involve some relative invariants that we first define."}
{"category": "Math", "title": "Infinite topology of curve complexes and non-Poincare duality of Teichmueller modular groups", "abstract": "In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex)."}
{"category": "Math", "title": "Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas", "abstract": "Motivated by the resemblance of a multivariate series identity and a finite analogue of Euler's pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas' formulas."}
{"category": "Math", "title": "Cobordisms of maps with singularities of a given class", "abstract": "Let P be a connected smooth p-manifold. We describe the group of all cobordism classes of smooth maps of n-manifolds to P with singularities of a given $cal K$-invariant class in terms of certain stable homotopy groups by applying the relative homotopy principle on the existence level. We also deal with the oriented version and construct a classifying space to which this oriented cobordism group is represented as the set of homotopy classes of P in the codimension n<p and n\\geqq p\\geqq 2."}
{"category": "Math", "title": "A new twist on Lorenz links", "abstract": "Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T--links coincide. Using this correspondence, we identify over half of the simplest hyperbolic knots as Lorenz knots. We show that both hyperbolic volume and the Mahler measure of Jones polynomials are bounded for infinite collections of hyperbolic Lorenz links. The correspondence provides unexpected symmetries for both Lorenz links and T-links, and establishes many new results for T-links, including new braid index formulas."}
{"category": "Math", "title": "The mapping class group and the Meyer function for plane curves", "abstract": "For each d>=2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d=4, using our Meyer function, we can define the local signature for 4-dimensional fiber spaces whose general fibers are non-hyperelliptic compact Riemann surfaces of genus 3. Some computations of our local signature will be given."}
{"category": "Math", "title": "Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities", "abstract": "We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\\R^\\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in $\\ell$. More precisely, we prove the following. Let $\\R$ be a real closed field and let \\[ {\\mathcal P} = \\{P_1,...,P_m\\} \\subset \\R[Y_1,...,Y_\\ell,X_1,...,X_k], \\] with ${\\rm deg}_Y(P_i) \\leq 2, {\\rm deg}_X(P_i) \\leq d, 1 \\leq i \\leq m$. Let $S \\subset \\R^{\\ell+k}$ be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms are of the form, $P \\geq 0, P\\leq 0, P \\in {\\mathcal P}$. Let $\\pi: \\R^{\\ell+k} \\to \\R^k$ be the projection on the last k co-ordinates. Then, the number of stable homotopy types amongst the fibers $S_{\\x} = \\pi^{-1}(\\x) \\cap S$ is bounded by \\[ (2^m\\ell k d)^{O(mk)}. \\]"}
{"category": "Math", "title": "Classification of free actions on complete intersections of four quadrics", "abstract": "In this paper we classify all free actions of finite groups on Calabi-Yau complete intersection of 4 quadrics in $\\PP^7$, up to projective equivalence. We get some examples of smooth Calabi-Yau threefolds with large nonabelian fundamental groups. We also observe the relation between some of these examples and moduli of polarized abelian surfaces."}
{"category": "Math", "title": "Holomorphic Motions, Fatou Linearization, and Quasiconformal Rigidity for Parabolic Germs", "abstract": "By applying holomorphic motions, we prove that a parabolic germ is quasiconformally rigid, that is, any two topologically conjugate parabolic germs are quasiconformally conjugate and the conjugacy can be chosen to be more and more near conformal as long as we consider these germs defined on smaller and smaller neighborhoods. Before proving this theorem, we use the idea of holomorphic motions to give a conceptual proof of the Fatou linearization theorem. As a by-product, we also prove that any finite number of analytic germs at different points in the Riemann sphere can be extended to a quasiconformal homeomorphism which can be more and more near conformal as as long as we consider these germs defined on smaller and smaller neighborhoods of these points."}
{"category": "Math", "title": "Moderate deviations and laws of the iterated logarithm for the local times of additive L\\'{e}vy processes and additive random walks", "abstract": "We study the upper tail behaviors of the local times of the additive L\\'{e}vy processes and additive random walks. The limit forms we establish are the moderate deviations and the laws of the iterated logarithm for the L_2-norms of the local times and for the local times at a fixed site."}
{"category": "Math", "title": "Exact Hausdorff measure on the boundary of a Galton--Watson tree", "abstract": "A necessary and sufficient condition for the almost sure existence of an absolutely continuous (with respect to the branching measure) exact Hausdorff measure on the boundary of a Galton--Watson tree is obtained. In the case where the absolutely continuous exact Hausdorff measure does not exist almost surely, a criterion which classifies gauge functions $\\phi$ according to whether $\\phi$-Hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given. Important examples are discussed in four additional theorems. In particular, Hawkes's conjecture in 1981 is solved. Problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved."}
{"category": "Math", "title": "Guaranteed Accuracy for Conic Programming Problems in Vector Lattices", "abstract": "This paper presents rigorous forward error bounds for linear conic optimization problems. The error bounds are formulated in a quite general framework; the underlying vector spaces are not required to be finite-dimensional, and the convex cones defining the partial ordering are not required to be polyhedral. In the case of linear programming, second order cone programming, and semidefinite programming specialized formulas are deduced yielding guaranteed accuracy. All computed bounds are completely rigorous because all rounding errors due to floating point arithmetic are taken into account. Numerical results, applications and software for linear and semidefinite programming problems are described."}
{"category": "Math", "title": "Denjoy constructions for fibred homeomorphisms of the torus", "abstract": "We construct different types of quasiperiodically forced circle homeomorphisms with transitive but non-minimal dynamics. Concerning the recent Poincar\\'e-like classification for this class of maps of Jaeger-Stark, we demonstrate that transitive but non-minimal behaviour can occur in each of the different cases. This closes one of the last gaps in the topological classification. Actually, we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets. For example, we show that, in some of the examples we construct, the unique minimal set is a Cantor set and its intersection with each vertical fibre is uncountable and nowhere dense (but may contain isolated points). We also prove that minimal sets of the later kind cannot occur when the dynamics are given by the projective action of a quasiperiodic SL(2,R)-cocycle. More precisely, we show that, for a quasiperiodic SL(2,R)-cocycle, any minimal strict subset of the torus either is a union of finitely many continuous curves, or contains at most two points on generic fibres."}
{"category": "Math", "title": "Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets", "abstract": "A new class of plurisubharmonic functions on the octonionic plane O^2= R^{16} is introduced. An octonionic version of theorems of A.D. Aleksandrov and Chern- Levine-Nirenberg, and Blocki are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of O^2=R^{16}. In particular a new example of Spin(9)-invariant valuation on R^{16} is given."}
{"category": "Math", "title": "Nonlinear Dirac equations on Riemann surfaces", "abstract": "We develop analytical methods for nonlinear Dirac equations. Examples of such equations include Dirac-harmonic maps with curvature term and the equations describing the generalized Weierstrass representation of surfaces in three-manifolds. We provide the key analytical steps, i.e., small energy regularity and removable singularity theorems and energy identities for solutions."}
{"category": "Math", "title": "Backward stochastic differential equations with random stopping time and singular final condition", "abstract": "In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type: \\[Y_t=\\xi -\\int_{t\\wedge \\tau}^{\\tau}Y_r|Y_r|^q dr-\\int_{t\\wedge \\tau}^{\\tau}Z_r dB_r,\\qquad t\\geq 0,\\] where $\\tau$ is a stopping time, $q$ is a positive constant and $\\xi$ is a $\\mathcal{F}_{\\tau}$-measurable random variable such that $\\mathbf{P}(\\xi =+\\infty)>0$. We study the link between these BSDE and the Dirichlet problem on a domain $D\\subset \\mathbb{R}^d$ and with boundary condition $g$, with $g=+\\infty$ on a set of positive Lebesgue measure. We also extend our results for more general BSDE."}
{"category": "Math", "title": "La $\\mathrm{Z}_l$-cohomologie du mod\\`ele de Deligne-Carayol est sans torsion", "abstract": "This article is the $\\mathrm{Z}_l$-version of my paper \"Monodromie du faisceau pervers des cycles \\'evanescents de quelques vari\\'et\\'es de Shimura simples\" in Invent. Math. 2009 vol 177 pp. 239-280, where we study the vanishing cycles of some unitary Shimura variety. The aim is to prove that the cohomology sheaves of this complexe are free so that, thanks to the main theorem of Berkovich on vanishing cycles, we can deduce that the $\\mathrm{Z}_l$-cohomology of the model of Deligne-Carayol is free. There will be a second article which will be the $\\mathrm{Z}_l$ version of my paper \"Conjecture de monodromie-poids pour quelques vari\\'t\\'es de Shimura unitaires\" in Compositio vol 146 part 2, pp. 367-403. The aim of this second article will be to study the torsion of the cohomology groups of these Shimura varieties."}
{"category": "Math", "title": "Enumerating the Saneblidze-Umble diagonal terms", "abstract": "The author presents a computer implementation, calculating the terms of the Saneblidze-Umble diagonals on the permutahedron and the associahedron. The code is analyzed for correctness and presented in the paper, the source code of which simultaneously represents both the paper and the program."}
{"category": "Math", "title": "Backlund Transformations and Darboux Integrability for Nonlinear Wave Equations", "abstract": "We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One direction of proof, proving Darboux integrability, follows the implications of the wave equation for the invariants of the G-structure associated to the Backlund transformation. The other direction constructs Backlund transformations for Darboux integrable equations as solutions of an involutive exterior differential system. Explicit transformations are given for several equations on the Goursat-Vessiot list of Darboux-integrable equations."}
{"category": "Math", "title": "On Shokurov-type b-divisorial algebras of higher rank", "abstract": "The purpose of this paper is to lay the foundations for the theory of higher rank b-divisorial algebras of Shokurov type. We develop techniques to deal with such objects and propose two natural conjectures regarding Shokurov algebras and adjoint algebras. We confirm these conjectures in the case of affine curves."}
{"category": "Math", "title": "An application of linear programming duality to discrete Fourier analysis and additive problems", "abstract": "Suppose that f is a function from Z_p -> [0,1] (Z_p is my notation for the integers mod p, not the p-adics), and suppose that a_1,...,a_k are some places in Z_p. In some additive number theory applications it would be nice to perturb f slightly so that Fourier transform f^ vanishes at a_1,...,a_k, while additive properties are left intact. In the present paper, we show that even if we are unsuccessful in this, we can at least say something interesting by using the principle of the separating hyperplane, a basic ingredient in linear programming duality."}
{"category": "Math", "title": "On the existence of A-loops with some commutative inner mappings and others of order 2", "abstract": "The existence of A$_\\rho$-loops, A$_\\lambda$-loops and A$_\\mu$-loops that are neither extra loops nor CC-loops such that any two of their inner mappings $R(x,y),L(x,y)$ and $T(x)$ commute while the other one is of order 2 is shown."}
{"category": "Math", "title": "Sur la d\\'efinissabilit\\'e existentielle de la non-nullit\\'e dans les anneaux", "abstract": "We investigate the rings in which the set of nonzero elements is positive-existential (i.e. a finite union of projections of \"algebraic\" sets). In the case of Noetherian domains, we prove in particular that this condition is satisfied whenever the ring in question is not local Henselian, while it is not satisfied for any excellent local Henselian domain which is not a field. As a byproduct, we obtain an answer to a question of Popescu on strong approximation for Henselian pairs."}
{"category": "Math", "title": "Free resolutions over short local rings", "abstract": "The structure of minimal free resolutions of finite modules M over commutative local rings (R,m,k) with m^3=0 and rank_k(m^2) < rank_k(m/m^2)is studied. It is proved that over generic R every M has a Koszul syzygy module. Explicit families of Koszul modules are identified. When R is Gorenstein the non-Koszul modules are classified. Structure theorems are established for the graded k-algebra Ext_R(k,k) and its graded module Ext_R(M,k)."}
{"category": "Math", "title": "A combinatorial basis for the free Lie algebra of the labelled rooted trees", "abstract": "The pre-Lie operad can be realized as a space T of labelled rooted trees. A result of F. Chapoton shows that the pre-Lie operad is a free twisted Lie algebra. That is, the S-module T is obtained as the plethysm of the S-module Lie with an S-module F. In the context of species, we construct an explicit basis of F. This allows us to give a new proof of Chapoton's results. Moreover it permits us to show that F forms a sub nonsymmetric operad of the pre-Lie operad T."}
{"category": "Math", "title": "Weak Convergence in the Prokhorov Metric of Methods for Stochastic Differential Equations", "abstract": "We consider the weak convergence of numerical methods for stochastic differential equations (SDEs). Weak convergence is usually expressed in terms of the convergence of expected values of test functions of the trajectories. Here we present an alternative formulation of weak convergence in terms of the well-known Prokhorov metric on spaces of random variables. For a general class of methods, we establish bounds on the rates of convergence in terms of the Prokhorov metric. In doing so, we revisit the original proofs of weak convergence and show explicitly how the bounds on the error depend on the smoothness of the test functions. As an application of our result, we use the Strassen - Dudley theorem to show that the numerical approximation and the true solution to the system of SDEs can be re-embedded in a probability space in such a way that the method converges there in a strong sense. One corollary of this last result is that the method converges in the Wasserstein distance, another metric on spaces of random variables. Another corollary establishes rates of convergence for expected values of test functions assuming only local Lipschitz continuity. We conclude with a review of the existing results for pathwise convergence of weakly converging methods and the corresponding strong results available under re-embedding."}
{"category": "Math", "title": "Note Integer Factoring Methods III", "abstract": "The best deterministic unconditionally proven integer factorization algorithms have exponential running time complexities of O(N^(1/4)) arithmetic operations, and conditional on the Riemann hypothesis, there is a deterministic algorithm of exponential running time complexity O(N^(1/5)). This note proposes a new deterministic integer factorization algorithm of deterministic exponential time complexity O(N^(1/6)). Furthermore, an algorithm for decomposing composite integers that have factor differences of the form q - p = (r - 1)N^(1/2) + u, where r > 1 is a fixed parameter, and | u | < N^(1/3), in deterministic logarithmic time and various other results are included."}
{"category": "Math", "title": "Large deviations for occupation times of Markov processes with $L_{\\mathbf{2}}$ semigroups", "abstract": "Our aim is to unify and extend the large deviation upper and lower bounds for the occupation times of a Markov process with $L_2$ semigroups under minimal conditions on the state space and the process trajectories; for example, no strong Markov property is needed. The methods used here apply in both continuous and discrete time. We present the proofs for continuous time only because of the inherent technical difficulties in that situation; the proofs can be adapted for discrete time in a straightforward manner."}
{"category": "Math", "title": "A signed analog of Euler's reduction formula for the double zeta function", "abstract": "The double zeta function is a function of two arguments defined by a double Dirichlet series, and was first studied by Euler in response to a letter from Goldbach in 1742. By calculating many examples, Euler inferred a closed form evaluation of the double zeta function in terms of values of the Riemann zeta function, in the case when the two arguments are positive integers with opposite parity. Here, we consider a signed analog of Euler's evaluation: namely a reduction formula for the signed double zeta function that reduces to Euler's evaluation when the signs are specialized to 1. This formula was first stated in a 1997 paper by Borwein, Bradley and Broadhurst and was subsequently proved by Flajolet and Salvy using contour integration. The purpose here is to give an elementary proof based on a partial fraction identity."}
{"category": "Math", "title": "Expected Utility Optimization - Calculus of Variations Approach", "abstract": "In this paper, I'll derive the Hamilton-Jacobi (HJ) equation for Merton's problem in Utility Optimization Theory using a Calculus of Variations (CoV) Approach. For stochastic control problems, Dynamic Programming (DP) has been used as a standard method. To the best of my knowledge, no one has used CoV for this problem. In addition, while the DP approach cannot guarantee that the optimum satisfies the HJ equation, the CoV approach does. Be aware that this is the first draft of this paper and many flaws might be introduced."}
{"category": "Math", "title": "On tight contact structures with negative maximal twisting number on small Seifert manifolds", "abstract": "We study some properties of transverse contact structures on small Seifert manifolds, and we apply them to the classification of tight contact structures on a family of small Seifert manifolds."}
{"category": "Math", "title": "On Quasiminimal Excellent Classes", "abstract": "A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the L_w1,w(Q)-definability assumption may be dropped, and each class is determined by its model of dimension aleph_0."}
{"category": "Math", "title": "A spectral condition for odd cycles in graphs", "abstract": "We give a sharp spectral condition for the existence of odd cycles in a graph of given order. We also prove a related stability result."}
{"category": "Math", "title": "The Analytic Classification of Plane Branches", "abstract": "In this paper we give a solution to Zariski's problem of analytic classification of plane branches."}
{"category": "Math", "title": "Criteria for virtual fibering", "abstract": "We prove that an irreducible 3-manifold whose fundamental group satisfies a certain group-theoretic property called RFRS is virtually fibered. As a corollary, we show that 3-dimensional reflection orbifolds and arithmetic hyperbolic orbifolds defined by a quadratic form virtually fiber. These include the Seifert Weber dodecahedral space and the Bianchi orbifolds. Moreover, we show that a taut sutured compression body has a finite-sheeted cover with a depth one taut-oriented foliation."}
{"category": "Math", "title": "Analytic classification of plane branches up to multiplicity 4", "abstract": "We perform the analytic classification of plane branches of multiplicity less or equal than four."}
{"category": "Math", "title": "Gr\\\"obner bases and Betti numbers of monoidal complexes", "abstract": "In this note we consider monoidal complexes and their associated algebras, called toric face rings. These rings generalize Stanley-Reisner rings and affine monoid algebras. We compute initial ideals of the presentation ideal of a toric face ring, and determine its graded Betti numbers. Our results generalize celebrated theorems of Hochster in combinatorial commutative algebra."}
{"category": "Math", "title": "Integration over complex manifolds via Hochschild homology", "abstract": "Given a holomorphic vector bundle $\\cale$ on a connected compact complex manifold X, [FLS] construct a $\\compl$-linear functional $I_{\\cale}$ on $\\hh{2n}{\\compl}$. This is done by constructing a linear functional on the 0-th completed Hochschild homology $\\choch{0}{(\\dif(\\cale))}$ of the sheaf of holomorphic differential operators on $\\cale$ using topological quantum mechanics. They show that this functional is $\\int_X$ if $\\cale$ has non zero Euler characteristic. They conjecture that this functional is $\\int_X$ for all $\\cale$. A subsequent work [Ram] by the author proved that the linear functional $I_{\\cale}$ is independent of the vector bundle $\\cale$. This note builds upon the work in [Ram] to prove that $I_{\\cale}=\\int_X$ for an arbitrary holomorphic vector bundle $\\cale$ on an arbitrary connected compact complex manifold X. This is done using an argument that is very natural from the geometric point of view. This argument enables us to extend the construction in [FLS] to a construction of a linear functional $I_{\\cale}$ on $\\text{H}^{2n}_{c}(Y,\\compl)$ for an arbitrary holomorphic vector bundle $\\cale$ on an arbitrary connected complex manifold Y and prove that $I_{\\cale} = \\int_Y$. We also generalize a result of [Ram] pertaining to \"cyclic homology analogs\" of $I_{\\cale}$."}
{"category": "Math", "title": "On the paper ``Weak convergence of some classes of martingales with jumps''", "abstract": "This note extends some results of Nishiyama [Ann. Probab. 28 (2000) 685--712]. A maximal inequality for stochastic integrals with respect to integer-valued random measures which may have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion is obtained; if the so-called quadratic modulus is bounded in probability and if a certain entropy condition on the parameter space is satisfied, then the tightness follows. Our approach is based on the entropy techniques developed in the modern theory of empirical processes."}
{"category": "Math", "title": "Structural properties of proportional fairness: stability and insensitivity", "abstract": "In this article we provide a novel characterization of the proportionally fair bandwidth allocation of network capacities, in terms of the Fenchel--Legendre transform of the network capacity region. We use this characterization to prove stability (i.e., ergodicity) of network dynamics under proportionally fair sharing, by exhibiting a suitable Lyapunov function. Our stability result extends previously known results to a more general model including Markovian users routing. In particular, it implies that the stability condition previously known under exponential service time distributions remains valid under so-called phase-type service time distributions. We then exhibit a modification of proportional fairness, which coincides with it in some asymptotic sense, is reversible (and thus insensitive), and has explicit stationary distribution. Finally we show that the stationary distributions under modified proportional fairness and balanced fairness, a sharing criterion proposed because of its insensitivity properties, admit the same large deviations characteristics. These results show that proportional fairness is an attractive bandwidth allocation criterion, combining the desirable properties of ease of implementation with performance and insensitivity."}
{"category": "Math", "title": "Good rough path sequences and applications to anticipating stochastic calculus", "abstract": "We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. We then show that this condition is satisfied by the Brownian motion. As application, we obtain rather flexible results such as support theorems, large deviation principles and Wong--Zakai approximations for SDEs driven by Brownian motion along anticipating vectorfields. In particular, this unifies many results on anticipative SDEs."}
{"category": "Math", "title": "Central limit theorem and almost sure central limit theorem for the product of some partial sums", "abstract": "In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables."}
{"category": "Math", "title": "Stationary distributions of a model of sympatric speciation", "abstract": "This paper deals with a model of sympatric speciation, that is, speciation in the absence of geographical separation, originally proposed by U. Dieckmann and M. Doebeli in 1999. We modify their original model to obtain a Fleming--Viot type model and study its stationary distribution. We show that speciation may occur, that is, the stationary distribution puts most of the mass on a configuration that does not concentrate on the phenotype with maximum carrying capacity, if competition between phenotypes is intense enough. Conversely, if competition between phenotypes is not intense, then speciation will not occur and most of the population will have the phenotype with the highest carrying capacity. The length of time it takes speciation to occur also has a delicate dependence on the mutation parameter, and the exact shape of the carrying capacity function and the competition kernel."}
{"category": "Math", "title": "Open Problems in Algebraic Statistics", "abstract": "Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field, with main emphasis on graphical models with hidden variables, maximum likelihood estimation, and multivariate Gaussian distributions. This article is based on a lecture presented at the IMA in Minneapolis during the 2006/07 program on Applications of Algebraic Geometry."}
{"category": "Math", "title": "A boundedness result for toric log Del Pezzo surfaces", "abstract": "In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index $\\ell$. This upper bound turns out to be a quadratic polynomial in the variable $\\ell$."}
{"category": "Math", "title": "Heat kernel estimates for the Grusin operator", "abstract": "We study the geometry associated to the Grusin operator G=\\Delta_{x}+|x|^{2}\\partial_{u}^{2} on \\mathbb{R}_{x}^{n}\\times\\mathbb{R}_{u}, to obtain heat kernel estimates for this operator. The main work is to find the shortest geodesics connecting two given points in $\\mathbb{R}^{n+1}$. This gives the Carnot-Caratheodory distance d_{CC}, associated to this operator. The main result in the second part is to give Gaussian bounds for the heat kernel K_{t} in terms of the Carnot-Caratheodory distance. In particular we obtain the following estimate |k_{t}(\\zeta,\\eta)|\\leq C t^{-\\frac{n}{2}-1}\\min(1+\\frac{d_{CC}(\\zeta,\\eta)} {|x+\\xi|},1+\\frac{d_{CC}(\\zeta,\\eta)^{2}}{4t})^{\\alpha}e^{-\\frac{1}{4t}d_{CC} (\\zeta,\\eta)^{2}} for all $\\zeta=(x,u_{1}), \\eta=(\\xi,u)\\in\\mathbb{R}^{n+1}$, where $\\alpha = \\max{\\frac{n}{2}-1,0}$. Here the homogeneous dimension is q=n+2, so that $\\frac{n}{2}-1=\\frac{q-4}{2}$. This shows that our result for $n\\geq2$ corresponds with the result on the Heisenberg group, which was given by Beals, Gaveau, Greiner in [1]."}
{"category": "Math", "title": "Probabilistic validation of homology computations for nodal domains", "abstract": "Homology has long been accepted as an important computable tool for quantifying complex structures. In many applications, these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurate the resulting homology computations are. In this paper, we present a probabilistic approach to quantifying the validity of homology computations for nodal domains of random fields in one and two space dimensions, which furnishes explicit probabilistic a priori bounds for the suitability of certain discretization sizes. We illustrate our results for the special cases of random periodic fields and random trigonometric polynomials."}
{"category": "Math", "title": "Einstein solvmanifolds with a simple Einstein derivation", "abstract": "The structure of a solvable Lie groups admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras)."}
{"category": "Math", "title": "Uniform convergence of exact large deviations for renewal reward processes", "abstract": "Let (X_n,Y_n) be i.i.d. random vectors. Let W(x) be the partial sum of Y_n just before that of X_n exceeds x>0. Motivated by stochastic models for neural activity, uniform convergence of the form $\\sup_{c\\in I}|a(c,x)\\operatorname {Pr}\\{W(x)\\gecx\\}-1|=o(1)$, $x\\to\\infty$, is established for probabilities of large deviations, with a(c,x) a deterministic function and I an open interval. To obtain this uniform exact large deviations principle (LDP), we first establish the exponentially fast uniform convergence of a family of renewal measures and then apply it to appropriately tilted distributions of X_n and the moment generating function of W(x). The uniform exact LDP is obtained for cases where X_n has a subcomponent with a smooth density and Y_n is not a linear transform of X_n. An extension is also made to the partial sum at the first exceedance time."}
{"category": "Math", "title": "Heavy traffic limit for a processor sharing queue with soft deadlines", "abstract": "This paper considers a GI/GI/1 processor sharing queue in which jobs have soft deadlines. At each point in time, the collection of residual service times and deadlines is modeled using a random counting measure on the right half-plane. The limit of this measure valued process is obtained under diffusion scaling and heavy traffic conditions and is characterized as a deterministic function of the limiting queue length process. As special cases, one obtains diffusion approximations for the lead time profile and the profile of times in queue. One also obtains a snapshot principle for sojourn times."}
{"category": "Math", "title": "Geometry of the theta divisor of a compactified jacobian", "abstract": "We study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves. Some applications on hyperelliptic stable curves are appended."}
{"category": "Math", "title": "Teaching the Kepler laws for freshmen", "abstract": "We present a natural proof of Kepler's law of ellipses in the spirit of Euclidean geometry. Moreover we discuss two existing Euclidean geometric proofs, one by Feynman in hist Lost Lecture from 1964 and the other by Newton in the Principia of 1687."}
{"category": "Math", "title": "LBB Stability of a Mixed Discontinuous/Continuous Galerkin Finite Element Pair", "abstract": "We introduce a new mixed discontinuous/continuous Galerkin finite element for solving the 2- and 3-dimensional wave equations and equations of incompressible flow. The element, which we refer to as P1dg-P2, uses discontinuous piecewise linear functions for velocity and continuous piecewise quadratic functions for pressure. The aim of introducing the mixed formulation is to produce a new flexible element choice for triangular and tetrahedral meshes which satisfies the LBB stability condition and hence has no spurious zero-energy modes. We illustrate this property with numerical integrations of the wave equation in two dimensions, an analysis of the resultant discrete Laplace operator in two and three dimensions, and a normal mode analysis of the semi-discrete wave equation in one dimension."}
{"category": "Math", "title": "Estimates for the maximal singular integral in terms of the singular integral:the case of even kernels", "abstract": "The purpose of this paper is to describe the smooth homogeneous Calderon-Zygmund operators for which the maximal singular integral T*f may be controlled by the singular integral Tf. We consider two types of control. The first is the L2 estimate of T*f by Tf, namely the estimate of the L2 norm of T*f by a constant times the L2 norm of Tf. The second is the pointwise estimate of T*f(x) by a constant times M(Tf)(x), where M denotes the Hardy-Littlewood maximal operator. Notice that this is an improved variant of Cotlar's inequality, because the term Mf(x) is missing on the right hand side. Our main result states that, for even operators, both are equivalent to a purely algebraic condition formulated in terms of the expansion of the kernel in spherical harmonics. The condition holds by higher order Riesz transforms, which then satisfy an improved version of Cotlar's inequality"}
{"category": "Math", "title": "Integrality of instanton numbers", "abstract": "We prove the results announced in a joint paper (arXiv:hep-th/0603106) with Maxim Kontsevich and Albert Schwarz."}
{"category": "Math", "title": "Semiparametrically efficient rank-based inference for shape I. optimal rank-based tests for sphericity", "abstract": "We propose a class of rank-based procedures for testing that the shape matrix $\\mathbf{V}$ of an elliptical distribution (with unspecified center of symmetry, scale and radial density) has some fixed value ${\\mathbf{V}}_0$; this includes, for ${\\mathbf{V}}_0={\\mathbf{I}}_k$, the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distribution-free when the center of symmetry is specified, and asymptotically so when it must be estimated. The multivariate ranks used throughout are those of the distances--in the metric associated with the null value ${\\mathbf{V}}_0$ of the shape matrix--between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and asymptotic relative efficiencies (AREs) are derived with respect to the adjusted Mauchly test (a modified version of the Gaussian likelihood ratio procedure proposed by Muirhead and Waternaux [Biometrika 67 (1980) 31--43]) or, equivalently, with respect to (an extension of) the test for sphericity introduced by John [Biometrika 58 (1971) 169--174]. For Gaussian scores, these AREs are uniformly larger than one, irrespective of the actual radial density. Necessary and/or sufficient conditions for consistency under nonlocal, possibly nonelliptical alternatives are given. Finite sample performances are investigated via a Monte Carlo study."}
{"category": "Math", "title": "Forbidden patterns and shift systems", "abstract": "The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. Due to its simple structure, shift systems are accessible to a more detailed analysis and, at the same time, exhibit all important properties of low-dimensional chaotic dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a dense set of periodic points), allowing to export the results to other dynamical systems via order-isomorphisms."}
{"category": "Math", "title": "Generating trees for permutations avoiding generalized patterns", "abstract": "We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation, which allows us to incorporate the adjacency condition about some entries in an occurrence of a generalized pattern. We use these trees to find functional equations for the generating functions enumerating these classes of permutations with respect to different parameters. In several cases we solve them using the kernel method and some ideas of Bousquet-M\\'elou. We obtain refinements of known enumerative results and find new ones."}
{"category": "Math", "title": "Special Fuzzy Matrices for Social Scientists", "abstract": "This book introduces special classes of Fuzzy and Neutrosophic matrices. These special classes of matrices are used in the construction of multi-expert special fuzzy models using FCM, FRM and FRE and their Neutorosophic analogues (simultaneous or otherwise, according to ones need). Using the six basic models, we have constructed a multi-expert multi-model called the Super Special Hexagonal Fuzzy and Neutrosophic model. Given any special input vector, these models can give the resultant using special operations. When coupled with computer programming, these operations can give the solution within a reasonable time period. Such multi-expert multi-model systems are not only a boon to social scientists, but also to anyone who wants to use Fuzzy or Neutrosophic models."}
{"category": "Math", "title": "Active Set and EM Algorithms for Log-Concave Densities Based on Complete and Censored Data", "abstract": "We develop an active set algorithm for the maximum likelihood estimation of a log-concave density based on complete data. Building on this fast algorithm, we indidate an EM algorithm to treat arbitrarily censored or binned data."}
{"category": "Math", "title": "On admissible rank one local systems", "abstract": "A rank one local system $\\LL$ on a smooth complex algebraic variety $M$ is 1-admissible if the dimension of the first cohomology group $H^1(M,\\LL)$ can be computed from the cohomology algebra $H^*(M,\\C)$ in degrees $\\leq 2$. Under the assumption that $M$ is 1-formal, we show that all local systems, except finitely many, on a non-translated irreducible component $W$ of the first characteristic variety $\\V_1(M)$ are 1-admissible, see Proposition 3.1. The same result holds for local systems on a translated component $W$, but now $H^*(M,\\C)$ should be replaced by $H^*(M_0,\\C)$, where $M_0$ is a Zariski open subset obtained from $M$ by deleting some hypersurfaces determined by the translated component $W$, see Theorem 4.3."}
{"category": "Math", "title": "On the space of morphisms between \\'etale groupoids", "abstract": "Given two \\'etale groupoids $\\Cal G$ and $\\Cal G'$, we consider the set of pointed morphisms from $\\Cal G$ to $\\Cal G'$. Under suitable hypothesis we introduce on this set a structure of Banach manifold which can be considered as the space of objects of an \\'etale groupoid whose space of orbits is the space of morphisms from $\\Cal G$ to $\\Cal G'$."}
{"category": "Math", "title": "q-Terms, singularities and the extended Bloch group", "abstract": "Our paper originated from a generalization of the Volume Conjecture to multisums of $q$-hypergeometric terms. This generalization was sketched by Kontsevich in a problem list in Aarhus University in 2006; \\cite{Ko}. We introduce the notion of a $q$-hypergeometric term (in short, $q$-term). The latter is a product of ratios of $q$-factorials in linear forms in several variables. In the first part of the paper, we show how to construct elements of the Bloch group (and its extended version) given a \\qterm. Their image under the Bloch-Wigner map or the Rogers dilogarithm is a finite set of periods of weight 2, in the sense of Kontsevich-Zagier. In the second part of the paper we introduce the notion of a special $q$-term, its corresponding sequence of polynomials, and its generating series. Examples of special $q$-terms come naturally from Quantum Topology, and in particular from planar projections of knots. The two parts are tied together by a conjecture that relates the singularities of the generating series of a special $q$-term with the periods of the corresponding elements of the extended Bloch group. In some cases (such as the $4_1$ knot), the conjecture is known."}
{"category": "Math", "title": "Semigroups of valuations on local rings", "abstract": "In this paper the question of which semigroups are realizable as the semigroup of values attained on a Noetherian local ring which is dominated by a valuation is considered. We give some striking examples, indicating that there may be no constraints on the semigroup beyond those known classically."}
{"category": "Math", "title": "A solid angle theory for real polytopes", "abstract": "We extend many theorems from the context of solid angle sums over rational polytopes to the context of solid angle sums over real polytopes. Moreover, we consider any real dilation parameter, as opposed to the traditional integer dilation parameters. One of the main results is an extension of Macdonald's solid angle quasipolynomial for rational polytopes to a real analytic function of the dilation parameter, for any real convex polytope."}
{"category": "Math", "title": "A renormalization approach to irrational rotations", "abstract": "We introduce a renormalization procedure which allows us to study in a unified and concise way different properties of the irrational rotations on the unit circle $\\beta \\mapsto \\set{\\alpha+\\beta}$, $\\alpha \\in \\R\\setminus \\Q$. In particular we obtain sharp results for the diffusion of the walk on $\\Z$ generated by the location of points of the sequence $\\{n\\alpha +\\beta\\}$ on a binary partition of the unit interval. Finally we give some applications of our method."}
{"category": "Math", "title": "Banach spaces with polynomial numerical index 1", "abstract": "We characterize Banach spaces with polynomial numerical index 1 when they have the Radon-Nikod\\'ym property. The holomorphic numerical index is introduced and the characterization of the Banach space with holomorphic numerical index 1 is obtained when it has the Radon-Nikod\\'ym property."}
{"category": "Math", "title": "Cross Validation for Comparing Multiple Density Estimation Procedures", "abstract": "We demonstrate the consistency of cross validation for comparing multiple density estimators using simple inequalities on the likelihood ratio. In nonparametric problems, the splitting of data does not require the domination of test data over the training/estimation data, contrary to Shao (1993). The result is complementary to that of Yang (2005) and Yang (2006)."}
{"category": "Math", "title": "Symmetries of spatial graphs and Simon invariants", "abstract": "An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientation-preserving or reversing symmetries of 2-component links, spatial complete graphs on 5 vertices and spatial complete bipartite graphs on 3+3 vertices in detail, and determine the necessary conditions on linking numbers and Simon invariants for such links and spatial graphs to be symmetric."}
{"category": "Math", "title": "Rational solutions of the Noumi and Yamada system of type $A_4^{(1)}$", "abstract": "In this paper, we completely classify the raional solutions of the Noumi and Yamada system of type A_4^{(1)}, which is a generalization of the forth Painlev\\'e equation. The rational solutions are classified to three types by the B\\\"acklund transformation group."}
{"category": "Math", "title": "Semiparametrically efficient rank-based inference for shape II. Optimal R-estimation of shape", "abstract": "A class of R-estimators based on the concepts of multivariate signed ranks and the optimal rank-based tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These R-estimators are root-n consistent under any radial density g, without any moment assumptions, and semiparametrically efficient at some prespecified density f. When based on normal scores, they are uniformly more efficient than the traditional normal-theory estimator based on empirical covariance matrices (the asymptotic normality of which, moreover, requires finite moments of order four), irrespective of the actual underlying elliptical density. They rely on an original rank-based version of Le Cam's one-step methodology which avoids the unpleasant nonparametric estimation of cross-information quantities that is generally required in the context of R-estimation. Although they are not strictly affine-equivariant, they are shown to be equivariant in a weak asymptotic sense. Simulations confirm their feasibility and excellent finite-sample performances."}
{"category": "Math", "title": "Farey Statistics in Time n^{2/3} and Counting Primitive Lattice Points in Polygons", "abstract": "We present algorithms for computing ranks and order statistics in the Farey sequence, taking time O (n^{2/3}). This improves on the recent algorithms of Pawlewicz [European Symp. Alg. 2007], running in time O (n^{3/4}). We also initiate the study of a more general algorithmic problem: counting primitive lattice points in planar shapes."}
{"category": "Math", "title": "2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization", "abstract": "Let $\\mathcal{F}$ be a class of measurable functions $f:S\\mapsto [0,1]$ defined on a probability space $(S,\\mathcal{A},P)$. Given a sample (X_1,...,X_n) of i.i.d. random variables taking values in S with common distribution P, let P_n denote the empirical measure based on (X_1,...,X_n). We study an empirical risk minimization problem $P_nf\\to \\min$, $f\\in \\mathcal{F}$. Given a solution $\\hat{f}_n$ of this problem, the goal is to obtain very general upper bounds on its excess risk \\[\\mathcal{E}_P(\\hat{f}_n):=P\\hat{f}_n-\\inf_{f\\in \\mathcal{F}}Pf,\\] expressed in terms of relevant geometric parameters of the class $\\mathcal{F}$. Using concentration inequalities and other empirical processes tools, we obtain both distribution-dependent and data-dependent upper bounds on the excess risk that are of asymptotically correct order in many examples. The bounds involve localized sup-norms of empirical and Rademacher processes indexed by functions from the class. We use these bounds to develop model selection techniques in abstract risk minimization problems that can be applied to more specialized frameworks of regression and classification."}
{"category": "Math", "title": "Equivariant Birch-Swinnerton-Dyer conjecture for the base change of elliptic curves: An example", "abstract": "Let E be an elliptic curved defined over $\\Q$ and let $K/\\Q$ be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for $h^1(E\\times_{\\Q} K)(1)$ viewed as a motive over $\\Q$ with coefficients in $\\Q[G]$ relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper we prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S_3-extension of $\\Q$."}
{"category": "Math", "title": "Note sur la d\\'etermination alg\\'ebrique du groupe fondamental pro-r\\'esoluble d'une courbe affine", "abstract": "Let X be a smooth projective algebraic curve of genus g minus $r\\geq 1$ points defined over an algebraically closed field k of characteristic $p\\geq 0$. The structure of the largest prime to p quotient of the \\'etale fundamental group is well known by transcendental methods : it is isomorphic to the largest prime to p quotient of a free pro-finite group on 2g+r-1 generators. We show that, with purely algebraic means, we can prove the corresponding result for the largest pro-solvable quotient of these groups."}
{"category": "Math", "title": "Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization'' by V. Koltchinskii", "abstract": "Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization'' by V. Koltchinskii [arXiv:0708.0083]"}
{"category": "Math", "title": "Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part II: proof of the existence result", "abstract": "We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroedinger Equation $- \\epsilon^2 \\Delta \\psi + V(x) \\psi = |\\psi|^{p-1} \\psi$ on a manifold or in the Euclidean space. Here V represents the potential, p is an exponent greater than 1 and $\\epsilon$ a small parameter corresponding to the Planck constant. As $\\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase in highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In the first part of this work we identified the limit set and constructed approximate solutions, while here we give the complete proof of our main existence result."}
{"category": "Math", "title": "Geometric construction of metaplectic covers of $\\GL_{n}$ in characteristic zero", "abstract": "This paper presents a new construction of the m-fold metaplectic cover of $\\GL_{n}$ over an algebraic number field k, where k contains a primitive m-th root of unity. A 2-cocycle on $\\GL_{n}(\\A)$ representing this extension is given and the splitting of the cocycle on $\\GL_{n}(k)$ is found explicitly. The cocycle is smooth at almost all places of k. As a consequence, a formula for the Kubota symbol on $\\SL_{n}$ is obtained. The construction of the paper requires neither class field theory nor algebraic K-theory, but relies instead on naive techniques from the geometry of numbers introduced by W. Habicht and T. Kubota. The power reciprocity law for a number field is obtained as a corollary."}
{"category": "Math", "title": "Principal values for Riesz transforms and rectifiability", "abstract": "Let $E\\subset R^d$ with $H^n(E)<\\infty$, where H^n stands for the $n$-dimensional Hausdorff measure. In this paper we prove that E is n-rectifiable if and only if the limit $$\\lim_{\\ve\\to0}\\int_{y\\in E:|x-y|>\\ve} \\frac{x-y}{|x-y|^{n+1}} dH^n(y)$$ exists H^n-almost everywhere in E. To prove this result we obtain precise estimates from above and from below for the $L^2$ norm of the n-dimensional Riesz transforms on Lipschitz graphs."}
{"category": "Math", "title": "Shintani Cocycles on $\\GL_{n}$", "abstract": "The aim of this paper is to define an n-1-cocycle $\\sigma$ on $\\GL_{n}(\\Q)$ with values in a certain space $\\hD$ of distributions on $\\A_f^{n}\\setminus\\{0\\}$. Here $\\A_f$ denotes the ring of finite ad\\`{e}les of $\\Q$, and the distributions take values in the Laurent series $\\C((z_{1},...,z_{n}))$. This cocycle can be used to evaluate special values of Artin L-functions on number fields at negative integers. The construction generalizes that of Solomon in the case n=2."}
{"category": "Math", "title": "Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions", "abstract": "We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \\epsilon^2 \\Delta \\psi + V(x) \\psi = |\\psi|^{p-1} \\psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an exponent greater than 1 and $\\epsilon$ a small parameter corresponding to the Planck constant. As $\\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase is highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In this first part we provide the characterization of the limit set, with natural stationarity and non-degeneracy conditions. We then construct an approximate solution up to order $\\epsilon^2$, showing that these conditions appear naturally in a Taylor expansion of the equation in powers of $\\epsilon$. Based on these, an existence result will be proved in the second part."}
{"category": "Math", "title": "Lowest Weights in Cohomology of Variations of Hodge Structure", "abstract": "Let X be a smooth complex projective variety, let $j:U\\into X$ an immersion of a Zariski open subset, and let V be a variation of Hodge structure of weight n over U. Then IH^k(X, j_*V) is known to carry a pure Hodge structure of weight k+n, while H^k(U,V) carries a mixed Hodge structure of weight $\\ge k+n$. In this note it is shown that the image of the natural map $IH^k(X,j_*V) \\to H^k(U,V)$ is the lowest weight part of this mixed Hodge structure. The proof uses Saito's theory of mixed Hodge modules."}
{"category": "Math", "title": "Rejoinder: 2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization", "abstract": "Rejoinder: 2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization [arXiv:0708.0083]"}
{"category": "Math", "title": "Harmonic morphisms from solvable Lie groups", "abstract": "In this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank $r\\ge 3$. The second method provides us with global solutions from any Damek-Ricci space and many non-compact Riemannian symmetric spaces. We then give a continuous family of 3-dimensional solvable Lie groups not admitting any complex valued harmonic morphisms, not even locally."}
{"category": "Math", "title": "Self-similar branching Markov chains", "abstract": "The main purpose of this work is to study self-similar branching Markov chains. First we will construct such a process. Then we will establish certain Limit Theorems using the theory of self-similar Markov processes."}
{"category": "Math", "title": "Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes", "abstract": "This paper deals with nonparametric maximum likelihood estimation for Gaussian locally stationary processes. Our nonparametric MLE is constructed by minimizing a frequency domain likelihood over a class of functions. The asymptotic behavior of the resulting estimator is studied. The results depend on the richness of the class of functions. Both sieve estimation and global estimation are considered. Our results apply, in particular, to estimation under shape constraints. As an example, autoregressive model fitting with a monotonic variance function is discussed in detail, including algorithmic considerations. A key technical tool is the time-varying empirical spectral process indexed by functions. For this process, a Bernstein-type exponential inequality and a central limit theorem are derived. These results for empirical spectral processes are of independent interest."}
{"category": "Math", "title": "Nonisomorphic Verdier octahedra on the same base", "abstract": "We show by an example that in a Verdier triangulated category, there may exist two mutually nonisomorphic Verdier octahedra containing the same commutative triangle."}
{"category": "Math", "title": "Robust estimates in generalized partially linear models", "abstract": "In this paper, we introduce a family of robust estimates for the parametric and nonparametric components under a generalized partially linear model, where the data are modeled by $y_i|(\\mathbf{x}_i,t_i)\\sim F(\\cdot,\\mu_i)$ with $\\mu_i=H(\\eta(t_i)+\\mathbf{x}_i^{$\\mathrm{T}$}\\beta)$, for some known distribution function F and link function H. It is shown that the estimates of $\\beta$ are root-n consistent and asymptotically normal. Through a Monte Carlo study, the performance of these estimators is compared with that of the classical ones."}
{"category": "Math", "title": "On the limiting distributions of multivariate depth-based rank sum statistics and related tests", "abstract": "A depth-based rank sum statistic for multivariate data introduced by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252--260] as an extension of the Wilcoxon rank sum statistic for univariate data has been used in multivariate rank tests in quality control and in experimental studies. Those applications, however, are based on a conjectured limiting distribution, provided by Liu and Singh [J. Amer. Statist. Assoc. 88 (1993) 252--260]. The present paper proves the conjecture under general regularity conditions and, therefore, validates various applications of the rank sum statistic in the literature. The paper also shows that the corresponding rank sum tests can be more powerful than Hotelling's T^2 test and some commonly used multivariate rank tests in detecting location-scale changes in multivariate distributions."}
{"category": "Math", "title": "Topological Free Entropy Dimension in Unital C^* algebras (II) : Orthogonal Sum of Unital C^*-algebras", "abstract": "In the paper, we obtain a formula for topological free entropy dimension in the orthogonal sum (or direct sum) of unital C^* algebras. As a corollary, we compute the topological free entropy dimension of any family of self-adjoint generators of a finite dimensional C^* algebra."}
{"category": "Math", "title": "Data-driven goodness-of-fit tests", "abstract": "We propose and study a general method for construction of consistent statistical tests on the basis of possibly indirect, corrupted, or partially available observations. The class of tests devised in the paper contains Neyman's smooth tests, data-driven score tests, and some types of multi-sample tests as basic examples. Our tests are data-driven and are additionally incorporated with model selection rules. The method allows to use a wide class of model selection rules that are based on the penalization idea. In particular, many of the optimal penalties, derived in statistical literature, can be used in our tests. We establish the behavior of model selection rules and data-driven tests under both the null hypothesis and the alternative hypothesis, derive an explicit detectability rule for alternative hypotheses, and prove a master consistency theorem for the tests from the class. The paper shows that the tests are applicable to a wide range of problems, including hypothesis testing in statistical inverse problems, multi-sample problems, and nonparametric hypothesis testing."}
{"category": "Math", "title": "Convergence rates for Bayesian density estimation of infinite-dimensional exponential families", "abstract": "We study the rate of convergence of posterior distributions in density estimation problems for log-densities in periodic Sobolev classes characterized by a smoothness parameter p. The posterior expected density provides a nonparametric estimation procedure attaining the optimal minimax rate of convergence under Hellinger loss if the posterior distribution achieves the optimal rate over certain uniformity classes. A prior on the density class of interest is induced by a prior on the coefficients of the trigonometric series expansion of the log-density. We show that when p is known, the posterior distribution of a Gaussian prior achieves the optimal rate provided the prior variances die off sufficiently rapidly. For a mixture of normal distributions, the mixing weights on the dimension of the exponential family are assumed to be bounded below by an exponentially decreasing sequence. To avoid the use of infinite bases, we develop priors that cut off the series at a sample-size-dependent truncation point. When the degree of smoothness is unknown, a finite mixture of normal priors indexed by the smoothness parameter, which is also assigned a prior, produces the best rate. A rate-adaptive estimator is derived."}
{"category": "Math", "title": "Asymptotically minimax Bayes predictive densities", "abstract": "Given a random sample from a distribution with density function that depends on an unknown parameter $\\theta$, we are interested in accurately estimating the true parametric density function at a future observation from the same distribution. The asymptotic risk of Bayes predictive density estimates with Kullback--Leibler loss function $D(f_{\\theta}||{\\hat{f}})=\\int{f_{\\theta} \\log{(f_{\\theta}/ hat{f})}}$ is used to examine various ways of choosing prior distributions; the principal type of choice studied is minimax. We seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk is minimax. A result resembling Stein's paradox for estimating normal means by the maximum likelihood holds for the uniform prior in the multivariate location family case: when the dimensionality of the model is at least three, the Jeffreys prior is minimax, though inadmissible. The Jeffreys prior is both admissible and minimax for one- and two-dimensional location problems."}
{"category": "Math", "title": "Semiparametric estimation of fractional cointegrating subspaces", "abstract": "We consider a common-components model for multivariate fractional cointegration, in which the $s\\geq1$ components have different memory parameters. The cointegrating rank may exceed 1. We decompose the true cointegrating vectors into orthogonal fractional cointegrating subspaces such that vectors from distinct subspaces yield cointegrating errors with distinct memory parameters. We estimate each cointegrating subspace separately, using appropriate sets of eigenvectors of an averaged periodogram matrix of tapered, differenced observations, based on the first $m$ Fourier frequencies, with $m$ fixed. The angle between the true and estimated cointegrating subspaces is $o_p(1)$. We use the cointegrating residuals corresponding to an estimated cointegrating vector to obtain a consistent and asymptotically normal estimate of the memory parameter for the given cointegrating subspace, using a univariate Gaussian semiparametric estimator with a bandwidth that tends to $\\infty$ more slowly than $n$. We use these estimates to test for fractional cointegration and to consistently identify the cointegrating subspaces."}
{"category": "Math", "title": "A frequency domain empirical likelihood for short- and long-range dependence", "abstract": "This paper introduces a version of empirical likelihood based on the periodogram and spectral estimating equations. This formulation handles dependent data through a data transformation (i.e., a Fourier transform) and is developed in terms of the spectral distribution rather than a time domain probability distribution. The asymptotic properties of frequency domain empirical likelihood are studied for linear time processes exhibiting both short- and long-range dependence. The method results in likelihood ratios which can be used to build nonparametric, asymptotically correct confidence regions for a class of normalized (or ratio) spectral parameters, including autocorrelations. Maximum empirical likelihood estimators are possible, as well as tests of spectral moment conditions. The methodology can be applied to several inference problems such as Whittle estimation and goodness-of-fit testing."}
{"category": "Math", "title": "Regular spanning subgraphs of bipartite graphs of high minimum degree", "abstract": "Let G be a simple balanced bipartite graph on $2n$ vertices, $\\delta = \\delta(G)/n$, and $\\rho={\\delta + \\sqrt{2 \\delta -1} \\over 2}$. If $\\delta > 1/2$ then it has a $\\lfloor \\rho n \\rfloor$-regular spanning subgraph. The statement is nearly tight."}
{"category": "Math", "title": "Algebraic K-theory and abstract homotopy theory", "abstract": "We decompose the K-theory space of a Waldhausen category in terms of its Dwyer-Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria appearing in the literature. We show that under mild hypotheses, a weakly exact functor that induces an equivalence of homotopy categories induces an equivalence of K-theory spectra."}
{"category": "Math", "title": "Triangulated categories of matrix factorizations for regular systems of weights with $\\epsilon=-1$", "abstract": "We construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a non-degenerate regular system of weights whose smallest exponents are equal to -1. In the associated Grothendieck group, the strongly exceptional collection defines a root basis of a generalized root system of sign (l,0,2) and a Coxeter element of finite order, whose primitive eigenvector is a regular element in the expanded symmetric domain of type IV with respect to the Weyl group."}
{"category": "Math", "title": "Permutations of Strongly Self-Absorbing C*-algebras", "abstract": "Let G be a finite group acting on {1,...,n}. For any C*-algebra A, this defines an action of \\alpha of G on A^{\\otimes n}. We show that if A tensorially absorbs a UHF algebra of infinite type, the Jiang-Su algebra, or is approximately divisible, then A \\times_{\\alpha} G has the corresponding property as well."}
{"category": "Math", "title": "Symmetric groups and conjugacy classes", "abstract": "Let S_n be the symmetric group on n-letters. Fix n>5. Given any nontrivial $\\alpha,\\beta\\in S_n$, we prove that the product $\\alpha^{S_n}\\beta^{S_n}$ of the conjugacy classes $\\alpha^{S_n}$ and $\\beta^{S_n}$ is never a conjugacy class. Furthermore, if n is not even and $n$ is not a multiple of three, then $\\alpha^{S_n}\\beta^{S_n}$ is the union of at least three distinct conjugacy classes. We also describe the elements $\\alpha,\\beta\\in S_n$ in the case when $\\alpha^{S_n}\\beta^{S_n}$ is the union of exactly two distinct conjugacy classes."}
{"category": "Math", "title": "Simplicity of C*-algebras associated to row-finite locally convex higher-rank graphs", "abstract": "In previous work, the authors showed that the C*-algebra C*(\\Lambda) of a row-finite higher-rank graph \\Lambda with no sources is simple if and only if \\Lambda is both cofinal and aperiodic. In this paper, we generalise this result to row-finite higher-rank graphs which are locally convex (but may contain sources). Our main tool is Farthing's \"removing sources\" construction which embeds a row-finite locally convex higher-rank graph in a row-finite higher-rank graph with no sources in such a way that the associated C*-algebras are Morita equivalent."}
{"category": "Math", "title": "Severe right Ore sets and universal localisation", "abstract": "We introduce the notion of a severe right Ore set in the main as a tool to study universal localisations of rings but also to provide a short proof of P. M. Cohn's classification of homomorphisms from a ring to a division ring. We prove that the category of finitely presented modules over a universal localisation is equivalent to a localisation at a severe right Ore set of the category of finitely presented modules over the original ring. This allows us to describe the structure of finitely presented modules over the universal localisation as modules over the original ring."}
{"category": "Math", "title": "Classifying the Unclassifiables", "abstract": "In 1955 George Mackey suggested that there is a fundamental dichotomy in the unitary representation theory of locally compact second countable groups. He felt that there cannnot be a reasonable classification theory for the unitary representations of a group G for which the dual is a non-smooth Borel space. Mackey's precise conjecture regarding when this is the case was subsequently verified by Glimm. This approach to \"classifiability\" can be applied in many other branches of mathematics. Included in this article is a sketch of some of the exciting new developments that have been made in this direction. Evidence is given that there should be extensions of Mackey's ideas to such \"finitistic\" problems as the classification of the finite p-groups. In a different direction, Mackey's thoughts about quantization are also briefly discussed."}
{"category": "Math", "title": "On algebraically integrable outer billiards", "abstract": "We prove that if the outer billiard map around a plane oval is algebraically integrable in a certain non-degenerate sense then the oval is an ellipse."}
{"category": "Math", "title": "Universal localisations of hereditary rings", "abstract": "We describe all possible universal localisations of a hereditary ring in terms of suitable full subcategories of the category of finitely presented modules. For these universal localisations we then identify the category of finitely presented bound modules over the universal localisation as being equivalent to a certain full subcategory of the category of finitely presented bound modules over the original ring. We also describe the abelian monoid of finitely generated projective modules over the universal localisation."}
{"category": "Math", "title": "Sharp k-order Sobolev inequalities in the hyperbolic space ${\\Bbb H}^n$", "abstract": "In this paper, we obtain the sharp $k$-th order Sobolev inequalities in the hyperbolic space ${\\H}^n$ for all $k=1,2,3,\\cdots$. This gives an answer to an open question raised by Aubin in [5, p.$\\;$176-177] for $W^{k,2}({\\H}^n)$ with $k>1$. In addition, we prove that the associated Sobolev constants are optimal."}
{"category": "Math", "title": "A simply connected surface of general type with p_g=0 and K^2=3", "abstract": "Motivated by a recent result of Y. Lee and the second author[7], we construct a simply connected minimal complex surface of general type with p_g=0 and K^2=3 using a rational blow-down surgery and Q-Gorenstein smoothing theory. In a similar fashion, we also construct a new simply connected symplectic 4-manifold with b_2^+=1 and K^2=4."}
{"category": "Math", "title": "Expert Elicitation for Reliable System Design", "abstract": "This paper reviews the role of expert judgement to support reliability assessments within the systems engineering design process. Generic design processes are described to give the context and a discussion is given about the nature of the reliability assessments required in the different systems engineering phases. It is argued that, as far as meeting reliability requirements is concerned, the whole design process is more akin to a statistical control process than to a straightforward statistical problem of assessing an unknown distribution. This leads to features of the expert judgement problem in the design context which are substantially different from those seen, for example, in risk assessment. In particular, the role of experts in problem structuring and in developing failure mitigation options is much more prominent, and there is a need to take into account the reliability potential for future mitigation measures downstream in the system life cycle. An overview is given of the stakeholders typically involved in large scale systems engineering design projects, and this is used to argue the need for methods that expose potential judgemental biases in order to generate analyses that can be said to provide rational consensus about uncertainties. Finally, a number of key points are developed with the aim of moving toward a framework that provides a holistic method for tracking reliability assessment through the design process."}
{"category": "Math", "title": "Stochastic Programming with Probability", "abstract": "In this work we study optimization problems subject to a failure constraint. This constraint is expressed in terms of a condition that causes failure, representing a physical or technical breakdown. We formulate the problem in terms of a probability constraint, where the level of \"confidence\" is a modelling parameter and has the interpretation that the probability of failure should not exceed that level. Application of the stochastic Arrow-Hurwicz algorithm poses two difficulties: one is structural and arises from the lack of convexity of the probability constraint, and the other is the estimation of the gradient of the probability constraint. We develop two gradient estimators with decreasing bias via a convolution method and a finite difference technique, respectively, and we provide a full analysis of convergence of the algorithms. Convergence results are used to tune the parameters of the numerical algorithms in order to achieve best convergence rates, and numerical results are included via an example of application in finance."}
{"category": "Math", "title": "Localizing the Elliott conjecture at strongly self-absorbing C*-algebras", "abstract": "We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C*-algebra $D$; we also explain how the known classification theorems for nuclear C*-algebras fit into this concept. As a new result in this direction, we employ recent results of Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C*-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang-Su algebra Z. Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition as well as the K-theory restriction entirely. When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense. The strategy to attack the general case aims at combining these two approaches. Our classification theorem covers simple ASH algebras for which projections separate traces (and the K-groups of which have finitely generated torsion part); it does, however, not at all depend on an inductive limit structure. Also, in the monotracial case it does not rely on the existence or absence of projections in any way. In fact, it is the first such result which, in a natural way, covers all known unital, separable, simple, nuclear and stably finite C*-algebras of real rank zero (the K-groups of which have finitely generated torsion part) as well as the (projectionless) Jiang-Su algebra itself."}
{"category": "Math", "title": "Uniqueness of ground states of some coupled nonlinear Schrodinger systems and their application", "abstract": "We establish the uniqueness of ground states of some coupled nonlinear Schrodinger systems in the whole space. We firstly use Schwartz symmetrization to obtain the existence of ground states for a more general case. To prove the uniqueness of ground states, we use the radial symmetry of the ground states to transform the systems into an ordinary differential system, and then we use the integral forms of the system. More interestingly, as an application of our uniqueness results, we derive a sharp vector-valued Gagliardo-Nirenberg inequality."}
{"category": "Math", "title": "Comment: Expert Elicitation for Reliable System Design", "abstract": "Comment: Expert Elicitation for Reliable System Design [arXiv:0708.0279]"}
{"category": "Math", "title": "Uniqueness of positive bound states to Schrodinger systems with critical exponents", "abstract": "We prove the uniqueness for the positive solutions of the following elliptic systems: \\begin{eqnarray*} \\left\\{\\begin{array}{ll} - \\lap (u(x)) = u(x)^{\\alpha}v(x)^{\\beta} - \\lap (v(x)) = u(x)^{\\beta} v(x)^{\\alpha} \\end{array} \\right. \\end{eqnarray*} Here $x\\in R^n$, $n\\geq 3$, and $1\\leq \\alpha, \\beta\\leq \\frac{n+2}{n-2}$ with $\\alpha+\\beta=\\frac{n+2}{n-2}$. In the special case when $n=3$ and $\\alpha =2, \\beta=3$, the systems come from the stationary Schrodinger system with critical exponents for Bose-Einstein condensate. As a key step, we prove the radial symmetry of the positive solutions to the elliptic system above with critical exponents."}
{"category": "Math", "title": "Comment: Expert Elicitation for Reliable System Design", "abstract": "Comment: Expert Elicitation for Reliable System Design [arXiv:0708.0279]"}
{"category": "Math", "title": "Comment: Expert Elicitation for Reliable System Design", "abstract": "Comment: Expert Elicitation for Reliable System Design [arXiv:0708.0279]"}
{"category": "Math", "title": "Rejoinder: Expert Elicitation for Reliable System Design", "abstract": "Rejoinder: Expert Elicitation for Reliable System Design [arXiv:0708.0279]"}
{"category": "Math", "title": "Reliability", "abstract": "This special volume of Statistical Sciences presents some innovative, if not provocative, ideas in the area of reliability, or perhaps more appropriately named, integrated system assessment. In this age of exponential growth in science, engineering and technology, the capability to evaluate the performance, reliability and safety of complex systems presents new challenges. Today's methodology must respond to the ever-increasing demands for such evaluations to provide key information for decision and policy makers at all levels of government and industry--problems ranging from international security to space exploration. We, the co-editors of this volume and the authors, believe that scientific progress in reliability assessment requires the development of processes, methods and tools that combine diverse information types (e.g., experiments, computer simulations, expert knowledge) from diverse sources (e.g., scientists, engineers, business developers, technology integrators, decision makers) to assess quantitative performance metrics that can aid decision making under uncertainty. These are highly interdisciplinary problems. The principal role of statistical sciences is to bring statistical rigor, thinking and methodology to these problems."}
{"category": "Math", "title": "Asymptotic behavior of flat surfaces in hyperbolic 3-space", "abstract": "In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H^3. Galvez, Martinez and Milan showed that when the singular set does not accumulate at an end, the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called \"pitch\" p) of the end determines the limiting shape, even when the singular set does accumulate at the end. If the singular set is bounded away from the end, we have -1<p<=0. If the singular set accumulates at the end, the pitch p is a positive rational number not equal to 1. Choosing appropriate positive integers n and m so that p=n/m, suitable slices of the end by horospheres are asymptotic to d-coverings (d-times wrapped coverings) of epicycloids or d-coverings of hypocycloids with 2n_0 cusps and whose normal directions have winding number m_0, where n=n_0d, m=m_0d (n_0, m_0 are integers or half-integers) and d is the greatest common divisor of m-n and m+n. Furthermore, it is known that the caustics of flat surfaces are also flat. So, as an application, we give a useful explicit formula for the pitch of ends of caustics of complete flat fronts."}
{"category": "Math", "title": "Monitoring Networked Applications With Incremental Quantile Estimation", "abstract": "Networked applications have software components that reside on different computers. Email, for example, has database, processing, and user interface components that can be distributed across a network and shared by users in different locations or work groups. End-to-end performance and reliability metrics describe the software quality experienced by these groups of users, taking into account all the software components in the pipeline. Each user produces only some of the data needed to understand the quality of the application for the group, so group performance metrics are obtained by combining summary statistics that each end computer periodically (and automatically) sends to a central server. The group quality metrics usually focus on medians and tail quantiles rather than on averages. Distributed quantile estimation is challenging, though, especially when passing large amounts of data around the network solely to compute quality metrics is undesirable. This paper describes an Incremental Quantile (IQ) estimation method that is designed for performance monitoring at arbitrary levels of network aggregation and time resolution when only a limited amount of data can be transferred. Applications to both real and simulated data are provided."}
{"category": "Math", "title": "Curvature of almost quaternion-Hermitian manifolds", "abstract": "We study the decomposition of the Riemannian curvature R tensor of an almost quaternion-Hermitian manifold under the action of its structure group Sp(n)Sp(1). Using the minimal connection, we show that most components are determined by the intrinsic torsion \\xi and its covariant derivative \\widetilde\\nabla\\xi and determine relations between the decompositions of \\xi\\otimes\\xi, \\widetilde\\nabla\\xi and R. We pay particular attention to the behaviour of the Ricci curvature and the q-Ricci curvature."}
{"category": "Math", "title": "Large inductive dimension of the Smirnov remainder", "abstract": "The purpose of this paper is to investigate the large inductive dimension of the remainder of the Smirnov compactification of the n-dimensional Euclidean space with the usual metric, and give an application of it."}
{"category": "Math", "title": "Comment: Monitoring Networked Applications With Incremental Quantile Estimation", "abstract": "Comment: Monitoring Networked Applications With Incremental Quantile Estimation [arXiv:0708.0302]"}
{"category": "Math", "title": "Global asymptotic stability for a class of nonlinear chemical equations", "abstract": "We consider a class of nonlinear differential equations that arises in the study of chemical reaction systems that are known to be locally asymptotically stable and prove that they are in fact globally asymptotically stable. More specifically, we will consider chemical reaction systems that are weakly reversible, have a deficiency of zero, and are equipped with mass action kinetics. We show that if for each $c \\in \\R_{> 0}^m$ the intersection of the stoichiometric compatibility class $c + S$ with the subsets on the boundary that could potentially contain equilibria, $L_W$, are at most discrete, then global asymptotic stability follows. Previous global stability results for the systems considered in this paper required $(c + S) \\cap L_W = \\emptyset$ for each $c \\in \\R^m_{> 0}$, and so this paper can be viewed as an extension of those works."}
{"category": "Math", "title": "Harmonic Analysis over adelic spaces", "abstract": "Extending ideas of A.N. Parshin and D.V. Osipov, Harmonic Analysis is developed for filtered infinite dimensional modules over a ring. We establish Pontryagin duality, the Fourier inversion formula, Plancherel formula and Poisson summation formula for all dimensions."}
{"category": "Math", "title": "Notes on the geometry of space of polynomials", "abstract": "We show that the symmetric injective tensor product space $\\hat{\\otimes}_{n,s,\\epsilon}E$ is not complex strictly convex if E is a complex Banach space of $\\dim E \\ge 2$ and if $n\\ge 2$ holds. It is also reproved that $\\ell_\\infty$ is finitely represented in $\\hat{\\otimes}_{n,s,\\epsilon}E$ if E is infinite dimensional and if $n\\ge 2$ holds, which was proved in the other way by Dineen."}
{"category": "Math", "title": "Comment: Monitoring Networked Applications With Incremental Quantile Estimation", "abstract": "Our comments are in two parts. First, we make some observations regarding the methodology in Chambers et al. [arXiv:0708.0302]. Second, we briefly describe another interesting network monitoring problem that arises in the context of assessing quality of service, such as loss rates and delay distributions, in packet-switched networks."}
{"category": "Math", "title": "Comment: Monitoring Networked Applications With Incremental Quantile Estimation", "abstract": "Comment: Monitoring Networked Applications With Incremental Quantile Estimation [arXiv:0708.0302]"}
{"category": "Math", "title": "Rejoinder: Monitoring Networked Applications With Incremental Quantile Estimation", "abstract": "Rejoinder: Monitoring Networked Applications With Incremental Quantile Estimation [arXiv:0708.0302]"}
{"category": "Math", "title": "Dynamic Modeling and Statistical Analysis of Event Times", "abstract": "This review article provides an overview of recent work in the modeling and analysis of recurrent events arising in engineering, reliability, public health, biomedicine and other areas. Recurrent event modeling possesses unique facets making it different and more difficult to handle than single event settings. For instance, the impact of an increasing number of event occurrences needs to be taken into account, the effects of covariates should be considered, potential association among the interevent times within a unit cannot be ignored, and the effects of performed interventions after each event occurrence need to be factored in. A recent general class of models for recurrent events which simultaneously accommodates these aspects is described. Statistical inference methods for this class of models are presented and illustrated through applications to real data sets. Some existing open research problems are described."}
{"category": "Math", "title": "Threshold Regression for Survival Analysis: Modeling Event Times by a Stochastic Process Reaching a Boundary", "abstract": "Many researchers have investigated first hitting times as models for survival data. First hitting times arise naturally in many types of stochastic processes, ranging from Wiener processes to Markov chains. In a survival context, the state of the underlying process represents the strength of an item or the health of an individual. The item fails or the individual experiences a clinical endpoint when the process reaches an adverse threshold state for the first time. The time scale can be calendar time or some other operational measure of degradation or disease progression. In many applications, the process is latent (i.e., unobservable). Threshold regression refers to first-hitting-time models with regression structures that accommodate covariate data. The parameters of the process, threshold state and time scale may depend on the covariates. This paper reviews aspects of this topic and discusses fruitful avenues for future research."}
{"category": "Math", "title": "Predictability of band-limited, high-frequency, and mixed processes in the presence of ideal low-pass filters", "abstract": "Pathwise predictability of continuous time processes is studied in deterministic setting. We discuss uniform prediction in some weak sense with respect to certain classes of inputs. More precisely, we study possibility of approximation of convolution integrals over future time by integrals over past time. We found that all band-limited processes are predictable in this sense, as well as high-frequency processes with zero energy at low frequencies. It follows that a process of mixed type still can be predicted if an ideal low-pass filter exists for this process."}
{"category": "Math", "title": "An application of Kapteyn series to a problem from queueing theory", "abstract": "We obtain exact solutions of a problem arising from queueing theory using properties of Kapteyn series."}
{"category": "Math", "title": "Advances in Data Combination, Analysis and Collection for System Reliability Assessment", "abstract": "The systems that statisticians are asked to assess, such as nuclear weapons, infrastructure networks, supercomputer codes and munitions, have become increasingly complex. It is often costly to conduct full system tests. As such, we present a review of methodology that has been proposed for addressing system reliability with limited full system testing. The first approaches presented in this paper are concerned with the combination of multiple sources of information to assess the reliability of a single component. The second general set of methodology addresses the combination of multiple levels of data to determine system reliability. We then present developments for complex systems beyond traditional series/parallel representations through the use of Bayesian networks and flowgraph models. We also include methodological contributions to resource allocation considerations for system relability assessment. We illustrate each method with applications primarily encountered at Los Alamos National Laboratory."}
{"category": "Math", "title": "Geometric proof of Thom conjecture", "abstract": "This paper has been withdrawn by the author due a crucial sign error in Theorem B. We present a geometric proof of Thom conjecture, which uses Khovanov homology. Our approach doesn't use any analytic methods and is quite different from proof given by Kronheimer and Mrowka in 1994."}
{"category": "Math", "title": "On the Statistical Modeling and Analysis of Repairable Systems", "abstract": "We review basic modeling approaches for failure and maintenance data from repairable systems. In particular we consider imperfect repair models, defined in terms of virtual age processes, and the trend-renewal process which extends the nonhomogeneous Poisson process and the renewal process. In the case where several systems of the same kind are observed, we show how observed covariates and unobserved heterogeneity can be included in the models. We also consider various approaches to trend testing. Modern reliability data bases usually contain information on the type of failure, the type of maintenance and so forth in addition to the failure times themselves. Basing our work on recent literature we present a framework where the observed events are modeled as marked point processes, with marks labeling the types of events. Throughout the paper the emphasis is more on modeling than on statistical inference."}
{"category": "Math", "title": "Cohomology and deformations of the infinite dimensional filiform Lie algebra m_2", "abstract": "Denote $\\fm_2$ the infinite dimensional $\\N$-graded Lie algebra defined by the basis $e_i$ for $i\\geq 1$ and by relations $[e_1,e_i]=e_{i+1}$ for all $i\\geq 2$, $[e_2,e_j]=e_{j+2}$ for all $j\\geq 3$. We compute in this article the bracket structure on $H^1(\\fm_2,\\fm_2)$, $H^2(\\fm_2,\\fm_2)$ and in relation to this, we establish that there are only finitely many true deformations of $\\fm_2$ in each weight by constructing them explicitely. It turns out that in weight 0 one gets as non-trivial deformation only one formal non-converging deformation."}
{"category": "Math", "title": "A Review of Accelerated Test Models", "abstract": "Engineers in the manufacturing industries have used accelerated test (AT) experiments for many decades. The purpose of AT experiments is to acquire reliability information quickly. Test units of a material, component, subsystem or entire systems are subjected to higher-than-usual levels of one or more accelerating variables such as temperature or stress. Then the AT results are used to predict life of the units at use conditions. The extrapolation is typically justified (correctly or incorrectly) on the basis of physically motivated models or a combination of empirical model fitting with a sufficient amount of previous experience in testing similar units. The need to extrapolate in both time and the accelerating variables generally necessitates the use of fully parametric models. Statisticians have made important contributions in the development of appropriate stochastic models for AT data [typically a distribution for the response and regression relationships between the parameters of this distribution and the accelerating variable(s)], statistical methods for AT planning (choice of accelerating variable levels and allocation of available test units to those levels) and methods of estimation of suitable reliability metrics. This paper provides a review of many of the AT models that have been used successfully in this area."}
{"category": "Math", "title": "Equilibrium states for potentials with $\\sup\\phi - \\inf\\phi < \\htop(f)$", "abstract": "In the context of smooth interval maps, we study an inducing scheme approach to prove existence and uniqueness of equilibrium states for potentials $\\phi$ with he `bounded range' condition $\\sup \\phi - \\inf \\phi < \\htop$, first used by Hofbauer and Keller. We compare our results to Hofbauer and Keller's use of Perron-Frobenius operators. We demonstrate that this `bounded range' condition on the potential is important even if the potential is H\\\"older continuous. We also prove analyticity of the pressure in this context."}
{"category": "Math", "title": "A Conversation With Harry Martz", "abstract": "Harry F. Martz was born June 16, 1942 and grew up in Cumberland, Maryland. He received a Bachelor of Science degree in mathematics (with a minor in physics) from Frostburg State University in 1964, and earned a Ph.D. in statistics at Virginia Polytechnic Institute and State University in 1968. He started his statistics career at Texas Tech University's Department of Industrial Engineering and Statistics right after graduation. In 1978, he joined the technical staff at Los Alamos National Laboratory (LANL) in Los Alamos, New Mexico after first working as Full Professor in the Department of Industrial Engineering at Utah State University in the fall of 1977. He has had a prolific 23-year career with the statistics group at LANL; over the course of his career, Martz has published over 80 research papers in books and refereed journals, one book (with co-author Ray Waller), and has four patents associated with his work at LANL. He is a fellow of the American Statistical Association and has received numerous awards, including the Technometrics Frank Wilcoxon Prize for Best Applications Paper (1996), Los Alamos National Laboratory Achievement Award (1998), R&D 100 Award by R&D Magazine (2003), Council for Chemical Research Collaboration Success Award (2004), and Los Alamos National Laboratory's Distinguished Licensing Award (2004). Since retiring as a Technical Staff member at LANL in 2001, he has worked as a LANL Laboratory Associate."}
{"category": "Math", "title": "Return time statistics for invariant measures for interval maps with positive Lyapunov exponent", "abstract": "We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around a.e. point. We also show a `polynomial Gibbs property' for these systems, and that the convergence to the entropy in the Ornstein-Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hoelder potentials."}
{"category": "Math", "title": "Propagation of Fluctuations in Biochemical Systems, II: Nonlinear Chains", "abstract": "We consider biochemical reaction chains and investigate how random external fluctuations, as characterized by variance and coefficient of variation, propagate down the chains. We perform such a study under the assumption that the number of molecules is high enough so that the behavior of the concentrations of the system is well approximated by differential equations. We conclude that the variances and coefficients of variation of the fluxes will decrease as one moves down the chain and, through an example, show that there is no corresponding result for the variances of the chemical species. We also prove that the fluctuations of the fluxes as characterized by their time averages decrease down reaction chains. The results presented give insight into how biochemical reaction systems are buffered against external perturbations solely by their underlying graphical structure and point out the benefits of studying the out-of-equilibrium dynamics of systems."}
{"category": "Math", "title": "On Sumsets and Spectral Gaps", "abstract": "It is well known that if S is a subset of the integers mod p, and if the second-largest Fourier coefficient is ``small'' relative to the largest coefficient, then the sumset S+S is much larger than S. We show in the present paper that if instead of having such a large ``spectral gap'' between the largest and second-largest Fourier coefficients, we had it between the kth largest and the (k+1)st largest, the same thing holds true, namely that |S+S| is appreciably larger than |S|. Well, we only do this for k < (log p)/(log 4). We also obtain analogous results for repeated sumsets S+S+...+S, and it turns out that the more terms one includes, the larger the index k that can be used."}
{"category": "Math", "title": "Tree-structured regression and the differentiation of integrals", "abstract": "This paper provides answers to questions regarding the almost sure limiting behavior of rooted, binary tree-structured rules for regression. Examples show that questions raised by Gordon and Olshen in 1984 have negative answers. For these examples of regression functions and sequences of their associated binary tree-structured approximations, for all regression functions except those in a set of the first category, almost sure consistency fails dramatically on events of full probability. One consequence is that almost sure consistency of binary tree-structured rules such as CART requires conditions beyond requiring that (1) the regression function be in ${\\mathcal {L}}^1$, (2) partitions of a Euclidean feature space be into polytopes with sides parallel to coordinate axes, (3) the mesh of the partitions becomes arbitrarily fine almost surely and (4) the empirical learning sample content of each polytope be ``large enough.'' The material in this paper includes the solution to a problem raised by Dudley in discussions. The main results have a corollary regarding the lack of almost sure consistency of certain Bayes-risk consistent rules for classification."}
{"category": "Math", "title": "On the maximum bias functions of MM-estimates and constrained M-estimates of regression", "abstract": "We derive the maximum bias functions of the MM-estimates and the constrained M-estimates or CM-estimates of regression and compare them to the maximum bias functions of the S-estimates and the $\\tau$-estimates of regression. In these comparisons, the CM-estimates tend to exhibit the most favorable bias-robustness properties. Also, under the Gaussian model, it is shown how one can construct a CM-estimate which has a smaller maximum bias function than a given S-estimate, that is, the resulting CM-estimate dominates the S-estimate in terms of maxbias and, at the same time, is considerably more efficient."}
{"category": "Math", "title": "A presentation for the mapping class group of the closed non-orientable surface of genus 4", "abstract": "Finite presentations for the mapping class group M(F) are known for arbitrary orientable compact surface F. If F is non-orientable, then such presentations are known only when F has genus at most 3 and few boundary components. In this paper we obtain finite presentation for the mapping class group of the closed non-orientable surface of genus 4 from its action on the so called ordered complex of curves."}
{"category": "Math", "title": "Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups", "abstract": "We consider the eigenvalue problem and the associated intersection theory of homogenous spaces for the symplectic and odd orthogonal groups. We solve the Horn and saturation problems for these classical groups."}
{"category": "Math", "title": "Algebraic Degeneracy of Non-Archimedean Analytic Maps", "abstract": "We prove non-Archimedean analogs of results of Noguchi and Winkelmann showing algebraic degeneracy of rigid analytic maps to projective varieties omitting an effective divisor with sufficiently many irreducible components relative to the rank of the group they generate in the Neron-Severi group of the variety."}
{"category": "Math", "title": "On the Geometry of the Moduli Space of Real Binary Octics", "abstract": "The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have 0, 1, ..., 4 complex-conjugate pairs of roots respectively. We show that the GIT-stable completion of each of these five components admits the structure of an arithmetic real hyperbolic orbifold. The corresponding monodromy groups are, up to commensurability, discrete hyperbolic reflection groups, and their Vinberg diagrams are computed. We conclude with a simple proof that the moduli space of GIT-stable real binary octics itself cannot be a real hyperbolic orbifold."}
{"category": "Math", "title": "On Emerton's $p$-adic Banach spaces", "abstract": "The purpose of the current paper is to introduce some new methods for studying the $p$-adic Banach spaces introduced by Emerton \\cite{emerton}. We first relate these spaces to more familiar sheaf cohomology groups. As an application, we obtain a more general version of Emerton's spectral sequence. We also calculate the spaces in some easy cases. As a consequence, we obtain a number of vanishing theorems."}
{"category": "Math", "title": "The Rank of the Cartier operator on cyclic covers of the projective line", "abstract": "We give a lower bound on the rank of the Cartier operator of Jacobian varieties of hyperelliptic and superelliptic curves in terms of their genus."}
{"category": "Math", "title": "Snapshot-based Balanced Truncation for Linear Time-periodic Systems", "abstract": "We introduce an algorithm based on a method of snapshots for computing approximate balanced truncations for discrete-time, stable, linear time-periodic systems. By construction, this algorithm is applicable to very high-dimensional systems, even with very high-dimensional outputs (or, alternatively, very high-dimensional inputs). An example is shown to validate the method."}
{"category": "Math", "title": "Avoiding small subgraphs in Achlioptas processes", "abstract": "For a fixed integer r, consider the following random process. At each round, one is presented with r random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected into a graph, which thus grows at the rate of one edge per round. This is a natural generalization of what is known in the literature as an Achlioptas process (the original version has r=2), which has been studied by many researchers, mainly in the context of delaying or accelerating the appearance of the giant component. In this paper, we investigate the small subgraph problem for Achlioptas processes. That is, given a fixed graph H, we study whether there is an online algorithm that substantially delays or accelerates a typical appearance of H, compared to its threshold of appearance in the random graph G(n, M). It is easy to see that one cannot accelerate the appearance of any fixed graph by more than the constant factor r, so we concentrate on the task of avoiding H. We determine thresholds for the avoidance of all cycles C_t, cliques K_t, and complete bipartite graphs K_{t,t}, in every Achlioptas process with parameter r >= 2."}
{"category": "Math", "title": "Finite dimensional Hecke algebras", "abstract": "These are notes prepared for ICRA workshop at Torun, Poland, August 2007. In the first part, we explain results on canonical basic sets by Geck and Jacon and propose a categorification framework which is suitable for our example of Hecke algebras. In the second part, we review basics of Kashiwara crystal and explain the Fock space theory of cyclotomic Hecke algebras and its applications. In the third part, we explain Rouquier's theory of quasihereditary covers of cyclotomic Hecke algebras. We add detailed explanation of the proofs here. The third part is based on my intensive course given at Nagoya university in January 2007."}
{"category": "Math", "title": "An Ozsvath-Szabo Floer homology invariant of knots in a contact manifold", "abstract": "Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston-Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than the three-sphere with negative maximal Thurston-Bennequin invariant. Perhaps more interesting, our invariant provides a criterion for an open book to induce a tight contact structure. A corollary is that if a manifold possesses contact structures with distinct non-vanishing Ozsvath-Szabo invariants, then any fibered knot can realize the classical Eliashberg-Bennequin bound in at most one of these contact structures."}
{"category": "Math", "title": "Stieltjes like functions and inverse problems for systems with Schr\\\"odinger operator", "abstract": "A class of scalar Stieltjes like functions is realized as linear-fractional transformations of transfer functions of conservative systems based on a Schr\\\"odinger operator T_h in $L_2[a,+\\infty)$ with a non-selfadjoint boundary condition. In particular it is shown that any Stieltjes function of this class can be realized in the unique way so that the main operator $\\bA$ of a system is an accretive (*)-extension of a Schr\\\"odinger operator T_h. We derive formulas that restore the system uniquely and allow to find the exact value of a non-real parameter h in the definition of T_h as well as a real parameter $\\mu$ that appears in the construction of the elements of the realizing system. An elaborate investigation of these formulas shows the dynamics of the restored parameters h and $\\mu$ in terms of the changing free term $\\gamma$ from the integral representation of the realizable function. It turns our that the parametric equations for the restored parameter h represent different circles whose centers and radii are determined by the realizable function. Similarly, the behavior of the restored parameter $\\mu$ are described by hyperbolas."}
{"category": "Math", "title": "Asymptotics for sliced average variance estimation", "abstract": "In this paper, we systematically study the consistency of sliced average variance estimation (SAVE). The findings reveal that when the response is continuous, the asymptotic behavior of SAVE is rather different from that of sliced inverse regression (SIR). SIR can achieve $\\sqrt{n}$ consistency even when each slice contains only two data points. However, SAVE cannot be $\\sqrt{n}$ consistent and it even turns out to be not consistent when each slice contains a fixed number of data points that do not depend on n, where n is the sample size. These results theoretically confirm the notion that SAVE is more sensitive to the number of slices than SIR. Taking this into account, a bias correction is recommended in order to allow SAVE to be $\\sqrt{n}$ consistent. In contrast, when the response is discrete and takes finite values, $\\sqrt{n}$ consistency can be achieved. Therefore, an approximation through discretization, which is commonly used in practice, is studied. A simulation study is carried out for the purposes of illustration."}
{"category": "Math", "title": "On the total curvatures of a tame function", "abstract": "Given a definable function f, enough differentiable, we study the continuity of the total curvature function t --> K(t), total curvature of the level {f=t}, and the total absolute curvature function t-->|K| (t), total absolute curvature of the level {f=t}. We show they admits at most finitely many discontinuities."}
{"category": "Math", "title": "Methodology and convergence rates for functional linear regression", "abstract": "In functional linear regression, the slope ``parameter'' is a function. Therefore, in a nonparametric context, it is determined by an infinite number of unknowns. Its estimation involves solving an ill-posed problem and has points of contact with a range of methodologies, including statistical smoothing and deconvolution. The standard approach to estimating the slope function is based explicitly on functional principal components analysis and, consequently, on spectral decomposition in terms of eigenvalues and eigenfunctions. We discuss this approach in detail and show that in certain circumstances, optimal convergence rates are achieved by the PCA technique. An alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view."}
{"category": "Math", "title": "Quantile regression with varying coefficients", "abstract": "Quantile regression provides a framework for modeling statistical quantities of interest other than the conditional mean. The regression methodology is well developed for linear models, but less so for nonparametric models. We consider conditional quantiles with varying coefficients and propose a methodology for their estimation and assessment using polynomial splines. The proposed estimators are easy to compute via standard quantile regression algorithms and a stepwise knot selection algorithm. The proposed Rao-score-type test that assesses the model against a linear model is also easy to implement. We provide asymptotic results on the convergence of the estimators and the null distribution of the test statistic. Empirical results are also provided, including an application of the methodology to forced expiratory volume (FEV) data."}
{"category": "Math", "title": "Asymptotic data analysis on manifolds", "abstract": "Given an m-dimensional compact submanifold $\\mathbf{M}$ of Euclidean space $\\mathbf{R}^s$, the concept of mean location of a distribution, related to mean or expected vector, is generalized to more general $\\mathbf{R}^s$-valued functionals including median location, which is derived from the spatial median. The asymptotic statistical inference for general functionals of distributions on such submanifolds is elaborated. Convergence properties are studied in relation to the behavior of the underlying distributions with respect to the cutlocus. An application is given in the context of independent, but not identically distributed, samples, in particular, to a multisample setup."}
{"category": "Math", "title": "Optimal Subgroups and Applications to Nilpotent Elements", "abstract": "Let G be a reductive group acting on an affine variety X, let x in X be a point whose G-orbit is not closed, and let S be a G-stable closed subvariety of X which meets the closure of the G-orbit of x but does not contain x. In this paper, we study G.R. Kempf's optimal class Omega_G(x,S) of cocharacters of G attached to the point x; in particular, we consider how this optimality transfers to subgroups of G. Suppose K is a G-completely reducible subgroup of G which fixes x, and let H = C_G(K)^0. Our main result says that the H-orbit of x is also not closed, and the optimal class Omega_H(x,S) for H simply consists of the cocharacters in Omega_G(x,S) which evaluate in H. We apply this result in the case that G acts on its Lie algebra via the adjoint representation to obtain some new information about cocharacters associated with nilpotent elements in good characteristic."}
{"category": "Math", "title": "Projective modules over discrete Hodge algebras", "abstract": "Let A be a Noetherian commutative ring. Assume that projective modules of rank r over polynomial extensions of A are extended from A. Then projective modules of rank r over discrete Hodge A-algebras are also extended from A. This result extends a result of T. Vorst."}
{"category": "Math", "title": "Outlier robust corner-preserving methods for reconstructing noisy images", "abstract": "The ability to remove a large amount of noise and the ability to preserve most structure are desirable properties of an image smoother. Unfortunately, they usually seem to be at odds with each other; one can only improve one property at the cost of the other. By combining M-smoothing and least-squares-trimming, the TM-smoother is introduced as a means to unify corner-preserving properties and outlier robustness. To identify edge- and corner-preserving properties, a new theory based on differential geometry is developed. Further, robustness concepts are transferred to image processing. In two examples, the TM-smoother outperforms other corner-preserving smoothers. A software package containing both the TM- and the M-smoother can be downloaded from the Internet."}
{"category": "Math", "title": "Asymptotic local efficiency of Cram\\'{e}r--von Mises tests for multivariate independence", "abstract": "Deheuvels [J. Multivariate Anal. 11 (1981) 102--113] and Genest and R\\'{e}millard [Test 13 (2004) 335--369] have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cram\\'{e}r--von Mises statistics derived from a M\\\"{o}bius decomposition of the empirical copula process. A result on the large-sample behavior of this process under contiguous sequences of alternatives is used here to give a representation of the limiting distribution of such test statistics and to compute their relative local asymptotic efficiency. Local power curves and asymptotic relative efficiencies are compared under familiar classes of copula alternatives."}
{"category": "Math", "title": "Convergence rates of posterior distributions for noniid observations", "abstract": "We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion. We then specialize our results to independent, nonidentically distributed observations, Markov processes, stationary Gaussian time series and the white noise model. We apply our general results to several examples of infinite-dimensional statistical models including nonparametric regression with normal errors, binary regression, Poisson regression, an interval censoring model, Whittle estimation of the spectral density of a time series and a nonlinear autoregressive model."}
{"category": "Math", "title": "Inference for mixtures of symmetric distributions", "abstract": "This article discusses the problem of estimation of parameters in finite mixtures when the mixture components are assumed to be symmetric and to come from the same location family. We refer to these mixtures as semi-parametric because no additional assumptions other than symmetry are made regarding the parametric form of the component distributions. Because the class of symmetric distributions is so broad, identifiability of parameters is a major issue in these mixtures. We develop a notion of identifiability of finite mixture models, which we call k-identifiability, where k denotes the number of components in the mixture. We give sufficient conditions for k-identifiability of location mixtures of symmetric components when k=2 or 3. We propose a novel distance-based method for estimating the (location and mixing) parameters from a k-identifiable model and establish the strong consistency and asymptotic normality of the estimator. In the specific case of L_2-distance, we show that our estimator generalizes the Hodges--Lehmann estimator. We discuss the numerical implementation of these procedures, along with an empirical estimate of the component distribution, in the two-component case. In comparisons with maximum likelihood estimation assuming normal components, our method produces somewhat higher standard error estimates in the case where the components are truly normal, but dramatically outperforms the normal method when the components are heavy-tailed."}
{"category": "Math", "title": "Zeta functions that hear the shape of a Riemann surface", "abstract": "To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose ``Riemannian'' aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson-Sullivan measure. We prove that the ergodic rigidity theorem for this boundary action implies that the zeta functions of the spectral triple suffice to characterize the (anti-)conformal isomorphism class of the corresponding Riemann surface. Thus, you can hear the shape of a Riemann surface, by listening to a suitable spectral triple."}
{"category": "Math", "title": "Nonparametric estimation in a nonlinear cointegration type model", "abstract": "We derive an asymptotic theory of nonparametric estimation for a time series regression model $Z_t=f(X_t)+W_t$, where \\ensuremath\\{X_t\\} and \\ensuremath\\{Z_t\\} are observed nonstationary processes and $\\{W_t\\}$ is an unobserved stationary process. In econometrics, this can be interpreted as a nonlinear cointegration type relationship, but we believe that our results are of wider interest. The class of nonstationary processes allowed for $\\{X_t\\}$ is a subclass of the class of null recurrent Markov chains. This subclass contains random walk, unit root processes and nonlinear processes. We derive the asymptotics of a nonparametric estimate of f(x) under the assumption that $\\{W_t\\}$ is a Markov chain satisfying some mixing conditions. The finite-sample properties of $\\hat{f}(x)$ are studied by means of simulation experiments."}
{"category": "Math", "title": "Nonparametric estimation when data on derivatives are available", "abstract": "We consider settings where data are available on a nonparametric function and various partial derivatives. Such circumstances arise in practice, for example in the joint estimation of cost and input functions in economics. We show that when derivative data are available, local averages can be replaced in certain dimensions by nonlocal averages, thus reducing the nonparametric dimension of the problem. We derive optimal rates of convergence and conditions under which dimension reduction is achieved. Kernel estimators and their properties are analyzed, although other estimators, such as local polynomial, spline and nonparametric least squares, may also be used. Simulations and an application to the estimation of electricity distribution costs are included."}
{"category": "Math", "title": "On representing claims for coherent risk measures", "abstract": "We consider the problem of representing claims for coherent risk measures. For this purpose we introduce the concept of (weak and strong) time-consistency with respect to a portfolio of assets, generalizing the one defined by Delbaen. In a similar way we extend the notion of m-stability, by introducing weak and strong versions. We then prove that the two concepts of m-stability and time-consistency are still equivalent, thus giving necessary and sufficient conditions for a coherent risk measure to be represented by a market with proportional transaction costs. We go on to deduce that, under a separability assumption, any coherent risk measure is strongly time-consistent with respect to a suitably chosen countable portfolio, and show the converse: that any market with proportional transaction costs is equivalent to a market priced by a coherent risk measure, essentially establishing the equivalence of the two concepts."}
{"category": "Math", "title": "Invariants of genus 2 mutants", "abstract": "Pairs of genus 2 mutant knots can have different Homfly polynomials, for example some 3-string satellites of Conway mutant pairs. We give examples which have different Kauffman 3-variable polynomials, answering a question raised by Dunfield et al in their study of genus 2 mutants. While pairs of genus 2 mutant knots have the same Jones polynomial, given from the Homfly polynomial by setting v=s^2, we give examples whose Homfly polynomials differ when v=s^3. We also give examples which differ in a Vassiliev invariant of degree 7, in contrast to satellites of Conway mutant knots."}
{"category": "Math", "title": "Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization", "abstract": "In this paper we discuss an abstract iteration scheme for the calculation of the smallest eigenvalue of an elliptic operator eigenvalue problem. A short and geometric proof based on the preconditioned inverse iteration (PINVIT) for matrices [Knyazev and Neymeyr, (2009)] is extended to the case of operators. We show that convergence is retained up to any tolerance if one only uses approximate applications of operators which leads to the perturbed preconditioned inverse iteration (PPINVIT). We then analyze the Besov regularity of the eigenfunctions of the Poisson eigenvalue problem on a polygonal domain, showing the advantage of an adaptive solver to uniform refinement when using a stable wavelet base. A numerical example for PPINVIT, applied to the model problem on the L-shaped domain, is shown to reproduce the predicted behaviour."}
{"category": "Math", "title": "Hazard models with varying coefficients for multivariate failure time data", "abstract": "Statistical estimation and inference for marginal hazard models with varying coefficients for multivariate failure time data are important subjects in survival analysis. A local pseudo-partial likelihood procedure is proposed for estimating the unknown coefficient functions. A weighted average estimator is also proposed in an attempt to improve the efficiency of the estimator. The consistency and asymptotic normality of the proposed estimators are established and standard error formulas for the estimated coefficients are derived and empirically tested. To reduce the computational burden of the maximum local pseudo-partial likelihood estimator, a simple and useful one-step estimator is proposed. Statistical properties of the one-step estimator are established and simulation studies are conducted to compare the performance of the one-step estimator to that of the maximum local pseudo-partial likelihood estimator. The results show that the one-step estimator can save computational cost without compromising performance both asymptotically and empirically and that an optimal weighted average estimator is more efficient than the maximum local pseudo-partial likelihood estimator. A data set from the Busselton Population Health Surveys is analyzed to illustrate our proposed methodology."}
{"category": "Math", "title": "Euler equations are not exactly controllable by a finite-dimensional external force", "abstract": "We show that the Euler system is not exactly controllable by a finite-dimensional external force. The proof is based on the comparison of the Kolmogorov epsilon-entropy for Holder spaces and for the class of functions that can be obtained by solving the 2D Euler equations with various right-hand sides."}
{"category": "Math", "title": "The supremum of conformally covariant eigenvalues in a conformal class", "abstract": "Let (M,g) be a compact Riemannian manifold of dimension >2. We show that there is a metric h conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with repect to h is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension >1."}
{"category": "Math", "title": "Discrete sums of classical symbols on Z^d and zeta functions associated with Laplacians on tori", "abstract": "We prove the uniqueness of a translation invariant extension to non integer order classical symbols of the ordinary discrete sum on $L^1$-symbols, which we then describe using an Hadamard finite part procedure for sums over integer points of infinite unions of nested convex polytopes in $\\R^d$. This canonical regularised sum is the building block to construct meromorphic extensions of the ordinary sum on holomorphic symbols. Explicit formulae for the complex residues at their poles are given in terms of noncommutative residues of classical symbols, thus extending results of Guillemin, Sternberg and Weitsman. These formulae are then applied to zeta functions associated with quadratic forms and with Laplacians on tori."}
{"category": "Math", "title": "G-linear sets and torsion points in definably compact groups", "abstract": "Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G X) < dim G for some definable X subset of G then X contains a torsion point of G. Along the way we develop a general theory for so-called G-linear sets, and investigate definable sets which contain abstract subgroups of G."}
{"category": "Math", "title": "Stable marked point processes", "abstract": "In many contexts such as queuing theory, spatial statistics, geostatistics and meteorology, data are observed at irregular spatial positions. One model of this situation involves considering the observation points as generated by a Poisson process. Under this assumption, we study the limit behavior of the partial sums of the marked point process $\\{(t_i,X(t_i))\\}$, where X(t) is a stationary random field and the points t_i are generated from an independent Poisson random measure $\\mathbb{N}$ on $\\mathbb{R}^d$. We define the sample mean and sample variance statistics and determine their joint asymptotic behavior in a heavy-tailed setting, thus extending some finite variance results of Karr [Adv. in Appl. Probab. 18 (1986) 406--422]. New results on subsampling in the context of a marked point process are also presented, with the application of forming a confidence interval for the unknown mean under an unknown degree of heavy tails."}
{"category": "Math", "title": "Lower bounds for Hilbert-Kunz multiplicities in local rings of fixed dimension", "abstract": "Let $(R,\\m)$ be a formally unmixed local ring of positive prime characteristic and dimension $d$. We examine the implications of having small Hilbert-Kunz multiplicity (i.e., close to 1). In particular, we show that if $R$ is not regular, there exists a lower bound, strictly greater than one, depending only on $d$, for its Hilbert-Kunz multiplicity."}
{"category": "Math", "title": "An invariant for flat virtual knots", "abstract": "The paper has been withdrawn by the author, due to a critical error stemming from the defined template."}
{"category": "Math", "title": "Unstable surface waves in running water", "abstract": "We consider the stability of periodic gravity free-surface water waves traveling downstream at a constant speed over a shear flow of finite depth. In case the free surface is flat, a sharp criterion of linear instability is established for a general class of shear flows with inflection points and the maximal unstable wave number is found. Comparison to the rigid-wall setting testifies that free surface has a destabilizing effect. For a class of unstable shear flows, the bifurcation of nontrivial periodic traveling waves of small-amplitude is demonstrated at any wave number. We show the linear instability of small nontrivial waves bifurcated at an unstable wave number of the background shear flow. The proof uses a new formulation of the linearized water-wave problem and a perturbation argument. An example of the background shear flow of unstable small-amplitude periodic traveling waves is constructed for an arbitrary vorticity strength and for an arbitrary depth, illustrating that vorticity has a subtle influence on the stability of water waves."}
{"category": "Math", "title": "The Laplacian on hyperbolic 3-manifolds with Dehn surgery type singularities", "abstract": "We study the spectrum of the Laplacian on hyperbolic 3-manifolds with Dehn surgery type singularities and its dependence on the generalized Dehn surgery coefficients."}
{"category": "Math", "title": "The Frobenius Structure of Local Cohomology", "abstract": "Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an anti-nilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the Ascending Chain Condition."}
{"category": "Math", "title": "Edge Flows in the Complete Random-Lengths Network", "abstract": "Consider the complete n-vertex graph whose edge-lengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some random total flow. In the $n \\to \\infty$ limit we find explicitly the empirical distribution of these edge-flows, suitably normalized."}
{"category": "Math", "title": "Multiplication operators on L(L_p) and $\\ell_p$-strictly singular operators", "abstract": "A classification of weakly compact multiplication operators on L(L_p), $1<p<\\infty$, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of $\\ell_p$-strictly singular operators, and we also investigate the structure of general $\\ell_p$-strictly singular operators on L_p. The main result is that if an operator T on L_p, 1<p<2, is $\\ell_p$-strictly singular and T_{|X} is an isomorphism for some subspace X of L_p, then X embeds into L_r for all r<2, but X need not be isomorphic to a Hilbert space. It is also shown that if T is convolution by a biased coin on L_p of the Cantor group, $1\\le p <2$, and $T_{|X}$ is an isomorphism for some reflexive subspace X of L_p, then X is isomorphic to a Hilbert space. The case p=1 answers a question asked by Rosenthal in 1976."}
{"category": "Math", "title": "Probabilistic implications of symmetries of q-Hermite and Al-Salam-Chihara polynomials", "abstract": "We prove the existence of stationary random fields with linear regressions for $q>1$ and thus close an open question posed by W. Bryc et al.. We prove this result by describing a discrete 1 dimensional conditional distribution and then checking Chapman-Kolmogorov equation. Support of this distribution consist of zeros of certain Al-Salam-Chihara polynomials. To find them we refer to and expose known result concerning addition of $q-$ exponential function. This leads to generalization of a well known formula $(x+y)^{n}% =\\sum_{i=0}^{n}\\binom{n}{k}i^{k}H_{n-k}(x) H_{k}(-iy) ,$ where $H_{k}(x) $ denotes $k-$th Hermite polynomial."}
{"category": "Math", "title": "Recursion formulae of higher Weil-Petersson volumes", "abstract": "In this paper we study effective recursion formulae for computing intersection numbers of mixed $\\psi$ and $\\kappa$ classes on moduli spaces of curves. By using the celebrated Witten-Kontsevich theorem, we generalize Mulase-Safnuk form of Mirzakhani's recursion and prove a recursion formula of higher Weil-Petersson volumes. We also present recursion formulae to compute intersection pairings in the tautological rings of moduli spaces of curves."}
{"category": "Math", "title": "Extended powers and Steenrod operations in algebraic geometry", "abstract": "Steenrod operations have been defined by Voedvodsky in motivic cohomology in order to show the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. We adapt the methods used by Bisson-Joyal in studying Steenrod and Dyer-Lashof operations in unoriented cobordism and mod 2 cohomology."}
{"category": "Math", "title": "Second Order Cumulants of products", "abstract": "We derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors. This extends to the second order case the formula of Krawczyk and Speicher. We apply our result to the problem of calculating the second order cumulants of a semi-circular and Haar unitary operator."}
{"category": "Math", "title": "The Coherence Theorem for Ann-Categories", "abstract": "This paper presents the proof of the coherence theorem for Ann-categories whose set of axioms and original basic properties were given in [9]. Let $$\\A=(\\A,{\\Ah},c,(0,g,d),a,(1,l,r),{\\Lh},{\\Rh})$$ be an Ann-category. The coherence theorem states that in the category $ \\A$, any morphism built from the above isomorphisms and the identification by composition and the two operations $\\tx$, $\\ts$ only depends on its source and its target. The first coherence theorems were built for monoidal and symmetric monoidal categories by Mac Lane [7]. After that, as shown in the References, there are many results relating to the coherence problem for certain classes of categories. For Ann-categories, applying Hoang Xuan Sinh's ideas used for Gr-categories in [2], the proof of the coherence theorem is constructed by faithfully ``embedding'' each arbitrary Ann-category into a quite strict Ann-category. Here, a {\\it quite strict} Ann-categogy is an Ann-category whose all constraints are strict, except for the commutativity and left distributivity ones. This paper is the work continuing from [9]. If there is no explanation, the terminologies and notations in this paper mean as in [9]."}
{"category": "Math", "title": "New Dirichlet Mean Identities", "abstract": "An important line of research is the investigation of the laws of random variables known as Dirichlet means as discussed in Cifarelli and Regazzini(1990). However there is not much information on inter-relationships between different Dirichlet means. Here we introduce two distributional operations, which consist of multiplying a mean functional by an independent beta random variable and an operation involving an exponential change of measure. These operations identify relationships between different means and their densities. This allows one to use the often considerable analytic work to obtain results for one Dirichlet mean to obtain results for an entire family of otherwise seemingly unrelated Dirichlet means. Additionally, it allows one to obtain explicit densities for the related class of random variables that have generalized gamma convolution distributions, and the finite-dimensional distribution of their associated L\\'evy processes. This has implications in, for instance, the explicit description of Bayesian nonparametric prior and posterior models, and more generally in a variety of applications in probability and statistics involving Levy processes."}
{"category": "Math", "title": "Lamperti-type laws", "abstract": "This paper explores various distributional aspects of random variables defined as the ratio of two independent positive random variables where one variable has an $\\alpha$-stable law, for $0<\\alpha<1$, and the other variable has the law defined by polynomially tilting the density of an $\\alpha$-stable random variable by a factor $\\theta>-\\alpha$. When $\\theta=0$, these variables equate with the ratio investigated by Lamperti [Trans. Amer. Math. Soc. 88 (1958) 380--387] which, remarkably, was shown to have a simple density. This variable arises in a variety of areas and gains importance from a close connection to the stable laws. This rationale, and connection to the $\\operatorname {PD}(\\alpha,\\theta)$ distribution, motivates the investigations of its generalizations which we refer to as Lamperti-type laws. We identify and exploit links to random variables that commonly appear in a variety of applications. Namely Linnik, generalized Pareto and $z$-distributions. In each case we obtain new results that are of potential interest. As some highlights, we then use these results to (i) obtain integral representations and other identities for a class of generalized Mittag--Leffler functions, (ii) identify explicitly the L\\'{e}vy density of the semigroup of stable continuous state branching processes (CSBP) and hence corresponding limiting distributions derived in Slack and in Zolotarev [Z. Wahrsch. Verw. Gebiete 9 (1968) 139--145, Teor. Veroyatn. Primen. 2 (1957) 256--266], which are related to the recent work by Berestycki, Berestycki and Schweinsberg, and Bertoin and LeGall [Ann. Inst. H. Poincar\\'{e} Probab. Statist. 44 (2008) 214--238, Illinois J. Math. 50 (2006) 147--181] on beta coalescents. (iii) We obtain explicit results for the occupation time of generalized Bessel bridges and some interesting stochastic equations for $\\operatorname {PD}(\\alpha,\\theta)$-bridges. In particular we obtain the best known results for the density of the time spent positive of a Bessel bridge of dimension $2-2\\alpha$."}
{"category": "Math", "title": "Gibbs Partitions (EPPF's) Derived From a Stable Subordinator are Fox H and Meijer G Transforms", "abstract": "This paper derives explicit results for the infinite Gibbs partitions generated by the jumps of an $\\alpha-$stable subordinator, derived in Pitman \\cite{Pit02, Pit06}. We first show that for general $\\alpha$ the conditional EPPF can be represented as ratios of Fox-$H$ functions, and in the case of rational $\\alpha,$ Meijer-G functions. Furthermore the results show that the resulting unconditional EPPF's, can be expressed in terms of H and G transforms indexed by a function h. Hence when h is itself a H or G function the EPPF is also an H or G function. An implication, in the case of rational $\\alpha,$ is that one can compute explicitly thousands of EPPF's derived from possibly exotic special functions. This would also apply to all $\\alpha$ except that computations for general Fox functions are not yet available. However, moving away from special functions, we demonstrate how results from probability theory may be used to obtain calculations. We show that a forward recursion can be applied that only requires calculation of the simplest components. Additionally we identify general classes of EPPF's where explicit calculations can be carried out using distribution theory."}
{"category": "Math", "title": "Lifting central invariants of quantized Hamiltonian actions", "abstract": "Let G be a connected reductive group over an algebraically closed field K of characteristic 0, X an affine symplectic variety equipped with a Hamiltonian action of G. Further, let * be a G-invariant Fedosov star-product on X such that the Hamiltonian action is quantized. We establish an isomorphism between the center of the associative algebra K[X][[h]]^G and the algebra of formal power series with coefficients in the Poisson center of K[X]^G."}
{"category": "Math", "title": "Universal representations of braid and braid-permutation groups", "abstract": "Drinfel'd used associators to construct families of universal representations of braid groups. We consider semi-associators (i.e., we drop the pentagonal axiom and impose a normalization in degree one). We show that the process may be reversed, to obtain semi-associators from universal representations of 3-braids. We view braid groups as subgroups of braid-permutation groups. We construct a family of universal representations of braid-permutation groups, without using associators. All representations in the family are faithful, defined over $\\bbQ$ by simple explicit formulae. We show that they give universal Vassiliev-type invariants for braid-permutation groups."}
{"category": "Math", "title": "A Schwarz lemma for a domain related to mu-synthesis", "abstract": "We prove a Schwarz lemma for a domain E in 3-dimensional complex space that arises in connection with a problem in H infinity control theory. We describe a class of automorphisms of E and determine the distinguished boundary of E. We obtain a type of Schwarz-Pick lemma for a two by two mu-synthesis problem."}
{"category": "Math", "title": "Quenched Limits for Transient, Ballistic, Sub-Gaussian One-Dimensional Random Walk in Random Environment", "abstract": "We consider a nearest-neighbor, one-dimensional random walk $\\{X_n\\}_{n\\geq 0}$ in a random i.i.d. environment, in the regime where the walk is transient with speed v_P > 0 and there exists an $s\\in(1,2)$ such that the annealed law of $n^{-1/s} (X_n - n v_P)$ converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {t_k} and {t_k'} depending on the environment only, such that a quenched central limit theorem holds along the subsequence t_k, but the quenched limiting distribution along the subsequence t_k' is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:0704.1778v1 [math.PR]) which handled the case when the parameter $s\\in(0,1)$."}
{"category": "Math", "title": "A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments", "abstract": "Let $f:[0,1)^d \\to {\\mathbb R}$ be an integrable function. An objective of many computer experiments is to estimate $\\int_{[0,1)^d} f(x) dx$ by evaluating f at a finite number of points in [0,1)^d. There is a design issue in the choice of these points and a popular choice is via the use of randomized orthogonal arrays. This article proves a multivariate central limit theorem for a class of randomized orthogonal array sampling designs [Owen (1992a)] as well as for a class of OA-based Latin hypercubes [Tang (1993)]."}
{"category": "Math", "title": "Exit paths and constructible stacks", "abstract": "For a Whitney stratification S of a space X (or more generally a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category $EP_{\\leq 2}(X,S)$, called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors from $EP_{\\leq 2}(X,S)$ to the 2-category of small categories."}
{"category": "Math", "title": "The automorphism group of the tetrablock", "abstract": "The tetrablock is a domain in 3-dimensional complex space that meets 3-dimensional Euclidean space in a regular tetrahedron. It is shown to be inhomogeneous and its automorphism group is determined. A type of Schwarz lemma for the tetrablock is proved. The action of the automorphism group is described in terms of a certain natural foliation of the tetrablock by complex geodesic discs."}
{"category": "Math", "title": "Asymptotics for rank partition functions", "abstract": "In this paper, we obtain asymptotic formulas for an infinite class of rank generating functions. As an application, we solve a conjecture of Andrews and Lewis on inequalities between certain ranks."}
{"category": "Math", "title": "Dyson's Rank, overpartitions, and weak Maass forms", "abstract": "In a series of papers the first author and Ono connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms. Naturally it is of wide interest to find other explicit examples of Maass forms. Here we construct a new infinite family of such forms, arising from overpartitions. As applications we obtain combinatorial decompositions of Ramanujan-type congruences for overpartitions as well as the modularity of rank differences in certain arithmetic progressions."}
{"category": "Math", "title": "Regularity of some class of nonlinear transformations", "abstract": "In this paper we consider quadratic stochastic operators designed on finite Abelian groups. It is proved that such operators have the property of regularity."}
{"category": "Math", "title": "On Colorings of Graph Powers", "abstract": "In this paper, some results concerning the colorings of graph powers are presented. The notion of helical graphs is introduced. We show that such graphs are hom-universal with respect to high odd-girth graphs whose $(2t+1)$st power is bounded by a Kneser graph. Also, we consider the problem of existence of homomorphism to odd cycles. We prove that such homomorphism to a $(2k+1)$-cycle exists if and only if the chromatic number of the $(2k+1)$st power of $S_2(G)$ is less than or equal to 3, where $S_2(G)$ is the 2-subdivision of $G$. We also consider Ne\\v{s}et\\v{r}il's Pentagon problem. This problem is about the existence of high girth cubic graphs which are not homomorphic to the cycle of size five. Several problems which are closely related to Ne\\v{s}et\\v{r}il's problem are introduced and their relations are presented."}
{"category": "Math", "title": "Convergence of adaptive mixtures of importance sampling schemes", "abstract": "In the design of efficient simulation algorithms, one is often beset with a poor choice of proposal distributions. Although the performance of a given simulation kernel can clarify a posteriori how adequate this kernel is for the problem at hand, a permanent on-line modification of kernels causes concerns about the validity of the resulting algorithm. While the issue is most often intractable for MCMC algorithms, the equivalent version for importance sampling algorithms can be validated quite precisely. We derive sufficient convergence conditions for adaptive mixtures of population Monte Carlo algorithms and show that Rao--Blackwellized versions asymptotically achieve an optimum in terms of a Kullback divergence criterion, while more rudimentary versions do not benefit from repeated updating."}
{"category": "Math", "title": "A Strict Inequality for a Minimal Degree of a Direct Product", "abstract": "The minimal faithful permutation degree of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright established conditions for when the minimal degree of a direct product equals the sum of the minimal degrees for two finite groups. Wright asked whether this is true for all finite groups. A counter- example of degree 15 was provided by the referee and was added as an addendum in a paper of Wright. Here we provide a counter-example of degree 12."}
{"category": "Math", "title": "Step-up simultaneous tests for identifying active effects in orthogonal saturated designs", "abstract": "A sequence of null hypotheses regarding the number of negligible effects (zero effects) in orthogonal saturated designs is formulated. Two step-up simultaneous testing procedures are proposed to identify active effects (nonzero effects) under the commonly used assumption of effect sparsity. It is shown that each procedure controls the experimentwise error rate at a given $\\alpha$ level in the strong sense."}
{"category": "Math", "title": "Hopf Bifurcation in a Model for Biological Control", "abstract": "In this paper we study the Lyapunov stability and Hopf bifurcation in a biological system which models the biological control of parasites of orange plantations."}
{"category": "Math", "title": "Some calculations on type II_1 unprojection", "abstract": "The type II_1 unprojection is, by definition, the generic complete intersection type II unprojection, in the sense of [Papadakis, Type II unprojection, J. Algebraic Geometry, 15 (2006) 399--414] Section 3.1, for the parameter value k = 1, and depends on a parameter n greater or equal than 2. Our main results are the explicit calculation of the linear relations of the type II_1 unprojection for any value of n greater or equal than 2 (Theorem 3.16) and the explicit calculation of the quadratic equation for the case n = 3 (Theorem 4.1). In addition, Section 5 contains applications to algebraic geometry while Section 6 contains the Macaulay 2 code for the type II_1 unprojection for the parameter value n = 3."}
{"category": "Math", "title": "Constant T-curvature conformal metrics on 4-manifolds with boundary", "abstract": "In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary there exists a metric conformal to g with constant T-curvature, zero Q-curvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to solving a fourth order nonlinear elliptic boundary value problem (BVP) with boundary conditions given by a third-order pseudodifferential operator, and homogeneous Neumann one. It has a variational structure, but since the corresponding Euler-Lagrange functional is in general unbounded from below, we look for saddle points. In order to do this, we use topological arguments and min-max methods combined with a compactness result for the corresponding BVP."}
{"category": "Math", "title": "Overview document for: A weight function theory of basis function interpolants and smoothers", "abstract": "This document is a brief overview of two documents which continue to develop the weight function theory of basis function smoothers and interpolants. One document considers the zero order theory and one considers the positive order theory."}
{"category": "Math", "title": "$\\infty$-jets of difeomorphisms preserving orbits of vector fields", "abstract": "Let $F$ be a smooth vector field defined in a neighborhood of the origin in $\\mathbb{R}^n$, $F(O)=0$, and let $F_t$ be its local flow. Denote by $E$ the set of germs of diffeomorphisms $h:\\mathbb{R}^n \\to \\mathbb{R}^n$ preserving orbits of $F$ and let $E_{\\mathrm{id}}^r$ be the identity component of $E$ with respect to $C^r$-topology. Then every $E_{\\mathrm{id}}^{r}$ contains a subset $Sh$ consisting of mappings of the form $F_{f(x)}(x)$, where $f: \\mathbb{R}^n \\to \\mathbb{R}$ is a smooth function. It was proved earlier by the author that if $F$ is a linear vector field, then $Sh=E_{\\mathrm{id}}^0$. In this paper we present a class of vector fields for which $Sh$ and $E_{\\mathrm{id}}^1$ coincide on the level of $\\infty$-jets. We also establish a parameter rigidity of linear vector fields and \"reduced\" Hamiltonian vector fields of real homogeneous polynomials in two variables."}
{"category": "Math", "title": "Prime ideals in the quantum grassmannian", "abstract": "We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a torus H that acts naturally on the quantum grassmannian and the cell decomposition of the set of H-primes leads to a parameterisation of the H-spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the non-negative cells in recent studies concerning the totally non-negative grassmannian. Finally, we use the cell decomposition to establish that the quantum grassmannian satisfies normal separation and catenarity."}
{"category": "Math", "title": "The decomposition of the hypermetric cone into L-domains", "abstract": "The hypermetric cone $\\HYP_{n+1}$ is the parameter space of basic Delaunay polytopes in n-dimensional lattice. The cone $\\HYP_{n+1}$ is polyhedral; one way of seeing this is that modulo image by the covariance map $\\HYP_{n+1}$ is a finite union of L-domains, i.e., of parameter space of full Delaunay tessellations. In this paper, we study this partition of the hypermetric cone into L-domains. In particular, it is proved that the cone $\\HYP_{n+1}$ of hypermetrics on n+1 points contains exactly {1/2}n! principal L-domains. We give a detailed description of the decomposition of $\\HYP_{n+1}$ for n=2,3,4 and a computer result for n=5 (see Table \\ref{TableDataHYPn}). Remarkable properties of the root system $\\mathsf{D}_4$ are key for the decomposition of $\\HYP_5$."}
{"category": "Math", "title": "Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption", "abstract": "We study the limit behaviour of solutions of a class of solutions of nonlinear parabolic equations with a degenerate strong absorption. We prove that two types of phenomena can occur: the pointwise singularity or the formation of razor blades (or persistent singularities)."}
{"category": "Math", "title": "Separable solutions of some quasilinear equations with source reaction", "abstract": "We study the existence of singular separable solutions to a class of quasilinear equations with reaction term. In the 2-dim case, we use a dynamical system approach to construct our solutions."}
{"category": "Math", "title": "A Subgroup of a Direct Product of Free Groups whose Dehn Function has a Cubic Lower Bound", "abstract": "We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of 3 free groups."}
{"category": "Math", "title": "Invariant rigid geometric structures and smooth projective factors", "abstract": "We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show the following. 1) Either the manifold admits a smooth equivariant map onto a homogeneous projective variety, defined on an open dense conull invariant set, or the Lie algebra of the Zariski closure of the Gromov representation of the fundamental group contains a Lie subalgebra isomorphic to the Lie algebra of the acting group. As a corollary, a smooth non-trivial homogeneous projective factor does exist whenever the fundamental group of $M$ admits only virtually solvable linear representations, and thus in particular when $M$ is simply connected, regardless of the real rank. 2) There exist explicit examples showing that analytic rigid actions of certain simple groups (of real rank one) may indeed fail to have a smooth projective factor. 3) It is possible to generalize Gromov's theorem on the algebraic hull of the representation of the fundamental group of the manifold to the case of analytic rigid non-unimodular structures, for actions of simple groups of any real rank. An important ingredient in the proofs is a generalization of Gromov's centralizer theorem beyond the case of invariant measures."}
{"category": "Math", "title": "Singular integral operators on variable Lebesgue spaces with radial oscillating weights", "abstract": "We prove a Fredholm criterion for operators in the Banach algebra of singular integral operators with matrix piecewise continuous coefficients acting on a variable Lebesgue space with a radial oscillating weight over a logarithmic Carleson curve. The local spectra of these operators are massive and have a shape of spiralic horns depending on the value of the variable exponent, the spirality indices of the curve, and the Matuszewska-Orlicz indices of the weight at each point. These results extend (partially) the results of A. B\\\"ottcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spaces to the case of variable Lebesgue spaces."}
{"category": "Math", "title": "A weight function theory of zero order basis function interpolants and smoothers", "abstract": "I develop a weight func theory of zero order basis func interpolants and smoothers.**Ch1 Basis funcs and data spaces are defined using wt funcs. Data (native)spaces are used to formulate the variational problems which define our interpolants /smoothers. Introduce the tensor prod extended B-splines.**Ch2 Prove the p'twise convergence of the minimal norm basis func interpol to its data func and obtain orders of converg. Data func spaces for the B-splines are locally Sobolev spaces.**Ch3 Another set of error estims for basis func interpol. Use distrib'n Taylor expansion of exp(i(a,x)).**Ch4 Derive local interpol errors for data funcs with bounded first derivs.**Ch5 Introduce class of tensor prod wt funcs which I call the central diff wt funcs - related to the B-splines. Apply theory to these wt funcs to obtain interpol converge results. The data func spaces are locally Sobolev spaces. **Ch6 A non-param variat smoothing problem studied with special interest in converge of smoother to its data func. This smoother is the min norm interpol stabilized by a smoothing coeff.**Ch7: A non-parametric, scalable, smoothing problem shown to converge to its data func. We discuss the software which implements these algorithms.**Ch8: Characterizes bounded linear functionals on data space.**Ch9 Bilinear form used to characterize the bounded linear functionals on the data spaces generated by the wt funcs.**Ch10 We derive an upper bound for the deriv of the 1-dim (scaled) hat basis func smoother assuming the data func has bounded derivs and large supp wrt. the data region.**Ch11: In one dim only; the local data funcs are assumed to have bounded derivs on the data region and consider a scaled hat basis func. If the basis func has large enough supp wrt. the data region then we show the order of converg of the interpol is 1. **Ch12: Exten ops derived based on Wloka; assumes rectangle condit."}
{"category": "Math", "title": "A new modified Galerkin method for the two-dimensional Navier-Stokes equations", "abstract": "We present a new type of modified Galerkin method. It is a construction with several (inductively defined) levels, that provides approximate solutions of increasing accuracy with every new level. These solutions are constructed as approximations of the so called induced trajectories (notion on which the definition of a class of approximate inertial manifolds used in the nonlinear and postprocessed Galerkin methods is based)."}
{"category": "Math", "title": "A weight function theory of positive order basis function interpolants and smoothers", "abstract": "In this document I develop a weight function theory of positive order basis function interpolants and smoothers. **In Chapter 1 the basis functions and data spaces are defined directly using weight functions. The data spaces are used to formulate the variational problems which define the interpolants and smoothers discussed in later chapters. The theory is illustrated using some standard examples of radial basis functions and two classes of weight functions I will call the tensor product extended B-splines and the central difference weight functions. **In Chapter 2 I derive modified inverse-Fourier transform formulas for the basis functions and the data functions (native spaces) and to use these formulas to obtain bounds for the rates of increase of these functions and their derivatives near infinity. **Chapter 3 shows how to prove functions are basis functions without using the awkward space of test functions $S_{0,n}$ which are infinitely smooth functions of rapid decrease with several zero-valued derivatives at the origin. Worked examples include several classes of well-known radial basis functions. **In Chapter 4 we prove the existence and uniqueness of a solution to the minimal seminorm interpolation problem. We then derive orders for the pointwise convergence of the interpolant to its data function as the density of the data increases. **In Chapter 5 a well-known non-parametric variational smoothing problem will be studied with special interest in the order of pointwise convergence of the smoother to its data function. This smoothing problem is the minimal norm interpolation problem stabilized by a smoothing coefficient. **In Chapter 6 a non-parametric, scalable, variational smoothing problem will be studied, with special interest in its order of pointwise convergence to its data function. **In Chapter 7 we study the bounded linear functionals on the data spaces."}
{"category": "Math", "title": "Cluster Complexes via Semi-Invariants", "abstract": "We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semi-invariants and the Tilting Triangulation of the (n-1)-sphere."}
{"category": "Math", "title": "Some pseudo-Anosov maps on punctured Riemann surfaces represented by multi-twists", "abstract": "This paper has been withdrawn by the author"}
{"category": "Math", "title": "Compositional Bernoulli numbers", "abstract": "We define and study the combinatorial properties of compositional Bernoulli numbers and polynomials within the framework of rational combinatorics."}
{"category": "Math", "title": "Values at s=-1 of L-functions for multi-quadratic extensions of number fields, and the fitting ideal of the tame kernel", "abstract": "Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E lying above those in S, and let O_E^S denote the ring of S_E-integers of E. We then compare the Fitting ideal of K_2(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption of the Birch-Tate conjecture, especially for biquadratic extensions, where we compute the index of the higher Stickelberger ideal. We find a sufficient condition for the Fitting ideal to contain the higher Stickelberger ideal in the case where E is a biquadratic extension of F containing the first layer of the cyclotomic Z_2-extension of F, and describe a class of biquadratic extensions of F=Q that satisfy this condition."}
{"category": "Math", "title": "Existence of quasi-arcs", "abstract": "We show that doubling, linearly connected metric spaces are quasi-arc connected. This gives a new and short proof of a theorem of Tukia."}
{"category": "Math", "title": "A separable non-remainder of H", "abstract": "We prove that there is a compact separable continuum that (consistently) is not a remainder of the real line."}
{"category": "Math", "title": "Triangularization of a Jordan Algebra of Schatten Operators", "abstract": "We show that a Jordan algebra of compact quasinilpotent operators which contains a nonzero trace class operator has a common invariant subspace. As a consequence of this result, we obtain that a Jordan algebra of quasinilpotent Schatten operators is simultaneously triangularizable."}
{"category": "Math", "title": "The Crepant Resolution Conjecture for 3-dimensional flags modulo an involution", "abstract": "After fixing a non-degenerate bilinear form on a vector space V we define an involution of the manifold of flags F in V by taking a flag to its orthogonal complement. When V is of dimension 3 we check that the Crepant Resolution Conjecture of J. Bryan and T. Graber holds: the genus zero (orbifold) Gromov-Witten potential function of [F / Z_2] agrees (up to unstable terms) with the genus zero Gromov-Witten potential function of a crepant resolution Y of the quotient scheme F / Z_2, after setting a quantum parameter to -1, making a linear change of variables, and analytically continuing coefficients. The crepant resolution Y (a hypersurface in the Hilbert scheme Hilb^2 P^2) is the projectivization of a novel rank 2 vector bundle over P^2."}
{"category": "Math", "title": "The mass-critical nonlinear Schr\\\"odinger equation with radial data in dimensions three and higher", "abstract": "We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\\\"odinger equation $iu_t + \\Delta u = \\pm |u|^{4/d} u$ for large spherically symmetric L^2_x(R^d) initial data in dimensions $d\\geq 3$. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time."}
{"category": "Math", "title": "Embeddings of discrete groups and the speed of random walks", "abstract": "For a finitely generated group G and a banach space X let \\alpha^*_X(G) (respectively \\alpha^#_X(G)) be the supremum over all \\alpha\\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G\\to X and c>0 such that for all x,y\\in G we have \\|f(x)-f(y)\\|\\ge c\\cdot d_G(x,y)^\\alpha. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is \\alpha^*(G)=\\alpha^*_{L_2}(G) (respectively \\alpha^#(G)= \\alpha_{L_2}^#(G)). We show that if X has modulus of smoothness of power type p, then \\alpha^#_X(G)\\le \\frac{1}{p\\beta^*(G)}. Here \\beta^*(G) is the largest \\beta\\ge 0 for which there exists a set of generators S of G and c>0 such that for all t\\in \\N we have \\E\\big[d_G(W_t,e)\\big]\\ge ct^\\beta, where \\{W_t\\}_{t=0}^\\infty is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X=L_p, generalizes a theorem of Guentner and Kaminker and answers a question posed by Tessera. We also show that if \\alpha^*(G)\\ge 1/2 then \\alpha^*(G\\bwr \\Z)\\ge \\frac{2\\alpha^*(G)}{2\\alpha^*(G)+1}. This improves the previous bound due to Stalder and ValetteWe deduce that if we write \\Z_{(1)}= \\Z and \\Z_{(k+1)}\\coloneqq \\Z_{(k)}\\bwr \\Z then \\alpha^*(\\Z_{(k)})=\\frac{1}{2-2^{1-k}}, and use this result to answer a question posed by Tessera in on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C_2\\bwr C_n embed into L_1 with uniformly bounded distortion, answering a question posed by Lee, Naor and Peres. Finally, we use these results to show that edge Markov type need not imply Enflo type."}
{"category": "Math", "title": "On the point spectrum of some perturbed differential operators with periodic coefficients", "abstract": "Finiteness of the point spectrum of linear operators acting in a Banach space is investigated from point of view of perturbation theory. In the first part of the paper we present an abstract result based on analytical continuation of the resolvent function through continuous spectrum. In the second part, the abstract result is applied to differential operators which can be represented as a differential operator with periodic coefficients perturbed by an arbitrary subordinated differential operator."}
{"category": "Math", "title": "The effect of memory on functional large deviations of infinite moving average processes", "abstract": "The large deviations of an infinite moving average process with exponentially light tails are very similar to those of an i.i.d. sequence as long as the coefficients decay fast enough. If they do not, the large deviations change dramatically. We study this phenomenon in the context of functional large, moderate and huge deviation principles."}
{"category": "Math", "title": "Asymptotic Curvature Decay and Removal of Singularities of Bach-Flat Metrics", "abstract": "We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special classes of Bach-flat metrics obtained in \\cite{TVMOD}. For the proof we analyze the decay rates of solutions to the Bach-flat equation linearized around a flat metric. This classification is used to prove that Bach-flat cones are in fact ALE of order $\\tau$ for any $\\tau < 2$. This result is then used to prove the removal of singularities theorem."}
{"category": "Math", "title": "Eriksson's numbers game on certain edge-weighted three-node cyclic graphs", "abstract": "The numbers game is a one-player game played on a finite simple graph with certain ``amplitudes'' assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many authors. Following Eriksson, we allow the amplitudes on graph edges to be certain real numbers. Games played on such graphs are ``E-games.'' We show that for certain such three-node cyclic graphs, any numbers game will diverge when played from an initial assignment of nonnegative real numbers not all zero. This result is a key step in a Dynkin diagram classification (obtained elsewhere) of all E-game graphs which meet a certain finiteness requirement."}
{"category": "Math", "title": "Volume-preserving mean curvature flow of revolution hypersurfaces in a Rotationally Symmetric Space", "abstract": "In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t, the evolving hypersurface M_t meets such tgh ortogonally, we prove that: a) the flow exists while M_t does not touch the axis of rotation; b) throughout the time interval of existence, b1) the generating curve of M_t remains a graph, and b2) the averaged mean curvature is double side bounded by positive constants; c) the singularity set (if non-empty) is finite and discrete along the axis; d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature."}
{"category": "Math", "title": "Operad of formal homogeneous spaces and Bernoulli numbers", "abstract": "It is shown that for any morphism, i: g --> h, of Lie algebras the vector space underlying the Lie algebra h is canonically a g-homogeneous formal manifold with the action of g being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from the study of a 2-coloured operad of formal homogeneous spaces and its minimal resolution, and is used to give a new conceptual explanation of both Ziv Ran's Jacobi-Bernoulli complex and Fiorenza-Manetti's L-infinity algebra structure on the mapping cone of a morphism of two Lie algebras. All these constructions are iteratively extended to the case of a morphism of arbitrary L-infinity algebras."}
{"category": "Math", "title": "Uniform Sobolev inequalities for manifolds evolving by Ricci flow", "abstract": "Let M be a compact n-dimensional manifold, $n\\ge 2$, with metric g(t) evolving by the Ricci flow $\\partial g_{ij}/\\partial t=-2R_{ij}$ in (0,T) for some $T\\in\\Bbb{R}^+\\cup\\{\\infty\\}$ with $g(0)=g_0$. Let $\\lambda_0(g_0)$ be the first eigenvalue of the operator $-\\Delta_{g_0} +\\frac{R(g_0)}{4}$ with respect to g_0. We extend a recent result of R. Ye and prove uniform logarithmic Sobolev inequality and uniform Sobolev inequalities along the Ricci flow for any $n\\ge 2$ when either $T<\\infty$ or $\\lambda_0(g_0)>0$. As a consequence we extend Perelman's local $\\kappa$-noncollapsing result along the Ricci flow for any $n\\ge 2$ in terms of upper bound for the scalar curvature when either $T<\\infty$ or $\\lambda_0(g_0)>0$."}
{"category": "Math", "title": "Self-similar carpets over finite fields", "abstract": "Some linear dynamical systems over finite fields are studied and the self-similar character of their development is proved. Connections with aperiodic tilings, Delanoy numbers and other topics are also proved. The prime fields F_p have a canonical presentation as sets of self-similar carpets. The multiplicative inverse corresponds to mirroring."}
{"category": "Math", "title": "Permanents of Circulants: a Transfer Matrix Approach (Expanded Version)", "abstract": "Calculating the permanent of a (0,1) matrix is a #P-complete problem but there are some classes of structured matrices for which the permanent is calculable in polynomial time. The most well-known example is the fixed-jump (0,1) circulant matrix which, using algebraic techniques, was shown by Minc to satisfy a constant-coefficient fixed-order recurrence relation. In this note we show how, by interpreting the problem as calculating the number of cycle-covers in a directed circulant graph, it is straightforward to reprove Minc's result using combinatorial methods. This is a two step process: the first step is to show that the cycle-covers of directed circulant graphs can be evaluated using a transfer matrix argument. The second is to show that the associated transfer matrices, while very large, actually have much smaller characteristic polynomials than would a-priori be expected. An important consequence of this new viewpoint is that, in combination with a new recursive decomposition of circulant-graphs, it permits extending Minc's result to calculating the permanent of the much larger class of circulant matrices with non-fixed (but linear) jumps. It also permits us to count other types of structures in circulant graphs, e.g., Hamiltonian Cycles."}
{"category": "Math", "title": "An explicit estimate on multiplicity truncation in the second main theorem for holomorphic curves encountering hypersurfaces in general position in projective space", "abstract": "Yan and Chen proved a weak Cartan-type second main theorem for holomorphic curves meeting hypersurfaces in projective space that included truncated counting functions. Here we give an explicit estimate for the level of truncation."}
{"category": "Math", "title": "Planar graphs and covers", "abstract": "Planar locally finite graphs which are almost vertex transitive are discussed. If the graph is 3-connected and has at most one end then the group of automorphisms is a planar discontinuous group and its structure is well-known. A general result is obtained for such graphs where no restriction is put on the number of ends. It is shown that such a graph can be built up from one ended or finite planar graphs in a precise way. The results give a classification of the finitely generated groups with planar Cayley graphs."}
{"category": "Math", "title": "Spherical Nilpotent Orbits in Positive Characteristic", "abstract": "Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we classify all the spherical nilpotent G-orbits in the Lie algebra of G. The classification is the same as in the characteristic zero case obtained by D.I. Panyushev in 1994: for e a nilpotent element in the Lie algebra of G, the G-orbit G.e is spherical if and only if the height of e is at most 3."}
{"category": "Math", "title": "Generalized Cherednik-Macdonald identities", "abstract": "We derive generalizations of the Cherednik-Macdonald constant term identities associated to root systems which depend, besides on the usual multiplicity function, symmetrically on two quasi-periods. They are natural analogues of the Cherednik-Macdonald constant term q-identities in which the deformation parameter q is allowed to have modulus one. They unite the Cherednik-Macdonald constant term q-identities with closely related Jackson p-integral identities due to Macdonald, where the deformation parameter p is related to q by modular inversion."}
{"category": "Math", "title": "An entire function with simply and multiply connected wandering domains", "abstract": "We modify a construction of Kisaka and Shishikura to show that there exists an entire function which has both a simply connected and a multiply connected wandering domain. Moreover, these domains are contained in the set of fast escaping points."}
{"category": "Math", "title": "An iterative tomogravity algorithm for the estimation of network traffic", "abstract": "This paper introduces an iterative tomogravity algorithm for the estimation of a network traffic matrix based on one snapshot observation of the link loads in the network. The proposed method does not require complete observation of the total load on individual edge links or proper tuning of a penalty parameter as existing methods do. Numerical results are presented to demonstrate that the iterative tomogravity method controls the estimation error well when the link data is fully observed and produces robust results with moderate amount of missing link data."}
{"category": "Math", "title": "Asymptotics for Hermite-Pade rational approximants for two analytic functions with separated pairs of branch points (case of genus 0)", "abstract": "We investigate the asymptotic behavior for type II Hermite-Pade approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite-Pade approximants are described by an algebraic function of order 3 and genus 0. This situation gives rise to a vector-potential equilibrium problem for three measures and the poles of the common denominator are asymptotically distributed like one of these measures. We also work out the strong asymptotics for the corresponding Hermite-Pade approximants by using a 3x3 Riemann-Hilbert problem that characterizes this Hermite-Pade approximation problem."}
{"category": "Math", "title": "Rational semigroup automata", "abstract": "We show that for any monoid M, the family of languages accepted by M-automata (or equivalently, generated by regular valence grammars over M) is completely determined by that part of M which lies outside the maximal ideal. Hence, every such family arises as the family of languages accepted by N-automata where N is a simple or 0-simple monoid. A consequence is that every such family is either the class of regular languages, contains all the blind one-counter languages, or is the family of languages accepted by G-automata for G a non-locally-finite torsion group. We consider a natural extension of the usual definition which permits the automata to utilise more of the structure of each monoid, and also allows us to define S-automata for S an arbitrary semigroup. In the monoid case, the resulting automata are equivalent to the valence automata with rational target sets} which arise in the theory of regulated rewriting systems. We study the case that the register semigroup is completely simple or completely 0-simple, obtaining a complete characterisation of the classes of languages corresponding to such semigroups in terms of their maximal subgroups. In the process, we obtain a number of results about rational subsets of Rees matrix semigroups which may be of independent interest."}
{"category": "Math", "title": "Pricing, Hedging and Optimally Designing Derivatives Via Minimization of Risk Measures", "abstract": "The question of pricing and hedging a given contingent claim has a unique solution in a complete market framework. When some incompleteness is introduced, the problem becomes however more difficult. Several approaches have been adopted in the literature to provide a satisfactory answer to this problem, for a particular choice criterion. In this paper, in order to price and hedge a non-tradable contingent claim, we first start with a (standard) utility maximization problem and end up with an equivalent risk measure minimization. This hedging problem can be seen as a particular case of a more general situation of risk transfer between different agents, one of them consisting of the financial market. In order to provide constructive answers to this general optimal risk transfer problem, both static and dynamic approaches are considered. When considering a dynamic framework, our main purpose is to find a trade-off between static and very abstract risk measures as we are more interested in tractability issues and interpretations of the dynamic risk measures we obtain rather than the ultimate general results. Therefore, after introducing a general axiomatic approach to dynamic risk measures, we relate the dynamic version of convex risk measures to BSDEs."}
{"category": "Math", "title": "Law of Large Numbers Limits for Many Server Queues", "abstract": "This work considers a many-server queueing system in which customers with i.i.d., generally distributed service times enter service in the order of arrival. The dynamics of the system is represented in terms of a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterised as the unique solution to a coupled pair of integral equations, which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, in the time-homogeneous setting, the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these systems is that they arise as models of computer data systems and call centers."}
{"category": "Math", "title": "Stanilov-Tsankov-Videv Theory", "abstract": "We survey some recent results concerning Stanilov-Tsankov-Videv theory, conformal Osserman geometry, and Walker geometry which relate algebraic properties of the curvature operator to the underlying geometry of the manifold."}
{"category": "Math", "title": "A flexible Bayesian generalized linear model for dichotomous response data with an application to text categorization", "abstract": "We present a class of sparse generalized linear models that include probit and logistic regression as special cases and offer some extra flexibility. We provide an EM algorithm for learning the parameters of these models from data. We apply our method in text classification and in simulated data and show that our method outperforms the logistic and probit models and also the elastic net, in general by a substantial margin."}
{"category": "Math", "title": "Overconvergence and classicality: the case of curves", "abstract": "Given our set-up of a system of curves and maps between them satisfying certain assumptions, we prove a classicality criterion for overconvergent sections of line bundles over these curves. As a result, we prove such criteria for overconvergent modular forms over various Shimura curves. In particular, we provide a classicality criterion for overconvergent modular forms studied in [Kassaei: P-adic modular forms over Shimura curves over totally real fields, Compositio Math. 140 (2004), no 2, 359-395] and their higher-level generalizations."}
{"category": "Math", "title": "Estimating the proportion of differentially expressed genes in comparative DNA microarray experiments", "abstract": "DNA microarray experiments, a well-established experimental technique, aim at understanding the function of genes in some biological processes. One of the most common experiments in functional genomics research is to compare two groups of microarray data to determine which genes are differentially expressed. In this paper, we propose a methodology to estimate the proportion of differentially expressed genes in such experiments. We study the performance of our method in a simulation study where we compare it to other standard methods. Finally we compare the methods in real data from two toxicology experiments with mice."}
{"category": "Math", "title": "A representative sampling plan for auditing health insurance claims", "abstract": "A stratified sampling plan to audit health insurance claims is offered. The stratification is by dollar amount of the claim. The plan is representative in the sense that with high probability for each stratum, the difference in the average dollar amount of the claim in the sample and the average dollar amount in the population, is ``small.'' Several notions of ``small'' are presented. The plan then yields a relatively small total sample size with the property that the overall average dollar amount in the sample is close to the average dollar amount in the population. Three different estimators and corresponding lower confidence bounds for over (under) payments are studied."}
{"category": "Math", "title": "Confidence distribution (CD) -- distribution estimator of a parameter", "abstract": "The notion of confidence distribution (CD), an entirely frequentist concept, is in essence a Neymanian interpretation of Fisher's Fiducial distribution. It contains information related to every kind of frequentist inference. In this article, a CD is viewed as a distribution estimator of a parameter. This leads naturally to consideration of the information contained in CD, comparison of CDs and optimal CDs, and connection of the CD concept to the (profile) likelihood function. A formal development of a multiparameter CD is also presented."}
{"category": "Math", "title": "Empirical Bayes methods for controlling the false discovery rate with dependent data", "abstract": "False discovery rate (FDR) has been widely used as an error measure in large scale multiple testing problems, but most research in the area has been focused on procedures for controlling the FDR based on independent test statistics or the properties of such procedures for test statistics with certain types of stochastic dependence. Based on an approach proposed in Tang and Zhang (2005), we further develop in this paper empirical Bayes methods for controlling the FDR with dependent data. We implement our methodology in a time series model and report the results of a simulation study to demonstrate the advantages of the empirical Bayes approach."}
{"category": "Math", "title": "A smoothing model for sample disclosure risk estimation", "abstract": "When a sample frequency table is published, disclosure risk arises when some individuals can be identified on the basis of their values in certain attributes in the table called key variables, and then their values in other attributes may be inferred, and their privacy is violated. On the basis of the sample to be released, and possibly some partial knowledge of the whole population, an agency which considers releasing the sample, has to estimate the disclosure risk. Risk arises from non-empty sample cells which represent small population cells and from population uniques in particular. Therefore risk estimation requires assessing how many of the relevant population cells are likely to be small. Various methods have been proposed for this task, and we present a method in which estimation of a population cell frequency is based on smoothing using a local neighborhood of this cell, that is, cells having similar or close values in all attributes. We provide some preliminary results and experiments with this method. Comparisons are made to two other methods: 1. a log-linear models approach in which inference on a given cell is based on a ``neighborhood'' of cells determined by the log-linear model. Such neighborhoods have one or some common attributes with the cell in question, but some other attributes may differ significantly. 2 The Argus method in which inference on a given cell is based only on the sample frequency in the specific cell, on the sample design and on some known marginal distributions of the population, without learning from any type of ``neighborhood'' of the given cell, nor from any model which uses the structure of the table."}
{"category": "Math", "title": "A note on the U,V method of estimation", "abstract": "The U,V method of estimation provides unbiased estimators or predictors of random quantities. The method was introduced by Robbins \\citer3 and subsequently studied in a series of papers by Robbins and Zhang. (See Zhang \\citer5.) Practical applications of the method are featured in these papers. We demonstrate that for one U function (one for which there is an important application) the V estimator is inadmissible for a wide class of loss functions. For another important U function the V estimator is admissible for the squared error loss function."}
{"category": "Math", "title": "Local polynomial regression on unknown manifolds", "abstract": "We reveal the phenomenon that ``naive'' multivariate local polynomial regression can adapt to local smooth lower dimensional structure in the sense that it achieves the optimal convergence rate for nonparametric estimation of regression functions belonging to a Sobolev space when the predictor variables live on or close to a lower dimensional manifold."}
{"category": "Math", "title": "Formal punctured ribbons and two-dimensional local fields", "abstract": "We investigate formal ribbons on curves. Roughly speaking, formal ribbon is a family of locally linearly compact vector spaces on a curve. We establish a one-to-one correspondence between formal ribbons on curves plus some geometric data and some subspaces of two-dimensional local field."}
{"category": "Math", "title": "Matrice magique associ\\'ee \\`a un germe de courbe plane et division par l'id\\'eal Jacobien", "abstract": "In the ring of holomorphic functions at the origin of C^2, we consider the equation uf'_x+vf'_y=wf where f and w are given. We introduce intersection multiplicities relative to w and f'_y along the branches of f, and we study the solutions (u,v) using these valuations. As an application, we construct an explicit functional equation satisfied by f."}
{"category": "Math", "title": "Prime numbers with Beatty sequences", "abstract": "A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\\alpha n]+1$, where $1<\\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately imply that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic formula for the number of primes $p=q[\\alpha n+\\beta]+a$ with $n\\le N$, where $\\alpha,\\beta$ are real numbers such that $\\alpha$ is positive and irrational of finite type (which is true for almost all $\\alpha$) and $a,q$ are integers with $0\\le a<q\\le N^\\kappa$ and $\\gcd(a,q)=1$, where $\\kappa>0$ depends only on $\\alpha$. We also prove a similar result for primes $p=[\\alpha n+\\beta]$ such that $p\\equiv a\\pmod q$."}
{"category": "Math", "title": "On the injectivity of the X-ray transform for Anosov thermostats", "abstract": "We consider Anosov thermostats on a closed surface and the X-ray transform on functions which are up to degree two in the velocities. We show that the subspace where the X-ray transform fails to be s-injective is finite dimensional. Furthermore, if the surface is negatively curved and the thermostat is pure Gaussian (i.e. no magnetic field is present), then the X-ray transform is s-injective."}
{"category": "Math", "title": "On deciding stability of multiclass queueing networks under buffer priority scheduling policies", "abstract": "One of the basic properties of a queueing network is stability. Roughly speaking, it is the property that the total number of jobs in the network remains bounded as a function of time. One of the key questions related to the stability issue is how to determine the exact conditions under which a given queueing network operating under a given scheduling policy remains stable. While there was much initial progress in addressing this question, most of the results obtained were partial at best and so the complete characterization of stable queueing networks is still lacking. In this paper, we resolve this open problem, albeit in a somewhat unexpected way. We show that characterizing stable queueing networks is an algorithmically undecidable problem for the case of nonpreemptive static buffer priority scheduling policies and deterministic interarrival and service times. Thus, no constructive characterization of stable queueing networks operating under this class of policies is possible. The result is established for queueing networks with finite and infinite buffer sizes and possibly zero service times, although we conjecture that it also holds in the case of models with only infinite buffers and nonzero service times. Our approach extends an earlier related work [Math. Oper. Res. 27 (2002) 272--293] and uses the so-called counter machine device as a reduction tool."}
{"category": "Math", "title": "Special K\\\"ahler-Ricci potentials and Ricci solitons", "abstract": "On a manifold of dimension at least six, let $(g,\\tau)$ be a pair consisting of a K\\\"ahler metric g which is locally K\\\"ahler irreducible, and a nonconstant smooth function $\\tau$. Off the zero set of $\\tau$, if the metric $\\hat{g}=g/\\tau^2$ is a gradient Ricci soliton which has soliton function $1/\\tau$, we show that $\\hat{g}$ is K\\\"ahler with respect to another complex structure, and locally of a type first described by Koiso. Moreover, $\\tau$ is a special K\\\"ahler-Ricci potential, a notion defined in earlier works of Derdzinski and Maschler. The result extends to dimension four with additional assumptions. We also discuss a Ricci-Hessian equation, which is a generalization of the soliton equation, and observe that the set of pairs $(g,\\tau)$ satisfying a Ricci-Hessian equation is invariant, in a suitable sense, under the map $(g,\\tau)\\to (\\hat{g},1/\\tau)$."}
{"category": "Math", "title": "Singular and tangent slit solutions to the Loewner equation", "abstract": "We consider the Loewner differential equation generating univalent maps of the unit disk (or of the upper half-plane) onto itself minus a single slit. We prove that the circular slits, tangent to the real axis are generated by H\\\"older continuous driving terms with exponent 1/3 in the Loewner equation. Singular solutions are described, and the critical value of the norm of driving terms generating quasisymmetric slits in the disk is obtained."}
{"category": "Math", "title": "Deconvolution by simulation", "abstract": "Given samples (x_1,...,x_m) and (z_1,...,z_n) which we believe are independent realizations of random variables X and Z respectively, where we further believe that Z=X+Y with Y independent of X, the problem is to estimate the distribution of Y. We present a new method for doing this, involving simulation. Experiments suggest that the method provides useful estimates."}
{"category": "Math", "title": "Shape restricted regression with random Bernstein polynomials", "abstract": "Shape restricted regressions, including isotonic regression and concave regression as special cases, are studied using priors on Bernstein polynomials and Markov chain Monte Carlo methods. These priors have large supports, select only smooth functions, can easily incorporate geometric information into the prior, and can be generated without computational difficulty. Algorithms generating priors and posteriors are proposed, and simulation studies are conducted to illustrate the performance of this approach. Comparisons with the density-regression method of Dette et al. (2006) are included."}
{"category": "Math", "title": "Non- and semi-parametric analysis of failure time data with missing failure indicators", "abstract": "A class of estimating functions is introduced for the regression parameter of the Cox proportional hazards model to allow unknown failure statuses on some study subjects. The consistency and asymptotic normality of the resulting estimators are established under mild conditions. An adaptive estimator which achieves the minimum variance-covariance bound of the class is constructed. Numerical studies demonstrate that the asymptotic approximations are adequate for practical use and that the efficiency gain of the adaptive estimator over the complete-case analysis can be quite substantial. Similar methods are also developed for the nonparametric estimation of the survival function of a homogeneous population and for the estimation of the cumulative baseline hazard function under the Cox model."}
{"category": "Math", "title": "Nonparametric estimation of a distribution function under biased sampling and censoring", "abstract": "This paper derives the nonparametric maximum likelihood estimator (NPMLE) of a distribution function from observations which are subject to both bias and censoring. The NPMLE is obtained by a simple EM algorithm which is an extension of the algorithm suggested by Vardi (Biometrika, 1989) for size biased data. Application of the algorithm to many models is discussed and a simulation study compares the estimator's performance to that of the product-limit estimator (PLE). An example demonstrates the utility of the NPMLE to data where the PLE is inappropriate."}
{"category": "Math", "title": "Estimating a Polya frequency function_2", "abstract": "We consider the non-parametric maximum likelihood estimation in the class of Polya frequency functions of order two, viz. the densities with a concave logarithm. This is a subclass of unimodal densities and fairly rich in general. The NPMLE is shown to be the solution to a convex programming problem in the Euclidean space and an algorithm is devised similar to the iterative convex minorant algorithm by Jongbleod (1999). The estimator achieves Hellinger consistency when the true density is a PFF_2 itself."}
{"category": "Math", "title": "A super Frobenius formula for the characters of Iwahori-Hecke algebras", "abstract": "We establish a super Frobenius formula for the characters of Iwahori-Hecke algebras. We show that the Hall-Littlewood sypersymmetric function, up to constant, generates the values of the irreducible characters of Iwahori-Hecke algebras at the elements corresponding to cycle permutations. Our formula in this article includes both the ordinary quantum case and the classical super case."}
{"category": "Math", "title": "A comparison of the accuracy of saddlepoint conditional cumulative distribution function approximations", "abstract": "Consider a model parameterized by a scalar parameter of interest and a nuisance parameter vector. Inference about the parameter of interest may be based on the signed root of the likelihood ratio statistic R. The standard normal approximation to the conditional distribution of R typically has error of order O(n^{-1/2}), where n is the sample size. There are several modifications for R, which reduce the order of error in the approximations. In this paper, we mainly investigate Barndorff-Nielsen's modified directed likelihood ratio statistic, Severini's empirical adjustment, and DiCiccio and Martin's two modifications, involving the Bayesian approach and the conditional likelihood ratio statistic. For each modification, two formats were employed to approximate the conditional cumulative distribution function; these are Barndorff-Nielson formats and the Lugannani and Rice formats. All approximations were applied to inference on the ratio of means for two independent exponential random variables. We constructed one and two-sided hypotheses tests and used the actual sizes of the tests as the measurements of accuracy to compare those approximations."}
{"category": "Math", "title": "Multivariate medians and measure-symmetrization", "abstract": "We discuss two research areas dealing respectively with (1) a class of multivariate medians and (2) a symmetrization algorithm for probability measures."}
{"category": "Math", "title": "Statistical thinking: From Tukey to Vardi and beyond", "abstract": "Data miners (minors?) and neural networkers tend to eschew modelling, misled perhaps by misinterpretation of strongly expressed views of John Tukey. I discuss Vardi's views of these issues as well as other aspects of Vardi's work in emision tomography and in sampling bias."}
{"category": "Math", "title": "Errors Theory using Dirichlet Forms, Linear Partial Differential Equations and Wavelets", "abstract": "We present an application of error theory using Dirichlet Forms in linear partial differential equations (LPDE). We study the transmission of an uncertainty on the terminal condition to the solution of the LPDE thanks to the decomposition of the solution on a wavelets basis. We analyze the basic properties and a particular class of LPDE where the wavelets bases show their powerful, the combination of error theory and wavelets basis justifies some hypotheses, helpful to simplify the computation."}
{"category": "Math", "title": "Statistical inverse problems in active network tomography", "abstract": "The analysis of computer and communication networks gives rise to some interesting inverse problems. This paper is concerned with active network tomography where the goal is to recover information about quality-of-service (QoS) parameters at the link level from aggregate data measured on end-to-end network paths. The estimation and monitoring of QoS parameters, such as loss rates and delays, are of considerable interest to network engineers and Internet service providers. The paper provides a review of the inverse problems and recent research on inference for loss rates and delay distributions. Some new results on parametric inference for delay distributions are also developed. In addition, a real application on Internet telephony is discussed."}
{"category": "Math", "title": "Densities for Ornstein-Uhlenbeck processes with jumps", "abstract": "We consider an Ornstein-Uhlenbeck process with values in R^n driven by a L\\'evy process (Z_t) taking values in R^d with d possibly smaller than n. The L\\'evy noise can have a degenerate or even vanishing Gaussian component. Under a controllability condition and an assumption on the L\\'evy measure of (Z_t), we prove that the law of the Ornstein-Uhlenbeck process at any time t>0 has a density on R^n. Moreover, when the L\\'evy process is of $\\alpha$-stable type, $\\alpha \\in (0,2)$, we show that such density is a $C^{\\infty}$-function."}
{"category": "Math", "title": "Using data network metrics, graphics, and topology to explore network characteristics", "abstract": "Yehuda Vardi introduced the term network tomography and was the first to propose and study how statistical inverse methods could be adapted to attack important network problems (Vardi, 1996). More recently, in one of his final papers, Vardi proposed notions of metrics on networks to define and measure distances between a network's links, its paths, and also between different networks (Vardi, 2004). In this paper, we apply Vardi's general approach for network metrics to a real data network by using data obtained from special data network tools and testing procedures presented here. We illustrate how the metrics help explicate interesting features of the traffic characteristics on the network. We also adapt the metrics in order to condition on traffic passing through a portion of the network, such as a router or pair of routers, and show further how this approach helps to discover and explain interesting network characteristics."}
{"category": "Math", "title": "Legendrian ribbons in overtwisted contact structures", "abstract": "We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies $\\sel(K,S)=-\\chi(S)$. In particular, every null-homologous topological knot type in an overtwisted contact manifold can be represented by the boundary of a Legendrian ribbon. Finally, we show that a contact structure is tight if and only if every Legendrian ribbon minimizes genus in its relative homology class."}
{"category": "Math", "title": "Semiclassical Limits of Quantum Affine Spaces", "abstract": "Semiclassical limits of generic multiparameter quantized coordinate rings A = O_q(k^n) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsionfree subgroup of k*. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R = O(k^n), and results of Oh, Park, Shin and the authors are used to construct homeomorphisms from the Poisson prime and Poisson primitive spectra of R onto the prime and primitive spectra of A. The Poisson primitive spectrum of R is then identified with the space of symplectic cores in k^n in the sense of Brown and Gordon, and an example is presented (over the complex numbers) for which the Poisson primitive spectrum of R is not homeomorphic to the space of symplectic leaves in k^n. Finally, these results are extended from quantum affine spaces to quantum affine toric varieties."}
{"category": "Math", "title": "Pseudo-Riemannian Jacobi-Videv Manifolds", "abstract": "We exhibit several families of Jacobi-Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi-Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure."}
{"category": "Math", "title": "Differential geometry of curves in Lagrange Grassmannians with given Young diagram", "abstract": "Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure. The Pontryagin extremals of this optimal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves produces the invariant of the original geometric structure. The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its kth column is equal to the rank of the kth derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construction of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical ones such as sub-Riemannian and sub-Finslerian structures, defined on nonholonomic distributions."}
{"category": "Math", "title": "Spectral geometry, homogeneous spaces, and differential forms with finite Fourier series", "abstract": "Let G be a compact Lie group acting transitively on Riemannian manifolds M and N. Let p be a G equivariant Riemannian submersion from M to N. We show that a smooth differential form on N has finite Fourier series if and only if the pull back has finite Fourier series on M"}
{"category": "Math", "title": "Additivity of Spin^c Quantization under Cutting", "abstract": "A G-equivariant spin^c structure on a manifold gives rise to a virtual representation of the group G, called the spin^c quantization of the manifold. We present a cutting construction for S^1-equivariant spin^c manifolds, and show that the quantization of the original manifold is isomorphic to the direct sum of the quantizations of the cut spaces. Our proof uses Kostant-type formulas, which express the quantization in terms of local data around the fixed point set of the S^1-action."}
{"category": "Math", "title": "Functional analysis via extensions of the band depth", "abstract": "The notion of data depth has long been in use to obtain robust location and scale estimates in a multivariate setting. The depth of an observation is a measure of its centrality, with respect to a data set or a distribution. The data depths of a set of multivariate observations translates to a center-outward ordering of the data. Thus, data depth provides a generalization of the median to a multivariate setting (the deepest observation), and can also be used to screen for extreme observations or outliers (the observations with low data depth). Data depth has been used in the development of a wide range of robust and non-parametric methods for multivariate data, such as non-parametric tests of location and scale [Li and Liu (2004)], multivariate rank-tests [Liu and Singh (1993)], non-parametric classification and clustering [Jornsten (2004)], and robust regression [Rousseeuw and Hubert (1999)]. Many different notions of data depth have been developed for multivariate data. In contrast, data depth measures for functional data have only recently been proposed [Fraiman and Muniz (1999), L\\'{o}pez-Pintado and Romo (2006a)]. While the definitions of both of these data depth measures are motivated by the functional aspect of the data, the measures themselves are in fact invariant with respect to permutations of the domain (i.e. the compact interval on which the functions are defined). Thus, these measures are equally applicable to multivariate data where there is no explicit ordering of the data dimensions. In this paper we explore some extensions of functional data depths, so as to take the ordering of the data dimensions into account."}
{"category": "Math", "title": "The hook fusion procedure and its generalisations", "abstract": "The fusion procedure provides a way to construct new solutions to the Yang-Baxter equation. In the case of the symmetric group the fusion procedure has been used to construct diagonal matrix elements using a decomposition of the Young diagram into its rows or columns. We present a new construction which decomposes the diagram into hooks, the great advantage of this is that it minimises the number of auxiliary parameters needed in the procedure. We go on to use the hook fusion procedure to find diagonal matrix elements computationally and calculate supporting evidence to a previous conjecture. We are motivated by the construction of certain elements that allow us to generate representations of the symmetric group and single out particular irreducible components. In this way we may construct higher representations of the symmetric group from elementary ones. We go some way to generalising the hook fusion procedure by considering other decompositions of Young diagrams, specifically into ribbons. Finally, we adapt our construction to the quantum deformation of the symmetric group algebra known as the Hecke algebra."}
{"category": "Math", "title": "Distribution of periodic torus orbits and Duke's theorem for cubic fields", "abstract": "We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke's theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular 5-fold SL_3(Z)\\SL_3(R)/SO_3(R). In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL_3(Z)\\SL_3(R)/SO_3(R) of volume less than V becomes equidistributed as V goes to infinity. The proof combines subconvexity estimates, measure classification, and local harmonic analysis."}
{"category": "Math", "title": "Remark on fundamental groups and effective Diophantine methods for hyperbolic curves", "abstract": "We discuss how non-commutative fundamental groups could eventually contribute to algorithms for finding rational points on hyperbolic curves."}
{"category": "Math", "title": "Complex Datasets and Inverse Problems. Tomography, Networks and Beyond", "abstract": "This book is a collection of papers dedicated to the memory of Yehuda Vardi. Yehuda was the chair of the Department of Statistics of Rutgers University when he passed away unexpectedly on January 13, 2005. On October 21--22, 2005, some 150 leading scholars from many different fields, including statistics, telecommunications, biomedical engineering, bioinformatics, biostatistics and epidemiology, gathered at Rutgers in a conference in his honor. This conference was on ``Complex Datasets and Inverse Problems: Tomography, Networks, and Beyond,'' and was organized by the editors. The present collection includes research work presented at the conference, as well as contributions from Yehuda's colleagues. The theme of the conference was networks and other important and emerging areas of research involving incomplete data and statistical inverse problems. Networks are abundant around us: communication, computer, traffic, social and energy are just a few examples. As enormous amounts of network data are collected in this information age, the field has attracted a great amount of attention from researchers in statistics and computer engineering as well as telecommunication providers and various government agencies. However, few statistical tools have been developed for analyzing network data as they are typically governed by time-varying and mutually dependent communication protocols sitting on complicated graph-structured network topologies. Many prototypical applications in these and other important technologies can be viewed as statistical inverse problems with complex, massive, high-dimensional and possibly biased/incomplete data. This unifying theme of inverse problems is particularly appropriate for a conference and volume dedicated to the memory of Yehuda. Indeed he made influential contributions to these fields, especially in medical tomography, biased data, statistical inverse problems, and network tomography."}
{"category": "Math", "title": "Automorphisms and derivations of free Poisson algebras in two variables", "abstract": "Let P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable."}
{"category": "Math", "title": "\"Boundary blowup\" type sub-solutions to semilinear elliptic equations with Hardy potential", "abstract": "Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential. The size of this potential effects the existence of a certain type of solutions (large solutions): if the potential is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a Phragmen-Lindelof type theorem which enables us to classify the solutions and sub-solutions according to their behavior near the boundary. Nonexistence follows from this principle together with the Keller-Osserman upper bound. The existence proofs rely on sub- and super-solution techniques and on estimates for the Hardy constant derived in Marcus, Mizel and Pinchover."}
{"category": "Math", "title": "Quaternion CR-submanifolds of a Locally Conformal Quaternion Kaehler Manifold", "abstract": "The purpose of the present paper is to study the differential geometric properties of a quaternion CR-submanifold in a locally conformal quaternion Kaehler manifold."}
{"category": "Math", "title": "Lieb-Thirring inequalities with improved constants", "abstract": "Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allow us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schroedinger operators."}
{"category": "Math", "title": "On the spinor structure of the homogeneous and isotropic universe in closed model", "abstract": "The closed homogeneous and isotropic universe is considered. The bundles of Weyl and Dirac spinors for this universe are explicitly described. Some explicit formulas for the basic fields and for the connection components in stereographic and in spherical coordinates are presented."}
{"category": "Math", "title": "On some lower bounds on the number of bicliques needed to cover a bipartite graph", "abstract": "The biclique covering number of a bipartite graph G is the minimum number of complete bipartite subgraphs (bicliques) whose union contains every edge of G. In this little note we compare three lower bounds on the biclique covering number: A bound jk(G) proposed by Jukna & Kulikov (Discrete Math. 2009); the well-known fooling set bound fool(G); the \"tensor-power\" fooling set bound fool^\\infty(G). We show jk \\le fool le fool^\\infty \\le min_Q (rk Q)^2, where the minimum is taken over all matrices with a certain zero/nonzero-pattern. Only the first inequality is really novel, the third one generalizes a result of Dietzfelbinger, Hromkovi\\v{c}, Schnitger (1994). We also give examples for which fool \\ge (rk)^{log_4 6} improving on Dietzfelbinger et al."}
{"category": "Math", "title": "The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories", "abstract": "We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand this correspondence fully we examine the totalities of such structures together with maps between them and higher maps between those. Categories naturally form a 2-category {\\bfseries Cat} so we take the full sub-2-category of this whose 0-cells are the degenerate categories. Monoids naturally form a category, but we regard this as a discrete 2-category to make the comparison. We show that this construction does not yield a biequivalence; to get an equivalence we ignore the natural transformations and consider only the {\\it category} of degenerate categories. A similar situation occurs for degenerate bicategories. The tricategory of such does not yield an equivalence with monoidal categories; we must consider only the categories of such structures. For doubly degenerate bicategories the tricategory of such is not naturally triequivalent to the category of commutative monoids (regarded as a tricategory). However in this case considering just the categories does not give an equivalence either; to get an equivalence we consider the {\\it bicategory} of doubly degenerate bicategories. We conclude with a hypothesis about how the above cases might generalise for n-fold degenerate n-categories."}
{"category": "Math", "title": "Trace fields and commensurability of link complements", "abstract": "This paper investigates the strength of the trace field as a commensurability invariant of hyperbolic 3-manifolds. We construct an infinite family of two-component hyperbolic link complements which are pairwise incommensurable and have the same trace field, and infinitely many 1-cusped finite volume hyperbolic 3-manifolds with the same property. We also show that the two-component link complements above have integral traces, but each has a mutant with a nonintegral trace."}
{"category": "Math", "title": "The hardness of computing an eigenform", "abstract": "In this article, we give evidence that computing Fourier coefficients of the Hecke eigenforms for composite indices is no easier than factoring integers. In particular, we show that the existence of a polynomial time algorithm that, given n, computes the n-th Fourier coefficient of a (fixed) Hecke eigenform implies that we can factor most RSA moduli (numbers that are products of two distinct primes) in polynomial time."}
{"category": "Math", "title": "A Selberg integral for the Lie algebra A_n", "abstract": "A new q-binomial theorem for Macdonald polynomials is employed to prove an A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra g."}
{"category": "Math", "title": "An Algebraic Chain Model of String Topology", "abstract": "A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes non simply-connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology."}
{"category": "Math", "title": "The symplectic structure of curves in three dimensional spaces of constant curvature and the equations of mathematical physics", "abstract": "This paper defines a symplectic form on the infinite dimensional Fr\\'echet manifold of framed curves of fixed length over a simply connected Riemannian manifold of constant curvature. The paper then considers Hamiltonian systems generated by the geometric invariants of the curves on the base manifold and relates them to the equations of mathematical physics."}
{"category": "Math", "title": "Projectively Osserman manifolds", "abstract": "One says that a smooth manifold M is a pseudo-Riemannian manifold of signature (p,q) if the tangent bundle TM is equipped with a smooth non-degenerate symmetric inner product g of signature (p,q). Similarly one says that M is an affine manifold if TM is equipped with a torsion free connection. One says g is Osserman if the eigenvalues of the Jacobi operator are constant on the pseudo-sphere bundles of unit timelike and spacelike vectors. We extend this concept from the pseudo-Riemannian to the affine setting to define the notion of a projectively Osserman manifold. This notion is the focus of the paper. We establish some basic results concerning projectively Osserman manifolds and exhibit examples of this structure which arise in several different geometrical contexts."}
{"category": "Math", "title": "On the collisions of singular points of complex algebraic plane curves", "abstract": "We study the \"generic\" degenerations of curves with two singular points when the points merge. First, the notion of generic degeneration is defined precisely. Then a method to classify the possible results of generic degenerations is proposed in the case of linear singularity types. We discuss possible bounds on the singularity invariants of the resulting type in terms of the initial types. In particular the strict upper bound on the resulting multiplicity is proved and a sufficient condition for $\\delta=const$ collision is given."}
{"category": "Math", "title": "On the enumeration of complex plane curves with two singular points", "abstract": "We study equi-singular strata of plane curves with two singular points of prescribed types. The method of the previous work [Kerner06] is generalized to this case. In particular we consider the enumerative problem for plane curves with two singular points of linear singularity types. First the problem for two ordinary multiple points of fixed multiplicities is solved. Then the enumeration for arbitrary linear types is reduced to the case of ordinary multiple points and to the understanding of \"merging\" of singular points. Many examples and numerical answers are given."}
{"category": "Math", "title": "Random permutations and unique fully supported ergodicity for the Euler adic transformation", "abstract": "There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations."}
{"category": "Math", "title": "Several Dirac Operator in parabolic geometry", "abstract": "In this thesis, we show the existence of a sequence of differential operators starting with with the Dirac operator in k Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\\sum_j e_j\\cdot \\partial_{ij}: C^\\infty((\\R^n)^k,\\S)\\to C^\\infty((\\R^n)^k,\\S)$ ($\\S$ is the spinor module). This operator is the Cauchy-Riemann operator for n=2 and its resolution is the Dolbeault complex. For higher n, the resolution of D is not known in general. While this problem was treated many times in the language of Clifford analysis and some partial results are known, we give a description of this operator in Parabolic geometry, which is a special type of Cartan geometry modeled on $G/P$, where P is a Parabolic subgroup of G. We construct sequences of invariant differential operators starting with the Dirac operator in several variables and assume that these sequences coinside in some cases with the resolution. We describe the structure of these sequences precisely in case the dimension $n$ is odd and give a conjecture that these sequences have similar structure for n even, $k\\leq n/2$ (the s.c. {\\it stable range}). We also give some information about these sequences in case n even, k>n/2. In the last chapter, explicite formulas for the operators are derived for the case k=2."}
{"category": "Math", "title": "Affineness of Deligne-Lusztig varieties for minimal length elements", "abstract": "We prove that the Deligne-Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik-Rapoport and it is inspired by the case of regular elements, which correspond to the varieties involved in Brou\\'e's conjectures."}
{"category": "Math", "title": "Smooth toric DM stacks", "abstract": "We give a new definition of smooth toric DM stacks in the same spirit of toric varieties. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks."}
{"category": "Math", "title": "On the number of stable quiver representations over finite fields", "abstract": "We prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail."}
{"category": "Math", "title": "Generic Continuous Spectrum for Ergodic Schr\"odinger Operators", "abstract": "We consider discrete Schr\"odinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon's Lemma that for a generic continuous sampling function, the associated Schr\"odinger operators have no eigenvalues in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus."}
{"category": "Math", "title": "Borel subalgebras of root-reductive Lie algebras", "abstract": "This paper generalizes the classification in a paper of Dimitrov and Penkov of Borel subalgebras of gl_infty. Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras along inclusions preserving the root spaces with respect to nested Cartan subalgebras. A Borel subalgebra of a root-reductive Lie algebra is by definition a maximal locally solvable subalgebra. The main general result of this paper is that a Borel subalgebra of an infinite-dimensional indecomposable root-reductive Lie algebra is the simultaneous stabilizer of a certain type of generalized flag in each of the standard representations. For the three infinite-dimensional simple root-reductive Lie algebras more precise results are obtained. The map sending a maximal closed (isotropic) generalized flag in the standard representation to its stabilizer hits Borel subalgebras, yielding a bijection in the cases of sl_infty and sp_infty; in the case of so_infty the fibers are of size one and two. A description is given of a nice class of toral subalgebras contained in any Borel subalgebra. Finally, certain Borel subalgebras of a general root-reductive Lie algebra are seen to correspond bijectively with Borel subalgebras of the commutator subalgebra, which are understood in terms of the special cases."}
{"category": "Math", "title": "Pre-quantization of the Moduli Space of Flat G-Bundles over a Surface", "abstract": "For a simply connected, compact, simple Lie group G, the moduli space of flat G-bundles over a closed surface is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction -- namely a certain cohomology class in H^3(G^2;Z) -- that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G."}
{"category": "Math", "title": "Generalisations of the Tits representation", "abstract": "We construct a group K_n with properties similar to infinite Coxeter groups. In particular, it has a geometric representation featuring hyperplanes and simplicial chambers. The generators of K_n are given by 2-element subsets of {0, .., n}. We give some easy combinatorial results on the finite residues of K_n."}
{"category": "Math", "title": "Calibrated associative and Cayley embeddings", "abstract": "Using the Cartan-Kahler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G_2-manifold as an associative submanifold. Second, the interior of a smooth, compact 4-manifold K, whose double has a trivial bundle of self-dual 2-forms, may be isometrically embedded into a Spin(7)-manifold as a Cayley submanifold. Along the way, we also show that Bochner's Theorem on real analytic approximation of smooth differential forms, can be obtained using real algebraic tools developed by Akbulut and King."}
{"category": "Math", "title": "Universal derived equivalences of posets of tilting modules", "abstract": "We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their tilting modules are related by a simple combinatorial construction, which we call flip-flop. We deduce that the posets of tilting modules of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent."}
{"category": "Math", "title": "On Almost Representations of Property (T) Groups", "abstract": "Property (T) for groups means a dichotomy: a representation either has an invariant vector or all vectors are far from being invariant. We show that, under a stronger condition of A.Zuk, a similar dichotomy holds for almost representations as well."}
{"category": "Math", "title": "Correction to: HZ-algebra spectra are differential graded algebras", "abstract": "This correction article is actually unnecessary. The proof of Theorem 1.2, concerning commutative HQ-algebra spectra and commutative differential graded algebras, in the author's paper [American Journal of Mathematics vol. 129 (2007) 351-379 (arxiv:math/0209215v4)] is correct as originally stated. Neil Strickland carefully proved that D is symmetric monoidal; so Proposition 4.7 and hence also Theorem 1.2 hold as stated. Strickland's proof will appear in joint work with Stefan Schwede; see related work in Strickland's [arxiv:0810.1747]. Note here D is defined as a colimit of chain complexes; in contrast, non-symmetric monoidal functors analogous to D are defined as homotopy colimits of spaces in previous work of the author."}
{"category": "Math", "title": "Parametrization of the regular equivalences of the canonical controller and its applications", "abstract": "We study control problems for linear systems in the behavioral framework. Our focus is a class of regular controllers that are equivalent to the canonical controller. The canonical controller is a particular controller that is guaranteed to be a solution whenever a solution exists. However, it has been shown that in most cases, the canonical controller is not regular. The main result of the paper is a parametrization of all regular controllers that are equivalent to the canonical controller. The parametrization is then used to solve two control problems. The first problem is related to designing a regular controller that uses as few control channels as possible. The second problem is to design a regular controller that satisfies a predefined input-output partitioning constraint. In both problems, based on the parametrization, we present algorithms that does the controller design."}
{"category": "Math", "title": "Graphical methods for efficient likelihood inference in Gaussian covariance models", "abstract": "In graphical modelling, a bi-directed graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bi-directed graph into a maximal ancestral graph that (i) represents the same independence structure as the original bi-directed graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i). Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi-directed edges. In Gaussian models, this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts."}
{"category": "Math", "title": "Limited scope adic transformations", "abstract": "We introduce a family of adic transformations on diagrams that are nonstationary and nonsimple. This family includes some previously studied adic transformations. We relate the dimension group of each these diagrams to the dynamical system determined by the adic transformation on the infinite edge paths, and we explicitly compute the dimension group for a subfamily. We also determine the ergodic adic invariant probability measures for this subfamily, and show that each system of the subfamily is loosely Bernoulli. We also give examples of particular adic transformations with roots of unity as well as one which is totally ergodic called the Euler adic. We also show that the Euler adic is loosely Bernoulli."}
{"category": "Math", "title": "Explicit gradient estimates for minimal Lagrangian surfaces of dimension two", "abstract": "We derive explicit, uniform, a priori interior Hessian and gradient estimates for special Lagrangian equations of all phases in dimension two."}
{"category": "Math", "title": "A Proof that Thompson's Groups have Infinitely Many Relative Ends", "abstract": "We show that each of Thompson's groups F, T, and V have infinitely many ends relative to certain subgroups. We go on to show that T and V both have Serre's property FA, i.e., any action of T or V on a tree will have a fixed point. (The proof of the latter statement was originally due to Ken Brown, and our proof is based on his notes.)"}
{"category": "Math", "title": "On Gr-Functors between Gr-Categories: Obstruction theory for Gr-Functors of the type $(\\varphi,f)$", "abstract": "Each Gr-functor of the type $(\\varphi,f)$ of a Gr-category of the type $(\\Pi,\\C)$ has the obstruction be an element $\\overline{k}\\in H^3(\\Pi,\\C).$ When this obstruction vanishes, there exists a bijection between congruence classes of Gr-functors of the type $(\\varphi,f)$ and the cohomology group $H^2(\\Pi,\\C).$ Then the relation of Gr-category theory and the group extension problem can be established and used to prove that each Gr-category is Gr-equivalent to a strict one."}
{"category": "Math", "title": "The marginalization paradox and the formal Bayes' law", "abstract": "It has recently been shown that the marginalization paradox (MP) can be resolved by interpreting improper inferences as probability limits. The key to the resolution is that probability limits need not satisfy the formal Bayes' law, which is used in the MP to deduce an inconsistency. In this paper, I explore the differences between probability limits and the more familiar pointwise limits, which do imply the formal Bayes' law, and show how these differences underlie some key differences in the interpretation of the MP."}
{"category": "Math", "title": "The theory of the exponential differential equations of semiabelian varieties", "abstract": "The complete first order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arises from an amalgamation-with-predimension construction in the style of Hrushovski. The theory includes necessary and sufficient conditions for a system of equations to have a solution. The necessary condition generalizes Ax's differential fields version of Schanuel's conjecture to semiabelian varieties. There is a purely algebraic corollary, the \"Weak CIT\" for semiabelian varieties, which concerns the intersections of algebraic subgroups with algebraic varieties."}
{"category": "Math", "title": "New equivalences for pattern avoiding involutions", "abstract": "We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S_5, S_6, and S_7 for both permutations and involutions."}
{"category": "Math", "title": "On Rational Nilpotent Orbits of $SL_{n}$ and $Sp_{2n}$ over a Local Non-Archimedean Field", "abstract": "We relate the partition-type parametrization of rational (arithmetic) nilpotent adjoint orbits of the split classical groups $SL_n$ and $Sp_{2n}$ over local non-Archimedean fields with a parametrization, introduced by DeBacker in 2002, which uses the associated Bruhat-Tits building to relate the question to one over the residue field."}
{"category": "Math", "title": "Homological algebra for affine Hecke algebras", "abstract": "In this paper we study homological properties of modules over an affine Hecke algebra H. In particular we prove a comparison result for higher extensions of tempered modules when passing to the Schwartz algebra S, a certain topological completion of the affine Hecke algebra. The proof is self-contained and based on a direct construction of a bounded contraction of certain standard resolutions of H-modules. This construction applies for all positive parameters of the affine Hecke algebra. This is an important feature since it is an ingredient to analyse how the irreducible discrete series representations of H arise in generic families over the parameter space of H. For irreducible non-simply laced affine Hecke algebras this will enable us to give a complete classification of the discrete series characters for all positive parameters (we will report on this application in a separate article)."}
{"category": "Math", "title": "Isogonal Conjugacy and Fermat Problems", "abstract": "We consider three types of isogonal conjugacy of two points with respect to a given triangle and characterize any of these types by a geometric equality. As an application to the Fermat problem with positive weights, we prove that in the general case the given weights determine uniquely a point X and the solution to the Fermat problem is the point Y, which is isogonally conjugate of type I to the point X. We obtain a similar characterization of the solution to the Fermat problem in the case of mixed weights as well."}
{"category": "Math", "title": "Remarks on $\\eta$-Einstein unit tangent bundles", "abstract": "We study the geometric properties of the base manifold for the unit tangent bundle satisfying the $\\eta$-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is $\\eta$-Einstein manifold if and only if base manifold is the space of constant sectional curvature 1 or 2."}
{"category": "Math", "title": "Data selection and confounding in the court case of Lucia de Berk", "abstract": "The nurse Lucia de Berk was convicted by the Dutch courts as a serial killer with 7 murders and 3 attempts at murder in three hospitals where she worked. The nurse however always professed her innocence and indeed was never observed in such an act of murder. The courts based their decision on circumstantial evidence and upon the use of statistics. In the appeal court, the use of statistical calculations was repealed but the use of \"data\" and \"statistical insights\" were not excluded. The trial hinged importantly on the role of statistics and data gathering. It appears that data selection and confounding feature strongly in this case. The notion of \"nominal correlation\" can be used to highlight those two features. This suggests a mistrial with the conviction of an innocent person."}
{"category": "Math", "title": "Four-dimensional almost Hermitian manifolds with vanishing Tricerri-Vanhecke Bochner curvature tensor", "abstract": "We study curvature properties of four-dimensional almost Hermitian manifolds with vanishing Bochner curvature tensor as defined by Tricerri and Vanhecke. We give local structure theorems for such Kaehler manifolds, and find out several examples related to the theorems."}
{"category": "Math", "title": "Which canonical algebras are derived equivalent to incidence algebras of posets?", "abstract": "We give a full description of all the canonical algebras over an algebraically closed field that are derived equivalent to incidence algebras of finite posets. These are the canonical algebras whose number of weights is either 2 or 3."}
{"category": "Math", "title": "Symplectic surgeries and normal surface singularities", "abstract": "We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of small exotic 4--manifolds."}
{"category": "Math", "title": "On Local Integrability Conditions Of Jet Groupoids", "abstract": "A Jet groupoid R_q over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms from X to X. As a subbundle of the q-th order jet bundle of the trivial bundle X times X, a jet groupoid can be considered as a nonlinear system of partial differential equations (PDE). This leads to the concept of formal integrability. On the other hand, each jet groupoid is the symmetry groupoid of a geometric object, modelled as a section of a natural bundle. Using the jet groupoids, we give a local characterisation of formal integrability for transitive jet groupoids in terms of their corresponding geometric objects."}
{"category": "Math", "title": "A property of cyclotomic polynomials", "abstract": "Given two cyclotomic polynomials $\\Phi_n(x)$ and $\\Phi_m(x)$, $n\\not= m$, we determine the minimal natural number k such that we can write $$k=a(x)\\Phi_n (x)+b(x)\\Phi_m(x),$$ with a(x) and b(x) integer polynomials."}
{"category": "Math", "title": "Qualitative properties of coupled parabolic systems of evolution equations", "abstract": "We apply functional analytical and variational methods in order to study well-posedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic problems of different nature: a coupled diffusive system arising in neurobiology, a strongly damped wave equation, a heat equation with dynamic boundary conditions, and a general semilinear Hodgkin--Huxley sytem."}
{"category": "Math", "title": "Determinants related to Dirichlet characters modulo 2, 4 and 8 of binomial coefficients and the algebra of recurrence matrices", "abstract": "Using recurrence matrices, defined and described with some details, we study a few determinants related to evaluations of binomial coefficients on Dirichlet characters modulo 2, 4 and 8."}
{"category": "Math", "title": "Gradient estimates for a degenerate parabolic equation with gradient absorption and applications", "abstract": "Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and the nonlinear absorption. In particular, the limit as time goes to infinity of the mass of integrable solutions is identified, together with the rate of expansion of the support for compactly supported initial data. The persistence of dead cores is also shown. The proof of these results strongly relies on gradient estimates which are first established."}
{"category": "Math", "title": "Twisted cohomology of the Hilbert schemes of points on surfaces", "abstract": "We calculate the cohomology spaces of the Hilbert schemes of points on surfaces with values in locally constant systems. For that purpose, we generalise I. Grojnoswki's and H. Nakajima's description of the ordinary cohomology in terms of a Fock space representation to the twisted case. We further generalise M. Lehn's work on the action of the Virasoro algebra to the twisted case. Building on work by M. Lehn and Ch. Sorger, we then give an explicit description of the cup-product in the twisted case whenever the surface has a numerically trivial canonical divisor. We formulate our results in a way that they apply to the projective and non-projective case in equal measure. As an application of our methods, we give explicit models for the cohomology rings of the generalised Kummer varieties and of a series of certain even dimensional Calabi--Yau manifolds."}
{"category": "Math", "title": "A note on the lambda-structure on the Burnside ring", "abstract": "Let G be a finite group and let S be a G-set. The Burnside ring of G has a natural structure of a lambda-ring. However, a priori the images of S under the lambda-operations can only be computed implicitly. In this paper we establish an explicit formula, expressing these images as linear combinations of classes of G-sets."}
{"category": "Math", "title": "Stanley Decompositions, Pretty Clean Filtrations and Reductions Modulo Regular Elements", "abstract": "We study the behavior of Stanley decompositions and of pretty clean filtrations under reduction modulo a regular element."}
{"category": "Math", "title": "Pathwise coordinate optimization", "abstract": "We consider ``one-at-a-time'' coordinate-wise descent algorithms for a class of convex optimization problems. An algorithm of this kind has been proposed for the $L_1$-penalized regression (lasso) in the literature, but it seems to have been largely ignored. Indeed, it seems that coordinate-wise algorithms are not often used in convex optimization. We show that this algorithm is very competitive with the well-known LARS (or homotopy) procedure in large lasso problems, and that it can be applied to related methods such as the garotte and elastic net. It turns out that coordinate-wise descent does not work in the ``fused lasso,'' however, so we derive a generalized algorithm that yields the solution in much less time that a standard convex optimizer. Finally, we generalize the procedure to the two-dimensional fused lasso, and demonstrate its performance on some image smoothing problems."}
{"category": "Math", "title": "Quasisymmetric and unipotent tensor categories", "abstract": "We classify braided tensor categories over C of exponential growth which are quasisymmetric, i.e., the squared braiding is the identity on the product of any two simple objects. This generalizes the classification results of Deligne on symmetric categories of exponential growth, and of Drinfeld on quasitriangular quasi-Hopf algebras. In particular, we classify braided categories of exponential growth which are unipotent, i.e., those whose only simple object is the unit object. We also classify fiber functors on such categories. Finally, using the Etingof-Kazhdan quantization theory of Poisson algebraic groups, we give a classification of coconnected Hopf algebras, i.e. of unipotent categories of exponential growth with a fiber functor."}
{"category": "Math", "title": "The reverse engineering problem with probabilities and sequential behavior: Probabilistic Sequential Networks", "abstract": "The reverse engineering problem with probabilities and sequential behavior is introducing here, using the expression of an algorithm. The solution is partially founded, because we solve the problem only if we have a Probabilistic Sequential Network. Therefore the probabilistic structure on sequential dynamical systems is introduced here, the new model will be called Probabilistic Sequential Network, PSN. The morphisms of Probabilistic Sequential Networks are defined using two algebraic conditions, whose imply that the distribution of probabilities in the systems are close. It is proved here that two homomorphic Probabilistic Sequential Networks have the same equilibrium or steady state probabilities. Additionally, the proof of the set of PSN with its morphisms form the category PSN, having the category of sequential dynamical systems SDS, as a full subcategory is given. Several examples of morphisms, subsystems and simulations are given."}
{"category": "Math", "title": "Sch'nol's Theorem For Strongly Local Forms", "abstract": "We prove a variant of Sch'nol's theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with $\\delta$- or Kirchhoff boundary conditions."}
{"category": "Math", "title": "Rankin-Cohen Deformations and Representation Theory", "abstract": "In this paper, we use the unitary representation theory of $SL_2(\\mathbb R)$ to understand the Rankin-Cohen brackets for modular forms. Then we use this interpretation to study the corresponding deformation problems that Paula Cohen, Yuri Manin and Don Zagier initiated. Two uniqueness results are established."}
{"category": "Math", "title": "Liouville-type theorems for foliations with complex leaves", "abstract": "We discuss various problems regarding the structure of the foliation of some foliated submanifolds S of C^n, in particular Levi flat ones. As a general scheme, we suppose that S is bounded along a coordinate (or a subset of coordinates), and prove that the leaves of its foliation are complex planes."}
{"category": "Math", "title": "Floer Homology for Symplectomorphism", "abstract": "Let (M,\\omega) be a compact symplectic manifold, and \\phi be a symplectic diffeomorphism on M, we define a Floer-type homology FH_*(\\phi) which is a gen- eralization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on \\phi up to a Hamiltonian isotopy."}
{"category": "Math", "title": "Seip's differentiability concepts as a particular case of the Bertram--Gloeckner--Neeb construction", "abstract": "From the point of view of unification of differentiation theory, it is of interest to note that the general construction principle of Bertram, Gloeckner and Neeb leading to a C^k differentiability concept from a given C^0 one, besides subsuming the Keller--Bastiani C_c^k differentiabilities on real Hausdorff locally convex spaces, also does the same to the \"arc-generated\" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl, and likewise to the \"compactly generated\" theory of Seip's continuous differentiabilities. In this article, we give the details of the proof for the assertion concerning Seip's theory. We also give an example indicating that the premises in Seip's various inverse and implicit function theorems may be too strong in order for these theorems to have much practical value. Also included is a presentation of the BGN--setting reformulated so as to be consistent with the Kelley--Morse--Godel--Bernays--von Neumann type approach to set theory, as well as a treatment of the function space constructions and development of their basic properties needed in the proof of the main result."}
{"category": "Math", "title": "Defining amalgams of compact Lie groups", "abstract": "For $n \\geq 2$ let $\\Delta$ be a Dynkin diagram of rank $n$ and let $I = {1, >..., n}$ be the set of labels of $\\Delta$. A group $G$ admits a weak Phan system of type $\\Delta$ over $\\mathbb{C}$ if $G$ is generated by subgroups $U_i$, $i \\in I$, which are central quotients of simply connected compact semisimple Lie groups of rank one, and contains subgroups $U_{i,j} = \\gen{U_i ,U_j}$, $i \\neq j \\in I$, which are central quotients of simply connected compact semisimple Lie groups of rank two such that $U_i$ and $U_j$ are rank one subgroups of $U_{i,j}$ corresponding to a choice of a maximal torus and a fundamental system of roots for $U_{i,j}$. It is shown in this article that $G$ then is a central quotient of the simply connected compact semisimple Lie group whose complexification is the simply connected complex semisimple Lie group of type $\\Delta$."}
{"category": "Math", "title": "Products in Residue Classes", "abstract": "We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results ``on average'' over moduli m and residue classes modulo m and somewhat stronger results when the average is restricted to prime moduli m = p. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results."}
{"category": "Math", "title": "Markov Chain Modelling for Reliability Estimation of Engineering Systems at Different Scales - Some Considerations", "abstract": "The concepts of probability, statistics and stochastic theory are being successfully used in structural engineering. Markov Chain modelling is a simple stochastic process model that has found its application in both describing stochastic evolution of system and in system reliability estimation. The recent developments in Markov Chain Monte Carlo and the possible integration of Bayesian theory within Markov Chain theory have enhanced its application possibilities. However, the application possibility can be furthered to range over wider scales of application (perhaps from nano- to macro-) by considering the developments in Physics (in particular Quantum Physics). This paper tries to present the results of quantum physics that would help in interpretation of transition probability matrix. However, care has to be taken in the choice of densities in computing the transition probability matrix. The paper is based on available literature, and the aim is only to make an attempt to show how Markov Chain can be used to model systems at various scales."}
{"category": "Math", "title": "Symmetries of quadratic forms classes and of quadratic surds continued fractions. Part I: A Poincar\\'e model for the de Sitter world", "abstract": "The problem of the classification of the indefinite binary quadratic forms with integer coefficients is solved introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Every class of indefinite forms, under the action of the special linear group acting on the integer plane lattice, has a finite and well defined number of representatives inside each one of such domains. This property belongs exclusively to rational points on the one-sheeted hyperboloid. In the second part we will show how to obtain the symmetry type of a class as well as its number of points in all domains from a sole representative of that class."}
{"category": "Math", "title": "Cartier isomorphism and Hodge Theory in the non-commutative case", "abstract": "These are lecture notes from Clay Summer School in Goettingen, in 2006; the lectures were an attempt at an elementary introduction to math.KT/0611623."}
{"category": "Math", "title": "Symmetric Homology of Algebras", "abstract": "In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed simplicial groups and the homological algebra of module-valued functors. The symmetric homology of group algebras is related to stable homotopy theory. Two spectral sequences for computing symmetric homology are constructed. The relation to cyclic homology is discussed and some conjectures and questions towards further work are discussed."}
{"category": "Math", "title": "Equivariant homotopy and deformations of diffeomorphisms", "abstract": "We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups."}
{"category": "Math", "title": "The equivariant cohomology of weighted projective space", "abstract": "We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. We deduce that the ring is a perfect invariant, and prove a Chern class formula for weighted projective bundles."}
{"category": "Math", "title": "A Littlewood-Richardson rule for Grassmannian Permutations", "abstract": "We give a combinatorial rule for computing intersection numbers on a flag manifold which come from products of Schubert classes pulled back from Grassmannian projections. This rule generalizes the known rule for Grassmannians."}
{"category": "Math", "title": "Intransitive geometries", "abstract": "A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion of an amalgam induced by a sufficiently transitive group of automorphisms of that geometry. In the present paper, we generalize this lemma to intransitive geometries, thus opening the door for numerous applications. We treat ourselves some amalgams related to intransitive actions of finite orthogonal groups, as a first class of examples."}
{"category": "Math", "title": "On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow", "abstract": "In this note we construct Nadel multiplier ideal sheaves using the Ricci flow on Fano manifolds. This extends a result of Phong, Sesum and Sturm. These sheaves, like their counterparts constructed by Nadel for the continuity method, can be used to obtain an existence criterion for Kahler-Einstein metrics."}
{"category": "Math", "title": "Every sum system is divisible", "abstract": "We show that every sum system is divisible. Combined with B. V. R. Bhat and R. Srinivasan's result, this shows that every product system arising from a sum system (and every generalized CCR flow) is either of type I or type III. A necessarily and sufficient condition for such a product system to be of type I is obtained."}
{"category": "Math", "title": "On similarity classes of well-rounded sublattices of $\\mathbb Z^2$", "abstract": "A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we study the similarity classes of well-rounded sublattices of ${\\mathbb Z}^2$. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has structure of a noncommutative infinitely generated monoid. We discuss the structure of a given similarity class, and define a zeta function corresponding to each similarity class. We relate it to Dedekind zeta of ${\\mathbb Z}[i]$, and investigate the growth of some related Dirichlet series, which reflect on the distribution of well-rounded lattices. Finally, we construct a sequence of similarity classes of well-rounded sublattices of ${\\mathbb Z}^2$, which gives good circle packing density and converges to the hexagonal lattice as fast as possible with respect to a natural metric we define."}
{"category": "Math", "title": "On the mean square of the divisor function in short intervals", "abstract": "We provide upper bounds for the mean square integral $$ \\int_X^{2X}(\\Delta_k(x+h) - \\Delta_k(x))^2 dx \\qquad(h = h(X)\\gg1, h = o(x) {\\roman{as}} X\\to\\infty) $$ where $h$ lies in a suitable range. For $k\\ge2$ a fixed integer, $\\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k(n)$, generated by $\\zeta^k(s)$."}
{"category": "Math", "title": "On determination of periods of geometric continued fractions for two-dimensional algebraic hyperbolic operators", "abstract": "For a given sequence of positive integers we make an explicit construction of a reduced hyperbolic operator in SL(2,z) with the sequence as a period of a geometric continued fraction in the sense of Klein. Further we experimentally study an algorithm to construct a period for an arbitrary operator of SL(2,z) (the Gauss Reduction Theory)."}
{"category": "Math", "title": "On families of $BS_{can}$ lagrangian tori in projective spaces", "abstract": "This small note contains two simple remarks about Bohr - Sommerfeld with respect to the anticanonical class lagrangian tori in the projective spaces, which are the most popular examples of monotone symplectic manifolds."}
{"category": "Math", "title": "Similarity classes of 3x3 matrices over a local principal ideal ring", "abstract": "In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly."}
{"category": "Math", "title": "The group of automorphisms of the first Weyl algebra in prime characteristic and the restriction map", "abstract": "Let $K$ be a {\\em perfect} field of characteristic $p>0$, $A_1:=K< x, \\der | \\der x- x\\der =1>$ be the first Weyl algebra and $Z:=K[X:=x^p, Y:=\\der^p]$ be its centre. It is proved that $(i)$ the restriction map $\\res :\\Aut_K(A_1)\\ra \\Aut_K(Z), \\s \\mapsto \\s|_Z$, is a monomorphism with $\\im (\\res) = \\G :=\\{\\tau \\in \\Aut_K(Z) | \\CJ (\\tau) =1\\}$ where $\\CJ (\\tau) $ is the Jacobian of $\\tau$ (note that $\\Aut_K(Z)=K^*\\ltimes \\G$ and if $K$ is {\\em not} perfect then $\\im (\\res) \\neq \\G$); $(ii)$ the bijection $\\res : \\Aut_K(A_1) \\ra \\G$ is a monomorphism of infinite dimensional algebraic groups which is {\\em not} an isomorphism (even if $K$ is algebraically closed); $(iii)$ an explicit formula for $\\res^{-1}$ is found via differential operators $\\CD (Z)$ on $Z$ and negative powers of the Fronenius map $F$. Proofs are based on the following (non-obvious) equality proved in the paper: $$ (\\frac{d}{dx}+f)^p= (\\frac{d}{dx})^p+\\frac{d^{p-1}f}{dx^{p-1}}+f^p, f\\in K[x].$$"}
{"category": "Math", "title": "Classifying finite localizations of quasi-coherent sheaves", "abstract": "Given a quasi-compact, quasi-separated scheme X, a bijection between the tensor localizing subcategories of finite type in Qcoh(X) and the set of all subsets $Y\\subseteq X$ of the form $Y=\\bigcup_{i\\in\\Omega}Y_i$, with $X\\setminus Y_i$ quasi-compact and open for all $i\\in\\Omega$, is established. As an application, there is constructed an isomorphism of ringed spaces (X,O_X)-->(Spec(Qcoh(X)),O_{Qcoh(X)}), where $(Spec(Qcoh(X)),O_{Qcoh(X)})$ is a ringed space associated to the lattice of tensor localizing subcategories of finite type. Also, a bijective correspondence between the tensor thick subcategories of perfect complexes $\\perf(X)$ and the tensor localizing subcategories of finite type in Qcoh(X) is established."}
{"category": "Math", "title": "Kontsevich formality and PBW algebras", "abstract": "This paper is based on the author's paper \"Koszul duality in deformation quantization, I\", with some improvements. In particular, an Introduction is added, and the convergence of the spectral sequence in Lemma 2.1 is rigorously proven. Some informal discussion in Section 1.5 is added."}
{"category": "Math", "title": "Erratum to: Deformation Quantization in Algebraic Geometry", "abstract": "This note contains a correction of the proofs of the main results of the paper [A. Yekutieli, Deformation quantization in algebraic geometry, Adv. Math. 198 (2005), 383-432]. The results are correct as originally stated."}
{"category": "Math", "title": "Some inequalities for $(\\alpha, \\beta)$-normal operators in Hilbert spaces", "abstract": "An operator $T$ acting on a Hilbert space is called $(\\alpha ,\\beta)$-normal ($0\\leq \\alpha \\leq 1\\leq \\beta $) if \\begin{equation*} \\alpha ^{2}T^{\\ast }T\\leq TT^{\\ast}\\leq \\beta ^{2}T^{\\ast}T. \\end{equation*} In this paper we establish various inequalities between the operator norm and its numerical radius of $(\\alpha ,\\beta)$-normal operators in Hilbert spaces. For this purpose, we employ some classical inequalities for vectors in inner product spaces."}
{"category": "Math", "title": "Counterexample to a geodesic length conjecture on the 2-sphere", "abstract": "Paper withdrawn by the author."}
{"category": "Math", "title": "The Egorov theorem for transverse Dirac type operators on foliated manifolds", "abstract": "Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations."}
{"category": "Math", "title": "Complex algebraic curves. Annuli", "abstract": "We provide the full classification of algebraic embeddings of $\\mathbb{C}^*$ into $\\mathbb{C}^2$ satisfying certain regularity condition, which conjecturally holds for all algebraic maps from $\\mathbb{C}^*$ into $\\mathbb{C}^2$. The resulting list comprises 1 smooth family, 18 discrete families and 4 special cases. Any embedding known to us can be reduced to one of this list by a de Jonqui\\`ere transform and a suitable change of variables. The classification uses in general tools from previous work \"Complex algebraic curves via Poincare--Hopf formula. I. Parametric lines.\" (Pacific. J. Math. 229 (2007) No. 2, 307--338): we carefully estimate Milnor numbers of singularities that may appear in the embedding of $\\mathbb{C}^*$. We use the regularity condition to bound the sum of so--called codimensions of singular points. The detailed discussion of this condition can be found in http://www.mimuw.edu.pl/~mcboro/pliki/artykuly/curv4.pdf"}
{"category": "Math", "title": "Harnack inequality and applications for stochastic generalized porous media equations", "abstract": "By using coupling and Girsanov transformations, the dimension-free Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations. As applications, explicit upper bounds of the $L^p$-norm of the density as well as hypercontractivity, ultracontractivity and compactness of the corresponding semigroup are derived."}
{"category": "Math", "title": "Major Indices and Perfect Bases for Complex Reflection Groups", "abstract": "It is shown that, under mild conditions, a complex reflection group $G(r,p,n)$ may be decomposed into a set-wise direct product of cyclic subgroups. This property is then used to extend the notion of major index and a corresponding Hilbert series identity to these and other closely related groups."}
{"category": "Math", "title": "Curve crossing for random walks reflected at their maximum", "abstract": "Let $R_n=\\max_{0\\leq j\\leq n}S_j-S_n$ be a random walk $S_n$ reflected in its maximum. Except in the trivial case when $P(X\\ge0)=1$, $R_n$ will pass over a horizontal boundary of any height in a finite time, with probability 1. We extend this by giving necessary and sufficient conditions for finiteness of passage times of $R_n$ above certain curved (power law) boundaries, as well. The intuition that a degree of heaviness of the negative tail of the distribution of the increments of $S_n$ is necessary for passage of $R_n$ above a high level is correct in most, but not all, cases, as we show. Conditions are also given for the finiteness of the expected passage time of $R_n$ above linear and square root boundaries."}
{"category": "Math", "title": "Possible connections between whiskered categories and groupoids, many object Leibniz algebras, automorphism structures and local-to-global questions", "abstract": "We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered $R$-categories, thus answering questions of what might be `commutative versions' of these theories. We relate these ideas to the theory of Leibniz algebras, but the commutator theory here does not satisfy the Leibniz identity. We also discuss potential applications and extensions, for example to resolutions of monoids."}
{"category": "Math", "title": "Vector bundles on degenerations of elliptic curves and Yang-Baxter equations", "abstract": "In this paper we introduce the notion of a geometric associative r-matrix attached to a genus one fibration with a section and irreducible fibres. It allows us to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. We also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstrass cubic curves."}
{"category": "Math", "title": "Derived Semidistributive Lattices", "abstract": "For L a finite lattice, let C(L) denote the set of pairs g = (g_0,g_1) such that g_0 is a lower cover of g_1 and order it as follows: g <= d iff g_0 <= d_0, g_1 <= d_1, but not g_1 <= d_0. Let C(L,g) denote the connected component of g in this poset. Our main result states that C(L,g) is a semidistributive lattice if L is semidistributive, and that C(L,g) is a bounded lattice if L is bounded. Let S_n be the permutohedron on n letters and T_n be the associahedron on n+1 letters. Explicit computations show that C(S_n,a) = S_{n-1} and C(T_n,a) = T_{n-1}, up to isomorphism, whenever a is an atom. These results are consequences of new characterizations of finite join semidistributive and finite lower bounded lattices: (i) a finite lattice is join semidistributive if and only if the projection sending g in C(L) to g_0 in L creates pullbacks, (ii) a finite join semidistributive lattice is lower bounded if and only if it has a strict facet labelling. Strict facet labellings, as defined here, are generalization of the tools used by Barbut et al. to prove that lattices of Coxeter groups are bounded."}
{"category": "Math", "title": "Weak Solutions of Stochastic Differential Equations over the Field of p-Adic Numbers", "abstract": "Study of stochastic differential equations on the field of p-adic numbers was initiated by the second author and has been developed by the first author, who proved several results for the p-adic case, similar to the theory of ordinary stochastic integral with respect to Levy processes on the Euclidean spaces. In this article, we present an improved definition of a stochastic integral on the field and prove the joint (time and space) continuity of the local time for p-adic stable processes. Then we use the method of random time change to obtain sufficient conditions for the existence of a weak solution of a stochastic differential equation on the field, driven by the p-adic stable process, with a Borel measurable coefficient."}
{"category": "Math", "title": "The Minimum Rank Problem: a counterexample", "abstract": "We provide a counterexample to a recent conjecture that the minimum rank of every sign pattern matrix can be realized by a rational matrix. We use one of the equivalences of the conjecture and some results from projective geometry. As a consequence of the counterexample, we show that there is a graph for which the minimum rank over the reals is strictly smaller than the minimum rank over the rationals. We also make some comments on the minimum rank of sign pattern matrices over different subfields of $\\mathbb R$."}
{"category": "Math", "title": "Generators of simple Lie algebras in arbitrary characteristics", "abstract": "In this paper we study the minimal number of generators for simple Lie algebras in characteristic 0 or p > 3. We show that any such algebra can be generated by 2 elements. We also examine the 'one and a half generation' property, i.e. when every non-zero element can be completed to a generating pair. We show that classical simple algebras have this property, and that the only simple Cartan type algebras of type W which have this property are the Zassenhaus algebras."}
{"category": "Math", "title": "Weighted projective spaces and minimal nilpotent orbits", "abstract": "We investigate (twisted) rings of differential operators on the resolution of singularities of a particular irreducible component of the (Zarisky) closure of the minimal orbit $\\bar O_{\\mathrm{min}}$ of $\\mathfrak{sp}_{2n}$, intersected with the Borel subalgebra $\\mathfrak n_+$ of $\\mathfrak{sp}_{2n}$, using toric geometry and show that they are homomorphic images of a subalgebra of the Universal Enveloping Algebra (UEA) of $\\mathfrak{sp}_{2n}$, which contains the maximal parabolic subalgebra $\\mathfrak p$ determining the minimal nilpotent orbit. Further, using Fourier transforms on Weyl algebras, we show that (twisted) rings of well-suited weighted projective spaces are obtained from the same subalgebra. Finally, investigating this subalgebra from the representation-theoretical point of view, we find new primitive ideals and rediscover old ones for the UEA of $\\mathfrak{sp}_{2n}$ coming from the aforementioned resolution of singularities."}
{"category": "Math", "title": "Dynamics of Jackson networks: perturbation theory", "abstract": "We introduce a new formalism for dealing with networks of queues. The formalism is based on the Doi-Peliti second quantization method for reaction diffusion systems. As a demonstration of the method's utility we compute perturbatively the different time busy-busy correlations between two servers in a Jackson network."}
{"category": "Math", "title": "On asymptotics of eigenvectors of large sample covariance matrix", "abstract": "Let \\{$X_{ij}$\\}, $i,j=...,$ be a double array of i.i.d. complex random variables with $EX_{11}=0,E|X_{11}|^2=1$ and $E|X_{11}|^4<\\infty$, and let $A_n=\\frac{1}{N}T_n^{{1}/{2}}X_nX_n^*T_n^{{1}/{2}}$, where $T_n^{{1}/{2}}$ is the square root of a nonnegative definite matrix $T_n$ and $X_n$ is the $n\\times N$ matrix of the upper-left corner of the double array. The matrix $A_n$ can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix $T_n$, or as a multivariate $F$ matrix if $T_n$ is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of $A_n$, a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if \\{$X_{ij}$\\} and $T_n$ are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of $A_n$ are proved to have Gaussian limits, which suggests that the eigenvector matrix of $A_n$ is nearly Haar distributed when $T_n$ is a multiple of the identity matrix, an easy consequence for a Wishart matrix."}
{"category": "Math", "title": "Differential inequalities of continuous functions and removing singularities of Rado type for J-holomorphic maps", "abstract": "We consider a continuous function $f$ on a domain in $\\mathbf C^n$ satisfying the inequality that $|\\bar \\partial f|\\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable singularity theorem of Rado type for J-holomorphic maps. Let $\\Omega$ be an open subset in $\\mathbf C$ and let $E$ be a closed polar subset of $\\Omega$. Let $u$ be a continuous map from $\\Omega$ into an almost complex manifold $(M,J)$ with $J$ of class $C^1$. We show that if $u$ is J-holomorphic on $\\Omega\\setminus E$ then it is J-holomorphic on $\\Omega$."}
{"category": "Math", "title": "Tropical bases by regular projections", "abstract": "We consider the tropical variety $\\mathcal{T}(I)$ of a prime ideal $I$ generated by the polynomials $f_1, ..., f_r$ and revisit the regular projection technique introduced by Bieri and Groves from a computational point of view. In particular, we show that $I$ has a short tropical basis of cardinality at most $r + \\codim I + 1$ at the price of increased degrees, and we provide a computational description of these bases."}
{"category": "Math", "title": "Conical limit sets and continued fractions", "abstract": "Inspired by questions of convergence in continued fraction theory, Erd\\H{o}s, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of M\\\"obius maps acting on the Riemann sphere, $S^2$. By identifying $S^2$ with the boundary of three-dimensional hyperbolic space, $H^3$, we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of $H^3$. Using hyperbolic geometry, we give simple geometric proofs of the theorems of Erd\\H{o}s, Piranian and Thron that generalise to arbitrary dimensions. New results are also obtained about the class of conical limit sets, for example, that it is closed under locally quasisymmetric homeomorphisms. Applications are given to continued fractions."}
{"category": "Math", "title": "A distance based test on random trees", "abstract": "In this paper, we address the question of comparison between populations of trees. We study an statistical test based on the distance between empirical mean trees, as an analog of the two sample z statistic for comparing two means. Despite its simplicity, we can report that the test is quite powerful to separate distributions with different means but it does not distinguish between different populations with the same mean, a more complicated test should be applied in that setting. The performance of the test is studied via simulations on Galton-Watson branching processes. We also show an application to a real data problem in genomics."}
{"category": "Math", "title": "The infinite valley for a recurrent random walk in random environment", "abstract": "We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time."}
{"category": "Math", "title": "The inner automorphism 3-group of a strict 2-group", "abstract": "Any group $G$ gives rise to a 2-group of inner automorphisms, $\\mathrm{INN}(G)$. It is an old result by Segal that the nerve of this is the universal $G$-bundle. We discuss that, similarly, for every 2-group $G_{(2)}$ there is a 3-group $\\mathrm{INN}(G_{(2)})$ and a slightly smaller 3-group $\\mathrm{INN}_0(G_{(2)})$ of inner automorphisms. We describe these for $G_{(2)}$ any strict 2-group, discuss how $\\mathrm{INN}_0(G_{(2)})$ can be understood as arising from the mapping cone of the identity on $G_{(2)}$ and show that its underlying 2-groupoid structure fits into a short exact sequence $G_{(2)} \\to \\mathrm{INN}_0(G_{(2)}) \\to \\Sigma G_{(2)}$. As a consequence, $\\mathrm{INN}_0(G_{(2)})$ encodes the properties of the universal $G_{(2)}$ 2-bundle."}
{"category": "Math", "title": "A universal deformation formula for Connes-Moscovici's Hopf algebra without any projective structure", "abstract": "We construct a universal deformation formula for Connes-Moscovici's Hopf algebra without any projective structure using Fedosov's quantization of symplectic diffeomorphisms."}
{"category": "Math", "title": "The multiplicity of weights in nonprimitive pairs of weights", "abstract": "For each type of classical Lie algebra, we list the dominant highest weights $\\zeta$ for which $(\\zeta;\\mu_i)$ is not a primitive pair and the weight space $V_{\\mu_i}$ has dimension one where $\\mu_i$ are the highest long and short roots in each case. These dimension one weight spaces lead to examples of nilmanifolds for which we cannot prove or disprove the density of closed geodesics."}
{"category": "Math", "title": "Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations", "abstract": "This paper concerns with numerical approximations of solutions of second order fully nonlinear partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for second order fully nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called vanishing moment method, hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods with \"guaranteed\" convergence. The main idea of the proposed vanishing moment method is to approximate a second order fully nonlinear PDE by a higher order, in particular, a fourth order quasilinear PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist."}
{"category": "Math", "title": "Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group", "abstract": "We consider the asymptotic behavior as $n\\to\\infty$ of the spectra of random matrices of the form \\[\\frac{1}{\\sqrt{n-1}}\\sum_{k=1}^{n-1}Z_{nk}\\rho_n ((k,k+1)),\\] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian and the matrices $\\rho_n((k,k+1))$ are obtained by applying an irreducible unitary representation $\\rho_n$ of the symmetric group on $\\{1,2,...,n\\}$ to the transposition $(k,k+1)$ that interchanges $k$ and $k+1$ [thus, $\\rho_n((k,k+1))$ is both unitary and self-adjoint, with all eigenvalues either +1 or -1]. Irreducible representations of the symmetric group on $\\{1,2,...,n\\}$ are indexed by partitions $\\lambda_n$ of $n$. A consequence of the results we establish is that if $\\lambda_{n,1}\\ge\\lambda_{n,2}\\ge...\\ge0$ is the partition of $n$ corresponding to $\\rho_n$, $\\mu_{n,1}\\ge\\mu_{n,2}\\ge >...\\ge0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of $\\mu_n$ is the transpose of the Young diagram of $\\lambda_n$), $\\lim_{n\\to\\infty}\\frac{\\lambda_{n,i}}{n}=p_i$ for each $i\\ge1$, and $\\lim_{n\\to\\infty}\\frac{\\mu_{n,j}}{n}=q_j$ for each $j\\ge1$, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean $\\theta Z$ and variance $1-\\theta^2$, where $\\theta$ is the constant $\\sum_ip_i^2-\\sum_jq_j^2$ and $Z$ is a standard Gaussian random variable."}
{"category": "Math", "title": "Extensions of McCoy Rings", "abstract": "A ring $R$ is said to be right McCoy if the equation $f(x)g(x)=0,$ where $f(x)$ and $g(x)$ are nonzero polynomials of $R[x],$ implies that there exists nonzero $s \\in R$ such that $f(x)s = 0$. It is proven that no proper (triangular) matrix ring is one-sided McCoy. If there exists the classical right quotient ring $Q$ of a ring $R$, then $R$ is right McCoy if and only if $Q$ is right McCoy. It is shown that for many polynomial extensions, a ring $R$ is right McCoy if and only if the polynomial extension over $R$ is right McCoy. Other basic extensions of right McCoy rings are also studied.\\leftskip0truemm \\rightskip0truemm \\{\\it Keywords}: matrix ring, McCoy ring, polynomial ring, upper triangular matrix ring."}
{"category": "Math", "title": "On Dissipative Quadratic Stochastic Operators", "abstract": "In present paper we introduce the notion of dissipative quadratic stochastic operator and cubic stochastic operator. We prove necessary conditions for dissipativity of quadratic stochastic operators. Besides, it is studied certain limit behavior of such operators. Finally we prove ergodic theorem for dissipative operators."}
{"category": "Math", "title": "Reducing variance in univariate smoothing", "abstract": "A variance reduction technique in nonparametric smoothing is proposed: at each point of estimation, form a linear combination of a preliminary estimator evaluated at nearby points with the coefficients specified so that the asymptotic bias remains unchanged. The nearby points are chosen to maximize the variance reduction. We study in detail the case of univariate local linear regression. While the new estimator retains many advantages of the local linear estimator, it has appealing asymptotic relative efficiencies. Bandwidth selection rules are available by a simple constant factor adjustment of those for local linear estimation. A simulation study indicates that the finite sample relative efficiency often matches the asymptotic relative efficiency for moderate sample sizes. This technique is very general and has a wide range of applications."}
{"category": "Math", "title": "Quasi-Nilpotent Operators on Locally Convex Spaces", "abstract": "In this article we extend the notion of quasi-nilpotent equivalent operators, introduced by Colojoara and Foias \\cite{co1} for Banach spaces, to the class of bounded operators on sequentially complete locally convex spaces."}
{"category": "Math", "title": "Confidence sets for split points in decision trees", "abstract": "We investigate the problem of finding confidence sets for split points in decision trees (CART). Our main results establish the asymptotic distribution of the least squares estimators and some associated residual sum of squares statistics in a binary decision tree approximation to a smooth regression curve. Cube-root asymptotics with nonnormal limit distributions are involved. We study various confidence sets for the split point, one calibrated using the subsampling bootstrap, and others calibrated using plug-in estimates of some nuisance parameters. The performance of the confidence sets is assessed in a simulation study. A motivation for developing such confidence sets comes from the problem of phosphorus pollution in the Everglades. Ecologists have suggested that split points provide a phosphorus threshold at which biological imbalance occurs, and the lower endpoint of the confidence set may be interpreted as a level that is protective of the ecosystem. This is illustrated using data from a Duke University Wetlands Center phosphorus dosing study in the Everglades."}
{"category": "Math", "title": "Fast rates for support vector machines using Gaussian kernels", "abstract": "For binary classification we establish learning rates up to the order of $n^{-1}$ for support vector machines (SVMs) with hinge loss and Gaussian RBF kernels. These rates are in terms of two assumptions on the considered distributions: Tsybakov's noise assumption to establish a small estimation error, and a new geometric noise condition which is used to bound the approximation error. Unlike previously proposed concepts for bounding the approximation error, the geometric noise assumption does not employ any smoothness assumption."}
{"category": "Math", "title": "Remarks on symplectic twistor spaces", "abstract": "We consider some classical fibre bundles furnished with almost complex structures of twistor type, deduce their integrability in some cases and study \\textit{self-holomorphic} sections of a \\textit{symplectic} twistor space. With these we define a moduli space of $\\omega$-compatible complex structures. We recall the theory of flag manifolds in order to study the Siegel domain and other domains alike, which is the fibre of the referred twistor space. Finally the structure equations for the twistor of a Riemann surface with the canonical symplectic-metric connection are deduced, based on a given conformal coordinate on the surface. We then relate with the moduli space defined previously."}
{"category": "Math", "title": "Cram\\'{e}r-type large deviations for samples from a finite population", "abstract": "Cram\\'{e}r-type large deviations for means of samples from a finite population are established under weak conditions. The results are comparable to results for the so-called self-normalized large deviation for independent random variables. Cram\\'{e}r-type large deviations for the finite population Student $t$-statistic are also investigated."}
{"category": "Math", "title": "Posterior convergence rates of Dirichlet mixtures at smooth densities", "abstract": "We study the rates of convergence of the posterior distribution for Bayesian density estimation with Dirichlet mixtures of normal distributions as the prior. The true density is assumed to be twice continuously differentiable. The bandwidth is given a sequence of priors which is obtained by scaling a single prior by an appropriate order. In order to handle this problem, we derive a new general rate theorem by considering a countable covering of the parameter space whose prior probabilities satisfy a summability condition together with certain individual bounds on the Hellinger metric entropy. We apply this new general theorem on posterior convergence rates by computing bounds for Hellinger (bracketing) entropy numbers for the involved class of densities, the error in the approximation of a smooth density by normal mixtures and the concentration rate of the prior. The best obtainable rate of convergence of the posterior turns out to be equivalent to the well-known frequentist rate for integrated mean squared error $n^{-2/5}$ up to a logarithmic factor."}
{"category": "Math", "title": "On the design-consistency property of hierarchical Bayes estimators in finite population sampling", "abstract": "We obtain a limit of a hierarchical Bayes estimator of a finite population mean when the sample size is large. The limit is in the sense of ordinary calculus, where the sample observations are treated as fixed quantities. Our result suggests a simple way to correct the hierarchical Bayes estimator to achieve design-consistency, a well-known property in the traditional randomization approach to finite population sampling. We also suggest three different measures of uncertainty of our proposed estimator."}
{"category": "Math", "title": "On rates of convergence for posterior distributions in infinite-dimensional models", "abstract": "This paper introduces a new approach to the study of rates of convergence for posterior distributions. It is a natural extension of a recent approach to the study of Bayesian consistency. In particular, we improve on current rates of convergence for models including the mixture of Dirichlet process model and the random Bernstein polynomial model."}
{"category": "Math", "title": "Resolvable designs with large blocks", "abstract": "Resolvable designs with two blocks per replicate are studied from an optimality perspective. Because in practice the number of replicates is typically less than the number of treatments, arguments can be based on the dual of the information matrix and consequently given in terms of block concurrences. Equalizing block concurrences for given block sizes is often, but not always, the best strategy. Sufficient conditions are established for various strong optimalities and a detailed study of E-optimality is offered, including a characterization of the E-optimal class. Optimal designs are found to correspond to balanced arrays and an affine-like generalization."}
{"category": "Math", "title": "Gegenbauer tau methods with and without spurious eigenvalues", "abstract": "It is proven that a class of Gegenbauer tau approximations to a 4th order differential eigenvalue problem of hydrodynamic type provide real, negative and distinct eigenvalues, as is the case for the exact solutions. This class of Gegenbauer tau methods includes Chebyshev and Legendre Galerkin and `inviscid' Galerkin but does not include Chebyshev and Legendre tau. Rigorous and numerical results show that the results are sharp: positive or complex eigenvalues arise outside of this class. The widely used modified tau approach is proved to be equivalent to the Galerkin method."}
{"category": "Math", "title": "On the number of support points of maximin and Bayesian optimal designs", "abstract": "We consider maximin and Bayesian $D$-optimal designs for nonlinear regression models. The maximin criterion requires the specification of a region for the nonlinear parameters in the model, while the Bayesian optimality criterion assumes that a prior for these parameters is available. On interval parameter spaces, it was observed empirically by many authors that an increase of uncertainty in the prior information (i.e., a larger range for the parameter space in the maximin criterion or a larger variance of the prior in the Bayesian criterion) yields a larger number of support points of the corresponding optimal designs. In this paper, we present analytic tools which are used to prove this phenomenon in concrete situations. The proposed methodology can be used to explain many empirically observed results in the literature. Moreover, it explains why maximin $D$-optimal designs are usually supported at more points than Bayesian $D$-optimal designs."}
{"category": "Math", "title": "A random walk approximation to fractional Brownian motion", "abstract": "We present a random walk approximation to fractional Brownian motion where the increments of the fractional random walk are defined as a weighted sum of the past increments of a Bernoulli random walk."}
{"category": "Math", "title": "Complete enumeration of two-Level orthogonal arrays of strength $d$ with $d+2$ constraints", "abstract": "Enumerating nonisomorphic orthogonal arrays is an important, yet very difficult, problem. Although orthogonal arrays with a specified set of parameters have been enumerated in a number of cases, general results are extremely rare. In this paper, we provide a complete solution to enumerating nonisomorphic two-level orthogonal arrays of strength $d$ with $d+2$ constraints for any $d$ and any run size $n=\\lambda2^d$. Our results not only give the number of nonisomorphic orthogonal arrays for given $d$ and $n$, but also provide a systematic way of explicitly constructing these arrays. Our approach to the problem is to make use of the recently developed theory of $J$-characteristics for fractional factorial designs. Besides the general theoretical results, the paper presents some results from applications of the theory to orthogonal arrays of strength two, three and four."}
{"category": "Math", "title": "Uniformly root-$N$ consistent density estimators for weakly dependent invertible linear processes", "abstract": "Convergence rates of kernel density estimators for stationary time series are well studied. For invertible linear processes, we construct a new density estimator that converges, in the supremum norm, at the better, parametric, rate $n^{-1/2}$. Our estimator is a convolution of two different residual-based kernel estimators. We obtain in particular convergence rates for such residual-based kernel estimators; these results are of independent interest."}
{"category": "Math", "title": "Metrics for sparse graphs", "abstract": "Recently, Bollob\\'as, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with $\\Theta(n)$ edges. Roughly speaking, there is one model for each {\\em kernel}, i.e., each symmetric measurable function from $[0,1]^2$ to the non-negative reals, although the details are much more complicated. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lov\\'asz, S\\'os, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with $n$ vertices and $\\Theta(n^2)$ edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of {\\em graphons}, which are essentially bounded kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasi-random graphs are in a sense completely general: any sequence of dense graphs contains such a subsequence. Our aim here is to briefly survey these results, and then to investigate to what extent they can be generalized to graphs with $o(n^2)$ edges. Although many of the definitions extend in a simple way, the connections between the various metrics, and between the metrics and random graph models, turn out to be much more complicated than in the dense case. We shall prove many partial results, and state even more conjectures and open problems, whose resolution would greatly enhance the currently rather unsatisfactory theory of metrics on sparse graphs. This paper deals mainly with graphs with $o(n^2)$ but $\\omega(n)$ edges: a companion paper [arXiv:0812.2656] will discuss the (more problematic still) case of {\\em extremely sparse} graphs, with O(n) edges."}
{"category": "Math", "title": "Cartesian Bicategories II", "abstract": "The notion of cartesian bicategory, introduced by Carboni and Walters for locally ordered bicategories, is extended to general bicategories. It is shown that a cartesian bicategory is a symmetric monoidal bicategory."}
{"category": "Math", "title": "Cross Curvature Flow on Locally Homogenous Three-manifolds (I)", "abstract": "Chow and Hamilton introduced the cross curvature flow on closed 3-manifolds with negative or positive sectional curvature. In this paper, we study the negative cross curvature flow in the case of locally homogenous metrics on 3-manifolds. In each case, we describe the long time behavior of the solutions of the corresponding ODE system."}
{"category": "Math", "title": "Some graph properties determined by edge zeta functions", "abstract": "Stark and Terras introduced the edge zeta function of a finite graph in 1996. The edge zeta function is the reciprocal of a polynomial in twice as many variables as edges in the graph and can be computed in polynomial time. We look at graph properties which we can determine using the edge zeta function. In particular, the edge zeta function is enough to deduce the clique number, the number of Hamiltonian cycles, and whether a graph is perfect or chordal. Actually computing these properties takes exponential time. Finally, we present a new example illustrating that the Ihara zeta function cannot necessarily do the same."}
{"category": "Math", "title": "Frobenius Objects in Cartesian Bicategories", "abstract": "Maps (left adjoint arrows) between Frobenius objects in a cartesian bicategory B are precisely comonoid homomorphisms and, for A Frobenius and any T in B, map(B)(T,A) is a groupoid."}
{"category": "Math", "title": "Stochastic bounds for two-layer loss systems", "abstract": "This paper studies multiclass loss systems with two layers of servers, where each server at the first layer is dedicated to a certain customer class, while the servers at the second layer can handle all customer classes. The routing of customers follows an overflow scheme, where arriving customers are preferentially directed to the first layer. Stochastic comparison and coupling techniques are developed for studying how the system is affected by packing of customers, altered service rates, and altered server configurations. This analysis leads to easily computable upper and lower bounds for the performance of the system."}
{"category": "Math", "title": "Rank-based estimation for all-pass time series models", "abstract": "An autoregressive-moving average model in which all roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa is called an all-pass time series model. All-pass models are useful for identifying and modeling noncausal and noninvertible autoregressive-moving average processes. We establish asymptotic normality and consistency for rank-based estimators of all-pass model parameters. The estimators are obtained by minimizing the rank-based residual dispersion function given by Jaeckel [Ann. Math. Statist. 43 (1972) 1449--1458]. These estimators can have the same asymptotic efficiency as maximum likelihood estimators and are robust. The behavior of the estimators for finite samples is studied via simulation and rank estimation is used in the deconvolution of a simulated water gun seismogram."}
{"category": "Math", "title": "A Two-Parameter Family of Infinite-Dimensional Diffusions in the Kingman Simplex", "abstract": "The aim of the paper is to introduce a two-parameter family of infinite-dimensional diffusion processes X(alpha,theta) related to Pitman's two-parameter Poisson-Dirichlet distributions PD(alpha,theta). The diffusions X(alpha,theta) are obtained in a scaling limit transition from certain finite Markov chains on partitions of natural numbers. The state space of X(alpha,theta) is an infinite-dimensional simplex called the Kingman simplex. In the special case when parameter alpha vanishes, our finite Markov chains are similar to Moran-type model in population genetics, and our diffusion processes reduce to the infinitely-many-neutral-alleles diffusion model studied by Ethier and Kurtz (1981). Our main results extend those of Ethier and Kurtz to the two-parameter case and are as follows: The Poisson-Dirichlet distribution PD(alpha,theta) is a unique stationary distribution for the corresponding process X(alpha,theta); the process is ergodic and reversible; the spectrum of its generator is explicitly described. The general two-parameter case seems to fall outside the setting of models of population genetics, and our approach differs in some aspects from that of Ethier and Kurtz. We also consider the case of degenerate series of parameters alpha and theta and conclude that the diffusions in finite-dimensional simplexes studied by Ethier and Kurtz (1981) arise as a special case of our two-parameter family of diffusions."}
{"category": "Math", "title": "The asymptotic geometry of right-angled Artin groups, I", "abstract": "We study atomic right-angled Artin groups -- those whose defining graph has no cycles of length less than five, and no separating vertices, separating edges, or separating vertex stars. We show that these groups are not quasi-isometrically rigid, but that an intermediate form of rigidity does hold. We deduce from this that two atomic groups are quasi-isometric iff they are isomorphic."}
{"category": "Math", "title": "Magnetic flows on Sol-manifolds: dynamical and symplectic aspects", "abstract": "We consider magnetic flows on compact quotients of the 3-dimensional solvable geometry Sol determined by the usual left-invariant metric and the distinguished monopole. We show that these flows have positive Liouville entropy and therefore are never completely integrable. This should be compared with the known fact that the underlying geodesic flow is completely integrable in spite of having positive topological entropy. We also show that for a large class of twisted cotangent bundles of solvable manifolds every compact set is displaceable."}
{"category": "Math", "title": "H\\\"older forms and integrability of invariant distributions", "abstract": "We prove an inequality for H\\\"older continuous differential forms on compact manifolds in which the integral of the form over the boundary of a sufficiently small, smoothly immersed disk is bounded by a certain multiplicative convex combination of the volume of the disk and the area of its boundary. This inequality has natural applications in dynamical systems, where H\\\"older continuity is ubiquitous. We give two such applications. In the first one, we prove a criterion for the existence of global cross sections to Anosov flows in terms of their expansion-contraction rates. The second application provides an analogous criterion for non-accessibility of partially hyperbolic diffeomorphisms."}
{"category": "Math", "title": "On the Hopf-Schur group of a field", "abstract": "Let k be any field. We consider the Hopf-Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a Hopf algebra over k, revealing so that the Hopf-Schur group can be much larger than the Schur group of k."}
{"category": "Math", "title": "On bilinear biquandles", "abstract": "We define a type of biquandle which is a generalization of symplectic quandles. We use the extra structure of these bilinear biquandles to define new knot and link invariants and give some examples."}
{"category": "Math", "title": "Tail Asymptotics and Estimation for Elliptical Distributions", "abstract": "Let (X,Y) be a bivariate elliptical random vector with associated random radius in the Gumbel max-domain of attraction. In this paper we obtain a second order asymptotic expansion of the joint survival probability P(X > x, Y> y) for x,y large. Further, based on the asymptotic bounds we discuss some aspects of the statistical modelling of joint survival probabilities and the survival conditional excess probability."}
{"category": "Math", "title": "Some quasinilpotent generators of the hyperfinite $\\mathrm{II}_1$ factor", "abstract": "For each sequence $\\{c_n\\}_n$ in $l_{1}(\\N)$ we define an operator $A$ in the hyperfinite $\\mathrm{II}_1$-factor $\\mathcal{R}$. We prove that these operators are quasinilpotent and they generate the whole hyperfinite $\\mathrm{II}_1$-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of $A$ are equally distributed, and we find a combinatorial formula as well as an analytical way to compute their moments. We present a combinatorial way of computing the moments of $A^{*}A$."}
{"category": "Math", "title": "Linear Algebraic Groups without the Normalizer Theorem", "abstract": "One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps in the standard development, including the self-normalizing property of Borel subgroups."}
{"category": "Math", "title": "Advances in Cardinal Arithmetic", "abstract": "If cf(kappa) = kappa, kappa^+< cf(lambda) = \\lambda, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing clubs and for each alpha<lambda we have: {C_delta \\cap alpha: alpha \\in nacc(C_delta)} has cardinality <lambda. Also, we prove that e.g. there is a stationary subset of S_{<aleph_1}(lambda) of cardinality cf(S_{<aleph_1}(lambda),subseteq) Then we prove the existence of nice filters when instead being normal filters on omega_1 they are normal filters with larger domains, which can increase during a play. They can help us transfer situation on aleph_1-complete filters to normal ones"}
{"category": "Math", "title": "The first almost free Whitehead group", "abstract": "Assume G.C.H. and kappa is the first uncountable cardinal such that there is a kappa-free abelian group which is not a Whitehead (abelian) group. We prove that kappa is necessarily an inaccessible cardinal"}
{"category": "Math", "title": "On some problems in general topology", "abstract": "We prove that Arhangelskii's problem has a consistent positive answer: if V\\models CH, then for some aleph_1-complete aleph_2-c.c. forcing notion P of cardinality aleph_2 we have that P forces ``CH and there is a Lindelof regular topological space of size aleph_2 with clopen basis with every point of pseudo-character aleph_0 (i.e. each singleton is the intersection of countably many open sets)''. Also, we prove the consistency of: CH+ 2^{aleph_1} > \\aleph_2 + \"there is no space as above with aleph_2 points\" (starting with a weakly compact cardinal)."}
{"category": "Math", "title": "Abelian and non-abelian second cohomologies of quantized enveloping algebras", "abstract": "For a class of pointed Hopf algebras including the quantized enveloping algebras, we discuss cleft extensions, cocycle deformations and the second cohomology. We present such a non-standard method of computing the abelian second cohomology that derives information from the non-abelian second cohomology classifying cleft extensions. As a sample computation, a quantum analogue of Whitehead's second lemma for Lie-algebra cohomology is proved."}
{"category": "Math", "title": "Density is at most the spread of the square", "abstract": "We show the following result: Assume B is an infinite Boolean Algebra and lambda=d(B). Then s(B*B)$, i.e. s(uf(B)xuf(B))>= lambda$ (if lambda limit - obtained)"}
{"category": "Math", "title": "Rational singularities associated to pairs", "abstract": "In this paper we introduce a notion of rational singularities associated to pairs $(X, \\ba^t)$ where $X$ is a variety, $\\ba$ is an ideal sheaf and $t$ is a nonnegative real number. We prove that most standard results about rational singularities extend to this context. We also show that some results commonly associated with log terminal pairs have analogs in this context, including results related to inversion of adjunction. A positive characteristic analogue of rational singularities of pairs is also defined and explored."}
{"category": "Math", "title": "Distance-regular graphs of $q$-Racah type and the $q$-tetrahedron algebra", "abstract": "In this paper we discuss a relationship between the following two algebras: (i) the subconstituent algebra $T$ of a distance-regular graph that has $q$-Racah type; (ii) the $q$-tetrahedron algebra $\\boxtimes_q$ which is a $q$-deformation of the three-point $sl_2$ loop algebra. Assuming that every irreducible $T$-module is thin, we display an algebra homomorphism from $\\boxtimes_q$ into $T$ and show that $T$ is generated by the image together with the center $Z(T)$."}
{"category": "Math", "title": "Bi-Legendrian manifolds and paracontact geometry", "abstract": "We study the interplays between paracontact geometry and the theory of bi-Legendrian manifolds. We interpret the bi-Legendrian connection of a bi-Legendrian manifold M as the paracontact connection of a canonical paracontact structure induced on M and then we discuss many consequences of this result both for bi-Legendrian and for paracontact manifolds. Finally new classes of examples of paracontact manifolds are presented."}
{"category": "Math", "title": "The logarithmic Sobolev inequality along the Ricci flow in dimension 2", "abstract": "In this paper we present our results on the logarithmic Sobolev inequality along the Ricci flow in dimension 2."}
{"category": "Math", "title": "The logarithmic Sobolev inequality along the Ricci flow: the case $\\lambda_0(g_0)=0$", "abstract": "We extend our previous results on the logarithmic Sobolev inequality along the Ricci flow in the case $\\lambda_0(g_0)>0$ to the case $\\lambda_0(g_0)=0$."}
{"category": "Math", "title": "The Log Entropy Functional Along the Ricci Flow", "abstract": "In this paper we introduce the log entropy functional and establish its monotonicity along the Ricci flow. One consequence of it is the monotonicity of the logarithmic Sobolev constant along the Ricci flow."}
{"category": "Math", "title": "Well-posedness and scattering for the KP-II equation in a critical space", "abstract": "The Cauchy problem for the Kadomtsev-Petviashvili-II equation (u_t+u_{xxx}+uu_x)_x+u_{yy}=0 is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space \\dot H^{-1/2,0}(R^2) is derived. Additionally, it is proved that for arbitrarily large initial data the Cauchy problem is locally well-posed in the homogeneous space \\dot H^{-1/2,0}(R^2) and in the inhomogeneous space H^{-1/2,0}(R^2), respectively."}
{"category": "Math", "title": "p-adic Monodromy of the Universal Deformation of a HW-cyclic Barsotti-Tate Group", "abstract": "Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the \"algebraic\" local moduli in characteristic p of $G_0$, by $G$ the universal deformation of $G_0$ over $S$, and by $U\\subset S$ the ordinary locus of $G$. The etale part of $G$ over $U$ gives rise to a monodromy representation $\\rho$ of the fundamental group of $U$ on the Tate module of $G$. Motivated by a famous theorem of Igusa, we prove in this article that $\\rho$ is surjective if $G_0$ is connected and HW-cyclic. This latter condition is equivalent to that Oort's $a$-number of $G_0$ equals 1, and it is satisfied by all connected one-dimensional Barsotti-Tate groups over $k$."}
{"category": "Math", "title": "Curvature flows on four manifolds with boundary", "abstract": "Given a compact four dimensional smooth Riemannian manifold $(M,g)$ with smooth boundary, we consider the evolution equation by $Q$-curvature in the interior keeping the $T$-curvature and the mean curvature to be zero and the evolution equation by $T$-curvature at the boundary with the condition that the $Q$-curvature and the mean curvature vanish. Using integral method, we prove global existence and convergence for the $Q$-curvature flow (resp $T$-curvature flow) to smooth metric of prescribed $Q$-curvature (resp $T$-curvature) under conformally invariant assumptions."}
{"category": "Math", "title": "Topological Andr\\'e-Quillen homology for cellular commutative $S$-algebras", "abstract": "Topological Andr\\'e-Quillen homology for commutative $S$-algebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW $S$-algebras and apply it to obtain results on minimal atomic $p$-local $S$-algebras which generalise those of Baker and May for $p$-local spectra and simply connected spaces. We exhibit some new examples of minimal atomic $S$-algebras."}
{"category": "Math", "title": "Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories", "abstract": "We introduce jump processes in R^k, called density-profile process, to model biological signaling networks. They describe the macroscopic evolution of finite-size spin-flip models with k types of spins interacting through a non-reversible Glauber dynamics. We focus on the the k-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation. As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model leading to a dynamical system with Hopf and pitchfork bifurcations; depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in R^k."}
{"category": "Math", "title": "Quelques courbes de Hecke se plongent dans l'espace de Colmez", "abstract": "Let p be a prime, C the p-adic Eigencurve (with tame level 1) and Z the blow-up of the Fredholm hypersurface of the U_p - operator at the special points. We show that for p = 2, 3, 5 and 7, the natural map C -> Z is a rigid-analytic isomorphism."}
{"category": "Math", "title": "Lefschetz fibrations, intersection numbers, and representations of the framed braid group", "abstract": "We examine the action of the fundamental group $\\Gamma$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles result in a nonlinear action of the framed braid group $\\widetilde{\\mathcal B}$ on $m$ strings on a suitable space of $m\\times m$ matrices. This action is determined by a family of cohomologous 1-cocycles ${\\mathcal S}_c:\\widetilde{\\mathcal B}\\to GL_m({\\mathbb{Z}}[\\Gamma])$ parametrized by distinguished configurations $c$ of embedded paths from the regular value to the critical values. In the case of the disc, we compare this family of cocycles with the Magnus cocycles given by Fox calculus and consider some abelian reductions giving rise to linear representations of braid groups. We also prove that, still in the case of the disc, the intersection numbers along straight lines, which conjecturally make sense in infinite dimensional situations, carry all the relevant information."}
{"category": "Math", "title": "Complex cobordism classes of homogeneous spaces", "abstract": "We consider compact homogeneous spaces G/H of positive Euler characteristic endowed with an invariant almost complex structure J and the canonical action \\theta of the maximal torus T ^{k} on G/H. We obtain explicit formula for the cobordism class of such manifold through the weights of the action \\theta at the identity fixed point eH by an action of the quotient group W_G/W_H of the Weyl groups for G and H. In this way we show that the cobordism class for such manifolds can be computed explicitly without information on their cohomology. We also show that formula for cobordism class provides an explicit way for computing the classical Chern numbers for (G/H, J). As a consequence we obtain that the Chern numbers for (G/H, J) can be computed without information on cohomology for G/H. As an application we provide an explicit formula for cobordism classes and characteristic numbers of the flag manifolds U(n)/T^n, Grassmann manifolds G_{n,k}=U(n)/(U(k)\\times U(n-k)) and some particular interesting examples."}
{"category": "Math", "title": "On (Enriched) Left Bousfield Localization of Model Categories", "abstract": "I verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and I prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. I also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopy-coherent descent condition."}
{"category": "Math", "title": "Symmetries of quadratic forms classes and of quadratic surds continued fractions. Part II: Classification of the periods' palindromes", "abstract": "The continue fractions of quadratic surds are periodic, according to a theorem by Lagrange. Their periods may have differing types of symmetries. This work relates these types of symmetries to the symmetries of the classes of the corresponding indefinite quadratic forms. This allows to classify the periods of quadratic surds and at the same time to find, for an arbitrary indefinite quadratic form, the symmetry type of its class and the number of integer points, for that class, contained in each domain of the Poincare' model of the de Sitter world, introduced in Part I. Moreover, we obtain the same information for every class of forms representing zero, by the finite continue fraction related to a special representative of that class. We will see finally the relation between the reduction procedure for indefinite quadratic forms, defined by the continued fractions, and the classical reduction theory, which acquires a geometrical description by the results of Part I."}
{"category": "Math", "title": "A hyperbolic approach to exp_3(S^1)", "abstract": "In this paper we investigate a new geometric method of studying exp_k(S^1), the set of all non-empty subsets of the circle of cardinality at most k. By considering the circle as the boundary of the hyperbolic plane we are able to use its group of isometries to determine explicitely the structure of its first few configuration spaces. We then study how these configuration spaces fit together in their union, exp_3(S^1), to reprove an old theorem of Bott as well as to offer a new proof (following that of E. Shchepin) of the fact that the embedding exp_1(S^1) into exp_3(S^1) is the trefoil knot."}
{"category": "Math", "title": "A version of the proof for Peres-Schlag's theorem on lacunary sequences", "abstract": "We present a proof of a multidimensional version of Peres-Schlag's theorem on Diophantine approximations with lacunary sequences."}
{"category": "Math", "title": "Gain of Regularity for the KP-I Equation", "abstract": "In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation's dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data $\\phi$ possesses certain regularity and sufficient decay as $x \\to \\infty$, then the solution $u(t)$ will be smoother than $\\phi$ for $0 < t \\leq T$ where $T$ is the existence time of the solution."}
{"category": "Math", "title": "Computing arithmetic invariants for hyperbolic reflection groups", "abstract": "We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $\\mathbb{H}^3$, the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such groups and the isomorphism class of the invariant quaternion algebra by analyzing its ramification. We present many computed examples of these invariants. This is enough to show that most of the groups that we consider are pairwise incommensurable. For pairs of groups with identical invariants, not all is lost: when both groups are arithmetic, having identical invariants guarantees commensurability. We discover many ``unexpected'' commensurable pairs this way. We also present a non-arithmetic pair with identical invariants for which we cannot determine commensurability."}
{"category": "Math", "title": "On the Number of Facets of Three-Dimensional Dirichlet Stereohedra IV: Quarter Cubic Groups", "abstract": "In this paper we finish the intensive study of three-dimensional Dirichlet stereohedra started by the second author and D. Bochis, who showed that they cannot have more than 80 facets, except perhaps for crystallographic space groups in the cubic system. Taking advantage of the recent, simpler classification of three-dimensional crystallographic groups by Conway, Delgado-Friedrichs, Huson and Thurston, in a previous paper we proved that Dirichlet stereohedra for any of the 27 \"full\" cubic groups cannot have more than 25 facets. Here we study the remaining \"quarter\" cubic groups. With a computer-assisted method, our main result is that Dirichlet stereohedra for the 8 quarter groups, hence for all three-dimensional crystallographic groups, cannot have more than 92 facets."}
{"category": "Math", "title": "A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow", "abstract": "This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation $u_t+\\De\\bigl(\\eps \\De u-\\eps^{-1} f(u)\\bigr)=0$. It is shown that the {\\it a posteriori} error bounds depends on $\\eps^{-1}$ only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm for computing the solution of the Cahn-Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm."}
{"category": "Math", "title": "Automorphisms of a polynomial ring which admit reductions of type I", "abstract": "Recently, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. To solve the conjecture, they defined notions called reductions of types I--IV for automorphisms of a polynomial ring. An automorphism admitting a reduction of type I was first found by Shestakov-Umirbaev. Using a computer, van den Essen--Makar-Limanov--Willems gave a family of such automorphisms. In this paper, we present a new construction of such automorphisms using locally nilpotent derivations. As a consequence, we discover that there exists an automorphism admitting a reduction of type I which satisfies some degree condition for each possible value."}
{"category": "Math", "title": "On The Solvability of Bilinear Equations in Finite Fields", "abstract": "We consider the equation $$ ab + cd = \\lambda, \\qquad a\\in A, b \\in B, c\\in C, d \\in D, $$ over a finite field $F_q$ of $q$ elements, with variables from arbitrary sets $ A, B, C, D \\subseteq F_q$. The question of solvability of such and more general equations has recently been considered by D. Hart and A. Iosevich, who, in particular, proved that if $$ #A #B #C #D \\gg q^3, $$ then above equation has a solution for any $\\lambda \\in F_q^*$. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property."}
{"category": "Math", "title": "A Short Study of Alexandroff Spaces", "abstract": "In this paper, we discuss the basic properties of Alexandroff spaces. Several examples of Alexandroff spaces are given. We show how to construct new Alexandroff spaces from given ones. Finally, two invariants for compact Alexandroff spaces are defined and calculated for the given examples."}
{"category": "Math", "title": "Torus quotients of homogeneous spaces of the general linear group and the standard representation of certain symmetric groups", "abstract": "We give a stratification of the GIT quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $SL_{n}(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $GL_{n}(k)/B_{n}$ can be obtained as a GIT quotient of $GL_{n+1}(k)/B_{n+1}$ modulo a maximal torus of $SL_{n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{n+1}(k)/B_{n+1}$."}
{"category": "Math", "title": "When do stepwise algorithms meet subset selection criteria?", "abstract": "Recent results in homotopy and solution paths demonstrate that certain well-designed greedy algorithms, with a range of values of the algorithmic parameter, can provide solution paths to a sequence of convex optimization problems. On the other hand, in regression many existing criteria in subset selection (including $C_p$, AIC, BIC, MDL, RIC, etc.) involve optimizing an objective function that contains a counting measure. The two optimization problems are formulated as (P1) and (P0) in the present paper. The latter is generally combinatoric and has been proven to be NP-hard. We study the conditions under which the two optimization problems have common solutions. Hence, in these situations a stepwise algorithm can be used to solve the seemingly unsolvable problem. Our main result is motivated by recent work in sparse representation, while two others emerge from different angles: a direct analysis of sufficiency and necessity and a condition on the mostly correlated covariates. An extreme example connected with least angle regression is of independent interest."}
{"category": "Math", "title": "Local partial likelihood estimation in proportional hazards regression", "abstract": "Fan, Gijbels and King [Ann. Statist. 25 (1997) 1661--1690] considered the estimation of the risk function $\\psi (x)$ in the proportional hazards model. Their proposed estimator is based on integrating the estimated derivative function obtained through a local version of the partial likelihood. They proved the large sample properties of the derivative function, but the large sample properties of the estimator for the risk function itself were not established. In this paper, we consider direct estimation of the relative risk function $\\psi (x_2)-\\psi (x_1)$ for any location normalization point $x_1$. The main novelty in our approach is that we select observations in shrinking neighborhoods of both $x_1$ and $x_2$ when constructing a local version of the partial likelihood, whereas Fan, Gijbels and King [Ann. Statist. 25 (1997) 1661--1690] only concentrated on a single neighborhood, resulting in the cancellation of the risk function in the local likelihood function. The asymptotic properties of our estimator are rigorously established and the variance of the estimator is easily estimated. The idea behind our approach is extended to estimate the differences between groups. A simulation study is carried out."}
{"category": "Math", "title": "Estimating the number of classes", "abstract": "Estimating the unknown number of classes in a population has numerous important applications. In a Poisson mixture model, the problem is reduced to estimating the odds that a class is undetected in a sample. The discontinuity of the odds prevents the existence of locally unbiased and informative estimators and restricts confidence intervals to be one-sided. Confidence intervals for the number of classes are also necessarily one-sided. A sequence of lower bounds to the odds is developed and used to define pseudo maximum likelihood estimators for the number of classes."}
{"category": "Math", "title": "Gain of analyticity for semilinear Schroedinger equations", "abstract": "We discuss gain of analyticity phenomenon of solutions to the initial value problem for semilinear Schroedinger equations with gauge invariant nonlinearity. We prove that if the initial data decays exponentially, then the solution becomes real-analytic in the space variable and a Gevrey function of order 2 in the time variable except in the initial plane. Our proof is based on the energy estimates developed in our previous work and on fine summation formulae concerned with a matrix norm."}
{"category": "Math", "title": "Strongly pseudoconvex domains as subvarieties of complex manifolds", "abstract": "In this paper (a sequel to B. Drinovec Drnovsek and F. Forstneric, Holomorphic curves in complex spaces, Duke Math. J. 139 (2007), 203-253) we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex Stein domains. Our sufficient condition for the existence of such subvarieties in a complex manifold is expressed in terms of the Morse indices and the number of positive Levi eigenvalues of an exhaustion function on the manifold. Examples show that our condition cannot be weakened in general. Optimal results are obtained for subvarieties of this type in complements of compact complex submanifolds with Griffiths positive normal bundle; in the projective case these results generalize classical theorems of Remmert, Bishop and Narasimhan concerning proper holomorphic maps and embeddings to complex Euclidean spaces."}
{"category": "Math", "title": "Rational algebraic K-theory of topological K-theory", "abstract": "We show that after rationalization there is a homotopy fiber sequence BBU -> K(ku) -> K(Z). We interpret this as a correspondence between the virtual 2-vector bundles over a space X and their associated anomaly bundles over the free loop space LX. We also rationally compute K(KU) by using the localization sequence, and K(MU) by a method that applies to all connective S-algebras."}
{"category": "Math", "title": "Normal Surface Theory in Link Diagrams", "abstract": "This paper has been withdrawn by the author, due to a significant error in section 4.3.1."}
{"category": "Math", "title": "Likelihood based inference for monotone response models", "abstract": "The behavior of maximum likelihood estimates (MLEs) and the likelihood ratio statistic in a family of problems involving pointwise nonparametric estimation of a monotone function is studied. This class of problems differs radically from the usual parametric or semiparametric situations in that the MLE of the monotone function at a point converges to the truth at rate $n^{1/3}$ (slower than the usual $\\sqrt{n}$ rate) with a non-Gaussian limit distribution. A framework for likelihood based estimation of monotone functions is developed and limit theorems describing the behavior of the MLEs and the likelihood ratio statistic are established. In particular, the likelihood ratio statistic is found to be asymptotically pivotal with a limit distribution that is no longer $\\chi^2$ but can be explicitly characterized in terms of a functional of Brownian motion. Applications of the main results are presented and potential extensions discussed."}
{"category": "Math", "title": "A simple proof of the matrix-valued Fej\\'er-Riesz theorem", "abstract": "A very short proof of the Fej\\'er-Riesz lemma is presented in the matrix case"}
{"category": "Math", "title": "A nonparametric approach to the estimation of lengths and surface areas", "abstract": "The Minkowski content $L_0(G)$ of a body $G\\subset{\\mathbb{R}}^d$ represents the boundary length (for $d=2$) or the surface area (for $d=3$) of $G$. A method for estimating $L_0(G)$ is proposed. It relies on a nonparametric estimator based on the information provided by a random sample (taken on a rectangle containing $G$) in which we are able to identify whether every point is inside or outside $G$. Some theoretical properties concerning strong consistency, $L_1$-error and convergence rates are obtained. A practical application to a problem of image analysis in cardiology is discussed in some detail. A brief simulation study is provided."}
{"category": "Math", "title": "Local rigidity in quaternionic hyperbolic space", "abstract": "In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane."}
{"category": "Math", "title": "Monte Carlo likelihood inference for missing data models", "abstract": "We describe a Monte Carlo method to approximate the maximum likelihood estimate (MLE), when there are missing data and the observed data likelihood is not available in closed form. This method uses simulated missing data that are independent and identically distributed and independent of the observed data. Our Monte Carlo approximation to the MLE is a consistent and asymptotically normal estimate of the minimizer $\\theta^*$ of the Kullback--Leibler information, as both Monte Carlo and observed data sample sizes go to infinity simultaneously. Plug-in estimates of the asymptotic variance are provided for constructing confidence regions for $\\theta^*$. We give Logit--Normal generalized linear mixed model examples, calculated using an R package."}
{"category": "Math", "title": "On combinatorial model categories", "abstract": "Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly generated. He has not published his results yet but proofs of some of them were presented by T. Beke or D. Dugger. We are contributing to this endeavour by proving that weak equivalences in a combinatorial model category form an accessible category. We also present some new results about weak equivalences and cofibrations in combinatorial model categories."}
{"category": "Math", "title": "Stochastic Variational Integrators", "abstract": "This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. The main result is to derive stochastic governing equations for such systems from a critical point of a stochastic action. Using this result the paper derives Langevin-type equations for constrained mechanical systems and implements a stochastic analog of Lagrangian reduction. These are easy consequences of the fact that the stochastic action is intrinsically defined. Stochastic variational integrators (SVIs) are developed using a discretized stochastic variational principle. The paper shows that the discrete flow of an SVI is a.s. symplectic and in the presence of symmetry a.s. momentum-map preserving. A first-order mean-square convergent SVI for mechanical systems on Lie groups is introduced. As an application of the theory, SVIs are exhibited for multiple, randomly forced and torqued rigid-bodies interacting via a potential."}
{"category": "Math", "title": "Comparison between Second Variation of Area and Second Variation of Energy of a Minimal Surface", "abstract": "The conformal parameterisation of a minimal surface is harmonic. Therefore, a minimal surface is a critical point of both the energy functional and the area functional. In this paper, we compare the Morse index of a minimal surface as a critical point of the area functional with its Morse index as a critical point of the energy functional. The difference between these indices is at most the real dimension of Teichmuller space. This comparison allows us to obtain surprisingly good upper bounds on the index of minimal surfaces of finite total curvature in Euclidean space of any dimension. We also bound the index of a minimal surface in an arbitrary Riemannian manifold by the area and genus of the surface, and the dimension and geometry of the ambient manifold."}
{"category": "Math", "title": "Primitive Divisors of some Lehmer-Pierce Sequences", "abstract": "We study the primitive divisors of the terms of $(\\Delta_n)_{n \\geq 1}$, where $\\Delta_n=N_{K/ \\mathbb{Q}}(u^n-1)$ for $K$ a real quadratic field, and $u>1$ a unit element of its ring of integers. The methods used allow us to find the terms of the sequence that do not have a primitive prime divisor."}
{"category": "Math", "title": "The global isoperimetric methodology applied to Kneser's Theorem", "abstract": "We give in the present work a new methodology that allows to give isoperimetric proofs, for Kneser's Theorem and Kemperman's structure Theory and most sophisticated results of this type. As an illustration we present a new proof of Kneser's Theorem."}
{"category": "Math", "title": "Focusing waves in unknown media by modified time reversal iteration", "abstract": "We study the wave equation in a bounded domain or on a compact Riemannian manifold with boundary. Assume that we are given the hyperbolic Neumann-to-Dirichlet map on the boundary corresponding to physical boundary measurements. We consider how to focus waves, that is, how to find Neumann boundary values so that at a given time the corresponding wave converges to a delta distribution $\\delta_y$ while the time derivative of the wave converges to zero. Such boundary value are generated by an iterative sequence of measurements. In each iteration step we apply time reversal and other simple operators to measured data and compute boundary values for the next iteration step. The key feature of the algorithm is that it does not require knowledge of the coefficients in the wave equation, that is, the material parameters inside the media. However, we assume that the point $y$ where the wave focuses is known in travel time coordinates."}
{"category": "Math", "title": "Connection between ordinary multinomials, generalized Fibonacci numbers, partial Bell partition polynomials and convolution powers of discrete uniform distribution", "abstract": "Using an explicit computable expression of ordinary multinomials, we establish three remarkable connections, with the q-generalized Fibonacci sequence, the exponential partial Bell partition polynomials and the density of convolution powers of the discrete uniform distribution. Identities and various combinatorial relations are derived."}
{"category": "Math", "title": "Piecewise linear regularized solution paths", "abstract": "We consider the generic regularized optimization problem $\\hat{\\mathsf{\\beta}}(\\lambda)=\\arg \\min_{\\beta}L({\\sf{y}},X{\\sf{\\beta}})+\\lambda J({\\sf{\\beta}})$. Efron, Hastie, Johnstone and Tibshirani [Ann. Statist. 32 (2004) 407--499] have shown that for the LASSO--that is, if $L$ is squared error loss and $J(\\beta)=\\|\\beta\\|_1$ is the $\\ell_1$ norm of $\\beta$--the optimal coefficient path is piecewise linear, that is, $\\partial \\hat{\\beta}(\\lambda)/\\partial \\lambda$ is piecewise constant. We derive a general characterization of the properties of (loss $L$, penalty $J$) pairs which give piecewise linear coefficient paths. Such pairs allow for efficient generation of the full regularized coefficient paths. We investigate the nature of efficient path following algorithms which arise. We use our results to suggest robust versions of the LASSO for regression and classification, and to develop new, efficient algorithms for existing problems in the literature, including Mammen and van de Geer's locally adaptive regression splines."}
{"category": "Math", "title": "$B_2$-crystals: axioms, structure, models", "abstract": "We present a list of ``local'' axioms and an explicit combinatorial construction for the regular $B_2$-crystals (crystal graphs of highest weight integrable modules over $U_q(sp_4)$). Also a new combinatorial model for these crystals is developed."}
{"category": "Math", "title": "Curves of given $p$-rank with trivial automorphism group", "abstract": "Let $k$ be an algebraically closed field of characteristic $p >0$. Suppose $g \\geq 3$ and $0 \\leq f \\leq g$. We prove there is a smooth projective $k$-curve of genus $g$ and $p$-rank $f$ with no non-trivial automorphisms. In addition, we prove there is a smooth projective hyperelliptic $k$-curve of genus $g$ and $p$-rank $f$ whose only non-trivial automorphism is the hyperelliptic involution. The proof involves computations about the dimension of the moduli space of (hyperelliptic) $k$-curves of genus $g$ and $p$-rank $f$ with extra automorphisms."}
{"category": "Math", "title": "Locally compact quantum groups. Radford's $S^4$ formula", "abstract": "Let $A$ be a finite-dimensional Hopf algebra. The left and the right integrals on $A$ are related by means of a distinguished group-like element $\\delta$ of $A$. Similarly, there is this element $\\hat\\delta$ in the dual Hopf algebra $\\hat A$. Radford showed that $$S^4(a)=\\delta^{-1}(\\hat\\delta\\triangleright a \\triangleleft \\hat\\delta^{-1})\\delta$$ for all $a$ in $A$ where $S$ is the antipode of $A$ and where $\\triangleright$ and $\\triangleleft$ are used to denote the standard left and right actions of $\\hat A$ on $A$. The formula still holds for multiplier Hopf algebras with integrals (algebraic quantum groups). In the theory of locally compact quantum groups, an analytical form of Radford's formula can be proven (in terms of bounded operators on a Hilbert space). In this talk, we do not have the intention to discuss Radford's formula as such, but rather to use it, together with related formulas, for illustrating various aspects of the road that takes us from the theory of Hopf algebras (including compact quantum groups) to multiplier Hopf algebras (including discrete quantum groups) and further to the more general theory of locally compact quantum groups."}
{"category": "Math", "title": "Statistical inferences for functional data", "abstract": "With modern technology development, functional data are being observed frequently in many scientific fields. A popular method for analyzing such functional data is ``smoothing first, then estimation.'' That is, statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another. However, little is known about this substitution effect on functional data analysis. In this paper this problem is investigated when the local polynomial kernel (LPK) smoothing technique is used for individual function reconstructions. We find that under some mild conditions, the substitution effect can be ignored asymptotically. Based on this, we construct LPK reconstruction-based estimators for the mean, covariance and noise variance functions of a functional data set and derive their asymptotics. We also propose a GCV rule for selecting good bandwidths for the LPK reconstructions. When the mean function also depends on some time-independent covariates, we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time. The LPK reconstruction-based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived. Moreover, we propose a $L^2$-norm-based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression. The effect of the bandwidths selected by the proposed GCV rule on the accuracy of the LPK reconstructions and the mean function estimator is investigated via a simulation study. The proposed methodologies are illustrated via an application to a real functional data set collected in climatology."}
{"category": "Math", "title": "Enumerative Properties of NC^B(p,q)", "abstract": "We determine the rank generating function, the zeta polynomial and the Moebius function for the poset NC^B(p,q) of annular non-crossing partitions of type B, where p and q are two positive integers. We give an alternative treatment of some of these results in the case q=1, for which this poset is a lattice. We also consider the general case of multiannular non-crossing partitions of type B, and prove that this reduces to the cases of non-crossing partitions of type B in the annulus and in the disc."}
{"category": "Math", "title": "On the p-th root of a p-adic number", "abstract": "We give a sufficient and necessary condition for a p-adic integer to have p-th root in the ring of p-adic integers. The same condition holds clearly for residues modulo p^k. We give a proof that Fermat's last theorem is false for p-adic integers and for residues mod p^k."}
{"category": "Math", "title": "Tensoring with infinite-dimensional modules in $\\scr O_0$", "abstract": "We show that the principal block $\\scr O_0$ of the BGG category $\\scr O$ for a semisimple Lie algebra $\\germ g$ acts faithfully on itself via exact endofunctors which preserve tilting modules, via right exact endofunctors which preserve projective modules and via left exact endofunctors which preserve injective modules. The origin of all these functors is tensoring with arbitrary (not necessarily finite-dimensional) modules in the category $\\scr O$. We study such functors, describe their adjoints and show that they give rise to a natural (co)monad structure on $\\scr O_0$. Furthermore, all this generalises to parabolic subcategories of $\\scr O_0$. As an example, we present some explicit computations for the algebra $\\germ{sl}_3$."}
{"category": "Math", "title": "On the $\\mathbb{L}_p$-error of monotonicity constrained estimators", "abstract": "We aim at estimating a function $\\lambda:[0,1]\\to \\mathbb {R}$, subject to the constraint that it is decreasing (or increasing). We provide a unified approach for studying the $\\mathbb {L}_p$-loss of an estimator defined as the slope of a concave (or convex) approximation of an estimator of a primitive of $\\lambda$, based on $n$ observations. Our main task is to prove that the $\\mathbb {L}_p$-loss is asymptotically Gaussian with explicit (though unknown) asymptotic mean and variance. We also prove that the local $\\mathbb {L}_p$-risk at a fixed point and the global $\\mathbb {L}_p$-risk are of order $n^{-p/3}$. Applying the results to the density and regression models, we recover and generalize known results about Grenander and Brunk estimators. Also, we obtain new results for the Huang--Wellner estimator of a monotone failure rate in the random censorship model, and for an estimator of the monotone intensity function of an inhomogeneous Poisson process."}
{"category": "Math", "title": "On equimultiple modules", "abstract": "We study the class of equimultiple modules. In particular, we prove several criteria for an equimultiple module to be a complete intersection and prove the openness of the equimultiple locus of an ideal module."}
{"category": "Math", "title": "On local $U$-statistic processes and the estimation of densities of functions of several sample variables", "abstract": "A notion of local $U$-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for $U$-processes that may be useful in other contexts. This local $U$-statistic process is based on an estimator of the density of a function of several sample variables proposed by Frees [J. Amer. Statist. Assoc. 89 (1994) 517--525] and, as a consequence, uniform in bandwidth central limit theorems in the sup and in the $L_p$ norms are obtained for these estimators."}
{"category": "Math", "title": "Sl(N) link homology using foams and the Kapustin-Li formula", "abstract": "We use foams to give a topological construction of a rational link homology categorifying the slN link invariant, for N>3. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky. We show that for any link our homology is isomorphic to Khovanov and Rozansky's."}
{"category": "Math", "title": "A complement to Le Cam's theorem", "abstract": "This paper examines asymptotic equivalence in the sense of Le Cam between density estimation experiments and the accompanying Poisson experiments. The significance of asymptotic equivalence is that all asymptotically optimal statistical procedures can be carried over from one experiment to the other. The equivalence given here is established under a weak assumption on the parameter space $\\mathcal{F}$. In particular, a sharp Besov smoothness condition is given on $\\mathcal{F}$ which is sufficient for Poissonization, namely, if $\\mathcal{F}$ is in a Besov ball $B_{p,q}^{\\alpha}(M)$ with $\\alpha p>1/2$. Examples show Poissonization is not possible whenever $\\alpha p<1/2$. In addition, asymptotic equivalence of the density estimation model and the accompanying Poisson experiment is established for all compact subsets of $C([0,1]^m)$, a condition which includes all H\\\"{o}lder balls with smoothness $\\alpha>0$."}
{"category": "Math", "title": "Realizing modules over the homology of a DGA", "abstract": "Let A be a DGA over a field and X a module over H_*(A). Fix an $A_\\infty$-structure on H_*(A) making it quasi-isomorphic to A. We construct an equivalence of categories between A_{n+1}-module structures on X and length n Postnikov systems in the derived category of A-modules based on the bar resolution of X. This implies that quasi-isomorphism classes of A_n-structures on X are in bijective correspondence with weak equivalence classes of rigidifications of the first n terms of the bar resolution of X to a complex of A-modules. The above equivalences of categories are compatible for different values of n. This implies that two obstruction theories for realizing X as the homology of an A-module coincide."}
{"category": "Math", "title": "Embedding group algebras into finite von Neumann regular rings", "abstract": "Let G be a group and let K be a field of characteristic zero. We shall prove that KG can be embedded into a von Neumann unit-regular ring. In the course of the proof, we shall obtain a result relevant to the Atiyah conjecture."}
{"category": "Math", "title": "Bridgeland-Stable Moduli Spaces for K-Trivial Surfaces", "abstract": "We give a natural family of Bridgeland stability conditions on the derived category of a smooth projective complex surface S and describe ``wall-crossing behavior'' for objects with the same invariants as $\\cO_C(H)$ when H generates Pic(S) and $C \\in |H|$. If, in addition, S is a K3 or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgeland-stable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural generalization of Thaddeus' stable pairs for curves embedded in the moduli spaces."}
{"category": "Math", "title": "Automorphisms and isomorphisms of Chevalley groups and algebras", "abstract": "An adjoint Chevalley group of rank at least 2 over a rational algebra (or a similar ring), its elementary subgroup, and the corresponding Lie ring have the same automorphism group. These automorphisms are explicitly described."}
{"category": "Math", "title": "Tropical fans and the moduli spaces of tropical curves", "abstract": "We give a rigorous definition of tropical fans (the \"local building blocks for tropical varieties\") and their morphisms. For such a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point - a statement that can be viewed as the beginning of a tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some R^r) together with the evaluation and forgetful morphisms. Using our results this gives new, easy, and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any R^r) through given points are independent of the points."}
{"category": "Math", "title": "Quasi-periodic stability of normally resonant tori", "abstract": "We study quasi-periodic tori under a normal-internal resonance, possibly with multiple eigenvalues. Two non-degeneracy conditions play a role. The first of these generalizes invertibility of the Floquet matrix and prevents drift of the lower dimensional torus. The second condition involves a Kolmogorov-like variation of the internal frequencies and simultaneously versality of the Floquet matrix unfolding. We focus on the reversible setting, but our results carry over to the Hamiltonian and dissipative contexts."}
{"category": "Math", "title": "Weighted hyperprojective spaces and homotopy invariance in orbifold cohomology", "abstract": "We show that Chen-Ruan cohomology is a homotopy invariant in certain cases. We introduce the notion of a T-representation homotopy, which is a stringent form of homotopy under which Chen-Ruan cohomology is invariant. We show that while hyperkahler quotients of the cotangent bundle to a complex vector space by a circle S^1 (here termed weighted hyperprojective spaces) are homotopy equivalent to weighted projective spaces, they are not S^1-representation homotopic. Indeed, we show that their Chen-Ruan cohomology rings (over the rationals) are distinct."}
{"category": "Math", "title": "Minimal Number of Generators and Minimum Order of a Non-Abelian Group whose Elements Commute with Their Endomorphic Images", "abstract": "A group in which every element commutes with its endomorphic images is called an $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-group is called a $pE$-group. Every abelian group is obviously an $E$-group. We prove that every 2-generator $E$-group is abelian and that all 3-generator $E$-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator $E$-group is abelian. We conjecture that every finite 3-generator $E$-group should be abelian. Moreover we show that the minimum order of a non-abelian $pE$-group is $p^8$ for any odd prime number $p$ and this order is $2^7$ for $p=2$. Some of these results are proved for a class wider than the class of $E$-groups."}
{"category": "Math", "title": "A lower bound for the number of conjugacy classes of finite groups", "abstract": "In 2000, L. H\\'{e}thelyi and B. K\\\"{u}lshammer proved that if $p$ is a prime number dividing the order of a finite solvable group $G$, then $G$ has at least $2\\sqrt{p-1}$ conjugacy classes. In this paper we show that if $p$ is large, the result remains true for arbitrary finite groups."}
{"category": "Math", "title": "Minimal blocking sets in PG(n,2) and covering groups by subgroups", "abstract": "In this paper we prove that a set of points $B$ of PG(n,2) is a minimal blocking set if and only if $<B>=PG(d,2)$ with $d$ odd and $B$ is a set of $d+2$ points of $PG(d,2)$ no $d+1$ of them in the same hyperplane. As a corollary to the latter result we show that if $G$ is a finite 2-group and $n$ is a positive integer, then $G$ admits a $\\mathfrak{C}_{n+1}$-cover if and only if $n$ is even and $G\\cong (C_2)^{n}$, where by a $\\mathfrak{C}_m$-cover for a group $H$ we mean a set $\\mathcal{C}$ of size $m$ of maximal subgroups of $H$ whose set-theoretic union is the whole $H$ and no proper subset of $\\mathcal{C}$ has the latter property and the intersection of the maximal subgroups is core-free. Also for all $n<10$ we find all pairs $(m,p)$ ($m>0$ an integer and $p$ a prime number) for which there is a blocking set $B$ of size $n$ in $PG(m,p)$ such that $<B>=PG(m,p)$."}
{"category": "Math", "title": "On quasi-Baer rings of Ore extensions", "abstract": "Let $R$ be a ring and $S=R[x;\\sigma,\\delta]$ its Ore extension. We prove under some conditions that $R$ is a quasi-Baer ring if and only if the Ore extension $R[x;\\sigma,\\delta]$ is a quasi-Baer ring. Examples are provided to illustrate and delimit our results."}
{"category": "Math", "title": "MASA's and certain type I closed faces of C*-algebras", "abstract": "A result of Akemann, Anderson, and Pedersen states that if a sequence of pure states of a C*-algebra A approaches infinity in a certain sense, then there is a MASA B such that each of the states has the unique extension property with respect to B. We generalize this in two ways: We prove that B can be required to contain an approximate identity of A, and we show that the discrete space which underlies the result cited can be replaced with a totally disconnected space. We consider two special kinds of type I closed faces, both related to the above, atomic closed faces and closed faces with nearly closed extreme boundary. One specific question is whether an atomic closed face always has an \"isolated point\". We give a counterexample for this and also show that the answer is yes if the the atomic face has nearly closed extreme boundary. We prove a complement to Glimm's theorem on type I C*-algebras which arises from the theory of type I closed faces. One of our examples is a type I closed face which is isomorphic to a closed face of every non-type I separable C*-algebra and which is not isomorphic to a closed face of any type I C*-algebra."}
{"category": "Math", "title": "Product-free subsets of groups, then and now", "abstract": "A subset of a group is product-free if it does not contain elements a, b, c such that ab = c. We review progress on the problem of determining the size of the largest product-free subset of an arbitrary finite group, including a lower bound due to the author, and a recent upper bound due to Gowers. The bound of Gowers is more general; it allows three different sets A, B, C such that one cannot solve ab = c with a in A, b in B, c in C. We exhibit a refinement of the lower bound construction which shows that for this broader question, the bound of Gowers is essentially optimal."}
{"category": "Math", "title": "Inverse problems for linear forms over finite sets of integers", "abstract": "Let f(x_1,x_2,...,x_m) = u_1x_1+u_2 x_2+... + u_mx_m be a linear form with positive integer coefficients, and let N_f(k) = min{|f(A)| : A \\subseteq Z and |A|=k}. A minimizing k-set for f is a set A such that |A|=k and |f(A)| = N_f(k). A finite sequence (u_1, u_2,...,u_m) of positive integers is called complete if {\\sum_{j\\in J} u_j : J \\subseteq {1,2,..,m}} = {0,1,2,..., U}, where $U = \\sum_{j=1}^m u_j.$ It is proved that if f is an m-ary linear form whose coefficient sequence (u_1,...,u_m) is complete, then N_f(k) = Uk-U+1 and the minimizing k-sets are precisely the arithmetic progressions of length k. Other extremal results on linear forms over finite sets of integers are obtained."}
{"category": "Math", "title": "Small value estimates for the additive group", "abstract": "We generalize Gel'fond's criterion of algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of the additive group of complex numbers, instead of just one point. We also provide one extension dealing with a subgroup of the multiplicative group of non-zero complex numbers."}
{"category": "Math", "title": "Markov bases for two-way subtable sum problems", "abstract": "It has been well-known that for two-way contingency tables with fixed row sums and column sums the set of square-free moves of degree two forms a Markov basis. However when we impose an additional constraint that the sum of a subtable is also fixed, then these moves do not necessarily form a Markov basis. Thus, in this paper, we show a necessary and sufficient condition on a subtable so that the set of square-free moves of degree two forms a Markov basis."}
{"category": "Math", "title": "Conformal representations of Leibniz algebras", "abstract": "In this note we present a more detailed and explicit exposition of the definition of a conformal representation of a Leibniz algebra. Recall (arXiv:math/0611501v3) that Leibniz algebras are exactly Lie dialgebras. The idea is based on the general fact that every dialgebra that belongs to a variety $\\Var $ can be embedded into a conformal algebra of the same variety. In particular, we prove that an arbitrary (finite dimensional) Leibniz algebra has a (finite) faithful conformal representation. As a corollary, we deduce the analogue of the PBW-theorem for Leibniz algebras."}
{"category": "Math", "title": "Generalized rational blow-down, torus knots, and Euclidean algorithm", "abstract": "We construct a Kirby diagram of the rational homology ball used in \"generalized rational blow-down\" developed by Jongil Park. The diagram consists of a dotted circle and a torus knot. The link is simpler, but the parameters are a little complicate. Euclidean Algorithm is used three times in the construction and the proof."}
{"category": "Math", "title": "The non-viscous Burgers equation associated with random positions in coordinate space: a threshold for blow up behaviour", "abstract": "It is well known that the solutions to the non-viscous Burgers equation develop a gradient catastrophe at a critical time provided the initial data have a negative derivative in certain points. We consider this equation assuming that the particle paths in the medium are governed by a random process with a variance which depends in a polynomial way on the velocity. Given an initial distribution of the particles which is uniform in space and with the initial velocity linearly depending on the position we show both analytically and numerically that there exists a threshold effect: if the power in the above variance is less than 1, then the noise does not influence the solution behavior, in the following sense: the mean of the velocity when we keep the value of position fixed goes to infinity outside the origin. If however the power is larger or equal 1, then this mean decays to zero as the time tends to a critical value."}
{"category": "Math", "title": "Fast learning rates for plug-in classifiers", "abstract": "It has been recently shown that, under the margin (or low noise) assumption, there exist classifiers attaining fast rates of convergence of the excess Bayes risk, that is, rates faster than $n^{-1/2}$. The work on this subject has suggested the following two conjectures: (i) the best achievable fast rate is of the order $n^{-1}$, and (ii) the plug-in classifiers generally converge more slowly than the classifiers based on empirical risk minimization. We show that both conjectures are not correct. In particular, we construct plug-in classifiers that can achieve not only fast, but also super-fast rates, that is, rates faster than $n^{-1}$. We establish minimax lower bounds showing that the obtained rates cannot be improved."}
{"category": "Math", "title": "Non-cyclic graph of a group", "abstract": "We associate a graph $\\Gamma_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\\{x\\in G | \\left<x,y\\right> \\text{is cyclic for all} y\\in G\\}$, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of $\\Gamma_G$ is finite if and only if $\\Gamma_G$ has no infinite clique. We prove that if $G$ is a finite nilpotent group and $H$ is a group with $\\Gamma_G\\cong\\Gamma_H$ and $|Cyc(G)|=|Cyc(H)|=1$, then $H$ is a finite nilpotent group. We give some examples of groups $G$ whose non-cyclic graphs are ``unique'', i.e., if $\\Gamma_G\\cong \\Gamma_H$ for some group $H$, then $G\\cong H$. In view of these examples, we conjecture that every finite non-abelian simple group has a unique non-cyclic graph. Also we give some examples of finite non-cyclic groups $G$ with the property that if $\\Gamma_G \\cong \\Gamma_H$ for some group $H$, then $|G|=|H|$. These suggest the question whether the latter property holds for all finite non-cyclic groups."}
{"category": "Math", "title": "Unimodality of ordinary multinomials and maximal probabilities of convolution powers of discrete uniform distribution", "abstract": "We establish the unimodality and the asymptotic strong unimodality of the ordinary multinomials and give their smallest mode leading to the expression of the maximal probability of convolution powers of the discrete uniform distribution. We conclude giving the generating functions of the sequence of generalized ordinary multinomials and for an extension of the sequence of maximal probabilities for convolution power of discrete uniform distribution."}
{"category": "Math", "title": "Complex dynamics in a nerve fiber model with periodic coefficients", "abstract": "We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x) F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell in the study of nerve fiber models. In some recent works we discussed the case of nonexistence of nontrivial solutions as well as the case in which many positive periodic solutions may arise, the different situations depending by threshold parameters related to the weight function $n(x).$ Here we show that for a step function $n(x)$ (or for small perturbations of it) it is possible to obtain infinitely many periodic solutions and chaotic dynamics, due to the presence of a topological horseshoe (according to Kennedy and Yorke)."}
{"category": "Math", "title": "Sums of products of generalized Fibonacci and Lucas numbers", "abstract": "In this paper, we establish several formulae for sums and alternating sums of products of generalized Fibonacci and Lucas numbers. In particular, we recover and extend all results of Z. Cerin and Z. Cerin & G. M. Gianella, more easily."}
{"category": "Math", "title": "Buchsteiner loops", "abstract": "Buchsteiner loops are those which satisfy the identity $x\\backslash (xy \\cdot z) = (y \\cdot zx)/ x$. We show that a Buchsteiner loop modulo its nucleus is an abelian group of exponent four, and construct an example where the factor achieves this exponent."}
{"category": "Math", "title": "A Note On The Kadison-Singer Problem", "abstract": "Let H be a separable Hilbert space with a fixed orthonormal basis (e_n), n>=1, and B(H) be the full von Neumann algebra of the bounded linear operators T: H -> H. Identifying l^\\infty = C(\\beta N) with the diagonal operators, we consider C(\\beta N) as a subalgebra of B(H). For each t in \\beta N, let [\\delta_t] be the set of the states of B(H) that extend the Dirac measure \\delta_t. Our main result shows that, for each t in \\beta N, this set either lies in a finite dimensional subspace of B(H)* or else it must contain a homeomorphic copy of \\beta N."}
{"category": "Math", "title": "Testing for change points in time series models and limiting theorems for NED sequences", "abstract": "This paper first establishes a strong law of large numbers and a strong invariance principle for forward and backward sums of near-epoch dependent sequences. Using these limiting theorems, we develop a general asymptotic theory on the Wald test for change points in a general class of time series models under the no change-point hypothesis. As an application, we verify our assumptions for the long-memory fractional ARIMA model."}
{"category": "Math", "title": "Singularities of quadratic differentials and extremal Teichm\\\"{u}ller mappings defined by Dehn twists", "abstract": "Let $S$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $\\omega$ be a pseudo-Anosov map of $S$ that is obtained from Dehn twists along two families $\\{A,B\\}$ of simple closed geodesics that fill $S$. Then $\\omega$ can be realized as an extremal Teichm\\\"{u}ller mapping on a surface of type $(p,n)$ which is also denoted by $S$. Let $\\phi$ be the corresponding holomorphic quadratic differential on $S$. In this paper, we compare the locations of some distinguished points on $S$ in the $\\phi$-flat metric to their locations with respect to the complete hyperbolic metric. More precisely, we show that all possible non-puncture zeros of $\\phi$ must stay away from all closures of once punctured disk components of $S\\backslash \\{A, B\\}$, and the closure of each disk component of $S\\backslash \\{A, B\\}$ contains at most one zero of $\\phi$. As a consequence of the result, we assert that the number of distinct zeros and poles of $\\phi$ is less than or equal to the number of components of $S\\backslash \\{A, B\\}$."}
{"category": "Math", "title": "Accumulated prediction errors, information criteria and optimal forecasting for autoregressive time series", "abstract": "The predictive capability of a modification of Rissanen's accumulated prediction error (APE) criterion, APE$_{\\delta_n}$, is investigated in infinite-order autoregressive (AR($\\infty$)) models. Instead of accumulating squares of sequential prediction errors from the beginning, APE$_{\\delta_n}$ is obtained by summing these squared errors from stage $n\\delta_n$, where $n$ is the sample size and $1/n\\leq \\delta_n\\leq 1-(1/n)$ may depend on $n$. Under certain regularity conditions, an asymptotic expression is derived for the mean-squared prediction error (MSPE) of an AR predictor with order determined by APE$_{\\delta_n}$. This expression shows that the prediction performance of APE$_{\\delta_n}$ can vary dramatically depending on the choice of $\\delta_n$. Another interesting finding is that when $\\delta_n$ approaches 1 at a certain rate, APE$_{\\delta_n}$ can achieve asymptotic efficiency in most practical situations. An asymptotic equivalence between APE$_{\\delta_n}$ and an information criterion with a suitable penalty term is also established from the MSPE point of view. This offers new perspectives for understanding the information and prediction-based model selection criteria. Finally, we provide the first asymptotic efficiency result for the case when the underlying AR($\\infty$) model is allowed to degenerate to a finite autoregression."}
{"category": "Math", "title": "On the geometric dependence of Riemannian Sobolev best constants", "abstract": "We concerns here with the continuity on the geometry of the second Riemannian L^p-Sobolev best constant B_0(p,g) associated to the AB program. Precisely, for 1 <= p <= 2, we prove that B_0(p,g) depends continuously on g in the C^2-topology. Moreover, this topology is sharp for p = 2. From this discussion, we deduce some existence and C^0-compactness results on extremal functions."}
{"category": "Math", "title": "Online Learning in Discrete Hidden Markov Models", "abstract": "We present and analyse three online algorithms for learning in discrete Hidden Markov Models (HMMs) and compare them with the Baldi-Chauvin Algorithm. Using the Kullback-Leibler divergence as a measure of generalisation error we draw learning curves in simplified situations. The performance for learning drifting concepts of one of the presented algorithms is analysed and compared with the Baldi-Chauvin algorithm in the same situations. A brief discussion about learning and symmetry breaking based on our results is also presented."}
{"category": "Math", "title": "Wishart distributions for decomposable graphs", "abstract": "When considering a graphical Gaussian model ${\\mathcal{N}}_G$ Markov with respect to a decomposable graph $G$, the parameter space of interest for the precision parameter is the cone $P_G$ of positive definite matrices with fixed zeros corresponding to the missing edges of $G$. The parameter space for the scale parameter of ${\\mathcal{N}}_G$ is the cone $Q_G$, dual to $P_G$, of incomplete matrices with submatrices corresponding to the cliques of $G$ being positive definite. In this paper we construct on the cones $Q_G$ and $P_G$ two families of Wishart distributions, namely the Type I and Type II Wisharts. They can be viewed as generalizations of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen [Ann. Statist. 21 (1993) 1272--1317]. We show that the Type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. Indeed, the inverse of the Type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the Type I Wishart is weak hyper Markov. Moreover, the inverse Type II Wishart as a conjugate family presents the advantage of having a multidimensional shape parameter, thus offering flexibility for the choice of a prior."}
{"category": "Math", "title": "Some explicit identities associated with positive self-similar Markov processes", "abstract": "We consider some special classes of L\\'evy processes with no gaussian component whose L\\'evy measure is of the type $\\pi(dx)=e^{\\gamma x}\\nu(e^x-1) dx$, where $\\nu$ is the density of the stable L\\'evy measure and $\\gamma$ is a positive parameter which depends on its characteristics. These processes were introduced in \\cite{CC} as the underlying L\\'evy processes in the Lamperti representation of conditioned stable L\\'evy processes. In this paper, we compute explicitly the law of these L\\'evy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points."}
{"category": "Math", "title": "On the capability of finite groups of class two and prime exponent", "abstract": "We consider the capability of $p$-groups of class two and odd prime exponent. The question of capability is shown to be equivalent to a statement about vector spaces and linear transformations, and using the equivalence we give proofs of some old results and several new ones. In particular, we establish a number of new necessary and new sufficient conditions for capability, including a sufficient condition based only on the ranks of $G/Z(G)$ and $[G,G]$. Finally, we characterise the capable groups among the 5-generated groups in this class."}
{"category": "Math", "title": "F-thresholds, tight closure, integral closure, and multiplicity bounds", "abstract": "The F-threshold $c^J(\\a)$ of an ideal $\\a$ with respect to the ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\\a$ with the Frobenius powers of $J$. We show that under mild assumptions, we can detect the containment in the integral closure or the tight closure of a parameter ideal using F-thresholds. We formulate a conjecture bounding $c^J(\\a)$ in terms of the multiplicities $e(\\a)$ and $e(J)$, when $\\a$ and $J$ are zero-dimensional ideals, and $J$ is generated by a system of parameters. We prove the conjecture when $J$ is a monomial ideal in a polynomial ring, and also when $\\a$ and $J$ are generated by homogeneous systems of parameters in a Cohen-Macaulay graded $k$-algebra."}
{"category": "Math", "title": "On rectangular diagrams, Legendrian knots and transverse knots", "abstract": "A correspondence is studied by H. Matsuda between front projections of Legendrian links in the standard contact structure for 3-space and rectangular diagrams. In this paper, we introduce braided rectangular diagrams, and study a relationship with Legendrian links in the standard contact structure for 3-space. We show Alexander and Markov Theorems for Legendrian links in 3-space."}
{"category": "Math", "title": "An extension of Boyd's $p$-adic algorithm for the harmonic series", "abstract": "In this paper we will extend a $p$-adic algorithm of Boyd in order to study the size of the set: \\[J_p(y)=\\left\\{n :\\sum_{j=1}^{n}\\frac{y^j}{j}\\equiv 0(\\mod p)\\right\\}.\\] Suppose that $p$ is one of the first 100 odd primes and $y\\in\\{1,2,...,p-1\\}$, then our calculations prove that $|J_p(y)|<\\infty$ in 24240 out of 24578 possible cases. Among other results we show that $|J_{13}(9)|=18763$. The paper concludes by discussing some possible applications of our method to sums involving Fibonacci numbers."}
{"category": "Math", "title": "Compactifications of Moduli Spaces and Cellular Decompositions", "abstract": "This paper studies compactifications of moduli spaces involving closed Riemann surfaces. The first main result identifies the homeomorphism types of these compactifications. The second main result introduces orbicell decompositions on these spaces using semistable ribbon graphs extending the earlier work of Looijenga."}
{"category": "Math", "title": "Nearly Ordinary Galois Deformations over Arbitrary Number Fields", "abstract": "Let K be an arbitrary number field, and let rho: Gal(Kbar/K) -> GL_2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of rho. When K is totally real and rho is modular, results of Hida imply that the nearly ordinary deformation space associated to rho contains a Zariski dense set of points corresponding to \"automorphic\" Galois representations. We conjecture that if K is_not_ totally real, then this is never the case, except in three exceptional cases, corresponding to (1) \"base change\", (2) \"CM\" forms, and (3) \"Even\" representations. The latter case conjecturally can only occur if the image of rho is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of Leopoldt's conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of \"many\" positive dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about \"p-adic functorality\", as well as some remarks on how our methods should apply to n-dimensional representations of Gal(Qbar/Q) when n > 2."}
{"category": "Math", "title": "The relative growth rate for partial quotients", "abstract": "We look at the rate of growth of the partial quotients of the infinite continued fraction expansion of an irrational number relative to the rate of approximation of the number by its convergents. In non-generic cases the Hausdorff dimension of some exceptional sets is computed."}
{"category": "Math", "title": "Tame Functions with strongly isolated singularities at infinity: a tame version of a Parusinski's Theorem", "abstract": "Let f be a definable function, enough differentiable. Under the condition of having strongly isolated singularities at infinity at a regular value c we give a sufficient condition expressed in terms of the total absolute curvature function to ensure the local triviality of the function f over a neighbourhood of c and doing so providing the tame version of Parusinski's Theorem on complex polynomials with isolated singularities at infinity."}
{"category": "Math", "title": "Descent for quasi-coherent sheaves on stacks", "abstract": "We give a homotopy theoretic characterization of sheaves on a stack and, more generally, a presheaf of groupoids on an arbitary small site C. We use this to prove homotopy invariance and generalized descent statements for categories of sheaves and quasi-coherent sheaves. As a corollary we obtain an alternate proof of a generalized change of rings theorem of Hovey."}
{"category": "Math", "title": "The Multidimensional Cube Recurrence", "abstract": "We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by Fomin and Zelevinsky, and Carroll and Speyer. The states of this recurrence are indexed by tilings of a polygon with rhombi, and the variables in the recurrence are indexed by vertices of these tilings. We travel from one state of the recurrence to another by performing elementary flips. We show that the values of the recurrence are independent of the order in which we perform the flips; this proof involves nontrivial combinatorial results about rhombus tilings which may be of independent interest. We then show that the multidimensional cube recurrence exhibits the Laurent phenomenon -- any variable is given by a Laurent polynomial in the other variables. We recognize a special case of the multidimensional cube recurrence as giving explicit equations for the isotropic Grassmannians IG(n-1,2n). Finally, we describe a tropical version of the multidimensional cube recurrence and show that, like the tropical octahedron recurrence, it propagates certain linear inequalities."}
{"category": "Math", "title": "Quasitriviality of the Forms of Segre Varieties", "abstract": "We prove the rationality of a $\\k$-form $X$ of the product $S$ of projective spaces provided the existence of a $\\k$-point on $X$. The method of the proof is to find a Galois-invariant birational projection of $S$ to the projective space. This method also allows to prove the quasitriviality of the forms of the hyperplane sections of some Segre varieties."}
{"category": "Math", "title": "A formula for the normal subgroup growth of Baumslag-Solitar groups", "abstract": "We give an exact formula for the number of normal subgroups of each finite index in the Baumslag-Solitar group BS(p,q) when p and q are coprime. Unlike the formula for all finite index subgroups, this one distinguishes different Baumslag-Solitar groups and is not multiplicative. This allows us to give an example of a finitely generated profinite group which is not virtually pronilpotent but whose zeta function has an Euler product."}
{"category": "Math", "title": "On Unirationality of Quartics over non Algebraically Closed Fields", "abstract": "We give examples of smooth $\\k$-unirational line-free quartic hypersurfaces over a non algebraically closed field $\\k$. Unlike other methods of proving unirationality, our method does not rely on existence of linear spaces on quartics."}
{"category": "Math", "title": "Duality and semi-group property for backward parabolic Ito equations", "abstract": "We study existence, uniqueness, semi-group property, and a priori estimates for solutions for backward parabolic Ito equations in domains with boundary. We study also duality between forward and backward equations. The semi-group for backward equations is established in the form of some anti-causality. The novelty is that the semi-group property involves the diffusion term that is a part of the solution."}
{"category": "Math", "title": "Variation of the unit roots along the Dwork family of Calabi-Yau varieties", "abstract": "We study the variation of unit roots of the Dwork families of Calabi-Yau varieties over a finite field by the method of Dwork-Katz and also from the point of view of formal group laws. A p-adic analytic formula for the unit roots away from the Hasse locus is obtained."}
{"category": "Math", "title": "Hoeffding's inequality in game-theoretic probability", "abstract": "This note makes the obvious observation that Hoeffding's original proof of his inequality remains valid in the game-theoretic framework. All details are spelled out for the convenience of future reference."}
{"category": "Math", "title": "A Characterization of the Angle Defect and the Euler Characteristic in Dimension 2 -- Preliminary Draft", "abstract": "The angle defect, which is the standard way to measure curvature at the vertices of polyhedral surfaces, goes back at least as far as Descartes. Although the angle defect has been widely studied, there does not appear to be in the literature an axiomatic characterization of the angle defect. We give a characterization of the angle defect for simplicial surfaces, and we show that variants of the same characterization work for two known approaches to generalizing the angle defect to arbitrary 2-dimensional simplicial complexes. Simultaneously, we give a characterization of the Euler characteristic on 2-dimensional simplicial complexes in terms of being geometrically locally determined."}
{"category": "Math", "title": "Invariants of Knot Diagrams", "abstract": "We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams."}
{"category": "Math", "title": "The Moduli Space of Polynomial Maps and Their Fixed-Point Multipliers", "abstract": "We consider the family $\\mathrm{MP}_d$ of affine conjugacy classes of polynomial maps of one complex variable with degree $d \\geq 2$, and study the map $\\Phi_d:\\mathrm{MP}_d\\to \\widetilde{\\Lambda}_d \\subset \\mathbb{C}^d / \\mathfrak{S}_d$ which maps each $f \\in \\mathrm{MP}_d$ to the set of fixed-point multipliers of $f$. We show that the local fiber structure of the map $\\Phi_d$ around $\\bar{\\lambda} \\in \\widetilde{\\Lambda}_d$ is completely determined by certain two sets $\\mathcal{I}(\\lambda)$ and $\\mathcal{K}(\\lambda)$ which are subsets of the power set of $\\{1,2,\\ldots,d \\}$. Moreover for any $\\bar{\\lambda} \\in \\widetilde{\\Lambda}_d$, we give an algorithm for counting the number of elements of each fiber $\\Phi_d^{-1}\\left(\\bar{\\lambda}\\right)$ only by using $\\mathcal{I}(\\lambda)$ and $\\mathcal{K}(\\lambda)$. It can be carried out in finitely many steps, and often by hand."}
{"category": "Math", "title": "Pointwise Estimates for Marginals of Convex Bodies", "abstract": "We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let X be an isotropic random vector in R^n with a log-concave density. For a typical subspace E in R^n of dimension n^c, consider the probability density of the projection of X onto E. We show that the ratio between this probability density and the standard gaussian density in E is very close to 1 in large parts of E. Here c > 0 is a universal constant. This complements a recent result by the second named author, where the total-variation metric between the densities was considered."}
{"category": "Math", "title": "Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds", "abstract": "We introduce certain homology and cohomology subgroups for any almost complex structure and study their pureness, fullness and duality properties. Motivated by a question of Donaldson, we use these groups to relate J-tamed symplectic cones and J-compatible symplectic cones over a large class of almost complex manifolds, including all Kahler manifolds, almost Kahler 4-manifolds and complex surfaces."}
{"category": "Math", "title": "Quantum $\\frak {gl}_\\infty$, infinite $q$-Schur algebras and their representations", "abstract": "In this paper, we investigate the structure and representations of the quantum group ${\\mathbf{U}(\\infty)}=\\mathbf U_\\upsilon(\\frak{gl}_\\infty)$. We will present a realization for $\\mathbf{U}(\\infty)$, following Beilinson--Lusztig--MacPherson (BLM) \\cite{BLM}, and show that the natural algebra homomorphism $\\zeta_r$ from $\\mathbf{U}(\\infty)$ to the infinite $q$-Schur algebra ${\\boldsymbol{\\mathcal S}}(\\infty,r)$ is not surjective for any $r\\geq 1$. We will give a BLM type realization for the image $\\mathbf{U}(\\infty,r):=\\zeta_r(\\mathbf{U}(\\infty))$ and discuss its presentation in terms of generators and relations. We further construct a certain completion algebra $\\hat{\\boldsymbol{\\mathcal K}}^\\dagger(\\infty)$ so that $\\zeta_r$ can be extended to an algebra epimorphism $\\tilde\\zeta_r:\\hat{\\boldsymbol{\\mathcal K}}^\\dagger(\\infty)\\to{\\boldsymbol{\\mathcal S}}(\\infty,r)$. Finally we will investigate the representation theory of ${\\bf U}(\\infty)$, especially the polynomial representations of ${\\bf U}(\\infty)$."}
{"category": "Math", "title": "A Remark on the Conjectures of Lang-Trotter and Sato-Tate on Average", "abstract": "We obtain new average results on the conjectures of Lang-Trotter and Sato-Tate about elliptic curves."}
{"category": "Math", "title": "The Romanov theorem revised", "abstract": "Let P be the set of all primes and P_2=P\\cup{p_1p_2: p_1,p_2\\in P}$. We prove that the sumset 2^P+P_2={2^p+q: p\\in P, q\\in P_2} has a positive lower density."}
{"category": "Math", "title": "The scalar curvature flow in Lorentzian manifolds", "abstract": "We prove the existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds provided there are barriers."}
{"category": "Math", "title": "Pursuit-Evasion Games with Incomplete Information in Discrete Time", "abstract": "Pursuit-Evasion Games (in discrete time) are stochastic games with nonnegative daily payoffs, with the final payoff being the cumulative sum of payoffs during the game. We show that such games admit a value even in the presence of incomplete information and that this value is uniform, i.e. there are epsilon-optimal strategies for both players that are epsilon-optimal in any long enough prefix of the game. We give an example to demonstrate that nonnegativity is essential and expand the results to leavable games."}
{"category": "Math", "title": "R-diagonal dilation semigroups", "abstract": "This paper addresses extensions of the complex Ornstein-Uhlenbeck semigroup to operator algebras in free probability theory. If $a_1,...,a_k$ are $\\ast$-free $\\mathscr{R}$-diagonal operators in a $\\mathrm{II}_1$ factor, then $D_t(a_{i_1}... a_{i_n}) = e^{-nt} a_{i_1}... a_{i_n}$ defines a dilation semigroup on the non-self-adjoint operator algebra generated by $a_1,...,a_k$. We show that $D_t$ extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by $a_1,...,a_k$. Moreover, we show that $D_t$ satisfies an optimal ultracontractive property: $\\|D_t\\colon L^2\\to L^\\infty\\| \\sim t^{-1}$ for small $t>0$."}
{"category": "Math", "title": "On the sum of the series formed from the prime numbers where the prime numbers of the form $4n-1$ have a positive sign and those of the form $4n+1$ a negative sign", "abstract": "This is an English translation of the Latin original \"De summa seriei ex numeris primis formatae ${1/3}-{1/5}+{1/7}+{1/11}-{1/13}-{1/17}+{1/19}+{1/23}-{1/29}+{1/31}-$ etc. ubi numeri primi formae $4n-1$ habent signum positivum formae autem $4n+1$ signum negativum\" (1775). E596 in the Enestrom index. Let $\\chi$ be the nontrivial character modulo 4. Euler wants to know what $\\sum_p \\chi(p)/p$ is, either an exact expression or an approximation. He looks for analogies to the harmonic series and the series of reciprocals of the primes. Another reason he is interested in this is that if this series has a finite value (which is does, the best approximation Euler gets is 0.3349816 in section 27) then there are infinitely many primes congruent to 1 mod 4 and infinitely many primes congruent to 3 mod 4. In section 15 Euler gives the Euler product for the L(chi,1). As a modern mathematical appendix appendix, I have written a proof following Davenport that the series $\\sum_p \\frac{\\chi(p)}{p}$ converges. This involves applications of summation by parts, and uses Chebyshev's estimate for the second Chebyshev function (summing the von Mangoldt function)."}
{"category": "Math", "title": "Algebraic Compactness OF $\\prod M_\\alpha / \\oplus M_\\alpha$", "abstract": "In this note, we are working within the category $\\rmod$ of (unitary, left) $R$-modules, where $R$ is a {\\bf countable} ring. It is well known (see e.g. Kie{\\l}pi\\'nski & Simson [5], Theorem 2.2) that the latter condition implies that the (left) pure global dimension of $R$ is at most 1. Given an infinite index set $A$, and a family $M_\\al\\in\\rmod$, $\\al\\in A$ we are concerned with the conditions as to when the $R$-module $$\\prod/\\coprod=\\prod_{\\al\\in A}M_\\al/\\bigoplus_{\\al\\in A}M_\\al$$ is or is not algebraically compact. There are a number of special results regarding this question and this note is meant to be an addition to and a generalization of the set of these results. Whether the module in the title is algebraically compact or not depends on the numbers of algebraically compact and non-compact modules among the components $M_\\al$."}
{"category": "Math", "title": "A Note on Surjective Inverse Systems", "abstract": "Given an upward directed set $I$ we consider surjective $I$-inverse systems $\\{X_\\al,f_{\\al\\be}:X_\\be\\lra X_\\al| \\al\\leq\\be\\in I\\}$, namely those inverse systems that have all $f_{\\al\\be}$ surjective. A number of properties of $I$-inverse systems have been investigated; such are the Mittag-Leffler condition, investigated by Grothendieck and flabby and semi-flabby $I$-inverse systems studied by Jensen. We note that flabby implies semi-flabby implies surjective implies Mittag-Leffler. Some of the results about surjective inverse systems have been known for some time. The aim of this note is to give a series of equivalent statements and implications involving surjective inverse systems and the systems satisfying the Mittag-Leffler condition, together with improvements of established results, as well as their relationships with the already known, but scattered facts. The most prominent results relate cardinalities of the index sets with right exactness of the inverse limit functor and the non-vanishing of the inverse limit -- connections related to cohomological dimensions."}
{"category": "Math", "title": "The Ratio Monotonicity of the $q$-Derangement Numbers", "abstract": "We show that the $q$-derangement numbers satisfy a ratio monotone property, which is analogous to the log-concavity and is stronger than the spiral property and the unimodality."}
{"category": "Math", "title": "The Limiting Distribution of the Coefficients of the $q$-Catalan Numbers", "abstract": "We show that the limiting distributions of the coefficients of the $q$-Catalan numbers and the generalized $q$-Catalan numbers are normal. Despite the fact that these coefficients are not unimodal for small $n$, we conjecture that for sufficiently large $n$, the coefficients are unimodal and even log-concave except for a few terms of the head and tail."}
{"category": "Math", "title": "Boundary Harnack Principle for Subordinate Brownian Motions", "abstract": "We establish a boundary Harnack principle for a large class of subordinate Brownian motion, including mixtures of symmetric stable processes, in bounded $\\kappa$-fat open set (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded $\\kappa$-fat open sets with respect to these processes with their Euclidean boundary."}
{"category": "Math", "title": "On manifolds satisfying stable systolic inequalities", "abstract": "We show that for closed orientable manifolds the $k$-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree $k$ that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles of dimension at least two. Additionally, we prove that the stable systolic constant depends only on the image of the fundamental class in a suitable Eilenberg-Mac Lane space. Consequently, the stable $k$-systolic constant is completely determined by the multilinear intersection form on $k$-dimensional cohomology."}
{"category": "Math", "title": "Boundedness and Compactness of Toeplitz operators with L^1 symbols on the Bergman space", "abstract": "We characterise the boundedness of a Toeplitz operator on the Bergman space with an L^1 symbol.We also prove that the compactness of a Toeplitz operator on the Bergman space with an L^1 symbol is completely determined by the boundary behaviour of itss Berezin transform. This result extends known results in the cases when the symbol is either a positive L^1 function, an L^\\infty function or a general BMO^1 function."}
{"category": "Math", "title": "Twisted conjugacy classes in Symplectic groups, Mapping class groups and Braid groups(including an Appendix written with Francois Dahmani)", "abstract": "We prove that the symplectic group $Sp(2n,\\mathbb Z)$ and the mapping class group $Mod_{S}$ of a compact surface $S$ satisfy the $R_{\\infty}$ property. We also show that $B_n(S)$, the full braid group on $n$-strings of a surface $S$, satisfies the $R_{\\infty}$ property in the cases where $S$ is either the compact disk $D$, or the sphere $S^2$. This means that for any automorphism $\\phi$ of $G$, where $G$ is one of the above groups, the number of twisted $\\phi$-conjugacy classes is infinite."}
{"category": "Math", "title": "Zonotopal algebra", "abstract": "A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\\cal H}(X)$. This well-known line of study is particularly interesting in case $n\\eqbd\\rank X \\ll N$. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\\it external, central, and internal.} Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in $n$ variables that are dual to each other: one encodes properties of the arrangement ${\\cal H}(X)$, while the other encodes by duality properties of the zonotope $Z(X)$. The algebraic structures are defined purely in terms of the combinatorial structure of $X$, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of $Z(X)$ or ${\\cal H}(X)$. The theory is universal in the sense that it requires no assumptions on the map $X$ (the only exception being that the algebro-analytic operations on $Z(X)$ yield sought-for results only in case $X$ is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory."}
{"category": "Math", "title": "Boundedness and Compactness of products of Toeplitz operators on the Bergman Space", "abstract": "In a celebrated conjecture D.Sarason stated a necessary and sufficient condition on the symbols f, g in the Bergman space, L^2_a(\\Delta) of the unit disk, \\Delta, for the products T_{f}T_{\\bar g} of associated Toeplitz operators to be bounded (respectively compact) on L^2_a(\\Delta) . K. Stroethoff and D. Zheng proved that these conditions are necessary. We prove the sufficiency of these conditions, thus solvind Sarason's conjecture."}
{"category": "Math", "title": "Genericity of supercuspidal representations of p-adic Sp(4)", "abstract": "We describe the supercuspidal representations of Sp(4,F), for F a non-archimedean local field of residual characteristic different from 2, and determine which are generic."}
{"category": "Math", "title": "On fixed points of permutations", "abstract": "The number of fixed points of a random permutation of 1,2,...,n has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. For most examples, they are trivial -- almost every permutation has no fixed points. For the usual action of the symmetric group on k-sets of 1,2,...,n, the limit is a polynomial in independent Poisson variables. This exhausts all cases. We obtain asymptotic estimates in some examples, and give a survey of related results."}
{"category": "Math", "title": "Optimal Monotonicity of $L^p$ Integral of Conformal Invariant Green Function", "abstract": "Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $\\Omega$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geometric isoperimetric inequalities under finiteness of the positive part of total Gauss curvature of a conformal metric on $\\Omega$. Consequently, new analytic and geometric isoperimetric-type inequalities are discovered. Furthermore, when applying the geometric principle to two-dimensional Riemannian manifolds, we find fortunately that $\\{0,1\\}$-form of the induced principle is midway between Moser-Trudinger's inequality and Nash-Sobolev's inequality on complete noncompact boundary-free surfaces, and yet equivalent to Nash-Sobolev's/Faber-Krahn's eigenvalue/Heat-kernel-upper-bound/Log-Sobolev's inequality on the surfaces with finite total Gauss curvature and quadratic area growth."}
{"category": "Math", "title": "Optimal L^p-Riemannian Gagliardo-Nirenberg inequalities", "abstract": "Let (M,g) be a compact Riemannian manifold of dimension n \\geq 2. In this work we prove the validity of the optimal L^p-Riemannian Gagliardo-Nirenberg inequality for 1 < p \\leq 2. Our proof relies strongly on a new distance lemma which. In particular, we extend L^p-Euclidean Gagliardo-Nirenberg inequalities due to Del Pino and Dolbeault and the optimal L^2-Riemannian Gagliardo-Nirenberg inequality due to Broutteland in a unified framework."}
{"category": "Math", "title": "Notes on the Jacobi equation", "abstract": "We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry."}
{"category": "Math", "title": "On positive opetopes, positive opetopic cardinals and positive opetopic set", "abstract": "We introduce the notion of a positive opetope and positive opetopic cardinals as certain finite combinatorial structures. The positive opetopic cardinals to positive-to-one polygraphs are like simple graphs to free omega-categories over omega-graphs, c.f. [MZ]. In particular, they allow us to give an explicit combinatorial description of positive-to-one polygraphs. Using this description we show, among other things, that positive-to-one polygraphs form a presheaf category with the exponent category being the category of positive opetopes. We also show that the category of omega-categories is monadic over the category of positive-to-one polygraphs with the `free functor' being an inclusion."}
{"category": "Math", "title": "On ordered face structures and many-to-one computads", "abstract": "We introduce the notion of an ordered face structure. The ordered face structures to many-to-one computads are like positive face structures to positive-to-one computads. This allow us to give an explicit combinatorial description of many-to-one computads in terms of ordered face structures."}
{"category": "Math", "title": "Hyperbolic cusps with convex polyhedral boundary", "abstract": "We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthemore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp. The proof uses the total scalar curvature functional on the space of ``cusps with particles'', which are hyperbolic cone-manifolds with the singular locus a union of half-lines. We prove, in addition, that convex polyhedral cusps with particles are rigid with respect to the induced metric on the boundary and the curvatures of the singular locus. Our main theorem is equivalent to a part of a general statement about isometric immersions of compact surfaces."}
{"category": "Math", "title": "Schubert calculus on the grassmannian of hermitian lagrangian spaces", "abstract": "The grassmannian of hermitian lagrangian spaces in $\\mathbb{C}^n\\oplus \\mathbb{C}^n$ is a natural compactification of the space of hermitian $n\\times n$ matrices. We describe a Schubert-like, Whitney regular stratification on this space which has a Morse theoretic origin. We prove that these strata define closed subanalytic currents \\`{a} la R. Hardt, generating the integral homology of this space, we investigate their intersection theoretic properties, and we prove certain odd (in K-theoretic sense) Thom-Porteous type theorems."}
{"category": "Math", "title": "On sequences of finitely generated discrete groups", "abstract": "We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. We show that, for algebraically convergent sequences (Gamma_i), unless Gamma_i's are (eventually) elementary or contain normal finite subgroups of arbitrarily high order, their algebraic limit is a discrete nonelementary subgroup of G. In the case of divergent sequences (Gamma_i) we show that the limiting action on a real tree T satisfies certain semistability condition, which generalizes the notion of stability introduced by Rips. We then verify that the group Gamma splits as an amalgam or HNN extension of finitely generated groups, so that the edge group has an amenable image in the isometry group of T."}
{"category": "Math", "title": "Automorphisms of the three-torus preserving a genus three Heegaard splitting", "abstract": "The mapping class group of a Heegaard splitting is the group of connected components in the set of automorphisms of the ambient manifold that map the Heegaard surface onto itself. For the genus three Heegaard splitting of the 3-torus, we find an eight element generating set for this group. Six of these generators induce generating elements of the mapping class group of the 3-torus and the remaining two are isotopy trivial in the 3-torus."}
{"category": "Math", "title": "A class of quantum doubles which are ribbon algebras", "abstract": "Andruskiewitsch and Schneider classify a large class of pointed Hopf algebras with abelian coradical. The quantum double of each such Hopf algebra is investigated. The quantum doubles of a family of Hopf algebras from the above classification are ribbon Hopf algebras."}
{"category": "Math", "title": "Fisher information of orthogonal polynomials I", "abstract": "Following the lead of J. Dehesa and his collaborators, we compute the Fisher information of the Meixner-Pollaczek, Meixner, Krawtchouk and Charlier polynomials."}
{"category": "Math", "title": "The mean-square of Dirichlet L-functions", "abstract": "We verify the conjecture of [CFKRS] for the mean square near the critical point of Dirichlet L-functions for a composite modulus q. We also prove a kind of reciprocity formula when the second moment for a prime modulus is twisted by a character evaluated at a different prime."}
{"category": "Math", "title": "Characterizing algebraic stacks", "abstract": "We extend the notion of algebraic stack to an arbitrary subcanonical site C. If the topology on C is local on the target and satisfies descent for morphisms, we show that algebraic stacks are precisely those which are weakly equivalent to representable presheaves of groupoids whose domain map is a cover. This leads naturally to a definition of algebraic n-stacks. We also compare different sites naturally associated to a stack."}
{"category": "Math", "title": "The quantization of a toric manifold is given by the integer lattice points in the moment polytope", "abstract": "We describe a very nice argument, which we learned from Sue Tolman, that the dimension of the quantization space of a toric manifold, using a Kaehler polarization, is given by the number of integer lattice points in the moment polytope."}
{"category": "Math", "title": "Directed random growth models on the plane", "abstract": "This is a brief survey of laws of large numbers, fluctuation results and large deviation principles for asymmetric interacting particle systems that represent moving interfaces on the plane. We discuss the exclusion process, the Hammersley process and the related last-passage growth models."}
{"category": "Math", "title": "Hochschild cohomology and Atiyah classes", "abstract": "In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich. Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well."}
{"category": "Math", "title": "Limits and C*-algebras of low rank or dimension", "abstract": "We explore various limit constructions for C*-algebras, such as composition series and inverse limits, in relation to the notions of real rank, stable rank, and extremal richness. We also consider extensions and pullbacks. We identify some conditions under which the constructions preserve low rank for the C*-algebras or their multiplier algebras. We also discuss the version of topological dimension theory appropriate for primitive ideal spaces of C*-algebras and provide an analogue for rank of the countable sum theorem of dimension theory. As an illustration of how the main results can be applied, we show that a CCR algebra has stable rank one if and only if it has topological dimension zero or one, and we characterize those sigma-unital CCR algebras whose multiplier algebras have stable rank one or extremal richness. (The real rank zero case was already known.)"}
{"category": "Math", "title": "Stabilization of an overloaded queueing network using measurement-based admission control", "abstract": "Admission control can be employed to avoid congestion in queueing networks subject to overload. In distributed networks the admission decisions are often based on imperfect measurements on the network state. This paper studies how the lack of complete state information affects the system performance by considering a simple network model for distributed admission control. The stability region of the network is characterized and it is shown how feedback signaling makes the system very sensitive to its parameters."}
{"category": "Math", "title": "Cubic points on cubic curves and the Brauer-Manin obstruction on K3 surfaces", "abstract": "We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the Brauer-Manin obstruction is not the only obstruction to the Hasse principle for K3 surfaces."}
{"category": "Math", "title": "Convergence of random zeros on complex manifolds", "abstract": "We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros of systems of m polynomials of degree N, orthonormalized on a regular compact subset K of C^m, almost surely converge to the equilibrium measure on K as the degree N goes to infinity."}
{"category": "Math", "title": "Twisted automorphisms of Hopf algebras", "abstract": "Twisted homomorphisms of bialgebras are bialgebra homomorphisms from the first into Drinfeld twistings of the second. They possess a composition operation extending composition of bialgebra homomorphisms. Gauge transformations of twists, compatible with adjacent homomorphisms, give rise to gauge transformation of twisted homomorphisms, which behave nicely with respect to compositions. Here we study (gauge classes of) twisted automorphisms of cocommutative Hopf algebras. After revising well-known relations between twists, twisted forms of bialgebras and $R$-matrices (for commutative bialgebras) we describe twisted automorphisms of universal enveloping algebras."}
{"category": "Math", "title": "Twisted automorphisms of group algebras", "abstract": "We continue the study of twisted automorphisms of Hopf algebras started in \"Twisted automorphisms of Hopf algebras\". In this paper we concentrate on the group algebra case. We describe the group of twisted automorphisms of the group algebra of a group of order coprime to 6. The description turns out to be very similar to the one for the universal enveloping algebra given in \"Twisted automorphisms of Hopf algebras\"."}
{"category": "Math", "title": "Nuclei of categories with tensor products", "abstract": "Following the analogy between algebras (monoids) and monoidal categories the construction of nucleus for non-associative algebras is simulated on the categorical level. Nuclei of categories of modules are considered as an example."}
{"category": "Math", "title": "Maxima of Moving Sums in a Poisson Random Field", "abstract": "The extremal tail probabilities of moving sums in a marked Poisson random field is examined here. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. Change of measure and analysis of local random fields are used to provide tail probabilities. The asymptotic constants are initially expressed in a form that seems hard to evaluate and do not seem to provide any additional information on the properties of the constants. A more sophisticated approach is then undertaken giving rise to an expression that is not only neater but also able to provide computable bounds. The technique used to obtain this constant can also be modified to work on continuous processes."}
{"category": "Math", "title": "Fields with several commuting derivations", "abstract": "For every natural number $m$, the existentially closed models of the theory of fields with $m$ commuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential fields has a model-companion. The axioms are that certain differential varieties determined by certain ordinary varieties are nonempty. There is no restriction on the characteristic of the underlying field."}
{"category": "Math", "title": "Examples of signature (2,2) manifolds with commuting curvature operators", "abstract": "We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A."}
{"category": "Math", "title": "Approximations to Euler's constant", "abstract": "We study a problem of finding good approximations to Euler's constant $\\gamma=\\lim_{n\\to\\infty}S_n,$ where $S_n=\\sum_{k=1}^n\\frac{1}{n}-\\log(n+1),$ by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow convergence of the sequence $S_n$ can be significantly improved if $S_n$ is replaced by linear combinations of $S_n$ with integer coefficients. In this paper, considering more general linear transformations of the sequence $S_n$ we establish new accelerating convergence formulae for $\\gamma.$ Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results."}
{"category": "Math", "title": "On the contact Ozsvath-Szabo invariant", "abstract": "Sarkar and Wang proved that the hat version of Heegaard Floer homology group of a closed oriented 3-manifold is combinatorial starting from an arbitrary nice Heegaard diagram and in fact every closed oriented 3-manifold admits such a Heegaard diagram. Plamenevskaya showed that the contact Ozsvath-Szabo invariant is combinatorial once we are given an open book decomposition compatible with a contact structure. The idea is to combine the algorithm of Sarkar and Wang with the recent description of the contact Ozsvath-Szabo invariant due to Honda, Kazez and Matic. Here we simply observe that the hat version of the Heegaard Floer homology group and the contact Ozsvath-Szabo invariant in this group can be combinatorially calculated starting from a contact surgery diagram. We give detailed examples pointing out to some shortcuts in the computations."}
{"category": "Math", "title": "Strongly r-matrix induced tensors, Koszul cohomology, and arbitrary-dimensional quadratic Poisson cohomology", "abstract": "We introduce the concept of strongly $r$-matrix induced ({\\small SRMI}) Poisson structure, report on the relation of this property with the stabilizer dimension of the considered quadratic Poisson tensor, and classify the Poisson structures of the Dufour-Haraki classification (DHC) according to their membership of the family of {\\small SRMI} tensors. One of the main results of our work is a generic cohomological procedure for {\\small SRMI} Poisson structures in arbitrary dimension. This approach allows decomposing Poisson cohomology into, basically, a Koszul cohomology and a relative cohomology. Moreover, we investigate this associated Koszul cohomology, highlight its tight connections with Spectral Theory, and reduce the computation of this main building block of Poisson cohomology to a problem of linear algebra. We apply these upshots to two structures of the DHC and provide an exhaustive description of their cohomology. We thus complete our list of data obtained in previous works, see \\cite{MP} and \\cite{AMPN}, and gain fairly good insight into the structure of Poisson cohomology."}
{"category": "Math", "title": "Antimagic labelings of regular bipartite graphs: An application of the Marriage Theorem", "abstract": "A labeling of a graph is a bijection from $E(G)$ to the set $\\{1, 2,..., |E(G)|\\}$. A labeling is \\textit{antimagic} if for any distinct vertices $u$ and $v$, the sum of the labels on edges incident to $u$ is different from the sum of the labels on edges incident to $v$. We say a graph is antimagic if it has an antimagic labeling. In 1990, Ringel conjectured that every connected graph other than $K_2$ is antimagic. In this paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem."}
{"category": "Math", "title": "A new metric between distributions of point processes", "abstract": "Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric $\\bar{d}_1$ that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about $\\bar{d}_1$ and its induced Wasserstein metric $\\bar{d}_2$ for point process distributions are given, including examples of useful $\\bar{d}_1$-Lipschitz continuous functions, $\\bar{d}_2$ upper bounds for Poisson process approximation, and $\\bar{d}_2$ upper and lower bounds between distributions of point processes of i.i.d. points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of $\\bar{d}_1$ in applications."}
{"category": "Math", "title": "The uniform order convergence structure on ML(X)", "abstract": "The aim of this paper is to set up appropriate uniform convergence spaces in which to reformulate and enrich the Order Completion Method for nonlinear PDEs. In this regard, we consider an appropriate space ML(X) of normal lower semi-continuous functions. The space ML(X)= appears in the ring theory of C(X), and its various extensions, as well as in the theory of nonlinear PDEs. We define a uniform convergence structure on ML(X) such that the induced convergence structure is the order convergence structure. The uniform convergence space completion of ML(X) is constructed as the set of normal lower semi-continuous functions. It is then shown how these ideas may be applied to solve nonlinear PDEs. In particular, we construct generalized solutions to the Navier-Stokes equations in three spatial dimensions, subject to an initial condition."}
{"category": "Math", "title": "Complete interpolating sequences, the discrete Muckenhoupt condition, and conformal mapping", "abstract": "We extend the parameterization of sine-type functions in terms of conformal mappings onto slit domains given by Eremenko and Sodin to the more general case of generating functions of real complete interpolating sequences. It turns out that the cuts have to fulfill the discrete Muckenhoupt condition studied earlier by Lyubarskii and Seip."}
{"category": "Math", "title": "Harmonic nets in metric spaces", "abstract": "We investigate harmonic maps from weighted graphs into metric spaces that locally admit unique centers of gravity, like Alexandrov spaces with upper curvature bounds. We prove an existence result by constructing an iterative geometric process that converges to such maps, called harmonic nets for short."}
{"category": "Math", "title": "On the cyclic subgroup separability of free products of two groups with amalgamated subgroup", "abstract": "Let $G$ be a free product of two groups with amalgamated subgroup, $\\pi$ be either the set of all prime numbers or the one-element set \\{$p$\\} for some prime number $p$. Denote by $\\Sigma$ the family of all cyclic subgroups of group $G$, which are separable in the class of all finite $\\pi$-groups. Obviously, cyclic subgroups of the free factors, which aren't separable in these factors by the family of all normal subgroups of finite $\\pi$-index of group $G$, the subgroups conjugated with them and all subgroups, which aren't $\\pi^{\\prime}$-isolated, don't belong to $\\Sigma$. Some sufficient conditions are obtained for $\\Sigma$ to coincide with the family of all other $\\pi^{\\prime}$-isolated cyclic subgroups of group $G$. It is proved, in particular, that the residual $p$-finiteness of a free product with cyclic amalgamation implies the $p$-separability of all $p^{\\prime}$-isolated cyclic subgroups if the free factors are free or finitely generated residually $p$-finite nilpotent groups."}
{"category": "Math", "title": "Characterizations of Hankel multipliers", "abstract": "We give characterizations of radial Fourier multipliers as acting on radial L^p-functions, 1<p<2d/(d+1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besides L^p-L^q bounds we also characterize weak type inequalities and intermediate inequalities involving Lorentz spaces. Applications include results on interpolation of multiplier spaces."}
{"category": "Math", "title": "On Reedy Model Categories", "abstract": "The sole purpose of this note is to introduce some elementary results on the structure and functoriality of Reedy model categories. In particular, I give a very useful little criterion to determine whether composition with a morphism of Reedy categories determines a left or right Quillen functor. I then give a number of useful inheritance results."}
{"category": "Math", "title": "CW type of inverse limits and function spaces", "abstract": "Given CW complexes X and Y, let map(X,Y) denote the space of continuous functions from X to Y with the compact open topology. The space map(X,Y) need not have the homotopy type of a CW complex. Here the results of an extensive investigation of various necessary and various sufficient conditions for map(X,Y) to have the homotopy type of a CW complex are exhibited. The results extend all previously known results on this topic. Moreover, appropriate converses are given for the previously known sufficient conditions. It is shown that this difficult question is related to well known problems in algebraic topology. For example, the geometric Moore conjecture (asserting that a simply connected finite complex admits an eventual geometric exponent at any prime if and only if it is elliptic) can be restated in terms of CW homotopy type of certain function spaces. Spaces of maps between CW complexes are a particular case of inverse limits of systems whose bonds are Hurewicz fibrations between spaces of CW homotopy type. Related problems concerning CW homotopy type of the limit space of such a system are also studied. In particular, an almost complete solution to a well known problem concerning towers of fibrations is presented."}
{"category": "Math", "title": "Efficient strong integrators for linear stochastic systems", "abstract": "We present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of Wiener processes. These schemes are based on the Neumann (stochastic Taylor) and Magnus expansions. Firstly, we consider the case when the governing linear diffusion vector fields commute with each other, but not with the linear drift vector field. We prove that numerical methods based on the Magnus expansion are more accurate in the mean-square sense than corresponding stochastic Taylor integration schemes. Secondly, we derive the maximal rate of convergence for arbitrary multi-dimensional stochastic integrals approximated by their conditional expectations. Consequently, for general nonlinear stochastic differential equations with non-commuting vector fields, we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order. Thirdly, we consider the consequences in two numerical studies, one of which is an application arising in stochastic linear-quadratic optimal control."}
{"category": "Math", "title": "Functoriality for Lagrangian correspondences in Floer theory", "abstract": "Using quilted Floer cohomology and relative quilt invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that this functor agrees with geometric composition in the case that the composition is smooth and embedded. As a consequence we obtain 'categorification commutes with composition' for Lagrangian correspondences."}
{"category": "Math", "title": "Dense sets of integers with prescribed representation functions", "abstract": "Let A be a set of integers and let h \\geq 2. For every integer n, let r_{A, h}(n) denote the number of representations of n in the form n=a_1+...+a_h, where a_1,...,a_h belong to the set A, and a_1\\leq ... \\leq a_h. The function r_{A,h} from the integers Z to the nonnegative integers N_0 U {\\infty} is called the representation function of order h for the set A. We prove that every function f from Z to N_0 U {\\infty} satisfying liminf_{|n|->\\infty} f (n)\\geq g is the representation function of order h for some sequence A of integers, and that A can be constructed so that it increases \"almost\" as slowly as any given B_h[g] sequence. In particular, for every epsilon >0 and g \\geq g(h,epsilon), we can construct a sequence A satisfying r_{A,h}=f and A(x)\\gg x^{(1/h)-epsilon}."}
{"category": "Math", "title": "Precovers, localizations and stable homotopy", "abstract": "We prove a new localization theorem for stable model categories if the localizing subcategory is generated by a precovering class in the model category. We use this to show how one may explicitly realize certain Bousfield localization functors that arise naturally in the study of relative homological algebra for group algebras."}
{"category": "Math", "title": "Sharpness of the Finsler-Hadwiger inequality", "abstract": "In this paper we shall prove a sharpened version of the Finsler-Hadwiger inequality which is a strong generalization of Weitzenbock inequality. After that we give another refinement of this inequality and in the final part we provide some basic applications."}
{"category": "Math", "title": "Positive bases in spaces of polynomials", "abstract": "For a nonempty compact set D of R we determine the maximal possible dimension of a subspace X of polynomial functions of degree at most m which possesses a positive bases (where positivity is understood on D). The exact value of this maximal dimension depends on topological features of the base set D. We show that at many cases dimension m can be achieved.Whereas only for low m or finite sets D it is possible to have m+1 (the full dimension of the space of polynomials of degree at most m) dimensional subspace X with positive basis. Therefore, it turns out that for no D it is possible to have a positive basis of the polynomial space of degree at most m for all m in N."}
{"category": "Math", "title": "Bordered Riemann surfaces in C^2", "abstract": "One of the oldest open problems in the classical function theory is whether every open Riemann surface admits a proper holomorphic embedding into C^2. In this paper we prove the following Theorem: If D is a bordered Riemann surface whose closure admits an injective immersion in C^2 that is holomorphic in D, then D admits a proper holomorphic embedding in C^2. The most general earlier results are due to J. Globevnik and B. Stensones (Math. Ann. 303 (1995), 579-597) and E. F. Wold (Internat. J. Math. 17 (2006), 963-974). We give an explicit and elementary construction that does not require the Teichmuller space theory, and we also indicate another possible proof using the latter theory."}
{"category": "Math", "title": "Linear maps preserving invariants", "abstract": "Let $G\\subset\\GL(V)$ be a complex reductive group. Let $G'$ denote $\\{\\phi\\in\\GL(V)\\mid p\\circ\\phi=p\\text{for all} p\\in\\C[V]^G\\}$. We show that, in general, $G'=G$. In case $G$ is the adjoint group of a simple Lie algebra $\\lieg$, we show that $G'$ is an order 2 extension of $G$. We also calculate $G'$ for all representations of $\\SL_2$."}
{"category": "Math", "title": "On the growth of the Bergman kernel near an infinite-type point", "abstract": "We study diagonal estimates for the Bergman kernels of certain model domains in $\\mathbb{C}^2$ near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range -- roughly speaking -- from being ``mildly infinite-type'' to very flat at the infinite-type points."}
{"category": "Math", "title": "Random Matrices: The circular Law", "abstract": "Let $\\a$ be a complex random variable with mean zero and bounded variance $\\sigma^{2}$. Let $N_{n}$ be a random matrix of order $n$ with entries being i.i.d. copies of $\\a$. Let $\\lambda_{1}, ..., \\lambda_{n}$ be the eigenvalues of $\\frac{1}{\\sigma \\sqrt n}N_{n}$. Define the empirical spectral distribution $\\mu_{n}$ of $N_{n}$ by the formula $$ \\mu_n(s,t) := \\frac{1}{n} # \\{k \\leq n| \\Re(\\lambda_k) \\leq s; \\Im(\\lambda_k) \\leq t \\}.$$ The Circular law conjecture asserts that $\\mu_{n}$ converges to the uniform distribution $\\mu_\\infty$ over the unit disk as $n$ tends to infinity. We prove this conjecture under the slightly stronger assumption that the $(2+\\eta)\\th$-moment of $\\a$ is bounded, for any $\\eta >0$. Our method builds and improves upon earlier work of Girko, Bai, G\\\"otze-Tikhomirov, and Pan-Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics."}
{"category": "Math", "title": "A rigidity theorem for holomorphic generators on the Hilbert ball", "abstract": "We present a rigidity property of holomorphic generators on the open unit ball $\\mathbb{B}$ of a Hilbert space $H$. Namely, if $f\\in\\Hol (\\mathbb{B},H)$ is the generator of a one-parameter continuous semigroup ${F_t}_{t\\geq 0}$ on $\\mathbb{B}$ such that for some boundary point $\\tau\\in \\partial\\mathbb{B}$, the admissible limit $K$-$\\lim\\limits_{z\\to\\tau}\\frac{f(x)}{\\|x-\\tau\\|^{3}}=0$, then $f$ vanishes identically on $\\mathbb{B}$."}
{"category": "Math", "title": "The geometry of Minkowski spaces -- a survey. Part I", "abstract": "We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equilateral sets, circles: intersection, circumscribed, characterizations, circumference and area, inscribed equilateral polygons."}
{"category": "Math", "title": "On the DDVV Conjecture and the Comass in Calibrated Geometry (II)", "abstract": "In this paper, we proved P(n,3), which is an important part of the DDVV conjecture. The general case will be treated in the next version of the paper."}
{"category": "Math", "title": "The reciprocity law for the twisted second moment of Dirichlet L-functions", "abstract": "We produce a new proof of the reciprocity law for the twisted second moment of Dirichlet L-functions that was recently proved by Conrey. Our method is to analyze certain two-variable sums where the variables satisfy a linear congruence. We show that these sums satisfy an elegant reciprocity formula. In the case that the modulus is prime, these sums are closely related to the twisted second moment, and the reciprocity formula for these sums implies Conrey's reciprocity formula. We also extend the range of uniformity of Conrey's formula."}
{"category": "Math", "title": "Neck Pinching Dynamics Under Mean Curvature Flow", "abstract": "In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a \"neck\" which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process."}
{"category": "Math", "title": "Hook modules for general linear groups", "abstract": "For an arbitrary infinite field k of characteristic p > 0, we describe the structure of a block of the algebraic monoid M_n(k) (all n x n matrices over k), or, equivalently, a block of the Schur algebra S(n,p), whose simple modules are indexed by p-hook partitions. The result is known; we give an elementary and self-contained proof, based only on a result of Peel and Donkin's description of the blocks of Schur algebras. The result leads to a character formula for certain simple GL_n(k)-modules, valid for all n and all p. This character formula is a special case of one found by Brundan, Kleshchev, and Suprunenko and, independently, by Mathieu and Papadopoulo."}
{"category": "Math", "title": "The K-Theory of Toeplitz C*-Algebras of Right-Angled Artin Groups", "abstract": "To a graph $\\Gamma$ one can associate a C^*-algebra $C^*(\\Gamma)$ generated by isometries. Such $C^*$-algebras were studied recently by Crisp and Laca. They are a special case of the Toeplitz C^*-algebras $\\mathcal{T}(G, P)$ associated to quasi-latice ordered groups (G, P) introduced by Nica. Crisp and Laca proved that the so called \"boundary quotients\" $C^*_q(\\Gamma)$ of $C^*(\\Gamma)$ are simple and purely infinite. For a certain class of finite graphs $\\Gamma$ we show that $C^*_q(\\Gamma)$ can be represented as a full corner of a crossed product of an appropriate C^*-subalgebra of $C^*_q(\\Gamma)$ built by using $C^*(\\Gamma')$, where $\\Gamma'$ is a subgraph of $\\Gamma$ with one less vertex, by the group $\\mathbb{Z}$. Using induction on the number of the vertices of $\\Gamma$ we show that $C^*_q(\\Gamma)$ are nuclear and belong to the small bootstrap class. This also enables us to use the Pimsner-Voiculescu exact sequence to find their K-theory. Finally we use the Kirchberg-Phillips classification theorem to show that those C^*-algebras are isomorphic to tensor products of $\\mathcal{O}_n$ for $1 \\leq n \\leq \\infty$."}
{"category": "Math", "title": "Energy of knots and the infinitesimal cross ratio", "abstract": "This is a survey article on two topics. The Energy E of knots can be obtained by generalizing an electrostatic energy of charged knots in order to produce optimal knots. It turns out to be invariant under Moebius transformations. We show that it can be expressed in terms of the infinitesimal cross ratio, which is a conformal invariant of a pair of 1-jets, and give two kinds of interpretations of the real part of the infinitesimal cross ratio."}
{"category": "Math", "title": "At Least Half Of All Graphs Satisfy \\chi \\leq {1/4}\\omega + {3/4}\\Delta + 1", "abstract": "We prove that for any graph G at least one of G or $\\bar{G}$ satisfies $\\chi \\leq {1/4}\\omega + {3/4}\\Delta + 1$. In particular, self-complementary graphs satisfy this bound."}
{"category": "Math", "title": "A quantization of the Hitchin hamiltonian system and the Beilinson-Drinfeld isomorphism", "abstract": "We will study the Hitchin's hamiltonian system for a modular stack of principal SL_2(C) bundle on a smooth projective curve which has a parabolic reduction at certain points. As an application we will obtain a generalization of the Beilinson-Drinfeld isomorphism, which is a quantization of the Hitchin's hamiltonian system."}
{"category": "Math", "title": "Rational Solutions of the Noumi and Yamada System of type $A_5^{(1)}$", "abstract": "We completely classify all of rational solutions of the Noumi and Yamada system of type $A_5^{(1)}$, which is a generalization of the fifth Painlev\\'e equation. The rational solutions are classified to five classes by the B\\\"acklund transformation group."}
{"category": "Math", "title": "Cancellation problem for projective modules over affine algebras", "abstract": "Let A be a ring of dimension d and let P be a projective A-module of rank d. We prove that if for every finite extension R of A, R^d is cancellative, then P is cancellative. This gives an alternate proof of Bhatwadekar's result: every projective module of rank d over an affine algebra of dimension d over a C_1-field of characteristic 0 is cancellative. Let P be a projective module of rank d-1 over an affine agebra of dimension d over an algebraically closed field. Then, it is not known if P is cancellative. We prove some results in this direction also."}
{"category": "Math", "title": "On the Low-lying zeros of Hasse-Weil L-functions for Elliptic Curves", "abstract": "In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate-Shafarevic group. Statements of this flavor were known previously under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper."}
{"category": "Math", "title": "On the conjecture of Kevin Walker", "abstract": "In 1985 Kevin Walker in his study of topology of polygon spaces raised an interesting conjecture in the spirit of the well-known question \"Can you hear the shape of a drum?\" of Marc Kac. Roughly, Walker's conjecture asks if one can recover relative lengths of the bars of a linkage from intrinsic algebraic properties of the cohomology algebra of its configuration space. In this paper we prove that the conjecture is true for polygon spaces in R^3. We also prove that for planar polygon spaces the conjecture holds is several modified forms: (a) if one takes into account the action of a natural involution on cohomology, (b) if the cohomology algebra of the involution's orbit space is known, or (c) if the length vector is normal. Some of our results allow the length vector to be non-generic, the corresponding polygon spaces have singularities. Our main tool is the study of the natural involution and its action on cohomology. A crucial role in our proof plays the solution of the isomorphism problem for monoidal rings due to J. Gubeladze."}
{"category": "Math", "title": "Topology of randon linkages", "abstract": "Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters -- the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R^3. We also prove results about higher moments of Betti numbers."}
{"category": "Math", "title": "Ihara's lemma for imaginary quadratic fields", "abstract": "An analogue over imaginary quadratic fields of a result in algebraic number theory known as Ihara's lemma is established. More precisely, we show that for a prime ideal P of the ring of integers of an imaginary quadratic field F, the kernel of the sum of the two standard P-degeneracy maps between the cuspidal sheaf cohomology H^1_!(X_0, M_0)^2 --> H^1_!(X_1, M_1) is Eisenstein. Here X_0 and X_1 are analogues over F of the modular curves X_0(N) and X_0(Np), respectively. To prove our theorem we use the method of modular symbols and the congruence subgroup property for the group SL(2) which is due to Serre."}
{"category": "Math", "title": "BMW algebra, quantized coordinate algebra and type C Schur--Weyl duality", "abstract": "We prove an integral version of the Schur--Weyl duality between the specialized Birman--Murakami--Wenzl algebra $B_n(-q^{2m+1},q)$ and the quantum algebra associated to the symplectic Lie algebra sp_{2m}. In particular, we deduce that this Schur--Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang [Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra (1) 306 (2006), 138--174] in the symplectic case. As a byproduct, we show that, as $Z[q,q^{-1}]$-algebra, the quantized coordinate algebra defined by Kashiwara is isomorphic to the quantized coordinate algebra arising from a generalized Faddeev--Reshetikhin--Takhtajan's construction."}
{"category": "Math", "title": "The Green's function and the Ahlfors map", "abstract": "The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected domain in the plane in terms of a single Ahlfors mapping of the domain, which is a proper holomorphic mapping of the domain onto the unit disc that is the analogue of the Riemann map in the multiply connected setting."}
{"category": "Math", "title": "Free 3-distributions: holonomy, Fefferman constructions and dual distributions", "abstract": "This paper analyses the parabolic geometries generated by a free 3-distribution in the tangent space of a manifold. It shows the existence of normal Fefferman constructions over CR and Lagrangian contact structures corresponding to holonomy reductions to SO(4,2) and SO(3,3), respectively. There is also a fascinating construction of a `dual' distribution when the holonomy reduces to $G_2'$. The paper concludes with some holonomy constructions for free $n$-distributions for $n>3$."}
{"category": "Math", "title": "On the algebraic classification of K-local spectra", "abstract": "In 1996, Jens Franke proved the equivalence of certain triangulated categories possessing an Adams spectral sequence. One particular application of this theorem is that the K_(p)-local stable homotopy category at an odd prime can be described as the derived category of an abelian category. We explain this proof from a topologist's point of view."}
{"category": "Math", "title": "Convergence Rate of K-Step Maximum Likelihood Estimate in Semiparametric Models", "abstract": "We suggest an iterative approach to computing K-step maximum likelihood estimates (MLE) of the parametric components in semiparametric models based on their profile likelihoods. The higher order convergence rate of K-step MLE mainly depends on the precision of its initial estimate and the convergence rate of the nuisance functional parameter in the semiparametric model. Moreover, we can show that the K-step MLE is as asymptotically efficient as the regular MLE after a finite number of iterative steps. Our theory is verified for several specific semiparametric models. Simulation studies are also presented to support these theoretical results."}
{"category": "Math", "title": "Stability conditions on generic complex tori", "abstract": "In this paper we describe a simply connected component of the complex manifold of stability conditions on the bounded derived category of a generic complex torus of any dimension. A torus is called generic if there are no nontrivial integral (p,p)-classes. We give an explicit description of all stability conditions in that connected component."}
{"category": "Math", "title": "Fourier transform on locally compact quantum groups", "abstract": "The notion of Fourier transform is among the more important tools in analysis, which has been generalized in abstract harmonic analysis to the level of abelian locally compact groups. The aim of this paper is to further generalize the Fourier transform: Motivated by some recent works by Van Daele in the multiplier Hopf algebra framework, and by using the Haar weights, we define here the (generalized) Fourier transform and the inverse Fourier transform, at the level of locally compact quantum groups. We will then consider the analogues of the Fourier inversion theorem, Plancherel theorem, and the convolution product. Along the way, we also obtain an alternative description of the dual pairing map between a quantum group and its dual."}
{"category": "Math", "title": "On the regularity of weak solutions of the 3D Navier-Stokes equations in $B^{-1}_{\\infty,\\infty}$", "abstract": "We show that if a Leray-Hopf solution $u$ to the 3D Navier-Stokes equation belongs to $C((0,T]; B^{-1}_{\\infty,\\infty})$ or its jumps in the $B^{-1}_{\\infty,\\infty}$-norm do not exceed a constant multiple of viscosity, then $u$ is regular on $(0,T]$. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya-Prodi-Serrin criterion."}
{"category": "Math", "title": "On the structure of Thom polynomials of singularities", "abstract": "Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a product rule. The product rule enables us to calculate the Thom polynomials of singularities if we know the Thom polynomial of the product singularity. As a special case of the product rule we define a formal power series (Thom series, Ts_Q) associated with a commutative, complex, finite dimensional local algebra Q, such that the Thom polynomial of {\\em every} singularity with local algebra Q can be recovered from Ts_Q."}
{"category": "Math", "title": "On higher real and stable ranks for CCR C*-algebras", "abstract": "We calculate the real rank and stable rank of CCR algebras which either have only finite dimensional irreducible representations or have finite topological dimension. We show that either rank of A is determined in a good way by the ranks of an ideal I and the quotient A/I in four cases: When A is CCR; when I has only finite dimensional irreducible representations; when I is separable, of generalized continuous trace and finite topological dimension, and all irreducible representations of I are infinite dimensional; or when I is separable, stable, has an approximate identity consisting of projections, and has the corona factorization property. We also present a counterexample on higher ranks of M(A), A subhomogeneous, and a theorem of P. Green on generalized continuous trace algebras."}
{"category": "Math", "title": "Defining the integers in large rings of number fields using one universal quantifier", "abstract": "Julia Robinson has given a first-order definition of the rational integers $\\mathbb Z$ in the rational numbers $\\mathbb Q$ by a formula $(\\forall \\exists \\forall \\exists)(F=0)$ where the $\\forall$-quantifiers run over a total of 8 variables, and where F is a polynomial. We show that for a large class of number fields, not including $\\mathbb Q$, for every $\\epsilon>0$, there exists a set of primes $\\cal S$ of natural density exceeding $1-\\epsilon$, such that $\\mathbb Z$ can be defined as a subset of the ``large'' subring $$\\{x \\in K : \\ord_{\\mathfrak p}x >0, \\forall \\mathfrak p \\not \\in \\cal S \\}$$ of K by a formula of the form $(\\exists \\forall \\exists)(F=0)$ where there is only one $\\forall$-quantifier, and where F is a polynomial."}
{"category": "Math", "title": "Non-stable K-theory and extremally rich C*-algebras", "abstract": "We consider the properties weak cancellation, K_1-surjectivity, good index theory, and K_1-injectivity for the class of extremally rich C*-algebras, and for the smaller class of isometrically rich C*-algebras. We establish all four properties for isometrically rich C*-algebras and for extremally rich C*-algebras that are either purely infinite or of real rank zero, K_1-injectivity in the real rank zero case following from a prior result of H. Lin. We also show that weak cancellation implies the other properties for extremally rich C*-algebras and that the class of extremally rich C*-algebras with weak cancellation is closed under extensions. Moreover, we consider analogous properties which replace the group K_1(A) with the extremal K-set K_e(A) as well as two versions of K_0-surjectivity."}
{"category": "Math", "title": "Weak subintegral closure of ideals", "abstract": "We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a geometric interpretation of the Reid-Roberts-Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal $I$ of a ring $A$ an ideal $I_>$, which consists of all elements of $A$ such that $v(a)>v(I)$, for all Rees valuations $v$ of $I$. The ideal $I_>$ plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition $A_f$ and is contained in every reduction of $I$. We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative closure."}
{"category": "Math", "title": "On the generalised Selberg integral of Richards and Zheng", "abstract": "In a recent paper Richards and Zheng compute the determinant of a matrix whose entries are given by beta-type integrals, thereby generalising an earlier result by Dixon and Varchenko. They then use their result to obtain a generalisation of the famous Selberg integral. In this note we point out that the Selberg-generalisation of Richards and Zheng is a special case of an integral over Jack polynomials due to Kadell. We then show how an integral formula for Jack polynomials of Okounkov and Olshanski may be applied to prove Kadell's integral along the lines of Richards and Zheng."}
{"category": "Math", "title": "Rogers-Szego polynomials and Hall-Littlewood symmetric functions", "abstract": "We use Rogers-Szego polynomials to unify some well-known identities for Hall-Littlewood symmetric functions due to Macdonald and Kawanaka."}
{"category": "Math", "title": "Cohen-Macaulay, Shellable and unmixed clutters with a perfect matching of K\\\"onig type", "abstract": "Let $\\mathcal{C}$ be a clutter with a perfect matching $e_1,...,e_g$ of K\\\"onig type and let $\\Delta_\\mathcal{C}$ be the Stanley-Reisner complex of the edge ideal of $\\mathcal{C}$. If all c-minors of $\\mathcal{C}$ have a free vertex and $\\mathcal{C}$ is unmixed, we show that $\\Delta_\\mathcal{C}$ is pure shellable. We are able to describe, in combinatorial and algebraic terms, when $\\Delta_\\mathcal{C}$ is pure. If $\\mathcal{C}$ has no cycles of length 3 or 4, then it is shown that $\\Delta_\\mathcal{C}$ is pure if and only if $\\Delta_\\mathcal{C}$ is pure shellable (in this case $e_i$ has a free vertex for all $i$), and that $\\Delta_\\mathcal{C}$ is pure if and only if for any two edges $f_1,f_2$ of $\\mathcal{C}$ and for any $e_i$, one has that $f_1\\cap e_i\\subset f_2\\cap e_i$ or $f_2\\cap e_i\\subset f_1\\cap e_i$. It is also shown that this ordering condition implies that $\\Delta_\\mathcal{C}$ is pure shellable, without any assumption on the cycles of $\\mathcal{C}$. Then we prove that complete admissible uniform clutters and their Alexander duals are unmixed. In addition, the edge ideals of complete admissible uniform clutters are facet ideals of shellable simplicial complexes, they are Cohen-Macaulay, and they have linear resolutions. Furthermore if $ \\mathcal{C}$ is admissible and complete, then $\\mathcal{C}$ is unmixed. We characterize certain conditions that occur in a Cohen-Macaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi--on the structure of unmixed simplicial trees--to clutters with the K\\\"onig property without 3-cycles or 4-cycles."}
{"category": "Math", "title": "Differential Twisted K-theory and Applications", "abstract": "In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory of compact simple Lie groups."}
{"category": "Math", "title": "An analog of the Iwasawa conjecture for a complete hyperbolic threefold of finite volume", "abstract": "For a unitary local system of rank one on a complete hyperbolic threefold of finite volume which has only one cusp, we will compare the order of the Alexander invariant at t=1 and one of Ruelle-Selberg L-function at s=0. Our result may be considered as a geometric analog of the Iwasawa main conjecture in the algebraic number theory."}
{"category": "Math", "title": "On a special value of the Ruelle L-function", "abstract": "For a unitary local system of rank one on a complete hyperbolic threefold of finite volume which has only one cusp, the Ruelle L-function is defined. We will show that if the first cohomology group of the local system vanishes its value at s=0 is equal to the square of the Franz-Reidemeister torsion."}
{"category": "Math", "title": "Semilattice Structures of Spreading Models", "abstract": "Given a Banach space X, denote by SP_{w}(X) the set of equivalence classes of spreading models of X generated by normalized weakly null sequences in X. It is known that SP_{w}(X) is a semilattice, i.e., it is a partially ordered set in which every pair of elements has a least upper bound. We show that every countable semilattice that does not contain an infinite increasing sequence is order isomorphic to SP_{w}(X) for some separable Banach space X."}
{"category": "Math", "title": "The u-invariant of the function fields of p-adic curves", "abstract": "The u-invariant of a field is the maximum dimension of ansiotropic quadratic forms over the field. It is an open question whether the u-invariant of function fields of p-aidc curves is 8. In this paper, we answer this question in the affirmative for function fields of non-dyadic p-adic curves."}
{"category": "Math", "title": "Controllability of Quantum Systems on the Lie Group SU(1,1)", "abstract": "This paper examines the controllability for quantum control systems with SU(1,1) dynamical symmetry, namely, the ability to use some electromagnetic field to redirect the quantum system toward a desired evolution. The problem is formalized as the control of a right invariant bilinear system evolving on the Lie group SU(1,1) of two dimensional special pseudo-unitary matrices. It is proved that the elliptic condition of the total Hamiltonian is both sufficient and necessary for the controllability. Conditions are also given for small time local controllability and strong controllability. The results obtained are also valid for the control systems on the Lie groups SO(2,1) and SL(2,R)."}
{"category": "Math", "title": "Convexity of Hypersurfaces in Spherical Spaces", "abstract": "A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a connected (n-1)-manifold into the n-sphere is a surjection onto the boundary of a convex set."}
{"category": "Math", "title": "Natural Frobenius Submanifolds", "abstract": "I.A.B. Strachan introduced the notion of a natural Frobenius submanifold of a Frobenius manifold and gave a sufficient but not necessary condition for a submanifold to be a natural Frobenius submanifold. This paper will give a necessary and sufficient condition and classify the natural Frobenius hypersurfaces."}
{"category": "Math", "title": "An asymptotically stable scheme for diffusive coagulation-fragmentation models", "abstract": "This paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation with diffusion in space. A finite volume scheme is developed, based on a conservative formulation of the space nonhomogeneous coagulation-fragmentation model, it is shown that the scheme preserves positivity, total volume and global steady states. Finally, several numerical simulations are performed to investigate the long time behavior of the solution."}
{"category": "Math", "title": "The Maslov cocycle, smooth structures and real-analytic complete integrability", "abstract": "This paper studies smooth obstructions to integrability and proves two main results. First, it is shown that if a smooth topological n-torus admits a real-analytically completely integrable convex hamiltonian on its cotangent bundle, then the torus is diffeomorphic to the standard n-torus. This is the first known result where the smooth structure of a manifold obstructs complete integrability. Second, it is proven that each one of the Witten-Kreck-Stolz 7-manifolds admit a real-analytically completely integrable geodesic flow on its cotangent bundle. This gives examples of topological manifolds all of whose smooth structures admit a real-analytically completely integrable convex hamiltonian on its cotangent bundle. Additional examples are provided by Eschenburgh and Aloff-Wallach spaces."}
{"category": "Math", "title": "Algebraic systems of matrices and Grobner basis", "abstract": "One studies a particular algebraic system where the unknowns are matrices. We solve this system according to the parameters values thanks to the theory of Grobner basis."}
{"category": "Math", "title": "A Generic Approach to Searching for Jacobians", "abstract": "We consider the problem of finding cryptographically suitable Jacobians. By applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family, we can search for Jacobians containing a large subgroup of prime order. For a suitable distribution of curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus 3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size, improving previous results. Our approach is particularly effective over low-degree extension fields, where in genus 2 we find Jacobians over F_{p^2) and trace zero varieties over F_{p^3} with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average time to find a group with 244-bit near-prime order is under an hour on a PC."}
{"category": "Math", "title": "On the topology of surface singularities {z^n=f(x,y)}, for f irreducible", "abstract": "The splice quotients are an interesting class of normal surface singularities with rational homology sphere links, defined by W. Neumann and J. Wahl. If Gamma is a tree of rational curves that satisfies certain combinatorial conditions, then there exist splice quotients with resolution graph Gamma. Suppose the equation z^n=f(x,y) defines a surface X_{f,n} with an isolated singularity at the origin in C^3. For f irreducible, we completely characterize, in terms of n and a variant of the Puiseux pairs of f, those X_{f,n} for which the resolution graph satisfies the combinatorial conditions that are necessary for splice quotients. This result is topological; whether or not X_{f,n} is analytically isomorphic to a splice quotient is treated separately."}
{"category": "Math", "title": "Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics background", "abstract": "We present a convergence result for infinite products of stochastic matrices with positive diagonals. We regard infinity of the product to the left. Such a product converges partly to a fixed matrix if the minimal positive entry of each matrix does not converge too fast to zero and if either zero-entries are symmetric in each matrix or the length of subproducts which reach the maximal achievable connectivity is bounded. Variations of this result have been achieved independently in Lorenz 2005, Moreau 2005 and Hendrickx 2005. We present briefly the opinion dynamics context, discuss the relations to infinite products where infinity is to the right (inhomogeneous Markov processes) and present a small improvement and sketch another."}
{"category": "Math", "title": "Curved Casimir Operators and the BGG Machinery", "abstract": "We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types of natural bundles. As a first application, we give a very general construction of splitting operators for parabolic geometries. Then we discuss the curved Casimir operators on differential forms with values in a tractor bundle, which nicely relates to the machinery of BGG sequences. This also gives a nice interpretation of the resolution of a finite dimensional representation by (spaces of smooth vectors in) principal series representations provided by a BGG sequence."}
{"category": "Math", "title": "Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups", "abstract": "We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions) is such that there exists a bounded set in the plane which contains any finite critical value of any map $g \\in G$. In general, the Julia set of such a semigroup $G$ may be disconnected, and each Fatou component of such $G$ is either simply connected or doubly connected (\\cite{Su01,Su9}). In this paper, we show that for any two distinct Fatou components of certain types (e.g., two doubly connected components of the Fatou set), the boundaries are separated by a Cantor set of quasicircles (with uniform dilatation) inside the Julia set of $G.$ Important in this theory is the understanding of various situations which can and cannot occur with respect to how the Julia sets of the maps $g \\in G$ are distributed within the Julia set of the entire semigroup $G$. We give several results in this direction and show how such results are used to generate (semi) hyperbolic semigroups possessing this postcritically boundedness condition."}
{"category": "Math", "title": "Strong wavefront lemma and counting lattice points in sectors", "abstract": "We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and, more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz."}
{"category": "Math", "title": "Detecting Infinitely Many Semisimple Representations in a Fixed Finite Dimension", "abstract": "Let $n$ be a positive integer, and let $k$ be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented $k$-algebra $R$ has infinitely many equivalence classes of semisimple representations $R \\to M_n(k')$, where $k'$ is the algebraic closure of $k$. The test reduces the problem to computational commutative algebra over $k$, via famous results of Artin, Procesi, and Shirshov. The test is illustrated by explicit examples, with $n = 3$."}
{"category": "Math", "title": "Cohomology and Support Varieties for Lie Superalgebras II", "abstract": "In \\cite{BKN} the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties for Kac supermodules for Type I Lie superalgebras and the simple supermodules for $\\mathfrak{gl}(m|n)$. The latter result verifies our earlier conjecture for $\\mathfrak{gl}(m|n)$. In our investigation we also delineate several of the major differences between Type I versus Type II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras."}
{"category": "Math", "title": "Uniformly perfect analytic and conformal attractor sets", "abstract": "Conditions are given which imply that analytic iterated function systems (IFS's) in the complex plane have uniformly perfect attractor sets. In particular, it is shown that the attractor set of a finitely generated conformal IFS is uniformly perfect when it contains two or more points. Also, an example of a finitely generated analytic attractor set which is not uniformly perfect is given."}
{"category": "Math", "title": "Recent developments of the DDVV Conjecture", "abstract": "In this paper, we give a survey of the recent develpoments of the DDVV conjecture."}
{"category": "Math", "title": "Potential confinement property in the Parabolic Anderson Model", "abstract": "We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in $Z^d$. We use i.i.d. potentials $\\xi: Z^d \\to \\R$ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, Konig and Morters [HKM06]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter $\\gamma=1$, but do not belong to the class of so-called double-exponentially distributed potentials studied by Gartner and Molchanov (PTRF 1998). In [HKM06] the asymptotics of the expected total mass was identified in terms of a variational problem that is closely connected to the well-known logarithmic Sobolev inequality and whose solution, unique up to spatial shifts, is a perfect parabola. In the present paper we show that those potentials whose shape (after appropriate vertical shifting and spatial rescaling) is away from that parabola contribute only negligibly to the total mass. The topology used is the strong $L^1$-topology on compacts for the exponentials of the potential. In the course of the proof, we show that any sequence of approximate minimisers of the above variational formula approaches some spatial shift of the minimiser, the parabola."}
{"category": "Math", "title": "Fictitious Play in 3x3 Games: the transition between periodic and chaotic behaviour", "abstract": "In the 60's Shapley provided an example of a two player fictitious game with periodic behaviour. In this game, player $A$ aims to copy $B$'s behaviour and player $B$ aims to play one ahead of player $A$. In this paper we generalize Shapley's example by introducing an external parameter. We show that the periodic behaviour in Shapley's example at some critical parameter value disintegrates into unpredictable (chaotic) behaviour, with players dithering a huge number of times between different strategies. At a further critical parameter the dynamics becomes periodic again and both players aim to play one ahead of the other. We study the dynamics of a two player continuous time bimatrix fictitious play with the dynamics of a one-parameter family of $3 \\times 3$ games that includes a well-known example of Shapley's as a special case. In this paper we adopt a geometric (dynamical systems) approach and study the bifurcations of simple periodic orbits. Here we concentrate on the periodic behaviour, while in a sequel we shall describe the chaotic behaviour."}
{"category": "Math", "title": "Real Zeros and Normal Distribution for statistics on Stirling permutations defined by Gessel and Stanley", "abstract": "We study Stirling permutations defined by Gessel and Stanley. We prove that their generating function according to the number of descents has real roots only. We use that fact to prove that the distribution of these descents, and other, equidistributed statistics on these objects converge to a normal distribution."}
{"category": "Math", "title": "Coxeter multiarrangements with quasi-constant multiplicities", "abstract": "We study structures of derivation modules of Coxeter multiarrangements with quasi-constant multiplicities by using the primitive derivation. As an application, we show that the characteristic polynomial of a Coxeter multiarrangement with quasi-constant multiplicity is combinatorially computable."}
{"category": "Math", "title": "Complexity results for CR mappings between spheres", "abstract": "Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here that there are infinitely many degrees for which the uniqueness of sharp examples fails. The proof uses a Pell equation and complicated explicit computations. We also show that the so-called gap phenomenon for proper mappings between balls does not occur beyond a certain target dimension. This proof uses the solution of the postage stamp problem."}
{"category": "Math", "title": "The Repetition Property for Sequences on Tori Generated by Polynomials or Skew-Shifts", "abstract": "The repetition property of a sequence in a metric space, a notion introduced by us in an earlier paper, is of importance in the spectral analysis of ergodic Schr\\\"odinger operators. It may be used to exclude eigenvalues for such operators. In this paper we study the question of when a sequence on a torus that is generated by a polynomial or a skew-shift has the repetition property. This provides classes of ergodic Schr\\\"odinger operators with potentials generated by skew-shifts on tori that have, contrary to earlier belief, no eigenvalues."}
{"category": "Math", "title": "On the Khovanov and knot Floer homologies of quasi-alternating links", "abstract": "Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are \"homologically thin\" for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander polynomial). The proofs use the exact triangles relating the homology of a link with the homologies of its two resolutions at a crossing."}
{"category": "Math", "title": "Applications of Wallis Theorem", "abstract": "In this paper we present theorems and applications of Wallis theorem related to trigonometric integrals."}
{"category": "Math", "title": "Tong's spectrum for Rosen continued fractions", "abstract": "The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approximation coefficients (in the sense of Diophantine approximation by continued fraction convergents). We also obtain metrical results for large blocks of ``bad'' approximations."}
{"category": "Math", "title": "Asymptotically Optimal Importance Sampling for Jackson Networks with a Tree Topology", "abstract": "Importance sampling (IS) is a variance reduction method for simulating rare events. A recent paper by Dupuis, Wang and Sezer (Ann. App. Probab. 17(4):1306- 1346, 2007) exploits connections between IS and stochastic games and optimal control problems to show how to design and analyze simple and efficient IS algorithms for various overflow events for tandem Jackson networks. The present paper uses the same approach to build asymptotically optimal IS schemes for stable open Jackson networks with a tree topology. Customers arrive at the single root of the tree. The rare overflow event we consider is the following: given that initially the network is empty, the system experiences a buffer overflow before returning to the empty state. Two types of buffer structures are considered: 1) A single system-wide buffer of size $n$ shared by all nodes, 2) each node $i$ has its own buffer of size $\\beta_i n$, $\\beta_i \\in (0,1)$."}
{"category": "Math", "title": "Complex Structures on Principal Bundles", "abstract": "Holomorphic principal G-bundles over a complex manifold M can be studied using non-abelian cohomology groups H^1(M,G). On the other hand, if M=\\Sigma is a closed Riemann surface, there is a correspondence between holomorphic principal G-bundles over \\Sigma and coadjoint orbits in the dual of a central extension of the Lie algebra C^\\infty(\\Sigma, \\g). We review these results and provide the details of an integrability condition for almost complex structures on smoothly trivial bundles. This article is a shortened version of the author's Diplom thesis."}
{"category": "Math", "title": "$G$-stable pieces and Lusztig's dimension estimates", "abstract": "We use $G$-stable pieces to construct some equidimensional varieties and as a consequence, obtain Lusztig's dimension estimates \\cite[section 4]{L2}. This is a generalization of \\cite{HL}."}
{"category": "Math", "title": "Convergence of freely decomposable Kleinian groups", "abstract": "We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressible, then the sequence has compact closure in the deformation space. As a consequence we generalise Thurston's double limit theorem and solve his conjecture on convergence of function groups affirmatively."}
{"category": "Math", "title": "Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains", "abstract": "We study Green's matrices for divergence form, second order strongly elliptic systems with bounded measurable coefficients in two dimensional domains. We establish existence, uniqueness, and pointwise estimates of the Green's matrices."}
{"category": "Math", "title": "Which weakly ramified group actions admit a universal formal deformation?", "abstract": "Consider a formal (mixed-characteristic) deformation functor D of a representation of a finite group G as automorphisms of a power series ring k[[t]] over a perfect field k of positive characteristic. Assume that the action of G is weakly ramified, i.e., the second ramification group is trivial. Examples of such representations are provided by a group action on an ordinary curve: the action of a ramification group on the completed local ring of any point on such a curve is weakly ramified. We prove that the only such D that are not pro-representable occur if k has characteristic two and G is of order two or isomorphic to a Klein group. Furthermore, we show that only the first of those has a non-pro-representable equicharacteristic deformation functor."}
{"category": "Math", "title": "Experiments with a Positivity Preserving Operator", "abstract": "We consider some multivariate rational functions which have (or are conjectured to have) only positive coefficients in their series expansion. We consider an operator that preserves positivity of series coefficients, and apply the inverse of this operator to the rational functions. We obtain new rational functions which seem to have only positive coefficients, whose positivity would imply positivity of the original series, and which, in a certain sense, cannot be improved any further."}
{"category": "Math", "title": "Optimal stability estimate of the inverse boundary value problem by partial measurements", "abstract": "In this work we establish log-type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homogeneous on the inaccessible part. This result, to some extent, improves our former result on the partial data problem in which log-log-type estimates were derived."}
{"category": "Math", "title": "Cesaro's integral formula for the Bell numbers (corrected)", "abstract": "M. E. Cesaro (1885) gave a quite remarkable expression for the Bell number --the number of partitions of an n-element set -- as a definite integral. This note is an exposition, correcting a typographical error in the original."}
{"category": "Math", "title": "Carlitz q-Bernoulli numbers and q-Stirling numbers", "abstract": "In this paper we consider carlitz q-Bernoulli numbers and q-stirling numbers of the first and the second kind. From these numbers we derive many interesting formulae associated with q-Bernoulli numbers."}
{"category": "Math", "title": "A p-adic analogue of a formula of Ramanujan", "abstract": "During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured $p$-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures."}
{"category": "Math", "title": "Structure of Ann-categories and Mac Lane - Shukla cohomology", "abstract": "In this paper we study the structure of a class of categories having two operations which satisfy axioms analoguos to that of rings. Such categories are called \"Ann - categories\". We obtain the classification theorems for regular Ann - categories and Ann - functors by using Mac Lane - Shukla cohomology of rings. These results give new interpretations of the cohomology groups and of the rings"}
{"category": "Math", "title": "A combinatorial realization of the Heisenberg action on the space of conformal blocks", "abstract": "This paper has been withdrawn by the author. Improved versions (arXiv:1109.5548 and arXiv:0708.4190) are accepted."}
{"category": "Math", "title": "Stability properties for the higher dimensional catenoid in $\\rr^{n+1}$", "abstract": "This paper concerns some stability properties of higher dimensional catenoids in $\\rr^{n+1}$ with $n\\ge 3$. We prove that higher dimensional catenoids have index one. We use $\\delta$-stablity for minimal hypersurfaces and show that the catenoid is $\\frac 2n$-stable and a complete $\\frac 2n$-stable minimal hypersurface is a catenoid or a hyperplane provided the second fundamental form satisfies some decay conditions."}
{"category": "Math", "title": "A construction of Horikawa surface via Q-Gorenstein smoothings", "abstract": "In this article we prove that Fintushel-Stern's construction of Horikawa surface, which is obtained from an elliptic surface via a rational blow-down surgery in smooth category, can be performed in complex category. The main technique involved is Q-Gorenstein smoothings."}
{"category": "Math", "title": "On the moduli stack of commutative, 1-parameter formal Lie groups", "abstract": "We attempt to develop a general algebro-geometric study of the moduli stack of commutative, 1-parameter formal Lie groups. We emphasize the pro-algebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of n-buds, and these latter stacks are algebraic. Our main results pertain to aspects of the height stratification relative to fixed prime p on the stacks of buds and formal Lie groups. We conclude with a largely expository account of some foundational material on limits in bicategories."}
{"category": "Math", "title": "Finite dimensional modules and perpendicular subcategories", "abstract": "We explain how, under some hypotheses, one can construct a sequence of finite dimensional $kG$-modules that lie in certain prescribed additive subcategories, but whose direct limits do not. We use these to show that many of the triangulated quotients of $\\Mod$ are not generated, as triangulated categories, by the corresponding quotient of $\\mod$ considered as a full subcategory."}
{"category": "Math", "title": "Existence and properties of geometric quotients", "abstract": "In this paper, we study quotients of groupoids and coarse moduli spaces of stacks in a general setting. Geometric quotients are not always categorical, but we present a natural topological condition under which a geometric quotient is categorical. We also show the existence of geometric quotients of finite flat groupoids and give explicit local descriptions. Exploiting similar methods, we give an easy proof of the existence of quotients of flat groupoids with finite stabilizers. As the proofs do not use noetherian methods and are valid for general algebraic spaces and algebraic stacks, we obtain a slightly improved version of Keel and Mori's theorem."}
{"category": "Math", "title": "Ergodic properties of a class of non-Markovian processes", "abstract": "We study a fairly general class of time-homogeneous stochastic evolutions driven by noises that are not white in time. As a consequence, the resulting processes do not have the Markov property. In this setting, we obtain constructive criteria for the uniqueness of stationary solutions that are very close in spirit to the existing criteria for Markov processes. In the case of discrete time, where the driving noise consists of a stationary sequence of Gaussian random variables, we give optimal conditions on the spectral measure for our criteria to be applicable. In particular, we show that under a certain assumption on the spectral density, our assumptions can be checked in virtually the same way as one would check that the Markov process obtained by replacing the driving sequence by a sequence of independent identically distributed Gaussian random variables is strong Feller and topologically irreducible. The results of the present article are based on those obtained previously in the continuous time context of diffusions driven by fractional Brownian motion."}
{"category": "Math", "title": "Lens spaces obtainable by surgery on doubly primitive knots", "abstract": "In this paper, we consider which lens spaces are obtainable by Dehn surgery described by Berge on doubly primitive knots. It is given an algorithm to decide whether a given lens space is obtainable by such surgery. Also included is a complete characterization of such surgery yielding lens spaces with Klein bottles."}
{"category": "Math", "title": "Some Non-Unimodal Level Algebras", "abstract": "In 2005, building on his own recent work and that of F. Zanello, A. Iarrobino discovered some constructions that, he conjectured, would yield level algebras with non-unimodal Hilbert functions. This thesis provides proofs of non-unimodality for Iarrobino's level algebras, as well as for other level algebras that the author has constructed along similar lines. The key technical contribution is to extend some results published by Iarrobino in 1984. Iarrobino's results provide insight into some naturally arising vector subspaces of the vector space R_d of forms of fixed degree in a polynomial ring in several variables. In this thesis, the problem is approached by combinatorial methods and results similar to Iarrobino's are proved for a different class of vector subspaces of R_d. The combinatorial methods involve the definition of a new class of matrices called L-Matrices, which have useful properties that are inherited by their submatrices. A particular class of square L-Matrices, associated with some specialized partially ordered sets having interesting combinatorial properties, is identified. For this class of L-Matrices, necessary and sufficient conditions are given that they be nonsingular. Several larger questions are discussed whose answers are incrementally improved by the knowledge that the new non-unimodal level algebras exist."}
{"category": "Math", "title": "An approximate version of the Loebl-Komlos-Sos conjecture", "abstract": "Loebl, Komlos, and Sos conjectured that if at least half of the vertices of a graph G have degree at least some natural number k, then every tree with at most k edges is a subgraph of G. Our main result is an approximate version of this conjecture for large enough n=|V(G)|, assumed that n=O(k). Our result implies an asymptotic bound for the Ramsey number of trees. We prove that r(T_k,T_m)\\leq k+m+o(k+m),as k+m tends to infinity."}
{"category": "Math", "title": "On the approximation by weighted ridge functions", "abstract": "We characterize the best $L_{2}$ approximation to a multivariate function by linear combinations of ridge functions multiplied by some fixed weight functions. In the special case when the weight functions are constants, we propose explicit formulas for both the best approximation and approximation error."}
{"category": "Math", "title": "On minimal norms on $M_n$", "abstract": "In this note, we show that for each minimal norm $N(\\cdot)$ on the algebra $M_n$ of all $n \\times n$ complex matrices, there exist norms $\\|\\cdot\\|_1$ and $\\|\\cdot\\|_2$ on ${\\mathbb C}^n$ such that $$N(A)=\\max\\{\\|Ax\\|_2: \\|x\\|_1=1, x\\in {\\mathbb C}^n\\}$$ for all $A \\in M_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms."}
{"category": "Math", "title": "Minimal Homogenous Liaison and Licci Ideals", "abstract": "We study the linkage classes of homogeneous ideals in polynomial rings. An ideal is said to be homogeneously licci if it can be linked to a complete intersection using only homogeneous regular sequences at each step. We ask a natural question: if $I$ is homogeneously licci, then can it be linked to a complete intersection by linking using regular sequences of forms of smallest possible degree at each step (we call such ideals minimally homogeneously licci)? In this paper we answer this question in the negative. In particular, for every $n\\geq 28$ we construct a set of $n$ points in $\\mathbb P^3$ which are homogeneously licci, but not minimally homogeneously licci. Moreover, we prove that one cannot distinguish between the classes of homogeneously licci and non-licci ideals based only on their Hilbert functions, nor distinguish between homogeneously licci and minimally homogeneously licci ideals based solely on the graded Betti numbers. Finally, by taking hypersurface sections, we show that the natural question has a negative answer whenever the height of the ideal is at least three."}
{"category": "Math", "title": "An algebraic formulation of Thurston's characterization of rational functions", "abstract": "Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston's characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy classes of simple closed curves induced by a rational function. The resulting finiteness results yield new information on the global dynamics of the pullback map on Teichm\\\"uller space used in the proof of the characterization theorem."}
{"category": "Math", "title": "On the almost sure spiraling of geodesics in negatively curved manifolds", "abstract": "Given a negatively curved geodesic metric space $M$, we study the statistical asymptotic penetration behavior of (locally) geodesic lines of $M$ in small neighborhoods of points, of closed geodesics, and of other compact (locally) convex subsets of $M$. We prove Khintchine-type and logarithme law-type results for the spiraling of geodesic lines around these objets. As a consequence in the tree setting, we obtain Diophantine approximation results of elements of non-archimedian local fields by quadratic irrational ones."}
{"category": "Math", "title": "On the symmetric subscheme of Hilbert scheme of points", "abstract": "We consider the Hilbert scheme Hilb^{d+1}(C^d) of (d+1) points in affine d-space C^d (d > 2), which includes the square of any maximal ideal. We describe equations for the most symmetric affine open subscheme of Hilb^{d+1}(C^d), in terms of Schur modules. In addition we prove that Hilb^{d+1}(C^d) is reducible for n>d>11."}
{"category": "Math", "title": "Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures", "abstract": "We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (and the cotensor product associative). Besides, for a number of technical reasons we mostly have to assume that the basic ring has a finite homological dimension (no such assumptions about the coring and semialgebra are made). In the final sections we construct model category structures on the categories of complexes of semi(contra)modules, and develop relative nonhomogeneous Koszul duality theory for filtered semialgebras and quasi-differential corings. Our motivating examples come from the semi-infinite cohomology theory. Comparison with the semi-infinite (co)homology of Tate Lie algebras and graded associative algebras is established in appendices, and the semi-infinite homology of a locally compact topological group relative to an open profinite subgroup is defined. An application to the correspondence between complexes of representations of an infinite-dimensional Lie algebra on the complementary central charge levels ($c$ and $26-c$ for the Virasoro) is worked out."}
{"category": "Math", "title": "The depth of a knot tunnel", "abstract": "The theory of tunnel number 1 knots detailed in our previous paper, The tree of knot tunnels, provides a non-negative integer invariant called the depth of the tunnel. We give various results related to the depth invariant. Noting that it equals the minimum number of Goda-Scharlemann-Thompson tunnel moves needed to construct the tunnel, we calculate the number of distinct minimal sequences of tunnel moves that can produce a given tunnel. Next, we give a recursion that tells the minimum bridge number of a knot having a tunnel of depth D. The rate of growth of this value improves the known estimates of the growth of bridge number as a function of the Hempel distance of the associated Heegaard splitting. We also give various upper bounds for bridge number in terms of the cabling constructions needed to produce a tunnel of a knot, showing in particular that the maximum bridge number of a knot produced by N cabling constructions is the (N+2)nd Fibonacci number. Finally, we explicitly compute the slope parameters for the \"short\" tunnels of torus knots. In particular, we find a sequence of such tunnels for which the bridge numbers of the associated knots, as a function of the depth, achieve the minimum growth rate. The actual minimum bridge number at a given depth cannot be achieved by a torus knot."}
{"category": "Math", "title": "Limit distribution theory for maximum likelihood estimation of a log-concave density", "abstract": "We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\\exp\\varphi_0$ where $\\varphi_0$ is a concave function on $\\mathbb{R}$. The pointwise limiting distributions depend on the second and third derivatives at 0 of $H_k$, the \"lower invelope\" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $\\varphi_0=\\log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values."}
{"category": "Math", "title": "Efficient computation of p-adic heights", "abstract": "We analyse and drastically improve the running time of the algorithm of Mazur, Stein and Tate for computing the canonical cyclotomic p-adic height of a point on an elliptic curve E/Q, where E has good ordinary reduction at p >= 5."}
{"category": "Math", "title": "Normal Hopf subalgebras in cocycle deformations of finite groups", "abstract": "Let $G$ be a finite group and let $\\pi: G \\to G'$ be a surjective group homomorphism. Consider the cocycle deformation $L = H^{\\sigma}$ of the Hopf algebra $H = k^G$ of $k$-valued linear functions on $G$, with respect to some convolution invertible 2-cocycle $\\sigma$. The (normal) Hopf subalgebra $k^{G'} \\subseteq k^G$ corresponds to a Hopf subalgebra $L' \\subseteq L$. Our main result is an explicit necessary and sufficient condition for the normality of $L'$ in $L$."}
{"category": "Math", "title": "Observability and nonlinear filtering", "abstract": "This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability in filter stability is explored. Finally, the results are partially extended to non-compact signal state spaces."}
{"category": "Math", "title": "Orbit semigroups and the representation type of quivers", "abstract": "We show that a finite, connected quiver Q without oriented cycles is a Dynkin or Euclidean quiver if and only if all orbit semigroups of representations of Q are saturated."}
{"category": "Math", "title": "Immersed Turnovers In Hyperbolic 3-Orbifolds", "abstract": "We show that any immersion, which is not a covering of an embedded 2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally geodesic boundary, called the \"turnover core,'' whose volume is bounded from above by a function depending only on the area of the given turnover. Furthermore, we show that, for a given type of turnover, there are only finitely many possibilities for the turnover core. As a corollary, if the volume of a complete orientable hyperbolic 3-orbifold is at least 2\\pi and if the fundamental group of the orbifold contains the fundamental group of a hyperbolic turnover (i.e., a triangle group), then the orbifold contains an embedded hyperbolic turnover."}
{"category": "Math", "title": "Quiver coefficients of Dynkin type", "abstract": "We study the Grothendieck classes of quiver cycles, i.e. invariant closed subvarieties of the representation space of a quiver. For quivers without oriented loops we show that the class of a quiver cycle is determined by quiver coefficients, which generalize the earlier studied quiver coefficients for equioriented quivers of type A. We conjecture that quiver coefficients satisfy positivity and finiteness properties. Our main result is a formula for the quiver coefficients for orbit closures of Dynkin type with rational singularities, which confirms the finiteness conjecture. This formula is based on Reineke's desingularization of such orbit closures. For quivers of type A3, we give positive combinatorial formulas for the quiver coefficients, which confirm the full conjecture. We also interpret quiver coefficients as formulas for degeneracy loci defined by quivers of vector bundle maps."}
{"category": "Math", "title": "Brownian-time Brownian motion SIEs on $\\Rp\\times\\Rd$: Ultra Regular direct and lattice-limits solutions and fourth order SPDEs links", "abstract": "We delve deeper into the compelling regularizing effect of the Brownian-time Brownian motion density, $\\KBtxy$, on the space-time-white-noise-driven stochastic integral equation we call BTBM SIE, which we recently introduced. In sharp contrast to second order heat-based SPDEs--whose real-valued mild solutions are confined to $d=1$--we prove the existence of solutions to the BTBM SIE in $d=1,2,3$ with dimension-dependent and striking Holder regularity, under both less than Lipschitz and Lipschitz conditions. In space, we show an unprecedented nearly local Lipschitz regularity for $d=1,2$--roughly, the SIE is spatially twice as regular as the Brownian sheet in these dimensions--and nearly local H\\\"older 1/2 regularity in d=3. In time, our solutions are locally H\\\"older continuous with exponent $\\gamma\\in(0,(4-d)/(8))$ for $1\\le d\\le3$. To investigate our SIE, we (a) introduce the Brownian-time random walk and we use it to formulate the spatial lattice version of the BTBM SIE; and (b) develop a delicate variant of Stroock-Varadhan martingale approach, the K-martingale approach, tailor-made for a wide variety of kernel SIEs including BTBM SIEs and the mild forms of many SPDEs of different orders on the lattice. Here, solutions types to our SIE are both direct and limits of their lattice version. The BTBM SIE is intimately connected to intriguing fourth order SPDEs in two ways. First, we show that it is connected to the diagonals of a new unconventional fourth order SPDE we call parametrized BTBM SPDE. Second, replacing $\\KBtxy$ by the intimately connected kernel of our recently introduced imaginary-Brownian time-Brownian-angle process (IBTBAP), our SIE becomes the mild form of a Kuramoto-Sivashinsky SPDE with linear PDE part. Ideas developed here are adapted in separate papers to give a new approach, via our explicit IBTBAP representation, to many KS-type SPDEs in multi spatial dimensions."}
{"category": "Math", "title": "A Remark on Hypercontractive Semigroups and Operator Ideals", "abstract": "In this note, we answer a question raised by Johnson and Schechtman \\cite{JS}, about the hypercontractive semigroup on $\\{-1,1\\}^{\\NN}$. More generally, we prove the folllowing theorem. Let $1<p<2$. Let $(T(t))_{t>0}$ be a holomorphic semigroup on $L_p$ (relative to a probability space). Assume the following mild form of hypercontractivity: for some large enough number $s>0$, $T(s)$ is bounded from $L_p$ to $L_2$. Then for any $t>0$, $T(t)$ is in the norm closure in $B(L_p)$ (denoted by $\\bar{\\Gamma_2}$) of the subset (denoted by ${\\Gamma_2}$) formed by the operators mapping $L_p$ to $L_2$ (a fortiori these operators factor through a Hilbert space)."}
{"category": "Math", "title": "A perfect stratification of M_g for g at most 5", "abstract": "We find for g at most 5 a stratification of depth g-2 of the moduli space of curves M_g with the property that its strata are affine and the classes of their closures provide a Q-basis for the Chow ring of M_g. The first property confirms a conjecture of one of us. The way we establish the second property yields new (and simpler) proofs of theorems of Faber and Izadi which, taken together, amount to the statement that in this range the Chow ring is generated by the lambda-class."}
{"category": "Math", "title": "Sally modules of rank one", "abstract": "The structure of Sally modules of $\\fkm$-primary ideals $I$ in a Cohen-Macaulay local ring $(A, \\m)$ satisfying the equality $\\e_1(I)=\\e_0(I)-\\ell_A(A/I)+1$ is explored, where $\\e_0(I)$ and $\\e_1(I)$ denote the first two Hilbert coefficients of $I$."}
{"category": "Math", "title": "Twelve bridges from a reductive group to its Langlands dual", "abstract": "These notes are based on lectures given in Wuhan (China) in July 2007. Their aim is to provide an introduction to Langlands philosophy."}
{"category": "Math", "title": "Toric dynamical systems", "abstract": "Toric dynamical systems are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established. They include as special cases all deficiency zero systems and all detailed balancing systems. One feature is that the steady state locus of a toric dynamical system is a toric variety, which has a unique point within each invariant polyhedron. We develop the basic theory of toric dynamical systems in the context of computational algebraic geometry and show that the associated moduli space is also a toric variety. It is conjectured that the complex balancing state is a global attractor. We prove this for detailed balancing systems whose invariant polyhedron is two-dimensional and bounded."}
{"category": "Math", "title": "Some counterexamples in dynamics of rational semigroups", "abstract": "We give an example of two rational functions with non-equal Julia sets that generate a rational semigroup whose completely invariant Julia set is a closed line segment. We also give an example of polynomials with unequal Julia sets that generate a non nearly Abelian polynomial semigroup with the property that the Julia set of one generator is equal to the Julia set of the semigroup. These examples show that certain conjectures in the field of dynamics of rational semigroups do not hold as stated and therefore require the allowance of certain exceptional cases."}
{"category": "Math", "title": "On the Dreaded Right Bousfield Localization", "abstract": "I verify the existence of right Bousfield localizations of right semimodel categories, and I apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right semimodel category."}
{"category": "Math", "title": "Generalized Cauchy identities, trees and multidimensional Brownian motions. Part II: Combinatorial differential calculus", "abstract": "We present an analogue of the differential calculus in which the role of polynomials is played by certain ordered sets and trees. Our combinatorial calculus has all nice features of the usual calculus and has an advantage that the elements of the considered ordered sets might carry some additional information. In this way an analytic proof of generalized Cauchy identities from the previous work of the second author can be directly reformulated in our new language of the combinatorial calculus; furthermore the additional information carried by vertices determines uniquely the bijections presented in Part I of this series."}
{"category": "Math", "title": "The special cuts of 600-cell", "abstract": "A polytope is called {\\em regular-faced} if every one of its facets is a regular polytope. The 4-dimensional regular-faced polytopes were determined by G. Blind and R. Blind \\cite{BlBl2,roswitha,roswitha2}. The last class of such polytopes is the one which consists of polytopes obtained by removing a set of non-adjacent vertices (an independent set) of the 600-cell. These independent sets are enumerated up to isomorphism and it is determined that the number of polytopes in this last class is $314,248,344$."}
{"category": "Math", "title": "On Local Behavior of Holomorphic Functions Along Complex Submanifolds of C^N", "abstract": "In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of $\\Co^{N}$. As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in $\\Co^{2}$."}
{"category": "Math", "title": "Diophantine subsets of function fields of curves", "abstract": "We consider diophantine subsets of function fields of curves and show, roughly speaking, that they are either very small or very large. In particular, this implies that the ring of polynomials $k[t]$ is a not a diophantine subset of the field of rational functions $k(t)$ for many fields $k$."}
{"category": "Math", "title": "Asymptotic behavior of the rate of adaptation", "abstract": "We consider the accumulation of beneficial and deleterious mutations in large asexual populations. The rate of adaptation is affected by the total mutation rate, proportion of beneficial mutations and population size $N$. We show that regardless of mutation rates, as long as the proportion of beneficial mutations is strictly positive, the adaptation rate is at least $\\mathcal{O}(\\log^{1-\\delta}N)$ where $\\delta$ can be any small positive number, if the population size is sufficiently large. This shows that if the genome is modeled as continuous, there is no limit to natural selection, that is, the rate of adaptation grows in $N$ without bound."}
{"category": "Math", "title": "On the Theory of Surfaces in the Four-dimensional Euclidean Space", "abstract": "For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are characterized by the equality kappa^2=k. The class of the surfaces with flat normal connection is characterized by the condition kappa = 0. For the surfaces of general type we obtain a geometrically determined orthonormal frame field at each point and derive Frenet-type derivative formulas. We apply our theory to the class of the rotational surfaces, which prove to be surfaces with flat normal connection, and describe the rotational surfaces with constant invariants."}
{"category": "Math", "title": "On topological spaces possessing uniformly distributed sequences", "abstract": "Two classes of topological spaces are introduced on which every probability Radon measure possesses a uniformly distributed sequence or a uniformly tight uniformly distributed sequence. It is shown that these classes are stable under multiplication by completely regular Souslin spaces"}
{"category": "Math", "title": "Extrema of low eigenvalues of the Dirichlet-Neumann Laplacian on a disk", "abstract": "We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet-Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition."}
{"category": "Math", "title": "On the theorem of M.Golomb", "abstract": "Let $X_{1},...,X_{n}$ be compact spaces and $X=X_{1}\\times ... \\times X_{n}.$ Consider the approximation of a function $f\\in C(X)$ by sums $g_{1}(x_{1})+... g_{n}(x_{n}),$ where $g_{i}\\in C(X_{i}),$ $i=1,...,n.$ In [8], M.Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of $X$, called \"projection cycles\". However, his proof had a gap, which was pointed out by Marshall and O'Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb's formula holds in a stronger form."}
{"category": "Math", "title": "Characteristic 2 approach to bivariate interpolation problems", "abstract": "We investigate bivariate interpolation problems in characteristic 2. Given a nonnegative integer $t$, we describe all the sub-linear systems generated by monomials, in which there is no curve passing through a general point with multiplicity at least $2^t$. As an application, we show that a certain linear system of plane curves with 10 base points is non-special."}
{"category": "Math", "title": "Intransitive geometries and fused amalgams", "abstract": "We study geometries that arise from the natural $G_2(K)$ action on the geometry of one-dimensional subspaces, of nonsingular two-dimensional subspaces, and of nonsingular three-dimensional subspaces of the building geometry of type $C_3(K)$ where $K$ is a perfect field of characteristic 2. One of these geometries is intransitive in such a way that the non-standard geometric covering theory by the first and the last author is not applicable. In this paper we introduce the concept of fused amalgams in order to extend the geometric covering theory so that it applies to that geometry. This yields an interesting new amalgamation result for the group $G_2(K)$."}
{"category": "Math", "title": "Sparse inverse covariance estimation with the lasso", "abstract": "We consider the problem of estimating sparse graphs by a lasso penalty applied to the inverse covariance matrix. Using a coordinate descent procedure for the lasso, we develop a simple algorithm that is remarkably fast: in the worst cases, it solves a 1000 node problem (~500,000 parameters) in about a minute, and is 50 to 2000 times faster than competing methods. It also provides a conceptual link between the exact problem and the approximation suggested by Meinhausen and Buhlmann (2006). We illustrate the method on some cell-signaling data from proteomics."}
{"category": "Math", "title": "A novel operation associated with Gauss' arithmetic-geometric means", "abstract": "The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an arithmetic operation such that AGM is the mean for this operation, it can be regarded as an intermediate operation between addition and multiplication in view of the meaning of AGM. In this paper such an operation is introduced and several of its algebraic properties are proved."}
{"category": "Math", "title": "Stability and convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains", "abstract": "The Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations. In this paper we investigate for the interior Helmholtz problem on analytic domains how the singularities (charge points) of the MFS basis functions have to be chosen such that approximate solutions can be represented by the MFS basis in a numerically stable way. For Helmholtz problems on the unit disc we give a full analysis which includes the high frequency (short wavelength) limit. For more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued, which in turn depends on both domain shape and boundary data. Using this we develop a recipe for locating charge points which allows us to reach error norms of typically 10^{-11} on a wide variety of analytic domains. At high frequencies of order only 3 points per wavelength are needed, which compares very favorably to boundary integral methods."}
{"category": "Math", "title": "Studies on the Chazy equations", "abstract": "In this paper, we study the Chazy III,IX and X equations. For the Chazy III equation, by making the birational transformations the Chazy III equation is transformed into a third-order ordinary differential equation of rational type. For this equation, we find its meromorphic solutions, whose free parameters are essentially two. We also show that the system associated with this equation admits new special solutions solved by $tanh(t)$. For the Chazy IX equation, we transform the Chazy IX equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two new birational B{\\\"a}cklund transformations. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we obtain a new partial differential system in two variables involving the Chazy IX equation, This system satisfies the compatibility condition, and admits a travelling wave solution. For the Chazy X equation, we transform the Chazy X equation to a system of the first-order ordinary differential equations by birational transformations. For this system, we give two birational B{\\\"a}cklund transformations. One of them is new. We also give the holomorphy condition of this system. Thanks to this holomorphy condition, we can recover this system."}
{"category": "Math", "title": "An EL-labeling of the subgroup lattice", "abstract": "In a 2001 paper, Shareshian conjectured that the subgroup lattice of a finite, solvable group has an EL-labeling. We construct such a labeling, and verify that our labeling has the expected properties."}
{"category": "Math", "title": "On the endomorphism algebra of modular Gelfand-Graev representations", "abstract": "We study the endomorphism algebras of a modular Gelfand-Graev representation of a finite reductive group by investigating modular properties of homomorphisms constructed by Curtis and Curtis-Shoji."}
{"category": "Math", "title": "Degenerations of quadratic differentials on CP1", "abstract": "We describe the connected components of the complement of a natural \"diagonal\" of real codimension 1 in a stratum of quadratic differentials on CP1. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such \"flat spheres\". We also prove that the stratum has only one topological end. Finally, we elaborate a necessary toolkit destined to evaluation of the Siegel-Veech constants."}
{"category": "Math", "title": "Applied statistics: A review", "abstract": "The main phases of applied statistical work are discussed in general terms. The account starts with the clarification of objectives and proceeds through study design, measurement and analysis to interpretation. An attempt is made to extract some general notions."}
{"category": "Math", "title": "An order result for the exponential divisor function", "abstract": "The integer $d=\\prod_{i=1}^s p_i^{b_i}$ is called an exponential divisor of $n=\\prod_{i=1}^s p_i^{a_i}>1$ if $b_i \\mid a_i$ for every $i\\in \\{1,2,...,s\\}$. Let $\\tau^{(e)}(n)$ denote the number of exponential divisors of $n$, where $\\tau^{(e)}(1)=1$ by convention. The aim of the present paper is to establish an asymptotic formula with remainder term for the $r$-th power of the function $\\tau^{(e)}$, where $r\\ge 1$ is an integer. This improves an earlier result of {\\sc M. V. Subbarao} [5]."}
{"category": "Math", "title": "Generalization error for multi-class margin classification", "abstract": "In this article, we study rates of convergence of the generalization error of multi-class margin classifiers. In particular, we develop an upper bound theory quantifying the generalization error of various large margin classifiers. The theory permits a treatment of general margin losses, convex or nonconvex, in presence or absence of a dominating class. Three main results are established. First, for any fixed margin loss, there may be a trade-off between the ideal and actual generalization performances with respect to the choice of the class of candidate decision functions, which is governed by the trade-off between the approximation and estimation errors. In fact, different margin losses lead to different ideal or actual performances in specific cases. Second, we demonstrate, in a problem of linear learning, that the convergence rate can be arbitrarily fast in the sample size $n$ depending on the joint distribution of the input/output pair. This goes beyond the anticipated rate $O(n^{-1})$. Third, we establish rates of convergence of several margin classifiers in feature selection with the number of candidate variables $p$ allowed to greatly exceed the sample size $n$ but no faster than $\\exp(n)$."}
{"category": "Math", "title": "On certain arithmetic functions involving exponential divisors, II", "abstract": "Asymptotic properties of certain arithmetic functions involving exponential divisors are investigated."}
{"category": "Math", "title": "First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes", "abstract": "We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to whole real line this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogeneous diffusions is given. These results are illustrated by numerical examples."}
{"category": "Math", "title": "Canonical Hilbert-Burch matrices for ideals of $k[x,y]$", "abstract": "An Artinian ideal $I$ of $k[x,y]$ has many Hilbert-Burch matrices. We show that there is a canonical choice. As an application, we determine the dimension of certain affine Gr\\\"obner cells and their Betti strata recovering results of Ellingsrud and Str{\\o}mme, G\\\"ottsche and Iarrobino."}
{"category": "Math", "title": "Relative Hyperbolicity, Trees of Spaces and Cannon-Thurston Maps", "abstract": "We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalises a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces."}
{"category": "Math", "title": "Hyper-atoms and the Kemperman's critical pair Theory", "abstract": "In the present work, we introduce the notion of a hyper-atom and prove their main structure theorem. We then apply the global isoperimetric methodology to give a new proof for Kemperman's structure Theory and a slight improvement."}
{"category": "Math", "title": "Defining Relations of Low Degree of Invariants of Two $4 \\times 4$ Matrices", "abstract": "Over a field K of characteristic 0, we study the algebra of invariants of the general linear group GL(4,K) acting by simultaneous conjugation on two matrices of order 4. It coincides with the trace algebra generated by all traces of products of two generic matrices of order 4. It is known that the minimal degree of the defining relations of any homogeneous minimal generating set of this algebra is equal to 12. Starting with the generating set given recently by Drensky and Sadikova, we have determined all relations of degree < 15. For this purpose we have developed further algorithms based on representation theory of the general linear group and easy computer calculations with standard functions of Maple."}
{"category": "Math", "title": "Globular realization and cubical underlying homotopy type of time flow of process algebra", "abstract": "We construct a small realization as flow of every precubical set (modeling for example a process algebra). The realization is small in the sense that the construction does not make use of any cofibrant replacement functor and of any transfinite construction. In particular, if the precubical set is finite, then the corresponding flow has a finite globular decomposition. Two applications are given. The first one presents a realization functor from precubical sets to globular complexes which is characterized up to a natural S-homotopy. The second one proves that, for such flows, the underlying homotopy type is naturally isomorphic to the homotopy type of the standard cubical complex associated with the precubical set."}
{"category": "Math", "title": "Fixed points of smooth varieties with Kodaira dimension zero", "abstract": "In this paper, we study the growth of the number of fixed points from iterating an endomorphism of an abelian variety. Using the estimates obtained on an abelian variety, we are able to extend the results to endomorphisms on varieties of Kodaira dimension zero and more generally their periodic subvarieties."}
{"category": "Math", "title": "Torus knots are Fourier-(1,1,2) knots", "abstract": "Every torus knot can be represented as a Fourier-(1,1,2) knot which is the simplest possible Fourier representation for such a knot. This answers a question of Kauffman and confirms the conjecture made by Boocher, Daigle, Hoste and Zheng. In particular, the torus knot T(p,q) can be parameterized as x(t)=cos(pt), y(t)=cos(qt+pi/(2p)), and z(t)=cos(pt+pi/2)\\cos((q-p)t+pi/(2p)-pi/(4q))."}
{"category": "Math", "title": "Non commutative functional calculus: bounded operators", "abstract": "In this paper we develop a functional calculus for bounded operators defined on quaternionic Banach spaces. This calculus is based on the notion of slice-regularity, see \\cite{gs}, and the key tools are a new resolvent operator and a new eigenvalue problem."}
{"category": "Math", "title": "Non commutative functional calculus: unbounded operators", "abstract": "In a recent work, \\cite{cgss}, we developed a functional calculus for bounded operators defined on quaternionic Banach spaces. In this paper we show how the results from \\cite{cgss} can be extended to the unbounded case, and we highlight the crucial differences between the two cases. In particular, we deduce a new eigenvalue equation, suitable for the construction of a functional calculus for operators whose spectrum is not necessarily real."}
{"category": "Math", "title": "A new functional calculus for non-commuting operators", "abstract": "In this paper we use the notion of slice monogenic functions \\cite{slicecss} to define a new functional calculus for an $n$-tuple $T$ of not necessarily commuting operators. This calculus is different from the one discussed in \\cite{jefferies} and it allows the explicit construction of the eigenvalue equation for the $n$-tuple $T$ based on a new notion of spectrum for $T$. Our functional calculus is consistent with the Riesz-Dunford calculus in the case of a single operator."}
{"category": "Math", "title": "Slice monogenic functions", "abstract": "In this paper we offer a definition of monogenicity for functions defined on $\\rr^{n+1}$ with values in the Clifford algebra $\\rr_n$ following an idea inspired by the recent papers \\cite{gs}, \\cite{advances}. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra $\\rr_n$. We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series."}
{"category": "Math", "title": "A correlated topic model of Science", "abstract": "Topic models, such as latent Dirichlet allocation (LDA), can be useful tools for the statistical analysis of document collections and other discrete data. The LDA model assumes that the words of each document arise from a mixture of topics, each of which is a distribution over the vocabulary. A limitation of LDA is the inability to model topic correlation even though, for example, a document about genetics is more likely to also be about disease than X-ray astronomy. This limitation stems from the use of the Dirichlet distribution to model the variability among the topic proportions. In this paper we develop the correlated topic model (CTM), where the topic proportions exhibit correlation via the logistic normal distribution [J. Roy. Statist. Soc. Ser. B 44 (1982) 139--177]. We derive a fast variational inference algorithm for approximate posterior inference in this model, which is complicated by the fact that the logistic normal is not conjugate to the multinomial. We apply the CTM to the articles from Science published from 1990--1999, a data set that comprises 57M words. The CTM gives a better fit of the data than LDA, and we demonstrate its use as an exploratory tool of large document collections."}
{"category": "Math", "title": "Continuous functions taking every value a given number of times", "abstract": "We give necessary and sufficient conditions on a function $f:[0,1]\\to {0,1,2,...,\\omega,\\continuum}$ under which there exists a continuous function $F:[0,1]\\to [0,1]$ such that for every $y\\in[0,1]$ we have $|F^{-1}(y)|=f(y)$."}
{"category": "Math", "title": "Thompson's Group F", "abstract": "We introduce forest diagrams and strand diagrams for elements of Thompson's group F. A forest diagram is a pair of infinite, bounded binary forests together with an order-preserving bijection of the leaves. Using forest diagrams, we derive a simple length formula for elements of F, and we discuss applications to the geometry of the Cayley graph, including a new upper bound on the isoperimetric constant (a.k.a. Cheeger constant) of F. Strand diagrams are similar to tree diagrams, but they can be concatenated like braids. Motivated by the fact that configuration spaces are classifying spaces for braid groups, we present a classifying space for F that is the ``configuration space'' of finitely many points on a line, with the points allowed to split and merge in pairs. Strand diagrams are related to a description of F as a groupoid, which we use to derive presentations for F, T, V, and the braided Thompson group BV. In addition to the new results, we include a thorough exposition of the basic theory of the group F. Highlights include a simplified proof that the commutator subgroup of F is simple, a discussion of open problems (with a focus on amenability), and a simplified derivation of the standard presentation and normal forms for F using forest diagrams."}
{"category": "Math", "title": "Explicit Evaluation of Certain Exponential Sums of Quadratic Functions over $\\Bbb F_{p^n}$, $p$ Odd", "abstract": "Let $p$ be an odd prime and let $f(x)=\\sum_{i=1}^ka_ix^{p^{\\alpha_i}+1}\\in\\Bbb F_{p^n}[x]$, where $0\\le \\alpha_1<...<\\alpha_k$. We consider the exponential sum $S(f,n)=\\sum_{x\\in\\Bbb F_{p^n}}e_n(f(x))$, where $e_n(y)=e^{2\\pi i\\text{Tr}_n(y)/p}$, $y\\in\\Bbb F_{p^n}$, $\\text{Tr}_n=\\text{Tr}_{\\Bbb F_{p^n}/\\Bbb F_p}$. There is an effective way to compute the nullity of the quadratic form $\\text{Tr}_{mn}(f(x))$ for all integer $m>0$. Assuming that all such nullities are known, we find relative formulas for $S(f,mn)$ in terms of $S(f,n)$ when $\\nu_p(m) \\le \\min\\{\\nu_p(\\alpha_i):1\\le i\\le k\\}$, where $\\nu_p$ is the $p$-adic order. We also find an explicit formula for $S(f,n)$ when $\\nu_2(\\alpha_1)=...= \\nu_2(\\alpha_k)<\\nu_2(n)$. These results generalize those by Carlitz and by Baumert and McEliece. Parallel results with $p=2$ were obtained in a previous paper by the second author."}
{"category": "Math", "title": "The $\\mathcal{Q}_p$ Carleson Measure Problem", "abstract": "Let $\\mu$ be a nonnegative Borel measure on the open unit disk $\\mathbb{D}\\subset\\mathbb{C}$. This note shows how to decide that the M\\\"obius invariant space $\\mathcal{Q}_p$, covering $\\mathcal{BMOA}$ and $\\mathcal{B}$, is boundedly (resp., compactly) embedded in the quadratic tent-type space $T^\\infty_p(\\mu)$. Interestingly, the embedding result can be used to determine the boundedness (resp., the compactness) of the Volterra-type and multiplication operators on $\\mathcal{Q}_p$."}
{"category": "Math", "title": "Derangements and Relative Derangements of Type $B$", "abstract": "By introducing the notion of relative derangements of type $B$, also called signed relative derangements, which are defined in terms of signed permutations, we obtain a type $B$ analogue of the well-known relation between relative derangements and the classical derangements. While this fact can be proved by using the principle of inclusion and exclusion, we present a combinatorial interpretation with the aid of the intermediate structure of signed skew derangements."}
{"category": "Math", "title": "Prediction of Fractional Processes with Long-range Dependence", "abstract": "We introduce a class of Gaussian processes with stationary increments which exhibit long-range dependence. The class includes fractional Brownian motion with Hurst parameter H>1/2 as a typical example. We establish infinite and finite past prediction formulas for the processes in which the predictor coefficients are given explicitly in terms of the MA and AR coefficients."}
{"category": "Math", "title": "Relationship Between Bicomplex Generalized Analytic Functions and Solutions of the Complexified Schr\\\"odinger Equation", "abstract": "Using three different representations of the bicomplex numbers $T\\cong Cl_{C}(1,0) \\cong Cl_{C}(0,1)$, which is a commutative ring with zero divisors defined by $T={w_0+w_1 {i_1}+w_2{i_2}+w_3 {j} | w_0,w_1,w_2,w_3 \\in{R}}$ where ${i_1^{2}}=-1, {i_2^{2}}=-1, {j^{2}}=1 and {i_1}{i_2}={j}={i_2}{i_1}$, we construct three classes of bicomplex pseudoanalytic functions. In particular, we obtain some specific systems of Vekua equations of two complex variables and we established some connections between one of these systems and the classical Vekua equations. We consider also the complexification of the real stationary two-dimensional Schr{\\\"o}dinger equation. With the aid of any of its particular solutions, we construct a specific bicomplex Vekua equation possessing the following special property. The scalar parts of its solutions are solutions of the original complexified Schr{\\\"o}dinger equation and the vectorial parts are solutions of another complexified Schr{\\\"o}dinger equation."}
{"category": "Math", "title": "Proof of Lemma 6.3 in ``The crossing number of $K_{4,n}$ on the torus and the Klein bottle\"", "abstract": "We found that Lemma 6.3 in the paper ``The crossing number of $K_{4,n}$ on the torus and the Klein bottle\" is wrong."}
{"category": "Math", "title": "Gradient regularity for elliptic equations in the Heisenberg Group", "abstract": "We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven by Domokos & Manfredi (Cont. Math. 2005). In turn, the a priori estimates found are shown to imply the suitable local Calderon-Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results of Iwaniec and Dibenedetto & Manfredi, and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations."}
{"category": "Math", "title": "On Kalai's conjectures concerning centrally symmetric polytopes", "abstract": "In 1989 Kalai stated the three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the ``$3^d$-conjecture''. It is well-known that the three conjectures hold in dimensions d \\leq 3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d \\geq 5."}
{"category": "Math", "title": "On a conjecture of Hacon and McKernan in dimension three", "abstract": "We prove that there exists a universal constant $r_3$ such that if $X$ is a smooth projective threefold over $\\mathbb{C}$ with non-negative Kodaira dimension, then the linear system $|r K_X|$ admits a fibration that is birational to the Iitaka fibration as soon as $r \\geq r_3$ and sufficiently divisible. This gives an affirmative answer to a conjecture of Hacon and McKernan in the case of threefolds. Viehweg and Zhang have posted a stronger result along these lines using different methods."}
{"category": "Math", "title": "Commutator maps, measure preservation, and T-systems", "abstract": "Let G be a finite simple group. We show that the commutator map $a : G \\times G \\to G$ is almost equidistributed as the order of G goes to infinity. This somewhat surprising result has many applications. It shows that for a subset X of G we have $a^{-1}(X)/|G|^2 = |X|/|G| + o(1)$, namely $a$ is almost measure preserving. From this we deduce that almost all elements $g \\in G$ can be expressed as commutators $g = [x,y]$ where x,y generate G. This enables us to solve some open problems regarding T-systems and the Product Replacement Algorithm (PRA) graph. We show that the number of T-systems in G with two generators tends to infinity as the order of G goes to infinity. This settles a conjecture of Guralnick and Pak. A similar result follows for the number of connected components of the PRA graph of G with two generators. Some of our results apply for more general finite groups, and more general word maps. Our methods are based on representation theory, combining classical character theory with recent results on character degrees and values in finite simple groups. In particular the so called Witten zeta function plays a key role in the proofs."}
{"category": "Math", "title": "A note on the convergence of renewal and regenerative processes to a Brownian bridge", "abstract": "The standard functional central limit theorem for a renewal process with finite mean and variance, results in a Brownian motion limit. This note shows how to obtain a Brownian bridge process by a direct procedure that does not involve conditioning. Several examples are also considered."}
{"category": "Math", "title": "A simple proof of the algebraic version of a conjecture by Vogan", "abstract": "In a recent manuscript, D.Vogan conjectures that four canonical globalizations of Harish-Chandra modules commute with certain n-cohomology groups. In this article we prove that Vogan's conjecture holds for one of the globalizations if and only if it holds for the dual. Using a previously published result of one of the authors, which establishes the conjecture for the minimal globalization, we can therefore deduce Vogan's conjecture for the maximal globalization."}
{"category": "Math", "title": "On the Cauchy problem for higher-order nonlinear dispersive equations", "abstract": "We study a class of higher-order KdV equations. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when in H^s(\\R), for any real s."}
{"category": "Math", "title": "On representations of certain pseudo-Anosov maps of Riemann surfaces with punctures", "abstract": "Let $S$ be a Riemann surface of type $(p,n)$ with $3p+n>4$ and $n\\geq 1$. Let $\\alpha_1,\\alpha_2\\subset S$ be two simple closed geodesics such that $\\{\\alpha_1, \\alpha_2\\}$ fills $S$. It was shown by Thurston that most maps obtained through Dehn twists along $\\alpha_1$ and $\\alpha_2$ are pseudo-Anosov. Let $a$ be a puncture. In this paper, we study the family $\\mathcal{F}(S,a)$ of pseudo-Anosov maps on $S$ that projects to the trivial map as $a$ is filled in, and show that there are infinitely many elements in $\\mathcal{F}(S,a)$ that cannot be obtained from Dehn twists along two filling geodesics. We further characterize all elements in $\\mathcal{F}(S,a)$ that can be constructed by two filling geodesics. Finally, for any point $b\\in S$, we obtain a family $\\mathcal{H}$ of pseudo-Anosov maps on $S\\backslash \\{b\\}$ that is not obtained from Thurston's construction and projects to an element $\\chi\\in \\mathcal{F}(S,a)$ as $b$ is filled in, some properties of elements in $\\mathcal{H}$ are also discussed."}
{"category": "Math", "title": "Arithmetic structures in smooth subsets of F_p", "abstract": "Fix integers a_1,...,a_d satisfying a_1 + ... + a_d = 0. Suppose that f : Z_N -> [0,1], where N is prime. We show that if f is ``smooth enough'' then we can bound from below the sum of f(x_1)...f(x_d) over all solutions (x_1,...,x_d) in Z_N to a_1 x_1 + ... + a_d x_d == 0 (mod N). Note that d = 3 and a_1 = a_2 = 1 and a_3 = -2 is the case where x_1,x_2,x_3 are in arithmetic progression. By ``smooth enough'' we mean that the sum of squares of the lower order Fourier coefficients of f is ``small'', a property shared by many naturally-occurring functions, among them certain ones supported on sumsets and on certain types of pseudoprimes. The paper can be thought of as a generalization of another result of the author, which dealt with a F_p^n analogue of the problem. It appears that the method in that paper, and to a more limited extent the present paper, uses ideas similar to those of B. Green's ``arithmetic regularity lemma'', as we explain in the paper."}
{"category": "Math", "title": "An Ergodic Result", "abstract": "A rather general ergodic type scheme is presented on arbitrary sets X, as they are generated by arbitrary mappings T : X \\longrightarrow X. The structures considered on X are given by suitable subsets of the set of all of its finite partitions. Ergodicity is studied not with respect to subsets of X, but with the {\\it inverse limits} of families of finite partitions."}
{"category": "Math", "title": "Homotopy characterization of ANR mapping spaces", "abstract": "Let Y be an absolute neighborhood retract (ANR) for the class of metric spaces and let X be a Hausdorff space. Let map(X,Y) denote the space of continuous maps from X to Y with the compact open topology. It is shown that if X is a CW complex then map(X,Y) is an ANR for the class of metric spaces if and only if map(X,Y) is metrizable and has the homotopy type of a CW complex. The same holds also when X is a compactly generated hemicompact space (metrizability assumption is void in this case)."}
{"category": "Math", "title": "About Goldbach strong conjecture", "abstract": "In this work we use the number classification in families of the form 6n+1, and 6n+5 with n integer (Such families contain all odd prime numbers greater than 3 and other compound numbers related with primes). We will use this kind of classification in order to attempt an approach to Goldbach strong conjecture. By means of a geometric method of binary bands of numbers we conceive a new form of study of the stated problem."}
{"category": "Math", "title": "An elementary sieve", "abstract": "In this paper we review the properties of families of numbers of the form $6n\\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new sieve that allows the separation of numbers $n$ that generate primes from those that only generate compounds. In principle, this can be used to find the amount of prime numbers up to a given number $h$; this means, $\\pi(h)$."}
{"category": "Math", "title": "Zero product preservers of C*-algebras", "abstract": "Let T be be a zero-product preserving bounded linear map between C*-algebras A and B. Here neither A nor B is necessarily unital. In this note, we investigate when T gives rise to a Jordan homomorphism. In particular, we show that A and B are isomorphic as Jordan algebras if T is bijective and sends zero products of self-adjoint elements to zero products. They are isomorphic as C*-algebras if T is bijective and preserves the full zero product structure."}
{"category": "Math", "title": "Dirac concentrations in Lotka-Volterra parabolic PDEs", "abstract": "We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained Hamilton-Jacobi equation. This extends several earlier results to the parabolic case and to general nonlinearities. Technical new ingredients are a $BV$ estimate in time on the non-local nonlinearity, a characterization of the concentration point (in a monomorphic situation) and, surprisingly, some counter-examples showing that jumps on the Dirac locations are indeed possible."}
{"category": "Math", "title": "Galois cohomology of completed link groups", "abstract": "In this paper we compute the Galois cohomology of the pro-p completion of primitive link groups. Here, a primitive link group is the fundamental group of a tame link in the 3-sphere whose linking number diagram is irreducible modulo p (e.g. none of the linking numbers is divisible by p). The result is that (with Z/pZ-coefficients) the Galois cohomology is naturally isomorphic to the Z/pZ-cohomology of the discrete link group. The main application of this result is that for such groups the Baum-Connes conjecture or the Atiyah conjecture are true for every finite extension (or even every elementary amenable extension), if they are true for the group itself."}
{"category": "Math", "title": "Densities for Rough Differential Equations under Hoermander's Condition", "abstract": "We consider stochastic differential equations dY=V(Y)dX driven by a multidimensional Gaussian process X in the rough path sense. Using Malliavin Calculus we show that Y(t) admits a density for t in (0,T] provided (i) the vector fields V=(V_1,...,V_d) satisfy Hoermander's condition and (ii) the Gaussian driving signal X satisfies certain conditions. Examples of driving signals include fractional Brownian motion with Hurst parameter H>1/4, the Brownian Bridge returning to zero after time T and the Ornstein-Uhlenbeck process."}
{"category": "Math", "title": "A convergent finite difference method for a nonlinear variational wave equation", "abstract": "We establish rigorously convergence of a semi-discrete upwind scheme for the nonlinear variational wave equation $u_{tt} - c(u)(c(u) u_x)_x = 0$ with $u|_{t=0}=u_0$ and $u_t|_{t=0}=v_0$. Introducing Riemann invariants $R=u_t+c u_x$ and $S=u_t-c u_x$, the variational wave equation is equivalent to $R_t-c R_x=\\tilde c (R^2-S^2)$ and $S_t+c S_x=-\\tilde c (R^2-S^2)$ with $\\tilde c=c'/(4c)$. An upwind scheme is defined for this system. We assume that the the speed $c$ is positive, increasing and both $c$ and its derivative are bounded away from zero and that $R|_{t=0}, S|_{t=0}\\in L^1\\cap L^3$ are nonpositive. The numerical scheme is illustrated on several examples."}
{"category": "Math", "title": "The correct relatively stable category for idempotent modules", "abstract": "We answer a question posed by Carlson, Peng, and Wheeler, and demonstrate that in general Rickard modules in relatively stable categories are not idempotent modules even if one localizes with respect to a tensor ideal subcategory. We also show that there is a modification one can make so as to recover the idempotent behaviour."}
{"category": "Math", "title": "Special correspondences and Chow traces of Landweber-Novikov operations", "abstract": "We prove that the function field of a variety which possesses a special correspondence in the sense of M. Rost preserves the rationality of cycles of small codimensions. This fact was proven by Vishik in the case of quadrics and played the crucial role in his construction of fields with $u$-invariant $2^r+1$. The main technical tools are algebraic cobordism of Levine-Morel, generalized Rost degree formula and divisibility of Chow traces of certain Landweber-Novikov operations. As a direct application of our methods we prove the Vishik's Theorem for all $F_4$-varieties."}
{"category": "Math", "title": "Contribution in combinatorics in commutative algebra.(ph-d thesis)", "abstract": "In the first chapter we present new results related on monomial ideals of Borel type. Also, we introduce a new class of monomial ideals, called $\\de$-fixed ideals, which generalize the class of $p$-Borel ideals and we extend several results to this new class. In the second chapter, we compute the generic initial ideal, with repect to the reverse lexicographic order, of an ideal which define a complete intersection of embedding dimension three with strong Lefschetz property and we show that it is an almost reverse lexicographic ideal. This enable us to give a proof for Moreno's conjecture in the case $n=3$ and characteristic zero. Also, we prove that the $d$-component of the generic initial ideal, with respect to the reverse lexicographic order, of an ideal generated by a regular sequence of homogeneous polynomials of degree $d$ is revlex, in a particular, but important, case."}
{"category": "Math", "title": "On derived equivalence classification of gentle two-cycle algebras", "abstract": "We classify, up to derived (equivalently, tilting-cotilting) equivalence all nondegenerate gentle two-cycle algebras. We also give a partial classification and formulate a conjecture in the degenerate case."}
{"category": "Math", "title": "Rate of Escape on Free Products", "abstract": "Suppose we are given the free product $V$ of a finite family of finite or countable sets $(V_i)_{i\\in\\mathcal{I}}$ and probability measures on each $V_i$, which govern random walks on it. We consider a transient random walk on the free product arising naturally from the random walks on the $V_i$. We prove the existence of the rate of escape with respect to the block length, that is, the speed, at which the random walk escapes to infinity, and furthermore we compute formulas for it. For this purpose, we present three different techniques providing three different, equivalent formulas."}
{"category": "Math", "title": "Rate of Escape on the Lamplighter Tree", "abstract": "Suppose we are given a homogeneous tree $\\mathcal{T}_q$ of degree $q\\geq 3$, where at each vertex sits a lamp, which can be switched on or off. This structure can be described by the wreath product $(\\mathbb{Z}/2)\\wr \\Gamma$, where $\\Gamma=\\ast_{i=1}^q \\mathbb{Z}/2$ is the free product group of $q$ factors $\\mathbb{Z}/2$. We consider a transient random walk on a Cayley graph of $(\\mathbb{Z}/2)\\wr \\Gamma$, for which we want to compute lower and upper bounds for the rate of escape, that is, the speed at which the random walk flees to infinity."}
{"category": "Math", "title": "Acceleration of Lamplighter Random Walks", "abstract": "Suppose we are given an infinite, finitely generated group $G$ and a transient random walk on the wreath product $(\\mathbb{Z}/ 2\\mathbb{Z})\\wr G$, such that its projection on $G$ is transient and has finite first moment. This random walk can be interpreted as a lamplighter random walk on $G$. Our aim is to show that the random walk on the wreath product escapes to infinity with respect to a suitable (pseudo-)metric faster than its projection onto $G$. We also address the case where the pseudo-metric is the length of a shortest ``travelling salesman tour''. In this context, and excluding some degenerate cases if $G=\\mathbb{Z}$, the linear rate of escape is strictly bigger than the rate of escape of the lamplighter random walk's projection on $G$."}
{"category": "Math", "title": "Fisher Lecture: Dimension Reduction in Regression", "abstract": "Beginning with a discussion of R. A. Fisher's early written remarks that relate to dimension reduction, this article revisits principal components as a reductive method in regression, develops several model-based extensions and ends with descriptions of general approaches to model-based and model-free dimension reduction in regression. It is argued that the role for principal components and related methodology may be broader than previously seen and that the common practice of conditioning on observed values of the predictors may unnecessarily limit the choice of regression methodology."}
{"category": "Math", "title": "Comment: Fisher Lecture: Dimension Reduction in Regression", "abstract": "Comment: Fisher Lecture: Dimension Reduction in Regression [arXiv:0708.3774]"}
{"category": "Math", "title": "Comment: Fisher Lecture: Dimension Reduction in Regression", "abstract": "Comment: Fisher Lecture: Dimension Reduction in Regression [arXiv:0708.3774]"}
{"category": "Math", "title": "Comment: Fisher Lecture: Dimension Reduction in Regression", "abstract": "Comment: Fisher Lecture: Dimension Reduction in Regression [arXiv:0708.3774]"}
{"category": "Math", "title": "Rejoinder: Fisher Lecture: Dimension Reduction in Regression", "abstract": "Rejoinder: Fisher Lecture: Dimension Reduction in Regression [arXiv:0708.3774]"}
{"category": "Math", "title": "Low upper bounds of ideals", "abstract": "We show that there is a low T-upper bound for the class of K-trivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in the T-degrees below 0' for which there is a low T-upper bound."}
{"category": "Math", "title": "Embedding Population Dynamics Models in Inference", "abstract": "Increasing pressures on the environment are generating an ever-increasing need to manage animal and plant populations sustainably, and to protect and rebuild endangered populations. Effective management requires reliable mathematical models, so that the effects of management action can be predicted, and the uncertainty in these predictions quantified. These models must be able to predict the response of populations to anthropogenic change, while handling the major sources of uncertainty. We describe a simple ``building block'' approach to formulating discrete-time models. We show how to estimate the parameters of such models from time series of data, and how to quantify uncertainty in those estimates and in numbers of individuals of different types in populations, using computer-intensive Bayesian methods. We also discuss advantages and pitfalls of the approach, and give an example using the British grey seal population."}
{"category": "Math", "title": "A General Framework for the Parametrization of Hierarchical Models", "abstract": "In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a clear qualitative understanding as to when centering and noncentering work well, and introduce theory concerning the convergence time complexity of Gibbs samplers using centered and noncentered parametrizations. We give general recipes for the construction of noncentered parametrizations, including an auxiliary variable technique called the state-space expansion technique. We also describe partially noncentered methods, and demonstrate their use in constructing robust Gibbs sampler algorithms whose convergence properties are not overly sensitive to the data."}
{"category": "Math", "title": "Twisted Alexander Polynomials and Representation Shifts", "abstract": "For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group."}
{"category": "Math", "title": "Derived classification of gentle algebras with two cycles", "abstract": "We classify gentle algebras defined by quivers with two cycles under derived equivalence in a non degenerate case, by using some combinatorial invariants constructed from the quiver with relations defining these algebras. We also present a list of normal forms; any such algebra is derived equivalent to one of the algebras in the list."}
{"category": "Math", "title": "Tropical Lines on Cubic Surfaces", "abstract": "Given a tropical line $L$ and a smooth tropical surface $X$, we look at the position of $L$ on $X$. We introduce its primal and dual motif which are respectively a decorated graph and a subcomplex of the dual triangulation of $X$. They encode the combinatorial position of $L$ on $X$. We classify all possible motifs of tropical lines on general smooth tropical surfaces. This classification allows to give an upper bound for the number of tropical lines on a general smooth tropical surface with a given subdivision. We focus in particular on surfaces of degree three. As a concrete example, we look at tropical cubic surfaces dual to a fixed honeycomb triangulation, showing that a general surface contains exactly $27$ tropical lines."}
{"category": "Math", "title": "Limits of zeros of polynomial sequences", "abstract": "In the present paper we consider $F_k(x)=x^{k}-\\sum_{t=0}^{k-1}x^t,$ the characteristic polynomial of the $k$-th order Fibonacci sequence, the latter denoted $G(k,l).$ We determine the limits of the real roots of certain odd and even degree polynomials related to the derivatives and integrals of $F_k(x),$ that form infinite sequences of polynomials, of increasing degree. In particular, as $k \\to \\infty,$ the limiting values of the zeros are determined, for both odd and even cases. It is also shown, in both cases, that the convergence is monotone for sufficiently large degree. We give an upper bound for the modulus of the complex zeros of the polynomials for each sequence. This gives a general solution related to problems considered by Dubeau 1989, 1993, Miles 1960, Flores 1967, Miller 1971 and later by the second author in the present paper, and Narayan 1997."}
{"category": "Math", "title": "Calculating the image of the second Johnson-Morita representation", "abstract": "Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $\\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping class group to ${1/2} \\wedge^3 H \\semi \\sp(H)$. This Johnson-Morita homomorphism is not surjective, but its image is finite index in ${1/2} \\wedge^3 H \\semi \\sp(H)$. Here we give a description of the exact image of Morita's homomorphism. Further, we compute the image of the handlebody subgroup of the mapping class group under the same map."}
{"category": "Math", "title": "Hilbert's Fifth Problem for Local Groups", "abstract": "We solve Hilbert's fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilbert's fifth problem for global groups by Hirschfeld."}
{"category": "Math", "title": "Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem", "abstract": "Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \\in C and d \\in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an interesting application of the famous Marriage Theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear groups, and by mentioning an open question related to Frobenius groups."}
{"category": "Math", "title": "Generically finite morphisms and formal neighborhoods of arcs", "abstract": "We show that given a dominant morphism between two smooth varieties of the same dimension, the induced morphism between the formal neighborhoods of two arcs on these varieties is a closed embedding, of codimension given by the order of vanishing along the ramification subscheme."}
{"category": "Math", "title": "Applications of a finite-dimensional duality principle to some prediction problems", "abstract": "Some of the most important results in prediction theory and time series analysis when finitely many values are removed from or added to its infinite past have been obtained using difficult and diverse techniques ranging from duality in Hilbert spaces of analytic functions (Nakazi, 1984) to linear regression in statistics (Box and Tiao, 1975). We unify these results via a finite-dimensional duality lemma and elementary ideas from the linear algebra. The approach reveals the inherent finite-dimensional character of many difficult prediction problems, the role of duality and biorthogonality for a finite set of random variables. The lemma is particularly useful when the number of missing values is small, like one or two, as in the case of Kolmogorov and Nakazi prediction problems. The stationarity of the underlying process is not a requirement. It opens up the possibility of extending such results to nonstationary processes."}
{"category": "Math", "title": "Nearly Optimal Patchy Feedbacks for Minimization Problems with Free Terminal Time", "abstract": "The paper is concerned with a general optimization problem for a nonlinear control system, in the presence of a running cost and a terminal cost, with free terminal time. We prove the existence of a patchy feedback whose trajectories are all nearly optimal solutions, with pre-assigned accuracy."}
{"category": "Math", "title": "Coarse categories I: foundations", "abstract": "Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is particularly suited to the development of the_Roe (C*-)algebras_ C*(X) and their K-theory on the analytic side; we also hope that it will be of use in the strictly geometric/algebraic setting of controlled topology and algebra. We leave these topics to future papers. Crucial to our approach are nonunital coarse spaces, and what we call _locally proper_ maps (which are actually implicit in [MR1988817]). Our_coarse category_ Crs generalizes the usual one: its objects are nonunital coarse spaces and its morphisms (locally proper) coarse maps modulo_closeness_. Crs is much richer than the usual unital coarse category. As such, it has all nonzero limits and all colimits. We examine various other categorical issues. E.g., Crs does not have a terminal object, so we substitute a_termination functor_ which will be important in the development of exponential objects (i.e., \"function spaces\") and also leads to a notion of_quotient coarse spaces_. To connect our methods with the standard methods, we also examine the relationship between Crs and the usual coarse category of Roe. Finally we briefly discuss some basic examples and applications. Topics include_metric coarse spaces_,_continuous control_ [MR1277522], metric and continuously controlled_coarse simplices_,_sigma-coarse spaces_ [MR2225040], and the relation between quotient coarse spaces and the K-theory of Roe algebras (of particular interest for continuously controlled coarse spaces)."}
{"category": "Math", "title": "Modules with reducible complexity", "abstract": "For a commutative Noetherian local ring we define and study the class of modules having reducible complexity, a class containing all modules of finite complete intersection dimension. Various properties of this class of modules are given, together with results on the vanishing of homology and cohomology."}
{"category": "Math", "title": "Is critical 2D percolation universal?", "abstract": "The aim of this paper is to explore possible ways of extending Smirnov's proof of Cardy's formula for critical site-percolation on the triangular lattice to other cases (such as bond-percolation on the square lattice); the main question we address is that of the choice of the lattice embedding into the plane which gives rise to conformal invariance in the scaling limit. Even though we were not able to produce a complete proof, we believe that the ideas presented here go in the right direction."}
{"category": "Math", "title": "On the Hochschild (co)homology of quantum exterior algebras", "abstract": "We compute the Hochschild cohomology and homology of a class of quantum exterior algebras, with coefficients in twisted bimodules. As a result we obtain several interesting examples of the homological behavior of these algebras."}
{"category": "Math", "title": "On support varieties for modules over complete intersections", "abstract": "We show that, over a local complete intersection, every possible variety is realized as the cohomological support variety of some module. Moreover, we show that the projective variety of a complete indecomposable maximal Cohen-Macaulay module is connected."}
{"category": "Math", "title": "Twisted support varieties", "abstract": "We define and study twisted support varieties for modules over an Artin algebra, where the twist is induced by an automorphism of the algebra. Under a certain finite generation hypothesis, we show that the twisted variety of a module satisfies Dade's Lemma and is one dimensional precisely when the module is periodic with respect to the twisting automorphism. As a special case we obtain results on DTr-periodic modules over Frobenius algebras."}
{"category": "Math", "title": "Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms", "abstract": "A skew Laurent polynomial ring R[x^{\\pm 1};\\alpha] is reversible if it has a reversing automorphism, that is, an automorphism of period two that transposes x and x^{-1} and restricts to an automorphism of R. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of two simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus. These two skew Laurent polynomial rings are deformations of Poisson algebras and we interpret their reversing automorphisms and their invariants as deformations of Poisson automorphisms and their invariants."}
{"category": "Math", "title": "Best constants for Lipschitz embeddings of metric spaces into c_0", "abstract": "We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into $c_0$ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical $\\ell_p-$spaces into $c_0$ and give other applications. We prove that if a Banach space embeds almost isometrically into $c_0$, then it embeds linearly almost isometrically into $c_0$. We also study Lipschitz embeddings into $c_0^+$."}
{"category": "Math", "title": "Finite-dimensional simple Poisson modules", "abstract": "We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to the Poisson subalgebra of invariants for the action of a finite group of Poisson automorphisms."}
{"category": "Math", "title": "The solution of the Minkowski problem for open surfaces in Riemannian space", "abstract": "Author reduces the Minkowski problem to the problem of construction the G-deformations preserving the product of principal curvatures for every point of surface in Riemannian space. G-deformation transfers every normal vector of surface in parallel along the path of the translation for each point of surface. The continuous G-deformations preserving the product of principal curvatures of surface with boundary are considered in this article. The equations of deformations which are obtained in this paper reduce to the nonlinear boundary-value problem. The method of construction continuous G-deformations preserving the product of principal curvatures of surface with boundary and its qualitative analysis are presented in this article"}
{"category": "Math", "title": "Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples", "abstract": "In Section 1, we present a number of classical results concerning the Generalized Gamma Convolution (:GGC) variables, their Wiener-Gamma representations, and relation with the Dirichlet processes.To a GGC variable, one may associate a unique Thorin measure. Let $G$ a positive r.v. and $\\Gamma_t(G)$ (resp. $\\Gamma_t(1/G))$ the Generalized Gamma Convolution with Thorin measure $t$-times the law of $G$ (resp. the law of $1/G$). In Section 2, we compare the laws of $\\Gamma_t(G)$ and $\\Gamma_t(1/G)$.In Section 3, we present some old and some new examples of GGC variables, among which the lengths of excursions of Bessel processes straddling an independent exponential time."}
{"category": "Math", "title": "Sharp Spectral Asymptotics for Dirac Energy", "abstract": "I derive sharp semiclassical asymptotics of $\\int |e_h(x,y,0)|^2\\omega(x,y) dx dy$ where $e_h(x,y,\\tau)$ is the Schwartz kernel of the spectral projector and $\\omega(x,y)$ is singular as $x=y$. I also consider asymptotics of more general expressions."}
{"category": "Math", "title": "A convenient category for directed homotopy", "abstract": "We propose a convenient category for directed homotopy consisting of preordered topological spaces generated by cubes. Its main advantage is that, like the category of topological spaces generated by simplices suggested by J. H. Smith, it is locally presentable."}
{"category": "Math", "title": "On the proximinality of ridge functions", "abstract": "Using two results of Garkavi, Medvedev and Khavinson, we give sufficient conditions for proximinality of sums of two ridge functions with bounded and continuous summands in the spaces of bounded and continuous multivariate functions respectively. In the first case, we give an example which shows that the corresponding sufficient condition cannot be made weaker for some subsets of $\\mathbb{R}^{n}$. In the second case, we obtain also a necessary condition for proximinality. All the results are furnished with plenty of examples. The results, examples and following discussions naturally lead us to a conjecture on the proximinality of the considered class of ridge functions. The main purpose of the paper is to draw readers' attention to this conjecture."}
{"category": "Math", "title": "Epidemics on random graphs with tunable clustering", "abstract": "In this paper, a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. It is investigated how these quantities varies with the clustering in the graph and it turns out for instance that, as the clustering increases, the epidemic threshold decreases. The network is modelled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if they share at least one group."}
{"category": "Math", "title": "On the modularity of supersingular elliptic curves over certain totally real number fields", "abstract": "We study generalisations to totally real fields of methods originating with Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over $\\mathbb{Q}(\\sqrt{2})$."}
{"category": "Math", "title": "Optimality and uniqueness of the (4,10,1/6) spherical code", "abstract": "Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use semidefinite programming bounds instead to show that the Petersen code, which consists of the midpoints of the edges of the regular simplex in dimension 4, is the unique (4,10,1/6) spherical code."}
{"category": "Math", "title": "Categorification of Wedderburn's basis for \\mathbb{C}[S_n]", "abstract": "M. Neunh{\\\"o}ffer studies in \\cite{Ne} a certain basis of $\\mathbb{C}[S_n]$ with the origins in \\cite{Lu} and shows that this basis is in fact Wedderburn's basis. In particular, in this basis the right regular representation of $S_n$ decomposes into a direct sum of irreducible representations (i.e. Specht or cell modules). In the present paper we rediscover essentially the same basis with a categorical origin coming from projective-injective modules in certain subcategories of the BGG-category $\\mathcal{O}$. An important role in our arguments is played by the dominant projective module in each of these categories. As a biproduct of the study of this dominant projective module we show that {\\it Kostant's problem} (\\cite{Jo}) has a negative answer for some simple highest weight module over the Lie algebra $\\mathfrak{sl}_4$, which disproves the general belief that Kostant's problem should have a positive answer for all simple highest weight modules in type $A$."}
{"category": "Math", "title": "Liftable D_4-Covers", "abstract": "Let k be an algebraically closed field of characteristic p and let G be a subgroup of Aut(k[[t]]) be a faithful action on a local power series ring over k. Let R be a discrete valuation ring of characteristic 0 with residue field k. One asks, whether it is possible to find a faithful action G inside Aut(R[[t]]) which reduces to the given action, i.e. a lift to characteristic 0. We show that liftable actions exists in the case that G = D_4 and p = 2. In fact we introduce a family, the supersimple D_4 -actions, which can always be lifted to characteristic 0."}
{"category": "Math", "title": "The asymptotic behaviour of recurrence coefficients for orthogonal polynomials with varying exponential weights", "abstract": "We consider orthogonal polynomials $\\{p_{n,N}(x)\\}_{n=0}^{\\infty}$ on the real line with respect to a weight $w(x)=e^{-NV(x)}$ and in particular the asymptotic behaviour of the coefficients $a_{n,N}$ and $b_{n,N}$ in the three term recurrence $x \\pi_{n,N}(x) = \\pi_{n+1,N}(x) + b_{n,N} \\pi_{n,N}(x) + a_{n,N} \\pi_{n-1,N}(x)$. For one-cut regular $V$ we show, using the Deift-Zhou method of steepest descent for Riemann-Hilbert problems, that the diagonal recurrence coefficients $a_{n,n}$ and $b_{n,n}$ have asymptotic expansions as $n \\to \\infty$ in powers of $1/n^2$ and powers of $1/n$, respectively."}
{"category": "Math", "title": "Chess, Chance and Conspiracy", "abstract": "Chess and chance are seemingly strange bedfellows. Luck and/or randomness have no apparent role in move selection when the game is played at the highest levels. However, when competition is at the ultimate level, that of the World Chess Championship (WCC), chess and conspiracy are not strange bedfellows, there being a long and colorful history of accusations levied between participants. One such accusation, frequently repeated, was that all the games in the 1985 WCC (Karpov vs Kasparov) were fixed and prearranged move by move. That this claim was advanced by a former World Champion, Bobby Fischer, argues that it ought be investigated. That the only published, concrete basis for this claim consists of an observed run of particular moves, allows this investigation to be performed using probabilistic and statistical methods. In particular, we employ imbedded finite Markov chains to evaluate run statistic distributions. Further, we demonstrate how both chess computers and game data bases can be brought to bear on the problem."}
{"category": "Math", "title": "Maty's Biography of Abraham De Moivre, Translated, Annotated and Augmented", "abstract": "November 27, 2004, marked the 250th anniversary of the death of Abraham De Moivre, best known in statistical circles for his famous large-sample approximation to the binomial distribution, whose generalization is now referred to as the Central Limit Theorem. De Moivre was one of the great pioneers of classical probability theory. He also made seminal contributions in analytic geometry, complex analysis and the theory of annuities. The first biography of De Moivre, on which almost all subsequent ones have since relied, was written in French by Matthew Maty. It was published in 1755 in the Journal britannique. The authors provide here, for the first time, a complete translation into English of Maty's biography of De Moivre. New material, much of it taken from modern sources, is given in footnotes, along with numerous annotations designed to provide additional clarity to Maty's biography for contemporary readers."}
{"category": "Math", "title": "PAC Fields over Finitely Generated Fields", "abstract": "We prove the following theorem for a finitely generated field $K$: Let $M$ be a Galois extension of $K$ which is not separably closed. Then $M$ is not PAC over $K$."}
{"category": "Math", "title": "A Conversation with Robert V. Hogg", "abstract": "Robert Vincent Hogg was born on November 8, 1924 in Hannibal, Missouri. He earned a Ph.D. in statistics at the University of Iowa in 1950, where his advisor was Allen Craig. Following graduation, he joined the mathematics faculty at the University of Iowa. He was the founding Chair when the Department of Statistics was created at Iowa in 1965 and he served in that capacity for 19 years. At Iowa he also served as Chair of the Quality Management and Productivity Program and the Hanson Chair of Manufacturing Productivity. He became Professor Emeritus in 2001 after 51 years on the Iowa faculty. He is a Fellow of the Institute of Mathematical Statistics and the American Statistical Association plus an Elected Member of the International Statistical Institute. He was President of the American Statistical Association (1988) and chaired two of its winter conferences (1992, 1994). He received the ASA's Founder's Award (1991) and the Gottfried Noether Award (2001) for contributions to nonparametric statistics. His publications through 1996 are described in Communications in Statistics--Theory and Methods (1996), 2467--2481. This interview was conducted on April 14, 2004 at the Department of Statistics, University of Florida, Gainesville, Florida, and revised in the summer of 2006."}
{"category": "Math", "title": "The Lie algebra perturbation lemma", "abstract": "Let g be a differential graded Lie algebra and suppose given a contraction of chain complexes of g onto a general chain complex M. We show that the data determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting cochain from the perturbed coalgebra S\" to the given Lie algebra g, and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg coalgebra on g onto S\" which is natural in the data. This extends a result established in a joint paper of the author with J. Stashef [Forum math. 14 (2002), 847-868, math.AG/9906036] where only the particular where M is the homology of g has been explored."}
{"category": "Math", "title": "Gaps in the differential forms spectrum on cyclic coverings", "abstract": "We are interested in the spectrum of the Hodge-de Rham operator on a cyclic covering $X$ over a compact manifold $M$ of dimension $n+1$. Let $\\Sigma$ be a hypersurface in $M$ which does not disconnect $M$ and such that $M-\\Sigma$ is a fundamental domain of the covering. If the cohomology group $H^{n/2 (\\Sigma)$ is trivial, we can construct for each $N \\in \\N$ a metric $g=g_N$ on $M$, such that the Hodge-de Rham operator on the covering $(X,g)$ has at least $N$ gaps in its (essential) spectrum. If $H^{n/2}(\\Sigma) \\ne 0$, the same statement holds true for the Hodge-de Rham operators on $p$-forms provided $p \\notin \\{n/2,n/2+1\\}$."}
{"category": "Math", "title": "Extreme values of zeta and L-functions", "abstract": "We introduce a \"resonance\" method to produce large values of $|\\zeta(1/2+it)|$ and large and small central values of $L$-functions."}
{"category": "Math", "title": "On ground fields of arithmetic hyperbolic reflection groups", "abstract": "Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing ground fields of arithmetic hyperbolic reflection groups are defined, and good bounds of their degrees (over Q) are obtained. For example, degree of the ground field of any arithmetic hyperbolic reflection group in dimension at least 6 is bounded by 56. These results could be important for further classification. We also formulate a mirror symmetric conjecture to finiteness of the number of arithmetic hyperbolic reflection groups which was established in full generality recently."}
{"category": "Math", "title": "The group of automorphisms of a real rational surface is n-transitive", "abstract": "Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n-transitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are isomorphic if and only if they are homeomorphic as topological surfaces."}
{"category": "Math", "title": "Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem", "abstract": "We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation $\\Delta u=4 e^{2u}$ and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary behaviour. For instance, we show that for every Blaschke sequence $\\{z_j\\}$ in the unit disk there is always a Blaschke product with $\\{z_j\\}$ as its set of critical points. Our work is closely related to the Berger-Nirenberg problem in differential geometry."}
{"category": "Math", "title": "The Drinfel'd Double and Twisting in Stringy Orbifold Theory", "abstract": "This paper exposes the fundamental role that the Drinfel'd double $\\dkg$ of the group ring of a finite group $G$ and its twists $\\dbkg$, $\\beta \\in Z^3(G,\\uk)$ as defined by Dijkgraaf--Pasquier--Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that $G$--Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of $\\dkg$--modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold $K$--theory of global quotient given by the inertia variety of a point with a $G$ action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full $K$--theory of the stack $[pt/G]$. Finally, we show how one can use the co-cycles $\\beta$ above to twist a) the global orbifold $K$--theory of the inertia of a global quotient and more importantly b) the stacky $K$--theory of a global quotient $[X/G]$. This corresponds to twistings with a special type of 2--gerbe."}
{"category": "Math", "title": "A critical constant for the k nearest neighbour model", "abstract": "Let P be a Poisson process of intensity one in a square S_n of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph G_{n,k}. We prove that there exists a critical constant c such that for c'<c, G_{n,c'log n} is disconnected with probability tending to 1 as n tends to infinity, and for c'>c G_{n,c'\\log n} is connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors."}
{"category": "Math", "title": "On the fifth order KdV equation: local well-posedness and lack of uniform continuity of the solution map", "abstract": "In this paper we prove that the fifth order equation arising from the KdV hierarchy $ \\partial_tu + \\partial_x^5u + c_1\\partial_x u\\partial_x^2u + c_2u\\partial_x^3u = 0 $ is locally well-posed in $ H^s(\\mathbb{R}) $ for $ s> 5/2. Also, we prove the solution map of the equation is not uniformly continuous for $s>0$."}
{"category": "Math", "title": "Comparing $GL_n$-Representations by Characteristic-Free Isomorphisms between Generalized Schur Algebras", "abstract": "Isomorphisms are constructed between generalized Schur algebras in different degrees. The construction covers both the classical case (of general linear groups over infinite fields of arbitrary characteristic) and the quantized case (in type $A$, for any non-zero value of the quantum parameter $q$). The construction does not depend on the characteristic of the underlying field or the choice of $q \\neq 0$. The proof combines a combinatorial construction with comodule structures and Ringel duality. Applications range from equivalences of categories to results on the structure and cohomology of Schur algebras to identities of decomposition numbers and also of $p$-Kostka numbers, in both cases reproving and generalizing row and column removal rules."}
{"category": "Math", "title": "Ultrapowers of Banach algebras and modules", "abstract": "The Arens products are the standard way of extending the product from a Banach algebra $\\mc A$ to its bidual $\\mc A''$. Ultrapowers provide another method which is more symmetric, but one that in general will only give a bilinear map, which may not be associative. We show that if $\\mc A$ is Arens regular, then there is at least one way to use an ultrapower to recover the Arens product, a result previously known for C$^*$-algebras. Our main tool is a Principle of Local Reflexivity result for modules and algebras."}
{"category": "Math", "title": "BDDC and FETI-DP under Minimalist Assumptions", "abstract": "The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETI-DP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example."}
{"category": "Math", "title": "Tate motives and the fundamental group", "abstract": "Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X."}
{"category": "Math", "title": "Whittaker unitary dual of affine graded Hecke algebras of type E", "abstract": "This paper gives the classification of the Whittaker unitary dual for affine graded Hecke algebras of type E. By the Iwahori-Matsumoto involution, this is equivalent also to the classification of the spherical unitary dual for type E. This work completes the classification of the Whittaker Iwahori-spherical unitary dual, or equivalently, the spherical unitary dual, of split linear algebraic p-adic groups."}
{"category": "Math", "title": "Combinatorial fiber bundles and fragmentation of a fiberwise PL-homeomorphism", "abstract": "With a compact PL manifold X we associate a category T(X). The objects of T(X) are all combinatorial manifolds of type X, and morphisms are combinatorial assemblies. We prove that the homotopy equivalence BT (X) \\approx BPL(X) holds, where PL(X) is the simplicial group of PL-homeomorphisms. Thus the space BT(X) is a canonical countable (as a CW-complex) model of BPL(X). As a result, we obtain functorial pure combinatorial models for PL fiber bundles with fiber X and a PL polyhedron B as the base. Such a model looks like a T(X)-coloring of some triangulation K of B. The vertices of K are colored by objects of T(X) and the arcs are colored by morphisms in such a way that the diagram arising from the 2-skeleton of K is commutative. Comparing with the classical results of geometric topology, we obtain combinatorial models of the real Grassmannian in small dimensions: BT(S^{n-1}) \\approx BO(n) for n=1,2,3,4. The result is proved in a sequence of results on similar models of B\\PL(X). Special attention is paid to the main noncompact case X=R^n and to the tangent bundle and Gauss functor of a combinatorial manifold. The trick that makes the proof possible is a collection of lemmas on \"fragmentation of a fiberwise homeomorphism\", a generalization of the folklore lemma on fragmentation of an isotopy."}
{"category": "Math", "title": "Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces", "abstract": "We prove effective equidistribution, with polynomial rate, for large closed orbits of semisimple groups on homogeneous spaces, under certain technical restrictions (notably, the acting group should have finite centralizer in the ambient group). The proofs make extensive use of spectral gaps, and also of a closing lemma for such actions."}
{"category": "Math", "title": "Moments of the critical values of families of elliptic curves, with applications", "abstract": "We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family. Furthermore, as arithmetical applications we make a conjecture on the distribution of a_p's amongst all rank 2 elliptic curves, and also show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec)."}
{"category": "Math", "title": "Stable rank for inclusions of C*-algebras", "abstract": "When a unital \\ca $A$ has topological stable rank one (write $\\tsr(A) = 1$), we know that $\\tsr(pAp) \\leq 1$ for a non-zero projection $p \\in A$. When, however, $\\tsr(A) \\geq 2$, it is generally faluse. We prove that if a unital C*-algebra $A$ has a simple unital C*-subalgebra $D$ of $A$ with common unit such that $D$ has \\PSP and $\\sup_{p\\in P(D)}\\tsr(pAp) < \\infty$, then $\\tsr(A) \\leq 2.$ As an application let $A$ be a simple unital \\ca with $\\tsr(A) = 1$ and \\PSP, $\\{G_k\\}_{k=1}^n$ finite groups, $\\af_k$ actions from $G_k$ to ${\\rm Aut}((...((A\\times_{\\af_1}G_1)\\times_{\\af_2} G_2)...)\\times_{\\af_{k-1}}G_{k-1}).$ $(G_0 = \\{1\\})$ Then $$ \\tsr((... ((A\\times_{\\af_1}G_1)\\times_{\\af_2} G_2)...)\\times_{\\af_n}G_n) \\leq 2. $$"}
{"category": "Math", "title": "Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands", "abstract": "The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings, computing the well known Erlang formula for blocking probability in these systems becomes intractable for larger resource capacities. Using compound point processes to capture stochastic variability in the request process, we generalize existing models in this framework and derive simple asymptotic expressions for blocking probabilities. In addition, we extend our model to incorporate reserving resources in advance. Although asymptotic, our experiments show an excellent match between derived formulas and simulation results even for relatively small resource capacities and relatively large values of blocking probabilities."}
{"category": "Math", "title": "Boundaries for algebras of holomorphic functions on Banach spaces", "abstract": "We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $\\lambda_{\\phi, w}$ is the Shilov boundary for algebras of holomorphic functions on $\\lambda_{\\phi, w}$ if $\\phi$ satisfies the $\\delta_2$-condition."}
{"category": "Math", "title": "Bishop's Theorem and Differentiability of a subspace of $C_b(K)$", "abstract": "Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the G\\^{a}teaux and Fr\\'echet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace $H$ of $C_b(K)$ is a dense $G_\\delta$ subset of $H$, if $K$ is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for $H$ is the smallest closed norming subset of $H$. The classical Bishop's theorem was proved for a separating subalgebra $H$ and a metrizable compact space $K$. In the case that $X$ is a complex Banach space with the Radon-Nikod\\'ym property, we show that the set of all strong peak functions in $A_b(B_X)=\\{f\\in C_b(B_X) : f|_{B_X^\\circ} {is holomorphic}\\}$ is dense. As an application, we show that the smallest closed norming subset of $A_b(B_X)$ is the closure of the set of all strong peak points for $A_b(B_X)$. This implies that the norm of $A_b(B_X)$ is G\\^{a}teaux differentiable on a dense subset of $A_b(B_X)$, even though the norm is nowhere Fr\\'echet differentiable when $X$ is nontrivial. We also study the denseness of norm attaining holomorphic functions and polynomials. Finally we investigate the existence of numerical Shilov boundary."}
{"category": "Math", "title": "The Loewy length of the descent algebra of type D", "abstract": "The Loewy length of the descent algebra of type D_{2m+1} is shown to be m+2, for m \\geq 2, by providing an upper bound that agrees with the lower bound in \\cite{BonnafePfeiffer2006}. The bound is obtained by showing that the length of the longest path in the quiver of the descent algebra of D_{2m+1} is at most m+1. To achieve this bound, the geometric approach to the descent algebra is used, in which the descent algebra of a finite Coxeter group is identified with an algebra associated to the reflection arrangement of the group."}
{"category": "Math", "title": "Classification of uniformly outer actions of Z^2 on UHF algebras", "abstract": "We give a complete classification up to cocycle conjugacy of uniformly outer actions of Z^2 on UHF algebras. In particular, it is shown that any two uniformly outer actions of Z^2 on a UHF algebra of infinite type are cocycle conjugate. We also classify them up to outer conjugacy."}
{"category": "Math", "title": "Classification of outer actions of Z^N on O_2", "abstract": "We will show that any two outer actions of Z^N on O_2 are cocycle conjugate."}
{"category": "Math", "title": "Holder stability of diffeomorphisms", "abstract": "We prove that a $C^2$ diffeomorphism $f$ of a compact manifold $M$ satisfies Axiom A and the strong transversality condition if and only if it is H\\\"{o}lder stable, that is, any $C^1$ diffeomorphism $g$ of $M$ sufficiently $C^1$ close to $f$ is conjugate to $f$ by a homeomorphism which is H\\\"{o}lder on the whole manifold."}
{"category": "Math", "title": "A recursive online algorithm for the estimation of time-varying ARCH parameters", "abstract": "In this paper we propose a recursive online algorithm for estimating the parameters of a time-varying ARCH process. The estimation is done by updating the estimator at time point $t-1$ with observations about the time point $t$ to yield an estimator of the parameter at time point $t$. The sampling properties of this estimator are studied in a non-stationary context -- in particular, asymptotic normality and an expression for the bias due to non-stationarity are established. By running two recursive online algorithms in parallel with different step sizes and taking a linear combination of the estimators, the rate of convergence can be improved for parameter curves from H\\\"{o}lder classes of order between 1 and 2."}
{"category": "Math", "title": "Limiting distributions of curves under geodesic flow on hyperbolic manifolds", "abstract": "We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold. Moreover, if this immersed submanifold is a proper subset, then a lift of the curve to the universal covering space $T^1(H^n)$ is mapped into a proper subsphere of the ideal boundary sphere $\\partial H^n$ under the visual map. This proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in $H^n$ covering the closed immersed submanifold. In particular, if the visual map does not send a lift of the curve into a proper subsphere of $\\partial H^n$, then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bundle of the manifold with respect to the normalized natural Riemannian measure. The proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of SO(n,1)."}
{"category": "Math", "title": "On dynamical smash product", "abstract": "In the theory of dynamical Yang-Baxter equation, with any Hopf algebra $H$ and a certain $H$-module and $H$-comodule algebra $L$ (base algebra) one associates a monoidal category. Given an algebra $A$ in that category, one can construct an associative algebra $A\\rtimes L$, which is a generalization of the ordinary smash product when $A$ is an ordinary $H$-algebra. We study this \"dynamical smash product\" and its modules induced from one-dimensional representation of the subalgebra $L$. In particular, we construct an analog of the Galois map $A\\otimes_{A^H} A\\to A\\otimes H^*$."}
{"category": "Math", "title": "Wavelet block thresholding for samples with random design: a minimax approach under the $L^p$ risk", "abstract": "We consider the regression model with (known) random design. We investigate the minimax performances of an adaptive wavelet block thresholding estimator under the $\\mathbb{L}^p$ risk with $p\\ge 2$ over Besov balls. We prove that it is near optimal and that it achieves better rates of convergence than the conventional term-by-term estimators (hard, soft,...)."}
{"category": "Math", "title": "Polynomial identities and noncommutative versal torsors", "abstract": "To any cleft Hopf Galois object, i.e., any algebra H[t] obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle t, we attach two \"universal algebras\" A(H,t) and U(H,t). The algebra A(H,t) is obtained by twisting the multiplication of H with the most general two-cocycle u formally cohomologous to t. The cocycle u takes values in the field of rational functions on H. By construction, A(H,t) is a cleft H-Galois extension of a \"big\" commutative algebra B(H,t). Any \"form\" of H[t] can be obtained from A(H,t) by a specialization of B(H,t) and vice versa. If the algebra H[t] is simple, then A(H,t) is an Azumaya algebra with center B(H,t). The algebra U(H,t) is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of H[t] are satisfied. We construct an embedding of U(H,t) into A(H,t); this embedding maps the center Z(H,t) of U(H,t) into B(H,t) when the algebra H[t] is simple. In this case, under an additional assumption, A(H,t) is isomorphic to B(H,t) \\otimes_{Z(H,t)} U(H,t), thus turning A(H,t) into a central localization of U(H,t). We work out these constructions in full detail for the four-dimensional Sweedler algebra."}
{"category": "Math", "title": "Full Bayesian analysis for a class of jump-diffusion models", "abstract": "A new Bayesian significance test is adjusted for jump detection in a diffusion process. This is an advantageous procedure for temporal data having extreme valued outliers, like financial data, pluvial or tectonic forces records and others."}
{"category": "Math", "title": "Exploring spatial nonlinearity using additive approximation", "abstract": "We propose to approximate the conditional expectation of a spatial random variable given its nearest-neighbour observations by an additive function. The setting is meaningful in practice and requires no unilateral ordering. It is capable of catching nonlinear features in spatial data and exploring local dependence structures. Our approach is different from both Markov field methods and disjunctive kriging. The asymptotic properties of the additive estimators have been established for $\\alpha$-mixing spatial processes by extending the theory of the backfitting procedure to the spatial case. This facilitates the confidence intervals for the component functions, although the asymptotic biases have to be estimated via (wild) bootstrap. Simulation results are reported. Applications to real data illustrate that the improvement in describing the data over the auto-normal scheme is significant when nonlinearity or non-Gaussianity is pronounced."}
{"category": "Math", "title": "Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions", "abstract": "In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given."}
{"category": "Math", "title": "Unipotent flows on products of $SL(2,K)/\\Gamma$'s", "abstract": "We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on $SL(2,K)/\\Gamma_1\\times ...\\times SL(2,K)/\\Gamma_n$, where $K$ is a locally compact field of characteristic 0 and each $\\Gamma_i$ is a cocompact discrete subgroup of $SL(2,K)$. This special case of Ratner's theorem plays a crucial role in the proofs of uniform distribution of Heegner points by Vatsal, and Mazur conjecture on Heegner points by C. Cornut; and their generalizations in their joint work on CM-points and quaternion algebras. A purpose of the article is to make the ergodic theoretic results accessible to a wide audience."}
{"category": "Math", "title": "Estimation in hidden Markov models via efficient importance sampling", "abstract": "Given a sequence of observations from a discrete-time, finite-state hidden Markov model, we would like to estimate the sampling distribution of a statistic. The bootstrap method is employed to approximate the confidence regions of a multi-dimensional parameter. We propose an importance sampling formula for efficient simulation in this context. Our approach consists of constructing a locally asymptotically normal (LAN) family of probability distributions around the default resampling rule and then minimizing the asymptotic variance within the LAN family. The solution of this minimization problem characterizes the asymptotically optimal resampling scheme, which is given by a tilting formula. The implementation of the tilting formula is facilitated by solving a Poisson equation. A few numerical examples are given to demonstrate the efficiency of the proposed importance sampling scheme."}
{"category": "Math", "title": "A limit result for a system of particles in random environment", "abstract": "We consider an infinite system of particles in one dimension, each particle performs independant Sinai's random walk in random environment. Considering an instant $t$, large enough, we prove a result in probability showing that the particles are trapped in the neighborhood of well defined points of the lattice depending on the random environment the time $t$ and the starting point of the particles."}
{"category": "Math", "title": "On the scaling limit of a singular integral operator", "abstract": "The scaling limit and Schauder bounds are derived for a singular integral operator arising from a difference equation approach to monodromy problems."}
{"category": "Math", "title": "Penalized nonparametric mean square estimation of the coefficients of diffusion processes", "abstract": "We consider a one-dimensional diffusion process $(X_t)$ which is observed at $n+1$ discrete times with regular sampling interval $\\Delta$. Assuming that $(X_t)$ is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms."}
{"category": "Math", "title": "Monogenic Functions in Conformal Geometry", "abstract": "Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these equations to a Riemannian spin manifold only one of which is conformally invariant. We present a straightforward exposition."}
{"category": "Math", "title": "Reflecting recollements", "abstract": "A recollement describes one triangulated category T as \"glued together\" from two others, S and U. The definition is not symmetrical in S and U, but this note shows how S and U can be interchanged when T has a Serre functor."}
{"category": "Math", "title": "Characterization of count data distributions involving additivity and binomial subsampling", "abstract": "In this paper we characterize all the $r$-parameter families of count distributions (satisfying mild conditions) that are closed under addition and under binomial subsampling. Surprisingly, few families satisfy both properties and the resulting models consist of the $r$th-order univariate Hermite distributions. Among these, we find the Poisson ($r=1$) and the ordinary Hermite distributions ($r=2$)."}
{"category": "Math", "title": "On martingale approximations", "abstract": "Consider additive functionals of a Markov chain $W_k$, with stationary (marginal) distribution and transition function denoted by $\\pi$ and $Q$, say $S_n=g(W_1)+...+g(W_n)$, where $g$ is square integrable and has mean 0 with respect to $\\pi$. If $S_n$ has the form $S_n=M_n+R_n$, where $M_n$ is a square integrable martingale with stationary increments and $E(R_n^2)=o(n)$, then $g$ is said to admit a martingale approximation. Necessary and sufficient conditions for such an approximation are developed. Two obvious necessary conditions are $E[E(S_n|W_1)^2]=o(n)$ and $\\lim_{n\\to \\infty}E(S_n^2)/n<\\infty$. Assuming the first of these, let $\\Vert g\\Vert^2_+=\\limsup_{n\\to \\infty}E(S_n^2)/n$; then $\\Vert\\cdot\\Vert_+$ defines a pseudo norm on the subspace of $L^2(\\pi)$ where it is finite. In one main result, a simple necessary and sufficient condition for a martingale approximation is developed in terms of $\\Vert\\cdot\\Vert_+$. Let $Q^*$ denote the adjoint operator to $Q$, regarded as a linear operator from $L^2(\\pi)$ into itself, and consider co-isometries ($QQ^*=I$), an important special case that includes shift processes. In another main result a convenient orthonormal basis for $L_0^2(\\pi)$ is identified along with a simple necessary and sufficient condition for the existence of a martingale approximation in terms of the coefficients of the expansion of $g$ with respect to this basis."}
{"category": "Math", "title": "The Laguerre process and generalized Hartman--Watson law", "abstract": "In this paper, we study complex Wishart processes or the so-called Laguerre processes $(X_t)_{t\\geq0}$. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semi-group. We also give absolute-continuity relations between different indices. Finally, we compute the density function of the so-called generalized Hartman--Watson law as well as the law of $T_0:=\\inf\\{t,\\det(X_t)=0\\}$ when the size of the matrix is 2."}
{"category": "Math", "title": "Constructive decomposition of a function of two variables as a sum of functions of one variable", "abstract": "Given a compact set $K$ in the plane, which does not contain any triple of points forming a vertical and a horizontal segment, and a map $f\\in C(K)$, we give a construction of functions $g,h\\in C(\\mathbb R)$ such that $f(x,y)=g(x)+h(y)$ for all $(x,y)\\in K$. This provides a constructive proof of a part of Sternfeld's theorem on basic embeddings in the plane. In our proof the set $K$ is approximated by a finite set of points."}
{"category": "Math", "title": "Rational maps between moduli spaces of curves and Gieseker-Petri divisors", "abstract": "We perform an intersection theoretic study of the rational map between two different moduli spaces of stable curves which associates to a curve its corresponding Brill-Noether locus (in the case this locus has virtual dimension 1). We then use these results to describe the cone of moving divisors on M_g. Several other applications to moduli spaces of Prym varieties are presented. In a different direction, we prove that the locus in M_g of curves failing to satisfy the Gieseker-Petri theorem is supported in codimension 1 for every possible type of linear series."}
{"category": "Math", "title": "Some algorithms for semi-invariants of quivers", "abstract": "We present some theorems and algorithms for calculating perpendicular categories and locally semi-simple decompositions. We implemented a computer program {\\sc TETIVA} based on these algorithms and we offer this program for everybody's use."}
{"category": "Math", "title": "External edge condition and group cohomologies associated with the quantum Clebsch-Gordan condition", "abstract": "In this article we determine the structure of a twisted first cohomology group of the first homology of a trivalent graph with a coefficient associated with the quantum Clebsch-Gordan condition. As an application we give a characterization of a combinatorial property, the external edge condition, which is defined by the author in the study of the Heisenberg representation on the TQFT-module."}
{"category": "Math", "title": "Partial magmatic bialgebras", "abstract": "A partial magmatic bialgebra, (T;S)-magmatic bialgebra where T \\subset S are subsets of the set of positive integers, is a vector space endowed with an n-ary operation for each n in S and an m-ary co-operation for each m in T satisfying some compatibility and unitary relations. We prove an analogue of the Poincar\\'e-Birkhoff-Witt theorem for these partial magmatic bialgebras."}
{"category": "Math", "title": "Amenability of ultraproducts of Banach algebras", "abstract": "We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of $\\mc A$ is Arens regular, and give some evidence that this is if and only if $\\mc A$ is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebra. We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of $\\mc A$ is amenable. We provide an abstract characterisation in terms of something like an approximate diagonal, and consider when every ultrapower of a C$^*$-algebra, or a group $L^1$-convolution algebra, is amenable."}
{"category": "Math", "title": "Braided-Lie bialgebras associated to Kac-Moody algebras", "abstract": "Braided-Lie bialgebras have been introduced by Majid, as the Lie versions of Hopf algebras in braided categories. In this paper we extend previous work of Majid and of ours to show that there is a braided-Lie bialgebra associated to each inclusion of Kac-Moody bialgebras. Doing so, we obtain many new examples of infinite-dimensional braided-Lie bialgebras. We analyze further the case of untwisted affine Kac-Moody bialgebras associated to finite-dimensional simple Lie algebras. The inclusion we study is that of the finite-type algebra in the affine algebra. This braided-Lie bialgebra is isomorphic to the current algebra over the simple Lie algebra, now equipped with a braided cobracket. We give explicit expressions for this braided cobracket for the simple Lie algebra $sl_3$."}
{"category": "Math", "title": "Global properties of indefinite metrics with parallel Weyl tensor", "abstract": "This is an exposition of some recent results on ECS manifolds, by which we mean pseudo-Riemannian manifolds of dimensions greater than 3 that are neither conformally flat nor locally symmetric, and have parallel Weyl tensor. All ECS metrics are indefinite. We state two classification theorems, describing the local structure of ECS manifolds, and outline an argument showing that compact ECS manifolds exist in infinitely many dimensions greater than 4. We also discuss some properties of compact manifolds that admit ECS metrics, and provide a list of open questions about compact ECS manifolds."}
{"category": "Math", "title": "On the quiver of the descent algebra", "abstract": "We study the quiver of the descent algebra of a finite Coxeter group W. The results include a derivation of the quiver of the descent algebra of types A and B. Our approach is to study the descent algebra as an algebra constructed from the reflection arrangement associated to W."}
{"category": "Math", "title": "Embedding $FD(\\omega)$ into $\\mathcal{P}_s$ densely", "abstract": "Let $\\mathcal{P}_s$ be the lattice of degrees of non-empty $\\Pi_1^0$ subsets of $2^\\omega$ under Medvedev reducibility. Binns and Simpson proved that $FD(\\omega)$, the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in $\\mathcal{P}_s$. Cenzer and Hinman proved that $\\mathcal{P}_s$ is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e.\\ Turing degrees. With a construction that is a modification of the one by Cenzer and Hinman, we improve on the result of Binns and Simpson by showing that for any $\\mathcal{U} <_s \\mathcal{V}$, we can lattice embed $FD(\\omega)$ into $\\mathcal{P}_s$ strictly between $deg_s(\\mathcal{U})$ and $deg_s(\\mathcal{V)}$. We also note that, in contrast to the infinite injury in the proof of the Sacks Density Theorem, in our proof all injury is finite, and that this is also true for the proof of Cenzer and Hinman, if a straightforward simplification is made."}
{"category": "Math", "title": "A note on the Markoff condition and central words", "abstract": "We define Markoff words as certain factors appearing in bi-infinite words satisfying the Markoff condition. We prove that these words coincide with central words, yielding a new characterization of Christoffel words."}
{"category": "Math", "title": "Quantum Structures for Lagrangian Submanifolds", "abstract": "We discuss various algebraic quantum structures associated to monotone Lagrangian submanifolds and we present a number of applications, computations and examples."}
{"category": "Math", "title": "Algebraic G-functions associated to matrices over a group-ring", "abstract": "Given a square matrix with elements in the group-ring of a group, one can consider the sequence formed by the trace (in the sense of the group-ring) of its powers. We prove that the corresponding generating series is an algebraic $G$-function (in the sense of Siegel) when the group is free of finite rank. Consequently, it follows that the norm of such elements is an exactly computable algebraic number, and their Green function is algebraic. Our proof uses the notion of rational and algebraic power series in non-commuting variables and is an easy application of a theorem of Haiman. Haiman's theorem uses results of linguistics regarding regular and context-free language. On the other hand, when the group is free abelian of finite rank, then the corresponding generating series is a $G$-function. We ask whether the latter holds for general hyperbolic groups. This version has an expanded introduction following suggestions from Lehner, Voiculescu and others."}
{"category": "Math", "title": "Minimal Distortion Bending and Morphing of Compact Manifolds", "abstract": "Let $M$ and $N$ be compact smooth oriented Riemannian $n$-manifolds without boundary embedded in $\\mathbb{R}^{n+1}$. Several problems about minimal distortion bending and morphing of $M$ to $N$ are posed. Cost functionals that measure distortion due to stretching or bending produced by a diffeomorphism $h:M \\to N$ are defined, and new results on the existence of minima of these cost functionals are presented. In addition, the definition of a morph between two manifolds $M$ and $N$ is given, and the theory of minimal distortion morphing of compact manifolds is reviewed."}
{"category": "Math", "title": "On the torsion of Drinfeld modules of rank two", "abstract": "We study rational points and torsion points on Drinfeld modular curves defined over rational function fields. As a consequence we derive a conjecture of Schweizer describing completely the torsion of Drinfeld modules of rank two over $\\Bbb F_2(T)$ implying Poonen's uniform boundedness conjecture in this particular case."}
{"category": "Math", "title": "Hurwitz-Hodge Integrals, the E6 and D4 root systems, and the Crepant Resolution Conjecture", "abstract": "Let G be the group A_4 or Z_2xZ_2. We compute the integral of \\lambda_g on the Hurwitz locus H_G\\subset M_g of curves admitting a degree 4 cover of P^1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E_6 and D_4 root systems respectively. As an application, we prove the Crepant Resolution Conjecture for the orbifolds [C^3/A_4] and [C^3/(Z_2xZ_2)]."}
{"category": "Math", "title": "Conjugacy and Dynamics in Thompson's Groups", "abstract": "We give a unified solution to the conjugacy problem for Thompson's groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson's groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson's groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results."}
{"category": "Math", "title": "Virtual Yang-Baxter cocycle invariants", "abstract": "We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links."}
{"category": "Math", "title": "On hitting times and fastest strong stationary times for skip-free and more general chains", "abstract": "An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues nu_j of the generator are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates nu_j. We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler. We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results. In the paper's final section we present extensions of our results to more general chains."}
{"category": "Math", "title": "The spectral sequence of an equivariant chain complex and homology with local coefficients", "abstract": "We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X. In the process, we identify the d^1 differential in terms of the coalgebra structure of H_*(X,\\k), and the \\k\\pi_1(X)-module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov, on the mod p cohomology of cyclic p-covers of aspherical complexes. This approach provides information on the homology of all Galois covers of X. It also yields computable upper bounds on the ranks of the cohomology groups of X, with coefficients in a prime-power order, rank one local system. When X admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cup-product structure of H^*(X,\\k), thereby generalizing a result of Cohen and Orlik."}
{"category": "Math", "title": "Canonical 2-forms on the moduli space of Riemann surfaces", "abstract": "As was shown by Harer the second homology of ${\\mathbb M}_g$, the moduli space of compact Riemann surfaces of genus $g$, is of rank 1, provided $g \\geq 3$. This means a nontrivial second de Rham cohomology class on ${\\mathbb M}_g$ is unique up to constant factor. But several canonical 2-forms on the moduli space have been constructed in various geometric contexts, and differ from each other. In this article we review some of constructions in order to provide materials for future research on \"secondary geometry\" of the moduli space ${\\mathbb M}_g$."}
{"category": "Math", "title": "Normal approximation for nonlinear statistics using a concentration inequality approach", "abstract": "Let $T$ be a general sampling statistic that can be written as a linear statistic plus an error term. Uniform and non-uniform Berry--Esseen type bounds for $T$ are obtained. The bounds are the best possible for many known statistics. Applications to U-statistics, multisample U-statistics, L-statistics, random sums and functions of nonlinear statistics are discussed."}
{"category": "Math", "title": "Continuation of solutions of coupled dynamical systems", "abstract": "Recently, the synchronization of coupled dynamical systems has been widely studied. Synchronization is referred to as a process wherein two (or many) dynamical systems are adjusted to a common behavior as time goes to infinity, due to coupling or forcing. Therefore, before discussing synchronization, a basic problem on continuation of the solution must be solved: For given initial conditions, can the solution of coupled dynamical systems be extended to the infinite interval $[0,+\\infty)$? In this paper, we propose a general model of coupled dynamical systems, which includes previously studied systems as special cases, and prove that under the assumption of QUAD, the solution of the general model exists on $[0,+\\infty)$."}
{"category": "Math", "title": "Lefschetz numbers for C*-algebras", "abstract": "Using Poincare duality, we formulate a formula of Lefschetz type which computes the Lefschetz number of an endomorphism of a separable, nuclear C*-algebra satisfying Poincare duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on K-theory tensored with the complex numbers, as in the classical case.) We then consider endomorphisms of Cuntz-Krieger algebras O_A. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of the endomorphism. Using this description we derive a closed polynomial formula for the Lefschetz number depending on the matrix A and the presentation of the endomorphism."}
{"category": "Math", "title": "A Lefschetz fixed-point formula for certain orbifold C*-algebras", "abstract": "Using Poincar\\'e duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of cross product C*-algebras $C_0(X)\\cross G$ coming from covariant pairs. Here $G$ is assumed countable, $X$ a manifold, and $X\\cross G$ cocompact and proper. The formula in question expresses the graded trace of the map on rationalized K-theory of $C_0(X)\\cross G$ induced by the endomorphism, \\emph{i.e.} the Lefschetz number, in terms of fixed orbits and representation-theoretic data connected with certain isotropy subgroups of the isotropy group at that point. The technique is to use noncommutative Poinca\\'e duality and the formal Lefschetz lemma of the second author."}
{"category": "Math", "title": "Correction to: The scaling limit behaviour of periodic stable-like processes", "abstract": "Correction to Bernoulli (2006), 12, 551--570 http://projecteuclid.org/euclid.bj/1151525136"}
{"category": "Math", "title": "Dynamical sensitivity of the infinite cluster in critical percolation", "abstract": "In dynamical percolation, the status of every bond is refreshed according to an independent Poisson clock. For graphs which do not percolate at criticality, the dynamical sensitivity of this property was analyzed extensively in the last decade. Here we focus on graphs which percolate at criticality, and investigate the dynamical sensitivity of the infinite cluster. We first give two examples of bounded degree graphs, one which percolates for all times at criticality and one which has exceptional times of nonpercolation. We then make a nearly complete analysis of this question for spherically symmetric trees with spherically symmetric edge probabilities bounded away from 0 and 1. One interesting regime occurs when the expected number of vertices at the nth level that connect to the root at a fixed time is of order n(\\log n)^\\alpha. R. Lyons (1990) showed that at a fixed time, there is an infinite cluster a.s. if and only if \\alpha >1. We prove that the probability that there is an infinite cluster at all times is 1 if \\alpha > 2, while this probability is 0 if 1<\\alpha \\le 2. Within the regime where a.s. there is an infinite cluster at all times, there is yet another type of ``phase transition'' in the behavior of the process: if the expected number of vertices at the nth level connecting to the root at a fixed time is of order n^\\theta with \\theta > 2, then the number of connected components of the set of times in [0,1] at which the root does not percolate is finite a.s., while if 1<\\theta < 2, then the number of such components is infinite with positive probability."}
{"category": "Math", "title": "Large sample asymptotics for the two-parameter Poisson--Dirichlet process", "abstract": "This paper explores large sample properties of the two-parameter $(\\alpha,\\theta)$ Poisson--Dirichlet Process in two contexts. In a Bayesian context of estimating an unknown probability measure, viewing this process as a natural extension of the Dirichlet process, we explore the consistency and weak convergence of the the two-parameter Poisson--Dirichlet posterior process. We also establish the weak convergence of properly centered two-parameter Poisson--Dirichlet processes for large $\\theta+n\\alpha.$ This latter result complements large $\\theta$ results for the Dirichlet process and Poisson--Dirichlet sequences, and complements a recent result on large deviation principles for the two-parameter Poisson--Dirichlet process. A crucial component of our results is the use of distributional identities that may be useful in other contexts."}
{"category": "Math", "title": "Dilatation structures in sub-riemannian geometry", "abstract": "Based on the notion of dilatation structure arXiv:math/0608536, we give an intrinsic treatment to sub-riemannian geometry, started in the paper arXiv:0706.3644 . Here we prove that regular sub-riemannian manifolds admit dilatation structures. From the existence of normal frames proved by Bellaiche we deduce the rest of the properties of regular sub-riemannian manifolds by using the formalism of dilatation structures."}
{"category": "Math", "title": "Vector Optimization by Two Objective Functions", "abstract": "Recently wide application in engineering-economic problems was received with problems of vector optimization. Development of methods of the decision of these problems it is executed in works A. Messac and others. Complexity of the offered methods consists in construction of an aggregate objective function. In the given work an algorithm of the solution of a vector optimization problem is suggested carry out by use analytical representation of Pareto cone."}
{"category": "Math", "title": "Z_2 actions on complexes with three non-trivial cells", "abstract": "In this paper, we study $Z_2$ actions on a cell complex X having the cohomology ring isomorphic to that of the wedge sum $P^2 (n) V S^{3n}$ or $S^n V S^{2n} V S^{3n}$. We determine the possible fixed point sets depending on whether or not X is totally non-homologous to zero in $X_{Z_2}$ and give examples realizing the possible cases."}
{"category": "Math", "title": "Coupling hidden Markov models for the discovery of Cis-regulatory modules in multiple species", "abstract": "Cis-regulatory modules (CRMs) composed of multiple transcription factor binding sites (TFBSs) control gene expression in eukaryotic genomes. Comparative genomic studies have shown that these regulatory elements are more conserved across species due to evolutionary constraints. We propose a statistical method to combine module structure and cross-species orthology in de novo motif discovery. We use a hidden Markov model (HMM) to capture the module structure in each species and couple these HMMs through multiple-species alignment. Evolutionary models are incorporated to consider correlated structures among aligned sequence positions across different species. Based on our model, we develop a Markov chain Monte Carlo approach, MultiModule, to discover CRMs and their component motifs simultaneously in groups of orthologous sequences from multiple species. Our method is tested on both simulated and biological data sets in mammals and Drosophila, where significant improvement over other motif and module discovery methods is observed."}
{"category": "Math", "title": "Group cocycles and the ring of affiliated operators", "abstract": "In this article we study cocycles of discrete countable groups with values in l^2(G) and the ring of affiliated operators UG. We clarify properties of the first cohomology of a group G with coefficients in l^2(G) and answer several questions from [CTV]. Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first l^2-Betti number. We give numerous applications and examples of groups which satisfy our assumptions."}
{"category": "Math", "title": "Structure and finiteness properties of subdirect products of groups", "abstract": "We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup."}
{"category": "Math", "title": "A statistical approach to simultaneous mapping and localization for mobile robots", "abstract": "Mobile robots require basic information to navigate through an environment: they need to know where they are (localization) and they need to know where they are going. For the latter, robots need a map of the environment. Using sensors of a variety of forms, robots gather information as they move through an environment in order to build a map. In this paper we present a novel sampling algorithm to solving the simultaneous mapping and localization (SLAM) problem in indoor environments. We approach the problem from a Bayesian statistics perspective. The data correspond to a set of range finder and odometer measurements, obtained at discrete time instants. We focus on the estimation of the posterior distribution over the space of possible maps given the data. By exploiting different factorizations of this distribution, we derive three sampling algorithms based on importance sampling. We illustrate the results of our approach by testing the algorithms with two real data sets obtained through robot navigation inside office buildings at Carnegie Mellon University and the Pontificia Universidad Catolica de Chile."}
{"category": "Math", "title": "Uniqueness of the 2-universality Criterion", "abstract": "Kim, Kim, and Oh gave a minimal criterion for the 2-universality of positive-definite integer-matrix quadratic forms. We show that this 2-universality criterion is unique in the sense of the uniqueness of the Conway-Schneeberger Fifteen Theorem."}
{"category": "Math", "title": "Continuation unique a partir de l'infini conforme pour les metriques d'Einstein", "abstract": "We prove the unique continuation property at the conformal infinity for asymptotically hyperbolic Einstein metrics."}
{"category": "Math", "title": "Random-set methods identify distinct aspects of the enrichment signal in gene-set analysis", "abstract": "A prespecified set of genes may be enriched, to varying degrees, for genes that have altered expression levels relative to two or more states of a cell. Knowing the enrichment of gene sets defined by functional categories, such as gene ontology (GO) annotations, is valuable for analyzing the biological signals in microarray expression data. A common approach to measuring enrichment is by cross-classifying genes according to membership in a functional category and membership on a selected list of significantly altered genes. A small Fisher's exact test $p$-value, for example, in this $2\\times2$ table is indicative of enrichment. Other category analysis methods retain the quantitative gene-level scores and measure significance by referring a category-level statistic to a permutation distribution associated with the original differential expression problem. We describe a class of random-set scoring methods that measure distinct components of the enrichment signal. The class includes Fisher's test based on selected genes and also tests that average gene-level evidence across the category. Averaging and selection methods are compared empirically using Affymetrix data on expression in nasopharyngeal cancer tissue, and theoretically using a location model of differential expression. We find that each method has a domain of superiority in the state space of enrichment problems, and that both methods have benefits in practice. Our analysis also addresses two problems related to multiple-category inference, namely, that equally enriched categories are not detected with equal probability if they are of different sizes, and also that there is dependence among category statistics owing to shared genes. Random-set enrichment calculations do not require Monte Carlo for implementation. They are made available in the R package allez."}
{"category": "Math", "title": "Jordan algebras over algebraic varieties", "abstract": "We construct Jordan algebras over a locally ringed space using generalizations of the Tits process and the first Tits construction by Achhammer. Some general results on the structure of these algebras are obtained. Examples of Albert algebras over a Brauer-Severi variety with associated central simple algebra of degree 3 are given."}
{"category": "Math", "title": "G-functions and multisum versus holonomic sequences", "abstract": "The purpose of the paper is three-fold: (a) we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of Gevrey type-1 with rational exponents, (b) we construct a class of $G$-functions that come from enumerative combinatorics, and (c) we give a counterexample to a question of Zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms. The proofs utilize the notion of a $G$-function, introduced by Siegel, and its analytic/arithmetic properties shown recently by Andr\\'e."}
{"category": "Math", "title": "Elevated soil lead: Statistical modeling and apportionment of contributions from lead-based paint and leaded gasoline", "abstract": "While it is widely accepted that lead-based paint and leaded gasoline are primary sources of elevated concentrations of lead in residential soils, conclusions regarding their relative contributions are mixed and generally study specific. We develop a novel nonlinear regression for soil lead concentrations over time. It is argued that this methodology provides useful insights into the partitioning of the average soil lead concentration by source and time over large residential areas. The methodology is used to investigate soil lead concentrations from the 1987 Minnesota Lead Study and the 1990 National Lead Survey. Potential litigation issues are discussed briefly."}
{"category": "Math", "title": "$G$-stable pieces and partial flag varieties", "abstract": "We will use the combinatorics of the $G$-stable pieces to describe the closure relation of the partition of partial flag varieties in \\cite[section 4]{L3}."}
{"category": "Math", "title": "The size of a pond in 2D invasion percolation", "abstract": "We consider invasion percolation on the square lattice. It has been proved by van den Berg, Peres, Sidoravicius and Vares, that the probability that the radius of a so-called pond is larger than n, differs at most a factor of order log n from the probability that in critical Bernoulli percolation the radius of an open cluster is larger than n. We show that these two probabilities are, in fact, of the same order. Moreover, we prove an analogous result for the volume of a pond."}
{"category": "Math", "title": "Fibonacci-like sequences and shift spaces in symbolic dynamics", "abstract": "We afford the problem of counting the blocks of a given length made with symbols drawn from an alphabet and relate this number to Fibonacci-like recurrent relations. The recurrence polynomia allows to calculate the limit ratio of two adjacent terms and this information is used to determine the topological entropy of a discrete numbers of associate shift spaces. We then describe a scheme to build a shift space with a pre-selected entropy."}
{"category": "Math", "title": "Bernoulli free-boundary problems in strip-like domains and a property of permanent waves in water of finite depth", "abstract": "We study weak solutions for a class of free boundary problems which includes as a special case the classical problem of traveling waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions. We also prove that in very general situations the free boundary is necessarily the graph of a function."}
{"category": "Math", "title": "Harmonic contact metric structures, and submersions", "abstract": "We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonicity to that of the almost Hermitian structure on the base or fibre."}
{"category": "Math", "title": "A rate-independent model for the isothermal quasi-static evolution of shape-memory materials", "abstract": "This note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics."}
{"category": "Math", "title": "Automorphisms of Galois Coverings of Generic $m$-Canonical Projections", "abstract": "The automorphism group of the Galois covering induced by a pluri-canonical generic covering of a projective space is investigated. It is shown that by means of such coverings one obtains, in dimensions one and two, serieses of specific actions of the symmetric groups $S_d$ on curves and surfaces not deformable to an action of $S_d$ which is not the full automorphism group. As an application, new DIF $\\ne$ DEF examples for $G$-varieties in complex and real geometry are given."}
{"category": "Math", "title": "A sufficient criterion for homotopy cartesianess", "abstract": "Suppose given a commutative quadrangle in a Verdier triangulated category such that there exists an induced isomorphism on the horizontally taken cones. Suppose that the endomorphism ring of the initial or the terminal corner object of this quadrangle satisfies a finiteness condition. Then this quadrangle is homotopy cartesian."}
{"category": "Math", "title": "On the Gr\\\"obner complexity of matrices", "abstract": "In this paper we show that if for an integer matrix A the universal Gr\\\"obner basis of the associated toric ideal \\Ideal_A coincides with the Graver basis of A, then the Gr\\\"obner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. We conclude that for the matrices A_{3\\times 3} and A_{3\\times 4}, defining the 3\\times 3 and 3\\times 4 transportation problems, we have u(A_{3\\times 3})=g(A_{3\\times 3})=9 and u(A_{3\\times 4})=g(A_{3\\times 4})\\geq 27. Moreover, we prove u(A_{a,b})=g(A_{a,b})=2(a+b)/\\gcd(a,b) for positive integers a,b and A_{a,b}=(\\begin{smallmatrix} 1 & 1 & 1 & 1 0 & a & b & a+b \\end{smallmatrix})."}
{"category": "Math", "title": "The computation of the classes of some tori in the Grothendieck ring of varieties", "abstract": "We establish a formula for the classes of certain tori in the Grothendieck ring of varieties, in terms of its lambda-structure. More explicitly, we will see that if L* is the torus of invertible elements in the n-dimensional separable k-algebra L, then the class of L* can be expressed as an alternating sum of the images of the spectrum of L under the lambda-operations, multiplied by powers of the Lefschetz class. This formula is suggested from the cohomology of the torus, illustrating a heuristic method that can be used in other situations. To prove the formula will require some rather explicit calculations in the Grothendieck ring. To be able to make these we introduce a homomorphism from the Burnside ring of the absolute Galois group of k, to the Grothendieck ring of varieties over k. In the process we obtain some information about the structure of the subring generated by zero-dimensional varieties."}
{"category": "Math", "title": "Cyclic odd degree base change lifting for unitary groups in three variables", "abstract": "Let E/F be a quadratic number (resp. p-adic) field extension, and F' an odd degree cyclic field extension of F. We establish a base-change functorial lifting of automorphic (resp. admissible) representations from the unitary group U(3,E/F) associated with E/F to the unitary group U(3,F'E/F'). As a consequence, we classify the invariant packets of U(3,F'E/F'), namely those which contain (irreducible) automorphic (resp. admissible) representations which are invariant under the action of the Galois group Gal(F'E/E). To do this we use the trace formula technique, and well-known results on the base-change lifting from U(3,E/F) to GL(3,E) and on the base-change lifting for the general linear groups. We also determine the invariance of individual representations, using Howe correspondence. This work is the first study of base change into an algebraic group whose packets are not all singletons, and which does not satisfy the \"strong multiplicity one theorem.\" Novel phenomena are encountered: e.g. there are invariant packets where not every irreducible automorphic (resp. admissible) member is Galois-invariant. We also obtain local twisted character identities with respect to this base-change lifting."}
{"category": "Math", "title": "2-gerbes and 2-Tate spaces", "abstract": "We construct a central extension of the group of automorphisms of a 2-Tate vector space viewed as a discrete 2-group. This is done using an action of this 2-group on a 2-gerbe of gerbel theories. This central extension is used to define central extensions of double loop groups."}
{"category": "Math", "title": "Mirror symmetry and tropical geometry", "abstract": "Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev and Borisov for Calabi-Yau complete intersections. We apply the construction to Pfaffian examples and recover the mirror given by Rodland for the degree 14 Calabi-Yau threefold in PP^6 defined by the Pfaffians of a general linear 7x7 skew-symmetric matrix. We provide the necessary background knowledge entering into the tropical mirror construction such as toric geometry, Groebner bases, tropical geometry, Hilbert schemes and deformations. The tropical approach yields an algorithm which we illustrate in a series of explicit examples."}
{"category": "Math", "title": "The largest component in a subcritical random graph with a power law degree distribution", "abstract": "It is shown that in a subcritical random graph with given vertex degrees satisfying a power law degree distribution with exponent $\\gamma>3$, the largest component is of order $n^{1/(\\gamma-1)}$. More precisely, the order of the largest component is approximatively given by a simple constant times the largest vertex degree. These results are extended to several other random graph models with power law degree distributions. This proves a conjecture by Durrett."}
{"category": "Math", "title": "Moments and distribution of the local times of a transient random walk on $\\Z^d$", "abstract": "Consider an arbitrary transient random walk on $\\Z^d$ with $d\\in\\N$. Pick $\\alpha\\in[0,\\infty)$ and let $L_n(\\alpha)$ be the spatial sum of the $\\alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for integers $\\alpha$, $L_n(\\alpha)$ is the number of the $\\alpha$-fold self-intersections of the walk. We prove a strong law of large numbers for $L_n(\\alpha)$ as $n\\to\\infty$. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by \\v{C}ern\\'y \\cite{Ce07}. Although these assertions are certainly known to experts, we could find no proof in the literature in this generality."}
{"category": "Math", "title": "Freeness with amalgamation, limit theorems and S-transform in non-commutative probability spaces of type B", "abstract": "The present material addresses several problems left open in the Trans. AMS paper \" Non-crossing cumulants of type B\" of P. Biane, F. Goodman and A. Nica. The main result is that a type B non-commutative probability space can be studied in the framework of freeness with amalgamation. This view allows easy ways of constructing a version of the S-transform as well as proving analogue results to Central Limit Theorem and Poisson Limit Theorem."}
{"category": "Math", "title": "On Selfadjoint Subspace of One-Speed Boltzmann Operator", "abstract": "The aim of the paper is to obtain a description of the selfadjoint subspace of the one-speed Boltzmann operator. It is proved that this subspace is nontrivial if the collision integral is polynomial and the multiplication coefficient has a lattice of gaps. A similar result is shown to hold for the 3-dimensional transport operator. Keywords: completely nonselfadjoint, Boltzmann operator, uncertainly principle."}
{"category": "Math", "title": "A note on the Verlinde bundles on elliptic curves", "abstract": "We study the splitting properties of the Verlinde bundles over elliptic curves. Our methods rely on the explicit description of the moduli space of semistable vector bundles on elliptic curves, and on the analysis of the symmetric powers of the Schrodinger representation of the Theta group."}
{"category": "Math", "title": "Sofic groups and profinite topology on free groups", "abstract": "We give a definition of weakly sofic groups (w-sofic groups). Our definition is rather natural extension of the definition of sofic groups where instead of Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non w-sofic groups is equivalent to some conjecture about profinite topology on free groups."}
{"category": "Math", "title": "Hidden Life of Riemann's Zeta Function 2. Electrons and Trains", "abstract": "The Riemann Hypothesis can be reformulated as statements about the eigenvalues of certain matrices whose entries are defined in terms of the Taylor coefficients of the zeta function. These eigenvalues exhibit interesting visual patterns allowing one to state a number of conjectures. The Hankel matrices introduced here are obtained, by rearranging of columns, from Toeplitz matrices whose eigenvalues were considered in arXiv:0707.1983 . The present paper is a continuation of that publication."}
{"category": "Math", "title": "Alexander Duality and Serre's Property $(S_i)$ for Square-free Monomial Ideals", "abstract": "In this note, we study Serre's property $(S_i)$, and its relation to Alexander duality for monomial ideals in a polynomial ring over a field. We describe ideals that define the non-Cohen-Macaulay- and the non-$(S_i)$-loci of finitely generated modules over regular rings, and show that minimal prime ideals in these loci are homogeneous, in the graded case. We show that a square-free monomial ideal has property $(S_i)$ if and only if its Alexander dual has a linear resolution up to homological degree $i-1$. We prove that for square-free monomial ideals, having property $(S_2)$ is equivalent to being locally connected in codimension 1."}
{"category": "Math", "title": "Chern classes of Deligne-Mumford stacks and their coarse moduli spaces", "abstract": "Let $X$ be a complex projective algebraic variety with Gorenstein quotient singularities and $\\X$ a smooth Deligne-Mumford stack having $X$ as its coarse moduli space. We show that the CSM class $c^{SM}(X)$ coincides with the pushforward to $X$ of the total Chern class $c(T_{I\\X})$ of the inertia stack $I\\X$. We also show that the stringy Chern class $c_{str}(X)$ of $X$, whenever is defined, coincides with the pushforward to $X$ of the total Chern class $c(T_{II\\X})$ of the double inertia stack $II\\X$. Some consequences concerning stringy/orbifold Hodge numbers are deduced."}
{"category": "Math", "title": "Circular law for non-central random matrices", "abstract": "Let $(X_{jk})_{j,k\\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean 0 and variance 1. Let $\\la_{n,1},...,\\la_{n,n}$ be the eigenvalues of $(\\frac{1}{\\sqrt{n}}X_{jk})_{1\\leq j,k\\leq n}$. The strong circular law theorem states that with probability one, the empirical spectral distribution $\\frac{1}{n}(\\de_{\\la_{n,1}}+...+\\de_{\\la_{n,n}})$ converges weakly as $n\\to\\infty$ to the uniform law over the unit disc $\\{z\\in\\dC;|z|\\leq1\\}$. In this short note, we provide an elementary argument that allows to add a deterministic matrix $M$ to $(X_{jk})_{1\\leq j,k\\leq n}$ provided that $\\mathrm{Tr}(MM^*)=O(n^2)$ and $\\mathrm{rank}(M)=O(n^\\al)$ with $\\al<1$. Conveniently, the argument is similar to the one used for the non-central version of Wigner's and Marchenko-Pastur theorems."}
{"category": "Math", "title": "Steiner-Minkowski Polynomials of Convex Sets in High Dimension, and Limit Entire Functions", "abstract": "For a convex set (K) of the (n)-dimensional Euclidean space, the Steiner-Minkowski polynomial (M_K(t)) is defined as the (n)-dimensional Euclidean volume of the neighborhood of the radius (t). Being defined for positive (t), the Steiner-Minkowski polynomials are considered for all complex (t). The renormalization procedure for Steiner polynomial is proposed. The renormalized Steiner-Minkowski polynomials corresponding to all possible solid convex sets in all dimensions form a normal family in the whole complex plane. For each of the four families of convex sets: the Euclidean balls, the cubes, the regular cross-polytopes and the regular symplexes of dimensions (n), the limiting entire functions, as (n) tends to infinity, are calculated explicitly."}
{"category": "Math", "title": "The Ramsey number for a triple of large cycles", "abstract": "We find the asymptotic value of the Ramsey number for a triple of long cycles, where the lengths of the cycles are large but may have different parity."}
{"category": "Math", "title": "On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension", "abstract": "Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class g^G of G we define its centralizing subgroup to be HC_G(g). Let K be such that H\\le K\\le G. We show that the G-conjugacy classes contained in K whose centralizing subgroup is K, are equally distributed between the cosets of H in K. The proof of this result is entirely elementary. As an application we find expressions for the number of conjugacy classes of K under its own action, in terms of quantities relating only to the action of G."}
{"category": "Math", "title": "Theta Series Associated with the Weil-Schroedinger Representation", "abstract": "The Weil representation discovered by Andre Weil plays an important role in the study of the tranformation properties of theta series. In this paper, we define the Weil-Schroedinger representation of the Jacobi group and prove that the theta series associated with the Weil-Schroedinger representation is a Jacobi form with respect to a suitable arithmetic subgroup of the Jacobi modular group."}
{"category": "Math", "title": "Quasi-Anosov diffeomorphisms of 3-manifolds", "abstract": "In 1969, Hirsch posed the following problem: given a diffeomorphism, and a compact invariant hyperbolic set, describe its topology and restricted dynamics. We solve the problem where the hyperbolic invariant set is a closed 3-manifold: if the manifold is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions. The dynamics of the diffeomorphisms restricted to these manifolds, called quasi-Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions."}
{"category": "Math", "title": "A Fixed Point Conjecture", "abstract": "Inverse limits, unlike direct limits, can in general be void, [1]. The existence of fixed points for arbitrary mappings $T : X \\longrightarrow X$ is conjectured to be equivalent with the fact that related direct limits of all finite partitions of X are not void."}
{"category": "Math", "title": "A wildland fire model with data assimilation", "abstract": "A wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to combustion corresponds to the fuel reaction rate. The resulting coupled partial differential equations have coefficients that can be approximated from prior measurements of wildfires. An ensemble Kalman filter technique with regularization is then used to assimilate temperatures measured at selected points into running wildfire simulations. The assimilation technique is able to modify the simulations to track the measurements correctly even if the simulations were started with an erroneous ignition location that is quite far away from the correct one."}
{"category": "Math", "title": "New q-Euler numbers and polynomials associated with p-adic q-integrals", "abstract": "In this paper we study q-Euler numbers and polynomials by using p-adic q-fermionic integrals on Z_p. The methods to study q-Euler numbers and polynomials in this paper are new."}
{"category": "Math", "title": "Theorie ergodique des fractions rationnelles sur un corps ultrametrique", "abstract": "We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure m_R which reprensents the asymptotic distribution of preimages of non-exceptional point. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of m_R, and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction."}
{"category": "Math", "title": "The magic functions and automorphisms of a domain", "abstract": "We introduce the notion of magic functions of a general domain in d-dimensional complex space and show that the set of magic functions of a given domain is an intrinsic complex-geometric object. We determine the set of magic functions of the symmetrised bidisc G, and thereby find all automorphisms of G and a formula for the Caratheodory distance on G."}
{"category": "Math", "title": "Heegaard genus and Property 'tau' for hyperbolic 3-manifolds", "abstract": "We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property $\\tau$. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive."}
{"category": "Math", "title": "Symmetry classes connected with the magnetic Heisenberg ring", "abstract": "We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups S_N such as decompositions of ideals of the group ring C[S_N], idempotents of C[S_N], discrete Fourier transforms of S_N, Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian of the magnet. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[S_N]. Let U be a subspace of eigenvectors of a a fixed eigenvalue \\mu of the Hamiltonian with weight (r_1,r_2). If one determines the smallest symmetry class for every v in U then one can observe jumps of the symmetry behaviour. For ''generic'' v all smallest symmetry classes have the same maximal dimension d and structure. But U can contain linear subspaces on which the dimension of the smallest symmetry class of v jumps to a value smaller than d. Then the stability subgroup of v can increase. We can calculate such jumps explicitely. In our investigations we use computer calculations by means of the Mathematica packages PERMS and HRing."}
{"category": "Math", "title": "Local convergence for alternating and averaged nonconvex projections", "abstract": "The idea of a finite collection of closed sets having \"strongly regular intersection\" at a given point is crucial in variational analysis. We show that this central theoretical tool also has striking algorithmic consequences. Specifically, we consider the case of two sets, one of which we assume to be suitably \"regular\" (special cases being convex sets, smooth manifolds, or feasible regions satisfying the Mangasarian-Fromovitz constraint qualification). We then prove that von Neumann's method of \"alternating projections\" converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having strongly regular intersection at some point, the method of \"averaged projections\" converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly."}
{"category": "Math", "title": "A new method for the estimation of variance matrix with prescribed zeros in nonlinear mixed effects models", "abstract": "We propose a new method for the Maximum Likelihood Estimator (MLE) of nonlinear mixed effects models when the variance matrix of Gaussian random effects has a prescribed pattern of zeros (PPZ). The method consists in coupling the recently developed Iterative Conditional Fitting (ICF) algorithm with the Expectation Maximization (EM) algorithm. It provides positive definite estimates for any sample size, and does not rely on any structural assumption on the PPZ. It can be easily adapted to many versions of EM."}
{"category": "Math", "title": "On the precision of the spectral profile", "abstract": "We examine the spectral profile bound of Goel, Montenegro and Tetali for the uniform mixing time of continuous-time random walk in reversible settings. We find that it is precise up to a log log factor, and that this log log factor cannot be improved."}
{"category": "Math", "title": "A remark on utility streams", "abstract": "We consider arbitrarily long, but finite utility streams, and some appropriate axioms."}
{"category": "Math", "title": "On Zariski's multiplicity conjecture", "abstract": "We discuss some features of the so-called Zariski's multiplicity problem especially the application of the work of A'Campo on the zeta function of a monodromy of an isolated singularity of a complex hypersurface to the problem."}
{"category": "Math", "title": "Pointed and copointed Hopf algebras as cocycle deformations", "abstract": "We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization of such Hopf algebras, which allows for the application of a result of Masuoka about Morita-Takeuchi equivalence and of Schauenburg about Hopf Galois extensions. The \"infinitesimal\" part of the deforming cocycle and of the deformation determine the deformed multiplication and can be described explicitly in terms of Hochschild cohomology. Applications to, and results for copointed Hopf algebras are also considered."}
{"category": "Math", "title": "Introduction to shape stability for a storage model", "abstract": "We consider a new idea for a storage model on n nodes, namely stability of shape. These nodes support K neighborhoods S_i \\subset {1, ..., n} and items arrive at the S_i as independent Poisson streams with rates lambda_i, i=1, ...,K. Upon arrival at S_i an item is stored at node j \\in S_i where j is determined by some policy. Under natural conditions on the lambda_i we exhibit simple local policies such that the multidimensional process describing the evolution of the number of items at each node is positive recurrent (stable) in shape."}
{"category": "Math", "title": "Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem", "abstract": "We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order error estimate. The full discrete backward Euler method and the Crank-Nicolson-Galerkin scheme are also considered. Finally, a simple algorithm for solving the fully discrete problem is proposed."}
{"category": "Math", "title": "On the index of the Heegner subgroup of elliptic curves", "abstract": "Let E be an elliptic curve of conductor N and rank one over Q. So there is a non-constant morphism X+0(N) --> E defined over Q, where X+0(N) = X0(N)/wN and wN is the Fricke involution of the modular curve X+0(N). Under this morphism the traces of the Heegner points of X+0(N) map to rational points on E. In this paper we study the index I of the subgroup generated by all these traces on E(Q). We propose and also discuss a conjecture that says that if N is prime and I > 1, then either the number of connected components of the real locus X+0(N)(R) is greater than 1 or (less likely) the order S of the Tate-Safarevich group is non-trivial. This conjecture is backed by computations performed on each E that satisfies the above hypothesis in the range N < 129999. This paper was prepared for the proceedings of the Conference on Algorithmic Number Theory, Turku, May 8-11, 2007. We tried to make the paper as self contained as possible."}
{"category": "Math", "title": "Harder-Narasimhan Filtrations and K-Groups of an Elliptic Curve", "abstract": "Let $X$ be an elliptic curve over an algebraically closed field. We prove that some exact sub-categories of the category of vector bundles over $X$, defined using Harder-Narasimhan filtrations, have the same K-groups as the whole category."}
{"category": "Math", "title": "On the Vertices of Indecomposable Modules Over Dihedral 2-Groups", "abstract": "Let $k$ be an algebraically closed field of characteristic 2. We compute the vertices of all indecomposable $kD_8$-modules for the dihedral group $D_8$ of order 8. We also give a conjectural formula of the induced module of a string module from $kT_0$ to $kG$ where $G$ is a dihedral group of order $\\geq 8$ and where $T_0$ is a dihedral subgroup of index 2 of $G$. Some cases where we verified this formula are given."}
{"category": "Math", "title": "Non-Regular Likelihood Inference for Seasonally Persistent Processes", "abstract": "The estimation of parameters in the frequency spectrum of a seasonally persistent stationary stochastic process is addressed. For seasonal persistence associated with a pole in the spectrum located away from frequency zero, a new Whittle-type likelihood is developed that explicitly acknowledges the location of the pole. This Whittle likelihood is a large sample approximation to the distribution of the periodogram over a chosen grid of frequencies, and constitutes an approximation to the time-domain likelihood of the data, via the linear transformation of an inverse discrete Fourier transform combined with a demodulation. The new likelihood is straightforward to compute, and as will be demonstrated has good, yet non-standard, properties. The asymptotic behaviour of the proposed likelihood estimators is studied; in particular, $N$-consistency of the estimator of the spectral pole location is established. Large finite sample and asymptotic distributions of the score and observed Fisher information are given, and the corresponding distributions of the maximum likelihood estimators are deduced. A study of the small sample properties of the likelihood approximation is provided, and its superior performance to previously suggested methods is shown, as well as agreement with the developed distributional approximations."}
{"category": "Math", "title": "Lens spaces given from L-space homology 3-spheres", "abstract": "We consider the problem when lens spaces are given from homology spheres, and demonstrate that many lens spaces are obtained from L-space homology sphere which the Ozsv\\'ath Szab\\'o's correction term $d(Y)$ is equal to 2. We show an inequality of slope and genus when $Y$ is L-space and $Y_p(K)$ is lens space."}
{"category": "Math", "title": "Realizability and exceptionality of candidate surface branched covers: methods and results", "abstract": "Given two closed orientable surfaces, the Hurwitz existence problem asks whether there exists a branched cover between them having prescribed global degree and local degrees over the branching points. The Riemann-Hurwitz formula gives a necessary condition, which was shown to be also sufficient when the base surface has positive genus. For the sphere one knows that for some data the cover exists and for some it does not, but the problem is still open in general. In this paper we will review five different techniques recently employed to attack it, and we will state the main results they have led to. To illustrate the techniques we will give five independent proofs of the fact that there is no branched cover of the sphere over itself with degree 4, three branching points, and local degrees (2,2), (2,2), and (3,1) over them (despite the fact that the Riemann-Hurwitz formula is satisfied)."}
{"category": "Math", "title": "The solution of the Minkowski problem for closed surfaces in Riemannian space", "abstract": "Author of this article created for the first time the method for finding solutions of the Minkowski problem for closed surfaces in Riemannian space."}
{"category": "Math", "title": "The Christoffel problem and two analogs of the Minkowski problem in Riemannian space", "abstract": "Author finds the solutions of the Christoffel problem for open and closed surfaces in Riemannian space. The Christoffel problem is reduced to the problem of construction the continuous G-deformations preserving the sum of principal radii of curvature for every point of surface in Riemannian space. G-deformation transfers every normal vector of surface in parallel along the path of the translation for each point of surface. The following analogs of the Minkowski problem for open and closed surfaces in Riemannian space are being considered in this article: 1) the problem of construction the surface with prescribed mean curvature and condition of G-deformation; 2) the problem of construction the deformations preserving the area of each arbitrary region of surface and condition of G-deformation."}
{"category": "Math", "title": "Of mice and men: Sparse statistical modeling in cardiovascular genomics", "abstract": "In high-throughput genomics, large-scale designed experiments are becoming common, and analysis approaches based on highly multivariate regression and anova concepts are key tools. Shrinkage models of one form or another can provide comprehensive approaches to the problems of simultaneous inference that involve implicit multiple comparisons over the many, many parameters representing effects of design factors and covariates. We use such approaches here in a study of cardiovascular genomics. The primary experimental context concerns a carefully designed, and rich, gene expression study focused on gene-environment interactions, with the goals of identifying genes implicated in connection with disease states and known risk factors, and in generating expression signatures as proxies for such risk factors. A coupled exploratory analysis investigates cross-species extrapolation of gene expression signatures--how these mouse-model signatures translate to humans. The latter involves exploration of sparse latent factor analysis of human observational data and of how it relates to projected risk signatures derived in the animal models. The study also highlights a range of applied statistical and genomic data analysis issues, including model specification, computational questions and model-based correction of experimental artifacts in DNA microarray data."}
{"category": "Math", "title": "Pairings on Jacobians of Hyperelliptic Curves", "abstract": "Consider the jacobian of a hyperelliptic genus two curve defined over a finite field. Under certain restrictions on the endomorphism ring of the jacobian we give an explicit description all non-degenerate, bilinear, anti-symmetric and Galois-invariant pairings on the jacobian. From this description it follows that no such pairing can be computed more efficiently than the Weil pairing. To establish this result, we need an explicit description of the representation of the Frobenius endomorphism on the l-torsion subgroup of the jacobian. This description is given. In particular, we show that if the characteristic polynomial of the Frobenius endomorphism splits into linear factors modulo l, then the Frobenius is diagonalizable. Finally, under the restriction that the Frobenius element is an element of a certain subring of the endomorphism ring, we prove that if the characteristic polynomial of the Frobenius endomorphism splits into linear factors modulo l, then the embedding degree and the total embedding degree of the jacobian with respect to l are the same number."}
{"category": "Math", "title": "Representations of orbifold groupoids", "abstract": "Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent to the translation groupoid of an action of a bundle of compact topological groups."}
{"category": "Math", "title": "Nahm's equations and free-boundary problems", "abstract": "This paper is a discussion of relations between some free-boundary problems and infinite dimensional Lie groups; particularly a version of Nahm's equations for the group of Hamiltonian diffeomorphisms in two dimensions."}
{"category": "Math", "title": "Some remarks about Cauchy integrals and fractal sets", "abstract": "Some aspects of Cauchy integrals on sets with dimension larger than 1 are briefly discussed."}
{"category": "Math", "title": "A canonical Frobenius structure", "abstract": "We show that it makes sense to speak of THE Frobenius manifold attached to a convenient and nondegenerate Laurent polynomial"}
{"category": "Math", "title": "Finite-dimensional modules for the polynomial ring in one variable as a vertex algebra", "abstract": "A commutative associative algebra $A$ over ${\\mathbb C}$ with a derivation is one of the simplest examples of a vertex algebra. However, the differences between the modules for $A$ as a vertex algebra and the modules for $A$ as an associative algebra are not well understood. In this paper, I give the classification of finite-dimensional indecomposable untwisted or twisted modules for the polynomial ring in one variable over ${\\mathbb C}$ as a vertex algebra."}
{"category": "Math", "title": "On generically stable types in dependent theories", "abstract": "We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent (NIP) theories."}
{"category": "Math", "title": "Kaehlerian reduction in steps", "abstract": "We study Hamiltonian actions of compact Lie groups K on Kaehler manifolds which extend to a holomorphic action of the complexified group K^C. For a closed normal subgroup L of K we show that the Kaehlerian reduction with respect to L is a stratified Hamiltonian Kaehler K^C/L^C-space whose Kaehlerian reduction with respect to K/L is naturally isomorphic to the Kaehlerian reduction of the original manifold with respect to K."}
{"category": "Math", "title": "Modular intersection cohomology complexes on flag varieties", "abstract": "We present a combinatorial procedure (based on the W-graph of the Coxeter group) which shows that the characters of many intersection cohomology complexes on low rank complex flag varieties with coefficients in an arbitrary field are given by Kazhdan-Lusztig basis elements. Our procedure exploits the existence and uniqueness of parity sheaves. In particular we are able to show that the characters of all intersection cohomology complexes with coefficients in a field on the flag variety of type A_n for n < 7 are given by Kazhdan-Lusztig basis elements. By results of Soergel, this implies a part of Lusztig's conjecture for SL(n) with n \\le 7. We also give examples where our techniques fail. In the appendix by Tom Braden examples are given of intersection cohomology complexes on the flag varities for SL(8) and SO(8) which have torsion in their stalks or costalks."}
{"category": "Math", "title": "Large deviation for return times in open sets for axiom A diffeomorphisms", "abstract": "For axiom A diffeomorphisms and equilibrium state, we prove a Large deviation result for the sequence of successive return times into a fixed open set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who where considering cylinder sets of a Markov partition."}
{"category": "Math", "title": "Percolation sur le syst\\`eme \\`a trois points", "abstract": "The three dot system is an action by homeomorphisms of $\\mathbb Z^2$ on a compact space, which invariant measures are supposed to satisfy rigidity properties analoguous to the ones of invariant measures of angle doubling and tripling on the circle. For such measures, we establish a dichotomy which is related to percolation properties."}
{"category": "Math", "title": "Analyse harmonique sur le graphe de Pascal", "abstract": "We prove a spectral decomposition theorem for a well-known self-similar graph, for some finite graphs which are quotients of this graph and for a compactification of it."}
{"category": "Math", "title": "The sharp $A_p$ constant for weights in a reverse-H\\\"older class", "abstract": "In a recent paper V. Vasyunin presented a proof of the reverse H\\\"older inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function technique to find the sharp $A_p$ constants for weights in a reverse-H\\\"older class on an interval; we also find the sharp constants for the higher-integrability result of Gehring. Additionally, we find bounds for the $A_p$ constants of reverse-H\\\"older-class weights defined on rectangles and on cubes in n dimensions."}
{"category": "Math", "title": "Homotopy theoretic models of identity types", "abstract": "This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of Martin-Loef type theory."}
{"category": "Math", "title": "Asymptotic analysis of the Bell polynomials by the ray method", "abstract": "We analyze the Bell polynomials $B_{n}(x)$ asymptotically as $n\\to\\infty$. We obtain asymptotic approximations from the differential-difference equation which they satisfy, using a discrete version of the ray method. We give some examples showing the accuracy of our formulas."}
{"category": "Math", "title": "Networks of Polynomial Pieces with Application to the Analysis of Point Clouds and Images", "abstract": "We consider Holder smoothness classes of surfaces for which we construct piecewise polynomial approximation networks, which are graphs with polynomial pieces as nodes and edges between polynomial pieces that are in `good continuation' of each other. Little known to the community, a similar construction was used by Kolmogorov and Tikhomirov in their proof of their celebrated entropy results for Holder classes. We show how to use such networks in the context of detecting geometric objects buried in noise to approximate the scan statistic, yielding an optimization problem akin to the Traveling Salesman. In the same context, we describe an alternative approach based on computing the longest path in the network after appropriate thresholding. For the special case of curves, we also formalize the notion of `good continuation' between beamlets in any dimension, obtaining more economical piecewise linear approximation networks for curves. We include some numerical experiments illustrating the use of the beamlet network in characterizing the filamentarity content of 3D datasets, and show that even a rudimentary notion of good continuity may bring substantial improvement."}
{"category": "Math", "title": "Strange non-chaotic attractors in quasiperiodically forced circle maps", "abstract": "The occurrence of strange non-chaotic attractors (SNA) in quasiperiodically forced systems has attracted considerable interest over the last two decades, in particular since it provides a rich class of examples for the possibility of complicated dynamics in the absence of chaos. Their existence was discovered in the early 1980's, independently by Herman for quasiperiodic SL(2,R)-cocycles and by Grebogi et al for so-called 'pinched skew products'. However, except for these two particular classes there are still hardly any rigorous results on the topic, despite a large number of numerical studies which all confirmed the widespread existence of SNA in quasiperiodically forced systems. Here, we prove the existence of SNA in quasiperiodically forced circle maps under rather general conditions, which can be stated in terms of C 1 -estimates. As a consequence, we obtain the existence of strange non-chaotic attractors for parameter sets of positive measure in suitable parameter families. Further, we show that the considered systems have minimal dynamics. The results apply in particular to a forced version of the Arnold circle map. For this particular example, we also describe how the first Arnold tongue collapses and looses its regularity due to the presence of strange non-chaotic attractors and a related unbounded mean motion property."}
{"category": "Math", "title": "Strong Law of Large Numbers for branching diffusions", "abstract": "Let $X$ be the branching particle diffusion corresponding to the operator $Lu+\\beta (u^{2}-u)$ on $D\\subseteq \\mathbb{R}^{d}$ (where $\\beta \\geq 0$ and $\\beta\\not\\equiv 0$). Let $\\lambda_{c}$ denote the generalized principal eigenvalue for the operator $L+\\beta $ on $D$ and assume that it is finite. When $\\lambda_{c}>0$ and $L+\\beta-\\lambda_{c}$ satisfies certain spectral theoretical conditions, we prove that the random measure $\\exp \\{-\\lambda_{c}t\\}X_{t}$ converges almost surely in the vague topology as $t$ tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions of \\cite{ET,EW}. We extend significantly the results in \\cite{AH76,AH77} and include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and `spine' decompositions or `immortal particle pictures'."}
{"category": "Math", "title": "Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below", "abstract": "We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motion in the extrinsic balls. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above."}
{"category": "Math", "title": "Phylogenetic networks form partial trees", "abstract": "A contemporary and fundamental problem faced by many evolutionary biologists is how to puzzle together a collection $\\mathcal P$ of partial trees (leaf-labelled trees whose leaves are bijectively labelled by species or, more generally, taxa, each supported by e. g. a gene) into an overall parental structure that displays all trees in $\\mathcal P$. This already difficult problem is complicated by the fact that the trees in $\\mathcal P$ regularly support conflicting phylogenetic relationships and are not on the same but only overlapping taxa sets. A desirable requirement on the sought after parental structure therefore is that it can accommodate the observed conflicts. Phylogenetic networks are a popular tool capable of doing precisely this. However, not much is known about how to construct such networks from partial trees, a notable exception being the $Z$-closure super-network approach and the recently introduced $Q$-imputation approach. Here, we propose the usage of closure rules to obtain such a network. In particular, we introduce the novel $Y$-closure rule and show that this rule on its own or in combination with one of Meacham's closure rules (which we call the $M$-rule) has some very desirable theoretical properties. In addition, we use the $M$- and $Y$-rule to explore the dependency of Rivera et al.'s ``ring of life'' on the fact that the underpinning phylogenetic trees are all on the same data set. Our analysis culminates in the presentation of a collection of induced subtrees from which this ring can be reconstructed."}
{"category": "Math", "title": "Oort groups and lifting problems", "abstract": "Let k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A_4 in characteristic 2. This proves one direction of a strong form of the Oort Conjecture."}
{"category": "Math", "title": "The Dimension of the Torelli group", "abstract": "We prove that the cohomological dimension of the Torelli group for a closed connected orientable surface of genus g at least 2 is equal to 3g-5. This answers a question of Mess, who proved the lower bound and settled the case of g=2. We also find the cohomological dimension of the Johnson kernel (the subgroup of the Torelli group generated by Dehn twists about separating curves) to be 2g-3. For g at least 2, we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli group in genus 2. The main tool is a new contractible complex, called the \"complex of cycles\", on which the Torelli group acts."}
{"category": "Math", "title": "$\\mathbb{Z}_{2}^{2}$-cordiality of complete and complete bipartite graphs", "abstract": "We prove that $K_{n}$ is $\\mathbb{Z}_{2}^{2}$-cordial if and only if $1 \\leq n \\leq 3$ and that $K_{m,n}$ is $\\mathbb{Z}_{2}^{2}$ if and only if it is false that $m=n=2$."}
{"category": "Math", "title": "Equivalences on Acyclic Orientations", "abstract": "The cyclic and dihedral groups can be made to act on the set Acyc(Y) of acyclic orientations of an undirected graph Y, and this gives rise to the equivalence relations ~kappa and ~delta, respectively. These two actions and their corresponding equivalence classes are closely related to combinatorial problems arising in the context of Coxeter groups, sequential dynamical systems, the chip-firing game, and representations of quivers. In this paper we construct the graphs C(Y) and D(Y) with vertex sets Acyc(Y) and whose connected components encode the equivalence classes. The number of connected components of these graphs are denoted kappa(Y) and delta(Y), respectively. We characterize the structure of C(Y) and D(Y), show how delta(Y) can be derived from kappa(Y), and give enumeration results for kappa(Y). Moreover, we show how to associate a poset structure to each kappa-equivalence class, and we characterize these posets. This allows us to create a bijection from Acyc(Y)/~kappa to the union of Acyc(Y')/~kappa and Acyc(Y'')/~kappa, Y' and Y'' denote edge deletion and edge contraction for a cycle-edge in Y, respectively, which in turn shows that kappa(Y) may be obtained by an evaluation of the Tutte polynomial at (1,0)."}
{"category": "Math", "title": "Somewhat stochastic matrices", "abstract": "The standard theorem for regular stochastic matrices is generalized to matrices with no sign restriction on the entries. The condition that column sums be equal to 1 is kept, but the regularity condition is replaced by a condition on the $\\ell_1$-distances between columns."}
{"category": "Math", "title": "On the uniformity of the Iitaka fibration", "abstract": "We study pluricanonical systems on smooth projective varieties of positive Kodaira dimension, following the approach of Hacon-McKernan, Takayama and Tsuji succesfully used in the case of varieties of general type. We prove a uniformity result for the Iitaka fibration of smooth projective varieties of positive Kodaira dimension, provided that the base of the Iitaka fibration is not uniruled, the variation of the fibration is maximal, and the generic fiber has a good minimal model."}
{"category": "Math", "title": "Lower bounds for the volume of hyperbolic $n$-orbifolds", "abstract": "In this paper an explicit formula for a lower bound on the volume of a hyperbolic orbifold, dependent on dimension and the maximal order of torsion in the orbifolds' fundamental group, is constructed."}
{"category": "Math", "title": "Geodesic cusp excursions and metric diophantine approximation", "abstract": "We derive several results that describe the rate at which a generic geodesic makes excursions into and out of a cusp on a finite area hyperbolic surface and relate them to approximation with respect to the orbit of infinity for an associated Fuchsian group. This provides proofs of some well known theorems from metric diophantine approximation in the context of Fuchsian groups. It also gives new results in the classical setting."}
{"category": "Math", "title": "Ore extensions of principally quasi-Baer rings", "abstract": "Let $R$ be a ring and $(\\sigma,\\delta)$ a quasi-derivation of $R$. In this paper, we show that if $R$ is an $(\\sigma,\\delta)$-skew Armendariz ring and satisfies the condition $(\\mathcal{C_{\\sigma}})$, then $R$ is right p.q.-Baer if and only if the Ore extension $R[x;\\sigma,\\delta]$ is right p.q.-Baer. As a consequence we obtain a generalization of \\cite{hong/2000}."}
{"category": "Math", "title": "Monomial bases for the centres of the group algebra and Iwahori--Hecke algebra of S_4", "abstract": "G. E. Murphy showed in 1983 that the centre of every symmetric group algebra has an integral basis consisting of a specific set of monomial symmetric polynomials in the Jucys--Murphy elements. While we have shown in earlier work that the centre of the group algebra of S_3 has exactly three additional such bases, we show in this paper that the centre of the group algebra of S_4 has infinitely many bases consisting of monomial symmetric polynomials in Jucys--Murphy elements, which we characterize completely. The proof of this result involves establishing closed forms for coefficients of class sums in the monomial symmetric polynomials in Jucys--Murphy elements, and solving several resulting exponential Diophantine equations with the aid of a computer. Our initial motivation was in finding integral bases for the centre of the Iwahori--Hecke algebra, and we address this question also, by finding several integral bases of monomial symmetric polynomials in Jucys--Murphy elements for the centre of the Iwahori--Hecke algebra of S_4."}
{"category": "Math", "title": "Effective base point free theorem for log canonical pairs--Koll\\'ar type theorem", "abstract": "We prove Koll\\'ar's effective base point free theorem for log canonical pairs."}
{"category": "Math", "title": "Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency", "abstract": "We study nonparametric maximum likelihood estimation of a log-concave probability density and its distribution and hazard function. Some general properties of these estimators are derived from two characterizations. It is shown that the rate of convergence with respect to supremum norm on a compact interval for the density and hazard rate estimator is at least $(\\log(n)/n)^{1/3}$ and typically $(\\log(n)/n)^{2/5}$, whereas the difference between the empirical and estimated distribution function vanishes with rate $o_{\\mathrm{p}}(n^{-1/2})$ under certain regularity assumptions."}
{"category": "Math", "title": "Effective base point free theorem for log canonical pairs II--Angehrn--Siu type theorems--", "abstract": "We prove Angehrn-Siu type effective base point freeness and point separation for log canonical pairs."}
{"category": "Math", "title": "Branson's Q-curvature in Riemannian and Spin Geometry", "abstract": "On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's Q-curvature. On a closed n-dimensional manifold, $n\\ge 5$, we compare the three basic conformally covariant operators : the Branson-Paneitz, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. Equality cases are also characterized."}
{"category": "Math", "title": "Resonances and O-curves in Hamiltonian systems", "abstract": "We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances."}
{"category": "Math", "title": "Cobordism invariants of fold maps", "abstract": "This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed normal bundles, detecting stable homotopy groups of spheres as direct summands of the cobordism semigroups of fold maps, Pontryagin-Thom type construction for -1 codimensional fold maps and estimations about the cobordism classes of manifolds which have fold maps into stably parallelizable manifolds. In the last section some of these results are extended and we show that our invariants also detect stable homotopy groups of the classifying spaces $BO(k)$ as direct summands of the cobordism semigroups of fold maps."}
{"category": "Math", "title": "A Problem in Categories", "abstract": "The problem is posed to find out for arbitrary nonvoid sets $X$ which are all the mappings $T : X \\longrightarrow X$ that can be defined and each separately identified through means of categories alone. As argued, this problem may have a certain foundational relevance."}
{"category": "Math", "title": "Control of the mean number of false discoveries, Bonferroni and stability of multiple testing", "abstract": "The Bonferroni multiple testing procedure is commonly perceived as being overly conservative in large-scale simultaneous testing situations such as those that arise in microarray data analysis. The objective of the present study is to show that this popular belief is due to overly stringent requirements that are typically imposed on the procedure rather than to its conservative nature. To get over its notorious conservatism, we advocate using the Bonferroni selection rule as a procedure that controls the per family error rate (PFER). The present paper reports the first study of stability properties of the Bonferroni and Benjamini--Hochberg procedures. The Bonferroni procedure shows a superior stability in terms of the variance of both the number of true discoveries and the total number of discoveries, a property that is especially important in the presence of correlations between individual $p$-values. Its stability and the ability to provide strong control of the PFER make the Bonferroni procedure an attractive choice in microarray studies."}
{"category": "Math", "title": "A long exact sequence in cohomology for deleted and restricted subspaces arrangements", "abstract": "The notions of deleted and restricted arrangements have been useful in the study of arrangements of hyperplanes. If A is an arrangement of hyperplanes, x in A and A', A'' the deleted and restricted arrangements, there is a formula connecting the Poincare polynomials of the complement spaces M(A), M(A') and M(A''). In this paper, we consider the extension of this formula to arbitrary subspaces arrangements. The main result is the existence of a long exact sequence connecting the rational cohomology of M(A), M(A') and M(A''). Using this sequence, we obtain new results connecting the Betti numbers and Poincare polynomials of deleted and restricted arrangements."}
{"category": "Math", "title": "An algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary", "abstract": "We provide an algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary. More generally, we supply an algorithm to determine (up to ambient isotopy) all the Heegaard splittings of any given genus for the manifold. As a consequence, the tunnel number of a hyperbolic link is algorithmically computable. Our techniques rely on Rubinstein's work on almost normal surfaces, and also a new structure called a partially flat angled ideal triangulation."}
{"category": "Math", "title": "Random walk local time approximated by a Wiener sheet combined with an independent Brownian motion", "abstract": "Let $\\xi(k,n)$ be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process $\\xi(k,n)-\\xi(0,n)$ in terms of a Wiener sheet and an independent Wiener process, time changed by an independent Brownian local time. Some related results and consequences are also established."}
{"category": "Math", "title": "Sobolev spaces and mappings with bounded (P,Q)-distortion on Carnot groups", "abstract": "We study mappings with bounded (p,q)-distortion associated to Sobolev spaces on Carnot groups. Mappings of such type have applications to the Sobolev type embedding theory and classification of manifolds. For this class of mappings, we obtain estimates of linear distortion, and a geometrical description. We prove also Liouville type theorems and give some sufficient conditions for removability of sets."}
{"category": "Math", "title": "Finite area and volume of pointed $k$-surfaces", "abstract": "We define the ``volume'' contained by pointed $k$-surfaces, first studied by the author in [9], and we show that this volume is always finite. Likewise, we show that the surface area of a pointed $k$-surface is always finite."}
{"category": "Math", "title": "Spatial variation of total column ozone on a global scale", "abstract": "The spatial dependence of total column ozone varies strongly with latitude, so that homogeneous models (invariant to all rotations) are clearly unsuitable. However, an assumption of axial symmetry, which means that the process model is invariant to rotations about the Earth's axis, is much more plausible and considerably simplifies the modeling. Using TOMS (Total Ozone Mapping Spectrometer) measurements of total column ozone over a six-day period, this work investigates the modeling of axially symmetric processes on the sphere using expansions in spherical harmonics. It turns out that one can capture many of the large scale features of the spatial covariance structure using a relatively small number of terms in such an expansion, but the resulting fitted model provides a horrible fit to the data when evaluated via its likelihood because of its inability to describe accurately the process's local behavior. Thus, there remains the challenge of developing computationally tractable models that capture both the large and small scale structure of these data."}
{"category": "Math", "title": "Approximation via regularization of the local time of semimartingales and Brownian motion", "abstract": "Through a regularization procedure, few approximation schemes of the local time of a large class of one dimensional processes are given. We mainly consider the local time of continuous semimartingales and reversible diffusions, and the convergence holds in ucp sense. In the case of standard Brownian motion, we have been able to determine a rate of convergence in $L^2$, and a.s. convergence of some of our schemes."}
{"category": "Math", "title": "A resampling-based test to detect person-to-person transmission of infectious disease", "abstract": "Early detection of person-to-person transmission of emerging infectious diseases such as avian influenza is crucial for containing pandemics. We developed a simple permutation test and its refined version for this purpose. A simulation study shows that the refined permutation test is as powerful as or outcompetes the conventional test built on asymptotic theory, especially when the sample size is small. In addition, our resampling methods can be applied to a broad range of problems where an asymptotic test is not available or fails. We also found that decent statistical power could be attained with just a small number of cases, if the disease is moderately transmissible between humans."}
{"category": "Math", "title": "Existence of Positive Solutions for Non Local p-Laplacian Thermistor Problems on Time Scales", "abstract": "We make use of the Guo-Krasnoselskii fixed point theorem on cones to prove existence of positive solutions to a non local p-Laplacian boundary value problem on time scales arising in many applications."}
{"category": "Math", "title": "Freiheitss\\\"{a}tze for one-relator quotients of surface groups and of limit groups", "abstract": "Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published corresponding statements for one-relator quotients of orientable surface groups. Two of the proofs are new even in that restricted context."}
{"category": "Math", "title": "SPM Bulletin 22", "abstract": "Contents: 2. Invited contribution: Ultrafilters and small sets 3. Research announcements 3.1. Inverse Systems and I-Favorable Spaces 3.2. Combinatorial and hybrid principles for sigma-directed families of countable sets modulo finite 3.3. A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension 3.4. A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension 3.5. Large continuum, oracles 3.6. Borel hierarchies in infinite products of Polish spaces 3.7. A game for the Borel functions 3.8. On some problems in general topology 4. Problem of the Issue"}
{"category": "Math", "title": "Probabilistic projections of HIV prevalence using Bayesian melding", "abstract": "The Joint United Nations Programme on HIV/AIDS (UNAIDS) has developed the Estimation and Projection Package (EPP) for making national estimates and short-term projections of HIV prevalence based on observed prevalence trends at antenatal clinics. Assessing the uncertainty about its estimates and projections is important for informed policy decision making, and we propose the use of Bayesian melding for this purpose. Prevalence data and other information about the EPP model's input parameters are used to derive a probabilistic HIV prevalence projection, namely a probability distribution over a set of future prevalence trajectories. We relate antenatal clinic prevalence to population prevalence and account for variability between clinics using a random effects model. Predictive intervals for clinic prevalence are derived for checking the model. We discuss predictions given by the EPP model and the results of the Bayesian melding procedure for Uganda, where prevalence peaked at around 28% in 1990; the 95% prediction interval for 2010 ranges from 2% to 7%."}
{"category": "Math", "title": "Asymptotics of eigenvalues of Sturm--Liouville problem with discrete self-similar weight", "abstract": "The Sturm--Liouville problem $-y''-\\lambda\\rho y=0$, $y(0)=y(1)=0$, where $\\rho$ is a generalized derivative of self-similar function $P\\in L_2[0,1]$ with spectral degree D=0, is studied. Asymptotic formulas for eigenvalues are obtained."}
{"category": "Math", "title": "A multivariate semiparametric Bayesian spatial modeling framework for hurricane surface wind fields", "abstract": "Storm surge, the onshore rush of sea water caused by the high winds and low pressure associated with a hurricane, can compound the effects of inland flooding caused by rainfall, leading to loss of property and loss of life for residents of coastal areas. Numerical ocean models are essential for creating storm surge forecasts for coastal areas. These models are driven primarily by the surface wind forcings. Currently, the gridded wind fields used by ocean models are specified by deterministic formulas that are based on the central pressure and location of the storm center. While these equations incorporate important physical knowledge about the structure of hurricane surface wind fields, they cannot always capture the asymmetric and dynamic nature of a hurricane. A new Bayesian multivariate spatial statistical modeling framework is introduced combining data with physical knowledge about the wind fields to improve the estimation of the wind vectors. Many spatial models assume the data follow a Gaussian distribution. However, this may be overly-restrictive for wind fields data which often display erratic behavior, such as sudden changes in time or space. In this paper we develop a semiparametric multivariate spatial model for these data. Our model builds on the stick-breaking prior, which is frequently used in Bayesian modeling to capture uncertainty in the parametric form of an outcome. The stick-breaking prior is extended to the spatial setting by assigning each location a different, unknown distribution, and smoothing the distributions in space with a series of kernel functions. This semiparametric spatial model is shown to improve prediction compared to usual Bayesian Kriging methods for the wind field of Hurricane Ivan."}
{"category": "Math", "title": "Enumeration of paths and cycles and e-coefficients of incomparability graphs", "abstract": "We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in terms of elementary symmetric functions. Analysis of some of the combinatorial implications of this expansion leads to three bijections involving acyclic orientations."}
{"category": "Math", "title": "Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form", "abstract": "Let $M$ be a real hypersurface of a complex space form with almost contact metric structure $(\\phi, \\xi, \\eta, g)$. In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_\\xi=R(\\cdot,\\xi)\\xi$ is $\\xi$-parallel. In particular, we prove that the condition $\\nabla_{\\xi} R_{\\xi}=0$ characterizes the homogeneous real hypersurfaces of type $A$ in a complex projective space or a complex hyperbolic space when $R_{\\xi} \\phi S=S \\phi R_{\\xi}$ holds on $M$, where $S$ denotes the Ricci tensor of type (1,1) on $M$."}
{"category": "Math", "title": "Assembly Maps for Group Extensions in $K$-Theory and $L$-Theory with Twisted Coefficients", "abstract": "In this paper we show that the Farrell-Jones isomorphism conjectures are inherited in group extensions for assembly maps in algebraic $K$-theory and $L$-theory with twisted coefficients."}
{"category": "Math", "title": "Log Minimal Model Program for the Kontsevich Space of Stable Maps $\\bar{\\mathcal M}_{0,0}(\\mathbb P^{3}, 3)$", "abstract": "This work is inspired by conversations with Izzet Coskun and Joe Harris. We run the log minimal model program for the Kontsevich space of stable maps $\\bar{\\mathcal M}_{0,0}(\\mathbb P^{3}, 3)$ and give modular interpretations to all the intermediate spaces appearing in the process. In particular, we show that one component of the Hilbert scheme $\\mathcal H_{3,0,3}$ is the flip of $\\bar{\\mathcal M}_{0,0}(\\mathbb P^{3}, 3)$ over the Chow variety. Finally as an easy corollary we obtain that $\\bar{\\mathcal M}_{0,0}(\\mathbb P^{3}, 3)$ is a Mori dream space."}
{"category": "Math", "title": "Local mixture models of exponential families", "abstract": "Exponential families are the workhorses of parametric modelling theory. One reason for their popularity is their associated inference theory, which is very clean, both from a theoretical and a computational point of view. One way in which this set of tools can be enriched in a natural and interpretable way is through mixing. This paper develops and applies the idea of local mixture modelling to exponential families. It shows that the highly interpretable and flexible models which result have enough structure to retain the attractive inferential properties of exponential families. In particular, results on identification, parameter orthogonality and log-concavity of the likelihood are proved."}
{"category": "Math", "title": "When is Eaton's Markov chain irreducible?", "abstract": "Consider a parametric statistical model $P(\\mathrm{d}x|\\theta)$ and an improper prior distribution $\\nu(\\mathrm{d}\\theta)$ that together yield a (proper) formal posterior distribution $Q(\\mathrm{d}\\theta|x)$. The prior is called strongly admissible if the generalized Bayes estimator of every bounded function of $\\theta$ is admissible under squared error loss. Eaton [Ann. Statist. 20 (1992) 1147--1179] has shown that a sufficient condition for strong admissibility of $\\nu$ is the local recurrence of the Markov chain whose transition function is $R(\\theta,\\mathrm{d}\\eta)=\\int Q(\\mathrm{d}\\eta|x)P(\\mathrm {d}x|\\theta)$. Applications of this result and its extensions are often greatly simplified when the Markov chain associated with $R$ is irreducible. However, establishing irreducibility can be difficult. In this paper, we provide a characterization of irreducibility for general state space Markov chains and use this characterization to develop an easily checked, necessary and sufficient condition for irreducibility of Eaton's Markov chain. All that is required to check this condition is a simple examination of $P$ and $\\nu$. Application of the main result is illustrated using two examples."}
{"category": "Math", "title": "Conformal dual of a quadruplet of points", "abstract": "Let $\\{P_1, P_2, P_3, P_4\\}$ be a quadruplet of points in $S^3$ . We define a ``dual'' quadruplet of it in a conformal geometric way. We show that the dual of a dual quadruplet coincides with the original one. We also show that the cross ratio of the dual quadruplet is equal to the complex conjugate of that of the original one."}
{"category": "Math", "title": "Sur les fonctions \\`a singularit\\'e de dimension 1", "abstract": "In this article we show that all results proved for a large class of holomorphic germs $f : (\\mathbb{C}^{n+1}, 0) \\to (\\mathbb{C}, 0)$ with a 1-dimension singularity in [B.II] are valid for an arbitrary such germ."}
{"category": "Math", "title": "Lie-algebra Dolbeault cohomology and small deformations of nilmanifolds", "abstract": "We consider nilmanifolds with left-invariant complex structure and prove that small deformations of such structures are again left invariant if the Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant forms. By a result of Console and Fino this is generically the case. Our main tool is an analog of Dolbeault-cohomology for Lie-algebras with complex structure."}
{"category": "Math", "title": "The Fr\\\"olicher spectral sequence can be arbitrarily non degenerate", "abstract": "The Fr\\\"olicher spectral sequence of a compact complex manifold $X$ measures the difference between Dolbeault cohomology and de Rham cohomology. We construct for $n\\geq 2$ nilmanifolds with left-invariant complex structure $X_n$ such that the $n$-th differential $d_n$ does not vanish. This replaces an earlier incorrect example by the second author."}
{"category": "Math", "title": "Springer correspondences for dihedral groups", "abstract": "Recent work by a number of people has shown that complex reflection groups give rise to many representation-theoretic structures (e.g., generic degrees and families of characters), as though they were Weyl groups of algebraic groups. Conjecturally, these structures are actually describing the representation theory of as-yet undescribed objects called ''spetses'', of which reductive algebraic groups ought to be a special case. In this paper, we carry out the Lusztig--Shoji algorithm for calculating Green functions for the dihedral groups. With a suitable set-up, the output of this algorithm turns out to satisfy all the integrality and positivity conditions that hold in the Weyl group case, so we may think of it as describing the geometry of the ''unipotent variety'' associated to a spets. From this, we determine the possible ''Springer correspondences'', and we show that, as is true for algebraic groups, each special piece is rationally smooth, as is the full unipotent variety."}
{"category": "Math", "title": "Enumeration formulas for Young tableaux in a diagonal strip", "abstract": "We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. Andre on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of a Schur function."}
{"category": "Math", "title": "Holomorphic harmonic analysis on complex reductive groups", "abstract": "We define the holomorphic Fourier transform of holomorphic functions on complex reductive groups, prove some properties like the Fourier inversion formula, and give some applications. The definition of the holomorphic Fourier transform makes use of the notion of $K$-admissible measures. We prove that $K$-admissible measures are abundant, and the definition of holomorphic Fourier transform is independent of the choice of $K$-admissible measures."}
{"category": "Math", "title": "Sobolev Inequalities, Riesz Transforms and the Ricci Flow", "abstract": "In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the $W^{1,2}$ Sobolev inequality along the Ricci flow to $W^{1,p}$ and $W^{2,p}$ Sobolev inequalities for general p."}
{"category": "Math", "title": "Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities", "abstract": "We construct six unitary trace invariants for 2 by 2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We prove two quaternionic versions of a well known characterization of triangularizable subalgebras of matrix algebras over an algebraically closed field. Finally we consider the problem of describing the semi-algebraic set of pairs (X,Y) of quaternionic n by n matrices which are simultaneously triangularizable. Even the case n=2, which we analyze in more detail, remains unsolved."}
{"category": "Math", "title": "Ore Extensions of Extended Symmetric and Reversible Rings", "abstract": "Let $\\sigma$ be an endomorphism and $\\delta$ an $\\sigma$-derivation of a ring $R$. In this paper, we show that if $R$ is $(\\sigma,\\delta)$-skew Armendariz and $a\\sigma(b)=0$ implies $ab=0$ for $a,b\\in R$. Then $R$ is symmetric (respectively, reversible) if and only if $R$ is $\\sigma$-symmetric (respectively, $\\sigma$-reversible) if and only if $R[x;\\sigma,\\delta]$ is symmetric (respectively, reversible). Moreover, we study on the relationship between the Baerness, quasi-Baerness and p.q.-Baerness of a ring $R$ and these of the Ore extension $R[x;\\sigma,\\delta]$. As a consequence we obtain a partial generalization of \\cite{hong/2000}."}
{"category": "Math", "title": "On the linear independence of spikes and sines", "abstract": "The purpose of this work is to survey what is known about the linear independence of spikes and sines. The paper provides new results for the case where the locations of the spikes and the frequencies of the sines are chosen at random. This problem is equivalent to studying the spectral norm of a random submatrix drawn from the discrete Fourier transform matrix. The proof involves depends on an extrapolation argument of Bourgain and Tzafriri."}
{"category": "Math", "title": "Constructing packings in Grassmannian manifolds via alternating projection", "abstract": "This paper describes a numerical method for finding good packings in Grassmannian manifolds equipped with various metrics. This investigation also encompasses packing in projective spaces. In each case, producing a good packing is equivalent to constructing a matrix that has certain structural and spectral properties. By alternately enforcing the structural condition and then the spectral condition, it is often possible to reach a matrix that satisfies both. One may then extract a packing from this matrix. This approach is both powerful and versatile. In cases where experiments have been performed, the alternating projection method yields packings that compete with the best packings recorded. It also extends to problems that have not been studied numerically. For example, it can be used to produce packings of subspaces in real and complex Grassmannian spaces equipped with the Fubini--Study distance; these packings are valuable in wireless communications. One can prove that some of the novel configurations constructed by the algorithm have packing diameters that are nearly optimal."}
{"category": "Math", "title": "Vector fields and foliations associated to groups of projective automorphisms", "abstract": "We introduce and give normal forms for (one-dimensional) Riccati foliations (vector fields) on $\\ov \\bc \\times \\bc P(2)$ and $\\ov \\bc \\times \\ov \\bc^n$. These are foliations are characterized by transversality with the generic fiber of the first projection and we prove they are conjugate {\\em in some invariant Zariski open subset} to the suspension of a group of automorphisms of the fiber, $\\bc P(2)$ or $\\ov \\bc^n$, this group called {\\it global holonomy}. Our main result states that given a finitely generated subgroup $G$ of $\\Aut(\\bc P (2))$, there is a Riccati foliation on $\\ov \\bc \\times \\bc P(2)$ for which the global holonomy is conjugate to $G$."}
{"category": "Math", "title": "C^*- Actions on Stein analytic spaces with isolated singularities", "abstract": "Let $V$ be an irreducible complex analytic space of dimension two with normal singularities and $\\vr:\\mathbb{C^*}\\times V\\to V$ a holomorphic action of the group $\\mathbb{C^*}$ on $V$. Denote by $\\fa_\\vr$ the foliation on $V$ induced by $\\vr$. The leaves of this foliation are the one-dimensional orbits of $\\vr$. %and its singularities are the fixed points of $\\vr$. We will assume that there exists a \\emph{dicritical} singularity $p\\in V$ for the $\\bc^*$-action, i.e. for some neighborhood $p\\in W\\subset V$ there are infinitely many leaves of $\\mathcal {F}_\\vr|_{W}$ accumulating only at $p$. The closure of such a local leaf is an invariant local analytic curve called a \\emph{separatrix} of $\\mathcal{F}_\\vr$ through $p$. In \\cite{Orlik} Orlik and Wagreich studied the 2-dimensional affine algebraic varieties embedded in $\\mathbb{C}^{n+1}$, with an isolated singularity at the origin, that are invariant by an effective action of the form $\\sigma_Q(t,(z_{0},...,z_{n}))=(t^{q_{0}}z_{0},..., t^{q_{n}}z_{n})$ where $Q=(q_0,...,q_n) \\in\\mathbb N^{n+1}$, i.e. all $q_{i}$ are positive integers. Such actions are called \\emph{good} actions. In particular they classified the algebraic surfaces embedded in $\\mathbb{C}^{3}$ endowed with such an action. It is easy to see that any good action on a surface embedded in $\\mathbb{C}^{n+1}$ has a dicritical singularity at $0\\in\\mathbb{C}^{n+1}$. Conversely, it is the purpose of this paper to show that good actions are the models for analytic $\\mathbb{C^*}$-actions on Stein analytic spaces of dimension two with a dicritical singularity."}
{"category": "Math", "title": "Explicit Connections with SU(2)-Monodromy", "abstract": "The pure braid group \\Gamma of a quadruply-punctured Riemann sphere acts on the SL(2,C)-moduli M of the representation variety of such sphere. The points in M are classified into \\Gamma-orbits. We show that, in this case, the monodromy groups of many explicit solutions to the Riemann-Hilbert problem are subgroups of SU(2). Most of these solutions are examples of representations that have dense images in SU(2), but with finite \\Gamma-orbits in M. These examples relate to explicit immersions of constant mean curvature surfaces."}
{"category": "Math", "title": "Applications of Cutoff Resolvent Estimates to the Wave Equation", "abstract": "We consider solutions to the linear wave equation on non-compact Riemannian manifolds without boundary when the geodesic flow admits a filamentary hyperbolic trapped set. We obtain a polynomial rate of local energy decay with exponent depending only on the dimension."}
{"category": "Math", "title": "General observations on series whose terms proceed as the sines and cosines of multiples of angles", "abstract": "Translated from the Latin original, \"Observationes generales circa series, quarum termini secundum sinus vel cosinus angulorum multiplorum progrediuntur\" (1777). E655 in the Enestrom index. Euler looks at the binomial expansion $(1+x)^n$ for $x=\\cos \\phi+i\\sin\\phi$. He does $n=$ positive and negative integers, and $n={1/2}, -{1/2}$, and particular values of $\\phi$."}
{"category": "Math", "title": "3-bounded property in a triangle-free distance-regular graph", "abstract": "Let $\\Gamma$ denote a distance-regular graph with classical parameters $(D, b, \\alpha, \\beta)$ and $D\\geq 3$. Assume the intersection numbers $a_1=0$ and $a_2\\not=0$. We show $\\Gamma$ is 3-bounded in the sense of the article [D-bounded distance-regular graphs, European Journal of Combinatorics(1997)18, 211-229]."}
{"category": "Math", "title": "Solutions of nonlinear PDEs in the completion of uniform convergence spaces", "abstract": "This paper deals with the solution of large classes of systems of nonlinear partial differential equations (PDEs) in spaces of generalized functions that are constructed as the completion of uniform convergence spaces. The existence result for the mentioned systems of equations are obtained as an application of a basic approximation result, which is formulated entirely in terms of usual real valued functions on open subsets of Euclidean n-space. The structure and regularity properties of the solutions are explained at the hand of suitable results relating to the structure of the completion of uniform convergence spaces that are defined as initial structures. In this regard, we include also a detailed discussion of the completion of initial uniform convergence spaces in general."}
{"category": "Math", "title": "Painlev\\'e scheme", "abstract": "In this note, we review the notion of Painlev\\'e scheme of the sixth Painlev\\'e equation from the viewpoint of accessible singular point and its local index in the Hirzebruch surface of degree two ${\\Sigma_2}$. The key method is Painlev\\'e $\\alpha$-method for each accessible singular point. Giving a Painlev\\'e scheme in the differential system satisfying certain conditions, we can recover the Painlev\\'e VI system with the polynomial Hamiltonian. We also consider the case of the Painlev\\'e V,IV and III systems, respectively. Finally, we study non-linear ordinary differential systems in dimension two with only simple accessible singular $(n+2)$-points in the Hirzebruch surface of degree $n$; ${\\Sigma_n}$. This equation has symmetry of symmetric group of degree $n+2$."}
{"category": "Math", "title": "Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes", "abstract": "Cohen, Guyon, Perrin and Pontier have given assumptions under which the second-order quadratic variations of a Gaussian process converge almost surely to a deterministic limit. In this paper we present two new convergence results about these variations: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we apply these results to identify two-parameter fractional Brownian motion and anisotropic fractional Brownian motion."}
{"category": "Math", "title": "Morita Type Equivalences and Reflexive Algebras", "abstract": "Two unital operator algebras A, B are called Delta-equivalent if there exists an equivalence functor between the categories A-mod and B-mod which \"extends\" to a *-functor implementing an equivalence between the categories A-dmod and B-dmod. Here A-mod denotes the category of normal representations of A and A-dmod denotes the category with the same objects as A-mod and D(A)-module maps (D(A) is the diagonal of A). We prove that any such functor maps completely isometric representations to completely isometric representations, \"respects\" the lattices of the algebras and maps reflexive algebras to reflexive algebras. We present applications to the class of CSL algebras."}
{"category": "Math", "title": "Asymptotic normality for the counting process of weak records and \\delta-records in discrete models", "abstract": "Let $\\{X_n,n\\ge1\\}$ be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call $X_n$ a $\\delta$-record if $X_n>\\max\\{X_1,...,X_{n-1}\\}+\\delta$, where $\\delta$ is an integer constant. We use martingale arguments to show that the counting process of $\\delta$-records among the first $n$ observations, suitably centered and scaled, is asymptotically normally distributed for $\\delta\\ne0$. In particular, taking $\\delta=-1$ we obtain a central limit theorem for the number of weak records."}
{"category": "Math", "title": "Poisson-type deviation inequalities for curved continuous-time Markov chains", "abstract": "In this paper, we present new Poisson-type deviation inequalities for continuous-time Markov chains whose Wasserstein curvature or $\\Gamma$-curvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and yield different deviation bounds. In the case of birth--death processes, we provide some conditions on the transition rates of the associated generator for such curvatures to be bounded below and we extend the deviation inequalities established [An\\'{e}, C. and Ledoux, M. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 (2000) 573--602] for continuous-time random walks, seen as models in null curvature. Some applications of these tail estimates are given for Brownian-driven Ornstein--Uhlenbeck processes and $M/M/1$ queues."}
{"category": "Math", "title": "Consistency and robustness of kernel-based regression in convex risk minimization", "abstract": "We investigate statistical properties for a broad class of modern kernel-based regression (KBR) methods. These kernel methods were developed during the last decade and are inspired by convex risk minimization in infinite-dimensional Hilbert spaces. One leading example is support vector regression. We first describe the relationship between the loss function $L$ of the KBR method and the tail of the response variable. We then establish the $L$-risk consistency for KBR which gives the mathematical justification for the statement that these methods are able to ``learn''. Then we consider robustness properties of such kernel methods. In particular, our results allow us to choose the loss function and the kernel to obtain computationally tractable and consistent KBR methods that have bounded influence functions. Furthermore, bounds for the bias and for the sensitivity curve, which is a finite sample version of the influence function, are developed, and the relationship between KBR and classical $M$ estimators is discussed."}
{"category": "Math", "title": "On It\\^{o}'s formula for elliptic diffusion processes", "abstract": "Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83--109] prove an extension of It\\^{o}'s formula for $F(X_t,t)$, where $F(x,t)$ has a locally square-integrable derivative in $x$ that satisfies a mild continuity condition in $t$ and $X$ is a one-dimensional diffusion process such that the law of $X_t$ has a density satisfying certain properties. This formula was expressed using quadratic covariation. Following the ideas of Eisenbaum [Potential Anal. 13 (2000) 303--328] concerning Brownian motion, we show that one can re-express this formula using integration over space and time with respect to local times in place of quadratic covariation. We also show that when the function $F$ has a locally integrable derivative in $t$, we can avoid the mild continuity condition in $t$ for the derivative of $F$ in $x$."}
{"category": "Math", "title": "Sample path properties of the local time of multifractional Brownian motion", "abstract": "We establish estimates for the local and uniform moduli of continuity of the local time of multifractional Brownian motion, $B^H=(B^{H(t)}(t),t\\in\\mathbb{R}^+)$. An analogue of Chung's law of the iterated logarithm is studied for $B^H$ and used to obtain the pointwise H\\\"{o}lder exponent of the local time. A kind of local asymptotic self-similarity is proved to be satisfied by the local time of $B^H$."}
{"category": "Math", "title": "Asymptotic behavior of grafting rays", "abstract": "In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmuller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmuller geodesics and lines of minima. We also show that the ray grafted along a weighted system of simple closed curves is at bounded distance from Teichmuller geodesic."}
{"category": "Math", "title": "Algebraic theta functions and Eisenstein-Kronecker numbers", "abstract": "In this paper, we give an overview of our previous paper concerning the investigation of the algebraic and $p$-adic properties of Eisenstein-Kronecker numbers using Mumford's theory of algebraic theta functions."}
{"category": "Math", "title": "Sharp constants related to the triangle inequality in Lorentz spaces", "abstract": "We study the Lorentz spaces $L^{p,s}(R,\\mu)$ in the range $1<p<s\\le \\infty$, for which the standard functional $$ ||f||_{p,s}=(\\int_0^\\infty (t^{1/p}f^*(t))^s\\frac{dt}{t})^{1/s} $$ is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: $$ ||f||_{(p,s)}=\\inf\\bigg\\{\\sum_{k}||f_k||_{p,s}\\bigg\\}, $$ where the infimum is taken over all finite representations $f=\\sum_{k}f_k. $ We also prove that the decomposition norm and the dual norm $$ ||f||_{p,s}'= \\sup\\left\\{\\int_R fg d\\mu: ||g||_{p',s'}=1\\right\\} $$ agree for all values $p,s>1$."}
{"category": "Math", "title": "Rearrangement transformations on general measure spaces", "abstract": "For a general set transformation ${\\cal R}$ between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of ${\\cal R}$, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees."}
{"category": "Math", "title": "Limit theorems for functionals on the facets of stationary random tessellations", "abstract": "We observe stationary random tessellations $X=\\{\\Xi_n\\}_{n\\ge1}$ in $\\mathbb{R}^d$ through a convex sampling window $W$ that expands unboundedly and we determine the total $(k-1)$-volume of those $(k-1)$-dimensional manifold processes which are induced on the $k$-facets of $X$ ($1\\le k\\le d-1$) by their intersections with the $(d-1)$-facets of independent and identically distributed motion-invariant tessellations $X_n$ generated within each cell $\\Xi_n$ of $X$. The cases of $X$ being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In both cases, however, we obtain that all of the total volumes measured in $W$ are approximately normally distributed when $W$ is sufficiently large. Structural formulae for mean values and asymptotic variances are derived and explicit numerical values are given for planar Poisson--Voronoi tessellations (PVTs) and Poisson line tessellations (PLTs)."}
{"category": "Math", "title": "K3 surfaces, rational curves, and rational points", "abstract": "We prove that for any of a wide class of elliptic surfaces $X$ defined over a number field $k$, if there is an algebraic point on $X$ that lies on only finitely many rational curves, then there is an algebraic point on $X$ that lies on no rational curves. In particular, our theorem applies to a large class of elliptic $K3$ surfaces, which relates to a question posed by Bogomolov in 1981. We apply our results to construct an explicit algebraic point on a $K3$ surface that does not lie on any smooth rational curves."}
{"category": "Math", "title": "Computation of weight lattices of G-varieties", "abstract": "Let G be a connected reductive group. To any irreducible G-variety one assigns the lattice generated by all weights of B-semiinvariant rational functions on X, where B$ is a Borel subgroup of G. This lattice is called the weight lattice of X. We establish algorithms for computing weight lattices for homogeneous spaces and affine homogeneous vector bundles. For affine homogeneous spaces of rank rk(G) we present a more or less explicit computation."}
{"category": "Math", "title": "Which Partial Sums of the Taylor Series for $e$ are Convergents to $e$? (and a Link to the Primes 2, 5, 13, 37, 463), II", "abstract": "This is an expanded version of our earlier paper. Let the $n$th partial sum of the Taylor series $e = \\sum_{r=0}^{\\infty} 1/r!$ be $A_n/n!$, and let $p_k/q_k$ be the $k$th convergent of the simple continued fraction for $e$. Using a recent measure of irrationality for $e$, we prove weak versions of our conjecture that only two of the partial sums are convergents to $e$. A related result about the denominators $q_k$ and powers of factorials is proved. We also show a surprising connection between the $A_n$ and the primes 2, 5, 13, 37, 463. In the Appendix, we give a conditional proof of the conjecture, assuming a second conjecture we make about the zeros of $A_n$ and $q_k$ modulo powers of 2. Tables supporting this Zeros Conjecture are presented and we discuss a 2-adic reformulation of it."}
{"category": "Math", "title": "Harmonic morphisms on heaven spaces", "abstract": "We prove that any (real or complex) analytic horizontally conformal submersion from a three-dimensional conformal manifold M to a two-dimensional conformal manifold N can be, locally, `extended' to a unique harmonic morphism from the heaven space of M to N."}
{"category": "Math", "title": "Constructing presentations of subgroups of right-angled Artin groups", "abstract": "Let $G$ be the right-angled Artin group associated to the flag complex $\\Sigma$ and let $\\pi:G\\to\\Z$ be its canonical height function. We investigate the presentation theory of the groups $\\Gamma_n=\\pi^{-1}(n\\Z)$ and construct an algorithm that, given $n$ and $\\Sigma$, outputs a presentation of optimal deficiency on a minimal generating set, provided $\\Sigma$ is triangle-free; the deficiency tends to infinity as $n\\to\\infty$ if and only if the corresponding Bestvina-Brady kernel $\\bigcap_n\\Gamma_n$ is not finitely presented, and the algorithm detects whether this is the case. We explain why there cannot exist an algorithm that constructs finite presentations with these properties in the absence of the triangle-free hypothesis. We explore what is possible in the general case, describing how to use the configuration of 2-simplices in $\\Sigma$ to simplify presentations and giving conditions on $\\Sigma$ that ensure that the deficiency goes to infinity with $n$. We also prove, for general $\\Sigma$, that the abelianized deficiency of $\\Gamma_n$ tends to infinity if and only if $\\Sigma$ is 1-acyclic, and discuss connections with the relation gap problem."}
{"category": "Math", "title": "Abundance of elliptic dynamics on conservative 3-flows", "abstract": "We consider a compact 3-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C1-residual such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M."}
{"category": "Math", "title": "Modules of covariants in modular invariant theory", "abstract": "Let the finite group $G$ act linearly on the vector space $V$ over the field $k$ of arbitrary characteristic. If $H<G$ is a subgroup the extension of invariant rings $k[V]^G\\subset k[V]^H$ is studied using modules of covariants. An example of our results is the following. Let $W$ be the subgroup of $G$ generated by the reflections in $G$. A classical theorem due to Serre says that if $k[V]$ is a free $k[V]^G$-module then $G=W$. We generalize this result as follows. If $k[V]^H$ is a free $k[V]^G$-module then $G$ is generated by $H$ and $W$, and the invariant ring $k[V]^{H\\cap W}$ is free over $k[V]^W$ and generated as an algebra by $H$-invariants and $W$-invariants."}
{"category": "Math", "title": "The extremal volume ellipsoids of convex bodies, their symmetry properties, and their determination in some special cases", "abstract": "A convex body K has associated with it a unique circumscribed ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum volume. We first give a unified, modern exposition of the basic theory of these extremal ellipsoids using the semi-infinite programming approach pioneered by Fritz John in his seminal 1948 paper. We then investigate the automorphism groups of convex bodies and their extremal ellipsoids. We show that if the automorphism group of a convex body K is large enough, then it is possible to determine the extremal ellipsoids CE(K) and IE(K) exactly, using either semi-infinite programming or nonlinear programming. As examples, we compute the extremal ellipsoids when the convex body K is the part of a given ellipsoid between two parallel hyperplanes, and when K is a truncated second order cone or an ellipsoidal cylinder."}
{"category": "Math", "title": "Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps", "abstract": "The Andreev-Thurston Circle Packing Theorem is generalized to packings of convex bodies in planar simply connected domains. This turns out to be a useful tool for constructing conformal and quasiconformal mappings with interesting geometric properties. We attempt to illustrate this with a few results about uniformizations of finitely connected planar domains. For example, the following variation of a theorem by Courant, Manel and Shiffman is proved and generalized. If $G$ is an $n+1$-connected bounded planar domain, $H$ is a simply connected bounded planar domain, and $P_1,P_2,...,P_n$ are (compact) planar convex bodies, then sets $P_j'$ can be found so that $G$ is conformally equivalent to $H-\\cup_{j=1}^n P_j'$, and each $P_j'$ is either a point, or is positively homothetic to $P_j$."}
{"category": "Math", "title": "Invariant theory of abelian transvection groups", "abstract": "Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called {\\em coregular} if the invariant ring is generated by algebraically independent homogeneous invariants and the {\\em direct summand property} holds if there is a surjective $k[V]^G$-linear map $\\pi:k[V]\\to k[V]^G$. The following Chevalley--Shephard--Todd type theorem is proved. Suppose $G$ is abelian, then the action is coregular if and only if $G$ is generated by pseudo-reflections and the direct summand property holds."}
{"category": "Math", "title": "On Chevalley-Shephard-Todd's theorem in positive characteristic", "abstract": "Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called coregular if the invariant ring is generated by algebraically independent homogeneous invariants and the direct summand property holds if there is a surjective $k[V]^G$-linear map $\\pi:k[V]\\to k[V]^G$. The following Chevalley-Shephard-Todd type theorem is proved. Suppose $V$ is an irreducible $kG$-representation, then the action is coregular if and only if $G$ is generated by pseudo-reflections and the direct summand property holds."}
{"category": "Math", "title": "Representation functions of bases for binary linear forms", "abstract": "Let F(x_1,...,x_m) = u_1 x_1 + ... + u_m x_m be a linear form with nonzero, relatively prime integer coefficients u_1,..., u_m. For any set A of integers, let F(A) = {F(a_1,...,a_m) : a_i in A for i=1,...,m}. The representation function associated with the form F is R_{A,F}(n) = card {(a_1,...,a_m) in A^m: F(a_1,..., a_m) = n}. The set A is a basis with respect to F for almost all integers the set Z\\F(A) has asymptotic density zero. Equivalently, the representation function of an asymptotic basis is a function f:Z -> N_0 U {\\infty} such that f^{-1}(0) has density zero. Given such a function, the inverse problem for bases is to construct a set A whose representation function is f. In this paper the inverse problem is solved for binary linear forms."}
{"category": "Math", "title": "On knot Floer width and Turaev genus", "abstract": "To each knot $K\\subset S^3$ one can associated its knot Floer homology $\\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizontal distance between two such lines. Also, for each diagram $D$ of $K$ there is an associated Turaev surface, and the Turaev genus is the minimum genus of all Turaev surfaces for $K$. We show that the width of knot Floer homology is bounded by Turaev genus plus one. Skein relations for genus of the Turaev surface and width of a complex that generates knot Floer homology are given."}
{"category": "Math", "title": "A remark on degenerate singularity in three dimensional Ricci flow", "abstract": "We show that a rescale limit at any degenerate singularity of Ricci flow in dimension 3 is a steady gradient soliton. In particular, we give a geometric description of type I and type II singularities."}
{"category": "Math", "title": "On the existence of tight contact structures on Seifert fibered 3-manifolds", "abstract": "We determine the closed, oriented Seifert fibered 3-manifolds which carry positive tight contact structures. Our main tool is a new non-vanishing criterion for the contact Ozsvath-Szabo invariant."}
{"category": "Math", "title": "Galois extensions and subspaces of bilinear forms with special rank properties", "abstract": "Let K be a field admitting a cyclic Galois extension of degree n. The main result of this paper is a decomposition theorem for the space of alternating bilinear forms defined on a vector space of odd dimension n over K. We show that this space of forms is the direct sum of (n-1)/2 subspaces, each of dimension n, and the non-zero elements in each subspace have constant rank defined in terms of the orders of the Galois automorphisms. Furthermore, if ordered correctly, for each integer k lying between 1 and (n-1)/2, the rank of any non-zero element in the sum of the first k subspaces is at most n-2k+1. Slightly less sharp similar results hold for cyclic extensions of even degree."}
{"category": "Math", "title": "Semitransitive and transitive subsemigroups of the inverse symmetric semigroups", "abstract": "We classify minimal transitive subsemigroups of the finitary inverse symmetric semigroup modulo the classification of minimal transitive subgroups of finite symmetric groups; and semitransitive subsemigroups of the finite inverse symmetric semigroup of the minimal cardinality modulo the classification of transitive subgroups of the minimal cardinality of finite symmetric groups."}
{"category": "Math", "title": "On Lyubeznik numbers of projective schemes", "abstract": "Let $X$ be an arbitrary projective scheme over a field $k$. Let $A$ be the local ring at the vertex of the affine cone for some embedding $\\iota: X\\hookrightarrow \\mathbb{P}^n_k$. G. Lyubeznik asked (in \\cite{l2}) whether the integers $\\lambda_{i,j}(A)$ (defined in \\cite{l1}), called the Lyubeznik numbers of $A$, depend only on $X$, but not on the embedding. In this paper, we make a big step toward a positive answer to this question by proving that in positive characteristic, for a fixed $X$, the Lyubezink numbers $\\lambda_{i,j}(A)$ of the local ring $A$, can only achieve finitely many possible values under all choices of embeddings."}
{"category": "Math", "title": "Morita equivalence of dual operator algebras", "abstract": "We consider a variant of the notion of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators, which we call {\\em weak Morita equivalence}. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint variants of aspects of Rieffel's $W^*$-algebraic Morita equivalence."}
{"category": "Math", "title": "On the ruin time distribution for a Sparre Andersen process with exponential claim sizes", "abstract": "We derive a closed-form (infinite series) representation for the distribution of the ruin time for the Sparre Andersen model with exponentially distributed claims. This extends a recent result of Dickson et al. (2005) for such processes with Erlang inter-claim times. We illustrate our result in the cases of gamma and mixed exponential inter-claim time distributions."}
{"category": "Math", "title": "Self-similar stable processes arising from high-density limits of occupation times of particle systems", "abstract": "We extend results on time-rescaled occupation time fluctuation limits of the $(d,\\alpha, \\beta)$-branching particle system $(0<\\alpha \\leq 2, 0<\\beta \\leq 1)$ with Poisson initial condition. The earlier results in the homogeneous case (i.e., with Lebesgue initial intensity measure) were obtained for dimensions $d>\\alpha / \\beta$ only, since the particle system becomes locally extinct if $d\\le \\alpha / \\beta$. In this paper we show that by introducing high density of the initial Poisson configuration, limits are obtained for all dimensions, and they coincide with the previous ones if $d>\\alpha/\\beta$. We also give high-density limits for the systems with finite intensity measures (without high density no limits exist in this case due to extinction); the results are different and harder to obtain due to the non-invariance of the measure for the particle motion. In both cases, i.e., Lebesgue and finite intensity measures, for low dimensions ($d<\\alpha(1+\\beta)/\\beta$ and $d<\\alpha(2+\\beta)/(1+\\beta)$, respectively) the limits are determined by non-L\\'evy self-similar stable processes. For the corresponding high dimensions the limits are qualitatively different: ${\\cal S}'(R^d)$-valued L\\'evy processes in the Lebesgue case, stable processes constant in time on $(0,\\infty)$ in the finite measure case. For high dimensions, the laws of all limit processes are expressed in terms of Riesz potentials. If $\\beta=1$, the limits are Gaussian. Limits are also given for particle systems without branching, which yields in particular weighted fractional Brownian motions in low dimensions. The results are obtained in the setup of weak convergence of S'(R^d)$-valued processes."}
{"category": "Math", "title": "Eigenvalues Estimates For The Dirac Operator In Terms Of Codazzi Tensors", "abstract": "We prove a lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold depending on the scalar curvature as well as a chosen Codazzi tensor. The inequality generalizes the classical estimate from [2]."}
{"category": "Math", "title": "Godel's theorem as a corollary of impossibility of complete axiomatization of geometry", "abstract": "Not any geometry can be axiomatized. The paradoxical Godel's theorem starts from the supposition that any geometry can be axiomatized and goes to the result, that not any geometry can be axiomatized. One considers example of two close geometries (Riemannian geometry and $\\sigma $-Riemannian one), which are constructed by different methods and distinguish in some details. The Riemannian geometry reminds such a geometry, which is only a part of the full geometry. Such a possibility is covered by the Godel's theorem."}
{"category": "Math", "title": "A description based on Schubert classes of cohomology of flag manifolds", "abstract": "We describe the integral cohomology rings of the flag manifolds of types B_n, D_n, G_2 and F_4 in terms of their Schubert classes. The main tool is the divided difference operators of Bernstein-Gelfand-Gelfand and Demazure. As an application, we compute the Chow rings of the corresponding complex algebraic groups, recovering thereby the results of R. Marlin."}
{"category": "Math", "title": "Laplacian comparison for Alexandrov spaces", "abstract": "We consider an infinitesimal version of the Bishop-Gromov relative volume comparison condition as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem."}
{"category": "Math", "title": "Quasisymmetric structures on surfaces", "abstract": "We show that a locally Ahlfors 2-regular and locally linearly locally contractible metric surface is locally quasisymmetrically equivalent to the disk. We also discuss an application of this result to the problem of characterizing surfaces in some Euclidean space that are locally bi-Lipschitz equivalent to an open subset of the plane."}
{"category": "Math", "title": "On frames in Hilbert modules over pro-C*-algebras", "abstract": "We introduce the concept of frame of multipliers in Hilbert modules over pro-C*-algebras and show that many properties of frames in Hilbert C*-modules are valid for frames of multipliers in Hilbert modules over pro-C*-algebras."}
{"category": "Math", "title": "Copies of a one-ended group in a Mapping Class Group", "abstract": "We establish that, given $\\Sigma$ a compact orientable surface, and $G$ a finitely presented one-ended group, the set of copies of $G$ in the mapping class group $\\mathcal{MCG}(\\Sigma)$ consisting of only pseudo-anosov elements except identity, is finite up to conjugacy. This relies on a result of Bowditch on the same problem for images of surfaces groups. He asked us whether we could reduce the case of one-ended groups to his result ; this is a positive answer. Our work involves analogues of Rips and Sela canonical cylinders in curve complexes, and the argument of Delzant to bound the number of images of a one-ended group in a hyperbolic group."}
{"category": "Math", "title": "Elation generalised quadrangles of order (s,p), where p is prime", "abstract": "We show that an elation generalised quadrangle which has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (i.e., is a flock quadrangle)."}
{"category": "Math", "title": "Lie admissible algebras are Volichenko", "abstract": "Withdrawn on account of a crucial error in the main theorem"}
{"category": "Math", "title": "Weak approximation of a fractional SDE", "abstract": "In this note, a diffusion approximation result is shown for stochastic differential equations driven by a (Liouville) fractional Brownian motion B with Hurst parameter H in (1/3,1/2). More precisely, we resort to the Kac-Stroock type approximation using a Poisson process studied in Bardina, Jolis and Tudor (2003) and Delgado and Jolis (2000), and our method of proof relies on the algebraic integration theory introduced by Gubinelli (2004)."}
{"category": "Math", "title": "Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces", "abstract": "In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on the compact hypersurfaces of ambient spaces with bounded sectional curvature. As application we deduce rigidity results for stable constant mean curvature hypersurfaces $M$ of these spaces $N$. Indeed, we prove that if $M$ is included in a ball of radius small enough then the Hausdorff-distance between $M$ and a geodesic sphere $S$ of $N$ is small. Moreover $M$ is diffeomorphic and quasi-isometric to $S$. As other application, we give rigidity results for almost umbilic hypersurfaces."}
{"category": "Math", "title": "Categories of categories", "abstract": "A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of finality (in particular terminal objects), discreteness and components, representability, colimits and universal arrows, seem to be best expressed in this very general setting. Furthermore, at this level we are in fact doing not only (E,M)-category theory but, in a sense, also (E,M)-topology. Other axioms, regarding power objects, duality, exponentials and the arrow object, are considered."}
{"category": "Math", "title": "Lattice cohomology of normal surface singularities", "abstract": "For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic can be compared with the analytic invariants. The Seiberg--Witten Invariant Conjecture is discussed in the light of this new object."}
{"category": "Math", "title": "On non-asymptotic bounds for estimation in generalized linear models with highly correlated design", "abstract": "We study a high-dimensional generalized linear model and penalized empirical risk minimization with $\\ell_1$ penalty. Our aim is to provide a non-trivial illustration that non-asymptotic bounds for the estimator can be obtained without relying on the chaining technique and/or the peeling device."}
{"category": "Math", "title": "Enveloping algebras of Hom-Lie algebras", "abstract": "Enveloping algebras of Hom-Lie and Hom-Leibniz algebras are constructed."}
{"category": "Math", "title": "On the Galois coverings of a cluster-tilted algebra", "abstract": "We study the module category of a certain Galois covering of a cluster-tilted algebra which we call the cluster repetitive algebra. Our main result compares the module categories of the cluster repetitive algebra of a tilted algebra C and the repetitive algebra of C, in the sense of Hughes and Waschbuesch."}
{"category": "Math", "title": "On the blocks of the walled Brauer algebra", "abstract": "We determine the blocks of the walled Brauer algebra in characteristic zero. These can be described in terms of orbits of the action of a Weyl group of type $A$ on a certain set of weights. In positive characteristic we give a linkage principle in terms of orbits of the corresponding affine Weyl group. We also classify the semisimple walled Brauer algebras in all characteristics."}
{"category": "Math", "title": "Diophantine exponents for mildly restricted approximation", "abstract": "We are studying the Diophantine exponent \\mu_{n,l}$ defined for integers 1 \\leq l < n and a vector \\alpha \\in \\mathbb{R}^n by letting \\mu_{n,l} = \\sup{\\mu \\geq 0: 0 < ||x \\cdot \\alpha|| < H(x)^{-\\mu} for infinitely many x \\in C_{n,l} \\cap \\mathbb{Z}^n}, where \\cdot is the scalar product and || . || denotes the distance to the nearest integer and C_{n,l} is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l+1, \\infty), with the value n attained for almost all \\alpha. We calculate the Hausdorff dimension of the set of vectors \\alpha with \\mu_{n,l} (\\alpha) = \\mu for \\mu \\geq n. Finally, letting w_n denote the exponent obtained by removing the restrictions on x, we show that there are vectors \\alpha for which the gaps in the increasing sequence \\mu_{n,1} (\\alpha) \\leq ... \\leq \\mu_{n,n-1} (\\alpha) \\leq w_n (\\alpha) can be chosen to be arbitrary."}
{"category": "Math", "title": "Ring geometries, Two-Weight Codes and Strongly Regular Graphs", "abstract": "It is known that a linear two-weight code $C$ over a finite field $\\F_q$ corresponds both to a multiset in a projective space over $\\F_q$ that meets every hyperplane in either $a$ or $b$ points for some integers $a<b$, and to a strongly regular graph whose vertices may be identified with the codewords of $C$. Here we extend this classical result to the case of a ring-linear code with exactly two nonzero homogeneous weights and multisets of points in an associated projective ring geometry. We will show that a two-weight code over a finite Frobenius ring gives rise to a strongly regular graph, and we will give some constructions of two-weight codes using ring geometries. These examples all yield infinite families of strongly regular graphs with non-trivial parameters."}
{"category": "Math", "title": "Asymptotic oracle properties of SCAD-penalized least squares estimators", "abstract": "We study the asymptotic properties of the SCAD-penalized least squares estimator in sparse, high-dimensional, linear regression models when the number of covariates may increase with the sample size. We are particularly interested in the use of this estimator for simultaneous variable selection and estimation. We show that under appropriate conditions, the SCAD-penalized least squares estimator is consistent for variable selection and that the estimators of nonzero coefficients have the same asymptotic distribution as they would have if the zero coefficients were known in advance. Simulation studies indicate that this estimator performs well in terms of variable selection and estimation."}
{"category": "Math", "title": "On binomial set-theoretic complete intersections in characteristic p", "abstract": "Using aritmethic conditions on affine semigroups we prove that for a simplicial toric variety of codimension 2 the property of being a set-theoretic complete intersection on binomials in characteristic $p$ holds either for all primes $p$, or for no prime $p$, or for exactly one prime $p$."}
{"category": "Math", "title": "On critical normal sections for two-dimensional immersions in R^{n+2}", "abstract": "We study orthonormal normal sections of two-dimensional immersions in $\\mathbb R^{n+2},$ $n\\ge 2$, at which these sections are critical for a functional of total torsion. In particular, we establish upper bounds for the torsion coefficients in the case of non-flat normal bundles. With these notes we continue a foregoing paper on surfaces in $\\mathbb R^4.$"}
{"category": "Math", "title": "New multivariate central limit theorems in linear structural and functional error-in-variables models", "abstract": "This paper deals simultaneously with linear structural and functional error-in-variables models (SEIVM and FEIVM), revisiting in this context generalized and modified least squares estimators of the slope and intercept, and some methods of moments estimators of unknown variances of the measurement errors. New joint central limit theorems (CLT's) are established for these estimators in the SEIVM and FEIVM under some first time, so far the most general, respective conditions on the explanatory variables, and under the existence of four moments of the measurement errors. Moreover, due to them being in Studentized forms to begin with, the obtained CLT's are a priori nearly, or completely, data-based, and free of unknown parameters of the distribution of the errors and any parameters associated with the explanatory variables. In contrast, in related CLT's in the literature so far, the covariance matrices of the limiting normal distributions are, in general, complicated and depend on various, typically unknown parameters that are hard to estimate. In addition, the very forms of the CLT's in the present paper are universal for the SEIVM and FEIVM. This extends a previously known interplay between a SEIVM and a FEIVM. Moreover, though the particular methods and details of the proofs of the CLT's in the SEIVM and FEIVM that are established in this paper are quite different, a unified general scheme of these proofs is constructed for the two models herewith."}
{"category": "Math", "title": "Pattern Avoiding Ballot Paths and Finite Operator Calculus", "abstract": "Counting pattern avoiding ballot paths begins with a careful analysis of the pattern. Not the length, but the characteristics of the pattern are responsible for the difficulties in finding explicit solutions. Certain features, like overlap and difference in number of right and up steps determine the solution of the recursion formula. If the recursion can be solved by a polynomial sequence, we apply the Finite Operator Calculus to find an explicit form of the solution in terms of binomial coefficients. Keywords: Pattern avoidance, ballot path, Dyck path, Finite Operator Calculus, Umbral Calculus"}
{"category": "Math", "title": "On the combinatorics of rigid objects in 2-Calabi-Yau categories", "abstract": "Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a cluster-tilting subcategory T' form a basis of the Grothendieck group of T and that, if T and T' are related by a mutation, then the indices with respect to T and T' are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given cluster-tilting subcategory T. Conjecturally, these indices coincide with Fomin-Zelevinsky's g-vectors."}
{"category": "Math", "title": "Two algorithms for evaluation of the Newman digit sum, and a new proof of Coquet's theorem", "abstract": "We give two simple algorithms for the evaluation of difference between the numbers of multiples of 3 with even and odd binary digit sums in interval [0,x), and give an elementary proof of Coquet's sharp estimates for it."}
{"category": "Math", "title": "Almost Euclidean subspaces of \\ell_1^N via expander codes", "abstract": "We give an explicit (in particular, deterministic polynomial time) construction of subspaces X of R^N of dimension (1-o(1))N such that for every element x in X, |x|_1 and N^{1/2} |x|_2 are equivalent up to a factor of (log N)^{log log log N}. If we are allowed to use N^{o(1)} random bits, this factor can be improved to poly(log N). Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of error-correcting codes."}
{"category": "Math", "title": "Additive isotone regression", "abstract": "This paper is about optimal estimation of the additive components of a nonparametric, additive isotone regression model. It is shown that asymptotically up to first order, each additive component can be estimated as well as it could be by a least squares estimator if the other components were known. The algorithm for the calculation of the estimator uses backfitting. Convergence of the algorithm is shown. Finite sample properties are also compared through simulation experiments."}
{"category": "Math", "title": "Symmetric Group Character Degrees and Hook Numbers", "abstract": "In this article we prove the following result: that for any two natural numbers k and j, and for all sufficiently large symmetric groups Sym(n), there are k disjoint sets of j irreducible characters of Sym(n), such that each set consists of characters with the same degree, and distinct sets have different degrees. In particular, this resolves a conjecture most recently made by Moret\\'o. The methods employed here are based upon the duality between irreducible characters of the symmetric groups and the partitions to which they correspond. Consequently, the paper is combinatorial in nature."}
{"category": "Math", "title": "On a class of metrics related to graph layout problems", "abstract": "We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the literature, and also to a class of combinatorial optimization problems known as graph layout problems. We prove several results about the structure of these metrics. In particular, it is shown that their convex hull is not closed in general. We then show that certain linear inequalities define facets of the closure of the convex hull. Finally, we characterise the unbounded edges of the convex hull and of its closure."}
{"category": "Math", "title": "Product Systems; a Survey with Commutants in View", "abstract": "The theory of product systems both of Hilbert spaces (Arveson systems) and product systems of Hilbert modules has reached a status where it seems appropriate to rest a moment and to have a look at what is known so far and what are open problems. However, the attempt to give an approximately complete account in view pages is destined to fail already for Arveson systems since Tsirelson, Powers and Liebscher have discovered their powerful methods to construct large classes of examples. In this survey we concentrate on that part of the theory that works also for Hilbert modules. This does not only help to make a selection among the possible topics, but it also helps to shed some new light on the case of Arveson systems. Often, proofs that work for Hilbert modules also lead to simpler proofs in the case of Hilbert spaces. We put emphasis on those aspects that arise from recent results about commutants of von Neumann correspondences, which, in the case of Hilbert spaces, explain the relation between the Arveson system and the Bhat system associated with an E_0--semigroup on B(H)."}
{"category": "Math", "title": "A comparison of duality and energy aposteriori estimates for L?(0,T;L2({\\Omega})) in parabolic problems", "abstract": "We use the elliptic reconstruction technique in combination with a duality approach to prove aposteriori error estimates for fully discrete back- ward Euler scheme for linear parabolic equations. As an application, we com- bine our result with the residual based estimators from the aposteriori esti- mation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the L \\infty (0, T ; L2({\\Omega})) norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson (1991) by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estima- tors. For comparison with previous results we derive also an energy-based aposteriori estimate for the L \\infty (0, T ; L2({\\Omega}))-error which simplifies a previous one given in Lakkis and Makridakis (2006). We then compare both estimators (duality vs. energy) in practical situations and draw conclusions."}
{"category": "Math", "title": "Note on pre-Courant algebroid structures for parabolic geometries", "abstract": "This note aims to demonstrate that every parabolic geometry has a naturally defined per-Courant algebro\\\"id structure. This structure is a Courant algebro\\\"id if and only if the the curvature $\\kappa$ of the Cartan connection vanishes. In all other cases, if the parabolic geometry is regular, there does not exist a natural universal expression for a Courant bracket."}
{"category": "Math", "title": "The Algebraic Complexity of Maximum Likelihood Estimation for Bivariate Missing Data", "abstract": "We study the problem of maximum likelihood estimation for general patterns of bivariate missing data for normal and multinomial random variables, under the assumption that the data is missing at random (MAR). For normal data, the score equations have nine complex solutions, at least one of which is real and statistically significant. Our computations suggest that the number of real solutions is related to whether or not the MAR assumption is satisfied. In the multinomial case, all solutions to the score equations are real and the number of real solutions grows exponentially in the number of states of the underlying random variables, though there is always precisely one statistically significant local maxima."}
{"category": "Math", "title": "On the Frobenius functor and colon ideals", "abstract": "In this paper we study the commutativity of the Frobenius functor and the colon operation of two ideals for Noetherian rings of positive characteristic $p$. New characterizations of regular rings and local UFDs are given."}
{"category": "Math", "title": "Fastest mixing Markov chain on graphs with symmetries", "abstract": "We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches."}
{"category": "Math", "title": "Counting and Locating the Solutions of Polynomial Systems of Maximum Likelihood Equations, II: The Behrens-Fisher Problem", "abstract": "Let $\\mu$ be a $p$-dimensional vector, and let $\\Sigma_1$ and $\\Sigma_2$ be $p \\times p$ positive definite covariance matrices. On being given random samples of sizes $N_1$ and $N_2$ from independent multivariate normal populations $N_p(\\mu,\\Sigma_1)$ and $N_p(\\mu,\\Sigma_2)$, respectively, the Behrens-Fisher problem is to solve the likelihood equations for estimating the unknown parameters $\\mu$, $\\Sigma_1$, and $\\Sigma_2$. We shall prove that for $N_1, N_2 > p$ there are, almost surely, exactly $2p+1$ complex solutions of the likelihood equations. For the case in which $p = 2$, we utilize Monte Carlo simulation to estimate the relative frequency with which a typical Behrens-Fisher problem has multiple real solutions; we find that multiple real solutions occur infrequently."}
{"category": "Math", "title": "The Morse-Bott inequalities via dynamical systems", "abstract": "Let $f:M \\to \\mathbb{R}$ be a Morse-Bott function on a compact smooth finite dimensional manifold $M$. The polynomial Morse inequalities and an explicit perturbation of $f$ defined using Morse functions $f_j$ on the critical submanifolds $C_j$ of $f$ show immediately that $MB_t(f) = P_t(M) + (1+t)R(t)$, where $MB_t(f)$ is the Morse-Bott polynomial of $f$ and $P_t(M)$ is the Poincar\\'e polynomial of $M$. We prove that $R(t)$ is a polynomial with nonnegative integer coefficients by showing that the number of gradient flow lines of the perturbation of $f$ between two critical points $p,q \\in C_j$ coincides with the number of gradient flow lines between $p$ and $q$ of the Morse function $f_j$. This leads to a relationship between the kernels of the Morse-Smale-Witten boundary operators associated to the Morse functions $f_j$ and the perturbation of $f$. This method works when $M$ and all the critical submanifolds are oriented or when $\\mathbb{Z}_2$ coefficients are used."}
{"category": "Math", "title": "Fault Tolerance in Cellular Automata at High Fault Rates", "abstract": "A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. We are concerned with the degree (the number of neighboring cells on which the state transition function depends) needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We also consider both a purely probabilistic fault model (in which the states of cells are perturbed at exactly the fault rate) and an adversarial model (in which the occurrence of a fault gives control of the state to an omniscient adversary). We show that there are cellular automata that can tolerate a fault rate $1/2 - \\xi$ (with $\\xi>0$) with degree $O((1/\\xi^2)\\log(1/\\xi))$, even with adversarial combined faults. The simplest such automata are based on infinite regular trees, but our results also apply to other structures (such as hyperbolic tessellations) that contain infinite regular trees. We also obtain a lower bound of $\\Omega(1/\\xi^2)$, even with purely probabilistic transient faults only."}
{"category": "Math", "title": "$k$-Ribbon Fibonacci Tableaux", "abstract": "We extend the notion of $k$-ribbon tableaux to the Fibonacci lattice, a differential poset defined by R. Stanley in 1975. Using this notion, we describe an insertion algorithm that takes $k$-colored permutations to pairs of $k$-ribbon Fibonacci tableaux of the same shape, and we demonstrate a color-to-spin property, similar to that described by Shimozono and White for ribbon tableaux. We give an evacuation algorithm which relates the pair of $k$-ribbon Fibonacci tableaux obtained through the insertion algorithm to the pair of $k$-ribbon Fibonacci tableaux obtained using Fomin's growth diagrams. In addition, we present an analogue of Knuth relations for $k$-colored permutations and $k$-ribbon Fibonacci tableaux."}
{"category": "Math", "title": "Multiloop realization of extended affine Lie algebras and Lie tori", "abstract": "An important theorem in the theory of infinite dimensional Lie algebras states that any affine Kac-Moody algebra can be realized (that is to say constructed explicitly) using loop algebras. In this paper, we consider the corresponding problem for a class of Lie algebras called extended affine Lie algebras (EALAs) that generalize affine algebras. EALAs occur in families that are constructed from centreless Lie tori, so the realization problem for EALAs reduces to the realization problem for centreless Lie tori. We show that all but one family of centreless Lie tori can be realized using multiloop algebras (in place of loop algebras). We also obtain necessary and sufficient for two centreless Lie tori realized in this way to be isotopic, a relation that corresponds to isomorphism of the corresponding families of EALAs."}
{"category": "Math", "title": "A characterization of the rational normal curve", "abstract": "We give a characterization of the rational normal curve in terms of the rank function associated to a curve."}
{"category": "Math", "title": "Self-injective algebras and the second Hochschild cohomology group", "abstract": "In this paper we study the second Hochschild cohomology group ${HH}^2(\\Lambda)$ of a finite dimensional algebra $\\Lambda$. In particular, we determine ${HH}^2(\\Lambda)$ where $\\Lambda$ is a finite dimensional self-injective algebra of finite representation type over an algebraically closed field $K$ and show that this group is zero for most such $\\Lambda$; we give a basis for ${HH}^2(\\Lambda)$ in the few cases where it is not zero."}
{"category": "Math", "title": "The combinatorics of associated Hermite polynomials", "abstract": "We develop a combinatorial model of the associated Hermite polynomials and their moments, and prove their orthogonality with a sign-reversing involution. We find combinatorial interpretations of the moments as complete matchings, connected complete matchings, oscillating tableaux, and rooted maps and show weight-preserving bijections between these objects. Several identities, linearization formulas, the moment generating function, and a second combinatorial model are also derived."}
{"category": "Math", "title": "On the Embeddability of Skeleta of Spheres", "abstract": "We consider a generalization of the van Kampen-Flores Theorem and relate it to the long-standing $g$-conjecture for simplicial spheres."}
{"category": "Math", "title": "Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I", "abstract": "In this article and in its sequel we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite-dimensional spaces. We illustrate the fruitfulness of this approach in the context of the Ricci flow, as well as another flow, in Kahler geometry. We introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics that arise as discretizations of these flows. We pose some problems regarding their dynamics. We point out a number of applications to well-studied objects in Kahler and conformal geometry such as constant scalar curvature metrics, Kahler-Ricci solitons, Nadel-type multiplier ideal sheaves, balanced metrics, the Moser-Trudinger-Onofri inequality, energy functionals and the geometry and structure of the space of Kahler metrics. E.g., we obtain a new sharp inequality strengthening the classical Moser-Trudinger-Onofri inequality on the two-sphere."}
{"category": "Math", "title": "Teichm\\\"uller Spaces and Bundles with Negatively Curved Fibers", "abstract": "We study the Teichm\\\"uller space of negatively curved metrics on a high dimensional manifold, with applications to bundles with negatively curved fibers."}
{"category": "Math", "title": "Classification of Toric log Del Pezzo Surfaces having Picard Number 1 and Index $\\le 3$", "abstract": "Toric log Del Pezzo surfaces with Picard number 1 have been completely classified whenever their index is $\\le 2$: In this paper we extend the classification for those having index 3: We prove that, up to isomorphism, there are exactly 18 surfaces of this kind."}
{"category": "Math", "title": "Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes", "abstract": "We reduce the construction of a weak solution of the Cauchy problem for the Navier-Stokes system to the construction of a solution to a stochastic problem. Namely, we construct diffusion processes which allow us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for the Navier- Stokes system."}
{"category": "Math", "title": "Empirical processes indexed by estimated functions", "abstract": "We consider the convergence of empirical processes indexed by functions that depend on an estimated parameter $\\eta$ and give several alternative conditions under which the ``estimated parameter'' $\\eta_n$ can be replaced by its natural limit $\\eta_0$ uniformly in some other indexing set $\\Theta$. In particular we reconsider some examples treated by Ghoudi and Remillard [Asymptotic Methods in Probability and Statistics (1998) 171--197, Fields Inst. Commun. 44 (2004) 381--406]. We recast their examples in terms of empirical process theory, and provide an alternative general view which should be of wide applicability."}
{"category": "Math", "title": "Evolution Strategies in Optimization Problems", "abstract": "Evolution Strategies are inspired in biology and part of a larger research field known as Evolutionary Algorithms. Those strategies perform a random search in the space of admissible functions, aiming to optimize some given objective function. We show that simple evolution strategies are a useful tool in optimal control, permitting to obtain, in an efficient way, good approximations to the solutions of some recent and challenging optimal control problems."}
{"category": "Math", "title": "Weyl asymptotics for magnetic Schr\\\"odinger operators and de Gennes' boundary condition", "abstract": "This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schr\\\"odinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof relies on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [A. Kachmar, J. Math. Phys. Vol. 47 (7) 072106 (2006)]."}
{"category": "Math", "title": "Critical scaling of stochastic epidemic models", "abstract": "In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent $p-$coin tosses. Spatial variants of these models are proposed, in which finite populations of size $N$ are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for both the mean-field and spatial models are given when the infection parameter $p$ is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift."}
{"category": "Math", "title": "A Dirac type result on Hamilton cycles in oriented graphs", "abstract": "We show that for each \\alpha>0 every sufficiently large oriented graph G with \\delta^+(G),\\delta^-(G)\\ge 3|G|/8+ \\alpha |G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. In fact, we prove the stronger result that G is still Hamiltonian if \\delta(G)+\\delta^+(G)+\\delta^-(G)\\geq 3|G|/2 + \\alpha |G|. Up to the term \\alpha |G| this confirms a conjecture of H\\\"aggkvist. We also prove an Ore-type theorem for oriented graphs."}
{"category": "Math", "title": "Introduction to Tropical Geometry (notes from the IMPA lectures in Summer 2007)", "abstract": "These condensed notes treat some basic notions in Tropical Geometry (varieties, cycles, modifications, equivalence). These topics are to be extended, illustrated and included to the upcoming book project http://www.math.toronto.edu/mikha/book.pdf ."}
{"category": "Math", "title": "Disproof of modularity of moduli space of CY 3-folds of double covers of P3 ramified along eight planes in general positions", "abstract": "We prove that the moduli space of Calabi-Yau 3-folds coming from eight planes of $P^3$ in general positions is not modular. In fact we show the stronger statement that the Zariski closure of the monodromy group is actually the whole $Sp(20,R)$. We construct an interesting submoduli, which we call \\emph{hyperelliptic locus}, over which the weight 3 $Q$-Hodge structure is the third wedge product of the weight 1 $Q$-Hodge structure on the corresponding hyperelliptic curve. The non-extendibility of the hyperelliptic locus inside the moduli space of a genuine Shimura subvariety is proved."}
{"category": "Math", "title": "Computational details on the disproof of modularity", "abstract": "The purpose of these notes is to provide the details of the Jacobian ring computations carried out in [1], based on the computer algebra system Magma [2]."}
{"category": "Math", "title": "A new semilocal convergence theorem for the Weierstrass method from data at one point", "abstract": "In this paper we present a new semilocal convergence theorem from data at one point for the Weierstrass iterative method for the simultaneous computation of polynomial zeros. The main result generalizes and improves all previous ones in this area."}
{"category": "Math", "title": "A complex semigroup approach to group algebras of infinite dimensional Lie groups", "abstract": "A host algebra of a topological group G is a C^*-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations of G. In this paper we present an approach to host algebras for infinite dimensional Lie groups which is based on complex involutive semigroups. Any locally bounded absolute value \\alpha on such a semigroup S leads in a natural way to a C^*-algebra C^*(S,\\alpha), and we describe a setting which permits us to conclude that this C^*-algebra is a host algebra for a Lie group G. We further explain how to attach to any such host algebra an invariant weak-*-closed convex set in the dual of the Lie algebra of G enjoying certain nice convex geometric properties. If G is the additive group of a locally convex space, we describe all host algebras arising this way. The general non-commutative case is left for the future."}
{"category": "Math", "title": "Lie groups of bundle automorphisms and their extensions", "abstract": "We describe natural abelian extensions of the Lie algebra $\\aut(P)$ of infinitesimal automorphisms of a principal bundle over a compact manifold $M$ and discuss their integrability to corresponding Lie group extensions. Already the case of a trivial bundle $P = M \\times K$ is quite interesting. In this case, we show that essentially all central extensions of the gauge algebra $C^\\infty(M,\\fk)$ can be obtained from three fundamental types of cocycles with values in one of the spaces $\\fz := C^\\infty(M,V)$, $\\Omega^1(M,V)$ and $\\Omega^1(M,V)/\\dd C^\\infty(M,V)$. These cocycles extend to $\\aut(P)$, and, under the assumption that $TM$ is trivial, we also describe the space $H^2({\\cal V}(M),\\fz)$ classifying the twists of these extensions. We then show that all fundamental types have natural generalizations to non-trivial bundles and explain under which conditions they extend to $\\aut(P)$ and integrate to global Lie group extensions."}
{"category": "Math", "title": "On convexity of the frequency response of a stable polynomial", "abstract": "In the complex plane, the frequency response of a univariate polynomial is the set of values taken by the polynomial when evaluated along the imaginary axis. This is an algebraic curve partitioning the plane into several connected components. In this note it is shown that the component including the origin is exactly representable by a linear matrix inequality if and only if the polynomial is stable, in the sense that all its roots have negative real parts."}
{"category": "Math", "title": "Semilocal convergence of two iterative methods for simultaneous computation of polynomial zeros", "abstract": "In this paper we study some iterative methods for simultaneous approximation of polynomial zeros. We give new semilocal convergence theorems with error bounds for Ehrlich's and Nourein's iterations. Our theorems generalize and improve recent results of Zheng and Huang [J. Comput. Math. 18 (2000), 113--122], Petkovi\\'c and Herceg [J. Comput. Appl. Math. 136 (2001), 283--307] and Nedi\\'c [Novi Sad J. Math. 31 (2001), 103--111]. We also present a new sufficient condition for simple zeros of a polynomial."}
{"category": "Math", "title": "Growth of Subharmonic Functions", "abstract": "In this course of lectures we give an account of the growth theory of subharmonic functions, which is directed towards its applications to entire functions of one and several complex variables."}
{"category": "Math", "title": "Two-scale homogenization of piezoelectric perforated structures", "abstract": "We are interested in the homogenization of elastic-electric coupling equation, with rapidly oscillating coefficients, in periodically perforated piezoelectric body. We justify the two first terms in the usual asymptotic development of the problem solution. For the main convergence results of this paper, we use the notion of {\\it two-scale convergence}. A two-scale homogenized system is obtained as the limit of the periodic problem. While in the static limit the method provides homogenized electroelastic coefficients whicht coincide with those deduced from other homogenization techniques (asymptotic homogenization, $\\Gamma$-convergence)."}
{"category": "Math", "title": "On some asymptotically flat manifolds with non maximal volume growth", "abstract": "Asymptotically flat manifolds with Euclidean volume growth are known to be ALE. In this paper, we consider a class of asymptotically flat manifolds with slower volume growth and prove that their asymptotic geometry is that of a fibration over an ALE manifold. In particular, we show that gravitational instantons with cubic volume growth are ALF."}
{"category": "Math", "title": "The Horn conjecture for compact selfadjoint operators", "abstract": "We determine the possible eigenvalues of compact selfadjoint operators A,B,C... with the property that A=B+C+... When all these operators are positive, the eigenvalues were known to be subject to certain inequalities which extend Horn's inequalities from the finite-dimensional case when A=B+C. We find the proper extension of the Horn inequalities and show that they, along with their reverse analogues, provide a complete characterization. Our results also allow us to discuss the more general situation where only some of the eigenvalues of the operators are specified. A special case is the requirement that B+C+... be positive of rank at most r."}
{"category": "Math", "title": "Geometry of invariant domains in complex semi-simple Lie groups", "abstract": "We investigate the joint action of two real forms of a semi-simple complex Lie group S by left and right multiplication. After analyzing the orbit structure, we study the CR structure of closed orbits. The main results are an explicit formula of the Levi form of closed orbits and the determination of the Levi cone of generic orbits. Finally, we apply these results to prove q-completeness of certain invariant domains in S."}
{"category": "Math", "title": "Characterizations of Sobolev inequalities on metric spaces", "abstract": "We present isocapacitary characterizations of Sobolev inequalities in very general metric measure spaces."}
{"category": "Math", "title": "CR-Invariants and the Scattering Operator for Complex Manifolds with Boundary", "abstract": "The purpose of this paper is to describe certain CR-covariant differential operators on a strictly pseudoconvex CR manifold $M$ as residues of the scattering operator for the Laplacian on an ambient complex K\\\"{a}hler manifold $X$ having $M$ as a `CR-infinity.' We also characterize the CR $Q$-curvature in terms of the scattering operator. Our results parallel earlier results of Graham and Zworski \\cite{GZ:2003}, who showed that if $X$ is an asymptotically hyperbolic manifold carrying a Poincar\\'{e}-Einstein metric, the $Q$-curvature and certain conformally covariant differential operators on the `conformal infinity' $M$ of $X$ can be recovered from the scattering operator on $X$. The results in this paper were announced in \\cite{HPT:2006}."}
{"category": "Math", "title": "Deformation quantization for actions of the affine group", "abstract": "We define a universal deformation formula (UDF) for the actions of the affine group on Frechet algebras. More precisely, starting with any associative Frechet algebra which the affine group acts on in a strongly continuous and isometrical manner, the UDF produces a family of topological associative algebra structures on the space of smooth vectors of the action deforming the initial product. The deformation field obtained is based over an infinite dimensional parameter space naturally associated with the space of pseudo-differential operators on the real line. This note also presents some geometrical aspects of the UDF and in particular its relation with hyperbolic geometry."}
{"category": "Math", "title": "Bifurcation of the ACT map", "abstract": "In this paper, we study the Arneodo-Coullet-Tresser map $ F(x,y,z)=(ax-b(y-z), bx+a(y-z), cx-dx^k+e z)$ where $a,b,c,d,e$ are real with $bd\\neq 0$ and $k>1$ is an integer. We obtain stability regions for fixed points of $F$ and symmetric period-2 points while $c$ and $e$ vary as parameters. Varying $a$ and $e$ as parameters, we show that there is a hyperbolic invariant set on which $F$ is conjugate to the full shift on two or three symbols. We also show that chaotic behaviors of $F$ while $c$ and $d$ vary as parameters and $F$ is near an anti-integrable limit. Some numerical results indicates $F$ has Hopf bifurcation, strange attractors, and nested structure of invariant tori."}
{"category": "Math", "title": "On the uniqueness of certain families of holomorphic disks", "abstract": "A Zoll metric is a Riemannian metric whose geodesics are all circles of equal length. Via the twistor correspondence of LeBrun and Mason, a Zoll metric on the 2 dimensional sphere corresponds to a family of holomorphic disks in CP_2 with boundary in a totally real submanifold P. In this paper, we show that for a fixed totally real submanifold P, such a family is unique if it exists, implying that the twistor correspondence of LeBrun and Mason is injective. One of the key ingredients in the proof is the blow-up and blow-down constructions in the sense of Melrose."}
{"category": "Math", "title": "Integral cohomology of certain Picard modular surfaces", "abstract": "Let Gamma be a congruence subgroup of the Picard modular group of an imaginary number field k, and let D be the associated symmetric space. We describe a method to compute the integral cohomology of the locally symmetric space Gamma\\D. The method is implemented for the case k=Q(i) and k=Q(sqrt(-3)), and the cohomology is computed for various Gamma."}
{"category": "Math", "title": "Takai Duality for Crossed Products by Hilbert C*-bimodules", "abstract": "We discuss the crossed product by the dual action of the circle on the crossed product of a C*-algebra A by a Hilbert C*-bimodule X. When X is an A-A Morita equivalence bimodule, the double crossed product is shown to be Morita equivalent to the C*-algebra A."}
{"category": "Math", "title": "Some Monotonicity Properties of Gamma and $q$-gamma Functions", "abstract": "We prove some properties of completely monotonic functions and apply them to obtain results on gamma and $q$-gamma functions."}
{"category": "Math", "title": "Spectral numbers in Floer theories", "abstract": "The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz, and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the ``nondegenerate spectrality'' axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties. The key ingredient is a theorem about the existence of best approximations of arbitrary elements of finitely generated free modules over Novikov rings by elements of prescribed submodules with respect to a certain family of non-Archimedean metrics."}
{"category": "Math", "title": "Analysis on disconnected sets", "abstract": "Some aspects of analysis on disconnected sets are briefly discussed, more along the lines of regions with infinitely many components than Cantor sets."}
{"category": "Math", "title": "Almost solutions of equations in permutations", "abstract": "We will say that the permutations f_1,...,f_n is an e-solution of an equation if the normalized Hamming distance between its l.h.p. and r.h.p. is less than e. We give a sufficient conditions when near to an e-solution exists an exact solution and some examples when there does not exist such a solution."}
{"category": "Math", "title": "Parameter estimation in diagonalizable bilinear stochastic parabolic equations", "abstract": "A parameter estimation problem is considered for a stochastic parabolic equation with multiplicative noise under the assumption that the equation can be reduced to an infinite system of uncoupled diffusion processes. From the point of view of classical statistics, this problem turns out to be singular not only for the original infinite-dimensional system but also for most finite-dimensional projections. This singularity can be exploited to improve the rate of convergence of traditional estimators as well as to construct completely new closed-form exact estimator."}
{"category": "Math", "title": "Characterizing Heavy-Tailed Distributions Induced by Retransmissions", "abstract": "Consider a generic data unit of random size L that needs to be transmitted over a channel of unit capacity. The channel availability dynamics is modeled as an i.i.d. sequence {A, A_i},i>0 that is independent of L. During each period of time that the channel becomes available, say A_i, we attempt to transmit the data unit. If L<A_i, the transmission was considered successful; otherwise, we wait for the next available period and attempt to retransmit the data from the beginning. We investigate the asymptotic properties of the number of retransmissions N and the total transmission time T until the data is successfully transmitted. In the context of studying the completion times in systems with failures where jobs restart from the beginning, it was shown that this model results in power law and, in general, heavy-tailed delays. The main objective of this paper is to uncover the detailed structure of this class of heavy-tailed distributions induced by retransmissions. More precisely, we study how the functional dependence between P[L>x] and P[A>x] impacts the distributions of N and T. In particular, we discover several functional criticality points that separate classes of different functional behavior of the distribution of N. We also discuss the engineering implications of our results on communication networks since retransmission strategy is a fundamental component of the existing network protocols on all communication layers, from the physical to the application one."}
{"category": "Math", "title": "Euler-Bernoulli beams from a symmetry standpoint-characterization of equivalent equations", "abstract": "We completely solve the equivalence problem for Euler-Bernoulli equation using Lie symmetry analysis. We show that the quotient of the symmetry Lie algebra of the Bernoulli equation by the infinite-dimensional Lie algebra spanned by solution symmetries is a representation of one of the following Lie algebras: $2A_1$, $A_1\\oplus A_2$, $3A_1$, or $A_{3,3}\\oplus A_1$. Each quotient symmetry Lie algebra determines an equivalence class of Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity, we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters (lineal mass density and flexural rigidity). For each class, we provide a simple representative and we explicitly construct transformations that maps a class member to its representative. The maximally symmetric class described by the four-dimensional quotient symmetry Lie algebra $A_{3,3}\\oplus A_1$ corresponds to Euler-Bernoulli equations homeomorphic to the uniform one (constant lineal mass density and flexural rigidity) . We rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations reducible to the uniform unit beam. Our models extend and emphasize the symmetry flavor of Gottlieb's iso-spectral beams (Proceedings of the Royal Society London A 413 (1987) 235-250)"}
{"category": "Math", "title": "Brauer-Manin obstructions to integral points", "abstract": "We study Brauer-Manin obstructions to integral points on open subsets of the projective plane."}
{"category": "Math", "title": "iso-spectral Euler-Bernoulli beams \\`a la Sophus Lie", "abstract": "We obtain iso-spectral Euler-Bernoulli beams by using factorization and Lie symmetry techniques. The canonical Euler-Bernoulli beam operator is factorized as the product of a second-order linear differential operator and its adjoint. The factors are then reversed to obtain iso-spectral beams. The factorization is possible provided the coefficients of the factors satisfy a system of non-linear ordinary differential equations . The uncoupling of this system yields a single non-linear third-order ordinary differential equation. This ordinary differential equation, refer to as the {\\it principal equation}, is analyzed and solved using Lie group methods. We show that the principal equation may admit a one-dimensional or three-dimensional symmetry Lie algebra. When the principal system admits a unique symmetry, the best we can do is to depress its order by one. We obtain a one-parameter family of solutions in this case. The maximally symmetric case is shown to be isomorphic to a Chazy equation which is solved in closed form to derive the general solution of the principal equation."}
{"category": "Math", "title": "On a class of weighted anisotropic Sobolev inequalities", "abstract": "In this article, motivated by a work of Caffarelli and Cordoba in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities, that is Sobolev type inequalities where different derivatives have different weight functions. The inequalities we are dealing with, are also intimately connected to weighted Sobolev inequalities for Grushin type operators, the weights being not necessarily Muckenhoupt. For example we consider here Sobolev inequalities on finite cylinders, the weights being different powers of the distance function from the top and the bottom of the cylinder. We also prove similar inequalities in the more general case in which the weight is the distance function from an higher codimension part of the boundary."}
{"category": "Math", "title": "On equitable zero sums", "abstract": "It is well-known that any sequence of at least N integers contains a subsequence whose sum is 0 (mod N). However, there can be very few subsequences with this property (e.g. if the initial sequence is just N 1's, then there is only one subsequence). When the length L of the sequence is much longer, we might expect that there are 2^L/N subsequences with this property (imagine the subsequences have sum-of-terms uniformly distributed modulo N -- the 0 class gets about 2^L/N subsequences); however, it is easy to see that this is actually false. Nonetheless, we are able to prove that if the initial sequence has length at least 4N, and N is odd, then there is a subsequence of length L > N, having at least 2^L/N subsequences that sum to 0 mod N."}
{"category": "Math", "title": "Isotopy for extended affine Lie algebras and Lie tori", "abstract": "Centreless Lie tori have been used by E. Neher to construct all extended affine Lie algebras (EALAs). In this article, we study notions of isotopes and isotopy for centreless Lie tori, and we use these notions, along with Neher's construction, to show that there is a 1-1 correspondence between centreless Lie tori up to isotopy and families of EALAs up to isomorphism. Also, centreless Lie tori can be coordinatized by unital algebras that are in general nonassociative, and, for many types of centreless Lie tori, there are classical notions of isotopes and isotopy for the coordinate algebras. We show for those types that an isotope of the Lie torus is coordinatized by an isotope of its coordinate algebra. In writing the article, we have not assumed prior knowledge of the theories of EALAs, Lie tori or isotopy. In fact, we hope that this article will help to introduce the reader to these theories and their interconnections."}
{"category": "Math", "title": "A refinement of Sharkovskii's theorem on orbit types characterized by two parameters", "abstract": "The so-called type problem or forcing problem is considered as a way to generalize Sharkovskii's theorem. In this paper, by focusing on certain types of orbits, we obtain a solution of the type problem, which gives a refinement of Sharkovskii's theorem on orbit types characterized by two parameters."}
{"category": "Math", "title": "Orbit equivalence of one-sided subshifts and the associated C^*-algebras", "abstract": "A $\\lambda$-graph system ${\\frak L}$ is a generalization of a finite labeled graph and presents a subshift. We will prove that the topological dynamical systems $(X_{{\\frak L}_1},\\sigma_{{\\frak L}_1})$ and $(X_{{\\frak L}_2},\\sigma_{{\\frak L}_2})$ for $\\lambda$-graph systems ${\\frak L}_1$ and ${\\frak L}_2$ are continuously orbit equivalent if and only if there exists an isomorphism between the associated $C^*$-algebras ${\\Cal O}_{{\\frak L}_1}$ and ${\\Cal O}_{{\\frak L}_2}$ keeping their commutative $C^*$-subalgebras $C(X_{{\\frak L}_1})$ and $C(X_{{\\frak L}_2})$. It is also equivalent to the condition that there exists a homeomorphism from $X_{{\\frak L}_1}$ to $X_{{\\frak L}_2}$ intertwining their topological full inverse semigroups. In particular, one-sided subshifts $X_{\\Lambda_1}$ and $X_{\\Lambda_2}$ are $\\lambda$-continuously orbit equivalent if and only if there exists an isomorphism between the associated $C^*$-algebras ${\\Cal O}_{\\Lambda_1}$ and ${\\Cal O}_{\\Lambda_2}$ keeping their commutative $C^*$-subalgebras $C(X_{\\Lambda_1})$ and $C(X_{\\Lambda_2})$."}
{"category": "Math", "title": "Thom polynomials of invariant cones, Schur functions, and positivity", "abstract": "We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the products of Schur functions of the bundles, have nonnegative coefficients. For classical Thom polynomials associated with maps of complex manifolds, this gives an extension of our former result for stable singularities to nonnecessary stable ones. We also discuss some related aspects of Thom polynomials, which makes the article expository to some extent."}
{"category": "Math", "title": "Harmonic ultrafilters", "abstract": "A set of natural numbers will be called \\emph{harmonic} if the reciprocals of its elements form a divergent series. An ultrafilter of the natural numbers will he called \\emph{harmonic} if all each members are harmonic sets. The harmonic ultrafilters are shown to constitute a compact semigroup under the Glazer addition and its smallest ideal is obtained. This paper is an extension of work begun by Hindman. Alle ingredients are found in the treatise by Hindman and Strauss."}
{"category": "Math", "title": "Primitive Characters and Permutation Characters of Solvable Groups", "abstract": "Let X be an irreducible, primitive complex character of the finite solvable group G, and let X* denote the complex conjugate character. If the degree X(1) is odd, then we show how to associate to X in a unique way, a conjugacy class of subgroups U of G for which X*X = (1_U)^G, the permutation character on the cosets of U. We investigate this situation and give a number of applications to properties of primitive characters of solvable and p-solvable groups."}
{"category": "Math", "title": "Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods", "abstract": "In this paper we derive uniform asymptotic expansions for the partial sums of the exponential series. We indicate how this information will be used in a later publication to obtain full and explicitly computable asymptotic expansions with error bounds for all zeros of the Taylor polynomials $p_{n-1}(z) = \\sum_{k=0}^{n-1} z^k/ k!$. Our proof is based on a representation of $p_{n-1}(nz)$ in terms of an integral of the form $\\int_{\\gamma} \\frac{e^{n\\phi(s)}}{s-z}ds$. We demonstrate how to derive uniform expansions for such integrals using a Riemann-Hilbert approach. A comparison with classical steepest descent analysis shows the advantages of the Riemann-Hilbert analysis in particular for points $z$ that are close to the critical points of $\\phi$."}
{"category": "Math", "title": "Group-theoretic Methods for Bounding the Exponent of Matrix Multiplication", "abstract": "The (asymptotic) complexity of matrix multiplication (over the complex field) is measured by a real parameter w > 0, called the exponent of matrix multiplication (over the complex field), which is defined to be the smallest real number w > 0 such that for an arbitrary degree of precision > 0, two n by n complex matrices can be multiplied using an algorithm using O(n^(w+\\epsilon)) number of non-division arithmetical operations. By the standard algorithm for multiplying two matrices, the trivial lower and upper bounds for the exponent w are 2 and 3 respectively. W. Strassen in 1969 obtained the first important result that w < 2.81 using his result that 2 by 2 matrix multiplication could be performed using 7 multiplications, not 8, as in the standard algorithm. In 1984, V. Pan improved this to 2.67, using a variant of Strassen's approach. It has been conjectured that w = 2, but the best known result is that w < 2.38, due to D. Coppersmith and S. Winograd. In all these approaches, estimates for w depend on the number of main running steps in their algorithms. In a recent series of papers in 2003 and 2005, H. Cohn and C. Umans put forward an entirely different approach using fairly elementary methods involving finite groups, group algebras and their representations. The author describes and proves their main results, and suggests possible ways of getting improved estimates for the exponent using their methods."}
{"category": "Math", "title": "Koszul duality for Operads", "abstract": "This is a copy of the article by the same authors published in Duke Math. J. (1994)."}
{"category": "Math", "title": "Outer Billiards on Kites", "abstract": "Outer billiards is a simple dynamical system based on a convex planar shape. The Moser-Neumann question, first posed by B.H. Neumann around 1960, asks if there exists a planar shape for which outer billiards has an unbounded orbit. The first half of this monograph proves that outer billiards has an unbounded orbit defined relative to any irrational kite. The second half of the monograph gives a very sharp description of the set of unbounded orbits, both in terms of the dynamics and the Hausdorff dimension. The analysis in both halves reveals a close connection between outer billiards on kites and the modular group, as well as connections to self-similar tilings, polytope exchange maps, Diophantine approximation, and odometers."}
{"category": "Math", "title": "Connexions with totally skew-symmetric torsion and nearly-Kaehler geometry", "abstract": "We study almost Hermitian structures admitting a Hermitian connexion with totally skew-symmetric torsion or equivalently, those almost Hermitian structures with totally skew-symmetric Nijenhuis tensor. We investigate up to what extent the Nijenhuis tensor fails to be parallel with respect to the characteristic connexion. This is naturally described by means of an extension of the notion of Killing form to almost Hermitian geometry. In this context, we also make an essentially self-contained survey of nearly-Kaehler geometry, but from the perspective of non-integrable holonomy systems."}
{"category": "Math", "title": "Monotonicity for entrywise functions of matrices", "abstract": "We characterize real functions $f$ on an interval $(-\\alpha,\\alpha)$ for which the entrywise matrix function $[a_{ij}] \\mapsto [f(a_{ij})]$ is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional power functions are exemplified and related weak majorizations are shown."}
{"category": "Math", "title": "Homotopically non-trivial maps with small k-dilation", "abstract": "We construct homotopically non-trivial maps from S^m to S^n with arbitrarily small 3-dilation for certain pairs (m,n). The simplest example is m=4, n=3. Other examples include arbitrarily large values of m and n. We show that a homotopy class in pi_7(S^4) can be represented by maps with arbitrarily small 4-dilation if and only if the class is torsion."}
{"category": "Math", "title": "The existence of two closed geodesics on every Finsler 2-sphere", "abstract": "In this paper, we prove that for every Finsler metric on the 2-dimensional sphere there exist at least two distinct prime closed geodesics. For the case of the two-sphere, this solves an open problem posed by D. V. Anosov in 1974."}
{"category": "Math", "title": "The Hopf invariant and simplex straightening", "abstract": "Let M be a closed 3-manifold which can be triangulated with N simplices. We prove that any map from M to a genus 2 surface has Hopf invariant at most C^N. Let X be a closed oriented hyperbolic 3-manifold with injectivity radius less than epsilon at one point. If there is a degree non-zero map from M to X, then we prove that epsilon is at least C^{-N}."}
{"category": "Math", "title": "The Geometry of Toric Hyperk\\\"ahler Varieties", "abstract": "In this survey article we describe the geometry of toric hyperk\\\"ahler varieties, which are hyperk\\\"ahler quotients of the quaternionic vector spaces by tori. In particular, we discuss the Betti numbers, the cohomology ring, and variation of hyperk\\\"ahler structures of these spaces with many improved results and proofs."}
{"category": "Math", "title": "The Entire Cyclic Cohomology of Noncommutative 2-Tori", "abstract": "Our aim in this paper is to compute the entire cyclic cohomology of noncommutative 2-tori. First of all, we clarify their algebraic structure of noncommutative 2-tori as a $F^*$-algebra, according to the idea of Elliott-Evans. Actually, they are the inductive limit of subhomogeneous $F^*$-algebras. Using such a result, we compute their entire cyclic cohomology, which is isomorphic to their periodic one as a complex vector space."}
{"category": "Math", "title": "Local structure of algebraic monoids", "abstract": "We describe the local structure of an irreducible algebraic monoid $M$ at an idempotent element $e$. When $e$ is minimal, we show that $M$ is an induced variety over the kernel $MeM$ (a homogeneous space) with fibre the two-sided stabilizer $M_e$ (a connected affine monoid having a zero element and a dense unit group). This yields the irreducibility of stabilizers and centralizers of idempotents when $M$ is normal, and criteria for normality and smoothness of an arbitrary $M$. Also, we show that $M$ is an induced variety over an abelian variety, with fiber a connected affine monoid having a dense unit group."}
{"category": "Math", "title": "L^2-spectral invariants and convergent sequences of finite graphs", "abstract": "Using the spectral theory of weakly convergent sequences of finite graphs, we prove the uniform existence of the integrated density of states for a large class of infinite graphs."}
{"category": "Math", "title": "An O(2,n) formulation of invariant theory for elliptic Weyl group", "abstract": "In this paper, we give a new formulation of invariant theory for elliptic Weyl group using the group O(2,n). As an elliptic Weyl group quotient, we define a suitable $\\C^*$-bundle. We show that it has a conformal Frobenius structure which we define in this paper. Then its good section could be identified with a Frobenius manifold which we constructed in arXiv:math/0611553."}
{"category": "Math", "title": "Discrete Monodromy, Pentagrams, and the Method of Condensation", "abstract": "This paper considers a simple geometric construction, called the Pentagram map. The pentagram map, performed on N-gons, gives rise to a birational mapping on the space of all N-gons. This paper finds what conjecturally are all the invariants for this map, and along the way relates the construction to the monodromy of 3rd order differential equations, and also to Dodgson's method of condensation for computing determinants."}
{"category": "Math", "title": "A criterion for the half-plane property", "abstract": "We establish a convenient necessary and sufficient condition for a multiaffine real polynomial to be stable, and use it to verify that the half-plane property holds for seven small matroids that resisted the efforts of Choe, Oxley, Sokal, and Wagner [5]."}
{"category": "Math", "title": "Divergence of a stationary random vector field can be always positive (a Weiss' phenomenon)", "abstract": "The divergence of a stationary random vector field at a given point is usually a centered (that is, zero mean) random variable. Strangely enough, it can be equal to 1 almost surely. This fact is another form of a phenomenon disclosed by B. Weiss in 1997."}
{"category": "Math", "title": "(GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F", "abstract": "Let F be an arbitrary local field. Consider the standard embedding of GL(n,F) into GL(n+1,F) and the two-sided action of GL(n,F) \\times GL(n,F) on GL(n+1,F). In this paper we show that any GL(n,F) \\times GL(n,F)-invariant distribution on GL(n+1,F) is invariant with respect to transposition. We show that this implies that the pair (GL(n+1,F),GL(n,F)) is a Gelfand pair. Namely, for any irreducible admissible representation $(\\pi,E)$ of (GL(n+1,F), $$dimHom_{GL(n,F)}(E,\\cc) \\leq 1.$$ For the proof in the archimedean case we develop several new tools to study invariant distributions on smooth manifolds."}
{"category": "Math", "title": "Dynamics of Symplectic SubVolumes", "abstract": "In this paper we will explore fundamental constraints on the evolution of certain symplectic subvolumes possessed by any Hamiltonian phase space. This research has direct application to optimal control and control of conservative mechanical systems. We relate geometric invariants of symplectic topology to computations that can easily be carried out with the state transition matrix of the flow map. We will show how certain symplectic subvolumes have a minimal obtainable volume; further if the subvolume dimension equals the phase space dimension, this constraint reduces to Liouville's Theorem. Finally we present a preferred basis that, for a given canonical transformation, has certain minimality properties with regards to the local volume expansion of phase space."}
{"category": "Math", "title": "Corners and Records of the Poisson Process in Quadrant", "abstract": "The scale-invariant spacings lemma due to Arratia, Barbour and Tavar{\\'e} establishes the distributional identity of a self-similar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the unit Poisson process in the positive quadrant."}
{"category": "Math", "title": "Study of a Z-form of the coordinate ring of a reductive group", "abstract": "We show how the theory of canonical bases in modified universal enveloping algebras can be used to develop the theory of Chevalley groups over any commutative ring with 1."}
{"category": "Math", "title": "On D.K. Biss' papers \"The homotopy type of the matroid Grassmannian\" and \"Oriented matroids, complex manifolds, and a combinatorial model for BU\"", "abstract": "We point out a flaw in papers by Daniel Biss devoted to homotopy type of matroid Grassmannians."}
{"category": "Math", "title": "A Rationality Problem of Some Cremona Transformation", "abstract": "We will show the rationality of the function field of two variables under certain Cremona transformation."}
{"category": "Math", "title": "Twisted Symmetric Group Actions", "abstract": "We will show the raitonality of some twisted symmetric group actions."}
{"category": "Math", "title": "Reduction theorems for Noether's problem", "abstract": "We will give several reduction theorems for Noether's problem."}
{"category": "Math", "title": "Staggered t-structures on derived categories of equivariant coherent sheaves", "abstract": "Let X be a scheme, and let G be an affine group scheme acting on X. Under reasonable hypotheses on X and G, we construct a t-structure on the derived category of G-equivariant coherent sheaves that in many ways resembles the perverse coherent t-structure, but which incorporates additional information from the G-action. Under certain circumstances, this t-structure, called the \"staggered t-structure,\" has an artinian heart, and its simple objects are particularly easy to describe. We also exhibit two small examples in which the staggered t-structure is better-behaved than the perverse coherent t-structure."}
{"category": "Math", "title": "MOST: detecting cancer differential gene expression", "abstract": "We propose a new statistics for the detection of differentially expressed genes, when the genes are activated only in a subset of the samples. Statistics designed for this unconventional circumstance has proved to be valuable for most cancer studies, where oncogenes are activated for a small number of disease samples. Previous efforts made in this direction include COPA, OS and ORT. We propose a new statistics called maximum ordered subset t-statistics (MOST) which seems to be natural when the number of activated samples is unknown. We compare MOST to other statistics and find the proposed method often has more power then its competitors."}
{"category": "Math", "title": "Bayes and empirical Bayes changepoint problems", "abstract": "We generalize the approach of Liu and Lawrence (1999) for multiple changepoint problems where the number of changepoints is unknown. The approach is based on dynamic programming recursion for efficient calculation of the marginal probability of the data with the hidden parameters integrated out. For the estimation of the hyperparameters, we propose to use Monte Carlo EM when training data are available. We argue that there is some advantages of using samples from the posterior which takes into account the uncertainty of the changepoints, compared to the traditional MAP estimator, which is also more expensive to compute in this context. The samples from the posterior obtained by our algorithm are independent, getting rid of the convergence issue associated with the MCMC approach. We illustrate our approach on limited simulations and some real data set."}
{"category": "Math", "title": "A mean ergodic theorem for actions of amenable quantum groups", "abstract": "We prove a weak form of the mean ergodic theorem for actions of amenable locally compact quantum groups in the von Neumann algebra setting."}
{"category": "Math", "title": "Mean time exit and isoperimetric inequalities \\\\for minimal submanifolds of $N\\times \\mathbb{R}$", "abstract": "Based on Markvorsen and Palmer's work on mean time exit and isoperimetric inequalities we establish slightly better isoperimetric inequalities and mean time exit estimates for minimal submanifolds of $N\\times\\mathbb{R}$. We also prove isoperimetric inequalities for submanifolds of Hadamard spaces with tamed second fundamental form."}
{"category": "Math", "title": "Le degre de la variete des courbes de Poncelet", "abstract": "We compute the degree of the projective variety of Poncelet curves of degree $n$. This variety is irreducible of dimension $2 n + 5$, and lies inside the projective space of degree $n$ plane curves. It is classically defined as the closure on this projective space of the locally closed subset of curves passing through the vertices of some nondegenerate $n$ sided polygone inscribed in some smooth conic (the polygone and the conic being variable). It is related to a specific class of semi-stable sheaves on the projective (dual) plane, named Poncelet sheaves. Using moduli spaces birational to the variety of Poncelet curves, we compute the requested degree. It involves quite cumbersome computations, and we obtain general formulas for $n \\geq 4$. We do numerical applications for $n \\leq 6$. For $n=4$ we find back the well known Donaldson number of the projective plane, 54, which is the degree of the hypersurface of Luroth quartics."}
{"category": "Math", "title": "Coincidence rotations of the root lattice $A_4$", "abstract": "The coincidence site lattices of the root lattice $A_4$ are considered, and the statistics of the corresponding coincidence rotations according to their indices is expressed in terms of a Dirichlet series generating function. This is possible via an embedding of $A_4$ into the icosian ring with its rich arithmetic structure, which recently (arXiv:math.MG/0702448) led to the classification of the similar sublattices of $A_4$."}
{"category": "Math", "title": "Conical square function estimates in UMD Banach spaces and applications to H-infinity functional calculi", "abstract": "We study conical square function estimates for Banach-valued functions, and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-McIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator (A) with certain off-diagonal bounds, such that (A) always has a bounded (H^{\\infty})-functional calculus on these spaces. This provides a new way of proving functional calculus of (A) on the Bochner spaces (L^p(\\R^n;X)) by checking appropriate conical square function estimates, and also a conical analogue of Bourgain's extension of the Littlewood-Paley theory to the UMD-valued context. Even when (X=\\C), our approach gives refined (p)-dependent versions of known results."}
{"category": "Math", "title": "Dynamics of Linear and Affine Maps", "abstract": "The well-known theory of \"rational canonical form of an operator\" describes the invariant factors, or elementary divisors, as a complete set of invariants of a similarity class of an operator on a finite-dimensional vector space $\\V$ over a given field $\\F$. A finer part of the theory is the contribution by Frobenius dealing with the structure of the centralizer of an operator. The viewpoint is that of finitely generated modules over a PID, cf. for example [J], ch. 3. In this paper we approach the issue from a \"dynamic\" viewpoint, as explained in [K]. We also extend the theory to affine maps. The formulation is in terms of the action of the geneal linear group, or the group of invertible affine maps, on the semigroup of all linear resp. affine maps by conjugacy. We describe a parametrization of orbits and orbit-classes under this action, and also provide a parametrization of all affine maps themselves."}
{"category": "Math", "title": "On the Wu metric in unbounded domains", "abstract": "We discuss the properties of the Wu pseudometric and present counterexamples for its upper semicontinuity that answers the question posed by Jarnicki and Pflug. We also give formulae for the Wu pseudometric in elementary Reinhardt domains."}
{"category": "Math", "title": "Self-interacting polynomials", "abstract": "We introduce a class of dynamical systems of algebraic origin, consisting of self-interacting irreducible polynomials over a field. A polynomial f is made to act on a polynomial g by mapping the roots of g. This action identifies a new polynomial h, as the minimal polynomial of the displaced roots. By allowing several polynomials to act on one another, we obtain a self-interacting system with a rich dynamics, which affords a fresh viewpoint on some algebraic dynamical constructs. We identify the basic invariant sets, and study in some detail the case of quadratic polynomials. We perform some experiments on self-interacting polynomials over finite fields."}
{"category": "Math", "title": "A nonlinear Poisson formula for the Schrodinger operator", "abstract": "We prove a nonlinear Poisson type formula for the Schrodinger group. Such a formula had been derived in a previous paper by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, energy-critical nonlinearities are allowed."}
{"category": "Math", "title": "One page proof of the Riemann hypothesis", "abstract": "We give a short Wiener measure proof of the Riemann hypothesis based on a surprising, unexpected and deep relation between the Riemann zeta $\\zeta(s)$ and the trivial zeta $\\zeta_{t}(s):=Im(s)(2Re(s)-1)$."}
{"category": "Math", "title": "Statistical stability of equilibrium states for interval maps", "abstract": "We consider families of multimodal interval maps with polynomial growth of the derivative along the critical orbits. For these maps Bruin and Todd have shown the existence and uniqueness of equilibrium states for the potential $\\phi_t:x\\mapsto-t\\log|Df(x)|$, for $t$ close to 1. We show that these equilibrium states vary continuously in the weak$^*$ topology within such families. Moreover, in the case $t=1$, when the equilibrium states are absolutely continuous with respect to Lebesgue, we show that the densities vary continuously within these families."}
{"category": "Math", "title": "A Brownian quasi-helix in IR4, built from an automatic sequence", "abstract": "We describe a brownian quasi-helix in R^4 by means of an automatic sequence"}
{"category": "Math", "title": "A topological characterisation of holomorphic parabolic germs in the plane", "abstract": "Gambaudo and P\\'ecou introduced the ``linking property'' to study the dynamics of germs of planar homeomorphims and provide a new proof of Naishul theorem in their paper \"A topological invariant for volume preserving diffeomorphisms\" (Ergodic Theory Dynam. Systems 15 (1995), no. 3, 535--541). In this paper we prove that the negation of Gambaudo-P\\'ecou property characterises the topological dynamics of holomorphic parabolic germs. As a consequence, a rotation set for germs of surface homeomorphisms around a fixed point can be defined, and it will turn out to be non trivial except for countably many conjugacy classes."}
{"category": "Math", "title": "A note on the component structure in random intersection graphs with tunable clustering", "abstract": "We study the component structure in random intersection graphs with tunable clustering, and show that the average degree works as a threshold for a phase transition for the size of the largest component. That is, if the expected degree is less than one, the size of the largest component is a.a.s. of logarithmic order, but if the average degree is greater than one, a.a.s. a single large component of linear order emerges, and the size of the second largest component is at most of logarithmic order."}
{"category": "Math", "title": "Jacobi fields along harmonic 2-spheres in $S^3$ and $S^4$ are not all integrable", "abstract": "In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4-sphere which have non-integrable Jacobi fields. This is particularly surprising in the case of the 3-sphere where the space of harmonic maps of any degree is a smooth manifold, each map having image in a totally geodesic 2-sphere."}
{"category": "Math", "title": "Compact differences of composition operators from Bloch space to bounded holomorphic function space in the Polydisc", "abstract": "Let $\\phi$ and $\\psi$ be holomorphic self-maps of the unit polydisc $U^n$ in the $n$-dimensional complex space, and denote by $C_{\\phi}$ and $C_{\\psi}$ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators $C_{\\phi}-C_{\\psi}$ from Bloch space to bounded holomorphic function space in the unit polydisc. Moreover the compactness of the difference is also characterized."}
{"category": "Math", "title": "Coherence in Linear Predicate Logic", "abstract": "Coherence with respect to Kelly-Mac Lane graphs is proved for categories that correspond to the multiplicative fragment without constant propositions of classical linear first-order predicate logic without or with mix. To obtain this result, coherence is first established for categories that correspond to the multiplicative conjunction-disjunction fragment with first-order quantifiers of classical linear logic, a fragment lacking negation. These results extend results published in previous two books by the authors, where coherence was established for categories of the corresponding fragments of propositional classical linear logic, which are related to proof nets, and which could be described as star-autonomous categories without unit objects."}
{"category": "Math", "title": "A higher order model for image restoration: the one dimensional case", "abstract": "The one-dimensional version of the higher order total variation-based model for image restoration proposed by Chan, Marquina, and Mulet in [4] is analyzed. A suitable functional framework in which the minimization problem is well posed is being proposed and it is proved analytically that the higher order regularizing term prevents the occurrence of the staircase effect."}
{"category": "Math", "title": "Computing Knot Floer Homology in Cyclic Branched Covers", "abstract": "We use grid diagrams to give a combinatorial algorithm for computing the knot Floer homology of the pullback of a knot K in its m-fold cyclic branched cover Sigma^m(K), and we give computations when m=2 for over fifty three-bridge knots with up to eleven crossings."}
{"category": "Math", "title": "The 191 orientable octahedral manifolds", "abstract": "We enumerate all spaces obtained by gluing in pairs the faces of the octahedron in an orientation-reversing fashion. Whenever such a gluing gives rise to non-manifold points, we remove small open neighbourhoods of these points, so we actually deal with three-dimensional manifolds with (possibly empty) boundary. There are 298 combinatorially inequivalent gluing patterns, and we show that they define 191 distinct manifolds, of which 132 are hyperbolic and 59 are not. All the 132 hyperbolic manifolds were already considered in different contexts by other authors, and we provide here their known ``names'' together with their main invariants. We also give the connected sum and JSJ decompositions for the 59 non-hyperbolic examples. Our arguments make use of tools coming from hyperbolic geometry, together with quantum invariants and more classical techniques based on essential surfaces. Many (but not all) proofs were carried out by computer, but they do not involve issues of numerical accuracy."}
{"category": "Math", "title": "Difference of composition operators in the Polydiscs", "abstract": "This paper gives some simple estimates of the essential norm for the difference of composition operators induced by $\\phi$ and $\\psi$ acting on bounded function space in the unit polydiscs $U^n$, where $\\phi(z)$ and $\\psi(z)$be holomorphic self-maps of $U^n$. As its applications, a characterization of compact difference is given for composition operators acting on the bounded function spaces."}
{"category": "Math", "title": "Essential Norms of Weighted Composition Operators between Hardy Spaces in the unit Ball", "abstract": "Let $\\phi(z)=(\\phi_1(z),...,\\phi_n(z))$ be a holomorphic self-map of $B_n$ and $\\psi(z)$ a holomorphic function on $B_n$, and $H(B_n)$ the class of all holomorphic functions on $B_n$, where $B_n$ is the unit ball of $C^n$, the weight composition operator $W_{\\psi,\\phi}$ is defined by $W_{\\psi,\\phi}=\\psi f(\\phi)$ for $f\\in H(B_n)$. In this paper we estimate the essential norm for the weighted composition operator $W_{\\psi,\\phi}$ acting from the Hardy space $H^p$ to $H^q$ ($0<p,q\\leq \\infty$). When $p=\\infty$ and $q=2$, we give an exact formula for the essential norm. As their applications, we also obtain some sufficient and necessary conditions for the bounded weighted composition operator to be compact from $H^p$ to $H^q$."}
{"category": "Math", "title": "Composition operators in the Lipschitz Space of the Polydiscs", "abstract": "In 1987, Shapiro shew that composition operator induced by symbol $\\phi$ is compact on the Lipschltz space if and only if the infinity norm of $\\phi$ is less than 1 by a spectral-theoretic argument, where $\\phi$ is a holomorphic self-map of the unit disk. In this paper, we shall generalize Shapiro's result to the $n$-dimensional case."}
{"category": "Math", "title": "Extended Ces$\\acute{a}$RO Operators on Zygmund Spaces in the Unit Ball", "abstract": "Let $g$ be a holomorphic function of the unit ball $B$ in the $n$-dimensional space, and denote by $T_g$ and $I_g$ the induced extended Ces$\\acute{a}$ro operator and another integral operator. The boundedness and compactness of $T_g$ and $I_g$ acting on the Zygmund spaces in the unit ball are discussed and necessary and sufficient conditions are given in this paper."}
{"category": "Math", "title": "Extended Ces$\\acute{a}$RO Operators between Generalized Besov Spaces and Bloch Type Spaces in the Unit Ball", "abstract": "Let $g$ be a holomorphic map of $B$, where $B$ is the unit ball of ${C}^n$. Let $0<p<+\\infty, -n-1<q<+\\infty$, $q>-1$ and $\\alpha>0$. This paper gives some necessary and sufficient conditions for the Extended Ces$\\acute{a}$ro Operators induced by $g$ to be bounded or compact between generalized Besov space $B(p,q)$ and $\\alpha$- Bloch space ${\\mathcal B}^\\alpha.$"}
{"category": "Math", "title": "A few remarks concerning complex-analytic metric spaces", "abstract": "The special case of closed subsets of C^n is briefly discussed."}
{"category": "Math", "title": "Pure Spinors on Lie groups", "abstract": "For any manifold M, the direct sum TM \\oplus T*M carries a natural inner product given by the pairing of vectors and covectors. Differential forms on M may be viewed as spinors for the corresponding Clifford bundle, and in particular there is a notion of \\emph{pure spinor}. In this paper, we study pure spinors and Dirac structures in the case when M=G is a Lie group with a bi-invariant pseudo-Riemannian metric, e.g. G semi-simple. The applications of our theory include the construction of distinguished volume forms on conjugacy classes in G, and a new approach to the theory of quasi-Hamiltonian G-spaces."}
{"category": "Math", "title": "Analysis of stochastic fluid queues driven by local time processes", "abstract": "We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a certain Markov process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is always singular with respect to the Lebesgue measure which in many applications is ``close'' to reality. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a L\\'evy process (a subordinator) hence making the theory of L\\'evy processes applicable. Another important ingredient in our approach is the Palm calculus coming from the point process point of view."}
{"category": "Math", "title": "Average twin prime conjecture for elliptic curves", "abstract": "Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over the finite field F_p is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves. Koblitz's conjecture is still widely open. In this paper we prove that Koblitz's conjecture is true on average over a two-parameter family of elliptic curves. One of the key ingredients in the proof is a short average distribution result in the style of Barban-Davenport-Halberstam, where the average is taken over twin primes and their differences."}
{"category": "Math", "title": "A sharp bound for the Stein-Wainger oscillatory integral", "abstract": "Let Pd denote the space of all real polynomials of degree at most d. It is an old result of Stein and Wainger that for every polynomial P in Pd: |p.v.\\int_R {e^{iP(t)} dt/t} | < C(d) for some constant C(d) depending only on d. On the other hand, Carbery, Wainger and Wright claim that the true order of magnitude of the above principal value integral is log d. We prove this conjecture."}
{"category": "Math", "title": "Random walks on quasisymmetric functions", "abstract": "Conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations. Spectral results are also obtained. Several well-studied random walks are now realized this way: Stanley's QS-distribution results from endomorphisms given by evaluation maps, a-shuffles result from the a-th convolution power of the universal character, and the Tchebyshev operator of the second kind introduced recently by Ehrenborg and Readdy yields traditional riffle shuffles. A conjecture of Ehrenborg regarding the spectra for a family of random walks on ab-words is proven. A theorem of Stembridge from the theory of enriched P-partitions is also recovered as a special case."}
{"category": "Math", "title": "Ricci iterations on Kahler classes", "abstract": "In this paper we consider the dynamical system involved by the Ricci operator on the space of K\\\"ahler metrics. A. Nadel has defined an iteration scheme given by the Ricci operator for Fano manifold and asked whether it has some nontrivial periodic points. First, we prove that no such periodic points can exist. We define the inverse of the Ricci operator and consider the dynamical behaviour of its iterates for a Fano K\\\"ahler-Einstein manifold. In particular we show that the iterates do converge to the existing K\\\"ahler-Ricci soliton on a toric manifold. Finally, we define a finite dimensional procedure to give an approximation of K\\\"ahler-Einstein metrics using this iterative procedure and apply it for $\\mathbb{CP}^2$ blown up in 3 points."}
{"category": "Math", "title": "Some remarks on conic degeneration and bending of Poincar\\'e-Einstein metrics", "abstract": "Let $(M,g)$ be a compact K\\\"ahler-Einstein manifold with $c_1 > 0$. Denote by $K\\to M$ the canonical line-bundle, with total space $X$, and $X_0$ the singular space obtained by blowing down $X$ along its zero section. We employ a construction by Page and Pope and discuss an interesting multi-parameter family of Poincar\\'e--Einstein metrics on $X$. One 1-parameter subfamily $\\{g_t\\}_{t>0}$ has the property that as $t\\searrow 0$, $g_t$ converges to a PE metric $g_0$ on $X_0$ with conic singularity, while $t^{-1}g_t$ converges to a complete Ricci-flat K\\\"ahler metric $\\hat{g}_0$ on $X$. Another 1-parameters subfamily has an edge singularity along the zero section of $X$, with cone angle depending on the parameter, but has constant conformal infinity. These illustrate some unexpected features of the Poincar\\'e-Einstein moduli space."}
{"category": "Math", "title": "Markoff Equation and Nilpotent Matrices", "abstract": "A triple (a,b,c) of positive integers is called a Markoff triple iff it satisfies the diophantine equation a2 + b2 + c2 = abc . Recasting the Markoff tree, whose vertices are Markoff triples, in the framework of intergral upper triangular 3x3 matrices, it will be shown that the largest member of such a triple determines the other two uniquely. This answers a question which has been open for almost 100 years."}
{"category": "Math", "title": "On the asymptotic of likelihood ratios for self-normalized large deviations", "abstract": "Motivated by multiple statistical hypothesis testing, we obtain the limit of likelihood ratio of large deviations for self-normalized random variables, specifically, the ratio of $P(\\sqrt{n}(\\bar X +d/n) \\ge x_n V)$ to $P(\\sqrt{n}\\bar X \\ge x_n V)$, as $n\\toi$, where $\\bar X$ and $V$ are the sample mean and standard deviation of iid $X_1, ..., X_n$, respectively, $d>0$ is a constant and $x_n \\toi$. We show that the limit can have a simple form $e^{d/z_0}$, where $z_0$ is the unique maximizer of $z f(x)$ with $f$ the density of $X_i$. The result is applied to derive the minimum sample size per test in order to control the error rate of multiple testing at a target level, when real signals are different from noise signals only by a small shift."}
{"category": "Math", "title": "Surgery description of colored knots", "abstract": "The pair (K,r) consisting of a knot K and a surjective map r from the knot group onto a dihedral group is said to be a p-colored knot. D. Moskovich conjectured that for any odd prime p there are exactly p equivalence classes of p-colored knots up to surgery along unknots in the kernel of the coloring. We show that there are at most 2p equivalence classes. This is a vast improvement upon the previous results by Moskovich for p=3, and 5, with no upper bound given in general. T. Cochran, A. Gerges, and K. Orr, in \"Dehn surgery equivalence relations of 3-manifolds\", define invariants of the surgery equivalence class of a closed 3-manifold M in the context of bordisms. By taking M to be 0-framed surgery of the 3-sphere along K we may define Moskovich's colored untying invariant in the same way as the Cochran-Gerges-Orr invariants. This bordism definition of the colored untying invariant will be then used to establish the upper bound."}
{"category": "Math", "title": "On well-posedness of the linear Cauchy problem with the distributional right-hand side and discontinuous coefficients", "abstract": "We prove the well-posedness of the Cauchy problem for the linear differential system of the form $x^{\\prime}-A(t)x=f$, where $f$ is a distribution and $A$ possesses at most first-kind discontinuities together with all its derivatives defined almost everywhere. The left-hand side of this system contains the product of a distribution and, in general, a discontinuous function, which is undefined in the classical space of the distributions with the smooth test functions $\\mathcal D'$, so the Cauchy problem has no solution in $\\mathcal D'$. In what follows, we cosider this system in the space of distributions with the discontinuous test functions, whose elements admit continuous and associative multiplication by functions possessing at most first-kind discontinuities (together with all their derivatives defined almost everywhere), and show that there exists the unique solution of the Cauchy problem which depends continuously on $f$."}
{"category": "Math", "title": "Ricci Flow with hyperbolic warped product metrics", "abstract": "In this short note, we show that the negative curvature is preserved in the deformation of hyperbolic warped product metrics under Ricci flow. It is also showed that the flow converges to a flat metric as time going to infinity."}
{"category": "Math", "title": "Tensor representations of classical locally finite Lie algebras", "abstract": "We study the structure of tensor representations of the classical infinite-dimensional locally finite Lie algebras $gl_\\infty$, $sl_\\infty$, $sp_\\infty$ and $so_\\infty$. In contrast with the finite-dimensional case, these tensor representations are not semisimple. We explicitly describe their Jordan-Holder constituents, socle filtrations, and indecomposable direct summands."}
{"category": "Math", "title": "The Existence of Pure Free Resolutions", "abstract": "Let d1,...,dn be a strictly increasing sequence of integers. Boij and S\\\"oderberg [arXiv:math/0611081] have conjectured the existence of a graded module M of finite length over any polynomial ring K[x_1,..., x_n], whose minimal free resolution is pure of type (d1,...,dn), in the sense that its i-th syzygies are generated in degree di. In this paper we prove a stronger statement, in characteristic zero: Such modules not only exist, but can be taken to be GL(n)-equivariant. In fact, we give two different equivariant constructions, and we construct pure resolutions over exterior algebras and Z/2-graded algebras as well. The constructions use the combinatorics of Schur functors and Bott's Theorem on the direct images of equivariant vector bundles on Grassmann varieties."}
{"category": "Math", "title": "Asymptotic Behaviour of Parameter Ideals in Generalized Cohen-Macaulay Modules", "abstract": "The purpose of this paper is to give affirmative answers to two open questions as follows. Let $(R, \\m)$ be a generalized Cohen-Macaulay Noetherian local ring. Both questions, the first question was raised by M. Rogers \\cite {R} and the second one is due to S. Goto and H. Sakurai \\cite {GS1}, ask whether for every parameter ideal $\\q$ contained in a high enough power of the maximal ideal $\\m $ the following statements are true: (1) The index of reducibility $N_R(\\q;R)$ is independent of the choice of $\\q$; and (2) $I^2=\\q I$, where $I=\\q:_R\\m$."}
{"category": "Math", "title": "The Set of Prime Numbers", "abstract": "In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime candidates are obtained in terms of the first perfect number. The asymptotic behaviour is also considered. We obtain for the first time an explicit relation for generating the full set P of prime numbers smaller than n or equal to n."}
{"category": "Math", "title": "Approximation of complex algebraic numbers by algebraic numbers of bounded degree", "abstract": "We investigate how well complex algebraic numbers can be approximated by algebraic numbers of degree at most n. We also investigate how well complex algebraic numbers can be approximated by algebraic integers of degree at most n+1. It follows from our investigations that for every positive integer n there are complex algebraic numbers of degree larger than n that are better approximable by algebraic numbers of degree at most n than almost all complex numbers. As it turns out, these numbers are more badly approximable by algebraic integers of degree at most n+1 than almost all complex numbers."}
{"category": "Math", "title": "The Szemeredi property in ergodic W*-dynamical systems", "abstract": "We study weak mixing of all orders for asymptotically abelian weakly mixing state preserving C*-dynamical systems, where the dynamics is given by the action of an abelian second countable locally compact group which contains a Folner sequence satisfying the Tempelman condition. For a smaller class of groups (which include Z^q and R^q) this is then used to show that an asymptotically abelian ergodic W*-dynamical system either has the \"Szemeredi property\" or contains a nontrivial subsystem (a \"compact factor\") that does. A van der Corput lemma for Hilbert space valued functions on the group is one of our main technical tools."}
{"category": "Math", "title": "Global phase-locking in finite populations of phase-coupled oscillators", "abstract": "We present new necessary and sufficient conditions for the existence of fixed points in a finite system of coupled phase oscillators on a complete graph. We use these conditions to derive bounds on the critical coupling."}
{"category": "Math", "title": "Koppelman formulas on Grassmannians", "abstract": "We construct Koppelman formulas on Grassmannians for forms with values in any holomorphic line bundle as well as in the tautological vector bundle and its dual. As a consequence we obtain some vanishing theorems of the Bott-Borel-Weil type. We also relate the projection part of our formulas to the Bergman kernels associated to the line bundles."}
{"category": "Math", "title": "On two notions of complexity of algebraic numbers", "abstract": "we derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the b-ary expansion of an irrational algebraic number. To this end, we apply a quantitative version of the Subspace Theorem due to Evertse and Schlickewei (2002)."}
{"category": "Math", "title": "Overpartition pairs and two classes of basic hypergeometric series", "abstract": "We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when specialized these series are also the generating functions for overpartition pairs with bounded successive ranks, overpartition pairs with conditions on their Durfee dissection, as well as certain lattice paths. When further specialized, the series become infinite products, leading to numerous identities for partitions, overpartitions, and overpartition pairs."}
{"category": "Math", "title": "Reciprocal cyclotomic polynomials", "abstract": "Let $\\Psi_n(x)$ be the monic polynomial having precisely all non-primitive $n$th roots of unity as its simple zeros. One has $\\Psi_n(x)=(x^n-1)/\\Phi_n(x)$, with $\\Phi_n(x)$ the $n$th cyclotomic polynomial. The coefficients of $\\Psi_n(x)$ are integers that like the coefficients of $\\Phi_n(x)$ tend to be surprisingly small in absolute value, e.g. for $n<561$ all coefficients of $\\Psi_n(x)$ are $\\le 1$ in absolute value. We establish various properties of the coefficients of $\\Psi_n(x)$."}
{"category": "Math", "title": "Groups which are not properly 3-realizable", "abstract": "A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it has {\\em pro-(finitely generated free) fundamental group at infinity} and {\\em semi-stable ends}. Conjecturally the quasi-simply filtration assumption is superfluous. Using these restrictions we provide the first examples of finitely presented groups which are not properly 3-realizable, for instance large families of Coxeter groups."}
{"category": "Math", "title": "Analytic Extension of a maximal surface in $\\Bbb L^3$ along its boudary", "abstract": "We prove that a maximal surface in Lorentz-Minkowski space $\\Bbb L^3$ can be extended analytically along its boundary if the boundary lies in a plane meeting the surface at a constant angle."}
{"category": "Math", "title": "Intersection homology D-Modules and Bernstein polynomials associated with a complete intersection", "abstract": "Let X be a complex analytic manifold. Given a closed subspace $Y\\subset X$ of pure codimension p>0, we consider the sheaf of local algebraic cohomology $H^p_{[Y]}({\\cal O}_X)$, and ${\\cal L}(Y,X)\\subset H^p_{[Y]}({\\cal O}_X)$ the intersection homology D_X-Module of Brylinski-Kashiwara. We give here an algebraic characterization of the spaces Y such that L(Y,X) coincides with $H^p_{[Y]}({\\cal O}_X)$, in terms of Bernstein-Sato functional equations."}
{"category": "Math", "title": "When will a One Parameter Family of Unimodal Maps Produce Finite Limit Cycles Monotonically with the Parameter?", "abstract": "In this note we consider a collection $\\cal{C}$ of one parameter families of unimodal maps of $[0,1].$ Each family in the collection has the form $\\{\\mu f\\}$ where $\\mu\\in [0,1].$ Denoting the kneading sequence of $\\mu f$ by $K(\\mu f)$, we will prove that for each member of $\\cal{C}$, the map $\\mu\\mapsto K(\\mu f)$ is monotone. It then follows that for each member of $\\cal{C}$ the map $\\mu\\mapsto h(\\mu f)$ is monotone, where $h(\\u{f})$ is the topological entropy of $\\mu f.$ For interest, $\\mu f(x)=4\\mu x(1-x)$ and $\\mu f(x)=\\mu\\sin(\\pi x)$ are shown to belong to $\\cal{C}.$ This extends the work of Masato Tsujii [1]."}
{"category": "Math", "title": "The Effects of Partial Crop Harvest on Biological Pest Control", "abstract": "In this paper, the effects of periodic partial harvesting of a continuously grown crop on augmentative biological control are analyzed. Partial harvesting can remove a proportion of both pests and biological control agents, so its influence on the control efficiency cannot be a priori neglected. An impulsive model consisting of a general predator-prey model in ODE, augmented by a discrete component to depict releases of biological control agents and the periodic partial harvesting is used. The periods are taken as integer multiples of each other. A stability condition for pest eradication is expressed as the minimal value of the budget per unit time to spend on predators. We consider the partial harvesting period to be fixed by both the plant's physiology and market forces so that the only manipulated variable is the release period. It is shown that varying the release period with respect to the harvest period influences the minimal budget value when the former is carried out more often than the latter and has no effect when releases take place as often as or less frequently than the partial harvests."}
{"category": "Math", "title": "Analytic continuation of residue currents", "abstract": "Let $X$ be a complex manifold and $f\\colon X\\to \\C^p$ a holomorphic mapping defining a complete intersection. We prove that the iterated Mellin transform of the residue integral associated to $f$ has an analytic continuation to a neighborhood of the origin in $\\C^p$."}
{"category": "Math", "title": "Linear convergence of iterative soft-thresholding", "abstract": "In this article a unified approach to iterative soft-thresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis. As main result we show that the algorithm converges with linear rate as soon as the underlying operator satisfies the so-called finite basis injectivity property or the minimizer possesses a so-called strict sparsity pattern. Moreover it is shown that the constants can be calculated explicitly in special cases (i.e. for compact operators). Furthermore, the techniques also can be used to establish linear convergence for related methods such as the iterative thresholding algorithm for joint sparsity and the accelerated gradient projection method."}
{"category": "Math", "title": "Local and global methods in arithmetic (in Russian)", "abstract": "Let $p$ be a prime. We discuss methods of solution of congruences modulo $p^n$ using $p$-adic numbers; these methods are similar to computations with real numbers (local methods). Examples of relations between local and global methods are given producing a passage from congruences to solutions in integers (global methods). The use of a computer is illistrated for the study of $p$-adic numbers and algebraic curves."}
{"category": "Math", "title": "The hyperbolic geometric flow on Riemann surfaces", "abstract": "In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ${\\mathbb{R}}^{2}$ in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric $g_{ij}$ keeps uniformly bounded for all time. If the initial velocity tensor does not satisfy the condition, then the solution blows up at a finite time, and the scalar curvature $R(t,x)$ goes to positive infinity as $(t,x)$ tends to the blowup points, and a flow with surgery has to be considered. The authors attempt to show that, comparing to Ricci flow, the hyperbolic geometric flow has the following advantage: the surgery technique may be replaced by choosing suitable initial velocity tensor. Some geometric properties of hyperbolic geometric flow on general open and closed Riemann surfaces are also discussed."}
{"category": "Math", "title": "On q-summation and confluence", "abstract": "This paper is divided in two parts. In the first part we consider irregular singular analytic q-difference equations, with q\\in ]0,1[, and we show how the Borel sum of a divergent solution of a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of such a q-difference equation. In the second part, we work under the assumption q\\in ]1,+\\infty[. In this case, at least four different q-Borel sums of a divergent solution of an irregular singular analytic q-difference equations are spread in the literature: under convenient assumptions we clarify the relations among them."}
{"category": "Math", "title": "Modular forms and $p$-adic numbers (in Russian)", "abstract": "Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of congruences between modular forms modulo $p$ and modulo $p^n$ are given, and the use of a computer for the study of modular forms is illistrated."}
{"category": "Math", "title": "On permutation polytopes", "abstract": "A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely classify permutation polytopes in dimensions 2,3,4, and the corresponding permutation groups up to a suitable notion of equivalence. We also provide a list of combinatorial types of possibly occuring faces of permutation polytopes up to dimension four."}
{"category": "Math", "title": "Bandwidth Selection for Weighted Kernel Density Estimation", "abstract": "In the this paper, the authors propose to estimate the density of a targeted population with a weighted kernel density estimator (wKDE) based on a weighted sample. Bandwidth selection for wKDE is discussed. Three mean integrated squared error based bandwidth estimators are introduced and their performance is illustrated via Monte Carlo simulation. The least-squares cross-validation method and the adaptive weight kernel density estimator are also studied. The authors also consider the boundary problem for interval bounded data and apply the new method to a real data set subject to informative censoring."}
{"category": "Math", "title": "Formulas of F-thresholds and F-jumping coefficients on toric rings", "abstract": "F-thresholds are defined by Mustata, Takagi and Watanabe in [F-thresholds and Bernstein-Sato polynomials], which are invariants of the pair of ideals on rings of characteristic $p$. In their paper, it is proved F-thresholds equal to jumping numbers for the test ideal on regular local rings. In this note, we give an formula of F-thresholds on toric rings. This formula is a generalization of the example in [Huneke, Mustata, Takagi and Watanabe:F-thresholds, tight closure, integral closure, and multiplicity bounds]. We prove that there exists an inequality between F-jumping coefficients and F-thresholds. In particular, we observe a comparison between F-pure thresholds and F-thresholds. As applications, we prove the characterization of regularity for toric rings defined by a simplicial cone, and the rationality of F-thresholds in some cases."}
{"category": "Math", "title": "Gaudin functions, and Euler-Poincar\\'e characteristics", "abstract": "Given two positive integers n,r, we define the Gaudin function of level r to be quotient of the numerator of the determinant det(1/ ((x_i-y_j)(x_i-ty_j) ... (x_i-t^r y_j)), i,j=1..n, by the two Vandermonde in x and y. We show that it can be characterized by specializing the x-variables into the y-variables, multiplied by powers of t. This allows us to obtain the Gaudin function of level 1 (due to Korepin and Izergin) as the image of a resultant under the the Euler-Poincar\\'e characteristics of the flag manifold. As a corollary, we recover a result of Warnaar about the generating function of Macdonald polynomials."}
{"category": "Math", "title": "Lefschetz fibrations and symplectic homology", "abstract": "We show that for each k > 3 there are infinitely many finite type Stein manifolds diffeomorphic to Euclidean space R^{2k} which are pairwise distinct as symplectic manifolds."}
{"category": "Math", "title": "Estimation of Missing Data Using Computational Intelligence and Decision Trees", "abstract": "This paper introduces a novel paradigm to impute missing data that combines a decision tree with an auto-associative neural network (AANN) based model and a principal component analysis-neural network (PCA-NN) based model. For each model, the decision tree is used to predict search bounds for a genetic algorithm that minimize an error function derived from the respective model. The models' ability to impute missing data is tested and compared using HIV sero-prevalance data. Results indicate an average increase in accuracy of 13% with the AANN based model's average accuracy increasing from 75.8% to 86.3% while that of the PCA-NN based model increasing from 66.1% to 81.6%."}
{"category": "Math", "title": "A devil's staircase from rotations and irrationality measures for Liouville numbers", "abstract": "From Sturmian and Christoffel words we derive a strictly increasing function $\\Delta:[0,\\infty)\\to\\mathbb{R}$. This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of $\\Delta$ distinguishes some irrationality measures of real numbers."}
{"category": "Math", "title": "On the Area Functional of the Second Fundamental Form of Ovaloids", "abstract": "The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called \"mean curvature of the second fundamental form\". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs, are given. In particular, it is shown that the spheres are the only ovaloids which are a critical point of the area functional of the second fundamental form under various constraints."}
{"category": "Math", "title": "On Zeta Functions and Families of Siegel Modular Forms", "abstract": "Let $p$ be a prime, and let $\\Gamma=\\Sp_g(\\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic $L$-functions attached to $\\Sp_g$, and their analytic properties are given. Critical values for the spinor $L$-functions and $p$-adic constructions are discussed. Rankin's lemma of higher genus is established. A general conjecture on a lifting from $ GSp_{2m} \\times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modular forms are given using Ikeda-Miyawaki constructions."}
{"category": "Math", "title": "Asymptotics: Particles, Processes and Inverse Problems. Festschrift for Piet Groeneboom", "abstract": "In September 2006, Piet Groeneboom officially retired as professor of statistics at Delft University of Technology and the Vrije Universiteit in Amsterdam. He did so by delivering his farewell lecture `Summa Cogitatio' ([42] in Piet's publication list) in the Aula of the university in Delft. To celebrate Piet's impressive contributions to statistics and probability, the workshop `Asymptotics: particles, processes and inverse problems' was held from July 10 until July 14, 2006, at the Lorentz Center in Leiden. Many leading researchers in the fields of probability and statistics gave talks at this workshop, and it became a memorable event for all who attended, including the organizers and Piet himself. This volume serves as a Festschrift for Piet Groeneboom. It contains papers that were presented at the workshop as well as some other contributions, and it represents the state of the art in the areas in statistics and probability where Piet has been (and still is) most active. Furthermore, a short CV of Piet Groeneboom and a list of his publications are included."}
{"category": "Math", "title": "Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds", "abstract": "Counterexamples to continuity of optimal transportation on Riemannian manifolds with everywhere positive sectional curvature are provided. These examples show that the condition A3w of Ma, Trudinger, & Wang is not guaranteed by positivity of sectional curvature."}
{"category": "Math", "title": "Localizing the Elliott Conjecture at Strongly Self-absorbing C*-algebras --An Appendix", "abstract": "This note provides some technical support to the proof of a result of W. Winter which shows that two unital separable simple amenable ${\\cal Z}$-absorbing C*-algebras with locally finite decomposition property satisfying the UCT whose projections separate the traces are isomorphic if their $K$-theory is finitely generated and their Elliott invariants are the same."}
{"category": "Math", "title": "On the localized phase of a copolymer in an emulsion: supercritical percolation regime", "abstract": "In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density 1/2. The emulsion is a random mixture of liquids of two types, $A$ and $B$, organised in large square blocks occurring with density $p$ and $1-p$, respectively, where $p \\in (0,1)$. The copolymer in the emulsion has an energy that is minus $\\alpha$ times the number of $AA$-matches minus $\\beta$ times the number of $BB$-matches, where without loss of generality the interaction parameters can be taken from the cone $\\{(\\alpha,\\beta)\\in\\R^2\\colon \\alpha\\geq |\\beta|\\}$. To make the model mathematically tractable, we assume that the copolymer is directed and can only enter and exit a pair of neighbouring blocks at diagonally opposite corners. In \\cite{dHW06}, it was found that in the supercritical percolation regime $p \\geq p_c$, with $p_c$ the critical probability for directed bond percolation on the square lattice, the free energy has a phase transition along a curve in the cone that is independent of $p$. At this critical curve, there is a transition from a phase where the copolymer is fully delocalized into the $A$-blocks to a phase where it is partially localized near the $AB$-interface. In the present paper we prove three theorems that complete the analysis of the phase diagram : (1) the critical curve is strictly increasing; (2) the phase transition is second order; (3) the free energy is infinitely differentiable throughout the partially localized phase."}
{"category": "Math", "title": "Partial Gr\\\"obner bases for multiobjective integer linear optimization", "abstract": "In this paper we present a new methodology for solving multiobjective integer linear programs using tools from algebraic geometry. We introduce the concept of partial Gr\\\"obner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of Gr\\\"obner basis for the single objective case, to the case of multiple objectives, i.e., a partial ordering instead of a total ordering over the feasible vectors. The main property of these bases is that the partial reduction of the integer elements in the kernel of the constraint matrix by the different blocks of the basis is zero. It allows us to prove that this new construction is a test family for a family of multiobjective programs. An algorithm '\\`a la Buchberger' is developed to compute partial Gr\\\"obner bases and two different approaches are derived, using this methodology, for computing the entire set of efficient solutions of any multiobjective integer linear problem (MOILP). Some examples illustrate the application of the algorithms and computational experiments are reported on several families of problems."}
{"category": "Math", "title": "Uniform Bahadur Representation for Local Polynomial Estimates of M-Regression and Its Application to The Additive Model", "abstract": "We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing stationary processes $\\{(Y_{i},\\underline{X}_{i})\\}$. We establish a strong uniform consistency rate for the Bahadur representation of estimators of the regression function and its derivatives. These results are fundamental for statistical inference and for applications that involve plugging in such estimators into other functionals where some control over higher order terms are required. We apply our results to the estimation of an additive M-regression model."}
{"category": "Math", "title": "On some new congruences for binomial coefficients", "abstract": "In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let $p$ be a prime and let $a$ be any positive integer. We determine $\\sum_{k=0}^{p^a-1}\\binom{2k}{k+d}$ mod $p^2$ for $d=0,...,p^a$ and $\\sum_{k=0}^{p^a-1}\\binom{2k}{k+\\delta}$ mod $p^3$ for $\\delta=0,1$. We also show that $$C_n^{-1}\\sum_{k=0}^{p^a-1}C_{p^an+k}=1-3(n+1)((p^a-1)/3) (mod p^2)$$ for every n=0,1,2,..., where $C_m$ is the Catalan number $\\binom{2m}{m}/(m+1)$, and (-) is the Legendre symbol."}
{"category": "Math", "title": "What is a metric space?", "abstract": "The question in the title is discussed briefly, with emphasis on a few basic examples and their properties."}
{"category": "Math", "title": "A remark on Frobenius descent for vector bundles", "abstract": "We give a class of examples of vector bundles on a relative smooth projective curve over Spec Z such that for infinitely many prime reductions the bundle has a Frobenius descent, but the restriction to the generic fiber in characteristic zero is not semistable. In the third section of the paper we prove for a large class of varieties (including abelian varieties) that any vector bundle with this Frobenius descent property is generically semistable."}
{"category": "Math", "title": "Asymptotic integration and dispersion for hyperbolic equations", "abstract": "The aim of this paper is to establish time decay properties and dispersive estimates for strictly hyperbolic equations with homogeneous symbols and with time-dependent coefficients whose derivatives are integrable. For this purpose, the method of asymptotic integration is developed for such equations and representation formulae for solutions are obtained. These formulae are analysed further to obtain time decay of Lp-Lq norms of propagators for the corresponding Cauchy problems. It turns out that the decay rates can be expressed in terms of certain geometric indices of the limiting equation and we carry out the thorough analysis of this relation. This provides a comprehensive view on asymptotic properties of solutions to time-perturbations of hyperbolic equations with constant coefficients. Moreover, we also obtain the time decay rate of the Lp-Lq estimates for equations of these kinds, so the time well-posedness of the corresponding nonlinear equations with additional semilinearity can be treated by standard Strichartz estimates."}
{"category": "Math", "title": "The extremal values of the Wiener index of a tree with given degree sequence", "abstract": "The Wiener index of a graph is the sum of the distances between all pairs of vertices, it has been one of the main descriptors that correlate achemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. The tree that minimizes the Wiener index among trees of given maximal degree was studied. We characterize trees that achieve the maximum and minimum Wiener index, given the number of vertices and the degree sequence."}
{"category": "Math", "title": "Fractional dynamical systems and applications in mechanics and economics", "abstract": "Using the fractional integration and differentiation on R we build the fractional jet fibre bundle on a differentiable manifold and we emphasize some important geometrical objects. Euler-Lagrange fractional equations are described. Some significant examples from mechanics and economics are presented."}
{"category": "Math", "title": "The Generalized Burnside Theorem", "abstract": "All groups have 2 generators. For every prime power q, the Generalized Burnside Theorem (Theorem GB) produces an infinite number of solvable groups, Some, such as groups of a prime power exponent, have only elements of finite order and are therefore finite groups. Others have elements of infinite order and are thus infinite groups. All these groups, even when infinite, are closely related to groups of exponent q. They have at least one generator of order q, and their commutator subgroup has exponent q."}
{"category": "Math", "title": "Seifert surfaces, Commutators and Vassiliev invariants", "abstract": "We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks \"simple\" (e.g. like the complement of a disc) to the lower central series of its fundamental group."}
{"category": "Math", "title": "Symplectic symmetries of 4-manifolds", "abstract": "A study of symplectic actions of a finite group $G$ on smooth 4-manifolds is initiated. The central new idea is the use of $G$-equivariant Seiberg-Witten-Taubes theory in studying the structure of the fixed-point set of these symmetries. The main result in this paper is a complete description of the fixed-point set structure (and the action around it) of a symplectic cyclic action of prime order on a minimal symplectic 4-manifold with $c_1^2=0$. Comparison of this result with the case of locally linear topological actions is made. As an application of these considerations, the triviality of many such actions on a large class of 4-manifolds is established. In particular, we show the triviality of homologically trivial symplectic symmetries of a $K3$ surface (in analogy with holomorphic automorphisms). Various examples and comments illustrating our considerations are also included."}
{"category": "Math", "title": "Symmetries and exotic smooth structures on a $K3$ surface", "abstract": "Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability of actions induced by a holomorphic automorphism of a prime order $\\geq 7$ is proved and nonexistence of smooth actions by several $K3$ groups is established (included among which is the binary tetrahedral group $T_{24}$ which has the smallest order). Concerning symplectic symmetries, the fixed-point set structure of a symplectic cyclic action of a prime order $\\geq 5$ is explicitly determined, provided that the action is homologically nontrivial."}
{"category": "Math", "title": "A Hyperelliptic View on Teichmuller Space. I", "abstract": "We explicitly describe the Teichmuller space TH_n of hyperelliptic surfaces in terms of natural and effective coordinates as the space of certain (2n-6)-tuples of distinct points on the ideal boundary of the Poincare disc. We essentially use the concept of a simple earthquake which is a particular case of a Fenchel-Nielsen twist deformation. Such earthquakes generate a group that acts transitively on TH_n. This fact can be interpreted as a continuous analog of the well-known Dehn theorem saying that the mapping class group is generated by Dehn twists. We find a simple and effective criterion that verifies if a given representation of the surface group \\pi_1\\Sigma in the group of isometries of the hyperbolic plane is faithful and discrete. The article also contains simple and elementary proofs of several known results, for instance, of W. M. Goldman's theorem [Gol1] characterizing the faithful discrete representations as having maximal Toledo invariant (which is essentially the area of the representation in the two-dimensional case)."}
{"category": "Math", "title": "Lattice and Schroder paths with periodic boundaries", "abstract": "We consider paths in the plane with $(1,0),$ $(0,1),$ and $(a,b)$-steps that start at the origin, end at height $n,$ and stay to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at most $b/a,$ then the ordinary generating function for the number of such paths ending at height $n$ is algebraic. Our argument is in two parts. We use a simple combinatorial decomposition to obtain an Appell relation or ``umbral'' generating function, in which the power $z^n$ is replaced by a power series of the form $z^n \\phi_n(z),$ where $\\phi_n(0) = 1.$ Then we convert (in an explicit way) the umbral generating function to an ordinary generating function by solving a system of linear equations and a polynomial equation. This conversion implies that the ordinary generating function is algebraic."}
{"category": "Math", "title": "Mean-field conditions for percolation on finite graphs", "abstract": "Let G_n be a sequence of finite transitive graphs with vertex degree d=d(n) and |G_n|=n. Denote by p^t(v,v) the return probability after t steps of the non-backtracking random walk on G_n. We show that if p^t(v,v) has quasi-random properties, then critical bond-percolation on G_n has a scaling window of width n^{-1/3}, as it would on a random graph. A consequence of our theorems is that if G_n is a transitive expander family with girth at least (2/3 + eps) \\log_{d-1} n, then the size of the largest component in p-bond-percolation with p={1 +O(n^{-1/3}) \\over d-1} is roughly n^{2/3}. In particular, bond-percolation on the celebrated Ramanujan graph constructed by Lubotzky, Phillips and Sarnak has the above scaling window. This provides the first examples of quasi-random graphs behaving like random graphs with respect to critical bond-percolation."}
{"category": "Math", "title": "Parallel marginalization Monte Carlo with applications to conditional path sampling", "abstract": "Monte Carlo sampling methods often suffer from long correlation times. Consequently, these methods must be run for many steps to generate an independent sample. In this paper a method is proposed to overcome this difficulty. The method utilizes information from rapidly equilibrating coarse Markov chains that sample marginal distributions of the full system. This is accomplished through exchanges between the full chain and the auxiliary coarse chains. Results of numerical tests on the bridge sampling and filtering/smoothing problems for a stochastic differential equation are presented."}
{"category": "Math", "title": "Statistical properties of one-dimensional maps with critical points and singularities", "abstract": "We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for H\\\"{o}lder observations."}
{"category": "Math", "title": "A Haar-like Construction for the Ornstein Uhlenbeck Process", "abstract": "The classical Haar construction of Brownian motion uses a binary tree of triangular wedge-shaped functions. This basis has compactness properties which make it especially suited for certain classes of numerical algorithms. We present a similar basis for the Ornstein-Uhlenbeck process, in which the basis elements approach asymptotically the Haar functions as the index increases, and preserve the following properties of the Haar basis: all basis elements have compact support on an open interval with dyadic rational endpoints; these intervals are nested and become smaller for larger indices of the basis element, and for any dyadic rational, only a finite number of basis elements is nonzero at that number. Thus the expansion in our basis, when evaluated at a dyadic rational, terminates in a finite number of steps. We prove the covariance formulae for our expansion and discuss its statistical interpretation and connections to asymptotic scale invariance."}
{"category": "Math", "title": "Residual Velocities in Steady Free Boundary Value Problems of Vector Laplacian Type", "abstract": "This paper describes a technique to determine the linear well-posedness of a general class of vector elliptic problems that include a steady interface, to be determined as part of the problem, that separates two subdomains. The interface satisfies mixed Dirichlet and Neumann conditions. We consider ``2+2'' models, meaning two independent variables respectively on each subdomain. The governing equations are taken to be vector Laplacian, to be able to make analytic progress. The interface conditions can be classified into four large categories, and we concentrate on the one with most physical interest. The well-posedness criteria in this case are particularly clear. In many physical cases, the movement of the interface in time-dependent situations can be reduced to a normal motion proportional to the residual in one of the steady state interface conditions (the elliptic interior problems and the other interface conditions are satisfied at each time). If only the steady state is of interest, one can consider using other residuals for the normal velocity. Our analysis can be extended to give insight into choosing residual velocities that have superior numerical properties. Hence, in the second part, we discuss an iterative method to solve free boundary problems. The advantages of the correctly chosen, non-physical residual velocities are demonstrated in a numerical example, based on a simplified model of two-phase flow with phase change in porous media."}
{"category": "Math", "title": "Undecidability in function fields of positive characteristic", "abstract": "We prove that the first-order theory of any function field K of characteristic p>2 is undecidable in the language of rings without parameters. When K is a function field in one variable whose constant field is algebraic over a finite field, we can also prove undecidability in characteristic 2. The proof uses a result by Moret-Bailly about ranks of elliptic curves over function fields."}
{"category": "Math", "title": "Non-semisimple Macdonald polynomials", "abstract": "The paper is mainly devoted to the irreducibility of the polynomial representation of the double affine Hecke algebra for an arbitrary reduced root systems and generic \"central charge\" q. The technique of intertwiners in the non-semisimple variant is the main tool. We introduce Macdonald's non-semisimple polynomials and use them to analyze the reducibility of the polynomial representation in terms of the affine exponents, counterparts of the classical Coxeter exponents. The focus is on the principal aspects of the technique of intertwiners, including related problems in the theory of reduced decompositions on affine Weyl groups."}
{"category": "Math", "title": "On exit times of Levy-driven Ornstein--Uhlenbeck processes", "abstract": "We prove two martingale identities which involve exit times of Levy-driven Ornstein--Uhlenbeck processes. Using these identities we find an explicit formula for the Laplace transform of the exit time under the assumption that positive jumps of the Levy process are exponentially distributed."}
{"category": "Math", "title": "On the Four Vertex Theorem on planes with radial density $e^{\\phi(r)}$", "abstract": "It is showed that on a plane with a radial density the Four Vertex Theorem holds for the class of all simple closed curves if and only if the density is constant. But for the class of simple closed curves that are invariant under a rotation about the origin, the Four Vertex Theorem holds for every radial density."}
{"category": "Math", "title": "Difference sets and Polynomials of prime variables", "abstract": "Let \\psi(x) be a polynomial with rational coefficients. Suppose that \\psi has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x,y\\in A and a prime p such that x-y=\\psi(p-1). Furthermore, if P be a set of primes with the positive relative upper density, then there exist x,y\\in P and a prime p such that x-y=\\psi(p-1)."}
{"category": "Math", "title": "Relative Singularity Categories and Gorenstein-Projective Modules", "abstract": "We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory $\\omega$ of an abelian category. We introduce the Frobenius category of $\\omega$-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of $\\omega$-Cohen-Macaulay objects is triangle-equivalent to the relative singularity category. As applications, we relate the stable category of (unnecessarily finitely-generated) Gorenstein-projective modules with singularity categories of rings. We prove that for a Gorenstein ring, the stable category of Gorenstein-projective modules is compactly generated and its compact objects coincide with finitely-generated Gorenstein-projective modules up to direct summands."}
{"category": "Math", "title": "K-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials", "abstract": "We have two constructions of the level-$(0,1)$ irreducible representation of the quantum toroidal algebra of type $A$. One is due to Nakajima and Varagnolo-Vasserot. They constructed the representation on the direct sum of the equivariant K-groups of the quiver varieties of type $\\hat{A}$. The other is due to Saito-Takemura-Uglov and Varagnolo-Vasserot. They constructed the representation on the q-deformed Fock space introduced by Kashiwara-Miwa-Stern. In this paper we give an explicit isomorphism between these two constructions. For this purpose we construct simultaneous eigenvectors on the q-Fock space using nonsymmetric Macdonald polynomials. Then the isomorphism is given by corresponding these vectors to the torus fixed points on the quiver varieties."}
{"category": "Math", "title": "Cohomology of $\\mathfrak {osp}(1|2)$ acting on linear differential operators on the supercircle $S^{1|1}", "abstract": "We compute the first cohomology spaces $H^1(\\mathfrak{osp}(1|2);\\mathfrak{D}_{\\lambda,\\mu})$ ($\\lambda, \\mu\\in\\mathbb{R}$) of the Lie superalgebra $\\mathfrak{osp}(1|2)$ with coefficients in the superspace $\\mathfrak{D}_{\\lambda,\\mu}$ of linear differential operators acting on weighted densities on the supercircle $S^{1|1}$. The structure of these spaces was conjectured in \\cite{gmo}. In fact, we prove here that the situation is a little bit more complicated. (To appear in LMP.)"}
{"category": "Math", "title": "A parametrized version of the Borsuk Ulam theorem", "abstract": "The main result of this note is a parametrized version of the Borsuk-Ulam theorem. We show that for a continuous family of Borsuk-Ulam situations, parameterized by points of a compact manifold W, its solution set also depends continuously on the parameter space W. Continuity here means that the solution set supports a homology class which maps onto the fundamental class of W. When W is a subset of Euclidean space, we also show how to construct such a continuous family starting from a family depending in the same way continuously on the points of the boundary of W. This solves a problem related to a conjecture which is relevant for the construction of equilibrium strategies in repeated two-player games with incomplete information. A new method (of independent interest) used in this context is a canonical symmetric squaring construction in Cech homology with coefficients in Z/2Z."}
{"category": "Math", "title": "Regularity of C^{1} smooth surfaces with prescribed p-mean curvature in the Heisenberg group", "abstract": "We consider a $C^{1}$ smooth surface with prescribed $p$(or $H$)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed $p$-mean curvature $H\\in C^{0},$ we show that any characteristic curve is $C^{2}$ smooth and its (line) curvature equals $-H$ in the nonsingular domain$.$ By introducing characteristic coordinates and invoking the jump formulas along characteristic curves, we can prove that the Legendrian (or horizontal) normal gains one more derivative. Therefore the seed curves are $C^{2}$ smooth. We also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. These results can be applied to more general situations."}
{"category": "Math", "title": "A local homology theory for linearly compact modules", "abstract": "We introduce a local homology theory for linearly compact modules which is in some sense dual to the local cohomology theory of A. Grothendieck. Some basic properties such as the noetherianness, the vanishing and non-vanishing of local homology modules of linearly compact modules are proved. A duality theory between local homology and local cohomology modules of linearly compact modules is developed by using Matlis duality and Macdonald duality. As consequences of the duality theorem we obtain some generalizations of well-known results in the theory of local cohomology for semi-discrete linearly compact modules."}
{"category": "Math", "title": "Dismantling sparse random graphs", "abstract": "We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n tending to infinity, then the number in question is essentially the same for all values of k such that k tends to infinity but k=o(n)."}
{"category": "Math", "title": "Leonhard Euler and a q-analogue of the logarithm", "abstract": "We study a q-logarithm which was introduced by Euler and give some of its properties. This q-logarithm did not get much attention in the recent literature. We derive basic properties, some of which were already given by Euler in a 1751-paper and 1734-letter to Daniel Bernoulli. The corresponding q-analogue of the dilogarithm is introduced. The relation to the values at 1 and 2 of a q-analogue of the zeta function is given. We briefly describe some other q-logarithms that have appeared in the recent literature."}
{"category": "Math", "title": "Symplectically aspherical manifolds", "abstract": "This is a survey article on symplectically aspherical manifolds. The paper contains a discussion on constructions of symplectically aspherical manifolds, their topological properties and the role of this class in symplectic topology. Research perspectives are discussed."}
{"category": "Math", "title": "Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes", "abstract": "In this paper, we study Gibbs point processes involving a hardcore interaction which is not necessarily hereditary. We first extend the famous Campbell equilibrium equation, initially proposed by Nguyen and Zessin [Math. Nachr. 88 (1979) 105--115], to the non-hereditary setting and consequently introduce the new concept of removable points. A modified version of the pseudo-likelihood estimator is then proposed, which involves these removable points. We consider the following two-step estimation procedure: first estimate the hardcore parameter, then estimate the smooth interaction parameter by pseudo-likelihood, where the hardcore parameter estimator is plugged in. We prove the consistency of this procedure in both the hereditary and non-hereditary settings."}
{"category": "Math", "title": "Tropical analysis of plurisubharmonic singularities", "abstract": "Some results on singularities of plurisubharmonic functions are put into the context of tropical mathematics."}
{"category": "Math", "title": "Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms", "abstract": "Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in $S^4$, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds interesting duality theorem. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori."}
{"category": "Math", "title": "Orthonormal dilations of Parseval wavelets", "abstract": "We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar group $$BS(1,2)=< u,t | utu^{-1}=t^2>.$$ We give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics. We show that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics. We give concrete examples of Parseval wavelets for which we compute the orthonormal dilations in detail; we show that there are examples of Parseval wavelet sets which have infinitely many non-isomorphic orthonormal dilations."}
{"category": "Math", "title": "Freeness of Conic-Line Arrangements in $\\mathbb P^2$", "abstract": "Let ${\\mathcal C}= \\bigcup_{i=1}^n C_i \\subseteq \\mathbb{P}^2$ be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a large class of arrangements of smooth rational curves, giving an inductive tool for understanding the freeness of the module $\\Omega^1({\\mathcal C})$ of logarithmic differential forms with pole along ${\\mathcal C}$. We also show that the analog of Terao's conjecture (freeness of $\\Omega^1({\\mathcal C})$ is combinatorially determined if ${\\mathcal C}$ is a union of lines) is false in this setting."}
{"category": "Math", "title": "Energy Optimal Control for Quantum System Evolving on SU(1,1)", "abstract": "This paper discusses the energy optimal control problem for the class of quantum systems that possess dynamical symmetry of SU(1,1), which are widely studied in various physical problems in the quantum theory. Based on the maximum principle on Lie group, the complete set of optimal controls are analytically obtained, including both normal and abnormal extremals. The results indicate that the normal extremal controls can be expressed by the Weierstrass elliptic function, while the abnormal extremal controls can only be constant functions of time t."}
{"category": "Math", "title": "On the lower central series of an associative algebra", "abstract": "This paper continues the study of the lower central series quotients of an associative algebra A, regarded as a Lie algebra, which was started in math/0610410 by Feigin and Shoikhet. Namely, it provides a basis for the second quotient in the case when A is the free algebra in n generators (note that the Hilbert series of this quotient was determined earlier in math/0610410). Further, it uses this basis to determine the structure of the second quotient in the case when A is the free algebra modulo the relations saying that the generators have given nilpotency orders. Finally, it determines the structure of the third and fourth quotient in the case of 2 generators, confirming an answer conjectured in math/0610410. Finally, in the appendix, the results of math/0610410 are generalized to the case when A is an arbitrary associative algebra (under certain conditions on $A$)."}
{"category": "Math", "title": "Asymptotics and Sequential Closures of Continued Fractions and Generalizations", "abstract": "Given a sequence of complex square matrices, $a_n$, consider the sequence of their partial products, defined by $p_n=p_{n-1}a_{n}$. What can be said about the asymptotics as $n\\to\\infty$ of the sequence $f(p_n)$, where $f$ is a continuous function? A special case of our most general result addresses this question under the assumption that the matrices $a_n$ are an $l_1$ perturbation of a sequence of matrices with bounded partial products. We apply our theory to investigate the asymptotics of the approximants of continued fractions. In particular, when a continued fraction is $l_1$ limit 1-periodic of elliptic or loxodromic type, we show that its sequence of approximants tends to a circle in $\\hat{\\mathbb{C}}$, or to a finite set of points lying on a circle. Our main theorem on such continued fractions unifies the treatment of the loxodromic and elliptic cases, which are convergent and divergent, respectively. When an approximating sequence tends to a circle, we obtain statistical information about the limiting distribution of the approximants. When the circle is the real line, the points are shown to have a Cauchy distribution with parameters given in terms of modifications of the original continued fraction. As an example of the general theory, a detailed study of a $q$-continued fraction in five complex variables is provided. The most general theorem in the paper holds in the context of Banach algebras. The theory is also applied to $(r,s)$-matrix continued fractions and recurrence sequences of Poincar\\'e type and compared with closely related literature."}
{"category": "Math", "title": "The K\\\"unneth formula for nuclear $DF$-spaces and Hochschild cohomology", "abstract": "We consider complexes $(\\X, d)$ of nuclear Fr\\'echet spaces and continuous boundary maps $d_n$ with closed ranges and prove that, up to topological isomorphism, $ (H_{n}(\\X, d))^*$ $\\iso$ $H^{n}(\\X^*,d^*),$ where $(H_{n}(\\X,d))^*$ is the strong dual space of the homology group of $(\\X,d)$ and $ H^{n}(\\X^*,d^*)$ is the cohomology group of the strong dual complex $(\\X^*,d^*)$. We use this result to establish the existence of topological isomorphisms in the K\\\"{u}nneth formula for the cohomology of complete nuclear $DF$-complexes and in the K\\\"{u}nneth formula for continuous Hochschild cohomology of nuclear $\\hat{\\otimes}$-algebras which are Fr\\'echet spaces or $DF$-spaces for which all boundary maps of the standard homology complexes have closed ranges. We describe explicitly continuous Hochschild and cyclic cohomology groups of certain tensor products of $\\hat{\\otimes}$-algebras which are Fr\\'echet spaces or nuclear $DF$-spaces."}
{"category": "Math", "title": "Sur la persistance des courbes invariantes pour les dynamiques holomorphes fibr\\'ees lisses", "abstract": "En s'appuyant sur un th\\'eor\\`me de fonctions implicites de Hamilton nous montrons la persistance d'une courbe invariante indiff\\'rente pour une dynamique holomorpphe fibr\\'ee en classe $C^{\\infty}$. Une condition diophantienne sur la paire de nombres de rotation est demand\\'ee. Nous montrons aussi que cette condition est optimale."}
{"category": "Math", "title": "Chaoticity for multi-class systems and exchangeability within classes", "abstract": "Classical results for exchangeable systems of random variables are extended to multi-class systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multi-class system, given the value of the vector of the empirical measures of its classes, corresponds to independent uniform orderings of the samples within each class, and that a family of such systems converges in law if and only if the corresponding empirical measure vectors converge in law. As a corollary, convergence within each class to an infinite i.i.d. system implies asymptotic independence between different classes. A result implying the Hewitt-Savage 0-1 Law is also extended."}
{"category": "Math", "title": "Symmetric $\\alpha$-stable subordinators and Cauchy problems", "abstract": "We survey the results in Nane (E. Nane, Higher order PDE's and iterated processes, Trans. American Math. Soc. (to appear)) and Baeumer, Meerschaert, and Nane (B. Baeumer, M.M. Meerschaert and E. Nane, Brownian subordinators and fractional Cauchy problems: Submitted (2007)) which deal with PDE connection of some iterated processes, and obtain a new probabilistic proof of the equivalence of the higher order PDE's and fractional in time PDE's."}
{"category": "Math", "title": "Clique-width of unit interval graphs", "abstract": "The clique-width is known to be unbounded in the class of unit interval graphs. In this paper, we show that this is a minimal hereditary class of unbounded clique-width, i.e., in every hereditary subclass of unit interval graphs the clique-width is bounded by a constant."}
{"category": "Math", "title": "On the Reduction Process of Nucci-Reduce Algorithm for Computing Nonlocal Symmetries of Dynamical Systems:A case study of the Kepler and Kepler-related problems", "abstract": "The snags in Nucci(1996)REDUCE algorithm are the intrinsic computational efforts and the ability to recognize the ignorable variable(s) during the reduction process of the algorithm. An inappropriate choice of the ignorable variable(s)may lead to an infinite loop. We construct an isomorphic transformation which ameliorates these problems, and with which a simple, definite steps of algebraic process, produced equivalent system of equations to that of Nucci that are easily solved by Lie point symmetry algorithm."}
{"category": "Math", "title": "Bessel potentials and optimal Hardy and Hardy-Rellich inequalities", "abstract": "We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R^n,$n \\geq 1$, so that the following inequalities hold for all $u \\in C_{0}^{\\infty}(B)$: $\\int_{B}V(x)|\\nabla u |^{2}dx \\geq \\int_{B} W(x)u^2dx$, and $\\int_{B}V(x)|\\Delta u |^{2}dx \\geq \\int_{B} W(x)|\\nabla u|^{2}dx+(n-1)\\int_{B}(\\frac{V(x)}{|x|^2}-\\frac{V_r(|x|)}{|x|})|\\nabla u|^2dx$. This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W) - that we call Bessel pairs -as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results -old and new- about Hardy and Hardy-Rellich type inequalities, such as those obtained by Caffarelli-Kohn-Nirenberg, Brezis-Vazquez, Wang-Willem, Adimurthi-Chaudhuri-Ramaswamy, Filippas-Tertikas, Adimurthi-Grossi -Santra, Tertikas-Zographopoulos, and Blanchet-Bonforte-Dolbeault-Grillo-Vasquez."}
{"category": "Math", "title": "Symplectic embeddings of polydisks", "abstract": "If P is a polydisk with radii R_1 < ... < R_n and P' is a polydisk with radii R'_1 < ... < R'_n, then we construct a symplectic embedding from P into P' provided that C(n) R_1 < R'_1 and C(n) R_1 ... R_n < C(n) R'_1 ... R'_n. Up to a constant factor, these conditions are optimal."}
{"category": "Math", "title": "Apery limits and special values of L-functions", "abstract": "We describe a general method to determine the Apery limits of a differential equation that have a modular-function origin. As a by-product of our analysis, we discover a family of identities involving the special values of L-functions associated with modular forms. The proof of these identities is independent of differential equations and Apery limits."}
{"category": "Math", "title": "Some new directions in p-adic Hodge theory", "abstract": "We recall some basic constructions from p-adic Hodge theory, then describe some recent results in the subject. We chiefly discuss the notion of B-pairs, introduced recently by Berger, which provides a natural enlargement of the category of p-adic Galois representations. (This enlargement, in a different form, figures in recent work of Colmez, Bellaiche, and Chenevier on trianguline representations.) We also discuss results of Liu that indicate that the formalism of Galois cohomology, including Tate local duality, extends to B-pairs."}
{"category": "Math", "title": "A combinatorial approach to functorial quantum sl(k) knot invariants", "abstract": "This paper contains a categorification of the sl(k) link invariant using parabolic singular blocks of category O. Our approach is intended to be as elementary as possible, providing combinatorial proofs of the main results of Sussan. We first construct an exact functor valued invariant of webs or 'special' trivalent graphs labelled with 1, 2, k-1, k satisfying the MOY relations. Afterwards we extend it to the sl(k)-invariant of links by passing to the derived categories. The approach using foams appears naturally in this context. More generally, we expect that our approach provides a representation theoretic interpretation of the sl(k)-homology, based on foams and the Kapustin-Lie formula. Conjecturally this implies that the Khovanov-Rozansky link homology is obtained from our invariant by restriction."}
{"category": "Math", "title": "Proper holomorphic mappings in the special class of Reinhardt domains", "abstract": "A complete characterization of proper holomorphic mappings between domains from the class of all pseudoconvex Reinhardt domains in $\\C^2$ with the logarithmic image equal to a strip or a half-plane is given."}
{"category": "Math", "title": "Universal local parametrizations via heat kernels and eigenfunctions of the Laplacian", "abstract": "We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with $\\mathcal{C}^\\alpha$ metric). These coordinates are bi-Lipschitz on embedded balls of the domain or manifold, with distortion constants that depend only on natural geometric properties of the domain or manifold. The proof of these results relies on estimates, from above and below, for the heat kernel and its gradient, as well as for the eigenfunctions of the Laplacian and their gradient. These estimates hold in the non-smooth category, and are stable with respect to perturbations within this category. Finally, these coordinate systems are intrinsic and efficiently computable, and are of value in applications."}
{"category": "Math", "title": "Factorial ratios, hypergeometric series, and a family of step functions", "abstract": "We give a complete classification of a certain family of step functions related to the Nyman--Beurling approach to the Riemann hypothesis and previously studied by V. I. Vasyunin. Equivalently, we completely describe when certain sequences of ratios of factorial products are always integral. Essentially, once certain observations are made, this comes down to an application of Beukers and Heckman's classification of the monodromy of the hypergeometric function nF_{n-1}. We also note applications to the classification of cyclic quotient singularities."}
{"category": "Math", "title": "On proofs of certain combinatorial identities", "abstract": "In this paper we formulate combinatorial identities that give representation of positive integers as linear combination of even powers of 2 with binomial coefficients. We present side by side combinatorial as well as computer generated proofs using the Wilf-Zeilberger(WZ) method."}
{"category": "Math", "title": "K3 surfaces of finite height over finite fields", "abstract": "Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for any such lifting, the endormorphism algebra of the transcendental cycles, as a Hodge module, is a CM field. The Tate conjecture for the product of certain two K3 surfaces is also proved. We illustrate by examples how to determine explicitly the formal Brauer group associated to a K3 surface over k. Examples discussed here are all of hypergeometric type."}
{"category": "Math", "title": "Improvement on Parameters of Algebraic-Geometry Codes from Hermitian Curves", "abstract": "Motivated by Xing's method [7], we show that there exist [n,k,d] linear Hermitian codes over F_{q^2} with k+d>=n-3 for all sufficiently large q. This improves the asymptotic bound of Algebraic-Geometry codes from Hermitian curves given in [9,10]."}
{"category": "Math", "title": "On Kummer type construction of supersingular K3 surfaces in characteristic 2", "abstract": "We show that every supersingular K3 surface in characteristic 2 with Artin invariant less than or equal to 2 is obtained by the Kummer type construction of Schroeer."}
{"category": "Math", "title": "Conjugacy, roots, and centralizers in Thompson's group $F$", "abstract": "We complete the program begun by Brin and Squier of characterising conjugacy in Thompson's group $F$ using the standard action of $F$ as a group of piecewise linear homeomorphisms of the unit interval."}
{"category": "Math", "title": "Generalized solutions to nonlinear first order Cauchy problems", "abstract": "The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and regularity of the generalized solutions to a family of nonlinear first order Cauchy problems. The spaces of generalized solutions are obtained as the completions of suitably constructed uniform convergence spaces."}
{"category": "Math", "title": "The geometry of fractional osculator bundle of higher order and applications", "abstract": "Using the reviewed Riemann-Liouville fractional derivative we introduce the fractional osculator Lagrange space of k order and the main structures on it. The results are applied at the k order fractional prolongation of Lagrange, Finsler and Riemann fractional structures."}
{"category": "Math", "title": "Sign changes of coefficients of half integral weight modular forms", "abstract": "For a half integral weight modular form $f$ we study the signs of the Fourier coefficients $a(n)$. If $f$ is a Hecke eigenform of level $ N$ with real Nebentypus character, and $t$ is a fixed square-free positive integer with $a(t)\\neq 0$, we show that for all but finitely many primes $p$ the sequence $(a(tp^{2m}))_{m}$ has infinitely many signs changes. Moreover, we prove similar (partly conditional) results for arbitrary cusp forms $f$ which are not necessarily Hecke eigenforms."}
{"category": "Math", "title": "On rate optimality for ill-posed inverse problems in econometrics", "abstract": "In this paper, we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildly ill-posed and severely ill-posed cases. We show that both a simple projection estimator for the NPIR model, and a sieve minimum distance estimator for the NPIV model, can achieve the minimax risk lower bounds, and are rate-optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases."}
{"category": "Math", "title": "Nonparametric estimation for L\\'evy processes from low-frequency observations", "abstract": "We suppose that a L\\'evy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the L\\'evy-Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific $C^2$-criterion this estimator is rate-optimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line."}
{"category": "Math", "title": "Continuity of the radius of convergence of p-adic differential equations on Berkovich analytic spaces", "abstract": "We consider a vector bundle with integrable connection (\\cE,\\na) on an analytic domain U in the generic fiber \\cX_{\\eta} of a smooth formal p-adic scheme \\cX, in the sense of Berkovich. We define the \\emph{diameter} \\delta_{\\cX}(\\xi,U) of U at \\xi\\in U, the \\emph{radius} \\rho_{\\cX}(\\xi) of the point \\xi\\in\\cX_{\\eta}, the \\emph{radius of convergence} of solutions of (\\cE,\\na) at \\xi, R(\\xi) = R_{\\cX}(\\xi, U,(\\cE, \\na)). We discuss (semi-) continuity of these functions with respect to the Berkovich topology. In particular, under we prove under certain assumptions that \\delta_{\\cX}(\\xi,U), \\rho_{\\cX}(\\xi) and R_{\\xi}(U,\\cE,\\na) are upper semicontinuous functions of \\xi; for Laurent domains in the affine space, \\delta_{\\cX}(-,U) is continuous. In the classical case of an affinoid domain U of the analytic affine line, R is a continuous function."}
{"category": "Math", "title": "Maximal Entropy Measures for Piecewise Affine Surface Homeomorphisms", "abstract": "We study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected \"good\" returns to dispense with Markov partitions. We control these good returns through some entropy and ergodic arguments."}
{"category": "Math", "title": "Equivalence and self-improvement of p-fatness and Hardy's inequality, and association with uniform perfectness", "abstract": "We present an easy proof that $p$--Hardy's inequality implies uniform $p$--fatness of the boundary when $p=n$. The proof works also in metric space setting and demonstrates the self--improving phenomenon of the $p$--fatness. We also explore the relationship between $p$-fatness, $p$-Hardy inequality, and the uniform perfectness for all $p\\ge 1$, and demonstrate that in the Ahlfors $Q$-regular metric measure space setting with $p=Q$, these three properties are equivalent."}
{"category": "Math", "title": "A Clark-Ocone formula in UMD Banach spaces", "abstract": "Let H be a separable real Hilbert space and let F = (F_t)_{t\\in [0,T]} be the augmented filtration generated by an H-cylindrical Brownian motion W_H on [0,T]. We prove that if E is a UMD Banach space, 1\\leq p<\\infty, and f\\in D^{1,p}(E) is F_T-measurable, then f = \\E f + \\int_0^T P_F(Df) dW_H where D is the Malliavin derivative and P_F is the projection onto the F-adapted elements in a suitable Banach space of L^p-stochastically integrable L(H,E)-valued processes."}
{"category": "Math", "title": "Biharmonic Hypersurfaces in 4-Dimensional Space Forms", "abstract": "We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms."}
{"category": "Math", "title": "Surface branched covers and geometric 2-orbifolds", "abstract": "For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whether a collection of partitions of an integer having the appropriate total length (that we call a candidate cover) always comes from some branched cover. The answer is known to be in the affirmative whenever the candidate base surface is not the 2-sphere, while for the 2-sphere exceptions do occur. A long-standing conjecture however asserts that when the candidate degree is a prime number, a candidate cover is always realizable. In this paper we analyze the question from the point of view of the geometry of 2-orbifolds, and we provide strong supporting evidence for the conjecture. In particular, we exhibit three different sequences of candidate covers, indexed by their degree, such that for each sequence: (1) The degrees giving realizable covers have asymptotically zero density in the naturals; (2) Each prime degree gives a realizable cover."}
{"category": "Math", "title": "Construction of covers in positive characteristic via degeneration", "abstract": "In this note we construct examples of covers of the projective line in positive characteristic such that every specialization is inseparable. The result illustrates that it is not possible to construct all covers of the generic r-pointed curve of genus zero inductively from covers with a smaller number of branch points."}
{"category": "Math", "title": "Geometrical embeddings for distributions into algebras of generalized functions", "abstract": "We use spectral theory to produce embeddings of distributions in the algebras of generalized functions on a closed Riemannian manifold. These embeddings are invariant under isometries and preserve the singularity structure of the distributions."}
{"category": "Math", "title": "Uniform limit laws of the logarithm for nonparametric estimators of the regression function in presence of censored data", "abstract": "In this paper, we establish uniform-in-bandwidth limit laws of the logarithm for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.) estimators of the multivariate regression function under random censorship. A similar result is deduced for estimators of the conditional distribution function. The uniform-in-bandwidth consistency for estimators of the conditional density and the conditional hazard rate functions are also derived from our main result. Moreover, the logarithm laws we establish are shown to yield almost sure simultaneous asymptotic confidence bands for the functions we consider. Examples of confidence bands obtained from simulated data are displayed."}
{"category": "Math", "title": "Sur une g\\'en\\'eralisation de la notion de syst\\`eme dynamique de rang un d\\'efinie par une propri\\'et\\'e de pistage (On a weak version of the rank one property defined by shadowing)", "abstract": "We investigate a shadowing property which appears naturally in the study of piecewise monotonic maps of the interval. It turns out to be a weak form of the rank one property, a well-known notion in abstract ergodic theory. We show that this new property is implied by finite or even local rank, but that it is logically independent of loose Bernoulliness. We give (counter)examples, including L.B. systems with arbitrarily high-order polynomial complexity. The shadowing property defines a small subset of all zero-entropy systems, in the sense that it defines a closed set with empty interior with respect to the $\\db$-metric, induced by the Hamming distance. We also make some remarks on the link between the shadowed system and the sequence assumed by the shadowing property."}
{"category": "Math", "title": "Vanishing Viscosity Limits and Boundary Layers for Circularly Symmetric 2D Flows", "abstract": "We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [LMN], on the vanishing viscosity limit of circularly symmetric viscous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in $L^2$-norm as long as the prescribed angular velocity $\\alpha(t)$ of the boundary has bounded total variation. Here we establish convergence in stronger $L^2$ and $L^p$-Sobolev spaces, allow for more singular angular velocities $\\alpha$, and address the issue of analyzing the behavior of the boundary layer. This includes an analysis of concentration of vorticity in the vanishing viscosity limit. We also consider such flows on an annulus, whose two boundary components rotate independently. [LMN] Lopes Filho, M. C., Mazzucato, A. L. and Nussenzveig Lopes, H. J., Vanishing viscosity limit for incompressible flow inside a rotating circle, preprint 2006."}
{"category": "Math", "title": "Global rigidity in CR geometry: the Schoen-Webster theorem", "abstract": "Schoen-Webster theorem asserts a pseudoconvex CR manifold whose automorphism group acts non properly is either the standard sphere or the Heisenberg space. The purpose of this paper is to survey successive works around this result and then provide a short geometric proof in the compact case."}
{"category": "Math", "title": "Semiclassical Resonances of Schr\\\"odinger operators as zeroes of regularized determinants", "abstract": "We prove that the resonances of long range perturbations of the (semiclassical) Laplacian are the zeroes of natural perturbation determinants. We more precisely obtain factorizations of these determinants of the form $ \\prod_{w = {\\rm resonances}}(z-w) \\exp (\\varphi_p(z,h)) $ and give semiclassical bounds on $ \\partial_z \\varphi_p $ as well as a representation of Koplienko's regularized spectral shift function. Here the index $ p \\geq 1 $ depends on the decay rate at infinity of the perturbation."}
{"category": "Math", "title": "On Kato's method for Navier--Stokes Equations", "abstract": "We investigate Kato's method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier-Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in ${\\mathbb R}^3$ and irregular domains in ${\\mathbb R}^n$."}
{"category": "Math", "title": "Strong asymptotics for Christoffel functions of planar measures", "abstract": "We prove a version of strong asymptotics of Christoffel functions with varying weights for a general class of sets E and measures in the complex plane. This class includes all regular measures in the sense of Stahl-Totik on regular compact sets E in the plane and even allows varying weights. Our main theorems cover some known results for subsets E of the real line R; in particular, we recover information in the case of E=R with Lebesgue measure dx and weight w(x) = exp(-Q(x)) where Q(x) is a nonnegative, even degree polynomial having positive leading coefficient."}
{"category": "Math", "title": "Localized non-diffusive asymptotic patterns for nonlinear parabolic equations with gradient absorption", "abstract": "We study the large-time behaviour of the solutions $u$ of the evolution equation involving nonlinear diffusion and gradient absorption $\\partial_t u - \\Delta_p u + |\\nabla u|^q=0$. We consider the problem posed for $x\\in {\\mathbb R}^N $ and $t>0$ with non-negative and compactly supported initial data. We take the exponent $p>2$ which corresponds to slow $p$-Laplacian diffusion, and the exponent $q$ in the superlinear range $1<q<p-1$. In this range the influence of the Hamilton-Jacobi term $ |\\nabla u|^q$ is determinant, and gives rise to the phenomenon of localization. The large time behaviour is described in terms of a suitable self-similar solution that solves a Hamilton-Jacobi equation. The shape of the corresponding spatial pattern is rather conical instead of bell-shaped or parabolic."}
{"category": "Math", "title": "The $K$-theory of toric varieties", "abstract": "Recent advances in computational techniques for $K$-theory allow us to describe the $K$-theory of toric varieties in terms of the $K$-theory of fields and simple cohomological data."}
{"category": "Math", "title": "Adding \\pm 1 to the argument of an Hall-Littlewood polynomial", "abstract": "Shifting by \\pm 1 powers sums: p_i \\to p_i \\pm 1 induces a transformation on symmetric functions that we detail in the case of Hall-Littlewood polynomials. By iteration, this gives a description of these polynomials in terms of plane partitions, as well as some generating functions. We recover in particular an identity of Warnaar related to Rogers-Ramanujan identities."}
{"category": "Math", "title": "Intersection Numbers of Polygon Spaces", "abstract": "We study the intersection ring of the space $\\M(\\alpha_1,...,\\alpha_m)$ of polygons in $\\R^3$. We find homology cycles dual to generators of this ring and prove a recursion relation in $m$ (the number of steps) for their intersection numbers. This result is analog of the recursion relation appearing in the work of Witten and Kontsevich on moduli spaces of punctured curves and on the work of Weitsman on moduli spaces of flat connections on two-manifolds of genus $g$ with $m$ marked points. Based on this recursion formula we obtain an explicit expression for the computation of the intersection numbers of polygon spaces and use it in several examples. Among others, we study the special case of equilateral polygon spaces (where all the $\\alpha_i$ are the same) and compare our results with the expressions for these particular spaces that have been determined by Kamiyama and Tezuka. Finally, we relate our explicit formula for the intersection numbers with the generating function for intersection pairings of the moduli space of flat connections of Yoshida, as well as with equivalent expressions for polygon spaces obtained by Takakura and Konno through different techniques."}
{"category": "Math", "title": "Representing simple d-dimensional polytopes by d polynomials", "abstract": "A polynomial representation of a convex d-polytope P is a finite set \\{p_1(x),...,p_n(x)\\} of polynomials over E^d such that P=\\setcond{x \\in \\E^d}{p_1(x) \\ge 0 {for every} 1 \\le i \\le n}. By s(d,P) we denote the least possible number of polynomials in a polynomial representation of P. It is known that d \\le s(d,P) \\le 2d-1. Moreover, it is conjectured that s(d,P)=d for all convex d-polytopes P. We confirm this conjecture for simple d-polytopes by providing an explicit construction of d polynomials that represent a given simple d-polytope P."}
{"category": "Math", "title": "Pluripolar hulls and fine analytic structure", "abstract": "We discuss the relation between pluripolar hulls and fine analytic structure. Our main result is the following. For each non polar subset $S$ of the complex plane $\\mathbb C$ we prove that there exists a pluripolar set $E \\subset (S \\times \\mathbb C)$ with the property that the pluripolar hull of $E$ relative to $\\mathbb C^2$ contains no fine analytic structure and its projection onto the first coordinate plane equals $\\mathbb C$."}
{"category": "Math", "title": "Homogeneous bundles and the first eigenvalue of symmetric spaces", "abstract": "We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type."}
{"category": "Math", "title": "The Mean Curvature of the Second Fundamental Form of a hypersurface", "abstract": "An expression for the first variation of the area functional of the second fundamental form is given for a hypersurface in a semi-Riemannian space. The concept of the \"mean curvature of the second fundamental form\" is then introduced. Some characterisations of extrinsic hyperspheres in terms of this curvature are given."}
{"category": "Math", "title": "Combination of quasiconvex subgroups of relatively hyperbolic groups", "abstract": "For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two quasiconvex subgroups $Q$ and $R$ is quasiconvex and isomorphic to $Q \\ast_{Q\\cap R} R$. Our results generalized known combination theorems for quasiconvex subgroups of word-hyperbolic groups. Some applications are presented."}
{"category": "Math", "title": "Manifolds of Hilbert space projections", "abstract": "The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(\\lambda) = \\exp(i(\\lambda_1\\psi_1 + ... + \\lambda_n\\psi_n)) for \\lambda in R^n, gives rise to a manifold M of orthogonal projections for the subspaces u(\\lambda)H^2(R) of L^2(R). For classes of admissible functions \\psi_i the strong operator topology closures of M and M \\cup M^\\perp are determined explicitly as various n-balls and n-spheres. The arguments used are direct and rely on the analysis of oscillatory integrals and Hilbert space geometry. Some classes of these closed projection manifolds are classified up to unitary equivalence. In particular the Fourier-Plancherel 2-sphere and the hyperbolic 3-sphere of Katavolos and Power appear as distinguished special cases admitting nontrivial unitary automorphisms groups which are explicitly described."}
{"category": "Math", "title": "Quasi-semi-stable representations", "abstract": "Fix K a p-adic field and denote by G_K its absolute Galois group. Let K_infty be the extension of K obtained by adding (p^n)-th roots of a fixed uniformizer, and G_\\infty its absolute Galois group. In this article, we define a class of p-adic torsion representations of G_\\infty, named quasi-semi-stable. We prove that these representations are \"explicitly\" described by a certain category of linear algebra objects. The results of this note should be consider as a first step in the understanding of the structure of quotients of two lattices in a crystalline (resp. semi-stable) Galois representation."}
{"category": "Math", "title": "\\'Etale cohomology of the complement of a linear subspace arrangement", "abstract": "We prove a formula for the cup product on the l-adic cohomology of the complement of a linear subspace arrangement."}
{"category": "Math", "title": "A skein approach to Bennequin type inequalities", "abstract": "We give a simple unified proof for several disparate bounds on Thurston-Bennequin number for Legendrian knots and self-linking number for transverse knots in R^3, and provide a template for possible future bounds. As an application, we give sufficient conditions for some of these bounds to be sharp."}
{"category": "Math", "title": "New Constructions of Slice Links", "abstract": "We use techniques of Freedman and Teichner to prove that, under certain circumstances, the multi-infection of a slice link is again slice (not necessarily smoothly slice). We provide a general context for proving links are slice that includes many of the previously known results."}
{"category": "Math", "title": "Rigid Dualizing Complexes via Differential Graded Algebras (Survey)", "abstract": "In this article we survey recent results on rigid dualizing complexes over commutative algebras. We begin by recalling what are dualizing complexes. Next we define rigid complexes, and explain their functorial properties. Due to the possible presence of torsion, we must use differential graded algebras in the constructions. We then discuss rigid dualizing complexes. Finally we show how rigid complexes can be used to understand Cohen-Macaulay homomorphisms and relative dualizing sheaves."}
{"category": "Math", "title": "Generalized Vandermonde's system and Lagrange's interpolation", "abstract": "We give explicit formulas as well as a quadratic time algorithm to solve (so called) generalized Vandermonde's systems of p linear equations and n variables. It allows in particular to find all (so called Lagrange's) interpolation polynoms with degree n-1 taking given values in p distinct points."}
{"category": "Math", "title": "A general learning algorithm for functions between metric spaces", "abstract": "In this paper we show how to approximate (\"learn\") a function f, where X and Y are metric spaces."}
{"category": "Math", "title": "The Candy-Passing Game for c\\geq3n-2", "abstract": "We determine the behavior of Tanton's candy-passing game for all distributions of at least 3n-2 candies, where n is the number of students. Specifically, we show that the configuration of candy in such a game eventually becomes fixed."}
{"category": "Math", "title": "Hyperbolic Unit Groups and Quaternion Algebras", "abstract": "We Classify the rational quadratic extensions K and the finite groups G for which the group ring R[G] of G over the ring R of integers of K has the property that the group of units of augmentation 1 of R[G] is hyperbolic. We also construct units in a non-split quaternion algebra over R."}
{"category": "Math", "title": "Inverse spectral problems on a closed manifold", "abstract": "In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $\\Sigma$ into two components and we know the eigenvalues $\\lambda_j$ of the Laplace operator on $(M,g)$ and also the Cauchy data, on $\\Sigma$, of the corresponding eigenfunctions $\\phi_j$, i.e. $\\phi_j|_{\\Sigma},\\partial_\\nu\\phi_j|_{\\Sigma}$, where $\\nu$ is the normal to $\\Sigma$. We prove that these data determine $(M,g)$ uniquely, i.e. up to an isometry. In the second problem we are given much less data, namely, $\\lambda_j$ and $\\phi_j|_{\\Sigma}$ only. However, if $\\Sigma$ consists of at least two components, $\\Sigma_1, \\Sigma_2$, we are still able to determine $(M,g)$ assuming some conditions on $M$ and $\\Sigma$. These conditions are formulated in terms of the spectra of the manifolds with boundary obtained by cutting $M$ along $\\Sigma_i$, $i=1,2$, and are of a generic nature. We consider also some other inverse problems on $M$ related to the above with data which is easier to obtain from measurements than the spectral data described."}
{"category": "Math", "title": "Stabilization for mapping class groups of 3-manifolds", "abstract": "We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks, and includes as particular cases homological stability for symmetric automorphisms of free groups, automorphisms of certain free products, and handlebody mapping class groups. Our methods also apply to manifolds of other dimensions in the case of stabilization by punctures."}
{"category": "Math", "title": "Topological rigidity for holomorphic foliations", "abstract": "We study analytic deformations and unfoldings of holomorphic foliations in complex projective plane $\\mathbb{C}P(2)$. Let $\\{\\mathcal{F}_t\\}_{t \\in \\mathbb{D}_{\\epsilon}}$ be topological trivial (in $\\mathbb{C}^2$) analytic deformation of a foliation $\\mathcal{F}_0$ on $\\mathbb{C}^2$. We show that under some dynamical restriction on $\\mathcal{F}_0$, we have two possibilities: $\\mathcal{F}_0$ is a Darboux (logarithmic) foliation, or $\\{\\mathcal{F}_t\\}_{t \\in \\mathbb{D}_{\\epsilon}}$ is an unfolding. We obtain in this way a link between the analytical classification of the unfolding and the one of its germs at the singularities on the infinity line. Also we prove that a finitely generated subgroup of $\\mathrm{Diff}(\\mathbb{C}^n,0)$ with polynomial growth is solvable."}
{"category": "Math", "title": "A probabilistic proof of Wallis's formula for pi", "abstract": "Using mostly elementary results and functions from probability, we prove Wallis's formula for pi: pi/2 = prod_n (2n * 2n) / ((2n-1) * (2n+1)). The proof involves normalization constants and the Gamma function, Standard normal, and the Student t-Distribution."}
{"category": "Math", "title": "Irrationality measure and lower bounds for pi(x)", "abstract": "In this note we show how the irrationality measure of $\\zeta(s) = \\pi^2/6$ can be used to obtain explicit lower bounds for $\\pi(x)$. We analyze the key ingredients of the proof of the finiteness of the irrationality measure, and show how to obtain good lower bounds for $\\pi(x)$ from these arguments as well. While versions of some of the results here have been done by other authors, our arguments are more elementary and yield a lower bound of order $x/\\log x$ as a natural boundary."}
{"category": "Math", "title": "L^p-estimates for the wave equation associated to the Grushin operator", "abstract": "Let G:=-((d/dx)^2+x^2(d/du)^2) denote the Grusin operator on R^2. Consider the Cauchy problem for the associated wave equation on R x R^2, given by ((d/dt)^2+G)v =0, v(0,.)=f, d/dt v(0,.)=g, where t denotes time and f, g are suitable functions. The focus of this thesis lies on smoothness properties of the solution v for fixed time t with respect to the initial data. Smoothness can be measured in terms of Sobolev norms |f|_Lp^\\alpha:=|(1+G)^{\\alpha/2}f|_Lp, defined in terms of the differential operator G. Let S_C denote the strip S_C:={(x,u) in R^2, |x|<=C} in R^2. We prove that for 1<=p<=\\infty the solution v is in L_p^{-\\alpha} if our initial data f and g are Lp-functions supported in a fixed strip S_C, C>0, and if \\alpha>|1/p-1/2| holds. In fact, we show that for every C>0 the operator \\exp(itG^{1/2})(1+G)^{-\\alpha/2}, defined for Schwartz functions, extends to a bounded operator from Lp(S_C) to Lp(R^2) for all \\alpha>|1/p-1/2|."}
{"category": "Math", "title": "Graphs of Transportation Polytopes", "abstract": "This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way transportation polytopes and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an $m\\times n$ transportation polytope is a multiple of the greatest common divisor of $m$ and $n$."}
{"category": "Math", "title": "Linear maps preserving fibers", "abstract": "Let $G\\subset\\GL(V)$ be a complex reductive group where $\\dim V<\\infty$, and let $\\pi\\colon V\\to\\quot VG$ be the categorical quotient. Let $\\NN:=\\pi\\inv\\pi(0)$ be the null cone of $V$, let $H_0$ be the subgroup of $\\GL(V)$ which preserves the ideal $\\I$ of $\\NN$ and let $H$ be a Levi subgroup of $H_0$ containing $G$. We determine the identity component of $H$. In many cases we show that $H=H_0$. For adjoint representations we have $H=H_0$ and we determine $H$ completely. We also investigate the subgroup $G_F$ of $\\GL(V)$ preserving a fiber $F$ of $\\pi$ when $V$ is an irreducible cofree $G$-module."}
{"category": "Math", "title": "Newton's method on Gra{\\ss}mann manifolds", "abstract": "A general class of Newton algorithms on Gra{\\ss}mann and Lagrange-Gra{\\ss}mann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition on the choice of coordinate systems. Our result extends and unifies previous convergence results for Newton's method on a manifold. Using special choices of the coordinates, new numerical algorithms are derived for principal component analysis and invariant subspace computations with improved computational complexity properties."}
{"category": "Math", "title": "Rational interpolation and mixed inverse spectral problem for finite CMV matrices", "abstract": "For finite dimensional CMV matrices the mixed inverse spectral problem of reconstruction the matrix by its submatrix and a part of its spectrum is considered. A general rational interpolation problem which arises in solving the mixed inverse spectral problem is studied, and the description of the space of its solutions is given. We apply the developed technique to give sufficient conditions for the uniqueness of the solution of the mixed inverse spectral problem."}
{"category": "Math", "title": "Measure of a 2-component link", "abstract": "A two-component link produces a torus as the product of the component knots in a two-point configuration space of a three-sphere. This space can be identified with a cotangent bundle and also with an indefinite Grassmannian. We show that the integration of the absolute value of the canonical symplectic form is equal to the area of the torus with respect to the pseudo-Riemannian structure, and that it attains the minimum only at the \"best\" Hopf links."}
{"category": "Math", "title": "A classification of prime-valent regular Cayley maps on some groups", "abstract": "A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are balanced and all prime-valent regular Cayley maps on abelian groups are either balanced or anti-balanced. Furthermore, we prove that there is no prime-valent regular Cayley map on any dicyclic group."}
{"category": "Math", "title": "Stochastic Variational Partitioned Runge-Kutta Integrators for Constrained Systems", "abstract": "Stochastic variational integrators for constrained, stochastic mechanical systems are developed in this paper. The main results of the paper are twofold: an equivalence is established between a stochastic Hamilton-Pontryagin (HP) principle in generalized coordinates and constrained coordinates via Lagrange multipliers, and variational partitioned Runge-Kutta (VPRK) integrators are extended to this class of systems. Among these integrators are first and second-order strongly convergent RATTLE-type integrators. We prove order of accuracy of the methods provided. The paper also reviews the deterministic treatment of VPRK integrators from the HP viewpoint."}
{"category": "Math", "title": "Pseudo-maximization and self-normalized processes", "abstract": "Self-normalized processes are basic to many probabilistic and statistical studies. They arise naturally in the the study of stochastic integrals, martingale inequalities and limit theorems, likelihood-based methods in hypothesis testing and parameter estimation, and Studentized pivots and bootstrap-$t$ methods for confidence intervals. In contrast to standard normalization, large values of the observations play a lesser role as they appear both in the numerator and its self-normalized denominator, thereby making the process scale invariant and contributing to its robustness. Herein we survey a number of results for self-normalized processes in the case of dependent variables and describe a key method called ``pseudo-maximization'' that has been used to derive these results. In the multivariate case, self-normalization consists of multiplying by the inverse of a positive definite matrix (instead of dividing by a positive random variable as in the scalar case) and is ubiquitous in statistical applications, examples of which are given."}
{"category": "Math", "title": "Convergence of weighted averages of relaxed projections", "abstract": "The convergence of the algorithm for solving convex feasibility problem is studied by the method of sequential averaged and relaxed projections. Some results of H. H. Bauschke and J. M. Borwein are generalized by introducing new methods. Examples illustrating these generalizations are given."}
{"category": "Math", "title": "Heegaard Splittings of Twisted Torus Knots", "abstract": "Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting. Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain \"twisted torus knots\" originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. We also conjecture that these manifolds possess irreducible yet weakly reducible splittings of genus three. There are no known examples of such Heegaard splittings."}
{"category": "Math", "title": "Non unique solutions to boundary value problems for non symmetric divergence form equations", "abstract": "We calculate explicitly solutions to the Dirichlet and Neumann boundary value problems in the upper half plane, for a family of divergence form equations with non symmetric coefficients with a jump discontinuity. It is shown that the boundary equation method and the Lax--Milgram method for constructing solutions may give two different solutions when the coefficients are sufficiently non symmetric."}
{"category": "Math", "title": "Parabolic Raynaud bundles", "abstract": "Let X be an irreducible smooth projective curve defined over complex numbers, S= {p_1, p_2,...,p_n} \\subset X$ a finite set of closed points and N > 1 a fixed integer. For any pair (r,d) in Z X Z/N, there exists a parabolic vector bundle R_{r,d,*} on X, with parabolic structure over S and all parabolic weights in Z/N, that has the following property: Take any parabolic vector bundle E_* of rank r on X whose parabolic points are contained in S, all the parabolic weights are in Z/N and the parabolic degree is d. Then E_* is parabolic semistable if and only if there is no nonzero parabolic homomorphism from R_{r,d,*} to E_*."}
{"category": "Math", "title": "Note on distortion and Bourgain $\\ell_1$ index", "abstract": "The relation between different notions measuring proximity to $\\ell_1$ and distortability of a Banach space is studied. The main result states that a Banach space, whose all subspaces have Bourgain $\\ell_1$ index greater than $\\omega^\\alpha$, $\\alpha<\\omega_1$, contains either an arbitrary distortable subspace or an $\\ell_1^\\alpha$-asymptotic subspace."}
{"category": "Math", "title": "Groupoid and Inverse Semigroup Presentations of Ultragraph $C^{*}$-Algebras", "abstract": "Inspired by the work of Paterson on $C^{\\ast}$-algebras of directed graphs, we show how to associate a groupoid $\\mathfrak{G}_{\\mathcal{G}}$ to an ultragraph $\\mathcal{G}$ in such a way that the $C^*$-algebra of $\\mathfrak{G}_{\\mathcal{G}}$ is canonically isomorphic to Tomforde's $C^*$-algebra $C^*(\\mathcal{G})$. The groupoid $\\mathfrak{G}_{\\mathcal{G}}$ is built from an inverse semigroup $S_{\\mathcal{G}}$ naturally associated to $\\mathcal{G}$."}
{"category": "Math", "title": "On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum", "abstract": "We deal with two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the $L^2$-norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity-high temperature regime. Our result generalizes that of [Finkelshtein D.L. et al., to appear in Random Oper. Stochastic Equations], which was proved for a special Glauber (Kawasaki, respectively) dynamics."}
{"category": "Math", "title": "A Poincar\\'e-Birkhoff-Witt criterion for Koszul operads", "abstract": "The aim of this article is to give a criterion, generalizing the criterion introduced by Priddy for algebras, to verify that an operad is Koszul. We define the notion of a Poincare-Birkhoff-Witt basis in the context of operads. Then we show that an operad having a Poincare-Birkhoff-Witt basis is Koszul. Besides, we obtain that the Koszul dual operad has also a Poincare-Birkhoff-Witt basis. We check that the classical examples of Koszul operads (commutative, associative, Lie) have a Poincare-Birkhoff-Witt basis."}
{"category": "Math", "title": "Topological complexity of motion planning and Massey products", "abstract": "We employ Massey products to find sharper lower bounds for the Schwarz genus of a fibration than those previously known. In particular we give examples of non-formal spaces $X$ for which the topological complexity $\\TC(X)$ (defined to be the genus of the free path fibration on $X$) is greater than the zero-divisors cup-length plus one."}
{"category": "Math", "title": "Asymptotic valuations of sequences satisfying first order recurrences", "abstract": "Let t[n] be a sequence that satisfies a first order homogeneous recurrence t[n] = Q[n]*t[n-1], where Q is a polynomial with integer coefficients. The asymptotic behavior of the p-adic valuation of t[n] is described under the assumption that all the roots of Q in Z/pZ have nonvanishing derivative."}
{"category": "Math", "title": "Mirror Symmetry via Logarithmic Degeneration Data II", "abstract": "This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B. We show that the Dolbeault cohomology groups of the Calabi-Yau associated to B are described in terms of some cohomology groups of sheaves on B, as expected. This is proved first by calculating the log de Rham and log Dolbeault cohomology groups on the log Calabi-Yau space associated to B, and then proving a base-change theorem for cohomology in our logarithmic setting. As applications, this shows that our mirror symmetry construction via Legendre duality of affine manifolds results in the usual interchange of Hodge numbers expected in mirror symmetry, and gives an explicit description of the monodromy of a smoothing."}
{"category": "Math", "title": "Groupoid Methods in Wavelet Analysis", "abstract": "We describe how the Deaconu-Renault groupoid may be used in the study of wavelets and fractals."}
{"category": "Math", "title": "Pairs of commuting nilpotent matrices, and Hilbert function", "abstract": "Let K be an infinite field and denote by H(n,K) the family of pairs (A,B) of commuting nilpotent n by n matrices with entries in K. There has been substantial recent study of the connection between H(n,K) and the fibre H[n] of the punctual Hilbert scheme of the plane, over an n-fold point of the symmetric product, by V. Baranovsky, R. Basili, and A. Premet. We study the stratification of H(n,K) by the Hilbert function of the Artinian ring K[A,B]. We show that when dim_K K[A,B] = n, then the generic element of the pencil A+\\lambda B, \\lambda \\in K, has Jordan partition the maximum partition P(H) whose diagonal lengths are the Hilbert function of K[A,B]. We denote by Q(P) the maximum Jordan partition of a nilpotent A commuting with a nilpotent B of Jordan partition P. We show that the stable partitions - those such that Q(P)=P - are those whose parts differ by at least two. In characteristic zero, the latter is a special case of a result of D. Panyushev. Our result on pencils shows that Q(P) has decreasing parts. In related work, T. Kosir and P. Oblak have shown further that Q(P) is itself stable."}
{"category": "Math", "title": "Module d'Alexander et repr\\'esentations m\\'etab\\'eliennes", "abstract": "It is known, since works of Burde and de Rham, that one can detect the roots of the Alexander polynomial of a knot by the study of the representations of the knot group into the group of the invertible upper triangular $2x2$ matrices. In this work, we propose to generalize this result by considering the representations of the knot group into the group of the invertible upper triangular $nxn$ matrices, $n\\geq 2$. This approach will enable us to find the decomposition of the Alexander module with complex coefficients."}
{"category": "Math", "title": "Albert algebras over curves of genus zero and one", "abstract": "Albert algebras and other Jordan algebras are constructed over curves of genus zero and one, using a generalization of the Tits process and the first Tits construction due to Achhammer."}
{"category": "Math", "title": "Adjusted Viterbi training for hidden Markov models", "abstract": "To estimate the emission parameters in hidden Markov models one commonly uses the EM algorithm or its variation. Our primary motivation, however, is the Philips speech recognition system wherein the EM algorithm is replaced by the Viterbi training algorithm. Viterbi training is faster and computationally less involved than EM, but it is also biased and need not even be consistent. We propose an alternative to the Viterbi training -- adjusted Viterbi training -- that has the same order of computational complexity as Viterbi training but gives more accurate estimators. Elsewhere, we studied the adjusted Viterbi training for a special case of mixtures, supporting the theory by simulations. This paper proves the adjusted Viterbi training to be also possible for more general hidden Markov models."}
{"category": "Math", "title": "On the relation between the WRT invariant and the Hennings invariant", "abstract": "The purpose of this note is to provide a simple relation between the Witten-Reshetikhin-Turaev SO(3) invariant and the Hennings invariant of 3-manifolds associated to quantum sl_2."}
{"category": "Math", "title": "Construction of Fredholm representations and a modification of the Higson-Roe corona", "abstract": "The Fredholm representation theory is well adapted to construction of homotopy invariants of non simply connected manifolds on the base of generalized Hirzebruch formula. Earlier a natural family of the Fredholm representations was constructed that lead to a symmetric vector bundle on completion of the fundamental group with a modification of the Higson-Roe corona when the completion is a closed manifold. Here we will discuss a homology version of symmetry in the case when completion with a modification of the Higson-Roe corona is a manifold with boundary. The results were developed during the visit of the first author in Ancona on March, 2007. The second version is supplemented by details of consideration the case of manifolds with boundary."}
{"category": "Math", "title": "Integrable operators and the squares of Hankel operators", "abstract": "Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distributions of large self-adjoint random matrices from the generalized unitary ensembles. This paper gives sufficient conditions for an integrable operator to be the square of a Hankel operator, and applies the condition to the Airy, associated Laguerre, modified Besses and Whittaker functions."}
{"category": "Math", "title": "Conjugate Points in Length Spaces", "abstract": "In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. Our versions apply in this more general setting. We next focus on ${\\rm CBA}(\\kappa)$ spaces, proving Rauch-type comparison theorems. In particular, much like the Riemannian setting, we prove an Alexander-Bishop theorem stating that there are no ultimate conjugate points less than $\\pi$ apart in a ${\\rm CBA}(1)$ space. We also prove a relative Rauch comparison theorem to precisely estimate the distance between nearby geodesics. We close with applications and open problems."}
{"category": "Math", "title": "Noetherian Hopf algebras", "abstract": "This short survey article reviews current understand- ing of the structure of noetherian Hopf algebras. The focus is on homological properties. A number of open problems are listed."}
{"category": "Math", "title": "Symplectic reflection algebras in positive characteristic", "abstract": "Basic properties of symplectic reflection algebras over an algebraically closed field k of positive characteristic are laid out. These algebras are always finite modules over their centres, in contrast to the situation in characteristic 0. For the subclass of rational Cherednik algebras, we determine the PI-degree and the Goldie rank, and show that the Azumaya and smooth loci of the centre coincide."}
{"category": "Math", "title": "Robust Dimension Reduction, Fusion Frames, and Grassmannian Packings", "abstract": "We consider estimating a random vector from its noisy projections onto low dimensional subspaces constituting a fusion frame. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first determine the minimum mean-squared error (MSE) in linearly estimating the random vector of interest from its fusion frame projections, in the presence of white noise. We show that MSE assumes its minimum value when the fusion frame is tight. We then analyze the robustness of the constructed linear minimum MSE (LMMSE) estimator to erasures of the fusion frame subspaces. We prove that tight fusion frames consisting of equi-dimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace, and that the optimal subspace dimension depends on signal-to-noise ratio (SNR). We also prove that tight fusion frames consisting of equi-dimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures. We call such fusion frames equi-distance tight fusion frames, and prove that the chordal distance between subspaces in such fusion frames meets the so-called simplex bound, and thereby establish connections between equi-distance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for construction of equi-distance tight fusion frames."}
{"category": "Math", "title": "The twisted fourth moment of the Riemann zeta function", "abstract": "We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length $T^{{1/11} - \\epsilon}$"}
{"category": "Math", "title": "Some remarks about Clifford analysis and fractal sets", "abstract": "Although Clifford analysis is like complex analysis in many ways, there are obvious differences related to noncommutativity, and a few aspects of this are considered here."}
{"category": "Math", "title": "Densities in Fabry's theorem", "abstract": "Fabry's theorem on the singularities of power series is improved: the maximum density in the assumptions of this theorem is replaced by an interior density of Beurling--Malliavin type."}
{"category": "Math", "title": "Simplified Proofs for the Pro-Lie Group Theorem and the One-Parameter Subgroup Lifting Lemma", "abstract": "This note is devoted to the theory of projective limits of finite-dimensional Lie groups, as developed in the recent monograph ``The Lie Theory of Connected Pro-Lie Groups'' by K.H. Hofmann and S.A. Morris. We replace the original, highly non-trivial proof of the One-Parameter Subgroup Lifting Lemma given in the monograph by a shorter and more elementary argument. Furthermore, we shorten (and correct) the proof of the so-called Pro-Lie Group Theorem, which asserts that pro-Lie groups and projective limits of Lie groups coincide."}
{"category": "Math", "title": "On rough isometries of Poisson processes on the line", "abstract": "Intuitively, two metric spaces are rough isometric (or quasi-isometric) if their large-scale metric structure is the same, ignoring fine details. This concept has proven fundamental in the geometric study of groups. Ab\\'{e}rt, and later Szegedy and Benjamini, have posed several probabilistic questions concerning this concept. In this article, we consider one of the simplest of these: are two independent Poisson point processes on the line rough isometric almost surely? Szegedy conjectured that the answer is positive. Benjamini proposed to consider a quantitative version which roughly states the following: given two independent percolations on $\\mathbb {N}$, for which constants are the first $n$ points of the first percolation rough isometric to an initial segment of the second, with the first point mapping to the first point and with probability uniformly bounded from below? We prove that the original question is equivalent to proving that absolute constants are possible in this quantitative version. We then make some progress toward the conjecture by showing that constants of order $\\sqrt{\\log n}$ suffice in the quantitative version. This is the first result to improve upon the trivial construction which has constants of order $\\log n$. Furthermore, the rough isometry we construct is (weakly) monotone and we include a discussion of monotone rough isometries, their properties and an interesting lattice structure inherent in them."}
{"category": "Math", "title": "KAM for the Non-Linear Schr\\\"odinger Equation", "abstract": "We consider the $d$-dimensional nonlinear Schr\\\"odinger equation under periodic boundary conditions: $-i\\dot u=-\\Delta u+V(x)*u+\\ep \\frac{\\p F}{\\p \\bar u}(x,u,\\bar u), \\quad u=u(t,x), x\\in\\T^d $ where $V(x)=\\sum \\hat V(a)e^{i\\sc{a,x}}$ is an analytic function with $\\hat V$ real, and $F$ is a real analytic function in $\\Re u$, $\\Im u$ and $x$. (This equation is a popular model for the `real' NLS equation, where instead of the convolution term $V*u$ we have the potential term $Vu$.) For $\\ep=0$ the equation is linear and has time--quasi-periodic solutions $u$, $$ u(t,x)=\\sum_{a\\in \\AA}\\hat u(a)e^{i(|a|^2+\\hat V(a))t}e^{i\\sc{a,x}} \\quad (|\\hat u(a)|>0), $$ where $\\AA$ is any finite subset of $\\Z^d$. We shall treat $\\omega_a=|a|^2+\\hat V(a)$, $a\\in\\AA$, as free parameters in some domain $U\\subset\\R^{\\AA}$. This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, under general conditions, will have the following consequence: If $|\\ep|$ is sufficiently small, then there is a large subset $U'$ of $U$ such that for all $\\omega\\in U'$ the solution $u$ persists as a time--quasi-periodic solution which has all Lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients."}
{"category": "Math", "title": "Canonical matrices of bilinear and sesquilinear forms", "abstract": "Canonical matrices are given for (a) bilinear forms over an algebraically closed or real closed field; (b) sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and (c) sesquilinear forms over a field F of characteristic different from 2 with involution (possibly, the identity) up to classification of Hermitian forms over finite extensions of F. A method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices. This method has its origins in representation theory and was devised in [V.V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501]."}
{"category": "Math", "title": "Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras", "abstract": "The aim of this paper is to generalize the concept of Lie-admissible coalgebra introduced by Goze and Remm to Hom-coalgebras and to introduce Hom-Hopf algebras with some properties. These structures are based on the Hom-algebra structures introduced by the authors."}
{"category": "Math", "title": "A Universal Property of the Groups Spin^c and Mp^c", "abstract": "It is well known that spinors on oriented Riemannian manifolds cannot be defined as sections of a vector bundle associated with the frame bundle. For this reason spin and spin^c structures are often introduced. In this paper we prove that spin^c structures have a universal property among all other structures that enable the construction of spinor bundles. We proceed to prove a similar result for metaplectic^c structures on symplectic manifolds."}
{"category": "Math", "title": "A new weak approximation scheme of stochastic differential equations and the Runge-Kutta method", "abstract": "In this paper, authors successfully construct a new algorithm for the new higher order scheme of weak approximation of SDEs. The algorithm presented here is based on [1][2]. Although this algorithm shares some features with the algorithm presented by [3], algorithms themselves are completely different and the diversity is not trivial. They apply this new algorithm to the problem of pricing Asian options under the Heston stochastic volatility model and obtain encouraging results. [1] Shigeo Kusuoka, \"Approximation of Expectation of Diffusion Process and Mathematical Finance,\" Advanced Studies in Pure Mathematics, Proceedings of Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp. 147--165. [2] Terry Lyons and Nicolas Victoir, \"Cubature on Wiener Space,\" Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 460 (2004), pp. 169--198. [3] Syoiti Ninomiya, Nicolas Victoir, \"Weak approximation of stochastic differential equations and application to derivative pricing,\" Applied Mathematical Finance, Volume 15, Issue 2 April 2008, pages 107--121"}
{"category": "Math", "title": "A Note on Singular Cardinals in Set Theory Without Choice", "abstract": "We discuss how singular can cardinals be in absence of the axiom of choice. We show that, contrasting with known negative consistency results (of Gitik and others), certain positive results are provable. Then we pose some problems."}
{"category": "Math", "title": "A characterization of Weingarten surfaces in hyperbolic 3-space", "abstract": "We study 2-dimensional submanifolds of the space ${\\mathbb{L}}({\\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral K\\\"ahler structure. Such a surface is Lagrangian iff there exists a surface in ${\\mathbb{H}}^3$ orthogonal to the geodesics of $\\Sigma$. We prove that the induced metric on a Lagrangian surface in ${\\mathbb{L}}({\\mathbb{H}}^3)$ has zero Gauss curvature iff the orthogonal surfaces in ${\\mathbb{H}}^3$ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in ${\\mathbb{L}}({\\mathbb{H}}^3)$ and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ${\\mathbb{H}}^3$."}
{"category": "Math", "title": "Complete intersection dimensions and Foxby classes", "abstract": "Let $R$ be a local ring and $M$ a finitely generated $R$-module. The complete intersection dimension of $M$--defined by Avramov, Gasharov and Peeva, and denoted $\\cidim_R(M)$--is a homological invariant whose finiteness implies that $M$ is similar to a module over a complete intersection. It is related to the classical projective dimension and to Auslander and Bridger's Gorenstein dimension by the inequalities $\\gdim_R(N)\\leq\\cidim_R(N)\\leq\\pd_R(N)$. Using Blanco and Majadas' version of complete intersection dimension for local ring homomorphisms, we prove the following generalization of a theorem of Avramov and Foxby: Given local ring homomorphisms $\\phi\\colon R\\to S$ and $\\psi\\colon S\\to T$ such that $\\phi$ has finite Gorenstein dimension, if $\\psi$ has finite complete intersection dimension, then the composition $\\psi\\circ\\phi$ has finite Gorenstein dimension. This follows from our result stating that, if $M$ has finite complete intersection dimension, then $M$ is $C$-reflexive and is in the Auslander class $\\catac(R)$ for each semidualizing $R$-complex $C$."}
{"category": "Math", "title": "Fractional Dynamical Systems on Fractional Leibniz Algebroids", "abstract": "The theory of derivative of noninteger order goes back to Leibniz, Liouville and Riemann. Derivatives of fractional order have found many applications in recent studies in mechanics, physics, economics. In this paper we define the fractional tangent bundle on a manifold, using a method of Radu Miron. The fractional Leibniz algebroids are investigated. The associated objects have an geometric character. Some fractional dynamical systems on a fractional Leibniz algebroid are disscussed."}
{"category": "Math", "title": "Unitary and Euclidean representations of a quiver", "abstract": "A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations in canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension."}
{"category": "Math", "title": "Symmetric symplectic homotopy K3 surfaces", "abstract": "A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is shown that an effective action by various maximal symplectic K3 groups forces the corresponding homotopy K3 surface to be minimally exotic with respect to our measure. (However, the standard K3 is the only known example of such minimally exotic homotopy K3 surfaces.) The possible structure of a finite group of symplectic symmetries of a minimally exotic homotopy K3 surface is determined and future research directions are indicated."}
{"category": "Math", "title": "Some classes of rational functions and related Banach spaces", "abstract": "For positive integers d, r, and M, we consider the class of rational functions on real d-dimensional space whose denominators are products of at most r functions of the form 1+Q(x) where each Q is a quadratic form with eigenvalues bounded above by M and below by 1/M. Each numerator is a monic monomial of the same degree as the corresponding denominator. Then we form the Banach space of countable linear combinations of such rational functions with absolutely summable coefficients, normed by the infimum of sums of absolute values of the coefficients. We show that for rational functions whose denominators are rth powers of a specific 1+Q, or differences of two such rational functions with the same numerator, the norm is achieved by and only by the obvious combination of one or two functions respectively. We also find bounds for coefficients in partial-fraction decompositions of some specific rational functions, which in some cases are quite sharp."}
{"category": "Math", "title": "Besov Spaces and Frames on Compact Manifolds", "abstract": "We show that one can characterize the Besov spaces on a smooth compact oriented Riemannian manifold, for the full range of indices, through a knowledge of the size of frame coefficients, using the frames we have constructed in [8]."}
{"category": "Math", "title": "Existence of a multiplicative basis for a finitely spaced module over an aggregate", "abstract": "By [R. Bautista, P. Gabriel, A.V Roiter., L. Salmeron, Representation-finite algebras and multiplicative basis. Invent. Math. 81 (1985) 217-285.], a finite-dimensional algebra having finitely many isoclasses of indecomposable representations admits a multiplicative basis. In Sections 4.10-4.12 of [P. Gabriel, A. V. Roiter, Representations of finite-dimensional algebras. Encyclopaedia of Math. Sci., vol. 73, Algebra 8, Springer-Verlag, 1992] an analogous hypothesis was formulated for finitely spaced modules over an aggregate. We prove this conjecture."}
{"category": "Math", "title": "Positivity criteria generalizing the leading principal minors criterion", "abstract": "For each Hermitian matrix, we prove that instead of the leading principal minors some of their sums can be used in the leading principal minors criterion and in other inertia problems."}
{"category": "Math", "title": "The problems of classifying pairs of forms and local algebras with zero cube radical are wild", "abstract": "We prove that over an algebraically closed field of characteristic not two the problems of classifying pairs of sesquilinear forms in which the second is Hermitian, pairs of bilinear forms in which the second is symmetric (skew-symmetric), and local algebras with zero cube radical and square radical of dimension 2 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity."}
{"category": "Math", "title": "A Logic of Injectivity", "abstract": "Injectivity of objects with respect to a set $\\ch$ of morphisms is an important concept of algebra, model theory and homotopy theory. Here we study the logic of injectivity consequences of $\\ch$, by which we understand morphisms $h$ such that injectivity with respect to $\\ch$ implies injectivity with respect to $h$. We formulate three simple deduction rules for the injectivity logic and for its finitary version where \\mor s between finitely ranked objects are considered only, and prove that they are sound in all categories, and complete in all \"reasonable\" categories."}
{"category": "Math", "title": "Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild", "abstract": "We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to congruence, local commutative associative algebras with zero cube radical and square radical of dimension 3, and Lie algebras with central commutator subalgebra of dimension 3 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity."}
{"category": "Math", "title": "An ultrametric version of the Maillet-Malgrange theorem for nonlinear q-difference equations", "abstract": "We prove an ultrametric q-difference version of the Maillet-Malgrange theorem, on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since \\deg_q and \\ord_q define two valuations on {\\mathbb C}(q), we obtain, in particular, a result on the growth of the degree in q and the order at q of formal solutions of nonlinear q-difference equations, when q is a parameter. We illustrate the main theorem by considering two examples: a q-deformation of ``Painleve' II'', for the nonlinear situation, and a q-difference equation satisfied by the colored Jones polynomials of the figure 8 knots, in the linear case. We consider also a q-analog of the Maillet-Malgrange theorem, both in the complex and in the ultrametric setting, under the assumption that |q|=1 and a classical diophantine condition."}
{"category": "Math", "title": "Biquandle longitude invariant of long virtual knots", "abstract": "It is known that the number of biquandle colorings of a long virtual knot diagram, with a fixed color of the initial arc, is a knot invariant. In this paper we describe a more subtle invariant: a family of biquandle endomorphisms obtained from the set of colorings and longitudinal information."}
{"category": "Math", "title": "Littlewood's algorithm and quaternion matrices", "abstract": "A strengthened form of Schur's triangularization theorem is given for quaternion matrices with real spectrum (for complex matrices it was given by Littlewood). Littlewood's algorithm for reducing a complex matrix to a canonical form under unitary similarity is extended to quaternion matrices whose eigenvalues have geometric multiplicity 1."}
{"category": "Math", "title": "Adjoint cohomology of graded Lie algebras of maximal class", "abstract": "We compute explicitly the adjoint cohomology of two N-graded Lie algebras of maximal class (infinite dimensional filiform Lie algebras) m_0 and m_2. It is known that up to an isomorphism there are only three N-graded Lie algebras of the maximal class. The third algebra from this list is the \"positive\" part L_1 of the Witt (or Virasoro) algebra and its adjoint cohomology was computed earlier by Feigin and Fukhs. We show that the total space H*(m_j,m_j) is \"almost\" isomorphic to the completed tensor product of the algebra m_j by scalar cohomology space H^*(m_j), j=0,2."}
{"category": "Math", "title": "Estimate of the number of one-parameter families of modules over a tame algebra", "abstract": "The problem of classifying modules over a tame algebra A reduces to a block matrix problem of tame type whose indecomposable canonical matrices are zero- or one-parameter. Respectively, the set of nonisomorphic indecomposable modules of dimension at most d divides into a finite number f(d,A) of modules and one-parameter series of modules. We prove that the number of m-by-n canonical parametric block matrices with a given partition into blocks is bounded by 4^s, where s is the number of free entries (which is at most mn), and estimate the number f(d,A)."}
{"category": "Math", "title": "Computation of the canonical form for the matrices of chains and cycles of linear mappings", "abstract": "Paul Van Dooren [Linear Algebra Appl. 27 (1979) 103-140] constructed an algorithm for the computation of all irregular summands in Kronecker's canonical form of a matrix pencil. The algorithm is numerically stable since it uses only unitary transformations. We extend Paul Van Dooren's algorithm to the matrices of a cycle of linear mappings."}
{"category": "Math", "title": "Canonical forms for complex matrix congruence and *congruence", "abstract": "Canonical forms for congruence and *congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347-353], based on Sergeichuk's paper [Math. USSR, Izvestiya 31 (3) (1988) 481-501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms. We use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew-symmetric matrices as well as canonical forms for simultaneous *congruence of pairs of complex Hermitian matrices."}
{"category": "Math", "title": "Sparse Representations for Structured Noise Filtering", "abstract": "The role of sparse representations in the context of structured noise filtering is discussed. A strategy, especially conceived so as to address problems of an ill posed nature, is presented. The proposed approach revises and extends the Oblique Matching Pursuit technique. It is shown that, by working with an orthogonal projection of the signal to be filtered, it is possible to apply orthogonal matching pursuit like strategies in order to accomplish the required signal discrimination"}
{"category": "Math", "title": "Normal form of m-by-n-by-2 matrices for equivalence", "abstract": "We give a canonical form of m-by-2-by-2 spatial matrices for equivalence over any field."}
{"category": "Math", "title": "Canonical matrices for linear matrix problems", "abstract": "We consider a large class of matrix problems, which includes the problem of classifying arbitrary systems of linear mappings. For every matrix problem from this class, we construct Belitskii's algorithm for reducing a matrix to a canonical form, which is the generalization of the Jordan normal form, and study the set C(m,n) of indecomposable canonical m-by-n matrices. Considering C(m,n) as a subset in the affine space of m-by-n matrices, we prove that either C(m,n) consists of a finite number of points and straight lines for every (m,n), or C(m,n) contains a 2-dimensional plane for a certain (m,n)."}
{"category": "Math", "title": "Complexity of matrix problems", "abstract": "In representation theory, the problem of classifying pairs of matrices up to simultaneous similarity is used as a measure of complexity; classification problems containing it are called wild problems. We show in an explicit form that this problem contains all classification matrix problems given by quivers or posets. Then we prove that it does not contain (but is contained in) the problem of classifying three-valent tensors. Hence, all wild classification problems given by quivers or posets have the same complexity; moreover, a solution of any one of these problems implies a solution of each of the others. The problem of classifying three-valent tensors is more complicated."}
{"category": "Math", "title": "Congruences of a square matrix and its transpose", "abstract": "It is known that any square matrix over any field F is congruent to its transpose. We show that they are also *congruent with respect to any nonidentity involution on F."}
{"category": "Math", "title": "Resolution of singularities, asymptotic expansions of oscillatory integrals, and related phenomena", "abstract": "The elementary resolution of singularities algorithm of the author's earlier paper (math.CA/0609217) is developed further, replacing the quasibump functions in the blown up coordinates with the characteristic function of a rectangle times a smooth function. Such functions are easier to deal with, and as application the existence of asymptotic expansions for oscillatory integrals and related objects is given an elementary proof. In addition, some more detailed information about these expansions is given."}
{"category": "Math", "title": "Rigidity, boundary interpolation and reproducing kernels", "abstract": "We use reproducing kernel methods to study various rigidity problems. The methods and setting allow us to also consider the non-positive case."}
{"category": "Math", "title": "Some aspects of calculus on non-smooth sets", "abstract": "Sets in R^n in which every pair of elements x, y can be connected by a path in the set of length bounded by a constant multiple of the distance between x and y are considered."}
{"category": "Math", "title": "Hamiltonian vector fields of homogeneous polynomials in two variables", "abstract": "Let $g:\\mathbb{R}^2\\to\\mathbb{R}$ be a homogeneous polynomial of degree $p>1$, $G=(-g'_{y}, g'_{x})$ be its Hamiltonian vector field, and $G_t$ be the local flow generated by $G$. Denote by $E(G,O)$ the space of germs of $C^{\\infty}$ diffeomorphisms $(\\mathbb{R}^2,O)\\to(\\mathbb{R}^2,O)$ that preserve orbits of $G$. Let also $E_{\\mathrm{id}}(G,O)$ be the identity component of $E(G,O)$ with respect to $C^1$-topology. Suppose that $g$ has no multiple prime factors. Then we prove that for every $h\\in E_{\\mathrm{id}}(G,O)$ there exists a germ of a smooth function $\\alpha:\\mathbb{R}^2\\to\\mathbb{R}$ at $O$ such that $h(z)=G_{\\alpha(z)}(z)$."}
{"category": "Math", "title": "Extension of the Weil-Petersson connection", "abstract": "Convexity properties of Weil-Petersson geodesics on the Teichm\\\"{u}ller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considerations are combined to establish convexity along Weil-Petersson geodesics of the functions the distance between horocycles for a hyperbolic metric."}
{"category": "Math", "title": "Time Optimal Attitude Control for a Rigid Body", "abstract": "A time optimal attitude control problem is studied for the dynamics of a rigid body. The objective is to minimize the time to rotate the rigid body to a desired attitude and angular velocity while subject to constraints on the control input. Necessary conditions for optimality are developed directly on the special orthogonal group using rotation matrices. They completely avoid singularities associated with local parameterizations such as Euler angles, and they are expressed as compact vector equations. In addition, a discrete control method based on a geometric numerical integrator, referred to as a Lie group variational integrator, is proposed to compute the optimal control input. The computational approach is geometrically exact and numerically efficient. The proposed method is demonstrated by a large-angle maneuver for an elliptic cylinder rigid body."}
{"category": "Math", "title": "A New Family of Somos-like Recurrences", "abstract": "We exhibit a three parameter infinite family of quadratic recurrence relations inspired by the well known Somos sequences. For one infinite subfamily we prove that the recurrence generates an infinite sequence of integers by showing that the same sequence is generated by a linear recurrence (with suitable initial conditions). We also give conjectured relations among the three parameters so that the quadratic recurrences generate sequences of integers."}
{"category": "Math", "title": "Dissipative Hyperbolic Geometric Flow", "abstract": "In this paper we introduce and study a new kind of hyperbolic geometric flows --dissipative hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear wave equations with dissipative terms. Some interesting exact solutions are given, in particular, a new concept-- hyperbolic Ricci soliton is introduced and some of its geometric properties are described. We also establish the short-time existence and uniqueness theorem for the dissipative hyperbolic geometric flow, and prove the nonlinear stability of the flow defined on the Euclidean space of dimension larger than 2. Wave character of the evolving metrics and curvatures is illustrated and the nonlinear wave equations satisfied by the curvatures are derived."}
{"category": "Math", "title": "Counting descent pairs with prescribed colors in the colored permutation groups", "abstract": "We define new statistics, (c, d)-descents, on the colored permutation groups Z_r \\wr S_n and compute the distribution of these statistics on the elements in these groups. We use some combinatorial approaches, recurrences, and generating functions manipulations to obtain our results."}
{"category": "Math", "title": "Towards the computation of the convex hull of a configuration from its corresponding separating matrix", "abstract": "In this paper, we cope with the following problem: compute the size of the convex hull of a configuration C, where the given data is the number of separating lines between any two points of the configuration (where the lines are generated by pairs of other points of the configuration). We give an algorithm for the case that the convex hull is of size 3, and a partial algorithm and some directions for the case that the convex hull is of size bigger than 3."}
{"category": "Math", "title": "SDLS: a Matlab package for solving conic least-squares problems", "abstract": "This document is an introduction to the Matlab package SDLS (Semi-Definite Least-Squares) for solving least-squares problems over convex symmetric cones. The package is shortly presented through the addressed problem, a sketch of the implemented algorithm, the syntax and calling sequences, a simple numerical example and some more advanced features. The implemented method consists in solving the dual problem with a quasi-Newton algorithm. We note that SDLS is not the most competitive implementation of this algorithm: efficient, robust, commercial implementations are available (contact the authors). Our main goal with this Matlab SDLS package is to provide a simple, user-friendly software for solving and experimenting with semidefinite least-squares problems. Up to our knowledge, no such freeware exists at this date."}
{"category": "Math", "title": "GloptiPoly 3: moments, optimization and semidefinite programming", "abstract": "We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming."}
{"category": "Math", "title": "Sur la cohomologie \\`a support des fibr\\'es en droites sur les vari\\'et\\'es sym\\'etriques compl\\`etes", "abstract": "Let $X$ be a complete symmetric variety i.e. the wonderful compactification of a symmetric $G-$homogeneous space (where $G$ is a simply-connected semi-simple linear algebraic group). If $L$ is a line bundle over $X$ and if $C$ is a Bialynicki-Birula cell of codimension $c$ in $X$, then the Lie algebra $\\mathfrak g$ of $G$ operates naturally on the cohomology group with support : $H^c_C(L)$. One gives here a necessary condition on the cell $C$ for that $\\mathfrak g-$module have a finite dimensional simple subquotient. As applications one calculates the Euler-Poincar\\'e characteristic of $L$ over $X$ and one estimates the higher cohomology group $H^d(X,L)$, $d \\ge 0$, with exact formulae in some cases among which the case of the complete conic variety. -- \\'Etant donn\\'e un groupe alg\\'ebrique lin\\'eaire semi-simple G, on s'int\\'eresse aux compactifications magnifiques des G?espaces homog\\`enes sym\\'etriques. Si X est une telle compactification, si L est un fibr\\'e en droites G?lin\\'earis\\'e sur X et si C est une cellule de Bialynicki-Birula de X de codimension c, alors l'alg\\`ebre de Lie g de G op\\`ere naturellement sur le groupe de cohomologie \\`a support Hc C (L). On donne ici une condition n\\'ecessaire, portant sur la cellule C, pour que ce g?module poss\\`ede un sous-quotient simple de dimension finie. On en d\\'eduit une formule pour la caract\\'eristique d'Euler-Poincar\\'e de L sur X et une estimation (exacte pour certains cas dont celui de la vari\\'et\\'e des coniques compl\\`etes) des groupes de cohomologie sup\\'erieure Hd(X,L), d ? 0."}
{"category": "Math", "title": "The isodiametric problem with lattice-point constraints", "abstract": "In this paper, the isodiametric problem for centrally symmetric convex bodies in the Euclidean d-space R^d containing no interior non-zero point of a lattice L is studied. It is shown that the intersection of a suitable ball with the Dirichlet-Voronoi cell of 2L is extremal, i.e., it has minimum diameter among all bodies with the same volume. It is conjectured that these sets are the only extremal bodies, which is proved for all three dimensional and several prominent lattices."}
{"category": "Math", "title": "The Riemann Zeta-Function and Hecke Congruence Subgroups. II", "abstract": "This is a rework of our old file, which has been left unpublished since September 1994, on an explicit spectral decomposition of the fourth power moment of the Riemann zeta-function against a weight which is the square of a Dirichlet polynomial. At this occasion we add an explicit treatment of generalized Kloosterman sums associated with arbitrary Hecke congruence subgroups (Section 15), which might have an independent interest. At the end (Section 36) of our discussion, we set out a few problems on the distribution of eigenvalues of the hyperbolic Laplacian, which appear to us to be related to the nature of the sixth power moment of the Riemann zeta-function. The contents of this work were presented in a worshop at RIMS Kyoto University on October 18, 2007. In this second version, some corrections are made in the part on generalized Kloosterman sums, and in Addendum a mention is made concerning a recent work by C.P. Hughes and M.P. Young (0709.2345)."}
{"category": "Math", "title": "Seshadri fibrations of algebraic surfaces", "abstract": "We refine results of Hwang, Keum and Szemberg, Tutaj-Gasinska which relate local invariants - Seshadri constants - of ample line bundles on surfaces to the global geometry - fibration structure. We show that the same picture emerges when looking at Seshadri constants measured at any finite subset of the given surface."}
{"category": "Math", "title": "Symmetric identities on Bernoulli polynomials", "abstract": "In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and Bernoulli numbers, including the Miki identity."}
{"category": "Math", "title": "Quantitative recurrence in two-dimensional extended processes", "abstract": "Under some mild condition, a random walk in the plane is recurrent. In particular each trajectory is dense, and a natural question is how much time one needs to approach a given small neighborhood of the origin. We address this question in the case of some extended dynamical systems similar to planar random walks, including $\\ZZ^2$-extension of hyperbolic dynamics. We define a pointwise recurrence rate and relate it to the dimension of the process, and establish a convergence in distribution of the rescaled return times near the origin."}
{"category": "Math", "title": "On uniform convergence in ergodic theorems for a class of skew product transformations", "abstract": "Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space (M, B, m) in the base and a semigroup of transformations on another probability space (W,F,P) in the fibre. Under suitable mixing conditions for the fibre transformation, we show that the properties ergodicity, weakly mixing, and strongly mixing are passed on from the base transformation to the skew product (with respect to the product measure). We derive ergodic theorems with respect to the skew product on the product space. The main aim of this paper is to establish uniform convergence with respect to the base variable for the series of ergodic averages of a function F on the product of the two probability spaces along the orbits of such a skew product. Assuming a certain growth condition for the coupling function, a strong mixing condition on the fibre transformation, and continuity andintegrability conditions for F, we prove uniform convergence in the base and L^p(P)-convergence in the fibre. Under an equicontinuity assumption on F we further show P-almost sure convergence in the fibre. Our work has an application in information theory: It implies convergence of the averages of functions on random fields restricted to parts of stair climbing patterns defined by a direction."}
{"category": "Math", "title": "Existence et equidistribution des matrices de denominateur n dans les groupes unitaires et orthogonaux", "abstract": "We study some subsets of rational points in an algebraic groups defined by open conditions on their projection in the finite adeles points. Using adelic mixing we are able to prove an equidistribution's result for the projection of these sets in the real points. As an application, we study the existence and the repartition of rational unitary matrices having a given denominator. We prove a local-global principle for this problem and the equirepartition of the sets of denominator n-matrices when they are not empty. Then we study the more complicated case of non simply-connected groups applying it to quadratic forms."}
{"category": "Math", "title": "$\\ell^1$ penalty for ill-posed inverse problems", "abstract": "We tackle the problem of recovering an unknown signal observed in an ill-posed inverse problem framework. More precisely, we study a procedure commonly used in numerical analysis or image deblurring: minimizing an empirical loss function balanced by an $l^1$ penalty, acting as a sparsity constraint. We prove that, by choosing a proper loss function, this estimation technique enables to build an adaptive estimator, in the sense that it converges at the optimal rate of convergence without prior knowledge of the regularity of the true solution"}
{"category": "Math", "title": "Curvature explosion in quotients and applications", "abstract": "We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points."}
{"category": "Math", "title": "On the topology of minimal orbits in complex flag manifolds", "abstract": "We compute the Euler-Poincar\\'e characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold."}
{"category": "Math", "title": "Characterization of substitution invariant 3iet words", "abstract": "We study infinite words coding an orbit under an exchange of three intervals which have full complexity $\\C(n)=2n+1$ for all $n\\in\\N$ (non-degenerate 3iet words). In terms of parameters of the interval exchange and the starting point of the orbit we characterize those 3iet words which are invariant under a primitive substitution. Thus, we generalize the result recently obtained for sturmian words."}
{"category": "Math", "title": "Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians", "abstract": "In this article, I classify the totally geodesic submanifolds in the complex 2-Grassmannians and in the quaternionic 2-Grassmannians. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank 2, published by Chen and Nagano (B.-Y. Chen, T. Nagano, \"Totally geodesic submanifolds of symmetric spaces, II\", Duke Math. J. 45 (1978), 405--425) is incomplete. For example, G_2(H^n) with n >= 7 contains totally geodesic submanifolds isometric to a HP^2, its metric scaled such that the minimal sectional curvature is 1/5; they are maximal in G_2(H^7). Also G_2(C^n) with n >= 6 contains totally geodesic submanifolds which are isometric to a CP^2 contained in the HP^2 mentioned above; they are maximal in G_2(C^6). Neither submanifolds are mentioned in the cited paper by Chen and Nagano."}
{"category": "Math", "title": "On homogeneous pinning models and penalizations", "abstract": "In this note, we show how the penalization method, introduced in order to describe some non-trivial changes of the Wiener measure, can be applied to the study of some simple polymer models such as the pinning model. The bulk of the analysis is then focused on the study of a martingale which has to be computed as a Markovian limit."}
{"category": "Math", "title": "A Note On Mixed Mean Inequalities", "abstract": "We give a simpler proof of a result of Holland concerning a mixed arithmetic-geometric mean inequality. We also prove a result of mixed mean inequality involving the symmetric means."}
{"category": "Math", "title": "Shannon-McMillan theorems for discrete random fields along curves and lower bounds for surface-order large deviations", "abstract": "The notion of a surface-order specific entropy h_c(P) of a two-dimensional discrete random field P along a curve c is introduced as the limit of rescaled entropies along lattice approximations of the blowups of c. Existence is shown by proving a corresponding Shannon-McMillan theorem. We obtain a representation of h_c(P) as a mixture of specific entropies along the tangent lines of c. As an application, the specific entropy along curves is used to refine Foellmer and Ort's lower bound for the large deviations of the empirical field of an attractive Gibbs measure from its ergodic behavior in the phase-transition regime."}
{"category": "Math", "title": "Regularity of the density for the stochastic heat equation", "abstract": "We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders."}
{"category": "Math", "title": "Description of two soliton collision for the quartic gKdV equation", "abstract": "This paper concerns the problem of collision of two solitons for the quartic generalized Korteweg-de Vries equation. We introduce a new framework to describe the collision in the special case where one soliton is small with respect to the other. We prove that the two soliton survive the collision, we describe the collision phenomenon (computation of the first order of the resulting shifts on the solitons). Moreover, we prove that in this situation, there does not exist pure two-soliton solutions."}
{"category": "Math", "title": "Spectra of alternating Hilbert operators", "abstract": "Spectra of real alternating operators seem to be quite interesting from the view point of explaining the Riemann Hypothesis for various zeta functions. Unfortunately we have not sufficient experiments concerning this theme. Necessary works would be to supply new examples of spectra related to zeros and poles of zeta functions. A century ago Hilbert (1907) considered a kind of operators representing quadratic forms of infinitely many variables. Demonstrating the calculation of spectra for alternating Hilbert operators we hope to present a novel scheme in this paper. Authors expect this study encourages experts for further studies."}
{"category": "Math", "title": "Stability of two soliton collision for nonintegrable gKdV equations", "abstract": "We continue our study of the collision of two solitons for the subcritical generalized KdV equations. In a previous paper, mainly devoted to the case of the quartic gKdV equation, we have introduced a new framework to understand the collision of two solitons in the case where one soliton is small with respect to the other. In this paper, we consider the case of a general nonlinearity $f(u)$ for which the two solitons are nonlinearly stable. We prove that in this situation the two solitons survive and we describe the collision at the main orders."}
{"category": "Math", "title": "Optimal test-configurations for toric varieties", "abstract": "On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle. We also show that if the Calabi flow exists for all time on a toric variety then it minimises the Calabi functional. In this case the infimum of the Calabi functional is given by the supremum of the normalised Futaki invariants over all destabilising test-configurations, as predicted by a conjecture of Donaldson."}
{"category": "Math", "title": "Sylow's theorem for Moufang loops", "abstract": "For finite Moufang loops, we prove an analog of the first Sylow theorem giving a criterion of the existence of a p-Sylow subloop. We also find the maximal order of p-subloops in the Moufang loops that do not possess p-Sylow subloops."}
{"category": "Math", "title": "Automorphisms of higher-dimensional right-angled Artin groups", "abstract": "We study the algebraic structure of the automorphism group of a general right-angled Artin group. We show that this group is virtually torsion-free and has finite virtual cohomological dimension. This generalizes results proved by the authors and John Crisp (arXiv:math/0610980) for two-dimensional right-angled Artin groups."}
{"category": "Math", "title": "Fourier series on fractals: a parallel with wavelet theory", "abstract": "We study orthogonality relations for Fourier frequencies and complex exponentials in Hilbert spaces $L^2(\\mu)$ with measures $\\mu$ arising from iterated function systems (IFS). This includes equilibrium measures in complex dynamics. Motivated by applications, we draw parallels between analysis of fractal measures on the one hand, and the geometry of wavelets on the other. We are motivated by spectral theory for commuting partial differential operators and related duality notions. While stated initially for bounded and open regions in $\\br^d$, they have since found reformulations in the theory of fractals and wavelets. We include a historical sketch with questions from early operator theory."}
{"category": "Math", "title": "On RSA Moduli with Almost Half of the Bits Prescribed", "abstract": "We show that using character sum estimates due to H. Iwaniec leads to an improvement of recent results about the distribution and finding RSA moduli $M=pl$, where $p$ and $l$ are primes, with prescribed bit patterns. We are now able to specify about $n$ bits instead of about $n/2$ bits as in the previous work. We also show that the same result of H. Iwaniec can be used to obtain an unconditional version of a combinatorial result of W. de Launey and D. Gordon that was originally derived under the Extended Riemann Hypothesis."}
{"category": "Math", "title": "Classification of connecting solutions of semilinear parabolic equations", "abstract": "For a given semilinear parabolic equation with polynomial nonlinearity, many solutions blow up in finite time. For a certain large class of these equations, we show that some of the solutions which do not blow up actually tend to equilibria. The characterizing property of such solutions is a finite energy constraint, which comes about from the fact that this class of equations can be written as the $L^2$ gradient of a certain functional."}
{"category": "Math", "title": "Extension of log pluricanonical forms from subvarieties", "abstract": "In this paper, I prove a very general extension theorem for log pluricanonical systems. The main application of this extension theorem is (together with Kawamata's subadjunction theorem) to give an optimal subadjunction theorem which relates the positivities of canonical bundle of the ambient projective manifold and that of the (maximal) center of log canonical singularities. This is an extension of the corresponding result in my previous work where I dealt with log pluricanonical systems of general type. This subadjunction theorem indicates an approach to solve the abundance conjecture for canonical divisors (or log canonical divisors) in terms of the induction in dimension."}
{"category": "Math", "title": "On the Signed Small Ball Inequality", "abstract": "This paper is a companion to our prior paper arXiv:0705.4619 on the `Small Ball Inequality in All Dimensions.' In it, we address a more restrictive inequality, and obtain a non-trivial, explicit bound, using a single essential estimate from our prior paper. The prior bound was not explicit and much more involved."}
{"category": "Math", "title": "Bounding surface actions on hyperbolic spaces", "abstract": "We give a diameter bound for fundamental domains for isometric actions of the fundamental group of a closed hyperbolic surface on a delta-hyperbolic space, where the bound depends on the hyperbolicity constant delta, the genus of the surface, and the injectivity radius of the action, which we assume to be strictly positive."}
{"category": "Math", "title": "Finding Almost Squares III", "abstract": "An almost square of type 2 is an integer $n$ that can be factored in two different ways as $n = a_1 b_1 = a_2 b_2$ with $a_1$, $a_2$, $b_1$, $b_2 \\approx \\sqrt{n}$. In this paper, we shall improve upon previous result on short intervals containing an almost square of type 2. This leads to an inquiry of finding a short interval around $x$ that contains an integer divisible by some integer in $[x^c, 2 x^c]$ with $0 < c < 1$."}
{"category": "Math", "title": "Entropy Functionals, Sobolev Inequalities and kappa-Noncollapsing Estimates along the Ricci Flow", "abstract": "In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries."}
{"category": "Math", "title": "Polynomial Approximation and $\\omega^r_\\phi (f,t)$ Twenty Years Later", "abstract": "About twenty years ago the measure of smoothness $\\omega ^r_\\phi (f,t)$ was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time."}
{"category": "Math", "title": "Convex-compactness and its applications", "abstract": "The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in lieu of compactness in a variety of cases. Specifically, we establish convex compactness for certain familiar classes of subsets of the set of positive random variables under the topology induced by convergence in probability. Two applications in infinite-dimensional optimization - attainment of infima and a version of the Minimax theorem - are given. Moreover, a new fixed-point theorem of the Knaster-Kuratowski-Mazurkiewicz-type is derived and used to prove a general version of the Walrasian excess-demand theorem."}
{"category": "Math", "title": "Different representations of Euclidean geometry and their application to the space-time geometry", "abstract": "Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point, segment, angle) and no additional structures. V-representation contains two basic elements (point, vector) and additional structure: linear vector space. In sigma-representation there is only one basic element and additional structure: world function \\sigma =\\rho^{2}/2, where \\rho is the distance. The concept of distance appears in all representations. However, as a structure, determining the geometry, the distance appears only in the sigma-representation. The sigma-representation is most appropriate for modification of the proper Euclidean geometry. Practically any modification of the proper Euclidean geometry turns it into multivariant geometry, where there are many vectors Q_0Q_1, Q_0Q_1^{\\prime},..., which are equal to the vector P_0P_1, but they are not equal between themselves, in general. Concept of multivariance is very important in application to the space-time geometry. The real space-time geometry is multivariant. Multivariance of the space-time geometry is responsible for quantum effects."}
{"category": "Math", "title": "Supervised Machine Learning with a Novel Kernel Density Estimator", "abstract": "In recent years, kernel density estimation has been exploited by computer scientists to model machine learning problems. The kernel density estimation based approaches are of interest due to the low time complexity of either O(n) or O(n*log(n)) for constructing a classifier, where n is the number of sampling instances. Concerning design of kernel density estimators, one essential issue is how fast the pointwise mean square error (MSE) and/or the integrated mean square error (IMSE) diminish as the number of sampling instances increases. In this article, it is shown that with the proposed kernel function it is feasible to make the pointwise MSE of the density estimator converge at O(n^-2/3) regardless of the dimension of the vector space, provided that the probability density function at the point of interest meets certain conditions."}
{"category": "Math", "title": "Subcritical Lp bounds on spectral clusters for Lipschitz metrics", "abstract": "We establish asymptotic bounds on the L^p norms of spectrally localized functions in the case of two-dimensional Dirichlet forms with coefficients of Lipschitz regularity. These bounds are new for the range p>6. A key step in the proof is bounding the rate at which energy spreads for solutions to hyperbolic equations with Lipschitz coefficients."}
{"category": "Math", "title": "A note on calculating autocovariances of periodic ARMA models", "abstract": "An analytically simple and tractable formula for the start-up autocovariances of periodic ARMA (PARMA) models is provided."}
{"category": "Math", "title": "Cubical convex ear decompositions", "abstract": "We consider the problem of constructing a convex ear decomposition for a poset. The usual technique, first used by Nyman and Swartz, starts with a CL-labeling and uses this to shell the `ears' of the decomposition. We axiomatize the necessary conditions for this technique as a \"CL-ced\" or \"EL-ced\". We find an EL-ced of the d-divisible partition lattice, and a closely related convex ear decomposition of the coset lattice of a relatively complemented group. Along the way, we construct new EL-labelings of both lattices. The convex ear decompositions so constructed are formed by face lattices of hypercubes. We then proceed to show that if two posets P_1 and P_2 have convex ear decompositions (CL-ceds), then their products P_1 \\times P_2, P_1 \\lrtimes P_2, and P_1 \\urtimes P_2 also have convex ear decompositions (CL-ceds). An interesting special case is: if P_1 and P_2 have polytopal order complexes, then so do their products."}
{"category": "Math", "title": "Polyhedral tori with minimal coordinates", "abstract": "We give explicit realizations with small integer coordinates for all triangulated tori with up to 12 vertices. In particular, we provide coordinate-minimal realizations in general position for all triangulations of the torus with 7, 8, 9, and 10 vertices. For the unique 7-vertex triangulation of the torus we show that all corresponding 72 oriented matroids are realizable in the 6x6x6-cube. Moreover, we present polyhedral tori with 8 vertices in the 2x2x2-cube, general position realizations of triangulated tori with 8 vertices in the 2x2x3-cuboid as well as polyhedral tori with 9 and 10 vertices in the 1x2x2-cuboid."}
{"category": "Math", "title": "Comments On \" Orbits of automorphism groups of fields\"", "abstract": "Let $R$ be a commutative $k-$algebra over a field $k$. Assume $R$ is a noetherian, infinite, integral domain. The group of $k-$automorphisms of $R$,i.e.$Aut_k(R)$ acts in a natural way on $(R-k)$.In the first part of this article, we study the structure of $R$ when the orbit space $(R-k)/Aut_k(R)$ is finite.We note that most of the results, not particularly relevent to fields, in [1,\\S 2] hold in this case as well. Moreover, we prove that $R$ is a field. In the second part, we study a special case of the Conjecture 2.1 in [1] : If $K/k$ is a non trivial field extension where $k$ is algebraically closed and $\\mid (K-k)/Aut_k(K) \\mid = 1$ then $K$ is algebraically closed. In the end, we give an elementary proof of [1,Theorem 1.1] in case $K$ is finitely generated over its prime subfield."}
{"category": "Math", "title": "The twist subgroup of the mapping class group of a nonorientable surface", "abstract": "Let T(N) be the subgroup of the mapping class group of a nonorientable surface N (possibly with punctures and/or boundary components) generated by twists about two-sided circles. We obtain a simple generating set for T(N). As an application we compute the first homology group (abelianization) of T(N)."}
{"category": "Math", "title": "Analogies between analysis on foliated spaces and arithmetic geometry", "abstract": "We point out analogies between (a) the explicit formulas in analytic number theory and transversal index theory, (b) Lichtenbaum's recent conjectures on special values of Hasse-Weil zeta functions and a formula for special values of Ruelle zeta functions, (c) work of Cramer on the zeroes of the Riemann zeta function and a result of Chazarain on the trace of a wave operator. These analogies suggest certain problems in the analysis on foliated spaces. The paper is mostly a review of previous work but the ideas concerning (c) are new."}
{"category": "Math", "title": "Pure Anderson Motives and Abelian \\tau-Sheaves", "abstract": "Pure t-motives were introduced by G. Anderson as higher dimensional generalizations of Drinfeld modules, and as the appropriate analogs of abelian varieties in the arithmetic of function fields. In order to construct moduli spaces for pure t-motives the second author has previously introduced the concept of abelian \\tau-sheaf. In this article we clarify the relation between pure t-motives and abelian \\tau-sheaves. We obtain an equivalence of the respective quasi-isogeny categories. Furthermore, we develop the elementary theory of both structures regarding morphisms, isogenies, Tate modules, and local shtukas. The later are the analogs of p-divisible groups."}
{"category": "Math", "title": "Pure Anderson Motives over Finite Fields", "abstract": "In the arithmetic of function fields Drinfeld modules play the role that elliptic curves take on in the arithmetic of number fields. As higher dimensional generalizations of Drinfeld modules, and as the appropriate analogues of abelian varieties, G. Anderson introduced pure t-motives. In this article we study the arithmetic of the later. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomorphism algebra, but also this fails over infinite fields. Still over finite fields we study the endomorphism rings of pure t-motives and criteria for the existence of isogenies. We obtain answers which are similar to Tate's famous results for abelian varieties."}
{"category": "Math", "title": "A priori bounds and a Liouville theorem on a half-space for higher order elliptic Dirichlet problems", "abstract": "We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\\Omega$ in $R^N$ with Dirichlet boundary conditions. The operator $L$ is a uniformly elliptic operator of order $2m$. We assume that for $s\\to \\pm\\infty$ the nonlinearity $f(x,s)$ behaves like $|s|^q$ multiplied by a continuous and positive function of $x$. Here the exponent $q$ is subcritical, i.e., $q>1$ if $N<=2m$, $1<q<\\frac{N+2m}{N-2m}$ if $N>2m$. We prove a priori bounds, i.e, we show that the $L^\\infty$-norm of every solution $u$ is bounded by a constant independent of $u$. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if $u$ is a classical, bounded, non-negative solution of $(-\\Delta)^m u = u^q$ in a half-space with Dirichlet boundary conditions and if $q>1$ is subcritical then $u$ vanishes identically."}
{"category": "Math", "title": "The Lyapunov spectrum of some parabolic systems", "abstract": "We study the Hausdorff dimension spectrum for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy."}
{"category": "Math", "title": "On the p-adic Leopoldt Transform of a power series", "abstract": "In this paper we give a bound for the Iwasawa lambda invariant of an abelian number field attached to the cyclotomic Z_p-extension of that field. We also give some properties of Iwaswa power series attached to p-adic L-functions."}
{"category": "Math", "title": "Regulator constants and the parity conjecture", "abstract": "The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/Q is abelian and K^\\infty is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of Gal(K^\\infty/Q). We also give analogous results when K/Q is non-abelian, the base field is not Q and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their \"regulator constants\", and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations."}
{"category": "Math", "title": "A Novel Solution to the Frenet-Serret Equations", "abstract": "A set of equations is developed to describe a curve in space given the curvature $\\kappa$ and the angle of rotation $\\theta$ of the osculating plane. The set of equations has a solution (in terms of $\\kappa$ and $\\theta$) that indirectly solves the Frenet-Serret equations, with a unique value of $\\theta$ for each specified value of $\\tau$. Explicit solutions can be generated for constant $\\theta$. The equations break down when the tangent vector aligns to one of the unit coordinate vectors, requiring a reorientation of the local coordinate system."}
{"category": "Math", "title": "Chern numbers and diffeomorphism types of projective varieties", "abstract": "In 1954 Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. We give a complete answer to this question in small dimensions, and also prove partial results without restrictions on the dimension."}
{"category": "Math", "title": "On Galois Groups of Prime Degree Polynomials with Complex Roots", "abstract": "Let $f$ be an irreducible polynomial of prime degree $p\\geq 5$ over $\\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens H\\\"{o}chsmann (1999), we show that if $p\\geq 4k+1$ then $\\Gal(f/\\QQ)$ is isomorphic to $A_{p}$ or $S_{p}$. This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T.Shaska. If such a polynomial $f$ is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of an irreducible polynomial of degree $p$ over $\\QQ$ having complex roots."}
{"category": "Math", "title": "Singular limits for the bi-laplacian operator with exponential nonlinearity in $\\R^4$", "abstract": "Let $\\Omega$ be a bounded smooth domain in $\\mathbb{R}^{4}$ such that for some integer $d\\geq1$ its $d$-th singular cohomology group with coefficients in some field is not zero, then problem {\\Delta^{2}u-\\rho^{4}k(x)e^{u}=0 & \\hbox{in}\\Omega, u=\\Delta u=0 & \\hbox{on}\\partial\\Omega, has a solution blowing-up, as $\\rho\\to0$, at $m$ points of $\\Omega$, for any given number $m$."}
{"category": "Math", "title": "A Generalization of a Result of Hardy and Littlewood", "abstract": "In this note we study the growth of \\sum_{m=1}^M\\frac1{\\|m\\alpha\\|} as a function of M for different classes of \\alpha\\in[0,1). Hardy and Littlewood showed that for numbers of bounded type, the sum is \\simeq M\\log M. We give a very simple proof for it. Further we show the following for generic \\alpha. For a non-decreasing function \\phi tending to infinity, \\limsup_{M\\to\\infty}\\frac1{\\phi(\\log M)}\\bigg[\\frac1{M\\log M}\\sum_{m=1}^M\\frac1{\\|m\\alpha\\|}\\bigg] is zero or infinity according as \\sum\\frac1{k\\phi(k)} converges or diverges."}
{"category": "Math", "title": "Hyperbolicity of general deformations", "abstract": "We present two methods of constructing low degree Kobayashi hyperbolic hypersurfaces in the projective space: the projection method and the deformation method. The talk is based on joint works of the speaker with B. Shiffman and C. Ciliberto."}
{"category": "Math", "title": "The group reduction for bounded cosine functions on UMD spaces", "abstract": "It is shown that if A generates a bounded cosine operator function on a UMD space X, then i(-A)^{1/2} generates a bounded C_0-group. The proof uses a transference principle for cosine functions."}
{"category": "Math", "title": "Selection principles and countable dimension", "abstract": "We characterize countable dimensionality and strong countable dimensionality by means of an infinite game."}
{"category": "Math", "title": "Products and selection principles", "abstract": "The product of a Sierpinski set and a Lusin set has Menger's property. The product of a gamma set and a Lusin set has Rothberger's property."}
{"category": "Math", "title": "Compactifying normal algebraic spaces", "abstract": "The author wrote this note after being asked about the existence of compactifications of algebraic spaces. Subsequent to posting the article to the math arXiv, the author learned from Yutakaa Matsuura that the results of this paper had been proved by Raoult in 1971, using the same techniques. Since Raoult's article may be unknown to those working in the field, the author is keeping this preprint on the arXiv server. However, he makes no claim of originality."}
{"category": "Math", "title": "Uniform estimates for some paraproducts", "abstract": "We establish $L^p\\times L^q$ to $L^r$ estimates for some paraproducts, which arise in the study of the bilinear Hilbert transform along curves."}
{"category": "Math", "title": "A bilinear oscillatory integral along parabolas", "abstract": "We establish an $L^\\infty\\times L^2 \\to L^2$ norm estimate for a bilinear oscillatory integral along parabolas incorporating oscillatory factors $e^{i|t|^{-\\beta}}$."}
{"category": "Math", "title": "Three lectures on elliptic surfaces and curves of high rank", "abstract": "Over the past two years we have improved several of the (Mordell-Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P_i in E(Q) (the previous record was 24, by R.Martin and W.McMillen 2000), and the first example of a curve over Q with Mordell-Weil group isomorphic with (Z/2Z) x Z^18 (the previous rank record for a curve with a 2-torsion point was 15, by Dujella 2002). In these lectures we give some of the background, theory, and computational tools that led to these new records and related applications. I Context and overview: the theorems of Mordell(-Weil) and Mazur; the rank problem; the approaches of Neron--Shioda and Mestre; elliptic surfaces and Neron specialization; fields other than Q. II Elliptic surfaces and K3 surfaces: the Mordell-Weil and Neron-Severi groups; K3 surfaces of high Neron-Severi rank and their moduli; an elliptic K3 surface over Q of Mordell-Weil rank 17. Some other applications of K3 surfaces of high rank and their moduli. III Computational issues, techniques, and results: slices of Niemeier lattices; finding and transforming models of K3 surfaces of high rank; searching for good specializations. Summary of new rank records for elliptic curves."}
{"category": "Math", "title": "Values of characters sums for finite unitary groups", "abstract": "A known result for the finite general linear group $\\GL(n,\\FF_q)$ and for the finite unitary group $\\U(n,\\FF_{q^2})$ posits that the sum of the irreducible character degrees is equal to the number of symmetric matrices in the group. Fulman and Guralnick extended this result by considering sums of irreducible characters evaluated at an arbitrary conjugacy class of $\\GL(n,\\FF_q)$. We develop an explicit formula for the value of the permutation character of $\\U(2n,\\FF_{q^2})$ over $\\Sp(2n,\\FF_q)$ evaluated an an arbitrary conjugacy class and use results concerning Gelfand-Graev characters to obtain an analogous formula for $\\U(n,\\FF_{q^2})$ in the case where $q$ is an odd prime. These results are also given as probabilistic statements."}
{"category": "Math", "title": "Hyperbolic geometry of multiply twisted knots", "abstract": "We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for the complements of these links, based only on a diagram. The results are obtained by finding geometric information on generalized augmentations of these links."}
{"category": "Math", "title": "From Littlewood-Richardson sequences to subgroup embeddings and back", "abstract": "Let $\\alpha$, $\\beta$, and $\\gamma$ be partitions describing the isomorphism types of the finite abelian $p$-groups $A$, $B$, and $C$. From theorems by Green and Klein it is well-known that there is a short exact sequence $0\\to A\\to B\\to C\\to 0$ of abelian groups if and only if there is a Littlewood-Richardson sequence of type $(\\alpha,\\beta,\\gamma)$. Starting from the observation that a sequence of partitions has the LR property if and only if every subsequence of length 2 does, we demonstrate how LR-sequences of length two correspond to embeddings of a $p^2$-bounded subgroup in a finite abelian $p$-group. Using the known classification of all such embeddings we derive short proofs of the theorems by Green and Klein."}
{"category": "Math", "title": "Shape and local growth for multidimensional branching random walks in random environment", "abstract": "We study branching random walks in random environment on the $d$-dimensional square lattice, $d \\geq 1$. In this model, the environment has finite range dependence, and the population size cannot decrease. We prove limit theorems (laws of large numbers) for the set of lattice sites which are visited up to a large time as well as for the local size of the population. The limiting shape of this set is compact and convex, though the local size is given by a concave growth exponent. Also, we obtain the law of large numbers for the logarithm of the total number of particles in the process."}
{"category": "Math", "title": "Asymptotic estimates for phi functions for subsets of {m+1, m+2,...,n}", "abstract": "Let f(m,n) denote the number of relatively prime subsets of {m+1,m+2,...,n}, and let Phi(m,n) denote the number of subsets A of {m+1,m+2,...,n} such that gcd(A) is relatively prime to n. Let f_k(m,n) and Phi_k(m,n) be the analogous counting functions restricted to sets of cardinality k. Simple explicit formulas and asymptotic estimates are obtained for these four functions."}
{"category": "Math", "title": "On the realization of double occurrence words", "abstract": "Let S be a double occurrence word, and let M_S be the word's interlacement matrix, regarded as a matrix over GF(2). Gauss addressed the question of which double occurrence words are realizable by generic closed curves in the plane. We reformulate answers given by Rosenstiehl and by de Fraysseix and Ossona de Mendez to give new graph-theoretic and algebraic characterizations of realizable words. Our algebraic characterization is especially pleasing: S is realizable if and only if there exists a diagonal matrix D_S such that M_S+D_S is idempotent over GF(2)."}
{"category": "Math", "title": "Bayesian Classification and Regression with High Dimensional Features", "abstract": "This thesis responds to the challenges of using a large number, such as thousands, of features in regression and classification problems. There are two situations where such high dimensional features arise. One is when high dimensional measurements are available, for example, gene expression data produced by microarray techniques. For computational or other reasons, people may select only a small subset of features when modelling such data, by looking at how relevant the features are to predicting the response, based on some measure such as correlation with the response in the training data. Although it is used very commonly, this procedure will make the response appear more predictable than it actually is. In Chapter 2, we propose a Bayesian method to avoid this selection bias, with application to naive Bayes models and mixture models. High dimensional features also arise when we consider high-order interactions. The number of parameters will increase exponentially with the order considered. In Chapter 3, we propose a method for compressing a group of parameters into a single one, by exploiting the fact that many predictor variables derived from high-order interactions have the same values for all the training cases. The number of compressed parameters may have converged before considering the highest possible order. We apply this compression method to logistic sequence prediction models and logistic classification models. We use both simulated data and real data to test our methods in both chapters."}
{"category": "Math", "title": "The p-harmonic boundary for finitely generated groups and the first reduced \\ell_p-cohomology", "abstract": "Let $p$ be a real number greater than one and let $G$ be a finitely generated, infinite group. In this paper we introduce the $p$-harmonic boundary of $G$. We then characterize the vanishing of the first reduced $\\ell^p$-cohomology of $G$ in terms of the cardinality of this boundary. Some properties of $p$-harmonic boundaries that are preserved under rough isometries are also given. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on $G$, the $p$-harmonic boundary of $G$ and the first reduced $\\ell^p$-cohomology of $G$."}
{"category": "Math", "title": "On Birnbaum-Saunders Inference", "abstract": "The Birnbaum-Saunders distribution, also known as the fatigue-life distribution, is frequently used in reliability studies. We obtain adjustments to the Birnbaum--Saunders profile likelihood function. The modified versions of the likelihood function were obtained for both the shape and scale parameters, i.e., we take the shape parameter to be of interest and the scale parameter to be of nuisance, and then consider the situation in which the interest lies in performing inference on the scale parameter with the shape parameter entering the modeling in nuisance fashion. Modified profile maximum likelihood estimators are obtained by maximizing the corresponding adjusted likelihood functions. We present numerical evidence on the finite sample behavior of the different estimators and associated likelihood ratio tests. The results favor the adjusted estimators and tests we propose. A novel aspect of the profile likelihood adjustments obtained in this paper is that they yield improved point estimators and tests. The two profile likelihood adjustments work well when inference is made on the shape parameter, and one of them displays superior behavior when it comes to performing hypothesis testing inference on the scale parameter. Two empirical applications are briefly presented."}
{"category": "Math", "title": "The relationships between Invertible Module Maps X and X_z", "abstract": "If $R$ and $M$ are Hilbert modules (in the sense of R. G. Douglas and V. I. Paulsen), we study the relationship between invertible module maps $X:R\\to{M}$ and $X_{z}:R/R_{z}\\to{M/M_{z}}$. In particular, for quasi-free Hilbert modules $R$ and $M$, we provide a condition of a module map $X:R\\to{M}$, such that if $X_{z}:R/R_{z}\\to{M/M_{z}}$ is invertible for every $z$ in a domain $\\Omega$ in the complex plane, then $X$ is also invertible."}
{"category": "Math", "title": "Sums of Products of Bernoulli numbers of the second kind", "abstract": "The Bernoulli numbers b_0,b_1,b_2,.... of the second kind are defined by \\sum_{n=0}^\\infty b_nt^n=\\frac{t}{\\log(1+t)}. In this paper, we give an explicit formula for the sum \\sum_{j_1+j_2+...+j_N=n, j_1,j_2,...,j_N>=0}b_{j_1}b_{j_2}...b_{j_N}. We also establish a q-analogue for \\sum_{k=0}^n b_kb_{n-k}=-(n-1)b_n-(n-2)b_{n-1}."}
{"category": "Math", "title": "Least squares volatility change point estimation for partially observed diffusion processes", "abstract": "A one dimensional diffusion process $X=\\{X_t, 0\\leq t \\leq T\\}$, with drift $b(x)$ and diffusion coefficient $\\sigma(\\theta, x)=\\sqrt{\\theta} \\sigma(x)$ known up to $\\theta>0$, is supposed to switch volatility regime at some point $t^*\\in (0,T)$. On the basis of discrete time observations from $X$, the problem is the one of estimating the instant of change in the volatility structure $t^*$ as well as the two values of $\\theta$, say $\\theta_1$ and $\\theta_2$, before and after the change point. It is assumed that the sampling occurs at regularly spaced times intervals of length $\\Delta_n$ with $n\\Delta_n=T$. To work out our statistical problem we use a least squares approach. Consistency, rates of convergence and distributional results of the estimators are presented under an high frequency scheme. We also study the case of a diffusion process with unknown drift and unknown volatility but constant."}
{"category": "Math", "title": "Structure of the string link concordance group and Hirzebruch-type invariants", "abstract": "We employ Hirzebruch-type invariants obtained from iterated p-covers to investigate concordance of links and string links. We show that the invariants naturally give various group homomorphisms of the string link concordance group into L-groups over number fields. We also obtain homomorphisms of successive quotients of the Cochran-Orr-Teichner filtration. As an application we show that the kernel of Harvey's $\\rho_n$-invariant is large enough to contain a subgroup with infinite rank abelianization, modulo local knots. As another application, we show that recently discovered nontrivial 2-torsion examples of iterated Bing doubles lying at an arbitrary depth of the Cochran-Orr-Teichner filtration are independent over $\\Z_2$ as links, in an appropriate sense. We also construct similar examples of infinite order links which are independent over $\\Z$."}
{"category": "Math", "title": "Cycle structures of autotopisms of the Latin squares of order up to 11", "abstract": "The cycle structure of a Latin square autotopism $\\Theta=(\\alpha,\\beta,\\gamma)$ is the triple $(\\mathbf{l}_{\\alpha},\\mathbf{l}_{\\beta},\\mathbf{l}_{\\gamma})$, where $\\mathbf{l}_{\\delta}$ is the cycle structure of $\\delta$, for all $\\delta\\in\\{\\alpha,\\beta,\\gamma\\}$. In this paper we study some properties of these cycle structures and, as a consequence, we give a classification of all autotopisms of the Latin squares of order up to 11."}
{"category": "Math", "title": "A Unified Approach to Stochastic Evolution Equations Using the Skorokhod Integral", "abstract": "We study stochastic evolution equations driven by Gaussian noise. The key features of the model are that the operators in the deterministic and stochastic parts can have the same order and the noise can be time-only, space-only, or space-time. Even the simplest equations of this kind do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition leads to natural weights and a natural replacement of the square integrability condition."}
{"category": "Math", "title": "Minimal surfaces in sub-Riemannian manifolds and structure of their singular sets in the (2,3) case", "abstract": "We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\\it horizontal} area functional associated to the canonical {\\it horizontal} area form. We derive the intrinsic equation in the general case and then consider in greater detail 2-dimensional surfaces in contact manifolds of dimension 3. We show that in this case minimal surfaces are projections of a special class of 2-dimensional surfaces in the horizontal spherical bundle over the base manifold. Generic singularities of minimal surfaces turn out the singularities of this projection, and we give a complete local classification of them. We illustrate our results by examples in the Heisenberg group and the group of roto-translations"}
{"category": "Math", "title": "The Groebner basis of the ideal of vanishing polynomials", "abstract": "We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner basis is independent of the monomial order and that the set of leading terms of the constructed Groebner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Groebner basis in Z/m[x_1,x_2,...,x_n] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip."}
{"category": "Math", "title": "Choice and Regularity: Common Consequences in Logic", "abstract": "It is well-known that Choice and Regularity are independent of each other but have important common consequences of logical character (reflection principles, representations of classes by sets, etc.). We explain this phenomenon by isolating their \"intersection\", a principle (called here Best-Foundedness) which is consistent with the negations of both axioms but implies all these consequences. Then we study relationships between these consequences (and near principles) in detail. Finally, we consider some arguments related to truth of various principles in set theory, especially arguments concerning the interpretability strength."}
{"category": "Math", "title": "Quasi-maximum likelihood estimation of periodic GARCH processes", "abstract": "This paper establishes the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for a GARCH process with periodically time-varying parameters. We first give a necessary and sufficient condition for the existence of a strictly periodically stationary solution for the periodic GARCH (P-GARCH) equation. As a result, it is shown that the moment of some positive order of the P-GARCH solution is finite, under which we prove the strong consistency and asymptotic normality (CAN) of the QMLE without any condition on the moments of the underlying process."}
{"category": "Math", "title": "On some probabilistic properties of periodic GARCH processes", "abstract": "This paper examines some probabilistic properties of the class of periodic GARCH processes (PGARCH) which feature periodicity in conditional heteroskedasticity. In these models, the parameters are allowed to switch between different regimes, so that their structure shares many properties with periodic ARMA process (PARMA). We examine the strict and second order periodic stationarities, the existence of higher-order moments, the covariance structure, the geometric ergodicity and -mixing of the PGARCH(p,q) process under general and tractable assumptions. Some examples are proposed to illustrate the various concepts."}
{"category": "Math", "title": "Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps", "abstract": "We study the moduli space of torsion-free G2-structures on a fixed compact manifold, and define its associated universal intermediate Jacobian J. We define the Yukawa coupling and relate it to a natural pseudo-Kahler structure on J. We consider natural Chern-Simons type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang-Mills connections), and also deformed Donaldson-Thomas connections. We show that the moduli spaces of these structures can be isotropically immersed in J by means of G2-analogues of Abel-Jacobi maps."}
{"category": "Math", "title": "Simulated Annealing: Rigorous finite-time guarantees for optimization on continuous domains", "abstract": "Simulated annealing is a popular method for approaching the solution of a global optimization problem. Existing results on its performance apply to discrete combinatorial optimization where the optimization variables can assume only a finite set of possible values. We introduce a new general formulation of simulated annealing which allows one to guarantee finite-time performance in the optimization of functions of continuous variables. The results hold universally for any optimization problem on a bounded domain and establish a connection between simulated annealing and up-to-date theory of convergence of Markov chain Monte Carlo methods on continuous domains. This work is inspired by the concept of finite-time learning with known accuracy and confidence developed in statistical learning theory."}
{"category": "Math", "title": "C*-pseudo-multiplicative unitaries", "abstract": "We introduce $C^*$-pseudo-multiplicative unitaries and (concrete) Hopf $C^*$-bimodules, which are $C^*$-algebraic variants of the pseudo-multiplicative unitaries on Hilbert spaces and the Hopf-von Neumann-bimodules studied by Enock, Lesieur, Vallin. Moreover, we associate to every regular $C^*$-pseudo-multiplicative unitary two Hopf-$C^*$-bimodules and discuss examples related to locally compact groupoids."}
{"category": "Math", "title": "Two polynomial representations of experimental design", "abstract": "In the context of algebraic statistics an experimental design is described by a set of polynomials called the design ideal. This, in turn, is generated by finite sets of polynomials. Two types of generating sets are mostly used in the literature: Groebner bases and indicator functions. We briefly describe them both, how they are used in the analysis and planning of a design and how to switch between them. Examples include fractions of full factorial designs and designs for mixture experiments."}
{"category": "Math", "title": "A Criterion for Precompactness in the Space of Hypermeasures", "abstract": "Let $Q$ denote the space of signed measures on the Borel $\\sigma$-algebra of a separable complete space $X$. We endow $Q$ with the norm $\\|q\\|=\\sup|\\int\\phi dq|$, where the supremum is taken over all Lipschitz with constant 1 functions whose module does not exceed unity. This normed space is incomplete provided $X$ is infinite and has at least one limit point. We call its completion the space of hypermeasures. Necessary and sufficient conditions for precompactness (=relative compactness) of a set of hypermeasures are found. They are similar to those of Prokhorov's and Fernique's theorems for measures."}
{"category": "Math", "title": "Some extensions of the uncertainty principle", "abstract": "We study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [1] and Zozor et al. [2]. Those inequalities can be considered as generalizations of the Heisenberg uncertainty principle, since they measure the mutual uncertainty of a wave function and its Fourier transform through their associated R\\'enyi entropies with conjugated indices. We consider here the general case where the entropic indices are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indices $\\alpha$ and $\\beta$ in the plane $(\\alpha, \\beta)$. Our results explain and extend a recent study by Luis [2], where states with quantum fluctuations below the Gaussian case are discussed at the single point $(2,2)$."}
{"category": "Math", "title": "Construction of a stationary queue with impatient customers", "abstract": "In this paper, we study the stability of queues with impatient customers. Under general stationary ergodic assumptions, we first provide some conditions for such a queue to be regenerative (i.e. to empty a.s. an infinite number of times). In the particular case of a single server operating in First in, First out, we prove the existence (in some cases, on an enlarged probability space) of a stationary workload. This is done by studying a non-monotonic stochastic recursion under the Palm settings, and by stochastic comparison of stochastic recursions."}
{"category": "Math", "title": "On the ideal $(v^0)$", "abstract": "The $\\sigma$-ideal $(v^0)$ is associated with the Silver forcing, see \\cite{bre}. Also, it constitutes the family of all completely doughnut null sets, see \\cite{hal}. We introduce segments and $*$-segments topologies, to state some resemblances of $(v^0)$ to the family of Ramsey null sets. To describe $add(v^0)$ we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen's conjecture $cov(v^0) = add(v^0)$ is confirmed under the hypothesis $t= \\min \\{\\cf (\\frak c), r\\} $. The hypothesis $h=\\omega_1$ implies that $(v^0)$ has the ideal type $(\\frak c, \\omega_1,\\frak c)$."}
{"category": "Math", "title": "Hankel hyperdeterminants, rectangular Jack polynomials and even powers of the Vandermonde", "abstract": "We investigate the link between rectangular Jack polynomials and Hankel hyperdeterminants. As an application we give an expression of the even power of the Vandermonde in term of Jack polynomials."}
{"category": "Math", "title": "Chern numbers and the geometry of partial flag manifolds", "abstract": "We calculate the Chern classes and Chern numbers for the natural almost Hermitian structures of the partial flag manifolds F_n=SU(n+2)/S(U(n)\\times U(1)\\times U(1)). For all n>1 there are two invariant complex algebraic structures, which arise from the projectivizations of the holomorphic tangent and cotangent bundles of complex projective spaces. The projectivization of the cotangent bundle is the twistor space of a Grassmannian considered as a quaternionic K\\\"ahler manifold. There is also an invariant nearly K\\\"ahler structure, because F_n is a 3-symmetric space. We explain the relations between the different structures and their Chern classes, and we prove that F_n is not geometrically formal."}
{"category": "Math", "title": "(Co)homology of quantum complete intersections", "abstract": "We construct a minimal projective bimodule resolution for every finite dimensional quantum complete intersection of codimension two. Then we use this resolution to compute both the Hochschild cohomology and homology for such an algebra. In particular, we show that the cohomology vanishes in high degrees, while the homology is always nonzero."}
{"category": "Math", "title": "The Tate conjecture over finite fields (AIM talk)", "abstract": "These are my notes for a talk at the The Tate Conjecture workshop at the American Institute of Mathematics in Palo Alto, CA, July 23--July 27, 2007, somewhat revised and expanded. The intent of the talk was to review what is known and to suggest directions for research. v2: Revised expanded (24 pages)."}
{"category": "Math", "title": "Determinants of (generalised) Catalan numbers", "abstract": "We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindstr\\\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a simple determinant lemma from [Manuscripta Math. 69 (1990), 173-202]. This approach leads also naturally to extensions and generalisations."}
{"category": "Math", "title": "Residue Classes Having Tardy Totients", "abstract": "We show, in an effective way, that there exists a sequence of congruence classes $a_k\\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $\\phi(n)\\equiv a_k\\pmod {m_k}$ exists and satisfies $\\log n_k/\\log m_k\\to\\infty $ as $k\\to\\infty$. Here, $\\phi(n)$ is the Euler function. This answers a question raised in \\cite{FS}. We also show that every congruence class containing an even integer contains infinitely many values of the Carmichael function $\\lambda(n)$ and the least such $n$ satisfies $n\\ll m^{13}$."}
{"category": "Math", "title": "Computing a pyramid partition generating function with dimer shuffling", "abstract": "We verify a recent conjecture of Kenyon/Szendroi, arXiv:0705.3419, by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson--Thomas theory of a non-commutative resolution of the conifold singularity {x1x2 -x3x4 = 0}. The proof does not require algebraic geometry; it uses a modified version of the domino shuffling algorithm of Elkies, Kuperberg, Larsen and Propp."}
{"category": "Math", "title": "The martingale problem for a class of stable-like processes", "abstract": "Let $\\alpha\\in (0,2)$ and consider the operator $$L f(x) =\\int [f(x+h)-f(x)-1_{(|h|\\leq 1)} \\nabla f(x)\\cdot h] \\frac{A(x,h)}{|h|^{d+\\alpha}} dh, $$ where the $\\nabla f(x)\\cdot h$ term is omitted if $\\alpha<1$. We consider the martingale problem corresponding to the operator $L$ and under mild conditions on the function $A$ prove that there exists a unique solution."}
{"category": "Math", "title": "Derived Algebraic Geometry IV: Deformation Theory", "abstract": "In this paper, we prove some foundational results on the deformation theory of E-infinity ring spectra."}
{"category": "Math", "title": "The fundamental form of a homogeneous Lagrangian in two independent variables", "abstract": "We construct, for a homogeneous Lagrangian of arbitrary order in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the `fundamental Lepage equivalent' associated with first-order Lagrangians defined on jets of sections of a fibred manifold."}
{"category": "Math", "title": "A tail inequality for suprema of unbounded empirical processes with applications to Markov chains", "abstract": "We present a tail inequality for suprema of empirical processes generated by variables with finite $\\psi_\\alpha$ norms and apply it to some geometrically ergodic Markov chains to derive similar estimates for empirical processes of such chains, generated by bounded functions. We also obtain a bounded difference inequality for symmetric statistics of such Markov chains."}
{"category": "Math", "title": "Cayley cones ruled by 2-planes: desingularization and implications of the twistor fibration", "abstract": "Cayley cones in the octonions $\\mathbb{O}$ that are ruled by oriented 2-planes are equivalent to pseudoholomorphic curves in the Grassmannian of oriented 2-planes G(2,8). The well known twistor fibration $G(2,8) -> S^6$ is used to prove the existence of immersed higher-genus pseudoholomorphic curves in $\\gro$. Equivalently, this produces Cayley cones whose links are $S^1$-bundles over genus-$g$ Riemann surfaces. When the degree of an immersed pseudoholomorphic curve is large enough, the corresponding 2-ruled Cayley cone is the asymptotic cone of a non-conical 2-ruled Cayley 4-fold."}
{"category": "Math", "title": "Normes invariantes et existence de filtrations admissibles", "abstract": "Let L be a finite extension of Q_p and d a positive integer. A conjecture, due to C. Breuil and P. Schneider, says that the existence of invariant norms on certain locally algebraic representations of GL_{d+1}(L) should be equivalent to the existence of certain (d+1)-dimensional de Rham representations of Gal(\\bar{L}/L). We prove the easy direction of this conjecture: the existence of invariant norms implies the existence of admissible filtrations, by generalizing an idea of M.Emerton."}
{"category": "Math", "title": "A general existence proof for non-linear elliptic equations in semi-Riemannian spaces", "abstract": "We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces with curvature prescribed in the tangent bundle of an ambient Riemannian manifold $N$ without supposing any sign condition on the sectional curvatures $K_N$. A curvature flow wouldn't work in this situation, neither the method of successive approximation."}
{"category": "Math", "title": "Induced forests in regular graphs with large girth", "abstract": "An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomised algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case."}
{"category": "Math", "title": "A Note on an Asymptotically Good Tame Tower", "abstract": "The explicit construction of function fields tower with many rational points relative to the genus in the tower play a key role for the construction of asymptotically good algebraic-geometric codes. In 1997 Garcia, Stichtenoth and Thomas [6] exhibited two recursive asymptotically good Kummer towers over any non-prime field. Wulftange determined the limit of one tower in his PhD thesis [13]. In this paper we determine the limit of another tower [14]."}
{"category": "Math", "title": "Twisting Elements in Homotopy G-algebras", "abstract": "We study the notion of twisting elements $da=a\\cup_1a$ with respect to $\\cup_1$ product when it is a part of homotopy Gerstenhaber algebra structure. This allows to bring to one context the two classical concepts, the theory of deformation of algebras of M. Gerstenhaber, and $A(\\infty)$-algebras of J. Stasheff."}
{"category": "Math", "title": "A comment on Ryser's conjecture for intersecting hypergraphs", "abstract": "Let $\\tau(\\mathcal{H})$ be the cover number and $\\nu(\\mathcal{H})$ be the matching number of a hypergraph $\\mathcal{H}$. Ryser conjectured that every $r$-partite hypergraph $\\mathcal{H}$ satisfies the inequality $\\tau(\\mathcal{H}) \\leq (r-1) \\nu (\\mathcal{H})$. This conjecture is open for all $r \\ge 4$. For intersecting hypergraphs, namely those with $\\nu(\\mathcal{H})=1$, Ryser's conjecture reduces to $\\tau(\\mathcal{H}) \\leq r-1$. Even this conjecture is extremely difficult and is open for all $ r \\ge 6$. For infinitely many $r$ there are examples of intersecting $r$-partite hypergraphs with $\\tau(\\mathcal{H})=r-1$, demonstrating the tightness of the conjecture for such $r$. However, all previously known constructions are not optimal as they use far too many edges. How sparse can an intersecting $r$-partite hypergraph be, given that its cover number is as large as possible, namely $\\tau(\\mathcal{H}) \\ge r-1$? In this paper we solve this question for $r \\le 5$, give an almost optimal construction for $r=6$, prove that any $r$-partite intersecting hypergraph with $\\tau(H) \\ge r-1$ must have at least $(3-\\frac{1}{\\sqrt{18}})r(1-o(1)) \\approx 2.764r(1-o(1))$ edges, and conjecture that there exist constructions with $\\Theta(r)$ edges."}
{"category": "Math", "title": "Some Relations between Rank, Chromatic Number and Energy of Graphs", "abstract": "The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $G$ be a graph of order $n$ and ${\\rm rank}(G)$ be the rank of the adjacency matrix of $G$. In this paper we characterize all graphs with $E(G)={\\rm rank}(G)$. Among other results we show that apart from a few families of graphs, $E(G)\\geq 2\\max(\\chi(G), n-\\chi(\\bar{G}))$, where $n$ is the number of vertices of $G$, $\\bar{G}$ and $\\chi(G)$ are the complement and the chromatic number of $G$, respectively. Moreover some new lower bounds for $E(G)$ in terms of ${\\rm rank}(G)$ are given."}
{"category": "Math", "title": "Inclusion Matrices and Chains", "abstract": "Given integers $t$, $k$, and $v$ such that $0\\leq t\\leq k\\leq v$, let $W_{tk}(v)$ be the inclusion matrix of $t$-subsets vs. $k$-subsets of a $v$-set. We modify slightly the concept of standard tableau to study the notion of rank of a finite set of positive integers which was introduced by Frankl. Utilizing this, a decomposition of the poset $2^{[v]}$ into symmetric skipless chains is given. Based on this decomposition, we construct an inclusion matrix, denoted by $W_{\\bar{t}k}(v)$, which is row-equivalent to $W_{tk}(v)$. Its Smith normal form is determined. As applications, Wilson's diagonal form of $W_{tk}(v)$ is obtained as well as a new proof of the well known theorem on the necessary and sufficient conditions for existence of integral solutions of the system $W_{tk}\\bf{x}=\\bf{b}$ due to Wilson. Finally we present anotherinclusion matrix with similar properties to those of $W_{\\bar{t}k}(v)$ which is in some way equivalent to $W_{tk}(v)$."}
{"category": "Math", "title": "On the properties of generalized harmonic and oscillatory numbers. Simple proof of the Prime Number Theorem", "abstract": "We derived the sum identities for generalized harmonic and corresponding oscillatory numbers for which a sieve procedure can be applied. The obtained results enable us to understand better the properties of these numbers and their asymptotic behavior. On the basis of these identities a simple proof of the Prime Number Theorem is represented."}
{"category": "Math", "title": "Local cohomology based on a nonclosed support defined by a pair of ideals", "abstract": "We introduce an idea for generalization of a local cohomology module, which we call a local cohomology module with respect to a pair of ideals (I,J), and study their various properties. Some vanishing and nonvanishing theorems are given for this generalized version of local cohomology. We also discuss its connection with the ordinary local cohomology."}
{"category": "Math", "title": "Morse-Novikov theory, Heegaard splittings and closed orbits of gradient flows", "abstract": "The works of Donaldson and Mark make the structure of the Seiberg-Witten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle-valued Morse map on a 3-manifold. We study these invariants using the Morse-Novikov theory and Heegaard splitting for sutured manifolds, and make detailed computations for knot complements."}
{"category": "Math", "title": "Restriction of stable bundles on a jacobian of genus 2 to an embedded curve", "abstract": "The aim of this note is to describe the restriction map from the moduli space of stable rank 2 bundle with small $c_2$ on a jacobian $X$ of dimension 2, to the moduli space of stable rank 2 bundles on the corresponding genus 2 curve $C$ embedded in $X$."}
{"category": "Math", "title": "Level sets of the stochastic wave equation driven by a symmetric L\\'{e}vy noise", "abstract": "We consider the solution $\\{u(t,x);t\\geq0,x\\in\\mathbf{R}\\}$ of a system of $d$ linear stochastic wave equations driven by a $d$-dimensional symmetric space-time L\\'{e}vy noise. We provide a necessary and sufficient condition on the characteristic exponent of the L\\'{e}vy noise, which describes exactly when the zero set of $u$ is non-void. We also compute the Hausdorff dimension of that zero set when it is non-empty. These results will follow from more general potential-theoretic theorems on the level sets of L\\'{e}vy sheets."}
{"category": "Math", "title": "Singular extensions and triangulated categories", "abstract": "We propose a new look on triangulated categories, which is based on the second Hochschild cohomology."}
{"category": "Math", "title": "Non-commutative Real Algebraic Geometry - Some Basic Concepts and First Ideas", "abstract": "We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A version of Stengle's Positivstellensatz for $n \\times n$ matrices of real polynomials is proved."}
{"category": "Math", "title": "Maximal regularity for stochastic convolutions driven by Levy noise", "abstract": "We show that the result from Da Prato and Lunardi is valid for stochastic convolutions driven by L\\'evy processes."}
{"category": "Math", "title": "Structure theorems for certain Gorenstein ideals", "abstract": "The main achievement of this paper is to provide a structure theorem for Artinian, Gorenstein local rings with the property that the square of the maximal ideal is generated by two elements. The moduli problem for this class of local algebras is also discussed. Finally, upper and lower bounds for the minimal number of generators of perfect ideals are given."}
{"category": "Math", "title": "Distribution functions of linear combinations of lattice polynomials from the uniform distribution", "abstract": "We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which include weighted sums, linear combinations of order statistics, and lattice polynomials, are actually those continuous functions that reduce to linear functions on each simplex of the standard triangulation of the unit cube. They are mainly used in aggregation theory, combinatorial optimization, and game theory, where they are known as discrete Choquet integrals and Lovasz extensions."}
{"category": "Math", "title": "3-Generator Groups whose Elements Commute with Their Endomorphic Images Are Abelian", "abstract": "A group in which every element commutes with its endomorphic images is called an $E$-group. Our main result is that all 3-generator $E$-groups are abelian. It follows that the minimal number of generators of a finitely generated non-abelian $E$-group is four."}
{"category": "Math", "title": "A Semismooth Newton Method for Tikhonov Functionals with Sparsity Constraints", "abstract": "Minimization problems in $\\ell^2$ for Tikhonov functionals with sparsity constraints are considered. Sparsity of the solution is ensured by a weighted $\\ell^1$ penalty term. The necessary and sufficient condition for optimality is shown to be slantly differentiable (Newton differentiable), hence a semismooth Newton method is applicable. Local superlinear convergence of this method is proved. Numerical examples are provided which show that our method compares favorably with existing approaches."}
{"category": "Math", "title": "Uniform Convergence of the Spectral Expansion for a Differential Operator with Periodic Matrix Coefficients", "abstract": "In this paper, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and the quasiperiodic boundary conditions. Using these asymptotic formulas, we find conditions on the coefficients for which the root functions of this operator form a Riesz basis. Then we obtain the uniformly convergent spectral expansion of the differential operators with the periodic matrix coefficients"}
{"category": "Math", "title": "A quantile-copula approach to conditional density estimation", "abstract": "We present a new non-parametric estimator of the conditional density of the kernel type. It is based on an efficient transformation of the data by quantile transform. By use of the copula representation, it turns out to have a remarkable product form. We study its asymptotic properties and compare its bias and variance to competitors based on nonparametric regression."}
{"category": "Math", "title": "Sur l'irr\\'eductibilit\\'e d'une induite parabolique", "abstract": "Let $F$ be a non-Archimedean locally compact field and let $D$ be a central division algebra over $F$. Let $\\pi_1$ and $\\pi_2$ be respectively two smooth irreducible representations of ${\\rm GL}(n_1,D)$ and ${\\rm GL}(n_2,F)$, $n_1, n_2 \\geq 0$. In this article, we give some sufficient conditions on $\\pi_1$ and $\\pi_2$ so that the parabolically induced representation of $\\pi_1 \\otimes \\pi_2$ to ${\\rm GL}(n_1+n_2,D)$ has a unique irreducible quotient. In the case where $\\pi_1$ is a cuspidal representation, we compute the Zelevinsky's parameters of such a quotient in terms of parameters of $\\pi_2$. This is the key point for making explicit Howe correspondence for dual pairs of type II."}
{"category": "Math", "title": "Two-scale numerical simulation of the weakly compressible 1D isentropic Euler equations", "abstract": "Motivated by the difficulty to solve numerically the weakly compressible 1D isentropic Euler equations with classical methods, we develop in this paper a two scale numerical method on this model. This method is based on two scale convergence theory developped by N'Guetseng and Allaire, and finite volume scheme. Furthermore, we do some numerical simulations in order to verify that the two-scale numerical method is more and more accurate when the Mach number diminishes."}
{"category": "Math", "title": "Fano manifolds and blow-ups of low-dimensional subvarieties", "abstract": "We study Fano manifolds of pseudoindex greater than one and dimension greater than five, which are blow-ups of smooth varieties along smooth centers of dimension equal to the pseudoindex of the manifold. We obtain a classification of the possible cones of curves of these manifolds, and we prove that there is only one such manifold without a fiber type elementary contraction."}
{"category": "Math", "title": "Local rigidity of surfaces in space forms", "abstract": "This paper is withdrawn."}
{"category": "Math", "title": "Infinitely divisible distributions over locally compact non-archimedean fields", "abstract": "The article is devoted to stochastic processes with values in finite-dimensional vector spaces over infinite locally compact fields with non-trivial non-archimedean valuations. Infinitely divisible distributions are investigated. Theorems about their characteristic functionals are proved. Particular cases are demonstrated."}
{"category": "Math", "title": "Positivity of Chern Classes for Reflexive Sheaves on P^N", "abstract": "It is well known that the Chern classes $c_i$ of a rank $n$ vector bundle on $\\PP^N$, generated by global sections, are non-negative if $i\\leq n$ and vanish otherwise. This paper deals with the following question: does the above result hold for the wider class of reflexive sheaves? We show that the Chern numbers $c_i$ with $i\\geq 4$ can be arbitrarily negative for reflexive sheaves of any rank; on the contrary for $i\\leq 3$ we show positivity of the $c_i$ with weaker hypothesis. We obtain lower bounds for $c_1$, $c_2$ and $c_3$ for every reflexive sheaf $\\FF$ which is generated by $H^0\\FF$ on some non-empty open subset and completely classify sheaves for which either of them reach the minimum allowed, or some value close to it."}
{"category": "Math", "title": "Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system", "abstract": "We describe the asymptotic behavior as time goes to infinity of solutions of the 2 dimensional corotational wave map system and of solutions to the 4 dimensional, radially symmetric Yang-Mills equation, in the critical energy space, with data of energy smaller than or equal to a harmonic map of minimal energy. An alternative holds: either the data is the harmonic map and the soltuion is constant in time, or the solution scatters in infinite time."}
{"category": "Math", "title": "On decomposition numbers and Alvis-Curtis duality", "abstract": "We show that for general linear groups ${\\rm GL}_n(q)$ as well as for $q$-Schur algebras the knowledge of the modular Alvis-Curtis duality over fields of characteristic $\\ell$, $\\ell \\nmid q$, is equivalent to the knowledge of the decomposition numbers."}
{"category": "Math", "title": "A one-parameter family of dendriform identities", "abstract": "We prove a q-identity in the dendriform dialgebra of colored free quasi-symmetric functions. For q=1, we recover identities due to Ebrahimi-Fard, Manchon, and Patras, in particular the noncommutative Bohnenblust-Spitzer identity."}
{"category": "Math", "title": "Analysis of two step nilsequences", "abstract": "Nilsequences arose in the study of the multiple ergodic averages associated to Furstenberg's proof of Szemer\\'edi's Theorem and have since played a role in problems in additive combinatorics. Nilsequences are a generalization of almost periodic sequences and we study which portions of the classical theory for almost periodic sequences can be generalized for two step nilsequences. We state and prove basic properties for 2-step nilsequences and give a classification scheme for them."}
{"category": "Math", "title": "Modular analogues of Jordan's theorem for finite linear groups", "abstract": "In 1878, Jordan showed that a finite subgroup of GL(n,C) contains an abelian normal subgroup whose index is bounded by a function of n alone. Previously, the author has given precise bounds. Here, we consider analogues for finite linear groups over algebraically closed fields of positive characteristic l. A larger normal subgroup must be taken, to eliminate unipotent subgroups and groups of Lie type and characteristic l, and we show that generically the bound is similar to that in characteristic 0 - being (n+1)!, or (n+2)! when l divides (n+2) - given by the faithful representations of minimal degree of the symmetric groups. A complete answer for the optimal bounds is given for all degrees n and every characteristic l."}
{"category": "Math", "title": "Bootstrap of means under stratified sampling", "abstract": "In a two-stage cluster sampling procedure, $n$ random populations are drawn independently from independent populations and a sub-sample of observations is taken in each of them. The estimator of the general mean of the observed variables is asymptotically Gaussian and the asymptotic distributions of several bootstrap versions of the normalized and studentized statistics are studied. A weighted population resampling provides a good approximation and its accuracy depends on the convergence rate of the sample size of the populations."}
{"category": "Math", "title": "Bruhat order, smooth Schubert varieties, and hyperplane arrangements", "abstract": "The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincare polynomial of the corresponding Schubert variety if and only if the Schubert variety is smooth. We give an explicit combinatorial formula for the Poincare polynomial. Our main technical tools are chordal graphs and perfect elimination orderings."}
{"category": "Math", "title": "Algebraic Shifting and f-Vector Theory", "abstract": "This thesis focuses on algebraic shifting and its applications to f-vector theory of simplicial complexes and more general graded posets. In particular, several approaches and partial results concerning the g-conjecture for simplicial spheres are presented here."}
{"category": "Math", "title": "Origami constructions", "abstract": "A characterization of real numbers constructible by paper folding."}
{"category": "Math", "title": "Galois groups of the basic hypergeometric equations", "abstract": "In this paper we compute the Galois groups of basic hypergeometric equations."}
{"category": "Math", "title": "Morse and Lyapunov Spectra and Dynamics on Flag Bundles", "abstract": "This paper studies characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group $G$. Projection against the Iwasawa decomposition $G = KAN$ defines an additive cocycle over the flow with values in $\\frak{a} = \\log A$. Its Lyapunov exponents (limits along trajectories) and Morse exponents (limits along chains) are studied. It is proved a symmetric property of these spectral sets, namely invariance under the Weyl group. It is proved also that these sets are located in certain Weyl chambers, defined from the dynamics on the associated flag bundles. As a special case linear flows on vector bundles are considered."}
{"category": "Math", "title": "A convexity theorem for the real part of a Borel invariant subvariety", "abstract": "M. Brion proved a convexity result for the moment map image of an irreducible subvariety of a compact integral Kaehler manifold preserved by the complexification of the Hamiltonian group action. V. Guillemin and R. Sjamaar generalized this result to irreducible subvarieties preserved only by a Borel subgroup. In another direction, L. O'Shea and R. Sjamaar proved a convexity result for the moment map image of the submanifold fixed by an antisymplectic involution. Analogous to Guillemin and Sjamaar's generalization of Brion's theorem, in this paper we generalize O'Shea and Sjamaar's result, proving a convexity theorem for the moment map image of the involution fixed set of an irreducible subvariety preserved by a Borel subgroup."}
{"category": "Math", "title": "A refinement of the Kushnirenko-Bernstein estimate", "abstract": "A theorem of Kushnirenko and Bernstein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry."}
{"category": "Math", "title": "Harmonic functions via restricted mean-value theorems", "abstract": "Let $f$ be a function on a bounded domain $\\Omega \\subseteq \\mathbb{R}^n$ and $\\delta$ be a positive function on $\\Omega$ such that $B(x,\\delta(x))\\subseteq \\Omega$. Let $\\sigma(f)(x)$ be the average of $f$ over the ball $B(x,\\delta(x))$. The restricted mean-value theorems discuss the conditions on $f,\\delta,$ and $\\Omega$ under which $\\sigma(f)=f$ implies that $f$ is harmonic. In this paper, we study the stability of harmonic functions with respect to the map $\\sigma$. One expects that, in general, the sequence $\\sigma^n(f)$ converges to a harmonic function. Among our results, we show that if $\\Omega$ is strongly convex (respectively $C^{2,\\alpha}$-smooth for some $\\alpha\\in [0,1]$), the function $\\delta(x)$ is continuous, and $f\\in C^0(\\bar \\Omega)$ (respectively, $f\\in C^{2,\\alpha}(\\bar \\Omega)$), then $\\sigma^n(f)$ converges to a harmonic function uniformly on $\\bar \\Omega$."}
{"category": "Math", "title": "Obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory", "abstract": "Branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory. Since all curves with Fredholm index one can never be regular for any choice of cylindrical almost complex structure, we generalize the obstruction bundle technique of Taubes for determining multiple cover contributions from Gromov-Witten theory to the case of moduli spaces with boundary. Our result proves that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration."}
{"category": "Math", "title": "Line crossing problem for biased monotonic random walks in the plane", "abstract": "In this paper, we study the problem of finding the probability that the two-dimensional (biased) monotonic random walk crosses the line $y=\\alpha x+d$, where $\\alpha,d \\geq 0$. A $\\beta$-biased monotonic random walk moves from $(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1/(\\beta + 1)$ and $\\beta/(\\beta + 1)$, respectively. Among our results, we show that if $\\beta \\geq \\lceil \\alpha \\rceil$, then the $\\beta$-biased monotonic random walk, starting from the origin, crosses the line $y=\\alpha x+d$ for all $d\\geq 0$ with probability 1."}
{"category": "Math", "title": "The Abelian Monodromy Extension Property for Families of Curves", "abstract": "Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset U of S to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne-Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show that the Baily-Borel compactification is the unique, maximal projective AME compactification of the moduli space of abelian varieties."}
{"category": "Math", "title": "Multiplier ideal sheaves, Nevanlinna theory, and diophantine approximation", "abstract": "This note states a conjecture for Nevanlinna theory or diophantine approximation, with a sheaf of ideals in place of the normal crossings divisor. This is done by using a correction term involving a multiplier ideal sheaf. This new conjecture trivially implies earlier conjectures in Nevanlinna theory or diophantine approximation, and in fact is equivalent to these conjectures. Although it does not provide anything new, it may be a more convenient formulation for some applications."}
{"category": "Math", "title": "Hochschild homology and cohomology of {\\ell}^1({\\mathbb Z}_+^k)", "abstract": "Building on the recent determination of the simplicial cohomology groups of the convolution algebra ${\\ell}^1({\\mathbb Z}_+^k)$ [Gourdeau, Lykova, White, 2005] we investigate what can be said for cohomology of this algebra with more general symmetric coefficients. Our approach leads us to a discussion of Harrison homology and cohomology in the context of Banach algebras, and a development of some of its basic features. As an application of our techniques we reprove some known results on second-degree cohomology."}
{"category": "Math", "title": "Rearrangements and radial graphs of constant mean curvature in hyperbolic space", "abstract": "We investigate the problem of finding smooth hypersurfaces of constant mean curvature in hyperbolic space, which can be represented as radial graphs over a subdomain of the upper hemisphere. Our approach is variational and our main results are proved via rearrangement techniques."}
{"category": "Math", "title": "The Classification of Higher Order Modular Forms and their Cohomology", "abstract": "We construct bases for the spaces of higher order modular forms of all orders and weights. We also provide a cohomological interpretation of these forms."}
{"category": "Math", "title": "On posterior distribution of Bayesian wavelet thresholding", "abstract": "We investigate the posterior rate of convergence for wavelet shrinkage using a Bayesian approach in general Besov spaces. Instead of studying the Bayesian estimator related to a particular loss function, we focus on the posterior distribution itself from a nonparametric Bayesian asymptotics point of view and study its rate of convergence. We obtain the same rate as in \\citet{abramovich04} where the authors studied the convergence of several Bayesian estimators."}
{"category": "Math", "title": "On the Schwartz space isomorphism theorem for rank one symmetric space", "abstract": "In this paper we give a simpler proof of the $L^p$-Schwartz space isomorphism $(0< p\\leq 2)$ under the Fourier transform for the class of functions of left $\\delta$-type on a Riemannian symmetric space of rank one. Our treatment rests on Anker's \\cite{A} proof of the corresponding result in the case of left $K$-invariant functions on $X$. Thus we give a proof which relies only on the Paley--Wiener theorem."}
{"category": "Math", "title": "On the cohomology of orbit space of free \\pmb{${\\ZZ}_{p}$}-actions on lens spaces", "abstract": "Let $G = \\ZZ_p$, $p$ an odd prime, act freely on a finite-dimensional CW-complex $X$ with mod $p$ cohomology isomorphic to that of a lens space $L^{2m-1} (p;q_1,...,q_m)$. In this paper, we determine the mod $p$ cohomology ring of the orbit space $X/G$, when $p^2\\nmid m$."}
{"category": "Math", "title": "Approximation of functions of two variables by certain linear positive operators", "abstract": "We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an $r$th order generalization of these operators and observe its approximation properties. Furthermore, we study the convergence of the linear positive operators in a weighted space of functions of two variables and find the rate of this convergence using weighted modulus of continuity."}
{"category": "Math", "title": "On an inequality concerning the polar derivative of a polynomial", "abstract": "In this paper, we present a correct proof of an $L_p$-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund's inequality to the polar derivative of a polynomial."}
{"category": "Math", "title": "Weighted composition operators from Bergman-type spaces into Bloch spaces", "abstract": "Let $\\phi$ be an analytic self-map and $u$ be a fixed analytic function on the open unit disk $D$ in the complex plane $\\CC.$ The weighted composition operator is defined\\break by \\begin{equation*} uC_\\phi f =u \\cdot (f\\circ \\phi), f \\in H(D). \\end{equation*} Weighted composition operators from Bergman-type spaces into Bloch spaces and little Bloch spaces are characterized by function theoretic properties of their inducing maps."}
{"category": "Math", "title": "A generalization of d'Alembert formula", "abstract": "In this paper we find a closed form of the solution for the factored inhomogeneous linear equation \\begin{equation*} \\prod_{j=1}^{n}(\\frac{\\hbox{d}}{\\hbox{d}t}-A_{j}) u(t) =f(t). \\end{equation*} Under the hypothesis $A_{1},A_{2}, ..., A_{n}$ are infinitesimal generators of mutually commuting strongly continuous semigroups of bounded linear operators on a Banach space $X$. Here we do not assume that $A_{j}$s are distinct and we offer the computational method to get explicit solutions of certain partial differential equations."}
{"category": "Math", "title": "A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces", "abstract": "Let $M$ be a closed hypersurface in a simply connected rank-1 symmetric space $\\olm$. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of $M$ in terms of the Ricci curvature of $\\olm$ and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of $M$."}
{"category": "Math", "title": "Equivariant embeddings of Hermitian symmetric spaces", "abstract": "We prove that equivariant, holomorphic embeddings of Hermitian symmetric spaces are totally geodesic (when the image is not of exceptional type)."}
{"category": "Math", "title": "Some notes on the equivalence of first-order rigidity in various geometries", "abstract": "These pages serve two purposes. First, they are notes to accompany the talk \"Hyperbolic and projective geometry in constraint programming for CAD\" by Walter Whiteley at the \"Janos Bolyai Conference on Hyperbolic Geometry\", 8--12 July 2002, in Budapest, Hungary. Second, they sketch results that will be included in a forthcoming paper that will present the equivalence of the first-order rigidity theories of bar-and-joint frameworks in various geometries, including Euclidean, hyperbolic and spherical geometry. The bulk of the theory is outlined here, with remarks and comments alluding to other results that will make the final version of the paper."}
{"category": "Math", "title": "On the Gauss Map with Vanishing Biharmonic stress-energy tensor", "abstract": "We study the biharmonic stress-energy tensor $S_2$ of Gauss map. Adding few assumptions, the Gauss map with vanishing $S_2$ would be harmonic."}
{"category": "Math", "title": "A non-monotone conservation law for dune morphodynamics", "abstract": "We investigate a non-local non linear conservation law, first introduced by A.C. Fowler to describe morphodynamics of dunes, see \\cite{Fow01, Fow02}. A remarkable feature is the violation of the maximum principle, which allows for erosion phenomenon. We prove well-posedness for initial data in $L^2$ and give explicit counterexample for the maximum principle. We also provide numerical simulations corroborating our theoretical results."}
{"category": "Math", "title": "On a property of the number 977731833235239280", "abstract": "We solve a theoretical arithmetics problem stated by Wac{\\l}aw Sierpi\\'nski. The problem has remained open for a couple of decades."}
{"category": "Math", "title": "Hyperbolicity of general deformations: proofs", "abstract": "We modify the deformation method explored previously in a joint work of B. Shiffman and the author, in order to construct further examples of Kobayashi hyperbolic surfaces in the projective 3-space of any even degree starting with degree 8."}
{"category": "Math", "title": "Local equivalence of symmetric hypersurfaces in $\\mathbb C^2$", "abstract": "The Chern-Moser normal form and its analog on finite type hypersurfaces in general do not respect symmetries. Extending the work of N. K. Stanton, we consider the local equivalence problem for symmetric Levi degenerate hypersurfaces of finite type in $\\mathbb C^2$. The results give for all such hypersurfaces a complete normalization which respects the symmetries. In particular, they apply to tubes and rigid hypersurfaces, providing an effective classification. The main tool is a complete normal form constructed for a general hypersurface with a tube model. As an application, we describe all biholomorphic maps between tubes, answering a question posed by N. Hanges."}
{"category": "Math", "title": "Compatible Geometric Matchings", "abstract": "This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are \\emph{compatible} if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings $M$ and $M'$ of the same set of $n$ points, for some $k\\in\\Oh{\\log n}$, there is a sequence of perfect matchings $M=M_0,M_1,...,M_k=M'$, such that each $M_i$ is compatible with $M_{i+1}$. This improves the previous best bound of $k\\leq n-2$. We then study the conjecture: \\emph{every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching}. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with $n$ edges has an edge-disjoint compatible matching with approximately $4n/5$ edges."}
{"category": "Math", "title": "Algebraic causality: Bayes nets and beyond", "abstract": "The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of Causal Bayesian Networks has also been investigated in this context. After reviewing these newer relationships, we proceed to demonstrate that many of the ideas embodied in the concept of a ``causal model'' can be more generally expressed directly in terms of a partial order and a family of polynomial maps. The more conventional graphical constructions, when available, remain a powerful tool."}
{"category": "Math", "title": "The causal manipulation of chain event graphs", "abstract": "Discrete Bayesian Networks have been very successful as a framework both for inference and for expressing certain causal hypotheses. In this paper we present a class of graphical models called the chain event graph (CEG) models, that generalises the class of discrete BN models. It provides a flexible and expressive framework for representing and analysing the implications of causal hypotheses, expressed in terms of the effects of a manipulation of the generating underlying system. We prove that, as for a BN, identifiability analyses of causal effects can be performed through examining the topology of the CEG graph, leading to theorems analogous to the back-door theorem for the BN."}
{"category": "Math", "title": "n-Dimensional geometric-shifted global bilinear correspondences of Langlands on mixed motives III", "abstract": "This third paper,devoted to global correspondences of Langlands,bears more particularly on geometric-shifted bilinear correspondences on mixed (bi)motives generated under the action of the products,right by left,of differential elliptic operators.The mathematical frame,underlying these correspondences,deals with the categories of the Suslin-Voevodsky mixed (bi)motives and of the Chow mixed (bi)motives which are both in one-to-one correspondence with the functional representation spaces of the shifted algebraic bilinear semigroups.A bilinear holomorphic and supercuspidal spectral representation of an elliptic bioperator is then developed."}
{"category": "Math", "title": "Non-commutative residue of projections in Boutet de Monvel's calculus", "abstract": "Using results by Melo, Nest, Schick, and Schrohe on the K-theory of Boutet de Monvel's calculus of boundary value problems, we show that the non-commutative residue introduced by Fedosov, Golse, Leichtnam, and Schrohe vanishes on projections in the calculus. This partially answers a question raised in a recent collaboration with Grubb, namely whether the residue is zero on sectorial projections for boundary value problems: This is confirmed to be true when the sectorial projections is in the calculus."}
{"category": "Math", "title": "Discrete Koenigs nets and discrete isothermic surfaces", "abstract": "We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by co-planarity of the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; this characterization is invariant with respect to projective transformations. Discrete isothermic nets are defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermic surfaces introduced previously as circular nets with factorized cross-ratios."}
{"category": "Math", "title": "Sure Wins, Separating Probabilities and the Representation of Linear Functionals", "abstract": "We discuss conditions under which a convex cone $\\K\\subset \\R^{\\Omega}$ admits a probability $m$ such that $\\sup_{k\\in \\K} m(k)\\leq0$. Based on these, we also characterize linear functionals that admit the representation as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions"}
{"category": "Math", "title": "Characterization of quasi-coherent modules that are module schemes", "abstract": "The R-module functors that are essential for the development of the theory of the linear representations of an affine R-group are the quasi-coherent R-modules and the R-module schemes. The aim of this paper is to study when a quasi-coherent R-module is an R-module scheme. We will prove that it is equivalent to giving a characterization of projective R-modules of finite type."}
{"category": "Math", "title": "Deconvolution for an atomic distribution", "abstract": "Let $X_1,...,X_n$ be i.i.d. observations, where $X_i=Y_i+\\sigma Z_i$ and $Y_i$ and $Z_i$ are independent. Assume that unobservable $Y$'s are distributed as a random variable $UV,$ where $U$ and $V$ are independent, $U$ has a Bernoulli distribution with probability of zero equal to $p$ and $V$ has a distribution function $F$ with density $f.$ Furthermore, let the random variables $Z_i$ have the standard normal distribution and let $\\sigma>0.$ Based on a sample $X_1,..., X_n,$ we consider the problem of estimation of the density $f$ and the probability $p.$ We propose a kernel type deconvolution estimator for $f$ and derive its asymptotic normality at a fixed point. A consistent estimator for $p$ is given as well. Our results demonstrate that our estimator behaves very much like the kernel type deconvolution estimator in the classical deconvolution problem."}
{"category": "Math", "title": "A normal distribution for the disturbance term in regression theory", "abstract": "In regression theory, it is stated that the disturbance term follows the normal distribution when the sample size is large. In Professor J.Johnston's words: \"In view of the many factors involved, an appeal to the Central Limit Theorem would further suggest a normal distribution for u.\" This paper includes an elementary proof that the disturbance term follows the normal distribution when n is large."}
{"category": "Math", "title": "Towards a good definition of algebraically overtwisted", "abstract": "Symplectic field theory (SFT) is a collection of homology theories that provide invariants for contact manifolds. We give a proof that vanishing of any one of either contact homology, rational SFT or (full) SFT are equivalent. We call a manifold for which these theories vanish \"algebraically overtwisted\"."}
{"category": "Math", "title": "G\\'eom\\'etrie, points entiers et courbes enti\\`eres", "abstract": "Let $X$ be a projective variety over a number field $K$ (resp. over $\\mathbb{C}$). Let $H$ be the sum of ``sufficiently many positive divisors'' on $X$. We show that any set of quasi-integral points (resp. any integral curve) in $X-H$ is not Zariski dense."}
{"category": "Math", "title": "Density modulo 1 of sublacunary sequences: application of Peres-Schlag's arguments", "abstract": "Let the sequence $\\{t_n\\}_{n=1}^{\\infty}$ of reals satisfy the condition $ \\frac{t_{n+1}}{t_n} \\ge 1+ \\frac{\\gamma}{n^\\beta},0\\le \\beta <1, \\gamma >0. $ Then the set $ \\{\\alpha \\in [0,1]: \\exists \\varkappa > 0 \\forall n \\in \\mathbb{N} ||t_n \\alpha || > \\frac{\\varkappa}{n^\\beta \\log (n+1)} \\} $ is uncountable. Moreover its Hausdorff dimension is equal to 1. Consider the set of naturals of the form $2^n3^m$ and let the sequence $ s_1=1, s_2=2, s_3=3, s_4=4, s_5=6, s_6 = 8,... $ performs this set as an increasing sequence. Then the set $ \\{\\alpha \\in [0,1]: \\exists \\varkappa > 0 \\forall n \\in \\mathbb{N} ||s_n \\alpha || > \\frac{\\varkappa}{\\sqrt{n}\\log (n+1)} \\} $ also has Hausdorff dimension equal to 1. The results obtained use an original approach due to Y. Peres and W. Schlag."}
{"category": "Math", "title": "The complexity of the envelope of line and plane arrangements", "abstract": "A facet of an hyperplane arrangement is called external if it belongs to exactly one bounded cell. The set of all external facets forms the envelope of the arrangement. The number of external facets of a simple arrangement defined by $n$ hyperplanes in dimension $d$ is hypothesized to be at least $d{n-2 \\choose d-1}$. In this note we show that, for simple arrangements of 4 lines or more, the minimum number of external facets is equal to $2(n-1)$, and for simple arrangements of 5 planes or more, the minimum number of external facets is between $\\frac{n(n-2)+6}{3}$ and $(n-4)(2n-3)+5$."}
{"category": "Math", "title": "Higher Homotopy Hopf Algebras Found: A Ten Year Retrospective", "abstract": "The search for higher homotopy Hopf algebras (known today as A_\\infty-bialgebras) began in 1996 during a conference at Vassar College honoring Jim Stasheff in the year of his 60th birthday. In a talk entitled \"In Search of Higher Homotopy Hopf Algebras\", I indicated that a DG Hopf algebra could be thought of as some (unknown) higher homotopy structure with trivial higher order structure and deformed using a graded version of Gerstenhaber and Schack's bialgebra deformation theory. In retrospect, the bi(co)module structure encoded in Gerstenhaber and Schack's differential defining deformation cohomology detects some (but not all) of the A_infty-bialgebra structure relations. Nevertheless, this motivated the discovery of A_infty-bialgebras by S. Saneblidze and myself in 2005."}
{"category": "Math", "title": "On period spaces for p-divisible groups", "abstract": "In their book Rapoport and Zink constructed rigid analytic period spaces for Fontaine's filtered isocrystals, and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. We determine the image of these period morphisms, thereby contributing to a question of Grothendieck. We give examples showing that only in rare cases the image is all of the Rapoport-Zink period space."}
{"category": "Math", "title": "The integrals in Gradshteyn and Ryzhik. Part 10: the digamma function", "abstract": "Many integrals in the classical table by Gradshteyn and Ryzhik can be evaluated in terms of the digamma function (= the logarithmic derivative of the gamma function). Some of them are presented here."}
{"category": "Math", "title": "Hodge cohomology of invertible sheaves", "abstract": "v2: We improved a little bit according to the referee's wishes. v1: On $X$ projective smooth over a field $k$, Pink and Roessler conjecture that the dimension of the Hodge cohomology of an invertible $n$-torsion sheaf $L$ is the same as the one of its $a$-th power $L^a$ if $a$ is prime to $n$, under the assumptions that $X$ lifts to $W_2(k)$ and $dim X\\le p$, if $k$ has characteristic $p>0$. They show this if $k$ has characteristic 0 and if $n$ is prime to $p$ in characteristic $p>0$. We show the conjecture in characteristic $p>0$ if $n=p$ assuming in addition that $X$ is ordinary (in the sense of Bloch-Kato)."}
{"category": "Math", "title": "On the auxiliary particle filter", "abstract": "In this article we study asymptotic properties of weighted samples produced by the auxiliary particle filter (APF) proposed by pitt and shephard (1999). Besides establishing a central limit theorem (CLT) for smoothed particle estimates, we also derive bounds on the Lp error and bias of the same for a finite particle sample size. By examining the recursive formula for the asymptotic variance of the CLT we identify first-stage importance weights for which the increase of asymptotic variance at a single iteration of the algorithm is minimal. In the light of these findings, we discuss and demonstrate on several examples how the APF algorithm can be improved."}
{"category": "Math", "title": "An Auslander-type result for Gorenstein-projective modules", "abstract": "An artin algebra $A$ is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective $A$-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type (\\cite{A,A1})."}
{"category": "Math", "title": "Iwasawa decompositions of split Kac-Moody groups", "abstract": "We characterize all fields F for which a group with an F-locally split root group datum admits an Iwasawa decomposition. This class of groups in particular includes the split semisimple algebraic groups and the split Kac-Moody groups."}
{"category": "Math", "title": "Ergodic behaviour of \"signed voter models\"", "abstract": "We consider some questions raised by the recent paper of Gantert, L\\\"owe and Steif (2005) concerning ``signed'' voter models on locally finite graphs. These are voter model like processes with the difference that the edges are considered to be either positive or negative. If an edge between a site $x$ and a site $y$ is negative (respectively positive) the site $y$ will contribute towards the flip rate of $x$ if and only if the two current spin values are equal (respectively opposed)."}
{"category": "Math", "title": "On geodesic homotopies of controlled width and conjugacies in isometry groups", "abstract": "We give an analytical proof of the Poincare-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space."}
{"category": "Math", "title": "On Brill-Noether loci over Quot schemes and a Torelli theorem", "abstract": "We prove a non abelian Torelli type result for smooth projective curves by working in the derived category of some associated polarized Quot schemes and defining Brill-Noether loci and Abel-Jacobi maps on them."}
{"category": "Math", "title": "Quasi-fractal sets in space", "abstract": "The focus here is on connected fractal sets with topological dimension 1 and a lot of topological activity, and their connections with analysis."}
{"category": "Math", "title": "Homotopy type and v1-periodic homotopy groups of p-compact groups", "abstract": "We determine the v1-periodic homotopy groups of all irreducible p-compact groups (BX,X). In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, if p is odd, every irreducible p-compact group has X of the homotopy type of a product of explicit spaces related to p-completed Lie groups."}
{"category": "Math", "title": "On the spectrum of infinite dimensional random products of compact operators", "abstract": "We consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f. Assuming that f acts in a compact Hausdorff space X and preserves a Borel regular ergodic measure which is positive on non-empty open sets, we conclude that there is a residual subset of cocycles within which, for almost every x, either the Oseledets-Ruelle's decomposition along the orbit of x is dominated or has a trivial spectrum."}
{"category": "Math", "title": "Distinguished tame supercuspidal representations", "abstract": "This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by (the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal representations of finite groups of Lie type are distinguished relative to particular quadratic characters. As an application of the main results, we obtain necessary and sufficient conditions for equivalence of two of Yu's supercuspidal representations associated to distinct G-data."}
{"category": "Math", "title": "Spherical characters: the supercuspidal case", "abstract": "We exhibit a basis for the space of spherical characters of a distinguished supercuspidal representation $\\pi$ of a connected reductive $p$-adic group, subject to the assumption that $\\pi$ is obtained via induction from a representation of an open compact mod centre subgroup. We derive an integral formula for each spherical character belonging to the basis. This formula involves integration of a particular kind of matrix coefficient of $\\pi$. We also obtain a similar formula for the function realizing the spherical character. In addition, we determine, subject to some conditions, which of these spherical characters vanish identically on an open neighbourhood of the identity. We verify that the requisite conditions are always satisfied for distinguished tame supercuspidal representations of groups that split over tamely ramified extensions."}
{"category": "Math", "title": "Estimates on Monge-Amp\\`ere operators derived from a local algebra inequality", "abstract": "The goal of this short note is to relate the integrability property of the exponential $e^{-2\\phi}$ of a plurisubharmonic function $\\phi$ with isolated or compactly supported singularities, to a priori bounds for the Monge-Amp\\`ere mass of $(dd^c\\phi)^n$. The inequality is valid locally or globally on an arbitrary open subset $\\Omega$ in $\\bC^n$. We show that $\\int_\\Omega(dd\\phi)^n<n^n$ implies $\\int_Ke^{-2\\phi}<+\\infty$ for every compact subset $K$ in $\\Omega$, while functions of the form $\\phi(z)=n\\log|z-z_0|$, $z_0\\in\\Omega$, appear as limit cases. The result is derived from an inequality of pure local algebra, which turns out a posteriori to be equivalent to it, proved by A.Corti in dimension $n=2$, and later extended by L.Ein, T.De Fernex and M.Musta\\c{t}\\v{a} to arbitrary dimensions."}
{"category": "Math", "title": "The log-linear group-lasso estimator and its asymptotic properties", "abstract": "We define the group-lasso estimator for the natural parameters of the exponential families of distributions representing hierarchical log-linear models under multinomial sampling scheme. Such estimator arises as the solution of a convex penalized likelihood optimization problem based on the group-lasso penalty. We illustrate how it is possible to construct an estimator of the underlying log-linear model using the blocks of nonzero coefficients recovered by the group-lasso procedure. We investigate the asymptotic properties of the group-lasso estimator as a model selection method in a double-asymptotic framework, in which both the sample size and the model complexity grow simultaneously. We provide conditions guaranteeing that the group-lasso estimator is model selection consistent, in the sense that, with overwhelming probability as the sample size increases, it correctly identifies all the sets of nonzero interactions among the variables. Provided the sequences of true underlying models is sparse enough, recovery is possible even if the number of cells grows larger than the sample size. Finally, we derive some central limit type of results for the log-linear group-lasso estimator."}
{"category": "Math", "title": "Autour de la cohomologie de Bott-Chern", "abstract": "The goal of the memoir is to develop a new cohomology theory which encompasses De Rham and Dolbeault cohomology as well as Deligne Beilinson cohomology, in the context of general complex analytic manifolds. The special case of the Iwasawa manifold is investigated as a typical example of what occurs in the non K\\\"ahler case. Elementary applications to the Kodaira-Spencer deformation theory and to the calculation of Chern classes are given."}
{"category": "Math", "title": "On the Hopf Lemma", "abstract": "The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,\\alpha}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators."}
{"category": "Math", "title": "Meridional Almost Normal Surfaces in Knot Complements", "abstract": "Suppose $K$ is a knot in a closed 3-manifold $M$ such that $\\bar{M-N(K)}$ is irreducible. We show that for any positive integer $b$ there exists a triangulation of $\\bar{M-N(K)}$ such that any weakly incompressible bridge surface for $K$ of $b$ bridges or fewer is isotopic to an almost normal bridge surface."}
{"category": "Math", "title": "Maximum Likelihood Estimation in Latent Class Models For Contingency Table Data", "abstract": "Statistical models with latent structure have a history going back to the 1950s and have seen widespread use in the social sciences and, more recently, in computational biology and in machine learning. Here we study the basic latent class model proposed originally by the sociologist Paul F. Lazarfeld for categorical variables, and we explain its geometric structure. We draw parallels between the statistical and geometric properties of latent class models and we illustrate geometrically the causes of many problems associated with maximum likelihood estimation and related statistical inference. In particular, we focus on issues of non-identifiability and determination of the model dimension, of maximization of the likelihood function and on the effect of symmetric data. We illustrate these phenomena with a variety of synthetic and real-life tables, of different dimension and complexity. Much of the motivation for this work stems from the \"100 Swiss Francs\" problem, which we introduce and describe in detail."}
{"category": "Math", "title": "Galois actions on torsion points of universal one-dimensional formal modules", "abstract": "Let $F$ be a local non-Archimedean field with ring of integers $o$. Let $\\bf X$ be a one-dimensional formal $o$-module of $F$-height $n$ over the algebraic closure of the residue field of $o$. By the work of Drinfeld, the universal deformation $X$ of $\\bf X$ is a formal group over a power series ring $R_0$ in $n-1$ variables over the completion of the maximal unramified extension of $o$. For $h \\in \\{0,...,n-1\\}$ let $U_h$ be the subscheme of $\\Spec(R_0)$ where the connected part of the associated divisible module of $X$ has height $h$. Using the theory of Drinfeld level structures we show that the representation of the fundamental group of $U_h$ on the Tate module of the etale quotient is surjective."}
{"category": "Math", "title": "Locally Adaptive Nonparametric Binary Regression", "abstract": "A nonparametric and locally adaptive Bayesian estimator is proposed for estimating a binary regression. Flexibility is obtained by modeling the binary regression as a mixture of probit regressions with the argument of each probit regression having a thin plate spline prior with its own smoothing parameter and with the mixture weights depending on the covariates. The estimator is compared to a single spline estimator and to a recently proposed locally adaptive estimator. The methodology is illustrated by applying it to both simulated and real examples."}
{"category": "Math", "title": "Euclidean Jordan Algebras and Generalized Krein parameters of a strongly regular graph", "abstract": "Let $\\tau$ be a strongly $(n,p;a,c)$ regular graph,such that $0<c<p<n-1,$ $A$ his matrix of adjacency and let ${\\cal V}_{n}$ be the Euclidean space spanned by the powers of $A$ over the reals where the scallar product $\\bullet|\\bullet$ is defined by $x|y={trace}(x \\cdot y).$ In this work ones proves that ${\\cal V}_{n}$ is an Euclidean Jordan algebra of rank 3 when one introduces in ${\\cal V}_{n}$ the usual product of matrices. In this Euclidean Jordan algebra one defines the modulus of a matrix, and afterwards one defines $|A|^x \\forall x\\in \\mathbb{R}.$ Working inside the Euclidean Jordan algebra ${\\cal V}_{n}$ and making use of the properties of $|A|^x$ one defines the generalized krein parameters of the strongly $(n,p;a,c)$ regular graph $\\tau$ and finally one presents necessary conditions over the parameters and the spectra of the $\\tau$ strongly $(n,p;a,c)$ regular graph."}
{"category": "Math", "title": "On The Density Estimation by Super-Parametric Method", "abstract": "The super-parametric density estimators and its related algorism were suggested by Y. -S. Tsai et al [7]. The number of parameters is unlimited in the super- parametric estimators and it is a general theory in sense of unifying or connecting nonparametric and parametric estimators. Before applying to numerical examples, we can not give any comment of the estimators. In this paper, we will focus on the implementation, the computer programming, of the algorism and strategies of choosing window functions. B-splines, Bezier splines and covering windows are studied as well. According to the criterion of the convergence conditions for Parzen window, the number of the window functions shall be, roughly, proportional to the number of samples and so is the number of the variables. Since the algorism is designed for solving the optimization of likelihood function, there will be a set of nonlinear equations with a large number of variables. The results show that algorism suggested by Y. -S. Tsai is very powerful and effective in the sense of mathematics, that is, the iteration procedures converge and the rates of convergence are very fast. Also, the numerical results of different window functions show that the approach of super-parametric density estimators has ushered a new era of statistics."}
{"category": "Math", "title": "The quantitative behaviour of polynomial orbits on nilmanifolds", "abstract": "A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\\ldots)$ on a nilmanifold $G/\\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\\Gamma$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \\epsilon g'\\gamma$, where $\\epsilon(n)$ is \"smooth\", $\\gamma(n)$ is periodic and \"rational\", and $(g'(a),g'(a+d),\\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $\\delta$) inside some subnilmanifold $G'/\\Gamma'$ of $G/\\Gamma$, for all sufficiently dense arithmetic progressions $a,a+d,\\ldots,a+d(l-1)$ inside $\\{1,..,N\\}$. Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper \"Linear equations in primes\"."}
{"category": "Math", "title": "The shape of hyperbolic Dehn surgery space", "abstract": "In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \\arctanh(1/\\sqrt{3}) \\approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing the previous restrictions on cone angles. We then apply this to obtain a new quantitative version of Thurston's hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery coefficients outside a disc of ``uniform'' size yield hyperbolic structures. Here the size of a surgery coefficient is measured using the Euclidean metric on a horospherical cross section to a cusp in the complete hyperbolic metric, rescaled to have area 1. We also obtain good estimates on the change in geometry (e.g. volumes and core geodesic lengths) during hyperbolic Dehn filling. This new harmonic deformation theory has also been used by Bromberg and his coworkers in their proofs of the Bers Density Conjecture for Kleinian groups."}
{"category": "Math", "title": "Torische Ideale von Flusspolytopen", "abstract": "In dieser Diplomarbeit werden einige Gradschranken f\\\"ur Erzeugendensysteme und Gr\\\"obnerbasen von torischen Idealen von Flusspolytopen bewiesen. Alle torischen Ideale von Flusspolytopen sind im Grad 3 erzeugt. Glatte (3x4)-Transportpolytope sind sogar im Grad 2 erzeugt. Die reduzierte Gr\\\"obnerbasis eines beliebigen (m \\times n)-Transportpolytops bez\\\"uglich einer beliebigen umgekehrt lexikographischen Termordnung hat h\\\"ochstens Grad mn/2. Wir konstruieren auch ein Beispiel, f\\\"ur das diese Schranke ann\\\"ahernd scharf ist. ----- In this Diplomarbeit (Master's thesis), we prove some degree bounds for generating sets and Gr\\\"obner bases of toric ideals of flow polytopes. All toric ideals of flow polytopes are generated in degree three. Smooth (3x4)-transportation polytopes are even generated in degree 2. The reduced Gr\\\"obner basis of an arbitrary (m \\times n)-transportation polytope with respect to an arbitrary reverse lexikographic term order has at most degree mn/2. We also construct an example, for which this bound is almost sharp."}
{"category": "Math", "title": "On Popoviciu type tormulas for generalized restricted partition function", "abstract": "Suppose that $a_1(n),a_2(n),...,a_s(n),m(n)$ are integer-valued polynomials in $n$ with positive leading coefficients. This paper presents Popoviciu type formulas for the generalized restricted partition function $$p_{A(n)}(m(n)):=#\\{(x_1,...,x_s)\\in \\mathbb{Z}^{s}: all x_j\\geqslant 0, x_1a_1(n)+...+x_sa_s(n)=m(n) \\}$$ when $s=2$ or 3. In either case, the formula implies that the function is an integer-valued quasi-polynomial. The main result is proved by a reciprocity law for a class of fractional part sums and the theory of generalized Euclidean division."}
{"category": "Math", "title": "A reciprocity map and the two variable p-adic L-function", "abstract": "For primes p greater than 3, we propose a conjecture that relates the values of cup products in the Galois cohomology of the maximal unramified outside p extension of a cyclotomic field on cyclotomic p-units to the values of p-adic L-functions of cuspidal eigenforms that satisfy mod p congruences with Eisenstein series. Passing up the cyclotomic and Hida towers, we construct an isomorphism of certain spaces that allows us to compare the value of a reciprocity map on a particular norm compatible system of p-units to what is essentially the two-variable p-adic L-function of Mazur and Kitagawa."}
{"category": "Math", "title": "Classical elliptic current algebras", "abstract": "In this paper we discuss classical elliptic current algebras and show that there are two different choices of commutative test function algebras on a complex torus leading to two different elliptic current algebras. Quantization of these classical current algebras give rise to two classes of quantized dynamical quasi-Hopf current algebras studied by Enriquez-Felder-Rubtsov and Arnaudon-Buffenoir-Ragoucy-Roche-Jimbo-Konno-Odake-Shiraishi. Different degenerations of the classical elliptic algebras are considered. They yield different versions of rational and trigonometric current algebras. We also review the averaging method of Faddeev-Reshetikhin, which allows to restore elliptic algebras from the trigonometric ones."}
{"category": "Math", "title": "Ubiquitous systems and metric number theory", "abstract": "We investigate the size and large intersection properties of $$E_{t}=\\{x\\in\\R^d \\:|\\: \\|x-k-x_{i}\\|<{r_{i}}^t\\text{for infinitely many}(i,k)\\in I^{\\mu,\\alpha}\\times\\Z^d\\},$$ where $d\\in\\N$, $t\\geq 1$, $I$ is a denumerable set, $(x_{i},r_{i})_{i\\in I}$ is a family in $[0,1]^d\\times (0,\\infty)$ and $I^{\\mu,\\alpha}$ denotes the set of all $i\\in I$ such that the $\\mu$-mass of the ball with center $x_{i}$ and radius $r_{i}$ behaves as ${r_{i}}^\\alpha$ for a given Borel measure $\\mu$ and a given $\\alpha>0$. We establish that the set $E_{t}$ belongs to the class $\\grint^h(\\R^d)$ of sets with large intersection with respect to a certain gauge function $h$, provided that $(x_{i},r_{i})_{i\\in I}$ is a heterogeneous ubiquitous system with respect to $\\mu$. In particular, $E_{t}$ has infinite Hausdorff $g$-measure for every gauge function $g$ that increases faster than $h$ in a neighborhood of zero. We also give several applications to metric number theory."}
{"category": "Math", "title": "Singularity sets of Levy processes", "abstract": "We completely describe the size and large intersection properties of the Holder singularity sets of Levy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Levy process. The Holder singularity sets of the sample paths of certain random wavelet series are investigated as well."}
{"category": "Math", "title": "Random wavelet series based on a tree-indexed Markov chain", "abstract": "We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Holder exponent form a set with large intersection."}
{"category": "Math", "title": "Random fractals and tree-indexed Markov chains", "abstract": "We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot's fractal percolation process."}
{"category": "Math", "title": "Liouville theorems for the Navier-Stokes equations and applications", "abstract": "We study bounded ancient solutions of the Navier-Stokes equations. These are the solutions which are defined for all past time. In two space dimensions we prove that such solutions are either constant or functions of time only, depending on the exact definition of admissible solutions. The general three dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axi-symmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions."}
{"category": "Math", "title": "Hurwitz numbers for regular coverings of surfaces by seamed surfaces and Cardy-Frobenius algebras of finite groups", "abstract": "Analogue of classical Hurwitz numbers is defined in the work for regular coverings of surfaces with marked points by seamed surfaces. Class of surfaces includes surfaces of any genus and orientability, with or without boundaries; coverings may have certain singularities over the boundary and marked points. Seamed surfaces introduced earlier are not actually surfaces. A simple example of seamed surface is book-like seamed surface: several rectangles glued by edges like sheets in a book. We prove that Hurwitz numbers for a class of regular coverings with action of fixed finite group $G$ on cover space such that stabilizers of generic points are conjugated to a fixed subgroup $K\\subset G$ defines a new example of Klein Topological Field Theory (KTFT). It is known that KTFTs are in one-to-one correspondence with certain class of algebras, called in the work Cardy-Frobenius algebras. We constructed a wide class of Cardy-Frobenius algebras, including particularly all Hecke algebras for finite groups. Cardy-Frobenius algebras corresponding to regular coverings of surfaces by seamed surfaces are described in terms of group $G$ and its subgroups. As a result, we give an algebraic formula for introduced Hurwitz numbers."}
{"category": "Math", "title": "On orbit closures for infinite type quivers", "abstract": "For the Kronecker quiver, Zwara has found an example of a representation whose orbit closure is neither unibranch nor Cohen-Macaulay. In this note, we explain how to extend this example to all infinite type quivers without oriented cycles."}
{"category": "Math", "title": "Lecture Notes on Equivariant Cohomology", "abstract": "These are the lecture notes for the introductory graduate course I taught at Yale during Spring 2007. I mostly followed [GS], [BGV], [AB], [Par2], and there are no original results in these notes."}
{"category": "Math", "title": "On maps which preserve equality of distance in F-spaces", "abstract": "In order to generalize the results of Mazur-Ulam and Vogt, we shall prove that any map T which preserves equality of distance with T(0)=0 between two F-spaces without surjective condition is linear. Then, as a special case linear isometries are characterized through a simple property of their range."}
{"category": "Math", "title": "\\ell-adic class field theory for regular local rings", "abstract": "In this paper, we prove the $\\ell$-adic abelian class field theory for henselian regular local rings of equi-characteristic assuming the surjectivity of Galois symbol maps, which is a $\\ell$-adic variant of a result of Matsumi [13]."}
{"category": "Math", "title": "A Functorial Approach to the Infinitesimal Theory of Groupoids", "abstract": "Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere's functorial notion of algebraic theory and Ehresmann's functorial notion of theory called sketches. We study totally intransitive Nishimura algebroids in detail. Finally we show that Nishimura algebroids naturally give rise to Lie algebroids."}
{"category": "Math", "title": "A convenient category of locally preordered spaces", "abstract": "As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is Cartesian closed and that the forgetful functor to the category of compactly generated spaces creates all limits and colimits."}
{"category": "Math", "title": "On the Correlations, Selberg Integral and Symmetry of Sieve Functions in Short Intervals", "abstract": "We study the arithmetic (real) function f=g*1, with g \"essentially bounded\" and supported over the integers of [1,Q]. In particular, we obtain non-trivial bounds, through f \"correlations\", for the \"Selberg integral\" and the \"symmetry integral\" of f in almost all short intervals [x-h,x+h], N<x<2N, beyond the \"classical\" level, up to level of distribution, say, lambda=log Q/log N < 2/3 (for enough large h). This time we don't apply Large Sieve inequality, as in our paper [C-S]. Precisely, our method is completely elementary."}
{"category": "Math", "title": "A Support Theorem For The Radiation Fields On Asymptotically Euclidean Manifolds", "abstract": "We prove a support theorem for the radiation fields on asymptotically Euclidean manifolds with metrics which are warped products near infinity. It generalizes to this setting the well known support theorem for the Radon transform on n-dimensional Euclidean space."}
{"category": "Math", "title": "Einstein's equations and the embedding of 3-dimensional CR manifolds", "abstract": "We prove several theorems concerning the connection between the local CR embeddability of 3-dimensional CR manifolds, and the existence of algebraically special Maxwell and gravitational fields. We reduce the Einstein equations for spacetimes associated with such fields to a system of CR invariant equations on a 3-dimensional CR manifold defined by the fields. Using the reduced Einstein equations we construct two independent CR functions for the corresponding CR manifold. We also point out that the Einstein equations, imposed on spacetimes associated with a 3-dimensional CR manifold, imply that the spacetime metric, after an appropriate rescaling, becomes well defined on a circle bundle over the CR manifold. The circle bundle itself emerges as a consequence of Einstein's equations."}
{"category": "Math", "title": "Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws", "abstract": "We analyze a class of weakly differentiable vector fields (\\FF \\colon \\rn \\to \\rn) with the property that (\\FF\\in L^{\\infty}) and (\\div \\FF) is a Radon measure. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field $\\FF$ over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green theorem. To achieve this, we establish a fundamental approximation theorem which states that, given a Radon measure $\\mu$ that is absolutely continuous with respect to $\\mathcal{H}^{N-1}$ on $\\rn$, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the measure $\\|\\mu\\|$. With this approximation theorem, we derive the normal trace of $\\FF$ on the boundary of any set of finite perimeter, (E), as the limit of the normal traces of $\\FF$ on the boundaries of the approximate sets with smooth boundary, so that the Gauss-Green theorem for $\\FF$ holds on (E). With these results, we analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface (i.e. an $(N-1)$-dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure-valued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws."}
{"category": "Math", "title": "Haagerup's Approximation Property and Relative Amenability", "abstract": "A finite von Neumann algebra $\\mathcal{M}$ with a faithful normal trace $% \\tau $ has Haagerup's approximation property (relative to a von Neumann subalgebra $\\mathcal{N}$) if there exists a net $(\\phi_{\\alpha})_{\\alpha\\in \\Lambda}$ of normal completely positive ($\\mathcal{N}$-bimodular) maps from $\\mathcal{M}$ to $\\mathcal{M}$ that satisfy the subtracial condition $% \\tau \\circ \\phi_{\\alpha}\\leq \\tau $, the extension operators $% T_{\\phi_{\\alpha}}$ are bounded compact operators (in $<\\mathcal{M%},e_{\\mathcal{N}}>$), and pointwise approximate the identity in the trace-norm, i.e., $\\lim_{\\alpha}||\\phi_{\\alpha}(x)-x||_{2}=0$ for all $% x\\in \\mathcal{M}$. We prove that the subtraciality condition can be removed, and provide a description of Haagerup's approximation property in terms of Connes's theory of correspondences. We show that if $\\mathcal{N}\\subseteq \\mathcal{M}$ is an amenable inclusion of finite von Neumann algebras and $% \\mathcal{N}$ has Haagerup's approximation property, then $\\mathcal{M}$ also has Haagerup's approximation property. This work answers two questions of Sorin Popa."}
{"category": "Math", "title": "Curvature estimates for minimal submanifolds of higher codimension", "abstract": "We derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau's results and Ecker-Huisken's results are generalized to higher codimension. In this way we improve Hildebrandt-Jost-Widman's result for the Bernstein type theorem."}
{"category": "Math", "title": "Flatness-based pre-compensation of laser diodes", "abstract": "A physical nonlinear dynamical model of a laser diode is considered. We propose a feed-forward control scheme based on differential flatness for the design of input-current modulations to compensate diode distortions. The goal is to transform without distortion a radio-frequency current modulation into a light modulation leaving the laser-diode and entering an optic fiber. We prove that standard physical dynamical models based on dynamical electron and photons balance are flat systems when the current is considered as control input, the flat output being the photon number (proportional to the light power). We prove that input-current is an affine map of the flat output, its logarithm and their time-derivatives up to order two. When the flat output is an almost harmonic signal with slowly varying amplitude and phase, these derivatives admit precise analytic approximations. It is then possible to design simple analogue electronic circuits to code approximations of the nonlinear computations required by our flatness-based approach. Simulations with the parameters of a commercial diode illustrate the practical interest of this pre-compensation scheme and its robustness versus modelling and analogue implementation errors."}
{"category": "Math", "title": "The Chow rings of the algebraic groups E_6, E_7, and E_8", "abstract": "We determine the Chow rings of the complex algebraic groups of the exceptional type E_6, E_7, and E_8, giving the explicit generators represented by the pull-back images of Schubert varieties of the corresponding flag varieties. This is a continuation of the work of R. Marlin on the computation of the Chow rings of SO_n, Spin_n, G_2, and F_4. Our method is based on Schubert calculus of the corresponding flag varieties, which has its own interest."}
{"category": "Math", "title": "First Steps in Tropical Intersection Theory", "abstract": "We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in R^n and then for \"abstract\" cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in R^n."}
{"category": "Math", "title": "Criteria for homotopic maps to be so along monotone homotopies", "abstract": "The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of abstract spacetime, we identify criteria for classically homotopic, monotone maps of pospaces to future homotope, or homotope along homotopies monotone in both coordinates, to a common map. We show that consequently, a hypercontinuous lattice equipped with its Lawson topology is future contractible, or contractible along a future homotopy, if its underlying space has connected CW type."}
{"category": "Math", "title": "Pulsating travelling fronts: Asymptotics and homogenization regimes", "abstract": "This paper is concerned with some nonlinear propagation phenomena for reaction-advection-diffusion equations with Kolmogrov-Petrovsky-Piskunov (KPP) type nonlinearities in general periodic domains or in infinite cylinders with oscillating boundaries. Having a variational formula for the minimal speed of propagation involving eigenvalue problems ( proved in Berestycki, Hamel and Nadirashvili \\cite{BHN1}), we consider the minimal speed of propagation as a function of diffusion factors, reaction factors and periodicity parameters. There we study the limits, the asymptotic behaviors and the variations of the considered functions with respect to these parameters. The last section treats a homogenization problem as an application of the results in the previous sections in order to find the limit of the minimal speed when the periodicity cell is very small."}
{"category": "Math", "title": "Functorial Cartier duality", "abstract": "In this paper we obtain the Cartier duality for k-schemes of commutative monoids functorially without providing the vector spaces of functions with a topology, generalizing a result for finite commutative algebraic groups by M. Demazure and P. Gabriel."}
{"category": "Math", "title": "Construction and analysis of approximate models for electromagnetic scattering from imperfectly conducting scatterers", "abstract": "This report is dedicated to the construction and analysis of so-called Generalized Impedance Boundary Conditions (GIBCs) used in electromagnetic scattering problems from imperfect conductors as higher order approximations of a perfect conductor condition. We consider here the 3-D case with Maxwell equations in a harmonic regime. The construction of GIBCs is based on a scaled asymptotic expansion with respect to the skin depth. The asymptotic expansion is theoretically justified at any order and we give explicit expressions till the third order. These expressions are used to derive the GIBCs. The associated boundary value problem is analyzed and error estimates are obtained in terms of the skin depth."}
{"category": "Math", "title": "On O^*-representability and C^*-representability of *-algebras", "abstract": "Characterization of the *-subalgebras in the algebra of bounded operators acting on Hilbert space is presented. Sufficient conditions for the existence of a faithful representation in pre-Hilbert space of a *-algebra in terms of its Groebner basis are given. These conditions are generalization of the unshrinkability of monomial *-algebras introduced by C. Lance and P. Tapper. The applications to *-doubles, monomial *-algebras and several other classes of *-algebras are presented."}
{"category": "Math", "title": "The Homogeneous Approximation Property and the Comparison Theorem for Coherent Frames", "abstract": "We show that the homogeneous approximation property and the comparison theorem hold for arbitrary coherent frames. This observation answers some questions about the density of frames that are not covered by the theory of Balan, Casazza, Heil, and Landau. The proofs are a variation of the method developed by Ramanathan and Steger."}
{"category": "Math", "title": "An elementary trigonometric equation", "abstract": "A systematic study of the trigonometric equation A tan a + B sin b = C, where A, B and C^2 are rational numbers. The special case tan Pi/11 + 4 sin 3 Pi/11 = sqrt[11] appears in the classical literature."}
{"category": "Math", "title": "Chen's double sieve, Goldbach's conjecture and the twin prime problem, 2", "abstract": "For every even integer N, denote by D_{1,2}(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D_{1,2}(N)."}
{"category": "Math", "title": "1953: An unrecognized summit in human genetic linkage analysis", "abstract": "This paper summarizes and discusses the methodological research in human genetic linkage analysis, leading up to and following from the paper of C. A. B. Smith presented as a Royal Statistical Society discussion paper in 1953. This paper was given as the Fisher XXVII Memorial Lecture, in Cambridge, December 4, 2006."}
{"category": "Math", "title": "On Deformations of Pasting Diagrams", "abstract": "We adapt the work of Power to describe general, not-necessarily composable, not-necessarily commutative 2-categorical pasting diagrams and their composable and commutative parts. We provide a deformation theory for pasting diagrams valued in $k$-linear categories, paralleling that provided for diagrams of algebras by Gerstenhaber and Schack, proving the standard results. Along the way, the construction gives rise to a bicategorical analog of the homotopy G-algebras of Gerstenhaber and Voronov."}
{"category": "Math", "title": "A Field of Generalised Puiseux Series for Tropical Geometry", "abstract": "In this paper we define a field K of characteristic zero with valuation whose value group is the real numbers, and we show that this field of generalised Puiseux series is algebraically closed and complete with respect to the norm induced by its valuation. We consider this field to be a good candidate for the base field for tropical geometry."}
{"category": "Math", "title": "The j-invariant of a plane tropical cubic", "abstract": "Several results in tropical geometry have related the j-invariant of an algebraic plane curve of genus one to the cycle length of a tropical curve of genus one. In this paper, we prove that for a plane cubic over the field of Puiseux series the negative of the generic valuation of the $j$-invariant is equal to the cycle length of the tropicalization of the curve, if there is a cycle at all."}
{"category": "Math", "title": "Geometric incidence theorems via Fourier analysis", "abstract": "We prove that non-trivial bounds for generalized Radon transforms imply correspondingly non-trivial discrete incidence theorems for manifolds and suitably regular point sets."}
{"category": "Math", "title": "On the path structure of a semimartingale arising from monotone probability theory", "abstract": "Let $X$ be the unique normal martingale such that $X_0=0$ and \\[\\mathrm{d}[X]_t=(1-t-X_{t-}) \\mathrm{d}X_t+\\mathrm{d}t\\] and let $Y_t:=X_t+t$ for all $t\\geq 0$; the semimartingale $Y$ arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of $Y$ are examined and various probabilistic properties are derived; in particular, the level set $\\{t\\geq 0\\dvt Y_t=1\\}$ is shown to be non-empty, compact, perfect and of zero Lebesgue measure. The local times of $Y$ are found to be trivial except for that at level 1; consequently, the jumps of $Y$ are not locally summable."}
{"category": "Math", "title": "On Anosov diffeomorphisms with asymptotically conformal periodic data", "abstract": "We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We establish various properties of such systems including strong pinching, C^{1+\\beta} smoothness of the Anosov splitting, and C^1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of a torus. As a corollary we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three."}
{"category": "Math", "title": "Global Structure of Locally Convex Hypersurfaces in Finsler-Hadamard Manifolds", "abstract": "Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. We prove that such hypersurfaces are embedded as the boundary of convex body under certain conditions on the normal curvatures"}
{"category": "Math", "title": "Morse theory and conjugacy classes of finite subgroups", "abstract": "We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type F_n inside mapping class groups, Aut(F), and Out(F) which have infinitely many conjugacy classes of finite-order elements."}
{"category": "Math", "title": "Complete Reducibility and Separability", "abstract": "Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module. Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and H is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1."}
{"category": "Math", "title": "Asymptotic estimates for the number of contingency tables, integer flows, and volumes of transportation polytopes", "abstract": "We prove an asymptotic estimate for the number of mxn non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer feasible flows in a network. Similarly, we estimate the volume of the polytope of mxn non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems and hence can be computed efficiently. As a corollary, we show that if row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n) with r_1 + ... + r_m =c_1 + ... +c_n =N are sufficiently far from constant vectors, then, asymptotically, in the uniform probability space of the mxn non-negative integer matrices with the total sum N of entries, the event consisting of the matrices with row sums R and the event consisting of the matrices with column sums C are positively correlated."}
{"category": "Math", "title": "New Digit Results and Several Problems", "abstract": "We give some new relations for Newman digit sums respectively different modulos and put some problems. In particular, for the odd prime modulos we put an important conjecture."}
{"category": "Math", "title": "An embedding theorem for automorphism groups of Cartan geometries", "abstract": "We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences of the theorem include general bounds on the rank and nilpotence degree of an automorphism group; a result asserting local homogeneity and completeness of parabolic geometries admitting a maximal-rank group of automorphisms; and a local freeness theorem for actions additionally preserving a continuous volume form."}
{"category": "Math", "title": "Estimating copula measure using ranks and subsampling: a simulation study", "abstract": "We describe here a new method to estimate copula measure. From N observations of two variables X and Y, we draw a huge number m of subsamples (size n<N), and we compute the joint ranks in these subsamples. Then, for each bivariate rank (p,q) (0<p,q<n+1), we count the number of subsamples such that there exist an observation of the subsample with bivariate rank (p,q). This counting gives an estimate of the density of the copula. The simulation study shows that this method seems to gives a better than the usual kernel method. The main advantage of this new method is then we do not need to choose and justify the kernel. In exchange, we have to choose a subsample size: this is in fact a problem very similar to the bandwidth choice. We have then reduced the overall difficulty."}
{"category": "Math", "title": "A lower bound on the subriemannian distance for H\\\"older distributions", "abstract": "Whereas subriemannian geometry usually deals with smooth horizontal distributions, partially hyperbolic dynamical systems provide many examples of subriemannian geometries defined by non-smooth (namely, H\\\"older continuous) distributions. These distributions are of great significance for the behavior of the parent dynamical system. The study of H\\\"older subriemannian geometries could therefore offer new insights into both dynamics and subriemannian geometry. In this paper we make a small step in that direction: we prove a H\\\"older-type lower bound on the subriemannian distance for H\\\"older continuous nowhere integrable codimension one distributions. This bound generalizes the well-known square root bound valid in the smooth case."}
{"category": "Math", "title": "Open-closed moduli spaces and related algebraic structures", "abstract": "We set up a Batalin-Vilkovisky Quantum Master Equation for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered J-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory and study the L_\\infty and A_\\infty algebraic structures associated to it."}
{"category": "Math", "title": "S-integral preperiodic points for dynamical systems over number fields", "abstract": "Given a rational map $\\phi: {\\mathbb P}^1\\to {\\mathbb P}^1$ defined over a number field $K$, we prove a finiteness result for $\\phi$-preperiodic points which are $S$-integral with respect to a non-preperiodic point $P$, provided $P$ satisfies a certain local condition at each place. This verifies a special case of a conjecture of S. Ih."}
{"category": "Math", "title": "Logarithmic vector fields and hyperbolicity", "abstract": "In this article we prove that the complement of a very generic curve of degree at least equal to 14 in the complex projective plane is hyperbolic in the sense of Kobayashi. Thus, using a new method, we improve the former known bound obtained by El Goul. We also improve the case of a very generic curve with two components."}
{"category": "Math", "title": "Hyperbolicit\\'e des vari\\'et\\'es complexes", "abstract": "These notes were written for the Cours Peccot given at the Coll\\`ege de France in 2007."}
{"category": "Math", "title": "Reduce Problems From Braid Groups To Braid Monoids", "abstract": "This paper proposes for every $n$, linear time reductions of the word and conjugacy problems on the braid groups $B_n$ to the corresponding problems on the braid monoids $B_n^+$ and moreover only using positive words representations."}
{"category": "Math", "title": "Mass transport and variants of the logarithmic Sobolev inequality", "abstract": "We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting."}
{"category": "Math", "title": "Linearization of skew-periodic loops and $\\mathbb S^1$-cocycles", "abstract": "We discuss linearization of skew-periodic loops. We generalize the situation to linearization of non-commutative loops and $\\mathbb S^1$-cocycles."}
{"category": "Math", "title": "Variations and estimators for the selfsimilarity order through Malliavin calculus", "abstract": "Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a specific non-Gaussian self-similar process, the Rosenblatt process. We apply our results to the design of strongly consistent statistical estimators for the self-similarity parameter $H$. Although, in the case of the Rosenblatt process, our estimator has non-Gaussian asymptotics for all $H>1/2$, we show the remarkable fact that the process's data at time 1 can be used to construct a distinct, compensated estimator with Gaussian asymptotics for $H\\in(1/2,2/3)$."}
{"category": "Math", "title": "Noncentral convergence of multiple integrals", "abstract": "Fix $\\nu>0$, denote by $G(\\nu/2)$ a Gamma random variable with parameter $\\nu/2$ and let $n\\geq2$ be a fixed even integer. Consider a sequence $\\{F_k\\}_{k\\geq1}$ of square integrable random variables belonging to the $n$th Wiener chaos of a given Gaussian process and with variance converging to $2\\nu$. As $k\\to\\infty$, we prove that $F_k$ converges in distribution to $2G(\\nu/2)-\\nu$ if and only if $E(F_k^4)-12E(F_k^3)\\to12\\nu^2-48\\nu$."}
{"category": "Math", "title": "On Voevodsky's algebraic K-theory spectrum BGL", "abstract": "Under a certain normalization assumption we prove that the $\\Pro^1$-spectrum $\\mathrm{BGL}$ of Voevodsky which represents algebraic $K$-theory is unique over $\\Spec(\\mathbb{Z})$. Following an idea of Voevodsky, we equip the $\\Pro^1$-spectrum $\\mathrm{BGL}$ with the structure of a commutative $\\Pro^1$-ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over $\\Spec(\\mathbb{Z})$. For an arbitrary Noetherian scheme $S$ of finite Krull dimension we pull this structure back to obtain a distinguished monoidal structure on $\\mathrm{BGL}$. This monoidal structure is relevant for our proof of the motivic Conner-Floyd theorem. It has also been used by Gepner and Snaith to obtain a motivic version of Snaith's theorem."}
{"category": "Math", "title": "Fast stable direct fitting and smoothness selection for Generalized Additive Models", "abstract": "Existing computationally efficient methods for penalized likelihood GAM fitting employ iterative smoothness selection on working linear models (or working mixed models). Such schemes fail to converge for a non-negligible proportion of models, with failure being particularly frequent in the presence of concurvity. If smoothness selection is performed by optimizing `whole model' criteria these problems disappear, but until now attempts to do this have employed finite difference based optimization schemes which are computationally inefficient, and can suffer from false convergence. This paper develops the first computationally efficient method for direct GAM smoothness selection. It is highly stable, but by careful structuring achieves a computational efficiency that leads, in simulations, to lower mean computation times than the schemes based on working-model smoothness selection. The method also offers a reliable way of fitting generalized additive mixed models."}
{"category": "Math", "title": "A Quiver Presentation for Solomon's Descent Algebra", "abstract": "The descent algebra $\\Sigma(W)$ is a subalgebra of the group algebra $\\Q W$ of a finite Coxeter group $W$, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of $W$. Thus $\\Sigma(W)$ is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct $\\Sigma(W)$ as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of $S$, the set of simple reflections in $W$. From this construction we obtain some general information about the quiver of $\\Sigma(W)$ and an algorithm for the construction of a quiver presentation for the descent algebra $\\Sigma(W)$ of any given finite Coxeter group $W$."}
{"category": "Math", "title": "Groebner bases for spaces of quadrics of codimension 3", "abstract": "Let $R=\\oplus_{i\\geq 0} R_i$ be an Artinian standard graded $K$-algebra defined by quadrics. Assume that $\\dim R_2\\leq 3$ and that $K$ is algebraically closed of characteristic $\\neq 2$. We show that $R$ is defined by a Gr\\\"obner basis of quadrics with, essentially, one exception. The exception is given by $K[x,y,z]/I$ where $I$ is a complete intersection of 3 quadrics not containing the square of a linear form."}
{"category": "Math", "title": "Blind Minimax Estimation", "abstract": "We consider the linear regression problem of estimating an unknown, deterministic parameter vector based on measurements corrupted by colored Gaussian noise. We present and analyze blind minimax estimators (BMEs), which consist of a bounded parameter set minimax estimator, whose parameter set is itself estimated from measurements. Thus, one does not require any prior assumption or knowledge, and the proposed estimator can be applied to any linear regression problem. We demonstrate analytically that the BMEs strictly dominate the least-squares estimator, i.e., they achieve lower mean-squared error for any value of the parameter vector. Both Stein's estimator and its positive-part correction can be derived within the blind minimax framework. Furthermore, our approach can be readily extended to a wider class of estimation problems than Stein's estimator, which is defined only for white noise and non-transformed measurements. We show through simulations that the BMEs generally outperform previous extensions of Stein's technique."}
{"category": "Math", "title": "Homotopy nilpotent groups", "abstract": "We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $\\pi_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $\\Omega F$ have values in homotopy n-nilpotent groups."}
{"category": "Math", "title": "On tameness and growth conditions", "abstract": "We study discrete subsets of C^d, relating \"tameness\" with growth conditions."}
{"category": "Math", "title": "On Meromorphic Functions which are Brody Curves", "abstract": "We discuss meromorphic functions on the complex plane which are Brody curves regarded as holomorphic maps to P_1, i.e., which have bounded spherical derivative."}
{"category": "Math", "title": "Discrete behavior of Seshadri constants on surfaces", "abstract": "Working over C, we show that, apart possibly from a unique limit point, the possible values of multi-point Seshadri constant for general points on smooth projective surfaces form a discrete set. In addition to its theoretical interest, this result is of practical value, which we demonstrate by giving significantly improved explicit lower bounds for Seshadri constants on P^2 and new results about ample divisors on blow ups of P^2 at general points."}
{"category": "Math", "title": "Lyapunov Coefficients for Degenerate Hopf Bifurcations", "abstract": "In this paper are studied the codimensions one, two, three and four Hopf bifurcations and the pertinent Lyapunov stability coefficients. Algebraic expressions obtained with computer assisted calculations are displayed."}
{"category": "Math", "title": "Intersecting Psi-classes on tropical M_{0,n}", "abstract": "We apply the tropical intersection theory developed by L. Allermann and J. Rau to compute intersection products of tropical Psi-classes on the moduli space of rational tropical curves. We show that in the case of zero-dimensional (stable) intersections, the resulting numbers agree with the intersection numbers of Psi-classes on the moduli space of n-marked rational curves computed in algebraic geometry."}
{"category": "Math", "title": "Combinatorial Gelfand Models", "abstract": "A combinatorial construction of a Gelafand model for the symmetric group and its Iwahori-Hecke algebra is presented."}
{"category": "Math", "title": "On Yamamuro's inverse and implicit function theorems in terms of calibrations", "abstract": "For the Frechet space E=C^{\\infty}(S^1) and for a smooth \\phi: R to R, we prove that the associated map E to E given by x mapsto\\phi\\circ x satisfies the continuous B\\Gamma--differentiability condition in Yamamuro's inverse function theorem only if \\phi is affine. Via more complicated examples, we also generally discuss the importance of testing the applicability of proposed inverse and implicit function theorems by this kind of simple maps."}
{"category": "Math", "title": "Plurisubharmonic polynomials and bumping", "abstract": "We wish to study the problem of bumping outwards a pseudoconvex, finite-type domain \\Omega\\subset C^n in such a way that pseudoconvexity is preserved and such that the lowest possible orders of contact of the bumped domain with bdy(\\Omega), at the site of the bumping, are explicitly realised. Generally, when \\Omega\\subset C^n, n\\geq 3, the known methods lead to bumpings with high orders of contact -- which are not explicitly known either -- at the site of the bumping. Precise orders are known for h-extendible/semiregular domains. This paper is motivated by certain families of non-semiregular domains in C^3. These families are identified by the behaviour of the least-weight plurisubharmonic polynomial in the Catlin normal form. Accordingly, we study how to perturb certain homogeneous plurisubharmonic polynomials without destroying plurisubharmonicity."}
{"category": "Math", "title": "The circular law for random matrices", "abstract": "We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices."}
{"category": "Math", "title": "Invariant Carnot-Caratheodory metrics on $S^3$, $SO(3)$, $SL(2)$ and lens spaces", "abstract": "In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric given on $SU(2)$ projects on the so called lens spaces $L(p,q)$. Also for lens spaces, we compute the cut loci (globally). For $SU(2)$ the cut locus is a maximal circle without one point. In all other cases the cut locus is a stratified set. To our knowledge, this is the first explicit computation of the whole cut locus in sub-Riemannian geometry, except for the trivial case of the Heisenberg group."}
{"category": "Math", "title": "Face enumeration - from spheres to manifolds", "abstract": "We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine $h$-vector of balanced semi-Eulerian complexes and the toric $h$-vector of semi-Eulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup's 3-dimensional constructions \\cite{Wal}, allow us to give a complete characterization of the $f$-vectors of arbitrary simplicial triangulations of $S^1 \\times S^3, \\C P^2,$ $ K3$ surfaces, and $(S^2 \\times S^2) # (S^2 \\times S^2).$ We also establish a principle which leads to a conjecture for homology manifolds which is almost logically equivalent to the $g$-conjecture for homology spheres. Lastly, we show that with sufficiently many vertices, every triangulable homology manifold without boundary of dimension three or greater can be triangulated in a 2-neighborly fashion."}
{"category": "Math", "title": "Automatically reduced degenerations of automatically normal varieties", "abstract": "Let F be a flat family of projective schemes, whose geometric generic fiber is reduced and irreducible. We give conditions on a special fiber (a \"limit\" of the family) to guarantee that it too is reduced. These conditions often imply also that the generic fiber is normal. The conditions are particularly easy to check in the setup of a \"geometric vertex decomposition\" [Knutson-Miller-Yong '07]. The primary tool used is the corresponding limit _branchvariety_ [Alexeev-Knutson '06], which is reduced by construction, and maps to the limit subscheme; our technique is to use normality to show that the branchvariety map must be an isomorphism. As a demonstration, we give an essentially naive proof that Schubert varieties in finite type are normal and Cohen-Macaulay. The proof does not involve any resolution of singularities or cohomology-vanishing techniques (e.g. appeal to characteristic p)."}
{"category": "Math", "title": "Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets", "abstract": "A set $S\\subseteq \\re^n$ is called to be {\\it Semidefinite (SDP)} representable if $S$ equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). The contributions of this paper are: (i) For bounded SDP representable sets $W_1,...,W_m$, we give an explicit construction of an SDP representation for $\\cv{\\cup_{k=1}^mW_k}$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set $S$, we prove sufficient condition: the boundary $\\bdS$ is positively curved, and necessary condition: $\\bdS$ has nonnegative curvature at smooth points and on nondegenerate corners. This amounts to the strict versus nonstrict quasi-concavity of defining polynomials on those points on $\\bdS$ where they vanish. The gaps between them are $\\bdS$ having positive versus nonnegative curvature and smooth versus nonsmooth points. A sufficient condition bypassing the gaps is when some defining polynomials of $S$ are sos-concave. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set $T$, we find that the critical object is $\\pt_cT$, the maximum subset of $\\pt T$ contained in $\\pt \\cv{T}$. We prove sufficient conditions for SDP representability: $\\pt_cT$ is positively curved, and necessary conditions: $\\pt_cT$ has nonnegative curvature at smooth points and on nondegenerate corners. The gaps between them are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition is also discussed."}
{"category": "Math", "title": "On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group", "abstract": "The following inequality \\cat X\\le \\cat Y+\\lceil\\frac{hd(X)-r}{r+1}\\rceil holds for every locally trivial fibration between $ANE$ spaces $f:X\\to Y$ which admits a section and has the $r$-connected fiber where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that \\cat X\\le \\lceil\\frac{\\dim X-1}{2}\\rceil+cd(\\pi_1(X)) for every complex $X$ with $cd(\\pi_1(X))\\le 2$."}
{"category": "Math", "title": "Infinite products with strongly $B$-multiplicative exponents", "abstract": "Let $N_{1,B}(n)$ denote the number of ones in the $B$-ary expansion of an integer $n$. Woods introduced the infinite product $P :=\\prod_{n \\geq 0} (\\frac{2n+1}{2n+2})^{(-1)^{N_{1,2}(n)}}$ and Robbins proved that $P = 1/\\sqrt{2}$. Related products were studied by several authors. We show that a trick for proving that $P^2 = 1/2$ (knowing that $P$ converges) can be extended to evaluating new products with (generalized) strongly $B$-multiplicative exponents. A simple example is $$ \\prod_{n \\geq 0} (\\frac{Bn+1}{Bn+2})^{(-1)^{N_{1,B}(n)}} = \\frac{1}{\\sqrt B}. $$"}
{"category": "Math", "title": "On Log Canonical Models of the Moduli Space of Stable Pointed Curves", "abstract": "We study the log canonical models of the moduli space MBar_{0,n} of pointed stable genus zero curves with respect to the standard log canonical divisors K+aD, where D denotes the boundary. In particular we show that, as a formal consequence of a conjecture by Fulton regarding the ample cone of MBar_{0,n}, these log canonical models are equal to certain of Hassett's weighted pointed curve spaces."}
{"category": "Math", "title": "A Generalization of De Vries Duality Theorem", "abstract": "Generalizing Duality Theorem of H. de Vries, we define a category which is dually equivalent to the category of all locally compact Hausdorff spaces and all perfect maps between them."}
{"category": "Math", "title": "Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation", "abstract": "We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a Lax-Richtmyer type equivalence theorem -- a consistent method is convergent if and only if it is stable. We define consistency as convergence on a dense subspace and stability as discrete well-posedness. In some applications convergence is harder to prove than consistency or stability since convergence requires knowledge of the solution. An equivalence theorem can be useful in such settings. We give concrete instances of equivalence theorems for polynomial interpolation, numerical differentiation, numerical integration using quadrature rules and Monte Carlo integration."}
{"category": "Math", "title": "Boring split links", "abstract": "Boring is an operation which converts a knot or two-component link in a 3--manifold into another knot or two-component link. It generalizes rational tangle replacement and can be described as a type of 2--handle attachment. Sutured manifold theory is used to study the existence of essential spheres and planar surfaces in the exteriors of knots and links obtained by boring a split link. It is shown, for example, that if the boring operation is complicated enough, a split link or unknot cannot be obtained by boring a split link. Particular attention is paid to rational tangle replacement. If a knot is obtained by rational tangle replacement on a split link, and a few minor conditions are satisfied, the number of boundary components of a meridional planar surface is bounded below by a number depending on the distance of the rational tangle replacement. This result is used to give new proofs of two results of Eudave-Mu\\~noz and Scharlemann's band sum theorem."}
{"category": "Math", "title": "Locally toric manifolds and singular Bohr-Sommerfeld leaves", "abstract": "When geometric quantization is applied to a manifold using a real polarization which is \"nice enough\", a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld leaves. Subsequently, several authors have taken this as motivation for counting Bohr-Sommerfeld leaves when studying the quantization of manifolds which are less \"nice\". In this paper, we examine the quantization of compact symplectic manifolds that can locally be modelled by a toric manifold, using a real polarization modelled on fibres of the moment map. We compute the results directly, and obtain a theorem similar to Sniatycki's, which gives the quantization in terms of counting Bohr-Sommerfeld leaves. However, the count does not include the Bohr-Sommerfeld leaves which are singular. Thus the quantization obtained is different from the quantization obtained using a Kaehler polarization."}
{"category": "Math", "title": "Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes", "abstract": "Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad hoc methods. In this note, we present a conjectural explanation for quasi-period collapse in rational polytopes. We show that this explanation applies to some previous cases appearing in the literature. We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope. Our approach depends on the invariance of the Ehrhart quasi-polynomial under the action of affine unimodular transformations. Motivated by the similarity of this idea to the scissors congruence problem, we explore the development of a Dehn-like invariant for rational polytopes in the lattice setting."}
{"category": "Math", "title": "The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system", "abstract": "We consider an Allen-Cahn type equation with a bistable nonlinearity associated to a double-well potential whose well-depths can be slightly unbalanced, and where the coefficient of the nonlinear reaction term is very small. Given rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within a small time, and we present an optimal estimate for its width. We then consider a class of reaction-diffusion systems which includes the FitzHugh-Nagumo system as a special case. Given rather general initial data, we show that the first component of the solution vector develops a steep transition layer and that all the results mentioned above remain true for this component."}
{"category": "Math", "title": "Stability of foliations induced by rational maps", "abstract": "We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space $\\mathscr F_q(r, d)$ of singular foliations of codimension $q$ and degree $d$ on the complex projective space $\\mathbb P^r$, when $1\\le q \\le r-2$. We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases."}
{"category": "Math", "title": "Local Floer Homology and the Action Gap", "abstract": "In this paper, we study the behavior of the local Floer homology of an isolated fixed point and the growth of the action gap under iterations. To be more specific, we prove that an isolated fixed point of a Hamiltonian diffeomorphism remains isolated for a certain class of iterations (the so-called admissible iterations) and that the local Floer homology groups for all such iterations are isomorphic to each other up to a shift of degree. Furthermore, we study the pair-of-pants product in local Floer homology, and characterize a particular class of isolated fixed points (the symplectically degenerate maxima), which plays an important role in the proof of the Conley conjecture. The proofs of these facts rely on an observation that for a general diffeomorphism, not necessarily Hamiltonian, an isolated fixed point remains isolated under all admissible iterations. Finally, we apply these results to show that for a quasi-arithmetic sequence of admissible iterations of a Hamiltonian diffeomorphism with isolated fixed points the minimal action gap is bounded from above when the ambient manifold is closed and symplectically aspherical. This theorem is a generalization of the Conley conjecture."}
{"category": "Math", "title": "Homological flat dimensions", "abstract": "For finitely generated module $M$ over a local ring $R$, the conventional notions of complete intersection dimension $\\cid_R M$ and Cohen-Macaulay dimension $\\cmdim_R M$ do not extend to cover the case of infinitely generated modules. In this paper we introduce similar invariants for not necessarily finitely generated modules, (namely, complete intersection flat and Cohen-Macaulay flat dimensions) which for finitely generated modules, coincide with the corresponding classical ones."}
{"category": "Math", "title": "Upper bounds of Hilbert coefficients and Hilbert functions", "abstract": "Let $(R, m)$ be a $d$-dimensional Cohen-Macaulay local ring. In this note we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a $m$-primary ideal $I\\subset R$ that improves all known upper bounds unless for a finite number of cases. We also provide new upper bounds of the Hilbert functions of $I$ extending the known bounds for the maximal ideal."}
{"category": "Math", "title": "An Extension of Mok's Theorem on the Generalized Frankel Conjecture", "abstract": "In this paper, we will give an extension of Mok's theorem on the generalized Frankel conjecture under the condition of the orthogonal bisectional curvature."}
{"category": "Math", "title": "The Infinite in Sciences and Arts", "abstract": "Actual infinity in its various forms is discussed, searched and not found."}
{"category": "Math", "title": "Capacitary estimates of solutions of semilinear parabolic equations", "abstract": "We prove an almoste representation formula for positive solutions of semilinear heat equations with power-type absorption the initial trace of which is the indicatrix function of a compact set. The representations involves a Wiener test via Bessel capacities."}
{"category": "Math", "title": "Factorization of Laurent series over commutative rings", "abstract": "We generalize the Wiener-Hopf factorization of Laurent series to more general commutative coefficient rings, and we give explicit formulas for the decomposition. We emphasize the algebraic nature of this factorization."}
{"category": "Math", "title": "Cyclotomy Primality Proofs and their Certificates", "abstract": "The first efficient general primality proving method was proposed in the year 1980 by Adleman, Pomerance and Rumely and it used Jacobi sums. The method was further developed by H. W. Lenstra Jr. and more of his students and the resulting primality proving algorithms are often referred to under the generic name of Cyclotomy Primality Proving (CPP). In the present paper we give an overview of the theoretical background and implementation specifics of CPP, such as we understand them in the year 2007."}
{"category": "Math", "title": "Dual Elliptic Primes and Applications to Cyclotomy Primality Proving", "abstract": "Two rational primes p, q are called dual elliptic if there is an elliptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primality proving algorithms. By extending to elliptic curves some notions of galois theory of rings used in the cyclotomy primality tests, one obtains a new algorithm which has heuristic cubic run time and generates certificates that can be verified in quadratic time. After the break through of Agrawal, Kayal and Saxena has settled the complexity theoretical problem of primality testing, some interest remains for the practical aspect of state of the art implementable proving algorithms."}
{"category": "Math", "title": "A universality theorem for Voevodsky's algebraic cobordism spectrum", "abstract": "An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P^1-spectra equipped with the symmetric monoidal structure described in arXiv:0709.3905v1 [math.AG]. The algebraic cobordism P^1-spectrum MGL is considered as a commutative monoid equipped with a canonical orientation. For a commutative monoid E in the category SH(k) we identify the set of monoid homomorphisms from MGL to E in the motivic stable homotopy category with the set of all orientations of E. This result was stated originally in a slightly different form by G. Vezzosi in arXiv:math/0004050v2 [math.AG]."}
{"category": "Math", "title": "Symmetries in projective multiresolution analyses", "abstract": "We give an equivariant version of Packer and Rieffel's theorem on sufficient conditions for the existence of orthonormal wavelets in projective multiresolution analyses. The scaling functions that generate a projective multiresolution analysis are supposed to be invariant with respect to some finite group action. We give sufficient conditions for the existence of wavelets with similar invariance."}
{"category": "Math", "title": "On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory", "abstract": "Quillen's algebraic K-theory is reconstructed via Voevodsky's algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P^1-spectrum MGL of Voevodsky is considered as a commutative P^1-ring spectrum. There is a unique ring morphism MGL^{2*,*}(k)--> Z which sends the class [X]_{MGL} of a smooth projective k-variety X to the Euler characteristic of the structure sheaf of X. Our main result states that there is a canonical grade preserving isomorphism of ring cohomology theories MGL^{*,*}(X,U) \\tensor_{MGL^{2*,*}(k)} Z --> K^{TT}_{- *}(X,U) = K'_{- *}(X-U)} on the category of smooth k-varieties, where K^{TT}_* is Thomason-Trobaugh K-theory and K'_* is Quillen's K'-theory. In particular, the left hand side is a ring cohomology theory. Moreover both theories are oriented and the isomorphism above respects the orientations. The result is an algebraic version of a theorem due to Conner and Floyd. That theorem reconstructs complex K-theory via complex cobordism."}
{"category": "Math", "title": "Global Square and Mutual Stationarity at the Aleph_n", "abstract": "We show using a proof of the Global Square property in Core Models below a measurable of Mitchell order o(kappa)=kappa^++ (a result originally due to Jensen & Zeman) that Foreman and Magidor's Mutual Stationarity property MS(Aleph_n (1<n<omega), Cof(omega_1)) implies the existence of inner models with measurables of high Mitchell order. This MS property states that any sequence of independently chosen stationary subsets S_n of the Aleph_n (of fixed cofinality omega_1) is mutually stationary below aleph_omega."}
{"category": "Math", "title": "Distances sets that are a shift of the integers and Fourier basis for planar convex sets", "abstract": "The aim of this paper is to prove that if a planar set $A$ has a difference set $\\Delta(A)$ satisfying $\\Delta(A)\\subset \\Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if $A$ is a set of exponentials mutually orthogonal with respect to any symmetric convex set $K$ in the plane with a smooth boundary and everywhere non-vanishing curvature, then $ # (A \\cap {[-q,q]}^2) \\leq C(K) q$ where $C(K)$ is a constant depending only on $K$. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from \\cite{IKP01} and \\cite{IKT01} that if $K$ is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then $L^2(K)$ does not possess an orthogonal basis of exponentials."}
{"category": "Math", "title": "On representation theory of affine Hecke algebras of type B", "abstract": "Ariki's and Grojnowski's approach to the representation theory of affine Hecke algebras of type $A$ is applied to type $B$ with unequal parameters to obtain -- under certain restrictions on the eigenvalues of the lattice operators -- analogous multiplicity-one results and a classification of irreducibles with partial branching rules as in type $A$."}
{"category": "Math", "title": "Stanley decompositions of squarefree modules and Alexander duality", "abstract": "In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and the exterior algebra behave with respect to Alexander duality."}
{"category": "Math", "title": "Uniqueness of solutions of stochastic differential equations", "abstract": "We consider a d-dimensional stochastic differential equation with additive noise and a drift coefficient which is assumed only to be a bounded Borel function. We show that, for almost all choices of the driving Brownian path, the equation has a unique solution."}
{"category": "Math", "title": "Lower bounds on the canonical height associated to the morphism \\phi(z)= z^d+c", "abstract": "Certain lower bounds are obtained on the canonical height associated to the morphism $\\phi(z)=z^d+c$."}
{"category": "Math", "title": "Timescale effect estimation in time-series studies of air pollution and health: A Singular Spectrum Analysis approach", "abstract": "A wealth of epidemiological data suggests an association between mortality/morbidity from pulmonary and cardiovascular adverse events and air pollution, but uncertainty remains as to the extent implied by those associations although the abundance of the data. In this paper we describe an SSA (Singular Spectrum Analysis) based approach in order to decompose the time-series of particulate matter concentration into a set of exposure variables, each one representing a different timescale. We implement our methodology to investigate both acute and long-term effects of $PM_{10}$ exposure on morbidity from respiratory causes within the urban area of Bari, Italy."}
{"category": "Math", "title": "Global existence for a kinetic model of chemotaxis via dispersion and Strichartz estimates", "abstract": "We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of 'run and tumble'. The existence of solutions has been obtained in several papers [Chalub et al. Monatsh. Math. 142, 123--141 (2004), Hwang et al. SIAM J. Math. Anal. 36, 1177--1199 (2005)] using direct and strong dispersive effects. Here, we use the weak dispersion estimates of [Castella et al. C. R. Acad. Sci. Paris 322, 535--540 (1996)] to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption."}
{"category": "Math", "title": "An Alternative Form of the Functional Equation for Riemann's Zeta Function", "abstract": "In this paper we present a simple method for deriving an alternative form of the functional equation for Riemann's Zeta function. The connections between some functional equations obtained implicitly by Leonhard Euler in his work \"Remarques sur un beau rapport entre les series des puissances tant directes que reciproques\" in Memoires de l'Academie des Sciences de Berlin 17, (1768), permit to define a special function, named A(s), which is fully symmetric and is similar to Riemann's \"XI\" function. To be complete we find several integral representations of the A(s) function and as a direct consequence of the second integral representation we obtain also an analytic continuation of the same function using an identity of Ramanujan."}
{"category": "Math", "title": "Fast Fourier Transforms for the Rook Monoid", "abstract": "We define the notion of the Fourier transform for the rook monoid (also called the symmetric inverse semigroup) and provide two efficient divide-and-conquer algorithms (fast Fourier transforms, or FFTs) for computing it. This paper marks the first extension of group FFTs to non-group semigroups."}
{"category": "Math", "title": "Threshold phenomena on product spaces: BKKKL revisited (once more)", "abstract": "We revisit the work of Bourgain, Kahn, Kalai, Katznelson and Linial (1992) -- referred to as ``BKKKL'' in the title -- about influences on Boolean functions in order to give a precise statement of threshold phenomenon on the product space $\\{1,..., r\\}^\\NN$, generalizing one of the main results of a paper by Talagrand (1994)."}
{"category": "Math", "title": "Strong homotopy inner product of an A-infinity algebra", "abstract": "We introduce a strong homotopy notion of a cyclic symmetric inner product of an A-infinity algebra and prove a characterization theorem in the formalism of the infinity inner products by Tradler. We also show that it is equivalent to the notion of a non-constant symplectic structure on the corresponding formal non-commutative supermanifold. We show that (open Gromov-Witten type) potential for a cyclic filtered A-infinity algebra is invariant under the cyclic filtered A-infinity homomorphism up to reparametrization, cyclization and a constant addition, generalizing the work of Kajiura."}
{"category": "Math", "title": "Local to Global Compatibility on the Eigencurve (l not equal p)", "abstract": "We generalise Coleman's construction of Hecke operators to define an action of GL_2(Q_l) on the space of finite slope overconvergent p-adic modular forms (l not equal p). In this way we associate to any C_p-valued point on the tame level N Coleman-Mazur eigencurve an admissible smooth representation of GL_2(Q_l) extending the classical construction. Using the Galois theoretic interpretation of the eigencurve we associate a 2-dimensional Weil-Deligne representation to such points and show that away from a discrete set they agree under the Local Langlands correspondence."}
{"category": "Math", "title": "On stable constant mean curvature surfaces in S2 X R and H2 X R", "abstract": "We study stable compact constant mean curvature surfaces in the product spaces S2 X R and H2 X R and in some other Riemannian 3-manifolds."}
{"category": "Math", "title": "Multiplicity one Theorems", "abstract": "In the local, characteristic 0, non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies that an admissible irreducible representation of GL(n+1), when restricted to GL(n) decomposes with multiplicity one. Similar Theorems are obtained for orthogonal or unitary groups."}
{"category": "Math", "title": "Archimedes' balance and Bianchi's Backlund transformation for quadrics", "abstract": "We establish a link between Archimedes' method of integration for calculating areas, volumes and centers of mass of segments of parabolas and quadrics of revolution by factorization via the moments of a balance and an integration technique for a particular integrable system, namely Bianchi's B\\\"{a}cklund transformation for quadrics."}
{"category": "Math", "title": "Tent Spaces Associated with Semigroups of Operators", "abstract": "We study tent spaces on general measure spaces $(\\Omega, \\mu)$. We assume that there exists a semigroup of positive operators on $L^p(\\Omega, \\mu)$ satisfying a monotone property but do not assume any geometric/metric structure on $\\Omega$. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's $H^1$-BMO duality theory. We also get a $H^1$-BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative $L^p$ spaces."}
{"category": "Math", "title": "An Extrapolation of Operator Valued Dyadic Paraproducts", "abstract": "We consider the dyadic paraproducts $\\pi_\\f$ on $\\T$ associated with an $\\M$-valued function $\\f.$ Here $\\T$ is the unit circle and $\\M$ is a tracial von Neumann algebra. We prove that their boundedness on $L^p(\\T,L^p(\\M))$ for some $1<p<\\infty $ implies their boundedness on $L^p(\\T,L^p(\\M))$ for all $1<p<\\infty$ provided $\\f$ is in an operator-valued BMO space. We also consider a modified version of dyadic paraproducts and their boundedness on $L^p(\\T,L^p(\\M))."}
{"category": "Math", "title": "Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II", "abstract": "Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v (x,t)| \\le C_*{|t|^{-1/2}} $ or, for some $\\e > 0$, $|v (x,t)| \\le C_* r^{-1+\\epsilon} |t|^{-\\epsilon /2}$ for $-T_0\\le t < 0$ and $0<C_*<\\infty$ allowed to be large. We prove that $v$ is regular at time zero."}
{"category": "Math", "title": "An introduction to the geometry of metric spaces", "abstract": "These informal notes deal with some basic properties of metric spaces, especially concerning lengths of curves."}
{"category": "Math", "title": "Permutahedra and generalized associahedra", "abstract": "Given a finite Coxeter system $(W,S)$ and a Coxeter element $c$, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan $F_c$, settling a conjecture of Reading that this is possible. We call this polytope the $c$-generalized associahedron. Our approach generalizes Loday's realization of the associahedron (a type $A$ $c$-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the $c$-singleton cones, the cones in the $c$-Cambrian fan which consist of a single maximal cone from the Coxeter fan. Moreover, if $W$ is a Weyl group and the vertices of the permutahedron are chosen in a lattice associated to $W$, then we show that our realizations have integer coordinates in this lattice."}
{"category": "Math", "title": "Direct and inverse theorems in the theory of approximation by the Ritz method", "abstract": "For an arbitrary self-adjoint operator $B$ in a Hilbert space $H$, we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector $x \\in H$ with respect to the operator $B$, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator $B$, and the $k$-modulus of continuity of the vector $x$ with respect to the operator $B$. The results are used for finding a priori estimates for the Ritz approximate solutions of operator equations in a Hilbert space."}
{"category": "Math", "title": "Finitistic dimension through infinite projective dimension", "abstract": "We show that an artin algebra having at most three radical layers of infinite projective dimension has finite finitistic dimension, generalizing the known result for algebras with vanishing radical cube."}
{"category": "Math", "title": "Arithmetic duality theorems for 1-motives over function fields", "abstract": "In this paper we obtain a Poitou-Tate exact sequence for finite and flat group schemes over a global function field. We also extend the duality theorems for 1-motives over number fields obtained by D.Harari and T.Szamuely to the function field case."}
{"category": "Math", "title": "Ullemar's formula for the moment map, II", "abstract": "We establish the complex analogue of Ullemar's formula for polynomial domains. We show that the Jacobian of the complex moment mapping is equal to the self-resultant of the defining polynomial."}
{"category": "Math", "title": "Weak convergence of Vervaat and Vervaat Error processes of long-range dependent sequences", "abstract": "Following Cs\\\"{o}rg\\H{o}, Szyszkowicz and Wang (Ann. Statist. {\\bf 34}, (2006), 1013--1044) we consider a long range dependent linear sequence. We prove weak convergence of the uniform Vervaat and the uniform Vervaat error processes, extending their results to distributions with unbounded support and removing normality assumption."}
{"category": "Math", "title": "Affine descents and the Steinberg torus", "abstract": "Let $W\\ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori."}
{"category": "Math", "title": "Modulated Branching Processes, Origins of Power Laws and Queueing Duality", "abstract": "Power law distributions have been repeatedly observed in a wide variety of socioeconomic, biological and technological areas. In many of the observations, e.g., city populations and sizes of living organisms, the objects of interest evolve due to the replication of their many independent components, e.g., births-deaths of individuals and replications of cells. Furthermore, the rates of the replication are often controlled by exogenous parameters causing periods of expansion and contraction, e.g., baby booms and busts, economic booms and recessions, etc. In addition, the sizes of these objects often have reflective lower boundaries, e.g., cities do not fall bellow a certain size, low income individuals are subsidized by the government, companies are protected by bankruptcy laws, etc. Hence, it is natural to propose reflected modulated branching processes as generic models for many of the preceding observations. Indeed, our main results show that the proposed mathematical models result in power law distributions under quite general polynomial Gartner-Ellis conditions, the generality of which could explain the ubiquitous nature of power law distributions. In addition, on a logarithmic scale, we establish an asymptotic equivalence between the reflected branching processes and the corresponding multiplicative ones. The latter, as recognized by Goldie (1991), is known to be dual to queueing/additive processes. We emphasize this duality further in the generality of stationary and ergodic processes."}
{"category": "Math", "title": "Confidence intervals for the normal mean utilizing prior information", "abstract": "Consider X_1,X_2,...,X_n that are independent and identically N(mu,sigma^2) distributed. Suppose that we have uncertain prior information that mu = 0. We answer the question: to what extent can a frequentist 1-alpha confidence interval for mu utilize this prior information?"}
{"category": "Math", "title": "Local density of diffeomorphisms with large centralizers", "abstract": "Given any compact manifold M, we construct a non-empty open subset O of the space of C^1-diffeomorphisms of M and a dense subset D of O such that the centralizer of every diffeomorphism in D is uncountable, hence non-trivial."}
{"category": "Math", "title": "Markov basis for design of experiments with three-level factors", "abstract": "We consider Markov basis arising from fractional factorial designs with three-level factors. Once we have a Markov basis, $p$ values for various conditional tests are estimated by the Markov chain Monte Carlo procedure. For designed experiments with a single count observation for each run, we formulate a generalized linear model and consider a sample space with the same sufficient statistics to the observed data. Each model is characterized by a covariate matrix, which is constructed from the main and the interaction effects we intend to measure. We investigate fractional factorial designs with $3^{p-q}$ runs noting correspondences to the models for $3^{p-q}$ contingency tables."}
{"category": "Math", "title": "Some properties of group-theoretical categories", "abstract": "We first show that every group-theoretical category is graded by a certain double coset ring. As a consequence, we obtain a necessary and sufficient condition for a group-theoretical category to be nilpotent. We then give an explicit description of the simple objects in a group-theoretical category (following Ostrik, arXiv:math/0202130) and of the group of invertible objects of a group-theoretical category, in group-theoretical terms. Finally, under certain restrictive conditions, we describe the universal grading group of a group-theoretical category."}
{"category": "Math", "title": "Sharp results in the integral-form John--Nirenberg inequality", "abstract": "We consider the strong form of the John-Nirenberg inequality for the $L^2$-based BMO. We construct explicit Bellman functions for the inequality in the continuous and dyadic settings and obtain the sharp constant as well as the precise bound on the inequality's range of validity, both previously unknown. The results for the two cases are substantially different. The paper not only gives another instance in the short list of such explicit calculations, but also presents the Bellman function method as a sequence of clear steps, adaptable to a wide variety of applications."}
{"category": "Math", "title": "Noncommutative Brownian motions with Kesten distributions and related Poisson processes", "abstract": "We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition $P$ we assign a weight depending on the numbers of disorders and orders in $P$ related to the natural partial order on the set of blocks of $P$ implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in non-crossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes."}
{"category": "Math", "title": "Transference of bilinear multiplier operators on Lorentz spaces", "abstract": "We prove a DeLeeuw type theorem of transference of boundedness for modulation invariant multiplier operators between the groups defined by the real line and the torus."}
{"category": "Math", "title": "On three approaches to conjugacy in semigroups", "abstract": "We compare three approaches to the notion of conjugacy for semigroups, the first one via the transitive closure of the $uv\\sim vu$ relation, the second one via an action of inverse semigroups on themselves by partial transformations, and the third one via characters of finite-dimensional representations."}
{"category": "Math", "title": "Boundedness from H^1 to L^1 of Riesz transforms on a Lie group of exponential growth", "abstract": "Let $G$ be the Lie group given by the semidirect product of $R^2$ and $R^+$ endowed with the Riemannian symmetric space structure. Let $X_0, X_1, X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $\\Delta=-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_i\\Delta^{-1/2}$ and $S_i=\\Delta^{-1/2}X_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second order Riesz transforms $T_{ij}=X_i\\Delta^{-1}X_j$ are bounded from $H^1$ to $L^1$, while the Riesz transforms $S_{ij}=\\Delta^{-1}X_iX_j$ and $R_{ij}=X_iX_j\\Delta^{-1}$ are not."}
{"category": "Math", "title": "The Hall algebra of a cyclic quiver at $q=0$", "abstract": "We show that the generic Hall algebra of nilpotent representations of an oriented cycle specialised at $q=0$ is isomorphic to the generic extension monoid in the sense of Reineke. This continues the work of Reineke."}
{"category": "Math", "title": "Which 3-manifold groups are K\\\"ahler groups?", "abstract": "The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if G can be realized as both the fundamental group of a closed 3-manifold and of a compact K\\\"ahler manifold, then G must be finite, and thus belongs to the well-known list of finite subgroups of O(4)."}
{"category": "Math", "title": "Un th\\'eor\\`eme de Helson pour des s\\'eries de Walsh", "abstract": "Extension to Walsh series of theorems of Helson and Katznelson on trigonometric series, saying that a trigonometric series whose partial sums are positive has its coefficients tend to zero but is not necessarily a Fourier-Lebesgue series"}
{"category": "Math", "title": "Hasard et d\\'eterminisme chez Laplace", "abstract": "Laplace's views on randomness and determinism. The paper was written for \"Cahiers rationalistes\" and addresses a rather wide audience. It contains large quotations of Laplace, most of them coming from his introduction to the book \"Analytical theory of probabilities\"."}
{"category": "Math", "title": "Mailles et ensembles de Sidon", "abstract": "Grid condition and Sidon sets. The grid condition stated by the author in 1957 as a necessary condition for a set of integers to be a Sidon set is neither improvable nor sufficient ; explanations and stronger statements are provided."}
{"category": "Math", "title": "Correspondance de Howe: paires duales de type II", "abstract": "In this article, we give a new method for proving Howe correspondence in the case of dual pairs of type $({\\rm GL}_n, {\\rm GL}_m)$ over a non-Archimedean locally compact field $F$. The proof consists in combining a study on Kudla's filtration with the results of a previous article of the autor about the irreducibility of a parabolically induced representation. The proof is valid for $F$ of any characteristic and allows us to make the correspondence explicit in terms of Langlands parameters."}
{"category": "Math", "title": "Ensembles quasi-ind\\'ependants et ensembles de Sidon, th\\'eorie de Bourgain (d'apr\\`es Myriam D\\'echamps, Li et Queffelec)", "abstract": "Exposition of the relation between quasi-independent and Sidon sets, theorems of Pisier according to the method of Bourgain"}
{"category": "Math", "title": "Graph-different permutations", "abstract": "We strengthen and put in a broader perspective previous results of the first two authors on colliding permutations. The key to the present approach is a new non-asymptotic invariant for graphs."}
{"category": "Math", "title": "Magnus subgroups of one-relator surface groups", "abstract": "A one-relator surface group is the quotient of an orientable surface group by the normal closure of a single relator. A Magnus subgroup is the fundamental group of a suitable incompressible sub-surface. A number of results are proved about the intersections of such subgroups and their conjugates, analogous to results of Bagherzadeh, Brodskii, and Collins in classical one-relator group theory."}
{"category": "Math", "title": "Minimal Affine Coordinates for SL(3,C) Character Varieties of Free Groups", "abstract": "Let X be the moduli of SL(3,C) representations of a rank r free group. In this paper we determine minimal generators of the coordinate ring of X. This at once gives explicit global coordinates for the moduli and determines the dimension of the moduli's minimal affine embedding. Along the way, we utilize results concerning the moduli of r-tuples of matrices in gl(3,C). Consequently, we also state general invariant theoretic correspondences between the coordinate rings of the moduli of r-tuples of elements in gl(n,C), sl(n,C), and SL(n,C)."}
{"category": "Math", "title": "Computing twisted conjugacy classes in free groups using nilpotent quotients", "abstract": "There currently exists no algebraic algorithm for computing twisted conjugacy classes in free groups. We propose a new technique for deciding twisted conjugacy relations using nilpotent quotients. Our technique is generalization of the common abelianization method, but admits significantly greater rates of success. We present experimental results demonstrating the efficacy of the technique, and detail how it can be applied the related settings of surface groups and doubly twisted conjugacy."}
{"category": "Math", "title": "Nested set complexes for posets and the Bier construction", "abstract": "We generalize the concept of combinatorial nested set complexes to posets and exhibit the topological relationship between the arising nested set complexes and the order complex of the underlying poset. In particular, a sufficient condition is given so that this relationship is actually a subdivision. We use the results to generalize the proof method of \\v{C}uki\\'c and Delucchi, so far restricted to semilattices, for a result of Bj\\\"orner, Paffenholz, Sj\\\"ostrand and Ziegler on the Bier construction on posets."}
{"category": "Math", "title": "Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations", "abstract": "In this paper we consider a class of planar autonomous systems having an isolated limit cycle x_0 of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x_0 as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem."}
{"category": "Math", "title": "Isometry classes of generalized associahedra", "abstract": "Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\\mathbb R$-Euclidean space with simple roots $\\Delta=\\{\\a_s | s\\in S\\}$ of the same length and fundamental weights $\\Delta^*=\\{v_s | s\\in S\\}$. We set $M(e)=\\sum_{s\\in S}\\kappa_s v_s$, $\\kappa_s>0$, and for $w\\in W$ we set $M(w)=w(M(e))$. The permutahedron $Perm(W)$ is the convex hull of the set $\\{M(w) | w\\in W\\}$. Given a Coxeter element $c\\in W$, we have defined in a previous work a generalized associahedron $Asso_c(W)$ whose normal fan is the corresponding $c$-Cambrian fan $F_c$ defined by N. Reading. By construction, $Asso_c(W)$ is obtained from $Perm(W)$ by removing some halfspaces according to a rule prescribed by $c$. In this work, we classify the isometry classes of these realizations. More precisely, for $(W,S)$ an irreducible finite Coxeter system and $c,c'$ two Coxeter elements in $W$, we have that $Asso_{c}(W)$ and $Asso_{c'}(W)$ are isometric if and only if $\\mu(c') = c$ or $\\mu(c')=w_0c^{-1}w_0$ for $\\mu$ an automorphism of the Coxeter graph of $W$ such that $\\kappa_s=\\kappa_{\\mu(s)}$ for all $s\\in S$. As a byproduct, we classify the isometric Cambrian fans of $W$."}
{"category": "Math", "title": "On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes", "abstract": "We consider an autonomous system in R^n having a limit cycle x_0 of period T>0 which is nondegenerate in a suitable sense. We then consider the perturbed system obtained by adding to the autonomous system a T-periodic, not necessarily differentiable, term whose amplitude tends to 0 as a small parameter e>0 tends to 0. Assuming the existence of a T-periodic solution x_e of the perturbed system and its convergence to x_0 as e->0, the paper establishes the existence of d_e->0 as e->0 such that \\|x_e(t+d_e)-x_0(t)\\|<=eM for some M>0 and any e>0 sufficiently small."}
{"category": "Math", "title": "On the maximal number of three-term arithmetic progressions in subsets of Z/pZ", "abstract": "Let a be a real number between 0 and 1. Ernie Croot showed that the quantity \\max_A #(3-term arithmetic progressions in A)/p^2, where A ranges over all subsets of Z/pZ of size at most a*p, tends to a limit as p tends to infinity through primes. Writing c(a) for this limit, we show that c(a) = a^2/2 provided that a is smaller than some absolute constant. In fact we prove rather more, establishing a structure theorem for sets having the maximal number of 3-term progressions amongst all subsets of Z/pZ of cardinality m, provided that m < c*p."}
{"category": "Math", "title": "Positive definite collections of disks", "abstract": "Let $Q(z,w)=-\\prod_{k=1}^n [(z-a_k)(\\bar{w}-\\bar{a}_k)-R_k^2]$. M. Putinar and B. Gustafsson proved recently that the matrix $Q(a_i,a_j)$, $1\\leq i,j\\leq n$, is positive definite if disks $|z-a_i|<R_i$ form a disjoint collection. We extend this result on symmetric collections of discs with overlapping. More precisely, we show that in the case when the nodes $a_j$ are situated at the vertices of a regular $n$-gon inscribed in the unit circle and $\\forall i: R_i\\equiv R$, the matrix $Q(a_i,a_j)$ is positive definite if and only if $R<\\rho_n$, where $z=2\\rho_n^2-1$ is the smallest $\\ne-1$ zero of the Jacobi polynomial $\\mathcal{P}^{n-2\\nu,-1}_\\nu(z)$, $\\nu=[n/2]$."}
{"category": "Math", "title": "Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator", "abstract": "In this paper we study the existence, uniqueness and asymptotic stability of the periodic solutions for a Lipschitz system with a small right hand side. Classical hypotheses in the periodic case of second Bogolyubov's theorem imply our ones. By means of the results established we construct the curves of dependence of the amplitude of asymptotically stable $2\\pi$--periodic solutions of the nonsmooth van der Pol oscillator on the detuning parameter and the amplitude of the perturbation. After, we compare the resonance curves obtained, with the resonance curves of the classical van der Pol oscillator which were first constructed by Andronov and Witt."}
{"category": "Math", "title": "Exchange moves and Fiedler polynomial", "abstract": "In order to obtain a Markov theorem without stabilization, Birman and Menasco introduced the notion of exchange related braids. In this paper I study the way the Fiedler polynomial distinguishes conjugacy classes of some particular braided knots. I introduce the Kauffman bracket in the solid torus. Its Taylor expansion give finite type invariants similar to the Fiedler polynomial. I investigate how these invariants distinguish exchange related braids."}
{"category": "Math", "title": "Embedding coproducts of partition lattices", "abstract": "We prove that the lattice Eq(X) of all equivalence relations on an infinite set X contains, as a 0,1-sublattice, the 0-coproduct of two copies of itself, thus answering a question by G.M. Bergman. Hence, by using methods initiated by de Bruijn and further developed by Bergman, we obtain that Eq(X) also contains, as a sublattice, the coproduct of 2^{card(X)} copies of itself."}
{"category": "Math", "title": "Generalized Descents and Normality", "abstract": "We use Janson's dependency criterion to prove that the distribution of $d$-descents of permutations of length $n$ converge to a normal distribution as $n$ goes to infinity. We show that this remains true even if $d$ is allowed to grow with $n$, up to a certain degree."}
{"category": "Math", "title": "Rank of divisors on tropical curves", "abstract": "We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, an elementary proof of the Riemann-Roch theorem for tropical curves, similar to the recent proof of the Riemann-Roch theorem for graphs by Baker and Norine, is presented. In addition, a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to the rank of D on the corresponding metric graph is confirmed, and an algorithm for computing the rank of a divisor on a tropical curve is constructed."}
{"category": "Math", "title": "Fun with \"Analysis I\": basic theorems in calculus revisited", "abstract": "This note tries to show that a re-examination of a first course in analysis, using the more sophisticated tools and approaches obtained in later stages, can be a real fun for experts, advanced students, etc. We start by going to the extreme, namely we present two proofs of the Extreme Value Theorem: \"the programmer proof\" that suggests a method (which is practical in down-to-earth settings) to approximate, to any required precision, the extreme values of the given function in a metric space setting, and an abstract space proof (\"the level-set proof\") for semicontinuous functions defined on compact topological spaces. Next, in the intermediate part, we consider the Intermediate Value Theorem, generalize it to a wide class of discontinuous functions, and re-examine the meaning of the intermediate value property. The trek reaches the final frontier when we discuss the Uniform Continuity Theorem, generalize it, re-examine the meaning of uniform continuity, and find the optimal delta of the given epsilon. Have fun!"}
{"category": "Math", "title": "Some Generalizations of Fedorchuk Duality Theorem -- I", "abstract": "Generalizing Duality Theorem of V. V. Fedorchuk, we prove Stone-type duality theorems for the following four categories: all of them have as objects the locally compact Hausdorff spaces, and their morphisms are, respectively, the continuous skeletal maps, the quasi-open perfect maps, the open maps, the open perfect maps. In particular, a Stone-type duality theorem for the category of all compact Hausdorff spaces and all open maps between them is proved. We also obtain equivalence theorems for these four categories. The versions of these theorems for the full subcategories of these categories having as objects all locally compact connected Hausdorff spaces are formulated as well."}
{"category": "Math", "title": "The Number of Periodic Orbits of a Rational Difference Equation", "abstract": "We show how a variant of the Lefschetz Fixed Point Theorem may be used to count the number of periodic orbits for certain rational difference equations."}
{"category": "Math", "title": "A recursion formula for k-Schur functions", "abstract": "The Bernstein operators allow to build recursively the Schur functions. We present a recursion formula for k-Schur functions at t=1 based on combinatorial operators that generalize the Bernstein operators. The recursion leads immediately to a combinatorial interpretation for the expansion coefficients of k-Schur functions at t=1 in terms of homogeneous symmetric functions."}
{"category": "Math", "title": "Quantum characteristic classes and the Hofer metric", "abstract": "Given a closed monotone symplectic manifold $M$, we define certain characteristic cohomology classes of the free loop space $L \\text {Ham}(M, \\omega)$ with values in $QH_* (M)$, and their $S^1$ equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern classes. In particular there is a Whitney sum formula, which gives rise to a graded ring homomorphism from the ring $H_{*} (L\\text {Ham}(M, \\omega), \\mathbb{Q})$, with its Pontryagin product to $QH_{2n+*} (M)$ with its quantum product. As an application we prove an extension of a theorem of McDuff and Slimowitz on minimality in the Hofer metric of a semifree Hamiltonian circle action, to higher dimensional geometry of the loop space $L \\text {Ham}(M, \\omega)$."}
{"category": "Math", "title": "Nilpotent orbits in classical Lie algebras over $\\textbf{F}_{2^n}$ and the Springer correspondence", "abstract": "We give the number of nilpotent orbits in the Lie algebras of orthogonal groups under the adjoint action of the groups over $\\tF_{2^n}$. Let $G$ be an adjoint algebraic group of type $B,C$ or $D$ defined over an algebraically closed field of characteristic 2. We construct the Springer correspondence for the nilpotent variety in the Lie algebra of $G$."}
{"category": "Math", "title": "Algebraic shifting of strongly edge decomposable spheres", "abstract": "Recently, Nevo introduced the notion of strongly edge decomposable spheres. In this paper, we characterize the algebraic shifted complex of those spheres. Algebraically, this result yields the characterization of the generic initial ideal of the Stanley--Reisner ideal of Gorenstein* complexes having the strong Lefschetz property in characteristic 0."}
{"category": "Math", "title": "A note on lattices in semi-stable representations", "abstract": "Let p>2 be a prime, K a finite extension over Q_p and G :=Gal(\\bar K/K). We extend Kisin's theory on \\phi-modules of finite E(u)-height to give a new classification of G-stable Z_p-lattices in semi-stable representations"}
{"category": "Math", "title": "Some results on Gaussian Besov-Lipschitz spaces and Gaussian Triebel-Lizorkin spaces", "abstract": "In this paper we define Besov-Lipschitz and Triebel-Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein-Uhlenbeck and the Poisson-Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces $L^p_\\alpha(\\gamma_d)$ are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusions semigroups."}
{"category": "Math", "title": "Hausdorff dimension of the set of singular pairs", "abstract": "In this paper we show that the Hausdorff dimension of the set of singular pairs is 4/3. We also show that the action of diag(e^t,e^t,e^{-2t}) on SL(3,R)/SL(3,Z) admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates, answering a question of A.N. Starkov. As a by-product of the analysis, we obtain a higher dimensional generalisation of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the techniques used to compute Hausdorff dimension, we show that the set of real numbers with divergent partial quotients has Hausdorff dimension 1/2."}
{"category": "Math", "title": "On Polar Legendre Polynomials", "abstract": "We introduce a new class of polynomials $\\{P_{n}\\}$, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with $n+1$ unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a differential operator and a discrete-continuous Sobolev type inner product."}
{"category": "Math", "title": "Piecewise linear density estimation for sampled data", "abstract": "Nonparametric density estimation is considered for a discretely observed stationary continuous-time process. For each of three given time sampling procedures either random or deterministic, we establish that histograms and frequency polygons can reach the same optimal $L_{2}$-rates as in the independent and identically distributed case. Moreover, thanks to a suitable \"high frequency\" sampling design, these rates are derived together with a minimized time of observation depending on the regularity of sample paths."}
{"category": "Math", "title": "A class of finite simple Bol loops of exponent 2", "abstract": "In this paper we give an infinite class of finite simple right Bol loops of exponent 2. The right multiplication group of these loops is an extension of an elementary Abelian 2-group by $S_5$. The construction uses the description of the structure of such loops given by M. Aschbacher. These results answer some questions of M. Aschbacher."}
{"category": "Math", "title": "On harmonic quasiconformal quasi-isometries", "abstract": "The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics."}
{"category": "Math", "title": "A model for the orbifold Chow ring of weighted projective spaces", "abstract": "We construct an isomorphism of graded Frobenius algebras between the orbifold Chow ring of weighted projective spaces and graded algebras of groups of roots of the unity."}
{"category": "Math", "title": "Classification problems of toric manifolds via topology", "abstract": "We propose some problems on the classification of toric manifolds from the viewpoint of topology and survey related results."}
{"category": "Math", "title": "Scalar Curvature Estimates by Parallel Alternating Torsion", "abstract": "We generalize Llarull's scalar curvature comparison to Riemannian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on 2-vectors. As a byproduct, we show that Euler number and signature of such manifolds are determined by their global holonomy representation. Our result holds in particular for all quotients of compact Lie groups of equal rank, equipped with a normal homogeneous metric. We also correct a mistake in the treatment of odd-dimensional spaces in arXiv:math/0010199 and arXiv:0705.0500"}
{"category": "Math", "title": "The isomorphism conjecture in L-theory: graphs of groups", "abstract": "We study the Fibered Isomorphism Conjecture of Farrell and Jones in L-theory for groups acting on trees. In several cases we prove the conjecture. This includes wreath products of abelian groups and free metabelian groups. We also deduce the conjecture in pseudoisotopy theory for these groups. Finally in B of Theorem 1.1 we prove the L-theory version of [[7], Theorem 1.2]."}
{"category": "Math", "title": "Flows of Spin(7)-structures", "abstract": "We consider flows of Spin(7)-structures. We use local coordinates to describe the torsion tensor of a Spin(7)-structure and derive the evolution equations for a general flow of a Spin(7)-structure on an 8-manifold M. Specifically, we compute the evolution of the metric and the torsion tensor. We also give an explicit description of the decomposition of the space of forms on a manifold with Spin(7)-structure, and derive an analogue of the second Bianchi identity in Spin(7)-geometry. This identity yields an explicit formula for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion."}
{"category": "Math", "title": "Braided differential structure on Weyl groups, quadratic algebras and elliptic functions", "abstract": "We discuss a class of generalized divided difference operators which give rise to a representation of Nichols-Woronowicz algebras associated to Weyl groups. For the root system of type $A,$ we also study the condition for the deformations of the Fomin-Kirillov quadratic algebra, which is a quadratic lift of the Nichols-Woronowicz algebra, to admit a representation given by generalized divided difference operators. The relations satisfied by the mutually commuting elements called Dunkl elements in the deformed Fomin-Kirillov algebra are determined. The Dunkl elements correspond to the truncated elliptic Dunkl operators via the representation given by the generalized divided difference operators."}
{"category": "Math", "title": "$C^*$-pseudo-Kac systems and duality for coactions of concrete Hopf $C^*$-bimodules", "abstract": "We study coactions of concrete Hopf $C^{*}$-bimodules in the framework of (weak) $C^{*}$-pseudo-Kac systems, define reduced crossed products and dual coactions, and prove an analogue of Baaj-Skandalis duality."}
{"category": "Math", "title": "Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations", "abstract": "We present an alpha-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We show that initially smooth self-avoiding vortex sheet remains smooth for all times under the alpha-regularized dynamics, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure."}
{"category": "Math", "title": "Periodic solutions of periodically perturbed planar autonomous systems: A topological approach", "abstract": "Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x_0 of least period T_0>0 when it is perturbed by a small parameter, T_1-periodic, perturbation. In the case when T_0/T_1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation e>0 sufficiently small, the existence of klT_0-periodic solutions x_e of the perturbed system which converge to the trajectory x_1 of the limit cycle as e->0. Moreover, we state conditions under which T=klT_0 is the least period of the periodic solutions x_e. We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when T_0/T_1 is an irrational number we show the nonexistence, whenever T>0 and e>0, of T-periodic solutions x_e of the perturbed system converging to x_1. The employed methods are based on the topological degree theory."}
{"category": "Math", "title": "Contracting Endomorphisms and Gorenstein Modules", "abstract": "We characterize Gorenstein modules over those local rings that admit a finite contracting endomorphism."}
{"category": "Math", "title": "Positve Entropy Geodesic Flows on Nilmanifolds", "abstract": "Let T be the nilpotent group of 4 x 4 real upper triangular matrices. In this note we show that the Euler equations of certain left-invariant riemannian metrics on T have a horseshoe. We also show, with the aid of a numerical computation of a Melnikov-type integral, that the Euler equations of the sub-riemannian Carnot metric on T has a horseshoe. This sharpens an earlier result of Montgomery, Shapiro and Stolin who had shown that the equations are algebraically non-integrable."}
{"category": "Math", "title": "An asymptotic-numerical approach for examining global solutions to an ordinary differential equation", "abstract": "Purely numerical methods do not always provide an accurate way to find all the global solutions to nonlinear ODE on infinite intervals. For example, finite-difference methods fail to capture the asymptotic behavior of solutions, which might be critical for ensuring global existence. We first show, by way of a detailed example, how asymptotic information alone provides significant insight into the structure of global solutions to a nonlinear ODE. Then we propose a method for providing this missing asymptotic data to a numerical solver, and show how the combined approach provides more detailed results than either method alone."}
{"category": "Math", "title": "Consequences of the Gross/Zagier formulae: Stability of average L-values, subconvexity, and non-vanishing mod p", "abstract": "In this paper we investigate some consequences of the Gross/Zagier types of formulae which were introduced by Gross and Zagier, and were then generalized in various directions by Hatcher, Zhang, Kudla and several other people. Working in the classical context of central values of L-series of holomorphic forms of prime level, we deduce an exact average formula for suitable twists of such L-values, with a relation to the class number of associated imaginary quadratic fieds, thereby strengthening a result of Duke. One also obtains a stability result, as well as subconvexity (in this setting), and certian non-vanishing assertions. This article is dedicated to the memory of Serge Lang."}
{"category": "Math", "title": "Binomial Coefficients and the Distribution of the Primes", "abstract": "Let omega(n) be the number of distinct prime factors dividing n and m > n natural numbers. We calculate a formula showing which prime numbers in which intervals divide a given binomial coefficient. From this formula we get an identity omega(binom(nk)(mk))=sum_i (pi(k/b(i))- pi(k/a(i))) + O(sqrt(k)). Erdoes mentioned that omega(binom(nk)(mk))= log n^n/(m^m (n-m)^(n-m)) k/log k + o(k/log k). As an application of the above identities, we conclude some well-known facts about the distribution of the primes and deduce for all natural numbers k an expression (also well-known) log k = sum_i a_k(i) which generalizes log 2 = sum_i^(infty) (-1)^(j+1) / j."}
{"category": "Math", "title": "A continuation principle for a class of periodically perturbed autonomous systems", "abstract": "In this paper we evaluate the topological index of periodic solutions otained via the Malkin-Loud bifurcation result. Incidentally, we do not assume that the perturbation is differentiale."}
{"category": "Math", "title": "The Dirichlet Markov Ensemble", "abstract": "We equip the polytope of $n\\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $\\mathbb{R}^{n^2}$. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean $(1/n,...,1/n)$. We show that if $\\bM$ is such a random matrix, then the empirical distribution built from the singular values of$\\sqrt{n} \\bM$ tends as $n\\to\\infty$ to a Wigner quarter--circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of $\\sqrt{n} \\bM$ tends as $n\\to\\infty$ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of $\\bM$ is of order $1-1/\\sqrt{n}$ when $n$ is large."}
{"category": "Math", "title": "Periodic bifurcation from families of periodic solutions for semilinear differential equations with Lipschitzian perturbations in Banach spaces", "abstract": "Let A:D(A)\\to E be an infinitesimal generator either of an analytic compact semigroup or of a contractive C_0-semigroup of linear operators acting in a Banach space E. In this paper we give both necessary and sufficient conditions for bifurcation of $T$-periodic solutions for the equation x'=Ax+f(t,x)+e g(t,x,e) from a k-parameterized family of T-periodic solutions of the unperturbed equation corresponding to e=0. We show that by means of a suitable modification of the classical Mel'nikov approach we can construct a bifurcation function and to formulate the conditions for the existence of bifurcation in terms of the topological index of the bifurcation function. To do this, since the perturbation term g is only Lipschitzian we need to extend the classical Lyapunov-Schmidt reduction to the present nonsmooth case."}
{"category": "Math", "title": "Regularity theory for fully nonlinear integro-differential equations", "abstract": "We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump L\\`evy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior $C^{1,\\alpha}$ regularity for general fully nonlinear integro-differential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations."}
{"category": "Math", "title": "Synchronization problems for unidirectional feedback coupled nonlinear systems", "abstract": "In this paper we consider three different synchronization problems consisting in designing a nonlinear feedback unidirectional coupling term for two (possibly chaotic) dynamical systems in order to drive the trajectories of one of them, the slave system, to a reference trajectory or to a prescribed neighborhood of the reference trajectory of the second dynamical system: the master system. If the slave system is chaotic then synchronization can be viewed as the control of chaos; namely the coupling term allows to suppress the chaotic motion by driving the chaotic system to a prescribed reference trajectory. Assuming that the entire vector field representing the velocity of the state can be modified, three different methods to define the nonlinear feedback synchronizing controller are proposed: one for each of the treated problems. These methods are based on results from the small parameter perturbation theory of autonomous systems having a limit cycle, from nonsmooth analysis and from the singular perturbation theory respectively. Simulations to illustrate the effectiveness of the obtained results are also presented."}
{"category": "Math", "title": "Poincare index and periodic solutions of perturbed autonomous systems", "abstract": "The basic tool of classical results by Malkin and Melnikov on bifurcation of periodic solutions from nondegenerate cycles of autonomous systems with periodic perturbations is an implicit function theorem. In this paper the Poincare index is used to avoid the requirement of nondegeneracity for the unperturbed cycles and to provide additional geometrical properties of periodic solutions of the perturbed system."}
{"category": "Math", "title": "On algebraic time-derivative estimation and deadbeat state reconstruction", "abstract": "This note places into perspective the so-called algebraic time-derivative estimation method recently introduced by Fliess and co-authors with standard results from linear state-space theory for control systems. In particular, it is shown that the algebraic method can in a sense be seen as a special case of deadbeat state estimation based on the reconstructibility Gramian of the considered system."}
{"category": "Math", "title": "Irrational Stable Commutator Length in Finitely Presented Groups", "abstract": "We give examples of finitely presented groups containing elements with irrational (in fact, transcendental) stable commutator length, thus answering in the negative a question of M. Gromov. Our examples come from 1-dimensional dynamics, and are related to the generalized Thompson groups studied by M. Stein, I. Liousse and others."}
{"category": "Math", "title": "Ramsey numbers of sparse hypergraphs", "abstract": "We give a short proof that any k-uniform hypergraph H on n vertices with bounded degree \\Delta has Ramsey number at most c(\\Delta, k)n, for an appropriate constant c(\\Delta, k). This result was recently proved by several authors, but those proofs are all based on applications of the hypergraph regularity method. Here we give a much simpler, self-contained proof which uses new techniques developed recently by the authors together with an argument of Kostochka and R\\\"odl. Moreover, our method demonstrates that, for k \\geq 4, c(\\Delta, k) \\leq 2^{2^{\\Ddots^{2^{c \\Delta}}}}, where the tower is of height k and the constant c depends on k. It significantly improves on the Ackermann-type upper bound that arises from the regularity proofs, and we present a construction which shows that, at least in certain cases, this bound is not far from best possible. Our methods also allows us to prove quite sharp results on the Ramsey number of hypergraphs with at most m edges."}
{"category": "Math", "title": "Extended TQFT's and Quantum Gravity", "abstract": "This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2-functor Z: nCob_2 -> 2Vect, by analogy with the description of a TQFT as a functor Z: nCob -> Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topological gauge theory, the Dijkgraaf-Witten model. To give this definition rigorously, we first define a bicategory of cobordisms between cobordisms. We also give some explicit description of a higher-categorical version of Vect, denoted 2Vect, a bicategory of \"2-vector spaces\". Along the way, we prove several results showing how to construct 2-vector spaces of \"Vect-valued presheaves\" on certain kinds of groupoids. In particular, we use the case when these are groupoids whose objects are connections, and whose morphisms are gauge transformations, on the manifolds on which the extended TQFT is to be defined. On cobordisms between these manifolds, we show how a construction of ``pullback and pushforward'' of presheaves gives both the morphisms and 2-morphisms in 2Vect for the extended TQFT, and that these satisfy the axioms for a weak 2-functor. Finally, we discuss the motivation for this research in terms of Quantum Gravity. If the results can be extended from a finite group G to a Lie group, then for some choices of G this theory will recover an existing theory of Euclidean quantum gravity in 3 dimensions. We suggest extensions of these ideas which may be useful to further this connection and apply it in higher dimensions."}
{"category": "Math", "title": "Developments in finite Phan theory", "abstract": "This is a final report on finite Phan theory, a project that has been concerned with a revision and generalisation of Phan's presentation results of twisted Chevalley groups over finite fields with simply laced diagrams by unitary subgroups of rank one and two. Finite Phan theory has been initiated on request by Richard Lyons and Ronald Solomon in order to support their revision of the classification of the finite simple groups."}
{"category": "Math", "title": "On a two variable class of Bernstein-Szego measures", "abstract": "The one variable Bernstein-Szego theory for orthogonal polynomials on the real line is extended to a class of two variable measures. The polynomials orthonormal in the total degree ordering and the lexicographical ordering are constructed and their recurrence coefficients discussed."}
{"category": "Math", "title": "Notes on a paper of Tyagi and Holm: A new integral representation for the Riemann Zeta function", "abstract": "It is shown that a new series representation of Riemann s Zeta function obtained by Tyagi and Holm leads to an interesting new recursion for Bernoulli numbers of even index as well as new representations of, and infinite series involving, Zeta functions of special (integer) argument."}
{"category": "Math", "title": "Characterization of the matrix whose norm is determined by its action on decreasing sequences:The exceptional cases", "abstract": "Let $A=(a_{j,k})_{j,k \\ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\\|A\\|_{\\ell_p,\\ell_q}$ are determined by their actions on non-negative decreasing sequences, where one of $p$ and $q$ is 1 or $\\infty$. The conditions forcing on $A$ are sufficient and they are also necessary for non-negative finite matrices."}
{"category": "Math", "title": "A modular absolute bound condition for primitive association schemes", "abstract": "The well-known absolute bound condition for a primitive symmetric association scheme (X,S) gives an upper bound for |X| in terms of |S| and the minimal non-principal multiplicity of the scheme. In this paper we prove another upper bounds for |X| for an arbitrary primitive scheme (X,S). They do not depend on |S| but depend on some invariants of its adjacency algebra KS where K is an algebraic number field or a finite field."}
{"category": "Math", "title": "The basis digraphs of p-schemes", "abstract": "It is proved that association schemes with bipartite basis graphs are exactly 2-schemes. This result follows from a characterization of p-schemes for an arbitrary prime p in terms of basis digraphs."}
{"category": "Math", "title": "Combinatorial Stokes formulas via minimal resolutions", "abstract": "We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Z_k of order k. We then demonstrate how such a chain map induces a \"Z_k-combinatorial Stokes theorem\", which in turn implies \"Dold's theorem\" that there is no equivariant map from an n-connected to an n-dimensional free Z_k-complex. Thus we build a combinatorial access road to problems in combinatorics and discrete geometry that have previously been treated with methods from equivariant topology. The special case k=2 for this is classical; it involves Tucker's (1949) combinatorial lemma which implies the Borsuk-Ulam theorem, its proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula of Fan (1967), and Meunier's work (2006)."}
{"category": "Math", "title": "Influence of a small perturbation on Poincare-Andronov operators with not well defined topological degree", "abstract": "Let $P_e\\in C^0(R^n,R^n)$ be the Poincare-Andronov operator over period $T>0$ of the $T$-periodically perturbed autonomous system $x'=f(x)+e g(t,x,e),$ where $e>0$ is small. Assuming that for $e=0$ this system has a $T$-periodic limit cycle $x_0$ we evaluate the topological degree $d(I-P_e,U)$ of $I-P_e$ on an open bounded set $U$ whose boundary contains $x_0([0,T])$ and does not contain other fixed points of $P_0.$ We give an explicit formula connecting $d(I-P_e,U)$ with topological indexes of zeros of the associated Malkin's bifurcation function. The goal of the paper is to prove Mawhin's conjecture which claims that $d(I-P_e,U)$ can be any integer in spite of the fact that the measure of the set of fixed points of $P_0$ on $\\partial U$ is zero."}
{"category": "Math", "title": "Periodic solutions for a class of singulary perturbated systems", "abstract": "In this paper we provide conditions to ensure the existence, for $e>0$ sufficiently small, of periodic solutions of given period $T>0$ in a prescribed domain $U$ for a class of singularly perturbed first order differential systems. Here $e>0$ is the perturbation parameter. Our approach, based on the topological degree theory and the averaging theory, permits to weaken the conditions in [K.R. Schneider, Vibrational control of singularly perturbed systems, in \"Lectures Notes in Control and Information Science\", 259, 397-408, Springer, London, 2001, Theorem 2] under which the existence of periodic solutions is proved."}
{"category": "Math", "title": "Small parameter perturbations of nonlinear periodic systems", "abstract": "In this paper we consider a class of nonlinear periodic differential systems perturbed by two nonlinear periodic terms with multiplicative different powers of a small parameter $e>0$. For such a class of systems we provide conditions which guarantee the existence of periodic solutions of given period $T>0$. These conditions are expressed in terms of the behaviour on the boundary of an open bounded set $U$ of $R^n$ of the solutions of suitably defined linearized systems. The approach is based on the classical theory of the topological degree for compact vector fields. An application to the existence of periodic solutions to the van der Pol equation is also presented."}
{"category": "Math", "title": "A new approach in the theory of ordinary differential equations with small parameter", "abstract": "A topological degree based averaging principle has been proposed by J. Mawhin in his PhD thesis [J. Mawhin, Le Probleme des Solutions Periodiques en Mecanique non Lineaire, These de doctorat en sciences, Universite de Liege, 1969]. In the present paper we extend this result for the case when the unperturbed system is nonlinear but possesses a set of T-periodic solutions whose initial conditions constitute a boundary of some open set."}
{"category": "Math", "title": "Methods of topological degree theory in Malkin I. G. - Melnikov V. K.'s problems for periodically perturbed systems", "abstract": "A topological degree based averaging principle has been proposed by J. Mawhin in his PhD thesis [J. Mawhin, Le Probleme des Solutions Periodiques en Mecanique non Lineaire, These de doctorat en sciences, Universite de Liege, 1969]. In his thesis the author gives analogous topological degree versions of classical bifurcation results due to I.G. Malkin and V.K. Melnikov, namely the conditions for bifurcation of periodic solutions from families are expressed in term of the topological degree of the bifurcation function. Moreover, the topological index of bifurcated periodic solutions is evaluated over that degree. A third part of the thesis is devoted to the rate the bifurcated periodic solutions converge when the perturbation vanishes. The differentiability of perturbed systems is not assumed."}
{"category": "Math", "title": "Higher Order Normalizations in the Generalized Photogravitational Restricted Three Body Problem with Poynting-Robertson Drag", "abstract": "Higher order normalizations are performed in the generalized photogravitational restricted three body problem with Poynting-Robertson drag. In this problem we have taken bigger primary as a source of radiation and smaller primary as an oblate spheroid. Whittaker method is used to transform the second order part of the Hamiltonian into the normal form. We have also performed Birkhoff's normalization of the Hamiltonian. For this we have tilized Henrard's method and expanded the coordinates of the infinitesimal body in double D'Alembert series. We have found the values of first and second order components. They are affected by radiation pressure, oblateness and P-R drag. Finally we obtained the third order part of the Hamiltonian zero. Keywords:Higher Order Normalization, Generalized Photogravitational, RTBP,P-R drag"}
{"category": "Math", "title": "Domain of attraction of asymptotically stable periodic solutions obtained via averaging principle", "abstract": "In this paper we propose an approach to evaluate the domain of attraction of asymptotically stable periodic solutions obtained via averaging principle (second Bogolubov's theorem or Mel'nikov's method). We discuss also how this result is extended in the case when the right hand part is nonsmooth."}
{"category": "Math", "title": "Commutativity and ideals in algebraic crossed products", "abstract": "We investigate properties of commutative subrings and ideals in non-commutative algebraic crossed products for actions by arbitrary groups. A description of the commutant of the base coefficient subring in the crossed product ring is given. Conditions for commutativity and maximal commutativity of the commutant of the base subring are provided in terms of the action as well as in terms of the intersection of ideals in the crossed product ring with the base subring, specially taking into account both the case of base rings without non-trivial zero-divisors and the case of base rings with non-trivial zero-divisors."}
{"category": "Math", "title": "Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index", "abstract": "We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, the asymptotic Maslov indices of periodic orbits are dense in the positive half line. Furthermore, if the Hamiltonian is the Fenchel dual of an electro-magnetic Lagrangian, every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of Mather's theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector."}
{"category": "Math", "title": "Casimir operators of Lie algebras with a nilpotent radical", "abstract": "We show that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. We give a different proof of this fact in the special and well-known case where the radical is abelian."}
{"category": "Math", "title": "A note on quantum 3-manifold invariants and hyperbolic volume", "abstract": "We investigate the conjectural relations between the Reshetikhin-Turaev-Witten quantum SU(2) invariants and the volume of hyperbolic 3-manifolds. Given a finite set of sufficiently large positive integers, say J, we construct examples of closed hyperbolic 3-manifolds with the same invariants at all levels in J and different volume."}
{"category": "Math", "title": "Equivalent metrics and compactifications", "abstract": "Let (X,d) be a metric space and m\\in X. Suppose that \\phi:X\\times X\\to\\mathbold{R} is a nonnegative symmetric function. We define a metric d^{\\phi,m} on X which is equivalent to d. If d^{\\phi,m} is totally bounded, its completion is a compactification of (X,d). As examples, we construct two compactifications of (\\mathhbold{R}^s,d_E), where d_E is the Euclidean metric and s\\geq 2."}
{"category": "Math", "title": "Permutative categories, multicategories, and algebraic K-theory", "abstract": "We show that the $K$-theory construction of arXiv:math/0403403, which preserves multiplicative structure, extends to a symmetric monoidal closed bicomplete source category, with the multiplicative structure still preserved. The source category of arXiv:math/0403403, whose objects are permutative categories, maps fully and faithfully to the new source category, whose objects are (based) multicategories."}
{"category": "Math", "title": "On rational blow-downs in Heegaard-Floer homology", "abstract": "Motivated by a result of L.P. Roberts on rational blow-downs in Heegaard-Floer homology, we study such operations along 3-manifolds that arise as branched double covers of $S^{3}$ along several non-alternating, slice knots."}
{"category": "Math", "title": "Generalized Dolbeault sequences in parabolic geometry", "abstract": "In this paper, we show the existence of a sequence of invariant differential operators on a particular homogeneous model $G/P$ of a Cartan geometry. The first operator in this sequence can be locally identified with the Dirac operator in $k$ Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\\sum_j e_j\\cdot \\partial_{ij}: C^\\infty((\\R^n)^k,\\S)\\to C^\\infty((\\R^n)^k,\\S)$. We describe the structure of these sequences in case the dimension $n$ is odd. It follows from the construction that all these operators are invariant with respect to the action of the group $G$. These results are obtained by constructing homomorphisms of generalized Verma modules, what are purely algebraic objects."}
{"category": "Math", "title": "Counting Nodal Lines Which Touch the Boundary of an Analytic Domain", "abstract": "We consider the zeros on the boundary $\\partial \\Omega$ of a Neumann eigenfunction $\\phi_{\\lambda}$ of a real analytic plane domain $\\Omega$. We prove that the number of its boundary zeros is $O (\\lambda)$ where $-\\Delta \\phi_{\\lambda} = \\lambda^2 \\phi_{\\lambda}$. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is $O(\\lambda)$. It follows that the number of nodal lines of $\\phi_{\\lambda}$ (components of the nodal set) which touch the boundary is of order $\\lambda$. This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal components in the interior. More generally, the results are proved for piecewise analytic domains."}
{"category": "Math", "title": "Representing Dehn twists with branched coverings", "abstract": "We show that any homologically non-trivial Dehn twist of a compact surface F with boundary is the lifting of a half-twist in the braid group B_n, with respect to a suitable branched covering p : F -> B^2. In particular, we allow the surface to have disconnected boundary. As a consequence, any allowable Lefschetz fibration on B^2 is a branched covering of B^2 x B^2."}
{"category": "Math", "title": "Fuzzy almost quadratic functions", "abstract": "We approximate a fuzzy almost quadratic function by a quadratic function in a fuzzy sense. More precisely, we establish a fuzzy Hyers--Ulam--Rassias stability of the quadratic functional equation $f(x+y)+f(x-y)=2f(x)+2f(y)$. Our result can be regarded as a generalization of the stability phenomenon in the framework of normed spaces. We also prove a generalized version of fuzzy stability of the Pexiderized quadratic functional equation $f(x+y)+f(x-y)=2g(x)+2h(y)$."}
{"category": "Math", "title": "A Mazur--Ulam theorem in non-Archimedean normed spaces", "abstract": "The classical Mazur--Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur--Ulam theorem in the non-Archimedean strictly convex normed spaces."}
{"category": "Math", "title": "On Isomorphism Classes and Invariants of Low Dimensional Complex Filiform Leibniz Algebras (PART 1)", "abstract": "The paper aims to investigate the classification problem of low dimensional complex none Lie filiform Leibniz algebras. There are two sources to get classification of filiform Leibniz algebras. The first of them is the naturally graded none Lie filiform Leibniz algebras and the another one is the naturally graded filiform Lie algebras. Here we do consider Leibniz algebras appearing from the naturally graded none Lie filiform Leibniz algebras. This class can be splited into two subclasses. However, isomorphisms within each class there were not investigated. Before U.D.Bekbaev and I.S.Rakhimov suggested an approach to the isomorphism problem in terms of invariants. This paper presents an implementation of their result in low dimensional cases. Here we give the complete classification of complex none Lie filiform Leibniz algebras in dimensions at most 8 from the first class of the above mentioned result and give a hypothetic formula for the number of isomorphism classes in finite dimensional case."}
{"category": "Math", "title": "A canonical bundle formular of projective Lagrangian fibrations", "abstract": "We classify singular fibres of a projective Lagrangian fibration over codimension one points. As an application, we obtain a canonical bundle formula for a projective Lagrangian fibration over a smooth manifold."}
{"category": "Math", "title": "On regular configurations and disjoint cycles in shift graphs", "abstract": "Configurations are necklaces with prescribed numbers of red and black beads. Among all possible configurations, the regular one plays an important role in many applications. In this paper, several aspects of regular configurations are discussed, including construction, uniqueness, symmetry group and the link with balanced words. Another model of configurations is the polygons formed by a given number of sides of two different lengths. In this context, regular configurations are used to obtain a lower bound for the cycles packing number of shift graphs, a subclass of the directed circulant graphs."}
{"category": "Math", "title": "Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case", "abstract": "Let $G\\subset \\O(n)$ be a compact group of isometries acting on $n$-dimensional Euclidean space $\\R^n$, and ${\\bf{X}}$ a bounded domain in $\\R^n$ which is transformed into itself under the action of $G$. Consider a symmetric, classical pseudodifferential operator $A_0$ in $\\L^2(\\R^n)$ that commutes with the regular representation of $G$, and assume that it is elliptic on $\\bf{X}$. We show that the spectrum of the Friedrichs extension $A$ of the operator $\\mathrm{res} \\circ A_0 \\circ \\mathrm{ext}: \\CT({\\bf{X}}) \\to \\L^2({\\bf{X}})$ is discrete, and using the method of the stationary phase, we derive asymptotics for the number $N_\\chi(\\lambda)$ of eigenvalues of $A$ equal or less than $\\lambda$ and with eigenfunctions in the $\\chi$-isotypic component of $\\L^2({\\bf{X}})$ as $\\lambda \\to \\infty$, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem."}
{"category": "Math", "title": "Existence and mutiplicity of solutions to elliptic equations of fourth order on compact manifolds", "abstract": "This paper deals with a fourth order elliptic equation on compact Riemannian manifolds.We establish the existence of solutions to the equation with critical Sobolev growth which is the subject of the first theorem. In the second one, we prove the multiplicity of solutions in the subcritical case."}
{"category": "Math", "title": "Tests \\`a la Hurewicz dans le plan", "abstract": "We give, for some Borel sets of a product of two Polish spaces, including the Borel sets with countable sections, a Hurewicz-like characterization of those which cannot become a transfinite difference of open sets by changing the two Polish topologies."}
{"category": "Math", "title": "Uniformisations partielles et crit\\`eres \\`a la Hurewicz dans le plan", "abstract": "We give characterizations of the Borel sets potentially in some Wadge class, among the Borel sets with countable vertical sections of a product of two Polish spaces. To do this, we use some partial uniformization results."}
{"category": "Math", "title": "Classes de Wadge potentielles des bor\\'eliens \\`a coupes d\\'enombrables", "abstract": "We give, for each non self-dual Wadge class C contained in the class of the Gdelta sets, a characterization of Borel sets which are not potentially in C, among Borel sets with countable vertical sections; to do this, we use results of partial uniformization."}
{"category": "Math", "title": "Classes de Wadge potentielles et th\\'eor\\`emes d'uniformisation partielle", "abstract": "We want to give a construction as simple as possible of a Borel subset of a product of two Polish spaces. This introduces the notion of potential Wadge class. Among other things, we study the non-potentially closed sets, by proving Hurewicz-like results. This leads to partial uniformization theorems, on big sets, in the sense of cardinality or Baire category."}
{"category": "Math", "title": "Complexit\\'e des bor\\'eliens \\`a coupes d\\'enombrables", "abstract": "We give, for each level of complexity L, a Hurewicz-like characterization of the Borel subsets with countable sections of a product of two Polish spaces that cannot become in L by changing the two Polish topologies."}
{"category": "Math", "title": "Generalized Taylor's Theorem", "abstract": "The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given multiplicities. Taken together with a simple expression for the remainder, this theorem becomes a powerful tool for approximation and interpolation in numerical analysis. We also have a corresponding theorem for rational approximation."}
{"category": "Math", "title": "Two Digit Theorems", "abstract": "We prove that if p is a prime with a primitive root 2 then S_p(2^p)=p and give a sufficient condition for an equality of kind S_p(2^p)=+or-p."}
{"category": "Math", "title": "Applications of integral transforms in fractional diffusion processes", "abstract": "The fundamental solution (Green function) for the Cauchy problem of the space-time fractional diffusion equation is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. Then, by using the Mellin transform, a general representation of the Green function in terms of Mellin-Barnes integrals in the complex plane is derived. This allows us to obtain a suitable computational form of the Green function in the space-time domain and to analyse its probability interpretation."}
{"category": "Math", "title": "On minimal non-potentially closed subsets of the plane", "abstract": "We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions. We show the existence of a perfect antichain made of minimal sets among non-potentially closed sets. We apply this result to graphs, quasi-orders and partial orders. We also give a non-potentially closed set minimum for another notion of comparison. Finally, we show that we cannot have injectivity in the Kechris-Solecki-Todorcevic dichotomy about analytic graphs."}
{"category": "Math", "title": "Omega-powers and descriptive set theory", "abstract": "We study the sets of the infinite sentences constructible with a dictionary over a finite alphabet, from the viewpoint of descriptive set theory. Among other things, this gives some true co-analytic sets. The case where the dictionary is finite is studied and gives a natural example of a set at the level omega of the Wadge hierarchy."}
{"category": "Math", "title": "Hurewicz-like tests for Borel subsets of the plane", "abstract": "Let xi be a non-null countable ordinal. We study the Borel subsets of the plane that can be made $\\bormxi$ by refining the Polish topology on the real line. These sets are called potentially $\\bormxi$. We give a Hurewicz-like test to recognize potentially $\\bormxi$ sets."}
{"category": "Math", "title": "How can we recover Baire class one functions?", "abstract": "Let X and Y be separable metrizable spaces, and f:X-->Y be a function. We want to recover f from its values on a small set via a simple algorithm. We show that this is possible if f is Baire class one, and in fact we get a characterization. This leads us to the study of sets of Baire class one functions and to a characterization of the separability of the dual space of an arbitrary Banach space."}
{"category": "Math", "title": "Sharp estimates for maximal operators associated to the wave equation", "abstract": "We give almost sharp conditions under which the maximal operator associated with the wave equation with initial data in Sobolev space H^s(R^n) is bounded from H^s(R^n) to L^q(R^n)."}
{"category": "Math", "title": "PCF and Abelian Groups", "abstract": "We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of aleph_omega-free abelian groups with trivial dual, i.e. no non-trivial homomorphisms to the integers. This relies on investigation of pcf; more specifically, for this we prove that \"almost always\" there are F subseteq lambda^kappa which are quite free and has black boxes. The \"almost always\" means that there are strong restrictions on cardinal arithmetic if the universe fails this, the restrictions are \"everywhere\". Also we replace Abelian groups by R-modules, so in some sense our advantage over earlier results becomes clearer."}
{"category": "Math", "title": "Affinity criterion for the quotient of an algebraic group by a one-dimensional subgroup", "abstract": "In this work we show that the homogeneous space of an affine algebraic group $G$ by a one-dimensional unipotent subgroup $H$ is affine if and only if the subgroup is not contained in any reductive subgroup of $G$."}
{"category": "Math", "title": "On ground fields of arithmetic hyperbolic reflection groups. II", "abstract": "This paper continues arXiv.org:math.AG/0609256 and arXiv:0708.3991 Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions at least 4 are defined, and good explicit bounds of their degrees (over Q) are obtained. This could be important for further classification. Thus, now, an explicit bound of degree of ground fields of arithmetic hyperbolic reflection groups is unknown in dimension 3 only."}
{"category": "Math", "title": "A few remarks about linear operators and disconnected open sets in the plane", "abstract": "Some aspects of analysis on disconnected open subsets of the plane with connected fractal boundary are discussed."}
{"category": "Math", "title": "Bochner-Kaehler metrics and connections of Ricci type", "abstract": "We apply the results from the article Cahen, Schwachh\\\"ofer: Special symplectic connections, to the case of Bochner-Kaehler metrics. We obtain a (local) classification of these based on the orbit types of the adjoint action in $su(n,1)$. The relation between Sasaki and Bochner-Kaehler metrics in cone and transveral metrics constructions is discussed. The connection of the special symplectic and Weyl connections is outlined. The duality between the Ricci-type and Bochner-Kaehler metrics is shown."}
{"category": "Math", "title": "Dominant K-theory and Integrable highest weight representations of Kac-Moody groups", "abstract": "We give a topological interpretation of the highest weight representations of Kac-Moody groups. Given the unitary form G of a Kac-Moody group (over C), we define a version of equivariant K-theory, K_G on the category of proper G-CW complexes. We then study Kac-Moody groups of compact type in detail (see Section 2 for definitions). In particular, we show that the Grothendieck group of integrable hightest weight representations of a Kac-Moody group G of compact type, maps isomorphically onto K_G^*(EG), where $EG$ is the classifying space of proper G-actions. For the affine case, this agrees very well with recent results of Freed-Hopkins-Teleman. We also explicitly compute K_G^*(EG) for Kac-Moody groups of extended compact type, which includes the Kac-Moody group E_{10}."}
{"category": "Math", "title": "Eriksson's numbers game and finite Coxeter groups", "abstract": "The numbers game is a one-player game played on a finite simple graph with certain ``amplitudes'' assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many authors. In particular, Eriksson connects certain geometric representations of Coxeter groups with games on graphs with certain real number amplitudes. Games played on such graphs are ``E-games.'' Here we investigate various finiteness aspects of E-game play: We extend Eriksson's work relating moves of the game to reduced decompositions of elements of a Coxeter group naturally associated to the game graph. We use Stembridge's theory of fully commutative Coxeter group elements to classify what we call here the ``adjacency-free'' initial positions for finite E-games. We characterize when the positive roots for certain geometric representations of finite Coxeter groups can be obtained from E-game play. Finally, we provide a new Dynkin diagram classification result of E-game graphs meeting a certain finiteness requirement."}
{"category": "Math", "title": "On a Conjecture about the Number of Solutions to Linear Diophantine Equations with a Positive Integer Parameter", "abstract": "Let A(n) be a $k\\times s$ matrix and $m(n)$ be a $k$ dimensional vector, where all entries of A(n) and $m(n)$ are integer-valued polynomials in $n$. Suppose that $$t(m(n)|A(n))=#\\{x\\in\\mathbb{Z}_{+}^{s}\\mid A(n)x=m(n)\\}$$ is finite for each $n\\in \\mathbb{N}$, where $Z_+$ is the set of nonnegative integers. This paper conjectures that $t(m(n)|A(n))$ is an integer-valued quasi-polynomial in $n$ for $n$ sufficiently large and verifies the conjecture in several cases."}
{"category": "Math", "title": "Quality assessment for short oligonucleotide microarray data", "abstract": "Quality of microarray gene expression data has emerged as a new research topic. As in other areas, microarray quality is assessed by comparing suitable numerical summaries across microarrays, so that outliers and trends can be visualized, and poor quality arrays or variable quality sets of arrays can be identified. Since each single array comprises tens or hundreds of thousands of measurements, the challenge is to find numerical summaries which can be used to make accurate quality calls. To this end, several new quality measures are introduced based on probe level and probeset level information, all obtained as a by-product of the low-level analysis algorithms RMA/fitPLM for Affymetrix GeneChips. Quality landscapes spatially localize chip or hybridization problems. Numerical chip quality measures are derived from the distributions of Normalized Unscaled Standard Errors and of Relative Log Expressions. Quality of chip batches is assessed by Residual Scale Factors. These quality assessment measures are demonstrated on a variety of datasets (spike-in experiments, small lab experiments, multi-site studies). They are compared with Affymetrix's individual chip quality report."}
{"category": "Math", "title": "Some Generalizations of Fedorchuk Duality Theorem -- II", "abstract": "As it was shown in the first part of this paper, there exists a duality between the category DSkeLC (introduced there) and the category SkeLC of locally compact Hausdorff spaces and continuous skeletal maps. We describe here the subcategories of the category DSkeLC which are dually equivalent to the following eight categories: all of them have as objects the locally compact Hausdorff spaces and their morphisms are, respectively, the injective (respectively, surjective) continuous skeletal maps, the injective (resp., surjective) open maps, the injective (resp., surjective) skeletal perfect maps, the injective (resp., surjective) open perfect maps. The particular cases of these theorems for the full subcategories of the last four categories having as objects all compact Hausdorff spaces are formulated and proved. The DSkeLC-morphisms which are LCA-embeddings and the dense homeomorphic embeddings are characterized through their dual morphisms. For any locally compact space X, a description of the frame of all open subsets of X in terms of the dual object of X is obtained. It is shown how one can build the dual object of an open subset (respectively, of a regular closed subset) of a locally compact Hausdorff space X directly from the dual object of X. Applying these results, a new description of the ordered set of all, up to equivalence, locally compact Hausdorff extensions of a locally compact Hausdorff space is obtained. Moreover, generalizing de Vries Compactification Theorem, we strengthen the Local Compactification Theorem of Leader. Some other applications are found."}
{"category": "Math", "title": "The spectrum of the Leray transform for convex Reinhardt domains in $\\mathbb C^2$", "abstract": "The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in $\\mathbb C^2$. Our class is self-dual; it contains some domains with less than $C^2$-smooth boundary and also some domains with smooth boundary and degenerate Levi form. $L^2$-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these results are new even for the classical case of smoothly bounded strongly convex Reinhardt domains."}
{"category": "Math", "title": "A local version of Gotzmann's Persistence", "abstract": "Gotzmann's Persistence states that the growth of an arbitrary ideal can be controlled by comparing it to the growth of the lexicographic ideal. This is used, for instance, in finding equations which cut out the Hilbert scheme (of subschemes of $\\mathbf{P}^n$ with fixed Hilbert polynomial) sitting inside an appropriate Grassmannian. We introduce the notion of an {\\it extremal ideal} which extends the notion of the lex ideal to other term orders. We then state and prove a version of Gotzmann's theorem for these ideals, valid in an open subset of a Grassmannian."}
{"category": "Math", "title": "Selfdecomposability and semi-selfdecomposability in subordination of cone-parameter convolution semigroups", "abstract": "Extension of two known facts concerning subordination is made. The first fact is that, in subordination of 1-dimensional Brownian motion with drift, selfdecomposability is inherited from subordinator to subordinated. This is extended to subordination of cone-parameter convolution semigroups. The second fact is that, in subordination of strictly stable cone-parameter convolution semigroups on $\\mathbb{R}^d$, selfdecomposability is inherited from subordinator to subordinated. This is extended to semi-selfdecomposability."}
{"category": "Math", "title": "Invariant chiral differential operators and the W_3 algebra", "abstract": "Attached to a vector space V is a vertex algebra S(V) known as the beta-gamma system or algebra of chiral differential operators on V. It is analogous to the Weyl algebra D(V), and is related to D(V) via the Zhu functor. If G is a connected Lie group with Lie algebra g, and V is a linear G-representation, there is an action of the corresponding affine algebra on S(V). The invariant space S(V)^{g[t]} is a commutant subalgebra of S(V), and plays the role of the classical invariant ring D(V)^G. When G is an abelian Lie group acting diagonally on V, we find a finite set of generators for S(V)^{g[t]}, and show that S(V)^{g[t]} is a simple vertex algebra and a member of a Howe pair. The Zamolodchikov W_3 algebra with c=-2 plays a fundamental role in the structure of S(V)^{g[t]}."}
{"category": "Math", "title": "A note on free quantum groups", "abstract": "We study the free complexification operation for compact quantum groups, $G\\to G^c$. We prove that, with suitable definitions, this induces a one-to-one correspondence between free orthogonal quantum groups of infinite level, and free unitary quantum groups satisfying $G=G^c$."}
{"category": "Math", "title": "An affine sphere equation associated to Einstein toric surfaces", "abstract": "As seen in the works of Calabi, Cheng-Yau and Loftin, affine sphere equations have a close relationship with Kaehler-Einstein metrics. The main purpose of this note is to show that an equation analogous to those of hyperbolic affine spheres arises naturally from Kaehler-Einstein metrics on Einstein toric surfaces. The case for the remaining toric surfaces with Kaehler-Ricci solitons will also be discussed."}
{"category": "Math", "title": "Dequantized Differential Operators between Tensor Densities as Modules over the Lie Algebra of Contact Vector Fields", "abstract": "In recent years, algebras and modules of differential operators have been extensively studied. Equivariant quantization and dequantization establish a tight link between invariant operators connecting modules of differential operators on tensor densities, and module morphisms that connect the corresponding dequantized spaces. In this paper, we investigate dequantized differential operators as modules over a Lie subalgebra of vector fields that preserve an additional structure. More precisely, we take an interest in invariant operators between dequantized spaces, viewed as modules over the Lie subalgebra of infinitesimal contact or projective contact transformations. The principal symbols of these invariant operators are invariant tensor fields. We first provide full description of the algebras of such affine-contact- and contact-invariant tensor fields. These characterizations allow showing that the algebra of projective-contact-invariant operators between dequantized spaces implemented by the same density weight, is generated by the vertical cotangent lift of the contact form and a generalized contact Hamiltonian. As an application, we prove a second key-result, which asserts that the Casimir operator of the Lie algebra of infinitesimal projective contact transformations, is diagonal. Eventually, this upshot entails that invariant operators between spaces induced by different density weights, are made up by a small number of building bricks that force the parameters of the source and target spaces to verify Diophantine-type equations."}
{"category": "Math", "title": "Sharp nonremovability examples for H\\\"older continuous quasiregular mappings in the plane", "abstract": "Let $\\alpha\\in(0,1)$, $K\\geq 1$, and $d=2\\frac{1+\\alpha K}{1+K}$. Given a compact set $E\\subset\\C$, it is known that if $\\H^d(E)=0$ then $E$ is removable for $\\alpha$-H\\\"older continuous $K$-quasiregular mappings in the plane. The sharpness of the index $d$ is shown with the construction, for any $t>d$, of a set $E$ of Hausdorff dimension $\\dim(E)=t$ which is not removable. In this paper, we improve this result and construct compact nonremovable sets $E$ such that $0<\\H^d(E)<\\infty$. For the proof, we give a precise planar $K$-quasiconformal mapping whose H\\\"older exponent is strictly bigger than $\\frac{1}{K}$, and that exhibits extremal distortion properties."}
{"category": "Math", "title": "Branching diffusions, superdiffusions and random media", "abstract": "Spatial branching processes became increasingly popular in the past decades, not only because of their obvious connection to biology, but also because superprocesses are intimately related to nonlinear partial differential equations. Another hot topic in today's research in probability theory is `random media', including the now classical problems on `Brownian motion among obstacles' and the more recent `random walks in random environment' and `catalytic branching' models. These notes aim to give a gentle introduction into some topics in spatial branching processes and superprocesses in deterministic environments (sections 2-6) and in random media (sections 7-11)."}
{"category": "Math", "title": "Fourier method for one dimensional Schr\\\"odinger operators with singular periodic potentials", "abstract": "By using quasi--derivatives, we develop a Fourier method for studying the spectral properties of one dimensional Schr\\\"odinger operators with periodic singular potentials."}
{"category": "Math", "title": "Crystal graphs for general linear Lie superalgebras and quasi-symmetric functions", "abstract": "We give a new representation theoretic interpretation of the ring of quasi-symmetric functions. This is obtained by showing that the super analogue of the Gessel's fundamental quasi-symmetric function can be realized as the character of an irreducible crystal for the Lie superalgebra $\\frak{gl}_{n|n}$ associated to its non-standard Borel subalgebra with a maximal number of odd isotropic simple roots. We also present an algebraic characterization of these super quasi-symmetric functions."}
{"category": "Math", "title": "Higher order group cohomology and the Eichler-Shimura map", "abstract": "Higher order group cohomology is defined and first properties are given. Using modular symbols, an Eichler-Shimura homomorphism is constructed mapping spaces of higher order cusp forms to higher order cohomology groups."}
{"category": "Math", "title": "Chain models on Hecke algebra for corner type representations", "abstract": "We consider the integrable open chain models formulated in terms of generators of the Hecke algebra. The spectrum of the Hamiltonians for the open Hecke chains of finite size with free boundary conditions is deduced for special (corner type) irreducible representations of the Hecke algebra."}
{"category": "Math", "title": "Two permanents in the universal enveloping algebras of the symplectic Lie algebras", "abstract": "This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent, and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by A. Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra."}
{"category": "Math", "title": "Uniqueness of real closure * of Baer regular rings", "abstract": "It was pointed out in my last paper that there are rings whose real closure * are not unique. In [4] we also discussed some example of rings by which there is a unique real closure * (mainly the real closed rings). Now we want to determine more classes of rings by which real closure * is unique. The main results involve characterisations of domains and Baer regular rings having unique real closure *, and an example showing that regular rings need not be f-rings in order to have a unique real closure *. The main objective here is to find characterisation for uniqueness of real closure * for real regular rings that will primarily only require information of the prime spectrum and the real spectrum of the ring."}
{"category": "Math", "title": "Schur type functions associated with polynomial sequences of binomial type", "abstract": "We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies some interesting expansion formulas, in which there is a curious duality. Moreover this class includes examples which are useful to describe the eigenvalues of Capelli type central elements of the universal enveloping algebras of classical Lie algebras."}
{"category": "Math", "title": "Zariski $k$-plets via dessins d'enfants", "abstract": "We construct exponentially large collections of pairwise distinct equisingular deformation families of irreducible plane curves sharing the same sets of singularities. The fundamental groups of all curves constructed are abelian."}
{"category": "Math", "title": "Heegner divisors, $L$-functions and harmonic weak Maass forms", "abstract": "Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as \"generating functions\" for central values and derivatives of quadratic twists of weight 2 modular $L$-functions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivatives of $L$-functions, and points in the Jacobian of modular curves is obtained by analyzing the properties of these differentials using works of Scholl, Waldschmidt, and Gross and Zagier."}
{"category": "Math", "title": "A geometric framework for the subfield problem of generic polynomials via Tschirnhausen transformation", "abstract": "Let $k$ be an arbitrary field. We study a general method to solve the subfield problem of generic polynomials for the symmetric groups over $k$ via Tschirnhausen transformation. Based on the general result in the former part, we give an explicit solution to the field isomorphism problem and the subfield problem of cubic generic polynomials for $\\frak{S}_3$ and $C_3$ over $k$. As an application of the cubic case, we also give several sextic generic polynomials over $k$."}
{"category": "Math", "title": "On the unipotent support of character sheaves", "abstract": "Let $G$ be a connected reductive group over $F_q$, where $q$ is large enough and the center of $G$ is connected. We are concerned with Lusztig's theory of {\\em character sheaves}, a geometric version of the classical character theory of the finite group $G(F_q)$. We show that under a certain technical condition, the restriction of a character sheaf to its {\\em unipotent support} (as defined by Lusztig) is either zero or an irreducible local system. As an application, the generalized Gelfand-Graev characters are shown to form a $\\Z$-basis of the $\\Z$-module of unipotently supported virtual characters of $G(F_q)$ (Kawanaka's conjecture)."}
{"category": "Math", "title": "GL(2,R) geometry of ODE's", "abstract": "We study five dimensional geometries associated with the 5-dimensional irreducible representation of GL(2,R). These are special Weyl geometries in signature (3,2) having the structure group reduced from CO(3,2) to GL(2,R). The reduction is obtained by means of a conformal class of totally symmetric 3-tensors. Among all GL(2,R) geometries we distinguish a subclass which we term `nearly integrable GL(2,R)geometries'. These define a unique gl(2,R) connection which has totally skew symmetric torsion. This torsion splits onto the GL(2,R) irreducible components having respective dimensions 3 and 7. We prove that on the solution space of every 5th order ODE satisfying certain three nonlinear differential conditions there exists a nearly integrable GL(2,R) geometry such that the skew symmetric torsion of its unique gl(2,R) connection is very special. In contrast to an arbitrary nearly integrable GL(2,R) geometry, it belongs to the 3-dimensional irreducible representation of GL(2,R). The conditions for the existence of the structure are lower order equivalents of the Doubrov-Wilczynski conditions found recently by Boris Doubrov [7]. We provide nontrivial examples of 5th order ODEs satisfying the three nonlinear differential conditions, which in turn provides examples of inhomogeneous GL(2,R) geometries in dimension five, with torsion in R^3. We also outline the theory and the basic properties of GL(2,R) geometries associated with n-dimensional irreducible representations of GL(2,R) in 5<n<10. In particular we give conditions for an n-th order ODE to possess this geometry on its solution space."}
{"category": "Math", "title": "Identification of the Isotherm Function in Chromatography Using CMA-ES", "abstract": "This paper deals with the identification of the flux for a system of conservation laws in the specific example of analytic chromatography. The fundamental equations of chromatographic process are highly non linear. The state-of-the-art Evolution Strategy, CMA-ES (the Covariance Matrix Adaptation Evolution Strategy), is used to identify the parameters of the so-called isotherm function. The approach was validated on different configurations of simulated data using either one, two or three components mixtures. CMA-ES is then applied to real data cases and its results are compared to those of a gradient-based strategy."}
{"category": "Math", "title": "Hyperplane Arrangements with Large Average Diameter", "abstract": "The largest possible average diameter of a bounded cell of a simple hyperplane arrangement is conjectured to be not greater than the dimension. We prove that this conjecture holds in dimension 2, and is asymptotically tight in fixed dimension. We give the exact value of the largest possible average diameter for all simple arrangements in dimension 2, for arrangements having at most the dimension plus 2 hyperplanes, and for arrangements having 6 hyperplanes in dimension 3. In dimension 3, we give lower and upper bounds which are both asymptotically equal to the dimension."}
{"category": "Math", "title": "Singular cohomology of the analytic Milnor fiber, and mixed Hodge structure on the nearby cohomology", "abstract": "We constructed the analytic Milnor fiber is a non-archimedean model of the classical topological Milnor fibration. In the present paper, we describe the homotopy type of the analytic Milnor fiber in terms of a strictly semi-stable model, and we show that its singular cohomology coincides with the weight zero part of the mixed Hodge structure on the nearby cohomology. We give a similar expression for Denef and Loeser's motivic Milnor fiber in terms of a strictly semi-stable model."}
{"category": "Math", "title": "A Generalization of A Leibniz Geometrical Theorem", "abstract": "In this article we present a generalization of a Leibniz's geometrical theorem and an application of it."}
{"category": "Math", "title": "An operational calculus for the Mould operad", "abstract": "The operad of moulds is realized in terms of an operational calculus of formal integrals (continuous formal power series). This leads to many simplifications and to the discovery of various suboperads. In particular, we prove a conjecture of the first author about the inverse image of non-crossing trees in the dendriform operad. Finally, we explain a connection with the formalism of noncommutative symmetric functions."}
{"category": "Math", "title": "On Floer homology and the Berge conjecture on knots admitting lens space surgeries", "abstract": "We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge's construction of knots in the three-sphere which admit lens space surgeries is complete. The first step, which we prove here, is to show that a knot in a lens space with a three-sphere surgery has simple (in the sense of rank) knot Floer homology. The second (conjectured) step involves showing that, for a fixed lens space, the only knots with simple Floer homology belong to a simple finite family. Using results of Baker, we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots. Coupled with work of Ni, these knots provide the first infinite family of non-trivial knots which are characterized by their knot Floer homology. As another application, we provide a Floer homology proof of a theorem of Berge."}
{"category": "Math", "title": "Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology", "abstract": "Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S^3 and the similarity of the combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S^3 admitting lens space surgeries."}
{"category": "Math", "title": "Central limits and homogenization in random media", "abstract": "We consider the perturbation of elliptic operators of the form $P(\\bx,\\bD)$ by random, rapidly varying, sufficiently mixing, potentials of the form $q(\\frac{\\bx}\\eps,\\omega)$. We analyze the source and spectral problems associated to such operators and show that the properly renormalized difference between the perturbed and unperturbed solutions may be written asymptotically as $\\eps\\to0$ as explicit Gaussian processes. Such results may be seen as central limit corrections to the homogenization (law of large numbers) process. Similar results are derived for more general elliptic equations in one dimension of space. The results are based on the availability of a rapidly converging integral formulation for the perturbed solutions and on the use of classical central limit results for random processes with appropriate mixing conditions."}
{"category": "Math", "title": "Consistent estimates of deformed isotropic Gaussian random fields on the plane", "abstract": "This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation $f:\\Bbb{R}^2\\to\\Bbb{R}^2$ when observing the deformed random field $Z\\circ f$ on a dense grid in a bounded, simply connected domain $\\Omega$, where $Z$ is assumed to be an isotropic Gaussian random field on $\\Bbb{R}^2$. The estimate $\\hat{f}$ is constructed on a simply connected domain $U$, such that $\\overline{U}\\subset\\Omega$ and is defined using kernel smoothed quadratic variations, Bergman projections and results from quasiconformal theory. We show, under mild assumptions on the random field $Z$ and the deformation $f$, that $\\hat{f}\\to R_{\\theta}f+c$ uniformly on compact subsets of $U$ with probability one as the grid spacing goes to zero, where $R_{\\theta}$ is an unidentifiable rotation and $c$ is an unidentifiable translation."}
{"category": "Math", "title": "Reduced zeta functions of Lie algebras", "abstract": "We define reduced zeta functions of Lie algebras, which can be derived from motivic zeta functions using the Euler characteristic. We show that reduced zeta functions of Lie algebras possessing a suitably well-behaved basis are easy to analyse. We prove that reduced zeta functions are multiplicative under certain conditions and investigate which reduced zeta functions have functional equations."}
{"category": "Math", "title": "Moving and ample cones of holomorphic symplectic fourfolds", "abstract": "We analyze the ample and moving cones of holomorphic symplectic manifolds, in light of recent advances in the minimal model program. As an application, we establish a numerical criterion for ampleness of divisors on fourfolds deformation-equivalent to punctual Hilbert schemes of K3 surfaces."}
{"category": "Math", "title": "Higman's PORC conjecture for a family of groups", "abstract": "We prove that the number of groups of order $p^n$ whose Frattini subgroup is central is for fixed $n$ a PORC (`polynomial on residue classes') function of $p$. This extends a result of G. Higman."}
{"category": "Math", "title": "Inhomogeneous Diophantine approximation of some Hurwitzian numbers", "abstract": "We continue the work of Takao Komatsu by considering the inhomogeneous approximation constant L(\\theta,\\phi) for Hurwitzian numbers \\theta, and rationally related \\phi(r \\theta +m)/n in Q(\\theta) +Q. The current work uses a compactness theorem to relate such inhomogeneous constants to the homogeneous approximation constants. Among the new results are: a characterization of such pairs \\theta,\\phi for which L(\\theta,\\phi) is zero; consideration of small values of n^2 L(e^{2/s},\\phi); and the proof of a conjecture of Komatsu."}
{"category": "Math", "title": "Area-expanding embeddings of rectangles", "abstract": "We estimate whether there is an embedding from one n-dimensional rectangle into another which expands every k-dimensional area. Our estimate is sharp up to a constant factor in each dimension."}
{"category": "Math", "title": "Optimal Transportation under Nonholonomic Constraints", "abstract": "We study Monge's optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantly the absence of sharp abnormal minimizers. In particular, this result is applicable in the case of subriemannian manifolds with a 2-generating distribution and cost given by $d^2$, where $d$ is the subriemannian distance. Also, we discuss some properties of the optimal plan when abnormal minimizers are present. Finally, we consider some examples of displacement interpolation in the case of Grushin plane."}
{"category": "Math", "title": "On potentially $K_{r+1}-U$-graphical Sequences", "abstract": "Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\\sigma(S)\\geq \\sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\\sigma (K_{r+1}-U, n)$ for $n\\geq 5r+18, r+1 \\geq k \\geq 7,$ $j \\geq 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3 \\bigcup P_3$ but not contains a cycle on 4 vertices and not contains $Z_4$. There are a number of graphs on $k$ vertices and $j$ edges which contains a graph $(K_{3} \\bigcup P_{3})$ but not contains a cycle on 4 vertices and not contains $Z_4$. (for example, $C_3\\bigcup C_{i_1} \\bigcup C_{i_2} \\bigcup >... \\bigcup C_{i_p}$ $(i_j\\neq 4, j=2,3,..., p, i_1 \\geq 5)$, $C_3\\bigcup P_{i_1} \\bigcup P_{i_2} \\bigcup ... \\bigcup P_{i_p}$ $(i_1 \\geq 3)$, $C_3\\bigcup P_{i_1} \\bigcup C_{i_2} \\bigcup >... \\bigcup C_{i_p}$ $(i_j\\neq 4, j=2,3,..., p, i_1 \\geq 3)$, etc)"}
{"category": "Math", "title": "Operated semigroups, Motzkin paths and rooted trees", "abstract": "Combinatorial objects such as rooted trees that carry a recursive structure have found important applications recently in both mathematics and physics. We put such structures in an algebraic framework of operated semigroups. This framework provides the concept of operated semigroups with intuitive and convenient combinatorial descriptions, and at the same time endows the familiar combinatorial objects with a precise algebraic interpretation. As an application, we obtain constructions of free Rota-Baxter algebras in terms of Motzkin paths and rooted trees."}
{"category": "Math", "title": "Differential Birkhoff decomposition and the renormalization of multiple zeta values", "abstract": "In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is views as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of multiple zeta values."}
{"category": "Math", "title": "Rota-Baxter operators on generalized power series rings", "abstract": "An important instance of Rota-Baxter algebras from their quantum field theory application is the ring of Laurent series with a suitable projection. We view the ring of Laurent series as a special case of generalized power series rings with exponents in an ordered monoid. We study when a generalized power series ring has a Rota-Baxter operator and how this is related to the ordered monoid."}
{"category": "Math", "title": "Application of Quantum Theory to Super-parametric Density Estimation", "abstract": "In this paper, we will discuss how to generalize nonparametric density estimators to MLE parametric estimators. Basing on the Parzen window theory and using the advantages of probability amplitude of quantum theory, we model a nonlinear optimization problem and it is very difficult, if not impossible, to solve the problem. A constructive procedure for solving the nonlinear programming problem is studied. Though it seems to be very complicated, the approach of this paper is simple and comprehensive. More precisely, the lemmas, the theorems and their proofs serve the purpose for mathematical rigor and practical computation. Instead of using techniques and terminologies of advanced mathematics, we use the popular techniques and terminologies of elementary calculus. From the numerical results of the paper by Y. --S. Tsai et al. [7], it shows that a new approach of density estimation, super-parametric density estimation, is established completely. Strictly speaking, the work of the paper is not confined in the category of statistics. It could be classified into nonlinear analysis such as optimization on linear space, or manifold, and the algorithm of computer science."}
{"category": "Math", "title": "Connectivity of the Product Replacement Graph of Simple Groups of Bounded Lie Rank", "abstract": "The Product Replacement Algorithm is a practical algorithm for generating random elements of a finite group. The algorithm can be described as a random walk on a graph whose vertices are the generating k-tuples of the group (for a fixed integer k). We show that there is a function c(r) such that for any finite simple group of Lie type, with Lie rank r, the product replacement graph of the generating k-tuples is connected for any k > c(r). The proof uses results of Larsen and Pink and does not rely on the classification of finite simple groups."}
{"category": "Math", "title": "Overlapping self-affine sets of Kakeya type", "abstract": "We compute the Minkowski dimension for a family of self-affine sets on the plane. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets."}
{"category": "Math", "title": "Permutation statistics related to a class of noncommutative symmetric functions and generalizations of the Genocchi numbers", "abstract": "We prove conjectures of the third author [L. Tevlin, Proc. FPSAC'07, Tianjin] on two new bases of noncommutative symmetric functions: the transition matrices from the ribbon basis have nonnegative integral coefficients. This is done by means of two composition-valued statistics on permutations and packed words, which generalize the combinatorics of Genocchi numbers."}
{"category": "Math", "title": "Differential Complexes and Stratified Pro-Modules", "abstract": "In this paper we introduce the category of stratified Pro-modules and the notion of induced object in this category. We propose a translation of a Morihiko Saito equivalence result using the dual language of Pro-objects. So we prove an equivalence between the derived category of stratified Pro-modules and the category of Pro-differential complexes. We also supply a comparison with the notion of Crystal in Pro-module (introduced by P. Deligne in 1960)."}
{"category": "Math", "title": "Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree", "abstract": "In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian excursion as $n\\to\\infty$. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton-Watson branching process, conditioned on the total population size."}
{"category": "Math", "title": "Human proofs of identities by Osburn and Schneider", "abstract": "Osburn and Schneider derived several combinatorial identities involving harmonic numbers using the computer programm Sigma. Here, they are derived by partial fraction decomposition and creative telescoping."}
{"category": "Math", "title": "A minimal set of generators for the ring of multisymmetric functions", "abstract": "The purpose of this article is to give, for any commutative ring A, an explicit minimal set of generators for the ring of multisymmetric functions TS^d_A(A[x_1,...,x_r]) as an A-algebra. In characteristic zero, i.e. when A is an algebra over the rational numbers, a minimal set of generators has been known since the 19th century. A rather small generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound on the generators, improving the degree bound previously obtained by Fleischmann."}
{"category": "Math", "title": "Fractional dynamical systems defined on fractional jet bundle and applications in economics", "abstract": "Using Caputo fractional derivative of order $\\alpha $ we build the fractional jet bundle of order $\\alpha $ and its main geometrical structures. Defined on that bundle, some fractional dynamical systems with applications to economics are studied."}
{"category": "Math", "title": "Analysis of Linear Difference Schemes in the Sparse Grid Combination Technique", "abstract": "Sparse grids are tailored to the approximation of smooth high-dimensional functions. On a $d$-dimensional tensor product space, the number of grid points is $N = \\mathcal O(h^{-1} |\\log h|^{d-1})$, where $h$ is a mesh parameter. The so-called combination technique, based on hierarchical decomposition and extrapolation, requires specific multivariate error expansions of the discretisation error on Cartesian grids to hold. We derive such error expansions for linear difference schemes through an error correction technique of semi-discretisations. We obtain overall error formulae of the type $\\epsilon = \\mathcal{O} (h^p |\\log h|^{d-1})$ and analyse the convergence, with its dependence on dimension and smoothness, by examples of linear elliptic and parabolic problems, with numerical illustrations in up to eight dimensions."}
{"category": "Math", "title": "On the second Paneitz-Branson invariant", "abstract": "We define the second Paneitz-Branson operator on a compact Einsteinian manifold of dimension $n\\geq 5$ and we give sufficient conditions that make it attained."}
{"category": "Math", "title": "Planar trees, free nonassociative algebras, invariants, and elliptic integrals", "abstract": "We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such algebras are described in terms of labeled reduced planar rooted trees. This allows to apply combinatorial techniques to study their Hilbert series and the asymptotics of their coefficients. These algebras satisfy the Nielsen-Schreier property and their subalgebras are also free. Then, over a field of characteristic 0, we investigate the subalgebras of invariants under the action of a linear group, their sets of free generators and their Hilbert series. It has turned out that, except in the trivial cases, the algebra of invariants is never finitely generated. In important partial cases the Hilbert series of the algebras of invariants and the generating functions of their sets of free generators are expressed in terms of elliptic integrals."}
{"category": "Math", "title": "Dynamical resonances and SSF singularities for a magnetic Schroedinger operator", "abstract": "We consider the Hamiltonian $H$ of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator $H$ has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb $H$ by appropriate scalar potentials $V$ and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic field, and obtain an asymptotic expansion of the resonances as the coupling constant $\\varkappa$ of the perturbation tends to zero. Further, under the assumption that the Fermi Golden Rule holds true, we deduce estimates for the time evolution of the resonance states with and without analyticity assumptions; in the second case we obtain these results as a corollary of suitable Mourre estimates and a recent article of Cattaneo, Graf and Hunziker \\cite{cgh}. Next, we describe sets of perturbations $V$ for which the Fermi Golden Rule is valid at each embedded eigenvalue of $H$; these sets turn out to be dense in various suitable topologies. Finally, we assume that $V$ decays fast enough at infinity and is of definite sign, introduce the Krein spectral shift function for the operator pair $(H+V, H)$, and study its singularities at the energies which coincide with eigenvalues of infinite multiplicity of the unperturbed operator $H$."}
{"category": "Math", "title": "Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane", "abstract": "We discuss generalizations of the well-known theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3-space. We show that this problem is expressed very naturally as the question of the existence of certain homotheties of reflective submanifolds of a symmetric space. As such, we conclude that the only other (non-compact) cases to which this theorem could generalize are the problem of isometric immersions with flat normal bundle of the hyperbolic space $H^n$ into a Euclidean space $E^{n+k}$, $n \\geq 2$, and the problem of Lagrangian isometric immersions of $H^n$ into $\\cc^n$, $n \\geq 2$. Moreover, there are natural compact counterparts to these problems, and for the compact cases we prove that the theorem does in fact generalize: local embeddings exist, but complete immersions do not."}
{"category": "Math", "title": "Structured variable selection in support vector machines", "abstract": "When applying the support vector machine (SVM) to high-dimensional classification problems, we often impose a sparse structure in the SVM to eliminate the influences of the irrelevant predictors. The lasso and other variable selection techniques have been successfully used in the SVM to perform automatic variable selection. In some problems, there is a natural hierarchical structure among the variables. Thus, in order to have an interpretable SVM classifier, it is important to respect the heredity principle when enforcing the sparsity in the SVM. Many variable selection methods, however, do not respect the heredity principle. In this paper we enforce both sparsity and the heredity principle in the SVM by using the so-called structured variable selection (SVS) framework originally proposed in Yuan, Joseph and Zou (2007). We minimize the empirical hinge loss under a set of linear inequality constraints and a lasso-type penalty. The solution always obeys the desired heredity principle and enjoys sparsity. The new SVM classifier can be efficiently fitted, because the optimization problem is a linear program. Another contribution of this work is to present a nonparametric extension of the SVS framework, and we propose nonparametric heredity SVMs. Simulated and real data are used to illustrate the merits of the proposed method."}
{"category": "Math", "title": "Constraints on exact Lagrangians in cotangent bundles of manifolds fibred over the circle", "abstract": "We give topological obstructions to the existence of a closed exact Lagrangian submanifold in the cotangent bundle of a closed manifold M which is the total space of a fibration over the circle. For instance we show that the fundamental group of such a Lagrangian submanifold cannot be the free product of two non-trivial groups and that in any finite presentation of this group the difference between the number of generators and the number of relations is less than two."}
{"category": "Math", "title": "Essential curves in handlebodies and topological contractions", "abstract": "If $X$ is a compact set, a {\\it topological contraction} is a self-embedding $f$ such that the intersection of the successive images $f^k(X)$, $k>0$, consists of one point. In dimension 3, we prove that there are smooth topological contractions of the handlebodies of genus $\\geq 2$ whose image is essential. Our proof is based on an easy criterion for a simple curve to be essential in a handlebody."}
{"category": "Math", "title": "Preliminary results on the homogenization of thin piezoelectric perforated shells", "abstract": "We consider a composite piezoelectric material whose reference configuration is a thin shell with fixed thickness. In this work, we give a new approach based on the periodic unfolding method to justify the modelling of a thin piezoelectric perforated shells and we establish the limit constitutive law by letting the size of holes is supposed to go to zero. This allows to use the homogenization technique to derive the limitting equations and the homogenizaed coefficients are explicity described."}
{"category": "Math", "title": "Investigation of one boundary-value problem for elliptic type equation", "abstract": "The goal of the paper is to study a spectral problem corresponding to the mixed problem for a sixth order weak parabolic equation."}
{"category": "Math", "title": "Trace ideals for pseudo-differential operators and their commutators with symbols in $\\alpha$-modulation spaces", "abstract": "In this paper, we consider the trace property of pseudo-differential operators with symbols in $\\alpha$-modulation spaces."}
{"category": "Math", "title": "Panel and Pseudo-Panel Estimation of Cross-Sectional and Time Series Elasticities of Food Consumption: The Case of American and Polish Data", "abstract": "The problem addressed in this article is the bias to income and expenditure elasticities estimated on pseudo-panel data caused by measurement error and unobserved heterogeneity. We gauge empirically these biases by comparing cross-sectional, pseudo-panel and true panel data from both Polish and American expenditure surveys. Our results suggest that unobserved heterogeneity imparts a downward bias to cross-section estimates of income elasticities of at-home food expenditures and an upward bias to estimates of income elasticities of away-from-home food expenditures. \"Within\" and first-difference estimators suffer less bias, but only if the effects of measurement error are accounted for with instrumental variables."}
{"category": "Math", "title": "The diagonal of the Stasheff polytope", "abstract": "We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and the operad A-infinity is a retract by deformation of it. We compare these constructions with analogous constructions due to Saneblidze-Umble and Markl-Shnider based on the Boardman-Vogt cubical decomposition of the Stasheff polytope."}
{"category": "Math", "title": "Combinatorial Aspects of Elliptic Curves II: Relationship between Elliptic Curves and Chip-Firing Games on Graphs", "abstract": "Let q be a power of a prime and E be an elliptic curve defined over F_q. In \"Combinatorial aspects of elliptic curves\" [17], the present author examined a sequence of polynomials which express the N_k's, the number of points on E over the field extensions F_{q^k}, in terms of the parameters q and N_1 = #E(F_q). These polynomials have integral coefficients which alternate in sign, and a combinatorial interpretation in terms of spanning trees of wheel graphs. In this sequel, we explore further ramifications of this connection. In particular, we highlight a relationship between elliptic curves and chip-firing games on graphs by comparing the groups structures of both. As a coda, we construct a cyclic rational language whose zeta function is dual to that of an elliptic curve."}
{"category": "Math", "title": "Strict essential extensions of C*-algebras and Hilbert C*-modules", "abstract": "In the present paper we develop both ideas of D. Baki\\'c and B. Gulja{\\v{s}} and the categorical approach to multipliers from E.C. Lance's book and publications of the second author, for the introduction and study of left multipliers of Hilbert $C^*$-modules. Some properties and, in particular, the property of maximality among all strictly essential extensions of a Hilbert $C^*$-module for left multipliers are proved. Also relations between left essential and left strictly essential extensions in different contexts are obtained. Left essential and left strictly essential extensions of matrix algebras are considered. In the final paragraph the topological approach to the left multiplier theory of Hilbert $C^*$-modules is worked out."}
{"category": "Math", "title": "Accuracy of approximation of subharmonic functions by logarithms of moduli of analytic ones in Chebyshev metrics", "abstract": "It is known that a subharmonic function of finite order $\\rho$ can be approximated by the logarithm of the modulus of an entire function at the point $z$ outside an exceptional set up to $C\\log|z|$. In this article we prove that if such an approximation is made more precise, i. e. a constant $C$ decreases, then, beginning with $C=\\rho/4$, the size of the exceptional set enlarges substantially. Similar results are proved for subharmonic functions of infinite order and functions subharmonic in the unit disk. These theorems improve and complement a result by Yulmukhametov."}
{"category": "Math", "title": "Commuting Families in Temperley-Lieb Algebras", "abstract": "We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group $U_h\\mathfrak{gl}_n$. We also give the explicit specializations of these results to the finite Temperley-Lieb algebra."}
{"category": "Math", "title": "Anosov AdS representations are quasi-Fuchsian", "abstract": "Let Gamma be a cocompact lattice in SO(1,n). A representation rho: Gamma \\to SO(2,n) is quasi-Fuchsian if it is faithfull, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space - a particular case is the case of Fuchsian representations, ie. composition of the inclusions of Gamma in SO(1,n) and of SO(1,n) in SO(2,n). We prove that if a representation is Anosov in the sense of Labourie then it is also quasi-Fuchsian. We also show that Fuchsian representations are Anosov : the fact that all quasi-Fuchsian representations are Anosov will be proved in a second part by T. Barbot. The study involves the geometry of locally anti-de Sitter spaces: quasi-Fuchsian representations are holonomy representations of globally hyperbolic spacetimes diffeomorphic to the product R \\times Gamma\\H^n and locally modeled on the anti-de Sitter space."}
{"category": "Math", "title": "Reflexive Ideals in Iwasawa Algebras", "abstract": "Let $G$ be a torsionfree compact $p$-adic analytic group. We give sufficient conditions on $p$ and $G$ which ensure that the Iwasawa algebra $\\Omega_G$ of $G$ has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every nonzero normal element in $\\Omega_G$ is a unit. We show that these conditions hold in the case when $G$ is an open subgroup of $\\SL_2(\\Zp)$ and $p$ is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in $\\Omega_G$ when $G$ is a congruence subgroup of $\\SL_2(\\Zp)$: the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown."}
{"category": "Math", "title": "Nonexistence of reflexive ideals in Iwasawa algebras of Chevalley type", "abstract": "Let $\\Phi$ be a root system and let $\\Phi(\\Zp)$ be the standard Chevalley $\\Zp$-Lie algebra associated to $\\Phi$. For any integer $t\\geq 1$, let $G$ be the uniform pro-$p$ group corresponding to the powerful Lie algebra $p^t \\Phi(\\Zp)$ and suppose that $p\\geq 5$. Then the Iwasawa algebra $\\Omega_G$ has no nontrivial reflexive two-sided ideals. This was previously proved by the authors for the root system $A_1$."}
{"category": "Math", "title": "Sheaves on abelian surfaces and Strange Duality", "abstract": "We formulate three versions of a strange duality conjecture for sections of the Theta bundles on the moduli spaces of sheaves on abelian surfaces. As supporting evidence, we check the equality of dimensions on dual moduli spaces, answering a question raised by Gottsche-Nakajima-Yoshioka."}
{"category": "Math", "title": "Dirac geometry, quasi-Poisson actions and D/G-valued moment maps", "abstract": "We study Dirac structures associated with Manin pairs (\\d,\\g) and give a Dirac geometric approach to Hamiltonian spaces with D/G-valued moment maps, originally introduced by Alekseev and Kosmann-Schwarzbach in terms of quasi-Poisson structures. We explain how these two distinct frameworks are related to each other, proving that they lead to isomorphic categories of Hamiltonian spaces. We stress the connection between the viewpoint of Dirac geometry and equivariant differential forms. The paper discusses various examples, including q-Hamiltonian spaces and Poisson-Lie group actions, explaining how presymplectic groupoids are related to the notion of \"double\" in each context."}
{"category": "Math", "title": "The Toda system and multiple-end solutions of autonomous planar elliptic problems", "abstract": "We construct a new class of positive solutions for a classical semilinear elliptic problem in the plane which arise for instance as the standing-wave problem for the standard nonlinear Schr\\\"odinger equation or in nonlinear models in Turing's theory biological theory of pattern formation such as the Gray-Scott or Gierer-Meinhardt systems. The solutions we construct have the property that their energy over a ball of radius R grows linearly with R as R tends to infinity. These solutions are strongly related to the solutions of a Toda system."}
{"category": "Math", "title": "Appendix to 'Roth's theorem on progressions revisited' by J Bourgain", "abstract": "We show two results. First, a refinement of Freiman's theorem: if A is a finite set of integers and |A+A| < K|A|, then A is contained in a multidimensional progression of dimension at most O(K^{7/4} log^3K) and size at most exp(O(K^{7/4} log^3K))|A|. Secondly, an improvement of a result of Konyagin and Laba: if A is a finite set of reals and a is a transcendental then |A+aA| >> |A|(log |A|)^{4/3-\\epsilon} for all \\epsilon>0."}
{"category": "Math", "title": "Chern character for twisted complexes", "abstract": "We construct a Chern character of a perfect complex of twisted modules over an algebroid stack."}
{"category": "Math", "title": "Difference sets and the primes", "abstract": "Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0."}
{"category": "Math", "title": "Matrix pairs over discrete valuation rings determine Littlewood-Richardson fillings", "abstract": "Let M and N be two r x r matrices over a discrete valuation ring of characteristic zero. The orders (with respect to a uniformizing parameter) of the invariant factors of M form a partition of non-negative integers, called the invariant partition of M. Let the invariant partition of M be mu, of N be nu, and of the product MN be lambda. In this paper we construct a Littlewood-Richardson filling of the skew shape lambda/mu with content nu, and show that this filling is an invariant of the orbit of the pair (M,N) with respect to a natural group action on the pair. We relate the algebraic combinatorics of Littlewood-Richardson fillings to a special semicanonical matrix in the orbit of (M,N), from which the Littlewood-Richardson filling, and other combinatorial invariants may be obtained."}
{"category": "Math", "title": "Steady-state analysis of a multi-server queue in the Halfin-Whitt regime", "abstract": "We consider a multi-server queue in the Halfin-Whitt regime: as the number of servers $n$ grows without a bound, the utilization approaches 1 from below at the rate $\\Theta(1/\\sqrt{n})$. Assuming that the service time distribution is lattice-valued with a finite support, we characterize the limiting stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting (scaled) steady-state queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime."}
{"category": "Math", "title": "Chaotic Period Doubling", "abstract": "The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C. Tresser in the nineteen-seventieth to study the asymptotic small scale geometry of the attractor of one-dimensional systems which are at the transition from simple to chaotic dynamics. This geometry turns out to not depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point which is also hyperbolic among generic smooth enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that in the space of $C^{2+\\alpha}$ unimodal maps, for $\\alpha$ close to one, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main results states that in the space of $C^2$ unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to get a priori bounds. In this smoother class, called $C^{2+|\\cdot|}$ the failure of hyperbolicity is tamer than in $C^2$. Things get much worse with just a bit less of smoothness than $C^2$ as then even the uniqueness is lost and other asymptotic behavior become possible. We show that the period doubling renormalization operator acting on the space of $C^{1+Lip}$ unimodal maps has infinite topological entropy."}
{"category": "Math", "title": "Remarks on the symmetric powers of cusp forms on GL(2)", "abstract": "In this paper we prove the following conditional result: Let F be a number field, and pi a cusp form on GL(2)/F which is not solvable polyhedral. Assume that all the symmetric powers sym^m(pi) are modular, i.e., define automorphic forms on GL(m+1)/F. If sym^6(pi) is cuspidal, then all the symmetric powers are cuspidal, for all m. Moreover, sym^6(pi) is Eisenteinian iff sym^5(pi) is an abelian twist of the functorial product of pi with the symmetric square of a cusp form pi' on GL(2)/F."}
{"category": "Math", "title": "The divided cell algorithm and the inhomogeneous Lagrange and Markoff spectra", "abstract": "The divided cell algorithm was introduced by Delone in 1947 to calculate the inhomogeneous minima of binary quadratic forms and developed further by E. S. Barnes and H. P. F. Swinnerton-Dyer in the 1950s. We show how advances of the past fifty years in both symbolic computation and our understanding of homogeneous spectra can be combined to make divided cells more useful for organizing information about inhomogeneous approximation problems. A crucial part of our analysis relies on work of Jane Pitman, who related the divided cell algorithm to the regular continued fraction algorithm. In particular, the relation to continued fractions allows two divided cells for the same problem to be compared without stepping through the chain of divided cells connecting them."}
{"category": "Math", "title": "Yang-Mills theory over surfaces and the Atiyah-Segal theorem", "abstract": "In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(\\Gamma) of a compact Lie group $\\Gamma$ to the complex K-theory of the classifying space $B\\Gamma$. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson's deformation $K$--theory spectrum $\\K (\\Gamma)$ (the homotopy-theoretical analogue of $R(\\Gamma)$). Our main theorem provides an isomorphism in homotopy $\\K_*(\\pi_1 \\Sigma)\\isom K^{-*}(\\Sigma)$ for all compact, aspherical surfaces $\\Sigma$ and all $*>0$. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces."}
{"category": "Math", "title": "Uniformly Balanced Repeated Measurements Designs in the Presence of Subject Dropout", "abstract": "Low, Lewis and Prescott (1999) showed that a crossover design based on a Williams Latin square of order 4 can suffer substantial loss of efficiency if some observations in the final period are unavailable. Indeed, if all observations are missing, the design becomes disconnected. We derive the information matrix for the direct effects of a Uniformly Balanced Repeated Measurements Design (UBRMD) in t periods when subjects may drop out before the end of the study and examine the maximum loss of information. The special case of loss of observations in the final period only is examined in detail. In particular we show that a UBRMD in t>= 5 periods remains connected when some or all observations in the final period are unavailable."}
{"category": "Math", "title": "Convexity and Cone-Vexing", "abstract": "The idea of convexity feeds generation, separation, calculus, and approximation. Generation appears as duality; separation, as optimality; calculus, as representation; and approximation, as stability. This is an overview of the origin, evolution, and trends of convexity. Study of convexity in the Sobolev Institute was initiated by Leonid Kantorovich (1912--1986) and Alexandr Alexandrov (1912--1999). This talk is a part of their memory."}
{"category": "Math", "title": "Deblurring of Motionally Averaged Images with Applications to Single-Particle Cryo-Electron Microscopy", "abstract": "This paper addresses the deconvolution of an image that has been obtained by superimposing many copies of an underlying unknown image of interest. The superposition is assumed to not be exact due to noise, and is described using an error distribution in position, orientation, or both. We assume that a good estimate of the error distribution is known. The most natural approach to take for the purely translational case is to apply the Fourier transform and use the classical convolution theorem together with a Weiner filter to invert. In the case of purely rotational deblurring, the similar Fourier analysis is applied. That is, for an blurred image function defined on polar coordinates, the Fourier series and the convolution theorem for the series can be applied. In the case of combinations of translational and rotational errors, the motion-group Fourier transform is used. In addition, for the three cases we present the alternative method using Hermite and Laguerre-Fourier expansion, which has a special property in Fourier transform. The problem that is solved here is motivated by one of the steps in the cryo-electron-tomographic reconstruction of biomolecular complexes such as viruses and ion channels."}
{"category": "Math", "title": "Toroidalization of generating sequences in dimension two function fields of positive characteristic", "abstract": "We give a characteristic free proof of the main result of our previous paper (math.AC/0509697) concerning toroidalization of generating sequences of valuations in dimension two function fields. We show that when an extension of two dimensional algebraic regular local rings $R\\subset S$ satisfies the conclusions of the Strong Monomialization theorem of Cutkosky and Piltant, the map between generating sequences in $R$ and $S$ has a toroidal structure."}
{"category": "Math", "title": "Area-Preserving Surface Dynamics and S. Saito's Fixed Point Formula", "abstract": "We show that S. Saito's fixed point formula serves as a powerful tool for counting the number of isolated periodic points of an area-preserving surface map admitting periodic curves. His notion of periodic curves of types I and II plays a central role in our discussion. We establish a Shub-Sullivan type result on the stability of local indices under iterations of the map, the finiteness of the number of periodic curves of type II, and the absence of periodic curves of type I. Combined with these results, Saito's formula implies the existence of infinitely many isolated periodic points whose cardinality grows exponentially as period tends to infinity."}
{"category": "Math", "title": "A characterization of all equilateral triangles in \\Bbb Z^3", "abstract": "This paper is a continuation of previous work of the authors. We extend one of the theorems that gave a way to construct equilateral triangles whose vertices have integer coordinates to the general situation. An approximate extrapolation formula for the sequence ET(n) of all equilateral triangles with vertices in $\\{0,1,2,...,n\\}^3$ (A 102698) is given and the asymptotic behavior of this sequence is analyzed."}
{"category": "Math", "title": "Symplectic resolutions, Lefschetz property and formality", "abstract": "We introduce a method to resolve a symplectic orbifold into a smooth symplectic manifold. Then we study how the formality and the Lefschetz property of the symplectic resolution are compared with that of the symplectic orbifold. We also study the formality of the symplectic blow-up of a symplectic orbifold along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov."}
{"category": "Math", "title": "Obstruction theory on 8-manifolds", "abstract": "This note gives a uniform, self-contained, and fairly direct approach to a variety of obstruction-theoretic problems on 8-manifolds. We give necessary and sufficient cohomological critera for the existence of almost complex and almost quaternionic structures on the tangent bundle and for the reduction of the structure group to U(3) by the homomorphism U(3) --> O(8) given by the Lie algebra representation of PU(3)."}
{"category": "Math", "title": "On 3-dimensional Asymptotically Harmonic Manifolds", "abstract": "Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature, provided M is asymptotically harmonic of constant h > 0."}
{"category": "Math", "title": "The foam and the matrix factorization sl3 link homologies are equivalent", "abstract": "We prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces."}
{"category": "Math", "title": "Differential equations driven by rough paths: an approach via discrete approximation", "abstract": "A theory of differential equations driven by a non-differentiable path has recently been developed by Lyons. We develop an alternative approach to this theory, using (modified Euler approximations), and investigate its applicability to stochastic differential equations driven by Brownian motion. We also give some other examples showing that the main results are reasonably sharp."}
{"category": "Math", "title": "On the cyclotomic Hecke algebras of complex reflection groups", "abstract": "Following the definition of Rouquier for the \"families of characters\" of a Weyl group and its generalization to the case of complex reflection groups, already appeared in the works of Broue-Kim and Malle-Rouquier, we show that these \"families\" depend on a numerical datum of the group, its \"essential hyperplanes\". Using this characterization, we provide the algorithm and the results of the determination of the Rouquier blocks of the cyclotomic Hecke algebras of all exceptional complex reflection groups."}
{"category": "Math", "title": "On Q-conic bundles, II", "abstract": "A $\\mathbb Q$-conic bundle germ is a proper morphism from a threefold with only terminal singularities to the germ $(Z \\ni o)$ of a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We obtain the complete classification of $\\mathbb Q$-conic bundle germs when the base surface germ is singular. This is a generalization of our previous paper math/0603736, which further assumed that the fiber over $o$ is irreducible."}
{"category": "Math", "title": "Global stability of travelling fronts for a damped wave equation with bistable nonlinearity", "abstract": "We consider the damped wave equation \\alpha u_tt + u_t = u_xx - V'(u) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u(x,t) = h(x-st) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V. We show that, if the initial data are sufficiently close to the profile of a front for large |x|, the solution of the damped wave equation converges uniformly on R to a travelling front as t goes to plus infinity. The proof of this global stability result is inspired by a recent work of E. Risler and relies on the fact that our system has a Lyapunov function in any Galilean frame."}
{"category": "Math", "title": "The eigenvalues of limits of radial Toeplitz operators", "abstract": "Let $A^2$ be the Bergman space on the unit disk. A bounded operator $S$ on $A^2$ is called radial if $Sz^n = \\lambda_n z^n$ for all $n\\ge 0$, where $\\lambda_n$ is a bounded sequence of complex numbers. We characterize the eigenvalues of radial operators that can be approximated by Toeplitz operators with bounded symbols."}
{"category": "Math", "title": "Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms", "abstract": "Tridiagonal canonical forms of square matrices under congruence or *congruence, pairs of symmetric or skew-symmetric matrices under congruence, and pairs of Hermitian matrices under *congruence are given over an algebraically closed field of characteristic different from 2."}
{"category": "Math", "title": "Valuation domains whose products of free modules are separable", "abstract": "It is proved that if $R$ is a valuation domain with maximal ideal $P$ and if $R_L$ is countably generated for each prime ideal $L$, then $R^R$ is separable if and only $R_J$ is maximal, where $J=\\cap_{n\\in\\mathbb{N}}P^n$."}
{"category": "Math", "title": "Canonical form of m-by-2-by-2 matrices over a field of characteristic other than two", "abstract": "We give a canonical form of m-by-2-by-2 matrices for equivalence over any field of characteristic not two."}
{"category": "Math", "title": "Permutahedra, HKR isomorphism and polydifferential Gerstenhaber-Schack complex", "abstract": "This paper aims to give a short but self-contained introduction into the theory of (wheeled) props, properads, dioperads and operads, and illustrate some of its key ideas in terms of a prop(erad)ic interpretation of simplicial and permutahedra cell complexes with subsequent applications to the Hochschild-Kostant-Rosenberg type isomorphisms."}
{"category": "Math", "title": "Three topics in additive prime number theory", "abstract": "This is an expository article to accompany my two lectures at the CDM conference. I have used this an excuse to make public two sets of notes I had lying around, and also to put together a short reader's guide to some recent joint work with T.Tao. Contents: 1. An exposition, without much detail, of the work of Goldston, Pintz and Yildirim on gaps between primes; 2. A detailed discussion of the work of Mauduit and Rivat establishing that 50 percent of the primes have odd digit sum when written in base 2; 3. A reader's guide to recent work of T.Tao and the author on linear equations in primes. The sections can be read independently."}
{"category": "Math", "title": "Miraculous Cancellation and Pick's Theorem", "abstract": "We show that the Cappell-Shaneson version of Pick's theorem for simple lattice polytopes is a consequence of a general relation between characteristic numbers of virtual submanifolds dual to the characteristic classes of a stably almost complex manifold. This relation is analogous to the miraculous cancellation formula of Alvarez-Gaume and Witten, and is imposed by the action of the Landweber-Novikov algebra in the complex cobordism ring of a point."}
{"category": "Math", "title": "Computing the Conditioning of the Components of a Linear Least Squares Solution", "abstract": "In this paper, we address the accuracy of the results for the overdetermined full rank linear least squares problem. We recall theoretical results obtained in Arioli, Baboulin and Gratton, SIMAX 29(2):413--433, 2007, on conditioning of the least squares solution and the components of the solution when the matrix perturbations are measured in Frobenius or spectral norms. Then we define computable estimates for these condition numbers and we interpret them in terms of statistical quantities. In particular, we show that, in the classical linear statistical model, the ratio of the variance of one component of the solution by the variance of the right-hand side is exactly the condition number of this solution component when perturbations on the right-hand side are considered. We also provide fragment codes using LAPACK routines to compute the variance-covariance matrix and the least squares conditioning and we give the corresponding computational cost. Finally we present a small historical numerical example that was used by Laplace in Theorie Analytique des Probabilites, 1820, for computing the mass of Jupiter and experiments from the space industry with real physical data."}
{"category": "Math", "title": "Congruence of multilinear forms", "abstract": "It is known that if A and B are two n-by-n complex matrices and (A,A^T) is simultaneously equivalent to (B,B^T), then A is congruent to B. We extend this statement to multilinear forms."}
{"category": "Math", "title": "The nested Chinese restaurant process and Bayesian nonparametric inference of topic hierarchies", "abstract": "We present the nested Chinese restaurant process (nCRP), a stochastic process which assigns probability distributions to infinitely-deep, infinitely-branching trees. We show how this stochastic process can be used as a prior distribution in a Bayesian nonparametric model of document collections. Specifically, we present an application to information retrieval in which documents are modeled as paths down a random tree, and the preferential attachment dynamics of the nCRP leads to clustering of documents according to sharing of topics at multiple levels of abstraction. Given a corpus of documents, a posterior inference algorithm finds an approximation to a posterior distribution over trees, topics and allocations of words to levels of the tree. We demonstrate this algorithm on collections of scientific abstracts from several journals. This model exemplifies a recent trend in statistical machine learning--the use of Bayesian nonparametric methods to infer distributions on flexible data structures."}
{"category": "Math", "title": "Formulas for Birkhoff-(Rota-Baxter) decompositions related to connected bialgebra", "abstract": "In recent years, The BPHZ algorithm for renormalization in quantum field theory has been interpreted, after dimensional regularization, as the Birkhoff-(Rota-Baxter) decomposition (BRB) of characters on the Hopf algebra of Feynmann graphs, with values in a Rota-Baxter algebra. We give in this paper formulas for the BRB decomposition in the group $\\mathcal{C}(H, A)$ of characters on a connected Hopf algebra $H$, with values in a Rota-Baxter (commutative) algebra $A$. To do so we first define the stuffle (or quasi-shuffle) Hopf algebra $A^{\\tmop{st}}$ associated to an algebra $A$. We prove then that for any connected Hopf algebra $H = k 1_H \\oplus H'$, there exists a canonical injective morphism from $H$ to $H'^{\\tmop{st}}$. This morphism induces an action of $\\mathcal{C}(A^{\\tmop{st}}, A)$ on $\\mathcal{C}(H, A)$ so that the BRB decomposition in $\\mathcal{C}(H, A)$ is determined by the action of a unique (universal) element of $\\mathcal{C}(A^{\\tmop{st}}, A)$."}
{"category": "Math", "title": "Decomposition of variance in terms of conditional means", "abstract": "We test against two different sets of data an apparently new approach to the analysis of the variance of a numerical variable which depends on qualitative characters. We suggest that this approach be used to complement other existing techniques to study the interdependence of the variables involved. According to our method the variance is expressed as a sum of orthogonal components, obtained as differences of conditional means, with respect to the qualitative characters. The resulting expression for the variance depends on the ordering in which the characters are considered. We suggest an algorithm which leads to an ordering which is deemed natural. The first set of data concerns the score achieved by a population of students, on an entrance examination, based on a multiple choice test with 30 questions. In this case the qualitative characters are dyadic and correspond to correct or incorrect answer to each question. The second set of data concerns the delay in obtaining the degree for a population of graduates of Italian universities. The variance in this case is analyzed with respect to a set of seven specific qualitative characters of the population studied (gender, previous education, working condition, parent's educational level, field of study, etc.)"}
{"category": "Math", "title": "Monte Carlo Methods and Path-Generation techniques for Pricing Multi-asset Path-dependent Options", "abstract": "We consider the problem of pricing path-dependent options on a basket of underlying assets using simulations. As an example we develop our studies using Asian options. Asian options are derivative contracts in which the underlying variable is the average price of given assets sampled over a period of time. Due to this structure, Asian options display a lower volatility and are therefore cheaper than their standard European counterparts. This paper is a survey of some recent enhancements to improve efficiency when pricing Asian options by Monte Carlo simulation in the Black-Scholes model. We analyze the dynamics with constant and time-dependent volatilities of the underlying asset returns. We present a comparison between the precision of the standard Monte Carlo method (MC) and the stratified Latin Hypercube Sampling (LHS). In particular, we discuss the use of low-discrepancy sequences, also known as Quasi-Monte Carlo method (QMC), and a randomized version of these sequences, known as Randomized Quasi Monte Carlo (RQMC). The latter has proven to be a useful variance reduction technique for both problems of up to 20 dimensions and for very high dimensions. Moreover, we present and test a new path generation approach based on a Kronecker product approximation (KPA) in the case of time-dependent volatilities. KPA proves to be a fast generation technique and reduces the computational cost of the simulation procedure."}
{"category": "Math", "title": "A regularization algorithm for matrices of bilinear and sesquilinear forms", "abstract": "We give an algorithm that uses only unitary transformations and for each square complex matrix constructs a *congruent matrix that is a direct sum of a nonsingular matrix and singular Jordan blocks."}
{"category": "Math", "title": "Lectures on two-dimensional critical percolation", "abstract": "These are the notes corresponding to the course given at the IAS-Park City graduate summer school in July 2007."}
{"category": "Math", "title": "Dynamic Programming Optimization over Random Data: the Scaling Exponent for Near-optimal Solutions", "abstract": "A very simple example of an algorithmic problem solvable by dynamic programming is to maximize, over sets A in {1,2,...,n}, the objective function |A| - \\sum_i \\xi_i 1(i \\in A,i+1 \\in A) for given \\xi_i > 0. This problem, with random (\\xi_i), provides a test example for studying the relationship between optimal and near-optimal solutions of combinatorial optimization problems. We show that, amongst solutions differing from the optimal solution in a small proportion \\delta of places, we can find near-optimal solutions whose objective function value differs from the optimum by a factor of order \\delta^2 but not smaller order. We conjecture this relationship holds widely in the context of dynamic programming over random data, and Monte Carlo simulations for the Kauffman-Levin NK model are consistent with the conjecture. This work is a technical contribution to a broad program initiated in Aldous-Percus (2003) of relating such scaling exponents to the algorithmic difficulty of optimization problems."}
{"category": "Math", "title": "A new technique for proving uniqueness for martingale problems", "abstract": "A new technique for proving uniqueness of martingale problems is introduced. The method is illustrated in the context of elliptic diffusions in $R^d$."}
{"category": "Math", "title": "Rigid systems of second-order linear differential equations", "abstract": "We say that a system of differential equations d^2x(t)/dt^2=Adx(t)/dt+Bx(t)+Cu(t), in which A and B are m-by-m complex matrices and C is an m-by-n complex matrix, is rigid if it can be reduced by substitutions x(t)=Sy(t), u(t)=Udy(t)/dt+Vy(t)+Pv(t) with nonsingular S and P to each system obtained from it by a small enough perturbation of its matrices A,B,C. We prove that there exists a rigid system if and only if m<n(1+square_root{5})/2, and describe all rigid systems."}
{"category": "Math", "title": "Peak points for pseudoconvex domains: a survey", "abstract": "This paper surveys results concerning peak points for pseudoconvex domains. It includes results of Laszlo that have not been published elsewhere."}
{"category": "Math", "title": "Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities", "abstract": "We prove a general duality result showing that a Brascamp--Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This open a new approach to the proof of Brascamp--Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on $\\R^n$, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp--Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard's inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schr\\\"odinger operators. Though we focus mainly on the case of random variables in $\\R^n$ in this paper, we discuss extensions to other settings as well."}
{"category": "Math", "title": "Stability of a Nonlinear Axially Moving String With the Kelvin-Voigt Damping", "abstract": "In this paper, a nonlinear axially moving string with the Kelvin-Voigt damping is considered. It is proved that the string is stable, i.e., its transversal displacement converges to zero when the axial speed of the string is less than a certain critical value. The proof is established by showing that a Lyapunov function corresponding to the string decays to zero exponentially. It is also shown that the string displacement is bounded when a bounded distributed force is applied to it transversally. Furthermore, a few open problems regarding the stability and stabilization of strings with the Kelvin-Voigt damping are stated."}
{"category": "Math", "title": "Ordered spanning sets for quasimodules for Mobius vertex algebras", "abstract": "Quasimodules for vertex algebras are generalizations of modules for vertex algebras. These new objects arise from a generalization of locality for fields. Quasimodules tie together module theory and twisted module theory, and both twisted and untwisted modules feature Poincare-Birkhoff-Witt-like spanning sets. This paper generalizes these spanning set results to quasimodules for certain Mobius vertex algebras. In particular this paper presents two spanning sets, one featuring a difference-zero ordering restriction on modes and another featuring a difference-one condition."}
{"category": "Math", "title": "Ordered spanning sets for vertex operator algebras and their modules", "abstract": "Moonshine relates three fundamental mathematical objects: the Monster sporadic simple group, the modular function j, and the moonshine module vertex operator algebra. Examining the relationship between modular functions and the representation theory of vertex operator algebras reveals rich structure. In particular, C2-cofiniteness (also called Zhu's finiteness condition) implies the existence of finite generating sets and Poincare-Birkhoff-Witt-like spanning sets for vertex operator algebras and their modules. These spanning sets feature desirable ordering restrictions, e.g., a difference-one condition."}
{"category": "Math", "title": "Some Properties of Hypergeometric Series Associated with Mirror Symmetry", "abstract": "We show that certain hypergeometric series used to formulate mirror symmetry for Calabi-Yau hypersurfaces, in string theory and algebraic geometry, satisfy a number of interesting properties. Many of these properties are used in separate papers to verify the BCOV prediction for the genus one Gromov-Witten invariants of a quintic threefold and more generally to compute the genus one Gromov-Witten invariants of any Calabi-Yau projective hypersurface."}
{"category": "Math", "title": "Rank 2 vector bundles on ind-Grassmannians", "abstract": "The simplest example of an ind-Grassmannian is the infinite projective space $\\mathbf P^\\infty$. The Barth-Van de Ven-Tyurin (BVT) Theorem, proved more than 30 years ago \\cite{BV}, \\cite{T}, \\cite{Sa} (see also a recent proof by A. Coand\\u{a} and G. Trautmann, \\cite{CT}), claims that any vector bundle of finite rank on $\\mathbf P^\\infty$ is isomorphic to a direct sum of line bundles. In the last decade natural examples of infinite flag varieties (or flag ind-varieties) have arisen as homogeneous spaces of locally linear ind-groups, \\cite{DPW}, \\cite{DiP}. In the present paper we concentrate our attention to the special case of ind-Grassmannians, i.e. to inductive limits of Grassmannians of growing dimension."}
{"category": "Math", "title": "Bounded generalized Harish-Chandra modules", "abstract": "Let $\\gg$ be a complex reductive Lie algebra and $\\kk\\subset\\gg$ be any reductive in $\\gg$ subalgebra. We call a $(\\gg,\\kk)$-module $M$ bounded if the $\\kk$-multiplicities of $M$ are uniformly bounded. In this paper we initiate a general study of simple bounded $(\\gg,\\kk)$-modules. We prove a strong necessary condition for a subalgebra $\\kk$ to be bounded (Corollary \\ref{cor1.6}), i.e. to admit an infinite-dimensional simple bounded $(\\gg,\\kk)$-module, and then establish a sufficient condition for a subalgebra $\\kk$ to be bounded (Theorem \\ref{thGroups2}). As a result we are able to classify all maximal bounded reductive subalgebras of $\\gg=\\sl(n)$. In the second half of the paper we describe in detail simple bounded infinite-dimensional $(\\gg,\\sl(2))$-modules, and in particular compute their characters and minimal $\\sl(2)$-types. We show that if $\\sl(2)$ is a bounded subalgebra of $\\gg$ which is not contained in a proper ideal of $\\gg$, then $\\gg\\simeq \\sl(2)\\oplus \\sl(2), \\sl(3),\\sp(4)$; alltogether, up to conjugation there are five possible embeddings of $\\sl(2)$ as a bounded subalgebra into $\\gg$ as above. In two of these cases $\\sl(2)$ is a symmetric subalgebra, and many results about simple bounded $(\\gg,\\sl(2))$-modules are known. A case where our results are entirely new is the case of a principal $\\sl(2)$-subalgebra in $\\sp(4)$."}
{"category": "Math", "title": "The Starting and Stopping Problem under Knightian Uncertainty and Related Systems of Reflected BSDEs", "abstract": "This article deals with the starting and stopping problem under Knightian uncertainty, i.e., roughly speaking, when the probability under which the future evolves is not exactly known. We show that the lower price of a plant submitted to the decisions of starting and stopping is given by a solution of a system of two reflected backward stochastic differential equations (BSDEs for short). We solve this latter system and we give the expression of the optimal strategy. Further we consider a more general system of $m$ ($m\\geq 2$) reflected BSDEs with interconnected obstacles. Once more we show existence and uniqueness of the solution of that system."}
{"category": "Math", "title": "Computation of expansions for the maximum likelihood estimator and its distribution function", "abstract": "In this paper, insight is given in the techniques used to compute asymptotic expansions. In a broad fashion the technique is described. Most of the results apply to the paper \"An expansion for the maximum likelihood estimator and its distribution function\", which will be submitted."}
{"category": "Math", "title": "Discrete sets with minimal moment of inertia", "abstract": "This paper has been withdrawn by the author due to a crucial error in the proof of Theorem 1."}
{"category": "Math", "title": "The ambient metric", "abstract": "This paper provides details of the construction, properties and some applications of the ambient metric associated to a conformal class of metrics on a smooth manifold. Existence and uniqueness of formal expansions defining such metrics are considered. Equivalence with the expansions of associated Poincare metrics is established. Definitions and properties of conformal curvature tensors defined by ambient metrics together with formulation and proof of a jet isomorphism theorem with application to the characterization of scalar conformal invariants are given."}
{"category": "Math", "title": "Characterizing Generic Global Rigidity", "abstract": "A d-dimensional framework is a graph and a map from its vertices to E^d. Such a framework is globally rigid if it is the only framework in E^d with the same graph and edge lengths, up to rigid motions. For which underlying graphs is a generic framework globally rigid? We answer this question by proving a conjecture by Connelly, that his sufficient condition is also necessary: a generic framework is globally rigid if and only if it has a stress matrix with kernel of dimension d+1, the minimum possible. An alternate version of the condition comes from considering the geometry of the length-squared mapping l: the graph is generically locally rigid iff the rank of l is maximal, and it is generically globally rigid iff the rank of the Gauss map on the image of l is maximal. We also show that this condition is efficiently checkable with a randomized algorithm, and prove that if a graph is not generically globally rigid then it is flexible one dimension higher."}
{"category": "Math", "title": "Classification of sesquilinear forms with the first argument on a subspace or a factor space", "abstract": "We give canonical matrices of bilinear or sesquilinear forms UxV-->C, (V/U)xV-->C, in which V is a vector space over the field C of complex numbers and U is its subspace."}
{"category": "Math", "title": "Canonical matrices of isometric operators on indefinite inner product spaces", "abstract": "We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases: (i) F is an algebraically closed field of characteristic different from 2 or a real closed field, and B is symmetric or skew-symmetric; (ii) F is an algebraically closed field or the skew field of quaternions over a real closed field, and B is Hermitian or skew-Hermitian with respect to any nonidentity involution on F. We use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver. This method was described in [V.V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501]."}
{"category": "Math", "title": "Partitioning 3-homogeneous latin bitrades", "abstract": "A latin bitrade $(T^{\\diamond}, T^{\\otimes})$ is a pair of partial latin squares which defines the difference between two arbitrary latin squares $L^{\\diamond} \\supseteq T^{\\diamond}$ and $L^{\\diamond} \\supseteq T^{\\otimes}$ of the same order. A 3-homogeneous bitrade $(T^{\\diamond}, T^{\\otimes})$ has three entries in each row, three entries in each column, and each symbol appears three times in $T^{\\diamond}$. Cavenagh (2006) showed that any 3-homogeneous bitrade may be partitioned into three transversals. In this paper we provide an independent proof of Cavenagh's result using geometric methods. In doing so we provide a framework for studying bitrades as tessellations of spherical, euclidean or hyperbolic space."}
{"category": "Math", "title": "Sharp asymptotics for the partition function of some continuous-time directed polymers", "abstract": "This paper is concerned with two related types of directed polymers in a random medium. The first one is a d-dimensional Brownian motion living in a random environment which is Brownian in time and homogeneous in space. The second is a continuous-time random walk on the lattice Z^d, in a random environment with similar properties as in continuous space. The case of a space-time white noise environment can be acheived in this second setting. By means of some Gaussian tools, we estimate the free energy of these models at low temperature, and give some further information on the strong disorder regime of the objects under consideration."}
{"category": "Math", "title": "On some properties of Riemann zeta function on critical line", "abstract": "The aim of this paper is to show further results following those published in [5], and to relate the Riemann zeta function to the relativistic cosmology."}
{"category": "Math", "title": "Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils", "abstract": "V. I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a simple normal form for a family of complex n-by-n matrices that smoothly depend on parameters with respect to similarity transformations that smoothly depend on the same parameters. We construct analogous normal forms for a family of real matrices and a family of matrix pencils that smoothly depend on parameters, simplifying their normal forms by D. M. Galin [Uspehi Mat. Nauk 27 (1) (1972) 241-242] and by A. Edelman, E. Elmroth, B. Kagstrom [Siam J. Matrix Anal. Appl. 18 (3) (1997) 653-692]."}
{"category": "Math", "title": "Generic families of matrix pencils and their bifurcation diagrams", "abstract": "V. I. Arnold [Russian Math. Surveys 26, no. 2, 1971, 29-43] constructed smooth generic families of matrices with respect to similarity transformations depending smoothly on the entries of matrices and got bifurcation diagrams of such families with a small number of parameters. We extend these results to pencils of matrices."}
{"category": "Math", "title": "If all geodesics are closed on the projective plane", "abstract": "The paper shows that the curvature of RP2 is constant iff all geodesics are closed. Therefore RP2 is the first known manifold with only one G-structure. It took quiete a long time to find such a manifold. The author shows only that if all geodesics are closed then there are infinitely many simple closed geodesics. This proof is based on the geodesic return map and the theory of topological dynamics. From important results of Green, Grove and Gromoll one can conclude the theorem."}
{"category": "Math", "title": "Classification of squared normal operators on unitary and Euclidean spaces", "abstract": "We give a canonical form for a complex matrix, whose square is normal, under transformations of unitary similarity as well as a canonical form for a real matrix, whose square is normal, under transformations of orthogonal similarity."}
{"category": "Math", "title": "Fair Triangulations", "abstract": "We describe the statistics of checkerboard triangulations obtained by colouring black every other triangle in triangulations of convex polygons."}
{"category": "Math", "title": "Correction of Errors in the First-Order Perturbation Expansions of Singular Vectors", "abstract": "This note corrects some serious errors in the first-order perturbation analysis of singular vectors as published by G.W. Stewart in his book \"Matrix Algorithms Volume II: Eigensystems\", SIAM, 2001."}
{"category": "Math", "title": "Quasi-Fuchsian AdS representations are Anosov", "abstract": "In a recent paper, Q. M\\'erigot proved that representations in SO(2,n) of uniform lattices of SO(1,n) which are Anosov in the sense of Labourie are quasi-Fuchsian, i.e. are faithfull, discrete, and preserve an acausal subset in the boundary of anti-de Sitter space. In the present paper, we prove the reverse implication. It also includes: -- A construction of Dirichlet domains in the context of anti-de Sitter geometry, -- A proof that spatially compact globally hyperbolic anti-de Sitter spacetimes with acausal limit set admit locally CAT(-1) Cauchy hypersurfaces."}
{"category": "Math", "title": "A Floer homology for exact contact embeddings", "abstract": "In this paper we construct the Floer homology for an action functional which was introduced by Rabinowitz and prove a vanishing theorem. As an application, we show that there are no displaceable exact contact embeddings of the unit cotangent bundle of a sphere of dimension greater than three into a convex exact symplectic manifold with vanishing first Chern class. This generalizes Gromov's result that there are no exact Lagrangian embeddings of a sphere into a complex vector space."}
{"category": "Math", "title": "Modulation invariant bilinear T(1) theorem", "abstract": "We prove a T(1) theorem for bilinear singular integral operators (trilinear forms) with a one-dimensional modulation symmetry."}
{"category": "Math", "title": "Bilinear multipliers on Lorenzt spaces", "abstract": "We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform."}
{"category": "Math", "title": "Poincar\\'e series associated with surface singularities", "abstract": "We unify and generalize formulas obtained by Campillo, Delgado and Gusein-Zade in their series of articles. Positive results are established for rational and minimally elliptic singularities. By examples and counterexamples we also try to find the `limits' of these identities. Connections with the Seiberg-Witten Invariant Conjecture and Semigroup Density Conjecture are discussed."}
{"category": "Math", "title": "A characterization of nef and good divisors by asymptotic multiplier ideals", "abstract": "A characterization of nef and good divisors is given: a divisor D on a smooth complex projective variety is nef and good if and only if the asymptotic multiplier ideals of sufficiently high multiples of e(D) D$ are trivial, where e(D) denotes the exponent of the divisor D. Some results of the same kind are proved in the analytic setting."}
{"category": "Math", "title": "Morita theory of comodules over corings", "abstract": "By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule $\\Sigma$ of an $A$-coring $\\cC$. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring $\\cC$ or the comodule $\\Sigma$ is finitely generated and projective as an $A$-module. That is, we obtain relations between the category of $\\cC$-comodules and the category of firm modules for a firm ring $R$, which is an ideal of the endomorphism algebra $^\\cC(\\Sigma)$. For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map."}
{"category": "Math", "title": "Generic fiber of power series ring extensions", "abstract": "Let D be a Noetherian domain containing a field, d a nonzero nonunit of D and z an indeterminate over D. We prove that the generic fiber of D[1/d][[z]] over D[[z]] has dimension greater than the dimension of D/dD."}
{"category": "Math", "title": "An elementary construction of Anick's fibration", "abstract": "Cohen, Moore, and Neisendorfer's work on the odd primary homotopy theory of spheres and Moore spaces, as well as the first author's work on the secondary suspension, predicted the existence of a p-local fibration S^2n-1 --> T --> \\Omega S^2n+1 whose connecting map is degree p^r. In a long and complex monograph, Anick constructed such a fibration for p>= 5 and r>= 1. Using new methods we give a much more conceptual construction which is also valid for p=3 and r>= 1. We go on to establish several properties of the space T."}
{"category": "Math", "title": "The Schur group of an abelian number field", "abstract": "We characterize the maximum $r$-local index of a Schur algebra over an abelian number field $K$ in terms of global information determined by the field $K$, for $r$ an arbitrary rational prime. This completes and unifies previous results of Janusz and Pendergrass."}
{"category": "Math", "title": "The gap between the Schur group and the subgroup generated by cyclic cyclotomic algebras", "abstract": "Let $K$ be an abelian extension of the rationals. Let $S(K)$ be the Schur group of $K$ and let $CC(K)$ be the subgroup of $S(K)$ generated by classes containing cyclic cyclotomic algebras. We characterize when $CC(K)$ has finite index in $S(K)$ in terms of the relative position of $K$ in the lattice of cyclotomic extensions of the rationals."}
{"category": "Math", "title": "Grassmannian spectral shooting", "abstract": "We present a new numerical method for computing the pure-point spectrum associated with the linear stability of coherent structures. In the context of the Evans function shooting and matching approach, all the relevant information is carried by the flow projected onto the underlying Grassmann manifold. We show how to numerically construct this projected flow in a stable and robust manner. In particular, the method avoids representation singularities by, in practice, choosing the best coordinate patch representation for the flow as it evolves. The method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed. For large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization. We demonstrate this by comparing the two methods in three applications: Boussinesq solitary waves, autocatalytic travelling waves and Ekman boundary layer."}
{"category": "Math", "title": "A Step Beyond Kemperman's Structure Theorem", "abstract": "A classical result of Kemperman gives a complete recursive description of the structure of those subsets $A$ and $B$ of an abelian group that fail to satisfy the triangle inequality, i.e., $|A+B|<|A|+|B|$. In this paper, we achieve the complete description in the case when equality holds: $|A+B|=|A|+|B|$."}
{"category": "Math", "title": "On three dimensional conformally flat almost cosymplectic manifolds", "abstract": "In the paper there are described new examples of conformally flat three dimensional almost cosymplectic manifolds. All these manifolds form a class which was completely characterized."}
{"category": "Math", "title": "Regularization independent of the noise level: an analysis of quasi-optimality", "abstract": "The quasi-optimality criterion chooses the regularization parameter in inverse problems without taking into account the noise level. This rule works remarkably well in practice, although Bakushinskii has shown that there are always counterexamples with very poor performance. We propose an average case analysis of quasi-optimality for spectral cut-off estimators and we prove that the quasi-optimality criterion determines estimators which are rate-optimal {\\em on average}. Its practical performance is illustrated with a calibration problem from mathematical finance."}
{"category": "Math", "title": "Blow-analytic equivalence of two variable real analytic function germs", "abstract": "Blow-analytic equivalence is a notion for real analytic function germs, introduced by Tzee-Char Kuo in order to develop real analytic equisingularity theory. In this paper we give complete characterisations of blow-analytic equivalence in the two dimensional case: in terms of the real tree model for the arrangement of real parts of Newton-Puiseux roots and their Puiseux pairs, and in terms of minimal resolutions. These characterisations show that in the two dimensional case the blow-analytic equivalence is a natural analogue of topological equivalence of complex analytic function germs. Moreover, we show that in the two-dimensional case the blow-analytic equivalence can be made cascade, and hence satisfies several geometric properties. It preserves, for instance, the contact orders of real analytic arcs. In the general $n$-dimensional case, we show that a singular real modification satisfies the arc-lifting property."}
{"category": "Math", "title": "Extensions for supersingular representations of $GL_2(Q_p)$", "abstract": "Let $p>2$ be a prime number. Let $G:=GL_2(Q_p)$ and $\\pi$, $\\tau$ smooth irreducible representations of $G$ on $\\bar{F}_p$-vector spaces with a central character. We show if $\\pi$ is supersingular then $Ext^1_G(\\tau,\\pi)\\neq 0$ implies $\\tau\\cong \\pi$. This answers affirmatively for $p>2$ a question of Colmez. We also determine $Ext^1_G(\\tau,\\pi)$, when $\\pi$ is the Steinberg representation. As a consequence of our results combined with those already in the literature one knows $Ext^1_G(\\tau,\\pi)$ for all irreducible representations of $G$."}
{"category": "Math", "title": "Sums of hermitian squares and the BMV conjecture", "abstract": "Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. This has previously been known only up to length 7. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators."}
{"category": "Math", "title": "One Dimensional Locally Connected S-spaces", "abstract": "We construct, assuming Jensen's principle diamond, a one-dimensional locally connected hereditarily separable continuum without convergent sequences. The construction is an inverse limit in omega_1 steps, and is patterned after the original Fedorchuk construction of a compact S-space. To make it one-dimensional, each space in the inverse limit is a copy of the Menger sponge."}
{"category": "Math", "title": "Real Zeuthen numbers for two lines", "abstract": "Given three natural numbers $k,l,d$ such that $k+l=d(d+3)/2$, the Zeuthen number $N_{d}(l)$ is the number of nonsingular complex algebraic curves of degree $d$ passing through $k$ points and tangent to $l$ lines in $\\PP^2$. It does not depend on the generic configuration $C$ of points and lines chosen. If the points and lines are real, the corresponding number $N_{d}^\\RR(l,C)$ of real curves usually depends on the configuration chosen. We use Mikhalkin's tropical correspondence theorem to prove that for two lines the real Zeuthen problem is maximal: there exists a configuration $C$ such that $N_{d}^\\RR(2,C)=N_{d}(2)$. The correspondence theorem reduces the computation to counting certain lattice paths with multiplicities."}
{"category": "Math", "title": "The Newton polygon of a rational plane curve", "abstract": "The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the Kushnirenko-Bernstein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic rational functions, with explicit genericity conditions. We also show that the variety of rational curves with given Newton polygon is unirational and we compute its dimension. As a consequence, we obtain that any convex lattice polygon with positive area is the Newton polygon of a rational plane curve."}
{"category": "Math", "title": "Smooth Volume Rigidity for Manifolds with Negatively Curved Targets", "abstract": "We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is one. The conditions hold when the volumes or entropy-volumes of the two manifolds differ by less than a uniform constant after an appropriate normalization of the metrics. The results are qualitatively sharp in the sense that the dependencies are necessary. We give a number of corollaries."}
{"category": "Math", "title": "On the Long Time Behavior of Second Order Differential Equations with Asymptotically Small Dissipation", "abstract": "We investigate the time-asymptotic properties of solutions of the differential equation x''(t) + a(t)x'(t) + g(x(t)) = 0 in a Hilbert space, where a(.) is non-increasing and g is the gradient of a potential G. If the coefficient a(.) is constant and positive, we recover the so-called ``Heavy Ball with Friction'' system. On the other hand, when a(t)=1/(t+1) we obtain the trajectories associated to some averaged gradient system. Our analysis is mainly based on the existence of some suitable energy function. When the potential G is convex and the coeffient a is non-integrable at infinity, the energy function converges to its minimum. A more stringent condition is required to obtain the convergence of the trajectories of toward some minimum point of the potential. In the one-dimensional setting, a precise description of the convergence of solutions is given for a general coercive non-convex potentials with many local minima and maxima. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense."}
{"category": "Math", "title": "Rough Isometries of Lipschitz Function Spaces", "abstract": "We show that rough isometries between metric spaces X, Y can be lifted to the spaces of real valued 1-Lipschitz functions over X and Y with supremum metric and apply this to their scaling limits. For the inverse, we show how rough isometries between X and Y can be reconstructed from structurally enriched rough isometries between their Lipschitz function spaces."}
{"category": "Math", "title": "Adapted Linear-Nonlinear Decomposition And Global Well-posedness For Solutions To The Defocusing Cubic Wave Equation On $\\mathbb{R}^{3}$", "abstract": "We prove global well-posedness for the defocusing cubic wave equation with data in $H^{s} \\times H^{s-1}$, $1>s>{13/18}$. The main task is to estimate the variation of an almost conserved quantity on an arbitrary long time interval. We divide it into subintervals. On each of these subintervals we write the solution as the sum of its linear part adapted to the subinterval and its corresponding npnlinear part. Some terms resulting from this decomposition have a controlled global variation and other terms have a slow local variation."}
{"category": "Math", "title": "Iterated distributive laws", "abstract": "We give a framework for combining $n$ monads on the same category via distributive laws satisfying Yang-Baxter equations, extending the classical result of Barr and Wells which combines two monads via one distributive law. We show that this corresponds to iterating $n$-times the process of taking the 2-category of monads in a 2-category, extending the result of Street characterising distributive laws. We show that this framework can be used to construct the free strict $n$-category monad on $n$-dimensional globular sets; we first construct for each $i$ a monad for composition along bounding $i$-cells, and then we show that the interchange laws define distributive laws between these monads, satisfying the necessary Yang-Baxter equations."}
{"category": "Math", "title": "Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials", "abstract": "This is a short survey of a class of functions introduces by Tom Apostol. The survey is focused on their relation to Eulerian polynomials, derivative polynomials, and also on some integral representations."}
{"category": "Math", "title": "The values of an Euler sum at negative integers and relation to a convolution of Bernoulli numbers", "abstract": "We study a special Dirichlet series studied before by Apostol and Matsuoka and specify its values at negative integers. These values are related to a certain convolution of Bernoulli numbers"}
{"category": "Math", "title": "On the Use of Integrals to Evaluate Series of Rational Terms", "abstract": "This article is written with the hope to draw attention to a method that uses integral transforms to find exact values for a large class of convergent series (and, in particular, series of rational terms). We apply the method to some series that generalize a problem published in `The College Mathematics Journal'. Some of the results thus obtained have not been founded in standard references."}
{"category": "Math", "title": "Some bijections on set partitions", "abstract": "This paper has been withdrawn by the author due to an error in Lemma 3, making the (bijective) proof of Theorem 4 and Corollary 5 invalid (symmetry of k-nonnesting and k-noncrossing set partitions)."}
{"category": "Math", "title": "Inverse problems for Einstein manifolds", "abstract": "We show that the Dirichlet-to-Neumann operator of the Laplacian on an open subset of the boundary of a connected compact Einstein manifold with boundary determines the manifold up to isometries. Similarly, for connected conformally compact Einstein manifolds of even dimension $n+1$, we prove that the scattering matrix at energy $n$ on an open subset of its boundary determines the manifold up to isometries."}
{"category": "Math", "title": "A uniform L^{\\infty} estimate for complex Monge-Ampere equations", "abstract": "We prove uniform sup-norm estimates for the Monge-Ampere equation with respect to a family of Kahler metrics which degenerate towards a pull-back of a metric from a lower dimensional manifold. This is then used to show the existence of generalized Kahler-Einstein metrics as the limits of the Kahler-Ricci flow for some holomorphic fibrations (in the spirit of Song and Tian \"The Kahler-Ricci flow on surfaces of positive Kodaira dimension\", arXiv:math/0602150)."}
{"category": "Math", "title": "Newton's method and Baker domains", "abstract": "We show that there exists an entire function f without zeros for which the associated Newton function N(z)=z-f(z)/f'(z) is a transcendental meromorphic functions without Baker domains. We also show that there exists an entire function f with exactly one zero for which the complement of the immediate attracting basin has at least two components and contains no invariant Baker domains of N. The second result answers a question of J. Rueckert and D. Schleicher while the first one gives a partial answer to a question of X. Buff."}
{"category": "Math", "title": "Optimal properties of some Bayesian inferences", "abstract": "Relative surprise regions are shown to minimize, among Bayesian credible regions, the prior probability of covering a false value from the prior. Such regions are also shown to be unbiased in the sense that the prior probability of covering a false value is bounded above by the prior probability of covering the true value. Relative surprise regions are shown to maximize both the Bayes factor in favor of the region containing the true value and the relative belief ratio, among all credible regions with the same posterior content. Relative surprise regions emerge naturally when we consider equivalence classes of credible regions generated via reparameterizations."}
{"category": "Math", "title": "Convexity properties of gradient maps", "abstract": "We consider the action of a real reductive group G on a Kaehler manifold Z which is the restriction of a holomorphic action of the complexified group G^C. We assume that the induced action of a compatible maximal compact subgroup U of G^C on Z is Hamiltonian. We have an associated gradient map obtained from a Cartan decomposition of G. For a G-stable subset Y of Z we consider convexity properties of the intersection of the image of Y under the gradient map with a closed Weyl chamber. Our main result is a Convexity Theorem for real semi-algebraic subsets Y of the projective space corresponding to a unitary representation of U."}
{"category": "Math", "title": "A note on local trigonal fibrations", "abstract": "We show that some of centeral fibers of degenerations of hyperelliptic curves are realized as those trigonal curves. In particular, any hyperelliptic curve can be the central fiber of a degeneration of trigonal curves."}
{"category": "Math", "title": "Bimonads and Hopf monads on categories", "abstract": "The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to {\\em monoidal} categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category $\\A$ and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on $\\A$: we define a {\\em bimonad} on $\\A$ as an endofunctor $B$ which is a monad and a comonad with an entwining $\\lambda:BB\\to BB$ satisfying certain conditions. This $\\lambda$ is also employed to define the category $\\A^B_B$ of (mixed) $B$-bimodules. In the classical situation, an entwining $\\lambda$ is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws $\\tau:BB\\to BB$ satisfying the Yang-Baxter equation ({\\em local prebraidings}) which induce an entwining $\\lambda$ and lead to an extension of the theory of {\\em braided Hopf algebras}. An antipode is defined as a natural transformation $S:B\\to B$ with special properties and for categories $\\A$ with limits or colimits and bimonads $B$ preserving them, the existence of an antipode is equivalent to $B$ inducing an equivalence between $\\A$ and the category $\\A^B_B$ of $B$-bimodules. This is a general form of the {\\em Fundamental Theorem} of Hopf algebras."}
{"category": "Math", "title": "Improved estimation of the MSEs and the MSE matrices for shrinkage estimators of multivariate normal means and their applications", "abstract": "In this article we provide some nonnegative and positive estimators of the mean squared errors(MSEs) for shrinkage estimators of multivariate normal means. Proposed estimators are shown to improve on the uniformly minimum variance unbiased estimator(UMVUE) under a quadratic loss criterion. A similar improvement is also obtained for the estimators of the MSE matrices for shrinkage estimators. We also apply the proposed estimators of the MSE matrix to form confidence sets centered at shrinkage estimators and show their usefulness through numerical experiments."}
{"category": "Math", "title": "Combinatorial Alexander Duality -- a Short and Elementary Proof", "abstract": "Let X be a simplicial complex with the ground set V. Define its Alexander dual as a simplicial complex X* = {A \\subset V: V \\setminus A \\notin X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V|-i-3)-th reduced cohomology group of X* (over a given commutative ring R). We give a self-contained proof."}
{"category": "Math", "title": "The structure of Sally modules of rank one", "abstract": "A complete structure theorem of Sally modules of $\\fkm$-primary ideals $I$ in a Cohen-Macaulay local ring $(A, \\m)$ satisfying the equality $\\e_1(I)=\\e_0(I)-\\ell_A(A/I)+1$ is given, where $\\e_0(I)$ and $\\e_1(I)$ denote the first two Hilbert coefficients of $I$."}
{"category": "Math", "title": "Connectivity of Addition Cayley Graphs", "abstract": "For any finite abelian group $G$ and any subset $S\\seq G$, we determine the connectivity of the addition Cayley graph induced by $S$ on $G$. Moreover, we show that if this graph is not complete, then it possesses a minimum vertex cut of a special, explicitly described form."}
{"category": "Math", "title": "From Bruhat intervals to intersection lattices and a conjecture of Postnikov", "abstract": "We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation $w\\in \\Sn$ is at most the number of elements below $w$ in the Bruhat order, and (B) that equality holds if and only if $w$ avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns."}
{"category": "Math", "title": "Euler Characteristic of real nondegenerate tropical complete intersections", "abstract": "We define nondegenerate tropical complete intersections imitating the corresponding definition in complex algebraic geometry. As in the complex situation, all nonzero intersection multiplicity numbers between tropical hypersurfaces defining a nondegenerate tropical complete intersection are equal to 1. The intersection multiplicity numbers we use are sums of mixed volumes of polytopes which are dual to cells of the tropical hypersurfaces. We show that the Euler characteristic of a real nondegenerate tropical complete intersection depends only on the Newton polytopes of the tropical polynomials which define the intersection. Basically, it is equal to the usual signature of a complex complete intersection with same Newton polytopes, when this signature is defined. The proof reduces to the toric hypersurface case, and uses the notion of $E$-polynomials of complex varieties."}
{"category": "Math", "title": "Quantum cohomology of minuscule homogeneous spaces III : semi-simplicity and consequences", "abstract": "We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at q=1, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in a previous paper. We deduce Vafa-Intriligator type formulas for the Gromov-Witten invariants."}
{"category": "Math", "title": "A Canonical Quadratic Form on the Determinant Line of a Flat Vector Bundle", "abstract": "We introduce and study a canonical quadratic form, called the torsion quadratic form, of the determinant line of a flat vector bundle over a closed oriented odd-dimensional manifold. This quadratic form caries less information than the refined analytic torsion, introduced in our previous work, but is easier to construct and closer related to the combinatorial Farber-Turaev torsion. In fact, the torsion quadratic form can be viewed as an analytic analogue of the Poincare-Reidemeister scalar product, introduced by Farber and Turaev. Moreover, it is also closely related to the complex analytic torsion defined by Cappell and Miller and we establish the precise relationship between the two. In addition, we show that up to an explicit factor, which depends on the Euler structure, and a sign the Burghelea-Haller complex analytic torsion, whenever it is defined, is equal to our quadratic form. We conjecture a formula for the value of the torsion quadratic form at the Farber-Turaev torsion and prove some weak version of this conjecture. As an application we establish a relationship between the Cappell-Miller and the combinatorial torsions."}
{"category": "Math", "title": "Hamming Distance for Conjugates", "abstract": "Let x, y be strings of equal length. The Hamming distance h(x,y) between x and y is the number of positions in which x and y differ. If x is a cyclic shift of y, we say x and y are conjugates. We consider f(x,y), the Hamming distance between the conjugates xy and yx. Over a binary alphabet f(x,y) is always even, and must satisfy a further technical condition. By contrast, over an alphabet of size 3 or greater, f(x,y) can take any value between 0 and |x|+|y|, except 1; furthermore, we can always assume that the smaller string has only one type of letter."}
{"category": "Math", "title": "On the computation of Galois representations associated to level one modular forms", "abstract": "In this paper we explicitly compute mod-l Galois representations associated to modular forms. To be precise, we look at cases with l<=23 and the modular forms considered will be cusp forms of level 1 and weight up to 22. We present the result in terms of polynomials associated to the projectivised representations. As an application, we will improve a known result on Lehmer's non-vanishing conjecture for Ramanujan's tau function."}
{"category": "Math", "title": "Realization of abstract convex geometries by point configurations, Part 1", "abstract": "The Edelman-Jamison problem is to characterize those abstract convex geometries that are representable by a set of points in the plane. We show that some natural modification of the Edelman-Jamison problem is equivalent to the well known NP-hard order type problem."}
{"category": "Math", "title": "The forgetful map in rational K-theory", "abstract": "Let G be a connected reductive algebraic group acting on a scheme X. Let R(G) denote the representation ring of G, and let I be the ideal in R(G) of virtual representations of rank 0. Let G(X) (resp. G(G,X)) denote the Grothendieck group of coherent sheaves (resp. G-equivariant coherent sheaves) on X. Merkurjev proved that if the fundamental group of G is torsion-free, then the map of G(G,X)/IG(G,X) to G(X) is an isomorphism. Although this map need not be an isomorphism if the fundamental group of G has torsion, we prove that without the assumption on the fundamental group of G, this map is an isomorphism after tensoring with the rational numbers."}
{"category": "Math", "title": "Minimization of convex functionals over frame operators", "abstract": "We present results about minimization of convex functionals defined over a finite set of vectors in a finite dimensional Hilbert space, that extend several known results for the Benedetto-Fickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original one."}
{"category": "Math", "title": "Algorithmically detecting the bridge number of hyperbolic knots", "abstract": "We exhibit an algorithm to determine the bridge number of a hyperbolic knot in the 3-sphere. The proof uses adaptations of almost normal surface theory for compact surfaces with boundary in ideally triangulated knot exteriors."}
{"category": "Math", "title": "Generic canonical form of pairs of matrices with zeros", "abstract": "We consider a family of pairs of m-by-p and m-by-q matrices, in which some entries are required to be zero and the others are arbitrary, with respect to transformations (A,B)--> (SAR,SBL) with nonsingular S, R, L. We prove that almost all of these pairs reduce to the same pair (C, D) from this family, except for pairs whose arbitrary entries are zeros of a certain polynomial. The polynomial and the pair (C D) are constructed by a combinatorial method based on properties of a certain graph."}
{"category": "Math", "title": "Riemannian groupoids and solitons for three-dimensional homogeneous Ricci and cross curvature flows", "abstract": "In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John Lott, allows us to investigate the long term behavior of collapsing solutions of the flow, producing soliton solutions in the limit. We also review Lott's results on the long term behavior of three-dimensional homogeneous solutions of Ricci flow, demonstrating the coordinates we choose and reviewing the groupoid technique. We find cross curvature soliton metrics on Sol and Nil, and show that the cross curvature flow of SL(2,R) limits to Sol."}
{"category": "Math", "title": "Miniversal deformations of chains of linear mappings", "abstract": "V.I. Arnold [Russian Math. Surveys, 26 (no. 2), 1971, pp. 29-43] gave a miniversal deformation of matrices of linear operators; that is, a simple canonical form, to which not only a given square matrix A, but also the family of all matrices close to A, can be reduced by similarity transformations smoothly depending on the entries of matrices. We study miniversal deformations of quiver representations and obtain a miniversal deformation of matrices of chains of linear mappings."}
{"category": "Math", "title": "Normal holomorphic curves from parabolic regions to projective spaces", "abstract": "A holomorphic map from the complex line to a complex projective space is called normal (a. k. a. Brody curve) if it is uniformly continuous from the Euclidean metric to the Fubini--Study metric. The paper contains a survey of known results about such maps, as well as some new theorems."}
{"category": "Math", "title": "Renewal-type Limit Theorem for the Gauss Map and Continued Fractions", "abstract": "In this paper we prove the following renewal-type limit theorem. Given an irrational $\\alpha$ in (0,1) and R>0, let $q_{n_R}$ be the first denominator of the convergents of $\\alpha$ which exceeds R. The main result in the paper is that the ratio $q_{n_R}/R$ has a limiting distribution as R tends to infinity. The existence of the limiting distribution uses mixing of a special flow over the natural extension of the Gauss map."}
{"category": "Math", "title": "A Limit Theorem for Birkoff sums of non-integrable functions over rotations", "abstract": "We consider Birkhoff sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let $S_N= S_N(\\alpha,x)$ be the N^th non-renormalized Birkhoff sum, where $x in [0,1)$ is the initial point, $\\alpha\\in [0,1)$ is the rotation number and $(\\alpha, x)$ are uniformly distributed. We prove that $S_N/N$ has a joint limiting distribution in $(\\alpha,x)$ as N tends to infinity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums."}
{"category": "Math", "title": "On Groups with a Supercomplemented Subgroup", "abstract": "Groups, in which every subgroup containing some fixed primary cyclic subgroup has a complement, are investigated."}
{"category": "Math", "title": "On freely indecomposable measures", "abstract": "We show that a probability measure is not a nontrivial free additive convolution if it puts no mass in an interval whose endpoints are atoms. The analogous results for free multiplicative convolutions are proved as well. The proofs use analytic subordination."}
{"category": "Math", "title": "On isogenous principally polarized abelian surfaces", "abstract": "We study a relationship between two genus 2 curves whose jacobians are isogenous with kernel equal to a maximal isotropic subspace of p-torsion points with respect to the Weil pairing. For p = 3 we find an explicit relationship between the set of Weierstrass points of the two curves extending the classical results of F. Richelot (1837) and G. Humbert (1901) in the case p = 2."}
{"category": "Math", "title": "Unique continuation results for Ricci curvature and applications", "abstract": "Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss-Codazzi constraint equations under deformations are also discussed."}
{"category": "Math", "title": "Elliptic nets and elliptic curves", "abstract": "An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose E is an elliptic curve over a field K, and P_1, ..., P_n are points on E defined over K. To this information we associate an n-dimensional array of values in K satisfying a nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. We demonstrate an explicit bijection between the set of elliptic nets and the set of elliptic curves with specified points. We also obtain Laurentness/integrality results for elliptic nets."}
{"category": "Math", "title": "The non-existence of certain mod 2 Galois representations of some small quadratic fields", "abstract": "For a few quadratic fields, the non-existence is proved of continuous irreducible mod 2 Galois representations of degree 2 unramified outside 2."}
{"category": "Math", "title": "Dedekind-Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra", "abstract": "We study higher-dimensional analogs of the Dedekind-Carlitz polynomials c(u,v;a,b) := sum_{k=1..b-1} u^[ka/b] v^(k-1), where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law (v-1) c(u,v;a,b) + (u-1) c(v,u;b,a) = u^(a-1) v^(b-1) - 1, from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that Dedekind-Carlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the Carlitz reciprocity law and various extensions of it. Our approach gives rise to new reciprocity theorems and computational complexity results for Dedekind-Carlitz polynomials, a characterization of Dedekind-Carlitz polynomials in terms of generating functions of lattice points in triangles, and a multivariate generalization of the Mordell-Pommersheim theorem on the appearance of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice polytopes."}
{"category": "Math", "title": "Traces of heat operators on Riemannian foliations", "abstract": "We consider the basic heat operator on functions on a Riemannian foliation of a compact, Riemannian manifold, and we show that the trace of this operator has a particular short time asymptotic expansion. The coefficients in this expansion are obtainable from local transverse geometric invariants - functions computable by analyzing the manifold in an arbitrarily small neighborhood of a leaf closure. Using this expansion, we prove some results about the spectrum of the basic Laplacian, such as the analogue of Weyl's asymptotic formula. Also, we explicitly calculate the first two nontrivial coefficients of the expansion for special cases such as codimension two foliations and foliations with regular closure."}
{"category": "Math", "title": "The Galois action on character tables", "abstract": "A geometric interpretation and generalisation for the Galois action on finite group character tables is sketched. The generalisation is a Galois action on the space Map_G(G^n,\\bar{Q})/S_n for each finite G, where G acts by simultaneous conjugation on the n-tuples G^n and the symmetric group S_n permutes the components."}
{"category": "Math", "title": "Conformal field theory and mapping class groups", "abstract": "Rational conformal field theories produce a tower of finite-dimensional representations of surface mapping class groups, acting on the conformal blocks of the theory. We review this formalism. We show that many recent mathematical developments can be fit into the first 2 floors of this tower. We also review what is known in higher genus."}
{"category": "Math", "title": "Polyexponentials", "abstract": "We discuss a special function (polyexponential) that extends the natural exponential function and also the exponential integral. The basic properties of the polyexponential are listed and some applications are given. In particular, it is shown that certain Mellin integrals can be evaluated in terms of polyexponentials. The polyexponential is related to the exponential polynomials, the Riemann zeta function, the Dirichlet eta function and the Lerch Transcendent."}
{"category": "Math", "title": "Dimension Reduction for the Hyperbolic Space", "abstract": "A dimension reduction for the hyperbolic space is established. When points are far apart an embedding with bounded distortion into the hyperbolic plane is achieved."}
{"category": "Math", "title": "On the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution", "abstract": "We consider $n\\times n$ real symmetric and hermitian random matrices $H_{n,m}$ equals the sum of a non-random matrix $H_{n}^{(0)}$ matrix and the sum of $m$ rank-one matrices determined by $m$ i.i.d. isotropic random vectors with log-concave probability law and i.i.d. random amplitudes $\\{\\tau_{\\alpha }\\}_{\\alpha =1}^{m}$. This is a generalization of the case of vectors uniformly distributed over the unit sphere, studied in [Marchenko-Pastur (1967)]. We prove that if $n\\to \\infty, m\\to \\infty, m/n\\to c\\in \\lbrack 0,\\infty)$ and that the empirical eigenvalue measure of $H_{n}^{(0)}$ converges weakly, then the empirical eigenvalue measure of $H_{n,m}$ converges in probability to a non-random limit, found in [Marchenko-Pastur (1967)]."}
{"category": "Math", "title": "The asymptotic Tian-Yau-Zelditch expansion on Riemann surfaces with Constant Curvature", "abstract": "Let $M$ be a regular Riemann surface with a metric which has constant scalar curvature $\\rho$. We give the asymptotic expansion of the sum of the square norm of the sections of the pluricanonical bundles $K_{M}^{m}$. That is, \\[\\sum_{i=0}^{d_{m}-1}\\|S_{i}(x_{0})\\|_{h_{m}}^{2} \\sim m(1+\\frac{\\rho}{2 m})+O(e^{-\\frac{(\\log m)^{2}}{8}}),\\] where $\\{S_{0},...,S_{d_{m}-1}\\}$ is an orthonormal basis for $H^{0}(M, K_{M}^{m})$ for sufficiently large $m$."}
{"category": "Math", "title": "On the Kirchheim-Magnani counterexample to metric differentiability", "abstract": "In this short note we give an interpretation of the Kirchheim-Magnani counterexample to metric differentiability in terms of dilatation structures."}
{"category": "Math", "title": "On the Newman sum over multiples of a prime with a primitive or semiprimitive root 2", "abstract": "We obtain a simple relations for the Newman sum over multiples of a prime with a primitive or semiprimitive root 2. We consider the case of p=17 as well."}
{"category": "Math", "title": "Studies on the Lorenz model", "abstract": "We study the Lorenz model from the viewpoint of its accessible singularities and local index."}
{"category": "Math", "title": "Nodal solutions to quasilinear elliptic equations on compact Riemannian manifolds", "abstract": "We show the existence of nodal solutions to perturbed quasilinear elliptic equations with critical Sobolev exponent on compact Riemannian manifolds. A nonexistence result is also given."}
{"category": "Math", "title": "Complementary self-similarity", "abstract": "A few aspects of self-similarity related to complementary components of closed subsets of R^n are briefly discussed."}
{"category": "Math", "title": "Matrices Totally Positive Relative to a Tree", "abstract": "It is known that for a totally positive (TP) matrix, the eigenvalues are positive and distinct and the eigenvector associated with the smallest eigenvalue is totally nonzero and has an alternating sign pattern. Here, a certain weakening of the TP hypothesis is shown to yield a similar conclusion."}
{"category": "Math", "title": "A problem of enumeration of two-color bracelets with several variations", "abstract": "We consider the problem of enumeration of incongruent two-color bracelets of $n$ beads, $k$ of which are black, and study several natural variations of this problem. We also give recursion formulas for enumeration of $t$-color bracelets, $t\\geq3."}
{"category": "Math", "title": "On the symplectic phase space of KdV", "abstract": "We prove that the Birkhoff map $\\Om$ for KdV constructed on $H^{-1}_0(\\T)$ can be interpolated between $H^{-1}_0(\\T)$ and $L^2_0(\\T)$. In particular, the symplectic phase space $H^{1/2}_0(\\T)$ can be described in terms of Birkhoff coordinates. As an application, we characterize the regularity of a potential $q\\in H^{-1}(\\T)$ in terms of the decay of the gap lengths of the periodic spectrum of Hill's operator on the interval $[0,2]$."}
{"category": "Math", "title": "Quasi-socle ideals in Gorenstein numerical semigroup rings", "abstract": "Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \\mathfrak{m}^q$ in Gorenstein numerical semigroup rings over fields are explored, where $Q$ is a parameter ideal, and $\\mathfrak{m}$ is the maximal ideal in the base local ring, and $q \\geq 1$ is an integer. The problems of when $I$ is integral over $Q$ and of when the associated graded ring $\\mathrm{G}(I) = \\bigoplus_{n \\geq 0}I^n/I^{n+1}$ of $I$ is Cohen-Macaulay are studied. The problems are rather wild; examples are given."}
{"category": "Math", "title": "Quasi-socle ideals in local rings with Gorenstein tangent cones", "abstract": "Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \\mathfrak{m}^q$ in a Noetherian local ring $(A, \\mathfrak{m})$ with the Gorenstein tangent cone $\\mathrm{G}(\\mathfrak{m}) = \\bigoplus_{n \\geq 0}{\\mathfrak{m}}^n/{\\mathfrak{m}}^{n+1}$ are explored, where $q \\geq 1$ is an integer and $Q$ is a parameter ideal of $A$ generated by monomials of a system $x_1, x_2, ..., x_d$ of elements in $A$ such that $(x_1, x_2, ..., x_d)$ is a reduction of $\\mathfrak{m}$. The questions of when $I$ is integral over $Q$ and of when the graded rings $\\mathrm{G}(I) = \\bigoplus_{n \\geq 0}I^n/I^{n+1}$ and $\\mathrm{F}(I) = \\bigoplus_{n \\ge 0}I^n/\\mathfrak{m} I^n$ are Cohen-Macaulay are answered. Criteria for $\\mathrm{G} (I)$ and $\\mathcal{R} (I) = \\bigoplus_{n \\geq 0}I^n$ to be Gorenstein rings are given."}
{"category": "Math", "title": "Patching over fields", "abstract": "We develop a new form of patching that is both far-reaching and more elementary than the previous versions that have been used in inverse Galois theory for function fields of curves. A key point of our approach is to work with fields and vector spaces, rather than rings and modules. After presenting a self-contained development of this form of patching, we obtain applications to other structures such as Brauer groups and differential modules."}
{"category": "Math", "title": "The isoperimetric profile of a compact Riemannian Manifold for small volumes", "abstract": "We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,\\alpha}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifold, there is a unique family concentric pseudo balls which contains all the pseudo balls $C^{2,\\alpha}$-close to small spheres. This allows us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension."}
{"category": "Math", "title": "Logic of Simultaneity", "abstract": "A logical model of spatiotemporal structures is pictured as a succession of processes in time. One usual way to formalize time structure is to assume the global existence of time points and then collect some of them to form time intervals of processes. Under this set-theoretic approach, the logic that governs the processes acquires a Boolean structure. However, in a real distributed system or a relativistic universe where the message-passing time between different locations is not negligible, the logic has no choice but to accept time interval instead of time point as a primitive concept. From this modeling process of spatiotemporal structures, orthologic, the most simplified version of quantum logic, emerges naturally."}
{"category": "Math", "title": "Covering an uncountable square by countably many continuous functions", "abstract": "We prove that there exists a countable family of continuous real functions whose graphs together with their inverses cover an uncountable square, i.e. a set of the form $X\\times X$, where $X$ is an uncountable subset of the real line. This extends Sierpi\\'nski's theorem from 1919, saying that $S\\times S$ can be covered by countably many graphs of functions and inverses of functions if and only if the size of $S$ does not exceed $\\aleph_1$. Our result is also motivated by Shelah's study of planar Borel sets without perfect rectangles."}
{"category": "Math", "title": "Equivariant representable K-theory", "abstract": "We interpret certain equivariant Kasparov groups as equivariant representable K-theory groups. We compute these groups via a classifying space and as K-theory groups of suitable sigma-C*-algebras. We also relate equivariant vector bundles to these sigma-C*-algebras and provide sufficient conditions for equivariant vector bundles to generate representable K-theory. Mostly we work in the generality of locally compact groupoids with Haar system."}
{"category": "Math", "title": "Conjugation-invariant norms on groups of geometric origin", "abstract": "A group is said to be bounded if it has a finite diameter with respect to any bi-invariant metric. In the present paper we discuss boundedness of various groups of diffeomorphisms."}
{"category": "Math", "title": "Quantum invariants and free Z_{p^2} -actions on 3-manifolds", "abstract": "We give a congruence for the quantum invariant of a Z_p-quotient of a 3$-manifold with a Z_{p^2} action. We show the congruence does not hold for quotients of 3--manifolds with a Z_{5}xZ_{5} action."}
{"category": "Math", "title": "Gelfand-Kirillov conjecture for symplectic reflection algebras", "abstract": "We construct functorially a class of algebras using the formalism of double derivations. These algebras extend to higher dimensions Crawley-Boevey and Holland's construction of deformed preprojective algebras and encompass symplectic reflection algebras associated to wreath products. We use this construction to show that the quotient field of a symplectic reflection algebra is \"rational\", confirming a pair of conjectures of Etingof and Ginzburg."}
{"category": "Math", "title": "Classification of 1st order symplectic spinor operators over contact projective geometries", "abstract": "We give a classification of $1^{st}$ order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so called higher symplectic, harmonic or generalized Kostant spinor modules. Higher symplectic spinor modules are arising from the Segal-Shale-Weil representation of the metaplectic group by tensoring it by finite dimensional modules. We show that for all pairs of the considered bundles, there is at most one $1^{st}$ order invariant differential operator up to a complex multiple and give an equivalence condition for the existence of such an operator. Contact projective analogues of the well known Dirac, twistor and Rarita-Schwinger operators appearing in Riemannian geometry are special examples of these operators."}
{"category": "Math", "title": "Automorphisms of non-spherical buildings have unbounded displacement", "abstract": "If f is a nontrivial automorphism of a thick building Delta of purely infinite type, we prove that there is no bound on the distance that f moves a chamber. This has the following group-theoretic consequence: If G is a group of automorphisms of Delta with bounded quotient, then the center of G is trivial."}
{"category": "Math", "title": "The accuracy of merging approximation in generalized St. Petersburg games", "abstract": "Merging asymptotic expansions of arbitrary length are established for the distribution functions and for the probabilities of suitably centered and normalized cumulative winnings in a full sequence of generalized St. Petersburg games, extending the short expansions due to Cs\\\"org\\H{o}, S., Merging asymptotic expansions in generalized St. Petersburg games, \\textit{Acta Sci. Math. (Szeged)} \\textbf{73} 297--331, 2007. These expansions are given in terms of suitably chosen members from the classes of subsequential semistable infinitely divisible asymptotic distribution functions and certain derivatives of these functions. The length of the expansion depends upon the tail parameter. Both uniform and nonuniform bounds are presented."}
{"category": "Math", "title": "Variations on themes of Kostant", "abstract": "Let g be a complex semisimple Lie algebra and let G' be the Langlands dual group. We give a description of the cohomology algebra of an arbitrary spherical Schubert variety in the loop Grassmannian for G' as a quotient of the form Sym(g^e)/J. Here, J is an appropriate ideal in the symmetric algebra of g^e, the centralizer of a principal nilpotent in g. We also discuss a `topological' proof of Kostant's famous result on the structure of the polynomial algebra on g."}
{"category": "Math", "title": "String topology prospectra and Hochschild cohomology", "abstract": "We study string topology for classifying spaces of connected compact Lie groups, drawing connections with Hochschild cohomology and equivariant homotopy theory. First, for a compact Lie group $G$, we show that the string topology prospectrum $LBG^{-TBG}$ is equivalent to the homotopy fixed-point prospectrum for the conjugation action of $G$ on itself, $G^{hG}$. Dually, we identify $LBG^{-ad}$ with the homotopy orbit spectrum $(DG)_{hG}$, and study ring and co-ring structures on these spectra. Finally, we show that in homology, these products may be identified with the Gerstenhaber cup product in the Hochschild cohomology of $C^*(BG)$ and $C_*(G)$, respectively. These, in turn, are isomorphic via Koszul duality."}
{"category": "Math", "title": "On some properties on bivariate Fibonacci and Lucas polynomials", "abstract": "In this paper we generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. We prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers satisfying remarkable recurrence relations."}
{"category": "Math", "title": "Geometry and complexity of O'Hara's algorithm", "abstract": "In this paper we analyze O'Hara's partition bijection. We present three type of results. First, we show that O'Hara's bijection can be viewed geometrically as a certain scissor congruence type result. Second, we obtain a number of new complexity bounds, proving that O'Hara's bijection is efficient in several special cases and mildly exponential in general. Finally, we prove that for identities with finite support, the map of the O'Hara's bijection can be computed in polynomial time, i.e. much more efficiently than by O'Hara's construction."}
{"category": "Math", "title": "A large deviation approach to optimal transport", "abstract": "A probabilistic method for solving the Monge-Kantorovich mass transport problem on $R^d$ is introduced. A system of empirical measures of independent particles is built in such a way that it obeys a doubly indexed large deviation principle with an optimal transport cost as its rate function. As a consequence, new approximation results for the optimal cost function and the optimal transport plans are derived. They follow from the Gamma-convergence of a sequence of normalized relative entropies toward the optimal transport cost. A wide class of cost functions including the standard power cost functions $|x-y|^p$ enter this framework."}
{"category": "Math", "title": "Convex minimization problems with weak constraint qualifications", "abstract": "One revisits the standard saddle-point method based on conjugate duality for solving convex minimization problems. Our aim is to reduce or remove unnecessary topological restrictions on the constraint set. Dual equalities and characterizations of the minimizers are obtained with weak or without constraint qualifications. The main idea is to work with intrinsic topologies which reflect some geometry of the objective function. The abstract results of this article are applied in other papers to the Monge-Kantorovich optimal transport problem and the minimization of entropy functionals."}
{"category": "Math", "title": "A note on the cone restriction conjecture in the cylindrically symmetric case", "abstract": "In this note, we present two arguments showing that the classical \\textit{linear adjoint cone restriction conjecture} holds for the class of functions supported on the cone and invariant under the spatial rotation in all dimensions. The first is based on a dyadic restriction estimate, while the second follows from a strengthening version of the Hausdorff-Young inequality and the H\\\"older inequality in the Lorentz spaces."}
{"category": "Math", "title": "Number of binomial coefficients divided by a fixed power of a prime", "abstract": "We state a general formula for the number of binomial coefficients $n$ choose $k$ that are divided by a fixed power of a prime $p$, i.e., the number of binomial coefficients divided by $p^j$ and not divided by $p^{j+1}$."}
{"category": "Math", "title": "Asymmetry of near-critical percolation interfaces", "abstract": "We study the possible scaling limits of percolation interfaces in two dimensions on the triangular lattice. When one lets the percolation parameter p(N) vary with the size N of the box that one is considering, three possibilities arise in the large-scale limit. It is known that when p(N) does not converge to 1/2 fast enough, then the scaling limits are degenerate, whereas if p(N) - 1 / 2 goes to zero quickly, the scaling limits are SLE(6) as when p=1/2. We study some properties of the (non-void) intermediate regime where the large scale behavior is neither SLE(6) nor degenerate. We prove that in this case, the law of any scaling limit is singular with respect to that of SLE(6), even if it is still supported on the set of curves with Hausdorff dimension equal to 7/4."}
{"category": "Math", "title": "Symplectic Connections of Ricci Type and Star Products", "abstract": "In this article we relate the construction of Ricci type symplectic connections by reduction to the construction of star product by reduction yielding rather explicit descriptions for the star product on the reduced space."}
{"category": "Math", "title": "The complex of pant decompositions of a surface", "abstract": "We exhibit a set of edges (moves) and 2-cells (relations) making the complex of pant decompositions on a surface a simply connected complex. Our construction, unlike the previous ones, keeps the arguments concerning the structural transformations independent from those deriving from the action of the mapping class group. The moves and the relations turn out to be supported in subsurfaces with 3g-3+n=1,2 (where g is the genus and n is the number of boundary components), illustrating in this way the so called Grothendieck principle."}
{"category": "Math", "title": "Limits of dihedral groups", "abstract": "We give a characterization of limits of dihedral groups in the space of finitely generated marked groups. We also describe the topological closure of dihedral groups in the space of marked groups on a fixed number of generators."}
{"category": "Math", "title": "Intermediate rank and property RD", "abstract": "We introduce concepts of intermediate rank for countable groups that \"interpolate\" between consecutive values of the classical (integer-valued) rank. Various classes of groups are proved to have intermediate rank behaviors. We are especially interested in interpolation between rank 1 and rank 2. For instance, we construct groups \"of rank 7/4\". Our setting is essentially that of non positively curved spaces, where concepts of intermediate rank include polynomial rank, local rank, and mesoscopic rank. The resulting framework has interesting connections to operator algebras. We prove property RD in many cases where intermediate rank occurs. This gives a new family of groups satisfying the Baum-Connes conjecture. We prove that the reduced $C^*$-algebras of groups of rank 7/4 have stable rank 1."}
{"category": "Math", "title": "Strong Approximations of BSDEs in a domain", "abstract": "We study the strong approximation of a Backward SDE with finite stopping time horizon, namely the first exit time of a forward SDE from a cylindrical domain. We use the Euler scheme approach of Bouchard and Touzi, Zhang 04}. When the domain is piecewise smooth and under a non-characteristic boundary condition, we show that the associated strong error is at most of order $h^{\\frac14-\\eps}$ where $h$ denotes the time step and $\\eps$ is any positive parameter. This rate corresponds to the strong exit time approximation. It is improved to $h^{\\frac12-\\eps}$ when the exit time can be exactly simulated or for a weaker form of the approximation error. Importantly, these results are obtained without uniform ellipticity condition."}
{"category": "Math", "title": "Multicolor urn models with reducible replacement matrices", "abstract": "Consider the multicolored urn model where, after every draw, balls of the different colors are added to the urn in a proportion determined by a given stochastic replacement matrix. We consider some special replacement matrices which are not irreducible. For three- and four-color urns, we derive the asymptotic behavior of linear combinations of the number of balls. In particular, we show that certain linear combinations of the balls of different colors have limiting distributions which are variance mixtures of normal distributions. We also obtain almost sure limits in certain cases in contrast to the corresponding irreducible cases, where only weak limits are known."}
{"category": "Math", "title": "Algebraic quantum permutation groups", "abstract": "We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If $K$ is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra $K^n$: this is a refinement of Wang's universality theorem for the (compact) quantum permutation group. We also prove a structural result for Hopf algebras having a non-ergodic coaction on the diagonal algebra $K^n$, on which we determine the possible group gradings when $K$ is algebraically closed and has characteristic zero."}
{"category": "Math", "title": "Canonical Forms for Unitary Congruence and *Congruence", "abstract": "We use methods of the general theory of congruence and *congruence for complex matrices--regularization and cosquares-to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that \\bar{A}A (respectively, A^2) is normal. As special cases of our canonical forms, we obtain-in a coherent and systematic way-known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A^3 is normal, and (b) unitary congruence when A\\bar{A}A is normal, are both unitarily wild, so there is no reasonable hope that a simple solution to them can be found."}
{"category": "Math", "title": "Symmetric bundles and representations of Lie triple systems", "abstract": "We define symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. We show that such a bundle has an underlying reflection space, and we investigate the corresponding forgetful functor both from the point of view of differential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing \"unusual\" symmetric bundle structures on the tangent bundles of certain symmetric spaces."}
{"category": "Math", "title": "The Inductive Kernels of Graphs", "abstract": "It is well known that kernels in graphs are powerful and useful structures, for instance in the theory of games. However, a kernel does not always exist and Chv\\'atal proved in 1973 that it is an NP-Complete problem to decide its existence. We present here an alternative definition of kernels that uses an inductive machinery : the inductive kernels. We prove that inductive kernels always exist and a particular one can be constructed in quadratic time. However, it is an NP-Complete problem to decide the existence of an inductive kernel including (resp. excluding) some fixed vertex."}
{"category": "Math", "title": "Manin's conjecture for a quartic del Pezzo surface with A_4 singularity", "abstract": "The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type A_4."}
{"category": "Math", "title": "Ballistic Transport at Uniform Temperature", "abstract": "A paradigm for isothermal, mechanical rectification of stochastic fluctuations is introduced in this paper. The central idea is to transform energy injected by random perturbations into rigid-body rotational kinetic energy. The prototype considered in this paper is a mechanical system consisting of a set of rigid bodies in interaction through magnetic fields. The system is stochastically forced by white noise and dissipative through mechanical friction. The Gibbs-Boltzmann distribution at a specific temperature defines the unique invariant measure under the flow of this stochastic process and allows us to define ``the temperature'' of the system. This measure is also ergodic and weakly mixing. Although the system does not exhibit global directed motion, it is shown that global ballistic motion is possible (the mean-squared displacement grows like t squared). More precisely, although work cannot be extracted from thermal energy by the second law of thermodynamics, it is shown that ballistic transport from thermal energy is possible. In particular, the dynamics is characterized by a meta-stable state in which the system exhibits directed motion over random time scales. This phenomenon is caused by interaction of three attributes of the system: a non flat (yet bounded) potential energy landscape, a rigid body effect (coupling translational momentum and angular momentum through friction) and the degeneracy of the noise/friction tensor on the momentums (the fact that noise is not applied to all degrees of freedom)."}
{"category": "Math", "title": "Manin's conjecture for a quintic del Pezzo surface with A_2 singularity", "abstract": "Manin's conjecture is proved for a split del Pezzo surface of degree 5 with a singularity of type A_2."}
{"category": "Math", "title": "Large Groups of Deficiency One", "abstract": "We prove that if a group possesses a deficiency 1 presentation where one of the relators is a commutator then it is the integers times the integers, is large, or is as far as possible from being residually finite. Then we use this to show that a mapping torus of an endomorphism of a finitely generated free group is large if it contains the integers times the integers as a subgroup of infinite index, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We give applications to free by cyclic groups, 1 relator groups and residually finite groups."}
{"category": "Math", "title": "SUSY Lattice Vertex Algebras", "abstract": "We construct and study SUSY lattice vertex algebras. As a simple example, we obtain the simple vertex algebra associated to the vertex algebra $V_c(N3)$ of central charge $c=3/2$, as the SUSY lattice vertex algebra associated to $\\mathbb{Z}$ with bilinear form $(a,b) = 2ab$."}
{"category": "Math", "title": "On the minimal free resolution for fat point schemes of multiplicity at most 3 in P^2", "abstract": "Let Z be a fat point scheme in P^2 supported on general points. Here we prove that if the multiplicities are at most 3 and the length of Z is sufficiently high then the number of generators of the homogeneous ideal I_Z in each degree is as small as numerically possible. Since it is known that Z has maximal Hilbert function, this implies that Z has the expected minimal free resolution."}
{"category": "Math", "title": "Bohr's Theorem for Monogenic Power Series", "abstract": "The main goal of this paper is to generalize Bohr's phenomenon from complex one-dimensional analysis to higher dimensions in the framework of Quaternionic Analysis."}
{"category": "Math", "title": "Transformations of Markov Processes and Classification Scheme for Solvable Driftless Diffusions", "abstract": "We propose a new classification scheme for diffusion processes for which the backward Kolmogorov equation is solvable in analytically closed form by reduction to hypergeometric equations of the Gaussian or confluent type. The construction makes use of transformations of diffusion processes to eliminate the drift which combine a measure change given by Doob's h-transform and a diffeomorphism. Such transformations have the important property of preserving analytic solvability of the process: the transition probability density for the driftless process can be expressed through the transition probability density of original process. We also make use of tools from the theory of ordinary differential equations such as Liouville transformations, canonical forms and Bose invariants. Beside recognizing all analytically solvable diffusion process known in the previous literature fall into this scheme and we also discover rich new families of analytically solvable processes."}
{"category": "Math", "title": "Borel-Carath\\'{e}odory Type Theorem for monogenic functions", "abstract": "In this paper we give a generalization of the classical Borel-Carath\\'{e}odory Theorem in complex analysis to higher dimensions in the framework of Quaternionic Analysis."}
{"category": "Math", "title": "Laplace Transforms for Integrals of Markov Processes", "abstract": "Laplace transforms for integrals of stochastic processes have been known in analytically closed form for just a handful of Markov processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In this paper, we construct broad extensions of these process classes. We show how the known models fit into a classification scheme for diffusion processes for which Laplace transforms for integrals of the diffusion processes and transitional probability densities can be evaluated as integrals of hypergeometric functions against the spectral measure for certain self-adjoint operators. We also extend this scheme to a class of finite-state Markov processes related to hypergeometric polynomials in the discrete series of the Askey classification tree."}
{"category": "Math", "title": "A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation", "abstract": "The global regularity problem for the periodic Navier-Stokes system asks whether to every smooth divergence-free initial datum $u_0: (\\R/\\Z)^3 \\to \\R^3$ there exists a global smooth solution u. In this note we observe (using a simple compactness argument) that this qualitative question is equivalent to the more quantitative assertion that there exists a non-decreasing function $F: \\R^+ \\to \\R^+$ for which one has a local-in-time \\emph{a priori} bound $$ \\| u(T) \\|_{H^1_x((\\R/\\Z)^3)} \\leq F(\\|u_0\\|_{H^1_x((\\R/\\Z)^3)})$$ for all $0 < T \\leq 1$ and all smooth solutions $u: [0,T] \\times (\\R/\\Z)^3 \\to \\R^3$ to the Navier-Stokes system. We also show that this local-in-time bound is equivalent to the corresponding global-in-time bound."}
{"category": "Math", "title": "Pseudoconvex regions of finite D'Angelo type in four dimensional almost complex manifolds", "abstract": "Let D be a J-pseudoconvex region in a smooth almost complex manifold (M,J) of real dimension four. We construct a local peak J-plurisubharmonic function at every boundary point p of finite D'Angelo type. As applications we give local estimates of the Kobayashi pseudometric, implying the local Kobayashi hyperbolicity of D at p. In case the point p is of D'Angelo type less than or equal to four, or the approach is nontangential, we provide sharp estimates of the Kobayashi pseudometric."}
{"category": "Math", "title": "Operator Methods, Abelian Processes and Dynamic Conditioning", "abstract": "A mathematical framework for Continuous Time Finance based on operator algebraic methods offers a new direct and entirely constructive perspective on the field and leads to new numerical analysis techniques. This is partly a review paper as it covers and expands on the mathematical framework underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the treatment self-contained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates for Markov chains on simplicial sequences. We emphasize high precision computability by numerical linear algebra methods as opposed to the ability of arriving to analytically closed form expressions in terms of special functions. Path dependent processes adapted to a given Markov filtration are associated to an operator algebra. If this algebra is commutative, the corresponding process is named Abelian, a concept which provides a far reaching extension of the notion of stochastic integral. We recover the classic Cameron-Dyson-Feynman-Girsanov-Ito-Kac-Martin theorem as a particular case of a broadly general block-diagonalization algorithm. This technique has many applications ranging from the problem of pricing cliquets to target-redemption-notes and volatility derivatives. Non-Abelian processes are also relevant and appear in several important applications to for instance snowballs and soft calls. We show that in these cases one can effectively use block-factorization algorithms. Finally, we discuss the method of dynamic conditioning that allows one to dynamically correlate over possibly even hundreds of processes in a numerically noiseless framework while preserving marginal distributions."}
{"category": "Math", "title": "Motives of Azumaya algebras", "abstract": "We study the slice filtration for the K-theory of a sheaf of Azumaya algebras A, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson-Lichtenbaum conjecture, we apply our results to show the vanishing of SK_2(A) for a central simple algebra A of square-free index."}
{"category": "Math", "title": "Unitarizablity of premodular categories", "abstract": "We study the unitarizability of premodular categories constructed from representations of quantum group at roots of unity. We introduce \\emph{Grothendieck unitarizability} as a natural generalization of unitarizability to any class of premodular categories with a common Grothendieck semiring. We obtain new results for quantum groups of Lie types $F_4$ and $G_2$, and improve the known results for Lie types $B$ and $C$."}
{"category": "Math", "title": "Linear sections of the Severi variety and moduli of curves", "abstract": "We study the Severi variety $V_{d,g}$ of plane curves of degree $d$ and geometric genus $g$. Corresponding to every such variety, there is a one-parameter family of genus $g$ stable curves whose numerical invariants we compute. Building on the work of Caporaso and Harris, we derive a recursive formula for the degrees of the Hodge bundle on the families in question. For $d$ large enough, these families induce moving curves in $\\bar{M}_g$. We use this to derive lower bounds for the slopes of effective divisors on $\\bar{M}_g$. Another application of our results is to various enumerative problems on $V_{d,g}$."}
{"category": "Math", "title": "Geometric cycles, index theory and twisted K-homology", "abstract": "We study twisted $Spin^c$-manifolds over a paracompact Hausdorff space $X$ with a twisting $\\alpha: X \\to K(\\ZZ, 3)$. We introduce the topological index and the analytical index on the bordism group of $\\alpha$-twisted $Spin^c$-manifolds over $(X, \\alpha)$, taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this paper is to establish the equality between the topological index and the analytical index. We also define a notion of geometric twisted K-homology, whose cycles are geometric cycles of $(X, \\a)$ analogous to Baum-Douglas's geometric cycles. As an application of our twisted index theorem, we discuss the twisted longitudinal index theorem for a foliated manifold $(X, F)$ with a twisting $\\alpha: X \\to K(\\ZZ, 3)$, which generalizes the Connes-Skandalis index theorem for foliations and the Atiyah-Singer families index theorem to twisted cases."}
{"category": "Math", "title": "Degree theorems and Lipschitz simplicial volume for non-positively curved manifolds of finite volume", "abstract": "We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively curved manifolds or locally symmetric spaces of non-compact type. In contrast, we provide vanishing results for the ordinary simplicial volume; for instance, we show that the ordinary simplicial volume of non-compact locally symmetric spaces with finite volume of Q-rank at least 3 is zero."}
{"category": "Math", "title": "Linear Recurrences in the Degree Sequences of Monomial Mappings", "abstract": "Let $A$ be an integer matrix, and let $f_A$ be the associated monomial map. We give a connection between the eigenvalues of $A$ and existence of a linear recurrence relation in the sequence of degrees."}
{"category": "Math", "title": "Derived equivalences of K3 surfaces and orientation", "abstract": "Every Fourier--Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive directions. This leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical Torelli theorem for K3 surfaces determining all Hodge isometries that are induced by automorphisms."}
{"category": "Math", "title": "Comment on GL(2,R) geometry of 4th order ODEs", "abstract": "We describe 4th order ODEs satisfying two contact invariant conditions of Bryant in terms of the Ricci tensor of a certain gl(2,R) valued connection. We also provide nonhomogeneous examples of such ODEs."}
{"category": "Math", "title": "On the 2-typical de Rham-Witt complex", "abstract": "In this paper we introduce the 2-typical de Rham-Witt complex for arbitrary commutative, unital rings and log-rings. We describe this complex for the rings \\Z and \\Z_{(2)}, for the log-ring (\\Z_{(2)},M) with the canonical log-structure, and we describe its behaviour under polynomial extensions. In an appendix we also describe the $p$-typical de Rham-Witt complex of (\\Z_{(p)},M) for p odd."}
{"category": "Math", "title": "Mordell-Lang and Skolem-Mahler-Lech theorems for endomorphisms of semiabelian varieties", "abstract": "Using the Skolem-Mahler-Lech theorem, we prove a dynamical Mordell-Lang conjecture for semiabelian varieties."}
{"category": "Math", "title": "Jet isomorphism for conformal geometry", "abstract": "Jet isomorphism theorems for conformal geometry are discussed. A new proof of the jet isomorphism theorem for odd-dimensional conformal geometry is outlined, using an ambient realization of the conformal deformation complex. An infinite order ambient lift for conformal densities in the case in which harmonic extension is obstructed is described. A jet isomorphism theorem for even dimensional conformal geometry is formulated using the inhomogeneous ambient metrics recently introduced by the author and K. Hirachi."}
{"category": "Math", "title": "On computations of Hurwitz-Hodge integrals", "abstract": "We describe a method to compute Hurwitz-Hodge integrals."}
{"category": "Math", "title": "Adaptive estimation of linear functionals by model selection", "abstract": "We propose an estimation procedure for linear functionals based on Gaussian model selection techniques. We show that the procedure is adaptive, and we give a non asymptotic oracle inequality for the risk of the selected estimator with respect to the $\\mathbb{L}_p$ loss. An application to the problem of estimating a signal or its $r^{th}$ derivative at a given point is developed and minimax rates are proved to hold uniformly over Besov balls. We also apply our non asymptotic oracle inequality to the estimation of the mean of the signal on an interval with length depending on the noise level. Simulations are included to illustrate the performances of the procedure for the estimation of a function at a given point. Our method provides a pointwise adaptive estimator."}
{"category": "Math", "title": "Inference in nonparametric current status models with covariates", "abstract": "In interval censored models with current status observations, the variables are indicators of the presence of individuals on observation intervals and covariates. When several individuals share the same observation interval, a simple procedure provides new estimators for the distribution of the observation times and their intensity, in a closed form. They are $n^{1/2}$-consistent for piece-wise constant covariates. Estimators of the sample-sizes are deduced and asymptotic $\\chi^2$ tests for independence of the observations on consecutive intervals and for independence between consecutive classes for the observed individuals are proposed."}
{"category": "Math", "title": "Rates of asymptotic regularity for Halpern iterations of nonexpansive mappings", "abstract": "In this paper we obtain new effective results on the Halpern iterations of nonexpansive mappings using methods from mathematical logic or, more specifically, proof-theoretic techniques. We give effective rates of asymptotic regularity for the Halpern iterations of nonexpansive self-mappings of nonempty convex sets in normed spaces. The paper presents another case study in the project of {\\em proof mining}, which is concerned with the extraction of effective uniform bounds from (prima-facie) ineffective proofs."}
{"category": "Math", "title": "Finitely generated lattice-ordered groups with soluble word problem", "abstract": "William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for lattice-ordered groups: Theorem: A finitely generated lattice-ordered group has soluble word problem if and only if it can be embedded in an simple lattice-ordered group that can be embedded in a finitely presented lattice-ordered group. The proof uses permutation groups and the ideas used to prove the lattice-ordered group analogue of Higman's Embedding Theorem."}
{"category": "Math", "title": "A remark on Fano 4-folds having (3,1)-type extremal contractions", "abstract": "Let X be the blow-up of a smooth projective 4-fold Y along a smooth curve C and let E be the exceptional divisor. Assume that X is a Fano manifold and has an elementary extremal contraction $\\phi: X \\to Z$ of (3,1)-type such that E is $\\phi$-ample (recall that a contraction map for a 4-fold is called (3,1)-type if the exceptional locus is a divisor and its image is a curve). We show that if the exceptional divisor of $\\phi$ is smooth, then Y is isomorphic to $\\mathbb{P}^{4}$ and C is an elliptic curve of degree 4."}
{"category": "Math", "title": "A regularity and compactness theory for immersed stable minimal hypersurfaces of multiplicity at most 2", "abstract": "We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior. Applications including a compactness theorem for a class of immersed stable minimal hypersurfaces and a pointwise curvature estimate for the hypersurfaces in this class in low dimensions are also discussed."}
{"category": "Math", "title": "A generalisation of the Cauchy-Kovalevskaia theorem", "abstract": "I prove, under mild assumptions, that solutions to linear evolution equations admit sectorial solutions. The size of the sector depends on the regularity of the initial data. If it is regular enough the solution is holomorphic and unique otherwise it is sectorial. I also prove that the result is optimal for many partial differential systems (which includes KdV and other examples)."}
{"category": "Math", "title": "Morin singularities and global geometry in a class of ordinary differential operators", "abstract": "We consider the operator $F(u) = u' + f(t,u(t))$ acting on periodic real valued functions. Generically, critical points of $F$ are infinite dimensional Morin-like singularities and we provide operational characterizations of the singularities of different orders. A global Lyapunov-Schmidt decomposition of $F$ converts $F$ into adapted coordinates, $\\Fbd(\\tilde v, \\overline u) = (\\tilde v, \\overline v)$, where $\\tilde v$ is a function of average zero and both $\\overline u$ and $\\overline v$ are numbers. Thus, global geometric aspects of $F$ reduce to the study of a family of one-dimensional maps: we use this approach to obtain normal forms for several nonlinearities $f$. For example, we characterize autonomous nonlinearities giving rise to global folds and, in general, we show that $F$ is a global fold if all critical points are folds. Also, $f(t,x) = x^3 - x$, or, more generally, the Cafagna-Donati nonlinearity, yield global cusps; for $F$ interpreted as a map between appropriate Hilbert spaces, the requested changes of variable to bring $F$ to normal form can be taken to be diffeomorphisms. A key ingredient in the argument is the contractibility of both the critical set and the set of non-folds for a generic autonomous nonlinearity. We also obtain a numerical example of a polynomial $f$ of degree 4 for which $F$ contains butterflies (Morin singularities of order 4)---% it then follows that $F(u) = v$ has six solutions for some $v$."}
{"category": "Math", "title": "Families intersecting on an interval", "abstract": "We shall be interested in the following Erdos-Ko-Rado-type question. Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic translate (modulo n) of B? Chung, Graham, Frankl and Shearer have proved that, in the case where B is a block of length t, we can do no better than to take A to consist of all supersets of B. We give an alternative proof of this result, which is in a certain sense more 'direct'."}
{"category": "Math", "title": "Crossings and Nestings of Two Edges in Set Partitions", "abstract": "Let $\\pi$ and $\\lambda$ be two set partitions with the same number of blocks. Assume $\\pi$ is a partition of $[n]$. For any integer $l, m \\geq 0$, let $\\mathcal{T}(\\pi, l)$ be the set of partitions of $[n+l]$ whose restrictions to the last $n$ elements are isomorphic to $\\pi$, and $\\mathcal{T}(\\pi, l, m)$ the subset of $\\mathcal{T}(\\pi,l)$ consisting of those partitions with exactly $m$ blocks. Similarly define $\\mathcal{T}(\\lambda, l)$ and $\\mathcal{T}(\\lambda, l,m)$. We prove that if the statistic $cr$ ($ne$), the number of crossings (nestings) of two edges, coincides on the sets $\\mathcal{T}(\\pi, l)$ and $\\mathcal{T}(\\lambda, l)$ for $l =0, 1$, then it coincides on $\\mathcal{T}(\\pi, l,m)$ and $\\mathcal{T}(\\lambda, l,m)$ for all $l, m \\geq 0$. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings."}
{"category": "Math", "title": "Computations with finite index subgroups of $PSL_2(\\mathbb Z)$ using Farey Symbols", "abstract": "Finite index subgroups of the modular group are of great arithmetic importance. Farey symbols, introduced by Ravi Kulkarni in 1991, are a tool for working with these groups. Given such a group $\\Gamma$, a Farey symbol for $\\Gamma$ is a certain finite sequence of rational numbers (representing vertices of a fundamental domain of $\\Gamma$) together with pairing information for the edges between the vertices. They are a compact way of encoding the information about the group and they provide a simple way to do calculations with the group. For example: calculating an independent set of generators and decomposing group elements into a word in these generators, finding coset representatives, elliptic points, and genus of the group, testing if the group is congruence, etc. In this expository article, we will discuss Farey Symbols and explicit algorithms for working with them."}
{"category": "Math", "title": "Killing graphs with prescribed mean curvature and Riemannian submersions", "abstract": "It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field."}
{"category": "Math", "title": "Smooth deformations of piecewise expanding unimodal maps", "abstract": "In the space of C^k piecewise expanding unimodal maps, k>=1, we characterize the C^1 smooth families of maps where the topological dynamics does not change (the \"smooth deformations\") as the families tangent to a continuous distribution of codimension-one subspaces (the \"horizontal\" directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of C^{k-1+Lip} deformations tangent to every given C^k horizontal direction, for k>=2."}
{"category": "Math", "title": "Regularity of solutions of the isoperimetric problem that are close to a smooth manifold", "abstract": "In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and $C^{2,\\alpha}$-close to the given sub manifold. We show also a version with variable metric on the manifold. The techniques used are, among other, the standards outils of linear elliptic analysis and comparison theorems of riemannian geometry, Allard's regularity theorem for minimizing varifolds, the isometric immersion theorem of Nash and a parametric version due to Gromov."}
{"category": "Math", "title": "Area distances of Convex Plane Curves and Improper Affine Spheres", "abstract": "The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. This link makes possible a better understanding of both theories. The concepts of the theory of affine spheres lead to a new definition of an area distance on the outer part of a convex plane arc. Also, based on the theory of discrete affine spheres, we propose fast algorithms to compute the area distances. On the other hand, area distances provide a good geometrical understanding of improper affine spheres."}
{"category": "Math", "title": "Traveling waves in a one-dimensional random medium", "abstract": "We consider solutions of a scalar reaction-diffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We also establish existence of generalized random traveling waves and of transition fronts in general heterogeneous media."}
{"category": "Math", "title": "Primes, Pi, and Irrationality Measure", "abstract": "A folklore proof of Euclid's theorem on the infinitude of primes uses the Euler product and the irrationality of $\\zeta(2) = \\pi^2/6$. A quantified form of Euclid's Theorem is Bertrand's postulate $p_{n+1} < 2p_n$. By quantifying the folklore proof using an irrationality measure for $6/\\pi^2$, we give a proof (communicated to Paulo Ribenboim in 2005) of a much weaker upper bound on $p_{n+1}$."}
{"category": "Math", "title": "Operator Theory and Complex Geometry", "abstract": "One approach to multivariate operator theory involves concepts and techniques from algebraic and complex geometry and is formulated in terms of Hilbert modules. In these notes we provide an introduction to this approach including many proofs. We are particularly interested in examples related to hermitian holomorphic vector bundles and we study submodules and reducing submodules in such cases. We go into some detail concerning a problem of Zhu on the reducing subspaces of powers of the Bergman shift as well as more recent work of the author and J. Sarkar on proper submodules which are unitarily equivalent to the orginal. Although the basic results are not new, there is some novelty in the details and the organization of the material."}
{"category": "Math", "title": "Descartes' Rule of Signs by an Easy Induction", "abstract": "If c is a positive number, Descartes' rule of signs implies that multiplying a polynomial f(x) by c - x introduces an odd number of changes of sign in the coefficients. We turn this around, proving this fact about sign changes inductively and deriving Descartes' rule from it."}
{"category": "Math", "title": "On the Spectrum of the Dirichlet Laplacian in a Narrow Strip, II", "abstract": "We consider the Dirichlet Laplacian in a family of narrow unbounded domains. As the width of these domains goes to 0, we study the asymptotic behavior of the eigenvalues that lie below the essential spectrum and the asymptotic behavior of the corresponding eigenfunctions."}
{"category": "Math", "title": "Conjugacy classes of solutions to equations and inequations over hyperbolic groups", "abstract": "We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results to enumerate the immutable subgroups of a torsion-free hyperbolic group."}
{"category": "Math", "title": "On moduli spaces of quiver representations associated with dimer models", "abstract": "We give a sufficient condition for the moduli space of quiver representations associated with a dimer model to be smooth for a general stability parameter. We also show that the moduli space in this case is a crepant resolution of the toric variety determined by the Newton polygon of the characteristic polynomial."}
{"category": "Math", "title": "Robin functions for complex manifolds and applications", "abstract": "We prove a generalization of the second variation formula of the Robin function associated to a smooth variation of domains in C^N to the case of the c-Robin function associated to a smooth variation of domains in a complex manifold M equipped with a Hermitian metric and a smooth, nonnegative function c. Our purpose is that, with this added flexibility, we are able to give a criterion for a bounded, smoothly bounded, pseudoconvex domain D in a complex homogeneous space to be Stein."}
{"category": "Math", "title": "Decompositions of Laurent polynomials", "abstract": "In the 1920's, Ritt studied the operation of functional composition g o h(x) = g(h(x)) on complex rational functions. In the case of polynomials, he described all the ways in which a polynomial can have multiple `prime factorizations' with respect to this operation. Despite significant effort by Ritt and others, little progress has been made towards solving the analogous problem for rational functions. In this paper we use results of Avanzi--Zannier and Bilu--Tichy to prove analogues of Ritt's results for decompositions of Laurent polynomials, i.e., rational functions with denominator a power of x."}
{"category": "Math", "title": "Triangulated categories of Gorenstein cyclic quotient singularities", "abstract": "We prove an equivalence of triangulated categories between Orlov's triangulated category of singularities for a Gorenstein cyclic quotient singularity and the derived category of representations of a quiver with relations which is obtained from the McKay quiver by removing one vertex and half of the arrows."}
{"category": "Math", "title": "Logarithmic vector fields along smooth plane cubic curves", "abstract": "We study the sheaves of logarithmic vector fields along smooth cubic curves in the projective plane, and prove a Torelli-type theorem in the sense of Dolgachev-Kapranov for those with non-vanishing j-invariants."}
{"category": "Math", "title": "Increasing power of the test through pre-test - a robust method", "abstract": "This paper develops robust test procedures for testing the intercept of a simple regression model when it is \\textit{apriori} suspected that the slope has a specified value. Defining unrestricted test (UT), restricted test (RT) and pre-test test (PTT) corresponding to the unrestricted (UE), restricted (RE), and preliminary test estimators (PTE) in the estimation case, the M-estimation methodology is used to formulate the M-tests and derive their asymptotic power functions. Analytical and graphical comparisons of the three tests are obtained by studying the power functions with respect to size and power of the tests. It is shown that PTT achieves a reasonable dominance over the others asymptotically."}
{"category": "Math", "title": "Polynomial splittings of metabelian von Neumann rho-invariants", "abstract": "We show that if the connected sum of two knots with coprime Alexander polynomials has vanishing von Neumann rho-invariants associated with certain metabelian representations then so do both knots. As an application, we give a new example of an infinite family of knots which are linearly independent in the knot concordance group."}
{"category": "Math", "title": "Regular integers modulo n", "abstract": "Let $n=p_1^{\\nu_1}... p_r^{\\nu_r} >1$ be an integer. An integer $a$ is called regular (mod $n$) if there is an integer $x$ such that $a^2x\\equiv a$ (mod $n$). Let $\\varrho(n)$ denote the number of regular integers $a$ (mod $n$) such that $1\\le a\\le n$. Here $\\varrho(n)=(\\phi(p_1^{\\nu_1})+1)... (\\phi(p_r^{\\nu_r})+1)$, where $\\phi(n)$ is the Euler function. In this paper we first summarize some basic properties of regular integers (mod $n$). Then in order to compare the rates of growth of the functions $\\varrho(n)$ and $\\phi(n)$ we investigate the average orders and the extremal orders of the functions $\\varrho(n)/\\phi(n)$, $\\phi(n)/\\varrho(n)$ and $1/\\varrho(n)$."}
{"category": "Math", "title": "Hirzebruch-Riemann-Roch theorem for DG algebras", "abstract": "For an arbitrary proper DG algebra A (i.e. DG algebra with finite dimensional total cohomology) we introduce a pairing on the Hochschild homology of A and present an explicit formula for a Chern-type character of an arbitrary perfect A-module (the Chern characters take values in the Hochschild homology of A). The Hirzebruch-Riemann-Roch formula in this context expresses the Euler characteristic of the Hom-complex between two perfect A-modules in terms of the pairing of their Chern characters. We mention two examples of proper DG algebras and the HRR formulas for them. The first example is Ringel's formula for quivers with relations. The second example is related to orbifold singularities of the form V/G where V is a complex vector space and G is a finite subgroup of SL(V). Furthermore, we prove that the above pairing on the Hochschild homology is non-degenerate when the DG algebra is smooth. We also formulate the conjecture that for a Calabi-Yau DG algebra A the pairing coincides with the one coming from the Topological Field Theory associated with A and verify it in the case of Frobenius algebras."}
{"category": "Math", "title": "A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides", "abstract": "We study the uniqueness of solutions of Helmholtz equation for a problem that concerns wave propagation in waveguides. The classical radiation condition does not apply to our problem because the inhomogeneity of the index of refraction extends to infinity in one direction. Also, because of the presence of a waveguide, some waves propagate in one direction with different propagation constants and without decaying in amplitude. Our main result provides an explicit condition for uniqueness which takes into account the physically significant components, corresponding to guided and non-guided waves; this condition reduces to the classical Sommerfeld-Rellich condition in the relevant cases. Finally, we also show that our condition is satisfied by a solution, already present in literature, of the problem under consideration."}
{"category": "Math", "title": "Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers", "abstract": "The rotated multipliers method is performed in the case of the boundary stabilization by means of a(linear or non-linear) Neumann feedback. this method leads to new geometrical cases concerning the \"active\" part of the boundary where the feedback is apllied. Due to mixed boundary conditions, these cases generate singularities. Under a simple geometrical conditon concerning the orientation of boundary, we obtain a stabilization result in both cases."}
{"category": "Math", "title": "Three manifolds as geometric branched coverings of the three sphere", "abstract": "One method for obtaining every closed orientable 3-manifold is as branched covering of the 3-sphere over a link. There is a classical topological result showing that the minimun possible number of sheets in the covering is three. In this paper we obtain a geometric version of this result. The interest is given by the growing importance of geometry in 3-manifolds theory."}
{"category": "Math", "title": "Pfaffian Systems from Twistor Fibrations", "abstract": "Canonical twistor fibrations lead to Pfaffian systems by means of their superhorizontal distribution. The aim of this note is to identify explicitly the Pfaffian systems of five or less variables that arise in this way in terms of the classification given by A.Awane and M.Goze."}
{"category": "Math", "title": "Zeta functions of 3-dimensional p-adic Lie algebras", "abstract": "We give an explicit formula for the subalgebra zeta function of a general 3-dimensional Lie algebra over the p-adic integers $\\mathbb{Z}_p$. To this end, we associate to such a Lie algebra a ternary quadratic form over $\\mathbb{Z}_p$. The formula for the zeta function is given in terms of Igusa's local zeta function associated to this form."}
{"category": "Math", "title": "Combinatorial Gelfand models for some semigroups and q-rook monoid algebras", "abstract": "Inspired by the results of [R. Adin, A. Postnikov, Y. Roichman, Combinatorial Gelfand model, preprint math.RT arXiv:0709.3962], we propose combinatorial Gelfand models for semigroup algebras of some finite semigroups, which include the symmetric inverse semigroup, the dual symmetric inverse semigroup, the maximal factorizable subsemigroup in the dual symmetric inverse semigroup, and the factor power of the symmetric group. Furthermore we extend the Gelfand model for the semigroup algebras of the symmetric inverse semigroup to a Gelfand model for the $q$-rook monoid algebra."}
{"category": "Math", "title": "The Levy-Gromov Isoperimetric Inequality in Convex Manifolds with Boundary", "abstract": "We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary."}
{"category": "Math", "title": "Groups with the Minimal Conditions for Subgroups and for Nonabelian Subgroups", "abstract": "For some very wide classes $\\mathfrak{D}$ and $\\mathfrak{B}\\subset\\mathfrak{D}$ of groups, the author proves that an arbitrary (nonabelian) group $G\\in \\mathfrak{D}$ (respectively $G\\in \\mathfrak{B}$) satisfies the minimal condition for (nonabelian) subgroups iff it is Chernikov"}
{"category": "Math", "title": "Oriented matroids and Ky Fan's theorem", "abstract": "L. Lovasz has shown that Sperner's combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. Inspired by this result we prove that classical Ky Fan's theorem admits an oriented matroid generalization of similar nature. Ky Fan's theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C^{m,r} ."}
{"category": "Math", "title": "Dynamics of Mandelbrot Cascades", "abstract": "Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures defined by inequalities on moments. To be more specific, beyond the first iteration, the trajectories take values in the set of fixed points of smoothing transformations (i.e., some generalized stable laws). Studying this system leads to a central limit theorem and to its functional version. The limit Gaussian process can also be obtained as limit of an `additive cascade' of independent normal variables."}
{"category": "Math", "title": "A dual eigenvector condition for strong lumpability of Markov chains", "abstract": "Necessary and sufficient conditions for identifying strong lumpability in Markov chains are presented. We show that the states in a lump necessarily correspond to identical elements in eigenvectors of the dual transition matrix. If there exist as many dual eigenvectors that respect the necessary condition as there are lumps in the aggregation, then the condition is also sufficient. The result is demonstrated with two simple examples."}
{"category": "Math", "title": "Banach algebras of pseudodifferential operators and their almost diagonalization", "abstract": "We define new symbol classes for pseudodifferntial operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra over a lattice we associate a symbol class. Then every operator with such a symbol is almost diagonal with respect to special wave packets (coherent states or Gabor frames), and the rate of almost diagonalization is described precisely by the underlying convolution algebra. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on $L^2 $. If a version of Wiener's lemma holds for the underlying convolution algebra, then the algebra of pseudodifferential operators is closed under inversion. The theory contains as a special case the fundamental results about Sj\\\"ostrand's class and yields a new proof of a theorem of Beals about the H\\\"ormander class of order 0."}
{"category": "Math", "title": "Metric Dichotomies", "abstract": "These are notes from talks given at ICMS, Edinburgh, 4/2007 (\"Geometry and Algorithms workshop\") and at Bernoulli Center, Lausanne 5/2007 (\"Limits of graphs in group theory and computer science\"). We survey the following type of dichotomies exhibited by certain classes X of finite metric spaces: For every host space H, either all metrics in X embed almost isometrically in H, or the distortion of embedding some metrics of X in H is unbounded."}
{"category": "Math", "title": "Structure and f-dependence of the a.c.i.m. for a unimodal map f of Misiurewicz type", "abstract": "By using a suitable Banach space on which we let the transfer operator act, we make a detailed study of the ergodic theory of a unimodal map $f$ of the interval in the Misiurewicz case. We show in particular that the absolutely continuous invariant measure $\\rho$ can be written as the sum of 1/square root spikes along the critical orbit, plus a continuous background. We conclude by a discussion of the sense in which the map $f\\mapsto\\rho$ may be differentiable."}
{"category": "Math", "title": "Decomposition of residue currents", "abstract": "Given a submodule $J\\subset \\mathcal O_0^{\\oplus r}$ and a free resolution of $J$ one can define a certain vector valued residue current whose annihilator is $J$. We make a decomposition of the current with respect to Ass$(J)$ that correspond to a primary decomposition of $J$. As a tool we introduce a class of currents that includes usual residue and principal value currents; in particular these currents admit a certain type of restriction to analytic varieties and more generally to constructible sets."}
{"category": "Math", "title": "Valiron's construction in higher dimension", "abstract": "We consider holomorphic self-maps $\\v$ of the unit ball $\\B^N$ in $\\C^N$ ($N=1,2,3,...$). In the one-dimensional case, when $\\v$ has no fixed points in $\\D\\defeq \\B^1$ and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map $\\phi$, and therefore, in this case, the dynamical properties of $\\phi$ are well understood. In what follows, we generalize the classical Valiron construction to higher dimensions under some weak assumptions on $\\v$ at its Denjoy-Wolff point. As a result, we construct a semi-conjugation $\\sigma$, which maps the ball into the right half plane of $\\C$, and solves the functional equation $\\sigma\\circ \\v=\\lambda \\sigma$, where $\\lambda>1$ is the (inverse of the) boundary dilation coefficient at the Denjoy-Wolff point of $\\v$."}
{"category": "Math", "title": "Ratios: A short guide to confidence limits and proper use", "abstract": "Researchers often calculate ratios of measured quantities. Specifying confidence limits for ratios is difficult and the appropriate methods are often unknown. Appropriate methods are described (Fieller, Taylor, special bootstrap methods). For the Fieller method a simple geometrical interpretation is given. Monte Carlo simulations show when these methods are appropriate and that the most frequently used methods (index method and zero-variance method) can lead to large liberal deviations from the desired confidence level. It is discussed when we can use standard regression or measurement error models and when we have to resort to specific models for heteroscedastic data. Finally, an old warning is repeated that we should be aware of the problems of spurious correlations if we use ratios."}
{"category": "Math", "title": "Concerning the Strauss conjecture and almost global existence for nonlinear Dirichlet-wave equations in 4-dimensions", "abstract": "We show the obstacle version of the Strauss conjecture holds when the spatial dimension is equal to 4. We also show that an almost global existence theorem of H\\\"ormander for (4+1)-dimensional Minkowski space holds in the obstacle setting. We use weighed space-time variants of the energy inequality and a variant of the classical Hardy inequality."}
{"category": "Math", "title": "Ricci curvature and conformality of Riemannian manifolds to spheres", "abstract": "In this paper we give bounds for the first eigenvalue of the conformal Laplacian and the Yamabe invariant of a compact Riemannian manifold, by using conditions on the Ricci curvature and the diameter and deduce certain conditions on the manifold to be conformal to a sphere."}
{"category": "Math", "title": "A method of moments estimator of tail dependence", "abstract": "In the world of multivariate extremes, estimation of the dependence structure still presents a challenge and an interesting problem. A procedure for the bivariate case is presented that opens the road to a similar way of handling the problem in a truly multivariate setting. We consider a semi-parametric model in which the stable tail dependence function is parametrically modeled. Given a random sample from a bivariate distribution function, the problem is to estimate the unknown parameter. A method of moments estimator is proposed where a certain integral of a nonparametric, rank-based estimator of the stable tail dependence function is matched with the corresponding parametric version. Under very weak conditions, the estimator is shown to be consistent and asymptotically normal. Moreover, a comparison between the parametric and nonparametric estimators leads to a goodness-of-fit test for the semiparametric model. The performance of the estimator is illustrated for a discrete spectral measure that arises in a factor-type model and for which likelihood-based methods break down. A second example is that of a family of stable tail dependence functions of certain meta-elliptical distributions."}
{"category": "Math", "title": "Roots, symmetries and conjugacy of pseudo-Anosov mapping classes", "abstract": "An algorithm is proposed that solves two decision problems for pseudo-Anosov elements in the mapping class group of a surface with at least one marked fixed point. The first problem is the root problem: decide if the element is a power and in this case compute the roots. The second problem is the symmetry problem: decide if the element commutes with a finite order element and in this case compute this element. The structure theorem on which this algorithm is based provides also a new solution to the conjugacy problem."}
{"category": "Math", "title": "Estimation of Gaussian graphs by model selection", "abstract": "We investigate in this paper the estimation of Gaussian graphs by model selection from a non-asymptotic point of view. We start from a n-sample of a Gaussian law P_C in R^p and focus on the disadvantageous case where n is smaller than p. To estimate the graph of conditional dependences of P_C, we introduce a collection of candidate graphs and then select one of them by minimizing a penalized empirical risk. Our main result assess the performance of the procedure in a non-asymptotic setting. We pay a special attention to the maximal degree D of the graphs that we can handle, which turns to be roughly n/(2 log p)."}
{"category": "Math", "title": "The volume and Chern-Simons invariant of a representation", "abstract": "We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal triangulation with no use of additional combinatorial topology. In particular, the Chern-Simons invariant is computed just as easily as the volume."}
{"category": "Math", "title": "Reduction of structure for torsors over semilocal rings", "abstract": "Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is semisimple or k is normal and noetherian. We show that G has a finite k-subgroup S such that the natural map H^1(R, S) --> H^1(R, G) is surjective for every semilocal ring R containing k. In other words, G-torsors over Spec(R) admit reduction of structure to S. We also show that the natural map H^1(X, S) --> H^1(X, G) is surjective in several other contexts, under suitable assumptions on the base ring k, the scheme X/k and the group scheme G/k. These results have already been used to study loop algebras as well as essential dimension of connected algebraic groups in prime characteristic. Additional applications are presented at the end of this paper."}
{"category": "Math", "title": "A Unified Spiral Chain Coloring Algorithm for Planar Graphs", "abstract": "In this paper we have given a unified graph coloring algorithm for planar graphs. The problems that have been considered in this context respectively, are vertex, edge, total and entire colorings of the planar graphs. The main tool in the coloring algorithm is the use of spiral chain which has been used in the non-computer proof of the four color theorem in 2004. A more precies explanation of the proof of the four color theorem by spiral chain coloring is also given in this paper. Then we continue to spiral-chain coloring solutions by giving the proof of other famous conjectures of Vizing's total coloring and planar graph conjectures of maximum vertex degree six. We have also given the proof of a conjecture of Kronk and Mitchem that any plane graph of maximum degree \"Delta\" is entirely (\"Delta\"+4)-colorable.The last part of the paper deals with the three colorability of planar graphs under the spiral chain coloring. We have given an efficient and short proof of the Groetzsch's Theorem that triangle-free planar graphs are 3-colorable."}
{"category": "Math", "title": "The sh-Lie algebra perturbation Lemma", "abstract": "Let R be a commutative ring which contains the rationals as a subring and let g be a chain complex. Suppose given an sh-Lie algebra structure on g, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra T' on the suspension of g and write the perturbed coalgebra as T\". Suppose, furthermore, given a contraction of g onto a chain complex M. We show that the data determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting cochain from the perturbed coalgebra S\" to the loop Lie algebra L on the perturbed coalgebra T\", and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg coalgebra on L onto S\" which is natural in the data. For the special case where M and g are connected we also construct an explicit extension of the perturbed retraction to an sh-Lie map. This approach includes a very general solution of the master equation."}
{"category": "Math", "title": "Generalized multiresolution analyses with given multiplicity functions", "abstract": "Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space $\\H$ that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis generated by a fixed scaling function. Previous authors have studied a multiplicity function $m$ which, loosely speaking, measures the failure of the GMRA to be an MRA. When the Hilbert space $\\H$ is $L^2(\\mathbb R^n)$, the possible multiplicity functions have been characterized by Baggett and Merrill. Here we start with a function $m$ satisfying a consistency condition which is known to be necessary, and build a GMRA in an abstract Hilbert space with multiplicity function $m$."}
{"category": "Math", "title": "On numerical averaging of the conductivity coefficient using two-scale extensions", "abstract": "In this article we compare solutions to elliptic problems having rapidly oscillated conductivity (permeability, etc) coefficient with solutions to corresponding homogenized problems obtained from two-scale extensions of the initial coefficient. The comparison is done numerically on several one and two dimensional test problems with randomly generated coefficients for different intensities of oscillation. The dependency of the approximation error on the size of averaging is investigated."}
{"category": "Math", "title": "The yoga of the Cassels-Tate pairing", "abstract": "Cassels has described a pairing on the 2-Selmer group of an elliptic curve which shares some properties with the Cassels-Tate pairing. In this article, we prove that the two pairings are the same."}
{"category": "Math", "title": "The classification of simple Jacobi--Ricci commuting algebraic curvature tensors", "abstract": "We classify algebraic curvature tensors such that the Ricci operator is simple (i.e. the Ricci operator is complex diagonalizable and either the complex spectrum consists of a single real eigenvalue or the complex spectrum consists of a pair of eigenvalues which are complex conjugates of each other) and which are Jacobi--Ricci commuting (i.e. the Ricci operator commutes with the Jacobi operator of any vector)."}
{"category": "Math", "title": "A semigroup approach to wreath-product extensions of Solomon's descent algebras", "abstract": "There is a well-known combinatorial definition, based on ordered set partitions, of the semigroup of faces of the braid arrangement. We generalize this definition to obtain a semigroup Sigma_n^G associated with G wr S_n, the wreath product of the symmetric group S_n with an arbitrary group G. Techniques of Bidigare and Brown are adapted to construct an anti-homomorphism from the S_n-invariant subalgebra of the semigroup algebra of Sigma_n^G into the group algebra of G wr S_n. The generalized descent algebras of Mantaci and Reutenauer are obtained as homomorphic images when G is abelian."}
{"category": "Math", "title": "Exponential stability of non-autonomous stochastic partial differential equations with finite memory", "abstract": "The exponential stability, in both mean square and almost sure senses, for energy solutions to a class of nonlinear and non-autonomous stochastic PDEs with finite memory is investigated. Various criteria for stability are obtained. An example is presented to demonstrate the main results."}
{"category": "Math", "title": "Multiplicative function instead of logarithm (an elementary approach)", "abstract": "V.I. Arnold has recently defined the complexity of finite sequences of zeroes and ones in terms of periods and preperiods of attractors of a dynamic system of the operator of finite differentiation. Arnold has set up a hypothesis that the sequence of the values of the logarithm is most complicated or almost most complicated. In this paper we obtain the necessary and sufficient conditions which make this sequence (supplemented with zero) most complicated for a more wide class of operators. We prove that a sequence of values of a multiplicative function in a finite field is most complicated or almost most complicated for any operator divisible by the differentiation operator."}
{"category": "Math", "title": "Dynamical systems with double recursion are undecidable", "abstract": "A primitive type of two-dimensional dynamic system is introduced. It is shown that there is no decision procedure able to answer if such a dynamical system is ultimately zero."}
{"category": "Math", "title": "A geometric approach to full Colombeau algebras", "abstract": "We present a geometric approach to diffeomorphism invariant full Colombeau algebras which allows a particularly clear view on the construction of the intrinsically defined algebra $\\hat{\\mathcal G}(M)$ on the manifold $M$ given in (Grosser et al., Adv. Math. 166 (2002), 179--206)."}
{"category": "Math", "title": "Order one invariants of spherical curves", "abstract": "We give a complete description of all order 1 invariants of spherical curves. We also identify the subspaces of all J-invariants and S-invariants, and present two equalities satisfied by any spherical curve."}
{"category": "Math", "title": "Large nearly regular induced subgraphs", "abstract": "For a real c \\geq 1 and an integer n, let f(n,c) denote the maximum integer f so that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in particular, every graph on n vertices contains a regular induced subgraph on at least f(n,1) vertices. The problem of estimating $(n,1) was posed long time ago by Erdos, Fajtlowicz and Staton. In this note we obtain the following upper and lower bounds for the asymptotic behavior of f(n,c): (i) For fixed c>2.1, n^{1-O(1/c)} \\leq f(n,c) \\leq O(cn/\\log n). (ii) For fixed c=1+\\epsilon with epsilon>0 sufficiently small, f(n,c) \\geq n^{\\Omega(\\epsilon^2/ \\ln (1/\\epsilon))}. (iii) \\Omega (\\ln n) \\leq f(n,1) \\leq O(n^{1/2} \\ln^{3/4} n). An analogous problem for not necessarily induced subgraphs is briefly considered as well."}
{"category": "Math", "title": "Whitehead moves for G-trees", "abstract": "We generalize the familiar notion of a Whitehead move from Culler and Vogtmann's Outer space to the setting of deformation spaces of G-trees. Specifically, we show that there are two moves, each of which transforms a reduced G-tree into another reduced G-tree, that suffice to relate any two reduced trees in the same deformation space. These two moves further factor into three moves between reduced trees that have simple descriptions in terms of graphs of groups. This result has several applications."}
{"category": "Math", "title": "On the isomorphism problem for generalized Baumslag-Solitar groups", "abstract": "Generalized Baumslag-Solitar groups (GBS groups) are groups that act on trees with infinite cyclic edge and vertex stabilizers. Such an action is described by a labeled graph (essentially, the quotient graph of groups). This paper addresses the problem of determining whether two given labeled graphs define isomorphic groups; this is the isomorphism problem for GBS groups. There are two main results and some applications. First, we find necessary and sufficient conditions for a GBS group to be represented by only finitely many reduced labeled graphs. These conditions can be checked effectively from any labeled graph. Then we show that the isomorphism problem is solvable for GBS groups whose labeled graphs have first Betti number at most one."}
{"category": "Math", "title": "A new proof of the Erd\\H{o}s-Ko-Rado theorem for intersecting families of permutations", "abstract": "Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations \\pi, \\sigma in S there is a point i in {1,...,n} such that \\pi(i)=\\sigma(i). Deza and Frankl \\cite{MR0439648} proved that if S a subset of S(n) is intersecting then |S| \\leq (n-1)!. Further, Cameron and Ku \\cite{MR2009400} show that the only sets that meet this bound are the cosets of a stabilizer of a point. In this paper we give a very different proof of this same result."}
{"category": "Math", "title": "Periodic automorphisms of Takiff algebras, contractions, and $\\theta$-groups", "abstract": "Let $q$ be an algebraic Lie algebra and $q<m>$ a (generalised) Takiff algebra. Any finite order automorphism $\\theta$ of $q$ induces an automorphisms of $q<m>$ of the same order, denoted $\\Theta$. We study invariant-theoretic properties of representations of the fixed point subalgebra of $\\Theta$ on other eigenspaces of $\\Theta$ in $q<m>$."}
{"category": "Math", "title": "A Note on the a-numbers and p-ranks of Kummer Covers", "abstract": "We study the a-numbers and p-ranks of Kummer covers of the projective line, and we give bounds for these numbers."}
{"category": "Math", "title": "Are There Infinitely Many Primes?", "abstract": "This paper is based on a talk given to motivated high school (and younger) students at a BAMA (Bay Area Math Adventure) event. Some of the methods used to study primes and twin primes are introduced."}
{"category": "Math", "title": "Twisted equivariant K-theory for proper actions of discrete groups", "abstract": "We give a construction for twisted equivariant K-theory in the case of a proper action of a discrete group using twisted bundles. Our construction uses results of Lueck and Oliver to extend a construction of Adem and Ruan. We also show the existence of a Chern character to twisted Bredon cohomology. This gives a partial answer to the question of when you can construct twisted equivariant K-theory out of finite rank twisted bundles."}
{"category": "Math", "title": "Right coideal subalgebras in $U_q(\\frak{sl}_{n+1}).$", "abstract": "We offer a complete classification of right coideal subalgebras which contain all group-like elements for the multiparameter version of the quantum group $U_q(\\mathfrak{sl}_{n+1})$ provided that the main parameter $q$ is not a root of 1. As a consequence, we determine that for each subgroup $\\Sigma $ of the group $G$ of all group-like elements the quantum Borel subalgebra $U_q^+ (\\mathfrak{sl}_{n+1})$ containes $(n+1)!$ different homogeneous right coideal subalgebras $U$ such that $U\\cap G=\\Sigma .$ If $q$ has a finite multiplicative order $t>2,$ the classification remains valid for homogeneous right coideal subalgebras of the multiparameter version of the Lusztig quantum group $u_q (\\frak{sl}_{n+1}).$ In the paper we consider the quantifications of Kac-Moody algebras as character Hopf algebras [V.K. Kharchenko, A combinatorial approach to the quantifications of Lie algebras, Pacific J. Math., 203(1)(2002), 191- 233]."}
{"category": "Math", "title": "Superspecies and their representations", "abstract": "Superspecies are introduced to provide the nice constructions of all finite-dimensional superalgebras. All acyclic superspecies, or equivalently all finite-dimensional (gr-basic) gr-hereditary superalgebras, are classified according to their graded representation types. To this end, graded equivalence, graded representation type and graded species are introduced for finite group graded algebras."}
{"category": "Math", "title": "On (Co-)morphisms of Lie Pseudoalgebras and Groupoids", "abstract": "We give a unified description of morphisms and comorphisms of Lie pseudoalgebras, showing that the both types of morphisms can be regarded as subalgebras of a Lie pseudoalgebra, called the $\\psi$-sum. We also provide similar descriptions for morphisms and comorphisms of Lie algebroids and groupoids."}
{"category": "Math", "title": "Construction and Analysis of Projected Deformed Products", "abstract": "We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are ``strictly preserved'' under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose projection to the last dcoordinates yields a neighborly cubical d-polytope. As an extension of thecubical case, we construct matrix representations of deformed products of(even) polygons (DPPs), which have a projection to d-space that retains the complete (\\lfloor \\tfrac{d}{2} \\rfloor - 1)-skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the ``projected deformed products of polygons'' that were announced by Ziegler (2004), a family of 4-polytopes whose ``fatness'' gets arbitrarily close to 9."}
{"category": "Math", "title": "Local torus actions modeled on the standard representation", "abstract": "We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants called a characteristic pair and an Euler class of the orbit map, and prove that local torus actions are classified topologically by them. As a corollary, we obtain a topological classification of locally standard torus actions, which is a generalization of the topological classification of quasi-toric manifolds by Davis and Januszkiewicz and of effective two-dimensional torus actions on four-dimensional manifolds without nontrivial finite stabilizers by Orlik and Raymond. We investigate locally toric Lagrangian fibrations from the viewpoint of local torus actions. We give a necessary and sufficient condition in order that a local torus action becomes a locally toric Lagrangian fibration. Locally toric Lagrangian fibrations are classified by Boucetta and Molino up to fiber-preserving symplectomorphisms. We shall reprove the classification theorem of locally toric Lagrangian fibrations by refining the proof of the classification theorem of local torus actions. We also investigate the topology of a manifold equipped with a local torus action when the Euler class of the orbit map vanishes."}
{"category": "Math", "title": "The (weak-$L^2$) Boundedness of The Quadratic Carleson Operator", "abstract": "We prove that the generalized Carleson operator with polynomial phase function of degree two is of weak type (2,2). For this, we introduce a new approach to the time-frequency analysis of the quadratic phase."}
{"category": "Math", "title": "On Liftings of Local Torus Actions to Fiber Bundles", "abstract": "In this note we define a lifting of a local torus action modeled on the standard representation (we call it a local torus action for simplicity) to a principal torus bundle, and show that there is an obstruction class for the existence of liftings in the first cohomology of the fundamental group of the orbit space with coefficients in a certain module."}
{"category": "Math", "title": "Strong consistency of the maximum likelihood estimator for finite mixtures of location-scale distributions when penalty is imposed on the ratios of the scale parameters", "abstract": "In finite mixtures of location-scale distributions, if there is no constraint or penalty on the parameters, then the maximum likelihood estimator does not exist because the likelihood is unbounded. To avoid this problem, we consider a penalized likelihood, where the penalty is a function of the minimum of the ratios of the scale parameters and the sample size. It is shown that the penalized maximum likelihood estimator is strongly consistent. We also analyze the consistency of a penalized maximum likelihood estimator where the penalty is imposed on the scale parameters themselves."}
{"category": "Math", "title": "Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates", "abstract": "In this paper we will review the main results concerning the issue of stability for the determination unknown boundary portion of a thermic conducting body from Cauchy data for parabolic equations. We give detailed and selfcontained proofs. We prove that such problems are severely ill-posed in the sense that under a priori regularity assumptions on the unknown boundaries, up to any finite order of differentiability, the continuous dependence of unknown boundary from the measured data is, at best, of logarithmic type."}
{"category": "Math", "title": "Degree one cohomology with twisted coefficients of the mapping class group", "abstract": "Let $\\Gamma$ be the mapping class group of an oriented surface $\\Sigma$ of genus g with r boundary components. We prove that the first cohomology group $H^1(\\Gamma, O(M_{SL(2, C)})^*)$ is non-trivial, where the coefficient module is the dual of the space of algebraic functions on the $SL(2, C)$ moduli space over $\\Sigma$."}
{"category": "Math", "title": "On Nurowski's conformal structure associated to a generic rank two distribution in dimension five", "abstract": "For a generic distribution of rank two on a manifold $M$ of dimension five, we introduce the notion of a generalized contact form. To such a form we associate a generalized Reeb field and a partial connection. From these data, we explicitly constructed a pseudo--Riemannian metric on $M$ of split signature. We prove that a change of the generalized contact form only leads to a conformal rescaling of this metric, so the corresponding conformal class is intrinsic to the distribution. In the second part of the article, we relate this conformal class to the canonical Cartan connection associated to the distribution. This is used to prove that it coincides with the conformal class constructed by Nurowski."}
{"category": "Math", "title": "A note on congruences for theta divisors", "abstract": "The classes of two theta divisors on an abelian variety in the naive Grothendieck ring of varieties need not be congruent modulo the class of the affine line."}
{"category": "Math", "title": "The dying rabbit problem revisited", "abstract": "In this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular we give a general recurrence relation for these sequences (improving the work in the paper Hoggatt, V. E., Jr.; Lind, D. A. \"The dying rabbit problem\". Fibonacci Quart. 7 1969 no. 5, 482--487) and we calculate explicitly their general term (extending the work in the paper Miles, E. P., Jr. Generalized Fibonacci numbers and associated matrices. Amer. Math. Monthly 67 1960 745--752). In passing, and as a technical requirement, we also study the behavior of the positive real roots of the characteristic polynomial of the considered sequences."}
{"category": "Math", "title": "Non-vanishing theorems for rank 2 bundles on P^3: a simple approach without the speciality lemma", "abstract": "The paper investigates vanishing conditions on the first cohomology module of a normalized rank 2 vector bundle E on P^3 which force E to split, and finds therefore strategic levels of non-vanishing for a non-split bundle. The present conditions improve other conditions known in the literature and are obtained with simple computations on the Euler characteristic function, avoiding the speciality lemma, Barth's restriction theorem, the discriminat property, and other heavy tools."}
{"category": "Math", "title": "Size, power and false discovery rates", "abstract": "Modern scientific technology has provided a new class of large-scale simultaneous inference problems, with thousands of hypothesis tests to consider at the same time. Microarrays epitomize this type of technology, but similar situations arise in proteomics, spectroscopy, imaging, and social science surveys. This paper uses false discovery rate methods to carry out both size and power calculations on large-scale problems. A simple empirical Bayes approach allows the false discovery rate (fdr) analysis to proceed with a minimum of frequentist or Bayesian modeling assumptions. Closed-form accuracy formulas are derived for estimated false discovery rates, and used to compare different methodologies: local or tail-area fdr's, theoretical, permutation, or empirical null hypothesis estimates. Two microarray data sets as well as simulations are used to evaluate the methodology, the power diagnostics showing why nonnull cases might easily fail to appear on a list of ``significant'' discoveries."}
{"category": "Math", "title": "McKay's correspondence for cocompact discrete subgroups of SU(1,1)", "abstract": "The classical McKay correspondence establishes an explicit link from the representation theory of a finite subgroup G of SU(2) and the geometry of the minimal resolution of the quotient of the affine plane by G. In this paper we discuss a possible generalization of the McKay correspondence to the case when G is replaced with a cocompact discrete subgroup of the universal cover of SU(1,1) such that its image in PSU(1,1) is a cocompact fuchsian group with quotient of genus 0. We establish a correspondence between a certain class of finite-dimensional unitary representations of G and vector bundles on an open algebraic surface with the trivial canonical class canonically associated to G."}
{"category": "Math", "title": "Adding inverses to diagrams II: Invertible homotopy theories are spaces", "abstract": "In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete Segal space model structure on the category of simplicial spaces. Here, we show that these results still hold if we instead use groupoid or \"invertible\" cases. Namely, we show that model structures on the categories of simplicial groupoids, Segal pregroupoids, and invertible simplicial spaces are all Quillen equivalent to one another and to the standard model structure on the category of spaces. We prove this result using two different approaches to invertible complete Segal spaces and Segal groupoids."}
{"category": "Math", "title": "Control of generalized error rates in multiple testing", "abstract": "Consider the problem of testing $s$ hypotheses simultaneously. The usual approach restricts attention to procedures that control the probability of even one false rejection, the familywise error rate (FWER). If $s$ is large, one might be willing to tolerate more than one false rejection, thereby increasing the ability of the procedure to correctly reject false null hypotheses. One possibility is to replace control of the FWER by control of the probability of $k$ or more false rejections, which is called the $k$-FWER. We derive both single-step and step-down procedures that control the $k$-FWER in finite samples or asymptotically, depending on the situation. We also consider the false discovery proportion (FDP) defined as the number of false rejections divided by the total number of rejections (and defined to be 0 if there are no rejections). The false discovery rate proposed by Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289--300] controls $E(FDP)$. Here, the goal is to construct methods which satisfy, for a given $\\gamma$ and $\\alpha$, $P\\{FDP>\\gamma\\}\\le \\alpha$, at least asymptotically. In contrast to the proposals of Lehmann and Romano [Ann. Statist. 33 (2005) 1138--1154], we construct methods that implicitly take into account the dependence structure of the individual test statistics in order to further increase the ability to detect false null hypotheses. This feature is also shared by related work of van der Laan, Dudoit and Pollard [Stat. Appl. Genet. Mol. Biol. 3 (2004) article 15], but our methodology is quite different. Like the work of Pollard and van der Laan [Proc. 2003 International Multi-Conference in Computer Science and Engineering, METMBS'03 Conference (2003) 3--9] and Dudoit, van der Laan and Pollard [Stat. Appl. Genet. Mol. Biol. 3 (2004) article 13], we employ resampling methods to achieve our goals. Some simulations compare finite sample performance to currently available methods."}
{"category": "Math", "title": "A Note on the Effective Non-vanishing Conjecture", "abstract": "We give a reduction of the irregular case for the effective non-vanishing conjecture by virtue of the Fourier-Mukai transform. As a consequence, we reprove that the effective non-vanishing conjecture holds on algebraic surfaces."}
{"category": "Math", "title": "On the Farrell-Jones and related Conjectures", "abstract": "These extended notes are based on a series of six lectures presented at the summer school ``Cohomology of groups and algebraic $K$-theory'' which took place in Hangzhou, China from July 1 until July 12 in 2007. They give an introduction to the Farrell-Jones and the Baum-Connes Conjecture."}
{"category": "Math", "title": "Unit Rectangle Visibility Graphs", "abstract": "Over the past twenty years, rectangle visibility graphs have generated considerable interest, in part due to their applicability to VLSI chip design. Here we study unit rectangle visibility graphs, with fixed dimension restrictions more closely modeling the constrained dimensions of gates and other circuit components in computer chip applications. A graph $G$ is a unit rectangle visibility graph (URVG) if its vertices can be represented by closed unit squares in the plane with sides parallel to the axes and pairwise disjoint interiors, in such a way that two vertices are adjacent if and only if there is a non-degenerate horizontal or vertical band of visibility joining the two rectangles. Our results include necessary and sufficient conditions for $K_n$, $K_{m,n}$, and trees to be URVGs, as well as a number of general edge bounds."}
{"category": "Math", "title": "On crossed product rings with twisted involutions, their module categories and L-theory", "abstract": "We study the Farrell-Jones Conjecture with coefficients in an additive G-category with involution. This is a variant of the L-theoretic Farrell-Jones Conjecture which originally deals with group rings with the standard involution. We show that this formulation of the conjecture can be applied to crossed product rings R*G equipped with twisted involutions and automatically implies the a priori more general fibered version."}
{"category": "Math", "title": "Constant curvature foliations on asymptotically hyperbolic spaces", "abstract": "Let $(M,g)$ be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on $\\del M$ and Weingarten foliations in some neighbourhood of infinity in $M$. We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant $\\sigma_k$-curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature. There is a subtle interplay between the precise terms in the expansion for $g$ and various properties of the foliation. Unlike other recent works in this area, by Rigger \\cite{Ri} and Neves-Tian \\cite{NT1}, \\cite{NT2}, we work in the context of conformally compact spaces, which are more general than perturbations of the AdS-Schwarzschild space, but we do assume a nondegeneracy condition."}
{"category": "Math", "title": "Freeness of equivariant cohomology and mutants of compactified representations", "abstract": "We survey generalisations of the Chang-Skjelbred Lemma for integral coefficients. Moreover, we construct examples of manifolds with actions of tori of rank > 2 whose equivariant cohomology is torsion-free, but not free. This answers a question of Allday's. The \"mutants\" we construct are obtained from compactified representations and involve Hopf bundles in a crucial way."}
{"category": "Math", "title": "Comment on the article arXiv 0710.0349 by Chapoton et al. titled ``An operational calculus for the Mould operad''", "abstract": "This paper has been withdrawn."}
{"category": "Math", "title": "The number of cliques in graphs of given order and size", "abstract": "Let k_r(n,m) denote the minimum number of r-cliques in graphs with n vertices and m edges. For r=3,4 we give a lower bound on k_r(n,m) that approximates k_r(n,m) with an error smaller than n^r/(n^2-2m). The solution is based on a constraint minimization of certain multilinear forms. In our proof, a combinatorial strategy is coupled with extensive analytical arguments."}
{"category": "Math", "title": "A stability version of H\\\"older's inequality", "abstract": "We present a stability version of H\\\"older's inequality, incorporating an extra term that measures the deviation from equality. Applications are given."}
{"category": "Math", "title": "Integrability of Rough Almost Complex Structures", "abstract": "We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures."}
{"category": "Math", "title": "Testing Benson's Regularity Conjecture", "abstract": "D. J. Benson conjectures that the Castelnuovo-Mumford regularity of a group cohomology ring is always zero. More generally he conjectures that the cohomology ring always has a system of parameters satisfying a property he calls very strong quasi-regular. Using computer calculations we find that the more general conjecture holds for all groups of order less than 256."}
{"category": "Math", "title": "The Complex Frobenius Theorem for Rough Involutive Structures", "abstract": "We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat CR-manifold, it suffices that S be Lipschitz for our results to apply. A principal tool in the analysis is a precise version of the Newlander-Nirenberg theorem with parameters, for integrable almost complex structures with minimal regularity, which builds on previous recent work of the authors."}
{"category": "Math", "title": "Weyl's law in the theory of automorphic forms", "abstract": "For a compact Riemannian manifold, Weyl's law describes the asymptotic behavior of the counting function of the eigenvalues of the associated Laplace operator. In this paper we discuss Weyl's law in the context of automorphic forms. The underlying manifolds are locally symmetric spaces of finite volume. In the non-compact case Weyl's law is closely related to the problem of existence of cusp forms."}
{"category": "Math", "title": "Random walk delayed on percolation clusters", "abstract": "We study a continuous time random walk on the $d$-dimensional lattice, subject to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we discuss the diffusive behavior in the case of zero drift and weak attraction."}
{"category": "Math", "title": "The resultant on compact Riemann surfaces", "abstract": "We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain."}
{"category": "Math", "title": "An approach to the finitistic dimension conjecture", "abstract": "Let $R$ be a finite dimensional $k$-algebra over an algebraically closed field $k$ and $\\mathrm{mod} R$ be the category of all finitely generated left $R$-modules. For a given full subcategory $\\mathcal{X}$ of $\\mathrm{mod} R,$ we denote by $\\pfd \\mathcal{X}$ the projective finitistic dimension of $\\mathcal{X}.$ That is, $\\pfd \\mathcal{X}:=\\mathrm{sup} \\{\\pd X : X\\in\\mathcal{X} \\text{and} \\pd X<\\infty\\}.$ \\ It was conjectured by H. Bass in the 60's that the projective finitistic dimension $\\pfd (R):=\\pfd (\\mathrm{mod} R)$ has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and J. Todorov defined a function $\\Psi:\\mathrm{mod} R\\to \\Bbb{N},$ which turned out to be useful to prove that $\\pfd (R)$ is finite for some classes of algebras. In order to have a different approach to the finitistic dimension conjecture, we propose to consider a class of full subcategories of $\\mathrm{mod} R$ instead of a class of algebras, namely to take the class of categories $\\F(\\theta)$ of $\\theta$-filtered $R$-modules for all stratifying systems $(\\theta,\\leq)$ in $\\mathrm{mod} R.$"}
{"category": "Math", "title": "On NIP and invariant measures", "abstract": "We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if $p = tp(b/A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd(A)$ (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ definably amenable, (iii) definitions, characterizations and properties of \"generically stable\" types and groups (iv) uniqueness of translation invariant Keisler measures on groups with finitely satisfiable generics (vi) A proof of the compact domination conjecture for definably compact commutative groups in $o$-minimal expansions of real closed fields."}
{"category": "Math", "title": "On ground fields of arithmetic hyperbolic reflection groups. III", "abstract": "This paper continues arXiv.org:math.AG/0609256, arXiv:0708.3991 and arXiv:0710.0162 . Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimension at least 3 are defined, and explicit bounds of their degrees (over Q) are obtained. Thus, now, explicit bound of degree of ground fields of arithmetic hyperbolic reflection groups is known in all dimensions. Thus, now, we can, in principle, obtain effective finite classification of arithmetic hyperbolic reflection groups in all dimensions together."}
{"category": "Math", "title": "Hilbert Functions of Filtered Modules", "abstract": "In this presentation we shall deal with some aspects of the theory of Hilbert functions of modules over local rings, and we intend to guide the reader along one of the possible routes through the last three decades of progress in this area of dynamic mathematical activity. Motivated by the ever increasing interest in this field, our goal is to gather together many new developments of this theory into one place, and to present them using a unifying approach which gives self-contained and easier proofs. In this text we shall discuss many results by different authors, following essentially the direction typified by the pioneering work of J. Sally. Our personal view of the subject is most visibly expressed by the presentation of Chapters 1 and 2 in which we discuss the use of the superficial elements and related devices. Basic techniques will be stressed with the aim of reproving recent results by using a more elementary approach. Over the past few years several papers have appeared which extend classical results on the theory of Hilbert functions to the case of filtered modules. The extension of the theory to the case of general filtrations on a module has one more important motivation. Namely, we have interesting applications to the study of graded algebras which are not associated to a filtration, in particular the Fiber cone and the Sally-module. We show here that each of these algebras fits into certain short exact sequences, together with algebras associated to filtrations. Hence one can study the Hilbert function and the depth of these algebras with the aid of the know-how we got in the case of a filtration."}
{"category": "Math", "title": "Ramsey degrees of finite ultrametric spaces, ultrametric Urysohn spaces and dynamics of their isometry groups", "abstract": "We study Ramsey-theoretic properties of several natural classes of finite ultrametric spaces, describe the corresponding Urysohn spaces and compute a dynamical invariant attached to their isometry groups."}
{"category": "Math", "title": "Big Ramsey degrees and divisibility in classes of ultrametric spaces", "abstract": "Given a countable set S of positive reals, we study finite-dimensional Ramsey-theoretic properties of the countable ultrametric Urysohn space with distances in S."}
{"category": "Math", "title": "A Criterion For Ergodicity of Non-uniformly hyperbolic Diffeomorphisms", "abstract": "In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the $C^1$ topology, to a conjecture of Pugh-Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle."}
{"category": "Math", "title": "Global duality, signature calculus and the discrete logarithm problem", "abstract": "We study the discrete logarithm problem for the multiplicative group and for elliptic curves over a finite field by using a lifting of the corresponding object to an algebraic number field and global duality. We introduce the \\textit{signature} of a Dirichlet character (in the multiplicative group case) or principal homogeneous space (in the elliptic curve case), which is a measure of the ramification at certain places. We then develop \\textit{signature calculus}, which generalizes and refines the index calculus method. Finally, we show the random polynomial time equivalence for these two cases between the problem of computing signatures and the discrete logarithm problem."}
{"category": "Math", "title": "Surgery, Yamabe invariant, and Seiberg-Witten theory", "abstract": "By using the gluing formula of the Seiberg-Witten invariant, we compute the Yamabe invariant Y(X) of 4-manifolds X obtained by performing surgeries along points, circles or tori on compact Kaehler surfaces. For instance, if M is a compact Kaehler surface of nonnegative Kodaira dimension, and N is a smooth closed oriented 4-manifold with b_2^+(N)=0 and Y(N)>= 0, then we show that Y(M # N)=Y(M)."}
{"category": "Math", "title": "Characteristic foliation on the discriminantal hypersurface of a holomorphic Lagrangian fibration", "abstract": "We give a Kodaira-type classification of general singular fibers of a holomorphic Lagrangian fibration in Fujiki's class $\\mathcal C$. Our approach is based on the study of the characteristic vector field of the discriminantal hypersurface, which naturally arises from the defining equation of the hypersurface via the symplectic form. As an application, we show that the characteristic foliation of the discriminantal hypersurface has algebraic leaves which are either rational curves or smooth elliptic curves."}
{"category": "Math", "title": "Connected sums with HP^n or CaP^2 and the Yamabe invariant", "abstract": "Let $M$ be a smooth closed $4k$-manifold whose Yamabe invariant $Y(M)$ is nonpositive. We show that $$Y(M\\sharp l \\Bbb HP^k\\sharp m \\bar{\\Bbb HP^k})=Y(M),$$ where $l,m$ are nonnegative integers, and $\\Bbb HP^k$ is the quaternionic projective space. When $k=4$, we also have $$Y(M\\sharp l CaP^2\\sharp m \\bar{CaP^2})=Y(M),$$ where $CaP^2$ is the Cayley plane."}
{"category": "Math", "title": "Long-time limit for a class of quadratic infinite-dimensional dynamical systems inspired by models of viscoelastic fluids", "abstract": "We study a class of quadratic, infinite-dimensional dynamical systems, inspired by models for viscoelastic fluids. We prove that these equations define a semi-flow on the cone of positive, essentially bounded functions. As time tends to infinity, the solutions tend to an equilibrium manifold in the $L^2$-norm. Convergence to a particular function on the equilibrium manifold is only proved under additional assumptions. We discuss several possible generalizations."}
{"category": "Math", "title": "Local Convexity Properties of j-metric Balls", "abstract": "This paper deals with local convexity properties of the j-metric. We consider convexity and starlikeness of the j-metric balls in convex, starlike and general subdomains of R^n."}
{"category": "Math", "title": "Characterizing arbitrarily slow convergence in the method of alternating projections", "abstract": "In 1997, Bauschke, Borwein, and Lewis have stated a trichotomy theorem that characterizes when the convergence of the method of alternating projections can be arbitrarily slow. However, there are two errors in their proof of this theorem. In this note, we show that although one of the errors is critical, the theorem itself is correct. We give a different proof that uses the multiplicative form of the spectral theorem, and the theorem holds in any real or complex Hilbert space, not just in a real Hilbert space."}
{"category": "Math", "title": "On Chern-Moser normal forms of strongly pseudoconvex hypersurfaces with high-dimensional stability group", "abstract": "We explicitly describe germs of strongly pseudoconvex non-spherical real-analytic hypersurfaces $M$ at the origin in $\\CC^{n+1}$ for which the group of local CR-automorphisms preserving the origin has dimension $d_0(M)$ equal to either $n^2-2n+1$ with $n\\ge 2$, or $n^2-2n$ with $n\\ge 3$. The description is given in terms of equations defining hypersurfaces near the origin, written in the Chern-Moser normal form. These results are motivated by the classification of locally homogeneous Levi non-degenerate hypersurfaces in $\\CC^3$ with $d_0(M)=1,2$ due to A. Loboda, and complement earlier joint work by V. Ezhov and the author for the case $d_0(M)\\ge n^2-2n+2$."}
{"category": "Math", "title": "Demailly-Semple jets of orders 4 and 5 in dimension 2", "abstract": "In view of Kobayashi's hyperbolicity conjecture, Demailly-Semple jets of orders 4 and 5 in dimension 2 are studied, some expectations about their algebraic tameness being (dis)proved, after systematic, substantial, formal, manual calculations."}
{"category": "Math", "title": "A peculiar two point boundary value problem", "abstract": "In this paper we consider a one-dimensional diffusion equation on the interval $[0,1]$ satisfying non-Feller boundary conditions. As a consequence, the initial value Cauchy problem fails to preserve nonnegativity or boundedness. Nonetheless, probability theory plays an interesting role in our analysis and understanding of solutions to this equation."}
{"category": "Math", "title": "Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging", "abstract": "The work treats systems combining slow and fast motions depending on each other where fast motions are perturbations of families of either dynamical systems or Markov processes with freezed slow variable. In the first case we consider hyperbolic dynamical systems and in the second case we deal with random evolutions which are combinations of diffusions and continuous time Markov chains. We study first large deviations of the slow motion from the averaged one and then use these results together with some Markov property type arguments in order to describe very long time behavior of the slow motion such as its transitions between attractors of the averaged system."}
{"category": "Math", "title": "The Kontsevich weight of a wheel with spokes pointing outward", "abstract": "This is a companion note to ``Hochschild cohomology and Atiyah classes'' by Damien Calaque and the author. Using elementary methods we compute the Kontsevich weight of a wheel with spokes pointing outward. The result is in terms of modified Bernouilli numbers. The same result had been obtained earlier by Torossian (unpublished) and also recently by Thomas Willmacher using more advanced methods."}
{"category": "Math", "title": "The Nicolas and Robin inequalities with sums of two squares", "abstract": "In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $\\sigma(n)<e^\\gamma n\\log\\log n$ holds for every integer $n>5040$, where $\\sigma(n)$ is the sum of divisors function, and $\\gamma$ is the Euler-Mascheroni constant. We exhibit a broad class of subsets $\\cS$ of the natural numbers such that the Robin inequality holds for all but finitely many $n\\in\\cS$. As a special case, we determine the finitely many numbers of the form $n=a^2+b^2$ that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality $n/\\phi(n)<e^{\\gamma}\\log \\log n$; since $\\sigma(n)/n<n/\\phi(n)$ for $n>1$ our results for the Robin inequality follow at once."}
{"category": "Math", "title": "A variational principle for hardening elastoplasticity", "abstract": "We present a variational principle governing the quasistatic evolution of a linearized elastoplastic material. In case of linear hardening, the novel characterization allows to recover and partly extend some known results and proves itself to be especially well-suited for discussing general approximation and convergence issues. In particular, the variational principle is exploited in order to prove in a novel setting the convergence of time and space-time discretizations as well as to provide some possible a posteriori error control."}
{"category": "Math", "title": "Isospectral orbifolds with different maximal isotropy orders", "abstract": "We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which also implies isospectrality on $p$-forms; here the orbifolds are quotients of certain compact normal homogeneous spaces. In another type of examples, the orbifolds are quotients of Euclidean $\\R^3$ and are shown to be isospectral on functions using dimension formulas for the eigenspaces. In the latter type of examples the orbifolds are not isospectral on 1-forms. Along the way we also give several additional examples of isospectral orbifolds which do not have maximal isotropy groups of different size but other interesting properties."}
{"category": "Math", "title": "Integrability of geodesic flows and isospectrality of Riemannian manifolds", "abstract": "We construct a pair of compact, eight-dimensional, two-step Riemannian nilmanifolds $M$ and $M'$ which are isospectral for the Laplace operator on functions and such that $M$ has completely integrable geodesic flow in the sense of Liouville, while $M'$ has not. Moreover, for both manifolds we analyze the structure of the submanifolds of the unit tangent bundle given by to maximal continuous families of closed geodesics with generic velocity fields. The structure of these submanifolds turns out to reflect the above (non)integrability properties. On the other hand, their dimension is larger than that of the Lagrangian tori in $M$, indicating a degeneracy which might explain the fact that the wave invariants do not distinguish an integrable from a nonintegrable system here. Finally, we show that for $M$, the invariant eight-dimensional tori which are foliated by closed geodesics are dense in the unit tangent bundle, and that both $M$ and $M'$ satisfy the so-called Clean Intersection Hypothesis."}
{"category": "Math", "title": "Two positivity conjectures for Kerov polynomials", "abstract": "Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the free cumulants of the associated Young diagram. We present two positivity conjectures for their coefficients. The latter are stronger than the positivity conjecture of Kerov-Biane, recently proved by Feray."}
{"category": "Math", "title": "On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity", "abstract": "We prove an inequality on the Wasserstein distance with quadratic cost between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, from which we deduce some uniqueness results. In particular, we obtain a local (in time) well-posedness result in the case of (possibly very) soft potentials. A global well-posedeness result is shown for all regularized hard and soft potentials without angular cutoff. Our uniqueness result seems to be the first one applying to a strong angular singularity, except in the special case of Maxwell molecules. Our proof relies on the ideas of Tanaka: we give a probabilistic interpretation of the Boltzmann equation in terms of a stochastic process. Then we show how to couple two such processes started with two different initial conditions, in such a way that they almost surely remain close to each other."}
{"category": "Math", "title": "A geometric construction for fractal sets and related linear operators", "abstract": "A way to add an extra dimension is briefly discussed."}
{"category": "Math", "title": "On the dynamics of (left) orderable groups", "abstract": "We study left orderable groups by using dynamical methods. We apply these techniques to study the space of orderings of these groups. We show for instance that for the case of (non-Abelian) free groups, this space is homeomorphic to the Cantor set. We also study the case of braid groups (for which the space of orderings has isolated points but contains homeomorphic copies of the Cantor set). To do this we introduce the notion of the Conradian soul of an order as the maximal subgroup which is convex and restricted to which the original ordering satisfies the so called conradian property, and we elaborate on this notion."}
{"category": "Math", "title": "Singularities of Brill-Noether loci for vector bundles on a curve", "abstract": "In this paper we consider the singularities of the varieties parameterizing stable vector bundles of fixed rank and degree with sections on a smooth curve of genus at least two. In particular, we extend results of Y. Laszlo, and of the second author, regarding the singularities of generalized theta divisors."}
{"category": "Math", "title": "On the arithmetical rank of a special class of minimal varieties", "abstract": "We study the arithmetical ranks and the cohomological dimensions of an infinite class of Cohen-Macaulay varieties of minimal degree. Among these we find, on the one hand, infinitely many set-theoretic complete intersections, on the other hand examples where the arithmetical rank is arbitrarily greater than the codimension."}
{"category": "Math", "title": "Submersions and effective descent of etale morphisms", "abstract": "Using the flatification by blow-up result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of etale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination of noetherian hypotheses in many instances."}
{"category": "Math", "title": "A Categorical Construction of Ultrafilters", "abstract": "Ultrafilters are useful mathematical objects having applications in nonstandard analysis, Ramsey theory, Boolean algebra, topology, and other areas of mathematics. In this note, we provide a categorical construction of ultrafilters in terms of the inverse limit of an inverse family of finite partitions; this is an elementary and intuitive presentation of a consequence of the profiniteness of Stone spaces. We then apply this construction to answer a question of Rosinger posed in arXiv:0709.0084v2 in the negative."}
{"category": "Math", "title": "Stochastic Integrals and Evolution Equations with Gaussian Random Fields", "abstract": "The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of the chaos space of the random field, combined with the Wick product, leads to the \\Ito-Skorokhod integral, and provides an efficient tool to study the integral, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields. Some examples of the corresponding stochastic differential equations are also considered."}
{"category": "Math", "title": "Locally Compact Objects in Exact Categories", "abstract": "We identify two categories of locally compact objects on an exact category A. They correspond to the well-known constructions of the Beilinson category lim A and the Kato category k(A). We study their mutual relations and compare the two constructions. We prove that lim A is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants. It is natural therefore to consider the Beilinson category lim A as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categories Ind_{aleph_0}(C), Pro_{aleph_0}(C) of countably indexed ind/pro-objects over any category C can be described as localizations of categories of diagrams over C."}
{"category": "Math", "title": "A formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles", "abstract": "We give a formula for the coincidence Reidemeister trace of selfmaps on bouquets of circles in terms of the Fox calculus. Our formula reduces the problem of computing the coincidence Reidemeister trace to the problem of distinguishing doubly twisted conjugacy classes in free groups."}
{"category": "Math", "title": "Lens space surgeries and L-space homology spheres", "abstract": "We describe necessary and sufficient conditions for a knot in an L-space to have an L-space homology sphere surgery. We use these conditions to reformulate a conjecture of Berge about which knots in S^3 admit lens space surgeries."}
{"category": "Math", "title": "Isoperimetric regions in spherical cones and Yamabe constants of $M\\times S^1$", "abstract": "Given closed Riemannian manifold $(M^n, g)$ of positive Ricci curvature $Ricci(g) \\geq (n-1)g$ we study isoperimetric regions on the spherical cone over $M$. When $g$ is Einstein we use this to compute the Yamabe constant of $(M \\times {\\bf R}, g + dt^2)$ and so to obtain lower bounds for the Yamabe invariant of $M\\times S^1$."}
{"category": "Math", "title": "A Freiman-type theorem for locally compact abelian groups", "abstract": "We prove a Freiman-type theorem for locally compact abelian groups. If A is a subset of a locally compact abelian group with Haar measure m and m(nA) < n^dm(A) for all n>d log d then we describe A in a way which is tight up to logarithmic factors in d."}
{"category": "Math", "title": "Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture", "abstract": "Let k be a positive integer divisible by 4, l>k a prime, and f an elliptic cuspidal eigenform of weight k-1, level 4, and non-trivial character. Let \\rho_f be the l-adic Galois representation attached to f. In this paper we provide evidence for the Bloch-Kato conjecture for a twist of the adjoint motif of \\rho_f in the following way. Let L(f,s) denote the symmetric square L-function of f. We prove that (under certain conditions) the l-adic valuation of the algebraic part of L(f, k) is no greater than the l-adic valuation of the order of S, where S is (the Pontryagin dual of) the Selmer group attached to the Galois module \\ad^0\\rho_f|_{G_K} (-1), and K= Q(i). Our method uses an idea of Ribet in that we introduce an intermediate step and produce congruences between CAP and non-CAP modular forms on the unitary group U(2,2)."}
{"category": "Math", "title": "Products in Hopf-Cyclic Cohomology", "abstract": "We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. The key ideas instrumental in constructing these pairings are the derived functor interpretation of Hopf-cyclic and equivariant cyclic (co)homology, and the Yoneda interpretation of Ext-groups. As a special case of one of these pairings, we recover the Connes-Moscovici characteristic map in Hopf-cyclic cohomology. We also prove that this particular pairing, along with few others, would stay the same if we replace the derived category of (co)cyclic modules with the homotopy category of (special) towers of $X$-complexes, or the derived category of mixed complexes."}
{"category": "Math", "title": "Rigidity of graph products of abelian groups", "abstract": "We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe."}
{"category": "Math", "title": "On the automorphisms of a graph product of abelian groups", "abstract": "We study the automorphisms of a graph product of finitely-generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = Aut W whenever vertex groups are finite and in a number of other cases. We prove a number of structure results, including a semi-direct product decomposition of Aut* W in which one of the factors is Inn W. We also give a number of applications, some of which are geometric in nature."}
{"category": "Math", "title": "A Comparison Theorem and A Sharp Bound via the Ricci Flow", "abstract": "We prove a comparison theorem for the compact surfaces with negative Euler characteristic via the Ricci flow."}
{"category": "Math", "title": "The representation dimension of quantum complete intersections", "abstract": "We study the representation dimension of the class of algebras known as quantum complete intersections. For such an algebra, we show that the representation dimension is at most twice its codimension. Moreover, we show that the representation dimension of a \"homogeneous\" quantum complete intersection is strictly larger than its codimension."}
{"category": "Math", "title": "On the conditional logistic estimator for repeated binary outcomes in two-arm experimental studies with non-compliance", "abstract": "The behavior of the conditional logistic estimator is analyzed under a causal model for two-arm experimental studies with possible non-compliance in which the effect of the treatment is measured by a binary response variable. We show that, when non-compliance may only be observed in the treatment arm, the effect (measured on the logit scale) of the treatment on compliers and that of the control on non-compliers can be identified and consistently estimated under mild conditions. The same does not happen for the effect of the control on compliers. A simple correction of the conditional logistic estimator is then proposed which allows us to considerably reduce its bias in estimating this quantity and the causal effect of the treatment over control on compliers. A two-step estimator results whose asymptotic properties are studied by exploiting the general theory on maximum likelihood estimation of misspecified models. Finite-sample properties of the estimator are studied by simulation and the extension to the case of missing responses is outlined. The approach is illustrated by an application to a dataset deriving from a study on the efficacy of a training course on the practise of breast self examination."}
{"category": "Math", "title": "Bijective 1-cocycles and classification of 3-dimensional Left-symmetric Algebras", "abstract": "Left-symmetric algebras have close relations with many important fields in mathematics and mathematical physics. Their classification is very complicated due to the nonassociativity. In this paper, we re-study the correspondence between left-symmetric algebras and the bijective 1-cocycles. Then a procedure is provided to classify left-symmetric algebras in terms of classification of equivalent classes of bijective 1-cocycles. As an example, the 3-dimensional complex left-symmetric algebras are classified."}
{"category": "Math", "title": "Involutions on tori with codimension one fixed point set", "abstract": "The standard P. A. Smith theory of p-group actions on spheres, disks, and euclidean spaces is extended to the case of p-group actions on tori (i.e., products of circles) and coupled with topological surgery theory to give a complete topological classification, valid in all dimensions, of the locally linear, orientation reversing, involutions on tori with fixed point set of codimension one."}
{"category": "Math", "title": "Multivariate integration of functions depending explicitly on the minimum and the maximum of the variables", "abstract": "By using some basic calculus of multiple integration, we provide an alternative expression of the integral $$ \\int_{]a,b[^n} f(\\mathbf{x},\\min x_i,\\max x_i) d\\mathbf{x}, $$ in which the minimum and the maximum are replaced with two single variables. We demonstrate the usefulness of that expression in the computation of orness and andness average values of certain aggregation functions. By generalizing our result to Riemann-Stieltjes integrals, we also provide a method for the calculation of certain expected values and distribution functions."}
{"category": "Math", "title": "Fundamental groups of topological stacks with slice property", "abstract": "The main result of the paper is a formula for the fundamental group of the coarse moduli space of a topological stack. As an application, we find simple general formulas for the fundamental group of the coarse quotient of a group action on a topological space in terms of the fixed point data. The formulas seem, surprisingly, to be new. In particular, we recover, and vastly generalize, results of Armstrong, Bass, Higgins, Rhodes."}
{"category": "Math", "title": "Classification of complete Finsler manifolds through a second order differential equation", "abstract": "By using a certain second order differential equation, the notion of adapted coordinates on Finsler manifolds is defined and some classifications of complete Finsler manifolds are found. Some examples of Finsler metrics, with positive constant sectional curvature, not necessarily of Randers type nor projectively flat, are found. This work generalizes some results in Riemannian geometry and open up, a vast area of research on Finsler geometry."}
{"category": "Math", "title": "Cup products in Hopf cyclic cohomology via cyclic modules I", "abstract": "This is the first one in a series of two papers on the continuation of our study in cup products in Hopf cyclic cohomology. In this note we construct cyclic cocycles of algebras out of Hopf cyclic cocycles of algebras and coalgebras. In the next paper we consider producing Hopf cyclic cocycle from \"equivariant\" Hopf cyclic cocycles. Our approach in both situations is based on (co)cyclic modules and bi(co)cyclic modules together with Eilenberg-Zilber theorem which is different from the old definition of cup products defined via traces and cotraces on DG algebras and coalgebras."}
{"category": "Math", "title": "On two problems concerning topological centers", "abstract": "Let G be an infinite discrete group and bG its Cech-Stone compactification. Using the well known fact that a free ultrafilter on an infinite set is nonmeasurable, we show that for each element p of the remainder bG G, left multiplication L_p:bG \\to bG is not Borel measurable. Next assume that G is abelian. Let D \\subset \\ell^\\infty(G)$ denote the subalgebra of distal functions on G and let G^D denote the corresponding universal distal (right topological group) compactification of G. Our second result is that the topological center of G^D (i.e. the set of p in G^D for which L_p:G^D \\to G^D is a continuous map) is the same as the algebraic center and that for G=Z (the group of integers) this center coincides with the canonical image of G in G^D."}
{"category": "Math", "title": "Branching integrals and Casselman phenomenon", "abstract": "Let $G$ be a real semisimple Lie group, $K$ its maximal complex subgroup, and $G_C$ its complexification. It is known that all the $K$-finite matrix elements on $G$ admit holomorphic continuation to branching functions on $G_C$ having singularities at the a prescribed divisor. We propose a geometric explanation of this phenomenon. The note also contsins a general survey of holomorphic continuations of infinite-dimensional representations."}
{"category": "Math", "title": "On cross-ratio distortion and Schwarz derivative", "abstract": "We prove asymptotic estimates for the cross-ratio distortion with respect to a smooth or holomorphic function in terms of its Schwarz derivative."}
{"category": "Math", "title": "Deformations of Border Bases", "abstract": "Here we study the problem of generalizing one of the main tools of Groebner basis theory, namely the flat deformation to the leading term ideal, to the border basis setting. After showing that the straightforward approach based on the deformation to the degree form ideal works only under additional hypotheses, we introduce border basis schemes and universal border basis families. With their help the problem can be rephrased as the search for a certain rational curve on a border basis scheme. We construct the system of generators of the vanishing ideal of the border basis scheme in different ways and study the question of how to minimalize it. For homogeneous ideals, we also introduce a homogeneous border basis scheme and prove that it is an affine space in certain cases. In these cases it is then easy to write down the desired deformations explicitly."}
{"category": "Math", "title": "Origins and breadth of the theory of higher homotopies", "abstract": "Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. We recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940's. Prompted by the failure of the Alexander-Whitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor's classifying space construction to associative H-spaces, and a careful examination of this extension led Stasheff to the discovery of An-spaces and Ainfty-spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950's, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic observation is that higher homotopy structures behave much better relative to homotopy than strict structures, and HPT enables one to exploit this observation in various concrete situations which, in particular, leads to the effective calculation of various invariants which are otherwise intractable. Higher homotopies abound but they are rarely recognized explicitly and their significance is hardly understood; at times, their appearance might at first glance even come as a surprise, for example in the Kodaira-Spencer approach to deformations of complex manifolds or in the theory of foliations."}
{"category": "Math", "title": "Hopf algebras and characters of classical groups", "abstract": "Schur functions provide an integral basis of the ring of symmetric functions. It is shown that this ring has a natural Hopf algebra structure by identifying the appropriate product, coproduct, unit, counit and antipode, and their properties. Characters of covariant tensor irreducible representations of the classical groups GL(n), O(n) and Sp(n) are then expressed in terms of Schur functions, and the Hopf algebra is exploited in the determination of group-subgroup branching rules and the decomposition of tensor products. The analysis is carried out in terms of n-independent universal characters. The corresponding rings, CharGL, CharO and CharSp, of universal characters each have their own natural Hopf algebra structure. The appropriate product, coproduct, unit, counit and antipode are identified in each case."}
{"category": "Math", "title": "Geometry of Multiplicative Preprojective Algebra", "abstract": "Crawley-Boevey and Shaw recently introduced a certain multiplicative analogue of the deformed preprojective algebra, which they called the multiplicative preprojective algebra. In this paper we study the moduli space of (semi)stable representations of such an algebra (the multiplicative quiver variety), which in fact has many similarities to the quiver variety. We show that there exists a complex analytic isomorphism between the nilpotent subvariety of the quiver variety and that of the multiplicative quiver variety (which can be extended to a symplectomorphism between these tubular neighborhoods). We also show that when the quiver is star-shaped, the multiplicative quiver variety parametrizes Simpson's (poly)stable filtered local systems on a punctured Riemann sphere with prescribed filtration type, weight and associated graded local system around each puncture."}
{"category": "Math", "title": "Groupoid Extensions of Mapping Class Representations for Bordered Surfaces", "abstract": "The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichm\\\"uller space, whose fundamental path groupoid is called the Ptolemy groupoid. It is natural to try to extend representations of the mapping class group to the mapping class groupoid, i.e., construct a homomorphism from the mapping class groupoid to the same target that extends the given representations arising from various choices of basepoint. Among others, we extend both aforementioned representations to the groupoid level in this sense, where the symplectic representation is lifted both rationally and integrally. The techniques of proof include several algorithms involving fatgraphs and chord diagrams. The former extension is given by explicit formulae depending upon six essential cases, and the kernel and image of the groupoid representation are computed. Furthermore, this provides groupoid extensions of any representation of the mapping class group that factors through its action on the fundamental group of the surface including, for instance, the Magnus representation and representations on the moduli spaces of flat connections."}
{"category": "Math", "title": "Catalog of dessins d'enfants with \\le 4 edges", "abstract": "In this work all the dessins d'enfant with no more than 4 edges are listed and their Belyi pairs are computed. In order to enumerate all dessins the technique of matrix model computations was used. The total number of dessins is 134; among them 77 are spherical, 53 of genus 1 and 4 of genus 2. The orders of automorphism groups of all the dessins are also found. Dessins are listed by the number of edges. Dessins with the same number of edges are ordered lexicographically by their lists of 0-valencies. The corresponding matrix model for any list of 0-valencies is given and computed. Complex matrix models for dessins with 1 -- 3 edges are used. For the dessins with 4 edges we use Hermitian matrix model, correlators for which are computed in [1]."}
{"category": "Math", "title": "Affine Geometry of Space Curves", "abstract": "This paper is devoted to the complete classification of space curves under affine transformations in the view of Cartan's theorem. Spivak has introduced the method but has not found the invariants. Furthermore, for the first time, we propound a necessary and sufficient condition for the invariants. Then, we study the shapes of space curves with constant curvatures in detail and suggest their applications in physics, computer vision and image processing."}
{"category": "Math", "title": "Affine Classification of n-Curves", "abstract": "Classification of curves up to affine transformation in a finite dimensional space was studied by some different methods. In this paper, we achieve the exact formulas of affine invariants via the equivalence problem and in the view of Cartan's lemma and then, state a necessary and sufficient condition for classification of n--Curves."}
{"category": "Math", "title": "A State Sum Link Invariant of Regular Isotopy", "abstract": "This paper has been withdrawn because there is a fundamental error in the computations; with the right computational scheme it seems to be just a version of the Jones polynomial"}
{"category": "Math", "title": "Lower bounds on the coefficients of Ehrhart polynomials", "abstract": "We present lower bounds for the coefficients of Ehrhart polynomials of convex lattice polytopes in terms of their volume. Concerning the coefficients of the Ehrhart series of a lattice polytope we show that Hibi's lower bound is not true for lattice polytopes without interior lattice points. The counterexample is based on a formula of the Ehrhart series of the join of two lattice polytope. We also present a formula for calculating the Ehrhart series of integral dilates of a polytope."}
{"category": "Math", "title": "Sets non-thin at \\infty in \\Bbb C ^m, and the growth of sequences of entire functions of genus zero", "abstract": "In this paper we define the notion of non-thin at $\\infty$ as follows: Let $E$ be a subset of $\\Bbb C^m$. For any $R>0$ define $E_R=E\\cap \\{z\\in \\Bbb C ^m :|z|\\leq R\\}$. We say that $E$ is non-thin at $\\infty$ if \\lim_{R\\to\\infty}V_{E_R}(z)=0 for all $z\\in \\Bbb C^m$, where $V_E$ is the pluricomplex Green function of $E$. This definition of non-thin at $\\infty$ has good properties: If $E\\subset \\Bbb C^m$ is non-thin at $\\infty$ and $A$ is pluripolar then $E\\backslash A$ is non-thin at $\\infty$, if $E\\subset \\Bbb C^m$ and $F\\subset \\Bbb C^n$ are closed sets non-thin at $\\infty$ then $E\\times F\\subset \\Bbb C^m\\times \\Bbb C^n$ is non-thin at $\\infty$ (see Lemma \\ref{Lem1}). Then we explore the properties of non-thin at $\\infty$ sets and apply this to extend the results in \\cite{mul-yav} and \\cite{trong-tuyen}."}
{"category": "Math", "title": "Adaptive Directional Subdivision Schemes and Shearlet Multiresolution Analysis", "abstract": "In this paper, we propose a solution for a fundamental problem in computational harmonic analysis, namely, the construction of a multiresolution analysis with directional components. We will do so by constructing subdivision schemes which provide a means to incorporate directionality into the data and thus the limit function. We develop a new type of non-stationary bivariate subdivision schemes, which allow to adapt the subdivision process depending on directionality constraints during its performance, and we derive a complete characterization of those masks for which these adaptive directional subdivision schemes converge. In addition, we present several numerical examples to illustrate how this scheme works. Secondly, we describe a fast decomposition associated with a sparse directional representation system for two dimensional data, where we focus on the recently introduced sparse directional representation system of shearlets. In fact, we show that the introduced adaptive directional subdivision schemes can be used as a framework for deriving a shearlet multiresolution analysis with finitely supported filters, thereby leading to a fast shearlet decomposition."}
{"category": "Math", "title": "Multiplicative properties of Morin maps", "abstract": "In the first part of the paper we construct a ring structure on the rational cobordism classes of Morin maps (i. e. smooth generic maps of corank 1). We show that associating to a Morin map its singular strata defines a ring homomorphism to $\\Omega_* \\otimes \\Q$, the rational oriented cobordism ring. This is proved by analyzing multiple-point sets of product immersion. Using these homomorphisms we are able to identify the ring of Morin maps. In the second part of the paper we compute the oriented Thom polynomial of the $\\Sigma^2$ singularity type with $\\Q$ coefficients. Then we provide a product formula for the $\\Sigma^2$ and the $\\Sigma^{1,1}$ singularities."}
{"category": "Math", "title": "Cuspidal representations of sl(n+1)", "abstract": "In this paper we study the subcategory of cuspidal modules of the category of weight modules over the Lie algebra sl(n+1). Our main result is a complete classification and explicit description of the indecomposable cuspidal modules."}
{"category": "Math", "title": "The Three Hat Problem", "abstract": "In this paper we study the Three Hat Problem which appeared in Puzzle Corner of the Technology Review magazine. This puzzle gives a scenario in which three players wearing hats are sitting together and each hat can be seen by everyone except the player that is wearing that hat. Each player is told that all of the hats contain a positive integer and that two of the integers add to the third. In an ordered, turn-wise, modular fashion, each player truthfully states whether or not he knows his integer. We give a strategy which allows for one of the players to solve for his integer for all possible integer configurations of the puzzle and prove it is the optimal such strategy."}
{"category": "Math", "title": "Another introduction to the geometry of metric spaces", "abstract": "Here Lipschitz conditions are used as a primary tool, for studying curves in metric spaces in particular."}
{"category": "Math", "title": "Modeling Wildland Fire Propagation with Level Set Methods", "abstract": "Level set methods are versatile and extensible techniques for general front tracking problems, including the practically important problem of predicting the advance of a firefront across expanses of surface vegetation. Given a rule, empirical or otherwise, to specify the rate of advance of an infinitesimal segment of firefront arc normal to itself (i.e., given the firespread rate as a function of known local parameters relating to topography, vegetation, and meteorology), level set methods harness the well developed mathematical machinery of hyperbolic conservation laws on Eulerian grids to evolve the position of the front in time. Topological challenges associated with the swallowing of islands and the merger of fronts are tractable. The principal goals of this paper are to: collect key results from the two largely distinct scientific literatures of level sets and firespread; demonstrate the practical value of level set methods to wildland fire modeling through numerical experiments; probe and address current limitations; and propose future directions in the simulation of, and the development of decision-aiding tools to assess countermeasure options for, wildland fires. In addition, we introduce a freely available two-dimensional level set code used to produce the numerical results of this paper and designed to be extensible to more complicated configurations."}
{"category": "Math", "title": "Lower bounds for warping functions on warped-product AHE manifolds", "abstract": "Let $[\\gamma]$ be the conformal boundary of a warped product $C^{3,\\alpha}$ AHE metric $g=g_M+u^2h$ on $N=M \\times F$, where $(F,h)$ is compact with unit volume and nonpositive curvature. We show that if $[\\gamma]$ has positive Yamabe constant, then $u$ has a positive lower bound that depends only on $[\\gamma]$."}
{"category": "Math", "title": "Kawamata-Viehweg Vanishing on Rational Surfaces in Positive Characteristic", "abstract": "We prove that the Kawamata-Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W_2(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata-Viehweg vanishing theorem holds on log del Pezzo surfaces in positive characteristic."}
{"category": "Math", "title": "Hodge-theoretic aspects of the Decomposition Theorem", "abstract": "Given a projective morphism of compact, complex, algebraic varieties and a relatively ample line bundle on the domain we prove that a suitable choice, dictated by the line bundle, of the decomposition isomorphism of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber, yields isomorphisms of pure Hodge structures. The proof is based on a new cohomological characterization of the decomposition isomorphism associated with the line bundle. We prove some corollaries concerning the intersection form in intersection cohomology, the natural map from cohomology to intersection cohomology, projectors and Hodge cycles, and induced morphisms in intersection cohomology."}
{"category": "Math", "title": "The structures of standard (g,K)-modules of SL(3,R)", "abstract": "We describe explicitely the structures of standard $(g,K)$-modules of $SL(3,R)$."}
{"category": "Math", "title": "Schubert Polynomials for the affine Grassmannian of the symplectic group", "abstract": "We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions, defined in terms of Schur's P and Q-functions. An explicit combinatorial description is obtained for the Schubert basis of the cohomology of Gr, and this is extended to a definition of the affine type C Stanley symmetric functions. A homology Pieri rule is also given for the product of a special Schubert class with an arbitrary one."}
{"category": "Math", "title": "A moduli scheme of embedded curve singularities", "abstract": "The main purpose of this paper is to prove the existence of the moduli space parameterizing the embedded curve singularities of $(k^N,0)$ with an admissible Hilbert polynomial and to study its basic properties."}
{"category": "Math", "title": "Log Structures on Generalized Semi-Stable Varieties", "abstract": "This is my PhD Thesis, part of it has published in Acta Mathematica Sinica. In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structures on this family of morphisms."}
{"category": "Math", "title": "Primes in Tuples II", "abstract": "We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken on by the differences between primes."}
{"category": "Math", "title": "Reliability of Module Based Software System", "abstract": "This paper consider the problem of determining the reliability of a software system which can be decomposed in a number of modules. We have derived the expression of the reliability of a system using the Markovian model for the transfer of control between modules in order. We have given the expression of reliability by considering both benign and catastrophic failure. The expression of reliability presented in this work is applicable for some control software which are designed to detect its own internal errors."}
{"category": "Math", "title": "Kullback Leibler property of kernel mixture priors in Bayesian density estimation", "abstract": "Positivity of the prior probability of Kullback-Leibler neighborhood around the true density, commonly known as the Kullback-Leibler property, plays a fundamental role in posterior consistency. A popular prior for Bayesian estimation is given by a Dirichlet mixture, where the kernels are chosen depending on the sample space and the class of densities to be estimated. The Kullback-Leibler property of the Dirichlet mixture prior has been shown for some special kernels like the normal density or Bernstein polynomial, under appropriate conditions. In this paper, we obtain easily verifiable sufficient conditions, under which a prior obtained by mixing a general kernel possesses the Kullback-Leibler property. We study a wide variety of kernel used in practice, including the normal, $t$, histogram, gamma, Weibull densities and so on, and show that the Kullback-Leibler property holds if some easily verifiable conditions are satisfied at the true density. This gives a catalog of conditions required for the Kullback-Leibler property, which can be readily used in applications."}
{"category": "Math", "title": "Algebraic curves for commuting elements in the q-deformed Heisenberg algebra", "abstract": "In this paper we extend the eliminant construction of Burchnall and Chaundy for commuting differential operators in the Heisenberg algebra to the q-deformed Heisenberg algebra and show that it again provides annihilating curves for commuting elements, provided q satisfies a natural condition. As a side result we obtain estimates on the dimensions of the eigenspaces of elements of this algebra in its faithful module of Laurent series."}
{"category": "Math", "title": "A note on mean volume and surface densities for a class of birth-and-growth stochastic processes", "abstract": "Many real phenomena may be modelled as locally finite unions of $d$-dimensional time dependent random closed sets in $\\mathbb{R}^d$, described by birth-and-growth stochastic processes, so that their mean volume and surface densities, as well as the so called mean \\emph{extended} volume and surface densities, may be studied in terms of relevant quantities characterizing the process. We extend here known results in the Poissonian case to a wider class of birth-and-growth stochastic processes, proving in particular the absolute continuity of the random time of capture of a point $x\\in\\R^d$ by processes of this class."}
{"category": "Math", "title": "Reduced branching processes with very heavy tails", "abstract": "The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter $\\alpha\\in(0,1]$. We turn to the case of very heavy tailed reproduction distribution $\\alpha=0$ assuming Zubkov's regularity condition with parameter $\\beta\\in(0,\\infty)$. Our main result gives a new asymptotic pattern for the reduced branching process conditioned on non-extinction during a long time interval."}
{"category": "Math", "title": "Threefolds with big and nef anticanonical bundles II", "abstract": "In continuation of our paper in Math. Ann. 333 we classify smooth complex projective threefolds X with -K_X big and nef but not ample and Picard number 2, whose anticanonical map is small. We assume also that the Mori contraction of X and of its flop are not both birational."}
{"category": "Math", "title": "Irreducible polynomials with prescribed trace and restricted norm", "abstract": "Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a coset of a subgroup <g^s>, s|(q-1). We give the number explicitly for s=2, 3, 4 when q=p, and for s|(p^e+1) when r=2en. Finally, we give explicit formulae for the number when both trace and norm are fixed, p=2 and m =< 30."}
{"category": "Math", "title": "On a problem of Juhasz and van Mill", "abstract": "A 27 years old and still open problem of Juhasz and van Mill asks whether there exists a cardinal kappa such that every regular dense in itself countably compact space has a dense in itself subset of cardinality at most kappa. We give a negative answer for the analogous question where_regular_ is weakened to_Hausdorff_, and_coutnably compact_ is strengthened to_sequentially compact_."}
{"category": "Math", "title": "Rhombus Filtrations and Rauzy Algebras", "abstract": "Peach introduced rhombal algebras associated to quivers given by tilings of the plane by rhombi. We develop general techniques to analyse rhombal algebras, including a filtration by what we call rhombus modules. We introduce a way to relate the infinite-dimensional rhombal algebra corresponding to a complete tiling of the plane to finite-dimensional algebras corresponding to finite portions of the tiling. Throughout, we apply our general techniques to the special case of the Rauzy tiling, which is built in stages reflecting an underlying self-similarity. Exploiting this self-similar structure allows us to uncover interesting features of the associated finite-dimensional algebras, including some of the tree classes in the stable Auslander-Reiten quiver."}
{"category": "Math", "title": "A weight two phenomenon for the moduli of rank one local systems on open varieties", "abstract": "The twistor space of representations on an open variety maps to a weight two space of local monodromy transformations around a divisor component at infinty. The space of $\\sigma$-invariant sections of this slope-two bundle over the twistor line is a real 3 dimensional space whose parameters correspond to the complex residue of the Higgs field, and the real parabolic weight of a harmonic bundle."}
{"category": "Math", "title": "A classification of some Finsler connections and their applications", "abstract": "Some general Finsler connections are defined. Emphasis is being made on the Cartan tensor and its derivatives. Vanishing of the hv-curvature tensors of these connections characterizes Landsbergian, Berwaldian as well as Riemannian structures. This view point makes it possible to give a smart representation of connection theory in Finsler geometry and yields to a classification of Finsler connections. Some practical applications of these connections are also considered."}
{"category": "Math", "title": "First Stiefel-Whitney class of real moduli spaces of stable maps to a convex surface", "abstract": "Let $(X,c_X)$ be a convex projective surface equipped with a real structure. The space of stable maps $\\bar{\\mathcal{M}}_{0,k}(X,d)$ carries different real structures induced by $c_X$ and any order two element $\\tau$ of permutation group $S_k$ acting on marked points. Each corresponding real part $\\R_{\\tau}\\bar{\\mathcal{M}}_{0,k}(X,d)$ is a real normal projective variety. As the singular locus is of codimension bigger than two, these spaces thus carry a first Stiefel-Whitney class for which we determine a representative in the case $k=c_1(X)d-1$ where $c_1(X)$ is the first Chern class of $X$. Namely, we give a homological description of these classes in term of the real part of boundary divisors of the space of stable maps."}
{"category": "Math", "title": "On the Whitehead spectrum of the circle", "abstract": "The seminal work of Waldhausen, Farrell and Jones, Igusa, and Weiss and Williams shows that the homotopy groups in low degrees of the space of homeomorphisms of a closed Riemannian manifold of negative sectional curvature can be expressed as a functor of the fundamental group of the manifold. To determine this functor, however, it remains to determine the homotopy groups of the topological Whitehead spectrum of the circle. The cyclotomic trace of B okstedt, Hsiang, and Madsen and a theorem of Dundas, in turn, lead to an expression for these homotopy groups in terms of the equivariant homotopy groups of the homotopy fiber of the map from the topological Hochschild T-spectrum of the sphere spectrum to that of the ring of integers induced by the Hurewicz map. We evaluate the latter homotopy groups, and hence, the homotopy groups of the topological Whitehead spectrum of the circle in low degrees. The result extends earlier work by Anderson and Hsiang and by Igusa and complements recent work by Grunewald, Klein, and Macko."}
{"category": "Math", "title": "Topological entropies of equivalent smooth flows", "abstract": "Two flows defined on a smooth manifold are equivalent if there exists a homeomorphism of the manifold that sends each orbit of one flow onto an orbit of the other flow while preserving the time orientation. The topological entropy of a flow is defined as the entropy of its time-1 map. While topological entropy is an invariant for equivalent homeomorphisms, finite non-zero topological entropy for a flow cannot be an invariant because its value is affected by time reparameterization. However, 0 and $\\infty$ topological entropy are invariants for equivalent flows without fixed points. In equivalent flows with fixed points there exists a counterexample, constructed by Ohno, showing that neither 0 nor $\\infty$ topological entropy is preserved by equivalence. The two flows constructed by Ohno are suspensions of a transitive subshift and thus are not differentiable. Note that a differentiable flow on a compact manifold cannot have $\\infty$ entropy. These facts led Ohno in 1980 to ask the following: \"Is 0 topological entropy an invariant for equivalent differentiable flows?\" In this paper, we construct two equivalent $C^\\infty$ smooth flows with a singularity, one of which has positive topological entropy while the other has zero topological entropy. This gives a negative answer to Ohno's question in the class $C^\\infty$."}
{"category": "Math", "title": "An asymptotic integral representation for Carleman orthogonal polynomials", "abstract": "In this paper we investigate the asymptotic behavior of polynomials that are orthonormal over the interior domain of an analytic Jordan curve L with respect to area measure. We prove that, inside L, these polynomials behave asymptotically like a sequence of certain integrals involving the canonical conformal map of the exterior of L onto the exterior of the unit circle and certain meromorphic kernel function defined in terms of a conformal map of the interior of L onto the unit disk. The error term in the integral representation is proven to decay geometrically and sufficiently fast, allowing us to obtain more precise asymptotic formulas for the polynomials under certain additional geometric considerations. These formulas yield, in turn, fine results on the location, limiting distribution and accumulation points of the zeros of the polynomials."}
{"category": "Math", "title": "Universal derived equivalences of posets of cluster tilting objects", "abstract": "We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flip-flop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian lattices corresponding to the various orientations of the same Dynkin diagram are universally derived equivalent."}
{"category": "Math", "title": "A Feynman-Kac-type formula for the deterministic and stochastic wave equations", "abstract": "We establish a probabilistic representation for a wide class of linear deterministic p.d.e.s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies to the heat equation, where it is related to the classical Feynman-Kac formula, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, then this formula leads to an expression for the moments of the solution."}
{"category": "Math", "title": "Canonical extensions of local systems", "abstract": "A local system H on a complex manifold M can be viewed in two ways--either as a locally free sheaf, or as a union of covering spaces T = T(H). When M is an open set in a bigger manifold, the local system will generally not extend, because of local monodromy. This paper proposes an extension of the local system as an analytic space, in the case when the complement of M has normal crossing singularities, and the local system is unipotent along the boundary divisor. The analytic space is obtained by taking the closure of T inside the total space of Deligne's canonical extension of the associated vector bundle. It is not normal, but its normalization is locally toric."}
{"category": "Math", "title": "On the \"pits effect\" of Littlewood and Offord", "abstract": "Suppose that the moduli of the coefficients of a power series are 1/n!, while the arguments are arbitrary. If an entire function f represented by such power series decreases exponentially on some ray, then it has to be an exponential. If the arguments of the coefficients are of the form 2pi n^2a, where a is irrational, then the function displays the so-called \"pits effect\". More precisely, under this condition, f is of completely regular growth with constant indicator."}
{"category": "Math", "title": "The Urysohn sphere is oscillation stable", "abstract": "We solve the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for the Hilbert space in the context of the Urysohn universal metric space. This is achieved by solving a purely combinatorial problem involving a family of countable homogeneous metric spaces with finitely many distances."}
{"category": "Math", "title": "Partition properties of the dense local order and a colored version of Milliken's theorem", "abstract": "We study the finite dimensional partition properties of the countable homogeneous dense local order. Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of Milliken's theorem on trees."}
{"category": "Math", "title": "A tour of theta dualities on moduli spaces of sheaves", "abstract": "The purpose of this paper is twofold. First, we survey known results about theta dualities on moduli spaces of sheaves on curves and surfaces. Secondly, we establish new such dualities in the surface case. Among others, the case of elliptic K3 surfaces is studied in detail; we propose further conjectures which are shown to imply strange duality."}
{"category": "Math", "title": "Spectral isolation of bi-invariant metrics on compact Lie groups", "abstract": "We show that a bi-invariant metric on a compact connected Lie group $G$ is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer $N$ such that, within a neighborhood of $g_0$ in the class of left-invariant metrics of at most the same volume, $g_0$ is uniquely determined by the first $N$ distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where $G$ is simple, $N$ can be chosen to be two."}
{"category": "Math", "title": "Updating Probabilities: An Econometric Example", "abstract": "We demonstrate how information in the form of observable data and moment constraints are introduced into the method of Maximum relative Entropy (ME). A general example of updating with data and moments is shown. A specific econometric example is solved in detail which can then be used as a template for real world problems. A numerical example is compared to a large deviation solution which illustrates some of the advantages of the ME method."}
{"category": "Math", "title": "Tight closure does not commute with localization", "abstract": "We give an example showing that tight closure does not commute with localization."}
{"category": "Math", "title": "Twisted Yangians and finite W-algebras", "abstract": "We construct an explicit set of generators for the finite W-algebras associated to nilpotent matrices in the symplectic or orthogonal Lie algebras whose Jordan blocks are all of the same size. We use these generators to show that such finite W-algebras are quotients of twisted Yangians."}
{"category": "Math", "title": "Total positivity for cominuscule Grassmannians", "abstract": "In this paper we explore the combinatorics of the non-negative part (G/P)+ of a cominuscule Grassmannian. For each such Grassmannian we define Le-diagrams -- certain fillings of generalized Young diagrams which are in bijection with the cells of (G/P)+. In the classical cases, we describe Le-diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Le-diagram. We give enumerative results and relate our Le-diagrams to other combinatorial objects. Surprisingly, the totally non-negative cells in the open Schubert cell of the odd and even orthogonal Grassmannians are (essentially) in bijection with preference functions and atomic preference functions respectively."}
{"category": "Math", "title": "Optimal design problems in rough inhomogeneous media. Existence theory", "abstract": "This paper settles the existence question for a rather general class of convex optimal design problems with a volume constraint. In low dimensions, we prove the existence of an optimal configuration for general convex minimization problems ruled by bounded measurable degenerate elliptic operators. Under a mild continuity assumption on the medium, the free boundary is proven to enjoy the appropriate weak geometry and we establish the existence of an optimal design for general convex optimal design problems with volume constraints for all dimensions."}
{"category": "Math", "title": "The R(S^1)-graded equivariant homotopy of THH(F_p)", "abstract": "The main result of this paper is the computation of TR^n_{\\alpha}(F_p;p) for \\alpha in R(S^1). These R(S^1)-graded TR-groups are the equivariant homotopy groups naturally associated to the S^1-spectrum THH(F_p), the topological Hochschild S^1-spectrum. This computation, which extends a partial result of Hesselholt and Madsen, provides the first example of the R(S^1)-graded TR-groups of a ring. These groups arise in algebraic K-theory computations, and are particularly important to the understanding of the algebraic K-theory of non-regular schemes."}
{"category": "Math", "title": "L-functions of Symmetric Products of the Kloosterman Sheaf over Z", "abstract": "The classical $n$-variable Kloosterman sums over the finite field ${\\bf F}_p$ give rise to a lisse $\\bar {\\bf Q}_l$-sheaf ${\\rm Kl}_{n+1}$ on ${\\bf G}_{m, {\\bf F}_p}={\\bf P}^1_{{\\bf F}_p}-\\{0,\\infty\\}$, which we call the Kloosterman sheaf. Let $L_p({\\bf G}_{m,{\\bf F}_p}, {\\rm Sym}^k{\\rm Kl}_{n+1}, s)$ be the $L$-function of the $k$-fold symmetric product of ${\\rm Kl}_{n+1}$. We construct an explicit virtual scheme $X$ of finite type over ${\\rm Spec} {\\bf Z}$ such that the $p$-Euler factor of the zeta function of $X$ coincides with $L_p({\\bf G}_{m,{\\bf F}_p}, {\\rm Sym}^k{\\rm Kl}_{n+1}, s)$. We also prove similar results for $\\otimes^k {\\rm Kl}_{n+1}$ and $\\bigwedge^k {\\rm Kl}_{n+1}$."}
{"category": "Math", "title": "Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians", "abstract": "We compute the initial ideals, with respect to certain conveniently chosen term orders, of ideals of tangent cones at torus fixed points to Schubert varieties in orthogonal Grassmannians. The initial ideals turn out to be square-free monomial ideals and therefore Stanley-Reisner face rings of simplicial complexes. We describe these complexes. The maximal faces of these complexes encode certain sets of non-intersecting lattice paths."}
{"category": "Math", "title": "Maximal regularity and Hardy spaces", "abstract": "In this work, we consider the Cauchy problem for $u' - Au = f$ with $A$ the Laplacian operator on some Riemannian manifolds or a sublapacian on some Lie groups or some second order elliptic operators on a domain. We show the boundedness of the operator of maximal regularity $f\\mapsto Au$ and its adjoint on appropriate Hardy spaces which we define and study for this purpose. As a consequence we reobtain the maximal $L^q$ regularity on $L^p$ spaces for $p,q$ between 1 and $\\infty$."}
{"category": "Math", "title": "Prime and zero distributions for meromorphic Euler products", "abstract": "The aim of the present paper is to study the relations between the prime distribution and the zero distribution for generalized zeta functions which are expressed by Euler products and is analytically continued as meromorphic functions of finite order. In this paper, we give an inequality between the order of the zeta function as a meromorphic function and the growth of the multiplicity in the prime distribution."}
{"category": "Math", "title": "Local Convexity Properties of Quasihyperbolic Balls in Punctured Space", "abstract": "This paper deals with local convexity properties of the quasihyperbolic metric in the punctured space. We consider convexity and starlikeness of quasihyperbolic balls."}
{"category": "Math", "title": "Separable p-harmonic functions in a cone and related quasilinear equations on manifolds", "abstract": "In considering a class of quasilinear elliptic equations on a Riemannian manifold with nonnegative Ricci curvature, we give a new proof of Tolksdorf's result on the construction of separable $p$-harmonic functions in a cone."}
{"category": "Math", "title": "An explicit formula for the Skorokhod map on $[0,a]$", "abstract": "The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map $\\Gamma_{0,a}$ on $[0,a]$ for any $a>0$ is derived. Specifically, it is shown that on the space $\\mathcal{D}[0,\\infty)$ of right-continuous functions with left limits taking values in $\\mathbb{R}$, $\\Gamma_{0,a}=\\Lambda_a\\circ \\Gamma_0$, where $\\Lambda_a:\\mathcal{D}[0,\\infty)\\to\\mathcal{D}[0,\\infty)$ is defined by \\[\\Lambda_a(\\phi)(t)=\\phi(t)-\\sup_{s\\in[0,t]}\\biggl[\\bigl(\\ phi(s)-a\\bigr)^+\\wedge\\inf_{u\\in[s,t]}\\phi(u)\\biggr]\\] and $\\Gamma_0:\\mathcal{D}[0,\\infty)\\to\\mathcal{D}[0,\\infty)$ is the Skorokhod map on $[0,\\infty)$, which is given explicitly by \\[\\Gamma_0(\\psi)(t)=\\psi(t)+\\sup_{s\\in[0,t]}[-\\psi(s)]^+.\\] In addition, properties of $\\Lambda_a$ are developed and comparison properties of $\\Gamma_{0,a}$ are established."}
{"category": "Math", "title": "An unconditionnally stable pressure correction scheme for compressible barotropic Navier-Stokes equations", "abstract": "We present in this paper a pressure correction scheme for barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution. The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the $L^2$-stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential. The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type to a space discretization associating low order non-conforming mixed finite elements and finite volumes. Numerical tests with an exact smooth solution show the convergence of the scheme."}
{"category": "Math", "title": "Toroidal automorphic forms for some function fields", "abstract": "Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL_2 is toroidal if all its right translates integrate to zero over all nonsplit tori in GL_2, and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one and genus g zero or one, and with a rational place. The space has dimension g and is spanned by the expected Eisenstein series. We deduce an \"automorphic\" proof for the Riemann hypothesis for the zeta function of those curves."}
{"category": "Math", "title": "On the growth rate of minor-closed classes of graphs", "abstract": "A minor-closed class of graphs is a set of labelled graphs which is closed under isomorphism and under taking minors. For a minor-closed class $C$, we let $c_n$ be the number of graphs in $C$ which have $n$ vertices. A recent result of Norine et al. shows that for all minor-closed class $C$, there is a constant $r$ such that $c_n < r^n n!$. Our main results show that the growth rate of $c_n$ is far from arbitrary. For example, no minor-closed class $C$ has $c_n= r^{n+o(n)} n!$ with $0 < r < 1$ or $1 < r < \\xi \\approx 1.76$."}
{"category": "Math", "title": "Weak convergence of measure-valued processes and $r$-point functions", "abstract": "We prove a sufficient set of conditions for a sequence of finite measures on the space of cadlag measure-valued paths to converge to the canonical measure of super-Brownian motion in the sense of convergence of finite-dimensional distributions. The conditions are convergence of the Fourier transform of the $r$-point functions and perhaps convergence of the ``survival probabilities.'' These conditions have recently been shown to hold for a variety of statistical mechanical models, including critical oriented percolation, the critical contact process and lattice trees at criticality, all above their respective critical dimensions."}
{"category": "Math", "title": "Greedy and quasi-greedy expansions in non-integer bases", "abstract": "We generalize several theorems of R\\'enyi, Parry, Dar\\'oczy and K\\'atai by characterizing the greedy and quasi-greedy expansions in non-integer bases."}
{"category": "Math", "title": "A polynomial oracle-time algorithm for convex integer minimization", "abstract": "In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. We extend these results to convex $N$-fold integer minimization problems and to convex 2-stage stochastic integer minimization problems. Finally, we present some applications of convex $N$-fold integer minimization problems for which our approach provides polynomial time solution algorithms."}
{"category": "Math", "title": "Inverse sequences, rooted trees and their end spaces", "abstract": "In this paper we prove that if we consider the standard real metric on simplicial rooted trees then the category Tower-Set of inverse sequences can be described by means of the bounded coarse geometry of the naturally associated trees. Using this we give a geometrical characterization of Mittag-Leffler property in inverse sequences in terms of the Lipschitz, and metrically proper, homotopy type of the corresponding tree and its maximal geodesically complete subtree. We also some consequences in shape theory, in particular we obtain some new representations of shape morphisms related to infinite brunches in trees."}
{"category": "Math", "title": "Compactified Picard stacks over $\\bar{\\mathcal M}_g$", "abstract": "We study algebraic (Artin) stacks over $\\bar{\\mathcal M}_g$ giving a functorial way of compactifying the relative degree $d$ Picard variety for families of stable curves. We also describe for every $d$ the locus of genus $g$ stable curves over which we get Deligne-Mumford stacks strongly representable over $\\bar{\\mathcal M}_g$."}
{"category": "Math", "title": "Approximation of definable sets by compact families, and upper bounds on homotopy and homology", "abstract": "We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by quantifier-free formulae, and obtain for the first time a singly exponential bound on Betti numbers of sub-Pfaffian sets."}
{"category": "Math", "title": "On a rigidity condition for Berwald Spaces", "abstract": "We show that which that for a Berwald structure, any Riemannian structure that is preserved by the Berwald connection leaves the indicatrix invariant under horizontal parallel transport. We also obtain the converse result: if $({\\bf M},F)$ is a Finsler structure such that there exists a Riemannian structure that leaves invariant the indicatrix under parallel transport of the associated Levi-Civita connection, then the structure $({\\bf M},F)$ is Berwald. As application, a necessary condition for pure Landsberg spaces is formulated. Using this criterion we provide an strategy to solve the existence or not of pure Landsberg surfaces"}
{"category": "Math", "title": "Hypergraph regularity and the multidimensional Szemer\\'edi theorem", "abstract": "We prove analogues for hypergraphs of Szemer\\'edi's regularity lemma and the associated counting lemma for graphs. As an application, we give the first combinatorial proof of the multidimensional Szemer\\'edi theorem of Furstenberg and Katznelson, and the first proof that provides an explicit bound. Similar results with the same consequences have been obtained independently by Nagle, R\\\"odl, Schacht and Skokan."}
{"category": "Math", "title": "On cardinality constrained cycle and path polytopes", "abstract": "Given a directed graph D = (N, A) and a sequence of positive integers 1 <= c_1 < c_2 < ... < c_m <= |N|, we consider those path and cycle polytopes that are defined as the convex hulls of simple paths and cycles of D of cardinality c_p for some p, respectively. We present integer characterizations of these polytopes by facet defining linear inequalities for which the separation problem can be solved in polynomial time. These inequalities can simply be transformed into inequalities that characterize the integer points of the undirected counterparts of cardinality constrained path and cycle polytopes. Beyond we investigate some further inequalities, in particular inequalities that are specific to odd/even paths and cycles."}
{"category": "Math", "title": "A Short Proof of the VPN Tree Routing Conjecture on Ring Networks", "abstract": "The VPN Tree Routing Conjecture states that there always exists an optimal solution to the symmetric Virtual Private Network Design (sVPND) problem where the paths between all terminals form a tree. Only recently, Hurkens, Keijsper, and Stougie gave a proof of this conjecture for the special case of ring networks. Their proof is based on a dual pair of linear programs and is somewhat in- volved. We present a short proof of a slightly stronger conjecture which might also turn out to be useful for proving the VPN Tree Routing Conjecture for general networks."}
{"category": "Math", "title": "A Unified Approach to Algebraic Set Theory", "abstract": "The paper provides an introduction to the field of Algebraic Set Theory (AST). AST is a flexible categorical framework for studying different kinds of set theories: both classical and constructive, predicative and impredicative. We discuss the basic results in this area, with a particular emphasis on applications to the constructive set theories IZF and CZF. (This paper is a summary of a tutorial on categorical logic given by the second named author at the Logic Colloquium 2006 in Nijmegen.)"}
{"category": "Math", "title": "$C^*$-envelopes of universal free products and semicrossed products for multivariable dynamics", "abstract": "We show that for a class of operator algebras satisfying a natural condition the $C^*$-envelope of the universal free product of operator algebras $A_i$ is given by the free product of the $C^*$-envelopes of the $A_i$. We apply this theorem to, in special cases, the $C^*$-envelope of the semicrossed products for multivariable dynamics in terms of the single variable semicrossed products of Peters."}
{"category": "Math", "title": "A curious example of two model categories and some associated differential graded algebras", "abstract": "The paper gives a new proof that the model categories of stable modules for the rings Z/(p^2) and (Z/p)[\\epsilon]/(\\epsilon^2) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with non-isomorphic K-theories."}
{"category": "Math", "title": "Separable states and positive maps", "abstract": "Using the natural duality between linear functionals on tensor products of C*-algebras with the trace class operators on a Hilbert space H and linear maps of the C*-algebra into B(H), we study the relationship between separability, entanglement and the Peres condition of states and positivity properties of the linear maps."}
{"category": "Math", "title": "Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles", "abstract": "Let $X$ a smooth quasi-projective algebraic surface, $L$ a line bundle on $X$. Let $X^{[n]}$ the Hilbert scheme of $n$ points on $X$ and $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We explicitely compute the image $\\bkrh(L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\\bkrh : \\B{D}^b(X^{[n]}) \\ra \\B{D}^b_{\\perm_n}(X^n)$ in terms of a complex $\\comp{\\mc{C}}_L$ of $\\perm_n$-equivariant sheaves in $\\B{D}^b_{\\perm_n}(X^n)$. We give, moreover, a characterization of the image $\\bkrh(L^{[n]} \\tens ... \\tens L^{[n]})$ in terms of of the hyperderived spectral sequence $E^{p,q}_1$ associated to the derived $k$-fold tensor power of the complex $\\comp{\\mc{C}}_L$. The study of the $\\perm_n$-invariants of this spectral sequence allows to get the derived direct images of the double tensor power and of the general $k$-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This yields easily the computation of the cohomology of $X^{[n]}$ with values in $L^{[n]} \\tens L^{[n]}$ and $\\Lambda^k L^{[n]}$."}
{"category": "Math", "title": "Aspects of Predicative Algebraic Set Theory I: Exact Completion", "abstract": "This is the first in a series of three papers on Algebraic Set Theory. Its main purpose is to lay the necessary groundwork for the next two parts, one on Realisability and the other on Sheaf Models in Algebraic Set Theory."}
{"category": "Math", "title": "Multivariable Wilson polynomials and degenerate Hecke algebras", "abstract": "We study a rational version of the double affine Hecke algebra associated to the nonreduced affine root system of type $(C^\\vee_n,C_n)$. A certain representation in terms of difference-reflection operators naturally leads to the definition of nonsymmetric versions of the multivariable Wilson polynomials. Using the degenerate Hecke algebra we derive several properties, such as orthogonality relations and quadratic norms, for the nonsymmetric and symmetric multivariable Wilson polynomials."}
{"category": "Math", "title": "Knot Concordance and Higher-Order Blanchfield Duality", "abstract": "In 1997, T. Cochran, K. Orr, and P. Teichner defined a filtration {F_n} of the classical knot concordance group C. The filtration is important because of its strong connection to the classification of topological 4-manifolds. Here we introduce new techniques for studying C and use them to prove that, for each natural number n, the abelian group F_n/F_{n.5} has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a long-standing question as to whether certain natural families of knots, first considered by Casson-Gordon and Gilmer, contain slice knots."}
{"category": "Math", "title": "Addendum to: Commensurations of the Johnson kernel", "abstract": "Let K(S) be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. In our earlier paper, we showed that Comm(K(S)) and Aut(K(S)) are both isomorphic to Mod(S) when S is a closed, connected, orientable surface of genus g at least 4. By modifying our original proof, we show that the same result holds for g at leat 3, thus confirming Farb's conjecture in all cases (the statement is not true for any g less than 3)."}
{"category": "Math", "title": "Zonotopes With Large 2D Cuts", "abstract": "There are d-dimensional zonotopes with n zones for which a 2-dimensional central section has \\Omega(n^{d-1}) vertices. For d=3 this was known, with examples provided by the \"Ukrainian easter eggs'' by Eppstein et al. Our result is asymptotically optimal for all fixed d>=2."}
{"category": "Math", "title": "Product groups acting on manifolds", "abstract": "We analyse volume-preserving actions of product groups on Riemannian manifolds. To this end, we establish a new superrigidity theorem for ergodic cocycles of product groups ranging in linear groups. There are no a priori assumptions on the acting groups, except a spectral gap assumption on their action. Our main application to manifolds concerns irreducible actions of Kazhdan product groups. We prove the following dichotomy: Either the action is infinitesimally linear, which means that the derivative cocycle arises from unbounded linear representations of all factors. Otherwise, the action is measurably isometric, in which case there are at most two factors in the product group. As a first application, this provides lower bounds on the dimension of the manifold in terms of the number of factors in the acting group. Another application is a strong restriction for actions of non-linear groups."}
{"category": "Math", "title": "Properties of two dimensional sets with small sumset", "abstract": "Let $A, B\\subseteq \\mathbb{R}^2$ be finite, nonempty subsets, let $s\\geq 2$ be an integer, and let $h_1(A,B)$ denote the minimal number $t$ such that there exist $2t$ (not necessarily distinct) parallel lines, $\\ell_1,...,\\ell_{t},\\ell'_1,...,\\ell'_{t}$, with $A\\subseteq \\bigcup_{i=1}^{t}\\ell_i$ and $B\\subseteq\\bigcup_{i=1}^{t}\\ell'_i$. Suppose $h_1(A,B)\\geq s$. Then we show that: (a) if $||A|-|B||\\leq s$ and $|A|+|B|\\geq 4s^2-6s+3$, then $$|A+B|\\geq (2-\\frac 1 s)(|A|+|B|)-2s+1;$$ (b) if $|A|\\geq |B|+s$ and $|B|\\geq 2s^2-{7/2}s+{3/2}$, then $$|A+B|\\geq |A|+(3-\\frac 2 s)|B|-s;$$ (c) if $|A|\\geq {1/2}s(s-1)|B|+s$ and either $|A|> {1/8}(2s-1)^2|B|-{1/4}(2s-1)+\\frac{(s-1)^2}{2(|B|-2)}$ or $|B|\\geq \\frac{2s+4}{3}$, then $$|A+B|\\geq |A|+s(|B|-1).$$ This extends the 2-dimensional case of the Freiman $2^d$--Theorem to distinct sets $A$ and $B$, and, in the symmetric case $A=B$, improves the best prior known bound for $|A|+|B|$ (due to Stanchescu, and which was cubic in $s$) to an exact value. As part of the proof, we give general lower bounds for two dimensional subsets that improve the 2-dimensional case of estimates of Green and Tao and of Gardner and Gronchi, and that generalize the 2-dimensional case of the Brunn-Minkowski Theorem."}
{"category": "Math", "title": "Poisson convergence for the largest eigenvalues of Heavy Tailed Random Matrices", "abstract": "We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in \\cite{Sos1}, we prove that, in the absence of the fourth moment, the top eigenvalues behave, in the limit, as the largest entries of the matrix."}
{"category": "Math", "title": "On gradient bounds for the heat kernel on the Heisenberg group", "abstract": "It is known that the couple formed by the two dimensional Brownian motion and its L\\'evy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel."}
{"category": "Math", "title": "Cuspidal representations which are not strongly cuspidal", "abstract": "We give a description of all the cuspidal representations of $\\mathrm{GL}_4(\\mathfrak{o}_2)$, where $\\mathfrak{o}_2$ is a finite ring coming from the ring of integers in a local field, modulo the square of its maximal ideal $\\mathfrak{p}$. This shows in particular the existence of representations which are cuspidal, yet are not strongly cuspidal, that is, do not have orbit with irreducible characteristic polynomial mod $\\mathfrak{p}$. It has been shown by Aubert, Onn, and Prasad that this phenomenon cannot occur for $\\mathrm{GL}_n$, when $n$ is prime."}
{"category": "Math", "title": "Local acyclic fibrations and the de Rham complex", "abstract": "We reinterpret algebraic de Rham cohomology for a possibly singular complex variety X as sheaf cohomology in the site of smooth schemes over X with Voevodsky's h-topology. Our results extend to the algebraic de Rham complex as well. Our main technique is to extend Cech cohomology of hypercovers to arbitrary local acyclic fibrations of simplicial presheaves."}
{"category": "Math", "title": "On cluster algebras with coefficients and 2-Calabi-Yau categories", "abstract": "Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in `Cluster algebras IV: Coefficients'. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not always hold."}
{"category": "Math", "title": "Dependency and false discovery rate: Asymptotics", "abstract": "Some effort has been undertaken over the last decade to provide conditions for the control of the false discovery rate by the linear step-up procedure (LSU) for testing $n$ hypotheses when test statistics are dependent. In this paper we investigate the expected error rate (EER) and the false discovery rate (FDR) in some extreme parameter configurations when $n$ tends to infinity for test statistics being exchangeable under null hypotheses. All results are derived in terms of $p$-values. In a general setup we present a series of results concerning the interrelation of Simes' rejection curve and the (limiting) empirical distribution function of the $p$-values. Main objects under investigation are largest (limiting) crossing points between these functions, which play a key role in deriving explicit formulas for EER and FDR. As specific examples we investigate equi-correlated normal and $t$-variables in more detail and compute the limiting EER and FDR theoretically and numerically. A surprising limit behavior occurs if these models tend to independence."}
{"category": "Math", "title": "Rigidity of gradient Ricci Solitons", "abstract": "We define a gradient Ricci soliton to be rigid if it is a flat bundle $% N\\times_{\\Gamma}\\mathbb{R}^{k}$ where $N$ is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section."}
{"category": "Math", "title": "Filling real hypersurfaces by pseudoholomorphic discs", "abstract": "We study pseudoholomorphic discs with boundaries attached to a real hypersurface in an almost complex manifold. We give sufficient conditions for filling a one sided neighborhood of the hypersurface by the discs."}
{"category": "Math", "title": "On Monotonic Strengthening of Newman-like Phenomenon on (2m+1)-multiples in Base 2m", "abstract": "We obtain exact and asymptotic expressions for the excess of nonnegative (2m+1)-multiples less than (2m)^k with even digit sums in the base 2m."}
{"category": "Math", "title": "Schubert polynomials and classes of Hessenberg varieties", "abstract": "Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a \"Giambelli formula\" expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and we give closed combinatorial formulas for the coefficients in many cases. We introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use our results to determine when such schemes are reduced."}
{"category": "Math", "title": "Probabilistic coherence and proper scoring rules", "abstract": "We provide self-contained proof of a theorem relating probabilistic coherence of forecasts to their non-domination by rival forecasts with respect to any proper scoring rule. The theorem appears to be new but is closely related to results achieved by other investigators."}
{"category": "Math", "title": "Powers of Coxeter elements in infinite groups are reduced", "abstract": "Let W be an infinite irreducible Coxeter group with (s_1, ..., s_n) the simple generators. We give a simple proof that the word s_1 s_2 ... s_n s_1 s_2 >... s_n ... s_1 s_2 ... s_n is reduced for any number of repetitions of s_1 s_2 >... s_n. This result was proved for simply-laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof only using basic facts about Coxeter groups and the geometry of root systems."}
{"category": "Math", "title": "On a classification of the gradient shrinking solitons", "abstract": "The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. In dimension three we also generalize the result by removing the $\\kappa$-non-collapsing assumption. In high dimension this new method allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature tensor, which includes the rotationally symmetric ones."}
{"category": "Math", "title": "On 4-dimensional gradient shrinking solitons", "abstract": "In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional gradient shrinking solitons to dimension four."}
{"category": "Math", "title": "Generalized Jacquet modules of parabolic induction", "abstract": "In this paper we study a generalization of the Jacquet module of a parabolic induction and construct a filtration on it. The successive quotient of the filtration is written by using the twisting functor."}
{"category": "Math", "title": "Knot homology via derived categories of coherent sheaves II, sl(m) case", "abstract": "Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu's by homological mirror symmetry."}
{"category": "Math", "title": "A natural prime-generating recurrence", "abstract": "For the sequence defined by a(n) = a(n-1) + gcd(n, a(n-1)) with a(1) = 7 we prove that a(n) - a(n-1) takes on only 1s and primes, making this recurrence a rare \"naturally occurring\" generator of primes. Toward a generalization of this result to an arbitrary initial condition, we also study the limiting behavior of a(n)/n and a transience property of the evolution."}
{"category": "Math", "title": "Representations of Temperley--Lieb Algebras", "abstract": "We define a commuting family of operators $T_0,T_1,...,T_n$ in the Temperley--Lieb algebra $\\mathcal{A}_n(x)$ of type $A_{n-1}$. Using an appropriate analogue to Murphy basis of the Iwahori--Hecke algebra of the symmetric group, we describe the eigenvalues arising from the triangular action of the said operators on the cell modules of $\\mathcal{A}_n(x)$. These results are used to provide the Temperley--Lieb algebras of type $A_{n-1}$ with a semi--normal form, together with a branching law, and explicit formulae for associated Gram determinants."}
{"category": "Math", "title": "On some explicit evaluations of multiple zeta-star values", "abstract": "In this paper, we give some explicit evaluations of multiple zeta-star values which are rational multiple of powers of $\\pi^2$."}
{"category": "Math", "title": "Lens sequences", "abstract": "A family of sequences produced by a non-homogeneous linear recurrence formula derived from the geometry of circles inscribed in lenses is introduced and studied. Mysterious ``underground'' sequences underlying them are discovered in this paper."}
{"category": "Math", "title": "Multi-peak solutions for magnetic NLS equations without non--degeneracy conditions", "abstract": "In the work we consider the magnetic NLS equation (\\frac{\\hbar}{i} \\nabla -A(x))^2 u + V(x)u - f(|u|^2)u = 0 \\quad {in} \\R^N where $N \\geq 3$, $A \\colon \\R^N \\to \\R^N$ is a magnetic potential, possibly unbounded, $V \\colon \\R^N \\to \\R$ is a multi-well electric potential, which can vanish somewhere, $f$ is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u\\colon \\R^N \\to \\C$, under conditions on the nonlinearity which are nearly optimal."}
{"category": "Math", "title": "Reflection Groups and Differential Forms", "abstract": "We study differential forms invariant under a finite reflection group over a field of arbitrary characteristic. In particular, we prove an analogue of Saito's freeness criterion for invariant differential 1-forms. We also discuss how twisted wedging endows the invariant forms with the structure of a free exterior algebra in certain cases. Some of the results are extended to the case of relative invariants with respect to a linear character."}
{"category": "Math", "title": "The inverse integrating factor and the Poincar\\'e map", "abstract": "This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincar\\'e return map in terms of the inverse integrating factor. In particular, we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it. We also apply this result to study the homoclinic loop bifurcation. We only consider homoclinic loops whose critical point is a hyperbolic saddle and whose Poincar\\'e return map is not the identity. A local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it. Our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable, that is, the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method."}
{"category": "Math", "title": "Compactness in vector-valued Banach function spaces", "abstract": "We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces $L_X^p$, where $X$ is a Banach space and $1\\le p<\\infty$, and extend the result to vector-valued Banach function spaces $E_X$, where $E$ is a Banach function space with order continuous norm."}
{"category": "Math", "title": "Cellular resolutions of Cohen-Macaulay monomial quotient rings", "abstract": "We investigate monomial labellings on cell complexes, giving a minimal cellular resolution of the ideal generated by these monomials, and such that the associated quotient ring is Cohen-Macaulay. We introduce a notion of such a labelling being maximal. There is only a finite number of maximal labellings for each cell complex, and we classify these for trees, partly for subdivisions of polygons, and for some classes of selfdual polytopes."}
{"category": "Math", "title": "A theorem about three quadratic forms", "abstract": "We prove a self-improvement property regarding quadratic forms on arbitrary vector spaces. We discuss several consequences of this result, in particular those concerning dimension-free L^p estimates of certain singular integral operators (Riesz transforms)."}
{"category": "Math", "title": "Algebraic dependence of commuting elements in algebras", "abstract": "The aim of this paper to draw attention to several aspects of the algebraic dependence in algebras. The article starts with discussions of the algebraic dependence problem in commutative algebras. Then the Burchnall-Chaundy construction for proving algebraic dependence and obtaining the corresponding algebraic curves for commuting differential operators in the Heisenberg algebra is reviewed. Next some old and new results on algebraic dependence of commuting q-difference operators and elements in q-deformed Heisenberg algebras are reviewed. The main ideas and essence of two proofs of this are reviewed and compared. One is the algorithmic dimension growth existence proof. The other is the recent proof extending the Burchnall-Chaundy approach from differential operators and the Heisenberg algebra to the q-deformed Heisenberg algebra, showing that the Burchnall-Chaundy eliminant construction indeed provides annihilating curves for commuting elements in the q-deformed Heisenberg algebras for q not a root of unity."}
{"category": "Math", "title": "Branching rules for unramified principal series representations of GL(3) over a p-adic field", "abstract": "On restriction to the maximal compact subgroup $\\mathrm{GL}(3,\\mathscr{R})$, an unramified principal series representation of the $p$-adic group $\\mathrm{GL}(3,F)$ decomposes into a direct sum of finite-dimensional irreducibles each appearing with finite multiplicity. We describe a coarser decomposition into components which, although reducible in general, capture the equivalences between the irreducible constituents."}
{"category": "Math", "title": "$L^1$ bounds in normal approximation", "abstract": "The zero bias distribution $W^*$ of $W$, defined though the characterizing equation $\\mathit{EW}f(W)=\\sigma^2Ef'(W^*)$ for all smooth functions $f$, exists for all $W$ with mean zero and finite variance $\\sigma^2$. For $W$ and $W^*$ defined on the same probability space, the $L^1$ distance between $F$, the distribution function of $W$ with $\\mathit{EW}=0$ and $Var(W)=1$, and the cumulative standard normal $\\Phi$ has the simple upper bound \\[\\Vert F-\\Phi\\Vert_1\\le2E|W^*-W|.\\] This inequality is used to provide explicit $L^1$ bounds with moderate-sized constants for independent sums, projections of cone measure on the sphere $S(\\ell_n^p)$, simple random sampling and combinatorial central limit theorems."}
{"category": "Math", "title": "Branching rules for ramified principal series representations of GL(3) over a p-adic field", "abstract": "We decompose the restriction of ramified principal series representations of the $p$-adic group $\\mathrm{GL}(3,\\mathrm{k})$ to its maximal compact subgroup $K=\\mathrm{GL}(3,\\mathscr{R})$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples."}
{"category": "Math", "title": "GAP Computations Concerning Probabilistic Generation of Finite Simple Groups", "abstract": "This is a collection of examples showing how the GAP system can be used to compute information about the probabilistic generation of finite almost simple groups. It includes all examples that were needed for the computational results in the paper \"Probabilistic generation of finite simple groups, II\" by Thomas Breuer, Robert M. Guralnick, and William M. Kantor. The purpose of this writeup is twofold. On the one hand, the computations are documented this way. On the other hand, the GAP code shown for the examples can be used as test input for automatic checking of the data and the functions used."}
{"category": "Math", "title": "Differential equation approximations for Markov chains", "abstract": "We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs."}
{"category": "Math", "title": "Geometric properties derived from generic initial spaces", "abstract": "For a vector space V of homogeneous forms of the same degree in a polynomial ring, we investigate what can be said about the generic initial ideal of the ideal generated by V, from the form of the generic initial space gin(V) for the revlex order. Our main result is a considerable generalisation of a previous result by the first author."}
{"category": "Math", "title": "Rigidity in the intersection of varieties does not imply rigidity in the intersecting varieties", "abstract": "This paper has been withdrawn by the author due to a crucial error in example."}
{"category": "Math", "title": "On the performance of FDR control: Constraints and a partial solution", "abstract": "The False Discovery Rate (FDR) paradigm aims to attain certain control on Type I errors with relatively high power for multiple hypothesis testing. The Benjamini--Hochberg (BH) procedure is a well-known FDR controlling procedure. Under a random effects model, we show that, in general, unlike the FDR, the positive FDR (pFDR) of the BH procedure cannot be controlled at an arbitrarily low level due to the limited evidence provided by the observations to separate false and true nulls. This results in a criticality phenomenon, which is characterized by a transition of the procedure's power from being positive to asymptotically 0 without any reduction in the pFDR, once the target FDR control level is below a positive critical value. To address the constraints on the power and pFDR control imposed by the criticality phenomenon, we propose a procedure which applies BH-type procedures at multiple locations in the domain of $p$-values. Both analysis and simulations show that the proposed procedure can attain substantially improved power and pFDR control."}
{"category": "Math", "title": "Smooth rationally connected threefolds contain all smooth curves", "abstract": "We show that if X is a smooth rationally connected threefold and C is a smooth projective curve then C can be embedded in X. Furthermore, a version of this property characterises rationally connected varieties of dimension at least 3. We give some details about the toric case."}
{"category": "Math", "title": "One more approach to the convergence of the empirical process to the Brownian bridge", "abstract": "A theorem of Donsker asserts that the empirical process converges in distribution to the Brownian bridge. The aim of this paper is to provide a new and simple proof of this fact."}
{"category": "Math", "title": "On a conjecture of Serre on abelian threefolds", "abstract": "In this article, we give a reformulation of a result from Howe, Leprevost and Poonen on a three dimensional family of abelian threefolds. We also link their result to a conjecture of Serre on a precise form of Torelli theorem for genus 3 curves."}
{"category": "Math", "title": "On C-fibrations over projective curves", "abstract": "The goal of this paper is to present a modified version (GML) of ML invariant which should take into account rulings over a projective base and allow further stratification of smooth affine rational surfaces. We provide a non-trivial example where GML invariant is computed for a smooth affine rational surface admitting no C-actions. We apply GML invariant to computation of ML invariant of some threefolds."}
{"category": "Math", "title": "A la Fock-Goncharov coordinates for PU(2,1)", "abstract": "We describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface $S$ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperpolic plane. We establish a bijection between a set of decorations of an ideal triangulation of $S$ and a subset of the PU(2,1)-representation variety of $\\pi_1(S)$."}
{"category": "Math", "title": "Description of Derivations on Measurable Operator Algebras of Type I", "abstract": "Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $\\tau,$ let $L(M, \\tau)$ be the algebra of all $\\tau$-measurable operators affiliated with $M.$ We give a complete description of all derivations on the algebra $L(M, \\tau).$ In particular, we prove that if $M$ is of type I$_\\infty$ then every derivation on $L(M, \\tau)$ is inner."}
{"category": "Math", "title": "Applications of Level Curves to Some Problems on Algebraic Surfaces", "abstract": "In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some topological features of surfaces (reality, compactness, connectivity) and to the problem of plotting."}
{"category": "Math", "title": "Freyd's generating hypothesis for groups with periodic cohomology", "abstract": "Let $G$ be a finite group and let $k$ be a field whose characteristic $p$ divides the order of $G$. Freyd's generating hypothesis for the stable module category of $G$ is the statement that a map between finite-dimensional $kG$-modules in the thick subcategory generated by $k$ factors through a projective if the induced map on Tate cohomology is trivial. We show that if $G$ has periodic cohomology then the generating hypothesis holds if and only if the Sylow $p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions that are equivalent to the GH for groups with periodic cohomology."}
{"category": "Math", "title": "Riemann surfaces and AF-algebras", "abstract": "For a generic set in the Teichmueller space, we construct a covariant functor with the range in a category of the AF-algebras; the functor maps isomorphic Riemann surfaces to the stably isomorphic AF-algebras. As a special case, one gets a categorical correspondence between complex tori and the so-called Effros-Shen algebras."}
{"category": "Math", "title": "A counter-example to the Andreotti-Grauert conjecture", "abstract": "Let $X$ be an analytic complex space which is $q$-complete. Then it follows from a theorem of Andreotti-Grauert [1] that $H^{p}(X, {\\mathcal{F}})=0$ for every coherent analytic sheaf ${\\mathcal{F}}$ on $X$ if $p\\geq q$. Until now it is not known if these two conditions are equivalent. The aim of this article is to give a counterexample to the converse of this statement. We show that there exist for each $n\\geq 3$ open sets $\\Omega\\subset\\complexes^{n}$ such that $H^{n-1}(\\Omega, {\\mathcal{F}})=0$ for every ${\\mathcal{F}}\\in coh(\\Omega)$ but $\\Omega$ is not $(n-1)$-complete."}
{"category": "Math", "title": "Non-linearizable Actions of Commutative Reductive Groups", "abstract": "We generalize a construction of Freudenburg and Moser-Jauslin in order to obtain an example of a non-linearizable action of a commutative reductive algebraic group on the affine space for every field of characteristic zero which admits a quadratic field extension."}
{"category": "Math", "title": "An arithmetic Riemann-Roch theorem for pointed stable curves", "abstract": "We prove an arithmetic Riemann-Roch theorem for pointed stable curves. We derive consequences for the Selberg zeta function of an open modular curve $Y_{1}(p)$ (resp. $Y_{0}(p)$), for a prime number $p\\geq 11$ (resp. congruent to 11 modulo 12)."}
{"category": "Math", "title": "Transient Random Walks in Random Environment on a Galton-Watson Tree", "abstract": "We consider a transient random walk $(X_n)$ in random environment on a Galton--Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that $X_n$ is of order of magnitude $n^{\\Lambda}$, with $\\Lambda \\in (0,1)$. We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio \\cite{Col06}."}
{"category": "Math", "title": "Positivity of Tur\\'an determinants for orthogonal polynomials", "abstract": "The orthogonal polynomials $p_n$ satisfy Tur\\'an's inequality if $p_n^2(x)-p_{n-1}(x)p_{n+1}(x)\\ge 0$ for $n\\ge 1$ and for all $x$ in the interval of orthogonality. We give general criteria for orthogonal polynomials to satisfy Tur\\'an's inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the $q$--ultraspherical polynomials."}
{"category": "Math", "title": "Calabi-Yau Frobenius algebras", "abstract": "We define Calabi-Yau and periodic Frobenius algebras over arbitrary base commutative rings. We define a Hochschild analogue of Tate cohomology, and show that the \"stable Hochschild cohomology\" of periodic CY Frobenius algebras has a Batalin-Vilkovisky and Frobenius algebra structure. Such algebras include (centrally extended) preprojective algebras of (generalized) Dynkin quivers, and group algebras of classical periodic groups. We use this theory to compute (for the first time) the Hochschild cohomology of many algebras related to quivers, and to simplify the description of known results. Furthermore, we compute the maps on cohomology from extended Dynkin preprojective algebras to the Dynkin ones, which relates our CY property (for Frobenius algebras) to that of Ginzburg (for algebras of finite Hochschild dimension)."}
{"category": "Math", "title": "Differential operators and BV structures in noncommutative geometry", "abstract": "We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck's notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DDer(A), the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A), a certain `Fock space' associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)), of differential operators, is filtered and gr D(F(A)), the associated graded algebra, is commutative in some `twisted' sense. The resulting double Poisson structure on gr D(F(A)) is closely related to the one introduced by Van den Bergh. Specifically, we prove that gr D(F(A))=F(T_A(DDer(A)), provided A is smooth. It is crucial for our construction that the Fock space F(A) carries an extra-structure of a wheelgebra, a new notion closely related to the notion of a wheeled PROP. There are also notions of Lie wheelgebras, and so on. In that language, D(F(A)) becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on DDer(A) gives rise to a second order (wheeled) differential operator, a noncommutative analogue of the BV-operator."}
{"category": "Math", "title": "Duality, Vector advection and the Navier-Stokes equations", "abstract": "In this article we show that three dimensional vector advection equation is self dual in certain sense defined below. As a consequence, we infer classical result of Serrin of existence of strong solution of Navier-Stokes equation. Also we deduce Feynman-Kac type formula for solution of the vector advection equation and show that the formula is not unique i.e. there exist flows which differ from standard flow along which vorticity is conserved."}
{"category": "Math", "title": "Desingularizations of some Weighted Projective Planes", "abstract": "In this paper we discuss the desingularization algorithm for a toric surface. In particular, we construct an iterable method of determining the Hirzebruch-Jung continued fraction decomposition. These results are then applied to weighted projective planes with at least one tivial weight, ${\\mathbb P}(1,m,n)$. The paper concludes with the development of a computer program that computes this continued fraction decomposition."}
{"category": "Math", "title": "Effective cone of $overline{M}_{0,n}$ for odd $n$", "abstract": "In this work, we compute the effective cone of the space of $n$ pointed genus 0 rational curves, $\\bar{M}_{0,n}$ for odd $n$. We will, in fact, use the equivalence of $\\bar{M}_{0,n}$ and the space of (semi) stable configurations of $n$ weighted points on $\\mathbb{CP}^{1}$ with special weight $(1,..., 1)$ up to symmetric group action."}
{"category": "Math", "title": "Moderate deviations for Poisson--Dirichlet distribution", "abstract": "The Poisson--Dirichlet distribution arises in many different areas. The parameter $\\theta$ in the distribution is the scaled mutation rate of a population in the context of population genetics. The limiting case of $\\theta$ approaching infinity is practically motivated and has led to new, interesting mathematical structures. Laws of large numbers, fluctuation theorems and large-deviation results have been established. In this paper, moderate-deviation principles are established for the Poisson--Dirichlet distribution, the GEM distribution, the homozygosity, and the Dirichlet process when the parameter $\\theta$ approaches infinity. These results, combined with earlier work, not only provide a relatively complete picture of the asymptotic behavior of the Poisson--Dirichlet distribution for large $\\theta$, but also lead to a better understanding of the large deviation problem associated with the scaled homozygosity. They also reveal some new structures that are not observed in existing large-deviation results."}
{"category": "Math", "title": "Quasidiagonality of crossed products", "abstract": "We prove that the crossed product A x G of a separable, unital, quasidiagonal C*- algebra A by a discrete, countable, amenable, maximally almost periodic group G is quasidiagonal, provided that the action is almost periodic."}
{"category": "Math", "title": "Some remarks on groupoids and small categories", "abstract": "This unpublished note contains some materials taken from my old study note on groupoids and small categories. It contains a proof for the fact that any groupoid is a group bundle over an equivalence relation. Moreover, the action of a category $G$ on a category $H$ as well as the resulting semi-direct product category $H\\times_\\alpha G$ will be defined (when either $G$ is a groupoid or $H^{(0)} = G^{(0)}$). If both $G$ and $H$ are groupoids, then $H\\times_\\alpha G$ is also a groupoid. The reason of producing this note is for people who want to check some details in a recent work of Li."}
{"category": "Math", "title": "Problems of Testology", "abstract": "Some problems of testology are discussed."}
{"category": "Math", "title": "Comlexity of prime-dimensional sequences over a finite field", "abstract": "V.I. Arnold has recently defined the complexity of a sequence of $n$ zeros and ones with the help of the operator of finite differences. In this paper we describe the results obtained for almost most complicated sequences of elements of a finite field, whose dimension $n$ is a prime number. We prove that with $n\\to \\infty$ this property is inherent in almost all sequences, while the values of multiplicative functions possess this property with any $n$ different from the characteristic of the field. We also describe the prime values of the parameter $n$ which make the logarithmic function almost most complicated. All these sequences reveal a stronger complexity; its algebraic sense is quite clear."}
{"category": "Math", "title": "On Bost-Connes type systems for number fields", "abstract": "We give a complete description of the phase transition of the Bost-Connes type systems for number fields recently introduced by Connes-Marcolli-Ramachandran and Ha-Paugam. We also introduce a notion of K-lattices and discuss an interpretation of these systems in terms of 1-dimensional K-lattices."}
{"category": "Math", "title": "On an extreme two-point distribution", "abstract": "A bound for functional $\\Delta(F)=\\sup_{x\\in\\mathbb R}|F(x)-\\Phi(x)|$ is obtained, which is uniform for all distribution functions $F$ of random variables with zero mean-value and unity variance. Moreover, a two-point distribution is found, for which this bound is reached."}
{"category": "Math", "title": "Bayesian variable selection for high dimensional generalized linear models: convergence rates of the fitted densities", "abstract": "Bayesian variable selection has gained much empirical success recently in a variety of applications when the number $K$ of explanatory variables $(x_1,...,x_K)$ is possibly much larger than the sample size $n$. For generalized linear models, if most of the $x_j$'s have very small effects on the response $y$, we show that it is possible to use Bayesian variable selection to reduce overfitting caused by the curse of dimensionality $K\\gg n$. In this approach a suitable prior can be used to choose a few out of the many $x_j$'s to model $y$, so that the posterior will propose probability densities $p$ that are ``often close'' to the true density $p^*$ in some sense. The closeness can be described by a Hellinger distance between $p$ and $p^*$ that scales at a power very close to $n^{-1/2}$, which is the ``finite-dimensional rate'' corresponding to a low-dimensional situation. These findings extend some recent work of Jiang [Technical Report 05-02 (2005) Dept. Statistics, Northwestern Univ.] on consistency of Bayesian variable selection for binary classification."}
{"category": "Math", "title": "Bounded step functions and factorial ratio sequences", "abstract": "We study certain step functions whose nonnegativity is related to the integrality of sequences of ratios of factorial products. In particular, we obtain a lower bound for the mean square of such step functions which allows us to give a restriction on when such a factorial ratio sequence can be integral. Additionally, we note that this work has applications to the classification of cyclic quotient singularities."}
{"category": "Math", "title": "Boundary $C^*$-algebras for acylindrical groups", "abstract": "Let $\\Delta$ be an infinite, locally finite tree with more than two ends. Let $\\Gamma<\\aut(\\Delta)$ be an acylindrical uniform lattice. Then the boundary algebra $\\cl A_\\Gamma = C(\\partial\\Delta)\\rtimes \\Gamma$ is a simple Cuntz-Krieger algebra whose K-theory is determined explicitly."}
{"category": "Math", "title": "Wahl's conjecture holds in odd characteristics for symplectic and orthogonal Grassmannians", "abstract": "It is shown that the proof by Mehta and Parameswaran of Wahl's conjecture for Grassmannians in positive odd characteristics also works for symplectic and orthogonal Grassmannians."}
{"category": "Math", "title": "Modelling the effects of air pollution on health using Bayesian Dynamic Generalised Linear Models", "abstract": "The relationship between short-term exposure to air pollution and mortality or morbidity has been the subject of much recent research, in which the standard method of analysis uses Poisson linear or additive models. In this paper we use a Bayesian dynamic generalised linear model (DGLM) to estimate this relationship, which allows the standard linear or additive model to be extended in two ways: (i) the long-term trend and temporal correlation present in the health data can be modelled by an autoregressive process rather than a smooth function of calendar time; (ii) the effects of air pollution are allowed to evolve over time. The efficacy of these two extensions are investigated by applying a series of dynamic and non-dynamic models to air pollution and mortality data from Greater London. A Bayesian approach is taken throughout, and a Markov chain monte carlo simulation algorithm is presented for inference. An alternative likelihood based analysis is also presented, in order to allow a direct comparison with the only previous analysis of air pollution and health data using a DGLM."}
{"category": "Math", "title": "Nonparametric estimation of a point-spread function in multivariate problems", "abstract": "The removal of blur from a signal, in the presence of noise, is readily accomplished if the blur can be described in precise mathematical terms. However, there is growing interest in problems where the extent of blur is known only approximately, for example in terms of a blur function which depends on unknown parameters that must be computed from data. More challenging still is the case where no parametric assumptions are made about the blur function. There has been a limited amount of work in this setting, but it invariably relies on iterative methods, sometimes under assumptions that are mathematically convenient but physically unrealistic (e.g., that the operator defined by the blur function has an integrable inverse). In this paper we suggest a direct, noniterative approach to nonparametric, blind restoration of a signal. Our method is based on a new, ridge-based method for deconvolution, and requires only mild restrictions on the blur function. We show that the convergence rate of the method is close to optimal, from some viewpoints, and demonstrate its practical performance by applying it to real images."}
{"category": "Math", "title": "Holomorphic Sobolev spaces, Hermite ans special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform", "abstract": "The images of Hermite and Laguerre Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterised. These are used to characterise the Schwartz class of rapidly decreasing functions. The image of the space of all tempered distributions is also considered and a Paley-Wiener theorem for the windowed Fourier transform is proved."}
{"category": "Math", "title": "A ridge-parameter approach to deconvolution", "abstract": "Kernel methods for deconvolution have attractive features, and prevail in the literature. However, they have disadvantages, which include the fact that they are usually suitable only for cases where the error distribution is infinitely supported and its characteristic function does not ever vanish. Even in these settings, optimal convergence rates are achieved by kernel estimators only when the kernel is chosen to adapt to the unknown smoothness of the target distribution. In this paper we suggest alternative ridge methods, not involving kernels in any way. We show that ridge methods (a) do not require the assumption that the error-distribution characteristic function is nonvanishing; (b) adapt themselves remarkably well to the smoothness of the target density, with the result that the degree of smoothness does not need to be directly estimated; and (c) give optimal convergence rates in a broad range of settings."}
{"category": "Math", "title": "Global Mixed Periods and local Klyachko models for the general linear group", "abstract": "We show that every irreducible representation in the discrete automorphic spectrum of GL(n) admits a non vanishing mixed (Whittaker-symplectic) period integral. The analog local problem is a study of models first considered by Klyachko over a finite field. Locally, we show that for a p-adic field F every irreducible, unitary representation of GL(n,F) has a Klyachko model."}
{"category": "Math", "title": "Small value probabilities via the branching tree heuristic", "abstract": "In the first part of this paper we give easy and intuitive proofs for the small value probabilities of the martingale limit of a supercritical Galton-Watson process in both the Schr\\\"oder and the B\\\"ottcher case. These results are well-known, but the most cited proofs rely on generating function arguments which are hard to transfer to other settings. In the second part we show that the strategy underlying our proofs can be used in the quite different context of self-intersections of stochastic processes. Solving a problem posed by Wenbo Li, we find the small value probabilities for intersection local times of several Brownian motions, as well as for self-intersection local times of a single Brownian motion."}
{"category": "Math", "title": "Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology", "abstract": "Here we study vector bundles $E$ on the Hirzebruch surface $F_e$ such that their twists by a spanned, but not ample, line bundle $M = \\mathcal {O}_{F_e}(h+ef)$ have natural cohomology, i.e. $h^0(F_e,E(tM)) >0$ implies $h^1(F_e,E(tM)) = 0$."}
{"category": "Math", "title": "Attractors in coherent systems of differential equations", "abstract": "Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property that no directed feedback loops are negative. Several new results for cooperative systems are obtained in the process."}
{"category": "Math", "title": "Wavelet sets without groups", "abstract": "Most of the examples of wavelet sets are for dilation sets which are groups. We find a necessary and sufficient condition under which subspace wavelet sets exist for dilation sets of the form $A B$, which are not necessarily groups. We explain the construction by a few examples."}
{"category": "Math", "title": "Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: nonparametric kernel estimation and hypotheses testing", "abstract": "Let $v$ be a vector field in a bounded open set $G\\subset {\\mathbb {R}}^d$. Suppose that $v$ is observed with a random noise at random points $X_i, i=1,...,n,$ that are independent and uniformly distributed in $G.$ The problem is to estimate the integral curve of the differential equation \\[\\frac{dx(t)}{dt}=v(x(t)),\\qquad t\\geq 0,x(0)=x_0\\in G,\\] starting at a given point $x(0)=x_0\\in G$ and to develop statistical tests for the hypothesis that the integral curve reaches a specified set $\\Gamma\\subset G.$ We develop an estimation procedure based on a Nadaraya--Watson type kernel regression estimator, show the asymptotic normality of the estimated integral curve and derive differential and integral equations for the mean and covariance function of the limit Gaussian process. This provides a method of tracking not only the integral curve, but also the covariance matrix of its estimate. We also study the asymptotic distribution of the squared minimal distance from the integral curve to a smooth enough surface $\\Gamma\\subset G$. Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches $\\Gamma$. The problems of this nature are of interest in diffusion tensor imaging, a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter, where the diffusion of water molecules is typically anisotropic. The diffusion tensor data is used to estimate the dominant orientations of the diffusion and to track white matter fibers from the initial location following these orientations. Our approach brings more rigorous statistical tools to the analysis of this problem providing, in particular, hypothesis testing procedures that might be useful in the study of axonal connectivity of the white matter."}
{"category": "Math", "title": "Deformations of metabelian representations of knot groups into $SL(3,\\mathbb{C})$", "abstract": "Let K be a knot in $S^3$ and $X$ its complement. We study deformations of reducible metabelian representations of the knot group $\\pi_1(X)$ into $SL(3,\\mathbb{C})$ which are associated to a double root of the Alexander polynomial. We prove that these reducible metabelian representations are smooth points of the representation variety and that they have irreducible non metabelian deformations."}
{"category": "Math", "title": "Coxeter Groups, Wavelets, Multiresolution and Sampling", "abstract": "In this short note we discuss the interplay between finite Coxeter groups and construction of wavelet sets, generalized multiresolution analysis and sampling."}
{"category": "Math", "title": "Remarks Concerning Lubotzky's Filtration", "abstract": "A discrete group which admits a faithful, finite dimensional, linear representation over a field $\\mathbb F$ of characteristic zero is called linear. This note combines the natural structure of semi-direct products with work of A. Lubotzky on the existence of linear representations to develop a technique to give sufficient conditions to show that a semi-direct product is linear. Let $G$ denote a discrete group which is a semi-direct product given by a split extension $1 \\to \\pi \\to G \\to \\Gamma \\to 1$. This note defines an additional type of structure for this semi-direct product called a stable extension below. The main results are as follows: 1. If $\\pi$ and $\\Gamma$ are linear, and the extension is stable, then $G$ is also linear. Restrictions concerning this extension are necessary to guarantee that $G$ is linear as seen from properties of the Formanek-Procesi \"poison group\". 2. If the action of $\\Gamma$ on $\\pi$ has a \"Galois-like\" property that it factors through the automorphisms of certain natural \"towers of groups over $\\pi$\" (to be defined below), then the associated extension is stable and thus $G$ is linear. 3. The condition of a stable extension also implies that $G$ admits filtration quotients which themselves give a natural structure of Lie algebra and which also imply earlier results of Kohno, and Falk-Randell on the Lie algebra attached to the descending central series associated to the fundamental groups of complex hyperplane complements. The methods here suggest that a possible technique for obtaining new linearity results may be to analyze automorphisms of towers of groups."}
{"category": "Math", "title": "Three-way tiling sets in two dimensions", "abstract": "In this article we show that there exist measurable sets W in the plane with finite measure that tile the plane in a measurable way under the action of a expansive matrix A, an affine Weyl group W, and a full rank lattice G. This note is follow-up research to the earlier article \"Coxeter groups and wavelet sets\" by the first and second authors, and is also relevant to the earlier article \"Coxeter groups, wavelets, multiresolution and sampling\" by M. Dobrescu and the third author. After writing these two articles, the three authors participated in a workshop at the Banff Center on \"Operator methods in fractal analysis, wavelets and dynamical systems,\" December 2 -- 7, 2006, organized by O. Bratteli, P. Jorgensen, D. Kribs, G. Olafsson, and S. Silvestrov, and discussed the interrelationships and differences between the articles, and worked on two open problems posed in the Larson-Massopust article. We solved part of Problem 2, including a surprising positive solution to a conjecture that was raised, and we present our results in this article."}
{"category": "Math", "title": "$C^*-$crossed product of groupoid actions on categories", "abstract": "Suppose that $G$ is a groupoid acting on a small category $H$ in the sense of \\cite[Definition 4]{NOT} and $H\\times_\\alpha G$ is the resulting semi-direct product category (as in \\cite[Proposition 8]{NOT}). We show that there exists a subcategory $H_r \\subseteq H$ satisfying some nice property called ``regularity'' such that $H_r \\times_\\alpha G = H\\times_\\alpha G$. Moreover, we show that there exists a so-called ``quasi action'' (see Definition \\ref{quasi}) $\\beta$ of $G$ on $C^*(H_r)$ (where $C^*(H_r)$ is the semigroupoid $C^*$-algebra as defined in \\cite{EXE}) such that $C^*(H_r\\times_\\alpha G) = C^*(H_r)\\times_\\beta G$ (where the crossed product for $\\beta$ is as defined in Definition \\ref{cross})."}
{"category": "Math", "title": "A combinatorial framework for RNA tertiary interaction", "abstract": "In this paper we show how to express RNA tertiary interactions via the concepts of tangled diagrams. Tangled diagrams allow to formulate RNA base triples and pseudoknot-interactions and to control the maximum number of mutually crossing arcs. In particular we study two subsets of tangled diagrams: 3-noncrossing tangled-diagrams with $\\ell$ vertices of degree two and 2-regular, 3-noncrossing partitions (i.e. without arcs of the form $(i,i+1)$). Our main result is an asymptotic formula for the number of 2-regular, 3-noncrossing partitions, denoted by $p_{3,2}(n)$, 3-noncrossing partitions over $[n]$. The asymptotic formula is derived by the analytic theory of singular difference equations due to Birkhoff-Trjitzinsky. Explicitly, we prove the formula $p_{3,2}(n+1)\\sim K 8^{n}n^{-7}(1+c_{1}/n+c_{2}/n^2+c_3/n^3)$ where $K,c_i$, $i=1,2,3$ are constants."}
{"category": "Math", "title": "$n$-blocks collections on Fano manifolds and sheaves with regularity $-\\infty$", "abstract": "Let $X$ be a smooth Fano manifold equipped with a `` nice '' $n$-blocks collection in the sense of \\cite{cm2} and $\\mathcal {F}$ a coherent sheaf on $X$. Assume that $X$ is Fano and that all blocks are coherent sheaves. Here we prove that $\\mathcal {F}$ has regularity $-\\infty$ in the sense of \\cite{cm2} if ${Supp}(\\mathcal {F})$ is finite, the converse being true under mild assumptions. The corresponding result is also true when $X$ has a geometric collection in the sense of \\cite{cm1}."}
{"category": "Math", "title": "Classical and new loglog-theorems", "abstract": "We present integral variants of results due to Carleman, Wolf, Levinson, Sjoberg, and Matsaev on majorants of analytic functions. Main ingredient is a complete description for radial projections of harmonic measures of strictly star-shaped domains in the plane."}
{"category": "Math", "title": "OCHA and the swiss-cheese operad", "abstract": "In this paper we show that the relation between Kajiura-Stasheff's OCHA and A. Voronov's swiss-cheese operad is analogous to the relation between SH Lie algebras and the little discs operad. More precisely, we show that the OCHA operad is quasi-isomorphic to the operad of top-dimensional homology classes of the swiss-cheese operad."}
{"category": "Math", "title": "Free frobenius algebra on the differential forms of a manifold", "abstract": "We construct an action of a free resolution of the Frobenius properad on the differential forms of a closed oriented manifold. As a consequence, the forms of a manifold with values in a semi-simple Lie algebra have an additional structure given by an action of a free resolution of the properad describing Lie di-algebras with module compatibility."}
{"category": "Math", "title": "Homotopical interpretation of globular complex by multipointed d-space", "abstract": "Globular complexes were introduced by E. Goubault and the author in arXiv:math/0107060 to model higher dimensional automata. Globular complexes are topological spaces equipped with a globular decomposition which is the directed analogue of the cellular decomposition of a CW-complex. We prove that there exists a combinatorial model category such that the cellular objects are exactly the globular complexes and such that the homotopy category is equivalent to the homotopy category of flows introduced in arXiv:math/0308054. The underlying category of this model category is a variant of M. Grandis' notion of d-space over a topological space colimit generated by simplices. This result enables us to understand the relationship between the framework of flows and other works in directed algebraic topology using d-spaces. It also enables us to prove that the underlying homotopy type functor of flows constructed in arXiv:math/0308063 can be interpreted up to equivalences of categories as the total left derived functor of a left Quillen adjoint."}
{"category": "Math", "title": "A criterion for an abelian variety to be simple", "abstract": "In this note we give a numerical criterion that expresses the condition that an abelian variety be simple in terms of an invariant that is closely related to the s-invariant of Ein-Cutkosky-Lazarsfeld. The criterion yields new examples where s-invariants are irrational. It may also be viewed as a statement about the geometry of the ample cone."}
{"category": "Math", "title": "On the Eisenstein ideal of Drinfeld modular curves", "abstract": "Let $\\goth E(\\goth p)$ denote the Eisenstein ideal in the Hecke algebra $\\Bbb T(\\goth p)$ of the Drinfeld modular curve $X_0(\\goth p)$ parameterizing Drinfeld modules of rank two over $\\Bbb F_q[T]$ of general characteristic with Hecke level $\\goth p$-structure, where $\\goth p\\triangleleft\\Bbb F_q[T]$ is a non-zero prime ideal. We prove that the characteristic $p$ of the field $\\Bbb F_q$ does not divide the order of the quotient $\\Bbb T(\\goth p)/\\goth E(\\goth p)$ and the Eisenstein ideal $\\goth E(\\goth p)$ is locally principal."}
{"category": "Math", "title": "Muntz type Theorems I", "abstract": "In this paper, we concentrate our attention on the Muntz problem in the univariate setting and for the uniform norm."}
{"category": "Math", "title": "On the failure of the Poincar\\'e Lemma for de-bar-sub-M II", "abstract": "We obtain very sharp results about the lack of validity of the Poincare lemma for the tangential Cauchy Riemann equations, acting on tangential forms, tangential to a CR manifold M of general CR dimension n, and general CR codimension k. This generalizes the classical nonsolvability example of H. Lewy. We also discuss the CR structure on the characteristic bundle to M, due to certain degeneracies in the Levi form. A number of naturally geometrically occuring examples are given."}
{"category": "Math", "title": "A graph theoretic expansion formula for cluster algebras of classical type", "abstract": "In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their perfect matchings encodes the numerator of the associated Laurent polynomial while decompositions of the graphs correspond to the denominator. This complements recent work by Schiffler and Carroll-Price for a cluster expansion formula for the A_n case while providing a novel interpretation for the B_n, C_n, and D_n cases."}
{"category": "Math", "title": "Holomorphic correspondences between CR manifolds", "abstract": "It is proved that a germ of a real analytic CR map from a smooth real-analytic minimal CR manifold M to an essentially finite real-algebraic generic submanifold M' of P^N of the same CR-dimension extends as a holomorphic correspondence along M. Applications are given for pseudoconcave submanifolds of P^N."}
{"category": "Math", "title": "Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions", "abstract": "This paper studies tilings related to the beta-transformation when beta is a Pisot number (that is not supposed to be a unit). Then it applies the obtained results to study the set of rational numbers having a purely periodic beta-expansion. Special focus is given to some quadratic examples."}
{"category": "Math", "title": "On gradient Ricci solitons with Symmetry", "abstract": "We study gradient Ricci solitons with maximal symmetry. First we show that there are no non-trivial homogeneous gradient Ricci solitons. Thus the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the main result in our paper \"Rigidity of gradient Ricci solitons\" to show that there are no noncompact cohomogeneity one shrinking gradient solitons with nonnegative curvature."}
{"category": "Math", "title": "Vector bundles on contractible smooth schemes", "abstract": "We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A^1-homotopy theory; when k = C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are that of Spec k. One might hope that furthermore, and in analogy with the classification of topological vector bundles on manifolds, algebraic vector bundles on such schemes are all isomorphic to trivial bundles; this is almost certainly true when the scheme is affine. However, in the non-affine case this is false: we show that (essentially) every smooth A^1-contractible strictly quasi-affine scheme that admits a U-torsor whose total space is affine, for U a unipotent group, possesses a non-trivial vector bundle. Indeed we produce explicit arbitrary dimensional families of non-isomorphic such schemes, with each scheme in the family equipped with \"as many\" (i.e., arbitrary dimensional moduli of) non-isomorphic vector bundles, of every sufficiently large rank n, as one desires; neither the schemes nor the vector bundles on them are distinguishable by algebraic K-theory. We also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible, but that are not necessarily A^1-contractible."}
{"category": "Math", "title": "Embedding Bratteli-Vershik systems in cellular automata", "abstract": "Many dynamical systems can be naturally represented as `Bratteli-Vershik' (or `adic') systems, which provide an appealing combinatorial description of their dynamics. If an adic system X satisfies two technical conditions (`focus' and `bounded width') then we show how to represent X using a two-dimensional subshift of finite type Y; each `row' in a Y-admissible configuration corresponds to an infinite path in the Bratteli diagram of X, and the vertical shift on Y corresponds to the `successor' map of X. Any Y-admissible configuration can then be recoded as the spacetime diagram of a one-dimensional cellular automaton F; in this way X is `embedded' in F (i.e. X is conjugate to a subsystem of F). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata."}
{"category": "Math", "title": "Subsonic Flows for the Full Euler Equations in Half Plane", "abstract": "We study the subsonic flows governed by full Euler equations in the half plane bounded below by a piecewise smooth curve asymptotically approaching x1-axis. Nonconstant conditions in the far field are prescribed to ensure the real Euler flows. The Euler system is reduced to a single elliptic equation for the stream function. The existence, uniqueness and asymptotic behaviors of the solutions for the reduced equation are established by Schauder fixed point argument and some delicate estimates. The existence of subsonic flows for the original Euler system is proved based on the results for the reduced equation, and their asymptotic behaviors in the far field are also obtained."}
{"category": "Math", "title": "Homotopy on spatial graphs and generalized Sato-Levine invariants", "abstract": "Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split links. In this paper, we construct some new edge (resp. vertex)-homotopy invariants of spatial graphs without any restriction of linking numbers of the constituent 2-component links by applying the generalized Sato-Levine invariant."}
{"category": "Math", "title": "Universal Baxterization for $\\mathbb{Z}$-graded Hopf algebras", "abstract": "We present a method for Baxterizing solutions of the constant Yang-Baxter equation associated with $\\mathbb{Z}$-graded Hopf algebras. To demonstrate the approach, we provide examples for the Taft algebras and the quantum group $U_q[sl(2)]$."}
{"category": "Math", "title": "Scattering for the non-radial 3D cubic nonlinear Schroedinger equation", "abstract": "Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schr\\\"odinger equation below a mass-energy threshold $M[u]E[u] < M[Q]E[Q]$ and satisfying an initial mass-gradient bound $\\|u_0\\|_{L^2} \\|\\nabla u_0 \\|_{L^2} < \\|Q\\|_{L^2} \\|\\nabla Q\\|_{L^2}$, where $Q$ is the ground state, was established in Holmer-Roudenko (2007). In this note, we extend the result in Holmer-Roudenko (2007) to non-radial $H^1$ data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle (2006), we control via momentum conservation the rate of divergence of the spatial translation parameter and by a convexity argument based on a local virial identity deduce scattering. An application to the defocusing case is also mentioned."}
{"category": "Math", "title": "2-pile Nim with a Restricted Number of Move-size Imitations", "abstract": "We study a variation of the combinatorial game of 2-pile Nim. Move as in 2-pile Nim but with the following constraint: Suppose the previous player has just removed say $x>0$ tokens from the shorter pile (either pile in case they have the same height). If the next player now removes $x$ tokens from the larger pile, then he imitates his opponent. For a predetermined natural number $p$, by the rules of the game, neither player is allowed to imitate his opponent on more than $p-1$ consecutive moves. We prove that the strategy of this game resembles closely that of a variant of Wythoff Nim--a variant with a blocking manoeuvre on $p-1$ diagonal positions. In fact, we show a slightly more general result in which we have relaxed the notion of what an imitation is."}
{"category": "Math", "title": "Dynamics in Thompson's Group F", "abstract": "We describe an explicit relationship between strand diagrams and piecewise-linear functions for elements of Thompson's group F. Using this correspondence, we investigate the dynamics of elements of F, and we show that conjugacy of one-bump functions can be described by a Mather-type invariant."}
{"category": "Math", "title": "A Rigidity Theorem for Affine K\\\"ahler-Ricci Flat Graph", "abstract": "It is shown that any smooth strictly convex global solution of $$\\det(\\frac{\\partial^{2}u}{\\partial \\xi_{i}\\partial \\xi_{j}}) = \\exp \\left\\{-\\sum_{i=1}^n d_i \\frac{\\partial u}{\\partial \\xi_{i}} - d_0\\right\\},$$ where $d_0$, $d_1$,...,$d_n$ are constants, must be a quadratic polynomial. This extends a well-known theorem of J\\\"{o}rgens-Calabi-Pogorelov."}
{"category": "Math", "title": "Nonparametric estimation of correlation functions in longitudinal and spatial data, with application to colon carcinogenesis experiments", "abstract": "In longitudinal and spatial studies, observations often demonstrate strong correlations that are stationary in time or distance lags, and the times or locations of these data being sampled may not be homogeneous. We propose a nonparametric estimator of the correlation function in such data, using kernel methods. We develop a pointwise asymptotic normal distribution for the proposed estimator, when the number of subjects is fixed and the number of vectors or functions within each subject goes to infinity. Based on the asymptotic theory, we propose a weighted block bootstrapping method for making inferences about the correlation function, where the weights account for the inhomogeneity of the distribution of the times or locations. The method is applied to a data set from a colon carcinogenesis study, in which colonic crypts were sampled from a piece of colon segment from each of the 12 rats in the experiment and the expression level of p27, an important cell cycle protein, was then measured for each cell within the sampled crypts. A simulation study is also provided to illustrate the numerical performance of the proposed method."}
{"category": "Math", "title": "A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options", "abstract": "Under the assumption of no-arbitrage, the pricing of American and Bermudan options can be casted into optimal stopping problems. We propose a new adaptive simulation based algorithm for the numerical solution of optimal stopping problems in discrete time. Our approach is to recursively compute the so-called continuation values. They are defined as regression functions of the cash flow, which would occur over a series of subsequent time periods, if the approximated optimal exercise strategy is applied. We use nonparametric least squares regression estimates to approximate the continuation values from a set of sample paths which we simulate from the underlying stochastic process. The parameters of the regression estimates and the regression problems are chosen in a data-dependent manner. We present results concerning the consistency and rate of convergence of the new algorithm. Finally, we illustrate its performance by pricing high-dimensional Bermudan basket options with strangle-spread payoff based on the average of the underlying assets."}
{"category": "Math", "title": "The Euler Characteristic Formula for Logarithmic Minimal Degenerations of Surfaces with Kodaira Dimension Zero and its application to Calabi-Yau Threefolds with a pencil", "abstract": "In this paper, the Euler characteristic formula for projective logarithmic minimal degenerations of surfaces with Kodaira dimension zero over a 1-dimensional complex disk is proved under a reasonable assumption and as its application, the singularity of logarithmic minimal degenerations are determined in the abelian or hyperelliptic case. By globalizing this local analysis of singular fibres via generalized canonical bundle formulae due to Fujino-Mori, we bound the number of singular fibres of abelian fibred Calabi-Yau threefolds from above,which was previously done by Oguiso in the potentially good reduction case."}
{"category": "Math", "title": "Asymptotic approximation of nonparametric regression experiments with unknown variances", "abstract": "Asymptotic equivalence results for nonparametric regression experiments have always assumed that the variances of the observations are known. In practice, however the variance of each observation is generally considered to be an unknown nuisance parameter. We establish an asymptotic approximation to the nonparametric regression experiment when the value of the variance is an additional parameter to be estimated or tested. This asymptotically equivalent experiment has two components: the first contains all the information about the variance and the second has all the information about the mean. The result can be extended to regression problems where the variance varies slowly from observation to observation."}
{"category": "Math", "title": "Linearization of holomorphic germs with quasi-Brjuno fixed points", "abstract": "Let $f$ be a germ of holomorphic diffeomorphism of $\\C^n$ fixing the origin $O$, with $df_O$ diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of $df_O$ and some restrictions on the resonances, $f$ is locally holomorphically linearizable if and only if there exists a particular $f$-invariant complex manifold. Most of the classical linearization results can be obtained as corollaries of our result."}
{"category": "Math", "title": "Time-Frequency Analysis of Fourier Integral Operators", "abstract": "We use time-frequency methods for the study of Fourier Integral operators (FIOs). In this paper we shall show that Gabor frames provide very efficient representations for a large class of FIOs. Indeed, similarly to the case of shearlets and curvelets frames, the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is well-organized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudo-differential operators with symbols in $M^{\\infty,1}$, for some unimodular Fourier multipliers and metaplectic operators."}
{"category": "Math", "title": "Aggregation for Gaussian regression", "abstract": "This paper studies statistical aggregation procedures in the regression setting. A motivating factor is the existence of many different methods of estimation, leading to possibly competing estimators. We consider here three different types of aggregation: model selection (MS) aggregation, convex (C) aggregation and linear (L) aggregation. The objective of (MS) is to select the optimal single estimator from the list; that of (C) is to select the optimal convex combination of the given estimators; and that of (L) is to select the optimal linear combination of the given estimators. We are interested in evaluating the rates of convergence of the excess risks of the estimators obtained by these procedures. Our approach is motivated by recently published minimax results [Nemirovski, A. (2000). Topics in non-parametric statistics. Lectures on Probability Theory and Statistics (Saint-Flour, 1998). Lecture Notes in Math. 1738 85--277. Springer, Berlin; Tsybakov, A. B. (2003). Optimal rates of aggregation. Learning Theory and Kernel Machines. Lecture Notes in Artificial Intelligence 2777 303--313. Springer, Heidelberg]. There exist competing aggregation procedures achieving optimal convergence rates for each of the (MS), (C) and (L) cases separately. Since these procedures are not directly comparable with each other, we suggest an alternative solution. We prove that all three optimal rates, as well as those for the newly introduced (S) aggregation (subset selection), are nearly achieved via a single ``universal'' aggregation procedure. The procedure consists of mixing the initial estimators with weights obtained by penalized least squares. Two different penalties are considered: one of them is of the BIC type, the second one is a data-dependent $\\ell_1$-type penalty."}
{"category": "Math", "title": "Coxeter Groups and Wavelet Sets", "abstract": "A traditional wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a system of unitary operators defined in terms of translation and dilation operations. A Coxeter/fractal-surface wavelet is obtained by defining fractal surfaces on foldable figures, which tesselate the embedding space by reflections in their bounding hyperplanes instead of by translations along a lattice. Although both theories look different at their onset, there exist connections and communalities which are exhibited in this semi-expository paper. In particular, there is a natural notion of a dilation-reflection wavelet set. We prove that dilation-reflection wavelet sets exist for arbitrary expansive matrix dilations, paralleling the traditional dilation-translation wavelet theory. There are certain measurable sets which can serve simultaneously as dilation-translation wavelet sets and dilation-reflection wavelet sets, although the orthonormal structures generated in the two theories are considerably different."}
{"category": "Math", "title": "Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers", "abstract": "All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in $\\mathbb{Q}(\\sqrt{-3})$ are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3."}
{"category": "Math", "title": "Asymptotics of eigenfunctions on plane domains", "abstract": "We consider a family of domains $(\\Omega_N)_{N>0}$ obtained by attaching an $N\\times 1$ rectangle to a fixed set $\\Omega_0 = \\{(x,y): 0<y<1, -\\phi(y)<x<0\\}$, for a Lipschitz function $\\phi\\geq 0$. We derive full asymptotic expansions, as $N\\to\\infty$, for the $m$th Dirichlet eigenvalue (for any fixed $m$) and for the associated eigenfunction on $\\Omega_N$. The second term involves a scattering phase arising in the Dirichlet problem on the infinite domain $\\Omega_\\infty$. We determine the first variation of this scattering phase, with respect to $\\phi$, at $\\phi\\equiv 0$. This is then used to prove sharpness of results, obtained previously by the same authors, about the location of extrema and nodal line of eigenfunctions on convex domains."}
{"category": "Math", "title": "Regression for partially observed variables and nonparametric quantiles of conditional probabilities", "abstract": "Efficient estimation under bias sampling, censoring or truncation is a difficult question which has been partially answered and the usual estimators are not always consistent. Several biased designs are considered for models with variables $(X,Y)$ where $Y$ is an indicator and $X$ an explanatory variable, or for continuous variables $(X,Y)$. The identifiability of the models are discussed. New nonparametric estimators of the regression functions and conditional quantiles are proposed."}
{"category": "Math", "title": "Conformal changes of generalized complex structures", "abstract": "A conformal change of $TM\\oplus T^*M$ is a morphism of the form $(X,\\alpha)\\mapsto(X,e^\\tau\\alpha)$ $(X\\in TM,\\alpha\\in T^*M,\\tau\\in C^\\infty(M))$. We characterize the generalized almost complex and almost Hermitian structures that are locally conformal to integrable and to generalized K\\\"ahler structures, respectively, and give examples of such structures."}
{"category": "Math", "title": "Harmonic sections of tangent bundles equipped with Riemannian $g$-natural metrics", "abstract": "Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, the only vector fields which define harmonic maps from $(M,g)$ to $(TM,g^s)$, are the parallel ones. The Sasaki metric, and other well known Riemannian metrics on $TM$, are particular examples of $g$-natural metrics. We equip $TM$ with an arbitrary Riemannian $g$-natural metric $G$, and investigate the harmonicity of a vector field $V$ of $M$, thought as a map from $(M,g)$ to $(TM,G)$. We then apply this study to the Reeb vector field and, in particular, to Hopf vector fields on odd-dimensional spheres."}
{"category": "Math", "title": "Alpha-determinant cyclic modules and Jacobi polynomials", "abstract": "We study the cyclic $U(\\mathfrak{gl}_n)$-module generated by the $l$-th power of the $\\alpha$-determinant. When $l$ is a non-negative integer, for all but finite exceptional values of $alpha$, one shows that this cyclic module is isomorphic to the $n$-th tensor space $(S^l(\\mathbb{C}^n))^{\\otimes n}$ of the symmetric $l$-th tensor space of $\\mathbb{C}^n$. If $alpha$ is exceptional, then the structure of the module changes drastically, i.e. some irreducible representations which are the irreducible components of the decomposition of $(S^l(\\mathbb{C}^n))^{\\otimes n}$ disappear in the decomposition of the cyclic module. The degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a Kostka number and entries are polynomials in $alpha$ with rational coefficients. As a special case, we determine the matrix in a full of the detail for the case where $n=2$; the matrix becomes a scalar and is essentially given by the classical Jacobi polynomial. Moreover, we prove that these polynomials are unitary."}
{"category": "Math", "title": "Semigroup algebras of submonoids of polycyclic-by-finite groups and maximal orders", "abstract": "Necessary and sufficient conditions are given for a prime Noetherian algebra K[S] of a submonoid S of a polycyclic-by-finite group G to be a maximal order. These conditions are entirely in terms of the monoid S. This extends earlier results of Brown concerned with the group ring case and of the authors for the case where K[S] satisfies a polynomial identity."}
{"category": "Math", "title": "Outliers in dynamic factor models", "abstract": "Dynamic factor models have a wide range of applications in econometrics and applied economics. The basic motivation resides in their capability of reducing a large set of time series to only few indicators (factors). If the number of time series is large compared to the available number of observations then most information may be conveyed to the factors. This way low dimension models may be estimated for explaining and forecasting one or more time series of interest. It is desirable that outlier free time series be available for estimation. In practice, outlying observations are likely to arise at unknown dates due, for instance, to external unusual events or gross data entry errors. Several methods for outlier detection in time series are available. Most methods, however, apply to univariate time series while even methods designed for handling the multivariate framework do not include dynamic factor models explicitly. A method for discovering outliers occurrences in a dynamic factor model is introduced that is based on linear transforms of the observed data. Some strategies to separate outliers that add to the model and outliers within the common component are discussed. Applications to simulated and real data sets are presented to check the effectiveness of the proposed method."}
{"category": "Math", "title": "An Extension of Alzer's Inequality", "abstract": "In this article, we obtain two interesting general inequalities concerning Riemman sums of convex functions, which in particular, sharpen Alzer's inequality and give a suitable converse for it."}
{"category": "Math", "title": "Bayesian inference with rescaled Gaussian process priors", "abstract": "We use rescaled Gaussian processes as prior models for functional parameters in nonparametric statistical models. We show how the rate of contraction of the posterior distributions depends on the scaling factor. In particular, we exhibit rescaled Gaussian process priors yielding posteriors that contract around the true parameter at optimal convergence rates. To derive our results we establish bounds on small deviation probabilities for smooth stationary Gaussian processes."}
{"category": "Math", "title": "Some New Inequalities Between Important Means", "abstract": "In this paper, mainly using the convexity of the function $\\frac{a^x-b^x}{c^x-d^x}$ and convexity or concavity of the function $\\ln\\frac{a^x-b^x}{c^x-d^x}$ on the real line, where $a>b\\geq c>d>0$ are fixed real numbers, we obtain some important relations between various important means of these numbers. Also, we apply the obtained results to Ky Fan type inequalities and get some new refinements."}
{"category": "Math", "title": "Weak convergence of Metropolis algorithms for non-i.i.d. target distributions", "abstract": "In this paper, we shall optimize the efficiency of Metropolis algorithms for multidimensional target distributions with scaling terms possibly depending on the dimension. We propose a method for determining the appropriate form for the scaling of the proposal distribution as a function of the dimension, which leads to the proof of an asymptotic diffusion theorem. We show that when there does not exist any component with a scaling term significantly smaller than the others, the asymptotically optimal acceptance rate is the well-known 0.234."}
{"category": "Math", "title": "On invariant measures of stochastic recursions in a critical case", "abstract": "We consider an autoregressive model on $\\mathbb{R}$ defined by the recurrence equation $X_n=A_nX_{n-1}+B_n$, where $\\{(B_n,A_n)\\}$ are i.i.d. random variables valued in $\\mathbb{R}\\times\\mathbb{R}^+$ and $\\mathbb {E}[\\log A_1]=0$ (critical case). It was proved by Babillot, Bougerol and Elie that there exists a unique invariant Radon measure of the process $\\{X_n\\}$. The aim of the paper is to investigate its behavior at infinity. We describe also stationary measures of two other stochastic recursions, including one arising in queuing theory."}
{"category": "Math", "title": "Rational subsets of polycyclic monoids and valence automata", "abstract": "We study the classes of languages defined by valence automata with rational target sets (or equivalently, regular valence grammars with rational target sets), where the valence monoid is drawn from the important class of polycyclic monoids. We show that for polycyclic monoids of rank 2 or more, such automata accept exactly the context-free languages. For the polycyclic monoid of rank 1 (that is, the bicyclic monoid), they accept a class of languages strictly including the partially blind one-counter languages. Key to the proof is a description of the rational subsets of polycyclic and bicyclic monoids, other consequences of which include the decidability of the rational subset membership problem for these monoids, and the closure of the class of rational subsets under intersection and complement."}
{"category": "Math", "title": "Weighted Sequences in Finite Cyclic Groups", "abstract": "Let $p>7$ be a prime, let $G=\\Z/p\\Z$, and let $S_1=\\prod_{i=1}^p g_i$ and $S_2=\\prod_{i=1}^p h_i$ be two sequences with terms from $G$. Suppose that the maximum multiplicity of a term from either $S_1$ or $S_2$ is at most $\\frac{2p+1}{5}$. Then we show that, for each $g\\in G$, there exists a permutation $\\sigma$ of $1,2,..., p$ such that $g=\\sum_{i=1}^{p}(g_i\\cdot h_{\\sigma(i)})$. The question is related to a conjecture of A. Bialostocki concerning weighted subsequence sums and the Erd\\H{o}s-Ginzburg-Ziv Theorem."}
{"category": "Math", "title": "Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations", "abstract": "This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category $\\Tops$ of stratified spaces, that are topological spaces $X$ endowed with a partition $\\cF$ and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element $(X,\\cF)$ of $\\Tops$ together with a class $\\cA$ of subsets of $X$; they are similar to invariants introduced by M. Clapp and D. Puppe. If $(X,\\cF)\\in\\Tops$, we define a transverse subset as a subspace $A$ of $X$ such that the intersection $S\\cap A$ is at most countable for any $S\\in \\cF$. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a $C^1$-foliation, the three previous definitions, with $\\cA$ the class of transverse subsets, coincide with the tangential category and are homotopical invariants."}
{"category": "Math", "title": "Linkages in Polytope Graphs", "abstract": "A graph is k-linked if any k disjoint vertex-pairs can be joined by k disjoint paths. We improve a lower bound on the linkedness of polytopes slightly, which results in exact values for the minimal linkedness of 7-, 10- and 13-dimensional polytopes. We analyze in detail linkedness of polytopes on at most (6d+7)/5 vertices. In that case, a sharp lower bound on minimal linkedness is derived, and examples meeting this lower bound are constructed. These examples contain a class of examples due to Gallivan."}
{"category": "Math", "title": "L1Packv2: A Mathematica package for minimizing an $\\ell_1$-penalized functional", "abstract": "L1Packv2 is a Mathematica package that contains a number of algorithms that can be used for the minimization of an $\\ell_1$-penalized least squares functional. The algorithms can handle a mix of penalized and unpenalized variables. Several instructive examples are given. Also, an implementation that yields an exact output whenever exact data are given is provided."}
{"category": "Math", "title": "(Non)Commutative Hopf algebras of trees and (quasi)symmetric functions", "abstract": "The Connes-Kreimer Hopf algebra of rooted trees, its dual, and the Foissy Hopf algebra of of planar rooted trees are related to each other and to the well-known Hopf algebras of symmetric and quasi-symmetric functions via a pair of commutative diagrams. We show how this point of view can simplify computations in the Connes-Kreimer Hopf algebra and its dual, particularly for combinatorial Dyson-Schwinger equations."}
{"category": "Math", "title": "Additional Gradings in Khovanov Homology", "abstract": "The main goal of the present paper is to construct new invariants of knots with additional structure by adding new gradings to the Khovanov complex. The ideas given below work in the case of virtual knots, closed braids and some other cases of knots with additional structure. The source of our additional grading may be topological or combinatorial; it is axiomatised for many partial cases. As a byproduct, this leads to a complex which in some cases coincides (up to grading renormalisation) with the usual Khovanov complex and in some other cases with the Lee-Rasmussen complex. The grading we are going to construct behaves well with respect to some generalisations of the Khovanov homology, e.g., Frobenius extensions. These new homology theories give sharper estimates for some knot characteristics, such as minimal crossing number, atom genus, slice genus, etc. Our gradings generate a natural filtration on the usual Khovanov complex. There exists a spectral sequence starting with our homology and converging to the usual Khovanov homology."}
{"category": "Math", "title": "Bayesian Online Changepoint Detection", "abstract": "Changepoints are abrupt variations in the generative parameters of a data sequence. Online detection of changepoints is useful in modelling and prediction of time series in application areas such as finance, biometrics, and robotics. While frequentist methods have yielded online filtering and prediction techniques, most Bayesian papers have focused on the retrospective segmentation problem. Here we examine the case where the model parameters before and after the changepoint are independent and we derive an online algorithm for exact inference of the most recent changepoint. We compute the probability distribution of the length of the current ``run,'' or time since the last changepoint, using a simple message-passing algorithm. Our implementation is highly modular so that the algorithm may be applied to a variety of types of data. We illustrate this modularity by demonstrating the algorithm on three different real-world data sets."}
{"category": "Math", "title": "Combinatorial Hopf algebras and Towers of Algebras", "abstract": "Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\\bigoplus_{n\\ge0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\\bigoplus_{n\\ge0}A_n$ gives rise to graded dual Hopf algebras then we must have $\\dim(A_n)=r^nn!$ where $r = \\dim(A_1)$."}
{"category": "Math", "title": "On Algebraic Shift Equivalence of Matrices over Polynomial Rings", "abstract": "The paper studies algebraic strong shift equivalence of matrices over $n$-variable polynomial rings over a principal ideal domain $D$($n\\leq 2$). It is proved that in the case $n=1$, every non-zero matrix over $D[x]$ has a full rank factorization and every non-nilpotent matrix over $D[x]$ is algebraically strong shift equivalent to a nonsingular matrix. In the case $n=2$, an example of non-nilpotent matrix over $\\mathbb{R}[x,y,z]=\\mathbb{R}[x][y,z]$, which can not be algebraically shift equivalent to a nonsingular matrix, is given."}
{"category": "Math", "title": "Completions of quantum coordinate rings", "abstract": "Given an iterated skew polynomial ring C[y_1;t_1,d_1]ldots [y_n;t_n,d_n] over a complete local ring C with maximal ideal m, we prove, under suitable assumptions, that the completion at the ideal m + < y_1,y_2,ldots,y_n> is an iterated skew power series ring. Under further conditions, this completion is a local, noetherian, Auslander regular domain. Applicable examples include quantum matrices, quantum symplectic spaces, and quantum Euclidean space."}
{"category": "Math", "title": "Determinants of rational knots", "abstract": "We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd."}
{"category": "Math", "title": "Equivariant $K$-theory of quaternionic flag manifolds", "abstract": "We consider the manifold $Fl_n(\\mathbb{H})=Sp(n)/Sp(1)^n$ of all complete flags in $\\mathbb{H}^n$, where $\\mathbb{H}$ is the skew-field of quaternions. We study its equivariant $K$-theory rings with respect to the action of two groups: $Sp(1)^n$ and a certain canonical subgroup $T:=(S^1)^n\\subset Sp(1)^n$ (a maximal torus). For the first group action we obtain a Goresky-Kottwitz-MacPherson type description. For the second one, we describe the ring $K_T(Fl_n(\\mathbb{H}))$ as a subring of $K_T(Sp(n)/T)$. This ring is well known, since $Sp(n)/T$ is a complex flag variety."}
{"category": "Math", "title": "Cohomogeneity one manifolds and selfmaps of nontrivial degree", "abstract": "We construct natural selfmaps of compact cohomgeneity one manifolds with finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields in certain cases relations between the order of the Weyl group and the Euler characteristic of a principal orbit. We apply our construction to the compact Lie group SU(3) where we extend identity and transposition to an infinite family of selfmaps of every odd degree. The compositions of these selfmaps with the power maps realize all possible degrees of selfmaps of SU(3)."}
{"category": "Math", "title": "On the localization theorem for F-pure rings", "abstract": "We solve Grothendieck's localization problem for certain class of rings arising from the tight closure theory. The idea of the proof depends heavily on the study of the relative version of the Frobenius map."}
{"category": "Math", "title": "Reducible And Finite Dehn Fillings", "abstract": "We show that the distance between a finite filling slope and a reducible filling slope on the boundary of a hyperbolic knot manifold is at most one."}
{"category": "Math", "title": "Approximating critical parameters of branching random walks", "abstract": "Given a branching random walk on a graph, we consider two kinds of truncations: by inhibiting the reproduction outside a subset of vertices and by allowing at most $m$ particles per site. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation."}
{"category": "Math", "title": "Curve complexes are rigid", "abstract": "Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface."}
{"category": "Math", "title": "Levi-flat hypersurfaces with real analytic boundary", "abstract": "Let $X$ be a Stein manifold of dimension at least 3. Given a compact codimension 2 real analytic submanifold $M$ of $X$, that is the boundary of a compact Levi-flat hypersurface $H$, we study the regularity of $H$. Suppose that the CR singularities of $M$ are an $\\mathcal{O}(X)$-convex set. For example, suppose $M$ has only finitely many CR singularities, which is a generic condition. Then $H$ must in fact be a real analytic submanifold. If $M$ is real algebraic, it follows that $H$ is real algebraic and in fact extends past $M$, even near CR singularities. To prove these results we provide two variations on a theorem of Malgrange, that a smooth submanifold contained in a real analytic subvariety of the same dimension is itself real analytic. We prove a similar theorem for submanifolds with boundary, and another one for subanalytic sets."}
{"category": "Math", "title": "On the number of tetrahedra with minimum, unit, and distinct volumes in three-space", "abstract": "We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by $n$ points in 3-space, and in general in $d$ dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by $n$ points in $\\RR^3$ is at most ${2/3}n^3-O(n^2)$, and there are point sets for which this number is ${3/16}n^3-O(n^2)$. We also present an $O(n^3)$ time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for every $k,d\\in \\NN$, $1\\leq k \\leq d$, the maximum number of $k$-dimensional simplices of minimum (nonzero) volume spanned by $n$ points in $\\RR^d$ is $\\Theta(n^k)$. (ii) The number of unit-volume tetrahedra determined by $n$ points in $\\RR^3$ is $O(n^{7/2})$, and there are point sets for which this number is $\\Omega(n^3 \\log \\log{n})$. (iii) For every $d\\in \\NN$, the minimum number of distinct volumes of all full-dimensional simplices determined by $n$ points in $\\RR^d$, not all on a hyperplane, is $\\Theta(n)$."}
{"category": "Math", "title": "Explicit approximation of the sum of the reciprocal of the imaginary parts of the zeta zeros", "abstract": "In this note, we give some explicit upper and lower bounds for the summation $\\sum_{0<\\gamma\\leq T}\\frac{1}{\\gamma}$, where $\\gamma$ is the imaginary part of nontrivial zeros $\\rho=\\beta+i\\gamma$ of $\\zeta(s)$."}
{"category": "Math", "title": "Trace ideals for Fourier integral operators with non-smooth symbols II", "abstract": "We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces."}
{"category": "Math", "title": "A new class of examples of group-valued moment maps", "abstract": "The purpose of this paper is to construct new examples of group-valued moment maps. As the main tool for construction of such examples we use quasi-symplectic implosion which was introduced in [HJS06]. More precisely we show that there are certain strata of $D{\\bf Sp}(n)_{\\rm impl}$, the universal imploded space, where it is singular but whose closure is a smooth quasi-Hamiltonian ${\\bf Sp}(n) \\times T$ space."}
{"category": "Math", "title": "Cells and Constructible Representations in type B", "abstract": "We examine the partition of a finite Coxeter group of type $B$ into cells determined by a weight function $L$. The main objective of these notes is to reconcile Lusztig's description of constructible representations in this setting with conjectured combinatorial descriptions of cells."}
{"category": "Math", "title": "Hopf modules for autonomous pseudomonoids and the monoidal centre", "abstract": "In this work we develop some aspects of the theory of Hopf algebras to the context of autonomous map pseudomonoids. We concentrate in the Hopf modules and the Centre or Drinfel'd double. If $A$ is a map pseudomonoid in a monoidal bicategory \\M, the analogue of the category of Hopf modules for $A$ is an Eilenberg-Moore construction for a certain monad in $\\mathbf{Hom}(\\M^{\\mathrm{op}},\\mathbf{Cat})$. We study the existence of the internalisation of this notion, called the Hopf module construction, by extending the completion under Eilenberg-Moore objects of a 2-category to a endo-homomorphism of tricategories on $\\mathbf{Bicat}$. Our main result is the equivalence between the existence of a left dualization for $A$ ({\\em i.e.}, $A$ is left autonomous) and the validity of an analogue of the structure theorem of Hopf modules. In this case the Hopf module construction for $A$ always exists. We use these results to study the lax centre of a left autonomous map pseudomonoid. We show that the lax centre is the Eilenberg-Moore construction for a certain monad on $A$ (one existing if the other does). If $A$ is also right autonomous, then the lax centre equals the centre. We look at the examples of the bicategories of \\V-modules and of comodules in \\V, and obtain the Drinfel'd double of a coquasi-Hopf algebra $H$ as the centre of $H$."}
{"category": "Math", "title": "Generalizations of Sch\\\"{o}bi's Tetrahedral Dissection", "abstract": "Let v_1, ..., v_n be unit vectors in R^n such that v_i . v_j = -w for i != j, where -1 <w < 1/(n-1). The points Sum_{i=1..n} lambda_i v_i, where 1 >= lambda_1 >= ... >= lambda_n >= 0, form a ``Hill-simplex of the first type'', denoted by Q_n(w). It was shown by Hadwiger in 1951 that Q_n(w) is equidissectable with a cube. In 1985, Sch\\\"{o}bi gave a three-piece dissection of Q_3(w) into a triangular prism c Q_2(1/2) X I, where I denotes an interval and c = sqrt{2(w+1)/3}. The present paper generalizes Sch\\\"{o}bi's dissection to an n-piece dissection of Q_n(w) into a prism c Q_{n-1}(1/(n-1)) X I, where c = sqrt{(n-1)(w+1)/n}. Iterating this process leads to a dissection of Q_n(w) into an n-dimensional rectangular parallelepiped (or ``brick'') using at most n! pieces. The complexity of computing the map from Q_n(w) to the brick is O(n^2). A second generalization of Sch\\\"{o}bi's dissection is given which applies specifically in R^4. The results have applications to source coding and to constant-weight binary codes."}
{"category": "Math", "title": "Tamagawa defect of Euler systems", "abstract": "As remarked in [Kolyvagin systems, by Barry Mazur and Karl Rubin] Proposition 6.2.6 and Buyukboduk[ arXiv:0706.0377v1 ] Remark 3.25 one does not expect the Kolyvagin system obtained from an Euler system for a p-adic Galois representation T to be primitive (in the sense of [Kolyvagin systems, by Barry Mazur and Karl Rubin] Definition 4.5.5) if p divides a Tamagawa number at a prime \\ell different from p; thus fails to compute the correct size of the relevant Selmer module. In this paper we obtain a lower bound for the size of the cokernel of the Euler system to Kolyvagin system map (see Theorem 3.2.4 of [Kolyvagin systems, by Barry Mazur and Karl Rubin] for a definition of this map) in terms of the Tamagawa numbers of T, refining [Kolyvagin systems, by Barry Mazur and Karl Rubin] Propostion 6.2.6. We show how this partially accounts for the missing Tamagawa factors in Kato's calculations with his Euler system."}
{"category": "Math", "title": "Prime and composite Laurent polynomials", "abstract": "In 1922 Ritt constructed the theory of functional decompositions of polynomials with complex coefficients. In particular, he described explicitly indecomposable polynomial solutions of the functional equation f(p(z))=g(q(z)). In this paper we study the equation above in the case when f,g,p,q are holomorphic functions on compact Riemann surfaces. We also construct a self-contained theory of functional decompositions of rational functions with at most two poles generalizing the Ritt theory. In particular, we give new proofs of the theorems of Ritt and of the theorem of Bilu and Tichy."}
{"category": "Math", "title": "Analysis of the convex hull of the attractor of an IFS", "abstract": "In this paper we will introduce the methodology of analysis of the convex hull of the attractors of iterated functional systems (IFS) - compact fixed sets of self-similarity mapping. The method is based on a function which for a direction, gives width in that direction. We can write the self similarity equation in terms of this function, solve and analyze them. Using this function we can quickly check if the distance from K of a given x is smaller than a given distance or even compute analytically convex hull area and the length of its boundary"}
{"category": "Math", "title": "Criteria for the density property of complex manifolds", "abstract": "In this paper we suggest new effective criteria for the density property. This enables us to give a trivial proof of the original Anders\\'en-Lempert result and to establish (almost free of charge) the algebraic density property for all linear algebraic groups whose connected components are different from tori or $\\C_+$. As another application of this approach we tackle the question (asked among others by F. Forstneri\\v{c}) about the density of algebraic vector fields on Euclidean space vanishing on a codimension 2 subvariety."}
{"category": "Math", "title": "The WKB method for conjugate points in the volumorphism group", "abstract": "In this paper, we are interested in the location of conjugate points along a geodesic in the volumorphism group of a compact three-dimensional manifold without boundary (the configuration space of an ideal fluid). As shown in the author's previous work, these are typically pathological, i.e., they can occur in clusters along a geodesic, unlike on finite-dimensional Riemannian manifolds. (This phenomenon does not occur for the volumorphism groups of two-dimensional manifolds, which are known to have discrete conjugate points along any geodesic by Ebin-Misiolek-Preston.) We give an explicit algorithm for finding them in terms of a certain ordinary differential equation, derived via the WKB-approximation methods of Lifschitz-Hameiri and Friedlander-Vishik. We prove that for a typical geodesic in the volumorphism group, there will be pathological conjugate point locations filling up closed intervals; hence typically the zeroes of Jacobi fields on the volumorphism group are dense in intervals."}
{"category": "Math", "title": "Algebraic Geometry over Free Metabelian Lie Algebra I: U-Algebras and Universal Classes", "abstract": "This paper is the first in a series of three, the aim of which is to lay the foundations of algebraic geometry over the free metabelian Lie algebra $F$. In the current paper we introduce the notion of a metabelian Lie $U$-algebra and establish connections between metabelian Lie $U$-algebras and special matrix Lie algebras. We define the $\\Delta $-localisation of a metabelian Lie $U$-algebra $A$ and the direct module extension of the Fitting's radical of $A$ and show that these algebras lie in the universal closure of $A$."}
{"category": "Math", "title": "Algebraic Geometry over Free Metabelian Lie Algebra II: Finite Field Case", "abstract": "This paper is the second in a series of three, the aim of which is to construct algebraic geometry over a free metabelian Lie algebra $F$. For the universal closure of free metabelian Lie algebra of finite rank $r \\ge 2$ over a finite field $k$ we find a convenient set of axioms in the language of Lie algebras $L$ and the language $L_{F}$ enriched by constants from $F$. We give a description of: * The structure of finitely generated algebras from the universal closure of $F_r$ in both $L$ and $L_{F_r}$ * The structure of irreducible algebraic sets over $F_r $ and respective coordinate algebras. We also prove that the universal theory of a free metabelian Lie algebra over a finite field is decidable in both languages."}
{"category": "Math", "title": "Semidomains and Metabelian Product of Metabelian Lie Algebras", "abstract": "This paper is the third in a series of papers, the aim of which is to construct algebraic geometry over metabelian Lie algebras."}
{"category": "Math", "title": "Quasirandom groups", "abstract": "Babai and S\\'os have asked whether there exists a constant c>0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy=z. In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset."}
{"category": "Math", "title": "Special types of intuitionistic fuzzy left h-ideals of hemirings", "abstract": "Characteristic, normal and completely normal intuitionistic fuzzy left $h$-ideals of hemirings are described."}
{"category": "Math", "title": "Fuzzy n-ary groups as a generalization of Rosenfeld's fuzzy groups", "abstract": "The notion of an $n$-ary group is a natural generalization of the notion of a group and has many applications in different branches. In this paper, the notion of (normal) fuzzy $n$-ary subgroup of an $n$-ary group is introduced and some related properties are investigated. Characterizations of fuzzy $n$-ary subgroups are given."}
{"category": "Math", "title": "Ultra LI-ideals in lattice implication algebras and MTL-algebras", "abstract": "A mistake concerning the ultra \\textit{LI}-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an \\textit{LI}-ideal to be an ultra \\textit{LI}-ideal are given. Moreover, the notion of an \\textit{LI}-ideal is extended to MTL-algebras, the notions of a (prime, ultra, obstinate, Boolean) \\textit{LI}-ideal and an \\textit{ILI}-ideal of an MTL-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in MTL-algebra: (1) prime proper \\textit{LI}-ideal and Boolean \\textit{LI}-ideal, (2) prime proper \\textit{LI}-ideal and \\textit{ILI}-ideal, (3) proper obstinate \\textit{LI}-ideal, (4) ultra \\textit{LI}-ideal."}
{"category": "Math", "title": "The genus of a curve of Fermat type", "abstract": "In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to the one from Fermat's last theorem. We begin by finding local coordinates for the standard affine cover of the plane, and then prove that the curve is smooth. This is done by pulling the curve up to the surface's desingularization. Then a map from the curve to ${\\mathbb P^1}$ is constructed, and it's ramification divisor is determined. We conclude by applying Hurwitz's theorem to this map to obtain $C$'s genus."}
{"category": "Math", "title": "Upper and lower bounds on resonances for manifolds hyperbolic near infinity", "abstract": "For a conformally compact manifold that is hyperbolic near infinity and of dimension $n+1$, we complete the proof of the optimal $O(r^{n+1})$ upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an $r^{n+1}$ lower bound on the counting function for scattering poles."}
{"category": "Math", "title": "Lie Rinehart Bialgebras for Crossed Products", "abstract": "In this paper, we study Lie Rinehart bialgebras, the algebraic generalization of Lie bialgebroids. More precisely, we analyze the structure of Lie Rinehart bialgebras for crossed products induced by actions of Lie algebras on K[t]."}
{"category": "Math", "title": "On the Existence of Global Bisections of Lie Groupoids", "abstract": "We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed curve. The existence of bisections through more than one prescribed points is also discussed. We give some interesting applications of these results."}
{"category": "Math", "title": "The sectional curvature remains positive when taking quotients by certain nonfree actions", "abstract": "We study some cases when the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups $S^1$ and $S^3$ such that the quotient space can be endowed with a smooth structure using the fibrations $S^3/S^1{\\simeq}S^2$ and $S^7/S^3\\simeq S^4$. We prove that the quotient space carries a metric of positive sectional curvature, provided that the original metric has positive sectional curvature on all 2-planes orthogonal to the orbits of the action."}
{"category": "Math", "title": "Inverse spectral results for Schr\\\"odinger operators on the unit interval with potentials in L^P spaces", "abstract": "We consider the Schr\\\"odinger operator on $[0,1]$ with potential in $L^1$. We prove that two potentials already known on $[a,1]$ ($a\\in(0,{1/2}]$) and having their difference in $L^p$ are equal if the number of their common eigenvalues is sufficiently large. The result here is to write down explicitly this number in terms of $p$ (and $a$) showing the role of $p$."}
{"category": "Math", "title": "Bounding sectional curvature along a K\\\"ahler-Ricci flow", "abstract": "If a normalized K\\\"{a}hler-Ricci flow $g(t),t\\in[0,\\infty),$ on a compact K\\\"{a}hler $n$-manifold, $n\\geq 3$, of positive first Chern class satisfies $g(t)\\in 2\\pi c_{1}(M)$ and has $L^{n}$ curvature operator uniformly bounded, then the curvature operator will also uniformly bounded along the flow. Consequently the flow will converge along a subsequence to a K\\\"{a}hler-Ricci soliton."}
{"category": "Math", "title": "An Introduction to Potential Theory in Calibrated Geometry", "abstract": "We introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. In particular, the notion of pseudo-convexity for a calibrated manifold (X,\\phi) is introduced and studied. Analogues of totally real submanifolds are also introduced and used to construct enormous families of strictly \\phi-convex spaces with every topological type allowed by Morse Theory. Specific calibrations are used as examples throughout."}
{"category": "Math", "title": "Duality of Positive Currents and Plurisubharmonic Functions in Calibrated Geometry", "abstract": "Recently the authors showed that there is a robust potential theory attached to any calibrated manifold (X,\\phi). In particular, on X there exist \\phi-plurisubharmonic functions, \\phi-convex domains, \\phi-convex boundaries, etc., all inter-related and having a number of good properties. In this paper we show that, in a strong sense, the plurisubharmonic functions are the polar duals of the \\phi-submanifolds, or more generally, the \\phi-currents studied in the original paper on calibrations. In particular, we establish an analogue of Duval-Sibony Duality which characterizes points in the \\phi-convex hull of a compact set K in X in terms of \\phi-positive Green's currents on X and Jensen measures on K. We also characterize boundaries of \\phi-currents entirely in terms of \\phi-plurisubharmonic functions. Specific calibrations are used as examples throughout. Analogues of the Hodge Conjecture in calibrated geometry are considered."}
{"category": "Math", "title": "Convexity properties for generalized moment maps I", "abstract": "We study generalized moment maps for a Hamiltonian action on a connected compact $H$-twisted generalized complex manifold introduced by Lin and Tolman and prove the convexity and connectedness properties of the generalized moment maps for a Hamiltonian torus action."}
{"category": "Math", "title": "An Analytic Proof of the Matrix Spectral Factorization Theorem", "abstract": "An analytic proof is proposed of Wiener's theorem on factorization of positive definite matrix-functions."}
{"category": "Math", "title": "First Eigenvalues of Geometric Operators under the Ricci Flow", "abstract": "In this paper, we prove that the first eigenvalues of $-\\Delta + cR$ ($c\\geq \\frac14$) is nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized flow for the case $c=1/4$, and $r\\le 0$."}
{"category": "Math", "title": "A note on pairs of metrics in a two-dimensional linear vector space", "abstract": "Pairs of metrics in a two-dimensional linear vector space are considered, one of which is a Minkowski type metric. Their simultaneous diagonalizability is studied and canonical presentations for them are suggested."}
{"category": "Math", "title": "The fractional stochastic heat equation on the circle: Time regularity and potential theory", "abstract": "We consider a system of $d$ linear stochastic heat equations driven by an additive infinite-dimensional fractional Brownian noise on the unit circle $S^1$. We obtain sharp results on the H\\\"older continuity in time of the paths of the solution $u=\\{u(t, x)\\}_{t \\in \\mathbb{R}_+, x \\in S^1}$. We then establish upper and lower bounds on hitting probabilities of $u$, in terms of respectively Hausdorff measure and Newtonian capacity."}
{"category": "Math", "title": "The solution of a memorable problem by a special artifice of calculation", "abstract": "E731 in the Enestrom index. Originally published as \"Solutio problematis ob singularia calculi artificia memorabilis\", Memoires de l'academie des sciences de St-Petersbourg 2 (1810), 3-9. For $z$ the distance from the origin, and $v$ a given function of $z$, Euler wants to find a curve $s$ such that the integral of $z$ over $s$ is a maximum or a minimum. He starts with the Euler-Lagrange equation, and does a lot of manipulations with polar coordinates."}
{"category": "Math", "title": "Non-archimedean equidistribution on elliptic curves with global applications", "abstract": "Let $E$ be an elliptic curve over an algebraically closed, complete, non-archimedean field $K$, and let ${\\mathsf E}$ denote the Berkovich analytic space associated to $E/K$. We study the $\\mu$-equidistribution of finite subsets of $E(K)$, where $\\mu$ is a certain canonical unit Borel measure on ${\\mathsf E}$. Our main result is an inequality bounding the error term when testing against a certain class of continuous functions on ${\\mathsf E}$. We then give two applications to elliptic curves over global function fields: we prove a function field analogue of the Szpiro-Ullmo-Zhang equidistribution theorem for small points, and a function field analogue of a result of Baker-Ih-Rumely on the finiteness of $S$-integral torsion points. Both applications are given in explicit quantitative form."}
{"category": "Math", "title": "A Kruskal-Katona Type Theorem for Graphs", "abstract": "A bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal-Katona theorem. A bound on non-consecutive clique numbers is also proven."}
{"category": "Math", "title": "A note on a curious formula for Euler's constant", "abstract": "In this short note we will use the residue theorem to establish a formula for Euler's constant. In particular, we offer a slightly generalized version of an interesting infinite series due to Flajolet, Gourdon, and Dumas."}
{"category": "Math", "title": "Quasisymmetric functions and Kazhdan-Lusztig polynomials", "abstract": "We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality."}
{"category": "Math", "title": "Associahedra, Cyclohedra and a Topological solution to the A_{\\infty}--Deligne conjecture", "abstract": "We give a topological solution to the $\\Ainf$ Deligne conjecture using associahedra and cyclohedra. For this we construct three CW complexes whose cells are indexed by products of polytopes. Giving new explicit realizations of the polytopes in terms of different types of trees, we are able to show that the CW complexes are cell models for the little discs. The cellular chains of one complex in particular, which is built out of associahedra and cyclohedra, naturally acts on the Hochschild cochains of an $\\Ainf$ algebra yielding an explicit, topological and minimal solution to the $\\Ainf$ Deligne conjecture. Along the way we obtain new results about the cyclohedra, such as a new decompositions into products of cubes and simplices, which can be used to realize them via a new iterated blow--up construction."}
{"category": "Math", "title": "Gromov-Witten invariants of blow-ups along submanifolds with convex normal bundles", "abstract": "Given a submanifold Z inside X, let Y be the blow-up of X along Z. When the normal bundle of Z in X is convex with a minor assumption, we prove that genus-zero GW-invariants of Y with cohomology insertions from X, are identical to GW-invariants of X. Under the same hypothesis, a vanishing theorem is also proved. An example to which these two theorems apply is when the normal bundle is generated by global sections. These two main theorems do not hold for arbitrary blow-ups, and counter-examples are included."}
{"category": "Math", "title": "Homotopy nilpotency in p-compact groups", "abstract": "A p-compact group is a mod p homotopy theoretical analogue of a compact Lie group. It is determined the homotopy nilpotency class of a p-compact group having the homotopy type of the $p$-completion of the direct product of spheres."}
{"category": "Math", "title": "Reconstruction of the Berger measure when the core is of tensor form", "abstract": "Let $\\mathfrak{H}_{0}$ denote the class of commuting pairs of subnormal operators on Hilbert space, and let $\\mathcal{TC}:=\\{\\mathbf{T}\\in \\mathfrak{% H}_{0}:c(\\mathbf{T)}$ is of tensor form$\\}$, where $c(\\mathbf{T})$ is the core of $\\mathbf{T}$. We obtain a concrete necessary and sufficient condition for the subnormality of $\\mathbf{T}\\equiv (T_{1},T_{2})\\in \\mathcal{TC}$ in terms of $c(\\mathbf{T})$, the marginal measures of $T_{1}$ and $T_{2}$, and the weight $\\alpha_{01}$."}
{"category": "Math", "title": "Some properties of minimizers for the Chan-Esedoglu L1TV functional", "abstract": "We present two results characterizing minimizers of the Chan-Esedoglu L1TV functional $F(u) \\equiv \\int |\\nabla u | dx + \\lambda \\int |u - f| dx $; $u,f:\\Bbb{R}^n \\to \\Bbb{R}$. If we restrict to $u = \\chi_{\\Sigma}$ and $f = \\chi_{\\Omega}$, $\\Sigma, \\Omega \\in \\Bbb{R}^n$, the $L^1$TV functional reduces to $E(\\Sigma) = \\Per(\\Sigma) + \\lambda |\\Sigma\\vartriangle \\Omega |$. We show that there is a minimizer $\\Sigma$ such that its boundary $\\partial\\Sigma$ lies between the union of all balls of radius $\\frac{n}{\\lambda}$ contained in $\\Omega$ and the corresponding union of $\\frac{n}{\\lambda}$-balls in $\\Omega^c$. We also show that if a ball of radius $\\frac{n}{\\lambda} + \\epsilon$ is almost contained in $\\Omega$, a slightly smaller concentric ball can be added to $\\Sigma$ to get another minimizer. Finally, we comment on recent results Allard has obtained on $L^1$TV minimizers and how these relate to our results."}
{"category": "Math", "title": "Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method", "abstract": "We study the two-scale asymptotics for a charged beam under the action of a rapidly oscillating external electric field. After proving the convergence to the correct asymptotic state, we develop a numerical method for solving the limit model involving two time scales and validate its efficiency for the simulation of long time beam evolution."}
{"category": "Math", "title": "C1-generic conservative diffeomorphisms have trivial centralizer", "abstract": "We prove that the spaces of C1 symplectomorphisms and of C1 volume-preserving diffeomorphisms of connected manifolds both contain residual subsets of diffeomorphisms whose centralizers are trivial. (Les diff\\'eomorphismes conservatifs C1-g\\'en\\'eriques ont un centralisateur trivial. Nous montrons que l'espace des symplectomorphismes de classe C1 et l'espace des diff\\'eomomorphismes de classe C1 pr\\'eservant une forme volume contiennent tous deux des sous-ensembles r\\'esiduels de diff\\'eomorphismes dont le centralisateur est trivial.)"}
{"category": "Math", "title": "Dirichlet Duality and the Nonlinear Dirichlet Problem", "abstract": "We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric ``F-convexity'' assumption on the boundary bdy(F). The topological structure of F-convex domains is also studied and a theorem of Andreotti-Frankel type is proved for them. Two key ingredients in the analysis are the use of subaffine functions and Dirichlet duality, both introduced here. Associated to F is a Dirichlet dual set F* which gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F* is F and in the analysis the roles of F and F* are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: All branches of the homogeneous Monge-Ampere equation over R, C and H; equations appearing naturally in calibrated geometry, Lagrangian geometry and p-convex riemannian geometry, and all branches of the Special Lagrangian potential equation."}
{"category": "Math", "title": "Zero modes for the magnetic Pauli operator in even-dimensional Euclidean space", "abstract": "We study the ground state of the Pauli Hamiltonian with a magnetic field in R^(2d). We consider the case where a scalar potential W is present and the magnetic field B is given by $B=2i\\partial\\bar\\partial W$. The main result is that there are no zero modes if the magnetic field decays faster than quadratically at infinity. If the magnetic field decays quadratically then zero modes may appear, and we give a lower bound for the number of them. The results in this paper partly correct a mistake in a paper from 1993."}
{"category": "Math", "title": "Reversibility in the group of homeomorphisms of the circle", "abstract": "We identify those elements of the homeomorphism group of the circle that can be expressed as a composite of two involutions."}
{"category": "Math", "title": "Chow--Kuenneth decomposition for special varieties", "abstract": "In this paper we investigate Murre's conjecture on the Chow--K\\\"unneth decomposition for two classes of examples. We look at the universal families of smooth curves over spaces which dominate the moduli space $\\cM_g$, in genus at most 8 and show existence of a Chow--K\\\"unneth decomposition. The second class of examples include the representation varieties of a finitely generated group with one relation. This is done in the setting of equivariant cohomology and equivariant Chow groups to get equivariant Chow--K\\\"unneth decompositions."}
{"category": "Math", "title": "The N-Vortex Problem on a Symmetric Ellipsoid: A Perturbation Approach", "abstract": "We consider the N-vortex problem on a ellipsoid of revolution. Applying standard techniques of classical perturbation theory we construct a sequence of conformal transformations from the ellipsoid into the complex plane. Using these transformations the equations of motion for the N-vortex problem on the ellipsoid are written as a formal series on the eccentricity of the ellipsoid's generating ellipse. First order equations are obtained explicitly. We show numerically that the truncated first order system for the three-vortices system on the symmetric ellipsoid is non-integrable."}
{"category": "Math", "title": "Dynamics of horizontal-like maps in higher dimension", "abstract": "We study the regularity of the Green currents and of the equilibrium measure associated to a horizontal-like map in C^k, under a natural assumption on the dynamical degrees. We estimate the speed of convergence towards the Green currents, the decay of correlations for the equilibrium measure and the Lyapounov exponents. We show in particular that the equilibrium measure is hyperbolic. We also show that the Green currents are the unique invariant vertical and horizontal positive closed currents. The results apply, in particular, to Henon-like maps, to regular polynomial automorphisms of C^k and to their small pertubations."}
{"category": "Math", "title": "The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field", "abstract": "Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let \\mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=\\mathbf{G}(k_v). Let \\Gamma be an arithmetic lattice in G and let C=C(\\Gamma) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for \\Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is \\hat{F}_{\\omega}, the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example \\Gamma=SL_2(\\mathcal{O}(S)), where \\mathcal{O}(S) is the ring of S-integers in k, with S=\\{v\\}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of \\Gamma on the Bruhat-Tits tree associated with G."}
{"category": "Math", "title": "Equicontinuous Geodesic Flows", "abstract": "The author shows that equicontinuous geodesic flows on surfaces are periodic. A similar result for flows on 3-manifolds is also proven. The idea of the proof is to show that the return map is recurrent and therefore periodic."}
{"category": "Math", "title": "Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume I", "abstract": "In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods."}
{"category": "Math", "title": "Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume II(a)", "abstract": "In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods."}
{"category": "Math", "title": "Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume II(b)", "abstract": "In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods."}
{"category": "Math", "title": "Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume III", "abstract": "In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods."}
{"category": "Math", "title": "Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume IV", "abstract": "In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods."}
{"category": "Math", "title": "Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume VI", "abstract": "In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods."}
{"category": "Math", "title": "Application of Groebner bases to the cup-length of oriented Grassmann manifolds", "abstract": "We determine the cup-length of some oriented Grassmann manifolds by finding a Groebner basis associated with a certain subring of the cohomology of them. As its applications, we provide not only a lower but also an upper bound for the LS-category of some oriented Grassmann manifolds. We also study the immersion problem of them."}
{"category": "Math", "title": "Area limit laws for symmetry classes of staircase polygons", "abstract": "We derive area limit laws for the various symmetry classes of staircase polygons on the square lattice, in a uniform ensemble where, for fixed perimeter, each polygon occurs with the same probability. This complements a previous study by Leroux and Rassart, where explicit expressions for the area and perimeter generating functions of these classes have been derived."}
{"category": "Math", "title": "Compactified Jacobians, Abel maps and Theta divisors", "abstract": "This is an expository paper about the topics listed in the title."}
{"category": "Math", "title": "Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume V", "abstract": "In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods."}
{"category": "Math", "title": "Crossings and nesting in tangled-diagrams", "abstract": "A tangled-diagram over $[n]=\\{1,...,n\\}$ is a graph of degree less than two whose vertices $1,...,n$ are arranged in a horizontal line and whose arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Generalizing the construction of Chen {\\it et.al.} we prove a bijection between generalized vacillating tableaux with less than $k$ rows and $k$-noncrossing tangled-diagrams and study their crossings and nestings. We show that the number of $k$-noncrossing and $k$-nonnesting tangled-diagrams are equal and enumerate tangled-diagrams."}
{"category": "Math", "title": "A Babylonian tower theorem for principal bundles over projective spaces", "abstract": "We generalise the variant of the Babylonian tower theorem for vector bundles on projective spaces proved by I. Coanda and G. Trautmann (2006) to the case of principal $G$-bundles over projective spaces, where $G$ is a linear algebraic group defined over an algebraically closed field. In course of the proofs some new insight into the structure of such principal $G$-bundles is obtained."}
{"category": "Math", "title": "Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups", "abstract": "In this version small mistakes are corrected and the exposition is changed as suggested by the referee (to appear in Canadian Journal of Mathematics). The first main result of the paper is a criterion for a partially commutative group $\\GG$ to be a domain. It allows us to reduce the study of algebraic sets over $\\GG$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\\GG$ (of a coordinate group over $\\GG$) to the elementary theories of the direct factors of $\\GG$ (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group $\\HH$. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\\HH$ has quantifier elimination and that arbitrary first-order formulas lift from $\\HH$ to $\\HH\\ast F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable."}
{"category": "Math", "title": "Slope Stability and Exceptional Divisors of High Genus", "abstract": "We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse direction we show that slope stability of surfaces can be tested with divisors, and prove that for surfaces with non-negative Kodaira dimension any destabilising divisor must have negative self-intersection and arithmetic genus at least two. We also prove that a destabilising divisor can never be nef, and as an application give an example of a surface that is slope stable but not K-stable."}
{"category": "Math", "title": "Geodesics in trees of hyperbolic and relatively hyperbolic groups", "abstract": "We present a careful approximation of the geodesics in trees of hyperbolic or relatively hyperbolic groups. As an application we prove a combination theorem for finite graphs of relatively hyperbolic groups, with both Farb's and Gromov's definitions."}
{"category": "Math", "title": "On the performance of algorithms for the minimization of $\\ell_1$-penalized functionals", "abstract": "The problem of assessing the performance of algorithms used for the minimization of an $\\ell_1$-penalized least-squares functional, for a range of penalty parameters, is investigated. A criterion that uses the idea of `approximation isochrones' is introduced. Five different iterative minimization algorithms are tested and compared, as well as two warm-start strategies. Both well-conditioned and ill-conditioned problems are used in the comparison, and the contrast between these two categories is highlighted."}
{"category": "Math", "title": "Minimal Gromov--Witten ring", "abstract": "We build the abstract theory of Gromov-Witten invariants of genus 0 for quantum minimal Fano varieties (a minimal natural (with respect to Gromov-Witten theory) class of varieties). In particular, we consider ``the minimal Gromov-Witten ring'', i. e. a commutative algebra with generators and relations of the form used in the Gromov-Witten theory of Fano variety (of unspecified dimension). Gromov-Witten theory of any quantum minimal variety is a homomorphism of this ring to $\\mathbb C$. We prove the Abstract Reconstruction Theorem which states the particular isomorphism of this ring with a free commutative ring generated by ``prime two-pointed invariants''. We also find the solutions of the differential equations of type DN for a Fano variety of dimension N in terms of generating series of one-pointed Gromov-Witten invariants."}
{"category": "Math", "title": "Solution of the polynomial moment problem", "abstract": "In this paper we give a complete solution of the following \"polynomial moment problem\" which arose about 10 years ago in connection with Poincare's center-focus problem. For a given polynomial P(z) to describe polynomials Q(z) orthogonal to all powers of P(z) on a segment [a,b]."}
{"category": "Math", "title": "Analysis and geometry on worm domains", "abstract": "We describe recent work on the Bergman kernel of the (non-smooth) worm domain in several complex variables. An asymptotic expansion is obtained for the Bergman kernel. Mapping properties of the Bergman projection are studied. Irregularity properties of the kernal at the boundary are established. This is an expository paper, and considerable background is provided. Discussion of the smooth worm is also included."}
{"category": "Math", "title": "Asymptotic vanishing conditions which force regularity in local rings of prime characteristic", "abstract": "Let $(R,\\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\\dt$ be a minimal resolution of the residue field $k$, and for each $i\\ge 0$, let $\\gothic t_i(R) = \\lim_{e\\to \\8} {\\length(H_i(F^e(G_\\dt)))}/{p^{ed}}$. We show that if $\\gothic t_i(R) = 0$ for some $i>0$, then $R$ is a regular local ring. Using the same method, we are also able to show that if $R$ is an excellent local domain and $\\Tor_i^R(k,R^+) = 0$ for some $i>0$, then $R$ is regular (where $R^+$ is the absolute integral closure of $R$). Both of the two results were previously known only for $i = 1$ or 2 via completely different methods."}
{"category": "Math", "title": "Decay of the Maxwell field on the Schwarzschild manifold", "abstract": "We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild coordinate $r$ ranges over $2M < r_1 < r < r_2$, we obtain a decay rate of $t^{-1}$ for all components of the Maxwell field. We use vector field methods and do not require a spherical harmonic decomposition. In outgoing regions, where the Regge-Wheeler tortoise coordinate is large, $r_*>\\epsilon t$, we obtain decay for the null components with rates of $|\\phi_+| \\sim |\\alpha| < C r^{-5/2}$, $|\\phi_0| \\sim |\\rho| + |\\sigma| < C r^{-2} |t-r_*|^{-1/2}$, and $|\\phi_{-1}| \\sim |\\underline{\\alpha}| < C r^{-1} |t-r_*|^{-1}$. Along the event horizon and in ingoing regions, where $r_*<0$, and when $t+r_*1$, all components (normalized with respect to an ingoing null basis) decay at a rate of $C \\uout^{-1}$ with $\\uout=t+r_*$ in the exterior region."}
{"category": "Math", "title": "Extremal problems on triangle areas in two and three dimensions", "abstract": "The study of extremal problems on triangle areas was initiated in a series of papers by Erd\\H{o}s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles. In the plane, our main result is an $O(n^{44/19}) =O(n^{2.3158})$ upper bound on the number of unit-area triangles spanned by $n$ points, which is the first breakthrough improving the classical bound of $O(n^{7/3})$ from 1992. We also make progress in a number of important special cases: We show that (i) For points in convex position, there exist $n$-element point sets that span $\\Omega(n\\log n)$ triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by $n$ points is at most ${2/3}(n^2-n)$; there exist $n$-element point sets (for arbitrarily large $n$) that span $(6/\\pi^2-o(1))n^2$ minimum-area triangles. (iii) The number of acute triangles of minimum area determined by $n$ points is O(n); this is asymptotically tight. (iv) For $n$ points in convex position, the number of triangles of minimum area is O(n); this is asymptotically tight. (v) If no three points are allowed to be collinear, there are $n$-element point sets that span $\\Omega(n\\log n)$ minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds). In 3-space we prove an $O(n^{17/7}\\beta(n))= O(n^{2.4286})$ upper bound on the number of unit-area triangles spanned by $n$ points, where $\\beta(n)$ is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, $O(n^{8/3})$, is an old result from 1971."}
{"category": "Math", "title": "Lower estimates on microstates free entropy dimension", "abstract": "By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_{1},...,X_{n}$: we show that Abstract. By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain n-tuples X_{1},...,X_{n}. In particular, we show that \\delta_{0}(X_{1},...,X_{n})\\geq\\dim_{M\\bar{\\otimes}M^{o}}V where M=W^{*}(X_{1},...,X_{n}) and V=\\{(\\partial(X_{1}),...,\\partial(X_{n})):\\partial\\in\\mathcal{C}\\} is the set of values of derivations A=\\mathbb{C}[X_{1},... X_{n}]\\to A\\otimes A with the property that \\partial^{*}\\partial(A)\\subset A. We show that for q sufficiently small (depending on n) and X_{1},...,X_{n} a q-semicircular family, \\delta_{0}(X_{1},...,X_{n})>1. In particular, for small q, q-deformed free group factors have no Cartan subalgebras. An essential tool in our analysis is a free analog of an inequality between Wasserstein distance and Fisher information introduced by Otto and Villani (and also studied in the free case by Biane and Voiculescu)."}
{"category": "Math", "title": "String Topology: Background and Present State", "abstract": "The data of a \"2D field theory with a closed string compactification\" is an equivariant chain level action of a cell decomposition of the union of all moduli spaces of punctured Riemann surfaces with each component compactified as a pseudomanifold with boundary. The axioms on the data are contained in the following assumptions. It is assumed the punctures are labeled and divided into nonempty sets of inputs and outputs. The inputs are marked by a tangent direction and the outputs are weighted by nonnegative real numbers adding to unity. It is assumed the gluing of inputs to outputs lands on the pseudomanifold boundary of the cell decomposition and the entire pseudomanifold boundary is decomposed into pieces by all such factorings. It is further assumed that the action is equivariant with respect to the toroidal action of rotating the markings. A main result of compactified string topology is the Theorem (closed strings): Each oriented smooth manifold has a 2D field theory with a closed string compactification on the equivariant chains of its free loop space mod constant loops. The sum over all surface types of the top pseudomanifold chain yields a chain X satisfying the master equation dX + X*X = 0 where * is the sum over all gluings. This structure is well defined up to homotopy. The genus zero parts yields an infinity Lie bialgebra on the equivariant chains of the free loop space mod constant loops. The higher genus terms provide further elements of algebraic structure called a \"quantum Lie bialgebra\" partially resolving the involutive identity. There is also a compactified discussion and a Theorem 2 for open strings as the first step to a more complete theory. We note a second step for knots."}
{"category": "Math", "title": "Tangle analysis of difference topology experiments: applications to a Mu protein-DNA complex", "abstract": "We develop topological methods for analyzing difference topology experiments involving 3-string tangles. Difference topology is a novel technique used to unveil the structure of stable protein-DNA complexes involving two or more DNA segments. We analyze such experiments for the Mu protein-DNA complex. We characterize the solutions to the corresponding tangle equations by certain knotted graphs. By investigating planarity conditions on these graphs we show that there is a unique biologically relevant solution. That is, we show there is a unique rational tangle solution, which is also the unique solution with small crossing number."}
{"category": "Math", "title": "Unsigned state models for the Jones polynomial", "abstract": "It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in the 3-sphere there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating the Jones and Bollobas-Riordan polynomials and show they arise from two different interpretations of the same embedded graph."}
{"category": "Math", "title": "Hook-content formulae for symplectic and orthogonal tableaux", "abstract": "By considering the specialisation $s_{\\lambda}(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $\\lambda$ in terms of two properties of the boxes in the diagram for $\\lambda$. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $\\lambda$-tableaux."}
{"category": "Math", "title": "A note on plurisubharmonic defining functions in $\\mathbb{C}^n$", "abstract": "Let D be a smoothly bounded domain in complex space of dimension larger than 2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis."}
{"category": "Math", "title": "Interference Cancelation in Coherent CDMA Systems Using Parallel Iterative Algorithms", "abstract": "Least mean square-partial parallel interference cancelation (LMS-PPIC) is a partial interference cancelation using adaptive multistage structure in which the normalized least mean square (NLMS) adaptive algorithm is engaged to obtain the cancelation weights. The performance of the NLMS algorithm is mostly dependent to its step-size. A fixed and non-optimized step-size causes the propagation of error from one stage to the next one. When all user channels are balanced, the unit magnitude is the principal property of the cancelation weight elements. Based on this fact and using a set of NLMS algorithms with different step-sizes, the parallel LMS-PPIC (PLMS-PPIC) method is proposed. In each iteration of the algorithm, the parameter estimate of the NLMS algorithm is chosen to match the elements' magnitudes of the cancelation weight estimate with unity. Simulation results are given to compare the performance of our method with the LMS-PPIC algorithm in three cases: balanced channel, unbalanced channel and time varying channel."}
{"category": "Math", "title": "Interference Cancelation in Non-coherent CDMA Systems Using Parallel Iterative Algorithms", "abstract": "Parallel least mean square-partial parallel interference cancelation (PLMS-PPIC) is a partial interference cancelation which employs adaptive multistage structure. In this algorithm the channel phases for all users are assumed to be known. Having only their quarters in (0,2\\pi), a modified version of PLMS-PPIC is proposed in this paper to simultaneously estimate the channel phases and the cancelation weights. Simulation examples are given in the cases of balanced, unbalanced and time varying channels to show the performance of the modified PLMS-PPIC method."}
{"category": "Math", "title": "Full MIMO Channel Estimation Using A Simple Adaptive Partial Feedback Method", "abstract": "Partial feedback in multiple-input multiple-output (MIMO) communication systems provides tremendous capacity gain and enables the transmitter to exploit channel condition and to eliminate channel interference. In the case of severely limited feedback, constructing a quantized partial feedback is an important issue. To reduce the computational complexity of the feedback system, in this paper we introduce an adaptive partial method in which at the transmitter, an easy to implement least square adaptive algorithm is engaged to compute the channel state information. In this scheme at the receiver, the time varying step-size is replied to the transmitter via a reliable feedback channel. The transmitter iteratively employs this feedback information to estimate the channel weights. This method is independent of the employed space-time coding schemes and gives all channel components. Simulation examples are given to evaluate the performance of the proposed method."}
{"category": "Math", "title": "Structural Conditions for Full MHD Equations", "abstract": "In this paper, we investigate the characteristic structure of the full equations of magnetohydrodynamics (MHD) and show that it satisfies the hypotheses of a general variable-multiplicity stability frame- work introduced by Metivier and Zumbrun, thereby extending to the general case various results obtained by Metivier and Zumbrun for the isentropic equations of MHD."}
{"category": "Math", "title": "Integral means spectrum of random conformal snowflakes", "abstract": "In this paper we construct random conformal snowflakes with large integral means spectrum at different points. These new estimates are significant improvement over previously known lower bound of the universal spectrum. Our estimates are within 5-10 percent from the conjectured value of the universal spectrum."}
{"category": "Math", "title": "Hochschild and cyclic (co)homology of preprojective algebras of quivers of type T", "abstract": "We calculate the additive and multiplicative structure (together with the grading) of the Hochschild homology and cohomology and the cyclic homology of preprojective algebras of types T. We also compute the calculus structure which is formed by the Hochschild homology/cohomology pair."}
{"category": "Math", "title": "Parameter estimation of ODE's via nonparametric estimators", "abstract": "Ordinary differential equations (ODE's) are widespread models in physics, chemistry and biology. In particular, this mathematical formalism is used for describing the evolution of complex systems and it might consist of high-dimensional sets of coupled nonlinear differential equations. In this setting, we propose a general method for estimating the parameters indexing ODE's from times series. Our method is able to alleviate the computational difficulties encountered by the classical parametric methods. These difficulties are due to the implicit definition of the model. We propose the use of a nonparametric estimator of regression functions as a first-step in the construction of an M-estimator, and we show the consistency of the derived estimator under general conditions. In the case of spline estimators, we prove asymptotic normality, and that the rate of convergence is the usual $\\sqrt{n}$-rate for parametric estimators. Some perspectives of refinements of this new family of parametric estimators are given."}
{"category": "Math", "title": "On the Hodge-Newton filtration for p-divisible O-modules", "abstract": "The notions Hodge-Newton decomposition and Hodge-Newton filtration for F-crystals are due to Katz and generalize Messing's result on the existence of the local-\\'etale filtration for p-divisible groups. Recently, some of Katz's classical results have been generalized by Kottwitz to the context of F-crystals with additional structures and by Moonen to $\\mu$-ordinary p-divisible groups. In this paper, we discuss further generalizations to the situation of crystals in characteristic p and of p-divisible groups with additional structure by endomorphisms."}
{"category": "Math", "title": "Simple helices on Fano threefolds", "abstract": "Building on the work of Nogin \\cite{Nogin}, we prove that the braid group $B_4$ acts transitively on full exceptional collections of vector bundles on Fano threefolds with $b_2=1$ and $b_3=0$. Equivalently, this group acts transitively on the set of simple helices (considered up to a shift in the derived category) on such a Fano threefold. We also prove that on threefolds with $b_2=1$ and very ample anticanonical class, every exceptional coherent sheaf is locally free."}
{"category": "Math", "title": "Low degree bounded cohomology and l^2-invariants for negatively curved groups", "abstract": "We study the subgroup structure of discrete groups which share cohomological properties which resemble non-negative curvature. Examples include all Gromov hyperbolic groups. We provide strong restrictions on the possible s-normal subgroups of a Gromov hyperbolic group, or more generally a 'negatively curved' group. Another result says that the image of a group, which is boundedly generated by a finite set of amenable subgroups, in a Gromov hyperbolic group has to be virtually cyclic. Moreover, we show that any homomorphic image of an analogue of a higher rank lattices in a Gromov hyperbolic group must be finite. These results extend to a certain class of randomorphisms in the sense of Monod. We study the class of groups which admit proper quasi-1-cocycles and show that it is closed under l2-orbit equivalence."}
{"category": "Math", "title": "Chow motives of universal families over some Shimura surfaces", "abstract": "We prove an absolute Chow-Kuenneth decomposition for the motive of universal families A of abelian varieties over some compact Shimura surface. We furthermore prove the Hodge conjecture for general fibres of A, extending results of Ribet."}
{"category": "Math", "title": "Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences", "abstract": "Let $(X_i)_{i=1,...,n}$ be a possibly nonstationary sequence such that $\\mathscr{L}(X_i)=P_n$ if $i\\leq n\\theta$ and $\\mathscr{L}(X_i)=Q_n$ if $i>n\\theta$, where $0<\\theta <1$ is the location of the change-point to be estimated. We construct a class of estimators based on the empirical measures and a seminorm on the space of measures defined through a family of functions $\\mathcal{F}$. We prove the consistency of the estimator and give rates of convergence under very general conditions. In particular, the $1/n$ rate is achieved for a wide class of processes including long-range dependent sequences and even nonstationary ones. The approach unifies, generalizes and improves on the existing results for both parametric and nonparametric change-point estimation, applied to independent, short-range dependent and as well long-range dependent sequences."}
{"category": "Math", "title": "Arithmetic of rationaly connected quintic 3-folds over finite and function fields", "abstract": "We prove that both classical Chevalley-Warning-Ax and Tsen theorems hold for the blowing up of a quintic 3-fold along a line of multiplicity 3. Both proofs, which are of the same spirit than the original ones, involve the description of this blowing-up as a subvariety of a toric variety."}
{"category": "Math", "title": "The dynamical rigid body with memory", "abstract": "In the present paper we describe the dynamics of the revised rigid body, the dynamics of the rigid body with distributed delays and the dynamics of the fractional rigid body. We analyze the stationary states for given values of the rigid body's parameters."}
{"category": "Math", "title": "Action of Hecke operators on Siegel theta series II", "abstract": "Given a Siegel theta series and a prime p not dividing the level of the theta series, we apply to the theta series the n+1 Hecke operators that generate the local Hecke algebra at p. We show that the average theta series is an eigenform and we compute the eigenvalues."}
{"category": "Math", "title": "Hecke operators on Hilbert-Siegel modular forms", "abstract": "We define Hilbert-Siegel modular forms and Hecke \"operators\" acting on them. As with Hilbert modular forms, these linear transformations are not linear operators until we consider a direct product of spaces of modular forms (with varying groups), modulo natural identifications we can make between certain spaces. With Hilbert-Siegel forms we identify several families of natural identifications between certain spaces of modular forms. We associate the Fourier coefficients of a form in our product space to even integral lattices, independent of a basis and choice of coefficient rings. We then determine the action of the Hecke operators on these Fourier coefficients, paralleling the result of Hafner and Walling for Siegel modular forms (where the number field is the field of rationals)."}
{"category": "Math", "title": "Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical model", "abstract": "Inference for Dirichlet process hierarchical models is typically performed using Markov chain Monte Carlo methods, which can be roughly categorised into marginal and conditional methods. The former integrate out analytically the infinite-dimensional component of the hierarchical model and sample from the marginal distribution of the remaining variables using the Gibbs sampler. Conditional methods impute the Dirichlet process and update it as a component of the Gibbs sampler. Since this requires imputation of an infinite-dimensional process, implementation of the conditional method has relied on finite approximations. In this paper we show how to avoid such approximations by designing two novel Markov chain Monte Carlo algorithms which sample from the exact posterior distribution of quantities of interest. The approximations are avoided by the new technique of retrospective sampling. We also show how the algorithms can obtain samples from functionals of the Dirichlet process. The marginal and the conditional methods are compared and a careful simulation study is included, which involves a non-conjugate model, different datasets and prior specifications."}
{"category": "Math", "title": "Trees, linear orders and G\\^ateaux smooth norms", "abstract": "We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a G\\^ateaux smooth norm on C(T), where T is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fr\\'echet smooth norms. In addition, we prove that if C(T) admits a G\\^ateaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm."}
{"category": "Math", "title": "Hamiltonian stationary Lagrangian tori contained in a hypersphere", "abstract": "The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using the Hopf fibration. In this paper, all Hamiltonian stationary Lagrangian tori which is contained in a hypersphere in the complex Euclidean plane are constructed explicitly. Then it is shown that they are homogeneous tori. For the construction, flat quaternionic connections of Hamiltonian stationary Lagrangian tori are considered and a spectral curve of an associated family of them is used."}
{"category": "Math", "title": "Stability of the Gibbs Sampler for Bayesian Hierarchical Models", "abstract": "We characterise the convergence of the Gibbs sampler which samples from the joint posterior distribution of parameters and missing data in hierarchical linear models with arbitrary symmetric error distributions. We show that the convergence can be uniform, geometric or sub-geometric depending on the relative tail behaviour of the error distributions, and on the parametrisation chosen. Our theory is applied to characterise the convergence of the Gibbs sampler on latent Gaussian process models. We indicate how the theoretical framework we introduce will be useful in analyzing more complex models."}
{"category": "Math", "title": "Adaptive Importance Sampling in General Mixture Classes", "abstract": "In this paper, we propose an adaptive algorithm that iteratively updates both the weights and component parameters of a mixture importance sampling density so as to optimise the importance sampling performances, as measured by an entropy criterion. The method is shown to be applicable to a wide class of importance sampling densities, which includes in particular mixtures of multivariate Student t distributions. The performances of the proposed scheme are studied on both artificial and real examples, highlighting in particular the benefit of a novel Rao-Blackwellisation device which can be easily incorporated in the updating scheme."}
{"category": "Math", "title": "Particle Filters for Partially Observed Diffusions", "abstract": "In this paper we introduce a novel particle filter scheme for a class of partially-observed multivariate diffusions. %continuous-time dynamic models where the %signal is given by a multivariate diffusion process. We consider a variety of observation schemes, including diffusion observed with error, observation of a subset of the components of the multivariate diffusion and arrival times of a Poisson process whose intensity is a known function of the diffusion (Cox process). Unlike currently available methods, our particle filters do not require approximations of the transition and/or the observation density using time-discretisations. Instead, they build on recent methodology for the exact simulation of the diffusion process and the unbiased estimation of the transition density as described in \\cite{besk:papa:robe:fear:2006}. %In particular, w We introduce the Generalised Poisson Estimator, which generalises the Poisson Estimator of \\cite{besk:papa:robe:fear:2006}. %Thus, our filters avoid the systematic biases caused by %time-discretisations and they have significant computational %advantages over alternative continuous-time filters. These %advantages are supported theoretically by a A central limit theorem is given for our particle filter scheme."}
{"category": "Math", "title": "There are non homotopic framed homotopies of long knots", "abstract": "Let $\\mathcal {M}$ be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let $cl(\\Sigma^{(1)}_{iness})$ be the closure of the union of all singular knots in $\\mathcal {M}$ with exactly one ordinary double point and such that the two resolutions represent the same (non singular) knot type. We call $\\Sigma^{(1)}_{iness}$ the {\\em inessential walls} and we call $\\mathcal {M}_{ess} = \\mathcal {M} \\setminus cl(\\Sigma^{(1)}_{iness})$ the {\\em essential diagram space}. We construct a non trivial class in $H^1(\\mathcal {M}_{ess}; \\mathbb{Z}[A, A^{-1}])$ by an extension of the Kauffman bracket. This implies in particular that there are loops in $\\mathcal {M}_{ess}$ which consist of regular isotopies of knots together with crossing changings and which are not contractible in $\\mathcal {M}_{ess}$ (leading to the title of the paper). We conjecture that our construction gives rise to a new knot polynomial for knots of unknotting number one."}
{"category": "Math", "title": "Ergodicity of Langevin Processes with Degenerate Diffusion in Momentums", "abstract": "This paper introduces a geometric method for proving ergodicity of degenerate noise driven stochastic processes. The driving noise is assumed to be an arbitrary Levy process with non-degenerate diffusion component (but that may be applied to a single degree of freedom of the system). The geometric conditions are the approximate controllability of the process the fact that there exists a point in the phase space where the interior of the image of a point via a secondarily randomized version of the driving noise is non void. The paper applies the method to prove ergodicity of a sliding disk governed by Langevin-type equations (a simple stochastic rigid body system). The paper shows that a key feature of this Langevin process is that even though the diffusion and drift matrices associated to the momentums are degenerate, the system is still at uniform temperature."}
{"category": "Math", "title": "Diophantine approximation, Khintchine's theorem, torus geometry and Hausdorff dimension", "abstract": "A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a relatively direct proof of the planar Groshev theorem for the set of $\\psi$-approximable points in the plane. The construction and use of Haudsorff measure and dimension are explained and the notion of ubiquity, which is effective in estimating the lower bound of the Hausdorff dimension for quite general lim sup sets, is described. An application is made to obtain the Hausdorff dimension of the set of approximable points in the plane when $\\psi(q)=q^{-v}$, $v>0$, corresponding to the planar Jarnik-Besicovich theorem."}
{"category": "Math", "title": "Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs", "abstract": "We define 2-decompositions of ribbon graphs, which generalise 2-sums and tensor products of graphs. We give formulae for the Bollobas-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams."}
{"category": "Math", "title": "Finite Field Experiments (with an Appendix by Stefan Wiedmann)", "abstract": "We explain how to use computer experiments over finite fields to gain heuristic information about the solution set of polynomial equations in characteristic zero. These are notes of a tutorial I gave at the NATO Advanced Study Institute on Higher-Dimensional Geometry over Finite Fields in G\"ottingen 2007."}
{"category": "Math", "title": "Spin-c Prequantization and Symplectic Cutting", "abstract": "We define spin-c prequantization of a symplectic manifold to be a spin-c structure and a connection which are compatible with the symplectic form. We describe the cutting of an S^1-equivariant spin-c prequantization. The cutting process involves a choice of a spin-c prequantization for the complex plane. We prove that the cutting is possible if and only if the moment map level set along which the cutting is done is compatible with this choice."}
{"category": "Math", "title": "Numerical obstructions to abelian surfaces in toric Fano 4-folds", "abstract": "Some of the 124 toric Fano 4-folds contain abelian surfaces but most do not: in a few cases it is not known whether they do or not. By elementary methods, with a little computer help, we exclude some more possibilities."}
{"category": "Math", "title": "Finite groups with an automorphism cubing a large fraction of elements", "abstract": "We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or have an abelian subgroup of index at most 2. For our second theorem we show that, if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A_5 shows that this result is best possible. Both our main findings closely parallel results of previous authors on finite groups possessing an automorphism which inverts many group elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations."}
{"category": "Math", "title": "A Class of Monotonic Quantities along the Ricci Flow", "abstract": "We construct a class of monotonic quantities along the normalized Ricci flow on closed n-dimensional manifolds."}
{"category": "Math", "title": "Partial Differential system in two variables with $W(D_6^{(1)})$-symmetry and the Garnier system in two variables", "abstract": "In this note, we will compare the Garnier system in two variables with four-dimensional partial differential system in two variables with $W(D_6^{(1)})$-symmetry. Both systems are different in each compactification in the variables $q_1,q_2$, however, has same five holomorphy conditions in the variables $p_1,p_2$."}
{"category": "Math", "title": "Odd Khovanov homology", "abstract": "We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There is a reduced version which is a link invariant whose graded Euler characteristic is the normalized Jones polynomial."}
{"category": "Math", "title": "Theorie homotopique des DG-categories", "abstract": "In this thesis we present several original contributions to the study of: - DG categories and their invariants; - Neeman's well-generated (algebraic) triangulated categories; - Fomin-Zelevinsky's cluster algebras approach via representation theory."}
{"category": "Math", "title": "The quermassintegral inequalities for starshaped domains", "abstract": "We give a simple proof of the insoperimetric inequality for quermassintegrals of non-convex starshaped domains, using a reslut of Gerhardt \\cite{G} and Urbas \\cite{U} on an expanding geometric curvature flow."}
{"category": "Math", "title": "Twisting on associative algebras and Rota-Baxter type operators", "abstract": "We will introduce an operation \"twisting\" on Hochschild complex by analogy with Drinfeld's twisting operations. By using the twisting and derived bracket construction, we will study differential graded Lie algebra structures associated with bi-graded Hochschild complex. We will show that Rota-Baxter type operators are solutions of Maurer-Cartan equations. As an application of twisting, we will give a construction of associative Nijenhuis operators."}
{"category": "Math", "title": "One dimensional conformal metric flow II", "abstract": "In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in [8]. In this part we mainly focus on evolution equations involving fourth order derivatives. The global existence and exponential convergence of metrics for the 1-Q and 4-Q flows are obtained."}
{"category": "Math", "title": "Liouville energy on a topological two sphere", "abstract": "In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below. Our proof does not rely on the uniformization theorem and the Onofri inequality, thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow. Such an analytic approach also sheds light on how to obtain the boundedness for E_1 energy in the study of general K\\\"ahler manifolds."}
{"category": "Math", "title": "A sharp inequality and its applications", "abstract": "We establish an analog Hardy inequality with sharp constant involving exponential weight function. The special case of this inequality (for n=2) leads to a direct proof of Onofri inequality on S^2."}
{"category": "Math", "title": "Some Asymptotic Behavior of the first Eigenvalue along the Ricci Flow", "abstract": "We study some asymptotic behavior of the first nonzero eigenvalue of the Lalacian along the normalized Ricci flow and give a direct short proof for an asymptotic upper limit estimate."}
{"category": "Math", "title": "On tilting modules over cluster-tilted algebras", "abstract": "In this paper, we show that the tilting modules over a cluster-tilted algebra $A$ lift to tilting objects in the associated cluster category $\\mathcal{C}_H$. As a first application, we describe the induced exchange relation for tilting $A$-modules arising from the exchange relation for tilting object in $\\mathcal{C}_H$. As a second application, we exhibit tilting $A$-modules having cluster-tilted endomorphism algebras."}
{"category": "Math", "title": "Denominators of cluster variables", "abstract": "Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional indecomposable objects in the cluster category inducing a correspondence between clusters and cluster-tilting objects. Fix a cluster-tilting object T and a corresponding initial cluster. By the Laurent phenomenon, every cluster variable can be written as a Laurent polynomial in the initial cluster. We give conditions on T equivalent to the fact that the denominator in the reduced form for every cluster variable in the cluster algebra has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T."}
{"category": "Math", "title": "Differential characters as stacks and prequantization", "abstract": "We generalize geometric prequantization of symplectic manifolds to differentiable stacks. Our approach is atlas-independent and provides a bijection between isomorphism classes of principal circle bundles (with or without connections) and second cohomology groups of certain chain complexes."}
{"category": "Math", "title": "Ehrhart polynomials of matroid polytopes and polymatroids", "abstract": "We investigate properties of Ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that for fixed rank their Ehrhart polynomials are computable in polynomial time. The proof relies on the geometry of these polytopes as well as a new refined analysis of the evaluation of Todd polynomials. In the second half we discuss two conjectures about the h^*-vector and the coefficients of Ehrhart polynomials of matroid polytopes; we provide theoretical and computational evidence for their validity."}
{"category": "Math", "title": "Multiplicative bijections of semigroups of interval-valued continuous functions", "abstract": "We characterize all compact and Hausdorff spaces $X$ which satisfy that for every multiplicative bijection $\\phi$ on $C(X, I)$, there exist a homeomorphism $\\mu : X \\to X$ and a continuous map $p: X \\to (0, +\\infty)$ such that $$\\phi (f) (x) = f(\\mu (x))^{p(x)}$$ for every $f \\in C(X,I)$ and $x \\in X$. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. {\\bf 134} (2006), 1065-1075). Some related results on other semigroups of functions are also given."}
{"category": "Math", "title": "Notes on axiomatic Gromov--Witten theory and applications", "abstract": "This an expository article on Givental's axiomatic Gromov--Witten theory and some of its applications."}
{"category": "Math", "title": "Geometrization of 3-dimensional Coxeter orbifolds and Singer's conjecture", "abstract": "Associated to any Coxeter system $(W,S)$, there is a labeled simplicial complex $L$ and a contractible CW-complex $\\Sigma_L$ (the Davis complex) on which $W$ acts properly and cocompactly. $\\Sigma_L$ admits a cellulation under which the nerve of each vertex is $L$. It follows that if $L$ is a triangulation of $\\mathbb{S}^{n-1}$, then $\\Sigma_L$ is a contractible $n$-manifold. In this case, the orbit space, $K_L:=\\Sigma_L/W$, is a \\emph{Coxeter orbifold}. We prove a result analogous to the JSJ-decomposition for 3-dimensional manifolds: Every 3-dimensional Coxeter orbifold splits along Euclidean suborbifolds into the \\emph{characteristic suborbifold} and simple (hyperbolic) pieces. It follows that every 3-dimensional Coxeter orbifold has a decomposition into pieces which have hyperbolic, Euclidean, or the geometry of $\\mathbb{H}^2\\times\\mathbb{R}$. (We leave out the case of spherical Coxeter orbifolds.) A version of Singer's conjecture in dimension 3 follows: That the reduced $\\ell^2$-homology of $\\Sigma_L$ vanishes."}
{"category": "Math", "title": "Conformally parametrized surfaces associated with CP^(N-1) sigma models", "abstract": "Two-dimensional conformally parametrized surfaces immersed in the su(N) algebra are investigated. The focus is on surfaces parametrized by solutions of the equations for the CP^(N-1) sigma model. The Lie-point symmetries of the CP^(N-1) model are computed for arbitrary N. The Weierstrass formula for immersion is determined and an explicit formula for a moving frame on a surface is constructed. This allows us to determine the structural equations and geometrical properties of surfaces in R^(N^2-1). The fundamental forms, Gaussian and mean curvatures, Willmore functional and topological charge of surfaces are given explicitly in terms of any holomorphic solution of the CP^2 model. The approach is illustrated through several examples, including surfaces immersed in low-dimensional su(N) algebras."}
{"category": "Math", "title": "Topological Hochschild homology of l and ko", "abstract": "We calculate the integral homotopy groups of THH(l) at any prime and of THH(ko) at p=2, where l is the Adams summand of the connective complex p-local K-theory spectrum and ko is the connective real K-theory spectrum."}
{"category": "Math", "title": "Bergman kernels and equilibrium measures for line bundles over projective manifolds", "abstract": "Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels (i.e. orthogonal projection kernels). The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h_e associated to h, as well as in terms of the Monge-Ampere measure of h on a certain support set. It is also shown that the equilibrium metric h_e is in the class C^{1,1} on the complement of the augmented base locus of L. For L ample these results give generalizations of well-known results concerning the case when the curvature of h is globally positive (then h_e=h). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of L."}
{"category": "Math", "title": "Castelnuovo-Mumford regularity and reduction number of smooth monomial curves", "abstract": "We compare, for smooth monomial projective curves, the Castel- nuovo-Mumford regularity and the reduction number; we present an example where these two numbers differ. However, we show they coin- cide for a certain class of monomial curves. Furthermore, for smooth monomial curves we prove an inequality which is stronger than the one from the Eisenbud-Goto conjecture."}
{"category": "Math", "title": "Solution and Asymptotic Behavior for a Nonlocal Coupled System of Reaction-Diffusion", "abstract": "This paper concerns with existence, uniqueness and asymptotic behavior of the solutions for a nonlocal coupled system of reaction-diffusion. We prove the existence and uniqueness of weak solutions by the Faedo-Galerkin method and exponential decay of solutions by the classic energy method. We improve the results obtained by Chipot-Lovato and Menezes for coupled systems. A numerical scheme is presented."}
{"category": "Math", "title": "Isotopies of Legendrian 1-knots and Legendrian 2-tori", "abstract": "We construct a Legendrian 2-torus in the 1-jet space of $S^1\\times\\R$ (or of $\\R^2$) from a loop of Legendrian knots in the 1-jet space of $\\R$. The differential graded algebra (DGA) for the Legendrian contact homology of the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit any augmentation (linearization) but which still has non-trivial homology, as well as two Legendrian tori with isomorphic linearized contact homologies but with distinct contact homologies."}
{"category": "Math", "title": "Some matrices associated with the split decomposition for a Q-polynomial distance-regular graph", "abstract": "We consider a $Q$-polynomial distance-regular graph $\\Gamma$ with vertex set $X$ and diameter $D \\geq 3$. For $\\mu, \\nu \\in \\lbrace \\downarrow, \\uparrow \\rbrace$ we define a direct sum decomposition of the standard module $V=\\C X$, called the $(\\mu,\\nu)$--split decomposition. For this decomposition we compute the complex conjugate and transpose of the associated primitive idempotents. Now fix $b,\\beta \\in \\mathbb C$ such that $b \\neq 1$ and assume $\\Gamma$ has classical parameters $(D,b,\\alpha,\\beta)$ with $\\alpha = b-1$. Under this assumption Ito and Terwilliger displayed an action of the $q$-tetrahedron algebra $\\boxtimes_q$ on the standard module of $\\Gamma$. To describe this action they defined eight matrices in $\\hbox{Mat}_X(\\mathbb C)$, called \\begin{eqnarray*} \\label{eq:list} A,\\quad A^*,\\quad B,\\quad B^*, \\quad K,\\quad K^*,\\quad \\Phi,\\quad \\Psi. \\end{eqnarray*} For each matrix in the above list we compute the transpose and complex conjugate. Using this information we compute the transpose and complex conjugate for each generator of $\\boxtimes_q$ on $V$."}
{"category": "Math", "title": "Non-measurable automorphisms of Lie groups relative to the real- and non-archimedean-valued measures", "abstract": "In this work the problem about an existence of non-measurable automorphisms of Lie groups finite and as well infinite dimensional over the field of real numbers and also over the non-archimedean local fields is investigated. Non-measurability of automorphisms is considered relative to real-valued measures and also measures with values in non-archimedean local fields. Their existence is proved and a procedure for their construction is given. Their application for a construction of non-measurable irreducible unitary representations is demonstrated."}
{"category": "Math", "title": "Dynamic importance sampling for queueing networks", "abstract": "Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The standard approach, which simulates the system using an a priori fixed change of measure suggested by large deviation analysis, has been shown to fail in even the simplest network setting (e.g., a two-node tandem network). Exploiting connections between importance sampling, differential games, and classical subsolutions of the corresponding Isaacs equation, we show how to design and analyze simple and efficient dynamic importance sampling schemes for general classes of networks. The models used to illustrate the approach include $d$-node tandem Jackson networks and a two-node network with feedback, and the rare events studied are those of large queueing backlogs, including total population overflow and the overflow of individual buffers."}
{"category": "Math", "title": "Kernel estimation of Greek weights by parameter randomization", "abstract": "A Greek weight associated to a parameterized random variable $Z(\\lambda)$ is a random variable $\\pi$ such that $\\nabla_{\\lambda}E[\\phi(Z(\\lambda))]=E[\\phi(Z(\\lambda))\\pi]$ for any function $\\phi$. The importance of the set of Greek weights for the purpose of Monte Carlo simulations has been highlighted in the recent literature. Our main concern in this paper is to devise methods which produce the optimal weight, which is well known to be given by the score, in a general context where the density of $Z(\\lambda)$ is not explicitly known. To do this, we randomize the parameter $\\lambda$ by introducing an a priori distribution, and we use classical kernel estimation techniques in order to estimate the score function. By an integration by parts argument on the limit of this first kernel estimator, we define an alternative simpler kernel-based estimator which turns out to be closely related to the partial gradient of the kernel-based estimator of $\\mathbb{E}[\\phi(Z(\\lambda))]$. Similarly to the finite differences technique, and unlike the so-called Malliavin method, our estimators are biased, but their implementation does not require any advanced mathematical calculation. We provide an asymptotic analysis of the mean squared error of these estimators, as well as their asymptotic distributions. For a discontinuous payoff function, the kernel estimator outperforms the classical finite differences one in terms of the asymptotic rate of convergence. This result is confirmed by our numerical experiments."}
{"category": "Math", "title": "The base components of the dualizing sheaf of a curve on a surface", "abstract": "This note studies the structure of the divisorial fixed part of the dualizing sheaf of a 1-connected curve D on a smooth surface S. It is shown that if the divisorial fixed part F of the dualizing sheaf is non empty then it has arithmetic genus<1 and each component of F is a smooth rational curve."}
{"category": "Math", "title": "A general dynamical statistical model with possible causal interpretation", "abstract": "We develop a general dynamical model as a framework for possible causal interpretation. We first state a criterion of local independence in terms of measurability of processes involved in the Doob-Meyer decomposition of stochastic processes, as in Aalen (1987); then we define direct and indirect influence. We propose a definition of causal influence using the concepts of ``physical system''. This framework makes it possible to link descriptive and explicative statistical models, and encompasses quantitative processes and events. One of the features of this paper is the clear distinction between the model for the system and the model for the observation. We give a dynamical representation of a conventional joint model for HIV load and CD4 counts. We show its inadequacy to capture causal influences while on the contrary known mechanisms of HIV infection can be expressed directly through a system of differential equations."}
{"category": "Math", "title": "Attrition and Non-Response in Panel Data: The Case of the Canadian Survey of Labor and Income Dynamics", "abstract": "This paper provides an analysis of the effects of attrition and non-response on employment and wages using the Canadian Survey of Labour and Income Dynamics. We consider a structural model composed of three freely correlated equations for nonattrition/response, employment and wages. The model is estimated using microdata from 22,990 individuals who provided sufficient information in the first wave of the 1996-2001 panel. The main findings of the paper are that attrition is not random. Attritors and non-respondents likely are less attached to employment and come from low-income population. The correlation between non-attrition and employment is positive and statistically significant, though small. Also, wage estimates are biased upwards. Observed wages are on average higher than wages that would be observed if all the individuals initially selected in the panel remained in the sample."}
{"category": "Math", "title": "Rigidity and non local connectivity of Julia sets of some quadratic polynomials", "abstract": "For an infinitely renormalizable quadratic map $f_c: z\\mapsto z^2+c$ with the sequence of renormalization periods ${k_m}$ and rotation numbers ${t_m=p_m/q_m}, we prove that if $\\limsup k_m^{-1}\\log |p_m|>0$, then the Mandelbrot set is locally connected at $c$. We prove also that if $\\limsup |t_{m+1}|^{1/q_m}<1$ and $q_m\\to \\infty$, then the Julia set of $f_c$ is not locally connected and the Mandelbrot set is locally connected at $c$ provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor. Abstract of the Addendum: We improve one of the main results of the above paper."}
{"category": "Math", "title": "The sharp lower bound for the volume of 3-folds of general type with \\chi(\\Co{X})=1", "abstract": "Let $V$ be a smooth projective 3-fold of general type. Denote by $K^{3}$, a rational number, the self-intersection of the canonical sheaf of any minimal model of $V$. One defines $K^{3}$ as a canonical volume of $V$. The paper is devoted to proving the sharp lower bound $K^{3}\\ge {1/420}$ which can be reached by an example: $X_{46}\\subseteq \\mathbb{P}(4,5,6,7,23)$."}
{"category": "Math", "title": "A remark on odd dimensional normalized Ricci flow", "abstract": "Let $(M^n,g_0)$ ($n$ odd) be a compact Riemannian manifold with $\\lambda(g_0)>0$, where $\\lambda(g_0)$ is the first eigenvalue of the operator $-4\\Delta_{g_0}+R(g_0)$, and $R(g_0)$ is the scalar curvature of $(M^n,g_0)$. Assume the maximal solution $g(t)$ to the normalized Ricci flow with initial data $(M^n,g_0)$ satisfies $|R(g(t))| \\leq C$ and $\\int_M |Rm(g(t))|^{n/2}d\\mu_t \\leq C$ uniformly for a constant $C$. Then we show that the solution sub-converges to a shrinking Ricci soliton. Moreover,when $n=3$, the condition $\\int_M |Rm(g(t))|^{n/2}d\\mu_t \\leq C$ can be removed."}
{"category": "Math", "title": "Valuations for matroid polytope subdivisions", "abstract": "We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes."}
{"category": "Math", "title": "Cohomological characterization of relative hyperbolicity and combination theorem", "abstract": "We give a cohomological characterization of Gromov relative hyperbolicity. As an application we prove a converse to the combination theorem for graphs of relatively hyperbolic groups given in a previous paper of the first author. We build upon, and follow the ideas of, the work of S. Gersten in ``Cohom. lower bounds for isoperim. funct. on groups'' (Topology 37, 1998) about the same topics in the classical Gromov hyperbolic setting."}
{"category": "Math", "title": "On the disconnection of a discrete cylinder by a biased random walk", "abstract": "We consider a random walk on the discrete cylinder $({\\mathbb{Z}}/N{\\mathbb{Z}})^d\\times{\\mathbb{Z}}$, $d\\geq3$ with drift $N^{-d\\alpha}$ in the $\\mathbb{Z}$-direction and investigate the large $N$-behavior of the disconnection time $T^{\\mathrm{disc}}_N$, defined as the first time when the trajectory of the random walk disconnects the cylinder into two infinite components. We prove that, as long as the drift exponent $\\alpha$ is strictly greater than 1, the asymptotic behavior of $T^{\\mathrm{disc}}_N$ remains $N^{2d+o(1)}$, as in the unbiased case considered by Dembo and Sznitman, whereas for $\\alpha<1$, the asymptotic behavior of $T^{\\mathrm{disc}}_N$ becomes exponential in $N$."}
{"category": "Math", "title": "Homotopy dimension of orbits of Morse functions on surfaces", "abstract": "Let $f$ be a real- or circle-valued Morse function on a compact surface M having exactly $n>0$ critical points. Denote by $O$ the orbit of $f$ with respect to the right action of the group of diffeomorphisms of $M$. We show that the connected components of $O$ have the homotopy type of a finite-dimensional CW-complex. Actually, these connected components are homotopy equivalent to a certain covering space of the $n$-th configuration space of the interior of $M$. As a consequence we obtain that the fundamental group of $O$ is a subgroup of the $n$-th braid group of $M$."}
{"category": "Math", "title": "Calabi-Yau cones from contact reduction", "abstract": "We consider a generalization of Einstein-Sasaki manifolds, which we characterize in terms both of spinors and differential forms, that in the real analytic case corresponds to contact manifolds whose symplectic cone is Calabi-Yau. We construct solvable examples in seven dimensions. Then, we consider circle actions that preserve the structure, and determine conditions for the contact reduction to carry an induced structure of the same type. We apply this construction to obtain a new hypo-contact structure on S^2\\times T^3."}
{"category": "Math", "title": "Non-rational configurations, polytopes, and surfaces", "abstract": "It is an amazing and a bit counter-intuitive discovery by Micha Perles from the sixties that there are ``non-rational polytopes'': combinatorial types of convex polytopes that cannot be realized with rational vertex coordinates. We describe a simple construction of non-rational polytopes that does not need duality (Perles' ``Gale diagrams''): It starts from a non-rational point configuration in the plane, and proceeds with so-called Lawrence extensions. We also show that there are non-rational polyhedral surfaces in 3-space, a discovery by Ulrich Brehm from 1997. His construction also starts from any non-rational point configuration in the plane, and then performs what one should call Brehm extensions, in order to obtain non-rational partial surfaces. These examples and objects are first mile stones on the way to the remarkable \"universality theorems'' for polytopes and for polyhedral surfaces by Mn\\\"ev (1986), Richter-Gebert (1994), and Brehm (1997)."}
{"category": "Math", "title": "Causality and Association: The Statistical and Legal Approaches", "abstract": "This paper discusses different needs and approaches to establishing ``causation'' that are relevant in legal cases involving statistical input based on epidemiological (or more generally observational or population-based) information. We distinguish between three versions of ``cause'': the first involves negligence in providing or allowing exposure, the second involves ``cause'' as it is shown through a scientifically proved increased risk of an outcome from the exposure in a population, and the third considers ``cause'' as it might apply to an individual plaintiff based on the first two. The population-oriented ``cause'' is that commonly addressed by statisticians, and we propose a variation on the Bradford Hill approach to testing such causality in an observational framework, and discuss how such a systematic series of tests might be considered in a legal context. We review some current legal approaches to using probabilistic statements, and link these with the scientific methodology as developed here. In particular, we provide an approach both to the idea of individual outcomes being caused on a balance of probabilities, and to the idea of material contribution to such outcomes. Statistical terminology and legal usage of terms such as ``proof on the balance of probabilities'' or ``causation'' can easily become confused, largely due to similar language describing dissimilar concepts; we conclude, however, that a careful analysis can identify and separate those areas in which a legal decision alone is required and those areas in which scientific approaches are useful."}
{"category": "Math", "title": "Composantes irr\\'eductibles de la vari\\'et\\'e commutante nilpotente d'une alg\\`ebre de Lie sym\\'etrique semi-simple", "abstract": "Let \\theta be an involution of the semisimple Lie algebra g and g=k+p be the associated Cartan decomposition. The nilpotent commuting variety of (g,\\theta) consists in pairs of nilpotent elements (x,y) of p such that [x,y]=0. It is conjectured that this variety is equidimensional and that its irreducible components are indexed by the orbits of p-distinguished elements. This conjecture was established by A. Premet in the case (g \\times g, \\theta) where \\theta(x,y)=(y,x). In this work we prove the conjecture in a significant number of other cases."}
{"category": "Math", "title": "LASSO, Iterative Feature Selection and the Correlation Selector: Oracle Inequalities and Numerical Performances", "abstract": "We propose a general family of algorithms for regression estimation with quadratic loss. Our algorithms are able to select relevant functions into a large dictionary. We prove that a lot of algorithms that have already been studied for this task (LASSO and Group LASSO, Dantzig selector, Iterative Feature Selection, among others) belong to our family, and exhibit another particular member of this family that we call Correlation Selector in this paper. Using general properties of our family of algorithm we prove oracle inequalities for IFS, for the LASSO and for the Correlation Selector, and compare numerical performances of these estimators on a toy example."}
{"category": "Math", "title": "Homomorphisms and Structural Properties of Relational Systems", "abstract": "Two main topics are considered: The characterisation of finite homomorphism dualities for relational structures, and the splitting property of maximal antichains in the homomorphism order."}
{"category": "Math", "title": "Innerness of Derivations on Subalgebras of Measurable Operators", "abstract": "Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $\\tau,$ let $L(M, \\tau)$ be the algebra of all $\\tau$-measurable operators affiliated with $M.$ We prove that if $A$ is a locally convex reflexive complete metrizable solid $\\ast$-subalgebra in $L(M, \\tau),$ which can be embedded into a locally bounded weak Fr\\'{e}chet $M$-bimodule, then any derivation on $A$ is inner."}
{"category": "Math", "title": "Correction to \"The divergence of Banach space valued random variables on Wiener space\", Prob. Th. Rel. Fields 132, 291-320 (2005)", "abstract": "As a result of some mistakes discovered in the paper mentioned in the title, Corollaries 3.5 and 3.17a) are withdrawn and a new proof is provided for Proposition 3.14, under the added assumption that the second dual of the underlying Banach space Y possesses the Radon-Nykodim property."}
{"category": "Math", "title": "Homogeneous Poisson structures on symmetric spaces", "abstract": "We calculate, in a relatively explicit way, the Hamiltonian systems which arise from the Evens-Lu construction of homogeneous Poisson structures on both compact and noncompact type symmetric spaces. A corollary is that the Hamiltonian system arising in the noncompact case is isomorphic to the generic Hamiltonian system arising in the compact case. In the group case these systems are also isomorphic to those arising from the Bruhat Poisson structure on the flag space, and hence, by results of Lu, can be completely factored."}
{"category": "Math", "title": "Equidistribution of Dense Subgroups on Nilpotent Lie Groups", "abstract": "Let $\\Gamma$ be a dense subgroup of a simply connected nilpotent Lie group $G$ generated by a finite symmetric set $S$. We consider the $n$-ball $S_n$ for the word metric induced by $S$ on $\\Gamma$. We show that $S_n$ (with uniform measure) becomes equidistributed on $G$ with respect to the Haar measure as n tends to infinity. We give rates and also prove the analogous result for random walk averages (i.e. the local limit theorem)."}
{"category": "Math", "title": "Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds", "abstract": "We study compact complex 3-manifolds admitting holomorphic Riemannian metrics. We prove a uniformization result: up to a finite unramified cover, such a manifold admits a holomorphic Riemannian metric of constant sectionnal curvature."}
{"category": "Math", "title": "Mean-value property on manifolds with minimal horospheres", "abstract": "Let (M,g) be a non-compact and complete Riemannian manifold with minimal horospheres and infinite injectivity radius. We prove that bounded functions on (M,g) satisfying the mean-value property are constant. We extend thus a result of A. Ranjan and H. Shah who proved a similar result for bounded harmonic functions on harmonic manifolds with minimal horospheres."}
{"category": "Math", "title": "The lonely runner with seven runners", "abstract": "Suppose $k+1$ runners having nonzero constant speeds run laps on a unit-length circular track starting at the same time and place. A runner is said to be lonely if she is at distance at least $1/(k+1)$ along the track to every other runner. The lonely runner conjecture states that every runner gets lonely. The conjecture has been proved up to six runners ($k\\le 5$). A formulation of the problem is related to the regular chromatic number of distance graphs. We use a new tool developed in this context to solve the first open case of the conjecture with seven runners."}
{"category": "Math", "title": "Topological types of 3-dimensional small covers", "abstract": "In this paper we study the (equivariant) topological types of a class of 3-dimensional closed manifolds (i.e., 3-dimensional small covers), each of which admits a locally standard $(\\mathbb{Z}_2)^3$-action such that its orbit space is a simple convex 3-polytope. We introduce six equivariant operations on 3-dimensional small covers. These six operations are interesting because of their combinatorial natures. Then we show that each 3-dimensional small cover can be obtained from $\\mathbb{R}P^3$ and $S^1\\times\\mathbb{R}P^2$ with certain $(\\mathbb{Z}_2)^3$-actions under these six operations. As an application, we classify all 3-dimensional small covers up to $({\\Bbb Z}_2)^3$-equivariant unoriented cobordism."}
{"category": "Math", "title": "Curvatures of Smooth and Discrete Surfaces", "abstract": "We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation."}
{"category": "Math", "title": "Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole", "abstract": "We say that two graphs are similar if their adjacency matrices are similar matrices. We show that the square grid $G_n$ of order $n$ is similar to the disjoint union of two copies of the quartered Aztec diamond $QAD_{n-1}$ of order $n-1$ with the path $P_n^{(2)}$ on $n$ vertices having edge weights equal to~2. Our proof is based on an explicit change of basis in the vector space on which the adjacency matrix acts. The arguments verifying that this change of basis works are combinatorial. In particular, this allows computing the number of spanning trees of quartered Aztec diamonds. We present and analyze three more families of graphs that share the above described ``linear squarishness'' property of square grids: odd Aztec diamonds, mixed Aztec diamonds, and Aztec pillowcases--graphs obtained from two copies of an Aztec diamond by identifying the corresponding vertices on their convex hulls. We apply the above results to enumerate all the symmetry classes of spanning trees of the even Aztec diamonds, and all the symmetry classes not involving rotations of the spanning trees of odd and mixed Aztec diamonds. We also enumerate all but the base case of the symmetry classes of perfect matchings of odd square grids with the central vertex removed. In addition, we obtain a product formula for the number of spanning trees of Aztec pillowcases."}
{"category": "Math", "title": "Global formality at the $G_\\infty$-level", "abstract": "In this paper we prove that the sheaf of $\\Lscr$-poly-differential operators for a locally free Lie algebroid $\\Lscr$ is formal when viewed as a sheaf of $G_\\infty$-algebras via Tamarkin's morphism of DG-operads $G_\\infty\\r B_\\infty$. In an appendix we prove a strengthening of Halbout's globalization result for Tamarkin's local quasi-isomorphism."}
{"category": "Math", "title": "The External Fundamental Group of an Algebraic Number Field", "abstract": "We associate to every algebraic number field a hyperbolic surface lamination and an external fundamental group: the latter a generalization of the fundamental germ that necessarily contains external (not first order definable) elements. The external fundamental group of the rationals is a split extension of the absolute Galois group, that conjecturally contains a subgroup whose abelianization is isomorphic to the idele class group."}
{"category": "Math", "title": "Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type", "abstract": "In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel'd double for a class of graded Hopf algebras is investigated. Various quantum algebras, including multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfel'd doubles Lusztig maps are defined. It is shown that these maps induce isomorphisms between doubles of Nichols algebras of diagonal type. Further, the obtained isomorphisms satisfy Coxeter type relations in a generalized sense. As an application, the Lusztig isomorphisms are used to give a characterization of Nichols algebras of diagonal type with finite arithmetic root system. Key words: Hopf algebra, quantum group, Weyl groupoid"}
{"category": "Math", "title": "Akivis Superalgebras and speciality", "abstract": "In this paper we define Akivis superalgebra and study enveloping superalgebras for this class of algebras, proving an analogous of the PBW Theorem. Lie and Malcev superalgebras are examples of Akivis superalgebras. For these particular superalgebras, we describe the connection between the classical enveloping superalgebras and the corresponding generalized concept defined in this work."}
{"category": "Math", "title": "Bayesian treed Gaussian process models with an application to computer modeling", "abstract": "Motivated by a computer experiment for the design of a rocket booster, this paper explores nonstationary modeling methodologies that couple stationary Gaussian processes with treed partitioning. Partitioning is a simple but effective method for dealing with nonstationarity. The methodological developments and statistical computing details which make this approach efficient are described in detail. In addition to providing an analysis of the rocket booster simulator, our approach is demonstrated to be effective in other arenas."}
{"category": "Math", "title": "Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions", "abstract": "In this work we introduce the use of powerful tools from geometric measure theory (GMT) to study problems related to the size and structure of sets of mutual absolute continuity for the inside and outside harmonic measures of domains in n-dimensional Euclidean space, with n bigger than or equal to 3."}
{"category": "Math", "title": "Quadratic Malcev Superalgebras with reductive even part", "abstract": "It is our goal to give an inductive description of quadratic Malcev superalgebras with reductive even part. We use the notion of double extension of Malcev superalgebras presented by H. Albuquerque and S. Benayadi and transfer to Malcev superalgebras the concept of generalized double extension for Lie superalgebras."}
{"category": "Math", "title": "The action of the Cremona group on the non-commutative ring", "abstract": "The Cremona group acts on the field of two independent commutative variables over complex numbers. We provide a non-commutative ring that is an analog of non-commutative field of two independent variables and prove that the Cremona group embeds in the group of outer automorphisms of this ring. First proof of this result is technical, the second one is conceptual and gives a way to obtain non-commutative rings from the bounded derived categories of coherent sheaves."}
{"category": "Math", "title": "Generalised Kummer constructions and Weil restrictions", "abstract": "We use a generalised Kummer construction to realise all but one known weight four newforms with complex multiplication and rational Fourier coefficients in smooth Calabi-Yau threefolds defined over the rational numbers. The Calabi-Yau manifolds are smooth models of quotients of the Weil restrictions of elliptic curves with CM of class number three."}
{"category": "Math", "title": "A Schottky decomposition theorem for complex projective structures", "abstract": "Let S be a closed orientable surface of genus at least two, and let C be an arbitrary (complex) projective structure on S. We show that there is a decomposition of S into pairs of pants and cylinders such that the restriction of C to each component has an injective developing map and a discrete and faithful holonomy representation. This decomposition implies that every projective structure can be obtained by the construction of Gallo, Kapovich, and Marden. Along the way, we show that there is an admissible loop on (S, C), along which a grafting can be done."}
{"category": "Math", "title": "Limits of Calabi-Yau metrics when the Kahler class degenerates", "abstract": "We study the behaviour of families of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-flat metric. The limit can also be understood from algebraic geometry."}
{"category": "Math", "title": "The Dynamics of Rabinovich system", "abstract": "The paper presents some dynamical aspects of Rabinovich type, with distributed delay and with fractional derivatives."}
{"category": "Math", "title": "A universal Stein-Tomas restriction estimate for measures in three dimensions", "abstract": "We study restriction estimates in R^3 for surfaces given as graphs of W^1_1(R^2) (integrable gradient) functions. We obtain a \"universal\" L^2(mu) -> L^4(R^3, L^2(SO(3))) estimate for the extension operator f -> \\hat{f mu} in three dimensions. We also prove that the three dimensional estimate holds for any Frostman measure supported on a compact set of Hausdorff dimension greater than two. The approach is geometric and is influenced by a connection with the Falconer distance problem."}
{"category": "Math", "title": "Limit theorems for maximum flows on a lattice", "abstract": "We independently assign a non-negative value, as a capacity for the quantity of flows per unit time, with a distribution F to each edge on the Z^d lattice. We consider the maximum flows through the edges of two disjoint sets, that is from a source to a sink, in a large cube. In this paper, we show that the ratio of the maximum flow and the size of source is asymptotic to a constant. This constant is denoted by the flow constant."}
{"category": "Math", "title": "A new proof of Gromov's theorem on groups of polynomial growth", "abstract": "We give a new proof of Gromov's theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups."}
{"category": "Math", "title": "A criterion for a proper rational map to be equivalent to a proper polynomial map", "abstract": "In this paper, we give an explicit criterion when a rational holomorphic map between balls is equivalent to a polynomial holomorphic map. Making use of this criterion, we show that any proper rational holomorphic map from B^2 into B^N of degree two is equivalent to a polynomial holomorphic map; we also construct rational holomorphic maps of degree 3 that are almost linear but are not equivalent to polynomial holomorphic maps."}
{"category": "Math", "title": "Noncommutative geometry and compactifications of the moduli space of curves", "abstract": "In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Witten's conjectures, can be described completely algebraically as the homology of a certain differential graded Lie algebra. This two-parameter family is constructed by using a Lie cobracket on the space of noncommutative 0-forms, a structure which corresponds to pinching simple closed curves on a Riemann surface, to deform the noncommutative symplectic geometry described by Kontsevich in his subsequent papers."}
{"category": "Math", "title": "Nathanson heights in finite vector spaces", "abstract": "Let $p$ be a prime, and let $\\mathbb{Z}_p$ denote the field of integers modulo $p$. The \\emph{Nathanson height} of a point $v \\in \\mathbb{Z}_p^n$ is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace $V \\subseteq \\mathbb{Z}_p^n$ is the least Nathanson height of any of its nonzero points. In this paper, we resolve a conjecture of Nathanson [M. B. Nathanson, Heights on the finite projective line, International Journal of Number Theory, to appear], showing that on subspaces of $\\mathbb{Z}_p^n$ of codimension one, the Nathanson height function can only take values about $p, p/2, p/3, ....$ We show this by proving a similar result for the coheight on subsets of $\\mathbb{Z}_p$, where the \\emph{coheight} of $A \\subseteq \\mathbb{Z}_p$ is the minimum number of times $A$ must be added to itself so that the sum contains 0. We conjecture that the Nathanson height function has a similar constraint on its range regardless of the codimension, and produce some evidence that supports this conjecture."}
{"category": "Math", "title": "Families of m-convex polygons: m = 2", "abstract": "Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', $m$. Such polygons are called \\emph{$m$-convex} polygons and are characterised by having up to $m$ indentations in the side. We use a `divide and conquer' approach, factorising 2-convex polygons by extending a line along the base of its indents. We then use the inclusion-exclusion principle, the Hadamard product and extensions to known methods to derive the generating functions for each case."}
{"category": "Math", "title": "Galois groups of Schubert problems via homotopy computation", "abstract": "Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in C^8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S_6006."}
{"category": "Math", "title": "A Family of Generalized Beta Distributions for Income", "abstract": "The mathematical properties of a family of generalized beta distribution, including beta-normal, skewed-t, log-F, beta-exponential, beta-Weibull distributions have recently been studied in several publications. This paper applies these distributions to the modeling of the size distribution of income and computes the maximum likelihood estimation estimates of parameters. Their performances are compared to the widely used generalized beta distributions of the first and second types in terms of measures of goodness of fit."}
{"category": "Math", "title": "Limit properties of the monotone rearrangement for density and regression function estimation", "abstract": "The monotone rearrrangement algorithm was introduced by Hardy, Littlewood and P\\'olya as a sorting device for functions. Assuming that $x$ is a monotone function and that an estimate $x_n$ of $x$ is given, consider the monotone rearrangement $\\hat{x}_n$ of $x_n$. This new estimator is shown to be uniformly consistent. Under suitable assumptions, pointwise limit distribution results for $\\hat{x}_n$ are obtained. The framework is general and allows for weakly dependent and long range dependent stationary data. Applications in monotone density and regression function estimation are detailed."}
{"category": "Math", "title": "The Use of Unlabeled Data in Predictive Modeling", "abstract": "The incorporation of unlabeled data in regression and classification analysis is an increasing focus of the applied statistics and machine learning literatures, with a number of recent examples demonstrating the potential for unlabeled data to contribute to improved predictive accuracy. The statistical basis for this semisupervised analysis does not appear to have been well delineated; as a result, the underlying theory and rationale may be underappreciated, especially by nonstatisticians. There is also room for statisticians to become more fully engaged in the vigorous research in this important area of intersection of the statistical and computer sciences. Much of the theoretical work in the literature has focused, for example, on geometric and structural properties of the unlabeled data in the context of particular algorithms, rather than probabilistic and statistical questions. This paper overviews the fundamental statistical foundations for predictive modeling and the general questions associated with unlabeled data, highlighting the relevance of venerable concepts of sampling design and prior specification. This theory, illustrated with a series of central illustrative examples and two substantial real data analyses, shows precisely when, why and how unlabeled data matter."}
{"category": "Math", "title": "Statistical and Clinical Aspects of Hospital Outcomes Profiling", "abstract": "Hospital profiling involves a comparison of a health care provider's structure, processes of care, or outcomes to a standard, often in the form of a report card. Given the ubiquity of report cards and similar consumer ratings in contemporary American culture, it is notable that these are a relatively recent phenomenon in health care. Prior to the 1986 release of Medicare hospital outcome data, little such information was publicly available. We review the historical evolution of hospital profiling with special emphasis on outcomes; present a detailed history of cardiac surgery report cards, the paradigm for modern provider profiling; discuss the potential unintended negative consequences of public report cards; and describe various statistical methodologies for quantifying the relative performance of cardiac surgery programs. Outstanding statistical issues are also described."}
{"category": "Math", "title": "Ahlfors theorems for differential forms", "abstract": "Some counterparts of theorems of Phragm\\'en-Lindel\\\"of and of Ahlfors are proved for differential forms of ${\\cal WT}$--classes.}"}
{"category": "Math", "title": "Recurrence for branching Markov chains", "abstract": "The question of recurrence and transience of branching Markov chains is more subtle than for ordinary Markov chains; they can be classified in transience, weak recurrence, and strong recurrence. We review criteria for transience and weak recurrence and give several new conditions for weak recurrence and strong recurrence. These conditions make a unified treatment of known and new examples possible and provide enough information to distinguish between weak and strong recurrence. This represents a step towards a general classification of branching Markov chains. In particular, we show that in \\emph{homogeneous} cases weak recurrence and strong recurrence coincide. Furthermore, we discuss the generalization of positive and null recurrence to branching Markov chains and show that branching random walks on $\\Z$ are either transient or positive recurrent."}
{"category": "Math", "title": "Bounding the number of rational places using Weierstrass semigroups", "abstract": "Let Lambda be a numerical semigroup. Assume there exists an algebraic function field over GF(q) in one variable which possesses a rational place that has Lambda as its Weierstrass semigroup. We ask the question as to how many rational places such a function field can possibly have and we derive an upper bound in terms of the generators of Lambda and q. Our bound is an improvement to a bound by Lewittes which takes into account only the multiplicity of Lambda and q. From the new bound we derive significant improvements to Serre's upper bound in the cases q=2, 3 and 4. We finally show that Lewittes' bound has important implications to the theory of towers of function fields."}
{"category": "Math", "title": "A generalization of Tverberg's Theorem", "abstract": "The well know theorem of Tverberg states that if n > (d+1)(r-1) then one can partition any set of n points in R^d to r disjoint subsets whose convex hulls have a common point. The numbers T(d,r) = (d + 1)(r - 1) + 1 are known as Tverberg numbers. Reay asks the following question: if we add an additional parameter k (1 < k < r+1) what is the minimal number of points we need in order to guarantee that there exists an r partition of them such that any k of the r convex hulls intersect. This minimal number is denoted by T(d,r,k). Reay conjectured that T(d,r,k) = T(d,r) for all d,r and k. In this article we prove that this is true for the following cases: when k > [ (d+3)/2 ]-1 or when d < rk/(r-k)-1 and for the specific values d = 3; r = 4; k = 2 and d = 5; r = 3; k = 2."}
{"category": "Math", "title": "A Conversation with Shoutir Kishore Chatterjee", "abstract": "Shoutir Kishore Chatterjee was born in Ranchi, a small hill station in India, on November 6, 1934. He received his B.Sc. in statistics from the Presidency College, Calcutta, in 1954, and M.Sc. and Ph.D. degrees in statistics from the University of Calcutta in 1956 and 1962, respectively. He was appointed a lecturer in the Department of Statistics, University of Calcutta, in 1960 and was a member of its faculty until his retirement as a professor in 1997. Indeed, from the 1970s he steered the teaching and research activities of the department for the next three decades. Professor Chatterjee was the National Lecturer in Statistics (1985--1986) of the University Grants Commission, India, the President of the Section of Statistics of the Indian Science Congress (1989) and an Emeritus Scientist (1997--2000) of the Council of Scientific and Industrial Research, India. Professor Chatterjee, affectionately known as SKC to his students and admirers, is a truly exceptional person who embodies the spirit of eternal India. He firmly believes that ``fulfillment in man's life does not come from amassing a lot of money, after the threshold of what is required for achieving a decent living is crossed. It does not come even from peer recognition for intellectual achievements. Of course, one has to work and toil a lot before one realizes these facts.''"}
{"category": "Math", "title": "Congruences between modular forms and related modules", "abstract": "We fix $\\ell$ a prime and let $M$ be an integer such that $\\ell\\not|M$; let $f\\in S_2(\\Gamma_1(M\\ell^2))$ be a newform supercuspidal of fixed type related to the nebentypus, at $\\ell$ and special at a finite set of primes. Let $\\TT^\\psi$ be the local quaternionic Hecke algebra associated to $f$. The algebra $\\TT^\\psi$ acts on a module $\\mathcal M^\\psi_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $\\TT^\\psi$ is the universal deformation ring of a global Galois deformation problem associated to $\\orho_f$. Moreover $\\mathcal M^\\psi_f$ is free of rank 2 over $\\TT^\\psi$. If $f$ occurs at minimal level, by a generalization of a Conrad, Diamond and Taylor's result and by the classical Ihara's lemma, we prove a theorem of raising the level and a result about congruence ideals. The extension of this results to the non minimal case is an open problem."}
{"category": "Math", "title": "Robust control of uncertain multi-inventory systems via Linear Matrix Inequality", "abstract": "We consider a continuous time linear multi inventory system with unknown demands bounded within ellipsoids and controls bounded within ellipsoids or polytopes. We address the problem of \"-stabilizing the inventory since this implies some reduction of the inventory costs. The main results are certain conditions under which \"-stabilizability is possible through a saturated linear state feedback control. All the results are based on a Linear Matrix Inequalities (LMIs) approach and on some recent techniques for the modeling and analysis of polytopic systems with saturations."}
{"category": "Math", "title": "A Conversation with Dorothy Gilford", "abstract": "In 1946, Public Law 588 of the 79th Congress established the Office of Naval Research (ONR). Its mission was to plan, foster and encourage scientific research in support of Naval problems. The establishment of ONR predates the National Science Foundation and initiated the refocusing of scientific infrastructure in the United States following World War II. At the time, ONR was the only source for federal support of basic research in the United States. Dorothy Gilford was one of the first Heads of the Probability and Statistics program at the Office of Naval Research (1955 to 1962), and she went on to serve as Director of the Mathematical Sciences Division (1962 to 1968). During her time at ONR, Dorothy influenced many areas of statistics and mathematics and was ahead of her time in promoting interdisciplinary projects. Dorothy continued her career at the National Center for Education Statistics (1969 to 1974). She was active in starting international comparisons of education outcomes in different countries, which has influenced educational policy in the United States. Dorothy went on to serve in many capacities at the National Academy of Sciences, including Director of Human Resources Studies (1975 to 1978), Senior Statistician on the Committee on National Statistics (1978 to 1988) and Director of the Board on International Comparative Studies in Education (1988 to 1994). The following is a conversation we had with Dorothy Gilford in March of 2004. We found her to be an interesting person and a remarkable statistician. We hope you agree."}
{"category": "Math", "title": "On the integrability of holomorphic vector fields", "abstract": "We determine topological and algebraic conditions for a germ of holomorphic foliation $\\mathcal F(X)$ induced by a generic vector field $X$ on $(\\mathbb{C}^{3},0)$ to have a holomorphic first integral, i.e., a germ of holomorphic map $F \\colon(\\mathbb{C}^{3},0)\\longrightarrow(\\mathbb{C}^{2},0)$ such that the leaves of $\\mathcal F(X)$ are contained in the level curves of $F$."}
{"category": "Math", "title": "The Hasse principle for pairs of diagonal cubic forms", "abstract": "By means of the Hardy-Littlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables."}
{"category": "Math", "title": "Linearization of fourth-order ordinary differential equations by point transformations", "abstract": "We present here the solution of the problem on linearization of fourth-order equations by means of point transformations. We show that all fourth-order equations that are linearizable by point transformations are contained in the class of equations which is linear in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation."}
{"category": "Math", "title": "Test vectors for trilinear forms, when two representations are unramified", "abstract": "Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three infinite dimensional, irreducible, admissible representations of G, the space of G-invariant linear forms has dimension 0 or 1. When a non-zero linear form exists, one wants to find an element of V which is not in its kernel : this is a test vector. Gross and Prasad found explicit test vectors when the three representations are unramified principal series, and when the three representations are unramified twists of the Steinberg representation. In this paper, we find an explicit test vector when two of the representations are unramified principal series and the third one has ramification at least 1."}
{"category": "Math", "title": "Sur une conjecture de Dehornoy", "abstract": "Let M_n be the n! * n! matrix indexed by permutations of S_n, defined by M_n(sigma,tau)=1 if every descent of tau^{-1} is also a descent of sigma, and M_n(sigma,tau)=0 otherwise. We prove the following result, conjectured by P. Dehornoy: the characteristic polynomial P_n(x)=|xI-M_n| of M_n divides P_{n+1}(x) in Z[x]."}
{"category": "Math", "title": "Van der Corput sets in Z^d", "abstract": "In this partly expository paper we study van der Corput sets in $\\Z^d$, with a focus on connections with harmonic analysis and recurrence properties of measure preserving dynamical systems. We prove multidimensional versions of some classical results obtained for $d=1$ in \\cite{K-MF} and \\cite{R}, establish new characterizations, introduce and discuss some modifications of van der Corput sets which correspond to various notions of recurrence, provide numerous examples and formulate some natural open questions."}
{"category": "Math", "title": "Intersective polynomials and polynomial Szemeredi theorem", "abstract": "Let $P=\\{p_{1},\\ld,p_{r}\\}\\subset\\Q[n_{1},\\ld,n_{m}]$ be a family of polynomials such that $p_{i}(\\Z^{m})\\sle\\Z$, $i=1,\\ld,r$. We say that the family $P$ has {\\it PSZ property} if for any set $E\\sle\\Z$ with $d^{*}(E)=\\limsup_{N-M\\ras\\infty}\\frac{|E\\cap[M,N-1]|}{N-M}>0$ there exist infinitely many $n\\in\\Z^{m}$ such that $E$ contains a polynomial progression of the form \\hbox{$\\{a,a+p_{1}(n),\\ld,a+p_{r}(n)\\}$}. We prove that a polynomial family $P=\\{p_{1},\\ld,p_{r}\\}$ has PSZ property if and only if the polynomials $p_{1},\\ld,p_{r}$ are {\\it jointly intersective}, meaning that for any $k\\in\\N$ there exists $n\\in\\Z^{m}$ such that the integers $p_{1}(n),\\ld,p_{r}(n)$ are all divisible by $k$. To obtain this result we give a new ergodic proof of the polynomial Szemer\\'{e}di theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If $p_{1},\\ld,p_{r}\\in\\Q[n]$ are jointly intersective integral polynomials, then for any finite partition of $\\Z$, $\\Z=\\bigcup_{i=1}^{k}E_{i}$, there exist $i\\in\\{1,\\ld,k\\}$ and $a,n\\in E_{i}$ such that $\\{a,a+p_{1}(n),\\ld,a+p_{r}(n)\\}\\sln E_{i}$."}
{"category": "Math", "title": "Elementary aspects of the geometry of metric spaces", "abstract": "The setting of metric spaces is very natural for numerous questions concerning manifolds, norms, and fractal sets, and a few of the main ingredients are surveyed here."}
{"category": "Math", "title": "The Lonely Vertex Problem", "abstract": "In a locally finite tiling of n-dim Euclidean space by convex polytopes, each point of the space is either a vertex of at least two tiles, or no vertex at all."}
{"category": "Math", "title": "A discrete version and stability of Brunn Minkowski inequality", "abstract": "In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for length spaces. Our new definition based only on distance properties allows us also to deal with discrete spaces. Then we show the stability of our new inequality under a convergence of metric measure spaces. This result gives as a corollary the stability of the classical Brunn-Minkowski inequality for geodesic spaces. The proof of this stability was done for different inequalities (curvature dimension inequality, metric contraction property) but as far as we know not for the Brunn-Minkowski one. In the second part of the paper, we show that every metric measure space satisfying classical Brunn-Minkowski inequality can be approximated by discrete spaces with some approximated Brunn-Minkowski inequalities."}
{"category": "Math", "title": "The Weight System of the Multivariable Alexander Polynomial", "abstract": "We derive a formula for the weight system of the multivariable Alexander polynomial using determinants, show that it obeys known relations, and satisfies some of the same relations as the single variable polynomial."}
{"category": "Math", "title": "On the quasi-derivation relation for multiple zeta values", "abstract": "Recently, Masanobu Kaneko introduced a conjecture on an extension of the derivation relation for multiple zeta values. The goal of the present paper is to present a proof of this conjecture by reducing it to a class of relations for multiple zeta values studied by Kawashima. In addition, some algebraic aspects of the quasi-derivation operator $\\partial_n^{(c)}$ on $\\mathbb{Q}< x,y>$, which was defined by modeling a Hopf algebra developed by Connes and Moscovici, will be presented."}
{"category": "Math", "title": "Extensions of discrete classical orthogonal polynomials beyond the orthogonality", "abstract": "It is well known that the family of Hahn polynomials $\\{h_n^{\\alpha,\\beta}(x;N)\\}_{n\\ge 0}$ is orthogonal with respect to a certain weight function up to $N$. In this paper we present a factorization for Hahn polynomials for a degree higher than $N$ and we prove that these polynomials can be characterized by a $\\Delta$-Sobolev orthogonality. We also present an analogous result for dual-Hahn, Krawtchouk, and Racah polynomials and give the limit relations between them for all $n\\in \\XX N_0$. Furthermore, in order to get this results for the Krawtchouk polynomials we will get a more general property of orthogonality for Meixner polynomials."}
{"category": "Math", "title": "The concentration index of subharmonic functions of infinite order", "abstract": "The purpose of this paper is to introduce into consideration an analogue of the concentration index in the class of subharmonic functions of infinite order. The one in the case of finite order is used in the interpolation theory."}
{"category": "Math", "title": "Degree Bounds for Gr\\\"obner Bases in Algebras of Solvable Type", "abstract": "We establish doubly-exponential degree bounds for Gr\\\"obner bases in certain algebras of solvable type over a field (as introduced by Kandri-Rody and Weispfenning). The class of algebras considered here includes commutative polynomial rings, Weyl algebras, and universal enveloping algebras of finite-dimensional Lie algebras. For the computation of these bounds, we adapt a method due to Dub\\'e based on a generalization of Stanley decompositions. Our bounds yield doubly-exponential degree bounds for ideal membership and syzygies, generalizing the classical results of Hermann and Seidenberg (in the commutative case) and Grigoriev (in the case of Weyl algebras)."}
{"category": "Math", "title": "A New Algorithm in Geometry of Numbers", "abstract": "A lattice Delaunay polytope P is called perfect if its Delaunay sphere is the only ellipsoid circumscribed about P. We present a new algorithm for finding perfect Delaunay polytopes. Our method overcomes the major shortcomings of the previously used method. We have implemented and used our algorithm for finding perfect Delaunay polytopes in dimensions 6, 7, 8. Our findings lead to a new conjecture that sheds light on the structure of lattice Delaunay tilings."}
{"category": "Math", "title": "A Symbolic Finite-state approach for Automated Proving of Theorems in Combinatorial Game Theory", "abstract": "We develop a finite-state automata approach, implemented in a Maple package {\\tt ToadsAndFrogs} available from our websites, for conjecturing, and then rigorously proving, values for large families of positions in Richard Guy's combinatorial game ``Toads and Frogs''. In particular, we prove a conjecture of Jeff Erickson."}
{"category": "Math", "title": "On descent theory and main conjectures in non-commutative Iwasawa theory", "abstract": "We prove a `Weierstrass Preparation Theorem' and develop an explicit descent formalism in the context of Whitehead groups of non-commutative Iwasawa algebras. We use these results to describe the precise connection between the main conjecture of non-commutative Iwasawa theory (in the sense of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture."}
{"category": "Math", "title": "q-Bernoulli numbers and Stirling numbers(2)", "abstract": "In this paper we study q-Bernoulli numbers and polynomials related to q-Stirling numbers. From thsese studying we investigate some interesting q-stirling numbers' identities related to q-Bernoulli numbers."}
{"category": "Math", "title": "Limits of log canonical thresholds", "abstract": "Let T_n denote the set of log canonical thresholds of pairs (X,Y), with X a nonsingular variety of dimension n, and Y a nonempty closed subscheme of X. Using non-standard methods, we show that every limit of a decreasing sequence in T_n lies in T_{n-1}, proving in this setting a conjecture of Koll\\'{a}r. We also show that T_n is a closed subset in the set of real numbers; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in order to check Shokurov's ACC Conjecture for all T_n, it is enough to show that 1 is not a point of accumulation from below of any T_n. In a different direction, we interpret the ACC Conjecture as a semi-continuity property for log canonical thresholds of formal power series."}
{"category": "Math", "title": "A note on the p-adic log-gamma functions", "abstract": "In this paper we prove that q-Euler numbers are occured in the coefficients of some stirling type seies for p-adic analytic q-log gamma function"}
{"category": "Math", "title": "The Fifteen Theorem for Universal Hermitian Lattices over Imaginary Quadratic Fields", "abstract": "We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over $\\mathbb{Q}(\\sqrt{-m})$ for all m. For each imaginary quadratic field $\\mathbb{Q}(\\sqrt{-m})$, we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13,14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13."}
{"category": "Math", "title": "Toeplitz Operators, K\\\"ahler Manifolds, and Line Bundles", "abstract": "This is a survey paper. We discuss Toeplitz operators in K\\\"ahler geometry, with applications to geometric quantization, and review some recent developments."}
{"category": "Math", "title": "Binary normal regular Hermitian lattices over imaginary quadratic fields", "abstract": "We call a positive definite Hermitian lattice regular if it represents all integers which can be represented locally by the lattice. We investigate binary regular Hermitian lattices over imaginary quadratic fields $\\mathbb{Q}(\\sqrt{-m})$ and provide a complete list of the (normal) Hermitian lattices."}
{"category": "Math", "title": "Computer-intensive rate estimation, diverging statistics and scanning", "abstract": "A general rate estimation method is proposed that is based on studying the in-sample evolution of appropriately chosen diverging/converging statistics. The proposed rate estimators are based on simple least squares arguments, and are shown to be accurate in a very general setting without requiring the choice of a tuning parameter. The notion of scanning is introduced with the purpose of extracting useful subsamples of the data series; the proposed rate estimation method is applied to different scans, and the resulting estimators are then combined to improve accuracy. Applications to heavy tail index estimation as well as to the problem of estimating the long memory parameter are discussed; a small simulation study complements our theoretical results."}
{"category": "Math", "title": "Struggles with Survey Weighting and Regression Modeling", "abstract": "The general principles of Bayesian data analysis imply that models for survey responses should be constructed conditional on all variables that affect the probability of inclusion and nonresponse, which are also the variables used in survey weighting and clustering. However, such models can quickly become very complicated, with potentially thousands of poststratification cells. It is then a challenge to develop general families of multilevel probability models that yield reasonable Bayesian inferences. We discuss in the context of several ongoing public health and social surveys. This work is currently open-ended, and we conclude with thoughts on how research could proceed to solve these problems."}
{"category": "Math", "title": "Comment: Struggles with Survey Weighting and Regression Modeling", "abstract": "Comment: Struggles with Survey Weighting and Regression Modeling [arXiv:0710.5005]"}
{"category": "Math", "title": "Comment: Struggles with Survey Weighting and Regression Modeling", "abstract": "Comment: Struggles with Survey Weighting and Regression Modeling [arXiv:0710.5005]"}
{"category": "Math", "title": "Comment: Struggles with Survey Weighting and Regression Modeling", "abstract": "Comment: Struggles with Survey Weighting and Regression Modeling [arXiv:0710.5005]"}
{"category": "Math", "title": "On $k$-noncrossing partitions", "abstract": "In this paper we prove a duality between $k$-noncrossing partitions over $[n]=\\{1,...,n\\}$ and $k$-noncrossing braids over $[n-1]$. This duality is derived directly via (generalized) vacillating tableaux which are in correspondence to tangled-diagrams \\cite{Reidys:07vac}. We give a combinatorial interpretation of the bijection in terms of the contraction of arcs of tangled-diagrams. Furthermore it induces by restriction a bijection between $k$-noncrossing, 2-regular partitions over $[n]$ and $k$-noncrossing braids without isolated points over $[n-1]$. Since braids without isolated points correspond to enhanced partitions this allows, using the results of \\cite{MIRXIN}, to enumerate 2-regular, 3-noncrossing partitions."}
{"category": "Math", "title": "Comment: Struggles with Survey Weighting and Regression Modeling", "abstract": "Comment: Struggles with Survey Weighting and Regression Modeling [arXiv:0710.5005]"}
{"category": "Math", "title": "Comment: Struggles with Survey Weighting and Regression Modeling", "abstract": "Comment: Struggles with Survey Weighting and Regression Modeling [arXiv:0710.5005]"}
{"category": "Math", "title": "Some remarks on Pr\\\"{u}fer $\\star $--multiplication domains and class groups", "abstract": "Let $D$ be an integral domain with quotient field $K$ and let $X$ be an indeterminate over $D$. Also, let $\\boldsymbol{\\mathcal{T}}:=\\{T_{\\lambda}\\mid \\lambda \\in \\Lambda \\}$ be a defining family of quotient rings of $D$ and suppose that $\\ast $ is a finite type star operation on $D$ induced by $\\boldsymbol{\\mathcal{T}}$. We show that $D$ is a P$ \\ast $MD (resp., P$v$MD) if and only if $(\\co_D(fg))^{\\ast}=(\\co_D(f)\\co_D(g))^{\\ast}$ (resp., $(\\co_D(fg))^{w}=(\\co_D(f)\\co_D(g))^{w}$) for all $0 \\ne f,g \\in K[X]$. A more general version of this result is given in the semistar operation setting. We give a method for recognizing P$v$MD's which are not P$\\ast $MD's for a certain finite type star operation $\\ast $. We study domains $D$ for which the $\\ast $--class group $\\Cl^{\\ast}(D)$ equals the $t$--class group $\\Cl^{t}(D)$ for any finite type star operation $\\ast $, and we indicate examples of P$v$MD's $D$ such that $\\Cl^{\\ast}(D)\\subsetneq \\Cl^{t}(D)$. We also compute $\\Cl^v(D)$ for certain valuation domains $D$."}
{"category": "Math", "title": "Rejoinder: Struggles with Survey Weighting and Regression Modeling", "abstract": "Rejoinder: Struggles with Survey Weighting and Regression Modeling [arXiv:0710.5005]"}
{"category": "Math", "title": "On the weight and the density of the space of order-preserving functionals", "abstract": "In the present paper it is proved that the functors $O_\\tau$ of $\\tau$-smooth order preserving functionals and $O_R$ of Radon order preserving functionals preserve the weight of infinite Tychonoff spaces. Moreover, it is established that the density and the weak density of infinite Tychonoff spaces do not increase under these functors."}
{"category": "Math", "title": "On the first integral conjecture of Rene Thom", "abstract": "More that half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise for what concerns the interpretation of the word \\textquotedblleft generically\\textquotedblright, this statement is ostensibly true and is nowadays commonly accepted. On the other hand, the (few) known formal proofs of Thom's conjecture are all relying to the classical Sard theorem and are thus requiring the technical assumption that first integrals should be of class $C^{k}$ with $k\\geq d,$ where $d$ is the dimension of the manifold. In this work, using a recent nonsmooth extension of Sard theorem we establish the validity of Thom's conjecture for locally Lipschitz first integrals, interpreting genericity in the $C^{1}$ sense."}
{"category": "Math", "title": "On fractional Ornstein-Uhlenbeck processes", "abstract": "In this paper we study Doob's transform of fractional Brownian motion (FBM). It is well known that Doob's transform of standard Brownian motion is identical in law with the Ornstein-Uhlenbeck diffusion defined as the solution of the (stochastic) Langevin equation where the driving process is a Brownian motion. It is also known that Doob's transform of FBM and the process obtained from the Langevin equation with FBM as the driving process are different. However, also the first one of these can be described as a solution of a Langevin equation but now with some other driving process than FBM. We are mainly interested in the properties of this new driving process denoted Y^{(1)}. We also study the solution of the Langevin equation with Y^{(1)} as the driving process. Moreover, we show that the covariance of Y^{(1)} grows linearly; hence, in this respect Y^{(1)} is more like a standard Brownian motion than a FBM. In fact, it is proved that a properly scaled version of Y^{(1)} converges weakly to Brownian motion."}
{"category": "Math", "title": "From the Pr\\'ekopa-Leindler inequality to modified logarithmic Sobolev inequality", "abstract": "We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux. Using the Pr\\'ekopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on $\\dR^n$, with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality for all uniform strictly convex potential as well as the Euclidean logarithmic Sobolev inequality."}
{"category": "Math", "title": "Elementary Pseudoconcavity and fields of CR meromorphic functions", "abstract": "Let M be a smooth CR manifold of CR dimension n and CR codimension k, which is not compact, but has the local extension property E. We introduce the notion of \"elementary pseudoconcavity\" for M, which extends to CR manifolds the concept of a \"pseudoconcave\" complex manifold. This notion is then used to obtain generalizations, to the noncompact case, of the results of our previous paper about algebraic dependence, transcendence degree and related matters for the field K(M) of CR meromorphic functions on M."}
{"category": "Math", "title": "A Remark on Almost Umbilical Hypersurfaces", "abstract": "In this article, we prove new stability results for almost-Einstein hypersurfaces of the Euclidean space, based on previous eigenvalue pinching results. Then, we deduce some comparable results for almost umbilical hypersurfaces."}
{"category": "Math", "title": "On the extendability of free multiarrangements", "abstract": "A free multiarrangement of rank $k$ is defined to be extendable if it is obtained from a simple rank $(k+1)$ free arrangement by the natural restriction to a hyperplane (in the sense of Ziegler). Not all free multiarrangements are extendable. We will discuss extendability of free multiarrangements for a special class. We also give two applications. The first is to produce totally non-free arrangements. The second is to give interpolating free arrangements between extended Shi and Catalan arrangements."}
{"category": "Math", "title": "Abstract Hermitian Algebras I. Spectral Resolution", "abstract": "We refer to the real Jordan Banach algebra of bounded Hermitian operators on a Hilbert space as a Hermitian algebra. We define an abstract Hermitian algebra (AH-algebra) to be the directed group of an e-ring that contains a semitransparent element, has the quadratic annihilation property, and satisfies a Vigier condition on pairwise commuting ascending sequences. All of this terminology is explicated in this article, where we launch a study of AH-algebras. Here we establish the fundamental properties of AH-algebras, including the existence of polar decompositions and spectral resolutions, and we show that two elements of an AH-algebra commute if and only if their spectral projections commute. We employ spectral resolutions to assess the structure of maximal pairwise commuting subsets of an AH-algebra."}
{"category": "Math", "title": "The William Kruskal Legacy: 1919--2005", "abstract": "William Kruskal (Bill) was a distinguished statistician who spent virtually his entire professional career at the University of Chicago, and who had a lasting impact on the Institute of Mathematical Statistics and on the field of statistics more broadly, as well as on many who came in contact with him. Bill passed away last April following an extended illness, and on May 19, 2005, the University of Chicago held a memorial service at which several of Bill's colleagues and collaborators spoke along with members of his family and other friends. This biography and the accompanying commentaries derive in part from brief presentations on that occasion, along with recollections and input from several others. Bill was known personally to most of an older generation of statisticians as an editor and as an intellectual and professional leader. In 1994, Statistical Science published an interview by Sandy Zabell (Vol. 9, 285--303) in which Bill looked back on selected events in his professional life. One of the purposes of the present biography and accompanying commentaries is to reintroduce him to old friends and to introduce him for the first time to new generations of statisticians who never had an opportunity to interact with him and to fall under his influence."}
{"category": "Math", "title": "On derived categories and derived functors", "abstract": "For an abelian category, a category equivalent to its derived category is constructed by means of specific projective (injective) multicomplexes, the so-called homological resolutions."}
{"category": "Math", "title": "Pointwise multipliers in Hardy-Orlicz spaces, and interpolation", "abstract": "We study multipliers of Hardy-Orlicz spaces $\\mH_{\\Phi}$ which are strictly contained between $\\bigcup_{p>0}H^p$ and so-called ``big'' Hardy-Orlicz spaces. Big Hardy-Orlicz spaces, carrying an algebraic structure, are equal to their multiplier algebra, whereas in classical Hardy spaces $H^p$, the multipliers reduce to $H^{\\infty}$. For Hardy-Orlicz spaces $\\mH_{\\Phi}$ between these two extremal situations and subject to some conditions, we exhibit multipliers that are in Hardy-Orlicz spaces the defining functions of which are related to $\\Phi$. Even if the results do not entirely characterize the multiplier algebra, some examples show that we are not very far from precise conditions. In certain situations we see how the multiplier algebra grows in a sense from $\\Hi$ to big Hardy-Orlicz spaces when we go from classical $H^p$ spaces to big Hardy-Orlicz spaces. However, the multiplier algebras are not always ordered as their underlying Hardy-Orlicz spaces. Such an ordering holds in certain situations, but examples show that there are large Hardy-Orlicz spaces for which the multipliers reduce to $\\Hi$ so that the multipliers do in general not conserve the ordering of the underlying Hardy-Orlicz spaces. We apply some of the multiplier results to construct Hardy-Orlicz spaces close to $\\bigcup_{p>0}H^p$ and for which the free interpolating sequences are no longer characterized by the Carleson condition which is well known to characterize free interpolating sequences in $H^p$, $p>0$."}
{"category": "Math", "title": "A Tribute to Bill Kruskal", "abstract": "Discussion of ``The William Kruskal Legacy: 1919--2005'' by Stephen E. Fienberg, Stephen M. Stigler and Judith M. Tanur [arXiv:0710.5063]"}
{"category": "Math", "title": "William H. Kruskal and the Development of Coordinate-Free Methods", "abstract": "Discussion of ``The William Kruskal Legacy: 1919--2005'' by Stephen E. Fienberg, Stephen M. Stigler and Judith M. Tanur [arXiv:0710.5063]"}
{"category": "Math", "title": "William Kruskal: My Scholarly and Scientific Model", "abstract": "Discussion of ``The William Kruskal Legacy: 1919--2005'' by Stephen E. Fienberg, Stephen M. Stigler and Judith M. Tanur [arXiv:0710.5063]"}
{"category": "Math", "title": "Working with Bill Kruskal: From 1950 Onward", "abstract": "Discussion of ``The William Kruskal Legacy: 1919--2005'' by Stephen E. Fienberg, Stephen M. Stigler and Judith M. Tanur [arXiv:0710.5063]"}
{"category": "Math", "title": "Automorphisms of order three on numerical Godeaux surfaces", "abstract": "We prove that a numerical Godeaux surface cannot have an automorphism of order three."}
{"category": "Math", "title": "Bill Kruskal and the Committee on National Statistics", "abstract": "Discussion of ``The William Kruskal Legacy: 1919--2005'' by Stephen E. Fienberg, Stephen M. Stigler and Judith M. Tanur [arXiv:0710.5063]"}
{"category": "Math", "title": "William Kruskal Remembered", "abstract": "Discussion of ``The William Kruskal Legacy: 1919--2005'' by Stephen E. Fienberg, Stephen M. Stigler and Judith M. Tanur [arXiv:0710.5063]"}
{"category": "Math", "title": "William H. Kruskal, Mentor and Friend", "abstract": "Discussion of ``The William Kruskal Legacy: 1919--2005'' by Stephen E. Fienberg, Stephen M. Stigler and Judith M. Tanur [arXiv:0710.5063]"}
{"category": "Math", "title": "Galois extensions of Lubin-Tate spectra", "abstract": "Let E_n be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E^{nr}_n whose coefficients are built from the coefficients of E_n and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E^{nr}_n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that there are no non-trivial connected Galois extensions of E^{nr}_n with Galois group a finite group G with cyclic quotient. Our results carry over to the K(n)-local context."}
{"category": "Math", "title": "Particle Filters for Multiscale Diffusions", "abstract": "We consider multiscale stochastic systems that are partially observed at discrete points of the slow time scale. We introduce a particle filter that takes advantage of the multiscale structure of the system to efficiently approximate the optimal filter."}
{"category": "Math", "title": "On fields of definition of arithmetic Kleinian reflection groups", "abstract": "We show that degrees of the real fields of definition of arithmetic Kleinian reflection groups are bounded by 35."}
{"category": "Math", "title": "Degenerate complex Monge-Amp\\`ere equations over compact K\\\"ahler manifolds", "abstract": "We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Amp\\`ere equations. This type of equations is precisely what is needed in order to construct K\\\"ahler-Einstein metrics over irreducible singular K\\\"ahler spaces with ample or trivial canonical sheaf and singular K\\\"ahler-Einstein metrics over varieties of general type."}
{"category": "Math", "title": "Congruences of lines in $\\mathbb{P}^5$, quadratic normality, and completely exceptional Monge-Amp\\`ere equations", "abstract": "The existence is proved of two new families of locally Cohen-Macaulay sextic threefolds in $\\mathbb{P}^5$, which are not quadratically normal. These threefolds arise naturally in the realm of first order congruences of lines as focal loci and in the study of the completely exceptional Monge-Amp\\`ere equations. One of these families comes from a smooth congruence of multidegree $(1,3,3)$ which is a smooth Fano fourfold of index two and genus 9."}
{"category": "Math", "title": "The Half-Perimeter Generating Function of Gated and Wicketed Ferrers diagrams", "abstract": "We show that the half-perimeter generating functions for the number of Wicketed and Gated Ferrers diagrams is algebraic. Furthermore, the generating function of the Wicketed Ferrers diagrams is closely related to the generating function of the Catalan numbers. The methodology of the experimentation as well as the proof is the umbral transfer matrix method."}
{"category": "Math", "title": "Linearizable ordinary differential equations", "abstract": "Our purpose in this paper is to study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse integrating factor which can be constructed by means of the solutions of linear differential equations. We give several families of differential systems which illustrate how the integrability of the system passes through the solutions of a linear differential equation. At the end of the work, we describe some families of differential systems which are Darboux integrable and whose inverse integrating factor is constructed using the solutions of a second--order linear differential equation defining a family of orthogonal polynomials."}
{"category": "Math", "title": "The Bramble-Hilbert Lemma", "abstract": "This is an introductory document surveying several results in polynomial approximation, known as the Bramble-Hilbert lemma."}
{"category": "Math", "title": "Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix", "abstract": "Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superalgebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described."}
{"category": "Math", "title": "One optional observation inflates $\\alpha$ by $100/\\sqrt{n}$ per cent", "abstract": "For one-sample level $\\alpha$ tests $\\psi_m$ based on independent observations $X_1,...,X_m$, we prove an asymptotic formula for the actual level of the test rejecting if at least one of the tests $\\psi_{n},...,\\psi_{n+k}$ would reject. For $k=1$ and usual tests at usual levels $\\alpha$, the result is approximately summarized by the title of this paper. Our method of proof, relying on some second order asymptotic statistics as developed by Pfanzagl and Wefelmeyer, might also be useful for proper sequential analysis. A simple and elementary alternative proof is given for $k=1$ in the special case of the Gauss test."}
{"category": "Math", "title": "The Calabi flow on K\\\"ahler surface with bounded Sobolev constant", "abstract": "We consider the formation of singularities along the Calabi flow with the assumption of the uniform Sobolev constant. In particular, on K\\\"ahler surface we show that any \"maximal bubble\" has to be a scalar flat ALE K\\\"ahler metric. In some certain classes on toric Fano surface, the Sobolev constant is a priori bounded along the Calabi flow with small Calabi energy. Also we can show in certain case no maximal bubble can form along the flow, it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently. To illustrate our results more clearly, we focus on an example on CP^2 blown up three points at generic position. Our result also implies existence of constant scalar curvature metrics on CP^2 blown up three points at generic position in the K\\\"ahler classes where the exceptional divisors have the same area."}
{"category": "Math", "title": "Fields of CR meromorphic functions", "abstract": "Let $M$ be a smooth compact $CR$ manifold of $CR$ dimension $n$ and $CR$ codimension $k$, which has a certain local extension property $E$. In particular, if $M$ is pseudoconcave, it has property $E$. Then the field $\\Cal K(M)$ of $CR$ meromorphic functions on $M$ has transcendence degree $d$, with $d\\leq n+k$. If $f_1, f_2, \\hdots , f_d$ is a maximal set of algebraically independent $CR$ meromorphic functions on $M$, then $\\Cal K(M)$ is a simple finite algebraic extension of the field $\\Bbb C(f_1, f_2, \\hdots, f_d)$ of rational functions of the $f_1, f_2, \\hdots , f_d$. When $M$ has a projective embedding, there is an analogue of Chow's theorem, and $\\Cal K(M)$ is isomorphic to the field $\\Cal R(Y)$ of rational functions on an irreducible projective algebraic variety $Y$, and $M$ has a $CR$ embedding in $\\roman{reg} Y$. The equivalence between algebraic dependence and analytic dependence fails when condition $E$ is dropped."}
{"category": "Math", "title": "The $X$-class and almost-increasing permutations", "abstract": "In this paper we give a bijection between the class of permutations that can be drawn on an X-shape and a certain set of permutations that appears in [Knuth] in connection to sorting algorithms. A natural generalization of this set leads us to the definition of almost-increasing permutations, which is a one-parameter family of permutations that can be characterized in terms of forbidden patterns. We find generating functions for almost-increasing permutations by using their cycle structure to map them to colored Motzkin paths. We also give refined enumerations with respect to the number of cycles, fixed points, excedances, and inversions."}
{"category": "Math", "title": "Stein fillability and the realization of contact manifolds", "abstract": "There is an intrinsic notion of what it means for a contact manifold to be the smooth boundary of a Stein manifold. The same concept has another more extrinsic formulation, which is often used as a convenient working hypothesis. We give a simple proof that the two are equivalent. Moreover it is shown that, even though a border always exists, it's germ is not unique; nevertheless the germ of the Dolbeault cohomology of any border is unique. We also point out that any Stein fillable compact contact 3- manifold has a geometric realization in C^4 via an embedding, or in C^3 via an immersion."}
{"category": "Math", "title": "The sixth power moment of Dirichlet L-functions", "abstract": "We prove a formula, with power savings, for the sixth moment of Dirichlet L-functions averaged over moduli $q$, over primitive characters $\\chi$ modulo $q$, and over the critical line. Our formula agrees precisely with predictions motivated by random matrix theory. In particular, the constant 42 appears as a factor in the leading order term, exactly as is predicted for the sixth moment of the Riemann zeta-function."}
{"category": "Math", "title": "Persistence of stratification of normally expanded laminations", "abstract": "This manuscript complements the Hirsch-Pugh-Shub (HPS) theory on persistence of normally hyperbolic laminations and the theorem of Robinson on the structural stability of diffeomorphisms that satisfy Axiom A and the strong transversality condition (SA). We generalize these results by introducing a geometric object: the stratification of laminations. It is a stratification whose strata are laminations. Our main theorem implies the persistence of some stratifications whose strata are normally expanded. The dynamics is a $C^r$-endomorphism of a manifold (which is possibly not invertible). The persistence means that for any $C^r$-perturbation of the dynamics, there exists a close $C^r$-stratification preserved by the perturbation. This theorem in its elementary statement (the stratification is constituted by a unique stratum) gives the persistence of normally expanded laminations by endomorphisms, generalizing HPS theory. Another application of this theorem is the persistence, as stratifications, of submanifolds with boundary or corners normally expanded. Moreover, we remark that SA diffeomorphism gives a canonical stratifications: the stratification whose strata are the stable sets of basic pieces of the spectral decomposition. Our Main theorem then implies the persistence of some ``normally SA'' laminations which are not normally hyperbolic."}
{"category": "Math", "title": "Ergodicity and hydrodynamic limits for an epidemic model", "abstract": "We consider two approaches to study the spread of infectious diseases within a spatially structured population distributed in social clusters. According whether we consider only the population of infected individuals or both populations of infected individuals and healthy ones, two models are given to study an epidemic phenomenon. Our first approach is at a microscopic level, its goal is to determine if an epidemic may occur for those models. The second one is the derivation of hydrodynamics limits. By using the relative entropy method we prove that the empirical measures of infected and healthy individuals converge to a deterministic measure absolutely continuous with respect to the Lebesgue measure, whose density is the solution of a system of reaction-diffusion equations."}
{"category": "Math", "title": "A regularity criterion for the dissipative quasi-geostrophic equations", "abstract": "We establish a regularity criterion for weak solutions of the dissipative quasi-geostrophic equations in mixed time-space Besov spaces."}
{"category": "Math", "title": "The category of 3-computads is not cartesian closed", "abstract": "We show, using Eckmann-Hilton argument, that the category of 3-computads is not cartesian closed. As a corollary we get that neither the category of all computads nor the category of n-computads, for n>2, do form locally cartesian closed categories, and hence elementary toposes."}
{"category": "Math", "title": "K\\\"ahler Ricci flow on Fano surfaces (I)", "abstract": "We show the properties of the blowup limits of \\KRf solutions on Fano surfaces if Riemannian curvature is unbounded. As an application, on every toric Fano surface, we prove that \\KRf converges to a K\\\"ahler Ricci soliton metric if the initial metric has toric symmetry. Therefore we give a new Ricci flow proof of existence of K\\\"ahler Ricci soliton metrics on toric surfaces."}
{"category": "Math", "title": "A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic", "abstract": "We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero."}
{"category": "Math", "title": "Anti-affine algebraic groups", "abstract": "We say that an algebraic group $G$ over a field is anti-affine if every regular function on $G$ is constant. We obtain a classification of these groups, with applications to the structure of algebraic groups in positive characteristics, and to the construction of many counterexamples to Hilbert's fourteenth problem."}
{"category": "Math", "title": "A note on singularity and non-proper value set of polynomial maps of $\\mathbb{C}^2$", "abstract": "Some properties of the relation between the singular point set and the non-proper value curve of polynomial maps of $\\mathbb{C}^2$ are expressed in terms of Newton-Puiseux expansions."}
{"category": "Math", "title": "Tensor factorization and Spin construction for Kac-Moody algebras", "abstract": "In this paper we discuss the \"Factorization phenomenon\" which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into a tensor product of smaller representations of the subalgebra. We analyze this phenomenon for symmetrizable Kac-Moody algebras (including finite-dimensional, semi-simple Lie algebras). We present a few factorization results for a general embedding of a symmetrizable Kac-Moody algebra into another and provide an algebraic explanation for such a phenomenon using Spin construction. We also give some application of these results for semi-simple finite dimensional Lie algebras. We extend the notion of Spin functor from finite-dimensional to symmetrizable Kac-Moody algebras, which requires a very delicate treatment. We introduce a certain category of orthogonal $\\g$-representations for which, surprisingly, the Spin functor gives a $\\g$-representation in Bernstein-Gelfand-Gelfand category $\\O$. Also, for an integrable representation $\\Spin$ produces an integrable representation. We give the formula for the character of Spin representation for the above category and work out the factorization results for an embedding of a finite dimensional semi-simple Lie algebra into its untwisted affine Lie algebra. Finally, we discuss classification of those representations for which $\\Spin$ is irreducible."}
{"category": "Math", "title": "Local linear regression for functional data", "abstract": "We study a non linear regression model with functional data as inputs and scalar response. We propose a pointwise estimate of the regression function that maps a Hilbert space onto the real line by a local linear method. We provide the asymptotic mean square error. Computations involve a linear inverse problem as well as a representation of the small ball probability of the data and are based on recent advances in this area. The rate of convergence of our estimate outperforms those already obtained in the literature on this model."}
{"category": "Math", "title": "Derivation of a Convection Process in a Steady Diffusion-Transfer Problem by Homogenization", "abstract": "We study the homogenization of a steady diffusion equation in a highly heterogeneous medium made of two subregions separated by a periodic barrier through which the flow is proportional to the jump of the temperature by a layer conductance of the same order of magnitude of the materials in consideration. The macroscopic governing equations and the effective conductivity of the homogenized model are obtained by means of the two scale convergence technique. We show that under some hypothesis the homogenized systems contain convective terms of order one."}
{"category": "Math", "title": "Another View on the Hoelder Inequality", "abstract": "Every diagonalmatrix D yields an endomorphism on the n-dimensional complex vectorspace. If one provides this space with Hoelder norms, we can compute the operator norm of D. We define homogeneous weighted spaces as a generalization of normed spaces. We generalize the Hoelder norms for negative values, this leads to a proof of an extented version of the Hoelder inequality. Finally, we formulate this version also for measurable functions."}
{"category": "Math", "title": "Abstract interpolation problem in Nevanlinna classes", "abstract": "The abstract interpolation problem (AIP) in the Schur class was posed V. Katznelson, A. Kheifets and P. Yuditskii in 1987 as an extension of the V.P. Potapov's approach to interpolation problems. In the present paper an analog of the AIP for Nevanlinna classes is considered. The description of solutions of the AIP is reduced to the description of L-resolvents of some model symmetric operator associated with the AIP. The latter description is obtained by using the M.G. Krein's theory of L-resolvent matrices. Both regular and singular cases of the AIP are treated. The results are illustrated by the following examples: bitangential interpolation problem, full and truncated moment problems. It is shown that each of these problems can be included into the general scheme of the AIP."}
{"category": "Math", "title": "Affine Demazure modules and $T$-fixed point subschemes in the affine Grassmannian", "abstract": "Let $G$ be a simple algebraic group of type $A$ or $D$ defined over $\\C$ and $T$ be a maximal torus of $G$. For a dominant coweight $\\lambda$ of $G$, the $T$-fixed point subscheme $(\\bar{Gr}_G^\\lambda)^T$ of the Schubert variety $\\bar{Gr}_G^\\lambda$ in the affine Grassmannian $Gr_G$ is a finite scheme. We prove that there is a natural isomorphism between the dual of the level one affine Demazure module corresponding to $\\lambda$ and the ring of functions (twisted by certain line bundle on $Gr_G$) of $(\\bar{Gr}_G^\\lambda)^T$. We use this fact to give a geometric proof of the Frenkel-Kac-Segal isomorphism between basic representations of affine algebras of $A,D,E$ type and lattice vertex algebras."}
{"category": "Math", "title": "Positivity of Thom polynomials II: the Lagrange singularities", "abstract": "We show that Thom polynomials of Lagrangian singularities have nonnegative coefficients in the basis consisting of Q-functions. The main tool in the proof is nonnegativity of cone classes for globally generated bundles."}
{"category": "Math", "title": "Cycle-free chessboard complexes and symmetric homology of algebras", "abstract": "Chessboard complexes and their relatives have been one of important recurring themes of topological combinatorics. Closely related ``cycle-free chessboard complexes'' have been recently introduced by Ault and Fiedorowicz as a tool for computing symmetric analogues of the cyclic homology of algebras. We study connectivity properties of these complexes and prove a result that confirms a strengthened conjecture of Ault and Fiedorowicz."}
{"category": "Math", "title": "Motivic L-Functions", "abstract": "This is the text of an introductory lecture delivered at the IHES summer school on motives in July, 2006."}
{"category": "Math", "title": "Diophantine Geometry as Galois Theory in the Mathematics of Serge Lang", "abstract": "A remark about the role of Galois theory in Diophantine geometry as reflected in the work of Serge Lang. An entry in `The mathematical contributions of Serge Lang.'"}
{"category": "Math", "title": "Geometry of Character Varieties of Surface Groups", "abstract": "This article is based on a talk delivered at the RIMS--OCAMI Joint International Conference on Geometry Related to Integrable Systems in September, 2007. Its aim is to review a recent progress in the Hitchin integrable systems and character varieties of the fundamental groups of Riemann surfaces. A survey on geometric aspects of these character varieties is also provided as we develop the exposition from a simple case to more elaborate cases."}
{"category": "Math", "title": "Goldman flows on a nonorientable surface", "abstract": "Given an embedded cylinder in an arbitrary surface, we give a gauge theoretic definition of the associated Goldman flow, which is a circle action on a dense open subset of the moduli space of equivalence classes of flat SU(2)-connections over the surface. A cylinder in a compact nonorientable surface lifts to two cylinders in the orientable double cover, and the \"composite flow\" is the composition of one of the associated flows with the inverse flow of the other. Providing explicit descriptions, we relate the flow on the moduli space of the nonorientable surface with the composite flow on the moduli space of the double cover. We prove that the composite flow preserves a certain Lagrangian submanifold."}
{"category": "Math", "title": "How to evaluate the calibration of a disease risk prediction tool", "abstract": "To evaluate the calibration of a disease risk prediction tool, the quantity $E/O$, i.e., the ratio of the expected number of events to the observed number of events, is generally computed. However, because of censoring, or more precisely because of individuals who drop out before the termination of the study, this quantity is generally unavailable for the complete population study and an alternative estimate has to be computed. In this paper, we present and compare four methods to do this. We show that two of the most commonly used methods generally lead to biased estimates. Our arguments are first based on some theoretic considerations. Then, we perform a simulation study to highlight the magnitude of the previously mentioned biases. As a concluding example, we evaluate the calibration of an existing predictive model for breast cancer on the E3N-EPIC cohort."}
{"category": "Math", "title": "Truncated decompositions and filtering methods with Reflective/Anti-Reflective boundary conditions: a comparison", "abstract": "The paper analyzes and compares some spectral filtering methods as truncated singular/eigen-value decompositions and Tikhonov/Re-blurring regularizations in the case of the recently proposed Reflective [M.K. Ng, R.H. Chan, and W.C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21 (1999), no. 3, pp.851-866] and Anti-Reflective [S. Serra Capizzano, A note on anti-reflective boundary conditions and fast deblurring models, SIAM J. Sci. Comput., 25-3 (2003), pp. 1307-1325] boundary conditions. We give numerical evidence to the fact that spectral decompositions (SDs) provide a good image restoration quality and this is true in particular for the Anti-Reflective SD, despite the loss of orthogonality in the associated transform. The related computational cost is comparable with previously known spectral decompositions, and results substantially lower than the singular value decomposition. The model extension to the cross-channel blurring phenomenon of color images is also considered and the related spectral filtering methods are suitably adapted."}
{"category": "Math", "title": "Differential equations associated with nonarithmetic Fuchsian groups", "abstract": "We describe globally nilpotent differential operators of rank 2 defined over a number field whose monodromy group is a nonarithmetic Fuchsian group. We show that these differential operators have an S-integral solution. These differential operators are naturally associated with Teichmueller curves in genus 2. They are counterexamples to conjectures by Chudnovsky--Chudnovsky and Dwork. We also determine the field of moduli of primitive Teichmueller curves in genus 2, and an explicit equation in some cases."}
{"category": "Math", "title": "Normal subgroups of the algebraic fundamental group of affine curves in positive characteristic", "abstract": "Let $\\pi_1(C)$ be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic $p>0$ of countable cardinality. Let $N$ be a normal (resp. characteristic) subgroup of $\\pi_1(C)$. Under the hypothesis that the quotient $\\pi_1(C)/N$ admits an infinitely generated Sylow $p$-subgroup, we prove that $N$ is indeed isomorphic to a normal (resp. characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of $N$ is a free profinite group of countable cardinality."}
{"category": "Math", "title": "Simplicial resolutions and Ganea fibrations", "abstract": "In this work, we compare the two approximations of a path-connected space $X$, by the Ganea spaces $G_n(X)$ and by the realizations $\\|\\Lambda_\\bullet X\\|_{n}$ of the truncated simplicial resolutions emerging from the loop-suspension cotriple $\\Sigma\\Omega$. For a simply connected space $X$, we construct maps $\\|\\Lambda_\\bullet X\\|_{n-1}\\to G_n(X)\\to \\|\\Lambda_\\bullet X\\|_{n}$ over $X$, up to homotopy. In the case $n=2$, we prove the existence of a map $G_2(X)\\to\\|\\Lambda_\\bullet X\\|_{1}$ over $X$ (up to homotopy) and conjecture that this map exists for any $n$."}
{"category": "Math", "title": "p-adic L-functions and Selmer varieties associated to elliptic curves with complex multiplication", "abstract": "We show how non-vanishing of p-adic L functions controls the dimensions of Selmer varieties associated to the complement of the origin in an elliptic curve with CM. As a corollary, one obtains a \\pi_1-proof of the theorem of Siegel for such curves."}
{"category": "Math", "title": "A note on Berestycki-Cazenave's classical instability result for nonlinear Schr\\\"odinger equations", "abstract": "In this note we give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schr\\\"odinger equations."}
{"category": "Math", "title": "Act globally, compute locally: group actions, fixed points, and localization", "abstract": "Localization is a topological technique that allows us to make global equivariant computations in terms of local data at the fixed points. For example, we may compute a global integral by summing integrals at each of the fixed points. Or, if we know that the global integral is zero, we conclude that the sum of the local integrals is zero. This often turns topological questions into combinatorial ones and vice versa. This expository article features several instances of localization that occur at the crossroads of symplectic and algebraic geometry on the one hand, and combinatorics and representation theory on the other. The examples come largely from the symplectic category, with particular attention to toric varieties. In the spirit of the 2006 International Conference on Toric Topology at Osaka City University, the main goal of this exposition is to exhibit toric techniques that arise in symplectic geometry."}
{"category": "Math", "title": "Comparison study for Level set and Direct Lagrangian methods for computing Willmore flow of closed planar curves", "abstract": "The main goal of this paper is to present results of comparison study for the level set and direct Lagrangian methods for computing evolution of the Willmore flow of embedded planar curves. To perform such a study we construct new numerical approximation schemes for both Lagrangian as well as level set methods based on semi-implicit in time and finite/complementary volume in space discretizations. The Lagrangian scheme is stabilized in tangential direction by the asymptotically uniform grid point redistribution. Both methods are experimentally second order accurate. Moreover, we show precise coincidence of both approaches in case of various elastic curve evolutions provided that solving the linear systems in semi-implicit level set method is done in a precise way, redistancing is performed occasionally and the influence of boundary conditions on the level set function is eliminated."}
{"category": "Math", "title": "Interpolation maps and congruence domains for wavelet sets", "abstract": "It is proven that if an interpolation map between two wavelet sets preserves the union of the sets, then the pair must be an interpolation pair. We also construct an example of a pair of wavelet sets for which the congruence domains of the associated interpolation map and its inverse are equal, and yet the pair is not an interpolation pair. The first result solves affirmatively a problem that the second author had posed several years ago, and the second result solves an intriguing problem of D. Han. The key to this counterexample is a special technical lemma on constructing wavelet sets. Several other applications of this result are also given. In addition, some problems are posed. We also take the opportunity to give some general exposition on wavelet sets and operator-theoretic interpolation of wavelets."}
{"category": "Math", "title": "On tangential stabilization in curvature driven flows of planar curves", "abstract": "We discuss the role of tangential stabilization in a curvature driven flow of planar curves. The governing system of nonlinear parabolic equations includes a nontrivial tangential velocity functional yielding a uniform redistribution of grid points along the evolving family of curves preventing numerically computed curves from forming various instabilities."}
{"category": "Math", "title": "All superconformal surfaces in \\R^4 in terms of minimal surfaces", "abstract": "We give an explicit construction of any simply-connected superconformal surface $\\phi\\colon M^2\\to \\R^4$ in Euclidean space in terms of a pair of conjugate minimal surfaces $g,h\\colon M^2\\to\\R^4$. That $\\phi$ is superconformal means that its ellipse of curvature is a circle at any point. We characterize the pairs $(g,h)$ of conjugate minimal surfaces that give rise to images of holomorphic curves by an inversion in $\\R^4$ and to images of superminimal surfaces in either a sphere $\\Sf^4$ or a hyperbolic space $\\Hy^4$ by an stereographic projection. We also determine the relation between the pairs $(g,h)$ of conjugate minimal surfaces associated to a superconformal surface and its image by an inversion. In particular, this yields a new transformation for minimal surfaces in $\\R^4$."}
{"category": "Math", "title": "An effective recursion formula for computing intersection numbers", "abstract": "We prove a new effective recursion formula for computing all intersection indices (integrals of $\\psi$ classes) on the moduli space of curves, inducting only on the genus."}
{"category": "Math", "title": "Multitopes are the same as principal ordered face structures", "abstract": "We show that the category of principal ordered face structures is equivalent to the category of multitopes. We show that the category of principal ordered face structures is equivalent to the category of multitopes. On the way we introduce the notion of a graded tensor theory to state the abstract properties of the category of ordered face structures and show how it fits into the recent work of T. Leinster and M. Weber concerning the nerve construction."}
{"category": "Math", "title": "Monotonicity formulas under rescaled Ricci flow", "abstract": "In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem \\ref{theo rescaled}."}
{"category": "Math", "title": "The moduli space of cubic threefolds via degenerations of the intermediate Jacobian", "abstract": "A well known result of Clemens and Griffiths says that a smooth cubic threefold can be recovered from its intermediate Jacobian. In this paper we discuss the possible degenerations of these abelian varieties, and thus give a description of the compactification of the moduli space of cubic threefolds obtained in this way. The relation between this compactification and those constructed in the work of Allcock-Carlson-Toledo and Looijenga-Swierstra is also considered, and is similar in spirit to the relation between the various compactifications of the moduli spaces of low genus curves."}
{"category": "Math", "title": "On injectivity of maps between Grothendieck groups induced by completion", "abstract": "We give an example of a local normal domain $R$ such that the map of Grothendieck groups $\\G(R) \\to \\G(\\hat R)$ is not injective. We also raise some questions about the kernel of that map."}
{"category": "Math", "title": "A geometric approach to maximum likelihood estimation of the functional principal components from sparse longitudinal data", "abstract": "In this paper, we consider the problem of estimating the eigenvalues and eigenfunctions of the covariance kernel (i.e., the functional principal components) from sparse and irregularly observed longitudinal data. We approach this problem through a maximum likelihood method assuming that the covariance kernel is smooth and finite dimensional. We exploit the smoothness of the eigenfunctions to reduce dimensionality by restricting them to a lower dimensional space of smooth functions. The estimation scheme is developed based on a Newton-Raphson procedure using the fact that the basis coefficients representing the eigenfunctions lie on a Stiefel manifold. We also address the selection of the right number of basis functions, as well as that of the dimension of the covariance kernel by a second order approximation to the leave-one-curve-out cross-validation score that is computationally very efficient. The effectiveness of our procedure is demonstrated by simulation studies and an application to a CD4 counts data set. In the simulation studies, our method performs well on both estimation and model selection. It also outperforms two existing approaches: one based on a local polynomial smoothing of the empirical covariances, and another using an EM algorithm."}
{"category": "Math", "title": "Two addition theorems on polynomials of prime variables", "abstract": "We extend a recent result of Khalfalah and Szemeredi to the polynomials of prime variables."}
{"category": "Math", "title": "Gr\\\"obner bases of simplicial toric ideals", "abstract": "Bounds for the maximal degree of certain Gr\\\"obner bases of simplicial toric ideals are given. These bounds are close to the bound stated in Eisenbud-Goto's Conjecture on the Castelnuovo-Mumford regularity."}
{"category": "Math", "title": "A scale-based approach to finding effective dimensionality in manifold learning", "abstract": "The discovering of low-dimensional manifolds in high-dimensional data is one of the main goals in manifold learning. We propose a new approach to identify the effective dimension (intrinsic dimension) of low-dimensional manifolds. The scale space viewpoint is the key to our approach enabling us to meet the challenge of noisy data. Our approach finds the effective dimensionality of the data over all scale without any prior knowledge. It has better performance compared with other methods especially in the presence of relatively large noise and is computationally efficient."}
{"category": "Math", "title": "Elementary evaluations of some Euler sums", "abstract": "This short note contains elementary evaluations of some Euler sums."}
{"category": "Math", "title": "On the bounded cohomology of semi-simple groups, S-arithmetic groups and products", "abstract": "We prove vanishing results for Lie groups and algebraic groups (over any local field) in bounded cohomology. The main result is a vanishing below twice the rank for semi-simple groups. Related rigidity results are established for S-arithmetic groups and groups over global fields. We also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices."}
{"category": "Math", "title": "Algebraic differential characters of flat connections with nilpotent residues", "abstract": "We construct unramified algebraic differential characters for flat connections with nilpotent residues along a strict normal crossings divisor."}
{"category": "Math", "title": "Measured Quantum Groupoids in action", "abstract": "Franck Lesieur had introduced in his thesis (now published in an expended and revised version in the {\\it M\\'emoires de la SMF} (2007)) a notion of measured quantum groupoid, in the setting of von Neumann algebras and a simplification of Lesieur's axioms is presented in an appendix of this article. We here develop the notions of actions, crossed-product, and obtain a biduality theorem, following what had been done by Stefaan Vaes for locally compact quantum groups. Moreover, we prove that the inclusion of the initial algebra into its crossed-product is depth 2, which gives a converse of a result proved by Jean-Michel Vallin and the author. More precisely, to any action of a measured quantum groupoid, we associate another measured quantum groupoid. In particular, starting from an action of a locally compact quantum group, we obtain a measured quantum groupoid canonically associated to this action; when the action is outer, this measured quantum groupoid is the initial locally compact quantum group"}
{"category": "Math", "title": "Coexistence in locally regulated competing populations and survival of branching annihilating random walk", "abstract": "We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which do not, at present, incorporate all of the competitive strategies that a population might adopt. The second is a simplification of the first, in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching annihilating random walk. For each model, using a comparison with oriented percolation, we show that for certain parameter values, both populations will coexist for all time with positive probability. As a corollary, we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates. We also present a number of conjectures relating to the r\\^{o}le of space in the survival probabilities for the two populations."}
{"category": "Math", "title": "Gruenhage compacta and strictly convex dual norms", "abstract": "We prove that if K is a Gruenhage compact space then C(K)* admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X* is the |.|-closed linear span of K, where K is a Gruenhage compact in the w*-topology and |.| is equivalent to a coarser, w*-lower semicontinuous norm on X*, then X* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if T is a tree, then C(T)* admits an equivalent, strictly convex dual norm if and only if T is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images."}
{"category": "Math", "title": "Characteristic varieties of nilpotent groups and applications", "abstract": "We compute the characteristic varieties and the Alexander polynomial of a finitely generated nilpotent group. We show that the first characteristic variety may be used to detect nilpotence. We use the Alexander polynomial to deduce that the only torsion-free, finitely generated nilpotent groups with positive deficiency are $\\Z$ and $\\Z^2$, extending a classical result on nilpotent link groups."}
{"category": "Math", "title": "Stationary distribution for dioecious branching particle systems with rapid stirring", "abstract": "We study dioecious (i.e., two-sex) branching particle system models, where there are two types of particles, modeling the male and female populations, and where birth of new particles requires the presence of both male and female particles. We show that stationary distributions of various dioecious branching particle models are nontrivial under certain conditions, for example, when there is sufficiently fast stirring."}
{"category": "Math", "title": "Large, global solutions to the Navier-Stokes equations, slowly varying in one direction", "abstract": "In to previous papers by the authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions, in the whole space. Contrary to the previous examples, the initial data has no particular oscillatory properties, but varies slowly in one direction. The proof uses the special structure of the nonlinear term of the equation."}
{"category": "Math", "title": "On the Hodge decomposition in R^n", "abstract": "We prove a version of the $L^p$ hodge decomposition for differential forms in Euclidean space and a generalization to the class of Lizorkin currents. We also compute the $L_{qp}-$cohomology of $\\mathbb{R}^n$."}
{"category": "Math", "title": "Generating functions for borders", "abstract": "We give the generating function for the index of integer lattice points, relative to a finite order ideal. The index is an important concept in the theory of border bases, an alternative to Gr\\\"obner bases."}
{"category": "Math", "title": "Another Look at AR(1)", "abstract": "Given a stationary first-order autoregressive process X_t (with lag-one correlation rho satisfying |rho|<1), we examine the Central Limit Theorem for (1/n)*ln |X_1...X_n| and compute variances to high precision. Given a nonstationary process X_t (with |rho|>1), we examine instead (1/n)*ln|X_n| and study the distribution of ln|X_n|-n*ln|rho|."}
{"category": "Math", "title": "On the existence of a v_2^32-self map on M(1,4) at the prime 2", "abstract": "Let M(1) be the mod 2 Moore spectrum. J.F. Adams proved that M(1) admits a minimal v_1-self map v_1^4: Sigma^8 M(1) -> M(1). Let M(1,4) be the cofiber of this self-map. The purpose of this paper is to prove that M(1,4) admits a minimal v_2-self map of the form v_2^32: Sigma^192 M(1,4) -> M(1,4). The existence of this map implies the existence of many 192-periodic families of elements in the stable homotopy groups of spheres."}
{"category": "Math", "title": "Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging", "abstract": "We propose a general method to study dependent data in a binary tree, where an individual in one generation gives rise to two different offspring, one of type 0 and one of type 1, in the next generation. For any specific characteristic of these individuals, we assume that the characteristic is stochastic and depends on its ancestors' only through the mother's characteristic. The dependency structure may be described by a transition probability $P(x,dy dz)$ which gives the probability that the pair of daughters' characteristics is around $(y,z)$, given that the mother's characteristic is $x$. Note that $y$, the characteristic of the daughter of type 0, and $z$, that of the daughter of type 1, may be conditionally dependent given $x$, and their respective conditional distributions may differ. We then speak of bifurcating Markov chains. We derive laws of large numbers and central limit theorems for such stochastic processes. We then apply these results to detect cellular aging in Escherichia Coli, using the data of Stewart et al. and a bifurcating autoregressive model."}
{"category": "Math", "title": "Parameter Estimation for Partially Observed Hypoelliptic Diffusions", "abstract": "Hypoelliptic diffusion processes can be used to model a variety of phenomena in applications ranging from molecular dynamics to audio signal analysis. We study parameter estimation for such processes in situations where we observe some components of the solution at discrete times. Since exact likelihoods for the transition densities are typically not known, approximations are used that are expected to work well in the limit of small inter-sample times $\\Delta t$ and large total observation times $N\\Delta t$. Hypoellipticity together with partial observation leads to ill-conditioning requiring a judicious combination of approximate likelihoods for the various parameters to be estimated. We combine these in a deterministic scan Gibbs sampler alternating between missing data in the unobserved solution components, and parameters. Numerical experiments illustrate asymptotic consistency of the method when applied to simulated data. The paper concludes with application of the Gibbs sampler to molecular dynamics data."}
{"category": "Math", "title": "Weak order for the discretization of the stochastic heat equation", "abstract": "In this paper we study the approximation of the distribution of $X_t$ Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as $$ dX_t+AX_t dt = Q^{1/2} d W_t, \\quad X_0=x \\in H, \\quad t\\in[0,T], $$ driven by a Gaussian space time noise whose covariance operator $Q$ is given. We assume that $A^{-\\alpha}$ is a finite trace operator for some $\\alpha>0$ and that $Q$ is bounded from $H$ into $D(A^\\beta)$ for some $\\beta\\geq 0$. It is not required to be nuclear or to commute with $A$. The discretization is achieved thanks to finite element methods in space (parameter $h>0$) and implicit Euler schemes in time (parameter $\\Delta t=T/N$). We define a discrete solution $X^n_h$ and for suitable functions $\\phi$ defined on $H$, we show that $$ |\\E \\phi(X^N_h) - \\E \\phi(X_T) | = O(h^{2\\gamma} + \\Delta t^\\gamma) $$ \\noindent where $\\gamma<1- \\alpha + \\beta$. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations."}
{"category": "Math", "title": "Integral elements of K-theory and products of modular curves II", "abstract": "We discuss the relationship between different notions of \"integrality\" in motivic cohomology/K-theory which arise in the Beilinson and Bloch-Kato conjectures, and prove their equivalence in some cases for products of curves (used in the authors' previous paper in this series), as well as obtaining a general result, first proved by Jannsen (unpublished), which reduces their equivalence to standard conjectures in arithmetic algebraic geometry."}
{"category": "Math", "title": "Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle", "abstract": "We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states), some non-monotone Lagrangian torus fibers. We also extend the results by Oh and the author about the computations of Floer cohomology of Lagrangian torus fibers to the case of all toric Fano manifolds, removing the convexity assumption in the previous work."}
{"category": "Math", "title": "Cubist Algebras", "abstract": "We construct algebras from rhombohedral tilings of Euclidean space obtained as projections of certain cubical complexes. We show that these `Cubist algebras' satisfy strong homological properties, such as Koszulity and quasi-heredity, reflecting the combinatorics of the tilings. We construct derived equivalences between Cubist algebras associated to local mutations in tilings. We recover as a special case the Rhombal algebras of Michael Peach and make a precise connection to weight 2 blocks of symmetric groups."}
{"category": "Math", "title": "Rock blocks", "abstract": "Consider representation theory associated to symmetric groups, or to Hecke algebras in type A, or to q-Schur algebras, or to finite general linear groups in non-describing characteristic. Rock blocks are certain combinatorially defined blocks appearing in such a representation theory, first observed by R. Rouquier. Rock blocks are much more symmetric than general blocks, and every block is derived equivalent to a Rock block. Motivated by a theorem of J. Chuang and R. Kessar in the case of symmetric group blocks of abelian defect, we pursue a structure theorem for these blocks."}
{"category": "Math", "title": "Local invariants attached to Weierstrass points", "abstract": "Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation of the hyperelliptic discriminant of X/S, and the valuation of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable these invariants are known to satisfy certain inequalities. We prove an exact formula relating the two invariants with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the semistable case. We also prove an exact formula for the stable Faltings height of an arbitrary curve over a number field, involving local contributions associated to its Weierstrass points."}
{"category": "Math", "title": "Projective bases of division algebras and groups of central type II", "abstract": "Let G be a finite group and let k be a field. We say that G is a projective basis of a k-algebra A if it is isomorphic to a twisted group algebra k^\\alpha G for some class \\alpha in H^2(G,k^\\times), where the action of G on k^\\times is trivial. In a preceding paper by Aljadeff, Haile and the author (Projective bases of division algebras and groups of central type, Israel J. Math. 146 (2005) 317-335) it was shown that if a group G is a projective basis in a k-central division algebra then G is nilpotent and every Sylow-p subgroup of G is on the short list of families of p-groups, denoted by \\Lambda. In this paper we complete the classification of projective bases of division algebras by showing that every group on that list is a projective basis for a suitable division algebra. We also consider the question of uniqueness of a projective basis of a k-central division algebra. We show that basically all groups on the list \\Lambda but one satisfy certain rigidity property."}
{"category": "Math", "title": "Centres of skewfields and completely faithful Iwasawa modules", "abstract": "Let H be a torsionfree compact p-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebra \\Lambda_H of H has trivial centre and use this result to classify the prime c-ideals in the Iwasawa algebra \\Lambda_G of G := H \\times \\Zp. We also show that a finitely generated torsion \\Lambda_G-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves."}
{"category": "Math", "title": "The work of Jesse Douglas on Minimal Surfaces", "abstract": "This paper describes the work of Jesse Douglas on the Plateau problem, work for which he was awarded a Fields Medal in 1936, and considers the contributions Tibor Rado made in the 1930s."}
{"category": "Math", "title": "Mild Solutions for a Class of Fractional SPDEs and Their Sample Paths", "abstract": "In this article we introduce and analyze a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\\subset\\mathbb{R}^{d}$ and driven by an infinite-dimensional fractional noise. The noise is derived from an $L^{2}(D)$-valued fractional Wiener process $W^{H}$ whose covariance operator satisfies appropriate restrictions; moreover, the Hurst parameter $H$ is subjected to constraints formulated in terms of $d$ and the H\\\"{o}lder exponent of the derivative $h^\\prime$ of the noise nonlinearity in the equations. We prove the existence of such solution, establish its relation with the variational solution introduced in \\cite{nuavu} and also prove the H\\\"{o}lder continuity of its sample paths when we consider it as an $L^{2}(D)$--valued stochastic processes. When $h$ is an affine function, we also prove uniqueness. The proofs are based on a relation between the notions of mild and variational solution established in Sanz-Sol\\'e and Vuillermot 2003, and adapted to our problem, and on a fine analysis of the singularities of Green's function associated with the class of parabolic problems we investigate. An immediate consequence of our results is the indistinguishability of mild and variational solutions in the case of uniqueness."}
{"category": "Math", "title": "Ricci Yang-Mills flow on surfaces", "abstract": "We study the behaviour of the Ricci Yang-Mills flow for U(1) bundles on surfaces. We show that existence for the flow reduces to a bound on the isoperimetric constant. In the presence of such a bound, we show that on $S^2$, if the bundle is nontrivial, the flow exists for all time. For higher genus surfaces the flow always exists for all time. The volume normalized flow always exists for all time and converges to a constant scalar curvature metric with the bundle curvature $F$ parallel. Finally, in an appendix we classify all gradient solitons of this flow on surfaces."}
{"category": "Math", "title": "Koszul Equivalences in $A_\\infty$-Algebras", "abstract": "We prove a version of Koszul duality and the induced derived equivalence for Adams connected $A_\\infty$-algebras that generalizes the classical Beilinson-Ginzburg-Soergel Koszul duality. As an immediate consequence, we give a version of the Bern\\v{s}te{\\u\\i}n-Gel'fand-Gel'fand correspondence for Adams connected $A_\\infty$-algebras. We give various applications. For example, a connected graded algebra $A$ is Artin-Schelter regular if and only if its Ext-algebra $\\Ext^\\ast_A(k,k)$ is Frobenius. This generalizes a result of Smith in the Koszul case. If $A$ is Koszul and if both $A$ and its Koszul dual $A^!$ are noetherian satisfying a polynomial identity, then $A$ is Gorenstein if and only if $A^!$ is. The last statement implies that a certain Calabi-Yau property is preserved under Koszul duality."}
{"category": "Math", "title": "Looking for rational curves on cubic hypersurfaces", "abstract": "These are the substantially expanded notes of the lectures of JK at the summer school \"Higher-Dimensional Geometry over Finite Fields\" in G\\\"ottingen, June 2007. The first part gives an overview of the methods. The main new result is the construction of rational curves passing through a given collection of points on smooth cubic hypersurfaces over finite fields."}
{"category": "Math", "title": "Metric properties of the braided Thompson's groups", "abstract": "Braided Thompson's groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups $B_n$, the finitary braid group $B_{\\infty}$ and Thompson's group $F$ as subgroups. We describe some of the metric properties of braided Thompson's groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams used to represent elements."}
{"category": "Math", "title": "Algebraic series and valuation rings over nonclosed fields", "abstract": "Suppose that $k$ is an arbitrary field. Consider the field $k((x_1,...,x_n))$, which is the quotient field of the ring $k[[x_1,...,x_n]]$ of formal power series in the variables $x_1,...,x_n$, with coefficients in $k$. Suppose that $\\sigma$ is a formal power series in $x_1,...,x_n$ with coefficints in the algebraic closure of $k$. We give a very simple necessary and sufficient condition for $\\sigma$ to be algebraic over $k((x_1,...,x_n))$. As an application of our methods, we give a characterization of valuation rings $V$ which dominate an excellent, Noetherian local domain $R$ of dimension two, and such that the rank increases after passing to the completion of a birational extension of $R$."}
{"category": "Math", "title": "Notes on formal smoothness", "abstract": "The definition of an S-category is proposed by weakening the axioms of a Q-category introduced by Kontsevich and Rosenberg. Examples of Q- and S-categories and (co)smooth objects in such categories are given."}
{"category": "Math", "title": "Extending a theorem of Herstein", "abstract": "Just infinite algebras have been considered from various perspectives; a common thread in these treatments is that the notion of just infinite is an extension of the notion of simple. We reinforce this generalization by considering some well-known results of Herstein regarding simple rings and their Lie and Jordan structures and extend these results to their just infinite analogues. In particular, we prove that if A is a just infinite associative algebra, of characteristic not 2,3, or 5, then the Lie algebra $[A,A]/(Z\\cap[A,A])$ is also just infinite (where Z denotes the center of A)."}
{"category": "Math", "title": "On measure-preserving ${\\mathcal C}^1$ transformations of compact-open subsets of non-archimedean local fields", "abstract": "We introduce the notion of a \\emph{locally scaling} transformation defined on a compact-open subset of a non-archimedean local field. We show that this class encompasses the Haar measure-preserving transformations defined by ${\\mathcal C}^1$ (in particular, polynomial) maps, and prove a structure theorem for locally scaling transformations. We use the theory of polynomial approximation on compact-open subsets of non-archimedean local fields to demonstrate the existence of ergodic Markov, and mixing Markov transformations defined by such polynomial maps. We also give simple sufficient conditions on the Mahler expansion of a continuous map $\\mathbb Z_p \\to \\mathbb Z_p$ for it to define a Bernoulli transformation."}
{"category": "Math", "title": "A Note on $\\aleph_{0}$-injective Rings", "abstract": "A ring $R$ is called right $\\aleph_{0}$-injective if every homomorphism from a countably generated right ideal of $R$ to $R_{R}$ can be extended to a homomorphism from $R_{R}$ to $R_{R}$. In this note, some characterizations of $\\aleph_{0}$-injective rings are given. It is proved that if $R$ is semilocal, then $R$ is right $\\aleph_{0}$-injective if and only if every homomorphism from a countably generated small right ideal of $R$ to $R_{R}$ can be extended to one from $R_{R}$ to $R_{R}$. It is also shown that if $R$ is right noetherian and left $\\aleph_{0}$-injective, then $R$ is \\emph{QF}. This result can be considered as an approach to the Faith-Menal conjecture."}
{"category": "Math", "title": "An infinite dimensional Schur-Horn theorem and majorization theory with applications to operator ideals", "abstract": "The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences x and y that converge to 0, there exists a compact operator A with eigenvalue list y and diagonal sequence x if and only if y majorizes x (\\sum_{j=1}^n x_j \\le \\sum_{j=1}^n y_j for all n) if and only if x = Qy for some orthostochastic matrix Q. The similar result requiring equality of the infinite series in the case that the sequences x and y are summable is an extension of a recent theorem by Arveson and Kadison. Our proof depends on the construction and analysis of an infinite product of T-transform matrices. Further results on majorization for infinite sequences providing \"intermediate\" sequences generalize known results from the finite case. Majorization properties and invariance under various classes of stochastic matrices are then used to characterize arithmetic mean closed operator ideals."}
{"category": "Math", "title": "A chain rule for Goodwillie derivatives of functors from spectra to spectra", "abstract": "We prove a chain rule for the Goodwillie calculus of functors from spectra to spectra. We show that the (higher) derivatives of a composite functor $FG$ at a base object $X$ are given by taking the composition product (in the sense of symmetric sequences) of the derivatives of $F$ at $G(X)$ with the derivatives of $G$ at $X$. We also consider the question of finding $P_n(FG)$, and give an explicit formula for this when $F$ is homogeneous."}
{"category": "Math", "title": "Graded identities of matrix algebras and the universal graded algebra", "abstract": "We consider fine G-gradings on M_n(C) (i.e. gradings of the matrix algebra over the complex numbers where each component is 1 dimensional). Groups which provide such a grading are known to be solvable. We consider the T-ideal of G-graded identities and show that it is generated by a special type of binomial identities which we call elementary. In particular we show that the ideal of graded identities is finitely generated as a T-ideal. Next, given such grading we construct a universal algebra U_{G,c} in two different ways: one by means of polynomial identities and the other one by means of a generic two-cocycle (this parallels the classical constructions in the non-graded case). We show that a suitable central localization of U_{G,c} is Azumaya over its center and moreover, its homomorphic images are precisely the G-graded forms of M_n(C). Finally, we consider the ring of central quotients Q(U_{G,c}) (this is an F-central simple algebra where F=Frac(Z) and Z is the center of of U_{G,c}). Using an earlier results of the authors (see E. Aljadeff, D. Haile and M. Natapov, Projective bases of division algebras and groups of central type, Israel J. Math.146 (2005) 317-335 and M. Natapov arXiv:0710.5468v1 [math.RA]) we show that this is a division algebra for a very explicit (and short) family of nilpotent groups. As a consequence, for groups G such that Q(U_{G,c}) is not a division algebra, one can find a non identity polynomial p(x_{i,g}) such that p(x_{i,g})^r is a graded identity for some integer r. We illustrate this phenomenon with a fine G-grading of M_6(C) where G is a semidirect product of S_3 and C_6."}
{"category": "Math", "title": "Ramsey-type problem for an almost monochromatic K_4", "abstract": "In this short note we prove that there is a constant $c$ such that every k-edge-coloring of the complete graph K_n with n > 2^{ck} contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and is the first exponential bound for this problem."}
{"category": "Math", "title": "Large deviations associated with Poisson--Dirichlet distribution and Ewens sampling formula", "abstract": "Several results of large deviations are obtained for distributions that are associated with the Poisson--Dirichlet distribution and the Ewens sampling formula when the parameter $\\theta$ approaches infinity. The motivation for these results comes from a desire of understanding the exact meaning of $\\theta$ going to infinity. In terms of the law of large numbers and the central limit theorem, the limiting procedure of $\\theta$ going to infinity in a Poisson--Dirichlet distribution corresponds to a finite allele model where the mutation rate per individual is fixed and the number of alleles going to infinity. We call this the finite allele approximation. The first main result of this article is concerned with the relation between this finite allele approximation and the Poisson--Dirichlet distribution in terms of large deviations. Large $\\theta$ can also be viewed as a limiting procedure of the effective population size going to infinity. In the second result a comparison is done between the sample size and the effective population size based on the Ewens sampling formula."}
{"category": "Math", "title": "Noncompact Shrinking 4-Solitons with Nonnegative Curvature", "abstract": "We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc>= 0."}
{"category": "Math", "title": "Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments", "abstract": "We introduce mountain-pass type arguments in the context of orbital instability for Klein-Gordon equations. Our aim is to illustrate on two examples how these arguments can be useful to simplify proofs and derive new results of orbital stability/instability. For a power-type nonlinearity, we prove that the ground states of the associated stationary equation are minimizers of the functional \"action\" on a wide variety of constraints. For a general nonlinearity, we extend to the dimension N=2 the classical instability result for stationary solutions of nonlinear Klein-Gordon equations proved in 1985 by Shatah in dimension N>2."}
{"category": "Math", "title": "Hall algebras associated to triangulated categories, II: almost associativity", "abstract": "By using the approach in \\cite{XX2006} to Hall algebras arising in homologically finite triangulated categories, we find an `almost' associative multiplication structure for indecomposable objects in a 2-periodic triangulated category. As an application, we give a new proof of the theorem of Peng and Xiao in \\cite{PX2000} which provides a way of realizing symmetrizable Kac-Moody algebras and elliptic Lie algebras via 2-periodic triangulated categories."}
{"category": "Math", "title": "Minimal $f^q$-martingale measures for exponential L\\'evy processes", "abstract": "Let $L$ be a multidimensional L\\'evy process under $P$ in its own filtration. The $f^q$-minimal martingale measure $Q_q$ is defined as that equivalent local martingale measure for $\\mathcal {E}(L)$ which minimizes the $f^q$-divergence $E[(dQ/dP)^q]$ for fixed $q\\in(-\\infty,0)\\cup(1,\\infty)$. We give necessary and sufficient conditions for the existence of $Q_q$ and an explicit formula for its density. For $q=2$, we relate the sufficient conditions to the structure condition and discuss when the former are also necessary. Moreover, we show that $Q_q$ converges for $q\\searrow1$ in entropy to the minimal entropy martingale measure."}
{"category": "Math", "title": "The two-type Richardson model with unbounded initial configurations", "abstract": "The two-type Richardson model describes the growth of two competing infections on $\\mathbb{Z}^d$ and the main question is whether both infection types can simultaneously grow to occupy infinite parts of $\\mathbb{Z}^d$. For bounded initial configurations, this has been thoroughly studied. In this paper, an unbounded initial configuration consisting of points $x=(x_1,...,x_d)$ in the hyperplane $\\mathcal{H}=\\{x\\in\\mathbb{Z}^d:x_1=0\\}$ is considered. It is shown that, starting from a configuration where all points in $\\mathcal{H} {\\mathbf{0}\\}$ are type 1 infected and the origin $\\mathbf{0}$ is type 2 infected, there is a positive probability for the type 2 infection to grow unboundedly if and only if it has a strictly larger intensity than the type 1 infection. If, instead, the initial type 1 infection is restricted to the negative $x_1$-axis, it is shown that the type 2 infection at the origin can also grow unboundedly when the infection types have the same intensity."}
{"category": "Math", "title": "Property (T) for non-unital C*-algebras", "abstract": "Inspired by the recent work of Bekka, we study two reasonable analogues of property (T) for not necessarily unital C*-algebras. The stronger one of the two is called ``property (T)'' and the weaker one is called ``property (T_{e})''. It is shown that all non-unital C*-algebras do not have property (T) (neither do their unitalizations). Moreover, all non-unital $\\sigma$-unital C*-algebras do not have property (T_e)."}
{"category": "Math", "title": "Universal cycles for permutations", "abstract": "A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n+1 distinct integers are used. This is best possible and proves a conjecture of Chung, Diaconis and Graham."}
{"category": "Math", "title": "Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials", "abstract": "Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmueller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely the linear involutions. These maps are related to general measured foliations on surfaces (orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with Z/2Z linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy-Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows, in particular, to classify the connected components of all exceptional strata."}
{"category": "Math", "title": "Kuranishi homology and Kuranishi cohomology: a User's Guide", "abstract": "A Kuranishi space is a topological space with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on moduli spaces of J-holomorphic curves in symplectic geometry. This paper is a brief introduction to the author's book arXiv:0707.3572. Let Y be an orbifold and R a Q-algebra. We define the Kuranishi homology KH_*(Y;R) of Y with coefficients in R. The chain complex KC_*(Y;R) defining KH_*(Y;R) is spanned over R by [X,f,G], for X a compact oriented Kuranishi space with corners, f : X --> Y smooth, and G \"gauge-fixing data\" which makes Aut(X,f,G) finite. Our main result is that KH_*(Y;R) is isomorphic to singular homology. We define a Poincare dual theory of Kuranishi cohomology KH^*(Y;R), isomorphic to compactly-supported cohomology, using a cochain complex KC^*(Y;R) spanned over R by [X,f,C], for X a compact Kuranishi space with corners, f : X --> Y a submersion, and C \"co-gauge-fixing data\". We also define simpler theories of Kuranishi bordism KB_*(Y;R) and Kuranishi cobordism KB^*(Y;R), for R a commutative ring. These are new topological invariants, and we show they are very large. These theories are powerful new tools in symplectic geometry. Defining virtual cycles and chains for moduli spaces of J-holomorphic curves is trivial in Kuranishi (co)homology. There is no need to perturb moduli spaces, and no problems with transversality. This gives major simplifications in Lagrangian Floer cohomology."}
{"category": "Math", "title": "Towards classification of simple finite dimensional modular Lie superalgebras", "abstract": "A way to construct (conjecturally all) simple finite dimensional modular Lie (super)algebras over algebraically closed fields of characteristic not 2 is offered. In characteristic 2, the method is supposed to give only simple Lie (super)algebras graded by integers and only some of the non-graded ones). The conjecture is backed up with the latest results computationally most difficult of which are obtained with the help of Grozman's software package SuperLie."}
{"category": "Math", "title": "Central and non-central limit theorems for weighted power variations of fractional Brownian motion", "abstract": "In this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q>=2 of the fractional Brownian motion with Hurst parameter H in (0,1), where q is an integer. The central limit holds for 1/(2q)<H<= 1-1/(2q), the limit being a conditionally Gaussian distribution. If H<1/(2q), we show the convergence in L^2 to a limit which only depends on the fractional Brownian motion, and if H> 1-1/(2q), we show the convergence in L^2 to a stochastic integral with respect to the Hermite process of order q."}
{"category": "Math", "title": "Tight homomorphisms and Hermitian symmetric spaces", "abstract": "We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetric spaces. We show that tight maps behave in a functorial way with respect to the Shilov boundary and use this to prove a general structure theorem for tight homomorphisms. Furthermore we classify all tight embeddings of the Poincare' disk."}
{"category": "Math", "title": "The open mapping theorem and the fundamental theorem of algebra", "abstract": "This note is devoted to two classical theorems: the open mapping theorem for analytic functions (OMT) and the fundamental theorem of algebra (FTA). We present a new proof of the first theorem, and then derive the second one by a simple topological argument. The proof is elementary in nature and does not use any kind of integration (neither complex nor real). In addition, it is also independent of the fact that the roots of an analytic function are isolated. The proof is based on either the Banach or Brouwer fixed point theorems. In particular, this shows that one can obtain a proof of the FTA (albeit indirect) which is based on the Brouwer fixed point theorem, an aim which was not reached in the past and later the possibility to achieve it was questioned. We close this note with a simple generalization of the FTA. A short review of certain issues related to the OMT and the FTA is also included."}
{"category": "Math", "title": "Core blocks of Ariki-Koike algebras II: the weight of a core block", "abstract": "We study combinatorial blocks of multipartitions, exploring further the notions of weight, hub and core block introduced by the author in earlier papers. We answer the question of which pairs (w,theta) occur as the weight and hub of a block, and we examine the action of the affine Weyl group on the set of blocks."}
{"category": "Math", "title": "Hopf algebras of diagrams", "abstract": "We investigate several Hopf algebras of diagrams related to Quantum Field Theory of Partitions and whose product comes from the Hopf algebras WSym or WQSym respectively built on integer set partitions and set compositions. Bases of these algebras are indexed either by bipartite graphs (labelled or unlabbeled) or by packed matrices (with integer or set coefficients). Realizations on biword are exhibited, and it is shown how these algebras fit into a commutative diagram. Hopf deformations and dendriform structures are also considered for some algebras in the picture."}
{"category": "Math", "title": "Graphs with extremal energy should have a small number of distinct eigenvalues", "abstract": "The sum of the absolute values of the eigenvalues of a graph is called the energy of the graph. We study the problem of finding graphs with extremal energy within specified classes of graphs. We develop tools for treating such problems and obtain some partial results. Using calculus, we show that an extremal graph ``should'' have a small number of distinct eigenvalues. However, we also present data that shows in many cases that extremal graphs can have a large number of distinct eigenvalues."}
{"category": "Math", "title": "An Elegant Method for Generating Multivariate Poisson Random Variable", "abstract": "Generating multivariate Poisson data is essential in many applications. Current simulation methods suffer from limitations ranging from computational complexity to restrictions on the structure of the correlation matrix. We propose a computationally efficient and conceptually appealing method for generating multivariate Poisson data. The method is based on simulating multivariate Normal data and converting them to achieve a specific correlation matrix and Poisson rate vector. This allows for generating data that have positive or negative correlations as well as different rates."}
{"category": "Math", "title": "Surgery and the spinorial tau-invariant", "abstract": "We associate to a compact spin manifold M a real-valued invariant \\tau(M) by taking the supremum over all conformal classes over the infimum inside each conformal class of the first positive Dirac eigenvalue, normalized to volume 1. This invariant is a spinorial analogue of Schoen's $\\sigma$-constant, also known as the smooth Yamabe number. We prove that if N is obtained from M by surgery of codimension at least 2, then $\\tau(N) \\geq \\min\\{\\tau(M),\\Lambda_n\\}$ with $\\Lambda_n>0$. Various topological conclusions can be drawn, in particular that \\tau is a spin-bordism invariant below $\\Lambda_n$. Below $\\Lambda_n$, the values of $\\tau$ cannot accumulate from above when varied over all manifolds of a fixed dimension."}
{"category": "Math", "title": "Nonparametric Conditional Inference for Regression Coefficients with Application to Configural Polysampling", "abstract": "We consider inference procedures, conditional on an observed ancillary statistic, for regression coefficients under a linear regression setup where the unknown error distribution is specified nonparametrically. We establish conditional asymptotic normality of the regression coefficient estimators under regularity conditions, and formally justify the approach of plugging in kernel-type density estimators in conditional inference procedures. Simulation results show that the approach yields accurate conditional coverage probabilities when used for constructing confidence intervals. The plug-in approach can be applied in conjunction with configural polysampling to derive robust conditional estimators adaptive to a confrontation of contrasting scenarios. We demonstrate this by investigating the conditional mean squared error of location estimators under various confrontations in a simulation study, which successfully extends configural polysampling to a nonparametric context."}
{"category": "Math", "title": "q-Hardy-Berndt type sums associated with q-Genocchi type zeta and l-functions", "abstract": "The aim of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, q-analogue Genocchi type l-function and two-variable q-Genocchi type l-function. Furthermore, we construct new genereting functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations related to q-Hardy-Berndt type sums and q-Genocchi zeta function as well."}
{"category": "Math", "title": "The random case of Conley's theorem: II. The complete Lyapunov function", "abstract": "Conley in \\cite{Con} constructed a complete Lyapunov function for a flow on compact metric space which is constant on orbits in the chain recurrent set and is strictly decreasing on orbits outside the chain recurrent set. This indicates that the dynamical complexity focuses on the chain recurrent set and the dynamical behavior outside the chain recurrent set is quite simple. In this paper, a similar result is obtained for random dynamical systems under the assumption that the base space $(\\Omega,\\mathcal F,\\mathbb P)$ is a separable metric space endowed with a probability measure. By constructing a complete Lyapunov function, which is constant on orbits in the random chain recurrent set and is strictly decreasing on orbits outside the random chain recurrent set, the random case of Conley's fundamental theorem of dynamical systems is obtained. Furthermore, this result for random dynamical systems is generalized to noncompact state spaces."}
{"category": "Math", "title": "Simultaneous inhomogeneous Diophantine approximation on manifolds", "abstract": "In 1998, Kleinbock & Margulis established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent $w_{0}(\\vv x) = 1/n$ for almost every point $\\vv x$ on a non-degenerate submanifold $\\cM$ of $\\R^n$. In this paper the simultaneous inhomogeneous analogue of Sprindzuk's conjecture is established. More precisely, for any `inhomogeneous' vector $\\bm\\theta\\in\\R^n$ we prove that the simultaneous inhomogeneous Diophantine exponent $w_{0}(\\vv x, \\bm\\theta)= 1/n$ for almost every point $\\vv x$ on $M$. The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent $w_0(\\vv x)=1/n$ for almost all $\\vv x\\in \\cM$ if and only if for any $\\bm\\theta\\in\\R^n$ the inhomogeneous exponent $w_0(\\vv x,\\bm\\theta)=1/n$ for almost all $\\vv x\\in \\cM$. The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered in \\cite{Beresnevich-Velani-new-inhom}. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas of \\cite{Beresnevich-Velani-new-inhom} while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory."}
{"category": "Math", "title": "The random case of Conley's theorem: III. Random semiflow case and Morse decomposition", "abstract": "In the first part of this paper, we generalize the results of the author \\cite{Liu,Liu2} from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose invariant random compact set (e.g. global random attractor) into random Morse sets and connecting orbits between them, which generalizes the Morse decomposition of invariant sets originated from Conley \\cite{Con} to the random semiflow setting and gives the positive answer to an open problem put forward by Caraballo and Langa \\cite{CL}."}
{"category": "Math", "title": "A simple proof of uniqueness of the particle trajectories for solutions of the Navier-Stokes equations", "abstract": "We give a simple proof of the uniqueness of fluid particle trajectories corresponding to: 1) the solution of the two-dimensional Navier Stokes equations with an initial condition that is only square integrable, and 2) the local strong solution of the three-dimensional equations with an $H^{1/2}$-regular initial condition i.e.\\ with the minimal Sobolev regularity known to guarantee uniqueness. This result was proved by Chemin & Lerner (J Diff Eq 121 (1995) 314-328) using the Littlewood-Paley theory for the flow in the whole space $\\R^d$, $d\\ge 2$. We first show that the solutions of the differential equation $\\dot{X}=u(X,t)$ are unique if $u\\in L^p(0,T;H^{(d/2)-1})$ for some $p>1$ and $\\sqrt{t}\\,u\\in L^2(0,T;H^{(d/2)+1})$. We then prove, using standard energy methods, that the solution of the Navier-Stokes equations with initial condition in $H^{(d/2)-1}$ satisfies these conditions. This proof is also valid for the more physically relevant case of bounded domains."}
{"category": "Math", "title": "Generating connected and biconnected graphs", "abstract": "We focus on the algorithm underlying the main result of [A. Mestre, R. Oeckl, Generating loop graphs via Hopf algebra in quantum field theory. J. Math. Phys., 47, 122302, 2006]. This is an algebraic formula to generate all connected graphs in a recursive and efficient manner. The key feature is that each graph carries a scalar factor given by the inverse of the order of its group of automorphisms. In the present paper, we revise that algorithm on the level of graphs. Moreover, we extend the result subsequently to further classes of connected graphs, namely, (edge) biconnected, simple and loopless graphs. Our method consists of basic graph transformations only."}
{"category": "Math", "title": "Complete families of linearly non-degenerate rational curves", "abstract": "We prove that a complete family of linearly non-degenerate rational curves of degree $e > 2$ in $\\mathbb{P}^n$ has at most $n-1$ moduli. For $e = 2$ we prove that such a family has at most $n$ moduli. It is unknown whether or not this is the best possible result. The general method involves exhibiting a map from the base of a family $X$ to the Grassmaninian of $e$-planes in $\\mathbb{P}^n$ and analyzing the resulting map on cohomology."}
{"category": "Math", "title": "Normal triangulations in o-minimal structures", "abstract": "We work over an o-minimal expansion of a real closed field R. Given a closed simplicial complex K and a finite number of definable subsets of its realization |K| in R we prove that there exists a triangulation (K',f) of |K| compatible with the definable subsets such that K' is a subdivision of K and f is definably homotopic to the identity on |K|."}
{"category": "Math", "title": "Existence of K\\\"ahler-Einstein metrics and multiplier ideal sheaves on del Pezzo surfaces", "abstract": "We apply Nadel's method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a K\\\"ahler-Einstein metric. In particular, all del Pezzo surfaces of degree $4,5$, or $6$ and certain special del Pezzo surfaces of lower degree are shown to have a K\\\"ahler-Einstein metric. This result is not new, but the proofs given in the present paper are less involved than earlier ones by Siu, Tian and Tian-Yau."}
{"category": "Math", "title": "Convergence of the K\\\"ahler-Ricci flow and multiplier ideal sheaves on del Pezzo surfaces", "abstract": "On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the K\\\"ahler-Ricci flow with a certain initial value converges in $C^\\infty$-norm exponentially fast to a K\\\"ahler-Einstein metric. The proof is based on the method of multiplier ideal sheaves."}
{"category": "Math", "title": "The H-principle and Pseudoconcave CR Manifolds", "abstract": "The H-principle, which is the analogue, for CR manifolds, of the classical Hartogs principle in several complex variables, is known to be valid in the small on a pseudoconcave CR manifold of any codimension. However it fails in the large, as has been shown by the counterexample found in [HN1]. Hence there is an underlying obstruction to the global H-principle on a pseudoconcave CR manifold. The purpose of this note is to take the first steps toward a deeper understanding of this obstruction."}
{"category": "Math", "title": "Mathematical Foundations of Supersymmetry", "abstract": "We lay down the foundations for a systematic study of differentiable and algebraic supervarieties, with a special attention to supergroups."}
{"category": "Math", "title": "Random Walk on a Surface Group: Boundary Behavior of the Green's Function at the Spectral Radius", "abstract": "It is proved that the Green's function of the simple random walk on a surface group of large genus decays exponentially at the spectral radius. It is also shown that Ancona's inequalities extend to the spectral radius R, and therefore that the Martin boundary for R-potentials coincides with the natural geometric boundary S^1. This implies that the Green's function obeys a power law with exponent 1/2 at the spectral radius."}
{"category": "Math", "title": "On the Laplace transform of some quadratic forms and the exact distribution of the sample variance from a gamma or uniform parent distribution", "abstract": "From a suitable integral representation of the Laplace transform of a positive semi-definite quadratic form of independent real random variables with not necessarily identical densities a univariate integral representation is derived for the cumulative distribution function of the sample variance of i.i.d. random variables with a gamma density, supplementing former formulas of the author. Furthermore, from the above Laplace transform Fourier series are obtained for the density and the distribution function of the sample variance of i.i.d. random variables with a uniform distribution. This distribution can be applied e.g. to a statistical test for a scale parameter."}
{"category": "Math", "title": "A Vanishing Result for Toric Varieties Associated with Root Systems", "abstract": "Consider a root system $R$ and the corresponding toric variety $V_R$ whose fan is the Weyl fan and whose lattice of characters is given by the root lattice for $R$. We prove the vanishing of the higher cohomology groups for certain line bundles on $V_R$ by proving a purely combinatorial result for root systems. These results are related to a converse to Mazur's Inequality for (simply-connected) split reductive groups."}
{"category": "Math", "title": "Some classifications of \\infty-Harmonic maps between Riemannian manifolds", "abstract": "$\\infty$-Harmonic maps are a generalization of $\\infty$-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic $\\infty$-harmonic maps from and into a sphere, quadratic $\\infty$-harmonic maps between Euclidean spaces. We describe all linear and quadratic $\\infty$-harmonic maps between Nil and Euclidean spaces, between Sol and Euclidean spaces. We also study holomorphic $\\infty$-harmonic maps between complex Euclidean spaces."}
{"category": "Math", "title": "Integrability and reduction of Poisson group actions", "abstract": "In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group $G$ with dual $G^\\star$ we obtain a suitably connected integrating symplectic double groupoid $\\calS$. As a consequence, the cotangent lift of a Poisson action on an integrable Poisson manifold $P$ can be integrated to a Poisson action of the symplectic groupoid $\\poidd{\\calS}{G^\\star}$ on the symplectic groupoid for $P$. Finally, we show that the quotient Poisson manifold $P/G$ is also integrable, giving an explicit construction of a symplectic groupoid for it, by a reduction procedure on an associated morphism of double Lie groupoids."}
{"category": "Math", "title": "On exotic modular tensor categories", "abstract": "It has been conjectured that every $(2+1)$-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair $(G,\\lambda)$, where $G$ is a compact Lie group, and $\\lambda \\in H^4(BG;Z)$ a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the quantum double of the even sectors of the $E_6$ subfactor, and the other is the quantum double of the even sectors of the Haagerup subfactor. We cannot prove mathematically that the two TQFTs are indeed counterexamples because CSW TQFTs, while physically defined, are not yet mathematically constructed for every pair $(G,\\lambda)$. The cases that are constructed mathematically include: 1. $G$ is a finite group--the Dijkgraaf-Witten TQFTs; 2. $G$ is torus $T^n$; 3. $G$ is a connected semi-simple Lie group--the Reshetikhin-Turaev TQFTs. We prove that the two TQFTs are not among those mathematically constructed TQFTs or their direct products. Both TQFTs are of the Turaev-Viro type: quantum doubles of spherical tensor categories. We further prove that neither TQFT is a quantum double of a braided fusion category, and give evidence that neither is an orbifold or coset of TQFTs above. Moreover, representation of the braid groups from the half $E_6$ TQFT can be used to build universal topological quantum computers, and the same is expected for the Haagerup case."}
{"category": "Math", "title": "Pre-Poisson submanifolds", "abstract": "This is an expository and introductory note on some results obtained in \"Coisotropic embeddings in Poisson manifolds\" (ArXiv math/0611480). Some original material is contained in the last two sections, where we consider linear Poisson structures."}
{"category": "Math", "title": "Geometric approach towards stable homotopy groups of spheres. The Hopf invariant", "abstract": "We develop a geometric approach to stable homotopy groups of spheres in the spirit of the work of Pontrjagin and Rokhlin. A new proof of the Hopf Invariant One Theorem by J.F.Adams is obtained in all dimensions except 15 and 31. To prove that the stable Hopf invariant H: \\Pi_n \\to Z/2 vanishes for n>31, we apply methods of geometric topology. The Pontrjagin-Thom construction along with Hirsch's compression lemma identify every \\alpha \\in \\Pi_n with the framed bordism class of a framed immersion of a closed n-manifold into R^{n+k}, for any given k>0. Its self-intersection M projects to an immersion f: M \\to R^n which is framed by k copies of a line bundle \\kappa. It is well-known that H(\\alpha) = <w_1(\\kappa)^{n-k}, [M]>. The self-intersection N of f is framed by k copies of a plane bundle with structure group D_4. We observe that H(\\alpha) = <w_1(i^*\\kappa)^{n-2k}, [\\bar N]>, where i immerses the double cover \\bar N of N into M. The hardest part of the proof is to show that, after modifying f in its skew-framed bordism class, the classifying map g: N \\to K(D_4,1) factors through K(Z/4,1), provided that n=2^l-1, l>5 and n-2k=15. This is achieved by analyzing immersions in the regular homotopy class of f that approximate the composition of the classifying map M \\to RP^{n-k}, the projection of RP^{n-k} onto the join of copies of S^1/(Z/4) (the standard sphere), and an embedding of this join in R^n. The last step is proved with the quaternions."}
{"category": "Math", "title": "Logarithmic singularities of Schwartz kernels and local invariants of conformal and CR structures", "abstract": "This paper consists of two parts. In the first part we show that in odd dimension, as well as in even dimension below the critical weight (i.e. half the dimension), the logarithmic singularities of Schwartz kernels and Green kernels of conformal invariant pseudodifferential operators are linear combinations of Weyl conformal invariants, i.e., of local conformal invariants arising from complete tensorial contractions of the covariant derivatives of the Lorentz ambient metric of Fefferman-Graham. In even dimension and above the critical weight exceptional local conformal invariants may further come into play. As a consequence, this allows us to get invariant expressions for the logarithmic singularities of the Green kernels of the GJMS operators (including the Yamabe and Paneitz operators). In the second part, we prove analogues of these results in CR geometry. Namely, we prove that the logarithmic singularities of Schwartz kernels and Green kernels of CR invariant Heisenberg pseudodifferential operators give rise to local CR invariants, and below the critical weight are linear combinations of complete tensorial contractions of the covariant derivatives of Fefferman's K\\\"alher-Lorentz ambient metric. As a consequence, we can obtain invariant descriptions of the logarithmic singularities of the Green kernels of the CR GJMS operators of Gover-Graham (including the CR Yamabe operator of Jerison-Lee)."}
{"category": "Math", "title": "On extremely amenable groups of homeomorphisms", "abstract": "A topological group $G$ is {\\em extremely amenable} if every compact $G$-space has a $G$-fixed point. Let $X$ be compact and $G\\subset{\\mathrm{Homeo}} (X)$. We prove that the following are equivalent: (1) $G$ is extremely amenable; (2) every minimal closed $G$-invariant subset of $\\exp R$ is a singleton, where $R$ is the closure of the set of all graphs of $g\\in G$ in the space $\\exp (X^2)$ ($\\exp$ stands for the space of closed subsets); (3) for each $n=1,2,...$ there is a closed $G$-invariant subset $Y_n$ of $(\\exp X)^n$ such that $\\cup_{n=1}^\\infty Y_n$ contains arbitrarily fine covers of $X$ and for every $n\\ge 1$ every minimal closed $G$-invariant subset of $\\exp Y_n$ is a singleton. This yields an alternative proof of Pestov's theorem that the group of all order-preserving self-homeomorphisms of the Cantor middle-third set (or of the interval $[0,1]$) is extremely amenable."}
{"category": "Math", "title": "The Selberg Trace Formula for Hecke operators on cocompact Kleinian groups", "abstract": "We compute the Selberg trace formula for Hecke operators (also called the trace formula for modular correspondences) in the context of cocompact Kleinian groups with finite-dimentional unitary representations. We give some applications to the distribution of Hecke eigenvalues, and give an analogue of Huber's theorem."}
{"category": "Math", "title": "Congruences for Andrews' Smallest Parts Partition Function and New Congruences for Dyson's Rank", "abstract": "Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt(n) mod p for p = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms."}
{"category": "Math", "title": "The distribution of maxima of approximately Gaussian random fields", "abstract": "Motivated by the problem of testing for the existence of a signal of known parametric structure and unknown ``location'' (as explained below) against a noisy background, we obtain for the maximum of a centered, smooth random field an approximation for the tail of the distribution. For the motivating class of problems this gives approximately the significance level of the maximum score test. The method is based on an application of a likelihood-ratio-identity followed by approximations of local fields. Numerical examples illustrate the accuracy of the approximations."}
{"category": "Math", "title": "Configurations of Rank-40r Extremal Even Unimodular Lattices (r=1,2,3)", "abstract": "We show that if L is an extremal even unimodular lattice of rank 40r with r=1,2,3 then L is generated by its vectors of norms 4r and 4r+2. Our result is an extension of Ozeki's result for the case r=1."}
{"category": "Math", "title": "Cuntz semigroups of ideals and quotients and a generalized Kasparov Stabilization Theorem", "abstract": "Let A be a C*-algebra and I a closed two-sided ideal of A. We use the Hilbert C*-modules picture of the Cuntz semigroup to investigate the relations between the Cuntz semigroups of I, A and A/I. We obtain a relation on two elements of the Cuntz semigroup of A that characterizes when they are equal in the Cuntz semigroup of A/I. As a corollary, we show that the Cuntz semigroup functor is exact. Replacing the Cuntz equivalence relation of Hilbert modules by their isomorphism, we obtain a generalization of Kasparov's Stabilization theorem."}
{"category": "Math", "title": "Estimating exposure response functions using ambient pollution concentrations", "abstract": "This paper presents an approach to estimating the health effects of an environmental hazard. The approach is general in nature, but is applied here to the case of air pollution. It uses a computer model involving ambient pollution and temperature inputs, to simulate the exposures experienced by individuals in an urban area, whilst incorporating the mechanisms that determine exposures. The output from the model comprises a set of daily exposures for a sample of individuals from the population of interest. These daily exposures are approximated by parametric distributions, so that the predictive exposure distribution of a randomly selected individual can be generated. These distributions are then incorporated into a hierarchical Bayesian framework (with inference using Markov Chain Monte Carlo simulation) in order to examine the relationship between short-term changes in exposures and health outcomes, whilst making allowance for long-term trends, seasonality, the effect of potential confounders and the possibility of ecological bias. The paper applies this approach to particulate pollution (PM$_{10}$) and respiratory mortality counts for seniors in greater London ($\\geq$65 years) during 1997. Within this substantive epidemiological study, the effects on health of ambient concentrations and (estimated) personal exposures are compared."}
{"category": "Math", "title": "Bernoulli-Taylor formula in the case of Q-umbral Calculus", "abstract": "In this note we derive the Q-difference Bernoulli-Taylor formula with the rest term of the Cauchy form by the Viskov's method. This is an extension of technique by the use of Q-extented Kwasniewski's *-product . The main theorems of Q-umbral calculus were given by G. Markowsky in 1968 and extented by A.K.Kwasniewski."}
{"category": "Math", "title": "Note on q-extensions of Euler numbers and polynomials of higher order", "abstract": "In [14] Ozden-Simsek-Cangul constructed generating functions of higher-order twisted $(h,q)$-extension of Euler polynomials and numbers, by using $p$-adic q-deformed fermionic integral on $\\Bbb Z_p$. By applying their generating functions, they derived the complete sums of products of the twisted $(h,q)$-extension of Euler polynomials and numbers, see[13, 14]. In this paper we cosider the new $q$-extension of Euler numbers and polynomials to be different which is treated by Ozden-Simsek-Cangul. From our $q$-Euler numbers and polynomials we derive some interesting identities and we construct $q$-Euler zeta functions which interpolate the new $q$-Euler numbers and polynomials at a negative integer. Furthermore we study Barnes' type $q$-Euler zeta functions. Finally we will derive the new formula for \" sums products of $q$-Euler numbers and polynomials\" by using fermionic $p$-adic $q$-integral on $\\Bbb Z_p$."}
{"category": "Math", "title": "Periodic cyclic homology of reductive p-adic groups", "abstract": "Let G be a reductive p-adic group, H(G) its Hecke algebra and S(G) its Schwartz algebra. We will show that these algebras have the same periodic cyclic homology. This might be used to provide an alternative proof of the Baum-Connes conjecture for G, modulo torsion. As preparation for our main theorem we prove two results that have independent interest. Firstly a general comparison theorem for the periodic cyclic homology of finite type algebras and certain Fr\\'echet completions thereof. Secondly a refined form of the Langlands classification for G, which clarifies the relation between the smooth spectrum and the tempered spectrum."}
{"category": "Math", "title": "Asymptotic behavior of Tor over complete intersections and applications", "abstract": "Let $R$ be a local complete intersection and $M,N$ are $R$-modules such that $\\ell(\\Tor_i^R(M,N))<\\infty$ for $i\\gg 0$. Imitating an approach by Avramov and Buchweitz, we investigate the asymptotic behavior of $\\ell(\\Tor_i^R(M,N))$ using Eisenbud operators and show that they have well-behaved growth. We define and study a function $\\eta^R(M,N)$ which generalizes Serre's intersection multiplicity $\\chi^R(M,N)$ over regular local rings and Hochster's function $\\theta^R(M,N)$ over local hypersurfaces. We use good properties of $\\eta^R(M,N)$ to obtain various results on complexities of $\\Tor$ and $\\Ext$, vanishing of $\\Tor$, depth of tensor products, and dimensions of intersecting modules over local complete intersections."}
{"category": "Math", "title": "On a random recursion related to absorption times of death Markov chains", "abstract": "Let $X_1,X_2,...$ be a sequence of random variables satisfying the distributional recursion $X_1=0$ and $X_n= X_{n-I_n}+1$ for $n=2,3,...$, where $I_n$ is a random variable with values in $\\{1,...,n-1\\}$ which is independent of $X_2,...,X_{n-1}$. The random variable $X_n$ can be interpreted as the absorption time of a suitable death Markov chain with state space ${\\mathbb N}:=\\{1,2,...\\}$ and absorbing state 1, conditioned that the chain starts in the initial state $n$. This paper focuses on the asymptotics of $X_n$ as $n$ tends to infinity under the particular but important assumption that the distribution of $I_n$ satisfies ${\\mathbb P}\\{I_n=k\\}=p_k/(p_1+...+p_{n-1})$ for some given probability distribution $p_k={\\mathbb P}\\{\\xi=k\\}$, $k\\in{\\mathbb N}$. Depending on the tail behaviour of the distribution of $\\xi$, several scalings for $X_n$ and corresponding limiting distributions come into play, among them stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is a coupling technique which relates the distribution of $X_n$ to a random walk, which explains, for example, the appearance of the Mittag-Leffler distribution in this context. The results are applied to describe the asymptotics of the number of collisions for certain beta-coalescent processes."}
{"category": "Math", "title": "On finite index subgroups of a universal group", "abstract": "The orbifold group of the Borromean rings with singular angle 90 degrees, $U$, is a universal group, because every closed oriented 3--manifold $M^{3}$ occurs as a quotient space $M^{3} = H^{3}/G$, where $G$ is a finite index subgroup of $U$. Therefore, an interesting, but quite difficult problem, is to classify the finite index subgroups of the universal group $U$. One of the purposes of this paper is to begin this classification. In particular we analyze the classification of the finite index subgroups of $U$ that are generated by rotations."}
{"category": "Math", "title": "On estimating covariances between many assets with histories of highly variable length", "abstract": "Quantitative portfolio allocation requires the accurate and tractable estimation of covariances between a large number of assets, whose histories can greatly vary in length. Such data are said to follow a monotone missingness pattern, under which the likelihood has a convenient factorization. Upon further assuming that asset returns are multivariate normally distributed, with histories at least as long as the total asset count, maximum likelihood (ML) estimates are easily obtained by performing repeated ordinary least squares (OLS) regressions, one for each asset. Things get more interesting when there are more assets than historical returns. OLS becomes unstable due to rank--deficient design matrices, which is called a \"big p small n\" problem. We explore remedies that involve making a change of basis, as in principal components or partial least squares regression, or by applying shrinkage methods like ridge regression or the lasso. This enables the estimation of covariances between large sets of assets with histories of essentially arbitrary length, and offers improvements in accuracy and interpretation. We further extend the method by showing how external factors can be incorporated. This allows for the adaptive use of factors without the restrictive assumptions common in factor models. Our methods are demonstrated on randomly generated data, and then benchmarked by the performance of balanced portfolios using real historical financial returns. An accompanying R package called monomvn, containing code implementing the estimators described herein, has been made freely available on CRAN."}
{"category": "Math", "title": "2-level fractional factorial designs which are the union of non trivial regular designs", "abstract": "Every fraction is a union of points, which are trivial regular fractions. To characterize non trivial decomposition, we derive a condition for the inclusion of a regular fraction as follows. Let $F = \\sum_\\alpha b_\\alpha X^\\alpha$ be the indicator polynomial of a generic fraction, see Fontana et al, JSPI 2000, 149-172. Regular fractions are characterized by $R = \\frac 1l \\sum_{\\alpha \\in \\mathcal L} e_\\alpha X^\\alpha$, where $\\alpha \\mapsto e_\\alpha$ is an group homeomorphism from $\\mathcal L \\subset \\mathbb Z_2^d$ into $\\{-1,+1\\}$. The regular $R$ is a subset of the fraction $F$ if $FR = R$, which in turn is equivalent to $\\sum_t F(t)R(t) = \\sum_t R(t)$. If $\\mathcal H = \\{\\alpha_1 >... \\alpha_k\\}$ is a generating set of $\\mathcal L$, and $R = \\frac1{2^k}(1 + e_1X^{\\alpha_1}) ... (1 + e_kX^{\\alpha_k})$, $e_j = \\pm 1$, $j=1 ... k$, the inclusion condition in term of the $b_\\alpha$'s is % \\begin{equation}b_0 + e_1 b_{\\alpha_1} + >... + e_1 ... e_k b_{\\alpha_1 + ... + \\alpha_k} = 1. \\tag{*}\\end{equation} % The last part of the paper will discuss some examples to investigate the practical applicability of the previous condition (*). This paper is an offspring of the Alcotra 158 EU research contract on the planning of sequential designs for sample surveys in tourism statistics."}
{"category": "Math", "title": "Algebras of unbounded operators over the ring of measurable functions and their derivations and automorphisms", "abstract": "In the present paper derivations and *-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all L^0-linear derivations and L^{0}-linear *-automorphisms are inner. Moreover, it is proved that each L^0-linear automorphism of the algebra of all linear operators on a bo-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial."}
{"category": "Math", "title": "Sturm numbers and substitution invariance of 3iet words", "abstract": "In this paper, we give a necessary condition for an infinite word defined by a non-degenerate interval exchange on three intervals (3iet word) to be invariant by a substitution: a natural parameter associated to this word must be a Sturm number. We deduce some algebraic consequences from this condition concerning the incidence matrix of the associated substitution. As a by-product of our proof, we give a combinatorial characterization of 3iet words."}
{"category": "Math", "title": "Geometric approach towards stable homotopy groups of spheres. Kervaire Invariant", "abstract": "It is proved that there exists an integer $L$ such that a framed manifold of dimension $2^l-2$, $l\\le L$ has the trivial Kervaire Invariant."}
{"category": "Math", "title": "Lingering random walks in random environment on a strip", "abstract": "We consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW exhibits the \"(log t)-squared\" asymptotic behaviour. The exceptional algebraic subsurface is described by an explicit system of algebraic equations. One-dimensional walks with bounded jumps in a RE are treated as a particular case of the strip model. If the one dimensional RE is i. i. d., then our approach leads to a complete and constructive classification of possible types of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the $(\\log t)^{2}$ asymptotic behaviour if the distribution of the RE is not supported by a hyperplane in the space of parameters which shall be explicitly described. And if the support of the RE belongs to this hyperplane then the corresponding RW is a martingale and its asymptotic behaviour is governed by the Central Limit Theorem."}
{"category": "Math", "title": "On reality property of Wronski maps", "abstract": "We prove that if all roots of the discrete Wronskian with step 1 of a set of quasi-exponentials with real bases are real, simple and differ by at least 1, then the complex span of this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This result generalizes the B. and M.Shapiro conjecture about spaces of polynomials. The proof is based on the Bethe ansatz method for the XXX model."}
{"category": "Math", "title": "An elementary proof of Grothendieck's Non-vanishing Theorem", "abstract": "We give an elementary proof of Grothendieck's non-vanishing Theorem: For a finitely generated non-zero module $M$ over a Noetherian local ring $A$ with maximal ideal $\\m$, the local cohomology module $H^{\\dim M}_{\\m}(M)$ is non-zero."}
{"category": "Math", "title": "Computing generators of free modules over orders in group algebras", "abstract": "Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition of E[G] is explicitly computable and each component is in fact a matrix ring over a field, this leads to an algorithm that either gives an A-basis for X or determines that no such basis exists. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d=[K:E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A of O_L in E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K=E=Q."}
{"category": "Math", "title": "Implementing Quasi-Monte Carlo Simulations with Linear Transformations", "abstract": "Pricing exotic multi-asset path-dependent options requires extensive Monte Carlo simulations. In the recent years the interest to the Quasi-monte Carlo technique has been renewed and several results have been proposed in order to improve its efficiency with the notion of effective dimension. To this aim, Imai and Tan introduced a general variance reduction technique in order to minimize the nominal dimension of the Monte Carlo method. Taking into account these advantages, we investigate this approach in detail in order to make it faster from the computational point of view. Indeed, we realize the linear transformation decomposition relying on a fast ad hoc QR decomposition that considerably reduces the computational burden. This setting makes the linear transformation method even more convenient from the computational point of view. We implement a high-dimensional (2500) Quasi-Monte Carlo simulation combined with the linear transformation in order to price Asian basket options with same set of parameters published by Imai and Tan. For the simulation of the high-dimensional random sample, we use a 50-dimensional scrambled Sobol sequence for the first 50 components, determined by the linear transformation method, and pad the remaining ones out by the Latin Hypercube Sampling. The aim of this numerical setting is to investigate the accuracy of the estimation by giving a higher convergence rate only to those components selected by the linear transformation technique. We launch our simulation experiment also using the standard Cholesky and the principal component decomposition methods with pseudo-random and Latin Hypercube sampling generators. Finally, we compare our results and computational times, with those presented in Imai and Tan."}
{"category": "Math", "title": "Graphical models for marked point processes based on local independence", "abstract": "A new class of graphical models capturing the dependence structure of events that occur in time is proposed. The graphs represent so-called local independences, meaning that the intensities of certain types of events are independent of some (but not necessarily all) events in the past. This dynamic concept of independence is asymmetric, similar to Granger non-causality, so that the corresponding local independence graphs differ considerably from classical graphical models. Hence a new notion of graph separation, called delta-separation, is introduced and implications for the underlying model as well as for likelihood inference are explored. Benefits regarding facilitation of reasoning about and understanding of dynamic dependencies as well as computational simplifications are discussed."}
{"category": "Math", "title": "Base loci of linear systems and the Waring problem", "abstract": "Waring problem for homogeneus forms asks for additive decomposition of a form $f$ into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this note I refine the work in arXiv:math/0406288v1 and answer this question under a divisibility assumption. To do this I translate the algebraic statement into a geometric one concerning the base loci of linear systems of $P^n$ with assigned singularities."}
{"category": "Math", "title": "Hecke-Clifford algebras and spin Hecke algebras II: the rational double affine type", "abstract": "The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl operator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras."}
{"category": "Math", "title": "Some aspects of extreme value theory under serial dependence", "abstract": "On the occasion of Laurens de Haan's 70th birthday, we discuss two aspects of the statistical inference on the extreme value behavior of time series with a particular emphasis on his important contributions. First, the performance of a direct marginal tail analysis is compared with that of a model-based approach using an analysis of residuals. Second, the importance of the extremal index as a measure of the serial extremal dependence is discussed by the example of solutions of a stochastic recurrence equation."}
{"category": "Math", "title": "Combinatorial interpretation and positivity of Kerov's character polynomials", "abstract": "Kerov's polynomials give irreducible character values in term of the free cumulants of the associated Young diagram. We prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov. Our method, through decomposition of maps, gives a description of the coefficients of the k-th Kerov's polynomials using permutations in S(k). We also obtain explicit formulas or combinatorial interpretations for some coefficients. In particular, we are able to compute the subdominant term for character values on any fixed permutation (it was known for cycles)."}
{"category": "Math", "title": "A version of Fabry's theorem for power series with regularly varying coefficients", "abstract": "For real power series whose non-zero coefficients satisfy $|a_m|^{1/m}\\to~1$ we prove a stronger version of Fabry theorem relating the frequency of sign changes in the coefficients and analytic continuation of the sum of the power series."}
{"category": "Math", "title": "Supervised Machine Learning with a Novel Pointwise Density Estimator", "abstract": "This article proposes a novel density estimation based algorithm for carrying out supervised machine learning. The proposed algorithm features O(n) time complexity for generating a classifier, where n is the number of sampling instances in the training dataset. This feature is highly desirable in contemporary applications that involve large and still growing databases. In comparison with the kernel density estimation based approaches, the mathe-matical fundamental behind the proposed algorithm is not based on the assump-tion that the number of training instances approaches infinite. As a result, a classifier generated with the proposed algorithm may deliver higher prediction accuracy than the kernel density estimation based classifier in some cases."}
{"category": "Math", "title": "Exponential inequalities for empirical unbounded context trees", "abstract": "In this paper we obtain non-uniform exponential upper bounds for the rate of convergence of a version of the algorithm Context, when the underlying tree is not necessarily bounded. The algorithm Context is a well-known tool to estimate the context tree of a Variable Length Markov Chain. As a consequence of the exponential bounds we obtain a strong consistency result. We generalize in this way several previous results in the field."}
{"category": "Math", "title": "Converse Sturm-Hurwitz-Kellogg theorem and related results", "abstract": "The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and a function with at least n+1 sign changes, there exists an orientation preserving diffeomorphism of the circle that takes this function to a function, orthogonal to the Chebyshev system. We also prove that if a function on the real projective line has at least four sign changes then there exists an orientation preserving diffeomorphism of the projective line that takes this function to the Schwarzian derivative of some function. These results extend the converse four vertex theorem of H. Gluck and B. Dahlberg: a function on a circle with at least two local maxima and two local minima is the curvature of a closed plane curve."}
{"category": "Math", "title": "On an extension of the Blaschke-Santalo inequality", "abstract": "Let $K$ be a convex body and $K^\\circ$ its polar body. Call $\\phi(K)=\\frac{1}{|K||K^\\circ|}\\int_K\\int_{K^\\circ}< x,y>^2 dxdy$. It is conjectured that $\\phi(K)$ is maximum when $K$ is the euclidean ball. In particular this statement implies the Blaschke-Santalo inequality. We verify this conjecture when $K$ is restricted to be a $p$--ball."}
{"category": "Math", "title": "The motivic zeta function and its smallest poles", "abstract": "Let f be a regular function on a nonsingular complex algebraic variety of dimension d. We prove a formula for the motivic zeta function of f in terms of an embedded resolution. This formula is over the Grothendieck ring itself, and specializes to the formula of Denef and Loeser over a certain localization. We also show that the space of n-jets satisfying f=0 can be partitioned into locally closed subsets which are isomorphic to a cartesian product of some variety with an affine space of dimension the round up of dn/2. Finally, we look at the consequences for the poles of the motivic zeta function."}
{"category": "Math", "title": "Dynamic of threshold solutions for energy-critical NLS", "abstract": "We consider the radial energy-critical non-linear focusing Schr\\\"odinger equation in dimension N=3,4,5. An explicit stationnary solution, W, of this equation is known. In a previous work by C. Carlos and F. Merle, the energy E(W) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article, we study the dynamics at the critical level E(u)=E(W) and classify the corresponding solutions. This gives in particular a dynamical characterization of W."}
{"category": "Math", "title": "A description of the outer automorphism of S_6, and the invariants of six points in projective space", "abstract": "We use a simple description of the outer automorphism of S_6 to cleanly describe the invariant theory of six points in P^1, P^2, and P^3."}
{"category": "Math", "title": "The modular variety of hyperelliptic curves of genus three", "abstract": "The modular variety of non singular and complete hyperelliptic curves with level-two structure of genus 3 is a 5-dimensional quasi projective variety which admits several standard compactifications. The first one, X, comes from the realization of this variety as a sub-variety of the Siegel modular variety of level two and genus three .We will be to describe the equations of X in a suitable projective embedding and its Hilbert function. It will turn out that X is normal. A further model comes from geometric invariant theory using so-called semistable degenerated point configurations in (P^1)^8 . We denote this GIT-compactification by Y. The equations of this variety in a suitable projective embedding are known. This variety also can by identified with a Baily-Borel compactified ball-quotient. We will describe these results in some detail and obtain new proofs including some finer results for them. We have a birational map between Y and X . In this paper we use the fact that there are graded algebras (closely related to algebras of modular forms) A,B such that X=proj(A) and Y=proj(B). This homomorphism rests on the theory of Thomae (19th century), in which the thetanullwerte of hyperelliptic curves have been computed. Using the explicit equations for $A,B$ we can compute the base locus of the map from Y to X. Blowing up the base locus and the singularity of Y, we get a dominant, smooth model {\\tilde Y}. We will see that {\\tilde Y} is isomorphic to the compactification of families of marked projective lines (P^1,x_1,...,x_8), usually denoted by {\\bar M_{0,8}}. There are several combinatorial similarities between the models X and Y. These similarities can be described best, if one uses the ball-model to describe Y."}
{"category": "Math", "title": "Mod 2 cohomology of 2-local finite groups of low rank", "abstract": "We determine the mod $2$ cohomology over the Steenrod algebra of the classifying spaces of the free loop groups $LG$ for compact groups $G=Spin(7)$, $Spin(8)$, $Spin(9)$, and $F_4$. Then, we show that they are isomorphic as algebras over the Steenrod algebra to the mod $2$ cohomology of the corresponding Chevalley groups of type $G(q)$, where $q$ is an odd prime power. In a similar manner, we compute the cohomology of the free loop space over $BDI(4)$ and show that it is isomorphic to that of $BSol(q)$ as algebras over the Steenrod algebra."}
{"category": "Math", "title": "Free Bessel laws", "abstract": "We introduce and study a remarkable family of real probability measures $\\pi_{st}$, that we call free Bessel laws. These are related to the free Poisson law $\\pi$ via the formulae $\\pi_{s1}=\\pi^{\\boxtimes s}$ and $\\pi_{1t}=\\pi^{\\boxplus t}$. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups."}
{"category": "Math", "title": "Dynamic of threshold solutions for energy-critical wave equation", "abstract": "We consider the energy-critical non-linear focusing wave equation in dimension N=3,4,5. An explicit stationnary solution, $W$, of this equation is known. The energy E(W,0) has been shown by C. Kenig and F. Merle to be a threshold for the dynamical behavior of solutions of the equation. In the present article we study the dynamics at the critical level E(u_0,u_1)=E(W,0) and classify the corresponding solutions. We show in particular the existence of two special solutions, connecting different behaviors for negative and positive times. Our results are analoguous to our previous work on radial Schr\\\"odinger equation, but without any radial assumption on the data. We also refine the understanding of the dynamical behavior of the special solutions."}
{"category": "Math", "title": "On Optimality Properties of the Shiryaev-Roberts Procedure", "abstract": "We consider the simple changepoint problem setting, where observations are independent, iid pre-change and iid post-change, with known pre- and post-change distributions. The Shiryaev-Roberts detection procedure is known to be asymptotically minimax in the sense of minimizing maximal expected detection delay subject to a bound on the average run length to false alarm, as the latter goes to infinity. Here we present other optimality properties of the Shiryaev-Roberts procedure."}
{"category": "Math", "title": "Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane", "abstract": "We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups $P_{n}(RP^2)$ of the projective plane. The maximal finite subgroups of $P_{n}(RP^2)$ are isomorphic to the quaternion group of order 8 if $n=3$, and to $\\Z_{4}$ if $n\\geq 4$. Further, for all $n\\geq 3$, up to isomorphism, the following groups are the infinite virtually cyclic subgroups of $P_{n}(RP^2)$: $\\Z$, $\\Z_{2} \\times \\Z$ and the amalgamated product $\\Z_{4} \\ast_{\\Z_{2}} \\Z_{4}$."}
{"category": "Math", "title": "Lectures on the stable homotopy of BG", "abstract": "This paper is a survey of the stable homotopy theory of BG for G a finite group. It is based on a series of lectures given at the Summer School associated with the Topology Conference at the Vietnam National University, Hanoi, August 2004."}
{"category": "Math", "title": "Some new characterizations of Banach spaces containing $\\ell^1$", "abstract": "Several new characterizations of Banach spaces containing a subspace isomorphic to $\\ell^1$, are obtained. These are applied to the question of when $\\ell^1$ embeds in the injective tensor product of two Banach spaces."}
{"category": "Math", "title": "Homology of the curve complex and the Steinberg module of the mapping class group", "abstract": "By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the homotopy type of a bouquet of spheres. Here, we give the first explicit homologically nontrivial sphere in the curve complex and show that under the action of the mapping class group, the orbit of this homology class generates the reduced homology of the curve complex."}
{"category": "Math", "title": "On fractional Brownian motion limits in one dimensional nearest-neighbor symmetric simple exclusion", "abstract": "A well-known result with respect to the one dimensional nearest-neighbor symmetric simple exclusion process is the convergence to fractional Brownian motion with Hurst parameter 1/4, in the sense of finite-dimensional distributions, of the subdiffusively rescaled current across the origin, and the subdiffusively rescaled tagged particle position. The purpose of this note is to improve this convergence to a functional central limit theorem, with respect to the uniform topology, and so complete the solution to a conjecture in the literature with respect to simple exclusion processes."}
{"category": "Math", "title": "Dualities in equivariant Kasparov theory", "abstract": "We study several duality isomorphisms between equivariant bivariant K-theory groups, generalising Kasparov's first and second Poincare duality isomorphisms. We use the first duality to define an equivariant generalisation of Lefschetz invariants of generalised self-maps. The second duality is related to the description of bivariant Kasparov theory for commutative C*-algebras by families of elliptic pseudodifferential operators. For many groupoids, both dualities apply to a universal proper G-space. This is a basic requirement for the dual Dirac method and allows us to describe the Baum-Connes assembly map via localisation of categories."}
{"category": "Math", "title": "Equivariant Lefschetz maps for simplicial complexes and smooth manifolds", "abstract": "Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers to equivariant K-homology classes. We compute the Lefschetz invariants for self-maps of finite-dimensional simplicial complexes and of self-maps of smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in these cases. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Luck and Rosenberg."}
{"category": "Math", "title": "Cohen-Macaulay modules and holonomic modules over filtered rings", "abstract": "We study Gorenstein dimension and grade of a module $M$ over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G-dim$M\\leq{G-dim gr}M$ and an equality ${\\rm grade}M={\\rm grade gr}M$, whenever Gorenstein dimension of ${\\rm gr}M$ is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We apply these results to a filtered ring with a Cohen-Macaulay or Gorenstein associated graded ring and study a Cohen-Macaulay, perfect or holonomic module."}
{"category": "Math", "title": "Schreier spectrum of the Hanoi Towers group on three pegs", "abstract": "Finite dimensional representations of the Hanoi Towers group are used to calculate the spectra of the finite graphs associated to the Hanoi Towers Game on three pegs (the group serves as a renorm group for the game). These graphs are Schreier graphs of the action of the Hanoi Towers group on the levels of the rooted ternary tree. The spectrum of the limiting graph (Schreier graph of the action on the boundary of the tree) is also provided."}
{"category": "Math", "title": "Mirkovic-Vilonen cycles and polytopes for a Symmetric pair", "abstract": "Let $G$ be a connected, simply-connected, and almost simple algebraic group, and let $\\sigma$ be a Dynkin automorphism on $G$. In this paper, we get a bijection between the set of $\\st$-invariant MV cycles (polytopes) for $G$ and the set of MV cycles (polytopes) for $G^\\st$, which is the fixed point subgroup of $G$; moreover, this bijection can be restricted to the set of MV cycles (polytopes) in irreducible representations. As an application, we obtain a new proof of the twining character formula."}
{"category": "Math", "title": "A modification of the Anderson-Mirkovic conjecture for Mirkovic-Vilonen polytopes in types B and C", "abstract": "We give an explicit description of the (lowering) Kashiwara operators on Mirkovi\\'c-Vilonen polytopes in types $B$ and $C$, which provides a simple method for generating Mirkovi\\'c-Vilonen polytopes inductively. This description can be thought of as a modification of the original Anderson-Mirkovi\\'c conjecture, which Kamnitzer proved in the case of type $A$, and presented a counterexample in the case of type $C_{3}$."}
{"category": "Math", "title": "Note on the question of Sikora", "abstract": "A natural topology on the set of left orderings on free abelian groups and free groups $F_n$, $n>1$ has studied in [1]. It has been proven already that in the abelian case the resulted topological space is a Cantor set. There was a conjecture: this is also true for the free group $F_n$ with $n>1$ generators. We point out the article dealing with equivalent questions. The answer is \"yes\"."}
{"category": "Math", "title": "Third moment of the remainder term for Heisenberg manifolds", "abstract": "Let R(t) be the remainder term in Weyl's law for a 3-dimensional Riemannian Heisenberg manifold with a certain arithmetic metric. We prove a third moment result stating that \\int_1^T R(t)^3 dt =d_3 T^(13/4)+O_\\delta(T^(45/14+\\delta)), where d_3 is a specific positive constant which can be evaluated explicitly. This proves the asymmetric behavior of R(t) about the t-axis. This result is consistent with the conjecture of Petridis and Toth stating that R(t)=O_\\delta(t^(3/4+\\delta)). Similar results hold for (2n+1)-dimensional Heisenberg manifolds with arithmetic metrics."}
{"category": "Math", "title": "Number of sets with small sumset and the clique number of random Cayley graphs", "abstract": "Let $G$ be a finite abelian group of order $n$. For any subset $B$ of $G$ with $B=-B$, the Cayley graph $G_B$ is a graph on vertex set $G$ in which $ij$ is an edge if and only if $i-j\\in B.$ It was shown by Ben Green that when $G$ is a vector space over a finite field $Z/pZ$, then there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than $clog nloglog n,$ where $c$ is an absolute constant. In this article we observe that a modification of his arguments shows that for an arbitrary finite abelian group of order $n$, there is a Cayley graph containing neither a complete subgraph nor an independent set of size more than $c(omega^3(n)log omega(n) +log nloglog n)$, where $c$ is an absolute constant and $omega(n)$ denotes the number of distinct prime divisors of $n$."}
{"category": "Math", "title": "Elliptic curves related to cyclic cubic extensions", "abstract": "The aim of this paper is to study certain family of elliptic curves $\\{\\mathscr{X}_H\\}_H$ defined over a number field $F$ arising from hyperplane sections of some cubic surface $\\mathscr{X}/F$ associated to a cyclic cubic extension $K/F$. We show that each $\\mathscr{X}_H$ admits a 3-isogeny $\\phi$ over $F$ and the dual Selmer group $S^{(\\hat{\\phi})}(\\hat{\\mathscr{X}_H}/F)$ is bounded by a kind of unit/class groups attached to $K/F$. This is proven via certain rational function on the elliptic curve $\\mathscr{X}_H$ with nice property. We also prove that the Shafarevich-Tate group $\\text{\\cyr X} (\\hat{\\mathscr{X}_H}/\\rat)[\\hat{\\phi}]$ coincides with a class group of $K$ as a special case."}
{"category": "Math", "title": "Twisted p-adic (h,q)-L-functions", "abstract": "By using q-Volkenborn integral on Z_{p}, we (simsek, simsekCanada) constructed new generating functions of the (h,q)-Bernoulli polynomials and numbers. By applying the Mellin transformation to the generating functions, we constructed integral representation of the twisted (h,q)-Hurwitz function and twisted (h,q)-two-variable L-function. By using these functions, we construct twisted new (h,q)-partial zeta function which interpolates the twisted (h,q)-Bernoulli polynomials and generalized twisted (h,q)-Bernoulli numbers at negative integers. We give relation between twisted (h,q)-partial zeta functions and twisted (h,q)-two-variable L-function. We find the value of this function at s=0. We also find residue of this function at s=1. We construct p-adic twisted (h,q)-L-function, which interpolates the twisted (h,q)-Bernoulli polynomials."}
{"category": "Math", "title": "The spectral flow, the Fredholm index, and the spectral shift function", "abstract": "We discuss the well known ``Fredholm index=spectral flow'' theorem and show that it can be interpreted as a limit case of an identity involving two spectral shift functions."}
{"category": "Math", "title": "Uniqueness of roots up to conjugacy for some affine and finite type Artin groups", "abstract": "Let $G$ be one of the Artin groups of finite type ${\\mathbf B}_n={\\mathbf C}_n$, and affine type $\\tilde{\\mathbf A}_{n-1}$ and $\\tilde{\\mathbf C}_{n-1}$. In this paper, we show that if $\\alpha$ and $\\beta$ are elements of $G$ such that $\\alpha^k=\\beta^k$ for some nonzero integer $k$, then $\\alpha$ and $\\beta$ are conjugate in $G$. For the Artin group of type $\\mathbf A_n$, this was recently proved by J. Gonz\\'alez-Meneses. In fact, we prove a stronger theorem, from which the above result follows easily by using descriptions of those Artin groups as subgroups of the braid group on $n+1$ strands. Let $P$ be a subset of $\\{1,...,n\\}$. An $n$-braid is said to be \\emph{$P$-pure} if its induced permutation fixes each $i\\in P$, and \\emph{$P$-straight} if it is $P$-pure and it becomes trivial when we delete all the $i$-th strands for $i\\not\\in P$. Exploiting the Nielsen-Thurston classification of braids, we show that if $\\alpha$ and $\\beta$ are $P$-pure $n$-braids such that $\\alpha^k=\\beta^k$ for some nonzero integer $k$, then there exists a $P$-straight $n$-braid $\\gamma$ with $\\beta=\\gamma\\alpha\\gamma^{-1}$. Moreover, if $1\\in P$, the conjugating element $\\gamma$ can be chosen to have the first strand algebraically unlinked with the other strands. Especially in case of $P=\\{1,...,n\\}$, our result implies the uniqueness of root of pure braids, which was known by V. G. Bardakov and by D. Kim and D. Rolfsen."}
{"category": "Math", "title": "Geometry of A_g and Its Compactifications", "abstract": "In this survey we give a brief introduction to, and review the progress made in the last decade in understanding the geometry of the moduli spaces A_g of principally polarized abelian varieties and its compactifications. Topics surveyed include: compactifications; birational geometry: nef and effective cones, canonical models; homology, Chow rings and intersection theory; and subvarieties of moduli spaces. We also discuss some open problems and possible further directions. This is an expanded and updated version of the talk given at the 2005 Summer Institute for Algebraic Geometry"}
{"category": "Math", "title": "Symmetric units in modular group algebras", "abstract": "Let p be a prime, G a locally finite p-group, K a commutative ring of characteristic p. The anti-automorphism g\\mapsto g\\m1 of G extends linearly to an anti-automorphism a\\mapsto a^* of KG. An element a of KG is called symmetric if a^*=a. In this paper we answer the question: for which G and K do the symmetric units of KG form a multiplicative group."}
{"category": "Math", "title": "Unitary units in modular group algebras", "abstract": "Let p be a prime, K a field of characteristic p, G a locally finite p-group, KG the group algebra, and V the group of the units of KG with augmentation 1. The anti-automorphism g\\mapsto g^{-1} of G extends linearly to KG; this extension leaves V setwise invariant, and its restriction to V followed by v\\mapsto v^{-1} lives an automorphism of V. The elements of V fixed by this automorphism are called unitary; they form a subgroup. Our first theorem describes the K and G for which this subgroup is normal in V. For each element g in G, let \\bar{g} denote the sum (in KG) of the distinct powers of g. The elements 1+(g-1)h\\bar{g} with g,h\\in G are the bicyclic units of KG. Our second theorem describes the K and G for which all bicyclic units are unitary."}
{"category": "Math", "title": "Homology of coloured posets: a generalisation of Khovanov's cube construction", "abstract": "We define a homology theory for a certain class of posets equipped with a representation. We show that when restricted to Boolean lattices this homology is isomorphic to the homology of the \"cube\" complex defined by Khovanov."}
{"category": "Math", "title": "Continuity of CP-semigroups in the point-strong operator topology", "abstract": "We prove that if $\\{\\phi_t\\}_{t \\geq 0}$ is a CP-semigroup acting on a von Neumann algebra $M \\subseteq B(H)$, then for every $A\\in M$ and $\\xi \\in H$, the map $t \\mapsto \\phi_t(A)\\xi$ is norm-continuous. We discuss the implications of this fact to the existence of dilations of CP-semigroups to semigroups of endomorphisms."}
{"category": "Math", "title": "Minimal types in super-dependent theories", "abstract": "We give necessary and sufficient geometric conditions for a theory definable in an o-minimal structure to interpret a real closed field. The proof goes through an analysis of thorn-minimal types in super-rosy dependent theories of finite rank. We prove that such theories are coordinatised by thorn-minimal types and that such a type is unstable if an only if every non-algebraic extension thereof is. We conclude that a type is stable if and only if it admits a coordinatisation in thorn-minimal stable types. We also show that non-trivial thorn-minimal stable types extend stable sets."}
{"category": "Math", "title": "Tensor products of maximal abelian subalgebras of C*-algebras", "abstract": "It is shown that if $C_1$ and $C_2$ are maximal abelian self-adjoint subalgebras (masas) of C*-algebras $A_1$ and $A_2$, respectively, then the completion $C_1\\otimes C_2$ of the algebraic tensor product $C_1\\odot C_2$ of $C_1$ and $C_2$ in any C*-tensor product $A_1\\otimes_{\\beta} A_2$ is maximal abelian provided that $C_1$ has the extension property of Kadison and Singer and $C_2$ contains an approximate identity for $A_2$. An example is given to show that $C_1\\otimes C_2$ can fail to be a masa in $A_1\\otimes_{\\beta} A_2$ with $A_1$ and $A_2$ unital if neither $C_1$ nor $C_2$ has the extension property. This gives an answer to a long-standing question, but leaves open some other interesting problems, one of which turns out to have a potentially intriguing implication for the Kadison-Singer extension problem."}
{"category": "Math", "title": "Kernel Convergence Estimates for Diffusions with Continuous Coefficients", "abstract": "We are interested in the kernel of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step $h>0$ in the limit as $h\\to0$. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step small enough for the method to be stable. We find sharp uniform bounds for the convergence rate as a function of the degree of smoothness which we conjecture. The bounds also apply to the time derivative of the kernel and its first two space derivatives. Our proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. Convergence rates depend on the degree of smoothness and H\\\"older differentiability of the coefficients. We find that the fastest convergence rate is of order $O(h^2)$ and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of H\\\"older differentiability except that the convergence rate is slower. H\\\"older continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity."}
{"category": "Math", "title": "Noncommutative Vitali-Hahn-Saks Theorem holds precisely for finite $W^\\ast$-algebras", "abstract": "It is shown that the bona fide generalization of the Vitali-Hahn-Saks Theorem to von Neumann algebras is possible if, and only if, the algebra is finite. This settles the problem on the noncommutative Vitali-Hahn-Saks Theorem completely and provides new means of characterizing finite von Neumann algebras."}
{"category": "Math", "title": "Analogues of the Smale and Hirsch Theorems for Cooperative Boolean and Other Discrete Systems", "abstract": "Discrete dynamical systems defined on the state space {0,1,...,p-1}^n have been used in multiple applications, most recently for the modeling of gene and protein networks. In this paper we study to what extent well-known theorems by Smale and Hirsch, which form part of the theory of (continuous) monotone dynamical systems, generalize or fail to do so in the discrete case. We show that that arbitrary m-dimensional systems cannot necessarily be embedded into n-dimensional cooperative systems for n=m+1, as in the Smale theorem for the continuous case, but we show that this is possible for n=m+2 as long as p is sufficiently large. We also prove that a natural discrete analogue of strong cooperativity implies nontrivial bounds on the lengths of periodic orbits and imposes a condition akin to Lyapunov stability on all attractors. Finally, we explore several natural candidates for definitions of irreducibility of a discrete system. While some of these notions imply the strong cooperativity of a given cooperative system and impose even tighter bounds on the lengths of periodic orbits than strong cooperativity alone, other plausible definitions allow the existence of exponentially long periodic orbits."}
{"category": "Math", "title": "The quenched critical point of a diluted disordered polymer model", "abstract": "We consider a model for a polymer interacting with an attractive wall through a random sequence of charges. We focus on the so-called diluted limit, when the charges are very rare but have strong intensity. In this regime, we determine the quenched critical point of the model, showing that it is different from the annealed one. The proof is based on a rigorous renormalization procedure. Applications of our results to the problem of a copolymer near a selective interface are discussed."}
{"category": "Math", "title": "Leibniz rules for enveloping algebras in symmetric ordering", "abstract": "Given a finite-dimensional Lie algebra, and a representation by derivations on the completed symmetric algebra of its dual, a number of interesting twisted constructions appear: certain twisted Weyl algebras, deformed Leibniz rules, quantized ``star'' product. We first illuminate a number of interrelations between these constructions and then proceed to study a special case in certain precise sense corresponding to the symmetric or Weyl ordering. This case has been known earlier to be related to computations with Hausdorff series, for example the expression for the star product is in such terms. For the deformed Leibniz rule, hence a coproduct, we present here a new nonsymmetric expression, which is then expanded into a sum of expressions labelled by a class of planar trees, and for a given tree evaluated by Feynman-like rules. These expressions are filtered by a bidegree and we show recursion formulas for the sums of expressions of a given bidegree, and compare the recursions to recursions for Hausdorff series, including the comparison of initial conditions. This way we show a direct corespondence between the Hausdorff series and the expression for twisted coproduct."}
{"category": "Math", "title": "Localized cohomology and some applications of Popa's cocycle super-rigidity theorem", "abstract": "We prove that orbit equivalence of measure preserving ergodic a.e. free actions of a countable group with the relative property (T) is a complete analytic equivalence relation."}
{"category": "Math", "title": "Definable Davies' Theorem", "abstract": "We prove the following analogue of a Theorem of R.O. Davies: Every $\\Sigma^1_2$ function $f:\\R\\times\\R\\to\\R$ can be represented as a sum of rectangular $\\Sigma^1_2$ functions if and only if all reals are constructible."}
{"category": "Math", "title": "Isoperimetry and Rough Path Regularity", "abstract": "Optimal sample path properties of stochastic processes often involve generalized H\\\"{o}lder- or variation norms. Following a classical result of Taylor, the exact variation of Brownian motion is measured in terms of $\\psi (x) \\equiv $ $x^{2}/\\log \\log (1/x) $ near $0+$. Such $\\psi $-variation results extend to classes of processes with values in abstract metric spaces. (No Gaussian or Markovian properties are assumed.) To establish integrability properties of the $\\psi $-variation we turn to a large class of Gaussian rough paths (e.g. Brownian motion and L\\'{e}vy's area viewed as a process in a Lie group) and prove Gaussian integrability properties using Borell's inequality on abstract Wiener spaces. The interest in such results is that they are compatible with rough path theory and yield certain sharp regularity and integrability properties (for iterated Stratonovich integrals, for example) which would be difficult to obtain otherwise. At last, $\\psi $-variation is identified as robust regularity property of solutions to (random) rough differential equations beyond semimartingales."}
{"category": "Math", "title": "A note on convergence of the equi-energy sampler", "abstract": "In a recent paper `The equi-energy sampler with applications statistical inference and statistical mechanics' [Ann. Stat. 34 (2006) 1581--1619], Kou, Zhou & Wong have presented a new stochastic simulation method called the equi-energy (EE) sampler. This technique is designed to simulate from a probability measure $\\pi$, perhaps only known up to a normalizing constant. The authors demonstrate that the sampler performs well in quite challenging problems but their convergence results (Theorem 2) appear incomplete. This was pointed out, in the discussion of the paper, by Atchad\\'e & Liu (2006) who proposed an alternative convergence proof. However, this alternative proof, whilst theoretically correct, does not correspond to the algorithm that is implemented. In this note we provide a new proof of convergence of the equi-energy sampler based on the Poisson equation and on the theory developed in Andrieu et al. (2007) for \\emph{Non-Linear} Markov chain Monte Carlo (MCMC). The objective of this note is to provide a proof of correctness of the EE sampler when there is only one feeding chain; the general case requires a much more technical approach than is suitable for a short note. In addition, we also seek to highlight the difficulties associated with the analysis of this type of algorithm and present the main techniques that may be adopted to prove the convergence of it."}
{"category": "Math", "title": "A slice theorem for quivers with an involution", "abstract": "We study the Luna slice theorem in the case of quivers with an involution or supermixed quivers as introduced by Zubkov. We construct an analogue to the notion of a local quiver setting. We use this technique to determine dimension vectors of simple supermixed representations."}
{"category": "Math", "title": "Spherical and hyperbolic lengths of images of arcs", "abstract": "Let $f$ be an analytic function on the unit disc which is in the Dirichlet class, so the Euclidean area of the image, counting multiplicity, is finite. The Euclidean length of a radial arc of hyperbolic length $\\rho$ is then $o(\\rho^1/2)$. In this note we consider the corresponding results when $f$ maps into the unit disc with the hyperbolic metric or the Riemann sphere with the spherical metric. Similar but not identical results hold."}
{"category": "Math", "title": "On the distribution of \\alpha p modulo one for primes p of a special form", "abstract": "A classical problem in analytic number theory is to study the distribution of $\\alpha p$ modulo 1, where $\\alpha$ is irrational and $p$ runs over the set of primes. We consider the subsequence generated by the primes $p$ such that $p+2$ is an almost-prime (the existence of infinitely many such $p$ is another topical result in prime number theory) and prove that its distribution has a similar property."}
{"category": "Math", "title": "Noncommutative Tangent Cones and Calabi Yau Algebras", "abstract": "We study the generalization of the idea of a local quiver of a representation of a formally smooth algebra, to broader classes of finitely generated algebras. In this new setting we can construct for every semisimple representation $M$ a local model and a non-commutative tangent cone. The representation schemes of these new algebras model the local structure and the tangent cone of the representation scheme of the original algebra at $M$. In this way one can try to classify algebras according to their local behavior. As an application we will show that the tangent cones of Calabi Yau 2 Algebras are always preprojective algebras. For Calabi Yau 3 Algebras the corresponding statement would be that the local model and the tangent cones derive from superpotentials. Although we do not have a proof in all cases, we will show that this will indeed hold in many cases."}
{"category": "Math", "title": "The true complexity of a system of linear equations", "abstract": "It is well-known that if a subset A of a finite Abelian group G satisfies a quasirandomness property called uniformity of degree k, then it contains roughly the expected number of arithmetic progressions of length k, that is, the number of progressions one would expect in a random subset of G of the same density as A. One is naturally led to ask which degree of uniformity is required of A in order to control the number of solutions to a general system of linear equations. Using so-called \"quadratic Fourier analysis\", we show that certain linear systems that were previously thought to require quadratic uniformity are in fact governed by linear uniformity. More generally, we conjecture a necessary and sufficient condition on a linear system L which guarantees that any subset A of F_p^n which is uniform of degree k contains the expected number of solutions to L."}
{"category": "Math", "title": "Population-Based Reversible Jump Markov Chain Monte Carlo", "abstract": "In this paper we present an extension of population-based Markov chain Monte Carlo (MCMC) to the trans-dimensional case. One of the main challenges in MCMC-based inference is that of simulating from high and trans-dimensional target measures. In such cases, MCMC methods may not adequately traverse the support of the target; the simulation results will be unreliable. We develop population methods to deal with such problems, and give a result proving the uniform ergodicity of these population algorithms, under mild assumptions. This result is used to demonstrate the superiority, in terms of convergence rate, of a population transition kernel over a reversible jump sampler for a Bayesian variable selection problem. We also give an example of a population algorithm for a Bayesian multivariate mixture model with an unknown number of components. This is applied to gene expression data of 1000 data points in six dimensions and it is demonstrated that our algorithm out performs some competing Markov chain samplers."}
{"category": "Math", "title": "Thick triangulations of hyperbolic n-manifolds", "abstract": "We show that a complete hyperbolic n-manifold has a geodesic triangulation such that the tetrahedra contained in the thick part are L-bilipschitz diffeomorphic to the standard Euclidean n-simplex, for some constant L depending only on the dimension and the constant used to define the thick-thin decomposition of M."}
{"category": "Math", "title": "A Geometric Approach to Confidence Sets for Ratios: Fieller's Theorem, Generalizations, and Bootstrap", "abstract": "We present a geometric method to determine confidence sets for the ratio E(Y)/E(X) of the means of random variables X and Y. This method reduces the problem of constructing confidence sets for the ratio of two random variables to the problem of constructing confidence sets for the means of one-dimensional random variables. It is valid in a large variety of circumstances. In the case of normally distributed random variables, the so constructed confidence sets coincide with the standard Fieller confidence sets. Generalizations of our construction lead to definitions of exact and conservative confidence sets for very general classes of distributions, provided the joint expectation of (X,Y) exists and the linear combinations of the form aX + bY are well-behaved. Finally, our geometric method allows to derive a very simple bootstrap approach for constructing conservative confidence sets for ratios which perform favorably in certain situations, in particular in the asymmetric heavy-tailed regime."}
{"category": "Math", "title": "Decomposing p-groups via Jordan algebras", "abstract": "For finite p-groups P of class 2 and exponent p the following are invariants of fully refined central decompositions of P: the number of members in the decomposition, the multiset of orders of the members, and the multiset of orders of their centers. Unlike for direct product decompositions, Aut P is not always transitive on the set of fully refined central decompositions, and the number of orbits can in fact be any positive integer. The proofs use the standard semi-simple and radical structure of Jordan algebras. These algebras also produce useful criteria for a p-group to be centrally indecomposable."}
{"category": "Math", "title": "Entropic Projections and Dominating Points", "abstract": "Generalized entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugate duality and functional analysis, criteria are derived for their existence. Representations of the generalized entropic projections are obtained: they are the ``measure component\" of some extended entropy minimization problem."}
{"category": "Math", "title": "Euclidean and hyperbolic lenghs of images of arcs", "abstract": "Let $f$ be a function that is analytic in the unit disc. We give new estimates, and new proofs of existing estimates, of the Euclidean length of the image under $f$ of a radial segment in the unit disc. Our methods are based on the hyperbolic geometry of plane domains, and we address some new questions that follow naturally from this approach."}
{"category": "Math", "title": "Remarks on weakly pseudoconvex boundaries: Erratum", "abstract": "We make two tiny corrections to our previous paper with the same title, and also obtain, as a bonus, something new."}
{"category": "Math", "title": "Remarks on weakly pseudoconvex boundaries", "abstract": "In this paper, we consider the boundary M of a weakly pseudoconvex domain in a Stein manifold. We point out a striking difference between the local cohomology and the global cohomology of M, and illustrate this with an example. We also discuss the first and second Cousin problems, and the strong Poincare problem for CR meromorphic functions on the weakly pseudoconvex boundary M."}
{"category": "Math", "title": "The Poincare lemma and local embeddability", "abstract": "For pseudoconvex abstract CR manifolds, the validity of the Poincare Lemma for (0,1) forms implies local embeddability in C^N. The two properties are equivalent for hypersurfaces of real dimension > or = 5. As a corollary we obtain a criterion for the non validity of the Poicare Lemma for (0,1) forms for a large class of abstract CR manifolds of CR codimension larger than one."}
{"category": "Math", "title": "Obstructions to generic embeddings", "abstract": "In Grauert's paper [G] it is noted that finite dimensionality of cohomology groups sometimes implies vanishing of these cohomomogy groups. Later on Laufer formulated a zero or infinity law for the cohomology groups of domains in Stein manifolds. In this paper we generalize Laufer's Theorem in [L] and its version for small domains of CR manifolds, proved in [Br], by considering Whitney cohomology on locally closed subsets and cohomology with supports for currents. With this approach we obtain a global result for CR manifolds generically embedded in a Stein manifold. Namely a necessary condition for global embedding into an open subset of a Stein manifold is that the de-bar-M-cohomology groups must be either zero or infinite dimensional."}
{"category": "Math", "title": "Multiply Connected Topological Economics, Confidence Relation and Political Economy", "abstract": "Using the similar formulas of the preference relation and the utility function, we propose the confidence relations and the corresponding influence functions that represent various interacting strengths of different families, cliques and systems of organization. Since they can affect products, profit and prices, etc., in an economic system, and are usually independent of economic results, therefore, the system can produce a multiply connected topological economics. If the political economy is an economy chaperoned polity, it will produce consequentially a binary economy. When the changes of the product and the influence are independent one another, they may be a node or saddle point. When the influence function large enough achieves a certain threshold value, it will form a wormhole with loss of capital. Various powers produce usually the economic wormhole and various corruptions."}
{"category": "Math", "title": "Superspecial Abelian Varieties and the Eichler Basis Problem for Hilbert Modular Forms", "abstract": "Let $p$ be an unramified prime in a totally real field $L$ such that $h^+(L)=1$. Our main result shows that Hilbert modular newforms of parallel weight two for $\\Gamma_0(p)$ can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This can be viewed as a geometric reinterpretation of the Eichler Basis Problem for Hilbert modular forms."}
{"category": "Math", "title": "Algebraic setup of non-strict multiple zeta values", "abstract": "In this article, we introduce an algebraic setup of non-strict multiple zeta values (NMZVs, for short) and prove some relations of NMZVs, which are analogous to Hoffman's relations of multiple zeta values, by using this algebraic setup of NMZVs."}
{"category": "Math", "title": "Lineage-through-time plots of birth-death processes", "abstract": "We calculate the density and expectation for the number of lineages in a reconstructed tree with $n$ extant species. This is done with conditioning on the age of the tree as well as with assuming a uniform prior for the age of the tree."}
{"category": "Math", "title": "Global existence for energy critical waves in 3-d domains : Neumann boundary conditions", "abstract": "We prove that the defocusing quintic wave equation, with Neumann boundary conditions, is globally wellposed on $H^1_N(\\Omega) \\times L^2(\\Omega)$ for any smooth (compact) domain $\\Omega \\subset \\mathbb{R}^3$. The proof relies on one hand on $L^p$ estimates for the spectral projector by Smith and Sogge, and on the other hand on a precise analysis of the boundary value problem, which turns out to be much more delicate than in the case of Dirichlet boundary conditions."}
{"category": "Math", "title": "\\Pi^0_1 classes, strong minimal covers and hyperimmune-free degrees", "abstract": "We investigate issues surrounding an old question of Yates' as to the existence of a minimal Turing degree with no strong minimal cover, specifically with respect to the hyperimmune-free degrees."}
{"category": "Math", "title": "On proper and exterior sequentiality", "abstract": "In this article a sequential theory in the category of spaces and proper maps is described and developed. As a natural extension a sequential theory for exterior spaces and maps is obtained."}
{"category": "Math", "title": "Continuum percolation at and above the uniqueness treshold on homogeneous spaces", "abstract": "We consider the Poisson Boolean model of continuum percolation on a homogeneous Riemannian manifold $M$. Let $lambda$ be intensity of the Poisson process in the model and let $lambda_u$ be the infimum of the set of intensities that a.s. produce a unique unbounded component. We show that above $\\lambda_u$ there is a.s. a unique unbounded component. We also study what happens at $\\lambda_u$ for some spaces. In particular, if $M$ is the product of the hyperbolic disc and the real line, then at $\\lambda_u$ there is a.s. not a unique unbounded component. The results are inspired by results for Bernoulli bond percolation on graphs due to Haggstrom, Peres and Schonmann."}
{"category": "Math", "title": "Period Lengths for Iterated Functions", "abstract": "For random maps, the expected value of the order (i.e. the period of the sequence of compositional iterates) is approximated asymptotically. It is much smaller than the expected value for the product of the cycle lengths."}
{"category": "Math", "title": "Tensor product for symmetric monoidal categories", "abstract": "We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2-category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2-cells monoidal natural transformations. Our tensor product together with a suitable unit is part of a structure on SMC that is a 2-categorical version of the symmetric monoidal closed categories. This structure is surprisingly simple. In particular the arrows involved in the associativity and symmetry laws for the tensor and in the unit cancellation laws are 2-natural and satisfy coherence axioms which are strictly commuting diagrams. We also show that the category quotient of SMC by the congruence generated by its 2-cells admits a symmetric monoidal closed structure."}
{"category": "Math", "title": "Ueber Eigenwerte, Integrale und pi^2/6: Die Idee der Spurformel (On eigenvalues, integrals and pi^2/6: The idea of the trace formula)", "abstract": "This is an expository article that results from a talk given to second year students at Oldenburg university. The aim of the talk was to show what beautiful and unexpected results may be obtained if one plays with daring analogies in a way that is usually not done in undergraduate education (unfortunately): We start from the fact that the sum of diagonal entries of a symmetric matrix equals the sum of its eigenvalues. We then guess an analogous formula where the matrix is replaced by a function of two real variables and sums are replaced by integrals in a systematic way. We show that this is indeed a worthwhile process: In a special case it yields that the sum of inverse squares of the positive integers is pi^2/6. Finally, an outline of the proof of the guessed formula is given, and further applications, for example to the connection between billiards and the frequencies of a drum, are explained."}
{"category": "Math", "title": "Twisted Gromov-Witten r-spin potential and Givental's quantization", "abstract": "The universal curve p:C->\\Mbar over the moduli space \\Mbar of stable r-spin maps to a target K\\\"ahler manifold X carries a universal spinor bundle L->C. Therefore the moduli space \\Mbar itself carries a natural K-theory class Rp_*L. We introduce a twisted r-spin Gromov-Witten potential of X enriched with Chern characters of Rp_*L. We show that the twisted potential can be reconstructed from the ordinary r-spin Gromov-Witten potential of X via an operator that assumes a particularly simple form in Givental's quantization formalism."}
{"category": "Math", "title": "Poincar\\'e duality for $K$-theory of equivariant complex projective spaces", "abstract": "We make explicit Poincar\\'{e} duality for the equivariant $K$-theory of equivariant complex projective spaces. The case of the trivial group provides a new approach to the $K$-theory orientation."}
{"category": "Math", "title": "The Twisted Higher Harmonic Signature for Foliations", "abstract": "We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation of a compact Riemannian manifold with coefficients in a leafwise U(p,q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences for the Novikov conjecture for foliations and for groups are investigated. Replaces The Higher Harmonic Signature for Foliations I: The Untwisted Case, and contains significant improvements."}
{"category": "Math", "title": "Critical values of moment maps on quantizable manifolds", "abstract": "Let $M$ be a quantizable symplectic manifold acted on by $T=(S^1)^r$ in a Hamiltonian fashion and $J$ a moment map for this action. Suppose that the set $M^{T}$ of fixed points is discrete and denote by ${\\alpha}_{pj}\\in{\\mathbb Z}^r$ the weights of the isotropy representation at $p$. By means of the $\\alpha_{pj}$'s we define a partition ${\\mathcal Q}_+$, ${\\mathcal Q}_-$ of $M^T$. (When $r=1$, ${\\mathcal Q}_{\\pm}$ will be the set of fixed points such that the half of the Morse index of $J$ at them is even (odd)). We prove the existence of a map $\\pi_{\\pm}:{\\mathcal Q}_{\\pm}\\to{\\mathcal Q}_{\\mp}$ such that $J(q)-J(\\pi_{\\pm}(q))\\in I_{\\mp}$, for all $q\\in {\\mathcal Q}_{\\pm}$, where $I_{\\pm}$ is the lattice generated by the $\\alpha_{pj}$'s with $p\\in{\\mathcal Q}_{\\pm}.$ We define partition functions $N_p$ similar to the ones of Kostant \\cite{Gui} and we prove that $\\sum_{p\\in{\\mathcal Q}_+}N_p(l)=\\sum_{p\\in{\\mathcal Q}_-}N_p(l)$, for any $l\\in{\\mathbb Z}^r$ with $|l|$ sufficiently large."}
{"category": "Math", "title": "On the integration of Poisson homogeneous spaces", "abstract": "We study a reduction procedure for describing the symplectic groupoid of a Poisson homogeneous space obtained by quotient of a coisotropic subgroup. We perform it as a reduction of the Lu-Weinstein symplectic groupoid integrating Poisson Lie groups, that is suitable even for the non complete case."}
{"category": "Math", "title": "Mixing Least-Squares Estimators when the Variance is Unknown", "abstract": "We propose a procedure to handle the problem of Gaussian regression when the variance is unknown. We mix least-squares estimators from various models according to a procedure inspired by that of Leung and Barron (2007). We show that in some cases the resulting estimator is a simple shrinkage estimator. We then apply this procedure in various statistical settings such as linear regression or adaptive estimation in Besov spaces. Our results provide non-asymptotic risk bounds for the Euclidean risk of the estimator."}
{"category": "Math", "title": "Geodesible contact structures on 3--manifolds", "abstract": "In this paper, we study and almost completely classify contact structures on closed 3--manifolds which are totally geodesic for some Riemannian metric. Due to previously known results, this amounts to classifying contact structures on Seifert manifolds which are transverse to the fibers. Actually, we obtain the complete classification of contact structures with negative (maximal) twisting number (which includes the transverse ones) on Seifert manifolds whose base is not a sphere, as well as partial results in the spherical case."}
{"category": "Math", "title": "Lawson homology, morphic cohomology and Chow motives", "abstract": "In this paper, the Lawson homology and morphic cohomology are defined on the Chow motives. We also define the rational coefficient Lawson homology and morphic cohomology of the Chow motives of finite quotient projective varieties. As a consequence, we obtain a formula for the Hilbert scheme of points on a smooth complex projective surface. Further discussion concerning generic finite maps is given. As a result, we give examples of self-product of smooth projective curves with nontrivial Griffiths groups by using a result of Ceresa."}
{"category": "Math", "title": "Uniform non-amenability, cost, and the first l^2-Betti number", "abstract": "It is shown that $2\\beta_1(\\G)\\leq h(\\G)$ for any countable group $\\G$, where $\\beta_1(\\G)$ is the first $\\ell^2$-Betti number and $h(\\G)$ the uniform isoperimetric constant. In particular, a countable group with non-vanishing first $\\ell^2$-Betti number is uniformly non-amenable. We then define isoperimetric constants in the framework of measured equivalence relations. For an ergodic measured equivalence relation $R$ of type $\\IIi$, the uniform isoperimetric constant $h(R)$ of $R$ is invariant under orbit equivalence and satisfies $$ 2\\beta_1(R)\\leq 2C(R)-2\\leq h(R), $$ where $\\beta_1(\\R)$ is the first $\\ell^2$-Betti number and $C(R)$ the cost of $R$ in the sense of Levitt (in particular $h(R)$ is a non-trivial invariant). In contrast with the group case, uniformly non-amenable measured equivalence relations of type $\\IIi$ always contain non-amenable subtreeings. An ergodic version $h_e(\\G)$ of the uniform isoperimetric constant $h(\\G)$ is defined as the infimum over all essentially free ergodic and measure preserving actions $\\alpha$ of $\\G$ of the uniform isoperimetric constant $h(\\R_\\alpha)$ of the equivalence relation $R_\\alpha$ associated to $\\alpha$. By establishing a connection with the cost of measure-preserving equivalence relations, we prove that $h_e(\\G)=0$ for any lattice $\\G$ in a semi-simple Lie group of real rank at least 2 (while $h_e(\\G)$ does not vanish in general)."}
{"category": "Math", "title": "Hyperbolic conservation laws and spacetimes with limited regularity", "abstract": "Hyperbolic conservation laws posed on manifolds arise in many applications to geophysical flows and general relativity. Recent work by the author and his collaborators attempts to set the foundations for a study of weak solutions defined on Riemannian or Lorentzian manifolds and includes an investigation of the existence and qualitative behavior of solutions. The metric on the manifold may either be fixed (shallow water equations on the sphere, for instance) or be one of the unknowns of the theory (Einstein-Euler equations of general relativity)."}
{"category": "Math", "title": "Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions", "abstract": "Solutions to conservation laws satisfy the monotonicity property: the number of local extrema is a non-increasing function of time, and local maximum/minimum values decrease/increase monotonically in time. This paper investigates this property from a numerical standpoint. We introduce a class of fully discrete in space and time, high order accurate, difference schemes, called generalized monotone schemes. Convergence toward the entropy solution is proven via a new technique of proof, assuming that the initial data has a finite number of extremum values only, and the flux-function is strictly convex. We define discrete paths of extrema by tracking local extremum values in the approximate solution. In the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum. As a corollary, we obtain a proof of convergence for a MUSCL-type scheme being second order accurate away from sonic points and extrema."}
{"category": "Math", "title": "Heat kernels and critical limits", "abstract": "This paper is an exposition of several questions linking heat kernel measures on infinite dimensional Lie groups, limits associated with critical Sobolev exponents, and Feynmann-Kac measures for sigma models."}
{"category": "Math", "title": "Conservation laws with vanishing nonlinear diffusion and dispersion", "abstract": "We study the limiting behavior of the solutions to a class of conservation laws with vanishing nonlinear diffusion and dispersion terms. We prove the convergence to the entropy solution of the first order problem under a condition on the relative size of the diffusion and the dispersion terms. This work is motivated by the pseudo-viscosity approximation introduced by Von Neumann in the 50's."}
{"category": "Math", "title": "Twists of genus three Jacobians", "abstract": "We give a criterion to distinguish between a genus three Jacobian and its [-1] twist in terms of the product of the 36 even theta nulls. We also express the product of the 36 theta nulls in terms of the discriminant of a genus three curve. The results are arithmetic in nature and thus add to previous work over C on the product of the even theta nulls. They generalize previous work of Ritzenthaler and Lachaud for Abelian threefolds which are (2,2,2) isogenous to a product of elliptic curves. This answers a 2003 of letter of Jean-Pierre Serre to Jaap Top."}
{"category": "Math", "title": "Spaces and groups with conformal dimension greater than one", "abstract": "We show that if a complete, doubling metric space is annularly linearly connected then its conformal dimension is greater than one, quantitatively. As a consequence, we answer a question of Bonk and Kleiner: if the boundary of a one-ended hyperbolic group has no local cut points, then its conformal dimension is greater than one."}
{"category": "Math", "title": "Equivalence of types and Catlin boundary systems", "abstract": "The D'Angelo finite type is shown to be equivalent to the Kohn finite ideal type on smooth, pseudoconvex domains in complex n space. This is known as the Kohn Conjecture. The argument uses Catlin's notion of a boundary system as well as methods from subanalytic and semialgebraic geometry. When a subset of the boundary contains only two level sets of the Catlin multitype, a lower bound for the subelliptic gain in the \\bar\\partial-Neumann problem is obtained in terms of the D'Angelo type, the dimension of the ambient space, and the level of forms."}
{"category": "Math", "title": "Non-adic formal schemes", "abstract": "Our purpose is to make a contribution to the foundation of the theory of formal scheme. We are interested particularly in non-Noetherian or non-adic formal schemes, which have been little studied. We redefine the formal scheme as a proringed space and study its basic properties. We also find several examples of non-adic formal schemes."}
{"category": "Math", "title": "A catalogue of singularities", "abstract": "This paper is an attempt to classify finite-time singularities of PDEs. Most of the problems considered describe free-surface flows, which are easily observed experimentally. We consider problems where the singularity occurs at a point, and where typical scales of the solution shrink to zero as the singularity is approached. Upon a similarity transformation, exact self-similar behaviour is mapped to the fixed point of a {\\it infinite dimensional dynamical system} representing the original dynamics. We show that the dynamics close to the fixed point is a useful way classifying the structure of the singularity. Specifically, we consider various types of stable and unstable fixed points, centre-manifold dynamics, limit cycles, and chaotic dynamics."}
{"category": "Math", "title": "A new family of maximal curves over a finite field", "abstract": "A new family of maximal curves over a finite field is presented and some of their properties are investigated."}
{"category": "Math", "title": "Asymptotic Properties of Hilbert Geometry", "abstract": "We show that the spheres in Hilbert geometry have the same volume growth entropy as those in the Lobachevsky space. We give the asymptotic estimates for the ratio of the volume of metric ball to the area of the metric sphere in Hilbert geometry. Derived estimates agree with the well-known fact in the Lobachevsky space"}
{"category": "Math", "title": "Publier sous l'Occupation I. Autour du cas de Jacques Feldbau et de l'Acad\\'emie des sciences", "abstract": "This is an article on mathematical publishing during the German occupation of France. Looking at the cases of four of them and especially at the case of Jacques Feldbau (one of the founders of the theory of fibre bundles), we investigate the way censorship struck the French mathematicians who declared jewish by the Statut des juifs of october 1940, and the strategies these mathematicians then developed. The way the Vichy laws have been discussed and applied at the Acad\\'emie des sciences is investigated."}
{"category": "Math", "title": "The singular points of self-similar functions with zero spectral degree. Stieltjes self-similar string", "abstract": "The self-similar functions of zero spectral degree are defined. The singular points of these functions are investigated and full classification of points is given. The connection with spectral problems (Stieltjes string) is pointed out."}
{"category": "Math", "title": "Bayesian finite mixtures: a note on prior specification and posterior computation", "abstract": "A new method for the computation of the posterior distribution of the number k of components in a finite mixture is presented. Two aspects of prior specification are also studied: an argument is made for the use of a Poisson(1) distribution as the prior for k; and methods are given for the selection of hyperparameter values in the mixture of normals model, with natural conjugate priors on the components parameters."}
{"category": "Math", "title": "On the structure and representations of the insertion-elimination Lie algebra", "abstract": "We examine the structure of the insertion-elimination Lie algebra on rooted trees introduced in \\cite{CK}. It possesses a triangular structure $\\g = \\n_+ \\oplus \\mathbb{C}.d \\oplus \\n_-$, like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a \"lowest weight\" $\\lambda \\in \\mathbb{C}$. We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible."}
{"category": "Math", "title": "Generic Properties of Homogeneous Ricci Solitons", "abstract": "We discuss the geometry of homogeneous Ricci solitons. After showing the nonexistence of compact homogeneous and noncompact steady homogeneous solitons, we concentrate on the study of left invariant Ricci solitons. We show that, in the unimodular case, the Ricci soliton equation does not admit solutions in the set of left invariant vector fields. We prove that a left invariant soliton of gradient type must be a Riemannian product with a nontrivial Euclidean de Rham factor. As an application of our results we prove that any generalized metric Heisenberg Lie group is a nongradient left invariant Ricci soliton of expanding type."}
{"category": "Math", "title": "Model Structures on the Category of Small Double Categories", "abstract": "In this paper we obtain several model structures on {\\bf DblCat}, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in {\\bf Cat} and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, {\\bf DblCat} inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, several nerves, and horizontal categorification."}
{"category": "Math", "title": "A pastiche on embeddings into simple groups (following P. E. Schupp)", "abstract": "Let lambda be an infinite cardinal number and let C = {H_i| i in I} be a family of nontrivial groups. Assume that |I|<=lambda, |H_i|<= lambda, for i in I, and at least one member of C achieves the cardinality lambda. We show that there exists a simple group S of cardinality lambda that contains an isomorphic copy of each member of C and, for all H_i, H_j in C with |H_j|=lambda, is generated by the copies of H_i and H_j in S. This generalizes a result of Paul E. Schupp (moreover, our proof follows the same approach based on small cancelation). In the countable case, we partially recover a much deeper embedding result of Alexander Yu. Ol'shanskii."}
{"category": "Math", "title": "Noncommutative geometry and motives (a quoi servent les endomotifs?)", "abstract": "This paper gives a short and historical survey on the theory of pure motives in algebraic geometry and reviews some of the recent developments of this theory in noncommutative geometry. The second part of the paper outlines the new theory of endomotives and some of its relevant applications in number-theory."}
{"category": "Math", "title": "Quasi-Sectorial Contractions", "abstract": "We revise the notion of the quasi-sectorial contractions. Our main theorem establishes a relation between semigroups of quasi-sectorial contractions and a class of m-sectorial generators. We discuss a relevance of this kind of contractions to the theory of operator-norm approximations of strongly continuous semigroups."}
{"category": "Math", "title": "On q-deformed Stirling numbers", "abstract": "The purpose of this article is to introduce q-deformed Stirling numbers of the first and second kinds. Relations between these numbers, Riemann zeta function and q-Bernoulli numbers of higher order are given. Some relations related to the classical Stirling numbers and Bernoulli numbers of higher order are found. By using derivative operator to the generating function of the q-deformed Stirling numbers of the second kinds, a new function is defined which interpolates the q-deformed Stirling numbers of the second kinds at negative integers. The recurrence relations of the Stirling numbers of the first and second kind are given. In addition, relation between q-deformed Stirling numbers and q-Bell numbers is obtained."}
{"category": "Math", "title": "Idempotent ultrafilters and polynomial recurrence", "abstract": "We give a new proof of a polynomial recurrence result due to Bergelson, Furstenberg, and McCutcheon, using idempotent ultrafilters instead of IP-limits."}
{"category": "Math", "title": "Markov processes with product-form stationary distribution", "abstract": "We study a class of Markov processes with finite state space and continuous time that have product form stationary distributions. We obtain a number of examples that can generate conjectures for diffusions with inert drift."}
{"category": "Math", "title": "Linear \\infty-Harmonic maps between Rienmannian manifolds", "abstract": "In this paper, we give complete classifications of linear $\\infty$-harmonic maps between Euclidean and Heisenberg spaces, between Nil and Sol spaces. We also classify all $\\infty$-harmonic linear endomorphisms of Sol space and show that there is a subgroup of $\\infty$-harmonic linear automorphisms in the group of linear automorphisms of Sol space."}
{"category": "Math", "title": "On relations among Dirichlet series whose coefficients are class numbers of binary cubic forms", "abstract": "We study the class numbers of integral binary cubic forms. For each $SL_2(Z)$ invariant lattice $L$, Shintani introduced Dirichlet series whose coefficients are the class numbers of binary cubic forms in $L$. We classify the invariant lattices, and investigate explicit relationships between Dirichlet series associated with those lattices. We also study the analytic properties of the Dirichlet series, and rewrite the functional equation in a self dual form using the explicit relationship."}
{"category": "Math", "title": "A new approach to strong embeddings", "abstract": "We revisit strong approximation theory from a new perspective, culminating in a proof of the Koml\\'os-Major-Tusn\\'ady embedding theorem for the simple random walk. The proof is almost entirely based on a series of soft arguments and easy inequalities. The new technique, inspired by Stein's method of normal approximation, is applicable to any setting where Stein's method works. In particular, one can hope to take it beyond sums of independent random variables."}
{"category": "Math", "title": "On time dynamics of coagulation-fragmentation processes", "abstract": "We establish a characterization of coagulation-fragmentation processes, such that the induced birth and death processes depicting the total number of groups at time $t\\ge 0$ are time homogeneous. Based on this, we provide a characterization of mean-field Gibbs coagulation-fragmentation models, which extends the one derived by Hendriks et al. As a by- product of our results, the class of solvable models is widened and a question posed by N. Berestycki and Pitman is answered, under restriction to mean-field models."}
{"category": "Math", "title": "Differential Invariants of SL(2) and SL(3)-ACTIONS on R^2", "abstract": "The main purpose of this paper is calculation of differential invariants which arise from prolonged actions of two Lie groups SL(2) and SL(3) on the $n$th jet space of $R^2$. It is necessary to calculate $n$th prolonged infenitesimal generators of the action."}
{"category": "Math", "title": "Maximum likelihood estimators and random walks in long memory models", "abstract": "We consider statistical models driven by Gaussian and non-Gaussian self-similar processes with long memory and we construct maximum likelihood estimators (MLE) for the drift parameter. Our approach is based on the approximation by random walks of the driving noise. We study the asymptotic behavior of the estimators and we give some numerical simulations to illustrate our results."}
{"category": "Math", "title": "Purely Algebraic Method to Construct Toric Schemes", "abstract": "In this article, we first give some elementary proprieties of monoids and fans, then construct a toric scheme over an arbitrary ring, from a given fan. Using Valuative Criterion, we prove that this scheme is separated and give the sufficient and necessary condition when it is proper. We also study the regularity and logarithmic regularity of it. Finally we study the morphisms of toric schemes induced by the homomorphisms of fans."}
{"category": "Math", "title": "Jacobi Forms of Degree One and Weil Representations", "abstract": "We discuss the notion of Jacobi forms of degree one with matrix index, we state dimension formulas, give explicit examples, and indicate how closely their theory is connected to the theory of invariants of Weil representations associated to finite quadratic modules."}
{"category": "Math", "title": "How many latin rectangles are there?", "abstract": "Until now the problem counting Latin rectangles m x n has been solved with an explicit formula for m = 2, 3 and 4 only. In the present paper an explicit formula is provided for the calculation of the number of Latin rectangles for any order m. The results attained up to now become particular cases of this new formula. Furthermore, putting m = n, the number of Latin squares of order n can also be obtained in an explicit form."}
{"category": "Math", "title": "Automorphisms of Chevalley groups of types B_2 and G_2 over local rings", "abstract": "In the paper we prove that every automorphism of any adjoint Chevalley group of types B_2 or G_2 is standard, i.e., it is a composition of the ``inner'' automorphism, ring automorphism and central automorphism."}
{"category": "Math", "title": "Automorphisms of the semigroup of invertible matrices with nonnegative elements over commutative partially ordered rings", "abstract": "Suppose that R is an ordered ring, G_n(R) is a subsemigroup of $GL_n(R)$, consisting of all matrices with nonnegative elements. A.V. Mikhalev and M.A. Shatalova described all automorphisms of G_n(R), where R is a linearly ordered skewfield and n>1. E.I. Bunina and A.V. Mikhalev found all automorphisms of G_n(R), if R is an arbitrary linearly ordered associative ring with 1/2, n>2. In this paper we describe automorphisms of G_n(R), if R is a commutative partially ordered ring, containing Q, n>2."}
{"category": "Math", "title": "Umkehr Maps", "abstract": "In this note we study umkehr maps in generalized (co)homology theories arising from the Pontrjagin-Thom construction, from integrating along fibers, pushforward homomorphisms, and other similar constructions. We consider the basic properties of these constructions and develop axioms which any umkehr homomorphism must satisfy. We use a version of Brown representability to show that these axioms completely characterize these homomorphisms, and a resulting uniqueness theorem follows. Finally, motivated by constructions in string topology, we extend this axiomatic treatment of umkehr homomorphisms to a fiberwise setting."}
{"category": "Math", "title": "Wall rational functions and Khrushchev's formula for orthogonal rational functions", "abstract": "We prove that the Nevalinna-Pick algorithm provides different homeomorphisms between certain topological spaces of measures, analytic functions and sequences of complex numbers. This algorithm also yields a continued fraction expansion of every Schur function, whose approximants are identified. The approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants are studied. The above results permit us to obtain a Khrushchev's formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is also presented."}
{"category": "Math", "title": "Tate Resolutions for Segre Embeddings", "abstract": "We give an explicit description of the terms and differentials of the Tate resolution of sheaves arising from Segre embeddings of $\\P^a\\times\\P^b$. We prove that the maps in this Tate resolution are either coming from Sylvester-type maps, or from Bezout-type maps arising from the so-called toric Jacobian."}
{"category": "Math", "title": "Gelfand-Tsetlin bases for representations of finite W-algebras and shifted Yangians", "abstract": "Remarkable subalgebras of the Yangian for gl_n called the shifted Yangians were introduced in a recent work by Brundan and Kleshchev in relation to their study of finite W-algebras. In particular, in that work a classification of finite-dimensional irreducible representations of the shifted Yangians and the associated finite W-algebras was given. We construct a class of these representations in an explicit form via bases of Gelfand-Tsetlin type."}
{"category": "Math", "title": "A note on pairs of metrics in a three-dimensional linear vector space", "abstract": "Pairs of metrics in a three-dimensional linear vector space are considered, one of which is a Minkowski type metric with the signature (+,-,-). Such metric pairs are classified and canonical presentations for them in each class are suggested."}
{"category": "Math", "title": "Reduced distance based at singular time in the Ricci flow", "abstract": "In this paper, we define a reduced distance function based at a point at the singular time $T<\\infty$ of a Ricci flow. We also show the monotonicity of the corresponding reduced volume based at time T, with equality iff the Ricci flow is a gradient shrinking soliton. Our curvature bound assumption is more general than the type I condition."}
{"category": "Math", "title": "On the large-distance asymptotics of steady state solutions of the Navier-Stokes equations in 3D exterior domains", "abstract": "We identify the leading term describing the behavior at large distances of the steady state solutions of the Navier-Stokes equations in 3D exterior domains with vanishing velocity at the spatial infinity."}
{"category": "Math", "title": "Inverse scattering for the nonlinear Schr\\\"{o}dinger equation with the Yukawa potential", "abstract": "We study the inverse scattering problem for the three dimensional nonlinear Schroedinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation."}
{"category": "Math", "title": "Normal generation of very ample line bundles on toric varieties", "abstract": "The article has been withdrawn by the author due to the existence of counterexamples."}
{"category": "Math", "title": "Confirmation of Matheron's conjecture on the covariogram of a planar convex body", "abstract": "The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that for d=2 the covariogram g_K determines K within the class of all planar convex bodies, up to translations and reflections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. In this paper we confirm Matheron's conjecture completely."}
{"category": "Math", "title": "Twisted Dedekind Type Sums Associated with Barnes' Type Multiple Frobenius-Euler l-Functions", "abstract": "The aim of this paper is to construct new Dedekind type sums. We construct generating functions of Barnes' type multiple Frobenius-Euler numbers and polynomials. By applying Mellin transformation to these functions, we define Barnes' type multiple l-functions, which interpolate Frobenius-Euler numbers at negative integers. By using generalizations of the Frobenius-Euler functions, we define generalized Dedekind type sums and prove corresponding reciprocity law. We also give twisted versions of the Frobenius-Euler polynomials and new Dedekind type sums and corresponding reciprocity law. Furthermore, by using p-adic q-Volkenborn integral and twisted (h,q)-Bernoulli functions, we construct p-adic (h,q)-higher order Dedekind type sums. By using relation between Bernoulli and Frobenius-Euler functions, we also define analogues of Hardy-Berndt type sums. We give some new relations related to to these sums as well."}
{"category": "Math", "title": "Non-abelian pseudomeasures and congruences between abelian Iwasawa L-functions", "abstract": "The paper starts out from pseudomeasures (in the sense of Serre) which hold the arithmetic properties of the abelian $l$-adic Artin $L$-functions over totally real number fields. In order to generalize to non-abelian $l$-adic $L$-functions, these abelian pseudomeasures must satisfy congruences which are introduced but not yet known to be true. The relation to the ``equivariant main conjecture'' of Iwasawa theory is discussed."}
{"category": "Math", "title": "Permutations defining convex permutominoes", "abstract": "A permutomino of size n is a polyomino determined by particular pairs (P1, P2) of permutations of size n, such that P1(i) is different from P2(i), for all i. Here we determine the combinatorial properties and, in particular, the characterization for the permutations defining convex permutominoes. Using such a characterization, these permutations can be uniquely represented in terms of the so called square permutations, introduced by Mansour and Severini. Then, we provide a closed formula for the number of these permutations with size n."}
{"category": "Math", "title": "Fluctuations for a conservative interface model on a wall", "abstract": "We consider an effective interface model on a hard wall in (1+1) dimensions, with conservation of the area between the interface and the wall. We prove that the equilibrium fluctuations of the height variable converge in law to the solution of a SPDE with reflection and conservation of the space average. The proof is based on recent results obtained with L. Ambrosio and G. Savare on stability properties of Markov processes with log-concave invariant measures."}
{"category": "Math", "title": "An effective and sharp lower bound on Seshadri constants on surfaces with Picard number 1", "abstract": "We study lower bounds on Seshadri constants at arbitrary points on surfaces with Picard number 1."}
{"category": "Math", "title": "Parametric estimation in noisy blind deconvolution model: a new estimation procedure", "abstract": "In a parametric framework, the paper is devoted to the study of a new estimation procedure for the inverse filter and the level noise in a complex noisy blind discrete deconvolution model. Our estimation method is a consequence of the sharp exploitation of the specifical properties of the Hankel forms. The distribution of the input signal is also estimated. The strong consistency and the asymptotic distribution of all estimates are established. A consistent simulation study is added in order to demonstrate empirically the computational performance of our estimation procedures."}
{"category": "Math", "title": "Congruences between abelian pseudomeasures", "abstract": "Following Deligne and Ribet (`Values of abelian $L$-functions at negative integers over totally real fields.' Invent. Math. 59 (1980), 227-286) we prove that the `torsion congruences' (as introduced in our paper `Non-abelian pseudomeasures and congruences between abelian Iwasawa $L$-functions.' To appear in Pure and Applied Mathematics Quarterly) hold and so reduce the `main conjecture' of equivariant Iwasawa theory to the integrality of the logarithmic pseudomeasure."}
{"category": "Math", "title": "Dimensional reduction for energies with linear growth involving the bending moment", "abstract": "A $\\Gamma$-convergence analysis is used to perform a 3D-2D dimension reduction of variational problems with linear growth. The adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external loadings inducing a bending moment, may depend on the average in the transverse direction of a Cosserat vector field, as well as on the deformation of the mid-plane. The assumption of linear growth on the energy leads to an asymptotic analysis in the spaces of measures and of functions with bounded variation."}
{"category": "Math", "title": "Controlled Synchronization Under Information Constraints", "abstract": "The class of controlled synchronization systems under information constraints imposed by limited information capacity of the coupling channel is analyzed. It is shown that the framework proposed in A. L. Fradkov, B. Andrievsky, R. J. Evans, Physical Review E 73, 066209 (2006) is suitable not only for observer-based synchronization but also for controlled master-slave synchronization via communication channel with limited information capacity. A simple first order coder-decoder scheme is proposed and a theoretical analysis for multi-dimensional master-slave systems represented in the Lurie form (linear part plus nonlinearity depending only on measurable outputs) is provided. An output feedback control law is proposed based on the Passification theorem. It is shown that the upper bound of the limit synchronization error is proportional to the upper bound of the transmission error. As a consequence, both upper and lower bounds of limit synchronization error are proportional to the maximum rate of the coupling signal and inversely proportional to the information transmission rate (channel capacity). The results are applied to controlled synchronization of two chaotic Chua systems coupled via a controller and a channel with limited capacity."}
{"category": "Math", "title": "(a, 1)f structures on product of spheres", "abstract": "Our aim in this paper is to give some examples of $(a, 1)f$ Riemannian structures (a generalization of an $r$-paracontact structure) induced on product of spheres of codimension $r$ ($r \\in \\{1,2\\} $) in an $m$-dimensional Euclidean space ($m>2$), endowed with an almost product structure."}
{"category": "Math", "title": "Normal domains with monomial presentations", "abstract": "Let A be a finitely generated commutative algebra over a field K with a presentation A=K < X_{1}, ..., X_{n} | R >, where R is a set of monomial relations in the generators X_{1}, ..., X_{n}. So A = K[S], the semigroup algebra of the monoid S=< X_{1}, ..., X_{n} | R >. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated."}
{"category": "Math", "title": "Numerical approximation of the thermistor problem", "abstract": "We use a finite element approach based on Galerkin method to obtain approximate steady state solutions of the thermistor problem with temperature dependent electrical conductivity."}
{"category": "Math", "title": "Multidimensional decay in van der Corput lemma", "abstract": "In this paper we present a multidimensional version of the van der Corput lemma where the decay of the oscillatory integral is gained with respect to all space variables, connecting the standard one-dimensional van der Corput lemma with the stationary phase method."}
{"category": "Math", "title": "The integral logarithm in Iwasawa theory: an exercise", "abstract": "Let $l$ be an odd prime number and $H$ a finite abelian $l$-group. We determine the unit group of $\\Lambda_\\wedge[H]$ (the completion of the localization at $l$ of $\\Bbb{Z}_l[[T]][H]$) as well as the kernel and cokernel of the integral logarithm $L:\\Lambda_\\wedge[H]^\\times\\to \\Lambda_\\wedge[H]$, which appears in non-commutative Iwasawa theory."}
{"category": "Math", "title": "Equivariant Iwasawa theory: an example", "abstract": "The equivariant `main conjecture' of Iwasawa theory is shown to hold for a Galois extension $K/k$ of number fields with Galois group an $l$-adic pro-$l$ Lie group of dimension 1 containing an abelian subgroup of index $l$, provided that Iwasawa's $\\mu$-invariant $\\mu(K/k)$ vanishes."}
{"category": "Math", "title": "An open problem in complex analytic geometry arising in harmonic analysis", "abstract": "In this paper, an open problem in the multidimensional complex analysis is pesented that arises in the investigation of the regularity properties of Fourier integral operators and in the regularity theory for hyperbolic partial differential equations. The problem is discussed in a self-contained elementary way and some results towards its resolution are presented. A conjecture concerning the structure of appearing affine fibrations is formulated."}
{"category": "Math", "title": "Multiplier ideal sheaves and integral invariants on toric Fano manifolds", "abstract": "We extend Nadel's results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal sheaves on toric del Pezzo surfaces which do not admit K\\\"ahler-Einstein metrics. We also show that one can define multiplier ideal sheaves for K\\\"ahler-Ricci solitons and extend the result of Nadel using the holomorphic invariant defined by Tian and Zhu."}
{"category": "Math", "title": "Billiard scattering on rough sets: two-dimensional case", "abstract": "The notion of a rough two-dimensional (convex) body is introduced, and to each rough body there is assigned a measure on $\\TTT^3$ describing billiard scattering on the body. The main result is characterization of the set of measures generated by rough bodies. This result can be used to solve various problems of least aerodynamical resistance."}
{"category": "Math", "title": "On the stochastic Burgers equation with some applications to turbulence and astrophysics", "abstract": "We summarise a selection of results on the inviscid limit of the stochastic Burgers equation emphasising geometric properties of the caustic, Maxwell set and Hamilton-Jacobi level surfaces and relating these results to a discussion of stochastic turbulence. We show that for small viscosities there exists a vortex filament structure near to the Maxwell set. We discuss how this vorticity is directly related to the adhesion model for the evolution of the early universe and include new explicit formulas for the distribution of mass within the shock."}
{"category": "Math", "title": "Reflected BSDE with quadratic growth and unbounded terminal value", "abstract": "In this paper we prove the existence of a solution for reflected BSDE's\\ whose coefficient is of quadratic growth in $z$ and of linear growth in $y$, with an unbounded terminal value."}
{"category": "Math", "title": "Lifting Measures to Inducing Schemes", "abstract": "In this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [PS05, PS06]. We show that under some natural assumptions on the inducing schemes - which hold for many known examples - any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [Buz99], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [Bru95] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multimodal maps) and for some multidimensional maps."}
{"category": "Math", "title": "Skorohod-reflection of Brownian Paths and BES^3", "abstract": "Let B(t), X(t) and Y(t) be independent standard 1d Borwnian motions. Define X^+(t) and Y^-(t) as the trajectories of the processes X(t) and Y(t) pushed upwards and, respectively, downwards by B(t), according to Skorohod-reflection. In a recent paper, Jon Warren proves inter alia that Z(t):= X^+(t)-Y^-(t) is a three-dimensional Bessel-process. In this note, we present an alternative, elementary proof of this fact."}
{"category": "Math", "title": "Memorandum on Dimension Formulas for Spaces of Jacobi Forms", "abstract": "We state ready to compute dimension formulas for the spaces of Jacobi cusp forms of integral weight $k$ and integral scalar index $m$ on subgroups of $\\SL$."}
{"category": "Math", "title": "Poincare polynomials of moduli spaces of stable bundles over curves", "abstract": "Given a curve over a finite field, we compute the number of stable bundles of not necessarily coprime rank and degree over it. We apply this result to compute the virtual Poincare polynomials of the moduli spaces of stable bundles over a curve. A similar formula for the virtual Hodge polynomials and motives is conjectured."}
{"category": "Math", "title": "Spectral flow and iteration of closed semi-Riemannian geodesics", "abstract": "We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior."}
{"category": "Math", "title": "HJB equations for certain singularly controlled diffusions", "abstract": "Given a closed, bounded convex set $\\mathcal{W}\\subset{\\mathbb {R}}^d$ with nonempty interior, we consider a control problem in which the state process $W$ and the control process $U$ satisfy \\[W_t= w_0+\\int_0^t\\vartheta(W_s) ds+\\int_0^t\\sigma(W_s) dZ_s+GU_t\\in \\mathcal{W},\\qquad t\\ge0,\\] where $Z$ is a standard, multi-dimensional Brownian motion, $\\vartheta,\\sigma\\in C^{0,1}(\\mathcal{W})$, $G$ is a fixed matrix, and $w_0\\in\\mathcal{W}$. The process $U$ is locally of bounded variation and has increments in a given closed convex cone $\\mathcal{U}\\subset{\\mathbb{R}}^p$. Given $g\\in C(\\mathcal{W})$, $\\kappa\\in{\\mathbb{R}}^p$, and $\\alpha>0$, consider the objective that is to minimize the cost \\[J(w_0,U)\\doteq\\mathbb{E}\\biggl[\\int_0^{\\infty}e^{-\\alpha s}g(W_s) ds+\\int_{[0,\\infty)}e^{-\\alpha s} d(\\kappa\\cdot U_s)\\biggr]\\] over the admissible controls $U$. Both $g$ and $\\kappa\\cdot u$ ($u\\in\\mathcal{U}$) may take positive and negative values. This paper studies the corresponding dynamic programming equation (DPE), a second-order degenerate elliptic partial differential equation of HJB-type with a state constraint boundary condition. Under the controllability condition $G\\mathcal{U}={\\mathbb{R}}^d$ and the finiteness of $\\mathcal{H}(q)=\\sup_{u\\in\\mathcal{U}_1}\\{-Gu\\cdot q-\\kappa\\cdot u\\}$, $q\\in {\\mathbb{R}}^d$, where $\\mathcal{U}_1=\\{u\\in\\mathcal{U}:|Gu|=1\\}$, we show that the cost, that involves an improper integral, is well defined. We establish the following: (i) the value function for the control problem satisfies the DPE (in the viscosity sense), and (ii) the condition $\\inf_{q\\in{\\mathbb{R}}^d}\\mathcal{H}(q)<0$ is necessary and sufficient for uniqueness of solutions to the DPE. The existence and uniqueness of solutions are shown to be connected to an intuitive ``no arbitrage'' condition. Our results apply to Brownian control problems that represent formal diffusion approximations to control problems associated with stochastic processing networks."}
{"category": "Math", "title": "Geodetic Line at Constant Altitude above the Ellipsoid", "abstract": "The two-dimensional surface of a bi-axial ellipsoid is characterized by the lengths of its major and minor axes. Longitude and latitude span an angular coordinate system across. We consider the egg-shaped surface of constant altitude above (or below) the ellipsoid surface, and compute the geodetic lines - lines of minimum Euclidean length - within this surface which connect two points of fixed coordinates. This addresses the common \"inverse\" problem of geodesics generalized to non-zero elevations. The system of differential equations which couples the two angular coordinates along the trajectory is reduced to a single integral, which is handled by Taylor expansion up to fourth power in the eccentricity."}
{"category": "Math", "title": "Survival and complete convergence for a spatial branching system with local regulation", "abstract": "We study a discrete time spatial branching system on $\\mathbb{Z}^d$ with logistic-type local regulation at each deme depending on a weighted average of the population in neighboring demes. We show that the system survives for all time with positive probability if the competition term is small enough. For a restricted set of parameter values, we also obtain uniqueness of the nontrivial equilibrium and complete convergence, as well as long-term coexistence in a related two-type model. Along the way we classify the equilibria and their domain of attraction for the corresponding deterministic coupled map lattice on $\\mathbb{Z}^d$."}
{"category": "Math", "title": "On the Distribution of Penalized Maximum Likelihood Estimators: The LASSO, SCAD, and Thresholding", "abstract": "We study the distributions of the LASSO, SCAD, and thresholding estimators, in finite samples and in the large-sample limit. The asymptotic distributions are derived for both the case where the estimators are tuned to perform consistent model selection and for the case where the estimators are tuned to perform conservative model selection. Our findings complement those of Knight and Fu (2000) and Fan and Li (2001). We show that the distributions are typically highly nonnormal regardless of how the estimator is tuned, and that this property persists in large samples. The uniform convergence rate of these estimators is also obtained, and is shown to be slower than 1/root(n) in case the estimator is tuned to perform consistent model selection. An impossibility result regarding estimation of the estimators' distribution function is also provided."}
{"category": "Math", "title": "Differential Equations Driven by Gaussian Signals II", "abstract": "Large classes of multi-dimensional Gaussian processes can be enhanced with stochastic Levy area(s). In a previous paper, we gave sufficient and essentially necessary conditions, only involving variational properties of the covariance. Following T. Lyons, the resulting lift to a \"Gaussian rough path\" gives a robust theory of (stochastic) differential equations driven by Gaussian signals with sample path regularity worse than Brownian motion. The purpose of this sequel paper is to establish convergence of Karhunen-Loeve approximations in rough path metrics. Particular care is necessary since martingale arguments are not enough to deal with third iterated integrals. An abstract support criterion for approximately continuous Wiener functionals then gives a description of the support of Gaussian rough paths as the closure of the (canonically lifted) Cameron-Martin space."}
{"category": "Math", "title": "On D. Haegele's approach to the Bessis-Moussa-Villani conjecture", "abstract": "The reformulation of the Bessis-Moussa-Villani conjecture given by Lieb and Seiringer asserts that the coefficient of t^r in the polynomial Trace[(A+tB)^p], with A and B positive semidefinite matrices, is nonnegative for all p and r. We propose a natural extension of a method of attack on this problem due to Haegele, and investigate for what values of p and r the method is successful, obtaining a complete determination when either p or r is odd."}
{"category": "Math", "title": "Colimits of representable algebra-valued functors", "abstract": "If C and D are varieties of algebras in the sense of general algebra, then by a representable functor C --> D we understand a functor which, when composed with the forgetful functor D --> Set, gives a representable functor in the classical sense; Freyd showed that these functors are determined by D-coalgebra objects of C. Let Rep(C,D) denote the category of all such functors, a full subcategory of Cat(C,D), opposite to the category of D-coalgebras in C. It is proved that Rep(C,D) has small colimits, and in certain situations, explicit constructions for the representing coalgebras are obtained. In particular, Rep(C,D) always has an initial object. This is shown to be \"trivial\" unless C and D either both have_no_ zeroary operations, or both have _more_than_one_ derived zeroary operation. In those two cases, the functors in question may have surprisingly opulent structures. It is also shown that every set-valued representable functor on C admits a universal morphism to a D-valued representable functor. Several examples are worked out in detail, and areas for further investigation noted."}
{"category": "Math", "title": "Failure of Wiener's property for positive definite periodic functions", "abstract": "We say that Wiener's property holds for the exponent $p>0$ if we have that whenever a positive definite function $f$ belongs to $L^p(-\\epsilon,\\epsilon)$ for some $\\epsilon>0$, then $f$ necessarily belongs to $L^p(\\TT)$, too. This holds true for $p\\in 2\\NN$ by a classical result of Wiener. Recently various concentration results were proved for idempotents and positive definite functions on measurable sets on the torus. These new results enable us to prove a sharp version of the failure of Wiener's property for $p\\notin 2\\NN$. Thus we obtain strong extensions of results of Wainger and Shapiro, who proved the negative answer to Wiener's problem for $p\\notin 2\\NN$."}
{"category": "Math", "title": "Restrictions of continuous functions", "abstract": "Given a continuous real-valued function on [0, 1], and a closed subset E \\subset [0, 1] we denote by f E the restriction of f to E, that is, the function defined only on E that takes the same values as f at every point of E >. The restriction f E will typically be \"better behaved\" than f . It may have bounded variation when f doesn't, it may have a better modulus of continuity than f, it may be monotone when f is not, etc. All this clearly depends on f and on E, and the questions that we discuss here are about the existence, for every f, or every f in some class, of \"substantial\" sets E such that f E has bounded total variation, is monotone, or satisfies a given modulus of continuity. The notion of \"substantial\" that we use is that of either Hausdorff or Minkowski dimensions."}
{"category": "Math", "title": "The strength of the Inner Model Hypothesis", "abstract": "The Inner Model Hypothesis (IMH) and the Strong Inner Model Hypothesis (SIMH) were introduced by the first author in ``Internal consistency and the inner model hypothesis'', Bulletin of Symbolic Logic, December 2006. In this article we establish some upper and lower bounds for their consistency strength."}
{"category": "Math", "title": "Finitely generated groups with polynomial index growth", "abstract": "We prove that a finitely generated soluble residually finite group has polynomial index growth if and only if it is a minimax group. We also show that if a finitely generated group with PIG is residually finite-soluble then it is a linear group. These results apply in particular to boundedly generated groups; they imply that every infinite BG residually finite group has an infinite linear quotient."}
{"category": "Math", "title": "Nonparametric Regression, Confidence Regions and Regularization", "abstract": "In this paper we offer a unified approach to the problem of nonparametric regression on the unit interval. It is based on a universal, honest and non-asymptotic confidence region which is defined by a set of linear inequalities involving the values of the functions at the design points. Interest will typically centre on certain simplest functions in that region where simplicity can be defined in terms of shape (number of local extremes, intervals of convexity/concavity) or smoothness (bounds on derivatives) or a combination of both. Once some form of regularization has been decided upon the confidence region can be used to provide honest non-asymptotic confidence bounds which are less informative but conceptually much simpler."}
{"category": "Math", "title": "Arithmetic and Differential Swan Conductors of rank one representations with finite local monodromy", "abstract": "We consider a complete discrete valuation field of characteristic p, with possibly non perfect residue field. Let V be a rank one continuous representation with finite local monodromy of its absolute Galois group. We will prove that the Arithmetic Swan conductor of V (defined after Kato in [Kat89] which fits in the more general theory of [AS02] and [AS06]) coincides with the Differential Swan conductor of the associated differential module $D^{\\dag}(V)$ defined by Kedlaya in [Ked]. This construction is a generalization to the non perfect residue case of the Fontaine's formalism as presented in [Tsu98a]. Our method of proof will allow us to give a new interpretation of the Refined Swan Conductor."}
{"category": "Math", "title": "An introduction to the geometry of ultrametric spaces", "abstract": "Some examples and basic properties of ultrametric spaces are briefly discussed."}
{"category": "Math", "title": "Inequalities for Integer and Fractional Parts", "abstract": "In this paper we present 43 new inequalities related to integer part and fractional part."}
{"category": "Math", "title": "Pointwise convergence of solutions to Schr\\\"odinger equations", "abstract": "We study pointwise convergence of the solutions to Schr\\\"odinger equations with initial datum $f\\in H^s(\\mathbb R^n)$. The conjecture is that the solution $e^{it\\Delta}f$ converges to $f$ almost everywhere for all $f\\in H^s(\\mathbb R^n)$ if and only if $s\\ge 1/4$. The conjecture is known true for one spatial dimension and the convergence when $s>1/2$ was verified for $n\\ge 2$. Recently, concrete progresses have been made in $\\mathbb R^2$ for some $s<1/2$. However, when $n\\ge 3$ no positive result is known for the initial datum $f\\in H^s(\\mathbb R^n)$, $s\\le 1/2$. We show that $\\lim_{t\\to 0} e^{it\\Delta}f= f$ a.e. for $f\\in H^s(\\mathbb R^3)$ whenever $s>1/2-1/{24}$."}
{"category": "Math", "title": "Decompounding under Gaussian noise", "abstract": "Assuming that a stochastic process $X=(X_t)_{t\\geq 0}$ is a sum of a compound Poisson process $Y=(Y_t)_{t\\geq 0}$ with known intensity $\\lambda$ and unknown jump size density $f,$ and an independent Brownian motion $Z=(Z_t)_{t\\geq 0},$ we consider the problem of nonparametric estimation of $f$ from low frequency observations from $X.$ The estimator of $f$ is constructed via Fourier inversion and kernel smoothing. Our main result deals with asymptotic normality of the proposed estimator at a fixed point."}
{"category": "Math", "title": "On the Evolution Equation for Magnetic Geodesics", "abstract": "In this paper we prove the existence of long time solutions for the parabolic equation for closed magnetic geodesics."}
{"category": "Math", "title": "A Character Formula for the Category \\~O", "abstract": "One may construct, for any function on the integers, an irreducible module of level zero for affine sl(2), using the values of the function as structure constants. The modules constructed using exponential-polynomial functions realise the irreducible modules with finite-dimensional weight spaces in the category \\~O of Chari. In this work, an expression for the formal character of such a module is derived using the highest-weight theory of truncations of the loop algebra."}
{"category": "Math", "title": "On a Bruhat-like poset", "abstract": "We investigate the poset of strata of a Schubert-like stratification of certain natural compactification of the space of hermitian $n\\times n$ matrices. We prove that this poset is a modular ortholattice, we compute its M\\\"{o}bius function and we describe the topology of its order intervals."}
{"category": "Math", "title": "Nested Hilbert schemes and the nested q,t-Catalan series", "abstract": "In this paper we study the tangent spaces of the smooth nested Hilbert scheme $ Hil{n,n-1}$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\\TT^2$-equivariant locally free sheaf on $\\Hil{n,n-1}$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients . We call this conjecturally positive polynomial as \\textsl{the nested $q,t$-Cat alan series}, for it has many conjectural properties similar to that of the $q,t $-Catalan series."}
{"category": "Math", "title": "Arrangements of curves and algebraic surfaces", "abstract": "We prove a strong relation between Chern and log Chern invariants of algebraic surfaces. For a given arrangement of curves, we find nonsingular projective surfaces with Chern ratio arbitrarily close to the log Chern ratio of the log surface defined by the arrangement. Our method is based on sequences of random p-th root covers, which exploit a certain large scale behavior of Dedekind sums and lengths of continued fractions. We show that randomness is necessary for our asymptotic result, providing another instance of \"randomness implies optimal\". As an application over the complex numbers, we construct nonsingular simply connected projective surfaces of general type with large Chern ratio. In particular, we improve the Persson-Peters-Xiao record for Chern ratios of such surfaces."}
{"category": "Math", "title": "Rigidity of polyhedral surfaces, II", "abstract": "We study the rigidity of polyhedral surfaces using variational principle. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a non-triangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover a uniform approach on several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, Bobenko-Springborn and Thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context."}
{"category": "Math", "title": "A remark on the enclosure method for a body with an unknown homogeneous background conductivity", "abstract": "Previous applications of the enclosure method with a finite set of observation data to a mathematical model of electrical impedance tomography are based on the assumption that the conductivity of the background body is homogeneous and known. This paper considers the case when the conductivity is homogeneous and unknown. It is shown that, in two dimensions if the domain occupied by the background body is enclosed by an ellipse, then it is still possible to extract some information about the location of unknown cavities or inclusions embedded in the body without knowing the background conductivity provided the Fourier series expansion of the voltage on the boundary does not contain high frequency parts (band limited) and satisfies a non vanishing condition of a quantity involving the Fourier coefficients."}
{"category": "Math", "title": "Integration Theory for Zero Sets of Polyfold Fredholm Sections", "abstract": "In this paper we develop an integration theory for zero sets of polyfold Fredholm sections. The results are needed in the application of the polyfold theory. We use it for example in the construction of symplectic field theory."}
{"category": "Math", "title": "Socles of Buchsbaum modules, complexes and posets", "abstract": "The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative K\\\"uhnel's conjecture for the maximum value of the Euler characteristic of a $2k$-dimensional simplicial manifold on $n$ vertices as well as Kalai's conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number."}
{"category": "Math", "title": "Operator algebra of foliations with projectively invariant transverse measure", "abstract": "We study the structure of operator algebras associated with the foliations which have projectively invariant measures. When a certain ergodicity condition on the measure preserving holonomies holds, the lack of holonomy invariant transverse measure can be established in terms of a cyclic cohomology class associated with the transverse fundamental cocycle and the modular automorphism group."}
{"category": "Math", "title": "On Moduli Spaces for Abelian Categories", "abstract": "We show that if A is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories."}
{"category": "Math", "title": "Finite-Dimensional Representations of Hyper Loop Algebras Over Non-Algebraically Closed Fields", "abstract": "We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change, tensor products of irreducible and Weyl modules, and the block decomposition of the underlying abelian category. Several results are interestingly related to the study of irreducible representations of polynomial algebras and Galois theory."}
{"category": "Math", "title": "Flattening Functions on Flowers", "abstract": "Let $T$ be an orientation-preserving Lipschitz expanding map of the circle $\\T$. A pre-image selector is a map $\\tau:\\T\\to\\T$ with finitely many discontinuities, each of which is a jump discontinuity, and such that $\\tau(x)\\in T^{-1}(x)$ for all $x\\in\\T$. The closure of the image of a pre-image selector is called a flower, and a flower with $p$ connected components is called a $p$-flower. We say that a real-valued Lipschitz function can be Lipschitz flattened on a flower whenever it is Lipschitz cohomologous to a constant on that flower. The space of Lipschitz functions which can be flattened on a given $p$-flower is shown to be of codimension $p$ in the space of all Lipschitz functions, and the linear constraints determining this subspace are derived explicitly. If a Lipschitz function $f$ has a maximizing measure $S$ which is Sturmian (i.e. is carried by a 1-flower), it is shown that $f$ can be Lipschitz flattened on some 1-flower carrying $S$."}
{"category": "Math", "title": "Correction. Perfect simulation for a class of positive recurrent Markov chains", "abstract": "Correction to Annals of Applied Probability 17 (2007) 781--808 [doi:10.1214/105051607000000032]."}
{"category": "Math", "title": "Functional approach for excess mass estimation in the density model", "abstract": "We consider a multivariate density model where we estimate the excess mass of the unknown probability density $f$ at a given level $\\nu>0$ from $n$ i.i.d. observed random variables. This problem has several applications such as multimodality testing, density contour clustering, anomaly detection, classification and so on. For the first time in the literature we estimate the excess mass as an integrated functional of the unknown density $f$. We suggest an estimator and evaluate its rate of convergence, when $f$ belongs to general Besov smoothness classes, for several risk measures. A particular care is devoted to implementation and numerical study of the studied procedure. It appears that our procedure improves the plug-in estimator of the excess mass."}
{"category": "Math", "title": "Geometric Linearization of Ordinary Differential Equations", "abstract": "The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable equations and even on systems of equations. However, little has been done in the way of providing explicit criteria to determine their linearizability. Using the connection between isometries and symmetries of the system of geodesic equations criteria were established for second order quadratically and cubically semi-linear equations and for systems of equations. The connection was proved for maximally symmetric spaces and a conjecture was put forward for other cases. Here the criteria are briefly reviewed and the conjecture is proved."}
{"category": "Math", "title": "A local-global problem for linear differential equations", "abstract": "An inhomogeneous linear differential equation Ly=f over a global differential field can have a formal solution for each place without having a global solution. The vector space lgl(L) measures this phenomenon. This space is interpreted in terms of cohomology of linear algebraic groups and is computed for abelian differential equations and for regular singular equations. An analogue of Artin reciprocity for abelian differential equations is given. Malgrange's work on irregularity is reproved in terms of cohomology of linear algebraic groups."}
{"category": "Math", "title": "Multidimensional Gauss Reduction Theory for conjugacy classes of SL(n,Z)", "abstract": "In this paper we describe the set of conjugacy classes in the group SL(n,Z). We expand geometric Gauss Reduction Theory that solves the problem for SL(2,Z) to the multidimensional case. Further we find complete invariant of classes in terms of multidimensional Klein-Voronoi continued fractions, where $\\varsigma$-reduce Hessenberg matrices play the role of reduced matrices."}
{"category": "Math", "title": "Duality for partial group actions", "abstract": "Given a finite group G acting as automorphisms on a ring A, the skew group ring A*G is an important tool for studying the structure of G-stable ideals of A. The ring A*G is G-graded, i.e.G coacts on A*G. The Cohen-Montgomery duality says that the smash product A*G#k[G]^* of A*G with the dual group ring k[G]^* is isomorphic to the full matrix ring M_n(A) over A, where n is the order of G. In this note we show how much of the Cohen-Montgomery duality carries over to partial group actions in the sense of R.Exel. In particular we show that the smash product (A*_\\alpha G)#k[G]^* of the partial skew group ring A*_\\alpha G and k[G]^* is isomorphic to a direct product of the form K x eM_n(A)e where e is a certain idempotent of M_n(A) and K is a subalgebra of (A *_\\alpha G)#k[G]^*. Moreover A*_\\alpha G is shown to be isomorphic to a separable subalgebra of eM_n(A)e. We also look at duality for infinite partial group actions and for partial Hopf actions."}
{"category": "Math", "title": "Weighted power variations of iterated Brownian motion", "abstract": "We characterize the asymptotic behaviour of the weighted power variation processes associated with iterated Brownian motion. We prove weak convergence results in the sense of finite dimensional distributions, and show that the laws of the limiting objects can always be expressed in terms of three independent Brownian motions X, Y and B, as well as of the local times of Y. In particular, our results involve ``weighted'' versions of Kesten and Spitzer's Brownian motion in random scenery. Our findings extend the theory initiated by Khoshnevisan and Lewis (1999), and should be compared with the recent result by Nourdin and R\\'eveillac (2008), concerning the weighted power variations of fractional Brownian motion with Hurst index H=1/4."}
{"category": "Math", "title": "On properness and related properties of quasilinear systems on unbounded domains", "abstract": "The purpose of this paper is to provide tools for analyzing the compactness of sequences in Sobolev spaces, in particular if the sequence gets mapped onto a compact set by some nonlinear operator. Here, our focus lies on a very general class of nonlinear operators arising in quasilinear systems of partial differential equations of second order, in divergence form. Our approach, based on a suitable decomposition lemma, admits the discussion of problems with some inherent loss of compactness, for example due to a domain with infinite measure or a lower order term with critical growth. As an application, we obtain a characterization of properness which is considerably easier to verify than the definition. The methods presented can also be used to check Palais--Smale conditions for variational problems."}
{"category": "Math", "title": "Decomposition into weight * level + jump and application to a new classification of primes", "abstract": "In this paper we introduce an Euclidean decomposition of elements a_n of an increasing sequence of natural numbers into weight * level + jump which we use to classify the numbers a_n either by weight or by level. We then show that this decomposition can be seen as a generalization of the sieve of Eratosthenes (which is the particular case of the whole sequence of natural numbers). We apply this decomposition to prime numbers in order to obtain a new classification of primes, we analyze a few properties of this classification and we make a series of conjectures based on numerical data. Finally we show how composite numbers and 2-almost primes behave under the decomposition."}
{"category": "Math", "title": "Sheaves on affine Schubert varieties, modular representations and Lusztig's conjecture", "abstract": "We relate a certain category of sheaves of k-vector spaces on a complex affine Schubert variety to modules over the k-Lie algebra (for ch k>0) or to modules over the small quantum group (for ch k=0) associated to the Langlands dual root datum. As an application we give a new proof of Lusztig's conjecture on quantum characters and on modular characters for almost all characteristics. Moreover, we relate the geometric and representation theoretic sides to sheaves on the underlying moment graph, which allows us to extend the known instances of Lusztig's modular conjecture in two directions: We give an upper bound on the exceptional characteristics and verify its multiplicity one case for all relevant primes."}
{"category": "Math", "title": "Bayesian nonparametric estimation of the spectral density of a long memory Gaussian time series", "abstract": "Let $\\mathbf {X}=\\{X_t, t=1,2,... \\}$ be a stationary Gaussian random process, with mean $EX_t=\\mu$ and covariance function $\\gamma(\\tau)=E(X_t-\\mu)(X_{t+\\tau}-\\mu)$. Let $f(\\lambda)$ be the corresponding spectral density; a stationary Gaussian process is said to be long-range dependent, if the spectral density $f(\\lambda)$ can be written as the product of a slowly varying function $\\tilde{f}(\\lambda)$ and the quantity $\\lambda ^{-2d}$. In this paper we propose a novel Bayesian nonparametric approach to the estimation of the spectral density of $\\mathbf {X}$. We prove that, under some specific assumptions on the prior distribution, our approach assures posterior consistency both when $f(\\cdot)$ and $d$ are the objects of interest. The rate of convergence of the posterior sequence depends in a significant way on the structure of the prior; we provide some general results and also consider the fractionally exponential (FEXP) family of priors (see below). Since it has not a well founded justification in the long memory set-up, we avoid using the Whittle approximation to the likelihood function and prefer to use the true Gaussian likelihood."}
{"category": "Math", "title": "Partial open book decompositions and the contact class in sutured Floer homology", "abstract": "We demonstrate how to combinatorially calculate the EH-class of a compatible contact structure in the sutured Floer homology group of a balanced sutured three manifold which is associated to an abstract partial open book decomposition. As an application we show that every contact three manifold (closed or with convex boundary) can be obtained by gluing tight contact handlebodies whose EH-classes are nontrivial."}
{"category": "Math", "title": "Data-driven wavelet-Fisz methodology for nonparametric function estimation", "abstract": "We propose a wavelet-based technique for the nonparametric estimation of functions contaminated with noise whose mean and variance are linked via a possibly unknown variance function. Our method, termed the data-driven wavelet-Fisz technique, consists of estimating the variance function via a Nadaraya-Watson estimator, and then performing a wavelet thresholding procedure which uses the estimated variance function and local means of the data to set the thresholds at a suitable level. We demonstrate the mean-square near-optimality of our wavelet estimator over the usual range of Besov classes. To achieve this, we establish an exponential inequality for the Nadaraya-Watson variance function estimator. We discuss various implementation issues concerning our wavelet estimator, and demonstrate its good practical performance. We also show how it leads to a new wavelet-domain data-driven variance-stabilising transform. Our estimator can be applied to a variety of problems, including the estimation of volatilities, spectral densities and Poisson intensities, as well as to a range of problems in which the distribution of the noise is unknown."}
{"category": "Math", "title": "Stable reduction of curves and tame ramification", "abstract": "We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to T. Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain purely geometric proof of Saito's criterion, avoiding the use of vanishing cycles."}
{"category": "Math", "title": "On sets represented by partitions", "abstract": "We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts) partitions of an integer n."}
{"category": "Math", "title": "Monomial bases related to the n! conjecture", "abstract": "The purpose of this paper is to find a new way to prove the $n!$ conjecture for particular partitions. The idea is to construct a monomial and explicit basis for the space $M_{\\mu}$. We succeed completely for hook-shaped partitions, i.e., $\\mu=(K+1,1^L)$. We are able to exhibit a basis and to verify that its cardinality is indeed $n!$, that it is linearly independent and that it spans $M_{\\mu}$. We derive from this study an explicit and simple basis for $I_{\\mu}$, the annihilator ideal of $\\Delta_{\\mu}$. This method is also successful for giving directly a basis for the homogeneous subspace of $M_{\\mu}$ consisting of elements of 0 $x$-degree."}
{"category": "Math", "title": "Bases explicites et conjecture n!", "abstract": "The aim of this work is to construct a monomial and explicit basis for the space $M_{\\mu}$ relative to the $n!$ conjecture. We succeed completely for hook-shaped partitions, i.e. $\\mu=(K+1,1^L)$. We are indeed able to exhibit a basis and to verify that its cardinality is $n!$, that it is linearly independent and that it spans $M_{\\mu}$. We deduce from this study an explicit and simple basis for $I_{\\mu}$, the annulator ideal of $\\Delta_{\\mu}$. This method is also successful for giving directly a basis for the homogeneous subspace of $M_{\\mu}$ consisting of elements of 0 $x$-degree."}
{"category": "Math", "title": "On certain spaces of lattice diagram polynomials", "abstract": "The aim of this work is to study some lattice diagram determinants $\\Delta_L(X,Y)$. We recall that $M_L$ denotes the space of all partial derivatives of $\\Delta_L$. In this paper, we want to study the space $M^k_{i,j}(X,Y)$ which is defined as the sum of $M_L$ spaces where the lattice diagrams $L$ are obtained by removing $k$ cells from a given partition, these cells being in the ``shadow'' of a given cell $(i,j)$ in a fixed Ferrers diagram. We obtain an upper bound for the dimension of the resulting space $M^k_{i,j}(X,Y)$, that we conjecture to be optimal. This dimension is a multiple of $n!$ and thus we obtain a generalization of the $n!$ conjecture. Moreover, these upper bounds associated to nice properties of some special symmetric differential operators (the ``shift'' operators) allow us to construct explicit bases in the case of one set of variables, i.e. for the subspace $M^k_{i,j}(X)$ consisting of elements of 0 $Y$-degree."}
{"category": "Math", "title": "On certain spaces of lattice diagram determinants", "abstract": "The aim of this work is to study some lattice diagram polynomials $\\Delta_D(X,Y)$. We recall that $M_D$ denotes the space of all partial derivatives of $\\Delta_D$. In this paper, we want to study the space $M^k_{i,j}(X,Y)$ which is the sum of $M_D$ spaces where the lattice diagrams $D$ are obtained by removing $k$ cells from a given partition, these cells being in the ``shadow'' of a given cell $(i,j)$ of the Ferrers diagram. We obtain an upper bound for the dimension of the resulting space $M^k_{i,j}(X,Y)$, that we conjecture to be optimal. These upper bounds allow us to construct explicit bases for the subspace $M^k_{i,j}(X)$ consisting of elements of 0 $Y$-degree."}
{"category": "Math", "title": "A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces", "abstract": "We study the nonlinear eigenvalue problem $-{\\rm div}(a(|\\nabla u|)\\nabla u)=\\lambda|u|^{q(x)-2}u$ in $\\Omega$, $u=0$ on $\\partial\\Omega$, where $\\Omega$ is a bounded open set in $\\RR^N$ with smooth boundary, $q$ is a continuous function, and $a$ is a nonhomogeneous potential. We establish sufficient conditions on $a$ and $q$ such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases $a(t)=t^{p-2}\\log (1+t^r)$ and $a(t)= t^{p-2} [\\log (1+t)]^{-1}$."}
{"category": "Math", "title": "Ideals and quotients of B-quasisymmetric functions", "abstract": "The space $QSym_n(B)$ of $B$-quasisymmetric polynomials in 2 sets of $n$ variables was recently studied by Baumann and Hohlweg. The aim of this work is a study of the ideal $<QSym_n(B)^+>$ generated by $B$-quasisymmetric polynomials without constant term. In the case of the space $QSym_n$ of quasisymmetric polynomials in 1 set of $n$ variables, Aval, Bergeron and Bergeron proved that the dimension of the quotient of the space of polynomials by the ideal $<QSym_n^+>$ is given by Catalan numbers $C_n=\\frac 1 {n+1} {2n \\choose n}$. In the case of $B$-quasisymmetric polynomials, our main result is that the dimension of the analogous quotient is equal to $\\frac{1}{2n+1}{3n\\choose n}$, the numbers of ternary trees with $n$ nodes. The construction of a Gr\\\"obner basis for the ideal, as well as of a linear basis for the quotient are interpreted by a bijection with lattice paths. These results are finally extended to $p$ sets of variables, and the dimension is in this case $\\frac{1}{pn+1}{(p+1)n\\choose n}$, the numbers of $p$-ary trees with $n$ nodes."}
{"category": "Math", "title": "Multivariate Fuss-Catalan numbers", "abstract": "Catalan numbers $C(n)=\\frac{1}{n+1}{2n\\choose n}$ enumerate binary trees and Dyck paths. The distribution of paths with respect to their number $k$ of factors is given by ballot numbers $B(n,k)=\\frac{n-k}{n+k}{n+k\\choose n}$. These integers are known to satisfy simple recurrence, which may be visualised in a ``Catalan triangle'', a lower-triangular two-dimensional array. It is surprising that the extension of this construction to 3 dimensions generates integers $B_3(n,k,l)$ that give a 2-parameter distribution of $C_3(n)=\\frac 1 {2n+1} {3n\\choose n}$, which may be called order-3 Fuss-Catalan numbers, and enumerate ternary trees. The aim of this paper is a study of these integers $B_3(n,k,l)$. We obtain an explicit formula and a description in terms of trees and paths. Finally, we extend our construction to $p$-dimensional arrays, and in this case we obtain a $(p-1)$-parameter distribution of $C_p(n)=\\frac 1 {(p-1)n+1} {pn\\choose n}$, the number of $p$-ary trees."}
{"category": "Math", "title": "Quasi-invariant and super-coinvariant polynomials for the generalized symmetric group", "abstract": "The aim of this work is to extend the study of super-coinvariant polynomials, to the case of the generalized symmetric group $G_{n,m}$, defined as the wreath product $C_m\\wr\\S_n$ of the symmetric group by the cyclic group. We define a quasi-symmetrizing action of $G_{n,m}$ on $\\Q[x_1,...,x_n]$, analogous to those defined by Hivert in the case of $\\S_n$. The polynomials invariant under this action are called quasi-invariant, and we define super-coinvariant polynomials as polynomials orthogonal, with respect to a given scalar product, to the quasi-invariant polynomials with no constant term. Our main result is the description of a Gr\\\"obner basis for the ideal generated by quasi-invariant polynomials, from which we dedece that the dimension of the space of super-coinvariant polynomials is equal to $m^n C_n$ where $C_n$ is the $n$-th Catalan number."}
{"category": "Math", "title": "Polyn\\^omes quasi-invariants et super-coinvariants pour le groupe sym\\'etrique g\\'en\\'eralis\\'e", "abstract": "A classical result of Artin states that the ideal generated by symmetric polynomials in $n$ variables is of codimension $n!$. The author, F. Bergeron and N. Bergeron have recently obtained a surprising analogous in the case of quasi-symmetric polynomials. In this case, the ideal is of codimension given by $C_n$, the $n$-th Catalan number. Quasi-symmetric polynomials are the invariants of a certain action of the symmetric group $S_n$ defined by F. Hivert. The aim of this work is to generalize these results to the wreath product $S_n\\wr \\Z_m$, also known as the generalized symmetric group $G\\nm$. We first define a quasi-symmetrizing action of $G\\nm$ on $\\C[x_1,...,x_n]$, then obtain a description of the invariants and the codimension of the associated ideal, which is $m^n C_n$."}
{"category": "Math", "title": "Generalizations of two theorems of Ritt on decompositions of polynomial maps", "abstract": "Two theorems of J. F. Ritt on decompositions of polynomials maps are generalized to a more general situation: for, so-called, reduction monoids ($(K[x], \\circ)$ and $(K[x^2]x, \\circ)$ are examples of reduction monoids). In particular, analogues of the two theorems of J. F. Ritt hold for the monoid $(K[x^2]x, \\circ)$ of odd polynomials. It is shown that, in general, the two theorems of J. F. Ritt fail for the cusp $(K+K[x]x^2, \\circ)$ but their analogues are still true for decompositions of maximal length of regular elements of the cusp."}
{"category": "Math", "title": "The Discrete Fundamental Group of the Order Complex of $B_n$", "abstract": "A few years ago Kramer and Laubenbacher introduced a discrete notion of homotopy for simplicial complexes. In this paper, we compute the discrete fundamental group of the order complex of the Boolean lattice. As it turns out, it is equivalent to computing the discrete homotopy group of the 1-skeleton of the permutahedron. To compute this group we introduce combinatorial techniques that we believe will be helpful in computing discrete fundamental groups of other polytopes. More precisely, we use the language of words, over the alphabet of simple transpositions, to obtain conditions that are necessary and sufficient to characterize the equivalence classes of cycles. The proof requires only simple combinatorial arguments. As a corollary, we also obtain a combinatorial proof of the fact that the first Betti number of the complement of the 3-equal arrangement is equal to $2^{n-3}(n^2-5n+8)-1.$ This formula was originally obtained by Bj\\\"orner and Welker in 1995."}
{"category": "Math", "title": "Commensurations and Subgroups of Finite Index of Thompson's Group F", "abstract": "We determine the abstract commensurator com(F) of Thompson's group F and describe it in terms of piecewise linear homeomorphisms of the real line and in terms of tree pair diagrams. We show com (F) is not finitely generated and determine which subgroups of finite index in F are isomorphic to F. We show that the natural map from the commensurator group to the quasi-isometry group of F is injective."}
{"category": "Math", "title": "Finiteness and vanishing results on weighted Poincare inequality of complete manifolds", "abstract": "We study manifolds satisfying a weighed Poincare inequality, which was first introduced by Li-Wang. We generalized one of their results by relaxing the Ricci curvature bound condition only being satisfied outside a compact set and established a finitely many ends result. We proved a vanishing result for $L^2$ harmonic 1-form provided that the weight function $\\rho$ is of sub-quadratic growth of the distance function."}
{"category": "Math", "title": "Infinite Viterbi alignments in the two state hidden Markov models", "abstract": "Since the early days of digital communication, Hidden Markov Models (HMMs) have now been routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. An HMM $(X_i,Y_i)_{i\\ge 1}$ assumes observations $X_1,X_2,...$ to be conditionally independent given an \"explanotary\" Markov process $Y_1,Y_2,...$, which itself is not observed; moreover, the conditional distribution of $X_i$ depends solely on $Y_i$. Central to the theory and applications of HMM is the Viterbi algorithm to find {\\em a maximum a posteriori} estimate $q_{1:n}=(q_1,q_2,...,q_n)$ of $Y_{1:n}$ given the observed data $x_{1:n}$. Maximum {\\em a posteriori} paths are also called Viterbi paths or alignments. Recently, attempts have been made to study the behavior of Viterbi alignments of HMMs with two hidden states when $n$ tends to infinity. It has indeed been shown that in some special cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions. This work proves the existence of infinite Viterbi alignments for virtually any HMM with two hidden states."}
{"category": "Math", "title": "The space of tropically collinear points is shellable", "abstract": "The space T_{d,n} of n tropically collinear points in a fixed tropical projective space TP^{d-1} is equivalent to the tropicalization of the determinantal variety of matrices of rank at most 2, which consists of real d x n matrices of tropical or Kapranov rank at most 2, modulo projective equivalence of columns. We show that it is equal to the image of the moduli space M_{0,n}(TP^{d-1},1) of n-marked tropical lines in TP^{d-1} under the evaluation map. Thus we derive a natural simplicial fan structure for T_{d,n} using a simplicial fan structure of M_{0,n}(TP^{d-1},1) which coincides with that of the space of phylogenetic trees on d+n taxa. The space of phylogenetic trees has been shown to be shellable by Trappmann and Ziegler. Using a similar method, we show that T_{d,n} is shellable with our simplicial fan structure and compute the homology of the link of the origin. The shellability of T_{d,n} has been conjectured by Develin in 2005."}
{"category": "Math", "title": "Rings whose modules are weakly supplemented are perfect", "abstract": "In this note we show that a ring R is left perfect if and only if every left R-module is weakly supplemented if and only if R is semilocal and the radical of the countably infinite free left R-module has a weak supplement."}
{"category": "Math", "title": "Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores", "abstract": "We provide the explicit solutions of linear, left-invariant, (convection)-diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2). These diffusion equations are forward Kolmogorov equations for stochastic processes for contour enhancement and completion. The solutions are group-convolutions with the corresponding Green's function, which we derive in explicit form. We mainly focus on the Kolmogorov equations for contour enhancement processes which, in contrast to the Kolmogorov equations for contour completion, do not include convection. The Green's functions of these left-invariant partial differential equations coincide with the heat-kernels on SE(2), which we explicitly derive. Then we compute completion distributions on SE(2) which are the product of a forward and a backward resolvent evolved from resp. source and sink distribution on SE(2). On the one hand, the modes of Mumford's direction process for contour completion coincide with elastica curves minimizing $\\int \\kappa^{2} + \\epsilon ds$, related to zero-crossings of 2 left-invariant derivatives of the completion distribution. On the other hand, the completion measure for the contour enhancement concentrates on geodesics minimizing $\\int \\sqrt{\\kappa^{2} + \\epsilon} ds$. This motivates a comparison between geodesics and elastica, which are quite similar. However, we derive more practical analytic solutions for the geodesics. The theory is motivated by medical image analysis applications where enhancement of elongated structures in noisy images is required. We use left-invariant (non)-linear evolution processes for automated contour enhancement on invertible orientation scores, obtained from an image by means of a special type of unitary wavelet transform."}
{"category": "Math", "title": "Singularities of admissible normal functions", "abstract": "In a recent paper, M. Green and P. Griffiths used R. Thomas' works on nodal hypersurfaces to establish the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we study normal functions using M. Saito's mixed Hodge modules and prove that the existence of singularities of the type considered by Griffiths and Green is equivalent to the Hodge conjecture. Several of the intermediate results, including a relative version of the weak Lefschetz theorem for perverse sheaves, are of independent interest."}
{"category": "Math", "title": "On a question of Landis and Oleinik", "abstract": "We answer in the affirmative a question posed by Landis and Oleinik on unique continuation of variable coefficients parabolic equations."}
{"category": "Math", "title": "Affine Hermitian-Einstein Metrics", "abstract": "We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence of Hermitian-Einstein metrics on K\\\"ahler manifolds, and the extension of this theorem by Li-Yau to the non-K\\\"ahler complex case of Gauduchon metrics. Our definition of stability involves only flat vector subbundles (and not singular subsheaves), and so is simpler than the complex case in some places."}
{"category": "Math", "title": "Minimal Homeomorphisms on Low-Dimension Tori", "abstract": "In this article we study minimal homeomorphisms(all orbits are dense) of the tori $T^{n},$ $n<5.$ The linear part of a homeomorphism $\\phi $ of $T^{n}$ is the linear mapping $L$ induced by $\\phi $ on the first homology group of $T^{n}$. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of $L$ if $\\phi $ minimal. We show that if $\\phi $ is minimal and $n<5$ then $L$ is quasi-unipontent, i.e., all the eigenvalues of $L$ are roots of unity and conversely if $L\\in GL(n,\\Z)$ is quasi-unipotent and 1 is an eigenvalue of $L$ then there exists a $ C^{\\infty}$ minimal skew-product diffeomorphism $\\phi $ of $T^{n}$ whose linear part is precisely $L.$ We do not know if these results are true for $n>4$. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation."}
{"category": "Math", "title": "First exit times for L\\'evy-driven diffusions with exponentially light jumps", "abstract": "We consider a dynamical system described by the differential equation $\\dot{Y}_t=-U'(Y_t)$ with a unique stable point at the origin. We perturb the system by the L\\'evy noise of intensity $\\varepsilon$ to obtain the stochastic differential equation $dX^{\\varepsilon}_t=-U'(X^{\\varepsilon}_{t-}) dt+\\varepsilon dL_t.$ The process $L$ is a symmetric L\\'evy process whose jump measure $\\nu$ has exponentially light tails, $\\nu([u,\\infty))\\sim\\exp(-u^{\\alpha})$, $\\alpha>0$, $u\\to \\infty$. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval $(-1,1)$. In the small noise limit $\\varepsilon\\to0$, the law of the first exit time $\\sigma_x$, $x\\in(-1,1)$, has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index $\\alpha=1$, namely, $\\ln\\mathbf{E}\\sigma\\sim\\varepsilon^{-\\alpha}$ for $0<\\alpha<1$, whereas $\\ln\\mathbf{E}\\sigma\\sim\\varepsilon^{- 1}|\\ln\\varepsilon|^{1-{1}/{\\alpha}}$ for $\\alpha>1$."}
{"category": "Math", "title": "Positivity in the cohomology of flag bundles (after Graham)", "abstract": "We give a short, geometric proof of Graham's theorem on positivity in the equivariant cohomology of a flag variety, based on a transversality argument."}
{"category": "Math", "title": "Constructing processes with prescribed mixing coefficients", "abstract": "The rate at which dependencies between future and past observations decay in a random process may be quantified in terms of mixing coefficients. The latter in turn appear in strong laws of large numbers and concentration of measure results for dependent random variables. Questions regarding what rates are possible for various notions of mixing have been posed since the 1960's, and have important implications for some open problems in the theory of strong mixing conditions. This paper deals with $\\eta$-mixing, a notion defined in [Kontorovich and Ramanan], which is closely related to $\\phi$-mixing. We show that there exist measures on finite sequences with essentially arbitrary $\\eta$-mixing coefficients, as well as processes with arbitrarily slow mixing rates."}
{"category": "Math", "title": "Obtaining Measure Concentration from Markov Contraction", "abstract": "Concentration bounds for non-product, non-Haar measures are fairly recent: the first such result was obtained for contracting Markov chains by Marton in 1996 via the coupling method. The work that followed, with few exceptions, also used coupling. Although this technique is of unquestionable utility as a theoretical tool, it is not always simple to apply. As an alternative to coupling, we use the elementary Markov contraction lemma to obtain simple, useful, and apparently novel concentration results for various Markov-type processes. Our technique consists of expressing probabilities as matrix products and applying Markov contraction to these expressions; thus it is fairly general and holds the potential to yield further results in this vein."}
{"category": "Math", "title": "A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group \\mathcal{M}_{g,*}", "abstract": "We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus >=2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution."}
{"category": "Math", "title": "Upper bounds on the minimum coverage probability of confidence intervals in regression after variable selection", "abstract": "We consider a linear regression model, with the parameter of interest a specified linear combination of the regression parameter vector. We suppose that, as a first step, a data-based model selection (e.g. by preliminary hypothesis tests or minimizing AIC) is used to select a model. It is common statistical practice to then construct a confidence interval for the parameter of interest based on the assumption that the selected model had been given to us a priori. This assumption is false and it can lead to a confidence interval with poor coverage properties. We provide an easily-computed finite sample upper bound (calculated by repeated numerical evaluation of a double integral) to the minimum coverage probability of this confidence interval. This bound applies for model selection by any of the following methods: minimum AIC, minimum BIC, maximum adjusted R-squared, minimum Mallows' Cp and t-tests. The importance of this upper bound is that it delineates general categories of design matrices and model selection procedures for which this confidence interval has poor coverage properties. This upper bound is shown to be a finite sample analogue of an earlier large sample upper bound due to Kabaila and Leeb."}
{"category": "Math", "title": "Continuous extension of a densely parameterized semigroup", "abstract": "Let S be a dense sub-semigroup of the positive real numbers, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over S can be extended to a weakly continuous semigroup over the positive real numbers. We obtain similar results for non-linear, non-expansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over the positive real numbers."}
{"category": "Math", "title": "Finite index subgroups of R. Thompson's group F", "abstract": "The authors classify the finite index subgroups of R. Thompson's group $F$. All such groups that are not isomorphic to $F$ are non-split extensions of finite cyclic groups by $F$. The classification describes precisely which finite index subgroups of $F$ are isomorphic to $F$, and also separates the isomorphism classes of the finite index subgroups of $F$ which are not isomorphic to $F$ from each other; characterizing the structure of the extensions using properties of the structure of the finite index subgroups of $Z\\times Z$."}
{"category": "Math", "title": "Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds", "abstract": "In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curvatures of the ambient symmetric spaces. Consequently, all of our solvmanifolds are Einstein, which provide a large number of new examples of noncompact homogeneous Einstein manifolds. We also show that our solvmanifolds are minimal, but not totally geodesic submanifolds of symmetric spaces."}
{"category": "Math", "title": "CoHochschild homology of chain coalgebras", "abstract": "Generalizing work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra $C$ is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex $\\cohoch (C)$ admits a natural comultiplicative structure. In particular, if $K$ is a reduced simplicial set and $C_{*}K$ is its normalized chain complex, then $\\cohoch (C_{*}K)$ is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on $\\cohoch (C_{*}K)$ when $K$ is a simplicial suspension. The coHochschild complex construction is topologically relevant. Given two simplicial maps $g,h:K\\to L$, where $K$ and $L$ are reduced, the homology of the coHochschild complex of $C_{*}L$ with coefficients in $C_{*}K$ is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of $g$ and $h$, and this isomorphism respects comultiplicative structure. In particular, there a isomorphism, respecting comultiplicative structure, from the homology of $\\cohoch(C_{*}K)$ to $H_{*}\\op L|K|$, the homology of the free loops on the geometric realization of $K$."}
{"category": "Math", "title": "Finite subsets of projective space, and their ideals", "abstract": "Let $\\mathscr{A}$ be a finite set of closed rational points in projective space, let $\\mathscr{I}$ be the vanishing ideal of $\\mathscr{A}$, and let $\\mathscr{D}(\\mathscr{A})$ be the set of exponents of those monomials which do not occur as leading monomials of elements of $\\mathscr{I}$. We show that the size of $\\mathscr{A}$ equals the number of axes contained in $\\mathscr{D}(\\mathscr{A})$. Furthermore, we present an algorithm for the construction of the Gr\\\"obner basis of $\\mathscr{I}(\\mathscr{A})$, hence also of $\\mathscr{D}(\\mathscr{A})$."}
{"category": "Math", "title": "Skew domino Schensted algorithm and sign-imbalance", "abstract": "Using growth diagrams, we define a skew domino Schensted correspondence which is a domino analogue of the skew Robinson-Schensted correspondence due to Sagan and Stanley. The color-to-spin property of Shimozono and White is extended. As an application, we give a simple generating function for the weighted sum of skew domino tableaux, which is a generalization of Stanley's sign-imbalance formula. The generating function gives a method to calculate the generalized sign-imbalance formula. We also extend Sj{\\\"o}strand's theorems on sign-imbalance of skew shapes."}
{"category": "Math", "title": "Confidence Sets Based on Sparse Estimators Are Necessarily Large", "abstract": "Confidence sets based on sparse estimators are shown to be large compared to more standard confidence sets, demonstrating that sparsity of an estimator comes at a substantial price in terms of the quality of the estimator. The results are set in a general parametric or semiparametric framework."}
{"category": "Math", "title": "Well-posedness and ill-posedness of the fifth order modifed KdV equation", "abstract": "We consider the initial value problem of the fifth order modified KdV equation on the Sobolev spaces. \\partial_t u - \\partial_x^5u + c_1\\partial_x^3(u^3) + c_2u\\partial_x u\\partial_x^2 u + c_3uu\\partial_x^3 u =0, u(x,0)= u_0(x) where $ u:R\\timesR \\to R $ and $c_j$'s are real. We show the local well-posedness in H^s(R) for s \\geq 3/4 via the contraction principle on $X^{s,b}$ space. Also, we show that the solution map from data to the solutions fails to be uniformly continuous below $H^{3/4}(R)$. The counter example is obtained by approximating the fifth order mKdV equation by the cubic NLS equation."}
{"category": "Math", "title": "A conservative evolution of the Brownian excursion", "abstract": "We consider the problem of conditioning the Brownian excursion to have a fixed time average over the interval [0,1] and we study an associated stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space-time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution."}
{"category": "Math", "title": "Adaptive Eigenvalue Computation - Complexity Estimates", "abstract": "This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on $\\ell_2$, the space of square summable sequences, the problem becomes sufficiently well conditioned so that a gradient type iteration can be shown to reduce the error by some fixed factor per step. It then remains to realize these (ideal) iterations within suitable dynamically updated error tolerances. It is shown under which circumstances the adaptive scheme exhibits in some sense asymptotically optimal complexity."}
{"category": "Math", "title": "Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition", "abstract": "In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein's method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a higher-dimensional space, we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to Stein's exchangeable pairs approach. To illustrate the method, we provide the examples of runs on the line as well as double-indexed permutation statistics."}
{"category": "Math", "title": "On function spaces on symmetric spaces", "abstract": "Let Y=G/H be a semisimple symmetric space. It is shown that the smooth vectors for the regular representation of G on L^p(Y) vanish at infinity."}
{"category": "Math", "title": "Approximating Perpetuities", "abstract": "We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be approximated well."}
{"category": "Math", "title": "Selection Principles and Baire spaces", "abstract": "We prove that if X is a separable metric space with the Hurewicz covering property, then the Banach-Mazur game played on X is determined. The implication is not true when \"Hurewicz covering property\" is replaced with \"Menger covering property\"."}
{"category": "Math", "title": "Evolution of convex lens-shaped networks under curve shortening flow", "abstract": "We consider convex symmetric lens-shaped networks in R^2 that evolve under curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks converge to a self-similarly shrinking network, which we prove to be unique in an appropriate class. We also include a classification result for some self-similarly shrinking networks."}
{"category": "Math", "title": "On the blow-up problem and new a priori estimates for the 3D Euler and the Navier-Stokes equations", "abstract": "We study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the 3D Euler equations, where the sense of convergence and self-similarity are considered in various sense. We extend much further, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities obtained in \\cite{cha1,cha2}. Some implications the notions for the 3D Navier-Stokes equations are also deduced. Generalization of the self-similar transforms is also considered, and by appropriate choice of the transform we obtain new \\textit{a priori} estimates for the 3D Euler and the Navier-Stokes equations."}
{"category": "Math", "title": "Goldbach Conjecture and First-Order Arithmetic", "abstract": "Using the concepts of Hyperbolic Classification of Natural Numbers, Essential Regions and Goldbach Conjecture Function we prove that the existence of a proof of the Goldbach Conjecture in First-Order Arithmetic would imply the existence of another proof in a certain extension that would not be valid in all states of time associated to natural numbers created by means of adequate dynamic processes."}
{"category": "Math", "title": "Local Geometric Langlands Correspondence: the Spherical Case", "abstract": "A module over an affine Kac--Moody algebra g^ is called spherical if the action of the Lie subalgebra g[[t]] on it integrates to an algebraic action of the corresponding group G[[t]]. Consider the category of spherical g^-modules of critical level. In this paper we prove that this category is equivalent to the category of quasi-coherent sheaves on the ind-scheme of opers on the punctured disc which are unramified as local systems. This result is a categorical version of the well-known description of spherical vectors in representations of groups over local non-archimedian fields. It may be viewed as a special case of the local geometric Langlands correspondence proposed in arXiv:math/0508382."}
{"category": "Math", "title": "Landweber exact formal group laws and smooth cohomology theories", "abstract": "The main aim of this paper is the construction of a smooth (sometimes called differential) extension \\hat{MU} of the cohomology theory complex cobordism MU, using cycles for \\hat{MU}(M) which are essentially proper maps W\\to M with a fixed U(n)-structure and U(n)-connection on the (stable) normal bundle of W\\to M. Crucial is that this model allows the construction of a product structure and of pushdown maps for this smooth extension of MU, which have all the expected properties. Moreover, we show, using the Landweber exact functor principle, that \\hat{R}(M):=\\hat{MU}(M)\\otimes_{MU^*}R defines a multiplicative smooth extension of R(M):=MU(M)\\otimes_{MU^*}R whenever R is a Landweber exact MU*-module. An example for this construction is a new way to define a multiplicative smooth K-theory."}
{"category": "Math", "title": "The rank of a quiver representation", "abstract": "We define a functor which gives the \"global rank of a quiver representation\" and prove that it has nice properties which make it a generalization of the rank of a linear map. We demonstrate how to construct other \"rank functors\" for a quiver Q, which induce ring homomorphisms (called \"rank functions\") from the representation ring of Q to Z. These rank functions give discrete numerical invariants of quiver representations, useful for computing tensor product multiplicities of representations and determining some structure of the representation ring. We also show that in characteristic 0, rank functors commute with the Schur operations on quiver representations, and the homomorphisms induced by rank functors are lambda-ring homomorphisms."}
{"category": "Math", "title": "On Enumeration of Conjugacy Classes of Coxeter Elements", "abstract": "In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group. We give a direct proof of a recursion for the number of equivalence classes of this relation for an arbitrary graph Y using edge deletion and edge contraction of non-bridge edges. We conclude by showing how this result may also be obtained through an evaluation of the Tutte polynomial as T(Y,1,0), and we provide bijections to two other classes of acyclic orientations that are known to be counted in the same way. A transversal of the set of equivalence classes is given."}
{"category": "Math", "title": "Modeling homophily and stochastic equivalence in symmetric relational data", "abstract": "This article discusses a latent variable model for inference and prediction of symmetric relational data. The model, based on the idea of the eigenvalue decomposition, represents the relationship between two nodes as the weighted inner-product of node-specific vectors of latent characteristics. This ``eigenmodel'' generalizes other popular latent variable models, such as latent class and distance models: It is shown mathematically that any latent class or distance model has a representation as an eigenmodel, but not vice-versa. The practical implications of this are examined in the context of three real datasets, for which the eigenmodel has as good or better out-of-sample predictive performance than the other two models."}
{"category": "Math", "title": "Stein or Milnor fillability and cohomology", "abstract": "I have withdrawn the paper, after having incorporated it into the paper arXiv:0712.3484. In the meantime I have discovered that one of the theorems proved in the paper had already been proved by Durfee & Hain."}
{"category": "Math", "title": "Projections, Entropy and Sumsets", "abstract": "In this paper we have shall generalize Shearer's entropy inequality and its recent extensions by Madiman and Tetali, and shall apply projection inequalities to deduce extensions of some of the inequalities concerning sums of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa. We shall also discuss projection and entropy inequalities and their connections."}
{"category": "Math", "title": "Estimating the trace-free Ricci tensor in Ricci flow", "abstract": "An important and natural question in the analysis of Ricci flow singularity formation in dimensions four and above is as follows: What are the weakest conditions that provide control of the norm of the Riemann curvature tensor? In this short note, we show that on a compact manifold, the trace-free Ricci tensor is controlled in a precise fashion by the other components of the irreducible decomposition of the curvature tensor, without any hypotheses on the initial data."}
{"category": "Math", "title": "L^2 Castelnuovo-de Franchis, the cup product lemma, and filtered ends of Kaehler manifolds", "abstract": "Simple approaches to the proofs of the L^2 Castelnuovo-de Franchis theorem and the cup product lemma which give new versions are developed. For example, suppose u and v are two linearly independent closed holomorphic 1-forms on a bounded geometry connected complete Kaehler manifold X with v in L^2. According to a version of the L^2 Castelnuovo-de Franchis theorem obtained in this paper, if u and v are pointwise linearly dependent, then there exists a surjective proper holomorphic mapping of X onto a Riemann surface for which u and v are pull-backs. Previous versions required both forms to be in L^2."}
{"category": "Math", "title": "Toward a Unit Distance Embedding for the Heawood graph", "abstract": "The unit distance embeddability of a graph, like planarity, involves a mix of constraints that are combinatorial and geometric. We construct a unit distance embedding for $H-e$ in the hope that it will lead to an embedding for $H$. We then investigate analytical methods for a general decision procedure for testing unit distance embeddability."}
{"category": "Math", "title": "On rigidity and the isomorphism problem for tree braid groups", "abstract": "We solve the isomorphism problem for braid groups on trees with $n = 4$ or 5 strands. We do so in three main steps, each of which is interesting in its own right. First, we establish some tools and terminology for dealing with computations using the cohomology of tree braid groups, couching our discussion in the language of differential forms. Second, we show that, given a tree braid group $B_nT$ on $n = 4$ or 5 strands, $H^*(B_nT)$ is an exterior face algebra. Finally, we prove that one may reconstruct the tree $T$ from a tree braid group $B_nT$ for $n = 4$ or 5. Among other corollaries, this third step shows that, when $n = 4$ or 5, tree braid groups $B_nT$ and trees $T$ (up to homeomorphism) are in bijective correspondence. That such a bijection exists is not true for higher dimensional spaces, and is an artifact of the 1-dimensionality of trees. We end by stating the results for right-angled Artin groups corresponding to the main theorems, some of which do not yet appear in the literature."}
{"category": "Math", "title": "Convergence of Diagonal Ergodic Averages", "abstract": "Tao has recently proved that if $T_1,...,T_l$ are commuting, invertible, measure-preserving transformations on a dynamical system then for any $L^\\infty$ functions $f_1,...,f_l$, the average $\\frac{1}{N}\\sum_{n=0}^{N-1}\\prod_{i\\leq l}f_i\\circ T^n_i$ converges in the $L^2$ norm. Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence \"backwards\". In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao's argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties."}
{"category": "Math", "title": "Gorenstein categories and Tate cohomology on projective schemes", "abstract": "We study Gorenstein categories. We show that such a category has Tate cohomological functors and Avramov-Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute and the Tate cohomological functors. We show that such a category has what Hovey calls an injective model structure and also a projective model structure in case the category has enough projectives. As examples we show that if X is a locally Gorenstein projective scheme then the category Qco(X) of quasi-coherent sheaves on $X$ is such a category and so has these features."}
{"category": "Math", "title": "Conformal Graph Directed Markov Systems", "abstract": "We present the main concepts and results for Graph Directed Markov Systems that have a finitely irreducible incidence matrix. We then see how these results change when the incidence matrix is not assumed to be finitely irreducible."}
{"category": "Math", "title": "Complete r-partite subgraphs of dense r-graphs", "abstract": "We determine how large r-partite graphs can be found in r-uniform graphs with n vertices and Cn^r edges, where C is a slowly decreasing function of n. This refines results of Erdos from 1964."}
{"category": "Math", "title": "Degree Complexity of a Family of Birational Maps", "abstract": "We compute the degree complexity of a family of birational mappings of the plane with high order singularities."}
{"category": "Math", "title": "Pushnitski's $\\mu$-invariant and Schr\\\"odinger operators with embedded eigenvalues", "abstract": "In this note, under a certain assumption on an affine space of operators, which admit embedded eigenvalues, it is shown that the singular part of the spectral shift function of any pair of operators from this space is an integer-valued function. The proof uses a natural decomposition of Pushnitski's $\\mu$-invariant into \"absolutely continuous\" and \"singular\" parts. As a corollary, the Birman-Krein formula follows."}
{"category": "Math", "title": "The classification of $\\bf Z$-graded modules of the intermediate series over the $q$-analog Virasoro-like algebra", "abstract": "In this paper, we complete the classification of the {\\bf Z}-graded modules of the intermediate series over the $q$-analog Virasoro-like algebra $L$. We first construct four classes of irreducible {\\bf Z}-graded $L$-modules of the intermediate series. Then we prove that any {\\bf Z}-graded $L$-modules of the intermediate series must be the direct sum of some trivial $L$-modules or one of the modules constructed by us."}
{"category": "Math", "title": "Introduction to the theory of quasi-log varieties", "abstract": "This paper is a gentle introduction to the theory of quasi-log varieties by Ambro. We explain the fundamental theorems for the log minimal model program for log canonical pairs. More precisely, we give a proof of the base point free theorem for log canonical pairs in the framework of the theory of quasi-log varieties."}
{"category": "Math", "title": "Rational Integrals of the second kind on a complex projective manifold and its primitive cohomology", "abstract": "Let X be a complex algebraic manifold of dimension n+1 embedded in a sufficiently higher dimensional complex projective space, and Y a generic hyperplane section of X. We describe the mixed Hodge structure on H^p(X-Y,C) and the Hodge filtration of the middle primitive cohomology group H^n(Y,C)_0 of Y in terms of rational integrals on X. (Key words: Primitive cohomology, Rational integral of the 2nd kind, Generalized Poincare residue map, Hodge filtration, Mixed Hodge structure)"}
{"category": "Math", "title": "Finite dimensional subspaces of noncommutative $L_p$ spaces", "abstract": "We prove the following noncommutative version of Lewis's classical result. Every n-dimensional subspace E of Lp(M) (1<p<\\infty) for a von Neumann algebra M satisfies d_{cb}(E, RC^n_{p'}) \\leq c_p n^{\\abs{1/2-1/p}} for some constant c_p depending only on $p$, where $1/p +1/p' =1$ and $RC^n_{p'} = [R_n\\cap C_n, R_n+C_n]_{1/p'}$. Moreover, there is a projection $P:Lp(M) --> Lp(M)$ onto E with $\\norm{P}_{cb} \\leq c_p n^{\\abs{1/2-1/p}}.$ We follow the classical change of density argument with appropriate noncommutative variations in addition to the opposite trick."}
{"category": "Math", "title": "Invariant Linearization Criteria for Systems of Cubically Semi-Linear Second-Order Ordinary Differential Equations", "abstract": "Invariant linearization criteria of square systems of second-order quadratically semi-linear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in the first derivatives. It is shown that there are two branches for the linearization problem via point transformations for an arbitrary system of second-order ODEs. One is when the system is at most cubic in the first derivatives. We solve this branch of the linearization problem by point transformations in the case of a square sytem of two second-order ODEs. Necessary and sufficient conditions for linearization by means of point transformations are given in terms of coefficient functions of the system of two second-order ODEs cubically nonlinear in the first derivatives. A consequence of our geometric approach of projection is a re-derivation of Lie's conditions for a single second-order ODE and sheds light on more recent results on them. In particular, we show here how one can construct point transformations for reduction to the simplest linear equation by going to the higher space and just utilising the coefficients of the original ODE. We also obtain invariant criteria for the reduction of a linear square system to the simplest system. Moreover, these results contain the quadratic case as a special case. Examples are given to illustrate our results."}
{"category": "Math", "title": "Linearizability Criteria for a Class of Third Order Semi-Linear ODEs", "abstract": "Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations, we extend to the third order by differentiating the second order equation. This yields criteria for linearizability of a class of third order semi-linear ordinary differential equations, which is distinct from the classes available in the literature. Some examples are given and discussed."}
{"category": "Math", "title": "Conditional linearizability criteria for a system of third-order ordinary differential equations", "abstract": "We provide linearizability criteria for a class of systems of third-order ordinary differential equations (ODEs) that is cubically semi-linear in the first derivative, by differentiating a system of second-order quadratically semi-linear ODEs and using the original system to replace the second derivative. The procedure developed splits into two cases, those where the coefficients are constant and those where they are variables. Both cases are discussed and examples given."}
{"category": "Math", "title": "Constructing a Space from the System of Geodesic Equations", "abstract": "Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from the Christoffel symbols. The procedure is extended for determining if a second order quadratically semi-linear system can be expressed as a system of geodesic equations, provided it has terms only quadratic in the first derivative apart from the second derivative term. A computer code has been developed for dealing with larger systems of geodesic equations."}
{"category": "Math", "title": "Conditional Linearizability Criteria for Scalar Fourth Order Semi-Linear Ordinary Differential Equations", "abstract": "Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations and third order quintically semi-linear ordinary differential equations, we extend to the fourth order by differentiating the third order equation. This yields criteria for linearizability of a class of fourth order semi-linear ordinary differential equations, which have not been discussed in the literature previously. It is shown that the procedure can be extended to higher order. Though the results for the higher orders are complicated, they are doable by algebraic computing. The standard Lie approach, as developed at present does not seem to be amenable to giving results that can be handled even by algebraic computing."}
{"category": "Math", "title": "A Zoll counterexample to a geodesic length conjecture", "abstract": "We construct a counterexample to a conjectured inequality L<2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. Thus the round metric is not optimal for the ratio L/D."}
{"category": "Math", "title": "Maximum principle and convergence of fundamental solutions for the Ricci flow", "abstract": "In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in $B_k\\times (0,T)$ to the minimal fundamental solution of the conjugate heat equation as $k\\to\\infty$. We will prove the uniqueness of the fundamental solution under some exponential decay assumption on the fundamental solution. We will also give a detail proof of the convergence of the fundamental solutions of the conjugate heat equation for a sequence of pointed Ricci flow $(M_k\\times (-\\alpha,0],x_k,g_k)$ to the fundamental solution of the limit manifold as $k\\to\\infty$ which was used without proof by Perelman in his proof of the pseudolocality theorem for Ricci flow."}
{"category": "Math", "title": "Contributions to Random Energy Models", "abstract": "In this thesis, we consider several Random Energy Models. This includes Derrida's Random Energy Model (REM) and Generalized Random Energy Model (GREM) and a nonhierarchical version (BK-GREM) by Bolthausen and Kistler. The limiting free energy in all these models along with Word GREM, a model proposed by us, turn out to be a cute consequence of large deviation principle (LDP). This LDP argument allows us to consider non-Gaussian driving distributions as well as external field. We could also consider random trees as the underlying tree structure in GREM. In all these models, as expected, limiting free energy is not 'universal' unlike the SK model. However it is 'rate specific'. Consideration of non-Gaussian driving distribution as well as different driving distributions for the different levels of the underlying trees in GREM leads to interesting phenomena. For example in REM, if the Hamiltonian is Binomial with parameter $N$ and $p$ then the existence of phase transition depends on the parameter $p$. More precisely, phase transition takes place only when $p>{1/2}$. For another example, consider a 2 level GREM with exponential driving distribution at the first level and Gaussian in the second with equal weights at both the levels. Then even if the limiting ratio for the second level particles, $p_2$ is 0.00001 (very small), the system reduces to a Gaussian REM. On the other hand, if we consider a 2 level GREM with Gaussian driving distribution at the first level and exponential in the second, the system will never reduce to a Gaussian REM. In either case, the system will never reduce to that of an exponential REM. etc."}
{"category": "Math", "title": "Convexity in locally conformally flat manifolds with boundary", "abstract": "Given a closed subset $\\La$ of the open unit ball $B_1\\subset \\real^n$, $n \\geq 3$, we will consider a complete Riemannian metric $g$ on $\\bar{B_1} \\setminus \\La$ of constant scalar curvature equal to $n(n-1)$ and conformally related to the Euclidean metric. In this paper we prove that every closed Euclidean ball $\\bar{B} \\subset B_1\\setminus \\La$ is convex with respect to the metric $g$, assuming the mean curvature of the boundary $\\partial B_1$ is nonnegative with respect to the inward normal."}
{"category": "Math", "title": "Bifurcations in the regularized Ericksen bar model", "abstract": "We consider the regularized Ericksen model of an elastic bar on an elastic foundation on an interval with Dirichlet boundary conditions as a two-parameter bifurcation problem. We explore, using local bifurcation analysis and continuation methods, the structure of bifurcations from double zero eigenvalues. Our results provide evidence in support of M\\\"uller's conjecture \\cite{Muller} concerning the symmetry of local minimizers of the associated energy functional and describe in detail the structure of the primary branch connections that occur in this problem. We give a reformulation of M\\\"uller's conjecture and suggest two further conjectures based on the local analysis and numerical observations. We conclude by analysing a ``loop'' structure that characterizes $(k,3k)$ bifurcations."}
{"category": "Math", "title": "The critical contact process in a randomly evolving environment dies out", "abstract": "Bezuidenhout and Grimmett proved that the critical contact process dies out. Here, we generalize the result to the so called contact process in a random evolving environment (CPREE), introduced by Erik Broman. This process is a generalization of the contact process where the recovery rate can vary between two values. The rate which it chooses is determined by a background process, which evolves independently at different sites. As for the contact process, we can similarly define a critical value in terms of survival for this process. In this paper we prove that this definition is independent of how we start the background process, that finite and infinite survival (meaning nontriviality of the upper invariant measure) are equivalent and finally that the process dies out at criticality."}
{"category": "Math", "title": "Inductive Methods and zero-sum free sequences", "abstract": "We obtain a decidability result for the Davenport constant."}
{"category": "Math", "title": "An Asymptotic Formalism for Reconstructing Small Perturbations of Scatterers from Electric or Acoustic Far-Field Measurements", "abstract": "We consider the problem of determining the boundary perturbations of an object from far-field electric or acoustic measurements. Assuming that the unknown object boundary is a small perturbation of a circle, we develop a linearized relation between the far-field data that result from fixed Dirichlet boundary conditions, entering as parameters, and the shape of the object, entering as variables. This relation is used to find the Fourier coefficients of the perturbation of the shape and makes use of an expansion of the Dirichlet-to-Neumann operator."}
{"category": "Math", "title": "Characterization of optimal Transport Plans for the Monge-Kantorovich-Problem", "abstract": "We prove that $c$-cyclically monotone transport plans $\\pi$ optimize the Monge-Kantorovich transportation problem under an additional measurability condition. This measurability condition is always satisfied for finitely valued, lower semi-continuous cost functions. In particular, this yields a positive answer to Problem 2.25 in C. Villani's book. We emphasize that we do not need any regularity conditions as were imposed in the previous literature."}
{"category": "Math", "title": "Hecke operators and Hilbert modular forms", "abstract": "Let F be a real quadratic field with ring of integers O and with class number 1. Let Gamma be a congruence subgroup of GL_2 (O). We describe a technique to compute the action of the Hecke operators on the cohomology H^3 (Gamma; C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way to compute the Hecke action on these Hilbert modular forms."}
{"category": "Math", "title": "A classification of spherical symmetric CR manifolds", "abstract": "In this paper we classify the simply connected, spherical pseudohermitian manifolds whose Webster metric is CR-symmetric."}
{"category": "Math", "title": "A Conformal de Rham Complex", "abstract": "We introduce the notion of a conformal de Rham complex of a Riemannian manifold. This is a graded differential Banach algebra and it is invariant under quasiconformal maps, in particular the associated cohomology is a new quasiconformal invariant."}
{"category": "Math", "title": "Convergence of Singular integrals with general measures", "abstract": "We show that L^2-bounded singular integral in metric spaces with respect to general measures and kernels converge weakly. This implies a kind of average convergence almost everywhere. For measures with zero density we prove the almost everywhere existence of principal values."}
{"category": "Math", "title": "Local probabilities for random walks conditioned to stay positive", "abstract": "Let S_0=0,{S_n, n>0} be a random walk generated by a sequence of i.i.d. random variables X_1,X_2,... and let \\tau^{-} be the first descending ladder epoch. Assuming that the distribution of X_1 belongs to the domain of attraction of an \\alpha-stable law we study the asymptotic behavior of the local probabilities P(\\tau ^{-}=n) and the conditional local probabilities P(S_n\\in [x,x+y)|\\tau^{-}>n) for fixed y and x=x(n)\\in (0,\\infty)."}
{"category": "Math", "title": "Exponential sums and rank of persymmetric matrices over F_2", "abstract": "Over the finite field with two elements, we present a method for obtaining explicit expressions for the number of rank i matrices of the form A above B, where A is persymmetric (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j = r+s)."}
{"category": "Math", "title": "A remark on singularities of primitive cohomology classes", "abstract": "Green and Griffiths have introduced several notions of singularities associated with normal functions, especially in connection with middle dimensional primitive Hodge classes. In this note, by using the more elementary aspects of the Decomposition Theorem, we define global and local singularities associated with primitive middle dimensional cohomology classes and by using the Relative Hard Lefschetz Theorem, we show that these singularities detect the global and local triviality of the primitive class. In a final section, we write-up a classical inductive argument relating the Hodge conjecture to the local non-vanishing of primitive classes."}
{"category": "Math", "title": "Fractional martingales and characterization of the fractional Brownian motion", "abstract": "In this paper we introduce the notion of fractional martingale as the fractional derivative of order $\\alpha$ of a continuous local martingale, where $\\alpha\\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation of order $\\frac{2}{1+2\\alpha}$, under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of L\\'evy's characterization theorem for the fractional Brownian motion."}
{"category": "Math", "title": "A study of Galois objects for algebraic quantum groups", "abstract": "We supplement the study of Galois theory for algebraic quantum groups started in the paper 'Galois Theory for Multiplier Hopf Algebras with Integrals' by A. Van Daele and Y.H. Zhang. We examine the structure of the Galois objects: algebras equipped with a Galois coaction such that only the scalars are coinvariants. We show how their structure is as rich as the one of the quantum groups themselves: there are two distinguished weak K.M.S. functionals, related by a modular element, and there is an analogue of the antipode squared. We also show how to reflect the quantum group across the Galois object to obtain a (possibly) new algebraic quantum group. We end by considering an example."}
{"category": "Math", "title": "Selective screenability in topological groups", "abstract": "We examine the selective screenability property in topological groups. In the metrizable case we also give characterizations in terms of the Haver property and finitary Haver property respectively relative to left-invariant metrics. We prove theorems stating conditions under which the properties are preserved by products. Among metrizable groups we characterize the ones of countable covering dimension by a natural game."}
{"category": "Math", "title": "The Fermat cubic and special Hurwitz loci in M_g", "abstract": "We compute the class of the locus in M_g of curves having a pencil with two unspecified triple ramification points. This is the first example of a geometric divisor on M_g which is not the pull-back of a divisor on the moduli space of pseudo-stable curves. This space, in which elliptic tails are replaced by cusps, appears as a result of the first divisorial contraction in the minimal model program for M_g. In particular, we show that our divisor picks-up the locus of Fermat cubic tails when restricted to the boundary divisor of elliptic tails. We also give various enumerative applications concerning coverings of the generic curve having special ramification behaviour."}
{"category": "Math", "title": "Homotopical equivalence of combinatorial and categorical semantics of process algebra", "abstract": "It is possible to translate a modified version of K. Worytkiewicz's combinatorial semantics of CCS (Milner's Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turns out that a satisfactory semantics in terms of flows requires to work directly in their homotopy category since such a semantics requires non-canonical choices for constructing cofibrant replacements, homotopy limits and homotopy colimits. No geometric information is lost since two precubical sets are isomorphic if and only if the associated flows are weakly equivalent. The interest of the categorical semantics is that combinatorics totally disappears. Last but not least, a part of the categorical semantics of CCS goes down to a pure homotopical semantics of CCS using A. Heller's privileged weak limits and colimits. These results can be easily adapted to any other process algebra for any synchronization algebra."}
{"category": "Math", "title": "Cellular structures, quasisymmetric mappings, and spaces of homogeneous type", "abstract": "A class of Cantor-type spaces and related geometric structures are discussed."}
{"category": "Math", "title": "Recursive state estimation for noncausal discrete-time descriptor systems under uncertainties", "abstract": "This paper describes a method for the online state estimation of systems described by a general class of linear noncausal time-varying difference descriptor equations subject to uncertainties. The method is based on the notions of a linear minimax estimation and an index of causality introduced here for singular difference equations. The online minimax estimator is derived by the application of the dynamical programming and Moore's pseudoinverse theory to the minimax estimation problem. It coincides with Kalman's filter for regular systems. A numerical example of the state estimation for 2D noncasual descriptor system is presented. Keywords: Kalman filtering, online state observer, guaranteed estimation, descriptor systems, singular systems, DAEs."}
{"category": "Math", "title": "Parabolic Iterated Function Systems with Applications to the Backward Continued Fractions", "abstract": "To the Renyi or backward continued fraction transformation we associate a parabolic iterated function system whose limit set has Hausdorff dimension 1. We show that the Texan Conjecture holds, i.e. for every t in1] there exists a subsystem whose limit set has Hausdorff dimension t"}
{"category": "Math", "title": "Random subgroups of Thompson's group $F$", "abstract": "We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of {\\em persistent} subgroups, whose isomorphism classes occur with positive density within the set of $k$-generator subgroups, for all sufficiently large $k$. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of $F$ are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite and not algebraic. We then use the asymptotic growth to prove our density results."}
{"category": "Math", "title": "Seifert Fibered Spaces: Notes for a course given in the Spring of 1993", "abstract": "Notes for a one semester course. The notes contain a description of compact three dimensional Seifert fibered spaces and a classification up to homeomorphism of compact three dimensional Seifert fibered spaces with non-empty boundary."}
{"category": "Math", "title": "Regular cell complexes in total positivity", "abstract": "This paper proves a conjecture of Fomin and Shapiro that their combinatorial model for any Bruhat interval is a regular CW complex which is homeomorphic to a ball. The model consists of a stratified space which may be regarded as the link of an open cell intersected with a larger closed cell, all within the totally nonnegative part of the unipotent radical of an algebraic group. A parametrization due to Lusztig turns out to have all the requisite features to provide the attaching maps. A key ingredient is a new, readily verifiable criterion for which finite CW complexes are regular involving an interplay of topology with combinatorics."}
{"category": "Math", "title": "Banach spaces without minimal subspaces", "abstract": "We prove three new dichotomies for Banach spaces \\`a la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size $\\aleph_1$ into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability."}
{"category": "Math", "title": "The generic isometry and measure preserving homeomorphism are conjugate to their powers", "abstract": "It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving homeomorphism is moreover conjugate to all of its powers. It follows that the generic measure preserving homeomorphism extends to an action of $(\\mathbb Q,+)$ by measure preserving homeomorphisms, and, in fact, to an action of the locally compact ring $\\mathfrak A$ of finite ad\\`eles. Similarly, S. Solecki has proved that there is a comeagre set of mutually conjugate isometries of the rational Urysohn metric space. We prove that these are all conjugate with their powers and therefore also embed into $\\mathbb Q$-actions. In fact, we extend these actions to actions of $\\mathfrak A$ as in the case of measure preserving homeomorphisms. We also consider a notion of topological similarity in Polish groups and use this to give simplified proofs of the meagreness of conjugacy classes in the automorphism group of the standard probability space and in the isometry group of the Urysohn metric space."}
{"category": "Math", "title": "On the Hausdorff Dimension of the Mather Quotient", "abstract": "Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of Hausdorff dimension. Then, we present applications in dynamics."}
{"category": "Math", "title": "Finitely approximable groups and actions Part I: The Ribes--Zalesski\\u\\i{} property", "abstract": "We investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces \\cite{solecki1} and obtain the following exact equivalence: any action of a discrete group $\\Gamma$ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of $\\Gamma$ is closed in the profinite topology on $\\Gamma$."}
{"category": "Math", "title": "Pattern recognition on random trees associated to protein functionality families", "abstract": "In this paper, we address the problem of identifying protein functionality using the information contained in its aminoacid sequence. We propose a method to define sequence similarity relationships that can be used as input for classification and clustering via well known metric based statistical methods. In our examples, we specifically address two problems of supervised and unsupervised learning in structural genomics via simple metric based techniques on the space of trees 1)Unsupervised detection of functionality families via K means clustering in the space of trees, 2)Classification of new proteins into known families via k nearest neighbour trees. We found evidence that the similarity measure induced by our approach concentrates information for discrimination. Classification has the same high performance than others VLMC approaches. Clustering is a harder task, though, but our approach for clustering is alignment free and automatic, and may lead to many interesting variations by choosing other clustering or classification procedures that are based on pre-computed similarity information, as the ones that performs clustering using flow simulation, see (Yona et al 2000, Enright et al, 2003)."}
{"category": "Math", "title": "An Indefinite Convection-Diffusion Operator With Real Spectrum", "abstract": "We confirm rigorously the conjecture, based on numerical and asymptotic evidence, that all the eigenvalues of a certain non-self-adjoint operator are real."}
{"category": "Math", "title": "Partition Polynomials: Asymptotics and Zeros", "abstract": "Let $F_n(x)$ be the partition polynomial $\\sum_{k=1}^n p_k(n) x^k$ where $p_k(n)$ is the number of partitions of $n$ with $k$ parts. We emphasize the computational experiments using degrees up to $70,000$ to discover the asymptotics of these polynomials. Surprisingly, the asymptotics of $F_n(x)$ have two scales of orders $n$ and $\\sqrt{n}$ and in three different regimes inside the unit disk. Consequently, the zeros converge to network of curves inside the unit disk given in terms of the dilogarithm."}
{"category": "Math", "title": "On weighted approximations in $D[0, 1]$ with applications to self-normalized partial sum processes", "abstract": "Let $X, X_1, X_2,...$ be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in $D[0, 1]$ for the partial sum processes $\\{S_{[nt]}, 0\\le t\\le 1\\}$, where $S_n=\\sum_{j=1}^nX_j$, under the assumption that $X$ belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes $\\{S_{[nt]}/V_n, 0\\le t\\le 1\\}$, where $V_n^2=\\sum_{j=1}^nX_j^2$. $L_p$ approximations of self-normalized partial sum processes are also discussed."}
{"category": "Math", "title": "Asymptotics of Studentized U-type processes for changepoint problems", "abstract": "This paper investigates weighted approximations for studentized $U$-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition $E|h(X_1,X_2)|^2 < \\infty$ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence."}
{"category": "Math", "title": "Leading coefficients of Kazhdan--Lusztig polynomials for Deodhar elements", "abstract": "We show that the leading coefficient of the Kazhdan--Lusztig polynomial $P_{x,w}(q)$ known as $\\mu(x,w)$ is always either 0 or 1 when $w$ is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance by Billey--Warrington (2001) and Billey--Jones (2007). In type $A$, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar's (1990) algorithm, we provide some combinatorial criteria to determine when $\\mu(x,w) = 1$ for such permutations $w$."}
{"category": "Math", "title": "Novikov-symplectic cohomology and exact Lagrangian embeddings", "abstract": "Let L be an exact Lagrangian submanifold inside the cotangent bundle of a closed manifold N. We prove that if N satisfies a mild homotopy assumption then the image of \\pi_2(L) in \\pi_2(N) has finite index. We make no assumption on the Maslov class of L, and we make no orientability assumptions. The homotopy assumption is either that N is simply connected, or more generally that \\pi_m(N) is finitely generated for each m \\geq 2. The result is proved by constructing the Novikov homology theory for symplectic cohomology and generalizing Viterbo's construction of a transfer map between the homologies of the free loopspaces of N and L."}
{"category": "Math", "title": "Polynomials associated with Partitions: Polynomials associated with Partitions: Their Asymptotics and Zeros", "abstract": "Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting behavior of their zeros as sets and densities."}
{"category": "Math", "title": "Boxicity of Halin Graphs", "abstract": "A k-dimensional box is the Cartesian product R_1 x R_2 x ... x R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G) is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K_4, then box(G)=2. In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then box(G)=2 unless G is isomorphic to K_4 (in which case its boxicity is 1)."}
{"category": "Math", "title": "Finite-dimensional Hopf C*-bimodules and C*-pseudo-multiplicative unitaries", "abstract": "Finite quantum groupoids can be described in many equivalent ways: In terms of the weak Hopf C*-algebras of B\\\"ohm, Nill, and Szlach\\'anyi or the finite-dimensional Hopf-von Neumann bimodules of Vallin, and in terms of finite-dimensional multiplicative partial isometries or the finite-dimensional pseudo-multiplicative unitaries of Vallin. In this short note, we show that in finite dimensions, the notions of a Hopf-von Neumann bimodule and of a pseudo-multiplicative unitary coincide with the notions of a concrete Hopf-C*-bimodule and of a C*-pseudo-multiplicative unitary, respectively."}
{"category": "Math", "title": "Composite Wavelet Transforms: Applications and Perspectives", "abstract": "We introduce a new concept of the so-called {\\it composite wavelet transforms}. These transforms are generated by two components, namely, a kernel function and a wavelet function (or a measure). The composite wavelet transforms and the relevant Calder\\'{o}n-type reproducing formulas constitute a unified approach to explicit inversion of the Riesz, Bessel, Flett, parabolic and some other operators of the potential type generated by ordinary (Euclidean) and generalized (Bessel) translations. This approach is exhibited in the paper. Another concern is application of the composite wavelet transforms to explicit inversion of the k-plane Radon transform on $\\bbr^n$. We also discuss in detail a series of open problems arising in wavelet analysis of $L_p$-functions of matrix argument."}
{"category": "Math", "title": "Spectrum of the Laplacian on manifolds with Spin(9) holonomy", "abstract": "We consider noncompact complete manifolds with Spin(9) holonomy and proved an one end result and a splitting type theorem under different conditions on the bottom of the spectrum. We proved that any harmonic functions with finite Dirichlet integral must be Cayley-harmonic, which allowed us to conclude an one end result. In the second part, we established a splitting type theorem by utilizing the Busemann function.."}
{"category": "Math", "title": "Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces", "abstract": "We consider weak convergence of the rescaled error processes arising from Riemann discretizations of certain stochastic integrals and relate the $L_p$-integrability of the weak limit to the fractional smoothness in the Malliavin sense of the stochastic integral."}
{"category": "Math", "title": "Ptolemy relations for punctured discs", "abstract": "We construct frieze patterns of type D_N with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram of type D_N, we show that the numbers in the pattern can be interpreted as specialisations of cluster variables in the corresponding Fomin-Zelevinsky cluster algebra."}
{"category": "Math", "title": "Instantaneous and lagged measurements of linear and nonlinear dependence between groups of multivariate time series: frequency decomposition", "abstract": "Measures of linear dependence (coherence) and nonlinear dependence (phase synchronization) between any number of multivariate time series are defined. The measures are expressed as the sum of lagged dependence and instantaneous dependence. The measures are non-negative, and take the value zero only when there is independence of the pertinent type. These measures are defined in the frequency domain and are applicable to stationary and non-stationary time series. These new results extend and refine significantly those presented in a previous technical report (Pascual-Marqui 2007, arXiv:0706.1776 [stat.ME], http://arxiv.org/abs/0706.1776), and have been largely motivated by the seminal paper on linear feedback by Geweke (1982 JASA 77:304-313). One important field of application is neurophysiology, where the time series consist of electric neuronal activity at several brain locations. Coherence and phase synchronization are interpreted as \"connectivity\" between locations. However, any measure of dependence is highly contaminated with an instantaneous, non-physiological contribution due to volume conduction and low spatial resolution. The new techniques remove this confounding factor considerably. Moreover, the measures of dependence can be applied to any number of brain areas jointly, i.e. distributed cortical networks, whose activity can be estimated with eLORETA (Pascual-Marqui 2007, arXiv:0710.3341 [math-ph])."}
{"category": "Math", "title": "(O(V+F), O(V)) is a Gelfand pair for any quadratic space V over a local field F", "abstract": "Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let $W=V \\oplus Fe$ with the form Q extending q with Q(e)=1. Consider the standard embedding of O(V) into O(W) and the two-sided action of $O(V)\\times O(V)$ on $O(W)$. In this note we show that any $O(V)\\times O(V)$-invariant distribution on O(W) is invariant with respect to transposition. This result was earlier proven in a bit different form in [vD] for F=R, in [AvD] for F=C and in [BvD] for p-adic fields. Here we give a different proof. Using results from [AGS], we show that this result on invariant distributions implies that the pair (O(V),O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Frechet representation E of O(W) we have $dim Hom_{O(V)}(E,C) \\leq 1.$ A stronger result for p-adic fields is obtained in [AGRS]."}
{"category": "Math", "title": "Radon transform on symmetric matrix domains", "abstract": "Let $\\bbK=\\mathbb R, \\mathbb C, \\mathbb H$ be the field of real, complex or quaternionic numbers and $M_{p, q}(\\bbK)$ the vector space of all $p\\times q$-matrices. Let $X$ be the matrix unit ball in $M_{n-r, r}(\\bbK)$ consisting of contractive matrices. As a symmetric space, $X=G/K=O(n-r, r)/O(n-r)\\times O(r)$, $U(n-r, r)/U(n-r)\\times U(r)$ and respectively $Sp(n-r, r)/Sp(n-r)\\times Sp(r)$. The matrix unit ball $y_0$ in $M_{r^\\prime-r, r}$ with $r^\\prime \\le n-1$ is a totally geodesic submanifold of $X$ and let $Y$ be the set of all $G$-translations of the submanifold $y_0$. The set $Y$ is then a manifold and an affine symmetric space. We consider the Radon transform $\\mathcal Rf(y)$ for functions $f\\in C_0^\\infty(X)$ defined by integration of $f$ over the subset $y$, and the dual transform $\\mathcal R^t F(x), x\\in X$ for functions $F(y)$ on $Y$. We find inversion formulas by constructing explicit certain invariant differential operators."}
{"category": "Math", "title": "On Weak Tail Domination of Random Vectors", "abstract": "Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular weak tail domination implies strong tail domination. In particular positive answer to Oleszkiewicz question would follow from the so-called Bernoulli conjecture."}
{"category": "Math", "title": "Degenerate principal series representations and their holomorphic extensions", "abstract": "Let $X=H/L$ be an irreducible real bounded symmetric domain realized as a real form in an Hermitian symmetric domain $D=G/K$. The intersection $S$ of the Shilov boundary of $D$ with $X$ defines a distinguished subset of the topological boundary of $X$ and is invariant under $H$ and can also be realized as $S=H/P$ for certain parabolic subgroup $P$ of $H$. We study the spherical representations $Ind_P^H(\\lam)$ of $H$ induced from $P$. We find formulas for the spherical functions in terms of the Macdonald ${}_2F_1$ hypergeometric function. This generalizes the earlier result of Faraut-Koranyi for Hermitian symmetric spaces $D$. We consider a class of $H$-invariant integral intertwining operators from the representations $Ind_P^H(\\lam)$ on $L^2(S)$ to the holomorphic representations of $G$ on $D$ restricted to $H$. We construct a new class of complementary series for the groups $H=SO(n, m)$, $SU(n, m)$ (with $n-m >2$) and $Sp(n, m)$ (with $n-m>1$). We realize them as a discrete component in the branching rule of the analytic continuation of the holomorphic discrete series of $G=SU(n, m)$, $SU(n, m)\\times SU(n, m)$ and $SU(2n, 2m)$ respectively."}
{"category": "Math", "title": "Osculating spaces and diophantine equations (with an appendix by Pietro Corvaja and Umberto Zannier)", "abstract": "This paper deals with some classical problems about the projective geometry of complex algebraic curves. We call \\textit{locally toric} a projective curve that in a neighbourhood of every point has a local analytical parametrization of type $(t^{a_1},...,t^{a_n})$, with $a_1,..., a_n$ relatively prime positive integers. In this paper we prove that the general tangent line to a locally toric curve in $\\bP^3$ meets the curve only at the point of tangency. This result extends and simplifies those of the paper \\cite{kaji} by H.Kaji where the same result is proven for any curve in $\\bP^3$ such that every branch is smooth. More generally, under mild hypotesis, up to a finite number of anomalous parametrizations $(t^{a_1},...,t^{a_n})$, the general osculating 2-space to a locally toric curve of genus $g<2$ in $\\bP^4$ does not meet the curve again. The arithmetic part of the proof of this result relies on the Appendix \\cite{cz:rk} to this paper. By means of the same methods we give some applications and we propose possible further developments."}
{"category": "Math", "title": "Rings graded equivalent to the Weyl algebra", "abstract": "We consider the first Weyl algebra, A, in the Euler gradation, and completely classify graded rings B that are graded equivalent to A: that is, the categories gr-A and gr-B are equivalent. This includes some surprising examples: in particular, we show that A is graded equivalent to an idealizer in a localization of A. We obtain this classification as an application of a general Morita-type characterization of equivalences of graded module categories."}
{"category": "Math", "title": "From Hopf to Neimark-Sacker bifurcation: a computational algorithm", "abstract": "We construct an algorithm for approximating the invariant tori created at a Neimark-Sacker bifurcation point. It is based on the same philosophy as many algorithms for approximating the periodic orbits created at a Hopf bifurcation point, i.e. a Fourier spectral method. For Neimark-Sacker bifurcation, however, we use a simple parametrisation of the tori in order to determine low-order approximations, and then utilise the information contained therein to develop a more general parametrisation suitable for computing higher-order approximations. Different algorithms, applicable to either autonomous or periodically-forced systems of differential equations, are obtained."}
{"category": "Math", "title": "Assouad-Nagata dimension of locally finite groups and asymptotic cones", "abstract": "In this work we study two problems about Assouad-Nagata dimension: 1) Is there a metric space of non zero Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes) 2) Suppose $G$ is a locally finite group with a proper left invariant metric $d_G$. If $\\dim_{AN}(G, d_G)>0$, is $\\dim_{AN} (G, d_G)$ infinite? (Brodskiy, Dydak and Lang) The first question is answered positively not only for general metric spaces but also for discrete groups with proper left invariant metrics. The second question has a negative solution. We show that for each $n$ there exists a locally finite group of Assouad-Nagata dimension $n$. A generalization to countable groups of arbitrary asymptotic dimension is given"}
{"category": "Math", "title": "Selective screenability and the Hurewicz property", "abstract": "We characterize the Hurewicz covering property in metrizable spaces in terms of properties of the metrics of the space. Then we show that a weak version of selective screenability, when combined with the Hurewicz property, implies selective screenability."}
{"category": "Math", "title": "Combinatorial polar orderings and recursively orderable arrangements", "abstract": "Polar orderings arose in recent work of Salvetti and the second author on minimal CW-complexes for complexified hyperplane arrangements. We study the combinatorics of these orderings in the classical framework of oriented matroids, and reach thereby a weakening of the conditions required to actually determine such orderings. A class of arrangements for which the construction of the minimal complex is particularly easy, called {\\em recursively orderable} arrangements, can therefore be combinatorially defined. We initiate the study of this class, giving a complete characterization in dimension 2 and proving that every supersolvable complexified arrangement is recursively orderable."}
{"category": "Math", "title": "Manin's conjecture on toric varieties with different heights", "abstract": "In this paper I verify Manin's conjecture for a class of rational projective toric varieties with a large class of heights other than the usual one that comes from the standard metric on projective space."}
{"category": "Math", "title": "Lawvere-Tierney sheaves in algebraic set theory", "abstract": "We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results."}
{"category": "Math", "title": "A Hofer-like metric on the group of symplectic diffeomorphisms", "abstract": "Using a \"Hodge decomposition\" of symplectic isotopies on a compact symplectic manifold $(M,\\omega)$, we construct a norm on the identity component in the group of all symplectic diffeomorphisms of $(M,\\omega)$ whose restriction to the group $Ham(M,\\omega)$ of hamiltonian diffeomorphisms is bounded from above by the Hofer norm. Moreover, $Ham(M,\\omega)$ is closed in $Symp(M,\\omega)$ equipped with the topology induced by the extended norm. We give an application to the $C^0$ symplectic topology. We also discuss extensions of Oh's spectral distance."}
{"category": "Math", "title": "Group Extensions and Automorphism Group Rings", "abstract": "The group ring of the automorphism group of a p-group is studied using the automorphism groups of subgroups and quotient groups of P."}
{"category": "Math", "title": "A bound for the \"torsion conductor\" of a non-CM elliptic curve", "abstract": "Given a non-CM elliptic curve E over Q, define the ``torsion conductor'' m_E to be the smallest positive integer so that the Galois representation on the torsion of E has image Pi^{-1}(Gal(Q(E[m_E])/Q), where Pi denotes the natural projection GL_2(\\hat{Z}) onto GL_2(Z/m_E Z). We show that, uniformly for semi-stable non-CM elliptic curves E over Q, m_E is less than a constant times the 5th power of the conductor of E."}
{"category": "Math", "title": "Weak pseudoconcavity and the maximum modulus principle", "abstract": "We discuss the maximum modulus principle, and weak unique continuation, for CR functions on an abstract almost CR manifold M. We investigate these matters under the assumption of weak pseudoconcavity, and obtain sharp results about propagation along Sussmann leaves."}
{"category": "Math", "title": "Homotopy methods for counting reaction network equilibria", "abstract": "Dynamical system models of complex biochemical reaction networks are usually high-dimensional, nonlinear, and contain many unknown parameters. In some cases the reaction network structure dictates that positive equilibria must be unique for all values of the parameters in the model. In other cases multiple equilibria exist if and only if special relationships between these parameters are satisfied. We describe methods based on homotopy invariance of degree which allow us to determine the number of equilibria for complex biochemical reaction networks and how this number depends on parameters in the model."}
{"category": "Math", "title": "The Hecke group algebra of a Coxeter group and its representation theory", "abstract": "Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, ...). In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. It turns out to be closely related to the monoid algebras of respectively nondecreasing functions and nondecreasing parking functions, the representation theory of which we describe as well. This defines three towers of algebras, and we give explicitly the Grothendieck algebras and coalgebras given respectively by their induction products and their restriction coproducts. This yields some new interpretations of the classical bases of quasi-symmetric and noncommutative symmetric functions as well as some new bases."}
{"category": "Math", "title": "Holomorphic dynamics, Painlev\\'e VI equation and Character Varieties", "abstract": "We study the monodromy of Painlev\\'e VI equation from a dynamical point of view. This is applied to the description of bounded orbits, and to a proof of the irreducibility of Painlev\\'e VI equation in the sens of Casale and Malgrange. On our way, we compute the entropy of each element of the monodromy group, and we precise the dictionary between character varieties and Painlev\\'e equations."}
{"category": "Math", "title": "Strong rigidity of constant curvature Finsler manifolds", "abstract": "Here, an extension of the Obata-Tanno's theorem to Finsler geometry is established and the following rigidity result is obtained; Every complete connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to an $n$-sphere equipped with a certain Finsler metric, and vise versa."}
{"category": "Math", "title": "Nombres de Bernoulli et une formule de Ramanujan", "abstract": "In the first part we establish a connection between the Euler-Maclaurin summation formula and the Rota-Baxter functional equation. In the second part we give a simple proof of a formula, due to Ramanujan, on the summation of certain exponential series."}
{"category": "Math", "title": "Discrete Kakeya-type problems and small bases", "abstract": "A subset U of a group G is called k-universal if U contains a translate of every k-element subset of G. We give several nearly optimal constructions of small k-universal sets, and use them to resolve an old question of Erdos and Newman on bases for sets of integers, and to obtain several extensions for other groups."}
{"category": "Math", "title": "In Ehresmann's footsteps: from Group Geometries to Groupoid Geometries", "abstract": "For a smooth (locally trivial) principal bundle in Ehresmann's sense, the relation between the commuting vertical and horizontal actions of the structural Lie group and the structural Lie groupoid (isomorphisms between vertical fibers) is regarded as a special case of a symmetrical concept of conjugation between \"principal\" Lie groupoid actions, allowing possibly non-locally trivial bundles. A diagrammatic description of this concept via a symmetric \"butterfly diagram\" allows its \"internalization\" in a wide class of categories (used by \"working mathematicians\") whenever they are endowed with two distinguished classes of monomorphisms and epimorphisms mimicking the properties of embeddings and surjective submersions. As an application, a general theorem of \"universal activation\" encompasses in a unified way such various situations as Palais' theory of globalization for partial action laws, the realization of non-abelian cocycles (including Haefliger cocycles for foliations) or the description of the \"homogeneous space\" attached to an embedding of Lie groups (still valid for Lie groupoids)."}
{"category": "Math", "title": "A conjugate prior for discrete hierarchical log-linear models", "abstract": "In Bayesian analysis of multi-way contingency tables, the selection of a prior distribution for either the log-linear parameters or the cell probabilities parameters is a major challenge. In this paper, we define a flexible family of conjugate priors for the wide class of discrete hierarchical log-linear models, which includes the class of graphical models. These priors are defined as the Diaconis--Ylvisaker conjugate priors on the log-linear parameters subject to \"baseline constraints\" under multinomial sampling. We also derive the induced prior on the cell probabilities and show that the induced prior is a generalization of the hyper Dirichlet prior. We show that this prior has several desirable properties and illustrate its usefulness by identifying the most probable decomposable, graphical and hierarchical log-linear models for a six-way contingency table."}
{"category": "Math", "title": "Sums of dilates", "abstract": "The lambda-dilate of a set A is lambda*A={lambda a : a \\in A}. We give an asymptotically sharp lower bound on the size of sumsets of the form lambda_1*A+...+lambda_k*A for arbitrary integers lambda_1,...,lambda_k and integer sets A. We also establish an upper bound for such sums, which is similar to, but often stronger than Plunnecke's inequality."}
{"category": "Math", "title": "Enhancing Sparsity by Reweighted L1 Minimization", "abstract": "It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained L1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms L1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted L1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations--not by reweighting the L1 norm of the coefficient sequence as is common, but by reweighting the L1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as compressed sensing."}
{"category": "Math", "title": "Homomorphisms of abelian varieties over finite fields", "abstract": "These are notes of my lectures at the summer school \"Higher-dimensional geometry over finite fields\" in Goettingen, June--July 2007. We present a proof of Tate's theorem on homomorphisms of abelian varieties over finite fields (including the $\\ell=p$ case) that is based on a \"quaternion trick\". In fact, a a slightly stronger version of those theorems with \"finite coefficients\" is proven."}
{"category": "Math", "title": "Arens Regularity of Module Actions and the Second Adjoit of a Derivation", "abstract": "In this paper, first we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions and then we investigate those conditions under which the second adjoint of a derivation into a dual Banach module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included."}
{"category": "Math", "title": "The weighted fusion category algebra and the q-Schur algebra for \\mathrm{GL}_2(q)", "abstract": "We show that the weighted fusion category algebra of the principal 2-block $b_0$ of $\\mathrm{GL}_2(q)$ is the quotient of the $q$-Schur algebra $\\mathcal{S}_2(q)$ by its socle, for $q$ an odd prime power. As a consequence, we get a canonical bijection between the set of isomorphism classes of simple $k\\mathrm{GL}_2(q)b_0$-modules and the set of conjugacy classes of $b_0$-weights in this case."}
{"category": "Math", "title": "Chevalley's ambiguous class number formula for an arbitrary torus", "abstract": "This version is a significant improvement of the original paper. It includes a new section where we discuss norm tori in some detail. The new abstract is the following: In this paper we obtain Chevalley's ambiguous class number formula for an arbitrary torus T over a global field. The classical formula of C.Chevalley may be recovered by setting T=G_{m} in our formula. As an illustration of the general result, we discuss norm tori in detail. A key ingredient of the proof of our main theorem is the work of X.Xarles on groups of components of Neron-Raynaud models of tori."}
{"category": "Math", "title": "Poincar\\'e's inequality and diffusive evolution equations", "abstract": "This paper addresses the question of change of decay rate from exponential to algebraic for diffusive evolution equations. We show how the behaviour of the spectrum of the Dirichlet Laplacian in the two cases yields the passage from exponential decay in bounded domains to algebraic decay or no decay at all in the case of unbounded domains. It is well known that such rates of decay exist: the purpose of this paper is to explain what makes the change in decay happen. We also discuss what kind of data is needed to obtain various decay rates."}
{"category": "Math", "title": "The Classification of Spun Torus Knots", "abstract": "S. Satoh has defined a construction to obtain a ribbon torus knot given a welded knot. This construction is known to be surjective. We show that it is not injective. Using the invariant of the peripheral structure, it is possible to provide a restriction on this failure of injectivity. In particular we also provide an algebraic classification of the construction when restricted to classical knots, where it is equivalent to the torus spinning construction."}
{"category": "Math", "title": "A Measurable-Group-Theoretic Solution to von Neumann's Problem", "abstract": "We give a positive answer, in the measurable-group-theory context, to von Neumann's problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors."}
{"category": "Math", "title": "Topological entropy and blocking cost for geodesics in riemannian manifolds", "abstract": "For a pair of points $x,y$ in a compact, riemannian manifold $M$ let $n_t(x,y)$ (resp. $s_t(x,y)$) be the number of geodesic segments with length $\\leq t$ joining these points (resp. the minimal number of point obstacles needed to block them). We study relationships between the growth rates of $n_t(x,y)$ and $s_t(x,y)$ as $t\\to\\infty$. We derive lower bounds on $s_t(x,y)$ in terms of the topological entropy $h(M)$ and its fundamental group. This strengthens the results of Burns-Gutkin \\cite{BG06} and Lafont-Schmidt \\cite{LS}. For instance, by \\cite{BG06,LS}, $h(M)>0$ implies that $s$ is unbounded; we show that $s$ grows exponentially, with the rate at least $h(M)/2$."}
{"category": "Math", "title": "Some comparison theorems in Finsler-Hadamard manifolds", "abstract": "We give upper and lower bounds for the ratio of the volume of metric ball to the area of the metric sphere in Finsler-Hadamard manifolds with pinched S-curvature. We apply these estimates to find the limit at the infinity for this ratio. Derived estimates are the generalization of the well-known result in Riemannian geometry. We also estimate the volume growth entropy for the balls in such manifolds."}
{"category": "Math", "title": "Diophantine Approximation of non-algebraic points on varieties II: Explicit estimates for arithmetic Hilbert Functions", "abstract": "Because of its ineffectiveness, the usual arithmetic Hilbert-Samuel formula is not applicable in the context of Diophantine Approximation. In order to overcome this difficulty, the present paper presents explicit estimates for arithmetic Hilbert Functions of closed subvarieties in projective space."}
{"category": "Math", "title": "On Optimal 4-Dimensional Metrics", "abstract": "We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-self-dual Riemannian metrics. The key new ingredient is a proof that the connected sum of five reverse-oriented complex projective planes admits such metrics."}
{"category": "Math", "title": "Length of parallel curves", "abstract": "We prove that the length difference between a closed periodic curve and its parallel curve at a sufficiently small distance is proportional to the rotation index. As an application, the rotation index of a curve could be estimated by means of Cauchy-Crofton formula."}
{"category": "Math", "title": "Quasiisometries between negatively curved Hadamard manifolds", "abstract": "Let X, Y be the universal covers of two compact Riemannian manifolds (with dimension not equal to 4) with negative sectional curvature. Then every quasiisometry between them lies at a finite distance from a bilipschitz homeomorphism."}
{"category": "Math", "title": "Fra\\\"iss\\'e sequences: category-theoretic approach to universal homogeneous structures", "abstract": "We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces."}
{"category": "Math", "title": "The decomposition of Global Conformal Invariants I: On a conjecture of Deser and Schwimmer", "abstract": "This is the first in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global confor- mal invariants\"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. In this paper we set up an iterative procedure that proves the decom- position. We then derive the iterative step in the first of two cases, subject to a purely algebraic result which is proven in [6, 7, 8]."}
{"category": "Math", "title": "Bounding Multiplicity by Shifts in the Taylor Resolution", "abstract": "A weaker form of the multiplicity conjecture of Herzog, Huneke, and Srinivasan is proven for two classes of monomial ideals: quadratic monomial ideals and squarefree monomial ideals with sufficiently many variables relative to the Krull dimension. It is also shown that tensor products, as well as Stanley-Reisner ideals of certain unions, satisfy the multiplicity conjecture if all the components do. Conditions under which the bounds are achieved are also studied."}
{"category": "Math", "title": "Minimum de Bruijn Sequence in a Language with Forbidden Substrings", "abstract": "Let be the following strategy to construct a walk in a labeled digraph: at each vertex, we follow the unvisited arc of minimum label. In this work we study for which languages, applying the previous strategy over the corresponding de Bruijn graph, we finish with an Eulerian cycle, in order to obtain the minimal de Bruijn sequence of the language."}
{"category": "Math", "title": "On dual quadri-algebras", "abstract": "This paper has been withdrawn"}
{"category": "Math", "title": "On the de Rham and p-adic realizations of the Elliptic Polylogarithm for CM elliptic curves", "abstract": "In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic analogue of the result of Beilinson and Levin expressing the complex elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. Our result is valid even if the elliptic curve has supersingular reduction at p."}
{"category": "Math", "title": "Cooperative Robot Control and Concurrent Synchronization of Lagrangian Systems", "abstract": "Concurrent synchronization is a regime where diverse groups of fully synchronized dynamic systems stably coexist. We study global exponential synchronization and concurrent synchronization in the context of Lagrangian systems control. In a network constructed by adding diffusive couplings to robot manipulators or mobile robots, a decentralized tracking control law globally exponentially synchronizes an arbitrary number of robots, and represents a generalization of the average consensus problem. Exact nonlinear stability guarantees and synchronization conditions are derived by contraction analysis. The proposed decentralized strategy is further extended to adaptive synchronization and partial-state coupling."}
{"category": "Math", "title": "Cutsets in infinite graphs", "abstract": "We answer three questions posed in a paper by Babson and Benjamini. They introduced a parameter $C_G$ for Cayley graphs $G$ that has significant application to percolation. For a minimal cutset of $G$ and a partition of this cutset into two classes, take the minimal distance between the two classes. The supremum of this number over all minimal cutsets and all partitions is $C_G$. We show that if it is finite for some Cayley graph of the group then it is finite for any (finitely generated) Cayley graph. Having an exponential bound for the number of minimal cutsets of size $n$ separating $o$ from infinity also turns out to be independent of the Cayley graph chosen. We show a 1-ended example (the lamplighter group), where $C_G$ is infinite. Finally, we give a new proof for a question of de la Harpe, proving that the number of $n$-element cutsets separating $o$ from infinity is finite unless $G$ is a finite extension of $Z$."}
{"category": "Math", "title": "Minimal tori with low nullity", "abstract": "The nullity of a minimal submanifold $M\\subset S^{n}$ is the dimension of the nullspace of the second variation of the area functional. That space contains as a subspace the effect of the group of rigid motions $SO(n+1)$ of the ambient space, modulo those motions which preserve $M$, whose dimension is the Killing nullity $kn(M)$ of $M$. In the case of 2-dimensional tori $M$ in $S^{3}$, there is an additional naturally-defined 2-dimensional subspace; the dimension of the sum of the action of the rigid motions and this space is the natural nullity $nnt(M)$. In this paper we will study minimal tori in $S^{3}$ with natural nullity less than 8. We construct minimal immersions of the plane $R^{2}$ in $S^{3}$ that contain all possible examples of tori with $nnt(M)<8$. We prove that the examples of Lawson and Hsiang with $kn(M)=5$ also have $nnt(M)=5$, and we prove that if the $nnt(M)\\le6$ then the group of isometries of $M$ is not trivial."}
{"category": "Math", "title": "Bondary-connectivity via graph theory", "abstract": "We generalize theorems of Kesten and Deuschel-Pisztora about the connectedness of the exterior boundary of a connected subset of $\\mathbb{Z}^d$, where \"connectedness\" and \"boundary\" are understood with respect to various graphs on the vertices of $\\mathbb{Z}^d$. We provide simple and elementary proofs of their results. It turns out that the proper way of viewing these questions is graph theory, instead of topology."}
{"category": "Math", "title": "Chern-Simons theory, analytic continuation and arithmetic", "abstract": "The purpose of the paper is to introduce some conjectures regarding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in a neighborhood of zero, and we postulate their arithmetic resurgence. By the latter term, we mean analytic continuation as a multivalued analytic function in the complex numbers minus a discrete set of points, with restricted singularities, local and global monodromy. We point out some key features of arithmetic resurgence in connection to various problems of asymptotic expansions of exact and perturbative Chern-Simons theory with compact or complex gauge group. Finally, we discuss theoretical and experimental evidence for our conjecture."}
{"category": "Math", "title": "Uniqueness Theorems for Meromorphic Mappings with Few Targets", "abstract": "The purpose of this article is to show uniqueness theorems for meromorphic mappings of C^m to CP^n with few hyperplanes H_j, j=1,...,q. It is well known that uniqueness theorems hold for q \\geq 3n+2. In this paper we show that for every nonnegative integer c there exists a positive integer N(c), depending only on c in an explicit way, such that uniqueness theorems hold if q\\geq (3n+2 -c) and n\\geq N(c). Furthermore, we also show that the coefficient of n in the formula of q can be replaced by a number which is strictly smaller than 3 for all n>>0. At the same time, a big number of recent uniqueness theorems are generalized considerably."}
{"category": "Math", "title": "Bers and H\\'enon, Painlev\\'e and Schroedinger", "abstract": "We study the dynamics of mapping class groups on 2-dimensional character varieties. We shall show that the dynamics of pseudo-Anosov mapping classes resembles in many ways the dynamics of H\\'enon mappings, and then apply this idea to answer open questions concerning the geometry of discrete and faithful representations, Painlev\\'e sixth equation, and discrete Schroedinger operators."}
{"category": "Math", "title": "Base change for semiorthogonal decompositions", "abstract": "Consider an algebraic variety $X$ over a base scheme $S$ and a faithful base change $T \\to S$. Given an admissible subcategory $\\CA$ in the bounded derived category of coherent sheaves on $X$, we construct an admissible subcategory in the bounded derived category of coherent sheaves on the fiber product $X\\times_S T$, called the base change of $\\CA$, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of the bounded derived category of $X$ is given then the base changes of its components form a semiorthogonal decomposition of the bounded derived category of the fiber product. As an intermediate step we construct a compatible system of semiorthogonal decompositions of the unbounded derived category of quasicoherent sheaves on $X$ and of the category of perfect complexes on $X$. As an application we prove that the projection functors of a semiorthogonal decomposition are kernel functors."}
{"category": "Math", "title": "Schwarz Reflection Principle, Boundary Regularity and Compactness for J-Complex Curves", "abstract": "We establish the Schwarz Reflection Principle for $J$-complex discs attached to a real analytic $J$-totally real submanifold of an almost complex manifold with real analytic $J$. We also prove the precise boundary regularity and derive the precise convergence in Gromov compactness theorem in $\\calc^{k,\\alpha}$-classes."}
{"category": "Math", "title": "Galois actions on models of curves", "abstract": "We study group actions on regular models of curves. If $X$ is a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, every tamely ramified extension $K'/K$ with Galois group $G$ induces a $G$-action on $X_{K'}$. In this paper we study the extension of this $G$-action to certain regular models of $X_{K'}$. In particular, we obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups of the structure sheaf of the special fiber of such a regular model. Inspired by this global study, we also consider similar questions for Galois actions on the exceptional locus of a tame cyclic quotient singularity. We apply these results to study a natural filtration of the special fiber of the N\\'eron model of the Jacobian of $X$ by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for $X$ over $\\Spec(R)$, and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber type for curves of genus 1 and 2."}
{"category": "Math", "title": "When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?", "abstract": "Given $\\{P_n \\}$ a sequence of monic orthogonal polynomials, we analyze their linear combinations $\\{Q_n \\}$with constant coefficients and fixed length $k+1$. Necessary and sufficient conditions are given for the orthogonality of the monic sequence $\\{Q_n \\}$ as well as an interesting interpretation in terms of the Jacobi matrices associated with $\\{P_n \\}$ and $\\{Q_n \\}$. Moreover, in the case $k=2$, we characterize the families $\\{P_n \\}$ such that the corresponding polynomials $\\{Q_n \\}$ are also orthogonal."}
{"category": "Math", "title": "Explicit matrices for Hecke operators on Siegel modular forms", "abstract": "We present an explicit set of matrices giving the action of the Hecke operators $T(p)$, $T_j(p^2)$ on Siegel modular forms."}
{"category": "Math", "title": "On small fractional parts of polynomials", "abstract": "We prove that for any real polynomial $f(x) \\in\\mathbb{R} [x]$ the set $$ \\{\\alpha \\in \\mathbb{R}: \\liminf_{n\\to \\infty} n\\log n ||\\alpha f(n)|| >0\\} $$ has positive Hausdorff dimension. Here $||\\xi ||$ means the distance from $\\xi $ to the nearest integer. Our result is based on an original method due to Y. Peres and W. Schlag."}
{"category": "Math", "title": "The Number of Different Effective Partitions and a Specific Goldbach Partition of Any Given Even Number Greater Than 6", "abstract": "A specific Goldbach partition of any given even number greater than 6 can be found definitely."}
{"category": "Math", "title": "The low-dimensional structures formed by tricategories", "abstract": "We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories."}
{"category": "Math", "title": "Second order differential operators having several families of orthogonal matrix polynomials as eigenfunctions", "abstract": "The aim of this paper is to bring into the picture a new phenomenon in the theory of orthogonal matrix polynomials satisfying second order differential equations. The last few years have witnessed some examples of a (fixed) family of orthogonal matrix polynomials whose elements are common eigenfunctions of several linearly independent second order differential operators. We show that the dual situation is also possible: there are examples of different families of matrix polynomials, each family orthogonal with respect to a different weight matrix, whose elements are eigenfunctions of a common second order differential operator. These examples are constructed by adding a discrete mass at certain point to a weight matrix: $\\widetilde{W}=W+\\delta_{t_0}M(t_0)$. Our method consists in showing how to choose the discrete mass $M(t_0)$, the point $t_0$ where the mass lives and the weight matrix $W$ so that the new weight matrix $\\widetilde{W}$ inherits some of the symmetric second order differential operators associated with $W$. It is well known that this situation is not possible for the classical scalar families of Hermite, Laguerre and Jacobi."}
{"category": "Math", "title": "Pictorial Representation for Antisymmetric Eigenfunctions of PS-3 Integral Equations", "abstract": "Eigenvalue problem for Poincare-Steklov-3 integral equation is reduced to the solution of three transcendential equations for three unknown numbers, moduli of pants. The complete list of antisymmetric eigenfunctions of integral equation in terms of Kleinian membranes is given."}
{"category": "Math", "title": "Constructible ideals", "abstract": "We introduce the concept of constructible ideal and we relate this concept with the notion of constructible simplicial complex. Several properties of constructible ideals are studied."}
{"category": "Math", "title": "Vanishing and Non-Vanishing Dirichlet Twists of L-Functions of Elliptic Curves", "abstract": "Let L(E/Q,s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L(E,1,\\chi) of the twisted L-function as \\chi ranges over Dirichlet characters of given order."}
{"category": "Math", "title": "Planar Contact Structures with Binding Number Three", "abstract": "In this article, we find the complete list of all contact structures (up to isotopy) on closed three-manifolds which are supported by an open book decomposition having planar pages with three (but not less) boundary components. We distinguish them by computing their first Chern classes and three dimensional invariants (whenever possible). Among these contact structures we also distinguish tight ones from those which are overtwisted."}
{"category": "Math", "title": "Renyi information for ergodic diffusion processes", "abstract": "In this paper we derive explicit formulas of the R\\'enyi information, Shannon entropy and Song measure for the invariant density of one dimensional ergodic diffusion processes. In particular, the diffusion models considered include the hyperbolic, the generalized inverse Gaussian, the Pearson, the exponential familiy and a new class of skew-$t$ diffusions."}
{"category": "Math", "title": "Arithmetic and Geometric Progressions in Productsets over Finite Fields", "abstract": "Given two sets $\\cA, \\cB \\subseteq \\F_q$ of elements of the finite field $\\F_q$ of $q$ elements, we show that the productset $$ \\cA\\cB = \\{ab | a \\in \\cA, b \\in\\cB\\} $$ contains an arithmetic progression of length $k \\ge 3$ provided that $k<p$, where $p$ is the characteristic of $\\F_q$, and $# \\cA # \\cB \\ge 3q^{2d-2/k}$. We also consider geometric progressions in a shifted productset $\\cA\\cB +h$, for $f \\in \\F_q$, and obtain a similar result."}
{"category": "Math", "title": "Spotlight Tiling", "abstract": "This article introduces spotlight tiling, a type of covering which is similar to tiling. The distinguishing aspects of spotlight tiling are that the \"tiles\" have elastic size, and that the order of placement is significant. Spotlight tilings are decompositions, or coverings, and can be considered dynamic as compared to typical static tiling methods. A thorough examination of spotlight tilings of rectangles is presented, including the distribution of such tilings according to size, and how the directions of the spotlights themselves are distributed. The spotlight tilings of several other regions are studied, and suggest that further analysis of spotlight tilings will continue to yield elegant results and enumerations."}
{"category": "Math", "title": "Extendable Cohomologies for Complex Analytic Varieties", "abstract": "We introduce a cohomology, called extendable cohomology, for abstract complex singular varieties based on suitable differential forms. Beside a study of the general properties of such a cohomology, we show that, given a complex vector bundle, one can compute its topological Chern classes using the extendable Chern classes, defined via a Chern-Weil type theory. We also prove that the localizations of the extendable Chern classes represent the localizations of the respective topological Chern classes, thus obtaining an abstract residue theorem for compact singular complex analytic varieties. As an application of our theory, we prove a Camacho-Sad type index theorem for holomorphic foliations of singular complex varieties."}
{"category": "Math", "title": "On the asymptotic behaviour of increasing positive self-similar Markov processes", "abstract": "We are interested by the rate of growth of increasing positive self-similar Markov processes (ipssMp) such that the subordinator associated to it via Lamperti's transformation has infinite mean. We prove that the logarithm of an ipssMp normalized by the logarithm of the time converges weakly, as the time tends to infinity, if and only if the Laplace exponent of the underlying subordinator is regularly varying at zero. Moreover, we prove that the regular variation at zero of the Laplace exponent is essentially nasc for the existence of a function that normalizes the logarithm of an ipssMp. We obtain a law of iterated logarithm for the liminf of the logarithm of an ipssMp and an integral test to study the upper envelope of it. Furthermore, results concerning the rate of growth of the random clock appearing in Lamperti's transformation are obtained."}
{"category": "Math", "title": "Number variance of random zeros on complex manifolds, II: smooth statistics", "abstract": "We consider the zero sets $Z_N$ of systems of $m$ random polynomials of degree $N$ in $m$ complex variables, and we give asymptotic formulas for the random variables given by summing a smooth test function over $Z_N$. Our asymptotic formulas show that the variances for these smooth statistics have the growth $N^{m-2}$. We also prove analogues for the integrals of smooth test forms over the subvarieties defined by $k<m$ random polynomials. Such linear statistics of random zero sets are smooth analogues of the random variables given by counting the number of zeros in an open set, which we proved elsewhere to have variances of order $N^{m-1/2}$. We use the variance asymptotics and off-diagonal estimates of Szego kernels to extend an asymptotic normality result of Sodin-Tsirelson to the case of smooth linear statistics for zero sets of codimension one in any dimension $m$."}
{"category": "Math", "title": "Branching properties for the groups G(de,e,r)", "abstract": "We study general properties of the restriction of the representations of the finite complex reflection groups $G(de,e,r+1)$ to their maximal parabolic subgroups of type $G(de,e,r)$, and focus notably on the multiplicity of components. In combinatorial terms, this amounts to the following question : which symmetries arise or disappear when one changes (exactly) one pearl in a combinatorial necklace ?"}
{"category": "Math", "title": "Homology of tropical varieties", "abstract": "Given a closed subvariety of an algebraic torus, the associated tropical variety is a polyhedral fan in the space of 1-parameter subgroups of the torus which describes the behaviour of the subvariety at infinity. We show that the link of the origin has only top rational homology if a genericity condition is satisfied. Our result is obtained using work of Tevelev and Deligne's theory of mixed Hodge structures."}
{"category": "Math", "title": "On the mu-bar invariant of rational surface singularities", "abstract": "We show that for rational surface singularities with odd determinant the mu-bar invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4-ball. We identify the mu-bar invariant with the corresponding correction term in Heegaard Floer theory."}
{"category": "Math", "title": "The Effros-Ruan conjecture for bilinear forms on C^*-algebras", "abstract": "In 1991 Effros and Ruan conjectured that a certain Grothendieck-type inequality for a bilinear form on C$^*$-algebras holds if (and only if) the bilinear form is jointly completely bounded. In 2002 Pisier and Shlyakhtenko proved that this inequality holds in the more general setting of operator spaces, provided that the operator spaces in question are exact. Moreover, they proved that the conjecture of Effros and Ruan holds for pairs of C$^*$-algebras, of which at least one is exact. In this paper we prove that the Effros-Ruan conjecture holds for general C$^*$-algebras, with constant one. More precisely, we show that for every jointly completely bounded (for short, j.c.b.) bilinear form on a pair of C$^*$-algebras $A$ and $B$, there exist states $f_1$, $f_2$ on $A$ and $g_1$, $g_2$ on $B$ such that for all $a\\in A$ and $b\\in B$, |u(a, b)| \\leq ||u||_{jcb}(f_1(aa^*)^{1/2}g_1(b^*b)^{1/2} + f_2(a^*a)^{1/2}g_2(bb^*)^{1/2}) . While the approach by Pisier and Shlyakhtenko relies on free probability techniques, our proof uses more classical operator algebra theory, namely, Tomita-Takesaki theory and special properties of the Powers factors of type III$_\\lambda$, $0< \\lambda< 1$ ."}
{"category": "Math", "title": "Doubles for monoidal categories", "abstract": "In a recent paper, Daisuke Tambara defined two-sided actions on an endomodule (= endodistributor) of a monoidal V-category A. When A is autonomous (= rigid = compact), he showed that the V-category (that we call Tamb(A)) of so-equipped endomodules (that we call Tambara modules) is equivalent to the monoidal centre Z[A,V] of the convolution monoidal V-category [A,V]. Our paper extends these ideas somewhat. For general A, we construct a promonoidal V-category DA (which we suggest should be called the double of A) with an equivalence [DA,V] \\simeq Tamb(A). When A is closed, we define strong (respectively, left strong) Tambara modules and show that these constitute a V-category Tamb_s(A) (respectively, Tamb_{ls}(A)) which is equivalent to the centre (respectively, lax centre) of [A,V]. We construct localizations D_s A and D_{ls} A of DA such that there are equivalences Tamb_s(A) \\simeq [D_s A,V] and Tamb_{ls}(A) \\simeq [D_{ls} A,V]. When A is autonomous, every Tambara module is strong; this implies an equivalence Z[A,V] \\simeq [DA,V]."}
{"category": "Math", "title": "New $L_p$ Affine Isoperimetric Inequalities", "abstract": "We prove new $L_p$ affine isoperimetric inequalities for all $ p \\in [-\\infty,1)$. We establish, for all $p\\neq -n$, a duality formula which shows that $L_p$ affine surface area of a convex body $K$ equals $L_\\frac{n^2}{p}$ affine surface area of the polar body $K^\\circ$."}
{"category": "Math", "title": "Musical Actions of Dihedral Groups", "abstract": "The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles."}
{"category": "Math", "title": "Dated ancestral trees from binary trait data and its application to the diversification of languages", "abstract": "Binary trait data record the presence or absence of distinguishing traits in individuals. We treat the problem of estimating ancestral trees with time depth from binary trait data. Simple analysis of such data is problematic. Each homology class of traits has a unique birth event on the tree, and the birth event of a trait visible at the leaves is biased towards the leaves. We propose a model-based analysis of such data, and present an MCMC algorithm that can sample from the resulting posterior distribution. Our model is based on using a birth-death process for the evolution of the elements of sets of traits. Our analysis correctly accounts for the removal of singleton traits, which are commonly discarded in real data sets. We illustrate Bayesian inference for two binary-trait data sets which arise in historical linguistics. The Bayesian approach allows for the incorporation of information from ancestral languages. The marginal prior distribution of the root time is uniform. We present a thorough analysis of the robustness of our results to model mispecification, through analysis of predictive distributions for external data, and fitting data simulated under alternative observation models. The reconstructed ages of tree nodes are relatively robust, whilst posterior probabilities for topology are not reliable."}
{"category": "Math", "title": "Duality of Schramm-Loewner Evolutions", "abstract": "In this note, we prove a version of the conjectured duality of Schramm-Loewner Evolutions, by establishing exact identities in distribution between some boundary arcs of chordal $\\SLE_\\kappa$, $\\kappa>4$, and appropriate versions of $\\SLE_{\\hat\\kappa}$, $\\hat\\kappa=16/\\kappa$."}
{"category": "Math", "title": "A Class of Infinite Dimensional Diffusion Processes with Connection to Population Genetics", "abstract": "Starting from a sequence of independent Wright-Fisher diffusion processes on $[0,1]$, we construct a class of reversible infinite dimensional diffusion processes on $\\DD_\\infty:= \\{{\\bf x}\\in [0,1]^\\N: \\sum_{i\\ge 1} x_i=1\\}$ with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence to the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space $S$. This provides a reasonable alternative to the Fleming-Viot process which does not satisfy the log-Sobolev inequality when $S$ is infinite as observed by W. Stannat \\cite{S}."}
{"category": "Math", "title": "Growth of the Number of Spanning Trees of the Erd\\\"os-R\\'enyi Giant Component", "abstract": "The number of spanning trees in the giant component of the random graph $\\G(n, c/n)$ ($c>1$) grows like $\\exp\\big\\{m\\big(f(c)+o(1)\\big)\\big\\}$ as $n\\to\\infty$, where $m$ is the number of vertices in the giant component. The function $f$ is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on $f'(c)$. A key lemma is the following. Let $\\PGW(\\lambda)$ denote a Galton-Watson tree having Poisson offspring distribution with parameter $\\lambda$. Suppose that $\\lambda^*>\\lambda>1$. We show that $\\PGW(\\lambda^*)$ conditioned to survive forever stochastically dominates $\\PGW(\\lambda)$ conditioned to survive forever."}
{"category": "Math", "title": "Spherical Means in Odd Dimensions and EPD equations", "abstract": "The paper contains a simple proof of the Finch-Patch-Rakesh inversion formula for the spherical mean Radon transform in odd dimensions. This transform arises in thermoacoustic tomography. Applications are given to the Cauchy problem for the Euler-Poisson-Darboux equation with initial data on the cylindrical surface. The argument relies on the idea of analytic continuation and known properties of Erdelyi-Kober fractional integrals."}
{"category": "Math", "title": "Dynamics with choice", "abstract": "Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evolution operator at every time step there is a choice of operators to transform the current state of the system. This notion is new and interesting from the mathematical point of view. At the same time, many real life processes studied in chemical physics, engineering, biology and medicine, from autocatalytic reaction systems to switched systems to cellular biochemical processes to malaria transmission in urban environments, exhibit the properties described by dynamics with choice. We study the long-term behavior in dynamics with choice."}
{"category": "Math", "title": "Smooth parametrized torsion -- a manifold approach", "abstract": "We give a construction of a torsion invariant of bundles of smooth manifolds which is based on the work of Dwyer, Weiss and Williams on smooth structures on fibrations."}
{"category": "Math", "title": "A Local time correspondence for stochastic partial differential equations", "abstract": "It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have random-field solutions only in spatial dimension one. Here we show that in many cases, where the ``spatial operator'' is the L^2-generator of a L\\'evy process X, a linear SPDE has a random-field solution if and only if the symmetrization of X possesses local times. This result gives a probabilistic reason for the lack of existence of random-field solutions in dimensions strictly bigger than one. In addition, we prove that the solution to the SPDE is [H\\\"older] continuous in its spatial variable if and only if the said local time is [H\\\"older] continuous in its spatial variable. We also produce examples where the random-field solution exists, but is almost surely unbounded in every open subset of space-time. Our results are based on first establishing a quasi-isometry between the linear L^2-space of the weak solutions of a family of linear SPDEs, on one hand, and the Dirichlet space generated by the symmetrization of X, on the other hand. We study mainly linear equations in order to present the local-time correspondence at a modest technical level. However, some of our work has consequences for nonlinear SPDEs as well. We demonstrate this assertion by studying a family of parabolic SPDEs that have additive nonlinearities. For those equations we prove that if the linearized problem has a random-field solution, then so does the nonlinear SPDE. Moreover, the solution to the linearized equation is [H\\\"older] continuous if and only if the solution to the nonlinear equation is. And the solutions are bounded and unbounded together as well. Finally, we prove that in the cases that the solutions are unbounded, they almost surely blow up at exactly the same points."}
{"category": "Math", "title": "The Residual Information Criterion, Corrected", "abstract": "Shi and Tsai (JRSSB, 2002) proposed an interesting residual information criterion (RIC) for model selection in regression. Their RIC was motivated by the principle of minimizing the Kullback-Leibler discrepancy between the residual likelihoods of the true and candidate model. We show, however, under this principle, RIC would always choose the full (saturated) model. The residual likelihood therefore, is not appropriate as a discrepancy measure in defining information criterion. We explain why it is so and provide a corrected residual information criterion as a remedy."}
{"category": "Math", "title": "Duality of Anderson T-motives", "abstract": "Let $M$ be a T-motive. We introduce the notion of duality for $M$. Main results of the paper (we consider uniformizable $M$ over $F_q[T]$ of rank $r$, dimension $n$, whose nilpotent operator $N$ is 0): 1. Algebraic duality implies analytic duality (Theorem 5). Explicitly, this means that the lattice of the dual of $M$ is the dual of the lattice of $M$, i.e. the transposed of a Siegel matrix of $M$ is a Siegel matrix of the dual of $M$. 2. Let $n=r-1$. There is a 1 -- 1 correspondence between pure T-motives (all they are uniformizable), and lattices of rank $r$ in $C^n$ having dual (Corollary 8.4)."}
{"category": "Math", "title": "Bootstrap Confidence Regions for Optimal Operating Conditions in Response Surface Methodology", "abstract": "This article concerns the application of bootstrap methodology to construct a likelihood-based confidence region for operating conditions associated with the maximum of a response surface constrained to a specified region. Unlike classical methods based on the stationary point, proper interpretation of this confidence region does not depend on unknown model parameters. In addition, the methodology does not require the assumption of normally distributed errors. The approach is demonstrated for concave-down and saddle system cases in two dimensions. Simulation studies were performed to assess the coverage probability of these regions."}
{"category": "Math", "title": "Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schroedinger equations", "abstract": "This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy-Littlewood-Sobolev (HLS) inequalities. In the same way we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schroedinger equation. The proof of the weighted HLS inequality for radially symmetric functions involves an application of the weighted inequality due to Stein and Weiss and the Hardy-Littlewood maximal inequality in the weighted Lebesgue space due to Muckenhoupt. Under radial symmetry we get significant gains over the usual HLS inequality and Strichartz estimate."}
{"category": "Math", "title": "Exponential sums and rank of double persymmetric matrices over F_2", "abstract": "We obtain, using exponential quadratic sums, explicit expressions for the number of double persymmetric matrices with entries in F_2 of given rank. (A matix [a(i,j)) is persymmetric if a(i,j) = a(r,s) for i+j = r+s)"}
{"category": "Math", "title": "The discrepancy of a needle on a checkerboard", "abstract": "Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. We show that for any such coloring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their black length, in absolute value, is at least the square root of their length, up to a multiplicative constant. For the corresponding ``finite'' problem ($N \\times N$ checkerboard) we also prove that we can color it in such a way that the above quantity is at most $C \\sqrt{N \\log N}$, for any placement of the line segment."}
{"category": "Math", "title": "Nonlinear Schroedinger equations with radially symmetric data of critical regularity", "abstract": "This paper is concerned with the global existence of small solutions to pure-power nonlinear Schroedinger equations subject to radially symmetric data with critical regularity. Under radial symmetry we focus our attention on the case where the power of nonlinearity is somewhat smaller than the pseudoconformal power and the initial data belong to the scale-invariant homogeneous Sobolev space. In spite of the negative-order differentiability of initial data the nonlinear Schroedinger equation has global in time solutions provided that the initial data have the small norm. The key ingredient in the proof of this result is an effective use of global weighted smoothing estimates specific to radially symmetric solutions."}
{"category": "Math", "title": "More constructing pairing-friendly elliptic curves for cryptography", "abstract": "The problem of constructing elliptic curves suitable for pairing applications has received a lot of attention. To solve this, we propose a variant algorithm of a known method by Brezing and Weng. We produce new families of parameters using our algorithm for pairing-friendly elliptic curves of embedding degree 8, and we actually compute some explicit curves as numerical examples."}
{"category": "Math", "title": "LULU operators for functions of continuous argument", "abstract": "The LULU operators, well known in the nonlinear multiresolution analysis of sequences, are extended to functions defined on a continuous domain, namely, a real interval. We show that the extended operators replicate the essential properties of their discrete counterparts. More precisely, they form a fully ordered semi-group of four elements, preserve the local trend and the total variation."}
{"category": "Math", "title": "Batalin-Vilkovisky algebra structures on Hochschild Cohomology", "abstract": "Let $M$ be any compact simply-connected $d$-dimensional smooth manifold and let $\\mathbb{F}$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$, $HH^*(S^*(M);S^*(M))$, extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjecturated to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$, $H_{*+d}(LM)$ introduced by Chas and Sullivan. We also show that the negative cyclic cohomology $HC^*_-(S^*(M))$ has a Lie bracket. Such Lie bracket is expected to coincide with the Chas-Sullivan string bracket on the equivariant homology $H_*^{S^1}(LM)$."}
{"category": "Math", "title": "Addendum to ``Canonical bases for quantum generalized Kac-Moody algebras''", "abstract": "We provide some necessary details to several arguments appearing in our previous paper ``Canonical bases for quantum generalized Kac-Moody algebras''. We also make the link with some other work on the same subject."}
{"category": "Math", "title": "Cohomology algebra of plane curves, weak combinatorial type, and formality", "abstract": "We determine an explicit presentation by generators and relations of the cohomology algebra $H^*(\\mathbb P^2\\setminus C,\\mathbb C)$ of the complement to an algebraic curve $C$ in the complex projective plane $\\mathbb P^2$, via the study of log-resolution logarithmic forms on $\\mathbb P^2$. As a first consequence, we derive that $H^*(\\mathbb P^2\\setminus C,\\mathbb C)$ depends only on the following finite pieces of data: the number of irreducible components of $C$ together with their degrees and genera, the number of local branches of each component at each singular point, and the intersection numbers of every two distinct local branches at each singular point of $C$. This finite set of data is referred to as the weak combinatorial type of $C$. A further corollary is that the twisted cohomology jumping loci of $H^*(\\mathbb P^2\\setminus C,\\mathbb C)$ containing the trivial character also depend on the weak combinatorial type of $C$. Finally, the explicit construction of the generators and relations allows us to prove that complements of plane projective curves are formal spaces in the sense of Sullivan."}
{"category": "Math", "title": "Irreducibility of the symmetric Yagzhev's maps", "abstract": "Let $F:\\Cn \\to \\Cn$ be a polynomial mapping in Yagzhev's form,i.e. $$F(x_1,\\ld,x_n)=(x_1+H_1(x_1,\\ld,x_n),\\ld,x_n+H_n(x_1,\\ld,x_n)),$$ where $H_i$ are homogenous polynomials of degree 3. In this paper we show that if $\\Jac(F) \\in \\mathbb{C}^*$ and the Jacobian matrix of $F$ is symmetric, then all the polynomials $x_i+H_i(x_1,\\ld,x_n)$ are irreducible as elements of the ring $\\mathbb{C}[x_1,\\ld,x_n]$."}
{"category": "Math", "title": "Good Banach spaces for piecewise hyperbolic maps via interpolation", "abstract": "We introduce a weak transversality condition for piecewise C^{1+\\alpha} and piecewise hyperbolic maps which admit a C^{1+\\alpha} stable distribution. We show good bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel-Lizorkin type. In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) and applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results."}
{"category": "Math", "title": "Tropical complete intersection curves", "abstract": "A tropical complete intersection curve C in R^(n+1) is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the genus of C (defined as the number of independent cycles of C) when C is smooth and connected."}
{"category": "Math", "title": "Decomposition of neuronal assembly activity via empirical de-Poissonization", "abstract": "Consider a compound Poisson process with jump measure $\\nu$ supported by finitely many positive integers. We propose a method for estimating $\\nu$ from a single, equidistantly sampled trajectory and develop associated statistical procedures. The problem is motivated by the question whether nerve cells in the brain exhibit higher-order interactions in their firing patterns. According to the neuronal assembly hypothesis (Hebb [13]), synchronization of action potentials across neurons of different groups is considered a signature of assembly activity, but it was found notoriously difficult to demonstrate it in recordings of neuronal activity. Our approach based on a compound Poisson model allows to detect the presence of joint spike events of any order using only population spike count samples, thus bypassing both the ``curse of dimensionality'' and the need to isolate single-neuron spike trains in population signals."}
{"category": "Math", "title": "On the infinitesimal rigidity of polyhedra with vertices in convex position", "abstract": "Let $P \\subset \\R^3$ be a polyhedron. It was conjectured that if $P$ is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. $P$ can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assumption of codecomposability. The proof relies on a result of independent interest concerning the Hilbert-Einstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In particular, if there are no interior vertices, then the Hessian is negative definite."}
{"category": "Math", "title": "Circular spectrum and bounded solutions of periodic evolution equations", "abstract": "We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form $u'=A(t)u+\\epsilon H(t,u)+f(t)$, where $A(t)$ is, in general, an unbounded operator depending 1-periodically on $t$, $H$ is 1-periodic in $t$, $\\epsilon$ is small, and $f$ is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of \"circular spectrum\" and then apply it to study the linear equations $u'=A(t)u+f(t)$ with general conditions on $f$. For small $\\epsilon$ we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of $f$, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of $f$."}
{"category": "Math", "title": "Analytic vectors in continuous p-adic representations", "abstract": "Given a compact p-adic Lie group G over a finite unramified extension L/Q_p let G_0 be the product over all Galois conjugates of G. We construct an exact and faithful functor from admissible G-Banach space representations to admissible locally L-analytic G_0-representations that coincides with passage to analytic vectors in case L=Q_p. On the other hand, we study the functor \"passage to analytic vectors\" and its derived functors over general basefields. As an application we determine the higher analytic vectors in certain locally analytic induced representations."}
{"category": "Math", "title": "A Constrained Nevanlinna-Pick Interpolation Problem", "abstract": "We obtain necessary and sufficient conditions for Nevanlinna-Pick interpolation on the unit disk with the additional restriction that all analytic interpolating functions satisfy $f'(0)=0.$ Alternatively, these results can be interpreted as interpolation results for $H^{\\infty}(V),$ where $V$ is the intersection of the bidisk with an algebraic variety. We use an analysis of C*-envelopes to show that these same conditions do not suffice for matrix interpolation."}
{"category": "Math", "title": "Solvmanifolds and noncommutative tori with real multiplication", "abstract": "We prove that the Shimizu L-function of a real quadratic field is obtained from a (Lorentzian) spectral triple on a noncommutative torus with real multiplication, as an adiabatic limit of the Dirac operator on a 3-dimensional solvmanifold. The Dirac operator on this 3-dimensional geometry gives, via the Connes-Landi isospectral deformations, a spectral triple for the noncommutative tori obtained by deforming the fiber tori to noncommutative spaces. The 3-dimensional solvmanifold is the homotopy quotient in the sense of Baum--Connes of the noncommutative space obtained as the crossed product of the noncommutative torus by the action of the units of the real quadratic field. This noncommutative space is identified with the twisted group C*-algebra of the fundamental group of the 3-manifold. The twisting can be interpreted as the cocycle arising from a magnetic field, as in the theory of the quantum Hall effect. We prove a twisted index theorem that computes the range of the trace on the K-theory of this noncommutative space and gives an estimate on the gaps in the spectrum of the associated Harper operator."}
{"category": "Math", "title": "Splitting for Rare Event Simulation: A Large Deviation Approach to Design and Analysis", "abstract": "Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set $B$ before another set $A$, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented."}
{"category": "Math", "title": "Poisson and symplectic functions in Lie algebroid theory", "abstract": "Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between non-degenerate Poisson functions and symplectic functions. We determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector."}
{"category": "Math", "title": "AP Theory II:Intrinsic 4D Quantum YM Theory with Mass Gap", "abstract": "We describe a sub-theory of Artin Presentation Theory (AP Theory), which has many genuine,discrete,group-theoretic,non-infinitesimal, qualitative analogues (including with the mass gap) of the main desiderata of the hypothetical quantitative infinitesimal '4D Quantum YM Theory' for the so-called Clay Millenium 'YM Existence and Mass Gap' problem. Our entirely mathematically rigorous theory is not a model, no new axioms or measures are introduced,does not rely on SUSY,is free of smooth 4D singularities,moduli spaces,path integrals, graph/lattice combinatorics and probabilistic,category,twistor or topos arguments and is intimately related to the theory of pure framed braids. Despite being based on a rigorous, radical,universal Holographic Principle,the theory still contains an analogue of Donaldson/Seiberg-Witten Theory, an infinitely generated, at each stage, graded group of topology-changing transitions and interactions and more. Our main contention is: the radical,universal AP-holography, with its strong topology changing interactions, which reach all the way to the 'vacuum' of discrete group theory, may destroy any infinitesimal, PDE based Field Theory,required for solving the actual Clay YM problem in its present quantitative form as a problem of so-called 'constructive' 3+1 QFT. More generally,due to the fact that AP Theory is not a model,e.g., does not introduce any new axioms,any rigorous axiomatic 3+1 QFT has to align itself with it in a mathematically rigorous fashion."}
{"category": "Math", "title": "Column and row operator spaces over QSL_p-spaces and their use in abstract harmonic analysis", "abstract": "The notions of column and row operator space were extended by A. Lambert from Hilbert spaces to general Banach spaces. In this paper, we use column and row spaces over quotients of subspaces of general $L_p$-spaces to equip several Banach algebras occurring naturally in abstract harmonic analysis with canonical, yet not obvious operator space structures that turn them into completely bounded Banach algebras. We use these operator space structures to gain new insights on those algebras."}
{"category": "Math", "title": "A class of residual distribution schemes and their relation to relaxation systems", "abstract": "Residual distributions (RD) schemes are a class of of high-resolution finite volume methods for unstructured grids. A key feature of these schemes is that they make use of genuinely multidimensional (approximate) Riemann solvers as opposed to the piecemeal 1D Riemann solvers usually employed by finite volume methods. In 1D, LeVeque and Pelanti [J. Comp. Phys. 172, 572 (2001)] showed that many of the standard approximate Riemann solver methods (e.g., the Roe solver, HLL, Lax-Friedrichs) can be obtained from applying an exact Riemann solver to relaxation systems of the type introduced by Jin and Xin [Comm. Pure Appl. Math. 48, 235 (1995)]. In this work we extend LeVeque and Pelanti's results and obtain a multidimensional relaxation system from which multidimensional approximate Riemann solvers can be obtained. In particular, we show that with one choice of parameters the relaxation system yields the standard N-scheme. With another choice, the relaxation system yields a new Riemann solver, which can be viewed as a genuinely multidimensional extension of the local Lax-Friedrichs scheme. This new Riemann solver does not require the use Roe-Struijs-Deconinck averages, nor does it require the inversion of an m-by-m matrix in each computational grid cell, where $m$ is the number of conserved variables. Once this new scheme is established, we apply it on a few standard cases for the 2D compressible Euler equations of gas dynamics. We show that through the use of linear-preserving limiters, the new approach produces numerical solutions that are comparable in accuracy to the N-scheme, despite being computationally less expensive."}
{"category": "Math", "title": "New constructions of Yang-Baxter systems", "abstract": "The quantum Yang-Baxter equation admits generalisations to systems of Yang-Baxter type equations called Yang-Baxter systems. Starting from algebra structures, we propose new constructions of some constant as well as the spectral-parameter dependent Yang-Baxter systems. Besides, we also present explicitly the commutation algebra structure associated to the constant type in dimension two."}
{"category": "Math", "title": "Lie Groupoids as generalized atlases", "abstract": "Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the \"virtual structure\" of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This \"structure\" keeps memory of the isotropy groups and of the smoothness as well. To take the smoothness into account, we claim that we can go very far by retaining just a few formal properties of embeddings and surmersions, yielding a very polymorphous unifying theory. We suggest further developments."}
{"category": "Math", "title": "On the 1-density of Unit Ball Covering", "abstract": "Motivated by modern applications like image processing and wireless sensor networks, we consider a variation of the famous Kepler Conjecture. Given any infinite set of unit balls covering the whole space, we want to know the optimal (lim sup) density of the volume which is covered by exactly one ball (i.e., the maximum such density over all unit ball covers, called the {\\em optimal 1-density} and denoted as $\\delta_d$, where $d$ is the dimension of the Euclidean space). We prove that in 2D the optimal 1-density $\\delta_2=(3\\sqrt(3)-\\pi)/\\pi \\approx 0.6539$, which is achieved through a regular hexagonal covering. In 3D, the problem is widely open and we present a Dodecehadral Cover Conjecture which states that the optimal 1-density in 3D, $\\delta_3$, is bounded from above by the 1-density of a unit ball whose Voronoi polyhedron is a regular dodecahedron of circum-radius one (determined by 12 extra unit balls). We show numerically that this 1-density $\\delta_3(dc)\\approx 0.315$."}
{"category": "Math", "title": "Metric sparsification and operator norm localization", "abstract": "We study an operator norm localization property and its applications to the coarse Novikov conjecture in operator K-theory. A metric space X is said to have operator norm localization property if there exists a positive number c such that for every r>0, there is R>0 for which, if m is a positive locally finite Borel measure on X, H is a separable infinite dimensional Hilbert space and T is a bounded linear operator acting on L^2(X,m) with propagation r, then there exists an unit vector v satisfying with support of diameter at most R and such that |Tv| is larger or equal than c|T|. If X has finite asymptotic dimension, then X has operator norm localization property. In this paper, we introduce a sufficient geometric condition for the operator norm localization property. This is used to give many examples of finitely generated groups with infinite asymptotic dimension and the operator norm localization property. We also show that any sequence of expanding graphs does not possess the operator norm localization property."}
{"category": "Math", "title": "Proper actions of Lie groups of dimension $n^2+1$ on $n$-dimensional complex manifolds", "abstract": "In this paper we continue to study actions of high-dimensional Lie groups on complex manifolds. We give a complete explicit description of all pairs $(M,G)$, where $M$ is a connected complex manifold $M$ of dimension $n\\ge 2$, and $G$ is a connected Lie group of dimension $n^2+1$ acting effectively and properly on $M$ by holomorphic transformations. This result complements a classification obtained earlier by the first author for $n^2+2\\le\\hbox{dim} G<n^2+2n$ and a classical result due to W. Kaup for the maximal group dimension $n^2+2n$."}
{"category": "Math", "title": "Goodness-of-fit Tests for high-dimensional Gaussian linear models", "abstract": "Let $(Y,(X_i)_{i\\in\\mathcal{I}})$ be a zero mean Gaussian vector and $V$ be a subset of $\\mathcal{I}$. Suppose we are given $n$ i.i.d. replications of the vector $(Y,X)$. We propose a new test for testing that $Y$ is independent of $(X_i)_{i\\in \\mathcal{I}\\backslash V}$ conditionally to $(X_i)_{i\\in V}$ against the general alternative that it is not. This procedure does not depend on any prior information on the covariance of $X$ or the variance of $Y$ and applies in a high-dimensional setting. It straightforwardly extends to test the neighbourhood of a Gaussian graphical model. The procedure is based on a model of Gaussian regression with random Gaussian covariates. We give non asymptotic properties of the test and we prove that it is rate optimal (up to a possible $\\log(n)$ factor) over various classes of alternatives under some additional assumptions. Besides, it allows us to derive non asymptotic minimax rates of testing in this setting. Finally, we carry out a simulation study in order to evaluate the performance of our procedure."}
{"category": "Math", "title": "Ergodic properties of sub-hyperbolic functions with polynomial Schwarzian derivative", "abstract": "The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is bounded and the map $f$ restricted to its closure is expanding, the property refered to as sub-expanding. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure $m$ with minimal exponent $h$; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a $\\sg$--finite invariant measure $\\mu$ absolutely continuous with respect to $m$. Our main result states that $\\mu$ is finite if and only if the order $\\rho$ of the function $f$ satisfies the condition $h>3\\frac{\\rho}{\\rho +1}$. When finite, this measure is shown to be metrically exact. We also establish a version of Bowen's formula showing that the exponent $h$ equals the Hausdorff dimension of the Julia set of $f$."}
{"category": "Math", "title": "Support varieties of non-restricted modules over Lie algebras of reductive groups: corrigenda and addenda", "abstract": "This paper fixes a gap in my article \"Support varieties of non-restricted modules over Lie algebras of reductive groups\" pointed out to me by J.C. Jantzen. It was written several years ago, but never widely circulated."}
{"category": "Math", "title": "Martingale dimensions for fractals", "abstract": "We prove that the martingale dimensions for canonical diffusion processes on a class of self-similar sets including nested fractals are always one. This provides an affirmative answer to the conjecture of S. Kusuoka [Publ. Res. Inst. Math. Sci. 25 (1989) 659--680]."}
{"category": "Math", "title": "Rank two filtered $(\\phi, N)$-modules with Galois descent data and coefficients", "abstract": "Let $K$ be any finite extension of $Q_{p}$, $L$ any finite Galois extension of $K$ and $E$ any finite large enough coefficient field containing $L$. We classify two-dimensional, F-semistable $E$-representations of $G_{K}$, by listing the isomorphism classes of rank two weakly admissible filtered $(\\phi,N,L/K,E)$-modules."}
{"category": "Math", "title": "Dispersive and Strichartz estimates for hyperbolic equations with constant coefficients", "abstract": "Dispersive and Strichartz estimates for solutions to general strictly hyperbolic partial differential equations with constant coefficients are considered. The global time decay estimates of $L^p-L^q$ norms of propagators are obtained, and it is shown how the time decay rates depend on the geometry of the problem. The frequency space is separated in several zones each giving a certain decay rate. Geometric conditions on characteristics responsible for the particular decay are identified and investigated. Thus, a comprehensive analysis is carried out for strictly hyperbolic equations of high orders with lower order terms of a general form. Results are applied to establish time decay estimates for the Fokker-Planck equation and for semilinear hyperbolic equations."}
{"category": "Math", "title": "A Singular Control Model with Application to the Goodwill Problem", "abstract": "We consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional It\\^{o} diffusion. The control effort that can be applied to this system takes the form that is associated with the so-called monotone follower problem of singular stochastic control. The control problem that we address aims at maximising a performance criterion that rewards high values of the utility derived from the system's controlled state but penalises any expenditure of control effort. This problem has been motivated by applications such as the so-called goodwill problem in which the system's state is used to represent the image that a product has in a market, while control expenditure is associated with raising the product's image, e.g., through advertising. We obtain the solution to the optimisation problem that we consider in a closed analytic form under rather general assumptions. Also, our analysis establishes a number of results that are concerned with analytic as well as probabilistic expressions for the first derivative of the solution to a second order linear non-homogeneous ordinary differential equation. These results have independent interest and can potentially be of use to the solution of other one-dimensional stochastic control problems."}
{"category": "Math", "title": "Reflecting Ornstein-Uhlenbeck processes on pinned path spaces", "abstract": "Consider a set of continuous maps from the interval $[0,1]$ to a domain in ${\\mathbb R}^d$. Although the topological boundary of this set in the path space is not smooth in general, by using the theory of functions of bounded variation (BV functions) on the Wiener space and the theory of Dirichlet forms, we can discuss the existence of the surface measure and the Skorokhod representation of the reflecting Ornstein-Uhlenbeck process associated with the canonical Dirichlet form on this set."}
{"category": "Math", "title": "Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds", "abstract": "Given a domain $\\Omega$ of $\\mathbb{R}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\\pa \\Omega$ with $1\\le k\\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\\partial \\Omega$ perpendicularly."}
{"category": "Math", "title": "Keys and alternating sign matrices", "abstract": "Lascoux and Sch\\\"utzenberger introduced a notion of key associated to any Young tableau. More recently Lascoux defined the key of an alternating sign matrix by recursively removing all -1's in such matrices. But alternating sign matrices are in bijection with monotone triangles, which form a subclass of Young tableaux. We show that in this case these two notions of keys coincide. Moreover we obtain an elegant and direct way to compute the key of any Young tableau, and discuss consequences of our result."}
{"category": "Math", "title": "$KK$-theory spectra for $C^\\ast$-categories and discrete groupoid $C^\\ast$-algebras", "abstract": "In this paper we refine a version of bivariant $K$-theory developed by Cuntz to define symmetric spectra representing the $KK$-theory of $C^\\ast$-categories and discrete groupoid $C^\\ast$-algebras. In both cases, the Kasparov product can be expressed as a smash product of spectra."}
{"category": "Math", "title": "Mean-field backward stochastic differential equations: A limit approach", "abstract": "Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution $(Y,Z)$ of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean--Vlasov type with solution $X$ we study a special approximation by the solution $(X^N,Y^N,Z^N)$ of some decoupled forward--backward equation which coefficients are governed by $N$ independent copies of $(X^N,Y^N,Z^N)$. We show that the convergence speed of this approximation is of order $1/\\sqrt{N}$. Moreover, our special choice of the approximation allows to characterize the limit behavior of $\\sqrt{N}(X^N-X,Y^N-Y,Z^N-Z)$. We prove that this triplet converges in law to the solution of some forward--backward stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field."}
{"category": "Math", "title": "Elliptic systems of pseudodifferential equations in a refined scale on a closed manifold", "abstract": "We study a system of pseudodifferential equations that is elliptic in the sense of Petrovskii on a closed compact smooth manifold. We prove that the operator generated by the system is Fredholm one on a refined two-sided scale of the functional Hilbert spaces. Elements of this scale are the special isotropic spaces of H\\\"{o}rmander--Volevich--Paneah."}
{"category": "Math", "title": "Mean-Field Backward Stochastic Differential Equations and Related Partial Differential Equations", "abstract": "In [5] the authors obtained Mean-Field backward stochastic differential equations (BSDE) associated with a Mean-field stochastic differential equation (SDE) in a natural way as limit of some highly dimensional system of forward and backward SDEs, corresponding to a large number of ``particles'' (or ``agents''). The objective of the present paper is to deepen the investigation of such Mean-Field BSDEs by studying them in a more general framework, with general driver, and to discuss comparison results for them. In a second step we are interested in partial differential equations (PDE) whose solutions can be stochastically interpreted in terms of Mean-Field BSDEs. For this we study a Mean-Field BSDE in a Markovian framework, associated with a Mean-Field forward equation. By combining classical BSDE methods, in particular that of ``backward semigroups\" introduced by Peng [14], with specific arguments for Mean-Field BSDEs we prove that this Mean-Field BSDE describes the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated to Mean-Field BSDEs one cannot expect to have uniqueness in a larger space of continuous functions."}
{"category": "Math", "title": "The Key Renewal Theorem for a Transient Markov Chain", "abstract": "We consider a time-homogeneous Markov chain $X_n$, $n\\ge0$, valued in ${\\bf R}$. Suppose that this chain is transient, that is, $X_n$ generates a $\\sigma$-finite renewal measure. We prove the key renewal theorem under condition that this chain has asymptotically homogeneous at infinity jumps and asymptotically positive drift."}
{"category": "Math", "title": "The $KH$-Isomorphism Conjecture and Algebraic $KK$-theory", "abstract": "In this article we prove that the $KH$-asembly map, as defined by Bartels and L{\\\"u}ck, can be described in terms of the algebraic $KK$-theory of Cortinas and Thom. The $KK$-theory description of the $KH$-assembly map is similar to that of the Baum-Connes assembly map. In very elementary cases, methods used to prove the Baum-Connes conjecture also apply to the $KH$-isomorphism conjecture."}
{"category": "Math", "title": "Algebraic $K$-theory Spectra and Factorisations of Analytic Assembly Maps", "abstract": "In this article we use existing machinery to define connective $K$-theory spectra associated to topological ringoids. Algebraic $K$-theory of discrete ringoids, and the analytic $K$-theory of Banach categories are obtained as special cases. As an application, we show how the analytic assembly maps featuring in the Novikov and Baum-Connes conjectures can be factorised into composites of assembly maps resembling those appearing in algebraic $K$-theory and maps coming from completions of certain topological ringoids into Banach categories. These factorisations are proved by using existing characterisations of assembly maps along with our unified picture of algebraic and analytic $K$-theory."}
{"category": "Math", "title": "Exact finite approximations of average-cost countable Markov Decision Processes", "abstract": "For a countable-state Markov decision process we introduce an embedding which produces a finite-state Markov decision process. The finite-state embedded process has the same optimal cost, and moreover, it has the same dynamics as the original process when restricting to the approximating set. The embedded process can be used as an approximation which, being finite, is more convenient for computation and implementation."}
{"category": "Math", "title": "The defect of Fano 3-folds", "abstract": "This paper studies the defect of terminal Gorenstein Fano 3 folds. I determine a bound on the defect of terminal Gorenstein Fano 3-folds of Picard rank 1 that do not contain a plane. I give a general bound for quartic 3-folds and indicate how to study the defect of terminal Gorenstein Fano 3-folds with Picard rank 1 that contain a plane."}
{"category": "Math", "title": "Efficient routing in heavy traffic under partial sampling of service times", "abstract": "We consider a queue with renewal arrivals and n exponential servers in the Halfin-Whitt heavy traffic regime, where n and the arrival rate increase without bound, so that a critical loading condition holds. Server k serves at rate $\\mu_k $, and the empirical distribution of the $\\mu_k $ is assumed to converge weakly. We show that very little information on the service rates is required for a routing mechanism to perform well. More precisely, we construct a routing mechanism that has access to a single sample from the service time distribution of each of $n$ to the power of $1/2 + \\epsilon $ randomly selected servers, but not to the actual values of the service rates, the performance of which is asymptotically as good as the best among mechanisms that have the complete information on $ \\mu_k $."}
{"category": "Math", "title": "Integer operators in finite von Neumann algebras", "abstract": "Motivated by the study of spectral properties of self-adjoint operators in the integral group ring of a sofic group, we define and study integer operators. We establish a relation with classical potential theory and in particular the circle of results obtained by M. Fekete and G. Szeg\"o. More concretely, we use results by R. Rumely on equidistribution of algebraic integers to obtain a description of those integer operator which have spectrum of logarithmic capacity less or equal to one. Finally, we relate the study of integer operators to a recent construction by B. and L. Petracovici and A. Zaharescu."}
{"category": "Math", "title": "Uniqueness of a constrained variational problem and large deviations of buffer size", "abstract": "We show global uniqueness of the solution to a class of constrained variational problems, using scaling properties. This is used to establish the essential uniqueness of solutions of a large deviations problem in multiple dimensions. The result is motivated by models of buffers, and in particular the probability of, and typical path to overflow in the limit of small buffers, which we analyze."}
{"category": "Math", "title": "Cyclic coverings, Calabi-Yau manifolds and Complex multiplication", "abstract": "We construct families of Calabi-Yau manifolds with dense set of complex multiplication fibers in an arbitrary dimension. We will also give explicite examples of complex multiplication fibers. For this construction we use families of curves with dense set of complex multiplication fibers. In addition, we give examples of such families for each genus less or equal 7 and we study the generic Hodge groups of families of cyclic covers of the projective line."}
{"category": "Math", "title": "Supercritical biharmonic equations with power-type nonlinearity", "abstract": "The biharmonic supercritical equation $\\Delta^2u=|u|^{p-1}u$, where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole space $\\mathbb{R}^n$ as well as in a modified form with $\\lambda(1+u)^p$ as right-hand-side with an additional eigenvalue parameter $\\lambda>0$ in the unit ball, in the latter case together with Dirichlet boundary conditions. As for entire regular radial solutions we prove oscillatory behaviour around the explicitly known radial {\\it singular} solution, provided $p\\in((n+4)/(n-4),p_c)$, where $p_c\\in ((n+4)/(n-4),\\infty]$ is a further critical exponent, which was introduced in a recent work by Gazzola and the second author. The third author proved already that these oscillations do not occur in the complementing case, where $p\\ge p_c$. Concerning the Dirichlet problem we prove existence of at least one singular solution with corresponding eigenvalue parameter. Moreover, for the extremal solution in the bifurcation diagram for this nonlinear biharmonic eigenvalue problem, we prove smoothness as long as $p\\in((n+4)/(n-4),p_c)$."}
{"category": "Math", "title": "Symplectic critical surfaces in K\\\"ahler surfaces", "abstract": "Let $M$ be a K\\\"ahler surface and $\\Sigma$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\\alpha$ be the K\\\"ahler angle of $\\Sigma$ in $M$. We first deduce the Euler-Lagrange equation of the functional $L=\\int_{\\Sigma}\\frac{1}{\\cos\\alpha}d\\mu$ in the class of symplectic surfaces. It is $\\cos^3\\alpha H=(J(J\\nabla\\cos\\alpha)^\\top)^\\bot$, where $H$ is the mean curvature vector of $\\Sigma$ in $M$, $J$ is the complex structure compatible with the K\\\"ahler form $\\omega$ in $M$, which is an elliptic equation. We then study the properties of the equation."}
{"category": "Math", "title": "Pattern avoidance and the Bruhat order on involutions", "abstract": "We show that the principal order ideal below an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin paths is demonstrated."}
{"category": "Math", "title": "A smooth foliation of convex hypersurfaces for quasi-hyper-Fuchsian manifolds", "abstract": "This submission has been withdrawn by the author and superseded by arXiv:0804.0744."}
{"category": "Math", "title": "Fourier-Mukai transforms of line bundles on derived equivalent abelian varieties", "abstract": "We study the Fourier-Mukai functor D(Y) -> D(X) induced by the universal family on a fine moduli space Y for simple semihomogeneous vector bundles on an abelian variety X. The main result is that the Fourier-Mukai transform of a very negative line bundle on Y is ample if and only if the bundles parametrized by Y are nef."}
{"category": "Math", "title": "A note on the least totient of a residue class", "abstract": "Let $q$ be a large prime number, $a$ be any integer, $\\epsilon$ be a fixed small positive quantity. Friedlander and Shparlinksi \\cite{FSh} have shown that there exists a positive integer $n\\ll q^{5/2+\\epsilon}$ such that $\\phi(n)$ falls into the residue class $a \\pmod q.$ Here, $\\phi(n)$ denotes Euler's function. In the present paper we improve this bound to $n\\ll q^{2+\\epsilon}.$"}
{"category": "Math", "title": "Bounds on the cardinality of partition", "abstract": "If A is infinite and well-ordered, then |2^A|<=|Part(A)|<=|A^A|."}
{"category": "Math", "title": "Random Homogenization of Fractional Obstacle Problems", "abstract": "We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains."}
{"category": "Math", "title": "Gradients of Laplacian Eigenfunctions on the Sierpinski Gasket", "abstract": "We use spectral decimation to provide formulae for computing the harmonic gradients of Laplacian eigenfunctions on the Sierpinski Gasket. These formulae are given in terms of special functions that are defined as infinite products."}
{"category": "Math", "title": "Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula", "abstract": "In the first part of this paper, we prove local interior and boundary gradient estimates for p-harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the 1/H (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp volume growth assumption. In the second part of this paper, we consider two parabolic analogues of the p-Laplace equation and prove sharp Li-Yau type gradient estimates for positive solutions to these equations on manifolds of nonnegative Ricci curvature. For one of these equations, we also prove an entropy monotonicity formula generalizing an earlier such formula of the second author for the linear heat equation. As an application of this formula, we show that a complete Riemannian manifold with non-negative Ricci curvature and sharp L^p-logarithmic Sobolev inequality must be isometric to Euclidean space."}
{"category": "Math", "title": "Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem", "abstract": "We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\\hat M, T^1\\mathcal{F})$ of a compact minimal lamination $(M,\\mathcal F)$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal, and examples where this action is not minimal. In the first case, we prove that if $\\mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologically transitive."}
{"category": "Math", "title": "Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra", "abstract": "This paper builds on the previous paper arXiv:math/0612730 by the author, where a relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra (DAHA) corresponding to the Askey-Wilson polynomials was established. It is shown here that the spherical subalgebra of this DAHA is isomorphic to AW(3) with an additional relation that the Casimir operator equals an explicit constant. A similar result with q-shifted parameters holds for the antispherical subalgebra. Some theorems on centralizers and centers for the algebras under consideration will finally be proved as corollaries of the characterization of the spherical and antispherical subalgebra."}
{"category": "Math", "title": "Rigidity and relative hyperbolicity of real hyperbolic hyperplane complements", "abstract": "For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf, biautomatic, residually hyperbolic, not K\\\"ahler, not isomorphic to lattices in virtually connected real Lie groups, have no nontrivial subgroups with property (T), have finite outer automorphism groups, satisfy Mostow-type Rigidity, have finite asymptotic dimension and rapid decay property, and satisfy Baum-Connes conjecture. We also characterize those lattices in real Lie groups that are isomorphic to relatively hyperbolic groups."}
{"category": "Math", "title": "Encouraging the grand coalition in convex cooperative games", "abstract": "A solution function for convex transferable utility games encourages the grand coalition if no player prefers (in a precise sense defined in the text) any coalition to the grand coalition. We show that the Shapley value encourages the grand coalition in all convex games and the tau-value encourages the grand coalitions in convex games up to three (but not more than three) players. Solution functions that encourage the grand coalition in convex games always produce allocations in the core, but the converse is not necessarily true."}
{"category": "Math", "title": "Adjoint ideals along closed subvarieties of higher codimension", "abstract": "In this paper, we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using characteristic $p$ methods. When $X$ is a normal Gorenstein closed subvariety of a smooth complex variety $A$, we formulate a restriction property of the adjoint ideal sheaf $\\adj_X(A)$ of $A$ along $X$ involving the l.c.i. ideal sheaf $\\mathcal{D}_X$ of $X$. The proof relies on a modification of generalized test ideals of Hara and Yoshida."}
{"category": "Math", "title": "Models for dependent extremes using stable mixtures", "abstract": "This paper unifies and extends results on a class of multivariate Extreme Value (EV) models studied by Hougaard, Crowder, and Tawn. In these models both unconditional and conditional distributions are EV, and all lower-dimensional marginals and maxima belong to the class. This leads to substantial economies of understanding, analysis and prediction. One interpretation of the models is as size mixtures of EV distributions, where the mixing is by positive stable distributions. A second interpretation is as exponential-stable location mixtures (for Gumbel) or as power-stable scale mixtures (for non-Gumbel EV distributions). A third interpretation is through a Peaks over Thresholds model with a positive stable intensity. The mixing variables are used as a modeling tool and for better understanding and model checking. We study extreme value analogues of components of variance models, and new time series, spatial, and continuous parameter models for extreme values. The results are applied to data from a pitting corrosion investigation."}
{"category": "Math", "title": "Robust model selection in generalized linear models", "abstract": "In this paper, we extend to generalized linear models (including logistic and other binary regression models, Poisson regression and gamma regression models) the robust model selection methodology developed by Mueller and Welsh (2005; JASA) for linear regression models. As in Mueller and Welsh (2005), we combine a robust penalized measure of fit to the sample with a robust measure of out of sample predictive ability which is estimated using a post-stratified m-out-of-n bootstrap. A key idea is that the method can be used to compare different estimators (robust and nonrobust) as well as different models. Even when specialized back to linear regression models, the methodology presented in this paper improves on that of Mueller and Welsh (2005). In particular, we use a new bias-adjusted bootstrap estimator which avoids the need to centre the explanatory variables and to include an intercept in every model. We also use more sophisticated arguments than Mueller and Welsh (2005) to establish an essential monotonicity condition."}
{"category": "Math", "title": "Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle", "abstract": "We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in $S^2$ and in $\\R^2$."}
{"category": "Math", "title": "Function spaces of variable smoothness and integrability", "abstract": "In this article we introduce Triebel--Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Applying it, we give molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions. As in the classical case, a unified scale of spaces permits clearer results in cases where smoothness and integrability interact, such as Sobolev embedding and trace theorems. As an application of our decomposition, we prove optimal trace theorems in the variable indices case."}
{"category": "Math", "title": "Every longest circuit of a 3-connected, $K_{3,3}$-minor free graph has a chord", "abstract": "Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no $K_{3,3}$ minor, and consequently for planar graphs."}
{"category": "Math", "title": "Forced Convex Mean Curvature Flow in Euclidean Spaces", "abstract": "In this paper, we consider the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. We show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can also converge to a round sphere for some special forcing term and initial hypersurface. Furthermore, the normalization of the flow is carried out so that long time existence and convergence of the rescaled flow are studied. Our work extends Huisken's well-known mean curvature flow and McCoy's mixed volume preserving mean curvature flow."}
{"category": "Math", "title": "Representations for the non-graded Virasoro-like algebra", "abstract": "It is proved that an irreducible module over the non-graded Virasoro-like algebra, which satisfies a natural condition, is a GHW module or uniformly bounded. Furthermore, the classification of some uniformly bounded modules is given."}
{"category": "Math", "title": "Braid groups and Artin groups", "abstract": "This article is a survey on the braid groups, the Artin groups, and the Garside groups. It is a presentation, accessible to non-experts, of various topological and algebraic aspects of these groups. It is also a report on three points of the theory: the faithful linear representations, the cohomology, and the geometrical representations."}
{"category": "Math", "title": "Urn-related random walk with drift $\\rho x^{\\alpha} / t^{\\beta}$", "abstract": "We study a one-dimensional random walk whose expected drift depends both on time and the position of a particle. We establish a non-trivial phase transition for the recurrence vs. transience of the walk, and show some interesting applications to Friedman's urn, as well as showing the connection with Lamperti's walk with asymptotically zero drift."}
{"category": "Math", "title": "Describing the set of words generated by interval exchange transformation", "abstract": "Let $W$ be an infinite word over finite alphabet $A$. We get combinatorial criteria of existence of interval exchange transformations that generate the word W."}
{"category": "Math", "title": "The induced capacity and Choquet integral monotone convergece", "abstract": "Given a probability measure over a state space, a partial collection (sub-$\\sigma$-algebra) of events whose probabilities are known, induces a capacity over the collection of all possible events. The \\emph{induced capacity} of an event $F$ is the probability of the maximal (with respect to inclusion) event contained in $F$ whose probability is known. The Choquet integral with respect to the induced capacity coincides with the integral with respect to a \\emph{probability specified on a sub-algebra} (Lehrer \\cite{Lehrer2}). We study Choquet integral monotone convergence and apply the results to the integral with respect to the induced capacity. The paper characterizes the properties of sub-$\\sigma$-algebras and of induced capacities which yield integral monotone convergence."}
{"category": "Math", "title": "Sufficient conditions for the convergence of the Magnus expansion", "abstract": "Two different sufficient conditions are given for the convergence of the Magnus expansion arising in the study of the linear differential equation $Y' = A(t) Y$. The first one provides a bound on the convergence domain based on the norm of the operator $A(t)$. The second condition links the convergence of the expansion with the structure of the spectrum of $Y(t)$, thus yielding a more precise characterization. Several examples are proposed to illustrate the main issues involved and the information on the convergence domain provided by both conditions."}
{"category": "Math", "title": "Poisson approximation for search of rare words in DNA sequences", "abstract": "Using recent results on the occurrence times of a string of symbols in a stochastic process with mixing properties, we present a new method for the search of rare words in biological sequences generally modelled by a Markov chain. We obtain a bound on the error between the distribution of the number of occurrences of a word in a sequence (under a Markov model) and its Poisson approximation. A global bound is already given by a Chen-Stein method. Our approach, the psi-mixing method, gives local bounds. Since we only need the error in the tails of distribution, the global uniform bound of Chen-Stein is too large and it is a better way to consider local bounds. We search for two thresholds on the number of occurrences from which we can regard the studied word as an over-represented or an under-represented one. A biological role is suggested for these over- or under-represented words. Our method gives such thresholds for a panel of words much broader than the Chen-Stein method. Comparing the methods, we observe a better accuracy for the psi-mixing method for the bound of the tails of distribution. We also present the software PANOW (available at http://stat.genopole.cnrs.fr/software/panowdir/) dedicated to the computation of the error term and the thresholds for a studied word."}
{"category": "Math", "title": "Local conjugacy classes for analytic torus flows", "abstract": "If a real-analytic flow on the multidimensional torus close enough to linear has a unique rotation vector which satisfies an arithmetical condition Y, then it is analytically conjugate to linear. We show this by proving that the orbit under renormalization of a constant Y vector field attracts all nearby orbits with the same rotation vector."}
{"category": "Math", "title": "Compact complete minimal immersions in R^3", "abstract": "In this paper we find, for any arbitrary finite topological type, a compact Riemann surface $\\mathcal{M},$ an open domain $M\\subset\\mathcal{M}$ with the fixed topological type, and a conformal complete minimal immersion $X:M\\to\\R^3$ which can be extended to a continuous map $X:\\bar{M}\\to\\R^3,$ such that $X_{|\\partial M}$ is an embedding and the Hausdorff dimension of $X(\\partial M)$ is $1.$ We also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in $\\R^3$, endowed with the topology of the Hausdorff distance."}
{"category": "Math", "title": "Drawing polytopal graphs with polymake", "abstract": "This note wants to explain how to obtain meaningful pictures of (possibly high-dimensional) convex polytopes, triangulated manifolds, and other objects from the realm of geometric combinatorics such as tight spans of finite metric spaces and tropical polytopes. In all our cases we arrive at specific, geometrically motivated, graph drawing problems. The methods displayed are implemented in the software system polymake."}
{"category": "Math", "title": "Constituting Atoms of a $\\sigma$ Algebra via Its Generator", "abstract": "To constitute atoms of a $\\sigma$ algebra is not a easy task due to the large number of its elements. However, determining them via generators seems a feasible and simple way since most $\\sigma$ algebras are generated by their smaller proper subsets. Precisely, under some conditions each atom of a $\\sigma$ algebra equals the intersection of the elements containing a point of the atom in the generator. In this paper, a very weak sufficient condition for determining atoms by the generator is presented. The condition, though not being a necessary one, is shown to be almost the weakest one in the sense that it can hardly be improved."}
{"category": "Math", "title": "Biorthogonal Expansion of Non-Symmetric Jack Functions", "abstract": "We find a biorthogonal expansion of the Cayley transform of the non-symmetric Jack functions in terms of the non-symmetric Jack polynomials, the coefficients being Meixner-Pollaczek type polynomials. This is done by computing the Cherednik-Opdam transform of the non-symmetric Jack polynomials multiplied by the exponential function."}
{"category": "Math", "title": "$C^m$-theory of damped wave equations with stabilisation", "abstract": "The aim of this note is to extend the energy decay estimates from [J. Wirth, J. Differential Equations 222 (2006) 487--514] to a broader class of time-dependent dissipation including very fast oscillations. This is achieved using stabilisation conditions on the coefficient in the spirit of [F. Hirosawa, Math. Ann. 339/4 (2007) 819--839]."}
{"category": "Math", "title": "Homogenization of spectral problems in bounded domains with doubly high contrasts", "abstract": "Homogenization of a spectral problem in a bounded domain with a high contrast in both stiffness and density is considered. For a special critical scaling, two-scale asymptotic expansions for eigenvalues and eigenfunctions are constructed. Two-scale limit equations are derived and relate to certain non-standard self-adjoint operators. In particular they explicitly display the first two terms in the asymptotic expansion for the eigenvalues, with a surprising bound for the error of order \\epsilon^{5/4} proved."}
{"category": "Math", "title": "On Surfaces of Prescribed Weighted Mean Curvature", "abstract": "Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature."}
{"category": "Math", "title": "Travelling waves for the Gross-Pitaevskii equation II", "abstract": "The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results, where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy."}
{"category": "Math", "title": "Copulas: compatibility and Fr\\'echet classes", "abstract": "We determine under which conditions three bivariate copulas are compatible, viz. they are the bivariate marginals of the same trivariate copula, and, then, construct the class of these copulas. In particular, the upper and lower bounds for this class of trivariate copulas are determined."}
{"category": "Math", "title": "Adjacency preserving mappings on real symmetric matrices", "abstract": "Let $S_{n}$ denote the space of all $n \\times n$ real symmetric matrices. For n=2 or n>2 we characterize maps F from $S_{n}$ to $S_{m}$ which preserve adjacency, i.e. if rank(A-B)=1, then rank(F(A)-F(B))=1."}
{"category": "Math", "title": "Ornstein-Uhlenbeck Processes on Lie Groups", "abstract": "We consider Ornstein-Uhlenbeck processes (OU-processes) associated to hypoelliptic diffusion processes on finite-dimensional Lie groups: let $ \\mathcal{L} $ be a hypoelliptic, left-invariant ``sum of the squares''-operator on a Lie group $ G $ with associated Markov process $ X $, then we construct OU-processes by adding negative horizontal gradient drifts of functions $ U $. In the natural case $ U(x) = - \\log p(1,x) $, where $ p(1,x) $ is the density of the law of $ X $ starting at identity $ e $ at time $ t =1 $ with respect to the right-invariant Haar measure on $G$, we show the Poincar\\'e inequality by applying the Driver-Melcher inequality for ``sum of the squares'' operators on Lie groups. The resulting Markov process is called the natural OU-process associated to the hypoelliptic diffusion on $ G $. We prove the global strong existence of these OU-type processes on $ G $ under an integrability assumption on $U$. The Poincar\\'e inequality for a large class of potentials $U$ is then shown by a perturbation technique. These results are applied to obtain a hypoelliptic equivalent of standard results on cooling schedules for simulated annealing on compact homogeneous spaces $M$."}
{"category": "Math", "title": "Two body systems from sl(2,C)-tops", "abstract": "It is shown that sl(2,$\\mathbb{C}$) Euler-Arnold tops are equivalent to the two-body systems of Calogero-Moser type. We prove that generic Hamiltonians of sl(2,$\\mathbb{C}$) tops are equivalent to one of three canonical Hamiltonians. For all canonical Hamiltonians the corresponding two-body system is found. Bosonisation formulas for each case are obtained explicitly. Relations with Antonov-Zabrodin-Hasegawa R-matrix are discussed."}
{"category": "Math", "title": "Variable importance in binary regression trees and forests", "abstract": "We characterize and study variable importance (VIMP) and pairwise variable associations in binary regression trees. A key component involves the node mean squared error for a quantity we refer to as a maximal subtree. The theory naturally extends from single trees to ensembles of trees and applies to methods like random forests. This is useful because while importance values from random forests are used to screen variables, for example they are used to filter high throughput genomic data in Bioinformatics, very little theory exists about their properties."}
{"category": "Math", "title": "Some Geometry of Nodal Curves", "abstract": "We find a geometrical method of analysing the singularities of a plane nodal curve. The main results will be used in a forthcoming paper on geometric Plucker formulas for such curves. Plane nodal curves, that is plane curves having at most nodes as singularities, form an important class of curves, as any projective algebraic curve is birational to a plane nodal curve."}
{"category": "Math", "title": "Uniqueness and factorization of Coleff-Herrera currents", "abstract": "We prove a uniqueness result for Coleff-Herrera currents which in particular means that if $f=(f_1,..., f_m)$ defines a complete intersection, then the classical Coleff-Herrera product associated to $f$ is the unique Coleff-Herrera current that is cohomologous to 1 with respect to the operator $\\delta_f-\\dbar$, where $\\delta_f$ is interior multiplication with $f$. From the uniqueness result we deduce that any Coleff-Herrera current on a variety $Z$ is a finite sum of products of residue currents with support on $Z$ and holomorphic forms."}
{"category": "Math", "title": "A note on polylogarithms on curves and abelian schemes", "abstract": "In this note we investigate the connection between polylogarithms on curves and abelian schemes. The main result shows that the polylogarithm on the abelian scheme can be obtained as the push-forward of the polylogarithm on a suitable sub-curve. If the abelian scheme is the Jacobian of a smooth projective curve, this push-forward can also be written as a cup-product with the fundamental class of the curve."}
{"category": "Math", "title": "Global regularity for the 3D Navier-Stokes and the 3D Euler equations", "abstract": "The article `Global regularity for the 3D Navier-Stokes and the 3D Euler equations'(arXiv:0711.2453) is withdrawn due to a serious error in the proof."}
{"category": "Math", "title": "Linear dimension-free estimates for the Hermite-Riesz transforms", "abstract": "We utilize the Bellman function technique to prove a bilinear dimension-free inequality for the Hermite operator. The Bellman technique is applied here to a non-local operator, which at first did not seem to be feasible. As a consequence of our bilinear inequality one proves dimension-free boundedness for the Riesz-Hermite transforms on L^p with linear growth in terms of p. A feature of the proof is a theorem establishing L^p(R^n) estimates for a class of spectral multipliers with bounds independent of n and p. Connections with known results on the Heisenberg group as well as with results for the Hilbert transform along the parabola are also explored. We believe our approach is quite universal in the sense that one could apply it to a whole range of Riesz transforms arising from various differential operators. As a first step towards this goal we prove our dimension-free bilinear embedding theorem for quite a general family of Schroedinger semigroups."}
{"category": "Math", "title": "Groupo\\\"ides de Lie et Feuilletages", "abstract": "This is a survey concerning the relationship between Lie Groupoids (and their morphisms) and singular foliations in the sense of Sussmann-Stefan (considered from a purely geometrical point of view). We focus on the interaction between the algebraic and differentiable structures underlying Lie groupoids, and between groups and graphs of equivalence relations, regarded as two basic degeneracies for groupoids. Historical remarks, motivations and examples are developed in five appendices."}
{"category": "Math", "title": "A two-dimensional ruin problem on the positive quadrant", "abstract": "In this paper we study the joint ruin problem for two insurance companies that divide between them both claims and premia in some specified proportions (modeling two branches of the same insurance company or an insurance and re-insurance company). Modeling the risk processes of the insurance companies by Cram\\'{e}r-Lundberg processes we obtain the Laplace transform in space of the probability that either of the insurance companies is ruined in finite time. Subsequently, for exponentially distributed claims, we derive an explicit analytical expression for this joint ruin probability by explicitly inverting this Laplace transform. We also provide a characterization of the Laplace transform of the joint ruin time."}
{"category": "Math", "title": "Representations of the quantum torus and applications to finitely presented groups", "abstract": "A structure theorem is proved for strongly holonomic modules over a quantum torus (a crossed product of a field with a free abelian group in which the field is central). This can be applied to give a structure theorem for finitely presented abelian-by-nilpotent groups."}
{"category": "Math", "title": "The automorphism groups of the groups of orders $16p$ and $16p^2$", "abstract": "Results of the computation of the automorphism groups for the groups of orders $16p$ and $16p^{2}$ are given. In some cases it has not been possible to give as complete a set of results as was done previously for the case of groups of order $8p^2$. Problems arise for those groups of the form ($C_{p} \\times C_{p}$) @ $\\G$[16] that occur in the orders $p\\equiv 1$ mod(8) and $p\\equiv 7$ mod(8), where $G$[16] means any group of order 16. The groups $G$[16] in question are $C_{16}$, $D_{8}$, $QD_{8}$, and $Q_{4}$. For the other cases, explicit presentations are presented for the automorphism groups of the groups of orders 16$p$ and 16$p^2$."}
{"category": "Math", "title": "Coefficient Quantization for Frames in Banach Spaces", "abstract": "Let $(e_i)$ be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces."}
{"category": "Math", "title": "Controllability properties of a class of systems modeling swimming microscopic organisms", "abstract": "We consider a finite-dimensional model for the motion of microscopic organisms whose propulsion exploits the action of a layer of cilia covering its surface. The model couples Newton's laws driving the organism, considered as a rigid body, with Stokes equations governing the surrounding fluid. The action of the cilia is described by a set of controlled velocity fields on the surface of the organism. The first contribution of the paper is the proof that such a system is generically controllable when the space of controlled velocity fields is at least three-dimensional. We also provide a complete characterization of controllable systems in the case in which the organism has a spherical shape. Finally, we offer a complete picture of controllable and non-controllable systems under the additional hypothesis that the organism and the fluid have densities of the same order of magnitude."}
{"category": "Math", "title": "Weak homogeneity in generalized function algebras", "abstract": "In this paper, weakly homogeneous generalized functions in the special Colombeau algebras are determined up to equality in the sense of generalized distributions. This yields characterizations that are formally similar to distribution theory. Further, we give several characterizations of equality in the sense of generalized distributions in these algebras."}
{"category": "Math", "title": "Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup", "abstract": "By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an expansion of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology."}
{"category": "Math", "title": "A singular stochastic differential equation driven by fractional Brownian motion", "abstract": "In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter $H>\\frac 12$. Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time $t>0$."}
{"category": "Math", "title": "On Iwahori--Hecke algebras with unequal parameters and Lusztig's isomorphism theorem", "abstract": "By Tits' deformation argument, a generic Iwahori--Hecke algebra $H$ associated to a finite Coxeter group $W$ is abstractly isomorphic to the group algebra of $W$. Lusztig has shown how one can construct an explicit isomorphism, provided that the Kazhdan--Lusztig basis of $H$ satisfies certain deep properties. If $W$ is crystallographic and $H$ is a one-parameter algebra, then these properties are known to hold thanks to a geometric interpretation. In this paper, we develop some new general methods for verifying these properties, and we do verify them for two-parameter algebras of type $I_2(m)$ and $F_4$ (where no geometric interpretation is available in general). Combined with previous work by Alvis, Bonnaf\\'e, DuCloux, Iancu and the author, we can then extend Lusztig's construction of an explicit isomorphism to all types of $W$, without any restriction on the parameters of $H$."}
{"category": "Math", "title": "Remarks on topological algebras", "abstract": "The note complements topological aspects of the theory of chiral algebras."}
{"category": "Math", "title": "Hecke duality relations of Jacobi forms", "abstract": "In this paper we introduce a new subspace of Jacobi forms of higher degree via certain relations among Fourier coefficients. We prove that this space can also be characterized by duality properties of certain distinguished embedded Hecke operators. We then show that this space is Hecke invariant with respect to all good Hecke operators. As explicit examples we give Eisenstein series. Conversely we show the existence of forms that are not contained in this space."}
{"category": "Math", "title": "Splitting formulas for certain Waldhausen Nil-groups", "abstract": "For a group G that splits as an amalgamation of A and B over a common subgroup C, there is an associated Waldhausen Nil-group, measuring the \"failure\" of Mayer-Vietoris for algebraic K-theory. Assume that (1) the amalgamation is acylindrical, and (2) the groups A,B,G satisfy the Farrell-Jones isomorphism conjecture. Then we show that the Waldhausen Nil-group splits as a direct sum of Nil-groups associated to certain (explicitly describable) infinite virtually cyclic subgroups of G. We note that a special case of an acylindrical amalgamation includes any amalgamation over a finite group C."}
{"category": "Math", "title": "Schubert calculus and cohomology of Lie groups. Part I. 1-connected Lie groups", "abstract": "Let $G$ be a compact and $1$--connected Lie group with a maximal torus $T$. Based on Schubert calculus on the flag manifold $G/T$ [15] we construct the integral cohomology ring $H^{\\ast}(G)$ uniformly for all $G$."}
{"category": "Math", "title": "General Connections, Exponential Maps, and Second-order Differential Equations", "abstract": "The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study a connection via a certain, closely associated second-order differential equation. One of the most important results is our extended Ambrose-Palais-Singer correspondence. We extend the theory of geodesic sprays to certain second-order differential equations, show that locally diffeomorphic exponential maps can be defined for all, and give a full theory of (possibly nonlinear) covariant derivatives for (possibly nonlinear) connections. In the process, we introduce vertically homogeneous connections. Unlike homogeneous connections, these complete our theory and allow us to include Finsler spaces in a completely consistent manner. This is an expanded version of the article published in Differ. Geom. Dyn. Syst. 13 (2011) 72--90. Included are the proof published in Nonlinear Anal. 63 (2005) e501--e510 (for the reader's convenience) and some new material on homogeneity."}
{"category": "Math", "title": "On the estimation of the convergence rate in the Janashia-Lagvilava spectral factorization algorithm", "abstract": "In the present paper, we estimate the convergence rate in the Janashia-Lagvilava spectral factorization algorithm (see Studia Mathematica, 137, 1999, 93-100) under the restriction on a spectral density matrix that its inverse is integrable."}
{"category": "Math", "title": "The Chains of Left-invariant CR-structures on SU(2)", "abstract": "We compute the chains associated to the left-invariant CR structures on the three-sphere. These structures are characterized by a single real modulus $a$. For the standard structure $a=1$, the chains are well-known and are closed curves. We show that for almost all other values of the modulus $a$ either two or three types of chains are simultaneously present : (I) closed curves, (II) quasi-periodic curves dense on two-torii, or (III) chains homoclinic between closed curves. For $1 < a < \\sqrt{3}$ no curves of the last type occur. A bifurcation occurs at $a = \\sqrt{3}$ and from that point on all three types of chains are guaranteed to exist, and exhaust all chains. The method of proof is to use the Fefferman metric characterization of chains, combined with tools from geometric mechanics. The key to the computation is a reduced Hamiltonian system, similar to Euler's rigid body system, and depending on $a$, which is integrable."}
{"category": "Math", "title": "Conditional independence relations and log-linear models for random permutations", "abstract": "We propose a new class of models for random permutations, which we call log-linear models, by the analogy with log-linear models used in the analysis of contingency tables. As a special case, we study the family of all Luce-decomposable distributions, and the family of those random permutations, for which the distribution of both the permutation and its inverse is Luce-decomposable. We show that these latter models can be described by conditional independence relations. We calculate the number of free parameters in these models, and describe an iterative algorithm for maximum likelihood estimation, which enables us to test if a set of data satisfies the conditional independence relations or not."}
{"category": "Math", "title": "Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity", "abstract": "In this paper we investigate a time dependent family of plane closed Jordan curves evolving in the normal direction with a velocity which is assumed to be a function of the curvature, tangential angle and position vector of a curve. We follow the direct approach and analyze the system of governing PDEs for relevant geometric quantities. We focus on a class of the so-called curvature adjusted tangential velocities for computation of the curvature driven flow of plane closed curves. Such a curvature adjusted tangential velocity depends on the modulus of the curvature and its curve average. Using the theory of abstract parabolic equations we prove local existence, uniqueness and continuation of classical solutions to the system of governing equations. We furthermore analyze geometric flows for which normal velocity may depend on global curve quantities like the length, enclosed area or total elastic energy of a curve. We also propose a stable numerical approximation scheme based on the flowing finite volume method. Several computational examples of various nonlocal geometric flows are also presented in this paper."}
{"category": "Math", "title": "On the Ramsey numbers for a combination of paths and Jahangirs", "abstract": "For given graphs $G$ and $H,$ the \\emph{Ramsey number} $R(G,H)$ is the least natural number $n$ such that for every graph $F$ of order $n$ the following condition holds: either $F$ contains $G$ or the complement of $F$ contains $H.$ In this paper, we improve the Surahmat and Tomescu's result \\cite{ST:06} on the Ramsey number of paths versus Jahangirs. We also determine the Ramsey number $R(\\cup G,H)$, where $G$ is a path and $H$ is a Jahangir graph."}
{"category": "Math", "title": "A short note on small deviations of sequences of i.i.d. random variables with exponentially decreasing weights", "abstract": "We obtain some new results concerning the small deviation problem for $S=\\sum_n q^n X_n$ and $M=\\sup_n q^n X_n$, where $0<q<1$ and $(X_n)$ are i.i.d. non-negative random variables. In particular, the asymptotics is shown to be the same for $S$ and $M$ in some cases."}
{"category": "Math", "title": "On multiply connected wandering domains of meromorphic functions", "abstract": "We describe conditions under which a multiply connected wandering domain of a transcendental meromorphic function with a finite number of poles must be a Baker wandering domain, and we discuss the possible eventual connectivity of Fatou components of transcendental meromorphic functions. We also show that if $f$ is meromorphic, $U$ is a bounded component of $F(f)$ and $V$ is the component of $F(f)$ such that $f(U)\\subset V$, then $f$ maps each component of $\\partial U$ onto a component of the boundary of $V$ in $\\hat{\\C}$. We give examples which show that our results are sharp; for example, we prove that a multiply connected wandering domain can map to a simply connected wandering domain, and vice versa."}
{"category": "Math", "title": "Large Cardinals and Definable Well-Orderings of the Universe", "abstract": "We use a reverse Easton forcing iteration to obtain a universe with a definable well-ordering, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle diamond star at a proper class of successor cardinals."}
{"category": "Math", "title": "A note on groups with the minimal conditions for nonabelian and abelian subgroups", "abstract": "We give a new proof of the known Shunkov's Theorem on locally finite groups with the minimal condition for nonabelian subgroups and also an extension of the known Suchkova-Shunkov Theorem on Shunkov groups with the minimal condition for abelian subgroups."}
{"category": "Math", "title": "Expressions of algebra elements and transcendental noncommutative calculus", "abstract": "Ideas from deformation quantization are applied to deform the expression of elements of an algebra. Extending these ideas to certain transcendental elements implies that $\\frac{1}{i\\h}uv$ in the Weyl algebra is naturally viewed as an indeterminate living in a discrete set $\\mathbb{N}{+}{1/2}$ {\\it or} ${-}(\\mathbb{N}{+}{1/2})$ . This may yield a more mathematical understanding of Dirac's positron theory."}
{"category": "Math", "title": "The VPN Tree Routing Conjecture for Outerplanar Networks", "abstract": "The VPN Tree Routing Conjecture is a conjecture about the Virtual Private Network Design problem. It states that the symmetric version of the problem always has an optimum solution which has a tree-like structure. In recent work, Hurkens, Keijsper and Stougie (Proc. IPCO XI, 2005; SIAM J. Discrete Math., 2007) have shown that the conjecture holds when the network is a ring. A shorter proof of the VPN Conjecture for rings was found a few months ago by Grandoni, Kaibel, Oriolo and Skutella (to appear in Oper. Res. Lett., 2008). In their paper, Grandoni et al. introduce another conjecture, called the Pyramidal Routing Conjecture (or simply PR Conjecture), which implies the VPN Conjecture. Here we consider a strengthened version of the PR Conjecture. First we establish several general tools which can be applied in arbitrary networks. Then we use them to prove that outerplanar networks satisfy the PR Conjecture."}
{"category": "Math", "title": "Delay equations driven by rough paths", "abstract": "In this article, we illustrate the flexibility of the algebraic integration formalism introduced by M. Gubinelli (2004), by establishing an existence and uniqueness result for delay equations driven by rough paths. We then apply our results to the case where the driving path is a fractional Brownian motion with Hurst parameter H>1/3."}
{"category": "Math", "title": "Localisations and Completions of Skew Power Series Rings", "abstract": "This paper is a natural continuation of the study of skew power series rings A initiated in [P. Schneider and O. Venjakob, On the codimension of modules over skew power series rings with applications to Iwasawa algebras, J. Pure Appl. Algebra 204 (2005), 349 - 367.]. We construct skew Laurent series rings B and show the existence of some canonical Ore sets S for the skew power series rings A such that a certain completion of the localisation A_S is isomorphic to B. This is applied to certain Iwasawa algebras. Finally we introduce subrings of overconvergent skew Laurent series rings."}
{"category": "Math", "title": "Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts", "abstract": "We prove that for meromorphic maps with logarithmic tracts (e.g. entire or meromorphic maps with a finite number of poles from class $\\mathcal B$), the Julia set contains a compact invariant hyperbolic Cantor set of Hausdorff dimension greater than 1. Hence, the hyperbolic dimension of the Julia set is greater than 1."}
{"category": "Math", "title": "Remarks on Congruence of 3-manifolds", "abstract": "We give two proofs that the 3-torus is not weakly d-congruent to the connected sum of three S^1xS^2's, if d>2. We study how cohomology ring structure relates to weak congruence. We give an example of three 3--manifolds which are weakly 5-congruent but are not 5-congruent."}
{"category": "Math", "title": "A note on random walks in a hypercube", "abstract": "We study a simple random walk on an n-dimensional hypercube. For any starting position we find the probability of hitting vertex a before hitting vertex b, whenever a and b share the same edge. This generalizes the model in Doyle, P., and Snell, J., \"Random Walks and Electric Networks\", Mathematical Association of America, 1984 (see Exercise 1.3.7 there)."}
{"category": "Math", "title": "Uniformizing Tropical Curves I: Genus Zero and One", "abstract": "In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus zero case and by non-archimedean elliptic functions in the genus one case. For genus zero curves, those graphs which can be lifted can be characterized in a completely combinatorial manner. For genus one curves, show that certain conditions identified by Mikhalkin are sufficient and we also identify a new necessary condition."}
{"category": "Math", "title": "Bijections from Dyck paths to 321-avoiding permutations revisited", "abstract": "There are (at least) three bijections from Dyck paths to 321-avoiding permutations in the literature, due to Billey-Jockusch-Stanley, Krattenthaler, and Mansour-Deng-Du. How different are they? Denoting them B,K,M respectively, we show that M = B \\circ L = K \\circ L' where L is the classical Kreweras-Lalanne involution on Dyck paths and L', also an involution, is a sort of derivative of L. Thus K^{-1} \\circ B, a measure of the difference between B and K, is the product of involutions L' \\circ L and turns out to be a very curious bijection: as a permutation on Dyck n-paths it is an nth root of the \"reverse path\" involution. The proof of this fact boils down to a geometric argument involving pairs of nonintersecting lattice paths."}
{"category": "Math", "title": "Regularity and the Cesaro-Nevai class", "abstract": "We consider OPRL and OPUC with measures regular in the sense of Ullman-Stahl-Totik and prove consequences on the Jacobi parameters or Verblunsky coefficients. For example, regularity on $[-2,2]$ implies $\\lim_{N\\to\\infty} N^{-1} [\\sum_{n=1}^N (a_n-1)^2 + b_n^2] =0$."}
{"category": "Math", "title": "On the Rank of Random Sparse Matrices", "abstract": "We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This allows us to obtain an exact estimate for the co-rank."}
{"category": "Math", "title": "Equilibrium measures and capacities in spectral theory", "abstract": "This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl-Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrodinger operators where one of our new results implies that, in complete generality, the spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential theory and on Fekete-Szego theory."}
{"category": "Math", "title": "Monotone Jacobi parameters and non-Szego weights", "abstract": "We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for $a_n\\equiv 1$, $b_n =-C n^{-\\beta}$ ($0<\\beta< \\frac23)$, one has $d\\mu(x)= w(x) dx$ on $(-2,2)$, and near $x=2$, $w(x)=e^{-2Q(x)}$ where \\[ Q(x)=\\beta^{-1} C^{\\frac{1}{\\beta}} \\frac{\\Gamma(\\frac32)\\Gamma(\\frac{1}\\beta}-\\frac12)(2-x)^{\\frac12 -\\frac{1}{\\beta}}}{\\Gamma(\\frac{1}{\\beta}+1)}(1+O((2-x))) \\]"}
{"category": "Math", "title": "The Analytic Theory of Matrix Orthogonal Polynomials", "abstract": "We give a survey of the analytic theory of matrix orthogonal polynomials."}
{"category": "Math", "title": "The fundamental group of random 2-complexes", "abstract": "We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\\H{o}s-R\\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log(n)/n. We use a variant of Gromov's local-to-global theorem for linear isoperimetric inequalities to show that when p = O(n^{-1/2 -\\epsilon}) the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2-dimensional simplicial complexes and establish isoperimetric inequalities for such complexes. These intermediate results do not involve randomness and may be of independent interest."}
{"category": "Math", "title": "On the universal Gr\\\"obner bases of varieties of minimal degree", "abstract": "A universal Gr\\\"obner basis of an ideal is the union of all its reduced Gr\\\"obner bases. It is contained in the Graver basis, the set of all primitive elements. Obtaining an explicit description of either of these sets, or even a sharp degree bound for their elements, is a nontrivial task. In their '95 paper, Graham, Diaconis and Sturmfels give a nice combinatorial description of the Graver basis for any rational normal curve in terms of primitive partition identities. Their result is extended here to rational normal scrolls. The description of the Graver bases is given in terms of colored partition identities. This leads to a sharp bound on the degree of Graver basis elements, which is always attained by a circuit. Finally, for any variety obtained from a scroll by a sequence of projections to some of the coordinate hyperplanes, the degree of any element in any reduced Gr\\\"obner basis is bounded by the degree of the variety."}
{"category": "Math", "title": "The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble", "abstract": "We solve the largest sample eigenvalue distribution problem in the rank 1 spiked model of the quaternionic Wishart ensemble, which is the first case of a statistical generalization of the Laguerre symplectic ensemble (LSE) on the soft edge. We observe a phase change phenomenon similar to that in the complex case, and prove that the new distribution at the phase change point is the GOE Tracy--Widom distribution."}
{"category": "Math", "title": "Free Martingale polynomials for stationary Jacobi processes", "abstract": "We generalize a previous result concerning free martingale polynomials for the stationary free Jacobi process of parameters $\\lambda \\in ]0.1], \\theta = 1/2$. Hopelessly, apart from the case $\\lambda = 1$, the polynomials we derive are no longer orthogonal with respect to the spectral measure. As a matter of fact, we use the multiplicative renormalization to write down the corresponding orthogonality measure."}
{"category": "Math", "title": "On the dimension of the sheets of a reductive Lie algebra", "abstract": "This note is a corrigendum to the previous version arXiv:0711.2735v3 published in J. Lie Theory. As it has been recently pointed out to me by Alexander Premet, Remark 3 of arXiv:0711.2735v3 is incorrect. We verify in this note thanks to recent results of Premet and Topley (see arXiv:1301.4653) that Theorem 25 of arXiv:0711.2735v3 remains correct in spite of this error."}
{"category": "Math", "title": "Invarianten zusammenh\\\"angender Gruppen und die Cohen-Macaulay Eigenschaft", "abstract": "For G=SL_n or GL_n we construct representations V such that the invariant ring K[V]^G is not Cohen-Macaulay."}
{"category": "Math", "title": "Konstruktion von Invariantenringen ohne die Cohen-Macaulay Eigenschaft", "abstract": "We give examples of Non-Cohen-Macaulay invariant rings."}
{"category": "Math", "title": "Asymptotic cohomology of circular units", "abstract": "Let $F$ be a number field, abelian over the rational field, and fix a odd prime number $p$. Consider the cyclotomic $Z_p$-extension $F_\\infty/F$ and denote $F_n$ the ${n}^{\\rm th}$ finite subfield and $C_n$ its group of circular units. Then the Galois groups $G_{m,n}=\\Gal(F_m/F_n)$ act naturally on the $C_m$'s (for any $m\\geq n>> 0$). We compute the Tate cohomology groups $\\Hha^i(G_{m,n}, C_m)$ for $i=-1,0$ without assuming anything else neither on $F$ nor on $p$."}
{"category": "Math", "title": "Actions of semisimple Lie groups preserving a degenerate Riemannian metric", "abstract": "We prove a rigidity of the lightcone in Minkowski space. It is essentially the unique space endowed with a degenerate Riemannian metric, of lightlike type, and supporting an isometric non-proper action of a semi-simple group."}
{"category": "Math", "title": "Rational curves of degree 11 on a general quintic threefold", "abstract": "We prove that the incidence scheme of rational curves of degree 11 on quintic threefolds is irreducible. This implies a strong form of the Clemens conjecture in degree 11. Namely, on a general quintic threefold $F$ in $\\mathbb{P}^4$, there are only finitely many smooth rational curves of degree 11, and each curve $C$ is embedded in $F$ with normal bundle $\\mathcal{O}(-1) \\oplus \\mathcal{O}(-1)$. Moreover, in degree 11, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components on $F$."}
{"category": "Math", "title": "Constructing quantized enveloping algebras via inverse limits of finite dimensional algebras", "abstract": "It is known that a generalized $q$-Schur algebra may be constructed as a quotient of a quantized enveloping algebra $\\UU$ or its modified form $\\dot{\\UU}$. On the other hand, we show here that both $\\UU$ and $\\dot{\\UU}$ may be constructed within an inverse limit of a certain inverse system of generalized $q$-Schur algebras. Working within the inverse limit $\\hat{\\UU}$ clarifies the relation between $\\dot{\\UU}$ and $\\UU$. This inverse limit is a $q$-analogue of the linear dual $R[G]^*$ of the coordinate algebra of a corresponding linear algebraic group $G$."}
{"category": "Math", "title": "Superconnections and Parallel Transport", "abstract": "This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold $M$. A superpath in $M$ is, loosely speaking, a path in $M$ together with an odd vector field in $M$ along the path. We also develop a notion of parallel transport associated with a connection (a.k.a. covariant derivative) on a vector bundle over a \\emph{supermanifold} which is a direct generalization of the classical notion of parallel transport for connections over manifolds."}
{"category": "Math", "title": "A new proof of a theorem of Mansour and Sun", "abstract": "We give a new proof of a theorem of Mansour and Sun by using number theory and Rothe's identity."}
{"category": "Math", "title": "Dynamical compactifications of C^2", "abstract": "We find good dynamical compactifications for arbitrary polynomial mappings of C^2 and use them to show that the degree growth sequence satisfies a linear integral recursion formula. For maps of low topological degree we prove that the Green function is well behaved. For maps of maximum topological degree, we give normal forms."}
{"category": "Math", "title": "Analytic approximation of matrix functions and dual extremal functions", "abstract": "We study the question of the existence of a dual extremal function for a bounded matrix function on the unit circle in connection with the problem of approximation by analytic matrix functions. We characterize the class of matrix functions, for which a dual extremal function exists in terms of the existence of a maximizing vector of the corresponding Hankel operator and in terms of certain special factorizations that involve thematic matrix functions."}
{"category": "Math", "title": "The special linear representations of compact Lie groups", "abstract": "The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to define the integral of linear representation and extend this concept to special linear representation for next using."}
{"category": "Math", "title": "Representation dimension and finitely generated cohomology", "abstract": "We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension."}
{"category": "Math", "title": "On Lie groups as quasi-K\\\"ahler manifolds with Killing Norden metric", "abstract": "A 6-parametric family of 6--dimensional quasi-K\\\"ahler manifolds with Norden metric is constructed on a Lie group. This family is characterized geometrically."}
{"category": "Math", "title": "Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure", "abstract": "Almost hypercomplex pseudo-Hermitian manifolds are considered. Isotropic hyper-K\\\"ahler manifolds are introduced. A 4-parametric family of 4-dimensional manifolds of this type is constructed on a Lie group. This family is characterized geometrically. The condition a 4-manifold to be isotropic hyper-K\\\"ahler is given."}
{"category": "Math", "title": "*-Doubles and embedding of associative algebras in B(H)", "abstract": "We study the *-double functor between the categories of associative and involutive algebras. It is proved that an associative algebra is isomorphic to a subalgebra of a $C\\sp*$-algebra if and only if its *-double is *-isomorphic to a *-subalgebra of a $C\\sp*$-algebra. Some applications in the theory of operator algebras are presented. In particular each operator algebra is shown to be completely boundedly isomorphic to an operator algebra $B$ with the greatest $C\\sp*$-subalgebra consisting of the multiples of the unit and such that each element in $B$ is determined by its module up to a scalar multiple. We also study the maximal subalgebras of an operator algebra $A$ which are mapped into $C\\sp*$-algebras under completely bounded faithful representations of $A$."}
{"category": "Math", "title": "The growth at infinity of a sequence of entire functions of bounded orders", "abstract": "In this paper we shall consider the growth at infinity of a sequence $(P_n)$ of entire functions of bounded orders. Our results extend the results in \\cite{trong-tuyen2} for the growth of entire functions of genus zero. Given a sequence of entire functions of bounded orders $P_n(z)$, we found a nearly optimal condition, given in terms of zeros of $P_n$, for which $(k_n)$ that we have \\begin{eqnarray*} \\limsup_{n\\to\\infty}|P_n(z)|^{1/k_n}\\leq 1 \\end{eqnarray*} for all $z\\in \\mathbb C$ (see Theorem \\ref{theo5}). Exploring the growth of a sequence of entire functions of bounded orders lead naturally to an extremal function which is similar to the Siciak's extremal function (See Section 6)."}
{"category": "Math", "title": "Root Systems for Levi Factors and Borel-de Siebenthal Theory", "abstract": "Let $\\frak{m}$ be a Levi factor of a proper parabolic subalgebra $\\frak{q}$ of a complex semisimple Lie algebra $\\frak{g}$. Let $\\frak{t} = cent \\frak{m}$. A nonzero element $\\nu \\in \\frak{t}^*$ is called a $\\frak {t}$-root if the corresponding adjoint weight space $\\frak{g}_{nu}$ is not zero. If $\\nu$ is a $\\frak{t}$-root, some time ago we proved that $\\frak{g}_{\\nu}$ is $ad \\frak{m}$ irreducible. Based on this result we develop in the present paper a theory of $\\frak{t}$-roots which replicates much of the structure of classical root theory (case where $\\frak{t}$ is a Cartan subalgebra). The results are applied to obtain new reults about the structure of the nilradical $\\frak{n}$ of $\\frak{q}$. Also applications in the case where $dim \\frak{t}=1$ are used in Borel-de Siebenthal theory to determine irreducibility theorems for certain equal rank subalgebras of $\\frak{g}$. In fact the irreducibility results readily yield a proof of the main assertions of the Borel-de Siebenthal theory."}
{"category": "Math", "title": "The Lie module structure on the Hochschild cohomology groups of monomial algebras with radical square zero", "abstract": "We study the Lie module structure given by the Gerstenhaber bracket on the Hochschild cohomology groups of a monomial algebra with radical square zero. The description of such Lie module structure will be given in terms of the combinatorics of the quiver. The Lie module structure will be related to the classification of finite dimensional modules over simple Lie algebras when the quiver is given by the two loops and the ground field is the complex numbers."}
{"category": "Math", "title": "On Baxter Q-operators And Their Arithmetic Implications", "abstract": "We consider Baxter Q-operators for various versions of quantum affine Toda chain. The interpretation of eigenvalues of the finite Toda chain Baxter operators as local Archimedean L-functions proposed recently is generalized to the case of affine Lie algebras. We also introduce a simple generalization of Baxter operators and local L-functions compatible with this identification. This gives a connection of the Toda chain Baxter Q-operators with an Archimedean version of the Polya-Hilbert operator proposed by Berry-Kitting. We also elucidate the Dorey-Tateo spectral interpretation of eigenvalues of Q-operators. Using explicit expressions for eigenfunctions of affine/relativistic Toda chain we obtain an Archimedean analog of Casselman-Shalika-Shintani formula for Whittaker function in terms of characters."}
{"category": "Math", "title": "On a constrained reaction-diffusion system related to multiphase problems", "abstract": "We solve and characterize the Lagrange multipliers of a reaction-diffusion system in the Gibbs simplex of R^{N+1} by considering strong solutions of a system of parabolic variational inequalities in R^N. Exploring properties of the two obstacles evolution problem, we obtain and approximate a N-system involving the characteristic functions of the saturated and/or degenerated phases in the nonlinear reaction terms. We also show continuous dependence results and we establish sufficient conditions of non-degeneracy for the stability of those phase subregions."}
{"category": "Math", "title": "On the geometry of the first and second Painlev\\'e equations", "abstract": "In this paper we \\emph{explicitly} compute the transformation that maps the generic second order differential equation $y''= f(x, y, y')$ to the Painlev\\'e first equation $y''=6y^2+x$ (resp. the Painlev\\'e second equation ${y''=2 y^{3}+yx+ \\alpha}$). This change of coordinates, which is function of $f$ and its partial derivatives, does not exist for every $f$; it is necessary that the function $f$ satisfies certain conditions that define the equivalence class of the considered Painlev\\'e equation. In this work we won't consider these conditions and the existence issue is solved \\emph{on line} as follows: If the input equation is known then it suffices to specialize the change of coordinates on this equation and test by simple substitution if the equivalence holds. The other innovation of this work lies in the exploitation of discrete symmetries for solving the equivalence problem."}
{"category": "Math", "title": "Automorphism Groups of Finite p-Groups: Structure and Applications", "abstract": "This thesis has three goals related to the automorphism groups of finite $p$-groups. The primary goal is to provide a complete proof of a theorem showing that, in some asymptotic sense, the automorphism group of almost every finite $p$-group is itself a $p$-group. We originally proved this theorem in a paper with Martin; the presentation of the proof here contains omitted proof details and revised exposition. We also give a survey of the extant results on automorphism groups of finite $p$-groups, focusing on the order of the automorphism groups and on known examples. Finally, we explore a connection between automorphisms of finite $p$-groups and Markov chains. Specifically, we define a family of Markov chains on an elementary abelian $p$-group and bound the convergence rate of some of those chains."}
{"category": "Math", "title": "Presentations of Finite Simple Groups: Profinite and Cohomological Approaches", "abstract": "We prove the following three closely related results. The first is that every finite simple group has a profinite presentation with 2 generators and at most 18 relations. The second is that if G is a finite simple group, F a field and M an FG-module, then the dimension of the second cohomology group of G with coefficients in M is at most 17.5 times the dimension of M. The third result is that we may replace 17.5 by 18.5 as long as M is faithful irreducible G-module. These last two results answer conjectures of Holt."}
{"category": "Math", "title": "Characterization of non-degenerate plane curve singularities", "abstract": "We characterize plane curve germes non-degenerate in Kouchnirenko's sense in terms of characteristics and intersection multiplicities of branches."}
{"category": "Math", "title": "G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty", "abstract": "We introduce a new notion of G-normal distributions. This will bring us to a new framework of stochastic calculus of Ito's type (Ito's integral, Ito's formula, Ito's equation) through the corresponding G-Brownian motion. We will also present analytical calculations and some new statistical methods with application to risk analysis in finance under volatility uncertainty. Our basic point of view is: sublinear expectation theory is very like its special situation of linear expectation in the classical probability theory. Under a sublinear expectation space we still can introduce the notion of distributions, of random variables, as well as the notions of joint distributions, marginal distributions, etc. A particularly interesting phenomenon in sublinear situations is that a random variable Y is independent to X does not automatically implies that X is independent to Y. Two important theorems have been proved: The law of large number and the central limit theorem."}
{"category": "Math", "title": "Graded Sparse Graphs and Matroids", "abstract": "Sparse graphs and their associated matroids play an important role in rigidity theory, where they capture the combinatorics of generically rigid structures. We define a new family called {\\bf graded sparse graphs}, arising from generically pinned (completely immobilized) bar-and-joint frameworks and prove that they also form matroids. We address five problems on graded sparse graphs: {\\bf Decision}, {\\bf Extraction}, {\\bf Components}, {\\bf Optimization}, and {\\bf Extension}. We extend our {\\bf pebble game algorithms} to solve them."}
{"category": "Math", "title": "Rainbow number of matchings in regular bipartite graphs", "abstract": "Given a graph $G$ and a subgraph $H$ of $G$, let $rb(G,H)$ be the minimum number $r$ for which any edge-coloring of $G$ with $r$ colors has a rainbow subgraph $H$. The number $rb(G,H)$ is called the rainbow number of $H$ with respect to $G$. Denote $mK_2$ a matching of size $m$ and $B_{n,k}$ a $k$-regular bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|=n$ and $k\\leq n$. In this paper we give an upper and lower bound for $rb(B_{n,k},mK_2)$, and show that for given $k$ and $m$, if $n$ is large enough, $rb(B_{n,k},mK_2)$ can reach the lower bound. We also determine the rainbow number of matchings in paths and cycles."}
{"category": "Math", "title": "On the existence of a rainbow 1-factor in proper coloring of K_{rn}^{(r)}", "abstract": "El-Zanati et al proved that for any 1-factorization $\\mathcal{F}$ of the complete uniform hypergraph $\\mathcal {G}=K_{rn}^{(r)}$ with $r\\geq 2$ and $n\\geq 3$, there is a rainbow 1-factor. We generalize their result and show that in any proper coloring of the complete uniform hypergraph $\\mathcal {G}=K_{rn}^{(r)}$ with $r\\geq 2$ and $n\\geq 3$, there is a rainbow 1-factor."}
{"category": "Math", "title": "Partitioning complete graphs by heterochromatic trees", "abstract": "A {\\it heterochromatic tree} is an edge-colored tree in which any two edges have different colors. The {\\it heterochromatic tree partition number} of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum positive integer $p$ such that whenever the edges of the graph $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $p$ vertex-disjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of an $r$-edge-colored complete graph."}
{"category": "Math", "title": "Lifting of nilpotent contractions", "abstract": "It is proved that that every nilpotent contraction in a quotient C*-algebra can be lifted to a nilpotent contraction. As a consequence we get that the universal C*-algebra generated by a nilpotent contraction is projective. This answers the question posed by T. Loring."}
{"category": "Math", "title": "Finiteness theorem on Blow-semialgebraic triviality for a family of 3-dimensional algebraic sets", "abstract": "In this paper we introduce the notion of Blow-semialgebraic triviality consistent with a compatible filtration for an algebraic family of algebraic sets, as an equisingularity for real algebraic singularities. Given an algebraic family of 3-dimensional algebraic sets defined over a nonsingular algebraic variety, we show that there is a finite subdivision of the parameter algebraic set into connected Nash manifolds over which the family admits a Blow-semialgebraic trivialisation consistent with a compatible filtration. We show a similar result on finiteness also for a Nash family of 3-dimensional Nash sets through the Artin-Mazur theorem. As a corollary of the arguments in their proofs, we have a finiteness theorem on semialgebraic types of polynomial mappings from the 2-dimensional Euclidean space to the p-diemnsional Euclidean space."}
{"category": "Math", "title": "A residue criterion for strong holomorphicity", "abstract": "We give a local criterion in terms of a residue current for strong holomorphicity of a meromorphic function on an arbitrary pure-dimensional analytic variety. This generalizes a result by A Tsikh for the case of a reduced complete intersection."}
{"category": "Math", "title": "Weighted Sobolev L2 estimates for a class of Fourier integral operators", "abstract": "In this paper we develop elements of the global calculus of Fourier integral operators in $R^n$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev $L^2$ estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of smoothing estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions."}
{"category": "Math", "title": "Monotone unitary families", "abstract": "A unitary family is a family of unitary operators $U(x)$ acting on a finite dimensional hermitian vector space, depending analytically on a real parameter $x$. It is monotone if $\\frac1i U'(x)U(x)^{-1}$ is a positive operator for each $x$. We prove a number of results generalizing standard theorems on the spectral theory of a single unitary operator $U_0$, which correspond to the 'commutative' case $U(x)=e^{ix}U_0$. Also, for a two-parameter unitary family -- for which there is no analytic perturbation theory -- we prove an implicit function type theorem for the spectral data under the assumption that the family is monotone in one argument."}
{"category": "Math", "title": "On the link pattern distribution of quarter-turn symmetric FPL configurations", "abstract": "We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions."}
{"category": "Math", "title": "What type of dynamics arise in E_0-dilations of commuting quantum Markov process?", "abstract": "Let H be a separable Hilbert space. Given two strongly commuting CP_0-semigroups $\\phi$ and $\\theta$ on B(H), there is a Hilbert space K containing H and two (strongly) commuting E_0-semigroups $\\alpha$ and $\\beta$ such that $\\phi_s \\circ \\theta_t (P_H A P_H) = P_H \\alpha_s \\circ \\beta_t (A) P_H$ for all s,t and all A in B(K). In this note we prove that if $\\phi$ is not an automorphism semigroup then $\\alpha$ is cocycle conjugate to the minimal *-endomorphic dilation of $\\phi$, and that if $\\phi$ is an automorphism semigroup then $\\alpha$ is also an automorphism semigroup. In particular, we conclude that if $\\phi$ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional) then $\\alpha$ is a type I E_0-semigroup."}
{"category": "Math", "title": "A family of martingales generated by a process with independent increments", "abstract": "An explicit procedure to construct a family of martingales generated by a process with independent increments is presented. The main tools are the polynomials that give the relationship between the moments and cumulants, and a set of martingales related to the jumps of the process called Teugels martingales"}
{"category": "Math", "title": "Uniqueness and disjointness of Klyachko models", "abstract": "We show the uniqueness and disjointness of Klyachko models for GL(n,F) over a non-archimedean local field F. This completes, in particular, the study of Klyachko models on the unitary dual. Our local results imply a global rigidity property for the discrete automorphic spectrum."}
{"category": "Math", "title": "E-dilation of strongly commuting CP-semigroups (the nonunital case)", "abstract": "In a previous paper, we showed that every strongly commuting pair of CP_0-semigroups on a von Neumann algebra (acting on a separable Hilbert space) has an E_0-dilation. In this paper we show that if one restricts attention to the von Neumann algebra B(H) then the unitality assumption can be dropped, that is, we prove that every pair of strongly commuting CP-semigroups on B(H) has an E-dilation. The proof is significantly different from the proof for the unital case, and is based on a construction of Ptak from the 1980's designed originally for constructing a unitary dilation to a two-parameter contraction semigroup."}
{"category": "Math", "title": "The Method of Normalized Correlations - A Fast Alternative to Maximum Likelihood Estimation for Random Processes and Isotropic Random Fields with Short-Range Dependence", "abstract": "This paper has been withdrawn by the authors, due the copyright policy of the journal it has been submited to."}
{"category": "Math", "title": "Simple Lie algebras of small characteristic VI. Completion of the classification", "abstract": "Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of F characteristic p>3. We prove that if the p-envelope of L in the derivation algebra of L contains nonstandard tori of maximal dimension, then p=5 and L is isomorphic to one of the Melikian algebras. Together with our earlier results this implies that any finite-dimensional simple Lie algebra over F is of classical, Cartan or Melikian type."}
{"category": "Math", "title": "Cubic-matrix splines and second-order matrix models", "abstract": "We discuss the direct use of cubic-matrix splines to obtain continuous approximations to the unique solution of matrix models of the type $Y''(x) = f(x,Y(x))$. For numerical illustration, an estimation of the approximation error, an algorithm for its implementation, and an example are given."}
{"category": "Math", "title": "Carmichael number variable relations: three-prime Carmichael numbers up to 10^24", "abstract": "Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding three-prime Carmichael numbers is described, with its implementation up to $10^{24}$. Statistics relevant to the distribution of three-prime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [A.Granville and C.Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comp. 71 (2001), 883-908]."}
{"category": "Math", "title": "Natural Number Arithmetic in the Theory of Finite Sets", "abstract": "We describe a theory of finite sets, and investigate the analogue of Dedekind's theory of natural number systems (simply infinite systems) in this theory. Unlike the infinitary case, in our theory, natural number systems come in differing lengths and with different closure properties. We give examples of natural number systems incomparable in length; we define hierarchies of natural number systems closed under increasingly powerful functions; and we describe a method by which to construct natural number systems with given closure properties. These natural number systems form natural models for various systems of weak arithmetic."}
{"category": "Math", "title": "Spectral asymptotics for arithmetic quotients of SL(n,R)/SO(n)", "abstract": "In this paper we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space S=SL(n,R)/SO(n). In particular, we obtain Weyl's law with an estimation on the remainder term. This extends results of Duistermaat-Kolk-Varadarajan on spectral asymptotics for compact locally symmetric spaces to this non-compact setting."}
{"category": "Math", "title": "The topology of symplectic circle bundles", "abstract": "We consider circle bundles over compact three-manifolds with symplectic total spaces. We show that the base of such a space must be irreducible or the product of the two-sphere with the circle. We then deduce that such a bundle admits a symplectic form if and only if it admits one that is invariant under the circle action in three special cases: namely if the base is Seifert fibered, has vanishing Thurston norm, or if the total space admits a Lefschetz fibration."}
{"category": "Math", "title": "Two way subtable sum problems and quadratic Groebner bases", "abstract": "Hara, Takemura and Yoshida discuss toric ideals arising from two way subtable sum problems and shows that these toric ideals are generated by quadratic binomials if and only if the subtables are either diagonal or triangular. In the present paper, we show that if the subtables are either diagonal or triangular, then their toric ideals possess quadratic Groebner bases."}
{"category": "Math", "title": "Holomorphic fillability and cohomology", "abstract": "I have withdrawn the paper, after having incorporated it into the paper arXiv:0712.3484. In the meantime I have discovered that the main theorem proved in the paper had already been proved by Bungart."}
{"category": "Math", "title": "Stochastic convergence of random search to fixed size Pareto set approximations", "abstract": "This paper presents the first convergence result for random search algorithms to a subset of the Pareto set of given maximum size k with bounds on the approximation quality. The core of the algorithm is a new selection criterion based on a hypothetical multilevel grid on the objective space. It is shown that, when using this criterion for accepting new search points, the sequence of solution archives converges with probability one to a subset of the Pareto set that epsilon-dominates the entire Pareto set. The obtained approximation quality epsilon is equal to the size of the grid cells on the finest level of resolution that allows an approximation with at most k points within the family of grids considered. While the convergence result is of general theoretical interest, the archiving algorithm might be of high practical value for any type iterative multiobjective optimization method, such as evolutionary algorithms or other metaheuristics, which all rely on the usage of a finite on-line memory to store the best solutions found so far as the current approximation of the Pareto set."}
{"category": "Math", "title": "On conjugacy of unipotent elements in finite groups of Lie type", "abstract": "Let $\\bfG$ be a connected reductive algebraic group defined over $\\F_q$, where $q$ is a power of a prime $p$ that is good for $\\bfG$. Let $F$ be the Frobenius morphism associated with the $\\FF_q$-structure on $\\bfG$ and set $G = \\bfG^F$, the fixed point subgroup of $F$. Let $\\bfP$ be an $F$-stable parabolic subgroup of $\\bfG$ and let $\\bfU$ be the unipotent radical of $\\bfP$; set $P = \\bfP^F$ and $U = \\bfU^F$. Let $G_\\uni$ be the set of unipotent elements in $G$. In this note we show that the number of conjugacy classes of $U$ in $G_\\uni$ is given by a polynomial in $q$ with integer coefficients."}
{"category": "Math", "title": "Free holomorphic functions and interpolation", "abstract": "In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna-Pick interpolation problem for analytic functions with positive real parts on the open unit disc. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball on C^n."}
{"category": "Math", "title": "Composition with a two variable function", "abstract": "We compute the motivic nearby cycles of functions obtained by composition of two functions with distinct sets of variables with a two variable function"}
{"category": "Math", "title": "Stochastic Integrals and Abelian Processes", "abstract": "We study triangulation schemes for the joint kernel of a diffusion process with uniformly continuous coefficients and an adapted, non-resonant Abelian process. The prototypical example of Abelian process to which our methods apply is given by stochastic integrals with uniformly continuous coeffcients. The range of applicability includes also a broader class of processes of practical relevance, such as the sup process and certain discrete time summations we discuss. We discretize the space coordinate in uniform steps and assume that time is either continuous or finely discretized as in a fully explicit Euler method and the Courant condition is satisfied. We show that the Fourier transform of the joint kernel of a diffusion and a stochastic integral converges in a uniform graph norm associated to the Markov generator. Convergence also implies smoothness properties for the Fourier transform of the joint kernel. Stochastic integrals are straightforward to define for finite triangulations and the convergence result gives a new and entirely constructive way of defining stochastic integrals in the continuum. The method relies on a reinterpretation and extension of the classic theorems by Feynman-Kac, Girsanov, Ito and Cameron-Martin, which are also re-obtained. We make use of a path-wise analysis without relying on a probabilistic interpretation. The Fourier representation is needed to regularize the hypo-elliptic character of the joint process of a diffusion and an adapted stochastic integral. The argument extends as long as the Fourier analysis framework can be generalized. This condition leads to the notion of non-resonant Abelian process."}
{"category": "Math", "title": "Almost positive curvature on the Gromoll-Meyer sphere", "abstract": "Gromoll and Meyer have represented a certain exotic 7-sphere $\\Sigma^7$ as a biquotient of the Lie group $G = Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on $\\Sigma^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $\\geq 1$ which we describe explicitly."}
{"category": "Math", "title": "Quasiperiodic Spectra and Orthogonality for Iterated Function System Measures", "abstract": "We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a ``small perturbation'' of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs $X$ admit spectral duality. When $X$ is given, we identify geometric conditions on $X$ for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices."}
{"category": "Math", "title": "An asymptotic behavior of the dilatation for a family of pseudo-Anosov braids", "abstract": "The dilatation of a pseudo-Anosov braid is a conjugacy invariant. In this paper, we study the dilatation of a special family of pseudo-Anosov braids. We prove an inductive formula to compute their dilatation, a monotonicity and an asymptotic behavior of the dilatation for this family of braids. We also give an example of a family of pseudo-Anosov braids with arbitrarily small dilatation such that the mapping torus obtained from such braid has 2 cusps and has an arbitrarily large volume."}
{"category": "Math", "title": "Absolutely Indecomposable Modules", "abstract": "A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more general result about R-modules over a large class of commutative rings R with endomorphism ring R which remains the same when passing to a generic extension of the universe. It turns out that `large' in this context has the precise meaning, namely being smaller then the first omega-Erdos cardinal defined below. We will first apply result on large rigid trees with a similar property established by Shelah in 1982, and will prove the existence of related ` R_omega-modules' (R-modules with countably many distinguished submodules) and finally pass to R-modules. The passage through R_omega-modules has the great advantage that the proofs become very transparent essentially using a few `linear algebra' arguments accessible also for graduate students. The result gives a new construction of indecomposable modules in general using a counting argument."}
{"category": "Math", "title": "Stacky Abelianization of an Algebraic Group", "abstract": "Let G be a connected algebraic group and let [G,G] be its commutator subgroup. We prove a conjecture of Drinfeld about the existence of a connected etale group cover H of [G,G], characterized by the following properties: every central extension of G, by a finite etale group scheme, splits over H, and the commutator map of G lifts to H. We prove, moreover, that the quotient stack of G by the natural action of H is the universal Deligne-Mumford Picard stack to which G maps."}
{"category": "Math", "title": "A noncommutative Atiyah-Patodi-Singer index theorem in KK-theory", "abstract": "We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Kreiger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C*-algebras."}
{"category": "Math", "title": "Diamonds", "abstract": "We prove, e.g., that if lambda=chi^+=2^chi and S subseteq {delta<lambda:cf(delta) neq cf(chi)} is stationary then diamondsuit_lambda holds true."}
{"category": "Math", "title": "Basic Subgroups and Freeness, A Counterexample", "abstract": "We construct a non-free but aleph_1-separable, torsion-free abelian group G with a pure free subgroup B such that all subgroups of G disjoint from B are free and such that G/B is divisible. This answers a question of Irwin and shows that a theorem of Blass and Irwin cannot be strengthened so as to give an exact analog for torsion-free groups of a result proved for p-groups by Benabdallah and Irwin."}
{"category": "Math", "title": "Pseudo focal points along Lorentzian geodesics and Morse index", "abstract": "Given a Lorentzian manifold $(M,g)$, a geodesic $\\gamma$ in $M$ and a timelike Jacobi field $\\mathcal Y$ along $\\gamma$, we introduce a special class of instants along $\\gamma$ that we call $\\mathcal Y$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\\mathcal Y$-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field $\\mathcal Y$ is obtained as the restriction of a globally defined timelike Killing vector field."}
{"category": "Math", "title": "Statistical Inference for Disordered Sphere Packings", "abstract": "Sphere packings are essential to the development of physical models for powders, composite materials, and the atomic structure of the liquid state. There is a strong scientific need to be able to assess the fit of packing models to data, but this is complicated by the lack of formal probabilistic models for packings. Without formal models, simulation algorithms and collections of physical objects must be used as models. Identification of common aspects of different realizations of the same packing process requires the use of new descriptive statistics, many of which have yet to be developed. Model assessment will require the use of large samples of independent and identically distributed realizations, rather than the large single stationary realizations found in conventional spatial statistics. The development of procedures for model assessment will resemble the development of thermodynamic models, and will be based on much exploration and experimentation rather than on extensions of established statistical methods."}
{"category": "Math", "title": "Tie-points and fixed-points in N^*", "abstract": "A point x is a (bow) tie-point of a space X if X setminus {x} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of betaN setminus N and in the recent study of (precisely) 2-to-1 maps on betaN setminus N . In these cases the tie-points have been the unique fixed point of an involution on betaN setminus N. This paper is motivated by the search for 2-to-1 maps and obtaining tie-points of strikingly differing characteristics."}
{"category": "Math", "title": "More on Tie-points and homeomorphism in N^*", "abstract": "A point x is a (bow) tie-point of a space X if X setminus {x} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of betaN setminus N= N^* and in the recent study of (precisely) 2-to-1 maps on N^*. In these cases the tie-points have been the unique fixed point of an involution on N^*. One application of the results in this paper is the consistency of there being a 2-to-1 continuous image of N^* which is not a homeomorph of N^* ."}
{"category": "Math", "title": "A free-boundary problem for the evolution $p$-Laplacian equation with a combustion boundary condition", "abstract": "We study the existence, uniqueness and regularity of solutions of the equation $f_t = \\Delta_p f = \\text{div} (|Df|^{p-2} Df)$ under over-determined boundary conditions $f = 0$ and $|Df| = 1$. We show that if the initial data is concave and Lipschitz with a bounded and convex support, then the problem admits a unique solution which exists until it vanishes identically. Furthermore, the free-boundary of the support of $f$ is smooth for all positive time."}
{"category": "Math", "title": "Karp height of models of stable theories", "abstract": "A trichotomy theorem for countable, stable, unsuperstable theories is offered. We develop the notion of a `regular ideal' of formulas and study types that are minimal with respect to such an ideal."}
{"category": "Math", "title": "Generalized E-Algebras via lambda-Calculus I", "abstract": "An R-algebra A is called E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra End_RA of the R-module {}_R A, taking any a in A to the right multiplication a_r in End_R A by a is an isomorphism of algebras. In this case {}_R A is called an E(R)-module. E(R)-algebras come up naturally in various topics of algebra, so it's not surprising that they were investigated thoroughly in the last decade. Despite some efforts it remained an open question whether proper generalized E(R)-algebras exist. These are R-algebras A isomorphic to End_R A but not under the above canonical isomorphism, so not E(R)-algebras. This question was raised about 30 years ago (for R=Z) by Phil Schultz and we will answer it. For PIDs R of characteristic 0 that are neither quotient fields nor complete discrete valuation rings - we will establish the existence of generalized E(R)-algebras. It can be shown that E(R)-algebras over rings R that are complete discrete valuation rings or fields must trivial (copies of R). The main tool is an interesting connection between lambda-calculus (used in theoretical computer sciences) and algebra. It seems reasonable to divide the work into two parts, in this paper we will work in V=L (Godel universe) hence stronger combinatorial methods make the final arguments more transparent. The proof based entirely on ordinary set theory (the axioms of ZFC) will appear in a subsequent paper."}
{"category": "Math", "title": "The inverse inertia problem for graphs", "abstract": "Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, and investigate restrictions on the inertia set of any graph."}
{"category": "Math", "title": "Ramsey properties of subsets of $\\mathbb{N}$", "abstract": "We associate ergodic properties to some subsets of the natural numbers. For any given family of subsets of the natural numbers one may study the question of occurrence of certain \"algebraic patterns\" in every subset in the family. By \"algebraic pattern\" we mean a set of solutions of a system of diophantine equations. In this work we investigate a concrete family of subsets - WM sets. These sets are characterized by the property that the dynamical systems associated to such sets are \"weakly mixing\", and as such they represent a broad family of randomly constructed subsets of (\\mathbb{N}). We find that certain systems of equations are solvable within every WM set, and our subject is to learn which systems have this property. We give a complete characterization of linear diophantine systems which are solvable within every WM set. In addition we study some non-linear equations and systems of equations with regard to the question of solvability within every WM set."}
{"category": "Math", "title": "Non-existence of unbounded Fatou components of a meromorphic function", "abstract": "This paper is devoted to establish sufficient conditions under which a transcendental meromorphic function has no unbounded Fatou components and to extend some results for entire functions to meromorphic functions. Actually, we shall mainly discuss non-existence of unbounded wandering domains of a meromorphic function. The case for a composition of finitely many meromorphic function with at least one of them being transcendental can be also investigated in the argument of this paper."}
{"category": "Math", "title": "The centers of spin symmetric group algebras and Catalan numbers", "abstract": "Generalizing the work of Farahat-Higman on symmetric groups, we describe the structures of the even centers Z_n of integral spin symmetric group superalgebras, which lead to universal algebras termed as the spin FH-algebras. A connection between the odd Jucys-Murphy elements and the Catalan numbers is developed and then used to determine the algebra generators of the spin FH-algebras and of the even centers Z_n."}
{"category": "Math", "title": "Cayley graphs formed by conjugate generating sets of S_n", "abstract": "We investigate subsets of the symmetric group with structure similar to that of a graph. The trees of these subsets correspond to minimal conjugate generating sets of the symmetric group. There are two main theorems in this paper. The first is a characterization of minimal conjugate generating sets of S_n. The second is a generalization of a result due to Feng characterizing the automorphism groups of the Cayley graphs formed by certain generating sets composed of cycles. We compute the full automorphism groups subject to a weak condition and conjecture that the characterization still holds without the condition. We also present some computational results in relation to hamiltonicity of Cayley graphs, including a generalization of the work on quasi-hamiltonicity by Gutin and Yeo to undirected graphs."}
{"category": "Math", "title": "Regular representations of the quantum groups at roots of unity", "abstract": "We study the bimodule structure of the quantum function algebra at roots of 1 and prove that it admits an increasing filtration with factors isomorphic to the tensor products of the dual of Weyl modules $V_\\lambda^* \\otimes V_{- \\omega_0 \\lambda}^*$. As an application we compute the 0-th Hochschild cohomology of the function algebra at roots of 1."}
{"category": "Math", "title": "Some algebraic invariants of mixed product ideals", "abstract": "We compute some algebraic invariants (e.g. depth, Castelnuovo - Mumford regularity) for a special class of monomial ideals, namely the ideals of mixed products. As a consequence, we characterize the Cohen-Macaulay ideals of mixed products."}
{"category": "Math", "title": "A plane sextic with finite fundamental group", "abstract": "We analyze irreducible plane sextics whose fundamental group factors to $D_{14}$. We produce explicit equations for all curves and show that, in the simplest case of the set of singularities $3A_6$, the group is $D_{14}\\times Z_3$."}
{"category": "Math", "title": "On irreducible sextics with non-abelian fundamental group", "abstract": "We calculate the fundamental groups $\\pi=\\pi_1(P^2\\setminus B)$ for all irreducible plane sextics $B\\subset\\P^2$ with simple singularities for which $\\pi$ is known to admit a dihedral quotient $D_{10}$. All groups found are shown to be finite, two of them being of large order: 960 and 21600."}
{"category": "Math", "title": "Robust Global Stabilizability by Means of Sampled-Data Control with Positive Sampling Rate", "abstract": "This work proposes a notion of robust reachability of one set from another set under constant control. This notion is used to construct a control strategy, involving sequential set-to-set reachability, which guarantees robust global stabilization of nonlinear sampled data systems with positive sampling rate. Sufficient conditions for robust reachability of one set from another under constant control are also provided. Finally, the proposed method is illustrated through two examples."}
{"category": "Math", "title": "On the semi-regular module and vertex operator algebras", "abstract": "We prove a conjecture stated in a previous paper by the author about the existence of canonical filtrations for a family of vertex operator algebras in rational levels."}
{"category": "Math", "title": "Width of l^p balls", "abstract": "We say a map f:X \\to Y is an \\epsilon-embedding if it is continuous and the diameter of the fibres is less than \\epsilon. This type of maps is used in the notion of Urysohn width (sometimes referred to as Alexandrov width), a_n(X). It is the smallest real number such that there exists an \\epsilon-embedding from X to a n-dimensional polyhedron. Surprisingly few estimations of these numbers can be found, and one of the aims of this paper is to present some. Following Gromov, we take the slightly different point of view by looking at the smallest dimension n for which there exists a \\epsilon-embedding to a polyhedron of dimension n. While bounds are obtained using Hadamard matrices, the Borsuk-Ulam theorem, the filling radius of spheres, and lower bounds for the diameter of sets of n+1 points not contained in a hemisphere (obtained by methods very close to those of Ivanov and Pichugov). We are also able to give a complete description in dimension 3 for 1 \\leq p \\leq 2."}
{"category": "Math", "title": "Control Lyapunov Functions and Stabilization by Means of Continuous Time-Varying Feedback", "abstract": "For a general time-varying system, we prove that existence of an \"Output Robust Control Lyapunov Function\" implies existence of continuous time-varying feedback stabilizer, which guarantees output asymptotic stability with respect to the resulting closed-loop system. The main results of the present work constitute generalizations of a well-known result towards feedback stabilization due to J. M. Coron and L. Rosier concerning stabilization of autonomous systems by means of time-varying periodic feedback."}
{"category": "Math", "title": "Krammer representations for complex braid groups", "abstract": "Let B be the generalized braid group associated to some finite complex reflection group. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer representation of the classical braid groups, and is thus a good candidate in view of proving the linearity of these groups. We decompose this representation in irreducible components and compute its Zariski closure, as well as its restriction to parabolic subgroups. We prove that it is faithful when W is a Coxeter group of type ADE and odd dihedral types, and conjecture its faithfulness when W has a single class of reflections. If true, this conjecture would imply various group-theoretic properties for these groups, that we prove separately to be true for the other groups."}
{"category": "Math", "title": "Long range scattering for the Maxwell-Schr\"odinger system with arbitrarily large asymptotic data", "abstract": "We review the proof of existence and uniqueness of solutions of the Maxwell-Schr\"odinger system in a neighborhood of infinity in time, with prescribed asymptotic behaviour defined in terms of asymptotic data, without any restriction on the size of those data. That result is the basic step in the construction of modified wave operators for the Maxwell-Schr\"odinger system."}
{"category": "Math", "title": "A Uniqueness and Periodicity Result for Solutions of Elliptic Equations in Unbounded Domains", "abstract": "We proof a uniqueness and periodicity theorem for bounded solutions of uniformly elliptic equations in certain unbounded domains."}
{"category": "Math", "title": "Poles of the topological zeta function associated to an ideal in dimension two", "abstract": "To an ideal in $\\mathbb{C}[x,y]$ one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an embedded resolution of the curve. In this paper we will study two questions about the poles of this zeta function. First, we will give a criterion to determine whether or not a candidate pole is a pole. It turns out that we can know this immediately by looking at the intersection diagram of the principalization, together with the numerical data of the exceptional curves. Afterwards we will completely describe the set of rational numbers that can occur as poles of a topological zeta function associated to an ideal in dimension two. The same results are valid for related zeta functions, as for instance the motivic zeta function."}
{"category": "Math", "title": "Positive association in the fractional fuzzy Potts model", "abstract": "A fractional fuzzy Potts measure is a probability distribution on spin configurations of a finite graph $G$ obtained in two steps: first a subgraph of $G$ is chosen according to a random cluster measure $\\phi_{p,q}$, and then a spin ($\\pm1$) is chosen independently for each component of the subgraph and assigned to all vertices of that component. We show that whenever $q\\geq1$, such a measure is positively associated, meaning that any two increasing events are positively correlated. This generalizes earlier results of H\\\"{a}ggstr\\\"{o}m [Ann. Appl. Probab. 9 (1999) 1149--1159] and H\\\"{a}ggstr\\\"{o}m and Schramm [Stochastic Process. Appl. 96 (2001) 213--242]."}
{"category": "Math", "title": "On pointed Hopf algebras associated with the Mathieu simple groups", "abstract": "Let G be a Mathieu simple group, s in G, O_s the conjugacy class of s and \\rho an irreducible representation of the centralizer of s. We prove that either the Nichols algebra B(O_s,\\rho) is infinite-dimensional or the braiding of the Yetter-Drinfeld module M(O_s, \\rho) is negative. We also show that if G=M22 or M24, then the group algebra of G is the only (up to isomorphisms) finite-dimensional complex pointed Hopf algebra with group-likes isomorphic to G."}
{"category": "Math", "title": "Object oriented data analysis: Sets of trees", "abstract": "Object oriented data analysis is the statistical analysis of populations of complex objects. In the special case of functional data analysis, these data objects are curves, where standard Euclidean approaches, such as principal component analysis, have been very successful. Recent developments in medical image analysis motivate the statistical analysis of populations of more complex data objects which are elements of mildly non-Euclidean spaces, such as Lie groups and symmetric spaces, or of strongly non-Euclidean spaces, such as spaces of tree-structured data objects. These new contexts for object oriented data analysis create several potentially large new interfaces between mathematics and statistics. This point is illustrated through the careful development of a novel mathematical framework for statistical analysis of populations of tree-structured objects."}
{"category": "Math", "title": "Klein paradox and Scattering theory for the semi-classical Dirac equation", "abstract": "We study the Klein paradox for the semi-classical Dirac operator on $\\R$ with potentials having constant limits, not necessarily the same at infinity. Using the complex WKB method, the time-independent scattering theory in terms of incoming and outgoing plane wave solutions is established. The corresponding scattering matrix is unitary. We obtain an asymptotic expansion, with respect to the semi-classical parameter $h$, of the scattering matrix in the cases of the Klein paradox, the total transmission and the total reflection. Finally, we treat the scattering problem in the zero mass case."}
{"category": "Math", "title": "Powers of sequences and recurrence", "abstract": "We study recurrence, and multiple recurrence, properties along the $k$-th powers of a given set of integers. We show that the property of recurrence for some given values of $k$ does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory concerning additive basis of natural numbers. Moreover, motivated by a result of Kamae and Mend\\`es-France, that links single recurrence with uniform distribution properties of sequences, we look for an analogous result dealing with higher order recurrence and make a related conjecture."}
{"category": "Math", "title": "Invariant functions in Denjoy-Carleman classes", "abstract": "Let $V$ be a real finite dimensional representation of a compact Lie group $G$. It is well-known that the algebra $\\mathbb R[V]^G$ of $G$-invariant polynomials on $V$ is finitely generated, say by $\\sigma_1,...,\\sigma_p$. Schwarz proved that each $G$-invariant $C^\\infty$-function $f$ on $V$ has the form $f=F(\\sigma_1,...,\\sigma_p)$ for a $C^\\infty$-function $F$ on $\\mathbb R^p$. We investigate this representation within the framework of Denjoy-Carleman classes. One can in general not expect that $f$ and $F$ lie in the same Denjoy-Carleman class $C_M$ (with $M=(M_k)$). For finite groups $G$ and (more generally) for polar representations $V$ we show that for each $G$-invariant $f$ of class $C_M$ there is an $F$ of class $C_N$ such that $f=F(\\sigma_1,...,\\sigma_p)$, if $N$ is strongly regular and satisfies $N_k \\ge M_{km} \\ep^{k+1}$, for all $k$, with $m$ an (explicitly known) integer depending only on the representation and $\\epsilon>0$ independent of $k$. In particular, each $G$-invariant $(1+\\delta)$-Gevrey function $f$ has the form $f=F(\\sigma_1,...,\\sigma_p)$ for a $(1+\\delta m)$-Gevrey function $F$. Applications to equivariant functions and basic differential forms are given."}
{"category": "Math", "title": "Hopf algebroids and secondary characteristic classes", "abstract": "We study a Hopf algebroid, $\\calh$, naturally associated to the groupoid $U_n^\\delta\\ltimes U_n$. We show that classes in the Hopf cyclic cohomology of $\\calh$ can be used to define secondary characteristic classes of trivialized flat $U_n$-bundles. For example, there is a cyclic class which corresponds to the universal transgressed Chern character and which gives rise to the continuous part of the $\\rho$-invariant of Atiyah-Patodi-Singer. Moreover, these cyclic classes are shown to extend to the K-theory of the associated $C^{*}$-algebra. This point of view gives leads to homotopy invariance results for certain characteristic numbers. In particular, we define a subgroup of the cohomology of a group analogous to the Gelfand-Fuchs classes described by Connes, \\cite{connes:transverse}, and show that the higher signatures associated to them are homotopy invariant."}
{"category": "Math", "title": "The distribution of polynomials over finite fields, with applications to the Gowers norms", "abstract": "In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P : F^n -> F is poorly-distributed only if P is determined by the values of a few polynomials of lower degree, in which case we say that P has small rank. We give several applications of this result, paying particular attention to consequences for the theory of the so-called Gowers norms. We establish an inverse result for the Gowers U^{d+1}-norm of functions of the form f(x)= e_F(P(x)), where P : F^n -> F is a polynomial of degree less than F, showing that this norm can only be large if f correlates with e_F(Q(x)) for some polynomial Q : F^n -> F of degree at most d. The requirement deg(P) < |F| cannot be dropped entirely. Indeed, we show the above claim fails in characteristic 2 when d = 3 and deg(P)=4, showing that the quartic symmetric polynomial S_4 in F_2^n has large Gowers U^4-norm but does not correlate strongly with any cubic polynomial. This shows that the theory of Gowers norms in low characteristic is not as simple as previously supposed. This counterexample has also been discovered independently by Lovett, Meshulam, and Samorodnitsky. We conclude with sundry other applications of our main result, including a recurrence result and a certain type of nullstellensatz."}
{"category": "Math", "title": "On the global well-posedness of the Boussinesq system with zero viscosity", "abstract": "In this paper we prove the global well-posedness of the two-dimensional Boussinesq system with zero viscosity for rough initial data."}
{"category": "Math", "title": "Towards a theory of classification", "abstract": "The well-known difficulties arising in a classification which is not set-theoretically trivial---involving what is sometimes called a non-smooth quotient---have been overcome in a striking way in the theory of operator algebras by the use of what might be called a classification functor---the very existence of which is already a surprise. Here the notion of such a functor is developed abstractly, and a number of examples are considered (including those which have arisen for various classes of operator algebras)."}
{"category": "Math", "title": "Polynomial largeness of sumsets and totally ergodic sets", "abstract": "We prove that a sumset of a TE subset of (\\N) (these sets can be viewed as \"aperiodic\" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \\subset \\N ) a TE set, for any (p(n) \\in \\Z[n]: \\deg{p(n)} > 0, p(n) \\to_{n \\to \\infty} \\infty ) and any subset (B \\subset \\N ) of positive upper density we have (R_p = A+B \\cap \\{p(n) | n \\in \\N \\} \\neq \\emptyset). For (A ) a WM set (subclass of TE sets) we prove that (R_p ) has lower density 1. In addition we obtain a generalization of the latter result to the case of several polynomials and several WM sets."}
{"category": "Math", "title": "An Integral Measure of Aging/Rejuvenation for Repairable and Non-repairable Systems", "abstract": "This paper introduces a simple index that helps to assess the degree of aging or rejuvenation of a (non)repairable system. The index ranges from -1 to 1 and is negative for the class of decreasing failure rate distributions (or deteriorating point processes) and is positive for the increasing failure rate distributions (or improving point processes). The introduced index is distribution free."}
{"category": "Math", "title": "Annihilators of permutation modules", "abstract": "Permutation modules are fundamental in the representation theory of symmetric groups $\\Sym_n$ and their corresponding Iwahori--Hecke algebras $\\He = \\He(\\Sym_n)$. We find an explicit combinatorial basis for the annihilator of a permutation module in the \"integral\" case -- showing that it is a cell ideal in G.E. Murphy's cell structure of $\\He$. The same result holds whenever $\\He$ is semisimple, but may fail in the non-semisimple case."}
{"category": "Math", "title": "Topological dynamics of the Weil-Petersson geodesic flow", "abstract": "We prove topological transitivity for the Weil Petersson geodesic flow for two-dimensional moduli spaces of hyperbolic structures. Our proof follows a new approach that exploits the density of singular unit tangent vectors, the geometry of cusps and convexity properties of negative curvature. We also show that the Weil Petersson geodesic flow has: horseshoes, invariant sets with positive topological entropy, and that there are infinitely many hyperbolic closed geodesics, whose number grows exponentially in length. Furthermore, we note that the volume entropy is infinite."}
{"category": "Math", "title": "Base manifolds for fibrations of projective irreducible symplectic manifolds", "abstract": "Given a projective irreducible symplectic manifold $M$ of dimension $2n$, a projective manifold $X$ and a surjective holomorphic map $f:M \\to X$ with connected fibers of positive dimension, we prove that $X$ is biholomorphic to the projective space of dimension $n$. The proof is obtained by exploiting two geometric structures at general points of $X$: the affine structure arising from the action variables of the Lagrangian fibration $f$ and the structure defined by the variety of minimal rational tangents on the Fano manifold $X$."}
{"category": "Math", "title": "Liv\\v{s}ic Theorems for Non-Commutative Groups including Diffeomorphism Groups and Results on the Existence of Conformal Structures for Anosov Systems", "abstract": "The celebrated Livsic theorem states that given M a manifold, a Lie group G, a transitive Anosov diffeomorphism f on M and a Holder function \\eta: M \\mapsto G whose range is sufficiently close to the identity, it is sufficient for the existence of \\phi :M \\mapsto G satisfying \\eta(x) = \\phi(f(x)) \\phi(x)^{-1} that a condition -- obviously necessary -- on the cocycle generated by \\eta restricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems."}
{"category": "Math", "title": "Confidence intervals in regression utilizing prior information", "abstract": "We consider a linear regression model with regression parameter beta=(beta_1,...,beta_p) and independent and identically N(0,sigma^2) distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau=c^T beta-t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. We present a new frequentist 1-alpha confidence interval for theta that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-alpha confidence interval when the data strongly contradicts this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when tau=0. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about tau is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2-by 2 factorial experiment with 20 replicates. Suppose that the parameter of interest theta is a specified simple effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for theta that utilizes this prior information."}
{"category": "Math", "title": "On the cluster multiplication theorem for acyclic cluster algebras", "abstract": "In \\cite{CK2005} and \\cite{Hubery2005}, the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the properties of 2--Calabi--Yau (Auslander--Reiten formula) and high order associativity."}
{"category": "Math", "title": "Harmonic analysis related to Schroedinger operators", "abstract": "In this article we give an overview on some recent development of Littlewood-Paley theory for Schr\\\"odinger operators. We extend the Littlewood-Paley theory for special potentials considered in the authors' previous work. We elaborate our approach by considering potential in $C^\\infty_0$ or Schwartz class in one dimension. In particular the low energy estimates are treated by establishing some new and refined asymptotics for the eigenfunctions and their Fourier transforms. We give maximal function characterization of the Besov spaces and Triebel-Lizorkin spaces associated with $H$. Then we prove a spectral multiplier theorem on these spaces and derive Strichartz estimates for the wave equation with a potential. We also consider similar problem for the unbounded potentials in the Hermite and Laguerre cases, whose potentials $V=a|x|^2+b|x|^{-2}$ are known to be critical in the study of perturbation of nonlinear dispersive equations. This improves upon the previous results when we apply the upper Gaussian bound for the heat kernel and its gradient."}
{"category": "Math", "title": "An Improved Procedure for Selecting the Profiles of Perfectly Matched Layers", "abstract": "The perfectly matched layers (PMLs), as a boundary termination over an unbounded spatial domain, are widely used in numerical simulations of wave propagation problems. Given a set of discretization parameters, a procedure to select the PML profiles based on minimizing the discrete reflectivity is established for frequency domain simulations. We, by extending the function class and adopting a direct search method, improve the former procedure for traveling waves."}
{"category": "Math", "title": "Computer model validation with functional output", "abstract": "A key question in evaluation of computer models is Does the computer model adequately represent reality? A six-step process for computer model validation is set out in Bayarri et al. [Technometrics 49 (2007) 138--154] (and briefly summarized below), based on comparison of computer model runs with field data of the process being modeled. The methodology is particularly suited to treating the major issues associated with the validation process: quantifying multiple sources of error and uncertainty in computer models; combining multiple sources of information; and being able to adapt to different, but related scenarios. Two complications that frequently arise in practice are the need to deal with highly irregular functional data and the need to acknowledge and incorporate uncertainty in the inputs. We develop methodology to deal with both complications. A key part of the approach utilizes a wavelet representation of the functional data, applies a hierarchical version of the scalar validation methodology to the wavelet coefficients, and transforms back, to ultimately compare computer model output with field output. The generality of the methodology is only limited by the capability of a combination of computational tools and the appropriateness of decompositions of the sort (wavelets) employed here. The methods and analyses we present are illustrated with a test bed dynamic stress analysis for a particular engineering system."}
{"category": "Math", "title": "The Atiyah conjecture and Artinian rings", "abstract": "Let G be a group such that its finite subgroups have bounded order, let d denote the lowest common multiple of the orders of the finite subgroups of G, and let K be a subfield of C that is closed under complex conjugation. Let U(G) denote the algebra of unbounded operators affiliated to the group von Neumann algebra N(G), and let D(KG,U(G)) denote the division closure of KG in U(G); thus D(KG,U(G)) is the smallest subring of U(G) containing KG that is closed under taking inverses. Suppose n is a positive integer, and \\alpha \\in \\Mat_n(KG). Then \\alpha induces a bounded linear map \\alpha: l^2(G)^n \\to \\l^2(G)^n, and \\ker\\alpha has a well-defined von Neumann dimension \\dim_{N(G)} (\\ker\\alpha). This is a nonnegative real number, and one version of the Atiyah conjecture states that d \\dim_{N(G)}(\\ker\\alpha) \\in Z. Assuming this conjecture, we shall prove that if G has no nontrivial finite normal subgroup, then D(KG,U(G)) is a d \\times d matrix ring over a skew field. We shall also consider the case when G has a nontrivial finite normal subgroup, and other subrings of U(G) that contain KG."}
{"category": "Math", "title": "Estimation of the Hurst parameter from discrete noisy data", "abstract": "We estimate the Hurst parameter $H$ of a fractional Brownian motion from discrete noisy data observed along a high frequency sampling scheme. The presence of systematic experimental noise makes recovery of $H$ more difficult since relevant information is mostly contained in the high frequencies of the signal. We quantify the difficulty of the statistical problem in a min-max sense: we prove that the rate $n^{-1/(4H+2)}$ is optimal for estimating $H$ and propose rate optimal estimators based on adaptive estimation of quadratic functionals."}
{"category": "Math", "title": "Boundary proximity of SLE", "abstract": "This paper examines how close the chordal $\\SLE_\\kappa$ curve gets to the real line asymptotically far away from its starting point. In particular, when $\\kappa\\in(0,4)$, it is shown that if $\\beta>\\beta_\\kappa:=1/(8/\\kappa-2)$, then the intersection of the $\\SLE_\\kappa$ curve with the graph of the function $y=x/(\\log x)^{\\beta}$, $x>e$, is a.s. bounded, while it is a.s. unbounded if $\\beta=\\beta_\\kappa$. The critical $\\SLE_4$ curve a.s. intersects the graph of $y=x^{-(\\log\\log x)^\\alpha}$, $x>e^e$, in an unbounded set if $\\alpha\\le 1$, but not if $\\alpha>1$. Under a very mild regularity assumption on the function $y(x)$, we give a necessary and sufficient integrability condition for the intersection of the $\\SLE_\\kappa$ path with the graph of $y$ to be unbounded. We also prove that the Hausdorff dimension of the intersection set of the $\\SLE_{\\kappa}$ curve and real axis is $2-8/\\kappa$ when $4<\\kappa<8$."}
{"category": "Math", "title": "On orbits of antichains of positive roots", "abstract": "For any finite poset P, there is a natural operator $X$ acting on the antichains of P. We discuss conjectural properties of this operator for some graded posets associated with irreducible root systems. In particular, if $\\Delta^+$ is the set of positive roots and $\\Pi$ is the set of simple roots in $\\Delta^+$, then we consider the cases $P=\\Delta^+$ and $\\Delta^+\\setminus \\Pi$. For the root system of type $A_n$, we consider an $X$-invariant integer-valued function on the set of antichains of $\\Delta^+$ and establish some properties of it."}
{"category": "Math", "title": "On Schr\\\"odinger operators with multisingular inverse-square anisotropic potentials", "abstract": "We study positivity, localization of binding and essential self-adjointness properties of a class of Schroedinger operators with many anisotropic inverse square singularities, including the case of multiple dipole potentials."}
{"category": "Math", "title": "Exponential sums: questions by Denef, Sperber, and Igusa", "abstract": "We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., \\textit{Exponential sums mod $p^n$ and {N}ewton polyhedra}, Bull. Belg. Math. Soc., {\\bf{suppl.}} (2001) 55-63] on nondegenerate local exponential sums modulo $p^m$. We generalize Igusa's conjecture of the introduction of [Igusa, J., \\textit{Lectures on forms of higher degree}, Lect. math. phys., Springer-Verlag, {\\bf{59}} (1978)] from the homogeneous to the quasi-homogeneous case and prove the nondegenerate case as well as the modulo $p$ case. We generalize some results by Katz of [Katz, N. M., \\textit{Estimates for \"singular\" exponential sums}, Internat. Math. Res. Notices (1999) no. 16, 875-899] on finite field exponential sums to the quasi-homogeneous case."}
{"category": "Math", "title": "Betti numbers of hypergraphs", "abstract": "In this paper we study some algebraic properties of hypergraphs, in particula their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge, are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions."}
{"category": "Math", "title": "Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics", "abstract": "Conformally compact asymptotically hyperbolic metrics have been intensively studied. The goal of this note is to understand what intrinsic conditions on a complete Riemannian manifold (M,g) will ensure that g is asymptotically hyperbolic in this sense. We use the geodesic compactification by asymptotic geodesic rays to compactify M and appropriate curvature decay conditions to study the regularity of the conformal compactification. We also present an interesting example that shows our conclusion is nearly optimal for our assumptions."}
{"category": "Math", "title": "Generalized test ideals, sharp F-purity, and sharp test elements", "abstract": "Consider a pair $(R, \\ba^t)$ where $R$ is a ring of positive characteristic, $\\ba$ is an ideal such that $a \\cap $R^{\\circ} \\neq \\emptyset$, and $t > 0$ is a real number. In this situation we have the ideal $\\tau_R(\\ba^t)$, the generalized test ideal associated to $(R, a^t)$ as defined by Hara and Yoshida. We show that $\\tau_R(a^t) \\cap R^{\\circ}$ is made up of appropriately defined generalized test elements which we call \\emph{sharp test elements}. We also define a variant of $F$-purity for pairs, \\emph{sharp $F$-purity}, which interacts well with sharp test elements and agrees with previously defined notions of $F$-purity in many common situations. We show that if $(R, \\ba^t)$ is sharply F-pure, then $\\tau_R(\\ba^t)$ is a radical ideal. Furthermore, by following an argument of Vassilev, we show that if $R$ is a quotient of an $F$-finite regular local ring and $(R, \\ba^t)$ is sharply $F$-pure, then $R/{\\tau_R(\\ba^t)}$ itself is $F$-pure. We conclude by showing that sharp $F$-purity can be used to define the $F$-pure threshold. As an application we show that the $F$-pure threshold must be a rational number under certain hypotheses."}
{"category": "Math", "title": "Solution of the Pompeiu problem (I)", "abstract": "This paper has been withdrawn by the author due to some errors."}
{"category": "Math", "title": "Global Attractor in Competitive Lotka-Volterra Systems", "abstract": "For autonomous Lotka-Volterra systems of differential equations modelling the dynamics of n competing species, new criteria are established for the existence of a single point global attractor. Under the conditions of these criteria, some of the species will survive and stabilise at a steady state whereas the others, if any, will die out."}
{"category": "Math", "title": "Converging to Gosper's Algorithm", "abstract": "Given two polynomials, we find a convergence property of the GCD of the rising factorial and the falling factorial. Based on this property, we present a unified approach to computing the universal denominators as given by Gosper's algorithm and Abramov's algorithm for finding rational solutions to linear difference equations with polynomial coefficients."}
{"category": "Math", "title": "Enumeration of some classes of words avoiding two generalized patterns of length three", "abstract": "The method we have applied in \"A. Bernini, L. Ferrari, R. Pinzani, Enumerating permutations avoiding three Babson-Steingrimsson patterns, Ann. Comb. 9 (2005), 137--162\" to count pattern avoiding permutations is adapted to words. As an application, we enumerate several classes of words simultaneously avoiding two generalized patterns of length 3."}
{"category": "Math", "title": "Inverse Conjecture for the Gowers norm is false", "abstract": "Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse Conjecture for the Gowers norm' states that if the \"$d$-th Gowers norm\" of a function $f:\\F_p^N \\to \\F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially approximated by a degree $d-1$ polynomial. The conjecture is known to hold for $d=2,3$ and for any prime $p$. In this paper we show the conjecture to be false for $p=2$ and for $d = 4$, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao \\cite{gt07}. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel \\cite{ab} to show inapproximability of certain functions by low-degree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime $p$, for $d = p^2$."}
{"category": "Math", "title": "Improved Poincare inequalities with weights", "abstract": "In this paper we prove that if $\\Omega\\in\\mathbb{R}^n$ is a bounded John domain, the following weighted Poincare-type inequality holds: $$ \\inf_{a\\in \\mathbb{R}}\\| (f(x)-a) w_1(x) \\|_{L^q(\\Omega)} \\le C \\|\\nabla f(x) d(x)^\\alpha w_2(x) \\|_{L^p(\\Omega)} $$ where $f$ is a locally Lipschitz function on $\\Omega$, $d(x)$ denotes the distance of $x$ to the boundary of $\\Omega$, the weights $w_1, w_2$ satisfy certain cube conditions, and $\\alpha \\in [0,1]$ depends on $p,q$ and $n$. This result generalizes previously known weighted inequalities, which can also be obtained with our approach."}
{"category": "Math", "title": "On the Non-degeneracy of Kendall's and Spearman's Correlation Coefficients", "abstract": "Hoeffding proved that Kendall's and Spearman's nonparametric measures of correlation between two continuous random variables X and Y are each asymptotically normal with an asymptotic variance of the form sigma^2/n -- provided the non-degeneracy condition sigma^2>0 holds, where sigma^2 is a certain (always nonnegative) expression which is determined by the joint distribution (say mu) of X and Y. Sufficient conditions for sigma^2>0 in terms of the support set (say S) of mu are given, the same for both correlation statistics. One of them is that there exist a rectangle with all its vertices in S, sides parallel to the X and Y axes, and an interior point also in S. Another sufficient condition is that the Lebesgue measure of S be nonzero."}
{"category": "Math", "title": "Thread-wire surfaces: Near-wire minimizers and topological finiteness (superseded)", "abstract": "(NOTE: per referee comments, this article has been split; it is now superseded by \"Existence of thread-wire minimizers\" and \"Near-wire thread-wire minimizers\"; please see http://www.bkstephens.net.) Alt's thread problem asks for least-area surfaces bounding a fixed \"wire\" curve and a movable \"thread\" curve of length L. We conjecture that if the wire has finitely many maxima of curvature, then its Alt minimizers have finitely many surface components. We show that this conjecture reduces to controlling near-wire minimizers, and thus begin a three paper series to understand them. In this paper we show they arise, show that they are embedded, and show that they have a nice parametrization in wire exponential coordinates. In doing so we prove tools of independent interest: a weighted isoperimetric inequality, a nonconvex enclosure theorem, and a classification of how Alt minimizers intersect planes. The last item reduces to a question about harmonic functions in the spirit of Rado's lemma."}
{"category": "Math", "title": "On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations", "abstract": "We prove new \\emph{a priori} estimates for the 3D Euler, the 3D Navier-Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms."}
{"category": "Math", "title": "On the generation of the coefficient field of a newform by a single Hecke eigenvalue", "abstract": "Let f be a non-CM newform of weight k > 1. Let L be a subfield of the coefficient field of f. We completely settle the question of the density of the set of primes p such that the p-th coefficient of f generates the field L. This density is determined by the inner twists of f. As a particular case, we obtain that in the absence of non-trivial inner twists, the density is 1 for L equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation."}
{"category": "Math", "title": "The $C^{\\a}$ regularity of a class of non-homogeneous ultraparabolic equations", "abstract": "We obtain the $C^{\\a}$ regularity for weak solutions of a class of non-homogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent $C^{\\a}$ regularity results of homogeneous ultraparabolic equation."}
{"category": "Math", "title": "On Ramanujan Cubic Polynomials", "abstract": "A polynomial x^3+px^2+qx+r with the condition pr^(1/3)+ 3r^(2/3)+q=0 we call a Ramanujan cubic polynomial (RCP). We study different interest properties of RCP, in particular, an important role of a parameter pq/r. We prove some new beautiful identities containing sums of 6 cubic radicals of values of trigonometrical functions as well."}
{"category": "Math", "title": "On beta-function of tube of light cone", "abstract": "We construct $B$-function of the Hermitian symmetric space $\\OO(n,2)/\\OO(n)\\times \\OO(2)$ or equivalently of the tube $(Re z_0)^2> (Re z_1)^2+...+ (Re z_n)^2$ in $C^{n+1}"}
{"category": "Math", "title": "On q-analogs of weight multiplicities for the Lie superalgebras gl(n,m) and spo(2n,M)", "abstract": "The paper is devoted to the generalization of Lusztig's q-analog of weight multiplicities to the Lie superalgebras gl(n,m) and spo(2n,M). We define such q-analogs K_{lambda,mu}(q) for the typical modules and for the irreducible covariant tensor gl(n,m)-modules of highest weight lambda. For gl(n,m), the defined polynomials have nonnegative integer coefficients if the weight mu is dominant. For spo(2n,M), we show that the positivity property holds when mu is dominant and sufficiently far from a specific wall of the fundamental chamber. We also establish that the q-analog associated to an irreducible covariant tensor gl(n,m)-module of highest weight lambda and a dominant weight mu is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape lambda and weight mu. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schutzenberger."}
{"category": "Math", "title": "Central extensions of groups of sections", "abstract": "If q : P -> M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact forms. In the present paper we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context we provide sufficient conditions for integrability in terms of data related only to the group K."}
{"category": "Math", "title": "Generation of polycyclic groups", "abstract": "In this note we give an alternative proof of a theorem of Linnell and Warhurst that the number of generators d(G) of a polycyclic group G is at most d(\\hat G), where d(\\hat G) is the number of generators of the profinite completion of G. While not claiming anything new we believe that our argument is much simpler that the original one. Moreover our result gives some sufficient condition when d(G)=d(\\hat G) which can be verified quite easily in the case when G is virtually abelian."}
{"category": "Math", "title": "On the existence of infinite energy solutions for nonlinear Schrodinger equations", "abstract": "We derive new results about existence and uniqueness of local and global solutions for nonlinear Schrodinger equation, including self-similar global solutions. Our analysis is performed in the framework of Marcinkiewicz spaces."}
{"category": "Math", "title": "Linear bound for the dyadic paraproduct on weighted Lebesgue space $L_2(w)$", "abstract": "The dyadic paraproduct is bounded in weighted Lebesgue spaces $L_p(w)$ if and only if the weight $w$ belongs to the Muckenhoupt class $A_p^d$. However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest $L_2(w)$ case. In this paper we prove the linear bound on the norm of the dyadic paraproduct in the weighted Lebesgue space $L_2(w)$ using Bellman function techniques and extrapolate this result to the $L_p(w)$ case."}
{"category": "Math", "title": "On Uniserial Modules in the Auslander-Reiten Quiver", "abstract": "This article begins the study of irreducible maps involving finite-dimensional uniserial modules over finite-dimensional associative algebras. We work on the classification of irreducible maps between two uniserials over triangular algebras, and give estimates for the number of middle terms of an almost split sequence with a uniserial end term."}
{"category": "Math", "title": "The sphericity of the complex of non-degenerate subspaces", "abstract": "We prove that the complex of proper non-trivial non-degenerate subspaces of a finite-dimensional vector space endowed with a non-degenerate sesquilinear form is homotopy equivalent to a wedge of spheres. Additionally, we show that the same is true for a slight generalization, the so-called generalized Phan geometries of type A_n. These generalized Phan geometries occur as relative links of certain filtrations. Their sphericity implies finiteness properties of suitable arithmetic groups and allows for a revision of Phan's group-theoretical local recognition of suitable finite groups of Lie type with simply laced diagram."}
{"category": "Math", "title": "Efficiently computing Groebner bases of ideals of points", "abstract": "We present an algorithm for computing Groebner bases of vanishing ideals of points that is optimized for the case when the number of points in the associated variety is less than the number of indeterminates. The algorithm first identifies a set of essential variables, which reduces the time complexity with respect to the number of indeterminates, and then uses PLU decompositions to reduce the time complexity with respect to the number of points. This gives a theoretical upper bound for its time complexity that is an order of magnitude lower than the known one for the standard Buchberger-Moeller algorithm if the number of indeterminates is much larger than the number of points. Comparison of implementations of our algorithm and the standard Buchberger-Moeller algorithm in Macaulay 2 confirm the theoretically predicted speedup. This work is motivated by recent applications of Groebner bases to the problem of network reconstruction in molecular biology."}
{"category": "Math", "title": "Generalized Koszul properties for augmented algebras", "abstract": "Under certain conditions, a filtration on an augmented algebra A admits a related filtration on the Yoneda algebra E(A) := Ext_A(K, K). We show that there exists a bigraded algebra monomorphism from gr E(A) to E_Gr(gr A), where E_Gr(gr A) is the graded Yoneda algebra of gr A. This monomorphism can be applied in the case where A is connected graded to determine that A has the K_2 property recently introduced by Cassidy and Shelton."}
{"category": "Math", "title": "Collapsing Manifolds with Boundary", "abstract": "This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result establishes a disc bundle structure for any manifold-with-boundary having two-sided bounds on sectional curvature and second fundamental form, and a lower bound on intrinsic injectivity radius, which is sufficiently close in the Gromov-Hausdorff topology to a closed manifold. The second main result identifies Gromov-Hausdorff limits of certain sequences of manifolds-with-boundary as Alexandrov spaces of curvature bounded below."}
{"category": "Math", "title": "Averages of elliptic curve constants", "abstract": "We compute the averages over elliptic curves of the constants occurring in the Lang-Trotter conjecture, the Koblitz conjecture, and the cyclicity conjecture. The results obtained confirm the consistency of these conjectures with the corresponding ``theorems on average'' obtained recently by various authors."}
{"category": "Math", "title": "A spectral stability theorem for large forbidden graphs", "abstract": "We extend the classical stability theorem of Erdos and Simonovits in two directions: first, we allow the order of the forbidden graph to grow as log of order of the host graph, and second, our extremal condition is on the spectral radius of the host graph."}
{"category": "Math", "title": "Spectral saturation: inverting the spectral Turan theorem", "abstract": "We prove that if the spectral radius of a graph G of order n is larger than the spectral radius of the r-partite Turan graph of the same order, then G contains various supergraphs of the complete graph of order r+1. In particular G contains a complete r-partite graph of size log n with one edge added to the first part. These results complete a project of Erdos from 1963. We also give corresponding stability results."}
{"category": "Math", "title": "A negative mass theorem for the 2-Torus", "abstract": "For a closed surface M with metric g, the Robin mass m(p) at the point p is the value of the Green function G(p,q) at p=q after the logarithmic singularity has been removed. The Laplacian-mass is the average value of the Robin mass, minus the value of the Robin mass for the round sphere of the same area. The Laplacian-mass is a spectral invariant which is a natural analog of the ADM mass for asymptotically flat manifolds. We show that if M is a torus, then the minimum value of the Laplacian-mass on the conformal class of g is negative. It is attained by a (smooth) metric for which one gets a sharp logarithmic Hardy-Littlewood-Sobolev inequality and Onofri-type inequality. If the flat metric in the conformal class is sufficiently long and thin, then the minimizer for the Laplacian-mass is non-flat."}
{"category": "Math", "title": "Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial", "abstract": "We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobas-Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobas-Riordan polynomials."}
{"category": "Math", "title": "Graphs with many copies of a given subgraph", "abstract": "We show that if a graph G of order n contains many copies of a given subgraph H, then it contains a blow-up of H of order log n."}
{"category": "Math", "title": "Van der Waerden/Schrijver-Valiant like Conjectures and Stable (aka Hyperbolic) Homogeneous Polynomials : One Theorem for all", "abstract": "Let $p$ be a homogeneous polynomial of degree $n$ in $n$ variables, $p(z_1,...,z_n) = p(Z)$, $Z \\in C^{n}$. We call such a polynomial $p$ {\\bf H-Stable} if $p(z_1,...,z_n) \\neq 0$ provided the real parts $Re(z_i) > 0, 1 \\leq i \\leq n$. This notion from {\\it Control Theory} is closely related to the notion of {\\it Hyperbolicity} used intensively in the {\\it PDE} theory. The main theorem in this paper states that if $p(x_1,...,x_n)$ is a homogeneous {\\bf H-Stable} polynomial of degree $n$ with nonnegative coefficients; $deg_{p}(i)$ is the maximum degree of the variable $x_i$, $C_i = \\min(deg_{p}(i),i)$ and $$ Cap(p) = \\inf_{x_i > 0, 1 \\leq i \\leq n} \\frac{p(x_1,...,x_n)}{x_1 ... x_n} $$ then the following inequality holds $$ \\frac{\\partial^n}{\\partial x_1... \\partial x_n} p(0,...,0) \\geq Cap(p) \\prod_{2 \\leq i \\leq n} (\\frac{C_i -1}{C_i})^{C_{i}-1}. $$ This inequality is a vast (and unifying) generalization of the Van der Waerden conjecture on the permanents of doubly stochastic matrices as well as the Schrijver-Valiant conjecture on the number of perfect matchings in $k$-regular bipartite graphs. These two famous results correspond to the {\\bf H-Stable} polynomials which are products of linear forms. Our proof is relatively simple and ``noncomputational''; it uses just very basic properties of complex numbers and the AM/GM inequality."}
{"category": "Math", "title": "The energy of C4-free graphs of bounded degree", "abstract": "Answering some questions of Gutman, we show that, except for four specific trees, every connected graph G of order n, with no cycle of order 4 and with maximum degree at most 3, has energy greater that its order. Here, the energy of a graph is the sum of the moduli of its eigenvalues. We give more general theorems and state two conjectures."}
{"category": "Math", "title": "Explicit Ramsey graphs and Erdos distance problem over finite Euclidean and non-Euclidean spaces", "abstract": "We study the Erdos distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These graphs are shown to be asymptotically Ramanujan graphs. The advantage of using these graphs is twofold. First, we can derive new lower bounds on the Erdos distance problems with explicit constants. Second, we can construct many explicit tough Ramsey graphs R(3,k)."}
{"category": "Math", "title": "Proof of the normal scalar curvature conjecture", "abstract": "In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture."}
{"category": "Math", "title": "Rankin-Cohen Brackets and van der Pol-Type Identities for the Ramanujan's Tau Function", "abstract": "We use Rankin-Cohen brackets for modular forms and quasimodular forms to give a different proof of the results obtained by D. Lanphier and D. Niebur on the van der Pol type identities for the Ramanujan's tau function. As consequences we obtain convolution sums and congruence relations involving the divisor functions."}
{"category": "Math", "title": "Galois groups of the Lie-irreducible generalized $q$-hypergeometric equations of order three with $q$-real parameters : an approach using a density theorem", "abstract": "In this paper we compute the difference Galois groups of the Lie-irreducible regular singular generalized q-hypergeometric equations of order 3 with q-real parameters by using a density theorem due to Sauloy. In contrast with the differential case, we show that these groups automatically contain the special linear group SL(3,C)."}
{"category": "Math", "title": "The ratio and generating function of cogrowth coefficients of finitely generated groups", "abstract": "Let G be a group generated by $r$ elements $g_1,g_2,..., g_r.$ Among the reduced words in $g_1,g_2,..., g_r$ of length $n$ some, say $\\gamma_n,$ represent the identity element of the group $G.$ It has been shown in a combinatorial way that the $2n$th root of $\\gamma_{2n}$ has a limit, called the cogrowth exponent with respect to generators $g_1,g_2,..., g_r.$ We show by analytic methods that the numbers $\\gamma_n$ vary regularly; i.e. the ratio $\\gamma_{2n+2}/\\gamma_{2n}$ is also convergent. Moreover we derive new precise information on the domain of holomorphy of $\\gamma(z),$ the generating function associated with the coefficients $\\gamma_n.$"}
{"category": "Math", "title": "Towards ISS disturbance attenuation for randomly switched systems", "abstract": "We are concerned with input-to-state stability (ISS) of randomly switched systems. We provide preliminary results dealing with sufficient conditions for stochastic versions of ISS for randomly switched systems without control inputs, and with the aid of universal formulae we design controllers for ISS-disturbance attenuation when control inputs are present. Two types of switching signals are considered: the first is characterized by a statistically slow-switching condition, and the second by a class of semi-Markov processes."}
{"category": "Math", "title": "Non-dense sets of subvarieties in a power of an elliptic curve", "abstract": "Let V be an algebraic variety embedded in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of V which are of bounded height and which satisfy certain algebraic conditions are a non-dense subset of V. This result has implications in the context of the Pink-Zilber Conjecture and Mordel-Lang plus Bogomolov Theorem."}
{"category": "Math", "title": "The intersection of a curve with a union of translated codimension 2 subgroups in a power of an elliptic curve", "abstract": "Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the Pink-Zilber Conjeture and the Mordel-Lang plus Bogomolov Theorem for curves."}
{"category": "Math", "title": "Applications de la bi-quantification \\`a la th\\'eorie de Lie", "abstract": "This article is a survey about applications of bi-quantization theory in Lie theory. We focus on a conjecture of M. Duflo. Most of the applications are coming from our article with Alberto Cattaneo and some extensions are relating discussions with my student. The end of the article is completely new. We prove that the conjecture E=1 implies the Kashiwara-Vergne conjecture. Our deformation is non geometric but uses a polynomial deformation of the coefficients."}
{"category": "Math", "title": "Bundles of C*-algebras and the KK(X;-,-)-bifunctor", "abstract": "An overview about C*-algebra bundles with a Z-grading is presented, with particular emphasis on classification questions. In particular, we discuss the role of the representable KK(X ; -, -)-bifunctor introduced by Kasparov. As an application, we consider Cuntz-Pimsner algebras associated with vector bundles, and give a classification in terms of K-theoretical invariants in the case in which the base space is an n-sphere."}
{"category": "Math", "title": "Rulings of Legendrian knots as spanning surfaces", "abstract": "Each ruling of a Legendrian link can be naturally treated as a surface. For knots, the ruling is 2-graded if and only if the surface is orientable. For 2-graded rulings of homogeneous (in particular, alternating) knots, we prove that the genus of this surface is at most the genus of the knot. While this is not true in general, we do prove that the canonical genus (a.k.a. diagram genus) of any knot is an upper bound for the genera of its 2-graded rulings."}
{"category": "Math", "title": "Transform martingale estimating functions", "abstract": "An estimation method is proposed for a wide variety of discrete time stochastic processes that have an intractable likelihood function but are otherwise conveniently specified by an integral transform such as the characteristic function, the Laplace transform or the probability generating function. This method involves the construction of classes of transform-based martingale estimating functions that fit into the general framework of quasi-likelihood. In the parametric setting of a discrete time stochastic process, we obtain transform quasi-score functions by projecting the unavailable score function onto the special linear spaces formed by these classes. The specification of the process by any of the main integral transforms makes possible an arbitrarily close approximation of the score function in an infinite-dimensional Hilbert space by optimally combining transform martingale quasi-score functions. It also allows an extension of the domain of application of quasi-likelihood methodology to processes with infinite conditional second moment."}
{"category": "Math", "title": "Discrete symmetry with compact fundamental domain, and geometric simple connectivity - A provisional Outline of work in Progress -", "abstract": "We show that a certain geometric property, the QSF introduced by S. Brick and M. Mihalik, is universally true for {\\ibf all} finitely presented groups $\\Gamma$. One way of defining this property is the existence of a smooth compact manifold $M$ with $\\pi_1 M = \\Gamma$, such that $\\tilde M$ is geometrically simply-connected ({\\it i.e.} without handles of index $\\lambda = 1$). There are also alternative, more group-theoretical definitions, which are presentation independent. But $\\Gamma \\in {\\rm QSF}$ is not only a universal property, it is quite highly non-trivial too; its very special case for $\\Gamma = \\pi_1 M^3$ (where it means $\\pi_1^{\\infty} \\tilde M^3 = 0$) is actually already known, as a corollary of G. Perelman's big breakthrough on the Geometrization of 3-Manifolds."}
{"category": "Math", "title": "Semi-classical calculus on manifolds with ends and weighted Lp estimates", "abstract": "For a class of non compact Riemannian manifolds with ends, we give pseudo-differential expansions of bounded functions of the semi-classical Laplacian and study related Lp boundedness properties."}
{"category": "Math", "title": "Littlewood-Paley decompositions on manifolds with ends", "abstract": "For certain non compact Riemannian manifolds with ends, we obtain Littlewood-Paley type estimates on (weighted) Lp spaces, using the usual square function defined by a dyadic partition."}
{"category": "Math", "title": "Strichartz estimates on asymptotically hyperbolic manifolds", "abstract": "We prove local in time Strichartz estimates without loss for the restriction of the solution of the Schroedinger equation, outside a large compact set, on a class of asymptotically hyperbolic manifolds."}
{"category": "Math", "title": "Representations of quivers, their generalizations and invariants", "abstract": "This paper is a survey on invariants of representations of quivers and their generalizations. We present the description of generating systems for invariants and relations between generators."}
{"category": "Math", "title": "Asymptotic theory of least squares estimators for nearly unstable processes under strong dependence", "abstract": "This paper considers the effect of least squares procedures for nearly unstable linear time series with strongly dependent innovations. Under a general framework and appropriate scaling, it is shown that ordinary least squares procedures converge to functionals of fractional Ornstein--Uhlenbeck processes. We use fractional integrated noise as an example to illustrate the important ideas. In this case, the functionals bear only formal analogy to those in the classical framework with uncorrelated innovations, with Wiener processes being replaced by fractional Brownian motions. It is also shown that limit theorems for the functionals involve nonstandard scaling and nonstandard limiting distributions. Results of this paper shed light on the asymptotic behavior of nearly unstable long-memory processes."}
{"category": "Math", "title": "Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment", "abstract": "The ordinary contact process is used to model the spread of a disease in a population. In this model, each infected individual waits an exponentially distributed time with parameter 1 before becoming healthy. In this paper, we introduce and study the contact process in a randomly evolving environment. Here we associate to every individual an independent two-state, $\\{0,1\\},$ background process. Given $\\delta_0<\\delta_1,$ if the background process is in state $0,$ the individual (if infected) becomes healthy at rate $\\delta_0,$ while if the background process is in state $1,$ it becomes healthy at rate $\\delta_1.$ By stochastically comparing the contact process in a randomly evolving environment to the ordinary contact process, we will investigate matters of extinction and that of weak and strong survival. A key step in our analysis is to obtain stochastic domination results between certain point processes. We do this by starting out in a discrete setting and then taking continuous time limits."}
{"category": "Math", "title": "Higher-order asymptotic normality of approximations to the modified signed likelihood ratio statistic for regular models", "abstract": "Approximations to the modified signed likelihood ratio statistic are asymptotically standard normal with error of order $n^{-1}$, where $n$ is the sample size. Proofs of this fact generally require that the sufficient statistic of the model be written as $(\\hat{\\theta},a)$, where $\\hat{\\theta}$ is the maximum likelihood estimator of the parameter $\\theta$ of the model and $a$ is an ancillary statistic. This condition is very difficult or impossible to verify for many models. However, calculation of the statistics themselves does not require this condition. The goal of this paper is to provide conditions under which these statistics are asymptotically normally distributed to order $n^{-1}$ without making any assumption about the sufficient statistic of the model."}
{"category": "Math", "title": "On eigenvalues of rectangular matrices", "abstract": "Given a $(k+1)$-tuple $A, B_1,...,B_k$ of $(m\\times n)$-matrices with $m\\le n$ we call the set of all $k$-tuples of complex numbers $\\{\\la_1,...,\\la_k\\}$ such that the linear combination $A+\\la_1B_1+\\la_2B_2+...+\\la_kB_k$ has rank smaller than $m$ the {\\it eigenvalue locus} of the latter pencil. Motivated primarily by applications to multi-parameter generalizations of the Heine-Stieltjes spectral problem, see \\cite{He} and \\cite{Vol}, we study a number of properties of the eigenvalue locus in the most important case $k=n-m+1$."}
{"category": "Math", "title": "A Representation of Multiplicative Arithmetic Functions by Symmetric Polynomials", "abstract": "We give a representation of the classical theory of multiplicative arithmetic functions (MF)in the ring of symmetric polynomials. The basis of the ring of symmetric polynomials that we use is the isobaric basis, a basis especially sensitive to the combinatorics of partitions of the integers. The representing elements are recursive sequences of Schur polynomials evaluated at subrings of the complex numbers. The multiplicative arithmetic functions are units in the Dirichlet ring of arithmetic functions, and their properties can be described locally, that is, at each prime number $p$. Our representation is, hence, a local representation. One such representing sequence is the sequence of generalized Fibonacci polynomials. In general the sequences consist of Schur-hook polynomials. This representation enables us to clarify and generalize classical results, e.g., the Busche-Ramanujan identity, as well as to give a richer structural description of the convolution group of multiplicative functions. It is a consequence of the representation that the MF's can be defined in a natural way on the negative powers of the prime $p$."}
{"category": "Math", "title": "On stability of randomly switched nonlinear systems", "abstract": "This article is concerned with stability analysis and stabilization of randomly switched nonlinear systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switches are triggered by a stochastic process which is independent of the state of the system, and between two consecutive switching instants the dynamics are deterministic. Our results provide sufficient conditions for almost sure global asymptotic stability using Lyapunov-based methods when individual subsystems are stable and a certain ``slow switching'' condition holds. This slow switching condition takes the form of an asymptotic upper bound on the probability mass function of the number of switches that occur between the initial and current time instants. This condition is shown to hold for switching signals coming from the states of finite-dimensional continuous-time Markov chains; our results therefore hold for Markov jump systems in particular. For systems with control inputs we provide explicit control schemes for feedback stabilization using the universal formula for stabilization of nonlinear systems."}
{"category": "Math", "title": "Uniformity seminorms on $\\ell^\\infty$ and applications", "abstract": "A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on $\\Z/N\\Z$ introduced by Gowers in his proof of Szemer\\'edi's Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg's proof of Szemer\\'edi's Theorem) defined by the authors. For each integer $k\\geq 1$, we define seminorms on $\\ell^\\infty(\\Z)$ analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem."}
{"category": "Math", "title": "Galois theory in bicategories", "abstract": "We develop a Galois (descent) theory for comonads within the framework of bicategories. We give generalizations of Beck's theorem and the Joyal-Tierney theorem. Many examples are provided, including classical descent theory, Hopf-Galois theory over Hopf algebras and Hopf algebroids, Galois theory for corings and group-corings, and Morita-Takeuchi theory for corings. As an application we construct a new type of comatrix corings based on (dual) quasi bialgebras."}
{"category": "Math", "title": "Stability of Bounded Solutions for Degenerate Complex Monge-Amp\\`ere equations", "abstract": "We generalize and strenghten Ko{\\l}odziej's stability theorem. In particular we give sharp stability exponent and treat the case with more singular right hand side of the Monge-Amp\\`ere equation."}
{"category": "Math", "title": "Diophantine Approximation on varieties III: Approximation of non-algebraic points by algebraic points", "abstract": "For $\\theta$ a non-algebraic point on a quasi projective variety over a number field, I prove that $\\theta$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications of this result will include a proof of a slightly strengthened version of the Philippon criterion, some new algebraic independence criteria, statements concerning metric transcendence theory on varieties of arbitrary dimension, and a rather accurate estimate for the number of algebraic points of bounded height and degree on quasi projective varieties over number fields."}
{"category": "Math", "title": "Lectures on zeta functions over finite fields", "abstract": "These are the notes from the summer school in G\\\"ottingen sponsored by NATO Advanced Study Institute on Higher-Dimensional Geometry over Finite Fields that took place in 2007. The aim was to give a short introduction on zeta functions over finite fields, focusing on moment zeta functions and zeta functions of affine toric hypersurfaces."}
{"category": "Math", "title": "Various analytic observations on combinations", "abstract": "E158 in the Enestrom index. Translation of the Latin original \"Observationes analyticae variae de combinationibus\" (1741). This paper introduces the problem of partitions, or partitio numerorum (the partition of integers). In the first part of the paper Euler looks at infinite symmetric functions. He defines three types of series: the first denoted with capital Latin letters are sums of powers, e.g. $A=a+b+c+...$, $B=a^2+b^2+c^3+...$, etc.; the second denoted with lower case Greek letters are the elementary symmetric functions; the third denoted with Germanic letters are sums of all combinations of $n$ symbols, e.g. $\\mathfrak{A}=a+b+c+...$ is the series for $n=1$, $\\mathfrak{B}=a^2+ab+b^2+ac+bc+c^2+...$ is the series for $n=2$, etc. Euler proves a lot of relations between these series. He defines some infinite products and proves some more relations between the products and these series. Then in \\S 17 he looks at the particular case where $a=n,b=n^2,c=n^3$ etc. In \\S 19 he says the Naud\\'e has proposed studying the number of ways to break an integer into a certain number of parts. Euler proves his recurrence relations for the number of partitions into a $\\mu$ parts with repetition and without repetition. Finally at the end of the paper Euler states the pentagonal number theorem, but says he hasn't been able to prove it rigorously."}
{"category": "Math", "title": "Bayesian Shrinkage Variable Selection", "abstract": "Withdrawn due to extensions and submission as another paper."}
{"category": "Math", "title": "Sur l'ind\\'ependance de l en cohomologie l-adique sur les corps locaux", "abstract": "Gabber deduced his theorem of independence of $l$ of intersection cohomology from a general stability result over finite fields. In this article, we prove an analogue of this general result over local fields. More precisely, we introduce a notion of independence of $l$ for systems of complexes of $l$-adic sheaves on schemes of finite type over a local field, equivariant under finite groups. We establish its stability by Grothendieck's six operations and the nearby cycle functor. Our method leads to a new proof of Gabber's theorem. We also give a generalization to algebraic stacks. ----- Gabber a d\\'eduit son th\\'eor\\`eme d'ind\\'ependance de $l$ de la cohomologie l'intersection d'un r\\'esultat g\\'en\\'eral de stabilit\\'e sur les corps finis. Dans cet article, nous d\\'emontrons un analogue sur les corps locaux de ce r\\'esultat g\\'en\\'eral. Plus pr\\'ecis\\'ement, nous introduisons une notion d'ind\\'ependance de $l$ pour les syst\\`emes de complexes de faisceaux $l$-adiques sur les sch\\'emas de type fini sur un corps local \\'equivariants sous des groupes finis et nous \\'etablissons sa stabilit\\'e par les six op\\'erations de Grothendieck et le foncteur des cycles proches. Notre m\\'ethode permet d'obtenir une nouvelle d\\'emonstration du th\\'eor\\`eme de Gabber. Nous donnons aussi une g\\'en\\'eralisation aux champs alg\\'ebriques."}
{"category": "Math", "title": "On the Axiomatics of Ann-Categories", "abstract": "In this paper, we have studied the axiomatics of {\\it Ann-categories} and {\\it categorical rings.} These are the categories with distributivity constraints whose axiomatics are similar with those of ring structures. The main result we have achieved is proving the independence of the axiomatics of Ann-category definition. And then we have proved that after adding an axiom into the definition of categorical rings, we obtain the new axiomatics which is equivalent to the one of Ann-categories."}
{"category": "Math", "title": "Geometric objects in an approach to quantum geometry", "abstract": "Ideas from deformation quantization applied to algebras with one generator lead to methods to treat a nonlinear flat connection. It provides us elements of algebras to be parallel sections. The moduli space of the parallel sections is studied as an example of bundle-like objects with discordant (sogo) transition functions, which suggests us to treat movable branching singularities."}
{"category": "Math", "title": "Star exponential functions as two-valued elements", "abstract": "We propose a relatively new notion of two-valued elements, which arises naturally in constructing the star exponential functions of the quad-ratics in the Weyl algebra over the complex number field. This notion enables us to describe the group like objects of the set of star exponential functions of quadratics in the Weyl algebra."}
{"category": "Math", "title": "Simplicial cohomology of augmentation ideals in ${\\ell}^1(G)$", "abstract": "Let $G$ be a discrete group. We give a decomposition theorem for the Hochschild cohomology of $\\ell^1(G)$ with coefficients in certain $G$-modules. Using this we show that if $G$ is commutative-transitive, the canonical inclusion of bounded cohomology of $G$ into simplicial cohomology of $\\ell^1(G)$ is an isomorphism."}
{"category": "Math", "title": "Strong invariance principles for dependent random variables", "abstract": "We establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds. Applications to linear and some nonlinear processes are discussed. Strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions."}
{"category": "Math", "title": "Nonexistence of triples of nonisomorphic connected graphs with isomorphic connected $P_3$-graphs", "abstract": "In the paper \"Broersma and Hoede, {\\it Path graphs}, J. Graph Theory {\\bf 13} (1989) 427-444\", the authors proposed a problem whether there is a triple of mutually nonisomorphic connected graphs which have an isomorphic connected $P_3$-graph. For a long time, this problem remains unanswered. In this paper, we give it a negative answer that there is no such triple, and thus completely solve this problem."}
{"category": "Math", "title": "Twisted character of a small representation", "abstract": "This paper has been withdrawn by the author as it has already been submitted under the title \"Twisted character of a small Representation of GL(4)\"."}
{"category": "Math", "title": "Biased random walks on a Galton-Watson tree with leaves", "abstract": "We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\\gamma= \\gamma(\\beta) \\in (0,1)$, depending on the bias $\\beta$, such that $X_n$ is of order $n^{\\gamma}$. Denoting $\\Delta_n$ the hitting time of level $n$, we prove that $\\Delta_n/n^{1/\\gamma}$ is tight. Moreover we show that $\\Delta_n/n^{1/\\gamma}$ does not converge in law (at least for large values of $\\beta$). We prove that along the sequences $n_{\\lambda}(k)=\\lfloor \\lambda \\beta^{\\gamma k}\\rfloor$, $\\Delta_n/n^{1/\\gamma}$ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables."}
{"category": "Math", "title": "Residual-based localization and quantification of peaks in x-ray diffractograms", "abstract": "We consider data consisting of photon counts of diffracted x-ray radiation as a function of the angle of diffraction. The problem is to determine the positions, powers and shapes of the relevant peaks. An additional difficulty is that the power of the peaks is to be measured from a baseline which itself must be identified. Most methods of de-noising data of this kind do not explicitly take into account the modality of the final estimate. The residual-based procedure we propose uses the so-called taut string method, which minimizes the number of peaks subject to a tube constraint on the integrated data. The baseline is identified by combining the result of the taut string with an estimate of the first derivative of the baseline obtained using a weighted smoothing spline. Finally, each individual peak is expressed as the finite sum of kernels chosen from a parametric family."}
{"category": "Math", "title": "Regularity, Local and Microlocal Analysis in Theories of Generalized Functions", "abstract": "We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the B-local analysis of sections of A. But the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a \"frequential microlocal analysis \" and into a \"microlocal asymptotic analysis\". The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. The microlocal asymptotic analysis can inherit from the algebraic structure of B some good properties with respect to nonlinear operations."}
{"category": "Math", "title": "A Short Proof of a Known Relation for Consecutive Power Sums", "abstract": "We give a new short proof of the most simple relation between consecutive power sums of the first m positive integers."}
{"category": "Math", "title": "Maximum Principle for Linear-Convex Boundary Control Problems applied to Optimal Investment with Vintage Capital", "abstract": "The paper concerns the study of the Pontryagin Maximum Principle for an infinite dimensional and infinite horizon boundary control problem for linear partial differential equations. The optimal control model has already been studied both in finite and infinite horizon with Dynamic Programming methods in a series of papers by the same author, or by Faggian and Gozzi. Necessary and sufficient optimality conditions for open loop controls are established. Moreover the co-state variable is shown to coincide with the spatial gradient of the value function evaluated along the trajectory of the system, creating a parallel between Maximum Principle and Dynamic Programming. The abstract model applies, as recalled in one of the first sections, to optimal investment with vintage capital."}
{"category": "Math", "title": "Harmonicity of sections of sphere bundles", "abstract": "We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold (M, <,>) equipped with the Sasaki metric and we discuss the characterising condition for critical points. Likewise, we provide a useful method for computing the tension field in some particular situations. Such a method is shown to be adequate for many tensor fields defined on manifolds M equipped with a G-structure compatible with <,>. This leads to the construction of a lot of new examples of differential forms which are harmonic sections or determine a harmonic map from (M,<,>) into its sphere bundle."}
{"category": "Math", "title": "Poincare's Conjecture for three manifolds", "abstract": "We prove Poincare's Conjecture that every simply connected, closed three-manifold is topologically equivalent to the three-sphere. The proof is founded on the algebraic formulation discovered by J. Stallings."}
{"category": "Math", "title": "The net created from the Penrose tiling is biLipschitz to the integer lattice", "abstract": "A separated net is a set of points which is relatively dense and uniformly discrete (another name for a Delone set). We are dealing with tilings and separated nets in Euclidean spaces and with the question whether a given separated net is biLipschitz to the integer lattice. In this paper we show, as an answer to a question of Burago and Kleiner, that the net that is obtained form the Penrose tiling is biLipschitz to the integer lattice."}
{"category": "Math", "title": "Equations in a free group. Elementary theory", "abstract": "We prove the decidability of the elementary theory of a free group."}
{"category": "Math", "title": "Orbits of parabolic subgroups on metabelian ideals", "abstract": "We consider the action of a parabolic subgroup of the General Linear Group on a metabelian ideal. For those actions, we classify actions with finitely many orbits using methods from representation theory."}
{"category": "Math", "title": "The Hunting of the Hopf Ring", "abstract": "We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E^*. Our description is as a graded and completed version of a Tall-Wraith monoid. The E^*-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another."}
{"category": "Math", "title": "The iterated Aluthge transforms of a matrix converge", "abstract": "Given an $r\\times r$ complex matrix $T$, if $T=U|T|$ is the polar decomposition of $T$, then, the Aluthge transform is defined by $$ \\Delta(T)= |T|^{1/2} U |T |^{1/2}. $$ Let $\\Delta^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, i.e. $\\Delta^{0}(T)=T$ and $\\Delta^{n}(T)=\\Delta(\\Delta^{n-1}(T))$, $n\\in\\mathbb{N}$. We prove that the sequence $\\{\\Delta^{n}(T)\\}_{n\\in\\mathbb{N}}$ converges for every $r\\times r$ matrix $T$. This result was conjecturated by Jung, Ko and Pearcy in 2003. We also analyze the regularity of the limit function."}
{"category": "Math", "title": "The fundamental group of period domains over finite fields", "abstract": "We determine the fundamental group of period domains over finite fields."}
{"category": "Math", "title": "The Chabauty space of closed subgroups of the three-dimensional Heisenberg group", "abstract": "When equipped with the natural topology first defined by Chabauty, the closed subgroups of a locally compact group $G$ form a compact space $\\Cal C(G)$. We analyse the structure of $\\Cal C(G)$ for some low-dimensional Lie groups, concentrating mostly on the 3-dimensional Heisenberg group $H$. We prove that $\\Cal C(H)$ is a 6-dimensional space that is path--connected but not locally connected. The lattices in $H$ form a dense open subset $\\Cal L(H) \\subset \\Cal C(H)$ that is the disjoint union of an infinite sequence of pairwise--homeomorphic aspherical manifolds of dimension six, each a torus bundle over $(\\bold S^3 \\smallsetminus T) \\times \\bold R$, where $T$ denotes a trefoil knot. The complement of $\\Cal L(H)$ in $\\Cal C(H)$ is also described explicitly. The subspace of $\\Cal C(H)$ consisting of subgroups that contain the centre $Z(H)$ is homeomorphic to the 4--sphere, and we prove that this is a weak retract of $\\Cal C(H)$."}
{"category": "Math", "title": "Simplicial Hochschild cochains as an Amitsur complex", "abstract": "It is shown that the cochain complex of relative Hochschild A-valued cochains of a depth two extension A | B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End {}_BA_B over the centralizer R = A^B with grouplike element 1_S, which itself is isomorphic to the Cartier complex of S with coefficients in the (S,S)-bicomodule R^e. This specializes to finite dimensional algebras, H-separable extensions and Hopf-Galois extensions."}
{"category": "Math", "title": "Dynamic balancing of planar mechanisms using toric geometry", "abstract": "In this paper, a new method to determine the complete set of dynamically balanced planar four-bar mechanims is presented. Using complex variables to model the kinematics of the mechanism, the dynamic balancing constraints are written as algebraic equations over complex variables and joint angular velocities. After elimination of the joint angular velocity variables, the problem is formulated as a problem of factorization of Laurent polynomials. Using toric polynomial division, necessary and sufficient conditions for dynamic balancing of planar four-bar mechanisms are derived."}
{"category": "Math", "title": "Symmetries and Invariant Differential Pairings", "abstract": "The purpose of this article is to motivate the study of invariant, and especially conformally invariant, differential pairings. Since a general theory is lacking, this work merely presents some interesting examples of these pairings, explains how they naturally arise, and formulates various associated problems."}
{"category": "Math", "title": "Osculating properties of decomposable scrolls", "abstract": "Osculating spaces of decomposable scrolls (of any genus and not necessarily normal)are studied and their inflectional loci are related to those of their generating curves by using systematically an idea introduced by Piene and Sacchiero in the setting of rational normal scrolls. In this broader setting the extra components of the second discriminant locus - deriving from flexes - are investigated and a new class of uninflected surface scrolls is presented and characterized. Further properties related to osculation are discussed for (not necessarily decomposable) scrolls."}
{"category": "Math", "title": "MCMC Inference for a Model with Sampling Bias: An Illustration using SAGE data", "abstract": "This paper explores Bayesian inference for a biased sampling model in situations where the population of interest cannot be sampled directly, but rather through an indirect and inherently biased method. Observations are viewed as being the result of a multinomial sampling process from a tagged population which is, in turn, a biased sample from the original population of interest. This paper presents several Gibbs Sampling techniques to estimate the joint posterior distribution of the original population based on the observed counts of the tagged population. These algorithms efficiently sample from the joint posterior distribution of a very large multinomial parameter vector. Samples from this method can be used to generate both joint and marginal posterior inferences. We also present an iterative optimization procedure based upon the conditional distributions of the Gibbs Sampler which directly computes the mode of the posterior distribution. To illustrate our approach, we apply it to a tagged population of messanger RNAs (mRNA) generated using a common high-throughput technique, Serial Analysis of Gene Expression (SAGE). Inferences for the mRNA expression levels in the yeast Saccharomyces cerevisiae are reported."}
{"category": "Math", "title": "Convex-transitivity and function spaces", "abstract": "If X is a convex-transitive Banach space and 1\\leq p\\leq \\infty then the closed linear span of the simple functions in the Bochner space L^{p}([0,1],X) is convex-transitive. If H is an infinite-dimensional Hilbert space and C_{0}(L) is convex-transitive, then C_{0}(L,H) is convex-transitive. Some new fairly concrete examples of convex-transitive spaces are provided."}
{"category": "Math", "title": "Finding rational points on elliptic curves using 6-descent and 12-descent", "abstract": "We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in the Stein-Watkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer."}
{"category": "Math", "title": "Graphs of 2-torus actions", "abstract": "It has been known that an effective smooth $({\\Bbb Z}_2)^k$-action on a smooth connected closed manifold $M^n$ fixing a finite set can be associated to a $({\\Bbb Z}_2)^k$-colored regular graph. In this paper, we consider abstract graphs $(\\Gamma,\\alpha)$ of $({\\Bbb Z}_2)^k$-actions, called abstract 1-skeletons. We study when an abstract 1-skeleton is a colored graph of some $({\\Bbb Z}_2)^k$-action. We also study the existence of faces of an abstract 1-skeleton (note that faces often have certain geometric meanings if an abstract 1-skeleton is a colored graph of some $({\\Bbb Z}_2)^k$-action)."}
{"category": "Math", "title": "Lelong-Skoda transform for compact Kaehler manifolds and self-intersection inequalities", "abstract": "Let $X$ be a compact Kaehler manifold of dimension $k$ and $T$ be a positive closed current on $X$ of bidimension $(p,p)$ ($1\\leq p < k-1$). We construct a continuous linear transform $\\mathcal{L}_p(T)$ of $T$ which is a positive closed current on $X$ of bidimension $(k-1,k-1)$ which has the same Lelong numbers as $T$. We deduce from that construction self-intersection inequalities for positive closed currents of any bidegree."}
{"category": "Math", "title": "The Lindelof Hypothesis for almost all Hurwitz's Zeta-Functions holds True", "abstract": "By Probability theory, that is, by a kind of quasi-law of the iterated logarithm, we prove the title claim."}
{"category": "Math", "title": "Unprovability results involving braids", "abstract": "We construct long sequences of braids that are descending with respect to the standard order of braids (``Dehornoy order''), and we deduce that, contrary to all usual algebraic properties of braids, certain simple combinatorial statements involving the braid order are true, but not provable in the subsystems ISigma1 or ISigma2 of the standard Peano system."}
{"category": "Math", "title": "Free Brownian motion and evolution towards boxplus-infinite divisibility for k-tuples", "abstract": "Let D be the space of non-commutative distributions of k-tuples of selfadjoints in a C*-probability space (for a fixed k). We introduce a semigroup of transformations B_t of D, such that every distribution in D evolves under the B_t towards infinite divisibility with respect to free additive convolution. The very good properties of B_t come from some special connections that we put into evidence between free additive convolution and the operation of Boolean convolution. On the other hand we put into evidence a relation between the transformations B_t and free Brownian motion. More precisely, we introduce a transformation Phi of D which converts the free Brownian motion started at an arbitrary distribution m in D into the process B_t (Phi(m)), t>0."}
{"category": "Math", "title": "Bernstein-Sato polynomials in positive characteristic", "abstract": "In characteristic zero, the Bernstein-Sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an Euler operator on a suitable D-module. We consider the analogous D-module in positive characteristic, and use it to define a sequence of Bernstein-Sato polynomials (corresponding to the fact that we need to consider also divided powers Euler operators). We show that the information contained in these polynomials is equivalent to that given by the F-jumping exponents of the hypersurface, in the sense of Hara and Yoshida."}
{"category": "Math", "title": "Descent constructions for central extensions of infinite dimensional Lie algebras", "abstract": "We use Galois descent to construct central extensions of twisted forms of split simple Lie algebras over rings. These types of algebras arise naturally in the construction of Extended Affine Lie Algebras. The construction also gives information about the structure of the group of automorphisms of such algebras."}
{"category": "Math", "title": "Geometric Intersection Number and analogues of the Curve Complex for free groups", "abstract": "For the free group $F_{N}$ of finite rank $N \\geq 2$ we construct a canonical Bonahon-type continuous and $Out(F_N)$-invariant \\emph{geometric intersection form} \\[ <, >: \\bar{cv}(F_N)\\times Curr(F_N)\\to \\mathbb R_{\\ge 0}. \\] Here $\\bar{cv}(F_N)$ is the closure of unprojectivized Culler-Vogtmann's Outer space $cv(F_N)$ in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that $\\bar{cv}(F_N)$ consists of all \\emph{very small} minimal isometric actions of $F_N$ on $\\mathbb R$-trees. The projectivization of $\\bar{cv}(F_N)$ provides a free group analogue of Thurston's compactification of the Teichm\\\"uller space. As an application, using the \\emph{intersection graph} determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter."}
{"category": "Math", "title": "Hyperfinite graph limits", "abstract": "G\\'abor Elek introduced the notion of a hyperfinite graph family: a collection of graphs is hypefinite if for every $\\epsilon>0$ there is some finite $k$ such that each graph $G$ in the collection can be broken into connected components of size at most $k$ by removing a set of edges of size at most $\\epsilon|V(G)|$. We presently extend this notion to a certain compactification of finite bounded-degree graphs, and show that if a sequence of finite graphs converges to a hyperfinite limit, then the sequence itself is hyperfinite."}
{"category": "Math", "title": "Least Squares Fitting of Low-Level Gamma-ray Spectra with B-Spline Basis Functions", "abstract": "In this paper, new methods for smoothing gamma-ray spectra measured by NaI detector are derived. Least squares fitting method with B-spline basis functions is used to reduce the influence of statistical fluctuations. The derived procedures are simple and automatic. The results show that this method is better than traditional method with a more complete reduction of staistical fluctuation."}
{"category": "Math", "title": "Quenched CLT for random toral automorphism", "abstract": "We establish a quenched Central Limit Theorem (CLT) for a smooth observable of random sequences of iterated linear hyperbolic maps on the torus. To this end we also obtain an annealed CLT for the same system. We show that, almost surely, the variance of the quenched system is the same as for the annealed system. Our technique is the study of the transfer operator on an anisotropic Banach space specifically tailored to use the cone condition satisfied by the maps."}
{"category": "Math", "title": "Codimensions of Newton Strata for SL_3 in the Iwahori Case", "abstract": "We study the Newton stratification on SL_3(F), where F is a Laurent power series field. We provide a formula for the codimensions of the Newton strata inside each component of the affine Bruhat decomposition on SL_3(F). These calculations are related to the study of certain affine Deligne-Lusztig varieties. In particular, we describe a method for determining which of these varieties is non-empty in the case of SL_3(F)."}
{"category": "Math", "title": "Unique ergodicity of circle and interval exchange transformations with flips", "abstract": "We study the existence of transitive exchange maps with flips defined on the unit circle. We provide a complete answer to the question of whether there exists a transitive exchange map of the unit circle defined on n subintervals and having f flips."}
{"category": "Math", "title": "Monochromatic and heterochromatic subgraph problems in a randomly colored graph", "abstract": "Let $K_n$ be the complete graph with $n$ vertices and $c_1, c_2, ..., c_r$ be $r$ different colors. Suppose we randomly and uniformly color the edges of $K_n$ in $c_1, c_2, ..., c_r$. Then we get a random graph, denoted by $\\mathcal{K}_n^r$. In the paper, we investigate the asymptotic properties of several kinds of monochromatic and heterochromatic subgraphs in $\\mathcal{K}_n^r$. Accurate threshold functions in some cases are also obtained."}
{"category": "Math", "title": "Symplectically hyperbolic manifolds", "abstract": "A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology class then it is hyperbolic. * The symplectic hyperbolicity is equivalent to a certain isoperimetric inequality. * The fundamental group of symplectically hyperbolic manifold is non-amenable. We also construct hyperbolic symplectic forms on certain bundles and Lefschetz fibrations, discuss the dependenc of the symplectic hyperbolicity on the fundamental group and discuss some properties of the group of symplectic diffeomorphisms of a symplectically hyperbolic manifold."}
{"category": "Math", "title": "Interpr\\'etation de l'Arithm\\'etique dans certains groupes de permutations affines par morceaux d'un intervalle", "abstract": "The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group $F$ and some of its generalizations interpret the Arithmetic without parameters."}
{"category": "Math", "title": "On the Analytic Wavelet Transform", "abstract": "An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of non-negligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias."}
{"category": "Math", "title": "On the Induction Operation for Shift Subspaces and Cellular Automata as Presentations of Dynamical Systems", "abstract": "We consider continuous, translation-commuting transformations of compact, translation-invariant families of mappingsfrom finitely generated groups into finite alphabets. It is well-known that such transformations and spaces can be described \"locally\" via families of patterns and finitary functions; such descriptions can be re-used on groups larger than the original, usually defining non-isomorphic structures. We show how some of the properties of the \"induced\" entities can be deduced from those of the original ones, and vice versa; then, we show how to \"simulate\" the smaller structure into the larger one, and obtain a characterization in terms of group actions for the dynamical systems admitting of presentations via structures as such. Special attention is given to the class of sofic shifts."}
{"category": "Math", "title": "Differential graded versus Simplicial categories", "abstract": "We construct a zig-zag of Quillen adjunctions between the homotopy theories of differential graded and simplicial categories. In an intermediate step we generalize Shipley-Schwede's work on connective DG algebras by extending the Dold-Kan correspondence to a Quillen equivalence between categories enriched over positive graded chain complexes and simplicial k-modules. As an application we obtain a conceptual explanation of Simpson's homotopy fiber construction."}
{"category": "Math", "title": "Relation of Orbital Integrals on SO(5) and PGL(2)", "abstract": "We relate the \"Fourier\" orbital integrals of corresponding spherical functions on the p-adic groups SO(5) and PGL(2). The correspondence is defined by a \"lifting\" of representations of these groups. This is a local \"fundamental lemma\" needed to compare the geometric sides of the global Fourier summation formulae (or relative trace formulae) on these two groups. This comparison leads to conclusions about a well known lifting of representations from PGL(2) to PGSp(4). This lifting produces counter examples to the Ramanujan conjecture."}
{"category": "Math", "title": "Periodic Chandrasekhar recursions", "abstract": "This paper extends the Chandrasekhar-type recursions due to Morf, Sidhu, and Kailath \"Some new algorithms for recursive estimation in constant, linear, discrete-time systems, IEEE Trans. Autom. Control 19 (1974) 315-323\" to the case of periodic time-varying state-space models. We show that the S-lagged increments of the one-step prediction error covariance satisfy certain recursions from which we derive some algorithms for linear least squares estimation for periodic state-space models. The proposed recursions may have potential computational advantages over the Kalman Filter and, in particular, the periodic Riccati difference equation."}
{"category": "Math", "title": "Convergence and stability of locally \\mathbb{R}^{N}-invariant solutions of Ricci flow", "abstract": "Important models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally G-invariant solutions on principal bundles, where G is a nilpotent Lie group. In this paper, we establish convergence and asymptotic stability, modulo smooth finite-dimensional center manifolds, of certain R^{N}-invariant solutions. When the dimension of the total space is three, these results are relevant to work of Lott classifying the asymptotic behavior of all 3-dimensional Ricci flow solutions whose sectional curvatures and diameters are respectively O(t^{-1}) and O(t^{1/2})."}
{"category": "Math", "title": "Computing Hilbert modular forms over fields with nontrivial class group", "abstract": "In previous work, the first author developed an algorithm for the computation of Hilbert modular forms. In this paper, we extend this to all totally real number fields of even degree and nontrivial class group. Using the algorithm over $\\Q(\\sqrt{10})$ and $\\Q(\\sqrt{85})$ and their Hilbert class fields, we present some new instances of the conjectural Eichler-Shimura construction for totally real fields, and in particular find new examples of modular abelian varieties with everywhere good reduction."}
{"category": "Math", "title": "Difference fields and descent in algebraic dynamics - I", "abstract": "We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line. The paper comes in three parts. This first part contains an exposition some of the main results of the model theory of difference fields, and their immediate connection to questions of descent in algebraic dynamics. We present the model-theoretic notion of internality in a context that does not require a universal domain with quantifier-elimination. We also note a version of canonical heights that applies well beyond polarized algebraic dynamics. Part II sharpens the structure theory to arbitrary base fields and constructible maps where in part I we emphasize finite base change and correspondences. Part III will include precise structure theorems related to the Galois theory considered here, and will enable a sharpening of the descent results for non-modular dynamics."}
{"category": "Math", "title": "Difference fields and descent in algebraic dynamics, II", "abstract": "This second part of the paper strengthens the descent theory described in the first part to rational maps, arbitrary base fields, and dynamics given by correspondences. We obtain in particular a decomposition of any difference field extension into a tower of finite, field-internal and one-based difference field extensions. This is needed in order to obtain the \"dynamical Northcott\" Theorem 1.11 of Part I in sharp form."}
{"category": "Math", "title": "The G-Fredholm Property of the \\bar\\partial-Neumann Problem", "abstract": "Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ be the total space of a principal bundle $G\\to M\\to X$ so that $M$ is also a strongly pseudoconvex complex manifold. In this work, we show that if $G$ acts by holomorphic transformations in $M$, then the complex Laplacian $\\square$ on $M$ has the following properties: The kernel of $\\square$ restricted to the forms $\\Lambda^{p,q}$ with $q$ positive is a closed, $G$-invariant subspace in $L^{2}(M,\\Lambda^{p,q})$ of finite $G$-dimension. Secondly, we show that if $q$ is positive, then the image of $\\square$ contains a closed, $G$-invariant subspace of finite codimension in $L^{2}(M,\\Lambda^{p,q})$. These two properties taken together amount to saying that $\\square$ is a $G$-Fredholm operator. The boundary Laplacian has similar properties."}
{"category": "Math", "title": "Packing 3-Vertex Paths in Claw-Free Graphs", "abstract": "An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) denote the number of vertices of G. A graph is called claw-free if it does not have a subgraph isomorphic to the graph with 4 vertices and 3 edges having a common vertex. Our results include the following. Let G$ be a 3-connected claw-free graph, x be a vertex, e = xy be an edge, and P be a 3-vertex path in G. Then (c1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (c2) if v(G) = 1 mod 3, then G - x has a L-factor, (c3) if v(G) = 2 mod 3, then G - x -y has an L-factor, (c4) if v(G) = 0 mod 3 and G is either cubic or 4-connected, then G - P has an L-factor, and (c5) if G is cubic and E is a set of three edges in G, then G - E has an L -factor if and only if the subgraph induced by E in G is not a claw and not a triangle. Keywords: claw-free graph, cubic graph, L-packing, L-factor."}
{"category": "Math", "title": "An Overview of Hopf Algebras of Trees and Their Actions on Functions", "abstract": "We provide an expository account of some of the Hopf algebras that can be defined using trees, labeled trees, ordered trees and heap ordered trees. We also describe some actions of these Hopf algebras on algebra of functions."}
{"category": "Math", "title": "Hopf-algebraic structures of families of trees", "abstract": "Description of cocommutative Hopf algebras associated with families of trees. Applications include Cayley's theorem on the number of rooted trees with n nodes, and Catalan's theorem on the number of rooted ordered trees with n nodes."}
{"category": "Math", "title": "Local discriminants, kummerian extensions, and elliptic curves", "abstract": "Some thoughts on the congruence D=0,1(mod 4) for the absolute discriminant D of a number field"}
{"category": "Math", "title": "Wilson's theorem", "abstract": "We show that there are four possibilities for the product of all elements in the multiplicative group of a quotient of the ring of integers in a number field, and give precise conditions for each of the possibilities to occur. This generalisation of Wilson's theorem turns out to have been first discovered by M. La\\v{s}\\v{s}\\'ak (2000), but our proof is simpler and more direct."}
{"category": "Math", "title": "On the universal abelian variety of dimension 4", "abstract": "Let A be the moduli space of principally polarized abelian varieties of dimension 4 over an algebraically closed field k of characteristic different from 2,3. It is proved that the universal principally polarized abelian variety over A, as well as the universal theta divisor over A, are unirational varieties."}
{"category": "Math", "title": "A Simple Proof of Sharkovsky's Theorem Rerevisited", "abstract": "Based on various strategies and a new general doubling operator, we obtain several simple proofs of the celebrated Sharkovsky's cycle coexistence theorem. A simple non-directed graph proof which is especially suitable for a calculus course right after the introduction of Intermediate Value Theorem is also given (in section 3)."}
{"category": "Math", "title": "Plane Jacobian conjecture for simple polynomials", "abstract": "A non-zero constant Jacobian polynomial map $F=(P,Q):\\mathbb{C}^2 \\longrightarrow \\mathbb{C}^2$ has a polynomial inverse if the component $P$ is a simple polynomial, i.e. if, when $P$ extended to a morphism $p:X\\longrightarrow \\mathbb{P}^1$ of a compactification $X$ of $\\mathbb{C}^2$, the restriction of $p$ to each irreducible component $C$ of the compactification divisor $D = X-\\mathbb{C}^2$ is either degree 0 or 1."}
{"category": "Math", "title": "Equivariant relative Thom forms and Chern characters", "abstract": "These notes are the first chapter of a monograph, dedicated to a detailed proof of the equivariant index theorem for transversally elliptic operators. In this preliminary chapter, we prove a certain number of natural relations in equivariant cohomology. These relations include the Thom isomorphism in equivariant cohomology, the multiplicativity of the relative Chern characters, and the Riemann-Roch relation between the relative Chern character of the Bott symbol and of the relative Thom class."}
{"category": "Math", "title": "Weighted Projective Lines Associated to Regular Systems of Weights of Dual Type", "abstract": "We associate to a regular system of weights a weighted projective line over an algebraically closed field of characteristic zero in two different ways. One is defined as a quotient stack via a hypersurface singularity for a regular system of weights and the other is defined via the signature of the same regular system of weights. The main result in this paper is that if a regular system of weights is of dual type then these two weighted projective lines have equivalent abelian categories of coherent sheaves. As a corollary, we can show that the triangulated categories of the graded singularity associated to a regular system of weights has a full exceptional collection, which is expected from homological mirror symmetries. Main theorem of this paper will be generalized to more general one, to the case when a regular system of weights is of genus zero, which will be given in the joint paper with Kajiura and Saito. Since we need more detailed study of regular systems of weights and some knowledge of algebraic geometry of Deligne--Mumford stacks there, the author write a part of the result in this paper to which another simple proof based on the idea by Geigle--Lenzing can be applied."}
{"category": "Math", "title": "A problem with Artin's Vanishing for torsion motivic homology", "abstract": "The paper is suspended. The reason: as was noted by prof. H. Esnault, Theorem 2.1.1 of the previous version (as well as the related Theorem 6.1.1 of http://arxiv.org/PS_cache/math/pdf/9908/9908037v2.pdf of D. Arapura and P. Sastry) is wrong unless one assumes H to be a generic hyperplane section. Hence the proofs of all results starting from 2.3 contain gaps. The author hopes to correct this (somehow) in a future version. At least, most of the results follow from certain \"standard\" motivic conjectures (see part 1 of Remark 3.2.4 in the previous version). If the author would not find a way to prove Theorems 2.3.1 and 2.3.2 (without 2.1.1), then in the next version of the preprint the results of section 4 will be deduced from certain conjectures; certainly this is not a very exiting result."}
{"category": "Math", "title": "Small Subspaces of L_p", "abstract": "We prove that if $X$ is a subspace of $L_p$ $(2<p<\\infty)$, then either $X$ embeds isomorphically into $\\ell_p \\oplus \\ell_2$ or $X$ contains a subspace $Y,$ which is isomorphic to $\\ell_p(\\ell_2)$. We also give an intrinsic characterization of when $X$ embeds into $\\ell_p \\oplus \\ell_2$ in terms of weakly null trees in $X$ or, equivalently, in terms of the \"infinite asymptotic game\" played in $X$. This solves problems concerning small subspaces of $L_p$ originating in the 1970's. The techniques used were developed over several decades, the most recent being that of weakly null trees developed in the 2000's."}
{"category": "Math", "title": "Moderate deviations for stationary sequences of bounded random variables", "abstract": "In this paper we derive the moderate deviation principle for stationary sequences of bounded random variables under martingale-type conditions. Applications to functions of $\\phi$-mixing sequences, contracting Markov chains, expanding maps of the interval, and symmetric random walks on the circle are given."}
{"category": "Math", "title": "Primitive cohomology and the tube mapping", "abstract": "Let X be a smooth complex projective variety of dimension d. We show that its primitive cohomology in degree d is generated by certain \"tube classes,\" constructed from the monodromy of the family of smooth hyperplane sections on X. The proof makes use of a result about the group cohomology of certain representations that may be of independent interest."}
{"category": "Math", "title": "A posteriori error estimates in the maximum norm for parabolic problems", "abstract": "We derive a posteriori error estimates in the $L_\\infty((0,T];L_\\infty(\\Omega))$ norm for approximations of solutions to linear para bolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estimates for linear parabolic pr oblems, we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations. We then establish a posteriori bounds for a fully discrete backward Euler finite element approximation. The elliptic reconstruction technique greatly simplifies our development by allow\\ ing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators."}
{"category": "Math", "title": "On a reduction procedure for Horn inequalities in finite von Neumann algebras", "abstract": "We consider the analogues of the Horn inequalities in finite von Neumann algebras, which concern the possible spectral distributions of sums $a+b$ of self--adjoint elements $a$ and $b$ in a finite von Neumann algebra. It is an open question whether all of these Horn inequalities must hold in all finite von Neumann algebras, and this is related to Connes' embedding problem. For each choice of integers $1\\le r\\le n$, there is a set $T^n_r$ of Horn triples, and the Horn inequalities are in one-to-one correspondence with $\\cup_{1\\le r\\le n}T^n_r$. We consider a property P$_n$, analogous to one introduced by Therianos and Thompson in the case of matrices, amounting to the existence of projections having certain properties relative to arbitrary flags, which guarantees that a given Horn inequality holds in all finite von Neumann algebras. It is an open question whether all Horn triples in $T^n_r$ have property P$_n$. Certain triples in $T^n_r$ can be reduced to triples in $T^{n-1}_r$ by an operation we call {\\em TT--reduction}. We show that property P$_n$ holds for the original triple if property P$_{n-1}$ holds for the reduced one. We then characterize the TT--irreducible Horn triples in $T^n_3$, for arbitrary $n$, and for those LR--minimal ones (namely, those having Littlewood--Richardson coefficient equal to 1), we perform a construction of projections with respect to flags in arbitrary von Neumann algebras in order to prove property P$_n$ for them. This shows that all LR--minimal triples in $\\cup_{n\\ge3}T^n_3$ have property P$_n$, and so that the corresponding Horn inequalities hold in all finite von Neumann algebras."}
{"category": "Math", "title": "The tube method for the moment index in projection pursuit", "abstract": "The projection pursuit index defined by a sum of squares of the third and the fourth sample cumulants is known as the moment index proposed by Jones and Sibson. Limiting distribution of the maximum of the moment index under the null hypothesis that the population is multivariate normal is shown to be the maximum of a Gaussian random field with a finite Karhunen-Loeve expansion. An approximate formula for tail probability of the maximum, which corresponds to the p-value, is given by virtue of the tube method through determining Weyl's invariants of all degrees and the critical radius of the index manifold of the Gaussian random field."}
{"category": "Math", "title": "Sparsistency and rates of convergence in large covariance matrix estimation", "abstract": "This paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions. Here, sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one. Depending on the case of applications, sparsity priori may occur on the covariance matrix, its inverse or its Cholesky decomposition. We study these three sparsity exploration problems under a unified framework with a general penalty function. We show that the rates of convergence for these problems under the Frobenius norm are of order $(s_n\\log p_n/n)^{1/2}$, where $s_n$ is the number of nonzero elements, $p_n$ is the size of the covariance matrix and $n$ is the sample size. This explicitly spells out the contribution of high-dimensionality is merely of a logarithmic factor. The conditions on the rate with which the tuning parameter $\\lambda_n$ goes to 0 have been made explicit and compared under different penalties. As a result, for the $L_1$-penalty, to guarantee the sparsistency and optimal rate of convergence, the number of nonzero elements should be small: $s_n'=O(p_n)$ at most, among $O(p_n^2)$ parameters, for estimating sparse covariance or correlation matrix, sparse precision or inverse correlation matrix or sparse Cholesky factor, where $s_n'$ is the number of the nonzero elements on the off-diagonal entries. On the other hand, using the SCAD or hard-thresholding penalty functions, there is no such a restriction."}
{"category": "Math", "title": "A Support Theorem for the Geodesic Ray Transform of Functions", "abstract": "Let $(M,g)$ be a simple Riemannian manifold. Under the assumption that the metric $g$ is real-analytic, it is shown that if the geodesic ray transform of a function $f\\in L^{2}(M)$ vanishes on an appropriate open set of geodesics, then $f=0$ on the set of points lying on these geodesics. The approach is based on a microlocal version of unique continuation of analytic functions."}
{"category": "Math", "title": "A recursion equation for prime numbers", "abstract": "It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\\lim\\limits_{s\\to +\\infty} [\\prod\\limits^n_{k=1} (1-\\frac{1}{p^s_k}) \\sum\\limits^\\infty_{j=1} \\frac{1}{j^s} -1]^{-1/s}. $$ The upper limit on the sum can be replaced by $2p_n -1$, and the result still holds."}
{"category": "Math", "title": "Tensor product of coherent systems", "abstract": "Let X be a smooth algebraic curve of genus g>=2. A stable vector bundle over X of degree d, rank n with at least k sections is called a Brill-Noether bundle of type (n,d,k). By tensoring coherent systems, we prove that most of the known Brill-Noether bundles define coherent systems of type (n,d,k) that are alpha-stables for all allowable alpha ."}
{"category": "Math", "title": "Multiple eigenvalues", "abstract": "The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each corresponding codimension is the number of conditions which a matrix of the given type must satisfy in order to have the specified multiplicities."}
{"category": "Math", "title": "Semi-parametric second-order efficient estimation of the period of a signal", "abstract": "This paper is concerned with the estimation of the period of an unknown periodic function in Gaussian white noise. A class of estimators of the period is constructed by means of a penalized maximum likelihood method. A second-order asymptotic expansion of the risk of these estimators is obtained. Moreover, the minimax problem for the second-order term is studied and an estimator of the preceding class is shown to be second order efficient."}
{"category": "Math", "title": "Moment estimation for ergodic diffusion processes", "abstract": "We investigate the moment estimation for an ergodic diffusion process with unknown trend coefficient. We consider nonparametric and parametric estimation. In each case, we present a lower bound for the risk and then construct an asymptotically efficient estimator of the moment type functional or of a parameter which has a one-to-one correspondence to such a functional. Next, we clarify a higher order property of the moment type estimator by the Edgeworth expansion of the distribution function."}
{"category": "Math", "title": "The classification and the conjugacy classes of the finite subgroups of the sphere braid groups", "abstract": "Let n\\geq 3. We classify the finite groups which are realised as subgroups of the sphere braid group B_n(S^2). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B_n(S^2): Z_{2(n-1)}; the dicyclic groups of order 4n and 4(n-2); the binary tetrahedral group T_1; the binary octahedral group O_1; and the binary icosahedral group I. We give geometric as well as some explicit algebraic constructions of these groups in B_n(S^2), and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi's classification of the torsion elements of B_n(S^2), and explain how the finite subgroups of B_n(S^2) are related to this classification, as well as to the lower central and derived series of B_n(S^2)."}
{"category": "Math", "title": "Remark to the paper Describing the set of words generated by interval exchange transformation, posted 15 November 2007", "abstract": "Let us call subdivision {\\it good}, if 1) set corresponding to each symbol is convex (i.e. interval or (semi)closed interval). 2) If points $A$ and $B$ corresponds to the some color and interval $(A,B)$ has discontinuity point, then $f(A)$ and $f(B)$ has different color. Every subdivision can be further divided into good subdivision, old superword can be obtained from new one by gluing letters. Hence in the section ``Equivalence of the set of uniformly recurrent words generated by piecewise-continuous transformation to the set of words generated by interval exchange transformation'' one can consider only good subdivision."}
{"category": "Math", "title": "Goussarov-Polyak-Viro combinatorial formulas for finite type invariants", "abstract": "Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type invariants. This article presents a simplified account of the original approach. The simplifications provide an easy generalization to the cases of pure tangles and pure braids. The associated problem on group algebras is introduced and used to prove the existence of ``multi-local word formulas'' for finite type invariants of pure braids."}
{"category": "Math", "title": "Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces", "abstract": "Ricci-type symplectic manifolds have been introduced and extensively studied by M. Cahen et al.. In this note, we describe their deformation quantizations in the split solvable symmetric case. In particular, we introduce the notion of non-formal tempered deformation quantization on such a space. We show that the set of tempered deformation quantizations is in one-to-one correspondence with the space of Schwartz operator multipliers on the real line. Moreover we prove that every invariant formal star product on a split Ricci-type solvable symmetric space is an asymptotic expansion of a tempered non-formal quantization. This note illustrates and partially reviews through an example a problematic studied by the author regarding non-formal quantization in presence of large groups of symmetries."}
{"category": "Math", "title": "Nonparametric deconvolution problem for dependent sequences", "abstract": "We consider the nonparametric estimation of the density function of weakly and strongly dependent processes with noisy observations. We show that in the ordinary smooth case the optimal bandwidth choice can be influenced by long range dependence, as opposite to the standard case, when no noise is present. In particular, if the dependence is moderate the bandwidth, the rates of mean-square convergence and, additionally, central limit theorem are the same as in the i.i.d. case. If the dependence is strong enough, then the bandwidth choice is influenced by the strength of dependence, which is different when compared to the non-noisy case. Also, central limit theorem are influenced by the strength of dependence. On the other hand, if the density is supersmooth, then long range dependence has no effect at all on the optimal bandwidth choice."}
{"category": "Math", "title": "The Kuramoto-Sivashinsky equation in R^1 and R^2: effective estimates of the high-frequency tails and higher Sobolev norms", "abstract": "We consider the Kuramoto-Sivashinsky (KS) equation in finite domains of the form $[-L,L]^d$. Our main result provides refined Gevrey estimates for the solutions of the one dimensional differentiated KS, which in turn imply effective new estimates for higher Sobolev norms of the solutions in terms of powers of $L$. We illustrate our method on a simpler model, namely the regularized Burger's equation. We also show local well-posedness for the two dimensional KS equation and provide an explicit criteria for (eventual) blow-up in terms of its $L^2$ norm. The common underlying idea in both results is that {\\it a priori} control of the $L^2$ norm is enough in order to conclude higher order regularity and allows one to get good estimates on the high-frequency tails of the solutions."}
{"category": "Math", "title": "On the detectability of different forms of interaction in regression models", "abstract": "We derive an asymptotic power function for a likelihood-based test for interaction in a regression model, with possibly misspecified alternative distribution. This allows a general investigation of types of interactions which are poorly or well detected via data. Principally we contrast pairwise-interaction models with `diffuse interaction models' as introduced in Gustafson, Kazi, and Levy (2005)."}
{"category": "Math", "title": "Pauli Pascal Pyramids, Pauli Fibonacci Numbers, and Pauli Jacobsthal Numbers", "abstract": "The three anti-commutative two-dimensional Pauli Pascal triangles can be generalized into multi-dimensional Pauli Pascal hyperpyramids. Fibonacci and Jacobsthal numbers are then generalized into Pauli Fibonacci numbers, Pauli Jacobsthal numbers, and Pauli Fibonacci numbers of higher order. And the question is: are Pauli rabbits killer rabbits?"}
{"category": "Math", "title": "Equations aux q-differences et fibres vectoriels holomorphes sur la courbe elliptique C^*/q^Z", "abstract": "Nous presentons diverses applications des fibres vectoriels aux equations aux q-differences, dans la lignee de la correspondance de Weil. (We present some applications of vector bundles to $q$-difference equtions, in continuation of Weil's correspondance.)"}
{"category": "Math", "title": "The q-analogue of the wild fundamental group (II)", "abstract": "In [RS1], we defined q-analogues of alien derivations and stated their basic properties. In this paper, we prove the density theorem and the freeness theorem announced in loc. cit. [RS1] Ramis J.-P. and Sauloy J., 2007. The q-analogue of the wild fundamental group (I)"}
{"category": "Math", "title": "Borovik-Poizat rank and stability", "abstract": "There is an axiomatic treatment of Morley rank in groups, due to Borovik and Poizat. These axioms form the basis of the algebraic treatment of groups of finite Morley rank which is common today. There are, however, ranked structures, i.e. structures on which a Borovik-Poizat rank function is defined, which are not $\\aleph_0$-stable. Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: ``un groupe de Borovik est une structure stable, alors qu'un univers rang\\'e n'a aucune raison de l'\\^etre ...''. Nonetheless, we show that a ranked structure is superstable."}
{"category": "Math", "title": "An introduction to upper half plane polynomials", "abstract": "This is a straightforward introduction to the properties of polynomials in many variables that do not vanish in the open upper half plane. Such polynomials generalize many of the well-known properties of polynomials with all real roots."}
{"category": "Math", "title": "Dimensional reduction and the long-time behavior of Ricci flow", "abstract": "If g(t) is a three-dimensional Ricci flow solution, with sectional curvatures that decay like the inverse of t and diameter that increases at most like the square root of t, then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton."}
{"category": "Math", "title": "Hausdorff dimension of the SLE curve intersected with the real line", "abstract": "We establish an upper bound on the asymptotic probability of an SLE(kappa) curve hitting two small intervals on the real line as the interval width goes to zero, for the range 4 < kappa < 8. As a consequence we are able to prove that the SLE curve intersected with the real line has Hausdorff dimension 2-8/kappa, almost surely."}
{"category": "Math", "title": "A weighted generalization of Gao's n+D-1 Theorem", "abstract": "Let $G$ denotes a finite abelian group of order $n$ and Davenport constant $D$, and put $m= n+D-1$. Let $x=(x_1, ..., x_m)\\in G^m$ be a sequence with a maximal repetition $\\ell$ attained by $x_m$ and put $r=\\min(D,\\ell)$. Let $w=(w_1, ..., w_{m-r})\\in \\Z^{m-r}.$ Then there are an $n$-subset $I\\subset [1,m-r]$ and an injection $f: I\\mapsto [1,m]$, such that $m\\in f(I)$ and $$\\sum_{i\\in I}w_{i}x_{f({i})}=(\\sum_{i\\in I}w_{i})x_{m}.$$"}
{"category": "Math", "title": "Counterexamples to Rational Dilation on Symmetric Multiply Connected Domains", "abstract": "We show that if R is a compact domain in the complex plane with two or more holes and an anticonformal involution onto itself (or equivalently a hyperelliptic Schottky double), then there is an operator T which has R as a spectral set, but does not dilate to a normal operator with spectrum on the boundary of R."}
{"category": "Math", "title": "Ideal boundary of 7-systolic complexes and groups", "abstract": "We prove that ideal boundary of a 7-systolic group is strongly hereditarily aspherical. For some class of 7-systolic groups we show their boundaries are connected and without local cut points, thus getting some results concerning splittings of those groups."}
{"category": "Math", "title": "Loops in the Hamiltonian group: a survey", "abstract": "This note describes some recent results about the homotopy properties of Hamiltonian loops in various manifolds, including toric manifolds and one point blow ups. We describe conditions under which a circle action does not contract in the Hamiltonian group, and construct an example of a loop $\\ga$ of diffeomorphisms of a symplectic manifold M with the property that none of the loops smoothly isotopic to $\\ga$ preserve any symplectic form on M. We also discuss some new conditions under which the Hamiltonian group has infinite Hofer diameter. Some of the methods used are classical (Weinstein's action homomorphism and volume calculations), while others use quantum methods (the Seidel representation and spectral invariants)."}
{"category": "Math", "title": "Connectedness at infinity of systolic complexes and groups", "abstract": "By studying connectedness at infinity of systolic groups we distinguish them from some other classes of groups, in particular from the fundamental groups of manifolds covered by euclidean space of dimension at least three. We also study semistability at infinity for some systolic groups."}
{"category": "Math", "title": "Cluster algebras and preprojective algebras : the non simply-laced case", "abstract": "We generalize to the non simply-laced case results of Gei\\ss, Leclerc and Schr\\\"oer about the cluster structure of the coordinate ring of the maximal unipotent subgroups of simple Lie groups. In this way, cluster structures in the non simply-laced case can be seen as projections of cluster structures in the simply-laced case. This allows us to prove that cluster monomials are linearly independent in the non simply-laced case."}
{"category": "Math", "title": "On the Convex Closure of the Graph of Modular Inversions", "abstract": "In this paper we give upper and lower bounds as well as a heuristic estimate on the number of vertices of the convex closure of the set $$ G_n=\\left\\{(a,b) : a,b\\in \\Z, ab \\equiv 1 \\pmod{n}, 1\\leq a,b\\leq n-1\\right\\}. $$ The heuristic is based on an asymptotic formula of R\\'{e}nyi and Sulanke. After describing two algorithms to determine the convex closure, we compare the numeric results with the heuristic estimate. The numeric results do not agree with the heuristic estimate -- there are some interesting peculiarities for which we provide a heuristic explanation. We then describe some numerical work on the convex closure of the graph of random quadratic and cubic polynomials over $\\mathbb{Z}_n$. In this case the numeric results are in much closer agreement with the heuristic, which strongly suggests that the the curve $xy=1\\pmod{n}$ is ``atypical''."}
{"category": "Math", "title": "Twisted Homology of Quantum SL(2) - Part II", "abstract": "We complete the calculation of the twisted cyclic homology of the quantised coordinate ring of SL(2) that we began in math.QA/0405249. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form."}
{"category": "Math", "title": "A Simplified Calculation for the Fundamental Solution to the Heat Equation on the Heisenberg Group", "abstract": "Let $L = -1/4 (\\sum_{j=1}^n(X_j^2+Y_j^2)+i\\gamma T)$ where $\\gamma$ is a complex number, $X_j$, $Y_j$, and $T$ are the left invariant vector fields of the Heisenberg group structure for $R^n \\times R^n \\times R$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the Heat Equation $\\partial_s\\rho = -L\\rho$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the $\\Box_b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted dbar-operator in $C^n$ with weight $\\exp(-\\tau P(z_1,...,z_n))$ where $P(z_1,...,z_n) = 1/2(x_1^2 + >... x_n^2)$, $z_j=x_j+iy_j$, and $\\tau\\in R$."}
{"category": "Math", "title": "Examples of Coorbit Spaces for Dual Pairs", "abstract": "In this paper we summarize and give examples of a generalization of the coorbit space theory initiated in the 1980's by H.G. Feichtinger and K.H. Gr\\\"ochenig. Coorbit theory has been a powerful tool in characterizing Banach spaces of distributions with the use of integrable representations of locally compact groups. Examples are a wavelet characterization of the Besov spaces and a characterization of some Bergman spaces by the discrete series representation of $\\mathrm{SL}_2(\\mathbb{R})$. We present examples of Banach spaces which could not be covered by the previous theory, and we also provide atomic decompositions for an example related to a non-integrable representation."}
{"category": "Math", "title": "An inequality for correlated measurable functions", "abstract": "A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply this result to a problem arising from probability theory."}
{"category": "Math", "title": "A Short Note on some open problems in the geometry of operator ideals", "abstract": "We list and discuss the background of some open problems, regarding the principle of local reflexivity for maximal Banach ideals."}
{"category": "Math", "title": "The Plateau problem for marginally outer trapped surfaces", "abstract": "We solve the Plateau problem for marginally outer trapped surfaces in general Cauchy data sets. We employ the Perron method and tools from geometric measure theory to force and control a blow-up of Jang's equation. Substantial new geometric insights regarding the lower order properties of marginally outer trapped surfaces are gained in the process. The techniques developed in this paper are flexible and can be adapted to other non-variational existence problems."}
{"category": "Math", "title": "On Haagerup's list of potential principal graphs of subfactors", "abstract": "We show that any graph, in the sequence given by Haagerup in 1991 as that of candidates of principal graphs of subfactors, is not realized as a principal graph except for the smallest two. This settles the remaining case of a previous work of the first author."}
{"category": "Math", "title": "Sylow 0-unipotent subgroups in groups of finite Morley rank", "abstract": "One of the central tools in the classification of simple algebraic groups is the distinction between semisimple subgroups and unipotent subgroups. It is not a priori clear how to make this distinction for torsion-free subgroups of a group of finite Morley rank. We exploit the ``graded'' notion of 0-unipotence to develop a Sylow theory for torsion-free subgroups of a solvable group of finite Morley rank. This Sylow theory provides a robust alternative to the usual theory of Carter subgroups, and will be used in the analysis of intersections of Borel subgroups in minimal simple groups."}
{"category": "Math", "title": "Grid graphs, Gorenstein polytopes, and domino stackings", "abstract": "We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Gorenstein. We also introduce the notion of domino stackings and present some results and several open questions. Our techniques use results from graph theory, polyhedral geometry, and enumerative combinatorics."}
{"category": "Math", "title": "The Bender method in groups of finite Morley rank", "abstract": "Jaligot's Lemma states that the Fitting subgroups of distinct Borel subgroups do not intersect in a tame minimal simple groups of finite Morley. Such a strong result appears hopeless without tameness. Here we use the 0-unipotence theory to build a toolkit for the analysis of nonabelian intersections of Borel subgroups. As a demonstration, we show that any connected nilpotent subgroup of an intersection of Borel subgroups, in a nontame minimal simple group, must actually be abelian."}
{"category": "Math", "title": "Two Coefficients of the Dyson Product", "abstract": "In this paper, the closed-form expressions for the coefficients of $\\frac{x_r^2}{x_s^2}$ and $\\frac{x_r^2}{x_sx_t}$ in the Dyson product are found by applying an extension of Good's idea. As onsequences, we find several interesting Dyson style constant term identities."}
{"category": "Math", "title": "Scaled Asymptotics For Some $q$-Series As $q$ Approaching Unit", "abstract": "In this work we investigate Plancherel-Rotach type asymptotics for some $q$-series as $q\\to1$. These $q$-series generalize Ramanujan function $A_{q}(z)$; Jackson's $q$-Bessel function $J_{\\nu}^{(2)}$(z;q), Ismail-Masson orthogonal polynomials($q^{-1}$-Hermite polynomials) $h_{n}(x|q)$, Stieltjes-Wigert orthogonal polynomials $S_{n}(x;q)$, $q$-Laguerre orthogonal polynomials $L_{n}^{(\\alpha)}(x;q)$ and confluent basic hypergeometric series."}
{"category": "Math", "title": "Minimal connected simple groups of finite Morley rank with strongly embedded subgroups", "abstract": "We show that a minimal nonalgebraic simple groups of finite Morley rank has Prufer rank at most 2, and eliminates tameness from Cherlin and Jaligot's past work on minimal simple groups. The argument given here begins with the strongly embedded minimal simple configuration of Borovik, Burdges and Nesin. The 0-unipotence machinery of Burdges's thesis is used to analyze configurations involving nonabelian intersections of Borel subgroups. The number theoretic punchline of Cherlin and Jaligot has been replaced with a new genericity argument."}
{"category": "Math", "title": "Involutions in groups of finite Morley rank of degenerate type", "abstract": "This article proves a version of the Feit-Thompson theorem for simple groups of finite Morley rank: a connected groups of finite Morley rank with a finite Sylow 2-subgroup has a trivial Sylow 2-subgroups."}
{"category": "Math", "title": "Uniqueness cases in odd type groups of finite Morley rank", "abstract": "Here we analyze a proper 2-generated core in a minimal counter example to the Cherlin-Zilber Algebraicity Conjecture for simple groups of finite Morley rank. We ultimately show that such a group is strongly embedded and the ambiant group is minimal connected simple."}
{"category": "Math", "title": "A New Trichotomy Theorem", "abstract": "We show that a minimal counter example to the Cherlin-Zilber Algebraicity Conjecture for simple groups of finite Morley rank has normal 2-rank at most two, which is a tameness free version of Borovik's original trichotomy theorem. This result serves as a bridge by showing that there are no groups found strictly between the generic and quasithin cases, i.e. between groups of Lie rank at least three, and groups of Lie rank one and two. Again this result depends upon previous work for the uniqueness case analysis."}
{"category": "Math", "title": "On Distributed Averaging Algorithms and Quantization Effects", "abstract": "We consider distributed iterative algorithms for the averaging problem over time-varying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of performance when only quantized information is available. We study a large and natural class of averaging algorithms, which includes the vast majority of algorithms proposed to date, and provide tight polynomial bounds on their convergence time. We also describe an algorithm within this class whose convergence time is the best among currently available averaging algorithms for time-varying topologies. We then propose and analyze distributed averaging algorithms under the additional constraint that agents can only store and communicate quantized information, so that they can only converge to the average of the initial values of the agents within some error. We establish bounds on the error and tight bounds on the convergence time, as a function of the number of quantization levels."}
{"category": "Math", "title": "Existence and Stability of Steady-State Solutions with Finite Energy for the Navier-Stokes equation in the Whole Space", "abstract": "We consider the steady-state Navier-Stokes equation in the whole space $\\mathbb{R}^3$ driven by a forcing function $f$. The class of source functions $f$ under consideration yield the existence of at least one solution with finite Dirichlet integral ($\\|\\nabla U\\|_2<\\infty$). Under the additional assumptions that $f$ is absent of low modes and the ratio of $f$ to viscosity is sufficiently small in a natural norm we construct solutions which have finite energy (finite $L^2$ norm). These solutions are unique among all solutions with finite energy and finite Dirichlet integral. The constructed solutions are also shown to be stable in the following sense: If $U$ is such a solution then any viscous, incompressible flow in the whole space, driven by $f$ and starting with finite energy, will return to $U$."}
{"category": "Math", "title": "Floer trajectories with immersed nodes and scale-dependent gluing", "abstract": "We define an enhanced compactification of Floer trajectories under Morse background using the adiabatic degeneration and the scale-dependent gluing techniques. The compactification reflects the 1-jet datum of the smooth Floer trajectories nearby the limiting nodal Floer trajectories arising from adiabatic degeneration of the background Morse function. This paper studies the gluing problem when the limiting gradient trajectories has length zero through a renomalization process. The case with limiting gradient trajectories of non-zero length will be treated elsewhere. An immediate application of our result is a proof of the isomorphism property of the PSS map : A proof of this isomorphism property was first outlined by P\\\"unihikin-Salamon-Schwarz \\cite{PSS} in a way somewhat different from the current proof in its details. This kind of scale-dependent gluing techniques was initiated in [FOOO07] in relation to the metamorphosis of holomorphic polygons under Lagrangian surgery and is expected to appear in other gluing and compactification problem of pseudo-holomorphic curves that involves `adiabatic' parameters or rescales the targets."}
{"category": "Math", "title": "On dispersion for Klein Gordon equation with periodic potential in 1D", "abstract": "By exploiting estimates on Bloch functions obtained in a previous paper, we prove decay estimates for Klein Gordon equations with a time independent potential periodic in space in 1D and with generic mass"}
{"category": "Math", "title": "A revision of \"On asymptotic stability in energy space of ground states of NLS in 1D\"", "abstract": "We transpose work by T.Mizumachi to prove smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schr\\\"odinger equation (NLS) in 1D. As an application we extend to dimension 1D a result on asymptotic stability of ground states of NLS proved by Cuccagna & Mizumachi for dimensions $\\ge 3$"}
{"category": "Math", "title": "On asymptotic stability in energy space of ground states for Nonlinear Schr\\\"odinger equations", "abstract": "We consider nonlinear Schr\\\"odinger equations in dimension 3 or higher. We prove that symmetric finite energy solutions close to orbitally stable ground states converge asymptotically to a sum of a ground state and a dispersive wave assuming the so called Fermi Golden Rule (FGR) hypothesis. We improve the sign condition required in a recent paper by Gang Zhou and I.M.Sigal"}
{"category": "Math", "title": "On a class of curved flag multipliers", "abstract": "We characterize a family of curved flag kernels in terms of their multipliers and prove L^p boundedness."}
{"category": "Math", "title": "Mean density of inhomogeneous Boolean models with lower dimensional typical grain", "abstract": "The mean density of a random closed set $\\Theta$ in $\\R^d$ with Hausdorff dimension $n$ is the Radon-Nikodym derivative of the expected measure $\\E[\\h^n(\\Theta\\cap\\cdot)]$ induced by $\\Theta$ with respect to the usual $d$-dimensional Lebesgue measure. We consider here inhomogeneous Boolean models with lower dimensional typical grain. Under general regularity assumptions on the typical grain, related to the existence of its Minkowski content, and on the intensity measure of the underlying Poisson point process, we prove an explicit formula for the mean density. The proof of such formula provides as by-product estimators for the mean density in terms of the empirical capacity functional, which turns to be closely related to the well known random variable density estimation by histograms in the extreme case $n=0$. Particular cases and examples are also discussed."}
{"category": "Math", "title": "Signalizers and balance in groups of finite Morley rank", "abstract": "We show that a minimal counter example to the Cherlin-Zilber Algebraicity Conjecture for simple groups of finite Morley rank has Prufer 2-rank at most two. This article covers the signalizer functor theory and identifies the groups of Lie rank at least three; leaving the uniqueness case analysis to previous articles. This result signifies the end of the general methods used to handle large groups; hereafter each individual group PSL$_2$, PSL$_3$, PSp$_4$, and G$_2$ will require its own identification theorem."}
{"category": "Math", "title": "Empirical likelihood based testing for regression", "abstract": "Consider a random vector $(X,Y)$ and let $m(x)=E(Y|X=x)$. We are interested in testing $H_0:m\\in {\\cal M}_{\\Theta,{\\cal G}}=\\{\\gamma(\\cdot,\\theta,g):\\theta \\in \\Theta,g\\in {\\cal G}\\}$ for some known function $\\gamma$, some compact set $\\Theta \\subset $IR$^p$ and some function set ${\\cal G}$ of real valued functions. Specific examples of this general hypothesis include testing for a parametric regression model, a generalized linear model, a partial linear model, a single index model, but also the selection of explanatory variables can be considered as a special case of this hypothesis. To test this null hypothesis, we make use of the so-called marked empirical process introduced by \\citeD and studied by \\citeSt for the particular case of parametric regression, in combination with the modern technique of empirical likelihood theory in order to obtain a powerful testing procedure. The asymptotic validity of the proposed test is established, and its finite sample performance is compared with other existing tests by means of a simulation study."}
{"category": "Math", "title": "Computing Humbert Surfaces", "abstract": "We describe an algorithm which computes components of Humbert surfaces in terms of Rosenhain invariants, based on Runge's method"}
{"category": "Math", "title": "On the Colored HOMFLY-PT, Multivariable and Kashaev Link Invariants", "abstract": "We study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of sl(m|n) super-modules previously defined by the authors contains both the multivariable Alexander polynomial and Kashaev's invariants. We conjecture these multivariable link invariants also specialize to the generalized multivariable Alexander invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki."}
{"category": "Math", "title": "Existence of Least-perimeter Partitions", "abstract": "We prove the existence of a perimeter-minimizing partition of R^n into regions of unit volume. We conclude with a short tribute to the late Manuel A. Fortes."}
{"category": "Math", "title": "Modified quantum dimensions and re-normalized link invariants", "abstract": "In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum dimensions. More interestingly we will give two examples where the usual quantum dimensions vanish but the modified quantum dimensions are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras previously defined by the first two authors. These link invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of link invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of knots."}
{"category": "Math", "title": "An invariant supertrace for the category of representations of Lie superalgebras", "abstract": "In this paper we give a re-normalization of the supertrace on the category of representations of Lie superalgebras of type I, by a kind of modified superdimension. The genuine superdimensions and supertraces are generically zero. However, these modified superdimensions are non-zero and lead to a kind of supertrace which is non-trivial and invariant. As an application we show that this new supertrace gives rise to a non-zero bilinear form on a space of invariant tensors of a Lie superalgebra of type I. The results of this paper are completely classical results in the theory of Lie superalgebras but surprisingly we can not prove them without using quantum algebra and low-dimensional topology."}
{"category": "Math", "title": "The Schur transformation for Nevanlinna functions: operator representations, resolvent matrices, and orthogonal polynomials", "abstract": "A Nevanlinna function is a function which is analytic in the open upper half plane and has a non-negative imaginary part there. In this paper we study a fractional linear transformation for a Nevanlinna function $n$ with a suitable asymptotic expansion at $\\infty$, that is an analogue of the Schur transformation for contractive analytic functions in the unit disc. Applying the transformation $p$ times we find a Nevanlinna function $n_p$ which is a fractional linear transformation of the given function $n$. The main results concern the effect of this transformation to the realizations of $n$ and $n_p$, by which we mean their representations through resolvents of self-adjoint operators in Hilbert space. Our tools are block operator matrix representations, $u$--resolvent matrices, and reproducing kernel Hilbert spaces."}
{"category": "Math", "title": "Infinite groups with fixed point properties", "abstract": "We construct finitely generated groups with strong fixed point properties. Let $\\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any action of $Q$ on any $X\\in \\mathcal{X}_{ac}$, there is a global fixed point. Moreover, $Q$ may be chosen to be simple and to have Kazhdan's property (T). We construct a finitely presented infinite group $P$ that admits no non-trivial action by diffeomorphisms on any smooth manifold in $\\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\\geq 1$ and each prime $p$, we construct a non-elementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$-group."}
{"category": "Math", "title": "Uniruled symplectic divisors", "abstract": "This is a paper devoted to the symplectic birational geometry program where many basic notions are defined in terms of genus 0 GW invariants. We show that the existence of a positive uniruled symplectic divisor often implies that the ambient manifold has a nonzero uniruled genus 0 GW invariant, hence is uniruled as well. This confirms a part of the dichotomy on uniruled symplectic divisors. In addition, it gives a rather general construction of uniruled symplectic manifolds, generalizing some beautiful results of McDuff."}
{"category": "Math", "title": "Simple Lie Algebras having Extremal Elements", "abstract": "Let L be a simple finite-dimensional Lie algebra of characteristic distinct from 2 and from 3. Suppose that L contains an extremal element that is not a sandwich, that is, an element x such that [x, [x, L]] is equal to the linear span of x in L. In this paper we prove that, with a single exception, L is generated by extremal elements. The result is known, at least for most characteristics, but the proofs in the literature are involved. The current proof closes a gap in a geometric proof that every simple Lie algebra containing no sandwiches (that is, ad-nilpotent elements of order 2) is in fact of classical type."}
{"category": "Math", "title": "Unipotent classes and special Weyl group representations", "abstract": "We show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the Weyl group."}
{"category": "Math", "title": "Notes on the Kazhdan-Lusztig theorem on equivalence of the Drinfeld category and the category of Uq(g)-modules", "abstract": "We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category D(g,h) of g-modules and the category of finite dimensional Uq(g)-modules, q=exp(\\pi ih), for h\\in C\\Q*. Aiming at operator algebraists the result is formulated as the existence for each h\\in iR of a normalized unitary 2-cochain F on the dual \\hat G of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by F is *-isomorphic to the convolution algebra of the q-deformation G_q of G, while the coboundary of F^{-1} coincides with Drinfeld's KZ-associator defined via monodromy of the Knizhnik-Zamolodchikov equations."}
{"category": "Math", "title": "Can B(l^p) ever be amenable?", "abstract": "It is known that ${\\cal B}(\\ell^p)$ is not amenable for $p =1,2,\\infty$, but whether or not ${\\cal B}(\\ell^p)$ is amenable for $p \\in (1,\\infty) \\setminus \\{2 \\}$ is an open problem. We show that, if ${\\cal B}(\\ell^p)$ is amenable for $p \\in (1,\\infty)$, then so are $\\ell^\\infty({\\cal B}(\\ell^p))$ and $\\ell^\\infty({\\cal K}(\\ell^p))$. Moreover, if $\\ell^\\infty({\\cal K}(\\ell^p))$ is amenable so is $\\ell^\\infty(\\mathbb{I},{\\cal K}(E))$ for any index set $\\mathbb I$ and for any infinite-dimensional ${\\cal L}^p$-space $E$; in particular, if $\\ell^\\infty({\\cal K}(\\ell^p))$ is amenable for $p \\in (1,\\infty)$, then so is $\\ell^\\infty({\\cal K}(\\ell^p \\oplus \\ell^2))$. We show that $\\ell^\\infty({\\cal K}(\\ell^p \\oplus \\ell^2))$ is not amenable for $p =1,\\infty$, but also that our methods fail us if $p \\in (1,\\infty)$. Finally, for $p \\in (1,2)$ and a free ultrafilter $\\cal U$ over $\\posints$, we exhibit a closed left ideal of $({\\cal K}(\\ell^p))_{\\cal U}$ lacking a right approximate identity, but enjoying a certain, very weak complementation property."}
{"category": "Math", "title": "Singular Moduli of Shimura Curves", "abstract": "The $j$-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over $\\mathbb{Q}$. This paper shows that the coordinate maps for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies the conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points on these Shimura curves with arbitrarily large negative discriminant."}
{"category": "Math", "title": "Sum-free subsets of finite abelian groups of type III", "abstract": "A finite abelian group $G$ of cardinality $n$ is said to be of type III if every prime divisor of $n$ is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian group $G$ of type III. This theorem, when taken together with known results, gives a complete characterisation of sum-free subsets of the largest cardinality in any finite abelian group $G$. We supplement this result with a theorem on the structure of sum-free subsets of cardinality \"close\" to the largest possible in a type III abelian group $G$. We then give two applications of these results. Our first application allows us to write down a formula for the number of orbits under the natural action of ${\\rm Aut}(G)$ on the set of sum-free subsets of $G$ of the largest cardinality when $G$ is of the form $({\\mathbf{Z}}/m{\\mathbf{Z}})^r$, with all prime divisors of $m$ congruent to 1 modulo 3, thereby extending a result of Rhemtulla and Street. Our second application provides an upper bound for the number of sum-free subsets of $G$. For finite abelian groups $G$ of type III and with {\\em a given exponent} this bound is substantially better than that implied by the bound for the number of sum-free subsets in an arbitrary finite abelian group, due to Green and Ruzsa."}
{"category": "Math", "title": "Group Bundle Duality", "abstract": "This paper introduces a generalization of Pontryagin duality for locally compact Hausdorff abelian groups to locally compact Hausdorff abelian group bundles."}
{"category": "Math", "title": "On Three Different Notions of Monotone Subsequences", "abstract": "We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and provide some new results related to limiting distributions."}
{"category": "Math", "title": "Existence and Uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds II", "abstract": "In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term is positive. We do this by combining the Kazdan-Warner obstructions with a theorem due to De Lellis and M\\\"uller."}
{"category": "Math", "title": "Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds", "abstract": "We construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3-manifold which does not have the convergence properties needed in order to prove a Penrose--type inequality. This contrasts sharply with the asymptotically flat case. The main idea consists in combining inverse mean curvature flow with work done by Shi--Tam regarding boundary behavior of compact manifolds. Assuming the Penrose inequality holds, we also derive a nontrivial inequality for functions on $S^2$."}
{"category": "Math", "title": "The real loci of Calogero-Moser spaces, representations of rational Cherednik algebras and the Shapiro conjecture", "abstract": "We prove a criterion for the reality of irreducible representations of the rational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterion for the real loci of the Calogero-Moser spaces C_n in terms of the Etingof-Ginzburg finite maps \\Upsilon \\colon C_n \\to C^n/S_n \\times C^n/S_n, recovering a result of Mikhin, Tarasov, and Varchenko [MTV2]. As a consequence we obtain a criterion for the real locus of the Wilson's adelic Grassmannian of rank one bispectral solutions of the KP hierarchy. Using Wilson's first parametrisation of the adelic Grassmannian, we give a new proof of a result of [MTV2] on real bases of spaces of quasi polynomials. The Shapiro Conjecture for Grassmannians is equivalent to a special case of our result for Calogero-Moser spaces, namely for the fibres of \\Upsilon over C^n/S_n \\times 0."}
{"category": "Math", "title": "Intersection form, laminations and currents on free groups", "abstract": "Let $F_N$ be a free group of rank $N\\ge 2$, let $\\mu$ be a geodesic current on $F_N$ and let $T$ be an $\\mathbb R$-tree with a very small isometric action of $F_N$. We prove that the geometric intersection number $<T, \\mu>$ is equal to zero if and only if the support of $\\mu$ is contained in the dual algebraic lamination $L^2(T)$ of $T$. Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. As another application, we define the notion of a \\emph{filling} element in $F_N$ and prove that filling elements are \"nearly generic\" in $F_N$. We also apply our results to the notion of \\emph{bounded translation equivalence} in free groups."}
{"category": "Math", "title": "Translating solutions to Lagrangian mean curvature flow", "abstract": "We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal."}
{"category": "Math", "title": "Perfect domination in regular grid graphs", "abstract": "We show there is an uncountable number of parallel total perfect codes in the integer lattice graph ${\\Lambda}$ of $\\R^2$. In contrast, there is just one 1-perfect code in ${\\Lambda}$ and one total perfect code in ${\\Lambda}$ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products $C_m\\times C_n$ with parallel total perfect codes, and the $d$-perfect and total perfect code partitions of ${\\Lambda}$ and $C_m\\times C_n$, the former having as quotient graph the undirected Cayley graphs of $\\Z_{2d^2+2d+1}$ with generator set $\\{1,2d^2\\}$. For $r>1$, generalization for 1-perfect codes is provided in the integer lattice of $\\R^r$ and in the products of $r$ cycles, with partition quotient graph $K_{2r+1}$ taken as the undirected Cayley graph of $\\Z_{2r+1}$ with generator set $\\{1,...,r\\}$."}
{"category": "Math", "title": "Perfect domination in rectangular grid graphs", "abstract": "A dominating set $S$ in a graph $G$ is said to be perfect if every vertex of $G$ not in $S$ is adjacent to just one vertex of $S$. Given a vertex subset $S'$ of a side $P_m$ of an $m\\times n$ grid graph $G$, the perfect dominating sets $S$ in $G$ with $S'=S\\cap V(P_m)$ can be determined via an exhaustive algorithm $\\Theta$ of running time $O(2^{m+n})$. Extending $\\Theta$ to infinite grid graphs of width $m-1$, periodicity makes the binary decision tree of $\\Theta$ prunable into a finite threaded tree, a closed walk of which yields all such sets $S$. The graphs induced by the complements of such sets $S$ can be codified by arrays of ordered pairs of positive integers via $\\Theta$, for the growth and determination of which a speedier %greedy algorithm exists. %and their periodic structure, further studied. A recent characterization of grid graphs having total perfect codes $S$ (with just 1-cubes as induced components), due to Klostermeyer and Goldwasser, is given in terms of $\\Theta$, which allows to show that these sets $S$ are restrictions of only one total perfect code $S_1$ in the integer lattice graph ${\\Lambda}$ of $\\R^2$. Moreover, the complement ${\\Lambda}-S_1$ yields an aperiodic tiling, like the Penrose tiling. In contrast, the parallel, horizontal, total perfect codes in ${\\Lambda}$ are in 1-1 correspondence with the doubly infinite $\\{0,1\\}$-sequences."}
{"category": "Math", "title": "Cohomology and Duality for (phi,Gamma)-modules over the Robba ring", "abstract": "Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated etale (phi,Gamma)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily etale (phi,Gamma)-modules over the Robba ring, of the Euler-Poincare characteristic formula and Tate local duality for p-adic representations. These results are expected to intervene in the duality theory for Selmer groups associated to de Rham representations."}
{"category": "Math", "title": "On fixed point sets of distinguished collections for groups of parabolic characteristic", "abstract": "We determine the nature of the fixed point sets of groups of order p, acting on complexes of distinguished p-subgroups (those p-subgroups containing p-central elements in their centers). The case when G has parabolic characteristic p is analyzed in detail."}
{"category": "Math", "title": "A note on the \\alpha-invariant of the Mukai-Umemura 3-fold", "abstract": "We give an elementary argument to compute the $\\alpha$-invariant of this Fano 3-fold, which implies the existence of a Kahler-Einstein metric."}
{"category": "Math", "title": "A characterization of two classes of locally truncated diagram geometries", "abstract": "We study locally truncated geometries that are parapolar spaces locally of type A_{n-1,j}(K) with n>6 and j=3,4. Residually connected sheaves over these geometries are constructed. It is proved that these geometries are residually connected diagram geometries whose universal 2-covers are truncations of buildings."}
{"category": "Math", "title": "The delta method for analytic functions of random operators with application to functional data", "abstract": "In this paper, the asymptotic distributions of estimators for the regularized functional canonical correlation and variates of the population are derived. The method is based on the possibility of expressing these regularized quantities as the maximum eigenvalue and the corresponding eigenfunctions of an associated pair of regularized operators, similar to the Euclidean case. The known weak convergence of the sample covariance operator, coupled with a delta-method for analytic functions of covariance operators, yields the weak convergence of the pair of associated operators. From the latter weak convergence, the limiting distributions of the canonical quantities of interest can be derived with the help of some further perturbation theory."}
{"category": "Math", "title": "Independence-friendly cylindric set algebras", "abstract": "Independence-friendly logic is a conservative extension of first-order logic that has the same expressive power as existential second-order logic. In her Ph.D. thesis, Dechesne introduces a variant of independence-friendly logic called IFG logic. We attempt to algebraize IFG logic in the same way that Boolean algebra is the algebra of propositional logic and cylindric algebra is the algebra of first-order logic. We define independence-friendly cylindric set algebras and prove two main results. First, every independence-friendly cylindric set algebra over a structure has an underlying Kleene algebra. Moreover, the class of such underlying Kleene algebras generates the variety of all Kleene algebras. Hence the equational theory of the class of Kleene algebras that underly an independence-friendly cylindric set algebra is finitely axiomatizable. Second, every one-dimensional independence-friendly cylindric set algebra over a structure has an underlying monadic Kleene algebra. However, the class of such underlying monadic Kleene algebras does not generate the variety of all monadic Kleene algebras. Finally, we offer a conjecture about which subvariety of monadic Kleene algebras the class of such monadic Kleene algebras does generate."}
{"category": "Math", "title": "Poincare isomorphism in K-theory on manifolds with edges", "abstract": "The aim of this paper is to construct the Poincare isomorphism in K-theory on manifolds with edges. We show that the Poincare isomorphism can naturally be constructed in the framework of noncommutative geometry. More precisely, to a manifold with edges we assign a noncommutative algebra and construct an isomorphism between the K-group of this algebra and the K-homology group of the manifold with edges viewed as a compact topological space."}
{"category": "Math", "title": "Weighted Ehrhart Theory and Orbifold Cohomology", "abstract": "We introduce the notion of a weighted $\\delta$-vector of a lattice polytope. Although the definition is motivated by motivic integration, we study weighted $\\delta$-vectors from a combinatorial perspective. We present a version of Ehrhart Reciprocity and prove a change of variables formula. We deduce a new geometric interpretation of the coefficients of the Ehrhart $\\delta$-vector. More specifically, they are sums of dimensions of orbifold cohomology groups of a toric stack."}
{"category": "Math", "title": "Atiyah-Bott index on stratified manifolds", "abstract": "We define Atiyah-Bott index on stratified manifolds and express it in topological terms. By way of example, we compute this index for geometric operators on manifolds with edges."}
{"category": "Math", "title": "Integrated Harnack inequalities on Lie groups", "abstract": "We show that the logarithmic derivatives of the convolution heat kernels on a uni-modular Lie group are exponentially integrable. This result is then used to prove an \"integrated\" Harnack inequality for these heat kernels. It is shown that this integrated Harnack inequality is equivalent to a version of Wang's Harnack inequality. (A key feature of all of these inequalities is that they are dimension independent.) Finally, we show these inequalities imply quasi-invariance properties of heat kernel measures for two classes of infinite dimensional \"Lie\" groups."}
{"category": "Math", "title": "Lattice points in Minkowski sums", "abstract": "Fakhruddin has proved that for two lattice polygons P and Q any lattice point in their Minkowski sum can be written as a sum of a lattice point in P and one in Q, provided P is smooth and the normal fan of P is a subdivision of the normal fan of Q. We give a shorter combinatorial proof of this fact that does not need the smoothness assumption on P."}
{"category": "Math", "title": "A note on a degree sum condition for long cycles in graphs", "abstract": "We conjecture that a 2-connected graph $G$ of order $n$, in which $d(x)+d(y)\\geq n-k$ for every pair of non-adjacent vertices $x$ and $y$, contains a cycle of length $n-k$ ($k<n/2$), unless $G$ is bipartite and $n-k$ is odd. This generalizes to long cycles a well-known degree sum condition for hamiltonicity of Ore. The conjecture is shown to hold for $k=1$."}
{"category": "Math", "title": "The Cuntz semigroup of some spaces of dimension at most two", "abstract": "It is shown that the Cuntz semigroup of a space with dimension at most two, and with second cohomology of its compact subsets equal to zero, is isomorphic to the ordered semigroup of lower semicontinuous functions on the space with values in the natural numbers with the infinity adjoined. This computation is then used to obtain the Cuntz semigroup of all compact surfaces. A converse to the first computation is also proven: if the Cuntz semigroup of a separable C*-algebra is isomorphic to the lower semicontinuous functions on a topological space with values in the extended natural numbers, then the C*-algebra is commutative up to stability, and its spectrum satisfies the dimensional and cohomological conditions mentioned above."}
{"category": "Math", "title": "The forcing partial order on a family of braids forced by pseudo-Anosov 3-braids", "abstract": "Li-York theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the set of all braid types Boyland's dynamical partial order can be defined. We describe the partial order on a family of braids and show that a period 3 orbit of pseudo-Anosov braid type implies the Smale-horseshoe map which is a factor possessing complicated chaotic dynamics."}
{"category": "Math", "title": "There is a Van Douwen MAD family", "abstract": "We prove in ZFC that there is a MAD family of functions in omega^omega which is also maximal with respect to infinite partial functions. This solves a 20 year old question of Van Douwen. We also strengthen a result of J. Steprans stating that strongly MAD families of functions cannot be analytic. We show that analytic MAD families of functions, if they exist, must satisfy some strong constraints."}
{"category": "Math", "title": "Sheaves as modules", "abstract": "We revisit sheaves on locales by placing them in the context of the theory of quantale modules. The local homeomorphisms $p:X\\to B$ are identified with the Hilbert $B$-modules that are equipped with a natural notion of basis. The homomorphisms of these modules are necessarily adjointable, and the resulting self-dual category yields a description of the equivalence between local homeomorphisms and sheaves whereby morphisms of sheaves arise as the ``operator adjoints'' of the maps of local homeomorphisms."}
{"category": "Math", "title": "On stable numerical differentiation", "abstract": "Based on a regularized Volterra equation, two different approaches for numerical differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete favorably with the variational regularization method for stable calculating the derivatives of noisy functions."}
{"category": "Math", "title": "A Modification of the Sarkar-Wang Algorithm and an Analysis of its Computational Complexity", "abstract": "The Sarkar-Wang algorithm computes the hat version of the Heegaard Floer homology of a closed oriented three manifold. This paper analyzes the computational complexity of the Sarkar-Wang algorithm; then the algorithm is modified to obtain a better bound. Then the computational complexity of calculating HFK hat from a Heegaard diagram by means of the modified Sarkar-Wang algorithm is also analyzed. Under certain assumptions it is shown that the modified Sarkar-Wang algorithm is faster than the Manolescu-Ozsvath-Sarkar algorithm."}
{"category": "Math", "title": "Mapping Incidences", "abstract": "We show that any finite set S in a characteristic zero integral domain can be mapped to the finite field of order p, for infinitely many primes p, preserving all algebraic incidences in S. This can be seen as a generalization of the well-known Freiman isomorphism lemma, and we give several combinatorial applications (such as sum-product estimates)."}
{"category": "Math", "title": "Spherical functors", "abstract": "This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give sufficient conditions for a collection of spherical functors to yield a weak representation of the category of tangles, and prove a structure theorem for such representations under certain restrictions."}
{"category": "Math", "title": "Stirling's formula derived simply", "abstract": "Stirling's formula, the asymptotic expansion of $n!$ for $n$ large, or of $\\Gamma(z)$ for $z\\to \\infty$, is derived directly from the recursion equation $\\Gamma(z+1) =z \\Gamma(s)$ and the normalization condition $\\Gamma ({1/2}) =\\sqrt{\\pi}$."}
{"category": "Math", "title": "Non-trivalent graph cocycle and cohomology of the long knot space", "abstract": "In this paper we show that via the configuration space integral construction a non-trivalent graph cocycle can also yield a non-zero cohomology class of the space of higher (and even) codimensional long knots. This simultaneously proves that the Browder operation induced by the operad action defined by R. Budney is not trivial."}
{"category": "Math", "title": "On the cycle structure of hamiltonian k-regular bipartite graphs of order 4k", "abstract": "It is shown that a hamiltonian $n/2$-regular bipartite graph $G$ of order $2n>8$ contains a cycle of length $2n-2$. Moreover, if such a cycle can be chosen to omit a pair of adjacent vertices, then $G$ is bipancyclic."}
{"category": "Math", "title": "Szemeredi-Trotter type theorem and sum-product estimate in finite fields", "abstract": "We study a Szemer\\'edi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields."}
{"category": "Math", "title": "Translating solitons to symplectic and Lagrangian mean curvature flows", "abstract": "In this paper, we construct finite blow-up examples for symplectic mean curvature flows and we study properties of symplectic translating solitons. We prove that, the K\\\"ahler angle $\\alpha$ of a symplectic translating soliton with $\\max |A|=1$ satisfies that $\\sup |\\alpha|>\\frac{\\pi}{4}\\frac{|T|}{|T|+1}$ where $T$ is the direction in which the surface transltes."}
{"category": "Math", "title": "Tate-Shafarevich groups and K3 surfaces", "abstract": "Following (and elaborating on) a method of Logan and van Luijk, we exhibit explicit genus-2 curves whose Jacobians have nontrivial 2-torsion in their Tate-Shafarevich groups."}
{"category": "Math", "title": "Holomorphic functions and regular quaternionic functions on the hyperk\\\"ahler space H", "abstract": "Let H be the space of quaternions, with its standard hypercomplex structure. Let R(D) be the module of regular functions on D. For every unitary vector p in S^2, R(D) contains the space of holomorphic functions w.r.t. the complex structure J_p induced by p. We prove the existence, on any bounded domain D, of regular functions that are not J_p-holomorphic for any p. Our starting point is a result of Chen and Li concerning maps between hyperkaehler manifolds, where a similar result is obtained for a less restricted class of quaternionic maps. We give a criterion, based on the energy-minimizing property of holomorphic maps, that distinguishes J_p-holomorphic functions among regular functions."}
{"category": "Math", "title": "Recurrent extensions of self-similar Markov processes and Cram\\'er's condition II", "abstract": "We prove that a positive self-similar Markov process $(X,\\mathbb{P})$ that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying L\\'{e}vy process satisfies Cram\\'{e}r's condition."}
{"category": "Math", "title": "On the symmetric square. Unstable Twisted Characters", "abstract": "We provide a purely local computation of the (elliptic) twisted (by \"transpose-inverse\") character of the representation \\pi=I(\\1) of PGL(3) over a p-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals."}
{"category": "Math", "title": "Uniformization of \\mathcal{G}-bundles", "abstract": "We show some of the conjectures of Pappas and Rapoport concerning the moduli stack of $\\mathcal{G}$-torsors on a curve C, where $\\mathcal{G}$ is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the uniformization theorem of Drinfeld-Simpson in this setting. Furthermore we apply this to compute the connected components of these moduli stacks and to calculate the Picard group of the stack of torsors in case $\\mathcal{G}$ is simply connected."}
{"category": "Math", "title": "Singular elliptic genus of normal surfaces", "abstract": "We define the singular elliptic genus for arbitrary normal surfaces, prove that it is a birational invariant, and show that it generalizes the singular elliptic genus of Borisov and Libgober and the stringy $\\chi_y$ genus of Batyrev and Veys."}
{"category": "Math", "title": "A Direct Proof of the Theorem on Formal Functions", "abstract": "We give a direct and elementary proof of the theorem on formal functions by studying the behaviour of the Godement resolution of a sheaf of modules under completion."}
{"category": "Math", "title": "Bounds for the covariance of functions of infinite variance stable random variables with applications to central limit theorems and wavelet-based estimation", "abstract": "We establish bounds for the covariance of a large class of functions of infinite variance stable random variables, including unbounded functions such as the power function and the logarithm. These bounds involve measures of dependence between the stable variables, some of which are new. The bounds are also used to deduce the central limit theorem for unbounded functions of stable moving average time series. This result extends the earlier results of Tailen Hsing and the authors on central limit theorems for bounded functions of stable moving averages. It can be used to show asymptotic normality of wavelet-based estimators of the self-similarity parameter in fractional stable motions."}
{"category": "Math", "title": "Transitive projective planes and 2-rank", "abstract": "Suppose that a group $G$ acts transitively on the points of a non-Desarguesian plane, $\\mathcal{P}$. We prove first that the Sylow 2-subgroups of $G$ are cyclic or generalized quaternion. We also prove that $\\mathcal{P}$ must admit an odd order automorphism group which acts transitively on the set of points of $\\mathcal{P}$."}
{"category": "Math", "title": "Homogeneous geodesics of left invariant Finsler metrics", "abstract": "In this paper, we study the set of homogeneous geodesics of a leftinvariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. As an application, we study some geometric properties of bi-invariant Finsler metrics on Lie groups. In particular a necessary and sufficient condition that left-invariant Randers metrics are of Berwald type is given. Finally a correspondence of homogeneous geodesics to critical points of restricted Finsler metrics is given. Then results concerning the existence homogeneous geodesics are obtained."}
{"category": "Math", "title": "Invariance property of orbifold elliptic genus for multi-fans", "abstract": "Multi-fan is an analogous notion of fan. As a fan is associated to a toric variety a multi-fan is associated to a torus orbifold. Orbifold elliptic class and orbifold elliptic genus are defined for a triple of a multi-fan, a set of generating integral vectors of one dimensional cones and a $\\mathbb{Q}$ divisor. They are shown to behave functorially with respect to birational morphisms between these triples. The result may be considered as a combinatorial or topological analogue of the main result of Borisov and Libgober, McKay correspondence for elliptic genera, Ann. of Math., 161 (2005)."}
{"category": "Math", "title": "Orbits of real forms in complex flag manifolds", "abstract": "We investigate the $CR$ geometry of the orbits $M$ of a real form $G_0$ of a complex simple group $G$ in a complex flag manifold $X=G/Q$. We are mainly concerned with finite type, Levi non-degeneracy conditions, canonical $G_0$-equivariant and Mostow fibrations, and topological properties of the orbits."}
{"category": "Math", "title": "Higher class field theory and the connected component", "abstract": "In this note we present a new self-contained approach to the class field theory of arithmetic schemes in the sense of Wiesend. Along the way we prove new results on space filling curves on arithmetic schemes and on the class field theory of local rings. We show how one can deduce the more classical version of higher global class field theory due to Kato and Saito from Wiesend's version. One of our new results says that the connected component of the identity element in Wiesend's class group is divisible if some obstruction is absent."}
{"category": "Math", "title": "Laws of large numbers in stochastic geometry with statistical applications", "abstract": "Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $\\nu_n=\\sum_i\\xi_i$, where $\\xi_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $\\xi_i$ is determined by the (suitably rescaled) set of points near $X_i$. For bounded test functions $f$ on $R^d$, we give weak and strong laws of large numbers for $\\nu_n(f)$. The general results are applied to demonstrate that an unknown set $A$ in $d$-space can be consistently estimated, given data on which of the points $X_i$ lie in $A$, by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze. Further applications are given concerning the Gamma statistic for estimating the variance in nonparametric regression."}
{"category": "Math", "title": "A quenched limit theorem for the local time of random walks on \\Z^2", "abstract": "Let $X$ and $Y$ be two independent random walks on $\\Z^2$ with zero mean and finite variances, and let $L_t(X,Y)$ be the local time of $X-Y$ at the origin at time $t$. We show that almost surely with respect to $Y$, $L_t(X,Y)/\\log t$ conditioned on $Y$ converges in distribution to an exponential random variable with the same mean as the distributional limit of $L_t(X,Y)/\\log t$ without conditioning. This question arises naturally from the study of the parabolic Anderson model with a single moving catalyst, which is closely related to a pinning model."}
{"category": "Math", "title": "Lower limits for distributions of randomly stopped sums", "abstract": "We study lower limits for the ratio $\\frac{\\bar{F^{*\\tau}}(x)}{\\bar F(x)}$ of tail distributions where $ F^{*\\tau}$ is a distribution of a sum of a random size $\\tau$ of i.i.d. random variables having a common distribution $F$, and a random variable $\\tau$ does not depend on summands."}
{"category": "Math", "title": "Horizontal Dehn Surgery and genericity in the curve complex", "abstract": "We introduce a general notion of \"genericity\" for countable subsets of a space with Borel measure, and apply it to the set of vertices in the curve complex of a surface S, interpreted as subset of the space of projective measured laminations in S, equipped with its natural Lebesgue measure. We prove that, for any 3-manifold M, the set of curves c on a Heegaard surface S in M, such that every non-trivial Dehn twist at c yields a Heegaard splitting of high distance, is generic in the set of all essential simple closed curves on S. Our definition of \"genericity\" is different and more intrinsic than alternative such existing notions, given e.g. via random walks or via limits of quotients of finite sets."}
{"category": "Math", "title": "Consistency and application of moving block bootstrap for non-stationary time series with periodic and almost periodic structure", "abstract": "The aim of this paper it to establish sufficient conditions for consistency of moving block bootstrap for non-stationary time series with periodic and almost periodic structure. The parameter of the study is the mean value of the expectation function. Consistency holds in quite general situations: if all joint distributions of the series are periodic, then it suffices to assume the central limit theorem and strong mixing property, together with summability of the autocovariance function. In the case where the mean function is almost periodic, we additionally need uniform boundedness of the fourth moments of the root statistics. It is shown that these theoretical results can be applied in statistical inference concerning the Fourier coefficients of periodically (PC) and almost periodically (APC) correlated time series. A simulation example shows how to use a graphical diagnostic test for significant frequencies and stationarity within these classes of time series."}
{"category": "Math", "title": "L'alg\\`ebre des invariants d'un groupe de Coxeter agissant sur un mutiple de sa repr\\'esentation standard", "abstract": "Let G be a Coxeter group of type A_n, B_n, D_n or I_2(N), or a complex reflection group of type G(de,e,n). Let V be its standard representation and let k be an integer greater than 2. Then G acts on S(V)^{\\otimes k}. We show that the algebra of invariants (S(V)^{\\otimes k})^G is a free (S(V)^G)^{\\otimes k}-module of rank |G|^{k-1}, and that S(V)^{\\otimes k} is not a free (S(V)^{\\otimes k})^G-module."}
{"category": "Math", "title": "On poles of twisted tensor L-functions", "abstract": "It is shown that the only possible pole of the twisted tensor L-functions in Re(s)\\geq 1 is located at s=1 for all quadratic extensions of global fields."}
{"category": "Math", "title": "DG-algebras and derived A-infinity algebras", "abstract": "A differential graded algebra can be viewed as an A-infinity algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A-infinity algebra. We introduce the notion of a derived A-infinity algebra and show that any dga A over an arbitrary commutative ground ring k is equivalent to a minimal derived A-infinity algebra. Such a minimal derived A-infinity algebra model for A is a k-projective resolution of the homology algebra of A together with a family of maps satisfying appropriate relations. As in the case of A-infinity algebras, it is possible to recover the dga up to quasi-isomorphism from a minimal derived A-infinity algebra model. Hence the structure we are describing provides a complete description of the quasi-isomorphism type of the dga."}
{"category": "Math", "title": "Adaptive optimal allocation in stratified sampling methods", "abstract": "In this paper, we propose a stratified sampling algorithm in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum. These proportions converge to the optimal allocation in terms of variance reduction. And our stratified estimator is asymptotically normal with asymptotic variance equal to the minimal one. Numerical experiments confirm the efficiency of our algorithm."}
{"category": "Math", "title": "Optimal transportation on non-compact manifolds", "abstract": "In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type $d^r,r>1$, where $d$ is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction."}
{"category": "Math", "title": "Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order", "abstract": "We address the problem of determining finite subsets of Delone sets $\\varLambda\\subset\\R^d$ with long-range order by $X$-rays in prescribed $\\varLambda$-directions, i.e., directions parallel to non-zero interpoint vectors of $\\varLambda$. Here, an $X$-ray in direction $u$ of a finite set gives the number of points in the set on each line parallel to $u$. For our main result, we introduce the notion of algebraic Delone sets $\\varLambda\\subset\\R^2$ and derive a sufficient condition for the determination of the convex subsets of these sets by $X$-rays in four prescribed $\\varLambda$-directions."}
{"category": "Math", "title": "Donaldson Thomas invariant of P^1 scroll", "abstract": "Let X be a P^1 scroll (a compactification of a line bundle L) over a complex surafce S and assume S has a global two form with zero loci a smooth curve C. The Donaldson Thomas invariants of X is shown to be zero if the curve class has is component on S not a multiple of [C]. For nonzero case, when the prime field insertion are above C, the invariant is shown to depend only on the analytic neighborhood of L in X."}
{"category": "Math", "title": "Foncteur de Picard d'un champ alg\\'ebrique", "abstract": "In this article we study the Picard functor and the Picard stack of an algebraic stack. We give a new and direct proof of the representability of the Picard stack. We prove that it is quasi-separated, and that the connected component of the identity is proper when the fibers of the stack are geometrically normal. We study some examples of Picard functors of classical stacks. In an appendix, we review the lisse-etale cohomology of abelian sheaves on an algebraic stack."}
{"category": "Math", "title": "Sparse Additive Models", "abstract": "We present a new class of methods for high-dimensional nonparametric regression and classification called sparse additive models (SpAM). Our methods combine ideas from sparse linear modeling and additive nonparametric regression. We derive an algorithm for fitting the models that is practical and effective even when the number of covariates is larger than the sample size. SpAM is closely related to the COSSO model of Lin and Zhang (2006), but decouples smoothing and sparsity, enabling the use of arbitrary nonparametric smoothers. An analysis of the theoretical properties of SpAM is given. We also study a greedy estimator that is a nonparametric version of forward stepwise regression. Empirical results on synthetic and real data are presented, showing that SpAM can be effective in fitting sparse nonparametric models in high dimensional data."}
{"category": "Math", "title": "The second type singularity of symplectic and Lagrangian mean curvature flows", "abstract": "In this paper we mainly study the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K \\\"ahler-Einstein surface. We show the relation between the maximum of the K\\\"ahler angle and the maximum of $|H|^2$ on the limit flow."}
{"category": "Math", "title": "Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space", "abstract": "Let $(M^{n},g)$ be a compact Riemannian manifold with $Ric\\geq-(n-1) $. It is well known that the bottom of spectrum $\\lambda_{0}$ of its unverversal covering satisfies $\\lambda_{0}\\leq(n-1) ^{2}/4 $. We prove that equality holds iff $M$ is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy."}
{"category": "Math", "title": "A Rigidity Theorem for the Hemisphere", "abstract": "We prove the following rigidity theorem: For an n-dimensional compact Riemannian manifold with boundary whose Ricci curvature is bounded by n-1 from below, if its boundary is isometric to the standard sphere of dimension n-1 and totally geodesic, then the manifold is isometric to the standard hemisphere."}
{"category": "Math", "title": "Pinned distance sets, Wolff's exponent in finite fields and improved sum-product estimates", "abstract": "An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $\\alpha>0$ such that $|\\Delta(E)| \\gtrsim q$ whenever $|E| \\gtrsim q^{\\alpha}$, where $E \\subset {\\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $\\Delta(E)=\\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \\in E\\}$. The second listed author and Misha Rudnev established the threshold $\\frac{d+1}{2}$, and the authors of this paper, Doowon Koh and Misha Rudnev proved that this exponent is sharp in even dimensions. In this paper we improve the threshold to $\\frac{d^2}{2d-1}$ under the additional assumption that $E$ has product structure. In particular, we obtain the exponent 4/3, consistent with the corresponding exponent in Euclidean space obtained by Wolff."}
{"category": "Math", "title": "W-algebra W(2,2) and the vertex operator algebra L(1/2,0)\\otimes L(1/2,0)", "abstract": "In this paper the W-algebra W(2,2) and its representation theory are studied. It is proved that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex operator algebra associated to a highest irreducible W(2,2)-module or a tensor product of two irreducible Virasoro vertex operator algebras. Furthermore, any rational, C_2-cofinite simple vertex operator algebra whose weight 1 subspace is zero and weight 2 subspace is 2-dimensional, and with central charge c=1 is isomorphic to L(1/2,0)\\otimes L(1/2,0)."}
{"category": "Math", "title": "Toward classfication of rational vertex operator algebras with central charges less than 1", "abstract": "The rational and C_2-cofinite simple vertex operator algebras whose effective central charges and the central charges c are equal and less than 1 are classified. Such a vertex operator algebra is zero if c<0 and C if c=0. If c>0, it is an extension of discrete Virasoro vertex operator algebra L(c_{p,q},0) by its irreducible modules. It is also proved that for any rational and C_2-cofinite simple vertex operator algebra whose effective central charge and central charge are equal, the vertex operator subalgebra generated by the Virasoro vector is simple."}
{"category": "Math", "title": "On Commutativity and Finiteness in Groups", "abstract": "The second author introduced notions of weak permutability and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite groups. Two groups H,K weakly commute provided there exists a bijection f: H -> K which fixes the identity element and such that h commutes with its image h^f for all h in H. The present paper gives support to conjectures about the nilpotency of groups generated by two weakly commuting finite abelian groups H,K."}
{"category": "Math", "title": "Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics", "abstract": "Let $Q^N_l\\subset \\bC\\bP^{N+1}$ denote the standard real, nondegenerate hyperquadric of signature $l$ and $M\\subset \\bC^{n+1}$ a real, Levi nondegenerate hypersurface of the same signature $l$. We shall assume that there is a holomorphic mapping $H_0\\colon U\\to \\bC\\bP^{N_0+1}$, where $U$ is some neighborhood of $M$ in $\\bC^{n+1}$, such that $H_0(M)\\subset Q^{N_0}_l$ but $H(U)\\not\\subset Q^{N_0}_l$. We show that if $N_0-n<l$ then, for any $N\\geq N_0$, any holomorphic mapping $H\\colon U\\to \\bC\\bP^{N+1}$ with $H(M)\\subset Q^{N}_l$ and $H(U)\\not\\subset Q^{N_0}_l$ must be the standard linear embedding of $Q^{N_0}_l$ into $Q^N_l$ up to conjugation by automorphisms of $Q^{N_0}_l$ and $Q^N_l$."}
{"category": "Math", "title": "A Remark on a Theorem by Kodama and Shimizu", "abstract": "We prove a characterization theorem for the unit polydisc $\\Delta^n\\subset\\CC^n$ in the spirit of a recent result due to Kodama and Shimizu. We show that if $M$ is a connected $n$-dimensional complex manifold such that (i) the group $\\hbox{Aut}(M)$ of holomorphic automorphisms of $M$ acts on $M$ with compact isotropy subgroups, and (ii) $\\hbox{Aut}(M)$ and $\\hbox{Aut}(\\Delta^n)$ are isomorphic as topological groups equipped with the compact-open topology, then $M$ is holomorphically equivalent to $\\Delta^n$."}
{"category": "Math", "title": "Icons", "abstract": "Categorical orthodoxy has it that collections of ordinary mathematical structures such as groups, rings, or spaces, form categories (such as the category of groups); collections of 1-dimensional categorical structures, such as categories, monoidal categories, or categories with finite limits, form 2-categories; and collections of 2-dimensional categorical structures, such as 2-categories or bicategories, form 3-categories. We describe a useful way in which to regard bicategories as objects of a 2-category. This is a bit surprising both for technical and for conceptual reasons. The 2-cells of this 2-category are the crucial new ingredient; they are the icons of the title. These can be thought of as ``the oplax natural transformations whose components are identities'', but we shall also give a more elementary description. We describe some properties of these icons, and give applications to monoidal categories, to 2-nerves of bicategories, to 2-dimensional Lawvere theories, and to bundles of bicategories."}
{"category": "Math", "title": "Caldero-Keller approach to the denominators of cluster variables", "abstract": "Buan, Marsh and Reiten proved that if a cluster-tilting object $T$ in a cluster category $\\mathcal C$ associated to an acyclic quiver $Q$ satisfies certain conditions with respect to the exchange pairs in $\\mathcal C$, then the denominator in its reduced form of every cluster variable in the cluster algebra associated to $Q$ has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of $T$. In this paper, we give an alternative proof of this result using the Caldero-Keller approach to acyclic cluster algebras and the work of Palu on cluster characters."}
{"category": "Math", "title": "Automorphism groups of root systems matroids", "abstract": "Given a root system $\\mathsf{R}$, the vector system $\\tilde{\\mathsf{R}}$ is obtained by taking a representative $v$ in each antipodal pair $\\{v, -v\\}$. The matroid $M(\\mathsf{R})$ is formed by all independent subsets of $\\tilde{\\mathsf{R}}$. The automorphism group of a matroid is the group of permutations preserving its independent subsets. We prove that the automorphism groups of all irreducible root systems matroids $M(\\mathsf{R})$ are uniquely determined by their independent sets of size 3. As a corollary, we compute these groups explicitly, and thus complete the classification of the automorphism groups of root systems matroids."}
{"category": "Math", "title": "Central extensions of Lax operator algebras", "abstract": "Lax operator algebras were introduced by Krichever and Sheinman as a further development of I.Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this article local cocycles and associated almost-graded central extensions are classified. It is shown that in the case that the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebra of meromorphic vector fields on the Lax operator algebra via suitable covariant derivatives."}
{"category": "Math", "title": "The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces", "abstract": "This article gives a natural decomposition of the suspension of generalized moment-angle complexes or {\\it partial product spaces} which arise as {\\it polyhedral product functors} described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in Goresky-MacPherson \\cite{goresky.macpherson}, Hochster\\cite{hochster}, Baskakov \\cite{baskakov}, Panov \\cite{panov}, and Buchstaber-Panov \\cite{buchstaber.panov}. Since the splitting is geometric, an analogous homological decomposition for a generalized moment-angle complex applies for any homology theory. This decomposition gives an additive decomposition for the Stanley-Reisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter \\cite{porter} and Ganea \\cite{ganea}. The spirit of the work here follows that of Denham-Suciu in \\cite{denham.suciu}."}
{"category": "Math", "title": "Elliptic polynomials orthogonal on the unit circle with a dense point spectrum", "abstract": "We introduce two explicit examples of polynomials orthogonal on the unit circle. Moments and the reflection coefficients are expressed in terms of Jacobi elliptic functions. We find explicit expression for these polynomials in terms of a new type of elliptic hypergeometric function. We show that obtained polynomials are orthogonal on the unit circle with respect to a dense point meausure, i.e. the spectrum consists from infinite number points of increase which are dense on the unit circle. We construct also corresponding explicit systems of polynomials orthogonal on the interval of the real axis with respect to a dense point measure. They can be considered as an elliptic generalization of the Askey-Wilson polynomials of a special type."}
{"category": "Math", "title": "H\\\"older-differentiability of Gibbs distribution functions", "abstract": "In this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil's staircases) supported on limit sets of finitely generated conformal iterated function systems in $\\R$. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not $\\alpha$-H\\\"older-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris, and Xiao. In particular, our results clearly show that the results of these authors have their natural home within thermodynamic formalism."}
{"category": "Math", "title": "Ranks of elliptic curves over function fields", "abstract": "We present experimental evidence to support the widely held belief that one half of all elliptic curves have infinitely many rational points. The method used to gather this evidence is a refinement of an algorithm due to the author which is based upon rigid and crystalline cohomology."}
{"category": "Math", "title": "Five Conferences on Undecidability", "abstract": "These five lectures on undecidability were given to students with a good level in mathematics but with no special knowledge on logic. The first conference presents the formalization of mathematics with a short historical survey, the language of first order predicates and the axioms of set theory. The second and third lectures explain the incompleteness phenomena from the Hilbert program until G\\\"odel's theorems with a presentation of the sequent calculus of Gentzen.The fourth talk deepens model theory reasoning in the case of the continuum hypothesis, and the last conference gives examples of effective computability results."}
{"category": "Math", "title": "Asymptotically efficient estimators for nonparametric heteroscedastic regression models", "abstract": "This paper concerns the estimation of the regression function at a given point in nonparametric heteroscedastic models with Gaussian noise or with noise having unknown distribution. In the two cases an asymptotically efficient kernel estimator is constructed for the minimax absolute error risk."}
{"category": "Math", "title": "Rotation set and Entropy", "abstract": "In 1991 Llibre and MacKay proved that if $f$ is a 2-torus homeomorphism isotopic to identity and the rotation set of $f$ has a non empty interior then $f$ has positive topological entropy. Here, we give a converselike theorem. We show that the interior of the rotation set of a 2-torus $C^{1+ \\alpha}$ diffeomorphism isotopic to identity of positive topological entropy is not empty, under the additional hypotheses that $f$ is topologically transitive and irreducible. We also give examples that show that these hypotheses are necessary."}
{"category": "Math", "title": "Ueber die Tiefe von Invariantenringen unendlicher Gruppen", "abstract": "Let K be an algebraically closed field. For a graded K-Algebra R, we write cmdef R:=dim R -depth R. We show that for each reductive group G (over K) which is not linearly reductive, there exists a faithful G-module V such that cmdef K[\\sum_i=1^k V]^G >= k-2 for all k. We will give such a V explicitly."}
{"category": "Math", "title": "Finite Gap Jacobi Matrices: An Announcement", "abstract": "We consider Jacobi matrices whose essential spectrum is a finite union of closed intervals. We focus on Szego's theorem, Jost solutions, and Szego asymptotics for this situation. This announcement describes talks the authors gave at OPSFA 2007."}
{"category": "Math", "title": "On the depth of invariant rings of infinite groups", "abstract": "Let K be an algebraically closed field. For a finitely generated graded K algebra R, let cmdef R := dim R - depth R denote the Cohen-Macaulay-defect of R. Let G be a linear algebraic group over K that is reductive but not linearly reductive. We show that there exists a faithful rational representation V of G (which we will give explicitly) such that cmdef K[\\sum_i=1^k V]^G >= k-2 for all k. We give refinements in the case G = SL2."}
{"category": "Math", "title": "A Note on 3-quasi-Sasakian Geometry", "abstract": "3-quasi-Sasakian manifolds were recently studied by the authors as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. In this paper some geometric properties of this class of almost 3-contact metric manifolds are briefly reviewed, with an emphasis on those more related to physical applications."}
{"category": "Math", "title": "Power series solution of the modified KdV equation", "abstract": "We prove local-wellposedness of the mKdV equation in $\\mathcal{F}L^{s,p}$ spaces using the new method of M. Christ."}
{"category": "Math", "title": "On a space related to the affine building of type E7", "abstract": "A locally truncated geometry with diagram of type affine E7 is studied. One considers a parapolar space, locally of type A_{7,4}, which is subject to an extra axiom. A covering of this space is constructed; it is proved that this covering space is a rank 6, residually connected, locally truncated diagram geometry which is a homomorphic image of a truncated building of affine type E7. Consequently, the initial parapolar space is also a homomorphic image of a truncated building."}
{"category": "Math", "title": "Projective relations for m-th root metric spaces", "abstract": "For Finsler spaces (M,F) endowed with m-th root metrics, we provide necessary and sufficient conditions in which they are projectively flat, or projectively related to Berwald/Riemann spaces. We also give a specific characterization for m-th root metrics spaces of Landsberg and of Berwald type."}
{"category": "Math", "title": "Conformal Powers of the Laplacian via Stereographic Projection", "abstract": "A new derivation is given of Branson's factorization formula for the conformally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the corresponding conformally invariant operator on Euclidean space (the k-th power of the Euclidean Laplacian) via conjugation by the stereographic projection mapping."}
{"category": "Math", "title": "Semicrossed products of simple C*-algebras", "abstract": "Let $(\\A, \\alpha)$ and $(\\B, \\beta)$ be C*-dynamical systems and assume that $\\A$ is a separable simple C*-algebra and that $\\alpha$ and $\\beta$ are *-automorphisms. Then the semicrossed products $\\A \\times_{\\alpha} \\bbZ^{+}$ and $\\B \\times_{\\beta} \\bbZ^{+}$ are isometrically isomorphic if and only if the dynamical systems $(\\A, \\alpha)$ and $(\\B, \\beta)$ are outer conjugate."}
{"category": "Math", "title": "The integral cohomology rings of some p-groups", "abstract": "We determine the integral cohomology rings of an infinite family of p-groups, for odd primes p, with cyclic derived subgroups. Our method involves embedding the groups in a compact Lie group of dimension one, and was suggested by P H Kropholler and J Huebschmann."}
{"category": "Math", "title": "Induced trees in triangle-free graphs", "abstract": "We prove that every connected triangle-free graph on $n$ vertices contains an induced tree on $\\exp(c\\sqrt{\\log n})$ vertices, where $c$ is a positive constant. The best known upper bound is $(2+o(1))\\sqrt n$. This partially answers questions of Erdos, Saks, and Sos and of Pultr."}
{"category": "Math", "title": "The mod-p cohomology rings of some p-groups", "abstract": "We determine the mod-p cohomology rings of an infinite family of p-groups, for odd primes p, with cyclic derived subgroups. Our method involves embedding the groups in a compact Lie group of dimension one, and was suggested by P. H. Kropholler and J. Huebschmann."}
{"category": "Math", "title": "Some examples in the integral and Brown-Peterson cohomology of p-groups", "abstract": "For each odd prime p, we exhibit p-groups G of p-rank two such that (suitably defined) Chern classes of unitary representations of G fail to generate the following rings: 1. The even degree integral cohomology of G; 2. The final page of the Atiyah-Hirzebruch spectral sequence for G; 3. The Brown-Peterson generalized cohomology of G. It follows that these groups afford counterexamples to conjectures of C. B. Thomas, M. F. Atiyah and P. Landweber."}
{"category": "Math", "title": "A differential in the Lyndon-Hochschild-Serre spectral sequence", "abstract": "We consider the Lyndon-Hochschild-Serre spectral sequence with mod-p coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. For this spectral sequence the second and third differentials are known, and we give a description for the fourth differential. Using this result we deduce a similar formula for the Serre spectral sequence for a principal fibration with fibre the classifying space of a cyclic p-group. The differential from odd rows to even rows involves a Massey triple product, so we describe the calculation of such products in the cohomology of a finite abelian group. As an example we determine the Poincare series for the mod-3 cohomology of various 3-groups. Remarks. 1) My definition of the higher differentials $d_i$ for $i\\geq 2$ in the spectral sequence for a double chain complex differs from the usual one by a factor of $(-1)^{i+1}$. Both conventions are consistent, but the usual definition has the advantage of agreeing with the ``obvious'' definition of the differentials in the spectral sequence for the associated filtered chain complex. All of the theorems in this paper remain true exactly as stated if the more usual definition of $d_i$ is taken. 2) Carles Broto found a small mistake in this paper: the result for fibrations with fibre the classifying space of a cyclic group is stated for arbitrary fibrations, although it is only proved for principal fibrations. Since it is apparent from the first sentence of the proof that only principal fibrations are being considered, I have not bothered to publish an erratum."}
{"category": "Math", "title": "Time averages of polynomials", "abstract": "We define and study when a polynomial mapping has a local or global time average. We conjecture that a polynomial f in the complex plane has a time average near a point z if and only if z is eventually mapped into a Siegel-disc of f. We prove that the conjecture holds generically, namely for those polynomials whose iterates have the maximal number of critical values. Important steps in the proofs rely on understanding the iterated monodromy groups. We also show that a polynomial automorphism of C^2 has a global time average if and only if the map is conjugate to an elementary mapping. The definition of a time average is motivated by an attempt to understand the polynomial automorphism groups in dimensions 3 and higher."}
{"category": "Math", "title": "Divisorial Cohomology Vanishing on Toric Varieties", "abstract": "This work discusses combinatorial and arithmetic aspects of cohomology vanishing for divisorial sheaves on toric varieties. We obtain a refined variant of the Kawamata-Viehweg theorem which is slightly stronger. Moreover, we prove a new vanishing theorem related to divisors whose inverse is nef and has small Kodaira dimension. Finally, we give a new criterion for divisorial sheaves for being maximal Cohen-Macaulay."}
{"category": "Math", "title": "p-Groups are not determined by their integral cohomology groups", "abstract": "For each prime p, we exhibit pairs of p-groups all of whose integral cohomology groups are isomorphic. The method used involves very little calculation. The groups are exhibited as kernels of homomorphisms from a compact Lie group G to U(1), and the main result is that kernels of `similar' elements of Hom(G,U(1)) have isomorphic integral cohomology groups. The 2-groups constructed in this version have been corrected (there was a mistake in the presentations given in the published paper)."}
{"category": "Math", "title": "3-groups are not determined by their integral cohomology rings", "abstract": "We compute the integral cohomology rings of a family of 3-groups. As a corollary, we exhibit, for each n greater than or equal to 5, a pair of groups of order 3^n whose integral cohomology rings are isomorphic."}
{"category": "Math", "title": "Sets, Lists and Noncrossing Partitions", "abstract": "Partitions of [n]={1,2,...,n} into sets of lists are counted by sequence number A000262 in the On-Line Encyclopedia of Integer Sequences. They are somewhat less numerous than partitions of [n] into lists of sets, A000670. Here we observe that the former are actually equinumerous with partitions of [n] into lists of *noncrossing* sets and give a bijective proof. We show that partitions of [n] into sets of noncrossing lists are counted by A088368 and generalize this result to introduce a transform on integer sequences that we dub the \"noncrossing partition\" transform. We also derive recurrence relations to count partitions of [n] into lists of noncrossing lists."}
{"category": "Math", "title": "A formula for the R-matrix using a system of weight preserving endomorphisms", "abstract": "We give a formula for the universal R-matrix of the quantized universal enveloping algebra $U_q(\\g).$ This is similar to a previous formula due to Kirillov-Reshetikhin and Levendorskii-Soibelman, except that where they use the action of the braid group element $T_{w_0}$ on each representation, we show that one can instead use a system of weight preserving endomorphisms. One advantage of our construction is that it is well defined for all symmetrizable Kac-Moody algebras. However we have only established that the result in equal to the universal R-matrix in finite type."}
{"category": "Math", "title": "Higher string topology operations", "abstract": "Chas and Sullivan have defined an intersection-type product on the homology of the free loop space LM of an oriented manifold M. In this paper we show how to extend this construction to a topological conformal field theory of degree d. In particular, we get operations on the homology of LM which are parameterized by the homology of the moduli space of open-closed Riemann surfaces."}
{"category": "Math", "title": "Infinitesimal Hecke algebra of sl_2 in positive characteristic", "abstract": "In this paper we consider an infinitesimal Hecke algebra of $sl_2$ in positive characteristic. We show that it is a finitely generated module over its center, and the smooth and the Azumaya loci of its center coincide."}
{"category": "Math", "title": "Time decay for Schroedinger equation with rough potentials", "abstract": "We obtain certain time decay and regularity estimates for 3D Schroedinger equation with a potential in the Kato class by using Besov spaces associated with Schroedinger operators."}
{"category": "Math", "title": "A random walk on Z with drift driven by its occupation time at zero", "abstract": "We consider a nearest neighbor random walk on the one-dimensional integer lattice with drift towards the origin determined by an asymptotically vanishing function of the number of visits to zero. We show the existence of distinct regimes according to the rate of decay of the drift. In particular, when the rate is sufficiently slow, the position of the random walk, properly normalized, converges to a symmetric exponential law. In this regime, in contrast to the classical case, the range of the walk scales differently from its position."}
{"category": "Math", "title": "Large deviations for random walk in a space--time product environment", "abstract": "We consider random walk $(X_n)_{n\\geq0}$ on $\\mathbb{Z}^d$ in a space--time product environment $\\omega\\in\\Omega$. We take the point of view of the particle and focus on the environment Markov chain $(T_{n,X_n}\\omega)_{n\\geq0}$ where $T$ denotes the shift on $\\Omega$. Conditioned on the particle having asymptotic mean velocity equal to any given $\\xi$, we show that the empirical process of the environment Markov chain converges to a stationary process $\\mu_{\\xi}^{\\infty}$ under the averaged measure. When $d\\geq3$ and $\\xi$ is sufficiently close to the typical velocity, we prove that averaged and quenched large deviations are equivalent and when conditioned on the particle having asymptotic mean velocity $\\xi$, the empirical process of the environment Markov chain converges to $\\mu_{\\xi}^{\\infty}$ under the quenched measure as well. In this case, we show that $\\mu_{\\xi}^{\\infty}$ is a stationary Markov process whose kernel is obtained from the original kernel by a Doob $h$-transform."}
{"category": "Math", "title": "The Lie-Poisson Structure of the Euler Equations of an Ideal Fluid", "abstract": "This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in R^n (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphism group. This is interesting and nontrivial because in Eulerian representation, the time t maps need not be C^1 from the Sobolev class H^s to itself (where s > (n/2) + 1). The idea of how this difficulty is overcome is to exploit the fact that one does have smoothness in the Lagrangian representation and then carefully perform a Lie-Poisson reduction procedure."}
{"category": "Math", "title": "Optimal Decompositions of Translations of $L^{2}$-functions", "abstract": "In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space $L^{2}(\\mathbb{R}^{n})$. Our approach applies more generally to families of $n$ arbitrary commuting unitary operators in a complex Hilbert space $\\mathcal{H}$, or equivalent the spectral theory of a unitary representation $U$ of the rank-$n$ lattice $\\mathbb{Z}^{n}$ in $\\mathbb{R}^{n}$. Starting with a non-zero vector $\\psi \\in \\mathcal{H}$, we look for relations among the vectors in the cyclic subspace in $\\mathcal{H}$ generated by $\\psi$. Since these vectors $\\{U(k)\\psi | k \\in \\mathbb{Z}^{n}\\}$ involve infinite ``linear combinations,\" the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name $L^{2}$-independence. This refers to \\textit{infinite} linear combinations of integral translates of a fixed function with $l^{2}$-coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic integrals."}
{"category": "Math", "title": "An upper bound on the reduction number of an ideal", "abstract": "Let A be a commutative ring and I an ideal of A with a reduction Q. In this paper we give an upper bound on the reduction number of I with respect to Q, when a suitable family of ideals in A is given. As a corollary it follows that if some ideal J containing I satisfies J^2 = QJ, then I^{v + 2} = QI^{v + 1}, where v denotes the number of generators of J / I as an A-module."}
{"category": "Math", "title": "Comparison of Spline with Kriging in an Epidemiological Problem", "abstract": "There are various methods to analyze different kinds of data sets. Spatial data is defined when data is dependent on each other based on their respective locations. Spline and Kriging are two methods for interpolating and predicting spatial data. Under certain conditions, these methods are equivalent, but in practice they show different behaviors. Amount of data can be observed only at some positions that are chosen as positions of sample points, therefore, prediction of data values in other positions is important. In this paper, the link between Spline and Kriging methods is described, then for an epidemiological two dimensional real data set, data is observed in geological longitude and in latitude dimensions, and behavior of these methods are investigated. Comparison of these performances show that for this data set, Kriging method has a better performance than Spline method."}
{"category": "Math", "title": "Values of coefficients of cyclotomic polynomials II", "abstract": "Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. The first two authors showed in part I that if m is a prime power and n and k range over the non-negative integers, then a(mn,k) assumes every integer value. Here this result is extended to the case where m is arbitrary. The proof use some properties of reciprocal cyclotomic polynomials (see arXiv:0709.1570)."}
{"category": "Math", "title": "The Categorification of a Symmetric Operad is Independent of Signature", "abstract": "Given a symmetric operad $P$, and a signature (or generating sequence) $\\Phi$ for $P$, we define a notion of the \"categorification\" (or \"weakening\") of $P$ with respect to $\\Phi$. When $P$ is the symmetric operad whose algebras are commutative monoids, with the standard signature, we recover the notion of symmetric monoidal categories. We then show that this categorification is independent (up to equivalence) of the choice of signature."}
{"category": "Math", "title": "Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - I: Ordinary Differential Equations", "abstract": "The Lie linearizability criteria are extended to complex functions for complex ordinary differential equations. The linearizability of complex ordinary differential equations is used to study the linearizability of corresponding systems of two real ordinary differential equations. The transformations that map a system of two nonlinear ordinary differential equations into systems of linear ordinary differential equations are obtained from complex transformations. Invariant criteria for linearization are given for second order complex ordinary differential equations in terms of the coefficients of the equations, as well as the corresponding real system, which provide procedures for writing down the solutions of the equations. Illustrative examples are given and discussed."}
{"category": "Math", "title": "Use of Complex Lie Symmetries for Linearization of Systems of Differential Equations - II: Partial Differential Equations", "abstract": "The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations implies the linearizability of systems of partial differential equations corresponding to those complex ordinary differential equations. The invertible complex transformations can be used to obtain invertible real transformations that map a system of nonlinear partial differential equations into a system of linear partial differential equation. Explicit invariant criteria are given that provide procedures for writing down the solutions of the linearized equations. A few non-trivial examples are mentioned."}
{"category": "Math", "title": "Moduli space of stable maps to projective space via GIT", "abstract": "We compare the Kontsevich moduli space of genus 0 stable maps to projective space with the quasi-map space when $d=3$. More precisely, we prove that when $d=3$, the obvious birational map from the quasi-map space to the moduli space of stable maps is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces for lower degrees. Using this, we calculate the Betti numbers, the integral Picard group, and the rational cohomology ring. The degree two case is worked out as a warm-up."}
{"category": "Math", "title": "Means and Hermite Interpolation", "abstract": "Let $m_{2}<m_{1}$ be two given nonnegative integers with $n=m_{1}+m_{2}+1$. For suitably differentiable $f$, we let $P,Q\\in \\pi_{n}$ be the Hermite polynomial interpolants to $f$ which satisfy $P^{(j)}(a)=f^{(j)}(a),j=0,1,...,m_{1}$ and $P^{(j)}(b)=f^{(j)}(b),j=0,1,...,m_{2},$ $Q^{(j)}(a)=f^{(j)}(a),j=0,1,...,m_{2}$ and $Q^{(j)}(b)=f^{(j)}(b),j=0,1,...,m_{1}$. Suppose that $f\\in C^{n+2}(I)$ with $f^{(n+1)}(x)\\neq 0$ for $x\\in (a,b)$. If $m_{1}-m_{2}$ is even, then there is a unique $x_{0},a<x_{0}<b,$ such that $P(x_{0})=Q(x_{0})$. If $m_{1}-m_{2}$ is odd, then there is a unique $x_{0},a<x_{0}<b,$ such that $f(x_{0})=\\tfrac{1}{2}(P(x_{0})+Q(x_{0})) $. $x_{0}$ defines a strict, symmetric mean, which we denote by $M_{f,m_{1},m_{2}}(a,b)$. We prove various properties of these means. In particular, we show that $f(x)=x^{m_{1}+m_{2}+2}$ yields the arithmetic mean, $f(x)=x^{-1}$ yields the harmonic mean, and $f(x)=x^{(m_{1}+m_{2}+1)/2}$ yields the geometric mean."}
{"category": "Math", "title": "Near-critical percolation in two dimensions", "abstract": "We give a self-contained and detailed presentation of Kesten's results that allow to relate critical and near-critical percolation on the triangular lattice. They constitute an important step in the derivation of the exponents describing the near-critical behavior of this model. For future use and reference, we also show how these results can be obtained in more general situations, and we state some new consequences."}
{"category": "Math", "title": "Irreducible representations of inner quasidiagonal C*-algebras", "abstract": "It is shown that a separable C*-algebra is inner quasidiagonal if and only if it has a separating family of quasidiagonal irreducible representations. As a consequence, a separable C*-algebra is a strong NF algebra if and only if it is nuclear and has a separating family of quasidiagonal irreducible representations. We also obtain some permanence properties of the class of inner quasidiagonal C*-algebras."}
{"category": "Math", "title": "Affine actions on Nilpotent Lie groups", "abstract": "To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G)=dim(N) less than 6. Finally, we specialize to the case of abelian simply transitive affine actions on a given connected and simply connected nilpotent Lie group. It turns out that such a simply transitive abelian affine action on N corresponds to a particular Lie compatible bilinear product on the Lie algebra of N, which we call an LR-structure."}
{"category": "Math", "title": "Bicartesian Coherence Revisited", "abstract": "A survey is given of results about coherence for categories with finite products and coproducts. For these results, which were published previously by the authors in several places, some formulations and proofs are here corrected, and matters are updated. The categories investigated in this paper formalize equality of proofs in classical and intuitionistic conjunctive-disjunctive logic without distribution of conjunction over disjunction."}
{"category": "Math", "title": "On a continuity theorem for constructive functions", "abstract": "One proves that any everywhere defined constructive mapping from a complete metric space into a complete metric space which preserves the property of precompacity of subsets is locally uniformly continuous. This fact can be viewed as interpretation of L. E. J. Brower's fan theorem in terms of A. A. Markov's constructive analysis."}
{"category": "Math", "title": "A Method for Compressing Parameters in Bayesian Models with Application to Logistic Sequence Prediction Models", "abstract": "Bayesian classification and regression with high order interactions is largely infeasible because Markov chain Monte Carlo (MCMC) would need to be applied with a great many parameters, whose number increases rapidly with the order. In this paper we show how to make it feasible by effectively reducing the number of parameters, exploiting the fact that many interactions have the same values for all training cases. Our method uses a single ``compressed'' parameter to represent the sum of all parameters associated with a set of patterns that have the same value for all training cases. Using symmetric stable distributions as the priors of the original parameters, we can easily find the priors of these compressed parameters. We therefore need to deal only with a much smaller number of compressed parameters when training the model with MCMC. The number of compressed parameters may have converged before considering the highest possible order. After training the model, we can split these compressed parameters into the original ones as needed to make predictions for test cases. We show in detail how to compress parameters for logistic sequence prediction models. Experiments on both simulated and real data demonstrate that a huge number of parameters can indeed be reduced by our compression method."}
{"category": "Math", "title": "Subalgebras of Lie algebras with non-degenerate restriction of the Killing form", "abstract": "Let $\\mf{g}$ be any finite-dimensional Lie algebra with Killling form $B$. Let $\\mf{h}$ be a subalgebra of $\\mf{g}$ on which the Killing form is non degenerate. Then $\\mf{h}$ is reductive."}
{"category": "Math", "title": "On highly transcendental quantities which cannot be expressed by integral formulas", "abstract": "E565 in the Enestrom index. Translated from the Latin original, \"De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet\" (1775). Euler does not prove any results in this paper. It seems to me like he is trying to develop some general ideas about special functions. He gives some examples of numbers he claims but does not prove cannot be represented by definite integrals of algebraic functions. Euler has the idea that if we knew more about the function with the power series $\\sum x^{t_n}$ where $t_n$ is the $n$th triangular number, this could lead to a proof of Fermat's theorem that every positive integer is the sum of three triangular numbers. This doesn't end of being fruitful for Euler, but in fact later Jacobi proves a lot of results like this with his theta functions. The last paragraph (\\S 9) is not clear to me. My best reading is that there are infinitely many \"levels\" of transcendental numbers and that this is unexpected or remarkable."}
{"category": "Math", "title": "On the Ramsey multiplicity of complete graphs", "abstract": "We show that, for $n$ large, there must exist at least \\[\\frac{n^t}{C^{(1+o(1))t^2}}\\] monochromatic $K_t$s in any two-colouring of the edges of $K_n$, where $C \\approx 2.18$ is an explicitly defined constant. The old lower bound, due to Erd\\H{o}s \\cite{E62}, and based upon the standard bounds for Ramsey's theorem, is \\[\\frac{n^t}{4^{(1+o(1))t^2}}.\\]"}
{"category": "Math", "title": "Complex hyperbolic hyperplane complements", "abstract": "We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volume A-regular Riemannian metric of negative sectional curvature. It follows that for n>1 the fundamental group of M-S satisfies Mostow-type Rigidity, has finite asymptotic dimension and rapid decay property, satisfies Borel and Baum-Connes conjectures, is co-Hopf and residually hyperbolic, has no nontrivial subgroups with property (T), and has finite outer automorphism group. Furthermore, if M is compact, then the fundamental group of M-S is biautomatic and satisfies Strong Tits Alternative."}
{"category": "Math", "title": "A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals", "abstract": "A poly-log time method to compute the truncated theta function, its derivatives, and integrals is presented. The method is elementary, rigorous, explicit, and suited for computer implementation. We repeatedly apply the Poisson summation formula to the truncated theta function while suitably normalizing the linear and quadratic arguments after each repetition. The method relies on the periodicity of the complex exponential, which enables the suitable normalization of the arguments, and on the self-similarity of the Gaussian, which ensures that we still obtain a truncated theta function after each application of the Poisson summation. In other words, our method relies on modular properties of the theta function. Applications to the numerical computation of the Riemann zeta function and to finding the number of solutions of Waring type Diophantine equations are discussed."}
{"category": "Math", "title": "Contact equations, Lipschitz extensions and isoperimetric inequalities", "abstract": "We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called Allcock groups, observing that their horizontal distribution may have any codimesion. Finally, we show how these Lipschitz extensions theorems lead us to quadratic isoperimetric inequalities in all Allcock groups."}
{"category": "Math", "title": "A note on lower bounds for hypergraph Ramsey numbers", "abstract": "We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \\[r_3 (l,l,l) \\geq 2^{l^{c \\log \\log l}}.\\] The old bound, due to Erd\\H{o}s and Hajnal, was \\[r_3 (l,l,l) \\geq 2^{c l^2 \\log^2 l}.\\]"}
{"category": "Math", "title": "Fast methods to compute the Riemann zeta function", "abstract": "The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Sch\\\"onhage's method, or Heath-Brown's method. The complexities of these methods have exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this paper, three new fast and potentially practical methods to compute zeta are presented. One method is very simple. Its complexity has exponent 2/5. A second method relies on this author's algorithm to compute quadratic exponential sums. Its complexity has exponent 1/3. The third method employs an algorithm, developed in this paper, to compute cubic exponential sums. Its complexity has exponent 4/13 (approximately, 0.307)."}
{"category": "Math", "title": "A bound on the exponent of the cohomology of BC-bundles", "abstract": "We give a lower bound for the exponent of certain elements in the integral cohomology of the total spaces of principal BC-bundles for C a finite cyclic group. As applications we give a proof of the theorem of A. Adem and H.-W. Henn that a p-group is elementary abelian if and only if its integral cohomology has exponent p, and we exhibit some infinite groups of finite virtual cohomological dimension whose Tate-Farrell cohomology contains torsion of order greater than the l.c.m. of the orders of their finite subgroups. We also give an upper bound for the exponent of all but finitely many of the integral cohomology groups of a finite group, in terms of the permutation representations of the group."}
{"category": "Math", "title": "Smoothness criteria for Navier-Stokes equations in terms of regularity along the steam lines", "abstract": "This article is devoted to a regularity criteria for solutions of the Navier-Stokes equations in terms of regularity along the stream lines. More precisely, we prove that a suitable weak solution for the Navier-Stokes equations is regular under some constraint on the second derivative of |u| along the stream lines."}
{"category": "Math", "title": "The Yagita invariant of general linear groups", "abstract": "We give a definition of the Yagita invariant at a prime p of an arbitrary group G, and compute the invariant for each prime for the general linear groups over any integrally closed subring of the complex numbers. We also compute the invariants for special linear groups over the same rings, except in some cases when both the degree of the linear group and the ring are `small' compared to p."}
{"category": "Math", "title": "On subgroups of Coxeter groups", "abstract": "A right-angled Coxeter group is a group with a given set of generators of order two, subject only to the relations that certain pairs of the generators commute. Various papers have shown how homological properties of the Coxeter group are related to homological properties of the simplicial complex whose simplices are the sets of commuting generators. Using these techniques, we construct torsion-free groups which are Poincare duality groups over some rings but not over others, and a group whose integral cohomological dimension is finite but strictly greater than its cohomological dimension over any field. We determine which Coxeter groups have finite virtual cohomological dimension (it is classical that all finitely generated Coxeter groups have finite vcd, but there are others). We also give minimal presentations for certain torsion-free finite-index subgroups of right-angled Coxter groups. Finally we give a `bare-hands' construction (using free products with amalgamation and HNN extensions) of a torsion-free group whose integral cohomological dimension is strictly greater than its rational cohomological dimension."}
{"category": "Math", "title": "Chern classes and extraspecial groups", "abstract": "The mod-p cohomology ring of the extraspecial p-group of exponent p is studied for odd p. We investigate the subquotient ch(G) generated by Chern classes modulo the nilradical. The subring of ch(G) generated by Chern classes of one-dimensional representations was studied by Tezuka and Yagita. The subring generated by the Chern classes of the faithful irreducible representations is a polynomial algebra. We study the interplay between these two families of generators, and obtain some relations between them."}
{"category": "Math", "title": "On universally stable elements", "abstract": "We show that certain subrings of the cohomology of a finite p-group P may be realised as the images of restriction from suitable virtually free groups. We deduce that the cohomology of P is a finite module for any such subring. Examples include the ring of `universally stable elements' defined by Evens and Priddy, and rings of invariants such as the mod-2 Dickson algebras."}
{"category": "Math", "title": "On the GL(V)-module structure of K(n)^*(BV)", "abstract": "We study the question of whether the Morava K-theory of the classifying space of an elementary abelian group V is a permutation module (in either of two distinct senses) for the automorphism group of V. We use Brauer characters and computer calculations. We construct and implement an algorithm for finding permutation submodules of maximal dimension inside modules for p-groups in characteristic p."}
{"category": "Math", "title": "On the integral cohomology of wreath products", "abstract": "Under mild conditions on the space X, we describe the additive structure of the integral cohomology of the space $X^p \\times_{C_p}EC_p$ in terms of the cohomology of X. We give weaker results for other similar spaces, and deduce various corollaries concerning the cohomology of finite groups."}
{"category": "Math", "title": "The cohomology of Bestvina-Brady groups", "abstract": "For each subcomplex of the standard CW-structure on any torus, we compute the homology of a certain infinite cyclic regular covering space. In all cases when the homology is finitely generated, we also compute the cohomology ring. For aspherical subcomplexes of the torus our computation gives the homology and cohomology of Bestvina-Brady groups. We compute the cohomological dimension of each of these groups over any field and over any subring of the rationals."}
{"category": "Math", "title": "The cohomology of certain groups", "abstract": "Computations in the cohomology of finite groups."}
{"category": "Math", "title": "The p-adic closure of a subgroup of rational points on a commutative algebraic group", "abstract": "Let G be a commutative algebraic group over Q. Let Gamma be a subgroup of G(Q) contained in the union of the compact subgroups of G(Q_p). We formulate a guess for the dimension of the closure of Gamma in G(Q_p), and show that its correctness for certain tori is equivalent to Leopoldt's conjecture."}
{"category": "Math", "title": "The Computation of the Logarithmic Cohomology for Plane Curves", "abstract": "We give algorithms of computing bases of logarithmic cohomology groups for square-free polynomials in two variables. (Fixed typos of v1)"}
{"category": "Math", "title": "New classification techniques for ordinary differential equations", "abstract": "The goal of the present paper is to propose an enhanced ordinary differential equations solver by exploitation of the powerful equivalence method of \\'Elie Cartan. This solver returns a target equation equivalent to the equation to be solved and the transformation realizing the equivalence. The target ODE is a member of a dictionary of ODE, that are regarded as well-known, or at least well-studied. The dictionary considered in this article are ODE in a book of Kamke. The major advantage of our solver is that the equivalence transformation is obtained without integrating differential equations. We provide also a theoretical contribution revealing the relationship between the change of coordinates that maps two differential equations and their symmetry pseudo-groups."}
{"category": "Math", "title": "The Structure of a Bernoulli Process Variation of the Fibonacci Sequence", "abstract": "We consider the structure of a variation of the Fibonacci sequence which is determined by a Bernoulli process. The associated structure of all Bernoulli variations of the Fibonacci sequence can be represented by a directed binary tree, which we denote X, with vertex labels representing the specific state of the recurrence variation. Since X is a binary tree, we can consider the term of a sequence variation given by a finite traversal of X represented by a binary code t. We then prove that the traversal of X that is the reflection of the digits of t gives exactly the integer term corresponding to t. We consider how to further this result with the statement of an additional conjecture. Finally, we give connections to Fibonacci expansions, the Stern-Brocot tree, and we apply our methods to the Three Hat Problem as seen in ``Puzzle Corner'' of the ``Technology Review'' magazine."}
{"category": "Math", "title": "Equilibrium points for Optimal Investment with Vintage Capital", "abstract": "The paper concerns the study of equilibrium points, namely the stationary solutions to the closed loop equation, of an infinite dimensional and infinite horizon boundary control problem for linear partial differential equations. Sufficient conditions for existence of equilibrium points in the general case are given and later applied to the economic problem of optimal investment with vintage capital. Explicit computation of equilibria for the economic problem in some relevant examples is also provided. Indeed the challenging issue here is showing that a theoretical machinery, such as optimal control in infinite dimension, may be effectively used to compute solutions explicitly and easily, and that the same computation may be straightforwardly repeated in examples yielding the same abstract structure. No stability result is instead provided: the work here contained has to be considered as a first step in the direction of studying the behavior of optimal controls and trajectories in the long run."}
{"category": "Math", "title": "A linear equation for Minkowski sums of polytopes relatively in general position", "abstract": "The objective of this paper is to study a special family of Minkowski sums, that is of polytopes relatively in general position. We show that the maximum number of faces in the sum can be attained by this family. We present a new linear equation that is satisfied by f-vectors of the sum and the summands. We study some of the implications of this equation."}
{"category": "Math", "title": "Pluripolarity of manifolds", "abstract": "By the classical result of E. Bedford, a real-analytic non-generic manifold is pluripolar. We extend this result for manifolds of the Gevrey class. This also gives a generalization of the recent result of D. Coman, N. Levenberg and E. Poletsky on pluripolarity of curves of the Gevrey class."}
{"category": "Math", "title": "Slider-pinning Rigidity: a Maxwell-Laman-type Theorem", "abstract": "We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley's direction networks."}
{"category": "Math", "title": "Fillings method in number theory", "abstract": "Number of results in number theory have been developed using a new method. The Goldbach binary conjecture in strengthened formulation have been among them."}
{"category": "Math", "title": "Periodic orbits of period 3 in the disc", "abstract": "Let f be an orientation preserving homeomorphism of the disc D2 which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi /3 or 4 pi /3, or f has a periodic point of least period n for each n in N*."}
{"category": "Math", "title": "Elliptic solutions of the Toda chain and a generalization of the Stieltjes-Carlitz polynomials", "abstract": "We construct new elliptic solutions of the restricted Toda chain. These solutions give rise to a new explicit class of orthogonal polynomials which can be considered as a generalization of the Stieltjes-Carlitz elliptic polynomials. Relations between characteristic (i.e. positive definite) functions, Toda chain and orthogonal polynomials are developed in order to derive main properties of these polynomials. The recurrence coefficients and the weight function of these polynomials are expressed explicitly. In the degenerated cases of the elliptic functions the modified Meixner polynomials and the Krall-Laguerre polynomials appear."}
{"category": "Math", "title": "Janet's Algorithm", "abstract": "We have introduced the Janet's algorithm for the Stanley decomposition of a monomial ideal I in a polynomial ring S = K[x_1,...,x_n] and prove that Janet's algorithm gives the squarefree Stanley decomposition of S/I for a squarefree monomial ideal I. We have also shown that the Janet's algorithm gives a partition of a simplicial complex."}
{"category": "Math", "title": "Generalized Bochner theorem: characterization of the Askey-Wilson polynomials", "abstract": "Assume that there is a set of monic polynomials $P_n(z)$ satisfying the second-order difference equation $$ A(s) P_n(z(s+1)) + B(s) P_n(z(s)) + C(s) P_n(z(s-1)) = \\lambda_n P_n(z(s)), n=0,1,2,..., N$$ where $z(s), A(s), B(s), C(s)$ are some functions of the discrete argument $s$ and $N$ may be either finite or infinite. The irreducibility condition $A(s-1)C(s) \\ne 0$ is assumed for all admissible values of $s$. In the finite case we assume that there are $N+1$ distinct grid points $z(s), \\: s=0,1,..., N$ such that $z(i) \\ne z(j), \\: i \\ne j$. If $N=\\infty$ we assume that the grid $z(s)$ has infinitely many different values for different values of $s$. In both finite and infinite cases we assume also that the problem is non-degenerate, i.e. $\\lambda_n \\ne \\lambda_m, n \\ne m$. Then we show that necessarily: (i) the grid $z(s)$ is at most quadratic or q-quadratic in $s$; (ii) corresponding polynomials $P_n(z)$ are at most the Askey-Wilson polynomials corresponding to the grid $z(s)$. This result can be considered as generalizing of the Bochner theorem (characterizing the ordinary classical polynomials) to generic case of arbitrary difference operator on arbitrary grids."}
{"category": "Math", "title": "Birational geometry of Fano double covers", "abstract": "We prove divisorial canonicity of Fano double hypersurfaces of general position."}
{"category": "Math", "title": "The hyperbolic mean curvature flow", "abstract": "We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established."}
{"category": "Math", "title": "Symplectic Jacobi diagrams and the Lie algebra of homology cylinders", "abstract": "Let S be a compact connected oriented surface, whose boundary is connected or empty. A homology cylinder over the surface S is a cobordism between S and itself, homologically equivalent to the cylinder over S. The Y-filtration on the monoid of homology cylinders over S is defined by clasper surgery. Using a functorial extension of the Le-Murakami-Ohtsuki invariant, we show that the graded Lie algebra associated to the Y-filtration is isomorphic to the Lie algebra of ``symplectic Jacobi diagrams.'' This Lie algebra consists of the primitive elements of a certain Hopf algebra whose multiplication is a diagrammatic analogue of the Moyal-Weyl product. The mapping cylinder construction embeds the Torelli group into the monoid of homology cylinders, sending the lower central series to the Y-filtration. We give a combinatorial description of the graded Lie algebra map induced by this embedding, by connecting Hain's infinitesimal presentation of the Torelli group to the Lie algebra of symplectic Jacobi diagrams. This Lie algebra map is shown to be injective in degree two, and the question of the injectivity in higher degrees is discussed."}
{"category": "Math", "title": "Zero diffusion-dispersion limits for scalar conservation laws", "abstract": "We consider solutions of hyperbolic conservation laws regularized with vanishing diffusion and dispersion terms. Following a pioneering work by Schonbek, we establish the convergence of the regularized solutions toward discontinuous solutions of the hyperbolic conservation law. The proof relies on the method of compensated compactness in the $L^2$ setting. Our result improves upon Schonbek's earlier results and provides an optimal condition on the balance between the relative sizes of the diffusion and the dispersion parameters. A convergence result is also established for multi-dimensional conservation laws by relying on DiPerna's uniqueness theorem for entropy measure-valued solutions."}
{"category": "Math", "title": "Auxiliary Information and A Priori Values in Construction of Improved Estimators", "abstract": "This volume is a collection of six papers on the use of auxiliary information and 'a priori' values in construction of improved estimators. The work included here will be of immense application for researchers and students who emply auxiliary information in any form."}
{"category": "Math", "title": "Global wellposedness in the energy space for the Maxwell-Schr\\\"odinger system", "abstract": "We prove that the Maxwell-Schr\\\"odinger system in $\\R^{3+1}$ is globally well-posed in the energy space. The key element of the proof is to obtain a short time wave packet parametrix for the magnetic Schr\\\"odinger equation, which leads to linear, bilinear and trilinear estimates. These, in turn, are extended to larger time scales via a bootstrap argument."}
{"category": "Math", "title": "Univoque numbers and an avatar of Thue-Morse", "abstract": "Univoque numbers are real numbers $\\lambda > 1$ such that the number 1 admits a unique expansion in base $\\lambda$, i.e., a unique expansion $1 = \\sum_{j \\geq 0} a_j \\lambda^{-(j+1)}$, with $a_j \\in \\{0, 1, ..., \\lceil \\lambda \\rceil -1\\}$ for every $j \\geq 0$. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called {\\em admissible sequences}. We show how a 1983 study of the first author gives both a result of Komornik and Loreti on the smallest admissible sequence on the set $\\{0, 1, >..., b\\}$, and a result of de Vries and Komornik (2007) on the smallest univoque number belonging to the interval $(b, b+1)$, where $b$ is any positive integer. We also prove that this last number is transcendental. An avatar of the Thue-Morse sequence, namely the fixed point beginning in 3 of the morphism $3 \\to 31$, $2 \\to 30$, $1 \\to 03$, $0 \\to 02$, occurs in a \"universal\" manner."}
{"category": "Math", "title": "Symmetric Systems and their Applications to Root Systems Extended by Abelian Groups", "abstract": "We investigate the class of root systems R obtained by extending an irreducible root system by a torsion-free group G. In this context there is a Weyl group W and a group U with the presentation by conjugation. We show under additional hypotheses that the kernel of the natural homomorphism from U to W is isomorphic to the kernel of the homomorphism from the abelianization of U to that of W. For this we introduce the concept of a symmetric system, a discrete version of the concept of a symmetric space. Mathematics Subject Classification 2000: 20F55, 17B65, 17B67, 22E65, 22E40. Key Words and Phrases: Weyl group, root system, presentation by conjugation, extended affine Weyl group (EAWeG), extended affine root system (EARS), irreducible root system extended by an abelian group."}
{"category": "Math", "title": "Hessian estimates for the sigma-2 equation in dimension three", "abstract": "We derive a priori interior Hessian estimates for the special Lagrangian equation $\\sigma_{2}=1$ in dimension three."}
{"category": "Math", "title": "Morse-Novikov cohomology of locally conformally K\\\"ahler manifolds", "abstract": "A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the Bott-Chern class, of an LCK-structure. These invariants together play the same role as the Kahler class in Kahler geometry. If these classes for two LCK-structures coincide, the difference between these structures can be expressed by a smooth potential, similar to the Kahler case. We show that the Morse-Novikov class and the Bott-Chern class of a Vaisman manifold vanishes. Moreover, for any LCK-structure on a Vaisman manifold, we prove that its Morse-Novikov class vanishes. We show that a compact LCK-manifold $M$ with vanishing Bott-Chern class admits a holomorphic embedding to a Hopf manifold, if $\\dim_\\C M \\geq 3$, a result which parallels the Kodaira embedding theorem."}
{"category": "Math", "title": "On the moduli of constant mean curvature cylinders of finite type in the 3-sphere", "abstract": "We show that one-sided Alexandrov embedded constant mean curvature cylinders of finite type in the 3-sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3-sphere are rotational."}
{"category": "Math", "title": "Random sampling of plane partitions", "abstract": "This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is $O(n (\\ln n)^3)$ in approximate-size sampling and $O(n^{4/3})$ in exact-size sampling (under a real-arithmetic computation model). To our knowledge, these are the first polynomial-time samplers for plane partitions according to the size (there exist polynomial-time samplers of another type, which draw plane partitions that fit inside a fixed bounding box). The same principles yield efficient samplers for $(a\\times b)$-boxed plane partitions (plane partitions with two dimensions bounded), and for skew plane partitions. The random samplers allow us to perform simulations and observe limit shapes and frozen boundaries, which have been analysed recently by Cerf and Kenyon for plane partitions, and by Okounkov and Reshetikhin for skew plane partitions."}
{"category": "Math", "title": "Similarity of operators and geometry of eigenvector bundles", "abstract": "We characterize the contractions that are similar to the backward shift in the Hardy space $H^2$. This characterization is given in terms of the geometry of the eigenvector bundles of the operators."}
{"category": "Math", "title": "Generalized Cohn's Theorem", "abstract": "We introduce the notion of a free associative $\\mathcal{Z}_2$-algebra on the union of two disjoint sets and prove a generalization of Cohn's Theorem on Jordan algebras."}
{"category": "Math", "title": "On the partition of numbers into parts of a given type and number", "abstract": "E394 in the Enestrom index. Translated from the Latin original, \"De partitione numerorum in partes tam numero quam specie datas\" (1768). Euler finds a lot of recurrence formulas for the number of partitions of $N$ into $n$ parts from some set like 1 to 6 (numbers on the sides of a die). He starts the paper talking about how many ways a number $N$ can be formed by throwing $n$ dice. There do not seem to be any new results or ideas here that weren't in \"Observationes analyticae variae de combinationibus\", E158 and \"De partitione numerorum\", E191. In this paper Euler just does a lot of special cases. My impression is that Euler is trying to make his theory of partitions more approachable,. Also, maybe for his own benefit he wants to say it all again in different words, to make it clear."}
{"category": "Math", "title": "Gorenstein Global Dimensions and Cotorsion Dimension of Rings", "abstract": "In this paper, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings. We use this result to compute the Gorenstein global dimension of some particular cases of trivial extensions of rings and of group rings."}
{"category": "Math", "title": "Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media", "abstract": "We consider a space-homogeneous gas of {\\it inelastic hard spheres}, with a {\\it diffusive term} representing a random background forcing (in the framework of so-called {\\em constant normal restitution coefficients} $\\alpha \\in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $\\alpha \\in [\\alpha_*,1)$ for some constructive $\\alpha_* \\in [0,1)$) we prove uniqueness of the stationary solution for given values of the restitution coefficient $\\alpha \\in [\\alpha_*,1)$, the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution."}
{"category": "Math", "title": "Global Gorenstein dimensions of polynomial rings and of direct product of rings", "abstract": "In this paper, we extend the well-known Hilbert's syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct product of rings, which gives examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension."}
{"category": "Math", "title": "Rings over which all modules are strongly Gorenstein projective", "abstract": "One of the main results of this paper is the characterization of the rings over which all modules are strongly Gorenstein projective. We show that these kinds of rings are very particular cases of the well-known quasi-Frobenius rings. We give examples of rings over which all modules are Gorenstein projective but not necessarily strongly Gorenstein projective."}
{"category": "Math", "title": "Pr\\\"ufer-Like Conditions in Subring Retracts and Applications", "abstract": "In this paper, we consider five possible extensions of the Pr\\\"ufer domain notion to the case of commutative rings with zero divisors. We investigate the transfer of these Pr\\\"ufer-like properties between a commutative ring and its subring retract. Our results generate new families of examples of rings subject to a given Pr\\\"ufer-like conditions."}
{"category": "Math", "title": "Symmetry p-adic invariant integral on Z_p for Bernoulli and euler polynomials", "abstract": "The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integral on Z_p."}
{"category": "Math", "title": "The Algebra of Graph Invariants - Lower and Upper Bounds for Minimal Generators", "abstract": "In this paper we study the algebra of graph invariants, focusing mainly on the invariants of simple graphs. All other invariants, such as sorted eigenvalues, degree sequences and canonical permutations, belong to this algebra. In fact, every graph invariant is a linear combination of the basic graph invariants which we study in this paper. To prove that two graphs are isomorphic, a number of basic invariants are required, which are called separator invariants. The minimal set of separator invariants is also the minimal basic generator set for the algebra of graph invariants. We find lower and upper bounds for the minimal number of generator/separator invariants needed for proving graph isomorphism. Finally we find a sufficient condition for Ulam's conjecture to be true based on Redfield's enumeration formula."}
{"category": "Math", "title": "Graded modules for Virasoro-like algebra", "abstract": "In this paper, we consider the classification of irreducible ${\\bf Z}$- and ${\\bf Z}^2$-graded modules with finite dimensional homogeneous subspaces over the Virasoro-like algebra. We first prove that such a module is a uniformly bounded module or a generalized highest weight module. Then we determine all generalized highest weight irreducible modules. As a consequence, we also determine all the modules with nonzero center. Finally, we prove that there does not exist any nontrivial ${\\bf Z}$-graded modules of intermediate series."}
{"category": "Math", "title": "The Ring of Graph Invariants - Graphic Values", "abstract": "The ring of graph invariants is spanned by the basic graph invariants which calculate the number of subgraphs isomorphic to a given graph in other graphs. These subgraphs counting invariants are not algebraically independent. In our view the most important problem in graph theory of unlabeled graphs is the problem of determining graphic values of arbitrary sets of graph invariants. This corresponds to explaining the syzygy of the graph invariants when the number of vertices is unbounded. We introduce two methods to explore this complicated structure. Sets of graphs with a small number of vertices impose constraints on larger sets. We describe families of inequalities of graph invariants. These inequalities allow to loop over all values of graph invariants which look like graphic from the small sets point of view. We also develop strong notion of graphic values where the existence of the corresponding graphs is guaranteed once the constraints are satisfied by the basic graph invariants. These constraints are necessary and sufficient for graphs whose local neighborhoods are generated by a finite set of locally connected graphs. The reconstruction of the graph from the basic graph invariants is shown to be NP-complete in this restricted case. Finally we apply these results to formulate the problem of Ramsey numbers as an integer polyhedron problem of moderate and adjustable dimension."}
{"category": "Math", "title": "Generating functions of Cauchy-Stieltjes type for orthogonal polynomials", "abstract": "We characterize by the use of free probability the family of measures for which the mulitiplicative renormalization method applies with $h(x) = (1-x)^_{-1}$. This provides a representation formula for their Voiculescu Transforms."}
{"category": "Math", "title": "Branching of Hitchin's Prym cover for SL(2)", "abstract": "It is shown that the map from the Jacobian of the spectral curve to the moduli of stable bundles of rank 2 is generically simply branched along an irreducible divisor. This observation falsifies the key step in the \"abelianization of the SU(2) WZW connection\" presented in a recent paper [Yoshida, Annals 2006]"}
{"category": "Math", "title": "The zero-product problem for Toeplitz operators with radial symbols", "abstract": "For any bounded measurable function $f$ on the unit ball $B_n$, let $T_f$ be the Toeplitz operator with symbol $f$ acting on the Bergman space $A^2(B_n)$. The Zero-Product Problem asks: if $f_1,..., f_N$ are bounded measurable functions such that $T_{f_1}... T_{f_N}=0$, does it follow that one of the functions must be zero almost everywhere? This paper give the affirmative answer to this question when all except possibly one of the symbols are radial functions. The answer in the general case remains unknown."}
{"category": "Math", "title": "A novel numerical technique used in the solution of ordinary differential equations with a mixture of integer and fractional derivatives", "abstract": "Using both fractional derivatives, defined in the Riemann-Liouville and Caputo senses, and classical derivatives of the integer order we examine different numerical approaches to ordinary differential equations. Generally we formulate some algorithms where four discrete forms of the Caputo derivative and three different numerical techniques of solving ordinary differential equations are proposed. We then illustrate how to introduce classical initial conditions into equations where the Riemann-Liouville derivative is included."}
{"category": "Math", "title": "Rates of convergence for minimal distances in the central limit theorem under projective criteria", "abstract": "In this paper, we give estimates of ideal or minimal distances between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary martingale difference sequences or stationary sequences satisfying projective criteria. Applications to functions of linear processes and to functions of expanding maps of the interval are given."}
{"category": "Math", "title": "Summarization and Classification of Non-Poisson Point Processes", "abstract": "Fitting models for non-Poisson point processes is complicated by the lack of tractable models for much of the data. By using large samples of independent and identically distributed realizations and statistical learning, it is possible to identify absence of fit through finding a classification rule that can efficiently identify single realizations of each type. The method requires a much wider range of descriptive statistics than are currently in use, and a new concept of model fitting which is derive from how physical laws are judged to fit data."}
{"category": "Math", "title": "The Non-Backtracking Spectrum of the Universal Cover of a Graph", "abstract": "A non-backtracking walk on a graph, $H$, is a directed path of directed edges of $H$ such that no edge is the inverse of its preceding edge. Non-backtracking walks of a given length can be counted using the non-backtracking adjacency matrix, $B$, indexed by $H$'s directed edges and related to Ihara's Zeta function. We show how to determine $B$'s spectrum in the case where $H$ is a tree covering a finite graph. We show that when $H$ is not regular, this spectrum can have positive measure in the complex plane, unlike the regular case. We show that outside of $B$'s spectrum, the corresponding Green function has ``periodic decay ratios.'' The existence of such a ``ratio system'' can be effectively checked, and is equivalent to being outside the spectrum. We also prove that the spectral radius of the non-backtracking walk operator on the tree covering a finite graph is exactly $\\sqrt\\gr$, where $\\gr$ is the growth rate of the tree. This further motivates the definition of the graph theoretical Riemann hypothesis proposed by Stark and Terras \\cite{ST}. Finally, we give experimental evidence that for a fixed, finite graph, $H$, a random lift of large degree has non-backtracking new spectrum near that of $H$'s universal cover. This suggests a new generalization of Alon's second eigenvalue conjecture."}
{"category": "Math", "title": "Some finiteness results for Fourier-Mukai partners", "abstract": "We develop some methods for studying the Fourier-Mukai partners of an algebraic variety. As applications we prove that abelian varieties have finitely many Fourier-Mukai partners and that they are uniquely determined by their derived category of coherent $D$-modules. We also generalize a famous theorem due to A. Bondal and D. Orlov."}
{"category": "Math", "title": "Vector bundles with Theta divisors I: Bundles on Castelnuovo curves", "abstract": "In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g >= 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g >= 2 which extend to semistable vector bundles on any Castelnuovo degeneration of the general curve admit a theta divisor."}
{"category": "Math", "title": "The limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions", "abstract": "It is shown that the limits of the nested subclasses of five classes of infinitely divisible distributions on $R^d$, which are the Jurek class, the Goldie-Steutel-Bondesson class, the class of selfdecomposable distributions, the Thorin class and the class of generalized type $G$ distributions, are identical with the closure of the class of stable distributions. More general results are also given."}
{"category": "Math", "title": "The String Topology Loop Coproduct and Cohomology Operations", "abstract": "This note explores the interaction between cohomology operations in a generalized cohomology theory and a string topology loop coproduct dual to the Chas--Sullivan loop product. More precisely, we ask for a description for the failure of a given operation to commute with the loop coproduct, and will obtain a satisfactory answer in the case where the operation preserves both sums and products. Examples of such operations include the total Steenrod square in ordinary mod 2 cohomology and the Adams operations in K-theory."}
{"category": "Math", "title": "Upper bound for isometric embeddings \\ell_2^m\\to\\ell_p^n", "abstract": "The isometric embeddings $\\ell_{2;K}^m\\to\\ell_{p;K}^n$ ($m\\geq 2$, $p\\in 2\\N$) over a field $K\\in{R, C, H}$ are considered, and an upper bound for the minimal $n$ is proved. In the commutative case ($K\\neq H$) the bound was obtained by Delbaen, Jarchow and Pe{\\l}czy{\\'n}ski (1998) in a different way."}
{"category": "Math", "title": "A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm", "abstract": "We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group $G$ and find that if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in $\\Theta(\\sqrt{|G|})$ steps. Moreover, for the parallelized distinguished points algorithm on $J$ processors we find that $\\Theta(\\sqrt{|G|}/J)$ steps suffices. These are the first proofs of the correct order bounds which do not assume that every step of the algorithm produces an i.i.d. sample from $G$."}
{"category": "Math", "title": "Product formulas for the cyclotomic v-Schur algebra and for the canonical bases of the Fock space", "abstract": "In our earlier work, we have proved a product formula for certain decomposition numbers of the cyclotomic v-Schur algebra associated to the Ariki-Koike algebra. It is conjectured by Yvonne that the decomposition numbers of this algebra can be described in terms of the canonical basis of the higher level Fock space studied by Uglov. In this paper we prove a product formula related to the canonical basis of the Fock space. In view of Yvonne's conjecture, this formula is regarded as a counter-part for the Fock space of our previous formula."}
{"category": "Math", "title": "On completing three cyclic transversals to a latin square", "abstract": "Let $P$ be a partial latin square of prime order $p>7$ consisting of three cyclically generated transversals. Specifically, let $P$ be a partial latin square of the form: \\[ P=\\{(i,c+i,s+i),(i,c'+i,s'+i),(i,c''+i,s''+i)\\mid 0 \\leq i< p\\} \\] for some distinct $c,c',c''$ and some distinct $s,s',s''$. In this paper we show that any such $P$ completes to a latin square which is diagonally cyclic."}
{"category": "Math", "title": "Lyapunov conditions for logarithmic Sobolev and Super Poincar\\'e inequality", "abstract": "We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincar\\'e inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincar\\'e and weak Poincar\\'e inequalities was studied in Bakry and al. This approach allows us to recover and extend in an unified way some known criteria in the euclidean case (Bakry-Emery, Wang, Kusuoka-Stroock ...)."}
{"category": "Math", "title": "The variational particle-mesh method for matching curves", "abstract": "Diffeomorphic matching (only one of several names for this technique) is a technique for non-rigid registration of curves and surfaces in which the curve or surface is embedded in the flow of a time-series of vector fields. One seeks the flow between two topologically-equivalent curves or surfaces which minimises some metric defined on the vector fields, \\emph{i.e.} the flow closest to the identity in some sense. In this paper, we describe a new particle-mesh discretisation for the evolution of the geodesic flow and the embedded shape. Particle-mesh algorithms are very natural for this problem because Lagrangian particles (particles moving with the flow) can represent the movement of the shape whereas the vector field is Eulerian and hence best represented on a static mesh. We explain the derivation of the method, and prove conservation properties: the discrete method has a set of conserved momenta corresponding to the particle-relabelling symmetry which converge to conserved quantities in the continuous problem. We also introduce a new discretisation for the geometric current matching condition of (Vaillant and Glaunes, 2005). We illustrate the method and the derived properties with numerical examples."}
{"category": "Math", "title": "On the Structure of Equidistant Foliations of Euclidean Space", "abstract": "This thesis is concerned with equidistant foliations of Euclidean space, i.e. partitions into complete, connected, properly embedded smooth submanifolds. The space of leaves is an Alexandrov space of nonnegative curvature and the canonical projection is a submetry. Generalizing a result of Gromoll and Walschap we show that an equidistant foliation always has an affine leaf and we prove homogeneneity of the foliation under certain additional assumptions. Moreover, we give several reducibility results and construct new (noncompact) inhomogeneous examples of equidistant foliations."}
{"category": "Math", "title": "Pac-Bayesian Supervised Classification: The Thermodynamics of Statistical Learning", "abstract": "This monograph deals with adaptive supervised classification, using tools borrowed from statistical mechanics and information theory, stemming from the PACBayesian approach pioneered by David McAllester and applied to a conception of statistical learning theory forged by Vladimir Vapnik. Using convex analysis on the set of posterior probability measures, we show how to get local measures of the complexity of the classification model involving the relative entropy of posterior distributions with respect to Gibbs posterior measures. We then discuss relative bounds, comparing the generalization error of two classification rules, showing how the margin assumption of Mammen and Tsybakov can be replaced with some empirical measure of the covariance structure of the classification model.We show how to associate to any posterior distribution an effective temperature relating it to the Gibbs prior distribution with the same level of expected error rate, and how to estimate this effective temperature from data, resulting in an estimator whose expected error rate converges according to the best possible power of the sample size adaptively under any margin and parametric complexity assumptions. We describe and study an alternative selection scheme based on relative bounds between estimators, and present a two step localization technique which can handle the selection of a parametric model from a family of those. We show how to extend systematically all the results obtained in the inductive setting to transductive learning, and use this to improve Vapnik's generalization bounds, extending them to the case when the sample is made of independent non-identically distributed pairs of patterns and labels. Finally we review briefly the construction of Support Vector Machines and show how to derive generalization bounds for them, measuring the complexity either through the number of support vectors or through the value of the transductive or inductive margin."}
{"category": "Math", "title": "Small overlap monoids: the word problem", "abstract": "We develop a combinatorial approach to the study of semigroups and monoids with finite presentations satisfying small overlap conditions. In contrast to existing geometric methods, our approach facilitates a sequential left-right analysis of words which lends itself to the development of practical, efficient computational algorithms. In particular, we obtain a highly practical linear time solution to the word problem for monoids and semigroups with finite presentations satisfying the condition C(4), and a polynomial time solution to the uniform word problem for presentations satisfying the same condition."}
{"category": "Math", "title": "Die lokale Struktur von T-Dualit\\\"atstripeln", "abstract": "We show that the $C^*$-algebraic approach to T-duality of Mathai and Rosenberg is equivalent to the topological approach of Bunke and Schick."}
{"category": "Math", "title": "Koszul complexes and fully faithful integral functors", "abstract": "We characterise those objects in the derived category of a scheme which are a sheaf supported on a closed subscheme in terms of Koszul complexes. This is applied to generalize to arbitrary schemes the fully faithfullness criteria of an integral functor."}
{"category": "Math", "title": "An Extension of the Classical Gauss Series-product Identity by Fermionic Construction of \\hat{sl}_n", "abstract": "The main result of this paper is two infinity classes of series-product identities which is based on classical Gauss identity and two different interpretations of character formula for irreducible highest weight modules of affine Lie algebras."}
{"category": "Math", "title": "Deformation of Brody curves and mean dimension", "abstract": "The main purpose of this paper is to show that ideas of deformation theory can be applied to \"infinite dimensional geometry\". We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves."}
{"category": "Math", "title": "Random graphs with forbidden vertex degrees", "abstract": "We study the random graph G_{n,\\lambda/n} conditioned on the event that all vertex degrees lie in some given subset S of the non-negative integers. Subject to a certain hypothesis on S, the empirical distribution of the vertex degrees is asymptotically Poisson with some parameter \\mux given as the root of a certain `characteristic equation' of S that maximises a certain function \\psis(\\mu). Subject to a hypothesis on S, we obtain a partial description of the structure of such a random graph, including a condition for the existence (or not) of a giant component. The requisite hypothesis is in many cases benign, and applications are presented to a number of choices for the set S including the sets of (respectively) even and odd numbers. The random \\emph{even} graph is related to the random-cluster model on the complete graph K_n."}
{"category": "Math", "title": "Heisenberg modules over real multiplication noncommutative tori and related algebraic structures", "abstract": "We review some aspects of the theory of noncommutative two-tori with real multiplication focusing on the role played by Heisenberg groups in the definition of algebraic structures associated to these noncommutative spaces."}
{"category": "Math", "title": "On the automorphism groups of q-enveloping algebras of nilpotent Lie algebras", "abstract": "We investigate the automorphism group of the quantised enveloping algebra U of the positive nilpotent part of certain simple complex Lie algebras g in the case where the deformation parameter q \\in \\mathbb{C}^* is not a root of unity. Studying its action on the set of minimal primitive ideals of U we compute this group in the cases where g=sl_3 and g=so_5 confirming a Conjecture of Andruskiewitsch and Dumas regarding the automorphism group of U. In the case where g=sl_3, we retrieve the description of the automorphism group of the quantum Heisenberg algebra that was obtained independently by Alev and Dumas, and Caldero. In the case where g=so_5, the automorphism group of U was computed in [16] by using previous results of Andruskiewitsch and Dumas. In this paper, we give a new (simpler) proof of the Conjecture of Andruskiewitsch and Dumas in the case where g=so_5 based both on the original proof and on graded arguments developed in [17] and [18]."}
{"category": "Math", "title": "Wavelet methods in statistics: Some recent developments and their applications", "abstract": "The development of wavelet theory has in recent years spawned applications in signal processing, in fast algorithms for integral transforms, and in image and function representation methods. This last application has stimulated interest in wavelet applications to statistics and to the analysis of experimental data, with many successes in the efficient analysis, processing, and compression of noisy signals and images. This is a selective review article that attempts to synthesize some recent work on ``nonlinear'' wavelet methods in nonparametric curve estimation and their role on a variety of applications. After a short introduction to wavelet theory, we discuss in detail several wavelet shrinkage and wavelet thresholding estimators, scattered in the literature and developed, under more or less standard settings, for density estimation from i.i.d. observations or to denoise data modeled as observations of a signal with additive noise. Most of these methods are fitted into the general concept of regularization with appropriately chosen penalty functions. A narrow range of applications in major areas of statistics is also discussed such as partial linear regression models and functional index models. The usefulness of all these methods are illustrated by means of simulations and practical examples."}
{"category": "Math", "title": "Forgetting of the initial condition for the filter in general state-space hidden Markov chain: a coupling approach", "abstract": "We give simple conditions that ensure exponential forgetting of the initial conditions of the filter for general state-space hidden Markov chain. The proofs are based on the coupling argument applied to the posterior Markov kernels. These results are useful both for filtering hidden Markov models using approximation methods (e.g., particle filters) and for proving asymptotic properties of estimators. The results are general enough to cover models like the Gaussian state space model, without using the special structure that permits the application of the Kalman filter."}
{"category": "Math", "title": "An algorithm for evaluating the Gamma function and ramifications", "abstract": "We provide a new algorithm for evaluating the gamma function at any (rational) point and a new infinite product representation free from the presence of Euler and Mascheroni constant.Formulae and inequalities seemingly new are obtained as byproducts of the method."}
{"category": "Math", "title": "Liapunov's direct method for Birkhoffian systems: Applications to electrical networks", "abstract": "In this paper, the concepts and the direct theorems of stability in the sense of Liapunov, within the framework of Birkhoffian dynamical systems on manifolds, are considered. The Liapunov-type functions are constructed for linear and nonlinear LC and RLC electrical networks, to prove stability under certain conditions."}
{"category": "Math", "title": "Constrained BSDE and Viscosity Solutions of Variation Inequalities", "abstract": "In this paper, we study the relation between the smallest $g$-supersolution of constraint backward stochastic differential equation and viscosity solution of constraint semilineare parabolic PDE, i.e. variation inequalities. And we get an existence result of variation inequalities via constraint BSDE, and prove a uniqueness result under certain condition."}
{"category": "Math", "title": "Difference Problems and Differential Problems", "abstract": "We state some elementary problems concerning the relation between difference calculus and differential calculus, and we try to convince the reader that, in spite of the simplicity of the statements, a solution of these problems would be a significant contribution to the understanding of the foundations of differential and integral calculus."}
{"category": "Math", "title": "Point fixe li\\'e \\`a une orbite p\\'eriodique d'un diff\\'eomorphisme de R2", "abstract": "Given a diffeomorphism of the plane, which has a periodic orbit, we show how Nielsen fixed point theory can be used to establish the existence of a fixed point which is linked with this periodic orbit."}
{"category": "Math", "title": "A new upper bound for the cross number of finite Abelian groups", "abstract": "In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite Abelian groups. Given a finite Abelian group, this upper bound appears to depend only on the rank and on the number of distinct prime divisors of the exponent. The main theorem of this paper allows us, among other consequences, to prove that a classical conjecture concerning the cross and little cross numbers of finite Abelian groups holds asymptotically in at least two different directions."}
{"category": "Math", "title": "Duality of Chordal SLE", "abstract": "We derive some geometric properties of chordal SLE$(\\kappa;\\vec{\\rho})$ processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE$(\\kappa;\\vec{\\rho})$ process has the same distribution as the image of a chordal SLE$(\\kappa';\\vec{\\rho'})$ trace, where $\\kappa>4$, $\\kappa'=16/\\kappa$, and the forces $\\vec{\\rho}$ and $\\vec{\\rho'}$ are suitably chosen. We find that for $\\kappa\\ge 8$, the boundary of a standard chordal SLE$(\\kappa)$ hull stopped on swallowing a fixed $x\\in\\R\\sem\\{0\\}$ is the image of some SLE$(16/\\kappa;\\vec{\\rho})$ trace started from $x$. Then we obtain a new proof of the fact that chordal SLE$(\\kappa)$ trace is not reversible for $\\kappa>8$. We also prove that the reversal of SLE$(4;\\vec{\\rho})$ trace has the same distribution as the time-change of some SLE$(4;\\vec{\\rho'})$ trace for certain values of $\\vec{\\rho}$ and $\\vec{\\rho'}$."}
{"category": "Math", "title": "A maximum principle for relaxed stochastic control of linear SDE's with application to bond portfolio optimization", "abstract": "We study relaxed stochastic control problems where the state equation is a one dimensional linear stochastic differential equation with random and unbounded coefficients. The two main results are existence of an optimal relaxed control and necessary conditions for optimality in the form of a relaxed maximum principle. The main motivation is an optimal bond portfolio problem in a market where there exists a continuum of bonds and the portfolio weights are modeled as measure-valued processes on the set of times to maturity."}
{"category": "Math", "title": "The Decomposition Theorem and the topology of algebraic maps", "abstract": "We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the Decomposition Theorem, indicate some important applications and examples."}
{"category": "Math", "title": "A Mathematical Proof of Dodgson's Algorithm", "abstract": "In this paper we give a mathematical proof of Dodgson algorithm [1]. Recently Zeilberger [2] gave a bijective proof. Our techniques are based on determinant properties and they are obtained by induction."}
{"category": "Math", "title": "Coincidences in 4 dimensions", "abstract": "The coincidence site lattices (CSLs) of prominent 4-dimensional lattices are considered. CSLs in 3 dimensions have been used for decades to describe grain boundaries in crystals. Quasicrystals suggest to also look at CSLs in dimensions $d>3$. Here, we discuss the CSLs of the root lattice $A_4$ and the hypercubic lattices, which are of particular interest both from the mathematical and the crystallographic viewpoint. Quaternion algebras are used to derive their coincidence rotations and the CSLs. We make use of the fact that the CSLs can be linked to certain ideals and compute their indices, their multiplicities and encapsulate all this in generating functions in terms of Dirichlet series. In addition, we sketch how these results can be generalised for 4--dimensional $\\Z$--modules by discussing the icosian ring."}
{"category": "Math", "title": "Functions with support in a lacunary system of intervals and cyclicity for the semi-group of left translations", "abstract": "We prove in the vector-valued spaces $L^2(\\mathbb{R}_+, X)$ (where $X$ is a finite dimensional Hilbert space) the cyclicity for the semi-group of left translations of some particular functions with support included in a lacunary system of intervals."}
{"category": "Math", "title": "Global regular solutions for the Navier-stokes system with small initial data in $\\Phi(2)$: an elementary approach", "abstract": "We show existence and regularity result for the Navier Stokes system for small data in the space $\\Phi(2)$, and we show relations with some classical results."}
{"category": "Math", "title": "The $(L^{p},L^{1})$ bilinear Hardy-Littlewood function and Furstenberg averages", "abstract": "More work needs to be done to move from the tail to the averages themselves. So at this time we prefer to withdraw the paper about the averages. However a previous version of the paper which deals with the tail has been checked and we believe it to be complete and correct."}
{"category": "Math", "title": "Rips Complexes of Planar Point Sets", "abstract": "Fix a finite set of points in Euclidean $n$-space $\\euc^n$, thought of as a point-cloud sampling of a certain domain $D\\subset\\euc^n$. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of $D$. There is a natural ``shadow'' projection map from the Rips complex to $\\euc^n$ that has as its image a more accurate $n$-dimensional approximation to the homotopy type of $D$. We demonstrate that this projection map is 1-connected for the planar case $n=2$. That is, for planar domains, the Rips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a Rips complex for a planar point set is a free group. We show that, in contrast, introducing even a small amount of uncertainty in proximity detection leads to `quasi'-Rips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasi-Rips complexes and using ideas from persistent topology. Finally, we show that the projection map does not preserve higher-order topological data for planar sets, nor does it preserve fundamental group data for point sets in dimension larger than three."}
{"category": "Math", "title": "Hopf algebras of dimension 16", "abstract": "We complete the classification of Hopf algebras of dimension 16 over an algebraically closed field of characteristic zero. We show that a non-semisimple Hopf algebra of dimension 16, has either the Chevalley property or its dual is pointed."}
{"category": "Math", "title": "Inverse Problems for Representation Functions in Additive Number Theory", "abstract": "For every positive integer h, the representation function of order h associated to a subset A of the integers or, more generally, of any group or semigroup X, counts the number of ways an element of X can be written as the sum (or product, if X is nonabelian) of h not necessarily distinct elements of X. The direct problem for representation functions in additive number theory begins with a subset A of X and seeks to understand its representation functions. The inverse problem for representation functions starts with a function f:X ->N_0 U {\\infty} and asks if there is a set A whose representation function is f, and, if the answer is yes, to classify all such sets. This paper is a survey of recent progress on the inverse representation problem."}
{"category": "Math", "title": "Structure theorems for embedded disks with mean curvature bounded in L^P", "abstract": "After appropriate normalizations an embedded disk whose second fundamental form has large norm contains a multi-valued graph, provided the L^P norm of the mean curvature is sufficiently small. This generalizes to non-minimal surfaces a well known result of Colding and Minicozzi."}
{"category": "Math", "title": "About the logarithm function over the matrices", "abstract": "We prove the following results: let x,y be (n,n) complex matrices such that x,y,xy have no eigenvalue in ]-infinity,0] and log(xy)=log(x)+log(y). If n=2, or if n>2 and x,y are simultaneously triangularizable, then x,y commute. In both cases we reduce the problem to a result in complex analysis."}
{"category": "Math", "title": "Sequential Tracking of a Hidden Markov Chain Using Point Process Observations", "abstract": "We study finite horizon optimal switching problems for hidden Markov chain models under partially observable Poisson processes. The controller possesses a finite range of strategies and attempts to track the state of the unobserved state variable using Bayesian updates over the discrete observations. Such a model has applications in economic policy making, staffing under variable demand levels and generalized Poisson disorder problems. We show regularity of the value function and explicitly characterize an optimal strategy. We also provide an efficient numerical scheme and illustrate our results with several computational examples."}
{"category": "Math", "title": "Closed categories, star-autonomy, and monoidal comonads", "abstract": "This paper determines what structure is needed for internal homs in a monoidal category C to be liftable to the category C^G of Eilenberg-Moore coalgebras for a monoidal comonad G on C. We apply this to lift star-autonomy with the view to recasting the definition of quantum groupoid."}
{"category": "Math", "title": "The number of hypergraphs and colored Hypergraphs with hereditary properties", "abstract": "As an application of Szemeredi's regularity lemma, Erdos-Frankl-Rodl (1986) showed that the number of graphs on vertex set {1,2,...n} with a monotone class P is $2^{(1+o(1))ex(n,P)n^2/2}$ where $ex(n,P)$ is the maximum number of edges of an n-vertex graph which has no subgraph in P. Kohayakawa et al. (2003) extended it from monotone to hereditary and from graphs to 3-uniform hypergraphs. We extend it to general hypergraphs. This may be a simple example illustrating how to apply a recent hypergraph regularity lemma by the author."}
{"category": "Math", "title": "Solvable automorphism groups of a compact Kaehler manifold", "abstract": "Let X be a compact Kaehler manifold of complex dimension n. Let G be a connected solvable subgroup of the automorphism group Aut(X), and let N(G) be the normal subgroup of G of elements of null entropy. One of the goals of this paper is to show that G/N(G) is a free abelian group of rank r(G) less than or equal to n-1 and that the rank estimate is optimal. This gives an alternative proof of the conjecture of Tits type. In addition, we show some non-obvious implications on the structure of solvable automorphism groups of compact Kaehler manifolds. Furthermore, we also show that if the rank r(G) of the quotient group G/N(G) is equal to n-1 and the identity component Aut_0(X) of Aut(X) is trivial, then N(G) is a finite set. The main strategy of this paper is to combine the method of Dinh and Sibony and the theorem of Birkhoff-Perron-Frobenius (or Lie-Kolchin type), and one argument of D.-Q. Zhang originated from the paper of Dinh and Sibony plays an important role."}
{"category": "Math", "title": "Projective normality of nonsingular toric varieties of dimension three", "abstract": "We show that if an ample line bundle L on a nonsingular toric 3-fold satisfies h^0(L+2K)=0, then L is normally generated. As an application, we show that the anti-canonical divisor on a nonsingular toric Fano 4-fold is normally generated."}
{"category": "Math", "title": "Large Deviations for Heavy-Tailed Factor Models", "abstract": "We study large deviation probabilities for a sum of dependent random variables from a heavy-tailed factor model, assuming that the components are regularly varying. We identify conditions where both the factor and the idiosyncratic terms contribute to the behaviour of the tail-probability of the sum. A simple conditional Monte Carlo algorithm is also provided together with a comparison between the simulations and the large deviation approximation. We also study large deviation probabilities for stochastic processes with factor structure. The processes involved are assumed to be Levy processes with regularly varying jump measures. Based on the results of the first part of the paper, we show that large deviations on a finite time interval are due to one large jump that can come from either the factor or the idiosyncratic part of the process."}
{"category": "Math", "title": "Multiple equilibria of nonhomogeneous Markov chains and self-validating web rankings", "abstract": "PageRank is a ranking of the web pages that measures how often a given web page is visited by a random surfer on the web graph, for a simple model of web surfing. It seems realistic that PageRank may also have an influence on the behavior of web surfers. We propose here a simple model taking into account the mutual influence between web ranking and web surfing. Our ranking, the T-PageRank, is a nonlinear generalization of the PageRank. It is defined as the limit, if it exists, of some nonlinear iterates. A positive parameter T, the temperature, measures the confidence of the web surfer in the web ranking. We prove that, when the temperature is large enough, the T-PageRank is unique and the iterates converge globally on the domain. But when the temperature is small, there may be several T-PageRanks, that may strongly depend on the initial ranking. Our analysis uses results of nonlinear Perron-Frobenius theory, Hilbert projective metric and Birkhoff's coefficient of ergodicity."}
{"category": "Math", "title": "An Exact Exprsssion of Pi(x)", "abstract": "The author states an exact expression of the distribution of primes."}
{"category": "Math", "title": "Integral means and boundary limits of Dirichlet series", "abstract": "We study the boundary behavior of functions in the Hardy spaces HD^p for ordinary Dirichlet series. Our main result, answering a question of H. Hedenmalm, shows that the classical F. Carlson theorem on integral means does not extend to the imaginary axis for functions in HD^\\infty, i.e., for ordinary Dirichlet series in H^\\infty of the right half-plane. We discuss an important embedding problem for HD^p, the solution of which is only known when p is an even integer. Viewing HD^p as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatou's theorem."}
{"category": "Math", "title": "Sharp Spectral Asymptotics for Dirac Energy. II. Magnetic Schroedinger operator", "abstract": "I derive sharp semiclassical asymptotics of \\int |e_h(x,y,0)|^2\\omega (x,y)dxdy where e_h(x,y,\\tau) is the Schwartz kernel of the spectral projector of Magnetic Schroedinger operator and \\omega (x,y) is singular as x=y. I also consider asymptotics of more general expressions."}
{"category": "Math", "title": "Rational Extensions of C(X) via Hausdorff Continuous Functions", "abstract": "The ring operations and the metric on $C(X)$ are extended to the set $\\mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $\\mathbb{H}_{nf}(X)$ is both rationally and topologically complete. Hence, the rings of quotients of $C(X)$ as well as their metric completions are represented as rings of Hausdorff continuous functions."}
{"category": "Math", "title": "Newhouse phenomenon and homoclinic classes", "abstract": "We show that for a $C^1$ residual subset of diffeomorphisms far away from tangency, every non-trivial chain recurrent class that is accumulated by sources ia a homoclinic class contains periodic points with index 1 and it's the Hausdorff limit of a family of sources."}
{"category": "Math", "title": "Lyapunov stable chain recurrent classes", "abstract": "We show that for a $C^1$ residual subset of diffeomorphisms far away from homoclinic tangency, the stable manifolds of periodic points cover a dense subset of the ambient manifold. This gives a partial proof to a conjecture of C. Bonatti."}
{"category": "Math", "title": "From Quantum Universal Enveloping Algebras to Quantum Algebras", "abstract": "The ``local'' structure of a quantum group G_q is currently considered to be an infinite-dimensional object: the corresponding quantum universal enveloping algebra U_q(g), which is a Hopf algebra deformation of the universal enveloping algebra of a n-dimensional Lie algebra g=Lie(G). However, we show how, by starting from the generators of the underlying Lie bialgebra (g,\\delta), the analyticity in the deformation parameter(s) allows us to determine in a unique way a set of n ``almost primitive'' basic objects in U_q(g), that could be properly called the ``quantum algebra generators''. So, the analytical prolongation (g_q,\\Delta) of the Lie bialgebra (g,\\delta) is proposed as the appropriate local structure of G_q. Besides, as in this way (g,\\delta) and U_q(g) are shown to be in one-to-one correspondence, the classification of quantum groups is reduced to the classification of Lie bialgebras. The su_q(2) and su_q(3) cases are explicitly elaborated."}
{"category": "Math", "title": "Bernoulli coding map and almost sure invariance principle for endomorphisms of $\\mathbb{P}^k$", "abstract": "Let $f$ be an holomorphic endomorphism of $\\mathbb{P}^k$ and $\\mu$ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems $(\\mathbb{P}^k,f,\\mu)$. Our class $\\cal{U}$ of observables includes the H\\\"older functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map $\\omega: (\\Sigma, s, \\nu) \\to (\\mathbb{P}^k,f,\\mu)$. We obtain the invariance principle for an observable $\\psi$ on $(\\mathbb{P}^k,f,\\mu)$ by applying Philipp-Stout's theorem for $\\chi = \\psi \\circ \\omega$ on $(\\Sigma, s, \\nu)$. The invariance principle implies the Central Limit Theorem as well as several statistical properties for the class $\\cal{U}$. As an application, we give a \\emph{direct} proof of the absolute continuity of the measure $\\mu$ when it satisfies Pesin's formula. This approach relies on the Central Limit Theorem for the unbounded observable $\\log \\textsf{Jac} f \\in \\cal{U}$."}
{"category": "Math", "title": "K-spectral sets and intersections of disks of the Riemann sphere", "abstract": "We prove that if two closed disks X_1 and X_2 of the Riemann sphere are spectral sets for a bounded linear operator A on a Hilbert space, then the intersection X_1\\cap X_2 is a complete (2+2/\\sqrt{3})-spectral set for A. When the intersection of X_1 and X_2 is an annulus, this result gives a positive answer to a question of A.L. Shields (1974)."}
{"category": "Math", "title": "Comparing Gr\\\"obner bases and word reversing", "abstract": "Gr\\\"obner bases, in their noncommutative version, and word reversing are methods for solving the word problem of a presented monoid, and both rely on iteratively completing the initial list of relations. Simple examples may suggest to conjecture that both completion procedures are closely related. Here we disprove this conjecture by exhibiting families of presentations for which they radically differ."}
{"category": "Math", "title": "Large deviations for return times in non-rectangle sets for axiom A diffeomorphisms", "abstract": "For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition."}
{"category": "Math", "title": "${\\cal T}$-class algorithms for pseudocontractions and $\\kappa$-strict pseudocontractions in Hilbert spaces", "abstract": "In this paper we study iterative algorithms for finding a common element of the set of fixed points of $\\kappa$-strict pseudocontractions or finding a solution of a variational inequality problem for a monotone, Lipschitz continuous mapping. The last problem being related to finding fixed points of pseudocontractions. These algorithms were already studied in [G.L. Acedo, H.-K. Xu] and [N. Nadezhkina, W. Takahashi] but our aim here is to provide the links between these know algorithms and the general framework of ${\\cal T}$-class algorithms studied in [H.H. Bauschke, P.L. Combettes]."}
{"category": "Math", "title": "A simplified multidimensional integral", "abstract": "We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained more quickly. We also give a characterization of the integrable functions and their primitives."}
{"category": "Math", "title": "Fragmenting random permutations", "abstract": "Problem 1.5.7 from Pitman's Saint-Flour lecture notes: Does there exist for each n a fragmentation process (\\Pi_{n,k}, 1 \\leq k \\leq n) taking values in the space of partitions of {1,2,...,n} such that \\Pi_{n,k} is distributed like the partition generated by cycles of a uniform random permutation of {1,2,...,n} conditioned to have k cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions."}
{"category": "Math", "title": "Quivers, Geometric Invariant Theory, and Moduli of Linear Dynamical Systems", "abstract": "We use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems. A general approach to this problem is presented and applied to two well known cases: We show how both Lomadze's and Helmke's compactification arises naturally as a geometric invariant theory quotient. Both moduli spaces are proven to be smooth projective manifolds. Furthermore, a description of Lomadze's compactification as a Quot scheme is given, whereas Helmke's compactification is shown to be an algebraic Grassmann bundle over a Quot scheme. This gives an algebro-geometric description of both compactifications. As an application, we determine the cohomology ring of Helmke's compactification and prove that the two compactifications are not isomorphic when the number of outputs is positive."}
{"category": "Math", "title": "Differential Equations in Metric Spaces with Applications", "abstract": "This paper proves the local well posedness of differential equations in metric spaces under assumptions that allow to comprise several different applications. We consider below a system of balance laws with a dissipative non local source, the Hille-Yosida Theorem, a generalization of a recent result on nonlinear operator splitting, an extension of Trotter formula for linear semigroups and the heat equation."}
{"category": "Math", "title": "Commensurability and QI classification of free products of finitely generated abelian groups", "abstract": "Suppose a group $G$ is quasi-isometric to a free product of a finite set $S$ of finitely generated abelian groups; let $S'$ denote the set of ranks of the free abelian parts of the groups in $S$. Then $G$ is commensurable with the free product of $\\Z$ with a $\\Z^n$ for each $n$ occurring in $S'$."}
{"category": "Math", "title": "Large deviations for local time fractional Brownian motion and applications", "abstract": "Let $W^H=\\{W^H(t), t \\in \\rr\\}$ be a fractional Brownian motion of Hurst index $H \\in (0, 1)$ with values in $\\rr$, and let $L = \\{L_t, t \\ge 0\\}$ be the local time process at zero of a strictly stable L\\'evy process $X=\\{X_t, t \\ge 0\\}$ of index $1<\\alpha\\leq 2$ independent of $W^H$. The $\\a$-stable local time fractional Brownian motion $Z^H=\\{Z^H(t), t \\ge 0\\}$ is defined by $Z^H(t) = W^H(L_t)$. The process $Z^H$ is self-similar with self-similarity index $H(1 - \\frac 1 \\alpha)$ and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps (\\cite{coupleCTRW,limitCTRW}). However, $Z^H$ does not have stationary increments and is non-Gaussian. In this paper we establish large deviation results for the process $Z^H$. As applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for $Z^H$."}
{"category": "Math", "title": "Inverse problems for regular variation of linear filters, a cancellation property for $\\sigma$-finite measures and identification of stable laws", "abstract": "In this paper, we consider certain $\\sigma$-finite measures which can be interpreted as the output of a linear filter. We assume that these measures have regularly varying tails and study whether the input to the linear filter must have regularly varying tails as well. This turns out to be related to the presence of a particular cancellation property in $\\sigma$-finite measures, which in turn, is related to the uniqueness of the solution of certain functional equations. The techniques we develop are applied to weighted sums of i.i.d. random variables, to products of independent random variables, and to stochastic integrals with respect to L\\'evy motions."}
{"category": "Math", "title": "Stochastic FitzHugh-Nagumo equations on networks with impulsive noise", "abstract": "We consider a system of nonlinear partial differential equations with stochastic dynamical boundary conditions that arises in models of neurophysiology for the diffusion of electrical potentials through a finite network of neurons. Motivated by the discussion in the biological literature, we impose a general diffusion equation on each edge through a generalized version of the FitzHugh-Nagumo model, while the noise acting on the boundary is described by a generalized stochastic Kirchhoff law on the nodes. In the abstract framework of matrix operators theory, we rewrite this stochastic boundary value problem as a stochastic evolution equation in infinite dimensions with a power-type nonlinearity, driven by an additive L\\'evy noise. We prove global well-posedness in the mild sense for such stochastic partial differential equation by monotonicity methods."}
{"category": "Math", "title": "Enhanced delay to Bifurcation", "abstract": "This article provides an example of fast-slow system such that most orbits remain as close as possible to the unstable manifold of the fast dynamics for an arbitrarily long time."}
{"category": "Math", "title": "Cardinal sequences of LCS spaces under GCH", "abstract": "We give full characterization of the sequences of regular cardinals that may arise as cardinal sequences of locally compact scattered spaces under GCH. The proofs are based on constructions of universal locally compact scattered spaces."}
{"category": "Math", "title": "Non group-theoretical semisimple Hopf algebras from group actions on fusion categories", "abstract": "Given an action of a finite group G on a fusion category C we give a criterion for the category of G-equivariant objects in C to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in math/0203060. Namely, we show that certain Z/2Z-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami are equivalent to representation categories of non group-theoretical semisimple Hopf algebras. We describe these Hopf algebras as extensions and show that they are upper and lower semisolvable."}
{"category": "Math", "title": "Maximal small extensions of o-minimal structures", "abstract": "A proper elementary extension of a model is called small if it realizes no new types over any finite set in the base model. We answer a question of Marker, and show that it is possible to have an o-minimal structure with a maximal small extension. Our construction yields such a structure for any cardinality. We show that in some cases, notably when the base structure is countable, the maximal small extension has maximal possible cardinality."}
{"category": "Math", "title": "Some families of increasing planar maps", "abstract": "Stack-triangulations appear as natural objects when one wants to define some increasing families of triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $2n$ faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by $n^{1/2}$, they converge for the Gromov-Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by $(6/11)\\log n$ converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations."}
{"category": "Math", "title": "Irreducible Representations of Groupoid $C^*$-algebras", "abstract": "If $G$ is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible."}
{"category": "Math", "title": "A folk model structure on omega-cat", "abstract": "We establish a model structure on the category of strict omega-categories. The constructions leading to the model structure in question are expressed entirely within the scope of omega-categories, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while cofibrant objects are exactly the free ones. Our model structure transfers to n-categories along right-adjoints, for each n, thus recovering the known cases n = 1 and n = 2."}
{"category": "Math", "title": "Filtrations", "abstract": "In this article, we define the notion of a filtration and then give the basic theorems on initial and progressive enlargements of filtrations."}
{"category": "Math", "title": "Modular unit and cuspidal divisor class groups of X_1(N)", "abstract": "In this article, we consider the group $F_1^\\infty(N)$ of modular units on $X_1(N)$ that have divisors supported on the cusps lying over $\\infty$ of $X_0(N)$, called the $\\infty$-cusps. For each positive integer $N$, we will give an explicit basis for the group $F_1^\\infty(N)$. This enables us to compute the group structure of the rational torsion subgroup $C_1^\\infty(N)$ of the Jacobian $J_1(N)$ of $X_1(N)$ generated by the differences of the $\\infty$-cusps. In addition, based on our numerical computation, we make a conjecture on the structure of the $p$-primary part of $C_1^\\infty(p^n)$ for a regular prime $p$."}
{"category": "Math", "title": "Overpartitions and class numbers of binary quadratic forms", "abstract": "We show that the Zagier-Eisenstein series shares its non-holomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact formulas, asymptotics, and congruences for the rank differences as well as $q$-series identities of the mock theta type."}
{"category": "Math", "title": "Topologically unique maximal elementary Abelian group actions on compact oriented surfaces", "abstract": "We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus $\\sigma\\geq 2$ which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations."}
{"category": "Math", "title": "Controlled Synchronization of One Class of Nonlinear Systems under Information Constraints", "abstract": "Output feedback controlled synchronization problems for a class of nonlinear unstable systems under information constraints imposed by limited capacity of the communication channel are analyzed. A binary time-varying coder-decoder scheme is described and a theoretical analysis for multi-dimensional master-slave systems represented in Lurie form (linear part plus nonlinearity depending only on measurable outputs) is provided. An output feedback control law is proposed based on the Passification Theorem. It is shown that the synchronization error exponentially tends to zero for sufficiantly high transmission rate (channel capacity). The results obtained for synchronization problem can be extended to tracking problems in a straightforward manner, if the reference signal is described by an {external} ({exogenious}) state space model. The results are applied to controlled synchronization of two chaotic Chua systems via a communication channel with limited capacity."}
{"category": "Math", "title": "The Natural Philosophy of Kazuo Kondo", "abstract": "Kazuo Kondo (1911-2001) was Chair of the Department of Mathematical Engineering at the University of Tokyo, Japan. Over a period of 50 years, he and a few colleagues wrote and published a voluminous series of papers and monographs on the applications of analytical geometry within a diverse range of subjects in the natural sciences. Inspired by Otto Fischer's attempt at a quaternionic unified theory in the late 1950's he adopted the mathematics of the revered Akitsugu Kawaguchi to produce his own speculative unified theory. The theory appears to successfully apply Kawaguchi's mathematics to the full range of natural phenomena, from the structure of fundamental particles to the geometry of living beings. The theories are testable and falsifiable. Kondo and his theories are now almost completely unknown and this paper serves as the barest introduction to his work"}
{"category": "Math", "title": "Free curves and periodic points for torus homeomorphisms", "abstract": "We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically nontrivial simple closed curve that is disjoint from its image. We prove that every rational point in the rotation set is realized by a periodic point provided that there is no free curve and the rotation set has empty interior. This gives a topological version of a theorem of Franks. Using this result, and inspired by a theorem of Guillou, we prove a version of the Poincar\\'e-Birkhoff Theorem for torus homeomorphisms: in the absence of free curves, either there is a fixed point or the rotation set has nonempty interior."}
{"category": "Math", "title": "A practical illustration of the importance of realistic individualized treatment rules in causal inference", "abstract": "The effect of vigorous physical activity on mortality in the elderly is difficult to estimate using conventional approaches to causal inference that define this effect by comparing the mortality risks corresponding to hypothetical scenarios in which all subjects in the target population engage in a given level of vigorous physical activity. A causal effect defined on the basis of such a static treatment intervention can only be identified from observed data if all subjects in the target population have a positive probability of selecting each of the candidate treatment options, an assumption that is highly unrealistic in this case since subjects with serious health problems will not be able to engage in higher levels of vigorous physical activity. This problem can be addressed by focusing instead on causal effects that are defined on the basis of realistic individualized treatment rules and intention-to-treat rules that explicitly take into account the set of treatment options that are available to each subject. We present a data analysis to illustrate that estimators of static causal effects in fact tend to overestimate the beneficial impact of high levels of vigorous physical activity while corresponding estimators based on realistic individualized treatment rules and intention-to-treat rules can yield unbiased estimates. We emphasize that the problems encountered in estimating static causal effects are not restricted to the IPTW estimator, but are also observed with the $G$-computation estimator, the DR-IPTW estimator, and the targeted MLE. Our analyses based on realistic individualized treatment rules and intention-to-treat rules suggest that high levels of vigorous physical activity may confer reductions in mortality risk on the order of 15-30%, although in most cases the evidence for such an effect does not quite reach the 0.05 level of significance."}
{"category": "Math", "title": "Least Area Planes in Hyperbolic 3-Space are Properly Embedded", "abstract": "We show that if P is an embedded least area (area minimizing) plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded."}
{"category": "Math", "title": "Quasi-kernels and quasi-sinks in infinite graphs", "abstract": "Given a directed graph G=(V,E) an independent set A of the vertices V is called quasi-kernel (quasi-sink) iff for each point v there is a path of length at most 2 from some point of A to v (from v to some point of A). Every finite directed graph has a quasi-kernel. The plain generalization for infinite graphs fails, even for tournaments. We investigate the following conjecture here: for any digraph G=(V,E) there is a a partition (V_0,V_1) of the vertex set such that the induced subgraph G[V_0] has a quasi-kernel and the induced subgraph G[V_1] has a quasi-sink."}
{"category": "Math", "title": "Modules with Finite Cousin Cohomologies Have Uniform Local Cohomological Annihilators", "abstract": "Let A be a Noetherian ring. It is shown that any finite A--module M of finite Krull dimension with finite Cousin complex cohomologies has a uniform local cohomological annihilator. The converse is also true for a finite module M satisfying (S_2) which is over a local ring with Cohen--Macaulay formal fibres."}
{"category": "Math", "title": "An Improved Error Bound for Multiquadric and Inverse Multiquadric Interpolation", "abstract": "A new error bound which is better than the current exponential-type error bound is presented in this paper."}
{"category": "Math", "title": "Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents", "abstract": "For an ergodic measure preserving action on a probability space, consider the corresponding crossed product von Neumann algebra. We calculate the Fuglede-Kadison determinant for a class of operators in this von Neumann algebra in terms of the Ljapunov exponents of an associated measurable cocycle. The proof is based on recent work of Dykema and Schultz. As an application one obtains formulas for the Fuglede-Kadison determinant of noncommutative polynomials in the von Neumann algebra of the discrete Heisenberg group. These had been previously obtained by Lind and Schmidt via entropy considerations."}
{"category": "Math", "title": "Integral Lattices of the SU(2)-TQFT-Modules", "abstract": "We find bases for naturally defined lattices over certain rings of integers in the SU(2)-TQFT-theory modules of surfaces. We consider the TQFT where the Kauffman's A variable is a root of unity of order four times an odd prime. As an application, we show that the Frohman Kania-Bartoszynska ideal invariant for 3-manifolds with boundary using the SU(2)-TQFT-theory is equal to the product of the ideals using the 2^{'}-theory and the SO(3)-TQFT-theory under a certain change of coefficients."}
{"category": "Math", "title": "Asymptotic normality of the Quasi Maximum Likelihood Estimator for multidimensional causal processes", "abstract": "Strong consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator (QMLE) are given for a general class of multidimensional causal processes. For particular cases already studied in the literature (for instance univariate or multivariate GARCH, ARCH, ARMA-GARCH processes) the assumptions required for establishing these results are often weaker than existing conditions. The QMLE asymptotic behavior is also given for numerous new examples of univariate or multivariate processes (for instance TARCH or NLARCH processes)."}
{"category": "Math", "title": "Compact embeddings of model subspaces of the Hardy space", "abstract": "We study embeddings of model (star-invariant) subspaces $K^p_{\\Theta}$ of the Hardy space $H^p$, associated with an inner function $\\Theta$. We obtain a criterion for the compactness of the embedding of $K^p_{\\Theta}$ into $L^p(\\mu)$ analogous to the Volberg--Treil theorem on bounded embeddings and answer a question posed by Cima and Matheson. The proof is based on Bernstein inequalities for functions in $K^p_{\\Theta}$. Also we study measures $\\mu$ such that the embedding operator belongs to a Schatten--von Neumann ideal."}
{"category": "Math", "title": "Ergodic Theory, Abelian Groups, and Point Processes Induced by Stable Random Fields", "abstract": "We consider a point process sequence induced by a stationary symmetric alpha-stable (0 < alpha < 2) discrete parameter random field. It is easy to prove, following the arguments in the one-dimensional case in Resnick and Samorodnitsky (2004), that if the random field is generated by a dissipative group action then the point process sequence converges weakly to a cluster Poisson process. For the conservative case, no general result is known even in the one-dimensional case. We look at a specific class of stable random fields generated by conservative actions whose effective dimensions can be computed using the structure theorem of finitely generated abelian groups. The corresponding point processes sequence is not tight and hence needs to be properly normalized in order to ensure weak convergence. This weak limit is computed using extreme value theory and some counting techniques."}
{"category": "Math", "title": "Bounds for Behrend's conjecture on the canonical reduction", "abstract": "We prove Behrend's conjecture on the rationality of the canonical reduction of principal bundles and reductive group schemes for classical groups and give new bounds for the conjecture for exceptional groups. However we find a counterexample in the case of G_2-bundles in characteristic 2."}
{"category": "Math", "title": "Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures", "abstract": "Given a positive function $F$ on $S^n$ which satisfies a convexity condition, for $1\\leq r\\leq n$, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. We prove that a compact embedded hypersurface without boundary in $\\R^{n+1}$ with $H^F_r={constant}$ is the Wulff shape, up to translations and homotheties. In case $r=1$, our result is the anisotropic version of Alexandrov Theorem, which gives an affirmative answer to an open problem of F. Morgan."}
{"category": "Math", "title": "Pontrjagin-Thom maps and the homology of the moduli stack of stable curves", "abstract": "We study the singular homology (with field coefficients) of the moduli stack of stable n-pointed complex curves of genus g (the Deligne-Mumford compactification). Each of its irreducible boundary components determines via the Pontrjagin-Thom construction a map to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This implies the existence of many new torsion classes in the homology of the moduli stack."}
{"category": "Math", "title": "Orbit projections of proper Lie groupoids as fibrations", "abstract": "Let $\\mathcal{G} \\rightrightarrows M$ be a source locally trivial proper Lie groupoid such that each orbit is of finite type. The orbit projection $M \\to M/\\mathcal{G}$ is a fibration if and only if $\\mathcal{G} \\rightrightarrows M$ is regular."}
{"category": "Math", "title": "System reliability and weighted lattice polynomials", "abstract": "The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases. The central object of interest is the cumulative distribution function of Y. In particular, numerous results are obtained for lattice polynomials and weighted lattice polynomials in case of independent arguments and in general. For the general case, the technique consists in considering the joint probability generating function of \"indicator\" variables. A connection is studied between Y and order statistics of the set of arguments."}
{"category": "Math", "title": "Transfer Principle for the Fundamental Lemma", "abstract": "The purpose of this paper is to explain how the identities of various fundamental lemmas fall within the scope of the transfer principle, a general result that allows to transfer theorems about identities of p-adic integrals from one collection of fields to others. In particular, once the fundamental lemma has been established for one collection of fields (for example, fields of positive characteristic), it is also valid for others (fields of characteristic zero)."}
{"category": "Math", "title": "Asymptotically optimal multistage tests of simple hypotheses", "abstract": "A family of variable stage size multistage tests of simple hypotheses is described, based on efficient multistage sampling procedures. Using a loss function that is a linear combination of sampling costs and error probabilities, these tests are shown to minimize the integrated risk to second order as the costs per stage and per observation approach zero. A numerical study shows significant improvement over group sequential tests in a binomial testing problem."}
{"category": "Math", "title": "Dominance of a Rational Map to the Coble Quartic", "abstract": "We show the dominance of the restriction map from a moduli space of stable sheaves on the projective plane to the Coble quartic. With the dominance and the interpretation of a stable sheaf on the plane in terms of the hyperplane arrangements, we expect to investigate the geometry of the Coble quartic."}
{"category": "Math", "title": "Understanding the small object argument", "abstract": "The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an \"algebraic\" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies."}
{"category": "Math", "title": "Asymptotics for first-passage times of L\\'evy processes and random walks", "abstract": "We study the exact asymptotics for the distribution of the first time $\\tau_x$ a L\\'evy process $X_t$ crosses a negative level $-x$. We prove that $\\mathbf P(\\tau_x>t)\\sim V(x)\\mathbf P(X_t\\ge 0)/t$ as $t\\to\\infty$ for a certain function $V(x)$. Using known results for the large deviations of random walks we obtain asymptotics for $\\mathbf P(\\tau_x>t)$ explicitly in both light and heavy tailed cases. We also apply our results to find asymptotics for the distribution of the busy period in an M/G/1 queue."}
{"category": "Math", "title": "The Neumann problem for singular fully nonlinear operators", "abstract": "We consider the Neumann problem in $C^2$ bounded domains for fully nonlinear second order operators which are elliptic, homogenous with lower order terms. Inspired by \\cite{bnv}, we define the concept of principal eigenvalue and we characterize it through the maximum principle. Moreover, Lipschitz regularity, uniqueness and existence results for solutions of the Neumann problem are given."}
{"category": "Math", "title": "An absorption theorem for minimal AF equivalence relations on Cantor sets", "abstract": "We prove that a `small' extension of a minimal AF equivalence relation on a Cantor set is orbit equivalent to the AF relation. By a `small' extension we mean an equivalence relation generated by the minimal AF equivalence relation and another AF equivalence relation which is defined on a closed thin subset. The result we obtain is a generalization of the main theorem in [GMPS2]."}
{"category": "Math", "title": "Asymptotic analysis of vibrating system containing stiff-heavy and flexible-light parts", "abstract": "A model of strongly inhomogeneous medium with simultaneous perturbation of rigidity and mass density is studied. The medium has strongly contrasting physical characteristics in two parts with the ratio of rigidities being proportional to a small parameter $\\epsilon$. Additionally, the ratio of mass densities is of order $\\epsilon^{-1}$. We investigate the asymptotic behaviour of spectrum and eigensubspaces as $\\epsilon\\to 0$. Complete asymptotic expansions of eigenvalues and eigenfunctions are constructed and justified. We show that the limit operator is nonself-adjoint in general and possesses two-dimensional Jordan cells in spite of the singular perturbed problem is associated with a self-adjoint operator in appropriated Hilbert space $L_\\epsilon$. This may happen if the metric in which the problem is self-adjoint depends on small parameter $\\epsilon$ in a singular way. In particular, it leads to a loss of completeness for the eigenfunction collection. We describe how root spaces of the limit operator approximate eigenspaces of the perturbed operator."}
{"category": "Math", "title": "About the stability of the tangent bundle restricted to a curve", "abstract": "Let C be a smooth projective curve with genus g>1 and Clifford index c(C) and let L be a line bundle on C generated by its global sections. The morphism i:C -->P(H^0(L))=P is well-defined and i*T is the restriction to C of the tangent bundle T of the projective space P. Sharpening a theorem by Paranjape, we show that if deg L>2g-c(C)-1 then i*T is semi-stable, specifying when it is also stable. We then prove the existence on many curves of a line bundle L of degree 2g-c(C)-1 such that i*T is not semi-stable. Finally, we completely characterize the (semi-)stability of i*T when C is hyperelliptic."}
{"category": "Math", "title": "Some nonasymptotic results on resampling in high dimension, I: Confidence regions, II: Multiple tests", "abstract": "We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality of the vector can possibly be much larger than the number of observations and we focus on a nonasymptotic control of the confidence level, following ideas inspired by recent results in learning theory. We consider two approaches, the first based on a concentration principle (valid for a large class of resampling weights) and the second on a resampled quantile, specifically using Rademacher weights. Several intermediate results established in the approach based on concentration principles are of interest in their own right. We also discuss the question of accuracy when using Monte Carlo approximations of the resampled quantities."}
{"category": "Math", "title": "Linear Complete Differential Resultants and the Implicitization of Linear DPPEs", "abstract": "The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of $n$ linear differential polynomial parametric equations in $n-1$ differential parameters. We give necessary conditions to ensure properness of the system of differential polynomial parametric equations."}
{"category": "Math", "title": "Hausdorff hyperspaces of $R^m$ and their dense subspaces", "abstract": "Let $CLB_H(X)$ denote the hyperspace of closed bounded subsets of a metric space $X$, endowed with the Hausdorff metric topology. We prove, among others, that natural dense subspaces of $CLB_H(R^m)$ of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space $\\ell_2$. Moreover, we investigate the hyperspace $CL_H(R)$ of all nonempty closed subsets of the real line $R$ with the Hausdorff (infinite-valued) metric. We show that a nonseparable component of $CL_H(R)$ is homeomorphic to the Hilbert space $\\ell_2(2^{\\aleph_0})$ as long as it does not contain any of the sets $R, [0,\\infty), (-\\infty,0]$."}
{"category": "Math", "title": "D-modules on the affine flag variety and representations of affine Kac-Moody algebras", "abstract": "We study the connection between the category of modules over the affine Kac-Moody Lie algebra at the critical level, and the category of D-modules on the affine flag scheme G((t))/I, where I is the Iwahori subgroup. We prove a localization-type result, which establishes an equivalence between certain subcategories on both sides. We also establish an equivalence between a certain subcategory of Kac-Moody modules, and the category of quasi-coherent sheaves on the scheme of Miura opers for the Langlands dual group, thereby proving a conjecture of [FG2]."}
{"category": "Math", "title": "Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability", "abstract": "We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For beta < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1-beta)]^{-1} n log n. For beta = 1, we prove that the mixing time is of order n^{3/2}. For beta > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n)."}
{"category": "Math", "title": "Tempered solutions of $\\mathcal D$-modules on complex curves and formal invariants", "abstract": "Let $X$ be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of $\\mathcal D_X$-modules induces a fully faithful functor on a subcategory of germs of formal holonomic $\\mathcal D_X$-modules. Further, given a germ $\\mathcal M$ of holonomic $\\mathcal D_X$-module, we obtain some results linking the subanalytic sheaf of tempered solutions of $\\mathcal M$ and the classical formal and analytic invariants of $\\mathcal M$."}
{"category": "Math", "title": "Conformal deformation of spacelike surfaces in Minkowski space", "abstract": "We address the problem of second order conformal deformation of spacelike surfaces in compactified Minkowski 4-space. We explain the construction of the exterior differential system of conformal deformations and discuss its general and singular solutions. In particular, we show that isothermic surfaces are singular solutions of the system, which implies that a generic second order deformable surface is not isothermic. This differs from the situation in 3-dimensional conformal geometry, where isothermic surfaces coincide with deformable surfaces."}
{"category": "Math", "title": "Log-average periodogram estimator of the memory parameter", "abstract": "This paper introduces a semiparametric regression estimator of the memory parameter for long-memory time series process. It is based on the regression in a neighborhood of the zero-frequency of the periodogram averaged over epochs. The proposed estimator is theoretically justified and empirical Monte Carlo investigation gives evidence that the method is very promising to estimate the long-memory parameter."}
{"category": "Math", "title": "Primitivity of finitely presented monomial algebras", "abstract": "We study prime monomial algebras. Our main result is that a prime finitely presented monomial algebra is either primitive or it has GK dimension one and satisfies a polynomial identity. More generally, we show this result holds for the class of \\emph{automaton algebras}; that is, monomial algebras that have a basis consisting of the set of words recognized by some finite state automaton. This proves a special case of a conjecture of the first author and Agata Smoktunowicz."}
{"category": "Math", "title": "Multiplicity of Codimension Three Almost Complete Intersections", "abstract": "We establish the upper bound in the multiplicity conjecture of Herzog, Huneke and Srinivasan for the codimension three almost complete intersections. We also give some partial results in the case where I is the aci linked to a complete intersection in one step."}
{"category": "Math", "title": "Spectra and semigroup smoothing for non-elliptic quadratic operators", "abstract": "We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space, intrinsically associated to the Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal complement. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the globally elliptic case. We also describe the large time behavior of contraction semigroups generated by these operators."}
{"category": "Math", "title": "Condensation of Determinants", "abstract": "In this paper we tried to condense the determinant of n square matrix to the determinant of (n - 1) square matrix with the mathematical proof."}
{"category": "Math", "title": "Volume growth and the topology of manifolds with nonnegative Ricci curvature", "abstract": "Let $M^n$ be a complete, open Riemannian manifold with $\\Ric \\geq 0$. In 1994, Grigori Perelman showed that there exists a constant $\\delta_{n}>0$, depending only on the dimension of the manifold, such that if the volume growth satisfies $\\alpha_M := \\lim_{r \\to \\infty} \\frac{\\Vol(B_p(r))}{\\omega_n r^n} \\geq 1-\\delta_{n}$, then $M^n$ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, $\\alpha(k,n)$, depending only on $k$ and $n$, which guarantee the individual $k$-homotopy group of $M^n$ is trivial."}
{"category": "Math", "title": "First variation of the Log Entropy functional along the Ricci flow", "abstract": "In this note, we establish the first variation formula of the adjusted log entropy functional $\\mathcal Y_a$ introduced by Ye in \\cite{Y2}. As a direct consequence, we also obtain the monotonicity of $\\mathcal Y_a$ along the Ricci flow."}
{"category": "Math", "title": "Projective equivalence of ideals in Noetherian integral domains", "abstract": "Let I be a nonzero proper ideal in a Noetherian integral domain R. In this paper we establish the existence of a finite separable integral extension domain A of R and a positive integer m such that all the Rees integers of IA are equal to m. Moreover, if R has altitude one, then all the Rees integers of J = Rad(IA) are equal to one and the ideals J^m and IA have the same integral closure. Thus Rad(IA) = J is a projectively full radical ideal that is projectively equivalent to IA. In particular, if R is Dedekind, then there exists a Dedekind domain A having the following properties: (i) A is a finite separable integral extension of R; and (ii) there exists a radical ideal J of A and a positive integer m such that IA = J^m."}
{"category": "Math", "title": "The Divisor Matrix, Dirichlet Series and SL(2,Z)", "abstract": "A representation of SL(2,Z) by integer matrices acting on the space of analytic ordinary Dirichlet series is constructed, in which the standard unipotent element acts as multiplication by the Riemann zeta function. It is then shown that the Dirichlet series in the orbit of the zeta function are related to it by algebraic equations."}
{"category": "Math", "title": "Reflected Brownian motion in a wedge: sum-of-exponential stationary densities", "abstract": "We give necessary and sufficient conditions for the stationary density of semimartingale reflected Brownian motion in a wedge to be written as a finite sum of terms of exponential product form. Relying on geometric ideas reminiscent of the reflection principle, we give an explicit formula for the density in such cases."}
{"category": "Math", "title": "An Improved Error Bound for Gaussian Interpolation", "abstract": "An error bound for Gaussian Interpolation which is better than the current exponential-type error bound is presented."}
{"category": "Math", "title": "A New Error Bound for Shifted Surface Spline Interpolation", "abstract": "A New Error Bound for shifted surface spline interpolation is presented. This error bound probably is the most powerful one up to now."}
{"category": "Math", "title": "Alexander polynomials and hyperbolic volume of arborescent links", "abstract": "We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperbolic link of at least 2 components. For given polynomial, we give also an upper bound on the minimal hyperbolic volume of knots/links, and contrarily, construct knots of arbitrarily large volume, which are arborescent, or have given free genus at least 2."}
{"category": "Math", "title": "Boundary layers and the vanishing viscosity limit for incompressible 2D flow", "abstract": "This manuscript is a survey on results related to boundary layers and the vanishing viscosity limit for incompressible flow. It is the lecture notes for a 10 hour minicourse given at the Morningside Center, Academia Sinica, Beijing, PRC from 11/28 to 12/07, 2007. The main topics covered are: a derivation of Prandtl's boundary layer equation; an outline of the rigorous theory of Prandtl's equation, without proofs; Kato's criterion for the vanishing viscosity limit; the vanishing viscosity limit with Navier friction condition; rigorous boundary layer theory for the Navier friction condition and boundary layers for flows in a rotating cylinder."}
{"category": "Math", "title": "On the \"degrees of freedom\" of the lasso", "abstract": "We study the effective degrees of freedom of the lasso in the framework of Stein's unbiased risk estimation (SURE). We show that the number of nonzero coefficients is an unbiased estimate for the degrees of freedom of the lasso--a conclusion that requires no special assumption on the predictors. In addition, the unbiased estimator is shown to be asymptotically consistent. With these results on hand, various model selection criteria--$C_p$, AIC and BIC--are available, which, along with the LARS algorithm, provide a principled and efficient approach to obtaining the optimal lasso fit with the computational effort of a single ordinary least-squares fit."}
{"category": "Math", "title": "On closure operators and reflections in Goursat categories", "abstract": "By defining a closure operator on effective equivalence relations in a regular category $C$, it is possible to establish a bijective correspondence between these closure operators and the regular epireflective subcategories $L$ of $C$. When $C$ is an exact Goursat category this correspondence restricts to a bijection between the Birkhoff closure operators on effective equivalence relations and the Birkhoff subcategories of $C$. In this case it is possible to provide an explicit description of the closure, and to characterise the congruence distributive Goursat categories."}
{"category": "Math", "title": "On surrogate dimension reduction for measurement error regression: An invariance law", "abstract": "We consider a general nonlinear regression problem where the predictors contain measurement error. It has been recently discovered that several well-known dimension reduction methods, such as OLS, SIR and pHd, can be performed on the surrogate regression problem to produce consistent estimates for the original regression problem involving the unobserved true predictor. In this paper we establish a general invariance law between the surrogate and the original dimension reduction spaces, which implies that, at least at the population level, the two dimension reduction problems are in fact equivalent. Consequently we can apply all existing dimension reduction methods to measurement error regression problems. The equivalence holds exactly for multivariate normal predictors, and approximately for arbitrary predictors. We also characterize the rate of convergence for the surrogate dimension reduction estimators. Finally, we apply several dimension reduction methods to real and simulated data sets involving measurement error to compare their performances."}
{"category": "Math", "title": "Optimal third root asymptotic bounds in the statistical estimation of thresholds", "abstract": "This paper is concerned with estimating the intersection point of two densities, given a sample of both of the densities. This problem arises in classification theory. The main results provide lower bounds for the probability of the estimation errors to be large on a scale determined by the inverse cube root of the sample size. As corollaries, we obtain probabilistic bounds for the prediction error in a classification problem. The key to the proof is an entropy estimate. The lower bounds are based on bounds for general estimators, which are applicable in other contexts as well. Furthermore, we introduce a class of optimal estimators whose errors asymptotically meet the border permitted by the lower bounds."}
{"category": "Math", "title": "Variance estimation in nonparametric regression via the difference sequence method", "abstract": "Consider a Gaussian nonparametric regression problem having both an unknown mean function and unknown variance function. This article presents a class of difference-based kernel estimators for the variance function. Optimal convergence rates that are uniform over broad functional classes and bandwidths are fully characterized, and asymptotic normality is also established. We also show that for suitable asymptotic formulations our estimators achieve the minimax rate."}
{"category": "Math", "title": "Iterative estimating equations: Linear convergence and asymptotic properties", "abstract": "We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size increases to infinity. Furthermore, we show that the limiting estimator is consistent and asymptotically efficient, as expected. The method applies to semiparametric regression models with unspecified covariances among the observations. In the special case of linear models, the procedure reduces to iterative reweighted least squares. Finite sample performance of the procedure is studied by simulations, and compared with other methods. A numerical example from a medical study is considered to illustrate the application of the method."}
{"category": "Math", "title": "On optimality of Bayesian testimation in the normal means problem", "abstract": "We consider a problem of recovering a high-dimensional vector $\\mu$ observed in white noise, where the unknown vector $\\mu$ is assumed to be sparse. The objective of the paper is to develop a Bayesian formalism which gives rise to a family of $l_0$-type penalties. The penalties are associated with various choices of the prior distributions $\\pi_n(\\cdot)$ on the number of nonzero entries of $\\mu$ and, hence, are easy to interpret. The resulting Bayesian estimators lead to a general thresholding rule which accommodates many of the known thresholding and model selection procedures as particular cases corresponding to specific choices of $\\pi_n(\\cdot)$. Furthermore, they achieve optimality in a rather general setting under very mild conditions on the prior. We also specify the class of priors $\\pi_n(\\cdot)$ for which the resulting estimator is adaptively optimal (in the minimax sense) for a wide range of sparse sequences and consider several examples of such priors."}
{"category": "Math", "title": "A Lohner-type algorithm for control systems and ordinary differential inclusions", "abstract": "We describe a Lohner-type algorithm for the computation of rigorous upper bounds for reachable set for control systems, solutions of ordinary differential inclusions and perturbations of ODEs."}
{"category": "Math", "title": "Differentiability of the volume of a region enclosed by level sets", "abstract": "The level of a function f on an n-dimensional space encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the function f is smooth, the volume function is again smooth for regular values of f. For critical values of f the volume function is only finitely differentiable. The initial motivation for this study comes from Radiotherapy, where such volume functions are used in an optimization process. Thus their differentiability properties become important."}
{"category": "Math", "title": "Choice Number and Energy of Graphs", "abstract": "The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. It is proved that E(G)>= 2(n-\\chi(\\bar{G}))>= 2(ch(G)-1) for every graph G of order n, and that E(G)>= 2ch(G) for all graphs G except for those in a few specified families, where \\bar{G}, \\chi(G), and ch(G) are the complement, the chromatic number, and the choice number of G, respectively."}
{"category": "Math", "title": "Transgression and Clifford algebras", "abstract": "Let $W$ be a differential (not necessarily commutative) algebra which carries a free action of a polynomial algebra $SP$ with homogeneous generators $p_1, >..., p_r$. We show that for $W$ acyclic, the cohomology of the quotient $H(W/<p_1, ..., p_r>)$ is isomorphic to a Clifford algebra $\\text{Cl}(P,B)$, where the (possibly degenerate) bilinear form $B$ depends on $W$. This observation is an analogue of an old result of Borel in a non-commutative context. As an application, we study the case of $W$ given by the quantized Weil algebra $\\qWg = \\Ug \\otimes \\Clg$ for $\\Lieg$ a reductive Lie algebra. The resulting cohomology of the canonical Weil differential gives a Clifford algebra, but the bilinear form vanishes on the space of primitive invariants of the semi-simple part. As an application, we consider the deformed Weil differential (following Freed, Hopkins and Teleman)."}
{"category": "Math", "title": "Galois actions on homotopy groups", "abstract": "We study the Galois actions on the l-adic schematic and Artin-Mazur homotopy groups of algebraic varieties. For proper varieties of good reduction over a local field K, we show that the l-adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever l is not equal to the residue characteristic p of K. For quasi-projective varieties of good reduction, there is a similar characterisation involving the Gysin spectral sequence. When l=p, a slightly weaker result is proved by comparing the crystalline and p-adic schematic homotopy types. Under favourable conditions, a comparison theorem transfers all these descriptions to the Artin-Mazur homotopy groups."}
{"category": "Math", "title": "Proof of the Somos-4 Hankel Determinants Conjecture", "abstract": "By considering the fundamental equation $x=y-y^2=z-z^3$, Somos conjectured that the Hankel determinants for the generating series $y(z)$ are the Somos-4 numbers. We prove this conjecture by using the quadratic transformation for Hankel determinants of Sulanke and Xin."}
{"category": "Math", "title": "The cycle problem: an intriguing periodicity to the zeros of the Riemann zeta function", "abstract": "Summing the values of the real portion of the logarithmic integral of n^rho, where rho is one of a consecutive series of zeros of the Riemann zeta function, reveals an unexpected periodicity to the sum. This is the cycle problem."}
{"category": "Math", "title": "Spatial aggregation of local likelihood estimates with applications to classification", "abstract": "This paper presents a new method for spatially adaptive local (constant) likelihood estimation which applies to a broad class of nonparametric models, including the Gaussian, Poisson and binary response models. The main idea of the method is, given a sequence of local likelihood estimates (``weak'' estimates), to construct a new aggregated estimate whose pointwise risk is of order of the smallest risk among all ``weak'' estimates. We also propose a new approach toward selecting the parameters of the procedure by providing the prescribed behavior of the resulting estimate in the simple parametric situation. We establish a number of important theoretical results concerning the optimality of the aggregated estimate. In particular, our ``oracle'' result claims that its risk is, up to some logarithmic multiplier, equal to the smallest risk for the given family of estimates. The performance of the procedure is illustrated by application to the classification problem. A numerical study demonstrates its reasonable performance in simulated and real-life examples."}
{"category": "Math", "title": "Schur-Weyl duality for orthogonal groups", "abstract": "We prove Schur--Weyl duality between the Brauer algebra $\\mathfrak{B}_n(m)$ and the orthogonal group $O_{m}(K)$ over an arbitrary infinite field $K$ of odd characteristic. If $m$ is even, we show that each connected component of the orthogonal monoid is a normal variety; this implies that the orthogonal Schur algebra associated to the identity component is a generalized Schur algebra. As an application of the main result, an explicit and characteristic-free description of the annihilator of $n$-tensor space $V^{\\otimes n}$ in the Brauer algebra $mathfrak{B}_n(m)$ is also given."}
{"category": "Math", "title": "Cardinalities of k-distance sets in Minkowski spaces", "abstract": "A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1)^d points, with equality iff the unit ball is a parallelotope. We solve this conjecture in the affirmative for all 2-dimensional spaces and for spaces where the unit ball is a parallelotope. For general spaces we find various weaker upper bounds for k-distance sets."}
{"category": "Math", "title": "Affine Algebraic Varieties", "abstract": "In this paper, we give new criteria for affineness of a variety defined over $\\Bbb{C}$. Our main result is that an irreducible algebraic variety $Y$ (may be singular) of dimension $d$ ($d\\geq 1$) defined over $\\Bbb{C}$ is an affine variety if and only if $Y$ contains no complete curves, $H^i(Y, {\\mathcal{O}}_Y)=0$ for all $i>0$ and the boundary $X-Y$ is support of a big divisor, where $X$ is a projective variety containing $Y$. We construct three examples to show that a variety is not affine if it only satisfies two conditions among these three conditions. We also give examples to demonstrate the difference between the behavior of the boundary divisor $D$ and the affineness of $Y$. If $Y$ is an affine variety, then the ring $\\Gamma (Y, {\\mathcal{O}}_Y)$ is noetherian. However, to prove that $Y$ is an affine variety, we do not start from this ring. We explain why we do not need to check the noetherian property of the ring $\\Gamma (Y, {\\mathcal{O}}_Y)$ directly but use the techniques of sheaf and cohomology."}
{"category": "Math", "title": "What is the difference between a square and a triangle?", "abstract": "We offer a reader-friendly introduction to the attracting edge problem (also known as the \"triangle conjecture\") and its most general current solution of Limic and Tarr\\`es (2007). Little original research is reported; rather this article ``zooms in'' to describe the essential characteristics of two different techniques/approaches verifying the almost sure existence of the attracting edge for the strongly edge reinforced random walk (SERRW) on a square. Both arguments extend straightforwardly to the SERRW on even cycles. Finally, we show that the case where the underlying graph is a triangle cannot be studied by a simple modification of either of the two techniques."}
{"category": "Math", "title": "A p-adic approach to local analytic dynamics: analytic flows and analytic maps tangent to the identity", "abstract": "In this note, we will consider the question of local equivalence of analytic functions which fix the origin and are tangent to the identity, as well as the question of flows of analytic vector fields. All mappings and equivalences are considered in the non-archimedean context e.g. all norms can be considered $p$-adic norms. We show that any two mappings $f$ and $g$ which are formally equivalent are also analytically equivalent, and we show that analytic vector fields generate analytic flows. We consider the related questions of roots and centralizers for analytic mappings. In this setting, anything which can be done formally can also be done analytically."}
{"category": "Math", "title": "High resolution quantization and entropy coding of jump processes", "abstract": "We study the quantization problem for certain types of jump processes. The probabilities for the number of jumps are assumed to be bounded by Poisson weights. Otherwise, jump positions and increments can be rather generally distributed and correlated. We show in particular that in many cases entropy coding error and quantization error have distinct rates. Finally, we investigate the quantization problem for the special case of $\\mathbb{R}^d$-valued compound Poisson processes."}
{"category": "Math", "title": "On the Dirichlet problem for prescribed mean curvature equation over general domains", "abstract": "We study and solve the Dirichlet problem for graphs of prescribed mean curvature in $\\mathbb R^{n+1}$ over general domains $\\Omega$ without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition."}
{"category": "Math", "title": "Projective structures, grafting, and measured laminations", "abstract": "We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmuller space, complementing a result of Scannell-Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems for the space of complex projective ($\\CP^1$) structures on a surface. We also study the rays in Teichmuller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays."}
{"category": "Math", "title": "Editorial: Statistics and forensic science", "abstract": "Forensic science is usually taken to mean the application of a broad spectrum of scientific tools to answer questions of interest to the legal system. Despite such popular television series as CSI: Crime Scene Investigation and its spinoffs--CSI: Miami and CSI: New York--on which the forensic scientists use the latest high-tech scientific tools to identify the perpetrator of a crime and always in under an hour, forensic science is under assault, in the public media, popular magazines [Talbot (2007), Toobin (2007)] and in the scientific literature [Kennedy (2003), Saks and Koehler (2005)]. Ironically, this growing controversy over forensic science has occurred precisely at the time that DNA evidence has become the ``gold standard'' in the courts, leading to the overturning of hundreds of convictions many of which were based on clearly less credible forensic evidence, including eyewitness testimony [Berger (2006)]."}
{"category": "Math", "title": "On separation of variables and completeness of the Bethe ansatz for quantum gl_N Gaudin model", "abstract": "In this note, we discuss implications of the results obtained in [MTV4]. It was shown there that eigenvectors of the Bethe algebra of the quantum gl_N Gaudin model are in a one-to-one correspondence with Fuchsian differential operators with polynomial kernel. Here, we interpret this fact as a separation of variables in the gl_N Gaudin model. Having a Fuchsian differential operator with polynomial kernel, we construct the corresponding eigenvector of the Bethe algebra. It was shown in [MTV4] that the Bethe algebra has simple spectrum if the evaluation parameters of the Gaudin model are generic. In that case, our Bethe ansatz construction produces an eigenbasis of the Bethe algebra."}
{"category": "Math", "title": "5-move equivalence classes of links and their algebraic invariants", "abstract": "We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S^3 along a link. We use also the fact that a 5-move is a composition of two rational \\pm (2,2)-moves (i.e. \\pm 5/2-moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2,2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings."}
{"category": "Math", "title": "On continuous state branching processes: conditioning and self-similarity", "abstract": "In this paper, for $\\alpha\\in (1, 2}$ we show that the $\\alpha$-stable continuous-state branching process and the associated process conditioned never to become extinct are positive self-similar Markov processes. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive self-similar Markov processes permits accessto a number of explicit results concerning the paths of stable-continuous branching processes and its conditioned version."}
{"category": "Math", "title": "A maximal inequality for the tail of the bilinear Hardy-Littlewood function", "abstract": "Let $(X,\\mathcal{B}, \\mu, T)$ be an ergodic dynamical system on a non-atomic finite measure space. We assume without loss of generality that $\\mu(X)=1.$ Consider the maximal function $\\dis R^*:(f, g) \\in L^p\\times L^q \\to R^*(f, g)(x) = \\sup_{n\\geq 1} \\frac{f(T^nx)g(T^{2n}x)}{n}.$ We obtain the following maximal inequality. For each $1<p\\leq \\infty$ there exists a finite constant $C_p$ such that for each $\\lambda >0,$ and nonnegative functions $f\\in L^p$ and $g\\in L^1$ \\mu\\{x: R^*(f,g)(x)>\\lambda\\} \\leq C_p \\bigg(\\frac{\\|f\\|_p\\|g\\|_1}{\\lambda}\\bigg)^{1/2}. We also show that for each $\\alpha>2$ the maximal function $R^*(f,g)$ is a.e. finite for pairs of functions $(f,g)\\in (L(\\log L)^{2\\alpha}, L^1)$."}
{"category": "Math", "title": "Formality of DG algebras (after Kaledin)", "abstract": "We provide proper foundations and proofs for the main results of [Ka]. The results include a flat base change for formality and behavior of formality in flat families of $A(\\infty)$ and DG algebras."}
{"category": "Math", "title": "The First-Order Genus of a Knot", "abstract": "We introduce a geometric invariant of knots in the three-sphere, called the first-order genus, that is derived from certain 2-complexes called gropes, and we show it is computable for many examples. While computing this invariant, we draw some interesting conclusions about the structure of a general Seifert surface for some knots."}
{"category": "Math", "title": "Hankel operators that commute with second-order differential operators", "abstract": "Suppose that $\\Gamma$ is a continuous and self-adjoint Hankel operator on $L^2(0, \\infty)$ and that $Lf=-(d/dx(a(x)df/dx))+b(x)f(x)$ with $a(0)=0$. If $a$ and $b$ are both quadratic, hyperbolic or trigonometric functions, and $\\phi$ satisfies a suitable form of Gauss's hypergeometric equation, or the confluent hypergeometric equation, then $L\\Gamma =\\Gamma L$. The paper catalogues the commuting pairs $\\Gamma$ and $L$, including important cases in random matrix theory. There are also results proving rapid decay of the singular numbers of Hankel integral operators with kernels that are analytic and of exponential decay in the right half plane."}
{"category": "Math", "title": "Kernels and Ensembles: Perspectives on Statistical Learning", "abstract": "Since their emergence in the 1990's, the support vector machine and the AdaBoost algorithm have spawned a wave of research in statistical machine learning. Much of this new research falls into one of two broad categories: kernel methods and ensemble methods. In this expository article, I discuss the main ideas behind these two types of methods, namely how to transform linear algorithms into nonlinear ones by using kernel functions, and how to make predictions with an ensemble or a collection of models rather than a single model. I also share my personal perspectives on how these ideas have influenced and shaped my own research. In particular, I present two recent algorithms that I have invented with my collaborators: LAGO, a fast kernel algorithm for unbalanced classification and rare target detection; and Darwinian evolution in parallel universes, an ensemble method for variable selection."}
{"category": "Math", "title": "Moonshine elements in elliptic cohomology", "abstract": "This is a historical talk about the recent confluence of two lines of research in equivariant elliptic cohomology, one concerned with connected Lie groups, the other with the finite case. These themes come together in (what seems to me remarkable) work of N. Ganter, relating replicability of McKay-Thompson series to the theory of exponential cohomology operations."}
{"category": "Math", "title": "The Corona Theorem on the Complements of Certain Square Cantor Sets", "abstract": "Let $K$ be a square Cantor set, i.e. the Cartesian product $K=E\\times E$ of two linear Cantor sets. Let $\\delta_n$ denote the proportion of the intervals removed in the $n$th stage of the construction of $E$. It is shown that if $\\delta_n=o(\\frac1{\\log\\log n})$ then the corona theorem holds on the domain $\\Omega=\\mathbb C^\\ast\\setminus K$."}
{"category": "Math", "title": "The sorting order on a Coxeter group", "abstract": "Let $(W,S)$ be an arbitrary Coxeter system. For each word $\\omega$ in the generators we define a partial order--called the {\\sf $\\omega$-sorting order}--on the set of group elements $W_\\omega\\subseteq W$ that occur as subwords of $\\omega$. We show that the $\\omega$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and Bruhat orders on the group. Moreover, the $\\omega$-sorting order is a \"maximal lattice\" in the sense that the addition of any collection of Bruhat covers results in a nonlattice. Along the way we define a class of structures called {\\sf supersolvable antimatroids} and we show that these are equivalent to the class of supersolvable join-distributive lattices."}
{"category": "Math", "title": "Solutions with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation", "abstract": "We study solutions of the 2D Ginzburg-Landau equation -\\Delta u+\\frac{1}{\\ve^2}u(|u|^2-1)=0 subject to \"semi-stiff\" boundary conditions: the Dirichlet condition for the modulus, |u|=1, and the homogeneous Neumann condition for the phase. The principal result of this work shows there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small epsilon. For the Dirichlet bondary condition (\"stiff\" problem), the existence of stable solutions with vortices, whose energy blows up as epsilon goes to 0, is well known. By contrast, stable solutions with vortices are not established in the case of the homogeneous Neumann (\"soft\") boundary condition. (nonexistence is proved for simply connected domains). In this work, we develop a variational method which allows one to construct local minimizers of the corresponding Ginzburg-Landau energy functional. We introduce an approximate bulk degree as the key ingredient of this method, and, unlike the standard degree over the curve, it is preserved in the weak H^1-limit."}
{"category": "Math", "title": "Strongly minimal PD4-complexes", "abstract": "We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $\\pi$ such that $c.d.\\pi=2$ and $\\pi$ has one end. Let $\\beta=\\beta_2(\\pi;F_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on $(\\pi,w)$) there are at most $2^\\beta$ orbits of $k$-invariants determining \"strongly minimal\" complexes (i.e., those with homotopy intersection pairing $\\lambda_X$ trivial). The homotopy type of a $PD_4$-complex $X$ with $\\pi$ a $PD_2$-group is determined by $\\pi$, $w$, $\\lambda_X$ and the $v_2$-type of $X$. Our result also implies that Fox's 2-knot with metabelian group is determined up to TOP isotopy and reflection by its group."}
{"category": "Math", "title": "Generalised morphisms of k-graphs: k-morphs", "abstract": "In a number of recent papers, (k+l)-graphs have been constructed from k-graphs by inserting new edges in the last l dimensions. These constructions have been motivated by C*-algebraic considerations, so they have not been treated systematically at the level of higher-rank graphs themselves. Here we introduce k-morphs, which provide a systematic unifying framework for these various constructions. We think of k-morphs as the analogue, at the level of k-graphs, of C*-correspondences between C*-algebras. To make this analogy explicit, we introduce a category whose objects are k-graphs and whose morphisms are isomorphism classes of k-morphs. We show how to extend the assignment \\Lambda \\mapsto C*(\\Lambda) to a functor from this category to the category whose objects are C*-algebras and whose morphisms are isomorphism classes of C*-correspondences."}
{"category": "Math", "title": "A characterisation of the Calabi product of hyperbolic affine spheres", "abstract": "There exists a well known construction which allows to associate with two hyperbolic affine spheres $f_i: M_i^{n_i} \\to \\mathbb R^{n_i+1}$ a new hyperbolic affine sphere immersion of $I \\times M_1 \\times M_2$ into $\\mathbb R^{n_1+n_2+3}$. In this paper we deal with the inverse problem: how to determine from properties of the difference tensor whether a given hyperbolic affine sphere immersion of a manifold $M^n \\to \\mathbb R^{n+1}$ can be decomposed in such a way."}
{"category": "Math", "title": "On sumsets of dissociated sets", "abstract": "In the paper we are studying some properties of subsets Q of sums of dissociated sets. The exact upper bound for the number of solutions of the following equation (1) q_1 + ... + q_p = q_{p+1} + ... + q_{2p}, q_i \\in Q in groups F_2^n is found. Using our approach, we easily prove a recent result of J. Bourgain on sets of large exponential sums and obtain a tiny improvement of his theorem. Besides an inverse problem is considered in the article. Let Q be a set belonging a sumset of two dissociated sets such that equation (1) has many solutions. We prove that in the case the large proportion of Q is highly structured."}
{"category": "Math", "title": "Representation theory of liftings of quantum planes", "abstract": "We systematically determine the regular representations, quivers and representation type of all liftings of two-dimensional quantum linear spaces."}
{"category": "Math", "title": "Orbit closures in the enhanced nilpotent cone", "abstract": "We study the orbits of $G=\\mathrm{GL}(V)$ in the enhanced nilpotent cone $V\\times\\mathcal{N}$, where $\\mathcal{N}$ is the variety of nilpotent endomorphisms of $V$. These orbits are parametrized by bipartitions of $n=\\dim V$, and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition analogues of Kostka polynomials, defined by Shoji. Finally, we make a connection with Kato's exotic nilpotent cone in type C, proving that the closure ordering is the same, and conjecturing that the intersection cohomology is the same but with degrees doubled."}
{"category": "Math", "title": "On the Distribution of Pseudopowers", "abstract": "An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\\le x$. We improve an upper bound for the least such number due to E. Bach, R. Lukes, J. Shallit, and H. C. Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of $g$ modulo prime numbers."}
{"category": "Math", "title": "On Pseudosquares and Pseudopowers", "abstract": "Introduced by Kraitchik and Lehmer, an $x$-pseudosquare is a positive integer $n\\equiv1\\pmod 8$ that is a quadratic residue for each odd prime $p\\le x$, yet is not a square. We use bounds of character sums to prove that pseudosquares are equidistributed in fairly short intervals. An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\\le x$. It is conjectured by Bach, Lukes, Shallit, and Williams that the least such number is at most $\\exp(a_g x/\\log x)$ for a suitable constant $a_g$. A bound of $\\exp(a_g x\\log\\log x/\\log x)$ is proved conditionally on the Riemann Hypothesis for Dedekind zeta functions, thus improving on a recent conditional exponential bound of Konyagin and the present authors. We also give a GRH-conditional equidistribution result for pseudopowers that is analogous to our unconditional result for pseudosquares."}
{"category": "Math", "title": "Polynomial Bridgeland stability conditions and the large volume limit", "abstract": "We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpson-stability, and large volume limits of Bridgeland stability conditions. We show that the PT/DT-correspondence relating stable pairs to Donaldson-Thomas invariants (conjectured by Pandharipande and Thomas) can be understood as a wall-crossing in our family of polynomial stability conditions. Similarly, we show that the relation between stable pairs and invariants of one-dimensional torsion sheaves (proven recently by the same authors) is a wall-crossing formula."}
{"category": "Math", "title": "A maximum principle for the Muskat problem for fluids with different densities", "abstract": "We consider the fluid interface problem given by two incompressible fluids with different densities evolving by Darcy's law. This scenario is known as the Muskat problem for fluids with the same viscosities, being in two dimensions mathematically analogous to the two-phase Hele-Shaw cell. We prove in the stable case (the denser fluid is below) a maximum principle for the $L^\\infty$ norm of the free boundary."}
{"category": "Math", "title": "The rarity of DNA profiles", "abstract": "It is now widely accepted that forensic DNA profiles are rare, so it was a surprise to some people that different people represented in offender databases are being found to have the same profile. In the first place this is just an illustration of the birthday problem, but a deeper analysis must take into account dependencies among profiles caused by family or population membership."}
{"category": "Math", "title": "Interpretation of interaction: A review", "abstract": "Several different types of statistical interaction are defined and distinguished, primarily on the basis of the nature of the factors defining the interaction. Illustrative examples, mostly epidemiological, are given. The emphasis is primarily on interpretation rather than on methods for detecting interactions."}
{"category": "Math", "title": "The pigeonhole bootstrap", "abstract": "Recently there has been much interest in data that, in statistical language, may be described as having a large crossed and severely unbalanced random effects structure. Such data sets arise for recommender engines and information retrieval problems. Many large bipartite weighted graphs have this structure too. We would like to assess the stability of algorithms fit to such data. Even for linear statistics, a naive form of bootstrap sampling can be seriously misleading and McCullagh [Bernoulli 6 (2000) 285--301] has shown that no bootstrap method is exact. We show that an alternative bootstrap separately resampling rows and columns of the data matrix satisfies a mean consistency property even in heteroscedastic crossed unbalanced random effects models. This alternative does not require the user to fit a crossed random effects model to the data."}
{"category": "Math", "title": "Law of the exponential functional of a new family of one-sided Levy processes via self-similar continuous state branching processes with immigration and the Wright hypergeometric functions", "abstract": "We first introduce and derive some basic properties of a two-parameters family of one-sided Levy processes. Their Laplace exponents are given in terms of the Pochhammer symbol. This family includes, in a limit case, the family of Brownian motion with drifts. Then, we proceed by computing the density of the law of the exponential functional associated to some elements of this family (and their dual) and some transformations of these elements. These densities are expressed in terms of the Wright hypergeometric functions. By means of probabilistic arguments, we derive some interesting properties enjoyed by these functions. On the way we also characterize explicitly the density of the semi-groups of the family of self-similar continuous state branching processes with immigration."}
{"category": "Math", "title": "Period doubling in the Rossler system - a computer assisted proof", "abstract": "The goal of this paper is to show how to produce a piece of rigorous bifurcation diagram of periodic orbits for an ODE. We study the Rossler system, one of the textbook examples of ODEs generating nontrivial dynamics, for the parameter range containing two period doubling bifurcations."}
{"category": "Math", "title": "A statistical analysis of memory CD8 T cell differentiation: An application of a hierarchical state space model to a short time course microarray experiment", "abstract": "CD8 T cells are specialized immune cells that play an important role in the regulation of antiviral immune response and the generation of protective immunity. In this paper we investigate the differentiation of memory CD8 T cells in the immune response using a short time course microarray experiment. Structurally, this experiment is similar to many in that it involves measurements taken on independent samples, in one biological group, at a small number of irregularly spaced time points, and exhibiting patterns of temporal nonstationarity. To analyze this CD8 T-cell experiment, we develop a hierarchical state space model so that we can: (1) detect temporally differentially expressed genes, (2) identify the direction of successive changes over time, and (3) assess the magnitude of successive changes over time. We incorporate hidden Markov models into our model to utilize the information embedded in the time series and set up the proposed hierarchical state space model in an empirical Bayes framework to utilize the population information from the large-scale data. Analysis of the CD8 T-cell experiment using the proposed model results in biologically meaningful findings. Temporal patterns involved in the differentiation of memory CD8 T cells are summarized separately and performance of the proposed model is illustrated in a simulation study."}
{"category": "Math", "title": "On operations and characteristic classes", "abstract": "In this paper exterior products are used to define operations and characteristic classes with values in the K-theory of an abelian category with tensor and exterior products. We apply the general construction to define Chern and Segre classes with values in algebraic K-theory and the K-theory of connections."}
{"category": "Math", "title": "Homogeneous quasimorphisms on the symplectic linear group", "abstract": "In this note, we show a uniqueness result of homogeneous quasimorphisms defined on the universal cover of the symplectic linear group."}
{"category": "Math", "title": "Interpolation with a function parameter and refined scale of spaces", "abstract": "The interpolation of couples of separable Hilbert spaces with a function parameter is studied. The main properties of the classic interpolation are proved. Some applications to the interpolation of isotropic H\\\"ormander spaces over a closed manifold are given."}
{"category": "Math", "title": "A Generic Framework for Diamond Lemmas", "abstract": "This paper gives a generic form of the diamond lemma, which includes support for additive and topological structures of the base set, and which does not require any further structure (e.g. an associative multiplication operation) to be present. This result is intended to be used as the core of diamond lemmas for particular algebraic structures, taking care of all the common technicalities. With this generic diamond lemma, the main steps needed to prove a specialised diamond lemma is to define the reduction maps and analyse the structure of critical ambiguities. The abstract machinery is backed up with concrete suggestions for how one should set things up in order to reproduce traditional results in the general setting. Several instances of the fundamental theorem of Groebner basis theory are derived as corollaries of the main result."}
{"category": "Math", "title": "Euler equation for incompressible non-Newtonian fluids: finite speed of propagations and asymptotic behavior of weak solutions", "abstract": "We investigate multidimensional model for incompressible non-Newtonian fluids. Using method of energy estimates we prove the property of finite speed of propagations of the solution support for this problem. We find sharp bounds of the propagations by $L^2$--norm and $L^1$--norm of initial data."}
{"category": "Math", "title": "Detecting changes in the fluctuations of a Gaussian process and an application to heartbeat time series", "abstract": "The aim of this paper is first the detection of multiple abrupt changes of the long-range dependence (respectively self-similarity, local fractality) parameters from a sample of a Gaussian stationary times series (respectively time series, continuous-time process having stationary increments). The estimator of the $m$ change instants (the number $m$ is supposed to be known) is proved to satisfied a limit theorem with an explicit convergence rate. Moreover, a central limit theorem is established for an estimator of each long-range dependence (respectively self-similarity, local fractality) parameter. Finally, a goodness-of-fit test is also built in each time domain without change and proved to asymptotically follow a Khi-square distribution. Such statistics are applied to heart rate data of marathon's runners and lead to interesting conclusions."}
{"category": "Math", "title": "Introduction to (generalized) Gibbs measures", "abstract": "These notes have been written to complete a mini-course \"Introduction to (generalized) Gibbs measures\" given at the universities UFMG (Universidade Federal de Minas Gerais, Belo Horizonte, Brasil) and UFRGS (Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil) during the first semester 2007. The main goal of the lectures was to describe Gibbs and generalized Gibbs measures on lattices at a rigorous mathematical level, as equilibirum states of systems of a huge number of particles in interaction. In particular, our main message is that although the historical approach based on potentials has been rather successful from a physical point of view, one has to insist on (almost sure) continuity properties of conditional probabilities to get a proper mathematical framework."}
{"category": "Math", "title": "Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces", "abstract": "Let $X$ be a real Banach space with a normalized duality mapping uniformly norm-to-weak$^\\star$ continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping $J_{\\Phi}$ with gauge $\\phi$. Let $f$ be an {\\em $\\alpha$-contraction} and $\\{T_n\\}$ a sequence of nonexpansive mapping, we study the strong convergence of explicit iterative schemes x_{n+1} = \\alpha_n f(x_n) + (1-\\alpha_n) T_n x_n with a general theorem and then recover and improve some specific cases studied in the literature"}
{"category": "Math", "title": "Compactified Jacobians of curves with spine decompositions", "abstract": "A curve, that is, a connected, reduced, projective scheme of dimension 1 over an algebraically closed field, admits two types of compactifications of its (generalized) Jacobian: the moduli schemes of P-quasistable torsion-free, rank-1 sheaves and Seshadri's moduli schemes of S-equivalence classes of semistable torsion-free, rank-1 sheaves. Both are constructed with respect to a choice of polarization. The former are fine moduli spaces which were shown to be complete; here we show that they are actually projective. The latter are just coarse moduli spaces. Here we give a sufficient condition for when these two types of moduli spaces are equal."}
{"category": "Math", "title": "Gamma-entropy cost for scalar conservation laws", "abstract": "We are concerned with a control problem related to the vanishing viscosity approximation to scalar conservation laws. We investigate the $\\Gamma$-convergence of the control cost functional, as the viscosity coefficient tends to zero. A first order $\\Gamma$-limit is established, which characterizes the measure-valued solutions to the conservation laws as the zeros of the $\\Gamma$-limit. A second order $\\Gamma$-limit is then investigated, providing a characterization of entropic solutions to conservation laws as the zeros of the $\\Gamma$-limit."}
{"category": "Math", "title": "The $\\overline\\partial$-cohomology groups, holomorphic Morse inequalities, and finite type conditions", "abstract": "We study spectral behavior of the complex Laplacian on forms with values in the $k^{\\text{th}}$ tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. In particular, we prove that in the two dimensional case, a pseudoconvex domain is of finite type if and only if for any positive constant $C$, the number of eigenvalues of the $\\overline\\partial$-Neumann Laplacian less than or equal to $Ck$ grows polynomially as $k$ tends to infinity."}
{"category": "Math", "title": "Finite volume schemes on Lorentzian manifolds", "abstract": "We investigate the numerical approximation of (discontinuous) entropy solutions to nonlinear hyperbolic conservation laws posed on a Lorentzian manifold. Our main result establishes the convergence of monotone and first-order finite volume schemes for a large class of (space and time) triangulations. The proof relies on a discrete version of entropy inequalities and an entropy dissipation bound, which take into account the manifold geometry accurately and generalize techniques and estimates that were known in the (flat) Euclidian setting, only. The strong convergence of the scheme then is then a consequence of the well-posed theory recently developed by Ben-Artzi and LeFloch for conservation laws on manifolds."}
{"category": "Math", "title": "Superinduction for pattern groups", "abstract": "It is well-known that the representation theory of the finite group of unipotent upper-triangular matrices $U_n$ over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies Diaconis--Isaacs' concept of superinduction in pattern groups. While superinduction shares many desirable properties with usual induction, it no longer takes characters to characters. We begin by finding sufficient conditions guaranteeing that super-induction is in fact induction. It turns out for natural embedding of $U_m$ in $U_n$, super-induction is induction. We conclude with an explicit combinatorial algorithm for computing this induction analogous to the Pieri-formulas for the symmetric group."}
{"category": "Math", "title": "A characterization and a generalization of W*-modules", "abstract": "We give a new Banach module characterization of $W^*$-modules, also known as selfdual Hilbert $C^*$-modules over a von Neumann algebra. This leads to a generalization of the notion, and the theory, of W*-modules, to the setting where the operator algebras are $\\sigma$-weakly closed algebras of operators on a Hilbert space. That is, we find the appropriate weak* topology variant of our earlier notion of {\\em rigged modules}, and their theory, which in turn generalizes the notions of C*-module, and Hilbert space, successively. Our {\\em w*-rigged modules} have canonical `envelopes' which are W*-modules. Indeed, w*-rigged modules may be defined to be a subspace of a W*-module possessing certain properties."}
{"category": "Math", "title": "Restricting supercharacters of the finite group of unipotent uppertriangular matrices", "abstract": "It is well-known that the representation theory of the finite group of unipotent upper-triangular matrices $U_n$ over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies the supercharacter theory of a family of subgroups that interpolate between $U_{n-1}$ and $U_n$. We supply several combinatorial indexing sets for the supercharacters, supercharacter formulas for these indexing sets, and a combinatorial rule for restricting supercharacters from one group to another. A consequence of this analysis is a Pieri-like restriction rule from $U_n$ to $U_{n-1}$ that can be described on set-partitions (analogous to the corresponding symmetric group rule on partitions)."}
{"category": "Math", "title": "Galerkin Methods for the Fully Nonlinear Monge-Amp\\`ere Equation", "abstract": "This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Amp\\`ere equation $\\det(D^2u^0)=f$ based on the vanishing moment method which was developed by the authors in \\cite{Feng2,Feng1}. In this approach, the Monge-Amp\\`ere equation is approximated by the fourth order quasilinear equation $-\\epsilon\\Delta^2 u^\\epsilon + \\det{D^2u^\\epsilon} =f$ accompanied by appropriate boundary conditions. This new approach allows one to construct convergent Galerkin numerical methods for the fully nonlinear Monge-Amp\\`ere equation, a task which has been impracticable before. In this paper, we first develop some finite element and spectral Galerkin methods for approximating the solution $u^\\epsilon$ of the regularized fourth order problem. We then derive optimal order error estimates for the proposed numerical methods. In particular, we track explicitly the dependence of the error bounds on the parameter $\\vepsi$, for the error $u^\\epsilon-u^\\epsilon_h$. Finally, using the Aygris finite element method as an example, we present a detailed numerical study of the rates of convergence in terms of powers of $\\vepsi$ for the error $u^0-u_h^\\vepsi$, and numerically examine what is the \"best\" mesh size $h$ in relation to $\\vepsi$ in order to achieve these rates."}
{"category": "Math", "title": "Mixed finite element methods for the fully nonlinear Monge-Amp\\`ere equation based on the vanishing moment method", "abstract": "This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Monge-Amp\\`ere equation $\\det(D^2u^0)=f$ based on the vanishing moment method which was proposed recently by the authors in \\cite{Feng2}. In this approach, the second order fully nonlinear Monge-Amp\\`ere equation is approximated by the fourth order quasilinear equation $-\\epsilon\\Delta^2 u^\\epsilon + \\det{D^2u^\\epsilon} =f$. It was proved in \\cite{Feng1} that the solution $u^\\epsilon$ converges to the unique convex viscosity solution $u^0$ of the Dirichlet problem for the Monge-Amp\\`ere equation. This result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations, a task which has been impracticable before. The goal of this paper is threefold. First, we develop a family of Hermann-Miyoshi type mixed finite element methods for approximating the solution $u^\\epsilon$ of the regularized fourth order problem, which computes simultaneously $u^\\vepsi$ and the moment tensor $\\sigma^\\vepsi:=D^2u^\\epsilon$. Second, we derive error estimates, which track explicitly the dependence of the error constants on the parameter $\\vepsi$, for the errors $u^\\epsilon-u^\\epsilon_h$ and $\\sigma^\\vepsi-\\sigma_h^\\vepsi$. Finally, we present a detailed numerical study on the rates of convergence in terms of powers of $\\vepsi$ for the error $u^0-u_h^\\vepsi$ and $\\sigma^\\vepsi-\\sigma_h^\\vepsi$, and numerically examine what is the \"best\" mesh size $h$ in relation to $\\vepsi$ in order to achieve these rates."}
{"category": "Math", "title": "Projectively full ideals in Noetherian rings, a survey", "abstract": "We discuss projective equivalence of ideals in Noetherian rings and the existence or failure of existence of projectively full ideals. We describe connections with the Rees valuations and Rees integers of an ideal, and consider the question of whether improvements can be made by passing to an integral extension ring."}
{"category": "Math", "title": "Orbit closures of directing modules are regular in codimension one", "abstract": "We show that the orbit closure of a directing module is regular in codimension one. In particular, this result gives information about a distinguished irreducible component of a module variety."}
{"category": "Math", "title": "Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones", "abstract": "Let G be a simple graph and let J be its ideal of vertex covers. We give a graph theoretical description of the irreducible b-vertex covers of G, i.e., we describe the minimal generators of the symbolic Rees algebra of J. Then we study the irreducible b-vertex covers of the blocker of G, i.e., we study the minimal generators of the symbolic Rees algebra of the edge ideal of G. We give a graph theoretical description of the irreducible binary b-vertex covers of the blocker of G. It is shown that they correspond to irreducible induced subgraphs of G. As a byproduct we obtain a method, using Hilbert bases, to obtain all irreducible induced subgraphs of G. In particular we obtain all odd holes and antiholes. We study irreducible graphs and give a method to construct irreducible b-vertex covers of the blocker of G with high degree relative to the number of vertices of G."}
{"category": "Math", "title": "Majoration du nombre de z\\'eros d'une fonction m\\'eromorphe en dehors d'une droite verticale et applications", "abstract": "We study the distribution of the zeros of functions of the form $f(s)=h(s) \\pm h(2a-s)$, where $h(s)$ is a meromorphic function, real on the real line, $a$ a real number. One of our results establishes sufficient conditions under which all but finitely many of the zeros of $f(s)$ lie on the line $\\Re s = a$, called the {\\it critical line} for the function $f(s)$, and be simple, given that all but finitely many of the zeros of $h(s)$ lie on the half-plane $\\Re s < a$. This results can be regarded as a generalization of the necessary condition of stability for the function $h(s)$, in the Hermite-Biehler theorem. We apply this results to the study of translations of the Riemann Zeta Function and $L$ functions, and integrals of Eisenstein Series, among others."}
{"category": "Math", "title": "On the facial structure of Symmetric and Graphical Traveling Salesman Polyhedra", "abstract": "The Symmetric Traveling Salesman Polytope $S_n$ for a fixed number $n$ of cities is a face of the corresponding Graphical Traveling Salesman Polyhedron $P_n$. This has been used to study facets of $S_n$ using $P_n$ as a tool. In this paper, we study the operation of \"rotating\" (or \"lifting\") valid inequalities for $S_n$ to obtain a valid inequalities for $P_n$. As an application, we describe a surprising relationship between (a) the parsimonious property of relaxations of the Symmetric Traveling Salesman Polytope and (b) a connectivity property of the ridge graph of the Graphical Traveling Salesman Polyhedron."}
{"category": "Math", "title": "Fundamentals for Symplectic $\\mathcal{A}$-modules", "abstract": "Sheaf theoretically based Abstract Differential Geometry incorporates and generalizes all the classical differential geometry. Here, we undertake to partially explore the implications of Abstract Differential Geometry to classical symplectic geometry. The full investigation will be presented elsewhere."}
{"category": "Math", "title": "Miyawaki's $F_{12}$ Spinor L-function Conjecture", "abstract": "In this paper we prove the Miyawaki conjecture related to the spinor $L$--function of a Siegel cusp form of weight 12 and degree 3 as a special example of results related to Miyawaki lifts of odd degree."}
{"category": "Math", "title": "The Walsh model for $M_2^{*}$ Carleson", "abstract": "We study the Walsh model of a certain maximal truncation of Carleson's operator, related to the Return Times Theorem from Ergodic Theory."}
{"category": "Math", "title": "A generalization of inversion formulas of Pestov and Uhlmann", "abstract": "In this note, we give a generalization of the inversion formulas of Pestov-Uhlmann for the geodesic ray transform of functions and vector fields on simple 2-dimensional manifolds of constant curvature. The inversion formulas given here hold for 2-dimensional simple manifolds whose curvatures close to a constant."}
{"category": "Math", "title": "On the classification of gradient Ricci solitons", "abstract": "We show that the only complete shrinking gradient Ricci solitons with vanishing Weyl tensor are quotients of the standard ones. This gives a new proof of the Hamilton-Ivey-Perel'man classification of 3-dimensional shrinking gradient solitons. We also prove a classification for expanding gradient Ricci solitons with constant scalar curvature and suitably decaying Weyl tensor."}
{"category": "Math", "title": "Equidistribution of Horocyclic Flows on Complete Hyperbolic Surfaces of Finite Area", "abstract": "We provide a self-contained, accessible introduction to Ratner's Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early 1980s by Dani and Smillie and later reappeared as an illustrative special case of Ratner's work on the equidistribution of unipotent flows in homogeneous spaces. We also prove an interesting probabilistic result due to Breuillard: on the modular surface an arbitrary uncentered random walk on the horocycle through almost any point will fail to equidistribute, even though the horocycles are themselves equidistributed. In many aspects of this exposition we are indebted to Bekka and Mayer's more ambitious survey, \"Ergodic Theory and Topological Dynamics for Group Actions on Homogeneous Spaces.\""}
{"category": "Math", "title": "The maximum spectral radius of C_4-free graphs of given order and size", "abstract": "Let G be a graph of n vertices and m edges, and let G has no cycles of length 4. We give upper bounds on the adjacency spectral radius of G in terms of n and m."}
{"category": "Math", "title": "Spectrum of the product of Toeplitz matrices with application in probability", "abstract": "We study the spectrum of the product of two Toeplitz operators. Assume that the symbols of these operators are continuous and real-valued and that one of them is non-negative. We prove that the spectrum of the product of finite section Toeplitz matrices converges to the spectrum of the product of the semi-infinite Toeplitz operators. We give an example showing that the supremum of this set is not always the supremum of the product of the two symbols. Finally, we provide an application in probability which is the first motivation of this study. More precisely, we obtain a large deviation principle for Gaussian quadratic forms."}
{"category": "Math", "title": "How to facet a gemstone: from potential modularity to the proof of Serre's modularity conjecture", "abstract": "In this survey paper we present recent results obtained by Khare, Wintenberger and the author that have led to a proof of Serre's conjecture, such as existence of compatible families, modular upper bounds for universal deformation rings and existence of minimal lifts, prime switching and modularity propagation, weight reduction (via existence of conjugates) and (iterated) killing ramification. The main tools used in the proof of these results are modularity lifting theorems a la Wiles and a result of potential modularity due to R. Taylor."}
{"category": "Math", "title": "A homotopy theory for enrichment in simplicial modules", "abstract": "We put a Quillen model structure on the category of small categories enriched in simplicial $k$-modules and non-negatively graded chain complexes of $k$-modules, where $k$ is a commutative ring. The model structure is obtained by transfer from the model structure on simplicial categories due to J. Bergner."}
{"category": "Math", "title": "A beginner's guide to forcing", "abstract": "This expository paper, aimed at the reader without much background in set theory or logic, gives an overview of Cohen's proof (via forcing) of the independence of the continuum hypothesis. It emphasizes the broad outlines and the intuitive motivation while omitting most of the proofs. The reader must of course consult standard textbooks for the missing details, but this article provides a map of the forest so that the beginner will not get lost while forging through the trees."}
{"category": "Math", "title": "Singularity theorems and the Lorentzian splitting theorem for the Bakry-Emery-Ricci tensor", "abstract": "We consider the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry-Emery-Ricci curvature $Ric_f^m$ in timelike directions. We prove that they still hold when $m$ is finite, and when $m$ is infinite, they hold under the additional assumption that $f$ is bounded from above."}
{"category": "Math", "title": "Koszul differential graded algebras and BGG correspondence", "abstract": "The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When the Koszul DG algebra $A$ is AS-regular, the Ext-algebra $E$ of $A$ is Frobenius. In this case, similar to the classical BGG correspondence, there is an equivalence between the stable category of finitely generated left $E$-modules, and the quotient triangulated category of the full triangulated subcategory of the derived category of right DG $A$-modules consisting of all compact DG modules modulo the full triangulated subcategory consisting of all the right DG modules with finite dimensional cohomology. The classical BGG correspondence can derived from the DG version."}
{"category": "Math", "title": "Classification of Cohomogeneity One Manifolds in Low Dimensions", "abstract": "A cohomogeneity one manifold is a manifold with the action of a compact Lie group, whose quotient is one dimensional. Such manifolds are of interest in Riemannian geometry, in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. In this paper we classify compact simply connected cohomogeneity one manifolds in dimensions 5, 6 and 7. We also show that all such manifolds admit metrics of nonnegative sectional curvature, with the possible exception of two families of manifolds."}
{"category": "Math", "title": "When Are Torsionless Modules Projective?", "abstract": "In this paper, we study the problem when a finitely generated torsionless module is projective. Let $\\Lambda$ be an Artinian local algebra with radical square zero. Then a finitely generated torsionless $\\Lambda$-module $M$ is projective if ${\\rm Ext^1_\\Lambda}(M,M)=0$. For a commutative Artinian ring $\\Lambda$, a finitely generated torsionless $\\Lambda$-module $M$ is projective if the following conditions are satisfied: (1) ${\\rm Ext}^i_{\\Lambda}(M,\\Lambda)=0$ for $i=1,2,3$; and (2) ${\\rm Ext}^i_{\\Lambda}(M,M)=0$ for $i=1,2$. As a consequence of this result, we have that for a commutative Artinian ring $\\Lambda$, a finitely generated Gorenstein projective $\\Lambda$-module is projective if and only if it is selforthogonal."}
{"category": "Math", "title": "Strong non-collapsing and uniform Sobolev inequalities for Ricci flow with surgeries", "abstract": "We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a non-collapsing result stronger than Perelman's $\\kappa$ non-collapsing with surgery is derived. The proof is shorter and seems more accessible. The result also improves some earlier ones where the Sobolev inequality depended on the number of surgeries."}
{"category": "Math", "title": "Super-linear elliptic equation for the Pucci operator without growth restrictions for the data", "abstract": "In this paper we deal with existence and uniqueness of solution to super-linear problems for the Pucci operator: $$ -\\M^+(D^2u)+|u|^{s-1}u=f(x) \\quad {in} \\RR^n, $$ where $s>1$ and $f$ satisfies only local integrability conditions. This result is well known when, instead of the Pucci operator, the Laplacian or a divergence form operator is considered. Our existence results use the Alexandroff-Bakelman-Pucci inequality since we cannot use any variational formulation. For radially symmetric $f$ we can prove our results under less local integrability assumptions, taking advantage of an appropriate variational formulation. We also obtain an existence result with boundary explosion in smooth domains."}
{"category": "Math", "title": "Ramanujan-type formulae for $1/\\pi$: A second wind?", "abstract": "In 1914 S. Ramanujan recorded a list of 17 series for $1/\\pi$. We survey the methods of proofs of Ramanujan's formulae and indicate recently discovered generalizations, some of which are not yet proven."}
{"category": "Math", "title": "Stochastic adaptation of importance sampler", "abstract": "Improving efficiency of importance sampler is at the center of research in Monte Carlo methods. While adaptive approach is usually difficult within the Markov Chain Monte Carlo framework, the counterpart in importance sampling can be justified and validated easily. We propose an iterative adaptation method for learning the proposal distribution of an importance sampler based on stochastic approximation. The stochastic approximation method can recruit general iterative optimization techniques like the minorization-maximization algorithm. The effectiveness of the approach in optimizing the Kullback divergence between the proposal distribution and the target is demonstrated using several simple examples."}
{"category": "Math", "title": "Connectivity of the Product Replacement Algorithm Graph of PSL(2,q)", "abstract": "The product replacement algorithm is a practical algorithm to construct random elements of a finite group G. It can be described as a random walk on a graph whose vertices are the generating k-tuples of G (for a fixed k). We show that if G is PSL(2,q) or PGL(2,q), where q is a prime power, then this graph is connected for any k>=4. This generalizes former results obtained by Gilman and Evans."}
{"category": "Math", "title": "Signal Recovery from Incomplete and Inaccurate Measurements via Regularized Orthogonal Matching Pursuit", "abstract": "We demonstrate a simple greedy algorithm that can reliably recover a d-dimensional vector v from incomplete and inaccurate measurements x. Here our measurement matrix is an N by d matrix with N much smaller than d. Our algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to close the gap between two major approaches to sparse recovery. It combines the speed and ease of implementation of the greedy methods with the strong guarantees of the convex programming methods. For any measurement matrix that satisfies a Uniform Uncertainty Principle, ROMP recovers a signal with O(n) nonzeros from its inaccurate measurements x in at most n iterations, where each iteration amounts to solving a Least Squares Problem. The noise level of the recovery is proportional to the norm of the error, up to a log factor. In particular, if the error vanishes the reconstruction is exact. This stability result extends naturally to the very accurate recovery of approximately sparse signals."}
{"category": "Math", "title": "An expansion for polynomials orthogonal over an analytic Jordan curve", "abstract": "We consider polynomials that are orthogonal over an analytic Jordan curve L with respect to a positive analytic weight, and show that each such polynomial of sufficiently large degree can be expanded in a series of certain integral transforms that converges uniformly in the whole complex plane. This expansion yields, in particular and simultaneously, Szego's classical strong asymptotic formula and a new integral representation for the polynomials inside L. We further exploit such a representation to derive finer asymptotic results for weights having finitely many singularities (all of algebraic type) on a thin neighborhood of the orthogonality curve. Our results are a generalization of those previously obtained in [7] for the case of L being the unit circle."}
{"category": "Math", "title": "Dynkin's Isomorphism with Sign Structure", "abstract": "The Dynkin isomorphism associates a Gaussian field to a Markov chain. These Gaussian fields can be used as priors for prediction and time series analysis. Dynkin's construction gives Gaussian fields with all non-negative covariances. We extend Dynkin's construction (by introducing a sign structure on the Markov chain) to allow general covariance sign patterns."}
{"category": "Math", "title": "Behavior of bounded solutions of quasilinear elliptic equations on Riemannian manifolds", "abstract": "This paper deals with bounded solutions of quasilinear elliptic equations on Riemannian manifolds satisfying special condition."}
{"category": "Math", "title": "Commutation relations and Markov chains", "abstract": "It is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain Markov chains. Examples studied include random walk on irreducible representations, a local random walk on partitions whose stationary distribution is the Ewens distribution, and some birth-death chains."}
{"category": "Math", "title": "Lyapunov Exponents of Free Operators", "abstract": "Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede-Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents of free variables are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko-Pastur law, and relate this example to C. M. Newman's \"triangle\" law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede-Kadison determinant and Voiculescu's S-transform."}
{"category": "Math", "title": "Cluster combinatorics of d-cluster categories", "abstract": "We study the cluster combinatorics of $d-$cluster tilting objects in $d-$cluster categories. By using mutations of maximal rigid objects in $d-$cluster categories which are defined similarly for $d-$cluster tilting objects, we prove the equivalences between $d-$cluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of $d+1$ triangles of $d-$cluster tilting objects in [IY], we prove that any almost complete $d-$cluster tilting object has exactly $d+1$ complements, compute the extension groups between these complements, and study the middle terms of these $d+1$ triangles. All results are the extensions of corresponding results on cluster tilting objects in cluster categories established in [BMRRT] to $d-$cluster categories. They are applied to the Fomin-Reading's generalized cluster complexes of finite root systems defined and studied in [FR2] [Th] [BaM1-2], and to that of infinite root systems [Zh3]."}
{"category": "Math", "title": "The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves", "abstract": "We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through \"congruence\" subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f(c,d)=c^2+d^2, but restrict (c,d) to the orbit O = (0,1).Gamma, where Gamma is an infinite-index non-elementary finitely-generated subgroup of SL(2,Z). Assume that the Reimann surface Gamma\\H^2 has a cusp at infinity. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R>4/(delta-theta), where theta>1/2 is the spectral gap and delta<1 is the Hausdorff dimension of the limit set of Gamma. If delta>149/150, then we can take theta=5/6, giving R=25. The limit of this method is R=9 for delta-theta>4/9. This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture."}
{"category": "Math", "title": "Local wellposedness for the 2+1 dimensional monopole equation", "abstract": "The space-time monopole equation on $\\R^{2+1}$ can be derived by a dimensional reduction of the anti-self-dual Yang Mills equations on $\\R^{2+2}$. It can be also viewed as the hyperbolic analog of Bogomolny equations. We uncover null forms in the nonlinearities and employ optimal bilinear estimates in the framework of Wave-Sobolev spaces. As a result, we show the equation is locally wellposed in the Coulomb gauge for initial data sufficiently small in $H^s$ for $s>{1/4}$."}
{"category": "Math", "title": "Prolongations of Lie algebras and applications", "abstract": "We study the skew-symmetric prolongation of a Lie subalgebra $\\g \\subseteq \\mathfrak{so}(n)$, in other words the intersection $\\Lambda^3 \\cap (\\Lambda^1 \\otimes \\g)$.We compute this space in full generality. Applications include uniqueness results for connections with skew-symmetric torsion and also the proof of the Euclidean version of a conjecture posed in \\cite{ofarill} concerning a class of Pl\\\"ucker-type embeddings. We also derive a classification of the metric k-Lie algebras (or Filipov algebras), in positive signature and finite dimension. Prolongations of Lie algebras can also be used to finish the classification, started in \\cite{datri}, of manifolds admitting Killing frames, or equivalently flat connections with 3-form torsion. Next we study specific properties of invariant 4-forms of a given metric representation and apply these considerations to classify the holonomy representation of metric connections with vectorial torsion, that is with torsion contained in $\\Lambda^1 \\subseteq \\Lambda^1 \\otimes \\Lambda^2$."}
{"category": "Math", "title": "One metric result about analytic continuation of some Dirichlet series", "abstract": "In this paper we consider certain 1-parametric family of Dirichlet series. For a particular value of the parameter the series turns into the Dirichlet series for the Riemann zeta function. We prove that almost every series of the family has analytic continuation to the half plane Re s > 1/2 where it doesn't vanish. The result was obtained before by different authors. We give its simple proof in terms of estimates of some trigonometric sums."}
{"category": "Math", "title": "Curve alignment by moments", "abstract": "A significant problem with most functional data analyses is that of misaligned curves. Without adjustment, even an analysis as simple as estimation of the mean will fail. One common method to synchronize a set of curves involves equating ``landmarks'' such as peaks or troughs. The landmarks method can work well but will fail if marker events can not be identified or are missing from some curves. An alternative approach, the ``continuous monotone registration'' method, works by transforming the curves so that they are as close as possible to a target function. This method can also perform well but is highly dependent on identifying an accurate target function. We develop an alignment method based on equating the ``moments'' of a given set of curves. These moments are intended to capture the locations of important features which may represent local behavior, such as maximums and minimums, or more global characteristics, such as the slope of the curve averaged over time. Our method works by equating the moments of the curves while also shrinking toward a common shape. This allows us to capture the advantages of both the landmark and continuous monotone registration approaches. The method is illustrated on several data sets and a simulation study is performed."}
{"category": "Math", "title": "C*-Algebras over Topological Spaces: The Bootstrap Class", "abstract": "We carefully define and study C*-algebras over topological spaces, possibly non-Hausdorff, and review some relevant results from point-set topology along the way. We explain the triangulated category structure on the bivariant Kasparov theory over a topological space. We introduce and describe an analogue of the bootstrap class for C*-algebras over a finite topological space."}
{"category": "Math", "title": "On contact equivalence of systems of ordinary differential equations", "abstract": "We consider a problem of equivalence of generic pairs $(X,V)$ on a manifold $M$, where $V$ is a distribution of rank $m$ and $X$ is a distribution of rank one. We construct a canonical bundle with a canonical frame. We prove that two pairs are equivalent if and only if the corresponding frames are diffeomorphic. As a particular case, with $V$ integrable, we provide a new solution to the problem of contact equivalence of systems of $m$ ordinary differential equations: $x^{(k+1)}=F(t,x,x',...,x^{(k)})$, where $k>2$ or $k=2$ and $m>1$."}
{"category": "Math", "title": "Detecting abrupt changes of the long-range dependence or the self-similarity of a Gaussian process", "abstract": "In this paper, an estimator of $m$ instants ($m$ is known) of abrupt changes of the parameter of long-range dependence or self-similarity is proved to satisfy a limit theorem with an explicit convergence rate for a sample of a Gaussian process. In each estimated zone where the parameter is supposed not to change, a central limit theorem is established for the parameter's (of long-range dependence, self-similarity) estimator and a goodness-of-fit test is also built. {\\it To cite this article: J.M. Bardet, I. Kammoun, C. R. Acad. Sci. Paris, Ser. I 340 (2007).}"}
{"category": "Math", "title": "Abel maps of Gorenstein curves", "abstract": "For a Gorenstein curve X and a nonsingular point P of X, we construct Abel maps A from X to J_X^1 and A_P from X to J_X^0, where J_X^i is the moduli scheme for simple, torsion-free, rank-1 sheaves on X of degree i. The image curves of A and A_P are shown to have the same arithmetic genus of X. Also, A and A_P are shown to be embeddings away from rational subcurves L of X meeting the closure of X-L in separating nodes. Finally, we establish a connection with Seshadri's moduli scheme U_X(1) for semistable, torsion-free, rank-1 sheaves on X, obtaining an embedding of A(X) into U_X(1)."}
{"category": "Math", "title": "Accounting for spatial correlation in the scan statistic", "abstract": "The spatial scan statistic is widely used in epidemiology and medical studies as a tool to identify hotspots of diseases. The classical spatial scan statistic assumes the number of disease cases in different locations have independent Poisson distributions, while in practice the data may exhibit overdispersion and spatial correlation. In this work, we examine the behavior of the spatial scan statistic when overdispersion and spatial correlation are present, and propose a modified spatial scan statistic to account for that. Some theoretical results are provided to demonstrate that ignoring the overdispersion and spatial correlation leads to an increased rate of false positives, which is verified through a simulation study. Simulation studies also show that our modified procedure can substantially reduce the rate of false alarms. Two data examples involving brain cancer cases in New Mexico and chickenpox incidence data in France are used to illustrate the practical relevance of the modified procedure."}
{"category": "Math", "title": "Bounds on Tur{\\'a}n determinants", "abstract": "Let \\mu denote a symmetric probability measure on [-1,1] and let (p_n) be the corresponding orthogonal polynomials normalized such that p_n(1)=1. We prove that the normalized Tur{\\'a}n determinant \\Delta_n(x)/(1-x^2), where \\Delta_n=p_n^2-p_{n-1}p_{n+1}, is a Tur{\\'a}n determinant of order n-1 for orthogonal polynomials with respect to (1-x^2)d\\mu(x). We use this to prove lower and upper bounds for the normalized Tur{\\'a}n determinant in the interval -1<x<1."}
{"category": "Math", "title": "Spectral Radius and Amenability in Hilbert Geometries", "abstract": "We study the bottom of the spectrum in Hilbert geometries, we show that it is zero if and only if the geometry is amenable, in other words if and only if it admits a F\\\"olner sequence. We also show that the bottom of the spectrum admits an upper bound, which depends only on the dimension and which is the bottom of the spectrum of the Hyperbolic geometry of the same dimension. Horoballs, from a purely metric point of view, and their relation with the bottom of the spectrum in Hilbert geometries are briefly studied."}
{"category": "Math", "title": "Binary and Ternary Quasi-perfect Codes with Small Dimensions", "abstract": "The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of infinite families of QP codes which includes all binary, ternary and quaternary codes known to is. We continue further with a list of sporadic examples of binary and ternary QP codes. Later we present the results of our investigation where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions up to 13 are classified."}
{"category": "Math", "title": "Classification of Connections on Higher-Dimensional Non-Commutative Tori", "abstract": "If X is a full, finitely generated, projective module over a non-commutative torus, the Yang-Mills functional attains its minimum exactly on the flat connections on X. We classify the flat connections on modules admitting integrable connections."}
{"category": "Math", "title": "Crossing paths in 2D Random Walks", "abstract": "We investigate crossing path probabilities for two agents that move randomly in a bounded region of the plane or on a sphere (denoted $R$). At each discrete time-step the agents move, independently, fixed distances $d_1$ and $d_2$ at angles that are uniformly distributed in $(0,2\\pi)$. If $R$ is large enough and the initial positions of the agents are uniformly distributed in $R$, then the probability of paths crossing at the first time-step is close to $ 2d_1d_2/(\\pi A[R])$, where $A[R]$ is the area of $R$. Simulations suggest that the long-run rate at which paths cross is also close to $2d_1d_2/(\\pi A[R])$ (despite marked departures from uniformity and independence conditions needed for such a conclusion)."}
{"category": "Math", "title": "Correction: A correlated topic model of Science", "abstract": "Correction to Annals of Applied Statistics 1 (2007) 17--35 [doi:10.1214/07-AOAS114]"}
{"category": "Math", "title": "How universal are asymptotics of disconnection times in discrete cylinders?", "abstract": "We investigate the disconnection time of a simple random walk in a discrete cylinder with a large finite connected base. In a recent article of A. Dembo and the author it was found that for large $N$ the disconnection time of $G_N\\times\\mathbb{Z}$ has rough order $|G_N|^2$, when $G_N=(\\mathbb{Z}/N\\mathbb{Z})^d$. In agreement with a conjecture by I. Benjamini, we show here that this behavior has broad generality when the bases of the discrete cylinders are large connected graphs of uniformly bounded degree."}
{"category": "Math", "title": "Curvature estimates for minimal surfaces with total boundary curvature less than 4\\pi", "abstract": "We establish a curvature estimate for classical minimal surfaces with total boundary curvature less than 4\\pi. The main application is a bound on the genus of these surfaces depending solely on the geometry of the boundary curve. We also prove that the set of simple closed curves with total curvature less than 4\\pi and which do not bound an orientable compact embedded minimal surface of genus greater than g, for any given g, is open in the C^{2,\\alpha} topology."}
{"category": "Math", "title": "Residues of Intertwining Operators for Classical Groups with an Appendix \"$L$-Functions and Poles of Intertwining Operators\"", "abstract": "Let $\\tilde{G}$ be a symplectic or even orthogonal group over a p-adic field $F$, and $M$ the Levi factor of a maximal parabolic subgroup of $\\tilde{G}$. Suppose that $M$ has the shape of three blocks of the same size. Let $\\pi$ be a supercuspidal representation of $M$. In this paper we give a simple explicit expression for the residue of the standard intertwining operator for the parabolic induction of $\\pi$ from $M$ to $G$."}
{"category": "Math", "title": "Completely positive maps and extremal K-set", "abstract": "We explain how to find the KK-theoretic counterpart of extremal K-set defined by Larry Brown and Gert Pedersen."}
{"category": "Math", "title": "Rigidity of representations in SO(4,1) for Dehn fillings on 2-bridge knots", "abstract": "We prove that, for a hyperbolic two bridge knot, infinitely many Dehn fillings are rigid in $SO_0(4,1)$. Here rigidity means that any discrete and faithful representation in $SO_0(4,1)$ is conjugate to the holonomy representation in $SO_0(3,1)$. We also show local rigidity for almost all Dehn fillings."}
{"category": "Math", "title": "Adding a uniton via the DPW method", "abstract": "In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces out of certain holomorphic one forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we prove that the Gauss bundle of a harmonic map of finite type into a Grassmannian is also of finite type."}
{"category": "Math", "title": "Hyperbolic Balance Laws with a Dissipative Non Local Source", "abstract": "This paper considers systems of balance law with a dissipative non local source. A global in time well posedness result is obtained. Estimates on the dependence of solutions from the flow and from the source term are also provided. The technique relies on a recent result on quasidifferential equations in metric spaces."}
{"category": "Math", "title": "On Transverse Knots and Branched Covers", "abstract": "We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic transverse knots with the same self-linking number. These pairs of knots include most of the non-transversely simple knots of Birman-Menasco and Ng-Ozsvath-Thurston."}
{"category": "Math", "title": "The Lefschetz property for barycentric subdivisions of shellable complexes", "abstract": "We show that an 'almost strong Lefschetz' property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the $h$-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its $g$-vector is an $M$-sequence. In particular, the (combinatorial) $g$-conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations, where permutations are grouped by the number of descents and the image of 1."}
{"category": "Math", "title": "Symplectic reflection algebras", "abstract": "We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on category O."}
{"category": "Math", "title": "Stochastic Completeness of Graphs", "abstract": "In this thesis, we analyze the stochastic completeness of a heat kernel on graphs which is a function of three variables: a pair of vertices and a continuous time, for infinite, locally finite, connected graphs. For general graphs, a sufficient condition for stochastic completeness is given in terms of the maximum valence on spheres about a fixed vertex. That this result is optimal is shown by studying a particular family of trees. We also prove a lower bound on the bottom of the spectrum for the discrete Laplacian and use this lower bound to show that in certain cases the Laplacian has empty essential spectrum."}
{"category": "Math", "title": "Good Reductions of Shimura Varieties of Hodge Type in Arbitrary Unramified Mixed Characteristic, Part II", "abstract": "We prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of Milne pertaining to integral canonical models of Shimura varieties of Hodge type."}
{"category": "Math", "title": "Character sums to smooth moduli are small", "abstract": "Recently, Granville and Soundararajan have made fundamental breakthroughs in the study of character sums. Building on their work and using estimates on short character sums developed by Graham-Ringrose and Iwaniec, we improve the Polya-Vinogradov inequality for characters with smooth conductor."}
{"category": "Math", "title": "A simple proof for the existence of Zariski decompositions on surfaces", "abstract": "In this note we give a quick and simple proof of the existence (and uniqueness) of Zariski decompositions on surfaces. While Zariski's original proof employs a rather sophisticated procedure to construct the negative part of the decomposition, the present approach is based on the idea that the positive part can be constructed from a maximality condition. It may also be useful that this approach yields a practical algorithm for the computation of the positive part."}
{"category": "Math", "title": "Regular elliptic boundary-value problem in a two-sided refined scale of spaces", "abstract": "A regular elliptic boundary-value problem over a bounded domain with a smooth boundary is studied. We prove that the operator of this problem is a Fredholm one in the two-sided refined scale of the functional Hilbert spaces and generates a complete collection of isomorphisms. Elements of this scale are the isotropic spaces of Hormander-Volevich-Paneah and some its modifications. A priori estimate for the solution is established and its regularity is investigated."}
{"category": "Math", "title": "On fibering and splitting of 5-manifolds over the circle", "abstract": "Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman--Quinn topological surgery. Indeed, our key technique is topological cobordism, which may not be the trace of surgeries."}
{"category": "Math", "title": "Lines of Curvature on Surfaces, Historical Comments and Recent Developments", "abstract": "This survey starts with the historical landmarks leading to the study of principal configurations on surfaces, their structural stability and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of the qualitative theory of differential equations (QTDE), founded by Poincar\\'e in 1881. Here are also outlined a number of recent results developed after the assimilation into the subject of concepts and problems from the QTDE and Dynamical Systems, such as Structural Stability, Bifurcations and Genericity, among others, as well as extensions to higher dimensions. References to original works are given and open problems are proposed at the end of some sections."}
{"category": "Math", "title": "Quivers and the Euclidean group", "abstract": "We show that the category of representations of the Euclidean group of orientation-preserving isometries of two-dimensional Euclidean space is equivalent to the category of representations of the preprojective algebra of infinite type A. We also consider the moduli space of representations of the Euclidean group along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. Using these identifications, we prove various results about the representation theory of the Euclidean group. In particular, we prove it is of wild representation type but that if we impose certain restrictions on weight decompositions, we obtain only a finite number of indecomposable representations."}
{"category": "Math", "title": "Around the Gysin triangle II", "abstract": "We study the construction and properties of the Gysin triangle in an axiomatic framework which covers triangulated mixed motives and MGl-modules over an arbitrary base S. This allows to define the Gysin morphism associated to a projective morphism between smooth S-schemes and prove duality for projective smooth S-schemes. As part of the construction, cobordism classes are considered and we give a proof of the Myschenko theorem generalized in our context - this in fact gives another proof of the latter theorem in classical stable homotopy through complex realization. Finally, these constructions apply to rigid cohomology through the notion of a mixed Weil theory introduced by D.-C. Cisinski and the author in another work."}
{"category": "Math", "title": "The structure of critical sets for F_p arithmetic progressions", "abstract": "Fix a prime p and a density 0 < d <= 1. Among all functions f : F_p -> [0,1], what can one say about those which assign minimal weight to three-term arithmetic progressions -- that is, the sum of f(a)f(a+x)f(a+2x) is minimal as we sum over all a and x -- subject to the density constraint that the expected value of f equals d? In the present paper we show three things about them: 1) Such f are nearly indicator functions; 2) They enjoy a certain ``local minimal'' property; and, 3) They are approximately indicator functions for certain sumsets A+B."}
{"category": "Math", "title": "Staggered sheaves on partial flag varieties", "abstract": "Staggered $t$-structures are a class of $t$-structures on derived categories of equivariant coherent sheaves. In this note, we show that the derived category of coherent sheaves on a partial flag variety, equivariant for a Borel subgroup, admits an artinian staggered $t$-structure. As a consequence, we obtain a basis for its equivariant $K$-theory consisting of simple staggered sheaves."}
{"category": "Math", "title": "James' Conjecture for Hecke algebras of exceptional type, I", "abstract": "In this paper, and a second part to follow, we complete the programme (initiated more than 15 years ago) of determining the decomposition numbers and verifying James' Conjecture for Iwahori--Hecke algebras of exceptional type. The new ingredients which allow us to achieve this aim are: - the fact, recently proved by the first author, that all Hecke algebras of finite type are cellular in the sense of Graham--Lehrer, and - the explicit determination of $W$-graphs for the irreducible (generic) representations of Hecke algebras of type $E_7$ and $E_8$ by Howlett and Yin. Thus, we can reduce the problem of computing decomposition numbers to a manageable size where standard techniques, e.g., Parker's {\\sf MeatAxe} and its variations, can be applied. In this part, we describe the theoretical foundations for this procedure."}
{"category": "Math", "title": "Unique Tournaments and Radar Tracking", "abstract": "The sequence counting the number of unique tournaments with n people is the same as the sequence counting non-tracking binary strings corresponding to n-2 radar observations with the tracking rule \"3 out of 5 with loss 2.\" This fact allows us to build a bijection between unique tournaments and non-tracking binary strings."}
{"category": "Math", "title": "Primitive decompositions of Johnson graphs", "abstract": "A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts."}
{"category": "Math", "title": "Bridge Number and Conway Products", "abstract": "Schubert proved that, given a composite link $K$ with summands $K_{1}$ and $K_{2}$, the bridge number of $K$ satisfies the following equation: $$\\beta(K)=\\beta(K_{1})+\\beta(K_{2})-1.$$ In ``Conway Produts and Links with Multiple Bridge Surfaces\", Scharlemann and Tomova proved that, given links $K_{1}$ and $K_{2}$, there is a Conway product $K_{1}\\times_{c}K_{2}$ such that $$\\beta(K_{1}\\times_{c} K_{2}) \\leq \\beta(K_{1}) + \\beta(K_{2}) - 1$$ In this paper, we define the generalized Conway product $K_{1}\\ast_{c}K_{2}$ and prove the lower bound $\\beta(K_{1}\\ast_{c}K_{2}) \\geq \\beta(K_{1})-1$ where $K_{1}$ is the distinguished factor of the generalized product. We go on to show this lower bound is tight for an infinite class of links with arbitrarily high bridge number."}
{"category": "Math", "title": "Extension Theorems for Spheres in the Finite Field Setting", "abstract": "In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields.In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a subset of spheres. As a result, we improve the Tomas-Stein exponents, our previous results. The analytic approach and the explicit formula for Fourier transform of the characteristic function on spheres play an important role to get good bounds for exponential sums."}
{"category": "Math", "title": "Cayley sum graphs and eigenvalues of $(3,6)$-fullerenes", "abstract": "We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, \"(3,6)-fullerenes\", have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form $\\{\\lambda,-\\lambda\\}$ except for the four eigenvalues $\\{3,-1,-1,-1\\}$. We exhibit other families of graphs which are \"spectrally nearly bipartite\" in this sense. Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs."}
{"category": "Math", "title": "Evaluations of multiple Dirichlet $L$-values via symmetric functions", "abstract": "We explicitly evaluate a special type of multiple Dirichlet $L$-values at positive integers in two different ways: One approach involves using symmetric functions, while the other involves using a generating function of the values. Equating these two expressions, we derive several summation formulae involving the Bernoulli and Euler numbers. Moreover, values at non-positive integers, called central limit values, are also studied."}
{"category": "Math", "title": "The joint distribution of occupation times of skip-free Markov processes and a class of multivariate exponential distributions", "abstract": "For a skip-free Markov process on non-negative integers with generator matrix Q, we evaluate the joint Laplace transform of the occupation times before hitting the state n (starting at 0). This Laplace transform has a very straightforward and familiar expression. We investigate the properties of this Laplace transform, especially the conditions under which the occupation times form a Markov chain."}
{"category": "Math", "title": "Galois extensions ramified only at one prime", "abstract": "This paper gives some restrictions on finite groups that can occur as Galois groups of extensions over $\\Q$ and over $\\F_q(t)$ ramified only at one finite prime. Over $\\Q$, we strengthen results of Jensen and Yui about dihedral extensions and rule out some non-solvable groups. Over $\\F_q(t)$ restrictions are given for symmetric groups and dihedral groups to occur as tamely ramified extension over $\\F_q(t)$ ramified only at one prime."}
{"category": "Math", "title": "Smoothing $\\ell_1$-penalized estimators for high-dimensional time-course data", "abstract": "When a series of (related) linear models has to be estimated it is often appropriate to combine the different data-sets to construct more efficient estimators. We use $\\ell_1$-penalized estimators like the Lasso or the Adaptive Lasso which can simultaneously do parameter estimation and model selection. We show that for a time-course of high-dimensional linear models the convergence rates of the Lasso and of the Adaptive Lasso can be improved by combining the different time-points in a suitable way. Moreover, the Adaptive Lasso still enjoys oracle properties and consistent variable selection. The finite sample properties of the proposed methods are illustrated on simulated data and on a real problem of motif finding in DNA sequences."}
{"category": "Math", "title": "Special Values of Generalized Polylogarithms", "abstract": "We study values of generalized polylogarithms at various points and relationships among them. Polylogarithms of small weight at the points 1/2 and -1 are completely investigated. We formulate a conjecture about the structure of the linear space generated by values of generalized polylogarithms."}
{"category": "Math", "title": "Efficient blind search: Optimal power of detection under computational cost constraints", "abstract": "Some astronomy projects require a blind search through a vast number of hypotheses to detect objects of interest. The number of hypotheses to test can be in the billions. A naive blind search over every single hypothesis would be far too costly computationally. We propose a hierarchical scheme for blind search, using various \"resolution\" levels. At lower resolution levels, \"regions\" of interest in the search space are singled out with a low computational cost. These regions are refined at intermediate resolution levels and only the most promising candidates are finally tested at the original fine resolution. The optimal search strategy is found by dynamic programming. We demonstrate the procedure for pulsar search from satellite gamma-ray observations and show that the power of the naive blind search can almost be matched with the hierarchical scheme while reducing the computational burden by more than three orders of magnitude."}
{"category": "Math", "title": "On the constant in the Mertens product for arithmetic progressions. II. Numerical values", "abstract": "We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions $a \\bmod q$, for $q \\in \\{3$, ..., $100\\}$ and $(a, q) = 1$."}
{"category": "Math", "title": "Comparison of some solution concepts for linear first-order hyperbolic differential equations with non-smooth coefficients", "abstract": "We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on refined techniques concerning energy estimates. We provide a series of examples both as simple illustrations of the notions and conditions involved but also to show logical independence among the concepts."}
{"category": "Math", "title": "Estimation in a class of nonlinear heteroscedastic time series models", "abstract": "Parameter estimation in a class of heteroscedastic time series models is investigated. The existence of conditional least-squares and conditional likelihood estimators is proved. Their consistency and their asymptotic normality are established. Kernel estimators of the noise's density and its derivatives are defined and shown to be uniformly consistent. A simulation experiment conducted shows that the estimators perform well for large sample size."}
{"category": "Math", "title": "Diamond-$\\alpha$ Jensen's Inequality on Time Scales", "abstract": "The theory and applications of dynamic derivatives on time scales has recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-$\\alpha$ derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen's inequality on time scales via the diamond-$\\alpha$ integral and present some corollaries, including H\\\"{o}lder's and Minkowski's diamond-$\\alpha$ integral inequalities."}
{"category": "Math", "title": "Lemma Poincar\\'e for L_infty,loc - forms", "abstract": "We show that every closed L_infty,loc - form on R^n is exact. Differential is understood in the sense of currents. The proof does not use any explicit geometric constructions. De Rham theorem follows."}
{"category": "Math", "title": "Random Cluster Tessellations", "abstract": "This article describes, in elementary terms, a generic approach to produce discrete random tilings and similar random structures by using point process theory. The standard Voronoi and Delone tilings can be constructed in this way. For this purpose, convex polytopes are replaced by their vertex sets. Three explicit constructions are given to illustrate the concept."}
{"category": "Math", "title": "Local tail bounds for functions of independent random variables", "abstract": "It is shown that functions defined on $\\{0,1,...,r-1\\}^n$ satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger ``local'' sub-Gaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand's [Ann. Probab. 22 (1994) 1576--1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on $\\{0,1,...,r-1\\}^n$ for $r\\ge2$."}
{"category": "Math", "title": "Random graph models of communication network topologies", "abstract": "We consider a variant of so called power-law random graph. A sequence of expected degrees corresponds to a power-law degree distribution with finite mean and infinite variance. In previous works the asymptotic picture with number of nodes limiting to infinity has been considered. It was found that an interesting structure appears. It has resemblance with such graphs like the Internet graph. Some simulations have shown that a finite sized variant has similar properties as well. Here we investigate this case in more analytical fashion, and, with help of some simple lower bounds for large valued expectations of relevant random variables, we can shed some light into this issue. A new term, 'communication range random graph' is introduced to emphasize that some further restrictions are needed to have a relevant random graph model for a reasonable sized communication network, like the Internet. In this case a pleasant model is obtained, giving the opportunity to understand such networks on an intuitive level. This would be beneficial in order to understand, say, how a particular routing works in such networks."}
{"category": "Math", "title": "Approximating Data with weighted smoothing Splines", "abstract": "Given a data set (t_i, y_i), i=1,..., n with the t_i in [0,1] non-parametric regression is concerned with the problem of specifying a suitable function f_n:[0,1] -> R such that the data can be reasonably approximated by the points (t_i, f_n(t_i)), i=1,..., n. If a data set exhibits large variations in local behaviour, for example large peaks as in spectroscopy data, then the method must be able to adapt to the local changes in smoothness. Whilst many methods are able to accomplish this they are less successful at adapting derivatives. In this paper we show how the goal of local adaptivity of the function and its first and second derivatives can be attained in a simple manner using weighted smoothing splines. A residual based concept of approximation is used which forces local adaptivity of the regression function together with a global regularization which makes the function as smooth as possible subject to the approximation constraints."}
{"category": "Math", "title": "PAC-Bayesian Bounds for Randomized Empirical Risk Minimizers", "abstract": "The aim of this paper is to generalize the PAC-Bayesian theorems proved by Catoni in the classification setting to more general problems of statistical inference. We show how to control the deviations of the risk of randomized estimators. A particular attention is paid to randomized estimators drawn in a small neighborhood of classical estimators, whose study leads to control the risk of the latter. These results allow to bound the risk of very general estimation procedures, as well as to perform model selection."}
{"category": "Math", "title": "Stokes matrices of hypergeometric integrals", "abstract": "In this work we compute the Stokes matrices of the ordinary differential equation satisfied by the hypergeometric integrals associated to an arrangement of hyperplanes in generic position. This generalizes the computation done by Ramis and Duval for confluent hypergeometric functions, which correspond to the arrangement of two points on the line. The proof is based on an explicit description of a base of canonical solutions as integrals on the cones of the arrangement, and combinatorial relations between integrals on cones and on domains."}
{"category": "Math", "title": "Explicit construction of manifolds realizing the prescribed homology classes", "abstract": "We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the fundamental class of a manifold. Our main result is an explicit purely combinatorial construction that for a given integral cycle provides a combinatorial manifold realizing a multiple of the homology class of this cycle. The construction is based on a local procedure for resolving singularities of a pseudo-manifold. We give an application of our result to the problem of constructing a combinatorial manifold with the prescribed set of links of vertices."}
{"category": "Math", "title": "The \\'Etale Homology and The Cycle Maps in Adic Coefficients", "abstract": "In this article, we define the l-adic homology for a morphism of schemes satisfying certain finiteness conditions. This homology has these functors similar to the Chow groups: proper push-forward, flat pull-back, base change, cap-product, etc. In particular on singular varieties, this kind of l-adic homology behaves much better that the classical l-adic cohomology. As an application, we give an much easier approach to construct the cycle maps for arbitrary algebraic schemes over fields of finite cohomology dimension. And we prove these cycle maps kill the algebraic equivalences and commute with the Chern action of locally free sheaves."}
{"category": "Math", "title": "Coset decomposition for semisimple Hopf algebras", "abstract": "The notion of double coset for semisimple finite dimensional Hopf algebras is introduced. This is done by considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. As an application formulae for the restriction of the irreducible characters to normal Hopf subalgebras are given."}
{"category": "Math", "title": "On small homotopies of loops", "abstract": "Two natural questions are answered in the negative: (1) If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic? (2) Can adding arcs to a space cause an essential curve to become nulhomotopic? The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and $\\pi_1$-shape injective."}
{"category": "Math", "title": "Vinberg's \\theta-groups in positive characteristic and Kostant-Weierstrass slices", "abstract": "We generalize the basic results of Vinberg's \\theta-groups, or periodically graded reductive Lie algebras, to fields of good positive characteristic. To this end we clarify the relationship between the little Weyl group and the (standard) Weyl group. We deduce that the ring of invariants associated to the grading is a polynomial ring. This approach allows us to prove the existence of a KW-section for a classical graded Lie algebra (in zero or good characteristic), confirming a conjecture of Popov in this case."}
{"category": "Math", "title": "Generalizations of Swierczkowski's lemma and the arity gap of finite functions", "abstract": "Swierczkowski's Lemma - as it is usually formulated - asserts that if f is an at least quaternary operation on a finite set A and every operation obtained from f by identifying a pair of variables is a projection, then f is a semiprojection. We generalize this lemma in various ways. First, it is extended to B-valued functions on A instead of operations on A and to essentially at most unary functions instead of projections. Then we characterize the arity gap of functions of small arities in terms of quasi-arity, which in turn provides a further generalization of Swierczkowski's Lemma. Moreover, we explicitly classify all pseudo-Boolean functions according to their arity gap. Finally, we present a general characterization of the arity gaps of B-valued functions on arbitrary finite sets A."}
{"category": "Math", "title": "An Upper Bound of the Total Q-Curvature and Its Isoperimetric Deficit for Higher-dimensional Conformal Euclidean Metrics", "abstract": "The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $\\mathbb R^n$, $n\\ge 3$ with scalar curvature being nonnegative near infinity and Q-curvature being absolutely convergent."}
{"category": "Math", "title": "Euler-Mahonian Statistics On Ordered Set Partitions (II)", "abstract": "We study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of \\stein (Arxiv:math.CO/0605670). Our basic idea is to encode ordered partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. We also give a partition version of MacMahon's theorem on the equidistribution of the statistics inversion number and major index on words."}
{"category": "Math", "title": "Lagrangian Klein bottles in R^{2n}", "abstract": "It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into R^{2n} if and only if n is odd."}
{"category": "Math", "title": "Irrationalit\\'e aux entiers impairs positifs d'un q-analogue de la fonction zeta de Riemann", "abstract": "In this paper, we focus on a q-analogue of the Riemann zeta function at positive integers, which can be written for s\\in\\N^* by \\zeta_q(s)=\\sum_{k\\geq 1}q^k\\sum_{d|k}d^{s-1}. We give a new lower bound for the dimension of the vector space over \\Q spanned, for 1/q\\in\\Z\\setminus\\{-1;1\\} and an even integer A, by 1,\\zeta_q(3),\\zeta_q(5),...,\\zeta_q(A-1). This improves a recent result of Krattenthaler, Rivoal and Zudilin (\\emph{S\\'eries hyperg\\'eom\\'etriques basiques, q-analogues des valeurs de la fonction zeta et s\\'eries d'Eisenstein}, J. Inst. Jussieu {\\bf 5}.1 (2006), 53-79). In particular, a consequence of our result is that for 1/q\\in\\Z\\setminus\\{-1;1\\}, at least one of the numbers \\zeta_q(3),\\zeta_q(5),\\zeta_q(7),\\zeta_q(9) is irrational."}
{"category": "Math", "title": "Sur le centralisateur d'une involution de 2E6(2)", "abstract": "In this paper we prove that $2^{20+1}.U_6(2)$, known as the centralizer of an involution in the group $2E_6(2)$ is a quotient of a Coxeter group. We obtain a presentation of $2^{20+1}.U_6(2)$ as a $Q_{222}$-group, which now resolve a long pending question."}
{"category": "Math", "title": "Finite Sections of Weighted Hardy's Inequality", "abstract": "We study finite sections of weighted Hardy's inequality following the approach of De Bruijn. Similar to the unweighted case, we obtain an asymptotic expression for the optimal constant."}
{"category": "Math", "title": "Homogenization of variational problems in manifold valued Sobolev spaces", "abstract": "Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna, Fonseca, Maly and Trivisa \\cite{DFMT}. For energies with superlinear or linear growth, a $\\Gamma$-convergence result is established in Sobolev spaces, the homogenization problem in the space of functions of bounded variation being the object of \\cite{BM}."}
{"category": "Math", "title": "Existence of rational points on smooth projective varieties", "abstract": "Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and also an algorithm for computing X(k) for any k-variety X for which X(k) is finite. The proof involves the construction of a one-parameter algebraic family of Chatelet surfaces such that exactly one of the surfaces fails to have a k-point."}
{"category": "Math", "title": "The set of non-squares in a number field is diophantine", "abstract": "Fix a number field k. We prove that k* - k*^2 is diophantine over k. This is deduced from a theorem that for a nonconstant separable polynomial P(x) in k[x], there are at most finitely many a in k* modulo squares such that there is a Brauer-Manin obstruction to the Hasse principle for the conic bundle X given by y^2 - az^2 = P(x)."}
{"category": "Math", "title": "On deep Frobenius descent and flat bundles", "abstract": "Let R be an integral domain of finite type over Z and let f:X --> Spec R be a smooth projective morphism of relative dimension d >= 1. We investigate, for a vector bundle E on the total space X, under what arithmetical properties of a sequence (p_n, e_n)_{n \\in \\NN}, consisting of closed points p_n in Spec R and Frobenius descent data E_{p_n} \\cong F^{e_n}^*(F) on the closed fibers X_{p_n}, the bundle E_0 on the generic fiber X_0 is semistable."}
{"category": "Math", "title": "Lens space surgeries & primitive/Seifert type constructions", "abstract": "We show that lens space surgeries on knots in $S^3$ which arise from the primitive/Seifert type construction also arise from the primitive/primitive construction. This is the first step of a three step program to prove the Berge conjecture for tunnel number one knots."}
{"category": "Math", "title": "Existence de points fixes enlac\\'es \\`a une orbite p\\'eriodique d'un hom\\'eomorphisme du plan", "abstract": "Let f be an orientation-preserving homeomorphism of the plane such that f-Id is contracting. Under these hypotheses, we establish the existence, for every periodic orbit, of a fixed point which has nonzero linking number with this periodic orbit."}
{"category": "Math", "title": "Simplicial complexes and Macaulay's inverse systems", "abstract": "Let $\\Delta$ be a simplicial complex on $V = \\{x_1,...,x_n\\}$, with Stanley-Reisner ideal $I_{\\Delta}\\subseteq R = k[x_1,...,x_n]$. The goal of this paper is to investigate the class of artinian algebras $A=A(\\Delta,a_1,...,a_n)= R/(I_{\\Delta},x_1^{a_1},...,x_n^{a_n})$, where each $a_i \\geq 2$. By utilizing the technique of Macaulay's inverse systems, we can explicitly describe the socle of $A$ in terms of $\\Delta$. As a consequence, we determine the simplicial complexes, that we will call {\\em levelable}, for which there exists a tuple $(a_1,...,a_n)$ such that $A(\\Delta,a_1,...,a_n)$ is a level algebra."}
{"category": "Math", "title": "Scattering for H^1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions", "abstract": "We show that if a solution of the defocusing cubic NLS in 3d remains bounded in the homogeneous Sobolev norm of order 1/2 in its maximal interval of existence, then the interval is infinite and the solution scatters. No radial assumption is made."}
{"category": "Math", "title": "Geometry of Shimura varieties of Hodge type over finite fields", "abstract": "We present a general and comprehensive overview of recent developments in the theory of integral models of Shimura varieties of Hodge type. The paper covers the following topics: construction of integral models, their possible moduli interpretations, their uniqueness, their smoothness, their properness, and basic stratifications of their special fibres."}
{"category": "Math", "title": "A Note on Kasparov Product and Duality", "abstract": "Using Paschke-Higson duality, we can get a natural index pairing $K_{i}(A) \\times K_{i+1}(D_{\\Phi}) \\to \\boldsymbol{Z} \\quad (i=0,1) (\\mbox{mod}2)$, where $A$ is a separable $C\\sp*$-algebra, and $\\Phi$ is a representation of $A$ on a separable infinite dimensional Hilbert space $H$. It is proved that this is a special case of the Kasparov Product. As a step, we show a proof of Bott-periodicity for KK-theory asserting that $\\mathbb{C}_1$ and $S$ are $KK$-equivalent using the odd index pairing."}
{"category": "Math", "title": "Betti Numbers of Graded Modules and Cohomology of Vector Bundles", "abstract": "Mats Boij and Jonas Soederberg (math.AC/0611081) have conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring can be decomposed in a certain way as a positive linear combination of Betti tables of modules with pure resolutions. We prove, over any field, a strengthened form of their conjecture. Applications include a proof of the Multiplicity Conjecture of Huneke and Srinivasan and a proof of the convexity of a fan naturally associated to the Young lattice. We also characterize the rational cone of all cohomology tables of vector bundles on projective spaces in terms of the cohomology tables of \"supernatural\" bundles. This characterization is dual, in a certain sense, to our characterization of Betti tables."}
{"category": "Math", "title": "A variation of multiple $L$-values arising from the spectral zeta function of the non-commutative harmonic oscillator", "abstract": "A variation of multiple $L$-values, which arises from the description of the special values of the spectral zeta function of the non-commutative harmonic oscillator, is introduced. In some special cases, we show that its generating function can be written in terms of the gamma functions. This result enables us to obtain explicit evaluations of them."}
{"category": "Math", "title": "On endomorphism algebras of separable monoidal functors", "abstract": "We show that the (co)endomorphism algebra of a sufficiently separable \"fibre\" functor into Vect_k, for k a field of characteristic 0, has the structure of what we call a \"unital\" von Neumann core in Vect_k. For Vect_k, this particular notion of algebra is weaker than that of a Hopf algebra, although the corresponding concept in Set is again that of a group."}
{"category": "Math", "title": "Coplanar k-unduloids are nondegenerate", "abstract": "We prove each embedded, constant mean curvature (CMC) surface in Euclidean space with genus zero and finitely many coplanar ends is nondegenerate: there is no nontrivial square-integrable solution to the Jacobi equation, the linearization of the CMC condition. This implies that the moduli space of such coplanar surfaces is a real-analytic manifold and that a neighborhood of these in the full CMC moduli space is itself a manifold. Nondegeneracy further implies (infinitesimal and local) rigidity in the sense that the asymptotes map is an analytic immersion on these spaces, and also that the coplanar classifying map is an analytic diffeomorphism."}
{"category": "Math", "title": "Poisson Matching", "abstract": "Suppose that red and blue points occur as independent homogeneous Poisson processes in R^d. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d=1,2, the matching distance X from a typical point to its partner must have infinite d/2-th moment, while in dimensions d>=3 there exist schemes where X has finite exponential moments. The Gale-Shapley stable marriage is one natural matching scheme, obtained by iteratively matching mutually closest pairs. A principal result of this paper is a power law upper bound on the matching distance X for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in d=1, but far from optimal in d>=3. For the problem of matching Poisson points of a single color to each other, in d=1 there exist schemes where X has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process then X must have infinite mean."}
{"category": "Math", "title": "Duality Theorem and Hom Functor in Braided Tensor Categories", "abstract": "Blatter-Montgomery duality theorem is generalized into braided tensor categories. It is shown that $Hom(V,W)$ is a braided Yetter-Drinfeld module for any two braided Yetter-Drinfeld modules $V$ and $W$."}
{"category": "Math", "title": "Supercritical general branching processes conditioned on extinction are subcritical", "abstract": "It is well known that a simple, supercritical Bienaym\\'e-Galton-Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction."}
{"category": "Math", "title": "On localizing subcategories of derived categories", "abstract": "Let A be a commutative noetherian ring. In this paper, we interpret localizing subcategories of the derived category of A by using subsets of Spec A and subcategories of the category of A-modules. We unify theorems of Gabriel, Neeman and Krause."}
{"category": "Math", "title": "Trigonometry in extended hyperbolic space and extended de Sitter space", "abstract": "We study the hyperbolic cosine and sine laws in the extended hyperbolic space which contains hyperbolic space as a subset and is an analytic continuation of the hyperbolic space. And we also study the spherical cosine and sine laws in the extended de Sitter space which contains de Sitter Space $S^n_1$ as a subset and is also an analytic continuation of de Sitter space. In fact, the extended hyperbolic space and extended de Sitter space are the same space only differ by -1 multiple in the metric. Hence these two extended spaces clearly show and apparently explain that why many corresponding formulas in hyperbolic and spherical space are very similar each other. From these extended trigonometry laws, we can give a coherent and geometrically simple explanation for the various relations between the lengths and angles of hyperbolic polygons and relations on de Sitter polygons which lie on $S^2_1$."}
{"category": "Math", "title": "Normal Forms, K3 Surface Moduli, and Modular Parametrizations", "abstract": "The geometric objects of study in this paper are K3 surfaces which admit a polarization by the unique even unimodular lattice of signature (1,17). A standard Hodge-theoretic observation about this special class of K3 surfaces is that their polarized Hodge structures are identical with the polarized Hodge structures of abelian surfaces that are cartesian products of elliptic curves. Earlier work of the first two authors gives an explicit normal form and construction of the moduli space for these surfaces. In the present work, this normal form is used to derive Picard-Fuchs differential equations satisfied by periods of these surfaces. We also investigate the subloci of the moduli space on which the polarization is enhanced. In these cases, we derive information about the Picard-Fuchs differential equations satisfied by periods of these subfamilies, and we relate this information to the theory of genus zero quotients of the upper half-plane by Moonshine groups. For comparison, we also examine the analogous theory for elliptic curves in Weierstrass form."}
{"category": "Math", "title": "A Criterion for Vertex Colorability of a Graph Stated in Terms of Edge Orientations", "abstract": "L.M.Vitaver [1962] and G.I.Minty [1962] suggested criteria for vertex colorability of a graph in at most a given number of colors; these criteria are stated in terms of the orientation of the edges. One additional criterion of this kind is given here."}
{"category": "Math", "title": "Determinants of finite-dimensional algebras", "abstract": "To each associative unitary finite-dimensional algebra over a normal base, we associative a canonical multiplicative function called its determinant. We give various properties of this construction, as well as applications to the topology of the moduli stack of n-dimensional algebras."}
{"category": "Math", "title": "Unitary processes with independent increments and representations of Hilbert tensor algebras", "abstract": "The aim of this article is to characterize unitary increment process by a quantum stochastic integral representation on symmetric Fock space. Under certain assumptions we have proved its unitary equivalence to a Hudson-Parthasarathy flow."}
{"category": "Math", "title": "Cycles of random permutations with restricted cycle lengths", "abstract": "We prove some general results about the asymptotics of the distribution of the number of cycles of given length of a random permutation whose distribution is invariant under conjugation. These results were first established to be applied in a forthcoming paper (Cycles of free words in several random permutations with restricted cycles lengths), where we prove results about cycles of random permutations which can be written as free words in several independent random permutations. However, we also apply them here to prove asymptotic results about random permutations with restricted cycle lengths. More specifically, for $A$ a set of positive integers, we consider a random permutation chosen uniformly among the permutations of $\\{1,..., n\\}$ which have all their cycle lengths in $A$, and then let $n$ tend to infinity. Improving slightly a recent result of Yakymiv (Random A-Permutations: Convergence to a Poisson Process), we prove that under a general hypothesis on $A$, the numbers of cycles with fixed lengths of this random permutation are asymptotically independent and distributed according to Poisson distributions. In the case where $A$ is finite, we prove that the behavior of these random variables is completely different: cycles with length $\\max A$ are predominant."}
{"category": "Math", "title": "Binary linear forms as sums of two squares", "abstract": "We revisit recent work of Heath-Brown on the average order of the quantity r(L_1)r(L_2)r(L_3)r(L_4), for suitable binary linear forms L_1,..., L_4, for integers ranging over quite general regions. In addition to improving the error term in Heath-Brown's estimate we generalise his result quite extensively."}
{"category": "Math", "title": "Prediction of long memory processes on same-realisation", "abstract": "For the class of stationary Gaussian long memory processes, we study some properties of the least-squares predictor of X_{n+1} based on (X_n, ..., X_1). The predictor is obtained by projecting X_{n+1} onto the finite past and the coefficients of the predictor are estimated on the same realisation. First we prove moment bounds for the inverse of the empirical covariance matrix. Then we deduce an asymptotic expression of the mean-squared error. In particular we give a relation between the number of terms used to estimate the coefficients and the number of past terms used for prediction, which ensures the L^2-sense convergence of the predictor. Finally we prove a central limit theorem when our predictor converges to the best linear predictor based on all the past."}
{"category": "Math", "title": "Proofs of the martingale FCLT", "abstract": "This is an expository review paper elaborating on the proof of the martingale functional central limit theorem (FCLT). This paper also reviews tightness and stochastic boundedness, highlighting one-dimensional criteria for tightness used in the proof of the martingale FCLT. This paper supplements the expository review paper Pang, Talreja and Whitt (2007) illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations."}
{"category": "Math", "title": "Semi-parametric estimation of shifts", "abstract": "We observe a large number of functions differing from each other only by a translation parameter. While the main pattern is unknown, we propose to estimate the shift parameters using $M$-estimators. Fourier transform enables to transform this statistical problem into a semi-parametric framework. We study the convergence of the estimator and provide its asymptotic behavior. Moreover, we use the method in the applied case of velocity curve forecasting."}
{"category": "Math", "title": "Designs, groups and lattices", "abstract": "We study the Grassmannian 4-designs contained in lattices, in connection with the local property of the Rankin constant. We prove that the sequence of Barnes-Wall lattices contain Grassmannian 6-designs."}
{"category": "Math", "title": "Improving population-specific allele frequency estimates by adapting supplemental data: an empirical Bayes approach", "abstract": "Estimation of the allele frequency at genetic markers is a key ingredient in biological and biomedical research, such as studies of human genetic variation or of the genetic etiology of heritable traits. As genetic data becomes increasingly available, investigators face a dilemma: when should data from other studies and population subgroups be pooled with the primary data? Pooling additional samples will generally reduce the variance of the frequency estimates; however, used inappropriately, pooled estimates can be severely biased due to population stratification. Because of this potential bias, most investigators avoid pooling, even for samples with the same ethnic background and residing on the same continent. Here, we propose an empirical Bayes approach for estimating allele frequencies of single nucleotide polymorphisms. This procedure adaptively incorporates genotypes from related samples, so that more similar samples have a greater influence on the estimates. In every example we have considered, our estimator achieves a mean squared error (MSE) that is smaller than either pooling or not, and sometimes substantially improves over both extremes. The bias introduced is small, as is shown by a simulation study that is carefully matched to a real data example. Our method is particularly useful when small groups of individuals are genotyped at a large number of markers, a situation we are likely to encounter in a genome-wide association study."}
{"category": "Math", "title": "Scaling limit and aging for directed trap models", "abstract": "We consider one-dimensional directed trap models and suppose that the trapping times are heavy-tailed. We obtain the inverse of a stable subordinator as scaling limit and prove an aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality described by Ben Arous and \\v{C}ern\\'y for a large class of graphs."}
{"category": "Math", "title": "Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms", "abstract": "An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer-Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points. ----- Une forme quadratique enti\\`ere peut \\^etre repr\\'esent\\'ee par une autre forme quadratique enti\\`ere sur tous les anneaux d'entiers p-adiques et sur les r\\'eels, sans l'\\^etre sur les entiers. On en trouve de nombreux exemples dans la litt\\'erature. Nous montrons qu'une partie de ces exemples s'explique au moyen d'une obstruction de type Brauer-Manin pour les points entiers. Pour plusieurs types d'espaces homog\\`enes de groupes alg\\'ebriques lin\\'eaires, cette obstruction est la seule obstruction \\`a l'existence d'un point entier."}
{"category": "Math", "title": "The BARISTA: A model for bid arrivals in online auctions", "abstract": "The arrival process of bidders and bids in online auctions is important for studying and modeling supply and demand in the online marketplace. A popular assumption in the online auction literature is that a Poisson bidder arrival process is a reasonable approximation. This approximation underlies theoretical derivations, statistical models and simulations used in field studies. However, when it comes to the bid arrivals, empirical research has shown that the process is far from Poisson, with early bidding and last-moment bids taking place. An additional feature that has been reported by various authors is an apparent self-similarity in the bid arrival process. Despite the wide evidence for the changing bidding intensities and the self-similarity, there has been no rigorous attempt at developing a model that adequately approximates bid arrivals and accounts for these features. The goal of this paper is to introduce a family of distributions that well-approximate the bid time distribution in hard-close auctions. We call this the BARISTA process (Bid ARrivals In STAges) because of its ability to generate different intensities at different stages. We describe the properties of this model, show how to simulate bid arrivals from it, and how to use it for estimation and inference. We illustrate its power and usefulness by fitting simulated and real data from eBay.com. Finally, we show how a Poisson bidder arrival process relates to a BARISTA bid arrival process."}
{"category": "Math", "title": "Foundations for abstract forcing", "abstract": "The foundations of forcing theory are reworked to streamline the presentation and to show how the most basic results are applicable in very general contexts."}
{"category": "Math", "title": "The Jiang-Su algebra does not always embed", "abstract": "We exhibit a unital simple nuclear non-type-I C*-algebra into which the Jiang-Su algebra does not embed unitally. This answers a question of M. R{\\o}rdam."}
{"category": "Math", "title": "Hypergeometric D-modules and twisted Gauss-Manin systems", "abstract": "The Euler-Koszul complex is the fundamental tool in the homological study of A-hypergeometric differential systems and functions. We compare Euler-Koszul homology with D-module direct images from the torus to the base space through orbits in the corresponding toric variety. Our approach generalizes a result by Gel'fand et al. and yields a simpler, more algebraic proof. In the process we extend the Euler-Koszul functor a category of infinite toric modules and describe multigraded localizations of Euler-Koszul homology."}
{"category": "Math", "title": "Constructing elliptic curves of prime order", "abstract": "We present a very efficient algorithm to construct an elliptic curve E and a finite field F such that the order of the point group E(F) is a given prime number N. Heuristically, this algorithm only takes polynomial time Otilde((\\log N)^3), and it is so fast that it may profitably be used to tackle the related problem of finding elliptic curves with point groups of prime order of prescribed size. We also discuss the impact of the use of high level modular functions to reduce the run time by large constant factors and show that recent gonality bounds for modular curves imply limits on the time reduction that can be obtained."}
{"category": "Math", "title": "Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue", "abstract": "We prove that among all doubly connected domains of $\\mathbb{R}^n$ bounded by two spheres of given radii, the second eigenvalue of the Dirichlet Laplacian achieves its maximum when the spheres are concentric (spherical shell). The corresponding result for the first eigenvalue has been established by Hersch in dimension 2, and by Harrell, Kr\\\"oger and Kurata and Kesavan in any dimension. We also prove that the same result remains valid when the ambient space $\\mathbb{R}^n$ is replaced by the standard sphere $\\mathbb{S}^n$ or the hyperbolic space $\\mathbb{H}^n$ ."}
{"category": "Math", "title": "Diviseur Theta et Formes Differentielles", "abstract": "In this papers, we study the geometric and arithmetic properties of the theta divisor associated to the sheaf of locally exact differential forms over a curve in positive characteristic. In this published version, we prove a stronger version of the main result of chapiter 5."}
{"category": "Math", "title": "Moving codimension-one subvarieties over finite fields", "abstract": "We give the first examples of nef line bundles on smooth projective varieties over finite fields which are not semi-ample. More concretely, we find smooth curves on smooth projective surfaces over finite fields such that the normal bundle has degree zero, but no positive multiple of the curve moves in a family of disjoint curves. This answers questions by Keel and Mumford. The proof uses an obstruction theory, in the spirit of homotopy theory, which links the infinitely many obstructions to moving higher and higher multiples of a given codimension-one subvariety. On 3-folds over a finite field, we find nef and big line bundles which are not semi-ample. Finally, we reprove some of the known positive results about semi-ampleness over finite fields."}
{"category": "Math", "title": "Isogenies of supersingular elliptic curves over finite fields and operations in elliptic cohomology", "abstract": "We investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple proof of an elliptic cohomology version of the Morava change of rings theorem and also gives models for explicit stable operations in terms of isogenies and morphisms in certain enlarged isogeny categories. We relate our work to that of G. Robert on the Hecke algebra structure of the ring of supersingular modular forms."}
{"category": "Math", "title": "Equations of the moduli of Higgs pairs and infinite Grassmannian", "abstract": "In this paper the moduli space of Higgs pairs over a fixed smooth projective curve with extra formal data is defined and it is endowed with a scheme structure. We introduce a relative version of the Krichever map using a fibration of Sato Grassmannians and show that this map is injective. This fact and the characterization of the points of the image of the Krichever map allow us to prove that this moduli space is a closed subscheme of the product of the moduli of vector bundles (with formal extra data) and a formal anologue of the Hitchin base. This characterization also provide us the method to compute explicitely KP-type equations which describe the moduli space of Higgs pairs. Finally, for the case where the spectral cover is totally ramified at a fixed point of the curve, these equations are given in terms of the characteristic coefficients of the Higgs field."}
{"category": "Math", "title": "A Not-so-Characteristic Equation: the Art of Linear Algebra", "abstract": "Can the cross product be generalized? Why are the trace and determinant so important in matrix theory? What do all the coefficients of the characteristic polynomial represent? This paper describes a technique for `doodling' equations from linear algebra that offers elegant solutions to all these questions. The doodles, known as trace diagrams, are graphs labeled by matrices which have a correspondence to multilinear functions. This correspondence permits computations in linear algebra to be performed using diagrams. The result is an elegant theory from which standard constructions of linear algebra such as the determinant, the trace, the adjugate matrix, Cramer's rule, and the characteristic polynomial arise naturally. Using the diagrams, it is easy to see how little structure gives rise to these various results, as they all can be `traced' back to the definition of the determinant and inner product."}
{"category": "Math", "title": "On Character Amenability of Banach Algebras", "abstract": "Associated to a nonzero homomorphism $\\varphi$ of a Banach algebra $A$, we regard special functionals, say $m_\\varphi$, on certain subspaces of $A^\\ast$ which provide equivalent statements to the existence of a bounded right approximate identity in the corresponding maximal ideal in $A$. For instance, applying a fixed point theorem yields an equivalent statement to the existence of a $m_\\varphi$ on $A^\\ast$; and, in addition we expatiate the case that if a functional $m_\\varphi$ is unique, then $m_\\varphi$ belongs to the topological center of the bidual algebra $A^{\\ast\\ast}$. An example of a function algebra, surprisingly, contradicts a conjecture that a Banach algebra $A$ is amenable if $A$ is $\\varphi$-amenable in every character $\\varphi$ and if functionals $m_\\varphi$ associated to the characters $\\varphi$ are uniformly bounded. Aforementioned are also elaborated on the direct sum of two given Banach algebras."}
{"category": "Math", "title": "($\\ell,0)$-Carter partitions, a generating function, and their crystal theoretic interpretation", "abstract": "In this paper we give an alternate combinatorial description of the \"$(\\ell,0)$-JM partitions\" (see \\cite{F}) that are also $\\ell$-regular. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas (\\cite{JM}). The condition of being an $(\\ell,0)$-JM partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an $\\ell$-regular partition is that it indicates the irreducibility of the corresponding specialized Specht module over the finite Hecke algebra (see \\cite{JM}). We use our result to find a generating series which counts the number of such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph $B(\\Lambda_0)$ of the basic representation of $\\hat{\\mathfrak{sl}_{\\ell}}$, whose nodes are labeled by $\\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $\\ell$-regular $(\\ell,0)$-JM partitions in the graph $B(\\Lambda_0)$. Finally, we mention how our construction can be generalized to recent results of M. Fayers (see \\cite{F}) and S. Lyle (see \\cite{L}) to count the total number of (not necessarily $\\ell$-regular) Specht modules which stay irreducible at a primitive $\\ell$th root of unity (for $\\ell >2$)."}
{"category": "Math", "title": "On the irreducible representations of a finite semigroup", "abstract": "Work of Clifford, Munn and Ponizovski{\\u\\i} parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovski{\\u\\i} result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring."}
{"category": "Math", "title": "Homotopy groups of the spaces of self-maps of Lie groups", "abstract": "We compute the homotopy groups of the spaces of self maps of Lie groups of rank 2, SU(3), Sp(2), and G_2. We use the cell structures of these Lie groups and the standard methods of homotopy theory."}
{"category": "Math", "title": "Riesz transforms in one dimension", "abstract": "We study the boundedness on $L^p$ of the Riesz transform $\\nabla L^{-1/2}$, where $L$ is one of several operators defined on $\\R$ or $\\R_+$, endowed with the measure $r^{d-1} dr$, $d > 1$, where $dr$ is Lebesgue measure. For integer $d$, this mimics the measure on Euclidean $d$-dimensional space, and in this case our setup is equivalent to looking at the Laplacian acting on radial functions on Euclidean space or variations of Euclidean space such as the exterior of a sphere (with either Dirichlet or Neumann boundary conditions), or the connected sum of two copies of $\\R^d$. In this way we illuminate some recent results on the Riesz transform on asymptotically Euclidean manifolds. We are however interested in all real values of $d > 1$, and another goal of our analysis is to study the range of boundedness as a function of $d$; it is particularly interesting to see the behaviour as $d$ crosses 2. For example, in one of our cases which models radial functions on the connected sum of two copies of $\\R^d$, the upper threshold for $L^p$ boundedness is $p=d$ for $d \\ge 2$ and $p=d/(d-1)$ for $d < 2$. Only in the case $d=2$ is the Riesz transform actually bounded on $L^p$ when $p$ is equal to the upper threshold. We also study the Riesz transform when we have an inverse square potential, or a delta function potential; these cases provide a simple model for recent results of the first author and Guillarmou. Finally we look at the Hodge projector in a slightly more general setup."}
{"category": "Math", "title": "On the Berstein-Svarc Theorem in dimension 2", "abstract": "We prove that for any group of the cohomological dimension $n$ the $n$th power of the Berstein class of the group is nontrivial. This allows to prove the following Berstein-Svarc theorem for all $n$: Theorem. For a connected complex $X$ with $\\dim X=\\cat X=n$, the $n$th power of the Berstein class of $X$ is nontrivial. Previously it was known for $n\\ge 3$. We also prove that, for every map $f: M \\to N$ of degree $\\pm 1$ of closed orientable manifolds, the fundamental group of $N$ is free provided that the fundamental group of $M$ is."}
{"category": "Math", "title": "Asymptotic distributions and chaos for the supermarket model", "abstract": "In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate \\lambda n, where 0<\\lambda <1. Each customer chooses d > 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n -> oo. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order n^{-1}; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most n^{-1}."}
{"category": "Math", "title": "Bessel models for $GSp(4)$", "abstract": "Methods of theta correspondence are used to analyze local and global Bessel models for $GSp(4)$ proving a conjecture of Gross and Prasad which describes these models in terms of local epsilon factors in the local case, and the non-vanishing of central critical $L$-value in the global case."}
{"category": "Math", "title": "An Example of Constructing Versal Deformation for Leibniz Algebras", "abstract": "In this work we compute a versal deformation of the three dimensional nilpotent Leibniz algebra over $\\mathbb{C}$, defined by the nontrivial brackets $[e_1,e_3]=e_2$ and $[e_3,e_3]=e_1$."}
{"category": "Math", "title": "Prime spectrum and primitive Leavitt path algebras", "abstract": "In this paper a bijection between the set of prime ideals of a Leavitt path algebra $L_K(E)$ and a certain set which involves maximal tails in $E$ and the prime spectrum of $K[x,x^{-1}]$ is established. Necessary and sufficient conditions on the graph $E$ so that the Leavitt path algebra $L_K(E)$ is primitive are also found."}
{"category": "Math", "title": "Symplectic Heegaard splittings and linked abelian groups", "abstract": "Let $f$ be the gluing map of a Heegaard splitting of a 3-manifold $W$. The goal of this paper is to determine the information about $W$ contained in the image of $f$ under the symplectic representation of the mapping class group. We prove three main results. First, we show that the first homology group of the three manifold together with Seifert's linking form provides a complete set of stable invariants. Second, we give a complete, computable set of invariants for these linking forms. Third, we show that a slight augmentation of Birman's determinantal invariant for a Heegaard splitting gives a complete set of unstable invariants."}
{"category": "Math", "title": "Non-special scrolls with general moduli", "abstract": "In this paper we study smooth, non-special scrolls S of degree d, genus g, with general moduli. In particular, we study the scheme of unisecant curves of a given degree on S. Our approach is mostly based on degeneration techniques."}
{"category": "Math", "title": "Brill-Noether theory and non-special scrolls", "abstract": "In this paper we study the Brill-Noether theory of sub-line bundles of a general, stable rank-two vector bundle on a curve C with general moduli. We relate this theory to the geometry of unisecant curves on smooth, non-special scrolls with hyperplane sections isomorphic to C. Most of our results are based on degeneration techniques."}
{"category": "Math", "title": "Dirichlet sets and Erdos-Kunen-Mauldin theorem", "abstract": "By a theorem proved by Erdos, Kunen and Mauldin, for any nonempty perfect set $P$ on the real line there exists a perfect set $M$ of Lebesgue measure zero such that $P+M=\\mathbb{R}$. We prove a stronger version of this theorem in which the obtained perfect set $M$ is a Dirichlet set. Using this result we show that for a wide range of familes of subsets of the reals, all additive sets are perfectly meager in transitive sense. We also prove that every proper analytic subgroup $G$ of the reals is contained in an F-sigma set $F$ such that $F+G$ is a meager null set."}
{"category": "Math", "title": "A statistical framework for the analysis of microarray probe-level data", "abstract": "In microarray technology, a number of critical steps are required to convert the raw measurements into the data relied upon by biologists and clinicians. These data manipulations, referred to as preprocessing, influence the quality of the ultimate measurements and studies that rely upon them. Standard operating procedure for microarray researchers is to use preprocessed data as the starting point for the statistical analyses that produce reported results. This has prevented many researchers from carefully considering their choice of preprocessing methodology. Furthermore, the fact that the preprocessing step affects the stochastic properties of the final statistical summaries is often ignored. In this paper we propose a statistical framework that permits the integration of preprocessing into the standard statistical analysis flow of microarray data. This general framework is relevant in many microarray platforms and motivates targeted analysis methods for specific applications. We demonstrate its usefulness by applying the idea in three different applications of the technology."}
{"category": "Math", "title": "On the existence of homomorphisms between principal series of complex semisimple Lie groups", "abstract": "We determine when there exists a nonzero homomorphism between principal series representations of a complex semisimple Lie group. We also determines the existence of homomorphisms between twisted Verma modules."}
{"category": "Math", "title": "Conformal Metrics with Constant Q-Curvature", "abstract": "We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold."}
{"category": "Math", "title": "Describing disability through individual-level mixture models for multivariate binary data", "abstract": "Data on functional disability are of widespread policy interest in the United States, especially with respect to planning for Medicare and Social Security for a growing population of elderly adults. We consider an extract of functional disability data from the National Long Term Care Survey (NLTCS) and attempt to develop disability profiles using variations of the Grade of Membership (GoM) model. We first describe GoM as an individual-level mixture model that allows individuals to have partial membership in several mixture components simultaneously. We then prove the equivalence between individual-level and population-level mixture models, and use this property to develop a Markov Chain Monte Carlo algorithm for Bayesian estimation of the model. We use our approach to analyze functional disability data from the NLTCS."}
{"category": "Math", "title": "On an identity by Chaundy and Bullard. I", "abstract": "An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the theory of wavelets of compact support. We discuss or survey many different proofs of the identity, and also its relationship with Gauss hypergeometric series. We also consider the extension to complex values of the two parameters which occur as summation bounds. The paper concludes with a discussion of a multivariable analogue of the identity, which was first given by Damjanovic, Klamkin and Ruehr. We give the relationship with Lauricella hypergeometric functions and corresponding PDE's. The paper ends with a new proof of the multivariable case by splitting up Dirichlet's multivariable beta integral."}
{"category": "Math", "title": "Distances in random Apollonian network structures", "abstract": "In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express all information on distances, and singularity analysis for evaluating the coefficients of these functions, we describe the distribution of distances to an outermost vertex, and show that the average value of the distance between any pair of vertices in a RANS of order n is asymptotically square root of n."}
{"category": "Math", "title": "Diverse correlation structures in gene expression data and their utility in improving statistical inference", "abstract": "It is well known that correlations in microarray data represent a serious nuisance deteriorating the performance of gene selection procedures. This paper is intended to demonstrate that the correlation structure of microarray data provides a rich source of useful information. We discuss distinct correlation substructures revealed in microarray gene expression data by an appropriate ordering of genes. These substructures include stochastic proportionality of expression signals in a large percentage of all gene pairs, negative correlations hidden in ordered gene triples, and a long sequence of weakly dependent random variables associated with ordered pairs of genes. The reported striking regularities are of general biological interest and they also have far-reaching implications for theory and practice of statistical methods of microarray data analysis. We illustrate the latter point with a method for testing differential expression of nonoverlapping gene pairs. While designed for testing a different null hypothesis, this method provides an order of magnitude more accurate control of type 1 error rate compared to conventional methods of individual gene expression profiling. In addition, this method is robust to the technical noise. Quantitative inference of the correlation structure has the potential to extend the analysis of microarray data far beyond currently practiced methods."}
{"category": "Math", "title": "Jacobi osculating rank and isotropic geodesics on naturally reductive 3-manifolds", "abstract": "We study the Jacobi osculating rank of geodesics on naturally reductive homogeneous manifolds and we apply this theory to the 3-dimensional case. Here, each non-symmetric, simply connected naturally reductive 3-manifold can be given as a principal bundle over a surface of constant curvature, such that the curvature of its horizontal distribution is also a constant. Then, we prove that the Jacobi osculating rank of every geodesic is two except for the Hopf fibers, where it is zero. Moreover, we determine all isotropic geodesics and the isotropic tangent conjugate locus."}
{"category": "Math", "title": "The Div-Curl Lemma Revisited", "abstract": "The Div-Curl Lemma, which is the basic result of the compensated compactness theory in Sobolev spaces, was introduced by F. Murat (1978) with distinct proofs for the $L^2(\\Omega)$ and $L^p(\\Omega)$, $p \\neq 2$, cases. In this note we present a slightly different proof, relying only on a Green-Gauss integral formula and on the usual Rellich-Kondrachov compactness properties."}
{"category": "Math", "title": "Complexes of Injective Words and Their Commutation Classes", "abstract": "Let $S$ be a finite alphabet. An injective word over $S$ is a word over $S$ such that each letter in $S$ appears at most once in the word. We study Boolean cell complexes of injective words over $S$ and their commutation classes. This generalizes work by Farmer and by Bj\\\"orner and Wachs on the complex of all injective words."}
{"category": "Math", "title": "An example of noncommutative deformations", "abstract": "We compute the noncommutative deformations of a family of modules over the first Weyl algebra. This example shows some important properties of noncommutative deformation theory that separates it from commutative deformation theory."}
{"category": "Math", "title": "Sobolev homeomorphisms and Poincare inequality", "abstract": "We study global regularity properties of Sobolev homeomorphisms on $n$-dimensional Riemannian manifolds under the assumption of $p$-integrability of its first weak derivatives in degree $p\\geq n-1$. We prove that inverse homeomorphisms have integrable first weak derivatives. For the case $p>n$ we obtain necessary conditions for existence of Sobolev homeomorphisms between manifolds. These necessary conditions based on Poincar\\'e type inequality: $$ \\inf_{c\\in \\mathbb R} \\|u-c\\mid L_{\\infty}(M)\\|\\leq K \\|u\\mid L^1_{\\infty}(M)\\|. $$ As a corollary we obtain the following geometrical necessary condition: {\\em If there exists a Sobolev homeomorphisms $\\phi: M \\to M'$, $\\phi\\in W^1_p(M, M')$, $p>n$, $J(x,\\phi)\\ne 0$ a. e. in $M$, of compact smooth Riemannian manifold $M$ onto Riemannian manifold $M'$ then the manifold $M'$ has finite geodesic diameter.}}"}
{"category": "Math", "title": "Chemical and forensic analysis of JFK assassination bullet lots: Is a second shooter possible?", "abstract": "The assassination of President John Fitzgerald Kennedy (JFK) traumatized the nation. In this paper we show that evidence used to rule out a second assassin is fundamentally flawed. This paper discusses new compositional analyses of bullets reportedly to have been derived from the same batch as those used in the assassination. The new analyses show that the bullet fragments involved in the assassination are not nearly as rare as previously reported. In particular, the new test results are compared to key bullet composition testimony presented before the House Select Committee on Assassinations (HSCA). Matches of bullets within the same box of bullets are shown to be much more likely than indicated in the House Select Committee on Assassinations' testimony. Additionally, we show that one of the ten test bullets is considered a match to one or more assassination fragments. This finding means that the bullet fragments from the assassination that match could have come from three or more separate bullets. Finally, this paper presents a case for reanalyzing the assassination bullet fragments and conducting the necessary supporting scientific studies. These analyses will shed light on whether the five bullet fragments constitute three or more separate bullets. If the assassination fragments are derived from three or more separate bullets, then a second assassin is likely, as the additional bullet would not easily be attributable to the main suspect, Mr. Oswald, under widely accepted shooting scenarios [see Posner (1993), Case Closed, Bantam, New York]."}
{"category": "Math", "title": "Small index subgroups of the mapping class group", "abstract": "We prove that the mapping class group of a closed oriented surface of genus $\\rho \\ge 3$ has no proper subgroup of index $\\le 4 \\rho +4$."}
{"category": "Math", "title": "Non-existence of polar factorisations and polar inclusion of a vector-valued mapping", "abstract": "This paper proves some results concerning the polar factorisation of an integrable vector-valued function u into the composition of the gradient of a convex function with a measure-preserving mapping. Not every integrable function has a polar factorisation; we extend the class of counterexamples. We introduce a generalisation: u has a polar inclusion if u(x) belongs to the subdifferential of the convex function at y for almost every pair (x,y) with respect to a measure-preserving plan. Given a regularity assumption, we show that such measure-preserving plans are exactly the minimisers of a Monge-Kantorovich optimisation problem."}
{"category": "Math", "title": "Existence of positive definite noncoercive sums of squares", "abstract": "Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root."}
{"category": "Math", "title": "Integration on valuation fields over local fields", "abstract": "We present elements of a theory of translation-invariant integration, measure, and harmonic analysis on a valuation field with local field as residue field. This extends the work of Fesenko. Applications to zeta integrals for two-dimensional local fields are then considered."}
{"category": "Math", "title": "Integration on product spaces and GL_n of a valuation field over a local field", "abstract": "We present elements of a theory of translation-invariant integration on finite dimensional vector spaces and on GL_n over a valuation field with local field as residue field. We then discuss the case of an arbitrary algebraic group. This extends the work of Fesenko."}
{"category": "Math", "title": "Fubini's theorem and non-linear change of variables over a two-dimensional local field", "abstract": "We consider non-linear changes of variables and Fubini's theorem for certain integrals over a two-dimensional local field. An interesting example is presented in which imperfectness of a finite characteristic local field causes Fubini's theorem to unexpectedly fail. The relationship to ramification theory is discussed."}
{"category": "Math", "title": "The ordered distribution of natural numbers on the square root spiral", "abstract": "Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the Square Root Spiral (also named as the Spiral of Theodorus or Wurzel Spirale or Einstein Spiral). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36,... form a highly three-symmetrical system of three spiral graphs, which divides the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the golden mean (or golden section), which behaves as a self-avoiding-walk-constant in the lattice-like structure of the square root spiral."}
{"category": "Math", "title": "Neumann problems associated to nonhomogeneous differential operators in Orlicz--Sobolev spaces", "abstract": "We study a nonlinear Neumann boundary value problem associated to a nonhomogeneous differential operator. Taking into account the competition between the nonlinearity and the bifurcation parameter, we establish sufficient conditions for the existence of nontrivial solutions in a related Orlicz--Sobolev space."}
{"category": "Math", "title": "Harmonic functions on R-covered foliations and group actions on the circle", "abstract": "Let (M, F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F . If every such function is constant on leaves we say that (M, F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established."}
{"category": "Math", "title": "Noncommutative Analogs of Monomial Symmetric Functions, Cauchy Identity, and Hall Scalar Product", "abstract": "This paper introduces noncommutative analogs of monomial symmetric functions and fundamental noncommutative symmetric functions. The expansion of ribbon Schur functions in both of these basis is nonnegative. With these functions at hand, one can derive a noncommutative Cauchy identity as well as study a noncommutative scalar product implied by Cauchy identity. This scalar product seems be the noncommutative analog of Hall scalar product in the commutative theory."}
{"category": "Math", "title": "Wrinkled fibrations on near-symplectic manifolds", "abstract": "Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and broken Lefschetz fibrations on them. We present a set of four moves which allow us to pass from any given fibration to any other broken fibration which is deformation equivalent to it. Moreover, we study the change of the near-symplectic geometry under each of these moves. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory.Finally, we illustrate these constructions by showing how one can merge components of the zero-set of the near-symplectic form. We also disprove a conjecture of Gay and Kirby by showing that any achiral broken Lefschetz fibration can be turned into a broken Lefschetz fibration by applying a sequence of our moves."}
{"category": "Math", "title": "Real and integral structures in quantum cohomology I: toric orbifolds", "abstract": "We study real and integral structures in the space of solutions to the quantum differential equations. First we show that, under mild conditions, any real structure in orbifold quantum cohomology yields a pure and polarized tt^*-geometry near the large radius limit. Secondly, we use mirror symmetry to calculate the \"most natural\" integral structure in quantum cohomology of toric orbifolds. We show that the integral structure pulled back from the singularity B-model is described only in terms of topological data in the A-model; K-group and a characteristic class. Using integral structures, we give a natural explanation why the quantum parameter should specialize to a root of unity in Ruan's crepant resolution conjecture."}
{"category": "Math", "title": "Chern classes in Deligne cohomology for coherent analytic sheaves", "abstract": "In this article, we construct Chern classes in rational Deligne cohomology for coherent sheaves on a smooth complex compact manifold. We prove that these classes verify the functoriality property under pullbacks, the Whitney formula and the Grothendieck-Riemann-Roch theorem for projective morphisms between smooth complex compact manifolds."}
{"category": "Math", "title": "On the Bakry-Emery criterion for linear diffusions and weighted porous media equations", "abstract": "The goal of this paper is to give a non-local sufficient condition for generalized Poincar\\'e inequalities, which extends the well-known Bakry-Emery condition. Such generalized Poincar\\'e inequalities have been introduced by W. Beckner in the gaussian case and provide, along the Ornstein-Uhlenbeck flow, the exponential decay of some generalized entropies which interpolate between the $L^2$ norm and the usual entropy. Our criterion improves on results which, for instance, can be deduced from the Bakry-Emery criterion and Holley-Stroock type perturbation results. In a second step, we apply the same strategy to non-linear equations of porous media type. This provides new interpolation inequalities and decay estimates for the solutions of the evolution problem. The criterion is again a non-local condition based on the positivity of the lowest eigenvalue of a Schr\\\"odinger operator. In both cases, we relate the Fisher information with its time derivative. Since the resulting criterion is non-local, it is better adapted to potentials with, for instance, a non-quadratic growth at infinity, or to unbounded perturbations of the potential."}
{"category": "Math", "title": "Large scale geometry of certain solvable groups", "abstract": "In this paper we provide the final steps in the proof, announced by Eskin-Fisher-Whyte, of quasi-isometric rigidity of a class of non-nilpotent polycyclic groups. To this end, we prove a rigidity theorem on the boundaries of certain negatively curved homogeneous spaces and combine it with work of Eskin-Fisher-Whyte and Peng on the structure of quasi-isometries of certain solvable Lie groups."}
{"category": "Math", "title": "On coherent systems of type (n,d,n+1) on Petri curves", "abstract": "We study coherent systems of type $(n,d,n+1)$ on a Petri curve $X$ of genus $g\\ge2$. We describe the geometry of the moduli space of such coherent systems for large values of the parameter $\\alpha$. We determine the top critical value of $\\alpha$ and show that the corresponding ``flip'' has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of $\\alpha$, proving in many cases that the condition for non-emptiness is the same as for large $\\alpha$. We give some detailed results for $g\\le5$ and applications to higher rank Brill-Noether theory and the stability of kernels of evaluation maps, thus proving Butler's conjecture in some cases in which it was not previously known."}
{"category": "Math", "title": "Representation theorems for backward doubly stochastic differential equations", "abstract": "In this paper we study the class of backward doubly stochastic differential equations (BDSDEs, for short) whose terminal value depends on the history of forward diffusion. We first establish a probabilistic representation for the spatial gradient of the stochastic viscosity solution to a quasilinear parabolic SPDE in the spirit of the Feynman-Kac formula, without using the derivatives of the coefficients of the corresponding BDSDE. Then such a representation leads to a closed-form representation of the martingale integrand of BDSDE, under only standard Lipschitz condition on the coefficients."}
{"category": "Math", "title": "On the cuspidality of pullbacks of Siegel Eisenstein series and applications to the Bloch-Kato conjecture", "abstract": "Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\\bar{\\rho}_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \\leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $\\rho_{f} \\otimes \\epsilon^{1-k}$ with $\\epsilon$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain the lower bound on the Selmer group."}
{"category": "Math", "title": "Conway classification of alternating knots", "abstract": "The alternating knots, links and twists projected on the $S_2$ sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond to the stable and unstable manifolds connecting the saddles. Each face is then oriented in one of two different senses determined by the direction of these manifolds. This correspondence can be also realized between the knot and the Poincar\\'e section of a two degrees of freedom integrable dynamical system. The crossings corresponding to unstable orbits, and the faces to foliated torus, around a stable orbit. The associated matrix to that connected graph was decomposed in two permutations. The separation was shown unique for knots not for links. The characteristic polynomial corresponding to some knot, link or twist families was explicitly computed in terms of Chebyschev polynomials. A classification of rational knots was formulated in terms of the first derivative of the polynomial of a knot computed in $x=2$, equal to the number of crossings of the knot multiplying the same number used previously by Conway for tabulation of knot properties. This leads to a classification of knots exemplified for the families having up to five ribbons. We subdivide the families of $N$ ribbons in subfamilies related to the prime knots of $N$ crossings."}
{"category": "Math", "title": "Eta forms and determinant lines", "abstract": "We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form."}
{"category": "Math", "title": "How to Create a New Integer Sequence", "abstract": "There are several standard procedures used to create new sequences from a given sequence or from a given pair of sequences. In this paper I discuss the most popular of these procedures. For each procedure, I give a definition and provide examples based on three famous sequences: the natural numbers, the prime numbers and the Fibonacci numbers. I also add my thoughts on what makes a sequence interesting. My goal is to help my readers invent new sequences, differentiate interesting sequences from boring ones, and better understand sequences they encounter."}
{"category": "Math", "title": "Non-commutative Schur-Horn theorems and extended majorization for hermitian matrices", "abstract": "Let $\\mathcal A\\subseteq \\mat$ be a unital $*$-subalgebra of the algebra $\\mat$ of all $n\\times n$ complex matrices and let $B$ be an hermitian matrix. Let $\\U_n(B)$ denote the unitary orbit of $B$ in $\\mat$ and let $\\mathcal E_\\mathcal A$ denote the trace preserving conditional expectation onto $\\mathcal A$. We give an spectral characterization of the set $$ \\mathcal E_\\mathcal A(\\U_n(B))=\\{\\mathcal E_\\mathcal A(U^* B U): U\\in \\mat,\\ \\text{unitary matrix}\\}.$$ We obtain a similar result for the contractive orbit of a positive semi-definite matrix $B$. We then use these results to extend the notions of majorization and submajorization between self-adjoint matrices to spectral relations that come together with extended (non-commutative) Schur-Horn type theorems."}
{"category": "Math", "title": "Multigraded regularity and the Koszul property", "abstract": "We give a criterion for the section ring of an ample line bundle to be Koszul in terms of multigraded regularity. We discuss an application to polytopal semigroup rings."}
{"category": "Math", "title": "Large Deviations for Random Trees", "abstract": "We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures."}
{"category": "Math", "title": "On free profinite subgroups of free profinite monoids", "abstract": "We answer a question of Margolis from 1997 by establishing that the maximal subgroup of the minimal ideal of a finitely generated free profinite monoid is a free profinite group. More generally if $\\mathbf H$ is variety of finite groups closed under extension and containing $\\mathbb Z/p\\mathbb Z$ for infinitely may primes $p$, the corresponding result holds for free pro-$\\bar{\\mathbf H}$ monoids."}
{"category": "Math", "title": "Subspace correction methods for total variation and $\\ell_1-$minimization", "abstract": "This paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a semi-norm for a subspace. The optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces. On each subspace an iterative proximity-map algorithm is implemented via \\emph{oblique thresholding}, which is the main new tool introduced in this work. We provide convergence conditions for the algorithm in order to compute minimizers of the target energy. Analogous results are derived for a parallel variant of the algorithm. Applications are presented in domain decomposition methods for singular elliptic PDE's arising in total variation minimization and in accelerated sparse recovery algorithms based on $\\ell_1$-minimization. We include numerical examples which show efficient solutions to classical problems in signal and image processing."}
{"category": "Math", "title": "A Realization of Measurable Sets as Limit Points", "abstract": "Starting with a sigma finite measure on an algebra, we define a pseudometric and show how measurable sets from the Caratheodory Extension Theorem can be thought of as limit points of Cauchy sequences in the algebra."}
{"category": "Math", "title": "A Cheerful Introduction to Forcing and the Continuum Hypothesis", "abstract": "This is an introduction to the set-theoretic method of forcing, including its application in proving the independence of the Continuum Hypothesis from the Zermelo-Fraenkel axioms of set theory. I presuppose no particular mathematical background beyond some familiarity with set theory and mathematical logic - in particular, no algebra is presupposed, though it can be useful. The goal is to have a document that makes this material accessible to mathematics graduate students in all fields, and to philosophers with an interest in set theory and mathematical logic but no other mathematical background."}
{"category": "Math", "title": "Irreducible plane sextics with large fundamental groups", "abstract": "We compute the fundamental group of the complement of each irreducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also give a detailed geometric description of sextics of weight eight and nine and of their moduli spaces and compute their Alexander modules; the latter are shown to be free over an appropriate ring."}
{"category": "Math", "title": "Representation theory of the $\\alpha$-determinant and zonal spherical functions", "abstract": "We prove that the multiplicity of each irreducible component in the $\\mathcal{U}(\\mathfrak{gl}_n)$-cyclic module generated by the $l$-th power $\\det^{(\\alpha)}(X)^l$ of the $\\alpha$-determinant is given by the rank of a matrix whose entries are given by a variation of the spherical Fourier transformation for $(\\mathfrak{S}_{nl},\\mathfrak{S}_l^n)$. Further, we calculate the matrix explicitly when $n=2$. This gives not only another proof of the result by Kimoto-Matsumoto-Wakayama (2007) but also a new aspect of the representation theory of the $\\alpha$-determinants."}
{"category": "Math", "title": "Lusztig's conjecture for finite classical groups with even characteristic", "abstract": "The determination of scalars involved in Lusztig's conjecture for finite reductive groups $G(F_q)$ was achieved by Waldspurger in the case of symplectic groups or orthogonal groups, under the condition that $p,q$ are large enough. Here $p$ is the characteristic of the finite field $F_q$. In this paper, we determine the scalars in the case of symplectic groups with $p = 2$, by applying the theory of symmetric spaces over a finite field due to Kawanaka and Lusztig. We also obtain a partial result in the case of special orthogonal groups with $p = 2$."}
{"category": "Math", "title": "What is a superrigid subgroup?", "abstract": "This is an expository paper. It is well known that a linear transformation can be defined to have any desired action on a basis. From this fact, one can show that every group homomorphism from Z^k to R^d extends to a homomorphism from R^k to R^d, and we will see other examples of discrete subgroups H of connected groups G, such that the homomorphisms defined on $H$ can (\"almost\") be extended to homomorphisms defined on all of G. This is related to a very classical topic in geometry, the study of linkages."}
{"category": "Math", "title": "On simultaneous rational approximations to a real number, its square, and its cube", "abstract": "We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of Davenport and Schmidt. As a consequence, we get a sharper lower bound on the exponent of approximation of such a number xi by algebraic integers of degree at most 4."}
{"category": "Math", "title": "How to compute the Stanley depth of a monomial ideal", "abstract": "Let $J\\subset I$ be monomial ideals. We show that the Stanley depth of $I/J$ can be computed in a finite number of steps. We also introduce the $\\fdepth$ of a monomial ideal which is defined in terms of prime filtrations and show that it can also be computed in a finite number of steps. In both cases it is shown that these invariants can be determined by considering partitions of suitable finite posets into intervals."}
{"category": "Math", "title": "The $n$-Queens Problem in Higher Dimensions", "abstract": "A well-known chessboard problem is that of placing eight queens on the chessboard so that no two queens are able to attack each other. (Recall that a queen can attack anything on the same row, column, or diagonal as itself.) This problem is known to have been studied by Gauss, and can be generalized to an (n \\times n) board, where (n \\geq 4). We consider this problem in $d$-dimensional chess spaces, where (d \\geq 3), and obtain the result that in higher dimensions, $n$ queens do not always suffice (in any arrangement) to attack all board positions. Our methods allow us to obtain the first lower bound on the number of queens that are necessary to attack all positions in a $d$-dimensional chess space of size $n$, and further to show that for any $k$, there are higher-dimensional chess spaces in which not all positions can be attacked by (n^k) queens."}
{"category": "Math", "title": "Conformal maps from a 2-torus to the 4-sphere", "abstract": "We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type operators parameterized over the complexified dual torus. The kernel line bundle of this family over the spectral curve describes the generalized Darboux transforms of the conformally immersed torus. If the spectral curve has finite genus the kernel bundle can be extended to the compactification of the spectral curve and we obtain a linear 2-torus worth of algebraic curves in projective 3-space. The original conformal immersion of the 2-torus is recovered as the orbit under this family of the point at infinity on the spectral curve projected to the 4-sphere via the twistor fibration."}
{"category": "Math", "title": "Real forms and finite order automorphisms of affine Kac-Moody algebras - an outline of a new approach", "abstract": "We outline a new approach to classify real forms and automorphisms of finite order of affine Kac-Moody algebras."}
{"category": "Math", "title": "$L^2$-stability of explicit schemes for incompressible Euler equations", "abstract": "We present an original study on the numerical stabiliy of explicit schemes solving the incompressible Euler equations on an open domain with slipping boundary conditions. Relying on the skewness property of the non-linear term, we demonstrate that some explicit schemes are numerically stable for small perturbations under the condition $\\delta t\\leq C \\delta x^{2r/(2r-1)}$ where $r$ is an integer, $\\delta t$ the time step and $\\delta x$ the space step."}
{"category": "Math", "title": "Fixed points of circle actions on spaces with rational cohomology of $S^n V S^{2n} V S^{3n}$ or $P^2(n) V S^{3n}$", "abstract": "Let $X$ be a finitistic space with its rational cohomology isomorphic to that of the wedge sum $P^2(n)\\vee S^{3n} $ or $S^{n} \\vee S^{2n}\\vee S^{3n}$. We study continuous $\\mathbb{S}^1$ actions on $X$ and determine the possible fixed point sets up to rational cohomology depending on whether or not $X$ is totally non-homologous to zero in $X_{\\mathbb{S}^1}$ in the Borel fibration $X\\hookrightarrow X_{\\mathbb{S}^1} \\longrightarrow B_{\\mathbb{S}^1}$. We also give examples realizing the possible cases."}
{"category": "Math", "title": "Application of a curvature adjusted method in image segmentation", "abstract": "This article deals with flow of plane curves driven by the curvature and external force. We make use of such a geometric flow for the purpose of image segmentation. A parametric model for evolving curves with uniform and curvature adjusted redistribution of grid points will be described and compared."}
{"category": "Math", "title": "Mould expansions for the saddle-node and resurgence monomials", "abstract": "This article is an introduction to some aspects of \\'Ecalle's mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector field which is formally conjugate to Euler's vector field $x^2\\frac{\\pa}{\\pa x}+(x+y)\\frac{\\pa}{\\pa y}$, and for which the formal normalisation is shown to be resurgent in $1/x$. Resurgence monomials adapted to alien calculus are also described as another application of mould calculus."}
{"category": "Math", "title": "The Dynamical Mordell-Lang Conjecture", "abstract": "We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $\\phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $\\phi$ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(\\bP^1)^g$ has only finite intersection with any curve contained in $(\\bP^1)^g$. We also show that our result holds for indecomposable polynomials $\\phi$ with coefficients in $\\bC$. Our proof uses results from $p$-adic dynamics together with an integrality argument. The extension to polynomials defined over $\\bC$ uses the method of specializations coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of $(\\phi,\\phi)$ on $\\bA^2$."}
{"category": "Math", "title": "Virtual Bridge Number One Knots", "abstract": "We define the virtual bridge number $vb(K)$ and the virtual unknotting number $vu(K)$ invariants for virtual knots. For ordinary knots $K$ they are closely related to the bridge number $b(K)$ and the unknotting number $u(K)$ and we have $vu(K)\\leq u(K), vb(K)\\leq b(K).$ There are no ordinary knots $K$ with $b(K)=1.$ We show there are infinitely many homotopy classes of virtual knots each of which contains infinitely many isotopy classes of $K$ with $vb(K)=1.$ In fact for each $i\\in \\N$ there exists $K$ virtually homotopic (but not virtually isotopic) to the unknot with $vb(K)=1$ and $vu(K)=i.$"}
{"category": "Math", "title": "Robustly estimating the flow direction of information in complex physical systems", "abstract": "We propose a new measure to estimate the direction of information flux in multivariate time series from complex systems. This measure, based on the slope of the phase spectrum (Phase Slope Index) has invariance properties that are important for applications in real physical or biological systems: (a) it is strictly insensitive to mixtures of arbitrary independent sources, (b) it gives meaningful results even if the phase spectrum is not linear, and (c) it properly weights contributions from different frequencies. Simulations of a class of coupled multivariate random data show that for truly unidirectional information flow without additional noise contamination our measure detects the correct direction as good as the standard Granger causality. For random mixtures of independent sources Granger Causality erroneously yields highly significant results whereas our measure correctly becomes non-significant. An application of our novel method to EEG data (88 subjects in eyes-closed condition) reveals a strikingly clear front-to-back information flow in the vast majority of subjects and thus contributes to a better understanding of information processing in the brain."}
{"category": "Math", "title": "Singularities of the Secant Variety", "abstract": "We give positivity conditions on the embedding of a smooth variety which guarantee the normality of the secant variety, generalizing earlier results of the author and others. We also give classes of secant varieties satisfying the Hodge conjecture as well as a result on the singular locus of degenerate secant varieties."}
{"category": "Math", "title": "Ternary cyclotomic polynomials having a large coefficient", "abstract": "Let $\\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\\Phi_n(x)$, satisfies $|a_n(k)|\\le (p+1)/2$ in case $n=pqr$ with $p<q<r$ primes (in this case $\\Phi_n(x)$ is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example $|a_n(k)|\\le 3p/4$). Here we show that, nevertheless, Beiter's conjecture is false for every $p\\ge 11$. We also prove that given any $\\epsilon>0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1<p_2<... $ consecutive primes such that $|a_{p_jq_jr_j}(n_j)|>(2/3-\\epsilon)p_j$ for $j\\ge 1$."}
{"category": "Math", "title": "Petri map for rank two bundles with canonical determinant", "abstract": "We prove the Bertram-Feinberg-Mukai conjecture for a generic curve $C$ of genus $g$ and a semistable vector bundle $E$ of rank two and determinant $K$ on $C$, namely we prove the injectivity of the Petri-canonical map $S^2(H^0(E))\\to H^0(S^2(E))$."}
{"category": "Math", "title": "Lifting to cluster-tilting objects in 2-Calabi-Yau triangulated categories", "abstract": "We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi-Yau triangulated category lifts to a cluster-tilting object in this 2-Calabi-Yau triangulated category. This generalizes a recent work of D. Smith for cluster categories."}
{"category": "Math", "title": "The Wickstead Problem", "abstract": "In 1977 Anthony Wickstead raised the question of the conditions for all band preserving linear operators to be order bounded in a vector lattice. This article overviews the main ideas and results on the Wickstead problem and its variations, focusing primarily on the case of band preserving operators in a universally complete vector lattice."}
{"category": "Math", "title": "A proof of Goldbach's conjecture that all even numbers greater than four are the sum of two primes", "abstract": "In this paper I introduce a model which allows one to prove Goldbachs hypothesis. The model is produced by studying Goldbach partitions as displayed by an inverted mirror image of all the primes up to some even number equal to the last prime plus three. The bottom half of the model is then moved to the right in steps of two which exhibit the Goldbach partitions for the next even number. As long as the model contains all the primes up to the resulting even number minus three, then Goldbachs hypothesis can be proven if it can be shown that each move must produce a Goldbach partition until one reaches the next prime plus one. I show that this must be the case."}
{"category": "Math", "title": "Particle Approximation of the Wasserstein Diffusion", "abstract": "We construct a system of interacting two-sided Bessel processes on the unit interval and show that the associated empirical measure process converges to the Wasserstein Diffusion, assuming that Markov uniqueness holds for the generating Wasserstein Dirichlet form. The proof is based on the variational convergence of an associated sequence of Dirichlet forms in the generalized Mosco sense of Kuwae and Shioya."}
{"category": "Math", "title": "General runner removal and the Mullineux map", "abstract": "We prove a new `runner removal theorem' for $q$-decomposition numbers of the level 1 Fock space of type $A^{(1)}_{e-1}$, generalising earlier theorems of James--Mathas and the author. By combining this with another theorem relating to the Mullineux map, we show that the problem of finding all $q$-decomposition numbers indexed by partitions of a given weight is a finite computation."}
{"category": "Math", "title": "Local Ramsey theory. An abstract approach", "abstract": "It is shown that the known notion of selective coideal can be extended to a family $\\mathcal{H}$ of subsets of $\\mathcal{R}$, where $(\\mathcal{R},\\leq,r)$ is a topological Ramsey space in the sense of Todorcevic (see \\cite{todo}). Then it is proven that, if $\\mathcal{H}$ selective, the $\\mathcal{H}$-Ramsey and $\\mathcal{H}$-Baire subsets of $\\mathcal{R}$ are equivalent. This extends the results of Farah in \\cite{farah} for semiselective coideals of $\\mathbb{N}$. Also, it is proven that the family of ${\\cal H}$--Ramsey subsets of ${\\cal R}$ is closed under the Souslin operation."}
{"category": "Math", "title": "Large Deviations for Riesz Potentials of Additive Processes", "abstract": "We study functionals of the form \\[\\zeta_{t}=\\int_0^{t}...\\int_0^{t} | X_1(s_1)+...+ X_p(s_p)|^{-\\sigma}ds_1... ds_p\\] where $X_1(t),..., X_p(t)$ are i.i.d. $d$-dimensional symmetric stable processes of index $0<\\bb\\le 2$. We obtain results about the large deviations and laws of the iterated logarithm for $\\zeta_{t}$."}
{"category": "Math", "title": "Degree k Linear Recursions Mod(p)", "abstract": "Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\\{F_{k,n}(t_1,...,t_k)\\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If $F_{k,n}(t_1,...,t_k) = f_n$, then $$f_n - \\sum_{j=1}^k t_j f_{n-j} = 0,$$ and $\\{f_n\\}$ is a linear recursion of degree $k$. On the one hand, the periodic properties of such sequences modulo a prime $p$ are discussed, and are shown to be rela ted to the prime structure of certain algebraic number fields; for example, the arithmetic properties of the period ar e shown to characterize ramification of primes in an extension field. On the other hand, the structure of the semiloca l rings associated with the number field is shown to be completely determined by Schur-hook polynomials."}
{"category": "Math", "title": "Uniqueness of a pre-generator for $C_0$-semigroup on a general locally convex vector space", "abstract": "The main purpose is to generalize a theorem of Arendt about uniqueness of $C_0$-semigroups from Banach space setting to the general locally convex vector spaces, more precisely, we show that cores are the only domains of uniqueness for $C_0$-semigroups on locally convex spaces. As an application, we find a necessary and sufficient condition for that the mass transport equation has one unique $L^1(\\R^d,dx)$ weak solution."}
{"category": "Math", "title": "A polynomial parametrization of torus knots", "abstract": "For every odd integer $N$ we give an explicit construction of a polynomial curve $\\cC(t) = (x(t), y (t))$, where $\\deg x = 3$, $\\deg y = N + 1 + 2\\pent N4$ that has exactly $N$ crossing points $\\cC(t_i)= \\cC(s_i)$ whose parameters satisfy $s_1 < ... < s_{N} < t_1 < ... < t_{N}$. Our proof makes use of the theory of Stieltjes series and Pad\\'e approximants. This allows us an explicit polynomial parametrization of the torus knot $K_{2,N}$."}
{"category": "Math", "title": "The general quadruple point formula", "abstract": "Maps between manifolds $M^m\\to N^{m+\\ell}$ ($\\ell>0$) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes represented by multisingularity loci, and the characteristic classes of the manifolds. These relations include the celebrated Thom polynomials of monosingularities. For multisingularities, however, only the form of these relations is clear in general (due to Kazarian), the concrete polynomials occurring in the relations are much less known. In the present paper we prove the first general such relation outside the region of Morin-maps: the general quadruple point formula. We apply this formula in enumerative geometry by computing the number of 4-secant linear spaces to smooth projective varieties. Some other multisingularity formulas are also studied, namely 5, 6, 7 tuple point formulas, and one corresponding to $\\Sigma^2\\Sigma^0$ multisingularities."}
{"category": "Math", "title": "Multi-linear multipliers associated to simplexes of arbitrary length", "abstract": "In this article we prove that the $n$-linear operator whose symbol is the characteristic function of the simplex $\\Delta_n = \\xi_1 < ... < \\xi_n$ is bounded from $L^2 \\times ... \\times L^2$ into $L^{2/n}$, generalizing in this way our previous work on the \"bi-est\" operator (which corresponds to the case $n=3$) as well as Lacey-Thiele theorem on the bi-linear Hilbert transform (which corresponds to the case $n=2$)."}
{"category": "Math", "title": "Towards BAD conjecture", "abstract": "For $\\alpha, \\beta, \\delta \\in [0,1], \\alpha +\\beta = 1 $ we consider sets $$ {\\rm BAD}^* (\\alpha, \\beta ;\\delta) = \\left\\{\\xi = (\\xi_1,\\xi_2) \\in [0,1]^2: ,\\inf_{p\\in \\mathbb{N}} \\max \\{(p\\log(p+1))^\\alpha ||p\\xi_1||, (p\\log (p+1))^\\beta ||p\\xi_2||\\} \\ge \\delta \\right\\}. $$ We prove that for different $(\\alpha_1,\\beta_1), (\\alpha_2,\\beta_2), \\alpha_1 +\\beta_1 = \\alpha_2 +\\beta_2 = 1 $ and $\\delta $ small enough $$ {\\rm BAD}^* (\\alpha_1, \\beta_1 ;\\delta) \\bigcap {\\rm BAD}^* (\\alpha_2, \\beta_2 ;\\delta) \\neq \\varnothing . $$ Our result is based on A. Khintchine's construction and an original method due to Y. Peres and W. Schlag."}
{"category": "Math", "title": "Positivity results on ribbon Schur function differences", "abstract": "There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. We consider the posets that result from ordering skew diagrams according to Schur positivity, before focussing on the convex subposets corresponding to ribbons. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those whose expansion as a linear combination of Schur functions has all coefficients either zero or one. In particular, we show that the poset that results from ordering such ribbons according to Schur-positivity is essentially a product of two chains."}
{"category": "Math", "title": "Limits Of One Dimensional Diffusions", "abstract": "In this paper we look at the properties of limits of a sequence of real valued time inhomogeneous diffusions. When convergence is only in the sense of finite-dimensional distributions then the limit does not have to be a diffusion. However, we show that as long as the drift terms satisfy a Lipschitz condition and the limit is continuous in probability, then it will lie in a class of processes that we refer to as almost-continuous diffusions. These processes are strong Markov and satisfy an `almost-continuity' condition. We also give a simple condition for the limit to be a continuous diffusion. These results contrast with the multidimensional case where, as we show with an example, a sequence of two dimensional martingale diffusions can converge to a process that is both discontinuous and non-Markov."}
{"category": "Math", "title": "An equivariant index formula for elliptic actions on contact manifolds", "abstract": "Given an elliptic action of a compact Lie group $G$ on a co-oriented contact manifold $(M,E)$ one obtains two naturally associated objects: A $G$-transversally elliptic operator $\\dirac$, and an equivariant differential form with generalised coefficients $\\mathcal{J}(E,X)$ defined in terms of a choice of contact form on $M$. We explain how the form $\\mathcal{J}(E,X)$ is natural with respect to the contact structure, and give a formula for the equivariant index of $\\dirac$ involving $\\mathcal{J}(E,X)$. A key tool is the Chern character with compact support developed by Paradan-Vergne \\cite{PV1,PV}."}
{"category": "Math", "title": "Morse Inequalities for Orbifold Cohomology", "abstract": "This paper begins the study of Morse theory for orbifolds, or more precisely for differentiable Deligne-Mumford stacks. The main result is an analogue of the Morse inequalities that relates the orbifold Betti numbers of an almost-complex orbifold to the critical points of a Morse function on the orbifold. We also show that a generic function on an orbifold is Morse. In obtaining these results we develop for differentiable Deligne-Mumford stacks those tools of differential geometry and topology -- flows of vector fields, the strong topology -- that are essential to the development of Morse theory on manifolds."}
{"category": "Math", "title": "Refinements of Milnor's Fibration Theorem for Complex Singularities", "abstract": "Let $X$ be an analytic subset of an open neighbourhood $U$ of the origin $\\underline{0}$ in $\\mathbb{C}^n$. Let $f\\colon (X,\\underline{0}) \\to (\\mathbb{C},0)$ be holomorphic and set $V =f^{-1}(0)$. Let $\\B_\\epsilon$ be a ball in $U$ of sufficiently small radius $\\epsilon>0$, centred at $\\underline{0}\\in\\mathbb{C}^n$. We show that $f$ has an associated canonical pencil of real analytic hypersurfaces $X_\\theta$, with axis $V$, which leads to a fibration $\\Phi$ of the whole space $(X \\cap \\mathbb{B}_\\epsilon) \\setminus V$ over $\\mathbb{S}^1 $. Its restriction to $(X \\cap \\mathbb{S}_\\epsilon) \\setminus V$ is the usual Milnor fibration $\\phi = \\frac{f}{|f|}$, while its restriction to the Milnor tube $f^{-1}(\\partial \\D_\\eta) \\cap \\mathbb{B}_\\epsilon$ is the Milnor-L\\^e fibration of $f$. Each element of the pencil $X_\\theta$ meets transversally the boundary sphere $\\mathbb{S}_\\epsilon = \\partial \\B_\\epsilon$, and the intersection is the union of the link of $f$ and two homeomorphic fibers of $\\phi$ over antipodal points in the circle. Furthermore, the space ${\\tilde X}$ obtained by the real blow up of the ideal $(Re(f), Im(f))$ is a fibre bundle over $\\mathbb{R} \\mathbb{P}^1$ with the $X_\\theta$ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities."}
{"category": "Math", "title": "A priori bounds for some infinitely renormalizable quadratics: III. Molecules", "abstract": "In this paper we prove {\\it a priori bounds} for infinitely renormalizable quadratic polynomials satisfying a ``molecule condition''. Roughly speaking, this condition ensures that the renormalization combinatorics stay away from the satellite types. These {\\it a priori bounds} imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values."}
{"category": "Math", "title": "Gluing of Surfaces with Polygonal Boundaries", "abstract": "By pairwise gluing of edges of a polygon, one produces two-dimensional surfaces with handles and boundaries. In this paper, we count the number ${\\cal N}_{g,L}(n_1, n_2, ..., n_L)$ of different ways to produce a surface of given genus $g$ with $L$ polygonal boundaries with given numbers of edges $n_1, n_2, >..., n_L$. Using combinatorial relations between graphs on real two-dimensional surfaces, we derive recursive relations between ${\\cal N}_{g,L}$. We show that Harer-Zagier numbers appear as a particular case of ${\\cal N}_{g,L}$ and derive a new explicit expression for them."}
{"category": "Math", "title": "Tame nonsmooth inverse mapping theorems", "abstract": "We give several versions of local and global inverse mapping theorem for tame non necessarily smooth, mappings. Here tame mapping means a mapping which is subanalytic or, more generally, definable in some o-minimal structure. Our sufficient conditions are formulated in terms of various properties (convexity, positivity of some principal minors, contractiblity) of the space of Jacobi's matrices at smooth points."}
{"category": "Math", "title": "Takacs' asymptotic theorem and its applications: A survey", "abstract": "The book of Lajos Tak\\'acs \\emph{Combinatorial Methods in the Theory of Stochastic Processes} has been published in 1967. It discusses various problems associated with $$ P_{k,i}=\\mathrm{P}{\\sup_{1\\leq n\\leq\\rho(i)}(N_n-n)<k-i},\\leqno(*) $$ where $N_n=\\nu_1+\\nu_2...+\\nu_n$ is a sum of mutually independent, nonnegative integer and identically distributed random variables, $\\pi_j=\\mathrm{P}\\{\\nu_k=j\\}$, $j\\geq0$, $\\pi_0>0$, and $\\rho(i)$ is the smallest $n$ such that $N_n=n-i$, $i\\geq1$. (If there is no such $n$, then $\\rho(i)=\\infty$.) (*) is a discrete generalization of the classic ruin probability, and its value is represented as $P_{k,i}={Q_{k-i}}/{Q_k}$, where the sequence $\\{Q_k\\}_{k\\geq0}$ satisfies the recurrence relation of convolution type: $Q_0\\neq0$ and $Q_k=\\sum_{j=0}^k\\pi_jQ_{k-j+1}$. Since 1967 there have been many papers related to applications of the generalized classic ruin probability. The present survey concerns only with one of the areas of application associated with asymptotic behavior of $Q_k$ as $k\\to\\infty$. The theorem on asymptotic behavior of $Q_k$ as $k\\to\\infty$ and further properties of that limiting sequence are given on pages 22-23 of the aforementioned book by Tak\\'acs. In the present survey we discuss applications of Tak\\'acs' asymptotic theorem and other related results in queueing theory, telecommunication systems and dams. Many of the results presented in this survey have appeared recently, and some of them are new. In addition, further applications of Tak\\'acs' theorem are discussed."}
{"category": "Math", "title": "Lie Group Action and Stability Analysis of Stationary Solutions for a Free Boundary Problem Modelling Tumor Growth", "abstract": "In this paper we study asymptotic behavior of solutions for a multidimensional free boundary problem modelling the growth of nonnecrotic tumors. We first establish a general result for differential equations in Banach spaces possessing a local Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either $D_A(\\theta)$ or $D_A(\\theta,\\infty)$ type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient $\\gamma$ is larger than a threshold value $\\gamma^\\ast$ then the unique stationary solution is asymptotically stable modulo translations, provided the constant $c$ representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small, whereas if $\\gamma< \\gamma^\\ast$ then this stationary solution is unstable."}
{"category": "Math", "title": "Asymptotic Stability of the Stationary Solution for a Hyperbolic Free Boundary Problem Modeling Tumor Growth", "abstract": "In this paper we study asymptotic behavior of solutions for a free boundary problem modeling the growth of tumors containing two species of cells: proliferating cells and quiescent cells. This tumor model was proposed by Pettet et al in {\\em Bull. Math. Biol.} (2001). By using a functional approach and the $C_0$ semigroup theory, we prove that the unique stationary solution of this model ensured by the work of Cui and Friedman ({\\em Trans. Amer. Math. Soc.}, 2003) is locally asymptotically stable in certain function spaces. Key techniques used in the proof include an improvement of the linear estimate obtained by the work of Chen et al ({\\em Trans. Amer. Math. Soc.}, 2005), and a similarity transformation."}
{"category": "Math", "title": "Divergence of combinatorial averages and the unboundedness of the trilinear Hilbert transform", "abstract": "We consider multilinear averages in ergodic theory and harmonic analysis and prove their divergence in some range of $L^p$ spaces, with $p$ close enough to 1. We also prove that the trilinear Hilbert transform is unbounded in a similar range of $L^p$ spaces."}
{"category": "Math", "title": "A Duality Result for Moduli Spaces of Semistable Sheaves Supported on Projective Curves", "abstract": "We show that the moduli space M(r,c) of semistable sheaves on n-dimensional projective space with support of dimension one, with multiplicity r and with Euler characteristic c is isomorphic to M(r,-c)."}
{"category": "Math", "title": "Nonsmoothable group actions on elliptic surfaces", "abstract": "Let G be a cyclic group of order 3, 5 or 7, and X=E(n) be the relatively minimal elliptic surface with rational base. In this paper, we prove that under certain conditions on n, there exists a locally linear G-action on X which is nonsmoothable with respect to infinitely many smooth structures on X. This extends the main result of our previous paper."}
{"category": "Math", "title": "Cyclotomic expansion of exceptional spectral measures", "abstract": "We find explicit formulae for the circular spectral measures of $E_7,E_8$. This leads to a number of general observations regarding the ADE circular measures: these are linear combinations of measures supported by the roots of unity, with real density given by certain degree 3 polynomials."}
{"category": "Math", "title": "Calculating Colimits Compositionally", "abstract": "We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages."}
{"category": "Math", "title": "Variational inference for large-scale models of discrete choice", "abstract": "Discrete choice models are commonly used by applied statisticians in numerous fields, such as marketing, economics, finance, and operations research. When agents in discrete choice models are assumed to have differing preferences, exact inference is often intractable. Markov chain Monte Carlo techniques make approximate inference possible, but the computational cost is prohibitive on the large data sets now becoming routinely available. Variational methods provide a deterministic alternative for approximation of the posterior distribution. We derive variational procedures for empirical Bayes and fully Bayesian inference in the mixed multinomial logit model of discrete choice. The algorithms require only that we solve a sequence of unconstrained optimization problems, which are shown to be convex. Extensive simulations demonstrate that variational methods achieve accuracy competitive with Markov chain Monte Carlo, at a small fraction of the computational cost. Thus, variational methods permit inferences on data sets that otherwise could not be analyzed without bias-inducing modifications to the underlying model."}
{"category": "Math", "title": "An invariant regarding Waring's problem for cubic polynomials", "abstract": "We compute the equation of the 7-secant variety to the Veronese variety (P^4,O(3)), its degree is 15. This is the last missing invariant in the Alexander-Hirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariantof plane cubics as a pfaffian."}
{"category": "Math", "title": "On $p$-harmonic map heat flows for {$1\\leq p< \\infty$} and their finite element approximations", "abstract": "Motivated by emerging applications from imaging processing, the heat flow of a generalized $p$-harmonic map into spheres is studied for the whole spectrum, $1\\leq p<\\infty$, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a $BV$-solution concept is introduced and the existence of such a solution is proved for the 1-harmonic map heat flow. The main idea used to develop such a theory is to exploit the properties of measures of the forms $\\cA\\cdot\\nab\\bv$ and $\\cA\\wedge\\nab\\bv$; which pair a divergence-$L^1$, or a divergence-measure, tensor field $\\cA$, and a $BV$-vector field $\\bv$. Based on these analytical results, a practical fully discrete finite element method is then proposed for approximating weak solutions of the $p$-harmonic map heat flow, and the convergence of the proposed numerical method is also established."}
{"category": "Math", "title": "The Viterbo Transfer as a Map of Spectra", "abstract": "Let $L$ and $N$ be two smooth manifolds of the same dimension. Let $j\\colon L\\to T^*N$ be an exact Lagrange embedding. We denote the free loop space of $X$ by $\\Lambda X$. C. Viterbo constructed a transfer map $(\\Lambda j)^! \\colon H^*(\\Lambda L) \\to H^*(\\Lambda N)$. This transfer was constructed using finite dimensional approximation of Floer homology. In this paper we define a family of finite dimensional approximations and realize this transfer as a map of Thom spectra: $(\\Lambda j)_! \\colon (\\Lambda N)^{-TN} \\to (\\Lambda L)^{-TL+\\eta}$, where $\\eta$ is a virtual vector bundle classified by the tangential information of $j$."}
{"category": "Math", "title": "Betti numbers of monomial ideals and shifted skew shapes", "abstract": "We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a fixed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coefficient field."}
{"category": "Math", "title": "An Extended Bracket Polynomial for Virtual Knots and Links", "abstract": "This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by <<K>> for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram for K. Each state is reduced to a virtual 4-regular graph in the plane and the polynomial takes values in the module generated by these reduced graphs over the ring Q[A,A^{-1}]. The paper is relatively self-contained, with background information about virtual knots and long virtual knots. We give numerous examples applying the extended bracket, including a new proof of the non-triviality of the Kishino diagram and the flat Kishino diagram and non-classicality of single crossing virtualizations. The paper has a section on the estimation of virtual crossing number using the extended bracket state sum. Examples are given of virtual knots with arbitrary minimal embedding genus and arbitrarily high positive difference between the virtual crossing number and the minimal embedding genus. A simplification of <<K>> is introduced and denoted by A[K]. This simplified extended bracket, the arrow polynomial, is a polynomial in an infinite set of variables. It is quite strong (detecting the flat Kishino diagram for example) and easily computable. The paper contains a description of a computer algorithm for A[K] and uses the arrow polynomial, in conjunction with the extended bracket polynomial to determine the minimum genus surfaces on which some virtual knots can be represented."}
{"category": "Math", "title": "Double Ore Extensions", "abstract": "A double Ore extension is a natural generalization of the Ore extension. We prove that a connected graded double Ore extension of an Artin-Schelter regular algebra is Artin-Schelter regular. Some other basic properties such as the determinant of the DE-data are studied. Using the double Ore extension, we construct 26 families of Artin-Schelter regular algebras of global dimension four in a sequel paper."}
{"category": "Math", "title": "Double Extension Regular Algebras of Type (14641)", "abstract": "We construct several families of Artin-Schelter regular algebras of global dimension four using double Ore extension and then prove that all these algebras are strongly noetherian, Auslander regular, Koszul and Cohen-Macaulay domains. Many regular algebras constructed in the paper are new and are not isomorphic to either a normal extension or an Ore extension of an Artin-Schelter regular algebra of global dimension three."}
{"category": "Math", "title": "Artin-Schelter Regular Algebras, Subalgebras, and Pushouts", "abstract": "Take $A$ to be a regular quadratic algebra of global dimension three. We observe that there are examples of $A$ containing a dimension three regular cubic algebra $C$. If $B$ is another dimension three regular quadratic algebra, also containing $C$ as a subalgebra, then we can form the pushout algebra $D$ of the inclusions $i_1:C\\hookrightarrow A$ and $i_2:C\\hookrightarrow B$. We show that for a certain class of regular algebras $C\\hookrightarrow A,B$, their pushouts $D$ are regular quadratic algebras of global dimension four. Furthermore, some of the point module structures of the dimension three algebras get passed on to the pushout algebra $D$."}
{"category": "Math", "title": "Leavitt path algebras and direct limits", "abstract": "An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplicity, characterizations of the exchange property, and cancellation conditions for the K-theoretic monoid of equivalence classes of idempotent matrices."}
{"category": "Math", "title": "Algebraic generality vs arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874)", "abstract": "Throughout the whole year of 1874, C. Jordan and L. Kronecker were quarrelling over two theorems. On the one hand, Jordan had stated in 1870 a canonical form theorem for substitutions of linear groups; on the other hand, Karl Weierstrass had introduced in 1868 the elementary divisors of non singular pairs of bilinear forms (P,Q) in stating a key theorem of the theory of bilinear and quadratic forms. Although they would be considered equivalent as regard to modern mathematics, not only had these two theorems been stated independently and for different purposes, they had also been lying within the distinct frameworks of two theories until some connections came to light in 1872-1873, breeding the 1874 quarrel and hence revealing an opposition over two practices relating to distinctive cultural features. As we will be looking into the 1874 quarrel, our purpose will be to show how the complex identities of practices such as Jordan s canonical reduction and Kronecker s invariant computation highlight some cultural issues such as tacit knowledge and perceptions of history peculiar to individuals or communities as well as some local ways of thinking such as disciplinary ideals and internal philosophies of generality and simplicity."}
{"category": "Math", "title": "Convex Entropy Decay via the Bochner-Bakry-Emery approach", "abstract": "We develop a method, based on a Bochner-type identity, to obtain estimates on the exponential rate of decay of the relative entropy from equilibrium of Markov processes in discrete settings. When this method applies the relative entropy decays in a convex way. The method is shown to be rather powerful when applied to a class of birth and death processes. We then consider other examples, including inhomogeneous zero-range processes and Bernoulli-Laplace models. For these two models, known results were limited to the homogeneous case, and obtained via the martingale approach, whose applicability to inhomogeneous models is still unclear."}
{"category": "Math", "title": "Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties", "abstract": "For the root system of type $A$ we introduce and study a certain extension of the quadratic algebra invented by S. Fomin and the first author, to construct a model for the equivariant cohomology ring of the corresponding flag variety. As an application of our construction we describe a generalization of the equivariant Pieri rule for double Schubert polynomials. For a general finite Coxeter system we construct an extension of the corresponding Nichols-Woronowicz algebra. In the case of finite crystallographic Coxeter systems we present a construction of extended Nichols-Woronowicz algebra model for the equivariant cohomology of the corresponding flag variety."}
{"category": "Math", "title": "Minima in branching random walks", "abstract": "Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\\mathbf{E}M_n$ to within O(1) and prove exponential tail bounds for $\\mathbf{P}\\{|M_n-\\mathbf{E}M_n|>x\\}$, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89--108], our results fully characterize the possible behavior of $\\mathbf {E}M_n$ when the branching random walk has bounded branching and step size."}
{"category": "Math", "title": "Local Lipschitz geometry of weighted homogeneous surfaces", "abstract": "We compute Hoelder Complexes,i.e. the complete bi-Lipschitz invariants, for germs of real weighed homogeneous algebraic or semialgebraic surfaces."}
{"category": "Math", "title": "Twisting quasi-alternating links", "abstract": "Quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasi-alternating link gives rise to an infinite family of quasi-alternating links obtained by replacing a crossing with an alternating rational tangle. Consequently, we show that many pretzel links are quasi-alternating, and we determine the thickness of Khovanov homology for \"most\" pretzel links with arbitrarily many strands."}
{"category": "Math", "title": "A_2-web immanants", "abstract": "We describe the rank 3 Temperley-Lieb-Martin algebras in terms of Kuperberg's A_2-webs. We define consistent labelings of webs, and use them to describe the coefficients of decompositions into irreducible webs. We introduce web immanants, inspired by Temperley-Lieb immanants of Rhoades and Skandera. We show that web immanants are positive when evaluated on totally positive matrices, and describe some further properties."}
{"category": "Math", "title": "Rate of relaxation for a mean-field zero-range process", "abstract": "We study the zero-range process on the complete graph. It is a Markov chain model for a microcanonical ensemble. We prove that the process converges to a fluid limit. The fluid limit rapidly relaxes to the appropriate Gibbs distribution."}
{"category": "Math", "title": "New directions in Nielsen-Reidemeister theory", "abstract": "The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\\infty$ \\emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology."}
{"category": "Math", "title": "Even perfect polynomials over $F_2$ with four prime factors", "abstract": "A perfect polynomial over the binary field $\\F_2$ is a polynomial $A \\in \\F_2[x]$ that equals the sum of all its divisors. If $\\gcd(A,x^2-x) \\neq 1$ then we call $A$ even. The list of all even perfect polynomials over $\\F_2$ with at most 3 prime factors in known. The object of this paper is to give the list of all even perfect polynomials over $\\F_2$ with four prime factors. These are all the known perfect polynomials with four prime factors over $\\F_2$."}
{"category": "Math", "title": "Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis", "abstract": "We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau's solution. We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn-Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by J.F. Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis."}
{"category": "Math", "title": "Vector spaces with an order unit", "abstract": "We develop a theory of ordered *-vector spaces with an order unit. We prove fundamental results concerning positive linear functionals and states, and we show that the order (semi)norm on the space of self-adjoint elements admits multiple extensions to an order (semi)norm on the entire space. We single out three of these (semi)norms for further study and discuss their significance for operator algebras and operator systems. In addition, we introduce a functorial method for taking an ordered space with an order unit and forming an Archimedean ordered space. We then use this process to describe an appropriate notion of quotients in the category of Archimedean ordered spaces."}
{"category": "Math", "title": "Characters of unipotent groups over finite fields", "abstract": "Let G be a connected unipotent group over a finite field F_q with q elements. In this article we propose a definition of L-packets of complex irreducible representations of the finite group G(F_q) and give an explicit description of L-packets in terms of the so-called \"admissible pairs\" for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension of every complex irreducible representation of G(F_q) is a power of q, confirming a conjecture of V. Drinfeld. This paper is the first in a series of three papers exploring the relationship between representations of a group of the form G(F_q) (where G is a unipotent algebraic group over F_q), the geometry of G, and the theory of character sheaves."}
{"category": "Math", "title": "On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges", "abstract": "The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turan numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k in {3,4}. We give a complete solution when k=3 and an almost complete solution (with eleven exceptions) when k=4."}
{"category": "Math", "title": "Generalized Complex and Dirac Structures on Homogeneous Spaces", "abstract": "We partially describe equivariant Dirac and generalized complex structures on a homogeneous space $G/K$ by giving equivalent data involving only the Lie algebra. We consider real semisimple adjoint orbits in any semisimple Lie algebra over $\\mathbb R$ and real nilpotent orbits in $sl_n (\\mathbb R)$. We give a complete classification for Riemannian symmetric spaces and for a compact group modulo a closed, connected subgroup containing a Cartan subgroup."}
{"category": "Math", "title": "A Brunn-Minkowski Inequality for Symplectic Capacities of Convex Domains", "abstract": "In this work we prove a Brunn-Minkowski-type inequality in the context of symplectic geometry and discuss some of its applications."}
{"category": "Math", "title": "Exponentiating $2\\times2$ and $3\\times3$ Matrices Done Right", "abstract": "We derive explicit formulas for calculating $e^A$, $\\cosh{A}$, $\\sinh{A}, \\cos{A}$ and $\\sin{A}$ for a given $2\\times2$ matrix $A$. We also derive explicit formulas for $e^A$ for a given $3\\times3$ matrix $A$. These formulas are expressed exclusively in terms of the characteristic roots of $A$ and involve neither the eigenvectors of $A$, nor the transition matrix associated with a particular canonical basis. We believe that our method has advantages (especially if applied by non-mathematicians or students) over the more conventional methods based on the choice of canonical bases. We support this point with several examples for solving first order linear systems of ordinary differential equations with constant coefficients."}
{"category": "Math", "title": "Two Categories of Dirac Manifolds", "abstract": "We define two categories of Dirac manifolds, i.e. manifolds with complex Dirac structures. The first notion of maps I call \\emph{Dirac maps}, and the category of Dirac manifolds is seen to contain the categories of Poisson and complex manifolds as full subcategories. The second notion, \\emph{dual-Dirac maps}, defines a \\emph{dual-Dirac category} which contains presymplectic and complex manifolds as full subcategories. The dual-Dirac maps are stable under B-transformations. In particular we get two structures of a category on Hitchin'sgeneralized complex manifolds, i.e., two reasonable notions of generalized complex maps. We also generalize further to get categories of Dirac manifolds for which the Dirac structures lie in arbitrary exact Courant algebroids. As an example, we consider the case of a Lie group with a complex Dirac structure and establish conditions for which multiplication is a Dirac map."}
{"category": "Math", "title": "On the Limit Law of a Random Walk Conditioned to Reach a High Level", "abstract": "We consider a random walk with a negative drift and with a jump distribution which under Cram\\'er's change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably scaled, this random walk converges in law to a nondecreasing Markov process which can be interpreted as a spectrally-positive L\\'evy %-Khinchin process conditioned not to overshoot level one."}
{"category": "Math", "title": "Chern class formulas for $G_2$ Schubert loci", "abstract": "We define degeneracy loci for vector bundles with structure group $G_2$, and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein-Gelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type $G_2$. We include explicit descriptions of the $G_2$ flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham. In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, clarifying a previous computation of Edidin and Graham."}
{"category": "Math", "title": "The Veronese Construction for Formal Power Series and Graded Algebras", "abstract": "Let $(a_n)_{n \\geq 0}$ be a sequence of complex numbers such that its generating series satisfies $\\sum_{n \\geq 0} a_nt^n = \\frac{h(t)}{(1-t)^d}$ for some polynomial $h(t)$. For any $r \\geq 1$ we study the transformation of the coefficient series of $h(t)$ to that of $h^{< r >}(t)$ where $\\sum_{n \\geq 0} a_{nr} t^n = \\frac{h^{< r >}(t)}{(1-t)^d}$. We give a precise description of this transformation and show that under some natural mild hypotheses the roots of $h^{< r >}(t)$ converge when $r$ goes to infinity. In particular, this holds if $\\sum_{n \\geq 0} a_n t^n$ is the Hilbert series of a standard graded $k$-algebra $A$. If in addition $A$ is Cohen-Macaulay then the coefficients of $h^{< r >}(t)$ are monotonely increasing with $r$. If $A$ is the Stanley-Reisner ring of a simplicial complex $\\Delta$ then this relates to the $r$th edgewise subdivision of $\\Delta$ which in turn allows some corollaries on the behavior of the respective $f$-vectors."}
{"category": "Math", "title": "Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations", "abstract": "In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations (\\textit{ANS}). In order to do so, we first introduce the scaling invariant Besov-Sobolev type spaces, $B^{-1+\\frac{2}{p},{1/2}}_{p}$ and $B^{-1+\\frac{2}{p},{1/2}}_{p}(T)$, $p\\geq2$. Then, we prove the global wellposedness for (\\textit{ANS}) provided the initial data are sufficient small compared to the horizontal viscosity in some suitable sense, which is stronger than $B^{-1+\\frac{2}{p},{1/2}}_{p}$ norm. In particular, our results imply the global wellposedness of (\\textit{ANS}) with high oscillatory initial data."}
{"category": "Math", "title": "Analysis of nonlinear modes of variation for functional data", "abstract": "A set of curves or images of similar shape is an increasingly common functional data set collected in the sciences. Principal Component Analysis (PCA) is the most widely used technique to decompose variation in functional data. However, the linear modes of variation found by PCA are not always interpretable by the experimenters. In addition, the modes of variation of interest to the experimenter are not always linear. We present in this paper a new analysis of variance for Functional Data. Our method was motivated by decomposing the variation in the data into predetermined and interpretable directions (i.e. modes) of interest. Since some of these modes could be nonlinear, we develop a new defined ratio of sums of squares which takes into account the curvature of the space of variation. We discuss, in the general case, consistency of our estimates of variation, using mathematical tools from differential geometry and shape statistics. We successfully applied our method to a motivating example of biological data. This decomposition allows biologists to compare the prevalence of different genetic tradeoffs in a population and to quantify the effect of selection on evolution."}
{"category": "Math", "title": "Uniqueness of real closure * of regular rings", "abstract": "In this paper we give a characterisation of real closure * of regular rings, which is quite similar to the characterisation of real closure * of Baer regular rings seen in [4]. We also characterize Baer-ness of regular rings using near-open maps. The last part of this work will concentrate on classifying the real closure * of Baer and non-Baer regular rings (upto isomorphisms) using continuous sections of the support map, we construct a topology on this set for the Baer case. For the case of non-Baer regular rings, it will be shown that almost no information of the ring structure of the Baer hull is necessary in order to study the real and prime spectra of the Baer hull. We shall make use of the absolutes of Hausdorff spaces in order to give a construction of the spectra of the Baer hulls of regular rings. Finally we give example of a Baer regular ring that is not rationally complete."}
{"category": "Math", "title": "Large Deviations Analysis for Distributed Algorithms in an Ergodic Markovian Environment", "abstract": "We provide a large deviations analysis of deadlock phenomena occurring in distributed systems sharing common resources. In our model transition probabilities of resource allocation and deallocation are time and space dependent. The process is driven by an ergodic Markov chain and is reflected on the boundary of the d-dimensional cube. In the large resource limit, we prove Freidlin-Wentzell estimates, we study the asymptotic of the deadlock time and we show that the quasi-potential is a viscosity solution of a Hamilton-Jacobi equation with a Neumann boundary condition. We give a complete analysis of the colliding 2-stacks problem and show an example where the system has a stable attractor which is a limit cycle."}
{"category": "Math", "title": "Poisson structures and generalized Kahler structures", "abstract": "Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler structures and provide non-trivial examples of generalized Kahler submanifolds arising as holomorphic Poisson submanifolds. We also obtain unobstructed deformations of bi-Hermitian structures constructed from Poisson structures."}
{"category": "Math", "title": "Cubicity, Boxicity and Vertex Cover", "abstract": "A $k$-dimensional box is the cartesian product $R_1 \\times R_2 \\times ... \\times R_k$ where each $R_i$ is a closed interval on the real line. The {\\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ is the intersection graph of a collection of $k$-dimensional boxes. A unit cube in $k$-dimensional space or a $k$-cube is defined as the cartesian product $R_1 \\times R_2 \\times ... \\times R_k$ where each $R_i$ is a closed interval on the real line of the form $[a_i, a_{i}+1]$. The {\\it cubicity} of $G$, denoted as $cub(G)$, is the minimum $k$ such that $G$ is the intersection graph of a collection of $k$-cubes. In this paper we show that $cub(G) \\leq t + \\left \\lceil \\log (n - t)\\right\\rceil - 1$ and $box(G) \\leq \\left \\lfloor\\frac{t}{2}\\right\\rfloor + 1$, where $t$ is the cardinality of the minimum vertex cover of $G$ and $n$ is the number of vertices of $G$. We also show the tightness of these upper bounds. F. S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph $G$, $box(G) \\leq \\left \\lfloor\\frac{n}{2} \\right \\rfloor$, where $n$ is the number of vertices of $G$, and this bound is tight. We show that if $G$ is a bipartite graph then $box(G) \\leq \\left \\lceil\\frac{n}{4} \\right\\rceil$ and this bound is tight. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to $\\frac{n}{4}$. Interestingly, if boxicity is very close to $\\frac{n}{2}$, then chromatic number also has to be very high. In particular, we show that if $box(G) = \\frac{n}{2} - s$, $s \\geq 0$, then $\\chi(G) \\geq \\frac{n}{2s+2}$, where $\\chi(G)$ is the chromatic number of $G$."}
{"category": "Math", "title": "Q-systems as cluster algebras", "abstract": "Q-systems first appeared in the analysis of the Bethe equations for the XXX-model and generalized Heisenberg spin chains. Such systems are known to exist for any simple Lie algebra and many other Kac-Moody algebras. We formulate the Q-system associated with any simple, simply-laced Lie algebras g in the language of cluster algebras, and discuss the relation of the polynomiality property of the solutions of the $Q$-system in the initial variables, which follows from the representation-theoretical interpretation, to the Laurent phenomenon in cluster algebras."}
{"category": "Math", "title": "The uniform primality conjecture for elliptic curves", "abstract": "An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techniques are shown to lead to explicit results where all the irreducible terms can be computed."}
{"category": "Math", "title": "Square root voting in the Council of the European Union: Rounding effects and the Jagiellonian Compromise", "abstract": "In recent years, enlargement of the European Union has brought with it renewed discussion of voting arrangements in the Council of the EU. During these negotiations, the Polish government proposed a voting scheme that gives each country a voting weight proportional to the square root of its population, and sets a quota according to an optimality condition (\"Jagiellonian Compromise\"). In this paper, the optimal quota is found exactly for the current population data from the 27 EU member states, and it is found that rounding of the voting weights can be used to improve the voting scheme."}
{"category": "Math", "title": "The Banff Challenge: Statistical Detection of a Noisy Signal", "abstract": "Particle physics experiments such as those run in the Large Hadron Collider result in huge quantities of data, which are boiled down to a few numbers from which it is hoped that a signal will be detected. We discuss a simple probability model for this and derive frequentist and noninformative Bayesian procedures for inference about the signal. Both are highly accurate in realistic cases, with the frequentist procedure having the edge for interval estimation, and the Bayesian procedure yielding slightly better point estimates. We also argue that the significance, or $p$-value, function based on the modified likelihood root provides a comprehensive presentation of the information in the data and should be used for inference."}
{"category": "Math", "title": "On effaceability of certain $\\delta$-functors", "abstract": "We prove a conjecture of the first author for $GL_2(F)$, where $F$ is a finite extension of $Q_p$."}
{"category": "Math", "title": "Some unbounded functions of intermittent maps for which the central limit theorem holds", "abstract": "We compute some dependence coefficients for the stationary Markov chain whose transition kernel is the Perron-Frobenius operator of an expanding map $T$ of $[0, 1]$ with a neutral fixed point. We use these coefficients to prove a central limit theorem for the partial sums of $f\\circ T^i$, when $f$ belongs to a large class of unbounded functions from $[0, 1]$ to ${\\mathbb R}$. We also prove other limit theorems and moment inequalities."}
{"category": "Math", "title": "Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations", "abstract": "Let there be given a probability measure $\\mu$ on the unit circle $\\TT$ of the complex plane and consider the inner product induced by $\\mu$. In this paper we consider the problem of orthogonalizing a sequence of monomials $\\{z^{r_k}\\}_k$, for a certain order of the $r_k\\in\\mathbb{Z}$, by means of the Gram-Schmidt orthogonalization process. This leads to a basis of orthonormal Laurent polynomials $\\{\\psi_k\\}_k$. We show that the matrix representation with respect to the basis $\\{\\psi_k\\}_k$ of the operator of multiplication by $z$ is an infinite unitary or isometric matrix allowing a 'snake-shaped' matrix factorization. Here the 'snake shape' of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of Delvaux and Van Barel. We show that the shape of the snake is determined by the order in which the monomials $\\{z^{r_k}\\}_k$ are orthogonalized, while the 'segments' of the snake are canonically determined in terms of the Schur parameters for $\\mu$. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism."}
{"category": "Math", "title": "Free pluriharmonic majorants and noncommutative interpolation", "abstract": "In this paper, we initiate the study of sub-pluriharmonic curves in Cuntz-Toeplitz algebras and free pluriharmonic majorants on noncommutative balls. We are lead to a characterization of the noncommutative Hardy space $H^2_{\\bf ball}$ in terms of free pluriharmonic majorants, and to a Schur type description of the unit ball of $H^2_{\\bf ball}$. These results are used to solve a multivariable commutant lifting problem and provide a description of all solutions."}
{"category": "Math", "title": "Graph limits and exchangeable random graphs", "abstract": "We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph limits (work of Lovasz and many coauthors). Along the way, we translate the graph theory into more classical probability."}
{"category": "Math", "title": "Solution of the Pompeiu problem (II)", "abstract": "This paper has been withdrawn by the author."}
{"category": "Math", "title": "Positive divisors on quotients of $\\bar{M}_{0,n}$ and the Mori cone of $\\bar{M}_{g,n}$", "abstract": "We prove that if $m \\ge n-3$ then every $S_m$-invariant F-nef divisor on the moduli space of stable $n$-pointed curves of genus zero is linearly equivalent to an effective combination of boundary divisors. As an application, we determine the Mori cone of the moduli spaces of stable curves of small genus with few marked points."}
{"category": "Math", "title": "Regularity of radial minimizers of reaction equations involving the p-Laplacian", "abstract": "We consider semi-stable, radially symmetric, and decreasing solutions of a reaction equation involving the p-Laplacian, where the reaction term is a locally Lipschitz function, and the domain is the unit ball. For this class of radial solutions, which includes local minimizers, we establish pointwise and Sobolev estimates which are optimal and do not depend on the specific nonlinear reaction term. Under standard assumptions we also prove the regularity of the corresponding extremal solution."}
{"category": "Math", "title": "The copies of any permutation pattern are asymptotically normal", "abstract": "We prove that the number of copies of any given permutation pattern $q$ has an asymptotically normal distribution in random permutations."}
{"category": "Math", "title": "Some Progress in Conformal Geometry", "abstract": "This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the $\\sigma_2$-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes."}
{"category": "Math", "title": "Central cross-sections make surfaces of revolution quadric", "abstract": "We prove here that when all planes transverse and nearly perpendicular to the axis of a surface of revolution intersect it in loops having central symmetry, the surface must be quadric. It follows that the quadrics are the only surfaces of revolution without skewloops. Similar statements hold for hypersurfaces of revolution in higher dimensions."}
{"category": "Math", "title": "A convergent Finite Element-Finite Volume scheme for the compressible Stokes problem Part I -- the isothermal case", "abstract": "In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with a linear equation of state $\\rho=p$, based on Crouzeix-Raviart elements. The approximation of the momentum balance is obtained by usual finite element techniques. Since the pressure is piecewise constant, the discrete mass balance takes the form of a finite volume scheme, in which we introduce an upwinding of the density, together with two additional stabilization terms. We prove {\\em a priori} estimates for the discrete solution, which yields its existence by a topological degree argument, and then the convergence of the scheme to a solution of the continuous problem."}
{"category": "Math", "title": "Heteroclinic Travelling Waves of Gradient Diffusion Systems", "abstract": "We establish existence of travelling waves to the gradient system $u_t = u_{zz} - \\nabla W(u)$ connecting two minima of $W$ when $u : \\R \\times (0,\\infty) \\larrow \\R^N$, that is, we establish existence of a pair $(U,c) \\in [C^2(\\R)]^N \\by (0,\\infty)$, satisfying \\[ \\{{array}{l} U_{xx} - \\nabla W (U) = - c U_x U(\\pm \\infty) = a^{\\pm}, {array}. \\] where $a^{\\pm}$ are local minima of the potential $W \\in C_{\\textrm{loc}}^2(\\R^N)$ with $W(a^-)< W(a^+)=0$ and $N \\geq 1$. Our method is variational and based on the minimization of the functional $E_c (U) = \\int_{\\R}\\Big\\{{1/2}|U_x|^2 + W(U) \\Big\\}e^{cx} dx$ in the appropriate space setup. Following Alikakos-Fusco \\cite{A-F}, we introduce an artificial constraint to restore compactness and force the desired asymptotic behavior, which we later remove. We provide variational characterizations of the travelling wave and the speed. In particular, we show that $E_c(U)=0$."}
{"category": "Math", "title": "Transversely non simple knots", "abstract": "By proving a connected sum formula for the Legendrian invariant $\\lambda_+$ in knot Floer homology we exhibit infinitely many transversely non simple knots."}
{"category": "Math", "title": "Nestings of Matchings and Permutations and North Steps in PDSAWs", "abstract": "We present a simple bijective proof of the fact that matchings of [2n] with N nestings are equinumerous to partially directed self avoiding walks confined to the symmetric wedge defined by y=+-x, with n east steps and N north steps. A very similar construction connects permutations with N nestings and PDSAWs remaining below the x-axis, again with N north steps. Furthermore, both bijections transport several combinatorially meaningful parameters."}
{"category": "Math", "title": "Prescribing valuations of the order of a point in the reductions of abelian varieties and tori", "abstract": "Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a K-rational point on G of infinite order. Call n_R the number of connected components of the smallest algebraic K-subgroup of G to which R belongs. We prove that n_R is the greatest positive integer which divides the order of (R mod p) for all but finitely many primes p of K. Furthermore, let m>0 be a multiple of n_R and let S be a finite set of rational primes. Then there exists a positive Dirichlet density of primes p of K such that for every l in S the l-adic valuation of the order of (R mod p) equals v_l(m)."}
{"category": "Math", "title": "On pairs of commuting nilpotent matrices", "abstract": "Let $B$ be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition $\\lambda$. Then it is known that its nilpotent commutator $N_B$ is an irreducible variety and that there is a unique partition $\\mu$ such that the intersection of the orbit of nilpotent matrices corresponding to $\\mu$ with $N_B$ is dense in $N_B$. We prove that map $D$ given by $D(\\lambda)=\\mu$ is an idempotent map. This answers a question of Basili and Iarrobino and gives a partial answer to a question of Panyushev. In the proof, we use the fact that for a generic matrix $A \\in N_B$ the algebra generated by $A$ and $B$ is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe $D(\\lambda)$ in terms of $\\lambda$ if $D(\\lambda)$ has at most two parts."}
{"category": "Math", "title": "Two variants of the support problem for products of abelian varieties and tori", "abstract": "Let G be the product of an abelian variety and a torus defined over a number field K. Let P and Q be K-rational points on G. Suppose that for all but finitely many primes p of K the order of (Q mod p) divides the order of (P mod p). Then there exist a K-endomorphism f of G and a non-zero integer c such that f(P)=cQ. Furthermore, we are able to prove the above result with weaker assumptions: instead of comparing the order of the points we only compare the radical of the order (radical support problem) or the l-adic valuation of the order for some fixed rational prime l (l-adic support problem)."}
{"category": "Math", "title": "Coverage processes on spheres and condition numbers for linear programming", "abstract": "This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,\\alpha)$ be the probability that $n$ spherical caps of angular radius $\\alpha$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,\\alpha)$ in the case $\\alpha\\in[\\pi/2,\\pi]$ and an upper bound for $p(n,m,\\alpha)$ in the case $\\alpha\\in [0,\\pi/2]$ which tends to $p(n,m,\\pi/2)$ when $\\alpha\\to\\pi/2$. In the case $\\alpha\\in[0,\\pi/2]$ this yields upper bounds for the expected number of spherical caps of radius $\\alpha$ that are needed to cover $S^m$. Secondly, we study the condition number ${\\mathscr{C}}(A)$ of the linear programming feasibility problem $\\exists x\\in\\mathbb{R}^{m+1}Ax\\le0,x\\ne0$ where $A\\in\\mathbb{R}^{n\\times(m+1)}$ is randomly chosen according to the standard normal distribution. We exactly determine the distribution of ${\\mathscr{C}}(A)$ conditioned to $A$ being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\\mathbf{E}(\\ln{\\mathscr{C}}(A))\\le2\\ln(m+1)+3.31$ for all $n>m$, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition."}
{"category": "Math", "title": "On the motivic spectra representing algebraic cobordism and algebraic K-theory", "abstract": "We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $\\mathbb{P}^\\infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $\\mathbb{C}$ and passing to spaces of $\\mathbb{C}$-valued points, we obtain new proofs of the topological versions of these theorems, originally due to the second author. We conclude with a couple of applications: first, we give a short proof of the motivic Conner-Floyd theorem, and second, we show that algebraic $K$-theory and periodic algebraic cobordism are $E_\\infty$ motivic spectra."}
{"category": "Math", "title": "Voting, the symmetric group, and representation theory", "abstract": "We show how voting may be viewed naturally from an algebraic perspective by viewing voting profiles as elements of certain well-studied $\\mathbb{Q}S_n$-modules. By using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schur's Lemma), this allows us to recast and extend some well-known results in the field of voting theory."}
{"category": "Math", "title": "Transfinite diameter notions in C^N and integrals of Vandermonde determinants", "abstract": "We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in C^N. An ingredient is a formula of Rumely which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in C^N. Our results extend to certain weights and measures defined on cones in R^N."}
{"category": "Math", "title": "On the Lego-Teichmuller game for finite $G$ cover", "abstract": "Given a smooth, oriented, closed surface $\\Sigma$ of genus zero, possibly with boundary, let $\\tilde{\\Sigma} \\longrightarrow \\Sigma$ be a given $G$-cover of $\\Sigma$, where $G$ is a given finite group. Let $S_{n}$ denote the standard sphere with $n$ holes. There are many ways of gluing together several $G$-cover of $S_{n}$ to construct the $G$-cover $\\ts \\longrightarrow \\Sigma$, of $\\Sigma$. We let $M(\\tilde{\\Sigma} ,\\Sigma)$ be the set of all ways to construct the given $G$-cover, $\\tilde{\\Sigma} \\longrightarrow \\Sigma$, of $\\Sigma$ from gluing of several $G$-covers of $S_{n}$, here $n$ may vary. In this paper, we define some simple moves and relation which will turn $M(\\tilde{\\Sigma} ,\\Sigma)$ into a connected and simply-connected complex. This will be used in the future paper to construct $G$-equivariant Modular Functor. This $G$-equivariant Modular Functor will be an extension of the usual Modular Functor."}
{"category": "Math", "title": "The Skorokhod problem in a time-dependent interval", "abstract": "We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We establish two sets of sufficient conditions on the moving boundaries that guarantee that the variation of the local time of the associated reflected Brownian motion is, respectively, finite and infinite. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions."}
{"category": "Math", "title": "On Ext-indices of ring extensions", "abstract": "In this paper we are concerned with the finiteness property of Ext-indices of several ring extensions. In this direction, we introduce some conjectures and discuss the relationship of them. Also we give affirmative answers to these conjectures in some special cases. Furthermore, we prove that the trivial extension of an Artinian local ring by its residue class field is always of finite Ext-index and we show that the Auslander-Reiten conjecture is true for this type of rings."}
{"category": "Math", "title": "Smooth and palindromic Schubert varieties in affine Grassmannians", "abstract": "Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical Grassmannian manifold. It has a Schubert cell decomposition indexed by the coroot lattice or equivalently by the minimal length coset representatives for the affine Weyl group modulo the Weyl group for G. The closure of an affine Schubert cell is a finite dimensional projective variety that we call an affine Schubert variety. In this paper we completely determine the smooth and palindromic (rationally smooth) affine Schubert varieties."}
{"category": "Math", "title": "Approximation of the joint spectral radius using sum of squares", "abstract": "We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices. Additionally, we present a comparison between our approximation scheme and earlier techniques, including the use of common quadratic Lyapunov functions and a method based on matrix liftings. Theoretical results and numerical investigations show that our approach yields tighter approximations."}
{"category": "Math", "title": "On homotopy of volterrian quadratic stochastic operators", "abstract": "In the present paper we introduce a notion of homotopy of two Volterra operators which is related to fixed points of such operators. It is establish a criterion when two Volterra operators are homotopic, as a consequence we obtain that the corresponding tournaments of that operators are the same. This, due to \\cite{Ga1}, gives us a possibility to know some information about the trajectory of homotopic Volterra operators. Moreover, it is shown that any Volterra q.s.o. given on a face has at least two homotopic extension to the whole simplex."}
{"category": "Math", "title": "Hilbert schemes of finite abelian group orbits and Grobner fans", "abstract": "Let $G$ be a finite abelian subgroup of $PGL(r-1,K)=\\mathrm{Aut}(\\P^{r-1}_K)$. In this paper, we prove that the normalization of the $G$-orbit Hilbert scheme $\\Hilb^G(\\P^{r-1})$ is described as a toric variety, which corresponds to the Gr\\\"obner fan for some homogeneous ideal $I$ of $K[x_1, ..., x_r]$."}
{"category": "Math", "title": "Random matrices, free probability, planar algebras and subfactors", "abstract": "Using a family of graded algebra structures on a planar algebra and a family of traces coming from random matrix theory, we obtain a tower of non-commutative probability spaces, naturally associated to a given planar algebra. The associated von Neumann algebras are II$_{1}$ factors whose inclusions realize the given planar algebra as a system of higher relative commutants. We thus give an alternative proof to a result of Popa that every planar algebra can be realized by a subfactor."}
{"category": "Math", "title": "The Cops & Robber game on series-parallel graphs", "abstract": "The Cops and Robber game is played on undirected finite graphs. $k$ cops and one robber are positioned on vertices and take turn in moving along edges. The cops win if, after a move, a cop and the robber are on the same vertex. A graph is called $k$-copwin, if the cops have a winning strategy. It is known that planar graphs are 3-copwin (Aigner & Fromme, 1984) and that outerplanar graphs are 2-copwin (Clarke, 2002). In this short note, we prove that series-parallel (i.e., graphs with no $K_4$ minor) graphs are 2-copwin. It is a well-known trick in the literature of cops & robber games to define variants of the game which impose restrictions on the possible strategies of the cops (see Clarke, 2002). For our proof, we define the ``cops & robber game with exits''. Our proof yields a winning strategy for the cops."}
{"category": "Math", "title": "Model selection for quantum homodyne tomography", "abstract": "This paper deals with a non-parametric problem coming from physics, namely quantum tomography. That consists in determining the quantum state of a mode of light through a homodyne measurement. We apply several model selection procedures: penalized projection estimators, where we may use pattern functions or wavelets, and penalized maximum likelihood estimators. In all these cases, we get oracle inequalities. In the former we also have a polynomial rate of convergence for the non-parametric problem. We finish the paper with applications of similar ideas to the calibration of a photocounter, a measurement apparatus counting the number of photons in a beam. Here the mathematical problem reduces similarly to a non-parametric missing data problem. We again get oracle inequalities, and better speed if the photocounter is good."}
{"category": "Math", "title": "C^0-rigidity of Poisson brackets", "abstract": "Consider a functional associating to a pair of compactly supported smooth functions on a symplectic manifold the maximum of their Poisson bracket. We show that this functional is lower semi-continuous with respect to the product uniform (C^0) norm on the space of pairs of such functions. This extends previous results of Cardin-Viterbo and Zapolsky. The proof involves theory of geodesics of the Hofer metric on the group of Hamiltonian diffeomorphisms. We also discuss a failure of a similar semi-continuity phenomenon for multiple Poisson brackets of three or more functions."}
{"category": "Math", "title": "Attractive nearest-neighbor spin systems on the integers in a randomly evolving environment", "abstract": "We consider spin systems on $\\Z$ (i.e.\\ interacting particle systems on $\\Z$ in which each coordinate only has two possible values and only one coordinate changes in each transition) whose rates are determined by another process, called a background process. A canonical example is the so called contact process in randomly evolving environment (CPREE), introduced and analysed by E. Broman and furthermore studied by J. Steif and the author, where the marginals of the background process independently evolve as 2-state Markov chains and determine the recovery rates for a contact process. We prove that under certain conditions on the rates there are at most two extremal stationary distributions. The proof follows closely the ideas of Liggett's proof of a corresponding theorem for spin systems on $\\Z$ without a background process."}
{"category": "Math", "title": "Stein's method on Wiener chaos", "abstract": "We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. We also prove results concerning random variables admitting a possibly infinite Wiener chaotic decomposition. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-It\\^o integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry-Ess\\'een bounds in the Breuer-Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler's formula for Ornstein-Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors."}
{"category": "Math", "title": "On A two-variable p-adic l_q function", "abstract": "We prove that a two-variable p-adic l_q-function has the series p-adic expansion which interpolates a linear combinations of terms of the generalized q-Euler polynomials at non positive integers. The proof of this original construction is due to Kubota and Leopoldt in 1964, although the method given this note is due to Washington"}
{"category": "Math", "title": "$\\R$-trees, dual laminations, and compact systems of partial isometries", "abstract": "Let $\\FN$ be a free group of finite rank $N \\geq 2$, and let $T$ be an $\\R$-tree with a very small, minimal action of $\\FN$ with dense orbits. For any basis $\\CA$ of $\\FN$ there exists a {\\em heart} $K_{\\CA} \\subset \\bar T$ (= the metric completion of $T$) which is a compact subtree that has the property that the dynamical system of partial isometries $a_{i} : K_{\\CA} \\cap a_{i} K_{\\CA} \\to a_{i}\\inv K_{\\CA} \\cap K_{\\CA}$, for each $a_{i} \\in \\CA$, defines a tree $T_{(K_{\\CA}, \\CA)}$ which contains an isometric copy of $T$ as minimal subtree."}
{"category": "Math", "title": "Finite unions of balls in C^n are rationally convex", "abstract": "It is shown that the rational convexity of any finite union of disjoint closed balls in C^n follows easily from the results of Duval and Sibony."}
{"category": "Math", "title": "The first Hochschild cohomology group of a schurian cluster-tilted algebra", "abstract": "Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH^1(B) with coefficients in the B-B-bimodule B. We find several consequences when B is representation-finite, and also in the case where B is cluster-tilted of type \\tilde{\\mathbb{A}}."}
{"category": "Math", "title": "Rigid objects in higher cluster categories", "abstract": "We study maximal $m$-rigid objects in the $m$-cluster category $\\mathcal C_H^m$ associated with a finite dimensional hereditary algebra $H$ with $n$ nonisomorphic simple modules. We show that all maximal $m$-rigid objects in these categories have exactly $n$ nonisomorphic indecomposable summands, and that any almost complete $m$-rigid object in $\\mathcal C_H^m$ has exactly $m+1$ nonisomorphic complements. We also show that the maximal $m$-rigid objects and the $m$-cluster tilting objects in these categories coincide, and that the class of finite dimensional algebras associated with maximal $m$-rigid objects is closed under certain factor algebras."}
{"category": "Math", "title": "An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $H^2 \\times R$", "abstract": "In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in $H^2\\times R$. As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary $C$ is a Jordan curve homologous to zero in the asymptotic boundary of $ H^2\\times R,$ say $\\partial_\\infty H^2\\times R$, such that $C$ is contained in a slab between two horizontal circles of $\\partial_\\infty H^2\\times R$ with width equal to $\\pi.$ We construct minimal vertical graphs in $H^2\\times R$ over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains $\\Om$ in $H^2\\times \\{0\\}$ are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition."}
{"category": "Math", "title": "On the rate of convergence and Berry-Esseen type theorems for a multivariate free central limit theorem", "abstract": "We address the question of a Berry Esseen type theorem for the speed of convergence in a multivariate free central limit theorem. For this, we estimate the difference between the operator-valued Cauchy transforms of the normalized partial sums in an operator-valued free central limit theorem and the Cauchy transform of the limiting operator-valued semicircular element."}
{"category": "Math", "title": "Homogenization of reflected semilinear PDE with nonlinear Neumann boundary condition", "abstract": "We study the homogenization problem of semi linear reflected partial differential equations (reflected PDEs for short) with nonlinear Neumann conditions. The non-linear term is a function of the solution but not of its gradient. The proof are fully probabilistic and uses weak convergence of associated reflected generalized backward differential stochastic equations (reflected GBSDEs in short)."}
{"category": "Math", "title": "Note sur la conjecture de Leopoldt", "abstract": "We prove that number fields with arbitrary degree but weak ramification satisfy the Leopoldt conjecture on the l-adic rank of the group of units"}
{"category": "Math", "title": "A general Lagrange Theorem", "abstract": "The ordinary continued fractions expansion of a real number is based on the Euclidean division. Variants of the latter yield variants of the former, all encompassed by a more general Dynamical Systems framework. For all these variants the Lagrange Theorem holds: a number has an eventually periodic expansion if and only if it is a quadratic irrational. This fact is surely known for specific expansions, but the only proof for the general case that I could trace in the literature follows as an implicit corollary from much deeper results by Boshernitzan and Carroll on interval exchange transformations. It may then be useful to have at hand a simple and virtually computation-free proof of a general Lagrange Theorem."}
{"category": "Math", "title": "Plongements l-adiques et l-nombres de Weil", "abstract": "We define l-adic analogs of classical Weil numbers in connexion both with complex or l-adic imbeddings of number fields and real or l-adic absolute values. As an application we give some consequences related to the Iwasawa theory of cyclotomic towers."}
{"category": "Math", "title": "Compactification l-adique de R", "abstract": "We construct a compact topological group Rl which contains both the real additive group R and the l-adic one Ql (for a given prime number l) as dense subgroups; thus we study some of its properties. This construction gives an arithmetic description of the so-called l-adic solenoid classically defined in terms of foliations."}
{"category": "Math", "title": "On tropical and Kapranov ranks of tropical matrices", "abstract": "We prove that, for any g greater or equal than 3, a matrix g x 5 with tropical rank 3 has Kapranov rank 3."}
{"category": "Math", "title": "A Spectral sequence for polynomially bounded cohomology", "abstract": "We construct an analogue of the Lyndon-Hochschild-Serre spectral sequence in the context of polynomial cohomology, for group extensions. If G is an extension of Q by H, then the spectral sequence converges to the polynomial cohomology of G. For the polynomial extensions of Noskov with normal subgroup isocohomological, the E_2 term is the polynomial cohomology of Q with coefficients in the polynomial cohomology of H. When both Q and H are isocohomological G must be as well. By referencing results of Connes-Moscovici and Noskov, if Q and H are both isocohomological and have the Rapid Decay property of Jolissaint, then G satisfies the Novikov Conjecture."}
{"category": "Math", "title": "SLE and the free field: Partition functions and couplings", "abstract": "Schramm-Loewner Evolutions ($\\SLE$) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of $\\SLE$ and the free field with appropriate boundary conditions; this involves $\\zeta$-regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of $\\SLE$ with the free field, showing that, in a precise sense, chordal $\\SLE$ is the solution of a stochastic \"differential\" equation driven by the free field. Existence, uniqueness in law, and pathwise uniqueness for these SDEs are proved for general $\\kappa>0$."}
{"category": "Math", "title": "When Do Random Subsets Decompose a Finite Group?", "abstract": "Let A,B be two random subsets of a finite group G. We consider the event that the products of elements from A and B span the whole group; i.e. (AB union BA) = G. The study of this event gives rise to a group invariant we call \\Theta(G). \\Theta(G) is between 1/2 and 1, and is 1 if and only if the group is abelian. We show that a phase transition occurs as the size of A and B passes \\sqrt{\\Theta(G)|G|\\log|G|}; i.e. for any c>0, if the size of A and B is less than (1-c)\\sqrt{\\Theta(G)|G|\\log|G|}, then with high probability (AB union BA) does not equal G. If A and B are larger than (1+c)\\sqrt{\\Theta(G)|G|\\log|G|} then (AB union BA) equals G with high probability."}
{"category": "Math", "title": "Modular classes of Lie algebroid morphisms", "abstract": "We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, including Morita equivalences, that act on the 1-cohomology, and observe that the relative modular class is a coboundary on the category of Lie algebroids and generalized morphisms with values in the 1-cohomology."}
{"category": "Math", "title": "Renormalisation-induced phase transitions for unimodal maps", "abstract": "The thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential $-t \\log|Df|$. Multiple and indeed infinitely many phase transitions at positive $t$ can occur for some quadratic maps. All unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum."}
{"category": "Math", "title": "Sets of double and triple weights of trees", "abstract": "Let T be a weighted tree with n leaves. Let D_{i,j} be the distance between the leaves i and j. Let D_{i,j,k}= (D_{i,j} + D_{j,k} +D_{i,k})/2. We will call such numbers \"triple weights\" of the tree. In this paper, we give a characterization, different from the previous ones, for sets indexed by 2-subsets of a $n$-set to be double weights of a tree. By using the same ideas,we find also necessary and sufficient conditions for a set of real numbers indexed by 3-subsets of an $n$-set to be the set of the triple weights of a tree with $n$ leaves. Besides we propose a slight modification of Saitou-Nei's Neighbour-Joining algorithm to reconstruct trees from the data D_{i,j}."}
{"category": "Math", "title": "Approximation of sets defined by polynomials with holomorphic coefficients", "abstract": "Let X be an analytic set defined by polynomials whose coefficients a_1,...,a_s are holomorphic functions. We formulate conditions such that for all sequences {a_(1,n)},...,{a_(s,n)} of holomorphic functions converging locally uniformly to a_1,...,a_s respectively the following holds true. If a_(1,n),...,a_(s,n) satisfy the conditions then the sequence of the sets {X_n} obtained by replacing a_j by a_(j,n) in the polynomials, converge to X."}
{"category": "Math", "title": "Stability of homogeneous bundles on P^3", "abstract": "We study the stability of some homogeneous bundles on P^3 by using their representations of the quiver associated to the homgeneous bundles on P^3. In particular we show that homogeneous bundles on P^3 whose support of the quiver representation is a parallelepiped are stable, for instance the bundles E whose minimal free resolution is of the kind 0 --> S^{l_1, l_2, l_3} V (t) --> S^{l_1 +s, l_2, l_3} V (t+s) --> E --> 0 are stable."}
{"category": "Math", "title": "Identities and Inequalities for Tree Entropy", "abstract": "The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses Fuglede-Kadison determinants, while another uses effective resistance. We use the latter to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case, a special case of which establishes Lueck's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affiliated to von Neumann algebras."}
{"category": "Math", "title": "A note on generalized equivariant homotopy groups", "abstract": "In this paper, we generalize the equivariant homotopy groups or equivalently the Rhodes groups. We establish a short exact sequence relating the generalized Rhodes groups and the generalized Fox homotopy groups and we introduce $\\Gamma$-Rhodes groups, where $\\Gamma$ admits a certain co-grouplike structure. Evaluation subgroups of $\\Gamma$-Rhodes groups are discussed."}
{"category": "Math", "title": "Gibbs Sampling for a Bayesian Hierarchical General Linear Model", "abstract": "We consider a Bayesian hierarchical version of the normal theory general linear model which is practically relevant in the sense that it is general enough to have many applications and it is not straightforward to sample directly from the corresponding posterior distribution. Thus we study a block Gibbs sampler that has the posterior as its invariant distribution. In particular, we establish that the Gibbs sampler converges at a geometric rate. This allows us to establish conditions for a central limit theorem for the ergodic averages used to estimate features of the posterior. Geometric ergodicity is also a key component for using batch means methods to consistently estimate the variance of the asymptotic normal distribution. Together, our results give practitioners the tools to be as confident in inferences based on the observations from the Gibbs sampler as they would be with inferences based on random samples from the posterior. Our theoretical results are illustrated with an application to data on the cost of health plans issued by health maintenance organizations."}
{"category": "Math", "title": "Almost all integer matrices have no integer eigenvalues", "abstract": "For a fixed $n\\ge2$, consider an $n\\times n$ matrix $M$ whose entries are random integers bounded by $k$ in absolute value. In this paper, we examine the probability that $M$ is singular (hence has eigenvalue 0), and the probability that $M$ has at least one rational eigenvalue. We show that both of these probabilities tend to 0 as $k$ increases. More precisely, we establish an upper bound of size $k^{-2+\\epsilon}$ for the probability that $M$ is singular, and size $k^{-1+\\epsilon}$ for the probability that $M$ has a rational eigenvalue. These results generalize earlier work by Kowalsky for the case $n=2$ and answer a question posed by Hetzel, Liew, and Morrison."}
{"category": "Math", "title": "Complete semi-conjugacies for psuedo-Anosov homeomorphisms", "abstract": "Suppose $S$ is a surface of genus $\\ge 2 $, $f: S \\to S$ is a surface homeomorphism isotopic to a pseudo-Anosov map $\\alpha$ and suppose $\\ti S$ is the universal cover of $S$ and $F$ and $A$ are lifts of $f$ and $\\alpha$ respectively. We show there is a semiconjugacy $\\Theta : \\ti S \\to \\bar \\L^s \\times \\bar \\L^u$ from $F$ to $\\bar A$, where $\\bar \\L^s$ ($\\bar \\L^u$) is the completion of the $R$-tree of leaves of the stable (resp. unstable) foliation for $A$ and $\\bar A$ is the map induced by $A$. We also generalize a result of Markovich and show that for any $g \\in Homeo(S)$ which commutes with $f$ and has identity lift $G : \\ti S \\to \\ti S$ and for any $(c,w)$ in the image of $\\Theta$ each component of $\\Theta^{-1}(c,w)$ is $G$-invariant."}
{"category": "Math", "title": "Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS", "abstract": "The singular parabolic problem $u_t=\\Delta u -\\frac{\\lambda f(x)}{(1+u)^2}$ on a bounded domain $\\Omega$ of $R^N$ with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical System (MEMS) device. In this paper, we analyze and estimate the quenching time of the elastic membrane in terms of the applied voltage --represented here by $\\lambda$. As a byproduct, we prove that for sufficiently large $\\lambda$, finite-time quenching must occur near the maximum point of the varying dielectric permittivity profile $f(x)$."}
{"category": "Math", "title": "C*-algebras associated to product systems of Hilbert bimodules", "abstract": "Let (G,P) be a quasi-lattice ordered group and let X be a compactly aligned product system over P of Hilbert bimodules. Under mild hypotheses we associate to X a C*-algebra which we call the Cuntz-Nica-Pimsner algebra of X. Our construction generalises a number of others: a sub-class of Fowler's Cuntz-Pimsner algebras for product systems of Hilbert bimodules; Katsura's formulation of Cuntz-Pimsner algebras of Hilbert bimodules; the C*-algebras of finitely aligned higher-rank graphs; and Crisp and Laca's boundary quotients of Toeplitz algebras. We show that for a large class of product systems X, the universal representation of X in its Cuntz-Nica-Pimsner algebra is isometric."}
{"category": "Math", "title": "Continuity, curvature, and the general covariance of optimal transportation", "abstract": "Let M and \\bar M be n-dimensional manifolds equipped with suitable Borel probability measures \\rho and \\bar\\rho. Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c \\in C^4(M \\times \\bar M) to guarantee smoothness of the optimal map pushing \\rho forward to \\bar\\rho; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M \\times \\bar M. H\\\"older continuity of optimal maps was established for rougher mass distributions by Loeper, still relying on a key result of Trudinger & Wang which required certain structure on the domains and the cost. We go on to develop this theory for mass distributions on differentiable manifolds -- recovering Loeper's Riemannian examples such as the round sphere as particular cases -- give a direct proof of the key result mentioned above, and revise Loeper's H\\\"older continuity argument to make it logically independent of all earlier works, while extending it to less restricted geometries and cost functions even for subdomains M and \\bar M of R^n. We also give new examples of geometries satisfying the hypotheses -- obtained using submersions and tensor products -- and some connections to spacelike Lagrangian submanifolds in symplectic geometry."}
{"category": "Math", "title": "Block-diagonal semidefinite programming hierarchies for 0/1 programming", "abstract": "Lovasz and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for general 0/1 linear programming problems. In this paper these two constructions are revisited and two new, block-diagonal hierarchies are proposed. They have the advantage of being computationally less costly while being at least as strong as the Lovasz-Schrijver hierarchy. Our construction is applied to the stable set problem and experimental results for Paley graphs are reported."}
{"category": "Math", "title": "Orthogonal polynomials and partial differential equations on the unit ball", "abstract": "Orthogonal polynomials of degree $n$ with respect to the weight function $W_\\mu(x) = (1-\\|x\\|^2)^\\mu$ on the unit ball in $\\RR^d$ are known to satisfy the partial differential equation $$ [ \\Delta - \\la x, \\nabla \\ra^2 - (2 \\mu +d) \\la x, \\nabla \\ra \\right ] P = -n(n+2 \\mu+d) P $$ for $\\mu > -1$. The singular case of $\\mu = -1,-2, ...$ is studied in this paper. Explicit polynomial solutions are constructed and the equation for $\\nu = -2,-3,...$ is shown to have complete polynomial solutions if the dimension $d$ is odd. The orthogonality of the solution is also discussed."}
{"category": "Math", "title": "On splitting polynomials with noncommutative coefficients", "abstract": "It is shown that for every splitting of a polynomial with noncommutative coefficients into linear factors $(X-a_{k})$ with $a_{k}$'s commuting with coefficients, any cyclic permutation of linear factors gives the same result and all $a_{k}$ are roots of that polynomial. Examples are given and analyzed from Galois theory point of view."}
{"category": "Math", "title": "Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle", "abstract": "Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid."}
{"category": "Math", "title": "Orbit decidability and the conjugacy problem for some extensions of groups", "abstract": "Given a short exact sequence of groups with certain conditions, $1\\to F\\to G\\to H\\to 1$, we prove that $G$ has solvable conjugacy problem if and only if the corresponding action subgroup $A\\leqslant Aut(F)$ is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form $\\mathbb{Z}^2\\rtimes F_m$, $F_2\\rtimes F_m$, $F_n \\rtimes \\mathbb{Z}$, and $\\mathbb{Z}^n \\rtimes_A F_m$ with virtually solvable action group $A\\leqslant GL_n(\\mathbb{Z})$. Also, we give an easy way of constructing groups of the form $\\mathbb{Z}^4\\rtimes F_n$ and $F_3\\rtimes F_n$ with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in $Aut(F_2)$ is given."}
{"category": "Math", "title": "Generalized Hasimoto Transform of One-Dimensional Dispersive Flows into Compact Riemann Surfaces", "abstract": "We study the structure of differential equations of one-dimensional dispersive flows into compact Riemann surfaces. These equations geometrically generalize two-sphere valued systems modeling the motion of vortex filament. We define a generalized Hasimoto transform by constructing a good moving frame, and reduce the equation with values in the induced bundle to a complex valued equation which is easy to handle. We also discuss the relationship between our reduction and the theory of linear dispersive partial differential equations."}
{"category": "Math", "title": "Universal lifts of chain complexes over non-commutative parameter algebras", "abstract": "We define the notion of universal lift of a projective complex based on non-commutative parameter algebras, and prove its existence and uniqueness. We investigate the properties of parameter algebras for universal lifts."}
{"category": "Math", "title": "A most general edge elimination graph polynomial", "abstract": "We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and deletion of edges together with their end points. Like in the case of deletion and contraction only (W. Tutte, 1954), it turns out that there is a most general polynomial satisfying such recurrence relations, which we call $\\xi(G,x,y,z)$. We show that the new polynomial simultaneously generalizes the Tutte polynomial, the matching polynomial, and the recent generalization of the chromatic polynomial proposed by K.Dohmen, A.P\\\"{o}nitz and P.Tittman (2003), including also the independent set polynomial of I. Gutman and F. Harary, (1983) and the vertex-cover polynomial of F,M. Dong, M.D. Hendy, K.T. Teo and C.H.C. Little (2002). We establish two definitions of the new polynomial: first, the most general confluent recursive definition, and then an explicit one, using a set expansion formula, and prove their identity. We further expand this result to edge-labeled graphs as was done for the Tutte polynomial by T. Zaslavsky (1992) and B. Bollob\\'as and O. Riordan (1999). The edge labeled polynomial $\\xi_{lab}(G,x,y,z, \\bar{t})$ also generalizes the chain polynomial of R.C. Read and E.G. Whitehead Jr. (1999). Finally, we discuss the complexity of computing $\\xi(G,x,y,z)$."}
{"category": "Math", "title": "New Abstract Hardy Spaces", "abstract": "The aim of this paper is to propose an abstract construction of spaces which keep the main properties of the (already known) Hardy spaces H^1. We construct spaces through an atomic (or molecular) decomposition. We prove some results about continuity from these spaces into L^1 and some results about interpolation between these spaces and the Lebesgue spaces. We also obtain some results on weighted norm inequalities. Then we apply this abstract theory to the L^p maximal regularity. Finally we present partial results in order to understand a characterization of the duals of Hardy spaces."}
{"category": "Math", "title": "A new approach to Kostant's problem", "abstract": "For every involution $\\mathbf{w}$ of the symmetric group $S_n$ we establish, in terms ofa special canonical quotient of the dominant Verma module associated with $\\mathbf{w}$, an effective criterion, which allows us to verify whether the universal enveloping algebra $U(\\mathfrak{sl}_n)$ surjects onto the space of all ad-finite linear transformations of the simple highest weight module $L(\\mathbf{w})$. An easy sufficient condition derived from this criterion admits a straightforward computational check for example using a computer. All this is applied to get some old and many new results, which answer the classical question of Kostant in special cases, in particular we give a complete answer for simple highest weight modules in the regular block of $\\mathfrak{sl}_n$, $n\\leq 5$."}
{"category": "Math", "title": "Equivariant stable stems for prime order groups", "abstract": "For groups of prime order, equivariant stable maps between equivariant representation spheres are investigated using the Borel cohomology Adams spectral sequence. Features of the equivariant stable homotopy category, such as stability and duality, are shown to lift to the category of modules over the associated Steenrod algebra. The dependence on the dimension functions of the representations is clarified."}
{"category": "Math", "title": "Optimal eigenvalue estimate for the Dirac-Witten operator on bounded domains with smooth boundary", "abstract": "Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for the usual Dirac operator. The limiting cases are also investigated."}
{"category": "Math", "title": "Notes on Formal Deformations of Hom-associative and Hom-Lie Algebras", "abstract": "The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra $sl_2$ and describe as formal deformations the q-deformed Witt algebra and Jackson $sl_2$."}
{"category": "Math", "title": "Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces", "abstract": "Let $X$ be a rational homogeneous space and let $QH^*(X)_{loc}^\\times$ be the group of invertible elements in the small quantum cohomology ring of $X$ localised in the quantum parameters. We generalise results of arXiv:math/0609796 and realise explicitly the map $\\pi_1({\\rm Aut}(X))\\to QH^*(X)_{loc}^\\times$ described in arXiv:dg-ga/9511011. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring $QH^*_T(X)_{loc}^\\times$. We give explicit formulas for the product by these elements. The proof relies on a generalisation, to a quotient of the equivariant homology ring of the affine Grassmannian, of a formula proved by Peter Magyar arXiv:0705.3826. It also uses Peterson's unpublished result -- recently proved by Lam and Shimozono in arXiv:0705.1386 -- on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring."}
{"category": "Math", "title": "On the spectrum of lamplighter groups and percolation clusters", "abstract": "Let $G$ be a finitely generated group and $X$ its Cayley graph with respect to a finite, symmetric generating set $S$. Furthermore, let $H$ be a finite group and $H \\wr G$ the lamplighter group (wreath product) over $G$ with group of \"lamps\" $H$. We show that the spectral measure (Plancherel measure) of any symmetric \"switch--walk--switch\" random walk on $H \\wr G$ coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on $X$ with parameter $p = 1/|H|$. The return probabilities of the lamplighter random walk coincide with the expected (annealed) return probabilites on the percolation cluster. In particular, if the clusters of percolation with parameter $p$ are almost surely finite then the spectrum of the lamplighter group is pure point. This generalizes results of Grigorchuk and Zuk, resp. Dicks and Schick regarding the case when $G$ is infinite cyclic. Analogous results relate bond percolation with another lamplighter random walk. In general, the integrated density of states of site (or bond) percolation with arbitrary parameter $p$ is always related with the Plancherel measure of a convolution operator by a signed measure on $H \\wr G$, where $H = Z$ or another suitable group."}
{"category": "Math", "title": "Harnack Inequality and Strong Feller Property for Stochastic Fast-Diffusion Equations", "abstract": "This paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results."}
{"category": "Math", "title": "Transportation Cost Inequality on Path Spaces with Uniform Distance", "abstract": "Starting from a sequence of independent Wright-Fisher diffusion processes on $[0,1]$, we construct a class of reversible infinite dimensional diffusion processes on $\\DD_\\infty:= \\{{\\bf x}\\in Let $M$ be a complete Riemnnian manifold and $\\mu$ the distribution of the diffusion process generated by $\\ff 1 2\\DD+Z$ where $Z$ is a $C^1$-vector field. When $\\Ric-\\nn Z$ is bounded below and $Z$ has, for instance, linear growth, the transportation-cost inequality with respect to the uniform distance is established for $\\mu$ on the path space over $M$. A simple example is given to show the optimality of the condition."}
{"category": "Math", "title": "From Super Poincar\\'e to Weighted Log-Sobolev and Entropy-Cost Inequalities", "abstract": "We derive weighted log-Sobolev inequalities from a class of super Poincar\\'e inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\\log^\\dd$-Sobolev inequality with $\\dd\\in (1,2)$ implies the $L^{2/(2-\\dd)}$-transportation cost inequality $$W^\\rr_{2/(2-\\dd)}(f\\mu,\\mu)^{2/(2-\\dd)}\\le C\\mu(f\\log f), \\mu(f)=1, f\\ge 0$$ for some constant $C>0$, and they are equivalent if the curvature of the corresponding generator is bounded below. Weighted log-Sobolev and entropy-cost inequalities are also derived for a large class of probability measures on $\\R^d$."}
{"category": "Math", "title": "Log-Sobolev inequalities: Different roles of Ric and Hess", "abstract": "Let $P_t$ be the diffusion semigroup generated by $L:=\\Delta +\\nabla V$ on a complete connected Riemannian manifold with $\\operatorname {Ric}\\ge-(\\sigma ^2\\rho_o^2+c)$ for some constants $\\sigma, c>0$ and $\\rho_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided $-\\operatorname {Hess}_V\\ge\\delta$ holds outside of a compact set for some constant $\\delta >(1+\\sqrt{2})\\sigma \\sqrt{d-1}.$ This indicates, at least in finite dimensions, that $\\operatorname {Ric}$ and $-\\operatorname {Hess}_V$ play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied."}
{"category": "Math", "title": "Intrinsic Ultracontractivity on Riemannian Manifolds with Infinite Volume Measures", "abstract": "By establishing the intrinsic super-Poincar\\'e inequality, some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the resulting uniform upper bounds on the intrinsic heat kernels, are sharp for some concrete examples."}
{"category": "Math", "title": "Metrics of constant curvature on a Riemann surface with two corners on the boundary", "abstract": "We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows that a disk with two corners admits a conformal metric with constant Gauss curvature and constant geodesic curvature on its boundary if and only if the two corners have the same angle. In fact, we can classify all the solutions in a more general situation, that of the 2-sphere cut by two planes."}
{"category": "Math", "title": "The parabolic-parabolic Keller-Segel model in $\\mathbb{R}^2$", "abstract": "This paper is devoted mainly to the global existence problem for the two-dimensional parabolic-parabolic Keller-Segel in the full space. We derive a critical mass threshold below which global existence is ensured. Using carefully energy methods and ad hoc functional inequalities we improve and extend previous results in this direction. The given threshold is supposed to be the optimal criterion, but this question is still open. This global existence result is accompanied by a detailed discussion on the duality between the Onofri and the logarithmic Hardy-Littlewood-Sobolev inequalities that underlie the following approach."}
{"category": "Math", "title": "General Dirichlet series, arithmetic convolution equations and Laplace transforms", "abstract": "In an earlier paper, we studied solutions g to convolution equations of the form a_d*g^{*d}+a_{d-1}*g^{*(d-1)}+...+a_1*g+a_0=0, where a_0, ..., a_d are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form \\sum_{x\\in X} f(x) e^{-sx} (s in C^k), where X is an additive subsemigroup of [0,\\infty)^k. If X is discrete and a certain solvability criterion is satisfied, we determine solutions by an elementary recursive approach, adapting an idea of Feckan. The solution of the general case leads us to a more comprehensive question: Let X be an additive subsemigroup of a pointed, closed convex cone C in R^k. Can we find a complex Radon measure on X whose Laplace transform satisfies a given polynomial equation whose coefficients are Laplace transforms of such measures?"}
{"category": "Math", "title": "Symmetric units in integral group rings", "abstract": "In this paper, we study the question of when the symmetric units in an integral group ring ZG form a multiplicative group. When G is periodic, necessary and sufficient conditions are given for this to occur."}
{"category": "Math", "title": "An open string analogue of Viterbo functoriality", "abstract": "Liouville domains are a special type of symplectic manifolds with boundary (they have an everywhere defined Liouville flow, pointing outwards along the boundary). Symplectic cohomology for Liouville domains was introduced by Cieliebak-Floer-Hofer-Wysocki and Vitero. The latter constructed a restriction (or transfer) map associated to an embedding of one Liouville domain into another. In this preprint, we look at exact Lagrangian submanifolds with Legendrian boundary inside a Liouville domain. The analogue of symplectic cohomology for such submanifolds is called \"wrapped Floer cohomology\". We construct an A_\\infty-structure on the underlying wrapped Floer complex, and (under suitable assumptions) an A_\\infty-homomorphism realizing the restriction to a Liouville subdomain. The construction of the A_\\infty-structure relies on an implementation of homotopy direct limits, and involves some new moduli spaces which are solutions of generalized continuation map equations."}
{"category": "Math", "title": "Global division of cohomology classes via injectivity", "abstract": "We note that the vanishing and injectivity theorems of Koll\\'ar and Esnault-Viehweg can be used to give a quick algebraic proof of a strengthening of the Ein-Lazarsfeld Skoda-type division theorem for global sections of adjoint line bundles vanishing along suitable multiplier ideal sheaves, and to extend it to higher cohomology classes as well. For global sections, this is also a slightly more general statement of the algebraic translation of an analytic result of Siu. Along the way we write down an injectivity statement for multiplier ideals, and its standard consequences."}
{"category": "Math", "title": "Generalized theta linear series on moduli spaces of vector bundles on curves", "abstract": "This article is based on lecture notes prepared for the August 2006 Cologne Summer School. The first part contains background material and references for beginners. The second (and main) part is a survey of the current status in the theory of pluri-theta linear series and generalized theta divisors on moduli spaces of vector bundles on curves. It emphasizes relatively new techniques employed in the analysis of linear series on these moduli spaces, namely the use of moduli spaces of stable maps for understanding Quot schemes, and the Fourier-Mukai functor in the study of coherent sheaves on abelian varieties. In addition, it briefly describes recent important developments, most significant of which is the proof of the Strange Duality conjecture due to Belkale and Marian-Oprea."}
{"category": "Math", "title": "Lie Algebroids and Classification Problems in Geometry", "abstract": "We show how one can associate to a given class of finite type G-structures a classifying Lie algebroid. The corresponding Lie groupoid gives models for the different geometries that one can find in the class, and encodes also the different types of symmetry groups."}
{"category": "Math", "title": "Kirwan surjectivity in K-theory for Hamiltonian loop group quotients", "abstract": "Let G be a compact Lie group and LG its associated loop group. The main result of this manuscript is a surjectivity theorem from the equivariant K-theory of a Hamiltonian LG-space onto the integral K-theory of its Hamiltonian LG-quotient. Our result is a K-theoretic analogue of previous work in rational Borel-equivariant cohomology of Bott, Tolman, and Weitsman. Our proof techniques differ from that of Bott, Tolman, and Weitsman in that they explicitly use the Borel construction, which we do not have at our disposal in equivariant K-theory; we instead directly construct G-equivariant homotopy equivalences to obtain the necessary isomorphisms in equivariant K-theory. The main theorem should also be viewed as a first step toward a similar theorem in K-theory for quasi-Hamiltonian G-spaces and their associated quasi-Hamiltonian quotients."}
{"category": "Math", "title": "Tropical theta characteristics", "abstract": "This note is a follow up of math.AG/0612267v2 and it is largely inspired by a beautiful description of Baker-Norine of non-effective degree (g-1) divisors via chip-firing game. We consider the set of all theta characteristics on a tropical curve and identify the Riemann constant as a unique non-effective one among them."}
{"category": "Math", "title": "Simulation of a Local Time Fractional Stable Motion", "abstract": "In this paper, we simulate sample paths of a class of symmetric $\\alpha$-stable processes using their series expression. We will develop a result in the approximation of shot-noise series. And finally, we will get a convergence rate for the approximation."}
{"category": "Math", "title": "The number of lattice paths below a cyclically shifting boundary", "abstract": "We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result extends well known enumerative formulae concerning lattice paths, and its derivation involves a classical reflection argument. A refinement allows for the counting of paths with a specified number of corners. We also apply the result to examine paths dominated by periodic boundaries."}
{"category": "Math", "title": "Probabilistic analysis of the upwind scheme for transport", "abstract": "We provide a probabilistic analysis of the upwind scheme for multi-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we prove that the scheme is of order 1/2 for an initial datum in BV and of order 1/2-a, for all a>0, for a Lipschitz continuous initial datum. Our analysis provides a new interpretation of the numerical diffusion phenomenon."}
{"category": "Math", "title": "An optimal extension of Perelman's comparison theorem for quadrangles and its applications", "abstract": "In this paper we discuss an extension of Perelman's comparison for quadrangles. Among applications of this new comparison theorem, we study the equidistance evolution of hypersurfaces in Alexandrov spaces with non-negative curvature. We show that, in certain cases, the equidistance evolution of hypersurfaces become totally convex relative to a bigger sub-domain. An optimal extension of 2nd variational formula for geodesics by Petrunin will be derived for the case of non-negative curvature. In addition, we also introduced the generalized second fundament forms for subsets in Alexandrov spaces. Using this new notion, we will propose an approach to study two open problems in Alexandrov geometry."}
{"category": "Math", "title": "On the ideals of equivariant tree models", "abstract": "We introduce equivariant tree models in algebraic statistics, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group based models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. The main novelty is our proof that this procedure yields the entire ideal, not just an ideal defining the model set-theoretically. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices."}
{"category": "Math", "title": "Weakly dependent chains with infinite memory", "abstract": "We prove the existence of a weakly dependent strictly stationary solution of the equation $ X_t=F(X_{t-1},X_{t-2},X_{t-3},...;\\xi_t)$ called {\\em chain with infinite memory}. Here the {\\em innovations} $\\xi_t$ constitute an independent and identically distributed sequence of random variables. The function $F$ takes values in some Banach space and satisfies a Lipschitz-type condition. We also study the interplay between the existence of moments and the rate of decay of the Lipschitz coefficients of the function $F$. With the help of the weak dependence properties, we derive Strong Laws of Large Number, a Central Limit Theorem and a Strong Invariance Principle."}
{"category": "Math", "title": "An essentially saturated surface not of Kaehler-type", "abstract": "It is shown that if $X$ is an Inoue surface of type $S_M$ then the irreducible components of the Douady space of $X^n$ are compact, for all $n>0$. This gives an example of an essentially saturated compact complex manifold (in the sense of model theory) that is not of Kaehler-type. Among the known compact complex surfaces without curves, it is shown that these are the only examples."}
{"category": "Math", "title": "Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds", "abstract": "We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration of M over the circle is the modular elliptic curve E=X_0(49), which admits multiplication by the ring of integers of Q[sqrt(-7)]. We first base change the holomorphic differential on E to a cusp form on GL(2) over K=Q[sqrt(-3)], and then transfer over to a quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal congruence subgroup of level 7 of the multiplicative group of a maximal order of D. To analyze the topological properties of M, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a non-compact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument."}
{"category": "Math", "title": "The cohomological crepant resolution conjecture for P(1,3,4,4)", "abstract": "We prove the cohomological crepant resolution conjecture of Ruan for the weighted projective space P(1,3,4,4). To compute the quantum corrected cohomology ring we combine the results of Coates-Corti-Iritani-Tseng on P(1,1,1,3) and our previous results."}
{"category": "Math", "title": "Schramm-Loewner Evolution", "abstract": "This is the first expository set of notes on SLE I have written since publishing a book two years ago [45]. That book covers material from a year-long class, so I cannot cover everything there. However, these notes are not just a subset of those notes, because there is a slight change of perspective. The main differences are: o I have defined SLE as a finite measure on paths that is not necessarily a probability measure. This seems more natural from the perspective of limits of lattice systems and seems to be more useful when extending SLE to non-simply connected domains. (However, I do not discuss non-simply connected domains in these notes.) o I have made more use of the Girsanov theorem in studying corresponding martingales and local martingales. As in [45], I will focus these notes on the continuous process SLE and will not prove any results about convergence of discrete processes to SLE. However, my first lecture will be about discrete processes -- it is very hard to appreciate SLE if one does not understand what it is trying to model."}
{"category": "Math", "title": "Algebraic approximation of analytic sets and mappings", "abstract": "Let {X_n} be a sequence of analytic sets converging to some analytic set X in the sense of holomorphic chains. We introduce a condition which implies that every irreducible component of X is the limit of a sequence of irreducible components of the sets from {X_n}. Next we apply the condition to approximate a holomorphic solution y=f(x) of a system Q(x,y)=0 of Nash equations by Nash solutions. Presented methods allow to construct an algorithm of approximation of the holomorphic solutions."}
{"category": "Math", "title": "Using Gradual Numbers to Analyze Non-Monotonic Functions of Fuzzy Intervals", "abstract": "Gradual numbers have been introduced recently as a means of extending standard interval computation methods to fuzzy intervals. The literature treats monotonic functions of fuzzy intervals. In this paper, we combine the concepts of gradual numbers and optimization, which allows for the evaluation of any differentiable function on fuzzy intervals, with no monotonicity requirement."}
{"category": "Math", "title": "Parameter curves for the regular representations of tame bimodules", "abstract": "We present results and examples which show that the consideration of a certain tubular mutation is advantageous in the study of noncommutative curves which parametrize the simple regular representations of a tame bimodule. We classify all tame bimodules where such a curve is actually commutative, or in different words, where the unique generic module has a commutative endomorphism ring. This extends results from [14] to arbitrary characteristic; in characteristic two additionally inseparable cases occur. Further results are concerned with autoequivalences fixing all objects but not isomorphic to the identity functor."}
{"category": "Math", "title": "Relations between semidualizing complexes", "abstract": "We study the following question: Given two semidualizing complexes B and C over a commutative noetherian ring R, does the vanishing of Ext^n_R(B,C) for n>>0 imply that B is C-reflexive? This question is a natural generalization of one studied by Avramov, Buchweitz, and Sega. We begin by providing conditions equivalent to B being C-reflexive, each of which is slightly stronger than the condition Ext^n_R(B,C)=0 for all n>>0. We introduce and investigate an equivalence relation \\approx on the set of isomorphism classes of semidualizing complexes. This relation is defined in terms of a natural action of the derived Picard group and is well-suited for the study of semidualizing complexes over nonlocal rings. We identify numerous alternate characterizations of this relation, each of which includes the condition Ext^n_R(B,C)=0 for all n>>0. Finally, we answer our original question in some special cases."}
{"category": "Math", "title": "On admissibility criteria for weak solutions of the Euler equations", "abstract": "We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension."}
{"category": "Math", "title": "Mixed Weil cohomologies", "abstract": "We define, for a regular scheme $S$ and a given field of characteristic zero $\\KK$, the notion of $\\KK$-linear mixed Weil cohomology on smooth $S$-schemes by a simple set of properties, mainly: Nisnevich descent, homotopy invariance, stability (which means that the cohomology of $\\GG_{m}$ behaves correctly), and K\\\"unneth formula. We prove that any mixed Weil cohomology defined on smooth $S$-schemes induces a symmetric monoidal realization of some suitable triangulated category of motives over $S$ to the derived category of the field $\\KK$. This implies a finiteness theorem and a Poincar\\'e duality theorem for such a cohomology with respect to smooth and projective $S$-schemes (which can be extended to smooth $S$-schemes when $S$ is the spectrum of a perfect field). This formalism also provides a convenient tool to understand the comparison of such cohomology theories. Our main examples are algebraic de Rham cohomology and rigid cohomology, and the Berthelot-Ogus isomorphism relating them."}
{"category": "Math", "title": "Local and stable homological algebra in Grothendieck abelian categories", "abstract": "We define model category structures on the category of chain complexes over a Grothendieck abelian category depending on the choice of a generating family, and we study their behaviour with respect to tensor products and stabilization. This gives convenient tools to construct and understand triangulated categories of motives and we consider here the case of mixed motives over a regular base scheme."}
{"category": "Math", "title": "Ricci and Scalar Curvature Rigidity of the Hemisphere", "abstract": "We retract the scalar curvature rigidity theorem as there is a mistake in the proof. We thank S. Montiel for pointing out the mistake."}
{"category": "Math", "title": "Lie algebra F-normalisers are intravariant", "abstract": "Let F be a saturated formation of soluble Lie algebras and let U be an F-normaliser of the soluble Lie algebra L. Then U is intravariant in L."}
{"category": "Math", "title": "Analysis of the optimal exercise boundary of American options for jump diffusions", "abstract": "In this paper we show that the optimal exercise boundary / free boundary of the American put option pricing problem for jump diffusions is continuously differentiable (except at the maturity). This differentiability result has been established by Yang et al. (European Journal of Applied Mathematics, 17(1):95-127, 2006) in the case where the condition $r\\geq q+ \\lambda \\int_{\\R_+} (e^z-1) \\nu(dz)$ is satisfied. We extend the result to the case where the condition fails using a unified approach that treats both cases simultaneously. We also show that the boundary is infinitely differentiable under a regularity assumption on the jump distribution."}
{"category": "Math", "title": "The center of some braid groups and the Farrell cohomology of certain pure mapping class groups", "abstract": "In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the $p$-primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the $p$-primary part of the Farrell cohomology of a few pure mapping class groups."}
{"category": "Math", "title": "Surfaces with maximal constant mean curvature", "abstract": "In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean curvature surface homologous to a subset of the interior boundary components. This bound allows us to find a maximizer for the constant mean curvature of a surface homologous to the inner boundary. With this maximizer at hand, we can construct an increasing family of sets with boundaries of increasing constant mean curvature. We interpret this familiy as a weak version of a CMC foliation."}
{"category": "Math", "title": "Rapid paths in von Neumann-Gale dynamical systems", "abstract": "The paper examines random dynamical systems related to the classical von Neumann and Gale models of economic growth. Such systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. A central role in the theory of von Neumann-Gale dynamics is played by a special class of paths called rapid (they maximize properly defined growth rates). Up to now the theory lacked quite satisfactory results on the existence of such paths. This work provides a general existence theorem holding under assumptions analogous to the standard deterministic ones. The result solves a problem that remained open for more than three decades."}
{"category": "Math", "title": "On some low dimensional quantum groups", "abstract": "This paper is an adaptation of a chapter from an upcoming monograph on noncommutative geometry and quantum groups. We present examples of non compact quantum groups which are deformations of low dimensional Lie groups. The paper is of expository nature and provides both particular examples and some general procedures for constructing them."}
{"category": "Math", "title": "Cubic polynomials with a parabolic point", "abstract": "We consider the family of cubic polynomials with a simple parabolic fixed point. We prove that the boundary of the immediate basin of attraction of the parabolic point is a Jordan curve and give a description of the dynamics."}
{"category": "Math", "title": "Fundamental groups of moduli stacks of smooth Weierstrass fibrations", "abstract": "We give finite presentations for the fundamental group of moduli stacks of smooth Weierstrass curves over complex projective space P^n which extend the classical result for elliptic curves to positive dimensional base. We thus get natural generalisations of SL_2(Z) and pave the way to understanding the fundamental group of moduli stacks of elliptic surfaces in general. Our approach exploits the natural involution on Weierstrass curves and the identification of its fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over P^n. The fundamental group of the corresponding discriminant complement can be presented in terms of finitely many generators and relations using methods in the Zariski tradition, which were sucessfully elaborated in mathAG/0602371."}
{"category": "Math", "title": "The Loebl-Komlos-Sos conjecture for trees of diameter 5 and for certain caterpillars", "abstract": "Loebl, Komlos, and Sos conjectured that if at least half the vertices of a graph G have degree at least some k, then every tree with at most k edges is a subgraph of G. We prove the conjecture for all trees of diameter at most 5 and for a class of caterpillars. Our result implies a bound on the Ramsey number r(T,F) of trees T, F from the above classes."}
{"category": "Math", "title": "Matsuki's double coset decomposition via gradient maps", "abstract": "Let $G$ be a real-reductive Lie group and let $G_1$ and $G_2$ be two subgroups given by involutions. We show how the technique of gradient maps can be used in order to obtain a new proof of Matsuki's parametrization of the closed double cosets $G_1\\backslash G/G_2$ by Cartan subsets. We also describe the elements sitting in non-closed double cosets."}
{"category": "Math", "title": "On the symmetry of minimizers", "abstract": "For a large class of variational problems we prove that minimizers are symmetric whenever they are $C^1$."}
{"category": "Math", "title": "C-Supplemented Subalgebras of Lie Algebras", "abstract": "A subalgebra $B$ of a Lie algebra $L$ is {\\em c-supplemented} in $L$ if there is a subalgebra $C$ of $L$ with $L = B + C$ and $B \\cap C \\leq B_L$, where $B_L$ is the core of $B$ in $L$. This is analogous to the corresponding concept of a c-supplemented subgroup in a finite group. We say that $L$ is {\\em c-supplemented} if every subalgebra of $L$ is c-supplemented in $L$. We give here a complete characterisation of c-supplemented Lie algebras over a general field."}
{"category": "Math", "title": "On some explicit semi-stable degenerations of toric varieties", "abstract": "We study semi-stable degenerations of toric varieties determined by certain partitions of their moment polytopes. Analyzing their defining equations we prove a property of uniqueness."}
{"category": "Math", "title": "Semistable reduction for overconvergent F-isocrystals, IV: Local semistable reduction at nonmonomial valuations", "abstract": "We complete our proof that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue for F-isocrystals overconvergent inside a partial compactification. By previous results, this reduces to solving a local problem in a neighborhood of a valuation of height 1 and residual transcendence degree 0. We do this by studying the variation of some numerical invariants attached to p-adic differential modules, analogous to the irregularity of a complex meromorphic connection. This allows for an induction on the transcendence defect of the valuation, i.e., the discrepancy between the dimension of the variety and the rational rank of the valuation."}
{"category": "Math", "title": "Projective manifolds containing a large linear subspace with nef normal bundle", "abstract": "We classify smooth complex projective varieties $X \\subset \\proj^N$ of dimension $2s+1$ containing a linear subspace $\\Lambda$ of dimension $s$ whose normal bundle $N_{\\Lambda/X}$ is numerically effective."}
{"category": "Math", "title": "Tauberian theorems and large deviations", "abstract": "The link between Tauberian theorems and large deviations is surveyed, with particular reference to regular variation."}
{"category": "Math", "title": "The Parabolic Two-Phase Membrane Problem: Regularity in Higher Dimensions", "abstract": "For the parabolic obstacle-problem-like equation $$\\Delta u - \\partial_t u = \\lambda_+ \\chi_{\\{u>0\\}} - \\lambda_- \\chi_{\\{u<0\\}} ,$$ where $\\lambda_+$ and $\\lambda_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary $\\partial\\{u>0\\} \\cup\\partial\\{u<0\\}$ is in a neighborhood of each ``branch point'' the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper \\cite{imrn} to the parabolic case. The result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example."}
{"category": "Math", "title": "A note on the supremum of a stable process", "abstract": "If $X$ is a spectrally positive stable process of index $\\alpha\\in(1,2)$ whose L\\'{e}vy measure has density $cx^{-\\alpha-1}$ on $(0,\\infty),$ and $S_1=\\sup_{0<t\\leq1}X_t,$ it is known that $P(S_1>x)\\backsim c\\alpha^{-1}x^{-\\alpha}$ as $x\\to\\infty.$ It is also known that $S_1$has a continuous density, $s$ say. The point of this note is to show that $s(x)\\backsim cx^{-(\\alpha+1)}$ as $x\\to\\infty.$"}
{"category": "Math", "title": "Predicting the Last Zero of Brownian Motion with Drift", "abstract": "Given a standard Brownian motion $B^{\\mu}=(B_t^{\\mu})_{0\\le t\\le T}$ with drift $\\mu \\in IR$ and letting $g$ denote the last zero of $B^{\\mu}$ before $T$, we consider the optimal prediction problem V_*=\\inf_{0\\le \\tau \\le T}\\mathsf {E}\\:|\\:g-\\tau | where the infimum is taken over all stopping times $\\tau$ of $B^{\\mu}$. Reducing the optimal prediction problem to a parabolic free-boundary problem and making use of local time-space calculus techniques, we show that the following stopping time is optimal: \\tau_*=\\inf {t\\in [0,T] | B_t^{\\mu} \\le b_-(t) or B_t^{\\mu} \\ge b_+(t)} where the function $t\\mapsto b_-(t)$ is continuous and increasing on $[0,T]$ with $b_-(T)=0$, the function $t\\mapsto b_+(t)$ is continuous and decreasing on $[0,T]$ with $b_+(T)=0$, and the pair $b_-$ and $b_+$ can be characterised as the unique solution to a coupled system of nonlinear Volterra integral equations. This also yields an explicit formula for $V_*$ in terms of $b_-$ and $b_+$. If $\\mu=0$ then $b_-=-b_+$ and there is a closed form expression for $b_{\\pm}$ as shown in [10] using the method of time change from [4]. The latter method cannot be extended to the case when $\\mu \\ne 0$ and the present paper settles the remaining cases using a different approach."}
{"category": "Math", "title": "Stochastic Homogenization of Reflected Diffusion Processes", "abstract": "We investigate a functional limit theorem (homogenization) for Reflected Stochastic Differential Equations on a half-plane with stationary coefficients when it is necessary to analyze both the effective Brownian motion and the effective local time. We prove that the limiting process is a reflected non-standard Brownian motion. Beyond the result, this problem is known as a prototype of non-translation invariant problem making the usual method of the \"environment as seen from the particle\" inefficient."}
{"category": "Math", "title": "Cumulative record times in a Poisson process", "abstract": "We obtain a strong law of large numbers and a functional central limit theorem, as $t\\to\\infty$, for the number of records up to time $t$ and the Lebesgue measure (length) of the subset of the time interval $[0,t]$ during which the Poisson process is in a record lifetime."}
{"category": "Math", "title": "Large deviations for directed percolation on a thin rectangle", "abstract": "Following the recent investigations of Baik and Suidan in \\cite{baik2005gcl} and Bodineau and Martin in \\cite{bodineau2005upl}, we prove large deviation properties for a last-passage percolation model in $\\mathbb{Z}^{2}_{+}$ whose paths are close to the axis. The results are mainly obtained when the random weights are Gaussian or have a finite moment-generating function and rely, as in \\cite{baik2005gcl} and \\cite{bodineau2005upl}, on an embedding in Brownian paths and the KMT approximation. The study of the subexponential case completes the exposition."}
{"category": "Math", "title": "An analysis of two modifications of the Petersburg game", "abstract": "Two modifications of the Petersburg game are considered: 1. Truncation, so that the player has a finite capital at his disposal. 2. A cost of borrowing capital, so that the player has to pay interest on the capital needed. In both cases limit theorems for the total net gain are derived, so that it is easy to judge if the game is favourable or not."}
{"category": "Math", "title": "Symmetry approaches for reductions of PDEs, differential constraints and Lagrange-Charpit method", "abstract": "Many methods for reducing and simplifying differential equations are known. They provide various generalizations of the original symmetry approach of Sophus Lie. Plenty of relations between them have been noticed and in this note a unifying approach will be discussed. It is rather close to the classical differential constraint method, but we provide certain rigorous results basing on recent advances in compatibility theory of non-linear overdetermined systems and homological methods for PDEs."}
{"category": "Math", "title": "A polymer in a multi-interface medium", "abstract": "We consider a model for a polymer chain interacting with a sequence of equispaced flat interfaces through a pinning potential. The intensity $\\delta \\in \\mathbb {R}$ of the pinning interaction is constant, while the interface spacing $T=T_N$ is allowed to vary with the size $N$ of the polymer. Our main result is the explicit determination of the scaling behavior of the model in the large $N$ limit, as a function of $(T_N)_N$ and for fixed $\\delta >0$. In particular, we show that a transition occurs at $T_N=O(\\log N)$. Our approach is based on renewal theory."}
{"category": "Math", "title": "Extending ring derivations to right and symmetric rings and modules of quotients", "abstract": "We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total and perfect right and symmetric rings of quotients."}
{"category": "Math", "title": "Statistical analysis of redundant systems with \"warm\" stand-by units", "abstract": "Mathematical formulation of fluent switching from \"warm\" to \"hot\" conditions of standby units is given using the well known Sedyakin's and accelerated failure time (AFT) models. Non-parametric estimators of cumulative distribution function and mean failure time of a redundant system with several stand-by units are proposed. Goodness-of-fit tests for two given models are given."}
{"category": "Math", "title": "Domains of attraction of the random vector $(X,X^2)$ and applications", "abstract": "Many statistics are based on functions of sample moments. Important examples are the sample variance $s_{n-1}^2$, the sample coefficient of variation SV(n), the sample dispersion SD(n) and the non-central $t$-statistic $t(n)$. The definition of these quantities makes clear that the vector defined by (\\sum_{i=1}^nX_i,\\sum_{i=1}^nX_i^2) plays an important role. In studying the asymptotic behaviour of this vector we start by formulating best possible conditions under which the vector $(X,X^2)$ belongs to a bivariate domain of attraction of a stable law. This approach is new, uniform and simple. Our main results include a full discussion of the asymptotic behaviour of SV(n), SD(n) and $t^2(n)$. For simplicity, in restrict ourselves to positive random variables $X$."}
{"category": "Math", "title": "Multivariate Regular Variation on Cones: Application to Extreme Values, Hidden Regular Variation and Conditioned Limit Laws", "abstract": "We attempt to bring some modest unity to three subareas of heavy tail analysis and extreme value theory: limit laws for componentwise maxima of iid random variables;hidden regular variation and asymptotic independence;conditioned limit laws when one component of a random vector is extreme. The common theme is multivariate regular variation on a cone and the three cases cited come from specifying the cones $[0,\\infty]^d\\setminus \\{\\boldsymbol 0\\};(0,\\infty]^d;$ and $[0,\\infty]\\times (0,\\infty]$."}
{"category": "Math", "title": "Partial monoids and Dold-Thom functors", "abstract": "Dold-Thom functors are generalizations of infinite symmetric products, where integer multiplicities of points are replaced by composable elements of a partial abelian monoid. It is well-known that for any connective homology theory, the machinery of $\\Gamma$-spaces produces the corresponding linear Dold-Thom functor. In this note we construct such functors directly from spectra by exhibiting a partial monoid corresponding to a spectrum."}
{"category": "Math", "title": "The elementary obstruction and the Weil restriction", "abstract": "In this text we investigate the good behaviour of the elementary obstruction, introduced by Colliot-Thelene and Sansuc. This is an obstruction to the existence of a rational points on certain algebraic varieties. Assuming some conditions on the Picard group, we prove that the elementary obstruction behaves well under the Weil restriction of a variety."}
{"category": "Math", "title": "Optimality of estimators for misspecified semi-Markov models", "abstract": "Suppose we observe a geometrically ergodic semi-Markov process and have a parametric model for the transition distribution of the embedded Markov chain, for the conditional distribution of the inter-arrival times, or for both. The first two models for the process are semiparametric, and the parameters can be estimated by conditional maximum likelihood estimators. The third model for the process is parametric, and the parameter can be estimated by an unconditional maximum likelihood estimator. We determine heuristically the asymptotic distributions of these estimators and show that they are asymptotically efficient. If the parametric models are not correct, the (conditional) maximum likelihood estimators estimate the parameter that maximizes the Kullback--Leibler information. We show that they remain asymptotically efficient in a nonparametric sense."}
{"category": "Math", "title": "Priscilla Greenwood: Queen of Probability", "abstract": "This article contains the introduction to the special volume of Stochastics dedicated to Priscilla Greenwood, her CV and her list of publications."}
{"category": "Math", "title": "On weak Lie 2-algebras", "abstract": "A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and 2-term \"homotopy everything\" Lie algebras; for strictly skew-symmetric Lie 2-algebras, these reduce to $L_\\infty$-algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer--Cartan equation in some differential graded Lie algebras and $L_\\infty$-algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Lie bialgebroids and Courant algebroids provide large classes of examples."}
{"category": "Math", "title": "Pointwise characterizations in generalized function algebras", "abstract": "We define the algebra of Colombeau generalized functions on the space of generalized points of {\\mathbb R}^d which naturally contains the tempered generalized functions. The subalgebra of \\mathscr{S}-regular generalized functions of this algebra is characterized by a pointwise property of the generalized functions and their Fourier transforms. We also characterize the equality in the sense of generalized tempered distributions for certain elements of this algebra (namely those with so-called slow scale support) by means of a pointwise property of their Fourier transforms. Further, we show that (contrary to what has been claimed in the literature) for an open subset \\Omega of {\\mathbb R}^d, the algebra of pointwise regular generalized functions \\dot{\\mathcal G}^\\infty(\\Omega) equals {\\mathcal G}^\\infty(\\Omega) and give several characterizations of pointwise {\\mathcal G}^\\infty-regular generalized functions."}
{"category": "Math", "title": "Martingales and first passage times of AR(1) sequences", "abstract": "Using the martingale approach we find sufficient conditions for exponential boundedness of first passage times over a level for ergodic first order autoregressive sequences (AR(1)). Further, we prove a martingale identity to be used in obtaining explicit bounds for the expectation of first passage times."}
{"category": "Math", "title": "Embedding compacta into products of curves", "abstract": "We present some results on n-dimensional compacta lying in n-dimensional products of compacta, in particular, in products of n 1-dimensional compacta. Most of our basic results are proven under the assumption that the compacta X admit essential maps into the n-sphere. The results of the present paper may be viewed as an extension of the theory developed so far by other authors. First, we prove that if X is an n-dimensional compactum with $H^n(X) \\neq 0$ that embeds in either a product of n curves or the nth symmetric product of a curve, then there exists an algebraically essential map of X into the n-torus. Consequently, there exist elements $a_1,...,a_n$ in $H^1(X)$ whose cup product is non-zero. Therefore, the rank of $H^1(X)$ is at least n, and the category of X exceeds n. Next, we introduce some new classes of n-dimensional continua and show that embeddability of locally connected quasi n-manifolds into products of n curves also implies that the rank of $H^1(X)$ is at least n. It follows that some 2-dimensional contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron can be embedded in a product of two trees. We answer a question posed by Cauty proving that closed surfaces embeddable in products of two curves can be also embedded in products of two graphs. We prove that no closed surface, different from the 2-torus, lying in a product of two curves is a retract of that product."}
{"category": "Math", "title": "Monotone versions of $\\delta$-normality", "abstract": "We continue the study of properties related to monotone countable paracompactness, investigating various monotone versions of $\\delta$-normality. We factorize monotone normality and stratifiability in terms of these weaker properties."}
{"category": "Math", "title": "On cyclic fixed points of spectra", "abstract": "For a finite p-group G and a bounded below G-spectrum X of finite type mod p, the G-equivariant Segal conjecture for X asserts that the canonical map X^G --> X^{hG} is a p-adic equivalence. Let C_{p^n} be the cyclic group of order p^n. We show that if the C_p Segal conjecture holds for a C_{p^n} spectrum X, as well as for each of its C_{p^e} geometric fixed points for 0 < e < n, then then C_{p^n} Segal conjecture holds for X. Similar results hold for weaker forms of the Segal conjecture, asking only that the canonical map induces an equivalence in sufficiently high degrees, on homotopy groups with suitable finite coefficients."}
{"category": "Math", "title": "A note on restricted X-ray transforms", "abstract": "We show how the techniques introduces by Christ can be employed to derive endpoint $L^p-L^q$ bounds for the X-ray transform associated to the line complex generated by the curve $t\\to(t,t^2,...,t^{d-1}).$ Almost-sharp Lorentz space estimates are produced as well."}
{"category": "Math", "title": "The Kadets 1/4 theorem for polynomials", "abstract": "We determine the maximal angular perturbation of the (n+1)th roots of unity permissible in the Marcinkiewicz-Zygmund theorem on L^p means of polynomials of degree at most n. For p=2, the result is an analogue of the Kadets 1/4 theorem on perturbation of Riesz bases of holomorphic exponentials."}
{"category": "Math", "title": "On the cohomology rings of holomorphically fillable manifolds", "abstract": "An odd-dimensional differentiable manifold is called \\emph{holomorphically fillable} if it is diffeomorphic to the boundary of a compact strongly pseudoconvex complex manifold, \\emph{Stein fillable} if this last manifold may be chosen to be Stein and \\emph{Milnor fillable} if it is diffeomorphic to the abstract boundary of an isolated singularity of normal complex analytic space. We show that the homotopical dimension of a manifold-with-boundary of dimension at least 4 restricts the cohomology ring (with any coefficients) of its boundary. This gives restrictions on the cohomology rings of Stein fillable manifolds, on the dimension of the exceptional locus of any resolution of a given isolated singularity, and on the topology of smoothable singularities. We give also new proofs of structure theorems of Durfee & Hain and Bungart about the cohomology rings of Milnor fillable and respectively holomorphically fillable manifolds. The various structure theorems presented in this paper imply that in dimension at least 5, the classes of Stein fillable, Milnor fillable and holomorphically fillable manifolds are pairwise different."}
{"category": "Math", "title": "Lacunarity and cyclic vectors for the Backward Shift", "abstract": "This article gives a description of invariant subspaces for the backward shift generated by vector valued lacunary series and by a class of lacunary power series in $H^2(\\mathbb{D}, X)$, (where $X$ is an Hilbert space). In particular, we show that these series $f$ in $H^2(\\mathbb{D}, X)$ are cyclic vectors if and only if the queue of Taylor coefficients $\\{\\hat{f}(k)$, $k>N\\}$ generates the whole space $X$. Analogues of this result are obtained for some functions whose spectrum is a finite union of lacunary sequences and in the polydisc. In the scalar case $H^2$, we give a criterion on the Fourier spectrum of the function to have cyclicity for any power of the backward shift."}
{"category": "Math", "title": "Combinatorial rigidity for some infinitely renormalizable unicritical polynomials", "abstract": "We prove combinatorial rigidity of infinitely renormalizable unicritical polynomials, P_c :z \\mapsto z^d+c, with complex c, under the a priori bounds and a certain \"combinatorial condition\". This implies the local connectivity of the connectedness loci (the Mandelbrot set when d = 2) at the corresponding parameters."}
{"category": "Math", "title": "Perfect Symmetric Rings of Quotients", "abstract": "Perfect Gabriel filters of right ideals and their corresponding right rings of quotients have the desirable feature that every module of quotients is determined solely by the right ring of quotients. On the other hand, symmetric rings of quotients have a symmetry that mimics the commutative case. In this paper, we study rings of quotients that combine these two desirable properties. We define the symmetric versions of a right perfect ring of quotients and a right perfect Gabriel filter -- the perfect symmetric ring of quotients and the perfect symmetric Gabriel filter and study their properties. Then we prove that the standard construction of the total right ring of quotients can be adapted to the construction of the largest perfect symmetric ring of quotients -- the total symmetric ring of quotients. We also demonstrate that Morita's construction of the total right ring of quotients can be adapted to the construction of the total symmetric ring of quotients."}
{"category": "Math", "title": "Invariants and submanifolds in almost complex geometry", "abstract": "In this paper we describe the algebra of differential invariants for GL(n,C)-structures. This leads to classification of almost complex structures of general positions. The invariants are applied to the existence problem of higher-dimensional pseudoholomorphic submanifolds."}
{"category": "Math", "title": "Anomaly of linearization and auxiliary integrals", "abstract": "In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility reductions of PDEs."}
{"category": "Math", "title": "Differential invariants of the motion group actions", "abstract": "Differential invariants of a (pseudo)group action can vary when restricted to invariant submanifolds (differential equations). The algebra is still governed by the Lie-Tresse theorem, but may change a lot. We describe in details the case of the motion group $O(n)\\ltimes\\R^n$ acting on the full (unconstraint) jet-space as well as on some invariant equations."}
{"category": "Math", "title": "Transformations of L\\'evy Processes", "abstract": "A L\\'evy process on a *-bialgebra is given by its generator, a conditionally positive hermitian linear functional vanishing at the unit element. A *-algebra homomorphism k from a *-bialgebra C to a *-bialgebra B with the property that k respects the counits maps generators on B to generators on C. A tranformation between the corrresponding two L\\'evy processes is given by forming infinitesimal convolution products. This general result is applied to various situations, e.g., to a *-bialgebra and its associated primitive tensor *-bialgebra (called \"generator process\") as well as its associated group-like *-bialgebra (called Weyl-*-bialgebra). It follows that a L\\'evy process on a *-bialgebra can be realized on Bose Fock space as the infinitesimal convolution product of its generator process such that the vacuum vector is cyclic for the L\\e'vy process. Moreover, we obtain convolution approximations of the Az\\'ema martingale by the Wiener process and vice versa."}
{"category": "Math", "title": "Negative correlation and log-concavity", "abstract": "We give counterexamples and a few positive results related to several conjectures of R. Pemantle and D. Wagner concerning negative correlation and log-concavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. We also give short proofs of a pair of results due to Pemantle and Borcea et al.; prove that \"almost exchangeable\" measures satisfy the \"Feder-Mihail\" property, thus providing a \"non-obvious\" example of a class of measures for which this important property can be shown to hold; and mention some further questions."}
{"category": "Math", "title": "9 Divides no Odd Fibonacci", "abstract": "I discuss numbers that divide no odd Fibonacci. Number 9 plays a special role among such numbers."}
{"category": "Math", "title": "Hom-algebras and homology", "abstract": "Classes of $G$-Hom-associative algebras are constructed as deformations of $G$-associative algebras along algebra endomorphisms. As special cases, we obtain Hom-associative and Hom-Lie algebras as deformations of associative and Lie algebras, respectively, along algebra endomorphisms. Chevalley-Eilenberg type homology for Hom-Lie algebras are also constructed."}
{"category": "Math", "title": "Extending higher derivations to rings and modules of quotients", "abstract": "A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsion theories are higher differential as well. Then, we study conditions under which a higher derivation extended to a right module of quotients extends also to a right module of quotients with respect to a larger torsion theory. Lastly, we define and study the symmetric version of higher differential torsion theories. We prove that the symmetric versions of the results on higher differential (one-sided) torsion theories hold for higher derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and any perfect torsion theories are higher differential."}
{"category": "Math", "title": "Asymptotic representations of the reduced C*-algebra of a free group: an example", "abstract": "We give an example of a non-trivial asymptotic representation of the reduced C*-algebra of a free group. This example allows to evaluate the asymptotic tensor C*-norm of some elements in tensor product C*-algebras and to show semi-invertibility of the non-invertible extension of $C^*_r(\\mathbb F_2)$ considered by Haagerup and Thorbjornsen."}
{"category": "Math", "title": "A priori estimate of gradient of a solution to certain differential inequality and quasiconformal mappings", "abstract": "We will prove a global estimate for the gradient of the solution to the {\\it Poisson differential inequality} $|\\Delta u(x)|\\le a|\\nabla u(x)|^2+b$, $x\\in B^{n}$, where $a,b<\\infty$ and $u|_{S^{n-1}}\\in C^{1,\\alpha}(S^{n-1}, \\Bbb R^m)$. If $m=1$ and $a\\le (n+1)/(|u|_\\infty4n\\sqrt n)$, then $|\\nabla u| $ is a priori bounded. This generalizes some similar results due to E. Heinz (\\cite{EH}) and Bernstein (\\cite{BS}) for the plane. An application of these results yields the theorem, which is the main result of the paper: A quasiconformal mapping of the unit ball onto a domain with $C^2$ smooth boundary, satisfying the Poisson differential inequality, is Lipschitz continuous. This extends some results of the author, Mateljevi\\'c and Pavlovi\\'c from the complex plane to the space."}
{"category": "Math", "title": "Special, conjugate and complete scale functions for spectrally negative L\\'evy processes", "abstract": "Following from recent developments by Hubalek and Kyprianou, the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative L\\'evy processes which are completely explicit. This is the result of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization for spectrally negative L\\'evy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed."}
{"category": "Math", "title": "Test configurations, large deviations and geodesic rays on toric varieties", "abstract": "This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \\kahler metrics associated to the geodesic ray of potentials are discontinuous across certain hypersurfaces and are degenerate on certain open sets. A novelty in the analysis is the connection between Bergman metrics, Bergman kernels and the theory of large deviations."}
{"category": "Math", "title": "Network Tomography: Identifiability and Fourier Domain Estimation", "abstract": "The statistical problem for network tomography is to infer the distribution of $\\mathbf{X}$, with mutually independent components, from a measurement model $\\mathbf{Y}=A\\mathbf{X}$, where $A$ is a given binary matrix representing the routing topology of a network under consideration. The challenge is that the dimension of $\\mathbf{X}$ is much larger than that of $\\mathbf{Y}$ and thus the problem is often called ill-posed. This paper studies some statistical aspects of network tomography. We first address the identifiability issue and prove that the $\\mathbf{X}$ distribution is identifiable up to a shift parameter under mild conditions. We then use a mixture model of characteristic functions to derive a fast algorithm for estimating the distribution of $\\mathbf{X}$ based on the General method of Moments. Through extensive model simulation and real Internet trace driven simulation, the proposed approach is shown to be favorable comparing to previous methods using simple discretization for inferring link delays in a heterogeneous network."}
{"category": "Math", "title": "Convergence Rates for Approximations of Functionals of SDEs", "abstract": "We consider upper bounds for the approximation error E|g(X)-g(\\hat X)|^p, where X and \\hat X are random variables such that \\hat X is an approximation of X in the L_p-norm, and the function g belongs to certain function classes, which contain e.g. functions of bounded variation. We apply the results to the approximations of a solution of a stochastic differential equation at time T by the Euler and Milstein schemes. For the Euler scheme we provide also a lower bound."}
{"category": "Math", "title": "On Lusternik-Schnirelmann category of SO(10)", "abstract": "Let $G$ be a compact connected Lie group and $p : E\\to \\Sigma A$ be a principal G-bundle with a characteristic map $\\alpha : A\\to G$, where $A=\\Sigma A_{0}$ for some $A_{0}$. Let $\\{K_{i}{\\to} F_{i-1}{\\hookrightarrow} F_{i} \\,|\\, 1{\\le} i {\\le} n,\\, F_{0}{=} \\{\\ast\\} \\; F_{1}{=} \\Sigma{K_{1}} \\; \\text{and}\\; F_{n}{\\simeq} G \\}$ be a cone-decomposition of $G$ of length $m$ and $F'_{1}=\\Sigma{K'_{1}} \\subset F_{1}$ with $K'_{1} \\subset K_{1}$ which satisfy $F_{i}F'_{1} \\subset F_{i+1}$ up to homotopy for any $i$. Our main result is as follows: we have $\\operatorname{cat}(X) \\le m{+}1$, if firstly the characteristic map $\\alpha$ is compressible into $F'_{1}$, secondly the Berstein-Hilton Hopf invariant $H_{1}(\\alpha)$ vanishes in $[A, \\Omega F'_1{\\ast}\\Omega F'_1]$ and thirdly $K_{m}$ is a sphere. We apply this to the principal bundle $\\mathrm{SO}(9)\\hookrightarrow\\mathrm{SO}(10)\\to S^{9}$ to determine L-S category of $\\mathrm{SO}(10)$."}
{"category": "Math", "title": "Subcritical regimes in some models of continuum percolation", "abstract": "We consider some continuum percolation models. We are mainly interested in giving some sufficient conditions for absence of percolation. We give some general conditions and then focuse on two examples. The first one is a multiscale percolation model based on the Boolean model. It was introduced by Meester and Roy and subsequently studied by Menshikov, Popov and Vachkovskaia. The second one is based on the stable marriage of Poisson and Lebesgue introduced by Hoffman, Holroyd and Peres and whose percolation properties have been studied by Freire, Popov and Vachkovskaia."}
{"category": "Math", "title": "Strichartz estimates for Schroedinger equations with periodic potential in 1D", "abstract": "This paper has been withdrawn by the author due to a crucial error in the Proof of Theorem 0.3"}
{"category": "Math", "title": "Asymptotic Lower Bounds for a class of Schroedinger Equations", "abstract": "We shall study the following initial value problem: \\begin{equation}{\\bf i}\\partial_t u - \\Delta u + V(x) u=0, \\hbox{} (t, x) \\in {\\mathbf R} \\times {\\mathbf R}^n, \\end{equation} $$u(0)=f,$$ where $V(x)$ is a real short--range potential, whose radial derivative satisfies some supplementary assumptions. More precisely we shall present a family of identities satisfied by the solutions to the previous Cauchy problem. As a by--product of these identities we deduce some uniqueness results and a lower bound for the so called local smoothing which becomes an identity in a precise asymptotic sense."}
{"category": "Math", "title": "A bijection for rooted maps on orientable surfaces", "abstract": "The enumeration of maps and the study of uniform random maps have been classical topics of combinatorics and statistical physics ever since the seminal work of Tutte in the sixties. Following the bijective approach initiated by Cori and Vauquelin in the eighties, we describe a bijection between rooted maps, or rooted bipartite quadrangulations, on a surface of genus g and some simpler objects that generalize plane trees. Thanks to a rerooting argument, our bijection allows to compute the generating series of rooted maps on a surface of genus g with respect to the number of edges, and to recover the asymptotic numbers of such maps. Our construction allows to keep track in a bipartite quadrangulation of the distances of all vertices to a random basepoint. This is an analog for higher genus surfaces of the basic result on which were built the recent advances in the comprehension of the intrinsec geometry of large random planar maps, hopefully opening the way to the study of a model of continuum random surfaces of genus g."}
{"category": "Math", "title": "Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems", "abstract": "We study sample covariance matrices of the form $W=\\frac 1n C C^T$, where $C$ is a $k\\times n$ matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of $C$ are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when $k$ is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of $W$ when either $k$ is fixed and $n\\to \\infty$, or $k_n\\to \\infty$ with $k_n=o(n/\\log\\log{n})$, in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving a.s. limits of the eigenvalues, only require finite fourth moments. Our most explicit results for $k$ large are for the case where the entries of $C$ are $\\pm1$ with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalue to the rate functions for independent and identically distributed standard normal entries of $C$. This case is of particular interest, since it is related to the problem of the decoding of a signal in a code division multiple access system arising in mobile communication systems. In this example, $k$ plays the role of the number of users in the system, and $n$ is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency, and the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users."}
{"category": "Math", "title": "Symmetry of singular solutions of degenerate quasilinear elliptic equations", "abstract": "We prove radial symmetry of singular solutions to an overdetermined boundary value problem for a class of degenerate quasilinear elliptic equations."}
{"category": "Math", "title": "Cellular structures on Hecke algebras of type B", "abstract": "The aim of this paper is to gather and (try to) unify several approaches for the modular representation theory of Hecke algebras of type $B$. We attempt to explain the connections between Geck's cellular structures (coming from Kazhdan-Lusztig theory with unequal parameters) and Ariki's Theorem on the canonical basis of the Fock spaces."}
{"category": "Math", "title": "Sharp tridiagonal pairs", "abstract": "Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of $K$-linear transformations $A:V \\to V$ and $A^*:V \\to V$ that satisfies the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering ${V_i}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \\subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \\leq i \\leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering ${V^*_i}_{i=0}^\\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \\subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \\leq i \\leq \\delta$, where $V^*_{-1}=0$ and $V^*_{\\delta+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \\subseteq W$, $A^* W \\subseteq W$, $W \\neq 0$, $W \\neq V$. We call such a pair a {\\em tridiagonal pair} on $V$. It is known that $d=\\delta$ and for $0 \\leq i \\leq d$ the dimensions of $V_i$, $V_{d-i}$, $V^*_i$, $V^*_{d-i}$ coincide. We say the pair $A,A^*$ is {\\em sharp} whenever $\\dim V_0=1$. A conjecture of Tatsuro Ito and the second author states that if $K$ is algebraically closed then $A,A^*$ is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs."}
{"category": "Math", "title": "Generalizing a theorem of P. Hall on finite-by-nilpotent groups", "abstract": "Let $\\gamma_i(G)$ and $Z_i(G)$ denote the $i$-th terms of the lower and upper central series of a group $G$, respectively. P. Hall showed that if $\\gamma_{i+1}(G)$ is finite then the index $|G:Z_{2i}(G)|$ is finite. We prove that the same result holds under the weaker hypothesis that $|\\gamma_{i+1}(G):\\gamma_{i+1}(G)\\cap Z_i(G)|$ is finite."}
{"category": "Math", "title": "Homological Algebra and Divergent Series", "abstract": "We study some features of infinite resolutions of Koszul algebras motivated by the developments in the string theory initiated by Berkovits."}
{"category": "Math", "title": "Monotone Numerical Schemes for a Dirichlet Problem for Elliptic Operators in Divergence Form", "abstract": "We consider a second order differential operator $A(\\msx) = -\\:\\sum_{i,j=1}^d \\partial_i a_{ij}(\\msx) \\partial_j \\:+\\: \\sum_{j=1}^d \\partial_j \\big(b_j(\\msx) \\cdot \\big)\\:+\\: c(\\msx)$ on ${\\bbR}^d$, on a bounded domain $D$ with Dirichlet boundary conditions on $\\partial D$, under mild assumptions on the coefficients of the diffusion tensor $a_{ij}$. The object is to construct monotone numerical schemes to approximate the solution to the problem $A(\\msx) u(\\msx) \\: = \\: \\mu(\\msx), \\quad \\msx \\in D$, where $\\mu$ is a positive Radon measure. We start by briefly mentioning questions of existence and uniqueness, introducing function spaces needed to prove convergence results. Then, we define non-standard stencils on grid-knots that lead to extended discretization schemes by matrices possesing compartmental structure. We proceed to discretization of elliptic operators, starting with constant diffusion tensor and ending with operators in divergence form. Finally, we discuss $W_2^1$-convergence in detail, and mention convergence in $C$ and $L_1$ spaces. We conclude by a numerical example illustarting the schemes and convergence results."}
{"category": "Math", "title": "Relative Zariski Open Objects", "abstract": "In [TV], Bertrand To\\\"en and Michel Vaqui\\'e define a scheme theory for a closed monoidal category $(\\mathcal{C},\\otimes,1)$. One of the key ingredients of this theory is the definition of a Zariski topology on the category of commutative monoids in $\\mathcal{C}$. The purpose of this article is to prove that under some hypotheses, Zariski open subobjects of affine schemes can be classified almost as in the usual case of rings $(Z-mod,\\otimes,Z)$. The main result states that for any commutative monoid $A$, the locale of Zariski open subobjects of the affine scheme $Spec(A)$ is associated to a topological space whose points are prime ideals of $A$ and open subsets are defined by the same formula as in rings. As a consequence, we compare the notions of scheme over $\\mathbb{F}_{1}$ of [D] and [TV]."}
{"category": "Math", "title": "Numerical approach to $L_1$-problems with the second order elliptic operators", "abstract": "For a second order differential operator $A(\\msx) =-\\nabla a(\\msx)\\nabla + b'(\\msx)\\nabla+ \\nabla \\big(\\msb''(\\msx) \\cdot\\big)$ on a bounded domain $D$ with the Dirichlet boundary conditions on $\\partial D$ there exists the inverse $T(\\lambda, A)= (\\lambda I+A)^{-1}$ in $L_1(D)$. If $\\mu$ is a Radon (probability) measure on Borel algebra of subsets of $D$, then $T(\\lambda, A)\\mu \\in L_p(D), p \\in [1, d/(d-1))$. We construct the numerical approximations to $u =T(\\lambda, A)\\mu$ in two steps. In the first one we construct grid-solutions ${\\bf u}_n$ and in the second step we embed grid-solutions into the linear space of hat functions $u(n) \\in \\dot{W}_p^1(D)$. The strong convergence to the original solutions $u$ is established in $L_p(D)$ and the weak convergence in $\\dot{W}_p^1(D)$."}
{"category": "Math", "title": "On limit theorems for continued fractions", "abstract": "It is shown that for sums of functionals of digits in continued fraction expansion the Kolmogorov-Feller weak laws of large numbers and the Khinchine-L\\'evy-Feller-Raikov characterization of the domain of attraction of the normal law hold."}
{"category": "Math", "title": "On the sphericity of scaling limits of random planar quadrangulations", "abstract": "We give a new proof of a theorem by Le Gall & Paulin, showing that scaling limits of random planar quadrangulations are homeomorphic to the 2-sphere. The main geometric tool is a reinforcement of the notion of Gromov-Hausdorff convergence, called 1-regular convergence, that preserves topological properties of metric surfaces."}
{"category": "Math", "title": "Tessellations of random maps of arbitrary genus", "abstract": "We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points are linked by a unique geodesic."}
{"category": "Math", "title": "Curvature of classifying spaces for Brieskorn lattices", "abstract": "We study tt*-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the holomorphic sectional curvature of this hermitian metric, which is the analogue of a similar result for classifying spaces of pure polarized Hodge structures."}
{"category": "Math", "title": "From Hopf C*-families to concrete Hopf C*-bimodules", "abstract": "In the setting of von Neumann algebras, measurable quantum groupoids have successfully been axiomatized and studied by Enock, Vallin, and Lesieur, whereas in the setting of $C^{*}$-algebras, a similar theory of locally compact quantum groupoids could not yet be developed. Some basic building blocks for such a theory, like analogues of a Hopf-von Neumann bimodule and of a pseudo-multiplicative unitary, were introduced in the thesis and a recent article by the author. That approach, however, is restricted to decomposable quantum groupoids which generalize $r$-discrete groupoids. Recently, we developed a general approach that covers all locally compact groupoids. In this article, we explain how the special theory of our thesis embeds into the general one."}
{"category": "Math", "title": "Central limit theorem for sampled sums of dependent random variables", "abstract": "We prove a central limit theorem for linear triangular arrays under weak dependence conditions. Our result is then applied to the study of dependent random variables sampled by a $\\bbZ$-valued transient random walk. This extends the results obtained by Guillotin-Plantard & Schneider (2003). An application to parametric estimation by random sampling is also provided."}
{"category": "Math", "title": "The Baum-Connes conjecture for countable subgroups of SL(2)", "abstract": "We present an alternative approach to the result of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for finitely generated subgroups of SL(2,C). Using finite-dimensional methods, we show that the Baum-Connes assembly map for such groups is an isomorphism."}
{"category": "Math", "title": "On a Hasse principle for Mordell-Weil groups", "abstract": "In this paper we establish a Hasse principle concerning the linear dependence over $\\Z$ of nontorsion points in the Mordell-Weil group of an abelian variety over a number field."}
{"category": "Math", "title": "On the realizations of high dimensional solenoids as attractors and as non-wondering sets", "abstract": "This paper will be splited into two papers and submited later."}
{"category": "Math", "title": "Picard groups of Siegel modular threefolds and theta lifting", "abstract": "We show that the Humbert surfaces rationally generate the Picard groups of Siegel modular threefolds."}
{"category": "Math", "title": "Isometries Groups and a Multiresolution Analysis on Sub-Riemannian Manifolds", "abstract": "In this letter we exhibit the relation between the isometries of a Riemannian contraction of a sub-Riemannian manifold and those of the sub-Riemannian metric, for to use this relation with two goals: establishing a result about the existence of fixed points of isometries groups; and the other, defining a Multiresolution Analysis (MRA) on sub-Riemannian manifolds that it will permit to obtain Haar's bases on the manifolds before mentioned. Keywords: Sub-Riemannian geometry, minimizing geodesic, Haar functions, self-similarity."}
{"category": "Math", "title": "The rate of convergence of spectra of sample covariance matrices", "abstract": "It is shown that the Kolmogorov distance between the spectral distribution function of a random covariance matrix $\\frac1p XX^T$, where $X$ is a $n\\times p$ matrix with independent entries and the distribution function of the Marchenko-Pastur law is of order $O(n^{-1/2})$. The bounds hold {\\it uniformly} for any $p$, including $\\frac pn$ equal or close to 1."}
{"category": "Math", "title": "Passive systems with a normal main operator and quasi-selfadjoint systems", "abstract": "Passive systems $\\tau={T,M,N,H}$ with $M$ and $N$ as an input and output space and $H$ as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system $\\tau$ with $M=N$ is said to be quasi-selfadjoint if $ran(T-T^*)\\subset N$. The subclass $S^{qs}$ of the Schur class $S$ is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass $S^{qs}$ is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass $S^{qs}$ and the $Q$-function of $T$ is given."}
{"category": "Math", "title": "Subelliptic Bourgain-Brezis Estimates on Groups", "abstract": "We show that divergence free vector fields which belong to L^1 on stratified, nilpotent groups are in the dual space of functions whose sub-gradient are L^Q integrable where Q is the homogeneous dimension of the group. This was first obtained on Euclidean space by Bourgain and Brezis."}
{"category": "Math", "title": "Elimination with applications to singularities in positive characteristic", "abstract": "We present an application of elimination theory to the study of singularities over arbitrary fields, particularly to the open problem of resolution. A partial extension of a function, defining resolution of singularities over fields of characteristic zero, is discussed here in positive characteristic."}
{"category": "Math", "title": "Nonparametric estimation for a stochastic volatility model", "abstract": "Consider discrete time observations (X_{\\ell\\delta})_{1\\leq \\ell \\leq n+1}$ of the process $X$ satisfying $dX_t= \\sqrt{V_t} dB_t$, with $V_t$ a one-dimensional positive diffusion process independent of the Brownian motion $B$. For both the drift and the diffusion coefficient of the unobserved diffusion $V$, we propose nonparametric least square estimators, and provide bounds for theirrisk. Estimators are chosen among a collection of functions belonging to a finite dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method works."}
{"category": "Math", "title": "A priori $L^{\\infty}$-estimates for degenerate complex Monge-Amp\\`ere equations", "abstract": "We study families of complex Monge-Amp\\`ere equations, focusing on the case where the cohomology classes degenerate to a non big class. We establish uniform a priori $L^{\\infty}$-estimates for the normalized solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has interesting consequences in the study of the K\\\"ahler-Ricci flow."}
{"category": "Math", "title": "Convergence properties of the expected improvement algorithm", "abstract": "This paper has been withdrawn from the arXiv. It is now published by Elsevier in the Journal of Statistical Planning and Inference, under the modified title \"Convergence properties of the expected improvement algorithm with fixed mean and covariance functions\". See http://dx.doi.org/10.1016/j.jspi.2010.04.018 An author-generated post-print version is available from the HAL repository of SUPELEC at http://hal-supelec.archives-ouvertes.fr/hal-00217562 Abstract : \"This paper deals with the convergence of the expected improvement algorithm, a popular global optimization algorithm based on a Gaussian process model of the function to be optimized. The first result is that under some mild hypotheses on the covariance function k of the Gaussian process, the expected improvement algorithm produces a dense sequence of evaluation points in the search domain, when the function to be optimized is in the reproducing kernel Hilbert space generated by k. The second result states that the density property also holds for P-almost all continuous functions, where P is the (prior) probability distribution induced by the Gaussian process.\""}
{"category": "Math", "title": "Galois theory for iterative connections and nonreduced Galois groups", "abstract": "This article presents a theory of modules with iterative connection. This theory is a generalisation of the theory of modules with connection in characteristic zero to modules over rings of arbitrary characteristic. We show that these modules with iterative connection (and also the modules with integrable iterative connection) form a Tannakian category, assuming some nice properties for the underlying ring, and we show how this generalises to modules over schemes. We also relate these notions to stratifications on modules, as introduced by A. Grothendieck in order to extend integrable (ordinary) connections to finite characteristic. Over smooth rings, we obtain an equivalence of stratifications and integrable iterative connections. Furthermore, over a regular ring in positive characteristic, we show that the category of modules with integrable iterative connection is also equivalent to the category of flat bundles as defined by D. Gieseker. In the second part of this article, we set up a Picard-Vessiot theory for fields of solutions. For such a Picard-Vessiot extension, we obtain a Galois correspondence, which takes into account even nonreduced closed subgroup schemes of the Galois group scheme on one hand and inseparable intermediate extensions of the Picard-Vessiot extension on the other hand. Finally, we compare our Galois theory with the Galois theory for purely inseparable field extensions."}
{"category": "Math", "title": "Covering link calculus and iterated Bing doubles", "abstract": "We give a new geometric obstruction to the iterated Bing double of a knot being a slice link: for n>1 the (n+1)-st iterated Bing double of a knot is rationally slice if and only if the n-th iterated Bing double of the knot is rationally slice. The main technique of the proof is a covering link construction simplifying a given link. We prove certain similar geometric obstructions for n <= 1 as well. Our results are sharp enough to conclude, when combined with algebraic invariants, that if the n-th iterated Bing double of a knot is slice for some n, then the knot is algebraically slice. Also our geometric arguments applied to the smooth case show that the Ozsvath-Szabo and Manolescu-Owens invariants give obstructions to iterated Bing doubles being slice. These results generalize recent results of Harvey, Teichner, Cimasoni, Cha and Cha-Livingston-Ruberman. As another application, we give explicit examples of algebraically slice knots with non-slice iterated Bing doubles by considering von Neumann rho-invariants and rational knot concordance. Refined versions of such examples are given, that take into account the Cochran-Orr-Teichner filtration."}
{"category": "Math", "title": "Cubature on Wiener space in infinite dimension", "abstract": "We prove a stochastic Taylor expansion for SPDEs and apply this result to obtain cubature methods, i. e. high order weak approximation schemes for SPDEs, in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak convergence for well-defined classes of test functions if the process starts at sufficiently regular initial values. We can also derive analogous results in the presence of L\\'evy processes of finite type, here the results seem to be new even in finite dimension. Several numerical examples are added."}
{"category": "Math", "title": "Vanishing of trace forms in low characteristics", "abstract": "Every finite-dimensional representation of an algebraic group G gives a trace symmetric bilinear form on the Lie algebra of G. We give criteria in terms of root system data for the existence of a representation such that this form is nonzero or nondegenerate. As a corollary, we show that a Lie algebra of type E8 over a field of characteristic 5 does not have a so-called \"quotient trace form\", answering a question posed in the 1960s."}
{"category": "Math", "title": "Convex Hulls of Orbits and Orientations of a Moving Protein Domain", "abstract": "We study the facial structure and Carath\\'eodory number of the convex hull of an orbit of the group of rotations in R^3 acting on the space of pairs of anisotropic symmetric 3\\times 3 tensors. This is motivated by the problem of determining the structure of some proteins in aqueous solution."}
{"category": "Math", "title": "The conjecture H: A lower bound of cohomologic dimension for an elliptic space", "abstract": "The goal of this paper is to ameliorate the sufficients conditions, already established by the first author so that the sum of the numbers of Betti, of 1-connected rational finite CW-complex, is higher than the dimension of his $\\mathbb Q$-vectorial space of homotopy, we will present it in two aspects, one algebraic and another geometrical."}
{"category": "Math", "title": "Simple algebras of Gelfand-Kirillov dimension two", "abstract": "Let $k$ be a field. We show that a finitely generated simple Goldie $k$-algebra of quadratic growth is noetherian and has Krull dimension 1. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if A is a finitely generated simple domain of quadratic growth then A is noetherian and by a result of Stafford every right and left ideal is generated by at most two elements. We conclude by posing questions and giving examples in which we consider what happens when the hypotheses are relaxed."}
{"category": "Math", "title": "A Counterexample to a conjecture of Bosio and Meersseman", "abstract": "In a paper of Bosio and Meersseman (Real quadrics in Cn, complex manifolds and convex polytopes) the following is conjectured: If P is dual neighborly, then Zp is diffeomorphic to the connected sum of products of spheres. In this paper a counterexample is provided."}
{"category": "Math", "title": "Rotation set for maps of degree 1 on the graph sigma", "abstract": "For a continuous map on a topological graph containing a unique loop S it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and for every rational r in this interval there exists a periodic point of rotation number r. The whole rotation set (i.e. the set of all rotation numbers) may not be connected and it is not known in general whether it is closed. The graph sigma is the space consisting in an interval attached by one of its endpoints to a circle. We show that, for a map of degree 1 on the graph sigma, the rotation set is closed and has finitely many connected components. Moreover, for all rational numbers r in the rotation set, there exists a periodic point of rotation number r."}
{"category": "Math", "title": "Serre's Condition R_l for Affine Semigroup Rings", "abstract": "In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_l of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of nonnormal affine semigroup rings that satisfy R_2."}
{"category": "Math", "title": "Lie Algebroids and generalized projective structures on Riemann surfaces", "abstract": "The space of generalized projective structures on a Riemann surface $\\Sigma$ of genus g with n marked points is the affine space over the cotangent bundle to the space of SL(N)-opers. It is a phase space of $W_N$-gravity on $\\Sigma\\times\\mathbb{R}$. This space is a generalization of the space of projective structures on the Riemann surface. We define the moduli space of $W_N$-gravity as a symplectic quotient with respect to the canonical action of a special class of Lie algebroids. This moduli space describes in particular the moduli space of deformations of complex structures on the Riemann surface by differential operators of finite order, or equivalently, by a quotient space of Volterra operators. We call these algebroids the Adler-Gelfand-Dikii (AGD) algebroids, because they are constructed by means of AGD bivector on the space of opers restricted on a circle. The AGD-algebroids are particular case of Lie algebroids related to a Poisson sigma-model. The moduli space of the generalized projective structure can be described by cohomology of a BRST-complex."}
{"category": "Math", "title": "The well-ordering of dual braid monoids", "abstract": "We describe the restriction of the Dehornoy ordering of braids to the dual braid monoids introduced by Birman, Ko and Lee: we give an inductive characterization of the ordering of the dual braid monoids and compute the corresponding ordinal type. The proof consists in introducing a new ordering on the dual braid monoid using the rotating normal form of arXiv:0811.3902 [math.GR], and then proving that this new ordering coincides with the standard ordering of braids."}
{"category": "Math", "title": "On the convergence to the multiple Wiener-Ito integral", "abstract": "We study the convergence to the multiple Wiener-It\\^{o} integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in $\\mathcal C_0([0,T])$. Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-It\\^{o} integral process of a function $f\\in L^2([0,T]^n)$. We prove also the weak convergence in the space $\\mathcal C_0([0,T])$ to the second order integral for two important families of processes that converge to a standard Brownian motion."}
{"category": "Math", "title": "Invertible harmonic mappings, beyond Kneser", "abstract": "We prove necessary and sufficient criteria of invertibility for planar harmonic mappings which generalize a classical result of H. Kneser, also known as the Rad\\'{o}-Kneser-Choquet theorem."}
{"category": "Math", "title": "Fermat's Four Squares Theorem", "abstract": "It is easy to find a right-angled triangle with integer sides whose area is 6. There is no such triangle with area 5, but there is one with rational sides (a `\\emph{Pythagorean triangle}'). For historical reasons, integers such as 6 or 5 that are (the squarefree part of) the area of some Pythagorean triangle are called `\\emph{congruent numbers}'. These numbers actually are interesting for the following reason: Notice the sequence $\\frac14$, $6\\frac14$, $12\\frac14$. It is an arithmetic progression with common difference 6, consisting of squares $(\\frac12)^2$, $(\\frac52)^2$, $(\\frac72)^2$ of rational numbers. Indeed the common difference of three rational squares in AP is a congruent number and every congruent number is the common difference of three rational squares in arithmetic progression. The triangle given by $9^{2}+40^{2}=41^{2}$ has area $180=5\\cdot6^{2}$ and the numbers $x-5$, $x$ and $x+5$ all are rational squares if $x=11{97/144}$. Recall one obtains all Pythagorean triangles with relatively prime integer sides by taking $x=4uv$, $y=\\pm(4u^{2}-v^{2})$, $z=4u^{2}+v^{2}$ where $u$ and $v$ are integers with $2u$ and $v$ relatively prime. Fermat proved that there is no AP of more than three squares of rationals."}
{"category": "Math", "title": "Topics in Special Functions", "abstract": "The authors survey recent results in special functions, particularly the gamma function and the Gaussian hypergeometric function."}
{"category": "Math", "title": "Classification of $k$-tangle projections using cascade representation", "abstract": "The paper addresses the $k$-tangle enumeration problem. We introduce a notion of cascade diagram for $k$-tangle projections. An effective enumeration algorithm for projections is proposed based on cascade representation. Tangles projections with up to 12 crossings are tabulated. We provide also pictures of alternating $k$-tangles with 5 crossing or less."}
{"category": "Math", "title": "Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry", "abstract": "A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of $G$-invariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperkahler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any $G$-invariant HKT-metric on a nilmanifold is balanced with respect to all associated complex structures."}
{"category": "Math", "title": "Jordan-Holder theorem for imprimitivity systems and maximal decompositions of rational functions", "abstract": "In this paper we prove several results about the lattice of imprimitivity systems of a permutation group containing a cyclic subgroup with at most two orbits. As an application we generalize the first Ritt theorem about functional decompositions of polynomials, and some other related results. Besides, we discuss examples of rational functions, related to finite subgroups of the automorphism group of the sphere for which the first Ritt theorem fails to be true."}
{"category": "Math", "title": "On the uniqueness of elliptic K3 surfaces with maximal singular fibre", "abstract": "We explicitly determine the elliptic K3 surfaces with a maximal singular fibre. If the characteristic of the ground field is different from 2, for each of the two possible maximal fibre types, $I_{19}$ and $I^*_{14}$, the surface is unique. In characteristic 2 the maximal fibre types are $I_{18}$ and $I^*_{13}$, and there exist two (resp. one) one-parameter families of such surfaces."}
{"category": "Math", "title": "A Unified Approach to Local Cohomology Modules Using Serre Classes", "abstract": "This paper discusses the connection between the local cohomology modules and the Serre classes of $R$-modules. Such connection provided a common language for expressing some results about the local cohomology $R$-modules, that has appeared in different papers."}
{"category": "Math", "title": "The Dehn function of Stallings' group", "abstract": "We prove that the Dehn function of a group of Stallings that is finitely presented but not of type F_3 is quadratic. To appear in Geometric and Functional Analysis."}
{"category": "Math", "title": "Trisecant Lemma for Non Equidimensional Varieties", "abstract": "The classic trisecant lemma states that if $X$ is an integral curve of $\\PP^3$ then the variety of trisecants has dimension one, unless the curve is planar and has degree at least 3, in which case the variety of trisecants has dimension 2. In this paper, our purpose is first to present another derivation of this result and then to introduce a generalization to non-equidimensional varities. For the sake of clarity, we shall reformulate our first problem as follows. Let $Z$ be an equidimensional variety (maybe singular and/or reducible) of dimension $n$, other than a linear space, embedded into $\\PP^r$, $r \\geq n+1$. The variety of trisecant lines of $Z$, say $V_{1,3}(Z)$, has dimension strictly less than $2n$, unless $Z$ is included in a $(n+1)-$dimensional linear space and has degree at least 3, in which case $\\dim(V_{1,3}(Z)) = 2n$. Then we inquire the more general case, where $Z$ is not required to be equidimensional. In that case, let $Z$ be a possibly singular variety of dimension $n$, that may be neither irreducible nor equidimensional, embedded into $\\PP^r$, where $r \\geq n+1$, and $Y$ a proper subvariety of dimension $k \\geq 1$. Consider now $S$ being a component of maximal dimension of the closure of $\\{l \\in \\G(1,r) \\vtl \\exists p \\in Y, q_1, q_2 \\in Z \\backslash Y, q_1,q_2,p \\in l\\}$. We show that $S$ has dimension strictly less than $n+k$, unless the union of lines in $S$ has dimension $n+1$, in which case $dim(S) = n+k$. In the latter case, if the dimension of the space is stricly greater then $n+1$, the union of lines in $S$ cannot cover the whole space. This is the main result of our work. We also introduce some examples showing than our bound is strict."}
{"category": "Math", "title": "Further results on the Craig-Sakamoto equation", "abstract": "In this paper necessary and sufficient conditions are stated for the Craig-Sakamoto equation det(I-sA-tB) = det(I-sA)det(I-tB), for all scalars s, t. Moreover, spectral properties for A and B are investigated."}
{"category": "Math", "title": "Arithmetic progressions in sets of fractional dimension", "abstract": "Let $E\\subset\\rr$ be a closed set of Hausdorff dimension $\\alpha$. We prove that if $\\alpha$ is sufficiently close to 1, and if $E$ supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then $E$ contains non-trivial 3-term arithmetic progressions."}
{"category": "Math", "title": "The generalized Levinger transformation", "abstract": "In this paper, we present new results relating the numerical range of a matrix $A$ with generalized Levinger transformation $\\mathcal{L}(A,\\alpha,\\beta) = \\alphaH_A +\\betaS_A$, where $H_A$ and $S_A$, are respectively the Hermitian and skew-hermitian parts of $A$. Using these results, we derive expressions for eigenvalues and eigenvectors of the perturbed matrix $A + \\mathcal{L}(E,\\alpha,\\beta)$, for a fixed matrix $E$ and $\\alpha, \\beta$ are real parameters."}
{"category": "Math", "title": "Calculation of UNil for the cyclic group of order two", "abstract": "Cappell's unitary nilpotent groups UNil(R;R,R) are calculated for the integral group ring R=Z[C_2] of the cyclic group C_2 of order two. Specifically, they are determined as modules over the Verschiebung algebra V using the Connolly--Ranicki isomorphism and the Connolly--Davis relations."}
{"category": "Math", "title": "Ultra-discretization of the G^(1)_2-Geometric Crystals to the D^(3)_4-Perfect Crystals", "abstract": "We obtain the affirmative answer to the conjecture in [15]. More precisely, let X be the affine geometric crystal of type G^(1)_2 in [15] and UD(X,T,\\theta) a ultra-discretization of X with respect to a certain positive structure \\theta. Then we show that UD(X,T,\\theta) is isomorphic to the limit of coherent family of perfect crystals of type D^(3)_4 in [7]."}
{"category": "Math", "title": "An Enumeration of Graphical Designs", "abstract": "Let $\\Psi(t,k)$ denote the set of pairs $(v,\\lambda)$ for which there exists a graphical $t$-$(v,k,\\lambda)$ design. Most results on graphical designs have gone to show the finiteness of $\\Psi(t,k)$ when $t$ and $k$ satisfy certain conditions. The exact determination of $\\Psi(t,k)$ for specified $t$ and $k$ is a hard problem and only $\\Psi(2,3)$, $\\Psi(2,4)$, $\\Psi(3,4)$, $\\Psi(4,5)$, and $\\Psi(5,6)$ have been determined. In this paper, we determine completely the sets $\\Psi(2,5)$ and $\\Psi(3,5)$. As a result, we find more than 270000 inequivalent graphical designs, and more than 8000 new parameter sets for which there exists a graphical design. Prior to this, graphical designs are known for only 574 parameter sets."}
{"category": "Math", "title": "Addition Theorems Via Continued Fractions", "abstract": "We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for Bessel functions and confluent hypergeometric functions. We also derive several additions theorems for basic hypergeometric functions. Applications to the evaluation of Hankel determinants are also given ."}
{"category": "Math", "title": "Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere", "abstract": "We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for general spectrally defined operators on the space of CR-pluriharmonic functions. We will then obtain the sharp Beckner-Onofri inequality for CR-pluriharmonic functions on the sphere, and, as a consequence, a sharp logarithmic Hardy-Littlewood-Sobolev inequality in the form given by Carlen and Loss."}
{"category": "Math", "title": "Stochastic integration based on simple, symmetric random walks", "abstract": "A new approach to stochastic integration is described, which is based on an a.s. pathwise approximation of the integrator by simple, symmetric random walks. Hopefully, this method is didactically more advantageous, more transparent, and technically less demanding than other existing ones. In a large part of the theory one has a.s. uniform convergence on compacts. In particular, it gives a.s. convergence for the stochastic integral of a finite variation function of the integrator, which is not c\\`adl\\`ag in general."}
{"category": "Math", "title": "Lusztig's conjecture as a moment graph problem", "abstract": "We show that Lusztig's conjecture on the irreducible characters of a reductive algebraic group over a field of positive characteristic is equivalent to the generic multiplicity conjecture, which gives a formula for the Jordan-H\"older multiplicities of baby Verma modules over the corresponding Lie algebra. Then we give a short overview of a recent proof of the latter conjecture for almost all base fields via the theory of sheaves on moment graphs."}
{"category": "Math", "title": "N-systems, class polynomials for double eta-quotients and singular values of J-invariant function", "abstract": "Enge and Schertz gave the method of using the double eta-quotient for the construction of elliptic curves over finite fields. In their method, it is necessary to count the number of rational points of elliptic curves corresponding to solutions of the modular equation over a finite field, because in advance we can not know which solution of the modular equation is that corresponding to the modular invariant. We give a condition that the modular invariant is a multiple root of the modular polynomial. Consequently, we give a method to reduce the amount of computation in the process of counting the number of rational points."}
{"category": "Math", "title": "A note on Stein fillings of contact manifolds", "abstract": "In this note we construct infinitely many distinct simply connected Stein fillings of a certain infinite family of contact 3--manifolds."}
{"category": "Math", "title": "Tangential symmetries of Darboux integrable systems", "abstract": "In this paper we analyze the tangential symmetries of Darboux integrable decomposable exterior differential systems. The decomposable systems generalize the notion of a hyperbolic exterior differential system and include the classic notion of Darboux integrability for first order systems and second order scalar equations. For Darboux integrable systems the general solution can be found by integration (solving ordinary differential equations). We show that this property holds for our generalized systems as well. We give a geometric construction of the Lie algebras of tangential symmetries associated to the Darboux integrable systems. This construction has the advantage over previous constructions that our construction does not require the use of adapted coordinates and works for arbitrary dimension of the underlying manifold. In particular it works for the prolongations of decomposable exterior differential systems."}
{"category": "Math", "title": "Short Pulses Approximations in Dispersive Media", "abstract": "We derive various approximations for the solutions of nonlinear hyperbolic systems with fastly oscillating initial data. We first provide error estimates for the so-called slowly varying envelope, full dispersion, and Schr\\\"odinger approximations in a Wiener algebra; this functional framework allows us to give precise conditions on the validity of these models; we give in particular a rigorous proof of the ``practical rule'' which serves as a criterion for the use of the slowly varying envelope approximation (SVEA). We also discuss the extension of these models to short pulses and more generally to large spectrum waves, such as chirped pulses. We then derive and justify rigorously a modified Schr\\\"odinger equation with improved frequency dispersion. Numerical computations are then presented, which confirm the theoretical predictions."}
{"category": "Math", "title": "A combinatorial interpretation for the identity Sum_{k=0}^{n} binom{n}{k} Sum_{j=0}^{k} binom{k}{j}^{3}= Sum_{k=0}^{n} binom{n}{k}^{2}binom{2k}{k}", "abstract": "The title identity appeared as Problem 75-4, proposed by P. Barrucand, in Siam Review in 1975. The published solution equated constant terms in a suitable polynomial identity. Here we give a combinatorial interpretation in terms of card deals."}
{"category": "Math", "title": "A characterization of regular tetrahedra in Z^3", "abstract": "In this note we characterize all regular tetrahedra whose vertices in R^3 have integer coordinates. The main result is a consequence of the characterization of all equilateral triangles having integer coordinates contained in previous work. Then we use this characterization to point out some corollaries. The number of such tetrahedra whose vertices are in the finite set {0,1,2,...,n}^3, n in N, is related to the sequence A103158 in the Online Encyclopedia of Integer Sequences."}
{"category": "Math", "title": "Noisy heteroclinic networks", "abstract": "We consider a white noise perturbation of dynamics in the neighborhood of a heteroclinic network. We show that under the logarithmic time rescaling the diffusion converges in distributon in a special topology to a piecewise constant process that jumps between saddle points along the heteroclinic orbits of the network. We also obtain precise asymptotics for the exit measure for a domain containing the starting point of the diffusion."}
{"category": "Math", "title": "On rime Ansatz", "abstract": "The ice Ansatz on matrix solutions of the Yang-Baxter equation is weakened to a condition which we call rime. Generic rime solutions of the Yang-Baxter equation are described. We prove that the rime non-unitary (respectively, unitary) R-matrix is equivalent to the Cremmer-Gervais (respectively, boundary Cremmer-Gervais) solution. Generic rime classical r-matices satisfy the (non-)homogeneous associative classical Yang-Baxter equation."}
{"category": "Math", "title": "Cyclic systems of simultaneous congruences", "abstract": "This paper considers solutions (x_1, x_2, ..., x_n) to the cyclic system of n simultaneous congruences r (x_1x_2 ...x_n)/x_i = s (mod |x_i|), for fixed nonzero integers r,s with r>0 and gcd(r,s)=1. It shows this system has a finite number of solutions in positive integers x_i >1 having gcd(x_1x_2...x_n, s)=1, obtaining a sharp upper bound on the maximal size of the solutions in many cases. This bound grows doubly-exponentially in n. It shows there are infinitely many such solutions when the positivity restriction is dropped, when r=1, and not otherwise. The problem is reduced to the study of integer solutions of a three parameter family of Diophantine equations r(1/x_1 + 1/x_2 + ...+ 1/x_n)- s/(x_1x_2...x_n) = m, with parameters (r,s,m)."}
{"category": "Math", "title": "On a Theorem of Sewell and Trotter", "abstract": "Sewell and Trotter [J. Combin. Theory Ser. B, 1993] proved that every connected alpha-critical graph that is not isomorphic to K_1, K_2 or an odd cycle contains a totally odd K_4-subdivision. Their theorem implies an interesting min-max relation for stable sets in graphs without totally odd K_4-subdivisions. In this note, we give a simpler proof of Sewell and Trotter's theorem."}
{"category": "Math", "title": "Characterization of the critical magnetic field in the Dirac-Coulomb equation", "abstract": "We consider a relativistic hydrogenic atom in a strong magnetic field. The ground state level depends on the strength of the magnetic field and reaches the lower end of the spectral gap of the Dirac-Coulomb operator for a certain critical value, the critical magnetic field. We also define a critical magnetic field in a Landau level ansatz. In both cases, when the charge Z of the nucleus is not too small, these critical magnetic fields are huge when measured in Tesla, but not so big when the equation is written in dimensionless form. When computed in the Landau level ansatz, orders of magnitude of the critical field are correct, as well as the dependence in Z. The computed value is however significantly too big for a large Z, and the wave function is not well approximated. Hence, accurate numerical computations involving the Dirac equation cannot systematically rely on the Landau level ansatz. Our approach is based on a scaling property. The critical magnetic field is characterized in terms of an equivalent eigenvalue problem. This is our main analytical result, and also the starting point of our numerical scheme."}
{"category": "Math", "title": "Multispace and Multilevel BDDC", "abstract": "BDDC method is the most advanced method from the Balancing family of iterative substructuring methods for the solution of large systems of linear algebraic equations arising from discretization of elliptic boundary value problems. In the case of many substructures, solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC has the same structure as the original problem, it is straightforward to apply the BDDC method recursively to solve the coarse problem only approximately. In this paper, we formulate a new family of abstract Multispace BDDC methods and give condition number bounds from the abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as a special case of the Multispace BDDC and abstract multilevel condition number bounds are given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary fixed number of levels and scalar elliptic problems discretized by finite elements in two and three spatial dimensions. Numerical experiments confirm the theory."}
{"category": "Math", "title": "On Dynamics of $\\ell$- Volterra Quadratic Stochastic Operators", "abstract": "We introduce a notion of $\\ell$-Volterra quadratic stochastic operator defined on $(m-1)$-dimensional simplex, where $\\ell\\in\\{0,1,...,m\\}$. The $\\ell$-Volterra operator is a Volterra operator iff $\\ell=m$. We study structure of the set of all $\\ell$-Volterra operators and describe their several fixed and periodic points. For $m=2$ and 3 we describe behavior of trajectories of $(m-1)$-Volterra operators. The paper also contains many remarks with comparisons of $\\ell$-Volterra operators and Volterra ones."}
{"category": "Math", "title": "Functional equations of the dilogarithm in motivic cohomology", "abstract": "We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH^2(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generating the integral motivic cohomology groups of some number fields."}
{"category": "Math", "title": "On bases of tropical Pl\\\"ucker functions", "abstract": "We consider functions $f:B\\to\\Rset$ that obey tropical analogs of classical Pl\\\"ucker relations on minors of a matrix. The most general set $B$ that we deal with in this paper is of the form $\\{x\\in \\Zset^n\\colon 0\\le x\\le a, m\\le x_1+...+x_n\\le m'\\}$ (a rectangular integer box ``truncated from below and above''). We construct a basis for the set $\\Tscr$ of tropical Pl\\\"ucker functions on $B$, a subset $\\Bscr\\subseteq B$ such that the restriction map $\\Tscr\\to\\Rset^\\Bscr$ is bijective. Also we characterize, in terms of the restriction to the basis, the classes of submodular, so-called skew-submodular, and discrete concave functions in $\\Tscr$, discuss a tropical analogue of the Laurentness property, and present other results."}
{"category": "Math", "title": "$\\Lambda$-adic modular symbols and several variable $p$-adic L-functions over totally real fields", "abstract": "We interpolate cohomology classes attached to families of Hilbert modular forms. Using this we construct a two variable $p$-adic L-function which interpolates one variable $p$-adic L-functions."}
{"category": "Math", "title": "On the irrationality of Ramanujan's mock theta functions and other q-series at an infinite number of points", "abstract": "We show that all of Ramanujan's mock theta functions of order 3, Watson's three additional mock theta functions of order 3, the Rogers-Ramanujan q-series, and 6 mock theta functions of order 5 take on irrational values at the points q=\\pm 1/2,\\pm 1/3,\\pm 1/4,..."}
{"category": "Math", "title": "On growth and torsion of groups", "abstract": "We give a subexponential upper bound and a superpolynomial lower bound on the growth function of the Fabrykowski-Gupta group. As a consequence, we answer negatively a question by Longobardi, Maj and Rhemtulla about characterizing groups containing no free subsemigroups on two generators."}
{"category": "Math", "title": "Small permutation classes", "abstract": "We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number $\\kappa$, approximately 2.20557, for which there are only countably many permutation classes of growth rate (Stanley-Wilf limit) less than $\\kappa$ but uncountably many permutation classes of growth rate $\\kappa$, answering a question of Klazar. We go on to completely characterize the possible sub-$\\kappa$ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced herein), partial well-order, and atomicity (also known as the joint embedding property)."}
{"category": "Math", "title": "A two-page disproof of the Borsuk partition conjecture", "abstract": "It is presented the simplest known disproof of the Borsuk conjecture stating that if a bounded subset of n-dimensional Euclidean space contains more than n points, then the subset can be partitioned into n+1 nonempty parts of smaller diameter. The argument is due to N. Alon and is a remarkable application of combinatorics and algebra to geometry. This note is purely expository and is accessible for students."}
{"category": "Math", "title": "Theta-functions on the Kodaira-Thurston manifold", "abstract": "The Kodaira--Thurston M manifold is a compact, 4-dimensional nilmanifold which is symplectic and complex but not Kaehler. We describe a construction of theta-functions associated to M which parallels the classical theory of theta-functions associated to the torus (from the point of view of representation theory and geometry), and yields pseudoperiodic complex-valued functions on R^4. There exists a three-step nilpotent Lie group G which acts transitively on the Kodaira--Thurston manifold M in a Hamiltonian fashion. The theta-functions discussed in this paper are intimately related to the representation theory of G in much the same way the classical theta-functions are related to the Heisenberg group. One aspect of our results which has not appeared in the classical theory is a connection between the representation theory of G and the existence of Lagrangian and special Lagrangian foliations and torus fibrations in M."}
{"category": "Math", "title": "Bregman distances and Chebyshev sets", "abstract": "A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given."}
{"category": "Math", "title": "Counting Labelled Trees with Given Indegree Sequence", "abstract": "For a labelled tree on the vertex set $[n]:=\\{1,2,..., n\\}$, define the direction of each edge $ij$ to be $i\\to j$ if $i<j$. The indegree sequence of $T$ can be considered as a partition $\\lambda \\vdash n-1$. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on $[n]$ with indegree sequence corresponding to a partition $\\lambda$. In this paper we give two proofs of Cotterill's conjecture: one is `semi-combinatorial\" based on induction, the other is a bijective proof."}
{"category": "Math", "title": "Plurisubharmonic functions in calibrated geometry and q-convexity", "abstract": "Let $(M,\\omega)$ be a Kahler manifold. An integrable function on M is called $\\omega^q$-plurisubharmonic if it is subharmonic on all q-dimensional complex subvarieties. We prove that a smooth $\\omega^q$-plurisubharmonic function is q-convex. A continuous $\\omega^q$-plurisubharmonic function admits a local approximation by smooth, $\\omega^q$-plurisubharmonic functions. For any closed subvariety $Z\\subset M$, $\\dim Z < q$, there exists a strictly $\\omega^q$-plurisubharmonic function in a neighbourhood of $Z$ (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony's lemma on integrability of positive closed (p,p)-forms which are integrable outside of a complex subvariety of codimension >p."}
{"category": "Math", "title": "Valuation bases for generalized algebraic series fields", "abstract": "We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed, or algebraically closed field F, we give a sufficient condition for a valued subfield of the field of generalized power series F((G)) to admit a K-valuation basis. We show that the field of rational functions F(G) and the field F(G) of power series in F((G)) algebraic over F(G) satisfy this condition. It follows that for archimedean F and divisible G the real closed field F(G) admits a restricted exponential function."}
{"category": "Math", "title": "On one polynomial $p$-adic dynamical system", "abstract": "In the paper we describe basin of attraction and the Siegel discs of the $p$-adic dynamical system $f(x)=x^{2n+1}+ax^{n+1}$ over complex $p$-adic field."}
{"category": "Math", "title": "Representations of $GL_2(\\Fq)$ and $SL_2(\\Fq)$, and some remarks about $GL_n(\\Fq)$", "abstract": "The goal of these notes is to give a self-contained account of the representation theory of $GL_2$ and $SL_2$ over a finite field, and to give some indication of how the theory works for $GL_n$ over a finite field."}
{"category": "Math", "title": "A class function on the mapping class group of an orientable surface and the Meyer cocycle", "abstract": "In this paper we define a $\\mathbf{QP}^1$-valued class function on the mapping class group $\\mathcal{M}_{g,2}$ of a surface $\\Sigma_{g,2}$ of genus $g$ with two boundary components. Let $E$ be a $\\Sigma_{g,2}$ bundle over a pair of pants $P$. Gluing to $E$ the product of an annulus and $P$ along the boundaries of each fiber, we obtain a closed surface bundle over $P$. We have another closed surface bundle by gluing to $E$ the product of $P$ and two disks. The sign of our class function cobounds the 2-cocycle on $\\mathcal{M}_{g,2}$ defined by the difference of the signature of these two surface bundles over $P$."}
{"category": "Math", "title": "Algebraic tori - thirty years after", "abstract": "This is an expanded version of my talk given at the International Conference ``Algebra and Number Theory'' dedicated to the 80th anniversary of V. E. Voskresenskii, which was held at the Samara State University in May 2007. The goal is to give an overview of results of V. E. Voskresenskii on arithmetic and birational properties of algebraic tori which culminated in his monograph \"Algebraic Tori\" published in Russian 30 years ago. I shall try to put these results and ideas into somehow broader context and also to give a brief digest of the relevant activity related to the period after the English version of the monograph \"Algebraic Groups and Their Birational Invariants\" appeared."}
{"category": "Math", "title": "The Bogomolov multiplier of finite simple groups", "abstract": "The subgroup of the Schur multiplier of a finite group G consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero is called the Bogomolov multiplier of G. We prove that if G is quasisimple or almost simple, its Bogomolov multiplier is trivial except for the case of certain covers of PSL(3,4)."}
{"category": "Math", "title": "Order one invariants of planar curves", "abstract": "We give a complete description of all order 1 invariants of planar curves."}
{"category": "Math", "title": "Bilinear virial identities and applications", "abstract": "We prove bilinear virial identities for the nonlinear Schrodinger equation, which are extensions of the Morawetz interaction inequalities. We recover and extend known bilinear improvements to Strichartz inequalities and provide applications to various nonlinear problems, most notably on domains with boundaries."}
{"category": "Math", "title": "On the difference of partial theta functions", "abstract": "Sums of the form add((-1)^n q^(n(n-1)/2) x^n, n>=0) are called partial theta functions. In his lost notebook, Ramanujan recorded many identities for those functions. In 2003, Warnaar found an elegant formula for a sum of two partial theta functions. Subsequently, Andrews and Warnaar established a similar result for the product of two partial theta functions. In this note, I discuss the relation between the Andrews-Warnaar identity and the (1986) product formula due to Gasper and Rahman. I employ nonterminating extension of Sears-Carlitz transformation for 3\\phi_2 to provide a new elegant proof for a companion identity for the difference of two partial theta series. This difference formula first appeared in the work of Schilling-Warnaar (2002). Finally, I show that Schilling-Warnnar (2002) and Warnaar (2003) formulas are, in fact, equivalent."}
{"category": "Math", "title": "On the role of Convexity in Isoperimetry, Spectral-Gap and Concentration", "abstract": "We show that for convex domains in Euclidean space, Cheeger's isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the a-priori weakest requirement that Lipschitz functions have \\emph{arbitrarily slow} uniform tail-decay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz'ya, Cheeger, Gromov--Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are ``on-average'' Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the ``worst'' subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne--Weinberger, Li--Yau, Kannan--Lov\\'asz--Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semi-group following Bakry--Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the $CD(0,\\infty)$ curvature-dimension condition of Bakry-\\'Emery."}
{"category": "Math", "title": "Twisted planes", "abstract": "Let k be a commutative ring. We find and characterize a new family of twisted planes (i. e. associative unitary k-algebra structures on the k-module k[X,Y], having k[X] and and k[Y] as subalgebras).Similar results are obtained for the k-module of two variables power series k[[X,Y]]."}
{"category": "Math", "title": "Signed-eliminable graphs and free multiplicities on the braid arrangement", "abstract": "We define specific multiplicities on the braid arrangement by using edge-bicolored graphs. To consider their freeness, we introduce the notion of bicolor-eliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of the deformation of the braid arrangement in terms of directed graphs."}
{"category": "Math", "title": "Image of Schwartz Space Under Spectral Projection", "abstract": "Let $X= G/K$ symmetric space of non compact type, where $G$ is a rank-one connected semisimple Lie group with finite center. We shall look at the transform $ P_\\lambda f(x) = f \\ast \\varphi_\\lambda(x)$, where, $\\lambda \\in \\mathbb C$ and $\\varphi_\\lambda$ is the elementary spherical function. We shall try to characterizes the image of the Schwartz spaces $S^p(X) $ where $0 < p \\leq 2$ under the above transform."}
{"category": "Math", "title": "A Characterization of Signed Graphs with Generalized Perfect Elimination Orderings", "abstract": "An important property of chordal graphs is that these graphs are characterized by existence of perfect elimination orderings on their vertex sets. In this paper, we generalize the notion of perfect elimination orderings to signed graphs, and give a characterization for graphs admitting such orderings, together with characterizations restricted to some subclasses and further properties of those graphs."}
{"category": "Math", "title": "Long-Run Accuracy of Variational Integrators in the Stochastic Context", "abstract": "This paper presents a Lie-Trotter splitting for inertial Langevin equations (Geometric Langevin Algorithm) and analyzes its long-time statistical properties. The splitting is defined as a composition of a variational integrator with an Ornstein-Uhlenbeck flow. Assuming the exact solution and the splitting are geometrically ergodic, the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial Langevin to within the accuracy of the variational integrator in representing the Hamiltonian. In particular, if the variational integrator admits no energy error, then the method samples the invariant measure of inertial Langevin without error. Numerical validation is provided using explicit variational integrators with first, second, and fourth order accuracy."}
{"category": "Math", "title": "On irreducible algebras of conformal endomorphisms over a linear algebraic group", "abstract": "We study the algebra of conformal endomorphisms $\\Cend^{G,G}_n$ of a finitely generated free module $M_n$ over the coordinate Hopf algebra $H$ of a linear algebraic group $G$. It is shown that a conformal subalgebra of $\\Cend_n$ acting irreducibly on $M_n$ generates an essential left ideal of $\\Cend^{G,G}_n$ if enriched with operators of multiplication on elements of $H$. In particular, we describe such subalgebras for the case when $G$ is finite."}
{"category": "Math", "title": "On cluster algebras arising from unpunctured surfaces", "abstract": "We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. In the special case where the cluster algebra is acyclic, we also give a formula for the expansion of cluster variables as a polynomial whose indeterminates are the cluster variables contained in the union of an arbitrary acyclic cluster and all its neighbouring clusters in the mutation graph."}
{"category": "Math", "title": "Orthogonal involution on algebras of degree 16 and the Killing form of E8", "abstract": "We exploit various inclusions of algebraic groups to give a new construction of groups of type E8, determine the Killing forms of the resulting E8's, and define an invariant of central simple algebras of degree 16 with orthogonal involution \"in I^3\", equivalently, groups of type D8 with a half-spin representation defined over the base field. The determination of the Killing form is done by restricting the adjoint representation to various twisted forms of PGL2 and requires very little computation. An appendix by Kirill Zainoulline contains a type of \"index reduction\" result for groups of type D."}
{"category": "Math", "title": "Uniqueness for the martingale problem associated with pure jump processes of variable order", "abstract": "Let $L$ be the operator defined on $C^2$ functions by $$L f(x)=\\int[f(x+h)-f(x)-1_{(|h|\\leq 1)}\\nabla f(x)\\cdot h]\\frac{n(x,h)}{|h|^{d+\\alpha(x)}}dh.$$ This is an operator of variable order and the corresponding process is of pure jump type. We consider the martingale problem associated with $L$. Sufficient conditions for existence and uniqueness are given. Transition density estimates for $\\alpha$-stable processes are also obtained."}
{"category": "Math", "title": "Surfaces in three-dimensional Lie groups in terms of spinors", "abstract": "This is a survey of results on surfaces in noncommutative three-dimensional Lie groups obtained by using the Weierstrass (spinor) representation of surfaces. It is based on the talk given at the conference \"Geometry related to the theory of integrable systems\" (RIMS, Kyoto, September 2007)."}
{"category": "Math", "title": "Analogues of the Jordan-Holder theorem for transitive G-sets", "abstract": "Let G be a transitive group of permutations of a finite set X, and suppose that some element of G has at most two orbits on X. We prove that any two maximal chains of groups between G and a point-stabilizer of G have the same length, and the same sequence of relative indices between consecutive groups (up to permutation). We also deduce the same conclusion when G has a transitive quasi-Hamiltonian subgroup."}
{"category": "Math", "title": "Packing 3-Vertex Paths in 2-Connected Graphs", "abstract": "We give a construction that provides infinitely many 2-connected, cubic, bipartite, and planar graphs G with 3k vertices and such that the number of disjoint copies of a 3-vertex path in G is less than k."}
{"category": "Math", "title": "Invariant measures on the space of horofunctions of a word hyperbolic group", "abstract": "We introduce a natural equivalence relation on the space $\\sH_0$ of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen-Marcus theorem. Furthermore, if $\\eta$ is such a measure and $G$ acts on a space $(X,\\mu)$ by p.m.p. transformations then $\\eta \\times \\mu$ is virtually ergodic with respect to a natural equivalence relation on $\\sH_0\\times X$. This is comparable to a special case of the Howe-Moore theorem. These results are applied to prove a new ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space."}
{"category": "Math", "title": "Quiver varieties and Beilinson-Drinfeld Grassmannians of type A", "abstract": "We construct Nakajima's quiver varieties of type A in terms of conjugacy classes of matrices and (non-Slodowy's) transverse slices naturally arising from affine Grassmannians. In full generality quiver varieties are embedded into Beilinson-Drinfeld Grassmannians of type A. Our construction provides a compactification of Nakajima's quiver varieties and a decomposition of an affine Grassmannian into a disjoint union of quiver varieties. As an application we provide a geometric version of skew and symmetric $(GL(m), GL(n))$ duality."}
{"category": "Math", "title": "On the Empirical Importance of the Conditional Skewness Assumption in Modelling the Relationship Between Risk and Return", "abstract": "The main goal of this paper is an application of Bayesian inference in testing the relation between risk and return on the financial instruments. On the basis of the Intertemporal CAPM model we built a general sampling model suitable in analysing such a relationship. The most important feature of our assumptions is that the skewness of the conditional distribution of returns is used as an alternative source of relation between risk and return. This general specification relates to GARCH-In-Mean model. In order to make conditional distribution of financial returns skewed we considered a constructive approach based on the inverse probability integral transformation. In particular, we apply the hidden truncation mechanism, two equivalent approaches of the inverse scale factors, order statistics concept, Beta and Bernstein distribution transformations, and also the constructive method. Based on the daily excess returns on the Warsaw Stock Exchange Index we checked the empirical importance of the conditional skewness assumption on the relation between risk and return on the Warsaw Stock Market. We present posterior probabilities of all competing specifications as well as the posterior analysis of positive sign of the tested relationship."}
{"category": "Math", "title": "Simulation of the matrix Bingham-von Mises-Fisher distribution, with applications to multivariate and relational data", "abstract": "Orthonormal matrices play an important role in reduced-rank matrix approximations and the analysis of matrix-valued data. A matrix Bingham-von Mises-Fisher distribution is a probability distribution on the set of orthonormal matrices that includes linear and quadratic terms, and arises as a posterior distribution in latent factor models for multivariate and relational data. This article describes rejection and Gibbs sampling algorithms for sampling from this family of distributions, and illustrates their use in the analysis of a protein-protein interaction network."}
{"category": "Math", "title": "Appell polynomials and their relatives II. Boolean theory", "abstract": "The Appell-type polynomial family corresponding to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolvent-type form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, in the Boolean and free theory, coincide. This is true even in the multivariate case. A number of applications of this fact are described, to the Belinschi-Nica and Bercovici-Pata maps, conditional freeness, and the Laha-Lukacs type characterization. A number of properties which hold for the Meixner class in the free and classical cases turn out to hold in general in the Boolean theory. Examples include the behavior of the Jacobi coefficients under convolution, the relationship between the Jacobi coefficients and cumulants, and an operator model for cumulants. Along the way, we obtain a multivariate version of the Stieltjes continued fraction expansion for the moment generating function of an arbitrary state with monic orthogonal polynomials."}
{"category": "Math", "title": "Edgeworth expansions in operator form", "abstract": "An operator form of asymptotic expansions for Markov chains is established. Coefficients are given explicitly. Such expansions require a certain modification of the classical spectral method. They prove to be extremely useful within the context of large deviations."}
{"category": "Math", "title": "Martingale proofs of many-server heavy-traffic limits for Markovian queues", "abstract": "This is an expository review paper illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations. Careful treatment is given to an elementary model -- the classical infinite-server model $M/M/\\infty$, but models with finitely many servers and customer abandonment are also treated. The Markovian stochastic process representing the number of customers in the system is constructed in terms of rate-1 Poisson processes in two ways: (i) through random time changes and (ii) through random thinnings. Associated martingale representations are obtained for these constructions by applying, respectively: (i) optional stopping theorems where the random time changes are the stopping times and (ii) the integration theorem associated with random thinning of a counting process. Convergence to the diffusion process limit for the appropriate sequence of scaled queueing processes is obtained by applying the continuous mapping theorem. A key FCLT and a key FWLLN in this framework are established both with and without applying martingales."}
{"category": "Math", "title": "A note on spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients", "abstract": "In this note, by constructing suitable approximate solutions, we prove the existence of global weak solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients in the whole space $\\mathbb{R}^N$, $N\\geq2$ (or exterior domain), when the initial data are spherically symmetric. In particular, we prove the existence of spherically symmetric solutions to the Saint-Venant model for shallow water in the whole space (or exterior domain)."}
{"category": "Math", "title": "The Cohomology of Transitive Lie Algebroids", "abstract": "For a transitive Lie algebroid A on a connected manifold M and its a representation on a vector bundle F, we study the localization map Y^1: H^1(A,F)-> H^1(L_x,F_x), where L_x is the adjoint algebra at x in M. The main result in this paper is that: Ker Y^1_x=Ker(p^{1*})=H^1_{deR}(M,F_0). Here p^{1*} is the lift of H^1(\\huaA,F) to its counterpart over the universal covering space of M and H^1_{deR}(M,F_0) is the F_0=H^0(L,F)-coefficient deRham cohomology. We apply these results to study the associated vector bundles to principal fiber bundles and the structure of transitive Lie bialgebroids."}
{"category": "Math", "title": "Generalized backscattering and the Lax-Phillips transform", "abstract": "Using the free-space translation representation (modified Radon transform) of Lax and Phillips in odd dimensions, it is shown that the generalized backscattering transform (so outgoing angle $\\omega =S\\theta$ in terms of the incoming angle with $S$ orthogonal and $\\Id-S$ invertible) may be further restricted to give an entire, globally Fredholm, operator on appropriate Sobolev spaces of potentials with compact support. As a corollary we show that the modified backscattering map is a local isomorphism near elements of a generic set of potentials."}
{"category": "Math", "title": "Virtually free pro-p groups whose torsion elements have finite centralizer", "abstract": "A finitely generated virtually free pro-p group with finite centralizers of its torsion elements is the free pro-p product of finite p-groups and a free pro-p factor."}
{"category": "Math", "title": "The one-dimensional stratum in the boundary of the moduli stack of stable curves", "abstract": "The moduli stack of Deligne-Mumford stable curves of genus g admits a stratification, so that the number of nodes of the curves belonging to one stratum is constant. The irreducible components of the stratum corresponding to curves with exactly 3g-4 nodes are one-dimensional substacks. We show how they can be related to moduli stacks of (permutation classes of) pointed stable curves. Using this, we construct all components of this stratum in a new way as quotient stacks."}
{"category": "Math", "title": "Determinants of elliptic hypergeometric integrals", "abstract": "We start from an interpretation of the $BC_2$-symmetric \"Type I\" (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation, and give an extension to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding elliptic beta integral and transformation formula in a new way, by proving both sides satisfy the same difference equations, and that the difference equations satisfy a Galois-theoretical condition that ensures uniqueness of simultaneous solution."}
{"category": "Math", "title": "Homology of the mapping class group for surfaces of genus 2 with boundary", "abstract": "We report on the computation of the integral homology of the mapping class group of genus g surfaces with one boundary curve and m punctures, when 2g + m is smaller than 6. In particular, it includes the genus 2 case with no or one puncture."}
{"category": "Math", "title": "On 3-decomposable geometric drawings of $K_n$", "abstract": "The point sets of all known optimal rectilinear drawings of $K_n$ share an unmistakeable clustering property, the so--called {\\em 3--decomposability}. It is widely believed that the underlying point sets of all optimal rectilinear drawings of $K_n$ are 3--decomposable. We give a lower bound for the minimum number of $(\\le k)$--sets in a 3--decomposable $n$--point set. As an immediate corollary, we obtain a lower bound for the crossing number $\\rcr(\\dd)$ of any rectilinear drawing $\\dd$ of $K_n$ with underlying 3--decomposable point set, namely $\\rcr(\\dd) > {2/27}(15-\\pi^{2})\\binom{n}{4}+\\Theta(n^{3}) \\approx 0.380029\\binom{n}{4} + \\Theta(n^3)$. This closes this gap between the best known lower and upper bounds for the rectilinear crossing number $\\rcr(K_n)$ of $K_n$ by over 40%, under the assumption of 3--decomposability."}
{"category": "Math", "title": "The skew spectrum of functions on finite groups and their homogeneous spaces", "abstract": "Whenever we have a group acting on a class of functions by translation, the bispectrum offers a principled and lossless way of representing such functions invariant to the action. Unfortunately, computing the bispectrum is often costly and complicated. In this paper we propose a unitarily equivalent, but easier to compute set of invariants, which we call the skew spectrum. For functions on homogeneous spaces the skew spectrum can be efficiently computed using some ideas from Clausen-type fast Fourier transforms."}
{"category": "Math", "title": "A singular perturbation problem for a quasilinear operator satisfying the natural growth condition of Lieberman", "abstract": "In this paper we study the following problem. For any $\\ep>0$, take $u^{\\ep}$ a solution of, $$ \\L u^{\\ep}:= {div}\\Big(\\di\\frac {g(|\\nabla \\uep|)}{|\\nabla \\uep|}\\nabla \\uep\\Big)=\\beta_{\\ep}(u^{\\ep}),\\quad u^{\\ep}\\geq 0. $$ A solution to $(P_{\\ep})$ is a function $u^{\\ep}\\in W^{1,G}(\\Omega)\\cap L^{\\infty}(\\Omega)$ such that $$ \\int_{\\Omega} g(|\\nabla u^{\\ep}|) \\frac{\\nabla u^{\\ep}}{|\\nabla u^{\\ep}|} \\nabla \\phi dx =-\\int_{\\Omega} \\phi \\beta_{\\ep}(u^{\\ep}) dx $$ for every $\\phi \\in C_0^{\\infty}(\\Omega)$. Here $\\beta_{\\ep}(s)= \\frac{1}{\\ep} \\beta(\\frac{s}{\\ep}), $ with $\\beta\\in {Lip}(\\R)$, $\\beta>0$ in $(0,1)$ and $\\beta=0$ otherwise. We are interested in the limiting problem, when $\\ep\\to 0$. As in previous work with $\\L=\\Delta$ or $\\L=\\Delta_p$ we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a $C^{1,\\alpha}$ surface. This result is new even for $\\Delta_p$. Throughout the paper we assume that $g$ satisfies the conditions introduced by G. Lieberman in \\cite{Li1}"}
{"category": "Math", "title": "Hyperbolic dimension and radial Julia sets of transcendental functions", "abstract": "We survey the definition of the radial Julia set of a meromorphic function (in fact, more generally, any \"Ahlfors islands map\"), and give a simple proof that the Hausdorff dimension of the reduced Julia set always coincides with the hyperbolic dimension."}
{"category": "Math", "title": "Excursion sets of stable random fields", "abstract": "Studying the geometry generated by Gaussian and Gaussian- related random fields via their excursion sets is now a well developed and well understood subject. The purely non-Gaussian scenario has, however, not been studied at all. In this paper we look at three classes of stable random fields, and obtain asymptotic formulae for the mean values of various geometric characteristics of their excursion sets over high levels. While the formulae are asymptotic, they contain enough information to show that not only do stable random fields exhibit geometric behaviour very different from that of Gaussian fields, but they also differ significantly among themselves."}
{"category": "Math", "title": "On 1-Harmonic Functions", "abstract": "Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1$-tension field are given with applications in geometry via transformation group theory. In particular, we prove that every level hypersurface of such a subsolution is calibrated and hence is area-minimizing over $\\mathbb{R}$; and every 7-dimensional $SO(2)\\times SO(6)$-invariant absolutely area-minimizing integral current in $\\mathbb{R}^8$ is real analytic. The assumption on the $SO(2) \\times SO(6)$-invariance cannot be removed, due to the first counter-example in $\\mathbb{R}^8$, proved by Bombieri, De Girogi and Giusti."}
{"category": "Math", "title": "Hyperbolic Geometry and Distance Functions on Discrete Groups", "abstract": "Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane induced by the action of $PSL(2,\\mathbb{Z})$. Chapter 3 gives background on symmetric spaces and word metrics. Chapter 4 then contains a careful proof of the following theorem of Lubotzky--Mozes--Raghunathan: the word metric on $PSL(2,\\mathbb{Z})$ is not Lipschitz equivalent to the metric induced by its action on the associated symmetric space (the upper half-plane), but for $n \\geq 3$, these two metrics on $PSL(n,\\mathbb{Z})$ are Lipschitz equivalent."}
{"category": "Math", "title": "Short Rational Generating Functions For Multiobjective Linear Integer Programming", "abstract": "This paper presents algorithms for solving multiobjective integer programming problems. The algorithm uses Barvinok's rational functions of the polytope that defines the feasible region and provides as output the entire set of nondominated solutions for the problem. Theoretical complexity results on the algorithm are provided in the paper. Specifically, we prove that encoding the entire set of nondominated solutions of the problem is polynomially doable, when the dimension of the decision space is fixed. In addition, we provide polynomial delay algorithms for enumerating this set. An implementation of the algorithm shows that it is useful for solving multiobjective integer linear programs."}
{"category": "Math", "title": "Letters from William Burnside to Robert Fricke: Automorphic Functions, and the Emergence of the Burnside Problem", "abstract": "Two letters from William Burnside have recently been found in the Nachlass of Robert Fricke that contain instances of Burnside's Problem prior to its first publication. We present these letters as a whole to the public for the first time. We draw a picture of these two mathematicians and describe their activities leading to their correspondence. We thus gain an insight into their respective motivations, reactions, and attitudes, which may sharpen the current understanding of professional and social interactions of the mathematical community at the turn of the 20th century."}
{"category": "Math", "title": "P-symbols, Heun Identities, and 3F2 Identities", "abstract": "The usefulness of Riemann P-symbols in deriving identities involving the parametrized special function Hl is explored. Hl is the analytic local solution of the Heun equation, the canonical second-order differential equation on the Riemann sphere with four regular singular points. The identities discussed include ones coming from Moebius automorphisms and F-homotopies, and also quadratic and biquadratic transformations. The case when Hl is identical to a generalized hypergeometric function of 3F2 type is examined, and Pfaff and Euler transformations of 3F2(a1,a2,e+1;b1,e;x) are derived. They extend several 3F2 identities of Bailey and Slater."}
{"category": "Math", "title": "Holomorphic L^{p}-functions on Coverings of Strongly Pseudoconvex Manifolds", "abstract": "In this paper we will show how to construct holomorphic L^{p}-functions on unbranched coverings of strongly pseudoconvex manifolds. Also, we prove some extension and approximation theorems for such functions."}
{"category": "Math", "title": "One way cuts in oriented graphs", "abstract": "This paper has been withdrawn by the author."}
{"category": "Math", "title": "A Cuspidality Criterion for the Exterior Square Transfer of Cusp Forms on GL(4)", "abstract": "For a cuspidal automorphic representation \\Pi of GL(4,A), H. Kim proved that the exterior square transfer \\wedge^2\\Pi is an isobaric automorphic representation of GL(6,A). In this paper we characterize those representations \\Pi for which \\wedge^2\\Pi is cuspidal."}
{"category": "Math", "title": "Dispersion Models for Extremes", "abstract": "We propose extreme value analogues of natural exponential families and exponential dispersion models, and introduce the slope function as an analogue of the variance function. The set of quadratic and power slope functions characterize well-known families such as the Rayleigh, Gumbel, power, Pareto, logistic, negative exponential, Weibull and Fr\\'echet. We show a convergence theorem for slope functions, by which we may express the classical extreme value convergence results in terms of asymptotics for extreme dispersion models. The main idea is to explore the parallels between location families and natural exponential families, and between the convolution and minimum operations."}
{"category": "Math", "title": "Deformations of Compact Coassociative 4-folds with Boundary", "abstract": "Coassociative 4-folds are a particular class of 4-dimensional submanifolds which are defined in a 7-dimensional manifold M with a G_2 structure given by a `positive' differential 3-form, sometimes called G_2-form. Assuming that a G_2-form on M is closed, we study deformations of a compact coassociative submanifold N with boundary contained in fixed, codimension 1 submanifold S of M with a compatible Hermitian symplectic structure. We show that `small' coassociative deformations of N with special Lagrangian boundary in S are unobstructed and form a smooth moduli space of finite dimension not greater than the first Betti number of the boundary of N. It is also shown that N is `stable' under small deformations of the closed G_2-form on the ambient 7-manifold M. The results can be compared to those for special Lagrangian submanifolds of Calabi--Yau manifolds proved by A.Butscher in math.DG/0110052."}
{"category": "Math", "title": "On the value-semigroup of a simple complete ideal in a two-dimensional regular local ring", "abstract": "Let R be a two-dimensional regular local ring with maximal ideal \\mathfrak m, and let \\wp be a simple complete \\mathfrak m-primary ideal which is residually rational. Let R_0:= R\\subsetneqq ...\\subsetneqq R_r be the quadratic sequence associated to \\wp, let \\Gamma_\\wp be the value-semigroup associated to \\wp, and let ((e_j(\\wp))_{0\\leq j\\leq r} be the multiplicity sequence of \\wp. We associate to \\wp a sequence of natural integers, the formal characteristic sequence of \\wp, and we show that the value-semigroup, the multiplicity sequence and the formal characteristic sequence are equivalent data. Furthermore, we give a new proof that \\Gamma_\\wp is symmetric, and give a formula for c_\\wp, the conductor of \\Gamma_\\wp, in terms of entries of the Hamburger-Noether tableau of \\wp."}
{"category": "Math", "title": "Self-similar tiling systems, topological factors and stretching factors", "abstract": "In this paper we prove that if two self-similar tiling systems, with respective stretching factors $\\lambda_1$ and $\\lambda_2$, have a common factor which is a non periodic tiling system, then $\\lambda_1$ and $\\lambda_2$ are multiplicatively dependent."}
{"category": "Math", "title": "Cobham-Semenov theorem and $\\NN^d$-subshifts", "abstract": "We give a new proof of the Cobham's first theorem using ideas from symbolic dynamics and of the Cobham-Semenov theorem (in the primitive case) using ideas from tiling dynamics."}
{"category": "Math", "title": "Recent Progress in Special Colombeau Algebras: Geometry, Topology, and Algebra", "abstract": "Over the past few years there has been considerable progress in the structural understanding of special Colombeau algebras. We present some of the main trends in this development: non-smooth differential geometry, locally convex theory of modules over the ring of generalized numbers, and algebraic aspects of Colombeau theory. Some open problems are given and directions of further research are outlined."}
{"category": "Math", "title": "An optimal boundedness on weak $\\bQ$-Fano threefolds", "abstract": "Let $X$ be a terminal weak $\\bQ$-Fano threefold. We prove that $P_{-6}(X)>0$ and $P_{-8}(X)>1$. We also prove that the anti-canonical volume has a universal lower bound $-K_X^3 \\geq 1/330$. This lower bound is optimal."}
{"category": "Math", "title": "Convolution type stochastic Volterra equations", "abstract": "The aim of this work is to present, in self-contained form, results concerning fundamental and the most important questions related to linear stochastic Volterra equations of convolution type. The paper is devoted to study the existence and some kind of regularity of solutions to stochastic Volterra equations in Hilbert space and the space of tempered distributions, as well. In recent years the theory of Volterra equations, particularly fractional ones, has undergone a big development. This is an emerging area of research with interesting mathematical questions and various important applications. The increasing interest in these equations comes from their applications to problems from physics and engeenering, particularly from viscoelasticity, heat conduction in materials with memory or electrodynamics with memory."}
{"category": "Math", "title": "Limit Theorems for Internal Aggregation Models", "abstract": "We study the scaling limits of three different aggregation models on the integer lattice Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains. In the special case when all particles start at a single site, we show that the scaling limit is a Euclidean ball in R^d, and give quantitative bounds on the rate of convergence to a ball. We also improve on the previously best known bounds of Le Borgne and Rossin in Z^2 and Fey and Redig in higher dimensions for the shape of the classical abelian sandpile model. Lastly, we study the sandpile group of a regular tree whose leaves are collapsed to a single sink vertex, and determine the decomposition of the full sandpile group as a product of cyclic groups. For the regular ternary tree of height n, for example, the sandpile group is isomorphic to (Z_3)^{2^{n-3}} x (Z_7)^{2^{n-4}} x ... x Z_{2^{n-1}-1} x Z_{2^n-1}. We use this result to prove that rotor-router aggregation on the regular tree yields a perfect ball."}
{"category": "Math", "title": "Bochner transforms, perturbations and amoebae of holomorphic almost periodic mappings in tube domains", "abstract": "We give an alternative representation of the closure of the Bochner transform of a holomorphic almost periodic mapping in a tube domain. For such mappings we introduce a new notion of amoeba and we show that, for mappings which are regular in the sense of Ronkin, this new notion agrees with Favorov's one. We prove that the amoeba complement of a regular holomorphic almost periodic mapping, defined on Cn and taking its values in Cm+1, is a Henriques m-convex subset of Rn. Finally, we compare some different notions of regularity."}
{"category": "Math", "title": "Factorization (Splitting)", "abstract": "We show some elementary facts about the semantical analogue of Parikh's Splitting, which we call Factorization."}
{"category": "Math", "title": "Similarity, Codepth Two Bicomodules and QF Bimodules", "abstract": "For any k-coalgebra C it is shown that similar quasi-finite C-comodules have strongly equivalent coendomorphism coalgebras; (the converse is in general not true). As an application we give a general result about codepth two coalgebra homomorphisms. Also a notion of codepth two bicomodule is introduced. The last section applies similarity to an endomorphism ring theorem for quasi-Frobenius (QF) bimodules and then to finite depth ring extensions. For QF extensions, we establish that left and right depth two are equivalent notions as well as a converse endomorphism theorem, and characterize depth three in terms of separability and depth two."}
{"category": "Math", "title": "A Note on Overtwisted Contact Structures", "abstract": "In this note, we use the recent work of Honda-Kazez-Matic [HKM] to prove that a closed contact 3-manifold admitting a compatible open book decomposition with a nontrivial monodromy which can be presented as a product of left handed Dehn twists is overtwisted."}
{"category": "Math", "title": "Dehn surgeries that yield fibred $3$--manifolds", "abstract": "We study Dehn surgeries on null-homotopic knots that yield fibred $3$--manifolds when an additional (but natural) homological restriction is imposed. The major tool used is Gabai's theory of sutured manifold decomposition. Such surgeries are negative examples to a question of Michel Boileau. Another result we will prove is about surgeries which reduce the Thurston norm of a fibred manifold."}
{"category": "Math", "title": "The integral Chow ring of the stack of hyperelliptic curves of even genus", "abstract": "Let $g$ be an even positive integer. In this paper we compute the integral Chow ring of the stack of smooth hyperelliptic curves of genus $g$."}
{"category": "Math", "title": "Goresky-MacPherson calculus for the affine flag varieties", "abstract": "We use the fixed point arrangement technique developed by Goresky-MacPherson to calculate the part of the equivariant cohomology of the affine flag varieties generated by degree 2. This turns out to be a quadric cone. We also describe the spectrum of the full equivariant cohomology ring as an explicit geometric object. We use our results to show that the vertices of the moment map images of the affine flag varieties lie on a paraboloid."}
{"category": "Math", "title": "On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator", "abstract": "We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach--based on commutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage of the Cauchy-Schwarz inequality--is used to produce ``parameter-free'', ``projection-free'' versions of their theorems. We also analyze the strength of the various inequalities that ensue. The results contain classical bounds for the eigenvalues. Extensions of a variety of inequalities \\`a la Harrell-Stubbe are illustrated for both geometric and physical problems."}
{"category": "Math", "title": "The inverse problem for representation functions for general linear forms", "abstract": "The inverse problem for representation functions takes as input a triple (X,f,L), where X is a countable semigroup, f : X --> N_0 \\cup {\\infty} a function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A \\subseteq X such that there are f(x) solutions (counted appropriately) to L(x_1,...,x_h) = x for every x \\in X, or a proof that no such subset exists. This paper represents the first systematic study of this problem for arbitrary linear forms when X = Z, the setting which in many respects is the most natural one. Having first settled on the \"right\" way to count representations, we prove that every primitive form has a unique representation basis, i.e.: a set A which represents the function f \\equiv 1. We also prove that a partition regular form (i.e.: one for which no non-empty subset of the coefficients sums to zero) represents any function f for which {f^{-1}(0)} has zero asymptotic density. These two results answer questions recently posed by Nathanson. The inverse problem for partition irregular forms seems to be more complicated. The simplest example of such a form is x_1 - x_2, and for this form we provide some partial results. Several remaining open problems are discussed."}
{"category": "Math", "title": "Geometric structure-preserving optimal control of the rigid body", "abstract": "In this paper we study a discrete variational optimal control problem for the rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange--d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra so(3). We use Lagrange's method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver."}
{"category": "Math", "title": "Linear forms and complementing sets of integers", "abstract": "Let $\\varphi(x_1,\\ldots,x_h,y) = u_1x_1 + \\cdots + u_hx_h+vy$ be a linear form with nonzero integer coefficients $u_1,\\ldots, u_h, v.$ Let $\\mathcal{A} = (A_1,\\ldots, A_h)$ be an $h$-tuple of finite sets of integers and let $B$ be an infinite set of integers. Define the representation function associated to the form $\\varphi$ and the sets \\mca\\ and $B$ as follows: $$ R^{(\\varphi)}_{\\mathcal{A},B}(n) = \\text{card}\\left( \\left\\{ (a_1,\\ldots, a_h,b) \\in A_1 \\times \\cdots \\times A_h \\times B: \\varphi(a_1, \\ldots , a_h,b ) = n \\right\\} \\right).$$ If this representation function is constant, then the set $B$ is periodic and the period of $B$ will be bounded in terms of the diameter of the finite set $\\{ \\varphi(a_1,\\ldots,a_h,0): (a_1,\\ldots, a_h) \\in A_1 \\times \\cdots \\times A_h\\}.$"}
{"category": "Math", "title": "A Ricci nilsoliton is nongradient", "abstract": "In this brief note, we clarify that a Ricci nilsoliton cannot be of gradient type."}
{"category": "Math", "title": "On the defining relations for generalized q-Schur algebras", "abstract": "We show that the defining relations needed to describe a generalized q-Schur algebra as a quotient of a quantized enveloping algebra are determined completely by the defining ideal of a certain finite affine variety, the points of which correspond bijectively to the set of weights. This explains, unifies, and extends previous results."}
{"category": "Math", "title": "One-point reductions of finite spaces, h-regular CW-complexes and collapsibility", "abstract": "We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular CW-complex, generalizing the concept of regular CW-complex, and prove that the h-regular CW-complexes, which are a sort of combinatorial-up-to-homotopy objects, are modeled (up to homotopy) by their associated finite spaces. This is accomplished by generalizing a classical result of McCord on simplicial complexes."}
{"category": "Math", "title": "2-filteredness and the point of every Galois topos", "abstract": "A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bi-limits of topoi, we show that every Galois topos has a point."}
{"category": "Math", "title": "On a generalization of Chen's iterated integrals", "abstract": "Chen's iterated integrals may be generalized by interpolation of functions of the positive integer number of times which particular forms are iterated in integrals along specific paths, to certain complex values. These generalized iterated integrals satisfy both an additive and a (non-classical) multiplicative iterative property, in addition to a comultiplication formula. This theory is developed in the first part of the paper, after which various applications are discussed, including the expression of certain zeta functions as complex iterated integrals (from which an obstruction to the existence of a contour integration proof of the functional equation for the Dedekind zeta function emerges); an elegant reformulation of a result of Gel'fand and Shilov in the theory of distributions which gives a way of thinking about complex iterated derivatives; and a direct topological proof of the monodromy of polylogarithms."}
{"category": "Math", "title": "Small primitive roots and malleability of RSA moduli", "abstract": "In a paper of P. Paillier and J. Villar a conjecture is made about the malleability of an RSA modulus. In this paper we present an explicit algorithm refuting the conjecture. Concretely we can factorize an RSA modulus n using very little information on the factorization of a concrete n' coprime to n. However, we believe the conjecture might be true, when imposing some extra conditions on the auxiliary n' allowed to be used. In particular, the paper shows how subtle the notion of malleability is."}
{"category": "Math", "title": "Examples of para-cocyclic objects induced by BD-laws", "abstract": "In a recent paper arXiv:0705.3190, we gave a general construction of a para-cocyclic structure on a cosimplex, associated to a so called admissible septuple -- consisting of two categories, three functors and two natural transformations, subject to compatibility relations. The main examples of such admissible septuples were induced by algebra homomorphisms. In this note we provide more general examples coming from appropriate (`locally braided') morphisms of monads."}
{"category": "Math", "title": "A Problem of Powers and the Product of Spatial Product Systems", "abstract": "In the 2002 AMS summer conference on ``Advances in Quantum Dynamics'' in Mount Holyoke Robert Powers proposed a sum operation for spatial E0-semigroups. Still during the conference Skeide showed that the Arveson system of that sum is the product of spatial Arveson systems. This product may but need not coincide with the tensor product of Arveson systems. The Powers sum of two spatial E0-semigroups is, therefore, up to cocycle conjugacy Skeide's product of spatial noises."}
{"category": "Math", "title": "A contact geometric proof of the Whitney-Graustein theorem", "abstract": "The Whitney-Graustein theorem states that regular closed curves in the 2-plane are classified, up to regular homotopy, by their rotation number. Here we give a simple proof based on contact geometry."}
{"category": "Math", "title": "Horizontal loops in Engel space", "abstract": "A simple proof is given of the following result first observed by J. Adachi: embedded circles tangent to the standard Engel structure on Euclidean 4-space are classified, up to isotopy via such embeddings, by their rotation number."}
{"category": "Math", "title": "The moments of Minkowski question mark function: the dyadic period function", "abstract": "The Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane C(0,infinity). The exponential generating function satisfies the integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert-Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern-Brocot tree. Surprisingly, the Eisenstein series G_1(z) does manifest in both real and p-adic cases."}
{"category": "Math", "title": "Statistical properties of the Calkin--Wilf tree: real an p-adic distribution", "abstract": "We examine statistical properties of the Calkin--Wilf tree and give number-theoretical applications."}
{"category": "Math", "title": "On connection between reducibility of an n-ary quasigroup and that of its retracts", "abstract": "An $n$-ary operation $Q:S^n\\to S$ is called an $n$-ary quasigroup of order $|S|$ if in the equation $x_0=Q(x_1,...,x_n)$ knowledge of any $n$ elements of $x_0,...,x_n$ uniquely specifies the remaining one. An $n$-ary quasigroup $Q$ is (permutably) reducible if $Q(x_1,...,x_n)=P(R(x_{s(1)},...,x_{s(k)}),x_{s(k+1)},...,x_{s(n)})$ where $P$ and $R$ are $(n-k+1)$-ary and $k$-ary quasigroups, $s$ is a permutation, and $1<k<n$. An $m$-ary quasigroup $R$ is called a retract of $Q$ if it can be obtained from $Q$ or one of its inverses by fixing $n-m>0$ arguments. We show that every irreducible $n$-ary quasigroup has an irreducible $(n-1)$-ary or $(n-2)$-ary retract; moreover, if the order is finite and prime, then it has an irreducible $(n-1)$-ary retract. We apply this result to show that all $n$-ary quasigroups of order 5 or 7 whose all binary retracts are isotopic to $Z_5$ or $Z_7$ are reducible for $n>3$. Keywords: $n$-ary quasigroups, retracts, reducibility, latin hypercubes"}
{"category": "Math", "title": "Generating and zeta functions, structure, spectral and analytic properties of the moments of Minkowski question mark function", "abstract": "In this paper we are interested in moments of Minkowski question mark function ?(x). It appears that, to certain extent, the results are analogous to the results obtained for objects associated with Maass wave forms: period functions, L-series, distributions, spectral properties. These objects can be naturally defined for ?(x) as well. Despite the fact that there are various nice results about the nature of ?(x), these investigations are mainly motivated from the perspective of metric number theory, Hausdorff dimension, singularity and generalizations. In this work it is shown that analytic and spectral properties of various integral transforms of ?(x) do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals involving ?(x), define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects. At the end of the paper it is shown that certain object associated with ?(x) establish a bridge between realms of imaginary and real quadratic irrationals."}
{"category": "Math", "title": "The Maximal Probability that k-wise Independent Bits are All 1", "abstract": "A k-wise independent distribution on n bits is a joint distribution of the bits such that each k of them are independent. In this paper we consider k-wise independent distributions with identical marginals, each bit has probability p to be 1. We address the following question: how high can the probability that all the bits are 1 be, for such a distribution? For a wide range of the parameters n,k and p we find an explicit lower bound for this probability which matches an upper bound given by Benjamini et al., up to multiplicative factors of lower order. The question we investigate can be seen as a relaxation of a major open problem in error-correcting codes theory, namely, how large can a linear error correcting code with given parameters be? The question is a type of discrete moment problem, and our approach is based on showing that bounds obtained from the theory of the classical moment problem provide good approximations for it. The main tool we use is a bound controlling the change in the expectation of a polynomial after small perturbation of its zeros."}
{"category": "Math", "title": "Factorization, Fibration and Torsion", "abstract": "A simple definition of torsion theory is presented, as a factorization system with both classes satisfying the 3--for--2 property. Comparisons with the traditional notion are given, as well as connections with the notions of fibration and of weak factorization system, as used in abstract homotopy theory."}
{"category": "Math", "title": "Krall--type Orthogonal Polynomials in several variables", "abstract": "For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials."}
{"category": "Math", "title": "$q$-Analogue of the Dunkl transform on the real line", "abstract": "In this paper, we consider a $q$-analogue of the Dunkl operator on $\\mathbb{R}$, we define and study its associated Fourier transform which is a $q$-analogue of the Dunkl transform. In addition to several properties, we establish an inversion formula and prove a Plancherel theorem for this $q$-Dunkl transform. Next, we study the $q$-Dunkl intertwining operator and its dual via the $q$-analogues of the Riemann-Liouville and Weyl transforms. Using this dual intertwining operator, we provide a relation between the $q$-Dunkl transform and the $q^2$-analogue Fourier transform introduced and studied by R. Rubin."}
{"category": "Math", "title": "On the Basis Polynomials in the Theory of Permutations with Prescribed Up-Down Structure", "abstract": "Let $\\pi=(\\pi_1,\\pi_2,\\hdots,\\pi_n)$ be permutation of the elements $1,2,\\hdots,n. $ Positive integer $k\\leq2^{n-1}$ we call index of $\\pi,$ if in its binary notation as $n$-digital binary number, the 1's correspond to the ascent points. We study behavior and properties of numbers of permutations of $n$ elements having index $k.$"}
{"category": "Math", "title": "Initiation to mould calculus through the example of saddle-node singularities", "abstract": "This article proposes an initiation to \\'Ecalle's mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field. This is illustrated on the case of saddle-node singularities, generated by two-dimensional vector fields which are formally conjugate to Euler's vector field $x^2\\frac{\\pa}{\\pa x}+(x+y)\\frac{\\pa}{\\pa y}$, and for which the formal normalisation proves to be resurgent in $1/x$."}
{"category": "Math", "title": "A new approach to temperate generalized functions", "abstract": "A new approach to the algebra G_{\\tau} of temperate nonlinear generalized functions is proposed, in which G_{\\tau} is based on the space O_{M} endowed with is natural topology in contrary to previous constructions. Thus, this construction fits perfectly in the general scheme of construction of Colombeau type algebras and reveals better properties of G_{\\tau}. This is illustrated by the natural introduction of a regularity theory in G_{\\tau}, of the Fourier transform, with the definition of G_{O_{C prime}}, the space of rapidly generalized distributions which is the Fourier image of G_{\\tau}."}
{"category": "Math", "title": "Non-existence of absolutely continuous invariant probabilities for exponential maps", "abstract": "We show that for entire maps of the form $z \\mapsto \\lambda \\exp(z)$ such that the orbit of zero is bounded and such that Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem."}
{"category": "Math", "title": "Degree complexity of a family of birational maps: II. Exceptional cases", "abstract": "We compute the degree complexity of the family of birational maps considered in \\cite{bedford-kim-tuyen-abarenkova-maillard} for all exceptional cases. Some interesting properties of the family are also given."}
{"category": "Math", "title": "Asymptotics of Convex sets in En and Hn", "abstract": "We study convex sets C of finite (but non-zero volume in Hn and En. We show that the intersection of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n-1)/2, and this bound is sharp. In the hyperbolic case we show that for any k <= (n-1)/2 there is a bounded section S of C through any prescribed point p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of convex body in En, and give asymptotic estimates as 1 << k << n."}
{"category": "Math", "title": "An Algorithm to Estimate Monotone Normal Means and its Application to Identify the Minimum Effective Dose", "abstract": "In the standard setting of one-way ANOVA with normal errors, a new algorithm, called the Step Down Maximum Mean Selection Algorithm (SDMMSA), is proposed to estimate the treatment means under an assumption that the treatment mean is nondecreasing in the factor level. We prove that i) the SDMMSA and the Pooled Adjacent Violator Algorithm (PAVA), a widely used algorithm in many problems, generate the same estimators for normal means, ii) the estimators are the mle's, and iii) the distribution of each of the estimators is stochastically nondecreasing in each of the treatment means. As an application of this stochastic ordering, a sequence of null hypotheses to identify the minimum effective dose (MED) is formulated under the assumption of monotone treatment(dose) means. A step-up testing procedure, which controls the experimentwise error rate in the strong sense, is constructed. When the MED=1, the proposed test is uniformly more powerful than Hsu and Berger's (1999)."}
{"category": "Math", "title": "A new extension of the Erdos-Heilbronn conjecture", "abstract": "Let A_1,...,A_n be finite subsets of a field F, and let f(x_1,...,x_n)=x_1^k+...+x_n^k+g(x_1,...,x_n)\\in F[x_1,...,x_n] with deg g<k. We obtain a lower bound for the cardinality of {f(x_1,...,x_n): x_1\\in A_1,...,x_n\\in A_n, and x_i\\not=x_j if i\\not=j}. The result extends the Erdos-Heilbronn conjecture in a new way."}
{"category": "Math", "title": "Invariant Functions on Grassmannians", "abstract": "It is known, that every function on the unit sphere in $\\bbr^n$, which is invariant under rotations about some coordinate axis, is completely determined by a function of one variable. Similar results, when invariance of a function reduces dimension of its actual argument, hold for every compact symmetric space and can be obtained in the framework of Lie-theoretic consideration. In the present article, this phenomenon is given precise meaning for functions on the Grassmann manifold $G_{n,i}$ of $i$-dimensional subspaces of $\\bbr^n$, which are invariant under orthogonal transformations preserving complementary coordinate subspaces of arbitrary fixed dimension. The corresponding integral formulas are obtained. Our method relies on bi-Stiefel decomposition and does not invoke Lie theory."}
{"category": "Math", "title": "Central Extensions of Gerbes", "abstract": "We introduce the notion of central extension of gerbes on a topological space. We then show that there are obstruction classes to lifting objects and isomorphisms in a central extension. We also discuss pronilpotent gerbes. These results are used in a subsequent paper to study twisted deformation quantization on algebraic varieties."}
{"category": "Math", "title": "Spectral convergence for high contrast elliptic periodic problems with a defect via homogenization", "abstract": "We consider an eigenvalue problem for a divergence form elliptic operator $A_\\epsilon$ with high contrast periodic coefficients with period $\\epsilon$ in each coordinate, where $\\epsilon$ is a small parameter. The coefficients are perturbed on a bounded domain of `order one' size. The local perturbation of coefficients for such operator could result in emergence of localized waves - eigenfunctions with corresponding eigenvalues lying in the gaps of the Floquet-Bloch spectrum. We prove that, for the so-called double porosity type scaling, the eigenfunctions decay exponentially at infinity, uniformly in $\\epsilon$. Then, using the tools of two-scale convergence for high contrast homogenization, we prove the strong two-scale compactness of the eigenfunctions of $A_\\epsilon$. This implies that the eigenfunctions converge in the sense of the strong two-scale convergence to the eigenfunctions of a two-scale limit homogenized operator $A_0$, consequently establishing `asymptotic one-to-one correspondence' between the eigenvalues and the eigenfunctions of these two operators. We also prove by direct means the stability of the essential spectrum of the homogenized operator with respect to the local perturbation of its coefficients. That allows us to establish not only the strong two-scale resolvent convergence of $A_\\epsilon$ to $A_0$ but also the Hausdorff convergence of the spectra of $A_\\epsilon$ to the spectrum of $A_0$, preserving the multiplicity of the isolated eigenvalues."}
{"category": "Math", "title": "Perturbation of the Wigner equation in inner product C*-modules", "abstract": "Let $\\A$ be a $C^*$-algebra and $\\B$ be a von Neumann algebra that both act on a Hilbert space $\\Ha$. Let $\\M$ and $\\N$ be inner product modules over $\\A$ and $\\B$, respectively. Under certain assumptions we show that for each mapping $f\\colon{\\mathcal M} \\to {\\mathcal N}$ satisfying $$\\||\\ip{f(x)}{f(y)}|-|\\ip{x}{y}| \\|\\leq\\phi(x,y)\\qquad (x,y\\in{\\mathcal M}),$$ where $\\phi$ is a control function, there exists a solution $I\\colon{\\mathcal M} \\to {\\mathcal N}$ of the Wigner equation $$|\\ip{I(x)}{I(y)}|=|\\ip{x}{y}|\\qquad (x, y \\in {\\mathcal M})$$ such that $$\\|f(x)-I(x)\\|\\leq\\sqrt{\\phi(x,x)} \\qquad (x\\in {\\mathcal M}).$$"}
{"category": "Math", "title": "A zero divisor graph determined by equivalence classes of zero divisors", "abstract": "We study the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring R. We demonstrate how to recover information about R from this structure. In particular, we determine how to identify associated primes from the graph."}
{"category": "Math", "title": "Necessary conditions for linear noncooperative N-player delta differential games on time scales", "abstract": "We present necessary conditions for linear noncooperative N-player delta dynamic games on a generic time scale. Necessary conditions for an open-loop Nash-equilibrium and for a memoryless perfect state Nash-equilibrium are proved."}
{"category": "Math", "title": "A deformation problem for Galois representations over imaginary quadratic fields", "abstract": "We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL_2(A_F) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists."}
{"category": "Math", "title": "Complete gradient shrinking Ricci solitons have finite topological type", "abstract": "We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-\\'{E}mery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero. Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded."}
{"category": "Math", "title": "Congruences between modular forms and lowering the level mod l^n", "abstract": "In this article we study the behavior of inertia groups for modular Galois mod l^n representations and in some cases we give a generalization of Ribet's lowering the level result"}
{"category": "Math", "title": "Lindel\\\"of's hypothesis is true and Riemann's one is not", "abstract": "We present an elementary, short and simple proof of the validity of the Lindel\\\"of hypothesis about the Riemann zeta-function. The obtained estimate and classical results by Bohr-Landau and Littlewood disprove Riemann's hypothesis."}
{"category": "Math", "title": "Shestakov-Umirbaev reductions and Nagata's conjecture on a polynomial automorphism", "abstract": "In 2003, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. In the present paper, we reconstruct their theory by using the \"generalized Shestakov-Umirbaev inequality\", which was recently given by the author. As a consequence, we obtain a more precise tameness criterion for polynomial automorphisms. In particular, we show that no tame automorphism of a polynomial ring admits a reduction of type IV."}
{"category": "Math", "title": "Combinatorics of the change-making problem", "abstract": "We investigate the structure of the currencies (systems of coins) for which the greedy change-making algorithm always finds an optimal solution (that is, a one with minimum number of coins). We present a series of necessary conditions that must be satisfied by the values of coins in such systems. We also uncover some relations between such currencies and their sub-currencies."}
{"category": "Math", "title": "Sums of primes and squares of primes in short intervals", "abstract": "Let $\\mathbf H_2$ denote the set of even integers $n \\not\\equiv 1 \\pmod 3$. We prove that when $H \\ge X^{0.33}$, almost all integers $n \\in \\mathbf H_2$, $X < n \\le X + H$ can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes."}
{"category": "Math", "title": "An Integral Inequality and the Riccati-Bernoulli Differential Equation", "abstract": "We apply an integral inequality to obtain a rigorous apriori estimate of the accuracy of the partial sum to the power series solution of the celebrated Riccati-Bernoulli differential equation"}
{"category": "Math", "title": "Star products and local line bundles", "abstract": "The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using ideas of Boutet de Monvel and Guillemin the corresponding twisted Toeplitz algebroid on any compact symplectic manifold is shown to yield the star products of Lecomte and DeWilde ([MR84g:17014]) see also Fedosov's construction in [MR92k:58267]. This also shows that the trace on the star algebra is identified with the residue trace of Wodzicki and Guillemin"}
{"category": "Math", "title": "Frequency estimation based on the cumulated Lomb-Scargle periodogram", "abstract": "We consider the problem of estimating the period of an unknown periodic function observed in additive noise sampled at irregularly spaced time instants in a semiparametric setting. To solve this problem, we propose a novel estimator based on the cumulated Lomb-Scargle periodogram. We prove that this estimator is consistent, asymptotically Gaussian and we provide an explicit expression of the asymptotic variance. Some Monte-Carlo experiments are then presented to support our claims."}
{"category": "Math", "title": "Proper actions, fixed-point algebras and naturality in nonabelian duality", "abstract": "Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let gamma be the induced action on C_0(X). We consider a category in which the objects are C*-dynamical systems (A, G, alpha) for which there is an equivariant homomorphism of (C_0(X), gamma) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra A^alpha which is Morita equivalent to A times_{alpha,r} G. We show that the assignment (A, alpha) maps to A^alpha is functorial, and that Rieffel's Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural."}
{"category": "Math", "title": "Affine Toric SL(2)-embeddings", "abstract": "In 1973 V.L.Popov classified affine SL(2)-embeddings. He proved that a locally transitive SL(2)-action on a normal affine three-dimensional variety X is uniquely determined by a pair (p/q, r), where 0<p/q<=1 is an uncancelled fraction and r is a positive integer. Here r is the order of the stabilizer of a generic point. In this paper we show that the variety X is toric, i.e. admits a locally transitive action of an algebraic torus, if and only if r is divisible by q-p. To do this we prove the following necessary and sufficient condition for an affine G/H-embedding to be toric. Suppose X is a normal affine variety, G is a simply connected semisimple algebraic group acting regularly on X, H is a closed subgroup of G such that the character group $\\mathfrak{X}(H)$ is finite and G/H -> X is a dense open equivariant embedding. Then X is toric if and only if there exist a quasitorus T and a $(G\\times T)$-module V such that $X\\stackrel{G}{\\cong} V//T$. The key role in the proof plays D. Cox's construction."}
{"category": "Math", "title": "A compactification for the spaces of convex projective structures on manifolds", "abstract": "In this paper we construct a compactification for the parameter space of convex projective structures on a fixed n-manifold M. This parameter space is a closed semi-algebraic subset of the variety of characters of representations of the fundamental group of M in SL_{n+1}(R). The boundary is the inverse limit of an inverse system of logarithmic limit sets of this semi-algebraic set, in a sense it is the tropicalization of the parameter space. The interpretation of the boundary points can also be given using tropical geometry. This construction is a generalization of the construction of compactification of the Teichm\\\"uller spaces."}
{"category": "Math", "title": "Orders of $\\pi$-bases", "abstract": "We extend the scope of B. Shapirovskii's results [B.E Shapirovskii, \"Cardinal invariants in Compact Hausdorff Spaces,\" Amer. Math. Soc. Transl. (2) Vol. 134, 1987, pp. 93-118] on the order of $\\pi$-bases in compact spaces and answer some questions of V. Tkachuk in [V.V. Tkachuk, \"Point-countable pi-bases in first-countable and similar spaces,\" Fund. Math. 186 (2005), pp.55-69]."}
{"category": "Math", "title": "On a class of non-self-adjoint periodic eigenproblems with boundary and interior singularities", "abstract": "We prove that the spectrum of a certain PT-symmetric periodic problem is purely real. Our results extend to a larger class of potentials those recently found by Brian Davies [math.SP/0702122] and John Weir [arXiv:0711.1371]."}
{"category": "Math", "title": "Exceptional sequences and derived autoequivalences", "abstract": "We prove a general theorem that gives a non trivial relation in the group of derived autoequivalences of a variety (or stack) X, under the assumption that there exists a suitable functor from the derived category of another variety Y admitting a full exceptional sequence. Applications include the case in which X is Calabi-Yau and either X is a hypersurface in Y (this extends a previous result by the author and R.L. Karp, where Y was a weighted projective space) or Y is a hypersurface in X. The proof uses a resolution of the diagonal of Y constructed from the exceptional sequence."}
{"category": "Math", "title": "String topology of classifying spaces", "abstract": "Let $G$ be a finite group or a compact connected Lie group and let $BG$ be its classifying space. Let $\\mathcal{L}BG:=map(S^1,BG)$ be the free loop space of $BG$ i.e. the space of continuous maps from the circle $S^1$ to $BG$. The purpose of this paper is to study the singular homology $H_*(\\mathcal LBG)$ of this loop space. We prove that when taken with coefficients in a field the homology of $\\mathcal LBG$ is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology $H^*(\\mathcal LBG)$. We also prove an algebraic version of this result by showing that the Hochschild cohomology $HH^*(S_* (G),S_*(G))$ of the singular chains of $G$ is a Batalin-Vilkovisky algebra."}
{"category": "Math", "title": "The Weil-Petersson metric geometry", "abstract": "A summary introduction of the Weil-Petersson metric space geometry is presented. Teichmueller space and its augmentation are described in terms of Fenchel-Nielsen coordinates. Formulas for the gradients and Hessians of geodesic-length functions are presented. Applications are considered. A description of the Weil-Petersson metric in Fenchel-Nielsen coordinates is presented. The Alexandrov tangent cone at points of the augmentation is described. A comparison dictionary is presented between the geometry of the space of flat tori and Teichmueller space with the Weil-Petersson metric."}
{"category": "Math", "title": "Homotopy theory of modules over operads and non-Sigma operads in monoidal model categories", "abstract": "This paper studies the existence of model category structures on algebras and modules over operads in monoidal model categories."}
{"category": "Math", "title": "Topology of broken Lefschetz fibrations and near-symplectic 4-manifolds", "abstract": "The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefschetz fibrations/pencils, while showing that all near-symplectic closed 4-manifolds can be supported by these a la Auroux, Donaldson, Katzarkov. Various constructions of broken Lefschetz fibrations and a generalization of the symplectic fiber sum operation to the near-symplectic setting are given. Extending the study of Lefschetz fibrations, we detect certain constraints on the symplectic fiber sum operation to result in a 4-manifold with nontrivial Seiberg-Witten invariant, as well as the self-intersection numbers that sections of broken Lefschetz fibrations can acquire."}
{"category": "Math", "title": "Homotopy theory of modules over operads in symmetric spectra", "abstract": "We establish model category structures on algebras and modules over operads in symmetric spectra, and study when a morphism of operads induces a Quillen equivalence between corresponding categories of algebras (resp. modules) over operads."}
{"category": "Math", "title": "Cohomologies of harmonic bundles on quasi-compact Kaehler manifolds", "abstract": "In this note, we survey our recent work concerning cohomologies of harmonic bundles on quasi-compact Kaehler manifolds."}
{"category": "Math", "title": "Standard representation of multivariate functions on a general probability space", "abstract": "It is well-known that a random variable, i.e., a function defined on a probability space, with values in a Borel space, can be represented on the special probability space consisting of the unit interval with Lebesgue measure. We show an extension of this to multivariate functions. This is motivated by some recent constructions of random graphs."}
{"category": "Math", "title": "Lawvere completion and separation via closure", "abstract": "For a quantale $\\V$, first a closure-theoretic approach to completeness and separation in $\\V$-categories is presented. This approach is then generalized to $\\Tth$-categories, where $\\Tth$ is a topological theory that entails a set monad $\\mT$ and a compatible $\\mT$-algebra structure on $\\V$."}
{"category": "Math", "title": "There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations of $K_{14}$", "abstract": "We establish by means of a computer search that a complete graph on 14 vertices has 98,758,655,816,833,727,741,338,583,040 distinct and 1,132,835,421,602,062,347 nonisomorphic one-factorizations. The enumeration is constructive for the 10,305,262,573 isomorphism classes that admit a nontrivial automorphism."}
{"category": "Math", "title": "Symplectic Homogenization", "abstract": "Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $\\gamma$ topology defined by the author, to $\\bar{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is the sequence $H(kx,y,q,p)$ where the limit is of the type ${\\bar H}(y,q,p)$ and thus yields an \"effective Hamiltonian\". We give here the proof of the convergence, and the first properties of the homogenization operator, and give some immediate consequences for solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that the function $\\bar H$ coincides with Mather's $\\alpha$ function which gives a new proof of its symplectic invariance proved by P. Bernard. A previous version of this paper relied on the former \"On the capacity of Lagrangians in $T^*T^n$ which has been withdrawn. The present version of Symplectic Homogenization does not rely on it anymore."}
{"category": "Math", "title": "Risk management for analytical methods: conciliating objectives of methods, validation phase and routine decision rules", "abstract": "In the industries that involved either chemistry or biology, such as pharmaceutical industries, chemical industries or food industry, the analytical methods are the necessary eyes and hear of all the material produced or used. If the quality of an analytical method is doubtful, then the whole set of decision that will be based on those measures is questionable. For those reasons, being able to assess the quality of an analytical method is far more than a statistical challenge; it's a matter of ethic and good business practices. Many regulatory documents have been releases, primarily ICH and FDA documents in the pharmaceutical industry (FDA, 1995, 1997, 2001) to address that issue."}
{"category": "Math", "title": "On the capacity of Lagrangians in the cotangent disc bundle of the torus", "abstract": "The paper is wihdrawn due to a critical error in the argument using the spectral sequence"}
{"category": "Math", "title": "Support of Non-separable Multivariate Scaling Function", "abstract": "We make an estimation of the support of a multivariable scaling function for an arbitrary dilation matrix. We give a method of calculating the values of the scaling function on a tight set using the knowledge of the size of the support."}
{"category": "Math", "title": "The Sasaki Cone and Extremal Sasakian Metrics", "abstract": "We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest $L^2$-norm among all Sasakian metrics of fixed volume that can represent vector fields in the cone. We use links of isolated hypersurface singularities to produce examples of manifolds of Sasaki type, many of these in dimension five, whose Sasaki cone coincides with the extremal set, and examples where the extremal set is empty. We end up by proving that a conjecture of Orlik concerning the torsion of the homology groups of these links holds in the five dimensional case."}
{"category": "Math", "title": "Characterizations of algebras of rapidly decreasing generalized functions", "abstract": "The well-known characterizations of Schwartz space $\\mathcal{S}$ of rapidly decreasing functions is extended to the algebra $\\mathcal{G}_{\\mathcal{S}}$ of rapidly decreasing generalized functions and to the algebra $\\mathcal{G}_{% \\mathcal{S}}^{\\infty}$ of regular rapidly decreasing generalized functions."}
{"category": "Math", "title": "Products of Factorial Schur Functions", "abstract": "The product of any finite number of factorial Schur functions can be expanded as a $Z[y]$-linear combination of Schur functions. We give a rule for computing the coefficients in such an expansion which generalizes a specialization of the Molev-Sagan rule, which in turn generalizes the classical Littlewood-Richardson rule."}
{"category": "Math", "title": "An Abelian Category of Motivic Sheaves", "abstract": "The goal of this paper is to construct a category of motivic \"sheaves\" on an algebraic variety defined over a subfield of C, using Nori's method. This categoryis abelian and it possesses faithful exact realization functors to the categoriesof constructible sheaves for the classical and etale topologies. Moreover, there is a tannakian subcategory of motivic local systems with a realization functor into the category of variations of mixed Hodge structures. Conversely, all basic geometric examples of the latter come from this motivic category."}
{"category": "Math", "title": "Partitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree condition", "abstract": "The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $n\\geq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$"}
{"category": "Math", "title": "On The Isoperimetric Spectrum of Graphs and Its Approximations", "abstract": "In this paper we consider higher isoperimetric numbers of a (finite directed) graph. In this regard we focus on the $n$th mean isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set of $n$ disjoint subsets of the vertex set of the graph. We show that the second mean isoperimetric constant in this general setting, coincides with (the mean version of) the classical Cheeger constant of the graph, while for the rest of the spectrum we show that there is a fundamental difference between the $n$th isoperimetric constant and the number obtained by taking the minimum over all $n$-partitions. In this direction, we show that our definition is the correct one in the sense that it satisfies a Federer-Fleming-type theorem, and we also define and present examples for the concept of a supergeometric graph as a graph whose mean isoperimetric constants are attained on partitions at all levels. Moreover, considering the ${\\bf NP}$-completeness of the isoperimetric problem on graphs, we address ourselves to the approximation problem where we prove general spectral inequalities that give rise to a general Cheeger-type inequality as well. On the other hand, we also consider some algorithmic aspects of the problem where we show connections to orthogonal representations of graphs and following J.~Malik and J.~Shi ($2000$) we study the close relationships to the well-known $k$-means algorithm and normalized cuts method."}
{"category": "Math", "title": "On Certain Hypotheses in Optimal Control Theory and the Relationship of the Maximum Principle with the Dynamic Programming Method Proposed by L. I. Rozonoer", "abstract": "In this paper we will study three hypotheses proposed by L. I. Rozonoer (Automation and Remote Control, 2003, vol.64, no.8, pp.1237--1240) in optimal control theory in order to derive conditions for the existence of an optimal control under all initial conditions, and the relationships between Pontryagin maximum principle and the dynamic programming method."}
{"category": "Math", "title": "Cayley 4-form, comass, and triality isomorphisms", "abstract": "Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) to study the comass of certain self-dual 4-forms on R^8. In particular, we prove that the Cayley 4-form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio is SO(8)-conjugate to the Cayley 4-form. We also use triality to prove that the stabilizer in SO(8) of the Cayley form is Spin(7). The results have applications in systolic geometry, calibrated geometry, and Spin(7) manifolds."}
{"category": "Math", "title": "Complex product manifolds cannot be negatively curved", "abstract": "We show that if $M = X \\times Y$ is the product of two complex manifolds (of positive dimensions), then $M$ does not admit any complete K\\\"ahler metric with bisectional curvature bounded between two negative constants. More generally, a locally-trivial holomorphic fibre-bundle does not admit such a metric."}
{"category": "Math", "title": "An elementary approach to some rigidity theorems", "abstract": "Using elementary comparison geometry, we prove: Let $(M,g)$ be a simply-connected complete Riemannian manifold of dimension $\\ge 3$. Suppose that the sectional curvature $K$ satisfies $ -1-s(r) \\le K \\le -1$, where $r$ denotes distance to a fixed point in $M$. If $\\lim_{r \\rt \\infty} e^{2r}s(r) =0$, then $(M,g)$ has to be isometric to ${\\mathbb H}^n$. The same proof also yields that if $K$ satisfies $-s(r) \\le K \\le 0$ where $\\lim_{r \\rt \\infty} r^2s(r)=0$, then $(M,g)$ is isometric to $\\R^n$, a result due to Greene and Wu. Our second result is a local one: Let $(M,g)$ be any Riemannian manifold. For $a \\in \\R$, if $K \\le a$ on a geodesic ball $B_p(R)$ in $M$ and $K = a$ on $\\partial B_p(R)$, then $K= a $ on $B_p(R)$."}
{"category": "Math", "title": "Walk versus Wait: The Lazy Mathematician Wins", "abstract": "In this recreational mathematics note, we address a simple, yet instructive question: Justin has to travel a distance of d miles along a bus route. Along this route, there are n bus stops i, each spaced at a distance of d_i from the starting point. At each bus stop, Justin is faced with a choice: to walk or to wait. If he walks on, he can still catch a bus at the next bus stop--but if a bus passes him while he walks, he is almost assured a longer wait. We model Justin's decision constraint and completely solve the model in a special case. The answer is intuitive: the optimal strategy is the laziest."}
{"category": "Math", "title": "Cellularity of Cyclotomic Birman--Wenzl--Murakami algebras", "abstract": "We show that the cyclotomic Birman-Wenzl-Murakami algebras are cellular by producing a cellular basis of affine tangle diagrams."}
{"category": "Math", "title": "The kernel and continuity ideals of homomorphisms from C_0(\\Omega)", "abstract": "We give a description of the continuity ideals and the kernels of homomorphisms from the algebras of continuous functions on locally compact spaces into Banach algebras."}
{"category": "Math", "title": "Uncountable families of prime z-ideals in C_0(R)", "abstract": "Denote by $\\continuum=2^{\\aleph_0}$ the cardinal of continuum. We construct an intriguing family $(P_\\alpha: \\alpha\\in\\continuum)$ of prime $z$-ideals in $\\C_0(\\reals)$ with the following properties: If $f\\in P_{i_0}$ for some $i_0\\in\\continuum$, then $f\\in P_i$ for all but finitely many $i\\in \\continuum$; $\\bigcap_{i\\neq i_0} P_i \\nsubset P_{i_0}$ for each $\\i_0\\in \\continuum$. We also construct a well-ordered increasing chain, as well as a well-ordered decreasing chain, of order type $\\kappa$ of prime $z$-ideals in $\\C_0(\\reals)$ for any ordinal $\\kappa$ of cardinality $\\continuum$."}
{"category": "Math", "title": "Nonsplitting in Kirchberg's ideal-related KK-theory", "abstract": "A universal coefficient theorem in the setting of Kirchberg's ideal-related KK-theory was obtained in the fundamental case of a C*-algebra with one specified ideal by Bonkat and proved there to split, unnaturally, under certain conditions. Employing certain K-theoretical information derivable from the given operator algebras in a way introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the K-theory which must be used to classify *-isomorphisms for purely infinite C*-algebras with one non-trivial ideal."}
{"category": "Math", "title": "Nonparametric sequential prediction of time series", "abstract": "Time series prediction covers a vast field of every-day statistical applications in medical, environmental and economic domains. In this paper we develop nonparametric prediction strategies based on the combination of a set of 'experts' and show the universal consistency of these strategies under a minimum of conditions. We perform an in-depth analysis of real-world data sets and show that these nonparametric strategies are more flexible, faster and generally outperform ARMA methods in terms of normalized cumulative prediction error."}
{"category": "Math", "title": "Euler Numbers and polynomials associated with zeta functions", "abstract": "In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers."}
{"category": "Math", "title": "Properties of Expectations of Functions of Martingale Diffusions", "abstract": "Given a real valued and time-inhomogeneous martingale diffusion X, we investigate the properties of functions defined by the conditional expectation f(t,X_t)=E[g(X_T)|F_t]. We show that whenever g is monotonic or Lipschitz continuous then f(t,x) will also be monotonic or Lipschitz continuous in x. If g is convex then f(t,x) will be convex in x and decreasing in t. We also define the marginal support of a process and show that it almost surely contains the paths of the process. Although f need not be jointly continuous, we show that it will be continuous on the marginal support of X. We prove these results for a generalization of diffusion processes that we call `almost-continuous diffusions', and includes all continuous and strong Markov processes."}
{"category": "Math", "title": "Explicit fibrant replacement for discrete G-spectra", "abstract": "If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete G-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete G-spectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G."}
{"category": "Math", "title": "Hyperelliptic plane curves of type (d,d-2)", "abstract": "In a previous paper, we classified and constructed all rational plane curves of type (d,d-2). In this paper, we generalize these results to irreducible plane curves of type (d,d-2) with positive genus."}
{"category": "Math", "title": "Near-ideal model selection by $\\ell_1$ minimization", "abstract": "We consider the fundamental problem of estimating the mean of a vector $y=X\\beta+z$, where $X$ is an $n\\times p$ design matrix in which one can have far more variables than observations, and $z$ is a stochastic error term--the so-called \"$p>n$\" setup. When $\\beta$ is sparse, or, more generally, when there is a sparse subset of covariates providing a close approximation to the unknown mean vector, we ask whether or not it is possible to accurately estimate $X\\beta$ using a computationally tractable algorithm. We show that, in a surprisingly wide range of situations, the lasso happens to nearly select the best subset of variables. Quantitatively speaking, we prove that solving a simple quadratic program achieves a squared error within a logarithmic factor of the ideal mean squared error that one would achieve with an oracle supplying perfect information about which variables should and should not be included in the model. Interestingly, our results describe the average performance of the lasso; that is, the performance one can expect in an vast majority of cases where $X\\beta$ is a sparse or nearly sparse superposition of variables, but not in all cases. Our results are nonasymptotic and widely applicable, since they simply require that pairs of predictor variables are not too collinear."}
{"category": "Math", "title": "Classical solvability of nonlinear initial-boundary problems for first-order hyperbolic systems", "abstract": "We prove the global classical solvability of initial-boundary problems for semilinear first-order hyperbolic systems subjected to local and nonlocal nonlinear boundary conditions. We also establish lower bounds for the order of nonlinearity demarkating a frontier between regular cases (classical solvability) and singular cases (blow-up of solutions)."}
{"category": "Math", "title": "Convolution-Dominated Operators on Discrete Groups", "abstract": "We study infinite matrices $A$ indexed by a discrete group $G$ that are dominated by a convolution operator in the sense that $|(Ac)(x)| \\leq (a \\ast |c|)(x)$ for $x\\in G$ and some $a\\in \\ell ^1(G)$. This class of \"convolution-dominated\" matrices forms a Banach-*-algebra contained in the algebra of bounded operators on $\\ell ^2(G)$. Our main result shows that the inverse of a convolution-dominated matrix is again convolution-dominated, provided that $G$ is amenable and rigidly symmetric. For abelian groups this result goes back to Gohberg, Baskakov, and others, for non-abelian groups completely different techniques are required, such as generalized $L^1$-algebras and the symmetry of group algebras."}
{"category": "Math", "title": "The Lex-Plus-Powers Conjecture holds for pure powers", "abstract": "We prove Evans' Lex-Plus-Powers Conjecture for ideals containing a monomial regular sequence."}
{"category": "Math", "title": "Old and new examples of scale functions for spectrally negative Levy processes", "abstract": "We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Levy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations."}
{"category": "Math", "title": "An exact minimum degree condition for Hamilton cycles in oriented graphs", "abstract": "We show that every sufficiently large oriented graph with minimum in- and outdegree at least (3n-4)/8 contains a Hamilton cycle. This is best possible and solves a problem of Thomassen from 1979."}
{"category": "Math", "title": "On a problem of Molluzzo concerning Steinhaus triangles in finite cyclic groups", "abstract": "Let $X$ be a finite sequence of length $m\\geq 1$ in $\\mathbb{Z}/n\\mathbb{Z}$. The \\textit{derived sequence} $\\partial X$ of $X$ is the sequence of length $m-1$ obtained by pairwise adding consecutive terms of $X$. The collection of iterated derived sequences of $X$, until length 1 is reached, determines a triangle, the \\textit{Steinhaus triangle $\\Delta X$ generated by the sequence $X$}. We say that $X$ is \\textit{balanced} if its Steinhaus triangle $\\Delta X$ contains each element of $\\mathbb{Z}/n\\mathbb{Z}$ with the same multiplicity. An obvious necessary condition for $m$ to be the length of a balanced sequence in $\\mathbb{Z}/n\\mathbb{Z}$ is that $n$ divides the binomial coefficient $\\binom{m+1}{2}$. It is an open problem to determine whether this condition on $m$ is also sufficient. This problem was posed by Hugo Steinhaus in 1963 for $n=2$ and generalized by John C. Molluzzo in 1976 for $n\\geq3$. So far, only the case $n=2$ has been solved, by Heiko Harborth in 1972. In this paper, we answer positively Molluzzo's problem in the case $n=3^k$ for all $k\\geq1$. Moreover, for every odd integer $n\\geq3$, we construct infinitely many balanced sequences in $\\mathbb{Z}/n\\mathbb{Z}$. This is achieved by analysing the Steinhaus triangles generated by arithmetic progressions. In contrast, for any $n$ even with $n\\geq4$, it is not known whether there exist infinitely many balanced sequences in $\\mathbb{Z}/n\\mathbb{Z}$. As for arithmetic progressions, still for $n$ even, we show that they are never balanced, except for exactly 8 cases occurring at $n=2$ and $n=6$."}
{"category": "Math", "title": "Calculating conjugacy classes in Sylow $p$-subgroups of finite Chevalley groups", "abstract": "In earlier work, the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow $p$-subgroup $U(q)$ of a finite Chevalley group $G(q)$, valid when $q$ is a power of a good prime for $G(q)$. In this paper we develop this algorithm and discuss an implementation in the computer algebra language {\\sf GAP}. Using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in $U(q)$, when $G(q)$ is of rank at most 6. In these cases, we observe that the number of conjugacy classes of $U(q)$ is given by a polynomial in $q$ with integer coefficients."}
{"category": "Math", "title": "Algebras that satisfy Auslander's condition on vanishing of cohomology", "abstract": "Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture - by a 2003 counterexample due to Jorgensen and Sega - motivates the consideration of the class of rings that do satisfy Auslander's condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander-Reiten Conjecture is proved for AC rings, and Auslander's G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex."}
{"category": "Math", "title": "Extended Hyperbolicity", "abstract": "Given a complex space $X$, we cosidered the problem of finding a {\\it hyperbolic model} of $X$. This is an object $\\ip(X)$ with a morphism $i:X\\to \\ip(X)$ in such a way that $\\ip(X)$ is ``hyperbolic'' in a suitable sense and $i$ is as close as possible to be an isomorphism. Using the theory of model categories, we found a definition of hyperbolic simplicial sheaf (for the strong topology) that extends the classical one of Brody for complex spaces. We prove the existence of hyperbolic models for any simplicial sheaf. Furthermore, the morphism $i$ can be taken to be a cofibration and an affine weak equivalence (in an algebraic setting, Morel and Voevodsky called it an $\\aff$ weak equivalence). Imitating one possible definition of homotopy groups for a topological space, we defined the {\\it holotopy} groups for a simplicial sheaf and showed that their vanishing in ``positive'' degrees is a necessary condition for a sheaf to be hyperbolic. We deduce that if $X$ is a complex space with a non zero holotopy group in positive degree, then its hyperbolic model (that in general will only be a simplicial sheaf) cannot be weakly equivalent to a hyperbolic complex space (in particular is not itself hyperbolic). We finish the manuscript by applying these results and a {\\it topological realization functor}, constructed in the previous section, to prove that the hyperbolic models of the complex projective spaces cannot be weakly equivalent to hyperbolic complex spaces."}
{"category": "Math", "title": "Invariant tensors and graphs", "abstract": "We describe a correspondence between GL_n-invariant tensors and graphs, and show how this correspondence accomodates various types of symmetries and orientations."}
{"category": "Math", "title": "Operator-valued Herglotz kernels and functions of positive real part on the ball", "abstract": "We describe several classes of holomorphic functions of positive real part on the unit ball; each is characterized by an operator-valued Herglotz formula. Motivated by results of J.E. McCarthy and M. Putinar, we define a family of weighted Cauchy-Fantappi\\`e pairings on the ball and establish duality relations between certain pairs of classes, and in particular we identify the dual of the positive Schur class. We also establish the existence of self-dual classes with respect to this pairing, and identify some extreme points of the positive Schur class."}
{"category": "Math", "title": "Le lemme fondamental pour les algebres de Lie", "abstract": "We propose a proof for conjectures of Langlands, Shelstad and Waldspurger known as the fundamental lemma for Lie algebras and the non-standard fundamental lemma. The proof is based on a study of the decomposition of the l-adic cohomology of the Hitchin fibration into direct sum of simple perverse sheaves."}
{"category": "Math", "title": "Semigroups of valuations dominating local domains", "abstract": "A new criterion is given for a semigroup to be the semigroup of a valuation dominating an equicharacteristic local domain. The criterion is used to construct examples of well ordered subsemigroups of the positive rational numbers which are of ordinal type omega, but are not the value semigroup of a valuation on an equicharacteristic noetherian local domain. This shows that the necessary conditions on value semigroups given in Appendix 3 to Zariski and Samuel's book ``Commutative Algebra'' are not sufficient."}
{"category": "Math", "title": "On the wonderful compactification", "abstract": "These lecture notes explain the construction and basic properties of the wonderful compactification of a complex semisimple group of adjoint type. An appendix discusses the more general case of a semisimple symmetric space."}
{"category": "Math", "title": "An Alternative Prior Process for Nonparametric Bayesian Clustering", "abstract": "Prior distributions play a crucial role in Bayesian approaches to clustering. Two commonly-used prior distributions are the Dirichlet and Pitman-Yor processes. In this paper, we investigate the predictive probabilities that underlie these processes, and the implicit \"rich-get-richer\" characteristic of the resulting partitions. We explore an alternative prior for nonparametric Bayesian clustering -- the uniform process -- for applications where the \"rich-get-richer\" property is undesirable. We also explore the cost of this process: partitions are no longer exchangeable with respect to the ordering of variables. We present new asymptotic and simulation-based results for the clustering characteristics of the uniform process and compare these with known results for the Dirichlet and Pitman-Yor processes. We compare performance on a real document clustering task, demonstrating the practical advantage of the uniform process despite its lack of exchangeability over orderings."}
{"category": "Math", "title": "Analytic continuation from a family of lines", "abstract": "Given a function f in the exterior of a convex curve in the real plane, we prove that if the restrictions of f to the tangent lines to the curve extend as entire functions, then the function f is an entire function of two variables."}
{"category": "Math", "title": "The representations of cyclotomic BMW algebras", "abstract": "In this paper, we prove that the cyclotomic BMW algebras B2p+1,n are cellular in the sense of [16]. We also classify the irreducible B2p+1,nmodules over a field."}
{"category": "Math", "title": "Minimality and nonergodicity on a family of flat surfaces in genus 3", "abstract": "We prove that a certain family of flat surfaces in genus 3 does not fulfill Veech's Dichotomy. These flat surfaces provide uncountably many minimal but nonergodic directions. The conditions on this family are a combinatorical one and an irrationality condition. The Arnoux-Yoccoz surface fulfills this conditions."}
{"category": "Math", "title": "Kazhdan-Lusztig Basis and A Geometric Filtration of an affine Hecke Algebra, II", "abstract": "An affine Hecke algebras can be realized as an equivariant K-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the corresponding Lie algebra. In this paper we will show that the two-sided ideals are in fact the two-sided ideals of the affine Hecke algebra defined through two-sided cells of the corresponding affine Weyl group after the two-sided ideals are tensored by rational numbers field. This proves a weak form of a conjecture of Ginzburg proposed in 1987."}
{"category": "Math", "title": "Analytic Continuation of q-Euler numbers and polynomials", "abstract": "In this paper we study that the $q$-Euler numbers and polynomials are analytically continued to $E_q(s)$. A new formula for the Euler's $q$-Zeta function $\\zeta_{E,q}(s)$ in terms of nested series of $\\zeta_{E,q}(n)$ is derived. Finally we introduce the new concept of the dynamics of analytically continued $q$-Euler numbers and polynomials."}
{"category": "Math", "title": "On Universal Binary Hermitian Forms", "abstract": "Earnest and Khosravani, Iwabuchi, and Kim and Park recently gave a complete classification of the universal binary Hermitian forms. We give a unified proof of the universalities of these Hermitian forms, relying primarily on Ramanujan's list of universal quadratic forms and on the Bhargava-Hanke 290-Theorem. Our methods bypass nearly all of the ad hoc universality arguments required in the original classification."}
{"category": "Math", "title": "Global Gorenstein dimensions of polynomial rings and of direct products of rings", "abstract": "In this paper, we extend the well-known Hilbert's syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct products of rings. Our results generate examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension."}
{"category": "Math", "title": "Toric Ideals of Flow Polytopes", "abstract": "A referee found an error in the proof of the Main Theorem (\"toric ideals of flow polytopes are generated in degree 3\") that we could not fix. More precisely, the proof of Lemma 4.2.(ii) is incorrect. The results on Gr\\\"obner bases are untouched by this. ----- We show that toric ideals of flow polytopes are generated in degree 3. This was conjectured by Diaconis and Eriksson for the special case of the Birkhoff polytope. Our proof uses a hyperplane subdivision method developed by Haase and Paffenholz. It is known that reduced revlex Gr\\\"obner bases of the toric ideal of the Birkhoff polytope $B_n$ have at most degree $n$. We show that this bound is sharp for some revlex term orders. For $(m\\times n)$-transportation polytopes, a similar result holds: they have Gr\\\"obner bases of at most degree $\\lfloor mn/2\\rfloor$. We construct a family of examples, where this bound is sharp."}
{"category": "Math", "title": "Some examples of absolute continuity of measures in stochastic fluid dynamics", "abstract": "A non linear Ito equation in a Hilbert space is studied by means of Girsanov theorem. We consider a non linearity of polynomial growth in suitable norms, including that of quadratic type which appears in the Kuramoto-Sivashinsky equation and in the Navier-Stokes equation. We prove that Girsanov theorem holds for the 1-dimensional stochastic Kuramoto-Sivashinsky equation and for a modification of the 2- and 3-dimensional stochastic Navier-Stokes equation. In this way, we prove existence and uniqueness of solutions for these stochastic equations. Moreover, the asymptotic behaviour for large time is characterized."}
{"category": "Math", "title": "Adjusted Bayesian inference for selected parameters", "abstract": "We address the problem of providing inference from a Bayesian perspective for parameters selected after viewing the data. We present a Bayesian framework for providing inference for selected parameters, based on the observation that providing Bayesian inference for selected parameters is a truncated data problem. We show that if the prior for the parameter is non-informative, or if the parameter is a \"fixed\" unknown constant, then it is necessary to adjust the Bayesian inference for selection. Our second contribution is the introduction of Bayesian False Discovery Rate controlling methodology,which generalizes existing Bayesian FDR methods that are only defined in the two-group mixture model.We illustrate our results by applying them to simulated data and data froma microarray experiment."}
{"category": "Math", "title": "Parametrizing recollement data", "abstract": "We give a general parametrization of all the recollement data for a triangulated category with a set of generators. From this we deduce a characterization of when a perfectly generated (or aisled) triangulated category is a recollement of triangulated categories generated by a single compact object. Also, we use homological epimorphisms of dg categories to give a complete and explicit description of all the recollement data for (or smashing subcategories of) the derived category of a k-flat dg category. In the final part we give a bijection between smashing subcategories of compactly generated triangulated categories and certain ideals of the subcategory of compact objects, in the spirit of Henning Krause's work. This bijection implies the following weak version of the Generalized Smashing Conjecture: in a compactly generated triangulated category every smashing subcategory is generated by a set of Milnor colimits of compact objects."}
{"category": "Math", "title": "Quantum triads: an algebraic approach", "abstract": "A concept of quantum triad and its solution is introduced. It represents a common framework for several situations where we have a quantale with a right module and a left module, provided with a bilinear inner product. Examples include Van den Bossche quantaloids, quantum frames, simple and Galois quantales, operator algebras, or orthomodular lattices."}
{"category": "Math", "title": "Sharp estimates of the Kobayashi metric and Gromov hyperbolicity", "abstract": "Let D be a smooth relatively compact and strictly J-pseudoconvex domain in a four dimensional almost complex manifold (M,J). We give sharp estimates of the Kobayashi metric. Our approach is based on an asymptotic quantitative description of both the domain D and the almost complex structure J near a boundary point. Following Z.M.Balogh and M.Bonk, these sharp estimates provide the Gromov hyperbolicity of the domain D."}
{"category": "Math", "title": "On torsion torsionfree triples", "abstract": "We study torsion torsionfree(=TTF) triples in abelian and triangulated categories. (Notice that TTF triples in a triangulated category are essentially in bijection with recollement data for this triangulated category.) In particular, we complete Jans' characterization of split TTF triples on a category of modules, prove a weak version of the Generalized Smashing Conjecture, use homological epimorphisms of differential graded(=dg) categories to give an explicite description of all the TTF triples in the derived category of a k-flat dg category and develope an unbounded approach to Koenig's theorem on recollements of right bounded derived categories of ordinary algebras."}
{"category": "Math", "title": "The quotient of a complete symmetric variety", "abstract": "We study the quotient of a completion of a symmetric variety G/H under the action of H. We prove that this is isomorphic to the closure of the image of an isotropic torus under the action of the restricted Weyl group. In the case the completion is smooth and toroidal we describe the set of semistable points."}
{"category": "Math", "title": "Twisted Yangians and Mickelsson Algebras II", "abstract": "We introduce a skew analogue of the composition of the Cherednik and Drinfeld functors for twisted Yangians. Our definition is based on the skew Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group."}
{"category": "Math", "title": "A remark on amoebas in higher codimensions", "abstract": "It is shown that tube sets over amoebas of algebraic varieties (and, more generally, of almost periodic holomorphic chains) of dimension q are q-pseudoconcave in the sense of Rothstein. This is a direct consequence of a representation of such sets as supports of positive closed currents."}
{"category": "Math", "title": "A New Approach on Constant Angle Surfaces in E^3", "abstract": "In this paper we study constant angle surfaces in Euclidean 3-space. Even that the result is a consequence of some classical results involving the Gauss map (of the surface), we give another approach to classify all surfaces for which the unit normal makes a constant angle with a fixed direction."}
{"category": "Math", "title": "On the Neron-Severi group of surfaces with many lines", "abstract": "For a binary quartic form $\\phi$ without multiple factors, we classify the quartic K3 surfaces $\\phi(x,y)=\\phi(z,t)$ whose Neron-Severi group is (rationally) generated by lines. For generic binary forms $\\phi$, $\\psi$ of prime degree without multiple factors, we prove that the Neron-Severi group of the surface $\\phi(x,y)=\\psi(z,t)$ is rationally generated by lines."}
{"category": "Math", "title": "Dimension vs. Genus: A surface realization of the little k-cubes and an E_{\\infty}-operad", "abstract": "We define a new $E_{\\infty}$ operad based on surfaces with foliations which contains $E_k$ sub-operads. We construct CW models for these operads and provide applications of these models by giving actions on Hochschild complexes -thus making contact with string topology-, by giving explicit cell representatives for the Dyer-Lashof-Cohen operations for the 2-cubes and by constructing new $\\Omega$ spectra. The underlying novel principle is that we can trade genus in the surface representation vs. the dimension $k$ of the little $k$-cubes."}
{"category": "Math", "title": "Polyhedral hyperbolic metrics on surfaces", "abstract": "In the last section of \\cite{CompHyp} it is proved that the map $\\mathcal{I}$ is a finite-sheeted covering map between $\\mathcal{P}$ and $\\mathcal{M}$. As $\\mathcal{M}$ is simply connected it is deduced that $\\mathcal{I}$ is a homeomorphism. The fact that $\\mathcal{P}$ is connected is missing. Here we provide a proof."}
{"category": "Math", "title": "Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension", "abstract": "We consider the real-valued centered Gaussian field on the four-dimensional integer lattice, whose covariance matrix is given by the Green's function of the discrete Bilaplacian. This is interpreted as a model for a semiflexible membrane. $d=4$ is the critical dimension for this model. We discuss the effect of a hard wall on the membrane, via a multiscale analysis of the maximum of the field. We use analytic and probabilistic tools to describe the correlation structure of the field."}
{"category": "Math", "title": "Sur les ensembles d'entiers reconnaissables", "abstract": "Let U and V be two Bertrand numeration systems, and, a and b the two Parry numbers there are naturally associated with. Suppose they are multiplicatively independent. We prove that, if E is a subset of positive integers which is both U and V recognizable, then E is a finite union of arithmetical progressions."}
{"category": "Math", "title": "Hamiltonian handleslides for Heegaard Floer homology", "abstract": "A $g$-tuple of disjoint, linearly independent circles in a Riemann surface of genus $g$ determines a `Heegaard torus' in its $g$-fold symmetric product. Changing the circles by a handleslide produces a new torus. It is proved that, for symplectic forms with certain properties, these two tori are Hamiltonian-isotopic Lagrangian submanifolds. This provides an alternative route to the handleslide-invariance of Ozsvath-Szabo's Heegaard Floer homology."}
{"category": "Math", "title": "Counting One-Vertex Maps", "abstract": "The number of distinct maps (pre-maps) with a single vertex and valence $d$ is computed for any value of $d$. The types of maps (pre-maps) that we consider depend on whether the underlaying graph (pre-graph) is signed or unsigned and directed or undirected."}
{"category": "Math", "title": "Sampling-Based Resolution-Complete Algorithms for Safety Falsification of Linear Systems", "abstract": "In this paper, we describe a novel approach for checking safety specifications of a dynamical system with exogenous inputs over infinite time horizon that is guaranteed to terminate in finite time with a conclusive answer. We introduce the notion of resolution completeness for analysis of safety falsification algorithms and propose sampling-based resolution-complete algorithms for safety falsification of linear time-invariant discrete time systems over infinite time horizon. The algorithms are based on deterministic incremental search procedures, exploring the reachable set for feasible counter examples to safety at increasing resolution levels of the input. Given a target resolution of inputs, the algorithms are guaranteed to terminate either with a reachable state that violates the safety specification, or prove that no input exists at the given resolution that violates the specification."}
{"category": "Math", "title": "Polar orthogonal representations of real reductive algebraic groups", "abstract": "We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generalizes some results of the structural theory of real reductive Lie algebras."}
{"category": "Math", "title": "Discrete bidding games", "abstract": "We study variations on combinatorial games in which, instead of alternating moves, the players bid with discrete bidding chips for the right to determine who moves next. We consider both symmetric and partisan games, and explore differences between discrete bidding games and Richman games, which allow real-valued bidding. Unlike Richman games, discrete bidding game variations of many familiar games, such as chess, Connect Four, and even Tic-Tac-Toe, are suitable for recreational play. We also present an analysis of Tic-Tac-Toe for both discrete and real-valued bidding."}
{"category": "Math", "title": "No Finite Invariant Density for Misiurewicz Exponential Maps", "abstract": "For exponential mappings such that the orbit of the only singular value 0 is bounded, it is shown that no integrable density invariant under the dynamics exists on the complex plane."}
{"category": "Math", "title": "Minimal polynomial of an exponential automorphism of C^n", "abstract": "We show that the minimal polynomial of a polynomial exponential automorphism F of C^n (i.e. F=exp(D), where D is a locally nilpotent derivation) is of the form \\mu_F(T)=(T-1)^d, d=min{m \\in N: D^m(X_i)=0 for i=1,...,n}."}
{"category": "Math", "title": "Towards a classification of the tridiagonal pairs", "abstract": "Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. Let $End(V)$ denote the $K$-algebra consisting of all $K$-linear transformations from $V$ to $V$. We consider a pair $A,A^* \\in End(V)$ that satisfy (i)--(iv) below: (i) Each of $A,A^*$ is diagonalizable. (ii) There exists an ordering $\\{V_i\\}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \\subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \\leq i \\leq d$, where $V_{-1}=0$ and $V_{d+1}=0$. (iii) There exists an ordering $\\{V^*_i\\}_{i=0}^\\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \\subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \\leq i \\leq \\delta$, where $V^*_{-1}=0$ and $V^*_{\\delta+1}=0$. (iv) There is no subspace $W$ of $V$ such that $AW \\subseteq W$, $A^* W \\subseteq W$, $W \\neq 0$, $W \\neq V$. We call such a pair a {\\em tridiagonal pair} on $V$. Let $E^*_0$ denote the element of $End(V)$ such that $(E^*_0-I)V^*_0=0$ and $E^*_0V^*_i=0$ for $1 \\leq i \\leq d$. Let $D$ (resp. $D^*$) denote the $K$-subalgebra of $End(V)$ generated by $A$ (resp. $A^*$). In this paper we prove that the span of $E^*_0 D D^*DE^*_0$ equals the span of $E^*_0D E^*_0DE^*_0$, and that the elements of $E^*_0 D E^*_0$ mutually commute. We relate these results to some conjectures of Tatsuro Ito and the second author that are expected to play a role in the classification of tridiagonal pairs."}
{"category": "Math", "title": "From Bombieri's Mean Value Theorem to the Riemann Hypothesis", "abstract": "From Bombieri's mean value theorem one can deduce the prime number theorem being equivalent to the Riemann hypothesis and the least prime P(q) satisfying P(q)= O(q^2 [ln q]^32) in any arithmetic progressions with common difference q."}
{"category": "Math", "title": "A proof of the DDVV conjecture and its equality case", "abstract": "In this paper, we give a proof of the DDVV conjecture which is a pointwise inequality involving the scalar curvature, the normal scalar curvature and the mean curvature on a submanifold of a real space form. Furthermore we solved the problem of its equality case."}
{"category": "Math", "title": "Auslander-Reiten theory for simply connected differential graded algebras", "abstract": "Peter Jorgensen introduced the Auslander-Reiten quiver of a simply connected Poincare duality space. He showed that its components are of the form ZA_infty and that the Auslander-Reiten quiver of a d-dimensional sphere consists of d-1 such components. In this thesis we show that this is the only case where finitely many components appear. More precisely, we construct families of modules, where for each family, each module lies in a different component. Depending on the cohomology dimensions of the differential graded algebras which appear, this is either a discrete family or an n-parameter family for all n."}
{"category": "Math", "title": "Covering theorems for Artinian rings", "abstract": "The covering properties of Artinian rings which depend on their additive structure only, are investigated."}
{"category": "Math", "title": "On Potentially 3-regular graph graphic Sequences", "abstract": "For given a graph $H$, a graphic sequence $\\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\\pi$ containing $H$ as a subgraph. In this paper, we characterize the potentially $H$-graphic sequences where $H$ denotes 3-regular graph with 6 vertices. In other words, we characterize the potentially $K_{3,3}$ and $K_6-C_6$-graphic sequences where $K_{r,r}$ is an $r\\times r$ complete bipartite graph. One of these characterizations implies a theorem due to Yin [25]."}
{"category": "Math", "title": "Structure of normal twisted group rings", "abstract": "In this paper we describe those groups G and commutative rings K for which the twisted group rings K*G are f-normal."}
{"category": "Math", "title": "Invariant boundary distributions for finite graphs", "abstract": "Let $\\Gamma$ be the fundamental group of a finite connected graph $\\mathcal G$. Let $\\mathfrak M$ be an abelian group. A {\\it distribution} on the boundary $\\partial\\Delta$ of the universal covering tree $\\Delta$ is an $\\mathfrak M$-valued measure defined on clopen sets. If $\\mathfrak M$ has no $\\chi(\\mathcal G)$-torsion then the group of $\\Gamma$-invariant distributions on $\\partial\\Delta$ is isomorphic to $H_1(\\mathcal G,\\mathfrak M)$."}
{"category": "Math", "title": "A stronger model for peg solitaire, II", "abstract": "The main problem addressed here is to decide whether it is possible or not to go from a given position on a peg-solitaire board to another one. No non-trivial sufficient conditions are known, but tests have been devised to show impossibility. We expose the way these tests work in a unified formalism and provide a new test which is strictly stronger than all previous ones."}
{"category": "Math", "title": "On Appell Sets and the Fueter-Sce Mapping", "abstract": "It is proved, that the recently discussed Appell polynomials in Clifford algebras are the Fueter-Sce extension of the complex monomials z^k. Furthermore, it is shown, for which complex functions the Fueter-Sce extension and the extension method using Appell polynomials coincide."}
{"category": "Math", "title": "The Goto numbers of parameter ideals", "abstract": "Let Q be a parameter ideal of a Noetherian local ring (R,m). The Goto number g(Q) of Q is the largest integer g such that Q:m^g is integral over Q. We examine the values of g(Q) as Q varies over the parameter ideals of R. We concentrate mainly on the case where dim R = 1, and many of our results concern parameter ideals of a numerical semigroup ring."}
{"category": "Math", "title": "Meromorphic functions of one complex variable. A survey", "abstract": "This is an appendix to the English translation of the book by A. A. Goldberg and I. V. Ostrovskii, Distribution of values of meromorphic functions, Moscow, Nauka, 1970. An English translation of this book is to be published soon by the AMS. In this appendix we survey the results obtained on the topics of the book after 1970."}
{"category": "Math", "title": "Some remarks on tangent martingale difference sequences in $L^1$-spaces", "abstract": "Let X be a Banach space. Suppose that for all $p\\in (1, \\infty)$ a constant $C_{p,X}$ depending only on X and p exists such that for any two X-valued martingales f and g with tangent martingale difference sequences one has \\[\\E\\|f\\|^p \\leq C_{p,X} \\E\\|g\\|^p (*).\\] This property is equivalent to the UMD condition. In fact, it is still equivalent to the UMD condition if in addition one demands that either f or g satisfy the so-called (CI) condition. However, for some applications it suffices to assume that (*) holds whenever g satisfies the (CI) condition. We show that the class of Banach spaces for which (*) holds whenever only g satisfies the (CI) condition is more general than the class of UMD spaces, in particular it includes the space L^1. We state several problems related to (*) and other decoupling inequalities."}
{"category": "Math", "title": "Combinatorics in affine flag varieties", "abstract": "The Littelmann path moel gives a realization of the crystals of integrable representations of symmetrizable Kac-Moody Lie algebras. Recent work of Gaussent-Littelmann and others has demonstrated a connection between this model and the geometry of the loop Grassmannian. The alcove walk model is a version of the path model which is intimately connected to the combinatorics of the affine Hecke algebra. In this paper we define a refined alcove walk model which encodes the points of the affine flag variety. We show that this combinatorial indexing naturally indexes the \"cells\" in generalized Mirkovic-Vilonen intersections."}
{"category": "Math", "title": "Koppelman formulas and the $\\dbar$-equation on an analytic space", "abstract": "Let $X$ be an analytic space of pure dimension. We introduce a formalism to generate intrinsic weighted Koppelman formulas on $X$ that provide solutions to the $\\dbar$-equation. We prove that if $\\phi$ is a smooth $(0,q+1)$-form on a Stein space $X$ with $\\dbar\\phi=0$, then there is a smooth $(0,q)$-form $\\psi$ on $X_{reg}$ with at most polynomial growth at $X_{sing}$ such that $\\dbar\\psi=\\phi$. The integral formulas also give other new existence results for the $\\dbar$-equation and Hartogs theorems, as well as new proofs of various known results."}
{"category": "Math", "title": "Hermitian integral geometry", "abstract": "We give in explicit form the principal kinematic formula for the action of the affine unitary group on $\\C^n$, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce also several other canonical bases for the algebra of unitary-invariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone elements."}
{"category": "Math", "title": "Nonparametric estimation of a convex bathtub-shaped hazard function", "abstract": "In this paper, we study the nonparametric maximum likelihood estimator (MLE) of a convex hazard function. We show that the MLE is consistent and converges at a local rate of $n^{2/5}$ at points $x_0$ where the true hazard function is positive and strictly convex. Moreover, we establish the pointwise asymptotic distribution theory of our estimator under these same assumptions. One notable feature of the nonparametric MLE studied here is that no arbitrary choice of tuning parameter (or complicated data-adaptive selection of the tuning parameter) is required."}
{"category": "Math", "title": "Convergence of multi-dimensional quantized $SDE$'s", "abstract": "We quantize a multidimensional $SDE$ (in the Stratonovich sense) by solving the related system of $ODE$'s in which the $d$-dimensional Brownian motion has been replaced by the components of functional stationary quantizers. We make a connection with rough path theory to show that the solutions of the quantized solutions of the $ODE$ converge toward the solution of the $SDE$. On our way to this result we provide convergence rates of optimal quantizations toward the Brownian motion for $\\frac 1q$-H\\\" older distance, $q>2$, in $L^p(\\P)$."}
{"category": "Math", "title": "Generalized incidence theorems, homogeneous forms, and sum-product estimates in finite fields", "abstract": "In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and volumes have also received much attention. In this paper we prove a relatively straightforward function version of an incidence results for points and planes previously established in \\cite{HI07} and \\cite{HIKR07}. As a consequence of our methods, we obtain sharp or near sharp results on the distribution of volumes determined by subsets of vector spaces over finite fields and the associated arithmetic expressions. In particular, our machinery enables us to prove that if $E \\subset {\\Bbb F}_q^d$, $d \\ge 4$, the $d$-dimensional vector space over a finite field ${\\Bbb F}_q$, of size much greater than $q^{\\frac{d}{2}}$, and if $E$ is a product set, then the set of volumes of $d$-dimensional parallelepipeds determined by $E$ covers ${\\Bbb F}_q$. This result is sharp as can be seen by taking $E$ to equal to $A \\times A \\times ... \\times A$, where $A$ is a sub-field of ${\\Bbb F}_q$ of size $\\sqrt{q}$. In three dimensions we establish the same result if $|E| \\gtrsim q^{{15/8}}$. We prove in three dimensions that the set of volumes covers a positive proportion of ${\\Bbb F}_q$ if $|E| \\ge Cq^{{3/2}}$. Finally we show that in three dimensions the set of volumes covers a positive proportion of ${\\Bbb F}_q$ if $|E| \\ge Cq^2$, without any further assumptions on $E$, which is again sharp as taking $E$ to be a 2-plane through the origin shows."}
{"category": "Math", "title": "Biflatness and Pseudo-Amenability of Segal algebras", "abstract": "We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, L1(G), and the Fourier algebra, A(G), of a locally compact group, G."}
{"category": "Math", "title": "Jumping Numbers on Algebraic Surfaces with Rational Singularities", "abstract": "In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored."}
{"category": "Math", "title": "Global fixed points for centralizers and Morita's Theorem", "abstract": "We prove a global fixed point theorem for the centralizer of a homeomorphism of the two dimensional disk $D$ that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers. As one application, we show that there is a finite index subgroup of the centralizer of a pseudo-Anosov homeomorphism with infinitely many global fixed points. As another application we give an elementary proof of Morita's Theorem, that the mapping class group of a closed surface $S$ of genus $g$ does not lift to the group of diffeormorphisms of $S$ and we improve the lower bound for $g$ from 5 to 3."}
{"category": "Math", "title": "On Existence and Uniqueness of Universal Enveloping Locally C*-Algebra for a Locally JB-Algebra", "abstract": "A theorem is presented on existence and uniqueness up to the topological *-isomorphism of universal locally C*-algebra for an arbitrary locally JB-algebra."}
{"category": "Math", "title": "Quantum invariants can provide sharp Heegaard genus bounds", "abstract": "Using Seifert fibered three-manifold examples of Boileau and Zieschang, we demonstrate that the Reshetikhin-Turaev quantum invariants may be used to provide a sharp lower bound on the Heegaard genus which is strictly larger than the rank of the fundamental group."}
{"category": "Math", "title": "On lattices and their ideal lattices, and posets and their ideal posets", "abstract": "For P a poset or lattice, let Id(P) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P)=Id(P)-\\{\\emptyset\\}. This note obtains various results to the effect that Id(P) is always, and id(P) often, \"essentially larger\" than P. In the first vein, we find that a poset P admits no \"<\"-respecting map (and so in particular, no one-to-one isotone map) from Id(P) into P, and, going the other way, that an upper semilattice S admits no semilattice homomorphism from any subsemilattice of itself onto Id(S). The slightly smaller object id(P) is known to be isomorphic to P if and only if P has ascending chain condition. This result is strengthened to say that the only posets P_0 such that for every natural number n there exists a poset P_n with id^n(P_n)\\cong P_0 are those having ascending chain condition. On the other hand, a wide class of cases is noted here where id(P) is embeddable in P. Counterexamples are given to many variants of the results proved."}
{"category": "Math", "title": "Simple Jordan conformal superalgebras", "abstract": "We classify simple finite Jordan conformal superalgebras and establish preliminary results for the classification of simple finite Jordan pseudoalgebras."}
{"category": "Math", "title": "Lifting Group Actions and Nonnegative Curvature", "abstract": "We show that all vector bundles over CP^2 which are not spin admit a complete metric with nonnegative sectional curvature. In the proof we construct a nonnegatively curved metric on the corresponding principle bundle by showing that it admits a cohomogeneity one action with singular orbits of codimension 2. This is closely related to the problem of when an action of G on the base of an L principle bundle lifts to the total space, such that the lift commutes with L. We solve this lifting problem for all SO(k) principle bundles over a 4-dimensional simply connected base B with G a cohomogeneity one action on B."}
{"category": "Math", "title": "A Note on Banach Principle for JW-algebras", "abstract": "In the sequel we establish the Banach Principle for semifinite JW-algebras without direct summand of type I sub 2, which extends the recent results of Chilin and Litvinov on the Banach Principle for semifinite von Neumann algebras to the case of JW-algebras."}
{"category": "Math", "title": "Sifting Function Partition for the Goldbach Problem", "abstract": "All sieve methods for the Goldbach problem sift out all the composite numbers; even though, strictly speaking, it is not necessary to do so and which is, in general, very difficult. Some new methods introduced in this paper show that the Goldbach problem can be solved under sifting out only some composite numbers. In fact, in order to prove the Goldbach conjecture, it is only necessary to show that there are prime numbers left in the residual integers after the initial sifting! This idea can be implemented by using one of the three methods called sifting function partition by integer sort, sifting function partition by intervals and comparative sieve method, respectively. These are feasible methods for solving both the Goldbach problem and the problem of twin primes. An added bonus of the above methods is the elimination of the indeterminacy of the sifting functions brought about by their upper and lower bounds."}
{"category": "Math", "title": "On the monochromatic Schur Triples type problem", "abstract": "We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $\\{x,y,x+ay\\}$ triples for $a \\geq 1$. We give a new proof of the original case of $a=1$. We show that the minimum number of such triples is at most $\\frac{n^2}{2a(a^2+2a+3)} + O(n)$ when $a \\geq 2$. We also find a new upper bound for the minimum number, over all $r$-colorings of $[1,n]$, of monochromatic Schur triples, for $r \\geq 3$."}
{"category": "Math", "title": "Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds", "abstract": "We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows that such actions on geometric 3-manifolds (in the sense of Thurston) are always geometric, i.e. there exist invariant locally homogeneous Riemannian metrics. This answers a question posed by Thurston in [32]."}
{"category": "Math", "title": "One curious proof of Fermat's little theorem", "abstract": "We give a proof of Fermat's little theorem which does not use nor arithmetic(Euclidean algorithm) neither algebra (group theory), but it rather employs the field of the formal power series Q((x)). The note is an example of a mathematical joke, though it contains a rigorous proof. (The paper will appear in print exactly as in the version v3)."}
{"category": "Math", "title": "Symmetric subgroups in modular group algebras", "abstract": "Let V(KG) be a normalised unit group of the modular group algebra of a finite p-group G over the field K of p elements. We introduce a notion of symmetric subgroups in V(KG) as subgroups invariant under the action of the classical involution of the group algebra KG. We study properties of symmetric subgroups and construct a counterexample to the conjecture by V.Bovdi, which states that V(KG)=<G,S*>, where S* is a set of symmetric units of V(KG)."}
{"category": "Math", "title": "Analysis of the physical Laplacian and the heat flow on a locally finite graph", "abstract": "We study the physical Laplacian and the corresponding heat flow on an infinite, locally finite graph with possibly unbounded valence."}
{"category": "Math", "title": "Classification problems for system of forms and linear mappings", "abstract": "We devise a method that reduces the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings. Canonical matrices of (i) bilinear or sesquilinear forms, (ii) pairs of symmetric, skew-symmetric, or Hermitian forms, (iii) isometric or selfadjoint operators on a space with nonsingular symmetric, or skew-symmetric, or Hermitian form are obtained over any field of characteristic not 2 up to classification of Hermitian forms over its finite extensions."}
{"category": "Math", "title": "A Modified Version of Free Orbit-Dimension of von Neumann Algebras", "abstract": "Based on the notion of free orbit-dimension introduced by D. Hadwin and J. Shen [4], we introduce a new invariant on finite von Neumann algebras that do not necessarily act on separable Hilbert space. We show that this invariant is independent on the generating set, and we extend some results in [4] to von Neumann algebras that are not finitely generated."}
{"category": "Math", "title": "Semiclassical second microlocal propagation of regularity and integrable systems", "abstract": "We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold $X$ with respect to a Lagrangian submanifold of $T^*X.$ The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the consideration of a model case, that of the zero section of $T^*\\mathbb{R}^n,$ and conjugation by appropriate Fourier integral operators. We prove a propagation theorem for the associated wavefront set analogous to H\\\"ormander's theorem for operators of real principal type. As an application, we consider the propagation of Lagrangian regularity on invariant tori for quasimodes (e.g. eigenfunctions) of an operator with completely integrable classical hamiltonian. We prove a secondary propagation result for second wavefront set which implies that even in the (extreme) case of Lagrangian tori with all frequencies rational, provided a nondegeneracy assumption holds, Lagrangian regularity either spreads to fill out a whole torus or holds nowhere locally on it."}
{"category": "Math", "title": "Rigidity of Conformally Compact Manifolds with the Round Sphere as the Conformal Infinity", "abstract": "In this paper we prove that under a lower bound on the Ricci curvature and an asymptotic assumption on the scalar curvature, a complete conformally compact manifold $(M^{n+1},g)$, with a pole $p$ and with the conformal infinity in the conformal class of the round sphere, has to be the hyperbolic space."}
{"category": "Math", "title": "Introduction to the Prisoners Versus Guards Game", "abstract": "We introduce a two-player game in which one and his/her opponent attempt to pack as many ``prisoners'' as possible on the squares of an n-by-n checkerboard; each prisoner has to be ``protected'' by at least as many guards as the number of the other prisoners adjacent. Initially, the board is covered entirely with guards. The players take turns adjusting the board configuration using one of the following rules in each turn: I. Replace one guard with a prisoner of the player's color. II. Replace one prisoner of either color with a guard and replace two other guards with prisoners of the player's color. We analyze winning strategies for small n (n<5) and the maximum number of prisoners in general. We show that this maximum is less than (7n^2+4n)/11 and conjecture it is more likely 3n^2/5+O(n)."}
{"category": "Math", "title": "Classification of 5-dimensional MD-algebras having commutive derived ideals", "abstract": "In this paper, we study a subclass of the class of MD-algebras, i.e., the class of solvable real Lie algebras such that the K-orbits of its corresponding connected and simply connected Lie groups are either orbits of dimension zero or orbits with maximal dimensions. Our main result is to classify, up to isomorphism, all the 5-dimensional MD-algebras having commutative derived ideals."}
{"category": "Math", "title": "Dynamics of Automorphisms of Compact Complex Manifolds", "abstract": "We give an algebro-geometric approach towards the dynamics of automorphisms/endomorphisms of projective varieties or compact K\\\"ahler manifolds, try to determine the building blocks of automorphisms /endomorphisms, and show the relation between the dynamics of automorphisms/endomorphisms and the geometry of the underlying manifolds."}
{"category": "Math", "title": "Batch kernel SOM and related Laplacian methods for social network analysis", "abstract": "Large graphs are natural mathematical models for describing the structure of the data in a wide variety of fields, such as web mining, social networks, information retrieval, biological networks, etc. For all these applications, automatic tools are required to get a synthetic view of the graph and to reach a good understanding of the underlying problem. In particular, discovering groups of tightly connected vertices and understanding the relations between those groups is very important in practice. This paper shows how a kernel version of the batch Self Organizing Map can be used to achieve these goals via kernels derived from the Laplacian matrix of the graph, especially when it is used in conjunction with more classical methods based on the spectral analysis of the graph. The proposed method is used to explore the structure of a medieval social network modeled through a weighted graph that has been directly built from a large corpus of agrarian contracts."}
{"category": "Math", "title": "A Note on Testing of Hypothesis", "abstract": "In this paper, a problem of testing is discussed when the samples have been drawn from the normal distribution. The study of hypothesis testing is also extended to Baye's set up."}
{"category": "Math", "title": "Inequalities and Ehrhart $\\delta$-Vectors", "abstract": "For any lattice polytope $P$, we consider an associated polynomial $\\bar{\\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known inequalities satisfied by the coefficients of the Ehrhart $\\delta$-vector of a lattice polytope. We also provide combinatorial proofs of two results of Stanley that were previously established using techniques from commutative algebra. Finally, we give a necessary numerical criterion for the existence of a regular unimodular lattice triangulation of the boundary of a lattice polytope."}
{"category": "Math", "title": "Cyclotomic Solomon Algebras", "abstract": "This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type $G(r,1,n)$. As with the Solomon descent algebra, our algebra has a basis given by sums of `distinguished' coset representatives for certain `reflection subgroups'. We explicitly describe the structure constants with respect to this basis and show that they are polynomials in $r$. This allows us to define a deformation, or $q$-analogue, of these algebras which depends on a parameter $q$. We determine the irreducible representations of all of these algebras and give a basis for their radicals. Finally, we show that the direct sum of cyclotomic Solomon algebras is canonically isomorphic to a concatenation Hopf algebra."}
{"category": "Math", "title": "Groups not acting on manifolds", "abstract": "In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that \"generic\" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the same property."}
{"category": "Math", "title": "Instant Evaluation and Demystification of $\\zeta(n),L(n,\\chi)$ that Euler,Ramanujan Missed - I", "abstract": "For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive integral argument in first variable.On using functionalequations this results in instant evaluation of Riemann Zeta function at positive even integral values of its argument and of Dirichlet L series at positive integral values of its argument,when the argument and the corresponding Dirichlet character are both even or both odd.We also obtain finite sum expression for any Dirichlet L series,when its argument is one.We also deal with Lerch's Zeta function on similar lines."}
{"category": "Math", "title": "A more accurate treatment of the problem of drawing the shortest line on a surface", "abstract": "E727 in the Enestrom index. This is a translation from the Latin original \"Accuratior evolutio problematis de linea brevissima in superficie quacunque ducenda\" (1779). Given a surface $pdx+qdy+rdz=0$, Euler wants to develop equations that give the geodesics on this surface. I am new to the calculus of variations, so it is not clear to me what steps follow from results that are previously known (like the Euler-Lagrange equation in the calculations) and what steps follow from earlier in this paper. I would appreciate comments from any readers who are familiar with calculus of variations."}
{"category": "Math", "title": "Gelfand-Kirillov Conjecture and Harish-Chandra Modules for Finite W-Algebras", "abstract": "We address two problems regarding the structure and representation theory of finite W-algebras associated with the general linear Lie algebras. Finite W-algebras can be defined either via the Whittaker model of Kostant or, equivalently, by the quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of the finite W-algebras. The second main result is a parametrization of finite families of irreducible Harish-Chandra modules by the characters of the Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Harish-Chandra modules for the finite W-algebras."}
{"category": "Math", "title": "Characteristic classes of $\\ai$-algebras", "abstract": "Standard combinatorial construction, due to Kontsevich, associates to any $\\ai$-algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We propose an alternative version of this construction based on noncommutative geometry and use it to prove that homotopy equivalent algebras give rise to the same cohomology classes. Along the way we re-prove Kontsevich's theorem relating graph homology to the homology of certain infinite dimensional Lie algebras. An application to topological conformal field theories is given."}
{"category": "Math", "title": "Diophantine inequality for equicharacteristic excellent Henselian local domains", "abstract": "G. Rond has proved a Diophantine type inequality for the field of quotients of the convergent or formal power series ring in multivariables. We generalize his theorem to the field of the quotients of an excellent Henselian local domain in equicharacteristic case."}
{"category": "Math", "title": "A BGG-type resolution for tensor modules over general linear superalgebra", "abstract": "We construct a Bernstein-Gelfand-Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique maximal submodule of a corresponding reducible Kac module is generated by its proper singular vector."}
{"category": "Math", "title": "Computation of 2-groups of narrow logarithmic divisor classes of number fields", "abstract": "We present an algorithm for computing the 2-group of narrow logarithmic divisor classes of degree 0 for number fields F. As an application, we compute in some cases the 2-rank of the wild kernel WK2(F)."}
{"category": "Math", "title": "Sur le sous-groupe des \\'el\\'ements de hauteur infinie du K2 d'un corps de nombres", "abstract": "By using the logarithmic approach of the classical kernels for the K2 of number fields, we compute the 2-rank of the wild kernel WK2(F) and the 2-rank of the subgroup of infinite heigh elements in K2(F) in terms of positive class groups for any number field F."}
{"category": "Math", "title": "Approche logarithmique des noyaux \\'etales sauvages des corps de nombres", "abstract": "We study the l-part of the the wild \\'etale kernels WK2i(F) of an arbitary number field F for a given prime l in connection with the logarithmic l-class groups. From the logarithmic arithmetic we deduce rank formulas, periodicity and reflection theorems, triviality characterizations and various consequences."}
{"category": "Math", "title": "G\\'en\\'eralisation d'un Th\\'eor\\`eme d'Iwasawa", "abstract": "We extend to convenient finite quotients of a noetherian Lambda-module the classical result of K. Iwasawa giving the asymptotic expression of the l-part of the number of ideal class groups in Zl-extensions of number fields. Then, in the arithmetic context, we compute the three characters associated by this way to the l-groups of T-infinitesimal S-classes in the cyclotomic tower and relate them to the classical invariants and the decomposition characters associated to the finite sets of places S and T. A main tool in this study is the so-called Spiegelungssatz of Georges Gras, which exchanges (wild or tame) ramification and decomposition. The main results of this arithmetical part extend those we obtained with Christian Maire in a previous article. The most intricate study of the wild contribution of the sets S and T involves a generalization of a classical result of R. Greenberg on the genus theory of cyclotomic towers."}
{"category": "Math", "title": "A new Algorithm for the Computation of logarithmic l-Class Groups of Number Fields", "abstract": "We present an algorithm for the computation of logarithmic l-class groups of number fields. Our principal motivation is the effective determination of the l-rank of the wild kernel in the K-theory of number fields."}
{"category": "Math", "title": "Estimating of $P(Y<X)$ in the Exponential case Based on Censored Samples", "abstract": "In this article, the estimation of reliability of a system is discussed $p(y<x)$ when strength, $X$, and stress, $Y$, are two independent exponential distribution with different scale parameters when the available data are type II Censored sample. Different methods for estimating the reliability are applied. The point estimators obtained are maximum likelihood estimator, uniformly minimum variance unbiased estimator, and Bayesian estimators based on conjugate and non informative prior distributions. A comparison of the estimates obtained is performed. Interval estimators of the reliability are also discussed."}
{"category": "Math", "title": "Groebner bases of nested configurations", "abstract": "In this paper we introduce a new and large family of configurations whose toric ideals possess quadratic Groebner bases. As an application, a generalization of algebras of Segre-Veronese type will be studied."}
{"category": "Math", "title": "Relative Hyperbolic Extensions of Groups and Cannon-Thurston Maps", "abstract": "Let $1\\to (K,K_1)\\to (G,N_G(K_1))\\to(Q,Q_1)\\to 1$ be a short exact sequence of pairs of finitely generated groups with $K$ strongly hyperbolic relative to proper subgroup $K_1$. Assuming that for all $g\\in G$ there exists $k\\in K$ such that $gK_1g^{-1}=kK_1k^{-1}$, we prove that there exists a quasi-isometric section $s\\colon Q \\to G$. Further we prove that if $G$ is strongly hyperbolic relative to the normalizer subgroup $N_G(K_1)$ and weakly hyperbolic relative to $K_1$, then there exists a Cannon-Thurston map for the inclusion $i\\colon\\Gamma_K\\to \\Gamma_G$."}
{"category": "Math", "title": "On the graph-connectivity of skeleta of convex polytopes", "abstract": "Given a $d$-dimensional convex polytope $P$ and nonnegative integer $k$ not exceeding $d-1$, let $G_k (P)$ denote the simple graph on the node set of $k$-dimensional faces of $P$ in which two such faces are adjacent if there exists a $(k+1)$-dimensional face of $P$ which contains them both. The graph $G_k (P)$ is isomorphic to the dual graph of the $(d-k)$-dimensional skeleton of the normal fan of $P$. For fixed values of $k$ and $d$, the largest integer $m$ such that $G_k (P)$ is $m$-vertex-connected for all $d$-dimensional polytopes $P$ is determined. This result generalizes Balinski's theorem on the one-dimensional skeleton of a $d$-dimensional convex polytope."}
{"category": "Math", "title": "On the extremal rays of the cone of positive, positive definite functions", "abstract": "The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\\R^d$. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there many other extremals than the gaussians, thus disproving a conjecture of G. Choquet and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of positive definite functions available in the literature are actually particular cases of the Choquet integral representations we obtain."}
{"category": "Math", "title": "The Riemannian manifolds with boundary and large symmetry", "abstract": "Sixty years ago, S. B. Myers and N. E. Steenrod ({\\it Ann. of Math.} {\\bf 40} (1939), 400-416) showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. Recently A. V. Bagaev and N. I. Zhukova ({\\it Siberian Math. J.} {\\bf 48} (2007), 579-592) proved the same result for a Riemannian orbifold. In this paper, we firstly show that the isometry group of a Riemannian manifold $M$ with boundary has dimension at most ${1/2} \\dim M (\\dim M-1)$. Then we completely classify such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension."}
{"category": "Math", "title": "A Survey of Simple Permutations", "abstract": "We survey the known results about simple permutations. In particular, we present a number of recent enumerative and structural results pertaining to simple permutations, and show how simple permutations play an important role in the study of permutation classes. We demonstrate how classes containing only finitely many simple permutations satisfy a number of special properties relating to enumeration, partial well-order and the property of being finitely based."}
{"category": "Math", "title": "Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation", "abstract": "We consider the standard first passage percolation model in $\\mathbb{Z}^d$ for $d\\geq 2$. We are interested in two quantities, the maximal flow $\\tau$ between the lower half and the upper half of the box, and the maximal flow $\\phi$ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for $\\tau$ in rational directions. Kesten and Zhang have proved the law of large numbers for $\\tau$ and $\\phi$ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly with the surface $s$ of the basis of the box, with the same deterministic speed. We study the probabilities that the rescaled variables $\\tau /s$ and $\\phi /s$ are abnormally small. For $\\tau$, the box can have any orientation, whereas for $\\phi$, we require either that the box is sufficiently flat, or that its sides are parallel to the coordinate hyperplanes. We show that these probabilities decay exponentially fast with $s$, when $s$ grows to infinity. Moreover, we prove an associated large deviation principle of speed $s$ for $\\tau /s$ and $\\phi /s$, and we improve the conditions required to obtain the law of large numbers for these variables."}
{"category": "Math", "title": "Detecting change-points in a discrete distribution via model selection", "abstract": "This paper is concerned with the detection of multiple change-points in the joint distribution of independent categorical variables. The procedures introduced rely on model selection and are based on a penalized least-squares criterion. Their performance is assessed from a nonasymptotic point of view. Using a special collection of models, a preliminary estimator is built. According to an existing model selection theorem, it satisfies an oracle-type inequality. Moreover, thanks to an approximation result demonstrated in this paper, it is also proved to be adaptive in the minimax sense. In order to eliminate some irrelevant change-points selected by that first estimator, a two-stage procedure is proposed, that also enjoys some adaptivity property. Besides, the first estimator can be computed with a complexity only linear in the size of the data. A heuristic method allows to implement the second procedure quite satisfactorily with the same computational complexity."}
{"category": "Math", "title": "On Tsfasman--Vl\\u{a}du\\c{t} Invariants of Infinite Global Fields", "abstract": "In this article we study certain asymptotic properties of global fields. We consider the set of Tsfasman-Vladuts invariants of infinite global fields and answer some natural questions arising from their work. In particular, we prove the existence of infinite global fields having finitely many strictly positive invariants at given places, and the existence of infinite number fields with certain prescribed invariants being zero. We also give precisions on the deficiency of infinite global fields and on the primes decomposition in those fields."}
{"category": "Math", "title": "Laws of Large Numbers for Continuous Belief Measures on Compact Spaces", "abstract": "We prove for outer continuous belief measures defined on compact spaces strong and weak laws of large numbers as Kolmogorov's one for measures. These results contribute to M. Marinacci's (Journal of Economic Theory 84 (1999) 145-195) though with different methods."}
{"category": "Math", "title": "On the multiple q-Genocchi and Euler numbers", "abstract": "The purpose of this paper is to present a systemic study of some families of multiple q-Genocchi and euler numbers by using multivariate q-Volkenborn integral. From the studies of those q-Genocchi numbers and polynomials of higher order we derive some interesting identities related to q-Genocchi numbers and polynomials of higher order."}
{"category": "Math", "title": "Imprecise Markov chains and their limit behaviour", "abstract": "When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to, and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-called lower and upper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at time n evolves as n goes to infinity: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalisation of the classical Perron-Frobenius Theorem to imprecise Markov chains."}
{"category": "Math", "title": "Law of large numbers for non-additive measures", "abstract": "Our aim is to give for some classes non-additive measures some limit theorems. For balanced games we obtain a weak and strong law of large numbers for bounded random variables, a sharper conclusion is obtain with exact games. We provide an extension to upper enveloppe measures."}
{"category": "Math", "title": "Hamilton-Pontryagin Integrators on Lie Groups: Introduction and Structure-Preserving Properties", "abstract": "In this paper structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton-Pontryagin variational principle. From this principle one can derive a novel class of variational partitioned Runge-Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and St\\\"{o}rmer-Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form. In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack."}
{"category": "Math", "title": "Paires de structures de contact sur les vari\\'et\\'es de dimension trois", "abstract": "We introduce a notion of positive pair of contact structures on a 3-manifold which generalizes a previous definition of Eliashberg-Thurston and Mitsumatsu. Such a pair gives rise to a locally integrable plane field $\\lambda$. We prove that if $\\lambda$ is uniquely integrable and if both structures of the pair are tight, then the integral foliation of $\\lambda$ doesn't contain any Reeb component whose core curve is homologous to zero. Moreover, the ambient manifold carries a Reebless foliation. We also show a stability theorem \"\\`a la Reeb\" for positive pairs of tight contact structures."}
{"category": "Math", "title": "Equilibrium States for Partially Hyperbolic Horseshoes", "abstract": "In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in \\cite{DHRS07}. Despite the fact that the non-wandering set is a horseshoe, it contains intervals. We prove that every recurrent point has non-zero Lyapunov exponents and all ergodic invariant measures are hyperbolic. As a consequence, we obtain the existence of equilibrium measures for any continuous potential. We also obtain an example of a family of $C^\\infty$ potentials with phase transition."}
{"category": "Math", "title": "New results on lower bounds for the number of (at most k)-facets", "abstract": "In this paper we present three different results dealing with the number of $(\\leq k)$-facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\\binom{k+2}{2}$ of $(\\leq k)$-edges for a fixed $0\\leq k\\leq \\lfloor n/3 \\rfloor -1$; 2. We give a simple construction showing that the lower bound $3\\binom{k+2}{2}+3\\binom{k-\\lfloor \\frac{n}{3} \\rfloor+2}{2}$ for the number of $(\\leq k)$-edges of a planar point set appeared in [Aichholzer et al. New lower bounds for the number of ($\\leq k$)-edges and the rectilinear crossing number of $K_n$. {\\em Disc. Comput. Geom.} 38:1 (2007), 1--14] is optimal in the range $\\lfloor n/3 \\rfloor \\leq k \\leq \\lfloor 5n/12 \\rfloor -1$; 3. We show that for $k < \\lfloor n/(d+1) \\rfloor$ the number of $(\\leq k)$-facets of a set of $n$ points in general position in $\\mathbb{R}^d$ is at least $(d+1)\\binom{k+d}{d}$, and that this bound is tight in the given range of $k$."}
{"category": "Math", "title": "Moderate deviations for random fields and random complex zeroes", "abstract": "Moderate deviations for random complex zeroes are deduced from a new theorem on moderate deviations for random fields."}
{"category": "Math", "title": "Conjectural estimates on the Mordell-Weil and Tate-Shafarevich groups of an abelian variety", "abstract": "We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the N\\'eron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators of the free part of the Mordell-Weil group. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals."}
{"category": "Math", "title": "Lower bounds for measurable chromatic numbers", "abstract": "The Lovasz theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10,..., 24, and we give a new proof that it grows exponentially with the dimension."}
{"category": "Math", "title": "On Wittgenstein's philosophy of mathematics", "abstract": "Mathematics cannot anymore be assimilated to a linguistic game, where formal proofs are strongly differentiated with conjectural thinking, without building any category of knowledge to understand the passage (Wittgenstein's gist). Nowadays, philosophy has to face with the growing, exponential ramified tree of speculative mathematical thinking. Our main (problematical) theses are: 1. In mathematics, there is no empirical automatism, and no separate, physical-like motricity. 2: The irreversible-synthetical must force to complexify the exegetical game of philosophy; numerical experiments in algebra and in number theory are a kind of letting blow up all possible problems; 4. The nature of mathematical questioning still remains in question."}
{"category": "Math", "title": "On the robustness of power-law random graphs in the finite mean, infinite variance region", "abstract": "We consider a conditionally Poissonian random graph model where the mean degrees, `capacities', follow a power-tailed distribution with finite mean and infinite variance. Such a graph of size $N$ has a giant component which is super-small in the sense that the typical distance between vertices is of the order of $\\log\\log N$. The shortest paths travel through a core consisting of nodes with high mean degrees. In this paper we derive upper bounds of the typical distance when an upper part of the core is removed, including the case that the whole core is removed."}
{"category": "Math", "title": "Stability in the Stefan problem with surface tension (I)", "abstract": "We develop a high-order energy method to prove asymptotic stability of flat steady surfaces for the Stefan problem with surface tension - also known as the Stefan problem with Gibbs-Thomson correction."}
{"category": "Math", "title": "Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations", "abstract": "We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport (and the related Monge-Ampere equation. This includes the 2D semi-geostrophic equations and some fully non-linear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel for Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology."}
{"category": "Math", "title": "Heat equation approach to index theorems on odd dimensional manifolds", "abstract": "D.Freed has formulated and proved an index theorem on odd dimensional spin manifolds with boundary. The proof is based on analysis by Calderon and Seeley. In this note we are going to give a proof of this theorem using the heat kernels methods for boundary conditions of Dirichlet and Von Neumann type. Moreover we consider also the Atiyah-Patodi-Singer spectral boundary condition which is not considered in Freed's paper. As a direct consequence of the method, we will obtain some information about isospectral invariants of the boundary conditions. This proof does not uses the cobordism invariance of index and are easily generalized to family case."}
{"category": "Math", "title": "Simultaneous analysis of Lasso and Dantzig selector", "abstract": "We exhibit an approximate equivalence between the Lasso estimator and Dantzig selector. For both methods we derive parallel oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the $\\ell_p$ estimation loss for $1\\le p\\le 2$ in the linear model when the number of variables can be much larger than the sample size."}
{"category": "Math", "title": "A Combinatorial Interpretation for Certain Relatives of the Conolly Sequence", "abstract": "For any integer s >= 0, we derive a combinatorial interpretation for the family of sequences generated by the recursion (parameterized by s) h_s(n) = h_s(n - s - h_s(n - 1)) + h_s(n - 2 - s - h_s(n - 3)), n > s + 3, with the initial conditions h_s(1) = h_s(2) = ... = h_s(s+2) = 1 and h_s(s+3) = 2. We show how these sequences count the number of leaves of a certain infinite tree structure. Using this interpretation we prove that h_s sequences are \"slowly growing\", that is, h_s sequences are monotone nondecreasing, with successive terms increasing by 0 or 1, so each sequence hits every positive integer. Further, for fixed s the sequence h_s(n) hits every positive integer twice except for powers of 2, all of which are hit s+2 times. Our combinatorial interpretation provides a simple approach for deriving the ordinary generating functions for these sequences."}
{"category": "Math", "title": "Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g=q=1$ and $K^2=4$", "abstract": "In this paper we study the minimal surfaces of general type with $p_g=q=1$ and $K^2=4$ whose Albanese general fibre has genus 2, classifying those such that the direct image (under the Albanese morphism) of the bicanonical sheaf is sum of line bundles. We find 8 unirational families, all of dimension strictly bigger than the expected one. These families are pairwise disjoint irreducible components of the moduli space of minimal surfaces of general type."}
{"category": "Math", "title": "$G$-Parking Functions, Acyclic Orientations and Spanning Trees", "abstract": "Given an undirected graph $G=(V,E)$, and a designated vertex $q\\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study properties of {\\em maximum} $G$-parking functions and provide a new bijection between them and the set of spanning trees of $G$ with no broken circuit. As a case study, we specialize some of our results to the graph corresponding to the discrete $n$-cube $Q_n$. We present the article in an expository self-contained form, since we found the combinatorial aspects of $G$-parking functions somewhat scattered in the literature, typically treated in conjunction with sandpile models and closely related chip-firing games."}
{"category": "Math", "title": "Decay and Continuity of Boltzmann Equation in Bounded Domains", "abstract": "Boundaries occur naturally in kinetic equations and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: inflow, bounce-back reflection, specular reflection, and diffuse reflection. We establish exponential decay in $L^{\\infty}$ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set of the velocity at the boundary. Our contribution is based on a new $L^{2}$ decay theory and its interplay with delicate $% L^{\\infty}$ decay analysis for the linearized Boltzmann equation, in the presence of many repeated interactions with the boundary."}
{"category": "Math", "title": "Hessian and gradient estimates for three dimensional special Lagrangian Equations with large phase", "abstract": "We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three."}
{"category": "Math", "title": "Monoidal categories of comodules for coquasi Hopf algebras and Radford's formula", "abstract": "We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford's $S^4$ formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a monoidal isomorphism between certain double dual functors."}
{"category": "Math", "title": "Hierarchical selection of variables in sparse high-dimensional regression", "abstract": "We study a regression model with a huge number of interacting variables. We consider a specific approximation of the regression function under two ssumptions: (i) there exists a sparse representation of the regression function in a suggested basis, (ii) there are no interactions outside of the set of the corresponding main effects. We suggest an hierarchical randomized search procedure for selection of variables and of their interactions. We show that given an initial estimator, an estimator with a similar prediction loss but with a smaller number of non-zero coordinates can be found."}
{"category": "Math", "title": "Projective normality of quotient varieties modulo finite groups", "abstract": "In this note, we prove that for any finite dimensional vector space $V$ over an algebraically closed field $k$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the $|G|$ is a unit in $k$, the projective variety $\\mathbb P(V)/G$ is projectively normal with respect to the descent of $\\mathcal O(1)^{\\otimes |G|}$."}
{"category": "Math", "title": "New developments in the theory of Groebner bases and applications to formal verification", "abstract": "We present foundational work on standard bases over rings and on Boolean Groebner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Groebner basis in the polynomial ring over Z/2n while the bit-level model leads to Boolean Groebner bases. In addition to the theoretical foundations of both approaches, the algorithms have been implemented. Using these implementations we show that special data structures and the exploitation of symmetries make Groebner bases competitive to state-of-the-art tools from formal verification but having the advantage of being systematic and more flexible."}
{"category": "Math", "title": "Rings over which the class of Gorenstein flat modules is closed under extensions", "abstract": "A ring $R$ is called left GF-closed, if the class of all Gorenstein flat left $R$-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension. In this paper, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension."}
{"category": "Math", "title": "Markov partitions reflecting the geometry of x2,x3", "abstract": "We give an explicit geometric description of the $\\times2,\\times3$ system, and use his to study a uniform family of Markov partitions related to those of Wilson and Abramov. The behaviour of these partitions is stable across expansive cones and transitions in this behaviour detects the non-expansive lines."}
{"category": "Math", "title": "Imprecise probability trees: Bridging two theories of imprecise probability", "abstract": "We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walley's behavioural theory of imprecise probabilities, and Shafer and Vovk's game-theoretic account of probability. We show that the two theories are more closely related than would be suspected at first sight, and we establish a correspondence between them that (i) has an interesting interpretation, and (ii) allows us to freely import results from one theory into the other. Our approach leads to an account of probability trees and random processes in the framework of Walley's theory. We indicate how our results can be used to reduce the computational complexity of dealing with imprecision in probability trees, and we prove an interesting and quite general version of the weak law of large numbers."}
{"category": "Math", "title": "Adjamagbo Determinant and Serre conjecture for linear groups over Weyl algebras", "abstract": "Thanks to the theory of determinants over an Ore domain, also called Adjamagbo determinant by the Russian school of non commutative algebra, we extend to any Weyl algebra over a field of characteristic zero Suslin theorem solving what Suslin himself called the $K_1$-analogue of the well-known Serre Conjecture and asserting that for any integer $n$ greater than 2, any $n$ by $n$ matrix with coefficients in any algebra of polynomials over a field and with determinant one is the product of elementary matrices with coefficients in this algebra"}
{"category": "Math", "title": "Stochastic processes and their spectral representations over non-archimedean fields", "abstract": "The article is devoted to stochastic processes with values in finite- and infinite-dimensional vector spaces over infinite fields $\\bf K$ of zero characteristics with non-trivial non-archimedean norms. For different types of stochastic processes controlled by measures with values in $\\bf K$ and in complete topological vector spaces over $\\bf K$ stochastic integrals are investigated. Vector valued measures and integrals in spaces over $\\bf K$ are studied. Theorems about spectral decompositions of non-archimedean stochastic processes are proved."}
{"category": "Math", "title": "The typical countable algebra", "abstract": "We argue that it makes sense to talk about ``typical'' properties of lattices, and then show that there is, up to isomorphism, a unique countable lattice L* (the Fraisse limit of the class of finite lattices) that has all ``typical'' properties. Among these properties are: L* is simple and locally finite, every order preserving function can be interpolated by a lattice polynomial, and every finite lattice or countable locally finite lattice embeds into L*. The same arguments apply to other classes of algebras assuming they have a Fraisse limit and satisfy the finite embeddability property."}
{"category": "Math", "title": "Sur le Th\\'eor\\`eme Principal de Zariski en G\\'eom\\'etrie Alg\\'ebrique et G\\'eom\\'etrie Analytique", "abstract": "On Zariski Main Theorem in Algebraic Geometry and Analytic Geometry. We fill a surprising gap of Complex Analytic Geometry by proving the analogue of Zariski Main Theorem in this geometry, i.e. proving that an holomorphic map from an irreducible analytic space to a normal irreducible one is an open embedding if and only if all its fibers are discrete and it induces a bimeromorphic map on its image. We prove more generally the \"Generalized Zariski Main Theorem for analytic spaces\", which claims that an holomorphic map from an irreducible analytic space to a irreducible locally irreducible one is an open embedding if and only if it is flat and induces a bimeromorphic map on its image. Thanks to the \"analytic criterion of regularity\" of Serre-Samuel in GAGA [12] and to \"Lefschetz Principle\", we finally deduce the \"Generalized Zariski Main Theorem for algebraic varieties of characteristical zero\", which claims that a morphism from such an irreducible variety to an irreducible unibranch one is an open immersion if and only if it is birational and flat. ----- Nous comblons une lacune \\'etonnante de la G\\'eom\\'etrie Analytique Complexe en prouvant l'analogue du Th\\'eor\\`eme Principal de Zariski dans cette g\\'eom\\'etrie, c'est-\\`a-dire en prouvant que toute application holomorphe d'un espace analytique irreductible dans un espace analytique normal et irreductible est un plongement ouvert si et seulement si toutes ses fibres sont discr\\`etes et si elle induit une application bim\\'eromorphe sur son image. Nous prouvons plus g\\'en\\'eralement le ``Th\\'eor\\`eme Principal de Zariski G\\'en\\'eralis\\'e pour les espaces analytiques'', qui affirme qu'une application holomorphe d'un espace analytique irreductible dans un espace analytique irreductible et localement irreductible est un plongement ouvert si et seulement si elle est plate et induit une application bim\\'eromorphe sur son image. Gr\\^ace au ``crit\\^ere analytique de r\\'egularit\\'e'' de Serre-Samuel dans GAGA \\cite{serre} et au ``Principe de Lefschetz'', nous en d\\'eduisons enfin le ``Th\\'eor\\`eme Principal de Zariski G\\'en\\'eralis\\'e pour les vari\\'et\\'es alg\\'ebriques de caract\\'eristique nulle'', qui affirme qu'un morphisme d'une telle vari\\'et\\'e irreductible dans une autre unibranche est une immersion ouverte si et seulement s'il est birationnel et plat."}
{"category": "Math", "title": "Optimal co-adapted coupling for the symmetric random walk on the hypercube", "abstract": "Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube. We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class."}
{"category": "Math", "title": "On the singularity of random matrices with independent entries", "abstract": "We consider n by n real matrices whose entries are non-degenerate random variables that are independent but non necessarily identically distributed, and show that the probability that such a matrix is singular is O(1/sqrt{n}). The purpose of this note is to provide a short and elementary proof of this fact using a Bernoulli decomposition of arbitrary non degenerate random variables."}
{"category": "Math", "title": "Arakelov theory of noncommutative arithmetic surfaces", "abstract": "The purpose of this paper is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with noncommutative arithmetic surfaces. We introduce a version of arithmetic intersection theory on noncommutative arithmetic surfaces and we prove an arithmetic Riemann-Roch theorem in this setup."}
{"category": "Math", "title": "Classification of two dimensional split trianguline representations of $p$-adic fields", "abstract": "The aim of this paper is to classify two dimensional split trianguline representations of $p$-adic fields. This is a generalization of a result of Colmez who classified two dimensional split trianguline representations of $\\mathrm{Gal}(\\bar{\\mathbb{Q}}_p/\\mathbb{Q}_p)$ by using $(\\phi,\\Gamma)$-modules over Robba ring. In this paper, we classify two dimensional split trianguline representations of $\\mathrm{Gal}(\\bar{K}/K)$ for general $p$-adic field $K$ by using $B$-pairs defined by Berger."}
{"category": "Math", "title": "A Galois correspondence for compact quantum group actions", "abstract": "We establish a Galois correspondence for a minimal action of a compact quantum group ${\\mathbb G}$ on a von Neumann factor $M$. This extends the result of Izumi, Longo and Popa who treated the case of a Kac algebra. Namely, there exists a one-to-one correspondence between the lattice of left coideals of ${\\mathbb G}$ and that of intermediate subfactors of $M^{\\mathbb G}\\subset M$."}
{"category": "Math", "title": "Packing 3-vertex paths in cubic 3-connected graphs", "abstract": "Let v(G) and p(G) be the number of vertices and the maximum number of disjoint 3-vertex paths in G, respectively. We discuss the following old Problem: Is the following claim (P) true ? (P) if G is a 3-connected and cubic graph, then p(G) = [v(G)/3], where [v(G)/3] is the floor of v(G)/3. We show, in particular, that claim (P) is equivalent to some seemingly stronger claims. It follows that if claim (P) is true, then Reed's dominating graph conjecture (see [14]) is true for cubic 3-connected graphs."}
{"category": "Math", "title": "Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three", "abstract": "We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces."}
{"category": "Math", "title": "Balanced routing of random calls", "abstract": "We consider an online network routing problem in continuous time, where calls have Poisson arrivals and exponential durations. The first-fit dynamic alternative routing algorithm sequentially selects up to $d$ random two-link routes between the two endpoints of a call, via an intermediate node, and assigns the call to the first route with spare capacity on each link, if there is such a route. The balanced dynamic alternative routing algorithm simultaneously selects $d$ random two-link routes, and the call is accepted on a route minimising the maximum of the loads on its two links, provided neither of these two links is saturated. We determine the capacities needed for these algorithms to route calls successfully and find that the balanced algorithm requires a much smaller capacity. In order to handle such interacting random processes on networks, we develop appropriate tools such as lemmas on biased random walks."}
{"category": "Math", "title": "Finitely Additive Supermartingales", "abstract": "The concept of finitely additive supermartingales, originally due to Bochner, is revived and developed. We exploit it to study measure decompositions over filtered probability spaces and the properties of the associated Dol\\'{e}ans-Dade measure. We obtain versions of the Doob Meyer decomposition and, as an application, we establish a version of the Bichteler and Dellacherie theorem with no exogenous probability measure."}
{"category": "Math", "title": "Exchangeable lower previsions", "abstract": "We extend de Finetti's (1937) notion of exchangeability to finite and countable sequences of variables, when a subject's beliefs about them are modelled using coherent lower previsions rather than (linear) previsions. We prove representation theorems in both the finite and the countable case, in terms of sampling without and with replacement, respectively. We also establish a convergence result for sample means of exchangeable sequences. Finally, we study and solve the problem of exchangeable natural extension: how to find the most conservative (point-wise smallest) coherent and exchangeable lower prevision that dominates a given lower prevision."}
{"category": "Math", "title": "On the remainder in the Taylor theorem", "abstract": "We give a short straightforward proof for the bound of the reminder term in the Taylor theorem. The proof uses only induction and the fact that $f'\\geq 0$ implies the monotonicity of $f$, so it might be an attractive proof to give to undergraduate students."}
{"category": "Math", "title": "On asymptotic stability in energy space of ground states of NLS in 2D", "abstract": "We transpose work by K.Yajima and by T.Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schr\\\"odinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2."}
{"category": "Math", "title": "LR-algebras", "abstract": "In the study of NIL-affine actions on nilpotent Lie groups we introduced so called LR-structures on Lie algebras. The aim of this paper is to consider the existence question of LR-structures, and to start a structure theory of LR-algebras. We show that any Lie algebra admitting an LR-structure is 2-step solvable. Conversely we find several classes of 2-step solvable Lie algebras admitting an LR-structure, but also classes not admitting such a structure. We study also ideals in LR-algebras, and classify low-dimensional real LR-algebras."}
{"category": "Math", "title": "GIT stability of weighted pointed curves", "abstract": "Here I give a direct proof that smooth curves with distinct marked points are asymptotically Hilbert stable with respect to a wide range of parameter spaces and linearizations. This result can be used to construct the coarse moduli space of Deligne-Mumford stable pointed curves \\bar M_g,n and Hassett's moduli spaces of weighted pointed curves \\bar M_g,A (though the full construction of the moduli spaces is not contained in this paper, only the stability proof). My proof follows Gieseker's approach to reduce to the GIT problem to a combinatorial problem, though the solution is very different. The action of any 1-PS lambda on a curve C in P^N gives rise to weighted filtrations of H^0 (C, O(1)) and H^0 (C, O(m)), and I give a recipe in terms of the combinatorics of the base loci of the stages of these filtrations for showing that C is stable with respect to lambda."}
{"category": "Math", "title": "Radial components, prehomogeneous vector spaces, and rational Cherednik algebras", "abstract": "Let V be a finite dimensional representation of the connected complex reductive group H. Denote by G the derived subgroup of H and assume that the categorical quotient of V by G is one dimensional. In this situation there exists a homomorphism, denoted by rad, from the algebra A of G-invariant differential operators on V to the first Weyl algebra. We show that the image of rad is isomorphic to the spherical subalgebra of a Cherednik algebra, whose parameters are determined by the b-function of the relative invariant associated to the prehomogeneous vector space (H : V). If (H : V) is furthemore assumed to be multiplicity free we obtain a Howe duality between a set of representations of G and modules over a subalgebra of the associative Lie algebra A. Some applications to holonomic modules and H-equivariant D-modules on V are also given."}
{"category": "Math", "title": "Game-theoretic Brownian motion", "abstract": "This paper suggests a perfect-information game, along the lines of Levy's characterization of Brownian motion, that formalizes the process of Brownian motion in game-theoretic probability. This is perhaps the simplest situation where probability emerges in a non-stochastic environment."}
{"category": "Math", "title": "Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm", "abstract": "We define and study the Plancherel-Hecke probability measure on Young diagrams; the Hecke algorithm of [Buch-Kresch-Shimozono-Tamvakis-Yong '06] is interpreted as a polynomial-time exact sampling algorithm for this measure. Using the results of [Thomas-Yong '07] on jeu de taquin for increasing tableaux, a symmetry property of the Hecke algorithm is proved, in terms of longest strictly increasing/decreasing subsequences of words. This parallels classical theorems of [Schensted '61] and of [Knuth '70], respectively, on the Schensted and Robinson-Schensted-Knuth algorithms. We investigate, and conjecture about, the limit typical shape of the measure, in analogy with work of [Vershik-Kerov '77], [Logan-Shepp '77] and others on the ``longest increasing subsequence problem'' for permutations. We also include a related extension of [Aldous-Diaconis '99] on patience sorting. Together, these results provide a new rationale for the study of increasing tableau combinatorics, distinct from the original algebraic-geometric ones concerning K-theoretic Schubert calculus."}
{"category": "Math", "title": "Monodromy in Hamiltonian Floer theory", "abstract": "Schwarz showed that when a closed symplectic manifold (M,\\om) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on \\pi_2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group Ham (M,\\om). In this note we describe less stringent conditions on the Chern class and quantum homology of M under which the (asymptotic) spectral invariants descend to Ham (M,\\om). For example, they descend if the quantum multiplication of M is undeformed and H_2(M) has rank >1, or if the minimal Chern number is at least n+1 (where \\dim M=2n) and the even cohomology of M is generated by divisors. The proofs are based on certain calculations of genus zero Gromov--Witten invariants. As an application, we show that the Hamiltonian group of the one point blow up of T^4 admits a Calabi quasimorphism. Moreover, whenever the (asymptotic) spectral invariants descend it is easy to see that Ham (M,\\om) has infinite diameter in the Hofer norm. Hence our results establish the infinite diameter of Ham in many new cases. We also show that the area pseudonorm -- a geometric version of the Hofer norm -- is nontrivial on the (compactly supported) Hamiltonian group for all noncompact manifolds as well as for a large class of closed manifolds."}
{"category": "Math", "title": "SL(n,Z[t]) is not FP_{n-1}", "abstract": "We prove the result from the title using the geometry of Euclidean buildings."}
{"category": "Math", "title": "Highest weight theory for finite W-algebras", "abstract": "We define analogues of Verma modules for finite W-algebras. By the usual ideas of highest weight theory, this is a first step towards the classification of finite dimensional irreducible modules. Motivated by known results in type A, we then formulate some precise conjectures in the case of nilpotent orbits of standard Levi type."}
{"category": "Math", "title": "A Beurling-Helson type theorem for modulation spaces", "abstract": "We prove a Beurling-Helson type theorem on modulation spaces. More precisely, we show that the only $\\mathcal{C}^{1}$ changes of variables that leave invariant the modulation spaces $\\M{p,q}(\\rd)$ are affine functions on $\\rd$. A special case of our result involving the Sj\\\"ostrand algebra was considered earlier by A. Boulkhemair."}
{"category": "Math", "title": "Homological algebra in bivariant K-theory and other triangulated categories. II", "abstract": "We use homological ideals in triangulated categories to get a sufficient criterion for a pair of subcategories in a triangulated category to be complementary. We apply this criterion to construct the Baum-Connes assembly map for locally compact groups and torsion-free discrete quantum groups. Our methods are related to the abstract version of the Adams spectral sequence by Brinkmann and Christensen."}
{"category": "Math", "title": "Cup products and L-values of cusp forms", "abstract": "In this note, we describe a conjecture, that, for an odd prime p, relates special values of a cup product pairing on cyclotomic p-units in the pth cyclotomic field to the L-values of newforms satisfying modulo p congruences with Eisenstein series."}
{"category": "Math", "title": "On maxima of periodograms of stationary processes", "abstract": "We consider the limit distribution of maxima of periodograms for stationary processes. Our method is based on $m$-dependent approximation for stationary processes and a moderate deviation result."}
{"category": "Math", "title": "On Galois groups of unramified pro-p extensions", "abstract": "Let p be an odd prime satisfying Vandiver's conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z_p-extensions of the pth cyclotomic field and the Galois group G of the unramified pro-p extension of the cyclotomic field of all p-power roots of unity. We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for G to be abelian. The bound and the condition in the two results are given in terms of the special values of a cup product pairing on cyclotomic p-units. We obtain, in particular, that for p less than 1000, Greenberg's conjecture on the pseudo-nullity of X holds and G is in fact abelian."}
{"category": "Math", "title": "Computation of 2-groups of positive classes of exceptional number fields", "abstract": "We present an algorithm for computing the 2-group of the positive divisor classes of a number field F in case F has exceptional dyadic places. As an application, we compute the 2-rank of the wild kernel WK2(F) in K2(F) for such number fields."}
{"category": "Math", "title": "Central and $L^p$-concentration of 1-Lipschitz maps into $\\mathbb{R}$-trees", "abstract": "In this paper, we study the L\\'{e}vy-Milman concentration phenomenon of 1-Lipschitz maps from mm-spaces to $\\mathbb{R}$-trees. Our main theorems assert that the concentration to $\\mathbb{R}$-trees is equivalent to the concentration to the real line."}
{"category": "Math", "title": "Multifractal analysis of non-uniformly hyperbolic systems", "abstract": "We prove a multifractal formalism for Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville--Pomeau map."}
{"category": "Math", "title": "The Eulerian distribution on self evacuated involutions", "abstract": "We present an extensive study of the Eulerian distribution on the set of self evacuated involutions, namely, involutions corresponding to standard Young tableaux that are fixed under the Sch$\\ddot{\\textrm{u}}$tzenberger map. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study for the subset of self evacuated involutions without fixed points."}
{"category": "Math", "title": "Rationally connected $3$-folds and symplectic geometry", "abstract": "We study the following question, asked to us By Pandharipande and Starr: Let $X$ be a rationally connected $3$-fold, and $Y$ be a compact Kaehler $3$-fold symplectically equivalent to it. Is $Y$ rationally connected? We show that the answer is positive if $X$ is Fano or $b_2(X)\\leq2$."}
{"category": "Math", "title": "On multilinearity and skew-symmetry of certain symbols in motivic cohomology of fields", "abstract": "The purpose of the present article is to show the multilinearity for symbols in Goodwillie-Lichtenbaum complex in two cases. The first case shown is where the degree is equal to the weight. In this case, the motivic cohomology groups of a field are isomorphic to the Milnor's K-groups as shown by Nesterenko-Suslin, Totaro and Suslin-Voevodsky for various motivic complexes, but we give an explicit isomorphism for Goodwillie-Lichtenbaum complex in a form which visibly carries multilinearity of Milnor's symbols to our multilinearity of motivic symbols. Next, we establish multilinearity and skew-symmetry for irreducible Goodwillie-Lichtenbaum symbols in H^{l-1} (Spec k, Z(l)). These properties have been expected to hold from the author's construction of a bilinear form of dilogarithm in case k is a subfield of the field of complex numbers and l=2. Next, we establish multilinearity and skew-symmetry for Goodwillie-Lichtenbaum symbols in H^{l-1} (Spec k, Z(l)). These properties have been expected to hold from the author's construction of a bilinear form of dilogarithm in case k is a subfield of the field of complex numbers and l=2. The multilinearity of symbols may be viewed as a generalization of the well-known formula det(AB) = det(A) det(B) for tuples of commuting matrices."}
{"category": "Math", "title": "Integral points on generic fibers", "abstract": "Let P(x,y) be a rational polynomial and k in Q be a generic value. If the curve (P(x,y)=k) is irreducible and admits an infinite number of points whose coordinates are integers then there exist algebraic automorphisms that send P(x,y) to the polynomial x or to x^2-dy^2. Moreover for such curves (and others) we give a sharp bound for the number of integral points (x,y) with x and y bounded."}
{"category": "Math", "title": "Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I", "abstract": "In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group of spheres $\\Pi_n$ there is no elements with Hopf invariant one. The new proof is based on geometric topology methods. The Pontrjagin-Thom Theorem (in the form proposed by R.Wells) about the representation of stable homotopy groups of the real projective infinite-dimensional space (this groups is mapped onto 2-components of stable homotopy groups of spheres by the Khan-Priddy Theorem) by cobordism classes of immersions of codimension 1 of closed manifolds (generally speaking, non-orientable) is considered. The Hopf Invariant is expressed as a characteristic number of the dihedral group for the self-intersection manifold of an immersed codimension 1 manifold that represents the given element in the stable homotopy group. In the new proof the Geometric Control Principle (by M.Gromov) for immersions in a given regular homotopy classes based on Smale-Hirsch Immersion Theorem is required."}
{"category": "Math", "title": "Geometric approach towards stable homotopy groups of spheres. The Kervaire invariant II", "abstract": "The notion of the geometrical $\\Z/2 \\oplus \\Z/2$--control of self-intersection of a skew-framed immersion and the notion of the $\\Z/2 \\oplus \\Z/4$-structure (the cyclic structure) on the self-intersection manifold of a $\\D_4$-framed immersion are introduced. It is shown that a skew-framed immersion $f:M^{\\frac{3n+q}{4}} \\looparrowright \\R^n$, $0 < q <<n$ (in the $\\frac{3n}{4}+\\epsilon$-range) admits a geometrical $\\Z/2 \\oplus \\Z/2$--control if the characteristic class of the skew-framing of this immersion admits a retraction of the order $q$, i.e. there exists a mapping $\\kappa_0: M^{\\frac{3n+q}{4}} \\to \\RP^{\\frac{3(n-q)}{4}}$, such that this composition $I \\circ \\kappa_0: M^{\\frac{3n+q}{4}} \\to \\RP^{\\frac{3(n-q)}{4}} \\to \\RP^{\\infty}$ is the characteristic class of the skew-framing of $f$. Using the notion of $\\Z/2 \\oplus \\Z/2$-control we prove that for a sufficiently great $n$, $n=2^l-2$, an arbitrary immersed $\\D_4$-framed manifold admits in the regular cobordism class (modulo odd torsion) an immersion with a $\\Z/2 \\oplus \\Z/4$-structure. In the last section we present an approach toward the Kervaire Invariant One Problem."}
{"category": "Math", "title": "Deformation data, Belyi maps, and the local lifting problem", "abstract": "We prove existence and nonexistence results for certain differential forms in positive characteristic, called {\\em good deformation data}. Some of these results are obtained by reduction modulo $p$ of Belyi maps. As an application, we solve the local lifting problem for groups with Sylow $p$-subgroup of order $p$."}
{"category": "Math", "title": "Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces", "abstract": "A discretisation scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied. The unknowns of this scheme are the values at the centre of the control volumes and at some internal interfaces which may for instance be chosen at the diffusion tensor discontinuities. The scheme is therefore completely cell centred if no edge unknown is kept. It is shown to be accurate on several numerical examples. Mathematical convergence of the approximate solution to the continuous solution is obtained for general (possibly discontinuous) tensors, general (possibly non-conforming) meshes, and with no regularity assumption on the solution. An error estimate is then drawn under sufficient regularity assumptions on the solution."}
{"category": "Math", "title": "Parameterizations and fitting of bi-directed graph models to categorical data", "abstract": "We discuss two parameterizations of models for marginal independencies for discrete distributions which are representable by bi-directed graph models, under the global Markov property. Such models are useful data analytic tools especially if used in combination with other graphical models. The first parameterization, in the saturated case, is also known as the multivariate logistic transformation, the second is a variant that allows, in some (but not all) cases, variation independent parameters. An algorithm for maximum likelihood fitting is proposed, based on an extension of the Aitchison and Silvey method."}
{"category": "Math", "title": "The distribution of prime numbers on the square root spiral", "abstract": "Prime Numbers clearly accumulate on defined spiral graphs,which run through the Square Root Spiral. These spiral graphs can be assigned to different spiral-systems, in which all spiral-graphs have the same direction of rotation and the same -- second difference -- between the numbers, which lie on these spiral-graphs. A mathematical analysis shows, that these spiral graphs are caused exclusively by quadratic polynomials. For example the well known Euler Polynomial x2+x+41 appears on the Square Root Spiral in the form of three spiral-graphs, which are defined by three different quadratic polynomials. All natural numbers,divisible by a certain prime factor, also lie on defined spiral graphs on the Square Root Spiral (or Spiral of Theodorus, or Wurzelspirale). And the Square Numbers 4, 9, 16, 25, 36 even form a highly three-symmetrical system of three spiral graphs, which divides the square root spiral into three equal areas. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. With the help of the Number-Spiral, described by Mr. Robert Sachs, a comparison can be drawn between the Square Root Spiral and the Ulam Spiral. The shown sections of his study of the number spiral contain diagrams, which are related to my analysis results, especially in regards to the distribution of prime numbers."}
{"category": "Math", "title": "Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise", "abstract": "The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework."}
{"category": "Math", "title": "Boundedness of Fourier Integral Operators on $\\mathcal{F} L^p$ spaces", "abstract": "We study the action of Fourier Integral Operators (FIOs) of H{\\\"o}rmander's type on ${\\mathcal{F}} L^p({\\mathbb {R}}^d_{comp}$, $1\\leq p\\leq\\infty$. We see, from the Beurling-Helson theorem, that generally FIOs of order zero fail to be bounded on these spaces when $p\\not=2$, the counterexample being given by any smooth non-linear change of variable. Here we show that FIOs of order $m=-d|1/2-1/p|$ are instead bounded. Moreover, this loss of derivatives is proved to be sharp in every dimension $d\\geq1$, even for phases which are linear in the dual variables. The proofs make use of tools from time-frequency analysis such as the theory of modulation spaces."}
{"category": "Math", "title": "Counting RSA-integers", "abstract": "In the RSA cryptosystem integers of the form n=p.q with p and q primes of comparable size (`RSA-integers') play an important role. It is a folklore result of cryptographers that C_r(x), the number of integers n<=x that are of the form n=pq with p and q primes such that p<q<rp, is for fixed r>1 asymptotically equal to c_r*x*log^{-2}x for some constant c_r>0. Here we prove this and show that c_r=2log r."}
{"category": "Math", "title": "A prime-to-p version of the Grothendieck anabelian conjecture for hyperbolic curves over finite fields of characteristic p>0", "abstract": "In this paper, we prove a prime-to-p version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic p>0, whose original (full profinite) version was proved by Tamagawa in the affine case and by Mochizuki in the proper case."}
{"category": "Math", "title": "The Quiver of Projectives in Hereditary Categories with Serre Duality", "abstract": "Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to rep_k Q for a so called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form \\mathbb{Z} Q."}
{"category": "Math", "title": "On classes defining a homological dimension", "abstract": "A class $\\mathcal F$ of objects of an abelian category $\\mathcal A$ is said to define a \\emph{homological dimension} if for any object in $\\mathcal A$ the length of any $\\mathcal F$-resolution is uniquely determined. In the present paper we investigate classes satisfying this property."}
{"category": "Math", "title": "Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems", "abstract": "Let C be a uniform clutter, i.e., all the edges of C have the same size, and let A be the incidence matrix of C. We denote the column vectors of A by v1,...,vq. The vertex covering number of C, denoted by g, is the smallest number of vertices in any minimal vertex cover of C. Under certain conditions we prove that C is vertex critical. If C satisfies the max-flow min-cut property, we prove that A diagonalizes over the integers to an identity matrix and that v1,...,vq is a Hilbert basis. It is shown that if C has a perfect matching such that C has the packing property and g=2, then A diagonalizes over the integers to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v1,...,vq is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Gr\\\"obner bases of toric ideals and to Ehrhart rings."}
{"category": "Math", "title": "On k-resonant fullerene graphs", "abstract": "A fullerene graph $F$ is a 3-connected plane cubic graph with exactly 12 pentagons and the remaining hexagons. Let $M$ be a perfect matching of $F$. A cycle $C$ of $F$ is $M$-alternating if the edges of $C$ appear alternately in and off $M$. A set $\\mathcal H$ of disjoint hexagons of $F$ is called a resonant pattern (or sextet pattern) if $F$ has a perfect matching $M$ such that all hexagons in $\\mathcal H$ are $M$-alternating. A fullerene graph $F$ is $k$-resonant if any $i$ ($0\\leq i \\leq k$) disjoint hexagons of $F$ form a resonant pattern. In this paper, we prove that every hexagon of a fullerene graph is resonant and all leapfrog fullerene graphs are 2-resonant. Further, we show that a 3-resonant fullerene graph has at most 60 vertices and construct all nine 3-resonant fullerene graphs, which are also $k$-resonant for every integer $k>3$. Finally, sextet polynomials of the 3-resonant fullerene graphs are computed."}
{"category": "Math", "title": "Differential Galois Theory of Linear Difference Equations", "abstract": "We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric functions."}
{"category": "Math", "title": "Theory of Bergman spaces (I)", "abstract": "These notes are part of the research seminar with title \"Theory of Bergman spaces and related function spaces\" that took place in the University of Crete, Department of Mathematics (September 2006- December 2007), in the framework of the research program PYTHAGORAS II(75% European funds--25% Greek national funds)."}
{"category": "Math", "title": "Symplectic 4-manifolds with a free circle action", "abstract": "The content of this unpublished paper is subsumed in successive work of the authors, in particular arXiv:1102.0820, arXiv:1102.0821 and arXiv:1205.2434."}
{"category": "Math", "title": "A new bound on the number of special fibers in a pencil of curves", "abstract": "In the previous paper by Pereira and the author, it was proved that any pencil of plane curves of degree greater than one with irreducible generic fiber can have at most five completely reducible fibers although no examples with five such fibers were ever found. Recently Janis Stipins has proved that if any two fibers of a pencil intersect transversally then it cannot have five completely reducible fibers. In this paper we generalize the Stipins result to arbitrary pencils. We also include into consideration more general special fibers that are the unions of lines and non-reduced curves. These fibers are important for characteristic varieties of line complements."}
{"category": "Math", "title": "Approximate stabilization of a quantum particle in a 1D infinite square potential well", "abstract": "We consider a non relativistic charged particle in a 1-dimensional infinite square potential well. This quantum system is subjected to a control, which is a uniform (in space) time depending electric field. It is represented by a complex probability amplitude solution of a Schrodinger equation on a 1-dimensional bounded domain, with Dirichlet boundary conditions. We prove the almost global approximate stabilization of the eigenstates by explicit feedback laws."}
{"category": "Math", "title": "Sparse Fourier Transform via Butterfly Algorithm", "abstract": "We introduce a fast algorithm for computing sparse Fourier transforms supported on smooth curves or surfaces. This problem appear naturally in several important problems in wave scattering and reflection seismology. The main observation is that the interaction between a frequency region and a spatial region is approximately low rank if the product of their radii are bounded by the maximum frequency. Based on this property, equivalent sources located at Cartesian grids are used to speed up the computation of the interaction between these two regions. The overall structure of our algorithm follows the recently-introduced butterfly algorithm. The computation is further accelerated by exploiting the tensor-product property of the Fourier kernel in two and three dimensions. The proposed algorithm is accurate and has an $O(N \\log N)$ complexity. Finally, we present numerical results in both two and three dimensions."}
{"category": "Math", "title": "Multiplicity matrices for the affine graded Hecke algebra", "abstract": "In this paper, we look at the problem of determining the composition factors for the affine graded Hecke algebra via the computation of Kazhdan-Lusztig type polynomials. We review the algorithms of \\cite{L1,L2}, and use them in particular to compute, at every real central character which admits tempered modules, the geometric parameterization, the Kazhdan-Lusztig polynomials, the composition series, and the Iwahori-Matsumoto involution for the representations with Iwahori fixed vectors of the split $p$-adic groups of type $G_2$ and $F_4$."}
{"category": "Math", "title": "A Lie-theoretic construction of representations of the degenerate affine and double affine Hecke algebras of type $BC_n$", "abstract": "Let G=GL(N), K=GL(p)xGL(q), where p+q=N, and n be a positive integer. We construct a functor from the category of Harish-Chandra modules for the pair (G,K) to the category of representations of the degenerate affine Hecke algebra of type B_n, and a functor from the category of K-monodromic twisted D-modules on G/K to the category of representations of the degenerate double affine Hecke algebra of type BC_n; the second functor is an extension of the first one. These functors are generalizations of the type A functors from q-alg/9710037 and math/0702670, respectively."}
{"category": "Math", "title": "Left inverses of matrices with polynomial decay", "abstract": "The algebra of Schur operators on l^2 is known not to be inverse-closed. When l^2=l^2(X) where X is a metric space, we can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra for a strictly polynomial weight is inverse-closed. Here, we prove a result dealing with left-invertibility. Namely, if such an operator is bounded below in l^p for some p, then it is bounded below for all q, and it admits a left-inverse in the weighted Schur algebra."}
{"category": "Math", "title": "The Higher Transvectants are Redundant", "abstract": "Let A, B denote generic binary forms, and let u_r = (A,B)_r denote their r-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the u_r. As a consequence, we show that each of the higher transvectants u_r, r>1, is redundant in the sense that it can be completely recovered from u_0 and u_1. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of SL_2-representations, and the notion of a 9-j symbol from the quantum theory of angular momentum. We give explicit computational examples for SL_3, g_2 and S_5 to show that this result has possible analogues for other categories of representations."}
{"category": "Math", "title": "Comments on \"Reverse auction: the lowest positive integer game\"", "abstract": "In Zeng et al. [Fluct. Noise Lett. 7 (2007) L439--L447] the analysis of the lowest unique positive integer game is simplified by some reasonable assumptions that make the problem tractable for arbitrary numbers of players. However, here we show that the solution obtained for rational players is not a Nash equilibrium and that a rational utility maximizer with full computational capability would arrive at a solution with a superior expected payoff. An exact solution is presented for the three- and four-player cases and an approximate solution for an arbitrary number of players."}
{"category": "Math", "title": "Razborov flag algebras as algebras of measurable functions", "abstract": "These are some brief notes on the translation from Razborov's recently-developed notion of flag algebra into the lexicon of functions and measures on certain abstract Cantor spaces (totally disconnected compact metric spaces)."}
{"category": "Math", "title": "The canonical sheaf of Du Bois singularities", "abstract": "We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if $\\pi_*\\omega_{X'}(G) \\simeq \\omega_X$ for a log resolution $\\pi: X' \\to X$, where $G$ is the reduced exceptional divisor of $\\pi$. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen-Macaulay case. It also follows that the Kodaira vanishing theorem holds for semi-log-canonical varieties and that Cohen-Macaulay semi-log-canonical singularities are cohomologically insignificant in the sense of Dolgachev."}
{"category": "Math", "title": "Compactification des vari\\'et\\'es de Deligne-Lusztig", "abstract": "We construct explicitly the normalisation of the Bott-Samelson-Demazure-Hansen compactification of Deligne-Lusztig varieties $X(w)$ in their covering $Y(w)$: we retrieve a result by Deligne-Lusztig about the local monodromy around the divisors of the compactification."}
{"category": "Math", "title": "Basic differential geometry as a sequence of interesting problems", "abstract": "This book is expository and is in Russian (sample English translation of two pages is given). It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear different notions of curvature, which distinguish given geometry from the 'ordinary' one. Direct elementary definitions of these notions are presented. The book is accessible for students familiar with analysis of several variables, and could be an interesting easy reading for professional mathematicians. The material is presented as a sequence of problems, which is peculiar not only to Zen monasteries but also to serious mathematical education."}
{"category": "Math", "title": "Bounds and asymptotic minimal growth for Gorenstein Hilbert functions", "abstract": "We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically. Our first main theorem is a lower bound for the degree $i+1$ entry of a Gorenstein $h$-vector, in terms of its entry in degree $i$. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given $r$ and $i$, all Gorenstein $h$-vectors of codimension $r$ and socle degree $e\\geq e_0=e_0(r,i)$ (this function being explicitly computed) are unimodal up to degree $i+1$. This immediately gives a new proof of a theorem of Stanley that all Gorenstein $h$-vectors in codimension three are unimodal. Our second main theorem is an asymptotic formula for the least value that the $i$-th entry of a Gorenstein $h$-vector may assume, in terms of codimension, $r$, and socle degree, $e$. This theorem broadly generalizes a recent result of ours, where we proved a conjecture of Stanley predicting that asymptotic value in the specific case $e=4$ and $i=2$, as well as a result of Kleinschmidt which concerned the logarithmic asymptotic behavior in degree $i= \\lfloor \\frac{e}{2} \\rfloor $."}
{"category": "Math", "title": "Homological properties of cochain Differential Graded algebras", "abstract": "Consider a local chain Differential Graded algebra, such as the singular chain complex of a pathwise connected topological group. In two previous papers, a number of homological results were proved for such an algebra: An Amplitude Inequality, an Auslander-Buchsbaum Equality, and a Gap Theorem. These were inspired by homological ring theory. By the so-called looking glass principle, one would expect that analogous results exist for simply connected cochain Differential Graded algebras, such as the singular cochain complex of a simply connected topological space. Indeed, this paper establishes such analogous results."}
{"category": "Math", "title": "Note on a Conjecture of Wegner", "abstract": "The optimal packings of n unit discs in the plane are known for those natural numbers n, which satisfy certain number theoretic conditions. Their geometric realizations are the extremal Groemer packings (or Wegner packings). But an extremal Groemer packing of n unit discs does not exist for all natural numbers n and in this case, the number n is called exceptional. We are interested in number theoretic characterizations of the exceptional numbers. A counterexample is given to a conjecture of Wegner concerning such a characterization. We further give a characterization of the exceptional numbers, whose shape is closely related to that of Wegner's conjecture."}
{"category": "Math", "title": "Reduction theory for mapping class groups and applications to moduli spaces", "abstract": "Let $S=S_{g,p}$ be a compact, orientable surface of genus $g$ with $p$ punctures and such that $d(S):=3g-3+p>0$. The mapping class group $\\textup{Mod}_S$ acts properly discontinuously on the Teichm\\\"uller space $\\mathcal T(S)$ of marked hyperbolic structures on $S$. The resulting quotient $\\mathcal M(S)$ is the moduli space of isometry classes of hyperbolic surfaces. We provide a version of precise reduction theory for finite index subgroups of $\\textup{Mod}_S$, i.e., a description of exact fundamental domains. As an application we show that the asymptotic cone of the moduli space $\\mathcal M(S)$ endowed with the Teichm\\\"uller metric is bi-Lipschitz equivalent to the Euclidean cone over the finite simplicial (orbi-) complex $ \\textup{Mod}_S\\backslash\\mathcal C(S)$, where $\\mathcal C(S)$ of $S$ is the complex of curves of $S$. We also show that if $d(S)\\geq 2$, then $\\mathcal M(S)$ does \\emph{not} admit a finite volume Riemannian metric of (uniformly bounded) positive scalar curvature in the bi-Lipschitz class of the Teichm\\\"uller metric. These two applications confirm conjectures of Farb."}
{"category": "Math", "title": "Generating function identities for $\\zeta(2n+2), \\zeta(2n+3)$ via the WZ method", "abstract": "Using the WZ method we present simpler proofs of Koecher's, Leshchiner's and Bailey-Borwein-Bradley's identities for generating functions of the sequences $\\{\\zeta(2n+2)\\}_{n\\ge 0}, \\{\\zeta(2n+3)\\}_{n\\ge 0}.$ By the same method we give several new representations for these generating functions yielding faster convergent series for values of the Riemann zeta function."}
{"category": "Math", "title": "Estimation of ordinal pattern probabilities in fractional Brownian motion", "abstract": "For equidistant discretizations of fractional Brownian motion (fBm), the probabilities of ordinal patterns of order d=2 are monotonically related to the Hurst parameter H. By plugging the sample relative frequency of those patterns indicating changes between up and down into the monotonic relation to H, one obtains the Zero Crossing (ZC) estimator of the Hurst parameter which has found considerable attention in mathematical and applied research. In this paper, we generally discuss the estimation of ordinal pattern probabilities in fBm. As it turns out, according to the sufficiency principle, for ordinal patterns of order d=2 any reasonable estimator is an affine functional of the sample relative frequency of changes. We establish strong consistency of the estimators and show them to be asymptotically normal for H<3/4. Further, we derive confidence intervals for the Hurst parameter. Simulation studies show that the ZC estimator has larger variance but less bias than the HEAF estimator of the Hurst parameter."}
{"category": "Math", "title": "Isomorphisms of unitary forms of Kac-Moody groups over finite fields", "abstract": "We use methods developed by Caprace and M\\\"uhlherr to solve the isomorphism problem of unitary forms of infinite split Kac-Moody groups over finite fields of square order."}
{"category": "Math", "title": "A Hadwiger-type theorem for the special unitary group", "abstract": "The dimension of the space of SU(n) and translation invariant continuous valuations on $\\mathbb{C}^n, n \\geq 2$ is computed. For even $n$, this dimension equals $(n^2+3n+10)/2$; for odd $n$ it equals $(n^2+3n+6)/2$. An explicit geometric basis of this space is constructed. The kinematic formulas for SU(n) are obtained as corollaries."}
{"category": "Math", "title": "Random subgraphs of the 2D Hamming graph: the supercritical phase", "abstract": "We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the edge probability, and write $p=\\frac{1+\\vep}{2(n-1)}$ for some $\\vep\\in \\R$. In Borgs et al., Random subgraphs of finite graphs: I. The scaling window under the triangle condition, Rand. Struct. Alg. (2005), and in Borgs et al., Random subgraphs of finite graphs: II. The lace expansion and the triangle condition, Ann. Probab. (2005), the size of the largest connected component was estimated precisely for a large class of graphs including H(2,n) for $\\vep\\leq \\Lambda V^{-1/3}$, where $\\Lambda > 0$ is a constant and $V=n^2$ denotes the number of vertices in H(2,n). Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when $\\vep\\gg (\\log{V})^{1/3} V^{-1/3}$, then the largest connected component has size close to $2\\vep V$ with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of $p$ are supercritical. Barring the factor $(\\log{\\chs{V}})^{1/3}$, this identifies the size of the largest connected component all the way down to the critical $p$ window."}
{"category": "Math", "title": "The second largest component in the supercritical 2D Hamming graph", "abstract": "The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices $(i,j)$, $1\\leq i,j\\leq n$, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability $p$, so that the average degree $2(n-1)p=1+\\epsilon$. Previous work by van der Hofstad and Luczak had shown that in the barely supercritical region $n^{-2/3}\\ln^{1/3}n\\ll \\epsilon \\ll 1$ the largest component has size $\\sim 2\\epsilon n$. Here we show that the second largest component has size close to $\\epsilon^{-2}$, so that the dominant component has emerged. This result also suggests that a {\\it discrete duality principle} might hold, whereby, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime."}
{"category": "Math", "title": "A geometric preferential attachment model with fitness", "abstract": "We study a random graph $G_n$, which combines aspects of geometric random graphs and preferential attachment. The resulting random graphs have power-law degree sequences with finite mean and possibly infinite variance. In particular, the power-law exponent can be any value larger than 2. The vertices of $G_n$ are $n$ sequentially generated vertices chosen at random in the unit sphere in $\\mathbb R^3$. A newly added vertex has $m$ edges attached to it and the endpoints of these edges are connected to old vertices or to the added vertex itself. The vertices are chosen with probability proportional to their current degree plus some initial attractiveness and multiplied by a function, depending on the geometry."}
{"category": "Math", "title": "Fourier transform, null variety, and Laplacian's eigenvalues", "abstract": "We consider a quantity $\\kappa(\\Omega)$ -- the distance to the origin from the null variety of the Fourier transform of the characteristic function of $\\Omega$. We conjecture, firstly, that $\\kappa(\\Omega)$ is maximized, among all convex balanced domains $\\Omega\\subset\\Rbb^d$ of a fixed volume, by a ball, and also that $\\kappa(\\Omega)$ is bounded above by the square root of the second Dirichlet eigenvalue of $\\Omega$. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between $\\kappa(\\Omega)$ and the eigenvalues of the Laplacians."}
{"category": "Math", "title": "On colored Turaev-Viro invariants for links in arbitrary 3-manifolds", "abstract": "We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M,L), where M is a closed oriented 3-manifold and L is an oriented link in M, by a triangulation of M such that each component of L is an edge. We analyze some basic properties of these invariants, including the behavior under connected sums of pairs away and along links. These properties allow us to provide examples of links in the three-sphere having the same HOMFLY polynomial and the same Kauffman polynomial but distinct Turaev-Viro invariants, and similar examples for the Alexander polynomial. We also investigate the relations between the Turaev-Viro invariants of (M,L) and those of the complement of L in M, showing that they are sometimes but not always determined by each other."}
{"category": "Math", "title": "On the structure of the necklace Lie algebra", "abstract": "In this note, we initiate the systematic study of the Lie algebra structure of the necklace Lie algebra n of a free algebra in 2d variables. We begin by giving a description of n as an sp(2d)-module. Specializing to d = 1, we decompose n into a direct sum of highest weight modules for sl_2, the coefficients of which are given by a closed formula. Next, we observe that n has a nontrivial center, which we link through the center C of the trace ring of couples of generic 2x2 matrices to the Poisson center of S(sl_2). The Lie algebra structure of n induces a Poisson structure on C, the symplectic leaves of which we are able to describe as coadjoint orbits for the Lie group of the semidirect product sl_2\\rtimes h of sl_2 with the Heisenberg Lie algebra h. Finally, we provide a link between double Poisson algebras on one hand and Poisson orders on the other hand, showing that all trace rings of a double Poisson algebra are Poisson orders over their center."}
{"category": "Math", "title": "The Calogero-Moser partition and Rouquier families for complex reflection groups", "abstract": "Let $W$ be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify the conjecture in the case that $W$ is a wreath product of a symmetric group with a cyclic group of order $l$."}
{"category": "Math", "title": "Uppers to zero in polynomial rings and Pr\\\"ufer-like domains", "abstract": "Let $D$ be an integral domain and $X$ an indeterminate over $D$. It is well known that (a) $D$ is quasi-Pr\\\"ufer (i.e, its integral closure is a Pr\\\"ufer domain) if and only if each upper to zero $Q$ in $D[X] $ contains a polynomial $g \\in D[X]$ with content $\\co_D(g) = D$; (b) an upper to zero $Q$ in $D[X]$ is a maximal $t$-ideal if and only if $Q$ contains a nonzero polynomial $g \\in D[X]$ with $\\co_D(g)^v = D$. Using these facts, the notions of UM$t$-domain (i.e., an integral domain such that each upper to zero is a maximal $t$-ideal) and quasi-Pr\\\"ufer domain can be naturally extended to the semistar operation setting and studied in a unified frame. In this paper, given a semistar operation $\\star$ in the sense of Okabe-Matsuda, we introduce the $\\star$-quasi-Pr\\\"ufer domains. We give several characterizations of these domains and we investigate their relations with the UM$t$-domains and the Pr\\\"ufer $v$-multiplication domains."}
{"category": "Math", "title": "Cohomological invariants of odd degree Jordan algebras", "abstract": "In this paper we determine all possible cohomological invariants of Aut(J)-torsors in Galois cohomology with mod 2 coefficients (characteristic of the base field not 2), for J a split central simple Jordan algebra of odd degree n>=3. This has already been done for J of orthogonal and exceptional type, and we extend these results to unitary and symplectic type. We will use our results to compute the essential dimensions of some groups, for example we show that ed(PSp(2n))=n+1 for n odd."}
{"category": "Math", "title": "Chain recurrence rates and topological entropy", "abstract": "We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using $\\ep$-chains, and the minimal lengths of these $\\ep$-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy."}
{"category": "Math", "title": "Itineraries of rigid rotations and diffeomorphisms of the circle", "abstract": "We examine the itinerary of $0\\in S^{1}=\\R/\\Z$ under the rotation by $\\alpha\\in\\R\\bs\\Q$. The motivating question is: if we are given only the itinerary of 0 relative to $I\\subset S^{1}$, a finite union of closed intervals, can we recover $\\alpha$ and $I$? We prove that the itineraries do determine $\\alpha$ and $I$ up to certain equivalences. Then we present elementary methods for finding $\\alpha$ and $I$. Moreover, if $g:S^{1}\\to S^{1}$ is a $C^{2}$, orientation preserving diffeomorphism with an irrational rotation number, then we can use the orbit itinerary to recover the rotation number up to certain equivalences."}
{"category": "Math", "title": "Leading coefficients of Kazhdan--Lusztig polynomials and fully commutative elements", "abstract": "Let $W$ be a Coxeter group of type $\\widetilde{A}_{n-1}$. We show that the leading coefficient, $\\mu(x, w)$, of the Kazhdan--Lusztig polynomial $P_{x, w}$ is always equal to 0 or 1 if $x$ is fully commutative (and $w$ is arbitrary)."}
{"category": "Math", "title": "A note on the relationship between the Graphical Traveling Salesman Polyhedron, the Symmetric Traveling Salesman Polytope, and the Metric Cone", "abstract": "In this short communication, we observe that the Graphical Traveling Salesman Polyhedron is the intersection of the positive orthant with the Minkowski sum of the Symmetric Traveling Salesman Polytope and the polar of the metric cone. This follows almost trivially from known facts. There are two reasons why we find this observation worth communicating none-the-less: It is very surprising; it helps to understand the relationship between these two important families of polyhedra."}
{"category": "Math", "title": "Courant morphisms and moment maps", "abstract": "We study Hamiltonian spaces associated with pairs (E,A), where E is a Courant algebroid and A\\subset E is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties are discussed, including a reduction procedure. This set-up encompasses familiar moment map theories, such as group-valued moment maps, and it provides an intrinsic approach from which different geometrical descriptions of moment maps can be naturally derived. As an application, we discuss the relationship between quasi-Poisson and presymplectic groupoids."}
{"category": "Math", "title": "The basic geometry of Witt vectors, I: The affine case", "abstract": "We give a concrete description of the category of etale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical p-typical and big Witt vector functors but also for variants of these functors which are in a certain sense their analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for the classical Witt vectors. The larger purpose of this paper is to provide the affine foundations for the algebraic geometry of generalized Witt schemes and arithmetic jet spaces. So the basics here are developed somewhat fully, with an eye toward future applications."}
{"category": "Math", "title": "On exchangeable random variables and the statistics of large graphs and hypergraphs", "abstract": "De Finetti's classical result of [18] identifying the law of an exchangeable family of random variables as a mixture of i.i.d. laws was extended to structure theorems for more complex notions of exchangeability by Aldous [1,2,3], Hoover [41,42], Kallenberg [44] and Kingman [47]. On the other hand, such exchangeable laws were first related to questions from combinatorics in an independent analysis by Fremlin and Talagrand [29], and again more recently in Tao [62], where they appear as a natural proxy for the `leading order statistics' of colourings of large graphs or hypergraphs. Moreover, this relation appears implicitly in the study of various more bespoke formalisms for handling `limit objects' of sequences of dense graphs or hypergraphs in a number of recent works, including Lov\\'{a}sz and Szegedy [52], Borgs, Chayes, Lov\\'{a}sz, S\\'{o}s, Szegedy and Vesztergombi [17], Elek and Szegedy [24] and Razborov [54,55]. However, the connection between these works and the earlier probabilistic structural results seems to have gone largely unappreciated. In this survey we recall the basic results of the theory of exchangeable laws, and then explain the probabilistic versions of various interesting questions from graph and hypergraph theory that their connection motivates (particularly extremal questions on the testability of properties for graphs and hypergraphs). We also locate the notions of exchangeability of interest to us in the context of other classes of probability measures subject to various symmetries, in particular contrasting the methods employed to analyze exchangeable laws with related structural results in ergodic theory, particular the Furstenberg-Zimmer structure theorem for probability-preserving $\\mathbb {Z}$-systems, which underpins Furstenberg's ergodic-theoretic proof of Szemer\\'{e}di's Theorem. The forthcoming paper [10]--hereditarytest will make a much more elaborate appeal to the link between exchangeable laws and dense (directed) hypergraphs to establish various results in property testing."}
{"category": "Math", "title": "Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with flag singular integrals", "abstract": "The main purpose of this paper is to develop a unified approach of multi-parameter Hardy space theory using the discrete Littlewood-Paley-Stein analysis in the setting of implicit multi-parameter structure. It is motivated by the goal to establish and develop the Hardy space theory for the flag singular integral operators studied by Muller-Ricci-Stein and Nagel-Ricci-Stein. This approach enables us to avoid the use of transference method of Coifman-Weiss as often used in the $L^p$ theory for $p>1$ and establish the Hardy spaces $H^p_F$ and its dual spaces associated with the flag singular integral operators for all $0<p\\leq 1$. We also prove the boundedness of flag singular integral operators on $BMO_F$ and $H^p_F$, and from $H^p_F$ to $L^p$ for all $0<p\\le 1$ without using the deep atomic decomposition. As a result, it bypasses the use of Journe's type covering lemma in this implicit multi-parameter structure. The method used here provides alternate approaches of those developed by Chang, R. Fefferman, Journe and Pipher in the pure product setting. A Calderon-Zygmund decomposition and interpolation theorem are also proved for the implicit multi-parameter Hardy spaces."}
{"category": "Math", "title": "A non-separable Christensen's theorem and set tri-quotient maps", "abstract": "For every space $X$ let $\\mathcal K(X)$ be the set of all compact subsets of $X$. Christensen \\cite{c:74} proved that if $X, Y$ are separable metrizable spaces and $F\\colon\\mathcal{K}(X)\\to\\mathcal{K}(Y)$ is a monotone map such that any $L\\in\\mathcal{K}(Y)$ is covered by $F(K)$ for some $K\\in\\mathcal{K}(X)$, then $Y$ is complete provided $X$ is complete. It is well known \\cite{bgp} that this result is not true for non-separable spaces. In this paper we discuss some additional properties of $F$ which guarantee the validity of Christensen's result for more general spaces."}
{"category": "Math", "title": "Extension of Matrices with Entries in H^{\\infty} on Coverings of Riemann Surfaces of Finite Type", "abstract": "In the present paper continuing our previous work we prove an extension theorem for matrices with entries in the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of finite type."}
{"category": "Math", "title": "Probability measures and Milyutin maps between metric spaces", "abstract": "We prove that the functor $\\Hat{P}$ of Radon probability measures transforms any open map between completely metrizable spaces into a soft map. This result is applied to establish some properties of Milyutin maps between completely metrizable spaces."}
{"category": "Math", "title": "Bilinear Mixed Effects Models For Relations Between Universities", "abstract": "this article illustrates the use of linear and bilinear random effects models to represent statistical dependencies that often characterize dyadic data such as international relations. In particular, we show how to estimate models for dyadic data that simultaneously take into account: regressor variables and third-order dependencies, such as transitivity, clustering, and balance. We apply this new approach to the relations among ph.d. of university in Iran over the period from 1991-2005, illustrating the presence and strength of second and third-order statistical dependencies in these data."}
{"category": "Math", "title": "Proof mining in ${\\mathbb R}$-trees and hyperbolic spaces", "abstract": "This paper is part of the general project of proof mining, developed by Kohlenbach. By \"proof mining\" we mean the logical analysis of mathematical proofs with the aim of extracting new numerically relevant information hidden in the proofs. We present logical metatheorems for classes of spaces from functional analysis and hyperbolic geometry, like Gromov hyperbolic spaces, ${\\mathbb R}$-trees and uniformly convex hyperbolic spaces. Our theorems are adaptations to these structures of previous metatheorems of Gerhardy and Kohlenbach, and they guarantee a-priori, under very general logical conditions, the existence of uniform bounds. We give also an application in nonlinear functional analysis, more specifically in metric fixed-point theory. Thus, we show that the uniform bound on the rate of asymptotic regularity for the Krasnoselski-Mann iterations of nonexpansive mappings in uniformly convex hyperbolic spaces obtained in a previous paper is an instance of one of our metatheorems."}
{"category": "Math", "title": "An explicit integral polynomial whose splitting field has Galois group W(E_8)", "abstract": "Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group over a number field typically have splitting field with Galois group isomorphic to its Weyl group, we construct an explicit monic integral polynomial of degree 240 whose splitting field has Galois group the Weyl group of the exceptional group of type E_8."}
{"category": "Math", "title": "On 4n-dimensional Lie groups as quasi-Kaehler manifolds with Killing Norden metric", "abstract": "A 4n-parametric family of 4n-dimensional quasi-Kaehler manifolds with Killing Norden metric is constructed on a Lie group. This family is characterized geometrically."}
{"category": "Math", "title": "Acyclic Edge Coloring of Graphs with Maximum Degree 4", "abstract": "An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \\emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks that for any simple and finite graph $G$, $a'(G)\\le \\Delta+2$, where $\\Delta =\\Delta(G)$ denotes the maximum degree of $G$. We prove the conjecture for connected graphs with $\\Delta(G) \\le 4$, with the additional restriction that $m \\le 2n-1$, where $n$ is the number of vertices and $m$ is the number of edges in $G $. Note that for any graph $G$, $m \\le 2n$, when $\\Delta(G) \\le 4$. It follows that for any graph $G$ if $\\Delta(G) \\le 4$, then $a'(G) \\le 7$."}
{"category": "Math", "title": "On the H^1-L^1 boundedness of operators", "abstract": "We prove that if q is in (1,\\infty), Y is a Banach space and T is a linear operator defined on the space of finite linear combinations of (1,q)-atoms in R^n which is uniformly bounded on (1,q)-atoms, then T admits a unique continuous extension to a bounded linear operator from H^1(R^n) to Y. We show that the same is true if we replace (1,q)-atoms with continuous (1,\\infty)-atoms. This is known to be false for (1,\\infty)-atoms."}
{"category": "Math", "title": "Preserving positive polynomials and beyond", "abstract": "Following the classical approach of P\\'olya-Schur theory we initiate in this paper the study of linear operators acting on $\\mathbb{R}[x]$ and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials."}
{"category": "Math", "title": "A new transform for solving the noisy complex exponentials approximation problem", "abstract": "The problem of estimating a complex measure made up by a linear combination of Dirac distributions centered on points of the complex plane from a finite number of its complex moments affected by additive i.i.d. Gaussian noise is considered. A random measure is defined whose expectation approximates the unknown measure under suitable conditions. An estimator of the approximating measure is then proposed as well as a new discrete transform of the noisy moments that allows to compute an estimate of the unknown measure. A small simulation study is also performed to experimentally check the goodness of the approximations."}
{"category": "Math", "title": "Scorza quartics of trigonal spin curves and their varieties of power sums", "abstract": "Our fundamental result is the construction of new subvarieties in the varieties of power sums for the Scorza quartic of any general pairs of trigonal curves and non-effective theta characteristics. This is a generalization of Mukai's description of smooth prime Fano threefolds of genus twelve as the varieties of power sums for plane quartics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartic for any general pairs of curves and non-effective theta characteristics."}
{"category": "Math", "title": "Convergence rates and source conditions for Tikhonov regularization with sparsity constraints", "abstract": "This paper addresses the regularization by sparsity constraints by means of weighted $\\ell^p$ penalties for $0\\leq p\\leq 2$. For $1\\leq p\\leq 2$ special attention is payed to convergence rates in norm and to source conditions. As main result it is proven that one gets a convergence rate in norm of $\\sqrt{\\delta}$ for $1\\leq p\\leq 2$ as soon as the unknown solution is sparse. The case $p=1$ needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed. For $p<1$ only preliminary results are shown. These results indicate that, different from $p\\geq 1$, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for $p=0$ shows that regularization need not to happen."}
{"category": "Math", "title": "Proximal alternating minimization and projection methods for nonconvex problems. An approach based on the Kurdyka-Lojasiewicz inequality", "abstract": "We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: $L(x,y)=f(x)+Q(x,y)+g(y)$, where $f:\\R^n\\rightarrow\\R\\cup{+\\infty}$ and $g:\\R^m\\rightarrow\\R\\cup{+\\infty}$ are proper lower semicontinuous functions, and $Q:\\R^n\\times\\R^m\\rightarrow \\R$ is a smooth $C^1$ function which couples the variables $x$ and $y$. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize $L$. We work in a nonconvex setting, just assuming that the function $L$ satisfies the Kurdyka-\\L ojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to \"metrically regular\" problems. Our main result can be stated as follows: If L has the Kurdyka-\\L ojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of $L$. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function $L$ around its critical points. When specialized to $Q(x,y)=|x-y|^2$ and to $f$, $g$ indicator functions, the algorithm is an alternating projection mehod (a variant of Von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with \"regular\" intersection. In order to illustrate our results with concrete problems, we provide a convergent proximal reweighted $\\ell^1$ algorithm for compressive sensing and an application to rank reduction problems."}
{"category": "Math", "title": "Extremal fullerene graphs with the maximum Clar number", "abstract": "A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let $F_n$ be a fullerene graph with $n$ vertices. A set $\\mathcal H$ of mutually disjoint hexagons of $F_n$ is a sextet pattern if $F_n$ has a perfect matching which alternates on and off each hexagon in $\\mathcal H$. The maximum cardinality of sextet patterns of $F_n$ is the Clar number of $F_n$. It was shown that the Clar number is no more than $\\lfloor\\frac {n-12} 6\\rfloor$. Many fullerenes with experimental evidence attain the upper bound, for instance, $\\text{C}_{60}$ and $\\text{C}_{70}$. In this paper, we characterize extremal fullerene graphs whose Clar numbers equal $\\frac{n-12} 6$. By the characterization, we show that there are precisely 18 fullerene graphs with 60 vertices, including $\\text{C}_{60}$, achieving the maximum Clar number 8 and we construct all these extremal fullerene graphs."}
{"category": "Math", "title": "Harmonic measure and SLE", "abstract": "In this paper we rigorously compute the average multifractal spectrum of harmonic measure on the boundary of SLE clusters."}
{"category": "Math", "title": "On the Spezialschar of Maass", "abstract": "Let $M_k^{(n)}$ be the space of Siegel modular forms of degree $n$ and even weight $k$. In this paper firstly a certain subspace $\\mathsf{Spez}(M_k^{(2n)})$ the Spezialschar of $M_k^{(2n)}$ is introduced. In the setting of the Siegel three-fold it is proven that this Spezialschar is the Maass Spezialschar. Secondly an embedding of $M_k^{(2)}$ into a direct sum $\\oplus_{\\nu = 0}^{\\lfloor \\frac{k}{10} \\rfloor} \\text{Sym}^2 M_{k + 2 \\nu}$ is given. This leads to a basic characterization of the Spezialschar property. The results of this paper are directly related to the non-vanishing of certain special values of L-functions related to the Gross-Prasad conjecture. This is illustrated by a significant example in the paper."}
{"category": "Math", "title": "\"Voici ce que j'ai trouve\": Sophie Germain's grand plan to prove Fermat's Last Theorem", "abstract": "A study of Sophie Germain's extensive manuscripts on Fermat's Last Theorem calls for a reassessment of her work in number theory. There is much in these manuscripts beyond the single theorem for Case 1 for which she is known from a published footnote by Legendre. Germain had a fully-fledged, highly developed, sophisticated plan of attack on Fermat's Last Theorem. The supporting algorithms she invented for this plan are based on ideas and results discovered independently only much later by others, and her methods are quite different from any of Legendre's. In addition to her program for proving Fermat's Last Theorem in its entirety, Germain also made major efforts at proofs for particular families of exponents. The isolation Germain worked in, due in substantial part to her difficult position as a woman, was perhaps sufficient that much of this extensive and impressive work may never have been studied and understood by anyone."}
{"category": "Math", "title": "A Strong Symmetry Property Of Eisenstein Series", "abstract": "In this paper we present a new method to study Fourier coefficients of holomorphic and non-holomorphic Eisenstein series simultaneously."}
{"category": "Math", "title": "Lineability of summing sets of homogeneous polynomials", "abstract": "Given a continuous $n$-homogeneous polynomial $P\\colon E\\longrightarrow F$ between Banach spaces and $1\\leq q\\leq p<\\infty$, in this paper we investigate some properties concerning lineability and spaceability of the $(p;q)$-summing set of $P$, defined by $S_{p;q}(P)=\\{a\\in E:P\\mathrm{is}% (p;q)\\mathrm{summing at}a\\}$."}
{"category": "Math", "title": "The geometry of 3-quasi-Sasakian manifolds", "abstract": "3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. This paper throws new light on their geometric structure which reveals to be generally richer compared to the 3-Sasakian subclass. In fact, it turns out that they are multiply foliated by four distinct fundamental foliations. The study of the transversal geometries with respect to these foliations allows us to link the 3-quasi-Sasakian manifolds to the more famous hyper-Kaehler and quaternionic-Kaehler geometries. Furthermore, we strongly improve the splitting results previously found; we prove that any 3-quasi-Sasakian manifold of rank 4l+1 is 3-cosymplectic and any 3-quasi-Sasakian manifold of maximal rank is 3-alpha-Sasakian."}
{"category": "Math", "title": "Freidlin-Wentzell's Large Deviations for Stochastic Evolution Equations", "abstract": "We prove a Freidlin-Wentzell large deviation principle for general stochastic evolution equations with small perturbation multiplicative noises. In particular, our general result can be used to deal with a large class of quasi linear stochastic partial differential equations, such as stochastic porous medium equations and stochastic reaction diffusion equations with polynomial growth zero order term and $p$-Laplacian second order term."}
{"category": "Math", "title": "Mapping class groups have finite asymptotic dimension", "abstract": "By recognizing them as fundamental groups of developable complexes of groups we prove that mapping class groups of compact orientable surfaces have finite asymptotic dimension."}
{"category": "Math", "title": "Adjoints of rationally induced composition operators", "abstract": "We give an elementary proof of a formula recently obtained by Hammond, Moorhouse, and Robbins for the adjoint of a rationally induced composition operator on the Hardy space H^2. We discuss some variants and implications of this formula, and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator."}
{"category": "Math", "title": "Vector-valued Riesz potentials: Cartan type estimates and related capacities", "abstract": "There are many interesting problems about the electrostatic potential of finitely many charges. We consider one of them concerning the intensity of the field, in other words, about the magnitude of the gradient of this potential. We want to give a sharp estimate of the size of the set of points where this gradient is large. Of course we want the estimate to be sharp in number $N$ of charges. The size will be measured by the Hausdorff content with various gauge functions. Such a setting allows us to consider a wide class of measures (not necessarily with finitely many charges). The main technique will be Calder\\'on-Zygmund capacities and nonhomogeneous Calder\\'on-Zygmund operators. Here we establish a relationship between various types of capacities with singular kernels (e. g. analytic capacity, lipschitz harmonic capacity, etc) and non-linear capacity from the theory of potential \\'a la Adams, Hedberg, Havin, Maz'ya, Wolff. \"Capacitary\" part of the paper extends the theorem of Mateu, Prat and Verdera [J. reine und angew. Math., 578 (2005), 201--223]. \"Size estimates\" part of the paper extends the theorem of M. Anderson and V. Eiderman [Annals of Math., 163 (2005), 1057--1076]. The difficulty lies in the fact that we cannot use Menger's curvature anymore because we are working in spaces of dimension bigger than two."}
{"category": "Math", "title": "Convexity properties of Thompson's group F", "abstract": "We prove that Thompson's group F is not minimally almost convex with respect to any generating set which is a subset of the standard infinite generating set for F and which contains x_1. We use this to show that F is not almost convex with respect to any generating set which is a subset of the standard infinite generating set."}
{"category": "Math", "title": "Adaptive Independent Metropolis-Hastings by Fast Estimation of Mixtures of Normals", "abstract": "We construct an adaptive independent Metropolis-Hastings sampler that uses a mixture of normals as a proposal distribution. To take full advantage of the potential of adaptive sampling our algorithm updates the mixture of normals frequently, starting early in the chain. The algorithm is built for speed and reliability and its sampling performance is evaluated with real and simulated examples. Our article outlines conditions for adaptive sampling to hold and gives a readily accessible proof that under these conditions the sampling scheme generates iterates that converge to the target distribution."}
{"category": "Math", "title": "Combined dynamic Gruss inequalities on time scales", "abstract": "We prove a more general version of the Gruss inequality by using the recent theory of combined dynamic derivatives on time scales and the more general notions of diamond-alpha derivative and integral. For the particular case when alpha = 1, one gets a delta-integral Gruss inequality on time scales; for alpha = 0 a nabla-integral Gruss inequality. If we further restrict ourselves by fixing the time scale to the real (or integer) numbers, then the standard continuous (discrete) inequalities are obtained."}
{"category": "Math", "title": "Identification of Boundary Conditions Using Natural Frequencies in Case of a Ring Membrane", "abstract": "The problem of finding boundary conditions for fastening of a ring membrane, which are inaccessible for direct observation from the natural frequencies of its flexural oscillations, is considered. Two theorems on the uniqueness of this problem are proved, and a method for establishing the unknown conditions for fastening of the membrane to the walls is indicated. An approximate formula for determining the unknown conditions is obtained, using first three natural frequencies. The method of approximate calculation of unknown boundary conditions, is explained with the help of an example. Keywords: Boundary conditions, inverse spectral problem, membrane, natural frequencies, Plucker coordinates, Plucker relation."}
{"category": "Math", "title": "Can One Hear Fastening of a Rod?", "abstract": "Rods are parts of various devices. If it is impossible to observe the rod directly, the only source of information about possible defects of its fastening can be the natural frequencies of its flexural vibrations. The question arises whether one would be able to detect damage in rod fastening by the natural frequencies of its flexural vibrations. This paper gives and substantiates a positive answer to this question."}
{"category": "Math", "title": "Noncommutative Riesz transforms -- a probabilistic approach", "abstract": "For $2\\le p<\\infty$ we show the lower estimates \\[ \\|A^{\\frac 12}x\\|_p \\kl c(p)\\max\\{\\pl \\|\\Gamma(x,x)^{{1/2}}\\|_p,\\pl \\|\\Gamma(x^*,x^*)^{{1/2}}\\|_p\\} \\] for the Riesz transform associated to a semigroup $(T_t)$ of completely positive maps on a von Neumann algebra with negative generator $T_t=e^{-tA}$, and gradient form \\[ 2\\Gamma(x,y)\\lel Ax^*y+x^*Ay-A(x^*y)\\pl .\\] As additional hypothesis we assume that $\\Gamma^2\\gl 0$ and the existence of a Markov dilation for $(T_t)$. We give applications to quantum metric spaces and show the equivalence of semigroup Hardy norms and martingale Hardy norms derived from the Markov dilation. In the limiting case we obtain a viable definition of BMO spaces for general semigroups of completely positive maps which can be used as an endpoint for interpolation. For torsion free ordered groups we construct a connection between Riesz transforms and the Hilbert transform induced by the order."}
{"category": "Math", "title": "The Lexicographic First Occurrence of a I-II-III pattern", "abstract": "Consider a random permutation $\\pi\\in{\\cal S}_n$. In this paper, perhaps best classified as a contribution to discrete probability distribution theory, we study the {\\it first} occurrence $X=X_n$ of a I-II-III-pattern, where \"first\" is interpreted in the lexicographic order induced by the 3-subsets of $[n]=\\{1,2,...,n\\}$. Of course if the permutation is I-II-III-avoiding then the first I-II-III-pattern never occurs, and thus $\\e(X)=\\infty$ for each $n$; to avoid this case, we also study the first occurrence of a I-II-III-pattern given a bijection $f:{\\bf Z}^+\\to{\\bf Z}^+$."}
{"category": "Math", "title": "Far field asymptotics of solutions to convection equation with anomalous diffusion", "abstract": "The initial value problem for the conservation law $\\partial_t u+(-\\Delta)^{\\alpha/2}u+\\nabla \\cdot f(u)=0$ is studied for $\\alpha\\in (1,2)$ and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as $|x|\\to \\infty$ of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity."}
{"category": "Math", "title": "Cut-disks for level spheres in link and tangle complements", "abstract": "Wu has shown that if a link or a knot $L$ in $S^3$ in thin position has thin spheres, then the thin sphere of lowest width is an essential surface in the link complement. In this paper we show that if we further assume that $L \\subset S^3$ is prime, then the thin sphere of lowest width also does not have any vertical cut-disks. We also prove the result for a specific kind of tangles in $S^2 \\times [-1,1]$."}
{"category": "Math", "title": "Positive forms on hyperkahler manifolds", "abstract": "Let $(M,I,J,K)$ be a hyperkaehler manifold, $\\dim_\\R M =4n$. We study positive, Dolbeault-closed $(2p,0)$-forms on $(M,I)$. These forms are quaternionic analogues of the positive $(p,p)$-forms. We construct an injective homomorphism mapping Dolbeault-closed $(2p,0)$-forms to closed $(n+p,n+p)$-forms, and positive $(2p,0)$-forms to positive $(n+p,n+p)$-forms. This construction is used to prove a hyperkaehler version of the classical Skoda-El Mir theorem, which says that a trivial extension of a closed, positive current over a pluripolar set is again closed. We also prove the hyperkaehler version of the Sibony's lemma, showing that a closed, positive $(2p,0)$-form defined outside of a compact complex subvariety $Z\\subset (M,I)$, $\\codim Z > 2p$ is locally integrable in a neighbourhood of $Z$. These results are used to prove polystability of derived direct images of certain coherent sheaves."}
{"category": "Math", "title": "Open maps having the Bula property", "abstract": "Every open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed subsets of X so that their image by f is Y provided all fibers of f are infinite and C*-embedded in X. Applications are demonstrated for the existence of \"disjoint\" usco multiselections of set-valued l.s.c. mappings defined on paracompact C-spaces, and for special type of factorizations of open continuous maps from metrizable spaces onto paracompact C-spaces. This settles several open questions."}
{"category": "Math", "title": "Twisting and Rieffel's deformation of locally compact quantum groups. Deformation of the Haar measure", "abstract": "We develop the twisting construction for locally compact quantum groups. A new feature, in contrast to the previous work of M. Enock and the second author, is a non-trivial deformation of the Haar measure. Then we construct Rieffel's deformation of locally compact quantum groups and show that it is dual to the twisting. This allows to give new interesting concrete examples of locally compact quantum groups, in particular, deformations of the classical $az+b$ group and of the Woronowicz' quantum $az+b$ group."}
{"category": "Math", "title": "A locally compact quantum group of triangular matrices", "abstract": "We construct a one parameter deformation of the group of $2\\times 2$ upper triangular matrices with determinant 1 using the twisting construction. An interesting feature of this new example of a locally compact quantum group is that the Haar measure is deformed in a non-trivial way. Also, we give a complete description of the dual $\\cs$-algebra and the dual comultiplication."}
{"category": "Math", "title": "Effective resistance of random trees", "abstract": "We investigate the effective resistance $R_n$ and conductance $C_n$ between the root and leaves of a binary tree of height $n$. In this electrical network, the resistance of each edge $e$ at distance $d$ from the root is defined by $r_e=2^dX_e$ where the $X_e$ are i.i.d. positive random variables bounded away from zero and infinity. It is shown that $\\mathbf{E}R_n=n\\mathbf{E}X_e-(\\operatorname {\\mathbf{Var}}(X_e)/\\mathbf{E}X_e)\\ln n+O(1)$ and $\\operatorname {\\mathbf{Var}}(R_n)=O(1)$. Moreover, we establish sub-Gaussian tail bounds for $R_n$. We also discuss some possible extensions to supercritical Galton--Watson trees."}
{"category": "Math", "title": "The origin of infinitely divisible distributions: from de Finetti's problem to Levy-Khintchine formula", "abstract": "The article provides an historical survey of the early contributions on infinitely divisible distributions starting from the pioneering works of de Finetti in 1929 up to the canonical forms developed in the thirties by Kolmogorov, Levy and Khintchine. Particular attention is paid to single out the personal contributions of the above authors that were published in Italian, French or Russian during the period 1929-1938. In Appendix we report the translation from the Russian into English of a fundamental paper by Khintchine published in Moscow in 1937."}
{"category": "Math", "title": "Large cardinals and gap-1 morasses", "abstract": "We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong (0<n<omega+1), hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; we refer to them as mangroves and prove that their existence is equivalent to the existence of morasses. Finally, we exhibit a partial order that forces universal morasses to exist at every regular uncountable cardinal, and use this to show that universal morasses are consistent with n-superstrong, hyperstrong, and 1-extendible cardinals. This all contributes to the second author's outer model programme, the aim of which is to show that L-like principles can hold in outer models which nevertheless contain large cardinals."}
{"category": "Math", "title": "Solutions to open problems in Neeb's recent survey on infinite-dimensional Lie groups", "abstract": "We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb: (1) There exists a subgroup of some infinite-dimensional Lie group G which does not admit an initial Lie subgroup structure; (2) The pathology cannot occur if G is a direct limit of an ascending sequence of finite-dimensional Lie groups; (3) Every such direct limit group is a ``topological group with Lie algebra'' in the sense of Hofmann and Morris. Moreover, we prove a version of Borel's Theorem announced in the survey, ensuring the existence of compactly supported smooth diffeomorphisms with given Taylor series around a fixed point p (provided the tangent map at p has positive determinant)."}
{"category": "Math", "title": "Bi-Hermitian gray surfaces II", "abstract": "The aim of this paper is to classify bi-Hermitian compact surfaces $(M,g)$ whose Ricci tensor $\\rho$ satisfies the relation $\\nabla_X\\rho(X,X) =\\frac13X\\tau g(X,X)$."}
{"category": "Math", "title": "Positively and negatively excited random walks on integers, with branching processes", "abstract": "We consider excited random walks on the integers with a bounded number of i.i.d. cookies per site which may induce drifts both to the left and to the right. We extend the criteria for recurrence and transience by M. Zerner and for positivity of speed by A.-L. Basdevant and A. Singh to this case and also prove an annealed central limit theorem. The proofs are based on results from the literature concerning branching processes with migration and make use of a certain renewal structure."}
{"category": "Math", "title": "The DNA Inequality in Non-Convex Regions", "abstract": "A simple plane closed curve $\\Gamma$ satisfies the DNA Inequality if the average curvature of any closed curve contained inside $\\Gamma$ exceeds the average curvature of $\\Gamma$. In 1997 Lagarias and Richardson proved that all convex curves satisfy the DNA Inequality and asked whether this is true for any non-convex curve. They conjectured that the DNA Inequality holds for certain L-shaped curves. In this paper, we disprove this conjecture for all L-Shapes and construct a large class of non-convex curves for which the DNA Inequality holds. We also give a polynomial-time procedure for determining whether any specific curve in a much larger class satisfies the DNA Inequality."}
{"category": "Math", "title": "Artin formalism for Selberg zeta functions of co-finite Kleinian groups", "abstract": "Let $\\Gamma\\backslash\\mathbb H^3$ be a finite-volume quotient of the upper-half space, where $\\Gamma\\subset {\\rm SL}(2,\\mathbb C)$ is a discrete subgroup. To a finite dimensional unitary representation $\\chi$ of $\\Gamma$ one associates the Selberg zeta function $Z(s;\\Gamma;\\chi)$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\\tilde\\Gamma$ is a finite index group extension of $\\Gamma$ in ${\\rm SL}(2,\\mathbb C)$, and $\\pi={\\rm Ind}_{\\Gamma}^{\\tilde\\Gamma}\\chi$ is the induced representation, then $Z(s;\\Gamma;\\chi)=Z(s;\\tilde\\Gamma;\\pi)$. In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely $\\phi(s;\\Gamma;\\chi)=\\phi(s;\\tilde\\Gamma;\\pi)$, for an appropriate normalization of the Eisenstein series."}
{"category": "Math", "title": "On smooth curves endowed with a large automorphism $p$-group in characteristic $p>0$", "abstract": "Let $k$ be an algebraically closed field of characteristic $p>0$ and $C$ a connected nonsingular projective curve over $k$ with genus $g \\geq 2$. This paper continues the work begun by Lehr and Matignon, namely the study of \"big actions\", i.e. the pairs $(C,G)$ where $G$ is a $p$-subgroup of the $k$-automorphism group of $C$ such that$\\frac{|G|}{g} >\\frac{2 p}{p-1}$. If $G_2$ denotes the second ramification group of $G$ at the unique ramification point of the cover $C \\to C/G$, we display necessary conditions on $G_2$ for $(C,G)$ to be a big action, which allows us to pursue the classification of big actions. Our main source of examples comes from the construction of curves with many rational points using ray class field theory for global function fields, as initiated by J-P. Serre and followed by Lauter and Auer. In particular, we obtain explicit examples of big actions with $G_2$ abelian of large exponent."}
{"category": "Math", "title": "On standard forms of 1--dominations between knots with same Gromov volumes", "abstract": "Let $k$ and $k'$ be two knots in 3-sphere. Say $k$ 1--dominates $k'$, if there is a proper degree 1 map $f\\co E(k)\\to E(k')$, between knot exterior of $k_i$. Theorem: Suppose that any companion of $k$ is prime. If $k$ 1--dominates $k'$ with the same Gromov volume, then $k'$ can be obtained from $k$ by finitely many de-satellizations. The condition of \"same Gromov volume\" clearly can not be removed. We also give a new construction of 1-domination between knots with same Gromov volume to show that the condition \"any companion of $k$ is prime\" can not be removed."}
{"category": "Math", "title": "Finiteness of mapping degrees and ${\\rm PSL}(2,{\\R})$-volume on graph manifolds", "abstract": "For given closed orientable 3-manifolds $M$ and $N$ let $\\c{D}(M,N)$ be the set of mapping degrees from $M$ to $N$. We address the problem: For which $N$, $\\c{D}(M,N)$ is finite for all $M$? The answer is known in Thurston's picture of closed orientable irreducible 3-manifolds unless the target is a non-trivial graph manifold. We prove that for each closed non-trivial graph manifold $N$, $\\c{D}(M,N)$ is finite for all graph manifold $M$. The proof uses a recently developed standard forms of maps between graph manifolds and the estimation of the $\\widetilde{\\rm PSL}(2,{\\R})$-volume for certain class of graph manifolds."}
{"category": "Math", "title": "On the connected components of moduli spaces of finite flat models", "abstract": "We prove that the non-ordinary component is connected in the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This was conjectured by Kisin. As an application to global Galois representations, we prove a theorem on the modularity comparing a deformation ring and a Hecke ring."}
{"category": "Math", "title": "Uniform asyptotic formulae for eigenfunctions of Sturm--Liouville operators with singular potentials", "abstract": "In this paper we study a Sturm--Liouville operator $Ly=-y''+q(x)y$ in the space $L_2[0,\\pi]$ with Direchlet boundary conditions. Here the potential $q$ is a fitst order distribution $q\\in W_2^{-1}[0,\\pi]$. Such operators were defined in our previous papers. Here we study an asymptotic behaviour of eigenfunctions with uniform estimates of rests. We obtain this estimates also for potentials from Sobolev spaces $q\\in W_2^{\\theta-1}$, where $\\theta\\in[0,1/2)$."}
{"category": "Math", "title": "Convexity and smoothness of scale functions and de Finetti's control problem", "abstract": "Under appropriate conditions, we obtain smoothness and convexity properties of $q$-scale functions for spectrally negative L\\'evy processes. Our method appeals directly to very recent developments in the theory of potential analysis of subordinators. As an application of the latter results to scale functions, we are able to continue the very recent work of \\cite{APP2007} and \\cite{Loe}. We strengthen their collective conclusions by showing, amongst other results, that whenever the L\\'evy measure has a density which is log convex then for $q>0$ the scale function $W^{(q)}$ is convex on some half line $(a^*,\\infty)$ where $a^*$ is the largest value at which $W^{(q)\\prime}$ attains its global minimum. As a consequence we deduce that de Finetti's classical actuarial control problem is solved by a barrier strategy where the barrier is positioned at height $a^*$."}
{"category": "Math", "title": "n-Monotone exact functionals", "abstract": "We study n-monotone functionals, which constitute a generalisation of n-monotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise the notions of coherence and natural extension in the behavioural theory of imprecise probabilities. We improve upon a number of results in the literature, and prove among other things a representation result for exact n-monotone functionals in terms of Choquet integrals."}
{"category": "Math", "title": "Asymptotics of semigroups generated by operator matrices", "abstract": "We survey some known results about operator semigroup generated by operator matrices with diagonal or coupled domain. These abstract results are applied to the characterization of well-/ill-posedness for a class of evolution equations with dynamic boundary conditions on domains or metric graphs. In particular, our ill-posedness results on the heat equation with general Wentzell-type boundary conditions complement those previously obtained by, among others, Bandle-von Below-Reichel and Vitillaro-V\\'azquez."}
{"category": "Math", "title": "Symmetry of models versus models of symmetry", "abstract": "A model for a subject's beliefs about a phenomenon may exhibit symmetry, in the sense that it is invariant under certain transformations. On the other hand, such a belief model may be intended to represent that the subject believes or knows that the phenomenon under study exhibits symmetry. We defend the view that these are fundamentally different things, even though the difference cannot be captured by Bayesian belief models. In fact, the failure to distinguish between both situations leads to Laplace's so-called Principle of Insufficient Reason, which has been criticised extensively in the literature. We show that there are belief models (imprecise probability models, coherent lower previsions) that generalise and include the Bayesian belief models, but where this fundamental difference can be captured. This leads to two notions of symmetry for such belief models: weak invariance (representing symmetry of beliefs) and strong invariance (modelling beliefs of symmetry). We discuss various mathematical as well as more philosophical aspects of these notions. We also discuss a few examples to show the relevance of our findings both to probabilistic modelling and to statistical inference, and to the notion of exchangeability in particular."}
{"category": "Math", "title": "Projective embeddings of homogeneous spaces with small boundary", "abstract": "We study open equivariant projective embeddings of homogeneous spaces such that the complement of the open orbit does not contain divisors. Criterions of existence of such an embedding are considered and finiteness of isomorphism classes of embeddings for a given homogeneous space is proved. Any embedding with small boundary is realized as a GIT-quotient associated with a linearization of the trivial line bundle on the space of the canonical embedding. The generalized Cox's construction and the theory of bunched rings allow us to describe basic geometric properties of embeddings with small boundary in combinatorial terms."}
{"category": "Math", "title": "Families index for manifolds with hyperbolic cusp singularities", "abstract": "Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the family index theorem of Melrose and Piazza and by treating the index of families of such operators. We also extend the index theorem of Moroianu and Leichtnam-Mazzeo-Piazza to families of perturbed Dirac-type operators associated to fibered cusp metrics (sometimes known as fibered boundary metrics)."}
{"category": "Math", "title": "Intertwining relations and extended eigenvalues for analytic Toeplitz operators", "abstract": "We study the intertwining relations between analytic Toeplitz operators induced on the Hardy space H^2 by analytic functions bounded on the open unit disc. Our work centers on the connection between intertwining between the Toeplitz operators the image containment between their symbols, as well as on the nature of the intertwining operator. We use our results to study the \"extended eigenvalues\" of analytic Toeplitz operators, i.e., the special case where the operator is intertwined with a scalar multiple of itself."}
{"category": "Math", "title": "An Enumerative Function", "abstract": "We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is an introduction. In the second section we derive an explicit formula for F. From the expression for the power function we obtain a number theory result. Then we derive a formula which shows that the case of arbitrary m may be reduced to the case m=0. This formula extends Vandermonde convolution. In the second section we describe F by the series of recurrence relations with respect to each of arguments k, n, and P. As a special case of the first recurrence relation we state a binomial identity. As a consequence of the second recurrence relation we obtain relation for coefficients of Chebyshev polynomial of both kind. This means that these polynomials might be defined in pure combinatorial way."}
{"category": "Math", "title": "Neighboring Fractions in Farey Subsequences", "abstract": "We present explicit formulas for the computation of the neighbors of several elements of Farey subsequences."}
{"category": "Math", "title": "Exponential estimates for plurisubharmonic functions and stochastic dynamics", "abstract": "We prove exponential estimates for plurisubharmonic functions with respect to Monge-Ampere measures with Holder continuous potential. As an application, we obtain several stochastic properties for the equilibrium measures associated to holomorphic maps on projective spaces. More precisely, we prove the exponential decay of correlations, the central limit theorem for general d.s.h. observables, and the large deviations theorem for bounded d.s.h. observables and Holder continuous observables."}
{"category": "Math", "title": "C*-algebras of labelled graphs II - Simplicity results", "abstract": "We prove simplicity and pure infiniteness results for a certain class of labelled graph $C^*$-algebras. We show, by example, that this class of unital labelled graph $C^*$-algebras is strictly larger than the class of unital graph $C^*$-algebras."}
{"category": "Math", "title": "On the characterization of expansion maps for self-affine tilings", "abstract": "We consider self-affine tilings in $\\R^n$ with expansion matrix $\\phi$ and address the question which matrices $\\phi$ can arise this way. In one dimension, $\\lambda$ is an expansion factor of a self-affine tiling if and only if $|\\lambda|$ is a Perron number, by a result of Lind. In two dimensions, when $\\phi$ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex $\\lambda$ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for $\\phi$ to be an expansion matrix for any $n$, assuming only that $\\phi$ is diagonalizable over the complex numbers. We conjecture that this condition on $\\phi$ is also sufficient for the existence of a self-affine tiling."}
{"category": "Math", "title": "An upper bound for the lower central series quotients of a free associative algebra", "abstract": "Feigin and Shoikhet conjectured in math/0610410 that successive quotients $B_m(A_n)$ of the lower central series filtration of a free associative algebra $A_n$ have polynomial growth. In this paper we give a proof of this conjecture, using the structure of $W_n$-representation on $B_m(A_n)$ which was defined in math/0610410 . We also prove that the number of squares in a Young diagram $D$ corresponding to an irreducible $W_n$-module in the Jordan-Holder series of $B_m(A_n)$ is bounded above by the integer $(m-1)^2+2[(n-2)/2](m-1)$. This bound combined with MAGMA computations by Rains in math/0610410 allows us to confirm the $W_n$-module structure of $B_3(A_3)$ conjectured in math/0610410 ."}
{"category": "Math", "title": "Geometry and rigidity of mapping class groups", "abstract": "We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication, and as a consequence obtain quasi-isometric rigidity for MCG(S), namely that groups quasi-isometric to MCG(S) are virtually equal to it. (The latter theorem was proved by Hamenstadt using different methods). As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG(S); a characterization of the image of the curve-complex projection map from MCG(S) to the product of the curve complexes of essential subsurfaces of S; and a construction of Sigma-hulls in MCG(S), an analogue of convex hulls."}
{"category": "Math", "title": "On Limit Aperiodic G-Sets", "abstract": "We prove that the property to be limit aperiodic is preserved by the standard construction with groups like extension, HNN extension and free product. We also construct a non-limit aperiodic G-space."}
{"category": "Math", "title": "Contracting an element from a cocircuit", "abstract": "We consider the situation that M and N are 3-connected matroids such that |E(N)| > 3 and C* is a cocircuit of M with the property that M/y has an N-minor for some y in C*. We show that either there is an element x in C* such that si(M/x) or co(si(M/x)) is 3-connected with an N-minor, or there is a four-element fan of M that contains two elements of C* and an element x such that si(M/x) is 3-connected with an N-minor."}
{"category": "Math", "title": "The linear flows in the space of Krichever-Lax matrices over an algebraic curve", "abstract": "In \\cite{kri02}, I. M. Krichever invented the space of matrices parametrizing the cotangent bundle of moduli space of stable vector bundles over a compact Riemann surface, which is named as the Hitchin system after the investigation \\cite{hit87}. We study a necessary and sufficient condition for the linearity of flows on the space of Krichever-Lax matrices in a Lax representation in terms of cohomological classes using the similar technique and analysis from the work \\cite{grif85} by P. A. Griffiths."}
{"category": "Math", "title": "Meet homological mirror symmetry", "abstract": "In this paper, we introduce the interested reader to homological mirror symmetry. After recalling a little background knowledge, we tackle the simplest cases of homological mirror symmetry: curves of genus zero and one. We close by outlining the current state of the field and mentioning what homo- logical mirror symmetry has to say about other aspects of mirror symmetry."}
{"category": "Math", "title": "Additive properties of product sets in an arbitrary finite field", "abstract": "It is proved that for any two subsets $A$ and $B$ of an arbitrary finite field $\\Fq$ such that $|A||B|>q$ the identity $16AB=\\Fq$ holds. Moreover, it is established that for every subsets $X, Y\\subset \\Fq$ with the property $|X||Y|\\geqslant 2q$ the equality $8XY=\\Fq$ holds."}
{"category": "Math", "title": "Absolutely summing linear operators into spaces with no finite cotype", "abstract": "Given an infinite-dimensional Banach space $X$ and a Banach space $Y$ with no finite cotype, we determine whether or not every continuous linear operator from $X$ to $Y$ is absolutely $(q;p)$-summing for almost all choices of $p$ and $q$, including the case $p=q$. If $X$ assumes its cotype, the problem is solved for all choices of $p$ and $q$. Applications to the theory of dominated multilinear mappings are also provided."}
{"category": "Math", "title": "Calabi-Yau categories and Poincare duality spaces", "abstract": "The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of Auslander-Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincare duality space gives rise to a Calabi-Yau category. This paper is a review of the theory."}
{"category": "Math", "title": "Pseudosymmetric braidings, twines and twisted algebras", "abstract": "A laycle is the categorical analogue of a lazy cocycle. Twines (as introduced by Bruguieres) and strong twines (as introduced by the authors) are laycles satisfying some extra conditions. If $c$ is a braiding, the double braiding $c^2$ is always a twine; we prove that it is a strong twine if and only if $c$ satisfies a sort of modified braid relation (we call such $c$ pseudosymmetric, as any symmetric braiding satisfies this relation). It is known that symmetric Yetter-Drinfeld categories are trivial; we prove that the Yetter-Drinfeld category $_H{\\cal YD}^H$ over a Hopf algebra $H$ is pseudosymmetric if and only if $H$ is commutative and cocommutative. We introduce as well the Hopf algebraic counterpart of pseudosymmetric braidings under the name pseudotriangular structures and prove that all quasitriangular structures on the $2^{n+1}$-dimensional pointed Hopf algebras E(n) are pseudotriangular. We observe that a laycle on a monoidal category induces a so-called pseudotwistor on every algebra in the category, and we obtain some general results (and give some examples) concerning pseudotwistors, inspired by properties of laycles and twines."}
{"category": "Math", "title": "n-Groupoids and Stacky Groupoids", "abstract": "We discuss two generalizations of Lie groupoids. One consists of Lie $n$-groupoids defined as simplicial manifolds with trivial $\\pi_{k\\geq n+1}$. The other consists of stacky Lie groupoids $\\cG\\rra M$ with $\\cG$ a differentiable stack. We build a 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to a certain Morita equivalence. We prove this in a general set-up so that the statement is valid in both differential and topological categories. \\Equivalences of higher groupoids in various categories are also described."}
{"category": "Math", "title": "Structure of the extended Schrodinger-Virasoro Lie algebra", "abstract": "In this paper, we study the derivations, the central extensions and the automorphism group of the extended Schrodinger-Virasoro Lie algebra, introduced by J. Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that the extended Schrodinger-Virasoro Lie algebra is an infinite-dimensional complete Lie algebra and the universal central extension of the extended Schrodinger-Virasoro Lie algebra in the category of Leibniz algebras is the same as that in the category of Lie algebras."}
{"category": "Math", "title": "On n-Perfect Rings and Cotorsion Dimension", "abstract": "A ring is called $n$-perfect ($n\\geq 0$), if every flat module has projective dimension less or equal than $n$. In this paper, we show that the $n$-perfectness relate, via homological approach, some homological dimension of rings. We study $n$-perfectness in some known ring constructions. Finally, several examples of $n$-perfect rings satisfying special conditions are given."}
{"category": "Math", "title": "Joint behaviour of semirecursive kernel estimators of the location and of the size of the mode of a probability density function", "abstract": "Let $\\theta$ and $\\mu$ denote the location and the size of the mode of a probability density. We study the joint convergence rates of semirecursive kernel estimators of $\\theta$ and $\\mu$. We show how the estimation of the size of the mode allows to measure the relevance of the estimation of its location. We also enlighten that, beyond their computational advantage on nonrecursive estimators, the semirecursive estimators are preferable to use for the construction on confidence regions."}
{"category": "Math", "title": "The optimality of the Boundedness Height Conjecture", "abstract": "We show that the Boundedness Height Conjecture is optimal; all varieties in a power of an elliptic curve which do not satisfy the hypothesis neither satisfy the thesis. The Bounded Height Conjecture is known to hold for varieties in a power of an elliptic curve. We also present some examples and remarks."}
{"category": "Math", "title": "Neutral bi-Hermitian Gray surfaces", "abstract": "The aim of this paper is to give examples of compact neutral 4-manifolds $(M,g)$ whose Ricci tensor $\\rho$ satisfies the relation $\\nabla_X\\rho(X,X) =\\frac13X\\tau g(X,X)$. We present also a family of new Einstein bi-Hermitian neutral metrics on ruled surfaces of genus $g>1$."}
{"category": "Math", "title": "Mertens' theorem for toral automorphisms", "abstract": "A dynamical Mertens' theorem for ergodic toral automorphisms with error term O(N^{-1}) is found, and the influence of resonances among the eigenvalues of unit modulus is examined. Examples are found with many more, and with many fewer, periodic orbits than expected."}
{"category": "Math", "title": "Geometric Gamma Max-Infinitely Divisible Models", "abstract": "A transformation of gamma max-infinitely divisible laws viz. geometric gamma max-infinitely divisible laws is considered in this paper. Some of its distributional and divisibility properties are discussed and a random time changed extremal process corresponding to this distribution is presented. A new kind of invariance (stability) under geometric maxima is proved and a max-AR(1) model corresponding to it is also discussed."}
{"category": "Math", "title": "Some optimization problems for nonlinear elastic membranes", "abstract": "In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem of optimizing the cost functional $\\J(u)=\\int_{\\partial\\Omega} f(x) u \\rd \\H^{N-1}$ over some admissible class of loads $f$ where $u$ is the (unique) solution to the problem $-\\Delta_p u + |u|^{p-2}u = 0$ in $\\Omega$ with $|\\nabla u|^{p-2}u_\\nu = f$ on $\\partial \\Omega$."}
{"category": "Math", "title": "Some explicit constructions of integral structures in quaternion algebras", "abstract": "Let B be an undefined quaternion algebra over Q. Following the explicit chacterization of some Eichler orders in B given by Hashimoto, we define explicit embeddings of these orders in some local rings of matrices; we describe the two natural inclusions of an Eichler order of leven Nq in an Eichler order of level N. Moreover we provide a basis for a chain of Eichler orders in B and prove results about their intersection."}
{"category": "Math", "title": "Hamiltonian approach to geodesic image matching", "abstract": "This paper presents a generalization to image matching of the Hamiltonian approach for planar curve matching developed in the context of group of diffeomorphisms. We propose an efficient framework to deal with discontinuous images in any dimension, for example 2D or 3D. In this context, we give the structure of the initial momentum (which happens to be decomposed in a smooth part and a singular part) thanks to a derivation lemma interesting in itself. The second part develops a Hamiltonian interpretation of the variational problem, derived from the optimal control theory point of view."}
{"category": "Math", "title": "The Recursion Theorem and Infinite Sequences", "abstract": "In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove that there exists an increasing sequence such that W_{e_n}={e_{n+1},e_{n+2},...} for every n. We call a nonempty computably enumerable set A self-constructing if W_e=A for every e in A. We show that every nonempty computable enumerable set which is disjoint from an infinite computable set is one-one equivalent to a self-constructing set"}
{"category": "Math", "title": "On the K(\\pi,1)-property for rings of integers in the mixed case", "abstract": "We investigate the Galois group G_S(p) of the maximal p-extension unramified outside a finite set S of primes of a number field in the (mixed) case, when there are primes dividing p inside and outside S. We show that the cohomology of G_S(p) is \"often\" isomorphic to the etale cohomology of the scheme Spec(O_k S), in particular, G_S(p) is of cohomological dimension 2 then. We deduce this from the results in our previous paper \"Rings of integers of type K(\\pi,1)\" (arXiv:0705.3372), which mainly dealt with the tame case."}
{"category": "Math", "title": "The Local Time of the Classical Risk Process", "abstract": "In this paper we give an explicit expression for the local time of the classical risk process and associate it with the density of an occupational measure. To do so, we approximate the local time by a suitable sequence of absolutely continuous random fields. Also, as an application, we analyze the mean of the times $s \\in [0,T]$ such that $0\\leq X_{s} \\leq X_{s+\\epsilon} $ for some given $\\epsilon>0$."}
{"category": "Math", "title": "Vanishing homology", "abstract": "In this paper we introduce a new homology theory devoted to the study of families such as semi-algebraic or subanalytic families and in general to any family definable in an o-minimal structure (such as Denjoy-Carleman definable or $ln-exp$ definable sets). The idea is to study the cycles which are vanishing when we approach a special fiber. This also enables us to derive local metric invariants for germs of definable sets. We prove that the homology groups are finitely generated."}
{"category": "Math", "title": "q-Invariant Functions for Some Generalizations of the Ornstein-Uhlenbeck Semigroup", "abstract": "We show that the multiplication operator associated to a fractional power of a Gamma random variable, with parameter q>0, maps the convex cone of the 1-invariant functions for a self-similar semigroup into the convex cone of the q-invariant functions for the associated Ornstein-Uhlenbeck (for short OU) semigroup. We also describe the harmonic functions for some other generalizations of the OU semigroup. Among the various applications, we characterize, through their Laplace transforms, the laws of first passage times above and overshoot for certain two-sided stable OU processes and also for spectrally negative semi-stable OU processes. These Laplace transforms are expressed in terms of a new family of power series which includes the generalized Mittag-Leffler functions."}
{"category": "Math", "title": "Bounds on the Poincare constant under negative dependence", "abstract": "We give bounds on the Poincare (inverse spectral gap) constant of a non-negative, integer-valued random variable W, under negative dependence assumptions such as ultra log-concavity and total negative dependence. We show that the bounds obtained compare well to others in the literature. Examples treated include some occupancy and urn models, a random graph model and small spacings on the circumference of a circle. Applications to Poisson convergence theorems are considered."}
{"category": "Math", "title": "Zero cycles on projective varieties and the norm principle", "abstract": "Using the Gille-Merkurjev norm principle we compute in a uniform way the image of the degree map for quadrics (Springer's theorem), for twisted forms of maximal orthogonal Grassmannians (theorem of Bayer-Fluckiger and Lenstra), for E6- (Rost's theorem), and E7-varieties."}
{"category": "Math", "title": "A study of counts of Bernoulli strings via conditional Poisson processes", "abstract": "We say that a string of length $d$ occurs, in a Bernoulli sequence, if a success is followed by exactly $(d-1)$ failures before the next success. The counts of such $d$-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic $d$-cycle counts in random permutations. In this note, we give a new framework, in terms of conditional Poisson processes, which allows for a quick characterization of the joint distribution of the counts of all $d$-strings, in a general class of Bernoulli sequences, as certain mixtures of the product of Poisson measures. This general class includes all Bernoulli sequences considered before, as well many new sequences."}
{"category": "Math", "title": "Dynamics of Twisted Alexander Invariants", "abstract": "The Pontryagin dual of the twisted Alexander module for a d-component link and GL(N,Z) representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. In the case of a knot, its associated topological entropy is the logarithmic growth rate of the number of torsion elements in the twisted first-homology group of r-fold cyclic covers of the knot complement, as r goes to infinity. Total twisted representations are introduced, and their properties are studied. The twisted Alexander polynomial obtained from any nonabelian parabolic SL(2,C) representation of a 2-bridge knot group is seen to be nontrivial. The zeros of any twisted Alexander polynomial of a torus knot corresponding to a parabolic SL(2,C) representation or a finite-image permutation representation are shown to be roots of unity."}
{"category": "Math", "title": "Necessary and sufficient conditions for local Pareto optimality on time scales", "abstract": "We study a multiobjective variational problem on time scales. For this problem, necessary and sufficient conditions for weak local Pareto optimality are given. We also prove a necessary optimality condition for the isoperimetric problem with multiple constraints on time scales."}
{"category": "Math", "title": "Exponential Bounds in the Law of Iterated Logarithm for Martingales", "abstract": "In this paper non-asymptotic exponential estimates are derived for tail of maximum martingale distribution by naturally norming in the spirit of the classical Law of Iterated Logarithm. Key words: Martingales, exponential estimations, moment, Banach spaces of random variables, tail of distribution, conditional expectation."}
{"category": "Math", "title": "A local families index formula for d-bar operators on punctured Riemann surfaces", "abstract": "Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of d-bar operators on the Teichmuller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space M{g,n} in the sense of orbifolds where it can be written in terms of Mumford-Morita-Miller classes. The degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the Quillen connection, a result originally obtained by Takhtajan and Zograf."}
{"category": "Math", "title": "Well-posedness of the Fifth Order Kadomtsev-Petviashvili I Equation in Anisotropic Sobolev Spaces with Nonnegative Indices", "abstract": "In this paper we establish the local and global well-posedness of the real valued fifth order Kadomstev-Petviashvili I equation in the anisotropic Sobolev spaces with nonnegative indices. In particular, our local well-posedness improves Saut-Tzvetkov's one and our global well-posedness gives an affirmative answer to Saut-Tzvetkov's $L^2$-data conjecture."}
{"category": "Math", "title": "On scatteredly continuous maps between topological spaces", "abstract": "A map $f:X\\to Y$ between topological spaces is defined to be {\\em scatteredly continuous} if for each subspace $A\\subset X$ the restriction $f|A$ has a point of continuity. We show that for a function $f:X\\to Y$ from a perfectly paracompact hereditarily Baire Preiss-Simon space $X$ into a regular space $Y$ the scattered continuity of $f$ is equivalent to (i) the weak discontinuity (for each subset $A\\subset X$ the set $D(f|A)$ of discontinuity points of $f|A$ is nowhere dense in $A$), (ii) the $\\sigma$-continuity ($X$ can be written as a countable union of closed subsets on which $f$ is continuous), (iii) the $G_\\delta$-measurability (the preimage of each open set is of type $G_\\delta$). Also under Martin Axiom, we construct a $G_\\delta$-measurable map $f:X\\to Y$ between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V.Vinokurov."}
{"category": "Math", "title": "The coarse classification of homogeneous ultra-metric spaces", "abstract": "We prove that two homogeneous ultra-metric spaces $X,Y$ are coarsely equivalent if and only if $\\mathrm{Ent}^\\sharp(X)=\\mathrm{Ent}^\\sharp(Y)$ where $\\mathrm{Ent}^\\sharp(X)$ is the so-called sharp entropy of $X$. This classification implies that each homogeneous proper ultra-metric space is coarsely equivalent to the anti-Cantor set $2^{<\\omega}$. For the proof of these results we develop a technique of towers which can have an independent interest."}
{"category": "Math", "title": "Maximization of the second positive Neumann eigenvalue for planar domains", "abstract": "We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of a twice smaller area. This estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Polya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres."}
{"category": "Math", "title": "On the oscillatory integration of some ordinary differential equations", "abstract": "Conditions are given for a class of nonlinear ordinary differential equations x''(t)+a(t)w(x)=0, t>=1, which includes the linear equation to possess solutions x(t) with prescribed oblique asymptote that have an oscillatory pseudo-wronskian x'(t)-x(t)/t."}
{"category": "Math", "title": "Dominated splitting and zero volume for incompressible three-flows", "abstract": "We prove that there exists an open and dense subset of the incompressible 3-flows of class C^2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincar\\'e flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi-Ma\\~n\\'e and of Newhouse for flows with singularities. That is we obtain for a residual subset of the C^1 incompressible flows on 3-manifolds that: (i) either all Lyapunov exponents are zero or the flow is Anosov, and (ii) either the flow is Anosov or else the elliptic periodic points are dense in the manifold."}
{"category": "Math", "title": "On a ramification bound of torsion semi-stable representations over a local field", "abstract": "For a rational prime p, let k be a perfect field of characteristic p, K be a finite totally ramified extension of Frac(W(k)) of degree e and r be a non-negative integer satisfying r<p-1. In this article, we prove the upper numbering ramification group G^(j) for j>u(K,r,n) acts trivially on the p^n-torsion semi-stable G_K-representations with the Hodge-Tate weights in {0,...,r}, where u(K,0,n)=0, u(K,1,n)=1+e(n+1/(p-1)) and u(K,r,n)=1-p^{-n}+e(n+r/(p-1)) for r>1."}
{"category": "Math", "title": "On existence and uniqueness of the carrying simplex for competitive dynamical systems", "abstract": "Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology."}
{"category": "Math", "title": "On the testability and repair of hereditary hypergraph properties", "abstract": "Recent works of Alon-Shapira and R\\\"odl-Schacht have demonstrated that every hereditary property of undirected graphs or hypergraphs is testable with one-sided error; informally, this means that if a graph or hypergraph satisfies that property \"locally\" with sufficiently high probability, then it can be perturbed (or \"repaired\") into a graph or hypergraph which satisfies that property \"globally\". In this paper we make some refinements to these results, some of which may be surprising. In the positive direction, we strengthen the results to cover hereditary properties of multiple directed polychromatic graphs and hypergraphs. In the case of undirected graphs, we extend the result to continuous graphs on probability spaces, and show that the repair algorithm is \"local\" in the sense that it only depends on a bounded amount of data; in particular, the graph can be repaired in a time linear in the number of edges. We also show that local repairability also holds for monotone or partite hypergraph properties (this latter result is also implicitly in work of Ishigami). In the negative direction, we show that local repairability breaks down for directed graphs, or for undirected 3-uniform hypergraphs. The reason for this contrast in behavior stems from (the limitations of) Ramsey theory."}
{"category": "Math", "title": "Theory of non-lc ideal sheaves -basic properties-", "abstract": "We introduce the notion of non-lc ideal sheaves. It is an analogue of the notion of multiplier ideal sheaves. We establish the restriction theorem, which seems to be the most important property of non-lc ideal sheaves."}
{"category": "Math", "title": "Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over the Twisted Schr\\\"{o}dinger-Virasoro Lie algebra", "abstract": "It is shown that the support of an irreducible weight module over the Schr\\\"{o}dinger-Virasoro Lie algebra with an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite-dimensional. As a side-product, it is obtained that every simple weight module over the Schr\\\"{o}dinger-Virasoro Lie algebra with a nontrivial finite-dimensional weight space, is a Harish-Chandra module."}
{"category": "Math", "title": "The derivation algebra and automorphism group of the twisted Schr\\\"{o}dinger-Virasoro algebra", "abstract": "In this article, we determine the derivation algebra and the automorphism group of the twisted Schr\\\"{o}dinger-Virasoro algebra."}
{"category": "Math", "title": "Representations of the Schr\\\"{o}dinger-Virasoro algebras", "abstract": "In this paper it is proved that an irreducible weight module with finite-dimensional weight spaces over the Schr\\\"{o}dinger-Virasoro algebras is a highest/lowest weight module or a uniformly bounded module. Furthermore, indecomposable modules of the intermediate series over these algebras are completely determined."}
{"category": "Math", "title": "2-Cocycles of Deformative Schr\\\"{o}dinger-Virasoro Algebras", "abstract": "In a series of papers by Henkel, Roger and Unterberger, Schr\\\"{o}dinger-Virasoro algebras and their deformations were introduced and investigated. In the present paper we determine the 2-cocycles of a class of deformative Schr\\\"{o}dinger-Virasoro algebras."}
{"category": "Math", "title": "Leibniz Central Extension on the Twisted Schr\\\"{o}dinger-Virasoro Algebra", "abstract": "In this paper we present all the Leibniz 2-cocycles of the twisted Schr\\\"{o}dinger-Virasoro algebra, which determine its second Leibniz cohomology group."}
{"category": "Math", "title": "Knots in Riemannian manifolds", "abstract": "In this paper we study submanifold with nonpositive extrinsic curvature in a positively curved manifold. Among other things we prove that, if $K\\subset (S^n, g)$ is a totally geodesic submanifold in a Riemannian sphere with positive sectional curvature where $n\\ge 5$, then $K$ is homeomorphic to $S^{n-2}$ and the fundamental group of the knot complement $\\pi_1(S^n-K)\\cong \\Bbb Z$."}
{"category": "Math", "title": "On the topology of manifolds with positive isotropic curvature", "abstract": "We show that a closed orientable Riemannian $n$-manifold, $n \\ge 5$, with positive isotropic curvature and free fundamental group is homeomorphic to the connected sum of copies of $S^{n-1} \\times S^1$."}
{"category": "Math", "title": "A powerful test based on tapering for use in functional data analysis", "abstract": "A test based on tapering is proposed for use in testing a global linear hypothesis under a functional linear model. The test statistic is constructed as a weighted sum of squared linear combinations of Fourier coefficients, a tapered quadratic form, in which higher Fourier frequencies are down-weighted so as to emphasize the smooth attributes of the model. A formula is $Q_n^{OPT}=n\\sum_{j=1}^{p_n}j^{-1/2}\\|\\boldsymbol{Y}_{n,j}\\|^2$. Down-weighting by $j^{-1/2}$ is selected to achieve adaptive optimality among tests based on tapering with respect to its ``rates of testing,'' an asymptotic framework for measuring a test's retention of power in high dimensions under smoothness constraints. Existing tests based on truncation or thresholding are known to have superior asymptotic power in comparison with any test based on tapering; however, it is shown here that high-order effects can be substantial, and that a test based on $Q_n^{OPT}$ exhibits better (non-asymptotic) power against the sort of alternatives that would typically be of concern in functional data analysis applications. The proposed test is developed for use in practice, and demonstrated in an example application."}
{"category": "Math", "title": "Spatial modelling for mixed-state observations", "abstract": "In several application fields like daily pluviometry data modelling, or motion analysis from image sequences, observations contain two components of different nature. A first part is made with discrete values accounting for some symbolic information and a second part records a continuous (real-valued) measurement. We call such type of observations \"mixed-state observations\". This paper introduces spatial models suited for the analysis of these kinds of data. We consider multi-parameter auto-models whose local conditional distributions belong to a mixed state exponential family. Specific examples with exponential distributions are detailed, and we present some experimental results for modelling motion measurements from video sequences."}
{"category": "Math", "title": "On Frenkel-Mukhin algorithm for q-character of quantum affine algebras", "abstract": "The q-character is a strong tool to study finite-dimensional representations of quantum affine algebras. However, the explicit formula of the q-character of a given representation has not been known so far. Frenkel and Mukhin proposed the iterative algorithm which generates the q-character of a given irreducible representation starting from its highest weight monomial. The algorithm is known to work for various classes of representations. In this note, however, we give an example in which the algorithm fails to generate the q-character."}
{"category": "Math", "title": "Cosets, genericity, and the Weyl group", "abstract": "We prove a non-generosity theorem for proper cosets in groups of finite Morley rank and elaborate on the theory of Weyl groups in this context."}
{"category": "Math", "title": "Cohomological properties of non-standard multigraded modules", "abstract": "In this paper we study some cohomological properties of non-standard multigraded modules and Veronese transforms of them. Among others numerical characters, we study the generalized depth of a module and we see that it is invariant by taking a Veronese transform. We prove some vanishing theorems for the local cohomology modules of a multigraded module; as a corollary of these results we get that the depth of a Veronese module is asymptotically constant."}
{"category": "Math", "title": "Free-energy-dissipative schemes for the Oldroyd-B model", "abstract": "In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a free energy dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by Fattal and Kupferman, which have been reported to be numerically more stable than discretizations of the usual formulation in some benchmark problems. Our analysis gives some tracks to understand these numerical observations."}
{"category": "Math", "title": "On Wasserstein geometry of the space of Gaussian measures", "abstract": "The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the $L^2$-Wasserstein space, we manage to provide detailed descriptions of the $L^2$-Wasserstein geometry from a Riemannian geometric viewpoint. We first construct a Riemannian metric which induces the $L^2$-Wasserstein distance. Then we obtain a formula for the sectional curvatures of the space of Gaussian measures, which is written out in terms of the eigenvalues of the covariance matrix."}
{"category": "Math", "title": "On the Number of Matchings in Regular Graphs", "abstract": "For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures for 2-regular bipartite graphs and for $m$-matchings with $m\\le 4$."}
{"category": "Math", "title": "The Jiang-Su algebra revisited", "abstract": "We give a number of new characterizations of the Jiang-Su algebra Z, both intrinsic and extrinsic, in terms of C*-algebraic, dynamical, topological and K-theoretic conditions. Along the way we study divisibility properties of C*-algebras, we give a precise characterization of those unital C*-algebras of stable rank one that admit a unital embedding of the dimension-drop C*-algebra Z_{n,n+1}, and we prove a cancellation theorem for the Cuntz semigroup of C*-algebras of stable rank one."}
{"category": "Math", "title": "A functional central limit theorem for regenerative chains", "abstract": "Using the regenerative scheme of Comets, Fern\\'andez and Ferrari (2002), we establish a functional central limit theorem (FCLT) for discrete time stochastic processes (chains) with summable memory decay. Furthermore, under stronger assumptions on the memory decay, we identify the limiting variance in terms of the process only. As applications, we define classes of binary autoregressive processes and power-law Ising chains for which the FCLT is fulfilled."}
{"category": "Math", "title": "Non-existence and splitting theorems for normal integral bases", "abstract": "We establish new conditions that prevent the existence of (weak) normal integral bases in tame Galois extensions of number fields. This leads to the following result: under appropriate technical hypotheses, the existence of a normal integral basis in the upper layer of an abelian tower Q \\subset K \\subset L forces the tower to be split in a very strong sense."}
{"category": "Math", "title": "Holomorphic maps from rational homogeneous spaces onto projective manifolds", "abstract": "Answering a problem raised by Lazarsfeld, Hwang and Mok proved that a surjective holomorphic map from a rational homogeneous space of Picard number 1 onto projective manifold different from projective space must be a biholomorphism. THe aim of this paper is to generalized this result to irreducible rational homogeneous space of higher Picard number."}
{"category": "Math", "title": "Formal Desingularization of Surfaces - The Jung Method Revisited -", "abstract": "In this paper we propose the concept of formal desingularizations as a substitute for the resolution of algebraic varieties. Though a usual resolution of algebraic varieties provides more information on the structure of singularities there is evidence that the weaker concept is enough for many computational purposes. We give a detailed study of the Jung method and show how it facilitates an efficient computation of formal desingularizations for projective surfaces over a field of characteristic zero, not necessarily algebraically closed. The paper includes a generalization of Duval's Theorem on rational Puiseux parametrizations to the multivariate case and a detailed description of a system for multivariate algebraic power series computations."}
{"category": "Math", "title": "Adjoint Computation for Hypersurfaces Using Formal Desingularizations", "abstract": "We show how to use formal desingularizations (defined earlier by the first author) in order to compute the global sections (also called adjoints) of twisted pluricanonical sheaves. These sections define maps that play an important role in the birational classification of schemes."}
{"category": "Math", "title": "Liouville type results for periodic and almost periodic linear operators", "abstract": "We are concerned with some extensions of the classical Liouville theorem for bounded harmonic functions to solutions of more general equations. We deal with entire solutions of periodic and almost periodic parabolic equations including the elliptic framework as a particular case. We derive a Liouville type result for periodic operators as a consequence of a result for operators periodic in just one variable, which is new even in the elliptic case. More precisely, we show that if $c\\leq0$ and $a_{ij}, b_i, c, f$ are periodic in the same space/time direction, with the same period, then any bounded solution $u$ of $$\\partial_t u-a_{ij}(x,t)\\partial_{ij}u-b_i(x,t)\\partial_iu-c(x,t)u=f(x,t),\\quad x\\in\\R^N,\\ t\\in\\R,$$ is periodic in that direction. We then derive the following Liouville type result: if $c\\leq0, f\\equiv0$ and $a_{ij}, b_i, c$ are periodic in all the space/time variables, with the same periods, then the space of bounded solutions of the above equation has at most dimension one. In the case of the equation $\\partial_t u-Lu=f(x,t)$, with $L$ periodic elliptic operator independent of $t$, the hypothesis $c\\leq0$ can be weaken by requiring that the periodic principal eigenvalue of $-L$ is nonnegative. Instead, the periodicity assumption cannot be relaxed, because we explicitly exhibit an almost periodic function $b$ such that the space of bounded solutions of $u''+b(x)u'=0$ in $\\R$ has dimension 2, and it is generated by the constant solution and a non-almost periodic solution. Next, a sufficient condition for any bounded solution to be almost periodicis derived. We also treat the case of periodic domains under either Dirichlet or Robin boundary conditions."}
{"category": "Math", "title": "Aspects of Predicative Algebraic Set Theory II: Realizability", "abstract": "This is the second in a series of papers on the relation between algebraic set theory and predicative formal systems. In part I, we introduced the notion of a predicative category of small maps and obtained the result that such categories always contain a model of set theory. In the present paper, we show that the familiar realizability models of the constructive set theories CZF and IZF can be obtained as an application of this result. For this purpose, we show that predicative categories with small maps are closed under an internal notion of realizability."}
{"category": "Math", "title": "Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions", "abstract": "This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with $d\\ge3$ and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass $M_c$ such that if $M \\in (0,M_c]$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for $M \\in (0,M_c)$. While characterising the eventual infinite time blowing-up profile for $M=M_c$, we observe that the long time asymptotics are much more complicated than in the classical Patlak-Keller-Segel system in dimension two."}
{"category": "Math", "title": "Operators on C_{0}(L,X) whose range does not contain c_{0}", "abstract": "This paper contains the following results: a) Suppose that X is a non-trivial Banach space and L is a non-empty locally compact Hausdorff space without any isolated points. Then each linear operator T: C_{0}(L,X)\\to C_{0}(L,X), whose range does not contain C_{00} isomorphically, satisfies the Daugavet equality ||I+T||=1+||T||. b) Let \\Gamma be a non-empty set and X, Y be Banach spaces such that X is reflexive and Y does not contain c_{0} isomorphically. Then any continuous linear operator T: c_{0}(\\Gamma,X)\\to Y is weakly compact."}
{"category": "Math", "title": "On the global well-posedness for the axisymmetric Euler equations", "abstract": "This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in critical Besov spaces $B_{p,1}^{1+3/p}$. In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity."}
{"category": "Math", "title": "Malliavin calculus for difference approximations of multidimensional diffusions: truncated local limit theorem", "abstract": "For a difference approximations of multidimensional diffusion, the truncated local limit theorem is proved. Under very mild conditions on the distribution of the difference terms, this theorem provides that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess a densities that converge uniformly to the transition probability density for the limiting diffusion and satisfy a uniform diffusion-type estimates. The proof is based on the new version of the Malliavin calculus for the product of finite family of measures, that may contain non-trivial singular components. An applications for uniform estimates for mixing and convergence rates for difference approximations to SDE's and for convergence of difference approximations for local times of multidimensional diffusions are given."}
{"category": "Math", "title": "A Note on the class of superreflexive almost transitive Banach spaces", "abstract": "The class J of simultaneously almost transitive, uniformly convex and uniformly smooth Banach spaces is characterized in terms of convex-transitivity and weak geometry of the norm."}
{"category": "Math", "title": "Spectra of symmetric powers of graphs and the Weisfeiler-Lehman refinements", "abstract": "The k-th power of a n-vertex graph X is the iterated cartesian product of X with itself. The k-th symmetric power of X is the quotient graph of certain subgraph of its k-th power by the natural action of the symmetric group. It is natural to ask if the spectrum of the k-th power --or the spectrum of the k-th symmetric power-- is a complete graph invariant for small values of k, for example, for k=O(1) or k=O(log n). In this paper, we answer this question in the negative: we prove that if the well known 2k-dimensional Weisfeiler-Lehman method fails to distinguish two given graphs, then their k-th powers --and their k-th symmetric powers-- are cospectral. As it is well known, there are pairs of non-isomorphic n-vertex graphs which are not distinguished by the k-dim WL method, even for k=Omega(n). In particular, this shows that for each k, there are pairs of non-isomorphic n-vertex graphs with cospectral k-th (symmetric) powers."}
{"category": "Math", "title": "Analysis of the stochastic FitzHugh-Nagumo system", "abstract": "In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall be mainly concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure $\\nu$ associated with the transition semigroup $P_t$; further, we identify its infinitesimal generator in the space $L^2(H;\\nu)$."}
{"category": "Math", "title": "A class of transversal polymatroids with Gorenstein base ring", "abstract": "In this paper, the principal tool to describe transversal polymatroids with Gorenstein base ring is polyhedral geometry, especially the $Danilov-Stanley$ theorem for the characterization of canonical module. Also, we compute the $a-invariant$ and the Hilbert series of base ring associated to this class of transversal polymatroids."}
{"category": "Math", "title": "The Einstein relation for random walks on graphs", "abstract": "This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic framework for the study of (sub-) diffusive behavior of the random walks on weighted graphs."}
{"category": "Math", "title": "A Lie-theoretic construction of spherical symplectic reflection algebras", "abstract": "We propose a construction of the spherical subalgebra of a symplectic reflection algebra of an arbitrary rank corresponding to a star-shaped affine Dynkin diagram. Namely, it is obtained from the universal enveloping algebra of a certain semi-simple Lie algebra by the process of quantum Hamiltonian reduction. As an application, we propose a construction of finite-dimensional representations of the spherical subalgebra."}
{"category": "Math", "title": "Upper bounds for transition probabilities on graphs and isoperimetric inequalities", "abstract": "In this paper necessary and sufficient conditions are presented for heat kernel upper bounds for random walks on weighted graphs. Several equivalent conditions are given in the form of isoperimetric inequalities."}
{"category": "Math", "title": "Random walks on graphs with volume and time doubling", "abstract": "This paper studies the on- and off-diagonal upper estimate and the two-sided transition probability estimate of random walks on weighted graphs."}
{"category": "Math", "title": "Galois theory and integral models of Lambda-rings", "abstract": "We show that any Lambda-ring, in the sense of Riemann-Roch theory, which is finite etale over the rational numbers and has an integral model as a Lambda-ring is contained in a product of cyclotomic fields. In fact, we show that the category of them is described in a Galois-theoretic way in terms of the monoid of pro-finite integers under multiplication and the cyclotomic character. We also study the maximality of these integral models and give a more precise, integral version of the result above. These results reveal an interesting relation between Lambda-rings and class field theory."}
{"category": "Math", "title": "Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result", "abstract": "We deal with symmetry properties for solutions of nonlocal equations of the type $(-\\Delta)^s v= f(v)\\qquad {in $\\R^n$,}$ where $s \\in (0,1)$ and the operator $(-\\Delta)^s$ is the so-called fractional Laplacian. The study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation ${-div (x^\\a \\nabla u)=0 \\qquad {on $\\R^n\\times(0,+\\infty)$} -x^\\a u_x = f(u) \\qquad {on $\\R^n\\times\\{0\\}$} $ where $\\a \\in (-1,1)$. This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator $\\Gamma_\\a: u|_{\\partial \\R^{n+1}_+} \\mapsto -x^\\a u_x |_{\\partial \\R^{n+1}_+} $ is $(-\\Delta)^{\\frac{1-\\a}{2}}$. This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator $\\Gamma_\\a: u|_{\\partial \\R^{n+1}_+} \\mapsto -x^\\a u_x |_{\\partial \\R^{n+1}_+} $ is $(-\\Delta)^{\\frac{1-\\a}{2}}$. More generally, we study the so-called boundary reaction equations given by ${-div (\\mu(x) \\nabla u)+g(x,u)=0 {on $\\R^n\\times(0,+\\infty)$} - \\mu(x) u_x = f(u) {on $\\R^n\\times{0}$}$ under some natural assumptions on the diffusion coefficient $\\mu$ and on the nonlinearities $f$ and $g$. We prove a geometric formula of Poincar\\'e-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi."}
{"category": "Math", "title": "Feasibly Reducing KAT Equations to KA Equations", "abstract": "Kleene algebra (KA) is the algebra of regular events. Familiar examples of Kleene algebras include regular sets, relational algebras, and trace algebras. A Kleene algebra with tests (KAT) is a Kleene algebra with an embedded Boolean subalgebra. The addition of tests allows one to encode {\\tt while} programs as KAT terms, thus the equational theory of KAT can express (propositional) program equivalence. More complicated statements about programs can be expressed in the Hoare theory of KAT, which suffices to encode Propositional Hoare Logic. That the equational theory of KAT reduces to the equational theory of KA has been shown by Cohen et al. Unfortunately, their reduction involves an exponential blowup in the size of the terms involved. Here we give an alternate feasible reduction."}
{"category": "Math", "title": "Jet Geometrical Objects Depending on a Relativistic Time", "abstract": "In this paper we study a collection of jet geometrical concepts, we refer to d-tensors, relativistic time dependent semisprays, harmonic curves and nonlinear connections on the 1-jet space J1(R;M), necessary to the construction of a Miron's-like geometrization for Lagrangians depending on a relativistic time. The geometrical relations between these jet geometrical objects are exposed."}
{"category": "Math", "title": "One-Parameter Toric Deformations of Cyclic Quotient Singularities", "abstract": "In the case of two-dimensional cyclic quotient singularities, we classify all one-parameter toric deformations in terms of certain Minkowski decompositions. In particular, we describe to which components each such deformation maps, show how to induce each deformation from a versal family, give explicit equations for each deformation, describe the singularities in the general fibers, and construct the corresponding partial simultaneous resolutions."}
{"category": "Math", "title": "A Riemann mapping theorem for two-connected domains in the plane", "abstract": "We show how to express a conformal map of a general two connected domain in the plane such that neither boundary component is a point to a representative domain which has the virtue of having an explicit algebraic Bergman kernel function. We shall explain why the representative domain is the best analogue of the unit disc in the two connected setting. The conformal map will be given as a simple and explicit algebraic function of an Ahlfors map of the domain associated to a specially chosen point. It will follow that the conformal map can be found by solving the same extremal problem that determines a Riemann map in the simply connected case."}
{"category": "Math", "title": "Splitting finite antichains in the homomorphism order", "abstract": "A structural condition is given for finite maximal antichains in the homomorphism order of relational structures to have the splitting property. It turns out that non-splitting antichains appear only at the bottom of the order. Moreover, we examine looseness and finite antichain extension property for some subclasses of the homomorphism poset. Finally, we take a look at cut-points in this order."}
{"category": "Math", "title": "Sublattices of the lattice of local clones", "abstract": "We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices, but that the class of lattices embeddable into the local clone lattice is strictly larger than that."}
{"category": "Math", "title": "The volume and time comparison principle and transition probability estimates for random walks", "abstract": "This paper presents necessary and sufficient conditions for on- and off-diagonal transition probability estimates for random walks on weighted graphs. On the integer lattice and on may fractal type graphs both the volume of a ball and the mean exit time from a ball is independent of the centre, uniform in space. Here the upper estimate is given without such restriction and two-sided estimate is given if uniformity in the space assumed only for the mean exit time."}
{"category": "Math", "title": "Quantum toroidal algebras and their representations", "abstract": "Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to whom they are related via Schur-Weyl duality. In this review paper, we give a glimpse on some aspects of their very rich representation theory in the context of general quantum affinizations. We illustrate with several examples. We also announce new results and explain possible further developments, in particular on finite dimensional representations at roots of unity."}
{"category": "Math", "title": "Towards an optimal algorithm for recognizing Laman graphs", "abstract": "Laman graphs are fundamental to rigidity theory. A graph G with n vertices and m edges is a generic minimally rigid graph (Laman graph), if m=2n-3 and every induced subset of k vertices spans at most 2k-3 edges. We consider the verification problem: Given a graph G with n vertices, decide if it is Laman. We present an algorithm that takes O(T(n)+n log n) time, where T(n) is the best time to extract two edge disjoint spanning trees from G or decide no such trees exist. Our algorithm exploits a known construction called red-black hierarchy (RBH), that is a certificate for Laman graphs. First, we show how to verify if G admits an RBH and argue this is enough to conclude whether G is Laman or not. Second, we show how to construct the RBH using a two steps procedure that is simple and easy to implement. Finally, we point out some difficulties in using red-black hierarchies to compute a Henneberg construction, which seem to imply super-quadratic time algorithms when used for embedding a planar Laman graph as a pointed pseudo-triangulation."}
{"category": "Math", "title": "Recurrence times and large deviations", "abstract": "We give a criterion to determine the large deviation rate functions for abstract dynamical systems on towers. As an application of this criterion we show the level 2 large deviation principle for some class of smooth interval maps with nonuniform hyperbolicity."}
{"category": "Math", "title": "Generalized permutation patterns -- a short survey", "abstract": "An occurrence of a classical pattern p in a permutation \\pi is a subsequence of \\pi whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance--or the prescribed number of occurrences--of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns."}
{"category": "Math", "title": "Abstract p-time proof nets for MALL: Conflict nets", "abstract": "This paper presents proof nets for multiplicative-additive linear logic (MALL), called conflict nets. They are efficient, since both correctness and translation from a proof are p-time (polynomial time), and abstract, since they are invariant under transposing adjacent &-rules. A conflict net on a sequent is concise: axiom links with a conflict relation. Conflict nets are a variant of (and were inspired by) combinatorial proofs introduced recently for classical logic: each can be viewed as a maximal map (homomorphism) of contractible coherence spaces (P_4-free graphs, or cographs), from axioms to sequent. The paper presents new results for other proof nets: (1) correctness and cut elimination for slice nets (Hughes / van Glabbeek 2003) are p-time, and (2) the cut elimination proposed for monomial nets (Girard 1996) does not work. The subtleties which break monomial net cut elimination also apply to conflict nets: as with monomial nets, existence of a confluent cut elimination remains an open question."}
{"category": "Math", "title": "Weak approximation on del Pezzo surfaces of degree 1", "abstract": "We study del Pezzo surfaces of degree 1 of the form w^2 = z^3 + Ax^6 + By^6 in the weighted projective space P_k(1,1,2,3), where k is a perfect field of characteristic not 2 or 3 and A,B \\in k^*. Over a number field, we exhibit an infinite family of (minimal) counterexamples to weak approximation amongst these surfaces, via a Brauer-Manin obstruction."}
{"category": "Math", "title": "Perturbing singular solutions of the Gelfand problem", "abstract": "he equation $-\\Delta u = \\lambda e^u$ posed in the unit ball $B \\subseteq \\R^N$, with homogeneous Dirichlet condition $u|_{\\partial B} = 0$, has the singular solution $U=\\log\\frac1{|x|^2}$ when $\\lambda = 2(N-2)$. If $N\\ge 4$ we show that under small deformations of the ball there is a singular solution $(u,\\lambda)$ close to $(U,2(N-2))$. In dimension $N\\ge 11$ it corresponds to the extremal solution -- the one associated to the largest $\\lambda$ for which existence holds. In contrast, we prove that if the deformation is sufficiently large then even when $N\\ge 10$, the extremal solution remains bounded in many cases."}
{"category": "Math", "title": "Staircase Macdonald polynomials and the $q$-Discriminant", "abstract": "We prove that a $q$-deformation $\\Disc k\\X q$ of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of $\\Disc k\\X q$ on different basis of symmetric functions. In particular, we show that its expansion on the monomial basis can be explicitly described in terms of standard tableaux and we generalize a result of King-Toumazet-Wybourne about the expansion of the $q$-discriminant on the Schur basis."}
{"category": "Math", "title": "Schubert presentation of the integral cohomology ring of the flag manifolds G/T", "abstract": "Let G be a compact connected Lie group with a maximal torus T\\subsetG. In the context of Schubert calculus we obtain a canonical presentation for the integral cohomology ring H^{\\ast}(G/T) of the complete flag manifold G/T. The result have been applied in [15] to construct the integral cohomology ring H^{\\ast}(G) in terms of Schubert classes on G/T, and in [16] to determine the structure of the modp cohomology H^{\\ast}(G;F_{p}) as a Hopf algebra over the Steenrod algebra."}
{"category": "Math", "title": "Stable solutions for the bilaplacian with exponential nonlinearity", "abstract": "Let $\\lambda^*>0$ denote the largest possible value of $\\lambda$ such that \\begin{align*} \\left\\{\\begin{aligned} \\Delta^2 u & = \\la e^u && \\text{in $B $} u &= \\pd{u}{n} = 0 && \\text{on $ \\pa B $} \\end{aligned} \\right. \\end{align*} has a solution, where $B$ is the unit ball in $\\R^N$ and $n$ is the exterior unit normal vector. We show that for $\\lambda=\\lambda^*$ this problem possesses a unique {\\em weak} solution $u^*$. We prove that $u^*$ is smooth if $N\\le 12$ and singular when $N\\ge 13$, in which case $ u^*(r) = - 4 \\log r + \\log (8(N-2)(N-4) / \\lambda^*) + o(1)$ as $r\\to 0$. We also consider the problem with general constant Dirichlet boundary conditions."}
{"category": "Math", "title": "Eulerian calculus for the displacement convexity in the Wasserstein distance", "abstract": "In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto-Westdickenberg and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space."}
{"category": "Math", "title": "Coding on countably infinite alphabets", "abstract": "This paper describes universal lossless coding strategies for compressing sources on countably infinite alphabets. Classes of memoryless sources defined by an envelope condition on the marginal distribution provide benchmarks for coding techniques originating from the theory of universal coding over finite alphabets. We prove general upper-bounds on minimax regret and lower-bounds on minimax redundancy for such source classes. The general upper bounds emphasize the role of the Normalized Maximum Likelihood codes with respect to minimax regret in the infinite alphabet context. Lower bounds are derived by tailoring sharp bounds on the redundancy of Krichevsky-Trofimov coders for sources over finite alphabets. Up to logarithmic (resp. constant) factors the bounds are matching for source classes defined by algebraically declining (resp. exponentially vanishing) envelopes. Effective and (almost) adaptive coding techniques are described for the collection of source classes defined by algebraically vanishing envelopes. Those results extend ourknowledge concerning universal coding to contexts where the key tools from parametric inference"}
{"category": "Math", "title": "Some Characterizations of VNL Rings", "abstract": "A ring R is said to be VNL if for any a in R, either a or 1-a is (von Neumann) regular. The class of VNL rings lies properly between the exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without infinite set of orthogonal idempotents; and also the VNL rings having primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a, b) in R^2, one of the a or b is regular in R. Formal triangular matrix rings that are VNL, are also characterized. As a corollary it is shown that an upper triangular matrix ring T_n(R) is VNL if and only if n=2 or 3 and R is a division ring."}
{"category": "Math", "title": "Stability and instability of weighted composition operators", "abstract": "Let $\\epsilon >0$. A continuous linear operator $T:C(X) \\ra C(Y)$ is said to be {\\em $\\epsilon$-disjointness preserving} if $\\vc (Tf)(Tg)\\vd_{\\infty} \\le \\epsilon$, whenever $f,g\\in C(X)$ satisfy $\\vc f\\vd_{\\infty} =\\vc g\\vd_{\\infty} =1$ and $fg\\equiv 0$. In this paper we address basically two main questions: 1.- How close there must be a weighted composition operator to a given $\\epsilon$-disjointness preserving operator? 2.- How far can the set of weighted composition operators be from a given $\\epsilon$-disjointness preserving operator? We address these two questions distinguishing among three cases: $X$ infinite, $X$ finite, and $Y$ a singleton ($\\epsilon$-disjointness preserving functionals). We provide sharp stability and instability bounds for the three cases."}
{"category": "Math", "title": "Lie algebras with S3 or S4-action, and generalized Malcev algebras", "abstract": "Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied. In case of S3-symmetry, the Lie algebras are coordinatized by some nonassociative systems, which are termed generalized Malcev algebras, as they extend the classical Malcev algebras. These systems are endowed with a binary and a ternary products, and include both the Malcev algebras and the Jordan triple systems."}
{"category": "Math", "title": "Isometric embeddings of compact spaces into Banach spaces", "abstract": "We show the existence of a compact metric space $K$ such that whenever $K$ embeds isometrically into a Banach space $Y$, then any separable Banach space is linearly isometric to a subspace of $Y$. We also address the following related question: if a Banach space $Y$ contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space $X$, does it necessarily contain a subspace isometric to $X$? We answer positively this question when $X$ is a polyhedral finite-dimensional space, $c_0$ or $\\ell_1$."}
{"category": "Math", "title": "Motives of hypersurfaces of very small degree", "abstract": "We study the Chow motive (with rational coefficients) of a hypersurface X in the projective space by using the variety F(X) of l-dimensional planes contained in X. If the degree of X is sufficiently small we show that the primitive part of the motive of X is the tensor product of a direct summand in the motive of a suitable complete intersection in F(X) and the l-th twist Q(-l) of the Lefschetz motive."}
{"category": "Math", "title": "A new approach to the representation theory of the symmetric groups. IV. $ \\Bbb Z_{2}$-graded groups and algebras", "abstract": "We start with definitions of the general notions of the theory of $\\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the spirit of approach of the papers \\cite{VO,OV} to representation theory of symmetric groups. The main example is the classical - theory of the projective representations of symmetric groups."}
{"category": "Math", "title": "Plane geometry and convexity of polynomial stability regions", "abstract": "The set of controllers stabilizing a linear system is generally non-convex in the parameter space. In the case of two-parameter controller design (e.g. PI control or static output feedback with one input and two outputs), we observe however that quite often for benchmark problem instances, the set of stabilizing controllers seems to be convex. In this note we use elementary techniques from real algebraic geometry (resultants and Bezoutian matrices) to explain this phenomenon. As a byproduct, we derive a convex linear matrix inequality (LMI) formulation of two-parameter fixed-order controller design problem, when possible."}
{"category": "Math", "title": "Uniformly spread measures and vector fields", "abstract": "We show that two different ideas of uniform spreading of locally finite measures in the d-dimensional Euclidean space are equivalent. The first idea is formulated in terms of finite distance transportations to the Lebesgue measure, while the second idea is formulated in terms of vector fields connecting a given measure with the Lebesgue measure."}
{"category": "Math", "title": "Characters of the Grothendieck-Teichmueller group through rigidity of the Burau representation", "abstract": "We present examples of characters of absolute Galois groups of number fields that can be recovered through their action by automorphisms on the profinite completion of the braid groups, using a ``rigidity'' approach. The way we use to recover them is through classical representations of the braid groups, and in particular through the Burau representation. This enables one to extend these characters to Grothendieck-Teichmueller groups."}
{"category": "Math", "title": "Baxter's inequality for fractional Brownian motion-type processes with Hurst index less than 1/2", "abstract": "The aim of this paper is to prove an analogue of Baxter's inequality for fractional Brownian motion-type processes with Hurst index less than 1/2. This inequality is concerned with the norm estimate of the difference between finite- and infinite-past predictor coefficients."}
{"category": "Math", "title": "A Pair of Smarandachely Isotopic Quasigroups and Loops of the Same Variety", "abstract": "The isotopic invariance or universality of types and varieties of quasigroups and loops described by one or more equivalent identities has been of interest to researchers in loop theory in the recent past. A variety of quasigroups(loops) that are not universal have been found to be isotopic invariant relative to a special type of isotopism or the other. Presently, there are two outstanding open problems on universality of loops: semi automorphic inverse property loops(1999) and Osborn loops(2005). Smarandache isotopism(S-isotopism) was originally introduced by Vasantha Kandasamy in 2002. But in this work, the concept is re-restructured in order to make it more explorable. As a result of this, the theory of Smarandache isotopy inherits the open problems as highlighted above for isotopy. In this short note, the question 'Under what type of S-isotopism will a pair of S-quasigroups(S-loops) form any variety?' is answered by presenting a pair of specially S-isotopic S-quasigroups(loops) that both belong to the same variety of S-quasigroups(S-loops). This is important because pairs of specially S-isotopic S-quasigroups(e.g Smarandache cross inverse property quasigroups) that are of the same variety are useful for applications(e.g cryptography)."}
{"category": "Math", "title": "Combinatorial Characterization of the Assur Graphs from Engineering", "abstract": "We introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. The paper translates the terminology, questions, methods and conjectures from the kinematics terminology for one degree of freedom linkages to the terminology of Assur graphs as graphs with special properties in rigidity theory. Exploiting recent works in combinatorial rigidity theory we provide mathematical characterizations of these graphs derived from minimal linkages. With these characterizations, we confirm a series of conjectures posed by Offer Shai, and offer techniques and algorithms to be exploited further in future work."}
{"category": "Math", "title": "The variance of the shock in the HAD process", "abstract": "We consider the Hammersley-Aldous-Diaconis (HAD) process with sinks and sources such that there is a microscopic shock at every time $t$; denote $Z(t)$ its position. We show that the mean and variance of $Z(t)$ are linear functions of $t$ and compute explicitely the respective constants in function of the left and right densities. Furthermore, we describe the dependence of $Z(t)$ on the initial configuration in the scale $\\sqrt t$ and, as a corollary, prove a central limit theorem."}
{"category": "Math", "title": "Une nouvelle analyse des mesures maximisant l'entropie des diff\\'eomorphismes d'Anosov de surfaces", "abstract": "This note illustrates the strategy of our paper on piecewise affine surface homeomorphisms by giving a new proof of the finite multiplicity of the maximum entropy measure of Anosov diffeomorphisms (here on surfaces). This approach avoids the explicit construction of Markov partitions and will be applied elsewhere to some non-uniformly hyperbolic diffeomorphisms."}
{"category": "Math", "title": "On the persistence of invariant curves for Fibered Holomorphic Transformations", "abstract": "We consider the problem of the persistence of invariant curves for analytical fibered holomorphic transformations. We define a fibered rotation number associated to an invariant curve. We show that an invariant curve with a prescribed fibered rotation number persists under small perturbations on the dynamics provided that the pair of rotation numbers verifies a Brjuno type arithmetical condition."}
{"category": "Math", "title": "Moduli spaces of framed instanton sheaves on projective spaces", "abstract": "This paper has been withdraw. A fully revised version with two new co-authors has been posted: \"ADHM construction of perverse instanton sheaves\", arXiv:1201.5657."}
{"category": "Math", "title": "Topologically Trivial Legendrian Knots", "abstract": "The paper deals with topologically trivial Legendrian knots in tight and overtwisted contact 3-manifolds. The first part contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e. Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds. This part was essentially written more than 10 years ago, but only a short version, without the detailed proofs, was published (in CRM Proc. Lecture Notes, Vol. 15, 1998). That paper also briefly discussed the overtwisted case. The final part of the present paper contains a more systematic discussion of Legendrian knots in overtwisted contact manifolds, and in particular, gives the coarse classification (i.e. classification up to a global contactomorphism) of topologically trivial Legendrian knots in overtwisted contact S^3."}
{"category": "Math", "title": "Betti number bounds for fewnomial hypersurfaces via stratified Morse theory", "abstract": "We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a hypersurface in R^n_> defined by a polynomial with n+l+1 terms."}
{"category": "Math", "title": "Penalized Clustering of Large Scale Functional Data with Multiple Covariates", "abstract": "In this article, we propose a penalized clustering method for large scale data with multiple covariates through a functional data approach. In the proposed method, responses and covariates are linked together through nonparametric multivariate functions (fixed effects), which have great flexibility in modeling a variety of function features, such as jump points, branching, and periodicity. Functional ANOVA is employed to further decompose multivariate functions in a reproducing kernel Hilbert space and provide associated notions of main effect and interaction. Parsimonious random effects are used to capture various correlation structures. The mixed-effect models are nested under a general mixture model, in which the heterogeneity of functional data is characterized. We propose a penalized Henderson's likelihood approach for model-fitting and design a rejection-controlled EM algorithm for the estimation. Our method selects smoothing parameters through generalized cross-validation. Furthermore, the Bayesian confidence intervals are used to measure the clustering uncertainty. Simulation studies and real-data examples are presented to investigate the empirical performance of the proposed method. Open-source code is available in the R package MFDA."}
{"category": "Math", "title": "Exts and Vertex Operators", "abstract": "The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext groups between the two ideal sheaves in question, twisted by a line bundle. We express the Chern classes of these virtual bundles in terms of Nakajima operators."}
{"category": "Math", "title": "Cohomology of Frobenius Algebras and the Yang-Baxter Equation", "abstract": "A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained."}
{"category": "Math", "title": "Hypergames and full completeness for system F (rough draft)", "abstract": "This paper reviews the fully complete hypergames model of system $F$, presented a decade ago in the author's thesis. Instantiating type variables is modelled by allowing ``games as moves''. The uniformity of a quantified type variable $\\forall X$ is modelled by copycat expansion: $X$ represents an unknown game, a kind of black box, so all the player can do is copy moves between a positive occurrence and a negative occurrence of $X$. This presentation is based on slides for a talk entitled ``Hypergame semantics: ten years later'' given at `Games for Logic and Programming Languages', Seattle, August 2006."}
{"category": "Math", "title": "A new proof of Roth's theorem on arithmetic progressions", "abstract": "We present a proof of Roth's theorem that follows a slightly different structure to the usual proofs, in that there is not much iteration. Although our proof works using a type of density increment argument (which is typical of most proofs of Roth's theorem), we do not pass to a progression related to the large Fourier coefficients of our set (as most other proofs of Roth do). Furthermore, in our proof, the density increment is achieved through an application of a quantitative version of Varnavides's theorem, which is perhaps unexpected."}
{"category": "Math", "title": "Non-Realizable Minimal Vertex Triangulations of Surfaces: Showing Non-Realizability using Oriented Matroids and Satisfiability Solvers", "abstract": "We show that no minimal vertex triangulation of a closed, connected, orientable 2-manifold of genus 6 admits a polyhedral embedding in R^3. We also provide examples of minimal vertex triangulations of closed, connected, orientable 2-manifolds of genus 5 that do not admit any polyhedral embeddings. We construct a new infinite family of non-realizable triangulations of surfaces. These results were achieved by transforming the problem of finding suitable oriented matroids into a satisfiability problem. This method can be applied to other geometric realizability problems, e.g. for face lattices of polytopes."}
{"category": "Math", "title": "Lyapunov exponents, bifurcation currents and laminations in bifurcation loci", "abstract": "Bifurcation loci in the moduli space of degree $d$ rational maps are shaped by the hypersurfaces defined by the existence of a cycle of period $n$ and multiplier 0 or $e^{i\\theta}$. Using potential-theoretic arguments, we establish two equidistribution properties for these hypersurfaces with respect to the bifurcation current. To this purpose we first establish approximation formulas for the Lyapunov function. In degree $d=2$, this allows us to build holomorphic motions and show that the bifurcation locus has a lamination structure in the regions where an attracting basin of fixed period exists."}
{"category": "Math", "title": "Enumerating (multiplex) juggling sequences", "abstract": "We consider the problem of enumerating periodic $\\sigma$-juggling sequences of length $n$ for multiplex juggling, where $\\sigma$ is the initial state (or {\\em landing schedule}) of the balls. We first show that this problem is equivalent to choosing 1's in a specified matrix to guarantee certain column and row sums, and then using this matrix, derive a recursion. This work is a generalization of earlier work of Fan Chung and Ron Graham."}
{"category": "Math", "title": "Derived categories of sheaves on singular schemes with an application to reconstruction", "abstract": "We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We introduce the notions of pseudo-adjoints and Rouquier functors and study them. As an application of these ideas and results, we extend the reconstruction result of Bondal and Orlov to Gorenstein projective varieties."}
{"category": "Math", "title": "Some thoughts on the asymptotics of the deconvolution kernel density estimator", "abstract": "Via a simulation study we compare the finite sample performance of the deconvolution kernel density estimator in the supersmooth deconvolution problem to its asymptotic behaviour predicted by two asymptotic normality theorems. Our results indicate that for lower noise levels and moderate sample sizes the match between the asymptotic theory and the finite sample performance of the estimator is not satisfactory. On the other hand we show that the two approaches produce reasonably close results for higher noise levels. These observations in turn provide additional motivation for the study of deconvolution problems under the assumption that the error term variance $\\sigma^2\\to 0$ as the sample size $n\\to\\infty.$"}
{"category": "Math", "title": "Classification of Harish-Chandra modules over the $W$-algebra W(2,2)", "abstract": "In this paper, we classify all irreducible weight modules with finite dimensional weight spaces over the $W$-algebra $W(2, 2)$. Meanwhile, all indecomposable modules with one dimensional weight spaces over the $W$-algebra $W(2, 2)$ are also determined."}
{"category": "Math", "title": "Classification of irreducible weight modules over $W$-algebra W(2,2)", "abstract": "We show that the support of an irreducible weight module over the $W$-algebra $W(2, 2)$, which has an infinite dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the the $W$-algebra $W(2, 2)$, having a nontrivial finite dimensional weight space, is a Harish-Chandra module (and hence is either an irreducible highest or lowest weight module or an irreducible module of the intermediate series)."}
{"category": "Math", "title": "Ricci flow unstable cell centered at an Einstein metric on the twistor space of positive quaternion K\\\"ahler manifolds of dimension $\\geq 8$", "abstract": "We show that a 1-parameter family Ricci flow ancient solutions arises from the natural collapsings of the twistor space of positive quaternion K\\\"ahler manifolds. We use these ancient solutions to show that a positive quaternion K\\\"ahler manifold is isometric to one of the Wolf spaces."}
{"category": "Math", "title": "The Combinatorial Norm of a Morphism of Schemes", "abstract": "In this paper we will prove that there exists a covariant functor from the category of schemes to the category of graphs. This functor provides a combination between algebraic varieties and combinatorial graphs so that the invariants defined on graphs can be introduced to algebraic varieties in a natural manner. By the functor, we will define the combinatorial norm of a morphism of schemes. Then we will obtain some properties of morphisms of norm not great than one. The topics discussed here can be applied to study the discrete Morse theory on arithmetic schemes and Kontsevich's theory of graph homology."}
{"category": "Math", "title": "General isotropic flags are general (for Grassmannian Schubert calculus)", "abstract": "We show that general isotropic flags for odd-orthogonal and symplectic groups are general for Schubert calculus on the classical Grassmannian in that Schubert cells defined by such flags meet transversally. This strengthens a result of Belkale and Kumar in arXiv:0708.0398."}
{"category": "Math", "title": "A Global Uniqueness for Formally Determined Inverse Electromagnetic Obstacle Scattering", "abstract": "It is proved that a general polyhedral perfect conducting obstacle in $\\mathbb{R}^3$, possibly consisting of finitely many solid polyhedra, is uniquely determined by the far-field pattern corresponding to a single incident wave. This improves earlier results in the literature to the formally determined case."}
{"category": "Math", "title": "Least squares type estimation of the transition density of a particular hidden Markov chain", "abstract": "In this paper, we study the following model of hidden Markov chain: $Y_i=X_i+\\epsilon_i$, $i=1,...,n+1$ with $(X_i)$ a real-valued stationary Markov chain and $(\\epsilon_i)_{1\\leq i\\leq n+1}$ a noise having a known distribution and independent of the sequence $(X_i)$. We present an estimator of the transition density obtained by minimization of an original contrast that takes advantage of the regressive aspect of the problem. It is selected among a collection of projection estimators with a model selection method. The $L^2$-risk and its rate of convergence are evaluated for ordinary smooth noise and some simulations illustrate the method. We obtain uniform risk bounds over classes of Besov balls. In addition our estimation procedure requires no prior knowledge of the regularity of the true transition. Finally, our estimator permits to avoid the drawbacks of quotient estimators."}
{"category": "Math", "title": "Total-variation cutoff in birth-and-death chains", "abstract": "The cutoff phenomenon describes a case where a Markov chain exhibits a sharp transition in its convergence to stationarity. In 1996, Diaconis surveyed this phenomenon, and asked how one could recognize its occurrence in families of finite ergodic Markov chains. In 2004, the third author noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing-time and spectral-gap tends to infinity, and conjectured that in many settings, this condition should also be sufficient. Diaconis and Saloff-Coste (2006) verified this conjecture for continuous-time birth-and-death chains, started at an endpoint, with convergence measured in separation. It is natural to ask whether the conjecture holds for these chains in the more widely used total-variation distance. In this work, we confirm the above conjecture for all continuous-time or lazy discrete-time birth-and-death chains, with convergence measured via total-variation distance. Namely, if the product of the mixing-time and spectral-gap tends to infinity, the chains exhibit cutoff at the maximal hitting time of the stationary distribution median, with a window of at most the geometric mean between the relaxation-time and mixing-time. In addition, we show that for any lazy (or continuous-time) birth-and-death chain with stationary distribution $\\pi$, the separation $1 - p^t(x,y)/\\pi(y)$ is maximized when $x,y$ are the endpoints. Together with the above results, this implies that total-variation cutoff is equivalent to separation cutoff in any family of such chains."}
{"category": "Math", "title": "Over-populated Tails for conservative-in-the-mean Inelastic Maxwell Models", "abstract": "We introduce and discuss spatially homogeneous Maxwell-type models of the nonlinear Boltzmann equation undergoing binary collisions with a random component. The random contribution to collisions is such that the usual collisional invariants of mass, momentum and energy do not hold pointwise, even if they all hold in the mean. Under this assumption it is shown that, while the Boltzmann equation has the usual conserved quantities, it possesses a steady state with power-like tails for certain random variables. A similar situation occurs in kinetic models of economy recently considered by two of the authors [24], which are conservative in the mean but possess a steady distribution with Pareto tails. The convolution-like gain operator is subsequently shown to have good contraction/expansion properties with respect to different metrics in the set of probability measures. Existence and regularity of isotropic stationary states is shown directly by constructing converging iteration sequences as done in [8]. Uniqueness, asymptotic stability and estimates of overpopulated high energy tails of the steady profile are derived from the basic property of contraction/expansion of metrics. For general initial conditions the solutions of the Boltzmann equation are then proved to converge with computable rate as t goes to infinity to the steady solution in these distances, which metricizes the weak convergence of measures. These results show that power-like tails in Maxwell models are obtained when the point-wise conservation of momentum and/or energy holds only globally."}
{"category": "Math", "title": "Stanley Depth of Multigraded Modules", "abstract": "The Stanley's Conjecture on Cohen-Macaulay multigraded modules is studied especially in dimension 2. In codimension 2 similar results were obtained by Herzog, Soleyman-Jahan and Yassemi. As a consequence of our results Stanley's Conjecture holds in 5 variables."}
{"category": "Math", "title": "Operators with Corener-degenerate Symbols", "abstract": "We establish elements of a new approch to ellipticity and parametrices within operator algebras on a manifold with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaces. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The \"full\" calculus is voluminous; so we content ourselves here with some typical aspects such as symbols in terms of order reducing families, classes of relevant examples, and operators near a corner point."}
{"category": "Math", "title": "Premi\\`ere valeur propre du laplacien, volume conforme et chirurgies", "abstract": "We define a new differential invariant a compact manifold by $V_{\\mathcal M}(M)=\\inf_g V_c(M,[g])$, where $V_c(M,[g])$ is the conformal volume of $M$ for the conformal class $[g]$, and prove that it is uniformly bounded above. The main motivation is that this bound provides a upper bound of the Friedlander-Nadirashvili invariant defined by $\\inf_g\\sup_{\\tilde g\\in[g]}\\lambda_1(M,\\tilde g)\\Vol(M,\\tilde g)^{\\frac 2n}$. The proof relies on the study of the behaviour of $V_{\\mathcal M}(M)$ when one performs surgeries on $M$."}
{"category": "Math", "title": "Embedding properties of endomorphism semigroups", "abstract": "Denote by PSelf(X) (resp., Self(X)) the partial (resp., full) transformation monoid over a set X, and by Sub(V) (resp., End(V)) the collection of all subspaces (resp., endomorphisms) of a vector space V. We prove various results that imply the following: (1) If X has at least two elements, then Self(X) has a semigroup embedding into the dual of Self(Y) iff card(Y) >= 2^card(X). In particular, if X has at least two elements, then there exists no semigroup embedding from Self(X) into the dual of PSelf(X). (2) If V is infinite-dimensional, then there are no embedding from (Sub(V),+) into (Sub(V),\\cap) and no semigroup embedding from End(V) into its dual. (3) Let F be an algebra freely generated by an infinite subset X. If F has less than 2^card(X) operations, then End(F) has no semigroup embedding into its dual. The cardinality bound 2^card(X) is optimal. (4) Let F be a free left module over a left aleph one - noetherian ring (i.e., a ring without strictly increasing chains, of length aleph one, of left ideals). Then End(F) has no semigroup embedding into its dual. (1) and (2) above solve questions proposed by B. M. Schein and G. M. Bergman. We also formalize our results in the settings of algebras endowed with a notion of independence (in particular independence algebras)."}
{"category": "Math", "title": "Equivalence relations for two variable real analytic function germs", "abstract": "For two variable real analytic function germs we compare the blow-analytic equivalence in the sense of Kuo to the other natural equivalence relations. Our main theorem states that $C^1$ equivalent germs are blow-analytically equivalent. This gives a negative answer to a conjecture of Kuo. In the proof we show that the Puiseux pairs of real Newton-Puiseux roots are preserved by the $C^1$ equivalence of function germs. The proof is achieved, being based on a combinatorial characterisation of blow-analytic equivalence in terms of the real tree model. We also give several examples of bi-Lipschitz equivalent germs that are not blow-analytically equivalent."}
{"category": "Math", "title": "Global and exponential attractors for the Penrose-Fife system", "abstract": "The Penrose-Fife system for phase transitions is addressed. Dirichlet boundary conditions for the temperature are assumed. Existence of global and exponential attractors is proved. Differently from preceding contributions, here the energy balance equation is both singular at 0 and degenerate at infinity. For this reason, the dissipativity of the associated dynamical process is not trivial and has to be proved rather carefully."}
{"category": "Math", "title": "Long time convergence for a class of variational phase field models", "abstract": "In this paper we analyze a class of phase field models for the dynamics of phase transitions which extend the well-known Caginalp and Penrose-Fife models. Existence and uniqueness of the solution to the related initial boundary value problem are shown. Further regularity of the solution is deduced by exploiting the so-called regularizing effect. Then, the large time behavior of such a solution is studied and several convergence properties of the trajectory as time tends to infinity are discussed."}
{"category": "Math", "title": "Regular sequences of symmetric polynomials", "abstract": "Denote by p_k the k-th power sum symmetric polynomial n variables. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets n powersums forming a regular sequence. A necessary condition is that n! divides the product of the degrees of the elements. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already in 3 variables. Given positive integers a<b<c with GCD(a,b,c)=1, we conjecture that p_a, p_b, p_c is a regular sequence for n=3 if and only if 6 divides abc. We provide evidence for the conjecture by proving it in several special instances."}
{"category": "Math", "title": "On the derived category of an algebra over an operad", "abstract": "We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad."}
{"category": "Math", "title": "Simple Modules for Groups with Abelian Sylow 2-Subgroups are Algebraic", "abstract": "Let G be a finite group and let p be a prime. A module for G over a field of characteristic p is called algebraic if it satisfies a polynomial, with addition and multiplication given by direct sum and tensor product. In some sense, having this property is equivalent to the tensor structure being 'nice' for that module. In this paper we prove that if G is a group with abelian Sylow 2-subgroups, and p=2, then all simple modules for G are algebraic. We include the conjecture that this result holds for all abelian 2-blocks."}
{"category": "Math", "title": "Poisson suspensions and infinite ergodic theory", "abstract": "We investigate ergodic theory of Poisson suspensions. In the process, we establish close connections between finite and infinite measure preserving ergodic theory. Poisson suspensions thus provide a new approach to infinite measure preserving ergodic theory. Fields investigated here are mixing properties, spectral theory, joinings. We also compare Poisson suspensions to the apparently similar looking Gaussian dynamical systems."}
{"category": "Math", "title": "Poincare Inequality on the Path Space of Poisson Point Processes", "abstract": "The quasi-invariance is proved for the distributions of Poisson point processes under a random shift map on the path space. This leads to a natural Dirichlet form of jump type on the path space. Differently from the O-U Dirichlet form on the Wiener space satisfying the log-Sobolev inequality, this Dirichlet form merely satisfies the Poincare inequality but not the log-Sobolev one."}
{"category": "Math", "title": "Super Fuzzy Matrices and Super Fuzzy Models for Social Scientists", "abstract": "This book introduces the concept of fuzzy super matrices and operations on them. This book will be highly useful to social scientists who wish to work with multi-expert models. Super fuzzy models using Fuzzy Cognitive Maps, Fuzzy Relational Maps, Bidirectional Associative Memories and Fuzzy Associative Memories are defined here. The authors introduce 13 multi-expert models using the notion of fuzzy supermatrices. These models are described with illustrative examples. This book has three chapters. In the first chaper, the basic concepts about super matrices and fuzzy super matrices are recalled. Chapter two introduces the notion of fuzzy super matrices adn their properties. The final chapter introduces many super fuzzy multi expert models."}
{"category": "Math", "title": "On asymptotic stability in 3D of kinks for the $\\phi ^4$ model", "abstract": "We add to a kink, which is a 1 dimensional structure, two transversal directions. We then check its asymptotic stability with respect to compactly supported perturbations in 3D and a time evolution under a Nonlinear Wave Equation (NLW). The problem is inspired by work by Jack Xin on asymptotic stability in dimension larger than 1 of fronts for reaction diffusion equations. The proof involves a separation of variables. The transversal variables are treated as in work on Nonlinear Klein Gordon Equation (NLKG) originating from Klainerman and from Shatah in a particular elaboration due to Delort and others. The longitudinal variable is treated by means of a result by Weder on dispersion for Schroedinger operators in 1D."}
{"category": "Math", "title": "New Classes of Codes for Cryptologists and Computer Scientists", "abstract": "In this book, we have introduced several new classes of codes to aid cryptologists and computer scientists. We have explained these codes very non-technically so that a strong mathematical foundation is not needed to understand them. This book also provides an easy method to detect and correct errors that occur during transmission. Further, some of the codes are constructed so as to mislead an intruder/ hacker. False n-codes, whole n-codes can serve this pupose. These codes can be used to ensure security in networks and safe transmission of identity. We have named a few new classes of codes after Periyar, the south-Indian social leader, to mark his services to humanity. This book is divided into three chapters. Chapter one is introductory in nature. The notion of bicodes and their generalization, and n-codes are introduced in the second chapter. Periyar linear codes are introduced in the third chapter. We have used two methods, viz. pseudo best n-approximations and n-coset leader properties to detect and correct errors."}
{"category": "Math", "title": "Control Theorems for Abelian Varieties over Function Fields", "abstract": "We prove control theorems for abelian varieties over function fields."}
{"category": "Math", "title": "Algebraic Modules and the Auslander--Reiten Quiver", "abstract": "Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander--Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity."}
{"category": "Math", "title": "On the Tensor Products of Modules for Dihedral 2-Groups", "abstract": "Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that if L is a component of the (stable) Auslander-Reiten quiver for a dihedral 2-group consisting of non-periodic modules, then there is at most one algebraic module on L."}
{"category": "Math", "title": "Schatten p-norm inequalities related to a characterization of inner product spaces", "abstract": "Let $A_1, ... A_n$ be operators acting on a separable complex Hilbert space such that $\\sum_{i=1}^n A_i=0$. It is shown that if $A_1, ... A_n$ belong to a Schatten $p$-class, for some $p>0$, then 2^{p/2}n^{p-1} \\sum_{i=1}^n \\|A_i\\|^p_p \\leq \\sum_{i,j=1}^n\\|A_i\\pm A_j\\|^p_p for $0<p\\leq 2$, and the reverse inequality holds for $2\\leq p<\\infty$. Moreover, \\sum_{i,j=1}^n\\|A_i\\pm A_j\\|^2_p \\leq 2n^{2/p} \\sum_{i=1}^n \\|A_i\\|^2_p for $0<p\\leq 2$, and the reverse inequality holds for $2\\leq p<\\infty$. These inequalities are related to a characterization of inner product spaces due to E.R. Lorch."}
{"category": "Math", "title": "Classification of solutions to the higher order Liouville's equation on R^{2m}", "abstract": "We classify the solutions to the equation (- \\Delta)^m u=(2m-1)!e^{2mu} on R^{2m} giving rise to a metric g=e^{2u}g_{R^{2m}} with finite total $Q$-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of \\Delta u(x) as |x|\\to \\infty. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric e^{2u}g_{R^{2m}} at infinity, and we observe that the pull-back of this metric to $S^{2m}$ via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round."}
{"category": "Math", "title": "A greedy approach to sparse canonical correlation analysis", "abstract": "We consider the problem of sparse canonical correlation analysis (CCA), i.e., the search for two linear combinations, one for each multivariate, that yield maximum correlation using a specified number of variables. We propose an efficient numerical approximation based on a direct greedy approach which bounds the correlation at each stage. The method is specifically designed to cope with large data sets and its computational complexity depends only on the sparsity levels. We analyze the algorithm's performance through the tradeoff between correlation and parsimony. The results of numerical simulation suggest that a significant portion of the correlation may be captured using a relatively small number of variables. In addition, we examine the use of sparse CCA as a regularization method when the number of available samples is small compared to the dimensions of the multivariates."}
{"category": "Math", "title": "Construction of an Edwards' probability measure on $\\mathcal{C}(\\mathbb{R}_+,\\mathbb{R})$", "abstract": "In this article, we prove that the measures $\\mathbb{Q}_T$ associated to the one-dimensional Edwards' model on the interval $[0,T]$ converge to a limit measure $\\mathbb{Q}$ when $T$ goes to infinity, in the following sense: for all $s\\geq0$ and for all events $\\Lambda_s$ depending on the canonical process only up to time $s$, $\\mathbb{Q}_T(\\Lambda_s)\\rightarrow\\mathbb{Q}(\\Lambda_s)$. Moreover, we prove that, if $\\mathbb{P}$ is Wiener measure, there exists a martingale $(D_s)_{s\\in\\mathbb{R}_+}$ such that $\\mathbb{Q}(\\Lambda_s) =\\mathbb{E}_{\\mathbb{P}}(\\mathbh{1}_{\\Lambda_s}D_s)$, and we give an explicit expression for this martingale."}
{"category": "Math", "title": "Discrete approximation of a stable self-similar stationary increments process", "abstract": "The aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known on the context in which such processes can arise. To our knowledge, discretisation and convergence theorems are available only in the case of stable L\\'evy motions and fractional Brownian motions. This paper yields new results in this direction. Our main result is the convergence of the random rewards schema, which was firstly introduced by Cohen and Samorodnitsky, and that we consider in a more general setting. Strong relationships with Kesten and Spitzer's random walk in random sceneries are evidenced. Finally, we study some path properties of the limit process."}
{"category": "Math", "title": "On positivity in T-equivariant K-theory of flag varieties", "abstract": "We prove some general results on the T-equivariant K-theory K_T(G/P) of the flag variety G/P, where G is a semisimple complex algebraic group, P is a parabolic subgroup and T$ is a maximal torus contained in P. In particular, we make a conjecture about a positivity phenomenon in K_T(G/P) for the product of two basis elements written in terms of the basis of K_T(G/P) given by the dual of the structure sheaf (of Schubert varieties) basis. (For the full flag variety G/B, this dual basis is closely related to the basis given by Kostant-Kumar.) This conjecture is parallel to (but different from) the conjecture of Griffeth-Ram for the structure constants of the product in the structure sheaf basis. We give explicit expressions for the product in the T-equivariant K-theory of projective spaces in terms of these bases. In particular, we establish our conjecture and the conjecture of Griffeth-Ram in this case."}
{"category": "Math", "title": "Computing L-series of hyperelliptic curves", "abstract": "We discuss the computation of coefficients of the L-series associated to a hyperelliptic curve over Q of genus at most 3, using point counting, generic group algorithms, and p-adic methods."}
{"category": "Math", "title": "Quantization of $r-Z$-quasi-Poisson manifolds and related modified classical dynamical $r$-matrices", "abstract": "Le $X$ be a $C^\\infty$-manifold and $\\g$ be a finite dimensional Lie algebra acting freely on $X$. Let $r \\in \\ve^2(\\g)$ be such that $Z=[r,r] \\in \\ve^3(\\g)^\\g$. In this paper we prove that every quasi-Poisson $(\\g,Z)$-manifold can be quantized. This is a generalization of the existence of a twist quantization of coboundary Lie bialgebras (\\cite{EH}) in the case $X=G$ (where $G$ is the simply connected Lie group corresponding to $\\g$). We deduce our result from a generalized formality theorem. In the case Z=0, we get a new proof of the existence of (equivariant) formality theorem and so (equivariant) quantization of Poisson manifold ({\\it cf.} \\cite{Ko,Do}). As a consequence of our results, we get quantization of modified classical dynamical $r$-matrices over abelian bases in the reductive case"}
{"category": "Math", "title": "Strongly Consistent Model Order Selection for Estimating 2-D Sinusoids in Colored Noise", "abstract": "We consider the problem of jointly estimating the number as well as the parameters of two-dimensional sinusoidal signals, observed in the presence of an additive colored noise field. We begin by elaborating on the least squares estimation of 2-D sinusoidal signals, when the assumed number of sinusoids is incorrect. In the case where the number of sinusoidal signals is under-estimated we show the almost sure convergence of the least squares estimates to the parameters of the dominant sinusoids. In the case where this number is over-estimated, the estimated parameter vector obtained by the least squares estimator contains a sub-vector that converges almost surely to the correct parameters of the sinusoids. Based on these results, we prove the strong consistency of a new model order selection rule."}
{"category": "Math", "title": "Sums with multiplicative functions over a Beatty sequence", "abstract": "We study sums with multiplicative functions that take values over a non-homogenous Beatty sequence. We then apply our result in a few special cases to obtain asymptotic formulas such as the number of integers in a Beatty sequence representable as a sum of two squares up to a given magnitude."}
{"category": "Math", "title": "Every Minor-Closed Property of Sparse Graphs is Testable", "abstract": "Suppose $G$ is a graph with degrees bounded by $d$, and one needs to remove more than $\\epsilon n$ of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of $G$ is far from the statistics of local neighborhoods around vertices of any planar graph $G'$ with the same degree bound. In fact, a similar result is proved for any minor-closed property of bounded degree graphs. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outer-planar, series-parallel, bounded genus, bounded tree-width and several others, are testable with a constant number of queries, where the constant may depend on $\\epsilon$ and $d$, but not on the graph size. None of these properties was previously known to be testable even with $o(n)$ queries."}
{"category": "Math", "title": "A preferential attachment model with Poisson growth for scale-free networks", "abstract": "We propose a scale-free network model with a tunable power-law exponent. The Poisson growth model, as we call it, is an offshoot of the celebrated model of Barab\\'{a}si and Albert where a network is generated iteratively from a small seed network; at each step a node is added together with a number of incident edges preferentially attached to nodes already in the network. A key feature of our model is that the number of edges added at each step is a random variable with Poisson distribution, and, unlike the Barab\\'{a}si-Albert model where this quantity is fixed, it can generate any network. Our model is motivated by an application in Bayesian inference implemented as Markov chain Monte Carlo to estimate a network; for this purpose, we also give a formula for the probability of a network under our model."}
{"category": "Math", "title": "On the conjecture of King for smooth toric Deligne-Mumford stacks", "abstract": "We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne-Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne-Mumford stacks."}
{"category": "Math", "title": "Spectral multiplicity and odd K-theory", "abstract": "In this paper we begin a study of the space of unbounded self-adjoint Fredholm operators as a classifying space for K^{1}(X), with the goal of incorporating the information in the eigenspaces and eigenvalues of the operators. In particular, the role that the multiplicity of eigenvalues plays is developed here."}
{"category": "Math", "title": "Compound basis arising from the basic $A^{(1)}_{1}$-module", "abstract": "A new basis for the polynomial ring of infinitely many variables is constructed which consists of products of Schur functions and Q-functions. The transition matrix from the natural Schur function basis is investigated."}
{"category": "Math", "title": "Equivariant Chern characters with generalized coefficients", "abstract": "These notes form the next episode in a series of articles dedicated to a detailed proof of a cohomological index formula for transversally elliptic pseudo-differential operators and applications. The first two chapters are already available as math.DG/0702575 and arXiv:0711.3898. In this episode, we construct the relative equivariant Chern character of a morphism of vector bundles, localized by a one form, and we prove a multiplicativity property of this generalized Chern character"}
{"category": "Math", "title": "Non-Cyclic Subgroups of Jacobians of Genus Two Curves with Complex Multiplication", "abstract": "Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field extension. We generalize this result to Jacobians of genus two curves with complex multiplication. In particular, we show that the Weil- and the Tate-pairing on such a Jacobian are non-degenerate over the same field extension of the ground field."}
{"category": "Math", "title": "Mirror symmetry for toric Fano manifolds via SYZ transformations", "abstract": "We construct and apply Strominger-Yau-Zaslow mirror transformations to understand the geometry of the mirror symmetry between toric Fano manifolds and Landau-Ginzburg models."}
{"category": "Math", "title": "Relative Proportionality for subvarieties of moduli spaces of K3 and abelian surfaces", "abstract": "The relative proportionality principle of Hirzebruch and H\\\"ofer was discovered in the case of compactified ball quotient surfaces X when studying curves C in X. It can be expressed as an inequality which attains equality precisely when C is an induced quotient of a subball. A similar inequality holds for curves on Hilbert modular surfaces. In this paper we prove a generalization of this result to subvarieties of Shimura varieties of orthogonal type, i.e. locally symmetric spaces for the Lie group SO(n,2). Furthermore we study the ''inverse problem'' of deciding when an arbitrary subvariety Z of M is of Hodge type, provided it contains sufficiently many divisors W_i which are of Hodge type and satisfy relative proportionality."}
{"category": "Math", "title": "Non-Cyclic Subgroups of Jacobians of Genus Two Curves", "abstract": "Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field extension. We generalize this result to Jacobians of genus two curves. In particular, we show that the Weil- and the Tate-pairing are non-degenerate over the same field extension of the ground field. From this generalization we get a complete description of the l-torsion subgroups of Jacobians of supersingular genus two curves. In particular, we show that for l>3, the l-torsion points are rational over a field extension of degree at most 24."}
{"category": "Math", "title": "Linearization of germs: regular dependence on the multiplier", "abstract": "We prove that the linearization of a germ of holomorphic map of the type $F_\\lambda(z)=\\lambda(z+O(z^2))$ has a $ C^1$--holomorphic dependence on the multiplier $\\lambda$. $C^1$--holomorphic functions are $ C^1$--Whitney smooth functions, defined on compact subsets and which belong to the kernel of the $\\bar{\\partial}$ operator. The linearization is analytic for $|\\lambda|\\not= 1$ and the unit circle $S^1$ appears as a natural boundary (because of resonances, i.e. roots of unity). However the linearization is still defined at most points of $S^1$, namely those points which lie ``far enough from resonances'', i.e. when the multiplier satisfies a suitable arithmetical condition. We construct an increasing sequence of compacts which avoid resonances and prove that the linearization belongs to the associated spaces of ${\\cal C}^1$--holomorphic functions. This is a special case of Borel's theory of uniform monogenic functions, and the corresponding function space is arcwise-quasianalytic. Among the consequences of these results, we can prove that the linearization admits an asymptotic expansion w.r.t. the multiplier at all points of the unit circle verifying the Brjuno condition: in fact the asymptotic expansion is of Gevrey type at diophantine points."}
{"category": "Math", "title": "The best polynomial bounds for the number of triangles in a simple arrangement of n pseudo-lines", "abstract": "It is well-known that affine (respectively projective) simple arrangements of n pseudo-lines may have at most n(n-2)/3 (respectively n(n-1)/3) triangles. However, these bounds are reached for only some values of n (mod 6). We provide the best polynomial bound for the affine and the projective case, and for each value of n (mod 6)."}
{"category": "Math", "title": "Basic deformation theory of smooth formal schemes", "abstract": "We provide the main results of a deformation theory of smooth formal schemes. First we deal with the case of global lifting of smooth morphisms. We prove that the obstruction to the existence of a global lifting lies in a Ext^1 group. Then we study uniqueness and existence of lifting of smooth formal schemes. The set of isomorphism classes of smooth liftings is classified by a Ext^1 group and there exists an obstruction in a Ext^2 group whose vanishing characterizes the existence of smooth liftings."}
{"category": "Math", "title": "The behaviour of solutions of the Gaussian curvature equation near an isolated boundary point", "abstract": "A classical result of Nitsche \\cite{Nit57} about the behaviour of the solutions to the Liouville equation $\\Delta u=4 e^{2u}$ near isolated singularities is generalized to solutions of the Gaussian curvature equation $\\Delta u=- \\kappa(z) e^{2u}$ where $\\kappa$ is a negative H\\\"older continuous function. As an application a higher--order version of the Yau--Ahlfors--Schwarz lemma for complete conformal Riemannian metrics is obtained."}
{"category": "Math", "title": "Realization of critical eigenvalues for scalar and symmetric linear delay-differential equations", "abstract": "This paper studies the link between the number of critical eigenvalues and the number of delays in certain classes of delay-differential equations. There are two main results. The first states that for k purely imaginary numbers which are linearly independent over the rationals, there exists a scalar delay-differential equation depending on k fixed delays whose spectrum contains those k purely imaginary numbers. The second result is a generalization of the first result for delay-differential equations which admit a characteristic equation consisting of a product of s factors of scalar type. In the second result, the k eigenvalues can be distributed amongst the different factors. Since the characteristic equation of scalar equations contain only exponential terms, the proof exploits a toroidal structure which comes from the arguments of the exponential terms in the characteristic equation. Our second result is applied to delay coupled D_n-symmetric cell systems with one-dimensional cells. In particular, we provide a general characterization of delay coupled D_n-symmetric systems with arbitrary number of delays and cell dimension."}
{"category": "Math", "title": "Integrability of exit times and ballisticity for random walks in Dirichlet environment", "abstract": "We consider random walks in Dirichlet environment, introduced by Enriquez and Sabot in 2006. As this distribution on environments is not uniformly elliptic, the annealed integrability of exit times out of a given finite subset is a non-trivial property. We provide here an explicit equivalent condition for this integrability to happen, on general directed graphs. Such integrability problems arise for instance from the definition of Kalikow auxiliary random walk. Using our condition, we prove a refined version of the ballisticity criterion given by Enriquez and Sabot."}
{"category": "Math", "title": "Continuous multilinear functionals on $C(K)$-spaces are integral", "abstract": "This paper has been withdrawn by the authors, due to a crucial error."}
{"category": "Math", "title": "Canonic form of linear quaternion functions", "abstract": "The general linear quaternion function of degree one is a sum of terms with quaternion coefficients on the left and right. The paper considers the canonic form of such a function, and builds on the recent work of Todd Ell, who has shown that any such function may be represented using at most four quaternion coefficients. In this paper, a new and simple method is presented for obtaining these coefficients numerically using a matrix approach which also gives an alternative proof of the canonic forms."}
{"category": "Math", "title": "Fermi-Dirac-Fokker-Planck equation: well-posedness and long-time asymptotics", "abstract": "A Fokker-Planck type equation for interacting particles with exclusion principle is analysed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asymptotics in $L^1$ are characterized by the Fermi-Dirac equilibrium with the same initial mass. This result is achieved without rate for any constructed global solution and with exponential rate due to entropy/entropy-dissipation arguments for initial data controlled by Fermi-Dirac distributions. Finally, initial data below radial solutions with suitable decay at infinity lead to solutions for which the relative entropy towards the Fermi-Dirac equilibrium is shown to converge to zero without decay rate."}
{"category": "Math", "title": "On the derivatives of the Lempert functions", "abstract": "We show that if the Kobayashi--Royden metric of a complex manifold is continuous and positive at a given point and any non-zero tangent vector, then the \"derivatives\" of the higher order Lempert functions exist and equal the respective Kobayashi metrics at the point. It is a generalization of a result by M. Kobayashi for taut manifolds."}
{"category": "Math", "title": "Remarks on Lempert functions of balanced domains", "abstract": "This note should clarify how the behavior of certain invariant objects reflects the geometric convexity of balanced domains."}
{"category": "Math", "title": "Malliavin calculus and decoupling inequalities in Banach spaces", "abstract": "We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Ito isometry to Banach spaces. In the white noise case we obtain two sided L^p-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Ito chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces."}
{"category": "Math", "title": "Calculating Milnor Numbers and Versal Component Dimensions from P-Resolution Fans", "abstract": "We use Altmann's toric fan description of P-resolutions to formulate a new description of deformation theory invariants for two-dimensional cyclic quotient singularities. In particular, we show how to calculate the dimensions of the (reduced) versal base space components as well as Milnor numbers of smoothings over them."}
{"category": "Math", "title": "An asymptotic version of Dumnicki's algorithm for linear systems in $\\mathbb{CP}^2$", "abstract": "Using Dumnicki's approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on $\\mathbb{P}^2$ we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on $\\mathbb{P}^2$. With this method we prove the lower bound 4/13 for 10 general points on $\\mathbb{P}^2$."}
{"category": "Math", "title": "Approximate Unitary Equivalence in Simple C^*-algebras of Tracial Rank One", "abstract": "Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple C*-algebra with tracial rank no more than one. Suppose that $\\phi, \\psi: C\\to A$ are two unital monomorphisms. With some restriction on $C,$ we show that $\\phi$ and $\\psi$ are approximately unitarily equivalent if and only if [\\phi]=[\\psi] in KL(C,A) \\tau\\circ \\phi=\\tau\\circ \\psi for all tracial states of A and \\phi^{\\ddag}=\\psi^{\\ddag}, here \\phi^{\\ddag} and \\psi^{\\ddag} are homomorphisms from $U(C)/CU(C)\\to U(A)/CU(A) induced by \\phi and \\psi, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B."}
{"category": "Math", "title": "P-values for classification", "abstract": "Let $(X,Y)$ be a random variable consisting of an observed feature vector $X\\in \\mathcal{X}$ and an unobserved class label $Y\\in \\{1,2,...,L\\}$ with unknown joint distribution. In addition, let $\\mathcal{D}$ be a training data set consisting of $n$ completely observed independent copies of $(X,Y)$. Usual classification procedures provide point predictors (classifiers) $\\widehat{Y}(X,\\mathcal{D})$ of $Y$ or estimate the conditional distribution of $Y$ given $X$. In order to quantify the certainty of classifying $X$ we propose to construct for each $\\theta =1,2,...,L$ a p-value $\\pi_{\\theta}(X,\\mathcal{D})$ for the null hypothesis that $Y=\\theta$, treating $Y$ temporarily as a fixed parameter. In other words, the point predictor $\\widehat{Y}(X,\\mathcal{D})$ is replaced with a prediction region for $Y$ with a certain confidence. We argue that (i) this approach is advantageous over traditional approaches and (ii) any reasonable classifier can be modified to yield nonparametric p-values. We discuss issues such as optimality, single use and multiple use validity, as well as computational and graphical aspects."}
{"category": "Math", "title": "Irreducible Boolean Functions", "abstract": "This paper is a contribution to the study of a quasi-order on the set $\\Omega$ of Boolean functions, the \\emph{simple minor} quasi-order. We look at the join-irreducible members of the resulting poset $\\tilde{\\Omega}$. Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs. We observe that among Steiner systems, those which yield join-irreducible members of $\\tilde{\\Omega}$ are the -2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members of $\\tilde{\\Omega}$."}
{"category": "Math", "title": "Summary Of Four Generalized Exponential Models (GEM) For Continuous Probability Distributions", "abstract": "Four new probability models are derived which generalize the common univariate continuous distributions. Classical distributional measures are derived from Hoel, et al., Introduction to Probability Theory, 1971. Measures include probability density function, moments generating function, cumulative distribution function,derivatives, inverse distributions, skewness, kurtosis, change of variable distributions, log distributions. Maximum likelihood estimation technique is briefly outlined. Appendices describe applications. Errata/addenda sheet included."}
{"category": "Math", "title": "The power law for the Buffon needle probability of the four-corner Cantor set", "abstract": "Let $C_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $K_n$ of $C_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $K_n$ is essentially the average length of the projections of $K_n$, also known as the Favard length of $K_n$. A classical theorem of Besicovitch implies that the Favard length of $K_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\\exp(- c\\log_* n)$, due to Peres and Solomyak. ($\\log_* n$ is the number of times one needs to take log to obtain a number less than 1 starting from $n$). We obtain a power law bound by combining analytic and combinatorial ideas."}
{"category": "Math", "title": "Synchronizing discrete-time neutrally stable linear systems via partial-state coupling", "abstract": "A basic result in synchronization of linear systems via output coupling is presented. For identical discrete-time linear systems that are detectable from their outputs and neutrally stable, it is shown that a linear output feedback law exists under which the coupled systems globally asymptotically synchronize for all fixed connected (asymmetrical) network topologies. An algorithm is provided to compute such feedback law based on individual system parameters. A dual problem is also presented and solved."}
{"category": "Math", "title": "The Topology of Foliations Formed by the Generic K-Orbits of a Subclass of the Indecomposable MD5-Groups", "abstract": "The present paper is a continuation of [13], [14] of the authors. Specifically, the paper considers the MD5-foliations associated to connected and simply connected MD5-groups such that their Lie algebras have 4-dimensional commutative derived ideal. In the paper, we give the topological classification of all considered MD5-foliations. A description of these foliations by certain fibrations or suitable actions of $\\mathbb{R}^{2}$ and the Connes' C*-algebras of the foliations which come from fibrations are also given in the paper."}
{"category": "Math", "title": "Mould Calculus for Hamiltonian Vector Fields", "abstract": "We present the general framework of \\'Ecalle's moulds in the case of linearization of a formal vector field without and within resonances. We enlighten the power of moulds by their universality, and calculability. We modify then \\'Ecalle's technique to fit in the seek of a formal normal form of a Hamiltonian vector field in cartesian coordinates. We prove that mould calculus can also produce successive canonical transformations to bring a Hamiltonian vector field into a normal form. We then prove a Kolmogorov theorem on Hamiltonian vector fields near a diophantine torus in action-angle coordinates using moulds techniques."}
{"category": "Math", "title": "Profinite completion and double-dual : isomorphisms and counter-examples", "abstract": "We define, for any group $G$, finite approximations ; with this tool, we give a new presentation of the profinite completion $\\hat{\\pi} : G \\to \\hat{G}$ of an abtract group $G$. We then prove the following theorem : if $k$ is a finite prime field and if $V$ is a $k$-vector space, then, there is a natural isomorphism between $\\hat{V}$ (for the underlying additive group structure) and the additive group of the double-dual $V^{**}$. This theorem gives counter-examples concerning the iterated profinite completions of a group. These phenomena don't occur in the topological case."}
{"category": "Math", "title": "Semilinear Schr\\\"odinger Flows on Hyperbolic Spaces: Scattering in H^1", "abstract": "We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schr\\\"{o}dinger equations \\begin{equation*} \\begin{cases} &(i\\partial_t+\\Delta_\\g)u=u|u|^{2\\sigma}; &u(0)=\\phi, \\end{cases} \\end{equation*} on the hyperbolic spaces $\\H^d$, $d\\geq 2$, for exponents $\\sigma\\in(0,2/(d-2))$. The main unexpected conclusion is scattering to linear solutions in the case of small exponents $\\sigma$; for comparison, on Euclidean spaces scattering in $H^1$ is not known for any exponent $\\sigma\\in(1/d,2/d]$ and is known to fail for $\\sigma\\in(0,1/d]$. Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities."}
{"category": "Math", "title": "The Z^d Alpern multi-tower theorem for rectangles: a tiling approach", "abstract": "We provide a proof of the Alpern multi-tower theorem for Z^d actions. We reformulate the theorem as a problem of measurably tiling orbits of a Z^d action by a collection of rectangles whose corresponding sides have no non-trivial common divisors. We associate to such a collection of rectangles a special family of generalized domino tilings. We then identify an intrinsic dynamic property of these tilings, viewed as symbolic dynamical systems, which allows for a multi-tower decomposition."}
{"category": "Math", "title": "On Besov regularity of Brownian motions in infinite dimensions", "abstract": "We extend to the vector-valued situation some earlier work of Ciesielski and Roynette on the Besov regularity of the paths of the classical Brownian motion. We also consider a Brownian motion as a Besov space valued random variable. It turns out that a Brownian motion, in this interpretation, is a Gaussian random variable with some pathological properties. We prove estimates for the first moment of the Besov norm of a Brownian motion. To obtain such results we estimate expressions of the form $\\E \\sup_{n\\geq 1}\\|\\xi_n\\|$, where the $\\xi_n$ are independent centered Gaussian random variables with values in a Banach space. Using isoperimetric inequalities we obtain two-sided inequalities in terms of the first moments and the weak variances of $\\xi_n$."}
{"category": "Math", "title": "$C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero Center Lyapunov Exponents", "abstract": "We prove that if $f$ is a $C^1$-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if $f$ is not Anosov then all the exponents in the center bundle vanish. This establishes in full a result announced by R. Ma\\~{n}\\'{e} in the ICM 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks."}
{"category": "Math", "title": "Generalized quandle polynomials", "abstract": "We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants which further generalize the quandle counting invariant."}
{"category": "Math", "title": "The minimum rank problem over finite fields", "abstract": "The structure of all graphs having minimum rank at most k over a finite field with q elements is characterized for any possible k and q. A strong connection between this characterization and polarities of projective geometries is explained. Using this connection, a few results in the minimum rank problem are derived by applying some known results from projective geometry."}
{"category": "Math", "title": "The divisibility modulo 24 of Kloosterman sums on $GF(2^m)$, $m$ even", "abstract": "In a recent work by Charpin, Helleseth, and Zinoviev Kloosterman sums $K(a)$ over a finite field $\\F_{2^m}$ were evaluated modulo 24 in the case $m$ odd, and the number of those $a$ giving the same value for $K(a)$ modulo 24 was given. In this paper the same is done in the case $m$ even. The key techniques used in this paper are different from those used in the aforementioned work. In particular, we exploit recent results on the number of irreducible polynomials with prescribed coefficients."}
{"category": "Math", "title": "A new concept of strong controllability via the Schur complement in adaptive tracking", "abstract": "We propose a new concept of strong controllability associated with the Schur complement of a suitable limiting matrix. This concept allows us to extend the previous results associated with multidimensional ARX models. On the one hand, we carry out a sharp analysis of the almost sure convergence for both least squares and weighted least squares algorithms. On the other hand, we also provide a central limit theorem and a law of iterated logarithm for these two stochastic algorithms. Our asymptotic results are illustrated by numerical simulations."}
{"category": "Math", "title": "Dimensions of compact invariant sets of some expanding maps", "abstract": "We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a \"general Sierpinski carpet\" was found by McMullen and Bedford and the uniqueness of the measure of full Hausdorff dimension in some cases was proved by Kenyon and Peres. We extend these results by using compensation functions to study a general Sierpinski carpet represented by a shift of finite type. We give some conditions under which a general Sierpinski carpet has a unique measure of full Hausdorff dimension, and study the properties of the unique measure."}
{"category": "Math", "title": "Using integrals of squares of certain real-valued special functions to prove that the P\\'olya \\Xi^*(z) function, the functions K_{iz}(a), a > 0, and some other entire functions have only real zeros", "abstract": "Analogous to the use of sums of squares of certain real-valued special functions to prove the reality of the zeros of the Bessel functions J_\\alpha(z) when \\alpha \\ge -1, confluent hypergeometric functions {}_0F_1(c; z) when c > 0 or 0 > c > -1, Laguerre polynomials L_n^\\alpha(z) when \\alpha \\ge -2, Jacobi polynomials P_n^{(\\alpha,\\beta)}(z) when \\alpha \\ge -1 and \\beta \\ge -1, and some other entire special functions considered in G. Gasper [Using sums of squares to prove that certain entire functions have only real zeros, in Fourier Analysis: Analytic and Geometric Aspects, W. O. Bray, P. S. Milojevi\\'c and C. V. Stanojevi\\'c, eds., Marcel Dekker, Inc., 1994, 171--186.], integrals of squares of certain real-valued special functions are used to prove the reality of the zeros of the P\\'olya \\Xi^*(z) function, the K_{iz}(a) functions when a > 0, and some other entire functions."}
{"category": "Math", "title": "Strategic Execution in the Presence of an Uninformed Arbitrageur", "abstract": "We consider a trader who aims to liquidate a large position in the presence of an arbitrageur who hopes to profit from the trader's activity. The arbitrageur is uncertain about the trader's position and learns from observed price fluctuations. This is a dynamic game with asymmetric information. We present an algorithm for computing perfect Bayesian equilibrium behavior and conduct numerical experiments. Our results demonstrate that the trader's strategy differs significantly from one that would be optimal in the absence of the arbitrageur. In particular, the trader must balance the conflicting desires of minimizing price impact and minimizing information that is signaled through trading. Accounting for information signaling and the presence of strategic adversaries can greatly reduce execution costs."}
{"category": "Math", "title": "The tri-pentagonal number theorem and related identities", "abstract": "I revisit an automated proof of Andrews' pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews' result along with a variety of new formulas of similar type. I reveal an interesting relation between the tri-pentagonal theorem and items (19), (20), (94), (98) on the celebrated Slater list. Finally, I establish a new infinite family of multiple series identities."}
{"category": "Math", "title": "An analogue of the Magnus problem for associative algebras", "abstract": "We prove an analogue of the Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by n+k generators and k relations and has an n-element system of generators, then this algebra is a free algebra of rank n."}
{"category": "Math", "title": "A partial $A_\\infty$-structure on the cohomology of $C_n\\times C_m$", "abstract": "Suppose $k$ is a field of characteristic 2, and $n,m\\geq 4$ powers of 2. Then the $A_\\infty$-structure of the group cohomology algebras $H^*(C_n,k)$ and $H^*(C_m,k)$ are well known. We give results characterizing an $A_\\infty$-structure on $H^*(C_n\\times C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations."}
{"category": "Math", "title": "The conjugacy problem for two by two matrices over polynomial rings", "abstract": "We give an effective solution of the conjugacy problem for two by two matrices over the polynomial ring in one variable over a finite field."}
{"category": "Math", "title": "Graphs of relations and Hilbert series", "abstract": "We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by n generators and n(n-1)/2 relations for n less or equal then 7. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic."}
{"category": "Math", "title": "Weighted pluripotential theory on complex K\\\"{a}hler manifolds", "abstract": "We introduce a weighted version of the pluripotential theory on complex K\\\"{a}hler manifolds developed by Guedj and Zeriahi. We give the appropriate definition of a weighted pluricomplex Green function, its basic properties and consider its behaviour under holomorphic maps. We also establish a generalization of Siciak's H-principle."}
{"category": "Math", "title": "Stacks similar to the stack of perverse sheaves", "abstract": "We introduce, on a topological space X, a class of stacks of abelian categories we call \"stacks of type P.\" This class of stacks includes the stack of perverse sheaves (of any perversity, constructible with respect to a fixed stratification), and is singled out by fairly innocuous axioms. We show that some basic structure theory for perverse sheaves holds for a general stack of type P: such a stack is locally equivalent to a MacPherson-Vilonen construction, and under certain connectedness conditions its category of global objects is equivalent to the category of modules over a finite-dimensional algebra. To prove these results we develop a rudimentary tilting formalism for stacks of type P -- another sense in which these stacks are \"similar to stacks of perverse sheaves.\""}
{"category": "Math", "title": "Towards optimal DRP scheme for linear advection", "abstract": "Finite difference schemes are here solved by means of a linear matrix equation. The theoretical study of the related algebraic system is exposed, and enables us to minimize the error due to a finite difference approximation, while building a new DRP scheme in the same time."}
{"category": "Math", "title": "The maximum number of singular points on rational homology projective planes", "abstract": "A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane $\\mathbb{C}\\mathbb{P}^2$. It is known that a rational homology projective plane with quotient singularities has at most 5 singular points. So far all known examples have at most 4 singular points. In this paper, we prove that a rational homology projective plane $S$ with quotient singularities such that $K_S$ is nef has at most 4 singular points except one case. The exceptional case comes from Enriques surfaces with a configuration of 9 smooth rational curves whose Dynkin diagram is of type $ 3A_1 \\oplus 2A_3$. We also obtain a similar result in the differentiable case and in the symplectic case under certain assumptions which all hold in the algebraic case."}
{"category": "Math", "title": "Involutions in $S_n$ and associated coadjoint orbits", "abstract": "In the paper we study the coadjoint orbits of the group $\\mathrm{UT}(n,K)$ associated with involutions. We obtain a formula for dimension of the orbit. We construct a polarization for the canonical element of orbit. We find a system of generators in the defining ideal of orbit."}
{"category": "Math", "title": "On index of certain nilpotent Lie algebras", "abstract": "We introduce the method of calculation of index of Lie algebras that are factors of the unitriangular Lie algebra with respect to ideals spanned by subsets of root vectors."}
{"category": "Math", "title": "Estimation of quadratic variation for two-parameter diffusions", "abstract": "In this paper we give a central limit theorem for the weighted quadratic variations process of a two-parameter Brownian motion. As an application, we show that the discretized quadratic variations $\\sum_{i=1}^{[n s]} \\sum_{j=1}^{[n t]} | \\Delta_{i,j} Y |^2$ of a two-parameter diffusion $Y=(Y_{(s,t)})_{(s,t)\\in[0,1]^2}$ observed on a regular grid $G_n$ is an asymptotically normal estimator of the quadratic variation of $Y$ as $n$ goes to infinity."}
{"category": "Math", "title": "Heat kernels on Euclidean complexes", "abstract": "In this thesis we describe a type of metric space called an Euclidean polyhedral complex. We define a Dirichlet form on it; this is used to give a corresponding heat kernel. We provide a uniform small time Poincare inequality for complexes with bounded geometry and use this to determine uniform small time heat kernel bounds via a theorem of Sturm. We then consider such complexes with an underlying finitely generated group structure. We use techniques of Saloff-Coste and Pittet to show a large time asymptotic equivalence for the heat kernel on the complex and the heat kernel on the group."}
{"category": "Math", "title": "Twisted stable maps to tame Artin stacks", "abstract": "This paper is a continuation of our earlier development of a theory of tame Artin stacks. Our main goal here is the construction of an appropriate analogue of Kontsevich's space of stable maps in the case where the target is a tame Artin stack. When the target is a tame Deligne--Mumford stack, the theory was developed by Abramovich and Vistoli, and found a number of applications. The theory for arbitrary tame Artin stacks developed here is very similar, but it is necessary to overcome a number of technical hurdles and to generalize a few questions of foundation."}
{"category": "Math", "title": "Interpolation by entire functions with growth conditions", "abstract": "Let $A_p(\\C)$ be the space of entire functions such that $| f(z)|\\le Ae^{Bp(z)}$ for some $A,B>0$ and let $V$ be a discrete sequence of complex numbers which is not a uniqueness set for $A_p(\\C)$. We use $L^2$ estimates for the $\\bar\\partial$ equation to charaterize the trace of $A_p(\\C)$ on $V$."}
{"category": "Math", "title": "On a Dynamical Brauer-Manin Obstruction", "abstract": "Let F : X --> X be a morphism of a variety defined over a number field K, let V be a K-subvariety of X, and let O_F(P)= {F^n(P) :n=0,1,2,...} be the orbit of a point P in X(K). We describe a local-global principle for the intersection of V and O_F(P). This principle may be viewed as a dynamical analog of the Brauer-Manin obstruction. We show that the rational points of V(K) are Brauer--Manin unobstructed for power maps on P^2 in two cases: (1) V is a translate of a torus. (2) V is a line and P has a preperiodic coordinate. A key tool in the proofs is the classical Bang-Zsigmondy theorem on primitive divisors in sequences. We also prove analogous local-global results for dynamical systems associated to endomoprhisms of abelian varieties."}
{"category": "Math", "title": "Differentiability of M-functionals of location and scatter based on t likelihoods", "abstract": "The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector m and scatter matrix S of an elliptically symmetric t distribution on d-dimensional space with degrees of freedom larger than 1 extends to an M-functional defined on all probability distributions P in a weakly open, weakly dense domain U. Here U consists of P not putting too much mass in hyperplanes of dimension < d, as shown for empirical measures by Kent and Tyler, Ann. Statist. 1991. It is shown here that (m,S) is analytic on U, for the bounded Lipschitz norm, or for d=1, for the sup norm on distribution functions. For k=1,2,..., and other norms, depending on k and more directly adapted to t functionals, one has continuous differentiability of order k, allowing the delta-method to be applied to (m,S) for any P in U, which can be arbitrarily heavy-tailed. These results imply asymptotic normality of the corresponding M-estimators (m_n,S_n). In dimension d=1 only, the t functionals extend to be defined and weakly continuous at all t."}
{"category": "Math", "title": "Markovian Memory Embedded in Two-State Natural Processes", "abstract": "Markovian memory embedded in a binary system is shaping its evolution on the basis of its current state and introduces either clustering or dispersion of binary states. The consequence is directly observed in the lengthening or shortening of the runs of the same binary state and also in the way the proportion of a state within a sequence of state measurements scatters about its true average, which is quantifiable through the Markovian self-transition probabilities. It is shown that the Markovian memory can even imitate the evolution of a random process, regarding the long-term behavior of the frequencies of its binary states. This situation occurs when the associated binary state self-transition probabilities are balanced. To exemplify the behavior of Markovian memory, two natural processes are selected. The first example is studying the preferences of nonhuman troglodytes regarding handedness. The Markovian model in this case assesses the extent of influence two contiguous individuals may have on each other. The other example studies the hindering of the quantum state transitions that rapid state measurements introduce, known as the Quantum Zeno effect (QZE). Based on the current methodology, simulations of the experimentally observed clustering of states allowed for the estimation of the two self-transition probabilities in this quantum system. Through these, one can appreciate how the particular hindering of the evolution of a quantum state may have originated. The aim of this work is to illustrate as merits of the current mathematical approach, its wide range applicability and its potential to provide a variety of information regarding the dynamics of the studied process."}
{"category": "Math", "title": "Connectivity of the space of ending laminations", "abstract": "We prove that for any closed surface of genus at least four, and any punctured surface of genus at least two, the space of ending laminations is connected. A theorem of E. Klarreich implies that this space is homeomorphic to the Gromov boundary of the complex of curves. It follows that the boundary of the complex of curves is connected in these cases, answering the conjecture of P. Storm. Other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate Kleinian groups."}
{"category": "Math", "title": "Newton series and extended derivation relations for multiple $L$-values", "abstract": "We investigate Newton series for truncated multiple $L$-values and thereby obtain a class of relations for multiple $L$-values. In addition, we give a formulation and a proof of extended derivation relations for multiple $L$-values."}
{"category": "Math", "title": "Hamiltonian elliptic dynamics on symplectic 4-manifolds", "abstract": "We consider C2 Hamiltonian functions on compact 4-dimensional symplectic manifolds to study elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set U intersecting a far from Anosov regular energy surface, there is a nearby Hamiltonian having an elliptic closed orbit through U. Moreover, this implies that for far from Anosov regular energy surfaces of a C2-generic Hamiltonian the elliptic closed orbits are generic."}
{"category": "Math", "title": "Irregular and singular loci of commuting varieties", "abstract": "We prove that the singular locus of the commuting variety of a noncommutative reductive Lie algebra is contained in the irregular locus and we compute the codimension of the latter. We prove that one of the irreducible components of the irregular locus has codimension 4. This yields the lower bound of the codimension of the singular locus, in particular, implies that it is at least 2. We also prove that the commuting variety is rational."}
{"category": "Math", "title": "Subregular characters of the unitriangular group over a finite field", "abstract": "We present an explicit formula for subregular characters (i.e, irreducible finite-dimensional complex characters of submaximal degree) of the unitriangular group over a finite field of sufficiently large characteristic."}
{"category": "Math", "title": "Stability of K\\\"ahler-Ricci flow", "abstract": "We prove the convergence of K\\\"ahler-Ricci flow with some small initial curvature conditions. As applications, we discuss the convergence of K\\\"ahler-Ricci flow when the complex structure varies on a K\\\"ahler-Einstein manifold."}
{"category": "Math", "title": "On Steepest-Descent-Kaczmarz Methods for Regularizing Systems of Nonlinear Ill-posed Equations", "abstract": "We investigate modified steepest descent methods coupled with a loping Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a linear problem related to photoacoustic tomography and a non-linear problem related to the testing of semiconductor devices."}
{"category": "Math", "title": "Induced quasi-actions: a remark", "abstract": "In this note we observe that the notion of an induced representation has an analog for quasi-actions. We then use induced quasi-actions to refine some earlier rigidity results for product spaces."}
{"category": "Math", "title": "A lower bound for the Chung-Diaconis-Graham random process", "abstract": "Chung, Diaconis, and Graham considered random processes of the form X_{n+1}=a_n X_n+b_n (mod p) where p is odd, X_0=0, a_n=2 always, and b_n are i.i.d. for n=0,1,2,... . In this paper, we show that if P(b_n=-1)=P{b_n=0)=P(b_n=1)=1/3, then there exists a constant c>1 such that c log_2 p steps are not enough to make X_n get close to uniformly distributed on the integers mod p."}
{"category": "Math", "title": "Bethe-Sommerfeld Conjecture", "abstract": "We consider Schroedinger operator $-\\Delta+V$ in $R^d$ ($d\\ge 2$) with smooth periodic potential $V$ and prove that there are only finitely many gaps in its spectrum."}
{"category": "Math", "title": "Gradient flow approach to geometric convergence analysis of preconditioned eigensolvers", "abstract": "Preconditioned eigenvalue solvers (eigensolvers) are gaining popularity, but their convergence theory remains sparse and complex. We consider the simplest preconditioned eigensolver--the gradient iterative method with a fixed step size--for symmetric generalized eigenvalue problems, where we use the gradient of the Rayleigh quotient as an optimization direction. A sharp convergence rate bound for this method has been obtained in 2001--2003. It still remains the only known such bound for any of the methods in this class. While the bound is short and simple, its proof is not. We extend the bound to Hermitian matrices in the complex space and present a new self-contained and significantly shorter proof using novel geometric ideas."}
{"category": "Math", "title": "Non-symplectic automorphisms of order 3 on K3 surfaces", "abstract": "In this paper we study K3 surfaces with a non-symplectic automorphism of order 3. In particular, we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology. This allows us to describe the structure of the moduli space and to show that it has three irreducible components."}
{"category": "Math", "title": "Categorification of acyclic cluster algebras: an introduction", "abstract": "This is a concise introduction to Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition of cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category."}
{"category": "Math", "title": "Numerical primary decomposition", "abstract": "Consider an ideal $I \\subset R = \\bC[x_1,...,x_n]$ defining a complex affine variety $X \\subset \\bC^n$. We describe the components associated to $I$ by means of {\\em numerical primary decomposition} (NPD). The method is based on the construction of {\\em deflation ideal} $I^{(d)}$ that defines the {\\em deflated variety} $\\dXd$ in a complex space of higher dimension. For every embedded component there exists $d$ and an isolated component $\\dYd$ of $\\dId$ projecting onto $Y$. In turn, $\\dYd$ can be discovered by existing methods for prime decomposition, in particular, the {\\em numerical irreducible decomposition}, applied to $\\dXd$. The concept of NPD gives a full description of the scheme $\\Spec(R/I)$ by representing each component with a {\\em witness set}. We propose an algorithm to produce a collection of witness sets that contains a NPD and that can be used to solve the {\\em ideal membership problem} for $I$."}
{"category": "Math", "title": "Approximation by light maps and parametric Lelek maps", "abstract": "The class of metrizable spaces $M$ with the following approximation property is introduced and investigated: $M\\in AP(n,0)$ if for every $\\e>0$ and a map $g\\colon\\I^n\\to M$ there exists a 0-dimensional map $g'\\colon\\I^n\\to M$ which is $\\e$-homotopic to $g$. It is shown that this class has very nice properties. For example, if $M_i\\in AP(n_i,0)$, $i=1,2$, then $M_1\\times M_2\\in AP(n_1+n_2,0)$. Moreover, $M\\in AP(n,0)$ if and only if each point of $M$ has a local base of neighborhoods $U$ with $U\\in AP(n,0)$. Using the properties of AP(n,0)-spaces, we generalize some results of Levin and Kato-Matsuhashi concerning the existence of residual sets of $n$-dimensional Lelek maps."}
{"category": "Math", "title": "Equivariant complex structures on homogeneous spaces and their cobordism classes", "abstract": "We consider compact homogeneous spaces G/H, where G is a compact connected Lie group and H is its closed connected subgroup of maximal rank. The aim of this paper is to provide an effective computation of the universal toric genus for the complex, almost complex and stable complex structures which are invariant under the canonical left action of the maximal torus T^k on G/H. As it is known, on G/H we may have many such structures and the computations of their toric genus in terms of fixed points for the same torus action give the constraints on possible collections of weights for the corresponding representations of T^k in the tangent spaces at the fixed points, as well as on the signs at these points. In that context, the effectiveness is also approached due to an explicit description of the relations between the weights and signs for an arbitrary couple of such structures. Special attention is devoted to the structures which are invariant under the canonical action of the group G. Using classical results, we obtain an explicit description of the weights and signs in this case. We consequently obtain an expression for the cobordism classes of such structures in terms of coefficients of the formal group law in cobordisms, as well as in terms of Chern numbers in cohomology. These computations require no information on the cohomology ring of the manifold G/H, but, on their own, give important relations in this ring. As an application we provide an explicit formula for the cobordism classes and characteristic numbers of the flag manifolds U(n)/T^n, Grassmann manifolds G_{n,k}=U(n)/(U(k)\\times U(n-k)) and some particular interesting examples."}
{"category": "Math", "title": "Long hitting time, slow decay of correlations and arithmetical properties", "abstract": "Let $\\tau_r(x,x_0)$ be the time needed for a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered in $x_0$, with small radius $r$. We construct a class of translations on the two torus having particular arithmetic properties (Liouville components with intertwined denominators of convergents) not satisfying a logarithm law, i.e. such that for generic $x,x_0$ \\liminf_{r\\to 0} \\frac{\\log \\tau_r(x,x_0)}{-\\log r} = \\infty. By considering a suitable reparametrization of the flow generated by a suspension of this translation, using a previous construction by Fayad, we show the existence of a mixing system on three torus having the same properties. The speed of mixing of this example must be subpolynomial, because we also show that: in a system having polynomial decay of correlations the above ratio of logarithms (which is also called the lower hitting time indicator) is bounded (it is a function of the local dimension and the speed of correlation decay). More generally, this shows that reparametrizations of torus translations having a Liouville component cannot be polynomially mixing."}
{"category": "Math", "title": "Existence of broken Lefschetz fibrations", "abstract": "We prove that every closed oriented smooth 4-manifold X admits a broken Lefschetz fibration (aka singular Lefschetz fibration) over the 2-sphere. Given any closed orientable surface F of square zero in X, we can choose the fibration so that F is a fiber. Moreover, we can arrange it so that there is only one Lefschetz critical point when the Euler characteristic e(X) is odd, and none when e(X) is even. We make use of topological modifications of smooth maps with fold and cusp singularities due to Saeki and Levine, and thus we get alternative proofs of previous existence results. Also shown is the existence of broken Lefschetz pencils with connected fibers on any near-symplectic 4-manifold."}
{"category": "Math", "title": "Approximate word matches between two random sequences", "abstract": "Given two sequences over a finite alphabet $\\mathcal{L}$, the $D_2$ statistic is the number of $m$-letter word matches between the two sequences. This statistic is used in bioinformatics for expressed sequence tag database searches. Here we study a generalization of the $D_2$ statistic in the context of DNA sequences, under the assumption of strand symmetric Bernoulli text. For $k<m$, we look at the count of $m$-letter word matches with up to $k$ mismatches. For this statistic, we compute the expectation, give upper and lower bounds for the variance and prove its distribution is asymptotically normal."}
{"category": "Math", "title": "Remarks on 1-motivic sheaves", "abstract": "We generalize the construction of the category of 1-motives with torsion ${}^tM_1$ (introduced by Barbieri-Viale, Rosenschon and Saito) as well as the construction of the category of 1-motivic sheaves ${\\rm Shv}_1$ (defined by Barbieri-Viale and Kahn) to perfect fields $k$ (without inverting the exponential characteristic). For $k$ transcendental over the prime field we extend a result of Barbieri-Viale and Kahn, showing that ${}^tM$ and ${\\rm Shv}_1$ have equivalent bounded derived categories."}
{"category": "Math", "title": "Poisson suspensions and entropy for infinite transformations", "abstract": "The Poisson entropy of an infinite-measure-preserving transformation is defined as the Kolmogorov entropy of its Poisson suspension. In this article, we relate Poisson entropy with other definitions of entropy for infinite transformations: For quasi-finite transformations we prove that Poisson entropy coincides with Krengel's and Parry's entropy. In particular, this implies that for null-recurrent Markov chains, the usual formula for the entropy $-\\sum q_i p_{i,j}\\log p_{i,j}$ holds in any of the definitions for entropy. Poisson entropy dominates Parry's entropy in any conservative transformation. We also prove that relative entropy (in the sense of Danilenko and Rudolph) coincides with the relative Poisson entropy. Thus, for any factor of a conservative transformation, difference of the Krengel's entropy is equal to the difference of the Poisson entropies. In case there exists a factor with zero Poisson entropy, we prove the existence of a maximum (Pinsker) factor with zero Poisson entropy. Together with the preceding results, this answers affirmatively the question raised in arXiv:0705.2148v3 about existence of a Pinsker factor in the sense of Krengel for quasi-finite transformations."}
{"category": "Math", "title": "Adaptive thresholding estimation of a Poisson intensity with infinite support", "abstract": "The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $\\tilde{f}_{n,\\gamma}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a logarithmic term. Oracle inequalities allow to derive the maxiset of $\\tilde{f}_{n,\\gamma}$. Then, minimax properties of $\\tilde{f}_{n,\\gamma}$ are established. We first prove that the rate of this estimator on Besov spaces ${\\cal B}^\\al_{p,q}$ when $p\\leq 2$ is $(\\ln(n)/n)^{\\al/(1+2\\al)}$. This result has two consequences. First, it establishes that the minimax rate of Besov spaces ${\\cal B}^\\al_{p,q}$ with $p\\leq 2$ when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. This result is new. Furthermore, $\\tilde{f}_{n,\\gamma}$ is adaptive minimax up to a logarithmic term. When $p>2$, the situation changes dramatically and the rate of $\\tilde{f}_{n,\\gamma}$ on Besov spaces ${\\cal B}^\\al_{p,q}$ is worse than $(\\ln(n)/n)^{\\al/(1+2\\al)}$. Finally, the random threshold depends on a parameter $\\gamma$ that has to be suitably chosen in practice. Some theoretical results provide upper and lower bounds of $\\gamma$ to obtain satisfying oracle inequalities. Simulations reinforce these results."}
{"category": "Math", "title": "Polygones de Hodge, de Newton et de l'inertie mod\\'er\\'ee des repr\\'esentations semi-stables", "abstract": "Let k be a perfect field, and K be a totally ramified extension of K_0 = Frac W(k) of degree e. To a semi-stable p-adic representation of G_K (the absolute Galois group of K), one can classicaly associate two polygons : the Hodge polygon et the Newton polygon. It is well known that the former lies below the latter, and that they have same endpoints. In this note, we introduce a third polygon gotten from the semi-simplification of the representation mod p, and, under some conditions on Hodge-Tate weights, we prove that it lies above the Hodge polygon again with same endpoint. We finally examine one exemple associated to a crystalline representation."}
{"category": "Math", "title": "The heavy traffic limit of an unbalanced generalized processor sharing model", "abstract": "This work considers a server that processes $J$ classes using the generalized processor sharing discipline with base weight vector $\\alpha=(\\alpha _1,...,\\alpha_J)$ and redistribution weight vector $\\beta=(\\beta_1,...,\\beta_J)$. The invariant manifold $\\mathcal{M}$ of the so-called fluid limit associated with this model is shown to have the form $\\mathcal{M}=\\{x\\in\\mathbb{R}_+^J:x_j=0 for j\\in\\mathcal{S}\\}$, where $\\mathcal{S}$ is the set of strictly subcritical classes, which is identified explicitly in terms of the vectors $\\alpha$ and $\\beta$ and the long-run average work arrival rates $\\gamma_j$ of each class $j$. In addition, under general assumptions, it is shown that when the heavy traffic condition $\\sum_{j=1}^J\\gamma_j=\\sum_{j=1}^J\\alpha_j$ holds, the functional central limit of the scaled unfinished work process is a reflected diffusion process that lies in $\\mathcal{M}$. The reflected diffusion limit is characterized by the so-called extended Skorokhod map and may fail to be a semimartingale. This generalizes earlier results obtained for the simpler, balanced case where $\\gamma_j=\\alpha_j$ for $j=1,...,J$, in which case $\\mathcal{M}=\\mathbb{R}_+^J$ and there is no state-space collapse. Standard techniques for obtaining diffusion approximations cannot be applied in the unbalanced case due to the particular structure of the GPS model. Along the way, this work also establishes a comparison principle for solutions to the extended Skorokhod map associated with this model, which may be of independent interest."}
{"category": "Math", "title": "Explicit calculations of automorphic forms for definite unitary groups", "abstract": "I give an algorithm for computing the full space of automorphic forms for definite unitary groups over Q, and apply this to calculate the automorphic forms of level $G(Z-hat)$ and various small weights for an example of a rank 3 unitary group. This leads to some examples of various types of endoscopic lifting from automorphic forms for U_1 x U_1 x U_1 and U_1 x U_2, and to an example of a non-endoscopic form of weight (3,3) corresponding to a family of 3-dimensional irreducible l-adic Galois representations. I also compute the 2-adic slopes of some automorphic forms with level structure at 2, giving evidence for the local constancy of the slopes."}
{"category": "Math", "title": "Conjugacy classes in parabolic subgroups of general linear groups", "abstract": "We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalise this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small dimensions may be calculated."}
{"category": "Math", "title": "SPM Bulletin 23", "abstract": "A surprising number of new results in \"core\" SPM in the last quarter of 2007, and some other beautiful fundamental results are announced."}
{"category": "Math", "title": "The expected duration of random sequential adsorption", "abstract": "When gas molecules bind to a surface they may do so in such a way that the adsorption of one molecule inhibits the arrival of others. We consider random sequential adsorption in which the empty sites of a graph are irreversibly occupied in random order by a variety of types of ``particles.'' In a finite region the process terminates when no more particles can arrive. A universal asymptotic formula for the mean duration is given."}
{"category": "Math", "title": "Synchronizing continuous-time neutrally stable linear systems via partial-state coupling", "abstract": "Synchronization of coupled continuous-time linear systems is studied in a general setting. For identical neutrally-stable linear systems that are detectable from their outputs, it is shown that a linear output feedback law exists under which the coupled systems globally asymptotically synchronize under all fixed (directed) connected network topologies. An algorithm is provided to compute one such feedback law based on individual system parameters. The dual case, where individual systems are neutrally stable and stabilizable from their inputs, is also considered and parallel results are established."}
{"category": "Math", "title": "Time discretization and Markovian iteration for coupled FBSDEs", "abstract": "In this paper we lay the foundation for a numerical algorithm to simulate high-dimensional coupled FBSDEs under weak coupling or monotonicity conditions. In particular, we prove convergence of a time discretization and a Markovian iteration. The iteration differs from standard Picard iterations for FBSDEs in that the dimension of the underlying Markovian process does not increase with the number of iterations. This feature seems to be indispensable for an efficient iterative scheme from a numerical point of view. We finally suggest a fully explicit numerical algorithm and present some numerical examples with up to 10-dimensional state space."}
{"category": "Math", "title": "Curvature invariants, Killing vector fields, connections and cohomogeneity", "abstract": "A direct, bundle-theoretic method for defining and extending local isometries out of curvature data is developed. As a by-product, conceptual direct proofs of a classical result of Singer and a recent result of the authors are derived."}
{"category": "Math", "title": "A Note on Affinely Regular Polygons", "abstract": "The affinely regular polygons in certain planar sets are characterized. It is also shown that the obtained results apply to cyclotomic model sets and, additionally, have consequences in the discrete tomography of these sets."}
{"category": "Math", "title": "Invariant Measures and Maximal L^2 Regularity for Nonautonomous Ornstein-Uhlenbeck Equations", "abstract": "We characterize the domain of the realization of the linear parabolic operator Gu := u_t + L(t)u (where, for each real t, L(t) is an Ornstein-Uhlenbeck operator), in L^2 spaces with respect to a suitable measure, that is invariant for the associated evolution semigroup. As a byproduct, we obtain optimal L^2 regularity results for evolution equations with time-depending Ornstein-Uhlenbeck operators."}
{"category": "Math", "title": "A New Perspective for an Existing Homology Theory of Links Embedded in I-Bundles", "abstract": "This paper introduces a homology theory for links in I-bundles over an orientable surface. The theory is unique in that the elements of the chain groups are surfaces instead of diagrams. It is then shown this theory yields the same results as the homology theory constructed by Asaeda, Przytycki and Sikora."}
{"category": "Math", "title": "Twisted Deformation Quantization of Algebraic Varieties (Survey)", "abstract": "Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization map from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on equivalence classes."}
{"category": "Math", "title": "Beilinson's Hodge Conjecture for K_1 revisited", "abstract": "Let U be a smooth quasiprojective complex variety and CH^r(U,1) a special instance of Bloch's higher Chow groups. Jannsen was the first to show that the cycle class map cl_{r,1} from CH^r(U,1) (tensored with Q) to hom_{MHS}(Q(0), H^{2r-1}(U,Q(r)) is not in general surjective, contradicting an earlier conjecture of Beilinson. In this paper, we give a refinement of Jannsen's counterexample, and further show that the aforementioned cycle class map becomes surjective at the generic point."}
{"category": "Math", "title": "Seshadri constants on surfaces of general type", "abstract": "We study Seshadri constants of the canonical bundle on minimal surfaces of general type. First, we prove that if the Seshadri constant $\\eps(K_X,x)$ is between 0 and 1, then it is of the form $(m-1)/m$ for some integer $m\\ge 2$. Secondly, we study values of $\\eps(K_X,x)$ for a very general point $x$ and show that small values of the Seshadri constant are accounted for by the geometry of $X$."}
{"category": "Math", "title": "Scalar Curvature Bound for K\\\"ahler-Ricci Flows over Minimal Manifolds of General Type", "abstract": "In this short note, we use classic computations for K\\\"ahler-Ricci flow to achieve scalar curvature bound for minimal manifold of general type."}
{"category": "Math", "title": "On a Basis for the Framed Link Vector Space Spanned by Chord Diagrams", "abstract": "In view of the result of Kontsevich, now often called ``the fundamental theorem of Vassiliev theory'', identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the ``4T-relation'' (and in the unframed case, the ``1T-'' or ``isolated chord relation''), it is a problem of some interest to provide a basis for the space of chord diagrams modulo the 4T-relation. We construct the basis for the vector space spanned by chord diagrams with n chords and m distinguishable link components, modulo 4T relations for n less than or equal to 5."}
{"category": "Math", "title": "Degenerate stochastic differential equations arising from catalytic branching networks", "abstract": "We establish existence and uniqueness for the martingale problem associated with a system of degenerate SDE's representing a catalytic branching network. For example, in the hypercyclic case: $$dX_{t}^{(i)}=b_i(X_t)dt+\\sqrt{2\\gamma_{i}(X_{t}) X_{t}^{(i+1)}X_{t}^{(i)}}dB_{t}^{i}, X_t^{(i)}\\ge 0, i=1,..., d,$$ where $X^{(d+1)}\\equiv X^{(1)}$, existence and uniqueness is proved when $\\gamma$ and $b$ are continuous on the positive orthant, $\\gamma$ is strictly positive, and $b_i>0$ on $\\{x_i=0\\}$. The special case $d=2$, $b_i=\\theta_i-x_i$ is required in work of Dawson-Greven-den Hollander-Sun-Swart on mean fields limits of block averages for 2-type branching models on a hierarchical group. The proofs make use of some new methods, including Cotlar's lemma to establish asymptotic orthogonality of the derivatives of an associated semigroup at different times,and a refined integration by parts technique from Dawson-Perkins]. As a by-product of the proof we obtain the strong Feller property of the associated resolvent."}
{"category": "Math", "title": "Vertex coalgebras, comodules, cocommutativity and coassociativity", "abstract": "We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on vertex coalgebras. The former two properties require grading. We then discuss comodule structure. We conclude by discussing instances where graded vertex coalgebras appear, particularly as related to Primc's vertex Lie algebra and (universal) enveloping vertex algebras."}
{"category": "Math", "title": "A note about conditional Ornstein-Uhlenbeck processes", "abstract": "In this short note, the identity in law, which was obtained by P. Salminen, between on one hand, the Ornstein-Uhlenbeck process with parameter gamma, killed when it reaches 0, and on the other hand, the 3-dimensional radial Ornstein-Uhlenbeck process killed exponentially at rate gamma and conditioned to hit 0, is derived from a simple absolute continuity relationship."}
{"category": "Math", "title": "Inverse Spectral Problem for Schr\\\"odinger Operators", "abstract": "In this article we improve some of the inverse spectral results proved by Guillemin and Uribe in \\cite{GU}. They proved that under some symmetry assumptions on the potential $V(x)$, the Taylor expansion of $V(x)$ near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schr\\\"odinger operator in $\\mathbb R^n$. We prove some similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of $V(x)$. We establish our results by finding some explicit formulas for wave invariants at the bottom of the well."}
{"category": "Math", "title": "A homology theory for Smale spaces: a summary", "abstract": "We consider Smale spaces, a particular class of hyperbolic topological dynamical systems, which include the basic sets for Smale's Axiom A systems. We present a homology theory for such systems which is based on the dimension group in the special case of shifts of finite type. This theory provides a Lefschetz formula relating trace data with the number of periodic points of the system."}
{"category": "Math", "title": "Universal moduli spaces of surfaces with flat connections and cobordism theory", "abstract": "Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space. Rationally this says that the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller kappa-classes, and the ring of characteristic classes of principal G-bundles, H^*(BG). We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. We also explain how these results may be generalized to arbitrary compact connected Lie groups. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces."}
{"category": "Math", "title": "A method for computing general automorphic forms on general groups", "abstract": "This article describes a general method for computing automorphic forms using Voronoi-type summation formulas. It gives a numerical example where the technique is successful in quickly finding a cusp form on GL(3,Z)\\GL(3,R), albeit one whose existence was already known as a Langlands lift."}
{"category": "Math", "title": "Chip-Firing and Rotor-Routing on Directed Graphs", "abstract": "We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems."}
{"category": "Math", "title": "Surgery on links with unknotted components and three-manifolds", "abstract": "It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link L with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on the same link L could give the same three-manifold M."}
{"category": "Math", "title": "Simultaneous generation for zeta values by the Markov-WZ method", "abstract": "By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher's and Almkvist-Granville's Ap\\'ery-like formulae for odd zeta values. As a consequence, we get a new identity producing Ap\\'ery-like series for all $\\zeta(2n+4m+3),$ $n,m\\ge 0,$ convergent at the geometric rate with ratio $2^{-10}.$"}
{"category": "Math", "title": "Occupation densities for certain processes related to fractional Brownian motion", "abstract": "In this paper we establish the existence of a square integrable occupation density for two classes of stochastic processes. First we consider a Gaussian process with an absolutely continuous random drift, and secondly we handle the case of a (Skorohod) integral with respect to the fractional Brownian motion with Hurst parameter $H>\\frac 12$. The proof of these results uses a general criterion for the existence of a square integrable local time, which is based on the techniques of Malliavin calculus."}
{"category": "Math", "title": "Warped Wavelet and Vertical Thresholding", "abstract": "Let $\\{(X_i,Y_i)\\}_{i\\in \\{1,..., n\\}}$ be an i.i.d. sample from the random design regression model $Y=f(X)+\\epsilon$ with $(X,Y)\\in [0,1]\\times [-M,M]$. In dealing with such a model, adaptation is naturally to be intended in terms of $L^2([0,1],G_X)$ norm where $G_X(\\cdot)$ denotes the (known) marginal distribution of the design variable $X$. Recently much work has been devoted to the construction of estimators that adapts in this setting (see, for example, [5,24,25,32]), but only a few of them come along with a easy--to--implement computational scheme. Here we propose a family of estimators based on the warped wavelet basis recently introduced by Picard and Kerkyacharian [36] and a tree-like thresholding rule that takes into account the hierarchical (across-scale) structure of the wavelet coefficients. We show that, if the regression function belongs to a certain class of approximation spaces defined in terms of $G_X(\\cdot)$, then our procedure is adaptive and converge to the true regression function with an optimal rate. The results are stated in terms of excess probabilities as in [19]."}
{"category": "Math", "title": "Real map germs and higher open books", "abstract": "We present a general criterion for the existence of open book structures defined by real map germs $(\\bR^m, 0) \\to (\\bR^p, 0)$, where $m> p \\ge 2$, with isolated critical point. We show that this is satisfied by weighted-homogeneous maps. We also derive sufficient conditions in case of map germs with isolated critical value."}
{"category": "Math", "title": "The lineage process in Galton--Watson trees and globally centered discrete snakes", "abstract": "We consider branching random walks built on Galton--Watson trees with offspring distribution having a bounded support, conditioned to have $n$ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of ``globally centered discrete snake'' that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when $n$ goes to $+\\infty$, ``globally centered discrete snakes'' converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton--Watson tree conditioned by the size, and their links with a multinomial process [the lineage of a node $u$ is the vector indexed by $(k,j)$ giving the number of ancestors of $u$ having $k$ children and for which $u$ is a descendant of the $j$th one]. Some consequences concerning Galton--Watson trees conditioned by the size are also derived."}
{"category": "Math", "title": "Holomorphie des op\\'erateurs d'entrelacement normalis\\'es \\`a l'aide des param\\`etres d'Arthur", "abstract": "In this paper we prove holomorphy for certain intertwining operators arising from the theory of Eisenstein series."}
{"category": "Math", "title": "Convexity, translation invariance and subadditivity for $g$-expectations and related risk measures", "abstract": "Under the continuous assumption on the generator $g$, Briand et al. [Electron. Comm. Probab. 5 (2000) 101--117] showed some connections between $g$ and the conditional $g$-expectation $({\\mathcal{E}}_g[\\cdot|{\\mathcal{F}}_t])_{t\\in[0,T]}$ and Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19--34] showed some connections between $g$ and the corresponding dynamic risk measure $(\\rho^g_t)_{t\\in[0,T]}$. In this paper we prove that, without the additional continuous assumption on $g$, a $g$-expectation ${\\mathcal{E}}_g$ satisfies translation invariance if and only if $g$ is independent of $y$, and ${\\mathcal{E}}_g$ satisfies convexity (resp. subadditivity) if and only if $g$ is independent of $y$ and $g$ is convex (resp. subadditive) with respect to $z$. By these conclusions we deduce that the static risk measure $\\rho^g$ induced by a $g$-expectation ${\\mathcal{E}}_g$ is a convex (resp. coherent) risk measure if and only if $g$ is independent of $y$ and $g$ is convex (resp. sublinear) with respect to $z$. Our results extend the results in Briand et al. [Electron. Comm. Probab. 5 (2000) 101--117] and Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19--34] on these subjects."}
{"category": "Math", "title": "The Hopf algebra structure of the Z$_3$-graded quantum supergroup GL$_{q,j}(1|1)$", "abstract": "In this work, we give some features of the Z$_3$-graded quantum supergroup."}
{"category": "Math", "title": "On the condensed density of the generalized eigenvalues of pencils of Hankel Gaussian random matrices and applications", "abstract": "Pencils of Hankel matrices whose elements have a joint Gaussian distribution with nonzero mean and not identical covariance are considered. An approximation to the distribution of the squared modulus of their determinant is computed which allows to get a closed form approximation of the condensed density of the generalized eigenvalues of the pencils. Implications of this result for solving several moments problems are discussed and some numerical examples are provided."}
{"category": "Math", "title": "$C^*$-algebras arising from substitutions", "abstract": "In this paper, we introduce a $C^{\\ast}$-algebra associated with a proper primitive substitution. We show that the $C^{\\ast}$-algebra is simple and purely infinite and contains the associated Cuntz-Krieger algebra and the crossed product $C^{\\ast}$-algebra of the corresponding Cantor minimal system. We calculate the $K$-groups."}
{"category": "Math", "title": "Analyticity of the SRB measure for holomorphic families of quadratic-like Collet-Eckmann maps", "abstract": "We show that if f_t is a holomorphic family of quadratic-like maps with all periodic orbits repelling so that for each real t the map f_t is a real Collet-Eckmann S-unimodal map then, writing m_t for the unique absolutely continuous invariant probability measure of f_t, the map t -> \\int g dm_t is real analytic for any real analytic function g."}
{"category": "Math", "title": "The Effect of Radiation Pressure on the Equilibrium Points in the Generalised Photogravitational Restricted Three Body Problem", "abstract": "The existence of equilibrium points and the effect of radiation pressure have been discussed numerically. The problem is generalized by considering bigger primary as a source of radiation and small primary as an oblate spheroid. We have also discussed the Poynting-Robertson(P-R) effect which is caused due to radiation pressure. It is found that the collinear points $L_1,L_2,L_3$ deviate from the axis joining the two primaries, while the triangular points $L_4,L_5$ are not symmetrical due to radiation pressure. We have seen that $L_1,L_2,L_3$ are linearly unstable while $L_4,L_5$ are conditionally stable in the sense of Lyapunov when P-R effect is not considered. We have found that the effect of radiation pressure reduces the linear stability zones while P-R effect induces an instability in the sense of Lyapunov."}
{"category": "Math", "title": "One-dimensional stepping stone models, sardine genetics and Brownian local time", "abstract": "Consider a one-dimensional stepping stone model with colonies of size $M$ and per-generation migration probability $\\nu$, or a voter model on $\\mathbb{Z}$ in which interactions occur over a distance of order $K$. Sample one individual at the origin and one at $L$. We show that if $M\\nu/L$ and $L/K^2$ converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0."}
{"category": "Math", "title": "A Geometrical Study of Matching Pursuit Parametrization", "abstract": "This paper studies the effect of discretizing the parametrization of a dictionary used for Matching Pursuit decompositions of signals. Our approach relies on viewing the continuously parametrized dictionary as an embedded manifold in the signal space on which the tools of differential (Riemannian) geometry can be applied. The main contribution of this paper is twofold. First, we prove that if a discrete dictionary reaches a minimal density criterion, then the corresponding discrete MP (dMP) is equivalent in terms of convergence to a weakened hypothetical continuous MP. Interestingly, the corresponding weakness factor depends on a density measure of the discrete dictionary. Second, we show that the insertion of a simple geometric gradient ascent optimization on the atom dMP selection maintains the previous comparison but with a weakness factor at least two times closer to unity than without optimization. Finally, we present numerical experiments confirming our theoretical predictions for decomposition of signals and images on regular discretizations of dictionary parametrizations."}
{"category": "Math", "title": "Integrally closed and componentwise linear ideals", "abstract": "In two dimensional regular local rings integrally closed ideals have a unique factorization property and have a Cohen-Macaulay associated graded ring. In higher dimension these properties do not hold for general integrally closed ideals and the goal of the paper is to identify a subclass of integrally closed ideals for which they do. We restrict our attention to 0-dimensional homogeneous ideals in polynomial rings $R$ of arbitrary dimension and identify a class of integrally closed ideals, the Goto-class $\\G^*$, that is closed under product and that has a suitable unique factorization property. Ideals in $\\G^*$ have a Cohen-Macaulay associated graded ring if either they are monomial or $\\dim R\\leq 3$. Our approach is based on the study of the relationship between the notions of integrally closed, contracted, full and componentwise linear ideals."}
{"category": "Math", "title": "Saddle-shaped solutions of bistable diffusion equations in all of $\\mathbb{R}^{2m}$", "abstract": "We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation $-\\Delta u = f(u)$ in the whole $\\R^{2m}$, where $f$ is of bistable type. It is known that in dimension $2m=2$ there exists a saddle-shaped solution. This is a solution which changes sign in $\\R^2$ and vanishes only on $\\{|x_1|=|x_2|\\}$. It is also known that this solution is unstable. In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension $2m=4$. More precisely, our main result establishes that if $2m=4$, every solution vanishing on the Simons cone $\\{(x^1,x^2)\\in\\R^m\\times\\R^m : |x^1|=|x^2|\\}$ is unstable outside of every compact set and, as a consequence, has infinite Morse index. These results are relevant in connection with a conjecture of De Giorgi extensively studied in recent years and for which the existence of a counter-example in high dimensions is still an open problem."}
{"category": "Math", "title": "Gerasimov's theorem and N-Koszul algebras", "abstract": "The paper is devoted to graded algebras having a single homogeneous relation. Using Gerasimov's theorem, a criterion to be N-Koszul is given, providing new examples. An alternative proof of Gerasimov's theorem for N=2 is given. Some related results on Calabi-Yau algebras are proved."}
{"category": "Math", "title": "LQR-based coupling gain for synchronization of linear systems", "abstract": "Synchronization control of coupled continuous-time linear systems is studied. For identical systems that are stabilizable, a linear feedback law obtained via algebraic Riccati equation is shown to synchronize any fixed directed network of any number of coupled systems provided that the coupling is strong enough. The strength of coupling is determined by the smallest distance of a nonzero eigenvalue of the coupling matrix to the imaginary axis. A dual problem where detectable systems that are coupled via their outputs is also considered and solved."}
{"category": "Math", "title": "Semi-regular Relative Difference Sets with Large Forbidden Subgroups", "abstract": "Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters $(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We prove that there does not exist a $(2p,p,2p,2)$ relative difference set in any group of order $2p^2$, and an abelian $(4p,p,4p,4)$ relative difference set can only exist in the group $\\Bbb{Z}_2^2\\times \\Bbb{Z}_3^2$. On the other hand, we construct a family of non-abelian relative difference sets with parameters $(4q,q,4q,4)$, where $q$ is an odd prime power greater than 9 and $q\\equiv 1$ (mod 4). When $q=p$ is a prime, $p>9$, and $p\\equiv$ 1 (mod 4), the $(4p,p,4p,4)$ non-abelian relative difference sets constructed here are genuinely non-abelian in the sense that there does not exist an abelian relative difference set with the same parameters."}
{"category": "Math", "title": "A local form for the automorphisms of the spectral unit ball", "abstract": "If F is an automorphism of the spectral unit ball, we show that, in a neighborhood of any cyclic (i.e. non-derogatory) matrix of the ball, the map F can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result."}
{"category": "Math", "title": "Instabilit\\'{e} des cocycles d'\\'{e}volution fortement mesurables dans des espaces de Banach", "abstract": "The aim of the paper is to present various asymptotic behaviors of skew-evolution semiflows in Banach spaces, as exponential decay, instability, exponential in- stability and integral instability. Relations between these asymptotic properties are also given. As main results, two Datko type theorems are proved. A unified nonuniform approach is provided."}
{"category": "Math", "title": "Tensor, Sobolev, Multiplicative and Convolution Operators in the Bide - Side Grand Lebesque Spaces", "abstract": "In this paper we study the multiplicative, tensor, Sobolev's and convolution inequalities in certain Banach spaces, the so-called Bide - Side Grand Lebesque Spaces, and give examples to show their sharpness."}
{"category": "Math", "title": "The rigidity of embedded constant mean curvature surfaces", "abstract": "We study the rigidity of complete, embedded constant mean curvature surfaces in R^3. Among other things, we prove that when such a surface has finite genus, then intrinsic isometries of the surface extend to isometries of R^3 or its isometry group contains an index two subgroup of isometries that extend."}
{"category": "Math", "title": "When every multilinear mapping is multiple summing", "abstract": "In this paper we give a systematized treatment to some coincidence situations for multiple summing multilinear mappings which extend, generalize and simplify the methods and results obtained thus far. The application of our general results to the pertinent particular cases gives several new coincidences as well as easier proofs of some known results."}
{"category": "Math", "title": "Finite speed of propagations of the electromagnetic field in nonlinear isotropic dispersive mediums", "abstract": "We propose some modification of Maxwell's equations describing mediums which electric and magnetic properties are changed essentially after interaction with outer electromagnetic field. We show for such mediums that electromagnetic waves have finite speed of propagations property for some time depending on initial energy of electromagnetic field and nonlinear parameters of the problem which are responsible for properties of medium."}
{"category": "Math", "title": "Approximate and pseudo-amenability of various classes of Banach algebras", "abstract": "We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity. Among our other results, it is shown that the Fourier algebra of the free group on two generators is not approximately amenable. Further examples are obtained of ${\\ell}^1$-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate amenability need not imply sequential approximate amenability. Results are also given for Segal subalgebras of $L^1(G)$, where $G$ is a locally compact group, and the algebras $PF_p(\\Gamma)$ of $p$-pseudofunctions on a discrete group $\\Gamma$ (of which the reduced $C^*$-algebra is a special case)."}
{"category": "Math", "title": "Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets", "abstract": "In this paper we state and prove a central limit theorem for the finite-dimensional laws of the quadratic variations process of certain fractional Brownian sheets. The main tool of this article is a method developed by Nourdin and Nualart based on the Malliavin calculus."}
{"category": "Math", "title": "Automorphism groups of algebraic curves with p-rank zero", "abstract": "In positive characteristic, algebraic curves can have many more automorphisms than expected from the classical Hurwitz's bound. There even exist algebraic curves of arbitrary high genus g with more than 16g^4 automorphisms. It has been observed on many occasions that the most anomalous examples invariably have zero p-rank. In this paper, the K-automorphism group Aut(X) of a zero 2-rank algebraic curve X defined over an algebraically closed field K of characteristic 2 is investigated. The main result is that if the curve has genus g greater than or equal to 2, and |Aut(X)|>24g^2, then Aut(X) has a fixed point on X, apart from few exceptions. In the exceptional cases the possibilities for Aut(X) and g are determined."}
{"category": "Math", "title": "The modified Calabi-Yau problems for CR-manifolds and applications", "abstract": "In this paper, we derive a partial result related to a question of Yau: \"Does a simply-connected complete K\\\"ahler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?\" Main Theorem. Let $M^{2n}$ be a simply-connected complete K\\\"ahler manifold M with negative sectional curvature $ \\le -1 $ and $S_\\infty(M)$ be the sphere at infinity of $M$. Then there is an explicit {\\it bounded} contact form $\\beta$ defined on the entire manifold $M^{2n}$. Consequently, the sphere $S_\\infty(M)$ at infinity of M admits a {\\it bounded} contact structure and a bounded pseudo-Hermitian metric in the sense of Tanaka-Webster. We also discuss several open modified problems of Calabi and Yau for Alexandrov spaces and CR-manifolds."}
{"category": "Math", "title": "Finding central decompositions of p-groups", "abstract": "Polynomial-time algorithms are given to find a central decomposition of maximum size for a finite p-group of class 2 and for a nilpotent Lie ring of class 2. The algorithms use Las Vegas probabilistic routines to compute the structure of finite *-rings and also the Las Vegas C-MeatAxe. When p is small, the probabilistic methods can be replaced by deterministic polynomial-time algorithms. The methods introduce new group isomorphism invariants including new characteristic subgroups."}
{"category": "Math", "title": "Bayesian models to adjust for response bias in survey data for estimating rape and domestic violence rates from the NCVS", "abstract": "It is difficult to accurately estimate the rates of rape and domestic violence due to the sensitive nature of these crimes. There is evidence that bias in estimating the crime rates from survey data may arise because some women respondents are \"gagged\" in reporting some types of crimes by the use of a telephone rather than a personal interview, and by the presence of a spouse during the interview. On the other hand, as data on these crimes are collected every year, it would be more efficient in data analysis if we could identify and make use of information from previous data. In this paper we propose a model to adjust the estimates of the rates of rape and domestic violence to account for the response bias due to the \"gag\" factors. To estimate parameters in the model, we identify the information that is not sensitive to time and incorporate this into prior distributions. The strength of Bayesian estimators is their ability to combine information from long observational records in a sensible way. Within a Bayesian framework, we develop an Expectation-Maximization-Bayesian (EMB) algorithm for computation in analyzing contingency table and we apply the jackknife to estimate the accuracy of the estimates. Our approach is illustrated using the yearly crime data from the National Crime Victimization Survey. The illustration shows that compared with the classical method, our model leads to more efficient estimation but does not require more complicated computation."}
{"category": "Math", "title": "Quenched convergence of a sequence of superprocesses in R^d among Poissonian obstacles", "abstract": "We prove a convergence theorem for a sequence of super-Brownian motions moving among hard Poissonian obstacles, when the intensity of the obstacles grows to infinity but their diameters shrink to zero in an appropriate manner. The superprocesses are shown to converge in probability for the law $\\mathbf{P}$ of the obstacles, and $\\mathbf{P}$-almost surely for a subsequence, towards a superprocess with underlying spatial motion given by Brownian motion and (inhomogeneous) branching mechanism $\\psi(u,x)$ of the form $\\psi(u,x)= u^2+ \\kappa(x)u$, where $\\kappa(x)$ depends on the density of the obstacles. This work draws on similar questions for a single Brownian motion. In the course of the proof, we establish precise estimates for integrals of functions over the Wiener sausage, which are of independent interest."}
{"category": "Math", "title": "A Characteristic Map for Symplectic Manifolds", "abstract": "We construct a local characteristic map to a symplectic manifold M via certain cohomology groups of Hamiltonian vector fields. For each p in M, the Leibniz cohomology of the Hamiltonian vector fields on R^{2n} maps to the Leibniz cohomology of all Hamiltonian vector fields on M. For a particular extension g_n of the symplectic Lie algebra, the Leibniz cohomology of g_n is shown to be an exterior algebra on the canonical symplectic two-form. The Leibniz homology of g_n then maps to the Leibniz homology of Hamiltonian vector fields on R^{2n}."}
{"category": "Math", "title": "Stability of bounded global solutions for Navier-Stokes equations", "abstract": "In this paper some kind of asymptotic behavior of the solutions for the Navier-Stokes system on abstract Banach spaces is studied under the existence of global in time solutions. The asymptotic stability of the zero solution is also shown."}
{"category": "Math", "title": "Group actions and rational ideals", "abstract": "We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting. Our main result concerns rational actions of an affine algebraic group G on R. Working over an algebraically closed base field, we prove an existence and uniqueness result for generic rational ideals: for every G-rational ideal I of R, the closed subset of the rational spectrum Rat R that is defined by I is the closure of a unique G-orbit in Rat R. Under additional Goldie hypotheses, this was established earlier by Moeglin and Rentschler (in characteristic zero) and by Vonessen (in arbitrary characteristic), answering a question of Dixmier."}
{"category": "Math", "title": "A Modified K\\\"ahler-Ricci Flow", "abstract": "In this note, a modified K\\\"ahler-Ricci flow is introduced and studied. The main point is to show the flexibility of K\\\"ahler-Ricci flow and summarize some useful techniques."}
{"category": "Math", "title": "Climbing a Legendrian mountain range without Stabilization", "abstract": "We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specific Legendrian and transversal MTWS by enhancing the Legendrian mountain range for the (2,3)-cable of a (2,3)-torus knot provided by Etnyre and Honda, and showing that elementary negative flypes allow us to move toward maximal tb value without having to use Legendrian stabilization. In doing so we obtain new ways to visualize convex tori and Legendrian divides and rulings, using tilings and braided rectangular diagrams."}
{"category": "Math", "title": "Rayleigh's Stretched String", "abstract": "We obtain rigorous a priori upper and lower bounds to the exact period of the celebrated Rayleigh stretched string differential equation."}
{"category": "Math", "title": "Beyond Thresholding: Analysis and Improvements for Deterministic Parameter Estimation", "abstract": "Hard-threshold estimators are popular in signal processing applications. We provide a detailed study of using hard-threshold estimators for estimating an unknown deterministic signal when additive white Gaussian noise corrupts observations. The analysis, depending heavily on Cram{\\'e}r-Rao bounds, motivates piecewise-linear estimation as a simple improvement to hard thresholding. We compare the performance of two piecewise-linear estimators to a hard-threshold estimator. When either piecewise-linear estimator is optimized for the decay rate of the basis coefficients, its performance is better than the best possible with hard thresholding."}
{"category": "Math", "title": "On the appearance of Eisenstein series through degeneration", "abstract": "Let $\\Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\\mathbb H$, and let $M = \\Gamma \\backslash \\mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $\\gamma$ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $\\gamma$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If $\\gamma \\in \\Gamma$ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface."}
{"category": "Math", "title": "Sheaves on local Calabi-Yau varieties", "abstract": "We investigate sheaves supported on the zero section of the total space of a locally-free sheaf E on a smooth, projective variety X when the top exterior power of E is isomorphic to the canonical bundle of X. We rephrase this construction using the language of A-infinity algebra and provide a simple characterisation of the case E is simply the canonical bundle itself."}
{"category": "Math", "title": "Perelman's W-functional and stability of K\\\"ahler-Ricci flow", "abstract": "In this expository note, we study the second variation of Perelman's entropy on the space of Kahler metrics at a K\\\"ahler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelman's entropy is stable along the K\\\"ahler-Ricci flow. The Chinese version of this note has appeared in a volume in honor of professor K.C.Chang (Scientia Sinica Math., 46 (2016), 685-696)."}
{"category": "Math", "title": "Harmonic Analysis of Stochastic Equations and Backward Stochastic Differential Equations", "abstract": "The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations (SEs) in $\\cR^p$ ($p\\in [1, \\infty)$) and backward stochastic differential equations (BSDEs) in $\\cR^p\\times \\cH^p$ ($p\\in (1, \\infty)$) and in $\\cR^\\infty\\times \\bar{\\cH^\\infty}^{BMO}$, with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman's inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear case for SDEs and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse H\\\"older inequality for some suitable exponent $p\\ge 1$. Finally, we establish some relations between Kazamaki's quadratic critical exponent $b(M)$ of a BMO martingale $M$ and the spectral radius of the solution operator for the $M$-driven SDE, which lead to a characterization of Kazamaki's quadratic critical exponent of BMO martingales being infinite."}
{"category": "Math", "title": "Some results for the Perelman LYH-type inequality", "abstract": "Let $(M,g(t))$, $0\\le t\\le T$, $\\partial M\\ne\\phi$, be a compact $n$-dimensional manifold, $n\\ge 2$, with metric $g(t)$ evolving by the Ricci flow such that the second fundamental form of $\\partial M$ with respect to the unit outward normal of $\\partial M$ is uniformly bounded below on $\\partial M\\times [0,T]$. We will prove a global Li-Yau gradient estimate for the solution of the generalized conjugate heat equation on $M\\times [0,T]$. We will give another proof of Perelman's Li-Yau-Hamilton type inequality for the fundamental solution of the conjugate heat equation on closed manifolds without using the properties of the reduced distance. We will also prove various gradient estimates for the Dirichlet fundamental solution of the conjugate heat equation."}
{"category": "Math", "title": "Admissible local systems for a class of line arrangements", "abstract": "A rank one local system $\\LL$ on a smooth complex algebraic variety $M$ is admissible roughly speaking if the dimension of the cohomology groups $H^m(M,\\LL)$ can be computed directly from the cohomology algebra $H^*(M,\\C)$. We say that a line arrangement $\\A$ is of type $\\CC_k$ if $k \\ge 0 $ is the minimal number of lines in $\\A$ containing all the points of multiplicity at least 3. We show that if $\\A$ is a line arrangement in the classes $\\CC_k$ for $k\\leq 2$, then any rank one local system $\\LL$ on the line arrangement complement $M$ is admissible. Partial results are obtained for the class $\\CC_3$."}
{"category": "Math", "title": "On some difficulties with a posterior probability approximation technique", "abstract": "In Scott (2002) and Congdon (2006), a new method is advanced to compute posterior probabilities of models under consideration. It is based solely on MCMC outputs restricted to single models, i.e., it is bypassing reversible jump and other model exploration techniques. While it is indeed possible to approximate posterior probabilities based solely on MCMC outputs from single models, as demonstrated by Gelfand and Dey (1994) and Bartolucci et al. (2006), we show that the proposals of Scott (2002) and Congdon (2006) are biased and advance several arguments towards this thesis, the primary one being the confusion between model-based posteriors and joint pseudo-posteriors. From a practical point of view, the bias in Scott's (2002) approximation appears to be much more severe than the one in Congdon's (2006), the later being often of the same magnitude as the posterior probability it approximates, although we also exhibit an example where the divergence from the true posterior probability is extreme."}
{"category": "Math", "title": "Nonlinear Schr\\\"odinger equation on real hyperbolic spaces", "abstract": "We consider the Schr\\\"odinger equation with no radial assumption on real hyperbolic spaces. We obtain sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we get strong well-posedness results for NLS. Specifically, for small intial data, we prove $L^2$ and $H^1$ global well-posedness for any subcritical nonlinearity (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearity $F$. On the other hand, if $F$ is gauge invariant, $L^2$ charge is conserved and hence, as in the Euclidean case, it is possible to extend local $L^2$ solutions to global ones. The corresponding argument in $H^1$ requires the conservation of energy, which holds under the stronger condition that $F$ is defocusing. Recall that global well-posedness in the gauge invariant case was already proved by Banica, Carles & Staffilani, for small radial $L^2$ data and for large radial $H^1$ data. The second important application of our global Strichartz estimates is \"scattering\" for NLS both in $L^2$ and in $H^1$, with no radial or gauge invariance assumption. Notice that, in the Euclidean case, this is only possible for the critical power $\\gamma=1+\\frac4n$ and can be false for subcritical powers while, on hyperbolic spaces, global existence and scattering of small $L^2$ solutions holds for all powers $1<\\gamma\\le1+\\frac4n$. If we restrict to defocusing nonlinearities $F$, we can extend the $H^1$ scattering results of Banica, Carles & Staffilani to the nonradial case. Also there is no distinction anymore between short range and long range nonlinearity : the geometry of hyperbolic spaces makes every power-like nonlinearity short range."}
{"category": "Math", "title": "On some smooth projective two-orbits varieties with Picard number 1", "abstract": "We classify all smooth projective horospherical varieties with Picard number 1. We prove that the automorphism group of any such variety X acts with at most two orbits and that this group still acts with only two orbits on X blown up at the closed orbit. We characterize all smooth projective two-orbits varieties with Picard number 1 that satisfy this latter property."}
{"category": "Math", "title": "Uniforming n-place functions on ds(alpha)", "abstract": "In this paper the Erdos-Rado theorem is generalized to the class of well founded trees."}
{"category": "Math", "title": "Geometric structures on loop and path spaces", "abstract": "Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic structure. This article shows that this structure is not likely to recover the underlying Riemannian metric by proving a result that is a strong indication of the \"almost\" independence of the quasi-symplectic structure with respect to the metric. Finally conditions to have contact structures on these spaces are studied."}
{"category": "Math", "title": "Motivic and quantum invariance under stratified Mukai flops", "abstract": "For stratified Mukai flops of type $A_{n,k}, D_{2k+1}$ and $E_{6,I}$, it is shown the fiber product induces isomorphisms on Chow motives. In contrast to (standard) Mukai flops, the cup product is generally not preserved. For $A_{n, 2}$, $D_5$ and $E_{6, I}$ flops, quantum corrections are found through degeneration/deformation to ordinary flops."}
{"category": "Math", "title": "A statistical analysis of probabilistic counting algorithms", "abstract": "This paper considers the problem of cardinality estimation in data stream applications. We present a statistical analysis of probabilistic counting algorithms, focusing on two techniques that use pseudo-random variates to form low-dimensional data sketches. We apply conventional statistical methods to compare probabilistic algorithms based on storing either selected order statistics, or random projections. We derive estimators of the cardinality in both cases, and show that the maximal-term estimator is recursively computable and has exponentially decreasing error bounds. Furthermore, we show that the estimators have comparable asymptotic efficiency, and explain this result by demonstrating an unexpected connection between the two approaches."}
{"category": "Math", "title": "Majorizing measures and proportional subsets of bounded orthonormal systems", "abstract": "In this article we prove that for any orthonormal system $(\\vphi_j)_{j=1}^n \\subset L_2$ that is bounded in $L_{\\infty}$, and any $1 < k <n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on $\\spa\\{\\vphi_i\\}_{i \\in I}$, the $L_1$ norm and the $L_2$ norm are equivalent up to a factor $\\mu (\\log \\mu)^{5/2}$, where $\\mu = \\sqrt{n/k} \\sqrt{\\log k}$. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures."}
{"category": "Math", "title": "Efficient l_{alpha} Distance Approximation for High Dimensional Data Using alpha-Stable Projection", "abstract": "In recent years, large high-dimensional data sets have become commonplace in a wide range of applications in science and commerce. Techniques for dimension reduction are of primary concern in statistical analysis. Projection methods play an important role. We investigate the use of projection algorithms that exploit properties of the alpha-stable distributions. We show that l_{alpha} distances and quasi-distances can be recovered from random projections with full statistical efficiency by L-estimation. The computational requirements of our algorithm are modest; after a once-and-for-all calculation to determine an array of length k, the algorithm runs in O(k) time for each distance, where k is the reduced dimension of the projection."}
{"category": "Math", "title": "Stability of hypersurfaces with constant $r$-th anisotropic mean curvature", "abstract": "Given a positive function $F$ on $S^n$ which satisfies a convexity condition, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. Let $X:M\\to \\mathbb{R}^{n+1}$ be an $n$-dimensional closed hypersurface with $H^F_{r+1}=$constant, for some $r$ with $0\\leq r\\leq n-1$, which is a critical point for a variational problem. We show that $X(M)$ is stable if and only if $X(M)$ is the Wulff shape."}
{"category": "Math", "title": "A concentration inequality for interval maps with an indifferent fixed point", "abstract": "For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^n\\to\\R$ which are componentwise Lipschitz. The proof is based on coupling and decay of correlation properties of the map. We then give various applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram."}
{"category": "Math", "title": "Detecting rigid convexity of bivariate polynomials", "abstract": "Given a polynomial $x \\in {\\mathbb R}^n \\mapsto p(x)$ in $n=2$ variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set ${\\mathcal P} = \\{x : p(x) \\geq 0\\}$ containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LMI) representation, or equivalently, if polynomial $p(x)$ is hyperbolic with respect to the origin. The problem boils down to checking whether a univariate polynomial matrix is positive semidefinite, an optimization problem that can be solved with eigenvalue decomposition. When the variety ${\\mathcal C} = \\{x : p(x) = 0\\}$ is an algebraic curve of genus zero, a second algorithm based on B\\'ezoutians is proposed to detect whether $\\mathcal P$ has an LMI representation and to build such a representation from a rational parametrization of $\\mathcal C$. Finally, some extensions to positive genus curves and to the case $n>2$ are mentioned."}
{"category": "Math", "title": "K-duality for stratified pseudomanifolds", "abstract": "This paper is devoted to the study of Poincar\\'e duality in K-theory for general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification $\\fS$ of a topological space $X$ and we define a groupoid $T^{\\fS}X$, called the $\\fS$-tangent space. This groupoid is made of different pieces encoding the tangent spaces of the strata, and these pieces are glued into the smooth noncommutative groupoid $T^{\\fS}X$ using the familiar procedure introduced by A. Connes for the tangent groupoid of a manifold. The main result is that $C^{*}(T^{\\fS}X)$ is Poincar\\'e dual to $C(X)$, in other words, the $\\fS$-tangent space plays the role in $K$-theory of a tangent space for $X$."}
{"category": "Math", "title": "ACM bundles on cubic surfaces", "abstract": "In this paper we prove that, for every $r \\geq 2$, the moduli space $M^s_X(r;c_1,c_2)$ of rank $r$ stable vector bundles with Chern classes $c_1=rH$ and $c_2=(3r^2-r)/2$ on a nonsingular cubic surface $X \\subset \\mathbb{P}^3$ contains a nonempty smooth open subset formed by ACM bundles, i.e. vector bundles with no intermediate cohomology. The bundles we consider for this study are extremal for the number of generators of the corresponding module (these are known as Ulrich bundles), so we also prove the existence of indecomposable Ulrich bundles of arbitrarily high rank on $X$."}
{"category": "Math", "title": "Escaping points of entire functions of small growth", "abstract": "Let $f$ be a transcendental entire function and let $I(f)$ denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, $I(f)$ is connected. In particular, we show that $I(f)$ is connected if $f$ has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko's conjecture that $I(f)$ has no bounded components is true. We also give a new criterion related to $I(f)$ which is sufficient to ensure that $f$ has no unbounded Fatou components."}
{"category": "Math", "title": "Functions of small growth with no unbounded Fatou components", "abstract": "We prove a form of the $\\cos \\pi \\rho$ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen's condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components, also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions."}
{"category": "Math", "title": "Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications", "abstract": "In the first part of this work, we consider second order supersymmetric differential operators in the semiclassical limit, including the Kramers-Fokker-Planck operator, such that the exponent of the associated Maxwellian $\\phi$ is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish the long time convergence to the equilibrium for the associated heat semigroup, with the rate given by the first non-vanishing, exponentially small, eigenvalue. In the second part of the paper, we consider the case when the function $\\phi$ has precisely one local minimum and one saddle point. We also discuss further examples of supersymmetric operators, including the Witten Laplacian and the infinitesimal generator for the time evolution of a chain of classical anharmonic oscillators."}
{"category": "Math", "title": "Index theory and Groupoids", "abstract": "This paper collects the notes of a serie of lectures given by the two authors during the summer school \"Geometric and topological methods for Quantum Field Theory\" at Villa de Leyva, Colombia, summer 2007. These lecture notes are mainly devoted to a proof using groupoids and $KK$-theory of Atiyah-Singer index theorem on compact smooth manifolds. We will present an elementary introduction to groupoids, $C^*$-algebras, $KK$-theory and pseudodifferential calculus on groupoids. We will finish by showing that the point of view adopted here generalizes to the case of conical pseudo-manifolds."}
{"category": "Math", "title": "Toric complexes and Artin kernels", "abstract": "A simplicial complex L on n vertices determines a subcomplex T_L of the n-torus, with fundamental group the right-angled Artin group G_L. Given an epimorphism \\chi\\colon G_L\\to \\Z, let T_L^\\chi be the corresponding cover, with fundamental group the Artin kernel N_\\chi. We compute the cohomology jumping loci of the toric complex T_L, as well as the homology groups of T_L^\\chi with coefficients in a field \\k, viewed as modules over the group algebra \\k\\Z. We give combinatorial conditions for H_{\\le r}(T_L^\\chi;\\k) to have trivial \\Z-action, allowing us to compute the truncated cohomology ring, H^{\\le r}(T_L^\\chi;\\k). We also determine several Lie algebras associated to Artin kernels, under certain triviality assumptions on the monodromy \\Z-action, and establish the 1-formality of these (not necessarily finitely presentable) groups."}
{"category": "Math", "title": "Area, capacity and diameter versions of Schwarz's Lemma", "abstract": "The now canonical proof of Schwarz's Lemma appeared in a 1907 paper of Carath\\'eodory, who attributed it to Erhard Schmidt. Since then, Schwarz's Lemma has acquired considerable fame, with multiple extensions and generalizations. Much less known is that, in the same year 1907, Landau and Toeplitz obtained a similar result where the diameter of the image set takes over the role of the maximum modulus of the function. We give a new proof of this result and extend it to include bounds on the growth of the maximum modulus. We also develop a more general approach in which the size of the image is estimated in several geometric ways via notions of radius, diameter, perimeter, area, capacity, etc..."}
{"category": "Math", "title": "On the Representation Theory of an Algebra of Braids and Ties", "abstract": "We consider the algebra ${\\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\\cal E}_n(u)$ and to classify its irreducible representations."}
{"category": "Math", "title": "Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics", "abstract": "Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthogonal, parallel Jacobi field. As applications we obtain (1) constraints on the behavior of quasi-isometries between complete, simply connected, NPCR manifolds, and (2) constraints on the NPCR metrics supported by certain manifolds, and (3) a correspondence between metric splittings of complete, simply connected NPCR manifolds, and metric splittings of its asymptotic cones. Furthermore, combining our results with the Ballmann-Burns-Spatzier rigidity theorem and the classic Mostow rigidity, we also obtain (4) a new proof of Gromov's rigidity theorem for higher rank locally symmetric spaces."}
{"category": "Math", "title": "The sum of irreducible fractions with consecutive denominators is never an integer in a very weak arithmetic", "abstract": "Two theorems of elmentary arithmetic, one stating that the sum of the reciprocals of any number of consecutive positive integers is never an integer, and a generalization thereof by Trygve Nagell, are shown to be provable inside a very weak arithmetic, Richard Kaye's $PA^-$, in which there is no induction axiom whatsoever."}
{"category": "Math", "title": "Dynatomic cycles for morphisms of projective varieties", "abstract": "We prove the effectivity of the dynatomic cycles for morphisms of projective varieties. We then analyze the degrees of the dynatomic cycles and multiplicities of formal periodic points and apply these results to the existence of periodic points with arbitrarily large primitive periods."}
{"category": "Math", "title": "Good Reduction of Periodic Points", "abstract": "We consider the dynamical system created by iterating a morphism of a projective variety defined over the field of fractions of a discrete valuation ring. We study the primitive period of a periodic point in this field in relation to the primitive period of the reduced point in the residue field, the order of the action on the cotangent space, and the characteristic of the residue field."}
{"category": "Math", "title": "All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron", "abstract": "The hexabasic book is the cone of the 1-dimensional skeleton of the union of two tetrahedra glued along a common face. The universal 3-dimensional polyhedron UP is the product of a segment and the hexabasic book. We show that any 2-dimensional link in 4-space is isotopic to a surface in UP. The proof is based on a representation of surfaces in 4-space by marked graphs, links with double intersections in 3-space. We construct a finitely presented semigroup whose central elements uniquely encode all isotopy classes of 2-dimensional links."}
{"category": "Math", "title": "Finding Rational Periodic Points on Wehler K3 Surfaces", "abstract": "This article examines dynamical systems on a class of K3 surfaces in $\\mathbb{P}^{2} \\times \\mathbb{P}^{2}$ with an infinite automorphism group. In particular, this article develops an algorithm to find $\\mathbb{Q}$-rational periodic points using information modulo $p$ for various primes $p$. The algorithm is applied to exhibit K3 surfaces with $\\mathbb{Q}$-rational periodic points of primitive period $1,...,16$. A portion of the algorithm is then used to determine the Riemann zeta function modulo 3 of a particular K3 surface and find a family of K3 surfaces with Picard number two."}
{"category": "Math", "title": "Link concordance and generalized doubling operators", "abstract": "We introduce a technique for showing classical knots and links are not slice. As one application we show that the iterated Bing doubles of many algebraically slice knots are not topologically slice. Some of the proofs do not use the existence of the Cheeger-Gromov bound, a deep analytical tool used by Cochran-Teichner. We define generalized doubling operators, of which Bing doubling is an instance, and prove our nontriviality results in this more general context. Our main examples are boundary links that cannot be detected in the algebraic boundary link concordance group."}
{"category": "Math", "title": "The smallest multistationary mass-preserving chemical reaction network", "abstract": "Biochemical models that exhibit bistability are of interest to biologists and mathematicians alike. Chemical reaction network theory can provide sufficient conditions for the existence of bistability, and on the other hand can rule out the possibility of multiple steady states. Understanding small networks is important because the existence of multiple steady states in a subnetwork of a biochemical model can sometimes be lifted to establish multistationarity in the larger network. This paper establishes the smallest reversible, mass-preserving network that admits bistability and determines the semi-algebraic set of parameters for which more than one steady state exists."}
{"category": "Math", "title": "Asymptotics of degenerating Eisenstein series", "abstract": "We give some estimates for the asymptotic orders of degenerating Eisenstein series for some families of degenerating punctured Riemann surfaces, which is motivated by the question identifying $L_{2}$-cohomology of the Takhtajan-Zograf metric that is originally asked by To and Weng."}
{"category": "Math", "title": "The two dimensional distribution of values of $\\zeta(1+it)$", "abstract": "We prove several results on the distribution function of $\\zeta(1+it)$ in the complex plane, that is the joint distribution function of $\\arg\\zeta(1+it)$ and $|\\zeta(1+it)|$. Similar results are also given for $L(1,\\chi)$ (as $\\chi$ varies over non-principal characters modulo a large prime $q$)."}
{"category": "Math", "title": "Stochastic Depletion Problems: Effective Myopic Policies for a class of Dynamic Optimization Problems", "abstract": "This paper presents a general class of dynamic stochastic optimization problems we refer to as Stochastic Depletion Problems. A number of challenging dynamic optimization problems of practical interest are stochastic depletion problems. Optimal solutions for such problems are difficult to obtain, both from a pragmatic computational perspective as also from a theoretical perspective. As such, simple heuristics are highly desirable. We isolate two simple properties that, if satisfied by a problem within this class, guarantee that a myopic policy incurs a performance loss of at most 50 % relative to the optimal adaptive control policy for that problem. We are able to verify that these two properties are satisfied for several interesting families of stochastic depletion problems and as a consequence identify efficient near-optimal control policies for a number of interesting dynamic stochastic optimization problems."}
{"category": "Math", "title": "Hyperbolic Graphs of Surface Groups", "abstract": "We give a sufficient condition under which the fundamental group of a reglued graph of surfaces is hyperbolic. A reglued graph of surfaces is constructed by cutting a fixed graph of surfaces along the edge surfaces, then regluing by pseudo-Anosov homeomorphisms of the edge surfaces. By carefully choosing the regluing homeomorphism, we construct an example of such a reglued graph of surfaces, whose fundamental group is not abstractly commensurate to any surface-by-free group, i.e., which is different from all the examples given in Mosher's paper 'A hyperbolic-by-hyperbolic hyperbolic group'."}
{"category": "Math", "title": "$q$-Chaos", "abstract": "We consider the $L_p$ norm estimates for homogeneous polynomials of $q$-gaussian variables ($-1\\leq q\\leq 1$). When $-1<q<1$ the $L_p$ estimates for $1\\leq p \\leq 2$ are essentially the same as the free case ($q=0$), whilst the $L_p$ estimates for $2\\leq p \\leq \\infty$ show a strong $q$-dependence. Moreover, the extremal cases $q = \\pm 1$ produce decisively different formulae."}
{"category": "Math", "title": "Spherical two-distance sets", "abstract": "A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical two-distance sets is not exceed n(n+3)/2. This upper bound is known to be tight for n=2,6,22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n)=n(n+1)/2 for g(n. In this paper using the so-called polynomial method it is proved that for nonnegative a+b the largest cardinality of S is not greater than L(n). For the case a+b<0 we propose upper bounds on |S| which are based on Delsarte's method. Using this we show that g(n)=L(n) for 6<n<22, 23<n<40, and g(23)=276 or 277."}
{"category": "Math", "title": "Deformations of nilpotent cones and Springer correspondences", "abstract": "Let G = Sp (2n) be the symplectic group over Z. We present a certain kind of deformation of the nilpotent cone of G with G-action. This enables us to make direct links between the Springer correspondence of sp_{2n} over C, that over characteristic two, and our exotic Springer correspondence. As a by-product, we obtain a complete description of our exotic Springer correspondence."}
{"category": "Math", "title": "Topology of polar weighted homogeneous hypersurfaces", "abstract": "Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\\sqrt{-1} y_i$, which enjoys a \"polar action\". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials."}
{"category": "Math", "title": "Construction and Uniqueness for reflected BSDE under linear increasing condition", "abstract": "In this paper, we study the uniqueness of the solution of reflected BSDE with one or two barriers, under continuous and linear increasing condition of generator $g$. Before that we study the construction of solution of of reflected BSDE with one or two barriers."}
{"category": "Math", "title": "Self-similar solutions and translating solitons for Lagrangian mean curvature flow", "abstract": "We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regularity of Lagrangian mean curvature flow. Given two transverse Lagrangian planes R^n in C^n with sum of characteristic angles less than pi, we show there exists a Lagrangian self-expander asymptotic to this pair of planes. The Maslov class of these self-expanders is zero. Thus they can serve as local models for surgeries on Lagrangian mean curvature flow. Families of self-shrinkers and self-expanders with different topologies are also constructed. This paper generalizes the work of Anciaux, Joyce, Lawlor, and Lee and Wang."}
{"category": "Math", "title": "A unifying formulation of the Fokker-Planck-Kolmogorov equation for general stochastic hybrid systems", "abstract": "A general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and the notion of probability current (in the case of forced jumps). This work unifies all previously known instances of the FPK equation for stochastic hybrid systems, and provides GSHS practitioners with a tool to derive the correct evolution equation for the probability law of the state in any given example."}
{"category": "Math", "title": "Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane", "abstract": "We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they have shown that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace."}
{"category": "Math", "title": "Inverse zero-sum problems II", "abstract": "Let $G$ be an additive finite abelian group. A sequence over $G$ is called a minimal zero-sum sequence if the sum of its terms is zero and no proper subsequence has this property. Davenport's constant of $G$ is the maximum of the lengths of the minimal zero-sum sequences over $G$. Its value is well-known for groups of rank two. We investigate the structure of minimal zero-sum sequences of maximal length for groups of rank two. Assuming a well-supported conjecture on this problem for groups of the form $C_m \\oplus C_m$, we determine the structure of these sequences for groups of rank two. Combining our result and partial results on this conjecture, yields unconditional results for certain groups of rank two."}
{"category": "Math", "title": "Bounded H_\\infty-calculus for pseudodifferential Douglis-Nirenberg systems of mild regularity", "abstract": "Parameter-ellipticity with respect to a closed subsector of the complex plane for pseudodifferential Douglis-Nirenberg systems is discussed and shown to imply the existence of a bounded H_\\infty-calculus in suitable scales of Sobolev, Besov, and Hoelder spaces. We also admit non pseudodifferential perturbations. Applications concern systems with coefficients of mild Hoelder regularity and the generalized thermoelastic plate equations."}
{"category": "Math", "title": "Higher Accuracy for Bayesian and Frequentist Inference: Large Sample Theory for Small Sample Likelihood", "abstract": "Recent likelihood theory produces $p$-values that have remarkable accuracy and wide applicability. The calculations use familiar tools such as maximum likelihood values (MLEs), observed information and parameter rescaling. The usual evaluation of such $p$-values is by simulations, and such simulations do verify that the global distribution of the $p$-values is uniform(0, 1), to high accuracy in repeated sampling. The derivation of the $p$-values, however, asserts a stronger statement, that they have a uniform(0, 1) distribution conditionally, given identified precision information provided by the data. We take a simple regression example that involves exact precision information and use large sample techniques to extract highly accurate information as to the statistical position of the data point with respect to the parameter: specifically, we examine various $p$-values and Bayesian posterior survivor $s$-values for validity. With observed data we numerically evaluate the various $p$-values and $s$-values, and we also record the related general formulas. We then assess the numerical values for accuracy using Markov chain Monte Carlo (McMC) methods. We also propose some third-order likelihood-based procedures for obtaining means and variances of Bayesian posterior distributions, again followed by McMC assessment. Finally we propose some adaptive McMC methods to improve the simulation acceptance rates. All these methods are based on asymptotic analysis that derives from the effect of additional data. And the methods use simple calculations based on familiar maximizing values and related informations. The example illustrates the general formulas and the ease of calculations, while the McMC assessments demonstrate the numerical validity of the $p$-values as percentage position of a data point. The example, however, is very simple and transparent, and thus gives little indication that in a wide generality of models the formulas do accurately separate information for almost any parameter of interest, and then do give accurate $p$-value determinations from that information. As illustration an enigmatic problem in the literature is discussed and simulations are recorded; various examples in the literature are cited."}
{"category": "Math", "title": "Representation of nonnegative convex polynomials", "abstract": "We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic subset K of Rn. Namely, they belong to a specific subset of the quadratic module generated by the concave polynomials that define K."}
{"category": "Math", "title": "Generalized iteration, catastrophes and generalized Sharkovsky's ordering", "abstract": "We define iteration of functions that map n-dimensional vector spaces into m-dimensional vector spaces (m at most equal to n). It happens that usual iteration and Fibonacci iterative methods become special cases of this generalized iteration. Mathematical objects such as orbits, bifurcations, chaos, Feigenbaum constant, (generalized) Sharkovsky ordering, (generalized) Julia and Mandelbrot sets and a new kind of catastrophe can be found and studied in this enlarged context."}
{"category": "Math", "title": "Counting cluster-tilted algebras of type $A_n$", "abstract": "The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type $A_n$, by counting the mutation class of any quiver with underlying graph $A_n$. It will also follow that if $T$ and $T'$ are cluster-tilting objects in a cluster category $\\mathcal{C}$, then $\\End_{\\mathcal{C}}(T)$ is isomorphic to $\\End_{\\mathcal{C}}(T')$ if and only if $T=\\tau^i T'$."}
{"category": "Math", "title": "On the Well-possedness of the Problem of Reconstruction of Non-separate Boundary Conditions", "abstract": "We consider an inverse spectral problem with the third-order differential equation and the non-separated boundary conditions. Two theorems on the uniqueness of the solution of this problem are proved, and a method for establishing the unknown conditions is obtained, using 19 eigenvalues. The method of approximate calculation of unknown boundary conditions is explained, with the help of an example."}
{"category": "Math", "title": "Maximal Crossed Product Orders over Discrete Valuation Rings", "abstract": "The problem of determining when a (classical) crossed product $T=S^f*G$ of a finite group $G$ over a discrete valuation ring $S$ is a maximal order, was answered in the 1960's for the case where $S$ is tamely ramified over the subring of invariants $S^G$. The answer was given in terms of the conductor subgroup (with respect to $f$) of the inertia. In this paper we solve this problem in general when $S/S^G$ is residually separable. We show that the maximal order property entails a restrictive structure on the sub-crossed product graded by the inertia subgroup. In particular, the inertia is abelian. Using this structure, one is able to extend the notion of the conductor. As in the tame case, the order of the conductor is equal to the number of maximal two sided ideals of $T$ and hence to the number of maximal orders containing $T$ in its quotient ring. Consequently, $T$ is a maximal order if and only if the conductor subgroup is trivial."}
{"category": "Math", "title": "Stable laws and products of positive random matrices", "abstract": "Let $S$ be the multiplicative semigroup of $q\\times q$ matrices with positive entries such that every row and every column contains a strictly positive element. Denote by $(X_n)_{n\\geq1}$ a sequence of independent identically distributed random variables in $S$ and by $X^{(n)} = X_n ... X_1$, $ n\\geq 1$, the associated left random walk on $S$. We assume that $(X_n)_{n\\geq1}$ verifies the contraction property $\\P(\\bigcup_{n\\geq1}[X^{(n)} \\in S^\\circ])>0$, where $S^\\circ $ is the subset of all matrices which have strictly positive entries. We state conditions on the distribution of the random matrix $X_1$ which ensure that the logarithms of the entries, of the norm, and of the spectral radius of the products $X^{(n)}$, $n\\ge 1$, are in the domain of attraction of a stable law."}
{"category": "Math", "title": "Hilbert's Nullstellensatz and an Algorithm for Proving Combinatorial Infeasibility", "abstract": "Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert's Nullstellensatz certificates for polynomial systems arising in combinatorics and on large-scale linear-algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges."}
{"category": "Math", "title": "Inverse Zero-Sum Problems III", "abstract": "Let $G$ be a finite abeilian group. A sequence $S$ with terms from $G$ is zero-sum if the sum of terms in $S$ equals zero. It is a minimal zero-sum sequence if no proper, nontrivial subsequence is zero-sum. The maximal length of a minimal zero-sum subsequence in $G$ is the Davenport constant, denoted $D(G)$. For a rank 2 group $G=C_n \\oplus C_n$, it is known that $D(G)=2n-1$. However, the structure of all maximal length minimal zero-sum sequences remains open. If every such sequence contains a term with multiplicity $n-1$, then $C_n \\oplus C_n$ is said to have Property B, and it is conjectured that this is true for all rank 2 groups $C_n \\oplus C_n$. In this paper, we show that Property B is multiplicative, namely, if $G=C_n \\oplus C_n$ and $G=C_m \\oplus C_m$ both satisfy Property B, with $m, n\\geq 3$ odd and $mn>9$, then $C_{mn}\\oplus C_{mn}$ satisfies Property B also. Combined with previous work in the literature, this reduces the question of establishing Property B to the prime cases, and in such case the complete structural description of the sequence follows."}
{"category": "Math", "title": "Novel Bounds on Marginal Probabilities", "abstract": "We derive two related novel bounds on single-variable marginal probability distributions in factor graphs with discrete variables. The first method propagates bounds over a subtree of the factor graph rooted in the variable, and the second method propagates bounds over the self-avoiding walk tree starting at the variable. By construction, both methods not only bound the exact marginal probability distribution of a variable, but also its approximate Belief Propagation marginal (``belief''). Thus, apart from providing a practical means to calculate bounds on marginals, our contribution also lies in an increased understanding of the error made by Belief Propagation. Empirically, we show that our bounds often outperform existing bounds in terms of accuracy and/or computation time. We also show that our bounds can yield nontrivial results for medical diagnosis inference problems."}
{"category": "Math", "title": "An existence result for the sandpile problem on flat tables with walls", "abstract": "We derive an existence result for solutions of a differential system which characterizes the equilibria of a particular model in granular matter theory, the so-called partially open table problem for growing sandpiles. Such result generalizes a recent theorem of Cannarsa and Cardaliaguet established for the totally open table problem. Here, due to the presence of walls at the boundary, the surface flow density at the equilibrium may result no more continuous nor bounded, and its explicit mathematical characterization is obtained by domain decomposition techniques. At the same time we show how these solutions can be numerically computed as stationary solutions of a dynamical two-layer model for growing sandpiles and we present the results of some simulations."}
{"category": "Math", "title": "Vari\\'et\\'es homog\\`enes sous $\\PGL_n$", "abstract": "Let $A$ be an Azumaya algebra over a field. If $G$ is the group of automorphisms of $A$ and $X$ denotes a projective homogeneous variety under $G$, we construct in a very explicit way and under suitable hypotheses a bundle $\\mathcal{V}$ on $S$, where $S$ is a (generalized) Severi-Brauer variety associated to $A$, and a canonical isomorphism between $X$ and a flag bundle on $\\mathcal{V}$. This allows to explicitely compute Chow groups of $X$ in terms of the Chow groups of $S$."}
{"category": "Math", "title": "Frame potential and finite abelian groups", "abstract": "This article continues a prior investigation of the authors with the goal of extending characterization results of convolutional tight frames from the context of cyclic groups to general finite abelian groups. The collections studied are formed by translating a number of \\emph{generators} by elements of a fixed subgroup and it is shown, under certain norm conditions, that tight frames with this structure are characterized as local minimizers of the frame potential. Natural analogs to the downsampling and upsampling operators of finite cyclic groups are studied for arbitrary subgroups of finite abelian groups. Directions of further study are also proposed."}
{"category": "Math", "title": "On cusps and flat tops", "abstract": "We develop non-invertible Pesin theory for a new class of maps called cusp maps. These maps may have unbounded derivative, but nevertheless verify a property analogous to $C^{1+\\epsilon}$. We do not require the critical points to verify a non-flatness condition, so the results are applicable to $C^{1+\\epsilon}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz."}
{"category": "Math", "title": "Algorithm for solving optimization problems with Interval Valued Probability Measure", "abstract": "We are concerned with three types of uncertainties: probabilistic, possibilitistic and interval. By using possibility and necessity measures as an Interval Valued Probability Measure (IVPM), we present IVPM's interval expected values whose possibility distributions are in the form of polynomials. By working with interval expected values of independent uncertainty coefficients in a linear optimization problem together with operations suggested in Lodwick and Jamison (2007), the problem after applying these operations becomes a linear programming problem with constant coefficients. This is achieved by the application of two functions. The first is applied to the interval coefficients, v: I -> R^k, where I= {[a,b] | a <= b}. The second is u: R^k -> R, applied to the product we got from a previous function. Similar concepts hold for any types of optimization problems with linear constraints. Moreover, it implied that optimization problems containing all three types of uncertainties in one problem can be solved as ordinary optimization problems."}
{"category": "Math", "title": "Symmetry of Reidemeister torsion on $SU_2$-representation spaces of knots", "abstract": "We study two sorts of actions on the space of conjugacy classes of irreducible $SU_2$-representations of a knot group. One of them is an involution which comes from the algebraic structure of $SU_2$ and the other is the action by the outer automorphism group of the knot group. In particular, we consider them on an 1-dimensional smooth part of the space, which is canonically oriented and metrized via a Reidemeister torsion volume form. As an application we show that the Reidemeister torsion function on the 1-dimensional subspace has symmetry about the metrization."}
{"category": "Math", "title": "Normal Toric Ideals of Low Codimension", "abstract": "Every normal toric ideal of codimension two is minimally generated by a Grobner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of fixed codimension is normal."}
{"category": "Math", "title": "Large p-groups actions with a p-elementary abelian second ramification group", "abstract": "Let $k$ be an algebraically closed field of characteristic $p>0$ and $C$ a connected nonsingular projective curve over $k$ with genus $g \\geq 2$. Let $(C,G)$ be a \"big action\", i.e. a pair $(C,G)$ where $G$ is a $p$-subgroup of the $k$-automorphism group of $C$ such that$\\frac{|G|}{g} >\\frac{2 p}{p-1}$. We denote by $G_2$ the second ramification group of $G$ at the unique ramification point of the cover $C \\to C/G$. The aim of this paper is to describe the big actions whose $G_2$ is $p$-elementary abelian. In particular, we obtain a structure theorem by considering the $k$-algebra generated by the additive polynomials. We more specifically explore the case where there is a maximal number of jumps in the ramification filtration of $G_2$. In this case, we display some universal families."}
{"category": "Math", "title": "Computing Arakelov class groups", "abstract": "Shanks's infrastructure algorithm and Buchmann's algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we describe Buchmann's algorithm in this context."}
{"category": "Math", "title": "Pseudodifferential multi-product representation of the solution operator of a parabolic equation", "abstract": "By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on $\\R^n$ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality."}
{"category": "Math", "title": "Four primality testing algorithms", "abstract": "In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time algorithm to prove that a given numer is either prime or composite. The third and fourth primality tests are at present most widely used in practice. Both tests are capable of proving that a given number is prime or composite, but neither algorithm is deterministic. The third algorithm exploits the arithmetic of cyclotomic fields. Its running time is almost, but not quite polynomial time. The fourth algorithm exploits elliptic curves. Its running time is difficult to estimate, but it behaves well in practice."}
{"category": "Math", "title": "The Classifying Space of a Topological 2-Group", "abstract": "Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the first Cech cohomology of M with coefficients in G, or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bokstedt-Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and Cech cohomology with coefficients in a topological 2-group C, also known as 'nonabelian cohomology'. Then we give an elementary proof that under mild conditions on M and C there is a bijection between the first Cech cohomology of M with coefficients in C and [M,B|C|] where B|C| is the classifying space of the geometric realization of the nerve of C. Applying this result to the 'string 2-group' String(G) of a simply-connected compact simple Lie group G, it follows that principal String(G)-2-bundles have rational characteristic classes coming from elements of the rational cohomology of BG modulo the ideal generated by c, where c is any nonzero element in the 4th cohomology of BG."}
{"category": "Math", "title": "Graphic Bernstein Results in Curved Pseudo-Riemannian Manifolds", "abstract": "We generalize a Bernstein-type result due to Albujer and Al\\'ias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $\\Sigma_1\\times \\mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic submanifold with parallel mean curvature, defined by a map $f: \\Sigma_1\\to \\Sigma_2$ between two Riemannian manifolds $(\\Sigma_1^m, g_1)$ and $(\\Sigma^n_2, g_2)$ of sectional curvatures $K_1$ and $K_2$, respectively. We take on $\\Sigma_1\\times \\Sigma_2$ the pseudo-Riemannian product metric $g_1-g_2$. Under the curvature conditions, $\\mathrm{Ricci}_1 \\geq 0$ and $K_1\\geq K_2$, we prove that, if the second fundamental form of $M$ satisfies an integrability condition, then $M$ is totally geodesic, and it is a slice if $\\mathrm{Ricci}_1(p)>0$ at some point. For bounded $K_1$, $K_2$ and hyperbolic angle $\\theta$, we conclude $M$ must be maximal. If $M$ is a maximal surface and $K_1\\geq K_2^+$, we show $M$ is totally geodesic with no need for further assumptions. Furthermore, $M$ is a slice if at some point $p\\in \\Sigma_1$, $K_1(p)> 0$, and if $\\Sigma_1$ is flat and $K_2<0$ at some point $f(p)$, then the image of $f$ lies on a geodesic of $\\Sigma_2$."}
{"category": "Math", "title": "On the expansion of the resolvent for elliptic boundary contact problems", "abstract": "Let $A$ be an elliptic operator on a compact manifold with boundary $M$, and let $\\wp : \\partial\\M \\to Y$ be a covering map, where $Y$ is a closed manifold. Let $A_C$ be a realization of $A$ subject to a coupling condition $C$ that is elliptic with parameter in the sector $\\Lambda$. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary. We prove that the resolvent trace $\\Tr_{L^2} (A_C-\\lambda)^{-N}$ for $N$ sufficiently large has a complete asymptotic expansion as $|\\lambda| \\to \\infty$, $\\lambda \\in \\Lambda$. In particular, the heat trace $\\Tr_{L^2}e^{-tA_C}$ has a complete asymptotic expansion as $t \\to 0^+$, and the $\\zeta$-function has a meromorphic extension to $\\C$."}
{"category": "Math", "title": "Long cycles in fullerene graphs", "abstract": "It is conjectured that every fullerene graph is hamiltonian. Jendrol' and Owens proved [J. Math. Chem. 18 (1995), pp. 83--90] that every fullerene graph on n vertices has a cycle of length at least 4n/5. In this paper, we improve this bound to 5n/6-2/3."}
{"category": "Math", "title": "The Range of Approximate Unitary Equivalence Classes of Homomorphisms from AH-algebras", "abstract": "Let $C$ be a unital AH-algebra and $A$ be a unital simple C*-algebra with tracial rank zero. It has been shown that two unital monomorphisms $\\phi, \\psi: C\\to A$ are approximately unitarily equivalent if and only if $$ [\\phi]=[\\psi] {\\rm in} KL(C,A) and \\tau\\circ \\phi=\\tau\\circ \\psi \\tforal \\tau\\in T(A), $$ where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $\\kappa\\in KL(C,A)$ with $\\kappa(K_0(C)_+\\setminus \\{0\\})\\subset K_0(A)_+\\setminus \\{0\\}$ and with $\\kappa([1_C])=[1_A]$ and a continuous affine map $\\lambda: T(A)\\to T_{\\mathtt{f}}(C)$ which is compatible with $\\kappa,$ where $T_{\\mathtt{f}}(C)$ is the convex set of all faithful tracial states, there exists a unital monomorphism $\\phi: C\\to A$ such that $$ [\\phi]=\\kappa\\andeqn \\tau\\circ \\phi(c)=\\lambda(\\tau)(c) $$ for all $c\\in C_{s.a.}$ and $\\tau\\in T(A).$ Denote by ${\\rm Mon}_{au}^e(C,A)$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$ \\Lambda: {\\rm Mon}_{au}^e (C,A)\\to KLT(C,A)^{++}, $$ where $KLT(C,A)^{++}$ is the set of compatible pairs of elements in $KL(C,A)^{++}$ and continuous affine maps from $T(A)$ to $T_{\\mathtt{f}}(C).$ Moreover, we realized that there are compact metric spaces $X$, unital simple AF-algebras $A$ and $\\kappa\\in KL(C(X), A)$ with $\\kappa(K_0(C(X))_+\\setminus\\{0\\})\\subset K_0(A)_+\\setminus \\{0\\}$ for which there is no \\hm $h: C(X)\\to A$ so that $[h]=\\kappa.$"}
{"category": "Math", "title": "Simultaneous preconditioning and symmetrization of non-symmetric linear systems", "abstract": "Motivated by the theory of self-duality which provides a variational formulation and resolution for non self-adjoint partial differential equations \\cite{G1, G2}, we propose new templates for solving large non-symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain ill-conditioned, and highly non-symmetric systems."}
{"category": "Math", "title": "Infinite Dimensional Multiplicity Free Spaces II: Limits of Commutative Nilmanifolds", "abstract": "We study direct limits $(G,K) = \\varinjlim (G_n,K_n)$ of Gelfand pairs of the form $G_n = N_n\\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. First, we extend the criterion of \\cite{W3} for a direct limit representation to be multiplicity free. Then we study direct limits $G/K = \\varinjlim G_n/K_n$ of commutative nilmanifolds and look to see when the regular representation of $G = \\varinjlim G_n$ on an appropriate Hilbert space $\\varinjlim L^2(G_n/K_n)$ is multiplicity free. One knows that the $N_n$ are commutative or 2--step nilpotent. In many cases where the derived algebras $[\\gn_n,\\gn_n]$ are of bounded dimension we construct $G_n$--equivariant isometric maps $\\zeta_n : L^2(G_n/K_n) \\to L^2(G_{n+1}/K_{n+1})$ and prove that the left regular representation of $G$ on the Hilbert space $L^2(G/K) := \\varinjlim \\{L^2(G_n/K_n),\\zeta_n\\}$ is a multiplicity free direct integral of irreducible unitary representations. The direct integral and its irreducible constituents are described explicitly. One constituent of our argument is an extension of the classical Peter--Weyl Theorem to parabolic direct limits of compact groups."}
{"category": "Math", "title": "Infinite Dimensional Multiplicity Free Spaces I: Limits of Compact Commutative Spaces", "abstract": "We study direct limits $(G,K) = \\varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \\varinjlim G_n/K_n$ of compact riemannian symmetric spaces, where we combine our criterion with the Cartan--Helgason Theorem to show in general that the regular representation of $G = \\varinjlim G_n$ on a certain function space $\\varinjlim L^2(G_n/K_n)$ is multiplicity free. That method is not applicable for direct limits of nonsymmetric Gelfand pairs, so we introduce two other methods. The first, based on ``parabolic direct limits'' and ``defining representations'', extends the method used in the symmetric space case. The second uses some (new) branching rules from finite dimensional representation theory. In both cases we define function spaces $\\cA(G/K)$, $\\cC(G/K)$ and $L^2(G/K)$ to which our multiplicity--free criterion applies."}
{"category": "Math", "title": "2-Cocycles of Original Deformative Schr\\\"{o}dinger-Virasoro Algebras", "abstract": "Both original and twisted Schr\\\"{o}dinger-Virasoro algebras also their deformations were introduced and investigated in a series of papers by Henkel, Roger and Unterberger. In the present paper we aim to determine the 2-cocycles of original deformative Schr\\\"{o}dinger-Virasoro algebras."}
{"category": "Math", "title": "Smooth Solutions of Non-linear Stochastic Partial Differential Equations", "abstract": "In this paper, we study the regularities of solutions of nonlinear stochastic partial differential equations in the framework of Hilbert scales. Then we apply our general result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau's equations on the real line, stochastic 2D Navier-Stokes equations in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their respectively smooth solutions."}
{"category": "Math", "title": "Properties of Nested Sampling", "abstract": "Nested sampling is a simulation method for approximating marginal likelihoods proposed by Skilling (2006). We establish that nested sampling has an approximation error that vanishes at the standard Monte Carlo rate and that this error is asymptotically Gaussian. We show that the asymptotic variance of the nested sampling approximation typically grows linearly with the dimension of the parameter. We discuss the applicability and efficiency of nested sampling in realistic problems, and we compare it with two current methods for computing marginal likelihood. We propose an extension that avoids resorting to Markov chain Monte Carlo to obtain the simulated points."}
{"category": "Math", "title": "An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise", "abstract": "We study a linear quadratic problem for a system governed by the heat equation on a halfline with Dirichlet boundary control and Dirichlet boundary noise. We show that this problem can be reformulated as a stochastic evolution equation in a certain weighted L2 space. An appropriate choice of weight allows us to prove a stronger regularity for the boundary terms appearing in the infinite dimensional state equation. The direct solution of the Riccati equation related to the associated non-stochastic problem is used to find the solution of the problem in feedback form and to write the value function of the problem."}
{"category": "Math", "title": "A Unified Quantum SO(3) Invariant for Rational Homology 3-Spheres", "abstract": "Given a rational homology 3-sphere M with the first integral homology of rank b and a link L inside M, colored by odd numbers, we construct a unified invariant I_{M,L} belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten-Reshetikhin-Turaev invariants of the pair (M,L). If b=1 and L is empty, I_M coincides with Habiro's invariant of integral homology 3-spheres. For b>1, the unified invariant defined by the third author is determined by I_M. One of the applications are the new Ohtsuki series (perturbative expansions of I_M at roots of unity) dominating all quantum SO(3) invariants."}
{"category": "Math", "title": "Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise", "abstract": "We consider two exit problems for the Korteweg-de Vries equation perturbed by an additive white in time and colored in space noise of amplitude a. The initial datum gives rise to a soliton when a=0. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of a^{-2}. We prove exponential upper and lower bounds for the small noise limit of the probability that the exit time from a neighborhood of this randomly modulated soliton is less than T, of the same order in a and T. We obtain that the time scale is exactly the right one. We also study the similar probability for the exit from a neighborhood of the deterministic soliton solution. We are able to quantify the gain of eliminating the secular modes to better describe the persistence of the soliton."}
{"category": "Math", "title": "On finite generation of symbolic Rees rings of space monomial curves and existence of negative curves", "abstract": "In this paper, we shall study finite generation of symbolic Rees rings of the defining ideal of the space monomial curves $(t^a, t^b, t^c)$ for pairwise coprime integers $a$, $b$, $c$ such that $(a,b,c) \\neq (1,1,1)$. If such a ring is not finitely generated over a base field, then it is a counterexample to the Hilbert's fourteenth problem. Finite generation of such rings is deeply related to existence of negative curves on certain normal projective surfaces. We study a sufficient condition (Definition 3.6) for existence of a negative curve. Using it, we prove that, in the case of $(a+b+c)^2 > abc$, a negative curve exists. Using a computer, we shall show that there exist examples in which this sufficient condition is not satisfied."}
{"category": "Math", "title": "Sequences of knots and their limits", "abstract": "Hyperfinite knots, or limits of equivalence classes of knots induced by a knot invariant taking values in a metric space, were introduced in a previous article by the author. In this article, we present new examples of hyperfinite knots stemming from sequences of torus knots."}
{"category": "Math", "title": "A Tribute to Ingram Olkin", "abstract": "It is with pleasure and pride that I introduce this special section in honor of Ingram Olkin. This tribute is especially fitting because, among the many profound and far-reaching contributions that he has made to our profession, Ingram Olkin was the key force behind the genesis of Statistical Science. As put so eloquently by Morrie DeGroot [1], the founding Executive Editor of Statistical Science."}
{"category": "Math", "title": "Classification of certain cellular classes of chain complexes", "abstract": "Let (R, m) be a local commutative ring. Suppose that m is principal and that m^2 = 0. We give a complete description of the cellular lattice of perfect chain complexes of modules over this ring."}
{"category": "Math", "title": "The derivations, central extensions and automorphism group of the Lie algebra $W$", "abstract": "In this paper, we study the derivations, central extensions and the automorphisms of the infinite-dimensional Lie algebra W which appeared in [8] and Dong-Zhang's recent work [22] on the classification of some simple vertex operator algebras."}
{"category": "Math", "title": "On the universal enveloping algebra of a Lie-Rinehart algebra", "abstract": "We review the extent to which the universal enveloping algebra of a Lie-Rinehart algebra resembles a Hopf algebra, and refer to this structure as a Rinehart bialgebra. We then prove a Cartier-Milnor-Moore type theorem for such Rinehart bialgebras."}
{"category": "Math", "title": "The countable Telescope Conjecture for module categories", "abstract": "By the Telescope Conjecture for Module Categories, we mean the following claim: \"Let R be any ring and (A, B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A, B) is of finite type.\" We prove a modification of this conjecture with the word 'finite' replaced by 'countable'. We show that a hereditary cotorsion pair (A, B) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in A in terms of morphisms between finitely presented modules, and show that (A, B) is cogenerated by a single pure-injective module provided that A is closed under direct limits. Then we move our attention to strong analogies between cotorsion pairs in module categories and localizing pairs in compactly generated triangulated categories."}
{"category": "Math", "title": "Special values of L-functions and false Tate curve extensions II", "abstract": "In this paper we show how one can combine the p-adic Rankin-Selberg product construction of Hida with freeness results of Hecke modules of Wiles to establish interesting congruences between special values of L-functions. These congruences is a part of some deep conjectural congruences that follow from the work of Kato on the non-commutative Iwasawa theory of the false Tate curve extension."}
{"category": "Math", "title": "An algebraic characterization of simple closed curves on surfaces with boundary", "abstract": "We characterize in terms of the Goldman Lie algebra which conjugacy classes in the fundamental group of a surface with non empty boundary are represented by simple closed curves. We prove the following: A non power conjugacy class X contains an embedded representative if and only if the Goldman Lie bracket of X with the third power of X is zero. The proof uses combinatorial group theory and Chas' combinatorial description of the bracket recast here in terms of an exposition of the Cohen-Lustig algorithm. Using results of Ivanov, Korkmaz and Luo there are corollaries characterizing which permutations of conjugacy classes are related to diffeomorphisms of the surfaces. The problem is motivated by a group theoretical statement from the sixties equivalent to the Poincare conjecture due to Jaco and Stallings and by a question of Turaev from the eighties. Our main theorem actually counts the minimal possible number of self-intersection points of representatives of a conjugacy class X in terms of the bracket of X with the third power of X."}
{"category": "Math", "title": "A refined version of the Lang-Trotter Conjecture", "abstract": "Let $E$ be an elliptic curve defined over the rational numbers and $r$ a fixed integer. Using a probabilistic model consistent with the Chebotarev theorem for the division fields of $E$ and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to $x$ which have Frobenius trace equal to $r$, where $r$ is a {\\it fixed} integer. However, as shown in this note, this asymptotic estimate cannot hold for {\\it all} $r$ in the interval $|r|\\le 2\\sqrt{x}$ with a uniform bound for the error term, because an estimate of this kind would contradict the Chebotarev density theorem as well as the Sato-Tate conjecture. The purpose of this note is to refine the Lang-Trotter conjecture, by taking into account the \"semicircular law\", to an asymptotic formula that conjecturally holds for arbitrary integers $r$ in the interval $|r|\\le 2\\sqrt{x}$, with a uniform error term. We demonstrate consistency of our refinement with the Chebotarev theorem for a fixed division field, and with the Sato-Tate conjecture. We also present numerical evidence for the refined conjecture."}
{"category": "Math", "title": "Symbolic dynamics for the geodesic flow on Hecke surfaces", "abstract": "In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross section for which the first return map factors through a simple (explicit) map given in terms of the generating map of a particular continued fraction expansion closely related to the Hecke triangle groups. We also obtain explicit expressions for the associated first return times."}
{"category": "Math", "title": "Semisimple torsion in groups of finite Morley rank", "abstract": "We prove several results about groups of finite Morley rank without unipotent p-torsion: p-torsion always occurs inside tori, Sylow p-subgroups are conjugate, and p is not the minimal prime divisor of our approximation to the ``Weyl group.'' These results are quickly finding extensive applications within the classification project."}
{"category": "Math", "title": "A generation theorem for groups of finite Morley rank", "abstract": "We deal with two forms of the \"uniqueness cases\" in the classification of large simple $K^*$-groups of finite Morley rank of odd type, where large means the $m_2(G)$ at least three. This substantially extends results known for even larger groups having \\Prufer 2-rank at least three, to cover the two groups $\\PSp_4$ and $\\G_2$. With an eye towards distant developments, we carry out this analysis for $L^*$-groups which is substantially broader than the $K^*$ setting."}
{"category": "Math", "title": "Linear groups of finite Morley rank", "abstract": "We show that a non-algebraic simple group of finite Morley rank with a definable representation over a field has no involutions, and otherwise resembles a bad group. In particular, the modern form of the Cherlin-Zilber alebaricity conjecture hold for such groups."}
{"category": "Math", "title": "The Hausdorff dimension of the set of dissipative points for a Cantor-like model set for singly cusped parabolic dynamics", "abstract": "In this paper we introduce and study a certain intricate Cantor-like set $C$ contained in unit interval. Our main result is to show that the set $C$ itself, as well as the set of dissipative points within $C$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk."}
{"category": "Math", "title": "Cluster multiplication in regular components via generalized Chebyshev polynomials", "abstract": "We introduce a multivariate generalization of normalized Chebyshev polynomials of the second kind. We prove that these polynomials arise in the context of cluster characters associated to Dynkin quivers of type $\\mathbb A$ and representation-infinite quivers. This allows to obtain a simple combinatorial description of cluster algebras of type $\\mathbb A$. We also provide explicit multiplication formulas for cluster characters associated to regular modules over the path algebra of any representation-infinite quiver."}
{"category": "Math", "title": "Stability conditions and Stokes factors", "abstract": "Let A be the category of modules over a complex, finite-dimensional algebra. We show that the space of stability conditions on A parametrises an isomonodromic family of irregular connections on P^1 with values in the Hall algebra of A. The residues of these connections are given by the holomorphic generating function for counting invariants in A constructed by D. Joyce."}
{"category": "Math", "title": "Multiple disjunction for spaces of Poincare embeddings", "abstract": "We obtain multirelative connectivity statements about spaces of Poincare embeddings, as precursors to analogous statements about spaces of smooth embeddings. The latter are the key to convergence results in the functor calculus approach to spaces of embeddings."}
{"category": "Math", "title": "A Volume Product Representation and its Ramifications in lp,", "abstract": "We represent the volume product for the unit p-ball in a a form free of its gamma symbolism;this will enable us to confirm Mahler's lower bound and Santalo's upper bound by the use of basic only gamma function theory and moderately advanced classical analysis."}
{"category": "Math", "title": "Continuous Dependence for Backward Parabolic Operators with Log-Lipschitz Coefficients", "abstract": "We prove continuous dependence on Cauchy data for a backward parabolic operator whose coefficients are Log-Lipschitz continuous in time."}
{"category": "Math", "title": "Elliptic fibrations and symplectic automorphisms on K3 surfaces", "abstract": "Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice $U^3\\oplus E_8(-1)^2$ depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces."}
{"category": "Math", "title": "Cox rings and combinatorics II", "abstract": "We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to modifications, e.g., blow ups, and the question how the Cox ring changes under such maps. We answer this question for a certain class of modifications induced from modifications of ambient toric varieties. Moreover, we show that every variety with finitely generated Cox ring can be explicitly constructed in a finite series of toric ambient modifications from a combinatorially minimal one."}
{"category": "Math", "title": "Automorphism groups of generalized Reed-Solomon codes", "abstract": "We look at AG codes associated to the projective line, re-examining the problem of determining their automorphism groups (originally investigated by Duer in 1987 using combinatorial techniques) using recent methods from algebraic geometry. We (re)classify those finite groups that can arise as the automorphism group of an AG code for the projective line and give an explicit description of how these groups appear. We also give examples of generalized Reed-Solomon codes with large automorphism groups G, such as G=PSL(2,q), and explicitly describe their G-module structure."}
{"category": "Math", "title": "A computation in Khovanov-Rozansky Homology", "abstract": "We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a \"local\" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the \"foam\" version of sl_3-homology."}
{"category": "Math", "title": "Higher-dimensional linking integrals", "abstract": "We derive an integral formula for the linking number of two submanifolds of the n-sphere S^n, of the product S^n x R^m, and of other manifolds which appear as \"nice\" hypersurfaces in Euclidean space. The formulas are geometrically meaningful in that they are invariant under the action of the special orthogonal group on the ambient space."}
{"category": "Math", "title": "Quantization of Projective Homogeneous Spaces and Duality Principle", "abstract": "We introduce a general recipe to construct quantum projective homogeneous spaces, with a particular interest for the examples of the quantum Grassmannians and the quantum generalized flag varieties. Using this construction, we extend the quantum duality principle to quantum projective homogeneous spaces."}
{"category": "Math", "title": "On the infimum convolution inequality", "abstract": "In the paper we study the infimum convolution inequalites. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC-inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure. In particular, we show the optimal IC-inequality for product log-concave measures and for uniform measures on the l_p^n balls. Such an optimal inequality implies, for a given measure, in particular the Central Limit Theorem of Klartag and the tail estimates of Paouris."}
{"category": "Math", "title": "Decompositions and statistics for beta(1,0)-trees and nonseparable permutations", "abstract": "The subject of pattern avoiding permutations has its roots in computer science, namely in the problem of sorting a permutation through a stack. A formula for the number of permutations of length n that can be sorted by passing it twice through a stack (where the letters on the stack have to be in increasing order) was conjectured by West, and later proved by Zeilberger. Goulden and West found a bijection from such permutations to nonseparable planar maps, and later, Jacquard and Schaeffer presented a bijection from these planar maps to certain labeled plane trees, called beta(1,0)-trees. Using generating trees, Dulucq, Gire and West showed that nonseparable planar maps are equinumerous with permutations avoiding the (classical) pattern 2413 and the barred pattern 41\\bar{3}52; they called these permutations nonseparable. We give a new bijection between beta(1,0)-trees and permutations avoiding the dashed patterns 3-1-4-2 and 2-41-3. These permutations can be seen to be exactly the reverse of nonseparable permutations. Our bijection is built using decompositions of the permutations and the trees, and it translates seven statistics on the trees into statistics on the permutations. Among the statistics involved are ascents, left-to-right minima and right-to-left maxima for the permutations, and leaves and the rightmost and leftmost paths for the trees. In connection with this we give a nontrivial involution on the beta(1,0)-trees, which specializes to an involution on unlabeled rooted plane trees, where it yields interesting results. Lastly, we conjecture the existence of a bijection between nonseparable permutations and two-stack sortable permutations preserving at least four permutation statistics."}
{"category": "Math", "title": "Carry Propagation in Multiplication by Constants", "abstract": "Suppose that a random n-bit number V is multiplied by an odd constant M, greater than or equal to 3, by adding shifted versions of the number V corresponding to the 1s in the binary representation of the constant M. Suppose further that the additions are performed by carry-save adders until the number of summands is reduced to two, at which time the final addition is performed by a carry-propagate adder. We show that in this situation the distribution of the length of the longest carry-propagation chain in the final addition is the same (up to terms tending to 0 as n tends to infinity) as when two independent n-bit numbers are added, and in particular the mean and variance are the same (again up to terms tending to 0). This result applies to all possible orders of performing the carry-save additions."}
{"category": "Math", "title": "On the solvability of systems of pseudodifferential operators", "abstract": "The paper studies the solvability for square systems of pseudodifferential operators. We assume that the system is of principal type, i.e., the principal symbol vanishes of first order on the kernel. We shall also assume that the eigenvalues of the principal symbol close to zero have constant multiplicity. We prove that local solvability for the system is equivalent to condition (PSI) on the eigenvalues of the principal symbol. This condition rules out any sign changes from - to + of the imaginary part of the eigenvalue when going in the positive direction on the bicharacteristics of the real part. Thus we need no conditions on the lower order terms. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of 3/2 derivatives (compared with the elliptic case)."}
{"category": "Math", "title": "About the logic of the prime number distribution", "abstract": "There are two basic number sequences which play a major role in the prime number distribution. The first Number Sequence SQ1 contains all prime numbers of the form 6n+5 and the second Number Sequence SQ2 contains all prime numbers of the form 6n+1. All existing prime numbers seem to be contained in these two number sequences, except of the prime numbers 2 and 3. Riemanns Zeta Function also seems to indicate, that there is a logical connection between the mentioned number sequences and the distribution of prime numbers. This connection is indicated by lines in the diagram of the Zeta Function, which are formed by the points s where the Zeta Function is real. Another key role in the distribution of the prime numbers plays the number 5 and its periodic occurrence in the two number sequences SQ1 and SQ2. All non-prime numbers in SQ1 and SQ2 are caused by recurrences of these two number sequences with increasing wave-lengths in themselves, in a similar fashion as Overtones (harmonics) or Undertones derive from a fundamental frequency. On the contrary prime numbers represent spots in these two basic Number Sequences SQ1 and SQ2 where there is no interference caused by these recurring number sequences. The distribution of the non-prime numbers and prime numbers can be described in a graphical way with a -Wave Model- (or Interference Model) -- see Table 2."}
{"category": "Math", "title": "Weak Hopf monoids in braided monoidal categories", "abstract": "We develop the theory of weak bimonoids in braided monoidal categories and show them to be quantum categories in a certain sense. Weak Hopf monoids are shown to be quantum groupoids. Each separable Frobenius monoid R leads to a weak Hopf monoid R \\otimes R."}
{"category": "Math", "title": "The morphology of infinite tournaments. Application to the growth of their profile", "abstract": "A tournament is \\emph{acyclically indecomposable} if no acyclic autonomous set of vertices has more than one element. We identify twelve infinite acyclically indecomposable tournaments and prove that every infinite acyclically indecomposable tournament contains a subtournament isomorphic to one of these tournaments. The {\\it profile} of a tournament $T$ is the function $\\phi_T$ which counts for each integer $n$ the number $\\phi_T(n)$ of tournaments induced by $T$ on the $n$-element subsets of $T$, isomorphic tournaments being identified. As a corollary of the result above we deduce that the growth of $\\phi_T$ is either polynomial, in which case $\\phi_T(n)\\simeq an^k$, for some positive real $a$, some non-negative integer $k$, or as fast as some exponential."}
{"category": "Math", "title": "A Simple Solution to a Major Problem: Proof of the Riemann Hypothesis", "abstract": "Starting from the symmetrical reflection functional equation of the zeta function, we have found that the sigma values satisfying zeta(s) = 0 must also satisfy both |zeta(s)| = |zeta(1 - s)| and |gamma(s/2)zeta(s)| = |gamma((1 - s)/2)zeta(1 - s)|. We have shown that sigma = 1/2 is the only numeric solution that satisfies this requirement."}
{"category": "Math", "title": "Exceptional symmetric domains", "abstract": "We give the presentation of exceptional bounded symmetric domains using the Albert algebra and exceptional Jordan triple systems. The first chapter is devoted to Cayley-Graves algebras, the second to exceptional Jordan triple systems. In the third chapter, we give a geometric description of the two exceptional bounded symmetric domains, their boundaries and their compactification."}
{"category": "Math", "title": "Local estimates and global continuities in Lebesgue spaces for bilinear operators", "abstract": "In this paper, we first prove some local estimates for bilinear operators (closely related to the bilinear Hilbert transform and similar singular operators) with truncated symbol. Such estimates, in accordance with the Heisenberg uncertainty principle correspond to a description of ``off-diagonal'' decay. In addition they allow us to prove global continuities in Lebesgue spaces for bilinear operators with spatial dependent symbol."}
{"category": "Math", "title": "A compactly supported formula for equivariant localization, and, simplicial complexes of Bialynicki-Birula decompositions", "abstract": "Let X be a projective scheme carrying a circle action S with isolated fixed points. We associate a simplicial complex Delta(X,S) of \"closure chains\" using a refinement of its Morse/Bialynicki-Birula decomposition. If this decomposition is a stratification (e.g. when X is a flag manifold), then Delta(X,S) is just the order complex of the poset of fixed points. For X a toric variety, Delta(X,S) is a triangulation of the moment polytope. We compute some other examples, including a Bott-Samelson manifold and the punctual Hilbert scheme of 4 points in the plane. Summing over the facets of Delta(X,S), we obtain a positive formula for the Duistermaat-Heckman measure on the moment polytope of X, defined for any torus action extending S. We explain how, through brutal use of partial fractions, this can be extended to an AB/BV-type formula for integrating general classes. Throughout we work with equivariant Chow groups, and do not make any smoothness requirements on X."}
{"category": "Math", "title": "A sampling inequality for fractional order Sobolev semi-norms using arbitrary order data", "abstract": "To improve convergence results obtained using a framework for unsymmetric meshless methods due to Schaback (Preprint G\\\"ottingen 2006), we extend, in two directions, the Sobolev bound due to Arcang\\'eli et al. (Numer Math 107, 181-211, 2007), which itself extends two others due to Wendland and Rieger (Numer Math 101, 643-662, 2005) and Madych (J. Approx Theory 142, 116-128, 2006). The first is to incorporate discrete samples of arbitrary order derivatives into the bound, which are used to obtain higher order convergence in higher order Sobolev norms. The second is to optimally bound fractional order Sobolev semi-norms, which are used to obtain more optimal convergence rates when solving problems requiring fractional order Sobolev spaces, notably inhomogeneous boundary value problems."}
{"category": "Math", "title": "Extensions of Lie-Rinehart algebras and cotangent bundle reduction", "abstract": "Let Q denote a smooth manifold acted upon smoothly by a Lie group G. The G-action lifts to an action on the total space T of the cotangent bundle of Q and hence on the standard symplectic Poisson algebra of smooth functions on T. The Poisson algebra of G-invariant functions on T yields a Poisson structure on the space T/G of G-orbits. We relate this Poisson algebra with extensions of Lie-Rinehart algebras and derive an explicit formula for this Poisson structure in terms of differentials. We then show, for the particular case where the G-action on Q is principal, how an explicit description of the Poisson algebra derived in the literature by an ad hoc construction is essentially a special case of the formula for the corresponding extension of Lie-Rinehart algebras. By means of various examples, we also show that this kind of description breaks down when the G-action does not define a principal bundle."}
{"category": "Math", "title": "On a formula for the spectral flow and its applications", "abstract": "We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index."}
{"category": "Math", "title": "Note on Frobenius monoidal functors", "abstract": "It is well known that strong monoidal functors preserve duals. In this short note we show that a slightly weaker version of functor, which we call \"Frobenius monoidal\", is sufficient."}
{"category": "Math", "title": "Geometric Properties of Assur Graphs", "abstract": "In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks - Assur graphs - which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static self-stresses. We provide a new geometric characterization of Assur graphs, based on special singular realizations. These singular positions are then related to dead-end positions in which an associated mechanism with an inserted driver will stop or jam."}
{"category": "Math", "title": "Schubert patches degenerate to subword complexes", "abstract": "We study the intersections of general Schubert varieties X_w with permuted big cells, and give an inductive degeneration of each such \"Schubert patch\" to a Stanley-Reisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using the results of [Knutson 2007] on automatically reduced degenerations, or through more standard cohomology-vanishing arguments. The underlying simplicial complex of the Stanley-Reisner scheme is a subword complex, as introduced for slightly different purposes in [Knutson-Miller 2004], and is homeomorphic to a ball. This gives a new proof of the Andersen-Jantzen-Soergel/Billey and Graham/Willems formulae for restrictions of equivariant Schubert classes to fixed points."}
{"category": "Math", "title": "A note on the Compound Burgers-Korteweg-de Vries Equation with higher-order nonlinearities and its traveling solitary waves", "abstract": "In this paper, we study a compound Korteweg-de Vries-Burgers equation with a higher-order nonlinearity. A class of solitary wave solutions is obtained by means of a series expansion."}
{"category": "Math", "title": "Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram", "abstract": "The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from the Satake diagram, in a way that is suited for the use with computer algebra systems. As an example application, the totally geodesic submanifolds of the Riemannian symmetric space SU(3)/SO(3) are classified. The submission also contains an example implementation of the algorithms and formulas of the paper as a package for Maple 10, the technical documentation for this implementation, and a worksheet carrying out the computations for the space SU(3)/SO(3) used in the proof of Proposition 6.1 of the paper."}
{"category": "Math", "title": "Causality, Modality and Explanation", "abstract": "We present a sequent calculus system for a modal reformulation of a system of nonmonotonic logic due to McCain and Turner: we prove cut elimination for our system. The proof system is in general infinitary: because we can prove cut elimination, many applications need, in practice, only the application of finitary rules. Consequently, nonmonotonic logic is, in many cases, much less scary than it might seem to be a priori. We derive from this a critique of Fodor's philosophical worries about the nonmonotonicity of human reasoning."}
{"category": "Math", "title": "Characteristic cycles of standard modules for the rational Cherednik algebra of type Z/lZ", "abstract": "We study the representation theory of the rational Cherednik algebra $H_\\kappa = H_\\kappa({\\mathbb Z}_l)$ for the cyclic group ${\\mathbb Z}_l = {\\mathbb Z} / l {\\mathbb Z}$ and its connection with the geometry of the quiver variety $M_\\theta(\\delta)$ of type $A_{l-1}^{(1)}$. We consider a functor between the categories of $H_\\kappa$-modules with different parameters, called the shift functor, and give the condition when it is an equivalence of categories. We also consider a functor from the category of $H_\\kappa$-modules with good filtration to the category of coherent sheaves on $M_\\theta(\\delta)$. We prove that the image of the regular representation of $H_\\kappa$ by this functor is the tautological bundle on $M_\\theta(\\delta)$. As a corollary, we determine the characteristic cycles of the standard modules. It gives an affirmative answer to a conjecture given in [Gordon, arXiv:math/0703150v1] in the case of ${\\mathbb Z}_l$."}
{"category": "Math", "title": "Continuous biorthogonality of the elliptic hypergeometric function", "abstract": "We construct a family of continuous biorthogonal functions related to an elliptic analogue of the Gauss hypergeometric function. The key tools used for that are the elliptic beta integral and the integral Bailey chain introduced earlier by the author. Relations to the Sklyanin algebra and elliptic analogues of the Faddeev modular double are discussed in detail."}
{"category": "Math", "title": "Plane Jacobian problem for rational polynomials", "abstract": "This paper has been withdrawn by the author due to a crucial error in the last lines in the proof of Lemma 3.3."}
{"category": "Math", "title": "Divergence in lattices in semisimple Lie groups and graphs of groups", "abstract": "Divergence functions of a metric space estimate the length of a path connecting two points $A$, $B$ at distance $\\le n$ avoiding a large enough ball around a third point $C$. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. By Olshanskii-Osin-Sapir, that property is weaker than the property of having Morse (rank 1) quasi-geodesics. Using our characterization of Morse quasi-geodesics, we give a new proof of the theorem of Farb-Kaimanovich-Masur that states that mapping class groups cannot contain copies of irreducible lattices in semi-simple Lie groups of higher ranks. It also gives a generalization of the result of Birman-Lubotzky-McCarthy about solvable subgroups of mapping class groups not covered by the Tits alternative of Ivanov and McCarthy. We show that any group acting acylindrically on a simplicial tree or a locally compact hyperbolic graph always has \"many\" periodic Morse quasi-geodesics (i.e. Morse elements), so its divergence functions are never linear. We also show that the same result holds in many cases when the hyperbolic graph satisfies Bowditch's properties that are weaker than local compactness. This gives a new proof of Behrstock's result that every pseudo-Anosov element in a mapping class group is Morse. On the other hand, we conjecture that lattices in semi-simple Lie groups of higher rank always have linear divergence. We prove it in the case when the $\\mathbb{Q}$-rank is 1 and when the lattice is $SL_n(\\mathcal{O}_S)$ where $n\\ge 3$, $S$ is a finite set of valuations of a number field $K$ including all infinite valuations, and $\\mathcal{O}_S$ is the corresponding ring of $S$-integers."}
{"category": "Math", "title": "Lie bialgebra structures on the $W$-algebra W(2,2)", "abstract": "Verma modules over the $W$-algebra W(2,2) were considered by Zhang and Dong, while the Harish-Chandra modules and irreducible weight modules over the same algebra were classified by Liu and Zhu etc. In the present paper we shall investigate the Lie bialgebra structures on the referred algebra, which are shown to be triangular coboundary."}
{"category": "Math", "title": "From Laplacian Transport to Dirichlet-to-Neumann (Gibbs) Semigroups", "abstract": "The paper gives a short account of some basic properties of \\textit{Dirichlet-to-Neumann} operators $\\Lambda_{\\gamma,\\partial\\Omega}$ including the corresponding semigroups motivated by the Laplacian transport in anisotropic media ($\\gamma \\neq I$) and by elliptic systems with dynamical boundary conditions. For illustration of these notions and the properties we use the explicitly constructed \\textit{Lax semigroups}. We demonstrate that for a general smooth bounded convex domain $\\Omega \\subset \\mathbb{R}^d$ the corresponding {Dirichlet-to-Neumann} semigroup $\\left\\{U(t):= e^{-t \\Lambda_{\\gamma,\\partial\\Omega}}\\right\\}_{t\\geq0}$ in the Hilbert space $L^2(\\partial \\Omega)$ belongs to the \\textit{trace-norm} von Neumann-Schatten ideal for any $t>0$. This means that it is in fact an \\textit{immediate Gibbs} semigroup. Recently Emamirad and Laadnani have constructed a \\textit{Trotter-Kato-Chernoff} product-type approximating family $\\left\\{(V_{\\gamma, \\partial\\Omega}(t/n))^n \\right\\}_{n \\geq 1}$ \\textit{strongly} converging to the semigroup $U(t)$ for $n\\to\\infty$. We conclude the paper by discussion of a conjecture about convergence of the \\textit{Emamirad-Laadnani approximantes} in the the {\\textit{trace-norm}} topology."}
{"category": "Math", "title": "Goodness of fit test for small diffusions by discrete observations", "abstract": "We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional small diffusions. Our test is based on discrete observation of the processes, and the diffusion coefficient is a nuisance function which is estimated in our testing procedure. We prove that the limit distribution of our test is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free. We also show that our test is consistent under any fixed alternatives."}
{"category": "Math", "title": "Critical percolation of virtually free groups and other tree-like graphs", "abstract": "This article presents a method for finding the critical probability $p_c$ for the Bernoulli bond percolation on graphs with the so-called tree-like structure. Such a graph can be decomposed into a tree of pieces, each of which has finitely many isomorphism classes. This class of graphs includes the Cayley graphs of amalgamated products, HNN extensions or general groups acting on trees. It also includes all transitive graphs with more than one end. The idea of the method is to find a multi-type Galton--Watson branching process (with a parameter $p$) which has finite expected population size if and only if the expected percolation cluster size is finite. This provides sufficient information about $p_c$. In particular, if the pairwise intersections of pieces are finite, then $p_c$ is the smallest positive $p$ such that $\\operatorname {det}(M-1)=0$, where $M$ is the first-moment matrix of the branching process. If the pieces of the tree-like structure are finite, then $p_c$ is an algebraic number and we give an algorithm computing $p_c$ as a root of some algebraic function. We show that any Cayley graph of a virtually free group (i.e., a group acting on a tree with finite vertex stabilizers) with respect to any finite generating set has a tree-like structure with finite pieces. In particular, we show how to compute $p_c$ for the Cayley graph of a free group with respect to any finite generating set."}
{"category": "Math", "title": "From combinatorics to large deviations for the invariant measures of some multiclass particle systems", "abstract": "We prove large deviation principles (LDP) for the invariant measures of the multiclass totally asymmetric simple exclusion process (TASEP) and the multiclass Hammersely-Aldous-Diaconis (HAD) process on a torus. The proof is based on a combinatorial representation of the measures in terms of a \\emph{collapsing procedure} introduced in \\cite{A} for the 2-class TASEP and then generalized in \\cite{FM1}, \\cite{FM2} and \\cite{FM3} to the multiclass TASEP and the multiclass HAD process. The rate functionals are written in terms of variational problems that we solve in the cases of 2-class processes."}
{"category": "Math", "title": "Distribution of twisted Kloosterman sums modulo prime powers", "abstract": "In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure."}
{"category": "Math", "title": "Computational aspects and applications of a new transform for solving the complex exponentials approximation problem", "abstract": "Many real life problems can be reduced to the solution of a complex exponentials approximation problem which is usually ill posed. Recently a new transform for solving this problem, formulated as a specific moments problem in the plane, has been proposed in a theoretical framework. In this work some computational issues are addressed to make this new tool useful in practice. An algorithm is developed and used to solve a Nuclear Magnetic Resonance spectrometry problem, two time series interpolation and extrapolation problems and a shape from moments problem."}
{"category": "Math", "title": "Sobolev of the Euler School", "abstract": "This is a short overview of the origins of distribution theory as well as the life of Sergei Sobolev (1908--1989) and his contribution to the formation of the modern outlook of mathematics."}
{"category": "Math", "title": "Lectures on Stability and Constant Scalar Curvature", "abstract": "An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite-dimensional GIT, and conditions on the closure of orbits of almost-complex structures under the diffeomorphism group. Related analytic methods are also discussed, including estimates for energy functionals, Tian-Yau-Zelditch approximations, estimates for moment maps, complex Monge-Ampere equations and pluripotential theory, and the Kaehler-Ricci flow"}
{"category": "Math", "title": "Symmetric and Asymptotically Symmetric Permutations", "abstract": "We consider two related problems arising from a question of R. Graham on quasirandom phenomena in permutation patterns. A ``pattern'' in a permutation $\\sigma$ is the order type of the restriction of $\\sigma : [n] \\to [n]$ to a subset $S \\subset [n]$. First, is it possible for the pattern counts in a permutation to be exactly equal to their expected values under a uniform distribution? Attempts to address this question lead naturally to an interesting number theoretic problem: when does $k!$ divide $\\binom{n}{k}$? Second, if the tensor product of a permutation with large random permutations is random-like in its pattern counts, what must the pattern counts of the original permutation be? A recursive formula is proved which uses a certain permutation ``contraction.''"}
{"category": "Math", "title": "On the inheriting of the property $C_\\pi$ by some normal subgroups", "abstract": "In the paper we prove that the Hall property $C_\\pi$ is inherited by normal subgroups which index is a $\\pi'$-number."}
{"category": "Math", "title": "Multivariate Meta-Analysis: Contributions of Ingram Olkin", "abstract": "The research on meta-analysis and particularly multivariate meta-analysis has been greatly influenced by the work of Ingram Olkin. This paper documents Olkin's contributions by way of citation counts and outlines several areas of contribution by Olkin and his academic descendants. An academic family tree is provided."}
{"category": "Math", "title": "Experiments with moduli of quadrilaterals II", "abstract": "The numerical performance of the AFEM method of K. Samuelsson is studied in the computation of moduli of quadrilaterals."}
{"category": "Math", "title": "Classification of irreducible Harish-Chandra modules over the loop-Virasoro algebra", "abstract": "The loop-Virasoro algebra is the Lie algebra of the tensor product of the Virasoro algebra and the Laurent polynomial algebra. This paper classifies irreducible Harish-Chandra modules over the loop-Virasoro algebra, which turn out to be highest weight modules, lowest weight modules and evaluation modules of the intermediate series (all wight spaces are 1-dimensional). As a by-product, we obtain a classification of irreducible Harish-Chandra modules over truncated Virasoro algebras. We also determine the necessary and sufficient conditions for highest weigh irreducible modules over the loop-Virasoro algebra to have all finite dimensional weight spaces, as well as the necessary and sufficient conditions for highest weigh Verma modules to be irreducible."}
{"category": "Math", "title": "Johnson's homomorphisms and the Arakelov-Green function", "abstract": "Let $\\pi: {\\mathbb C}_g \\to {\\mathbb M}_g$ be the universal family of compact Riemann surfaces of genus $g \\geq 1$. We introduce a real-valued function on the moduli space ${\\mathbb M}_g$ and compute the first and the second variations of the function. As a consequence we relate the Chern form of the relative tangent bundle $T_{{\\mathbb C}_g/{\\mathbb M}_g}$ induced by the Arakelov-Green function with differential forms on ${\\mathbb C}_g$ induced by a flat connection whose holonomy gives Johnson's homomorphisms on the mapping class group."}
{"category": "Math", "title": "Majorization: Here, There and Everywhere", "abstract": "The appearance of Marshall and Olkin's 1979 book on inequalities with special emphasis on majorization generated a surge of interest in potential applications of majorization and Schur convexity in a broad spectrum of fields. After 25 years this continues to be the case. The present article presents a sampling of the diverse areas in which majorization has been found to be useful in the past 25 years."}
{"category": "Math", "title": "Generalization of Jeffreys' divergence based priors for Bayesian hypothesis testing", "abstract": "In this paper we introduce objective proper prior distributions for hypothesis testing and model selection based on measures of divergence between the competing models; we call them divergence based (DB) priors. DB priors have simple forms and desirable properties, like information (finite sample) consistency; often, they are similar to other existing proposals like the intrinsic priors; moreover, in normal linear models scenarios, they exactly reproduce Jeffreys-Zellner-Siow priors. Most importantly, in challenging scenarios such as irregular models and mixture models, the DB priors are well defined and very reasonable, while alternative proposals are not. We derive approximations to the DB priors as well as MCMC and asymptotic expressions for the associated Bayes factors."}
{"category": "Math", "title": "A permutation model for free random variables and its classical analogue", "abstract": "In this paper, we generalize a permutation model for free random variables which was first proposed by Biane in \\cite{biane}. We also construct its classical probability analogue, by replacing the group of permutations with the group of subsets of a finite set endowed with the symmetric difference operation. These constructions provide new discrete approximations of the respective free and classical Wiener chaos. As a consequence, we obtain explicit examples of non random matrices which are asymptotically free or independent. The moments and the free (resp. classical) cumulants of the limiting distributions are expressed in terms of a special subset of (noncrossing) pairings. At the end of the paper we present some combinatorial applications of our results."}
{"category": "Math", "title": "Solvable Subgroups of Locally Compact Groups", "abstract": "It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry."}
{"category": "Math", "title": "Geometric approach to Ending Lamination Conjecture", "abstract": "We present a new proof of the bi-Lipschitz model theorem, which occupies the main part of the Ending Lamination Conjecture proved by Minsky and Brock-Canary-Minsky. Our proof is done by using techniques of standard hyperbolic geometry as much as possible."}
{"category": "Math", "title": "On instability of excited states of the nonlinear Schr\\\"odinger equation", "abstract": "We introduce a new notion of linear stability for standing waves of the nonlinear Schr\\\"odinger equation (NLS) which requires not only that the spectrum of the linearization be real, but also that the generalized kernel be not degenerate and that the signature of all the positive eigenvalues be positive. We prove that excited states of the NLS are not linearly stable in this more restrictive sense. We then give a partial proof that this more restrictive notion of linear stability is a necessary condition to have orbital stability."}
{"category": "Math", "title": "Analyticity and propagation of plurisubharmonic singularities", "abstract": "A variant of Siu's analyticity theorem is proved for relative types of plurisubharmonic functions. Some results on propagation of plurisubharmonic singularities and maximality of pluricomplex Green functions with analytic singularities are derived."}
{"category": "Math", "title": "Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models", "abstract": "We consider the linear dissipative Boltzmann equation describing inelastic interactions of particles with a fixed background. For the simplified model of Maxwell molecules first, we give a complete spectral analysis, and deduce from it the optimal rate of exponential convergence to equilibrium. Moreover we show the convergence to the heat equation in the diffusive limit and compute explicitely the diffusivity. Then for the physical model of hard spheres we use a suitable entropy functional for which we prove explicit inequality between the relative entropy and the production of entropy to get exponential convergence to equilibrium with explicit rate. The proof is based on inequalities between the entropy production functional for hard spheres and Maxwell molecules. Mathematical proof of the convergence to some heat equation in the diffusive limit is also given. From the last two points we deduce the first explicit estimates on the diffusive coefficient in the Fick's law for (inelastic hard-spheres) dissipative gases."}
{"category": "Math", "title": "A Gr\\\"obner basis proof of the flat extension theorem for moment matrices", "abstract": "This paper has been withdrawn by the author since $U$ in Lemma 2 is in general not a subspace."}
{"category": "Math", "title": "On non-formality of a simply-connected symplectic 8-manifold", "abstract": "We show an alternative construction of the first example of a simply-connected compact symplectic non-formal 8-manifold given in arXiv:math/0506449. We also give an alternative proof of its non-formality using higher order Massey products."}
{"category": "Math", "title": "Inclusions and positive cones of von Neumann algebras", "abstract": "We consider cones in a Hilbert space associated to two von Neumann algebras and determine when one algebra is included in the other. If a cone is assocated to a von Neumann algebra, the Jordan structure is naturally recovered from it and we can characterize projections of the given von Neumann algebra with the structure in some special situations."}
{"category": "Math", "title": "Lower bounds for transition probabilities on graphs", "abstract": "The paper presents two results. The first one provides separate conditions for the upper and lower estimate of the distribution of the exit time from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk."}
{"category": "Math", "title": "A.-M. Guerry's Moral Statistics of France: Challenges for Multivariable Spatial Analysis", "abstract": "Andr\\'{e}-Michel Guerry's (1833) Essai sur la Statistique Morale de la France was one of the foundation studies of modern social science. Guerry assembled data on crimes, suicides, literacy and other ``moral statistics,'' and used tables and maps to analyze a variety of social issues in perhaps the first comprehensive study relating such variables. Indeed, the Essai may be considered the book that launched modern empirical social science, for the questions raised and the methods Guerry developed to try to answer them. Guerry's data consist of a large number of variables recorded for each of the d\\'{e}partments of France in the 1820--1830s and therefore involve both multivariate and geographical aspects. In addition to historical interest, these data provide the opportunity to ask how modern methods of statistics, graphics, thematic cartography and geovisualization can shed further light on the questions he raised. We present a variety of methods attempting to address Guerry's challenge for multivariate spatial statistics."}
{"category": "Math", "title": "Iterates of the Schur class operator-valued function and their conservative realizations", "abstract": "Let $\\mathfrak M$ and $\\mathfrak N$ be separable Hilbert spaces and let $\\Theta(\\lambda)$ be a function from the Schur class ${\\bf S}(\\mathfrak M,\\mathfrak N)$ of contractive functions holomorphic on the unit disk. The operator generalization of the classical Schur algorithm associates with $\\Theta$ the sequence of contractions (the Schur parameters of $\\Theta$) $\\Gamma_0=\\Theta(0)\\in \\bL(\\sM,\\sN), \\Gamma_n\\in\\bL(\\sD_{\\Gamma_{n-1}}, \\sD_{\\Gamma^*_{n-1}}) $ and the sequence of functions $\\Theta_0 = \\Theta$, $\\Theta_n\\in {\\bf S}(\\sD_{\\Gamma_n},\\sD_{\\Gamma^*_n})$ $ n=1,...$ (the Schur iterares of $\\Theta$) connected by the relations \\[ \\Gamma_n=\\Theta_n(0), \\Theta_n(\\lambda) = \\Gamma_n+\\lambda D_{\\Gamma^*_n} \\Theta_{n+1}(\\lambda) (I + \\lambda\\Gamma^*_n\\Theta_{n+1} (\\lambda))^{-1}D_{\\Gamma_n}, |\\lambda|<1. \\] The function $\\Theta(\\lambda)\\in {\\bf S}(\\sM,\\sN)$ can be realized as the transfer function \\[ \\Theta(\\lambda)=D+\\lambda C(I-\\lambda A)^{-1}B \\] of a linear conservative and simple discrete-time system $\\tau = {\\begin{bmatrix}D & C \\cr B & A\\end{bmatrix}; \\mathfrak M, \\mathfrak N,\\mathfrak H}$ with the state space $\\mathfrak H$ and the input and output spaces $\\mathfrak M$ and $\\mathfrak N $, respectively. In this paper we give a construction of conservative and simple realizations of the Schur iterates $\\Theta_n$ by means of the conservative and simple realization of $\\Theta$."}
{"category": "Math", "title": "Surfaces with Many Solitary Points", "abstract": "It is classically known that a real cubic surface in the real projective 3-space cannot have more than one solitary point (locally given by x^2+y^2+z^2=0) whereas it can have up to four nodes (x^2+y^2-z^2=0). We show that on any surface of degree at least 3 in the real projective 3-space, the maximum possible number of solitary points is strictly smaller than the maximum possible number of nodes. Conversely, we adapt a construction of Chmutov to obtain surfaces with many solitary points by using a refined version of Brusotti's theorem. Finally, we adapt this construction to get real algebraic surfaces with many singular points of type $A_{2k-1}^\\smbullet$ for all $k\\ge 1$."}
{"category": "Math", "title": "A general necessary and sufficient optimality conditions for singular control problems", "abstract": "We consider a stochastic control problem where the set of strict (classical) controls is not necessarily convex and the the variable control has two components, the first being absolutely continuous and the second singular. The system is governed by a nonlinear stochastic differential equation, in which the absolutely continuous component of the control enters both the drift and the diffusion coefficients. By introducing a new approach, we establish necessary and sufficient optimality conditions for two models. The first concerns the relaxed-singular controls, who are a pair of processes whose first component is a measure-valued processes. The second is a particular case of the first and relates to strict-singular control problems. These results are given in the form of global stochastic maximum principle by using only the first order expansion and the associated adjoint equation. This improves and generalizes all the previous works on the maximum principle of controlled stochastic differential equations."}
{"category": "Math", "title": "Complete intersections on general hypersurfaces", "abstract": "We ask when certain complete intersections of codimension $r$ can lie on a generic hypersurface in $\\PP^n$. We give a complete answer to this question when $2r \\leq n+2$ in terms of the degrees of the hypersurfaces and of the degrees of the generators of the complete intersection."}
{"category": "Math", "title": "A geometric Schur-Weyl duality for quotients of affine Hecke algebras", "abstract": "After establishing a geometric Schur-Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases."}
{"category": "Math", "title": "On the naturality of the exterior differential", "abstract": "We give sufficient conditions for the naturallity of the exterior differential under Sobolev mappings. In other words we study the validity of the equation $d f^* \\alpha = f^* d\\alpha$ for a smooth form $\\alpha$ and a Sobolev map $f$."}
{"category": "Math", "title": "Continuous first order logic and local stability", "abstract": "We develop continuous first order logic, a variant of the logic described in \\cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson's logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited."}
{"category": "Math", "title": "An Improved Construction of Progression-Free Sets", "abstract": "The problem of constructing dense subsets S of {1,2,..,n} that contain no arithmetic triple was introduced by Erdos and Turan in 1936. They have presented a construction with |S| = \\Omega(n^{\\log_3 2}) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is |S| = Omega({n \\over {2^{2 \\sqrt{2} \\sqrt{\\log_2 n}} \\cdot \\log^{1/4} n}}). Since then the problem became one of the most central, most fundamental, and most intensively studied problems in additive number theory. Nevertheless, no improvement of the lower bound of Behrend was reported since 1946. In this paper we present a construction that improves the result of Behrend by a factor of Theta(\\sqrt{\\log n}), and shows that |S| = Omega({n \\over {2^{2 \\sqrt{2} \\sqrt{\\log_2 n}}}} \\cdot \\log^{1/4} n). In particular, our result implies that the construction of Behrend is not optimal. Our construction and proof are elementary and self-contained."}
{"category": "Math", "title": "Two finiteness theorem for $(a,b)$-module", "abstract": "We prove the following two results 1. For a proper holomorphic function $ f : X \\to D$ of a complex manifold $X$ on a disc such that $\\{df = 0 \\} \\subset f^{-1}(0)$, we construct, in a functorial way, for each integer $p$, a geometric (a,b)-module $E^p$ \\ associated to the (filtered) Gauss-Manin connexion of $f$. This first theorem is an existence/finiteness result which shows that geometric (a,b)-modules may be used in global situations. 2. For any regular (a,b)-module $E$ we give an integer $N(E)$, explicitely given from simple invariants of $E$, such that the isomorphism class of $E\\big/b^{N(E)}.E$ determines the isomorphism class of $E$. This second result allows to cut asymptotic expansions (in powers of $b$) \\ of elements of $E$ without loosing any information."}
{"category": "Math", "title": "A general stochastic maximum principle for optimal control problems of forward-backward systems", "abstract": "Stochastic maximum principle of nonlinear controlled forward-backward systems, where the set of strict (classical) controls need not be convex and the diffusion coefficient depends explicitly on the variable control, is an open problem impossible to solve by the classical method of spike variation. In this paper, we introduce a new approach to solve this open problem and we establish necessary as well as sufficient conditions of optimality, in the form of global stochastic maximum principle, for two models. The first concerns the relaxed controls, who are a measure-valued processes. The second is a restriction of the first to strict control problems."}
{"category": "Math", "title": "Estimators of Long-Memory: Fourier versus Wavelets", "abstract": "There have been a number of papers written on semi-parametric estimation methods of the long-memory exponent of a time series, some applied, others theoretical. Some using Fourier methods, others using a wavelet-based technique. In this paper, we compare the Fourier and wavelet approaches to the local regression method and to the local Whittle method. We provide an overview of these methods, describe what has been done, indicate the available results and the conditions under which they hold. We discuss their relative strengths and weaknesses both from a practical and a theoretical perspective. We also include a simulation-based comparison. The software written to support this work is available on demand and we illustrate its use at the end of the paper."}
{"category": "Math", "title": "Necessary Optimality Condition for a Discrete Dead Oil Isotherm Optimal Control Problem", "abstract": "We obtain necessary optimality conditions for a semi-discretized optimal control problem for the classical system of nonlinear partial differential equations modelling the water-oil (isothermal dead-oil model)."}
{"category": "Math", "title": "Parametric Integer Programming in Fixed Dimension", "abstract": "We consider the following problem: Given a rational matrix $A \\in \\setQ^{m \\times n}$ and a rational polyhedron $Q \\subseteq\\setR^{m+p}$, decide if for all vectors $b \\in \\setR^m$, for which there exists an integral $z \\in \\setZ^p$ such that $(b, z) \\in Q$, the system of linear inequalities $A x \\leq b$ has an integral solution. We show that there exists an algorithm that solves this problem in polynomial time if $p$ and $n$ are fixed. This extends a result of Kannan (1990) who established such an algorithm for the case when, in addition to $p$ and $n$, the affine dimension of $Q$ is fixed. As an application of this result, we describe an algorithm to find the maximum difference between the optimum values of an integer program $\\max \\{c x : A x \\leq b, x \\in \\setZ^n \\}$ and its linear programming relaxation over all right-hand sides $b$, for which the integer program is feasible. The algorithm is polynomial if $n$ is fixed. This is an extension of a recent result of Ho\\c{s}ten and Sturmfels (2003) who presented such an algorithm for integer programs in standard form."}
{"category": "Math", "title": "Limit leaves of a CMC lamination are stable", "abstract": "Suppose ${\\cal L}$ is a lamination of a Riemannian manifold by hypersurfaces with the same constant mean curvature. We prove that every limit leaf of ${\\cal L}$ is stable for the Jacobi operator. A simple but important consequence of this result is that the set of stable leaves of ${\\cal L}$ has the structure of a lamination."}
{"category": "Math", "title": "Scott's problem for proper Scott sets", "abstract": "I show that assuming PFA, every proper Scott set is the standard system of a model of PA. A Scott set X is proper if it is arithmetically closed and the quotient Boolean algebra X/Fin is a proper partial order."}
{"category": "Math", "title": "The stability of conditional Markov processes and Markov chains in random environments", "abstract": "We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of $\\sigma$-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365--393]. A similar result is obtained also in the continuous time setting. The proofs are based on an ergodic theorem for Markov chains in random environments in a general state space."}
{"category": "Math", "title": "Knotted surfaces in 4-manifolds", "abstract": "Fintushel and Stern have proved that if S \\subset X is a symplectic surface in a symplectic 4-manifold such that S has simply-connected complement and nonnegative self-intersection, then there are infinitely many topologically equivalent but smoothly distinct embedded surfaces homologous to S. Here we extend this result to include symplectic surfaces whose self-intersection is bounded below by 2-2g, where g is the genus of S. We make use of tools from Heegaard Floer theory, and include several results that may be of independent interest. Specifically we give an analogue for Ozsvath-Szabo invariants of the Fintushel-Stern knot surgery formula for Seiberg-Witten invariants, both for closed 4-manifolds and manifolds with boundary. This is based on a formula for the Ozsvath-Szabo invariants of the result of a logarithmic transformation, analogous to one obtained by Morgan-Mrowka-Szab\\'o for Seiberg-Witten invariants, and the results on Ozsvath-Szabo invariants of fiber sums due to the author and Jabuka. In addition, we give a calculation of the twisted Heegaard Floer homology of circle bundles of \"large\" degree over Riemann surfaces."}
{"category": "Math", "title": "Proper and piecewise proper families of reals", "abstract": "I introduced the notions of proper and piecewise proper families of reals to make progress on an open question in the field of models of PA about whether every Scott set is the standard system of a model of PA. A family of reals X is proper if it is arithmetically closed and the quotient Boolean algebra X/fin is a proper poset. A family is piecewise proper if it is the union of a chain of proper families of size $\\leq\\omega_1$. Here, I explore the question of the existence of proper and piecewise proper families of reals of different cardinalities."}
{"category": "Math", "title": "Incidence Modules for Symplectic Spaces in Characteristic Two", "abstract": "We study the permutation action of a finite symplectic group of characteristic 2 on the set of subspaces of its standard module which are either totally isotropic or else complementary to totally isotropic subspaces with respect to the alternating form. A general formula is obtained for the 2-rank of the incidence matrix for the inclusion of one-dimensional subspaces in the distinguished subspaces of a fixed dimension."}
{"category": "Math", "title": "Symmetric and Quasi-Symmetric Functions associated to Polymatroids", "abstract": "To every subspace arrangement X we will associate symmetric functions P[X] and H[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant H[X] specializes to the Tutte polynomial T[X]. Billera, Jia and Reiner recently introduced a quasi-symmetric function F[X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function G[X] for polymatroids which has this property as well. Moreover, G[X] specializes to P[X], H[X], T[X] and F[X]."}
{"category": "Math", "title": "On bicycle tire tracks geometry, hatchet planimeter, Menzin's conjecture and oscillation of unicycle tracks", "abstract": "The model of a bicycle is a unit segment AB that can move in the plane so that it remains tangent to the trajectory of point A (the rear wheel is fixed on the bicycle frame); the same model describes the hatchet planimeter. The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. According to R. Foote's theorem, this mapping of a circle to a circle is a Moebius transformation. We extend this result to multi-dimensional setting. Moebius transformations belong to one of the three types: elliptic, parabolic and hyperbolic. We prove a 100 years old Menzin's conjecture: if the front wheel track is an oval with area at least pi then the respective monodromy is hyperbolic. We also study bicycle motions introduced by D. Finn in which the rear wheel follows the track of the front wheel. Such a ''unicycle\" track becomes more and more oscillatory in forward direction. We prove that it cannot be infinitely extended backward and relate the problem to the geometry of the space of forward semi-infinite equilateral linkages."}
{"category": "Math", "title": "Canonical moments and random spectral measures", "abstract": "We study some connections between the random moment problem and the random matrix theory. A uniform draw in a space of moments can be lifted into the spectral probability measure of the pair (A,e) where A is a random matrix from a classical ensemble and e is a fixed unit vector. This random measure is a weighted sampling among the eigenvalues of A. We also study the large deviations properties of this random measure when the dimension of the matrix grows. The rate function for these large deviations involves the reversed Kullback information."}
{"category": "Math", "title": "Some relational structures with polynomial growth and their associated algebras II: Finite generation", "abstract": "The profile of a relational structure $R$ is the function $\\varphi_R$ which counts for every integer $n$ the number, possibly infinite, $\\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $A(R)$, introduced by P.~J.~Cameron. In a previous paper, we studied the relationship between the properties of a relational structure and those of their algebra, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. In this paper, we investigate how far the well know algebraic properties of those rings extend to age algebras. The main result is a combinatorial characterization of when the age algebra is finitely generated. In the special case of tournaments, we show that the age algebra is finitely generated if and only if the profile is bounded. We explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra."}
{"category": "Math", "title": "Fully Bayes factors with a generalized g-prior", "abstract": "For the normal linear model variable selection problem, we propose selection criteria based on a fully Bayes formulation with a generalization of Zellner's $g$-prior which allows for $p>n$. A special case of the prior formulation is seen to yield tractable closed forms for marginal densities and Bayes factors which reveal new model evaluation characteristics of potential interest."}
{"category": "Math", "title": "Hilbert modular forms with prescribed ramification", "abstract": "Let $K$ be a totally real field. In this article we present an asymptotic formula for the number of Hilbert modular cusp forms $f$ with given ramification at every place $v$ of $K$. When $v$ is an infinite place, this means specifying the weight of $f$ at $k$, and when $v$ is finite, this means specifying the restriction to inertia of the local Weil-Deligne representation attached to $f$ at $v$. Our formula shows that with essentially finitely many exceptions, the cusp forms of $K$ exhibit every possible sort of ramification behavior, thus generalizing a theorem of Khare and Prasad. From this fact we compute the minimal field over which a modular Jacobian becomes semi-stable."}
{"category": "Math", "title": "The distribution of natural numbers divisible by 2,3,5,11,13 and 17 on the Square Root Spiral", "abstract": "The natural numbers divisible by the Prime Factors 2, 3, 5, 11, 13 and 17 lie on defined spiral graphs, which run through the Square Root Spiral. A mathematical analysis shows, that these spiral graphs are defined by specific quadratic polynomials. Basically all natural number which are divisible by the same prime factor lie on such spiral graphs. And these spiral graphs can be assigned to a certain number of Spiral Graph Systems, which have a defined spatial orientation to each other. This document represents a supplementation to my detailed introduction study to the Square Root Spiral, and it contains the missing diagrams and analyses, showing the distribution of the natural numbers divisible by 2, 3, 5, 11, 13 and 17 on the Square Root Spiral. My introduction study to the Square Root Spiral can be found in the arxiv-archive. The title of this study : The ordered distribution of the natural numbers on the Square Root Spiral."}
{"category": "Math", "title": "Invariant manifolds for a singular ordinary differential equation", "abstract": "We study the singular ordinary differential equation $$ \\frac{d U}{d t} = f (U) / z (U) + g (U), $$ where $U \\in R^N$, the functions $f \\in R^N $ and $g \\in R^N $ are of class $C^2$ and $z $ is a real valued $C^2$ function. The equation is singular in the sense that $z (U)$ can attain the value 0. We focus on the solutions of the singular ODE that belong to a small neighborhood of a point $\\bar U$ such that $f (\\bar U) = g (\\bar U) = \\vec 0$, $z (\\bar U) =0$. We investigate the existence of manifolds that are locally invariant for the singular ODE and that contain orbits with a suitable prescribed asymptotic behaviour. Under suitable hypotheses on the set $\\{U: z (U) = 0 \\}$, we extend to the case of the singular ODE the definitions of center manifold, center stable manifold and of uniformly stable manifold. An application of our analysis concerns the study of the viscous profiles with small total variation for a class of mixed hyperbolic-parabolic systems in one space variable. Such a class includes the compressible Navier Stokes equation."}
{"category": "Math", "title": "$L^2$-torsion invariants and the Magnus representation of the mapping class group", "abstract": "In this paper, we study a series of $L^2$-torsion invariants from the viewpoint of the mapping class group of a surface. We establish some vanishing theorems for them. Moreover we explicitly calculate the first two invariants and compare them with hyperbolic volumes."}
{"category": "Math", "title": "A bijective proof for a theorem of Ehrhart", "abstract": "We give a new proof for a theorem of Ehrhart regarding the quasi-polynomiality of the function that counts the number of integer points in the integral dilates of a rational polytope. The proof involves a geometric bijection, inclusion-exclusion, and recurrence relations, and we also prove Ehrhart reciprocity using these methods."}
{"category": "Math", "title": "An Algorithm for Finding Symmetric Gr\\\"obner Bases in Infinite Dimensional Rings", "abstract": "A \\textit{symmetric ideal} $I \\subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr\\\"obner bases for symmetric ideals in the infinite dimensional polynomial ring $R$. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of $R$."}
{"category": "Math", "title": "The Synthesis of Regression Slopes in Meta-Analysis", "abstract": "Research on methods of meta-analysis (the synthesis of related study results) has dealt with many simple study indices, but less attention has been paid to the issue of summarizing regression slopes. In part this is because of the many complications that arise when real sets of regression models are accumulated. We outline the complexities involved in synthesizing slopes, describe existing methods of analysis and present a multivariate generalized least squares approach to the synthesis of regression slopes."}
{"category": "Math", "title": "Conjugacy Classes of 3-Braid Group B_3", "abstract": "In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three."}
{"category": "Math", "title": "Antisymmetric Elements in Group Rings II", "abstract": "Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let $\\vp : RG\\to RG$ denote the $R$-linear extension of an involution $\\vp$ defined on $G$. An element $x$ in $RG$ is said to be $\\vp$-antisymmetric if $\\vp (x) = -x$. A characterization is given of when the $\\vp$-antisymmetric elements of $RG$ commute. This is a completion of earlier work."}
{"category": "Math", "title": "Integral Points on Hyperelliptic Curves", "abstract": "We give a completely explicit upper bound for integral points on (standard) affine models of hyperelliptic curves, provided we know at least one rational point and a Mordell-Weil basis of the Jacobian. We also explain a powerful refinement of the Mordell--Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on a two genus 2 hyperelliptic curves with Mordell--Weil Jacobian ranks of 3 and 6."}
{"category": "Math", "title": "Lipschitzness of the Lempert and Green functions", "abstract": "Necessary and sufficient conditions for Lipschitzness of the Lempert and Green functions are found in terms of their boundary behaviors."}
{"category": "Math", "title": "The simplest and fastest method of solving Exact Differential Equation by developing the idea of Basic Functions", "abstract": "This paper concerns exact differential equations. First, I define two types of functions which I have named Basic Function of Type One and Basic Function of Type Two. I then derive the property and theorems of these functions. Finally, by applying the property and theorems of Basic Functions, I establish the method of solving exact differential equation with n variables which is significantly simpler and faster than the standard method."}
{"category": "Math", "title": "On the birth-and-assassination process, with an application to scotching a rumor in a network", "abstract": "We give new formulas on the total number of born particles in the stable birth-and-assassination process, and prove that it has an heavy-tailed distribution. We also establish that this process is a scaling limit of a process of rumor scotching in a network, and is related to a predator-prey dynamics."}
{"category": "Math", "title": "On the motion under focal attraction in a rotating medium", "abstract": "New results are established here on the phase portraits and bifurcations of the kinematic model in a system of ODE's, first presented by H.K. Wilson in his 1971 book, and by him attributed to L. Markus (unpublished). A new, self-sufficient, study which extends Wilson's result and allows an essential conclusion for the applicability of the model is reported here."}
{"category": "Math", "title": "Non-archimedean canonical measures on abelian varieties", "abstract": "For a closed d-dimensional subvariety X of an abelian variety A and a canonically metrized line bundle L on A, Chambert-Loir has introduced measures $c_1(L|_X)^{\\wedge d}$ on the Berkovich analytic space associated to A with respect to the discrete valuation of the ground field. In this paper, we give an explicit description of these canonical measures in terms of convex geometry. We use a generalization of the tropicalization related to the Raynaud extension of A and Mumford's construction. The results have applications to the equidistribution of small points."}
{"category": "Math", "title": "Equidistribution over function fields", "abstract": "We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry."}
{"category": "Math", "title": "The Inverse Simpson Paradox (How To Win Without Overtly Cheating)", "abstract": "Given two sets of data which lead to a similar statistical conclusion, the Simpson Paradox describes the tactic of combining these two sets and achieving the opposite conclusion. Depending upon the given data, this may or may not succeed. Inverse Simpson is a method of decomposing a given set of comparison data into two disjoint sets and achieving the opposite conclusion for each one. This is always possible; however, the statistical significance of the conclusions does depend upon the details of the given data."}
{"category": "Math", "title": "Homotopy theory of Spectral categories", "abstract": "We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories."}
{"category": "Math", "title": "A connection between viscous profiles and singular ODEs", "abstract": "We deal with the viscous profiles for a class of mixed hyperbolic-parabolic systems. We focus, in particular, on the case of the compressible Navier Stokes equation in one space variable written in Eulerian coordinates. We describe the link between these profiles and a singular ordinary differential equation in the form $$ dV / dt = F(V) / z (V) . $$ Here $V \\in R^d$ and the function F takes values into $R^d$ and is smooth. The real valued function z is as well regular: the equation is singular in the sense that z (V) can attain the value 0."}
{"category": "Math", "title": "Groups that do and do not have context-sensitive word problem", "abstract": "We prove that a group has word problem that is a growing context-sensitive language precisely if its word problem can be solved using a non-deterministic Cannon's algorithm (the deterministic algorithms being defined by Goodman and Shapiro). We generalise their results to find many examples of groups not admitting non-deterministic Cannon's algorithms. This adds to the examples of Kambites and Otto of groups separating context-sensitive and growing context-sensitive word problems, and provides a new language-theoretic separation result."}
{"category": "Math", "title": "The cluster category of a canonical algebra", "abstract": "We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan-Iyama-Reiten-Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently, if A is of tame (domestic or tubular) representation type."}
{"category": "Math", "title": "Macroscopic dimension of the $\\ell^p$-ball with respect to the $\\ell^q$-norm", "abstract": "We show estimates of the \"macroscopic dimension\" of the $\\ell^p$-ball with respect to the $\\ell^q$-norm."}
{"category": "Math", "title": "Subordinated discrete semigroups of operators", "abstract": "Given a power-bounded linear operator T in a Banach space and a probability F on the non-negative integers, one can form a `subordinated' operator S = \\sum_k F(k) T^k. We obtain asymptotic properties of the subordinated discrete semigroup (S^n: n=1,2,...) under certain conditions on F. In particular, we study probabilities F with the property that S satisfies the Ritt resolvent condition whenever T is power-bounded. Examples and counterexamples of this property are discussed. The hypothesis of power-boundedness of T can sometimes be replaced by the weaker Kreiss resolvent condition."}
{"category": "Math", "title": "A proof of the Gordon Conjecture", "abstract": "A combinatorial proof of the Gordon Conjecture: The sum of two Heegaard splittings is stabilized if and only if one of the two summands is stabilized."}
{"category": "Math", "title": "Twistorial maps between quaternionic manifolds", "abstract": "We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: 1) A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. 2) A map between quaternionic manifolds endowed with the nonintegrable almost twistorial structures is twistorial if and only if it is quaternionic and totally-geodesic. As an application, we describe the quaternionic maps between open sets of quaternionic projective spaces."}
{"category": "Math", "title": "Counting lattice points in the moduli space of curves", "abstract": "We show how to define and count lattice points in the moduli space $\\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and tautological intersection numbers on the compactified moduli space."}
{"category": "Math", "title": "Characteristic varieties for a class of line arrangements", "abstract": "Let $\\mathcal{A}$ be a line arrangement in the complex projective plane $\\mathbb{P}^2$, having the points of multiplicity $\\geq 3$ situated on two lines in $\\mathcal{A}$, say $H_0$ and $H_{\\infty}$. Then we show that the non-local irreducible components of the first resonance variety $\\mathcal{R}_1(\\mathcal{A})$ are 2-dimensional and correspond to parallelograms $\\mathcal{P}$ in $\\mathbb{C}^2=\\mathbb{P}^2 \\setminus H_{\\infty}$ whose sides are in $\\mathcal{A}$ and for which $H_0$ is a diagonal."}
{"category": "Math", "title": "Relative hyperbolicity and relative quasiconvexity for countable groups", "abstract": "We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely generated). We prove the equivalence of Gromov, Osin, and Bowditch's definitions of relative hyperbolicity for countable groups. We then give several equivalent definitions of relatively quasiconvex subgroups in terms of various natural geometries on a relatively hyperbolic group. We show that each relatively quasiconvex subgroup is itself relatively hyperbolic, and that the intersection of two relatively quasiconvex subgroups is again relatively quasiconvex. In the finitely generated case, we prove that every undistorted subgroup is relatively quasiconvex, and we compute the distortion of a finitely generated relatively quasiconvex subgroup."}
{"category": "Math", "title": "C$^{*}$-bialgebra defined by the direct sum of Cuntz-Krieger algebras", "abstract": "Let ${\\sf CK}_{*}$ denote the C$^{*}$-algebra defined by the direct sum of all Cuntz-Krieger algebras. We introduce a comultiplication $\\Delta_{\\phi}$ and a counit $\\epsilon$ on ${\\sf CK}_{*}$ such that $\\Delta_{\\phi}$ is a nondegenerate $*$-homomorphism from ${\\sf CK}_{*}$ to ${\\sf CK}_{*}\\otimes {\\sf CK}_{*}$ and $\\epsilon$ is a $*$-homomorphism from ${\\sf CK}_{*}$ to ${\\bf C}$. From this, ${\\sf CK}_{*}$ is a counital non-commutative non-cocommutative C$^{*}$-bialgebra. Furthermore, C$^{*}$-bialgebra automorphisms, a tensor product of representations and C$^{*}$-subbialgebras of ${\\sf CK}_{*}$ are investigated."}
{"category": "Math", "title": "Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras", "abstract": "This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with the Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on the Cuntz algebra. We introduce a modified $K_1$-group of the Cuntz algebra so as to pair with this twisted cocycle. As a corollary we obtain a noncommutative geometry interpretation for Araki's notion of relative entropy in this example. We also note the connection of this example to the theory of noncommutative manifolds."}
{"category": "Math", "title": "Quotients by non-reductive algebraic group actions", "abstract": "Given a suitable action on a complex projective variety X of a non-reductive affine algebraic group H, this paper considers how to choose a reductive group G containing H and a projective completion of G x_H X which is a reductive envelope in the sense of math.AG/0703131. In particular it studies the family of examples given by moduli spaces of hypersurfaces in the weighted projective plane P(1,1,2) obtained as quotients by linear actions of the (non-reductive) automorphism group of P(1,1,2)."}
{"category": "Math", "title": "Finsleroid-regular space. Landsberg-to-Berwald implication", "abstract": "By performing required evaluations, we show that in the Finsleroid-regular space the Landsberg-space condition just degenerates to the Berwald-space condition (at any dimension number $N\\ge2$). Simple and clear expository representations are obtained. Due comparisons with the Finsleroid-Finsler space are indicated. Keywords: Finsler metrics, spray coefficients, curvature tensors."}
{"category": "Math", "title": "Un scindage de l'application de Frobenius sur toute l'alg\\`ebre des distributions de SL_2", "abstract": "We study a splitting of the Frobenius map on the whole algebra of distributions of SL_2 (over a finite field) and its relation with the explicit Frobenius descent on arithmetic D-modules over the projective line"}
{"category": "Math", "title": "Sup-norm convergence rate and sign concentration property of Lasso and Dantzig estimators", "abstract": "We derive the $l_{\\infty}$ convergence rate simultaneously for Lasso and Dantzig estimators in a high-dimensional linear regression model under a mutual coherence assumption on the Gram matrix of the design and two different assumptions on the noise: Gaussian noise and general noise with finite variance. Then we prove that simultaneously the thresholded Lasso and Dantzig estimators with a proper choice of the threshold enjoy a sign concentration property provided that the non-zero components of the target vector are not too small."}
{"category": "Math", "title": "Nonisomorphic curves that become isomorphic over extensions of coprime degrees", "abstract": "We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite extensions of K whose degrees over K are coprime to one another. More specifically, let K_0 be an arbitrary prime field and let r and s be integers greater than 1 that are coprime to one another. We show that one can find a finite extension K of K_0, a degree-r extension L of K, a degree-s extension M of K, and two curves C and D over K such that C and D become isomorphic to one another over L and over M, but not over any proper subextensions of L/K or M/K. We show that such C and D can never have genus 0, and that if K is finite, C and D can have genus 1 if and only if {r,s} = {2,3} and K is an odd-degree extension of F_3. On the other hand, when {r,s}={2,3} we show that genus-2 examples occur in every characteristic other than 3. Our detailed analysis of the case {r,s} = {2,3} shows that over every finite field K there exist nonisomorphic curves C and D that become isomorphic to one another over the quadratic and cubic extensions of K. Most of our proofs rely on Galois cohomology. Without using Galois cohomology, we show that two nonisomorphic genus-0 curves over an arbitrary field remain nonisomorphic over every odd-degree extension of the base field."}
{"category": "Math", "title": "Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems", "abstract": "The class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitution subshifts and odometers measure--theoretical and continuous eigenvalues are the same. It is natural to ask whether this rigidity property remains true for the class of linearly recurrent Cantor systems. We give partial answers to this question."}
{"category": "Math", "title": "Necessary and sufficient conditions to be an eigenvalue for linearly recurrent dynamical Cantor systems", "abstract": "We give necessary and sufficient conditions to have measurable and continuous eigenfunctions for linearly recurrent Cantor dynamical systems. We also construct explicitly an example of linearly recurrent system with nontrivial Kronecker factor and a trivial maximal equicontinuous factor."}
{"category": "Math", "title": "Convex ordering for random vectors using predictable representation", "abstract": "We prove convex ordering results for random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. Our method uses forward-backward stochastic calculus and extends previous results in the one-dimensional case. We also study a geometric interpretation of convex ordering for discrete measures in connection with the conditions set on the jump heights and intensities of the considered processes."}
{"category": "Math", "title": "Optimal regularity for planar mappings of finite distortion", "abstract": "Let $f:\\Omega\\to\\IR^2$ be a mapping of finite distortion, where $\\Omega\\subset\\IR^2 .$ Assume that the distortion function $K(x,f)$ satisfies $e^{K(\\cdot, f)}\\in L^p_{loc}(\\Omega)$ for some $p>0.$ We establish optimal regularity and area distortion estimates for $f$. Especially, we prove that $|Df|^2 \\log^{\\beta -1}(e + |Df|) \\in L^1_{loc}(\\Omega) $ for every $\\beta <p.$ This answers positively well known conjectures due to Iwaniec and Martin \\cite{IMbook} and to Iwaniec, Koskela and Martin \\cite{IKM}."}
{"category": "Math", "title": "On the Distribution of the Adaptive LASSO Estimator", "abstract": "We study the distribution of the adaptive LASSO estimator (Zou (2006)) in finite samples as well as in the large-sample limit. The large-sample distributions are derived both for the case where the adaptive LASSO estimator is tuned to perform conservative model selection as well as for the case where the tuning results in consistent model selection. We show that the finite-sample as well as the large-sample distributions are typically highly non-normal, regardless of the choice of the tuning parameter. The uniform convergence rate is also obtained, and is shown to be slower than $n^{-1/2}$ in case the estimator is tuned to perform consistent model selection. In particular, these results question the statistical relevance of the `oracle' property of the adaptive LASSO estimator established in Zou (2006). Moreover, we also provide an impossibility result regarding the estimation of the distribution function of the adaptive LASSO estimator.The theoretical results, which are obtained for a regression model with orthogonal design, are complemented by a Monte Carlo study using non-orthogonal regressors."}
{"category": "Math", "title": "Transversality and Lefschetz numbers for foliation maps", "abstract": "Let $F$ be a smooth foliation on a closed Riemannian manifold $M$, and let $\\Lambda$ be a transverse invariant measure of $F$. Suppose that $\\Lambda$ is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the $\\Lambda$-Lefschetz number of any leaf preserving diffeomorphism $(M,F)\\to(M,F)$ is given. For this purpose, standard results about smooth approximation and transversality are extended to the case of foliation maps. It is asked whether this topological $\\Lambda$-Lefschetz number is equal to the analytic $\\Lambda$-Lefschetz number defined by Heitsch and Lazarov which would be a version of the Lefschetz trace formula. Heitsch and Lazarov have shown such a trace formula when the fixed point set is transverse to $F$."}
{"category": "Math", "title": "Recursive Bias Estimation and $L_2$ Boosting", "abstract": "This paper presents a general iterative bias correction procedure for regression smoothers. This bias reduction schema is shown to correspond operationally to the $L_2$ Boosting algorithm and provides a new statistical interpretation for $L_2$ Boosting. We analyze the behavior of the Boosting algorithm applied to common smoothers $S$ which we show depend on the spectrum of $I-S$. We present examples of common smoother for which Boosting generates a divergent sequence. The statistical interpretation suggest combining algorithm with an appropriate stopping rule for the iterative procedure. Finally we illustrate the practical finite sample performances of the iterative smoother via a simulation study. simulations."}
{"category": "Math", "title": "Rational periodic points for quadratic maps", "abstract": "Let $K$ be a number field. Let $S$ be a finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we consider endomorphisms of $\\pro$ of degree 2, defined over $K$, with good reduction outside $S$. We prove that there exist only finitely many such endomorphisms, up to conjugation by ${\\rm PGL}_2(R_S)$, admitting a periodic point in $\\po$ of order $>3$. Also, all but finitely many classes with a periodic point in $\\po$ of order 3 are parametrized by an irreducible curve."}
{"category": "Math", "title": "Connectedness in the Pluri-fine Topology", "abstract": "We study connectedness in the pluri-fine topology on $\\CC^n$ and obtain the following results. If $\\Omega$ is a pluri-finely open and pluri-finely connected set in $\\CC^n$ and $E\\subset\\CC^n$ is pluripolar, then $\\Omega\\setminus E$ is pluri-finely connected. The proof hinges on precise information about the structure of open sets in the pluri-fine topology: Let $\\Omega$ be a pluri-finely open subset of $\\CC^{n}$. If $z$ is any point in $\\Omega$, and $L$ is a complex line passing through $z$, then obviously $\\Omega \\cap L$ is a finely open neighborhood of $z$ in $L$. Now let $C_L$ denote the finely connected component of $z$ in $\\Omega\\cap L$. Then $\\cup_{L\\ni z} C_L$ is a pluri-finely connected neighborhood of $z$. As a consequence we find that if $v$ is a finely plurisubharmonic function defined on a pluri-finely connected pluri-finely open set, then $v= -\\infty$ on a pluri-finely open subset implies $v\\equiv -\\infty$."}
{"category": "Math", "title": "Refracted Levy processes", "abstract": "Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted L\\'evy processes. The latter is a L\\'evy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted L\\'evy process is described by the unique strong solution to the stochastic differential equation \\[ \\D U_t = - \\delta \\mathbf{1}_{\\{U_t >b\\}}\\D t + \\D X_t \\] where $X=\\{X_t :t\\geq 0\\}$ is a L\\'evy process with law $\\mathbb{P}$ and $b, \\delta\\in \\mathbb{R}$ such that the resulting process $U$ may visit the half line $(b,\\infty)$ with positive probability. We consider in particular the case that $X$ is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the $q$-scale function of the driving L\\'evy process and its perturbed version describing motion above the level $b$. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes."}
{"category": "Math", "title": "Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials", "abstract": "Hubbard trees are invariant trees connecting the points of the critical orbits of postcritically finite polynomials. Douady and Hubbard \\cite{Orsay} introduced these trees and showed that they encode the essential information of Julia sets in a combinatorial way. The itinerary of the critical orbit within the Hubbard tree is encoded by a (pre)periodic sequence on $\\{\\0,\\1\\}$ called \\emph{kneading sequence}. We prove that the kneading sequence completely encodes the Hubbard tree and its dynamics, and we show how to reconstruct the tree and in particular its branch points (together with their periods, their relative posititions, their number of arms and their local dynamics) in terms of the kneading sequence alone. Every kneading sequence gives rise to an abstract Hubbard tree, but not every kneading sequence occurs in real dynamics or in complex dynamics. Milnor and Thurston \\cite{MT} classified which kneading sequences occur in real dynamics; we do the same for complex dynamics in terms of a complex \\emph{admissibility condition}. This complex admissibility condition fails if and only if the abstract Hubbard tree has a so-called \\emph{evil} periodic branch point that is incompatible with local homeomorphic dynamics on the plane."}
{"category": "Math", "title": "Symplectic embeddings of 4-dimensional ellipsoids", "abstract": "We show how to reduce the problem of symplectically embedding one 4-dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into appropriate blow ups of \\C P^2. For example, the problem of embedding the ellipsoid E(1,k) into a ball B is equivalent to that of embedding k disjoint equal balls into \\C P^2, and so can be solved by the work of Gromov, McDuff--Polterovich and Biran. (Here k is the ratio of the area of the major axis to that of the minor axis.) As a consequence we show that the ball may be fully filled by the ellipsoid E(1,k) for k=1,4 and all k\\ge 9, thus answering a question raised by Hofer."}
{"category": "Math", "title": "Stochastic maximum principle for optimal control problem of backward systems with terminal condition in L1", "abstract": "We consider a stochastic control problem, where the control domain is convex and the system is governed by a nonlinear backward stochastic differential equation. With a L1 terminal data, we derive necessary optimality conditions in the form of stochastic maximum principle."}
{"category": "Math", "title": "The strict and relaxed stochastic maximum principle for optimal control problem of backward systems", "abstract": "We consider a stochastic control problem where the set of controls is not necessarily convex and the system is governed by a nonlinear backward stochastic differential equation. We establish necessary as well as sufficient conditions of optimality for two models. The first concerns the strict (classical) controls. The second is an extension of the first to relaxed controls, who are a measure valued processes."}
{"category": "Math", "title": "A general stochastic maximum principle for mixed relaxed-singular control problems", "abstract": "We consider in this paper, mixed relaxed-singular stochastic control problems, where the control variable has two components, the first being measure-valued and the second singular. The control domain is not necessarily convex and the system is governed by a nonlinear stochastic differential equation, in which the measure-valued part of the control enters both the drift and the diffusion coefficients. We establish necessary optimality conditions, of the Pontryagin maximum principle type, satisfied by an optimal relaxed-singular control, which exist under general conditions on the coefficients. The proof is based on the strict singular stochastic maximum principle established by Bahlali-Mezerdi, Ekeland's variational principle and some stability properties of the trajectories and adjoint processes with respect to the control variable."}
{"category": "Math", "title": "Gamma-reduction for smooth orbifolds", "abstract": "The aim of this short note is to show how to construct a rationnal Remmert reduction for the universal cover of a smooth K\\\"ahler orbifold. Doins this, we are led to introduce some singular K\\\"ahler metrics adapted to the orbifold structure."}
{"category": "Math", "title": "Stagnation zones for $\\mathcal{A}$-harmonic functions on canonical domains", "abstract": "We study stagnation zones of $\\mathcal{A}$-harmonic functions on canonical domains in the Euclidean $n$-dimensional space. Phragmen-Lindel\\\"of type theorems are proved."}
{"category": "Math", "title": "A Modular Curve of Level 9 and the Class Number One Problem", "abstract": "In this note we give an explicit parametrization of the modular curve associated to the normalizer of a non-split Cartan subgroup of level 9. We determine all integral points of this modular curve. As an application, we give an alternative solution to the class number one problem."}
{"category": "Math", "title": "Asymptotic behavior for dissipative Korteweg-de Vrie equations", "abstract": "We study the large time behavior of solutions to the dissipative Korteweg-de Vrie equations $u_t+u_{xxx}+|D|^{\\alpha}u+uu_x=0$ with $0<\\alpha<2$. We find $v$ such that $u-v$ decays like $t^{-r(\\alpha)}$ as $t\\to\\infty$ in various Sobolev norm."}
{"category": "Math", "title": "Hyperbolicity of arborescent tangles and arborescent links", "abstract": "In this paper, we study the hyperbolicity of arborescent tangles and arborescent links. We will explicitly determine all essential surfaces in arborescent tangle complements with non-negative Euler characteristic, and show that given an arborescent tangle T, the complement X(T) is non-hyperbolic if and only if T is a rational tangle, T=Q_m * T' for some m greater than or equal to 1, or T contains Qn for some n greater than or equal to 2. We use these results to prove a theorem of Bonahon and Seibenmann which says that a large arborescent link L is non-hyperbolic if and only if it contains Q2."}
{"category": "Math", "title": "Gradient Estimate and Harnack Inequality on Non-Compact Riemannian Manifolds", "abstract": "A new type of gradient estimate is established for diffusion semigroups on non-compact complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived for diffusion semigroups on arbitrary complete Riemannian manifolds."}
{"category": "Math", "title": "Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions", "abstract": "Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well posed, so we find an additional assumption on the state-dependent delay function to guarantee the well posedness. For the constructed dynamical system we study the long-time asymptotic behavior and prove the existence of a compact global attractor."}
{"category": "Math", "title": "Necessary and Sufficient Lyapunov-Like Conditions for Robust Nonlinear Stabilization", "abstract": "In this work, we propose a methodology for the expression of necessary and sufficient Lyapunov-like conditions for the existence of stabilizing feedback laws. The methodology is an extension of the well-known Control Lyapunov Function (CLF) method and can be applied to very general nonlinear time-varying systems with disturbance and control inputs, including both finite- and infinite-dimensional systems. The generality of the proposed methodology is also reflected upon by the fact that partial stability with respect to output variables is addressed. In addition, it is shown that the generalized CLF method can lead to a novel tool for the explicit design of robust nonlinear controllers for a class of time-delay nonlinear systems with a triangular structure."}
{"category": "Math", "title": "Ramsey-like cardinals", "abstract": "One of the numerous characterizations of a Ramsey cardinal kappa involves the existence of certain types of elementary embeddings for transitive sets of size \\kappa satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V=L."}
{"category": "Math", "title": "The Bernoulli sieve revisited", "abstract": "We consider an occupancy scheme in which \"balls\" are identified with $n$ points sampled from the standard exponential distribution, while the role of \"boxes\" is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behavior of five quantities: the index $K_n^*$ of the last occupied box, the number $K_n$ of occupied boxes, the number $K_{n,0}$ of empty boxes whose index is at most $K_n^*$, the index $W_n$ of the first empty box and the number of balls $Z_n$ in the last occupied box. It is shown that the limiting distribution of properly scaled and centered $K_n^*$ coincides with that of the number of renewals not exceeding $\\log n$. A similar result is shown for $K_n$ and $W_n$ under a side condition that prevents occurrence of very small boxes. The condition also ensures that $K_{n,0}$ converges in distribution. Limiting results for $Z_n$ are established under an assumption of regular variation."}
{"category": "Math", "title": "Stochastic extrema as stationary phases of characteristic functions", "abstract": "The paper is dealing with semi-classical asymptotics of a characteristic function for a stochastic process. The main technical tool is provided by the stationary phase method. The extremal range for a stochastic process is defined by limit values of the complex logarithm of the characteristic function. The paper also outlines a numerical method for calculating stochastic extrema."}
{"category": "Math", "title": "Stabilization of nonlinear systems with semi-quadratic cost", "abstract": "The paper addresses the stabilization of nonlinear systems with semi-quadratic cost: quadratic with respect to controls and nonlinear for state variables. Paper presents the effective new feedback synthesis procedure. The novel feedback design procedure is based on the ideas borrowed from nonlinear optics and the theory of semi-classical asymptotics."}
{"category": "Math", "title": "Yang-Mills theory and Tamagawa numbers", "abstract": "Atiyah and Bott used equivariant Morse theory applied to the Yang-Mills functional to calculate the Betti numbers of moduli spaces of vector bundles over a Riemann surface, rederiving inductive formulae obtained from an arithmetic approach which involved the Tamagawa number of SL_n. This article surveys this link between Yang-Mills theory and Tamagawa numbers, and explains how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth complex projective curve can be adapted to the setting of A^1-homotopy theory to study the motivic cohomology of these moduli spaces."}
{"category": "Math", "title": "Brill-Noether Theory for stable vector bundles", "abstract": "This paper gives an overview of the main results of Brill-Noether Theory for vector bundles on algebraic curves."}
{"category": "Math", "title": "Construction of combinatorial manifolds with the prescribed sets of links of vertices", "abstract": "To each oriented closed combinatorial manifold we assign the set (with repetitions) of isomorphism classes of links of its vertices. The obtained transformation L is the main object of study of the present paper. We pose a problem on the inversion of the transformation L. We shall show that this problem is closely related to N.Steenrod's problem on realization of cycles and to the Rokhlin-Schwartz-Thom construction of combinatorial Pontryagin classes. It is easy to obtain a condition of balancing that is a necessary condition for a set of isomorphism classes of combinatorial spheres to belong to the image of the transformation L. In the present paper we give an explicit construction providing that each balanced set of isomorphism classes of combinatorial spheres gets into the image of L after passing to a multiple set and adding several pairs of the form (Z,-Z), where -Z is the sphere Z with the orientation reversed. This construction enables us, for a given singular simplicial cycle of a space R, to construct explicitly a combinatorial manifold M and a mapping $\\phi:M\\to R$ such that $\\phi_*[M]=r[\\xi]$ for some positive integer r. The construction is based on resolving singularities of the cycle $\\xi$. We give applications of our main construction to cobordisms of manifolds with singularities and cobordisms of simple cells. In particular, we prove that every rational additive invariant of cobordisms of manifolds with singularities admits a local formula. Another application is the construction of explicit (though inefficient) local combinatorial formulae for polynomials in the rational Pontryagin classes of combinatorial manifolds."}
{"category": "Math", "title": "Lower bounds for the number of semidualizing complexes over a local ring", "abstract": "We investigate the set S(R) of shift-isomorphism classes of semidualizing R-complexes, ordered via the reflexivity relation, where R is a commutative noetherian local ring. Specifically, we study the question of whether S(R$ has cardinality 2^n for some n. We show that, if there is a chain of length n in S(R) and if the reflexivity ordering on S(R) is transitive, then S(R) has cardinality at least 2^n, and we explicitly describe some of its order-structure. We also show that, given a local ring homomorphism f: R\\to S of finite flat dimension, if R and S admit dualizing complexes and if f is not Gorenstein, then the cardinality of S(S) is at least twice the cardinality of S(R)."}
{"category": "Math", "title": "Remarks on derived equivalences of Ricci-flat manifolds", "abstract": "We present results indicating that the decomposition of a Ricci-flat manifold in its irreducible factors is reflected by the derived category of coherent sheaves. More precisely, we prove that a smooth projective variety that is derived equivalent to an abelian variety resp. an irreducible symplectic variety is of the same type. The paper also contains a proof of a conjecure of Caldararu for manifolds with trivial canonical bundle saying that the modified HKR isomorphism for Hochschild homology is compatible with the module structure."}
{"category": "Math", "title": "Toward Best Isoperimetric Constants for $(H^1,BMO)$-Normal Conformal Metrics on $\\mathbb R^n$, $n\\ge 3$", "abstract": "The aim of this article is: (a) To establish the existence of the best isoperimetric constants for the $(H^1,BMO)$-normal conformal metrics $e^{2u}|dx|^2$ on $\\mathbb R^n$, $n\\ge 3$, i.e., the conformal metrics with the Q-curvature orientated conditions $$ (-\\Delta)^{n/2}u\\in H^1(\\mathbb R^n) & \\ u(x)=\\hbox{const.}+\\frac{\\int_{\\mathbb R^n}(\\log\\frac{|\\cdot|}{|x-\\cdot|})(-\\Delta)^{n/2} u(\\cdot) d\\mathcal{H}^n(\\cdot)}{2^{n-1}\\pi^{n/2}\\Gamma(n/2)}; $$ (b) To prove that $(n\\omega_n^\\frac1n)^\\frac{n}{n-1}$ is the optimal upper bound of the best isoperimetric constants for the complete $(H^1,BMO)$-normal conformal metrics with nonnegative scalar curvature; (c) To find the optimal upper bound of the best isoperimetric constants via the quotients of two power integrals of Green's functions for the $n$-Laplacian operators $-\\hbox{div}(|\\nabla u|^{n-2}\\nabla u)$."}
{"category": "Math", "title": "Effective structure theorems for symplectic spaces via height", "abstract": "Given a $2k$-dimensional symplectic space $(Z,F)$ in $N$ variables, $1 < 2k \\leq N$, over a global field $K$, we prove the existence of a symplectic basis for $(Z,F)$ of bounded height. This can be viewed as a version of Siegel's lemma for a symplectic space. As corollaries of our main result, we prove the existence of a small-height decomposition of $(Z,F)$ into hyperbolic planes, as well as the existence of two generating flags of totally isotropic subspaces. These present analogues of known results for quadratic spaces. A distinctive feature of our argument is that it works simultaneously for essentially any field with a product formula, algebraically closed or not. In fact, we prove an even more general version of these statements, where canonical height is replaced with twisted height. All bounds on height are explicit."}
{"category": "Math", "title": "Cryptanalysis of Anshel-Anshel-Goldfeld-Lemieux key agreement protocol", "abstract": "The Anshel-Anshel-Goldfeld-Lemieux (abbreviated AAGL) key agreement protocol is proposed to be used on low-cost platforms which constraint the use of computational resources. The core of the protocol is the concept of an Algebraic Eraser (abbreviated AE) which is claimed to be a suitable primitive for use within lightweight cryptography. The AE primitive is based on a new and ingenious idea of using an action of a semidirect product on a (semi)group to obscure involved algebraic structures. The underlying motivation for AAGL protocol is the need to secure networks which deploy Radio Frequency Identification (RFID) tags used for identification, authentication, tracing and point-of-sale applications. In this paper we revisit the computational problem on which AE relies and heuristically analyze its hardness. We show that for proposed parameter values it is impossible to instantiate the secure protocol. To be more precise, in 100% of randomly generated instances of the protocol we were able to find a secret conjugator z generated by TTP algorithm (part of AAGL protocol)."}
{"category": "Math", "title": "Asymptotic behavior of global solutions of the $u_t=\\Delta u + u^{p}$", "abstract": "We study the asymptotic behavior of nonnegative solutions of the semilinear parabolic problem {u_t=\\Delta u + u^{p}, x\\in\\mathbb{R}^{N}, t>0 u(0)=u_{0}, x\\in\\mathbb{R}^{N}, t=0. It is known that the nonnegative solution $u(t)$ of this problem blows up in finite time for $1<p\\leq 1+ 2/N$. Moreover, if $p> 1+ 2/N$ and the norm of $u_{0}$ is small enough, the problem admits global solution. In this work, we use the entropy method to obtain the decay rate of the global solution $u(t)$."}
{"category": "Math", "title": "Constructing Seifert surfaces from n-bridge link projections", "abstract": "This paper presents a new algorithm \"A\" for constructing Seifert surfaces from n-bridge projections of links. The algorithm produces minimal complexity surfaces for large classes of braids and alternating links. In addition, we consider a family of knots for which the canonical genus is strictly greater than the genus, (g_c(K) > g(K)), and show that A builds surfaces realizing the knot genus g(K). We also present a generalization of Seifert's algorithm which may be used to construct surfaces representing arbitrary relative second homology classes in a link complement."}
{"category": "Math", "title": "The Principal Element of a Frobenius Lie Algebra", "abstract": "We introduce the notion of the \\textit{principal element} of a Frobenius Lie algebra $\\f$. The principal element corresponds to a choice of $F\\in \\f^*$ such that $F[-,-]$ non-degenerate. In many natural instances, the principal element is shown to be semisimple, and when associated to $\\sl_n$, its eigenvalues are integers and are independent of $F$. For certain ``small'' functionals $F$, a simple construction is given which readily yields the principal element. When applied to the first maximal parabolic subalgebra of $\\sl_n$, the principal element coincides with semisimple element of the principal three-dimensional subalgebra. We also show that Frobenius algebras are stable under deformation."}
{"category": "Math", "title": "Equidistribution of Dynamically Small Subvarieties over the Function Field of a Curve", "abstract": "For a projective variety X defined over a field K, there is a special class of self-morphisms of X called algebraic dynamical systems. In this paper we take K to be the function field of a smooth curve and prove that at each place of K, subvarieties of X of dynamically small height are equidistributed on the associated Berkovich analytic space. We carefully develop all of the arithmetic intersection theory needed to state and prove this theorem, and we present several applications on the non-Zariski density of preperiodic points and of points of small height in field extensions of bounded degree."}
{"category": "Math", "title": "Commensurators of cusped hyperbolic manifolds", "abstract": "This paper describes a general algorithm for finding the commensurator of a non-arithmetic cusped hyperbolic manifold, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this to find the commensurators of all non-arithmetic hyperbolic once-punctured torus bundles over the circle. For hyperbolic 3-manifolds, the algorithm has been implemented using Goodman's computer program Snap. We use this to determine the commensurability classes of all cusped hyperbolic 3-manifolds triangulated using at most 7 ideal tetrahedra, and for the complements of hyperbolic knots and links with up to 12 crossings."}
{"category": "Math", "title": "A formula for Pl\\\"ucker coordinates associated with a planar network", "abstract": "For a planar directed graph G, Postnikov's boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. We establish an explicit formula for Postnikov's map by expressing each Pluecker coordinate as a ratio of two combinatorially defined polynomials in the edge weights, with positive integer coefficients. In the non-planar setting, we show that a similar formula holds for special choices of Pluecker coordinates."}
{"category": "Math", "title": "From Continuous-Time Design to Sampled-Data Design of Nonlinear Observers", "abstract": "In this work, a sampled-data nonlinear observer is designed using a continuous-time design coupled with an inter-sample output predictor. The proposed sampled-data observer is a hybrid system. It is shown that under certain conditions, the robustness properties of the continuous-time design are inherited by the sampled-data design, as long as the sampling period is not too large. The approach is applied to linear systems and to triangular globally Lipschitz systems."}
{"category": "Math", "title": "A representation formula for indefinite improper affine spheres", "abstract": "We construct a new representation formula for indefinite improper affine spheres in terms of two para-holomorphic functions and study singularities which appear in this representation formula. As a result, it follows that cuspidal cross caps never appear as the singularities on indefinite improper affine spheres and so on. Comparison with other representation formulae are also studied."}
{"category": "Math", "title": "Tropical and Ordinary Convexity Combined", "abstract": "A polytrope is a tropical polytope which at the same time is convex in the ordinary sense. A $d$-dimensional polytrope turns out to be a tropical simplex, that is, it is the tropical convex hull of $d+1$ points. This statement is equivalent to the known fact that the Segre product of two full polynomial rings (over some field $K$) has the Gorenstein property if and only if the factors are generated by the same number of indeterminates. The combinatorial types of polytropes up to dimension three are classified."}
{"category": "Math", "title": "Sparse permutation invariant covariance estimation", "abstract": "The paper proposes a method for constructing a sparse estimator for the inverse covariance (concentration) matrix in high-dimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lasso-type penalty. We establish a rate of convergence in the Frobenius norm as both data dimension $p$ and sample size $n$ are allowed to grow, and show that the rate depends explicitly on how sparse the true concentration matrix is. We also show that a correlation-based version of the method exhibits better rates in the operator norm. We also derive a fast iterative algorithm for computing the estimator, which relies on the popular Cholesky decomposition of the inverse but produces a permutation-invariant estimator. The method is compared to other estimators on simulated data and on a real data example of tumor tissue classification using gene expression data."}
{"category": "Math", "title": "Counting growth types of automorphisms of free groups", "abstract": "Given an automorphism of a free group $F_n$, we consider the following invariants: $e$ is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy classes); $d$ is the maximal degree of polynomial growth of conjugacy classes; $R$ is the rank of the fixed subgroup. We determine precisely which triples $(e,d,R)$ may be realized by an automorphism of $F_n$. In particular, the inequality $e\\le (3n-2)/4}$ (due to Levitt-Lustig) always holds. In an appendix, we show that any conjugacy class grows like a polynomial times an exponential under iteration of the automorphism."}
{"category": "Math", "title": "Cohomology rings and formality properties of nilpotent groups", "abstract": "We introduce partial formality and relate resonance with partial formality properties. For instance, we show that for finitely generated nilpotent groups that are k-formal, the resonance varieties are trivial up to degree k. We also show that the cohomology ring of a nilpotent k-formal group is generated in degree 1, up to degree k+1; this criterion is necessary and sufficient for 2-step nilpotent groups to be k-formal. We compute resonance varieties for Heisenberg-type groups and deduce the degree of partial formality for this class of groups."}
{"category": "Math", "title": "Region of Variability for Spirallike Functions with Respect to a Boundary Point", "abstract": "In this paper we determine the region of variability for spirallike funcions with respect to a boundary point. In the final section we graphically illustrate the region of variability for several sets of parameters."}
{"category": "Math", "title": "On algebras generated by inner derivations", "abstract": "We look for an effective description of the algebra D_{Lie}(X,B) of operators on a bimodule X over an algebra B, generated by inner derivations. It is shown that in some important examples D_{Lie}(X,B) consists of all elementary operators x\\to \\sum_i a_ixb_i satisfying the conditions $\\sum_i a_ib_i = \\sum_i b_ia_i = 0. The Banach algebraic versions of these results are also obtained and applied to the description of closed Lie ideals in some Banach algebras, and to the proof of a density theorem for Lie algebras of operators on Hilbert space."}
{"category": "Math", "title": "Hilbert Coefficients and Depth of the Associated Graded Ring of an Ideal", "abstract": "In this expository paper we survey results that relate Hilbert coefficients of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) with depth of the associated graded ring G(I). Several results in this area follow from two theorems of S. Huckaba and T. Marley. These were proved using homological techniques. We provide simple proofs using superficial sequences."}
{"category": "Math", "title": "Transseries for beginners", "abstract": "From the simplest point of view, transseries are a new kind of expansion for real-valued functions. But transseries constitute much more than that--they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries is a large ordered field, extending the real number field, and endowed with additional operations such as exponential, logarithm, derivative, integral, composition. Over the course of the last 20 years or so, transseries have emerged in several areas of mathematics: asymptotic analysis, model theory, computer algebra, surreal numbers. This paper is an exposition for the non-specialist mathematician. All a mathematician needs to know in order to apply transseries."}
{"category": "Math", "title": "Some applications of Ricci flow to 3-manifolds", "abstract": "We decribe and announce some results (joint with G. Besson, L. Bessieres, M. Boileau and J.Porti) about the geometry and topology of 3-manifolds. Most of the article is primarily intended as an introduction for nonexperts to geometrization of 3-manifolds, Ricci flow with surgery, and the simplicial volume approach to collapsing theorems. In the last section, Ricci flow with surgery on open 3-manifolds and obstructions to positive scalar curvature are discussed."}
{"category": "Math", "title": "A theorem on the cores of partitions", "abstract": "If s and t are relatively prime positive integers we show that the s-core of a t-core partition is again a t-core partition"}
{"category": "Math", "title": "Pointwise Green function bounds and long-time stability of large-amplitude noncharacteristic boundary layers", "abstract": "Using pointwise semigroup techniques of Zumbrun--Howard and Mascia--Zumbrun, we obtain sharp global pointwise Green function bounds for noncharacteristic boundary layers of arbitrary amplitude. These estimates allow us to analyze linearized and nonlinearized stability of noncharacteristic boundary layers of one-dimensional systems of conservation laws, showing that both are equivalent to a numerically checkable Evans function condition. Our results extend to the large-amplitude case results obtained for small amplitudes by Matsumura, Nishihara and others using energy estimates."}
{"category": "Math", "title": "Quantum Teichmuller theory and representations of the pure braid group", "abstract": "We adapt some of the methods of quantum Teichm\\\"uller theory to construct a family of representations of the pure braid group of the sphere."}
{"category": "Math", "title": "The dual geometry of Boolean semirings", "abstract": "It is well known that the variety of Boolean semirings, which is generated by the three element semiring S, is dual to the category of partially Stone spaces. We place this duality in the context of natural dualities. We begin by introducing a topological structure \\uS and obtain an optimal natural duality between the quasi-variety ISP(S) and the category IS_cP+(\\uS). Then we construct an optimal and very small structure \\uS_os that yields a strong duality. The geometry of some of the partially Stone spaces that take part in these dualities is presented, and we call them \"hairy cubes\", as they are n-dimensional cubes with unique incomparable covers for each element of the cube. We also obtain a polynomial representation for the elements of the hairy cube."}
{"category": "Math", "title": "Bijections between pattern-avoiding fillings of Young diagrams", "abstract": "The pattern-avoiding fillings of Young diagrams we study arose from Postnikov's work on positive Grassman cells. They are called Le-diagrams, and are in bijection with decorated permutations. Other closely-related diagrams are interpreted as acyclic orientations of some bipartite graphs. The definition of the diagrams is the same but the avoided patterns are different. We give here bijections proving that the number of pattern-avoiding filling of a Young diagram is the same, for these two different sets of patterns. The result was obtained by Postnikov via a reccurence relation. This relation was extended by Spiridonov to obtain more general results about other patterns and other polyominoes than Young diagrams, and we show that our bijections also extend to more general polyominoes."}
{"category": "Math", "title": "An invariant set in energy space for supercritical NLS in 1D", "abstract": "We consider radial solutions of a mass supercritical monic NLS and we prove the existence of a set, which looks like a hypersurface, in the space of finite energy functions, invariant for the flow and formed by solutions which converge to ground states."}
{"category": "Math", "title": "Incompressible flow around a small obstacle and the vanishing viscosity limit", "abstract": "In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition at the surface of the obstacle. We seek conditions under which solutions of the Navier-Stokes system in the exterior domain converge to solutions of the Euler system in the full space when both viscosity and the size of the obstacle vanish. We prove that this convergence is true assuming two hypothesis: first, that the initial exterior domain velocity converges strongly in $L^2$ to the full-space initial velocity and second, that the diameter of the obstacle is smaller than a suitable constant times viscosity, or, in other words, that the obstacle is sufficiently small. The convergence holds as long as the solution to the limit problem is known to exist and stays sufficiently smooth. This work complements the study of incompressible flow around small obstacles, which has been carried out in [1,2,3] [1] D. Iftimie and J. Kelliher, {\\it Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid.} Preprint available at http://math.univ-lyon1.fr/~iftimie/ARTICLES/viscoushrink3d.pdf . [2] D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. {\\it Two dimensional incompressible ideal flow around a small obstacle.} Comm. Partial Differential Equations {\\bf 28} (2003), no. 1-2, 349--379. [3] D. Iftimie, M. C. Lopes Filho, and H. J. Nussenzveig Lopes. {\\it Two dimensional incompressible viscous flow around a small obstacle.} Math. Ann. {\\bf 336} (2006), no. 2, 449--489."}
{"category": "Math", "title": "On mutation and Khovanov homology", "abstract": "It is conjectured that the Khovanov homology of a knot is invariant under mutation. In this paper, we review the spanning tree complex for Khovanov homology, and reformulate this conjecture using a matroid obtained from the Tait graph (checkerboard graph) G of a knot diagram K. The spanning trees of G provide a filtration and a spectral sequence that converges to the reduced Khovanov homology of K. We show that the E_2-term of this spectral sequence is a matroid invariant and hence invariant under mutation."}
{"category": "Math", "title": "Bispectral commuting difference operators for multivariable Askey-Wilson polynomials", "abstract": "We construct a commutative algebra A_z, generated by d algebraically independent q-difference operators acting on variables z_1, z_2,..., z_d, which is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered by Gasper and Rahman [6]. Iterating Sears' transformation formula, we show that the polynomials P_n(z) possess a certain duality between z and n. Analytic continuation allows us to obtain another commutative algebra A_n, generated by d algebraically independent difference operators acting on the discrete variables n_1, n_2,..., n_d, which is also diagonalized by P_n(z). This leads to a multivariable q-Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions."}
{"category": "Math", "title": "Trivial centralizers for Axiom A diffeomorphisms", "abstract": "We show there is a residual set of non-Anosov $C^{\\infty}$ Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. If $M$ is a surface and $2\\leq r\\leq \\infty$, then we will show there exists an open and dense set of of $C^r$ Axiom A diffeomorphisms with the no cycles property whose elements have trivial centralizer. Additionally, we examine commuting diffeomorphisms preserving a compact invariant set $\\Lambda$ where $\\Lambda$ is a hyperbolic chain recurrent class for one of the diffeomorphisms."}
{"category": "Math", "title": "Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator", "abstract": "We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at $\\pm \\infty$. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of the eigenvalues numerically. We compare these to earlier calculations by other authors."}
{"category": "Math", "title": "On ramification filtrations and $p$-adic differential modules, I: equal characteristic case", "abstract": "Let $k$ be a complete discretely valued field of equal characteristic $p > 0$ with possibly imperfect residue field and let $G_k$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on $G_k$ coincide with the differential Artin conductors and Swan conductors of Galois representations of $G_k$. As a consequence, we give a Hasse-Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse-Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors and Borger's conductors."}
{"category": "Math", "title": "Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion", "abstract": "We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H>1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration, and the classical Ito stochastic calculus. The existence result is based on the Yamada-Watanabe theorem."}
{"category": "Math", "title": "A simple algorithm for extending the identities for quantum minors to the multiparametric case", "abstract": "For any homogeneous identity between $q$-minors, we provide an identity between $P,Q$-minors."}
{"category": "Math", "title": "A Note on Boolean Lattices and Farey Sequences II", "abstract": "We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices."}
{"category": "Math", "title": "Covering maps for locally path-connected spaces", "abstract": "We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces. Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow. If $X$ is path-connected, then every Peano covering map is equivalent to the projection $\\widetilde X/H\\to X$, where $H$ is a subgroup of the fundamental group of $X$ and $\\widetilde X$ equipped with the basic topology. The projection $\\widetilde X/H\\to X$ is a Peano covering map if and only if it has the unique path lifting property. We define a new topology on $\\widetilde X$ for which one has a characterization of $\\widetilde X/H\\to X$ having the unique path lifting property if $H$ is a normal subgroup of $\\pi_1(X)$. Namely, $H$ must be closed in $\\pi_1(X)$. Such groups include $\\pi(\\mathcal{U},x_0)$ ($\\mathcal{U}$ being an open cover of $X$) and the kernel of the natural homomorphism from the fundamental group to the Cech fundamental group."}
{"category": "Math", "title": "Normal Factorization in $SL(2,Z)$ and the Confluence of Singular Fibers in Elliptic Fibrations", "abstract": "In this article we obtain a result about the uniqueness of factorization in terms of conjugates of the matrix $U=(\\xymatrix{1 & 1 0 & 1})$, of some matrices representing the conjugacy classes of those elements of $SL(2,Z)$ arising as the monodromy around a singular fiber in an elliptic fibration (i.e. those matrices that appear in Kodaira's list)."}
{"category": "Math", "title": "Instability of nonlinear dispersive solitary waves", "abstract": "We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to properties of solitary waves. A continuation argument with a moving kernel formula are used to find the instability criteria. Recently, these techniques have also been extended to study instability of periodic waves and to the full water wave problem."}
{"category": "Math", "title": "The dimensions of LU(3,q) codes", "abstract": "A family of LDPC codes, called LU(3,q) codes, has been constructed from q-regular bipartite graphs. Recently, P. Sin and Q. Xiang determined the dimensions of these codes in the case that q is a power of an odd prime. They also obtained a lower bound for the dimension of an LU(3,q) code when q is a power of 2. In this paper we prove that this lower bound is the exact dimension of the LU(3,q) code. The proof involves the geometry of symplectic generalized quadrangles, the representation theory of Sp(4,q), and the ring of polynomials."}
{"category": "Math", "title": "Time series analysis via mechanistic models", "abstract": "The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae."}
{"category": "Math", "title": "A moment problem for pseudo-positive definite functionals", "abstract": "A moment problem is presented for a class of signed measures which are termed pseudo-positive. Our main result says that for every pseudo-positive definite functional (subject to some reasonable restrictions) there exists a representing pseudo-positive measure. The second main result is a characterization of determinacy in the class of equivalent pseudo-positive representation measures. Finally the corresponding truncated moment problem is discussed."}
{"category": "Math", "title": "Towards the Carpenter's Theorem", "abstract": "Let M be a II_1 factor, A a masa in M and E the unique conditional expectation on A. Under some technical assumptions on the inclusion of A in M, which hold true for any semiregular masa of a separable factor, we show that for every discrete a in the positive part of the unit ball of A it is possible to find a projection p in M such that E(p)=a$. We also show an example of a diffuse operator x in A such that there exists a projection q in M with E(q)=x. These results show a new family of instances of a conjecture by Kadison, the so-called \"Carpenter's Theorem\"."}
{"category": "Math", "title": "Intersections and joins of free groups", "abstract": "Let H and K be subgroups of a free group of ranks h and k \\geq h. We prove the following strong form of Burns' inequality: rank(H \\cap K) - 1 \\leq 2(h-1)(k-1) - (h-1)(rank(H \\vee K) -1). A corollary of this, also obtained by L. Louder and D. B. McReynolds, has been used by M. Culler and P. Shalen to obtain information regarding the volumes of hyperbolic 3-manifolds. We also prove the following particular case of the Hanna Neumann Conjecture, which has also been obtained by Louder. If the join of H and K has rank at least (h + k + 1)/2, then the intersection of H and K has rank no more than (h-1)(k-1) + 1."}
{"category": "Math", "title": "Degenerate Stochastic Differential Equations for Catalytic Branching Networks", "abstract": "Uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks is proven. This work is an extension of a paper by Dawson and Perkins to arbitrary catalytic branching networks. As part of the proof estimates on the corresponding semigroup are found in terms of weighted Holder norms for arbitrary networks, which are proven to be equivalent to the semigroup norm for this generalized setting. ----- On prouve l'unicite d'un probleme de martingale correspondant a une EDS degeneree, qui apparait comme un modele de reseaux avec branchement catalytique. Ce travail est une extension des resultats de Dawson et Perkins au cas de reseaux generaux. On obtient en particulier des estimees pour le semi-groupe des reseaux generaux, sous forme de normes de Holder ponderees; et on etablit l'equivalence de ces normes avec des normes de semi-groupe dans ce contexte general."}
{"category": "Math", "title": "An introduction to the volume conjecture and its generalizations", "abstract": "In this paper we give an introduction to the volume conjecture and its generalizations. Especially we discuss relations of the asymptotic behaviors of the colored Jones polynomials of a knot with different parameters to representations of the fundamental group of the knot complement at the special linear group over complex numbers by taking the figure-eight knot and torus knots as examples."}
{"category": "Math", "title": "Existence of global invariant jet differentials on projective hypersurfaces of high degree", "abstract": "Let $X\\subset\\mathbb P^{n+1}$ be a smooth complex projective hypersurface. In this paper we show that, if the degree of $X$ is large enough, then there exist global sections of the bundle of invariant jet differentials of order $n$ on $X$, vanishing on an ample divisor. We also prove a logarithmic version, effective in low dimension, for the log-pair $(\\mathbb P^n,D)$, where $D$ is a smooth irreducible divisor of high degree. Moreover, these result are sharp, \\emph{i.e.} one cannot have such jet differentials of order less than $n$."}
{"category": "Math", "title": "Classifying Brumer's quintic polynomials by weak Mordell-Weil groups", "abstract": "We develop a general classification theory for Brumer's dihedral quintic polynomials by means of Kummer theory arising from certain elliptic curves. We also give a similar theory for cubic polynomials."}
{"category": "Math", "title": "Weighted Strichartz Estimates with Angular Regularity and their Applications", "abstract": "In this paper, we establish an optimal dual version of trace estimate involving angular regularity. Based on this estimate, we get the generalized Morawetz estimates and weighted Strichartz estimates for the solutions to a large class of evolution equations, including the wave and Schr\\\"{o}dinger equation. As applications, we prove the Strauss' conjecture with a kind of mild rough data for $2\\le n\\le 4$, and a result of global well-posedness with small data for some nonlinear Schr\\\"{o}dinger equation with $L^2$-subcritical nonlinearity."}
{"category": "Math", "title": "V-Variable Fractals: Fractals with Partial Self Similarity", "abstract": "We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V-variable fractal sets or measures. These V-variable fractals can be obtained from the points on the attractor of a single deterministic iterated function system. Existence, uniqueness and approximation results are established under average contractive assumptions. We also obtain extensions of some basic results concerning iterated function systems."}
{"category": "Math", "title": "Quantizations of the $W$ Algebra W(2,2)", "abstract": "We quantize the $W$-algebra W(2,2), whose Verma modules, Harish-Chandra modules, irreducible weight modules and Lie bialgebra structures have been investigated and determined in a series of papers recently."}
{"category": "Math", "title": "Renormalization for a Class of Dynamical Systems: some Local and Global Properties", "abstract": "We study the period doubling renormalization operator for dynamics which present two coupled laminar regimes with two weakly expanding fixed points. We focus our analysis on the potential point of view, meaning we want to solve $$V=\\mathcal{R} (V):=V\\circ f\\circ h+V \\circ h,$$ where $f$ and $h$ are naturally defined. Under certain hypothesis we show the existence of a explicit ``attracting'' fixed point $V^*$ for $\\mathcal{R} $. We call $\\mathcal{R}$ the renormalization operator which acts on potentials $V$. The log of the derivative of the main branch of the Manneville-Pomeau map appears as a special ``attracting'' fixed point for the local doubling period renormalization operator. We also consider an analogous definition for the one-sided 2-full shift $\\S$ (and also for the two-sided shift) and we obtain a similar result. Then, we consider global properties and we prove two rigidity results. Up to some weak assumptions, we get the uniqueness for the renormalization operator in the shift. In the last section we show (via a certain continuous fraction expansion) a natural relation of the two settings: shift acting on the Bernoulli space $\\{0,1\\}^\\mathbb{N}$ and Manneville-Pomeau-like map acting on an interval."}
{"category": "Math", "title": "Continuous and Random Vapnik-Chervonenkis Classes", "abstract": "We show that if $T$ is a dependent theory then so is its Keisler randomisation $T^R$. In order to do this we generalise the notion of a Vapnik-Chervonenkis class to families of $[0,1]$-valued functions (a \\emph{continuous} Vapnik-Chervonenkis class), and we characterise families of functions having this property via the growth rate of the mean width of an associated family of convex compacts."}
{"category": "Math", "title": "Nonparametric Bayesian model selection and averaging", "abstract": "We consider nonparametric Bayesian estimation of a probability density $p$ based on a random sample of size $n$ from this density using a hierarchical prior. The prior consists, for instance, of prior weights on the regularity of the unknown density combined with priors that are appropriate given that the density has this regularity. More generally, the hierarchy consists of prior weights on an abstract model index and a prior on a density model for each model index. We present a general theorem on the rate of contraction of the resulting posterior distribution as $n\\to \\infty$, which gives conditions under which the rate of contraction is the one attached to the model that best approximates the true density of the observations. This shows that, for instance, the posterior distribution can adapt to the smoothness of the underlying density. We also study the posterior distribution of the model index, and find that under the same conditions the posterior distribution gives negligible weight to models that are bigger than the optimal one, and thus selects the optimal model or smaller models that also approximate the true density well. We apply these result to log spline density models, where we show that the prior weights on the regularity index interact with the priors on the models, making the exact rates depend in a complicated way on the priors, but also that the rate is fairly robust to specification of the prior weights."}
{"category": "Math", "title": "On the Supremum of Some Random Dirichlet Polynomials", "abstract": "We study the supremum of some random Dirichlet polynomials with independent coefficients and obtain sharp upper and lower bounds for supremum expectation thus extending the results from our previous work (see http://arXiv.org/abs/math/0703691). Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method."}
{"category": "Math", "title": "Motzkin numbers, central trinomial coefficients and hybrid polynomials", "abstract": "We show that the formalism of hybrid polynomials, interpolating between Hermite and Laguerre polynomials, is very useful in the study of Motzkin numbers and central trinomial coefficients. These sequences are identified as special values of hybrid polynomials, a fact which we use to derive their generalized forms and new identities satisfied by them."}
{"category": "Math", "title": "Central limit theorem for Hotelling's $T^2$ statistic under large dimension", "abstract": "In this paper we prove the central limit theorem for Hotelling's $T^2$ statistic when the dimension of the random vectors is proportional to the sample size."}
{"category": "Math", "title": "On principally generated Q-modules in general, and skew local homeomorphisms in particular", "abstract": "Ordered sheaves on a small quantaloid Q have been defined in terms of Q-enriched categorical structures; they form a locally ordered category Ord(Q). The free-cocompletion KZ-doctrine on Ord(Q) has Mod(Q), the quantaloid of Q-modules, as category of Eilenberg-Moore algebras. In this paper we give an intrinsic description of the Kleisli algebras: we call them the 'locally principally generated Q-modules'. We deduce that Ord(Q) is biequivalent to the 2-category of locally principally generated Q-modules and left adjoint module morphisms. The example of locally principally generated modules on a locale X is worked out in full detail: relating X-modules to objects of the slice category Loc/X, we show that ordered sheaves on X correspond with 'skew local homeomorphisms into X' (like sheaves on X correspond with local homeomorphisms into X)."}
{"category": "Math", "title": "A direct proof of one Gromov's theorem", "abstract": "We give a new proof of the Gromov theorem: For any $C>0$ and integer $n>1$ there exists a function $\\Delta_{C,n}$ such that if the Gromov--Hausdorff distance between complete Riemannian $n$-manifolds $V$ and $W$ is not greater than $\\delta$, absolute values of their sectional curvatures $|K_{\\sigma}|\\leq C$, and their injectivity radii $\\geq 1/C$, then the Lipschitz distance between $V$ and $W$ is less than $\\Delta_{C,n}(\\delta)$ and $\\Delta_{C,n}\\to 0$ as $\\delta\\to 0$."}
{"category": "Math", "title": "The Riemann hypothesis for Weng's zeta function of $Sp(4)$ over $\\mathbb{Q}$", "abstract": "As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple poles. The rank one zeta function is the Dedekind zeta function. For the rank two case, the Riemann hypothesis is proved for a general number field. Recently, he defined more general new zeta function associated to a pair of reductive group and its maximal parabolic subgroup. As well as high rank zeta functions, the new zeta function satisfies standard properties of zeta functions.In this paper, we prove that the Riemann hypothesis of Weng's zeta function attached to the sympletic group of degree four.This paper includes an appendix written by L. Weng, in which he explains a general construction for zeta functions associated to $Sp(2n)$."}
{"category": "Math", "title": "Zeta functions for $G_2$ and their zeros", "abstract": "The exceptional group $G_2$ has two maximal parabolic subgroups $P_{long}$, $P_{short}$ corresponding to the so-called long root and short root. In this paper, the second author introduces two zeta functions associated to $(G_2,P_{long})$ and $(G_2,P_{short})$ respectively, and the first author proves that these zetas satisfy the Riemann Hypothesis."}
{"category": "Math", "title": "Properties of cellular classes of chain complexes", "abstract": "In this paper we prove certain properties of cellular and acyclic classes of chain complexes of modules over a commutative Noetherian ring. In particular we show that if X is finite and belongs to some cellular class C then \\Sigma^n H_X also belongs to C, for every n."}
{"category": "Math", "title": "Quadratic enhancements of surfaces: two vanishing results", "abstract": "This note records two results which were inexplicably omitted from our paper on Pin structures on low dimensional manifolds, [KT]. Kirby chose not to be listed as a coauthor. A Pin^- structure on a surface F induces a quadratic enhancement of the mod 2 intersection form, q: H_1(F;Z/2Z) -> Z/4Z Theorem 1.1 says that q vanishes on the kernel of the map in homology to a bounding 3-manifold. This is used by Kreck and Puppe (arXiv:0707.1599 [math.AT]) who refer for a proof to an email of the author to Kreck. A more polished and public proof seems desirable. In [KT], section 6, a Pin^- structure is constructed on a surface F dual to w_2 in an oriented 4-manifold M^4. Theorem 2.1 says that q vanishes on the Poincare dual to the image of H^1(M^4;Z/2Z) in H^1(F;Z/2Z)."}
{"category": "Math", "title": "Air Traffic Flow Management", "abstract": "Air Traffic Flow Management is the regulation of air traffic in order to avoid exceeding airport or flight sector capacity in handling traffic, and to ensure that available capacity is used efficiently. We have tried to explore the logic behind the claims by Bertsimas et.al about integral solutions to the LP relaxation of the Traffic Flow Management Problem(TFMP). Polyhedral theory only indicates that the stronger TFMP formulation of Bertsimas et.al might lead to integral solutions in some cases. Our computations indicate that the encouraging results reported by Bertsimas et.al are not merely fortuitous or due to their specific data set. Indeed, we found that the TFMP had integral solutions even in case of artificial data sets generated to include severe conflicts in the flight schedules. In our limited tests with 4-5 scenarios, we obtained non-integral solutions only once. This is of significant practical importance because, the LP relaxation can be solved even on small machines with low memory and processor speed."}
{"category": "Math", "title": "Groups with the same cohomology as their profinite completions", "abstract": "For any positive integer $n$, $\\mathcal{A}_n$ is the class of all groups $G$ such that, for $0\\leq i\\leq n$, $H^i(\\hat{G},A)\\cong H^i(G,A)$ for every finite discrete $\\hat{G}$-module $A$. We describe certain types of free products with amalgam and HNN extensions that are in some of the classes $\\mathcal{A}_n$. In addition, we investigate the residually finite groups in the class $\\mathcal{A}_2$."}
{"category": "Math", "title": "The escaping set of a quasiregular mapping", "abstract": "We show that if the maximum modulus of a quasiregular mapping f grows sufficiently rapidly then there exists a non-empty escaping set I(f) consisting of points whose forward orbits under iteration tend to infinity. This set I(f) has an unbounded component but, in contrast to the case of entire functions on the complex plane, the closure of I(f) may have a bounded component."}
{"category": "Math", "title": "Analytic torsions on contact manifolds", "abstract": "We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-type trace formulae."}
{"category": "Math", "title": "Generic separable metric structures", "abstract": "We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in descriptive set theoretic (topological) sense and in measure theoretic sense. In particular, it gives a new perspective on Vershik's theorems on genericity and randomness of Urysohn's space among separable metric spaces."}
{"category": "Math", "title": "Cohomology and removable subsets", "abstract": "Let X be a (connected and reduced) complex space. A q-collar of X is a bounded domain whose boundary is a union of a strongly q-pseudoconvex, a strongly q-pseudoncave and two flat (i.e. locally zero sets of pluriharmonic functions) hypersurfaces. Finiteness and vanishing cohomology theorems obtained in math/0503490 and math/0701549 for semi q-coronae are generalized in this context and lead to results on extension problem and removable sets for sections of coherent sheaves and analytic subsets."}
{"category": "Math", "title": "Superpotentials and Higher Order Derivations", "abstract": "We consider algebras defined from quivers with relations that are k-th order derivations of a superpotential, generalizing results of Dubois-Violette to the quiver case. We give a construction compatible with Morita equivalence, and show that many important algebras arise in this way, including McKay correspondence algebras for GL_n for all n, and four-dimensional Sklyanin algebras. More generally, we show that any N-Koszul, (twisted) Calabi-Yau algebra must have a (twisted) superpotential, and construct its minimal resolution in terms of derivations of the (twisted) superpotential. This yields an equivalence between N-Koszul twisted Calabi-Yau algebras A and algebras defined by a superpotential such that an associated complex is a bimodule resolution of A. Finally, we apply these results to give a description of the moduli space of four-dimensional Sklyanin algebras using the Weil representation of SL_2(Z/4)."}
{"category": "Math", "title": "Connections with skew-symmetric Ricci tensor on surfaces", "abstract": "Some known results on torsionfree connections with skew-symmetric Ricci tensor on surfaces are extended to connections with torsion, and Wong's canonical coordinate form of such connections is simplified."}
{"category": "Math", "title": "Elliptic periods for finite fields", "abstract": "We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Basis in the second family, the so-called normal elliptic basis are normal basis and allow fast (quasi linear) arithmetic. We prove that all extensions admit models of this kind."}
{"category": "Math", "title": "Two versions of a specific natural extension", "abstract": "We give two versions of the natural extension of a specific greedy beta-transformation with deleted digits. We use the natural extension to obtain an explicit expression for the invariant measure, equivalent to the Lebesgue measure, of this beta-transformation."}
{"category": "Math", "title": "On vertex algebras and their modules associated with even lattices", "abstract": "We study vertex algebras and their modules associated with possibly degenerate even lattices, using an approach somewhat different from others. Several known results are recovered and a number of new results are obtained. We also study modules for Heisenberg algebras and we classify irreducible modules satisfying certain conditions and obtain a complete reducibility theorem."}
{"category": "Math", "title": "On sumfree subsets of hypercubes", "abstract": "We consider the possible sizes of large sumfree sets contained in the discrete hypercube $\\{1,...,n\\}^k$, and we determine upper and lower bounds for the maximal size as $n$ becomes large. We also discuss a continuous analogue in which our lower bound remains valid and our upper bound can be strengthened, and we consider the generalization of both problems to $l$-fold-sumfree sets."}
{"category": "Math", "title": "L^p Estimates for Maximal Averages Along One-variable Vector Fields in R^2", "abstract": "We prove a conjecture of Lacey and Li in the case that the vector field depends only on one variable. Specifically: let v be a vector field defined on the unit square such that v(x,y) = (1,u(x)) for some measurable u from [0,1] to [0,1]. Fix a small parameter delta and let Z be the collection of rectangles R of a fixed width such that delta much of the vector field inside R is pointed in (approximately) the same direction as R. We show that the maximal averaging operator associated to the collection Z is bounded on L^p for p>1 with constants comparable to (delta)^(-1) ."}
{"category": "Math", "title": "Free Groups in Lattices", "abstract": "Let G be any locally compact, unimodular, metrizable group. The main result of this paper, roughly stated, is that if F<G is any finitely generated free group and \\Gamma < G any lattice, then up to a small perturbation and passing to a finite index subgroup, F is a subgroup of \\Gamma. If G/\\Gamma is noncompact then we require additional hypotheses that include G=SO(n,1)."}
{"category": "Math", "title": "Occupation time fluctuation limits of infinite variance equilibrium branching systems", "abstract": "We establish limit theorems for the fluctuations of the rescaled occupation time of a $(d,\\alpha,\\beta)$-branching particle system. It consists of particles moving according to a symmetric $\\alpha$-stable motion in $\\mathbb{R}^d$. The branching law is in the domain of attraction of a (1+$\\beta$)-stable law and the initial condition is an equilibrium random measure for the system (defined below). In the paper we treat separately the cases of intermediate $\\alpha/\\beta<d<(1+\\beta)\\alpha/\\beta$, critical $d=(1+\\beta)\\alpha/\\beta$ and large $d>(1+\\beta)\\alpha/\\beta $ dimensions. In the most interesting case of intermediate dimensions we obtain a version of a fractional stable motion. The long-range dependence structure of this process is also studied. Contrary to this case, limit processes in critical and large dimensions have independent increments."}
{"category": "Math", "title": "Dynamics on an infinite surface with the lattice property", "abstract": "Dynamical systems on an infinite translation surface with the lattice property are studied. The geodesic flow on this surface is found to be recurrent in all but countably many rational directions. Hyperbolic elements of the affine automorphism group are found to be nonrecurrent, and a precise formula regarding their action on cylinders is proven. A deformation of the surface in the space of translation surfaces is found, which \"behaves nicely\" with the geodesic flow and action of the affine automorphism group."}
{"category": "Math", "title": "Covariance estimation for multivariate conditionally Gaussian dynamic linear models", "abstract": "In multivariate time series, the estimation of the covariance matrix of the observation innovations plays an important role in forecasting as it enables the computation of the standardized forecast error vectors as well as it enables the computation of confidence bounds of the forecasts. We develop an on-line, non-iterative Bayesian algorithm for estimation and forecasting. It is empirically found that, for a range of simulated time series, the proposed covariance estimator has good performance converging to the true values of the unknown observation covariance matrix. Over a simulated time series, the new method approximates the correct estimates, produced by a non-sequential Monte Carlo simulation procedure, which is used here as the gold standard. The special, but important, vector autoregressive (VAR) and time-varying VAR models are illustrated by considering London metal exchange data consisting of spot prices of aluminium, copper, lead and zinc."}
{"category": "Math", "title": "Finite rank Bergman-Toeplitz and Bargmann-Toeplitz operators in many dimensions", "abstract": "The recent theorem by D. Luecking that finite rank Toeplitz-Bergman operators must be generated by a measure consisting of finitely many point masses is carried over to the many-dimensional case."}
{"category": "Math", "title": "Enumeration of totally real number fields of bounded root discriminant", "abstract": "We enumerate all totally real number fields F with root discriminant delta_F <= 14. There are 1229 such fields, each with degree [F:QQ] <= 9."}
{"category": "Math", "title": "SOS model partition function and the elliptic weight functions", "abstract": "We generalize a recent observation [arXiv:math/0610433] that the partition function of the 6-vertex model with domain-wall boundary conditions can be obtained by computing the projections of the product of the total currents in the quantum affine algebra $U_{q}(\\hat{\\mathfrak{sl}}_{2})$ in its current realization. A generalization is proved for the the elliptic current algebra [arXiv:q-alg/9703018,arXiv:q-alg/9601022]. The projections of the product of total currents are calculated explicitly and are represented as integral transforms of the product of the total currents. We prove that the kernel of this transform is proportional to the partition function of the SOS model with domain-wall boundary conditions."}
{"category": "Math", "title": "Computing fundamental domains for Fuchsian groups", "abstract": "We exhibit an algorithm to compute a Dirichlet domain for a cofinite Fuchsian group Gamma. As a consequence, we compute the invariants of Gamma, including an explicit finite presentation for Gamma."}
{"category": "Math", "title": "The Chern coefficients of local rings", "abstract": "The Chern numbers of the title are the first coefficients (after the multiplicities) of the Hilbert functions of various filtrations of ideals of a local ring $(R, \\mathfrak{m})$. For a Noetherian (good) filtration $\\mathcal{A}$ of $\\mathfrak{m}$-primary ideals, the positivity and bounds for $e_1(\\mathcal{A})$ are well-studied if $R$ is Cohen-Macaulay, or more broadly, if $R$ is a Buchsbaum ring or mild generalizations thereof. For arbitrary geometric local domains, we introduce techniques based on the theory of maximal Cohen-Macaulay modules and of extended multiplicity functions to establish the meaning of the positivity of $e_1(\\mathcal{A})$, and to derive lower and upper bounds for $e_1(\\mathcal{A})$."}
{"category": "Math", "title": "Limits of Solutions to a Parabolic Monge-Ampere Equation", "abstract": "We present the results from our earlier paper (arXiv:math/0602484) on the affine normal flow on noncompact convex hypersurfaces in affine space from a more PDE point of view, emphasizing the estimates involved. Our results concern the limits of solutions to a parabolic Monge-Ampere equation on $S^n$, where a sequence of smooth strictly convex initial value functions increase monotonically to a limiting initial value function which is infinite on at least a hemisphere of $S^n$. We prove long-time existence and instantaneous smoothing for quite general initial data, and we characterize ancient solutions as ellipsoids or paraboloids. We make essential use of estimates of Andrews and Gutierrez-Huang, and barriers due to Calabi."}
{"category": "Math", "title": "On noetherianity for logical formulas over fields", "abstract": "In this paper we consider noetherianity for formulas of propositional and predicate calculus over different fields. Three types of noetherianity are considered: standard noetherianity, logical noetherianity and denumerable noetherianity."}
{"category": "Math", "title": "Posterior mean and variance approximation for regression and time series problems", "abstract": "This paper develops a methodology for approximating the posterior first two moments of the posterior distribution in Bayesian inference. Partially specified probability models, which are defined only by specifying means and variances, are constructed based upon second-order conditional independence, in order to facilitate posterior updating and prediction of required distributional quantities. Such models are formulated particularly for multivariate regression and time series analysis with unknown observational variance-covariance components. The similarities and differences of these models with the Bayes linear approach are established. Several subclasses of important models, including regression and time series models with errors following multivariate $t$, inverted multivariate $t$ and Wishart distributions, are discussed in detail. Two numerical examples consisting of simulated data and of US investment and change in inventory data illustrate the proposed methodology."}
{"category": "Math", "title": "Real mixed Hodge structures", "abstract": "We identify the category of real mixed Hodge structures with the category of vector bundles with connections (not necessarily flat) on C, equivariant with respect to C^*. Here C is the complex plane considered as a 2-dimensional real manifold, and C^* is the multiplicative group of complex numbers considered as a real group."}
{"category": "Math", "title": "Multivariate control charts based on Bayesian state space models", "abstract": "This paper develops a new multivariate control charting method for vector autocorrelated and serially correlated processes. The main idea is to propose a Bayesian multivariate local level model, which is a generalization of the Shewhart-Deming model for autocorrelated processes, in order to provide the predictive error distribution of the process and then to apply a univariate modified EWMA control chart to the logarithm of the Bayes' factors of the predictive error density versus the target error density. The resulting chart is proposed as capable to deal with both the non-normality and the autocorrelation structure of the log Bayes' factors. The new control charting scheme is general in application and it has the advantage to control simultaneously not only the process mean vector and the dispersion covariance matrix, but also the entire target distribution of the process. Two examples of London metal exchange data and of production time series data illustrate the capabilities of the new control chart."}
{"category": "Math", "title": "Dynamic generalized linear models for non-Gaussian time series forecasting", "abstract": "The purpose of this paper is to provide a discussion, with illustrating examples, on Bayesian forecasting for dynamic generalized linear models (DGLMs). Adopting approximate Bayesian analysis, based on conjugate forms and on Bayes linear estimation, we describe the theoretical framework and then we provide detailed examples of response distributions, including binomial, Poisson, negative binomial, geometric, normal, log-normal, gamma, exponential, Weibull, Pareto, beta, and inverse Gaussian. We give numerical illustrations for all distributions (except for the normal). Putting together all the above distributions, we give a unified Bayesian approach to non-Gaussian time series analysis, with applications from finance and medicine to biology and the behavioural sciences. Throughout the models we discuss Bayesian forecasting and, for each model, we derive the multi-step forecast mean. Finally, we describe model assessment using the likelihood function, and Bayesian model monitoring."}
{"category": "Math", "title": "Spherical Stein manifolds and the Weyl involution", "abstract": "It is proved that a Stein manifold acted on by a connected compact Lie group is spherical if and only if there exists an antiholomorphic involution preserving each orbit of the action. This involution can be chosen equivariant with respect to a Weyl involution of the group."}
{"category": "Math", "title": "Simplified proof of the Theorem of Varopoulos in the commutative case", "abstract": "We give continuity properties of bitraces on (possibly non-commutative) Banach *-algebras based on the Closed Graph Theorem, leading to a simplified proof of the Theorem of Varopoulos in the commutative case."}
{"category": "Math", "title": "Integral means of the derivatives of Blaschke products", "abstract": "We study the rate of growth of some integral means of the derivatives of a Blaschke product and we generalize several classical results. Moreover, we obtain the rate of growth of integral means of the derivative of functions in the model subspace $K_B$ generated by the Blaschke product $B$"}
{"category": "Math", "title": "Equivariant Primary Decomposition and Toric Sheaves", "abstract": "We study global primary decompositions in the category of sheaves on a scheme which are equivariant under the action of an algebraic group. We show that equivariant primary decompositions exist if the group is connected. As main application we consider the case of varieties which are quotients of a quasi-affine variety by the action of a diagonalizable group and thus admit a homogeneous coordinate ring, such as toric varieties. Comparing these decompositions with primary decompositions of graded modules over the homogeneous coordinate ring, we show that these are equivalent if the action of the diagonalizable group is free. We give some specific examples for the case of toric varieties."}
{"category": "Math", "title": "Geodesic excursions into an embedded disc on a hyperbolic Riemann surface", "abstract": "We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point."}
{"category": "Math", "title": "Geometrical representations of equiaffine curvature operators", "abstract": "We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent bundle of some smooth manifold."}
{"category": "Math", "title": "A quadratic regression problem for two-state algebras with application to the Central Limit Theorem", "abstract": "We extend a free version of the Laha-Lukacs theorem to probability spaces with two-states. We then use this result to generalize a noncommutative CLT of Kargin to the two-state setting."}
{"category": "Math", "title": "Generic exponential sums associated to Laurent polynomials in one variable", "abstract": "Generic Newton polygons for L-functions of exponential sums associated to Laurent polynomials in one variable are determined. The corresponding Hasse polynomials are also determined."}
{"category": "Math", "title": "The linear dual of the derived category of a scheme", "abstract": "This paper has been withdrawn because Proposition 2.2 (c) is false. This invalids the main results of section 2 and 3. We thank A. Canonaco for pointing us the error."}
{"category": "Math", "title": "Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of C$^*$-algebras", "abstract": "In the paper, we introduce a new concept of topological orbit dimension of $n$-tuples of elements in a unital C$^*$ algebra. Using this concept, we conclude that the Voiculescu's topological free entropy dimension of any family of self-adjoint generators of a nuclear C$^*$ algebra is less than or equal to 1. We also show that the topological free entropy dimension is additive in the full free products of unital C$^*$ algebras. In the appendix, we show that unital full free product of Blackadar and Kirchberg's unital MF algebras is also MF algebra."}
{"category": "Math", "title": "Galois invariant smoothness basis", "abstract": "This text answers a question raised by Joux and the second author about the computation of discrete logarithms in the multiplicative group of finite fields. Given a finite residue field $\\bK$, one looks for a smoothness basis for $\\bK^*$ that is left invariant by automorphisms of $\\bK$. For a broad class of finite fields, we manage to construct models that allow such a smoothness basis. This work aims at accelerating discrete logarithm computations in such fields. We treat the cases of codimension one (the linear sieve) and codimension two (the function field sieve)."}
{"category": "Math", "title": "Harnack Inequality and Applications for Stochastic Evolution Equations with Monotone Drifts", "abstract": "In this paper, the dimension-free Harnack inequality is proved for the associated transition semigroups to a large class of stochastic evolution equations with monotone drifts. As applications, the ergodicity, hyper-(or ultra-)contractivity and compactness are established for the corresponding transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence of a spectral gap are also derived. The main results are applied to many concrete stochastic evolution equations such as stochastic reaction-diffusion equations, stochastic porous media equations and the stochastic p-Laplace equation in Hilbert space."}
{"category": "Math", "title": "An Elementary Proof of the Free-additivity of Voiculescu's Free Entropy", "abstract": "D. Voiculescu [2] proved that a standard family of independent random unitary k by k matrices and a constant k by k unitary matrix is asymtotically free as k goes to infinity. This result was a key ingredient in Voiculescu's proof [3] that his free entropy is additive when the variables are free. In this paper, we give a very elementary proof of a more detailed version of this result [2]."}
{"category": "Math", "title": "On the conformal scalar curvature equation and related problems", "abstract": "We review recent compactness and non-compactness results for the Yamabe equation. We also discuss the asymptotic behavior of the parabolic Yamabe flow."}
{"category": "Math", "title": "A characterization of Koiso's typed solitons", "abstract": "By extending Koiso's examples to the non-compact case, we construct complete gradient Kahler-Ricci solitons of various types on certain holomorphic line bundles over compact Kahler-Einstein manifolds. Moreover, a uniformization result on steady gradient Kahler-Ricci solitons with non-negative Ricci curvature is obtained under additional assumptions."}
{"category": "Math", "title": "A Symplectic Isotopy of a Dehn Twist on CP^n x CP^{n+1}", "abstract": "The complex manifold CP^n x CP^{n+1} with symplectic form \\sigma_\\mu=\\sigma_{CP^n}+\\mu\\sigma_{CP^{n+1}}, where \\sigma_{CP^n} and \\sigma_{CP^{n+1}} are normalized Fubini-Study forms, n a natural number and \\mu>1 a real number, contains a natural Lagrangian sphere L^{\\mu}. We prove that the Dehn twist along L^{\\mu} is symplectically isotopic to the identity for all \\mu>1. This isotopy can be chosen so that it pointwise fixes a complex hypersurface in CP^n x CP^{n+1} and lifts to the blow-up of CP^n x CP^{n+1} along a complex n-dimensional submanifold."}
{"category": "Math", "title": "On a characterization of the complex hyperbolic space", "abstract": "Consider a compact K\\\"{a}hler manifold $M^m$ with Ricci curvature lower bound $Ric_M\\geq -2(m+1) .$ Assume that its universal cover $% \\widetilde{M}$ has maximal bottom of spectrum $\\lambda_1(\\widetilde{M}%) =m^2.$ Then we prove that $\\widetilde{M}$ is isometric to the complex hyperbolic space $\\Bbb{CH}^m.$"}
{"category": "Math", "title": "Fourier series and approximation on hexagonal and triangular domains", "abstract": "Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Ces\\`aro summability of Fourier series, degree of approximation and best approximation by trigonometric functions, both direct and inverse theorems. One of the objective of this study is to demonstrate that Fourier series on spectral sets enjoy a rich structure that allow an extensive theory for Fourier expansions and approximation."}
{"category": "Math", "title": "Spectral flow invariants and twisted cyclic theory from the Haar state on SU_q(2)", "abstract": "In [CPR2], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SU_q(2), and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [CPR2], the computations are considerably more complex and interesting, because there are nontrivial `eta' contributions to this index."}
{"category": "Math", "title": "Linking integrals in the n-sphere", "abstract": "Let K and L be disjoint closed oriented submanifolds of the n-sphere, with dimensions adding up to n-1. We define a map from their join K*L to the n-sphere whose degree up to sign equals their linking number, and then use this to find the desired linking integral."}
{"category": "Math", "title": "Factorization of the Indefinite Convection-Diffusion Operator", "abstract": "We prove that some non-self-adjoint differential operator admits factorization and apply this new representation of the operator to construct explicitly its domain. We also show that this operator is J-self-adjoint in some Krein space."}
{"category": "Math", "title": "Nondifferentiable functions of one-dimensional semimartingales", "abstract": "We consider decompositions of processes of the form $Y=f(t,X_t)$ where $X$ is a semimartingale. The function $f$ is not required to be differentiable, so It\\^{o}'s lemma does not apply. In the case where $f(t,x)$ is independent of $t$, it is shown that requiring $f$ to be locally Lipschitz continuous in $x$ is enough for an It\\^{o}-style decomposition to exist. In particular, $Y$ will be a Dirichlet process. We also look at the case where $f(t,x)$ can depend on $t$, possibly discontinuously. It is shown, under some additional mild constraints on $f$, that the same decomposition still holds. Both these results follow as special cases of a more general decomposition which we prove, and which applies to nondifferentiable functions of Dirichlet processes. Possible applications of these results to the theory of one-dimensional diffusions are briefly discussed."}
{"category": "Math", "title": "Cobordisms of fold maps of 4-manifolds into the space", "abstract": "We compute the oriented cobordism group of fold maps of 4-manifolds into the space with all the possible restrictions (and also with no restriction) to the singular fibers. We also give geometric invariants which describe completely the cobordism group of fold maps."}
{"category": "Math", "title": "Homeomorphism and diffeomorphism groups of non-compact manifolds with the Whitney topology", "abstract": "For a non-compact n-manifold M let H(M) denote the group of homeomorphisms of M endowed with the Whitney topology and H_c(M) the subgroup of H(M) consisting of homeomorphisms with compact support. It is shown that the group H_c(M) is locally contractible and the identity component H_0(M) of H(M) is an open normal subgroup in H_c(M). This induces the topological factorization H_c(M) \\approx H_0(M) \\times \\M_c(M) for the mapping class group \\M_c(M) = H_c(M)/H_0(M) with the discrete topology. Furthermore, for any non-compact surface M, the pair (H(M), H_c(M)) is locally homeomorphic to (\\square^w l_2,\\cbox^w l_2) at the identity id_M of M. Thus the group H_c(M) is an (l_2 \\times R^\\infty)-manifold. We also study topological properties of the group D(M) of diffeomorphisms of a non-compact smooth n-manifold M endowed with the Whitney C^\\infty-topology and the subgroup D_c(M) of D(M) consisting of all diffeomorphisms with compact support. It is shown that the pair (D(M),D_c(M)) is locally homeomorphic to (\\square^w l_2, \\cbox^w l_2) at the identity id_M of M. Hence the group D_c(M) is a topological (l_2 \\times R^\\infty)-manifold for any dimension n."}
{"category": "Math", "title": "Improved mixing time bounds for the Thorp shuffle and L-reversal chain", "abstract": "We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models. E. Thorp introduced the following card shuffling model in 1973. Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We obtain a mixing time bound of O(log^4 n). Previously, the best known bound was O(log^{29} n) and previous proofs were only valid for n a power of 2. We also analyze the following model, called the L-reversal chain, introduced by Durrett. There are n cards arrayed in a circle. Each step, an interval of cards of length at most L is chosen uniformly at random and its order is reversed. Durrett has conjectured that the mixing time is O(max(n, n^3/L^3) log n). We obtain a bound that is within a factor O(log^2 n) of this,the first bound within a poly log factor of the conjecture."}
{"category": "Math", "title": "Approximate innerness and central triviality of endomorphisms", "abstract": "We introduce the notions of approximate innerness and central triviality for endomorphisms on separable von Neumann factors, and we characterize them for hyperfinite factors by Connes-Takesaki modules of endomorphisms and modular endomorphisms which are introduced by Izumi. Our result is a generalization of the corresponding result obtained by Kawahigashi-Sutherland-Takesaki in automorphism case."}
{"category": "Math", "title": "Exact exponential bounds for the random field maximum distribution via the majoring measures (generic chaining)", "abstract": "In this paper non-asymptotic exact exponential estimates are derived for the tail of maximum distribution of random field in the terms of majoring measures or, equally, generic chaining."}
{"category": "Math", "title": "Polynomial Poisson structures on affine solvmanifolds", "abstract": "A $n$-dimensional Lie group $G$ equipped with a left invariant symplectic form $\\om^+$ is called a symplectic Lie group. It is well-known that $\\om^+$ induces a left invariant affine structure on $G$. Relatively to this affine structure we show that the left invariant Poisson tensor $\\pi^+$ corresponding to $\\om^+$ is polynomial of degree 1 and any right invariant $k$-multivector field on $G$ is polynomial of degree at most $k$. If $G$ is unimodular, the symplectic form $\\om^+$ is also polynomial and the volume form $\\wedge^{\\frac{n}2}\\om^+$ is parallel. We show also that any left invariant tensor field on a nilpotent symplectic Lie group is polynomial, in particular, any left invariant Poisson structure on a nilpotent symplectic Lie group is polynomial. Because many symplectic Lie groups admit uniform lattices, we get a large class of polynomial Poisson structures on compact affine solvmanifolds."}
{"category": "Math", "title": "Hamiltonian stationary cones and self-similar solutions in higher dimension", "abstract": "In [LW], we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak formulation of the mean curvature flow. Moreover, there is no mass loss along the Brakke flow. In this paper, we generalize these results to higher dimension. We construct new higher dimensional Hamiltonian stationary cones of different topology as generalizations of the Schoen-Wolfson cones. Hamiltonian stationary self-shrinkers and self-expanders that are asymptotic to these Hamiltonian stationary cones are also constructed. They can also be glued together to produce eternal solutions of the Brakke flow without mass loss. Finally, we show the same conclusion holds for those Lagrangian self-similar examples recently found by Joyce, Tsui and the first author in [JLT]."}
{"category": "Math", "title": "Automorphic forms of higher order", "abstract": "In this paper a theory of Hecke operators for higher order modular forms is established. The definition of cusp forms and attached L-functions is extended beyond the realm of parabolic invariants. The role of representation theoretic methods is clarified and, motivated by higher order forms, new convolution products of L-functions are introduced."}
{"category": "Math", "title": "Symplectic Automorphisms on Kummer Surfaces", "abstract": "Nikulin proved that the isometries induced on the second cohomology group of a K3 surface $X$ by a finite abelian group $G$ of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of $H^2(X, \\Z)$ which is fixed by the isometries induced by $G$. However for certain groups these discriminants are not the same of those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference."}
{"category": "Math", "title": "Jensen's Inequality for g-Convex Function under g-Expectation", "abstract": "A real valued function defined on}$\\mathbb{R}$ {\\small is called}$g${\\small --convex if it satisfies the following \\textquotedblleft generalized Jensen's inequality\\textquotedblright under a given}$g${\\small -expectation, i.e., }$h(\\mathbb{E}^{g}[X])\\leq \\mathbb{E}% ^{g}[h(X)]${\\small, for all random variables}$X$ {\\small such that both sides of the inequality are meaningful. In this paper we will give a necessary and sufficient conditions for a }$C^{2}${\\small -function being}$% g ${\\small -convex. We also studied some more general situations. We also studied}$g${\\small -concave and}$g${\\small -affine functions."}
{"category": "Math", "title": "The obstacle problem for nonlinear elliptic equations with variable growth and L^1-data", "abstract": "The aim of this paper is twofold: to prove, for L^1-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy--Stampacchia inequalities to the general framework of L^1."}
{"category": "Math", "title": "A generalized Fourier inversion theorem", "abstract": "In this work we define operator-valued Fourier transforms for suitable integrable elements with respect to the Plancherel weight of a (not necessarily Abelian) locally compact group. Our main result is a generalized version of the Fourier inversion Theorem for strictly-unconditionally integrable Fourier transforms. Our results generalize and improve those previously obtained by Ruy Exel in the case of Abelian groups."}
{"category": "Math", "title": "Absolute continuity and singularity of two probability measures on a filtered space", "abstract": "Let $\\mu$ and $\\nu$ be fixed probability measures on a filtered space $(\\Omega, {\\cal F}, ({\\cal F}_t)_{t\\in {\\bf R}^{+}})$. Denote by $\\mu_T $ and $\\nu_T $ (respectively, $\\mu_{T-} $ and $\\nu_{T-} $) the restrictions of the measures $\\mu$ and $\\nu$ on ${\\cal F}_T $ (respectively, on ${\\cal F}_{T-} $) for a stopping time $T$. We find the Hahn decomposition of $\\mu_T $ and $\\nu_T $ using the Hahn decomposition of the measures $\\mu$, $\\nu$, and the Hellinger process $h_t$ in the strict sense of order 1/2. The norm of the absolutely continuous component of $\\mu_{T-} $ with respect to $\\nu_{T-} $ is computed in terms of density processes and Hellinger integrals."}
{"category": "Math", "title": "On bounded solutions of a problem of R. Schilling", "abstract": "The paper deals with locally bounded solutions of a Schilling's problem."}
{"category": "Math", "title": "CAT(0) groups and Coxeter groups whose boundaries are scrambled sets", "abstract": "In this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group $G$ acts geometrically (i.e. properly and cocompactly by isometries) on a CAT(0) space $X$. (Such group $G$ is called a {\\it CAT(0) group}.) Then the group $G$ acts by homeomorphisms on the boundary $\\partial X$ of $X$ and we can define a metric $d_{\\partial X}$ on the boundary $\\partial X$. The boundary $\\partial X$ is called a {\\it scrambled set} if for any $\\alpha,\\beta\\in\\partial X$ with $\\alpha\\neq\\beta$, (1) $\\limsup\\{d_{\\partial X}(g\\alpha,g\\beta) | g\\in G\\}>0$ and (2) $\\liminf\\{d_{\\partial X}(g\\alpha,g\\beta) | g\\in G\\}=0$. We investigate when are boundaries of CAT(0) groups (and Coxeter groups) scrambled sets."}
{"category": "Math", "title": "Generalised energy conservation law for the wave equations with variable propagation speed", "abstract": "We investigate the long time behaviour of the $L^2$-energy of solutions to wave equations with variable speed. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property."}
{"category": "Math", "title": "On principal hook length partitions and durfee sizes in skew characters", "abstract": "In this paper we construct for a given arbitrary skew diagram A all partitions nu with maximal principal hook lengths among all partitions with the character [nu] appearing in the skew character [A]. Furthermore we show that these are also partitions with minimal Durfee size. This we use to give the maximal Durfee size for [nu] appearing in [A] for the cases when A decays into two partitions and for some special cases of A. Also this gives conditions for two skew diagrams to represent the same skew character."}
{"category": "Math", "title": "On nondegeneracy of curves", "abstract": "A curve is called nondegenerate if it can be modeled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We show that up to genus 4, every curve is nondegenerate. We also prove that the locus of nondegenerate curves inside the moduli space of curves of fixed genus g > 1 is min(2g+1,3g-3)-dimensional, except in case g=7 where it is 16-dimensional."}
{"category": "Math", "title": "Une note \\`a propos du Jacobien de $n$ fonctions holomorphes \\`a l'origine de $\\mathbb{C}^n$", "abstract": "Let $f_1,...,f_n$ be $n$ germs of holomorphic functions at the origin of $\\mathbb{C}^n$ such that $f_i(0)=0$, $1\\leq i\\leq n$. We give a proof based on the J. Lipman's theory of residues via Hochschild Homology that the Jacobian of $f_1,...,f_n$ belongs to the ideal generated by $f_1,...,f_n$ belongs to the ideal generated by $f_1,...,f_n$ if and only if the dimension ot the germ of common zeos of $f_1,...,f_n$ is sttrictly positive. In fact we prove much more general results which are relatives versions of this result replacing the field $\\mathbb{C}$ by convenient noetherian rings $\\mathbf{A}$ (c.f. Th. 3.1 and Th. 3.3). We then show a \\L ojasiewicz inequality for the jacobian analogous to the classical one by S. \\L ojasiewicz for the gradient."}
{"category": "Math", "title": "Rank and regularity for averages over submanifolds", "abstract": "This paper establishes endpoint $L^p-L^q$ and Sobolev mapping properties of Radon-like operators which satisfy a homogeneity condition (similar to semiquasihomogeneity) and a condition on the rank of a matrix related to rotational curvature. For highly degenerate operators, the rank condition is generically satisfied for algebraic reasons, similar to an observation of Greenleaf, Pramanik, and Tang concerning oscillatory integral operators."}
{"category": "Math", "title": "A regional Bayesian POT model for flood frequency analysis", "abstract": "Flood frequency analysis is usually based on the fitting of an extreme value distribution to the local streamflow series. However, when the local data series is short, frequency analysis results become unreliable. Regional frequency analysis is a convenient way to reduce the estimation uncertainty. In this work, we propose a regional Bayesian model for short record length sites. This model is less restrictive than the index flood model while preserving the formalism of \"homogeneous regions\". The performance of the proposed model is assessed on a set of gauging stations in France. The accuracy of quantile estimates as a function of the degree of homogeneity of the pooling group is also analysed. The results indicate that the regional Bayesian model outperforms the index flood model and local estimators. Furthermore, it seems that working with relatively large and homogeneous regions may lead to more accurate results than working with smaller and highly homogeneous regions."}
{"category": "Math", "title": "Modeling All Exceedances Above a Threshold Using an Extremal Dependence Structure: Inferences on Several Flood Characteristics", "abstract": "Flood quantile estimation is of great importance for many engineering studies and policy decisions. However, practitioners must often deal with small data available. Thus, the information must be used optimally. In the last decades, to reduce the waste of data, inferential methodology has evolved from annual maxima modeling to peaks over a threshold one. To mitigate the lack of data, peaks over a threshold are sometimes combined with additional information - mostly regional and historical information. However, whatever the extra information is, the most precious information for the practitioner is found at the target site. In this study, a model that allows inferences on the whole time series is introduced. In particular, the proposed model takes into account the dependence between successive extreme observations using an appropriate extremal dependence structure. Results show that this model leads to more accurate flood peak quantile estimates than conventional estimators. In addition, as the time dependence is taken into account, inferences on other flood characteristics can be performed. An illustration is given on flood duration. Our analysis shows that the accuracy of the proposed models to estimate the flood duration is related to specific catchment characteristics. Some suggestions to increase the flood duration predictions are introduced."}
{"category": "Math", "title": "A bilinear pseudodifferential calculus", "abstract": "In this paper, we are interested in the construction of a bilinear pseudodifferential calculus. We define some symbolic classes which contains those of Coifman-Meyer. These new classes allow us to consider operators closely related to the bilinear Hilbert transform. We give a description of the action of our bilinear operators on Sobolev spaces. These classes also have a ``nice'' behavior through the transposition and the composition operations that we will present."}
{"category": "Math", "title": "Exceptional sets for the derivatives of Blaschke products", "abstract": "We obtain growth estimates for the logarithmic derivative $B'(z)/B(z)$ of a Blaschke product as $|z| \\to 1$ and $z$ avoids some exceptional sets."}
{"category": "Math", "title": "Invariant differential operators and an infinite dimensional Howe-type correspondence. Part I: Structure of the associated algebras of differential operators", "abstract": "If $Q$ is a non degenerate quadratic form on ${\\bb C}^n$, it is well known that the differential operators $X=Q(x)$, $Y=Q(\\partial)$, and $H=E+\\frac{n}{2}$, where $E$ is the Euler operator, generate a Lie algebra isomorphic to ${\\go sl}_{2}$. Therefore the associative algebra they generate is a quotient of the universal enveloping algebra ${\\cal U}({\\go sl}_{2})$. This fact is in some sense the foundation of the metaplectic representation. The present paper is devoted to the study of the case where $Q(x)$ is replaced by $\\Delta_{0}(x)$, where $\\Delta_{0}(x)$ is the relative invariant of a prehomogeneous vector space of commutative parabolic type ($ {\\go g},V $), or equivalently where $\\Delta_{0}$ is the \"determinant\" function of a simple Jordan algebra $V$ over ${\\bb C}$. In this Part I we show several structure results for the associative algebra generated by $X=\\Delta_{0}(x)$, $Y=\\Delta_{0}(\\partial)$. Our main result shows that if we consider this algebra as an algebra over a certain commutative ring ${\\bf A}$ of invariant differential operators it is isomorphic to the quotient of what we call a generalized Smith algebra $S(f, {\\bf A}, n)$ where $f\\in {\\bf A}[t]$. The Smith algebras (over ${\\bb C}$) were introduced by P. Smith as \"natural\" generalizations of ${\\cal U}({\\go sl}_{2})$."}
{"category": "Math", "title": "The infinitesimal Hopf algebra and the poset of planar forests", "abstract": "We introduce an infinitesimal Hopf algebra of planar trees, generalising the construction of the non-commutative Connes-Kreimer Hopf algebra. A non-degenerate pairing and a dual basis are defined, and a combinatorial interpretation of the pairing in terms of orders on the vertices of planar forests is given. Moreover, the coproduct and the pairing can also be described with the help of a partial order on the set of planar forests, making it isomorphic to the Tamari poset. As a corollary, the dual basis can be computed with a M\\\"obius inversion."}
{"category": "Math", "title": "Global Sensitivity Analysis of Stochastic Computer Models with joint metamodels", "abstract": "The global sensitivity analysis method, used to quantify the influence of uncertain input variables on the response variability of a numerical model, is applicable to deterministic computer code (for which the same set of input variables gives always the same output value). This paper proposes a global sensitivity analysis methodology for stochastic computer code (having a variability induced by some uncontrollable variables). The framework of the joint modeling of the mean and dispersion of heteroscedastic data is used. To deal with the complexity of computer experiment outputs, non parametric joint models (based on Generalized Additive Models and Gaussian processes) are discussed. The relevance of these new models is analyzed in terms of the obtained variance-based sensitivity indices with two case studies. Results show that the joint modeling approach leads accurate sensitivity index estimations even when clear heteroscedasticity is present."}
{"category": "Math", "title": "Usefulness of the Reversible Jump Markov Chain Monte Carlo Model in Regional Flood Frequency Analysis", "abstract": "Regional flood frequency analysis is a convenient way to reduce estimation uncertainty when few data are available at the gauging site. In this work, a model that allows a non-null probability to a regional fixed shape parameter is presented. This methodology is integrated within a Bayesian framework and uses reversible jump techniques. The performance on stochastic data of this new estimator is compared to two other models: a conventional Bayesian analysis and the index flood approach. Results show that the proposed estimator is absolutely suited to regional estimation when only a few data are available at the target site. Moreover, unlike the index flood estimator, target site index flood error estimation seems to have less impact on Bayesian estimators. Some suggestions about configurations of the pooling groups are also presented to increase the performance of each estimator."}
{"category": "Math", "title": "Non-commutative connections of the second kind", "abstract": "A connection-like objects, termed {\\em hom-connections} are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature for a hom-connection is defined, and it is shown that flat hom-connections give rise to a chain complex."}
{"category": "Math", "title": "Jack polynomials and free cumulants", "abstract": "We study the coefficients in the expansion of Jack polynomials in terms of power sums. We express them as polynomials in the free cumulants of the transition measure of an anisotropic Young diagram. We conjecture that such polynomials have nonnegative integer coefficients. This extends recent results about normalized characters of the symmetric group."}
{"category": "Math", "title": "Hierarchical Additive Modeling of Nonlinear Association with Spatial Correlations-An Application to Relate Alcohol Outlet Density and Neighborhood Assault Rates", "abstract": "Previous studies have suggested a link between alcohol outlets and assaultive violence. In this paper, we explore the effects of alcohol availability on assault crimes at the census tract level over time. The statistical analysis is challenged by several features of the data: (1) the effects of possible covariates (for example, the alcohol outlet density of each census tract) on the assaultive crime rates may be complex; (2) the covariates may be highly correlated with each other; (3) there are a lot of missing inputs in the data; and (4) spatial correlations exist in the outcome assaultive crime rates. We propose a hierarchical additive model, where the nonlinear correlations and the complex interaction effects are modeled using the multiple additive regression trees (MART) and the spatial variances in the assaultive rates that cannot be explained by the specified covariates are smoothed trough the Conditional Autoregressive (CAR) model. We develop a two-stage algorithm that connect the non-parametric trees with CAR to look for important variables covariates associated with the assaultive crime rates, while taking account of the spatial correlations among adjacent census tracts. The proposed methods are applied to the Los Angeles assaultive data (1990-1999) and compared with traditional method."}
{"category": "Math", "title": "Qregularity and an Extension of Evans-Griffiths Criterion to Vector Bundles on Quadrics", "abstract": "Here we define the concept of Qregularity for coherent sheaves on quadrics. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on $\\Q_n\\subset \\mathbb P^{n+1}$ with the Castelnuovo-Mumford regularity of their extension by zero in $\\mathbb P^{n+1}$. We also classify the coherent sheaves with Qregularity $-\\infty$. We use our notion of Qregularity in order to prove an extension of Evans-Griffiths criterion to vector bundles on Quadrics. In particular we get a new and simple proof of the Kn\\\"{o}rrer's characterization of ACM bundles."}
{"category": "Math", "title": "Principal eigenvalues for Isaacs operators with Neumann boundary conditions", "abstract": "In this paper we show the existence of two principal eigenvalues associated to general non-convex fully nonlinear elliptic operators with Neumann boundary conditions in a bounded $C^2$ domain. We study these objects and we establish some of their basic properties. Finally, Lipschitz regularity, uniqueness and existence results for the solution of the Neumann problem are given."}
{"category": "Math", "title": "Geometric theta-lifting for the dual pair GSp_{2n}, GSO_{2m}", "abstract": "Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Consider the dual pair H=GSO_{2m}, G=GSp_{2n} over X, where H splits over an etale two-sheeted covering of X. Write Bun_G and Bun_H for the stacks of G-torsors and H-torsors on X. We show that for m\\le n (respectively, for m>n) the theta-lifting functor from D(Bun_H) to D(Bun_G) (respectively, from D(Bun_G) to D(Bun_H)) commutes with Hecke functors with respect to a morphism of the corresponding L-groups involving the SL_2 of Arthur. So, they realize the geometric Langlands functoriality for the corresponding morphisms of L-groups. As an application, we prove a particular case of the geometric Langlands conjectures for GSp_4. Namely, we construct the automorphic Hecke eigensheaves on Bun_{GSp_4} corresponding to the endoscopic local systems on X."}
{"category": "Math", "title": "The multi-dimensional pencil phenomenon for Laguerre heat-diffusion maximal operators", "abstract": "We investigate in detail the mapping properties of the maximal operator associated with the heat-diffusion semigroup corresponding to expansions with respect to multi-dimensional standard Laguerre functions."}
{"category": "Math", "title": "Linear progress in the complex of curves", "abstract": "We show that a random walk on the mapping class group of an orientable surface of finite type makes linear progress in the relative metric, which is quasi-isometric to the complex of curves."}
{"category": "Math", "title": "Bound on the multiplicity of almost complete intersections", "abstract": "Let $R$ be a polynomial ring over a field of characteristic zero and let $I \\subset R$ be a graded ideal of height $N$ which is minimally generated by $N+1$ homogeneous polynomials. If $I=(f_1,...,f_{N+1})$ where $f_i$ has degree $d_i$ and $(f_1,...,f_N)$ has height $N$, then the multiplicity of $R/I$ is bounded above by $\\prod_{i=1}^N d_i - \\max\\{1, \\sum_{i=1}^N (d_i-1) - (d_{N+1}-1) \\}$."}
{"category": "Math", "title": "Riesz transforms for the Dunkl harmonic oscillator", "abstract": "We define and investigate a system of Riesz transforms related to the Dunkl harmonic oscillator."}
{"category": "Math", "title": "Generalized Helmholtz-Kirchhoff model for two dimensional distributed vortex motion", "abstract": "The two-dimensional Navier-Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the moments of an expansion of the vorticity with respect to Hermite functions and of the centers of vorticity concentrations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit we formally recover the Helmholtz-Kirchhoff model for the evolution of point-vortices. The present expansion systematically incorporates the effects of both viscosity and finite vortex core size. We also show that a low-order truncation of our expansion leads to the representation of the flow as a system of interacting Gaussian (i.e. Oseen) vortices which previous experimental work has shown to be an accurate approximation to many important physical flows [9]."}
{"category": "Math", "title": "Complex Interpolation between Hilbert, Banach and Operator spaces", "abstract": "Motivated by a question of Vincent Lafforgue, we study the Banach spaces $X$ satisfying the following property: there is a function $\\vp\\to \\Delta_X(\\vp)$ tending to zero with $\\vp>0$ such that every operator $T\\colon L_2\\to L_2$ with $\\|T\\|\\le \\vp$ that is simultaneously contractive (i.e. of norm $\\le 1$) on $L_1$ and on $L_\\infty$ must be of norm $\\le \\Delta_X(\\vp)$ on $L_2(X)$. We show that $\\Delta_X(\\vp)\\in O(\\vp^\\alpha)$ for some $\\alpha>0$ iff $X$ is isomorphic to a quotient of a subspace of an ultraproduct of $\\theta$-Hilbertian spaces for some $ \\theta>0$ (see Corollary \\ref{comcor4.3}), where $\\theta$-Hilbertian is meant in a slightly more general sense than in our previous paper \\cite{P1}. Let $B_{r}(L_2(\\mu))$ be the space of all regular operators on $L_2(\\mu)$. We are able to describe the complex interpolation space \\[ (B_{r}(L_2(\\mu), B(L_2(\\mu))^\\theta. \\] We show that $T\\colon L_2(\\mu)\\to L_2(\\mu)$ belongs to this space iff $T\\otimes id_X$ is bounded on $L_2(X)$ for any $\\theta$-Hilbertian space $X$. More generally, we are able to describe the spaces $$ (B(\\ell_{p_0}), B(\\ell_{p_1}))^\\theta {\\rm or} (B(L_{p_0}), B(L_{p_1}))^\\theta $$ for any pair $1\\le p_0,p_1\\le \\infty$ and $0<\\theta<1$. In the same vein, given a locally compact Abelian group $G$, let $M(G)$ (resp. $PM(G)$) be the space of complex measures (resp. pseudo-measures) on $G$ equipped with the usual norm $\\|\\mu\\|_{M(G)} = |\\mu|(G)$ (resp. \\[ \\|\\mu\\|_{PM(G)} = \\sup\\{|\\hat\\mu(\\gamma)| \\big| \\gamma\\in\\hat G\\}). \\] We describe similarly the interpolation space $(M(G), PM(G))^\\theta$. Various extensions and variants of this result will be given, e.g. to Schur multipliers on $B(\\ell_2)$ and to operator spaces."}
{"category": "Math", "title": "On a recent generalization of semiperfect rings", "abstract": "It follows from a recent paper by Ding and Wang that any ring which is generalized supplemented as left module over itself is semiperfect. The purpose of this note is to show that Ding and Wang's claim is not true and that the class of generalized supplemented rings lies properly between the class of semilocal and semiperfect rings. Moreover we rectify their claim by introducing a wider notion of local submodules."}
{"category": "Math", "title": "Measuring the roughness of random paths by increment ratios", "abstract": "A statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and multiplicative trends and applicability to infinite variance processes. The existence of the IR statistic limit (called the IR-roughness below) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. Firstly, for a diffusion process with smooth diffusion and drift coefficients, the IR-roughness coincides with the IR-roughness of a Brownian motion and its convergence rate is obtained. Secondly, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Thirdly, the IR-roughness of a L\\'evy process with $\\alpha-$stable tangent process is established and can be used to estimate the fractional parameter $\\alpha \\in (0,2)$ following a central limit theorem."}
{"category": "Math", "title": "Fredholm equations for non-symmetric kernels, with applications to iterated integral operators", "abstract": "We give the Jordan form and the Singular Value Decomposition for an integral operator ${\\cal N}$ with a non-symmetric kernel $N(y,z)$. This is used to give solutions of Fredholm equations for non-symmetric kernels, and to determine the behaviour of ${\\cal N}^n$ and $({\\cal N}{\\cal N^*})^n$ for large $n$."}
{"category": "Math", "title": "Causal Models for Estimating the Effects of Weight Gain on Mortality", "abstract": "Suppose, contrary to fact, in 1950, we had put the cohort of 18 year old non-smoking American men on a stringent mandatory diet that guaranteed that no one would ever weigh more than their baseline weight established at age 18. How would the counter-factual mortality of these 18 year olds have compared to their actual observed mortality through 2007? We describe in detail how this counterfactual contrast could be estimated from longitudinal epidemiologic data similiar to that stored in the electronic medical records of a large HMO by applying g-estimation to a novel structural nested model. Our analytic approach differs from any alternative approach in that in that, in the abscence of model misspecification, it can successfully adjust for (i) measured time-varying confounders such as exercise, hypertension and diabetes that are simultaneously intermediate variables on the causal pathway from weight gain to death and determinants of future weight gain, (ii) unmeasured confounding by undiagnosed preclinical disease (i.e reverse causation) that can cause both poor weight gain and premature mortality [provided an upper bound can be specified for the maximum length of time a subject may suffer from a subclinical illness severe enough to affect his weight without the illness becomes clinically manifest], and (iii) the prescence of particular identifiable subgroups, such as those suffering from serious renal, liver, pulmonary, and/or cardiac disease, in whom confounding by unmeasured prognostic factors so severe as to render useless any attempt at direct analytic adjustment."}
{"category": "Math", "title": "A guide to telescopic functors", "abstract": "In the mid 1980's, Pete Bousfield and I constructed certain p--local `telescopic' functors Phi_n from spaces to spectra, for each prime p and each positive integer n. These have striking properties that relate the chromatic approach to homotopy theory to infinite loopspace theory: roughly put, the spectrum Phi_n(Z) captures the v_n periodic homotopy of a space Z. Recently there have been a variety of new uses of these functors, suggesting that they have a central role to play in calculations of periodic phenomena. Here I offer a guide to their construction, characterization, application, and computation."}
{"category": "Math", "title": "Connected components of the compactification of representation spaces of surface groups", "abstract": "The Thurston compactification of Teichmuller spaces has been generalized to many different representation spaces by J. Morgan, P. Shalen, M. Bestvina, F. Paulin, A. Parreau and others. In the simplest case of representations of fundamental groups of closed hyperbolic surfaces in PSL(2,R), we prove that this compactification is very degenerated: the nice behaviour of the Thurston compactification of the Teichmuller space contrasts with wild phenomena happening on the boundary of the other connected components of these representation spaces. We prove that it is more natural to consider a refinement of this compactification, which remembers the orientation of the hyperbolic plane. The ideal points of this compactification are fat R-trees, i.e., R-trees equipped with a planar structure."}
{"category": "Math", "title": "Multifractal analysis of Birkhoff averages on \"self-affine\" symbolic spaces", "abstract": "We achieve on self-affine Sierpinski carpets the multifractal analysis of the Birkhoff averages of potentials satisfying a Dini condition. Given such a potential, the corresponding Hausdorff spectrum cannot be deduced from that of the associated Gibbs measure by a simple transformation. Indeed, these spectra are respectively obtained as the Legendre transform of two distinct concave differentiable functions that cannot be deduced from one another by a dilation and a translation. This situation is in contrast with what is observed in the familiar self-similar case. Our results are presented in the framework of almost-multiplicative functions on products of two distinct symbolic spaces and their projection on the associated self-affine carpets."}
{"category": "Math", "title": "The distribution of the maximum of a first order moving average: the continuous case", "abstract": "We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables have an absolutely continuous density. When the correlation is positive, $$ P(M_n %\\max^n_{i=1} X_i \\leq x) =\\ \\sum_{j=1}^\\infty \\beta_{jx} \\nu_{jx}^{n} \\approx B_{x} \\nu_{1x}^{n} $$ where %$\\{X_i\\}$ is a moving average of order 1 with positive correlation, and $\\{\\nu_{jx}\\}$ are the eigenvalues (singular values) of a Fredholm kernel and $\\nu_{1x}$ is the eigenvalue of maximum magnitude. A similar result is given when the correlation is negative. The result is analogous to large deviations expansions for estimates, since the maximum need not be standardized to have a limit. % there are more terms, and $$P(M_n <x) \\approx B'_{x}\\ (1+\\nu_{1x})^n.$$ For the continuous case the integral equations for the left and right eigenfunctions are converted to first order linear differential equations. The eigenvalues satisfy an equation of the form $$\\sum_{i=1}^\\infty w_i(\\lambda-\\theta_i)^{-1}=\\lambda-\\theta_0$$ for certain known weights $\\{w_i\\}$ and eigenvalues $\\{\\theta_i\\}$ of a given matrix. This can be solved by truncating the sum to an increasing number of terms."}
{"category": "Math", "title": "Conservative Properties of the Variational Free-Lagrange Method for Shallow Water", "abstract": "The variational free-Lagrange (VFL) method for shallow water is a free-Lagrange method with the additional property that it preserves the variational structure of shallow water. The VFL method was first derived in this context by \\cite{AUG84} who discretized Hamilton's action principle with a free-Lagrange data structure. The purpose of this article is to assess the long-time conservation properties of the VFL method for regularized shallow water which are useful for climate simulation. Long-time regularized shallow water simulations show that the VFL method exhibits no secular drift in the (i) energy error through the application of symplectic integrators; and (ii) the potential vorticity error through the construction of discrete curl, divergence and gradient operators which satisfy semi-discrete divergence and potential vorticity conservation laws. These diagnostic semi-discrete equations augment the description of the VFL method by characterizing the evolution of its respective irrotational and solenoidal components in the Lagrangian frame. Like the continuum equations, the former exhibits a $\\text{div}^2\\mathbf{U}$ term which indicates that the flow has a very strong tendency towards a purely rotational state. Numerical results show (i) the preservation of shape and strength of an initially radially symmetric vortex pair in purely rotational regularized shallow water and (ii) how the Voronoi diagram retains the history of the flow field and (iii) that energy is conserved to $\\mathcal{O}(\\Delta^2)$ and potential vorticity error to within 5% with no secular growth over a 50 year period."}
{"category": "Math", "title": "The distribution of the maximum of a first order moving average: the discrete case", "abstract": "We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables are discrete. When the correlation is positive, $$ P(M_n \\max^n_{i=1} X_i \\leq x) = \\sum_{j=1}^\\infty \\beta_{jx} \\nu_{jx}^{n} \\approx B_{x} r{1x}^{n} $$ where $\\{\\nu_{jx}\\}$ are the eigenvalues of a certain matrix, $r_{1x}$ is the maximum magnitude of the eigenvalues, and $I$ depends on the number of possible values of the underlying random variables. The eigenvalues do not depend on $x$ only on its range."}
{"category": "Math", "title": "Sobolev and Schwartz: Two Fates and Two Fames", "abstract": "This is a brief overview of the lives and contributions of S.L. Sobolev and L. Schwartz, the cofounders of distribution theory."}
{"category": "Math", "title": "On the Asymptotic Normality of the Conditional Maximum Likelihood Estimators for the Truncated Regression Model and the Tobit Model", "abstract": "In this paper, we study the asymptotic normality of the conditional maximum likelihood (ML) estimators for the truncated regression model and the Tobit model. We show that under the general setting assumed in his book, the conjectures made by Hayashi (2000) \\footnote{see page 516, and page 520 of Hayashi (2000).} about the asymptotic normality of the conditional ML estimators for both models are true, namely, a sufficient condition is the nonsingularity of $\\mathbf{x_tx'_t}$."}
{"category": "Math", "title": "On Using (Z^2, +) Homomorphisms to Generate Pairs of Coprime Integers", "abstract": "We use the group $(\\Z^2,+)$ and two associated homomorphisms, $\\tau_0, \\tau_1$, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote $T$. While this idea is related to the Stern-Brocot tree and the map of relatively prime pairs, the parents of an integer pair these trees do not necessarily correspond to the parents of the same integer pair in $T$. Our main result is a proof that for $x_i \\in \\{0,1\\}$, the sum of the pair $\\tau_{x_1}\\tau_{x_2}... \\tau_{x_n} [1,2]$ is equal to the sum of the pair $\\tau_{x_n}\\tau_{x_{n-1}} ... \\tau_{x_1} [1,2]$. Further, we give a conjecture as to the well-ordering of the sums of these integers."}
{"category": "Math", "title": "A Conversation with Ingram Olkin", "abstract": "Ingram Olkin was born on July 23, 1924 in Waterbury, Connecticut. His family moved to New York in 1934 and he graduated from DeWitt Clinton High School in 1941. He served three years in the Air Force during World War II and obtained a B.S. in mathematics at the City College of New York in 1947. After receiving an M.A. in mathematical statistics from Columbia in 1949, he completed his graduate studies in the Department of Statistics at the University of North Carolina in 1951. His dissertation was written under the direction of S. N. Roy and Harold Hotelling. He joined the Department of Mathematics at Michigan State University in 1951 as an Assistant Professor, subsequently being promoted to Professor. In 1960, he took a position as Chair of the Department of Statistics at the University of Minnesota. He moved to Stanford University in 1961 to take a joint position as Professor of Statistics and Professor of Education; he was also Chair of the Department of Statistics from 1973--1976. In 2007, Ingram became Professor Emeritus. Ingram was Editor of the Annals of Mathematical Statistics (1971--1972) and served as the first editor of the Annals of Statistics from 1972--1974. He was a primary force in the founding of the Journal of Educational Statistics, for which he was also Associate Editor during 1977--1985. In 1984, he was President of the Institute of Mathematical Statistics. Among his many professional activities, he has served as Chair of the Committee of Presidents of Statistical Societies (COPSS), Chair of the Committee on Applied and Theoretical Statistics of the National Research Council, Chair of the Management Board of the American Education Research Association, and as Trustee for the National Institute of Statistical Sciences. He has been honored by the American Statistical Association (ASA) with a Wilks Medal (1992) and a Founder's Award (1992). The American Psychological Association gave him a Lifetime Contribution Award (1997) and he was elected to the National Academy of Education in 2005. He received the COPSS Elizabeth L. Scott Award in 1998 and delivered the R. A. Fisher Lecture in 2000. In 2003, the City University of New York gave him a Townsend Harris Medal. An author of 5 books, an editor of 10 books, and an author of more than 200 publications, Ingram has made major contributions to statistics and education. His research has focused on multivariate analysis, majorization and inequalities, distribution theory, and meta-analysis. A volume in celebration of Ingram's 65th birthday contains a brief biography and an interview [Gleser, Perlman, Press and Sampson (1989)]. Ingram was chosen in 1997 to participate in the American Statistical Association Distinguished Statistician Video Series and a videotaped conversation and a lecture (Olkin, 1997) are available from the ASA (1997, DS041, DS042)."}
{"category": "Math", "title": "On continuous solutions of a problem of R.Schilling", "abstract": "The paper deals with continuous solutions of a Schilling's problem."}
{"category": "Math", "title": "Criteria for Bochner's extension problem", "abstract": "A necessary and sufficient condition for the resolution of the weak extension problem is given. This criterion is applied to also give a criterion for the solvability of the classical Bochner's extension problem in the $L^p$-category. The solution of the $L^p$-extension problem by Bochner giving the relation between the order of the operator, the dimension, and index $p$, for which the $L^p$-extension property holds, can be viewed as a subcritical case of the general $L^p$-extension problem. In general, this property fails in some critical and in all supercritical cases. In this paper, the $L^p$-extension problem is investigated for operators of all orders and for all $1\\leq p\\leq\\infty$. Necessary and sufficient conditions on the subset of $L^p$ are given for which the $L^p$-extension property still holds, in the critical and supercritical cases."}
{"category": "Math", "title": "V-fold cross-validation improved: V-fold penalization", "abstract": "We study the efficiency of V-fold cross-validation (VFCV) for model selection from the non-asymptotic viewpoint, and suggest an improvement on it, which we call ``V-fold penalization''. Considering a particular (though simple) regression problem, we prove that VFCV with a bounded V is suboptimal for model selection, because it ``overpenalizes'' all the more that V is large. Hence, asymptotic optimality requires V to go to infinity. However, when the signal-to-noise ratio is low, it appears that overpenalizing is necessary, so that the optimal V is not always the larger one, despite of the variability issue. This is confirmed by some simulated data. In order to improve on the prediction performance of VFCV, we define a new model selection procedure, called ``V-fold penalization'' (penVF). It is a V-fold subsampling version of Efron's bootstrap penalties, so that it has the same computational cost as VFCV, while being more flexible. In a heteroscedastic regression framework, assuming the models to have a particular structure, we prove that penVF satisfies a non-asymptotic oracle inequality with a leading constant that tends to 1 when the sample size goes to infinity. In particular, this implies adaptivity to the smoothness of the regression function, even with a highly heteroscedastic noise. Moreover, it is easy to overpenalize with penVF, independently from the V parameter. A simulation study shows that this results in a significant improvement on VFCV in non-asymptotic situations."}
{"category": "Math", "title": "On the number of collinear triples in permutations", "abstract": "Let $\\alpha:\\mathbb{Z}_n\\to\\mathbb{Z}_n$ be a permutation and $\\Psi(\\alpha)$ be the number of collinear triples modulo $n$ in the graph of $\\alpha$. Cooper and Solymosi had given by induction the bound $\\min_{\\alpha}\\Psi(\\alpha)\\geq\\lceil(n-1)/4\\rceil$ when $n$ is a prime number. The main purpose of this paper is to give a direct proof of that bound. Besides, the expected number of collinear triples a permutation can have is also been determined."}
{"category": "Math", "title": "On the existence of ground state solutions to nonlinear Schoedinger equations with multisingular inverse-square anisotropic potentials", "abstract": "A class of nonlinear Schroedinger equations with critical power-nonlinearities and potentials exhibiting multiple anisotropic inverse square singularities is investigated. Conditions on strength, location, and orientation of singularities are given for the minimum of the associated Rayleigh quotient to be achieved, both in the whole $\\R^N$ and in bounded domains."}
{"category": "Math", "title": "Von Neumann Stability Analysis of Finite Difference Schemes for Maxwell--Debye and Maxwell--Lorentz Equations", "abstract": "This technical report yields detailed calculations of the paper [1] (B. Bid\\'egaray-Fesquet, \"Stability of FD-TD schemes for Maxwell-Debye and Maxwell-Lorentz equations\", Technical report, LMC-IMAG, 2005) which have been however automated since (see http://ljk.imag.fr/membres/Brigitte.Bidegaray/NAUtil/). It deals with the stability analysis of various finite difference schemes for Maxwell--Debye and Maxwell--Lorentz equations. This work gives a systematic and rigorous continuation to Petropoulos previous work [5] (P.G. Petropoulos.,\"Stability and phase error analysis of FD-TD in dispersive dielectrics\", IEEE Transactions on Antennas and Propagation, 42(1):62--69, 1994)."}
{"category": "Math", "title": "On several problems about automorphisms of the free group of rank two", "abstract": "Let $F_n$ be a free group of rank $n$. In this paper we discuss three algorithmic problems related to automorphisms of $F_2$. A word $u$ of $F_n$ is called positive if $u$ does not have negative exponents. A word $u$ in $F_n$ is called potentially positive if $\\phi(u)$ is positive for some automorphism $\\phi$ of $F_n$. We prove that there is an algorithm to decide whether or not a given word in $F_2$ is potentially positive, which gives an affirmative solution to problem F34a in [1] for the case of $F_2$. Two elements $u$ and $v$ in $F_n$ are said to be boundedly translation equivalent if the ratio of the cyclic lengths of $\\phi(u)$ and $\\phi(v)$ is bounded away from 0 and from $\\infty$ for every automorphism $\\phi$ of $F_n$. We provide an algorithm to determine whether or not two given elements of $F_2$ are boundedly translation equivalent, thus answering question F38c in the online version of [1] for the case of $F_2$. We further prove that there exists an algorithm to decide whether or not a given finitely generated subgroup of $F_2$ is the fixed point group of some automorphism of $F_2$, which settles problem F1b in [1] in the affirmative for the case of $F_2$."}
{"category": "Math", "title": "Positive divisors in symplectic geometry", "abstract": "In this paper, we gave some explicit relations between absolute and relative Gromov-Witten invariants. We proved that a symplectic manifold is symplectic rationally connected if it contains a positive divisor symplectomorphic to $P^n$."}
{"category": "Math", "title": "Towards an analogue of Ihara's lemma for Shimura curves", "abstract": "The object of this work is to present the status of art of an open problem: to provide an analogue for Shimura curves of the Ihara's lemma \\cite{Ihara73} which holds for modular curves. We will describe our direct result towards the \"Problem of Ihara\" and we will present some possible approaches to it, giving a formulation of our conjecture in terms of congruence subgroup problem for quaternion algebras. Since some modular forms can be reinterpreted as elements of the cohomology of Shimura curves, we will describe a consequence of the \"Problem of Ihara\" about congruence modules of modular forms and a consequence of it about the problem of raising the level of modular forms."}
{"category": "Math", "title": "Boundedness of multidimensional Hausdorff operators on $L^1$ and $H^1$ spaces", "abstract": "For a wide family of multivariate Hausdorff operators, a new stronger condition for the boundedness of an operator from this family on the real Hardy space $H^1$ by means of atomic decomposition."}
{"category": "Math", "title": "Balanced category theory", "abstract": "Some aspects of basic category theory are developed in a finitely complete category $\\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this axiomatization of final and initial functors and discrete (op)fibrations, concepts such as components, slices and coslices, colimits and limits, left and right adjunctible maps, dense maps and arrow intervals, can be naturally defined in $\\C$, and several classical properties concerning them can be effectively proved. For any object $X$ of $\\C$, by restricting $\\C/X$ to the slices or to the coslices of $X$, two dual \"underlying categories\" are obtained. These can be enriched over internal sets (discrete objects) of $\\C$: internal hom-sets are given by the components of the pullback of the corresponding slice and coslice of $X$. The construction extends to give functors $\\C\\to\\Cat$, which preserve (or reverse) slices and adjunctible maps and which can be enriched over internal sets too."}
{"category": "Math", "title": "The pseudo-index of horospherical Fano varieties", "abstract": "We prove a conjecture of L.Bonavero, C. Casagrande, O. Debarre and S. Druel, on the pseudo-index of smooth Fano varieties, in the special case of horospherical varieties."}
{"category": "Math", "title": "A New Family of Random Graphs for Testing Spatial Segregation", "abstract": "We discuss a graph-based approach for testing spatial point patterns. This approach falls under the category of data-random graphs, which have been introduced and used for statistical pattern recognition in recent years. Our goal is to test complete spatial randomness against segregation and association between two or more classes of points. To attain this goal, we use a particular type of parametrized random digraph called proximity catch digraph (PCD) which is based based on relative positions of the data points from various classes. The statistic we employ is the relative density of the PCD. When scaled properly, the relative density of the PCD is a $U$-statistic. We derive the asymptotic distribution of the relative density, using the standard central limit theory of $U$-statistics. The finite sample performance of the test statistic is evaluated by Monte Carlo simulations, and the asymptotic performance is assessed via Pitman's asymptotic efficiency, thereby yielding the optimal parameters for testing. Furthermore, the methodology discussed in this article is also valid for data in multiple dimensions."}
{"category": "Math", "title": "A uniqueness theorem for solution of BSDEs", "abstract": "In this note, we prove that if $g$ is uniformly continuous in $z$, uniformly with respect to $(\\oo,t)$ and independent of $y$, the solution to the backward stochastic differential equation (BSDE) with generator $g$ is unique."}
{"category": "Math", "title": "On the Distribution of the Domination Number of a New Family of Parametrized Random Digraphs", "abstract": "We derive the asymptotic distribution of the domination number of a new family of random digraph called proximity catch digraph (PCD), which has application to statistical testing of spatial point patterns and to pattern recognition. The PCD we use is a parametrized digraph based on two sets of points on the plane, where sample size and locations of the elements of one is held fixed, while the sample size of the other whose elements are randomly distributed over a region of interest goes to infinity. PCDs are constructed based on the relative allocation of the random set of points with respect to the Delaunay triangulation of the other set whose size and locations are fixed. We introduce various auxiliary tools and concepts for the derivation of the asymptotic distribution. We investigate these concepts in one Delaunay triangle on the plane, and then extend them to the multiple triangle case. The methods are illustrated for planar data, but are applicable in higher dimensions also."}
{"category": "Math", "title": "On Summatory Totient Functions", "abstract": "The lower and upper bounds are found for the leading term of summatory totient function $\\sum_{k\\leq N}k^u\\phi^v(k)$ in various ranges of $u\\in{\\mathbb R}$ and $v\\in{\\mathbb Z}$."}
{"category": "Math", "title": "Relative Density of the Random $r$-Factor Proximity Catch Digraph for Testing Spatial Patterns of Segregation and Association", "abstract": "Statistical pattern classification methods based on data-random graphs were introduced recently. In this approach, a random directed graph is constructed from the data using the relative positions of the data points from various classes. Different random graphs result from different definitions of the proximity region associated with each data point and different graph statistics can be employed for data reduction. The approach used in this article is based on a parameterized family of proximity maps determining an associated family of data-random digraphs. The relative arc density of the digraph is used as the summary statistic, providing an alternative to the domination number employed previously. An important advantage of the relative arc density is that, properly re-scaled, it is a $U$-statistic, facilitating analytic study of its asymptotic distribution using standard $U$-statistic central limit theory. The approach is illustrated with an application to the testing of spatial patterns of segregation and association. Knowledge of the asymptotic distribution allows evaluation of the Pitman and Hodges-Lehmann asymptotic efficacies, and selection of the proximity map parameter to optimize efficiency. Furthermore the approach presented here also has the advantage of validity for data in any dimension."}
{"category": "Math", "title": "Heegaard Floer invariants of Legendrian knots in contact three--manifolds", "abstract": "We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. Moreover, we apply the invariants to find transversely non--simple knot types in many overtwisted contact 3--manifolds."}
{"category": "Math", "title": "A problem on polynomial maps over finite fields", "abstract": "This is a short note that explains a problem on polynomial maps over finite fields for non-experts. The problem is: Do there exist odd polynomial automorphisms over the finite fields with 4,8,16,32,64,... elements? The explanation is very, very basic. Also, the background of the problem is given, and why it is of such importance. This all with the idea that the problem enters the world of discrete mathematics, and can be approached from completely different angles than normally used by people working in Affine Algebraic Geometry."}
{"category": "Math", "title": "Region of variability for certain classes of univalent functions satisfying differential inequalities", "abstract": "In this paper we determine the region of variability for certain subclasses of univalent functions satisfying differential inequalities. In the final section we graphically illustrate the region of variability for several sets of parameters."}
{"category": "Math", "title": "Time--space white noise eliminates global solutions in reaction diffusion equations", "abstract": "We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} + (u_+)^p$ ($p>1$), with time--space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists."}
{"category": "Math", "title": "A note on families of hyperelliptic curves", "abstract": "We give a stack-theoretic proof for some results on families of hyperelliptic curves."}
{"category": "Math", "title": "The Use of Domination Number of a Random Proximity Catch Digraph for Testing Spatial Patterns of Segregation and Association", "abstract": "Priebe et al. (2001) introduced the class cover catch digraphs and computed the distribution of the domination number of such digraphs for one dimensional data. In higher dimensions these calculations are extremely difficult due to the geometry of the proximity regions; and only upper-bounds are available. In this article, we introduce a new type of data-random proximity map and the associated (di)graph in $\\mathbb R^d$. We find the asymptotic distribution of the domination number and use it for testing spatial point patterns of segregation and association."}
{"category": "Math", "title": "The hypertoric intersection cohomology ring", "abstract": "We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset."}
{"category": "Math", "title": "Semi-classical analysis of a random walk on a manifold", "abstract": "We prove a sharp rate of convergence to stationarity for a natural random walk on a compact Riemannian manifold $(M,g)$. The proof includes a detailed study of the spectral theory of the associated operator."}
{"category": "Math", "title": "Multifractional, multistable, and other processes with prescribed local form", "abstract": "We present a general method for constructing stochastic processes with prescribed local form. Such processes include variable amplitude multifractional Brownian motion, multifractional $\\alpha$-stable processes, and multistable processes, that is processes that are locally $\\alpha(t)$-stable but where the stability index $\\alpha(t)$ varies with $t$. In particular we construct multifractional multistable processes where both the local self-similarity and stability indices vary."}
{"category": "Math", "title": "Optimal and better transport plans", "abstract": "We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value infty. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided that {c=infty} is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a \"better\" notion of optimality called robust optimality."}
{"category": "Math", "title": "Stabilization and limit theorems for geometric functionals of Gibbs point processes", "abstract": "Given a Gibbs point process $\\P^{\\Psi}$ on $\\R^d$ having a weak enough potential $\\Psi$, we consider the random measures $\\mu_\\la := \\sum_{x \\in \\P^{\\Psi} \\cap Q_\\la} \\xi(x, \\P^{\\Psi} \\cap Q_\\la) \\delta_{x/\\la^{1/d}}$, where $Q_{\\la} := [-\\la^{1/d}/2,\\la^{1/d}/2]^d$ is the volume $\\la$ cube and where $\\xi(\\cdot,\\cdot)$ is a translation invariant stabilizing functional. Subject to $\\Psi$ satisfying a localization property and translation invariance, we establish weak laws of large numbers for $\\la^{-1} \\mu_\\la(f)$, $f$ a bounded test function on $\\R^d$, and weak convergence of $\\la^{-1/2} \\mu_\\la(f),$ suitably centered, to a Gaussian field acting on bounded test functions. The result yields limit laws for geometric functionals on Gibbs point processes including the Strauss and area interaction point processes as well as more general point processes defined by the Widom-Rowlinson and hard-core model. We provide applications to random sequential packing on Gibbsian input, to functionals of Euclidean graphs, networks, and percolation models on Gibbsian input, and to quantization via Gibbsian input."}
{"category": "Math", "title": "A second-order identity for the Riemann tensor and applications", "abstract": "A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-recurrency naturally emerges from an invariance property of an old identity by Lovelock."}
{"category": "Math", "title": "A family of local rings with rational Poincar\\'e Series", "abstract": "In this note we compute the Poincare Series of almost stretched Gorenstein local rings. It turns out that it is rational"}
{"category": "Math", "title": "Wrap groups of fiber bundles over quaternions and octonions", "abstract": "This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real $\\bf R$, complex $\\bf C$ numbers, the quaternion skew field $\\bf H$ and the octonion algebra $\\bf O$. These groups are constructed with mild conditions on fibers. Their examples are given. It is shown, that these groups exist and for differentiable fibers have the infinite dimensional Lie groups structure, that is, they are continuous or differentiable manifolds and the composition $(f,g)\\mapsto f^{-1}g$ is continuous or differentiable depending on a class of smoothness of groups. Moreover, it is demonstrated that in the cases of real, complex, quaternion and octonion manifolds these groups have structures of real, complex, quaternion or octonion manifolds respectively. Nevertheless, it is proved that these groups does not necessarily satisfy the Campbell-Hausdorff formula even locally."}
{"category": "Math", "title": "Smooth Functors vs. Differential Forms", "abstract": "We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in several fields, namely as connections on (non-abelian) gerbes, as curvatures of smooth functors and as critical points in BF theory. We demonstrate further that our dictionary provides a powerful tool to discuss the transgression of geometric objects to loop spaces."}
{"category": "Math", "title": "Vogan Diagrams of Twisted Affine Kac-Moody Lie Algebras", "abstract": "A Vogan diagram is a Dynkin diagram of a Kac-Moody Lie algebra of finite or affine type overlayed with additional structures. This paper develops the theory of Vogan diagrams for almost compact real forms of indecomposable twisted affine Kac- Moody Lie algebras and shows that equivalence classes of Vogan diagrams correspond to isomorphism classes of almost compact real forms of twisted affine Kac-Moody Lie algebras as given by H. Ben Messaoud and G. Rousseau."}
{"category": "Math", "title": "Absence of line fields and Mane's theorem for non-recurrent transcendental functions", "abstract": "Let f be a transcendental meromorphic function. Suppose that the finite part of the postsingular set of f is bounded, that f has no recurrent critical points or wandering domains, and that the degree of pre-poles of f is uniformly bounded. Then we show that f supports no invariant line fields on its Julia set. We prove this by generalizing two results about rational functions to the transcendental setting: a theorem of Mane about the branching of iterated preimages of disks, and a theorem of McMullen regarding absence of invariant line fields for \"measurably transitive\" functions. Both our theorems extend results previously obtained by Graczyk, Kotus and Swiatek."}
{"category": "Math", "title": "Fractional Cauchy problems on bounded domains", "abstract": "Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain $D\\subset\\mathbb{R}^d$ with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brownian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time."}
{"category": "Math", "title": "General definitions of chaos for continuous and discrete-time processes", "abstract": "A precise definition of chaos for discrete processes based on iteration already exists. We shall first reformulate it in a more general frame, taking into account the fact that discrete chaotic behavior is neither necessarily based on iteration nor strictly related to compact metric spaces or to bounded functions. Then we shall apply the central idea of this definition to continuous processes. We shall try to see what chaos is, regardless of the way it is generated."}
{"category": "Math", "title": "Boundary behavior of functions in the de Branges--Rovnyak spaces", "abstract": "This paper deals with the boundary behavior of functions in the de Branges--Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges--Rovnyak spaces. This criterion generalizes a result of Ahern-Clark. Then we prove that the continuity of all functions in a de Branges--Rovnyak space on an open arc $I$ of the boundary is enough to ensure the analyticity of these functions on $I$. We use this property in a question related to Bernstein's inequality."}
{"category": "Math", "title": "Homoscedastic controlled calibration model", "abstract": "In the context of the usual calibration model, we consider the case in which the independent variable is unobservable, but a pre-fixed value on its surrogate is available. Thus, considering controlled variables and assuming that the measurement errors have equal variances we propose a new calibration model. Likelihood based methodology is used to estimate the model parameters and the Fisher information matrix is used to construct a confidence interval for the unknown value of the regressor variable. A simulation study is carried out to asses the effect of the measurement error on the estimation of the parameter of interest. This new approach is illustrated with an example."}
{"category": "Math", "title": "Prime path coalgebras", "abstract": "We use prime coalgebras as a generalization of simple coalgebras, and observe that prime subcoalgebras represent the structure of the coalgebra in a more efficient way than simple coalgebras. In particular, in this work we focus our attention on the study and characterization of prime subcoalgebras of path coalgebras of quivers and, by extension, of prime pointed coalgebras."}
{"category": "Math", "title": "A new approach to solvability of some elliptic pde's with square integrable boundary data", "abstract": "This paper has been withdrawn, and is replaced with paper \"Solvability of elliptic systems with square integrable boundary data\" by the same authors."}
{"category": "Math", "title": "Gonality, apolarity and hypercubics", "abstract": "We show that any Fermat hypercubic is apolar to a trigonal curve, and vice versa. We show also that the Waring number of the polar hypercubic associated to a tetragonal curve of genus $g$ is at most $\\lceil 3/2g - 7/2\\rceil$, and for a large class of them is at most $4/3g - 3$."}
{"category": "Math", "title": "Residual finiteness, QCERF, and fillings of hyperbolic groups", "abstract": "We prove that if every hyperbolic group is residually finite, then every quasi-convex subgroup of every hyperbolic group is separable. The main tool is relatively hyperbolic Dehn filling."}
{"category": "Math", "title": "A new solution representation for the BBM equation in a quarter plane and the eventual periodicity", "abstract": "The initial- and boundary-value problem for the Benjamin-Bona-Mahony (BBM) equation is studied in this paper. The goal is to understand the periodic behavior (termed as eventual periodicity) of its solutions corresponding to periodic boundary condition or periodic forcing. To this aim, we derive a new formula representing solutions of this initial- and boundary-value problem by inverting the operator $\\partial_t +\\alpha \\partial_x -\\gamma\\partial_{xxt}$ defined in the space-time quarter plane. The eventual periodicity of the linearized BBM equation with periodic boundary data and forcing term is established by combining this new representation formula and the method of stationary phase."}
{"category": "Math", "title": "Relative Asymptotic of Multiple Orthogonal Polynomials for Nikishin Systems", "abstract": "We prove relative asymptotic for the ratio of two sequences of multiple orthogonal polynomials with respect to Nikishin system of measures. The first Nikishin system ${\\mathcal{N}}(\\sigma_1,...,\\sigma_m)$ is such that for each $k$, $\\sigma_k$ has constant sign on its compact support $\\supp {\\sigma_k} \\subset \\mathbb{R}$ consisting of an interval $\\widetilde{\\Delta}_k$, on which $|\\sigma_k^{\\prime}| > 0$ almost everywhere, and a discrete set without accumulation points in $\\mathbb{R} \\setminus \\widetilde{\\Delta}_k$. If ${Co}(\\supp {\\sigma_k}) = \\Delta_k$ denotes the smallest interval containing $\\supp {\\sigma_k}$, we assume that $\\Delta_k \\cap \\Delta_{k+1} = \\emptyset$, $k=1,...,m-1$. The second Nikishin system ${\\mathcal{N}}(r_1\\sigma_1,...,r_m\\sigma_m)$ is a perturbation of the first by means of rational functions $r_k$, $k=1,...,m,$ whose zeros and poles lie in $\\mathbb{C} \\setminus \\cup_{k=1}^m \\Delta_k$."}
{"category": "Math", "title": "Schlicht envelopes of holomorphy and foliations by lines", "abstract": "Given a domain Y in a complex manifold X, it is a difficult problem with no general solution to determine whether Y has a schlicht envelope of holomorphy in X, and if it does, to describe the envelope. The purpose of this paper is to tackle the problem with the help of a smooth 1-dimensional foliation F of X with no compact leaves. We call a domain Y in X an interval domain with respect to F if Y intersects every leaf of F in a nonempty connected set. We show that if X is Stein and if F satisfies a new property called quasiholomorphicity, then every interval domain in X has a schlicht envelope of holomorphy, which is also an interval domain. This result is a generalization and a global version of a well-known lemma from the mid-1980s. We illustrate the notion of quasiholomorphicity with sufficient conditions, examples, and counterexamples, and present some applications, in particular to a little-studied boundary regularity property of domains called local schlichtness."}
{"category": "Math", "title": "Laminating lattices with symmetrical glue", "abstract": "We use the automorphism group $Aut(H)$, of holes in the lattice $L_8=A_2\\oplus A_2\\oplus D_4$, as the starting point in the construction of sphere packings in 10 and 12 dimensions. A second lattice, $L_4=A_2\\oplus A_2$, enters the construction because a subgroup of $Aut(L_4)$ is isomorphic to $Aut(H)$. The lattices $L_8$ and $L_4$, when glued together through this relationship, provide an alternative construction of the laminated lattice in twelve dimensions with kissing number 648. More interestingly, the action of $Aut(H)$ on $L_4$ defines a pair of invariant planes through which dense, non-lattice packings in 10 dimensions can be constructed. The most symmetric of these is aperiodic with center density 1/32. These constructions were prompted by an unexpected arrangement of 378 kissing spheres discovered by a search algorithm."}
{"category": "Math", "title": "Contractively complemented subspaces of pre-symmetric spaces", "abstract": "In 1965, Ron Douglas proved that if $X$ is a closed subspace of an $L^1$-space and $X$ is isometric to another $L^1$-space, then $X$ is the range of a contractive projection on the containing $L^1$-space. In 1977 Arazy-Friedman showed that if a subspace $X$ of $C_1$ is isometric to another $C_1$-space (possibly finite dimensional), then there is a contractive projection of $C_1$ onto $X$. In 1993 Kirchberg proved that if a subspace $X$ of the predual of a von Neumann algebra $M$ is isometric to the predual of another von Neumann algebra, then there is a contractive projection of the predual of $M$ onto $X$. We widen significantly the scope of these results by showing that if a subspace $X$ of the predual of a $JBW^*$-triple $A$ is isometric to the predual of another $JBW^*$-triple $B$, then there is a contractive projection on the predual of $A$ with range $X$, as long as $B$ does not have a direct summand which is isometric to a space of the form $L^\\infty(\\Omega,H)$, where $H$ is a Hilbert space of dimension at least two. The result is false without this restriction on $B$."}
{"category": "Math", "title": "Mordell-Weil Problem for Cubic Surfaces, Numerical Evidence", "abstract": "Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset P \\subset V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents through pairs of previously constructed points and consecutively adding their new intersection points with V. In this paper we present numerical data regarding the analogous statement for cubic surfaces. For the surfaces examined, we also test Manin's conjecture relating the asymptotics of rational points of bounded height on a Fano variety with the rank of the Picard group of the surface."}
{"category": "Math", "title": "Bayesian Checking of the Second Levels of Hierarchical Models", "abstract": "Hierarchical models are increasingly used in many applications. Along with this increased use comes a desire to investigate whether the model is compatible with the observed data. Bayesian methods are well suited to eliminate the many (nuisance) parameters in these complicated models; in this paper we investigate Bayesian methods for model checking. Since we contemplate model checking as a preliminary, exploratory analysis, we concentrate on objective Bayesian methods in which careful specification of an informative prior distribution is avoided. Numerous examples are given and different proposals are investigated and critically compared."}
{"category": "Math", "title": "Comment: Bayesian Checking of the Second Levels of Hierarchical Models", "abstract": "We discuss the methods of Evans and Moshonov [Bayesian Analysis 1 (2006) 893--914, Bayesian Statistics and Its Applications (2007) 145--159] concerning checking for prior-data conflict and their relevance to the method proposed in this paper. [arXiv:0802.0743]"}
{"category": "Math", "title": "Comment: Bayesian Checking of the Second Levels of Hierarchical Models", "abstract": "Comment: Bayesian Checking of the Second Levels of Hierarchical Models [arXiv:0802.0743]"}
{"category": "Math", "title": "Comment: Bayesian Checking of the Second Levels of Hierarchical Models", "abstract": "Comment: Bayesian Checking of the Second Levels of Hierarchical Models [arXiv:0802.0743]"}
{"category": "Math", "title": "Comment: Bayesian Checking of the Second Level of Hierarchical Models: Cross-Validated Posterior Predictive Checks Using Discrepancy Measures", "abstract": "Comment: Bayesian Checking of the Second Level of Hierarchical Models [arXiv:0802.0743]"}
{"category": "Math", "title": "Rejoinder: Bayesian Checking of the Second Levels of Hierarchical Models", "abstract": "Rejoinder: Bayesian Checking of the Second Levels of Hierarchical Models [arXiv:0802.0743]"}
{"category": "Math", "title": "On Ricci solitons of cohomogeneity one", "abstract": "We analyse some properties of the cohomogeneity one Ricci soliton equations, and use Ansatze of cohomogeneity one type to produce new explicit examples of complete Kahler Ricci solitons of expanding, steady and shrinking types. These solitons are foliated by hypersurfaces which are circle bundles over a product of Fano Kahler-Einstein manifolds or over coadjoint orbits of a compact connected semisimple Lie group."}
{"category": "Math", "title": "On factoriality of Cox rings", "abstract": "Generalized Cox's construction associates with an algebraic variety a remarkable invariant -- its total coordinate ring, or Cox ring. In this note we give a new proof of factoriality of the Cox ring when the divisor class group of the variety is finitely generated and free. The proof is based on a notion of graded factoriality. We show that if the divisor class group has torsion, then the Cox ring is again factorially graded, but factoriality may be lost."}
{"category": "Math", "title": "The Steinberg Variety and Representations of Reductive Groups", "abstract": "We give an overview of some of the main results in geometric representation theory that have been proved by means of the Steinberg variety. Steinberg's insight was to use such a variety of triples in order to prove a conjectured formula by Grothendieck. The Steinberg variety was later used to give an alternative approach to Springer's representations and played a central role in the proof of the Deligne-Langlands conjecture for Hecke algebras by Kazhdan and Lusztig."}
{"category": "Math", "title": "On the local time of the asymmetric Bernoulli walk", "abstract": "We study some properties of the local time of the asymmetric Bernoulli walk on the line. These properties are very similar to the corresponding ones of the simple symmetric random walks in higher ($d\\geq3$) dimension, which we established in the recent years. The goal of this paper is to highlight these similarities."}
{"category": "Math", "title": "Transient nearest neighbor random walk and Bessel process", "abstract": "We prove strong invariance principle between a transient Bessel process and a certain nearest neighbor (NN) random walk that is constructed from the former by using stopping times. It is also shown that their local times are close enough to share the same strong limit theorems. It is shown furthermore, that if the difference between the distributions of two NN random walks are small, then the walks themselves can be constructed so that they are close enough. Finally, some consequences concerning strong limit theorems are discussed."}
{"category": "Math", "title": "Weighted norm inequalities for de Branges--Rovnyak spaces and their applications", "abstract": "Let $\\mathcal{H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^\\infty(\\mathbb{C}_+)$. We study the boundary behavior of the derivatives of functions in $\\mathcal{H}(b)$ and obtain weighted norm estimates of the form $\\|f^{(n)}\\|_{L^2(\\mu)} \\le C\\|f\\|_{\\mathcal{H}(b)}$, where $f \\in \\mathcal{H}(b)$ and $\\mu$ is a Carleson-type measure on $\\mathbb{C}_+\\cup\\mathbb{R}$. We provide several applications of these inequalities. We apply them to obtain embedding theorems for $\\mathcal{H}(b)$ spaces. These results extend Cohn and Volberg--Treil embedding theorems for the model (star-invariant) subspaces which are special classes of de Branges--Rovnyak spaces. We also exploit the inequalities for the derivatives to study stability of Riesz bases of reproducing kernels $\\{k^b_{\\lambda_n}\\}$ in $\\mathcal{H}(b)$ under small perturbations of the points $\\lambda_n$."}
{"category": "Math", "title": "Integral representation of the $n$-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel", "abstract": "In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges--Rovnyak spaces $\\HH(b)$, where $b$ is in the unit ball of $H^\\infty(\\CC_+)$. In particular, we generalize a result of Ahern--Clark obtained for functions of the model spaces $K_b$, where $b$ is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel $k_{\\omega,n}^b$ of the evaluation of $n$-th derivative of elements of $\\HH(b)$ at the point $\\omega$ as it tends radially to a point of the real axis."}
{"category": "Math", "title": "A multiple covariance approach to PLS regression with several predictor groups: Structural Equation Exploratory Regression", "abstract": "A variable group Y is assumed to depend upon R thematic variable groups X 1, >..., X R . We assume that components in Y depend linearly upon components in the Xr's. In this work, we propose a multiple covariance criterion which extends that of PLS regression to this multiple predictor groups situation. On this criterion, we build a PLS-type exploratory method - Structural Equation Exploratory Regression (SEER) - that allows to simultaneously perform dimension reduction in groups and investigate the linear model of the components. SEER uses the multidimensional structure of each group. An application example is given."}
{"category": "Math", "title": "Mod\\'elisation factorielle des interactions entre deux ensembles d'observations : la m\\'ethode PLS-FILM (Partial Least Squares Factor Interaction Linear Modelling)", "abstract": "In this work, we consider a data array encoding interactions between two sets of observations respectively referred to as \"subjects\" and \"objects\". Besides, descriptions of subjects and objects are available through two variable sets. We propose a geometrically grounded exploratory technique to analyze the interactions using descriptions of subjects and objects: interactions are modelled using a hierarchy of subject-factors and object-factors built up from these descriptions. Our method bridges the gap between those of Chessel (RLQ analysis) and Martens (L-PLS), although it only has rank 1 components in common with them."}
{"category": "Math", "title": "The ziqqurath of exact sequences of n-groupoids", "abstract": "Higher Dimensional Categories are showing relevant implications in several fields of mathematical research. Nevertheless basic algebraic tools, in order to further develop the theory, are far from being established. In this thesis we introduce a notion of exactness for exact sequences of pointed n-groupoids. Furthermore we test it generalizing a well known result for (fibrations of) groupoids [R.Brown, 1970]. Namely, given a fibration F of (pointed) groupoids and its strict kernel it is possible to obtain a 6-term exact sequence of groups (of loops) and pointed sets (iso classes of objects). The ziqqurath, aka step-pyramid, comes out from iterating this construction, and it consists in several sequences of n-groupoids, (n-1)-groupoids and so on up to pointed sets (0-groupoids), of increasing length."}
{"category": "Math", "title": "Submanifolds of codimension two attaining equality in an extrinsic inequality", "abstract": "We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the length of the mean curvature vector."}
{"category": "Math", "title": "Relative Pro-$\\ell$ Completions of Mapping Class Groups", "abstract": "Fix a prime number ell. In this paper we develop the theory of relative pro-ell completion of discrete and profinite groups -- a natural generalization of the classical notion of pro-ell completion -- and show that the pro-ell completion of the Torelli group does not inject into the relative pro-ell completion of the corresponding mapping class group when the genus is at least 3. As an application, we prove that when g > 2, the action of the pro-ell completion of the Torelli group T_{g,1} on the pro-ell fundamental group of a pointed genus g surface is not faithful. The choice of a first-order deformation of a maximally degenerate stable curve of genus g determines an action of the absolute Galois group G_Q on the relative pro-ell completion of the corresponding mapping class group. We prove that for all g all such representations are unramified at all primes \\neq ell when the first order deformation is suitably chosen. This proof was communicated to us by Mochizuki and Tamagawa."}
{"category": "Math", "title": "On the weak K\\\"ahler-Ricci flow", "abstract": "In this note, we define and study K\\\"ahler-Ricci flow with initial data not being smooth with some natural applications."}
{"category": "Math", "title": "On the relative Giroux correspondence", "abstract": "Recently, Honda, Kazez and Matic described an adapted partial open book of a compact contact 3-manifold with convex boundary by generalizing the work of Giroux in the closed case. They also implicitly established a one-to-one correspondence between isomorphism classes of partial open book decompositions modulo positive stabilization and isomorphism classes of compact contact 3-manifolds with convex boundary. In this expository article we explicate the relative version of Giroux correspondence."}
{"category": "Math", "title": "The Kauffman skein algebra of a surface at $\\sqrt{-1}$", "abstract": "We study the structure of the Kauffman algebra of a surface with parameter equal to sqrt(-1). We obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat connections in a trivial SU(2)-bundle over the surface. We analyse the asymptotics of traces of curve-operators in TQFT in non standard regimes where the root of unity parametrizing the TQFT accumulates to a root of unity. We interpret the case of sqrt(-1) in terms of parallel transport operators."}
{"category": "Math", "title": "Relative Weight Filtrations on Completions of Mapping Class Groups", "abstract": "This paper gives an exposition of relative weight filtrations on completions of mapping class groups associated to a stable degeneration of marked genus g curves. These relative weight filtrations have been constructed using Galois theory (with Matsumoto) and Hodge theory (with Pearlstein and Terasoma). It is shown that the level 0 part of the relative weight filtration is an analogue of a parabolic subalgebra of a Kac-Moody Lie algebra. It is shown that all such subalgebras correspond to equivalence classes of pants decompositions of the surface -- two being equivalent if and only if they determine the same handlebody that the reference surface bounds. One application is to show that handlebody subgroups of mapping class groups contain elements arbitrarily far down the lower central series of Torelli groups. (This result was also obtained independently by Jamie Jorgensen.)"}
{"category": "Math", "title": "Asymptotic normality of the mixture density estimator in a disaggregation scheme", "abstract": "The paper concerns the asymptotic distribution of the mixture density estimator, proposed by Oppenheim et al 2006, in the aggregation/disaggregation problem of random parameter AR(1) process. We prove that, under mild conditions on the (semiparametric) form of the mixture density, the estimator is asymptotically normal. The proof is based on the limit theory for the quadratic form in linear random variables developed by Bhansali et al 2007. The moving average representation of the aggregated process is investigated. A small simulation study illustrates the result."}
{"category": "Math", "title": "Characterizations of Lojasiewicz inequalities and applications", "abstract": "The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka-Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by $-\\partial f$ are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka-Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of $f$- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C^2 function in in the plane is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka-Lojasiewicz inequality."}
{"category": "Math", "title": "Joinings of W*-dynamical systems", "abstract": "We study the notion of joinings of W*-dynamical systems, building on ideas from measure theoretic ergodic theory. In particular we prove sufficient and necessary conditions for ergodicity in terms of joinings, and also briefly look at conditional expectation operators associated with joinings."}
{"category": "Math", "title": "Data-driven calibration of penalties for least-squares regression", "abstract": "Penalization procedures often suffer from their dependence on multiplying factors, whose optimal values are either unknown or hard to estimate from the data. We propose a completely data-driven calibration algorithm for this parameter in the least-squares regression framework, without assuming a particular shape for the penalty. Our algorithm relies on the concept of minimal penalty, recently introduced by Birge and Massart (2007) in the context of penalized least squares for Gaussian homoscedastic regression. On the positive side, the minimal penalty can be evaluated from the data themselves, leading to a data-driven estimation of an optimal penalty which can be used in practice; on the negative side, their approach heavily relies on the homoscedastic Gaussian nature of their stochastic framework. The purpose of this paper is twofold: stating a more general heuristics for designing a data-driven penalty (the slope heuristics) and proving that it works for penalized least-squares regression with a random design, even for heteroscedastic non-Gaussian data. For technical reasons, some exact mathematical results will be proved only for regressogram bin-width selection. This is at least a first step towards further results, since the approach and the method that we use are indeed general."}
{"category": "Math", "title": "Isomorphism classes of certain Artinian Gorenstein algebras", "abstract": "In this paper we classify, up to analytic isomorphism, the family of almost stretched Artinian complete intersection A=R/I with a given Hilbert function, in the case R is a power series ring with an arbitrary number of variables."}
{"category": "Math", "title": "On multiwell Liouville theorems in higher dimension", "abstract": "We consider certain subsets of the space of $n\\times n$ matrices of the form $K = \\cup_{i=1}^m SO(n)A_i$, and we prove that for $p>1, q \\geq 1$ and for connected $\\Omega'\\subset\\subset\\Omega\\subset \\R^n$, there exists positive constant $a<1$ depending on $n,p,q, \\Omega, \\Omega'$ such that for $ \\veps=\\| {dist}(Du, K)\\|_{L^p(\\Omega)}^p$ we have $\\inf_{R\\in K}\\|Du-R\\|^p_{L^p(\\Omega')}\\leq M\\veps^{1/p}$ provided $u$ satisfies the inequality $\\| D^2 u\\|_{L^q(\\Omega)}^q\\leq a\\veps^{1-q}$. Our main result holds whenever $m=2$, and also for {\\em generic} $m\\le n$ in every dimension $n\\ge 3$, as long as the wells $SO(n)A_1,..., SO(n)A_m$ satisfy a certain connectivity condition. These conclusions are mostly known when $n=2$, and they are new for $n\\ge 3$."}
{"category": "Math", "title": "On the Lamperti stable processes", "abstract": "We consider a new family of $\\R^d$-valued L\\'{e}vy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance \\cite{cc}, \\cite{ckp}, \\cite{kp} and \\cite{pp}). This family of processes shares many properties with the tempered stable and the layered stable processes, defined in Rosi\\'nski \\cite{ro} and Houdr\\'e and Kawai \\cite{hok} respectively, for instance their short and long time behaviour. Additionally, in the real valued case we find a series representation which is used for sample paths simulation. In this work we find general properties of this class and we also provide many examples, some of which appear in recent literature."}
{"category": "Math", "title": "Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson form", "abstract": "We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex 'modulus' and a complex 'argument'. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of -1), but the complex phase is multiplied by a different complex root of -1 in the exponential function. We show how to calculate the amplitude and phase from an arbitrary quaternion in Cartesian form."}
{"category": "Math", "title": "The moduli space of \\'etale double covers of genus 5 curves is unirational", "abstract": "We show that the coarse moduli space $\\cR_5$ of \\'etale double covers of curves of genus~5 over the complex numbers is unirational. We give two slightly different arguments, one purely geometric and the other more computational."}
{"category": "Math", "title": "Horizontal Heegaard splittings of Seifert fibered spaces", "abstract": "We show that if an orientable Seifert fibered space $M$ with an orientable genus $g$ base space admits a strongly irreducible horizontal Heegaard splitting then there is a one-to-one correspondence between isotopy classes of strongly irreducible horizontal Heegaard splittings and elements of $\\mathbf{Z}^{2g}$. The correspondence is determined by the slopes of intersection of each Heegaard splitting with a collection of $2g$ incompressible tori in $M$. We also show that there are Seifert fibered spaces with infinitely many non-isotopic Heegaard splittings that determine Nielsen equivalent generating systems for the fundamental group of $M$."}
{"category": "Math", "title": "Optimal control of impulsive Volterra equations with variable impulse times", "abstract": "We obtain necessary conditions of optimality for impulsive Volterra integral equations with switching and impulsive controls, with variable impulse time-instants. The present work continues and complements our previous work on impulsive Volterra control with fixed impulse times."}
{"category": "Math", "title": "Jumping numbers of hyperplane arrangements", "abstract": "M. Saito recently proved that the jumping numbers of a hyperplane arrangement depend only on the combinatorics of the arrangement. However, a formula in terms of the combinatorial data was still missing. In this note, we give a formula and a different proof of the fact that the jumping numbers of a hyperplane arrangement depend only on the combinatorics. We also give a combinatorial formula for part of the Hodge spectrum and for the inner jumping multiplicities."}
{"category": "Math", "title": "Residually free 3-manifolds", "abstract": "We classify those compact 3-manifolds with incompressible toral boundary whose fundamental groups are residually free. For example, if such a manifold $M$ is prime and orientable and the fundamental group of $M$ is non-trivial then $M \\cong \\Sigma\\times S^1$, where $\\Sigma$ is a surface."}
{"category": "Math", "title": "On the geography of threefolds of general type", "abstract": "Let $X$ be a complex nonsingular projective 3-fold of general type. We show that there are positive constants $c$, $c'$ and $m_1$ such that $\\chi (\\omega _X)\\geq -c\\Vol (X)$ and $P_m(X)\\geq c'm^3\\Vol (X)$ for all $m\\geq m_1$."}
{"category": "Math", "title": "The totally nonnegative part of G/P is a CW complex", "abstract": "The totally nonnegative part of a partial flag variety G/P has been shown by the first author to be a union of semi-algebraic cells. Moreover she showed that the closure of a cell is the union of smaller cells. In this note we provide glueing maps for each of the cells to prove that the totally nonnegative part of G/P is a CW complex. This generalizes a result of Postnikov, Speyer and the second author for Grassmannians."}
{"category": "Math", "title": "Closed ideals in some algebras of analytic functions", "abstract": "We obtain a complete description of closed ideals of the algebra $\\mathcal{D}\\cap \\mathrm{lip}_\\alpha},$ $0<\\alpha\\leq{1/2},$ where $\\mathcal{D}$ is the Dirichlet space and $\\mathrm{lip}_\\alpha}$ is the algebra of analytic functions satisfying the Lipschitz condition of order $\\alpha.$"}
{"category": "Math", "title": "Closed ideals in analytic weighted Lipschitz algebras", "abstract": "We obtain a complete description of closed ideals in weighted Lipschitz algebras $\\Lambda_\\omega$ of analytic functions on the unit disk satisfying the following condition $$\\frac{|f(z)-f(w)|}{\\omega(|z-w|)}=o(1)\\qquad(as |z-w| \\longrightarrow 0),$$ where $\\omega$ is a modulus of continuity satisfying some regularity conditions."}
{"category": "Math", "title": "Review of \"Garden of integrals\"", "abstract": "This is a review of the book \"Garden of integrals\" by Frank Burk."}
{"category": "Math", "title": "Ideaux fermes d'algebres de Beurling analytiques sur le bidisque", "abstract": "We study the closed ideal in the Beurling algebras $\\mathcal{A}^{+}_{\\alpha,\\beta}$ of holomorphic function $f$ in the bidisc such that $\\sum_{n,m\\geq 0}|\\hat{f}(n,m)|(1+n)^{\\alpha}(1+m)^\\beta<+\\infty$. We determine the function $f\\in\\mathcal{A}^{+}_{\\alpha,\\beta}$ such that the ideals generated by $f$ coincide with the ideal generated by their zeros set."}
{"category": "Math", "title": "Shimura curves of genus at most two", "abstract": "In this article, we enumerate all Shimura curves X^D_0(N) of genus at most two."}
{"category": "Math", "title": "Finiteness results for flat surfaces: large cusps and short geodesics", "abstract": "For fixed g and T we show that finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech groups contain a cusp of hyperbolic co-area less than T. We obtain new restrictions on Veech groups: we show that any non-elementary Veech group can appear only finitely many times in a fixed stratum, that any non-elementary Veech group is of finite index in its normalizer, and that the quotient of the upper half plane by a non-lattice Veech group contains arbitrarily large embedded disks. These are proved using the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a hyperbolic element with eigenvalue less than T."}
{"category": "Math", "title": "Interpolation of Sobolev spaces, Littlewood-Paley inequalities and Riesz transforms on graphs", "abstract": "Let $\\Gamma$ be a graph endowed with a reversible Markov kernel $p$, and $P$ the associated operator, defined by $Pf(x)=\\sum_y p(x,y)f(y)$. Denote by $\\nabla$ the discrete gradient. We give necessary and/or sufficient conditions on $\\Gamma$ in order to compare $\\Vert \\nabla f \\Vert_{p}$ and $\\Vert (I-P)^{1/2}f \\Vert_{p}$ uniformly in $f$ for $1<p<+\\infty$. These conditions are different for $p<2$ and $p>2$. The proofs rely on recent techniques developed to handle operators beyond the class of Calder\\'on-Zygmund operators. For our purpose, we also prove Littlewood-Paley inequalities and interpolation results for Sobolev spaces in this context, which are of independent interest."}
{"category": "Math", "title": "Regularity and non-emptyness of linear systems in $\\mathbb P^n$", "abstract": "The main goal of this paper is to present a new algorithm bounding the regularity and ``alpha'' (the lowest degree of existing hypersurface) of a linear system of hypersurfaces (in $\\mathbb P^n$) passing through multiple points in general position. To do the above we formulate and prove new theorem, which allows to show non-specialty of linear system by splitting it into non-special (and simpler) systems. As a result we give new bounds for multiple point Seshadri constants on $\\PP^2$."}
{"category": "Math", "title": "Continuous local time of a purely atomic immigration superprocess with dependent spatial motion", "abstract": "A purely atomic immigration superprocess with dependent spatial motion in the space of tempered measures is constructed as the unique strong solution of a stochastic integral equation driven by Poisson processes based on the excursion law of a Feller branching diffusion, which generalizes the work of Dawson and Li (2003). As an application of the stochastic equation, it is proved that the superprocess possesses a local time which is Holder continuous of order $\\alpha$ for every $\\alpha< 1/2$. We establish two scaling limit theorems for the immigration superprocess, from which we derive scaling limits for the corresponding local time."}
{"category": "Math", "title": "Weak Solutions for Dislocation Type Equations", "abstract": "We describe recent results obtained by G. Barles, P. Cardaliaguet, R. Monneau and the author recently. They are concerned with nonlocal Eikonal equations arising in the study of the dynamics of dislocation lines in crystals. These equations are nonlocal but also non monotone. We use a notion of weak solution to provide solutions for all time. Then, we discuss the link between these weak solutions and the classical viscosity solutions, and state some uniqueness results in particular cases. A counter-example to uniqueness is given."}
{"category": "Math", "title": "Homogenization of monotone systems of Hamilton-Jacobi equations", "abstract": "In this paper we study homogenization for a class of monotone systems of first-order time-dependent periodic Hamilton-Jacobi equations. We characterize the Hamiltonians of the limit problem by appropriate cell problems. Hence we show the uniform convergence of the solution of the oscillating systems to the bounded uniformly continuous solution of the homogenized system."}
{"category": "Math", "title": "Stochastic equations of non-negative processes with jumps", "abstract": "We study stochastic equations of non-negative processes with jumps. The existence and uniqueness of strong solutions are established under Lipschitz and non-Lipschitz conditions. The comparison property of two solutions are proved under suitable conditions. The results are applied to stochastic equations driven by one-sided Levy processes and those of continuous state branching processes with immigration."}
{"category": "Math", "title": "Bernstein-Heinz-Chern results in calibrated manifolds", "abstract": "Given $(\\bar{M},\\Omega)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $\\cos \\theta$ is bounded away from zero, where $\\theta$ is the $\\Omega$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricc^M\\geq 0$ we may replace the boundedness condition on $\\cos \\theta$ by $\\cos \\theta\\geq Cr^{-\\beta}$, when $r\\to +\\infty$, where $ 0\\leq\\beta <1 $ and $C > 0$ are constants and $r$ is the distance function to a point in $M$. Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on a estimation of $\\|H\\|$ in terms of $\\cos\\theta$ and an isoperimetric inequality. We also give some conditions to conclude $M$ is totally geodesic. We study some particular cases."}
{"category": "Math", "title": "Iteration of the rational function z-1/z and a Hausdorff moment sequence", "abstract": "In a previous paper we considered a positive function f, uniquely determined for s>0 by the requirements f(1)=1, log(1/f) is convex and the functional equation f(s)=psi(f(s+1)) with psi(s)=s-1/s. We prove that the meromorphic extension of f to the whole complex plane is given by the formula f(z)=lim_{n\\to\\infty}psi^{\\circ n}(lambda_n(lambda_{n+1}/lambda_n)^z), where the numbers lambda_n are defined by lambda_0=0 and the recursion lambda_{n+1}=(1/2)(lambda_n+sqrt{lambda_n^2+4}). The numbers m_n=1/lambda_{n+1} form a Hausdorff moment sequence of a probability measure \\mu such that \\int t^{z-1}d\\mu(t)=1/f(z)"}
{"category": "Math", "title": "The curvature of contact structure on 3-manifolds", "abstract": "We study the sectional curvature of plane distributions on 3-manifolds. We show that if the distribution is a contact structure it is easy to manipulate this curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3-dimensional manifold $M$ there is a metric, such that the sectional curvature of the contact distribution is equal to -1. We also introduce the notion of Gaussian curvature of the plane distribution. For this notion of curvature we get the similar results."}
{"category": "Math", "title": "Dimensions of triangulated categories via Koszul objects", "abstract": "Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras."}
{"category": "Math", "title": "The Topological Directional Entropy of Z^2-actions Generated by Linear Cellular Automata", "abstract": "In this paper we study the topological and metric directional entropy of $\\mathbb{Z}^2$-actions by generated additive cellular automata (CA hereafter), defined by a local rule $f[l, r]$, $l, r\\in \\mathbb{Z}$, $l\\leq r$, i.e. the maps $T_{f[l, r]}: \\mathbb{Z}^\\mathbb{Z}_{m} \\to \\mathbb{Z}^\\mathbb{Z}_{m}$ which are given by $T_{f[l, r]}(x) =(y_n)_ {-\\infty}^{\\infty}$, $y_{n} = f(x_{n+l}, ..., x_{n+r}) = \\sum_{i=l}^r\\lambda_{i}x_{i+n}(mod m)$, $x=(x_n)_ {n=-\\infty}^{\\infty}\\in \\mathbb{Z}^\\mathbb{Z}_{m}$, and $f: \\mathbb{Z}_{m}^{r-l+1}\\to \\mathbb{Z}_{m}$, over the ring $\\mathbb{Z}_m (m \\geq 2)$, and the shift map acting on compact metric space $\\mathbb{Z}^\\mathbb{Z}_{m}$, where $m$ $(m \\geq2)$ is a positive integer. Our main aim is to give an algorithm for computing the topological directional entropy of the $\\mathbb{Z}^2$-actions generated by the additive CA and the shift map. Thus, we ask to give a closed formula for the topological directional entropy of $\\mathbb{Z}^2$-action generated by the pair $(T_{f[l, r]}, \\sigma)$ in the direction $\\theta$ that can be efficiently and rightly computed by means of the coefficients of the local rule f as similar to [Theor. Comput. Sci. 290 (2003) 1629-1646]. We generalize the results obtained by Ak\\i n [The topological entropy of invertible cellular automata, J. Comput. Appl. Math. 213 (2) (2008) 501-508] to the topological entropy of any invertible linear CA."}
{"category": "Math", "title": "Rational Equivariant Spectra", "abstract": "We develop model categories of rational equivariant spectra whose homotopy categories are equivalent to the category of rational equivariant cohomology theories. We prove that given an orthogonal decomposition of the unit in the rational Burnside ring, the model category of rational equivariant spectra decomposes into a product of localisations. We use this result to reprove the classification of rational equivariant cohomology theories for finite groups and to study such cohomology theories for the group O(2). We then concentrate on a split piece of the O(2) case and relate it to rational SO(2) equivariant spectra."}
{"category": "Math", "title": "Multiwell rigidity in nonlinear elasticity", "abstract": "We derive a quantitative rigidity estimate for a multiwell problem in nonlinear elasticity. Precisely, we show that if a gradient field is L^1-close to a set of the form SO(n)U_1 \\cup ... \\cup SO(n)U_l, and an appropriate bound on the length of the interfaces holds, then the gradient field is actually close to only one of the wells SO(n)U_i. The estimate holds for any connected subdomain, and has the optimal scaling."}
{"category": "Math", "title": "A geometrical approach to Gordan--Noether's and Franchetta's contributions to a question posed by Hesse", "abstract": "Hesse claimed that an irreducible projective hypersurface in $\\PP^n$ defined by an equation with vanishing hessian determinant is necessarily a cone. Gordan and Noether proved that this is true for $n\\leq 3$ and constructed counterexamples for every $n\\geq 4$. Gordan and Noether and Franchetta gave classification of hypersurfaces in $\\PP^4$ with vanishing hessian and which are not cones. Here we translate in geometric terms Gordan and Noether approach, providing direct geometrical proofs of these results."}
{"category": "Math", "title": "Regularity and Cohomological Splitting Conditions for Vector Bundles on Multiprojective Spaces", "abstract": "Here we give a definition of regularity on multiprojective spaces which is different from the definitions of Hoffmann-Wang and Costa-Mir\\'o Roig. By using this notion we prove some splitting criteria for vector bundles."}
{"category": "Math", "title": "Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues", "abstract": "For integers $k\\geq 2$, we study two differential operators on harmonic weak Maass forms of weight $2-k$. The operator $\\xi_{2-k}$ (resp. $D^{k-1}$) defines a map to the space of weight $k$ cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are \"dual\" under $\\xi_{2-k}$ to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of $D^{k-1}$."}
{"category": "Math", "title": "Least angle and $\\ell_1$ penalized regression: A review", "abstract": "Least Angle Regression is a promising technique for variable selection applications, offering a nice alternative to stepwise regression. It provides an explanation for the similar behavior of LASSO ($\\ell_1$-penalized regression) and forward stagewise regression, and provides a fast implementation of both. The idea has caught on rapidly, and sparked a great deal of research interest. In this paper, we give an overview of Least Angle Regression and the current state of related research."}
{"category": "Math", "title": "Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature", "abstract": "The Liouville property of a complete Riemannian manifold (i.e., the question whether there exist non-trivial bounded harmonic functions) attracted a lot of attention. For Cartan-Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan-Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan-Hadamard manifolds is much more complicated than one might have expected."}
{"category": "Math", "title": "Actions of vanishing homogeneity rank on quaternionic-Kaehler projective spaces", "abstract": "We classify isometric actions of compact Lie groups on quaternionic-K\\\"ahler projective spaces with vanishing homogeneity rank. We also show that they are not in general quaternion-coisotropic."}
{"category": "Math", "title": "Branching Laws for Some Unitary Representations of SL(4,R)", "abstract": "In this paper we consider the restriction of a unitary irreducible representation of type $A_{\\mathfrak q}(\\lambda)$ of $GL(4,{\\mathbb R})$ to reductive subgroups $H$ which are the fixpoint sets of an involution. We obtain a formula for the restriction to the symplectic group and to $GL(2,{\\mathbb C})$, and as an application we construct in the last section some representations in the cuspidal spectrum of the symplectic and the complex general linear group. In addition to working directly with the cohmologically induced module to obtain the branching law, we also introduce the useful concept of pseudo dual pairs of subgroups in a reductive Lie group."}
{"category": "Math", "title": "Minimal classes on the intermediate Jacobian of a generic cubic threefold", "abstract": "Let X be a smooth cubic threefold. We can associate two objects to X: the intermediate Jacobian J and the Fano surface F parametrising lines on X. By a theorem of Clemens and Griffiths, the Fano surface can be embedded in the intermediate Jacobian and the cohomology class of its image is minimal. In this paper we show that if X is generic, the Fano surface is the only surface of minimal class in J."}
{"category": "Math", "title": "Computing Hilbert Class Polynomials", "abstract": "We present and analyze two algorithms for computing the Hilbert class polynomial $H_D$ . The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for computing $H_D$, and we show that all methods have comparable run times."}
{"category": "Math", "title": "A Unified Theory on Some Basic Topological Concepts", "abstract": "Several mathematicians, including myself, have studied some unifications in general topological spaces as well as in fuzzy topological spaces. For instance in our earlier works, using operations on topological spaces, we have tried to unify some concepts similar to continuity, openness, closedness of functions, compactness, filter convergence, closedness of graphs, countable compactness and Lindelof property. In this article, to obtain further unifications, we will study $\\phi_{1,2}$-compactness and relations between $\\phi_{1,2}$-compactness, filters and $\\phi_{1,2}$% -closure operator."}
{"category": "Math", "title": "On the structure of Hardy-Sobolev-Maz'ya inequalities", "abstract": "In this article we establish new improvements of the optimal Hardy inequality in the half space. We first add all possible linear combinations of Hardy type terms thus revealing the structure of this type of inequalities and obtaining best constants. We then add the critical Sobolev term and obtain necessary and sufficient conditions for the validity of Hardy-Sobolev-Maz'ya type inequalities."}
{"category": "Math", "title": "Image Deconvolution Under Poisson Noise Using Sparse Representations and Proximal Thresholding Iteration", "abstract": "We propose an image deconvolution algorithm when the data is contaminated by Poisson noise. The image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transform. Our key innovations are: First, we handle the Poisson noise properly by using the Anscombe variance stabilizing transform leading to a non-linear degradation equation with additive Gaussian noise. Second, the deconvolution problem is formulated as the minimization of a convex functional with a data-fidelity term reflecting the noise properties, and a non-smooth sparsity-promoting penalties over the image representation coefficients (e.g. l1-norm). Third, a fast iterative backward-forward splitting algorithm is proposed to solve the minimization problem. We derive existence and uniqueness conditions of the solution, and establish convergence of the iterative algorithm. Experimental results are carried out to show the striking benefits gained from taking into account the Poisson statistics of the noise. These results also suggest that using sparse-domain regularization may be tractable in many deconvolution applications, e.g. astronomy or microscopy."}
{"category": "Math", "title": "Explicit a priori bounds on transfer operator eigenvalues", "abstract": "We provide explicit bounds on the eigenvalues of transfer operators defined in terms of holomorphic data."}
{"category": "Math", "title": "Topological 4-manifolds with geometrically 2-dimensional fundamental groups", "abstract": "Closed oriented 4-manifolds with the same geometrically 2-dimensional fundamental group (satisfying certain properties) are classified up to $s$-cobordism by their $w_2$-type, equivariant intersection form and the Kirby-Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed oriented 4-manifolds with solvable Baumslag-Solitar fundamental groups, including a precise realization result."}
{"category": "Math", "title": "Quasi-coherent sheaves on the moduli stack of formal groups", "abstract": "The central aim of this monograph is to provide decomposition results for quasi-coherent sheaves on the moduli stack of one-dimensional formal groups. These results will be based on the geometry of the stack itself, particularly the height filtration and an analysis of the formal neighborhoods of the geometric points. The main theorems are algebraic chromatic convergence results and fracture square decompositions. There is a major technical hurdle in this story, as the moduli stack of formal groups does not have the finitness properties required of an algebraic stack as usually defined. This is not a conceptual problem, but in order to be clear on this point and to write down a self-contained narrative, I have included a great deal of discussion of the geometry of the stack itself, giving various equivalent descriptions."}
{"category": "Math", "title": "Quotient groups of the fundamental groups of certain strata of the moduli space of quadratic differentials", "abstract": "In this paper, we study fundamental groups of strata of the moduli space of quadratic differentials. We use certain properties of the Abel-Jacobi map, combined with local surgeries on quadratic differentials, to construct quotient groups of the fundamental groups for a particular family of strata."}
{"category": "Math", "title": "The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres", "abstract": "These are notes for a five lecture series intended to uncover large-scale phenomena in the homotopy groups of spheres using the Adams-Novikov Spectral Sequence. The lectures were given in Strasbourg, May 7-11, 2007."}
{"category": "Math", "title": "Calculations of Sobol indices for the Gaussian process metamodel", "abstract": "Global sensitivity analysis of complex numerical models can be performed by calculating variance-based importance measures of the input variables, such as the Sobol indices. However, these techniques, requiring a large number of model evaluations, are often unacceptable for time expensive computer codes. A well known and widely used decision consists in replacing the computer code by a metamodel, predicting the model responses with a negligible computation time and rending straightforward the estimation of Sobol indices. In this paper, we discuss about the Gaussian process model which gives analytical expressions of Sobol indices. Two approaches are studied to compute the Sobol indices: the first based on the predictor of the Gaussian process model and the second based on the global stochastic process model. Comparisons between the two estimates, made on analytical examples, show the superiority of the second approach in terms of convergence and robustness. Moreover, the second approach allows to integrate the modeling error of the Gaussian process model by directly giving some confidence intervals on the Sobol indices. These techniques are finally applied to a real case of hydrogeological modeling."}
{"category": "Math", "title": "Global sensitivity analysis of computer models with functional inputs", "abstract": "Global sensitivity analysis is used to quantify the influence of uncertain input parameters on the response variability of a numerical model. The common quantitative methods are applicable to computer codes with scalar input variables. This paper aims to illustrate different variance-based sensitivity analysis techniques, based on the so-called Sobol indices, when some input variables are functional, such as stochastic processes or random spatial fields. In this work, we focus on large cpu time computer codes which need a preliminary meta-modeling step before performing the sensitivity analysis. We propose the use of the joint modeling approach, i.e., modeling simultaneously the mean and the dispersion of the code outputs using two interlinked Generalized Linear Models (GLM) or Generalized Additive Models (GAM). The ``mean'' model allows to estimate the sensitivity indices of each scalar input variables, while the ``dispersion'' model allows to derive the total sensitivity index of the functional input variables. The proposed approach is compared to some classical SA methodologies on an analytical function. Lastly, the proposed methodology is applied to a concrete industrial computer code that simulates the nuclear fuel irradiation."}
{"category": "Math", "title": "On the distribution of the free path length of the linear flow in a honeycomb", "abstract": "Let $\\ell \\geq 2$ be an integer. For each $\\eps >0$ remove from $\\R^2$ the union of discs of radius $\\eps$ centered at the integer lattice points $(m,n$, with $m\\nequiv n\\mod{\\ell}$. Consider a point-like particle moving linearly at unit speed, with velocity $\\omega$, along a trajectory starting at the origin, and its free path length $\\tau_{\\ell,\\eps} (\\omega)\\in [0,\\infty]$. We prove the weak convergence of the probability measures associated with the random variables $\\eps \\tau_{\\ell,\\eps}$ as $\\eps \\to 0^+$ and explicitly compute the limiting distribution. For $\\ell=3$ this leads to an asymptotic formula for the length of the trajectory of a billiard in a regular hexagon, starting at the center, with circular pockets of radius $\\eps\\to 0^+$ removed from the corners. For $\\ell=2$ this corresponds to the trajectory of a billiard in a unit square with circular pockets removed from the corners and trajectory starting at the center of the square. The limiting probability measures on $[0,\\infty)$ have a tail at infinity, which contrasts with the case of a square with pockets and trajectory starting from one of the corners, where the limiting probability measure has compact support."}
{"category": "Math", "title": "Regularity on abelian varieties III: relationship with Generic Vanishing and applications", "abstract": "We describe the relationship between the notions of $M$-regular sheaf and $GV$-sheaf in the case of abelian varieties. The former is a natural strengthening of the latter, and we provide an algebraic criterion characterizing it among the larger class. Based on this we deduce new basic properties of both $M$-regular and $GV$-sheaves. In the second part we give a number of applications of generation criteria for $M$-regular sheaves to the study of Seshadri constants, Picard bundles, pluricanonical maps on irregular varieties, and semihomogeneous vector bundles. This second part of the paper is based on our earlier preprint math.AG/0306103, with some improved statements and shortened arguments."}
{"category": "Math", "title": "Reduction principles for quantile and Bahadur-Kiefer processes of long-range dependent linear sequences", "abstract": "In this paper we consider quantile and Bahadur-Kiefer processes for long range dependent linear sequences. These processes, unlike in previous studies, are considered on the whole interval $(0,1)$. As it is well-known, quantile processes can have very erratic behavior on the tails. We overcome this problem by considering these processes with appropriate weight functions. In this way we conclude strong approximations that yield some remarkable phenomena that are not shared with i.i.d. sequences, including weak convergence of the Bahadur-Kiefer processes, a different pointwise behavior of the general and uniform Bahadur-Kiefer processes, and a somewhat \"strange\" behavior of the general quantile process."}
{"category": "Math", "title": "The lollipop graph is determined by its spectrum", "abstract": "An even (resp. odd) lollipop is the coalescence of a cycle of even (resp. odd) length and a path with pendant vertex as distinguished vertex. It is known that the odd lollipop is determined by its spectrum and the question is asked by W. Haemers, X. Liu and Y. Zhang for the even lollipop. We revisit the proof for odd lollipop, generalize it for even lollipop and therefore answer the question. Our proof is essentially based on a method of counting closed walks."}
{"category": "Math", "title": "Constructing dynamical twists over a non-abelian base", "abstract": "We give examples of dynamical twists in finite-dimensional Hopf algebras over an arbitrary Hopf subalgebra. The construction is based on the categorical approach of dynamical twists introduced by Donin and Mudrov."}
{"category": "Math", "title": "Equivariant cohomology and tensor categories", "abstract": "We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of results in equivariant cohomology, including the Chern-Weil theorem for an arbitrary rigid Lie algebra object. For a quadratic Lie algebra object we obtain a proof of the Duflo isomorphism along the lines of Alekseev-Meinrenken, thereby generalizing their result to Lie superalgebras."}
{"category": "Math", "title": "Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations", "abstract": "We study the Cauchy problem for the dissipative Benjamin-Ono equations $u_t+\\H u_{xx}+|D|^\\alpha u+uu_x=0$ with $0\\leq\\alpha\\leq 2$. When $0\\leq\\alpha< 1$, we show the ill-posedness in $H^s(\\R)$, $s\\in\\R$, in the sense that the flow map $u_0\\mapsto u$ (if it exists) fails to be $\\C^2$ at the origin. For $1<\\alpha\\leq 2$, we prove the global well-posedness in $H^s(\\R)$, $s>-\\alpha/4$. It turns out that this index is optimal."}
{"category": "Math", "title": "Finite size scaling for homogeneous pinning models", "abstract": "Pinning models are built from discrete renewal sequences by rewarding (or penalizing) the trajectories according to their number of renewal epochs up to time $N$, and $N$ is then sent to infinity. They are statistical mechanics models to which a lot of attention has been paid both because they are very relevant for applications and because of their {\\sl exactly solvable character}, while displaying a non-trivial phase transition (in fact, a localization transition). The order of the transition depends on the tail of the inter-arrival law of the underlying renewal and the transition is continuous when such a tail is sufficiently heavy: this is the case on which we will focus. The main purpose of this work is to give a mathematical treatment of the {\\sl finite size scaling limit} of pinning models, namely studying the limit (in law) of the process close to criticality when the system size is proportional to the correlation length."}
{"category": "Math", "title": "Testing additivity in nonparametric regression under random censorship", "abstract": "In this paper, we are concerned with nonparametric estimation of the multivariate regression function in the presence of right censored data. More precisely, we propose a statistic that is shown to be asymptotically normally distributed under the additive assumption, and that could be used to test for additivity in the censored regression setting."}
{"category": "Math", "title": "Halphen pencils on quartic threefolds", "abstract": "For every smooth quartic threefold, we classify all pencils on it whose general element is an irreducible surface birational to a smooth surface of Kodaira dimension zero."}
{"category": "Math", "title": "Two universal 3-quantifier representations of recursively enumerable sets", "abstract": "It is proved that all recursively enumerable sets of natural numbers can be represented by arithmetic formulas (of two kinds) with only 3 quantifiers."}
{"category": "Math", "title": "Lefschetz Properties and Basic Constructions on Simplicial Spheres", "abstract": "The well known $g$-conjecture for homology spheres follows from the stronger conjecture that the face ring over the reals of a homology sphere, modulo a linear system of parameters, admits the strong-Lefschetz property. We prove that the strong-Lefschetz property is preserved under the following constructions on homology spheres: join, connected sum, and stellar subdivisions. The last construction is a step towards proving the $g$-conjecture for piecewise-linear spheres."}
{"category": "Math", "title": "On Ueno's Conjecture K", "abstract": "We show that if $X$ is a smooth complex projective variety with Kodaira dimension $0$ then the Kodaira dimension of a general fiber of its Albanese map is at most $h^0(\\Omega ^1 _X)$."}
{"category": "Math", "title": "A Few Splitting Criteria for Vector Bundles", "abstract": "We prove a few splitting criteria for vector bundles on a quadric hypersurface and Grassmannians. We give also some cohomological splitting conditions for rank 2 bundles on multiprojective spaces. The tools are monads and a Beilinson's type spectral sequence generalized by Costa and Mir\\'o-Roig."}
{"category": "Math", "title": "Affine tangles and irreducible exotic sheaves", "abstract": "We construct a weak representation of the category of framed affine tangles on a disjoint union of triangulated categories ${\\mathcal D}_{2n}$. The categories we use are that of coherent sheaves on Springer fibers over a nilpotent element of $sl_{2n}$ with two equal Jordan blocks. This representation allows us to enumerate the irreducible objects in the heart of the exotic $t$-structure on ${\\mathcal D}_{2n}$ by crossingless matchings of $2n$ points on a circle. We also describe the algebra of endomorphisms of the direct sum of the irreducible objects."}
{"category": "Math", "title": "A note on closed 3-braids", "abstract": "Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which, with luck, a researcher may be able to test various conjectures. The goal of this review article is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases. We also use it as an opportunity to review what is known and suggest some open questions."}
{"category": "Math", "title": "Surfaces obtained from CP^(N-1) sigma models", "abstract": "In this paper, the Weierstrass technique for harmonic maps S^2 -> CP^(N-1) is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the CP^(N-1) model equations are defined on the sphere S^2 and the associated action functional of this model is finite, then the generalized Weierstrass formula for immersion describes conformally parametrized surfaces in the su(N) algebra. In particular, for any holomorphic or antiholomorphic solution of this model the associated surface can be expressed in terms of an orthogonal projector of rank (N-1). The implementation of this method is presented for two-dimensional conformally parametrized surfaces immersed in the su(3) algebra. The usefulness of the proposed approach is illustrated with examples, including the dilation-invariant meron-type solutions and the Veronese solutions for the CP^2 model. Depending on the location of the critical points (zeros and poles) of the first fundamental form associated with the meron solution, it is shown that the associated surfaces are semi-infinite cylinders. It is also demonstrated that surfaces related to holomorphic and mixed Veronese solutions are immersed in R^8 and R^3, respectively."}
{"category": "Math", "title": "$p$-Adic multiresolution analysis and wavelet frames", "abstract": "We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating MRAs (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions. We also suggest a method for the construction of wavelet functions and prove that any wavelet function generates a $p$-adic wavelet frame."}
{"category": "Math", "title": "Ahlfors' currents in higher dimension", "abstract": "We consider a nondegenerate holomorphic map $f: V \\mapsto X$ where $(X, \\omega)$ is a compact hermitian manifold of dimension higher or equal to $k$ and $V$ is an open connected complex manifold of dimension $k$. In this article we give criteria which permit to construct Ahlfors' currents in $X$."}
{"category": "Math", "title": "On the Y555 complex reflection group", "abstract": "We give a computer-free proof of a theorem of Basak, describing the group generated by 16 complex reflections of order 3, satisfying the braid and commutation relations of the Y555 diagram. The group is the full isometry group of a certain lattice of signature (13,1) over the Eisenstein integers Z[cube root of 1]. Along the way we enumerate the cusps of this lattice and classify the root and Niemeier lattices over this ring."}
{"category": "Math", "title": "The Gram determinant of the type B Temperley-Lieb algebra", "abstract": "In this paper, we solve a problem posed by Rodica Simion regarding type B Gram determinants. We present this in a fashion influenced by the work of W.B.R.Lickorish on Witten-Reshetikhin-Turaev invariants of 3-manifolds. The roots of the determinant were predicted by Dabkowski and Przytycki, and the complete factorization was conjectured by Gefry Barad. We will give a detailed history of this problem in a sequel paper in which we also plan to address other related questions by Simion, and connect the problem to Frenkel-Khovanov's work."}
{"category": "Math", "title": "Auslander Bounds and Homological Conjectures", "abstract": "Inspired by recent works on rings satisfying Auslander's conjecture, we study invariants, which we call Auslander bounds, and prove that they have strong relations to some homological conjectures."}
{"category": "Math", "title": "Carter subgroups of finite groups", "abstract": "In the paper it is proven that Carter subgroups of a finite group are conjugate. A complete classification of Carter subgroups in finite almost simple groups is also obtained."}
{"category": "Math", "title": "Properties of element orders in covers for L(n,q) and U(n,q)", "abstract": "We show that if a finite simple group G isomorphic to PSL(n,q) or PSU(n,q), where either $n\\ne 4$, or q is prime or even, acts on a vector space over a field of the defining characteristic of G, then the corresponding semidirect product contains an element whose order is distinct from every element order of G. As a consequence, we prove that the group PSL(n,q), where $n\\ne 4$ or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka notebook."}
{"category": "Math", "title": "The structure of a tridiagonal pair", "abstract": "Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \\to V$ and $A^*:V \\to V$ that satisfy the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\\{V_i\\}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \\subseteq V_{i-1} + V_i + V_{i+1}$ for $0 \\leq i \\leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\\{V^*_i\\}_{i=0}^\\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \\subseteq V^*_{i-1} + V^*_i + V^*_{i+1}$ for $0 \\leq i \\leq \\delta$, where $V^*_{-1}=0$ and $V^*_{\\delta+1}=0$; (iv)there is no subspace $W$ of $V$ such that $AW \\subseteq W$, $A^* W \\subseteq W$, $W \\neq 0$, $W \\neq V$. We call such a pair a tridiagonal pair on $V$. It is known that $d=\\delta$ and for $0 \\leq i \\leq d$ the dimensions of $V_i, V_{d-i}, V^*_i, V^*_{d-i}$ coincide. In this paper we show that the following (i)--(iv) hold provided that $K$ is algebraically closed: (i) Each of $V_0$, $V^*_0$, $V_d$, $V^*_d$ has dimension 1. (ii) There exists a nondegenerate symmetric bilinear form $(,)$ on $V$ such that $(Au,v)=(u,Av)$ and $(A^*u,v)=(u,A^*v)$ for all $u,v \\in V$. (iii) There exists a unique anti-automorphism of $End(V)$ that fixes each of $A,A^*$. (iv) The pair $A,A^*$ is determined up to isomorphism by the data $(\\{\\th_i\\}_{i=0}^d; \\{\\th^*_i\\}_{i=0}^d; \\{\\zeta_i\\}_{i=0}^d)$, where $\\th_i$ (resp. $\\th^*_i$) is the eigenvalue of $A$ (resp. $A^*$) on $V_i$ (resp. $V^*_i$), and $\\{\\zeta_i\\}_{i=0}^d$ is the split sequence of $A,A^*$ corresponding to $\\{\\th_i\\}_{i=0}^d$ and $\\{\\th^*_i\\}_{i=0}^d$."}
{"category": "Math", "title": "An efficient methodology for modeling complex computer codes with Gaussian processes", "abstract": "Complex computer codes are often too time expensive to be directly used to perform uncertainty propagation studies, global sensitivity analysis or to solve optimization problems. A well known and widely used method to circumvent this inconvenience consists in replacing the complex computer code by a reduced model, called a metamodel, or a response surface that represents the computer code and requires acceptable calculation time. One particular class of metamodels is studied: the Gaussian process model that is characterized by its mean and covariance functions. A specific estimation procedure is developed to adjust a Gaussian process model in complex cases (non linear relations, highly dispersed or discontinuous output, high dimensional input, inadequate sampling designs, ...). The efficiency of this algorithm is compared to the efficiency of other existing algorithms on an analytical test case. The proposed methodology is also illustrated for the case of a complex hydrogeological computer code, simulating radionuclide transport in groundwater."}
{"category": "Math", "title": "On doubly stochastic quadratic operators and Birkhoff's problem", "abstract": "In the present paper we introduce a concept of doubly stochastic quadratic operator. We prove necessary and sufficient conditions for doubly stochasticity of operator. Besides, we prove that the set of all doubly stochastic operators forms convex polytope. Finally, we study analogue of Birkhoff's theorem for the class of doubly stochastic quadratic operators."}
{"category": "Math", "title": "Ising Problem on Simple Cubic Lattice", "abstract": "Simple cubic lattice (SC lattice) can be viewed as plane triangular lattice (PT lattice) by viewing it along its principle diagonal lines. By viewing thus we establish the exact one-to-one correspondence between the closed graphs on SC lattice and the corresponding closed graphs on PT lattice. We thus see that the propagator for PT lattice (with suitable modifications) can be used to solve, at least in principle, the 3D Ising problem for SC lattice in the absence of external magnetic field. A new method is then proposed to generate high temperature expansion for the partition function. This method is applicable to 2D as well as 3D lattices. This method does not require explicit counting of closed graphs and this counting is achieved in an indirect way and thus exact series expansion can be achieved up to any sufficiently large order."}
{"category": "Math", "title": "Testing polynomial covariate effects in linear and generalized linear mixed models", "abstract": "An important feature of linear mixed models and generalized linear mixed models is that the conditional mean of the response given the random effects, after transformed by a link function, is linearly related to the fixed covariate effects and random effects. Therefore, it is of practical importance to test the adequacy of this assumption, particularly the assumption of linear covariate effects. In this paper, we review procedures that can be used for testing polynomial covariate effects in these popular models. Specifically, four types of hypothesis testing approaches are reviewed, i.e. R tests, likelihood ratio tests, score tests and residual-based tests. Derivation and performance of each testing procedure will be discussed, including a small simulation study for comparing the likelihood ratio tests with the score tests."}
{"category": "Math", "title": "Extremal orders of compositions of certain arithmetical functions", "abstract": "We study the exact extremal orders of compositions $f(g(n))$ of certain arithmetical functions, including the functions $\\sigma(n)$, $\\phi(n)$, $\\sigma^*(n)$ and $\\phi^*(n)$, representing the sum of divisors of $n$, Euler's function and their unitary analogues, respectively. Our results complete, generalize and refine known results."}
{"category": "Math", "title": "Banach spaces with projectional skeletons", "abstract": "A projectional skeleton in a Banach space is a sigma-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Sigma-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton."}
{"category": "Math", "title": "Exponential decay phenomenon of the principal eigenvalue of an elliptic operator with a large drift term of gradient type", "abstract": "We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the velocity potential well condition as the parameter tends to infinity."}
{"category": "Math", "title": "Bornes pour la r\\'egularit\\'e de Castelnuovo-Mumford des sch\\'emas non lisses", "abstract": "We show that bounds on the Castelnuovo-Mumford regularity of singular schemes, as a function of the degrees of the equations defining the shceme, of its dimension and of the dimension of their singular space. In the case where the singularities are isolated, we improve the bound given by Chardin and Ulrich, and in the general case we establish a bound doubly exponential in the dimension of the singular space. -- Nous montrons dans cet article des bornes pour la regularite de Castelnuovo-Mumford d'un schema admettant des singularites, en fonction des degres des equations definissant le schema, de sa dimension et de la dimension de son lieu singulier. Dans le cas ou les singularites sont isolees, nous ameliorons la borne fournie par Chardin et Ulrich et dans le cas general, nous etablissons une borne doublement exponentielle en la dimension du lieu singulier."}
{"category": "Math", "title": "On Riemannian almost product manifolds with nonintegrable structure", "abstract": "The class of the Riemannian almost product manifolds with nonintegrable structure is considered. Some identities for curvature tensor as certain invariant tensors and quantities are obtained."}
{"category": "Math", "title": "Spectra of weighted algebras of holomorphic functions", "abstract": "We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operators and algebra homomorphisms, in particular to investigate how their induced mappings act on the analytic structure of the spectrum. Moreover, a Banach-Stone type question is addressed."}
{"category": "Math", "title": "Non maximal cyclically monotone graphs and construction of a bipotential for the Coulomb's dry friction law", "abstract": "We show a surprising connexion between a property of the inf convolution of a family of convex lower semicontinuous functions and the fact that the intersection of maximal cyclically monotone graphs is the critical set of a bipotential. We then extend the results from arXiv:math/0608424v4 to bipotentials convex covers, generalizing the notion of a bi-implicitly convex lagrangian cover. As an application we prove that the bipotential related to Coulomb's friction law is related to a specific bipotential convex cover with the property that any graph of the cover is non maximal cyclically monotone."}
{"category": "Math", "title": "On Hunting for Taxicab Numbers", "abstract": "In this article, we make use of some known method to investigate some properties of the numbers represented as sums of two equal odd powers, i.e., the equation $x^n+y^n=N$ for $n\\ge3$. It was originated in developing algorithms to search new taxicab numbers (i.e., naturals that can be represented as a sum of positive cubes in many different ways) and to verify their minimality. We discuss properties of diophantine equations that can be used for our investigations. This techniques is applied to develop an algorithm allowing us to compute new taxicab numbers (i.e., numbers represented as sums of two positive cubes in $k$ different ways), for $k=7...14$."}
{"category": "Math", "title": "Hiding a drift", "abstract": "In this article we consider a Brownian motion with drift of the form \\[dS_t=\\mu_t dt+dB_t\\qquadfor t\\ge0,\\] with a specific nontrivial $(\\mu_t)_{t\\geq0}$, predictable with respect to $\\mathbb{F}^B$, the natural filtration of the Brownian motion $B=(B_t)_{t\\ge0}$. We construct a process $H=(H_t)_{t\\ge0}$, also predictable with respect to $\\mathbb{F}^B$, such that $((H\\cdot S)_t)_{t\\ge 0}$ is a Brownian motion in its own filtration. Furthermore, for any $\\delta>0$, we refine this construction such that the drift $(\\mu_t)_{t\\ge0}$ only takes values in $]\\mu-\\delta,\\mu+\\delta[$, for fixed $\\mu>0$."}
{"category": "Math", "title": "Pfister's Theorem for orthogonal involutions of degree 12", "abstract": "We use the fact that a projective half-spin representation of $Spin_{12}$ has an open orbit to generalize Pfister's result on quadratic forms of dimension 12 in $I^3$ to orthogonal involutions."}
{"category": "Math", "title": "Finite type coarse expanding conformal dynamics", "abstract": "We continue the study of non-invertible topological dynamical systems with expanding behavior. We introduce the class of {\\em finite type} systems which are characterized by the condition that, up to rescaling and uniformly bounded distortion, there are only finitely many iterates. We show that subhyperbolic rational maps and finite subdivision rules (in the sense of Cannon, Floyd, Kenyon, and Parry) with bounded valence and mesh going to zero are of finite type. In addition, we show that the limit dynamical system associated to a selfsimilar, contracting, recurrent, level-transitive group action (in the sense of V. Nekrashevych) is of finite type. The proof makes essential use of an analog of the finiteness of cone types property enjoyed by hyperbolic groups."}
{"category": "Math", "title": "First derivatives estimates for finite-difference schemes", "abstract": "We give sufficient conditions under which solutions of discretized in space second-order parabolic and elliptic equations, perhaps degenerate, admit estimates of the first derivatives in the space variables independent of the mesh size."}
{"category": "Math", "title": "Extra cancellation of even Calderon-Zygmund operators and quasiconformal mappings", "abstract": "We discuss a special class of Beltrami coefficients whose associated quasiconformal mapping is bilipschitz. These are of the form the characteristic function of a planar bounded domain with smooth boundary of class C 1+epsilon times a density of class Lip epsilon on the domain. The crucial fact in the argument is the special extracancellation property of even Calderon-Zygmund kernels, namely that they have zero integral on half the unit ball. This property is expressed in a particularly suggestive way and is shown to have far-reaching consequences. The main result may also be viewed as a Lipschitz regularity result for the Beltrami equation, and so for certain planar second order elliptic equations in divergence form."}
{"category": "Math", "title": "Numerical Solution of the Beltrami Equation", "abstract": "An effective algorithm is presented for solving the Beltrami equation fzbar = mu fz in a planar disk. The algorithm involves no evaluation of singular integrals. The strategy, working in concentric rings, is to construct a piecewise linear mu-conformal mapping and then correct the image using a known algorithm for conformal mappings. Numerical examples are provided and the computational complexity is analyzed."}
{"category": "Math", "title": "Socle theory for Leavitt path algebras of arbitrary graphs", "abstract": "The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate the socle of a Leavitt path algebra, extending so the results for row-finite graphs in the previous paper [12] (but with different methods). A concrete description of the socle of a Leavitt path algebra is obtained: it is a direct sum of matrix rings (of finite or infinite size) over the base field. New proofs of the Graded Uniqueness and of the Cuntz-Krieger Uniqueness Theorems are given, shorthening significantly the original ones."}
{"category": "Math", "title": "Essentialit\\'e dans les bases additives", "abstract": "In this article we study the notion of essential subset of an additive basis, that is to say the minimal finite subsets $P$ of a basis $A$ such that $A \\setminus P$ doesn't remains a basis. The existence of an essential subset for a basis is equivalent for this basis to be included, for almost all elements, in an arithmetic non-trivial progression. We show that for every basis $A$ there exists an arithmetic progression with a biggest common difference containing $A$. Having this common difference $a$ we are able to give an upper bound to the number of essential subsets of $A$: this is the radical's length of $a$ (in particular there is always many finite essential subsets in a basis). In the case of essential subsets of cardinality 1 (essential elements) we introduce a way to \"dessentialize\" a basis. As an application, we definitively improve the earlier result of Deschamps and Grekos giving an upper bound of the number of the essential elements of a basis. More precisely, we show that for all basis $A$ of order $h$, the number $s$ of essential elements of $A$ satisfy $s\\leq c\\sqrt{\\frac{h}{\\log h}}$ where $c=30\\sqrt{\\frac{\\log 1564}{1564}}\\simeq 2,05728$, and we show that this inequality is best possible."}
{"category": "Math", "title": "Solving Fermat-type equations x^5+y^5=dz^p", "abstract": "In this paper we are interested in solving the Fermat-type equations x^5+y^5=dz^p where d is a positive integer and p a prime number $\\ge 7$. We describe a new method based on modularity theorems which allows us to improve all the results in a previous paper of the first author. We finally discuss the present limitations of the method by looking at the case d=3."}
{"category": "Math", "title": "Weak type estimates for spherical multipliers on noncompact symmetric spaces", "abstract": "In this paper we prove sharp weak type 1 estimates for spherical Fourier multipliers on symmetric spaces of the noncompact type. This complements earlier results of J.-Ph. Anker and A.D. Ionescu."}
{"category": "Math", "title": "A generalization of the duality for multiple harmonic sums", "abstract": "The duality is a fundamental property of the finite multiple harmonic sums (MHS). In this paper, we prove a duality result for certain generalizations of MHS which appear naturally as the differences of MHS. We also prove a formula for the differences of these generalized MHS."}
{"category": "Math", "title": "On the spectrum of unitary finite-Euclidean graphs", "abstract": "We consider unitary graphs attached to Z_d^n using an analogue of the Euclidean distance. These graphs are shown to be integral when n is odd or the dimension d is even."}
{"category": "Math", "title": "Algebraic Degree of Polynomial Optimization", "abstract": "Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the optimal coordinates. The derivation of the algebraic degree is equivalent to counting the number of all complex critical points. As special cases, we obtain the algebraic degrees of quadratically constrained quadratic programming (QCQP), second order cone programming (SOCP) and $p$-th order cone programming (pOCP), in analogy to the algebraic degree of semidefinite programming."}
{"category": "Math", "title": "Function spaces and capacity related to a Sublinear Expectation: application to G-Brownian Motion Pathes", "abstract": "In this paper we give some basic and important properties of several typical Banach spaces of functions of $G$-Brownian motion pathes induced by a sublinear expectation--G-expectation. Many results can be also applied to more general situations. A generalized version of Kolmogorov's criterion for continuous modification of a stochastic process is also obtained. The results can be applied to continuous time dynamic and coherent risk measures in finance in particular for path-dependence risky positions under situations of volatility model uncertainty."}
{"category": "Math", "title": "Deformation quantization modules I:Finiteness and duality", "abstract": "We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\\hbar]$-modules on a topological space. Then we consider a $Z[\\hbar]$-algebra satisfying some suitable conditions and prove coherency results by using the property of being cohomologically complete. We apply these results to the study of modules over deformation quantization algebroids on complex Poisson manifolds. We prove in particular that under a natural properness condition, the convolution of two coherent kernels over such algebroids is coherent. We also construct the dualizing complexes in this framework and show that the convolution of kernels commutes with duality."}
{"category": "Math", "title": "Adiabatic limits on Riemannian Sol-manifolds", "abstract": "We obtain an asymptotic formula for the spectrum distribution function of the Laplace operator on a compact Riemannian Sol-manifold in the adiabatic limit determined by a one-dimensional foliation defined by the orbits of a left-invariant flow."}
{"category": "Math", "title": "The correspondence between a plane curve and its complement", "abstract": "Given two irreducible curves of the plane which have isomorphic complements, it is natural to ask whether there exists an automorphism of the plane that sends one curve on the other. This question has a positive answer for a large family of curves and H.Yoshihara conjectured that it is true in general. We exhibit counterexamples to this conjecture, over any ground field. In some of the cases, the curves are isomorphic and in others not; this provides counterexamples of two different kinds. Finally, we use our construction to find the existence of surprising non-linear automorphisms of affine surfaces."}
{"category": "Math", "title": "On ergodic properties of convolution operators associated with compact quantum groups", "abstract": "Recent results of M.Junge and Q.Xu on the ergodic properties of the averages of kernels in noncommutative L^p-spaces are applied to the analysis of the almost uniform convergence of operators induced by the convolutions on compact quantum groups."}
{"category": "Math", "title": "Universal unfoldings of Laurent polynomials and tt* structures", "abstract": "This article surveys the relations between harmonic Higgs bundles and Saito structures which lead to tt* geometry on Frobenius manifolds. We give the main lines of the proof of the existence of a canonical tt* structure on the base space of the universal unfolding of convenient and nondegenerate Laurent polynomials."}
{"category": "Math", "title": "A characterization of the overcoherence", "abstract": "Let $\\mathcal{P}$ be a proper smooth formal $\\mathcal{V}$-scheme, $X$ a closed subscheme of the special fiber of $\\mathcal{P}$, $\\mathcal{E} \\in F\\text{-}D ^\\mathrm{b}_\\mathrm{coh} (\\D ^\\dag_{\\mathcal{P},\\mathbb{Q}})$ with support in $X$. We check that $\\mathcal{E}$ is $\\D ^\\dag _{\\mathcal{P},\\mathbb{Q}}$-overcoherent if and only if, for any morphism $f : \\mathcal{P}' \\to \\mathcal{P}$ of smooth formal $\\mathcal{V}$-schemes, $f ^! (\\mathcal{E}) $ is $\\D ^\\dag_{\\mathcal{P}', \\mathbb{Q}}$-coherent."}
{"category": "Math", "title": "Leibniz algebra deformations of a Lie algebra", "abstract": "In this note we compute Leibniz algebra deformations of the 3-dimensional nilpotent Lie algebra $\\mathfrak{n}_3$ and compare it with its Lie deformations. It turns out that there are 3 extra Leibniz deformations. We also describe the versal Leibniz deformation of $\\mathfrak{n}_3$ with the versal base."}
{"category": "Math", "title": "Rational approximations to $\\sqrt[3]{2}$ and other algebraic numbers revisited", "abstract": "In this paper, we establish improved effective irrationality measures for certain numbers of the form $\\sqrt[3]{n}$, using approximations obtained from hypergeometric functions. These results are very close to the best possible using this method. We are able to obtain these results by determining very precise arithmetic information about the denominators of the coefficients of these hypergeometric functions. Improved bounds for $\\theta(k,l;x)$ and $\\psi(k,l;x)$ for $k=1,3,4,6$ are also presented."}
{"category": "Math", "title": "Jet Berwald-Riemann-Lagrange Geometrization for Affine Maps between Finsler Manifolds", "abstract": "In this paper we introduce a natural definition for the affine maps between two Finsler manifolds $(M, F)$ and $(N,\\tilde F)$ and we give some geometrical properties of these affine maps. Starting from the equations of the affine maps, we construct a natural Berwald-Riemann-Lagrange geometry on the 1-jet space $J^1(TM;N)$, in the sense of a Berwald nonlinear connection $\\Gamma^b_jet$, a Berwald $\\Gamma^b_jet$-linear d-connection $B\\Gamma^b_jet$, together with its d-torsions and d-curvatures, which geometrically characterizes the initial affine maps between Finsler manifolds."}
{"category": "Math", "title": "Properties of lexsegment ideals", "abstract": "We show that any lexsegment ideal with linear resolution has linear quotients with respect to a suitable ordering of its minimal monomial generators. For completely lexsegment ideals with linear resolution we show that the decomposition function is regular. For arbitrary lexsegment ideals we compute the depth and the dimension. As application we characterize the Cohen-Macaulay lexsegment ideals."}
{"category": "Math", "title": "Absence of eigenvalues for integro-differential operators with periodic coefficients", "abstract": "Applying perturbation theory methods, the absence of the point spectrum for some nonselfadjoint integro-differential operators is investigated. The considered differential operators are of arbitrary order and act in either $\\mathbf{L}_{p}(\\mathbb{R}_{+})$ or $\\mathbf{L}_{p}(\\mathbb{R}) (1\\leq p<\\infty)$. As an application of general results, new spectral properties of the perturbed Hill operator are derived."}
{"category": "Math", "title": "On the multiplicity conjecture for non-Cohen-Macaulay simplicial complexes", "abstract": "We prove a reformulation of the multiplicity upper bound conjecture and use that reformulation to prove it for three-dimensional simplicial complexes and homology manifolds with many vertices. We provide necessary conditions for a Cohen-Macaulay complex with many vertices to have a pure minimal free resolution and a characterization of flag complexes whose minimal free resolution is pure."}
{"category": "Math", "title": "Deformations of associative submanifolds with boundary", "abstract": "Let $M$ be a topological $G_2$-manifold. We prove that the space of infinitesimal associative deformations of a compact associative submanifold $Y$ with boundary in a coassociative submanifold $X$ is the solution space of an elliptic problem. For a connected boundary $\\partial Y$ of genus $g$, the index is given by $\\int_{\\partial Y}c_1(\\nu_X)+1-g$, where $\\nu_X$ denotes the orthogonal complement of $T\\partial Y$ in $TX_{|\\partial Y}$ and $c_1(\\nu_X)$ the first Chern class of $\\nu_X$ with respect to its natural complex structure. Further, we exhibit explicit examples of non-trivial index."}
{"category": "Math", "title": "A poset structure on quasifibonacci partitions", "abstract": "In this paper, we study partitions of positive integers into distinct quasifibonacci numbers. A digraph and poset structure is constructed on the set of such partitions. Furthermore, we discuss the symmetric and recursive relations between these posets. Finally, we prove a strong generalization of Robbins' result on the coefficients of a quasifibonacci power series."}
{"category": "Math", "title": "The advanced maximum principle for parabolic systems on manifolds with boundary", "abstract": "In this short note we extend Chow and Lu's advanced maximum principles for parabolic systems on closed manifolds to the case of compact manifolds with boundary, which also generalizes a Hopf type theorem of Pulemotov."}
{"category": "Math", "title": "Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19", "abstract": "It is known that K3 surfaces S whose Picard number rho (= rank of the Neron-Severi group of S) is at least 19 are parametrized by modular curves X, and these modular curves X include various Shimura modular curves associated with congruence subgroups of quaternion algebras over Q. In a family of such K3 surfaces, a surface has rho=20 if and only if it corresponds to a CM point on X. We use this to compute equations for Shimura curves, natural maps between them, and CM coordinates well beyond what could be done by working with the curves directly as we did in ``Shimura Curve Computations'' (1998) = <http://arxiv.org/abs/math/0005160>"}
{"category": "Math", "title": "On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels I", "abstract": "We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In Part I, we will consider the general theory of modular functions for the normalizers."}
{"category": "Math", "title": "A generalization of Doob's maximal identity", "abstract": "In this paper, using martingale techniques, we prove a generalization of Doob's maximal identity in the setting of continuous nonnegative local submartingales $(X_{t})$ of the form: $X_{t}=N_{t}+A_{t}$, where the measure $(dA_{t})$ is carried by the set $\\left\\{t: X_{t}=0\\right\\}$. In particular, we give a multiplicative decomposition for the Az\\'ema supermartingale associated with some last passage times related to such processes and we prove that these non-stopping times contain very useful information. As a consequence, we obtain the law of the maximum of a continuous nonnegative local martingale $(M_t)$ which satisfies $M_\\infty=\\psi(\\sup_{t\\geq0}M_t)$ for some measurable function $\\psi$ as well as the law of the last time this maximum is reached."}
{"category": "Math", "title": "Asymptotic efficiency of simple decisions for the compound decision problem", "abstract": "We consider the compound decision problem of estimating a vector of $n$ parameters, known up to a permutation, corresponding to $n$ independent observations, and discuss the difference between two symmetric classes of estimators. The first and larger class is restricted to the set of all permutation invariant estimators. The second class is restricted further to simple symmetric procedures. That is, estimators such that each parameter is estimated by a function of the corresponding observation alone. We show that under mild conditions, the minimal total squared error risks over these two classes are asymptotically equivalent up to essentially O(1) difference."}
{"category": "Math", "title": "The Complex of Non-Crossing Diagonals of a Polygon", "abstract": "Given a convex n-gon P in the Euclidean plane, it is well known that the simplicial complex \\theta(P) with vertex set given by diagonals in P and facets given by triangulations of P is the boundary complex of a polytope of dimension n-3. We prove that for any non-convex polygonal region P with n vertices and h+1 boundary components, \\theta(P) is a ball of dimension n+3h-4. We also provide a new proof that \\theta(P) is a sphere when P is convex."}
{"category": "Math", "title": "The quadratic isoperimetric inequality for mapping tori of free group automorphisms", "abstract": "If F is a finitely generated free group and \\phi is an automorphism of F then F \\rtimes_\\phi Z satisfties a quadratic isoperimetric inequality."}
{"category": "Math", "title": "An equivalent form of Young's inequality with upper bound", "abstract": "Young's integral inequality is complemented with an upper bound to the remainder. The new inequality turns out to be equivalent to Young's inequality, and the cases in which the equality holds become particularly transparent in the new formulation."}
{"category": "Math", "title": "Large deviations for the Boussinesq Equations under Random Influences", "abstract": "A Boussinesq model for the Benard convection under random influences is considered as a system of stochastic partial differential equations. This is a coupled system of stochastic Navier-Stokes equations and the transport equation for temperature. Large deviations are proved, using a weak convergence approach based on a variational representation of functionals of infinite dimensional Brownian motion."}
{"category": "Math", "title": "Noncommutative Riemannian Geometry and Diffusion on Ultrametric Cantor Sets", "abstract": "An analogue of the Riemannian Geometry for an ultrametric Cantor set (C, d) is described using the tools of Noncommutative Geometry. Associated with (C, d) is a weighted rooted tree, its Michon tree. This tree allows to define a family of spectral triples giving the Cantor set the structure of a noncommutative Riemannian manifold. The family of spectral triples is indexed by the space of choice functions which is shown to be the analogue of the sphere bundle of a Riemannian manifold. The Connes metric coming from the Dirac operator D then allows to recover the metric on C. The corresponding zeta function is shown to have abscissa of convergence equal to the upper box dimension of (C, d). Taking the residue at this singularity leads to the definition of a canonical probability measure on C which in certain cases coincides with the Hausdorff measure. This measure in turns induces a measure on the space of choices. Given a choice, the commutator of D with a Lipschitz continuous function can be intepreted as a directional derivative. By integrating over all choices, this leads to the definition of an analogue of the Laplace-Beltrami operator. This operator has compact resolvent and generates a Markov semigroup which plays the role of a Brownian motion on C. This construction is applied to the simplest case, the triadic Cantor set."}
{"category": "Math", "title": "A remark on Frobenius characters for set representations of symmetric groups", "abstract": "For any set representation (permutation representation) of the symmetric group $S_n$, we give combinatorial interpretation for coefficients of its Frobenius character expanded in the basis of monomial symmetric functions."}
{"category": "Math", "title": "Topology of generalized complex quotients", "abstract": "Consider the Hamiltonian action of a torus on a compact twisted generalized complex manifold $M$. We first observe that Kirwan injectivity and surjectivity hold for ordinary equivariant cohomology in this setting. Then we prove that these two results hold for the twisted equivariant cohomology as well."}
{"category": "Math", "title": "Wave decay on convex co-compact hyperbolic manifolds", "abstract": "For convex co-compact hyperbolic quotients $X=\\Gamma\\backslash\\hh^{n+1}$, we analyze the long-time asymptotic of the solution of the wave equation $u(t)$ with smooth compactly supported initial data $f=(f_0,f_1)$. We show that, if the Hausdorff dimension $\\delta$ of the limit set is less than $n/2$, then $u(t) = C_\\delta(f) e^{(\\delta-\\ndemi)t} / \\Gamma(\\delta-n/2+1) + e^{(\\delta-\\ndemi)t} R(t)$ where $C_{\\delta}(f)\\in C^\\infty(X)$ and $||R(t)||=\\mc{O}(t^{-\\infty})$. We explain, in terms of conformal theory of the conformal infinity of $X$, the special cases $\\delta\\in n/2-\\nn$ where the leading asymptotic term vanishes. In a second part, we show for all $\\eps>0$ the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip $\\{-n\\delta-\\eps<\\Re(\\la)<\\delta\\}$. As a byproduct we obtain a lower bound on the remainder $R(t)$ for generic initial data $f$."}
{"category": "Math", "title": "Fixed points in the family of convex representations of a maximal monotone operator", "abstract": "Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exist a convex representation of the operator which is a fixed point of this conjugation."}
{"category": "Math", "title": "Stable commutator length is rational in free groups", "abstract": "For any group, there is a natural (pseudo-)norm on the vector space B1 of real (group) 1-boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov (pseudo)-norm on (ordinary) homology. We show that for a free group, the unit ball of this pseudo-norm is a rational polyhedron. It follows that stable commutator length in free groups takes on only rational values. Moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group. The proof of these facts yields an algorithm to compute stable commutator length in free groups. Using this algorithm, we answer a well-known question of Bavard in the negative, constructing explicit examples of elements in free groups whose stable commutator length is not a half-integer."}
{"category": "Math", "title": "A Bayesian reassessment of nearest-neighbour classification", "abstract": "The k-nearest-neighbour procedure is a well-known deterministic method used in supervised classification. This paper proposes a reassessment of this approach as a statistical technique derived from a proper probabilistic model; in particular, we modify the assessment made in a previous analysis of this method undertaken by Holmes and Adams (2002,2003), and evaluated by Manocha and Girolami (2007), where the underlying probabilistic model is not completely well-defined. Once a clear probabilistic basis for the k-nearest-neighbour procedure is established, we derive computational tools for conducting Bayesian inference on the parameters of the corresponding model. In particular, we assess the difficulties inherent to pseudo-likelihood and to path sampling approximations of an intractable normalising constant, and propose a perfect sampling strategy to implement a correct MCMC sampler associated with our model. If perfect sampling is not available, we suggest using a Gibbs sampling approximation. Illustrations of the performance of the corresponding Bayesian classifier are provided for several benchmark datasets, demonstrating in particular the limitations of the pseudo-likelihood approximation in this set-up."}
{"category": "Math", "title": "Asymptotics of the Spectral Gap for the Interchange Process on Large Hypercubes", "abstract": "We consider the interchange process (IP) on the $d$-dimensional, discrete hypercube of side-length $n$. Specifically, we compare the spectral gap of the IP to the spectral gap of the random walk (RW) on the same graph. We prove that the two spectral gaps are asymptotically equivalent, in the limit $n \\to \\infty$. This result gives further supporting evidence for a conjecture of Aldous, that the spectral gap of the IP equals the spectral gap of the RW on all finite graphs. Our proof is based on an argument invented by Handjani and Jungreis, who proved Aldous's conjecture for all trees. This also has implications for the spectral gap of the quantum Heisenberg ferromagnet."}
{"category": "Math", "title": "Asymptotics of Weil-Petersson geodesics I: ending laminations, recurrence, and flows", "abstract": "We define an ending lamination for a Weil-Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil-Petersson metric, these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem 1.1) for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex. As an application, we establish fundamentals of the topological dynamics of the Weil-Petersson geodesic flow, showing density of closed orbits and topological transitivity."}
{"category": "Math", "title": "Autoconjugate representers for linear monotone operators", "abstract": "Monotone operators are of central importance in modern optimization and nonlinear analysis. Their study has been revolutionized lately, due to the systematic use of the Fitzpatrick function. Pioneered by Penot and Svaiter, a topic of recent interest has been the representation of maximal monotone operators by so-called autoconjugate functions. Two explicit constructions were proposed, the first by Penot and Zalinescu in 2005, and another by Bauschke and Wang in 2007. The former requires a mild constraint qualification while the latter is based on the proximal average. We show that these two autoconjugate representers must coincide for continuous linear monotone operators on reflexive spaces. The continuity and the linearity assumption are both essential as examples of discontinuous linear operators and of subdifferential operators illustrate. Furthermore, we also construct an infinite family of autoconjugate representers for the identity operator on the real line."}
{"category": "Math", "title": "Estimation non-param\\'etrique de la densit\\'e spectrale d'un processus gaussien \\'echantillonn\\'e al\\'eatoirement", "abstract": "From a wavelet analysis, one derives a nonparametrical estimator for the spectral density of a Gaussian process with stationary increments. First, the idealistic case of a continuous time path of the process is considered. A punctual Central Limit Theorem (CLT) and an estimation of the Mean Integrate Square Error (MISE) are established. Next, to fit the applications, one considers the case where one observes a path at random times. One built a second estimator obtained by replacing the wavelet coefficients by their discretizations. A second CLT and the corresponding estimation of the MISE are provided. Finally, simulation results and an application on the heartbeat time series of marathon runners are presented."}
{"category": "Math", "title": "Groups of finite Morley rank with solvable local subgroups", "abstract": "We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these groups. We prove the main Uniqueness Theorem, analogous to the Bender method in finite group theory, and derive its corollaries. We also consider homogeneous cases as well as torsion."}
{"category": "Math", "title": "An arithmetic Riemann-Roch theorem in higher degrees", "abstract": "We prove an analogue in Arakelov geometry of the Grothendieck-Riemann-Roch theorem."}
{"category": "Math", "title": "About Stable Periodic Helixes, L-iteration and Chaos Generated by Unbounded Functions", "abstract": "We consider stable periodic helixes as a generalization of stable periodic orbits. We see that in the studied class of iterated functions Chaos always arise suddenly. Therefore, we shall study the route from chaos to order rather than the route from order to chaos. We show that, paradoxically, genuine Chaos may look as much like Order and during as many iteration steps as one may wish. Then, we shall propose a generalization of the idea of map iteration that do not imply the existence of periodic orbits. We shall show that, within a strictly deterministic context, unpredictability, aperiodic order, sensitive dependence and chaos are completely different concepts and we shall try to show what this difference is made of. We shall also propose an example of non chaotic aperiodic order."}
{"category": "Math", "title": "Two simple sufficient conditions for FDR control", "abstract": "We show that the control of the false discovery rate (FDR) for a multiple testing procedure is implied by two coupled simple sufficient conditions. The first one, which we call ``self-consistency condition'', concerns the algorithm itself, and the second, called ``dependency control condition'' is related to the dependency assumptions on the $p$-value family. Many standard multiple testing procedures are self-consistent (e.g. step-up, step-down or step-up-down procedures), and we prove that the dependency control condition can be fulfilled when choosing correspondingly appropriate rejection functions, in three classical types of dependency: independence, positive dependency (PRDS) and unspecified dependency. As a consequence, we recover earlier results through simple and unifying proofs while extending their scope to several regards: weighted FDR, $p$-value reweighting, new family of step-up procedures under unspecified $p$-value dependency and adaptive step-up procedures. We give additional examples of other possible applications. This framework also allows for defining and studying FDR control for multiple testing procedures over a continuous, uncountable space of hypotheses."}
{"category": "Math", "title": "Generalized induction of Kazhdan-Lusztig cells", "abstract": "Following Lusztig, we consider a Coxeter group $W$ together with a weight function. Geck showed that the Kazhdan-Lusztig cells of $W$ are compatible with parabolic subgroups. In this paper, we generalize this argument to some subsets of $W$ which may not be parabolic subgroups. We obtain two applications: we show that under specific technical conditions on the parameters, the cells of a certain finite parabolic subgroup of $W$ are cells in the whole group, and we decompose the affine Weyl group $\\tilde{G}_{2}$ into left and two-sided cells for a whole class of weight functions."}
{"category": "Math", "title": "An holomorphic study of Smarandache automorphic and cross inverse ploperty loops", "abstract": "By studying the holomorphic structure of automorphic inverse property quasigroups and loops[AIPQ and (AIPL)] and cross inverse property quasigroups and loops[CIPQ and (CIPL)], it is established that the holomorph of a loop is a Smarandache; AIPL, CIPL, K-loop, Bruck-loop or Kikkawa-loop if and only if its Smarandache automorphism group is trivial and the loop is itself is a Smarandache; AIPL, CIPL, K-loop, Bruck-loop or Kikkawa-loop."}
{"category": "Math", "title": "A Double Cryptography Using The Smarandache Keedwell Cross Inverse Quasigroup", "abstract": "The present study further strengthens the use of the Keedwell CIPQ against attack on a system by the use of the Smarandache Keedwell CIPQ for cryptography in a similar spirit in which the cross inverse property has been used by Keedwell. This is done as follows. By constructing two S-isotopic S-quasigroups(loops) $U$ and $V$ such that their Smarandache automorphism groups are not trivial, it is shown that $U$ is a SCIPQ(SCIPL) if and only if $V$ is a SCIPQ(SCIPL). Explanations and procedures are given on how these SCIPQs can be used to double encrypt information."}
{"category": "Math", "title": "Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets", "abstract": "H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal real mutually unbiased bases. These schemes also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes."}
{"category": "Math", "title": "Cartan Subalgebras of Leibniz $n$-Algebras", "abstract": "The present paper is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz $n$-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz $n$-algebra and Cartan subalgebras and regular elements of the corresponding factor $n$-Lie algebra is established."}
{"category": "Math", "title": "A Double Cryptography Using The Keedwell Cross Inverse Quasigroup", "abstract": "The present study further strenghtens the use of the Keedwell CIPQ against attack on a system. This is done as follows. The holomorphic structure of AIPQs(AIPLs) and CIPQs(CIPLs) are investigated. Necessary and sufficient conditions for the holomorph of a quasigroup(loop) to be an AIPQ(AIPL) or CIPQ(CIPL) are established. It is shown that if the holomorph of a quasigroup(loop) is a AIPQ(AIPL) or CIPQ(CIPL), then the holomorph is isomorphic to the quasigroup(loop). Hence, the holomorph of a quasigroup(loop) is an AIPQ(AIPL) or CIPQ(CIPL) if and only if its automorphism group is trivial and the quasigroup(loop) is a AIPQ(AIPL) or CIPQ(CIPL). Furthermore, it is discovered that if the holomorph of a quasigroup(loop) is a CIPQ(CIPL), then the quasigroup(loop) is a flexible unipotent CIPQ(flexible CIPL of exponent 2). By constructing two isotopic quasigroups(loops) $U$ and $V$ such that their automorphism groups are not trivial, it is shown that $U$ is a AIPQ or CIPQ(AIPL or CIPL) if and only if $V$ is a AIPQ or CIPQ(AIPL or CIPL). Explanations and procedures are given on how these CIPQs can be used to double encrypt information."}
{"category": "Math", "title": "On A Cryptographic Identity In Osborn Loops", "abstract": "This study digs out some new algebraic properties of an Osborn loop that will help in the future to unveil the mystery behind the middle inner mappings $T_{(x)}$ of an Osborn loop. These new algebraic properties, will open our eyes more to the study of Osborn loops like CC-loops which has received a tremendious attention in this $21^\\textrm{st}$ and VD-loops whose study is yet to be explored. In this study, some algebraic properties of non-WIP Osborn loops have been investigated in a broad manner. Huthnance was able to deduce some algebraic properties of Osborn loops with the WIP i.e universal weak WIPLs. So this work exempts the WIP. Two new loop identities, namely left self inverse property loop(LSIPL) identity and right self inverse property loop(RSLPL) are introduced for the first time and it is shown that in an Osborn loop, they are equivalent. A CC-loop is shown to be power associative if and only if it is a RSLPL or LSIPL. Among the few identities that have been established for Osborn loops, one of them is recognized and recommended for cryptography in a similar spirit in which the cross inverse property has been used by Keedwell following the fact that it was observed that Osborn loops that do not have the LSIP or RSIP or 3-PAPL or weaker forms of inverse property, power associativity and diassociativity to mention a few, will have cycles(even long ones). These identity is called an Osborn cryptographic identity(or just a cryptographic identity)."}
{"category": "Math", "title": "Notes on the biextension of Chow groups", "abstract": "The paper discusses four approaches to the biextension of Chow groups and their equivalences. These are the following: an explicit construction given by S.Bloch, a construction in terms of the Poincare biextension of dual intermediate Jacobians, a construction in terms of K-cohomology, and a construction in terms of determinant of cohomology of coherent sheaves. A new approach to J.Franke's Chow categories is given. An explicit formula for the Weil pairing of algebraic cycles is obtained."}
{"category": "Math", "title": "Some open 3-manifolds and 3-orbifolds without locally finite canonical decompositions", "abstract": "We give examples of open 3-manifolds and 3-orbifolds that exhibit pathological behavior with respect to splitting along surfaces (2-suborbifolds) with nonnegative Euler characteristic."}
{"category": "Math", "title": "Determination of some generalised Euler sums involving the digamma function", "abstract": "This paper evaluates some generalised Euler sums involving the digamma function."}
{"category": "Math", "title": "A JSJ splitting for triangulated open 3-manifolds", "abstract": "We give a sufficient condition for an open 3-manifold to admit a decomposition along properly embedded open annuli and tori, generalizing the toric splitting of Jaco-Shalen and Johannson."}
{"category": "Math", "title": "Generators of Jacobians of Genus Two Curves", "abstract": "We prove that in most cases relevant to cryptography, the Frobenius endomorphism on the Jacobian of a genus two curve is represented by a diagonal matrix with respect to an appropriate basis of the subgroup of l-torsion points. From this fact we get an explicit description of the Weil-pairing on the subgroup of l-torsion points. Finally, the explicit description of the Weil-pairing provides us with an efficient, probabilistic algorithm to find generators of the subgroup of l-torsion points on the Jacobian of a genus two curve."}
{"category": "Math", "title": "Prime numbers of a kind x^2+1", "abstract": "The number of primes of a kind x^2+1 is infinite."}
{"category": "Math", "title": "Macdonald polynomials at $t=q^k$", "abstract": "We investigate the homogeneous symmetric Macdonald polynomials $P_\\lambda(\\X;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_\\lambda(\\X;q,q^k)$ and $P_\\lambda(\\frac{1-q}{1-q^k}\\X;q,q^k)$. As a consequence, we describe an operator whose eigenvalues characterize the polynomials $P_\\lambda(\\X;q,q^k)$."}
{"category": "Math", "title": "Comparison Principles for subelliptic equations of Monge-Ampere type", "abstract": "We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvature in a Carnot group."}
{"category": "Math", "title": "On the Ruelle eigenvalue sequence", "abstract": "For certain real analytic data, we show that the eigenvalue sequence of the associated transfer operator L is insensitive to the holomorphic function space on which L acts. Explicit bounds on this eigenvalue sequence are established."}
{"category": "Math", "title": "Morphisms of Cartan connections", "abstract": "We define what we call morphisms of Cartan connections. We generalize the main theorems on Cartan connections to theorems on morphisms. Many of the known constructions involving Cartan connections turn out to be examples of morphisms. We prove some basic results concerning completeness of Cartan connections. We provide a new method to prove completeness of Cartan connections using families of morphisms."}
{"category": "Math", "title": "Spectral Properties of Matrices Associated with Some Directed Graphs", "abstract": "We study the spectral properties of certain non-self-adjoint matrices associated with large directed graphs. Asymptotically the eigenvalues converge to certain curves, apart from a finite number that have limits not on these curves."}
{"category": "Math", "title": "Laws of large numbers for epidemic models with countably many types", "abstract": "In modeling parasitic diseases, it is natural to distinguish hosts according to the number of parasites that they carry, leading to a countably infinite type space. Proving the analogue of the deterministic equations, used in models with finitely many types as a \"law of large numbers\" approximation to the underlying stochastic model, has previously either been done case by case, using some special structure, or else not attempted. In this paper we prove a general theorem of this sort, and complement it with a rate of convergence in the $\\ell_1$-norm."}
{"category": "Math", "title": "Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras", "abstract": "Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero. Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem. Denote by ${{KK}}_e(C,A)^{++}$ the set of those $\\kappa$ for which $\\kappa(K_0(C)_+\\setminus\\{0\\})\\subset K_0(A)_+\\setminus\\{0\\}$ and $\\kappa([1_C])=[1_A]$. Suppose that $\\kappa\\in {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $\\phi: C\\to A$ such that $[\\phi]=\\kappa.$ Suppose that $C$ is a unital AH-algebra and $\\lambda: \\mathrm{T}(A)\\to \\mathrm{T}_{\\mathtt{f}}(C)$ is a continuous affine map for which $\\tau(\\kappa([p]))=\\lambda(\\tau)(p)$ for all projections $p$ in all matrix algebras of $C$ and any $\\tau\\in \\mathrm{T}(A),$ where $\\mathrm{T}(A)$ is the simplex of tracial states of $A$ and $\\mathrm{T}_{\\mathtt{f}}(C)$ is the convex set of faithful tracial states of $C.$ We prove that there is a unital monomorphism $\\phi: C\\to A$ such that $\\phi$ induces both $\\kappa$ and $\\lambda.$ Suppose that $h: C\\to A$ is a unital monomorphism and $\\gamma \\in \\mathrm{Hom}(\\Kone(C), \\aff(A)).$ We show that there exists a unital monomorphism $\\phi: C\\to A$ such that $[\\phi]=[h]$ in ${KK}(C,A),$ $\\tau\\circ \\phi=\\tau\\circ h$ for all tracial states $\\tau$ and the associated rotation map can be given by $\\gamma.$ Applications to classification of simple C*-algebras are also given."}
{"category": "Math", "title": "Sunyer-i-Balaguer's Almost Elliptic Functions and Yosida's Normal Functions", "abstract": "We study the properties of two classes of meromorphic functions in the complex plane. The first one is the class of almost elliptic functions in the sense of Sunyer-i-Balaguer. This is the class of meromorphic functions f such that the family of shifts f(z+h) (h are complex numbers) is normal with respect to the uniform convergence in the whole complex plane. Given two sequences of complex numbers, we provide sufficient conditions for them to be zeros and poles of some almost elliptic function. These conditions enable one to give (for the first time) explicit non-trivial examples of almost elliptic functions. The second class was introduced by K.Yosida, who called it a class of normal functions of the first category. This is the class of meromorphic functions f such that the family of shifts f(z+h)is normal with respect to the uniform convergence on compacta in the complex plane and no limit point of the family is a constant function. We give necessary and sufficient conditions for two sequences of complex numbers to be zeros and poles of some normal function of the first category and obtain a parametric representation for this class in terms of zeros and poles."}
{"category": "Math", "title": "A characterisation of the Z^n + Z(\\delta) lattice and definite nonunimodular intersection forms", "abstract": "We prove a generalisation of Elkies' theorem to nonunimodular definite forms (and lattices). Combined with inequalities of Froyshov and of Ozsvath and Szabo, this gives a simple test of whether a rational homology 3-sphere may bound a definite four-manifold. As an example we show that small positive surgeries on torus knots do not bound negative-definite four-manifolds."}
{"category": "Math", "title": "Lie-like Algebras (Superalgeras)", "abstract": "We introduce the notion of a Lie-like algebra$^{\\diamond}$ (superalgebra$^{\\diamond}$) for $\\diamond\\in\\{^{1-st}, ^{2-nd}, ^{3-rd} \\}$."}
{"category": "Math", "title": "Helicoid-Like Minimal Disks and Uniqueness", "abstract": "We show that an embedded minimal disk in R^3 with large curvature is bilipschitz with a piece of a helicoid. Additionally, a simplified proof of the uniqueness of the helicoid is provided."}
{"category": "Math", "title": "The Bousfield lattice for truncated polynomial algebras", "abstract": "The global structure of the unbounded derived category of a truncated polynomial ring on countably many generators is investigated, via its Bousfield lattice. The Bousfield lattice is shown to have cardinality larger than that of the real numbers, and objects with large tensor-nilpotence height are constructed."}
{"category": "Math", "title": "On the $\\ell_1-\\ell_q$ Regularized Regression", "abstract": "In this paper we consider the problem of grouped variable selection in high-dimensional regression using $\\ell_1-\\ell_q$ regularization ($1\\leq q \\leq \\infty$), which can be viewed as a natural generalization of the $\\ell_1-\\ell_2$ regularization (the group Lasso). The key condition is that the dimensionality $p_n$ can increase much faster than the sample size $n$, i.e. $p_n \\gg n$ (in our case $p_n$ is the number of groups), but the number of relevant groups is small. The main conclusion is that many good properties from $\\ell_1-$regularization (Lasso) naturally carry on to the $\\ell_1-\\ell_q$ cases ($1 \\leq q \\leq \\infty$), even if the number of variables within each group also increases with the sample size. With fixed design, we show that the whole family of estimators are both estimation consistent and variable selection consistent under different conditions. We also show the persistency result with random design under a much weaker condition. These results provide a unified treatment for the whole family of estimators ranging from $q=1$ (Lasso) to $q=\\infty$ (iCAP), with $q=2$ (group Lasso)as a special case. When there is no group structure available, all the analysis reduces to the current results of the Lasso estimator ($q=1$)."}
{"category": "Math", "title": "Stochastic Algorithm For Parameter Estimation For Dense Deformable Template Mixture Model", "abstract": "Estimating probabilistic deformable template models is a new approach in the fields of computer vision and probabilistic atlases in computational anatomy. A first coherent statistical framework modelling the variability as a hidden random variable has been given by Allassonni\\`ere, Amit and Trouv\\'e in [1] in simple and mixture of deformable template models. A consistent stochastic algorithm has been introduced in [2] to face the problem encountered in [1] for the convergence of the estimation algorithm for the one component model in the presence of noise. We propose here to go on in this direction of using some \"SAEM-like\" algorithm to approximate the MAP estimator in the general Bayesian setting of mixture of deformable template model. We also prove the convergence of this algorithm toward a critical point of the penalised likelihood of the observations and illustrate this with handwritten digit images."}
{"category": "Math", "title": "Blueprint for a Classic Proof of the Four Colour Theorem", "abstract": "The proof uses the property that the vertices of a triangulated planar graph can be four coloured if the triangles can have a +1 or -1 orientation so that the sum of the triangle orientations around each vertex is a multiple of 3. Such orientation is first used separately on one of the two triangulated polygons resulting from a Hamilton circuit in a triangulated planar graph with v vertices. The graph is then reconstructed by adding the triangles of the other polygon one by one. When the graph is totally reconstructed there is always a combination for the orientations of the triangles for which their sum around each of v-2 successive vertices in the Hamilton circuit is a multiple of 3. It is then provable that the sum of the triangle orientations around the two remaining vertices must also be a multiple of 3."}
{"category": "Math", "title": "Mollifiers in Clifford Analysis", "abstract": "We introduce mollifiers in Clifford analysis setting and construct a sequence of $\\C^{\\infinity}$-functions that approximate a $\\gamma$-regular function and a solution to a non homogeneous BVP of an in homogeneous Dirac like operator in certain Sobolev spaces over bounded domains whose boundary is not that wild. One can extend the smooth functions upto the boundary if the domain has a $\\C^1$-boundary and this is the case in the paper as we consider a domain whose boundary is a $\\C^2$-hyper surface."}
{"category": "Math", "title": "Representation theory of Jordanian algebra", "abstract": "We describe the complete set of pairwise non-isomorphic irreducible modules S(a) over the algebra R given by the defining relation xy-yx=yy, and the rule how they could be glued to indecomposables. Namely, we show that Ext_k^1(S(a),S(b))=0, if a not equal to b. Also the set of all representations is described subject to the Jordan normal form of Y. We study then properties of the image algebras in the endomorphism ring. Among facts we prove is that they are all basic algebras. Along this line we establish an analogue of the Gerstenhaber-Taussky-Motzkin theorem on the dimension of algebras generated by two commuting matrices. All image algebras of indecomposable modules turned out to be local complete algebras. We compare them with the Ringel's classification by means of finding relations of image algebras. As a result we derive that all image algebras of n-dimensional representations with full block Y are tame for n smaller or equal then 4 and wild for n starting from 5. We suggest a stratification of representation space of R related to the partitions of n defined by the Jordan normal form of Y. We give a complete classification by parameters for some strata and present examples of tame (up to automorphisms) strata, while the generic strata is wild."}
{"category": "Math", "title": "A nonholonomic Moser theorem and optimal transport", "abstract": "We prove the following nonholonomic version of the classical Moser theorem: given a bracket-generating distribution on a connected compact manifold (possibly with boundary), two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution. We describe formal solutions of the corresponding nonholonomic mass transport problem and present the Hamiltonian framework for both the Otto calculus and its nonholonomic counterpart as infinite-dimensional Hamiltonian reductions on diffeomorphism groups. Finally, we define a nonholonomic analog of the Wasserstein (or, Kantorovich) metric on the space of densities and prove that the subriemannian heat equation defines a gradient flow on the nonholonomic Wasserstein space with the potential given by the Boltzmann relative entropy functional."}
{"category": "Math", "title": "Fast Computation of Partial Fourier Transforms", "abstract": "We introduce two efficient algorithms for computing the partial Fourier transforms in one and two dimensions. Our study is motivated by the wave extrapolation procedure in reflection seismology. In both algorithms, the main idea is to decompose the summation domain of into simpler components in a multiscale way. Existing fast algorithms are then applied to each component to obtain optimal complexity. The algorithm in 1D is exact and takes $O(N\\log^2 N)$ steps. Our solution in 2D is an approximate but accurate algorithm that takes $O(N^2 \\log^2 N)$ steps. In both cases, the complexities are almost linear in terms of the degree of freedom. We provide numerical results on several test examples."}
{"category": "Math", "title": "Generators for Vector Spaces Spanned by Double Zeta Values with Even Weight", "abstract": "Let $\\mathcal{DZ}_k$ be the $\\mathbb{Q}$-vector space spanned by double zeta values with weight $k$, and $\\mathcal{DM}_k$ be its quotient space divided by the space $\\mathcal{PZ}_k$ spanned by the zeta value $\\zeta(k)$ and products of two zeta values with total weight $k$. When $k$ is even, an upper bound for the dimension of $\\mathcal{DM}_k$ is known. By adding the dimensions of $\\mathcal{DM}_k$ and $\\mathcal{PZ}_k$, an upper bound of $\\mathcal{DZ}_k$ which equals $k/2$ minus the dimension of the space of modular forms of weight $k$ on the modular group is given. In this note, we obtain some specific sets of generators for $\\mathcal{DM}_k$ which represent the upper bound. These yield the corresponding sets and the upper bound for $\\mathcal{DZ}_k$."}
{"category": "Math", "title": "Modular varieties of D-elliptic sheaves and the Weil-Deligne bound", "abstract": "We compare the asymptotic grows of the number of rational points on modular varieties of D-elliptic sheaves over finite fields to the grows of their Betti numbers as the degree of the level tends to infinity. This is a generalization to higher dimensions of a well-known result for modular curves. As a consequence of the main result, we also produce a new asymptotically optimal sequence of curves."}
{"category": "Math", "title": "On the eigenvalues of p-adic curvature", "abstract": "We determine the maximal eigenvalue of the p-adic curvature transformations on Bruhat-Tits buildings, and we give an essentially optimal upper bound on the minimal non-zero eigenvalue of these transformations."}
{"category": "Math", "title": "Sobolev Institute of Mathematics Celebrates its Fiftieth Anniversary", "abstract": "This paper describes briefly history and current state of the Sobolev Institute of Mathematics, the biggest research mathematical institute of the Russian Academy of Sciences located east to Ural mountains."}
{"category": "Math", "title": "Stability of the Hartree-Fock model with temperature", "abstract": "This paper is devoted to the Hartree-Fock model with temperature in the euclidean space. For large classes of free energy functionals, minimizers are obtained as long as the total charge of the system does not exceed a threshold which depends on the temperature. The usual Hartree-Fock model is recovered in the zero temperature limit. An orbital stability result for the Cauchy problem is deduced from the variational approach."}
{"category": "Math", "title": "Poisson Hopf algebras associated to quantized enveloping algebras", "abstract": "We study certain Poisson structures related to quantized enveloping algebras. In particular, we give a description of the Poisson structure of a certain manifold associated to the ring of differential operators."}
{"category": "Math", "title": "Strongly nondegenerate Lie algebras", "abstract": "Let $A$ be a semiprime 2 and 3-torsion free non-commutative associative algebra. We show that the Lie algebra $\\der(A)$ of (associative) derivations of $A$ is strongly non-degenerate, which is a strong form of semiprimeness for Lie algebras, under some additional restrictions on the center of $A$. This result follows from a description of the quadratic annihilator of a general Lie algebra inside appropriate Lie overalgebras. Similar results are obtained for an associative algebra $A$ with involution and the Lie algebra $\\sder(A)$ of involution preserving derivations of $A$."}
{"category": "Math", "title": "A transference method in quantum probability", "abstract": "Working with a rather general notion of independence, we provide a transference method which allows to compare the p-norm of sums of independent copies with the p-norm of sums of free copies. Our main technique is to construct explicit operator space Lp embeddings preserving independence to reduce the problem to L1, where some recent results by the first-named author can be used. We find applications for noncommutative Khincthine/Rosenthal type inequalities and for noncommutative Lp embedding theory."}
{"category": "Math", "title": "Partial Regularity for Stationary Solutions to Liouville-Type Equation in dimension 3", "abstract": "In dimension $n=3$, we prove that the singular set of any stationary solution to the Liouville equation $-\\Delta u=e^u$, which belongs to $W^{1,2}$, has Hausdorff dimension at most 1."}
{"category": "Math", "title": "Properties of the density for a three dimensional stochastic wave equation", "abstract": "We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let $p_{t,x}(y)$ be the density of the law of the solution $u(t,x)$ of such an equation at points $(t,x)\\in]0,T]\\times \\IR^3$. We prove that the mapping $(t,x)\\mapsto p_{t,x}(y)$ owns the same regularity as the sample paths of the process $\\{u(t,x), (t,x)\\in]0,T]\\times \\mathbbR^3\\}$ established Dalang and Sanz-Sol\\'e [Memoirs of the AMS, to appear]. The proof relies on Malliavin calculus and more explicitely, Watanabe's integration by parts formula and estimates derived form it."}
{"category": "Math", "title": "Hardy's Uncertainty Principle, Convexity and Schr\\\"odinger Evolutions", "abstract": "We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schr\\\"odinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy's version of the uncertainty principle. We also obtain corresponding results for heat evolutions."}
{"category": "Math", "title": "A valuation criterion for normal basis generators in local fields of characteristic $p$", "abstract": "Let $K$ be a complete local field of characteristic $p$ with perfect residue field. Let $L/K$ be a finite, fully ramified, Galois $p$-extension. If $\\pi_L\\in L$ is a prime element, and $p'(x)$ is the derivative of $\\pi_L$'s minimal polynomial over $K$, then the relative different $\\euD_{L/K}$ is generated by $p'(\\pi_L)\\in L$. Let $v_L$ be the normalized valuation normalized with $v_L(L)=\\mathbb{Z}$. We show that any element $\\rho\\in L$ with $v_L(\\rho)\\equiv -v_L(p'(\\pi_L))-1\\bmod[L:K]$ generates a normal basis, $K[{Gal}(L/K)]\\cdot\\rho=L$. This criterion is tight: Given any integer $i$ such that $i\\not\\equiv -v_L(p'(\\pi_L))-1\\bmod[L:K]$, there is a $\\rho_i\\in L$ with $v_L(\\rho_i)=i$ such that $K[{Gal}(L/K)]\\cdot\\rho_i\\subsetneq L$."}
{"category": "Math", "title": "Asymptotic equivalence and contiguity of some random graphs", "abstract": "We show that asymptotic equivalence, in a strong form, holds between two random graph models with slightly differing edge probabilities under substantially weaker conditions than what might naively be expected. One application is a simple proof of a recent result by van den Esker, van der Hofstad and Hooghiemstra on the equivalence between graph distances for some random graph models."}
{"category": "Math", "title": "Explicit eigenvalue estimates for transfer operators", "abstract": "We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if D is any open set in C^d, and L is a suitable transfer operator acting on Bergman space A^2(D), its eigenvalue sequence lambda_n(L) is bounded by |lambda_n(L)| \\leq A\\exp(-a n^{1/d}), where a, A are explicitly given."}
{"category": "Math", "title": "Enumerative Geometry of Calabi-Yau 5-Folds", "abstract": "Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 5-folds. We find recursions for meeting numbers of genus 0 curves, and we determine the contributions of moving multiple covers of genus 0 curves to the genus 1 Gromov-Witten invariants. The resulting invariants, conjectured to be integral, are analogous to the previously defined BPS counts for Calabi-Yau 3 and 4-folds. We comment on the situation in higher dimensions where new issues arise. Two main examples are considered: the local Calabi-Yau P^2 with balanced normal bundle 3O(-1) and the compact Calabi-Yau hypersurface X_7 in P^6. In the former case, a closed form for our integer invariants has been conjectured by G. Martin. In the latter case, we recover in low degrees the classical enumeration of elliptic curves by Ellingsrud and Stromme."}
{"category": "Math", "title": "On Limiting Distributions Of Estimation Of Central Moments", "abstract": "This paper has been withdrawn at the author's request."}
{"category": "Math", "title": "On Jacobi Sums in $\\mathbb Q(\\zeta_p)$", "abstract": "We study the p-adic behavior of Jacobi Sums for $\\mathbb Q(\\zeta_p)$ and link this study to the p-Sylow subgroup of the ideal class group of $\\mathbb Q(\\zeta_p\\`a^+$"}
{"category": "Math", "title": "Mirabolic Robinson-Schensted-Knuth correspondence", "abstract": "The set of orbits of $GL(V)$ in $Fl(V)\\times Fl(V)\\times V$ is finite, and is parametrized by the set of certain decorated permutations in a work of Solomon. We describe a Mirabolic RSK correspondence (bijective) between this set of decorated permutations and the set of triples: a pair of standard Young tableaux, and an extra partition. It gives rise to a partition of the set of orbits into combinatorial cells. We prove that the same partition is given by the type of a general conormal vector to an orbit. We conjecture that the same partition is given by the bimodule Kazhdan-Lusztig cells in the bimodule over the Iwahori-Hecke algebra of $GL(V)$ arising from $Fl(V)\\times Fl(V)\\times V$. We also give conjectural applications to the classification of unipotent mirabolic character sheaves on $GL(V)\\times V$."}
{"category": "Math", "title": "Mirabolic affine Grassmannian and character sheaves", "abstract": "We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite field. The answer is given in terms of the character values of general linear groups over the finite field, and the structure constants of multiplication in the mirabolic Hall-Littlewood basis of symmetric functions, introduced by Shoji."}
{"category": "Math", "title": "Maximal monotonicity, conjugation and the duality product", "abstract": "Recently, the authors studied the connection between each maximal monotone operator T and a family H(T) of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities. The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a unique maximal monotone operator."}
{"category": "Math", "title": "M-decomposability, elliptical unimodal densities, and applications to clustering and kernel density estimation", "abstract": "Chia and Nakano (2009) introduced the concept of M-decomposability of probability densities in one-dimension. In this paper, we generalize M-decomposability to any dimension. We prove that all elliptical unimodal densities are M-undecomposable. We also derive an inequality to show that it is better to represent an M-decomposable density via a mixture of unimodal densities. Finally, we demonstrate the application of M-decomposability to clustering and kernel density estimation, using real and simulated data. Our results show that M-decomposability can be used as a non-parametric criterion to locate modes in probability densities."}
{"category": "Math", "title": "Equivariant cohomology of incidence Hilbert schemes and loop algebras", "abstract": "Let $S$ be the affine plane $\\C^2$ together with an appropriate $\\mathbb T = \\C^*$ action. Let $\\hil{m,m+1}$ be the incidence Hilbert scheme. Parallel to \\cite{LQ}, we construct an infinite dimensional Lie algebra that acts on the direct sum $$\\Wft = \\bigoplus_{m=0}^{+\\infty}H^{2(m+1)}_{\\mathbb T}(S^{[m,m+1]})$$ of the middle-degree equivariant cohomology group of $\\hil{m,m+1}$. The algebra is related to the loop algebra of an infinite dimensional Heisenberg algebra. In addition, we study the transformations among three different linear bases of $\\Wft$. Our results are applied to the ring structure of the ordinary cohomology of $\\hil{m,m+1}$ and to the ring of symmetric functions in infinitely many variables."}
{"category": "Math", "title": "Symmetric tensors and symmetric tensor rank", "abstract": "A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a sum of symmetric outer product of vectors. A rank-1 order-k tensor is the outer product of k non-zero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1."}
{"category": "Math", "title": "Differentiable Conjugacy of Anosov Diffeomorphisms on Three Dimensional Torus", "abstract": "We consider two C^2 Anosov diffeomorphisms in a C^1 neighborhood of a linear hyperbolic automorphism of three dimensional torus with real spectrum. We prove that they are C^1+ conjugate if and only if the differentials of the return maps at corresponding periodic points have the same eigenvalues."}
{"category": "Math", "title": "Helical CR Structures and Sub-Riemannian Geodesics", "abstract": "A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step two Carnot groups equipped with a distinguished normal geodesic, and also between such structures and smooth real curves whose derivatives have constant Euclidean norm. As a consequence, we relate step two Carnot groups equipped with sub-Riemannian geodesics with this family of curves. The restriction to the unit circle of certain planar homogeneous polynomial mappings gives an instructive class of examples. We describe these examples in detail."}
{"category": "Math", "title": "Artin HNN-extensions virtually embed in Artin groups", "abstract": "An Artin HNN-extension is an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups."}
{"category": "Math", "title": "On the complexity of proper holomorphic mappings between balls", "abstract": "We make several new contributions to the study of proper holomorphic mappings between balls. Our results include a degree estimate for rational proper maps, a new gap phenomenon for convex families of arbitrary proper maps, and an interesting result about inverse images."}
{"category": "Math", "title": "Riemann Hypothesis may be proved by induction", "abstract": "The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At least one of these identities may be applied to prove the Riemann Hypothesis by induction. Additionally using this approach, the new series for Euler's constant gamma has been found."}
{"category": "Math", "title": "Structured Semidefinite Representation of Some Convex Sets", "abstract": "Linear matrix Inequalities (LMIs) have had a major impact on control but formulating a problem as an LMI is an art. Recently there is the beginnings of a theory of which problems are in fact expressible as LMIs. For optimization purposes it can also be useful to have \"lifts\" which are expressible as LMIs. We show here that this is a much less restrictive condition and give methods for actually constructing lifts and their LMI representation."}
{"category": "Math", "title": "The conjugacy problem in right-angled Artin groups and their subgroups", "abstract": "We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperbolic groups, and it coincides with the class of fundamental groups of ``special cube complexes'' studied independently by Haglund and Wise."}
{"category": "Math", "title": "Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations", "abstract": "Solutions of the qKZ equation associated with the quantum affine algebra $U_q(\\hat{sl}_2)$ and its two dimensional evaluation representation are studied. The integral formulae derived from the free field realization of intertwining operators of $q$-Wakimoto modules are shown to coincide with those of Tarasov and Varchenko."}
{"category": "Math", "title": "Geometric structure of class two nilpotent groups and subgroup growth", "abstract": "In this paper we derive an explicit expression for the normal zeta function of class two nilpotent groups whose associated Pfaffian hypersurface is smooth. In particular, we show how the local zeta function depends on counting mod p rational points on related varieties, and we describe the varieties that can appear in such a decomposition. As a corollary, we also establish explicit results on the degree of polynomial subgroup growth in these groups, and we study the behaviour of poles of this zeta function. Under certain geometric conditions, we also confirm that these functions satisfy a functional equation."}
{"category": "Math", "title": "Groebner deformations, connectedness and cohomological dimension", "abstract": "This paper is an outcome of the author's master thesis written under the supervision of Aldo Conca. We prove some results relating connectedness properties with (local) cohomological dimension. As an interesting corollary we have that every initial complex of a Cohen-Macaulay ideal is strongly connected."}
{"category": "Math", "title": "The semiflow of a reaction diffusion equation with a singular potential", "abstract": "We study the semiflow $\\mathcal{S}(t)$ defined by a semilinear parabolic equation with a singular square potential $V(x)=\\frac{\\mu}{|x|^2}$. It is known that the Hardy-Poincar\\'{e} inequality and its improved versions, have a prominent role on the definition of the natural phase space. Our study concerns the case $0<\\mu\\leq\\mu^*$, where $\\mu^*$ is the optimal constant for the Hardy-Poincar\\'{e} inequality. On a bounded domain of $\\mathbb{R}^N$, we justify the global bifurcation of nontrivial equilibrium solutions for a reaction term $f(s)=\\lambda s-|s|^{2\\gamma}s$, with $\\lambda$ as a bifurcation parameter. The global bifurcation result is used to show that any solution $\\phi(t)=\\mathcal{S}(t)\\phi_0$, initiating form initial data $\\phi_0\\geq 0$ ($\\phi_0\\leq 0$), $\\phi_0\\not\\equiv 0$, tends to the unique nonnegative (nonpositive) equilibrium."}
{"category": "Math", "title": "Lectures on the Routh-Hurwitz problem", "abstract": "The notes contain a streamlined account on stability of univariate polynomials and related problems"}
{"category": "Math", "title": "An Algebra Containing the Two-Sided Convolution Operators", "abstract": "We present an intrinsically defined algebra of operators containing the right and left invariant Calder\\'on-Zygmund operators on a stratified group. The operators in our algebra are pseudolocal and bounded on L^p (1<p<\\infty). This algebra provides an example of an algebra of singular integrals that falls outside of the classical Calder\\'on-Zygmund theory."}
{"category": "Math", "title": "Conditions for stability and instability of retrial queueing systems with general retrial times", "abstract": "We study the stability of single server retrial queues under general distribution for retrial times and stationary ergodic service times, for three main retrial policies studied in the literature: classical linear, constant and control policies. The approach used is the renovating events approach to obtain sufficient stability conditions by strong coupling convergence of the process modeling the dynamics of the system to a unique stationary ergodic regime. We also obtain instability conditions by convergence in distribution to improper limiting sequences."}
{"category": "Math", "title": "Natural Lie Algebra bundles on rank two s-K\\\"ahler manifolds, abelian varieties and moduli of curves", "abstract": "We prove that one can obtain natural bundles of Lie algebras on rank two s-K\\\"ahler manifolds, whose fibres are isomorphic to so(s+1,s+1), su(s+1,s+1) and sl(2s + 2,\\R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have natural flat connections, whose flat global sections act naturally on cohomology. We also present several natural examples of manifolds which can be equipped with an s-K\\\"ahler structure with various levels of rigidity: complex tori and abelian varieties, cotangent bundles of smooth manifolds and moduli of pointed elliptic curves."}
{"category": "Math", "title": "The Longstaff--Schwartz algorithm for L\\'{e}vy models: Results on fast and slow convergence", "abstract": "We investigate the Longstaff--Schwartz algorithm for American option pricing assuming that both the number of regressors and the number of Monte Carlo paths tend to infinity. Our main results concern extensions, respectively, applications of results by Glasserman and Yu [Ann. Appl. Probab. 14 (2004) 2090--2119] and Stentoft [Manag. Sci. 50 (2004) 1193--1203] to several L\\'{e}vy models, in particular the geometric Meixner model. A convenient setting to analyze this convergence problem is provided by the L\\'{e}vy--Sheffer systems introduced by Schoutens and Teugels."}
{"category": "Math", "title": "An Inhomogeneous Transference Principle and Diophantine Approximation", "abstract": "In a landmark paper, D.Y. Kleinbock and G.A. Margulis established the fundamental Baker-Sprindzuk conjecture on homogeneous Diophantine approximation on manifolds. Subsequently, there has been dramatic progress in this area of research. However, the techniques developed to date do not seem to be applicable to inhomogeneous approximation. Consequently, the theory of inhomogeneous Diophantine approximation on manifolds remains essentially non-existent. In this paper we develop an approach that enables us to transfer homogeneous statements to inhomogeneous ones. This is rather surprising as the inhomogeneous theory contains the homogeneous theory and so is more general. As a consequence, we establish the inhomogeneous analogue of the Baker-Sprindzuk conjecture. Furthermore, we prove a complete inhomogeneous version of the profound theorem of Kleinbock, Lindenstrauss & Weiss on the extremality of friendly measures. The results obtained in this paper constitute the first step towards developing a coherent inhomogeneous theory for manifolds in line with the homogeneous theory."}
{"category": "Math", "title": "On the Schrodinger equation in $R^N$ under the effect of a general nonlinear term", "abstract": "In this paper we prove the existence of a positive solution to the equation $-\\Delta u + V(x)u=g(u)$ in $R^N,$ assuming the general hypotheses on the nonlinearity introduced by Berestycki & Lions. Moreover we show that a minimizing problem, related to the existence of a ground state, has no solution."}
{"category": "Math", "title": "Algebra in superextensions of groups, I: zeros and commutativity", "abstract": "Given a group $X$ we study the algebraic structure of its superextension $\\lambda(X)$. This is a right-topological semigroup consisting of all maximal linked systems on $X$ endowed with the operation $$\\mathcal A\\circ\\mathcal B=\\{C\\subset X:\\{x\\in X:x^{-1}C\\in\\mathcal B\\}\\in\\mathcal A\\}$$ that extends the group operation of $X$. We characterize right zeros of $\\lambda(X)$ as invariant maximal linked systems on $X$ and prove that $\\lambda(X)$ has a right zero if and only if each element of $X$ has odd order. On the other hand, the semigroup $\\lambda(X)$ contains a left zero if and only if it contains a zero if and only if $X$ has odd order $|X|\\le5$. The semigroup $\\lambda(X)$ is commutative if and only if $|X|\\le4$. We finish the paper with a complete description of the algebraic structure of the semigroups $\\lambda(X)$ for all groups $X$ of cardinality $|X|\\le5$."}
{"category": "Math", "title": "Algebra in superextension of groups, II: cancelativity and centers", "abstract": "Given a countable group $X$ we study the algebraic structure of its superextension $\\lambda(X)$. This is a right-topological semigroup consisting of all maximal linked systems on $X$ endowed with the operation $$\\mathcal A\\circ\\mathcal B=\\{C\\subset X:\\{x\\in X:x^{-1}C\\in\\mathcal B\\}\\in\\mathcal A\\}$$ that extends the group operation of $X$. We show that the subsemigroup $\\lambda^\\circ(X)$ of free maximal linked systems contains an open dense subset of right cancelable elements. Also we prove that the topological center of $\\lambda(X)$ coincides with the subsemigroup $\\lambda^\\bullet(X)$ of all maximal linked systems with finite support. This result is applied to show that the algebraic center of $\\lambda(X)$ coincides with the algebraic center of $X$ provided $X$ is countably infinite. On the other hand, for finite groups $X$ of order $3\\le|X|\\le5$ the algebraic center of $\\lambda(X)$ is strictly larger than the algebraic center of $X$."}
{"category": "Math", "title": "Right-topological semigroup operations on inclusion hyperspaces", "abstract": "We show that for any discrete semigroup $X$ the semigroup operation can be extended to a right-topological semigroup operation on the space $G(X)$ of inclusion hyperspaces on $X$. We detect some important subsemigroups of $G(X)$, study the minimal ideal, the (topological) center, left cancelable elements of $G(X)$, and describe the structure of the semigroups $G(\\IZ_n)$ for small numbers $n$."}
{"category": "Math", "title": "A sharp estimate and change on the dimension of the attractor for Allen-Cahn equations", "abstract": "We consider the semilinear reaction diffusion equation $\\partial_t\\phi-\\nu\\Delta\\phi-V(x)\\phi+f(\\phi)=0$, $\\nu>0$ in a bounded domain $\\Omega\\subset\\mathbb{R}^N$. We assume the standard Allen-Cahn-type nonlinearity, while the potential $V$ is either the inverse square potential $V(x)=\\delta |x|^{-2}$ or the borderline potential $V(x)=\\delta \\mathrm{dist}(x,\\partial\\Omega)^{-2}$, $\\delta\\geq 0$ (thus including the classical Allen-Cahn equation as a special case when $\\delta=0$). In the subcritical cases $\\delta=0$, $N\\geq 1$ and $0<\\mu:=\\frac{\\delta}{\\nu}<\\mu^*$, $N\\geq 3$ (where $\\mu^*$ is the optimal constant of Hardy and Hardy-type inequalities), we present a new estimate on the dimension of the global attractor. This estimate comes out by an improved lower bound for sums of eigenvalues of the Laplacian by A. D. Melas (Proc. Amer. Math. Soc. \\textbf{131} (2003), 631-636). The estimate is sharp, revealing the existence of (an explicitly given) threshold value for the ratio of the volume to the moment of inertia of $\\Omega$ on which the dimension of the attractor may considerably change. Consideration is also given on the finite dimensionality of the global attractor in the critical case $\\mu=\\mu^*$."}
{"category": "Math", "title": "The realization problem for von Neumann regular rings", "abstract": "We survey recent progress on the realization problem for von Neumann regular rings, which asks whether every countable conical refinement monoid can be realized as the monoid of isoclasses of finitely generated projective right $R$-modules over a von Neumann regular ring $R$."}
{"category": "Math", "title": "Abelian Varieties with Prescribed Embedding Degree", "abstract": "We present an algorithm that, on input of a CM-field $K$, an integer $k\\ge1$, and a prime $r \\equiv 1 \\bmod k$, constructs a $q$-Weil number $\\pi \\in \\O_K$ corresponding to an ordinary, simple abelian variety $A$ over the field $\\F$ of $q$ elements that has an $\\F$-rational point of order $r$ and embedding degree $k$ with respect to $r$. We then discuss how CM-methods over $K$ can be used to explicitly construct $A$."}
{"category": "Math", "title": "Induced Measures on \"Mu**\"- measurable Sets", "abstract": "We investigate extension of a measure to a very general set of undetermined structure. Structure may be imposed on this set in special cases"}
{"category": "Math", "title": "Bronsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces", "abstract": "In this work we are concerned with maximality of monotone operators representable by certain convex functions in non-reflexive Banach spaces. We also prove that these maximal monotone operators satisfy a Bronsted-Rockafellar type property. We show that if a function in XxX^* and its conjugate are above the duality product in their respective domains, then this function represents a maximal monotone operator."}
{"category": "Math", "title": "Markovian embeddings of general random strings", "abstract": "Let A be a finite set and X a sequence of A-valued random variables. We do not assume any particular correlation structure between these random variables; in particular, X may be a non-Markovian sequence. An adapted embedding of X is a sequence of the form R(X_1), R(X_1,X_2), R(X_1,X_2,X_3), etc where R is a transformation defined over finite length sequences. In this extended abstract we characterize a wide class of adapted embeddings of X that result in a first-order homogeneous Markov chain. We show that any transformation R has a unique coarsest refinement R' in this class such that R'(X_1), R'(X_1,X_2), R'(X_1,X_2,X_3), etc is Markovian. (By refinement we mean that R'(u)=R'(v) implies R(u)=R(v), and by coarsest refinement we mean that R' is a deterministic function of any other refinement of R in our class of transformations.) We propose a specific embedding that we denote as R^X which is particularly amenable for analyzing the occurrence of patterns described by regular expressions in X. A toy example of a non-Markovian sequence of 0's and 1's is analyzed thoroughly: discrete asymptotic distributions are established for the number of occurrences of a certain regular pattern in X_1,...,X_n, as n tends to infinity, whereas a Gaussian asymptotic distribution is shown to apply for another regular pattern."}
{"category": "Math", "title": "On a theorem of V. Bernik in the metrical theory of Diophantine approximation", "abstract": "This paper goes back to a famous problem of Mahler in metrical Diophantine approximation. The problem has been settled by Sprindzuk and subsequently improved by Alan Baker and Vasili Bernik. In particular, Bernik's result establishes a convergence Khintchine type theorem for Diophantine approximation by polynomials, that is it allows arbitrary monotonic error of approximation. In the present paper the monotonicity assumption is completely removed."}
{"category": "Math", "title": "Vectored Route-length Minimization - A Heuristic and An Open Conjecture", "abstract": "We propose a simple but interesting graph theoretic problem and posited a heuristic solution procedure, which we have christened as Vectored Route-length Minimization Search (VeRMinS). Basically, it constitutes of a re-casting of the classical 'shortest route' problem within a strictly Euclidean space. We have only presented a heuristic solution process with the hope that a formal proof will eventually emerge as the problem receives wider exposure within mathematical circles."}
{"category": "Math", "title": "Universal subspaces for compact Lie groups", "abstract": "For a representation of a connected compact Lie group G in a finite dimensional real vector space U and a subspace V of U, invariant under a maximal torus of G, we obtain a sufficient condition for V to meet all G-orbits in U, which is also necessary in certain cases. The proof makes use of the cohomology of flag manifolds and the invariant theory of Weyl groups. Then we apply our condition to the conjugation representations of U(n), Sp(n), and SO(n) in the space of $n\\times n$ matrices over C, H, and R, respectively. In particular, we obtain an interesting generalization of Schur's triangularization theorem."}
{"category": "Math", "title": "On approximation of p-adic numbers by p-adic algebraic numbers", "abstract": "A complete p-adic Khintchine type theorem for approximation by p-adic algebraic numbers is established."}
{"category": "Math", "title": "Bass' $NK$ groups and $cdh$-fibrant Hochschild homology", "abstract": "The $K$-theory of a polynomial ring $R[t]$ contains the $K$-theory of $R$ as a summand. For $R$ commutative and containing $\\Q$, we describe $K_*(R[t])/K_*(R)$ in terms of Hochschild homology and the cohomology of K\\\"ahler differentials for the $cdh$ topology. We use this to address Bass' question, on whether $K_n(R)=K_n(R[t])$ implies $K_n(R)=K_n(R[t_1,t_2])$. The answer is positive over fields of infinite transcendence degree; the companion paper arXiv:1004.3829 provides a counterexample over a number field."}
{"category": "Math", "title": "A Note on Chromatic Sum", "abstract": "The chromatic sum $\\Sigma(G)$ of a graph $G$ is the smallest sum of colors among of proper coloring with the natural number. In this paper, we introduce a necessary condition for the existence of graph homomorphisms. Also, we present $\\Sigma(G)<\\chi_f(G)|G|$ for every graph $G$."}
{"category": "Math", "title": "Functional interpretation and inductive definitions", "abstract": "Extending G\\\"odel's \\emph{Dialectica} interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees."}
{"category": "Math", "title": "On a conjectured inequality in convex analysis in the case of the unit ball of lp^n, 1<= p<= infinity", "abstract": "We re-confirm, for the case of the unit p-ball of R^n, one of recent conjectures of G.Kuperberg on centrally symmetric convex bodies.This conjecture was very recently confirmrd for this particular case by D.A.Gutierrez using polygamma functions and convexity theory.We present another proof using only the basic properties of gamma function and mildly advanced classical analysis tools."}
{"category": "Math", "title": "2-block Springer fibers: convolution algebras and coherent sheaves", "abstract": "For a fixed 2-block Springer fiber, we describe the structure of its irreducible components and their relation to the Bialynicki-Birula paving, following work of Fung. That is, we consider the space of complete flags in C^n preserved by a fixed nilpotent matrix with 2 Jordan blocks, and study the action of diagonal matrices commuting with our fixed nilpotent. In particular, we describe the structure of each component, its set of torus fixed points, and prove a conjecture of Fung describing the intersection of any pair. Then we define a convolution algebra structure on the direct sum of the cohomologies of pairwise intersections of irreducible components and closures of C^*-attracting sets (that is, Bialynicki-Birula cells), and show this is isomorphic to a generalization of the arc algebra of Khovanov defined by the first author. We investigate the connection of this algebra to Cautis & Kamnitzer's recent work on link homology via coherent sheaves and suggest directions for future research."}
{"category": "Math", "title": "Infinitesimal deformation of p-adic differential equations on Berkovich curves", "abstract": "We show that if a differential equations $\\mathscr{F}$ over a quasi-smooth Berkovich curve $X$ has a certain compatibility condition with respect to an automorphism $\\sigma$ of $X$, and if the automorphism is sufficiently close to the identity, then $\\mathscr{F}$ acquires a semi-linear action of $\\sigma$ (i.e. lifting that on $X$). This generalizes the previous works of Yves Andr\\'e, Lucia Di Vizio, and the author about $p$-adic $q$-difference equations. We also obtain an application to Morita's $p$-adic Gamma function, and to related values of $p$-adic $L$-functions."}
{"category": "Math", "title": "Note on the construction of free monoids", "abstract": "We construct free monoids in a monoidal category with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains."}
{"category": "Math", "title": "Transfer of ideals and quantization of small nilpotent orbits", "abstract": "We introduce and study a transfer map between ideals of the universal enveloping algebras of two members of a reductive dual pair of Lie algebras. Its definition is motivated by the approach to the real Howe duality through the theory of Capelli identities. We prove that this map provides a lower bound on the annihilators of theta lifts of representations with a fixed annihilator ideal. We also show that in the algebraic stable range, transfer respects the class of quantizations of nilpotent orbit closures. As an application, we explicitly describe quantizations of small nilpotent orbits of general linear and orthogonal Lie algebras and give presentations of certain rings of algebraic differential operators. We consider two algebraic versions of Howe duality and reformulate our results in terms of noncommutative Capelli identities."}
{"category": "Math", "title": "Diffeomorphisms of the circle and Brownian motions on an infinite-dimensional symplectic group", "abstract": "An embedding of the group $\\Diff(S^{1})$ of orientation preserving diffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplectic group, $\\Sp(\\infty)$, is studied. The authors prove that this embedding is not surjective. A Brownian motion is constructed on $\\Sp(\\infty)$. This study is motivated by recent work of H. Airault, S. Fang and P. Malliavin."}
{"category": "Math", "title": "Classification of non-symplectic automorphisms of order 3 on $K3$ surfaces", "abstract": "In this paper, we study non-symplectic automorphisms of order 3 on algebraic $K3$ surface over $\\mathbb{C}$ which act trivially on the N\\'{e}ron-Severi lattice. In particular we shall characterize their fixed locus in terms of the invariants of 3-elementary lattices."}
{"category": "Math", "title": "A Hochschild-cyclic approach to additive higher Chow cycles", "abstract": "Over a field of characteristic zero, we introduce two motivic operations on additive higher Chow cycles: analogues of the Connes boundary $B$ operator and the shuffle product on Hochschild complexes. The former allows us to apply the formalism of mixed complexes to additive Chow complexes building a bridge between additive higher Chow theory and additive $K$-theory. The latter induces a wedge product on additive Chow groups for which we show that the Connes operator is a graded derivation for the wedge product using a variation of a Totaro's cycle. Hence, the additive higher Chow groups with the wedge product and the Connes operator form a commutative differential graded algebra. On zero-cycles, they induce the wedge product and the exterior derivation on the absolute K\\\"ahler differentials, answering a question of S. Bloch and H. Esnault."}
{"category": "Math", "title": "Logarithmic nonabelian Hodge theory in characteristic p", "abstract": "Given a morphism $X \\to S$ of log schemes of characteristic $p > 0$ and a lifting of $X'$ over $S$ modulo $p^2$, we use Lorenzon's indexed algebras $A_X^{gp}$ and $B_{X/S}$ to construct an equivalence between $O_X$-modules with nilpotent integrable connection and indexed $B_{X/S}$-modules with nilpotent $B_{X/S}$-linear Higgs field. If either satisfies a stricter nilpotence condition, we find an isomorphism between the de Rham cohomology of the connection and the Higgs cohomology of the Higgs field."}
{"category": "Math", "title": "The number of small covers over cubes", "abstract": "In the present paper we find a bijection between the set of small covers over an $n$-cube and the set of acyclic digraphs with $n$ labeled nodes. Using this, we give a formula of the number of small covers over an $n$-cube (generally, a product of simplices) up to Davis-Januszkiewicz equivalence classes and $\\mathbf{Z}^n$-equivariant diffeomorphism classes. Moreover we prove that the number of acyclic digraphs with $n$ unlabeled nodes is an upper bound of the number of small covers over an $n$-cube up to diffeomorphism."}
{"category": "Math", "title": "Quantitative uniqueness for second order elliptic operators with strongly singular coefficients", "abstract": "In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen phases. A key strategy in the proof is to derive doubling inequalities via three-sphere inequalities. Our method can also be applied to certain elliptic systems with similar singular coefficients."}
{"category": "Math", "title": "Dispersive estimates for the Schrodinger equation in dimensions four and five", "abstract": "We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrodinger group $e^{it(-\\Delta+V)}$ for a class of real-valued potentials $V\\in C^k(R^n)$ with $k>(n-3)/2$, where $n=4,5$."}
{"category": "Math", "title": "Unbounded Viscosity Solutions of Hybrid Control Systems", "abstract": "We study a hybrid control system in which both discrete and continuous controls are involved. The discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set $A$ or a controlled jump set $C$ where controller can choose to jump or not. At each jump, trajectory can move to a different Euclidean space. We allow the cost functionals to be unbounded with certain growth and hence the corresponding value function can be unbounded. We characterize the value function as the unique viscosity solution of the associated quasivariational inequality in a suitable function class. We also consider the evolutionary, finite horizon hybrid control problem with similar model and prove that the value function is the unique viscosity solution in the continuous function class while allowing cost functionals as well as the dynamics to be unbounded."}
{"category": "Math", "title": "Logarithmic Combinatorial Differentials", "abstract": "Given a morphism $X \\to S$ of fine log schemes, we develop a geometric description of the sheaves of higher-order differentials $\\Omega^n_{X/S}$ for $n > 1$, as well as a definition of the de Rham complex in terms of this description."}
{"category": "Math", "title": "On the stability of a singular vortex dynamics", "abstract": "In this paper we address the question of the singular vortex dynamics exhibited in [15], which generates a corner in finite time. The purpose is to prove that under some appropriate small regular perturbation the corner still remains. Our approach uses the Hasimoto transform and deals with the long range scattering properties of a Gross-Pitaevski equation with time-variable coefficients."}
{"category": "Math", "title": "A criterion of convergence in the augmented Teichmueller space", "abstract": "We prove a criterion of convergence in the augmented Teichmueller space that can be phrased in terms of convergence of the hyperbolic metrics or of quasiconformal convergence away from the nodes."}
{"category": "Math", "title": "Bimodality, prion aggregates infectivity and prediction of strain phenomenon", "abstract": "We consider a model for the polymerization (fragmentation) process involved in infectious prion self-replication and study both its dynamics and non-zero steady state. We address several issues. Firstly, we give conditions leading to size repartitions of PrPsc aggregates that exhibit bimodal distributions, as indicated by recent experimental studies of prion aggregates distribution. Secondly, we show stability results for this steady state for general coefficients where reduction to a system of differential equations is not possible. We use a duality method based on recent ideas developed for population models. These results underline the potential influence of the amyloid precursor production rate in promoting amyloidogenic diseases. Finally, we numerically investigate the influence of different parameters of the model on PrPsc accumulation kinetics, in the aim to study specific features of prion strains."}
{"category": "Math", "title": "Free involutions on S^1xS^n", "abstract": "Topological free involutions on S^1xS^n are classified up to conjugation. As a byproduct we obtain a new computation of the group of concordance classes of homeomorphisms of the projective space RP^n."}
{"category": "Math", "title": "Miniscule representations, Gauss sum and modular invariance", "abstract": "After explaining the concepts of Langlands dual and miniscule representations, we define an analog of the Gauss sum for any compact, simple Lie group with a simply laced Lie algebra. We then show a reciprocity property when a Lie group is exchanged with its Langlands dual. We also explore the relation with theta functions and modular transformations. In the non-simply laced case, we construct a unitary representation of the Hecke group which involves interesting new phase factors."}
{"category": "Math", "title": "Generalized Andr\\'{e}-Quillen Cohomology", "abstract": "We explain how the approach of Andre and Quillen to defining cohomology and homology as suitable derived functors extends to generalized (co)homology theories, and how this identification may be used to study the relationship between them. As a side benefit, we clarify exactly what assumptions on an (algebraic) category are needed in order for the approach of Beck and Andre-Quillen to work. We also show how the description may be applied to construct universal coefficient and reverse Adams spectral sequences."}
{"category": "Math", "title": "Solution non universelle pour le probl\\`eme KV-78", "abstract": "In 78' M. Kashiwara and Vergne conjectured some property on the Campbell-Hausdorff series in such way a trace formula is satisfied. They proposed an explicit solution in the case of solvable Lie algebras. In this note we prove that this \"solvable solution\" is not universal. Our method is based on computer calculation. Furthermore our programs prove up to degree 16, Drinfeld's Lie algebra $\\mathfrak{grt}_1$ coincides with the Lie algebra $\\hat{kv_2}$ defined in \\cite{AT}."}
{"category": "Math", "title": "FINE: Fisher Information Non-parametric Embedding", "abstract": "We consider the problems of clustering, classification, and visualization of high-dimensional data when no straightforward Euclidean representation exists. Typically, these tasks are performed by first reducing the high-dimensional data to some lower dimensional Euclidean space, as many manifold learning methods have been developed for this task. In many practical problems however, the assumption of a Euclidean manifold cannot be justified. In these cases, a more appropriate assumption would be that the data lies on a statistical manifold, or a manifold of probability density functions (PDFs). In this paper we propose using the properties of information geometry in order to define similarities between data sets using the Fisher information metric. We will show this metric can be approximated using entirely non-parametric methods, as the parameterization of the manifold is generally unknown. Furthermore, by using multi-dimensional scaling methods, we are able to embed the corresponding PDFs into a low-dimensional Euclidean space. This not only allows for classification of the data, but also visualization of the manifold. As a whole, we refer to our framework as Fisher Information Non-parametric Embedding (FINE), and illustrate its uses on a variety of practical problems, including bio-medical applications and document classification."}
{"category": "Math", "title": "Uniform continuity over locally compact quantum groups", "abstract": "We define, for a locally compact quantum group $G$ in the sense of Kustermans--Vaes, the space of $LUC(G)$ of left uniformly continuous elements in $L^\\infty(G)$. This definition covers both the usual left uniformly continuous functions on a locally compact group and Granirer's uniformly continuous functionals on the Fourier algebra. We show that $LUC(G)$ is an operator system containing the $C^*$-algebra $C_0(G)$ and contained in its multiplier algebra $M(C_0(G))$. We use this to partially answer an open problem by Bedos--Tuset: if $G$ is co-amenable, then the existence of a left invariant mean on $M(C_0(G))$ is sufficient for $G$ to be amenable. Furthermore, we study the space $WAP(G)$ of weakly almost periodic elements of $L^\\infty(G)$: it is a closed operator system in $L^\\infty(G)$ containing $C_0(G)$ and--for co-amenable $G$--contained in $LUC(G)$. Finally, we show that--under certain conditions, which are always satisfied if $G$ is a group--the operator system $LUC(G)$ is a $C^*$-algebra."}
{"category": "Math", "title": "On strong ergodic properties of quantum dynamical systems", "abstract": "We show that the the shift on the reduced C*--algebras of RD--groups, including the free group on infinitely many generators, and the amalgamated free product C*--algebras, enjoys the very strong ergodic property of the convergence to the equilibrium. Namely, the free shift converges, pointwise in the weak topology, to the conditional expectation onto the fixed--point subalgebra. Provided the invariant state is unique, we also show that such an ergodic property cannot be fulfilled by any classical dynamical system, unless it is conjugate to the trivial one--point dynamical system."}
{"category": "Math", "title": "Moduli spaces of irreducible symplectic manifolds", "abstract": "We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of 2d polarised (split type) symplectic manifolds which are deformation equivalent to degree 2 Hilbert schemes of a K3 surface is of general type if d is at least 12."}
{"category": "Math", "title": "Zero dimensional arc valuations on smooth varieties", "abstract": "For a normalized transcendence degree zero arc valuation v on a nonsingular variety X (with dim X > 1), we describe the maximal irreducible subset C(v) of the arc space of X such that the valuation given by the order of vanishing along a general arc of C(v) equals v. We describe C(v) both algebraically, in terms of the sequence of valuation ideals of v, and geometrically, in terms of the sequence of infinitely near points associated to v. When X is a surface, our construction also applies to any divisorial valuation v, and in this case C(v) coincides with a subset Ein, Lazarsfeld, and Mustata associate to v."}
{"category": "Math", "title": "Why stratification may hurt, & how much", "abstract": "There are circumstances under which stratified sampling is worse than simple random sampling, even if the allocation of the sample sizes is optimal. This phenomenon was discovered more than sixty years ago, but is not as widely known as one might expect. We provide it with lower and upper bounds for its badness as well as with an explanation."}
{"category": "Math", "title": "Toric surface codes and Minkowski length of polygons", "abstract": "In this paper we prove new lower bounds for the minimum distance of a toric surface code defined by a convex lattice polygon P. The bounds involve a geometric invariant L(P), called the full Minkowski length of P which can be easily computed for any given P."}
{"category": "Math", "title": "The Bernstein-Gelfand-Gelfand complex and Kasparov theory for SL(3,C)", "abstract": "Let $G=\\mathrm{SL}(3,\\mathbb{C})$. We construct an element of $G$-equivariant $K$-homology from the Bernstein-Gelfand-Gelfand complex for $G$. This furnishes an explicit splitting of the restriction map from the Kasparov representation ring $R(G)$ to the representation ring $R(K)$ of its maximal compact subgroup, and the splitting factors through the equivariant $K$-homology of the flag variety of $G$. In particular, we obtain a new model for the gamma element of $G$. The proof makes extensive use of earlier results of the author concerning harmonic analysis of longitudinal psuedodifferential operators on the flag variety."}
{"category": "Math", "title": "Maass relations in higher genus", "abstract": "For an arbitrary even genus $2n$ we show that the subspace of Siegel cusp forms of degree $2n$ generated by Ikeda lifts of elliptic cusp forms can be characterized by certain linear relations among Fourier coefficients. This generatizes the work of Kohnen and Kojima."}
{"category": "Math", "title": "On slicing invariants of knots", "abstract": "The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many knots, previous bounds on unknotting number obtained by Ozsvath and Szabo and by the author in fact give bounds on the slicing number. Livingston defined another invariant $U_s(K)$ which takes into account signs of crossings changed to get a slice knot, and which is bounded above by the slicing number and below by the slice genus. We exhibit an infinite family of knots $K_n$ with slice genus $n$ and Livingston invariant greater than $n$. Our bounds are based on restrictions (using Donaldson's diagonalisation theorem or Heegaard Floer homology) on the intersection forms of four-manifolds bounded by the double branched cover of a knot."}
{"category": "Math", "title": "Holomorphic Motions and Related Topics", "abstract": "In this article we give an expository account of the holomorphic motion theorem based on work of M\\`a\\~n\\'e-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have $|\\epsilon \\log \\epsilon|$ moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarz's lemma and integration over the holomorphic variable to produce H\\\"older continuity on the mappings. We also prove, by using holomorphic motions, that Kobayashi's and Teichm\\\"uller's metrics on the Teichm\\\"uller space of a Riemann surface coincide. Finally, we present an application of holomorphic motions to complex dynamics, that is, we prove the Fatou linearization theorem for parabolic germs by involving holomorphic motions."}
{"category": "Math", "title": "Geometric realizations of the multiplihedron and its complexification", "abstract": "We realize Stasheff's multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves."}
{"category": "Math", "title": "Preservation of stability properties near fixed points of linear hamiltonian systems by symplectic integrators", "abstract": "Based on reasonable testing model problems, we study the preservation by symplectic Runge-Kutta method (SRK) and symplectic partitioned Runge-Kutta method (SPRK) of structures for fixed points of linear Hamiltonian systems. The structure-preservation region provides a practical criterion for choosing step-size in symplectic computation. Examples are given to justify the investigation."}
{"category": "Math", "title": "Perfect IFG-formulas", "abstract": "IFG logic is a variant of the independence-friendly logic of Hintikka and Sandu. We answer the question: ``Which IFG-formulas are equivalent to ordinary first-order formulas?'' We use the answer to show that the ordinary cylindric set algebra over a structure can be embedded into a reduct of the IFG-cylindric set algebra over the structure."}
{"category": "Math", "title": "Logarithmic vector fields along smooth divisors in projective spaces", "abstract": "We show that a smooth divisor in a projective space can be reconstructed from the isomorphism class of the sheaf of logarithmic vector fields along it if and only if its defining equation is of Sebastiani-Thom type."}
{"category": "Math", "title": "Einstein solvmanifolds and the pre-Einstein derivation", "abstract": "An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable extension. For every nilpotent Lie algebra, we construct an (essentially unique) derivation, the pre-Einstein derivation, the solvable extension by which may carry an Einstein inner product. Using the pre-Einstein derivation, we then give a variational characterization of Einstein nilradicals. As an application, we prove an easy-to-check convex geometry condition for a nilpotent Lie algebra with a nice basis to be an Einstein nilradical and also show that a typical two-step nilpotent Lie algebra is an Einstein nilradical."}
{"category": "Math", "title": "Jacobi forms of degree one", "abstract": "We show that a certain subspace of space of elliptic cusp forms is isomorphic as a Hecke module to a certain subspace of space of Jacobi cusp forms of degree one with matrix index by constructing an explicit lifting. This is a partial generalization of the work of Skoruppa and Zagier. This lifting is also related with the Ikeda lifting."}
{"category": "Math", "title": "On local and global regularity of Fourier integral operators", "abstract": "The aim of this paper is to give a review of local and global properties of Fourier integral operators with real and complex phases, in local $L^p$, global $L^2$, and in Colombeau's spaces."}
{"category": "Math", "title": "Stabilization of Heegaard splittings", "abstract": "For each g greater than one there is a 3-manifold with two genus g Heegaard splittings that require g stabilizations to become equivalent. Previously known examples required at most one stabilization. Control of families of Heegaard surfaces is obtained through a deformation to harmonic maps."}
{"category": "Math", "title": "Moduli of representations of quivers", "abstract": "An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is motivated, and the construction of moduli spaces is reviewed. Topological, arithmetic and algebraic methods for the study of moduli spaces are discussed."}
{"category": "Math", "title": "A New Approach of Point Estimation from Truncated or Grouped and Censored Data", "abstract": "We propose a new approach for estimating the parameters of a probability distribution. It consists on combining two new methods of estimation. The first is based on the definition of a new distance measuring the difference between variations of two distributions on a finite number of points from their support and on using this measure for estimation purposes by the method of minimum distance. For the second method, given an empirical discrete distribution, we build up an auxiliary discrete theoretical distribution having the same support of the first and depending on the same parameters of the parent distribution of the data from which the empirical distribution emanated. We estimate then the parameters from the empirical distribution by the usual statistical methods. In practice, we propose to compute the two estimations, the second based on maximum likelihood principle of known theoretical properties, and the first being as a control of the effectiveness of the obtained estimation, and for which we prove the convergence in probability, so we have also a criterion on the quality of the information contained in the observations. We apply the approach to truncated or grouped and censored data situations to give the flavour on the effectiveness of the approach. We give also some interesting perspectives of the approach including model selection from truncated data, estimation of the initial trial value in the celebrate EM algorithm in the case of truncation and merged normal populations, a test of goodness of fit based on the new distance, quality of estimations and data."}
{"category": "Math", "title": "Gray identities, canonical connection and integrability", "abstract": "We characterize quasi K\\\"ahler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related with the third Gray identity and in the almost K\\\"ahler case implies the integrability. Our main tool is the existence of generalized holomorphic frames introduced by the second author previously. By using such frames we also give a simpler and shorter proof of a Theorem of Goldberg. Furthermore we study almost Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. We show some explicit examples of quasi K\\\"ahler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity."}
{"category": "Math", "title": "Stability of PID-Controlled Linear Time-Delay Feedback Systems", "abstract": "The stability of feedback systems consisting of linear time-delay plants and PID controllers has been investigated for many years by means of several methods, of which the Nyquist criterion, a generalization of the Hermite-Biehler Theorem, and the root location method are well known. The main purpose of these researches is to determine the range of controller parameters that allow stability. Explicit and complete expressions of the boundaries of these regions and computation procedures with a finite number of steps are now available only for first-order plants, provided with one time delay. In this note, the same results, based on Pontryagin's studies, are presented for arbitrary-order plants."}
{"category": "Math", "title": "Weak convergence of the supremum distance for supersmooth kernel deconvolution", "abstract": "We derive the asymptotic distribution of the supremum distance of the deconvolution kernel density estimator to its expectation for certain supersmooth deconvolution problems. It turns out that the asymptotics are essentially different from the corresponding results for ordinary smooth deconvolution."}
{"category": "Math", "title": "Telescope conjecture, idempotent ideals, and the transfinite radical", "abstract": "We show that for an artin algebra $\\Lambda$, the telescope conjecture for module categories is equivalent to certain idempotent ideals of mod-$\\Lambda$ being generated by identity morphisms. As a consequence, we prove the conjecture for domestic standard selfinjective algebras and domestic special biserial algebras. We achieve this by showing that in any Krull-Schmidt category with local d.c.c. on ideals, any idempotent ideal is generated by identity maps and maps from the transfinite radical."}
{"category": "Math", "title": "On the Invariant Theory of Weingarten Surfaces in Euclidean Space", "abstract": "We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential equation. We apply these results to the special Weingarten surfaces: minimal surfaces, surfaces of constant mean curvature and surfaces of constant Gauss curvature."}
{"category": "Math", "title": "Arithmetic $\\D$-modules and Representations", "abstract": "We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's theory of arithmetic $D$-modules should give a $p$-adic analogue of Kashiwara's theory of $D$-modules for real Lie groups i.e. it should give a realization of the $p$-adic representations of a $p$-adic Lie group as spaces of overconvergent solutions of arithmetic $D$-modules which will come equipped with an action of the Galois group. We shall discuss the case of Siegel modular varieties as a possible testing ground for the proposal."}
{"category": "Math", "title": "Deviations of Riesz projections of Hill operators with singular potentials", "abstract": "It is shown that the deviations $P_n -P_n^0$ of Riesz projections $$ P_n = \\frac{1}{2\\pi i} \\int_{C_n} (z-L)^{-1} dz, \\quad C_n=\\{|z-n^2|= n\\}, $$ of Hill operators $L y = - y^{\\prime \\prime} + v(x) y, x \\in [0,\\pi],$ with zero and $H^{-1}$ periodic potentials go to zero as $n \\to \\infty $ even if we consider $P_n -P_n^0$ as operators from $L^1$ to $L^\\infty. $ This implies that all $L^p$-norms are uniformly equivalent on the Riesz subspaces $Ran P_n. $"}
{"category": "Math", "title": "A Brief Note on Foliations of Constant Gaussian Curvature", "abstract": "This note provides an alternative proof of a result of Labourie. We show that the two complements of the convex core of a three dimensional quasi-fuchsian hyperbolic manifold may be foliated by embedded hypersurfaces of constant Gaussian curvature."}
{"category": "Math", "title": "Analytical and numerical aspects on motion of polygonal curves with constant area speed", "abstract": "General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while solution polygonal curves keep belonging to the polygonal class."}
{"category": "Math", "title": "Boundary effects on the dynamics of chains of coupled oscillators", "abstract": "We study the dynamics of a chain of coupled particles subjected to a restoring force (Klein-Gordon lattice) in the cases of either periodic or Dirichlet boundary conditions. Precisely, we prove that, when the initial data are of small amplitude and have long wavelength, the main part of the solution is interpolated by a solution of the nonlinear Schr\\\"odinger equation, which in turn has the property that its Fourier coefficients decay exponentially. The first order correction to the solution has Fourier coefficients that decay exponentially in the periodic case, but only as a power in the Dirichlet case. In particular our result allows one to explain the numerical computations of the paper \\cite{BMP07}."}
{"category": "Math", "title": "Comparative Smootheology", "abstract": "We compare various different definitions of \"the category of smooth objects\". The definitions compared are due to Chen, Fr\\\"olicher, Sikorski, Smith, and Souriau. The method of comparison is to construct functors between the categories that enable us to see how the categories relate to each other. This produces a diagram of categories with the category of Fr\\\"olicher spaces sitting at its centre. Our method of study involves finding a general context into which these categories can be placed. This involves considering categories wherein objects are considered in relation to a certain collection of standard test objects. This therefore applies beyond the question of categories of smooth spaces."}
{"category": "Math", "title": "Explicit parametrix and local limit theorems for some degenerate diffusion processes", "abstract": "For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of the density from which we derive some explicit Gaussian controls that characterize the additional singularity induced by the degeneracy. We then give a local limit theorem with the usual convergence rate for an associated Markov chain approximation. The key point is that the \"weak\" degeneracy allows to exploit the techniques first introduced by Konakov and Molchanov and then developed by Konakov and Mammen that rely on Gaussian approximations."}
{"category": "Math", "title": "Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds", "abstract": "If $Y$ is a properly embedded minimal surface in a convex cocompact hyperbolic 3-manifold $M$ with boundary at infinity an embedded curve $\\gamma$, then Graham and Witten showed how to define a renormalized area $\\calA$ of $Y$ via Hadamard regularization. We study renormalized area as a functional on the space of all such minimal surfaces. This requires a closer examination of these moduli spaces; following White and Coskunuzer, we prove these are Banach manifolds and that the natural map taking $Y$ to $\\gamma$ is Fredholm of index zero and proper, which leads to the existence of a $\\ZZ$-valued degree theory for this mapping. We show that $\\calA(Y)$ can be expressed as a sum of the Euler characteristic of $Y$ and the total integral of norm squared of the trace-free second fundamental form of $Y$. An extension of renormalized area to a wider class of nonminimal surfaces has a similar formula also involving the integral of mean curvature squared. We prove a formula for the first variation of renormalized area, and characterize the critical points when $M = \\HH^3$ and $\\gamma$ has a single component. All of these results have analogues for 4-dimensional Poincar\\'e-Einstein metrics. We conclude by discussing the relationship of $\\calA$ to the Willmore functional."}
{"category": "Math", "title": "Cylindrical Wiener processes", "abstract": "In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this definition is a straightforward extension of the real-valued situation which results now in simple conditions on the integrand. In particular, we do not have to put any geometric constraints on the Banach space under consideration. Finally, we relate this integral to well-known stochastic integrals in literature."}
{"category": "Math", "title": "Proof of the Main Conjecture of Noncommutative Iwasawa Theory for Totally Real Number Fields in Certain Cases", "abstract": "Fix an odd prime $p$. Let $G$ be a compact $p$-adic Lie group containing a closed, normal, pro-$p$ subgroup $H$ which is abelian and such that $G/H$ is isomorphic to the additive group of $p$-adic integers $\\mathbbZ_p$ . First we assume that $H$ is finite and compute the Whitehead group of the Iwasawa algebra, $\\Lambda(G)$, of $G$. We also prove some results about certain localisation of $\\Lambda(G)$ needed in Iwasawa theory. Let $F$ be a totally real number field and let $F_{\\infty}$ be an admissible $p$-adic Lie extension of $F$ with Galois group $G$. The computation of the Whitehead groups are used to show that the Main Conjecture for the extension $F_{\\infty}/F$ can be deduced from certain congruences between abelian $p$-adic zeta functions of Delige and Ribet. We prove these congruences with certain assumptions on $G$. This gives a proof of the Main Conjecture in many interesting cases such as $\\mathbb{Z}_p\\rtimes"}
{"category": "Math", "title": "Pattern avoidance in \"flattened\" partitions", "abstract": "To flatten a set partition (with apologies to Mathematica) means to form a permutation by erasing the dividers between its blocks. Of course, the result depends on how the blocks are listed. For the usual listing--increasing entries in each block and blocks arranged in increasing order of their first entries--we count the partitions of [n] whose flattening avoids a single 3-letter pattern. Five counting sequences arise: a null sequence, the powers of 2, the Fibonacci numbers, the Catalan numbers, and the binomial transform of the Catalan numbers."}
{"category": "Math", "title": "Fixed Points of Generalized Conjugations", "abstract": "Conjugation, or Legendre transformation, is a basic tool in convex analysis, rational mechanics, economics and optimization. It maps a function on a linear topological space into another one, defined in the dual of the linear space by coupling these space by meas of the duality product. Generalized conjugation extends classical conjugation to any pair of domains, using an arbitrary coupling function between these spaces. This generalization of conjugation is now being widely used in optima transportation problems, variational analysis and also optimization. If the coupled spaces are equal, generalized conjugations define order reversing maps of a family of functions into itself. In this case, is natural to ask for the existence of fixed points of the conjugation, that is, functions which are equal to their (generalized) conjugateds. Here we prove that any generalized symmetric conjugation has fixed points. The basic tool of the proof is a variational principle involving the order reversing feature of the conjugation. As an application of this abstract result, we will extend to real linear topological spaces a fixed-point theorem for Fitzpatrick's functions, previously proved in Banach spaces."}
{"category": "Math", "title": "A refined Jones polynomial for symmetric unions", "abstract": "Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams we develop a two-variable refinement $W_D(s,t)$ of the Jones polynomial that is invariant under symmetric Reidemeister moves. Here the two variables $s$ and $t$ are associated to the two types of crossings, respectively on and off the symmetry axis. From sample calculations we deduce that a ribbon knot can have essentially distinct symmetric union presentations even if the partial knots are the same. If $D$ is a symmetric union diagram representing a ribbon knot $K$, then the polynomial $W_D(s,t)$ nicely reflects the geometric properties of $K$. In particular it elucidates the connection between the Jones polynomials of $K$ and its partial knots $K_\\pm$: we obtain $W_D(t,t) = V_K(t)$ and $W_D(-1,t) = V_{K_-}(t) \\cdot V_{K_+}(t)$, which has the form of a symmetric product $f(t) \\cdot f(t^{-1})$ reminiscent of the Alexander polynomial of ribbon knots."}
{"category": "Math", "title": "The Jones polynomial of ribbon links", "abstract": "For every n-component ribbon link L we prove that the Jones polynomial V(L) is divisible by the polynomial V(O^n) of the trivial link. This integrality property allows us to define a generalized determinant det V(L) := [V(L)/V(O^n)]_(t=-1), for which we derive congruences reminiscent of the Arf invariant: every ribbon link L = (K_1,...,K_n) satisfies det V(L) = det(K_1) >... det(K_n) modulo 32, whence in particular det V(L) = 1 modulo 8. These results motivate to study the power series expansion V(L) = \\sum_{k=0}^\\infty d_k(L) h^k at t=-1, instead of t=1 as usual. We obtain a family of link invariants d_k(L), starting with the link determinant d_0(L) = det(L) obtained from a Seifert surface S spanning L. The invariants d_k(L) are not of finite type with respect to crossing changes of L, but they turn out to be of finite type with respect to band crossing changes of S. This discovery is the starting point of a theory of surface invariants of finite type, which promises to reconcile quantum invariants with the theory of Seifert surfaces, or more generally ribbon surfaces."}
{"category": "Math", "title": "Cocycle Deformations of Algebraic Identities and R-matrices", "abstract": "For an arbitrary identity L=R between compositions of maps L and R on tensors of vector spaces V, a general construction of a 2-cocycle condition is given. These 2-cocycles correspond to those obtained in deformation theories of algebras. The construction is applied to a canceling pairings and copairings, with explicit examples with calculations. Relations to the Kauffman bracket and knot invariants are discussed."}
{"category": "Math", "title": "Bar constructions and Quillen homology of modules over operads", "abstract": "We show that topological Quillen homology of algebras and modules over operads in symmetric spectra can be calculated by realizations of simplicial bar constructions. Working with several model category structures, we give a homotopical proof after showing that certain homotopy colimits in algebras and modules over operads can be easily understood. A key result here, which lies at the heart of this paper, is showing that the forgetful functor commutes with certain homotopy colimits. We also prove analogous results for algebras and modules over operads in unbounded chain complexes."}
{"category": "Math", "title": "Equivariant classification of 2-torus manifolds", "abstract": "A 2-torus manifold is a closed smooth manifold of dimension $n$ with an effective action of a 2-torus group $(\\Z_2)^n$ of rank $n$, and it is said to be locally standard if it is locally isomorphic to a faithful representation of $(\\Z_2)^n$ on $\\R^n$. This paper studies the equivariant classification of locally standard 2-torus manifolds."}
{"category": "Math", "title": "Injectivity on the set of conjugacy classes of some monomorphisms between Artin groups", "abstract": "There are well-known monomorphisms between the Artin groups of finite type $\\arA_n$, $\\arB_n=\\arC_n$ and affine type $\\tilde \\arA_{n-1}$, $\\tilde\\arC_{n-1}$. The Artin group $A(\\arA_n)$ is isomorphic to the $(n+1)$-strand braid group $B_{n+1}$, and the other three Artin groups are isomorphic to some subgroups of $B_{n+1}$. The inclusions between these subgroups yield monomorphisms $A(\\arB_n)\\to A(\\arA_n)$, $A(\\tilde \\arA_{n-1})\\to A(\\arB_n)$ and $A(\\tilde \\arC_{n-1})\\to A(\\arB_n)$. There are another type of monomorphisms $A(\\arB_d)\\to A(\\arA_{md-1})$, $A(\\arB_d)\\to A(\\arB_{md})$ and $A(\\arB_d)\\to A(\\arA_{md})$ which are induced by isomorphisms between Artin groups of type $\\arB$ and centralizers of periodic braids. In this paper, we show that the monomorphisms $A(\\arB_d)\\to A(\\arA_{md-1})$, $A(\\arB_d)\\to A(\\arB_{md})$ and $A(\\arB_d)\\to A(\\arA_{md})$ induce injective functions on the set of conjugacy classes, and that none of the monomorphisms $A(\\arB_n)\\to A(\\arA_n)$, $A(\\tilde \\arA_{n-1})\\to A(\\arB_n)$ and $A(\\tilde \\arC_{n-1})\\to A(\\arB_n)$ does so."}
{"category": "Math", "title": "Global existence for the kinetic chemotaxis model without pointwise memory effects, and including internal variables", "abstract": "This paper is concerned with the kinetic model of Othmer-Dunbar-Alt for bacterial motion. Following a previous work, we apply the dispersion and Strichartz estimates to prove global existence under several borderline growth assumptions on the turning kernel. In particular we study the kinetic model with internal variables taking into account the complex molecular network inside the cell."}
{"category": "Math", "title": "Bregman distances and Klee sets", "abstract": "In 1960, Klee showed that a subset of a Euclidean space must be a singleton provided that each point in the space has a unique farthest point in the set. This classical result has received much attention; in fact, the Hilbert space version is a famous open problem. In this paper, we consider Klee sets from a new perspective. Rather than measuring distance induced by a norm, we focus on the case when distance is meant in the sense of Bregman, i.e., induced by a convex function. When the convex function has sufficiently nice properties, then - analogously to the Euclidean distance case - every Klee set must be a singleton. We provide two proofs of this result, based on Monotone Operator Theory and on Nonsmooth Analysis. The latter approach leads to results that complement work by Hiriart-Urruty on the Euclidean case."}
{"category": "Math", "title": "Improvement of graph theory Wei`s inequality", "abstract": "In this paper we give a generalization of a result of Wei."}
{"category": "Math", "title": "Solitons and affine projectively flat surfaces", "abstract": "The aim of this paper is to give a local description of affine surfaces, whose induced Blaschke structure is projectively flat. We show that such affine surfaces with constant Gauss affine curvature and indefinite induced Blaschke metric are described by soliton equations."}
{"category": "Math", "title": "Hilbert functions of multigraded algebras, mixed multiplicities of ideals and their applications", "abstract": "This paper is a survey on major results on Hilbert functions of multigraded algebras and mixed multiplicities of ideals, including their applications to the computation of Milnor numbers of complex analytic hypersurfaces with isolated singularity, multiplicities of blowup algebras and mixed volumes of polytopes."}
{"category": "Math", "title": "An Infinite Dimensional Approach to the Third Fundamental Theorem of Lie", "abstract": "We revisit the third fundamental theorem of Lie (Lie III) for finite dimensional Lie algebras in the context of infinite dimensional matrices."}
{"category": "Math", "title": "On fixed point sets and Lefschetz modules for sporadic simple groups", "abstract": "We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component of the centralizer. For odd primes, fixed point sets are computed for sporadic groups having an extraspecial Sylow p-subgroup of order p^3, acting on the complex of those p-radical subgroups containing a p-central element in their centers. Vertices for summands of the associated reduced Lefschetz modules are described."}
{"category": "Math", "title": "Stable symmetries of plane sextics", "abstract": "We classify projective symmetries of irreducible plane sextics with simple singularities which are stable under equivariant deformations. We also outline a connection between order~2 stable symmetries and maximal trigonal curves."}
{"category": "Math", "title": "On non Fundamental Group Equivalent Surfaces", "abstract": "In this paper we present an example of two polarized K3 surfaces which are not Fundamental Group Equivalent (their fundamental groups of the complement of the branch curves are not isomorphic; denoted by FGE) but the fundamental groups of their related Galois covers are isomorphic. For each surface, we consider a generic projection to CP^2 and a degenerations of the surface into a union of planes - the \"pillow\" degeneration for the non-prime surface and the \"magician\" degeneration for the prime surface. We compute the Braid Monodromy Factorization (BMF) of the branch curve of each projected surface, using the related degenerations. By these factorizations, we compute the above fundamental groups. It is known that the two surfaces are not in the same component of the Hilbert scheme of linearly embedded K3 surfaces. Here we prove that furthermore they are not FGE equivalent, and thus they are not of the same Braid Monodromy Type (BMT) (which implies that they are not a projective deformation of each other"}
{"category": "Math", "title": "Partial profiles of quasi-complete graphs", "abstract": "We enumerate graph homomorphisms to quasi-complete graphs, i.e., graphs obtained from complete graphs by removing one edge. The source graphs are complete graphs, quasi-complete graphs, cycles, paths, wheels and broken wheels. These enumerations give rise to sequences of integers with two indices; one of the indices is the number of vertices of the source graph, and the other index is the number of vertices of the target graph."}
{"category": "Math", "title": "Jet Riemann-Lagrange Geometry and Some Applications in Theoretical Biology", "abstract": "The aim of this paper is to construct a natural Riemann-Lagrange differential geometry on 1-jet spaces, in the sense of nonlinear connections, generalized Cartan connections, d-torsions, d-curvatures, jet electromagnetic fields and jet electromagnetic Yang-Mills energies, starting from some given nonlinear evolution ODEs systems modelling biologic phenomena like the cancer cell population model or the infection by human immunodeficiency virus-type 1 (HIV-1) model."}
{"category": "Math", "title": "Two-dimensional metrics admitting precisely one projective vector field", "abstract": "We give a complete list of two-dimensional metrics that admit an essential projective vector field. This solves a problem explicitly posed by Sophus Lie in 1882."}
{"category": "Math", "title": "Trace Ideals for Fourier Integral Operators with Non-Smooth Symbols III", "abstract": "We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces."}
{"category": "Math", "title": "Ergodic Subequivalence Relations Induced by a Bernoulli Action", "abstract": "Let $\\Gamma$ be a countable group and denote by $\\Cal S$ the equivalence relation induced by the Bernoulli action $\\Gamma\\curvearrowright [0,1]^{\\Gamma}$, where $[0,1]^{\\Gamma}$ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation $\\Cal R$ of $\\Cal S$, there exists a partition $\\{X_i\\}_{i\\geq 0}$ of $[0,1]^{\\Gamma}$ with $\\Cal R$-invariant measurable sets such that $\\Cal R_{|X_0}$ is hyperfinite and $\\Cal R_{|X_i}$ is strongly ergodic (hence ergodic), for every $i\\geq 1$."}
{"category": "Math", "title": "Quasiconformal mappings and singularity of boundary distortion", "abstract": "We extend a well-known theorem by Jones and Makarov [JM] on the singularity of boundary distortion of planar conformal mappings. We use a different technique to recover the previous result and, moreover, generalize the result for quasiconformal mappings of the unit ball $\\B^n\\subset \\mathbb{R}^n$, $n\\ge 2$. We also establish an estimate on the Hausdorff (gauge) dimension of the boundary of the image domain outside an exceptional set of given size on the sphere $\\partial \\B^n$. Furthermore, we show that this estimate is essentially sharp. [JM] P. W. Jones and N. Makarov: Density properties of harmonic measure. Ann. Math. 142 (1995), 427--455."}
{"category": "Math", "title": "Riesz transforms for Jacobi expansions", "abstract": "We define and study Riesz transforms and conjugate Poisson integrals associated with multi-dimensional Jacobi expansions."}
{"category": "Math", "title": "Rational functions associated with the white noise space and related topics", "abstract": "Motivated by the hyper-holomorphic case we introduce and study rational functions in the setting of Hida's white noise space. The Fueter polynomials are replaced by a basis computed in terms of the Hermite functions, and the Cauchy-Kovalevskaya product is replaced by the Wick product."}
{"category": "Math", "title": "Canonical Weierstrass Representation of Minimal Surfaces in Euclidean Space", "abstract": "Using the fact that any minimal strongly regular surface carries locally canonical principal parameters, we obtain a canonical representation of these surfaces, which makes more precise the Weierstrass representation in canonical principal parameters. This allows us to describe locally the solutions of the natural partial differential equation of minimal surfaces."}
{"category": "Math", "title": "Higher-Order Properties of Analytic Wavelets", "abstract": "The influence of higher-order wavelet properties on the analytic wavelet transform behavior is investigated, and wavelet functions offering advantageous performance are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a two-parameter family of exactly analytic continuous wavelets. The degree of time/frequency localization, the existence of a mapping between scale and frequency, and the bias involved in estimating properties of modulated oscillatory signals, are proposed as important considerations. Wavelet behavior is found to be strongly impacted by the degree of asymmetry of the wavelet in both the frequency and the time domain, as quantified by the third central moments. A particular subset of the generalized Morse wavelets, recognized as deriving from an inhomogeneous Airy function, emerge as having particularly desirable properties. These \"Airy wavelets\" substantially outperform the only approximately analytic Morlet wavelets for high time localization. Special cases of the generalized Morse wavelets are examined, revealing a broad range of behaviors which can be matched to the characteristics of a signal."}
{"category": "Math", "title": "On Some Weighted Average Values of L-functions", "abstract": "Let $q\\ge 2$ and $N\\ge 1$ be integers. W. Zhang (2008) has shown that for any fixed $\\epsilon> 0$, and $q^{\\epsilon} \\le N \\le q^{1/2 -\\epsilon}$, $$ \\sum_{\\chi \\ne \\chi_0} |\\sum_{n=1}^N \\chi(n)|^2 |L(1, \\chi)|^2 = (1 + o(1)) \\alpha_q q N $$ where the sum is take over all nonprincipal characters $\\chi$ modulo $q$, $L(s, \\chi)$ is the $L$-functions $L(1, \\chi)$ corresponding to $\\chi$ and $\\alpha_q = q^{o(1)}$ is some explicit function of $q$. Here we show that the same formula holds in the range $q^{\\epsilon} \\le N \\le q^{1 -\\epsilon}$."}
{"category": "Math", "title": "Homological stability of series of groups", "abstract": "``What aspects of a group are unchanged, or stable, under homology equivalences''? The model theorem in this regard is the 1963 result of J. Stallings that the lower central series is preserved under any integral homological equivalence of groups. Various other theorems of this nature have since appeared. Stallings himself proved similar theorems for homology with rational or mod p coefficients. These involved different series of groups- variations of the lower central series. W. Dwyer generalized Stallings' integral results to larger classes of maps, work that was completed in the other cases by the authors. More recently the authors proved analogues of the theorems of Stallings and Dwyer for variations of the derived series. The above theorems are all different but clearly have much in common. Here we present a new concept, that of the stability of a subgroup, or a series of subgroups under a class of maps, that offers a framework in which all of these theorems can be viewed. We contrast it with homological localization of groups, which is a previously well-studied framework that might also be applied to these questions."}
{"category": "Math", "title": "Combinatorics of least squares trees", "abstract": "A recurring theme in the least squares approach to phylogenetics has been the discovery of elegant combinatorial formulas for the least squares estimates of edge lengths. These formulas have proved useful for the development of efficient algorithms, and have also been important for understanding connections among popular phylogeny algorithms. For example, the selection criterion of the neighbor-joining algorithm is now understood in terms of the combinatorial formulas of Pauplin for estimating tree length. We highlight a phylogenetically desirable property that weighted least squares methods should satisfy, and provide a complete characterization of methods that satisfy the property. The necessary and sufficient condition is a multiplicative four point condition that the the variance matrix needs to satisfy. The proof is based on the observation that the Lagrange multipliers in the proof of the Gauss--Markov theorem are tree-additive. Our results generalize and complete previous work on ordinary least squares, balanced minimum evolution and the taxon weighted variance model. They also provide a time optimal algorithm for computation."}
{"category": "Math", "title": "Why are solitons stable?", "abstract": "The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, \\emph{solitary wave} or \\emph{soliton} solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions is relatively straightforward, but the fact that they are \\emph{stable} requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which create degeneracy in the stability analysis. The theory is particularly difficult in the \\emph{critical} case in which the nonlinearity is at exactly the right power to potentially allow for a self-similar blowup. In this article we survey some of the highlights of this theory, from the more classical orbital stability analysis of Weinstein and Grillakis-Shatah-Strauss, to the more recent asymptotic stability and blowup analysis of Martel-Merle and Merle-Raphael, as well as current developments in using this theory to rigorously demonstrate controlled blowup for several key equations."}
{"category": "Math", "title": "On the Scarf-Hirota model in the price-scaled price adjustment process", "abstract": "Hirota's results given in (Hirota.M.,1981) on the asymptotically stability are generalized to the price-scaled price adjustment process."}
{"category": "Math", "title": "Thresholding methods to estimate the copula density", "abstract": "This paper deals with the problem of the multivariate copula density estimation. Using wavelet methods we provide two shrinkage procedures based on thresholding rules for which the knowledge of the regularity of the copula density to be estimated is not necessary. These methods, said to be adaptive, are proved to perform very well when adopting the minimax and the maxiset approaches. Moreover we show that these procedures can be discriminated in the maxiset sense. We produce an estimation algorithm whose qualities are evaluated thanks some simulation. Last, we propose a real life application for financial data."}
{"category": "Math", "title": "Controlled stratification for quantile estimation", "abstract": "In this paper we propose and discuss variance reduction techniques for the estimation of quantiles of the output of a complex model with random input parameters. These techniques are based on the use of a reduced model, such as a metamodel or a response surface. The reduced model can be used as a control variate; or a rejection method can be implemented to sample the realizations of the input parameters in prescribed relevant strata; or the reduced model can be used to determine a good biased distribution of the input parameters for the implementation of an importance sampling strategy. The different strategies are analyzed and the asymptotic variances are computed, which shows the benefit of an adaptive controlled stratification method. This method is finally applied to a real example (computation of the peak cladding temperature during a large-break loss of coolant accident in a nuclear reactor)."}
{"category": "Math", "title": "Peterson's Deformations of Higher Dimensional Quadrics", "abstract": "We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in $\\mathbb{C}^3$ of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere $\\mathbb{S}^2\\subset\\mathbb{C}^3$ to an explicit $(n-1)$-dimensional family of deformations in $\\mathbb{C}^{2n-1}$ of $n$-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere $\\mathbb{S}^n\\subset\\mathbb{C}^{n+1}$ and non-degenerate joined second fundamental forms. It is then proven that this family is maximal."}
{"category": "Math", "title": "Fermats Last Theorem on Topological Fields", "abstract": "Even though flt is a number theoretic result we prove that the result depends on the topological as well as the field structure of the underlying space."}
{"category": "Math", "title": "Variations on Log Sarkisov Program for Surfaces", "abstract": "Let (S, BS) be the log-pair associated with a compactification of a given smooth quasi-projective surface V . Under the assumption that the boundary BS is irreducible, we propose an algorithm, in the spirit of the (log) Sarkisov program, to factorize any automorphism of V into a sequence of elementary links in the framework of the logarithmic Mori theory. The new noteworthy feature of our algorithm is that all the blow-ups and contractions involved in the process occur on the boundary."}
{"category": "Math", "title": "On the mass center of the tent map", "abstract": "It is well known that the time average or the center of mass for generic orbits of the standard tent map is 0.5. In this paper we show some interesting properties of the exceptional orbits, including periodic orbits, orbits without mass center, and orbits with mass centers different from 0.5. We prove that for any positive integer $n$, there exist $n$ distinct periodic orbits for the standard tent map with the same center of mass, and the set of mass centers of periodic orbits is a dense subset of $[0,2/3]$. Considering all possible orbits, then the set of mass centers is the interval $[0,2/3]$. Moreover, for every $x$ in $[0,2/3]$, there are uncountably many orbits with mass center $x$. We also show that there are uncountably many orbits without mass center."}
{"category": "Math", "title": "On A-tensors in Riemannian geometry", "abstract": "We present examples, both compact and non-compact complete, of lo- cally non-homogeneous proper A-manifolds."}
{"category": "Math", "title": "Folding = Colouring", "abstract": "The foldings of a connected graph $G$ are defined as follows. First, $G$ is a folding of itself. Let $G'$ be a graph obtained from $G$ by identifying two vertices at distance 2 in $G$. Then every folding of $G'$ is a folding of $G$. The folding number of $G$ is the minimum order of a complete folding of $G$. Theorem: The folding number of every graph equals its chromatic number."}
{"category": "Math", "title": "On the distribution of eigenvalues of non-selfadjoint operators", "abstract": "We prove quantitative bounds on the eigenvalues of non-selfadjoint bounded and unbounded operators. We use the perturbation determinant to reduce the problem to one of studying the zeroes of a holomorphic function."}
{"category": "Math", "title": "A remark on the Herzlich volume of asymptotically complex hyperbolic Einstein manifolds", "abstract": "We observe inequalities involving the Herzlich volume of a 4-dimensional asymptotically complex hyperbolic Einstein manifold and its Euler characteristic provided the metrics is either Kaehler or selfdual. In the selfdual case we have to assume furthermore that the Kronheimer-Mrowka invariant is non vanishing."}
{"category": "Math", "title": "A Note on Generating Functions for Hausdorff Moment Sequences", "abstract": "For functions $f$ whose Taylor coefficients at the origin form a Hausdorff moment sequence we study the behaviour of $w(y):=|f(\\gamma+iy)|$ for $y>0$ ($\\gamma\\leq1$ fixed)."}
{"category": "Math", "title": "Markov loops and renormalization", "abstract": "We study Poissonian ensembles of Markov loops and the associated renormalized self-intersection local times."}
{"category": "Math", "title": "K3-surfaces with special symmetry: An example of classification by Mori-reduction", "abstract": "The classification problem for K3-surfaces equipped with finite groups $H$ of symplectic symmetry centralized by an antisymplectic involution is considered. An approach via equivariant Mori-reduction is employed. This method, which has proved to be successful even for rather small groups, is exemplified here by giving a complete classification in the case $H = C_3 \\ltimes C_7$. The consideration of this particular group is related to the study of K3-surfaces with maximal finite groups of symplectic automorphisms. Applications to the case $L_2(7)$ are given."}
{"category": "Math", "title": "Ballot theorems for random walks with finite variance", "abstract": "We prove an analogue of the classical ballot theorem that holds for any random walk in the range of attraction of the normal distribution. Our result is best possible: we exhibit examples demonstrating that if any of our hypotheses are removed, our conclusions may no longer hold."}
{"category": "Math", "title": "Construction of a stationary FIFO queue with impatient customers", "abstract": "In this paper, we study the stability of queues with impatient customers. Under general stationary ergodic assumptions, we first provide some conditions for such a queue to be regenerative (i.e. to empty a.s. an infinite number of times). In the particular case of a single server operating in First in, First out, we prove the existence (in some cases, on an enlarged probability space) of a stationary workload. This is done by studying stochastic recursions under the Palm settings, and by stochastic comparison of stochastic recursions."}
{"category": "Math", "title": "Approximation using scattered shifts of a multivariate function", "abstract": "The approximation of a general $d$-variate function $f$ by the shifts $\\phi(\\cdot-\\xi)$, $\\xi\\in\\Xi\\subset \\Rd$, of a fixed function $\\phi$ occurs in many applications such as data fitting, neural networks, and learning theory. When $\\Xi=h\\Z^d$ is a dilate of the integer lattice, there is a rather complete understanding of the approximation problem \\cite{BDR,Johnson1} using Fourier techniques. However, in most applications the {\\it center} set $\\Xi$ is either given, or can be chosen with complete freedom. In both of these cases, the shift-invariant setting is too restrictive. This paper studies the approximation problem in the case $\\Xi$ is arbitrary. It establishes approximation theorems whose error bounds reflect the local density of the points in $\\Xi$. Two different settings are analyzed. The first is when the set $\\Xi$ is prescribed in advance. In this case, the theorems of this paper show that, in analogy with the classical univariate spline approximation, improved approximation occurs in regions where the density is high. The second setting corresponds to the problem of non-linear approximation. In that setting the set $\\Xi$ can be chosen using information about the target function $f$. We discuss how to `best' make these choices and give estimates for the approximation error."}
{"category": "Math", "title": "Coloring the 600 Cell", "abstract": "The 600 cell S has exactly 10 5-colorings. From these colorings we can construct the space of colorings $B(S)$. This complex has 1344 colorings, and is isomorphic to the space of 5 by 5 Latin Squares. These simplices split into 4 copies of a quotient of S by an involution, and two copies of a space made up of even Latin Squares."}
{"category": "Math", "title": "Coarse embeddings into a Hilbert space, Haagerup Property and Poincare inequalities", "abstract": "We prove that a metric space does not coarsely embed into a Hilbert space if and only if it satisfies a sequence of Poincar\\'e inequalities, which can be formulated in terms of (generalized) expanders. We also give quantitative statements, relative to the compression. In the equivariant context, our result says that a group does not have the Haagerup property if and only if it has relative property T with respect to a family of probabilities whose supports go to infinity. We give versions of this result both in terms of unitary representations, and in terms of affine isometric actions on Hilbert spaces."}
{"category": "Math", "title": "Obtainable Sizes of Topologies on Finite Sets", "abstract": "We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a topology with a prescribed size, we show that this number has a logarithmic upper bound. We deduce that there exists a topology on n points having k open sets, for all k in an interval which is exponentially large in n. The construction algorithms can be modified to produce topologies where the smallest neighborhood of each point has a minimal size, and we give a range of obtainable sizes for such topologies."}
{"category": "Math", "title": "Free subgroups in groups acting on rooted trees", "abstract": "We show that if a group $G$ acting faithfully on a rooted tree $T$ has a free subgroup, then either there exists a point $w$ of the boundary $\\partial T$ and a free subgroup of $G$ with trivial stabilizer of $w$, or there exists $w\\in\\partial T$ and a free subgroup of $G$ fixing $w$ and acting faithfully on arbitrarily small neighborhoods of $w$. This can be used to prove absence of free subgroups for different known classes of groups. For instance, we prove that iterated monodromy groups of expanding coverings have no free subgroups and give another proof of a theorem by S. Sidki."}
{"category": "Math", "title": "On Nichols algebras over PGL(2,q) and PSL(2,q)", "abstract": "We compute necessary conditions on Yetter-Drinfeld modules over the groups $\\mathbf{PGL}(2,q)=\\mathbf{PGL}(2,\\FF_q)$ and $\\mathbf{PSL}(2,q)=\\mathbf{PSL}(2,\\FF_q)$ to generate finite dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that there is no non-trivial finite-dimensional pointed Hopf algebra over the Mathieu groups $M_{20}$ and $M_{21}=\\mathbf{PSL}(3,4)$."}
{"category": "Math", "title": "Canonical measures and Kahler-Ricci flow", "abstract": "We show that the Kahler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under birational transformations under the assumption of the finite generation of canonical rings."}
{"category": "Math", "title": "Pants immersed in hyperbolic 3-manifolds", "abstract": "We show that an immersed thrice-punctured sphere in a cusped orientable hyperbolic 3-manifold is either embedded or has a single clasp in a manifold obtained by hyperbolic Dehn filling on a cusp of the Whitehead link complement."}
{"category": "Math", "title": "Simplicial matrix-tree theorems", "abstract": "We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $\\Delta$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $\\Delta$ and replacing the entries of the Laplacian with Laurent monomials. When $\\Delta$ is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes."}
{"category": "Math", "title": "3-Dimensional Schlaefli Formula and Its Generalization", "abstract": "Several identities similar to the Schlaefli formula are established for tetrahedra in a space of constant curvature."}
{"category": "Math", "title": "A Localization Approach to Improve Iterative Proportional Scaling in Gaussian Graphical Models", "abstract": "We discuss an efficient implementation of the iterative proportional scaling procedure in the multivariate Gaussian graphical models. We show that the computational cost can be reduced by localization of the update procedure in each iterative step by using the structure of a decomposable model obtained by triangulation of the graph associated with the model. Some numerical experiments demonstrate the competitive performance of the proposed algorithm."}
{"category": "Math", "title": "Connect sum and transversely non simple knots", "abstract": "We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Vertesi's result."}
{"category": "Math", "title": "T-adic exponential sums over finite fields", "abstract": "$T$-adic exponential sums associated to a Laurent polynomial $f$ are introduced. They interpolate all classical $p^m$-power order exponential sums associated to $f$. The Hodge bound for the Newton polygon of $L$-functions of $T$-adic exponential sums is established. This bound enables us to determine, for all $m$, the Newton polygons of $L$-functions of $p^m$-power order exponential sums associated to an $f$ which is ordinary for $m=1$. Deeper properties of $L$-functions of $T$-adic exponential sums are also studied. Along the way, new open problems about the $T$-adic exponential sum itself are discussed."}
{"category": "Math", "title": "On the Shuffling Algorithm for Domino Tilings", "abstract": "We study the dynamics of a certain discrete model of interacting particles that comes from the so called shuffling algorithm for sampling a random tiling of an Aztec diamond. It turns out that the transition probabilities have a particularly convenient determinantal form. An analogous formula in a continuous setting has recently been obtained by Jon Warren studying certain model of interlacing Brownian motions which can be used to construct Dyson's non-intersecting Brownian motion. We conjecture that Warren's model can be recovered as a scaling limit of our discrete model and prove some partial results in this direction. As an application to one of these results we use it to rederive the known result that random tilings of an Aztec diamond, suitably rescaled near a turning point, converge to the GUE minor process."}
{"category": "Math", "title": "Coarse differentiation and quasi-isometries of a class of solvable Lie groups I", "abstract": "This is the first of two papers which aim to understand quasi-isometries of a subclass of unimodular split solvable Lie groups. In the present paper, we show that locally (in a coarse sense), a quasi-isometry between two groups in this subclass is close to a map that respects their group structures."}
{"category": "Math", "title": "The eigenvalues of the Laplacian on domains with small slits", "abstract": "We introduce a small slit into a planar domain and study the resulting effect upon the eigenvalues of the Laplacian. In particular, we show that as the length of the slit tends to zero, each real-analytic eigenvalue branch tends to an eigenvalue of the original domain. By combining this with our earlier work (arXiv:math/0703616), we obtain the following application: The generic multiply connected polygon has simple spectrum."}
{"category": "Math", "title": "A Markov Basis for Conditional Test of Common Diagonal Effect in Quasi-Independence Model for Square Contingency Tables", "abstract": "In two-way contingency tables we sometimes find that frequencies along the diagonal cells are relatively larger(or smaller) compared to off-diagonal cells, particularly in square tables with the common categories for the rows and the columns. In this case the quasi-independence model with an additional parameter for each of the diagonal cells is usually fitted to the data. A simpler model than the quasi-independence model is to assume a common additional parameter for all the diagonal cells. We consider testing the goodness of fit of the common diagonal effect by Markov chain Monte Carlo (MCMC) method. We derive an explicit form of a Markov basis for performing the conditional test of the common diagonal effect. Once a Markov basis is given, MCMC procedure can be easily implemented by techniques of algebraic statistics. We illustrate the procedure with some real data sets."}
{"category": "Math", "title": "Comparison of estimates for dispersive equations", "abstract": "This paper describes a new comparison principle that can be used for the comparison of space-time estimates for dispersive equations. In particular, results are applied to the global smoothing estimates for several classes of dispersive partial differential equations."}
{"category": "Math", "title": "A note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation", "abstract": "We present in this paper a detailed note on the computation of Puiseux series solutions of the Riccatti equation associated with a homogeneous linear ordinary differential equation. This paper is a continuation of [1] which was on the complexity of solving arbitrary ordinary polynomial differential equations in terms of Puiseux series."}
{"category": "Math", "title": "Canonical Weierstrass Representation of Minimal and Maximal Surfaces in the Three-dimensional Minkowski Space", "abstract": "We prove that any minimal (maximal) strongly regular surface in the three-dimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like surfaces, which makes more precise the Weierstrass representation and shows more precisely the correspondence between these surfaces and holomorphic functions (in the Gauss plane). We also find a canonical representation of maximal strongly regular space-like surfaces, which makes more precise the Weierstrass representation and shows more precisely the correspondence between these surfaces and holomorphic functions (in the Lorentz plane). This allows us to describe locally the solutions of the corresponding natural partial differential equations."}
{"category": "Math", "title": "A limited in bandwidth uniformity for the functional limit law of the increments of the empirical process", "abstract": "Consider the following local empirical process indexed by $K\\in \\mathcal{G}$, for fixed $h>0$ and $z\\in \\mathbb{R}^d$: $$G_n(K,h,z):=\\sum_{i=1}^n K \\Bigl(\\frac{Z_i-z}{h^{1/d}}\\Big) - \\mathbbE \\Bigl(K \\Bigl(\\frac{Z_i-z}{h^{1/d}}\\Big)\\Big),$$ where the $Z_i$ are i.i.d. on $\\mathbb{R}^d$. We provide an extension of a result of Mason (2004). Namely, under mild conditions on $\\mathcal{G}$ and on the law of $Z_1$, we establish a uniform functional limit law for the collections of processes $\\bigl\\{G_n(\\cdot,h_n,z), z\\in H, h\\in [h_n,\\mathfrak{h}_n]\\big\\}$, where $H\\subset \\mathbb{R}^d$ is a compact set with nonempty interior and where $h_n$ and $\\mathfrak{h}_n$ satisfy the Cs\\\"{o}rg\\H{o}-R\\'{e}v\\'{e}sz-Stute conditions."}
{"category": "Math", "title": "Separation and coupling cutoffs for tuples of independent Markov processes", "abstract": "We consider an $n$-tuple of independent ergodic Markov processes, each of which converges (in the sense of separation distance) at an exponential rate, and obtain a necessary and sufficient condition for the $n$-tuple to exhibit a separation cutoff. We also provide general bounds on the (asymmetric) window size of the cutoff, and indicate links to classical extreme value theory."}
{"category": "Math", "title": "The normal distribution in some constrained sample spaces", "abstract": "Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data and with compositional data, like percentages and the like. If the natural measure of difference is not the absolute one, it is possible to use simple algebraic properties to show that it is more convenient to work with a geometry that is not the usual Euclidean geometry in real space, and with a measure which is not the usual Lebesgue measure, leading to alternative models which better fit the phenomenon under study. The general approach is presented and illustrated both on the positive real line and on the D-part simplex."}
{"category": "Math", "title": "Complete moment and integral convergence for sums of negatively associated random variables", "abstract": "For a sequence of identically distributed negatively associated random variables $\\{X_n; n\\geq 1\\}$ with partial sums $S_n=\\sum_{i=1}^nX_i, n\\geq 1$, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form $$ \\sum_{n \\ge n_0} n^{r -2 -\\frac{1}{pq}} a_n E(\\max_{1 \\le k \\le n}|S_k|^{\\frac{1}{q}} - \\epsilon b_n^{\\frac{1}{pq}})^+ < \\infty $$ to hold where $r>1, q>0$ and either $n_0=1, 0<p<2, a_n=1, b_n=n$ or $n_0=3, p=2, a_n=(\\log n)^{-\\frac{1}{2q}}, b_n=n\\log n$. These results extend results of Chow (1988) and Li and Sp\\u{a}taru (2005) from the independent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence."}
{"category": "Math", "title": "Multiples of integral points on elliptic curves", "abstract": "If $E$ is a minimal elliptic curve defined over $\\ZZ$, we obtain a bound $C$, depending only on the global Tamagawa number of $E$, such that for any point $P\\in E(\\QQ)$, $nP$ is integral for at most one value of $n>C$. As a corollary, we show that if $E/\\QQ$ is a fixed elliptic curve, then for all twists $E'$ of $E$ of sufficient height, and all torsion-free, rank-one subgroups $\\Gamma\\subseteq E'(\\QQ)$, $\\Gamma$ contains at most 6 integral points. Explicit computations for congruent number curves are included."}
{"category": "Math", "title": "Odd Entries in Pascal's Trinomial Triangle", "abstract": "The nth row of Pascal's trinomial triangle gives coefficients of (1+x+x^2)^n. Let g(n) denote the number of such coefficients that are odd. We review Moshe's algorithm for evaluating asymptotics of g(n) -- this involves computing the Lyapunov exponent for certain 2x2 random matrix products -- and then analyze further examples with more terms and higher powers of x."}
{"category": "Math", "title": "Micro-Macro Modelling of an Array of Spheres Interacting Through Lubrication Forces", "abstract": "We consider here a discrete system of spheres interacting through a lubrication force. This force is dissipative, and singular near contact: it behaves like the reciprocal of interparticle distance. We propose a macroscopic constitutive equation which is built as the natural continuous counterpart of this microscopic lubrication model. This model, which is of the newtonian type, relies on an elongational viscosity, which is proportional to the reciprocal of the local fluid fraction. We then establish the convergence in a weak sense of solutions to the discrete problem towards the solution to the partial differential equation which we identified as the macroscopic constitutive equation."}
{"category": "Math", "title": "Existence of travelling-wave solutions and local well-posedness of the Fowler equation", "abstract": "We study the existence of travelling-waves and local well-posedness in a subspace of $C_b^1(\\mathbb{R})$ for a nonlinear evolution equation recently proposed by Andrew C. Fowler to study the dynamics of dunes."}
{"category": "Math", "title": "Minimal positive stencils in meshfree finite difference methods for the Poisson equation", "abstract": "Meshfree finite difference methods for the Poisson equation approximate the Laplace operator on a point cloud. Desirable are positive stencils, i.e. all neighbor entries are of the same sign. Classical least squares approaches yield large stencils that are in general not positive. We present an approach that yields stencils of minimal size, which are positive. We provide conditions on the point cloud geometry, so that positive stencils always exist. The new discretization method is compared to least squares approaches in terms of accuracy and computational performance."}
{"category": "Math", "title": "Estimation and Test for Multidimensional Regression Models", "abstract": "This work is concerned with the estimation of multidimensional regression and the asymptotic behaviour of the test involved in selecting models. The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. We show in this paper that if we choose to minimise the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable and regular regression model. Numerical experiments confirm the theoretical results."}
{"category": "Math", "title": "Gromov-Witten theory of A_n-resolutions", "abstract": "We give a complete solution for the reduced Gromov-Witten theory of resolved surface singularities of type A_n, for any genus, with arbitrary descendent insertions. We also present a partial evaluation of the T-equivariant relative Gromov-Witten theory of the threefold A_n x P^1 which, under a nondegeneracy hypothesis, yields a complete solution for the theory. The results given here allow comparison of this theory with the quantum cohomology of the Hilbert scheme of points on the A_n surfaces. We discuss generalizations to linear Hodge insertions and to surface resolutions of type D,E. As a corollary, we present a new derivation of the stationary Gromov-Witten theory of P^1."}
{"category": "Math", "title": "Sous-groupes alg\\'ebriques du groupe de Cremona", "abstract": "We give a complete classification of maximal algebraic subgroups of the Cremona group of the plane and provide algebraic varieties that parametrize the conjugacy classes. ----- Nous donnons une classification compl\\`ete des sous-groupes alg\\'ebriques maximaux du groupe de Cremona du plan et explicitons les vari\\'et\\'es qui param\\`etrent les classes de conjugaison."}
{"category": "Math", "title": "Proof(s) of the Lamperti representation of Continuous-State Branching Processes", "abstract": "This paper uses two new ingredients, namely stochastic differential equations satisfied by continuous-state branching processes (CSBPs), and a topology under which the Lamperti transformation is continuous, in order to provide self-contained proofs of Lamperti's 1967 representation of CSBPs in terms of spectrally positive L\\'evy processes. The first proof is a direct probabilistic proof, and the second one uses approximations by discrete processes, for which the Lamperti representation is evident."}
{"category": "Math", "title": "The cobordism class of the moduli space of polygons in $\\mathbb{R}^3$", "abstract": "For any vector $r=(r_1,..., r_n)$, let $M_r$ denote the moduli space (under rigid motions) of polygons in $\\mathbb{R}^3$ with $n$-sides whose lengths are $r_1,...,r_n$. We give an explicit characterization of the oriented $S^1$-cobordism class of $M_r$ which depends uniquely on the length vector $r$."}
{"category": "Math", "title": "Measures and their random reals", "abstract": "We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-computable real is non-trivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every non-hyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for any continuous measure can be found throughout the hyperarithmetical Turing degrees."}
{"category": "Math", "title": "Nonsmoothable, locally indicable group actions on the interval", "abstract": "By the Thurston stability theorem, a group of C^1 orientation-preserving diffeomorphisms of the closed unit interval is locally indicable. We show that the local order structure of orbits gives a stronger criterion for nonsmoothability that can be used to produce new examples of locally indicable groups of homeomorphisms of the interval that are not conjugate to groups of C^1 diffeomorphisms."}
{"category": "Math", "title": "Descent Systems for Bruhat Posets", "abstract": "Let $(W,S)$ be a finite Weyl group and let $w\\in W$. It is widely appreciated that the descent set D(w)=\\{s\\in S | l(ws)<l(w)\\} determines a very large and important chapter in the study of Coxeter groups. In this paper we generalize some of those results to the situation of the Bruhat poset $W^J$ where $J\\subseteq S$. Our main results here include the identification of a certain subset $S^J\\subseteq W^J$ that convincingly plays the role of $S\\subseteq W$, at least from the point of view of descent sets and related geometry. The point here is to use this resulting {\\em descent system} $(W^J,S^J)$ to explicitly encode some of the geometry and combinatorics that is intrinsic to the poset $W^J$. In particular, we arrive at the notion of an {\\em augmented poset}, and we identify the {\\em combinatorially smooth} subsets $J\\subseteq S$ that have special geometric significance in terms of a certain corresponding torus embedding $X(J)$. The theory of $\\mathscr{J}$-irreducible monoids provides an essential tool in arriving at our main results."}
{"category": "Math", "title": "Asymptotic formula for the moments of Minkowski question mark function in the interval [0,1]", "abstract": "In this paper we prove the asymptotic formula for the moments of Minkowski question mark function, which describes the distribution of rationals in the Farey tree. The main idea is to demonstrate that certain a variation of a Laplace method is applicable in this problem, hence the task reduces to a number of technical calculations."}
{"category": "Math", "title": "Zeta functions and monodromy for surfaces that are general for a toric idealistic cluster", "abstract": "In this article we consider surfaces that are general with respect to a 3- dimensional toric idealistic cluster. In particular, this means that blowing up a toric constellation provides an embedded resolution of singularities for these surfaces. First we give a formula for the topological zeta function directly in terms of the cluster. Then we study the eigenvalues of monodromy. In particular, we derive a useful criterion to be an eigenvalue. In a third part we prove the monodromy and the holomorphy conjecture for these surfaces."}
{"category": "Math", "title": "Enumerating Palindromes and Primitives in Rank Two Free Groups", "abstract": "Let $F= < a,b>$ be a rank two free group. A word $W(a,b)$ in $F$ is {\\sl primitive} if it, along with another group element, generates the group. It is a {\\sl palindrome} (with respect to $a$ and $b$) if it reads the same forwards and backwards. It is known that in a rank two free group any primitive element is conjugate either to a palindrome or to the product of two palindromes, but known iteration schemes for all primitive words give only a representative for the conjugacy class. Here we derive a new iteration scheme that gives either the unique palindrome in the conjugacy class or expresses the word as a unique product of two unique palindromes. We denote these words by $E_{p/q}$ where $p/q$ is rational number expressed in lowest terms. We prove that $E_{p/q}$ is a palindrome if $pq$ is even and the unique product of two unique palindromes if $pq$ is odd. We prove that the pairs $(E_{p/q},E_{r/s})$ generate the group when $|ps-rq|=1$. This improves the previously known result that held only for $pq$ and $rs$ both even. The derivation of the enumeration scheme also gives a new proof of the known results about primitives."}
{"category": "Math", "title": "A Hardy field extension of Szemeredi's Theorem", "abstract": "In 1975 Szemer\\'edi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form $p(n)$ where $p(n)$ is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference of the progression in Szemer\\'edi's theorem can be of the form $[n^\\delta]$ where $\\delta$ is any positive real number and $[x]$ denotes the integer part of $x$. More generally, the common difference can be of the form $[a(n)]$ where $a(x)$ is any function that is a member of a Hardy field and satisfies $a(x)/x^k\\to \\infty$ and $a(x)/x^{k+1}\\to 0$ for some non-negative integer $k$. The proof combines a new structural result for Hardy sequences, techniques from ergodic theory, and some recent equidistribution results of sequences on nilmanifolds."}
{"category": "Math", "title": "Quantum cohomology of the Hilbert scheme of points on A_n-resolutions", "abstract": "We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the action of the affine Lie algebra \\hat{gl}(n+1) on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of A_n x P^1. We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of type D and E."}
{"category": "Math", "title": "Donaldson-Thomas theory of A_n x P^1", "abstract": "We study the relative Donaldson-Thomas theory of A_n x P^1, where A_n is the surface resolution of a type A_n singularity. The action of divisor operators in the theory is expressed in terms of operators of the affine algebra \\hat{gl}(n+1) on Fock space. Assuming a nondegeneracy conjecture, this gives a complete solution for the theory. The results complete the comparison of this theory with the Gromov-Witten theory of A_n x P^1 and the quantum cohomology of the Hilbert scheme of points on A_n."}
{"category": "Math", "title": "Uniform (m)-condition and Strong Milnor fibrations", "abstract": "In this paper we study the Milnor fibrations associated to real analytic map germs $\\psi:(\\mathbb{R}^{m},0) \\to (\\mathbb{R}^2,0)$ with isolated critical point at $0\\in \\mathbb{R}^{m}$. The main result relates the existence of called Strong Milnor fibrations with a transversality condition of a convenient family of analytic varieties with isolated critical points at the origin $0\\in \\mathbb{R}^{m}$, obtained by projecting the map germ $\\psi$ in the family $L_{-\\theta}$ of all lines through the origin in the plane $\\mathbb R^{2}.$"}
{"category": "Math", "title": "Labelling Algorithms for Paired-domination Problems in Block and Interval Graphs", "abstract": "Let $G=(V,E)$ be a graph without isolated vertices. A set $S\\subseteq V$ is a paired-domination set if every vertex in $V-S$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang and Ng [ Paired domination on interval and circular-arc graphs, Disc. Appl. Math. 155(2007),2077-2086], we present two linear time algorithms to find a minimum cardinality paired-dominating set in block and interval graphs. In addition, we prove that paired-domination problem is {\\em NP}-complete for bipartite graphs, chordal graphs, even split graphs."}
{"category": "Math", "title": "Ellipticity and Ergodicity", "abstract": "Let $S=\\{S_t\\}_{t\\geq0}$ be the submarkovian semigroup on $L_2(\\Ri^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients $c_{ij}$. Further let $\\Omega$ be an open subset of $\\Ri^d$. Under the assumption that $C_c^\\infty(\\Ri^d)$ is a core for $H$ we prove that $S$ leaves $L_2(\\Omega)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $Y_i=\\sum^d_{j=1}c_{ij}\\partial_j$."}
{"category": "Math", "title": "Heat Kernel and Essential Spectrum of Infinite Graphs", "abstract": "We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed vertex. A sufficient condition for non-uniqueness is also presented. Furthermore, we give a lower bound on the bottom of the spectrum of the discrete Laplacian and use this bound to give a condition ensuring that the essential spectrum of the Laplacian is empty."}
{"category": "Math", "title": "Equivalence of real Milnor's fibrations for quasi homogeneous singularities", "abstract": "We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a \"thin\" hollowed tube involving the zero level and the fibration in the complement of \"link\" in sphere are equivalents, since they exist. Moreover, in order to do that, we explicitly characterize the critical points of projection $\\frac{f}{\\|f\\|}:S_{\\epsilon}^{m}\\setminus K_{\\epsilon}\\to S^{1}$, where $K_{\\epsilon}$ is the link of singularity."}
{"category": "Math", "title": "A Chord Diagrammatic Presentation of the Mapping Class Group of a Once Bordered Surface", "abstract": "The Ptolemy groupoid is a combinatorial groupoid generated by elementary moves on marked trivalent fatgraphs with three types of relations. Through the fatgraph decomposition of Teichm\\\"uller space, the Ptolemy groupoid is a mapping class group equivariant subgroupoid of the fundamental path groupoid of Teichm\\\"uller space with a discrete set objects. In particular, it leads to an infinite, but combinatorially simple, presentation of the mapping class group of an orientable surface. In this note, we give a presentation of a full mapping class group equivariant subgroupoid of the Ptolemy groupoid of an orientable surface with one boundary component in terms of marked linear chord diagrams, with chord slides as generators and five types of relations. We also introduce a dual version of this presentation which has advantages for certain applications, one of which is given."}
{"category": "Math", "title": "Morita equivalent subalgebras of irrational rotation algebras and real quadratic fields", "abstract": "In this paper, we determine the isomorphic classes of Morita equivalent subalgebras of irrational rotation algebras. It is based on the solution of the quadratic Diophantine equations. We determine the irrational rotation algebras that have locally trivial inclusions. We compute the index of the locally trivial inclusions of irrational rotation algebras."}
{"category": "Math", "title": "Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners", "abstract": "In this paper we describe and continue the study begun by the author, Jones, and Segal, of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain complex of a CW -spectrum or pro-spectrum, where the attaching maps are determined by the compactified moduli spaces of connecting orbits. The basic obstructions to the existence of this realization are the smoothness of these moduli spaces, and the existence of compatible collections of framings of their stable tangent bundles. In this note we describe a generalization of this, to show that when these moduli spaces are smooth, and are oriented with respect to a generalized cohomology theory E^*, then a Floer E_* -homology theory can be defined. In doing this we describe a functorial viewpoint on how chain complexes can be realized by E -module spectra, generalizing the stable homotopy realization criteria given earlier by the author, Jones, and Segal. Since these moduli spaces, if smooth, will be manifolds with corners, we give a discussion about the appropriate notion of orientations of manifolds with corners."}
{"category": "Math", "title": "Time Varying Undirected Graphs", "abstract": "Undirected graphs are often used to describe high dimensional distributions. Under sparsity conditions, the graph can be estimated using $\\ell_1$ penalization methods. However, current methods assume that the data are independent and identically distributed. If the distribution, and hence the graph, evolves over time then the data are not longer identically distributed. In this paper, we show how to estimate the sequence of graphs for non-identically distributed data, where the distribution evolves over time."}
{"category": "Math", "title": "Classification of pairs of rotations in finite-dimensional Euclidean space", "abstract": "A rotation in a Euclidean space V is an orthogonal map on V which acts locally as a plane rotation with some fixed angle. We give a classification of all pairs of rotations in finite-dimensional Euclidean space, up to simultaneous conjugation with orthogonal maps."}
{"category": "Math", "title": "Skeletons of monomial ideals", "abstract": "In analogy to the skeletons of a simplicial complex and their Stanley--Reisner ideals we introduce the skeletons of an arbitrary monomial ideal $I\\subset S=K[x_1,...,x_n]$. This allows us to compute the depth of $S/I$ in terms of its skeleton ideals. We apply these techniques to show that Stanley's conjecture on Stanley decompositions of $S/I$ holds provided it holds whenever $S/I$ is Cohen--Macaulay. We also discuss a conjecture of Soleyman-Jahan and show that it suffices to prove his conjecture for monomial ideals with linear resolution."}
{"category": "Math", "title": "The Auslander-Reiten translate on monomial quotient rings", "abstract": "For a multidegree t in N^n, E.Miller has defined a category of positively t-determined modules over the polynomial ring S in n variables. We consider the Auslander-Reiten translate, Na_t, on the (derived) category of such modules. A monomial ideal I is positively t-determined if every generator x^a has a \\leq t. We compute the multigraded cohomology- and betti spaces of Na_t^k(S/I) for every iterate k, and also the S-module structure of these cohomology modules. This comprehensively generalizes results of Hochster and Gr\\\"abe on local cohomology of Stanley-Reisner rings."}
{"category": "Math", "title": "Eigenvalues estimate for the Neumann problem on bounded domains", "abstract": "In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we use test functions for the Rayleigh quotient subordinated to a family of open sets constructed in a general metric way, interesting for itself. As application, we get upper bounds for the Neumann spectrum which is clearly in agreement with the Weyl law and which is analogous to Buser's upper bounds of the spectrum of a closed Riemannian manifold with lower bound on the Ricci curvature."}
{"category": "Math", "title": "On pseudo-differential operators on group SU(2)", "abstract": "In this paper we will outline elements of the global calculus of seudo-differential operators on the group SU(2). This is a part of a more general approach to pseudo-differential operators on compact Lie groups that will appear in the forthcoming book by the authors."}
{"category": "Math", "title": "Two dimensional Berezin-Li-Yau inequalities with a correction term", "abstract": "We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is almost optimal. This improves a previous result by Melas."}
{"category": "Math", "title": "On Border Basis and Groebner Basis Schemes", "abstract": "Hilbert schemes of zero-dimensional ideals in a polynomial ring can be covered with suitable affine open subschemes whose construction is achieved using border bases. Moreover, border bases have proved to be an excellent tool for describing zero-dimensional ideals when the coefficients are inexact. And in this situation they show a clear advantage with respect to Groebner bases which, nevertheless, can also be used in the study of Hilbert schemes, since they provide tools for constructing suitable stratifications. In this paper we compare Groebner basis schemes with border basis schemes. It is shown that Groebner basis schemes and their associated universal families can be viewed as weighted projective schemes. A first consequence of our approach is the proof that all the ideals which define a Groebner basis scheme and are obtained using Buchberger's Algorithm, are equal. Another result is that if the origin (i.e. the point corresponding to the unique monomial ideal) in the Groebner basis scheme is smooth, then the scheme itself is isomorphic to an affine space. This fact represents a remarkable difference between border basis and Groebner basis schemes. Since it is natural to look for situations where a Groebner basis scheme and the corresponding border basis scheme are equal, we address the issue, provide an answer, and exhibit some consequences. Open problems are discussed at the end of the paper."}
{"category": "Math", "title": "Kernel regression uniform rate estimation for censored data under $\\alpha$-mixing condition", "abstract": "In this paper, we study the behavior of a kernel estimator of the regression function in the right censored model with $\\alpha$-mixing data . The uniform strong consistency over a real compact set of the estimate is established along with a rate of convergence. Some simulations are carried out to illustrate the behavior of the estimate with different examples for finite sample sizes."}
{"category": "Math", "title": "Remarks on Fourier multipliers and applications to the Wave equation", "abstract": "Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [3]. The same arguments may apply to obtain local wellposedness for the NLKG equation."}
{"category": "Math", "title": "Quiver representations of maximal rank type and an application to representations of a quiver with three vertices", "abstract": "We introduce the notion of ''maximal rank type'' for representations of quivers, which requires certain collections of maps involved in the representation to be of maximal rank. We show that real root representations of quivers are of maximal rank type. By using the maximal rank type property and universal extension functors we construct all real root representations of a particular wild quiver with three vertices. From this construction it follows that real root representations of this quiver are tree modules. Moreover, formulae given by Ringel can be applied to compute the dimension of the endomorphism ring of a given real root representation."}
{"category": "Math", "title": "Symbolic lumping of some catenary, mamillary and circular compartmental systems", "abstract": "Some of the most important compartmental systems, such as irreversible catenary, mamillary and circular systems are symbolically simplified by the method of exact linear lumping. A few symbolically unmanageable systems are numerically lumped. Transformation of the qualitative properties under lumping are also traced."}
{"category": "Math", "title": "On the Fredholm Solvability for a Class of Multidimensional Hyperbolic Problems", "abstract": "We prove the Fredholm alternative for a class of two-dimensional first-order hyperbolic systems with periodic-Dirichlet boundary conditions. Our approach is based on a regularization via a right parametrix."}
{"category": "Math", "title": "Numerical Simulation of Gluey Particles", "abstract": "We propose here a model and a numerical scheme to compute the motion of rigid particles interacting through the lubrication force. In the case of a particle approaching a plane, we propose an algorithm and prove its convergence towards the solutions to the gluey particle model proposed by B. Maury. We propose a multi-particle version of this gluey model which is based on the projection of the velocities onto a set of admissible velocities. Then, we describe a multi-particle algorithm for the simulation of such systems and present numerical results."}
{"category": "Math", "title": "A Gelfand Model for Wreath Products", "abstract": "A Gelafand model for wreath products $\\Z_r\\wr S_n$ is constructed. The proof relies on a combinatorial interpretation of the characters of the model, extending a classical result of Frobenius and Schur."}
{"category": "Math", "title": "On amenability of automata groups", "abstract": "We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups (\"Mother groups\") which is done by analyzing the asymptotic properties of random walks on these groups."}
{"category": "Math", "title": "The universal sl(2) cohomology via webs and foams", "abstract": "We construct the universal sl(2)-tangle cohomology using an approach with webs and dotted foams. This theory depends on two parameters, and for the case of links it is a categorification of the unnormalized Jones polynomial of the link."}
{"category": "Math", "title": "Floer homology and splicing knot complements", "abstract": "We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_i\\subset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also present a few applications. If $h_n^i$ denotes the rank of the Heegaard Floer group $\\widehat{\\mathrm{HFK}}$ for the knot obtained by $n$-surgery over $K_i$ we show that the rank of $\\widehat{\\mathrm{HF}}(Y(K_1,K_2))$ is bounded below by $$\\big|(h_\\infty^1-h_1^1)(h_\\infty^2-h_1^2)- (h_0^1-h_1^1)(h_0^2-h_1^2)\\big|.$$ We also show that if splicing the complement of a knot $K\\subset Y$ with the trefoil complements gives a homology sphere $L$-space then $K$ is trivial and $Y$ is a homology sphere $L$-space."}
{"category": "Math", "title": "Unital ${A}_\\infty$-categories", "abstract": "We prove that three definitions of unitality for A-infinity-categories suggested by the first author, by Kontsevich and Soibelman, and by Fukaya are equivalent."}
{"category": "Math", "title": "q-Abel polynomials", "abstract": "This note gives a simple approach to q-analogues of some results associated with Abel polynomials."}
{"category": "Math", "title": "On Cartan Spaces with the $m$-th Root Metric $K(x,p)=\\sqrt[m]{a^{i_{1}i_{2}...i_{m}}(x)p_{i_{1}}p_{i_{2}}...p_{i_{m}}}$", "abstract": "The aim of this paper is to expose some geometrical properties of the locally Minkowski-Cartan space with the Berwald-Moor metric of momenta. This space is regarded as a particular case of the $m$-th root Cartan space. Thus, Section 2 studies the $v$-covariant derivation components of the $m$-th root Cartan space. Section 3 computes the $v$-curvature d-tensor $S^{hijk}$ of the m-th root Cartan space and studies conditions for $S3$-likeness. Section 4 computes the $T$-tensor $T^{hijk}$ of the m-th root Cartan space. Section 5 particularizes the preceding geometrical results for the Berwald-Moor metric of momenta."}
{"category": "Math", "title": "Invariance de la Gamma-dimension pour certaines familles k\\\"ahl\\'eriennes de dimension 3", "abstract": "In this article, we study some properties of deformation invariance of the Gamma-dimension (defined for X a compact k\\\"ahler manifold). This birational invariant is defined as the codimension of the maximal compact subvarieties in the universal cover of X. In the surface case, the deformation invariance is a straightforward consequence of a theorem of Y.-T. Siu. Using some results from F. Campana et Q. Zhang, we settle this invariance for certain type of K\\\"ahler families of dimension 3."}
{"category": "Math", "title": "The inverse problem of differential Galois theory over the field R(z)", "abstract": "We describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extensions, we develop a Galois descent theory. We utilize this theory to prove that every linear algebraic group $G$ over $\\mathbb{R}$ occurs as a differential Galois group over $\\mathbb{R}(z)$. The main ingredient of the proof is the Riemann-Hilbert correspondence for regular singular differential equations over $\\mathbb{C}(z)$."}
{"category": "Math", "title": "Lyapunov Functionals and Local Dissipativity for the Vorticity Equation in L^p and Besov Spaces", "abstract": "In this paper we establish the local Lyapunov property of certain L^p and Besov norms of the vorticity fields. We have resolved in part, a certain open problem posed by Tosio Kato for the three dimensional Navier Stokes equation by studying the vorticity equation. The local dissipativity of the sum of linear and non-linear operators of the vorticity equation is established. One of the main techniques used here is Littlewood-Paley analysis."}
{"category": "Math", "title": "Stochastic 2-D Navier-Stokes Equation with Artificial Compressibility", "abstract": "In this paper we study the stochastic Navier-Stokes equation with artificial compressibility. The main results of this work are the existence and uniqueness theorem for strong solutions and the limit to incompressible flow. These results are obtained by utilizing a local monotonicity property of the sum of the Stokes operator and the nonlinearity."}
{"category": "Math", "title": "On antipodal spherical t-designs of degree s with $t\\geq 2s-3$", "abstract": "We prove that if X is a spherical t-design and s-distance set with $t\\geq 2s-3$, then X has the structure of Q-polynomial association scheme of class s. Also, we describe the parameters of the association scheme."}
{"category": "Math", "title": "Classification Constrained Dimensionality Reduction", "abstract": "Dimensionality reduction is a topic of recent interest. In this paper, we present the classification constrained dimensionality reduction (CCDR) algorithm to account for label information. The algorithm can account for multiple classes as well as the semi-supervised setting. We present an out-of-sample expressions for both labeled and unlabeled data. For unlabeled data, we introduce a method of embedding a new point as preprocessing to a classifier. For labeled data, we introduce a method that improves the embedding during the training phase using the out-of-sample extension. We investigate classification performance using the CCDR algorithm on hyper-spectral satellite imagery data. We demonstrate the performance gain for both local and global classifiers and demonstrate a 10% improvement of the $k$-nearest neighbors algorithm performance. We present a connection between intrinsic dimension estimation and the optimal embedding dimension obtained using the CCDR algorithm."}
{"category": "Math", "title": "Constructions for infinitesimal group schemes", "abstract": "Let G be an infinitesimal group scheme over a field k of positive characteristic p. We introduce the global p-nilpotent operator $\\Theta_G: k[G] \\to k[V(G)]$, where V(G) is the scheme which represents 1-parameter subgroups of G. This operator applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain G-modules (including those of constant Jordan type), we employ the global p-nilpotent operator to associate various algebraic vector bundles on the projective scheme $\\bP(G)$, the projectivization of the scheme of one-parameter subgroups of G. These vector bundles not only distinguish certain representations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on $\\bP(G)$."}
{"category": "Math", "title": "On Normalized Table Algebras Generated by a Faithful Non-real Element of Degree 3-II", "abstract": "The concept of \"table algebra\" was introduced by Z Arad anf H. Blau in order to study in a uniform way properties of products of conjugacy classes and of irreducible characters of a finite group, Except for certain cases which remain open, Normalized Table Algebras Generated by a Faithful Non- real Element of degree 3. As application we classified finite groups with a faithful non-real irreducible character of dimension 3 without using character theory of finite groups."}
{"category": "Math", "title": "Rank one Eisenstein cohomology of local systems on the moduli space of abelian varieties", "abstract": "We give a formula for the Eisenstein cohomology of local systems on the partial compactification of the moduli of principally polarized abelian varieties given by rank 1 degenerations. For genus 2 we give a formula for the full Eisenstein cohomology."}
{"category": "Math", "title": "Duke's Theorem and Continued Fractions", "abstract": "For uniformly chosen random $\\alpha \\in [0,1]$, it is known the probability the $n^{\\rm th}$ digit of the continued-fraction expansion, $[\\alpha]_n$ converges to the Gauss-Kuzmin distribution $\\mathbb{P}([\\alpha]_n = k) \\approx \\log_2 (1 + 1/ k(k+2))$ as $n \\to \\infty$. In this paper, we show the continued fraction digits of $\\sqrt{d}$, which are eventually periodic, also converge to the Gauss-Kuzmin distribution as $d \\to \\infty$ with bounded class number, $h(d)$. The proof uses properties of the geodesic flow in the unit tangent bundle of the modular surface, $T^1(\\text{SL}_2 \\mathbb{Z}\\backslash \\mathbb{H})$."}
{"category": "Math", "title": "Essentialities in additive bases", "abstract": "Let A be an asymptotic basis for N_0 of some order. By an essentiality of A one means a subset P such that A\\P is no longer an asymptotic basis of any order and such that P is minimal among all subsets of A with this property. A finite essentiality of A is called an essential subset. In a recent paper, Deschamps and Farhi asked the following two questions : (i) does every asymptotic basis of N_0 possess some essentiality ? (ii) is the number of essential subsets of size at most k of an asymptotic basis of order h bounded by a function of k and h only (they showed the number is always finite) ? We answer the latter question in the affirmative, and the former in the negative by means of an explicit construction, for every integer h >= 2, of an asymptotic basis of order h with no essentialities."}
{"category": "Math", "title": "Twisted conjugacy classes for polyfree groups", "abstract": "Let $G$ be a finitely generated polyfree group. If $G$ has nonzero Euler characteristic then we show that $Aut(G)$ has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain $G$ of length 2, we show that the number of Reidemeister classes of every automorphism is infinite."}
{"category": "Math", "title": "Power maps and subvarieties of the complex algebraic $n$--torus", "abstract": "Given a subvariety $V$ of the complex algebraic torus ${\\mathbb G}_{\\rm m}^n$ defined by polynomials of total degree at most $d$ and a power map $\\phi: {\\mathbb G}_{\\rm m}^n \\to {\\mathbb G}_{\\rm m}^n$, the points ${\\bf x}$ whose forward orbits ${\\mathcal O}_\\phi({\\bf x})$ belong to $V$ form its {\\em stable} subvariety $S(V,\\phi)$. The main result of the paper provides an upper bound $T=T(n,d,\\phi)$ for the number of iterations of the power map $\\phi$ required to ``cut off'' the points of $V$ that do not belong to $S$."}
{"category": "Math", "title": "Heat Content, Heat Trace, and Isospectrality", "abstract": "We study the heat content function, the heat trace function, and questions of isospectrality for the Laplacian with Dirichlet boundary conditions on a compact manifold with smooth boundary in the context of finite coverings and warped products."}
{"category": "Math", "title": "A remark on the boundedness and convergence properties of smooth sliding mode controllers", "abstract": "Conventional sliding mode controllers are based on the assumption of switching control but a well-known drawback of this approach is the chattering phenomenon. To overcome the undesirable chattering effects, the discontinuity in the control law can be smoothed out in a thin boundary layer neighboring the switching surface. In this work, rigorous proofs of the boundedness and convergence properties of smooth sliding mode controllers are presented. This result corrects flawed conclusions previously reached in the literature."}
{"category": "Math", "title": "Weyl groups, lattices and geometric manifolds", "abstract": "By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction of hyperbolic manifolds of very small volume in up to 8 dimensions."}
{"category": "Math", "title": "Lie group extensions associated to projective modules of continuous inverse algebras", "abstract": "We call a unital locally convex algebra $A$ a continuous inverse algebra if its unit group $A^\\times$ is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group $G$ on a continuous inverse algebra $A$ by automorphisms and any finitely generated projective right $A$-module $E$, we construct a Lie group extension $\\hat G$ of $G$ by the group $\\GL_A(E)$ of automorphisms of the $A$-module $E$. This Lie group extension is a ``non-commutative'' version of the group $\\Aut(\\V)$ of automorphism of a vector bundle over a compact manifold $M$, which arises for $G = \\Diff(M)$, $A = C^\\infty(M,\\C)$ and $E = \\Gamma\\V$. We also identify the Lie algebra $\\hat\\g$ of $\\hat G$ and explain how it is related to connections of the $A$-module $E$."}
{"category": "Math", "title": "Palindromic Saturation", "abstract": "We consider two {seemingly} different definitions of infinite words which contain {the} utmost number of palindromes. We show that these two definitions coincide. {The keynote of the proof is a meticulous inspection of properties of complete return words and the application of some basic graph theory.} In fact, we provide another proof of the result announced in \\cite{Zamboni}."}
{"category": "Math", "title": "Central extensions of the Ptolemy-Thompson group and quantized Teichmuller theory", "abstract": "The central extension of the Thompson group $T$ that arises in the quantized Teichm\\\"uller theory is 12 times the Euler class. This extension is obtained by taking a (partial) abelianization of the so-called braided Ptolemy-Thompson group introduced and studied in \\cite{FK2}. We describe then the cyclic central extensions of $T$ by means of explicit presentations."}
{"category": "Math", "title": "Spectral representation of some non stationary alpha-stable processes", "abstract": "In this paper, we give a new covariation spectral representation of some non stationary symmetric $\\alpha$-stable processes (S$\\alpha$S). This representation is based on a weaker covariation pseudo additivity condition which is more general than the condition of independence. This work can be seen as a generalization of the covariation spectral representation of processes expressed as stochastic integrals with respect to independent increments S$\\alpha$S processes (see Cambanis (1983)) or with respect to the general concept of independently scattered S$\\alpha$S measures (Samorodnitsky and Taqqu 1994). Relying on this result we investigate the non stationarity structure of some harmonisable S$\\alpha$S processes especially those having periodic or almost-periodic covariation functions."}
{"category": "Math", "title": "The mapping class group orbit of a multicurve", "abstract": "Given a set equipped with a transitive action of a group, we define the notion of an almost invariant coloring of the set. We consider the mapping class group orbit of a multicurve on a compact surface, and prove that in the case of genus at least two, no such almost invariant coloring exists. Conversely, in the case of a closed torus, one may find almost invariant colorings using arbitrarily many colors."}
{"category": "Math", "title": "The operad Lie is free", "abstract": "We show that the operad Lie is free as a non-symmetric operad. Then we study the generating series counting the operadic generators, finding a recursive formula for its coefficients, and showing that the asymptotic density of the operadic generators is 1/e."}
{"category": "Math", "title": "Thomae's formulae for non-hyperelliptic curves and spinorial square roots of theta-constants on the moduli space of curves", "abstract": "Determinantal formulae for Jacobian theta functions that go back to Klein are elaborated, via an idea due to Matone and Volpato. Also, the natural square roots of theta constants on the moduli space of curves whose existence was shown by Tsuyumine are proved to have a spinorial structure."}
{"category": "Math", "title": "A remark on the constructibility of real root representations of quivers using universal extension functors", "abstract": "In this paper we consider the following question: Is it possible to construct all real root representations of a given quiver Q by using universal extension functors, starting with a real Schur representation? We give a concrete example answering this question negatively."}
{"category": "Math", "title": "Complexity of planar and spherical curves", "abstract": "We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n+2 crossings."}
{"category": "Math", "title": "Equivalences entre conjectures de Soergel", "abstract": "Soergel's category B_k(V) over a field k is defined from a Coxeter system (W,S) and a k-linear representation V of W. It's a categorification of the Hecke algebra of (W,S). In this article we prove that for some representations V and V' of W, Soergel's conjecture over B_k(V') is equivalent to that over B_k(V). In particular, when k=IR we can choose V' to be the geometric representation."}
{"category": "Math", "title": "A conjectural presentation of fusion algebras", "abstract": "Let g be a semisimple Lie algebra over the complex numbers. Fix a positive integer l (called the level). Let R(l,g) be the fusion algebra at level l. Then, there is an algebra homomorphism from the representation ring R(g) of g to R(l,g). We study a presentation of its kernel. The generators for the kernel were given by Gepner, Gepner-Schwimmer, Bourdeau-Mlawer-Riggs-Schnitzer for g of type A and C series. We make a conjecture for other classical groups and also for g of type G2. We also have some partial results for F4 and E series."}
{"category": "Math", "title": "Nonlinear stability of stationary solutions for curvature flow with triple junction", "abstract": "In this paper we analyze the motion of a network of three planar curves with a speed proportional to the curvature of the arcs, having perpendicular intersections with the outer boundary and a common intersection at a triple junction. As a main result we show that a linear stability criterion due to Ikota and Yanagida is also sufficient for nonlinear stability. We also prove local and global existence of classical smooth solutions as well as various energy estimates. Finally, we prove exponential stabilization of an evolving network starting from the vicinity of a linearly stable stationary network."}
{"category": "Math", "title": "McShane's identity, using elliptic elements", "abstract": "We introduce a new method to establish McShane's Identity, based upon the fact that elliptic elements of order two in the Fuchsian group uniformizing the quotient of a fixed once-punctured hyperbolic torus act so as to exclude points as being highest points of geodesics."}
{"category": "Math", "title": "Commuting holonomies and rigidity of holomorphic foliations", "abstract": "In this article we study deformations of a holomorphic foliation with a generic non-rational first integral in the complex plane. We consider two vanishing cycles in a regular fiber of the first integral with a non-zero self intersection and with vanishing paths which intersect each other only at their start points. It is proved that if the deformed holonomies of such vanishing cycles commute then the deformed foliation has also a first integral. Our result generalizes a similar result of Ilyashenko on the rigidity of holomorphic foliations with a persistent center singularity. The main tools of the proof are Picard-Lefschetz theory and the theory of iterated integrals for such deformations."}
{"category": "Math", "title": "Estim\\'ees des noyaux de Green et de la chaleur sur les espaces sym\\'etriques", "abstract": "We obtain an off-diagonal upper bound for Green and heat kernel of Laplace type operator on symmetric spaces."}
{"category": "Math", "title": "Multiple Stratonovich integral and Hu--Meyer formula for L\\'{e}vy processes", "abstract": "In the framework of vector measures and the combinatorial approach to stochastic multiple integral introduced by Rota and Wallstrom [Ann. Probab. 25 (1997) 1257--1283], we present an It\\^{o} multiple integral and a Stratonovich multiple integral with respect to a L\\'{e}vy process with finite moments up to a convenient order. In such a framework, the Stratonovich multiple integral is an integral with respect to a product random measure whereas the It\\^{o} multiple integral corresponds to integrate with respect to a random measure that gives zero mass to the diagonal sets. A general Hu--Meyer formula that gives the relationship between both integrals is proved. As particular cases, the classical Hu--Meyer formulas for the Brownian motion and for the Poisson process are deduced. Furthermore, a pathwise interpretation for the multiple integrals with respect to a subordinator is given."}
{"category": "Math", "title": "Moduli of polarized Hodge structures", "abstract": "Around 1970 Griffiths introduced the moduli of polarized Hodge structures/the period domain $D$ and described a dream to enlarge $D$ to a moduli space of degenerating polarized Hodge structures. Since in general $D$ is not a Hermitian symmetric domain, he asked for the existence of a certain automorphic cohomology theory for $D$, generalizing the usual notion of automorphic forms on symmetric Hermitian domains. Since then there have been many efforts in the first part of Griffith's dream but the second part still lives in darkness. The objective of the present text is two-folded. First, we give an exposition of the subject. Second, we give another formulation of the Griffiths problem, based on the classical Weierstrass uniformization theorem."}
{"category": "Math", "title": "Penalized model-based clustering with cluster-specific diagonal covariance matrices and grouped variables", "abstract": "Clustering analysis is one of the most widely used statistical tools in many emerging areas such as microarray data analysis. For microarray and other high-dimensional data, the presence of many noise variables may mask underlying clustering structures. Hence removing noise variables via variable selection is necessary. For simultaneous variable selection and parameter estimation, existing penalized likelihood approaches in model-based clustering analysis all assume a common diagonal covariance matrix across clusters, which however may not hold in practice. To analyze high-dimensional data, particularly those with relatively low sample sizes, this article introduces a novel approach that shrinks the variances together with means, in a more general situation with cluster-specific (diagonal) covariance matrices. Furthermore, selection of grouped variables via inclusion or exclusion of a group of variables altogether is permitted by a specific form of penalty, which facilitates incorporating subject-matter knowledge, such as gene functions in clustering microarray samples for disease subtype discovery. For implementation, EM algorithms are derived for parameter estimation, in which the M-steps clearly demonstrate the effects of shrinkage and thresholding. Numerical examples, including an application to acute leukemia subtype discovery with microarray gene expression data, are provided to demonstrate the utility and advantage of the proposed method."}
{"category": "Math", "title": "$L^p$ Boundedness of Commutators of Riesz Transforms associated to Schr\\\"{o}dinger Operator", "abstract": "In this paper we consider $L^p$ boundedness of some commutators of Riesz transforms associated to Schr\\\"{o}dinger operator $P=-\\Delta+V(x)$ on $\\mathbb{R}^n, n\\geq 3$. We assume that $V(x)$ is non-zero, nonnegative, and belongs to $B_q$ for some $q \\geq n/2$. Let $T_1=(-\\Delta+V)^{-1}V,\\ T_2=(-\\Delta+V)^{-1/2}V^{1/2}$ and $T_3=(-\\Delta+V)^{-1/2}\\nabla$. We obtain that $[b,T_j] (j=1,2,3)$ are bounded operators on $L^p(\\mathbb{R}^n)$ when $p$ ranges in a interval, where $b \\in \\mathbf{BMO}(\\mathbb{R}^n)$. Note that the kernel of $T_j (j=1,2,3)$ has no smoothness."}
{"category": "Math", "title": "An explicit finite difference scheme for the Camassa-Holm equation", "abstract": "We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general $H^1$ initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in $H^1$ towards a dissipative weak solution of Camassa-Holm equation."}
{"category": "Math", "title": "Joyce invariants for K3 surfaces and mock theta functions", "abstract": "We will discuss Joyce invariants of stability conditions for K3 surfaces and mock theta functions."}
{"category": "Math", "title": "Convergence of weighted polynomial multiple ergodic averages", "abstract": "We study here weighted polynomial multiple ergodic averages. A sequence of weights is called universally good if any polynomial multiple ergodic average with this sequence of weights converges in $L^{2}$. We find a necessary condition and show that for any bounded measurable function $\\phi$ on an ergodic system, the sequence $\\phi(T^{n}x)$ is universally good for almost every $x$. The linear case was understood by Host and Kra."}
{"category": "Math", "title": "Testing the number of parameters with multidimensional MLP", "abstract": "This work concerns testing the number of parameters in one hidden layer multilayer perceptron (MLP). For this purpose we assume that we have identifiable models, up to a finite group of transformations on the weights, this is for example the case when the number of hidden units is know. In this framework, we show that we get a simple asymptotic distribution, if we use the logarithm of the determinant of the empirical error covariance matrix as cost function."}
{"category": "Math", "title": "Efficient Estimation of Multidimensional Regression Model with Multilayer Perceptron", "abstract": "This work concerns estimation of multidimensional nonlinear regression models using multilayer perceptron (MLP). The main problem with such model is that we have to know the covariance matrix of the noise to get optimal estimator. however we show that, if we choose as cost function the logarithm of the determinant of the empirical error covariance matrix, we get an asymptotically optimal estimator."}
{"category": "Math", "title": "Estimation of linear autoregressive models with Markov-switching, the E.M. algorithm revisited", "abstract": "This work concerns estimation of linear autoregressive models with Markov-switching using expectation maximisation (E.M.) algorithm. Our method generalise the method introduced by Elliot for general hidden Markov models and avoid to use backward recursion."}
{"category": "Math", "title": "The Virgin Island Model", "abstract": "A continuous mass population model with local competition is constructed where every emigrant colonizes an unpopulated island. The population founded by an emigrant is modeled as excursion from zero of an one-dimensional diffusion. With this excursion measure, we construct a process which we call Virgin Island Model. Furthermore, a necessary and sufficient condition for extinction of the total population is obtained for finite initial total mass."}
{"category": "Math", "title": "On the Linear Combinants of a Binary Pencil", "abstract": "Let A,B denote binary forms of order d, and let C_{2r-1} = (A,B)_{2r-1} be the sequence of their linear combinants for r between 1 and (d+1)/2. It is known that C_1 and C_3 together determine the pencil generated by A and B, and hence indirectly the higher C_{2r-1}. In this paper we exhibit explicit formulae for all r>2, which allow us to recover C_{2r-1} from the knowledge of C_1 and C_3. The calculations make use of the symbolic method of classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the second exterior power representation of S_d, for the group SL_2. We give an example for the group SL_3 to show that such a result may hold for other categories of representations."}
{"category": "Math", "title": "Self Organizing Map algorithm and distortion measure", "abstract": "We study the statistical meaning of the minimization of distortion measure and the relation between the equilibrium points of the SOM algorithm and the minima of distortion measure. If we assume that the observations and the map lie in an compact Euclidean space, we prove the strong consistency of the map which almost minimizes the empirical distortion. Moreover, after calculating the derivatives of the theoretical distortion measure, we show that the points minimizing this measure and the equilibria of the Kohonen map do not match in general. We illustrate, with a simple example, how this occurs."}
{"category": "Math", "title": "Efficient Estimation of Multidimensional Regression Model using Multilayer Perceptrons", "abstract": "This work concerns the estimation of multidimensional nonlinear regression models using multilayer perceptrons (MLPs). The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. However, we show in this paper that if we choose as the cost function the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable MLP. Numerical experiments confirm the theoretical results."}
{"category": "Math", "title": "A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable", "abstract": "We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally H\\\"older continuous with H\\\"older exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse H\\\"older inequality."}
{"category": "Math", "title": "Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction", "abstract": "We consider a random field $\\varphi:\\{1,...,N\\}\\to \\mathbb{R}$ with Laplacian interaction of the form $\\sum_iV(\\Delta\\varphi_i)$, where $\\Delta$ is the discrete Laplacian and the potential $V(\\cdot)$ is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward $\\varepsilon\\ge0$ each time it touches the x-axis, that plays the role of a defect line. It is known that this model exhibits a phase transition between a delocalized regime $(\\varepsilon<\\varepsilon_c)$ and a localized one $(\\varepsilon>\\varepsilon_c)$, where $0<\\varepsilon_c<\\infty$. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. We show, in particular, that in the delocalized regime the field wanders away from the defect line at a typical distance $N^{3/2}$, while in the localized regime the distance is just $O((\\log N)^2)$. A subtle scenario shows up in the critical regime $(\\varepsilon=\\varepsilon_c)$, where the field, suitably rescaled, converges in distribution toward the derivative of a symmetric stable L\\'evy process of index 2/5. Our approach is based on Markov renewal theory."}
{"category": "Math", "title": "Tridiagonal pairs of shape (1,2,1)", "abstract": "Let $\\mathbb F$ denote a field and let $V$ denote a vector space over $\\mathbb F$ with finite positive dimension. We consider a pair of linear transformations $A:V\\to V$ and $A^*:V\\to V$ that satisfies the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\\lbrace V_i \\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \\subseteq V_{i-1}+V_i+V_{i+1}$ for $0 \\leq i \\leq d$, where $V_{-1} = 0$ and $V_{d+1} = 0$; (iii) there exists an ordering $\\lbrace V^*_i \\rbrace_{i=0}^{\\delta}$ of the eigenspaces of $A^*$ such that $AV^*_i \\subseteq V^*_{i-1}+V^*_i+V^*_{i+1}$ for $0 \\leq i \\leq \\delta $, where $V^*_{-1} = 0$ and $V^*_{\\delta+1} = 0$; (iv) there is no subspace $W$ of $V$ such that $AW\\subseteq W$, $A^*W\\subseteq W$, $W \\neq 0, W \\neq V$. We call such a pair a {\\it tridiagonal pair} on $V$. It is known that $d = \\delta$ and that for $0 \\leq i \\leq d$ the dimensions of $V_i, V_{d-i}, V^*_i, V^*_{d-i}$ coincide; we denote this common value by $\\rho_i$. The sequence $\\lbrace \\rho_i\\rbrace_{i=0}^d$ is called the {\\it shape} of the pair. In this paper we assume the shape is $(1,2,1)$ and obtain the following results. We describe six bases for $V$; one diagonalizes $A$, another diagonalizes $A^*$, and the other four underlie the split decompositions for $A,A^*$. We give the action of $A$ and $A^*$ on each basis. For each ordered pair of bases among the six, we give the transition matrix. At the end we classify the tridiagonal pairs of shape $(1,2,1)$ in terms of a sequence of scalars called the parameter array."}
{"category": "Math", "title": "Decay estimates for a class of wave equations", "abstract": "In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by $e^{it\\phi(\\sqrt{-\\Delta})}$, where $\\phi: \\mathbb{R}^+\\to \\mathbb{R}$ is smooth away from the origin. Especially, the decay estimates for the solutions of the Klein-Gordon equation and the beam equation are simplified and slightly improved."}
{"category": "Math", "title": "Spectrum of the Lichnerowicz Laplacian on asymptotically hyperbolic surfaces", "abstract": "We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian $\\Delta_L$ contains the ray $[{1/4},+\\infty[$. If moreover the scalar curvature is constant then -2 and 0 are infinite dimensional eigenvalues. If, in addition, the inequality $<\\Delta u, u>_{L^2}\\geq \\frac14||u||^2_{L^2}$ holds for all smooth compactly supported function $u$, then there is no other value in the spectrum."}
{"category": "Math", "title": "Higher Extension Modules and the Yoneda Product", "abstract": "A chain of c submodules E =: E_0 >= E_1 >= ... >= E_c >= E_{c+1} := 0 gives rise to c composable 1-cocycles in Ext^1(E_{i-1}/E_i,E_i/E_{i+1}), i=1,...,c. In this paper we follow the converse question: When are c composable 1-cocycles induced by a module E together with a chain of submodules as above? We call such modules c-extension modules. The case c=1 is the classical correspondence between 1-extensions and 1-cocycles. For c=2 we prove an existence theorem stating that a 2-extension module exists for two composable 1-cocycles eta^M_L in Ext^1(M,L) and eta^L_N in Ext^1(L,N), if and only if their Yoneda product eta^M_L o eta^L_N in Ext^2(M,N) vanishes. We further prove a modelling theorem for c=2: In case the set of all such 2-extension modules is non-empty it is an affine space modelled over the abelian group that we call the first extension group of 1-cocycles, Ext^1(eta^M_L,eta^L_N) := Ext^1(M,N)/(Hom(M,L) o eta^L_N + eta^M_L o Hom(L,N))."}
{"category": "Math", "title": "$L^2$ extension of adjoint line bundle sections", "abstract": "We prove an $L^2$ extension theorem of Ohsawa-Takegoshi type for extending holomorphic sections of line bundles from a subvariety which is given as a maximal log-canonical center of a pair and is of general codimension in a projective variety. Our method of proof indicates that such a setting is the most natural one in a sense, for general $L^2$ extension of line bundle sections."}
{"category": "Math", "title": "Consistance d'un estimateur de minimum de variance \\'etendue", "abstract": "We consider a generalization of the criterion minimized by the K-means algorithm, where a neighborhood structure is used in the calculus of the variance. Such tool is used, for example with Kohonen maps, to measure the quality of the quantification preserving the neighborhood relationships. If we assume that the parameter vector is in a compact Euclidean space and all it components are separated by a minimal distance, we show the strong consistency of the set of parameters almost realizing the minimum of the empirical extended variance."}
{"category": "Math", "title": "Estimation consistante de l'architecture des perceptrons multicouches", "abstract": "We consider regression models involving multilayer perceptrons (MLP) with one hidden layer and a Gaussian noise. The estimation of the parameters of the MLP can be done by maximizing the likelihood of the model. In this framework, it is difficult to determine the true number of hidden units because the information matrix of Fisher is not invertible if this number is overestimated. However, if the parameters of the MLP are in a compact set, we prove that the minimization of a suitable information criteria leads to consistent estimation of the true number of hidden units."}
{"category": "Math", "title": "On Katz's bound for number of elements with given trace and norm", "abstract": "In this note an improvement of the Katz's bound on the number of elements in a finite field with given trace and norm is given. The improvement is obtained by reducing the problem to estimating the number of rational points on certain toric Calabi-Yau hypersurface, and then to use detailed cohomological calculations by Rojas-Leon and the second author for such toric hypersurfaces."}
{"category": "Math", "title": "A collection of sharp dilation invariant inequalities for differentiable functions", "abstract": "We find best constants in several dilation invariant integral inequalities involving derivatives of functions. Some of these inequalities are new and some were known without best constants. The contents: 1. Estimate for a quadratic form of the gradient, 2. Weighted G{\\aa}rding inequality for the biharmonic operator, 3.Dilation invariant Hardy's inequality with remainder term, 4. Generalized Hardy-Sobolev inequality with sharp constant, 5. Hardy's inequality with sharp Sobolev remainder term."}
{"category": "Math", "title": "Laplacians on the basilica Julia set", "abstract": "We consider the basilica Julia set of the polynomial $P(z)=z^{2}-1$ and construct all possible resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the effective resistance metric coincides with the usual topology. Then we concentrate on two particular cases. One is a self-similar harmonic structure, for which the energy renormalization factor is 2, the spectral dimension is $\\log9/\\log6$, and we can compute all the eigenvalues and eigenfunctions by a spectral decimation method. The other is graph-directed self-similar under the map $z\\mapsto P(z)$; it has energy renormalization factor $\\sqrt2$ and spectral dimension 4/3, but the exact computation of the spectrum is difficult. The latter Dirichlet form and Laplacian are in a sense conformally invariant on the basilica Julia set."}
{"category": "Math", "title": "Relationship between stochastic flows and connection forms", "abstract": "In this article I will prove new representation for the Levi-Civita connection in terms of the stochastic flow corresponding to Brownian motion on manifold."}
{"category": "Math", "title": "Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties", "abstract": "We investigate Siegel modular varieties in positive characteristic with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz-Rapoport stratification; one would like to understand how these stratifications are related to each other. We give a simple description of all KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne-Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties."}
{"category": "Math", "title": "Plurisubharmonic functions with weak singularities", "abstract": "We study the complex Monge-Amp\\`ere operator in bounded hyperconvex domains of $\\C^n$. We introduce a scale of classes of weakly singular plurisubharmonic functions : these are functions of finite weighted Monge-Amp\\`ere energy. They generalize the classes introduced by U.Cegrell, and give a stratification of the space of (almost) all unbounded plurisubharmonic functions. We give an interpretation of these classes in terms of the speed of decreasing of the Monge-Amp\\`ere capacity of sublevel sets and solve associated complex Monge-Amp\\`ere equations."}
{"category": "Math", "title": "Relative radial mass and rigidity of some warped product manifolds", "abstract": "We give a Riccati type formula adapted for two metrics having the same geodesics rays starting from a point or orthogonal to an hypersurface, one of these metrics being a warped product if the dimension $n$ is greater than or equal to 3. This formula has non-trivial geometric consequences such as a positive mass type theorem and other rigidity results. We also apply our result to some standard models."}
{"category": "Math", "title": "The Supermagic Square in characteristic 3 and Jordan superalgebras", "abstract": "Recently, the classical Freudenthal Magic Square has been extended over fields of characteristic 3 with two more rows and columns filled with (mostly simple) Lie superalgebras specific of this characteristic. This Supermagic Square will be reviewed and some of the simple Lie superalgebras that appear will be shown to be isomorphic to the Tits-Kantor-Koecher Lie superalgebras of some Jordan superalgebras."}
{"category": "Math", "title": "Adaptive Confidence Sets for the Optimal Approximating Model", "abstract": "In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive point estimation, the construction of adaptive confidence regions is severely limited (cf. Li, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence sets for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral chi-squared distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one."}
{"category": "Math", "title": "Automorphic properties of generating functions for generalized rank moments and Durfee symbols", "abstract": "We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews' smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta functions."}
{"category": "Math", "title": "Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation", "abstract": "We show that for almost all points on any analytic curve on R^{k} which is not contained in a proper affine subspace, the Dirichlet's theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in R^{k+1}. The proof involves ergodic properties of unipotent flows on homogeneous spaces."}
{"category": "Math", "title": "Iterated Grafting and Holonomy Lifts of Teichmueller space", "abstract": "Let $X$ be a closed hyperbolic surface and $\\lambda, \\eta$ be weighted geodesic multicurves which are short on X. We show that the iterated grafting along $\\lambda$ and $\\eta$ is close in the Teichmueller metric to grafting along a single multicurve which can be given explicitly in terms of $\\lambda$ and $\\eta$. Using this result, we study the holonomy lifts $gr_{\\lambda}\\rho_{X,\\lambda}$ of Teichmueller geodesics $\\rho_{X,\\lambda}$ for integral laminations $\\lambda$ and show that all of them have bounded Teichmueller distance to the geodesic $\\rho_{X,\\lambda}$. We obtain analogous results for grafting rays. Finally we consider the asymptotic behaviour of iterated grafting sequences $\\gr_{n\\lambda}X$ and show that they converge geometrically to a punctured surface."}
{"category": "Math", "title": "A model of continuous time polymer on the lattice", "abstract": "In this article, we try to give a rather complete picture of the behavior of the free energy for a model of directed polymer in a random environment, in which the polymer is a simple symmetric random walk on the lattice $\\Z^d$, and the environment is a collection $\\{W(t,x);t\\ge 0, x\\in \\Z^d\\}$ of i.i.d. Brownian motions."}
{"category": "Math", "title": "De-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds", "abstract": "In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute cohomologies of the Lie algebra of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case is known as Shapiro lemma."}
{"category": "Math", "title": "Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case $H=1/4$", "abstract": "We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion $B$ with Hurst index $H=1/4$. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C. A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to $B$."}
{"category": "Math", "title": "Mathematical study of resonant wind-driven oceanic motions", "abstract": "We are interested here in describing the linear response of the ocean to some wind forcing, which admits fast time oscillations and may be resonant with the Coriolis force. In addition to the usual Ekman layer, we exhibit another - much larger - boundary layer, and some global vertical profile. That means in particular that the wind effect is no longer localized in the vicinity of the surface. From a mathematical point of view, the main novelty here is to introduce some systematic approach for the study of boundary effects."}
{"category": "Math", "title": "Finsler conformal Lichnerowicz-Obata conjecture", "abstract": "We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture showing that a complete and essential conformal vector field on a non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski metric."}
{"category": "Math", "title": "A characterization of quadric constant mean curvature hypersurfaces of spheres", "abstract": "Let $\\phi:M\\to\\mathbb{S}^{n+1}\\subset\\mathbb{R}^{n+2}$ be an immersion of a complete $n$-dimensional oriented manifold. For any $v\\in\\mathbb{R}^{n+2}$, let us denote by $\\ell_v:M\\to\\mathbb{R}$ the function given by $\\ell_v(x)=\\phi(x),v$ and by $f_v:M\\to\\mathbb{R}$, the function given by $f_v(x)=\\nu(x),v$, where $\\nu:M\\to\\mathbb{S}^{n}$ is a Gauss map. We will prove that if $M$ has constant mean curvature, and, for some $v\\ne{\\bf 0}$ and some real number $\\lambda$, we have that $\\ell_v=\\lambda f_v$, then, $\\phi(M)$ is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface $M^n$ in $\\mathbb{S}^{n+1}$ which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to $2n+4$."}
{"category": "Math", "title": "Bifurcations, Schwarzian derivatives and Feigenbaum constant revisited", "abstract": "The main purpose is to show that Feigenbaum delta constant is much more universal than believed. The paper is mainly devoted to period-doubling processes in families in one parameter of endomorphisms of the interval and consider generalizations of the Feigenbaum delta constant. We formulate the so-called parenthesis permeability hypothesis, a conjecture that holds for all types of bifurcation (i.e. for flip, fold, pitchfork and transcritical bifurcations, which states that under some conditions two or three different functions may have exactly the same bifurcation points. We propose a conjecture that considerably relaxes David Singer conditions for endomorphism families to generate at most one stable orbit, showing that Feigenbaum constant appears also in some classes of functions that have more than one maximum and have positive Schwarzian in at least one sub-interval. This version contains more arguments in favor of an even greater Feiganbaum constant universality."}
{"category": "Math", "title": "The subelliptic heat kernel on SU(2): Representations, Asymptotics and Gradient bounds", "abstract": "The Lie group SU(2) endowed with its canonical subriemannian structure appears as a three-dimensional model of a positively curved subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities."}
{"category": "Math", "title": "Consistent estimation of the architecture of multilayer perceptrons", "abstract": "We consider regression models involving multilayer perceptrons (MLP) with one hidden layer and a Gaussian noise. The estimation of the parameters of the MLP can be done by maximizing the likelihood of the model. In this framework, it is difficult to determine the true number of hidden units using an information criterion, like the Bayesian information criteria (BIC), because the information matrix of Fisher is not invertible if the number of hidden units is overestimated. Indeed, the classical theoretical justification of information criteria relies entirely on the invertibility of this matrix. However, using recent methodology introduced to deal with models with a loss of identifiability, we prove that suitable information criterion leads to consistent estimation of the true number of hidden units."}
{"category": "Math", "title": "Generalized manifolds in products of curves", "abstract": "The intent of this article is to study some special $n$-dimensional continua lying in products of $n$ curves. (The paper is an improved version of a portion of \\cite{K-K-S}.) We show that if $X$ is a locally connected, so-called, quasi $n$-manifold lying in a product of $n$ curves then rank of $H^1(X)\\ge n$. Moreover, if $\\rank H^1(X)<2n$ then $X$ can be represented as a product of an $m$-torus and a quasi $(n-m)$-manifold, where $m\\ge2n-\\rank H^1(X)$. It follows that certain 2-dimensional contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron can be embedded in a product of two trees. We answer a question of R. Cauty proving that closed surfaces embeddable in products of two curves can be also embedded in products of two graphs. On the other hand, we construct an example of a 2-dimensional polyhedron which can be embedded in a product of two curves though it is not embeddable in any product of two graphes. This solves in the negative another problem of Cauty."}
{"category": "Math", "title": "A change of variable formula with It\\^{o} correction term", "abstract": "We consider the solution $u(x,t)$ to a stochastic heat equation. For fixed $x$, the process $F(t)=u(x,t)$ has a nontrivial quartic variation. It follows that $F$ is not a semimartingale, so a stochastic integral with respect to $F$ cannot be defined in the classical It\\^{o} sense. We show that for sufficiently differentiable functions $g(x,t)$, a stochastic integral $\\int g(F(t),t)\\,dF(t)$ exists as a limit of discrete, midpoint-style Riemann sums, where the limit is taken in distribution in the Skorokhod space of cadlag functions. Moreover, we show that this integral satisfies a change of variable formula with a correction term that is an ordinary It\\^{o} integral with respect to a Brownian motion that is independent of $F$."}
{"category": "Math", "title": "An abstract setting for hamiltonian actions", "abstract": "In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain $\\omega$ on a Lie algebra $h$ with values in an $h$-module $V$, we associate subalgebras $sp(h,\\omega) \\supeq ham(h,\\omega)$ of symplectic, resp., hamiltonian elements. Then $ham(h,\\omega)$ has a natural central extension which in turn is contained in a larger abelian extension of $sp(h,\\omega)$. In this setting, we study linear actions of a Lie group $G$ on $V$ which are compatible with a homomorphism $g \\to ham(h,\\omega)$, i.e. abstract hamiltonian actions, corresponding central and abelian extensions of $G$ and momentum maps $J : g \\to V$."}
{"category": "Math", "title": "A remark on primality testing and decimal expansions", "abstract": "We show that for any fixed base $a$, a positive proportion of primes have the property that they become composite after altering any one of their digits in the base $a$ expansion; the case $a=2$ was already established by Cohen-Selfridge and Sun, using some covering congruence ideas of Erd\\H{o}s. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base $a$ expansion without reading all of its digits. We also present some slight generalisations of these results."}
{"category": "Math", "title": "Evaluation and selection of models for out-of-sample prediction when the sample size is small relative to the complexity of the data-generating process", "abstract": "In regression with random design, we study the problem of selecting a model that performs well for out-of-sample prediction. We do not assume that any of the candidate models under consideration are correct. Our analysis is based on explicit finite-sample results. Our main findings differ from those of other analyses that are based on traditional large-sample limit approximations because we consider a situation where the sample size is small relative to the complexity of the data-generating process, in the sense that the number of parameters in a `good' model is of the same order as sample size. Also, we allow for the case where the number of candidate models is (much) larger than sample size."}
{"category": "Math", "title": "Elliptic surfaces without 1-handles", "abstract": "Harer-Kas-Kirby conjectured that every handle decomposition of the elliptic surface E(1)_{2,3} requires both 1- and 3-handles. We prove that the elliptic surface E(n)_{p,q} has a handle decomposition without 1-handles for $n\\geq 1$ and (p,q)=(2,3),(2,5),(3,4),(4,5)."}
{"category": "Math", "title": "Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals", "abstract": "Let (P,<) be a finite poset and let G be its comparability graph. If cl(G) is the clutter of maximal cliques of G, we prove that cl(G) satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. We prove that edge ideals of complete admissible uniform clutters are normally torsion free. The normality of a monomial ideal is expressed in terms of blocking polyhedra and the integer decomposition property. For edge ideals of clutters this property completely determine their normality"}
{"category": "Math", "title": "On secant varieties of Compact Hermitian Symmetric Spaces", "abstract": "We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three - with one exception, the secant variety of the $21$-dimensional spinor variety in $\\pp{63}$ where we show the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces."}
{"category": "Math", "title": "Goldman flows on the Jacobian", "abstract": "We show that the Goldman flows preserve the holomorphic structure on the moduli space of homomorphisms of the fundamental group of a Riemann surface into U(1), in other words the Jacobian."}
{"category": "Math", "title": "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations", "abstract": "This survey article begins with a discussion of the original form of the Strominger-Yau-Zaslow conjecture, surveys the state of knowledge concering this conjecture, and explains how thinking about this conjecture naturally leads to the program initiated by the author and Bernd Siebert to study mirror symmetry via degenerations of Calabi-Yau manifolds and log structures."}
{"category": "Math", "title": "Moment problems and boundaries of number triangles", "abstract": "The boundary problem for graphs like Pascal's but with general multiplicities of edges is related to a `backward' problem of moments of the Hausdorff type."}
{"category": "Math", "title": "Structure of T modules and restricted duals: the classical and the quantum case", "abstract": "A concrete realization of Enright's $T$ modules is obtained. This is used to show their self-duality. As a consequence, the restricted duals of Verma modules are also identified."}
{"category": "Math", "title": "On multiplicity problems for finite-dimensional representations of hyper loop algebras", "abstract": "Given a hyper loop algebra over a non-algebraically closed field, we address multiplicity problems in the underlying abelian tensor category of finite-dimensional representations. Namely, we give formulas for the l-characters of the simple objects, the Jordan-Holder multiplicities of the Weyl modules, and the Clebsch-Gordan coefficients."}
{"category": "Math", "title": "The sutured Floer homology polytope", "abstract": "In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decomposition, then a natural map projects P(M',g') onto a face of P(M,g); moreover, if H_2(M) = 0, then every face of P(M,g) can be obtained in this way for some surface decomposition. We show that if (M,g) is reduced, horizontally prime, and H_2(M) = 0, then P(M,g) is maximal dimensional in H^2(M,\\partial M; R). This implies that if rk(SFH(M,g)) < 2^{k+1} then (M,g) has depth at most 2k. Moreover, SFH acts as a complexity for balanced sutured manifolds. In particular, the rank of the top term of knot Floer homology bounds the topological complexity of the knot complement, in addition to simply detecting fibred knots."}
{"category": "Math", "title": "A New Proof for Classification of Irreducible Modules of a Hecke Algebra of Type $ A_n$", "abstract": "In this paper we give a new proof for the classification of irreducible modules of an affine Hecke algebra of type $A_n$, which was obtained by G. E. Murphy in 1995."}
{"category": "Math", "title": "An operator approach to multipoint Pade approximations", "abstract": "First, an abstract scheme of constructing biorthogonal rational systems related to some interpolation problems is proposed. We also present a modification of the famous step-by-step process of solving the Nevanlinna-Pick problems for Nevanlinna functions. The process in question gives rise to three-term recurrence relations with coefficients depending on the spectral parameter. These relations can be rewritten in the matrix form by means of two Jacobi matrices. As a result, a convergence theorem for multipoint Pad\\'e approximants to Nevanlinna functions is proved."}
{"category": "Math", "title": "On Khintchine exponents and Lyapunov exponents of continued fractions", "abstract": "Assume that $x\\in [0,1) $ admits its continued fraction expansion $x=[a_1(x), a_2(x),...]$. The Khintchine exponent $\\gamma(x)$ of $x$ is defined by $\\gamma(x):=\\lim\\limits_{n\\to \\infty}\\frac{1}{n}\\sum_{j=1}^n \\log a_j(x)$ when the limit exists. Khintchine spectrum $\\dim E_\\xi$ is fully studied, where $ E_{\\xi}:=\\{x\\in [0,1):\\gamma(x)=\\xi\\} (\\xi \\geq 0)$ and $\\dim$ denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum $\\dim E_{\\xi}$, as function of $\\xi \\in [0, +\\infty)$, is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by $\\gamma^{\\phi}(x):=\\lim\\limits_{n\\to\\infty}\\frac{1}{\\phi(n)} \\sum_{j=1}^n \\log a_j(x)$ are also studied, where $\\phi (n)$ tends to the infinity faster than $n$ does. Under some regular conditions on $\\phi$, it is proved that the fast Khintchine spectrum $\\dim (\\{x\\in [0,1]: \\gamma^{\\phi}(x)= \\xi \\}) $ is a constant function. Our method also works for other spectra like the Lyapunov spectrum and the fast Lyapunov spectrum."}
{"category": "Math", "title": "Generic points in systems of specification and Banach valued Birkhoff ergodic average", "abstract": "We prove that systems satisfying the specification property are saturated in the sense that the topological entropy of the set of generic points of any invariant measure is equal to the measure-theoretic entropy of the measure. We study Banach valued Birkhoff ergodic averages and obtain a variational principle for its topological entropy spectrum. As application, we examine a particular example concerning with the set of real numbers for which the frequencies of occurrences in their dyadic expansions of infinitely many words are prescribed. This relies on our explicit determination of a maximal entropy measure."}
{"category": "Math", "title": "Quotients of fake projective planes", "abstract": "Recently, Prasad and Yeung classified all possible fundamental groups of fake projective planes. According to their result, many fake projective planes admit a nontrivial group of automorphisms, and in that case it is isomorphic to $\\bbZ/3\\bbZ$, $\\bbZ/7\\bbZ$, $7:3$, or $(\\bbZ/3\\bbZ)^2$, where $7:3$ is the unique non-abelian group of order 21. Let $G$ be a group of automorphisms of a fake projective plane $X$. In this paper we classify all possible structures of the quotient surface $X/G$ and its minimal resolution."}
{"category": "Math", "title": "On a nonhierarchical version of the Generalized Random Energy Model. II. Ultrametricity", "abstract": "We study the Gibbs measure of the nonhierarchical versions of the Generalized Random Energy Models introduced in previous work. We prove that the ultrametricity holds only provided some nondegeneracy conditions on the hamiltonian are met."}
{"category": "Math", "title": "On peak phenomena for non-commutative $H^\\infty$", "abstract": "A non-commutative extension of Amar and Lederer's peak set result is given. As its simple applications it is shown that any non-commutative $H^\\infty$-algebra $H^\\infty(M,\\tau)$ has unique predual,and moreover some restriction in some of the results of Blecher and Labuschagne are removed, making them hold in full generality."}
{"category": "Math", "title": "H\\\"ormander type pseudodifferential calculus on homogeneous groups", "abstract": "We produce, on general homogeneous groups, an analogue of the usual H\\\"ormander pseudodifferential calculus on Euclidean space, at least as far as products and adjoints are concerned. In contrast to earlier works, we do not limit ourselves to analogues of classical symbols, nor to the Heisenberg group. The key technique is to understand ``multipliers'' of any given order j, and the operators of convolution with their inverse Fourier transforms, which we here call convolution operators of order j. (Here a ``multiplier'' is an analogue of a H\\\"ormander-type symbol a(x,\\xi), which is independent of x.) Specifically, we characterize the space of inverse Fourier transforms of multipliers of any order j, and use this characterization to show that the composition of convolution operators of order j_1 and j_2 is a convolution operator of order j_1+j_2."}
{"category": "Math", "title": "A Truncation Approach for Fast Computation of Distribution Functions", "abstract": "In this paper, we propose a general approach for improving the efficiency of computing distribution functions. The idea is to truncate the domain of summation or integration."}
{"category": "Math", "title": "Confidence Interval for the Mean of a Bounded Random Variable and Its Applications in Point Estimation", "abstract": "In this article, we derive an explicit formula for computing confidence interval for the mean of a bounded random variable. Moreover, we have developed multistage point estimation methods for estimating the mean value with prescribed precision and confidence level based on the proposed confidence interval."}
{"category": "Math", "title": "Estimating Traffic Parameters with Rigorous Error Control", "abstract": "To perform a queuing analysis or design in a communications context, we need to estimate the values of the input parameters, specifically the mean of the arrival rate and service time. In this paper, we propose an approach for estimating the arrival rate of Poisson processes and the average service time for servers under the assumption that the service time is exponential. In particular, we derive sample size (i.e., the number of i.i.d. observations) required to obtain an estimate satisfying a pre-specified relative accuracy with a given confidence level. A remarkable feature of this approach is that no a priori information about the parameter is needed. In contrast to conventional methods such as, standard error estimation and confidence interval construction, which only provides post-experimental evaluations of the estimate, this approach allows experimenters to rigorously control the error of estimation."}
{"category": "Math", "title": "Infinite class field towers", "abstract": "This paper studies infinite class field towers of number fields $K$ that are ramified over $\\Q$ only at one finite prime. In particular, we show the existence of such towers for a general family of primes including $p=2$, 3 and 5."}
{"category": "Math", "title": "Asymptotic Normality of the Additive Regression Components for Continuous Time Processes", "abstract": "In multivariate regression estimation, the rate of convergence depends on the dimension of the regressor. This fact, known as the curse of the dimensionality, motivated several works. The additive model, introduced by Stone (10), offers an efficient response to this problem. In the setting of continuous time processes, using the marginal integration method, we obtain the quadratic convergence rate and the asymptotic normality of the components of the additive model."}
{"category": "Math", "title": "Reconstruction of Random Colourings", "abstract": "Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random $k$-colourings on the $\\Delta$-ary tree for large $k$. Bhatnagar et. al. showed non-reconstruction when $\\Delta \\leq \\frac12 k\\log k - o(k\\log k)$ and reconstruction when $\\Delta \\geq k\\log k + o(k\\log k)$. We tighten this result and show non-reconstruction when $\\Delta \\leq k[\\log k + \\log \\log k + 1 - \\ln 2 -o(1)]$ and reconstruction when $\\Delta \\geq k[\\log k + \\log \\log k + 1+o(1)]$."}
{"category": "Math", "title": "Classification of Quiver Hopf Algebras and Pointed Hopf Algebras of Type One", "abstract": "The quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one."}
{"category": "Math", "title": "Efficient Counting and Asymptotics of $k$-noncrossing tangled-diagrams", "abstract": "In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $\\le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Our main result is the asymptotic formula for the number of $k$-noncrossing tangled-diagrams $T_{k}(n) \\sim c_k n^{-((k-1)^2+(k-1)/2)} (4(k-1)^2+2(k-1)+1)^n$ for some $c_k>0$."}
{"category": "Math", "title": "Mapping properties of fundamental operators in harmonic analysis related to Bessel operators", "abstract": "We prove sharp power-weighted strong type, weak type and restricted weak type inequalities for the heat and Poisson integral maximal operators, Riesz transform and a Littlewood-Paley type square function, emerging naturally in the harmonic analysis related to Bessel operators."}
{"category": "Math", "title": "Quasi-nearly subharmonicity and separately quasi-nearly subharmonic functions", "abstract": "Wiegerinck has shown that a separately subharmonic function need not be subharmonic. Improving previous results of Lelong, of Avanissian, of Arsove and of us, Armitage and Gardiner gave an almost sharp integrability condition which ensures a separately subharmonic function to be subharmonic. Completing now our recent counterparts to the cited results of Lelong, Avanissian and Arsove for so called quasi-nearly subharmonic functions, we present a counterpart to the cited result of Armitage and Gardiner for separately quasi-nearly subharmonic functions. This counterpart enables us to slightly improve Armitage's and Gardiner's original result, too. The method we use is a rather straightforward and technical, but still by no means easy, modification of Armitage's and Gardiner's argument combined with an old argument of Domar."}
{"category": "Math", "title": "Feynman diagrams and minimal models for operadic algebras", "abstract": "We construct an explicit minimal model for an algebra over the cobar-construction of a differential graded operad. The structure maps of this minimal model are expressed in terms of sums over decorated trees. We introduce the appropriate notion of a homotopy equivalence of operadic algebras and show that our minimal model is homotopy equivalent to the original algebra. All this generalizes and gives a conceptual explanation of well-known results for A-infinity algebras. Further, we show that these results carry over to the case of algebras over modular operads; the sums over trees get replaced by sums over general Feynman graphs. As a by-product of our work we prove gauge-independence of Kontsevich's `dual construction' producing graph cohomology classes from contractible differential graded Frobenius algebras."}
{"category": "Math", "title": "Coherent presentations of structure monoids and the Higman-Thompson groups", "abstract": "Structure monoids and groups are algebraic invariants of equational varieties. We show how to construct presentations of these objects from coherent categorifications of equational varieties, generalising several results of Dehornoy. We subsequently realise the higher Thompson groups $F_{n,1}$ and the Higman-Thompson groups $G_{n,1}$ as structure groups. We go on to obtain presentations of these groups via coherent categorifications of the varieties of higher-order associativity and of higher-order associativity and commutativity, respectively. These categorifications generalise Mac Lane's pentagon and hexagon conditions for coherently associative and commutative bifunctors."}
{"category": "Math", "title": "Maximum vertex occupation time and inert fugitive: recontamination does help", "abstract": "Given a simple graph $G$, we consider the node search problem with inert fugitive. We are interested in minimizing the maximum vertex occupation time, i.e. the maximum number of steps in which a vertex is occupied by a searcher during a search of $G$. We prove that a search program which does not allow a recontamination may not find an optimal solution to this problem, and the difference between the maximum vertex occupation time computed by a monotone search program and a program without such restriction may be arbitrarily large."}
{"category": "Math", "title": "On the locality of the Pr\\\"ufer code", "abstract": "The Pr\\\"ufer code is a bijection between trees on the vertex set $[n]$ and strings on the set $[n]$ of length $n-2$ (Pr\\\"ufer strings of order $n$). In this paper we examine the `locality' properties of the Pr\\\"ufer code, i.e. the effect of changing an element of the Pr\\\"ufer string on the structure of the corresponding tree. Our measure for the distance between two trees $T,T^*$ is $\\Delta(T,T^*)=n-1-| E(T)\\cap E(T^*)|$. We randomly mutate the $\\mu$th element of the Pr\\\"ufer string of the tree $T$, changing it to the tree $T^*$, and we asymptotically estimate the probability that this results in a change of $\\ell$ edges, i.e. $P(\\Delta=\\ell | \\mu).$ We find that P(\\Delta=\\ell | \\mu)$ is on the order of $ n^{-1/3+o(1)}$ for any integer $\\ell>1,$ and that $P(\\Delta=1 | \\mu)=(1-\\mu/n)^2+o(1).$ This result implies that the probability of a `perfect' mutation in the Pr\\\"ufer code (one for which $\\Delta(T,T^*)=1$) is $1/3.$"}
{"category": "Math", "title": "Moufang symmetry II. Moufang-Mal'tsev pairs and triality", "abstract": "A concept of the Moufang-Malt'tsev pair is elaborated. This concept is based on the generalized Maurer-Cartan equations of a local analytic Moufang loop. Triality can be seen as a fundamental property of such pairs. Based on triality, the Yamagutian is constructed. Properties of the Yamagutian are studied."}
{"category": "Math", "title": "The time constant vanishes only on the percolation cone in directed first passage percolation", "abstract": "We consider the directed first passage percolation model on ${\\bf Z}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. We denote by $\\vec{T}({\\bf 0}, (r,\\theta))$ the passage time from the origin to $(r, \\theta)$ by a northeast path for $(r, \\theta)\\in {\\bf R}^+\\times [0,\\pi/2]$. It is known that $\\vec{T}({\\bf 0}, (r, \\theta))/r$ converges to a time constant $\\vec{\\mu}_F (\\theta)$. Let $\\vec{p}_c$ denote the critical probability for oriented percolation. In this paper, we show that the time constant has a phase transition divided by $\\vec{p}_c$, as follows: (1) If $F(0) < \\vec{p}_c$, then $\\vec{\\mu}_F(\\theta) >0$ for all $0\\leq \\theta\\leq \\pi/2$. (2) If $F(0) = \\vec{p}_c$, then $\\vec{\\mu}_F(\\theta) >0$ if and only if $\\theta\\neq \\pi/4$. (3) If $F(0)=p > \\vec{p}_c$, then there exists a percolation cone between $\\theta_p^-$ and $\\theta_p^+$ for $0\\leq \\theta^-_p< \\theta^+_p \\leq \\pi/2$ such that $\\vec{\\mu} (\\theta) >0$ if and only if $\\theta\\not\\in [\\theta_p^-, \\theta^+_p]$. Furthermore, all the moments of $\\vec{T}({\\bf 0}, (r, \\theta))$ converge whenever $\\theta\\in [\\theta_p^-, \\theta^+_p]$. As applications, we describe the shape of the directed growth model on the distribution of $F$. We give a phase transition for the shape divided by $\\vec{p}_c$."}
{"category": "Math", "title": "Complex interpolation of compact operators mapping into lattice couples", "abstract": "Suppose that (A_0,A_1) and (B_0,B_1) are Banach couples, and that T is a linear operator which maps A_0 compactly into B_0 and A_1 boundedly (or even compactly) into B_1. Does this imply that T maps [A_0,A_1]_s to [B_0,B_1]_s compactly for 0<s<1 ? (Here, as usual, [A_0,A_1]_s denotes the complex interpolation space of Alberto Calderon.) This question has been open for 44 years. Affirmative answers are known for it in many special cases. We answer it affirmatively in the case where (A_0,A_1) is arbitrary and (B_0,B_1) is a couple of Banach lattices having absolutely continuous norms or the Fatou property. Our result has some overlap with a recent result by Evgeniy Pustylnik."}
{"category": "Math", "title": "On linear versions of some addition theorems", "abstract": "Let K \\subset L be a field extension. Given K-subspaces A,B of L, we study the subspace spanned by the product set AB = {ab | a \\in A, b \\in B}. We obtain some lower bounds on the dimension of this subspace and on dim B^n in terms of dim A, dim B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson."}
{"category": "Math", "title": "On contracting hyperplane elements from a 3-connected matroid", "abstract": "Let $\\tilde{K}_{3,n}$, $n\\geq 3$, be the simple graph obtained from $K_{3,n}$ by adding three edges to a vertex part of size three. We prove that if $H$ is a hyperplane of a 3-connected matroid $M$ and $M \\not\\cong M^*(\\tilde{K}_{3,n})$, then there is an element $x$ in $H$ such that the simple matroid associated with $M/x$ is 3-connected."}
{"category": "Math", "title": "A Remark on Triangle-Critical Graphs", "abstract": "A connected $k$-chromatic graph $G$ with $k \\geq 3$ is said to be triangle-critical, if every edge of $G$ is contained in an induced triangle of $G$ and the removal of any triangle from $G$ decreases the chromatic number of $G$ by three. B. Toft posed the problem of showing that the complete graphs on more than two vertices are the only triangle-critical graphs. By applying a method of M. Stiebitz [Discrete Math. 64 (1987), 91--93], we answer the problem affirmatively for triangle-critical $k$-chromatic graphs with $k \\leq 6$."}
{"category": "Math", "title": "Asymptotically Conical Associative 3-folds", "abstract": "Given an associative 3-fold in R^7 which is asymptotically conical with generic rate less than 1, we show that its moduli space of deformations is locally homeomorphic to the kernel of a smooth map between smooth manifolds. Moreover, the virtual dimension of the moduli space is computed and shown to be non-negative for rates greater than -1, whereas the associative 3-fold is expected to be isolated for rates less than or equal to -1."}
{"category": "Math", "title": "Interval Estimation of Bounded Variable Means via Inverse Sampling", "abstract": "In this paper, we develop interval estimation methods for means of bounded random variables based on a sequential procedure such that the sampling is continued until the sample sum is no less than a prescribed threshold."}
{"category": "Math", "title": "Algorithmic and combinatorial methods for enumerating the relators of a group presentation", "abstract": "The main achievement of this thesis is an algorithm which given a finite group presentation and natural numbers n and k, computes all the relators of length and area up to n and k respectively. The complexity of this algorithm is better by a factor which is over-exponential than that of classical methods using van Kampen diagrams."}
{"category": "Math", "title": "On $l^2$ norms of some weighted mean matrices", "abstract": "We give another proof of a result of Bennett on the $l^{p}$ operator norms of some weighted mean matrices for the case $p=2$ and we also present some related results."}
{"category": "Math", "title": "Almost Periodic Szeg\\H{o} Cocycles with Uniformly Positive Lyapunov Exponents", "abstract": "We exhibit examples of almost periodic Verblunsky coefficients for which Herman's subharmonicity argument applies and yields that the associated Lyapunov exponents are uniformly bounded away from zero."}
{"category": "Math", "title": "Type II$_1$ von Neumann algebra representations of Hecke operators on Maass forms and the Ramanujan-Petersson conjectures", "abstract": "Classical Hecke operators on Maass forms are unitarely equivalent, up to a commuting phase, to completely positive maps on II$_1$ factors, associated to a pair of isomorphic subfactors, and an intertwining unitary. This representation is obtained through a quantized representation of the Hecke operators. in this representation, the Hecke operators act on the Berezin's quantization, deformation algebra of the fundamental domain of $\\PSL(2,\\Z)$ in the upper halfplane. The Hecke operators are inheriting from the ambient, non-commutative algebra on which they act, a rich structure of matrix inequalities. Using this construction we obtain that, for every prime $p$, the essential spectrum of the classical Hecke operator $T_p$ is contained in the interval $[-2\\sqrt p, 2\\sqrt p]$, predicted by the Ramanujan Petersson conjectures. In particular, given an open interval containing $[-2\\sqrt p, 2\\sqrt p]$, there are at most a finite number of possible exceptional eigenvalues lying outside this interval. The main tool for obtaining this representation of the Hecke operators (unitarely equivalent to the classical representation, up to commuting phase) is a Schurr type, positive \"square root\" of the state on $\\PGL(2,\\Q)$, measuring the displacement of fundamental domain by translations in $\\PGL(2,\\Q)$. The \"square root\" is obtained from the matrix coefficients of the discrete series representations of $\\PSL(2,\\R)$ restricted to $\\PGL(2, \\Q)$. The methods in this paper may also be applied to any finite index, modular subgroup $\\Gamma_0(p^n)$, $n\\geq 1$, of $\\PSL(2,\\Z)$. In this case the essential norm of the Hecke operator is equal to the norm of the corresponding convolution operator on the cosets Hilbert space $\\ell^2((\\Gamma_0(p^n))\\backslash \\PSL(2,\\Z[1/p])$."}
{"category": "Math", "title": "Directional isoperimetric inequalities and rational homotopy invariants", "abstract": "We estimate the second order linking invariants of Lipschitz maps from an n-dimensional ellipse. The estimate uses a new directionally-dependent version of the isoperimetric inequality for cycles inside the ellipse. Using this work, we prove new lower bounds for the k-dilation of maps from one ellipse to another."}
{"category": "Math", "title": "Isoperimetric inequalities and rational homotopy invariants", "abstract": "We estimate the linear isoperimetric constants of an n-dimensional ellipse. Using these estimates and a technique of Gromov, we estimate the Hopf and linking invariants of Lipschitz maps from ellipses to round spheres. Using these estimates, we give a lower bound for the k-dilation of degree non-zero maps between ellipses."}
{"category": "Math", "title": "A natural extension for the greedy beta-transformation with three deleted digits", "abstract": "We give an explicit expression for the invariant measure, absolutely continuous with respect to the Lebesgue measure, of the greedy beta-transformation with three deleted digits. We define a version of the natural extension of the transformation to obtain this expression. We get that the transformation is exact and weakly Bernoulli."}
{"category": "Math", "title": "Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations", "abstract": "In this work we continue the study of the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random perturbations, by treating the case of multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects."}
{"category": "Math", "title": "A Systematic Study of Frame Sequence Operators and their Pseudoinverses", "abstract": "In this note we investigate the operators associated with frame sequences in a Hilbert space $H$, i.e., the synthesis operator $T:\\ell ^{2}(\\mathbb{N}) \\to H$, the analysis operator $T^{\\ast}:H\\to $ $% \\ell ^{2}(\\mathbb{N}) $ and the associated frame operator $S=TT^{\\ast}$ as operators defined on (or to) the whole space rather than on subspaces. Furthermore, the projection $P$ onto the range of $T$, the projection $Q$ onto the range of $T^{\\ast}$ and the Gram matrix $G=T^{\\ast}T$ are investigated. For all these operators, we investigate their pseudoinverses, how they interact with each other, as well as possible classification of frame sequences with them. For a tight frame sequence, we show that some of these operators are connected in a simple way."}
{"category": "Math", "title": "Moufang symmetry III. Integrability of generalized Lie equations", "abstract": "Integrability of generalized Lie equations of a local analytic Moufang loop is inquired."}
{"category": "Math", "title": "Strong solutions for stochastic porous media equations with jumps", "abstract": "We prove global well-posedness in the strong sense for stochastic generalized porous media equations driven by square integrable martingales with stationary independent increments."}
{"category": "Math", "title": "A Schwartz type algebra for the Tangent Groupoid", "abstract": "We construct an algebra of smooth functions over the tangent groupoid associated to any Lie groupoid. This algebra is a field of algebras over the closed interval [0, 1] which fiber at zero is the algebra of Schwartz functions over the Lie algebroid, whereas any fiber out of zero is the convolution algebra of the initial groupoid. Our motivation comes from index theory for Lie groupoids. In fact, our construction gives an intermediate algebra between the enveloping C-algebra and the convolution algebra of compactly supported functions of the tangent groupoid; and it will allows us, in a further work, to define other analytic index morphisms as a sort of deformations."}
{"category": "Math", "title": "Modeling and Optimal Control of Networks of Pipes and Canals", "abstract": "This paper deals with the optimal control of systems governed by nonlinear systems of conservation laws at junctions. The applications considered range from gas compressors in pipelines to open channels management. The existence of an optimal control is proved. From the analytical point of view, these results are based on the well posedness of a suitable initial boundary value problem and on techniques for quasidifferential equations in a metric space."}
{"category": "Math", "title": "Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs", "abstract": "The paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n-dimensional ODEs. We discuss the implementation and the performance of the algorithm in several examples, including an extended Lorenz-84 model and a laser system."}
{"category": "Math", "title": "A combinatorial approach to surgery formulas in Heegaard Floer homology", "abstract": "Using the combinatorial approach to Heegaard Floer homology we obtain a relatively easy formula for computation of hat Heegaard Floer homology for the three-manifold obtained by rational surgery on a knot K inside a homology sphere Y."}
{"category": "Math", "title": "Moufang symmetry IV. Reductivity and hidden associativity", "abstract": "It is shown how integrability of the generalized Lie equations of a local analytic Moufang loop is related to the reductivity conditions and Sagle-Yamaguti identity."}
{"category": "Math", "title": "Veraverbeke's theorem at large - On the maximum of some processes with negative drift and heavy tail innovations", "abstract": "Veraverbeke's (1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke's theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process, conditionally on a large maximum. Those results are obtained by using a general scheme of proof which we present in some detail and should be of value in other related problems."}
{"category": "Math", "title": "Strongly-Representable Operators", "abstract": "Recently in [1] a new class of maximal monotone operators has been introduced. In this note we study domain range properties as well as connections with other classes and calculus rules for these operators we called strongly-representable. While not every maximal monotone operator is strongly-representable, every maximal monotone NI operator is strongly-representable, and every strongly representable operator is locally maximal monotone, maximal monotone locally, and ANA. As a consequence the conjugate of the Fitzpatrick function of a maximal monotone operator is not necessarily a representative function."}
{"category": "Math", "title": "Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold", "abstract": "Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c_1=0 which are never Kahler; e.g., we produce such manifolds with b_1=0=b_3 and also with c_2.omega <0, answering questions posed by Smith-Thomas-Yau. Examples come from Riemannian geometry, via the Levi-Civita connection on Lambda^+. The underlying six-manifold is then the twistor space and often the symplectic structure tames the Eells-Salamon twistor almost complex structure. Our second aim is to exploit this to deduce new results about minimal surfaces: if a certain curvature inequality holds, it follows that the space of minimal surfaces (with fixed topological invariants) is compactifiable; the minimal surfaces must also satisfy an adjunction inequality, unifying and generalising results of Chen--Tian. One metric satisfying the curvature inequality is hyperbolic four-space H^4. Our final aim is to show that the corresponding symplectic manifold is symplectomorphic to the small resolution of the conifold xw-yz=0 in C^4. We explain how this fits into a hyperbolic description of the conifold transition, with isometries of H^4 acting symplectomorphically on the resolution and isometries of H^3 acting biholomorphically on the smoothing."}
{"category": "Math", "title": "Presque $\\mathbf{C}_p$-repr\\'esentations et $(\\phi,\\Gamma)$-modules", "abstract": "We associate two almost $C_p$-representations to a $(\\phi,\\Gamma)$-module, and we compute their dimensions and heights. As a corollary, we get a full faithfulness result for $B_e$-representations."}
{"category": "Math", "title": "Cohomology of diagrams of algebras", "abstract": "We consider cohomology of diagrams of algebras by Beck's approach, using comonads. We then apply this theory to computing the cohomology of $\\Psi$-rings. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of an algebra to the cohomology of the underlying algebra."}
{"category": "Math", "title": "Poincare duality complexes in dimension four", "abstract": "We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of Poincare duality complexes of dimension 4. Generalizing Turaev's fundamental triples of Poincare duality complexes of dimension 3, we introduce fundamental triples for Poincare duality complexes of dimension n > 2 and show that two Poincare duality complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjecture of Turaev and obtain a criterion for the existence of degree 1 maps between n-dimensional manifolds."}
{"category": "Math", "title": "A Note on Walking Versus Waiting", "abstract": "This mathematical recreation extends the analysis of a recent paper, asking when a traveller at a bus stop and not knowing the time of the next bus is best advised to wait or to start walking toward the destination. A detailed analysis and solution is provided for a very general class of probability distributions of bus arrival time, and the solution characterised in terms of a function analogous to the hazard rate in reliability theory. The note also considers the question of intermediate stops. It is found that the optimal strategy is not always the laziest, even when headways are not excessively long. For the common special case where one knows the (uniform) headway but not the exact timetable, it is shown that one should wait if the headway is less than the walking time (less bus travel time), and walk if the headway is more than twice this much. In between it may be better to wait or to walk, depending on one's confidence in being able to catch up to a passing bus."}
{"category": "Math", "title": "Random walk on a discrete torus and random interlacements", "abstract": "We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps, u > 0, and the model of random interlacements recently introduced by Sznitman. In particular, we show that for large N, the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to time u N^d converges to independent copies of the random interlacement at level u."}
{"category": "Math", "title": "Curves over higher local fields", "abstract": "In this work, we prove the vanishing of the two cohomological group of the higher local field. This generalize the well-known propriety of finite field and one dimensional local field. We apply this result to study the arithmetic of curve defined over higher local field."}
{"category": "Math", "title": "Extension of C*-bundles", "abstract": "Different (fibrewise) amalgamated products of continuous C*-bundles have been studied over the last years, one of the main question being to know when these amalgamated products are continuous C*-bundles. In order to gather these approaches in a joint framework, we first recall a few definitions from the theory of deformations of C*-algebras and we fix several notations that will be used in the sequel. Then we characterise the continuity properties of different amalgamated products of (continuous) C(X)-algebras."}
{"category": "Math", "title": "Coarse embeddability into Banach spaces", "abstract": "The main purposes of this paper are (1) To survey the area of coarse embeddability of metric spaces into Banach spaces, and, in particular, coarse embeddability of different Banach spaces into each other; (2) To present new results on the problems: (a) Whether coarse non-embeddability into $\\ell_2$ implies presence of expander-like structures? (b) To what extent $\\ell_2$ is the most difficult space to embed into?"}
{"category": "Math", "title": "A cascade of determinantal Calabi--Yau threefolds", "abstract": "We study Kustin--Miller unprojections of Calabi--Yau threefolds. As an application we work out the geometric properties of Calabi--Yau threefolds defined as linear sections of determinantal varieties. We compute their Hodge numbers and describe the morphisms corresponding to the faces of their K\\\"{a}hler--Mori cone."}
{"category": "Math", "title": "A Cyclic Operad in the Category of Artin Stacks and Gravitational Correlators", "abstract": "We define an Artin stack which may be considered as a substitute for the non-existing (or empty) moduli space of stable two-pointed curves of genus zero. We show that this Artin stack can be viewed as the first term of a cyclic operad in the category of stacks. Applying the homology functor we obtain a linear cyclic operad. We formulate conjectures which assert that cohomology of a smooth projective variety has the structure of an algebra over this homology operad and that gravitational quantum cohomology can naturally expressed in terms of this algebra. As a test for these conjectures we show how certain well-known relations between gravitational correlators can be deduced from them."}
{"category": "Math", "title": "Cohomological support loci for Abel-Prym curves", "abstract": "For an Abel-Prym curve contained in a Prym variety, we determine the cohomological support loci of its twisted ideal sheaves and the dimension of its theta-dual."}
{"category": "Math", "title": "On variance stabilisation by double Rao-Blackwellisation", "abstract": "Population Monte Carlo has been introduced as a sequential importance sampling technique to overcome poor fit of the importance function. In this paper, we compare the performances of the original Population Monte Carlo algorithm with a modified version that eliminates the influence of the transition particle via a double Rao-Blackwellisation. This modification is shown to improve the exploration of the modes through an large simulation experiment on posterior distributions of mean mixtures of distributions."}
{"category": "Math", "title": "A Characterization of Jacobians by the Existence of Picard Bundles", "abstract": "Based on the Matsusaka-Ran criterion we give a criterion to characterize when a principal polarized abelian variety is a Jacobian by the existence of Picard bundles."}
{"category": "Math", "title": "A Torelli theorem for curves over finite fields", "abstract": "We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry."}
{"category": "Math", "title": "Loop groups and string topology", "abstract": "Survey article on loop groups and their representations, following a course of three lectures held at the summer school \"algebraic groups\" at the Georg-August-Universitaet zu Goettingen, June 27--July 13, 2005. We discuss loop groups, their central extensions, and positive energy representations."}
{"category": "Math", "title": "Some arithmetic properties of matroidal ideals", "abstract": "In this paper, we study various properties of matroidal ideals."}
{"category": "Math", "title": "Heterodimensional tangencies on cycles leading to strange attractors", "abstract": "In this paper, we study heterodimensional cycles of two-parameter families of 3-dimensional diffeomorphisms some element of which contains nondegenerate heterodimensional tangencies of the stable and unstable manifolds of two saddle points with different indexes, and prove that such diffeomorphisms can be well approximated by another element which has a quadratic homoclinic tangency associated to one of these saddle points. Moreover, it is shown that the tangency unfolds generically with respect to the family. This result together with some theorem in Viana, we detect strange attractors appeared arbitrarily close to the original element with the heterodimensional cycle."}
{"category": "Math", "title": "Lower bound theorem for normal pseudomanifolds", "abstract": "In this paper we present a self-contained combinatorial proof of the lower bound theorem for normal pseudomanifolds, including a treatment of the cases of equality in this theorem. We also discuss McMullen and Walkup's generalised lower bound conjecture for triangulated spheres in the context of the lower bound theorem. Finally, we pose a new lower bound conjecture for non-simply connected triangulated manifolds."}
{"category": "Math", "title": "Fiber products of elliptic surfaces with section and associated Kummer fibrations", "abstract": "We investigate Calabi--Yau three folds which are small resolutions of fiber products of elliptic surfaces with section admitting reduced fibers. We start by the classification of all fibers that can appear on such varieties. Then, we find formulas to compute the Hodge numbers of obtained three folds in terms of the types of singular fibers of the elliptic surfaces. Next we study Kummer fibrations associated to these fiber products."}
{"category": "Math", "title": "Asymptotically optimal quantization schemes for Gaussian processes", "abstract": "We describe quantization designs which lead to asymptotically and order optimal functional quantizers. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions. Furthermore we derive a high-resolution formula for the $L^2$-quantization errors of Riemann-Liouville processes."}
{"category": "Math", "title": "Correspondences between modular Calabi--Yau fiber products", "abstract": "We describe two ways to construct finite rational morphisms between fiber products of rational elliptic surfaces with section and some Calabi--Yau manifolds. We use them to construct correspondences between such fiber products that admit at most five singular fibers and rigid Calabi--Yau threefolds."}
{"category": "Math", "title": "An algorithmic implementation of the Pi function based on a new sieve", "abstract": "In this paper we propose an algorithm that correctly discards a set of numbers (from a previously defined sieve) with an interval of integers. Leopoldo's Theorem states that the remaining integer numbers will generate and count the complete list of primes of absolute value greater than 3 in the interval of interest. This algorithm avoids the problem of generating large lists of numbers, and can be used to compute (even in parallel) the prime counting function $\\pi(h)$."}
{"category": "Math", "title": "The 2/3 - convergence rate for the Poisson bracket", "abstract": "In this paper we introduce a new method for approaching the C^0 - rigidity results for the Poisson bracket. Using this method, we provide a different proof for the lower semi-continuity under C^0 perturbations, for the uniform norm of the Poisson bracket. We find the precise rate for the modulus of the semi-continuity. This extends the previous results of Cardin-Viterbo, Zapolsky, Entov and Polterovich. Using our method, we prove a C^0 - rigidity result in the spirit of the work of Humiliere. We also discuss a general question of the C^0 - rigidity for multilinear differential operators."}
{"category": "Math", "title": "Connectedness in graph limits", "abstract": "We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is well-known, graph limits may be represented by symmetric functions on a probability space; there are natural definitions of direct sums and connectedness for such functions, and there is a perfect correspondence with the corresponding properties of the graph limit. Similarly, every graph limit determines an infinite random graph, which is a.s. connected if and only if the graph limit is connected. There are also characterizations in terms of the asymptotic size of the largest component in the corresponding finite random graphs, and of minimal cuts in sequences of graphs converging to a given limit."}
{"category": "Math", "title": "Moufang symmetry XI. Integrability of generalized Lie equations of continuous Moufang transformations", "abstract": "Integrability of generalized Lie equations of continuous Moufang transformations is inquired."}
{"category": "Math", "title": "Moufang symmetry XII. Reductivity and hidden associativity of infinitesimal Moufang transformations", "abstract": "It is shown how integrability of the generalized Lie equations of continous Moufang transformatiosn is related to the reductivity conditions and Sagle-Yamaguti identity."}
{"category": "Math", "title": "On the flat remainder in normal forms of families of analytic planar saddles", "abstract": "We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter."}
{"category": "Math", "title": "Representability of Hilbert schemes and Hilbert stacks of points", "abstract": "We show that the Hilbert functor of points on an arbitrary separated algebraic stack is an algebraic space. We also show the algebraicity of the Hilbert stack of points on an algebraic stack and the algebraicity of the Weil restriction of an algebraic stack along a finite flat morphism. For the latter two results, no separation assumptions are necessary."}
{"category": "Math", "title": "A Weak Chevalley-Warning Theorem for Quasi-finite Fields", "abstract": "There exists a function f: N -> N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension at least f(d), the set X(K) is non-empty. This is a special case of a more general result about intersections of hypersurfaces of fixed degree in projective spaces of sufficiently high dimension over fields with finitely generated Galois groups."}
{"category": "Math", "title": "Clones from ideals", "abstract": "On an infinite base set X, every ideal of subsets of X can be associated with the clone of those operations on X which map small sets to small sets. We continue earlier investigations on the position of such clones in the clone lattice."}
{"category": "Math", "title": "Pro-p groups of positive deficiency", "abstract": "Let \\Gamma be a finitely presentable pro-p group with a nontrivial finitely generated closed normal subgroup N of infinite index. Then def(\\Gamma)\\leq 1, and if def(\\Gamma)=1 then \\Gamma is a pro-p duality group of dimension 2, N is a free pro-p group and \\Gamma/N is virtually free. In particular, if the centre of \\Gamma is nontrivial and def(\\Gamma)\\geq 1, then def(\\Gamma)=1, cd G \\leq 2 and \\Gamma is virtually a direct product F \\times Z_p, with F a finitely generated free pro-p group."}
{"category": "Math", "title": "Searching for Strange Hypergeometric Identities By Sheer Brute Force", "abstract": "We describe a systematic search for all strange hypergeometric identities up to a certain complexity with sheer brute force that lead us to the discovery of two new infinite families of closed-form evaluations."}
{"category": "Math", "title": "Khovanov homology and tight contact structures", "abstract": "Using the relation between Khovanov homology and the Heegaard Floer homology of branched double covers, we show how Khovanov homology can be used to establish tightness of branched double covers of certain transverse knots. We give examples of several infinite families of knots whose branched covers are tight for Khovanov-homological reasons, and show that some of these branched covers are not Stein fillable."}
{"category": "Math", "title": "The universal Hopf algebra associated with a Hopf-Lie-Rinehart algebra", "abstract": "We introduce a notion of Hopf-Lie-Rinehart algebra and show that the universal algebra of a Hopf-Lie-Rinehart algebra acquires an ordinary Hopf algebra structure."}
{"category": "Math", "title": "Quadratic equations over free groups are NP-complete", "abstract": "We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete."}
{"category": "Math", "title": "A note on three dimensional good sets", "abstract": "We show that as in the case of n- fold Cartesian product for n greater than or equal to 4, even in 3-fold Cartesian product, a related component of a good set need not be a full component."}
{"category": "Math", "title": "Automatic Transversality and Orbifolds of Punctured Holomorphic Curves in Dimension Four", "abstract": "We derive a numerical criterion for J-holomorphic curves in 4-dimensional symplectic cobordisms to achieve transversality without any genericity assumption. This generalizes results of Hofer-Lizan-Sikorav and Ivashkovich-Shevchishin to allow punctured curves with boundary that generally need not be somewhere injective or immersed. As an application, we combine this with the intersection theory of punctured holomorphic curves to prove that certain geometrically natural moduli spaces are globally smooth orbifolds, consisting generically of embedded curves, plus unbranched multiple covers that form isolated orbifold singularities."}
{"category": "Math", "title": "Computational class field theory", "abstract": "Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions."}
{"category": "Math", "title": "A homotopical algebra of graphs related to zeta series", "abstract": "The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equivalences for this model structure are the Acyclics (graph morphisms which preserve cycles). The cofibrations and fibrations for the model are determined from the class of Whiskerings (graph morphisms produced by grafting trees). Our model structure seems to fit well with the importance of acyclic directed graphs in many applications. In addition to the weak factorization systems which form this model structure, we also describe two Freyd-Kelly factorization systems based on Folding, Injecting, and Covering graph morphisms."}
{"category": "Math", "title": "The open mapping theorem for regular quaternionic functions", "abstract": "The basic results of a new theory of regular functions of a quaternionic variable have been recently stated, following an idea of Cullen. In this paper we prove the minimum modulus principle and the open mapping theorem for regular functions. The proofs involve some peculiar geometric properties of such functions which are of independent interest."}
{"category": "Math", "title": "Generalization of One of Lie's Theorems", "abstract": "We prove a generalization of one of Lie's Theorems in the context of Lie-like algebras$^{2-nd}$."}
{"category": "Math", "title": "Gradient NLW on curved background in 4+1 dimensions", "abstract": "We obtain a sharp local well-posedness result for the Gradient Nonlinear Wave Equation on a nonsmooth curved background. In the process we introduce variable coefficient versions of Bourgain's $X^{s,b}$ spaces, and use a trilinear multiscale wave packet decomposition in order to prove a key trilinear estimate."}
{"category": "Math", "title": "On the endomorphisms of Weyl modules over affine Kac-Moody algebras at the critical level", "abstract": "We present an independent short proof of the main result of arXiv:0706.3725 that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. We derive this from the results of arXiv:0712.1183 about the shift of argument subalgebras."}
{"category": "Math", "title": "On low rank perturbation of matrices", "abstract": "The article is devoted to different aspects of the question \"What can be done with a matrix by low rank perturbation?\" It is proved that one can change a geometrically simple spectrum drastically by a rank 1 permutation, but the situation is quite different if one restricts oneself to normal matrices. Also, the Jordan normal form of a perturbed matrix is considered. It is proved that with respect to rank as a distance all almost unitary matrices are near unitary."}
{"category": "Math", "title": "Polygones de Newton de certaines sommes de caract\\`eres et s\\'eries de Poincar\\'e", "abstract": "In this paper, we shall precise the asymptotic behaviour of Newton polygons of $L$ functions associated to character sums, coming from some $n$ variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum $\\Delta=\\Delta_1\\oplus \\Delta_2$ when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behaviour of Newton polygons for multivariable polynomials."}
{"category": "Math", "title": "PGA Tour Scores as a Gaussian Random Variable", "abstract": "In this paper it is demonstrated that the scoring at each PGA Tour stroke play event can be reasonably modeled as a Gaussian random variable. All 46 stroke play events in the 2007 season are analyzed. The distributions of scores are favorably compared with a Gaussian distribution using the Kolmogorov-Smirnov test. This observation suggests performance tracking on the PGA tour should be done in terms of the z-score, calculated by subtracting the mean from the raw score and dividing by the standard deviation. This methodology measures performance relative to the field of competitors, independent of the venue, and in terms of a statistic that has quantitative meaning. Several examples of the use of this scoring methodology are provided, including a calculation of the probability that Tiger Woods will break Byron Nelson's record of eleven consecutive PGA Tour victories."}
{"category": "Math", "title": "The commutants of certain Toeplitz operators on weighted Bergman spaces", "abstract": "For $\\alpha>-1$, let $A^2_{\\alpha}$ be the corresponding weighted Bergman space of the unit ball in $\\mathbb{C}^n$. For a bounded measurable function $f$, let $T_f$ be the Toeplitz operator with symbol $f$ on $A^2_{\\alpha}$. This paper describes all the functions $f$ for which $T_f$ commutes with a given $T_g$, where $g(z)=z_{1}^{L_1}... z_{n}^{L_n}$ for strictly positive integers $L_1,..., L_n$, or $g(z)=|z_1|^{s_1}... |z_n|^{s_n}h(|z|)$ for non-negative real numbers $s_1,..., s_n$ and a bounded measurable function $h$ on $[0,1)$."}
{"category": "Math", "title": "Constrained Consensus", "abstract": "We present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time-varying connectivity. Our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem, where the global objective function is a combination of local agent objective functions. Our main focus is on constrained problems where the estimate of each agent is restricted to lie in a different constraint set. To highlight the effects of constraints, we first consider a constrained consensus problem and present a distributed ``projected consensus algorithm'' in which agents combine their local averaging operation with projection on their individual constraint sets. This algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents. We establish convergence and convergence rate results for the projected consensus algorithm. We next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets. We present a distributed ``projected subgradient algorithm'' which involves each agent performing a local averaging operation, taking a subgradient step to minimize its own objective function, and projecting on its constraint set. We show that, with an appropriately selected stepsize rule, the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal, and when the weights are time-varying but all agents have the same constraint set."}
{"category": "Math", "title": "A refined Luecking's theorem and finite-rank products of Toeplitz operators", "abstract": "For any function $f$ in $L^{\\infty}(\\mathbb{D})$, let $T_f$ denote the corresponding Toeplitz operator the Bergman space $A^2(\\mathbb{D})$. A recent result of D. Luecking shows that if $T_f$ has finite rank then $f$ must be the zero function. Using a refined version of this result, we show that if all except possibly one of the functions $f_1,..., f_{m}$ are radial and $T_{f_1}... T_{f_m}$ has finite rank, then one of these functions must be zero."}
{"category": "Math", "title": "Growth Gap vs. smoothness for diffeomorphisms of the interval", "abstract": "Given a diffeomorphism of the interval, consider the uniform norm of the derivative of its n-th iteration. We get a sequence of real numbers called the growth sequence. Its asymptotic behavior is an invariant which naturally appears both in smooth dynamics and in geometry of the diffeomorphisms groups. We find sharp estimates for the growth sequence of a given diffeomorphism in terms of the modulus of continuity of its derivative. These estimates extend previous results of Polterovich and Sodin, and Borichev."}
{"category": "Math", "title": "Dynamical systems gradient method for solving ill-conditioned linear algebraic systems", "abstract": "A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\\it a priori} and {\\it a posteriori} stopping rules are justified. An algorithm for computing the solution using a spectral decomposition of the left-hand side matrix is proposed. Numerical results show that when a spectral decompositon of the left-hand side matrix is available or not computationally expensive to obtain the new method can be considered as an alternative to the Variational Regularization."}
{"category": "Math", "title": "Stochastic Tamed 3D Navier-Stokes Equations: Existence, Uniqueness and Ergodicity", "abstract": "In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly \\cite{Ha-Ma}, we prove the uniqueness of invariant measures for the corresponding transition semigroup."}
{"category": "Math", "title": "Localization theorems in topological Hochschild homology and topological cyclic homology", "abstract": "We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a global construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason-Trobaugh in K-theory. We also deduce versions of Thomason's blow-up formula and the projective bundle formula for THH and TC."}
{"category": "Math", "title": "Strichartz estimates on Schwarzschild black hole backgrounds", "abstract": "We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of earlier work of Metcalfe-Tataru, in order to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere."}
{"category": "Math", "title": "A Link between Binomial Parameters and Means of Bounded Random Variables", "abstract": "In this paper, we establish a fundamental connection between binomial parameters and means of bounded random variables. Such connection finds applications in statistical inference of means of bounded variables."}
{"category": "Math", "title": "The smallest singular value of a random rectangular matrix", "abstract": "We prove an optimal estimate on the smallest singular value of a random subgaussian matrix, valid for all fixed dimensions. For an N by n matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order \\sqrt{N} - \\sqrt{n-1} with high probability. A sharp estimate on the probability is also obtained."}
{"category": "Math", "title": "A generalization of adjoint crystals for the quantized affine algebras of type $A\\sb{n}\\sp{(1)}$, $C\\sb{n}\\sp{(1)}$ and $D\\sb{n+1}\\sp{(2)}$", "abstract": "We propose to generalize Benkart-Frenkel-Kang-Lee's adjoint crystals and describe their crystal structure for type $A\\sb{n}\\sp{(1)}$, $C\\sb{n}\\sp{(1)}$ and $D\\sb{n+1}\\sp{(2)}$."}
{"category": "Math", "title": "A 24-h forecast of ozone peaks and exceedance levels using neural classifiers and weather predictions", "abstract": "A neural network combined to a neural classifier is used in a real time forecasting of hourly maximum ozone in the centre of France, in an urban atmosphere. This neural model is based on the MultiLayer Perceptron (MLP) structure. The inputs of the statistical network are model output statistics of the weather predictions from the French National Weather Service. With this neural classifier, the Success Index of forecasting is 78% whereas it is from 65% to 72% with the classical MLPs. During the validation phase, in the Summer of 2003, six ozone peaks above the threshold were detected. They actually were seven."}
{"category": "Math", "title": "The Levi Problem On Strongly Pseudoconvex $G$-Bundles", "abstract": "Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ the total space of a principal bundle $G\\to M\\to X$ so that $M$ is also a strongly pseudoconvex complex manifold. In this work, we show that if $G$ acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on $M$ is infinite $G$-dimensional."}
{"category": "Math", "title": "Frobenius polynomials for Calabi-Yau equations", "abstract": "We describe a variation of Dworks unit-root method to determine the degree four Frobenius polynomial for members of a 1-modulus Calabi-Yau family over $\\mathbb{P}^1$ in terms of the holomorphic period near a point of maximal unipotent monodromy. The method is illustrated on a couple of examples."}
{"category": "Math", "title": "The elliptic Hall algebra, Cherednick Hecke algebras and Macdonald polynomials", "abstract": "We show that the Hall algebra of the category of coherent sheaves on an elliptic curve (or, equivalently, the algebra of unramified automorphic forms for GL(n) for all n) is equal to the stable limit of spherical double affine Hecke algebras for GL(k) as k goes to infinity. The two parameters correspond to the size of the finite field and the modulus of the elliptc curve. Under this isomorphism the Hecke operators are mapped to the Macdonald operators. This allows us to give a geometric construction of Macdonald polynomials (eigenvectors for the Macdonald operator) in terms of a suitable specialization of Eisenstein series (eigenvectors for the Hecke operators)."}
{"category": "Math", "title": "Discretization of transfer operators using a sparse hierarchical tensor basis - the Sparse Ulam method", "abstract": "The global macroscopic behaviour of a dynamical system is encoded in the eigenfunctions of a certain transfer operator associated to it. For systems with low dimensional long term dynamics, efficient techniques exist for a numerical approximation of the most important eigenfunctions, cf. DeJu99a. They are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor - Ulam's method. In this paper we develop a numerical technique which makes Ulam's approach applicable to systems with higher dimensional long term dynamics. It is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids. We develop the technique, establish statements about its complexity and convergence and present two numerical examples."}
{"category": "Math", "title": "Moufang symmetry V. Triple closure", "abstract": "Triple closure of the infinitesimal translations of an analytic Moufang loop is inquired. This property is equivalent to reductivity and relates Mal'tsev algebras to the Lie triple systems."}
{"category": "Math", "title": "Ekedahl-Oort strata in the supersingular locus", "abstract": "We give a description of the individual Ekedahl-Oort strata contained in the supersingular locus in terms of Deligne-Lusztig varieties, refining a result of Harashita."}
{"category": "Math", "title": "Rational points in periodic analytic sets and the Manin-Mumford conjecture", "abstract": "We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analytic variety supposed invariant for translations by a full lattice, which is a topic with some independent motivation."}
{"category": "Math", "title": "Jacobians among Abelian threefolds: a formula of Klein and a question of Serre", "abstract": "Let k be a field and f be a Siegel modular form of weight h \\geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is the Jacobian of a smooth plane curve, we show how to associate to f a classical plane invariant. As straightforward consequences of these constructions, when g=3 and k is a subfield of the complex field, we obtain (i) a new proof of a formula of Klein linking the modular form \\chi_{18} to the square of the discriminant of plane quartics ; (ii) a proof that one can decide when (A,a) is a Jacobian over k by looking whether the value of \\chi_{18} at (A,a) is a square in k. This answers a question of J.-P. Serre. Finally, we study the possible generalizations of this approach for g>3."}
{"category": "Math", "title": "Elliptic curves with all quadratic twists of positive rank", "abstract": "We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour, and characterise them in terms of the Galois action on the Tate module. In particular, their existence shows that Goldfeld's conjecture does not extend directly to elliptic curves over number fields."}
{"category": "Math", "title": "Noncommutative variations on Laplace's equation", "abstract": "As a first step at developing a theory of noncommutative nonlinear elliptic partial differential equations, we analyze noncommutative analogues of Laplace's equation and its variants (some of the them nonlinear) over noncommutative tori. Along the way we prove noncommutative analogues of many results in classical analysis, such as Wiener's Theorem on functions with absolutely convergent Fourier series, and standard existence and non-existence theorems on elliptic functions. We show that many many classical methods, including the Maximum Principle, the direct method of the calculus of variations, the use of the Leray-Schauder Theorem, etc., have analogues in the noncommutative setting."}
{"category": "Math", "title": "On Casson-type instanton moduli spaces over negative definite four-manifolds", "abstract": "Recently Andrei Teleman considered instanton moduli spaces over negative definite four-manifolds $X$ with $b_2(X) \\geq 1$. If $b_2(X)$ is divisible by four and $b_1(X) =1$ a gauge-theoretic invariant can be defined; it is a count of flat connections modulo the gauge group. Our first result shows that if such a moduli space is non-empty and the manifold admits a connected sum decomposition $X \\cong X_1 # X_2$ then both $b_2(X_1)$ and $b_2(X_2)$ are divisible by four; this rules out a previously natural appearing source of 4-manifolds with non-empty moduli space. We give in some detail a construction of negative definite 4-manifolds which we expect will eventually provide examples of manifolds with non-empty moduli space."}
{"category": "Math", "title": "Observability and Detectability of Linear Switching Systems: A Structural Approach", "abstract": "We define observability and detectability for linear switching systems as the possibility of reconstructing and respectively of asymptotically reconstructing the hybrid state of the system from the knowledge of the output for a suitable choice of the control input. We derive a necessary and sufficient condition for observability that can be verified computationally. A characterization of control inputs ensuring observability of switching systems is given. Moreover, we prove that checking detectability of a linear switching system is equivalent to checking asymptotic stability of a suitable switching system with guards extracted from it, thus providing interesting links to Kalman decomposition and the theory of stability of hybrid systems."}
{"category": "Math", "title": "Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results", "abstract": "Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cram\\'{e}r light-tail assumption on the claim size distribution."}
{"category": "Math", "title": "Frobenius algebras and skein modules of surfaces in 3-manifolds", "abstract": "For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skein module of the 3-ball is isomorphic to the ground ring of the Frobenius algebra. We prove a presentation theorem for the skein module with generators incompressible surfaces colored by elements of a generating set of the Frobenius algebra, and with relations determined by tubing geometry in the manifold and relations of the algebra."}
{"category": "Math", "title": "The non-commutative $A$-polynomial of twist knots", "abstract": "The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative $A$-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form $J(n)=\\sum_k c(n,k) \\hatJ (k)$ given a recursion relation for $(\\hatJ(n))$ a the hypergeometric kernel $c(n,k)$. As an application of our method, we explicitly compute the non-commutative $A$-polynomial for twist knots with -8 and 11 crossings. The non-commutative $A$-polynomial of a knot encodes the monic, linear, minimal order $q$-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to $q=1$ is conjectured to be the better-known $A$-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the $A$-polynomial is harder to compute and already unknown for some knots with 12 crossings."}
{"category": "Math", "title": "Notes on Measure and Integration", "abstract": "This text grew out of notes I have used in teaching a one quarter course on integration at the advanced undergraduate level. My intent is to introduce the Lebesgue integral in a quick, and hopefully painless, way and then go on to investigate the standard convergence theorems and a brief introduction to the Hilbert space of $L^2$ functions on the interval. The actual construction of Lebesgue measure and proofs of its key properties are relegated to an appendix. Instead the text introduces Lebesgue measure as a generalization of the concept of length and motivates its key properties: monotonicity, countable additivity, and translation invariance."}
{"category": "Math", "title": "Crystals, quiver varieties and coboundary categories for Kac-Moody algebras", "abstract": "Henriques and Kamnitzer have defined a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra that gives it the structure of a coboundary category (somewhat analogous to a braided monoidal category). Kamnitzer and Tingley then gave an alternative definition of the crystal commutor, using Kashiwara's involution on Verma crystals, that generalizes to the setting of symmetrizable Kac-Moody algebras. In the current paper, we give a geometric interpretation of the crystal commutor using quiver varieties. Equipped with this interpretation we show that the commutor endows the category of crystals of a symmetrizable Kac-Moody algebra with the structure of a coboundary category, answering in the affirmative a question of Kamnitzer and Tingley."}
{"category": "Math", "title": "Metaplectic Tori over Local Fields", "abstract": "Smooth irreducible representations of tori over local fields have been parameterized by Langlands, using class field theory and Galois cohomology. This paper extends this parameterization to central extensions of such tori, which arise naturally in the setting of nonlinear covers of reductive groups."}
{"category": "Math", "title": "On local-to-global spectral sequences for the cohomology of diagrams", "abstract": "The aim of this paper is to construct and examine three candidates for local-to-global spectral sequences for the cohomology of diagrams of algebras with directed indexing. In each case, the $E^2$ -terms can be viewed as a type of local cohomology relative to a map or an object in the diagram."}
{"category": "Math", "title": "Fast Directional Computation for the High Frequency Helmholtz Kernel in Two Dimensions", "abstract": "This paper introduces a directional multiscale algorithm for the two dimensional $N$-body problem of the Helmholtz kernel with applications to high frequency scattering. The algorithm follows the approach in [Engquist and Ying, SIAM Journal on Scientific Computing, 29 (4), 2007] where the three dimensional case was studied. The main observation is that, for two regions that follow a directional parabolic geometric configuration, the interaction between the points in these two regions through the Helmholtz kernel is approximately low rank. We propose an improved randomized procedure for generating the low rank representations. Based on these representations, we organize the computation of the far field interaction in a multidirectional and multiscale way to achieve maximum efficiency. The proposed algorithm is accurate and has the optimal $O(N\\log N)$ complexity for problems from two dimensional scattering applications. We present numerical results for several test examples to illustrate the algorithm and its application to two dimensional high frequency scattering problems."}
{"category": "Math", "title": "Inhomogeneous Strichartz estimates", "abstract": "We present abstract inhomogeneous Strichartz estimates for dispersive operators, extending previous work by M. Keel and T. Tao on the one hand, and generalising results of D. Foschi, M. Vilela, M. Nakamura and T. Ozawa on the other hand. It is shown that these abstract estimates imply new inhomogeneous Strichartz estimates for the wave equation and some Schr\\\"odinger equations involving potentials."}
{"category": "Math", "title": "On the period-index problem in light of the section conjecture", "abstract": "Period and index of a curve $X/K$ over a $p$-adic local field $K$ such that the fundamental group $\\pi_1(X/K)$ admits a splitting are shown to be powers of $p$. As a consequence, examples of curves over number fields are constructed where having sections is obstructed locally at a $p$-adic place. Hence the section conjecture holds for these curves as there are neither sections nor rational points."}
{"category": "Math", "title": "Moufang symmetry VI. Reductivity and hidden associativity in Mal'tsev algebras", "abstract": "Reductivity in the Ma'tsev algebras is inquired. This property relates the Mal'tsev algebras to the general Lie triple systems."}
{"category": "Math", "title": "Global Stabilization by Means of Discrete-Delay Output Feedback", "abstract": "In the present work, sufficient conditions for global stabilization of nonlinear uncertain systems by means of discrete-delay static output feedback are presented. Illustrating examples show the efficiency of the proposed control strategy."}
{"category": "Math", "title": "The topology of syntax relations of a formal language", "abstract": "The method of constructing of Grothendieck's topology basing on a neighbourhood grammar, defined on the category of syntax diagrams is described in the article. Syntax diagrams of a formal language are the multigraphs with nodes, signed by symbols of the language's alphabet. The neighbourhood grammar allows to select correct syntax diagrams from the set of all syntax diagrams on the given alphabet by mapping an each correct diagram to the cover consisted of the grammar's neighbourhoods. Such the cover gives rise to Grothendieck's topology on category of correct syntax diagrams extended by neighbourhoods' diagrams. An each object of the category may be mapped to the set of meanings (abstract senses) of this syntax construction. So, the contrvariant functor from category of correct syntax diagrams to category of sets is defined. The given category of contravariant functors likes to be seen as the convenient means to think about relations between syntax and semantic of a formal language. The sheaves of set defined on category of contravariant functors are the objects that satisfy of compositionality principle defined in the semantic analysis."}
{"category": "Math", "title": "Orbit measures and interlaced determinantal point processes", "abstract": "We study some random interlaced configurations considering the eigenvalues of the main minors of Hermitian random matrices of the classical complex Lie algebras. We claim that these random configurations are determinantal and give their correlation kernels."}
{"category": "Math", "title": "Depth of Boolean algebras", "abstract": "We prove a Theorem about the relationship between the Depth of the ultraproduct of Boolean algebras, divided by an ultrafilter, and the products of the depths of each component. This answers (partly) an open problem of Monk."}
{"category": "Math", "title": "On the product of vector spaces in a commutative field extension", "abstract": "Let $K \\subset L$ be a commutative field extension. Given $K$-subspaces $A,B$ of $L$, we consider the subspace $<AB>$ spanned by the product set $AB=\\{ab \\mid a \\in A, b \\in B\\}$. If $\\dim_K A = r$ and $\\dim_K B = s$, how small can the dimension of $<AB>$ be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on $\\dim_K < AB>$ turns out, in this case, to be provided by the numerical function $$ \\kappa_{K,L}(r,s) = \\min_{h} (\\lceil r/h\\rceil + \\lceil s/h\\rceil -1)h, $$ where $h$ runs over the set of $K$-dimensions of all finite-dimensional intermediate fields $K \\subset H \\subset L$. This bound is closely related to one appearing in additive number theory."}
{"category": "Math", "title": "The concept of bounded mean motion for toral homeomorphisms", "abstract": "A conservative irrational pseudo-rotation of the two-torus is semi-conjugate to the irrational rotation if and only if it has the property of bounded mean motion [10]. (Here 'irrational pseudo-rotation' means a toral homeomorphism with uniquely defined, totally irrational rotation vector.) The aim of this note is to explore this concept some further. For instance, we provide an example which shows that the preceding statement does not hold in the non-conservative case. Further, we collect a number of observations concerning the case where the bounded mean motion property fails. In particular, we show that a non-wandering irrational pseudo-rotation of the two-torus with unbounded mean motion has sensitive dependence on initial conditions."}
{"category": "Math", "title": "Linear free divisors and Frobenius manifolds", "abstract": "We study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the Gauss-Manin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure."}
{"category": "Math", "title": "Bayesian Estimation of Inequalities with Non-Rectangular Censored Survey Data", "abstract": "Synthetic indices are used in Economics to measure various aspects of monetary inequalities. These scalar indices take as input the distribution over a finite population, for example the population of a specific country. In this article we consider the case of the French 2004 Wealth survey. We have at hand a partial measurement on the distribution of interest consisting of bracketed and sometimes missing data, over a subsample of the population of interest. We present in this article the statistical methodology used to obtain point and interval estimates taking into account the various uncertainties. The inequality indices being nonlinear in the input distribution, we rely on a simulation based approach where the model for the wealth per household is multivariate. Using the survey data as well as matched auxiliary tax declarations data, we have at hand a quite intricate non-rectangle multidimensional censoring. For practical issues we use a Bayesian approach. Inference using Monte-Carlo approximations relies on a Monte-Carlo Markov chain algorithm namely the Gibbs sampler. The quantities interesting to the decision maker are taken to be the various inequality indices for the French population. Their distribution conditional on the data of the subsample are assumed to be normal centered on the design-based estimates with variance computed through linearization and taking into account the sample design and total nonresponse. Exogeneous selection of the subsample, in particular the nonresponse mechanism, is assumed and we condition on the adequate covariates."}
{"category": "Math", "title": "Maxisets for Model Selection", "abstract": "We address the statistical issue of determining the maximal spaces (maxisets) where model selection procedures attain a given rate of convergence. By considering first general dictionaries, then orthonormal bases, we characterize these maxisets in terms of approximation spaces. These results are illustrated by classical choices of wavelet model collections. For each of them, the maxisets are described in terms of functional spaces. We take a special care of the issue of calculability and measure the induced loss of performance in terms of maxisets."}
{"category": "Math", "title": "Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds", "abstract": "A quaternionic version of the Calabi problem on Monge-Ampere equation is introduced. It is a quaternionic Monge-Ampere equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n;H), uniqueness (up to a constant) of a solution is proven, as well as the zero order a priori estimate. The existence of solution is conjectured, similar to Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of a complex Hessian equation, making sense on any complex manifold."}
{"category": "Math", "title": "Uniformity and Functional Equations for Local Zeta Functions of $\\mathfrak{K}$-Split Algebraic Groups", "abstract": "We study the local zeta functions of an algebraic group $\\mathcal{G}$ defined over $\\mathfrak{K}$ together with a faithful $\\mathfrak{K}$-rational representation $\\rho$ for a finite extension $\\mathfrak{K}$ of $\\mathbb{Q}$. These are given by integrals over $\\mathfrak{p}$-adic points of $\\mathcal{G}$ determined by $\\rho$ for a prime $\\mathfrak{p}$ of $\\mathfrak{K}$. We prove that the local zeta functions are almost uniform for all $\\mathfrak{K}$-split groups whose unipotent radical satisfies a certain lifting property. This property is automatically satisfied if $\\mathcal{G}$ is reductive. We provide a further criterion in terms of invariants of $\\mathcal{G}$ and $\\rho$ which guarantees that the local zeta functions satisfy functional equations for almost all primes of $\\mathfrak{K}$. We obtain these results by using a $\\mathfrak{p}$-adic Bruhat decomposition of Iwahori and Matsumoto [IM] to express the zeta function as a weighted sum over the Weyl group $W$ associated to $\\mathcal{G}$ of generating functions over lattice points of a polyhedral cone. The functional equation reflects an interplay between symmetries of the Weyl group and reciprocities of the combinatorial object. We construct families of groups with representations violating our second structural criterion whose local zeta functions do not satisfy functional equations. Our work generalizes results of Igusa [Igu] and du Sautoy and Lubotzky [dSL] and has implications for zeta functions of finitely generated torsion-free nilpotent groups."}
{"category": "Math", "title": "Affine interval exchange transformations with flips and wandering intervals", "abstract": "There exist uniquely ergodic affine interval exchange transformations of [0,1] with flips having wandering intervals and such that the support of the invariant measure is a Cantor set."}
{"category": "Math", "title": "Local analytic classification of $q$-difference equations with $|q|=1$", "abstract": "In this paper, we establish, under convenient diophantine assumptions, a complete analytic classification of $q$-difference modules over the field of germs of meromorphic functions at zero, proving some analytic analogs of the results by Soibelman and Vologodsky, cf. math.AG/0205117, and by Baranovsky and Ginzburg, cf. alg-geom/9607008."}
{"category": "Math", "title": "Symplectic Reduction of Sheaves of $\\mathcal{A}$-modules", "abstract": "Given an arbitrary sheaf $\\mathcal{E}$ of $\\mathcal{A}$-modules (or $\\mathcal{A}$-module in short) on a topological space $X$, we define \\textit{annihilator sheaves} of sub-$\\mathcal{A}$-modules of $\\mathcal{E}$ in a way similar to the classical case, and obtain thereafter the analog of the \\textit{main theorem}, regarding classical annihilators in module theory, see Curtis[\\cite{curtis}, pp. 240-242]. The familiar classical properties, satisfied by annihilator sheaves, allow us to set clearly the \\textit{sheaf-theoretic version} of \\textit{symplectic reduction}, which is the main goal in this paper."}
{"category": "Math", "title": "Limits Laws for Geometric Means of Free Random Variables", "abstract": "Let $\\{T_{k}\\}_{k=1}^{\\infty}$ be a family of *--free identically distributed operators in a finite von Neumann algebra. In this work we prove a multiplicative version of the free central limit Theorem. More precisely, let $B_{n}=T_{1}^{*}T_{2}^{*}... T_{n}^{*}T_{n}... T_{2}T_{1}$ then $B_{n}$ is a positive operator and $B_{n}^{1/2n}$ converges in distribution to an operator $\\Lambda$. We completely determine the probability distribution $\\nu$ of $\\Lambda$ from the distribution $\\mu$ of $|T|^{2}$. This gives us a natural map $\\mathcal{G}:\\mathcal{M_{+}}\\to \\mathcal{M_{+}}$ with $\\mu\\mapsto \\mathcal{G}(\\mu)=\\nu.$ We study how this map behaves with respect to additive and multiplicative free convolution. As an interesting consequence of our results, we illustrate the relation between the probability distribution $\\nu$ and the distribution of the Lyapunov exponents for the sequence $\\{T_{k}\\}_{k=1}^{\\infty}$ introduced in \\cite{LyaV}."}
{"category": "Math", "title": "Note on Integer Factoring Methods IV", "abstract": "This note continues the theoretical development of deterministic integer factorization algorithms based on systems of polynomials equations. The main result establishes a new deterministic time complexity bench mark in integer factorization."}
{"category": "Math", "title": "The quadratic character experiment", "abstract": "A fast new algorithm is used compute the zeros of the quadratic character L-functions for all negative fundamental discriminants with absolute value 10^12<d<10^12+10^7. These are compared to the 1-level density, including various lower order terms. These terms come from, on the one hand the Explicit Formula, and on the other the L-functions Ratios Conjecture. The latter give a much better fit to the data, providing numerical evidence for the conjecture."}
{"category": "Math", "title": "Hypercontractivity for log-subharmonic functions", "abstract": "We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on $\\RR^n$ and different classes of measures: Gaussian measures on $\\RR^n$, symmetric Bernoulli and symmetric uniform probability measures on $\\RR$, as well as their convolutions. Surprisingly, a slightly weaker strong hypercontractivity property holds for {\\em any} symmetric measure on $\\RR$. For all measures on $\\R$ for which we know the (SHC) holds, we prove that a log--Sobolev inequality holds in the log-subharmonic category with a constant {\\em smaller} than the one for Gaussian measure in the classical context. This result is extended to all dimensions for compactly-supported measures."}
{"category": "Math", "title": "Group action on bimodule categories", "abstract": "We study group action on bimodules and bimodule categories and prove for them analogues of the results known for representations of skew group algebras, mainly in the case, when the action is separable."}
{"category": "Math", "title": "On the dynamics of certain homoclinic tangles", "abstract": "In this paper we study homoclinic tangles formed by transversal intersections of the stable and the unstable manifold of a {\\it non-resonant, dissipative} homoclinic saddle point in periodically perturbed second order equations. We prove that the dynamics of these homoclinic tangles are that of {\\it infinitely wrapped horseshoe maps}. Using $\\mu$ as a parameter representing the magnitude of the perturbations, we prove that (a) there exist infinitely many disjoint open intervals of $\\mu$, accumulating at $\\mu = 0$, such that the entire homoclinic tangle of the perturbed equation consists of one single horseshoe of infinitely many symbols, (b) there are parameters in between each of these parameter intervals, such that the homoclinic tangle contains attracting periodic solutions, and (c) there are also parameters in between where the homoclinic tangles admit non-degenerate transversal homoclinic tangency of certain dissipative hyperbolic periodic solutions. In particular, (c) implies the existence of strange attractors with SRB measures for a positive measure set of parameters."}
{"category": "Math", "title": "Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties", "abstract": "These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology."}
{"category": "Math", "title": "On the dynamics of a time-periodic equation", "abstract": "In this paper we use the second order equation $\\frac{d^2 q}{dt^2} + (\\lambda - \\gamma q^2) \\frac{d q}{dt} - q + q^3 = \\mu q^2 \\sin \\omega t$ as a demonstrative example to illustrate how to apply the analysis of \\cite{WO} and \\cite{WOk} to the studies of concrete equations. We prove, among many other things, that there are positive measure sets of parameters $(\\lambda, \\gamma, \\mu, \\omega)$ corresponding to the case of intersected and the case of separated stable and unstable manifold of the solution $q(t) = 0$, $t \\in \\mathbb R$ respectively, so that the corresponding equations admit strange attractors with SRB measures."}
{"category": "Math", "title": "Dissipative homoclinic loops and rank one chaos", "abstract": "We prove that when subjected to periodic forcing of the form $p_{\\mu, \\rh, \\om} (t) = \\mu (\\rh h(x,y) + \\sin (\\om t))$, certain second order systems of differential equations with dissipative homoclinic loops admit strange attractors with SRB measures for a set of forcing parameters $(\\mu, \\rh, \\om)$ of positive measure. Our proof applies the recent theory of rank one maps, developed by Wang and Young based on the analysis of strongly dissipative H\\'enon maps by Benedicks and Carleson."}
{"category": "Math", "title": "Modular functionals and perturbations of Nakano spaces", "abstract": "We settle several questions regarding the model theory of Nakano spaces left open by the PhD thesis of Pedro Poitevin \\cite{Poitevin:PhD}. We start by studying isometric Banach lattice embeddings of Nakano spaces, showing that in dimension two and above such embeddings have a particularly simple and rigid form. We use this to show show that in the Banach lattice language the modular functional is definable and that complete theories of atomless Nakano spaces are model complete. We also show that up to arbitrarily small perturbations of the exponent Nakano spaces are $\\aleph_0$-categorical and $\\aleph_0$-stable. In particular they are stable."}
{"category": "Math", "title": "Definability of groups in $\\aleph_0$-stable metric structures", "abstract": "We prove that in a continuous $\\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \\begin{enumerate} \\item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from \\cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to prove the theorem in case the metric is invariant under the group action; and \\item Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones. \\end{enumerate}"}
{"category": "Math", "title": "A measure-conjugacy invariant for free group actions", "abstract": "This paper introduces a new measure-conjugacy invariant for actions of free groups. Using this invariant, it is shown that two Bernoulli shifts over a finitely generated free group are measurably conjugate if and only if their base measures have the same entropy. This answers a question of Ornstein and Weiss."}
{"category": "Math", "title": "BDDC by a frontal solver and the stress computation in a hip joint replacement", "abstract": "A parallel implementation of the BDDC method using the frontal solver is employed to solve systems of linear equations from finite element analysis, and incorporated into a standard finite element system for engineering analysis by linear elasticity. Results of computation of stress in a hip replacement are presented. The part is made of titanium and loaded by the weight of human body. The performance of BDDC with added constraints by averages and with added corners is compared."}
{"category": "Math", "title": "The Kashiwara-Vergne conjecture and Drinfeld's associators", "abstract": "The Kashiwara-Vergne (KV) conjecture is a property of the Campbell-Hausdorff series put forward in 1978. It has been settled in the positive by E. Meinrenken and the first author in 2006. In this paper, we study the uniqueness issue for the KV problem. To this end, we introduce a family of infinite dimensional groups KV_n, and an extension \\hat{KV}_2 of the group KV_2. We show that the group \\hat{KV}_2 contains the Grothendieck-Teichmueller group GRT as a subgroup, and that it acts freely and transitively on the set of solutions of the KV problem Sol(KV). Furthermore, we prove that Sol(KV) is isomorphic to a direct product of a line \\k (\\k being a field of characteristic zero) and the set of solutions of the pentagon equation with values in the group KV_3. The latter contains the set of Drinfeld's associators as a subset. As a by-product, we obtain a new proof of the Kashiwara-Vergne conjecture based on the Drinfeld's theorem on existence of associators."}
{"category": "Math", "title": "Frobenius splittings of toric varieties", "abstract": "We discuss a characteristic free version of Frobenius splittings for toric varieties and give a polyhedral criterion for a toric variety to be diagonally split. We apply this criterion to show that section rings of nef line bundles on diagonally split toric varieties are normally presented and Koszul, and that Schubert varieties are not diagonally split in general."}
{"category": "Math", "title": "Group Actions and Covering Maps in the Uniform Category", "abstract": "In Rips Complexes and Covers in the Uniform Category (arXiv:0706.3937) we define, following James, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. In this paper we investigate when these covering maps are induced by group actions. Also, as an application of our results we present an exposition of Prajs' homogeneous curve that is path-connected but not locally connected."}
{"category": "Math", "title": "Degree and valuation of the Schur elements of cyclotomic Hecke algebras", "abstract": "Following the generalization of the notion of families of characters, defined by Lusztig for Weyl groups, to the case of complex reflection groups, thanks to the definition given by Rouquier, we show that the degree and the valuation of the Schur elements (functions A and a) remain constant on the \"families\" of the cyclotomic Hecke algebras of the exceptional complex reflection groups. The same result has already been obtained for the groups of the infinite series and for some special cases of exceptional groups."}
{"category": "Math", "title": "On types for unramified p-adic unitary groups", "abstract": "Folloing the methods of Bushnell-Kutzko for general linear groups, we construct simple types attached to certain skew simple strata for a symplectic group and an unramified unitary group over a non-archimedean local field."}
{"category": "Math", "title": "Bayesball: A Bayesian hierarchical model for evaluating fielding in major league baseball", "abstract": "The use of statistical modeling in baseball has received substantial attention recently in both the media and academic community. We focus on a relatively under-explored topic: the use of statistical models for the analysis of fielding based on high-resolution data consisting of on-field location of batted balls. We combine spatial modeling with a hierarchical Bayesian structure in order to evaluate the performance of individual fielders while sharing information between fielders at each position. We present results across four seasons of MLB data (2002--2005) and compare our approach to other fielding evaluation procedures."}
{"category": "Math", "title": "Determinant Expansions of Signed Matrices and of Certain Jacobians", "abstract": "This paper treats two topics: matrices with sign patterns and Jacobians of certain mappings. The main topic is counting the number of plus and minus coefficients in the determinant expansion of sign patterns and of these Jacobians. The paper is motivated by an approach to chemical networks initiated by Craciun and Feinberg. We also give a graph-theoretic test for determining when the Jacobian of a chemical reaction dynamics has a sign pattern."}
{"category": "Math", "title": "Non-singular affine surfaces with self-maps", "abstract": "We classify surjective self-maps (of degree at least two) of affine surfaces according to the log Kodaira dimension."}
{"category": "Math", "title": "On the Equivalence of Primal and Dual Substructuring Preconditioners", "abstract": "After a short historical review, we present four popular substructuring methods: FETI-1, BDD, FETI-DP, BDDC, and derive the primal versions to the two FETI methods, called P-FETI-1 and P-FETI-DP, as proposed by Fragakis and Papadrakakis. The formulation of the BDDC method shows that it is the same as P-FETI-DP and the same as a preconditioner introduced by Cros. We prove the equality of eigenvalues of a particular case of the FETI-1 method and of the BDD method by applying a recent abstract result by Fragakis."}
{"category": "Math", "title": "A note on evaluations of multiple zeta values", "abstract": "Multiple zeta values (MZVs) with certain repeated arguments or certain sums of cyclically generated MZVs are evaluated as rational multiple of powers of $\\pi^2$. In this paper, we give a short and simple proof of the remarkable evaluations of MZVs established by D. Borman and D. M. Bradley."}
{"category": "Math", "title": "Constructions of regular algebras $L_p^w(G)$", "abstract": "Criterion of (Shilov) regularity for weighted algebras $L_1^w(G)$ on a locally compact abelian group $G$ is known by works of Beurling (1949) and Domar (1956). In the present paper this criterion is extended to translation invariant weighted algebras $L_p^w(G)$ with $p>1$. Regular algebras $L_p^w(G)$ are constructed on any sigma-compact abelian group $G$. It was proved earlier by the author that sigma-compactness is necessary (in the abelian case) for the existence of weighted algebras $L_p^w(G)$ with $p>1$."}
{"category": "Math", "title": "Algebraization of bundles on non-proper schemes", "abstract": "We consider the algebraization problem for principal bundles with reductive structure group, defined on the complement of a closed subset Z in a proper formal scheme. We show that, when Z is of codimension at least 3, an algebraization always exists. For codimension 2 we show that an algebraization exists precisely when a certain additional condition is satisfied."}
{"category": "Math", "title": "Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups", "abstract": "The classifying space of a crossed complex generalises the construction of Eilenberg-Mac Lane spaces. We show how the theory of fibrations of crossed complexes allows the analysis of homotopy classes of maps from a free crossed complex to such a classifying space. This gives results on the homotopy classification of maps from a CW-complex to the classifying space of a crossed module and also, more generally, of a crossed complex whose homotopy groups vanish in dimensions between 1 and n. The results are analogous to those for the obstruction to an abstract kernel in group extension theory."}
{"category": "Math", "title": "On the spectrum of the Stokes operator", "abstract": "We prove Li--Yau-type lower bounds for the eigenvalues of the Stokes operator and give applications to the attractors of the Navier--Stokes equations."}
{"category": "Math", "title": "A palindromization map for the free group", "abstract": "We define a self-map Pal: F_2 --> F_2 of the free group on two generators a, b, using automorphisms of F_2 that form a group isomorphic to the braid group B_3. The map Pal restricts to de Luca's right iterated palindromic closure on the submonoid generated by a, b, and is continuous for the profinite topology on F_2. The values of Pal are palindromes and coincide with the elements g of F_2 such that abg is conjugate to bag."}
{"category": "Math", "title": "Non-abelian extensions of minimal rotations", "abstract": "We consider continuous extensions of minimal rotations on a locally connected compact group X by arbitrary locally compact Lie groups and prove regularity (i.e. the existence of orbit closures which project onto the whole basis X) in certain special situations beyond the already known nilpotent case. We further discuss an open question on cocycles acting on homogeneous spaces which seems to be the missing key for a general regularity theorem."}
{"category": "Math", "title": "On additive doubling and energy", "abstract": "We show that if A is a set having small subtractive doubling in an abelian group, that is |A-A|< K|A|, then there is a polynomially large subset B of A-A so that the additive energy of B is large than (1/K)^{1 - \\epsilon) where epsilon is a positive, universal exponent. (1/37 seems to suffice.)"}
{"category": "Math", "title": "The first cohomology of the mapping class group with coefficients in algebraic functions on the SL(2, C) moduli space", "abstract": "Consider a compact surface of genus at least two. We prove that the first cohomology group of the mapping class group with coefficients in the space of algebraic functions on the SL(2, C) moduli space vanishes."}
{"category": "Math", "title": "Support theorems for the Radon transform and Cram\\'er-Wold theorems", "abstract": "This article presents extensions of the Cram{\\'e}r-Wold theorem to measures that may have infinite mass near the origin. Corresponding results for sequences of measures are presented together with examples showing that the assumptions imposed are sharp. The extensions build on a number of results and methods concerned with injectivity properties of the Radon transform. Using a few tools from distribution theory and Fourier analysis we show that the presented injectivity results for the Radon transform lead to Cram{\\'e}r-Wold type results for measures. One purpose of this article is to contribute to making known to probabilists interesting results for the Radon transform that have been developed essentially during the 1980ies and 1990ies."}
{"category": "Math", "title": "Unitary super perfect numbers", "abstract": "We shall show that 9, 165 are all of the odd unitary super perfect numbers."}
{"category": "Math", "title": "Optimal experimental design and some related control problems", "abstract": "This paper traces the strong relations between experimental design and control, such as the use of optimal inputs to obtain precise parameter estimation in dynamical systems and the introduction of suitably designed perturbations in adaptive control. The mathematical background of optimal experimental design is briefly presented, and the role of experimental design in the asymptotic properties of estimators is emphasized. Although most of the paper concerns parametric models, some results are also presented for statistical learning and prediction with nonparametric models."}
{"category": "Math", "title": "Asymptotic behaviour of a family of gradient algorithms in R^d and Hilbert spaces", "abstract": "The asymptotic behaviour of a family of gradient algorithms (including the methods of steepest descent and minimum residues) for the optimisation of bounded quadratic operators in R^d and Hilbert spaces is analyzed. The results obtained generalize those of Akaike (1959) in several directions. First, all algorithms in the family are shown to have the same asymptotic behaviour (convergence to a two-point attractor), which implies in particular that they have similar asymptotic convergence rates. Second, the analysis also covers the Hilbert space case. A detailed analysis of the stability property of the attractor is provided."}
{"category": "Math", "title": "On perturbations of continuous structures", "abstract": "We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately $\\aleph_0$-saturated up to arbitrarily small perturbations are isomorphic up to arbitrarily small perturbations (where the notion of perturbation is part of the data). As a corollary, we obtain a Ryll-Nardzewski style characterisation of complete theories all of whose separable models are isomorphic up to arbitrarily small perturbations."}
{"category": "Math", "title": "Tits geometry on ideal boundaries of Busemann non-positively curved space", "abstract": "Let $X$ be a non-compact proper Busemann space. We introduce a collection of binary relations on its ideal boundaries generalizing comparison of Tits metric with two key values $\\pi$ and $\\pi/2$. This allows to use properties of Tits metric known for CAT(0)-space without metric itself."}
{"category": "Math", "title": "Cycle Equivalence of Graph Dynamical Systems", "abstract": "Graph dynamical systems (GDSs) can be used to describe a wide range of distributed, nonlinear phenomena. In this paper we characterize cycle equivalence of a class of finite GDSs called sequential dynamical systems SDSs. In general, two finite GDSs are cycle equivalent if their periodic orbits are isomorphic as directed graphs. Sequential dynamical systems may be thought of as generalized cellular automata, and use an update order to construct the dynamical system map. The main result of this paper is a characterization of cycle equivalence in terms of shifts and reflections of the SDS update order. We construct two graphs C(Y) and D(Y) whose components describe update orders that give rise to cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper bound for the number of cycle equivalence classes one can obtain, and we enumerate these quantities through a recursion relation for several graph classes. The components of these graphs encode dynamical neutrality, the component sizes represent periodic orbit structural stability, and the number of components can be viewed as a system complexity measure."}
{"category": "Math", "title": "Semigroup cohomology as a derived functor", "abstract": "In this work we construct an extension for the category of 0-modules by analogy with [H.-J. Baues and G. Wirshing, Cohomology of small categories, J. Pure Appl. Algebra, 38(1985), 187-211]. The 0-cohomology functor becomes a derived functor in the extended category. As an application of this construction we calculate the cohomological dimension of so-called 0-free monoids."}
{"category": "Math", "title": "A characterization of Schur multipliers between character-automorphic Hardy spaces", "abstract": "We give a new characterization of character-automorphic Hardy spaces of order 2 and of their contractive multipliers in terms of de Branges Rovnyak spaces. Keys tools in our arguments are analytic extension and a factorization result for matrix-valued analytic functions due to Leech."}
{"category": "Math", "title": "On the Brauer monoid for finite fields", "abstract": "The Brauer monoid is studied by the notion of 0-cohomology. We investigate the impact of invertible elements of modifications on the structure of the Brauer monoid, especially for finite fields."}
{"category": "Math", "title": "Automorphisms of wonderful varieties", "abstract": "Let G be a complex semisimple linear algebraic group, and X a wonderful G-variety. We determine the connected automorphism group of X and we calculate Luna's invariants of X under its action."}
{"category": "Math", "title": "Uniformisation of foliations by curves", "abstract": "These are lecture notes for a course to be held. They provide a full discussion of certain analytic aspects of the uniformisation theory of (singular) holomorphic foliations by curves on compact Kaehler manifolds, with emphasis on their consequences on positivity properties of the corresponding canonical bundles."}
{"category": "Math", "title": "Asymptotic analysis of a fluid model modulated by an $M/M/1$ queue", "abstract": "We analyze asymptotically a differential-difference equation, that arises in a Markov-modulated fluid model. We use singular perturbation methods to analyze the problem with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used."}
{"category": "Math", "title": "Krasinkiewicz spaces and parametric Krasinkiewicz maps", "abstract": "We say that a metrizable space $M$ is a Krasinkiewicz space if any map from a metrizable compactum $X$ into $M$ can be approximated by Krasinkiewicz maps (a map $g\\colon X\\to M$ is Krasinkiewicz provided every continuum in $X$ is either contained in a fiber of $g$ or contains a component of a fiber of $g$). In this paper we establish the following property of Krasinkiewicz spaces: Let $f\\colon X\\to Y$ be a perfect map between metrizable spaces and $M$ a Krasinkiewicz complete $ANR$-space. If $Y$ is a countable union of closed finite-dimensional subsets, then the function space $C(X,M)$ with the source limitation topology contains a dense $G_{\\delta}$-subset of maps $g$ such that all restrictions $g|f^{-1}(y)$, $y\\in Y$, are Krasinkiewicz maps. The same conclusion remains true if $M$ is homeomorphic to a closed convex subset of a Banach space and $X$ is a $C$-space."}
{"category": "Math", "title": "Hopf Bifurcations in a Watt Governor With a Spring", "abstract": "This paper pursues the study carried out by the authors in \"Stability and Hopf bifurcation in a hexagonal governor system\", focusing on the codimension one Hopf bifurcations in the hexagonal Watt governor differential system. Here are studied the codimension two, three and four Hopf bifurcations and the pertinent Lyapunov stability coefficients and bifurcation diagrams, ilustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is found an open region in the parameter space where two attracting periodic orbits coexist with an attracting equilibrium point."}
{"category": "Math", "title": "Poisson geometry and first integrals of geostrophic equations", "abstract": "We describe first integrals of geostrophic equations, which are similar to the enstrophy invariants of the Euler equation for an ideal incompressible fluid. We explain the geometry behind this similarity, give several equivalent definitions of the Poisson structure on the space of smooth densities on a symplectic manifold, and show how it can be obtained via the Hamiltonian reduction from a symplectic structure on the diffeomorphism group."}
{"category": "Math", "title": "On the cohomology of certain non-compact Shimura varieties (with an appendix by Robert Kottwitz)", "abstract": "The goal of this paper is to calculate the trace of the composition of a Hecke correspondence and a (high enough) power of the Frobenius at a good place on the intersection cohomology of the Satake-Baily-Borel compactification of certain Shimura varieties, to stabilize the result for Shimura varieties associated to unitary groups over $\\mathbb{Q}$ and to give applications of this calculations using base change from these unitary groups to $GL_n$. ----- Le but de ce texte est de calculer la trace d'une correspondance de Hecke composee avec une puissance (assez grande) du Frobenius en une bonne place sur la cohomologie d'intersection de la compactification de Satake-Baily-Borel de certaines varietes de Shimura, de stabiliser le resultat obtenu pour les varietes de Shimura associees aux groupes unitaires sur $\\mathbb{Q}$, et de donner des applications de ces calculs en utilisant le changement de base de ces groupes unitaires a $GL_n$."}
{"category": "Math", "title": "Topometric spaces and perturbations of metric structures", "abstract": "We develop the general theory of \\emph{topometric spaces}, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric function. Spaces of global and local types in continuous logic are the motivating examples for the study of such spaces. In particular, we develop a theory of Cantor-Bendixson analysis of topometric spaces, which can serve as a basis for the study of local stability (extending the \\textit{ad hoc} development from \\cite{BenYaacov-Usvyatsov:CFO}), as well as of global $\\aleph_0$-stability. We conclude with a study of perturbation systems (see \\cite{BenYaacov:Perturbations}) in the formalism of topometric spaces. In particular, we show how the abstract development applies to $\\aleph_0$-stability up to perturbation."}
{"category": "Math", "title": "Renewal-type Limit Theorem for Continued Fractions with Even Partial Quotients", "abstract": "We prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by F. Schweiger and studied also by C. Kraaikamp and A. Lopes. Our main result is proven following the strategy used by Ya. Sinai and C. Ulcigrai in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow."}
{"category": "Math", "title": "Update on Modular Non-Rigid Calabi-Yau Threefolds", "abstract": "We review some recent results on the modularity of non-rigid Calabi-Yau threefolds."}
{"category": "Math", "title": "Derived Functors Related to Wall Crossing", "abstract": "The setting is the representation theory of a simply connected, semisimple algebraic group over a field of positive characteristic. There is a natural transformation from the wall-crossing functor to the identity functor. The kernel of this transformation is a left exact functor. This functor and its first derived functor are evaluated on the global sections of a line bundle on the flag variety. It is conjectured that the derived functors of order greater than one annihilate the global sections. Also, the principal indecomposable modules for the Frobenius subgroups are shown to be acyclic."}
{"category": "Math", "title": "Essential dimension of abelian varieties over number fields", "abstract": "We affirmatively answer a conjecture in the preprint ``Essential dimension and algebraic stacks,'' proving that the essential dimension of an abelian variety over a number field is infinite."}
{"category": "Math", "title": "Homotopical Intersection Theory, II: equivariance", "abstract": "This is the second in a series of papers. Here we develop here an intersection theory for manifolds equipped with an action of a finite group. As in our previous paper, our approach will be homotopy theoretic, enabling us to circumvent the specter of equivariant transversality. This theory has applications to embedding problems, equivariant fixed point theory and the problem of enumerating the periodic points of a self map of a compact smooth manifold."}
{"category": "Math", "title": "What to expect from $U(n)$ Seiberg-Witten monopoles for $n > 1$", "abstract": "We study generalisations to the structure groups U(n) of the familiar (abelian) Seiberg-Witten monopole equations on a four-manifold $X$ and their moduli spaces. For $n=1$ one obtains the classical monopole equations. For $n > 1$ our results indicate that there should not be any non-trivial gauge-theoretical invariants which are obtained by the scheme `evaluation of cohomology classes on the fundamental cycle of the moduli space'. For, if $b_2^+$ is positive the moduli space should be `cobordant' to the empty space because we can deform the equations so as the moduli space of the deformed equations is generically empty. Furthermore, on K\\\"ahler surfaces with $b_2^+ > 1$, the moduli spaces become empty as soon as we perturb with a non-vanishing holomorphic 2-form."}
{"category": "Math", "title": "$PU(N)$ monopoles, higher rank instantons, and the monopole invariants", "abstract": "A famous conjecture in gauge theory mathematics, attributed to Witten, suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg-Witten invariants if the underlying four-manifold is of simple type. Mathematicians have sought a proof of the conjecture by means of a `cobordism program' involving $PU(2)$ monopoles. A higher rank version of the Donaldson invariants was recently introduced by Kronheimer. Before being defined, the physicists Mari\\~no and Moore had already suggested that there should be a generalisation of Witten's conjecture to this type of invariants. We adopt a generalisation of the cobordism program to the higher rank situation by studying $PU(N)$ monopoles. We analyse the differences to the $PU(2)$ situation, yielding evidence that a generalisation of Witten's conjecture should hold."}
{"category": "Math", "title": "Generators for Rational Loop Groups and Geometric Applications", "abstract": "Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n,C) that satisfy the U(n)-reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables us to write down a natural set of simple elements. Using these simple elements we prove generator theorems for the fundamental representations of the remaining neo-classical groups and most of their symmetric spaces. In order to apply our theorems to submanifold geometry we also obtain explicit dressing and permutability formulae. We introduce a new submanifold geometry associated to G_2/SO(4) to which our theory applies."}
{"category": "Math", "title": "A remark on minimal Fano threefolds", "abstract": "We prove in the case of minimal Fano threefolds a conjecture stated by Dubrovin at the ICM 1998 in Berlin. The conjecture predicts that the symmetrized/alternated Euler characteristic pairing on $K_0$ of a Fano variety with an exceptional collection expressed in the basis of the classes of the exceptional objects coincides with the intersection pairing of the vanishing cycles in Dubrovin's second connection. We show that the conjecture holds for $V_{22}$, a minimal Fano threefold of anticanonical degree~22, and for $V_5$, the minimal Fano threefold of anticanonical degree~40, by applying the modularity result for rank 1 Fano threefolds. The truth of the conjecture for $\\P ^3$ and the three--dimensional quadric is known; we consider these cases for the sake of completeness."}
{"category": "Math", "title": "Cauchy_Riemann Equations for Cayley Numbers` Functions", "abstract": "Since the discovery of octonions in 1843 we seem to be still lacking a satisfactory if any theory of octave valued functions satisfactory according to standard requirements or expectation from the side of a theory like a one might look for. Here is a proposal coming back to my twentieth century presentation of a perhaps nonstandard idea hoping to be coping with nonassociativity by an invention."}
{"category": "Math", "title": "Banach Spaces with respect to Operator-Valued Norms", "abstract": "We introduce the notions of L(H)-valued norms and Banach spaces with respect to L(H)-valued norms. In particular, we introduce Hilbert spaces with respect to L(H)-valued inner products. In addition, we provide several fundamental examples of Hilbert spaces with respect to L(H)-valued inner products."}
{"category": "Math", "title": "Some graphs related to Thompson's group F", "abstract": "The Schreier graphs of Thompson's group F with respect to the stabilizer of 1/2 and generators x_0 and x_1, and of its unitary representation in L_2([0,1]) induced by the standard action on the interval [0,1] are explicitly described. The coamenability of the stabilizers of any finite set of dyadic rational numbers is established. The induced subgraph of the right Cayley graph of the positive monoid of F containing all the vertices of the form x_nv, where n>=0 and v is any word over the alphabet {x_0, x_1}, is constructed. It is proved that the latter graph is non-amenable."}
{"category": "Math", "title": "Direct limits of infinite-dimensional Lie groups", "abstract": "Many infinite-dimensional Lie groups of interest can be expressed as a union of an ascending sequence of (finite- or infinite-dimensional) Lie groups. In this survey article, we compile general results concerning such ascending unions, describe the main classes of examples, and explain what the general theory tells us about these. In particular, we discuss: (1) Direct limit properties of ascending unions of Lie groups in the relevant categories; (2) Regularity in Milnor's sense; (3) Homotopy groups of direct limit groups and of Lie groups containing a dense union of Lie groups; (4) Subgroups of direct limit groups; (5) Constructions of Lie group structures on ascending unions of Lie groups."}
{"category": "Math", "title": "On Fuglede's conjecture for three intervals", "abstract": "In this paper we prove the \"Tiling implies Spectral\" part of Fuglede's paper for the case of three intervals. Then we prove the \"Spectral implies Tiling\" part of the conjecture for the case of three equal intervals as also when the intervals have lengths 1/2, 1/4, 1/4. For the general case we change our approach to get information on the structure of the spectrum for the n-interval case. Finally, we use symbolic computations on Mathematica, and prove this part of the conjecture with an additional assumption on the spectrum."}
{"category": "Math", "title": "Adaptive methods for sequential importance sampling with application to state space models", "abstract": "In this paper we discuss new adaptive proposal strategies for sequential Monte Carlo algorithms--also known as particle filters--relying on criteria evaluating the quality of the proposed particles. The choice of the proposal distribution is a major concern and can dramatically influence the quality of the estimates. Thus, we show how the long-used coefficient of variation of the weights can be used for estimating the chi-square distance between the target and instrumental distributions of the auxiliary particle filter. As a by-product of this analysis we obtain an auxiliary adjustment multiplier weight type for which this chi-square distance is minimal. Moreover, we establish an empirical estimate of linear complexity of the Kullback-Leibler divergence between the involved distributions. Guided by these results, we discuss adaptive designing of the particle filter proposal distribution and illustrate the methods on a numerical example."}
{"category": "Math", "title": "An explicit classification for the cyclic rational torsion subgroups of odd order of elliptic curves over $ \\Q $", "abstract": "This paper has been withdrawn by the author due to an uninteresting calculation."}
{"category": "Math", "title": "Q-valued functions revisited", "abstract": "In this note we revisit Almgren's theory of Q-valued functions, that are functions taking values in the space of unordered Q-tuples of points in R^n. In particular: 1) we give shorter versions of Almgren's proofs of the existence of Dir-minimizing Q-valued functions, of their Hoelder regularity and of the dimension estimate of their singular set; 2) we propose an alternative intrinsic approach to these results, not relying on Almgren's biLipschitz embedding; 3) we improve upon the estimate of the singular set of planar Dir-minimizing functions by showing that it consists of isolated points."}
{"category": "Math", "title": "Homological properties of Orlik-Solomon algebras", "abstract": "The Orlik-Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first we study homological properties of E-modules as e.g. complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik-Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik-Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the ideal."}
{"category": "Math", "title": "On perfect colorings of the halved 24-cube", "abstract": "A vertex 2-coloring of a graph is said to be perfect with parameters $(a_{ij})_{i,j=1}^k$ if for every $i,j\\in\\{1,...,k\\}$ every vertex of color $i$ is adjacent with exactly $a_{ij}$ vertices of color $j$. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube $\\{0,1\\}^{24}$ with parameters $((20+c,256-c)(c,276-c))$ (i.e., with eigenvalue 20). We prove that such colorings exist for all $c$ from 1 to 128 except 1, 2, 4, 5, 7, 10, 13 and do not exist for $c=1, 2, 4, 5, 7$. Keywords: perfect coloring, equitable partition, hypercube, halved n-cube"}
{"category": "Math", "title": "Finitary incidence algebras", "abstract": "We consider the functions in two variables on an arbitrary poset, for which the convolution operation is defined. We obtain the generalization of incidence algebra and describe its properties: invertibility, the Jackobson radical, idempotents, regular elements. As a consequence a positive solution of the isomorphism problem for such algebras is obtained."}
{"category": "Math", "title": "On quasi-Frobenius semigroup algebras", "abstract": "We define quasi-Frobenius semigroups and find necessary and sufficient conditions under which a semigroup algebra of a 0-cancellative semigroup is quasi-Frobenius."}
{"category": "Math", "title": "Parabolic polygons", "abstract": "Main Theorem. Two parabols have four common points. There exists a circle tangent to the sides of the obtained parabolic quadrilateral if and only if the diagonals of this quadrilateral are orthogonal. The proof of the Main Theorem is elementary and purely synthetic. It is based on the following lemma. Assume that a parabola is tangent to a circle at points A and B. A point P of the plane lyes on the parabola if and only if the distance from the point P to the line AB equals to the length of the tangent from P to the circle. We present some beautiful elementary corollaries of the Main Theorem."}
{"category": "Math", "title": "A note on noncommutative unique ergodicity and weighted means", "abstract": "In this paper we study unique ergodicity of $C^*$-dynamical system $(\\ga,T)$, consisting of a unital $C^*$-algebra $\\ga$ and a Markov operator $T:\\ga\\mapsto\\ga$, relative to its fixed point subspace, in terms of Riesz summation which is weaker than Cesaro one. Namely, it is proven that $(\\ga,T)$ is uniquely ergodic relative to its fixed point subspace if and only if its Riesz means {equation*} \\frac{1}{p_1+...+p_n}\\sum_{k=1}^{n}p_kT^kx {equation*} converge to $E_T(x)$ in $\\ga$ for any $x\\in\\ga$, as $n\\to\\infty$, here $E_T$ is an projection of $\\ga$ to the fixed point subspace of $T$. It is also constructed a uniquely ergodic entangled Markov operator relative to its fixed point subspace, which is not ergodic."}
{"category": "Math", "title": "A codimension two CR singular submanifold that is formally equivalent to a symmetric quadric", "abstract": "Let $M\\subset \\mathbb{C}^{n+1}$ ($n\\geq 2$) be a real analytic submanifold defined by an equation of the form: $w=|z|^2+O(|z|^3)$, where we use $(z,w)\\in \\mathbb{C}^{n}\\times \\mathbb{C}$ for the coordinates of $\\mathbb{C}^{n+1}$. We first derive a pseudo-normal form for $M$ near 0. We then use it to prove that $(M,0)$ is holomorphically equivalent to the quadric $(M_\\infty: w=|z|^2,0)$ if and only if it can be formally transformed to $(M_\\infty,0)$. We also use it to give a necessary and sufficient condition when $(M,0)$ can be formally flattened. The result is due to Moser for the case of $n=1$."}
{"category": "Math", "title": "A complex surface of general type with p_g=0, K^2=2 and H_1=Z/2Z, Z/3Z", "abstract": "As the sequel to [3], we construct a minimal complex surface of general type with p_g=0, K^2=2 and H_1=Z/2Z using a rational blow-down surgery and Q-Gorenstein smoothing theory. We also present an example of p_g = 0,K^2 = 2 and H_1 = Z/3Z."}
{"category": "Math", "title": "Friedrichs' extension lemma with boundary values and applications in complex analysis", "abstract": "Let $Q$ be a first-order differential operator on a compact, smooth oriented Riemannian manifold with smooth boundary. Then, Friedrichs' extension lemma states that the minimal closed extension $Q_{min}$ (the closure of the graph) and the maximal closed extension $Q_{max}$ (in the sense of distributions) of $Q$ in $L^p$-spaces ($1\\leq p<\\infty$) coincide. In the present paper, we show that the same is true for boundary values with respect to $Q_{min}$ and $Q_{max}$. This gives a useful characterization of weak boundary values, particularly for $Q=d-bar$ the Cauchy-Riemann operator. As an application, we derive the Bochner-Martinelli-Koppelman formula for $L^p$-forms with weak d-bar-boundary values."}
{"category": "Math", "title": "Meridian twisting of closed braids and the Homfly polynomial", "abstract": "Let $\\beta$ be a braid on $n$ strands, with exponent sum $w$. Let $\\Delta$ be the Garside half-twist braid. We prove that the coefficient of $v^{w-n+1}$ in the Homfly polynomial of the closure of $\\beta$ agrees with $(-1)^{n-1}$ times the coefficient of $v^{w+n^2-1}$ in the Homfly polynomial of the closure of $\\beta\\Delta^2$. This coincidence implies that the lower Morton--Franks-Williams estimate for the $v$--degree of the Homfly polynomial of $\\hat\\beta$ is sharp if and only if the upper MFW estimate is sharp for the $v$--degree of the Homfly polynomial of $\\hat{\\beta\\Delta^2}$."}
{"category": "Math", "title": "A note on the rank of Heegaard Floer homology", "abstract": "We show that if K is a non-trivial knot inside a homology sphere Y, then the rank of knot Floer homology associated with K is strictly bigger than the rank of Heegaard Floer homology of Y."}
{"category": "Math", "title": "Generalized local cohomology modules and homological Gorenstein dimensions", "abstract": "Let \\fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \\cd_{\\fa}(M,N) denote the supremum of the i's such that H^i_{\\fa}(M,N)\\neq 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for \\cd_{\\fa}(M,N). Next, over a Cohen-Macaulay local ring (R,\\fm), we show that \\cd_{\\fm}(M,N)=\\dim R-\\grade(\\Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context of generalized local cohomology modules."}
{"category": "Math", "title": "A Construction of Complete Ricci-flat K\\\"ahler Manifolds", "abstract": "We consider an extension of the results of S. Bando, R. Kobyashi, G. Tian, and S. T. Yau on the existence of Ricci-flat K\\\"{a}hler metrics on quasi-projective varieties Y=X\\D with \\alpha[D]=c_1(X), \\alpha >1. The requirement that D admit a K\\\"{a}hler-Einstein metric is generalized to the condition that the link S in the normal bundle of D admits a Sasaki-Einstein structure in the Sasaki-cone of the usual Sasaki structure provided the embedding D\\subset X satisfies an additional holomorphic condition. If D is a toric variety, then S always admits a Sasaki-Einstein metric. As an application we prove that every small smooth deformation of a toric Gorenstein singularity admits a complete Ricci-flat K\\\"{a}hler metric asymptotic to a Calabi ansatz metric. Some examples are given which were not previously known."}
{"category": "Math", "title": "Maximal Cohen-Macaulay modules over surface singularities", "abstract": "This is a survey article about properties of Cohen-Macaulay modules over surface singularities. We discuss various results on the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay Correspondence. Finally, we describe matrix factorizations corresponding to the indecomposable Cohen-Macaulay modules over the non-isolated singularities $A_\\infty$ and $D_\\infty$."}
{"category": "Math", "title": "On the unitary subgroup of modular group algebras", "abstract": "It this note we investigate the structure of the group of \\sigma-unitary units in some noncommutative modular group algebras KG, where \\sigma is a non-classical ring involution of KG."}
{"category": "Math", "title": "Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics", "abstract": "Before we dive into the accessibility stream of nowadays indicatory applications of octonions to computer and other sciences and to quantum physics let us focus for a while on the crucially relevant events for todays revival on interest to nonassociativity. Our reflections keep wandering back to the $Brahmagupta$ $Fibonacc$ two square identity and then via the $Euler$ four square identity up to the $Degen$ $Ggraves$ $Cayley$ eight square identity. These glimpses of history incline and invite us to retell the story on how about one month after quaternions have been carved on the $Broughamian$ bridge octonions were discovered by $John$ $Thomas$ $Ggraves$, jurist and mathematician, a friend of $William$ $Rowan$ $Hamilton$. As for today we just mention en passant quaternionic and octonionic quantum mechanics, generalization of $Cauchy$ $Riemann$ equations for octonions and triality principle and $G_2$ group in spinor language in a descriptive way in order not to daunt non specialists. Relation to finite geometries is recalled and the links to the 7stones of seven sphere, seven imaginary octonions units in out of the $Plato$ cave reality applications are appointed . This way we are welcomed back to primary ideas of $Heisenberg$, $Wheeler$ and other distinguished fathers of quantum mechanics and quantum gravity foundations."}
{"category": "Math", "title": "Geometric torsions and invariants of manifolds with triangulated boundary", "abstract": "Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a manifold with a triangulated boundary. These invariants can be naturally united in a vector, and a change of the boundary triangulation corresponds to a linear transformation of this vector. Moreover, when two manifolds are glued by their common boundary, these vectors undergo scalar multiplication, i.e., they work according to M. Atiyah's axioms for a topological quantum field theory."}
{"category": "Math", "title": "On the mean square of the Riemann zeta-function in short intervals", "abstract": "It is proved that, for $T^\\epsilon\\le G = G(T) \\le {1\\over2}\\sqrt{T}$, $$ \\int_T^{2T}\\Bigl(I_1(t+G)-I_1(t)\\Bigr)^2 dt = TG\\sum_{j=0}^3a_j\\log^j \\Bigl({\\sqrt{T}\\over G}\\Bigr) + O_\\epsilon(T^{1+\\epsilon}+ T^{1/2+\\epsilon}G^2) $$ with some explicitly computable constants $a_j (a_3>0)$ where, for a fixed natural number $k$, $$I_k(t,G) = {1\\over\\sqrt{\\pi}}\\int_{-\\infty}^\\infty |\\zeta(1/2+it+iu)|^{2k} {\\rm e}^{-(u/G)^2} du. $$ The generalizations to the mean square of $I_1(t+U,G) - I_1(t,G)$ over $[T, T+H]$ and the estimation of the mean square of $I_2(t+U,G)-I_2(t,G)$ are also discussed."}
{"category": "Math", "title": "Semisimplicity of cellular algebras over positive characteristic fields", "abstract": "In this paper, we investigate semisimplicity of cellular algebras over positive characteristic fields. Our main result shows that the Frame number of cellular algebras characterizes semisimplicity of it. In a sense, this is a generalization of Maschke's theorem."}
{"category": "Math", "title": "Towards new schemes: A Lie-group approach of the CBKDV and its derived equations", "abstract": "The aim of this paper is to propose methods that enable us to build new numerical schemes, which preserve the Lie symmetries of the original differential equations. To this purpose, the compound Burgers-Korteweg-de Vries (\\textit{CBKDV}) equation is considered. The particular case of the Burgers equation is taken as a numerical example, and the resulting semi-invariant scheme is exposed."}
{"category": "Math", "title": "An explicit d-bar-integration formula for weighted homogeneous varieties", "abstract": "Let Y be a weighted homogeneous (singular) subvariety of C^n. The main objective of this paper is to present an explicit formula for solving the d-bar-equation $f=\\dbar{g}$ on the regular part of Y, where $f$ is a d-bar-closed $(0,1)$-form with compact support. This formula will then be used to give H\\\"older estimates for the solution in case $Y$ is homogeneous (a cone) with an isolated singularity. Finally, a slight modification of our formula also gives an $L^2$-bounded solution operator in case Y is pure dimensional and homogeneous."}
{"category": "Math", "title": "A d-bar-theoretical proof of Hartogs' Extension Theorem on Stein spaces with isolated singularities", "abstract": "Let X be a connected normal Stein space of pure dimension d>=2 with isolated singularities only. By solving a weighted d-bar-equation with compact support on a desingularization of X, we derive Hartogs' Extension Theorem on X by the d-bar-idea due to Ehrenpreis."}
{"category": "Math", "title": "Teichmuller geometry of moduli space, I: Distance minimizing rays and the Deligne-Mumford compactification", "abstract": "Let $S$ be a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we relate two important but disparate topics in the study of the moduli space $\\M(S)$ of Riemann surfaces: Teichm\\\"{u}ller geometry and the Deligne-Mumford compactification. We reconstruct the Deligne-Mumford compactification (as a metric stratified space) purely from the intrinsic metric geometry of $\\M(S)$ endowed with the Teichm\\\"{u}ller metric. We do this by first classifying (globally) geodesic rays in $\\M(S)$ and determining precisely how pairs of rays asymptote. We construct an \"iterated EDM ray space\" functor, which is defined on a quite general class of metric spaces. We then prove that this functor applied to $\\M(S)$ produces the Deligne-Mumford compactification."}
{"category": "Math", "title": "On q-deformed gl(l+1)-Whittaker function", "abstract": "We propose new explicit form of q-deformed Whittaker functions solving q-deformed gl(l+1)-Toda chains. In the limit q->1 constructed solutions reduce to classical class one gl(l+1)-Whittaker functions in the form proposed by Givental. An important property of the proposed expression for the q-deformed gl(l+1)-Whittaker function is that it can be represented as a character of C*x GL(l+1). This provides a q-version of the Shintani-Casselman-Shalika formula for p-adic Whittaker function. The Shintani-Casselman-Shalika formula is recovered in the limit q->0 when the q-deformed Whittaker function is reduced to a character of a finite-dimensional representation of gl(l+1) expressed through Gelfand-Zetlin bases."}
{"category": "Math", "title": "Global analytic geometry", "abstract": "We define a new type of valuation of a ring that combines the notion of Krull valuation with that of multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic space. After defining natural generalized valuation spectra and computing the spectrum of Z and Z[X], we define analytic spectra and sheaves of analytic functions on them."}
{"category": "Math", "title": "Perelman, Poincare, and the Ricci Flow", "abstract": "In this expository article, we introduce the topological ideas and context central to the Poincare Conjecture. Our account is intended for a general audience, providing intuitive definitions and spatial intuition whenever possible. We define surfaces and their natural generalizations, manifolds. We then discuss the classification of surfaces as it relates to the Poincare and Thurston Geometrization conjectures. Finally, we survey Perelman's results on Ricci flows with surgery."}
{"category": "Math", "title": "About the d-bar-equation at isolated singularities with regular exceptional set", "abstract": "Let Y be a pure dimensional analytic variety in C^n with an isolated singularity at the origin such that the exceptional set X of a desingularization of Y is regular. The main objective of this paper is to present a technique which allows to determine obstructions to the solvability of the d-bar-equation in the $L^2$ respectively $L^\\infty$ sense on Y^*=Y-{0} in terms of certain cohomology classes on X. More precisely, let D be a Stein domain, relatively compact in Y, containing the origin, D^*=D-{0}. We give a sufficient condition for the solvability of the d-bar-equation in the $L^2$-sense on D^*; and in the $L^\\infty$ sense, if D is in addition strongly pseudoconvex. If Y is an irreducible cone, we also give some necessary conditions and obtain optimal Hoelder estimates for solutions of the d-bar-equation."}
{"category": "Math", "title": "The conditioned reconstructed process", "abstract": "We investigate a neutral model for speciation and extinction, the constant rate birth-death process. The process is conditioned to have $n$ extant species today, we look at the tree distribution of the reconstructed trees-- i.e. the trees without the extinct species. Whereas the tree shape distribution is well-known and actually the same as under the pure birth process, no analytic results for the speciation times were known. We provide the distribution for the speciation times and calculate the expectations analytically. This characterizes the reconstructed trees completely. We will show how the results can be used to date phylogenies."}
{"category": "Math", "title": "How to Play Dundee", "abstract": "We consider the following one-player game called Dundee. We are given a deck consisting of s_i cards of Value i, where i=1,...,v, and an integer m\\le s_1+...+s_v. There are m rounds. In each round, the player names a number between 1 and v and draws a random card from the deck. The player loses if the named number coincides with the drawn value in at least one round. The famous Problem of Thirteen, proposed by Monmort in 1708, asks for the probability of winning in the case when v=13, s_1=...=s_{13}=4, m=13, and the player names the sequence 1,...,13. This problem and its various generalizations were studied by numerous mathematicians, including J. and N. Bernoulli, De Moivre, Euler, Catalan, and others. However, it seems that nobody has considered which strategies of the player maximize the probability of winning. We study two variants of this problem. In the first variant, the player's bid in Round i may depend on the values of the random cards drawn in the previous rounds. We completely solve this version. In the second variant, the player has to specify the whole sequence of m bids in advance, before turning any cards. We are able to solve this problem when s_1=...=s_v and m is arbitrary."}
{"category": "Math", "title": "A Bound for Orders in Differential Nullstellensatz", "abstract": "We give the first known bound for orders of differentiations in differential Nullstellensatz for both partial and ordinary algebraic differential equations. This problem was previously addressed by A. Seidenberg but no complete solution was given. Our result is a complement to the corresponding result in algebraic geometry, which gives a bound on degrees of polynomial coefficients in effective Nullstellensatz."}
{"category": "Math", "title": "Application of Synchronization to Formation Flying Spacecraft: Lagrangian Approach", "abstract": "This article presents a unified synchronization framework with application to precision formation flying spacecraft. Central to the proposed innovation, in applying synchronization to both translational and rotational dynamics in the Lagrangian form, is the use of the distributed stability and performance analysis tool, called contraction analysis that yields exact nonlinear stability proofs. The proposed decentralized tracking control law synchronizes the attitude of an arbitrary number of spacecraft into a common time-varying trajectory with global exponential convergence. Moreover, a decentralized translational tracking control law based on phase synchronization is presented, thus enabling coupled translational and rotational maneuvers. While the translational dynamics can be adequately controlled by linear control laws, the proposed method permits highly nonlinear systems with nonlinearly coupled inertia matrices such as the attitude dynamics of spacecraft whose large and rapid slew maneuvers justify the nonlinear control approach. The proposed method integrates both the trajectory tracking and synchronization problems in a single control framework."}
{"category": "Math", "title": "Rank Two Sheaves on K3 Surfaces: A Special Construction", "abstract": "Let X be a K3 surface of degree 8 in P^5 with hyperplane section H. We associate to it another K3 surface M which is a double cover of P^2 ramified on a sextic curve C. In the generic case when X is smooth and a complete intersection of three quadrics, there is a natural correspondence between M and the moduli space M' of rank two vector bundles on X with Chern classes c_1=H and c_2=4. We build on previous work of Mukai and others, giving conditions and examples where M' is fine, compact, non-empty; and birational or isomorphic to M. We also present an explicit calculation of the Fourier-Mukai transform when X contains a line and has Picard number two."}
{"category": "Math", "title": "The weight in a Serre-type conjecture for tame n-dimensional Galois representations", "abstract": "We formulate a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for n = 4 using automorphic inductions of Hecke characters."}
{"category": "Math", "title": "Spectral symmetries of zeta functions", "abstract": "We define, answering a question of Sarnak in his letter to Bombieri, a symplectic pairing on the spectral interpretation (due to Connes and Meyer) of the zeroes of Riemann's zeta function. This pairing gives a purely spectral formulation of the proof of the functional equation due to Tate, Weil and Iwasawa, which, in the case of a curve over a finite field, corresponds to the usual geometric proof by the use of the Frobenius-equivariant Poincar\\'e duality pairing in etale cohomology. We give another example of a similar construction in the case of the spectral interpretation of the zeroes of a cuspidal automorphic $L$-function, but this time of an orthogonal nature. These constructions are in adequation with Deninger's conjectural program and the arithmetic theory of random matrices."}
{"category": "Math", "title": "Fine properties of self-similar solutions of the Navier-Stokes equations", "abstract": "We study the solutions of the nonstationary incompressible Navier--Stokes equations in $\\R^d$, $d\\ge2$, of self-similar form $u(x,t)=\\frac{1}{\\sqrt t}U\\bigl(\\frac{x}{\\sqrt t}\\bigr)$, obtained from small and homogeneous initial data $a(x)$. We construct an explicit asymptotic formula relating the self-similar profile $U(x)$ of the velocity field to its corresponding initial datum $a(x)$."}
{"category": "Math", "title": "Iwasawa theory of totally real fields for certain non-commutative $p$-extensions", "abstract": "In this paper, we prove the Iwasawa main conjecture of totally real fields for certain specific non-commutative $p$-adic Lie extensions, using the integral logarithms introduced by Oliver and Taylor. Our result gives certain generalization of Kazuya Kato's proof of the main conjecture for Galois extensions of Heisenberg type."}
{"category": "Math", "title": "The order completion method for systems of nonlinear PDEs revisited", "abstract": "In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion. In particular, spaces of generalized functions are constructed that contain generalized solutions to all systems of continuous, nonlinear PDEs. In terms of the existence and uniqueness results previously obtained for such systems of equations, one may interpret the existence of generalized solutions presented here as a regularity result."}
{"category": "Math", "title": "Newtonian limit for weakly viscoelastic fluid flows of Olroyds' type", "abstract": "This paper is concerned with regular flows of incompressible weakly viscoelastic fluids which obey a differential constitutive law of Oldroyd type. We study the newtonian limit for weakly viscoelastic fluid flows in $\\R^N$ or $\\T^N$ for $N=2, 3$, when the Weissenberg number (relaxation time measuring the elasticity effect in the fluid) tends to zero. More precisely, we prove that the velocity field and the extra-stress tensor converge in their existence spaces (we examine the Sobolev-$H^s$ theory and the Besov-$B^{s,1}_2$ theory to reach the critical case $s= N/2$) to the corresponding newtonian quantities. These convergence results are established in the case of \"ill-prepared\"' data.We deduce, in the two-dimensional case, a new result concerning the global existence of weakly viscoelastic fluids flow. Our approach makes use of essentially two ingredients : the stability of the null solution of the viscoelastic fluids flow and the damping effect,on the difference between the extra-stress tensor and the tensor of rate of deformation, induced by the constitutive law of the fluid."}
{"category": "Math", "title": "Cutting Sequences and Palindromes", "abstract": "We give a unified geometric approach to some theorems about primitive elements and palindromes in free groups of rank 2. The geometric treatment gives new proofs of the theorems. Dedicated to Bill Harvey on his 65th birthday."}
{"category": "Math", "title": "Monodromy Groups of Hurwitz-type Problems", "abstract": "We solve the Hurwitz monodromy problem for degree-4 covers. That is, the Hurwitz space H_{4,g} of all simply branched covers of P^1 of degree 4 and genus g is an unramified cover of the space P_{2g+6} of (2g+6)-tuples of distinct points in P^1. We determine the monodromy of pi_1(P_{2g+6}) on the points of the fiber. This turns out to be the same problem as the action of pi_1(P_{2g+6}) on a certain local system of Z/2-vector spaces. We generalize our result by treating the analogous local system with Z/N coefficients, gcd(3,N)=1, in place of Z/2. This in turn allows us to answer a question of Ellenberg concerning families of Galois covers of P^1 with deck group (Z/N)^2:S_3."}
{"category": "Math", "title": "Moufang symmetry VII. Moufang transformations", "abstract": "Concept of a birepresentation for the Moufang loops is elaborated."}
{"category": "Math", "title": "Moufang symmetry VIII. Reconstruction of Moufang loops", "abstract": "Reconstruction theorem for the Moufang loops is proved."}
{"category": "Math", "title": "Representation of mean-periodic functions in series of exponential polynomials", "abstract": "Let $\\theta$ be a Young function and consider the space $\\mathcal{F}_{\\theta}(\\C)$ of all entire functions with $\\theta$-exponential growth. In this paper, we are interested in the solutions $f\\in \\mathcal{F}_{\\theta}(\\C)$ of the convolution equation $T\\star f=0$, called mean-periodic functions, where $T$ is in the topological dual of $\\mathcal{F}_{\\theta}(\\C)$. We show that each mean-periodic function can be represented in an explicit way as a convergent series of exponential polynomials."}
{"category": "Math", "title": "Secondary multiplication in Tate cohomology of certain p-groups", "abstract": "Let k be a field and let G be a finite group. By a theorem of D.Benson, H.Krause and S.Schwede, there is a canonical element in the Hochschild cohomology of the Tate cohomology HH^{3,-1} H*G with the following property: Given any graded H*G-module X, the image of the canonical element in Ext^{3,-1}(X,X) is zero if and only if X is isomorphic to a direct summand of H*(G,M) for some kG-module M. We investigate this canonical element in certain special cases, namely that of (finite) abelian p-groups and the quaternion group. In case of non-triviality of the canonical element, we also give examples of non-realizable modules X."}
{"category": "Math", "title": "Stability of multipeakons", "abstract": "The Camassa-Holm equation possesses well-known peaked solitary waves that are called peakons. Their orbital stability has been established by Constantin and Strauss (2000). We prove here the stability of ordered trains of peakons. We also establish a result on the stability of multipeakons."}
{"category": "Math", "title": "Number of \"udu\" of a Dyck path and ad-nilpotent ideals of parabolic subalgebras of sl_{l+1}(C)", "abstract": "For an ad-nilpotent ideal $i$ of a Borel subalgebra of $sl_{l+1}(C)$, we denote by $I_i$ the maximal subset $I$ of the set of simple roots such that $i$ is an ad-nilpotent ideal of the standard parabolic subalgebra $p_I$. We use the bijection given by G.E. Andrews, C. Krattenthaler, L. Orsina and P. Papi between the set of ad-nilpotent ideals of a Borel subalgebra in $sl_{l+1}(C)$ and the set of Dyck paths of length $2l+2$, to explicit a bijection between ad-nilpotent ideals $i$ of the Borel subalgebra such that the cardinality of $I_i$ is equal to $r$ and the Dyck paths of length $2l+2$ having $r$ occurence \"udu\". We obtain also a duality between antichains of cardinality $p$ and $l-p$ in the set of positive roots."}
{"category": "Math", "title": "Autobiographical Numbers", "abstract": "I introduce autobiographical numbers as defined in A046043 (see Online Encyclopedia of Integer Sequences). I continue by defining and analyzing biographies, curricula vitae and complete life stories of numbers. I end with the definition of mutually-praising number pairs."}
{"category": "Math", "title": "Generalized Fourier Integral Operators on spaces of Colombeau type", "abstract": "Generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is based on a theory of generalized oscillatory integrals (OIs) whose phase functions as well as amplitudes may be generalized functions of Colombeau type. The mapping properties of these FIOs are studied as the composition with a generalized pseudodifferential operator. Finally, the microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets are investigated. This theory of generalized FIOs is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distributional data."}
{"category": "Math", "title": "Surprising properties of centralisers in classical Lie algebras", "abstract": "Let $g$ be a classical Lie algebra, i.e., either $gl_n$, $sp_n$, or $so_n$ and let $e\\in g$ be a nilpotent element. We study various properties of centralisers $g_e$. The first four sections deal with rather elementary questions, like the centre of $g_e$, commuting varieties associated with $g_e$, or centralisers of commuting pairs. The second half of the paper addresses problems related to different Poisson structures on $g_e^*$ and symmetric invariants of $g_e$."}
{"category": "Math", "title": "Aspects of stable polynomials", "abstract": "This note is an introduction to the properties of stable polynomials in several variables with real or complex coefficients. These polynomials are defined in terms of where the polynomial is non-vanishing. We do not cover well-known topics in one variable such as Routh-Hurwitz, the Edge theorem, and Kharitonov theory."}
{"category": "Math", "title": "Wild twistor D-modules", "abstract": "We propose a definition of (polarized) wild twistor D-modules, generalizing to objects with irregular singularities that of (polarized) regular twistor D-modules. We give a precise analysis in dimension one."}
{"category": "Math", "title": "Nontrivial lower bounds for the least common multiple of some finite sequences of integers", "abstract": "We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form $u_n = a n (n + t) + b$ with $(a, t, b) \\in {\\mathbb{Z}}^3, a \\geq 5, t \\geq 0, \\rm{gcd}(a, b) = 1$. From this, we deduce for instance the lower bound: $\\mathrm{lcm}\\{1^2 + 1, 2^2 + 1, ..., n^2 + 1\\} \\geq 0,32 (1,442)^n$ (for all $n \\geq 1$). In the last part of this article, we study the integer $\\mathrm{lcm}(n, n + 1, ..., n + k)$ $(k \\in \\mathbb{N}, n \\in {\\mathbb{N}}^*)$. We show that it has a divisor $d_{n, k}$ simple in its dependence on $n$ and $k$, and a multiple $m_{n, k}$ also simple in its dependence on $n$. In addition, we prove that both equalities: $\\mathrm{lcm}(n, n + 1, ..., n + k) = d_{n, k}$ and $\\mathrm{lcm}(n, n + 1, ..., n + k) = m_{n, k}$ hold for an infinitely many pairs $(n, k)$."}
{"category": "Math", "title": "Hodge correlators", "abstract": "Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by an afine line. We call them twistor connections. Generalising this, we suggest a DG enhancement of the subcategory of Saito's Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin-Selberg integrals. Examples of the motivic correlators include Beilinson's elements in the motivic cohomology, e.g. the ones delivering the Beilinson - Kato Euler system on modular curves."}
{"category": "Math", "title": "Spectral measures on toric varieties and the asymptotic expansion of Tian-Yau-Zelditch", "abstract": "We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety. We show how this extended result together with the Tian-Yau-Zelditch asymptotic expansion can be used to deduce Abreu's formula for the scalar curvature of a toric metric on a toric variety in terms of polytope data."}
{"category": "Math", "title": "Counting Defective Parking Functions", "abstract": "Suppose that $n$ drivers each choose a preferred parking space in a linear car park with $m$ spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with larger number (if any). If all drivers park successfully, the sequence of choices is called a parking function. In general, if $k$ drivers fail to park, we have a \\emph{defective parking function} of \\emph{defect} $k$. Let $\\cp(n,m,k)$ be the number of such functions. In this paper, we establish a recurrence relation for the numbers $\\cp(n,m,k)$, and express this as an equation for a three-variable generating function. We solve this equation using the kernel method, and extract the coefficients explicitly: it turns out that the cumulative totals are partial sums in Abel's binomial identity. Finally, we compute the asymptotics of $\\cp(n,m,k)$. In particular, for the case $m=n$, if choices are made independently at random, the limiting distribution of the defect (the number of drivers who fail to park), scaled by the square root of $n$, is the Rayleigh distribution. On the other hand, in case $m=\\omega(n)$, the probability that all spaces are occupied tends asymptotically to one."}
{"category": "Math", "title": "Remarks on the faithfulness of the Jones representations", "abstract": "We consider the linear representations of the mapping class group of an n-punctured 2-sphere constructed by V. F. R. Jones using Iwahori-Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori-Hecke algebra representation of Artin's braid group of n-1 strands. In the case of n=6, we provide a further restriction for the kernel using our previous result, as well as a certain relation to the Burau representation of degree 4."}
{"category": "Math", "title": "A class of variational functionals in conformal geometry", "abstract": "We derive a class of variational functionals which arise naturally in conformal geometry. In the special case when the Riemannian manifold is locally conformal flat, the functional coincides with the well studied functional which is the integration over the manifold of the k-symmetric function of the Schouten tensor of the metric on the manifold."}
{"category": "Math", "title": "The \\bar{\\partial}_b Neumann problem on noncharacteristic domains", "abstract": "We study the $\\bar{\\partial}_b$-Neumann problem for domains $\\Omega$ contained in a strictly pseudoconvex manifold M^{2n+1} whose boundaries are noncharacteristic and have defining functions depending solely on the real and imaginary parts of a single CR function w. When the Kohn Laplacian is a priori known to have closed range in L^2, we prove sharp regularity and estimates for solutions. We establish a condition on the boundary which is sufficient for the Kohn Laplacian to be Fredholm on $L^2_{(0,q)}(\\Omega)$ and show that this condition always holds when M is embedded as a hypersurface in C^{n+1}. We present examples where the inhomogeneous $\\bar{\\partial}_b$ equation can always be solved smoothly up to the boundary on (p,q)-forms with 0<q<n-1."}
{"category": "Math", "title": "Hilbert schemes of 8 points", "abstract": "The Hilbert scheme H^d_n of n points in A^d contains an irreducible component R^d_n which generically represents n distinct points in A^d. We show that when n is at most 8, the Hilbert scheme H^d_n is reducible if and only if n = 8 and d >= 4. In the simplest case of reducibility, the component R^4_8 \\subset H^4_8 is defined by a single explicit equation which serves as a criterion for deciding whether a given ideal is a limit of distinct points. To understand the components of the Hilbert scheme, we study the closed subschemes of H_n^d which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multigraded Hilbert schemes, and we describe their components when the colength is at most 8. In particular, we show that the scheme corresponding to the Hilbert function (1,3,2,1) is the minimal reducible example."}
{"category": "Math", "title": "Cox rings of degree one del Pezzo surfaces", "abstract": "Let X be a del Pezzo surface of degree one over an algebraically closed field (of any characteristic), and let Cox(X) be its total coordinate ring. We prove the missing case of a conjecture of Batyrev and Popov, which states that Cox(X) is a quadratic algebra. We use a complex of vector spaces whose homology determines part of the structure of the minimal free Pic(X)-graded resolution of Cox(X) over a polynomial ring. We show that sufficiently many Betti numbers of this minimal free resolution vanish to establish the conjecture."}
{"category": "Math", "title": "The uniform Korn - Poincar\\'e inequality in thin domains", "abstract": "We study the Korn-Poincar\\'e inequality: \\|u\\|_{W^{1,2}(S^h)} < C_h \\|D(u)\\|_{L^2(S^h)}, in domains S^h that are shells of small thickness of order h, around an arbitrary smooth and closed hypersurface S in R^n. By D(u) we denote the symmetric part of the gradient \\nabla u, and we assume the tangential boundary conditions: u\\vec n^h = 0 on \\partial S^h. We prove that C_h remains uniformly bounded as h tends to 0, for vector fields u in any family of cones (with angle <\\pi/2, uniform in h) around the orthogonal complement of extensions of Killing vector fields on S. We also show that this condition is optimal, as in turn every Killing field admits a family of extensions u^h, for which the ratio: \\|u^h\\|_{W^{1,2}(S^h)} / \\|D(u^h)\\|_{L^2(S^h)} blows up as h tends to 0, even if the domains S^h are not rotationally symmetric."}
{"category": "Math", "title": "Maximaland Primitive Elements in Baby Verma Modules for Type $B_2$", "abstract": "The purpose of this paper is to find maximal and primitive elements of baby Verma modules for a quantum group of type $B_2$. As a consequence the composition factors of the baby Verma modules are determined. Similar approach can be used to find find maximal and primitive elements of Weyl modules for type $B_2$."}
{"category": "Math", "title": "Convergence of adaptive finite element methods for eigenvalue problems", "abstract": "In this article we prove convergence of adaptive finite element methods for second order elliptic eigenvalue problems. We consider Lagrange finite elements of any degree and prove convergence for simple as well as multiple eigenvalues under a minimal refinement of marked elements, for all reasonable marking strategies, and starting from any initial triangulation."}
{"category": "Math", "title": "A coarse classification of countable abelian groups", "abstract": "We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H where H is a direct sum (possibly infinite) of cyclic groups. A partial generalization to countable abelian groups of the Gromov rigidity theorem for abelian groups is shown."}
{"category": "Math", "title": "Some results on cosymplectic manifolds", "abstract": "We obtain a generalization of the Kodaira-Morrow stability theorem for cosymplectic structures. We investigate cosymplectic geometry on Lie groups and on their compact quotients by uniform discrete subgroups. In this way we show that a compact solvmanifold admits a cosymplectic structure if and only if it is a finite quotient of a torus."}
{"category": "Math", "title": "Contact homology of left-handed stabilizations and plumbing of open books", "abstract": "We show that on any closed contact manifold of dimension greater than 1 a contact structure with vanishing contact homology can be constructed. The basic idea for the construction comes from Giroux. We use a special open book decomposition for spheres. The page is the cotangent bundle of a sphere and the monodromy is given by a left-handed Dehn twist. In the resulting contact manifold we exhibit a closed Reeb orbit that bounds a single finite energy plane. As a result, the unit element of the contact homology algebra is exact and so the contact homology vanishes. This result can be extended to other contact manifolds by using connected sums. The latter is related to the plumbing- or 2-Murasugi sum of the contact open books. We shall give a possible description of this construction and some conjectures about the plumbing operation."}
{"category": "Math", "title": "Multiscale Inference for High-Frequency Data", "abstract": "This paper proposes a novel multiscale estimator for the integrated volatility of an Ito process, in the presence of market microstructure noise (observation error). The multiscale structure of the observed process is represented frequency-by-frequency and the concept of the multiscale ratio is introduced to quantify the bias in the realized integrated volatility due to the observation error. The multiscale ratio is estimated from a single sample path, and a frequency-by-frequency bias correction procedure is proposed, which simultaneously reduces variance. We extend the method to include correlated observation errors and provide the implied time domain form of the estimation procedure. The new method is implemented to estimate the integrated volatility for the Heston and other models, and the improved performance of our method over existing methods is illustrated by simulation studies."}
{"category": "Math", "title": "Asymptotic behaviour of a rapidly rotating fluid with random stationary surface stress", "abstract": "The goal of this paper is to describe in mathematical terms the effect on the ocean circulation of a random stationary wind stress at the surface of the ocean. In order to avoid singular behaviour, non-resonance hypotheses are introduced, which ensure that the time frequencies of the wind-stress are different from that of the Earth rotation. We prove a convergence result for a three-dimensional Navier-Stokes-Coriolis system in a bounded domain, in the asymptotic of fast rotation and vanishing vertical viscosity, and we exhibit some random and stationary boundary layer profiles. At last, an average equation is derived for the limit system in the case of the non-resonant torus."}
{"category": "Math", "title": "Semicosimplicial DGLAs in deformation theory", "abstract": "We identify Cech cocycles in nonabelian (formal) group cohomology with Maurer-Cartan elements in a suitable L-infinity algebra. Applications to deformation theory are described."}
{"category": "Math", "title": "Bijective counting of plane bipolar orientations and Schnyder woods", "abstract": "A bijection $\\Phi$ is presented between plane bipolar orientations with prescribed numbers of vertices and faces, and non-intersecting triples of upright lattice paths with prescribed extremities. This yields a combinatorial proof of the following formula due to R. Baxter for the number $\\Theta_{ij}$ of plane bipolar orientations with $i$ non-polar vertices and $j$ inner faces: $\\Theta_{ij}=2\\frac{(i+j)!(i+j+1)!(i+j+2)!}{i!(i+1)!(i+2)!j!(j+1)!(j+2)!}$. In addition, it is shown that $\\Phi$ specializes into the bijection of Bernardi and Bonichon between Schnyder woods and non-crossing pairs of Dyck words."}
{"category": "Math", "title": "A note on sensitivity of principal component subspaces and the efficient detection of influential observations in high dimensions", "abstract": "In this paper we introduce an influence measure based on second order expansion of the RV and GCD measures for the comparison between unperturbed and perturbed eigenvectors of a symmetric matrix estimator. Example estimators are considered to highlight how this measure compliments recent influence analysis. Importantly, we also show how a sample based version of this measure can be used to accurately and efficiently detect influential observations in practice."}
{"category": "Math", "title": "Hyperbolic mean curvature flow: Evolution of plane curves", "abstract": "In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. We show that there exists a class of initial velocities such that the solution of the corresponding initial value problem exists only at a finite time interval $[0,T_{\\max})$ and when $t$ goes to $T_{\\max}$, the solution converges to a point. We also discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time $\\mathbb{R}^{1,1}$."}
{"category": "Math", "title": "New Semifield Planes of order 81", "abstract": "A finite semifield is a finite nonassociative ring with identity such that the set of its nonzero elements is closed under the product. From any finite semifield a projective plane can be constructed. In this paper we obtain new semifield planes of orders 81 by means of computational methods. These computer-assisted results yield to a complete classification (up to isotopy) of 81-element finite semifields."}
{"category": "Math", "title": "The Mahler measure and the L-series of a singular K3-surface", "abstract": "We present the first example of a polynomial defining a singular K3-surface whose Mahler measure is expressed in terms of the Mahler measure of the faces of its Newton polyhedron and the L-series of the K3-surface."}
{"category": "Math", "title": "A series whose sum range is an arbitrary finite set", "abstract": "In finitely-dimensional spaces the sum range of a series has to be an affine subspace. It is long known this is not the case in infinitely dimensional Banach spaces. In particular in 1984 M.I. Kadets and K. Wo\\`{z}niakowski obtained an example of a series the sum range of which consisted of two points, and asked whether it is possible to obtain more than two, but finitely many points. This paper answers the question positively, by showing how to obtain an arbitrary finite set as the sum range of a series in any infinitely dimensional Banach space."}
{"category": "Math", "title": "A Conjecture about the Density of Prime Numbers", "abstract": "We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemann's function in the domain that goes from 2 to 1010 at least. Instead of using a constant as was done by Legendre and others in the formula of Gauss, we try to adjust the data through a function. This function has the remarkable property: its points of discontinuity are the prime numbers."}
{"category": "Math", "title": "Combinatorial Koszul Homology: Computations and Applications", "abstract": "With a particular focus on explicit computations and applications of the Koszul homology and Betti numbers of monomial ideals, the main goals of this thesis are the following: Analyze the Koszul homology of monomial ideals and apply it to describe the structure of monomial ideals. Describe algorithms to perform efficient computations of the homological invariants of monomial ideals. Apply the theory and computations on monomial ideals to problems inside and outside mathematics The thesis introduces as a main tool Mayer-Vietoris trees of monomial ideals."}
{"category": "Math", "title": "Curvature of a class of indefinite globally framed $f$-manifolds", "abstract": "We present a compared analysis of some properties of indefinite almost $\\mathcal{S}$-manifolds and indefinite $\\mathcal{S}$-manifolds. We give some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields. We study the sectional and ${\\phi}$-sectional curvature of indefinite almost $\\mathcal{S}$-manifolds and state an expression of the curvature tensor field for the indefinite $\\mathcal{S}$-space forms. We analyse the sectional curvature of indefinite $\\mathcal{S}$-manifold in which the number of the spacelike characteristic vector fields is equal to that of the timelike characteristic vector fields. Some examples are also described."}
{"category": "Math", "title": "The square negative correlation property for generalized Orlicz balls", "abstract": "Antilla, Ball and Perissinaki proved that the squares of coordinate functions in $\\ell_p^n$ are negatively correlated. This paper extends their results to balls in generalized Orlicz norms on R^n. From this, the concentration of the Euclidean norm and a form of the Central Limit Theorem for the generalized Orlicz balls is deduced. Also, a counterexample for the square negative correlation hypothesis for 1-symmetric bodies is given. Currently the CLT is known in full generality for convex bodies (see the paper \"Power-law estimates for the central limit theorem for convex sets\" by B. Klartag), while for generalized Orlicz balls a much more general result is true (see \"The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball\" by M. Pilipczuk and J. O. Wojtaszczyk). While, however, both aforementioned papers are rather long, complicated and technical, this paper gives a simple and elementary proof of, eg., the Euclidean concentration for generalized Orlicz balls."}
{"category": "Math", "title": "The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball", "abstract": "Random variables equidistributed on convex bodies have received quite a lot of attention in the last few years. In this paper we prove the negative association property (which generalizes the subindependence of coordinate slabs) for generalized Orlicz balls. This allows us to give a strong concentration property, along with a few moment comparison inequalities. Also, the theory of negatively associated variables is being developed in its own right, which allows us to hope more results will be available. Moreover, a simpler proof of a more general result for $\\ell_p^n$ balls is given."}
{"category": "Math", "title": "On finiteness of odd superperfect numbers", "abstract": "Some new results concerning the equation $\\sigma(N)=aM, \\sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors."}
{"category": "Math", "title": "Multivariate integration in C^\\infty([0,1]^d) is not strongly tractable", "abstract": "It has long been known that the multivariate integration problem for the unit ball in $C^r([0,1]^d)$ is intractable for fixed finite $r$. H. Wo\\'zniakowski has recently conjectured that this is true even if $r=\\infty$. This paper establishes a partial result in this direction. We prove that the multivariate integration problem, for infinitely differential functions all of whose variables are bounded by one, is not strongly tractable."}
{"category": "Math", "title": "Individual Risk and Lebesgue Extension without Aggregate Uncertainty", "abstract": "Many economic models include random shocks imposed on a large number (continuum) of economic agents with individual risk. In this context, an exact law of large numbers and its converse is presented in Sun (2006) to characterize the cancelation of individual risk via aggregation. However, it is well known that the Lebesgue unit interval is not suitable for modeling a continuum of agents in the particular setting. The purpose of this paper is to show that an extension of the Lebesgue unit interval does work well as an agent space with various desirable properties associated with individual risk."}
{"category": "Math", "title": "Vertex degrees of Steiner Minimal Trees in $\\ell_p^d$ and other smooth Minkowski spaces", "abstract": "We find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees in the d-dimensional Banach spaces \\ell_p^d independent of d. This is in contrast to Minimal Spanning Trees, where the maximum degree of vertices grows exponentially in d (Robins and Salowe, 1995). Our upper bounds follow from characterizations of singularities of SMT's due to Lawlor and Morgan (1994), which we extend, and certain \\ell_p-inequalities. We derive a general upper bound of d+1 for the degree of vertices of an SMT in an arbitrary smooth d-dimensional Banach space; the same upper bound for Steiner points having been found by Lawlor and Morgan. We obtain a second upper bound for the degrees of vertices in terms of 1-summing norms."}
{"category": "Math", "title": "On the characterization of Hilbertian fields", "abstract": "The main goal of this work is to answer a question of P. D`ebes and D. Haran by relaxing the condition for Hilbertianity. Namely we prove that for a field K to be Hilbertian it suffices that K has the irreducible specialization property merely for absolutely irreducible polynomials."}
{"category": "Math", "title": "Stein's method and exact Berry--Esseen asymptotics for functionals of Gaussian fields", "abstract": "We show how to detect optimal Berry--Esseen bounds in the normal approximation of functionals of Gaussian fields. Our techniques are based on a combination of Malliavin calculus, Stein's method and the method of moments and cumulants, and provide de facto local (one-term) Edgeworth expansions. The findings of the present paper represent a further refinement of the main results proven in Nourdin and Peccati [Probab. Theory Related Fields 145 (2009) 75--118]. Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan [Probab. Theory Related Fields 100 (1994) 395--406] and Ginovyan and Sahakyan [Probab. Theory Related Fields 138 (2007) 551--579]); (ii) to ``exploding'' quadratic functionals of a Brownian sheet; and (iii) to a continuous-time version of the Breuer--Major CLT for functionals of a fractional Brownian motion."}
{"category": "Math", "title": "Balancing unit vectors", "abstract": "Theorem A. Let $x_1,...,x_{2k+1}$ be unit vectors in a normed plane. Then there exist signs $\\epsi_1,...,\\epsi_{2k+1}\\in\\{\\pm 1\\}$ such that $\\norm{\\sum_{i=1}^{2k+1}\\epsi_i x_i}\\leq 1$. We use the method of proof of the above theorem to show the following point facility location result, generalizing Proposition 6.4 of Y. S. Kupitz and H. Martini (1997). Theorem B. Let $p_0,p_1,...,p_n$ be distinct points in a normed plane such that for any $1\\leq i<j\\leq n$ the closed angle $\\angle p_ip_0p_j$ contains a ray opposite some $\\overrightarrow{p_0p_k}, 1\\leq k\\leq n$. Then $p_0$ is a Fermat-Toricelli point of $\\{p_0,p_1,...,p_n\\}$, i.e. $x=p_0$ minimizes $\\sum_{i=0}^n\\norm{x-p_i}$. We also prove the following dynamic version of Theorem A. Theorem C. Let $x_1,x_2,...$ be a sequence of unit vectors in a normed plane. Then there exist signs $\\epsi_1,\\epsi_2,...\\in\\{\\pm 1\\}$ such that $\\norm{\\sum_{i=1}^{2k}\\epsi_i x_i}\\leq 2$ for all $k\\in\\N$. Finally we discuss a variation of a two-player balancing game of J. Spencer (1977) related to Theorem C."}
{"category": "Math", "title": "The finiteness dimension of local cohomology modules and its dual notion", "abstract": "Let \\fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{\\fa}(M), the finiteness dimension of M with respect to \\fa, and, its dual notion q_{\\fa}(M), the Artinianess dimension of M with respect to \\fa. When (R,\\fm) is local and r:=f_{\\fa}(M) is less than f_{\\fa}^{\\fm}(M), the \\fm-finiteness dimension of M relative to \\fa, we prove that H^r_{\\fa}(M) is not Artinian, and so the filter depth of \\fa on M doesn't exceeds f_{\\fa}(M). Also, we show that if M has finite dimension and H^i_{\\fa}(M) is Artinian for all i>t, where t is a given positive integer, then H^t_{\\fa}(M)/\\fa H^t_{\\fa}(M) is Artinian. It immediately implies that if q:=q_{\\fa}(M)>0, then H^q_{\\fa}(M) is not finitely generated, and so f_{\\fa}(M)\\leq q_{\\fa}(M)."}
{"category": "Math", "title": "Semigroup cohomology and applications", "abstract": "This article is a survey of the author's research. It consists of three sections concerned three kinds of cohomologies of semigroups. Section 1 considers `classic' cohomology as it was introduced by Eilenberg and MacLane. Here the attention is concentrated mainly on semigroups having cohomological dimension 1. In Section 2 a generalization of the Eilenberg-MacLane cohomology is introduced, the so-called 0-cohomology, which appears in applied topics (projective representations of semigroups, Brauer monoids). At last Section 3 is devoted to further generalizing: partial cohomology defined and discussed in it are used then for calculation of the classic cohomology for some semigroups."}
{"category": "Math", "title": "Using the smoothness of p-1 for computing roots modulo p", "abstract": "We prove, without recourse to the Extended Riemann Hypothesis, that the projection modulo $p$ of any prefixed polynomial with integer coefficients can be completely factored in deterministic polynomial time if $p-1$ has a $(\\ln p)^{O(1)}$-smooth divisor exceeding $(p-1)^{{1/2}+\\delta}$ for some arbitrary small $\\delta$. We also address the issue of computing roots modulo $p$ in deterministic time."}
{"category": "Math", "title": "On decomposition of commutative Moufang groupoids", "abstract": "We prove that every commutative Moufang groupoid is a semilattice of Archimedean subgroupoids."}
{"category": "Math", "title": "Minimal Niven numbers", "abstract": "Define a(k,q) to be the smallest positive multiple of k such that the sum of its digits in base q is equal to k. The asymptotic behavior, lower and upper bound estimates of a(k,q) are investigated. A characterization of the minimality condition is also considered."}
{"category": "Math", "title": "$t$-periodic light rays in conformally stationary spacetimes via Finsler geometry", "abstract": "In this paper we prove several multiplicity results of $t$-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of $t$-periodic light rays and compute a lower bound for the period of the light rays when the flag curvature of the Fermat metric is $\\eta$-pinched."}
{"category": "Math", "title": "Towards Theory of Piecewise Linear Dynamical Systems", "abstract": "In this paper, we consider a planar dynamical system with a piecewise linear function containing an arbitrary number (but finite) of dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system with $k$ dropping sections and $2k+1$ singular points can have at most $k+2$ limit cycles, $k+1$ of which surround the foci one by one and the last, $(k+2)$-th, limit cycle surrounds all of the singular points of this system."}
{"category": "Math", "title": "Bruhat-Chevalley order on the rook monoid", "abstract": "The rook monoid $R_n$ is the finite monoid whose elements are the 0-1 matrices with at most one nonzero entry in each row and column. The group of invertible elements of $R_n$ is isomorphic to the symmetric group $S_n$. The natural extension to $R_n$ of the Bruhat-Chevalley ordering on the symmetric group is defined in \\cite{Renner86}. In this paper, we find an efficient, combinatorial description of the Bruhat-Chevalley ordering on $R_n$. We also give a useful, combinatorial formula for the length function on $R_n$."}
{"category": "Math", "title": "Injective Simplicial Maps of the Arc Complex on Nonorientable Surfaces", "abstract": "We prove that each injective simplicial map from the arc complex of a compact, connected, nonorientable surface with nonempty boundary to itself is induced by a homeomorphism of the surface. We also prove that the automorphism group of the arc complex is isomorphic to the quotient of the mapping class group of the surface by its center."}
{"category": "Math", "title": "Abelian Hurwitz-Hodge integrals", "abstract": "Hodge classes on the moduli space of admissible covers with monodromy group G are associated to irreducible representations of G. We evaluate all linear Hodge integrals over moduli spaces of admissible covers with abelian monodromy in terms of multiplication in an associated wreath group algebra. In case G is cyclic and the representation is faithful, the evaluation is in terms of double Hurwitz numbers. In case G is trivial, the formula specializes to the well-known result of Ekedahl-Lando-Shapiro-Vainshtein for linear Hodge integrals over the moduli space of curves in terms of single Hurwitz numbers."}
{"category": "Math", "title": "Ozsvath-Szabo and Rasmussen invariants of cable knots", "abstract": "We study the behavior of the Ozsvath-Szabo and Rasmussen knot concordance invariants tau and s on K(m,n), the (m,n)-cable of a knot K where m and n are relatively prime. We show that for every knot K and for any fixed positive integer m, both of the invariants evaluated on K(m,n) differ from their value on the torus knot T(m,n) by a fixed constant for all but finitely many n>0. Combining this result together with Hedden's extensive work on the behavior of tau on (m,mr+1)-cables yields bounds on the value of tau on any (m,n)-cable of K. In addition, several of Hedden's obstructions for cables bounding complex curves are extended."}
{"category": "Math", "title": "Normal scalar curvature conjecture and its applications", "abstract": "In this paper, we proved the Normal Scalar Curvature Conjecture and the Bottcher-Wenzel Conjecture. We also established some new pinching theorems for minimal submanifolds in spheres."}
{"category": "Math", "title": "Discrete Fourier analysis on a dodecahedron and a tetrahedron", "abstract": "A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron is deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the tetrahedron is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\\log n)^3$."}
{"category": "Math", "title": "On a class of hypoelliptic operators with unbounded coefficients in ${\\matbb R}^N$", "abstract": "We consider a class of non-trivial perturbations ${\\mathscr A}$ of the degenerate Ornstein-Uhlenbeck operator in ${\\mathbb R}^N$. In fact we perturb both the diffusion and the drift part of the operator (say $Q$ and $B$) allowing the diffusion part to be unbounded in ${\\mathbb R}^N$. Assuming that the kernel of the matrix $Q(x)$ is invariant with respect to $x\\in {\\mathbb R}^N$ and the Kalman rank condition is satisfied at any $x\\in{\\mathbb R}^N$ by the same $m<N$, and developing a revised version of Bernstein's method we prove that we can associate a semigroup $\\{T(t)\\}$ of bounded operators (in the space of bounded and continuous functions) with the operator ${\\mathscr A}$. Moreover, we provide several uniform estimates for the spatial derivatives of the semigroup $\\{T(t)\\}$ both in isotropic and anisotropic spaces of (H\\\"older-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator ${\\mathscr A}$."}
{"category": "Math", "title": "Spectral stability of weak relaxation shock profiles", "abstract": "Using a combination of Kawashima- and Goodman-type energy estimates, we establish spectral stability of general small-amplitude relaxation shocks of symmetric dissipative systems. This extends previous results obtained by Plaza and Zumbrun by singular perturbation techniques under an additional technical assumption, namely, that the background equation be noncharacteristic with respect to the shock."}
{"category": "Math", "title": "Average performance of the sparsest approximation using a general dictionary", "abstract": "We consider the minimization of the number of non-zero coefficients (the $\\ell_0$ \"norm\") of the representation of a data set in terms of a dictionary under a fidelity constraint. (Both the dictionary and the norm defining the constraint are arbitrary.) This (nonconvex) optimization problem naturally leads to the sparsest representations, compared with other functionals instead of the $\\ell_0$ \"norm\". Our goal is to measure the sets of data yielding a $K$-sparse solution--i.e. involving $K$ non-zero components. Data are assumed uniformly distributed on a domain defined by any norm--to be chosen by the user. A precise description of these sets of data is given and relevant bounds on the Lebesgue measure of these sets are derived. They naturally lead to bound the probability of getting a $K$-sparse solution. We also express the expectation of the number of non-zero components. We further specify these results in the case of the Euclidean norm, the dictionary being arbitrary."}
{"category": "Math", "title": "An EM algorithm for estimation in the Mixture Transition Distribution model", "abstract": "The Mixture Transition Distribution (MTD) model was introduced by Raftery to face the need for parsimony in the modeling of high-order Markov chains in discrete time. The particularity of this model comes from the fact that the effect of each lag upon the present is considered separately and additively, so that the number of parameters required is drastically reduced. However, the efficiency for the MTD parameter estimations proposed up to date still remains problematic on account of the large number of constraints on the parameters. In this paper, an iterative procedure, commonly known as Expectation-Maximization (EM) algorithm, is developed cooperating with the principle of Maximum Likelihood Estimation (MLE) to estimate the MTD parameters. Some applications of modeling MTD show the proposed EM algorithm is easier to be used than the algorithm developed by Berchtold. Moreover, the EM Estimations of parameters for high-order MTD models led on DNA sequences outperform the corresponding fully parametrized Markov chain in terms of Bayesian Information Criterion. A software implementation of our algorithm is available in the library seq++ at http://stat.genopole.cnrs.fr/seqpp"}
{"category": "Math", "title": "Formulas for primitive Idempotents in Frobenius Algebras and an Application to Decomposition maps", "abstract": "In the first part of this paper we present explicit formulas for primitive idempotents in arbitrary Frobenius algebras using the entries of representing matrices coming from projective indecomposable modules with respect to a certain choice of basis. The proofs use a generalisation of the well known Frobenius-Schur relations for semisimple algebras. The second part of this paper considers $\\Oh$-free $\\Oh$-algebras of finite $\\Oh$-rank over a discrete valuation ring $\\Oh$ and their decomposition maps under modular reduction modulo the maximal ideal of $\\Oh$, thereby studying the modular representation theory of such algebras. Using the formulas from the first part we derive general criteria for such a decomposition map to be an isomorphism that preserves the classes of simple modules involving explicitly known matrix representations on projective indecomposable modules. Finally we show how this approach could eventually be used to attack a conjecture by Gordon James in the formulation of Meinolf Geck for Iwahori-Hecke-Algebras, provided the necessary matrix representations on projective indecomposable modules could be constructed explicitly."}
{"category": "Math", "title": "A note on the Cops & Robber game on graphs embedded in non-orientable surfaces", "abstract": "The Cops and Robber game is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if they can catch the robber. The minimum number of cops needed to win on a graph is called its cop number. It is known that the cop number of a graph embedded on a surface $X$ of genus $g$ is at most $3g/2 + 3$, if $X$ is orientable (Schroeder 2004), and at most $2g+1$, otherwise (Nowakowski & Schroeder 1997). We improve the bounds for non-orientable surfaces by reduction to the orientable case using covering spaces. As corollaries, using Schroeder's results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3; the cop number of graphs embeddable in the Klein Bottle is at most 4, and an upper bound is $3g/2 + 3/2$ for all other $g$."}
{"category": "Math", "title": "The abelianization of the level L mapping class group", "abstract": "We calculate the abelianizations of the level $L$ subgroup of the genus $g$ mapping class group and the level $L$ congruence subgroup of the $2g \\times 2g$ symplectic group for $L$ odd and $g \\geq 3$."}
{"category": "Math", "title": "Rough evolution equations", "abstract": "We generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise. As an illustration, we discuss a class of linear and nonlinear 1d SPDEs driven by a space--time Gaussian noise with singular space covariance and Brownian time dependence."}
{"category": "Math", "title": "n-ary associative algebras, cohomology, free algebras and coalgebras", "abstract": "When $n$ is odd, a cohomology of type Hochschild for $n$-ary partially associative algebras has been defined in Gnedbaye's thesis. Unfortunately, the cohomology definition is not valid when $n$ is even. This fact is found again in the computations of the $n$-ary partially associative free algebra. In this work, we define in a first time two approachs of an Hochschild cohomology for $n$-ary partially associative algebras. First by reducing the space of cochains, secondly by using a graded version. Next we compute the free $n$-ary algebra, giving a basis of this algebra. At last we extend the notion of coalgebras to $n$-ary algebras."}
{"category": "Math", "title": "Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions", "abstract": "Inverse problems of recovering the coefficients of Sturm-Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained."}
{"category": "Math", "title": "Almost-rainbow edge-colorings of some small subgraphs", "abstract": "Let $f(n,p,q)$ be the minimum number of colors necessary to color the edges of $K_n$ so that every $K_p$ is at least $q$-colored. We improve current bounds on the {7/4}n-3$, slightly improving the bound of Axenovich. We make small improvements on bounds of Erd\\H os and Gy\\'arf\\'as by showing ${5/6}n+1\\leq f(n,4,5)$ and for all even $n\\not\\equiv 1 \\pmod 3$, $f(n,4,5)\\leq n-1$ . For a complete bipartite graph $G=K_{n,n}$, we show an n-color construction to color the edges of $G$ so that every $C_4\\subseteq G$ is colored by at least three colors. This improves the best known upper bound of M. Axenovich, Z. F\\\"uredi, and D. Mubayi."}
{"category": "Math", "title": "The implicitization problem for $\\phi: P^n --> (P^1)^{n+1}$", "abstract": "We develop in this paper some methods for studying the implicitization problem for a rational map $\\phi: \\mathbb{P}^n \\to (\\mathbb{P}^1)^{n+1}$ defining a hypersurface in $(\\mathbb{P}^1)^{n+1}$, based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay Resultants and Koszul complexes coincides, in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions. Under suitable hypotheses, these techniques enable us to obtain the implicit equation, up to a power, and up to some other extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of $\\phi$, and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants."}
{"category": "Math", "title": "Projections of a learning space", "abstract": "Any subset Q' of the domain Q of a learning space defines a projection of that learning space on Q' which is itself a learning space consistent with the original one. Moreover, such a construction defines a partition of Q having each of its classes defining a learning space also consistent with the original learning space. We give a direct proof of these facts which are instrumental in parsing large learning spaces."}
{"category": "Math", "title": "On mirabolic D-modules", "abstract": "Let an algebraic group G act on X, a connected algebraic manifold, with finitely many orbits. For any Harish-Chandra pair (D,G) where D is a sheaf of twisted differential operators on X, we form a left ideal D.g in D generated by the Lie algebra g, of G. Then, D/D.g is a holonomic D-module and its restriction to a unique Zariski open dense G-orbit in X is a G-equivariant local system. We prove a criterion saying that the D-module D/D.g is isomorphic, under certain (quite restrictive) conditions, to a direct image of that local system to X. We apply this criterion in the special case where the group G=SL(n) acts diagonally on X = F \\times F \\times P^{n-1}, a triple product where F is the flag manifold for SL(n) and P^{n-1} is the projective space. We further relate D-modules on F \\times F \\times P^{n-1} to D-modules on the space SL(n) \\times P^{n-1} via a pair, CH, HC, of adjoint functors, analogous to those used in Lusztig's theory of character sheaves. A second important result of the paper provides an explicit description of these functors showing that the functor HC gives an exact functor on the abelian category of mirabolic D-modules."}
{"category": "Math", "title": "Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs", "abstract": "We interpret mathematically the pair (master equation, solution of master equation) up to equivalence, as the pair (a presentation of a free triangular dga T over a combination operad O, dga map of T into C, a dga over O) up to homotopy equivalence of dgOa maps, see Definition 1. We sketch two general applications: I to the theory of the definition and homotopy theory of infinity versions of general algebraic structures including noncompact frobenius algebras and Lie bialgebras. Here the target C would be the total Hom complex between various tensor products of another chain complex B, C = HomB, O describes combinations of operations like composition and tensor product sufficient to describe the algebraic structure and one says that B has the algebraic structure in question. II to geometric systems of moduli spaces up to deformation like the moduli of J holomorphic curves. Here C is some geometric chain complex containing the fundamental classes of the moduli spaces of the geometric problem. We also discuss analogues of homotopy groups and Postnikov systems for maps and impediments to using them related to linear terms in the master equation called anomalies."}
{"category": "Math", "title": "A non-Archimedean analogue of the Hodge-D-conjecture for products of elliptic curves", "abstract": "In this paper we show that the map % $$\\partial:CH^2(E_1 \\times E_2,1)\\otimes \\Q \\longrightarrow PCH^1(\\XX_v)$$ % is surjective, where $E_1$ and $E_2$ are two non-isogenous semistable elliptic curves over a local field, $CH^2(E_1 \\times E_2,1)$ is one of Bloch's higher Chow groups and $PCH^1(\\XX_v)$ is a certain subquotient of a Chow group of the special fibre $\\XX_{v}$ of a semi-stable model $\\XX$ of $E_1 \\times E_2$. On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-$\\D$-conjecture of Beilinson - which is known to be true in this case by the work of Chen and Lewis \\cite{lech}, and on the other, an analogue of the works of Spei{\\ss} \\cite{spie}, Mildenhall \\cite{mild} and Flach \\cite{flac} in the case when the elliptic curves have split multiplicative reduction."}
{"category": "Math", "title": "The Relationship Between a Function, a Functions Inverse, and their Antiderivatives with an Emphasis in Finding Exact Roots with the Technique of Integration", "abstract": "Using a new technique involving integration it is possible to find the exact roots of simple functions. In this case, simple functions are defined as smooth functions having an inverse, and that inverse having an antiderivative. This technique now makes it possible to find the exact roots of certain functions without the use of numerical or iterative methods."}
{"category": "Math", "title": "Quantum group structure of the q-deformed $W$ algebra $\\WW_q$", "abstract": "In this paper the q-deformed $W$ algebra $\\WW_q$ is constructed, whose nontrivial quantum group structure is presented."}
{"category": "Math", "title": "Superposition rules and stochastic Lie-Scheffers systems", "abstract": "This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. When stated in the particular case of standard deterministic systems, our main theorem improves various aspects of the classical Lie-Scheffers result. We show that the stochastic analog of the classical Lie-Scheffers systems can be reduced to the study of Lie group valued stochastic Lie-Scheffers systems; those systems, as well as those taking values in homogeneous spaces are studied in detail. The developments of the paper are illustrated with several examples."}
{"category": "Math", "title": "The Smooth Structure of the Space of Piecewise-Smooth Loops", "abstract": "We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite dimensional manifold. We succeed for a particular type of piecewise-smooth loops. We also examine the action of the diffeomorphism group of the circle. It is not a useful action on the manifold that we define. We consider how one might fix this problem and conclude that it can only be done by completing to the space of loops of bounded variation."}
{"category": "Math", "title": "Families of zero cycles and divided powers: I. Representability", "abstract": "For any separated algebraic space $X/S$ we construct a separated algebraic space $\\Gamma^d(X/S)$ -- the space of divided powers -- which parameterizes zero cycles of degree $d$ on $X$. The space of divided powers for an affine scheme is given by the spectrum of the algebra of divided powers. In characteristic zero or when $X/S$ is flat, the constructed space coincides with the symmetric product $Sym^d(X/S)$. We also prove several fundamental results on the kernels of multiplicative polynomial laws necessary for the construction of $\\Gamma^d(X/S)$."}
{"category": "Math", "title": "Positivity of cotangent bundles", "abstract": "In this paper we prove a generalization of a theorem of Schneider, which gives a criterion for a projective surface over the complex numbers to have an ample cotangent bundle. After reviewing different notions of positivity, we introduce a slightly weaker notion of ampleness, which we call quasi-ample, and then are able to extend Schneider's result to higher dimensions."}
{"category": "Math", "title": "Polling systems with parameter regeneration, the general case", "abstract": "We consider a polling model with multiple stations, each with Poisson arrivals and a queue of infinite capacity. The service regime is exhaustive and there is Jacksonian feedback of served customers. What is new here is that when the server comes to a station it chooses the service rate and the feedback parameters at random; these remain valid during the whole stay of the server at that station. We give criteria for recurrence, transience and existence of the $s$th moment of the return time to the empty state for this model. This paper generalizes the model, when only two stations accept arriving jobs, which was considered in [Ann. Appl. Probab. 17 (2007) 1447--1473]. Our results are stated in terms of Lyapunov exponents for random matrices. From the recurrence criteria it can be seen that the polling model with parameter regeneration can exhibit the unusual phenomenon of null recurrence over a thick region of parameter space."}
{"category": "Math", "title": "The tiered Aubry set for autonomous Lagrangian functions", "abstract": "If L is a Tonelli Lagrangian defined on the tangent bundle of a compact and connected manifold whose dimension is at least 2, we associate to L the tiered Aubry set and the tiered Mane set (defined in the article). We prove that the tiered Mane set is closed, connected, chain transitive and that if L is generic in the Mane sense, the tiered Mane set has no interior. Then, we give an example of such an explicit generic Tonelli Lagrangian function and an example proving that when M is the torus, the closure of the tiered Aubry set and the closure of the union of the K.A.M. tori may be different."}
{"category": "Math", "title": "Non-proper helicoid-like limits of closed minimal surfaces in 3-manifolds", "abstract": "We show that there exists a metric with positive scalar curvature on S2xS1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two helicoid-like singularities on the 2-sphere. The construction is inspired by a recent example by D. Hoffman and B. White."}
{"category": "Math", "title": "The Theory of Fallible Probability and The Dynamics of Degrees of Belief", "abstract": "This monograph is an account of the theory of fallible probability and of the dynamics of degrees of belief. It discusses the first order subjective theory in which first order degrees of belief are expressed by subjective probabilities and are updated by conditionalization (Bayes, 1764; Ramsey, 1926), gives an improved exposition of the greater part of the author's theory of Probability Dynamics (Nathan, 2006) which should replace the so-called Probability Kinematics (Jeffrey, 1965), resolves the problem of New Explanation of Old Evidence (Jeffrey, 1995), provides a Theory of Confirmation, and refutes the Principle of Reflection (Van Fraassen, 1984)."}
{"category": "Math", "title": "Constrained Willmore Tori in the 4--Sphere", "abstract": "We prove that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere is either of \"finite type\", that is, has a spectral curve of finite genus, or is of \"holomorphic type\" which means that it is super conformal or Euclidean minimal with planar ends. This implies that all constrained Willmore tori in the 4-sphere can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of Hitchin's approach to the study of harmonic tori in the 3-sphere."}
{"category": "Math", "title": "Metric properties of Outer Space", "abstract": "We define metrics on Culler-Vogtmann space, which are an analogue of the Teichmuller metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies of both choices. We show how to compute stretching factors between marked metric graphs in an easy way and we discuss the behaviour of stretching factors under iterations of automorphisms. We study metric properties of folding paths, showing that they are geodesic for the non-symmetric metric and, if they do not enter the thin part of Outer space, quasi-geodesic for the symmetric metric."}
{"category": "Math", "title": "Arithmetic of a fake projective plane and related elliptic surfaces", "abstract": "The purpose of the present paper is to explain the fake projective plane constructed by J.H. Keum from the point of view of arithmetic ball quotients. Beside the ball quotient associated with the fake projective plane, we also analize two further naturally related ball quotients whose minimal desingularizations lead to two elliptic surfaces, one already considered by J.H. Keum as well as the one constructed by M.N. Ishida in terms of p-adic uniformization."}
{"category": "Math", "title": "The defining ideals of conjugacy classes of nilpotent matrices and a conjecture of Weyman", "abstract": "Tanisaki introduced generating sets for the defining ideals of the schematic intersections of the closure of conjugacy classes of nilpotent matrices with the set of diagonal matrices. These ideals are naturally labeled by integer partitions. Given such a partition $\\lambda$, we define several methods to produce a reduced generating set for the associated ideal $I_{\\lambda}$. For particular shapes we find nice generating sets. By comparing our sets with some generating sets of $I_{\\lambda}$ arising from a work of Weyman, we find a counterexample to a related conjecture of Weyman."}
{"category": "Math", "title": "Examples of smooth maps with finitely many critical points in dimensions $(4,3)$, $(8,5)$ and $(16,9)$", "abstract": "We consider manifolds $M^{2n}$ which admit smooth maps into a connected sum of $S^1\\times S^n$ with only finitely many critical points, for $n\\in\\{2,4,8\\}$, and compute the minimal number of critical points."}
{"category": "Math", "title": "Regularization with the Smooth-Lasso procedure", "abstract": "We consider the linear regression problem. We propose the S-Lasso procedure to estimate the unknown regression parameters. This estimator enjoys sparsity of the representation while taking into account correlation between successive covariates (or predictors). The study covers the case when $p\\gg n$, i.e. the number of covariates is much larger than the number of observations. In the theoretical point of view, for fixed $p$, we establish asymptotic normality and consistency in variable selection results for our procedure. When $p\\geq n$, we provide variable selection consistency results and show that the S-Lasso achieved a Sparsity Inequality, i.e., a bound in term of the number of non-zero components of the oracle vector. It appears that the S-Lasso has nice variable selection properties compared to its challengers. Furthermore, we provide an estimator of the effective degree of freedom of the S-Lasso estimator. A simulation study shows that the S-Lasso performs better than the Lasso as far as variable selection is concerned especially when high correlations between successive covariates exist. This procedure also appears to be a good challenger to the Elastic-Net (Zou and Hastie, 2005)."}
{"category": "Math", "title": "Heat content", "abstract": "We study the heat content asymptotics with either Dirichlet or Robin boundary conditions where the initial temperature exhibits radial blowup near the boundary. We show that there is a complete small-time asymptotic expansion and give explicit geometrical formulas for the first few terms in the expansion."}
{"category": "Math", "title": "Spaces of $\\mathbb R$ - places of rational function fields", "abstract": "In the paper an answer to a problem \"When different orders of R(X) (where R is a real closed field) lead to the same real place ?\" is given. We use this result to show that the space of $\\mathbb R$-places of the field $\\textbf{R}(Y)$ (where \\textbf{R} is any real closure of $\\mathbb R(X)$) is not metrizable space. Thus the space $M(\\mathbb R(X,Y))$ is not metrizable, too."}
{"category": "Math", "title": "On the Duality between l^1-Homology and Bounded Cohomology", "abstract": "We modify the definition of l^1-homology and argue why our definition is more adequate than the classical one. While we cannot reconstruct the classical l^1-homology from the new definition for various reasons, we can reconstruct its Hausdorffification so that no information concerning semi-norms is lost. We obtain an axiomatic characterization of our l^1-homology as a universal delta-functor and prove that it is pre-dual to our definition of bounded cohomology. We thus answer a question raised by Loeh in her thesis. Moreover, we prove Gromov's theorem and the Matsumoto-Morita conjecture in our context."}
{"category": "Math", "title": "Pseudo-unitarizable weight modules over generalized Weyl algebras", "abstract": "We define a notion of pseudo-unitarizability for weight modules over a generalized Weyl algebra (of rank one, with commutative coeffiecient ring $R$), which is assumed to carry an involution of the form $X^*=Y$, $R^*\\subseteq R$. We prove that a weight module $V$ is pseudo-unitarizable iff it is isomorphic to its finitistic dual $V^\\sharp$. Using the classification of weight modules by Drozd, Guzner and Ovsienko, we obtain necessary and sufficient conditions for an indecomposable weight module to be isomorphic to its finitistic dual, and thus to be pseudo-unitarizable. Some examples are given, including $U_q(\\mathfrak{sl}_2)$ for $q$ a root of unity."}
{"category": "Math", "title": "Loewner's torus inequality with isosystolic defect", "abstract": "We show that Bonnesen's isoperimetic defect has a systolic analog for Loewner's torus inequality. The isosystolic defect is expressed in terms of the probabilistic variance of the conformal factor of the metric g with respect to the flat metric of unit area in the conformal class of g."}
{"category": "Math", "title": "Constructing Weyl group multiple Dirichlet series", "abstract": "Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, that has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. A heuristic definition of such series was given in [2], and they have been investigated in certain special cases in [2-6, 11-14]. In this paper we generalize results in [13] to construct Weyl group multiple Dirichlet series by a uniform method, and show in all cases that they have the expected properties."}
{"category": "Math", "title": "Quantum Monodromy and Non-concentration near a Closed Semi-Hyperbolic Orbit", "abstract": "For a large class of semiclassical operators $P(h)-z$ which includes Schr\\\"odinger operators on manifolds with boundary, we construct the Quantum Monodromy operator $M(z)$ associated to a periodic orbit $\\gamma$ of the classical flow. Using estimates relating $M(z)$ and $P(h)-z$, we prove semiclassical estimates for small complex perturbations of $P(h) -z$ in the case $\\gamma$ is semi-hyperbolic. As our main application, we give logarithmic lower bounds on the mass of eigenfunctions away from semi-hyperbolic orbits of the associated classical flow. As a second application of the Monodromy Operator construction, we prove if $\\gamma$ is an elliptic orbit, then $P(h)$ admits quasimodes which are well-localized near $\\gamma$."}
{"category": "Math", "title": "A Generalization of Siegel's Theorem and Hall's Conjecture", "abstract": "Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is placed upon the number of prime factors dividing a fixed coordinate? If the bound is zero, then Siegel's Theorem guarantees that there are only finitely many such points. We consider, theoretically and computationally, two conjectures: one is a generalization of Siegel's Theorem and the other is a refinement which resonates with Hall's conjecture."}
{"category": "Math", "title": "Riemann Surface Laminations with Singularities", "abstract": "In these introductory notes we give the basics of the theory of holomorphic foliations and laminations. The emphasis is on the theory of harmonic currents and unique ergodicity for laminations transversally Lipschitz in CP^2 and for generic holomorphic foliations in CP^2."}
{"category": "Math", "title": "Annular embeddings of permutations for arbitrary genus", "abstract": "In the symmetric group on a set of size 2n, let P_{2n} denote the conjugacy class of involutions with no fixed points (equivalently, we refer to these as ``pairings'', since each disjoint cycle has length 2). Harer and Zagier explicitly determined the distribution of the number of disjoint cycles in the product of a fixed cycle of length 2n and the elements of P_{2n}. Their famous result has been reproved many times, primarily because it can be interpreted as the genus distribution for 2-cell embeddings in an orientable surface,of a graph with a single vertex attached to n loops. In this paper we give a new formula for the cycle distribution when a fixed permutation with two cycles (say the lengths are p,q, where p+q=2n) is multiplied by the elements of P_{2n}. It can be interpreted as the genus distribution for 2-cell embeddings in an orientable surface, of a graph with two vertices, of degrees p and q. In terms of these graphs, the formula involves a parameter that allows us to specify, separately, the number of edges between the two vertices and the number of loops at each of the vertices. The proof is combinatorial, and uses a new algorithm that we introduce to create all rooted forests containing a given rooted forest."}
{"category": "Math", "title": "Discrete holomorphic geometry I. Darboux transformations and spectral curves", "abstract": "Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic line bundles over \"discrete surfaces\", by which we mean the vertex sets of triangulated surfaces with bi-colored set of faces. The resulting theory of discrete conformality is simultaneously Moebius invariant and based on linear equations. In the special case of maps into the 2-sphere we obtain a reinterpretation of the theory of complex holomorphic functions on discrete surfaces introduced by Dynnikov and Novikov. As an application of our theory we introduce a Darboux transformation for discrete surfaces in the conformal 4-sphere. This Darboux transformation can be interpreted as the space- and time-discrete Davey-Stewartson flow of Konopelchenko and Schief."}
{"category": "Math", "title": "Immersed Lagrangian Floer Theory", "abstract": "Let (M,w) be a compact symplectic manifold, and L a compact, embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian Floer cohomology for such M,L, yielding groups HF^*(L,b;\\Lambda) for one Lagrangian or HF^*((L,b),(L',b');\\Lambda) for two, where b,b' are choices of bounding cochains, and exist if and only if L,L' have unobstructed Floer cohomology. These are independent of choices up to canonical isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M,w), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich. The goal of this paper is to extend all this to immersed Lagrangians L in M with immersion i : L --> M, with transverse self-intersections. In the embedded case, Floer cohomology HF^*(L,b;\\Lambda) is a modified, 'quantized' version of cohomology H^*(L;\\Lambda) over the Novikov ring \\Lambda. In our immersed case, HF^*(L,b;\\Lambda) turns out to be a quantized version of the sum of H^*(L;\\Lambda) with a \\Lambda-module spanned by pairs (p,q) for p,q distinct points of L with i(p)=i(q) in M. The theory becomes simpler and more powerful for graded Lagrangians in Calabi-Yau manifolds, when we can work over a smaller Novikov ring \\Lambda_{CY}. The proofs involve associating a gapped filtered A-infinity algebra over \\Lambda or \\Lambda_{CY} to i : L --> M, which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called A_{N,0} algebras for N=0,1,2,..."}
{"category": "Math", "title": "On two examples by Iyama and Yoshino", "abstract": "In the recent paper \"Mutation in triangulated categories and rigid Cohen-Macaulay modules\" Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen-Macaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov's result on the graded singularity category. We obtain some new results on the singularity category of isolated singularities which may be interesting in their own right."}
{"category": "Math", "title": "Some remarks on varieties of pairs of commuting upper triangular matrices and an interpretation of commuting varieties", "abstract": "It is known that the variety of pairs of n x n commuting upper triangular matrices isn't a complete intersection for infinitely many values of n; we show that there exists m such that this happens if and only if n > m. We also show that m < 18 and that it could be found by determining the dimension of the variety of pairs of commuting strictly upper triangular matrices. Then we define a natural map from the variety of pairs of commuting n x n matrices onto a subvariety defined by linear equations of the grassmannian of subspaces of codimension 2 of a vector space of dimension n x n."}
{"category": "Math", "title": "Recurrence of the twisted planar random walk", "abstract": "We show that the \"twisted\" planar random walk - which results by summing up stationary increments rotated by multiples of a fixed angle - is recurrent under diverse assumptions on the increment process. For example, if the increment process is alpha-mixing and of finite second moment, then the twisted random walk is recurrent for every angle fixed choice of the angle out of a set of full Lebesgue measure, no matter how slow the mixing coefficients decay."}
{"category": "Math", "title": "A note on optimal probability lower bounds for centered random variables", "abstract": "In this note we obtain lower bounds for $\\P(\\xi\\geq 0)$ and $\\P(\\xi>0)$ under assumptions on the moments of a centered random variable $\\xi$. The obtained estimates are shown to be optimal and improve results from the literature. The results are applied to obtain probability lower bounds for second order Rademacher chaos."}
{"category": "Math", "title": "The elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences", "abstract": "We define three hard problems in the theory of elliptic divisibility sequences (EDS Association, EDS Residue and EDS Discrete Log), each of which is solvable in sub-exponential time if and only if the elliptic curve discrete logarithm problem is solvable in sub-exponential time. We also relate the problem of EDS Association to the Tate pairing and the MOV, Frey-R\\\"{u}ck and Shipsey EDS attacks on the elliptic curve discrete logarithm problem in the cases where these apply."}
{"category": "Math", "title": "Isomorphisms between Algebras of Semiclassical Pseudodifferential Operators", "abstract": "Following the work of Duistermaat-Singer \\cite{DS} on isomorphisms of algebras of global pseudodifferential operators, we classify isomorphisms of algebras of microlocally defined semiclassical pseudodifferential operators. Specifically, we show that any such isomorphism is given by conjugation by $A = BF$, where $B$ is a microlocally elliptic semiclassical pseudodifferential operator, and $F$ is a microlocal $h$-FIO associated to the graph of a local symplectic transformation."}
{"category": "Math", "title": "A Sequence of Degree One Vassiliev Invariants for Virtual Knots", "abstract": "For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual strings to the class of singular virtual knots with one double-point."}
{"category": "Math", "title": "Fibered Transverse Knots and the Bennequin Bound", "abstract": "We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold $(M,\\xi)$ with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact structure it supports (since it is also the binding of an open book) is $\\xi.$ This gives a geometric reason for the non-sharpness of the Bennequin bound for fibered links. We also note that this allows the classification, up to contactomorphism, of maximal self-linking number links in these knot types. Moreover, in the standard tight contact structure on $S^3$ we classify, up to transverse isotopy, transverse knots with maximal self-linking number in the knots types given as closures of positive braids and given as fibered strongly quasi-positive knots. We derive several braid theoretic corollaries from this. In particular. we give conditions under which quasi-postitive braids are related by positive Markov stabilizations and when a minimal braid index representative of a knot is quasi-positive. In the new version we also prove that our main result can be used to show, and make rigorous the statement, that contact structures on a given manifold are in a strong sense classified by the transverse knot theory they support."}
{"category": "Math", "title": "Eisenstein cohomology for congruence subgroups of SO(n,2)", "abstract": "The automorphic cohomology of a connected reductive algebraic group defined over Q decomposes as a direct algebraic sum of cuspidal and Eisenstein cohomology. In the present paper we construct regular Eisenstein cohomology classes for congruence subgroups of a rational form G of Q-rank 2 of SO(n,2)."}
{"category": "Math", "title": "Generalized Frobenius Algebras and the Theory of Hopf Algebras", "abstract": "\"Co-Frobenius\" coalgebras were introduced as dualizations of Frobenius algebras. Recently, it was shown in \\cite{I} that they admit left-right symmetric characterizations analogue to those of Frobenius algebras: a coalgebra $C$ is co-Frobenius if and only if it is isomorphic to its rational dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; in the first main result we show that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently right) rational dual $Rat(C^*)$, in the sense that certain coproduct or product powers of these objects are isomorphic. These show that QcF coalgebras can be viewed as generalizations of both co-Frobenius coalgebras and Frobenius algebras. Surprisingly, these turn out to have many applications to fundamental results of Hopf algebras. The equivalent characterizations of Hopf algebras with left (or right) nonzero integrals as left (or right) co-Frobenius, or QcF, or semiperfect or with nonzero rational dual all follow immediately from these results. Also, the celebrated uniqueness of integrals follows at the same time as just another equivalent statement. Moreover, as a by-product of our methods, we observe a short proof for the bijectivity of the antipode of a Hopf algebra with nonzero integral. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras."}
{"category": "Math", "title": "A statistic on the roots of a finite reflection group and a correspondence between the height function and Bruhat order", "abstract": "The action of a finite reflection group (type A) on its set of roots is understood as a permutation representation or group action. We show that this representation is an induced representation from a certain kind of parabolic subgroup. Furthermore, we use this representation to define a statistic (derived from the length function) on the set of roots. A possible application to Costas Arrays is hinted at in a proposition."}
{"category": "Math", "title": "The Rank of the Covariance Matrix of an Evanescent Field", "abstract": "Evanescent random fields arise as a component of the 2-D Wold decomposition of homogenous random fields. Besides their theoretical importance, evanescent random fields have a number of practical applications, such as in modeling the observed signal in the space time adaptive processing (STAP) of airborne radar data. In this paper we derive an expression for the rank of the low-rank covariance matrix of a finite dimension sample from an evanescent random field. It is shown that the rank of this covariance matrix is completely determined by the evanescent field spectral support parameters, alone. Thus, the problem of estimating the rank lends itself to a solution that avoids the need to estimate the rank from the sample covariance matrix. We show that this result can be immediately applied to considerably simplify the estimation of the rank of the interference covariance matrix in the STAP problem."}
{"category": "Math", "title": "Mutual Absolute Continuity of Harmonic and Surface Measures for Hormander Type Operators", "abstract": "In this paper, we consider the Sub-Laplacian L which consists of sum of squares of smooth vector fields that satisfy Hormander's finite rank condition. We study the Dirichlet problem for this operator on domains that satisfy certain geometric conditions. For such domains, several key results are established. These results consist of 1) A reversed Holder inequality for the Poisson kernel 2) Harmonic measure (corresponding to L) and surface measure (as well as the H-Perimeter measure) are mutually absolutely continuous 3) A representation (hence solvability of the Dirichlet problem) for solutions to the Dirichlet problem."}
{"category": "Math", "title": "Composition of Haar Paraproducts: The Random Case", "abstract": "When is the composition of paraproducts bounded? This is an important, and difficult question, related to to a question of Sarason on composition of Hankel matrices, and the two-weight problem for the Hilbert transform. We consider randomized variants of this question, finding non-classical characterizations, for dyadic paraproducts."}
{"category": "Math", "title": "A criteria for correct solvability in L_p(R) of a general Sturm-Liouville equation", "abstract": "We give criteria for correct solvability in L_p(R) of a general Sturm-Liouville equation"}
{"category": "Math", "title": "On periodic $p$-harmonic functions on Cayley tree", "abstract": "We show that any periodic with respect to normal subgroups (of the group representation of the Cayley tree) of finite index $p$-harmonic function is a constant. For some normal subgroups of infinite index we describe a class of (non-constant) periodic $p$-harmonic functions. If $p\\neq2$, the $p$-harmonicity is non-linear, i.e., the linear combination of $p$-harmonic functions need not be $p$-harmonic. In spite of this, we show that linear combinations of the $p$-harmonic functions described for normal subgroups of infinite index are also $p$-harmonic."}
{"category": "Math", "title": "Archimedean Type Conditions in Categories", "abstract": "Two concepts of being Archimedean are defined for arbitrary categories."}
{"category": "Math", "title": "Geometric quantization of weak-Hamiltonian functions", "abstract": "The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the corresponding principal circle bundle and we extend the notion of a polarization."}
{"category": "Math", "title": "Time--space harmonic polynomials relative to a L\\'{e}vy process", "abstract": "In this work, we give a closed form and a recurrence relation for a family of time--space harmonic polynomials relative to a L\\'{e}vy process. We also state the relationship with the Kailath--Segall (orthogonal) polynomials associated to the process."}
{"category": "Math", "title": "On the ternary Goldbach problem with primes in independent arithmetic progressions", "abstract": "We show that for every fixed $A>0$ and $\\theta>0$ there is a $\\vartheta=\\vartheta(A,\\theta)>0$ with the following property. Let $n$ be odd and sufficiently large, and let $Q_{1}=Q_{2}:=n^{\\h}(\\log n)^{-\\vartheta}$ and $Q_{3}:=(\\log n)^{\\theta}$. Then for all $q_{3}\\leq Q_{3}$, all reduced residues $a_{3}$ mod $q_{3}$, almost all $q_{2}\\leq Q_{2}$, all admissible residues $a_{2}$ mod $q_{2}$, almost all $q_{1}\\leq Q_{1}$ and all admissible residues $a_{1}$ mod $q_{1}$, there exists a representation $n=p_{1}+p_{2}+p_{3}$ with primes $p_{i}\\equiv a_{i} (q_{i})$, $i=1,2,3$."}
{"category": "Math", "title": "Goodness-of-fit tests for Markovian time series models: Central limit theory and bootstrap approximations", "abstract": "New goodness-of-fit tests for Markovian models in time series analysis are developed which are based on the difference between a fully nonparametric estimate of the one-step transition distribution function of the observed process and that of the model class postulated under the null hypothesis. The model specification under the null allows for Markovian models, the transition mechanisms of which depend on an unknown vector of parameters and an unspecified distribution of i.i.d. innovations. Asymptotic properties of the test statistic are derived and the critical values of the test are found using appropriate bootstrap schemes. General properties of the bootstrap for Markovian processes are derived. A new central limit theorem for triangular arrays of weakly dependent random variables is obtained. For the proof of stochastic equicontinuity of multidimensional empirical processes, we use a simple approach based on an anisotropic tiling of the space. The finite-sample behavior of the proposed test is illustrated by some numerical examples and a real-data application is given."}
{"category": "Math", "title": "Moufang symmetry X. Generalized Lie and Maurer-Cartan equations of continuous Moufang transformations", "abstract": "The differential equations for a continuous birepresentation of a local analytic Moufang loop are established. The commutation relations for the infinitesimal operators of the representation are found. These commutation relations can be seen as a (minimal) generalization of the Maurer-Cartan equations and do not depend on the particular birepresentation."}
{"category": "Math", "title": "Leading coefficients and cellular bases of Hecke algebras", "abstract": "Let $\\bH$ be the generic Iwahori--Hecke algebra associated with a finite Coxeter group $W$. Recently, we have shown that $\\bH$ admits a natural cellular basis in the sense of Graham--Lehrer, provided that $W$ is a Weyl group and all parameters of $\\bH$ are equal. The construction involves some data arising from the Kazhdan--Lusztig basis $\\{\\bC_w\\}$ of $\\bH$ and Lusztig's asymptotic ring $\\bJ$. This article attemps to study $\\bJ$ and its representation theory from a new point of view. We show that $\\bJ$ can be obtained in an entirely different fashion from the generic representations of $\\bH$, without any reference to $\\{\\bC_w\\}$. Then we can extend the construction of the cellular basis to the case where $W$ is not crystallographic. Furthermore, if $\\bH$ is a multi-parameter algebra, we will see that there always exists at least one cellular structure on $\\bH$. Finally, one may also hope that the new construction of $\\bJ$ can be extended to Hecke algebras associated to complex reflection groups."}
{"category": "Math", "title": "A simple adaptive estimator of the integrated square of a density", "abstract": "Given an i.i.d. sample $X_1,...,X_n$ with common bounded density $f_0$ belonging to a Sobolev space of order $\\alpha$ over the real line, estimation of the quadratic functional $\\int_{\\mathbb{R}}f_0^2(x) \\mathrm{d}x$ is considered. It is shown that the simplest kernel-based plug-in estimator \\[\\frac{2}{n(n-1)h_n}\\sum_{1\\leq i<j\\leq n}K\\biggl(\\frac{X_i-X_j}{h_n}\\biggr)\\] is asymptotically efficient if $\\alpha>1/4$ and rate-optimal if $\\alpha\\le1/4$. A data-driven rule to choose the bandwidth $h_n$ is then proposed, which does not depend on prior knowledge of $\\alpha$, so that the corresponding estimator is rate-adaptive for $\\alpha \\leq1/4$ and asymptotically efficient if $\\alpha>1/4$."}
{"category": "Math", "title": "Asymptotic analysis of $k$-noncrossing matchings", "abstract": "In this paper we study $k$-noncrossing matchings. A $k$-noncrossing matching is a labeled graph with vertex set $\\{1,...,2n\\}$ arranged in increasing order in a horizontal line and vertex-degree 1. The $n$ arcs are drawn in the upper halfplane subject to the condition that there exist no $k$ arcs that mutually intersect. We derive: (a) for arbitrary $k$, an asymptotic approximation of the exponential generating function of $k$-noncrossing matchings $F_k(z)$. (b) the asymptotic formula for the number of $k$-noncrossing matchings $f_{k}(n) \\sim c_k n^{-((k-1)^2+(k-1)/2)} (2(k-1))^{2n}$ for some $c_k>0$."}
{"category": "Math", "title": "Lindenmayer systems and primes", "abstract": "We study the surprising discrepancy between the number of primes corresponding, respectively, to the two letters of an infinite word engendered by one of the simplest Lindenmayer systems. We formulate a conjecture concerning the rate of growth of this discrepancy, which seems to tend to e for every two sufficiently high consecutive even rank iterates of the Lindenmayer system."}
{"category": "Math", "title": "Girard couples of quantales", "abstract": "We introduce the concept of a Girard couple, which consists of two (not necessarily unital) quantales linked by a strong form of duality. The two basic examples of Girard couples arise in the study of endomorphism quantales and of the spectra of operator algebras. We construct, for an arbitrary sup-lattice $S$, a Girard quantale whose right-sided part is isomorphic to $S$."}
{"category": "Math", "title": "The Geometric Bogomolov Conjecture for Small Genus Curves", "abstract": "The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 defined over a function field of characteristic zero. We recover the known result for genus 2 curves and in many cases improve upon the known bound for genus 3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved."}
{"category": "Math", "title": "L\\'{e}vy-based growth models", "abstract": "In the present paper, we give a condensed review, for the nonspecialist reader, of a new modelling framework for spatio-temporal processes, based on L\\'{e}vy theory. We show the potential of the approach in stochastic geometry and spatial statistics by studying L\\'{e}vy-based growth modelling of planar objects. The growth models considered are spatio-temporal stochastic processes on the circle. As a by product, flexible new models for space--time covariance functions on the circle are provided. An application of the L\\'{e}vy-based growth models to tumour growth is discussed."}
{"category": "Math", "title": "An upper bound for the number of perfect matchings in graphs", "abstract": "We give an upper bound on the number of perfect matchings in an undirected simple graph $G$ with an even number of vertices, in terms of the degrees of all the vertices in $G$. This bound is sharp if $G$ is a union of complete bipartite graphs. This bound is a generalization of the upper bound on the number of perfect matchings in bipartite graphs on $n+n$ vertices given by the Bregman-Minc inequality for the permanents of $(0,1)$ matrices."}
{"category": "Math", "title": "Symmetry Coefficients of Semilinear Partial Differential Equations", "abstract": "We show that for any semilinear partial differential equation of order m, the infinitesimals of the independent variables depend only on the independent variables and, if m>1 and the equation is also linear in its derivatives of order m-1 of the dependent variable, then the infinitesimal of the dependent variable is at most linear on the dependent variable. Many examples of important partial differential equations in Analysis, Geometry and Mathematical - Physics are given in order to enlighten the main result."}
{"category": "Math", "title": "Selection from a stable box", "abstract": "Let $\\{X_j\\}$ be independent, identically distributed random variables. It is well known that the functional CUSUM statistic and its randomly permuted version both converge weakly to a Brownian bridge if second moments exist. Surprisingly, an infinite-variance counterpart does not hold true. In the present paper, we let $\\{X_j\\}$ be in the domain of attraction of a strictly $\\alpha$-stable law, $\\alpha\\in(0,2)$. While the functional CUSUM statistics itself converges to an $\\alpha$-stable bridge and so does the permuted version, provided both the $\\{X_j\\}$ and the permutation are random, the situation turns out to be more delicate if a realization of the $\\{X_j\\}$ is fixed and randomness is restricted to the permutation. Here, the conditional distribution function of the permuted CUSUM statistics converges in probability to a random and nondegenerate limit."}
{"category": "Math", "title": "Statistical analysis of self-similar conservative fragmentation chains", "abstract": "We explore statistical inference in self-similar conservative fragmentation chains when only approximate observations of the sizes of the fragments below a given threshold are available. This framework, introduced by Bertoin and Martinez [Adv. Appl. Probab. 37 (2005) 553--570], is motivated by mineral crushing in the mining industry. The underlying object that can be identified from the data is the step distribution of the random walk associated with a randomly tagged fragment that evolves along the genealogical tree representation of the fragmentation process. We compute upper and lower rates of estimation in a parametric framework and show that in the nonparametric case, the difficulty of the estimation is comparable to ill-posed linear inverse problems of order 1 in signal denoising."}
{"category": "Math", "title": "Sagbi Bases of Cox-Nagata Rings", "abstract": "We degenerate Cox-Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev-Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective n-space at n+3 points, sagbi bases of Cox-Nagata rings establish a link between the Verlinde formula and phylogenetic algebraic geometry, and we use this to answer questions due to D'Cruz-Iarobbino and Buczynska-Wisniewski. Inspired by the zonotopal algebras of Holtz and Ron, our study emphasizes explicit computations, and offers a new approach to Hilbert functions of fat points."}
{"category": "Math", "title": "Admissibility and Controllability of diagonal Volterra equations with scalar inputs", "abstract": "This article studies Volterra evolution equations from the point of view of control theory, in the case that the generator of the underlying semigroup has a Riesz basis of eigenvectors. Conditions for admissibility of the system's control operator are given in terms of the Carleson embedding properties of certain discrete measures. Moreover, exact and null controllability are expressed in terms of a new interpolation question for analytic functions, providing a generalization of results known to hold for the standard Cauchy problem. The results are illustrated by examples involving heat conduction with memory."}
{"category": "Math", "title": "Asymptotic link invariants for ergodic vector fields", "abstract": "We study the asymptotics of a family of link invariants on the orbits of a smooth volume-preserving ergodic vector field on a compact domain of the 3-space. These invariants, called linear saddle invariants, include many concordance invariants and generate an infinite-dimensional vector space of link invariants. In contrast, the vector space of asymptotic linear saddle invariants is 1-dimensional, generated by the asymptotic signature. We also relate the asymptotic slice genus to the asymptotic signature."}
{"category": "Math", "title": "Series of Reciprocal Powers of k-almost Primes", "abstract": "Sums over inverse s-th powers of semiprimes and k-almost primes are transformed into sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the role of expansion coefficients. Founded on a known convergence acceleration for the ordinary prime zeta functions, the sums and first derivatives are tabulated with high precision for indices k=2,...,6 and integer powers s=2,...,8."}
{"category": "Math", "title": "A Numerical Approach to the Estimation of the Solutions of some Variational Problems with Convexity Costraints", "abstract": "We present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions. The main motivator behind this numerical method is estimating solutions to Adverse Selection problems within a Principal-Agent framework. Problems such as product lines design, optimal taxation, structured derivatives design, etc. can be studied through the scope of these models. We develop a method to estimate their optimal pricing schedules."}
{"category": "Math", "title": "On the ruin problem in the renewal risk processes perturbed by diffusion", "abstract": "In this paper, we consider the perturbed renewal risk process. Systems of integro-differential equations for the Gerber-Shiu functions at ruin caused by a claim and oscillation are established, respectively. The explicit Laplase transforms of Gerber-Shiu functions are obtained, while the closed form expressions for the Gerber-Shiu functions are derived when the claim amount distribution is from the rational family. Finally, we present numerical examples intended to illustrate the main results."}
{"category": "Math", "title": "Uniform partitions of frames of exponentials into Riesz sequences", "abstract": "The Feichtinger Conjecture, if true, would have as a corollary that for each set $E\\subset \\T$ and $\\Lambda \\subset \\Z$, there is a partition $\\Lambda_1,...,\\Lambda_N$ of $\\Z$ such that for each $1\\le i \\le N$, $\\{\\exp(2\\pi i x\\lambda): \\lambda \\in \\Lambda_i\\}$ is a Riesz sequence. In this paper, sufficient conditions on sets $E\\subset \\T$ and $\\Lambda\\subset \\R$ are given so that $\\{\\exp(2\\pi i x\\lambda) 1_E: \\lambda \\in \\Lambda\\}$ can be uniformly partitioned into Riesz sequences."}
{"category": "Math", "title": "Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures", "abstract": "We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the \\'etale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the contribution of the Eisenstein cohomology together with its S_6-action for the full level 2 structure and on the basis of our computations we make precise conjectures on the endoscopic contribution. We also make a prediction about the existence of a vector-valued analogue of the Saito-Kurokawa lift. Assuming these conjectures that are based on ample numerical evidence, we obtain the traces of the Hecke-operators T(p) for p < 41 on the remaining spaces of `genuine' Siegel modular forms. We present a number of examples of 1-dimensional spaces of eigenforms where these traces coincide with the Hecke eigenvalues. We hope that the experts on lifting and on endoscopy will be able to prove our conjectures."}
{"category": "Math", "title": "Robust Smoothed Analysis of a Condition Number for Linear Programming", "abstract": "We perform a smoothed analysis of the GCC-condition number C(A) of the linear programming feasibility problem \\exists x\\in\\R^{m+1} Ax < 0. Suppose that \\bar{A} is any matrix with rows \\bar{a_i} of euclidean norm 1 and, independently for all i, let a_i be a random perturbation of \\bar{a_i} following the uniform distribution in the spherical disk in S^m of angular radius \\arcsin\\sigma and centered at \\bar{a_i}. We prove that E(\\ln C(A)) = O(mn / \\sigma). A similar result was shown for Renegar's condition number and Gaussian perturbations by Dunagan, Spielman, and Teng [arXiv:cs.DS/0302011]. Our result is robust in the sense that it easily extends to radially symmetric probability distributions supported on a spherical disk of radius \\arcsin\\sigma, whose density may even have a singularity at the center of the perturbation. Our proofs combine ideas from a recent paper of B\\\"urgisser, Cucker, and Lotz (Math. Comp. 77, No. 263, 2008) with techniques of Dunagan et al."}
{"category": "Math", "title": "On the Second-Order Correlation Function of the Characteristic Polynomial of a Real Symmetric Wigner Matrix", "abstract": "We consider the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble essentially continues to hold for a general real symmetric Wigner matrix."}
{"category": "Math", "title": "Dynamics of meromorphic maps with small topological degree I: from cohomology to currents", "abstract": "We consider the dynamics of a meromorphic map on a compact kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which its iterates expand the cohomology class of a kahler form. Our goal in this article and its sequels is to carry out a conjectural program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, converting information about the linear action of the map on cohomology to invariant currents with special geometric structure. We also give some examples and identify some additional properties of maps on irrational surfaces and of maps whose invariant cohomology classes have vanishing self-intersection."}
{"category": "Math", "title": "The mixed problem for harmonic functions in polyhedra", "abstract": "R. M. Brown's theorem on mixed Dirichlet and Neumann boundary conditions is extended in two ways for the special case of polyhedral domains. A (1) more general partition of the boundary into Dirichlet and Neumann sets is used on (2) manifold boundaries that are not locally given as the graphs of functions. Examples are constructed to illustrate necessity and other implications of the geometric hypotheses."}
{"category": "Math", "title": "On one-sided Lie nilpotent ideals of associative rings", "abstract": "We prove that a Lie nilpotent one-sided ideal of an associative ring $R$ is contained in a Lie solvable two-sided ideal of $R$. An estimation of derived length of such Lie solvable ideal is obtained depending on the class of Lie nilpotency of the Lie nilpotent one-sided ideal of $R.$ One-sided Lie nilpotent ideals contained in ideals generated by commutators of the form $[... [ [r_1, r_{2}], ... ], r_{n-1}], r_{n}]$ are also studied."}
{"category": "Math", "title": "On q-deformed gl(l+1)-Whittaker function II", "abstract": "A representation of a specialization of a q-deformed class one lattice gl(\\ell+1}-Whittaker function in terms of cohomology groups of line bundles on the space QM_d(P^{\\ell}) of quasi-maps P^1 to P^{\\ell} of degree d is proposed. For \\ell=1, this provides an interpretation of non-specialized q-deformed gl(2)-Whittaker function in terms of QM_d(\\IP^1). In particular the (q-version of) Mellin-Barnes representation of gl(2)-Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Gamma-function as a substitute of topological genus in semi-infinite geometry. A relation with Givental-Lee universal solution (J-function) of q-deformed gl(2)-Toda chain is also discussed."}
{"category": "Math", "title": "The Chabauty-Coleman bound at a prime of bad reduction", "abstract": "We extend the refined version of the Chabauty-Coleman bound on the number of rational points on a curve of genus g>1 to the case of bad reduction."}
{"category": "Math", "title": "L^{2}-restriction bounds for eigenfunctions along curves in the quantum completely integrable case", "abstract": "We show that for a quantum completely integrable system in two dimensions,the $L^{2}$-normalized joint eigenfunctions of the commuting semiclassical pseudodifferential operators satisfy restriction bounds ofthe form $ \\int_{\\gamma} |\\phi_{j}^{\\hbar}|^2 ds = {\\mathcal O}(|\\log \\hbar|)$ for generic curves $\\gamma$ on the surface. We also prove that the maximal restriction bounds of Burq-Gerard-Tzvetkov are always attained for certain exceptional subsequences of eigenfunctions."}
{"category": "Math", "title": "The Corona theorem and stable rank for the algebra $\\mC +BH^\\infty$", "abstract": "Let $B$ be a Blaschke product. We prove in several different ways the corona theorem for the algebra $H^\\infty_B:=\\mC+BH^\\infty$. That is, we show the equivalence of the classical {\\em corona condition} on data $f_1, ..., f_n \\in H^\\infty_B$: \\[ \\forall z \\in \\mD, \\sum_{k=1}^{n} |f_k(z)| \\geq \\delta >0, \\] and the {\\em solvability of the Bezout equation} for $g_1, ..., g_n \\in H^\\infty_B$: \\[ \\forall z\\in \\mD, \\sum_{k=1}^n g_k (z)f_k(z)=1. \\] Estimates on solutions to the Bezout equation are also obtained. We also show that the Bass stable rank of $H^\\infty_B$ is 1. Let $A(\\mD)_B$ be the subalgebra of all elements from $H^\\infty_B$ having a continuous extension to the closed unit disk $\\bar{\\mD}$. Analogous results are obtained also for $A(\\mD)_B$."}
{"category": "Math", "title": "Kahler geometry on toric manifolds, and some other manifolds with large symmetry", "abstract": "This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-fold"}
{"category": "Math", "title": "Numerical approximations to extremal metrics on toric surfaces", "abstract": "We give some detailed numerical information about extremal metrics on four different toric surfaces. These are sample of many other cases which can be treated using a computer programme outlined in the paper."}
{"category": "Math", "title": "Easy Proofs of Some Borwein Algorithms for $\\pi$", "abstract": "In 1987 Jonathan and Peter Borwein, inspired by the works of Ramanujan, derived many efficient algorithms for computing $\\pi$. We will see that by using only a formula of Gauss's and elementary algebra we are able to prove the correctness of two of them."}
{"category": "Math", "title": "AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules", "abstract": "We investigate the properties of categories of G_C-flat R-modules where C is a semidualizing module over a commutative noetherian ring R. We prove that the category of all G_C-flat R-modules is part of a weak AB-context, in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander-Buchweitz approximations for R-modules of finite G_C-flat dimension. We also prove that two procedures for building R-modules from complete resolutions by certain subcategories of G_C-flat R-modules yield only the modules in the original subcategories."}
{"category": "Math", "title": "Lipschitz perturbations of differentiable implicit functions", "abstract": "Let $y=f(x)$ be a continuously differentiable implicit function solving the equation $F(x,y)=0$ with continuously differentiable $F.$ In this paper we show that if $F_\\eps$ is a Lipschitz function such that the Lipschitz constant of $F_\\eps-F$ goes to 0 as $\\eps\\to 0$ then the equation $F_\\eps(x,y)=0$ has a Lipschitz solution $y=f_\\eps(x)$ such that the Lipschitz constant of $f_\\eps-f$ goes to 0 as $\\eps\\to 0$ either. As an application we evaluate the length of time intervals where the right hand parts of some nonautonomous discontinuous systems of ODEs are continuously differentiable with respect to state variables. The latter is done as a preparatory step toward generalizing the second Bogolyubov's theorem for discontinuous systems."}
{"category": "Math", "title": "On Mitropol'skii Yu.A.'s Theorem on Periodic Solutions of Systems of Nonlinear Differential Equations with Non-Differentiable Right-Hand-Sides", "abstract": "The smooth second Bogolyubov's theorem is generalized for Lipschitz systems."}
{"category": "Math", "title": "Quantitative uniqueness for the power of Laplacian with singular coefficients", "abstract": "In this paper we study the local behavior of a solution to the $l$th power of Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order."}
{"category": "Math", "title": "Positroids and Schubert matroids", "abstract": "Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroid. We prove his conjecture that a positroid is exactly an intersection of permuted Schubert matroids. This leads to a nice combinatorial description of positroids that is easily computable."}
{"category": "Math", "title": "Geometric second derivative estimates in Carnot groups and convexity", "abstract": "We prove some new a priori estimates for H_2-convex functions which are zero on the boundary of a bounded smooth domain \\Omega in a Carnot group G. Such estimates are global and are geometric in nature as they involve the horizontal mean curvature \\mathcal H of the boundary of \\Omega. As a consequence of our bounds we show that if G has step two, then for any smooth $H_2$-convex function in \\Omega \\subset G vanishing on the boundary of \\Omega one has \\sum_{i,j=1}^m \\int_\\Omega ([X_i,X_j]u)^2 dg \\leq {4/3} \\int_{\\partial \\Omega} \\mathcal H |\\nabla_H u|^2 d\\sigma_H ."}
{"category": "Math", "title": "Multiple integral representation for functionals of Dirichlet processes", "abstract": "We point out that a proper use of the Hoeffding--ANOVA decomposition for symmetric statistics of finite urn sequences, previously introduced by the author, yields a decomposition of the space of square-integrable functionals of a Dirichlet--Ferguson process, written $L^2(D)$, into orthogonal subspaces of multiple integrals of increasing order. This gives an isomorphism between $L^2(D)$ and an appropriate Fock space over a class of deterministic functions. By means of a well-known result due to Blackwell and MacQueen, we show that each element of the $n$th orthogonal space of multiple integrals can be represented as the $L^2$ limit of $U$-statistics with degenerate kernel of degree $n$. General formulae for the decomposition of a given functional are provided in terms of linear combinations of conditioned expectations whose coefficients are explicitly computed. We show that, in simple cases, multiple integrals have a natural representation in terms of Jacobi polynomials. Several connections are established, in particular with Bayesian decision problems, and with some classic formulae concerning the transition densities of multiallele diffusion models, due to Littler and Fackerell, and Griffiths. Our results may also be used to calculate the best approximation of elements of $L^2(D)$ by means of $U$-statistics of finite vectors of exchangeable observations."}
{"category": "Math", "title": "Gorenstein polytopes obtained from bipartite graphs", "abstract": "Beck et. al. characterized the grid graphs whose perfect matching polytopes are Gorenstein and they also showed that for some parameters, perfect matching polytopes of torus graphs are Gorenstein. In this paper, we complement their result, that is, we characterize the torus graphs whose perfect matching polytopes are Gorenstein. Beck et. al. also gave a method to construct an infinite family of Gorenstein polytopes. In this paper, we introduce a new class of polytopes obtained from graphs and we extend their method to construct many more Gorenstein polytopes."}
{"category": "Math", "title": "TQFT string operations in open-closed string topology", "abstract": "To open-closed cobordism surfaces, open-closed string topology associates topological quantum field theory (TQFT) operations, namely string operations, which depend only on homeomorphism types of surfaces and which satisfy the sewing property. We show that most TQFT string operations vanish in open-closed string topology. We describe those open-closed cobordisms with vanishing string operations, and give a short list of open-closed cobordisms with possibly nontrivial string operations."}
{"category": "Math", "title": "Open-Closed TQFT String Operations for Disc Cobordisms, Simultaneous Saddle Interactions, and Constant Homology Classes", "abstract": "Previously, we showed that most of the open-closed topological quantum field theory (TQFT) string operations vanish including all the higher genus TQFT operations, and we described a small list of genus zero open-closed TQFT string operations which can be nontrivial. In this paper, we consider open-closed string operations associated to open-closed cobordisms homeomorphic to discs. These operations constitute the main part of genus zero string operations, and they include the saddle string operation of two open strings interacting at their internal points. We show not only that disc string operations are independent of their half-pair-of-pants decompositions but also that these disc string operations can be computed by \\emph{simultaneous} saddle interactions of incoming open strings at the same point, and they take values in homology classes of constant open strings on some closed orientable submanifolds, which we will precisely determine. We will also discuss a role played by fundamental constant homology classes in open-closed string topology. Our main tools are saddle interaction diagrams and their deformations."}
{"category": "Math", "title": "Conformal arc-length as $\\frac12$ dimensional length of the set of osculating circles", "abstract": "The set of osculating circles of a given curve in $\\SS^3$ forms a curve in the set of oriented circles in $\\SS^3$. We show that its \"${\\frac12}$-dimensional measure\" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century."}
{"category": "Math", "title": "Random motion with gamma-distributed alternating velocities in biological modeling", "abstract": "Motivated by applications in mathematical biology concerning randomly alternating motion of micro-organisms, we analyze a generalized integrated telegraph process. The random times between consecutive velocity reversals are gamma-distributed, and perform an alternating renewal process. We obtain the probability law and the mean of the process."}
{"category": "Math", "title": "Moduli spaces of reflexive sheaves of rank 2", "abstract": "Let \\sF be a coherent rank 2 sheaf on a scheme Y \\subset \\proj{n} of dimension at least two. In this paper we study the relationship between the functor which deforms a pair (\\sF,\\sigma), \\sigma \\in H^0(\\sF), and the functor which deforms the corresponding pair (X,\\xi) given as in the Serre correspondence. We prove that the scheme structure of e.g. the moduli scheme M_Y(P) of stable sheaves on a threefold Y at (\\sF), and the scheme structure at (X) of the Hilbert scheme of curves on Y are closely related. Using this relationship we get criteria for the dimension and smoothness of M_Y(P) at (\\sF), without assuming Ext^2(\\sF,\\sF) = 0. For reflexive sheaves on Y = \\proj{3} whose deficiency module M = H_{*}^1(\\sF) satisfies Ext^2(M,M) = 0 in degree zero (e.g. of diameter at most 2), we get necessary and sufficient conditions of unobstructedness which coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of H_{*}^0(\\sF). It follows that every irreducible component of M_{\\proj{3}}(P) containing a reflexive sheaf of diameter one is reduced (generically smooth). We also determine a good lower bound for the dimension of any component of M_{\\proj{3}}(P) which contains a reflexive stable sheaf with \"small\" deficiency module M."}
{"category": "Math", "title": "Depth of segments and circles through points enclosing many points: a note", "abstract": "Neumann-Lara and Urrutia showed in 1985 that in any set of n points in the plane in general positionthere is always a pair of points such that any circle through them contains at least (n-2)/60 points. In a series of papers, this result was subsequently improved till n/4.7, which is currently the best known lower bound. In this paper we propose a new approach to the problem that allows us, by using known results about j-facets of sets of points in $R^3$, to give a simple proof of a somehow stronger result: there is always a pair of points such that any circle through them has, both inside and outside, at least n/4.7 points."}
{"category": "Math", "title": "A family of representations of braid groups on surfaces", "abstract": "We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation."}
{"category": "Math", "title": "Quelques in\\'egalit\\'es effectives entre des fonctions arithm\\'etiques usuelles", "abstract": "Let us denote by $\\tau(n)$ and $\\si(n)$ the number and the sum of the divisors of $n$ and by $\\vfi$ Euler's function. We give effective upper bounds for $\\frac{n}{\\vfi(n)}$ in terms of $\\vfi(n)$, and for $\\frac{\\si(n)}{n}$ in terms of $\\tau(n)$."}
{"category": "Math", "title": "Single-index Regression models with right-censored responses", "abstract": "In this article, we propose some new generalizations of M-estimation procedures for single-index regression models in presence of randomly right-censored responses. We derive consistency and asymptotic normality of our estimates. The results are proved in order to be adapted to a wide range of techniques used in a censored regression framework (e.g. synthetic data or weighted least squares). As in the uncensored case, the estimator of the single-index parameter is seen to have the same asymptotic behavior as in a fully parametric scheme. We compare these new estimators with those based on the average derivative technique of Burke and Lu (2005) through a simulation study."}
{"category": "Math", "title": "On (twisted) Lawrence-Krammer representations", "abstract": "Lawrence-Krammer representations (LK-representations for short) are linear representations of Artin-Tits groups of small type, which are of importance since they are known to be faithful when the type is spherical, or more generally when restricted to the monoid. If the construction is essentially unique for a given small and spherical type, the structure of the set of LK-representations for a given small type is not understood in general. Another important question is to ask if there exists an analogue of this construction in the non-small cases ; a first answer is given in [Digne, On the linearity of Artin Braid groups. J. Algebra 268, (2003) 39-57], where is constructed a faithful ``twisted'' LK-representation for the spherical, non-small and crystallographic types. The aim of this paper is to continue the investigations on those two topics. Regarding the first one, we classify the LK-representations of the Artin-Tits monoids and groups of small and affine type. Concerning the second one, we generalize the construction of op.cit. to any Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under the action of graph automorphisms."}
{"category": "Math", "title": "0-Cohomology of semigroups", "abstract": "This article is a survey of 0-cohomology of semigroups. The main attention is devoted to applications."}
{"category": "Math", "title": "The Divisor Matrix, Dirichlet Series and SL(2,Z), II", "abstract": "We examine an elliptic curve constructed in an earlier paper from a certain representation of $\\SL(2,\\Z)$ on the space of convergent Dirichlet series. The curve is observed to be a modular curve for $\\Gamma^1(15)$ and a certain orbit of modular functions is thereby associated with the Riemann zeta function. Explicit descriptions are given of these functions and of the permutation action of $\\SL(2,\\Z)$ on them."}
{"category": "Math", "title": "A note on Larsen's conjecture and ranks of elliptic curves", "abstract": "Let E be an elliptic curve defined over a number field K. Michael Larsen conjectured that for any finitely generated subgroup G of Gal(\\bar K/K), the Mordell-Weil rank of E is unbounded in number fields fixed by G. We prove that the conjecture holds over K=Q for both the analytic rank and the p-infinity Selmer rank of E for every odd prime p. For arbitrary E/K, we show that Larsen's conjecture follows from the standard conjectures for ranks of elliptic curves, provided K has a real place or E has non-integral j-invariant."}
{"category": "Math", "title": "Equivalence for Differential Equations", "abstract": "We shall study the equivalence problem for ordinary differential equations with respect to the affine transformations group."}
{"category": "Math", "title": "Plane recursive trees, Stirling permutations and an urn model", "abstract": "We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also prove results on the joint distribution of the numbers of ascents, descents and plateaux in a random Stirling permutation. The proof uses an interesting generalized Polya urn"}
{"category": "Math", "title": "Opposite relation on dual polar spaces and half-spin Grassmann spaces", "abstract": "We characterize the collinearity (adjacency) relation of half-spin Grassmann spaces in terms of the relation to be opposite in the corresponding collinearity graphs. Also we show that this characterization does not hold for dual polar spaces."}
{"category": "Math", "title": "On the Gorenstein locus of some punctual Hilbert schemes", "abstract": "Let $k$ be an algebraically closed field and let $\\Hilb_{d}^{G}(\\p{N})$ be the open locus of the Hilbert scheme $\\Hilb_{d}(\\p{N})$ corresponding to Gorenstein subschemes. We prove that $\\Hilb_{d}^{G}(\\p{N})$ is irreducible for $d\\le9$, we characterize geometrically its singularities for $d\\le 8$ and we give some results about them when $d=9$ which give some evidence to a conjecture on the nature of the singular points in $\\Hilb_{d}^{G}(\\p{N})$."}
{"category": "Math", "title": "A short proof of nonhomogeneity of the pseudo-circle", "abstract": "The pseudo-circle is known to be nonhomogeneous. The original proofs of this fact were discovered independently by L. Fearnley and J.T. Rogers, Jr. The purpose of this paper is to provide an alternative, very short proof based on a result of D. Bellamy and W. Lewis."}
{"category": "Math", "title": "A combinatorial formula for Macdonald polynomials", "abstract": "In this paper we use the combinatorics of alcove walks to give a uniform combinatorial formula for Macdonald polynomials for all Lie types. These formulas are generalizations of the formulas of Haglund-Haiman-Loehr for Macdonald polynoimals of type GL(n). At q=0 these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at q=t=0 these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent-Littelmann)."}
{"category": "Math", "title": "Regular Points of a Subcartesian Space", "abstract": "We discuss properties of the regular part $S_{reg}$ of a subcartesian space $S$. We show that $S_{reg}$ is open and dense in $S$ and the restriction to $S_{reg}$ of the tangent bundle of $S$ is locally trivial."}
{"category": "Math", "title": "Plactic relations for $r$-domino tableaux", "abstract": "The work of C. Bonnaf{\\'e}, M. Geck, L. Iancu and T. Lam \\cite{Geck-Lam} shows through two conjectures that $r$-domino tableaux have an important role in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we provide plactic relations on signed permutations which determine whether given two signed permutations have the same insertion $r$-domino tableaux in Garfinkle's algorithm \\cite{Garfinkle1}."}
{"category": "Math", "title": "Pre-Hausdorff Spaces", "abstract": "This paper introduces three separation conditions for topological spaces, called T_{0,1}, T_{0,2} (\"pre-Hausdorff\"), and T_{1,2}. These conditions generalize the classical T_(1) and T_(2) separation axioms, and they have advantages over them topologically which we discuss. We establish several different characterizations of pre-Hausdorff spaces, and a characterization of Hausdorff spaces in terms of pre-Hausdorff. We also discuss some classical Theorems of general topology which can or cannot be generalized by replacing the Hausdorff condition by pre-Hausdorff."}
{"category": "Math", "title": "On Lagrangian fibrations by Jacobians I", "abstract": "Let Y->P^n be a flat family of integral Gorenstein curves, such that the compactified relative Jacobian X=\\bar{J}^d(Y/P^n) is a Lagrangian fibration. We prove that the degree of the discriminant locus Delta in P^n is at least 4n+2, and we prove that X is a Beauville-Mukai integrable system if the degree of Delta is greater than 4n+20."}
{"category": "Math", "title": "The Dolbeault complex with weights according to normal crossings", "abstract": "In this paper, we define a Dolbeault complex with weights according to normal crossings, which is a useful tool for studying the d-bar-equation on singular complex spaces by resolution of singularities (where normal crossings appear naturally). The major difficulty is to prove that this complex is locally exact. We do that by constructing a local d-bar-solution operator which involves only Cauchy's Integral Formula (in one complex variable) and behaves well for L^p-forms with weights according to normal crossings."}
{"category": "Math", "title": "The d-bar-equation on homogeneous varieties with an isolated singularity", "abstract": "Let X be a regular irreducible variety in CP^{n-1}, Y the associated homogeneous variety in C^n, and N the restriction of the universal bundle of CP^{n-1} to X. In the present paper, we compute the obstructions to solving the d-bar-equation in the L^p-sense on Y for 1<=p<=\\infty in terms of cohomology groups H^q(X,O(N^m)). That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to CP^1 or an elliptic curve."}
{"category": "Math", "title": "Stability and instability results in a model of Fermi acceleration", "abstract": "We consider the static wall approximation to the dynamics of a particle bouncing on a periodically oscillating infinitely heavy plate while subject to a potential force. We assume the case of a potential given by a power of the particle's height and sinusoidal motions of the plate. We find that for powers smaller than 1 the set of escaping orbits has full Hausdorff dimension for all motions and obtain existence of elliptic island of period 2 for arbitrarily high energies for a full-measure set of motions. Moreover we obtain conditions on the potential to ensure that the total (Lebesgue) measure of elliptic islands of period 2 is either finite or infinite."}
{"category": "Math", "title": "The finiteness result for Khovanov homology and localization in monoidal categories", "abstract": "In the previous paper we constructed the local system of Khovanov complexes on the Vassiliev space of knots and extended it to the singular locus. In this paper we introduce the definition of the homology theory (local system) of finite type and prove the first finiteness result: the Khovanov local system restricted to the subcategory of knots of the crossing number at most n is the theory of type less or equal to n. This result can be further generalized to the categorification of Birman-Lin theorem."}
{"category": "Math", "title": "Distributed Subgradient Methods and Quantization Effects", "abstract": "We consider a convex unconstrained optimization problem that arises in a network of agents whose goal is to cooperatively optimize the sum of the individual agent objective functions through local computations and communications. For this problem, we use averaging algorithms to develop distributed subgradient methods that can operate over a time-varying topology. Our focus is on the convergence rate of these methods and the degradation in performance when only quantized information is available. Based on our recent results on the convergence time of distributed averaging algorithms, we derive improved upper bounds on the convergence rate of the unquantized subgradient method. We then propose a distributed subgradient method under the additional constraint that agents can only store and communicate quantized information, and we provide bounds on its convergence rate that highlight the dependence on the number of quantization levels."}
{"category": "Math", "title": "Picard number of the generic fiber of an abelian fibered hyperkaehler manifold", "abstract": "We shall show that the Picard number of the generic fiber, in the sense of scheme, of an abelian fibered hyperk\\\"ahler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O'Grady's 10-dimensional manifold, we construct an abelian fibered hyperk\\\"ahler manifold of Mordell-Weil rank 20, which is the maximum possible among all known ones."}
{"category": "Math", "title": "Systems with the integer rounding property in normal monomial subrings", "abstract": "Let C be a clutter and let A be its incidence matrix. If the linear system x>=0;xA<=1 has the integer rounding property, we give a description of the canonical module and the a-invariant of certain normal subrings associated to C. If the clutter is a connected graph, we describe when the aforementioned linear system has the integer rounding property in combinatorial and algebraic terms using graph theory and the theory of Rees algebras. As a consequence we show that the extended Rees algebra of the edge ideal of a bipartite graph is Gorenstein if and only if the graph is unmixed."}
{"category": "Math", "title": "A stochastic-Lagrangian particle system for the Navier-Stokes equations", "abstract": "This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (\\texttt{arxiv:math.PR/0511067}, to appear), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $\\frac{1}{N}$ times the sum over these $N$ copies. (We remark that our formulation requires one to keep track of $N$ stochastic flows of diffeomorphisms, and not just the motion of $N$ particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space $\\holderspace{1}{\\alpha}$ which consists of differentiable functions whose first derivative is $\\alpha$ H\\\"older continuous (see Section \\ref{sGexist} for the precise definition). Further, we show that as $N \\to \\infty$ the system converges to the solution of Navier-Stokes equations on any finite interval $[0,T]$. However for fixed $N$, we prove that this system retains roughly $O(\\frac{1}{N})$ times its original energy as $t \\to \\infty$. Hence the limit $N \\to \\infty$ and $T\\to \\infty$ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as $t \\to \\infty$ explicitly."}
{"category": "Math", "title": "Parameter Collapse due to the Zeros in the Inverse Condition", "abstract": "Helton, Lasserre, and Putinar (2008, Ann. Probability; arXiv:math/0702314) expose the relationship between three properties of a measure: the conditional triangularity property of the associated orthogonal polynomials, the zeros in the inverse condition of the truncated moment matrix, and conditional independence. The purpose of this article is to provide examples of parameter collapse to product structure given that the zeros in the inverse condition holds up to some degree d. Specifically, start with a parameterized family of probability density functions; require that the zeros in the inverse condition up to degree d holds; and validate that imposing this restriction on the parameterized family results in a measure with product structure, or at least that conditional independence holds. Algorithms related to parameter collapse are supplied, including the computation of the zeros in the inverse condition up to degree d."}
{"category": "Math", "title": "A new class of transport distances", "abstract": "We introduce a new class of distances between nonnegative Radon measures in Euclidean spaces. They are modeled on the dynamical characterization of the Kantorovich-Rubinstein-Wasserstein distances proposed by Benamou-Brenier and provide a wide family interpolating between the Wasserstein and the homogeneous (dual) Sobolev distances. From the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established Kantorovich-Rubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given."}
{"category": "Math", "title": "On the Hard Lefschetz property of stringy Hodge numbers", "abstract": "For projective varieties with a certain class of 'mild' isolated singularities and for projective threefolds with arbitrary Gorenstein canonical singularities, we show that the stringy Hodge numbers satisfy the Hard Lefschetz property. This result fits nicely with a 6-dimensional counterexample of Mustata and Payne for the Hard Lefschetz property for stringy Hodge numbers in general. We also give such an example, ours is a hypersurface singularity."}
{"category": "Math", "title": "Moments of Two-Variable Functions and the Uniqueness of Graph Limits", "abstract": "For a symmetric bounded measurable function W on [0,1]^2, \"moments\" of W can be defined as values t(F,W) indexed by simple graphs. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. This implies that the limit of a convergent dense graph sequence is unique up to measure preserving transformation."}
{"category": "Math", "title": "Testing properties of graphs and functions", "abstract": "We define an analytic version of the graph property testing problem, which can be formulated as studying an unknown 2-variable symmetric function through sampling from its domain and studying the random graph obtained when using the function values as edge probabilities. We give a characterization of properties testable this way, and extend a number of results about ``large graphs'' to this setting. These results can be applied to the original graph-theoretic property testing."}
{"category": "Math", "title": "Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties", "abstract": "We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the C^2 topology by harmonic maps into the spaces of Bergman metrics. In particular, WZW maps, or equivalently solutions of a homogeneous Monge-Ampere equation on the product of the manifold with a Riemann surface with S^1 boundary admit such approximations. We also show that the Eells-Sampson flow on the space of Kahler potentials is transformed to the usual heat flow on the space of symplectic potentials under the Legendre transform."}
{"category": "Math", "title": "The Three Gap Theorem and Riemannian Geometry", "abstract": "The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find results in a similar spirit pertaining to isometries of compact Riemannian manifolds and the distribution of points along their geodesics."}
{"category": "Math", "title": "Contact surgeries and the transverse invariant in knot Floer homology", "abstract": "We study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg-Chekanov twist knots E_n are not transversely simple for n odd and n>3."}
{"category": "Math", "title": "Geometry and Rank of Fibered Hyperbolic 3-Manifolds", "abstract": "Assume that M is a closed hyperbolic 3-manifold fibering over the circle with fiber a closed orientable surface of genus g. We show that if M has large diameter and its injectivity radius is bounded below, then the rank of the fundamental group of M is 2g+1."}
{"category": "Math", "title": "Two paradigms for topological quantum computation", "abstract": "We present two paradigms relating algebraic, topological and quantum computational statistics for the topological model for quantum computation. In particular we suggest correspondences between the computational power of topological quantum computers, computational complexity of link invariants and images of braid group representations. While at least parts of these paradigms are well-known to experts, we provide supporting evidence for them in terms of recent results. We give a fairly comprehensive list of known examples and formulate two conjectures that would further support the paradigms."}
{"category": "Math", "title": "On approximation of continuous functions by entire functions on subsets of the real line", "abstract": "We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type on unbounded closed proper subsets of the real line is studded."}
{"category": "Math", "title": "On the two dimensional Bilinear Hilbert Transform", "abstract": "We investigate the Bilinear Hilbert Transform in the plane and the pointwise convergence of bilinear averages in Ergodic theory, arising from $\\Z^2$ actions. Our techniques combine novel one and a half dimensional phase-space analysis with more standard one dimensional theory."}
{"category": "Math", "title": "Symmetries and the Riemann Hypothesis", "abstract": "Associated to classical semi-simple groups and their maximal parabolics are genuine zeta functions. Naturally related to Riemann's zeta and governed by symmetries, including that of Weyl, these zetas are expected to satisfy the Riemann hypothesis."}
{"category": "Math", "title": "Repeat distributions from unequal crossovers", "abstract": "It is a well-known fact that genetic sequences may contain sections with repeated units, called repeats, that differ in length over a population, with a length distribution of geometric type. A simple class of recombination models with single crossovers is analysed that result in equilibrium distributions of this type. Due to the nonlinear and infinite-dimensional nature of these models, their analysis requires some nontrivial tools from measure theory and functional analysis, which makes them interesting also from a mathematical point of view. In particular, they can be viewed as quadratic, hence nonlinear, analogues of Markov chains."}
{"category": "Math", "title": "An extended class of minimax generalized Bayes estimators of regression coefficients", "abstract": "We derive minimax generalized Bayes estimators of regression coefficients in the general linear model with spherically symmetric errors under invariant quadratic loss for the case of unknown scale. The class of estimators generalizes the class considered in Maruyama and Strawderman (2005) to include non-monotone shrinkage functions."}
{"category": "Math", "title": "Nevanlinna-Pick interpolation for $C+BH^\\infty$", "abstract": "Given an inner function $B$ we classify the invariant subspaces of the algebra $H^\\infty_B:=\\mathbb{C}+BH^\\infty$. We derive a formula in terms of these invariant subspaces for the distance of an element in $L^\\infty$ to a certain weak*-closed ideal in $H^\\infty_B$ and use this to prove an analogue of the Nevanlinna-Pick interpolation theorem."}
{"category": "Math", "title": "Chernoff and Trotter type product formulas", "abstract": "We consider the abstract Cauchy problem x'=Ax, x(0)=x_0\\in D(A) for linear operators A on a Banach space X. We prove uniqueness of the (local) solution of this problem for a natural class of operators A. Moreover, we establish that the solution x(\\cdot) can be represented as a limit of sequence F(t/n)^{n} as n\\to\\infty in the weak operator topology, where a function F:[0,\\infty)\\to L(X) satisfies F'(0)y=Ay, y\\in D(A). As a consequence, we deduce necessary and sufficient conditions that a linear operator C is closable and its closure is a generator of C_0-semigroup. We also obtain some criteria for the sum of two generators of C_0-semigroups to be a generator of C_0-semigroup such that the Trotter formula is valid."}
{"category": "Math", "title": "Isoparametric hypersurfaces with four principal curvatures revisited", "abstract": "The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and M\\\"{u}nzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles. In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the Cartan-M\\\"{u}nzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear."}
{"category": "Math", "title": "Relative regular modules. Applications to von Neumann regular rings", "abstract": "We use the concept of a regular object with respect to another object in an arbitrary category, defined in \\cite{dntd}, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories $R-$mod and $S-$mod, when $S$ is an excellent extension of the ring $R$. Consequently, we obtain a result of \\cite{ps}: if $S$ is an excellent extension of the ring $R$, then $S$ is von Neumann regular ring if and only if $R$ is also von Neumann regular ring. In the second part, using relative regular modules, we give a new proof of a classical result: the von Neumann regular property of a ring is Morita invariant. Finally, the von Neumann regularity of the Morita ring is investigated."}
{"category": "Math", "title": "Inverse scattering on conformally compact manifolds", "abstract": "We study inverse scattering for $\\Delta_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature $-\\alf^2(y)$ at the boundary and $V\\in C^\\infty(X)$ not vanishing at the boundary. We prove that the scattering matrix at a fixed energies $(\\lambda_1,$ $\\lambda_2)$ in a suitable subset of $\\mc$, determines $\\alf,$ and the Taylor series of both the potential and the metric at the boundary."}
{"category": "Math", "title": "Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links", "abstract": "Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is uniquely determined by $(B,T)$ and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead an algebraic structure which is isomorphic to the rational numbers. We introduce a new knot and link class, algebraically alternating knots and links, roughly speaking which are constructed from alternating knots and links by replacing some crossings with algebraic tangles. We give a necessary and sufficient condition for a closed surface to be incompressible and meridionally incompressible in the complement of an algebraically alternating knot or link $K$, in particular we show that if $K$ is a knot, then the complement of $K$ does not contain such a surface."}
{"category": "Math", "title": "Euler-Hurwitz series and non-linear Euler sums", "abstract": "In this paper we derive two expressions for the Hurwitz zeta function involving the complete Bell polynomials in the restricted case where q is a positive integer greater than 1. The arguments of the complete Bell polynomials comprise the generalised harmonic number functions. These in turn give rise to Euler-Hurwitz series which may then be used to determine identities for combinations of both linear and non-linear Euler sums."}
{"category": "Math", "title": "Note sur les corps 2-rationnels", "abstract": "We compute the Galois group of the maximal 2-ramified pro-2-extension of a 2-rational number field"}
{"category": "Math", "title": "A proof by calibration of an isoperimetric inequality in the Heisenberg group H^n", "abstract": "Let $D$ be a closed disk centered at the origin in the horizontal hyperplane $\\{t=0\\}$ of the sub-Riemannian Heisenberg group $\\hh^n$, and $C$ the vertical cylinder over $D$. We prove that any finite perimeter set $E$ such that $D\\subset E\\subset C$ has perimeter larger than or equal to the one of the rotationally symmetric sphere with constant mean curvature of the same volume, and that equality holds only for the spheres using a recent result by Monti and Vittone [12]."}
{"category": "Math", "title": "Examples of area-minimizing surfaces in the subriemannian Heisenberg group H^1 with low regularity", "abstract": "We give new examples of entire area-minimizing t-graphs in the subriemannian Heisenberg group H^1. Most of the examples are locally lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given point. Amongst them, a large family of area-minimizing cones is obtained."}
{"category": "Math", "title": "Simplices and spectra of graphs", "abstract": "In this note we show the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent functions of the lengths of edges. In order to prove this we compute the complete spectrum of a combinatorially interesting graph."}
{"category": "Math", "title": "Global regularity for a modified critical dissipative quasi-geostrophic equation", "abstract": "In this paper, we consider the modified quasi-geostrophic equation \\begin{gather*} \\del_t \\theta + (u \\cdot \\grad) \\theta + \\kappa \\Lambda^\\alpha \\theta = 0 u = \\Lambda^{\\alpha - 1} R^{\\perp}\\theta. \\end{gather*} with $\\kappa > 0$, $\\alpha \\in (0,1]$ and $\\theta_0 \\in \\lp{2}(\\R^2)$. We remark that the extra $\\Lambda^{\\alpha - 1}$ is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence and regularity of strong solutions to this system."}
{"category": "Math", "title": "Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology", "abstract": "We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the \\'etale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line."}
{"category": "Math", "title": "A complex surface of general type with p_g=0, K^2=3 and H_1=Z/2Z", "abstract": "This paper is an addendum to [4], in which the authors constructed a simply connected minimal complex surface of general type with p_g=0 and K^2=3. In this paper we construct a new non-simply connected minimal surface of general type with p_g=0, K^2=3 and H_1=Z/2Z using a rational blow-down surgery and Q-Gorenstein smoothing theory."}
{"category": "Math", "title": "Covariants and the no-name lemma", "abstract": "A close connection between the no-name lemma (concerning algebraic groups acting on vector bundles) and the existence of sufficiently many independent rational covariants is pointed out. In particular, this leads to a new natural proof of the no-name lemma. For linearly reductive groups, the approach has a refined variant based on integral covariants. This fits into the usual context of invariant theory, and yields a version of the no-name lemma that has a constructive nature."}
{"category": "Math", "title": "Quivers with potentials associated to triangulated surfaces", "abstract": "We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin-Shapiro-Thurston, and quivers with potentials and their mutations introduced by Derksen-Weyman-Zelevinsky. To each ideal triangulation of a bordered surface with marked points we associate a quiver with potential, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective quivers with potentials are related by a mutation with respect to the flipped arc. We prove that if the surface has non-empty boundary, then the quivers with potentials associated to its triangulations are rigid and hence non-degenerate."}
{"category": "Math", "title": "Arithmetic Groups Have Rational Representation Growth", "abstract": "Let G be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if G has the congruence subgroup property, then the number of n-dimensional irreducible representations of G grows like n^a, where a is a rational number."}
{"category": "Math", "title": "Cohen-Macaulay admissible clutters", "abstract": "There is a one-to-one correspondence between square-free monomial ideals and clutters, which are also known as simple hypergraphs. It was conjectured that unmixed admissible clutters are Cohen-Macaulay. We prove the conjecture for uniform admissible clutters of heights 2 and 3. For admissible clutters of greater heights, we give a family of examples to show that the conjecture may fail. When the height is 4, we give an additional condition under which unmixed admissible clutters are Cohen-Macaulay."}
{"category": "Math", "title": "Minimality of the well-rounded retract", "abstract": "We prove that the well-rounded retract of SO_n\\SL_n(R) is a minimal SL_n(Z)-invariant spine."}
{"category": "Math", "title": "Primitive Central Idempotents of the Group Algebra", "abstract": "An approach to representations of finite groups is presented without recourse to character theory. Considering the group algebra C[G] as an algebra of linear maps on C[G] (by left multiplication), we derive the primitive central idempotents as a simultaneous eigenbasis of the centre, Z(C[G]). We apply this framework to obtain the irreducible representations of a class of finite meta-abelian groups. In particular, we give a general construction of the isomorphism between simple blocks of C[G] and the corresponding matrix algebra where G can be any finite group."}
{"category": "Math", "title": "Generating function for GL_n-invariant differential operators in the skew Capelli identity", "abstract": "Let Alt_n be the vector space of all alternating n-by-n complex matrices, on which the complex general linear group GL_n acts by $x \\mapsto g x g^t$. The aim of this paper is to show that Pfaffian of a certain matrix whose entries are multiplication operators or derivations acting on polynomials on Alt_n provides a generating function for the GL_n-invariant differential operators that play a role in the skew Capelli identity, with coefficients the Hermite polynomials."}
{"category": "Math", "title": "Wild Harmonic Bundles and Wild Pure Twistor D-modules", "abstract": "We study (i) asymptotic behaviour of wild harmonic bundles, (ii) the relation between semisimple meromorphic flat connections and wild harmonic bundles, (iii) the relation between wild harmonic bundles and polarized wild pure twistor $D$-modules. As an application, we show the hard Lefschetz theorem for algebraic semisimple holonomic $D$-modules, conjectured by M. Kashiwara."}
{"category": "Math", "title": "Good formal structure for meromorphic flat connections on smooth projective surfaces", "abstract": "We prove the algebraic version of a conjecture of C. Sabbah on the existence of the good formal structure for meromorphic flat connections on surfaces after some blow up."}
{"category": "Math", "title": "A report on \"Regulators of canonical extension are torsion; the smooth divisor case\"", "abstract": "In this note, we report on a work jointly done with C. Simpson on a generalization of Reznikov's theorem which says that the Chern-Simons classes and in particular the Deligne Chern classes (in degrees $> 1$) are torsion, of a flat vector bundle on a smooth complex projective variety. We consider the case of a smooth quasi--projective variety with an irreducible smooth divisor at infinity. We define the Chern-Simons classes of the Deligne's \\textit{canonical extension} of a flat vector bundle with unipotent monodromy at infinity, which lift the Deligne Chern classes and prove that these classes are torsion. The details of the proof can be found in arxiv:0707.0372 [math.AG]."}
{"category": "Math", "title": "On flows associated to Sobolev vector fields in Wiener spaces: an approach \\`a la DiPerna-Lions", "abstract": "In this paper we extend the DiPerna-Lions theory on ODEs with Sobolev vector fields to the setting of abstract Wiener spaces."}
{"category": "Math", "title": "Dilational Hilbert Scales and Deconvolutional Sharpening", "abstract": "Operationally, index functions of variable Hilbert scales can be viewed as generators for families of spaces and norms. Using a one parameter family of index functions based on the dilations of a given index function, a new class of scales (dilational Hilbert scales (DHS)) is derived which generates new interpolatory inequalities (dilational interpolatory inequalities (DII)) which have the ordinary Hilbert scales (OHS) interpolatory inequalities as special cases. They therefore represent a one-parameter family generalization of OHS, and are a precise and concise subset of VHS approriate for deriving error estimates for deconvolution. The role of the Hilbert scales in deriving error estimates for the approximate solution of inverse problems is discussed along with an application of DHS to deconvolution sharpening."}
{"category": "Math", "title": "The Frobenius action on rank 2 vector bundles over curves in small genus and small characteristic", "abstract": "Let X be a general proper and smooth curve of genus 2 (resp. of genus 3) defined over an algebraically closed field of characteristic p. When 3\\leq p \\leq 7, the action of Frobenius on rank 2 semi-stable vector bundles with trivial determinant is completely determined by its restrictions to the 30 lines (resp. the 126 Kummer surfaces) that are invariant under the action of some order 2 line bundle over X. Those lines (resp. those Kummer surfaces) are closely related to the elliptic curves (resp. the abelian varieties of dimension 2) that appear as the Prym varieties associated to double \\'etale coverings of X. We are therefore able to compute explicit equations of this action in these cases. We perform some of these computations and draw some consequences."}
{"category": "Math", "title": "Rigidity results for some boundary quasilinear phase transitions", "abstract": "We consider a quasilinear equation given in the half-space, i.e. a so called boundary reaction problem. Our concerns are a geometric Poincar\\'e inequality and, as a byproduct of this inequality, a result on the symmetry of low-dimensional bounded stable solutions, under some suitable assumptions on the nonlinearities. More precisely, we analyze the following boundary problem $$ \\left\\{\\begin{matrix} -{\\rm div} (a(x,|\\nabla u|)\\nabla u)+g(x,u)=0 \\qquad {on $\\R^n\\times(0,+\\infty)$} -a(x,|\\nabla u|)u_x = f(u) \\qquad{\\mbox{on $\\R^n\\times\\{0\\}$}}\\end{matrix} \\right.$$ under some natural assumptions on the diffusion coefficient $a(x,|\\nabla u|)$ and the nonlinearities $f$ and $g$. Here, $u=u(y,x)$, with $y\\in\\R^n$ and $x\\in(0,+\\infty)$. This type of PDE can be seen as a nonlocal problem on the boundary $\\partial \\R^{n+1}_+$. The assumptions on $a(x,|\\nabla u|)$ allow to treat in a unified way the $p-$laplacian and the minimal surface operators."}
{"category": "Math", "title": "Quasi-Minimal, Pseudo-Minimal Systems and Dense Orbits", "abstract": "We give a short discussion about a weaker form of minimality (called quasi-minimality). We call a system quasi-minimal if all dense orbits form an open set. It is hard to find examples which are not already minimal. Since elliptic behaviour makes them minimal, these systems are regarded as parabolic systems. Indeed, we show that a quasi-minimal homeomorphism on a manifold is not expansive (hyperbolic)."}
{"category": "Math", "title": "Tate modules of universal p-divisible groups", "abstract": "A p-divisible group over a complete local domain determines a Galois representation on the Tate module of its generic fibre. We determine the image of this representation for the universal deformation in mixed characteristic of a bi-infinitesimal group and for the p-rank strata of the universal deformation in positive characteristic of an infinitesimal group. The method is a reduction to the known case of one-dimensional groups by a deformation argument based on properties of the stratification by Newton polygons."}
{"category": "Math", "title": "Exponential sums and rank of triple persymmetric matrices over F_2", "abstract": "We obtain using exponential quadratic sums, explicit expressions for the number of triple persymmetric matrices over F_2 of given rank. (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j = r+s)"}
{"category": "Math", "title": "Pseudoknot RNA structures with arc-length $\\ge 4$", "abstract": "In this paper we study $k$-noncrossing RNA structures with minimum arc-length 4 and at most $k-1$ mutually crossing bonds. Let ${\\sf T}_{k}^{[4]}(n)$ denote the number of $k$-noncrossing RNA structures with arc-length $\\ge 4$ over $n$ vertices. We prove (a) a functional equation for the generating function $\\sum_{n\\ge 0}{\\sf T}_{k}^{[4]}(n)z^n$ and (b) derive for $k\\le 9$ the asymptotic formula ${\\sf T}_{k}^{[4]}(n)\\sim c_k n^{-((k-1)^2+(k-1)/2)} \\gamma_k^{-n}$. Furthermore we explicitly compute the exponential growth rates $\\gamma_k^{-1}$ and asymptotic formulas for $4\\le k\\le 9$."}
{"category": "Math", "title": "On some transformations of bilateral birth-and-death processes with applications", "abstract": "A method yielding simple relationships among bilateral birth-and-death processes is outlined. This allows one to relate birth and death rates of two processes in such a way that their transition probabilities, first-passage-time densities and ultimate crossing probabilities are mutually related by some product-form expressions."}
{"category": "Math", "title": "Infinitesimal or cocommutative dipterous bialgebras and good triples of operads", "abstract": "The works of Poincare, Birkhoff, Witt and Cartier, Milnor, Moore on the connected cocommutative Hopf algebras translated in the language of operads means that the triple of operads (Com, As, Lie) endowed with the Hopf compatiblity relation is good. In this paper, we focus on left dipterous (resp. right dipterous) algebras which are associative algebras with an extra left (resp. right) module on themselves and look for good triples were $As$ is replaced by the dipterous operad Dipt. Since the work of Loday and Ronco, the triple of operads (As, Dipt, B_\\infty) endowed with the semi-Hopf compatibility relations is known to be good. In this paper, we prove that the triple of operads (As, Dipt, Grove) endowed with the so-called (nonunital) semi-infinitesimal compatibility relations is good. For that, explicit constructions of the free dipterous algebra and the free grove-algebra over a K-vector space V are given. These constructions turn out to be related to rooted planar trees and the little an large Schroeder numbers. Many examples of dipterous algebras are given, notably the free L-dipterous algebras. As a corollary of our results, we also recover that the triple of operads (2As, Dipt, Vect) endowed both with the unital semi-Hopf and with the unital semi-infinitesimal compatibility relations is good, where 2As denotes the operad of 2-associative algebras. We also open this paper on a good triple, related to the Connes-Kreimer Hopf algebra in quantum field theory, (Com, Dipt, Prim_{Com} Dipt) endowed with the Hopf compatibility relations and also present a general theorem giving good triples from entangled dipterous like operads named associative molecules."}
{"category": "Math", "title": "Bases in Lie and Quantum Algebras", "abstract": "Applications of algebras in physics are related to the connection of measurable observables to relevant elements of the algebras, usually the generators. However, in the determination of the generators in Lie algebras there is place for some arbitrary conventions. The situation is much more involved in the context of quantum algebras, where inside the quantum universal enveloping algebra, we have not enough primitive elements that allow for a privileged set of generators and all basic sets are equivalent. In this paper we discuss how the Drinfeld double structure underlying every simple Lie bialgebra characterizes uniquely a particular basis without any freedom, completing the Cartan program on simple algebras. By means of a perturbative construction, a distinguished deformed basis (we call it the analytical basis) is obtained for every quantum group as the analytical prolongation of the above defined Lie basis of the corresponding Lie bialgebra. It turns out that the whole construction is unique, so to each quantum universal enveloping algebra is associated one and only one bialgebra. In this way the problem of the classification of quantum algebras is moved to the classification of bialgebras. In order to make this procedure more clear, we discuss in detail the simple cases of su(2) and su_q(2)."}
{"category": "Math", "title": "A Note on Coseparable Coalgebras", "abstract": "Given a coalgebra $C$ over a commutative ring $R,$ we show that $C$ can be considered as a (not necessarily counital) $C^{\\ast op}$-coring. Moreover, we show that this coring has a left (right) counity if and only if $C$ is coseparable as an $R$-coalgebra."}
{"category": "Math", "title": "Mass transport generated by a flow of Gauss maps", "abstract": "Let $A \\subset \\mathbb{R}^d$, $d\\ge 2$, be a compact convex set and let $\\mu = \\varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $\\nu = \\varrho_1 dx$ be a probability measure on $B_r := \\{x\\colon |x| \\le r\\}$ equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping $T$ such that $\\nu = \\mu \\circ T^{-1}$ and $T = \\phi \\cdot {\\rm n}$, where $\\phi\\colon A \\to [0,r]$ is a continuous potential with convex sub-level sets and ${\\rm n}$ is the Gauss map of the corresponding level sets of $\\phi$. Moreover, $T$ is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth $\\phi$ the level sets of $\\phi$ are driven by the Gauss curvature flow $\\dot{x}(s) = -s^{d-1} \\frac{\\varrho_1(s {\\rm n})}{\\varrho_0(x)} K(x) \\cdot {\\rm n}(x)$, where $K$ is the Gauss curvature. As a by-product one can reprove the existence of weak solutions of the classical Gauss curvature flow starting from a convex hypersurface."}
{"category": "Math", "title": "Estimation of Wiener--Ito integrals and polynomials of independent Gaussian random variables", "abstract": "In this paper I prove good estimates on the moments and tail distribution of $k$-fold Wiener--It\\^o integrals and also present their natural counterpart for polynomials of independent Gaussian random variables. The proof is based on the so-called diagram formula for Wiener--It\\^o integrals which yields a good representation for their products as a sum of such integrals. I intend to show in a subsequent paper that this method also yields good estimates for degenerate $U$-statistics. The main result of this paper is a generalization of the estimates of Hanson and Wright about bilinear forms of independent standard normal random variables. On the other hand, it is a weaker estimate than the main result of a paper of Lata{\\l}a [6]. But that paper contains an error, and it is not clear whether its result is true. This question is also discussed here."}
{"category": "Math", "title": "Acyclic edge coloring", "abstract": "This paper has been withdrawn by the author due to an error in the proof."}
{"category": "Math", "title": "Search for primes of the form $m^2+1$", "abstract": "The results of the computer hunt for the primes of the form $q = m^2+1$ up to $10^{20}$ are reported. The number of sign changes of the difference $\\pi_q(x) - \\frac{C_q}{2}\\int_2^x{du \\over \\sqrt{u}\\log(u)}$ and the error term for this difference is investigated. The analogs of the Brun's constant and the Skewes number are calculated. An analog of the B conjecture of Hardy--Littlewood is formulated. It is argued that there is no Chebyshev bias for primes of the form $q=m^2+1$. All encountered integrals we were able to express by the logarithmic integral."}
{"category": "Math", "title": "Optimization of periodic composite structures for sub-wavelength focusing", "abstract": "Recently, there has been plenty of work in designing and fabricating materials with an effective negative refractive index. Veselago realized that a slab of material with a refractive index of -1 would act as a lens. Pendry suggested that the Veselago lens would act as a superlens, providing a perfect image of an object in contrast to conventional lenses which are only able to focus a point source to an image having a diameter of the order of the wavelength of the incident field. Recent work has shown that similar focusing effects can be obtained with certain slabs of ``conventional'' periodic composite materials: photonic crystals. The present work seeks to answer the question of what periodic dielectric composite medium (described by dielectric coefficient with positive real part) gives an optimal image of a point source. An optimization problem is formulated and it is shown that a solution exists provided the medium has small absorption. Solutions are characterized by an adjoint-state gradient condition, and several numerical examples illustrate both the plausibility of this design approach, and the possibility of obtaining smaller image spot sizes than with typical photonic crystals."}
{"category": "Math", "title": "Lyapunov exponents for the one-dimensional parabolic Anderson model with drift", "abstract": "We consider the solution $u$ to the one-dimensional parabolic Anderson model with homogeneous initial condition $u(0, \\cdot) \\equiv 1$, arbitrary drift and a time-independent potential bounded from above. Under ergodicity and independence conditions we derive representations for both the quenched Lyapunov exponent and, more importantly, the $p$-th annealed Lyapunov exponents for {\\it all} $p \\in (0, \\infty).$ These results enable us to prove the heuristically plausible fact that the $p$-th annealed Lyapunov exponent converges to the quenched Lyapunov exponent as $p \\downarrow 0.$ Furthermore, we show that $u$ is $p$-intermittent for $p$ large enough. As a byproduct, we compute the optimal quenched speed of the random walk appearing in the Feynman-Kac representation of $u$ under the corresponding Gibbs measure. In this context, depending on the negativity of the potential, a phase transition from zero speed to positive speed appears."}
{"category": "Math", "title": "The similarity problem for $J$-nonnegative Sturm-Liouville operators", "abstract": "Sufficient conditions for the similarity of the operator $A := 1/r(x) (-d^2/dx^2 +q(x))$ with an indefinite weight $r(x)=(\\sgn x)|r(x)|$ are obtained. These conditions are formulated in terms of Titchmarsh-Weyl $m$-coefficients. Sufficient conditions for the regularity of the critical points 0 and $\\infty$ of $J$-nonnegative Sturm-Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm-Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case $r(x)=\\sgn x$ and $q\\in L^1(R, (1+|x|)dx)$, we prove that $A$ is similar to a self-adjoint operator if and only if $A$ is $J$-nonnegative. The latter condition on $q$ is sharp, i.e., we construct $q\\in \\cap_{\\gamma <1} L^1(R, (1+|x|)^\\gamma dx)$ such that $A$ is $J$-nonnegative with the singular critical point 0. Hence $A$ is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that $J$-positivity is sufficient for the similarity of $A$ to a self-adjoint operator. In the case $q\\equiv 0$, we prove the regularity of the critical point 0 for a wide class of weights $r$. This yields new results for \"forward-backward\" diffusion equations."}
{"category": "Math", "title": "Cyclic Approximation to K-Stasis", "abstract": "If a linear combination of k smooth vector fields is zero at a point, then, generically, near this point there are small cycles comprised of segments from the flow of each vector field. This answers a question posed in arXiv:math/0504365."}
{"category": "Math", "title": "Recurrence relations for characters of affine Lie algebra $A_{\\ell}^{(1)}$", "abstract": "By using the known description of combinatorial bases for Feigin-Stoyanovsky's type subspaces of standard modules for affine Lie algebra $\\mathfrak{sl}(l+1,\\mathbb{C})^{\\widetilde{}}$, as well as certain intertwining operators between standard modules, we obtain exact sequences of Feigin-Stoyanovsky's type subspaces at fixed level $k$. This directly leads to systems of recurrence relations for formal characters of those subspaces."}
{"category": "Math", "title": "Discrete Affine Minimal Surfaces with Indefinite Metric", "abstract": "Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that they are critical points of an affine area functional defined on the space of quadrangular discrete surfaces. The construction makes use of asymptotic coordinates and allows defining the discrete analogs of some differential geometric objects, such as the normal and co normal vector fields, the cubic form and the compatibility equations."}
{"category": "Math", "title": "Global Symplectic Uncertainty Propagation on SO(3)", "abstract": "This paper introduces a global uncertainty propagation scheme for rigid body dynamics, through a combination of numerical parametric uncertainty techniques, noncommutative harmonic analysis, and geometric numerical integration. This method is distinguished from prior approaches, as it allows one to consider probability densities that are global, and are not supported on only a single coordinate chart on the manifold. The use of Lie group variational integrators, that are symplectic and stay on the Lie group, as the underlying numerical propagator ensures that the advected probability densities respect the geometric properties of uncertainty propagation in Hamiltonian systems, which arise as consequence of the Gromov nonsqueezing theorem from symplectic geometry. We also describe how the global uncertainty propagation scheme can be applied to the problem of global attitude estimation."}
{"category": "Math", "title": "Derived Arithmetic Fuchsian Groups of Genus Two", "abstract": "We classify all torsion-free derived arithmetic Fuchsian groups of genus two by commensurability class. In particular, we show that there exist no such groups arising from quaternion algebras over number fields of degree greater than 5. We also prove some results on the existence and form of maximal orders for a class of quaternion algebras related to these groups. Using these results in conjunction with a computer program, one can determine an explicit set of generators for each derived arithmetic Fuchsian group containing a torsion-free subgroup of genus two. We show this for a number of examples."}
{"category": "Math", "title": "Birkhoff spectra for one-dimensional maps with some hyperbolicity", "abstract": "We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing $C^2$ map modelled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension."}
{"category": "Math", "title": "Equality of pressures for diffeomorphisms preserving hyperbolic measures", "abstract": "For a diffeomorphism which preserves a hyperbolic measure the potential $\\phi^u=-\\log|{\\rm Jac} df|_{E^u}|$ is studied. Various types of pressure of $\\phi^u$ are introduced. It is shown that these pressures satisfy a corresponding variational principle."}
{"category": "Math", "title": "Mukai duality for gerbes with connection", "abstract": "We study gerbes with connection over an etale stack via noncommutative algebras of differential forms on a groupoid presenting the stack. We then describe a dg-category of modules over any such algebra, which we claim represents a dg-enhancement of the derived category of coherent analytic sheaves on the gerbe in question. This category can be used to phrase and prove Fourier-Mukai type dualities between gerbes and other noncommutative spaces. As an application of the theory, we show that a gerbe with flat connection on a torus is dual (in a sense analogous to Fourier-Mukai duality or T-duality) to a noncommutative holomorphic dual torus."}
{"category": "Math", "title": "Equations of 2-linear ideals and arithmetical rank", "abstract": "In this paper we consider reduced homogeneous ideals $\\Jcal\\subset S$ of a polynomial ring $S$, having a 2-linear resolution. 1. We study systems of generators of $\\Jcal\\subset S$. 2. We compute the arithmetical rank for a large class of projective curves having a 2-linear resolution. 3. We show that the fiber cone $\\proj \\Fcal(I_{\\Lcal})$ of a lattice ideal $I_{\\Lcal}$ of codimension two is a set theoretical complete intersection."}
{"category": "Math", "title": "The Hochschild cohomology ring of a class of special biserial algebras", "abstract": "We consider a class of self-injective special biserial algebras $\\Lambda_N$ over a field $K$ and show that the Hochschild cohomology ring of $\\Lambda_N$ is a finitely generated $K$-algebra. Moreover the Hochschild cohomology ring of $\\Lambda_N$ modulo nilpotence is a finitely generated commutative $K$-algebra of Krull dimension two. As a consequence the conjecture of Snashall-Solberg \\cite{SS}, concerning the Hochschild cohomology ring modulo nilpotence, holds for this class of algebras."}
{"category": "Math", "title": "On the Gromov hyperbolicity of the Kobayashi metric on strictly pseudoconvex regions in the almost complex case", "abstract": "We prove that every bounded strictly $J$-convex region equipped with the Kobayashi metric is hyperbolic in the sense of Gromov. We apply this result to the study of the dynamics of pseudo-holomorphic maps."}
{"category": "Math", "title": "Bijections for Baxter Families and Related Objects", "abstract": "The Baxter number can be written as $B_n = \\sum_0^n \\Theta_{k,n-k-1}$. These numbers have first appeared in the enumeration of so-called Baxter permutations; $B_n$ is the number of Baxter permutations of size $n$, and $\\Theta_{k,l}$ is the number of Baxter permutations with $k$ descents and $l$ rises. With a series of bijections we identify several families of combinatorial objects counted by the numbers $\\Theta_{k,l}$. Apart from Baxter permutations, these include plane bipolar orientations with $k+2$ vertices and $l+2$ faces, 2-orientations of planar quadrangulations with $k+2$ white and $l+2$ black vertices, certain pairs of binary trees with $k+1$ left and $l+1$ right leaves, and a family of triples of non-intersecting lattice paths. This last family allows us to determine the value of $\\Theta_{k,l}$ as an application of the lemma of Gessel and Viennot. The approach also allows us to count certain other subfamilies, e.g., alternating Baxter permutations, objects with symmetries and, via a bijection with a class of plan bipolar orientations also Schnyder woods of triangulations, which are known to be in bijection with 3-orientations."}
{"category": "Math", "title": "A sharp Wirtinger inequality and some related functional spaces", "abstract": "We consider the generalized Wirtinger inequality \\[ (\\int_{0}^{T} a |u|^q )^{1/q} \\le C \\biggm(\\int_{0}^{T} a^{1-p} |u'|^{p}\\biggm)^{1/p}, \\] with $p,q>1$, $T>0$, $a\\in L^1[0,T]$, $a\\ge0$, $a\\not\\equiv0$ and where $u$ is a $T$-periodic function satisfying the constraint \\[ \\int_{0}^{T} a |u|^{q-2}u =0. \\] We provide the best constant $C>0$ as well as all extremals. Furthermore, we characterize the natural functional space where the inequality is defined."}
{"category": "Math", "title": "Nonlinear optimal control synthesis via occupation measures", "abstract": "We consider nonlinear optimal control problems (OCPs) for which all problem data are polynomial. In the first part of the paper, we review how occupation measures can be used to approximate pointwise the optimal value function of a given OCP, using a hierarchy of linear matrix inequality (LMI) relaxations. In the second part, we extend the methodology to approximate the optimal value function on a given set and we use such a function to constructively and computationally derive an almost optimal control law. Numerical examples show the effectiveness of the approach."}
{"category": "Math", "title": "A new test procedure of independence in copula models via chi-square-divergence", "abstract": "We introduce a new test procedure of independence in the framework of parametric copulas with unknown marginals. The method is based essentially on the dual representation of $\\chi^2$-divergence on signed finite measures. The asymptotic properties of the proposed estimate and the test statistic are studied under the null and alternative hypotheses, with simple and standard limit distributions both when the parameter is an interior point or not."}
{"category": "Math", "title": "Images directes et fonctions L en cohomologie rigide", "abstract": "Let $k$ be a perfect field of characteristic $p>0$, $\\mathcal{V}$ a complete discrete valuation ring with residue field $k$ and field of fractions $K$ of characteristic 0, and $S$ a separated $k$-scheme of finite type. When $S$ is smooth over $k$, we partially prove here a conjecture of Berthelot about the overconvergence of the higher direct images of the structure sheaf under a proper smooth morphism $f:X\\to S$; when $k$ is perfect and $\\mathcal{V}$ is tamely ramified such direct images are always convergent, not only for the structure sheaf but also for (almost) every convergent $F$-isocrystals. More generally, we prove this overconvergence when $f$ is liftable over $\\mathcal{V}$, or when $X$ is a relative complete intersection in some projective spaces over $S$, and taking as coefficients any overconvergent isocrystals. We then apply these results to $L$-functions with coefficients such direct images with Frobenius structure: we derive rationality or meromorphy for these $L$-functions (Dwork's conjecture), and we study their $p$-adic unit zeroes and poles (Katz's conjecture) ; and explicit case concerns the ordinary abelian schemes. A more precise presentation of results by chapters is given in the introduction."}
{"category": "Math", "title": "A class of statistical models to weaken independence in two-way contingency tables", "abstract": "In this paper we study a new class of statistical models for contingency tables. We define this class of models through a subset of the binomial equations of the classical independence model. We use some notions from Algebraic Statistics to compute their sufficient statistic, and to prove that they are log-linear. Moreover, we show how to compute maximum likelihood estimates and to perform exact inference through the Diaconis-Sturmfels algorithm. Examples show that these models can be useful in a wide range of applications."}
{"category": "Math", "title": "Distribution of Angles in Hyperbolic Lattices", "abstract": "We prove an effective equidistribution result about angles in a hyperbolic lattice. We use this to generalize a result due to F. P. Boca."}
{"category": "Math", "title": "On some generalized reinforced random walks on integers", "abstract": "We consider Reinforced Random Walks where transition probabilities are a function of the proportion of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to Pemantle \\cite{Pem000} on trees."}
{"category": "Math", "title": "On the non--existence of certain hyperovals in dual Andr\\'e planes of order $2^{2h}$", "abstract": "No regular hyperoval of the Desarguesian affine plane $AG(2,2^{2h})$, with $h>1$, is inherited by a dual Andr\\'e plane of order $2^{2h}$ with dimension 2 over its centre."}
{"category": "Math", "title": "Symmetric Numerical Semigroups Generated by Fibonacci and Lucas Triples", "abstract": "The symmetric numerical semigroups S(F_a,F_b,F_c) and S(L_k,L_m,L_n) generated by three Fibonacci (F_a,F_b,F_c) and Lucas (L_k,L_m,L_n) numbers are considered. Based on divisibility properties of the Fibonacci and Lucas numbers we establish necessary and sufficient conditions for both semigroups to be symmetric and calculate their Hilbert generating series, Frobenius numbers and genera."}
{"category": "Math", "title": "Solution to the Burnside Problem", "abstract": "The Burnside Problem asks whether a finitely generated group of exponent n is finite. We present a solution for 2-generator groups of prime power exponent. Results of P. Hall and G. Higman extends the finiteness conclusion to groups having composite exponents. Our main result, called the Generalized Burnside Theorem, is a solvability theorem that applies to a family of groups called GB (Generalized Burnside) groups that contain infinite as well as finite groups. The final section discusses the extension to k-generator groups although details are left for another time."}
{"category": "Math", "title": "The Kahler-Ricci flow and K-stability", "abstract": "We consider the K\\\"ahler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a K\\\"ahler-Einstein metric. The main ingredient is a result that says that a sufficiently small perturbation of a cscK manifold admits a cscK metric if it is K-polystable."}
{"category": "Math", "title": "On 0-homology of categorical at zero semigroups", "abstract": "The isomorphism of 0-homology groups of a categorical at zero semigroup and homology groups of its 0-reflector is proved. Some applications of 0-homology to Eilenberg-MacLane homology of semigroups are given."}
{"category": "Math", "title": "Monodromy of a family of hypersurfaces", "abstract": "Let $Y$ be an $(m+1)$-dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$, and $\\delta$ be a positive integer such that $\\mathcal I_{Z,Y}(\\delta)$ is generated by global sections. Fix an integer $d\\geq \\delta +1$, and assume the general divisor $X \\in |H^0(Y,\\ic_{Z,Y}(d))|$ is smooth. Denote by $H^m(X;\\mathbb Q)_{\\perp Z}^{\\text{van}}$ the quotient of $H^m(X;\\mathbb Q)$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$. In the present paper we prove that the monodromy representation on $H^m(X;\\mathbb Q)_{\\perp Z}^{\\text{van}}$ for the family of smooth divisors $X \\in |H^0(Y,\\ic_{Z,Y}(d))|$ is irreducible."}
{"category": "Math", "title": "Component models for large networks", "abstract": "Being among the easiest ways to find meaningful structure from discrete data, Latent Dirichlet Allocation (LDA) and related component models have been applied widely. They are simple, computationally fast and scalable, interpretable, and admit nonparametric priors. In the currently popular field of network modeling, relatively little work has taken uncertainty of data seriously in the Bayesian sense, and component models have been introduced to the field only recently, by treating each node as a bag of out-going links. We introduce an alternative, interaction component model for communities (ICMc), where the whole network is a bag of links, stemming from different components. The former finds both disassortative and assortative structure, while the alternative assumes assortativity and finds community-like structures like the earlier methods motivated by physics. With Dirichlet Process priors and an efficient implementation the models are highly scalable, as demonstrated with a social network from the Last.fm web site, with 670,000 nodes and 1.89 million links."}
{"category": "Math", "title": "Poisson (co)homology of polynomial Poisson algebras in dimension four : Sklyanin's case", "abstract": "In this paper, we compute the Poisson (co)homology of a polynomial Poisson structure given by two Casimir polynomial functions which define a complete intersection with an isolated singularity."}
{"category": "Math", "title": "Large induced trees in K_r-free graphs", "abstract": "For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K_r on r vertices. This problem was posed twenty years ago by Erdos, Saks, and Sos. Substantially improving earlier results of various researchers, we prove that every connected triangle-free graph on n vertices contains an induced tree of order \\sqrt{n}. When r >= 4, we also show that t(G) >= (\\log n)/(4 \\log r) for every connected K_r-free graph G of order n. Both of these bounds are tight up to small multiplicative constants, and the first one disproves a recent conjecture of Matousek and Samal."}
{"category": "Math", "title": "Algebraic K-theory over the infinite dihedral group: an algebraic approach", "abstract": "We prove that the Waldhausen nilpotent class group of an injective index 2 amalgamated free product is isomorphic to the Farrell-Bass nilpotent class group of a twisted polynomial extension. As an application, we show that the Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups."}
{"category": "Math", "title": "Some remarks on Nil groups in algebraic K-theory", "abstract": "This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group."}
{"category": "Math", "title": "Categorifying Coloring Numbers", "abstract": "Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots and links with free ends? Indeed we can, once we categorify. Starting from the definition of coloring numbers, we will categorify them and establish this extension to tangles. Then, decategorifying will leave us with matrix representations of the monoidal category of tangles."}
{"category": "Math", "title": "Betti numbers of graded modules and the Multiplicity Conjecture in the non-Cohen-Macaulay case", "abstract": "We use the results by Eisenbud and Schreyer to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the Multiplicity Conjecture of Herzog, Huneke and Srinivasan for modules that are not necessarily Cohen-Macaulay. We give a combinatorial proof of the convexity of the simplicial fan spanned by the pure diagrams."}
{"category": "Math", "title": "Logics preserving degrees of truth from varieties of residuated lattices", "abstract": "Let K be a variety of (commutative, integral) residuated lattices. The substructural logic usually associated with K is an algebraizable logic that has K as its equivalent algebraic semantics, and is a logic that preserves truth, i.e., 1 is the only truth value preserved by the inferences of the logic. In this paper we introduce another logic associated with K, namely the logic that preserves degrees of truth, in the sense that it preserves lower bounds of truth values in inferences. We study this second logic mainly from the point of view of abstract algebraic logic. We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth. We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus, and by a set of conditions on its closure operator. Concerning the relation between the two logics, we prove that the truth preserving logic is the purely inferential extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective."}
{"category": "Math", "title": "Spherical Spectral Synthesis and Two-Radius Theorems on Damek-Ricci Spaces", "abstract": "We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems."}
{"category": "Math", "title": "Weighted enumeration of spanning subgraphs with degree constraints", "abstract": "The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial $\\sum_k m_k(G) y^k$ has only real nonpositive zeros, in which $m_k(G)$ is the number of $k$-edge matchings of a graph $G$. There is a stronger multivariate version of this theorem. We provide a general method by which ``theorems of Heilmann-Lieb type'' can be proved for a wide variety of polynomials attached to the graph $G$. These polynomials are multivariate generating functions for spanning subgraphs of $G$ with certain weights and constraints imposed, and the theorems specify regions in which these polynomials are nonvanishing. Such theorems have consequences for the absence of phase transitions in certain probabilistic models for spanning subgraphs of $G$."}
{"category": "Math", "title": "Dynamics of the Nearly Parametric Pendulum", "abstract": "Dynamically stable periodic rotations of a driven pendulum provide a unique mechanism for generating a uniform rotation from bounded excitations. This paper studies the effects of a small ellipticity of the driving, perturbing the classical parametric pendulum. The first finding is that the region in the parameter plane of amplitude and frequency of excitation where rotations are possible increases with the ellipticity. Second, the resonance tongues, which are the most characteristic feature of the classical bifurcation scenario of a parametrically driven pendulum, merge into a single region of instability."}
{"category": "Math", "title": "A note on multi-type cookie random walk on integers", "abstract": "We consider a random walk on integers where at the first visits to a site the walker gets a positive drift, but where after a certain number of visits the walker gets a negative drift. We prove that the walker is almost surely transient to the left with positive speed."}
{"category": "Math", "title": "Cobordism of fold maps, stably framed manifolds and immersions", "abstract": "We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works."}
{"category": "Math", "title": "Geodesic Equations on Diffeomorphism Groups", "abstract": "We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant $L^2$ or $H^1$ metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the right logarithmic derivative of a geodesic in Lie groups with right invariant metrics."}
{"category": "Math", "title": "On Bass' question for finitely generated algebras over large fields", "abstract": "Recently Cortinas-Haesemayer-Walker-Weibel gave affirmative answer to Bass' 1972 question on NK-groups for algebras of essentially finite type over large fields of characteristic 0. Here we give an alternative short proof of this result for algebras of finite type over such fields. Our approach is based on classical techniques in higher K-theory of rings and a direct K_i-analogue of an old observation of Murthy-Pedrini, dating back from the same 1972."}
{"category": "Math", "title": "An open image theorem for a general class of abelian varieties", "abstract": "Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Neron model of A/K has at least one fiber with potential toric dimension one, then for almost all rational primes ell, the Galois group of the splitting field of the ell-torsion of A is GSp_{2g}(Z/ell)."}
{"category": "Math", "title": "Ordinary differential equations in Banach spaces and the spectral flow", "abstract": "We give a definition of the spectral flow for continuous paths in the space of bounded and essentially hyperbolic operators. We provide a homotopical characterization of the spectral flow in terms of a group homomorphism of the fundamental group of the projectors of the Calkin algebra with the infinite cyclic group Z. This characterization helps us to exhibit examples of infinite-dimensional Banach spaces where the spectral flow is not injective nor surjective. We prove that a path with spectral flow equal to an integer m exists if and only if there exists a projector P connected by an arc to a projector Q such that Range(Q) has co-dimension m in Range(P). We prove that if A is an asymptotically hyperbolic and essentially splitting path the differential operator F(u) = du/dt - Au is Fredholm. Moreover if A is also essentially hyperbolic the Fredholm index coincides with minus the spectral flow of A."}
{"category": "Math", "title": "Finite generation of the log canonical ring in dimension four", "abstract": "We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs. (a) Finite generation of the log canonical ring in dimension four. (b) Abundance theorem for irregular fourfolds. We obtain (a) as a direct consequence of the existence of four-dimensional log minimal models by using Fukuda's theorem on the four-dimensional log abundance conjecture. We can prove (b) only by using traditional arguments. More precisely, we prove the abundance conjecture for irregular $(n+1)$-folds on the assumption that the minimal model conjecture and the abundance conjecture hold in dimension $\\leq n$."}
{"category": "Math", "title": "Markov convexity and local rigidity of distorted metrics", "abstract": "It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type p if and only if it is Markov p-convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity of metric spaces, showing in particular that tree metrics fail to have the dichotomy property."}
{"category": "Math", "title": "Random fields of multivariate test statistics, with applications to shape analysis", "abstract": "Our data are random fields of multivariate Gaussian observations, and we fit a multivariate linear model with common design matrix at each point. We are interested in detecting those points where some of the coefficients are nonzero using classical multivariate statistics evaluated at each point. The problem is to find the $P$-value of the maximum of such a random field of test statistics. We approximate this by the expected Euler characteristic of the excursion set. Our main result is a very simple method for calculating this, which not only gives us the previous result of Cao and Worsley [Ann. Statist. 27 (1999) 925--942] for Hotelling's $T^2$, but also random fields of Roy's maximum root, maximum canonical correlations [Ann. Appl. Probab. 9 (1999) 1021--1057], multilinear forms [Ann. Statist. 29 (2001) 328--371], $\\bar{\\chi}^2$ [Statist. Probab. Lett 32 (1997) 367--376, Ann. Statist. 25 (1997) 2368--2387] and $\\chi^2$ scale space [Adv. in Appl. Probab. 33 (2001) 773--793]. The trick involves approaching the problem from the point of view of Roy's union-intersection principle. The results are applied to a problem in shape analysis where we look for brain damage due to nonmissile trauma."}
{"category": "Math", "title": "Rodeo: Sparse, greedy nonparametric regression", "abstract": "We present a greedy method for simultaneously performing local bandwidth selection and variable selection in nonparametric regression. The method starts with a local linear estimator with large bandwidths, and incrementally decreases the bandwidth of variables for which the gradient of the estimator with respect to bandwidth is large. The method--called rodeo (regularization of derivative expectation operator)--conducts a sequence of hypothesis tests to threshold derivatives, and is easy to implement. Under certain assumptions on the regression function and sampling density, it is shown that the rodeo applied to local linear smoothing avoids the curse of dimensionality, achieving near optimal minimax rates of convergence in the number of relevant variables, as if these variables were isolated in advance."}
{"category": "Math", "title": "Disproof of the Continuum Hypothesis and Determination of the Cardinality of Continuum by Approximations of Sets", "abstract": "A set theory is developed based on the approximations of sets and denoted by AS. In AS the set of all sets exists but the argument for Russell's and Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are not valid. All the other axioms of ZF are valid and all the basic sets, such as complement, intersection and cartesian product, exist although complement is not quite the same set as in ZF. The set of all sets can be equipped with the topology of approximations (Ta). Every set is closed and every function is continuous in Ta. This implies that the Continuum Hypothesis is false. The sets containing a subset which is perfect in Ta are of the greatest cardinality. A simple observation shows that the concept of well-ordering must be defined in a slightly different way than in ZF. We prove that a set can be well-ordered if and only if it has no perfect subset. Therefore the cardinalities of arbitrary sets are always comparable without assuming the Axiom of Choice. The cardinals following the smallest infinite cardinal w are 2w, 3w, ..., w^2, 2w^2, 3w^2, >..., w^3.... each being of greater cardinality than the previous one, which is not the case in ZF. Immediately after these cardinals does not follow w^w which is not a well-orderable set but some well-ordered cardinal k, and this one is followed by the cardinals 2k, 3k, ..., k^2, 2k^2, 3k^2, ..., k^3...., etc. The greatest cardinal is P(w) and is not a well-orderable set. The cofinality of a well-ordered set is either 2 or w. The only regular cardinals are 0, 1, 2, w and P(w). All other cardinals are singular. The only strong limit cardinal is w. The only inaccessible cardinal is P(w). Strongly inaccessible cardinals do not exist."}
{"category": "Math", "title": "Approximation and learning by greedy algorithms", "abstract": "We consider the problem of approximating a given element $f$ from a Hilbert space $\\mathcal{H}$ by means of greedy algorithms and the application of such procedures to the regression problem in statistical learning theory. We improve on the existing theory of convergence rates for both the orthogonal greedy algorithm and the relaxed greedy algorithm, as well as for the forward stepwise projection algorithm. For all these algorithms, we prove convergence results for a variety of function classes and not simply those that are related to the convex hull of the dictionary. We then show how these bounds for convergence rates lead to a new theory for the performance of greedy algorithms in learning. In particular, we build upon the results in [IEEE Trans. Inform. Theory 42 (1996) 2118--2132] to construct learning algorithms based on greedy approximations which are universally consistent and provide provable convergence rates for large classes of functions. The use of greedy algorithms in the context of learning is very appealing since it greatly reduces the computational burden when compared with standard model selection using general dictionaries."}
{"category": "Math", "title": "On the LMO conjecture", "abstract": "We give a proof of the LMO conjecture which say that for any simply connectd simple Lie group $G$, the LMO invariant of rational homology 3-spheres recovers the perturvative invariant $\\tau^{PG}$. By Habiro-Le theorem, this implies that the LMO invariant is the universal quantum invariant of integral homology 3-spheres."}
{"category": "Math", "title": "High breakdown point robust regression with censored data", "abstract": "In this paper, we propose a class of high breakdown point estimators for the linear regression model when the response variable contains censored observations. These estimators are robust against high-leverage outliers and they generalize the LMS (least median of squares), S, MM and $\\tau$-estimators for linear regression. An important contribution of this paper is that we can define consistent estimators using a bounded loss function (or equivalently, a redescending score function). Since the calculation of these estimators can be computationally costly, we propose an efficient algorithm to compute them. We illustrate their use on an example and present simulation studies that show that these estimators also have good finite sample properties."}
{"category": "Math", "title": "Primes in the form $[\\alpha p+\\beta]$", "abstract": "Let \\beta be a real number. Then for almost all irrational \\alpha>0 (in the sense of Lebesgue measure) \\limsup_{x\\to\\infty}\\pi_{\\alpha,\\beta}^*(x)(\\log x)^2/x>=1, where \\pi_{\\alpha,\\beta}^*(x)={p<=x: both p and [\\alpha p+\\beta] are primes}."}
{"category": "Math", "title": "Poincare series of filtrations corresponding to ideals on surfaces", "abstract": "Earlier the authors considered and, in some cases, computed Poincare series of two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular on the complex plane). A filtration from the first class was defined by a curve (with several branches) on the surface singularity. The other one (so called divisorial filtration) was defined by a set of components of the exceptional divisor of a modification of the surface singularity. Here we define a filtration corresponding to an ideal or to a set of ideals in the ring of germs of functions on a surface singularity and compute the corresponding Poincare series in some cases. For the complex plane this notion unites the two classes of filtrations described above."}
{"category": "Math", "title": "Equivalences between fusion systems of finite groups of Lie type", "abstract": "We prove, for certain pairs G,G of finite groups of Lie type, that the p-fusion systems for G and G' are equivalent. In other words, there is an isomorphism between a Sylow p-subgroup of G and one of G' which preserves p-fusion. This occurs, for example, when G=H(q) and G'=H(q') for a simple Lie type H, and q and q' are prime powers, both prime to p, which generate the same closed subgroup of the p-adic units. Our proof uses homotopy theoretic properties of the p-completed classifying spaces of G and G', and we know of no purely algebraic proof of this result."}
{"category": "Math", "title": "An isomorphism between the completion of an Algebra and its Caratheodory Extension", "abstract": "Let $\\Omega$ denote an algebra of sets and $\\mu$ a $\\sigma$-finite measure. We then prove that the completion of $\\Omega$ under the pseudometric $d(A,B)$ = $\\mu^{\\ast}(A \\triangle B)$ is $\\sigma$-algebra isomorphic and isometric to the Caratheodory Extension of $\\Omega$ under the equivalence relation $\\sim$."}
{"category": "Math", "title": "Weighted empirical likelihood in some two-sample semiparametric models with various types of censored data", "abstract": "In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator $(\\tilde{\\theta}_n,\\tilde{F}_n)$ for the underlying parameter $\\theta_0$ and distribution $F_0$ is derived, and the strong consistency of $(\\tilde{\\theta}_n,\\tilde{F}_n)$ and the asymptotic normality of $\\tilde{\\theta}_n$ are established. Under biased sampling models, the weighted empirical log-likelihood ratio is shown to have an asymptotic scaled chi-squared distribution for censored data aforementioned. For right censored data, doubly censored data and partly interval-censored data, it is shown that $\\sqrt{n}(\\tilde{F}_n-F_0)$ weakly converges to a centered Gaussian process, which leads to a consistent goodness-of-fit test for the case-control logistic regression models."}
{"category": "Math", "title": "On the performances of a new thresholding procedure using tree structure", "abstract": "This paper deals with the problem of function estimation. Using the white noise model setting, we provide a method to construct a new wavelet procedure based on thresholding rules which takes advantage of the dyadic structure of the wavelet decomposition. We prove that this new procedure performs very well since, on the one hand, it is adaptive and near-minimax over a large class of Besov spaces and, on the other hand, the maximal functional space (maxiset) where this procedure attains a given rate of convergence is very large. More than this, by studying the shape of its maxiset, we prove that the new procedure outperforms the hard thresholding procedure."}
{"category": "Math", "title": "Equisingularity of families of hypersurfaces and applications to mappings", "abstract": "In the study of equisingularity of isolated singularities we have the classical theorem of Briancon, Speder and Teissier which states that a family of isolated hypersurface singularities is Whitney equisingular if and only if the mu^*-sequence for a hypersurface is constant in the family. In this paper we generalize to non-isolated hypersurface singularities. By assuming non-contractibility of strata of a Whitney stratification of the non-isolated singularities outside the origin we show that Whitney equisingularity of a family is equivalent to constancy of a certain selection of invariants from two distinct generalizations of the mu^*-sequence. Applications of this theorem to equisingularity of more general mappings are given."}
{"category": "Math", "title": "Unimodular L-infinity algebras", "abstract": "We give a new short proof that the wheeled operad of unimodular Lie algebras is Koszul and use this to explicitly construct its minimal resolution. A representation of this resolution in a finite dimensional vector space V we call a unimodular L-infinity algebra. Such a structure corresponds to a homological vector field on V together with an invariant measure. We present explicit formulae for homotopy transferred structures, define the deformation complex and give a cohomological obstruction to the extension of an arbitrary structure of finite dimensional L-infinity algebra to a structure of unimodular L-infinity algebra."}
{"category": "Math", "title": "Existence and uniqueness results for a nonlinear stationary system", "abstract": "We prove a few existence results of a solution for a static system with a coupling of thermoviscoelastic type. As this system involves $L^1$ coupling terms we use the techniques of renormalized solutions for elliptic equations with $L^1$ data. We also prove partial uniqueness results."}
{"category": "Math", "title": "Nonlinear and non-coercive elliptic problems with integrable data", "abstract": "In this paper we study existence and uniqueness of renormalized solution to the following problem $\\lambda (x,u) -div a(x,Du) +\\Phi (x,u)) =f$ with $f$ in $L^1$ and with Dirichlet-Neumann boundary condition. The main difficulty in this task is that in general the operator entering in the above equation is not coercive in a Sobolev space. Moreover, the possible degenerate character of $\\lambda$ with respect to $u$ renders more complex the proof of uniqueness for integrable data $f$."}
{"category": "Math", "title": "Remarks on the uniqueness of comparable renormalized solutions of elliptic equations with measure data", "abstract": "We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem $-\\diw(\\aop(x,Du))=\\mu$ in $\\Omega$, $u=0$ on $\\partial\\Omega$, where $\\mu$ is a Radon measure with bounded variation on $\\Omega$."}
{"category": "Math", "title": "On the blow-up problem for the axisymmetric 3D Euler equations", "abstract": "In this paper we study the finite time blow-up problem for the axisymmetric 3D incompressible Euler equations with swirl. The evolution equations for the deformation tensor and the vorticity are reduced considerably in this case. Under the assumption of local minima for the pressure on the axis of symmetry with respect to the radial variations we show that the solution blows-up in finite time. If we further assume that the second radial derivative vanishes on the axis, then system reduces to the form of Constantin-Lax-Majda equations, and can be integrated explicitly."}
{"category": "Math", "title": "The finite time blow-up for the Euler-Poisson equations in $\\Bbb R^n$", "abstract": "We prove the finite time blow-up for $C^1$ solutions to the Euler-Poisson equations in $\\Bbb R^n$, $n\\geq 1$, with/without background density for initial data satisfying suitable conditions. We also find a sufficient condition for the initial data such that $C^3$ solution breaks down in finite time for the compressible Euler equations for polytropic gas flows."}
{"category": "Math", "title": "Superization and (q,t)-specialization in combinatorial Hopf algebras", "abstract": "We extend a classical construction on symmetric functions, the superization process, to several combinatorial Hopf algebras, and obtain analogs of the hook-content formula for the (q,t)-specializations of various bases. Exploiting the dendriform structures yields in particular (q,t)-analogs of the Bjorner-Wachs q-hook-length formulas for binary trees, and similar formulas for plane trees."}
{"category": "Math", "title": "On the algebraic geometry of polynomial dynamical systems", "abstract": "This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks. It is shown that several problems relating to their structure and dynamics, as well as control theory, can be formulated and solved in the language of algebraic geometry."}
{"category": "Math", "title": "Compressed Sensing with Cross Validation", "abstract": "Compressed Sensing decoding algorithms can efficiently recover an N dimensional real-valued vector x to within a factor of its best k-term approximation by taking m = 2klog(N/k) measurements y = Phi x. If the sparsity or approximate sparsity level of x were known, then this theoretical guarantee would imply quality assurance of the resulting compressed sensing estimate. However, because the underlying sparsity of the signal x is unknown, the quality of a compressed sensing estimate x* using m measurements is not assured. Nevertheless, we demonstrate that sharp bounds on the error || x - x* ||_2 can be achieved with almost no effort. More precisely, we assume that a maximum number of measurements m is pre-imposed; we reserve 4log(p) of the original m measurements and compute a sequence of possible estimates (x_j)_{j=1}^p to x from the m - 4log(p) remaining measurements; the errors ||x - x*_j ||_2 for j = 1, ..., p can then be bounded with high probability. As a consequence, numerical upper and lower bounds on the error between x and the best k-term approximation to x can be estimated for p values of k with almost no cost. Our observation has applications outside of compressed sensing as well."}
{"category": "Math", "title": "Polynomial solutions of nonlinear integral equations", "abstract": "We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of C. Bender and E. Ben-Naim. We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials."}
{"category": "Math", "title": "Strong law of large numbers with concave moments", "abstract": "In this note not intended for publication, it is observed that a wellnigh trivial application of the ergodic theorem of Karlsson-Ledrappier yields a strong LLN for arbitrary concave moments."}
{"category": "Math", "title": "Euler sums and a prime numbers conundrum", "abstract": "This note highlights an interesting connection between Euler sums of even weight and prime numbers."}
{"category": "Math", "title": "Two remarks on the Burr-Erdos conjecture", "abstract": "The Ramsey number r(H) of a graph H is the minimum positive integer N such that every two-coloring of the edges of the complete graph K_N on N vertices contains a monochromatic copy of H. A graph H is d-degenerate if every subgraph of H has minimum degree at most d. Burr and Erd\\H{o}s in 1975 conjectured that for each positive integer d there is a constant c_d such that r(H) \\leq c_dn for every d-degenerate graph H on n vertices. We show that for such graphs r(H) \\leq 2^{c_d\\sqrt{\\log n}}n, improving on an earlier bound of Kostochka and Sudakov. We also study Ramsey numbers of random graphs, showing that for d fixed, almost surely the random graph G(n,d/n) has Ramsey number linear in n. For random bipartite graphs, our proof gives nearly tight bounds."}
{"category": "Math", "title": "Divergence of Teichmueller Geodesics", "abstract": "We study the asymptotic geometry of Teichmueller geodesic rays. We show that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, then the rays diverge in Teichmueller space."}
{"category": "Math", "title": "Spacings between integers having typically many prime factors", "abstract": "We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\\de$ arbitrarily small and positive, the nearest neighbor spacings between integers $n$ with $|\\om(n)-\\log_2 n|\\le (\\log_2 n)^{\\de}$ obey the Poisson distribution law."}
{"category": "Math", "title": "Non-vanishing of the symmetric square $L$-function at the central point", "abstract": "Using the mollifier method, we show that for a positive proportion of holomorphic Hecke eigenforms of level one and weight bounded by a large enough constant, the associated symmetric square $L$-function does not vanish at the central point of its critical strip. We note that our proportion is the same as that found by other authors for other families of $L$-functions also having symplectic symmetry type."}
{"category": "Math", "title": "Conformal invariance of the writhe of a knot", "abstract": "We give a new proof of an old theorem by Banchoff and White 1975 that claims that the writhe of a knot is conformally invariant."}
{"category": "Math", "title": "Monotonicity for excited random walk in high dimensions", "abstract": "We prove that the drift $\\theta(d,\\beta)$ for excited random walk in dimension $d$ is monotone in the excitement parameter $\\beta \\in[0, 1]$, when $d\\ge 9$."}
{"category": "Math", "title": "The Minimal Degree for a Class of Finite Complex Reflection Groups", "abstract": "We calculate the minimal degree for a class of finite complex reflection groups $G(p,p,q)$, for $p$ and $q$ primes and establish relationships between minimal degrees when these groups are taken in a direct product."}
{"category": "Math", "title": "Partition statistics and quasiweak Maass forms", "abstract": "Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating their generating functions as well as to prove Ramanujan-type congruences for these objects and find relations between. In this paper we show that the hypergeometric generating functions for these objects are natural examples of quasimock theta functions, which are defined as the holomorphic parts of weak Maass forms and their derivatives. In particular, these generating functions may be viewed as analogs of Ramanujan's mock theta functions with arbitrarily high weight. We use the automorphic properties to prove the existence of infinitely many congruences for the Durfee symbols. Furthermore, we show that as k varies, the modularity of the k-marked Durfee symbols is precisely dictated by the case k=2. Finally, we use this relation in order to prove the existence of general congruences for rank moments in terms of level one modular forms of bounded weight."}
{"category": "Math", "title": "The necessary and sufficient condition for solvability of a partial integral equation", "abstract": "Let $T_1: L_2(\\Omega^2) \\to L_2(\\Omega^2)$ be a partial integral operator with the kernel from $C(\\Omega^3)$ where $\\Omega=[a,b ]^\\nu.$ In this paper we investigate solvability of a partial integral equation $f-\\varkappa T_1 f=g_0$ in the space $L_2(\\Omega^2)$ in the case when $\\varkappa$ is a characteristic number. We proved the theorem describing the necessary and sufficient condition for solvability of the partial integral equation $f-\\varkappa T_1 f=g_0.$"}
{"category": "Math", "title": "Mathematical Aspects and Numerical Computations of an Inverse Boundary Value Identification Using the Adjoint Method", "abstract": "The purpose of this study is to show some mathematical aspects of the adjoint method that is a numerical method for the Cauchy problem, an inverse boundary value problem. The adjoint method is an iterative method based on the variational formulation, and the steepest descent method minimizes an objective functional derived from our original problem. The conventional adjoint method is time-consuming in numerical computations because of the Armijo criterion, which is used to numerically determine the step size of the steepest descent method. It is important to find explicit conditions for the convergence and the optimal step size. Some theoretical results about the convergence for the numerical method are obtained. Through numerical experiments, it is concluded that our theories are effective."}
{"category": "Math", "title": "Method of Fundamental Solutions with Optimal Regularization Techniques for the Cauchy Problem of the Laplace Equation with Singular Points", "abstract": "The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularization parameter of the Tikhonov regularization technique and some parameters of MFS. The L-curve determines a suitable regularization parameter for obtaining an accurate solution. Numerical experiments show that such a suitable regularization parameter coincides with the optimal one. Moreover, a better choice of the parameters of MFS is numerically observed. It is noteworthy that a problem whose solution has singular points can successfully be solved. It is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain, singular points, and the Cauchy data with high noises."}
{"category": "Math", "title": "Regularized estimation of large covariance matrices", "abstract": "This paper considers estimating a covariance matrix of $p$ variables from $n$ observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance. We show that these estimates are consistent in the operator norm as long as $(\\log p)/n\\to0$, and obtain explicit rates. The results are uniform over some fairly natural well-conditioned families of covariance matrices. We also introduce an analogue of the Gaussian white noise model and show that if the population covariance is embeddable in that model and well-conditioned, then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix. The results can be extended to smooth versions of banding and to non-Gaussian distributions with sufficiently short tails. A resampling approach is proposed for choosing the banding parameter in practice. This approach is illustrated numerically on both simulated and real data."}
{"category": "Math", "title": "Smooth backfitting in generalized additive models", "abstract": "Generalized additive models have been popular among statisticians and data analysts in multivariate nonparametric regression with non-Gaussian responses including binary and count data. In this paper, a new likelihood approach for fitting generalized additive models is proposed. It aims to maximize a smoothed likelihood. The additive functions are estimated by solving a system of nonlinear integral equations. An iterative algorithm based on smooth backfitting is developed from the Newton--Kantorovich theorem. Asymptotic properties of the estimator and convergence of the algorithm are discussed. It is shown that our proposal based on local linear fit achieves the same bias and variance as the oracle estimator that uses knowledge of the other components. Numerical comparison with the recently proposed two-stage estimator [Ann. Statist. 32 (2004) 2412--2443] is also made."}
{"category": "Math", "title": "Weak solution for compressible fluid models of Korteweg type", "abstract": "This work is devoted to proving existence of global weak solutions for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) \\cite{3DS}, which can be used as a phase transition model. We distinguish two cases when the dimension N=2 and N=1, in the first case we need that $\\frac{1}{\\rho}\\in L^{\\infty}$, when N=1 we get a weak solution in finite time in the energy space."}
{"category": "Math", "title": "Existence of global weak solution for compressible fluid models with a capillary tensor for discontinuous interfaces", "abstract": "This work is devoted to the global existence of weak solution for a general isothermal model of capillary fluids derived by C. Rohde, which can be used as a phase transition model. This article is structured in the following way: first of all inspired by the result by P.-L. Lions on the Navier-Stokes compressible system we will show the global stability of weak solutions for our system with isentropic pressure and next with general pressure. Next we will consider perturbations close to a stable equilibrium as in the case of strong solutions."}
{"category": "Math", "title": "Variable selection in semiparametric regression modeling", "abstract": "In this paper, we are concerned with how to select significant variables in semiparametric modeling. Variable selection for semiparametric regression models consists of two components: model selection for nonparametric components and selection of significant variables for the parametric portion. Thus, semiparametric variable selection is much more challenging than parametric variable selection (e.g., linear and generalized linear models) because traditional variable selection procedures including stepwise regression and the best subset selection now require separate model selection for the nonparametric components for each submodel. This leads to a very heavy computational burden. In this paper, we propose a class of variable selection procedures for semiparametric regression models using nonconcave penalized likelihood. We establish the rate of convergence of the resulting estimate. With proper choices of penalty functions and regularization parameters, we show the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures perform as well as an oracle procedure. A semiparametric generalized likelihood ratio test is proposed to select significant variables in the nonparametric component. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null distribution follows a chi-square distribution which is independent of the nuisance parameters. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedures."}
{"category": "Math", "title": "Existence of solutions for compressible fluid models of Korteweg type", "abstract": "This work is devoted to the study of the initial boundary value problem for a general non isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985), which can be used as a phase transition model. We distinguish two cases, when the physical coefficients depend only on the density, and the general case. In the first case we can work in critical scaling spaces, and we prove global existence of solution and uniqueness for data close to a stable equilibrium. For general data, existence and uniqueness is stated on a short time interval. In the general case with physical coefficients depending on density and on temperature, additional regularity is required to control the temperature in $L^{\\infty}$ norm. We prove global existence of solution close to a stable equilibrium and local in time existence of solution with more general data. Uniqueness is also obtained."}
{"category": "Math", "title": "Cauchy problem for viscous shallow water equations with a term of capillarity", "abstract": "In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients and a term of capillarity introduced by Coquel et al in \\cite{5CR}. This model includes at the same time the barotropic Navier-Stokes equations with variable viscosity coefficients, shallow-water system and the model of Rohde. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possibleto the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence for solutions with general initial data. Uniqueness is also obtained."}
{"category": "Math", "title": "Mixed-rates asymptotics", "abstract": "A general method is presented for deriving the limiting behavior of estimators that are defined as the values of parameters optimizing an empirical criterion function. The asymptotic behavior of such estimators is typically deduced from uniform limit theorems for rescaled and reparametrized criterion functions. The new method can handle cases where the standard approach does not yield the complete limiting behavior of the estimator. The asymptotic analysis depends on a decomposition of criterion functions into sums of components with different rescalings. The method is explained by examples from Lasso-type estimation, $k$-means clustering, Shorth estimation and partial linear models."}
{"category": "Math", "title": "The generic points for the horocycle flow on a class of hyperbolic surfaces with infinite genus", "abstract": "A point is called generic for a flow preserving an infinite ergodic invariant Radon measure, if its orbit satisfies the conclusion of the ratio ergodic theorem for every pair of continuous functions with compact support and non-zero integrals. The generic points for horocycle flows on hyperbolic surfaces of finite genus are understood, but there are no results in infinite genus. We give such a result, by characterizing the generic points for $\\Z^d$--covers."}
{"category": "Math", "title": "SPM Bulletin 24", "abstract": "1. A Wikipedia entry on topological games 2. On a fragment of the universal Baire property for sigma^1_2 sets 3. The coarse classification of homogeneous ultra-metric spaces 4. Ramsey-like embeddings 5. Proper and piecewise proper families of reals 6. Measures and their random reals 7. Obtainable Sizes of Topologies on Finite Sets 8. Spaces of R-places of rational function fields 9. All properties in the Scheepers Diagram are linearly-sigma-additive."}
{"category": "Math", "title": "Growth of balls of holomorphic sections and energy at equilibrium", "abstract": "Let X be a compact complex manifold endowed with a big line bundle L. We define the energy at equilibrium of a weighted subset as the Monge-Ampere energy of the associated extremal plurisubharmonic weight. We prove the differentiability of the energy at equilibrium with respect to the weight, and show that this energy describes the asymptotic behaviour as k goes to infinity of the volume of the induced sup-norm unit ball in the space of global sections of kL. As a consequence of these results, we extend Rumely's Robin-type formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion and extend Yuan's equidistribution theorem for algebraic points of small height to the case of a big line bundle."}
{"category": "Math", "title": "Mixed 3-Sasakian structures and curvature", "abstract": "In this paper we deal with two classes of mixed metric 3-structures, namely the mixed 3-Sasakian structures and the mixed metric 3-contact structures. Firstly we study some properties of the curvature of mixed 3-Sasakian structures, proving that any manifold endowed with such a structure is Einstein. Then we prove the identity between the class of mixed 3-Sasakian structures and the class of mixed metric 3-contact structures."}
{"category": "Math", "title": "Nonlinear estimation for linear inverse problems with error in the operator", "abstract": "We study two nonlinear methods for statistical linear inverse problems when the operator is not known. The two constructions combine Galerkin regularization and wavelet thresholding. Their performances depend on the underlying structure of the operator, quantified by an index of sparsity. We prove their rate-optimality and adaptivity properties over Besov classes."}
{"category": "Math", "title": "Generalizing Simes' test and Hochberg's stepup procedure", "abstract": "In a multiple testing problem where one is willing to tolerate a few false rejections, procedure controlling the familywise error rate (FWER) can potentially be improved in terms of its ability to detect false null hypotheses by generalizing it to control the $k$-FWER, the probability of falsely rejecting at least $k$ null hypotheses, for some fixed $k>1$. Simes' test for testing the intersection null hypothesis is generalized to control the $k$-FWER weakly, that is, under the intersection null hypothesis, and Hochberg's stepup procedure for simultaneous testing of the individual null hypotheses is generalized to control the $k$-FWER strongly, that is, under any configuration of the true and false null hypotheses. The proposed generalizations are developed utilizing joint null distributions of the $k$-dimensional subsets of the $p$-values, assumed to be identical. The generalized Simes' test is proved to control the $k$-FWER weakly under the multivariate totally positive of order two (MTP$_2$) condition [J. Multivariate Analysis 10 (1980) 467--498] of the joint null distribution of the $p$-values by generalizing the original Simes' inequality. It is more powerful to detect $k$ or more false null hypotheses than the original Simes' test when the $p$-values are independent. A stepdown procedure strongly controlling the $k$-FWER, a version of generalized Holm's procedure that is different from and more powerful than [Ann. Statist. 33 (2005) 1138--1154] with independent $p$-values, is derived before proposing the generalized Hochberg's procedure. The strong control of the $k$-FWER for the generalized Hochberg's procedure is established in situations where the generalized Simes' test is known to control its $k$-FWER weakly."}
{"category": "Math", "title": "On false discovery control under dependence", "abstract": "A popular framework for false discovery control is the random effects model in which the null hypotheses are assumed to be independent. This paper generalizes the random effects model to a conditional dependence model which allows dependence between null hypotheses. The dependence can be useful to characterize the spatial structure of the null hypotheses. Asymptotic properties of false discovery proportions and numbers of rejected hypotheses are explored and a large-sample distributional theory is obtained."}
{"category": "Math", "title": "Optimal control of unilateral obstacle problem with a source term", "abstract": "We consider an optimal control problem for the obstacle problem with an elliptic variational inequality. The obstacle function which is the control function is assumed in $H^{2}$. We use an approximate technique to introduce a family of problems governed by variational equations. We prove optimal solutions existence and give necessary optimality conditions."}
{"category": "Math", "title": "Subsampling Algorithms for Semidefinite Programming", "abstract": "We derive a stochastic gradient algorithm for semidefinite optimization using randomization techniques. The algorithm uses subsampling to reduce the computational cost of each iteration and the subsampling ratio explicitly controls granularity, i.e. the tradeoff between cost per iteration and total number of iterations. Furthermore, the total computational cost is directly proportional to the complexity (i.e. rank) of the solution. We study numerical performance on some large-scale problems arising in statistical learning."}
{"category": "Math", "title": "Rational points on homogeneous varieties and Equidistribution of Adelic periods", "abstract": "Let U:=L\\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(K_v) acts transitively on U(K_v) for almost all places v of K, we obtain the asymptotic of the number of rational points in U(K) with height bounded by T, and settle new cases of Manin's conjecture for many wonderful varieties. The main ingredient of our approach is the equidistribution of semisimple adelic periods, which is established using the theory of unipotent flows."}
{"category": "Math", "title": "Traces of H\\\"ormander algebras on discrete sequences", "abstract": "We show that a discrete sequence $\\Lambda$ of the complex plane is the union of $n$ interpolating sequences for the H\\\"ormander algebras $A_p$ if and only if the trace of $A_p$ on $\\Lambda$ coincides with the space of functions on $\\Lambda$ for which the divided differences of order $n-1$ are uniformly bounded. The analogous result holds in the unit disk for Korenblum-type algebras."}
{"category": "Math", "title": "Constants of concentration for a simple recurrent random walk on random environment", "abstract": "We clarify the asymptotic of the limsup of the size of the neighborhood of concentration of Sinai's walk improving the result in \\cite{Pierre3}. Also we get the almost sure limit of the number of points visited more than a small but fixed proportion of a given amount of time."}
{"category": "Math", "title": "Geometry of obstructed equisingular families of projective hypersurfaces", "abstract": "We study geometric properties of certain obstructed equisingular families of projective hypersurfaces with emphasis on smoothness, reducibility, being reduced, and having expected dimension. In the case of minimal obstructness, we give a detailed description of such families corresponding to quasihomogeneous singularities. Next we study the behavior of these properties with respect to stable equivalence of singularities. We show that under certain conditions, stabilization of singularities ensures the existence of a reduced component of expected dimension. For minimally obstructed families the whole family becomes irreducible. As an application we show that if the equisingular family of a projective hypersurface H has a reduced component of expected dimension then the deformation of H induced by the linear system |H| is complete with respect to one-parameter deformations."}
{"category": "Math", "title": "Integration of Holomorphic Lie Algebroids", "abstract": "We prove that a holomorphic Lie algebroid is integrable if, and only if, its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic-Fernandes do also apply in the holomorphic context without any modification. As a consequence we give another proof of the following theorem: a holomorphic Poisson manifold is integrable if, and only if, its real (or imaginary) part is integrable as a real Poisson manifold."}
{"category": "Math", "title": "Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I", "abstract": "We show that certain representations of graphs by operators on Hilbert space have uses in signal processing and in symbolic dynamics. Our main result is that graphs built on automata have fractal characteristics. We make this precise with the use of Representation Theory and of Spectral Theory of a certain family of Hecke operators. Let G be a directed graph. We begin by building the graph groupoid G induced by G, and representations of G. Our main application is to the groupoids defined from automata. By assigning weights to the edges of a fixed graph G, we give conditions for G to acquire fractal-like properties, and hence we can have fractaloids or G-fractals. Our standing assumption on G is that it is locally finite and connected, and our labeling of G is determined by the \"out-degrees of vertices\". From our labeling, we arrive at a family of Hecke-type operators whose spectrum is computed. As applications, we are able to build representations by operators on Hilbert spaces (including the Hecke operators); and we further show that automata built on a finite alphabet generate fractaloids. Our Hecke-type operators, or labeling operators, come from an amalgamated free probability construction, and we compute the corresponding amalgamated free moments. We show that the free moments are completely determined by certain scalar-valued functions."}
{"category": "Math", "title": "Modular Classes of Loday Algebroids", "abstract": "We introduce the concept of Loday algebroids, a generalization of Courant algebroids. We define the naive cohomology and modular class of a Loday algebroid, and we show that the modular class of the double of a Lie bialgebroid vanishes. For Courant algebroids, we describe the relation between the naive and standard cohomologies and we conjecture that they are isomorphic when the Courant algebroid is transitive."}
{"category": "Math", "title": "The Kuranishi-space of complex parallelisable nilmanifolds", "abstract": "We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic symplectic in even dimension. We describe the Kuranishi space in detail in several examples and also analyse when small deformations remain complex parallelisable"}
{"category": "Math", "title": "Two-cover descent on hyperelliptic curves", "abstract": "We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a relatively efficiently tested criterion for solvability of hyperelliptic curves. We also discuss applications of this algorithm to curves of genus 1 and to curves with rational points."}
{"category": "Math", "title": "On the energy of inviscid singular flows", "abstract": "It is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space $B^{1/3}_{3,\\infty}$. When the singular set of the solution is (or belongs to) a smooth manifold, we derive various $L^p$-space regularity criteria dimensionally equivalent to the critical one. In particular, if the singular set is a hypersurface the energy of $u$ is conserved provided the one sided non-tangential limits to the surface exist and the non-tangential maximal function is $L^3$ integrable, while the maximal function of the pressure is $L^{3/2}$ integrable. The results directly apply to prove energy conservation of the classical vortex sheets in both 2D and 3D at least in those cases where the energy is finite."}
{"category": "Math", "title": "Schwarz Lemma for the tetrablock", "abstract": "We describe all complex geodesics in the tetrablock passing through the origin thus obtaining the form of all extremals in the Schwarz Lemma for the tetrablock. Some other extremals for the Lempert function and geodesics are also given. The paper may be seen as a continuation of the results Abouhajar, White and Young. The proofs rely on a necessary form of complex geodesics in general domains which is also proven in the paper."}
{"category": "Math", "title": "Compactly supported analytic indices for Lie groupoids", "abstract": "For any Lie groupoid we construct an analytic index morphism taking values in a modified $K-theory$ group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in \\cite{Ca2}. This allows in particular to prove a more primitive version of the Connes-Skandalis Longitudinal index Theorem for foliations, that is, an index theorem taking values in a group which pairs with Cyclic cocycles. As other application, for $D$ a $\\gr-$PDO elliptic operator with associated index $ind D\\in K_0(\\ci_c (\\gr))$, we prove that the pairing $$<ind D,\\tau>,$$ with $\\tau$ a bounded continuous cyclic cocycle, only depends on the principal symbol class $[\\sigma(D)]\\in K^0(A^*\\gr)$. The result is completely general for {\\'E}tale groupoids. We discuss some potential applications to the Novikov's conjecture."}
{"category": "Math", "title": "Actions of automorphism groups of free groups on homology spheres and acyclic manifolds", "abstract": "For n at least 3, let SAut(F_n) denote the unique subgroup of index two in the automorphism group of a free group. The standard linear action of SL(n,Z) on R^n induces non-trivial actions of SAut(F_n) on R^n and on S^{n-1}. We prove that SAut(F_n) admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension. Indeed, SAut(F_n) cannot act non-trivially on any generalized Z_2-homology sphere of dimension less than n-1, nor on any Z_2-acyclic Z_2-homology manifold of dimension less than n. It follows that SL(n,Z) cannot act non-trivially on such spaces either. When n is even, we obtain similar results with Z_3 coefficients."}
{"category": "Math", "title": "Alternatives for Testing Total Dual Integrality", "abstract": "In this paper we provide characterizing properties of TDI systems, among others the following: a system of linear inequalities is TDI if and only if its coefficient vectors form a Hilbert basis, and there exists a test-set for the system's dual integer programs where all test vectors have positive entries equal to 1. Reformulations of this provide relations between computational algebra and integer programming and they contain Applegate, Cook and McCormick's sufficient condition for the TDI property and Sturmfels' theorem relating toric initial ideals generated by square-free monomials to unimodular triangulations. We also study the theoretical and practical efficiency and limits of the characterizations of the TDI property presented here. In the particular case of set packing polyhedra our results correspond to endowing the weak perfect graph theorem with an additional, computationally interesting, geometric feature: the normal fan of the stable set polytope of a perfect graph can be refined into a regular triangulation consisting only of unimodular cones."}
{"category": "Math", "title": "Optimal two-value zero-mean disintegration of zero-mean random variables", "abstract": "For any continuous zero-mean random variable (r.v.) X, a reciprocating function r is constructed, based only on the distribution of X, such that the conditional distribution of X given the (at-most-)two-point set {X,r(X)} is the zero-mean distribution on this set; in fact, a more general construction without the continuity assumption is given in this paper, as well as a large variety of other related results, including characterizations of the reciprocating function and modeling distribution asymmetry patterns. The mentioned disintegration of zero-mean r.v.'s implies, in particular, that an arbitrary zero-mean distribution is represented as the mixture of two-point zero-mean distributions; moreover, this mixture representation is most symmetric in a variety of senses. Somewhat similar representations -- of any probability distribution as the mixture of two-point distributions with the same skewness coefficient (but possibly with different means) -- go back to Kolmogorov; very recently, Aizenman et al. further developed such representations and applied them to (anti-)concentration inequalities for functions of independent random variables and to spectral localization for random Schroedinger operators. One kind of application given in the present paper is to construct certain statistical tests for asymmetry patterns and for location without symmetry conditions. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition."}
{"category": "Math", "title": "Matrix factorizations and colored MOY graphs", "abstract": "The contents of this 98-page paper have been subsumed into the 191-page paper \"A colored sl(N)-homology for links in S^3\" (arXiv:0907.0695v1 [math.GT]), in which we further develop the theory and use it to construct a colored link homology."}
{"category": "Math", "title": "Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II", "abstract": "Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Koll\\'ar proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Our Main Theorem, answering in the affirmative three questions of Koll\\'ar, gives sharp estimates on the number and the multiplicities of the Seifert fibres and on the number and the torsions of the lens spaces when X is a geometrically rational surface. When N is Seifert fibred over a base orbifold F, our result generalizes Comessatti's theorem on smooth real rational surfaces: F cannot be simultaneously orientable and of hyperbolic type. We show as a surprise that, unlike in Comessatti's theorem, there are examples where F is non orientable, of hyperbolic type, and X is minimal. The technique we use is to construct Seifert fibrations as projectivized tangent bundles of Du Val surfaces."}
{"category": "Math", "title": "Koszul duality and modular representations of semi-simple Lie algebras", "abstract": "In this paper we prove that if G is a connected, simply-connected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)_0 of the Lie algebra g of G can be endowed with a Koszul grading (extending results of Andersen, Jantzen and Soergel). We also give information about the Koszul dual rings. Our main tool is the localization theory in positive characteristic developed by Bezrukavnikov, Mirkovic and Rumynin."}
{"category": "Math", "title": "A special value of Ruelle L-function and the theorem of Cheeger and Muller", "abstract": "We will show a theorem of a type of Cheeger and Muller for a noncompact complete hyperbolic threefold of finite vulume. As an application we will compute a special value of Ruelle L-function at the origin for a unitary local system which is cuspidal."}
{"category": "Math", "title": "Finite-type invariants for curves on surfaces", "abstract": "This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-order invariants for plane immersed curves. The invariants in this theory are developed using the word theory proposed by Turaev."}
{"category": "Math", "title": "On Algebraic Expressions of Sigma Functions for (n,s) Curves", "abstract": "An expression of the multivariate sigma function associated with a (n,s)-curve is given in terms of algebraic integrals. As a corollary the first term of the series expansion around the origin of the sigma function is directly proved to be the Schur function determined from the gap sequence at infinity."}
{"category": "Math", "title": "Tangential Convergence of bounded harmonic functions on Generalized Siegel domains", "abstract": "We extend a result of Hulanicki- Ricci to all H-type groups."}
{"category": "Math", "title": "Properties and applications of dual reduction", "abstract": "The dual reduction process, introduced by Myerson, allows to reduce a finite game into a smaller dimensional game such that any equilibrium of the reduced game is an equilibrium of the original game. This holds both for Nash equilibrium and correlated equilibrium. We present examples of applications of dual reduction and argue that this is a useful tool to study Nash equilibria and correlated equilibria. We then investigate its properties."}
{"category": "Math", "title": "Properties of higher criticism under strong dependence", "abstract": "The problem of signal detection using sparse, faint information is closely related to a variety of contemporary statistical problems, including the control of false-discovery rate, and classification using very high-dimensional data. Each problem can be solved by conducting a large number of simultaneous hypothesis tests, the properties of which are readily accessed under the assumption of independence. In this paper we address the case of dependent data, in the context of higher criticism methods for signal detection. Short-range dependence has no first-order impact on performance, but the situation changes dramatically under strong dependence. There, although higher criticism can continue to perform well, it can be bettered using methods based on differences of signal values or on the maximum of the data. The relatively inferior performance of higher criticism in such cases can be explained in terms of the fact that, under strong dependence, the higher criticism statistic behaves as though the data were partitioned into very large blocks, with all but a single representative of each block being eliminated from the dataset."}
{"category": "Math", "title": "Smallness of fundamental groups for arithmetic schemes", "abstract": "The smallness is proved of fundamental groups for arithmetic schemes. This is a higher dimensional analogue of the Hermite-Minkowski theorem. We also refer to the case of varieties over finite fields. As an application, we prove certain finiteness results of representations of the fundamental groups over algebraically closed fields."}
{"category": "Math", "title": "Hyperspaces with the Attouch-Wets topology homeomorphic to $l_2$", "abstract": "It is shown that the hyperspace of all nonempty closed subsets $\\Cld_{AW}(X)$ of a separable metric space $X$ endowed with the Attouch-Wets topology is homeomorphic to a separable Hilbert space if and only if the completion of $X$ is proper, locally connected and contains no bounded connected component, $X$ is topologically complete and not locally compact at infinity."}
{"category": "Math", "title": "Endogenous post-stratification in surveys: classifying with a sample-fitted model", "abstract": "Post-stratification is frequently used to improve the precision of survey estimators when categorical auxiliary information is available from sources outside the survey. In natural resource surveys, such information is often obtained from remote sensing data, classified into categories and displayed as pixel-based maps. These maps may be constructed based on classification models fitted to the sample data. Post-stratification of the sample data based on categories derived from the sample data (``endogenous post-stratification'') violates the standard post-stratification assumptions that observations are classified without error into post-strata, and post-stratum population counts are known. Properties of the endogenous post-stratification estimator are derived for the case of a sample-fitted generalized linear model, from which the post-strata are constructed by dividing the range of the model predictions into predetermined intervals. Design consistency of the endogenous post-stratification estimator is established under mild conditions. Under a superpopulation model, consistency and asymptotic normality of the endogenous post-stratification estimator are established, showing that it has the same asymptotic variance as the traditional post-stratified estimator with fixed strata. Simulation experiments demonstrate that the practical effect of first fitting a model to the survey data before post-stratifying is small, even for relatively small sample sizes."}
{"category": "Math", "title": "Formal meromorphic functions on manifolds of finite type", "abstract": "It is shown that a real-valued formal meromorphic function on a formal generic submanifold of finite Kohn-Bloom-Graham type is necessarily constant."}
{"category": "Math", "title": "Overholonomicity of overconvergent $F$-isocrystals over smooth varieties", "abstract": "We prove the overholonomicity of overconvergent $F$-isocrystals over smooth varieties. This implies that the notions of overholonomicity and devissability in overconvergent $F$-isocrystals are equivalent. Then the overholonomicity is stable under tensor products. So, the overholonomicity gives a $p$-adic cohomology stable under Grothendieck's cohomological operations."}
{"category": "Math", "title": "Locally D-optimal designs based on a class of composed models resulted from blending Emax and one-compartment models", "abstract": "A class of nonlinear models combining a pharmacokinetic compartmental model and a pharmacodynamic Emax model is introduced. The locally D-optimal (LD) design for a four-parameter composed model is found to be a saturated four-point uniform LD design with the two boundary points of the design space in the LD design support. For a five-parameter composed model, a sufficient condition for the LD design to require the minimum number of sampling time points is derived. Robust LD designs are also investigated for both models. It is found that an LD design with $k$ parameters is equivalent to an LD design with $k-1$ parameters if the linear parameter in the two composed models is a nuisance parameter. Assorted examples of LD designs are presented."}
{"category": "Math", "title": "Monodromy and Tangential Center Problems", "abstract": "We consider families of Abelian integrals arising from perturbations of planar Hamiltonian systems. The tangential center focus problem asks for the conditions under which these integrals vanish identically. The problem is closely related to the monodromy problem, which asks when the monodromy of a vanishing cycle generates the whole homology of the level curves of the Hamiltonian. We solve both these questions for the case when the Hamiltonian is hyperelliptic. As a side-product, we solve the corresponding problems for the \"0-dimensional Abelian integrals\" defined by Gavrilov and Movasati."}
{"category": "Math", "title": "Asymptotic properties of false discovery rate controlling procedures under independence", "abstract": "We investigate the performance of a family of multiple comparison procedures for strong control of the False Discovery Rate ($\\mathsf{FDR}$). The $\\mathsf{FDR}$ is the expected False Discovery Proportion ($\\mathsf{FDP}$), that is, the expected fraction of false rejections among all rejected hypotheses. A number of refinements to the original Benjamini-Hochberg procedure [1] have been proposed, to increase power by estimating the proportion of true null hypotheses, either implicitly, leading to one-stage adaptive procedures [4, 7] or explicitly, leading to two-stage adaptive (or plug-in) procedures [2, 21]. We use a variant of the stochastic process approach proposed by Genovese and Wasserman [11] to study the fluctuations of the $\\mathsf{FDP}$ achieved with each of these procedures around its expectation, for independent tested hypotheses. We introduce a framework for the derivation of generic Central Limit Theorems for the $\\mathsf{FDP}$ of these procedures, characterizing the associated regularity conditions, and comparing the asymptotic power of the various procedures. We interpret recently proposed one-stage adaptive procedures [4, 7] as fixed points in the iteration of well known two-stage adaptive procedures [2, 21]."}
{"category": "Math", "title": "Combinatorial properties of the numbers of tableaux of bounded height", "abstract": "We introduce an infinite family of lower triangular matrices $\\Gamma^{(s)}$, where $\\gamma_{n,i}^s$ counts the standard Young tableaux on $n$ cells and with at most $s$ columns on a suitable subset of shapes. We show that the entries of these matrices satisfy a three-term row recurrence and we deduce recursive and asymptotic properties for the total number $\\tau_s(n)$ of tableaux on $n$ cells and with at most $s$ columns."}
{"category": "Math", "title": "A complementary design theory for doubling", "abstract": "Chen and Cheng [Ann. Statist. 34 (2006) 546--558] discussed the method of doubling for constructing two-level fractional factorial designs. They showed that for $9N/32\\le n\\le 5N/16$, all minimum aberration designs with $N$ runs and $n$ factors are projections of the maximal design with $5N/16$ factors which is constructed by repeatedly doubling the $2^{5-1}$ design defined by $I=ABCDE$. This paper develops a general complementary design theory for doubling. For any design obtained by repeated doubling, general identities are established to link the wordlength patterns of each pair of complementary projection designs. A rule is developed for choosing minimum aberration projection designs from the maximal design with $5N/16$ factors. It is further shown that for $17N/64\\le n\\le 5N/16$, all minimum aberration designs with $N$ runs and $n$ factors are projections of the maximal design with $N$ runs and $5N/16$ factors."}
{"category": "Math", "title": "Jump estimation in inverse regression", "abstract": "We consider estimation of a step function $f$ from noisy observations of a deconvolution $\\phi*f$, where $\\phi$ is some bounded $L_1$-function. We use a penalized least squares estimator to reconstruct the signal $f$ from the observations, with penalty equal to the number of jumps of the reconstruction. Asymptotically, it is possible to correctly estimate the number of jumps with probability one. Given that the number of jumps is correctly estimated, we show that the corresponding parameter estimates of the jump locations and jump heights are $n^{-1/2}$ consistent and converge to a joint normal distribution with covariance structure depending on $\\phi$, and that this rate is minimax for bounded continuous kernels $\\phi$. As special case we obtain the asymptotic distribution of the least squares estimator in multiphase regression and generalisations thereof. In contrast to the results obtained for bounded $\\phi$, we show that for kernels with a singularity of order $O(| x|^{-\\alpha}),1/2<\\alpha<1$, a jump location can be estimated at a rate of $n^{-1/(3-2\\alpha)}$, which is again the minimax rate. We find that these rate do not depend on the spectral information of the operator rather on its localization properties in the time domain. Finally, it turns out that adaptive sampling does not improve the rate of convergence, in strict contrast to the case of direct regression."}
{"category": "Math", "title": "Asymptotic inference in some heteroscedastic regression models with long memory design and errors", "abstract": "This paper discusses asymptotic distributions of various estimators of the underlying parameters in some regression models with long memory (LM) Gaussian design and nonparametric heteroscedastic LM moving average errors. In the simple linear regression model, the first-order asymptotic distribution of the least square estimator of the slope parameter is observed to be degenerate. However, in the second order, this estimator is $n^{1/2}$-consistent and asymptotically normal for $h+H<3/2$; nonnormal otherwise, where $h$ and $H$ are LM parameters of design and error processes, respectively. The finite-dimensional asymptotic distributions of a class of kernel type estimators of the conditional variance function $\\sigma^2(x)$ in a more general heteroscedastic regression model are found to be normal whenever $H<(1+h)/2$, and non-normal otherwise. In addition, in this general model, $\\log(n)$-consistency of the local Whittle estimator of $H$ based on pseudo residuals and consistency of a cross validation type estimator of $\\sigma^2(x)$ are established. All of these findings are then used to propose a lack-of-fit test of a parametric regression model, with an application to some currency exchange rate data which exhibit LM."}
{"category": "Math", "title": "Algorithmic barriers from phase transitions", "abstract": "For many random Constraint Satisfaction Problems, by now, we have asymptotically tight estimates of the largest constraint density for which they have solutions. At the same time, all known polynomial-time algorithms for many of these problems already completely fail to find solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms already achieve this goal. Given the simplicity of those algorithms, one would expect there is a lot of room for improvement. Yet, to date, no algorithm is known that uses $(2-\\epsilon) \\chi$ colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we believe it is natural to inquire into its origin. We do so by analyzing the evolution of the set of $k$-colorings of a random graph, viewed as a subset of $\\{1,...,k\\}^{n}$, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly, the set of $k$-colorings looks like a giant ball for $k \\ge 2 \\chi$, but like an error-correcting code for $k \\le (2-\\epsilon) \\chi$. We prove that a completely analogous phase transition also occurs both in random $k$-SAT and in random hypergraph 2-coloring. And that for each problem, its location corresponds precisely with the point were all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to prove rigorously much of the celebrated 1-step Replica-Symmetry-Breaking hypothesis of statistical physics for random CSPs."}
{"category": "Math", "title": "The signed Eulerian numbers on involutions", "abstract": "We define an analogue of signed Eulerian numbers $f_{n,k}$ for involutions of the symmetric group and derive some combinatorial properties of this sequence. In particular, we exhibit both an explicit formula and a recurrence for $f_{n,k}$ arising from the properties of its generating function."}
{"category": "Math", "title": "Extensions of an $AC(\\sigma)$ functional calculus", "abstract": "On a reflexive Banach space $X$, if an operator $T$ admits a functional calculus for the absolutely continuous functions on its spectrum $\\sigma(T) \\subseteq \\mathbb{R}$, then this functional calculus can always be extended to include all the functions of bounded variation. This need no longer be true on nonreflexive spaces. In this paper, it is shown that on most classical separable nonreflexive spaces, one can construct an example where such an extension is impossible. Sufficient conditions are also given which ensure that an extension of an $\\AC$ functional calculus is possible for operators acting on families of interpolation spaces such as the $L^p$ spaces."}
{"category": "Math", "title": "Uniform saddlepoint approximations for ratios of quadratic forms", "abstract": "Ratios of quadratic forms in correlated normal variables which introduce noncentrality into the quadratic forms are considered. The denominator is assumed to be positive (with probability 1). Various serial correlation estimates such as least-squares, Yule--Walker and Burg, as well as Durbin--Watson statistics, provide important examples of such ratios. The cumulative distribution function (c.d.f.) and density for such ratios admit saddlepoint approximations. These approximations are shown to preserve uniformity of relative error over the entire range of support. Furthermore, explicit values for the limiting relative errors at the extreme edges of support are derived."}
{"category": "Math", "title": "Minimal Stable Sets in Tournaments", "abstract": "We propose a systematic methodology for defining tournament solutions as extensions of maximality. The central concepts of this methodology are maximal qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy of tournament solutions, which encompasses the top cycle, the uncovered set, the Banks set, the minimal covering set, the tournament equilibrium set, the Copeland set, and the bipartisan set. Moreover, the hierarchy includes a new tournament solution, the minimal extending set, which is conjectured to refine both the minimal covering set and the Banks set."}
{"category": "Math", "title": "Local Indices of a Vector Field at an Isolated Zero on the Boundary", "abstract": "We define two types of local indices of a vector field at an isolated zero on the boundary, and prove Poincare-Hopf-type index theorems for certain vector fields on a compact smooth manifold which have only isolated zeros."}
{"category": "Math", "title": "Graph products of right cancellative monoids", "abstract": "Our first main result shows that a graph product of right cancellative monoids is itself right cancellative. If each of the component monoids satisfies the condition that the intersection of two principal left ideals is either principal or empty, then so does the graph product. Our second main result gives a presentation for the inverse hull of such a graph product. We then specialise to the case of the inverse hulls of graph monoids, obtaining what we call polygraph monoids. Among other properties, we observe that polygraph monoids are F*-inverse. This follows from a general characterisation of those right cancellative monoids with inverse hulls that are F*-inverse."}
{"category": "Math", "title": "Ergodicity and mixing of W*-dynamical systems in terms of joinings", "abstract": "We study characterizations of ergodicity, weak mixing and strong mixing of W*-dynamical systems in terms of joinings and subsystems of such systems. Ergodic joinings and Ornstein's criterion for strong mixing are also discussed in this context."}
{"category": "Math", "title": "A class of hypergraphs that generalizes chordal graphs", "abstract": "In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given in \\cite{VT}, so that a triangulated hypergraph, according to our definition, is a natural generalization of a chordal (rigid circuit) graph. In \\cite{F1}, Fr\\\"oberg shows that the chordal graphs corresponds to graph algebras, $R/I(\\mc{G})$, with linear resolutions. We extend Fr\\\"oberg's method and show that the hypergraph algebras of generalized chordal hypergraphs, a class of hypergraphs that includes the chordal hypergraphs, have linear resolutions. The definitions we give, yield a natural higher dimensional version of the well known flag property of simplicial complexes. We obtain what we call $d$-flag complexes."}
{"category": "Math", "title": "Landau's function for one million billions", "abstract": "Let ${\\mathfrak S}_n$ denote the symmetric group with $n$ letters, and $g(n)$ the maximal order of an element of ${\\mathfrak S}_n$. If the standard factorization of $M$ into primes is $M=q_1^{\\al_1}q_2^{\\al_2}... q_k^{\\al_k}$, we define $\\ell(M)$ to be $q_1^{\\al_1}+q_2^{\\al_2}+... +q_k^{\\al_k}$; one century ago, E. Landau proved that $g(n)=\\max_{\\ell(M)\\le n} M$ and that, when $n$ goes to infinity, $\\log g(n) \\sim \\sqrt{n\\log(n)}$. There exists a basic algorithm to compute $g(n)$ for $1 \\le n \\le N$; its running time is $\\co(N^{3/2}/\\sqrt{\\log N})$ and the needed memory is $\\co(N)$; it allows computing $g(n)$ up to, say, one million. We describe an algorithm to calculate $g(n)$ for $n$ up to $10^{15}$. The main idea is to use the so-called {\\it $\\ell$-superchampion numbers}. Similar numbers, the {\\it superior highly composite numbers}, were introduced by S. Ramanujan to study large values of the divisor function $\\tau(n)=\\sum_{d\\dv n} 1$."}
{"category": "Math", "title": "Statistics of extremes under random censoring", "abstract": "We investigate the estimation of the extreme value index when the data are subject to random censorship. We prove, in a unified way, detailed asymptotic normality results for various estimators of the extreme value index and use these estimators as the main building block for estimators of extreme quantiles. We illustrate the quality of these methods by a small simulation study and apply the estimators to medical data."}
{"category": "Math", "title": "Tropicalization and irreducibility of Generalized Vandermonde Determinants", "abstract": "We find geometric and arithmetic conditions in order to characterize the irreducibility of the determinant of the generic Vandermonde matrix over the algebraic closure of any field k. We also characterize those determinants whose tropicalization with respect to the variables of a row is irreducible."}
{"category": "Math", "title": "On the asymptotic joint distribution of sample space--time covariance estimators", "abstract": "We study the asymptotic joint distribution of sample space--time covariance estimators of strictly stationary random fields. We do this without any marginal or joint distributional assumptions other than mild moment and mixing conditions. We consider several situations depending on whether the observations are regularly or irregularly spaced and whether one part or the whole domain of interest is fixed or increasing. A simulation experiment illustrates the theoretical results."}
{"category": "Math", "title": "About Substitution Tilings with Statistical Circular Symmetry", "abstract": "Two results about equidistribution of tile orientations in primitive substitution tilings are stated, one for finitely many, one for infinitely many orientations. Furthermore, consequences for the associated diffraction spectra and the dynamical systems are discussed."}
{"category": "Math", "title": "Adaptive Ridge Selector (ARiS)", "abstract": "We introduce a new shrinkage variable selection operator for linear models which we term the \\emph{adaptive ridge selector} (ARiS). This approach is inspired by the \\emph{relevance vector machine} (RVM), which uses a Bayesian hierarchical linear setup to do variable selection and model estimation. Extending the RVM algorithm, we include a proper prior distribution for the precisions of the regression coefficients, $v_{j}^{-1} \\sim f(v_{j}^{-1}|\\eta)$, where $\\eta$ is a scalar hyperparameter. A novel fitting approach which utilizes the full set of posterior conditional distributions is applied to maximize the joint posterior distribution $p(\\boldsymbol\\beta,\\sigma^{2},\\mathbf{v}^{-1}|\\mathbf{y},\\eta)$ given the value of the hyper-parameter $\\eta$. An empirical Bayes method is proposed for choosing $\\eta$. This approach is contrasted with other regularized least squares estimators including the lasso, its variants, nonnegative garrote and ordinary ridge regression. Performance differences are explored for various simulated data examples. Results indicate superior prediction and model selection accuracy under sparse setups and drastic improvement in accuracy of model choice with increasing sample size."}
{"category": "Math", "title": "Strongly Clean Matrix Rings Over Commutative Rings", "abstract": "A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute. By {\\rm SRC} factorization, Borooah, Diesl, and Dorsey \\cite{BDD051} completely determined when ${\\mathbb M}_n(R)$ over a commutative local ring $R$ is strongly clean. We generalize the notion of {\\rm SRC} factorization to commutative rings, prove that commutative $n$-{\\rm SRC} rings $(n\\ge 2)$ are precisely the commutative local rings over which ${\\mathbb M}_n(R)$ is strongly clean, and characterize strong cleanness of matrices over commutative projective-free rings having {\\rm ULP}. The strongly $\\pi$-regular property (hence, strongly clean property) of ${\\mathbb M}_n(C(X,{\\mathbb C}))$ with $X$ a {\\rm P}-space relative to ${\\mathbb C}$ is also obtained where $C(X,{\\mathbb C})$ is the ring of complex valued continuous functions."}
{"category": "Math", "title": "Partial differential equations driven by rough paths", "abstract": "We study a class of linear first and second order partial differential equations driven by weak geometric $p$-rough paths, and prove the existence of a unique solution for these equations. This solution depends continuously on the driving rough path. This allows a robust approach to stochastic partial differential equations. In particular, we may replace Brownian motion by more general Gaussian and Markovian noise. Support theorems and large deviation statements all became easy corollaries of the corresponding statements of the driving process. In the case of first order equations with Gaussian noise, we discuss the existence of a density with respect to the Lebesgue measure for the solution."}
{"category": "Math", "title": "On Singular Poisson Sternberg Spaces", "abstract": "We obtain a theory of stratified Sternberg spaces thereby extending the theory of cotangent bundle reduction for free actions to the singular case where the action on the base manifold consists of only one orbit type. We find that the symplectic reduced spaces are stratified topological fiber bundles over the cotangent bundle of the orbit space. We also obtain a Poisson stratification of the Sternberg space. To construct the singular Poisson Sternberg space we develop an appropriate theory of singular connections for proper group actions on a single orbit type manifold including a theory of holonomy extending the usual Ambrose-Singer theorem for principal bundles."}
{"category": "Math", "title": "Laws of the single logarithm for delayed sums of random fields", "abstract": "We extend a law of the single logarithm for delayed sums by Lai to delayed sums of random fields. A law for subsequences, which also includes the one-dimensional case, is obtained in passing."}
{"category": "Math", "title": "A unified approach on Springer fibers in the hook, two-row and two-column cases", "abstract": "We consider the Springer fiber over a nilpotent endomorphism. Fix a Jordan basis and consider the standard torus relative to this. We deal with the problem to describe the flags fixed by the torus which belong to a given component of the Springer fiber. We solve the problem in the hook, two-row and two-column cases. We provide two main characterizations which are common to the three cases, and which involve dominance relations between Young diagrams and combinatorial algorithms. Then, for these three cases, we deduce topological properties of the components and their intersections."}
{"category": "Math", "title": "A tropical interpretation of m-dissimilarity maps", "abstract": "Let T be a weighted tree with n numbered leaves and let D be its distance matrix, so D(i,j) is the distance between the leaves i and j. If m is an integer between 2 and n, we prove a tropical formula to compute the m-dissimilarity map of T (i.e. the weights of the subtrees of T with m leaves), given D. For m equal to 3, we present a tropical description of the set of m-dissimilarity maps of trees. For m equal to 4, a partial result is given."}
{"category": "Math", "title": "Singular components of the Springer fiber in the two-columns case", "abstract": "We consider the Springer fiber ${\\mathcal B}_u$ corresponding to a nilpotent endomorphism $u$ of nilpotent order 2. As a first result, we give a description of the elements of a given component of ${\\mathcal B}_u$ which are fixed by the action of the standard torus relative to some Jordan basis of $u$. By using this result, we establish a necessary and sufficient condition of singularity for the components of ${\\mathcal B}_u$."}
{"category": "Math", "title": "Estimates for periodic Zakharov-Shabat operators", "abstract": "We consider the periodic Zakharov-Shabat operators on the real line. The spectrum of this operator consists of intervals separated by gaps with the lengths $|g_n|\\ge 0, n\\in \\Z$. Let $\\m_n^\\pm$ be the corresponding effective masses and let $h_n$ be heights of the corresponding slits in the quasimomentum domain. We obtain a priori estimates of sequences $g=(|g_n|)_{n\\in \\Z},\\m^\\pm=(\\m_n^\\pm)_{n\\in \\Z}, h=(h_n)_{n\\in \\Z}$ in terms of weighted $\\ell^p-$norms at $p\\ge 1$. The proof is based on the analysis of the quasimomentum as the conformal mapping."}
{"category": "Math", "title": "When is a Bol loop Moufang?", "abstract": "There are a number of identities which, if satisfied by a Bol loop, imply that the loop is actually Moufang. In this paper we show that in a number of cases, the Moufang identity is also forced not by a single identity, but by giving elements a choice of equations to satisfy."}
{"category": "Math", "title": "Upper bounds for the number of limit cycles of some planar polynomial differential systems", "abstract": "We give an effective method for controlling the maximum number of limit cycles of some planar polynomial systems. It is based on a suitable choice of a Dulac function and the application of the well-known Bendixson-Dulac Criterion for multiple connected regions. The key point is a new approach to control the sign of the functions involved in the criterion. The method is applied to several examples."}
{"category": "Math", "title": "On Conditions for Convergence to Consensus", "abstract": "A new theorem on conditions for convergence to consensus of a multiagent time-dependent time-discrete dynamical system is presented. The theorem is build up on the notion of averaging maps. We compare this theorem to results by Moreau (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005) about set-valued Lyapunov theory and convergence under switching communication topologies. We give examples that point out differences of approaches including examples where Moreau's theorem is not applicable but ours is. Further on, we give examples that demonstrate that the theory of convergence to consensus is still not complete."}
{"category": "Math", "title": "Automorphisms of Partially Commutative Groups I: Linear Subgroups", "abstract": "The goal of this paper is to construct and describe certain arithmetic subgroups of the automorphism group of a partially commutative group. More precisely, given an arbitrary finite graph $\\Gamma$ we construct an arithmetic subgroup $St(L(G))$, represented as a subgroup of $GL(n,Z)$, where $n$ is the number of vertices of the graph $\\Gamma$. In the last section of the paper we give a description of the decomposition of the group of automorphisms $St^{conj}(L(G))$ as a semidirect product of the group of conjugating automorphisms $Conj(G)$ and $St(L(G))$. This result is closely related to Theorem 1.4 of the paper arXiv:0710.2573v1."}
{"category": "Math", "title": "The Gauss Map of Hypersurfaces in 2-Step Nilpotent Lie Groups", "abstract": "In this paper we consider smooth oriented hypersurfaces in 2-step nilpotent Lie groups with a left invariant metric and derive an expression for the Laplacian of the Gauss map for such hypersurfaces in the general case and in some particular cases. In the case of CMC-hypersurface in the (2m+1)-dimensional Heisenberg group we also derive necessary and sufficient conditions for the Gauss map to be harmonic and prove that for m=1 all CMC-surfaces with the harmonic Gauss map are \"cylinders\"."}
{"category": "Math", "title": "Integral group ring of the Suzuki sporadic simple group", "abstract": "Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Suzuki sporadic simple group Suz. As a consequence, for this group we confirm the Kimmerle's conjecture on prime graphs."}
{"category": "Math", "title": "Submanifolds with the Harmonic Gauss Map in Lie Groups", "abstract": "In this paper we find a criterion for the Gauss map of an immersed smooth submanifold in some Lie group with left invariant metric to be harmonic. Using the obtained expression we prove some necessary and sufficient conditions for the harmonicity of this map in the case of totally geodesic submanifolds in Lie groups admitting biinvariant metrics. We show that, depending on the structure of the tangent space of a submanifold, the Gauss map can be harmonic in all biinvariant metrics or non-harmonic in some metric. For 2-step nilpotent groups we prove that the Gauss map of a geodesic is harmonic if and only if it is constant."}
{"category": "Math", "title": "Duality of Chordal SLE, II", "abstract": "We improve the geometric properties of SLE$(\\kappa;\\vec{\\rho})$ processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for $\\kappa\\in (4,8)$, the boundary of a standard chordal SLE$(\\kappa)$ hull stopped on swallowing a fixed $x\\in\\R\\sem\\{0\\}$ is the image of some SLE$(16/\\kappa;\\vec{\\rho})$ trace started from a random point. Using this fact together with a similar proposition in the case that $\\kappa\\ge 8$, we obtain a description of the boundary of a standard chordal SLE$(\\kappa)$ hull for $\\kappa>4$, at a finite stopping time. Finally, we prove that for $\\kappa>4$, in many cases, the limit of a chordal or strip SLE$(\\kappa;\\vec{\\rho})$ trace exists."}
{"category": "Math", "title": "A decomposition of the bifractional Brownian motion and some applications", "abstract": "In this paper we show a decomposition of the bifractional Brownian motion with parameters H,K into the sum of a fractional Brownian motion with Hurst parameter HK plus a stochastic process with absolutely continuous trajectories. Some applications of this decomposition are discussed."}
{"category": "Math", "title": "Vertical Ends of Constant Mean Curvature H=1/2 in H^2\\times R", "abstract": "We prove a vertical halfspace theorem for surfaces with constant mean curvature $H={1/2},$ properly immersed in the product space $\\h^2\\times\\re,$ where $\\h^2$ is the hyperbolic plane and $\\re$ is the set of real numbers. The proof is a geometric application of the classical maximum principle for second order elliptic PDE, using the family of non compact rotational $H=1/2$ surfaces in $\\h^2\\times\\re.$"}
{"category": "Math", "title": "Monge--Amp\\`ere equation and Bellman optimization of Carleson Embedding Theorems", "abstract": "Monge--Amp\\`ere equation plays an important part in Analysis. For example, it is instrumental in mass transport problems. On the other hand, the Bellman function technique appeared recently as a way to consider certain Harmonic Analysis problems as the problems of Stochastic Optimal Control. This brings us to Bellman PDE, which in stochastic setting is often a Monge--Amp\\`ere equation or its close relative. We explore the way of solving Monge--Amp\\`ere equation by a sort of method of characteristics to find the Bellman function of certain classical Harmonic Analysis problems, and, therefore, of finding full structure of sharp constants and extremal sequences for those problems."}
{"category": "Math", "title": "Crossed interval groups and operations on the Hochschild cohomology", "abstract": "We prove that the operad B of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that B is a certain crossed interval extension of an operad T whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades."}
{"category": "Math", "title": "Diagonal vectors of shifted Young tableaux", "abstract": "We study vectors formed by entries on the diagonal of standard Young tableaux of shifted shapes. Such vectors are in bijection with integer lattice points of certain integral polytopes, which are Minkowski sums of simplices. We also describe vertices of these polytopes, and construct corresponding shifted Young tableaux."}
{"category": "Math", "title": "On Gautschi's conjecture for generalized Gauss-Radau and Gauss-Lobatto formulae", "abstract": "Recently, Gautschi introduced so-called generalized Gauss-Radau and Gauss-Lobatto formulae which are quadrature formulae of Gaussian type involving not only the values but also the derivatives of the function at the endpoints. In the present note we show the positivity of the corresponding weights; this positivity has been conjectured already by Gautschi. As a consequence, we establish several convergence theorems for these quadrature formulae."}
{"category": "Math", "title": "A Hausdorff-Young inequality for measured groupoids", "abstract": "The classical Hausdorff-Young inequality for locally compact abelian groups states that, for $1\\le p\\le 2$, the $L^p$-norm of a function dominates the $L^q$-norm of its Fourier transform, where $1/p+1/q=1$. By using the theory of non-commutative $L^p$-spaces and by reinterpreting the Fourier transform, R. Kunze (1958) [resp. M. Terp (1980)] extended this inequality to unimodular [resp. non-unimodular] groups. The analysis of the $L^p$-spaces of the von Neumann algebra of a measured groupoid provides a further extension of the Hausdorff-Young inequality to measured groupoids."}
{"category": "Math", "title": "Cartan subalgebras in C*-algebras", "abstract": "According to J. Feldman and C. Moore's well-known theorem on Cartan subalgebras, a variant of the group measure space construction gives an equivalence of categories between twisted countable standard measured equivalence relations and Cartan pairs, i.e. a von Neumann algebra (on a separable Hilbert space) together with a Cartan subalgebra. A. Kumjian gave a C*-algebraic analogue of this theorem in the early eighties. After a short survey of maximal abelian self-adjoint subalgebras in operator algebras, I present a natural definition of a Cartan subalgebra in a C*-algebra and an extension of Kumjian's theorem which covers graph algebras and some foliation algebras."}
{"category": "Math", "title": "Mirror fibrations and root stacks of weighted projective spaces", "abstract": "We show that the orbifold Chow ring of a root stack over a well-formed weighted projective space can be naturally seen as the Jacobian algebra of a function on a singular variety given by a partial compactification of its Ginzburg-Landau model."}
{"category": "Math", "title": "Generalized retarded integral inequalities", "abstract": "We prove some new retarded integral inequalities. The results generalize those in [J. Math. Anal. Appl. 301 (2005), no. 2, 265--275]."}
{"category": "Math", "title": "Every 4-Manifold is BLF", "abstract": "Here we show that every compact smooth 4-manifold X has a structure of a Broken Lefschetz Fibration (BLF in short). Furthermore, if b_{2}^{+}(X)> 0 then it also has a Broken Lefschetz Pencil structure (BLP) with nonempty base locus. This imroves a Theorem of Auroux, Donaldson and Katzarkov, and our proof is topological (i.e. uses 4-dimensional handlebody theory)."}
{"category": "Math", "title": "K\\\"ahler and Sasakian-Einstein Quotients", "abstract": "We construct symplectic and K\\\"ahler ray reduced spaces and discuss their relation with the Marsden-Weinstein (point) reduction. This K\\\"ahler reduction is well defined even when the momentum value is not totally isotropic. The compatibility of the ray reduction with the cone construction and the Boothby-Wang fibration is presented. Using the compatibility with the cone construction we provide the exact description of ray quotients of cotangent bundles. Some applications of the ray reduction to the study of conformal Hamiltonian systems are described. We also give necessary and sufficient conditions for the (ray) quotients of K\\\"ahler (Sasakian)-Einstein manifolds to be again K\\\"ahler (Sasakian)-Einstein."}
{"category": "Math", "title": "The nontrivial zeros of the Zeta Function lie on the Critical Line", "abstract": "In this paper is stablished a characterization of the solutions of the equation: zeta(z) = 0. Then such a characterization is used to give a proof for Riemann is Conjecture."}
{"category": "Math", "title": "A bijective proof of a factorization formula for Macdonald polynomials at roots of unity", "abstract": "We give a combinatorial proof of the factorization formula of modified Macdonald polynomials when the parameter t is specialized at a primitive root of unity. Our proof is restricted to the special case of partitions with 2 columns. We mainly use the combinatorial interpretation of Haglund, Haiman and Loehr giving the expansion of the modified Macdonald polynomials on the monomial basis."}
{"category": "Math", "title": "Representations of dynamical systems on Banach spaces not containing $l_1$", "abstract": "For a topological group G, we show that a compact metric G-space is tame if and only if it can be linearly represented on a separable Banach space which does not contain an isomorphic copy of $l_1$ (we call such Banach spaces, Rosenthal spaces). With this goal in mind we study tame dynamical systems and their representations on Banach spaces."}
{"category": "Math", "title": "When is a symmetric pin-jointed framework isostatic?", "abstract": "Maxwell's rule from 1864 gives a necessary condition for a framework to be isostatic in 2D or in 3D. Given a framework with point group symmetry, group representation theory is exploited to provide further necessary conditions. This paper shows how, for an isostatic framework, these conditions imply very simply stated restrictions on the numbers of those structural components that are unshifted by the symmetry operations of the framework. In particular, it turns out that an isostatic framework in 2D can belong to one of only six point groups. Some conjectures and initial results are presented that would give sufficient conditions (in both 2D and 3D) for a framework that is realized generically for a given symmetry group to be an isostatic framework."}
{"category": "Math", "title": "Decomposition numbers for perverse sheaves", "abstract": "The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial orbit in a simple Lie algebra. This work has applications to modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra, and for reductive algebraic group schemes using the affine Grassmannian of the Langlands dual group."}
{"category": "Math", "title": "Twisted K-theory and finite-dimensional approximation", "abstract": "We provide a finite-dimensional model of the twisted K-group twisted by any degree three integral cohomology class of a CW complex. One key to the model is Furuta's generalized vector bundle, and the other is a finite-dimensional approximation of Fredholm operators."}
{"category": "Math", "title": "A Taylor expansion theorem for an elliptic extension of the Askey-Wilson operator", "abstract": "We establish Taylor series expansions in rational (and elliptic) function bases using E. Rains' elliptic extension of the Askey-Wilson divided difference operator. The expansion theorem we consider extends M.E.H. Ismail's expansion for the Askey-Wilson monomial basis. Three immediate applications (essentially already due to Rains) include simple proofs of Frenkel and Turaev's elliptic extensions of Jackson's 8-phi-7 summation and of Bailey's 10-phi-9 transformation, and the computation of the connection coefficients of Spiridonov's elliptic extension of Rahman's biorthogonal rational functions. We adumbrate other examples including the nonterminating extension of Jackson's 8-phi-7 summation and a quadratic expansion."}
{"category": "Math", "title": "Consistent Computation of First- and Second-Order Differential Quantities for Surface Meshes", "abstract": "Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce inconsistent results and require ad hoc fixes. In this paper, we show that the computation of the gradient and Hessian of a height function provides the foundation for consistently computing the differential quantities. We derive simple, explicit formulas for the transformations between the first- and second-order differential quantities (i.e., normal vector and principal curvature tensor) of a smooth surface and the first- and second-order derivatives (i.e., gradient and Hessian) of its corresponding height function. We then investigate a general, flexible numerical framework to estimate the derivatives of the height function based on local polynomial fittings formulated as weighted least squares approximations. We also propose an iterative fitting scheme to improve accuracy. This framework generalizes polynomial fitting and addresses some of its accuracy and stability issues, as demonstrated by our theoretical analysis as well as experimental results."}
{"category": "Math", "title": "On the size of Kakeya sets in finite fields", "abstract": "A Kakeya set is a subset of F^n, where F is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least C_n * q^n, where C_n depends only on n. This improves the previously best lower bound for general n of ~q^{4n/7}."}
{"category": "Math", "title": "Interval valued $(\\in,\\ivq)$-fuzzy filters of pseudo $BL$-algebras", "abstract": "We introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. By using this new idea, we introduce the notions of interval valued $(\\in,\\ivq)$-fuzzy filters of pseudo $BL$-algebras and investigate some of their related properties. Some characterization theorems of these generalized interval valued fuzzy filters are derived. The relationship among these generalized interval valued fuzzy filters of pseudo $BL$-algebras is considered. Finally, we consider the concept of implication-based interval valued fuzzy implicative filters of pseudo $BL$-algebras, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed."}
{"category": "Math", "title": "Studies on the equations of Ince's table", "abstract": "We study the phase space of the equations of Ince's table from the viewpoint of its accessible singularities and local index."}
{"category": "Math", "title": "On H.Weyl and J.Steiner polynomials", "abstract": "The paper deals with root problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set V in R^n. A polynomial of this class describes the volume of the set V+tB^n as a function of t, where t is a positive number and B^n denotes the unit ball in R. The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold M, where M} is isometrically embedded with positive codimension in R^n. A Weyl polynomial describes the volume of a tubular neighborhood of its associated M as a function of the tube's radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of R^n into R^m for m>n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial's roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold."}
{"category": "Math", "title": "Classical metric Diophantine approximation revisited", "abstract": "The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation, a branch of Number Theory which draws on a rich and broad variety of mathematics. We discuss some recent progress and open problems concerning this classical theory. In particular, generalisations of the Duffin-Schaeffer and Catlin conjectures are formulated and explored."}
{"category": "Math", "title": "Hybrid moments of the Riemann zeta-function", "abstract": "The \"hybrid\" moments $$ \\int_T^{2T}|\\zeta(1/2+it)|^k{(\\int_{t-G}^{t+G}|\\zeta(1/2+ix)|^\\ell dx)}^m dt $$ of the Riemann zeta-function $\\zeta(s)$ on the critical line $\\Re s = 1/2$ are studied. The expected upper bound for the above expression is $O_\\epsilon(T^{1+\\epsilon}G^m)$. This is shown to be true for certain specific values of the natural numbers $k,\\ell,m$, and the explicitly determined range of $G = G(T;k,\\ell,m)$. The application to a mean square bound for the Mellin transform function of $|\\zeta(1/2+ix)|^4$ is given."}
{"category": "Math", "title": "A note on zero-one laws in metrical Diophantine approximation", "abstract": "In this paper we discuss a general problem on metrical Diophantine approximation associated with a system of linear forms. The main result is a zero-one law that extends one-dimensional results of Cassels and Gallagher. The paper contains a discussion on possible generalisations including a selection of various open problems."}
{"category": "Math", "title": "Limit stable objects on Calabi-Yau 3-folds", "abstract": "In this paper, we introduce new enumerative invariants of curves on Calabi-Yau 3-folds via certain stable objects in the derived category of coherent sheaves. We introduce the notion of limit stability on the category of perverse coherent sheaves, a subcategory in the derived category, and construct the moduli spaces of limit stable objects. We then define the counting invariants of limit stable objects using Behrend's constructible functions on that moduli spaces. It will turn out that our invariants are generalizations of counting invariants of stable pairs introduced by Pandharipande and Thomas. We will also investigate the wall-crossing phenomena of our invariants under change of stability conditions."}
{"category": "Math", "title": "Arithmetic Riemann-Roch and Hilbert-Samuel formulae for pointed stable curves", "abstract": "We prove arithmetic Riemann-Roch and Hilbert-Samuel type formulae for pointed stable curves. We give applications to volumes of lattices of integral cusps forms for pointed stable curves of genus 0."}
{"category": "Math", "title": "Cl\\^oture int\\'egrale des id\\'eaux et \\'equisingularit\\'e", "abstract": "This text has two parts; the first is the essentially unmodified text of the 1973-74 seminar of M. Lejeune-Jalabert and B. Teissier on integral dependence in complex analytic geometry with J-J. Risler's appendix on the Lojasiewicz exponents in the real analytic framework. The second part consists of seven complements written in 2007 surveying recent results directly connected to the content of the seminar. The main results of the first part concern the asymptotic order function with respect to an ideal and in particular its connection with the Lojasiewicz exponent. Another aspect concerns the finiteness properties of the graded algebra associated with the filtration by the asymptotic order function."}
{"category": "Math", "title": "Structured matrices and inverses", "abstract": "A matrix (and any associated linear system) will be referred to as structured if it has a small displacement rank. It is known that the inverse of a structured matrix is structured, which allows fast inversion (or solution), and reduced storage requirements. According to two definitions of displacement structure of practical interest, it is shown here that several types of inverses are also structured, including the Moore-Penrose inverse of rank-deficient matrices."}
{"category": "Math", "title": "A convergence analysis of the iteratively regularized Gauss-Newton method under Lipschitz condition", "abstract": "In this paper we consider the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed inverse problems. Under merely Lipschitz condition, we prove that this method together with an a posteriori stopping rule defines an order optimal regularization method if the solution is regular in some suitable sense."}
{"category": "Math", "title": "Limits of groupoid C*-algebras arising from open covers", "abstract": "I. Raeburn and J. Taylor have constructed continuous-trace C*-algebras with a prescribed Dixmier-Douady class, which also depend on the choice of an open cover of the spectrum. We study the asymptotic behavior of these algebras with respect to certain refinements of the cover and appropriate extension of cocycles. This leads to the analysis of a limit groupoid G and a cocycle \\sigma, and the algebra C*(G, \\sigma) may be regarded as a generalized direct limit of the Raeburn-Taylor algebras. As a special case, all UHF C*-algebras arise from this limit construction."}
{"category": "Math", "title": "Unavoidable patterns", "abstract": "Let \\mathcal{F}_k denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollob\\'as conjectured that for every \\epsilon>0 and positive integer k there is an n(k,\\epsilon) such that every 2-edge-coloring of the complete graph of order n \\geq n(k,\\epsilon) which has at least \\epsilon {n \\choose 2} edges in each color contains a member of \\mathcal{F}_k. This conjecture was proved by Cutler and Mont\\'agh, who showed that n(k,\\epsilon)<4^{k/\\epsilon}. We give a much simpler proof of this conjecture which in addition shows that n(k,\\epsilon)<\\epsilon^{-ck} for some constant c. This bound is tight up to the constant factor in the exponent for all k and \\epsilon. We also discuss similar results for tournaments and hypergraphs."}
{"category": "Math", "title": "Dirac generating operators and Manin triples", "abstract": "Given a pair of (real or complex) Lie algebroid structures on a vector bundle $A$ (over $M$) and its dual $A^*$, and a line bundle $\\module$ such that $\\module\\otimes\\module=(\\wedge^{\\TOP} A^*\\otimes\\wedge^{\\TOP} T^*M)$, there exist two canonically defined differential operators $\\bdees$ and $\\bdel$ on $\\sections{\\wedge A\\otimes\\module}$. We prove that the pair $(A,A^*)$ constitutes a Lie bialgebroid if, and only if, the square of $\\bdirac =\\bdees+\\bdel$ is the multiplication by a function on $M$. As a consequence, we obtain that the pair $(A,A^*)$ is a Lie bialgebroid if, and only if, $\\bdirac$ is a Dirac generating operator as defined by Alekseev & Xu \\cite{AlekseevXu}. Our approach is to establish a list of new identities relating the Lie algebroid structures on $A$ and $A^*$ (Theorem \\ref{Thm:C})."}
{"category": "Math", "title": "Effective computation of knot Floer homology", "abstract": "We extend an approach of Beliakova for computing knot Floer homology and implement it in a publicly available computer program. We review the main programming and optimization methods used. Our program is then used to check that the Floer homology of a prime non-alternating knot with less than 12 crossings has no torsion."}
{"category": "Math", "title": "Class field theory for curves over $p$-adic fields", "abstract": "We develop class field theory of curves over $p$-adic fields which extends the unramified theory of S. Saito. The class groups which approximate abelian \\'etale fundamental groups of such curves are introduced in the terms of algebraic $K$-groups by imitating G. Wiesend's class group for curves over finite fields."}
{"category": "Math", "title": "Absolute continuity and convergence in variation for distributions of functionals of Poisson point measure", "abstract": "General sufficient conditions are given for absolute continuity and convergence in variation of the distributions of the unctionals on a probability space, generated by a Poisson point measure. The phase space of the Poisson point measure is supposed to be of the form (0,\\infty)\\times U, and its intensity measure to be equal dt\\Pi(du). We introduce the family of time stretching transformations of the configurations of the point measure. The sufficient conditions for absolute continuity and convergence in variation are given in the terms of the time stretching transformations and the relative differential operators. These conditions are applied to solutions of SDE's driven by Poisson point measures, including an SDE's with non-constant jump rate."}
{"category": "Math", "title": "Orbits of s-representations with degenerate Gauss mappings", "abstract": "In this paper we study tangentially degeneracy of the orbits of s-representations in the sphere. We show that an orbit of an s-representation is tangentially degenerate if and only if it is through a long root, or a short root of restricted root system of type G_2. Moreover these orbits provide many new examples of tangentially degenerate submanifolds which satisfy the Ferus equality."}
{"category": "Math", "title": "The adjusted Viterbi training for hidden Markov models", "abstract": "The EM procedure is a principal tool for parameter estimation in the hidden Markov models. However, applications replace EM by Viterbi extraction, or training (VT). VT is computationally less intensive, more stable and has more of an intuitive appeal, but VT estimation is biased and does not satisfy the following fixed point property. Hypothetically, given an infinitely large sample and initialized to the true parameters, VT will generally move away from the initial values. We propose adjusted Viterbi training (VA), a new method to restore the fixed point property and thus alleviate the overall imprecision of the VT estimators, while preserving the computational advantages of the baseline VT algorithm. Simulations elsewhere have shown that VA appreciably improves the precision of estimation in both the special case of mixture models and more general HMMs. However, being entirely analytic, the VA correction relies on infinite Viterbi alignments and associated limiting probability distributions. While explicit in the mixture case, the existence of these limiting measures is not obvious for more general HMMs. This paper proves that under certain mild conditions, the required limiting distributions for general HMMs do exist."}
{"category": "Math", "title": "On Connection between the Numbers of Permutations and Full Cycles with Some Restrictions on Positions and Up-Down Structure", "abstract": "We discuss both simple and more subtle connections between the numbers of permutations and full cycles with some restrictions,in particular, between the numbers of permutations and full cycles with prescribed up-down structure."}
{"category": "Math", "title": "Lattice of Triangulations: the proof and an algorithm", "abstract": "In this paper, we prove that the set of triangulations of a polygon can be equipped with an order to become a lattice. First, we define this order. In [HN99], authors defined the flip operator and then prove some properties of the graph of triangulations. We use their theorems and extend them to construct the lattice of triangulations. We prove this lattice property and introduce an elegant algorithm which correctness is induced from the proof. The complexity of this algorithm will be considered. This algorithm is efficient to find the infimum of a pair of triangulations."}
{"category": "Math", "title": "Results about persymmetric matrices over F_2 and related exponential sums", "abstract": "In this paper we expose our main results about rank problems concerning persymmetric matrices over F_2 associated to some exponential sums."}
{"category": "Math", "title": "Standard Character Condition for C-algebras", "abstract": "It is well known that the adjacency algebra of an association scheme has the standard character. In this paper we first define the concept of standard character for C-algebras and we say that a C-algebra has the {\\it standard character condition} if it has the standard character. Then we investigate some properties of C-algebras which have the standard character condition and prove that under some conditions a C-algebra has an adjacency algebra homomorphic image. In particular, we obtain a necessary and sufficient condition for which a commutative table algebra comes from an association scheme."}
{"category": "Math", "title": "Ergodic Theory: Nonsingular Transformations", "abstract": "This survey is a 2022 update of the 2008 version, with recent developments and new references."}
{"category": "Math", "title": "Linearisation of conservative toral homeomorphisms", "abstract": "We give an equivalent characterisation for the existence of a semi-conjugacy to an irrational rotation for conservative homeomorphisms of the two-torus. This leads to an analogue of Poincare's classification of circle homeomorphisms for conservative toral homeomorphisms with unique rotation vector and a certain bounded mean motion property. For minimal toral homeomorphisms, the result extends to arbitrary dimensions. Further, we provide a basic classification for the dynamics of non-wandering toral homeomorphisms homotopic to the identitiy."}
{"category": "Math", "title": "On partita doppia", "abstract": "We give a precise mathematical account of partita doppia in terms of an algebraic structure on the bicategory of spans of reflexive graphs. (The paper was written in 1998.)"}
{"category": "Math", "title": "The Nichols algebra of a semisimple Yetter-Drinfeld module", "abstract": "We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants such as real roots. The crucial ingredient is a `reflection' in the class of such Nichols algebras. We conclude the classifications of finite-dimensional pointed Hopf algebras over S_3, and of finite-dimensional Nichols algebras over S_4. The revised version contains an extended introduction with references to recent applications, and a simplified definition of the Weyl groupoid of a semisimple Yetter-Drinfeld module. Key words: Hopf algebras, quantum groups, Weyl groupoid"}
{"category": "Math", "title": "Acyclic Edge colorings of 2-degenerate graphs", "abstract": "An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \\emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors and is denoted by $a'(G)$. A graph is called 2-$degenerate$ if any of its induced subgraph has a vertex of degree at most 2. The class of 2-$degenerate graphs$ properly conta in $series$-$parallel graphs$, $outerplanar graphs$, \\emph{non-regular subcubic graphs}, \\emph{planar graphs of girth at least 6} and \\emph{circle graphs of girth at least 5} as subclasses. It was conjectur ed by Alon, Sudakov and Zaks (and earlier by Fiamcik) that $a'(G)\\le \\Delta+2$, where $\\Delta =\\Delta(G)$ denotes the maximum deg ree of the graph. We prove the conjecture for 2-$degenerate$ graphs: in fact we prove a stronger bound . We prove that if $G$ is a 2-degenerate graph with maximum degree $\\Delta$, then $a'(G)\\le \\Delta + 1$."}
{"category": "Math", "title": "Global stucture of webs in codimension one", "abstract": "We describe the global structure of holomorphic webs in codimension one, and in particular their singularity (caustic). Various concepts are introduced, which have no interest locally near a regular point, such as the type, the reducibility, the quasi-smoothness, the CI property (complete intersection), the dicriticity... We prove for instance that the algebraicity of a web globally defined on a complex projective space may be readen on its caustic (dicriticity), at least if each irreducible component is CI, and the web quasi-smooth. ."}
{"category": "Math", "title": "On a Szego Type Limit Theorem, the Holder-Young-Brascamp-Lieb Inequality, and the Asymptotic Theory of Integrals and Quadratic Forms of Stationary Fields", "abstract": "Many statistical applications require establishing central limit theorems for sums, integrals, or for quadratic forms of functions of a stationary process. A particularly important case is that of Appell polynomials, since the Appell expansion rank\" determines typically the type of central limit theorem satisfied by these functionals. We review and extend here to multidimensional indices a functional analysis approach to this problem proposed by Avram and Brown (1989), based on the method of cumulants and on integrability assumptions in the spectral domain; several applications are presented as well."}
{"category": "Math", "title": "Global well posedness and inviscid limit for the Korteweg-de Vries-Burgers equation", "abstract": "Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation \\begin{eqnarray*} u_t+u_{xxx}+\\epsilon |\\partial_x|^{2\\alpha}u+(u^2)_x=0, \\ u(0)=\\phi, \\end{eqnarray*} where $0<\\epsilon,\\alpha\\leq 1$ and $u$ is a real-valued function, we show that it is globally well-posed in $H^s\\ (s>s_\\alpha)$, and uniformly globally well-posed in $H^s (s>-3/4)$ for all $\\epsilon \\in (0,1)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the KdV equation if $\\epsilon$ tends to 0."}
{"category": "Math", "title": "Finite Generation of Canonical Ring by Analytic Method", "abstract": "In the 80th birthday conference for Professor LU Qikeng in June 2006 I gave a talk on the analytic approach to the finite generation of the canonical ring for a compact complex algebraic manifold of general type. This article is my contribution to the proceedings of that conference from my talk. In this article I give an overview of the analytic proof and focus on explaining how the analytic method handles the problem of infinite number of interminable blow-ups in the intuitive approach to prove the finite generation of the canonical ring. The proceedings of the LU Qikeng conference will appear as Issue No. 4 of Volume 51 of Science in China Series A: Mathematics (www.springer.com/math/applications/journal/11425)."}
{"category": "Math", "title": "A Duality Exact Sequence for Legendrian Contact Homology", "abstract": "We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact homology CH_* which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_*\\to H_*) and the cokernel of the map (H_* \\to CH^*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a in CH_* maps to a class \\alpha in CH^* such that \\alpha(a)=1. The exact sequence generalizes the duality for Legendrian knots in Euclidean 3-space [24] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [6]."}
{"category": "Math", "title": "Stability of a processor sharing queue with varying throughput", "abstract": "In this paper, we present a stability criterion for Processor Sharing queues, in which the throughput may depend on the number of customers in the system (in such cases such as interferences between the users). Such a system is represented by a point measure-valued stochastic recursion keeping track of the remaining processing times of the customers."}
{"category": "Math", "title": "An entropy preserving finite-element/finite-volume pressure correction scheme for the drift-flux model", "abstract": "We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of a Darcy-like relation, the drift term becomes dissipative. Finally, the present algorithm preserves a constant pressure and a constant velocity through moving interfaces between phases. To ensure the stability as well as to obtain this latter property, a key ingredient is to couple the mass balance and the transport equation for the dispersed phase in an original pressure correction step. The existence of a solution to each step of the algorithm is proven; in particular, the existence of a solution to the pressure correction step is derived as a consequence of a more general existence result for discrete problems associated to the drift-flux model. Numerical tests show a near-first-order convergence rate for the scheme, both in time and space, and confirm its stability."}
{"category": "Math", "title": "Triviality of a trace on the space of commuting trace-class self-adjoint operators", "abstract": "In the present article, we investigate a possibility of a real-valued map on the space of tuples of commuting trace-class self-adjoint operators, which behaves like the usual trace map on the space of trace-class linear operators. It turns out that such maps are related with continuous group homomorphisms from the Milnor's $K$-group of the real numbers into the additive group of real numbers. Using this connection, it is shown that any such trace map must be trivial."}
{"category": "Math", "title": "Twisting the Baum-Connes morphism by a non-unitary representation", "abstract": "Let G be a locally compact group and rho a non-unitary finite dimensional representation of G. We consider tensor products of rho by some unitary representations of G in order to define two Banach algebras analogous to the group C*-algebras, C*(G) and C*_r(G). We calculate the K-theory of such algebras for a large class of groups satisfying the Baum-Connes conjecture."}
{"category": "Math", "title": "Quantile Estimation of A general Single-Index Model", "abstract": "The single-index model is one of the most popular semiparametric models in Econometrics. In this paper, we define a quantile regression single-index model, which includes the single-index structure for conditional mean and for conditional variance."}
{"category": "Math", "title": "Joint differential resolvents for pseudopolynomials", "abstract": "The existence of linear differential resolvents for z^alpha for any root z of an ordinary polynomial with coefficients in a given ordinary differential field has been established, where alpha is an indeterminate constant with respect to the derivation of the given field. In this paper we consider several alphas. We will call a finite sum of indeterminate powers of a variable v a pseudopolynomial in v. We will generalize the definition of a differential resolvent of a single polynomial for a single monomial z^alpha to the definition of a differential resolvent of several polynomials for a pseudopolynomial in the roots. We will also generalize the definition of a resolvent to have non-consecutive derivatives. We will show that the authors powersum formula may be used to compute this more general differential resolvent."}
{"category": "Math", "title": "A Kobayashi metric version of Bun Wong's theorem", "abstract": "We prove that a strongly pseudoconvex domain with noncompact group of Kobayashi/Royden metric isometries must be biholomorphic to the unit ball."}
{"category": "Math", "title": "Uniqueness Results for Nonlocal Hamilton-Jacobi Equations", "abstract": "We are interested in nonlocal Eikonal Equations describing the evolution of interfaces moving with a nonlocal, non monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts."}
{"category": "Math", "title": "Value distribution of cyclotomic polynomial coefficients", "abstract": "Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the integers, a_n(k) assumes only finitely many values. For any such value v we determine the density of integers n such that a_n(k)=v. Also we study the average of the a_n(k). We derive analogous results for the kth Taylor coefficient of 1/Phi_n(x) (taken around x=0), the kth coefficient of the nth reciprocal cyclotomic polynomial. We formulate various open problems."}
{"category": "Math", "title": "Problems on automorphism groups of nonpositively curved polyhedral complexes and their lattices", "abstract": "The goal of this paper is to present a number of problems about automorphism groups of nonpositively curved polyhedral complexes and their lattices, meant to highlight possible directions for future research."}
{"category": "Math", "title": "On the least squares estimator in a nearly unstable sequence of stationary spatial AR models", "abstract": "A nearly unstable sequence of stationary spatial autoregressive processes is investigated, when the sum of the absolute values of the autoregressive coefficients tends to one. It is shown that after an appropriate norming the least squares estimator for these coefficients has a normal limit distribution. If none of the parameters equals zero than the typical rate of convergence is n."}
{"category": "Math", "title": "Instability of Hopf vector fields on Lorentzian Berger spheres", "abstract": "In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. The Hessians of the functionals are negative when they act on these particular vector fields and then Hopf vector fields are unstable. Moreover, we use this technique to study some of the open problems in the Riemannian case."}
{"category": "Math", "title": "A Ruelle Operator for continuous time Markov Chains", "abstract": "We consider a generalization of the Ruelle theorem for the case of continuous time problems. We present a result which we believe is important for future use in problems in Mathematical Physics related to $C^*$-Algebras We consider a finite state set $S$ and a stationary continuous time Markov Chain $X_t$, $t\\geq 0$, taking values on S. We denote by $\\Omega$ the set of paths $w$ taking values on S (the elements $w$ are locally constant with left and right limits and are also right continuous on $t$). We consider an infinitesimal generator $L$ and a stationary vector $p_0$. We denote by $P$ the associated probability on ($\\Omega, {\\cal B}$). This is the a priori probability. All functions $f$ we consider bellow are in the set ${\\cal L}^\\infty (P)$. From the probability $P$ we define a Ruelle operator ${\\cal L}^t, t\\geq 0$, acting on functions $f:\\Omega \\to \\mathbb{R}$ of ${\\cal L}^\\infty (P)$. Given $V:\\Omega \\to \\mathbb{R}$, such that is constant in sets of the form $\\{X_0=c\\}$, we define a modified Ruelle operator $\\tilde{{\\cal L}}_V^t, t\\geq 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability $\\nu_V$ on $\\Omega$ associated to $\\tilde{{\\cal L}}^t_V, t\\geq 0$. We also show the following property for the probability $\\nu_V$: for any integrable $g\\in {\\cal L}^\\infty (P)$ and any real and positive $t$ $$ \\int e^{-\\int_0^t (V \\circ \\Theta_s)(.) ds} [ (\\tilde{{\\cal L}}^t_V (g)) \\circ \\theta_t ] d \\nu_V = \\int g d \\nu_V$$ This equation generalize, for the continuous time Markov Chain, a similar one for discrete time systems (and which is quite important for understanding the KMS states of certain $C^*$-algebras)."}
{"category": "Math", "title": "Semiclassical Asymptotics on Manifolds with Boundary", "abstract": "We discuss semiclassical asymptotics for the eigenvalues of the Witten Laplacian for compact manifolds with boundary in the presence of a general Riemannian metric. To this end, we modify and use the variational method suggested by Kordyukov, Mathai and Shubin (2005), with a more extended use of quadratic forms instead of the operators. We also utilize some important ideas and technical elements from Helffer and Nier (2006), who were the first to supply a complete proof of the full semi-classical asymptotic expansions for the eigenvalues with fixed numbers."}
{"category": "Math", "title": "On the geometry of SL(2)-equivariant flips", "abstract": "In this paper, we show that any 3-dimensional normal affine quasihomogeneous SL(2)-variety can be described as a categorical quotient of a 4-dimensional affine hypersurface. Moreover, we show that the Cox ring of an arbitrary 3-dimensional normal affine quasihomogeneous SL(2)-variety has a unique defining equation. This allows us to construct SL(2)-equivariant flips by different GIT-quotients of hypersurfaces. Using the theory of spherical varieties, we describe SL(2)-flips by means of 2-dimensional colored cones."}
{"category": "Math", "title": "All cyclic p-roots of index 3, found by symmetry-preserving calculations", "abstract": "When using a Groebner basis to solve the highly symmetric system of algebraic equations defining the cyclic p-roots, one has the feeling that much of the advantage of computerized symbolic algebra over hand calculation is lost through the fact that the symmetry is immediately ``thrown out'' by the calculations. In this paper, the problem of finding (for all relevant primes p) all cyclic p-roots of index 3 is treated with the symmetry preserved through the calculations. Once we had found the relevant formulas, using MAPLE and MATHEMATICA, the calculations could even be made by hand. On the other hand, with respect to a straightforward attack with Groebner basis, it is not even clear how this could be organized for a general p. In other terminologies, our results involve listings of all bi-unimodular sequences constant on the cosets of the group G_0 of cubic residues, or equivalently all circulant complex Hadamard matrices related to G_0. The corresponding problem for bi-unimodular sequences of index 2 was solved by the first named author in 1989 and shortly after solved independently by de la Harpe and Jones in the case p = 1 (mod 4) and by Munemasa and Watatani in the case p = 3 (mod 4)."}
{"category": "Math", "title": "Iterated Integrals and higher order automorphic forms", "abstract": "Higher order automorphic forms have recently been introduced to study important questions in number theory and mathematical physics. We investigate the connection between these functions and Chen's iterated integrals. Then using Chen's theory, we prove a structure theorem for automorphic forms of all orders. This allows us to define an analogue of a mixed Hodge structure on a space of higher order automorphic forms."}
{"category": "Math", "title": "When is Group Cohomology Finitary?", "abstract": "If $G$ is a group, then we say that the functor $H^n(G,-)$ is finitary if it commutes with all filtered colimit systems of coefficient modules. We investigate groups with cohomology almost everywhere finitary; that is, groups with $n$th cohomology functors finitary for all sufficiently large $n$. We establish sufficient conditions for a group $G$ possessing a finite dimensional model for $e.g.$ to have cohomology almost everywhere finitary. We also prove a stronger result for the subclass of groups of finite virtual cohomological dimension, and use this to answer a question of Leary and Nucinkis. Finally, we show that if $G$ is a locally (polycyclic-by-finite) group, then $G$ has cohomology almost everywhere finitary if and only if $G$ has finite virtual cohomological dimension and the normalizer of every non-trivial finite subgroup of $G$ is finitely generated."}
{"category": "Math", "title": "Finitary Group Cohomology and Group Actions on Spheres", "abstract": "We show that if G is an infinitely generated locally (polycyclic-by-finite) group with cohomology almost everywhere finitary, then every finite subgroup of G acts freely and orthogonally on some sphere."}
{"category": "Math", "title": "Finitary Group Cohomology and Eilenberg-Mac Lane Spaces", "abstract": "We say that a group G has cohomology almost everywhere finitary if and only if the nth cohomology functors of G commute with filtered colimits for all sufficiently large n. In this paper, we show that if G is a group in Kropholler's class LHF with cohomology almost everywhere finitary, then G has an Eilenberg--Mac Lane space K(G,1) which is dominated by a CW-complex with finitely many n-cells for all sufficiently large n. It is an open question as to whether this holds for arbitrary G. We also remark that the converse holds for any group G."}
{"category": "Math", "title": "Heteroscedastic controlled calibration model applied to analytical chemistry", "abstract": "In chemical analysis made by laboratories one has the problem of determining the concentration of a chemical element in a sample. In order to tackle this problem the guide EURACHEM/CITAC recommends the application of the linear calibration model, so implicitly assume that there is no measurement error in the independent variable $X$. In this work, it is proposed a new calibration model assuming that the independent variable is controlled. This assumption is appropriate in chemical analysis where the process tempting to attain the fixed known value $X$ generates an error and the resulting value is $x$, which is not an observable. However, observations on its surrogate $X$ are available. A simulation study is carried out in order to verify some properties of the estimators derived for the new model and it is also considered the usual calibration model to compare it with the new approach. Three applications are considered to verify the performance of the new approach."}
{"category": "Math", "title": "On the nature of ill-posedness of the forward-backward heat equation", "abstract": "We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by the J-self-adjoint linear operator L depending on a small parameter. The problem has been originated from the lubrication approximation of a viscous fluid film on the inner surface of the rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on the numerical evidence, that the set of eigenvectors of the operator $L$ does not form a Riesz basis in $\\L^2 (-\\pi,\\pi)$. Our method can be applied to a wide range of the evolutional problems given by $PT-$symmetric operators."}
{"category": "Math", "title": "A circle of interacting servers; spontaneous collective behavior in case of large fluctuations", "abstract": "We consider large fluctuations, namely overload of servers, in a network with dynamic routing of messages. The servers form a circle. The number of input flows is equal to the number of servers, the messages of any flow are distributed between two neighboring servers, upon its arrival a message is directed to the least loaded of these servers. Under the condition that at least two servers are overloaded the number of overloaded servers in such network depends on the rate of input flows. In particular there exists critical level of input rate that in case of higher rate most probable that all servers are overloaded."}
{"category": "Math", "title": "The maximum number of perfect matchings in graphs with a given degree sequence", "abstract": "We show that the number of perfect matching in a simple graph $G$ with an even number of vertices and degree sequence $d_1,d_2, ..., d_n$ is at most $\\prod_{i=1}^n (d_i !)^{\\frac{1}{2d_i}}$. This bound is sharp if and only if $G$ is a union of complete balanced bipartite graphs."}
{"category": "Math", "title": "Positive degree and arithmetic bigness", "abstract": "We establish, for a generically big Hermitian line bundle, the convergence of truncated Harder-Narasimhan polygons and the uniform continuity of the limit. As applications, we prove a conjecture of Moriwaki asserting that the arithmetic volume function is actually a limit instead of a sup-limit, and we show how to compute the asymptotic polygon of a Hermitian line bundle, by using the arithmetic volume function."}
{"category": "Math", "title": "Cumulative distribution function estimation under interval censoring case 1", "abstract": "We consider projection methods for the estimation of the cumulative distribution function under interval censoring, case 1. Such censored data also known as current status data, arise when the only information available on the variable of interest is whether it is greater or less than an observed random time. Two types of adaptive estimators are investigated. The first one is a two-step estimator built as a quotient estimator. The second estimator results from a mean square regression contrast. Both estimators are proved to achieve automatically the standard optimal rate associated with the unknown regularity of the function, but with some restriction for the quotient estimator. Simulation experiments are presented to illustrate and compare the methods."}
{"category": "Math", "title": "Categories of Fractions Revisited", "abstract": "The theory of categories of fractions as originally developed by Gabriel and Zisman is reviewed in a pedagogical manner giving detailed proofs of all statements. A weakening of the category of fractions axioms used by Higson is discussed and shown to be equivalent to the original axioms."}
{"category": "Math", "title": "Fixed point properties in the space of marked groups", "abstract": "We explain, following Gromov, how to produce uniform isometric actions of groups starting from isometric actions without fixed point, using common ultralimits techniques. This gives in particular a simple proof of a result by Shalom: Kazhdan's property (T) defines an open subset in the space of marked finitely generated groups."}
{"category": "Math", "title": "Markov Chains Approximations of jump-Diffusion Quantum Trajectories", "abstract": "\"Quantum trajectories\" are solutions of stochastic differential equations also called Belavkin or Stochastic Schr\\\"odinger Equations. They describe random phenomena in quantum measurement theory. Two types of such equations are usually considered, one is driven by a one-dimensional Brownian motion and the other is driven by a counting process. In this article, we present a way to obtain more advanced models which use jump-diffusion stochastic differential equations. Such models come from solutions of martingale problems for infinitesimal generators. These generators are obtained from the limit of generators of classical Markov chains which describe discrete models of quantum trajectories. Furthermore, stochastic models of jump-diffusion equations are physically justified by proving that their solutions can be obtained as the limit of the discrete trajectories."}
{"category": "Math", "title": "Twisted Group Rings Whose Units Form an FC-Group", "abstract": "Let U be the group of units of an infinite twisted group algebra K_\\lambda G over a field K. We describe the maximal FC-subgroup of U and give a characterization of U with finitely conjugacy classes. In the case of group algebras we obtain the Cliff-Sehgal-Zassenhaus' theorem."}
{"category": "Math", "title": "On Extending Pollard's Theorem for t-Representable Sums", "abstract": "Let $t\\geq 1$, let $A$ and $B$ be finite, nonempty subsets of an abelian group $G$, and let $A\\pp{i} B$ denote all the elements $c$ with at least $i$ representations of the form $c=a+b$, with $a\\in A$ and $b\\in B$. For $|A|, |B|\\geq t$, we show that either \\be\\label{almost}\\Sum{i=1}{t}|A\\pp{i} B|\\geq t|A|+t|B|-2t^2+1,\\ee or else there exist $A'\\subseteq A$ and $B'\\subseteq B$ with \\ber \\nn l&:=&|A\\setminus A'|+|B\\setminus B'|\\leq t-1, \\nn A'\\pp{t}B'&=&A'+B'=A\\pp{t}B,{and} \\nn \\Sum{i=1}{t}|A\\pp{i}B|&\\geq& t|A|+t|B|-(t-l)(|H|-\\rho)-tl\\geq t|A|+t|B|-t|H|,\\eer where $H$ is the (nontrivial) stabilizer of $A\\pp{t} B$ and $\\rho=|A'+H|-|A'|+|B'+H|-|B'|$. In the case $t=2$, we improve (\\ref{almost}) to $|A\\pp{1}B|+|A\\pp{2}B|\\geq 2|A|+2|B|-4$."}
{"category": "Math", "title": "The canonical fractional Galois ideal at s=0", "abstract": "The Stickelberger elements attached to an abelian extension of number fields conjecturally participate, under certain conditions, in annihilator relations involving higher algebraic K-groups. In [Victor P. Snaith, Stark's conjecture and new Stickelberger phenomena, Canad. J. Math. 58 (2) (2006) 419--448], Snaith introduces canonical Galois modules hoped to appear in annihilator relations generalising and improving those involving Stickelberger elements. In this paper we study the first of these modules, corresponding to the classical Stickelberger element, and prove a connection with the Stark units in a special case."}
{"category": "Math", "title": "Typical Dispersion and Generalized Lyapunov Exponents", "abstract": "Let f(n) denote the number of odd entries in the nth row of Pascal's binomial triangle. We study \"average dispersion\" and \"typical dispersion\" of f(n) -- the latter involves computing a generalized Lyapunov exponent -- and then turn to numerical analysis of higher dimensional examples."}
{"category": "Math", "title": "Morphism of T*-Representations", "abstract": "Importance of theorem dedicated to isomorphisms consist in statement that they allow to identify different mathematical objects which have something common from the point of view of certain model. This paper considers morphisms of \\Ts representation of $\\mathfrak{F}$\\Hyph algebra and morphisms of \\Ts representation of fibered $\\mathfrak{F}$\\Hyph algebra."}
{"category": "Math", "title": "Isometric Immersions of Hypersurfaces in 4-dimensional Manifolds via Spinors", "abstract": "We give a spinorial characterization of isometrically immersed hypersurfaces into 4-dimensional space forms and product spaces $\\M^3(\\kappa)\\times\\R$, in terms of the existence of particular spinor fields, called generalized Killing spinors or equivalently solutions of a Dirac equation. This generalizes to higher dimensions several recent results for surfaces by T. Friedrich, B. Morel and the two authors. The main argument is the interpretation of the energy-momentum tensor of a generalized Killing spinor as the second fundamental form, possibly up to a tensor depending on the ambient space. As an application, we deduce a non-existence result for hypersurfaces in the 4-dimensional Euclidean space."}
{"category": "Math", "title": "Deconvolution of confocal microscopy images using proximal iteration and sparse representations", "abstract": "We propose a deconvolution algorithm for images blurred and degraded by a Poisson noise. The algorithm uses a fast proximal backward-forward splitting iteration. This iteration minimizes an energy which combines a \\textit{non-linear} data fidelity term, adapted to Poisson noise, and a non-smooth sparsity-promoting regularization (e.g $\\ell_1$-norm) over the image representation coefficients in some dictionary of transforms (e.g. wavelets, curvelets). Our results on simulated microscopy images of neurons and cells are confronted to some state-of-the-art algorithms. They show that our approach is very competitive, and as expected, the importance of the non-linearity due to Poisson noise is more salient at low and medium intensities. Finally an experiment on real fluorescent confocal microscopy data is reported."}
{"category": "Math", "title": "A proximal iteration for deconvolving Poisson noisy images using sparse representations", "abstract": "We propose an image deconvolution algorithm when the data is contaminated by Poisson noise. The image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms. Our key contributions are: First, we handle the Poisson noise properly by using the Anscombe variance stabilizing transform leading to a {\\it non-linear} degradation equation with additive Gaussian noise. Second, the deconvolution problem is formulated as the minimization of a convex functional with a data-fidelity term reflecting the noise properties, and a non-smooth sparsity-promoting penalties over the image representation coefficients (e.g. $\\ell_1$-norm). Third, a fast iterative backward-forward splitting algorithm is proposed to solve the minimization problem. We derive existence and uniqueness conditions of the solution, and establish convergence of the iterative algorithm. Finally, a GCV-based model selection procedure is proposed to objectively select the regularization parameter. Experimental results are carried out to show the striking benefits gained from taking into account the Poisson statistics of the noise. These results also suggest that using sparse-domain regularization may be tractable in many deconvolution applications with Poisson noise such as astronomy and microscopy."}
{"category": "Math", "title": "Cyclic p-roots of prime lengths p and related complex Hadamard matrices", "abstract": "In this paper it is proved, that for every prime number p, the set of cyclic p-roots in C^p is finite. Moreover the number of cyclic p-roots counted with multiplicity is equal to (2p-2)!/(p-1)!^2. In particular, the number of complex circulant Hadamard matrices of size p, with diagonal entries equal to 1, is less or equal to (2p-2)!/(p-1)!^2."}
{"category": "Math", "title": "Global well posedness and scattering for the elliptic and non-elliptic derivative nonlinear Schrodinger equations with small data", "abstract": "We study the Cauchy problem for the generalized elliptic and non-elliptic derivative nonlinear Schrodinger equations, the existence of the scattering operators and the global well posedness of solutions with small data in Besov spaces and in modulation spaces are obtained. In one spatial dimension, we get the sharp well posedness result with small data in critical homogeneous Besov spaces. As a by-product, the existence of the scattering operators with small data is also shown. In order to show these results, the global versions of the estimates for the maximal functions on the elliptic and non-elliptic Schrodinger groups are established."}
{"category": "Math", "title": "Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers", "abstract": "In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume $E_2$-values; i.e., values that are products of exactly two primes. We use that result to prove that there are inifinitely many integers $x$ that simultaneously satisfy $$\\omega(x)=\\omega(x+1)=4, \\Omega(x)=\\Omega(x+1)=5, \\text{and} d(x)=d(x+1)=24.$$ Here, $\\omega(x), \\Omega(x), d(x)$ represent the number of prime divisors of $x$, the number of prime power divisors of $x$, and the number of divisors of $x$, respectively. We also prove similar theorems where $x+1$ is replaced by $x+b$ for an arbitrary positive integer $b$. Our results sharpen earlier work of Heath-Brown, Pinner, and Schlage-Puchta."}
{"category": "Math", "title": "Aubry sets vs Mather sets in two degrees of freedom", "abstract": "We study autonomous Tonelli Lagrangians on closed surfaces. We aim to clarify the relationship between the Aubry set and the Mather set, when the latter consists of periodic orbits which are not fixed points. Our main result says that in that case the Aubry set and the Mather set almost always coincide."}
{"category": "Math", "title": "Asymptotic behavior and hypercontractivity in nonautonomous Ornstein-Uhlenbeck equations", "abstract": "In this paper we investigate a class of nonautonomous linear parabolic problems with time depending Ornstein-Uhlenbeck operators. We study the asymptotic behavior of the associated evolution operator and evolution semigroup in the periodic and non-periodic situation. Moreover, we show that the associated evolution operator is hypercontractive."}
{"category": "Math", "title": "Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps", "abstract": "We prove existence of finitely many ergodic equilibrium states for a large class of non-uniformly expanding local homeomorphisms on compact manifolds and Holder continuous potentials with not very large oscillation. No Markov structure is assumed. If the transformation is topologically mixing there is a unique equilibrium state, it is exact and satisfies a non-uniform Gibbs property. Under mild additional assumptions we also prove that the equilibrium states vary continuously with the dynamics and the potentials (statistical stability) and are also stable under stochastic perturbations of the transformation."}
{"category": "Math", "title": "A New Central Limit Theorem under Sublinear Expectations", "abstract": "We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that mean-uncertainty can be also described. W present our new result of central limit theorem under sublinear expectation. This theorem can be also regarded as a generalization of the law of large number in the case of mean-uncertainty."}
{"category": "Math", "title": "On the cohomology of Young modules for the symmetric group", "abstract": "The main result of this paper is an application of the topology of the space $Q(X)$ to obtain results for the cohomology of the symmetric group on $d$ letters, $\\Sigma_d$, with `twisted' coefficients in various choices of Young modules and to show that these computations reduce to certain natural questions in representation theory. The authors extend classical methods for analyzing the homology of certain spaces $Q(X)$ with mod-$p$ coefficients to describe the homology $\\HH_{\\bullet}(\\Sigma_d, V^{\\otimes d})$ as a module for the general linear group $GL(V)$ over an algebraically closed field $k$ of characteristic $p$. As a direct application, these results provide a method of reducing the computation of $\\text{Ext}^{\\bullet}_{\\Sigma_{d}}(Y^{\\lambda},Y^{\\mu})$ (where $Y^{\\lambda}$, $Y^{\\mu}$ are Young modules) to a representation theoretic problem involving the determination of tensor products and decomposition numbers. In particular, in characteristic two, for many $d$, a complete determination of $\\Hs Y^\\lambda)$ can be found. This is the first nontrivial class of symmetric group modules where a complete description of the cohomology in all degrees can be given. For arbitrary $d$ the authors determine $\\HH^i(\\Sigma_d,Y^\\lambda)$ for $i=0,1,2$. An interesting phenomenon is uncovered--namely a stability result reminiscent of generic cohomology for algebraic groups. For each $i$ the cohomology $\\HH^i(\\Sigma_{p^ad}, Y^{p^a\\lambda})$ stabilizes as $a$ increases. The methods in this paper are also powerful enough to determine, for any $p$ and $\\lambda$, precisely when $\\HH^{\\bullet}(\\sd,Y^\\lambda)=0$. Such modules with vanishing cohomology are of great interest in representation theory because their support varieties constitute the representation theoretic nucleus."}
{"category": "Math", "title": "Statistical aspects of birth--and--growth stochastic processes", "abstract": "The paper considers a particular family of set--valued stochastic processes modeling birth--and--growth processes. The proposed setting allows us to investigate the nucleation and the growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set--valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived."}
{"category": "Math", "title": "Quotients of the Multiplihedron as Categorified Associahedra", "abstract": "We describe a new sequence of polytopes which characterize A_infinity maps from a topological monoid to an A_infinity space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Later term(s) in our sequence of polytopes are demonstrated not to be combinatorially equivalent to the associahedron, as was previously assumed. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence of composihedra, that is, the nth composihedron."}
{"category": "Math", "title": "A property of dominance of partitions", "abstract": "Given an integer partition $\\la=(\\la_1, ..., \\la_\\ell)$ and an integer k, denote by $\\la^{(k)}$ the sequence of length $\\ell$ obtained by reordering the values $|\\la_i-k|$ in non-increasing order. If $\\la$ dominates $\\mu$ and has the same weight, then $\\la^{(k)}$ dominates $\\mu^{(k)}$."}
{"category": "Math", "title": "Sortable elements in infinite Coxeter groups", "abstract": "In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection \\pidown^c mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of \\pidown^c essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of \\pidown^c. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans."}
{"category": "Math", "title": "Special Lagrangian fibrations, mirror symmetry and Calabi-Yau double covers", "abstract": "The first part of this paper is a review of the Strominger-Yau-Zaslow conjecture in various settings. In particular, we summarize how, given a pair (X,D) consisting of a Kahler manifold and an anticanonical divisor, families of special Lagrangian tori in X-D and weighted counts of holomorphic discs in X can be used to build a Landau-Ginzburg model mirror to X. In the second part we turn to more speculative considerations about Calabi-Yau manifolds with holomorphic involutions and their quotients. Namely, given a hypersurface H representing twice the anticanonical class in a Kahler manifold X, we attempt to relate special Lagrangian fibrations on X-H and on the (Calabi-Yau) double cover of X branched along H; unfortunately, the implications for mirror symmetry are far from clear."}
{"category": "Math", "title": "Integration of e^(x^n) and e^(-x^n) in forms of series, their applications in the field of differential equation; introducing generalized form of Skewness and Kurtosis; extension of starling's approximation", "abstract": "In this paper we tried a different approach to work out the integrals of e^(x^n) and e^(-x^n). Integration by parts shows a nice pattern which can be reduced to a form of series. We have shown both the indefinite and definite integrals of the functions mentioned along with some essential properties e.g. conditions of convergence of the series. Further more, we used the integrals in form of series to find out series solution of differential equations of the form x[(d^2 y)/(dx^2)]-(n-1)(dy/dx)-n^2 x^(2n-1)y-nx^n=0 and x[(d^2 y)/(dx^2)] -(n-1)(dy/dx)-n^2x^(2n-1)y+(n-1)=0, using some non standard method. We introduced modified Normal distribution incorporating some properties derived from the above integrals and defined a generalized version of Skewness and Kurtosis. Finally we extended Starling's approximation to limit [n to infinity ] (2n)! ~ 2n * \\sqrt{(2\\pi)} [(2n/e)]^(2n)."}
{"category": "Math", "title": "Higher topological cyclic homology and the Segal conjecture for tori", "abstract": "We investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory. Higher topological cyclic homology is constructed from the fixed points of a version of topological Hochschild homology based on the n-dimensional torus, and we propose it as a computationally tractable cousin of n-fold iterated algebraic K-theory. The fixed points of toral topological Hochschild homology are related to one another by restriction and Frobenius operators. We introduce two additional families of operators on fixed points, the Verschiebung, indexed on self-isogenies of the n-torus, and the differentials, indexed on n-vectors. We give a detailed analysis of the relations among the restriction, Frobenius, Verschiebung, and differentials, producing a higher analog of the structure Hesselholt and Madsen described for 1-dimensional topological cyclic homology. We calculate two important pieces of higher topological cyclic homology, namely topological restriction homology and topological Frobenius homology, for the sphere spectrum. The latter computation allows us to establish the Segal conjecture for the torus, which is to say to completely compute the cohomotopy type of the classifying space of the torus."}
{"category": "Math", "title": "Vector spaces as unions of proper subspaces", "abstract": "In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of partitioning V into subspaces."}
{"category": "Math", "title": "Quasitoric manifolds over a product of simplices", "abstract": "A quasitoric manifold (resp. a small cover) is a $2n$-dimensional (resp. an $n$-dimensional) smooth closed manifold with an effective locally standard action of $(S^1)^n$ (resp. $(\\mathbb Z_2)^n$) whose orbit space is combinatorially an $n$-dimensional simple convex polytope $P$. In this paper we study them when $P$ is a product of simplices. A generalized Bott tower over $\\F$, where $\\F=\\C$ or $\\R$, is a sequence of projective bundles of the Whitney sum of $\\F$-line bundles starting with a point. Each stage of the tower over $\\F$, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when $\\F=\\C$) and small covers (when $\\F=\\R$) over a product of simplices. It turns out that every small cover over a product of simplices is equivalent (in the sense of Davis and Januszkiewicz \\cite{DJ}) to a generalized Bott manifold. But this is not the case for quasitoric manifolds and we show that a quasitoric manifold over a product of simplices is equivalent to a generalized Bott manifold if and only if it admits an almost complex structure left invariant under the action. Finally, we show that a quasitoric manifold $M$ over a product of simplices is homeomorphic to a generalized Bott manifold if $M$ has the same cohomology ring as a product of complex projective spaces with $\\Q$ coefficients."}
{"category": "Math", "title": "Hyperbolic volume and Heegaard distance", "abstract": "We prove (Theorem~1.5) that there exists a constant $\\Lambda > 0$ so that if $M$ is a $(\\mu,d)$-generic complete hyperbolic 3-manifold of volume $\\vol[M] < \\infty$ and $\\Sigma \\subset M$ is a Heegaard surface of genus $g(\\Sigma) > \\Lambda \\vol[M]$, then $d(\\Sigma) \\leq 2$, where $d(\\Sigma)$ denotes the distance of $\\Sigma$ as defined by Hempel. The key for the proof of the main result is Theorem~1.8 which is on independent interest. There we prove that if $M$ is a compact 3-manifold that can be triangulated using at most $t$ tetrahedra (possibly with missing or truncated vertices), and $\\Sigma$ is a Heegaard surface for $M$ with $g(\\Sigma) \\geq 76t+26$, then $d(\\Sigma) \\leq 2$."}
{"category": "Math", "title": "Periodicity of non-central integral arrangements modulo positive integers", "abstract": "An integral coefficient matrix determines an integral arrangement of hyperplanes in R^m. After modulo q reduction, the same matrix determines an arrangement A_q of \"hyperplanes\" in Z^m. In the special case of central arrangements, Kamiya, Takemura and Terao [J. Algebraic Combin., to appear] showed that the cardinality of the complement of A_q in Z_q^m is a quasi-polynomial in q. Moreover, they proved in the central case that the intersection lattice of A_q is periodic from some q on. The present paper generalizes these results to the case of non-central arrangements. The paper also studies the arrangement B_m^{[0,a]} of Athanasiadis [J. Algebraic Combin. Vol.10 (1999), 207-225] to illustrate our results."}
{"category": "Math", "title": "Uniform value in Dynamic Programming", "abstract": "We consider dynamic programming problems with a large time horizon, and give sufficient conditions for the existence of the uniform value. As a consequence, we obtain an existence result when the state space is precompact, payoffs are uniformly continuous and the transition correspondence is non expansive. In the same spirit, we give an existence result for the limit value. We also apply our results to Markov decision processes and obtain a few generalizations of existing results."}
{"category": "Math", "title": "(l,k)-Routing on Plane Grids", "abstract": "The packet routing problem plays an essential role in communication networks. It involves how to transfer data from some origins to some destinations within a reasonable amount of time. In the $(\\ell,k)$-routing problem, each node can send at most $\\ell$ packets and receive at most $k$ packets. Permutation routing is the particular case $\\ell=k=1$. In the $r$-central routing problem, all nodes at distance at most $r$ from a fixed node $v$ want to send a packet to $v$. In this article we study the permutation routing, the $r$-central routing and the general $(\\ell,k)$-routing problems on plane grids, that is square grids, triangular grids and hexagonal grids. We use the \\emph{store-and-forward} $\\Delta$-port model, and we consider both full and half-duplex networks. We first survey the existing results in the literature about packet routing, with special emphasis on $(\\ell,k)$-routing on plane grids. Our main contributions are the following: 1. Tight permutation routing algorithms on full-duplex hexagonal grids, and half duplex triangular and hexagonal grids. 2. Tight $r$-central routing algorithms on triangular and hexagonal grids. 3. Tight $(k,k)$-routing algorithms on square, triangular and hexagonal grids. 4. Good approximation algorithms (in terms of running time) for $(\\ell,k)$-routing on square, triangular and hexagonal grids, together with new lower bounds on the running time of any algorithm using shortest path routing. These algorithms are all completely distributed, i.e., can be implemented independently at each node. Finally, we also formulate the $(\\ell,k)$-routing problem as a \\textsc{Weighted Edge Coloring} problem on bipartite graphs."}
{"category": "Math", "title": "An update on semisimple quantum cohomology and F-manifolds", "abstract": "In the first section of this note we show that the Theorem 1.8.1 of Bayer--Manin ([BaMa]) can be strengthened in the following way: {\\it if the even quantum cohomology of a projective algebraic manifold $V$ is generically semi--simple, then $V$ has no odd cohomology and is of Hodge--Tate type.} In particular, this addressess a question in [Ci]. In the second section, we prove that {\\it an analytic (or formal) supermanifold $M$ with a given supercommutative associative $\\Cal{O}_M$--bilinear multiplication on its tangent sheaf $\\Cal{T}_M$ is an $F$--manifold in the sense of [HeMa], iff its spectral cover as an analytic subspace of the cotangent bundle $T^*_M$ is coisotropic of maximal dimension.} This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau--Ginzburg models for Fano varieties."}
{"category": "Math", "title": "Hausdorff property of the Neron models of Green, Griffiths and Kerr", "abstract": "We prove the Hausdorff property of the Neron modle of the family of intermediate Jacobians which is recently defined by Green, Griffiths and Kerr assuming that the divisor at infinity is smooth. Using their result, this implies in this case the analyticity of the closure of the zero locus of an admissible normal function. The last assertion is also obtained by Brosnan and Pearlstein generalizing their method in the curve case."}
{"category": "Math", "title": "An explicit construction of a maximal relative symplectic packing of the Clifford torus", "abstract": "In this paper we present an explicit construction of a relative symplectic packing. This confirms the sharpness of the upper bound for the relative packing of a ball into the pair (CP^2, T^2) of the standard complex projective plane and the Clifford torus, obtained by Biran and Cornea."}
{"category": "Math", "title": "q-Pascal's triangle and irreducible representations of the braid group B_3 in arbitrary dimension", "abstract": "We construct a [(n+1)/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension n\\in N, using a q-deformation of the Pascal triangle. This construction extends in particular results by S.P.Humphries [8], who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E.Ferrand [7] obtained an equivalent representation of B_3 by considering two special operators in the space C^n[X]. Slightly more general representations were given by I.Tuba and H.Wenzl [11]. They involve [(n+1)/2] parameters (and also use the classical Pascal triangle). The latter authors also gave the complete classification of all simple representations of B_3 for dimension n\\leq 5. Our construction generalize all mentioned results and throws a new light on some of them. We also study the irreducibility and the equivalence of the representations. In [17] we establish the connection between the constructed representation of the braid group B_3 and the highest weight modules of U(sl_2) and quantum group U_q(sl_2)."}
{"category": "Math", "title": "Functoriality of the canonical fractional Galois ideal", "abstract": "The fractional Galois ideal of [Victor P. Snaith, Stark's conjecture and new Stickelberger phenomena, Canad. J. Math. 58 (2) (2006) 419--448] is a conjectural improvement on the higher Stickelberger ideals defined at negative integers, and is expected to provide non-trivial annihilators for higher K-groups of rings of integers of number fields. In this article, we extend the definition of the fractional Galois ideal to arbitrary (possibly infinite and non-abelian) Galois extensions of number fields under the assumption of Stark's conjectures, and prove naturality properties under canonical changes of extension. We discuss applications of this to the construction of ideals in non-commutative Iwasawa algebras."}
{"category": "Math", "title": "Representations of the braid group B_n and the highest weight modules of U(sl_{n-1}) and U_q(sl_{n-1})", "abstract": "In [1] we have constructed a [n+1/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension using a $q-$deformation of the Pascal triangle. This construction extends in particular results by S.P. Humphries (2000), who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E. Ferrand (2000) obtained an equivalent representation of B_3 by considering two special operators in the space ${\\mathbb C}^n[X].$ Slightly more general representations were given by I. Tuba and H. Wenzl (2001). They involve [n+1/2] parameters (and also use the classical Pascal's triangle). The latter authors also gave the complete classification of all simple representations of $B_3$ for dimension $n\\leq 5$. Our construction generalize all mentioned results and throws a new light on some of them. We also study the irreducibility and equivalence of the constructed representations. In the present article we show that all representations constructed in [1] may be obtained by taking exponent of the highest weight modules of U(sl}_2 and U_q(sl_2). We generalize these connections between the representation of the braid group $B_n$ and the highest weight modules of the U_q(sl_{n-1}) for arbitrary} n using the well-known reduced Burau representation."}
{"category": "Math", "title": "Multiplication for solutions of the equation $\\grad{f} = M\\grad{g}$", "abstract": "Linear first order systems of partial differential equations of the form $\\nabla f = M\\nabla g,$ where $M$ is a constant matrix, are studied on vector spaces over the fields of real and complex numbers, respectively. The Cauchy--Riemann equations belong to this class. We introduce a bilinear $*$-multiplication on the solution space, which plays the role of a nonlinear superposition principle, that allows for algebraic construction of new solutions from known solutions. The gradient equations $\\nabla f = M\\nabla g$ constitute only a simple special case of a much larger class of systems of partial differential equations which admit a bilinear multiplication on the solution space, but we prove that any gradient equation has the exceptional property that the general analytic solution can be expressed through power series of certain simple solutions, with respect to the $*$-multiplication."}
{"category": "Math", "title": "Lie algebras of smooth sections", "abstract": "Lie algebras of smooth sections are Lie algebras obtained from bundles of Lie algebras, where the latter are vector bundles of which the fibers are Lie algebras. We also consider the $\\operatorname{C}^k$-sections for $k \\in \\mathbb{N}$. This paper studies the derivations, the centroid and the isomorphisms of such Lie algebras and generalizes some facts from Pierre Lecomte's publications in 1979 and 1980 to the case where the fiber is perfect or centerfree and it gives some more explicit proofs."}
{"category": "Math", "title": "An Identity of Andrews and a New Method for the Riordan Array Proof of Combinatorial Identities", "abstract": "We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical arguments. We present a new proof, quite simple and based on a Riordan array argument. The main point of the proof is the construction of a new Riordan array from a given Riordan array, by the elimination of elements. We extend the method and as an application we obtain other identities, some of which are new. An important feature of our construction is that it establishes a nice connection between the generating function of the $A-$sequence of a certain class of Riordan arrays and hypergeometric functions."}
{"category": "Math", "title": "On some congruence properties of elliptic curves", "abstract": "In this paper, as a result of a theorem of Serre on congruence properties, a complete solution is given for an open question (see the text) presented recently by Kim, Koo and Park. Some further questions and results on similar types of congruence properties of elliptic curves are also presented and discussed."}
{"category": "Math", "title": "Paradan's wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator", "abstract": "We give an elementary algebraic proof of Paradan's wall crossing formulae for partition functions. We also express such jumps in volume and partition functions by one dimensional residue formulae. Subsequently we reprove the relation between them as given by the application of a generalized Khovanskii-Pukhlikov differential operator."}
{"category": "Math", "title": "Traffic Grooming in Unidirectional WDM Rings with Bounded Degree Request Graph", "abstract": "Traffic grooming is a major issue in optical networks. It refers to grouping low rate signals into higher speed streams, in order to reduce the equipment cost. In SONET WDM networks, this cost is mostly given by the number of electronic terminations, namely ADMs. We consider the case when the topology is a unidirectional ring. In graph-theoretical terms, the traffic grooming problem in this case consists in partitioning the edges of a request graph into subgraphs with a maximum number of edges, while minimizing the total number of vertices of the decomposition. We consider the case when the request graph has bounded maximum degree $\\Delta$, and our aim is to design a network being able to support any request graph satisfying the degree constraints. The existing theoretical models in the literature are much more rigid, and do not allow such adaptability. We formalize the problem, and solve the cases $\\Delta=2$ (for all values of $C$) and $\\Delta = 3$ (except the case C=4). We also provide lower and upper bounds for the general case."}
{"category": "Math", "title": "Mean-periodicity and zeta functions", "abstract": "This paper establishes new bridges between number theory and modern harmonic analysis, namely between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the zeta function of arithmetic scheme with its expected analytic shape is shown to correspond to mean-periodicity of a certain explicitly defined function associated to the zeta function. This correspondence can be viewed as an extension of the Hecke--Weil correspondence. The case of elliptic curves over number fields and their regular models is treated in more details, and many other examples are included as well."}
{"category": "Math", "title": "On Monge-Kantorovich Problem in the Plane", "abstract": "We transfer the celebrating Monge-Kontorovich problem in a bounded domain of Euclidean plane into a Dirichlet boundary problem associated to a quasi-linear elliptic equation with $0-$order term missing in its diffusion coefficients: \\begin{eqnarray*} A(x, F'_x)F''_{xx}+B(y, F'_y)F''_{yy}&=&C(x, y, F'_x, F'_y) \\end{eqnarray*} where $A(.,.)>0, B(.,.)>0$ and $C$ are functions based on the initial distributions, $F$ is an unknown probability distribution function and therefore closed the former problem."}
{"category": "Math", "title": "A prime sensitive Hankel determinant of Jacobi symbol enumerators", "abstract": "We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes iff n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p-1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of -1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums and orthogonality of trigonometric functions."}
{"category": "Math", "title": "Rational 6-cycles under iteration of quadratic polynomials", "abstract": "We present a proof, which is conditional on the Birch and Swinnerton-Dyer Conjecture for a specific abelian variety, that there do not exist rational numbers x and c such that x has exact period N = 6 under the iteration x |-> x^2 + c. This extends earlier results by Morton for N = 4 and by Flynn, Poonen and Schaefer for N = 5."}
{"category": "Math", "title": "Aggregation by exponential weighting, sharp PAC-Bayesian bounds and sparsity", "abstract": "We study the problem of aggregation under the squared loss in the model of regression with deterministic design. We obtain sharp PAC-Bayesian risk bounds for aggregates defined via exponential weights, under general assumptions on the distribution of errors and on the functions to aggregate. We then apply these results to derive sparsity oracle inequalities."}
{"category": "Math", "title": "Auslander-Reiten theory revisited", "abstract": "We recall several results in Auslander-Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce $n$-cluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained."}
{"category": "Math", "title": "Self-repelling random walk with directed edges on Z", "abstract": "We consider a variant of self-repelling random walk on the integer lattice Z where the self-repellence is defined in terms of the local time on oriented edges. The long-time asymptotic scaling of this walk is surprisingly different from the asymptotics of the similar process with self-repellence defined in terms of local time on unoriented edges. We prove limit theorems for the local time process and for the position of the random walker. The main ingredient is a Ray-Knight-type of approach. At the end of the paper, we also present some computer simulations which show the strange scaling behaviour of the walk considered."}
{"category": "Math", "title": "Note on constancy of some formal meromorphic maps", "abstract": "Segre sets are superfluous to verify that quotients of formal holomorphic maps that are real on a minimal generic submanifold of positive CR dimension must necessarily be constant."}
{"category": "Math", "title": "A mass for ALF manifolds", "abstract": "We prove positive mass theorems on ALF manifolds, i.e. complete noncompact manifolds that are asymptotic to a circle fibration over a Euclidean base, with fibers of asymptotically constant length."}
{"category": "Math", "title": "Proofs of two conjectures on ternary weakly regular bent functions", "abstract": "We study ternary monomial functions of the form $f(x)=\\Tr_n(ax^d)$, where $x\\in \\Ff_{3^n}$ and $\\Tr_n: \\Ff_{3^n}\\to \\Ff_3$ is the absolute trace function. Using a lemma of Hou \\cite{hou}, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising from \\cite{hk1} are weakly regular bent, settling a conjecture of Helleseth and Kholosha \\cite{hk1}. We also prove that the Coulter-Matthews bent functions are weakly regular."}
{"category": "Math", "title": "Validation of a Model of the Domino Effect?", "abstract": "A recent paper proposing a model of the limiting speed of the domino effect is discussed with reference to its need and the need of models in general for validation against experimental data. It is shown that the proposed model diverges significantly from experimentally derived speed estimates over a significant range of domino spacing using data from the existing literature and this author's own measurements, hence if its use had had economic importance its use outside its range of validity could have led to loses of one sort or another to its users."}
{"category": "Math", "title": "Eigenvalues of the Derangement Graph", "abstract": "We consider the Cayley graph on the symmetric group Sn generated by derangements. It is well known that the eigenvalues of this grpah are indexed by partitions of n. We investigate how these eigenvalues are determined by the shape of their corresponding partitions. In particular, we show that the sign of an eigenvalue is the parity of the number of cells below the first row of the corresponding Ferrers diagram. We also provide some lower and upper bounds for the absolute values of these eigenvalues."}
{"category": "Math", "title": "Asymptotics of coefficients of multivariate generating functions: improvements for smooth points", "abstract": "Let $\\sum_{\\beta\\in\\nats^d} F_\\beta x^\\beta$ be a multivariate power series. For example $\\sum F_\\beta x^\\beta$ could be a generating function for a combinatorial class. Assume that in a neighbourhood of the origin this series represents a nonentire function $F=G/H^p$ where $G$ and $H$ are holomorphic and $p$ is a positive integer. Given a direction $\\alpha\\in\\pnats^d$ for which the asymptotics are controlled by a smooth point of the singular variety $H = 0$, we compute the asymptotics of $F_{n \\alpha}$ as $n\\to\\infty$. We do this via multivariate singularity analysis and give an explicit formula for the full asymptotic expansion. This improves on earlier work of R. Pemantle and the second author and allows for more accurate numerical approximation, as demonstrated by our examples."}
{"category": "Math", "title": "Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies", "abstract": "In this paper, we study a two-parameter family of two-dimensional diffeomorphisms such that it has a cubic homoclinic tangency unfolding generically which is associated with a dissipative saddle point. Our first theorem presents an open set in the parameter-plane such that, for any parameter value in the open set, there exists a one-parameter subfamily through this value exhibiting cubically related persistent contact-making and contact-breaking quadratic tangencies. Moreover, the second theorem shows that any such two-parameter family satisfies Wang-Young's conditions which guarantee that it exhibits a cubic polynomial-like strange attractor with an SRB measure."}
{"category": "Math", "title": "Mass Transportation on Sub-Riemannian Manifolds", "abstract": "We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a weak form of the Monge-Amp\\`ere equation."}
{"category": "Math", "title": "Inductive Algebras for Finite Heisenberg Groups", "abstract": "A characterization of the maximal abelian sub-algebras of matrix algebras that are normalized by the canonical representation of a finite Heisenberg group is given. Examples are constructed using a classification result for finite Heisenberg groups."}
{"category": "Math", "title": "Eigenfunctions of the Laplace-Beltrami Operator on Hyperboloids", "abstract": "Eigenfunctions of the Laplace-Beltrami operator on a hyperboloid are studied in the spirit of the treatment of the spherical harmonics by Stein and Weiss. As a special case, a simple self-contained proof of Laplace's integral for a Legendre function is obtained."}
{"category": "Math", "title": "Extensions of smoothing via taut strings", "abstract": "Suppose that we observe independent random pairs $(X_1,Y_1)$, $(X_2,Y_2)$, >..., $(X_n,Y_n)$. Our goal is to estimate regression functions such as the conditional mean or $\\beta$--quantile of $Y$ given $X$, where $0<\\beta <1$. In order to achieve this we minimize criteria such as, for instance, $$ \\sum_{i=1}^n \\rho(f(X_i) - Y_i) + \\lambda \\cdot \\mathop TV\\nolimits (f) $$ among all candidate functions $f$. Here $\\rho$ is some convex function depending on the particular regression function we have in mind, $\\mathop {\\rm TV}\\nolimits (f)$ stands for the total variation of $f$, and $\\lambda >0$ is some tuning parameter. This framework is extended further to include binary or Poisson regression, and to include localized total variation penalties. The latter are needed to construct estimators adapting to inhomogeneous smoothness of $f$. For the general framework we develop noniterative algorithms for the solution of the minimization problems which are closely related to the taut string algorithm (cf. Davies and Kovac, 2001). Further we establish a connection between the present setting and monotone regression, extending previous work by Mammen and van de Geer (1997). The algorithmic considerations and numerical examples are complemented by two consistency results."}
{"category": "Math", "title": "Stepup procedures controlling generalized FWER and generalized FDR", "abstract": "In many applications of multiple hypothesis testing where more than one false rejection can be tolerated, procedures controlling error rates measuring at least $k$ false rejections, instead of at least one, for some fixed $k\\ge 1$ can potentially increase the ability of a procedure to detect false null hypotheses. The $k$-FWER, a generalized version of the usual familywise error rate (FWER), is such an error rate that has recently been introduced in the literature and procedures controlling it have been proposed. A further generalization of a result on the $k$-FWER is provided in this article. In addition, an alternative and less conservative notion of error rate, the $k$-FDR, is introduced in the same spirit as the $k$-FWER by generalizing the usual false discovery rate (FDR). A $k$-FWER procedure is constructed given any set of increasing constants by utilizing the $k$th order joint null distributions of the $p$-values without assuming any specific form of dependence among all the $p$-values. Procedures controlling the $k$-FDR are also developed by using the $k$th order joint null distributions of the $p$-values, first assuming that the sets of null and nonnull $p$-values are mutually independent or they are jointly positively dependent in the sense of being multivariate totally positive of order two (MTP$_2$) and then discarding that assumption about the overall dependence among the $p$-values."}
{"category": "Math", "title": "A non-commutative Sobolev estimate and its application to spectral synthesis", "abstract": "In [M. K. Vemuri, Realizations of the canonical representation], it was shown that the spectral synthesis problem for the Alpha transform is closely related to the problem of classifying realizations of the canonical representation (of the Heisenberg group). In this paper, we show that discrete sets are sets of spectral synthesis for the Alpha transform."}
{"category": "Math", "title": "Generalizations of Popoviciu's inequality", "abstract": "We establish a general criterion for the validity of inequalities of the following form: A certain convex combination of the values of a convex function at n points and of its value at a weighted mean of these n points is always greater or equal to a convex combination of the values of the function at some other weighted means of these points. Here, the left hand side contains only one weighted mean, while the right hand side may contain as many as possible, as long as there are finitely many. The weighted mean on the left hand side must have positive weights, while those on the right hand side must have nonnegative weights. The most prominent example of such kind of inequalities, Popoviciu's inequality in its most general form, follows from the general criterion. As another application, a result by Vasile Cirtoaje is sharpened."}
{"category": "Math", "title": "Addendum to: A simultaneous Frobenius splitting for closures of conjugacy classes of nilpotent matrices, by V. B. Mehta and Wilberd van der Kallen", "abstract": "Prompted by an exercise in the Brion Kumar book on Frobenius splittings, we show compatibility with a few more subvarieties of the Frobenius splitting from our 1992 paper."}
{"category": "Math", "title": "On the genus of a cyclic plane curve over a finite field", "abstract": "Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer type is computed."}
{"category": "Math", "title": "Consistency of cross validation for comparing regression procedures", "abstract": "Theoretical developments on cross validation (CV) have mainly focused on selecting one among a list of finite-dimensional models (e.g., subset or order selection in linear regression) or selecting a smoothing parameter (e.g., bandwidth for kernel smoothing). However, little is known about consistency of cross validation when applied to compare between parametric and nonparametric methods or within nonparametric methods. We show that under some conditions, with an appropriate choice of data splitting ratio, cross validation is consistent in the sense of selecting the better procedure with probability approaching 1. Our results reveal interesting behavior of cross validation. When comparing two models (procedures) converging at the same nonparametric rate, in contrast to the parametric case, it turns out that the proportion of data used for evaluation in CV does not need to be dominating in size. Furthermore, it can even be of a smaller order than the proportion for estimation while not affecting the consistency property."}
{"category": "Math", "title": "Mixed metric 3-contact manifolds and paraquaternionic K\\\"ahler manifolds", "abstract": "We study manifolds endowed with mixed metric 3--contact structures, proving that the distribution spanned by the Reeb vector fields is integrable, with totally geodesic integral manifolds, of constant sectional curvature $k=\\pm1$. We also prove a result of projectability of such structures onto paraquaternionic K\\\"ahlerian structures."}
{"category": "Math", "title": "A Markov dilation for self-adjoint Schur multipliers", "abstract": "We give a formula for Markov dilation in the sense of Anantharaman-Delaroche for real positive Schur multipliers on $\\B(H)$"}
{"category": "Math", "title": "Conditional density estimation in a regression setting", "abstract": "Regression problems are traditionally analyzed via univariate characteristics like the regression function, scale function and marginal density of regression errors. These characteristics are useful and informative whenever the association between the predictor and the response is relatively simple. More detailed information about the association can be provided by the conditional density of the response given the predictor. For the first time in the literature, this article develops the theory of minimax estimation of the conditional density for regression settings with fixed and random designs of predictors, bounded and unbounded responses and a vast set of anisotropic classes of conditional densities. The study of fixed design regression is of special interest and novelty because the known literature is devoted to the case of random predictors. For the aforementioned models, the paper suggests a universal adaptive estimator which (i) matches performance of an oracle that knows both an underlying model and an estimated conditional density; (ii) is sharp minimax over a vast class of anisotropic conditional densities; (iii) is at least rate minimax when the response is independent of the predictor and thus a bivariate conditional density becomes a univariate density; (iv) is adaptive to an underlying design (fixed or random) of predictors."}
{"category": "Math", "title": "Perturbation selection and influence measures in local influence analysis", "abstract": "Cook's [J. Roy. Statist. Soc. Ser. B 48 (1986) 133--169] local influence approach based on normal curvature is an important diagnostic tool for assessing local influence of minor perturbations to a statistical model. However, no rigorous approach has been developed to address two fundamental issues: the selection of an appropriate perturbation and the development of influence measures for objective functions at a point with a nonzero first derivative. The aim of this paper is to develop a differential--geometrical framework of a perturbation model (called the perturbation manifold) and utilize associated metric tensor and affine curvatures to resolve these issues. We will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model. Moreover, we will introduce new influence measures that are applicable to objective functions at any point. Examples including linear regression models and linear mixed models are examined to demonstrate the effectiveness of using new influence measures for the identification of influential observations."}
{"category": "Math", "title": "Examples of Calabi-Yau 3-manifolds with complex multiplication", "abstract": "We give some concrete examples of Calabi-Yau 3-manifolds with complex multiplication."}
{"category": "Math", "title": "A derived approach to geometric McKay correspondence in dimension three", "abstract": "We propose a three dimensional generalization of the geometric McKay correspondence described by Gonzales-Sprinberg and Verdier in dimension two. We work it out in detail when G is abelian and C^3/G has a single isolated singularity. More precisely, we show that the Bridgeland-King-Reid derived category equivalence induces a natural geometric correspondence between irreducible representations of G and subschemes of the exceptional set of G-Hilb (C^3). This correspondence appears to be related to Reid's recipe."}
{"category": "Math", "title": "Rate-optimal estimation for a general class of nonparametric regression models with unknown link functions", "abstract": "This paper discusses a nonparametric regression model that naturally generalizes neural network models. The model is based on a finite number of one-dimensional transformations and can be estimated with a one-dimensional rate of convergence. The model contains the generalized additive model with unknown link function as a special case. For this case, it is shown that the additive components and link function can be estimated with the optimal rate by a smoothing spline that is the solution of a penalized least squares criterion."}
{"category": "Math", "title": "Minors in random regular graphs", "abstract": "We show that there is a constant c>0 so that for any fixed r which is at least 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c n^{1/2} vertices as a minor. This confirms a conjecture of Markstrom. Since any minor of an r-regular graph on n vertices has at most rn/2 edges, our bound is clearly best possible up to the value of the constant c. As a corollary, we also obtain the likely order of magnitude of the largest complete minor in a random graph G(n,p) during the phase transition (i.e. when pn is close to 1)."}
{"category": "Math", "title": "Fundamental group for the complement of the Cayley's singularities", "abstract": "Given a singular surface X, one can extract information on it by investigating the fundamental group $\\pi_1(X - Sing_X)$. However, calculation of this group is non-trivial, but it can be simplified if a certain invariant of the branch curve of X - called the braid monodromy factorization - is known. This paper shows, taking the Cayley cubic as an example, how this fundamental group can be computed by using braid monodromy techniques and their liftings. This is one of the first examples that uses these techniques to calculate this sort of fundamental group."}
{"category": "Math", "title": "Testing the suitability of polynomial models in errors-in-variables problems", "abstract": "A low-degree polynomial model for a response curve is used commonly in practice. It generally incorporates a linear or quadratic function of the covariate. In this paper we suggest methods for testing the goodness of fit of a general polynomial model when there are errors in the covariates. There, the true covariates are not directly observed, and conventional bootstrap methods for testing are not applicable. We develop a new approach, in which deconvolution methods are used to estimate the distribution of the covariates under the null hypothesis, and a ``wild'' or moment-matching bootstrap argument is employed to estimate the distribution of the experimental errors (distinct from the distribution of the errors in covariates). Most of our attention is directed at the case where the distribution of the errors in covariates is known, although we also discuss methods for estimation and testing when the covariate error distribution is estimated. No assumptions are made about the distribution of experimental error, and, in particular, we depart substantially from conventional parametric models for errors-in-variables problems."}
{"category": "Math", "title": "A characterization of surfaces whose universal cover is the bidisk", "abstract": "We show that the universal cover of a compact complex surface $X$ is the bidisk $\\HH \\times \\HH$, or $X$ is biholomorphic to $\\PP^1 \\times \\PP^1$, if and only if $K_X^2 > 0$ and there exists an invertible sheaf $\\eta$ such that $\\eta^2\\cong \\hol_X$ and $H^0(X, S^2\\Omega^1_X (-K_X) \\otimes \\eta) \\neq 0$. The two cases are distinguished by the second plurigenus, $P_2(X)\\geq 2$ in the former case, $P_2(X)= 0$ in the latter. We also discuss related questions."}
{"category": "Math", "title": "Coxeter covers of the classical Coxeter groups", "abstract": "Let $C(T)$ be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either $B_n$ or $D_n$. Let $C_Y(T)$ be a natural quotient of $C(T)$, and if $C(T)$ is simply-laced (which means all the relations between the generators has order 2 or 3), $C_Y(T)$ is a generalized Coxeter group, too . Let $A_{t,n}$ be a group which contains $t$ Abelian groups generated by $n$ elements. The main result in this paper is that $C_Y(T)$ is isomorphic to $A_{t,n} \\semidirect B_n$ or $A_{t,n} \\semidirect D_n$, depends on whether the signed graph $T$ contains loops or not, or in other words C(T) is simply-laced or not, and $t$ is the number of the cycles in $T$. This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxeter groups."}
{"category": "Math", "title": "Accelerated convergence for nonparametric regression with coarsened predictors", "abstract": "We consider nonparametric estimation of a regression function for a situation where precisely measured predictors are used to estimate the regression curve for coarsened, that is, less precise or contaminated predictors. Specifically, while one has available a sample $(W_1,Y_1),...,(W_n,Y_n)$ of independent and identically distributed data, representing observations with precisely measured predictors, where $\\mathrm{E}(Y_i|W_i)=g(W_i)$, instead of the smooth regression function $g$, the target of interest is another smooth regression function $m$ that pertains to predictors $X_i$ that are noisy versions of the $W_i$. Our target is then the regression function $m(x)=E(Y|X=x)$, where $X$ is a contaminated version of $W$, that is, $X=W+\\delta$. It is assumed that either the density of the errors is known, or replicated data are available resembling, but not necessarily the same as, the variables $X$. In either case, and under suitable conditions, we obtain $\\sqrt{n}$-rates of convergence of the proposed estimator and its derivatives, and establish a functional limit theorem. Weak convergence to a Gaussian limit process implies pointwise and uniform confidence intervals and $\\sqrt{n}$-consistent estimators of extrema and zeros of $m$. It is shown that these results are preserved under more general models in which $X$ is determined by an explanatory variable. Finite sample performance is investigated in simulations and illustrated by a real data example."}
{"category": "Math", "title": "Product-form stationary distributions for deficiency zero chemical reaction networks", "abstract": "We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg's deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics."}
{"category": "Math", "title": "Limit Cycles of a Quadratic System with Two Parallel Straight Line-Isoclines", "abstract": "In this paper, a quadratic system with two parallel straight line-isoclines is considered. This system corresponds to the system of class II in the classification of Ye Yanqian. Using the field rotation parameters of the constructed canonical system and geometric properties of the spirals filling the interior and exterior domains of its limit cycles, we prove that the maximum number of limit cycles in a quadratic system with two parallel straight line-isoclines and two finite singular points is equal to two. Besides, we obtain the same result in a different way: applying the Wintner-Perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed earlier by the author."}
{"category": "Math", "title": "Edge connectivity in graphs: an expansion theorem", "abstract": "We show that if a graph is k-edge-connected, and we adjoin to it another graph satisfying a \"contracted diameter less or equal to 2\" condition, with minimal degree greater or equal to k, and some natural hypothesis on the edges connecting one graph to the other, the resulting graph is also k-edge-connected."}
{"category": "Math", "title": "A support theorem for the geodesic ray transform of symmetric tensor fields", "abstract": "Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown that there exists a vector field $v$ satisfying $f=dv$ on the set of points lying on these geodesics and $v=0$ on the intersection of this set with the boundary $\\PD M$ of the manifold $M$. Using this result, a Helgason's type of a support theorem for the geodesic ray transform is proven. The approach is based on analytic microlocal techniques."}
{"category": "Math", "title": "Arithmetic E_8 lattices with maximal Galois action", "abstract": "We construct explicit examples of E_8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E_8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E_8 and have maximal Galois action. Our main objects of study are del Pezzo surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice via the negative of the intersection pairing, contains a sublattice isomorphic to E_8. We construct examples of such surfaces for which the action of Galois on the geometric Picard group is maximal."}
{"category": "Math", "title": "Robin-to-Robin Maps and Krein-Type Resolvent Formulas for Schr\\\"odinger Operators on Bounded Lipschitz Domains", "abstract": "We study Robin-to-Robin maps, and Krein-type resolvent formulas for Schr\\\"odinger operators on bounded Lipschitz domains in $\\bbR^n$, $n\\ge 2$, with generalized Robin boundary conditions."}
{"category": "Math", "title": "Three notions of effective computation on $\\mathbb{R}$", "abstract": "We compare three notions of effectiveness on uncountable structures. The first notion is that of a $\\real$-computable structure, based on a model of computation proposed by Blum, Shub, and Smale, which uses full-precision real arithmetic. The second notion is that of an $F$-parameterizable structure, defined by Morozov and based on Mal'tsev's notion of a constructive structure. The third is $\\Sigma$-definability over $HF(\\real)$, defined by Ershov as a generalization of the observation that the computably enumerable sets are exactly those $\\Sigma_1$-definable in $HF(\\mathbb{N})$. We show that every $\\real$-computable structure has an $F$-parameterization, but that the expansion of the real field by the exponential function is $F$-parameterizable but not $\\real$-computable. We also show that the structures with $\\real$-computable copies are exactly the structures with copies $\\Sigma$-definable over $HF(\\real)$. One consequence of this equivalence is a method of approximating certain $\\real$-computable structures by Turing computable structures."}
{"category": "Math", "title": "Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms", "abstract": "We study an equation lying `mid-way' between the periodic Hunter-Saxton and Camassa-Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped, as well as smooth, traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three."}
{"category": "Math", "title": "Graph polynomials and their applications I: The Tutte polynomial", "abstract": "In this survey of graph polynomials, we emphasize the Tutte polynomial and a selection of closely related graph polynomials. We explore some of the Tutte polynomial's many properties and applications and we use the Tutte polynomial to showcase a variety of principles and techniques for graph polynomials in general. These include several ways in which a graph polynomial may be defined and methods for extracting combinatorial information and algebraic properties from a graph polynomial. We also use the Tutte polynomial to demonstrate how graph polynomials may be both specialized and generalized, and how they can encode information relevant to physical applications. We conclude with a brief discussion of computational complexity considerations."}
{"category": "Math", "title": "Bounds on exceptional Dehn filling II", "abstract": "We show that there are at most finitely many one cusped orientable hyperbolic 3-manifolds which have more than eight non-hyperbolic Dehn fillings. Moreover, we show that determining these finitely many manifolds is decidable."}
{"category": "Math", "title": "Function Theory in Real Hardy Spaces", "abstract": "We show that many classical results in Hardy space theory have exact analogues when the Fourier coefficients are allowed only to be real."}
{"category": "Math", "title": "Nonuniform measure rigidity", "abstract": "We consider an ergodic invariant measure $\\mu$ for a smooth action of $Z^k$, $k \\ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \\ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $\\mu$ is hyperbolic with the Lyapunov hyperplanes in general position and if one element of the action has positive entropy, then $\\mu$ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups."}
{"category": "Math", "title": "Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions", "abstract": "We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford's sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford's trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urn-model description of sampling from Dirichlet random distributions."}
{"category": "Math", "title": "A Note on Walk versus Wait: Lazy Mathematician Wins", "abstract": "Points out the errors in the paper \"J.G. Chen, S.D. Kominers, and R.W. Sinnott. Walk versus wait: The lazy mathematician wins. arXiv.org Mathematics 2008. arXiv:0801.0297\""}
{"category": "Math", "title": "Rigidity of compact Riemannian spin Manifolds with Boundary", "abstract": "In this article, we prove new rigidity results for compact Riemannian spin manifolds with boundary whose scalar curvature is bounded from below by a non-positive constant. In particular, we obtain generalizations of a result of Hang-Wang \\cite{hangwang1} based on a conjecture of Schroeder and Strake \\cite{schroeder}."}
{"category": "Math", "title": "A few remarks on the operator norm of random Toeplitz matrices", "abstract": "We present some results concerning the almost sure behaviour of the operator norm or random Toeplitz matrices, including the law of large numbers for the norm, normalized by its expectation (in the i.i.d. case). As tools we present some concentration inequalities for suprema of empirical processes, which are refinements of recent results by Einmahl and Li."}
{"category": "Math", "title": "Unconditional Proof of the Boltzmann-Sinai Ergodic Hypothesis", "abstract": "We consider the system of $N$ ($\\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\\Bbb T^\\nu$, $\\nu\\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis."}
{"category": "Math", "title": "Adams operations and power structures", "abstract": "We construct a family of additive endomorphisms $\\Psi_k, k=1, 2...$ of the Grothendieck ring of quasiprojective varieties and the Grothendieck ring of Chow motives similar to the Adams operations in the K-theory. The speciality of the $\\lambda$-structure on the Grothendieck ring of motives (proved by F. Heinloth) gives a set of natural equations for these operations. We discuss this construction in a general setting and relate it to the concept of power structures introduced by S. Gusein-Zade, I. Luengo and A. Melle-Hernandez. Some interpretation of the E. Getzler's formula for the equivariant Hodge-Deligne polynomial of the configuration spaces is also discussed."}
{"category": "Math", "title": "The property of the set of the real numbers generated by a Gelfond-Schneider operator and the countability of all real numbers", "abstract": "Considered will be properties of the set of real numbers $\\Re$ generated by an operator that has form of an exponential function of Gelfond-Schneider type with rational arguments. It will be shown that such created set has cardinal number equal to ${\\aleph_0}^{\\aleph_0}=c$. It will be also shown that the same set is countable. The implication of this contradiction to the countability of the set of real numbers will be discussed."}
{"category": "Math", "title": "An asymptotic variant of the Fubini theorem for maps into CAT(0)-spaces", "abstract": "The classical Fubini theorem asserts that the multiple integral is equal to the repeated one for any integrable function on a product measure space. In this paper, we derive an asymptotic variant of the Fubini theorem for maps into CAT$(0)$-spaces from the $L^1$ and $L^2$-concentration of the maps."}
{"category": "Math", "title": "Discussion: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$", "abstract": "Discussion of \"The Dantzig selector: Statistical estimation when $p$ is much larger than $n$\" [math/0506081]"}
{"category": "Math", "title": "Discussion: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$", "abstract": "Discussion of ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' [math/0506081]"}
{"category": "Math", "title": "Discussion: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$", "abstract": "Discussion of ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' [math/0506081]"}
{"category": "Math", "title": "Discussion: The Dantzig selector: statistical estimation when $p$ is much larger than $n$", "abstract": "Discussion of ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' [math/0506081]"}
{"category": "Math", "title": "Countable Choice and Compactness", "abstract": "We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p greater or equal to 1 (resp. . p = 0), and some closed subset F of [0, 1]^I which is a bounded subset of l^p(I), we show that AC(N) (resp. DC, the axiom of Dependent Choices) implies the compactness of F."}
{"category": "Math", "title": "Effect of linear lumping on controllability and observability", "abstract": "The effect of linear lumping, linear transformation to reduce the number of state variables on controllability and observability of linear differential equations has been studied. Controllability of the original system implies the controllability of the lumped system. Examples taken from reaction kinetics illustrate our results."}
{"category": "Math", "title": "Discussion: A tale of three cousins: Lasso, L2Boosting and Dantzig", "abstract": "Discussion of ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' by Emmanuel Candes and Terence Tao [math/0506081]"}
{"category": "Math", "title": "Discussion: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$", "abstract": "Discussion of ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' [math/0506081]"}
{"category": "Math", "title": "Rejoinder: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$", "abstract": "Rejoinder to ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' [math/0506081]"}
{"category": "Math", "title": "Counting Schr\\\"odinger boundstates: semiclassics and beyond", "abstract": "This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schr\\\"odinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and slowly decaying potentials, for which the semiclassical rules are violated. Some new results are presented, concerning operators on product manifolds and graphs."}
{"category": "Math", "title": "Sharpness of some properties of Wiener amalgam and modulation spaces", "abstract": "We prove sharp estimates for the dilation operator $f(x)\\longmapsto f(\\lambda x)$, when acting on Wiener amalgam spaces $W(L^p,L^q)$. Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces $M^{p,q}$, as well as the optimality of an estimate for the Schr\\\"odinger propagator on modulation spaces."}
{"category": "Math", "title": "Finite sets of $d$-planes in affine space", "abstract": "Let $A$ be a subvariety of affine space $\\mathbb{A}^n$ whose irreducible components are $d$-dimensional linear or affine subspaces of $\\mathbb{A}^n$. Denote by $D(A)\\subset\\mathbb{N}^n$ the set of exponents of standard monomials of $A$. We show that the combinatorial object $D(A)$ reflects the geometry of $A$ in a very direct way. More precisely, we define a $d$-plane in $\\mathbb{N}^n$ as being a set $\\gamma+\\oplus_{j\\in J}\\mathbb{N}e_{j}$, where $#J=d$ and $\\gamma_{j}=0$ for all $j\\in J$. We call the $d$-plane thus defined to be parallel to $\\oplus_{j\\in J}\\mathbb{N}e_{j}$. We show that the number of $d$-planes in $D(A)$ equals the number of components of $A$. This generalises a classical result, the finiteness algorithm, which holds in the case $d=0$. In addition to that, we determine the number of all $d$-planes in $D(A)$ parallel to $\\oplus_{j\\in J}\\mathbb{N}e_{j}$, for all $J$. Furthermore, we describe $D(A)$ in terms of the standard sets of the intersections $A\\cap\\{X_{1}=\\lambda\\}$, where $\\lambda$ runs through $\\mathbb{A}^1$."}
{"category": "Math", "title": "A note on minimal finite quotients of mapping class groups", "abstract": "We prove that the minimal nontrivial finite quotient group of the mapping class group M_g of a closed orientable surface of genus g is the symplectic group PSp(2g,Z_2), for g = 3 and 4 (this might remain true, however, for arbitrary genus g > 2). We discuss also some results for arbitrary genus g."}
{"category": "Math", "title": "Symmetric jump processes: localization, heat kernels, and convergence", "abstract": "We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the H\\\"older continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes."}
{"category": "Math", "title": "Documentation for the ratpoints program", "abstract": "This note explains how to obtain, install, and use the ratpoints program. The program finds rational points up to a specified height on hyperelliptic curves using a highly optimized quadratic sieving algorithm."}
{"category": "Math", "title": "Equiconvergence theorems for Sturm--Liouville operators with distribution potentials^ the rate of equiconvergence", "abstract": "We consider a Sturm--Liouville operator $Ly=-y''+qy$ in $L_2[0,\\pi]$ with Dirichlet boundary conditions. We assume, that the potential $q$ is complex valued and belongs to Sobolev space $W_2^\\theta[0,\\pi]$, $\\theta\\in(-1,-1/2$. This operators were successfully defined in papers of Savchuk A.M. and Shkalikov A.A. There were also shown, that theese operators have a discrete spectrum, which we denote by $\\{\\lambda_n\\}$, and $\\lim\\lambda_n=+\\infty$. All but finitely many of them are simple. The eigenfunctions form the Riesz basis in $L_2[0,\\pi]$. We investigate a uniform on $[0,\\pi]$ equiconvergence of series for this system and for trigonometric system $\\{\\sin(nt)\\}_1^\\infty$. We obtain not only a theorems of equiconvergence, but also estimate a rate of this equiconvergence."}
{"category": "Math", "title": "Bari-Markus property for Riesz projections of Hill operators with singular potentials", "abstract": "The Hill operators $L y = - y^{\\prime \\prime} + v(x) y, x \\in [0,\\pi],$ with $H^{-1}$ periodic potentials, considered with periodic, antiperiodic or Dirichlet boundary conditions, have discrete spectrum, and therefore, for sufficiently large $N,$ the Riesz projections $$ P_n = \\frac{1}{2\\pi i} \\int_{C_n} (z-L)^{-1} dz, \\quad C_n=\\{z: |z-n^2|= n\\} $$ are well defined. It is proved that $$\\sum_{n>N} \\|P_n - P_n^0\\|^2_{HS} < \\infty, $$ where $P_n^0$ are the Riesz projection of the free operator and $\\|\\cdot\\|_{HS}$ is the Hilbert--Schmidt norm."}
{"category": "Math", "title": "Generalized Robin Boundary Conditions, Robin-to-Dirichlet Maps, and Krein-Type Resolvent Formulas for Schr\\\"odinger Operators on Bounded Lipschitz Domains", "abstract": "We study generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schr\\\"odinger operators on bounded Lipschitz domains in $\\bbR^n$, $n\\ge 2$. We also discuss the case of bounded $C^{1,r}$-domains, $(1/2)<r<1$."}
{"category": "Math", "title": "Polynomial representation for long knots", "abstract": "We discuss the polynomial representation for long knots and elaborate on how to obtain them with a bound on degrees of the defining polynomials, for any knot-type."}
{"category": "Math", "title": "Maximalite des varietes toriques de dimension 4", "abstract": "A complex algebraic variety defined over the reals is maximal when the sum of its Betti numbers for Borel Moore homology with $\\zz$ coefficients coincides with the sum of the Betti numbers of its real part. We will show in this paper that toric varieties of dimension 4 are maximal."}
{"category": "Math", "title": "On some results of Cufaro Petroni about Student t-processes", "abstract": "This paper deals with Student t-processes as studied in (Cufaro Petroni N 2007 J. Phys. A, Math. Theor. 40(10), 2227-2250). We prove and extend some conjectures expressed by Cufaro Petroni about the asymptotical behavior of a Student t-process and the expansion of its density. First, the explicit asymptotic behavior of any real positive convolution power of a Student t-density with any real positive degrees of freedom is given in the multivariate case; then the integer convolution power of a Student t-distribution with odd degrees of freedom is shown to be a convex combination of Student t-densities with odd degrees of freedom. At last, we show that this result does not extend to the case of non-integer convolution powers."}
{"category": "Math", "title": "Anick's fibration and the odd primary homotopy exponent of spheres", "abstract": "For primes p>=3, Cohen, Moore, and Neisendorfer showed that the exponent of the p-torsion in the homotopy groups of S^2n+1 is p^n. This was obtained as a consequence of a thorough analysis of the homotopy theory of Moore spaces. Anick further developed this for p>=5 by constructing a homotopy fibration S^2n-1 --> T^2n+1(p^r) --> Loop S^2n+1 whose connecting map is degree p^r on the bottom cell. A much simpler construction of such a fibration for p>=3 was given by Gray and the author using new methods. In this paper the new methods are used to start over, first constructing Anick's fibration for p>=3, and then using it to obtain the exponent result for spheres."}
{"category": "Math", "title": "Scattering theory for the Gross-Pitaevskii equation in three dimensions", "abstract": "We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. Translated to the defocusing nonlinear Schr\\\"odinger equation, this implies asymptotic stability of all plane wave solutions for such disturbances. We also prove that every linearized solution with finite energy has a nonlinear solution which is asymptotic to it. The key ingredients are: (1) some quadratic transforms of the solutions, which effectively linearize the nonlinear energy space, (2) a bilinear Fourier multiplier estimate, which allows irregular denominators due to a degenerate non-resonance property of the quadratic interactions, and (3) geometric investigation of the degeneracy in the Fourier space to minimize its influence."}
{"category": "Math", "title": "Invariant subspaces of subgraded Lie algebras of compact operators", "abstract": "We show that finitely subgraded Lie algebras of compact operators have invariant subspaces when conditions of quasinilpotence are imposed on certain components of the subgrading. This allows us to obtain some useful information about the structure of such algebras. As an application, we prove a number of results on the existence of invariant subspaces for algebraic structures of compact operators. Along the way we obtain new criteria for the triangularizability of a Lie algebra of compact operators."}
{"category": "Math", "title": "Fundamental groups of symmetric sextics", "abstract": "We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with at least two type $\\bold{E}_6$ singular points. As a simple application, we compute the fundamental groups of 125 other sextics, most of which are new."}
{"category": "Math", "title": "Characters of prime degree", "abstract": "Let $G$ be a finite nilpotent group, $\\chi$ and $\\psi$ be irreducible complex characters of $G$ of prime degree. Assume that $\\chi(1)=p$. Then either the product $\\chi\\psi$ is a multiple of an irreducible character or $\\chi\\psi$ is the linear combination of at least $\\frac{p+1}{2}$ distinct irreducible characters."}
{"category": "Math", "title": "Equivariant K-theory, groupoids and proper actions", "abstract": "In this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite equivariant CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed."}
{"category": "Math", "title": "On semi-infinite cohomology of finite dimensional graded algebras", "abstract": "We describe a general setting for the definition of semi-infinite cohomology of finite dimensional algebras, and provide its categorical interpretation. We apply this interpretation to compute semi-infinite cohomology of some modules over the small group at a root of unity, generalizing an earlier result of S. Arkhipov (conjectured by B. Feigin)."}
{"category": "Math", "title": "Relative Cohen--Macaulayness of bigraded modules", "abstract": "In this paper we study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals."}
{"category": "Math", "title": "Les quartiers de la lune de Troie ou le bouclier d'Achille, essai sur le corps de classe", "abstract": "This is a text in french about classical class field theory, readable with a minimum of prerequisites"}
{"category": "Math", "title": "Remarks on modular symbols for Maass wave forms", "abstract": "In this paper I introduce modular symbols for Maass wave cusp forms. They appear in the guise of finitely additive functions on the Boolean algebra generated by intervals with non--positive rational ends, with values in analytic functions (pseudo--measures in the sense of [MaMar2]). After explaining the basic issues and analogies in the extended Introduction, I construct modular symbols in the sec. 1 and the related L\\'evy--Mellin transforms in the sec. 2. The whole paper is an extended footnote to the Lewis--Zagier fundamental study [LZ2]."}
{"category": "Math", "title": "Bimonotone enumeration", "abstract": "Solutions of a diophantine equation $f(a,b) = g(c,d)$, with $a,b,c,d$ in some finite range, can be efficiently enumerated by sorting the values of $f$ and $g$ in ascending order and searching for collisions. This article considers functions that are bimonotone in the sense that $f(a,b) \\le f(a',b')$ whenever $a \\le a'$ and $b \\le b'$. A two-variable polynomial with non-negative coefficients is a typical example. The problem is to efficiently enumerate all pairs $(a,b)$ such that the values $f(a,b)$ appear in increasing order. We present an algorithm that is memory-efficient and highly parallelizable. In order to enumerate the first $n$ values of $f$, the algorithm only builds up a priority queue of length at most $\\sqrt{2n}+1$. In terms of bit-complexity this ensures that the algorithm takes time $O(n \\log^2 n)$ and requires memory $O(\\sqrt{n} \\log n)$, which considerably improves on the memory bound $\\Theta(n \\log n)$ provided by a naive approach, and extends the semimonotone enumeration algorithm previously considered by R.L. Ekl and D.J. Bernstein."}
{"category": "Math", "title": "Recurrence and transience of a multi-excited random walk on a regular tree", "abstract": "We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition of the recurrence/transience property of the walk. In particular, we prove that the asymptotic behavior of the walk depends on the order of the excitations, which contrasts with the one dimensional setting studied by Zerner (2005). We also consider the limiting speed of the walk in the transient regime and conjecture that it is not a monotonic function of the environment."}
{"category": "Math", "title": "Branching Process approach for 2-SAT thresholds", "abstract": "It is well known that, as $n$ tends to infinity, the probability of satisfiability for a random 2-SAT formula on $n$ variables, where each clause occurs independently with probability $\\alpha/2n$, exhibits a sharp threshold at $\\alpha=1$. We study a more general 2-SAT model in which each clause occurs independently but with probability $\\alpha_i/2n$ where $i \\in \\{0,1,2\\}$ is the number of positive literals in that clause. We generalize branching process arguments by Verhoeven(99) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix."}
{"category": "Math", "title": "Dimension quotients", "abstract": "We present two approaches, one homological and the other simplicial, for the investigation of dimension quotients of groups. The theory is illustrated, in particular, with a conceptual discussion of the fourth and fifth dimension quotients."}
{"category": "Math", "title": "Comparing Classes of Finite Structures", "abstract": "We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This reducibility is calibrated by comparing several classes of structures. The class of cyclic graphs and the class of finite prime fields are equivalent, and are properly below the class of arbitrary finite graphs. The class of finite graphs and the class of finite linear orders are maximal among all classes of finite structures. We also prove some general characterizations of reducibility to certain classes. Examples of large chains and antichains of classes are constructed."}
{"category": "Math", "title": "Classification from a Computable Viewpoint", "abstract": "Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work, showing that certain classes of mathematical structures admit classification, while others do not. In the present paper, we describe some recent work on classification in computable structure theory."}
{"category": "Math", "title": "Index Sets of Computable Structures", "abstract": "The \\emph{index set} of a computable structure $\\mathcal{A}$ is the set of indices for computable copies of $\\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, $\\mathbb{Q}$-vector spaces, Archimedean real closed ordered fields, reduced Abelian $p$-groups of length less than $\\omega^{2}$, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be $m$-complete $\\Pi_{n}^{0}$, $d$-$\\Sigma_{n}^{0}$, or $\\Sigma_{n}^{0}$, for various $n$. In each case, the calculation involves finding an \\textquotedblleft optimal\\textquotedblright% \\ sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable $\\Pi_{n}$, $d$-$\\Sigma_{n}$, or $\\Sigma_{n}$) yields a bound on the complexity of the index set. When we show $m$% -completeness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey theory."}
{"category": "Math", "title": "Structures in Familiar Classes Which Have Scott Rank $\\omega_1^{CK}$", "abstract": "There are familiar examples of computable structures having various computable Scott ranks. There are also familiar structures, such as the Harrison ordering, which have Scott rank $\\omega_1^{CK}+1$. Makkai produced a structure of Scott rank $\\omega_1^{CK}$, which can be made computable, and simplified so that it is just a tree. In the present paper, we show that there are further computable structures of Scott rank $\\omega_1^{CK}$ in the following classes: undirected graphs, fields of any characteristic, and linear orderings. The new examples share with the Harrison ordering, and the tree just mentioned, a strong approximability property."}
{"category": "Math", "title": "$L_{p,q}$-Cohomology of Warped Cylinders", "abstract": "We extend some results by Gol'dshtein, Kuz'minov, and Shvedov about the $L_p$-cohomology of warped cylinders to $L_{p,q}$-cohomology for different $p$ and $q$. As an application, we establish some sufficient conditions for the nontriviality of the $L_{p,q}$-torsion of a surface of revolution in terms of some Hardy constants."}
{"category": "Math", "title": "Factorial threefold hypersurfaces", "abstract": "Let $X$ be a hypersurface in $\\mathbb{P}^{4}$ of degree $d$ that has at most isolated ordinary double points. We prove that $X$ is factorial in the case when $X$ has at most $(d-1)^{2}-1$ singular points."}
{"category": "Math", "title": "A Generalized Backward Equation For One Dimensional Processes", "abstract": "Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f, the backward Kolmogorov equation gives a condition for f(t,X) to be a local martingale. We generalize the backward equation in two main ways. First, it is extended to non-differentiable functions. Second, the process X is not required to satisfy an SDE. Instead, it is only required to be a quasimartingale satisfying an integrability condition, and the martingale condition for f(t,X) is then expressed in terms of the marginal distributions, drift measure and jumps of X. The proof involves the stochastic calculus of Dirichlet processes and a time-reversal argument. These results are then applied to show that a continuous and strong Markov martingale is uniquely determined by its marginal distributions."}
{"category": "Math", "title": "Higher Order Riesz Transforms for Laguerre Expansions", "abstract": "In this paper we investigate Lp-boundedness properties for the higher order Riesz transforms associated with Laguerre operators. Also we prove that the k-th Riesz transform is a principal value singular integral operator (modulus a constant times of the function when k is even). To establish our results we exploit a new identity connecting Riesz transforms in the Hermite and Laguerre settings."}
{"category": "Math", "title": "A connection whose curvature is the Lie bracket", "abstract": "Let G be a Lie group. On the trivial principal G-bundle over the Lie algebra of G there is a natural connection whose curvature is the Lie bracket. The exponential map is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold, the curvature of the natural connection is the Lie bracket of vectorfields on the manifold. The motion of a ball rolling on an oriented surface is the parallel transport of a similar connection on the trivial SO(3)-bundle over the surface. If the surface is a plane or a sphere, then the curvature of the connection is a scalar multiple of the Lie bracket in the Lie algebra of SO(3)."}
{"category": "Math", "title": "On Sp_4 modularity of Picard--Fuchs differential equations for Calabi--Yau threefolds (with an appendix by Vicentiu Pasol)", "abstract": "Motivated by the relationship of classical modular functions and Picard--Fuchs linear differential equations of order 2 and 3, we present an analogous concept for equations of order 4 and 5."}
{"category": "Math", "title": "Cocompact imbeddings and structure of weakly convergent sequences", "abstract": "Concentration compactness method is a powerful techniques for establishing existence of minimizers for inequalities and of critical points of functionals in general. The paper gives a functional-analytic formulation for the method in Banach space, generalizing the Hilbert space case elaborated in \\cite{ccbook}. The key object is a dislocation space - a triple $(X,F,D)$, where $F$ is a convex functional that defines a norm on Banach space $X$, and $D$ is a group of isometries on $X$. Bounded sequences in dislocation spaces admit a decomposition into an asymptotic sum \"profiles\" $w^{(n)}\\in X$ dislocated by actions of $D$, that is, a sum of the form $\\sum_ng^{(n)}_kw^{(n)}$, $g^{(n)}_k\\in D$, while the remainder term converges weakly under actions of any sequence $g_k\\in D$ ({\\em $D$-weak convergence}). This decomposition allows to extend the weak convergence argument from variational problems with compactness to problems where $X$ is {\\em cocompactly} (relatively to the group $D$) imbedded into a Banach space $Y$, that is, when every sequence $D$-weakly convergent in $X$ is convergent in the norm of $Y$. We prove a general statement on existence of minimizers in cocompact imbeddings that applies, in particular to Sobolev imbeddings which lack compactness (unbounded domain, critical exponent) including the subelliptic Sobolev spaces and spaces over Riemannian manifolds."}
{"category": "Math", "title": "Moduli of twisted orbifold sheaves", "abstract": "We study stacks of slope-semistable twisted sheaves on orbisurfaces with projective coarse spaces and prove that in certain cases they have many of the asymptotic properties enjoyed by the moduli of slope-semistable sheaves on smooth projective surfaces."}
{"category": "Math", "title": "Knuth Relations for the Hyperoctahedral Groups", "abstract": "C. Bonnaf{\\'e}, M. Geck, L. Iancu, and T. Lam have conjectured a description of one-sided cells in unequal parameter Hecke algebras of type $B$ which is based on domino tableaux of arbitrary rank. In the integer case, this generalizes the work of D. Garfinkle whose methods we adapt to construct a family of operators which generate the conjectured combinatorial description."}
{"category": "Math", "title": "The Canonical Model of a Singular Curve", "abstract": "We give refined statements and modern proofs of Rosenlicht's results about the canonical model C' of an arbitrary complete integral curve C. Notably, we prove that C and C' are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C' is equal to the blowup of C with respect to the canonical sheaf \\omega. We also prove some new results: we determine just when C' is rational normal, arithmetically normal, projectively normal, and linearly normal."}
{"category": "Math", "title": "Type ${\\rm III_1}$ factors generated by regular representations of infinite dimensional nilpotent group $B_0^{\\mathbb N}$", "abstract": "We study the von Neumann algebra, generated by the unitary representations of infinite-dimensional groups nilpotent group $B_0^{\\mathbb N}$. The conditions of the irreducibility of the regular and quasiregular representations of infinite-dimensional groups (associated with some quasi-invariant measures) are given by the so-called Ismagilov conjecture (see [1,2,9-11]). In this case the corresponding von Neumann algebra is type ${\\rm I}_\\infty$ factor. When the regular representation is reducible we find the sufficient conditions on the measure for the von Neumann algebra to be factor (see [13,14]). In the present article we determine the type of corresponding factors. Namely we prove that the von Neumann algebra generated by the regular representations of infinite-dimensional nilpotent group $B_0^{\\mathbb N}$ is type ${\\rm III}_1$ hyperfinite factor. The case of the nilpotent group $B_0^{\\mathbb Z}$ of infinite in both directions matrices will be studied in [6]."}
{"category": "Math", "title": "Geometric Interpretation of Second Elliptic Integrable System", "abstract": "In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4-symmetric spaces. We first show that a 4-symmetric space $G/G_0$ can be embedded into the twistor space of the corresponding symmetric space $G/H$. Then we prove that the second elliptic system is equivalent to the vertical harmonicity of an admissible twistor lift $J$ taking values in $G/G_0 \\hookrightarrow \\Sigma(G/H)$. We begin the paper by an example: $G/H=\\R^4$. We study also the structure of 4-symmetric bundles over Riemannian symmetric spaces."}
{"category": "Math", "title": "Alternative proofs of linear response for piecewise expanding unimodal maps", "abstract": "We give two new proofs that the SRB measure of a C^2 path f_t of unimodal piecewise expanding C^3 maps is differentiable at 0 if f_t is tangent to the topological class of f_0. The arguments are more conceptual than the one in our previous paper, but require proving Holder continuity of the infinitesimal conjugacy (a new result, of independent interest) and using spaces of bounded p-variation. The first new proof gives differentiability of higher order if f_t is smooth enough and stays in the topological class of f_0 and if the observable smooth enough (a new result). In addition, this proof does not require any information on the decomposition of the SRB measure into regular and singular terms, making it potentially amenable to extensions to higher dimensions. The second new proof allows us to recover the linear response formula (i.e., the formula for the derivative at 0) obtained in our previous paper and gives additional information on this formula."}
{"category": "Math", "title": "The value of Repeated Games with an informed controller", "abstract": "We consider the general model of zero-sum repeated games (or stochastic games with signals), and assume that one of the players is fully informed and controls the transitions of the state variable. We prove the existence of the uniform value, generalizing several results of the literature. A preliminary existence result is obtained for a certain class of stochastic games played with pure strategies."}
{"category": "Math", "title": "Counting points of homogeneous varieties over finite fields", "abstract": "Let $X$ be an algebraic variety over a finite field $\\bF_q$, homogeneous under a linear algebraic group. We show that the number of rational points of $X$ over $\\bF_{q^n}$ is a periodic polynomial function of $q^n$ with integer coefficients. Moreover, the shifted periodic polynomial function, where $q^n$ is formally replaced with $q^n + 1$, is shown to have non-negative coefficients."}
{"category": "Math", "title": "Differential operators and Cherednik algebras", "abstract": "We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction, used in [GS]; the other involving quantum hamiltonian reduction of an algebra of differential operators, used in [GG]. In the present paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on D-modules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result, [GS, Theorem 1.4] without recourse to Haiman's deep results on the n! theorem. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG]."}
{"category": "Math", "title": "Actions of F_\\infty whose II_1 factors and orbit equivalence relations have prescribed fundamental group", "abstract": "We show that given any subgroup F of R_+ which is either countable or belongs to a certain \"large\" class of uncountable subgroups, there exist continuously many free ergodic probability measure preserving actions \\sigma_i of the free group with infinitely many generators such that their associated group measure space II_1 factors M_i and orbit equivalence relations R_i have fundamental group equal to F and with M_i (respectively R_i) stably non-isomorphic. Moreover, these actions can be taken so that R_i has no outer automorphisms and any automorphism of M_i is unitary conjugate to an automorphism that acts trivially on $L^\\infty(X_i) \\subset M_i$."}
{"category": "Math", "title": "On strongly $g(x)$-clean rings", "abstract": "Let $R$ be an associative ring with identity, $C(R)$ denote the center of $R$, and $g(x)$ be a polynomial in the polynomial ring $C(R)[x]$. $R$ is called strongly $g(x)$-clean if every element $r \\in R$ can be written as $r=s+u$ with $g(s)=0$, $u$ a unit of $R$, and $su=us$. The relation between strongly $g(x)$-clean rings and strongly clean rings is determined, some general properties of strongly $g(x)$-clean rings are given, and strongly $g(x)$-clean rings generated by units are discussed."}
{"category": "Math", "title": "A Payne-Weinberger eigenvalue estimate for wedge domains on spheres", "abstract": "A Faber-Krahn type argument gives a sharp lower estimate for the first Dirichlet eigenvalue for subdomains of wedge domains in spheres, generalizing the inequality in the plane, found by Payne and Weinberger. An application is an alternative proof to the finiteness of a Brownian motion capture time estimate."}
{"category": "Math", "title": "On the zeta function of divisors for projective varieties with higher rank divisor class group", "abstract": "Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of C_p. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case. In particular, we prove a p-adic meromorphic continuation theorem which applies to a large class of varieties. Examples of such varieties are projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular)."}
{"category": "Math", "title": "The centers of Iwahori-Hecke algebras are filtered", "abstract": "We show that the center of the Iwahori--Hecke algebra of the symmetric group $S_n$ carries a natural filtered algebra structure, and that the structure constants of the associated graded algebra are independent of $n$. A series of conjectures and open problems are also included."}
{"category": "Math", "title": "Finitistic and Representation Dimensions", "abstract": "We prove that the finitistic dimension conjecture, the Gorenstein Symmetry Conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for artin algebras which can be realized as endomorphism algebras of modules over representation-finite algebras. Note it is a question whether or not all artin algebras have such realizations. It was also shown that if every quasi-hereditary algebras has a left idealized extension which is a monomial algebra or an algebra whose representation dimension is not more than 3, then the finitistic dimension conjecture holds."}
{"category": "Math", "title": "Zero loci of admissible normal functions with torsion singularities", "abstract": "We show that the zero locus of a normal function on a smooth complex algebraic variety S is algebraic provided that the normal function extends to a admissible normal function on a smooth compactification of S with torsion singularity. This result generalizes our previous result for admissible normal functions on curves [arxiv:math/0604345 [math.AG]]. It has also been obtained by M. Saito using a different method in a recent preprint [arXiv:0803.2771v2]."}
{"category": "Math", "title": "Maximal contact and normal crossings in resolution of singularities", "abstract": "This paper has been withdrawn by the author"}
{"category": "Math", "title": "On monotonicity of F-blowup sequences", "abstract": "For each variety in positive characteristic, there is a series of canonically defined blowups, called F-blowups. We are interested in the question of whether the $e+1$-th blowup dominates the $e$-th, locally or globally. It is shown that the answer is affirmative (globally for any $e$) when the given variety is F-pure. As a corollary, we obtain some result on the stability of the sequence of F-blowups. We also give a sufficient condition for local domination."}
{"category": "Math", "title": "Notes on uniform exponential growth and Tits alternative", "abstract": "These notes contain results concerning uniform exponential growth which were obtained in collaborations with E. Breuillard and A. Salehi-Golsefidy, mostly during 2005, improving Eskin-Mozes-Oh theorem \\cite{EMO}, as well as a uniform uniform version of Tits alternative improving \\cite{uti}."}
{"category": "Math", "title": "Generalized Harish-Chandra descent and applications to Gelfand pairs", "abstract": "In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over arbitrary local field F of characteristic zero. Our main tool is Luna slice theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pair (GL(n,C),GL(n,R)) is a Gelfand pair. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-archimedean F the later is a classical theorem of Gelfand and Kazhdan. We use the techniques developed here in our subsequent work [AG3] where we prove an archimedean analog of the theorem on uniqueness of linear periods by H. Jacquet and S. Rallis."}
{"category": "Math", "title": "An archimedean analog of Jacquet - Rallis theorem", "abstract": "In this paper we prove that the symmetric pair $(GL_{n+k}(F),GL_n(F) \\times GL_k(F))$ is a Gelfand pair for any local field F of characteristic 0. For non-archimedean F it has been proven in [JR]. We use techniques developed in [AG2] to generalize their proof to general local fields."}
{"category": "Math", "title": "Dynamics of tuples of matrices", "abstract": "In this article we answer a question raised by N. Feldman in \\cite{Feldman} concerning the dynamics of tuples of operators on $\\mathbb{R}^n$. In particular, we prove that for every positive integer $n\\geq 2$ there exist $n$ tuples $(A_1, A_2, ..., A_n)$ of $n\\times n$ matrices over $\\mathbb{R}$ such that $(A_1, A_2, ..., A_n)$ is hypercyclic. We also establish related results for tuples of $2\\times 2$ matrices over $\\mathbb{R}$ or $\\mathbb{C}$ being in Jordan form."}
{"category": "Math", "title": "Ambitable topological groups", "abstract": "A topological group is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the group with its right uniformity is contained in an ambit. For n=0,1,2,..., every locally aleph_n bounded topological group is either precompact or ambitable. In the familiar semigroups constructed over ambitable groups, topological centres have an effective characterization."}
{"category": "Math", "title": "Factors in random graphs", "abstract": "Let $H$ be a fixed graph on $v$ vertices. For an $n$-vertex graph $G$ with $n$ divisible by $v$, an $H$-{\\em factor} of $G$ is a collection of $n/v$ copies of $H$ whose vertex sets partition $V(G)$. In this paper we consider the threshold $th_{H} (n)$ of the property that an Erd\\H{o}s-R\\'enyi random graph (on $n$ points) contains an $H$-factor. Our results determine $th_{H} (n)$ for all strictly balanced $H$. The method here extends with no difficulty to hypergraphs. As a corollary, we obtain the threshold for a perfect matching in random $k$-uniform hypergraph, solving the well-known \"Shamir's problem.\""}
{"category": "Math", "title": "Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy--Widom limits and rates of convergence", "abstract": "Let $A$ and $B$ be independent, central Wishart matrices in $p$ variables with common covariance and having $m$ and $n$ degrees of freedom, respectively. The distribution of the largest eigenvalue of $(A+B)^{-1}B$ has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that $m$ and $n$ grow in proportion to $p$. We show that after centering and scaling, the distribution is approximated to second-order, $O(p^{-2/3})$, by the Tracy--Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role."}
{"category": "Math", "title": "The strength of the Weak Lefschetz Property", "abstract": "We study a number of conditions on the Hilbert function of a level artinian algebra which imply the Weak Lefschetz Property (WLP). Possibly the most important open case is whether a codimension 3 SI-sequence forces the WLP for level algebras. In other words, does every codimension 3 Gorenstein algebra have the WLP? We give some new partial answers to this old question: we prove an affirmative answer when the initial degree is 2, or when the Hilbert function is relatively small. Then we give a complete answer to the question of what is the largest socle degree forcing the WLP."}
{"category": "Math", "title": "Separable d-permutations and guillotine partitions", "abstract": "We characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula and explicit formula. We find a connection between multidimensional permutations and guillotine partitions of a box. In particular, a bijection between $d$-dimensional permutations and guillotine partitions of a $2^{d-1}$-dimensional box is constructed. We also study enumerating problems related to guillotine partitions under certain restrictions revealing connections to other combinatorial structures. This allows us to obtain results on avoided patterns in permutations."}
{"category": "Math", "title": "Cohomology of Flag Varieties and the Brylinski-Kostant Filtration", "abstract": "Let G be a semisimple complex algebraic group with Borel subgroup B and let P be a parabolic subgroup of G. Let T*(G/P) denote the cotangent bundle of G/P. Ranee Brylinski discovered a connection between cohomology of G-equivariant line bundles on T*(G/B) and the so-called Brylinski-Kostant filtration, which describes the action of principal sl_2 triples on G-representations. In this paper we generalize these results to a larger class of sl_2 triples. Along the way we also obtain generalizations of results due to Broer on cohomology of G-equivariant bundles on T*(G/P) for various parabolics P."}
{"category": "Math", "title": "Persistence of normally expanded submanifolds with boundary or corners", "abstract": "We show that invariant submanifolds with boundary, and more generally with corners which are normally expanded by an endomorphism are persistent as $a$-regular stratifications. This result will be shown in class $C^s$, for $s\\ge 1$. We present also a simple example of a submanifold with boundary which is normally expanded but non-persistent as a differentiable submanifold."}
{"category": "Math", "title": "Influence of Speed Limit on Roadway Safety in Indiana", "abstract": "The influence of speed limits on roadway safety is an extremely important social issue and is subject to an extensive debate in the State of Indiana and nationwide. With around 800-900 fatalities and thousands of injuries annually in Indiana, traffic accidents place an incredible social and economic burden on the state. Still, speed limits posted on highways and other roads are routinely exceeded as individual drivers try to balance safety and mobility (speed). This research explores the relationship between speed limits and roadway safety. Namely, the research focuses on the influence of the posted speed limit on the causation and severity of accidents. Data on individual accidents from the Indiana Electronic Vehicle Crash Record System is used in the research, and appropriate statistical models are estimated for causation and severity of different types of accidents on all road classes. The results of the modeling show that speed limits do not have a statistically significant adverse effect on unsafe-speed-related causation of accidents on all roads, but generally increase the severity of accidents on the majority of roads other than highways (the accident severity on highways is unaffected by speed limits). Our findings can perhaps save both lives and travel time by helping the Indiana Department of Transportation determine optimal speed limit policies in the state."}
{"category": "Math", "title": "The Equations of Singular Loci of Ample Divisors on (Subvarieties of) Abelian Varieties", "abstract": "In this paper we consider ideal sheaves associated to the singular loci of a divisor in a linear system $|L|$ of an ample line bundle on a complex abelian variety. We prove an effective result on their (continuous) global generation, after suitable twists by powers of $L$. Moreover we show that similar results hold for subvarieties of a complex abelian variety."}
{"category": "Math", "title": "Local theta correspondence and the lifting of Duke, Imamoglu and Ikeda", "abstract": "We use results on the local theta correspondence to prove that for large degrees the Duke-Imamoglu-Ikeda lifting of an elliptic modular form is not a linear combination of theta series."}
{"category": "Math", "title": "Periods for flat algebraic connections", "abstract": "In previous work, we established a duality between the algebraic de Rham cohomology of a flat algebraic connection on a smooth quasi-projective surface over the complex numbers and the rapid decay homology of the dual connection relying on a conjecture by C. Sabbah, which has been proved recently by T. Mochizuki for algebraic connections in any dimension. In the present article, we verify that Mochizuki's results allow to generalize these duality results to arbitrary dimensions also."}
{"category": "Math", "title": "Canonical metrics of commuting maps", "abstract": "In the present work we establish the equality of the canonical metric of two commuting maps on an algebraic variety X. As a consequence the canonical height and measure associated to both maps are identical."}
{"category": "Math", "title": "Changes of variables in modulation and Wiener amalgam spaces", "abstract": "In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of modulation and Wiener amalgam spaces and, as a consequence, we obtain several versions of local and global Beurling-Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on Fourier Lebesgue spaces. Finally, counterparts of these results are discussed for spaces on the torus as well as for weighted spaces."}
{"category": "Math", "title": "New techniques for pointed Hopf algebras", "abstract": "We present techniques that allow to decide that the dimension of some pointed Hopf algebras associated with non-abelian groups is infinite. These results are consequences of arXiv:0803.2430v1. We illustrate each technique with applications."}
{"category": "Math", "title": "On a Generalised Lehmer Problem for Arbitrary Powers", "abstract": "We consider a generalisation of the classical Lehmer problem about the parity distribution of an integer and its modular inverse. We use some known estimates of exponential sums to study a more general question of simultaneous distribution of the residues of any fixed number of negative and positive powers of integers in prescribed arithmetic progressions. In particular, we improve and generalise a recent result of Y. Yi and W. Zhang."}
{"category": "Math", "title": "Obtaining the One-Holed Torus from Pants: Duality in an SL(3,C)-Character Variety", "abstract": "The SL(3,C)-representation variety R of a free group F arises naturally by considering surface group representations for a surface with boundary. There is a SL(3,C)-action on the coordinate ring of R. The geometric points of the subring of invariants of this action is an affine variety X. The points of X parametrize isomorphism classes of completely reducible representations. The coordinate ring C[X] is a complex Poisson algebra with respect to a presentation of F imposed by the surface. In previous work, we have worked out the bracket on all generators when the surface is a three-holed sphere and when the surface is a one-holed torus. In this paper, we show how the symplectic leaves corresponding to these two different Poisson structures on X relate to each other. In particular, they are symplectically dual at a generic point. Moreover, the topological gluing map which turns the three-holed sphere into the one-holed torus induces a rank preserving Poisson map on C[X]."}
{"category": "Math", "title": "Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. I", "abstract": "We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces."}
{"category": "Math", "title": "Homogenization of semi-linear PDEs with discontinuous effective coefficients", "abstract": "We study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in \\`{C}esaro sense. In such a case, the averaged coefficients could be discontinuous. We use probabilistic approach based on weak convergence for the associated backward stochastic differential equation in the S-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of \"$L^p-$viscosity solution\" introduced in \\cite{CCKS}. We use BSDEs techniques to establish the existence of $L^p-$viscosity solution for the averaged PDE. We establish weak continuity for the flow of the limit diffusion process and related the PDE limit to the backward stochastic differential equation via the representation of $L^p$-viscosity solution."}
{"category": "Math", "title": "Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease", "abstract": "We analyze a finite volume scheme for nonlocal SIR model, which is a nonlocal reaction-diffusion system modeling an epidemic disease. We establish existence solutions to the finite volume scheme, and show that it converges to a weak solution. The convergence proof is based on deriving series of a priori estimates and using a general $L^p$ compactness criterion."}
{"category": "Math", "title": "Local Polynomial Estimation for Sensitivity Analysis on Models With Correlated Inputs", "abstract": "Sensitivity indices when the inputs of a model are not independent are estimated by local polynomial techniques. Two original estimators based on local polynomial smoothers are proposed. Both have good theoretical properties which are exhibited and also illustrated through analytical examples. They are used to carry out a sensitivity analysis on a real case of a kinetic model with correlated parameters."}
{"category": "Math", "title": "Limiting Carleman weights and anisotropic inverse problems", "abstract": "In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic X-ray transform. Earlier results in dimension $n \\geq 3$ were restricted to real-analytic metrics."}
{"category": "Math", "title": "Maxima of Dirichlet and triangular arrays of gamma variables", "abstract": "Consider a rowwise independent triangular array of gamma random variables with varying parameters. Under several different conditions on the shape parameter, we show that the sequence of row-maximums converges weakly after linear or power transformation. Depending on the parameter combinations, we obtain both Gumbel and non-Gumbel limits. The weak limits for maximum of the coordinates of certain Dirichlet vectors of increasing dimension are also obtained using the gamma representation."}
{"category": "Math", "title": "On the gap between representability and collapsibility", "abstract": "A simplicial complex K is called d-representable if it is the nerve of a collection of convex sets in R^d; K is d-collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d-1 that is contained in a unique maximal face; and K is d-Leray if every induced subcomplex of K has vanishing homology of dimension d and larger. It is known that d-representable implies d-collapsible implies d-Leray, and no two of these notions coincide for d greater or equal to 2. The famous Helly theorem and other important results in discrete geometry can be regarded as results about d-representable complexes, and in many of these results \"d-representable\" in the assumption can be replaced by \"d-collapsible\" or even \"d-Leray\". We investigate \"dimension gaps\" among these notions, and we construct, for all positive integers d, a 2d-Leray complex that is not (3d-1)-collapsible and a d-collapsible complex that is not (2d-2)-representable. In the proofs we obtain two results of independent interest: (i) The nerve of every finite family of sets, each of size at most d, is d-collapsible. (ii) If the nerve of a simplicial complex K is d-representable, then K embeds in R^d."}
{"category": "Math", "title": "Integration with respect to local time and Ito's formula for smooth nondegenerate martingales", "abstract": "We show an It\\^ o's formula for nondegenerate Brownian martingales $X_t=\\int_0^t u_s dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in It\\^o's s formula as an integral over space and time with respect to local time."}
{"category": "Math", "title": "On the size of Nikodym sets in finite fields", "abstract": "Let $\\mathbb{F}_q$ denote a finite field of $q$ elements. Define a set $B\\subset\\mathbb{F}_q^n$ to be Nikodym if for each $x\\in B^{c}$, there exists a line $L$ such that $L\\cap B^c=\\{x\\}.$ The main purpose of this note is to show that the size of every Nikodym set is at least $C_n\\cdot q^n$, where $C_n$ depends only on $n$."}
{"category": "Math", "title": "Landau's necessary density conditions for LCA groups", "abstract": "H. Landau's necessary density conditions for sampling and interpolation may be viewed as a general principle resting on a basic fact of Fourier analysis: The complex exponentials $e^{i kx}$ ($k$ in $\\mathbb{Z}$) constitute an orthogonal basis for $L^2([-\\pi,\\pi])$. The present paper extends Landau's conditions to the setting of locally compact abelian (LCA) groups, relying in an analogous way on the basics of Fourier analysis. The technicalities--in either case of an operator theoretic nature--are however quite different. We will base our proofs on the comparison principle of J. Ramanathan and T. Steger."}
{"category": "Math", "title": "Symplectic maps of complex domains into complex space forms", "abstract": "Let $M\\subset{\\complex}^n$ be a complex domain of ${\\complex}^n$ endowed with a rotation invariant \\K form $\\omega_{\\Phi}= \\frac{i}{2} \\partial\\bar\\partial\\Phi$. In this paper we describe sufficient conditions on the \\K potential $\\Phi$ for $(M, \\omega_{\\Phi})$ to admit a symplectic embedding (explicitely described in terms of $\\Phi$) into a complex space form of the same dimension of $M$. In particular we also provide conditions on $\\Phi$ for $(M, \\omega_{\\Phi})$ to admit global symplectic coordinates. As an application of our results we prove that each of the Ricci flat (but not flat) \\K forms on ${\\complex}^2$ constructed by LeBrun (Taub-NUT metric) admits explicitely computable global symplectic coordinates."}
{"category": "Math", "title": "Riemannian geometry of Hartogs domains", "abstract": "Let $D_F = \\{(z_0, z) \\in {\\C}^{n} | |z_0|^2 < b, \\|z\\|^2 < F(|z_0|^2) \\}$ be a strongly pseudoconvex Hartogs domain endowed with the \\K metric $g_F$ associated to the \\K form $\\omega_F = -\\frac{i}{2} \\partial \\bar{\\partial} \\log (F(|z_0|^2) - \\|z\\|^2)$. This paper contains several results on the Riemannian geometry of these domains. In the first one we prove that if $D_F$ admits a non special geodesic (see definition below) through the origin whose trace is a straight line then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that all the geodesics through the origin of $D_F$ do not self-intersect, we find necessary and sufficient conditions on $F$ for $D_F$ to be geodesically complete and we prove that $D_F$ is locally irreducible as a Riemannian manifold. Finally, we compare the Bergman metric $g_B$ and the metric $g_F$ in a bounded Hartogs domain and we prove that if $g_B$ is a multiple of $g_F$, namely $g_B=\\lambda g_F$, for some $\\lambda\\in \\R^+$, then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space."}
{"category": "Math", "title": "Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field", "abstract": "We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann Hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval I will contain 2g|I| angles as the genus grows. We show that for the variance of number of angles in I is asymptotically a constant multiple of log(2g|I|) and prove a central limit theorem: The normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g|I| tends to infinity."}
{"category": "Math", "title": "Symplectic duality between complex domains", "abstract": "In this paper after extending the definition of symplectic duality (given by the first two authors in arXiv:math/0603141 for bounded symmetric domains) to arbitrary complex domains of ${\\C}^n$ centered at the origin we generalize some of the results proved in arXiv:math/0603141 and arXiv:0707.2125 to those domains."}
{"category": "Math", "title": "Lindelof type of generalization of separability in Banach spaces", "abstract": "We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: For each subset \\mathcal{F} of X^{\\ast}, which separates X, there exists a countable separating subset \\mathcal{F}_{0} of \\mathcal{F}. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Markucevic-bases, Corson property and related geometric issues are discussed."}
{"category": "Math", "title": "Hochschild homology and global dimension", "abstract": "We prove that for certain classes of graded algebras (Koszul, local, cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided the characteristic of the ground field is zero. Our proof uses Igusa's formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant."}
{"category": "Math", "title": "$N=1$ formal genus $0$ Gromov-Witten theories and Givental's formalism", "abstract": "In [Gi3] Givental introduced and studied a space of formal genus zero Gromov-Witten theories $GW_0$, i.e. functions satisfying string and dilaton equations and topological recursion relations. A central role in the theory plays the geometry of certain Lagrangian cones and a twisted symplectic group of hidden symmetries. In this note we show that the Lagrangian cones description of the action of this group coincides with the genus zero part of Givental's quantum Hamiltonian formalism. As an application we identify explicitly the space of $N=1$ formal genus zero GW theories with lower-triangular twisted symplectic group modulo the string flow."}
{"category": "Math", "title": "Classification of groups generated by 3-state automata over a 2-letter alphabet", "abstract": "This article contains most of the known results on the classification of groups generated by 3-state automata over a 2-letter alphabet, extending the previous papers 0704.3876 and math/0612178."}
{"category": "Math", "title": "Positive Realness of a Transfer Function Neither Implies Nor is Implied by the External Positivity of their Associate Realizations", "abstract": "This letter discusses the differences between the properties of positive realness of transfer functions and external positivity in linear time-invariant dynamic systems. It is proved that each one of both properties does not imply to each other."}
{"category": "Math", "title": "Lecture notes on duality and interpolation spaces", "abstract": "Known or essentially known results about duals of interpolation spaces are presented, taking a point of view sometimes slightly different from the usual one. Particular emphasis is placed on Alberto Calderon's theorem describing the duals of his complex interpolation spaces [A_0,A_1]_\\theta. The pace is slow, since these notes are intended for graduate students who have just begun to study interpolation spaces. This second version corrects some small misprints. It also draws attention to a convenient norming subspace of the dual of a complex interpolation space, and to the slight difference between the spaces \\mathcal{G}(X_0,X_1) introduced by Calderon and by Stafney."}
{"category": "Math", "title": "Definably complete and Baire structures and Pfaffian closure", "abstract": "We consider definably complete and Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain can not be written as the union of a definable increasing family of nowhere dense sets. Every expansion of the real field is definably complete and Baire. So is every o-minimal expansion of a field. However, unlike the o-minimal case, the structures considered form an elementary class. In this context we prove a version of Kuratowski-Ulam's Theorem and some restricted version of Sard's Lemma. We use the above results to prove the following version of Wilkie's Theorem of the Complement: given a definably complete Baire expansion K of an ordered field with a family of smooth functions, if there are uniform bounds on the number of definably connected components of quantifier free definable sets, then K is o-minimal. We further generalize the above result, along the line of Speissegger's theorem, and prove the o-minimality of the relative Pfaffian closure of an o-minimal structure inside a definably complete Baire structure."}
{"category": "Math", "title": "The bigger Brauer group and twisted sheaves", "abstract": "Given an algebraic stack with quasiaffine diagonal, we show that each G_m-gerbe comes from a central separable algebra. In other words, Taylor's bigger Brauer group equals the etale cohomology in degree two with coefficients in G_m. This gives new results also for schemes. We use the method of twisted sheaves explored by de Jong and Lieblich."}
{"category": "Math", "title": "Whittaker Modules for Generalized Weyl Algebras", "abstract": "We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of sl_2 and of Heisenberg Lie algebras, Smith's generalizations of U(sl_2), various quantum analogues of these algebras, and many others. We show that the Whittaker modules V = Aw of the generalized Weyl algebra A = R(phi,t) are in bijection with the phi-stable left ideals of R. We determine the annihilator Ann_A(w) of the cyclic generator w of V. We also describe the annihilator ideal Ann_A(V) under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant's well-known results on Whittaker modules and their associated annihilators for U(sl_2)."}
{"category": "Math", "title": "The stable free rank of symmetry of products of spheres", "abstract": "A well known conjecture in the theory of transformation groups states that if p is a prime and (Z/p)^r acts freely on a product of k spheres, then r is less than or equal to k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non simply connected spaces."}
{"category": "Math", "title": "Harmonic maps from degenerating Riemann surfaces", "abstract": "We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy and show the so-called generalized energy identity. We find conditions that are both necessary and sufficient for the compactness in $W^{1,2}$ and $C^{0}$ modulo bubbles of sequences of such maps."}
{"category": "Math", "title": "On Morita theory for self-dual modules", "abstract": "Let $G$ be a finite group and let $k$ be a field of characteristic $p$. It is known that a $kG$-module $V$ carries a non-degenerate $G$-invariant bilinear form $b$ if and only if $V$ is self-dual. We show that whenever a Morita bimodule $M$ which induces an equivalence between two blocks $B(kG)$ and $B(kH)$ of group algebras $kG$ and $kH$ is self-dual then the correspondence preserves self-duality. Even more, if the bilinear form on $M$ is symmetric then for $p$ odd the correspondence preserves the geometric type of simple modules. In characteristic 2 this holds also true for projective modules."}
{"category": "Math", "title": "On the zeros of certain modular functions for the normalizers of congruence subgroups of low levels II", "abstract": "We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In Part II, we will observe the location of the zeros of the above functions by numerical calculation."}
{"category": "Math", "title": "Differentiating polynomials, and zeta(2)", "abstract": "We study the derivatives of polynomials with equally spaced zeros and find connections to the values of the Riemann zeta-function at the positive even integers."}
{"category": "Math", "title": "Algebras of higher operads as enriched categories", "abstract": "We decribe the correspondence between normalised $\\omega$-operads and certain lax monoidal structures on the category of globular sets. As with ordinary monoidal categories, one has a notion of category enriched in a lax monoidal category. Within the aforementioned correspondence, we provide also an equivalence between the algebras of a given normalised $\\omega$-operad, and categories enriched in globular sets for the induced lax monoidal structure."}
{"category": "Math", "title": "On continuous fields of JB-algebras", "abstract": "We introduce and study continuous fields of JB-algebras (which are real non-associate analogues of C*-algebras). In particular, we show that for the universal enveloping C*-algebra C*sub-u(B) for the JB-algebra B defined by a continuous field of JB-algebras A-sub-t, t belongs to T, on a locally compact space T, there exists a decomposition of C*-sub-u(B) into a continuous field of C*-algebras C*u(A-sub-t), t belongs to T, on the same space T, composed entirely of the universal enveloping C*-algebras of the corresponding JB-algebras from the aforementioned decomposition of the algebra B."}
{"category": "Math", "title": "On Intrinsic Characterization of Real Locally C*- and Locally JB-Algebras", "abstract": "In the present paper we obtain an intrinsic characterization of real locally C*-algebras (projective limits of projective families of real C*-algebras) among complete real lmc *-algebras, and of locally JB-algebras (projective limits of projective families of JB-algebras) among complete fine Jordan locally multiplicatively-convex topological algebras."}
{"category": "Math", "title": "Braid group B_3 irreducibles, a DIY guide", "abstract": "This note tells you how to construct a k(n)-dimensional family of (isomorphism classes of) irreducible representations of dimension n for the three string braid group B_3, where k(n) is an admissible function of your choosing; for example take k(n) = [ n/2 ] +1 as in arXiv:0803.2778 and arXiv:0803.2785."}
{"category": "Math", "title": "Certain Properties of Pythagorean Triangles involving the interior diameter, and the exterior diameters", "abstract": "There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is Pythagorean, the four diameters are integers. Consider a Pythagorean triangle with the property that one leglength is a perfect(or integer)square, and with one of the four diameters also a integer square.Of the eight resulting combinations, we prove that only six are possible or can occur. We then completely parametrically describe the six families; each corresponding to one of the six combinations."}
{"category": "Math", "title": "M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems", "abstract": "In this paper, we combine results on extensions of operators with recent results on the relation between the M-function and the spectrum, to examine the spectral behaviour of boundary value problems. M-functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that can not be detected by the M-function."}
{"category": "Math", "title": "Parabolic induction and restriction functors for rational Cherednik algebras", "abstract": "We introduce parabolic induction and restriction functors for rational Cherednik algebras, and study their basic properties. Then we discuss applications of these functors to representation theory of rational Cherednik algebras. In particular, we prove the Gordon-Stafford theorem about Morita equivalence of the rational Cherednik algebra for type A and its spherical subalgebra, without the assumption that c is not a half-integer, which was required up to now. Also, we classify representations from category O over the rational Cherednik algebras of type A which do not contain an S_n-invariant vector, and confirm a conjecture of Okounkov and the first author on the number of such representations. In the second version we have added a result on the simplicity of the spherical Cherednik algebra of type A for -1<c<0, and a strengthened version of the main result of arXiv:math/0312474, as well as an appendix by the second author containing arXiv:0706.4308, on the reducibility of the polynomial representation of the trigonometric Cherednik algebra."}
{"category": "Math", "title": "Characterizing Hilbert spaces using Fourier transform over the field of p-adic numbers", "abstract": "We characterize Hilbert spaces in the class of all Banach spaces using Fourier transform of vector-valued functions over the field $Q_p$ of $p$-adic numbers. Precisely, Banach space $X$ is isomorphic to a Hilbert one if and only if Fourier transform $F: L_2(Q_p,X)\\to L_2(Q_p,X)$ in space of functions, which are square-integrable in Bochner sense and take value in $X$, is a bounded operator."}
{"category": "Math", "title": "Affine partitions and affine Grassmannians", "abstract": "We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of Bousquet-Melou-Eriksson, Eriksson-Eriksson and Reiner. In other types the identities appear to be new. For type $A_{n}$, the affine colored partitions form another family of combinatorial objects in bijection with $n+1$-core partitions and $n$-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmannians in terms of affine partitions and a generalization of Young's lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted."}
{"category": "Math", "title": "A categorification of quantum sl(2)", "abstract": "We categorify Lusztig's version of the quantized enveloping algebra for sl(2). Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to Lusztig's algebra. The indecomposable morphisms of this 2-category lift Lusztig's canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form defined on quantum sl(2). Graded lifts of various homomorphisms and antihomomorphisms of Lusztig's algebra arise naturally in the context of our graphical calculus. Using iterated flag varieties, a representation of the 2-category is constructed for each positive integer N. This representation categorifies the irreducible (N+1)-dimensional representation of quantum sl(2)."}
{"category": "Math", "title": "Free products, cyclic homology, and the Gauss-Manin connection", "abstract": "We present a new approach to cyclic homology that does not involve the Connes differential and is based on a `noncommutative equivariant de Rham complex' of an associative algebra. The differential in that complex is a sum of the Karoubi-de Rham differential, which replaces the Connes differential, and another operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit construction of the Gauss-Manin connection, introduced earlier by E. Getzler, on the relative cyclic homology of a flat family of associative algebras over a central base ring. We introduce and study `free-product deformations' of an associative algebra, a new type of deformation over a not necessarily commutative base ring. Natural examples of free-product deformations arise from preprojective algebras and group algebras for compact surface groups."}
{"category": "Math", "title": "The Equivalence between Uniqueness and Continuous Dependence of Solution for BSDEs with Continuous Coefficient", "abstract": "In this paper, we will prove that, if the coefficient $g=g(t,y,z)$ of a BSDE is assumed to be continuous and linear growth in $(y,z)$, then the uniqueness of solution and continuous dependence with respect to $g$ and the terminal value $\\xi$ are equivalent."}
{"category": "Math", "title": "Unramified extensions and geometric $\\mathbb{Z}_p$-extensions of global function fields", "abstract": "We study on finite unramified extensions of global function fields (function fields of one valuable over a finite field). We show two results. One is an extension of Perret's result about the ideal class group problem. Another is a construction of a geometric $\\mathbb{Z}_p$-extension which has a certain property."}
{"category": "Math", "title": "Integration with respect to fractional local times with Hurst index $H$ greater than 1/2", "abstract": "Let ${\\mathscr L}^H(x,t)=2H\\int_0^t\\delta(B^H_s-x)s^{2H-1}ds$ be the weighted local time of fractional Brownian motion $B^H$ with Hurst index $1/2<H<1$. In this paper, we use Young integration to study the integral of determinate functions $\\int_{\\mathbb R}f(x){\\mathscr L}^H(dx,t)$. As an application, we investigate the {\\it weighted quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t:=\\lim_{n\\to \\infty}2H\\sum_{k=0}^{n-1} k^{2H-1}\\{f(B^H_{t_{k+1}})-f(B^H_{t_{k}})\\}(B^H_{t_{k+1}}-B^H_{t_{k}}), $$ where the limit is uniform in probability and $t_k=kt/n$. We show that it exists and $$ [f(B^H),B^H]^{(W)}_t=-\\int_{\\mathbb R}f(x){\\mathscr L}^H(dx,t), $$ provided $f$ is of bounded $p$-variation with $1\\leq p<\\frac{2H}{1-H}$. Moreover, we extend this result to the time-dependent case. These allow us to write the fractional It\\^{o} formula for new classes of functions."}
{"category": "Math", "title": "A simply connected surface of general type with p_g=0 and K^2=4", "abstract": "As the sequel to [5, 7], we construct a simply connected minimal complex surface of general type with p_g = 0 and K^2 = 4 by using a rational blow-down surgery and Q-Gorenstein smoothing theory."}
{"category": "Math", "title": "Categorification of integrable representations of quantum groups", "abstract": "We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As byproducts, we obtain a geometric realization of Lusztig's canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties."}
{"category": "Math", "title": "Deforming motivic theories I: Pure weight perfect Modules on divisorial schemes", "abstract": "In this paper, we introduce a notion of weight r pseudo-coherent Modules associated to a regular closed immersion i:Y -> X of codimension r, and prove that there is a canonical derived Morita equivalence between the DG-category of perfect complexes on a divisorial scheme X whose cohomological support are in Y and the DG-category of bounded complexes of weight r pseudo-coherent O_X-Modules supported on Y. The theorem implies that there is the canonical isomorphism between the Bass-Thomason-Trobaugh non-connected K-theory [TT90], [Sch06] (resp. the Keller-Weibel cyclic homology [Kel98], [Wei96]) for the immersion and the Schlichting non-connected K-theory [Sch04] associated to (resp. that of) the exact category of weight r pseudo-coherent Modules. For the connected K-theory case, this result is just Exercise 5.7 in [TT90]. As its application, we will decide on a generator of the topological filtration on the non-connected K-theory (resp. cyclic homology theory) for affine Cohen-Macaulay schemes."}
{"category": "Math", "title": "A new stochastic process to model Heart Rate series during exhaustive run and an estimator of its fractality parameter", "abstract": "In order to interpret and explain the physiological signal behaviors, it can be interesting to find some constants among the fluctuations of these data during all the effort or during different stages of the race (which can be detected using a change points detection method). Several recent papers have proposed the long-range dependence (Hurst) parameter as such a constant. However, their results induce two main problems. Firstly, DFA method is usually applied for estimating this parameter. Clearly, such a method does not provide the most efficient estimator and moreover it is not at all robust even in the case of smooth trends. Secondly, this method often gives estimated Hurst parameters larger than 1, which is the larger possible value for long memory stationary processes. In this article we propose solutions for both these problems and we define a new model allowing such estimated parameters."}
{"category": "Math", "title": "Variable selection for the multicategory SVM via adaptive sup-norm regularization", "abstract": "The Support Vector Machine (SVM) is a popular classification paradigm in machine learning and has achieved great success in real applications. However, the standard SVM can not select variables automatically and therefore its solution typically utilizes all the input variables without discrimination. This makes it difficult to identify important predictor variables, which is often one of the primary goals in data analysis. In this paper, we propose two novel types of regularization in the context of the multicategory SVM (MSVM) for simultaneous classification and variable selection. The MSVM generally requires estimation of multiple discriminating functions and applies the argmax rule for prediction. For each individual variable, we propose to characterize its importance by the supnorm of its coefficient vector associated with different functions, and then minimize the MSVM hinge loss function subject to a penalty on the sum of supnorms. To further improve the supnorm penalty, we propose the adaptive regularization, which allows different weights imposed on different variables according to their relative importance. Both types of regularization automate variable selection in the process of building classifiers, and lead to sparse multi-classifiers with enhanced interpretability and improved accuracy, especially for high dimensional low sample size data. One big advantage of the supnorm penalty is its easy implementation via standard linear programming. Several simulated examples and one real gene data analysis demonstrate the outstanding performance of the adaptive supnorm penalty in various data settings."}
{"category": "Math", "title": "Linearity defects of modules over commutative rings", "abstract": "This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the applications, it is proved that a local ring is Koszul if and only if it admits a Koszul module that is Cohen-Macaulay of minimal degree. An injective analogue of the linearity defect is introduced and studied. The main results express this new invariant in terms of linearity defects of free resolutions, and relate it to other ring theoretic and homological invariants of the module."}
{"category": "Math", "title": "Editorial: Statistics and \"The lost tomb of Jesus\"", "abstract": "What makes a problem suitable for statistical analysis? Are historical and religious questions addressable using statistical calculations? Such issues have long been debated in the statistical community and statisticians and others have used historical information and texts to analyze such questions as the economics of slavery, the authorship of the Federalist Papers and the question of the existence of God. But what about historical and religious attributions associated with information gathered from archeological finds? In 1980, a construction crew working in the Jerusalem neighborhood of East Talpiot stumbled upon a crypt. Archaeologists from the Israel Antiquities Authority came to the scene and found 10 limestone burial boxes, known as ossuaries, in the crypt. Six of these had inscriptions. The remains found in the ossuaries were reburied, as required by Jewish religious tradition, and the ossuaries were catalogued and stored in a warehouse. The inscriptions on the ossuaries were catalogued and published by Rahmani (1994) and by Kloner (1996) but there reports did not receive widespread public attention. Fast forward to March 2007, when a television ``docudrama'' aired on The Discovery Channel entitled ``The Lost Tomb of Jesus'' touched off a public and religious controversy--one only need think about the title to see why there might be a controversy! The program, and a simultaneously published book [Jacobovici and Pellegrino (2007)], described the ``rediscovery'' of the East Talpiot archeological find and they presented interpretations of the ossuary inscriptions from a number of perspectives. Among these was a statistical calculation attributed to the statistician Andrey Feuerverger: ``that the odds that all six names would appear together in one tomb are 1 in 600, calculated conservatively--or possibly even as much as one in one million.''"}
{"category": "Math", "title": "The generality of the zero-one laws", "abstract": "We prove game-theoretic generalizations of some well known zero-one laws. Our proofs make the martingales behind the laws explicit, and our results illustrate how martingale arguments can have implications going beyond measure-theoretic probability."}
{"category": "Math", "title": "Asymptotic stability of solitons for the Benjamin-Ono equation", "abstract": "In this paper, we prove the asymptotic stability of the family of solitons of the Benjamin-Ono equation in the energy space. The proof is based on a Liouville property for solutions close to the solitons for this equation, in the spirit of Martel and Merle (arXiv:0706.1174v2). As a corollary of the proofs, we obtain the asymptotic stability of exact multi-solitons."}
{"category": "Math", "title": "Cohomology of twisted tensor products", "abstract": "It is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies. We look at twisted tensor products, and investigate to which extend this is still true. We give an explicit description of the $\\Ext$-algebra of the tensor product of two modules, and under certain additional conditions, describe an essential part of the Hochschild cohomology ring of a twisted tensor product. As an application, we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the Hochschild cohomology ring. Moreover, both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra."}
{"category": "Math", "title": "Principal 2-bundles and their gauge 2-groups", "abstract": "In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Cech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups possess a natural smooth structure. In the last section we provide some explicit examples."}
{"category": "Math", "title": "Simple proofs for universal binary Hermitian lattices", "abstract": "If a positive definite Hermitian lattice represents all positive integers, we call it universal. Several mathematicians, including the author, found 25 universal binary Hermitian lattices. But their ad hoc proofs are complicated. We give simple and unified proofs."}
{"category": "Math", "title": "In-season prediction of batting averages: A field test of empirical Bayes and Bayes methodologies", "abstract": "Batting average is one of the principle performance measures for an individual baseball player. It is natural to statistically model this as a binomial-variable proportion, with a given (observed) number of qualifying attempts (called ``at-bats''), an observed number of successes (``hits'') distributed according to the binomial distribution, and with a true (but unknown) value of $p_i$ that represents the player's latent ability. This is a common data structure in many statistical applications; and so the methodological study here has implications for such a range of applications. We look at batting records for each Major League player over the course of a single season (2005). The primary focus is on using only the batting records from an earlier part of the season (e.g., the first 3 months) in order to estimate the batter's latent ability, $p_i$, and consequently, also to predict their batting-average performance for the remainder of the season. Since we are using a season that has already concluded, we can then validate our estimation performance by comparing the estimated values to the actual values for the remainder of the season. The prediction methods to be investigated are motivated from empirical Bayes and hierarchical Bayes interpretations. A newly proposed nonparametric empirical Bayes procedure performs particularly well in the basic analysis of the full data set, though less well with analyses involving more homogeneous subsets of the data. In those more homogeneous situations better performance is obtained from appropriate versions of more familiar methods. In all situations the poorest performing choice is the na\\\"{{\\i}}ve predictor which directly uses the current average to predict the future average."}
{"category": "Math", "title": "Models of Z/p^2 Z over a d.v.r. of unequal characteristic", "abstract": "Let R be a discrete valuation ring of unequal characteristic which contains a primitive p^2-th root of unity. If K is the fraction field of R, it is well known that (Z/p^2 Z)_K is isomorphic to \\mu_{p^2,K}. We prove that any finite and flat R-group scheme of order p^2 isomorphic to (Z/p^2 Z)_K on the generic fiber (i.e. a model of (Z/p^2 Z)_K), is the kernel in a short exact sequence which generically coincides with the Kummer sequence. We will explicitly describe and classify such models."}
{"category": "Math", "title": "On bijections between 231-avoiding permutations and Dyck paths", "abstract": "We construct a bijection between 231-avoiding permutations and Dyck paths that sends the sum of the major index and the inverse major index of a 231-avoiding permutation to the major index of the corresponding Dyck path. Furthermore, we relate this bijection to others and exhibit a bistatistic on 231-avoiding permutations which is related to the q,t-Catalan numbers."}
{"category": "Math", "title": "An equivariant version of the monodromy zeta function", "abstract": "We offer an equivariant version of the classical monodromy zeta function of a singularity as a series with coefficients from the Grothendieck ring of finite G-sets tensored by the field of rational numbers. Main two ingredients of the definition are equivariant Lefschetz numbers and the lambda-structure on the Grothendieck ring of finite G-sets. We give an A'Campo type formula for the equivariant zeta function."}
{"category": "Math", "title": "Radial Balanced metrics on the unit disk", "abstract": "Let $\\Phi$ be a strictly plurisubharmonic and radial function on the unit disk ${\\cal D}\\subset {\\complex}$ and let $g$ be the \\K metric associated to the \\K form $\\omega =\\frac{i}{2}\\partial\\bar\\partial\\Phi$. We prove that if $g$ is $g_{eucl}$-balanced of height 3 (where $g_{eucl}$ is the standard Euclidean metric on ${\\complex}={\\real}^2$), and the function $h(x)=e^{-\\Phi (z)}$, $x=|z|^2$, extends to an entire analytic function on ${\\real}$, then $g$ equals the hyperbolic metric. The proof of our result is based on a interesting characterization of the function $f(x)=1-x$."}
{"category": "Math", "title": "Numerical Algorithms and Simulations for Reflected Backward Stochastic Differential Equations with two Continuous Barriers", "abstract": "In this paper we study different algorithms for reflected backward stochastic differential equations (BSDE in short) with two continuous barriers basing on random work framework. We introduce different numerical algorithms by penalization method and reflected method. At last simulation results are also presented."}
{"category": "Math", "title": "A new principle for choosing regularization parameter in certain inverse problems", "abstract": "A new parameter choice rule for inverse problems is introduced. This parameter choice rule was developed for total variation regularization in electron tomography and might in general be useful for $L^1$ regularization of inverse problems with high levels of noise in the data."}
{"category": "Math", "title": "On distributional properties of perpetuities", "abstract": "We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of $p$-moments, $p>0$ as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article."}
{"category": "Math", "title": "Dirac-harmonic maps from degenerating spin surfaces I: the Neveu-Schwarz case", "abstract": "We study Dirac-harmonic maps from degenerating spin surfaces with uniformly bounded energy and show the so-called generalized energy identity in the case that the domain converges to a spin surface with only Neveu-Schwarz type nodes. We find condition that is both necessary and sufficient for the $W^{1,2} \\times L^{4}$ modulo bubbles compactness of a sequence of such maps."}
{"category": "Math", "title": "Dynamic Physical Systems: Energy Balances and Stability Issues", "abstract": "This paper discusses some links properties of operators with the well- known physical concepts of hyperstability, passivity, energy dissipativeness and conservativeness with positive realness properties of the transfer functions in linear dynamic systems."}
{"category": "Math", "title": "Effective models and extension of torsors over a discrete valuation ring of unequal characteristic", "abstract": "Let R be a discrete valuation ring of unequal characteristic with fraction field K which contains a primitive p^2-th root of unity. Let X be a faithfully flat R-scheme and G be a finite abstract group. Let us consider a G-torsor Y_K\\to X_K and let Y be the normalization of X_K in Y. If G=Z/p^n Z, n<3, under some hypothesis on X, we attach some invariants to Y_K \\to X_K. If p>2, we determine, through these invariants, when Y\\to X has a structure of torsor which extends that of Y_K\\to X_K. Moreover we explicitly calculate the effective model (defined by Romagny) of the action of G on Y."}
{"category": "Math", "title": "Structure of the cuspidal rational torsion subgroup of J_1(p^n)", "abstract": "In this article, we determine the structure of the $p$-primary subgroup of the cuspidal rational torsion subgroup of the Jacobian $J_1(p^n)$ of the modular curve $X_1(p^n)$ for a regular prime $p$."}
{"category": "Math", "title": "On sums of primes and triangular numbers", "abstract": "We study whether sufficiently large integers can be written in the form cp+T_x, where p is either zero or a prime congruent to r mod d, and T_x=x(x+1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2^ap-r)/m+T_x with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with related analysis and numerical data. One of our conjectures states that each natural number not equal to 216 can be written in the form p+T_x with x an integer and p a prime or zero; another conjecture asserts that any odd integer n>3 can be written in the form p+x(x+1) with p a prime and x a positive integer."}
{"category": "Math", "title": "Eigenvalue and Dirichlet problem for fully-nonlinear operators in non smooth domains", "abstract": "In this paper we study the maximum principle, the existence of eigenvalue and the existence of solution for the Dirichlet problem for operators which are fully-nonlinear, elliptic but presenting some singularity or degeneracy which are similar to those of the p-Laplacian, the novelty resides in the fact that we consider the equations in bounded domains which only satisfy the exterior cone condition."}
{"category": "Math", "title": "False discovery rate analysis of brain diffusion direction maps", "abstract": "Diffusion tensor imaging (DTI) is a novel modality of magnetic resonance imaging that allows noninvasive mapping of the brain's white matter. A particular map derived from DTI measurements is a map of water principal diffusion directions, which are proxies for neural fiber directions. We consider a study in which diffusion direction maps were acquired for two groups of subjects. The objective of the analysis is to find regions of the brain in which the corresponding diffusion directions differ between the groups. This is attained by first computing a test statistic for the difference in direction at every brain location using a Watson model for directional data. Interesting locations are subsequently selected with control of the false discovery rate. More accurate modeling of the null distribution is obtained using an empirical null density based on the empirical distribution of the test statistics across the brain. Further, substantial improvements in power are achieved by local spatial averaging of the test statistic map. Although the focus is on one particular study and imaging technology, the proposed inference methods can be applied to other large scale simultaneous hypothesis testing problems with a continuous underlying spatial structure."}
{"category": "Math", "title": "From cardinal spline wavelet bases to highly coherent dictionaries", "abstract": "Wavelet families arise by scaling and translations of a prototype function, called the {\\em {mother wavelet}}. The construction of wavelet bases for cardinal spline spaces is generally carried out within the multi-resolution analysis scheme. Thus, the usual way of increasing the dimension of the multi-resolution subspaces is by augmenting the scaling factor. We show here that, when working on a compact interval, the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter. By such a procedure we generate a redundant frame, called a {\\em{dictionary}}, spanning the same spaces as a wavelet basis but with wavelets of broader support. We characterise the correlation of the dictionary elements by measuring their `coherence' and produce examples illustrating the relevance of highly coherent dictionaries to problems of sparse signal representation."}
{"category": "Math", "title": "Einstein Metrics, Complex Surfaces, and Symplectic 4-Manifolds", "abstract": "Which smooth compact 4-manifolds admit an Einstein metric with non-negative Einstein constant? A complete answer is provided in the special case of 4-manifolds that also happen to admit either a complex structure or a symplectic structure."}
{"category": "Math", "title": "Transcience/recurrence for normally reflected Brownian motion in unbounded domains", "abstract": "Let $D\\subsetneq R^d$ be an unbounded domain and let $B(t)$ be a Brownian motion in $D$ with normal reflection at the boundary. We study the transcience/recurrence dichotomy, focusing mainly on domains of the form $D=\\{(x,z)\\in R^{l+m}:|z|<H(|x|)\\}$, where $d=l+m$ and $H$ is a sufficiently regular function. This class of domains includes various horn-shaped domains and generalized slab domains."}
{"category": "Math", "title": "Conditional Haar measures on classical compact groups", "abstract": "We give a probabilistic proof of the Weyl integration formula on U(n), the unitary group with dimension $n$. This relies on a suitable definition of Haar measures conditioned to the existence of a stable subspace with any given dimension $p$. The developed method leads to the following result: for this conditional measure, writing $Z_U^{(p)}$ for the first nonzero derivative of the characteristic polynomial at 1, \\[\\frac{Z_U^{(p)}}{p!}\\stackrel{\\mathrm{law}}{=}\\prod_{\\ell =1}^{n-p}(1-X_{\\ell}),\\] the $X_{\\ell}$'s being explicit independent random variables. This implies a central limit theorem for $\\log Z_U^{(p)}$ and asymptotics for the density of $Z_U^{(p)}$ near 0. Similar limit theorems are given for the orthogonal and symplectic groups, relying on results of Killip and Nenciu."}
{"category": "Math", "title": "On regression adjustments in experiments with several treatments", "abstract": "Regression adjustments are often made to experimental data. Since randomization does not justify the models, bias is likely; nor are the usual variance calculations to be trusted. Here, we evaluate regression adjustments using Neyman's nonparametric model. Previous results are generalized, and more intuitive proofs are given. A bias term is isolated, and conditions are given for unbiased estimation in finite samples."}
{"category": "Math", "title": "Stability of Delayed Systems Modeled by Fractional Models", "abstract": "The paper discusses linear fractional representations of parameter-dependent nonlinear systems with dynamics defined by real rational nonlinearities and a finite set of point delays. The global asymptotic stability is investigated via linear matrix inequalities."}
{"category": "Math", "title": "Cohomology and generic cohomology of Specht modules for the symmetric group", "abstract": "Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group results. Using work of Doty on the submodule structure of symmetric powers of the natural GL_d(k) module together with work of Andersen on cohomology for B and its Frobenius kernels, we prove new results about H^i(\\Sigma_d, S^\\lambda). We recover work of James in the case i=0. Then we prove two stability theorems, one of which is a \"generic cohomology\" result for Specht modules equating cohomology of S^{p\\lambda} with S^{p^2\\lambda}. This is the first theorem we know relating Specht modules S^\\lambda and S^{p\\lambda}. The second result equates cohomology of S^\\lambda with S^{\\lambda + p^a\\mu} for large a."}
{"category": "Math", "title": "Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras", "abstract": "Let $\\alpha,\\beta\\in(0,1)$ and \\[ K^{\\alpha,\\beta}:=\\left\\{a\\in L^\\infty(\\T): \\sum_{k=1}^\\infty |\\hat{a}(-k)|^2 k^{2\\alpha}<\\infty, \\sum_{k=1}^\\infty |\\hat{a}(k)|^2 k^{2\\beta}<\\infty \\right\\}. \\] Mark Krein proved in 1966 that $K^{1/2,1/2}$ forms a Banach algebra. He also observed that this algebra is important in the asymptotic theory of finite Toeplitz matrices. Ten years later, Harold Widom extended earlier results of Gabor Szeg\\H{o} for scalar symbols and established the asymptotic trace formula \\[ \\operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1) \\quad\\text{as}\\ n\\to\\infty \\] for finite Toeplitz matrices $T_n(a)$ with matrix symbols $a\\in K^{1/2,1/2}_{N\\times N}$. We show that if $\\alpha+\\beta\\ge 1$ and $a\\in K^{\\alpha,\\beta}_{N\\times N}$, then the Szeg\\H{o}-Widom asymptotic trace formula holds with $o(1)$ replaced by $o(n^{1-\\alpha-\\beta})$."}
{"category": "Math", "title": "Quantum Bounded Symmetric Domains", "abstract": "This is Leonid Vaksman's monograph \"Quantum bounded symmetric domains\" (in Russian), preceded with an English translation of the table of contents and (a part) of the introduction. Quantum bounded symmetric domains are interesting from several points of view. In particular, they provide interesting examples for noncommutative complex analysis (i.e., the theory of subalgebras of C^*-algebars) initiated by W. Arveson."}
{"category": "Math", "title": "Triangle angles and sides in progression and the diophantine equation x^2+3y^2 =z^2", "abstract": "The main result of this paper, is the complete parametric description of the family of triangles which have integer sidelengths and with one angle being sixty degrees."}
{"category": "Math", "title": "Fold maps, framed immersions and smooth structures", "abstract": "For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q, the splitting theorem implies that the cobordism group of fold maps to a manifold N is a sum of q/2 cobordism groups of framed immersions to N and a group related to diffeomorphism groups of manifolds of dimension q+1. Similarly, in the case of odd q, the cobordism group of fold maps splits off (q-1)/2 cobordism groups of framed immersions. The proof of the splitting theorem gives a partial splitting of the homotopy cofiber sequence of Thom spectra in the Madsen-Weiss approach to diffeomorphism groups of manifolds."}
{"category": "Math", "title": "Analytic Functions of a Quaternionic Variable", "abstract": "Here we follow the basic analysis that is common for real and complex variables and find how it can be applied to a quaternionic variable. Non-commutativity of the quaternion algebra poses obstacles for the usual manipulations; but we show how many of those obstacles can be overcome. After a tiny bit of linear algebra we look at the beginnings of differential calculus. The surprising result is that the first order term in the expansion of F(x+delta) is a compact formula involving both F'(x) and [F(x) - F(x*)]/(x-x*)."}
{"category": "Math", "title": "Stability in $H^{1/2}$ of the sum of $K$ solitons for the Benjamin-Ono equation", "abstract": "This note proves the orbital stability in the energy space $H^{1/2}$ of the sum of widely-spaced 1-solitons for the Benjamin-Ono equation, with speeds arranged so as to avoid collisions."}
{"category": "Math", "title": "New Proof of the Equation $\\sum_{k=1}^\\infty \\frac{\\mu(k)}{k}=0.$", "abstract": "This is an English translation of Edmund Landau's Doctoral Dissertation."}
{"category": "Math", "title": "Hilbert modular forms of weight 1/2 and theta functions", "abstract": "Serre and Stark found a basis for the space of modular forms of weight 1/2 in terms of theta series. In this paper, we generalize their result - under certain mild restrictions on the level and character - to the case of weight 1/2 Hilbert modular forms over a totally real field of narrow class number 1. The methods broadly follow those of Serre-Stark; however we are forced to overcome technical difficulties which arise when we move out of Q."}
{"category": "Math", "title": "On metrics of positive Ricci curvature conformal to MxR^m", "abstract": "Let (M, g) be a closed Riemannian manifold and gE the Euclidean metric. We show that for m > 1, (M x R^m, (g + gE)) is not conformal to a positive Einstein manifold. Moreover, (M x R^m, (g + gE)) is not conformal to a Riemannian manifold of positive Ricci curvature, through a smooth, radial, positive, integrable function of R^m, for m > 1. These results are motivated by some recent questions on Yamabe constants."}
{"category": "Math", "title": "Convergence of the Eilenberg-Moore spectral sequence for generalized cohomology theories", "abstract": "We prove that the Morava-$K$-theory-based Eilenberg-Moore spectral sequence has good convergence properties whenever the base space is a $p$-local finite Postnikov system with vanishing $(n+1)$st homotopy group."}
{"category": "Math", "title": "Largeness of LERF and 1-relator groups", "abstract": "We consider largeness of groups given by a presentation of deficiency 1, where the group is respectively free-by-cyclic, LERF or 1-relator. We give the first examples of (finitely generated free)-by-(infinite cyclic) word hyperbolic groups which are large, show that a LERF deficiency 1 group with first Betti number at least 2 is large or the integers times the integers, and show that 2-generator 1-relator groups where the relator has height 1 obey the dichotomy that either the group is large or all its finite images are metacyclic."}
{"category": "Math", "title": "The elliptic GL(n) dynamical quantum group as an h-Hopf algebroid", "abstract": "Using the language of h-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group, F_ell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra sl_n. We apply the generalized FRST construction and obtain an h-bialgebroid F_ell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the h-Hopf algebroid F_ell(GL(n))."}
{"category": "Math", "title": "Spanning sets for Moebius vertex algebras satisfying arbitrary difference conditions", "abstract": "Spanning sets for vertex operator algebras satisfying difference-zero and difference-one conditions have been extensively studied in the recent years. In this paper, we extend these results. More specifically, we show that for a suitably chosen generating set, any N-graded Moebius vertex algebra is spanned by monomials satisfying a difference-N ordering condition."}
{"category": "Math", "title": "A conjectured lower bound for the cohomological dimension of elliptic spaces. Some results in some simple cases", "abstract": "Here we prove some special cases of the following conjecture: that the sum of the Betti numbers of a 1-connected elliptic space is greater than the total rank of its homotopy groups. Our main tool is Sullivan's minimal model."}
{"category": "Math", "title": "Convergence rates for adaptive finite elements", "abstract": "In this article we prove that it is possible to construct, using newest-vertex bisection, meshes that equidistribute the error in $H^1$-norm, whenever the function to approximate can be decomposed as a sum of a regular part plus a singular part with singularities around a finite number of points. This decomposition is usual in regularity results of Partial Differential Equations (PDE). As a consequence, the meshes turn out to be quasi-optimal, and convergence rates for adaptive finite element methods (AFEM) using Lagrange finite elements of any polynomial degree are obtained."}
{"category": "Math", "title": "The homology of the stable non-orientable mapping class group", "abstract": "Combining results of Wahl, Galatius--Madsen--Tillmann--Weiss and Korkmaz one can identify the homotopy-type of the classifying space of the stable non-orientable mapping class group $N_\\infty$ (after plus-construction). At odd primes p, the F_p-homology coincides with that of $Q_0(HP^\\infty_+)$, but at the prime 2 the result is less clear. We identify the F_2-homology as a Hopf algebra in terms of the homology of well-known spaces. As an application we tabulate the integral stable homology of $N_\\infty$ in degrees up to six. As in the oriented case, not all of these cohomology classes have a geometric interpretation. We determine a polynomial subalgebra of $H^*(N_\\infty ; F_2)$ consisting of geometrically-defined characteristic classes."}
{"category": "Math", "title": "Set families with a forbidden subposet", "abstract": "We asymptotically determine the size of the largest family F of subsets of {1,...,n} not containing a given poset P if the Hasse diagram of P is a tree. This is a qualitative generalization of several known results including Sperner's theorem."}
{"category": "Math", "title": "The congruence subgroup property for the hyperelliptic modular group: the open surface case", "abstract": "Let ${\\cal M}_{g,n}$ and ${\\cal H}_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with $\\Gamma_{g,n}$ and $H_{g,n}$, the so called Teichm{\\\"u}ller modular group and hyperelliptic modular group. A choice of base point on ${\\cal H}_{g,n}$ defines a monomorphism $H_{g,n}\\hookrightarrow\\Gamma_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichm\\\"uller group $\\Gamma_{g,n}$ is the group of isotopy classes of diffeomorphisms of the surface $S_{g,n}$ which preserve the orientation and a given order of the punctures. As a subgroup of $\\Gamma_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\\hookrightarrow\\operatorname{Out}(\\pi_1(S_{g,n}))$. The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^\\lambda$ of $H_{g,n}$, there exists a finite index characteristic subgroup $K$ of $\\pi_1(S_{g,n})$ such that the kernel of the induced representation $H_{g,n}\\to\\operatorname{Out}(\\pi_1(S_{g,n})/K)$ is contained in $H^\\lambda$. The main result of the paper is an affirmative answer to this question for $n\\geq 1$."}
{"category": "Math", "title": "Combinatorics of binomial primary decomposition", "abstract": "An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup. These results are intimately connected to hypergeometric differential equations in several variables."}
{"category": "Math", "title": "Categorified quantum sl(2) and equivariant cohomology of iterated flag varieties", "abstract": "A 2-category was introduced in arXiv:0803.3652 [math.QA] that categorifies Lusztig's integral version of quantum sl(2). Here we construct for each positive integer N arepresentation of this 2-category using the equivariant cohomology of iterated flag varieties. This representation categorifies the irreducible (N+1)-dimensional representation of quantum sl(2)."}
{"category": "Math", "title": "Large intersection properties in Diophantine approximation and dynamical systems", "abstract": "We investigate the large intersection properties of the set of points that are approximated at a certain rate by a family of affine subspaces. We then apply our results to various sets arising in the metric theory of Diophantine approximation, in the study of the homeomorphisms of the circle and in the perturbation theory for Hamiltonian systems."}
{"category": "Math", "title": "Simultaneous inference: When should hypothesis testing problems be combined?", "abstract": "Modern statisticians are often presented with hundreds or thousands of hypothesis testing problems to evaluate at the same time, generated from new scientific technologies such as microarrays, medical and satellite imaging devices, or flow cytometry counters. The relevant statistical literature tends to begin with the tacit assumption that a single combined analysis, for instance, a False Discovery Rate assessment, should be applied to the entire set of problems at hand. This can be a dangerous assumption, as the examples in the paper show, leading to overly conservative or overly liberal conclusions within any particular subclass of the cases. A simple Bayesian theory yields a succinct description of the effects of separation or combination on false discovery rate analyses. The theory allows efficient testing within small subclasses, and has applications to ``enrichment,'' the detection of multi-case effects."}
{"category": "Math", "title": "Irreducible Representations of C*-crossed products by Finite Groups", "abstract": "We describe the structure of the irreducible representations of crossed products of unital C*-algebras by actions of finite groups in terms of irreducible representations of the C*-algebras on which the groups act. We then apply this description to derive a characterization of irreducible representations of crossed-products by finite cyclic groups in terms of representations of the C*-algebra and its fixed point subalgebra. These results are applied to crossed-products by the permutation group on three elements and illustrated by various examples."}
{"category": "Math", "title": "Existence criterion for Hall subgroups of finite groups", "abstract": "In the paper we obtain an existence criterion for Hall subgroups of finite groups in terms of a composition series."}
{"category": "Math", "title": "Sparse estimation of large covariance matrices via a nested Lasso penalty", "abstract": "The paper proposes a new covariance estimator for large covariance matrices when the variables have a natural ordering. Using the Cholesky decomposition of the inverse, we impose a banded structure on the Cholesky factor, and select the bandwidth adaptively for each row of the Cholesky factor, using a novel penalty we call nested Lasso. This structure has more flexibility than regular banding, but, unlike regular Lasso applied to the entries of the Cholesky factor, results in a sparse estimator for the inverse of the covariance matrix. An iterative algorithm for solving the optimization problem is developed. The estimator is compared to a number of other covariance estimators and is shown to do best, both in simulations and on a real data example. Simulations show that the margin by which the estimator outperforms its competitors tends to increase with dimension."}
{"category": "Math", "title": "Skip sequencing: A decision problem in questionnaire design", "abstract": "This paper studies questionnaire design as a formal decision problem, focusing on one element of the design process: skip sequencing. We propose that a survey planner use an explicit loss function to quantify the trade-off between cost and informativeness of the survey and aim to make a design choice that minimizes loss. We pose a choice between three options: ask all respondents about an item of interest, use skip sequencing, thereby asking the item only of respondents who give a certain answer to an opening question, or do not ask the item at all. The first option is most informative but also most costly. The use of skip sequencing reduces respondent burden and the cost of interviewing, but may spread data quality problems across survey items, thereby reducing informativeness. The last option has no cost but is completely uninformative about the item of interest. We show how the planner may choose among these three options in the presence of two inferential problems, item nonresponse and response error."}
{"category": "Math", "title": "Coordinate descent algorithms for lasso penalized regression", "abstract": "Imposition of a lasso penalty shrinks parameter estimates toward zero and performs continuous model selection. Lasso penalized regression is capable of handling linear regression problems where the number of predictors far exceeds the number of cases. This paper tests two exceptionally fast algorithms for estimating regression coefficients with a lasso penalty. The previously known $\\ell_2$ algorithm is based on cyclic coordinate descent. Our new $\\ell_1$ algorithm is based on greedy coordinate descent and Edgeworth's algorithm for ordinary $\\ell_1$ regression. Each algorithm relies on a tuning constant that can be chosen by cross-validation. In some regression problems it is natural to group parameters and penalize parameters group by group rather than separately. If the group penalty is proportional to the Euclidean norm of the parameters of the group, then it is possible to majorize the norm and reduce parameter estimation to $\\ell_2$ regression with a lasso penalty. Thus, the existing algorithm can be extended to novel settings. Each of the algorithms discussed is tested via either simulated or real data or both. The Appendix proves that a greedy form of the $\\ell_2$ algorithm converges to the minimum value of the objective function."}
{"category": "Math", "title": "A statistical framework for testing functional categories in microarray data", "abstract": "Ready access to emerging databases of gene annotation and functional pathways has shifted assessments of differential expression in DNA microarray studies from single genes to groups of genes with shared biological function. This paper takes a critical look at existing methods for assessing the differential expression of a group of genes (functional category), and provides some suggestions for improved performance. We begin by presenting a general framework, in which the set of genes in a functional category is compared to the complementary set of genes on the array. The framework includes tests for overrepresentation of a category within a list of significant genes, and methods that consider continuous measures of differential expression. Existing tests are divided into two classes. Class 1 tests assume gene-specific measures of differential expression are independent, despite overwhelming evidence of positive correlation. Analytic and simulated results are presented that demonstrate Class 1 tests are strongly anti-conservative in practice. Class 2 tests account for gene correlation, typically through array permutation that by construction has proper Type I error control for the induced null. However, both Class 1 and Class 2 tests use a null hypothesis that all genes have the same degree of differential expression. We introduce a more sensible and general (Class 3) null under which the profile of differential expression is the same within the category and complement. Under this broader null, Class 2 tests are shown to be conservative. We propose standard bootstrap methods for testing against the Class 3 null and demonstrate they provide valid Type I error control and more power than array permutation in simulated datasets and real microarray experiments."}
{"category": "Math", "title": "Integral geometry under $G_2$ and $Spin(7)$", "abstract": "A Hadwiger-type theorem for the exceptional Lie groups $G_2$ and $Spin(7)$ is proved. The algebras of $G_2$ or $Spin(7)$ invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formulas for these groups are determined."}
{"category": "Math", "title": "Julia and John revisited", "abstract": "We show that Fatou components of a semi-hyperbolic rational map are John domains and that the converse does not hold. This generalizes a famous result of Carleson, Jones and Yoccoz. We show that a connected Julia set is locally connected for a large class of non-uniformly hyperbolic rational maps. This class is more general than semi-hyperbolicity and includes Collet-Eckmann and Topological Collet-Eckmann maps and maps verifying a summability condition (as considered by Graczyk and Smirnov)."}
{"category": "Math", "title": "Accounting for self-protective responses in randomized response data from a social security survey using the zero-inflated Poisson model", "abstract": "In 2004 the Dutch Department of Social Affairs conducted a survey to assess the extent of noncompliance with social security regulations. The survey was conducted among 870 recipients of social security benefits and included a series of sensitive questions about regulatory noncompliance. Due to the sensitive nature of the questions the randomized response design was used. Although randomized response protects the privacy of the respondent, it is unlikely that all respondents followed the design. In this paper we introduce a model that allows for respondents displaying self-protective response behavior by consistently giving the nonincriminating response, irrespective of the outcome of the randomizing device. The dependent variable denoting the total number of incriminating responses is assumed to be generated by the application of randomized response to a latent Poisson variable denoting the true number of rule violations. Since self-protective responses result in an excess of observed zeros in relation to the Poisson randomized response distribution, these are modeled as observed zero-inflation. The model includes predictors of the Poisson parameters, as well as predictors of the probability of self-protective response behavior."}
{"category": "Math", "title": "More Discriminants with the Brezing-Weng Method", "abstract": "The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number."}
{"category": "Math", "title": "Genus and braid index associated to sequences of renormalizable Lorenz maps", "abstract": "We describe the Lorenz links generated by renormalizable Lorenz maps with reducible kneading invariant $(K_f^-,K_f^+)=(X,Y)*(S,W)$, in terms of the links corresponding to each factor. This gives one new kind of operation that permits us to generate new knots and links from old. Using this result we obtain explicit formulas for the genus and the braid index of this renormalizable Lorenz knots and links. Then we obtain explicit formulas for sequences of these invariants, associated to sequences of renormalizable Lorenz maps with kneading invariant $(X,Y)*(S,W)^{*n}$, concluding that both grow exponentially. This is specially relevant, since it is known that topological entropy is constant on the archipelagoes of renormalization."}
{"category": "Math", "title": "Lightlike hypersurfaces in indefinite $\\mathcal{S}$-manifolds", "abstract": "In a metric $g.f.f$-manifold we study lightlike hypersurfaces $M$ tangent to the characteristic vector fields, and owing to the presence of the $f$-structure, we determine some decompositions of $TM$ and of a chosen screen distribution obtaining two distributions invariant with respect to the structure. We discuss the existence of a $g.f.f$-structure on a lightlike hypersurface and, under suitable hypotheses, we obtain an indefinite $\\mathcal{S}$-structure on the leaves of an integrable distribution. The existence of totally umbilical lightlike hypersurfaces of an indefinite $\\mathcal{S}$-space form is also discussed. Finally, we explicitely describe a lightlike hypersurface of an indefinite $\\mathcal{S}$-manifold."}
{"category": "Math", "title": "Some addition formulae for Abelian functions for elliptic and hyperelliptic curves of cyclotomic type", "abstract": "We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of genus one and two with many automorphisms. In the genus one case we find new addition formulae for the equianharmonic and lemniscate cases, and in genus two we find some new addition formulae for a number of curves, including the Burnside curve."}
{"category": "Math", "title": "Transcription factor binding site prediction with multivariate gene expression data", "abstract": "Multi-sample microarray experiments have become a standard experimental method for studying biological systems. A frequent goal in such studies is to unravel the regulatory relationships between genes. During the last few years, regression models have been proposed for the de novo discovery of cis-acting regulatory sequences using gene expression data. However, when applied to multi-sample experiments, existing regression based methods model each individual sample separately. To better capture the dynamic relationships in multi-sample microarray experiments, we propose a flexible method for the joint modeling of promoter sequence and multivariate expression data. In higher order eukaryotic genomes expression regulation usually involves combinatorial interaction between several transcription factors. Experiments have shown that spacing between transcription factor binding sites can significantly affect their strength in activating gene expression. We propose an adaptive model building procedure to capture such spacing dependent cis-acting regulatory modules. We apply our methods to the analysis of microarray time-course experiments in yeast and in Arabidopsis. These experiments exhibit very different dynamic temporal relationships. For both data sets, we have found all of the well-known cis-acting regulatory elements in the related context, as well as being able to predict novel elements."}
{"category": "Math", "title": "Grothendieck Group and Generalized Mutation Rule for 2-Calabi--Yau Triangulated Categories", "abstract": "We compute the Grothendieck group of certain 2-Calabi--Yau triangulated categories appearing naturally in the study of the link between quiver representations and Fomin--Zelevinsky's cluster algebras. In this setup, we also prove a generalization of Fomin--Zelevinsky's mutation rule."}
{"category": "Math", "title": "Optimal factorial designs for cDNA microarray experiments", "abstract": "We consider cDNA microarray experiments when the cell populations have a factorial structure, and investigate the problem of their optimal designing under a baseline parametrization where the objects of interest differ from those under the more common orthogonal parametrization. First, analytical results are given for the $2\\times 2$ factorial. Since practical applications often involve a more complex factorial structure, we next explore general factorials and obtain a collection of optimal designs in the saturated, that is, most economic, case. This, in turn, is seen to yield an approach for finding optimal or efficient designs in the practically more important nearly saturated cases. Thereafter, the findings are extended to the more intricate situation where the underlying model incorporates dye-coloring effects, and the role of dye-swapping is critically examined."}
{"category": "Math", "title": "The Reverse of The Law of Large Numbers", "abstract": "The Law of Large Numbers tells us that as the sample size (N) is increased, the sample mean converges on the population mean, provided that the latter exists. In this paper, we investigate the opposite effect: keeping the sample size fixed while increasing the number of outcomes (M) available to a discrete random variable. We establish sufficient conditions for the variance of the sample mean to increase monotonically with the number of outcomes, such that the sample mean ``diverges'' from the population mean, acting like an ``reverse'' to the law of large numbers. These results, we believe, are relevant to many situations which require sampling of statistics of certain finite discrete random variables."}
{"category": "Math", "title": "Assessing surrogate endpoints in vaccine trials with case-cohort sampling and the Cox model", "abstract": "Assessing immune responses to study vaccines as surrogates of protection plays a central role in vaccine clinical trials. Motivated by three ongoing or pending HIV vaccine efficacy trials, we consider such surrogate endpoint assessment in a randomized placebo-controlled trial with case-cohort sampling of immune responses and a time to event endpoint. Based on the principal surrogate definition under the principal stratification framework proposed by Frangakis and Rubin [Biometrics 58 (2002) 21--29] and adapted by Gilbert and Hudgens (2006), we introduce estimands that measure the value of an immune response as a surrogate of protection in the context of the Cox proportional hazards model. The estimands are not identified because the immune response to vaccine is not measured in placebo recipients. We formulate the problem as a Cox model with missing covariates, and employ novel trial designs for predicting the missing immune responses and thereby identifying the estimands. The first design utilizes information from baseline predictors of the immune response, and bridges their relationship in the vaccine recipients to the placebo recipients. The second design provides a validation set for the unmeasured immune responses of uninfected placebo recipients by immunizing them with the study vaccine after trial closeout. A maximum estimated likelihood approach is proposed for estimation of the parameters. Simulated data examples are given to evaluate the proposed designs and study their properties."}
{"category": "Math", "title": "Dress induction and the Burnside quotient Green ring", "abstract": "We define and study the Burnside quotient Green ring of a Mackey functor. Some refinements of Dress induction theory are presented, together with applications to computation results for $K$-theory and $L$-theory of finite and infinite groups."}
{"category": "Math", "title": "Hyperfocused arcs in PG(2,32)", "abstract": "In PG(2,32) the following two results are proven by a computer aided search. (i) Uniqueness of hyperfocused 12-arcs, up to projectivities; (ii) Non-existence of hyperfocused 14-arcs. The existence problem for hyperfocused 16-arcs remains open."}
{"category": "Math", "title": "Convex Hypersurfaces in Hadamard Manifolds", "abstract": "We prove a theorem about an extremal property of Lobachevsky space among simply connected Riemannian manifolds of nonpositive curvature"}
{"category": "Math", "title": "A hidden spatial-temporal Markov random field model for network-based analysis of time course gene expression data", "abstract": "Microarray time course (MTC) gene expression data are commonly collected to study the dynamic nature of biological processes. One important problem is to identify genes that show different expression profiles over time and pathways that are perturbed during a given biological process. While methods are available to identify the genes with differential expression levels over time, there is a lack of methods that can incorporate the pathway information in identifying the pathways being modified/activated during a biological process. In this paper we develop a hidden spatial-temporal Markov random field (hstMRF)-based method for identifying genes and subnetworks that are related to biological processes, where the dependency of the differential expression patterns of genes on the networks are modeled over time and over the network of pathways. Simulation studies indicated that the method is quite effective in identifying genes and modified subnetworks and has higher sensitivity than the commonly used procedures that do not use the pathway structure or time dependency information, with similar false discovery rates. Application to a microarray gene expression study of systemic inflammation in humans identified a core set of genes on the KEGG pathways that show clear differential expression patterns over time. In addition, the method confirmed that the TOLL-like signaling pathway plays an important role in immune response to endotoxins."}
{"category": "Math", "title": "On the Global Structure of Hopf Hypersurfaces in Complex Space Form", "abstract": "It is known that a tube over a Kahler submanifold in a complex form is a Hopf hypersurface. In some sense the reverse statement is true: a connected compact generic immersed C^(2n-1) regular Hopf hypersurface in the complex projective plane is a tube iver an irreducible algebraic variety. In the complex hyperbolic space a connected compact generic immersed C^(2n-1) regular Hopf hypersurface is a geodesic hypersphere"}
{"category": "Math", "title": "An approximation algorithm for counting contingency tables", "abstract": "We present a randomized approximation algorithm for counting contingency tables, mxn non-negative integer matrices with given row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We define smooth margins (R,C) in terms of the typical table and prove that for such margins the algorithm has quasi-polynomial N^{O(ln N)} complexity, where N=r_1+...+r_m=c_1+...+c_n. Various classes of margins are smooth, e.g., when m=O(n), n=O(m) and the ratios between the largest and the smallest row sums as well as between the largest and the smallest column sums are strictly smaller than the golden ratio (1+sqrt{5})/2 = 1.618. The algorithm builds on Monte Carlo integration and sampling algorithms for log-concave densities, the matrix scaling algorithm, the permanent approximation algorithm, and an integral representation for the number of contingency tables."}
{"category": "Math", "title": "Dynamics of rational symplectic mappings and difference Galois theory", "abstract": "In this paper we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We present a Galoisian condition, of Morales-Ramis type, ensuring the non-integrability of a rational symplectic map in the non-commutative sense (Mishchenko-Fomenko). As a particular case, we obtain a com- plete discrete analogue of Morales-Ramis Theorems for non-integrabi- lity in the sense of Liouville."}
{"category": "Math", "title": "A solution to a problem and the Diophantine equation X^2+bX+c=Y^2", "abstract": "We prove that for given integers b and c, the diophantine equation x^2+bx+c=y^2, has finitely many integer solutions(i.e. pairs in ZxZ),in fact an even number of such solutions(including the zero or no solutions case).We also offer an explicit description of the solution set. Such a description depends on the form of the integer b^2-4c. Some Corollaries do follow. Furthermore, we show that the said equation has exactly two integer solutions, precisely when b^2-4c= 1,4,16,-4,or-16. In each case we list the two solutions in terms of the coefficients b and c."}
{"category": "Math", "title": "A sharp stability estimate in tensor tomography", "abstract": "We prove a sharp stability estimate for the problem of reconstructing a symmetric 2-tensor from its integrals along all maximal geodesics on a simple manifold."}
{"category": "Math", "title": "Invariant Krein subspaces, regular irreducibility and integral representations", "abstract": "We study unitary representations of groups in Krein spaces, irreducibility criteria and integral decompositions. Our main tool is the theory of Krein subspaces and their (reproducing) kernels and a variant of Choquet's theorem."}
{"category": "Math", "title": "The KP hierarchy, branched covers, and triangulations", "abstract": "The KP hierarchy is a completely integrable system of quadratic, partial differential equations that generalizes the KdV hierarchy. A linear combination of Schur functions is a solution to the KP hierarchy if and only if its coefficients satisfy the Plucker relations from geometry. We give a solution to the Plucker relations involving products of variables marking contents for a partition, and thus give a new proof of a content product solution to the KP hierarchy, previously given by Orlov and Shcherbin. In our main result, we specialize this content product solution to prove that the generating series for a general class of transitive ordered factorizations in the symmetric group satisfies the KP hierarchy. These factorizations appear in geometry as encodings of branched covers, and thus by specializing our transitive factorization result, we are able to prove that the generating series for two classes of branched covers satisfies the KP hierarchy. For the first of these, the double Hurwitz series, this result has been previously given by Okounkov. The second of these, that we call the m-hypermap series, contains the double Hurwitz series polynomially, as the leading coefficient in m. The m-hypermap series also specializes further, first to the series for hypermaps and then to the series for maps, both in an orientable surface. For the latter series, we apply one of the KP equations to obtain a new and remarkably simple recurrence for triangulations in a surface of given genus, with a given number of faces. This recurrence leads to explicit asymptotics for the number of triangulations with given genus and number of faces, in recent work by Bender, Gao and Richmond."}
{"category": "Math", "title": "{Spaces of Infinite Measure and Pointwise Convergence of the Bilinear Hilbert and Ergodic Averages Defined by $L^{p}$-Isometries", "abstract": "We generalize the respective ``double recurrence'' results of Bourgain and of the second author, which established for pairs of $L^{\\infty}$ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages and of the discrete bilinear Hilbert averages defined by invertible measure-preserving point transformations. Our generalizations are set in the context of arbitrary sigma-finite measure spaces and take the form of a.e. convergence of such discrete averages, as well as of their continuous variable counterparts, when these averages are defined by Lebesgue space isometries and act on $L^{p_{1}}\\times L^{p_{2}}$ ($ 1<p_{1},p_{2}<\\infty $, $p_{1}^{-1}+p_{2}^{-1}<3/2$). In the setting of an arbitrary measure space, this yields the a.e. convergence of these discrete bilinear averages when they act on $L^{p_{1}}\\times L^{p_{2}}$ and are defined by an invertible measure-preserving point transformation."}
{"category": "Math", "title": "On the Fundamental Properties of Linear Parameter-Varying Dynamic Systems Under Parametrical Multi-Perturbations. Applications to Time-Delay Systems. Preliminary Results", "abstract": "This paper deals with a unifying approach to the problems of computing the admissible sets of parametrical multi perturbations in appropriate bounded sets such that some fundamental properties of parameter-varying linear dynamic systems are maintained provided that the so-called (i.e. perturbation-free) nominal system possesses such properties. The sets of parametrical multi perturbations include any combinations of parametrical multi perturbations in the matrix of dynamics as well as in the control, output and input-output interconnection matrices which belong to some prescribed bounded domain in the complex space. The various properties which are investigated are controllability, observability, output controllability and existence of minimal state-space realizations together with the associate existence or not of associate decoupling, transmission and invariant zeros. All the matrices of parameters including the nominal and the disturbed ones which parameterize the dynamic system may be real or complex. The radii of the multi- parametrical perturbations are calculated in a simple way. The obtained results are then applied to systems subject to a finite number of point internal delays and parametrical multi perturbations by comparing the state- space descriptions of such systems with the general descriptions previously investigated. In particular, the contributions of the delays to the spectral descriptions are assimilated to the contributions of a set of varying parameters in a domain for the general description."}
{"category": "Math", "title": "How many distribution functions are there? Bracketing entropy bounds for high dimensional distribution functions", "abstract": "This paper has been withdrawn by the authors due to a crucial error in a bound on page 19 and some other errors earlier in the paper."}
{"category": "Math", "title": "On the Asymptotic Expansions for Time-Varying Scalar Differential Equations Possesssing Limiting Differential Equations by Application of the Residue Theorem to their Discretized Counterparts. Preliminary Results", "abstract": "In this paper, it is proved that, in a dual context, asymptotic expansions of ordinary linear time-differential equations which possess limiting equations to their limiting equations might be obtained by first discretizing them and then using residues calculation for such discrete- time counterparts"}
{"category": "Math", "title": "Local approximation of the solutions of algebraic equations", "abstract": "A method of local approximation of holomorphic solutions of algebraic equations is discussed"}
{"category": "Math", "title": "Fuzzy Statistical Limits", "abstract": "Statistical limits are defined relaxing conditions on conventional convergence. The main idea of the statistical convergence of a sequence l is that the majority of elements from l converge and we do not care what is going on with other elements. At the same time, it is known that sequences that come from real life sources, such as measurement and computation, do not allow, in a general case, to test whether they converge or statistically converge in the strict mathematical sense. To overcome these limitations, fuzzy convergence was introduced earlier in the context of neoclassical analysis and fuzzy statistical convergence is introduced and studied in this paper. We find relations between fuzzy statistical convergence of a sequence and fuzzy statistical convergence of its subsequences (Theorem 2.1), as well as between fuzzy statistical convergence of a sequence and conventional convergence of its subsequences (Theorem 2.2). It is demonstrated what operations with fuzzy statistical limits are induced by operations on sequences (Theorem 2.3) and how fuzzy statistical limits of different sequences influence one another (Theorem 2.4). In Section 3, relations between fuzzy statistical convergence and fuzzy convergence of statistical characteristics, such as the mean (average) and standard deviation, are studied (Theorems 3.1 and 3.2)."}
{"category": "Math", "title": "Description of the inelastic collision of two solitary waves for the BBM equation", "abstract": "We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves)."}
{"category": "Math", "title": "The tropical $j$-invariant", "abstract": "If (Q,A) is a marked polygon with one interior point, then a general polynomial f in K[x,y] with support A defines an elliptic curve C on the toric surface X_A. If K has a non-archimedean valuation into the real numbers we can tropicalize C to get a tropical curve Trop(C). If the Newton subdivision induced by f is a triangulation, then Trop(C) will be a graph of genus one and we show that the lattice length of the cycle of that graph is the negative of the valuation of the j-invariant of C."}
{"category": "Math", "title": "Dynamical systems method for solving linear finite-rank operator equations", "abstract": "A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\\it a priori} and {\\it a posteriori} stopping rules are justified. An iterative scheme is constructed for solving ill-conditioned linear algebraic systems."}
{"category": "Math", "title": "Super-Mathematics Functions", "abstract": "In this paper we talk about the so-called SuperMathematics Functions (SMF), which often constiture the base for generating technical neo-geometrical objects."}
{"category": "Math", "title": "Free algebras and the Freiheitssatz", "abstract": "There have been gaps found in the proofs. The paper is withdrawn until further notice."}
{"category": "Math", "title": "On semistable principal bundles over a complex projective manifold", "abstract": "Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character \\chi of P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle vanishes in rational cohomology if and only if the line bundle over E/P defined by \\chi is numerically effective. Similar results remain valid for principal bundles with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface."}
{"category": "Math", "title": "Ergodic Properties of a Class of Discrete Abelian Group Extensions of Rank-One Transformations", "abstract": "We define a class of discrete abelian group extensions of rank-one transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then study conditions sufficient for showing that cartesian products of transformations are conservative for a class of invertible infinite measure-preserving transformations and provide examples of these transformations."}
{"category": "Math", "title": "The invertible double of elliptic operators", "abstract": "First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept of the Calderon projection. Then we summarize a recent construction of a canonical invertible double for general first order elliptic differential operators over smooth compact manifolds with boundary. We derive a natural formula for the Calderon projection which yields a generalization of the famous Cobordism Theorem. We provide a list of assumptions to obtain a continuous variation of the Calderon projection under smooth variation of the coefficients. That yields various new spectral flow theorems. Finally, we sketch a research program for confining, respectively closing, the last re- maining gaps between the geometric Dirac operator type situation and the general linear elliptic case."}
{"category": "Math", "title": "Involutive Yang-Baxter Groups", "abstract": "In 1992 Drinfeld posed the question of finding the set theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate solutions on finite sets and groups of $I$-type. A group $\\mathcal{G}$ of $I$-type is a group isomorphic to a subgroup of the natural semidirect product of $Fa_n$, the free abelian group of rank $n$, by $Sym_n$, the symmetric group on $n$ letters, so that the projection onto $Fa_n$ is a bijective map. The projection of $\\mathcal{G}$ onto $Sym_n$ we call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfeld's problem for involutive non-degenerate set theoretic solutions. First classify the IYB groups and second, for a given IYB group $G$, classify the groups of $I$-type with $G$ as associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse of this property are obtained. More precisely, we show that some classes of groups are IYB groups. We also give a non-obvious method to construct infinitely many groups of $I$-type (and hence infinitely many involutive non-degenerate set theoretic solutions of the Yang-Baxter equation) with a prescribed associated IYB group."}
{"category": "Math", "title": "Letter to the editor. NAA and JFK: Can revisionism take us home?", "abstract": "Occasionally during the course of the human learning experience we are faced with an anomaly. An aberration of sorts, which try as we might, defies appropriate classification. The recent paper by Spiegelman et al.--Chemical and forensic analysis of JFK assassination bullet lots: Is a second shooter possible?--is one such aberration. It is riddled with both misconceptions and errors of fact. Purporting to cast doubt on the NAA (neutron activation analysis) work conducted by Dr. Vincent Guinn in the investigation of the assassination of President John F. Kennedy, it fails miserably. The paper offers two central conclusions, one which is demonstrably false, and the other which is specious. The authors opine; ``If the assassination fragments are derived from three or more separate bullets, then a second assassin is likely, as the additional bullet would not be attributable to the main suspect, Mr. Oswald.'' This statement relating to the likelihood of a second assassin based on the premise of three or more separate bullets is demonstrably false. The available evidence indicates that Oswald fired three shots, one of which is believed to have missed. However, on the off chance that all three shots hit (even though there is absolutely no other supporting forensic evidence for such a notion) those three shots alone in no way would indicate then that ``a second assassin is likely.'' The authors' erroneous conclusion was achieved because they have either been misled (which I personally believe is the case) or they simply aren't familiar with the evidence."}
{"category": "Math", "title": "Response to the Letter to the Editor", "abstract": "This paper has attracted interest around the world from the media (both TV and newspapers). In addition, we have received letters, emails and telephone calls. One of our favorites was a voicemail message asking us to return a call to Australia at which point we would learn who really killed JFK. We welcome the opportunity to respond to the letter to the editor from Mr. Fiorentino. Mr. Fiorentino claims that our ``statement relating to the likelihood of a second assassin based on the premise of three or more separate bullets is demonstrably false.'' In response we would like to simply quote from page 327 of Gerald Posner's book Case Closed, one of the most well known works supporting the single assassin theory: ``If Connally was hit by another bullet, it had to be fired from a second shooter, since the Warren Commission's own reconstructions showed that Oswald could not have operated the bolt and refired in 1.4 seconds.'' Mr. Fiorentino also claims that the ``second fatal flaw is the use of a rather uncomplicated formula based on Bayes Theorem.'' Let $E$ denote the evidence and $T$ denote the theory that there were just two bullets (and hence a single shooter). We used Bayes Theorem to hypothetically calculate $P(T|E)$ from $P(E|T)$ and the prior probability $P(T)$. In order to make $P(T|E)$ ten times more likely than $P(\\bar{T}|E)$, the ratio of the prior probabilities [i.e., $P(T) / P(\\bar{T})$] would have to be greater than 15. Thus, we again conclude that this casts serious doubt on Dr. Guinn's conclusion that the evidence supported just two bullets. Sadly, this is far from the first time that probability has been misunderstood and/or misapplied in a case of public interest. A notable British example is the Clark case. See Nobles and Schiff (2005) for details. Finally, we welcome and, in fact, encourage members of the scientific community to provide alternative analyses of the data."}
{"category": "Math", "title": "On the Berg--Chen--Ismail theorem and the Nevanlinna-Pick problem", "abstract": "In 2002 C. Berg, Y. Chen, and M. Ismail found a nice relation between the determinancy of the Hamburger moment problem and asymptotic behavior of the smallest eigenvalues of the corresponding Hankel matrices. We investigate whether an analog of this statement holds for the Nevanlinna--Pick interpolation problem."}
{"category": "Math", "title": "Statistical advances and challenges for analyzing correlated high dimensional SNP data in genomic study for complex diseases", "abstract": "Recent advances of information technology in biomedical sciences and other applied areas have created numerous large diverse data sets with a high dimensional feature space, which provide us a tremendous amount of information and new opportunities for improving the quality of human life. Meanwhile, great challenges are also created driven by the continuous arrival of new data that requires researchers to convert these raw data into scientific knowledge in order to benefit from it. Association studies of complex diseases using SNP data have become more and more popular in biomedical research in recent years. In this paper, we present a review of recent statistical advances and challenges for analyzing correlated high dimensional SNP data in genomic association studies for complex diseases. The review includes both general feature reduction approaches for high dimensional correlated data and more specific approaches for SNPs data, which include unsupervised haplotype mapping, tag SNP selection, and supervised SNPs selection using statistical testing/scoring, statistical modeling and machine learning methods with an emphasis on how to identify interacting loci."}
{"category": "Math", "title": "A Note on Ternary Sequences of Strings of 0 and 1", "abstract": "B. D. Acharya has conjectured that if $\\bigl(A_i: i=1, 2, ..., 2^{|X|}-1\\bigr)$ is a permutation of all nonempty subsets of a set $X$ with at least two elements such that for each even positive integer $j<2^{|X|}-1$, $A_{j-1}\\triangle A_j\\triangle A_{j+1}=\\emptyset$, then $|X|=2$. In this article, we show that if the cardinality of a set $X$ is more than four, then a permutation as described above indeed exists."}
{"category": "Math", "title": "On Central Automorphisms Fixing the Center Element-wise", "abstract": "Let $G$ be a finite $p$-group of nilpotency class 2. We find necessary and sufficient conditions on $G$ such that each central automorphism of $G$ fixes the center of $G$ element-wise."}
{"category": "Math", "title": "Profinite homotopy theory", "abstract": "We construct a model structure on simplicial profinite sets such that the homotopy groups carry a natural profinite structure. This yields a rigid profinite completion functor for spaces and pro-spaces. One motivation is the \\'etale homotopy theory of schemes in which higher profinite \\'etale homotopy groups fit well with the \\'etale fundamental group which is always profinite. We show that the profinite \\'etale topological realization functor is a good object in several respects."}
{"category": "Math", "title": "Analysis of boosting algorithms using the smooth margin function", "abstract": "We introduce a useful tool for analyzing boosting algorithms called the ``smooth margin function,'' a differentiable approximation of the usual margin for boosting algorithms. We present two boosting algorithms based on this smooth margin, ``coordinate ascent boosting'' and ``approximate coordinate ascent boosting,'' which are similar to Freund and Schapire's AdaBoost algorithm and Breiman's arc-gv algorithm. We give convergence rates to the maximum margin solution for both of our algorithms and for arc-gv. We then study AdaBoost's convergence properties using the smooth margin function. We precisely bound the margin attained by AdaBoost when the edges of the weak classifiers fall within a specified range. This shows that a previous bound proved by R\\\"{a}tsch and Warmuth is exactly tight. Furthermore, we use the smooth margin to capture explicit properties of AdaBoost in cases where cyclic behavior occurs."}
{"category": "Math", "title": "K-stability of constant scalar curvature K\\\"ahler manifolds", "abstract": "We show that a polarised manifold with a constant scalar curvature K\\\"ahler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S. K. Donaldson."}
{"category": "Math", "title": "Brill-Noether theory of curves on Enriques surfaces I: the positive cone and gonality", "abstract": "We study the existence of linear series on curves lying on an Enriques surface and general in their complete linear system. Using a method that works also below the Bogomolov-Reider range, we compute, in all cases, the gonality of such curves. We also give a new result about the positive cone of line bundles on an Enriques surface and we show how this relates to the gonality."}
{"category": "Math", "title": "Measuring and testing dependence by correlation of distances", "abstract": "Distance correlation is a new measure of dependence between random vectors. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but unlike the classical definition of correlation, distance correlation is zero only if the random vectors are independent. The empirical distance dependence measures are based on certain Euclidean distances between sample elements rather than sample moments, yet have a compact representation analogous to the classical covariance and correlation. Asymptotic properties and applications in testing independence are discussed. Implementation of the test and Monte Carlo results are also presented."}
{"category": "Math", "title": "Estimation of the covariance matrix of random effects in longitudinal studies", "abstract": "Longitudinal studies are often conducted to explore the cohort and age effects in many scientific areas. The within cluster correlation structure plays a very important role in longitudinal data analysis. This is because not only can an estimator be improved by incorporating the within cluster correlation structure into the estimation procedure, but also the within cluster correlation structure can sometimes provide valuable insights in practical problems. For example, it can reveal the correlation strengths among the impacts of various factors. Motivated by data typified by a set from Bangladesh pertinent to the use of contraceptives, we propose a random effect varying-coefficient model, and an estimation procedure for the within cluster correlation structure of the proposed model. The estimation procedure is optimization-free and the proposed estimators enjoy asymptotic normality under mild conditions. Simulations suggest that the proposed estimation is practicable for finite samples and resistent against mild forms of model misspecification. Finally, we analyze the data mentioned above with the new random effect varying-coefficient model together with the proposed estimation procedure, which reveals some interesting sociological dynamics."}
{"category": "Math", "title": "Classifying Hilbert functions of fat point subschemes in $\\mathbb P^2$", "abstract": "A recent paper by the first and third authors together with Sabourin raised the question of what the possible Hilbert functions are for fat point subschemes of the form $2p_1+...+2p_r$, for all possible choices of $r$ distinct points in the projective plane. We study this problem for $r$ points in the plane over an algebraically closed field $k$ of arbitrary characteristic in case either $r \\le 8$ or the points lie on a (possibly reducible) conic. In either case, it follows from work of the second author that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We say $p_1,...,p_r$ and $p'_1,...,p'_r$ have the same {\\it configuration type} if for all choices of nonnegative integers $m_i$, $Z=m_1p_1+...+m_rp_r$ and $Z'=m_1p'_1+...+m_rp'_r$ have the same Hilbert function.) Assuming either that $7 \\le r\\le 8$ (see recent work of Guardo and the second author for the cases $r\\le 6$) or that the points $p_i$ lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients $m_i$ determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of $Z=m_1p_1+...+m_rp_r$. We demonstrate our results by explicitly listing all Hilbert functions for schemes of $r\\le 8$ double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function."}
{"category": "Math", "title": "Examples of cohomology manifolds which are not homologically locally connected", "abstract": "Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions $n \\ge 3$."}
{"category": "Math", "title": "Fr\\'{e}chet-Urysohn fans in free topological groups", "abstract": "In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fr\\'echet-Urysohn fan $S_\\w$ in a topological group $G$ admitting a functorial embedding $[0,1]\\subset G$. The latter means that each autohomeomorphism of $[0,1]$ extends to a continuous homomorphism of $G$. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space $X$ containing a topological copy of the space $\\IQ$ of rationals give topological groups containing $S_\\w$."}
{"category": "Math", "title": "Correction. Strong invariance principles for sequential Bahadur--Kiefer and Vervaat error processes of long-range dependent sequences", "abstract": "Correction to The Annals of Statistics (2006) 34, 1013--1044 [URL: http://projecteuclid.org/euclid.aos/1151418250]"}
{"category": "Math", "title": "A diagrammatic approach to categorification of quantum groups I", "abstract": "To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify $U^-_q(\\mathfrak{g})$, where $\\mathfrak{g}$ is the Kac-Moody Lie algebra associated with the graph."}
{"category": "Math", "title": "Preserving $Z$-sets by Dranishnikov's resolution", "abstract": "We prove that Dranishnikov's $k$-dimensional resolution $d_k\\colon \\mu^k\\to Q$ is a UV$^{n-1}$-divider of Chigogidze's $k$-dimensional resolution $c_k$. This fact implies that $d_k^{-1}$ preserves $Z$-sets. A further development of the concept of UV$^{n-1}$-dividers permits us to find sufficient conditions for $d_k^{-1}(A)$ to be homeomorphic to the N\\\"{o}beling space $\\nu^k$ or the universal pseudoboundary $\\sigma^k$. We also obtain some other applications."}
{"category": "Math", "title": "Regularity of conjugacies of algebraic actions of Zariski dense groups", "abstract": "Let \\alpha_0 be an affine action of a discrete group \\Gamma on a compact homogeneous space X and \\alpha_1 a smooth action of \\Gamma on X which is C^1-close to \\alpha_0. We show that under some conditions, every topological conjugacy between \\alpha_0 and \\alpha_1 is smooth. In particular, our results apply to Zariski dense subgroups of SL_d(Z) acting on the torus T^d and Zariski dense subgroups of a simple noncompact Lie group G acting on a compact homogeneous space X of G with an invariant measure."}
{"category": "Math", "title": "On extending actions of groups", "abstract": "Problems of dense and closed extension of actions of compact transformation groups are solved. The method developed in the paper is applied to problems of extension of equivariant maps and of construction of equivariant compactifications."}
{"category": "Math", "title": "Homological stability for certain classical groups", "abstract": "We prove homological stability for standard unitary groups over R, C and H and for general linear groups over skew-fields with infinite centre. We focus on the similarities and differences of these proofs. Both proofs are due to Chih-Han Sah (Homology of classical Lie groups made discrete I: Stability theorems and Schur multipliers. Comment. Math. Helv. 61(2), 1986)."}
{"category": "Math", "title": "Surface subgroups from homology", "abstract": "Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on H_2(G;R) is a finite-sided rational polyhedron."}
{"category": "Math", "title": "Principalization of ideals in abelian extensions of number fields", "abstract": "We give the complete proof of a conjecture of Georges Gras which claims that, for any extension $K/k$ of number fields in which at least one infinite place is totally split, every ideal $I$ of $K$ principalizes in the compositum $Kk^{ab}$ of $K$ with the maximal abelian extension $k^{ab}$ of $k$"}
{"category": "Math", "title": "The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices", "abstract": "This paper extends the work of El Karoui [Ann. Probab. 35 (2007) 663--714] which finds the Tracy--Widom limit for the largest eigenvalue of a nonsingular $p$-dimensional complex Wishart matrix $W_{\\mathbb{C}}(\\Omega_p,n)$ to the case of several of the largest eigenvalues of the possibly singular $(n<p)$ matrix $W_{\\mathbb{C}}(\\Omega_p,n).$ As a byproduct, we extend all results of Baik, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643--1697] to the singular Wishart matrix case. We apply our findings to obtain a 95% confidence set for the number of common risk factors in excess stock returns."}
{"category": "Math", "title": "The Calderon Projection: New Definition and Applications", "abstract": "We consider an arbitrary linear elliptic first--order differential operator A with smooth coefficients acting between sections of complex vector bundles E,F over a compact smooth manifold M with smooth boundary N. We describe the analytic and topological properties of A in a collar neighborhood U of N and analyze various ways of writing A|U in product form. We discuss the sectorial projections of the corresponding tangential operator, construct various invertible doubles of A by suitable local boundary conditions, obtain Poisson type operators with different mapping properties, and provide a canonical construction of the Calderon projection. We apply our construction to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderon projection and of well--posed selfadjoint Fredholm extensions under continuous variation of the data."}
{"category": "Math", "title": "Strong approximation methods in group theory, an LMS/EPSRC Short course lecture notes", "abstract": "These are the lecture notes for the LMS/EPSRC short course on strong approximation methods in linear groups organized by Dan Segal in Oxford in September 2007."}
{"category": "Math", "title": "Relative differential K-characters", "abstract": "We define a group of relative differential K-characters associated with a smooth map between two smooth compact manifolds. We show that this group fits into a short exact sequence as in the non-relative case. Some secondary geometric invariants are expressed in this theory."}
{"category": "Math", "title": "Average growth of the spectral function on a Riemannian manifold", "abstract": "We study average growth of the spectral function of the Laplacian on a Riemannian manifold. Two types of averaging are considered: with respect to the spectral parameter and with respect to a point on a manifold. We obtain as well related estimates of the growth of the pointwise zeta-function along vertical lines in the complex plane. Some examples and open problems regarding almost periodic properties of the spectral function are also discussed."}
{"category": "Math", "title": "Parity patterns associated with lifts of Hecke groups", "abstract": "Let $q$ be an odd prime, $m$ a positive integer, and let $\\Ga_m(q)$ be the group generated by two elements $x$ and $y$ subject to the relations $x^{2m}=y^{qm}=1$ and $x^2=y^q$; that is, $\\Ga_m(q)$ is the free product of two cyclic groups of orders $2m$ respectively $qm$, amalgamated along their subgroups of order $m$. Our main result determines the parity behaviour of the generalized subgroup numbers of $\\Ga_m(q)$ which were defined in [T. W. M\\\"uller, Adv. in Math. 153 (2000), 118-154], and which count all the homomorphisms of index $n$ subgroups of $\\Ga_m(q)$ into a given finite group $H$, in the case when $\\gcd(m,| H|)=1$. This computation depends upon the solution of three counting problems in the Hecke group $\\mathfrak H(q)=C_2*C_q$: (i) determination of the parity of the subgroup numbers of $\\mathfrak H(q)$; (ii) determination of the parity of the number of index $n$ subgroups of $\\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_2$ and of $C_\\infty$; (iii) determination of the parity of the number of index $n$ subgroups in $\\mathfrak H(q)$ which are isomorphic to a free product of copies of $C_q$. The first problem has already been solved in [T. W. M\\\"uller, in: {\\it Groups: Topological, Combinatorial and Arithmetic Aspects}, (T. W. M\\\"uller ed.), LMS Lecture Notes Series 311, Cambridge University Press, Cambridge, 2004, pp. 327-374]. The bulk of our paper deals with the solution of Problems (ii) and (iii)."}
{"category": "Math", "title": "An approximation formula for holomorphic functions by interpolation on the ball", "abstract": "We deal with a problem of the reconstruction of any holomorphic function $f$ on the unit ball of $\\mathbb{C}^2$ from its restricions on a union of complex lines. We give an explicit formula of Lagrange interpolation's type that is constructed from the knowledge of $f$ and its derivatives on these lines. We prove that this formula approximates any function when the number of lines increases. The motivation of this problem comes also from possible applications in mathematical economics and medical imaging."}
{"category": "Math", "title": "Two-sided Grassmann-Rayleigh quotient iteration", "abstract": "The two-sided Rayleigh quotient iteration proposed by Ostrowski computes a pair of corresponding left-right eigenvectors of a matrix $C$. We propose a Grassmannian version of this iteration, i.e., its iterates are pairs of $p$-dimensional subspaces instead of one-dimensional subspaces in the classical case. The new iteration generically converges locally cubically to the pairs of left-right $p$-dimensional invariant subspaces of $C$. Moreover, Grassmannian versions of the Rayleigh quotient iteration are given for the generalized Hermitian eigenproblem, the Hamiltonian eigenproblem and the skew-Hamiltonian eigenproblem."}
{"category": "Math", "title": "A normalization formula for the Jack polynomials in superspace and an identity on partitions", "abstract": "We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of admissible tableaux of the non-symmetric Jack polynomials. In the final step of the proof appears an identity on weighted sums of partitions that we demonstrate using the methods of Gessel-Viennot."}
{"category": "Math", "title": "When is sinx+cosx+tanx+cotx+secx+cscx an integer ?", "abstract": "In this paper, we investigate the one-variable equation, sinx+cosx+tanx+cotx+secx+cscx=n, where n is an integer. We prove that if n lies between(inclusively) -1 and 6; then the above equation has no real number solutions. While if n is greater than or equal to 7; or less than or equal to -2, then the said equation has a nonempty solution set which we describe."}
{"category": "Math", "title": "Morphisms between spaces of leaves viewed as fractions", "abstract": "Reprint of a 1989 paper including minor corrections of misprints. Added comments (11 pages) about later related papers in the literature concerning comparison of Gabriel-Zisman calculus of (right) fractions and the use of generalized morphims in the sense of Haefliger-Skandalis-Hilsum for inverting differentiable equivalences between Lie groupoids."}
{"category": "Math", "title": "Toroidalization of Locally Toroidal Morphisms from N-folds to Surfaces", "abstract": "The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a toroidal morphism. Following a suggestion by Dale Cutkosky, we define the notion of \\emph{locally toroidal} morphisms and ask whether any locally toroidal morphism can be modified into a toroidal morphism. In this paper, we answer the question in the affirmative when the morphism is between any arbitrary variety and a surface."}
{"category": "Math", "title": "Local moduli of holomorphic bundles", "abstract": "We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds."}
{"category": "Math", "title": "A New two-dimensional Second Order Non-oscillatory Central Scheme Applied to multiphase flows in heterogeneous porous media", "abstract": "We compare the Kurganov-Tadmor (KT) two-dimensional second order semi-discrete central scheme in dimension by dimension formulation with a new two-dimensional approach introduced here and applied in numerical simulations for two-phase, two-dimensional flows in heterogeneous formations. This semi-discrete central scheme is based on the ideas of Rusanov's method using a more precise information about the local speeds of wave propagation computed at each Riemann Problem in two-space dimensions. We find the KT dimension by dimension has a much simpler mathematical description than the genuinely two-dimensional one with a little more numerical diffusion, particularly in the presence of viscous fingers. Unfortunately, as one can see, the KT with the dimension by dimension approach might produce incorrect boundary behavior in a typical geometry used in the study of porous media flows: the quarter of a five spot. This problem has been corrected by the authors with the new semi-discrete scheme proposed here. We conclude with numerical examples of two-dimensional, two-phase flow associated with two distinct flooding problems: a two-dimensional flow in a rectangular heterogeneous reservoir (called slab geometry) and a two-dimensional flow in a 5-spot geometry homogeneous reservoir."}
{"category": "Math", "title": "Analytic Subordination for Free Compression", "abstract": "We extend the free difference quotient coalgebra approach to analytic subordination to the case of a free compression in free probability."}
{"category": "Math", "title": "Koszulity of algebras with non-pure resolutions", "abstract": "We discuss certain homological properties of graded algebras whose trivial modules admit non-pure resolutions. Such algebras include both of Artin-Schelter regular algebras of types (12221) and (13431). Under certain conditions, a module with non-pure resolution is decomposed to form an extension by two modules with pure resolutions."}
{"category": "Math", "title": "Variational Principles on Triangulated Surfaces", "abstract": "We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles."}
{"category": "Math", "title": "A Bijection Between Partially Directed Paths in the Symmetric Wedge and Matchings", "abstract": "We give a bijection between partially directed paths in the symmetric wedge y= +/-x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of Prellberg et al. that was first discovered through the corresponding generating functions: the number of partially directed paths starting at the origin confined to the symmetric wedge y= +/-x with k north steps is equal to the number of matchings on [2n] with k nestings."}
{"category": "Math", "title": "Small Deviations of Smooth Stationary Gaussian Processes", "abstract": "We investigate the small deviation probabilities of a class of very smooth stationary Gaussian processes playing an important role in Bayesian statistical inference. Our calculations are based on the appropriate modification of the entropy method due to Kuelbs, Li, and Linde as well as on classical results about the entropy of classes of analytic functions. They also involve Tsirelson's upper bound for small deviations and shed some light on the limits of sharpness for that estimate."}
{"category": "Math", "title": "A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions", "abstract": "We prove the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (the space itself was constructed in one of our earlier papers). We also indicate some relations between this space and the well-known Griffiths' space from the 1950's."}
{"category": "Math", "title": "Some isoperimetric inequalities with application to the Stekloff problem", "abstract": "In this paper we establish isoperimetric inequalities for the product of some moments of inertia. As an application, we obtain an isoperimetric inequality for the product of the $N$ first nonzero eigenvalues of the Stekloff problem in $\\mathbb{R}^N$."}
{"category": "Math", "title": "Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces", "abstract": "Our main result states that the hyperspace of convex compact subsets of a compact convex subset $X$ in a locally convex space is an absolute retract if and only if $X$ is an absolute retract of weight $\\le\\omega_1$. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube $I^{\\omega_1}$ is homeomorphic to $I^{\\omega_1}$. An analogous result is also proved for the cone over $I^{\\omega_1}$. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved."}
{"category": "Math", "title": "Constructing near-embeddings of codimension one manifolds with countable dense singular sets", "abstract": "The purpose of this paper is to present, for all $n\\ge 3$, very simple examples of continuous maps $f:M^{n-1} \\to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an embedding, i.e. $f$ is a {\\sl near-embedding}."}
{"category": "Math", "title": "Spaces of idempotent measures of compact metric spaces", "abstract": "We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube."}
{"category": "Math", "title": "Hereditary invertible linear surjections and splitting problems for selections", "abstract": "Let $A+B$ be the pointwise (Minkowski) sum of two convex subsets $A$ and $B$ of a Banach space. Is it true that every continuous mapping $h:X \\to A+B$ splits into a sum $h=f+g$ of continuous mappings $f:X \\to A$ and $g:X \\to B$? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear surjections between Banach spaces. Some affirmative and negative results on such invertibility with respect to an appropriate class of convex compacta are presented. As a corollary, a positive answer to the above question is obtained for strictly convex finite-dimensional precompact spaces."}
{"category": "Math", "title": "Non-crossing Knight's Tour in 3-Dimension", "abstract": "Non-crossing knight's tours in 3-dimension is a new field of research. The author has shown its possibility in small cuboids and in cubes up to 8x8x8 size. It can also be extended to larger size cubes and cuboids. The author has achieved jumps of length 15, 46, 88, 159, 258 and 395 in cubes of size 3x3x3, 4x4x4, 5x5x5, 6x6x6, 7x7x7 and 8x8x8 respectively. This amounts to covering 59%, 73%, 71%, 74%, 76% and 77% cells in these cubes."}
{"category": "Math", "title": "Appell polynomials and their relatives III. Conditionally free theory", "abstract": "We extend to the multivariate non-commutative context the descriptions of a \"once-stripped\" probability measure in terms of Jacobi parameters, orthogonal polynomials, and the moment generating function. The corresponding map Phi on states was introduced previously by Belinschi and Nica. We then relate these constructions to the c-free probability theory, which is a version of free probability for algebras with two states, introduced by Bozejko, Leinert, and Speicher. This theory includes as two extreme cases the free and Boolean probability theories. The main objects in the paper are the analogs of the Appell polynomial families in the two state context. They arise as fixed points of the transformation which takes a polynomial family to the associated polynomial family (in several variables), and their orthogonality is also related to the map Phi above. In addition, we prove recursions, generating functions, and factorization and martingale properties for these polynomials, and describe the c-free version of the Kailath-Segall polynomials, their combinatorics, and Hilbert space representations."}
{"category": "Math", "title": "Free evolution on algebras with two states", "abstract": "The key result in the paper concerns two transformations, Phi(rho, psi) and B_t(psi) on states on the algebra of non-commutative polynomials, or equivalently on joint distributions of d-tuples of non-commuting operators. These transformations are related to free probability: Phi intertwines the action of B_t and the free convolution with the semigroup {rho_t}. The maps {B_t} were introduced by Belinschi and Nica as a semigroup of transformations such that B_1 is the bijection between infinitely divisible distributions in the Boolean and free probability theories. They proved the intertwining property above for a single-variable version of the map Phi and the particular case of the free heat semigroup. The more general two-variable map Phi comes, not from free probability, but from the theory of two-state algebras, also called the conditionally free probability theory, introduced by Bozejko, Leinert, and Speicher. Orthogonality of the c-free versions of the Appell polynomials, investigated in arXiv:0803.4279, is closely related to the single-variable map Phi. On the other hand, more general free Meixner families behave well under all the transformations above, and provide clues to their general behavior. Besides the evolution equation, other results include the positivity of the map Phi and descriptions of its fixed points and range."}
{"category": "Math", "title": "C^1 actions of the mapping class group on the circle", "abstract": "Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z."}
{"category": "Math", "title": "Homotopy type of the complement of an immersion and classification of embeddings of tori", "abstract": "This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\\times S^q\\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high dimension, namely in the metastable range $m\\ge p+3q/2+2$, $p\\le q$, which is a natural limit for the classical methods of embedding theory, has been explicitely described earlier. The aim of this note is to present an approach which allows for results in lower dimension."}
{"category": "Math", "title": "Overview of some general results in combinatorial enumeration", "abstract": "This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of context-free languages, 3. holonomicity (i.e., P-recursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn^{-3/2}r^n and 5. ultimate modular periodicity of numbers of MSOL-definable structures."}
{"category": "Math", "title": "Singularities of projected immersions revisited", "abstract": "Sz\\H ucs proved in 2000 that the $r$-tuple-point manifold of a generic immersion is cobordant to the $\\Sigma^{1_{r-1}}$-point manifold of its generic projection. Here we slightly extend this by showing that the natural mappings of these manifolds are bordant to each other. The main novelty of our approach is that we construct the bordism explicitly."}
{"category": "Math", "title": "Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of the Urysohn space", "abstract": "We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces $\\Bbb U$, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry $ISO(\\Bbb U)$ contains everywhere dense locally finite subgroup; Theorem G(Globalization): For each finite metric space $F$ there exists another finite metric space $\\bar F$ and isometric imbedding $j$ of $F$ to $\\bar F$ such that isometry $j$ induces the imbedding of the group monomorphism of the group of isometries of the space $F$ to the group of isometries of space $\\bar F$ and each partial isometry of $F$ can be extended up to global isometry in $\\bar F$. The fact that theorem $G$, is true was announced in 2005 by author without proof, and was proved by S.Solecki in \\cite{Sol} (see also \\cite{P,P1}) based on the previous complicate results of other authors. The theorem is generalization of the Hrushevski's theorem about the globalization of the partial isomorphisms of finite graphs. We intend to give a constructive proof in the same spirit for metric spaces elsewhere. We also give the strengthening of homogeneity of Urysohn space and in the last paragraph we gave a short survey of the various constructions of Urysohn space including the new proof of the construction of shift invariant universal distance matrix from \\cite{CV}."}
{"category": "Math", "title": "Sums of Hecke eigenvalues over quadratic polynomials", "abstract": "Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \\leq X} a(q(n)) = cX + O(X^{6/7+eps}) for some constant c depending on f and q. The constant vanishes in many cases, for example if k is even. On the way a Kuznetsov formula for half-integral weight and entries having different sign is derived."}
{"category": "Math", "title": "Metabelian SL(n,C) representations of knot groups", "abstract": "We give a classification of irreducible metabelian representations from a knot group into SL(n,C) and GL(n,C). If the homology of the n-fold branched cover of the knot is finite, we show that every irreducible metabelian SL(n,C) representation is conjugate to a unitary representation and that the set of conjugacy classes of such representations is finite. In that case, we give a formula for this number in terms of the Alexander polynomial of the knot. These results are the higher rank generalizations of a result of Nagasato, who recently studied irreducible, metabelian SL(2,C) representations of knot groups. Finally we deduce the existence irreducible metabelian SL(n,C) representations of the knot group for any knot with nontrivial Alexander polynomial."}
{"category": "Math", "title": "A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)", "abstract": "This is the original manuscript dated March 9th 1983, typeset by the Editors for the Proceedings of the Midwest Geometry Conference 2007 held in memory of Thomas Branson. Stephen Paneitz passed away on September 1st 1983 while attending a conference in Clausthal and the manuscript was never published. For more than 20 years these few pages were circulated informally. In November 2004, as a service to the mathematical community, Tom Branson added a scan of the manuscript to his website. Here we make it available more formally. It is surely one of the most cited unpublished articles. The differential operator defined in this article plays a key role in conformal differential geometry in dimension 4 and is now known as the Paneitz operator."}
{"category": "Math", "title": "Real and complex zeros of Riemannian random waves", "abstract": "We consider Riemannian random waves, i.e. Gaussian random linear combination of eigenfunctions of the Laplacian on a compact Riemannian manifold with frequencies from a short interval (`asymptotically fixed frequency'). We first show that the expected limit distribution of the real zero set of a is uniform with respect to the volume form of a compact Riemannian manifold $(M, g)$. We then show that the complex zero set of the analytic continuations of such Riemannian random waves to a Grauert tube in the complexification of $M$ tends to a limit current."}
{"category": "Math", "title": "Products and Factors of Banach function spaces", "abstract": "Given two Banach function spaces we study the pointwise product space E.F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E . M(E, F)=F, where M(E,F) denotes the space of multiplication operators from E into F."}
{"category": "Math", "title": "Commutators of contactomorphisms", "abstract": "The group of volume preserving diffeomorphisms, the group of symplectomorphisms and the group of contactomorphisms constitute the classical groups of diffeomorphisms. The first homology groups of the compactly supported identity components of the first two groups have been computed by Thurston and Banyaga, respectively. In this paper we solve the long standing problem on the algebraic structure of the third classical diffeomorphism group, i.e. the contactomorphism group. Namely we show that the compactly supported identity component of the group of contactomorphisms is perfect and simple (if the underlying manifold is connected). The result could be applied in various ways."}
{"category": "Math", "title": "Locally constant functors", "abstract": "We study locally constant coefficients. We first study the theory of homotopy Kan extensions with locally constant coefficients in model categories, and explain how it characterizes the homotopy theory of small categories. We explain how to interpret this in terms of left Bousfield localization of categories of diagrams with values in a combinatorial model category. At last, we explain how the theory of homotopy Kan extensions in derivators can be used to understand locally constant functors."}
{"category": "Math", "title": "On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian", "abstract": "Let $\\lambda$ be a positive number, and let $(x_j:j\\in\\mathbb Z)\\subset\\mathbb R$ be a fixed Riesz-basis sequence, namely, $(x_j)$ is strictly increasing, and the set of functions $\\{\\mathbb R\\ni t\\mapsto e^{ix_jt}:j\\in\\mathbb Z\\}$ is a Riesz basis ({\\it i.e.,} unconditionalbasis) for $L_2[-\\pi,\\pi]$. Given a function $f\\in L_2(\\mathbb R)$ whose Fourier transform is zero almost everywhere outside the interval $[-\\pi,\\pi]$, there is a unique square-summable sequence $(a_j:j\\in\\mathbb Z)$, depending on $\\lambda$ and $f$, such that the function$$I_\\lambda(f)(x):=\\sum_{j\\in\\mathbb Z}a_je^{-\\lambda(x-x_j)^2}, \\qquad x\\in\\mathbb R, $$ is continuous and square integrable on $(-\\infty,\\infty)$, and satisfies the interpolatory conditions $I_\\lambda (f)(x_j)=f(x_j)$, $j\\in\\mathbb Z$. It is shown that $I_\\lambda(f)$ converges to $f$ in $L_2(\\mathbb R)$, and also uniformly on $\\mathbb R$, as $\\lambda\\to0^+$. A multidimensional version of this result is also obtained. In addition, the fundamental functions for the univariate interpolation process are defined, and some of their basic properties, including their exponential decay for large argument, are established. It is further shown that the associated interpolation operators are bounded on $\\ell_p(\\mathbb Z)$ for every $p\\in[1,\\infty]$."}
{"category": "Math", "title": "Birational automorphisms of nodal quartic threefolds", "abstract": "It is well-known that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by standard relations between reflections on an elliptic curve. It is also known that a factorial nodal quartic threefold is birationally rigid and its group of birational selfmaps is generated by biregular ones and certain birational involutions. We prove that all relations between these involutions are implied by standard relations on elliptic curves, complete the proof of birational rigidity over a non-closed field and describe the situations when some of the birational involutions in question become regular (and, in particular, complete the proof of the initial theorem on birational rigidity, since some details were not established in the original paper of M.Mella)."}
{"category": "Math", "title": "The identity type weak factorisation system", "abstract": "We show that the classifying category C(T) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids."}
{"category": "Math", "title": "Spectral conditions on Lie and Jordan algebras of compact operators", "abstract": "We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable."}
{"category": "Math", "title": "Clifford-Wolf homogeneous Riemannian manifolds", "abstract": "In this paper, using connections between Clifford-Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford-Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing property defined and studied previously by J.E. D'Atri and H.K. Nickerson. In the last part of the paper we study properties of Clifford-Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres, and discuss numerous connections between these spaces and various classical objects."}
{"category": "Math", "title": "Transfer maps and nonexistence of joint determinant", "abstract": "Transfer Maps, sometimes called norm maps, for Milnor's $K$-theory were first defined by Bass and Tate (1972) for simple extensions of fields via tame symbol and Weil's reciprocity law, but their functoriality had not been settled until Kato (1980). On the other hand, functorial transfer maps for the Goodwillie group are easily defined. We show that these natural transfer maps actually agree with the classical but difficult transfer maps by Bass and Tate. With this result, we build an isomorphism from the Goodwillie groups to Milnor's $K$-groups of fields, which in turn provides a description of joint determinants for the commuting invertible matrices. In particular, we explicitly determine certain joint determinants for the commuting invertible matrices over a finite field, the field of rational numbers, real numbers and complex numbers into the respective group of units of given field."}
{"category": "Math", "title": "Regularity of monoids under Schutzenberger products", "abstract": "In this paper we give a partial answer to the problem which is about the regularity of Schutzenberger products in semigroups asked by Gallagher in his thesis [Problem 6.1.6]{Gallagher} and, also, we investigate the regularity for the new version of the Schutzenberger product which was defined in [Ates]."}
{"category": "Math", "title": "The Kostant form of $\\mathfrak{U}(sl_n^+)$ and the Borel subalgebra of the Schur algebra S(n,r)", "abstract": "Let $A_n(K)$ be the Kostant form of $\\mathfrak{U}(sl_n^+)$ and $\\Gamma$ the monoid generated by the positive roots of $sl_n$. For each $\\lambda\\in \\Lambda(n,r)$ we construct a functor $F_{\\lambda}$ from the category of finitely generated $\\Gamma$-graded $A_n(K)$-modules to the category of finite dimensional $S^+(n,r)$-modules, with the property that $F_{\\lambda}$ maps (minimal) projective resolutions of the one-dimensional $A_n(K)$-module $K_{A}$ to (minimal) projective resolutions of the simple $S^+(n,r)$-module $K_{\\lambda}$."}
{"category": "Math", "title": "Solution to a combinatorial puzzle arising from Mayer's theory of cluster integrals", "abstract": "Mayer's theory of cluster integrals allows one to write the partition function of a gas model as a generating function of weighted graphs. Recently, Labelle, Leroux and Ducharme have studied the graph weights arising from the one-dimensional hard-core gas model and noticed that the sum of the weights over all connected graphs with $n$ vertices is $(-n)^{n-1}$. This is, up to sign, the number of rooted Cayley trees on $n$ vertices and the authors asked for a combinatorial explanation. The main goal of this article is to provide such an explanation."}
{"category": "Math", "title": "A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers", "abstract": "In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers. An explicit formulae for hyperharmonic numbers, general generating functions of the Fibonacci- and Lucas numbers are obtained. Besides we define \"hyperfibonacci numbers\", \"hyperlucas numbers\". Using these new concepts, some relations between ordinary and incomplete Fibonacci- and Lucas numbers are investigated."}
{"category": "Math", "title": "On the image of code polynomials under theta map", "abstract": "The theta map sends code polynomials into the ring of Siegel modular forms of even weights. Explicit description of the image is known for $g\\leq 3$ and the surjectivity of the theta map follows. Instead it is known that this map is not surjective for $g\\geq 5$. In this paper we discuss the possibility of an embedding between the associated projective varieties. We prove that this is not possible for $g\\geq 4$ and consequently we get the non surjectivity of the graded rings for the remaining case $g=4$."}
{"category": "Math", "title": "On Riemann sums and maximal functions in $\\ZR^n$", "abstract": "In this paper we investigate problems on almost everywhere convergence of subsequences of Riemann sums \\md0 R_nf(x)=\\frac{1}{n}\\sum_{k=0}^{n-1}f\\bigg(x+\\frac{k}{n}\\bigg),\\quad x\\in \\ZT. \\emd We establish a relevant connection between Riemann and ordinary maximal functions, which allows to use techniques and results of the theory of differentiations of integrals in $\\ZR^n$ in mentioned problems. In particular, we prove that for a definite sequence of infinite dimension $n_k$ Riemann sums $R_{n_k}f(x)$ converge almost everywhere for any $f\\in L^p$ with $p>1$."}
{"category": "Math", "title": "Elementary proof of Rayleigh formula for graphs", "abstract": "The Rayleigh monotonicity is a principle from the theory of electrical networks. Its combinatorial interpretation says for each two edges of a graph G, that the presence of one of them in a random spanning tree of G is negatively correlated with the presence of the other edge. In this paper we give a self-contained (inductive) proof of Rayleigh monotonicity for graphs."}
{"category": "Math", "title": "Quelques calculs d'espaces $\\R^i f_* G$ sur des courbes", "abstract": "We give some properties (cancellation, representability, stratification) of the sheaf R^i f_* G for an affine relative curve f:U -> S admitting a smooth compactification and G a solvable group."}
{"category": "Math", "title": "Rademacher averages on noncommutative symmetric spaces", "abstract": "Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let $(\\epsilon_k)_k$ be a Rademacher sequence, on some probability space $\\Omega$. For finite sequences $(x_k)_k of E(M), we consider the Rademacher averages $\\sum_k \\epsilon_k\\otimes x_k$ as elements of the noncommutative function space $E(L^\\infty(\\Omega)\\otimes M)$ and study estimates for their norms $\\Vert \\sum_k \\epsilon_k \\otimes x_k\\Vert_E$ calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, the latter norm is equivalent to the infimum of $\\Vert (\\sum y_k^*y_k)^{{1/2}}\\Vert + \\Vert (\\sum z_k z_k^*)^{{1/2}}\\Vert$ over all $y_k,z_k$ in E(M) such that $x_k=y_k+z_k$ for any k. Dual estimates are given when E is 2-convex and has a non trivial upper Boyd index. We also study Rademacher averages for doubly indexed families of E(M)."}
{"category": "Math", "title": "Completely 1-complemented subspaces of Schatten spaces", "abstract": "We consider the Schatten spaces S^p in the framework of operator space theory and for any $1\\leq p\\not=2<\\infty$, we characterize the completely 1-complemented subspaces of S^p. They turn out to be the direct sums of spaces of the form S^p(H,K), where H,K are Hilbert spaces. This result is related to some previous work of Arazy-Friedman giving a description of all 1-complemented subspaces of S^p in terms of the Cartan factors of types 1-4. We use operator space structures on these Cartan factors regarded as subspaces of appropriate noncommutative L^p-spaces. Also we show that for any $n\\geq 2$, there is a triple isomorphism on some Cartan factor of type 4 and of dimension 2n which is not completely isometric, and we investigate L^p-versions of such isomorphisms."}
{"category": "Math", "title": "Dilations and rigid factorisations on noncommutative L^p-spaces", "abstract": "We study some factorisation and dilation properties of completely positive maps on noncommutative L^p-spaces. We show that Akcoglu's dilation theorem for positive contractions on classical (=commutative) L^p-spaces has no reasonable analog in the noncommutative setting. Our study relies on non symmetric analogs of Pisier's operator space valued noncommutative L^p-spaces that we investigate in the first part of the paper."}
{"category": "Math", "title": "Separable states and positive maps II", "abstract": "Using the natural duality between linear functionals on tensor products of C*-algebras with the trace class operators on a Hilbert space H and linear maps of the C*-algebra into B(H), we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF C*-algebras"}
{"category": "Math", "title": "On the number of graphs not containing $K_{3,3}$ as a minor", "abstract": "We derive precise asymptotic estimates for the number of labelled graphs not containing $K_{3,3}$ as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random $K_{3,3}$-minor-free graphs, like the expected number of edges. To establish these results, we translate a decomposition for the corresponding graph class into equations for generating functions and use singularity analysis. We also find a precise estimate for the number of graphs not containing the graph $K_{3,3}$ plus an edge as a minor."}
{"category": "Math", "title": "Quantum Isometry Group of the n tori", "abstract": "We show that the Quantum Isometry Group(as introduced in \\cite{goswami}) of the n tori is the classical isometry group. Moreover, using a result in \\cite{bhowmick goswami}, we conclude that the Quantum Isometry group of the noncommutative n tori is a Rieffel deformation of the Quantum Isometry Group of the commutative n tori."}
{"category": "Math", "title": "Yang-Mills equation for stable Higgs sheaves", "abstract": "We establish a Kobayashi-Hitchin correspondence for the stable Higgs sheaves on a compact Kaehler manifold. Using it, we also obtain a Kobayashi-Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group."}
{"category": "Math", "title": "Extension of the SAEM algorithm for nonlinear mixed models with two levels of random effects", "abstract": "This article focuses on parameter estimation of multi-levels nonlinear mixed effects models (MNLMEMs). These models are used to analyze data presenting multiple hierarchical levels of grouping (cluster data, clinical trials with several observation periods,...). The variability of the individual parameters of the regression function is thus decomposed as a between-sub ject variability and higher levels of variability (for example within-sub ject variability). We propose maximum likelihood estimates of parameters of those MNLMEMs with two levels of random effects, using an extension of the SAEM-MCMC algorithm. The extended SAEM algorithm is split into an explicit direct EM algorithm and a stochastic EM part. Compared to the original algorithm, additional sufficient statistics have to be approximated by relying on the conditional distribution of the second level of random effects. This estimation method is evaluated on pharmacokinetic cross-over simulated trials, mimicking theophyllin concentration data. Results obtained on those datasets with either the SAEM algorithm or the FOCE algorithm (implemented in the nlme function of R software) are compared: biases and RMSEs of almost all the SAEM estimates are smaller than the FOCE ones. Finally, we apply the extended SAEM algorithm to analyze the pharmacokinetic interaction of tenofovir on atazanavir, a novel protease inhibitor, from the ANRS 107-Puzzle 2 study. A significant decrease of the area under the curve of atazanavir is found in patients receiving both treatments."}
{"category": "Math", "title": "Periodic unique beta-expansions: the Sharkovskii ordering", "abstract": "Let $\\beta\\in(1,2)$. Each $x\\in[0,\\frac{1}{\\beta-1}]$ can be represented in the form \\[ x=\\sum_{k=1}^\\infty \\epsilon_k\\beta^{-k}, \\] where $\\epsilon_k\\in\\{0,1\\}$ for all $k$ (a $\\beta$-expansion of $x$). If $\\beta>\\frac{1+\\sqrt5}{2}$, then, as is well known, there always exist $x\\in(0,\\frac1{\\beta-1})$ which have a unique $\\be$-expansion. In the present paper we study (purely) periodic unique $\\beta$-expansions and show that for each $n\\ge2$ there exists $\\beta_n\\in[\\frac{1+\\sqrt5}{2},2)$ such that there are no unique periodic $\\beta$-expansions of smallest period $n$ for $\\beta\\le\\beta_n$ and at least one such expansion for $\\beta>\\beta_n$. Furthermore, we prove that $\\beta_k<\\beta_m$ if and only if $k$ is less than $m$ in the sense of the Sharkovski\\u{\\i} ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps."}
{"category": "Math", "title": "Covering functors without groups", "abstract": "Coverings in the representation theory of algebras were introduced for the Auslander-Reiten quiver of a representation finite algebra by Riedtmann and later for finite dimensional algebras by Bongartz and Gabriel, R. Martinez-Villa and de la Pe\\~na. The best understood class covering functors is that of Galois covering functors F: A -> B determined by the action of a group of automorphisms of A. In this work we introduce the balanced covering functors which include the Galois class and for which classical Galois covering-type results still hold. For instance, if F:A -> B is a balanced covering functor, where A and B are linear categories over an algebraically closed field, and B is tame, then A is tame."}
{"category": "Math", "title": "Comptage de courbes sur le plan projectif \\'eclat\\'e en trois points align\\'es", "abstract": "We prove a version of Manin's conjecture for the projective plane blown up in three collinear points, the base field being a global field of positive characteristic."}
{"category": "Math", "title": "On finite simple and nonsolvable groups acting on closed 4-manifolds", "abstract": "We show that the only finite nonabelian simple groups which admit a locally linear, homologically trivial action on a closed simply connected 4-manifold $M$ (or on a 4-manifold with trivial first homology) are the alternating groups $A_5$, $A_6$ and the linear fractional group PSL(2,7) (we note that for homologically nontrivial actions all finite groups occur). The situation depends strongly on the second Betti number $b_2(M)$ of $M$ and has been known before if $b_2(M)$ is different from two, so the main new result of the paper concerns the case $b_2(M)=2$. We prove that the only simple group that occurs in this case is $A_5$, and then give a short list of finite nonsolvable groups which contains all candidates for actions of such groups."}
{"category": "Math", "title": "Stochastic solution of a nonlinear fractional differential equation", "abstract": "A stochastic solution is constructed for a fractional generalization of the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a fractional generalization of the branching exponential process and propagation processes which are spectral integrals of Levy processes"}
{"category": "Math", "title": "Hyperelliptic curves, L-polynomials, and random matrices", "abstract": "We analyze the distribution of unitarized L-polynomials Lp(T) (as p varies) obtained from a hyperelliptic curve of genus g <= 3 defined over Q. In the generic case, we find experimental agreement with a predicted correspondence (based on the Katz-Sarnak random matrix model) between the distributions of Lp(T) and of characteristic polynomials of random matrices in the compact Lie group USp(2g). We then formulate an analogue of the Sato-Tate conjecture for curves of genus 2, in which the generic distribution is augmented by 22 exceptional distributions, each corresponding to a compact subgroup of USp(4). In every case, we exhibit a curve closely matching the proposed distribution, and can find no curves unaccounted for by our classification."}
{"category": "Math", "title": "On the strength of dependent products in the type theory of Martin-L\\\"of", "abstract": "One may formulate the dependent product types of Martin-L\\\"of type theory either in terms of abstraction and application operators like those for the lambda-calculus; or in terms of introduction and elimination rules like those for the other constructors of type theory. It is known that the latter rules are at least as strong as the former: we show that they are in fact strictly stronger. We also show, in the presence of the identity types, that the elimination rule for dependent products--which is a \"higher-order\" inference rule in the sense of Schroeder-Heister--can be reformulated in a first-order manner. Finally, we consider the principle of function extensionality in type theory, which asserts that two elements of a dependent product type which are pointwise propositionally equal, are themselves propositionally equal. We demonstrate that the usual formulation of this principle fails to verify a number of very natural propositional equalities; and suggest an alternative formulation which rectifies this deficiency."}
{"category": "Math", "title": "Ernest Michael and theory of continuous selections", "abstract": "This is the introductory paper to the special issue of Topology and Its Applications entirely dedicated to the theory of continuous selections of multivalued mappings. Since the pioneering work of Ernest Michael from 1956 can rightfully be considered as the year of birth of this theory, this issue of the journal is in fact dedicated to the 50th anniversary of the theory of continuous selections. At the same time the papers of this issue are all dedicated to the 80th anniversary of the founder of this theory - Ernest Michael."}
{"category": "Math", "title": "Quotients of bounded homogeneous domains by cyclic groups", "abstract": "Let $D$ be a bounded homogeneous domain in $\\mbb{C}^n$ and let $\\Gamma$ be a cyclic discrete subgroup of the automorphism group of $D$. It is shown that the complex space $D/\\Gamma$ is Stein."}
{"category": "Math", "title": "G\\'en\\'eralisation de l'homologie d'Heegaard-Floer aux entrelacs singuliers & Raffinement de l'homologie de Khovanov aux entrelacs restreints", "abstract": "A categorification of a polynomial link invariant is an homological invariant which contains the polynomial one as its graded Euler characteristic. This field has been initiated by Khovanov categorification of the Jones polynomial. Later, P. Ozsvath and Z. Szabo gave a categorification of Alexander polynomial. Besides their increased abilities for distinguishing knots, this new invariants seem to carry many geometrical informations. On the other hand, Vassiliev works gives another way to study link invariant, by generalizing them to singular links i.e. links with a finite number of rigid transverse double points. The first part of this thesis deals with a possible relation between these two approaches in the case of the Alexander polynomial. To this purpose, we extend grid presentation for links to singular links. Then we use it to generalize Ozsvath and Szabo invariant to singular links. Besides the consistency of its definition, we prove that this invariant is acyclic under some conditions which naturally make its Euler characteristic vanish. This work can be considered as a first step toward a categorification of Vassiliev theory. In a second part, we give a refinement of Khovanov homology to restricted links. Restricted links are link diagrams up to a restricted set of Reidemeister moves. In particular, closed braids can be seen as a subset of them. Such a refinement give then a new tool for studying knots and their deformations."}
{"category": "Math", "title": "On the maximum size of a $(k,l)$-sum-free subset of an abelian group", "abstract": "A subset $A$ of a given finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ does not equal the sum of $l$ (not necessarily distinct) elements of $A$. We are interested in finding the maximum size $\\lambda_{k,l}(G)$ of a $(k,l)$-sum-free subset in $G$. A $(2,1)$-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group $\\mathbb{Z}_n$ was found almost forty years ago by Diamanda and Yap; the general case for arbitrary finite abelian groups was recently settled by Green and Ruzsa. Here we find the value of $\\lambda_{3,1}(\\mathbb{Z}_n)$. More generally, a recent paper of Hamidoune and Plagne examines $(k,l)$-sum-free sets in $G$ when $k-l$ and the order of $G$ are relatively prime; we extend their results to see what happens without this assumption."}
{"category": "Math", "title": "A class of Lorentzian manifolds with indecomposable holonomy groups", "abstract": "We consider a class of $S^{1}$-bundles whose total space admits a nowhere vanishing recurrent lightlike vector field with respect to a Lorentzian metric. This metric can be modified such that its restricted holonomy group is indecomposable and reducible. We apply Hodge theory to construct examples with Hermitian screen holonomy. Finally, we construct complete pp-waves."}
{"category": "Math", "title": "Poisson Cluster Measures: Quasi-invariance, Integration by Parts and Equilibrium Stochastic Dynamics", "abstract": "The distribution $\\mu_{cl}$ of a Poisson cluster process in $X=\\mathbb{R}^{d}$ (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in $\\mathfrak{X}=\\sqcup_{n} X^n$, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure $\\mu_{cl}$ is quasi-invariant with respect to the group of compactly supported diffeomorphisms of $X$ and prove an integration-by-parts formula for $\\mu_{cl}$. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms."}
{"category": "Math", "title": "Geometry and intersection theory on Hilbert schemes of families of nodal curves", "abstract": "We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We work out the action of the blowup or 'discriminant' polarization on some natural cycles in the Hilbert scheme, including generalized diagonals and cycles, called 'node scrolls', parametrizing schemes supported on singular points. We derive an intersection calculus for Chern classes of tautological vector bundles, which are closely related to enumerative geometry."}
{"category": "Math", "title": "Triangle Area Numbers and Solid Rectangular Numbers", "abstract": "In this work, we define a triangle area number to be the area number of a triangle whose sides have integer lengths, and whose area is a rational number. In Result 3, on page 17, we prove that every triangle area number is in fact an integer which is a multiple of 6. Certain divisibility and other conditions and formulas are also derived, which the three integer sidelengths must satisfy. On pages 20 and 21, we list all the triangle area numbers not exceeding 999."}
{"category": "Math", "title": "Distortion of Mappings and $L_{q,p}$-Cohomology", "abstract": "We study some relation between some geometrically defined classes of diffeomorphisms between manifolds and the $L_{q,p}$-cohomology of these manifolds. Some applications to vanishing and non vanishing results in $L_{q,p}$-cohomology are given."}
{"category": "Math", "title": "Discrete series characters for affine Hecke algebras and their formal degrees", "abstract": "We introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types E) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases)."}
{"category": "Math", "title": "A combinatorial proof of Rayleigh monotonicity for graphs", "abstract": "We give an elementary, self-contained, and purely combinatorial proof of the Rayleigh monotonicity property of graphs."}
{"category": "Math", "title": "Nuclearity-related properties for nonselfadjoint algebras", "abstract": "In analogy with the C*-algebra theory, we study variants appropriate to nonselfadjoint algebras of nuclearity, the local lifting property, exactness, and the weak expectation property. In addition, we study the relationships between these notions, and how they are connected with the classical C*-algebra theory through the use of C*-algebras generated by the algebra."}
{"category": "Math", "title": "A hierarchical eigenmodel for pooled covariance estimation", "abstract": "While a set of covariance matrices corresponding to different populations are unlikely to be exactly equal they can still exhibit a high degree of similarity. For example, some pairs of variables may be positively correlated across most groups, while the correlation between other pairs may be consistently negative. In such cases much of the similarity across covariance matrices can be described by similarities in their principal axes, the axes defined by the eigenvectors of the covariance matrices. Estimating the degree of across-population eigenvector heterogeneity can be helpful for a variety of estimation tasks. Eigenvector matrices can be pooled to form a central set of principal axes, and to the extent that the axes are similar, covariance estimates for populations having small sample sizes can be stabilized by shrinking their principal axes towards the across-population center. To this end, this article develops a hierarchical model and estimation procedure for pooling principal axes across several populations. The model for the across-group heterogeneity is based on a matrix-valued antipodally symmetric Bingham distribution that can flexibly describe notions of ``center'' and ``spread'' for a population of orthonormal matrices."}
{"category": "Math", "title": "Strichartz estimates for Schr\\\"odinger operators with a non-smooth magnetic potential", "abstract": "We prove Strichartz estimates for the absolutely continuous evolution of a Schr\\\"odinger operator $H = (i\\nabla + A)^2 + V$ in $\\R^n$, $n > 2$. Both the magnetic and electric potentials are time-independent and satisfy pointwise polynomial decay bounds. The vector potential $A(x)$ is assumed to be continuous but need not possess any Sobolev regularity. This work is a refinement of previous methods, which required extra conditions on ${\\rm div} A$ or $|\\nabla|^{\\frac12}A$ in order to place the first order part of the perturbation within a suitable class of pseudo-differential operators."}
{"category": "Math", "title": "Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic", "abstract": "In analogy with the Riemann zeta function at positive integers, for each finite field F_p^r with fixed characteristic p we consider Carlitz zeta values zeta_r(n) at positive integers n. Our theorem asserts that among the zeta values in {zeta_r(1), zeta_r(2), zeta_r(3), ... | r = 1, 2, 3, ...}, all the algebraic relations are those algebraic relations within each individual family {zeta_r(1), zeta_r(2), zeta_r(3), ...}. These are the algebraic relations coming from the Euler-Carlitz relations and the Frobenius relations. To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed."}
{"category": "Math", "title": "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", "abstract": "We prove that the Whitehead link complement and the (-2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 x Catalan's constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound."}
{"category": "Math", "title": "Trigonometric Cherednik algebra at critical level and quantum many-body problems", "abstract": "For any module over the affine Weyl group we construct a representation of the associated trigonometric Cherednik algebra $A(k)$ at critical level in terms of Dunkl type operators. Under this representation the center of $A(k)$ produces quantum conserved integrals for root system generalizations of quantum spin-particle systems on the circle with delta function interactions. This enables us to translate the spectral problem of such a quantum spin-particle system to questions in the representation theory of $A(k)$. We use this approach to derive the associated Bethe ansatz equations. They are expressed in terms of the normalized intertwiners of $A(k)$."}
{"category": "Math", "title": "Transversely Lie holomorphic foliations on projective spaces", "abstract": "We prove that a one-dimensional foliation with generic singularities on a projective space, exhibiting a Lie group transverse structure in the complement of some codimension one algebraic subset is logarithmic, i.e., it is the intersection of codimension one foliations given by closed one-forms with simple poles. If there is only one singularity in a suitable affine space, then the foliation is induced by a linear diagonal vector field."}
{"category": "Math", "title": "Measured foliations and Hilbert 12th problem", "abstract": "Yu. I. Manin conjectured that the maximal abelian extensions of the real quadratic number fields are generated by the pseudo-lattices with real multiplication. We prove this conjecture using theory of measured foliations on the modular curves."}
{"category": "Math", "title": "Virtual Morse theory on $\\Omega Ham(M,\\omega)$", "abstract": "We relate previously defined quantum characteristic classes to Morse theoretic aspects of the Hofer length functional on $\\ls$. As an application we prove a theorem which can be interpreted as stating that this functional behaves \"virtually\" as a perfect Morse-Bott functional with a flow. This can be applied to study topology and Hofer geometry of $ \\text {Ham}(M, \\omega)$. We also use this to give a prediction for the index of some geodesics for this functional, which was recently partially verified by Yael Karshon and Jennifer Slimowitz."}
{"category": "Math", "title": "A Theoretical Study of Mafia Games", "abstract": "Mafia can be described as an experiment in human psychology and mass hysteria, or as a game between informed minority and uninformed majority. Focus on a very restricted setting, Mossel et al. [to appear in Ann. Appl. Probab. Volume 18, Number 2] showed that in the mafia game without detectives, if the civilians and mafias both adopt the optimal randomized strategy, then the two groups have comparable probabilities of winning exactly when the total player size is R and the mafia size is of order Sqrt(R). They also proposed a conjecture which stated that this phenomenon should be valid in a more extensive framework. In this paper, we first indicate that the main theorem given by Mossel et al. [to appear in Ann. Appl. Probab. Volume 18, Number 2] can not guarantee their conclusion, i.e., the two groups have comparable winning probabilities when the mafia size is of order Sqrt(R). Then we give a theorem which validates the correctness of their conclusion. In the last, by proving the conjecture proposed by Mossel et al. [to appear in Ann. Appl. Probab. Volume 18, Number 2], we generalize the phenomenon to a more extensive framework, of which the mafia game without detectives is only a special case."}
{"category": "Math", "title": "Statistical analysis of an archeological find", "abstract": "In 1980, a burial tomb was unearthed in Jerusalem containing ossuaries (limestone coffins) bearing such inscriptions as Yeshua son of Yehosef, Marya, Yoseh--names which match those of New Testament (NT) figures, but were otherwise in common use. This paper discusses certain statistical aspects of authenticating or repudiating links between this find and the NT family. The available data are laid out, and we examine the distribution of names (onomasticon) of the era. An approach is proposed for measuring the ``surprisingness'' of the observed outcome relative to a ``hypothesis'' that the tombsite belonged to the NT family. On the basis of a particular--but far from uncontested--set of assumptions, our measure of ``surprisingness'' is significantly high."}
{"category": "Math", "title": "Discussion of: Statistical analysis of an archeological find", "abstract": "Discussion of ``Statistical analysis of an archeological find'' by Andrey Feuerverger [arXiv:0804.0079]"}
{"category": "Math", "title": "Discussion of: Statistical analysis of an archeological find", "abstract": "Discussion of ``Statistical analysis of an archeological find'' by Andrey Feuerverger [arXiv:0804.0079]"}
{"category": "Math", "title": "Schlomilch and Bell Series for Bessel's Functions, with Probabilistic Applications", "abstract": "We have introduced and investigated so-called Shlomilchs and Bells series for modified Bessel's functions, namely, their asymptotic and non-asymptotic properties, connection with Stirling's and Bell's numbers etc. We have obtained exact constants in the moment inequalities for sums of centered independent random variables, improved their asymptotical properties, found lower and upper bounds, calculated a more exact approximation, elaborated the numerical algorithm for their calculation, studied the class of smoothing, etc."}
{"category": "Math", "title": "Discussion of: Statistical analysis of an archaeological find", "abstract": "Discussion of ``Statistical analysis of an archeological find'' by Andrey Feuerverger [arXiv:0804.0079]"}
{"category": "Math", "title": "A family of conformally flat Hamiltonian-minimal Lagrangian tori in $\\mathbb{CP}^3$", "abstract": "In this paper by reduction we construct a family of conformally flat Hamiltonian-minimal Lagrangian tori in $\\mathbb{CP}^3$ as the image of the composition of the Hopf map $\\mathcal{H}: \\mathbb{S}^7\\to \\mathbb{CP}^3$ and a map $\\psi:\\mathbb{R}^3 \\to \\mathbb{S}^7$ with certain conditions."}
{"category": "Math", "title": "Discussion of: Statistical analysis of an archeological find--skeptical counting challenges to an archaeological find", "abstract": "Discussion of ``Statistical analysis of an archeological find'' by Andrey Feuerverger [arXiv:0804.0079]"}
{"category": "Math", "title": "Discussion of: Statistical analysis of an archeological find", "abstract": "Discussion of ``Statistical analysis of an archeological find'' by Andrey Feuerverger [arXiv:0804.0079]"}
{"category": "Math", "title": "Discussion of: Statistical analysis of an archaeological find", "abstract": "We critique the analysis by A. Feuerverger of an archaeological find [arXiv:0804.0079] that has been alleged by some to be the tomb of Jesus of Nazareth. We show that his analysis rests on six faulty assumptions that have been severely criticized by historians, archaeologists, and scholars in related disciplines. We summarize the results of an alternative computation using Bayes' theorem that estimates a probability of less than 2% that the Talpiot tomb belongs to Jesus of Nazareth."}
{"category": "Math", "title": "A splitting criterion for vector bundles on blowing ups of the plane", "abstract": "Let $f_s: X_s \\to {\\bf {P}}^2$ be the blowing-up of $s$ distinct points and $E$ a vector bundle on $X_s$. Here we give a cohomological criterio which is equivalent to $E \\cong f_s^\\ast (A)$ with $A$ a direct sum of line bundles. We also some cohomological characterizations of very particular rank 2 vector bundles on ${\\bf {P}}^2$."}
{"category": "Math", "title": "Discussion of: Statistical analysis of an archeological find", "abstract": "Discussion of ``Statistical analysis of an archeological find'' by Andrey Feuerverger [arXiv:0804.0079]"}
{"category": "Math", "title": "Vector Bundles on Products of Varieties with $n$-blocks Collections", "abstract": "Here we consider the product of varieties with $n$-blocks collections . We give some cohomological splitting conditions for rank 2 bundles. A cohomological characterization for vector bundles is also provided. The tools are Beilinson's type spectral sequences generalized by Costa and Mir\\'o-Roig. Moreover we introduce a notion of Castelnuovo-Mumford regularity on a product of finitely many projective spaces and smooth quadric hypersurfaces in order to prove two splitting criteria for vector bundle with arbitrary rank."}
{"category": "Math", "title": "Grassmannians of classical buildings", "abstract": "This book is dedicated to Grassmannians associated with buildings of classical types: usual, polar, and half-spin Grassmannians. Grassmannians of vector spaces and Grassmannians consisting of totally isotropic subspaces of non-degenerate alternating, Hermitian, and symmetric forms are special cases of these \"building\" Grassmannians."}
{"category": "Math", "title": "Discussion of: Bayesian views of an archaeological find", "abstract": "Discussion of ``Statistical analysis of an archeological find'' by Andrey Feuerverger [arXiv:0804.0079]"}
{"category": "Math", "title": "Rejoinder of: Statistical analysis of an archeological find", "abstract": "Rejoinder of ``Statistical analysis of an archeological find'' [arXiv:0804.0079]"}
{"category": "Math", "title": "Prescription de la multiplicit\\'e des valeurs propres du laplacien de Hodge-de Rham", "abstract": "On any compact manifold of dimension greater than 6, we prescribe the volume and any finite part of the spectrum Hodge Laplacian acting on $p$-form for $1\\leq p<\\frac n2$. In particular, we prescribe the multiplicity of the first eigenvalues."}
{"category": "Math", "title": "Commensurability classes of (-2,3,n) pretzel knot complements", "abstract": "Let K be a hyperbolic (-2,3,n) pretzel knot and M = S^3 K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M. Indeed, if n \\neq 7, we show that M is the unique knot complement in its class. We include examples to illustrate how our methods apply to a broad class of Montesinos knots."}
{"category": "Math", "title": "Estimates of tempered stable densities", "abstract": "Estimates of densities of convolution semigroups of probability measures are given under specific assumptions on the corresponding L\\'evy measure and the L\\'evy--Khinchin exponent. The assumptions are satisfied, e.g., by tempered stable semigroups of J. Rosi{\\'n}ski."}
{"category": "Math", "title": "Spectral data for Hamiltonian-minimal Lagrangian tori in $CP^2$", "abstract": "In this work, we find spectral data that allow to find Hamiltonian-minimal Lagrangian tori in $CP^2$ in terms of theta functions of spectral curves."}
{"category": "Math", "title": "Weak existence of the squared Bessel process and CIR process with skew reflection on a deterministic time dependent curve", "abstract": "Using the technique of moving domains, and classical direct stochastic calculus, we construct the Cox-Ingersoll-Ross process, as well as its square root, with additional skew reflection on a deterministic time dependent curve."}
{"category": "Math", "title": "Proof of W.M.Schmidt's conjecture concerning successive minima of a lattice", "abstract": "For a real $N\\ge 1$ and a vector $\\xi =(1,\\xi_1,...,\\xi_n)$ define a matrix $$ {\\cal A} (\\xi, N) = ({array}{ccccc} N^{-1} & 0& 0& ... &0 \\cr N^{\\frac{1}{n}} \\xi_1 & -N^{\\frac{1}{n}} & 0&... & 0 \\cr N^{\\frac{1}{n}} \\xi_2 &0& -N^{\\frac{1}{n}} & ... & 0 \\cr ... &... &... &... \\cr N^{\\frac{1}{n}} \\xi_n &0&0&... &- N^{\\frac{1}{n}} {array}) $$ and a lattice $$ \\Lambda (\\xi, N) = {\\cal A} (\\xi, N)\\mathbb{Z}^{n+1}. $$ Consider a convex 0-symmetric body $${\\cal W} = \\{z= (x,y_1,...,y_n)\\in \\mathbb{R}^{n+1}: \\max (|x|, |y|)\\le 1 \\} >.$$ For a natural $l, 1\\le l \\le n+1$ let $\\mu_l (\\xi, N)$ be the $l$-th successive minimum of ${\\cal W}$ with respect to $ \\Lambda (\\xi, N)$. We prove that there exist real numbers $\\xi_1,...,\\xi_n$ linearly independent together with 1 over $\\mathbb{Z}$, such that $\\mu_k (\\xi, N) \\to 0$ as $ N\\to \\infty$ and $\\mu_{k+2} (\\xi, N) \\to \\infty$ as $ N\\to \\infty$."}
{"category": "Math", "title": "Basic properties of nonlinear stochastic Schr\\\"{o}dinger equations driven by Brownian motions", "abstract": "The paper is devoted to the study of nonlinear stochastic Schr\\\"{o}dinger equations driven by standard cylindrical Brownian motions (NSSEs) arising from the unraveling of quantum master equations. Under the Born--Markov approximations, this class of stochastic evolutions equations on Hilbert spaces provides characterizations of both continuous quantum measurement processes and the evolution of quantum systems. First, we deal with the existence and uniqueness of regular solutions to NSSEs. Second, we provide two general criteria for the existence of regular invariant measures for NSSEs. We apply our results to a forced and damped quantum oscillator."}
{"category": "Math", "title": "Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve", "abstract": "We investigate pathwise uniqueness for the squared Bessel and Cox-Ingersoll-Ross processes with additional reflection term that is multiplied by some real number strictly between minus one and one. The reflection term is the symmetric local time of the corresponding processes at a deterministic time dependent curve."}
{"category": "Math", "title": "Symmetry of local minimizers for the three dimensional Ginzburg-Landau functional", "abstract": "We classify nonconstant entire local minimizers of the standard Ginzburg-Landau functional for maps in $H^{1}_{loc}(R^3;R^3)$ satisfying a natural energy bound. Up to translations and rotations, such solutions of the Ginzburg-Landau system are given by an explicit solution equivariant under the action of the orthogonal group."}
{"category": "Math", "title": "Infinitesimal affine geometry of metric spaces endowed with a dilatation structure", "abstract": "We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry, endowed with a noncommutative vector addition operation and with a modified version of ratio of three collinear points. This is the geometry of normed affine group spaces, a category which contains the ones of homogeneous groups, Carnot groups or contractible groups. In this category group operations are not fundamental, but derived objects, and the generalization of affine geometry is not based on incidence relations."}
{"category": "Math", "title": "Phase Transitions in Partially Structured Random Graphs", "abstract": "We study a one parameter family of random graph models that spans a continuum between traditional random graphs of the Erd\\H{o}s-R\\'enyi type, where there is no underlying structure, and percolation models, where the possible edges are dictated exactly by a geometry. We find that previously developed theories in the fields of random graphs and percolation have, starting from different directions, covered almost all the models described by our family. In particular, the existence or not of a phase transition where a giant cluster arises has been proved for all values of the parameter but one. We prove that the single remaining case behaves like a random graph and has a single linearly sized cluster when the expected vertex degree is greater than one."}
{"category": "Math", "title": "A criterion for topological equivalence of two variable complex analytic function germs", "abstract": "We show that two analytic function germs $(\\C^2,0) \\to (\\C,0)$ are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets that preserves the multiplicites of these components, their Puiseux pairs, and the intersection numbers of any pairs of distinct components."}
{"category": "Math", "title": "Estimates of Newman Sum over Numbers Multiple of a Fixed Integer", "abstract": "We prove that the ratio of the Newman sum over numbers multiple of a fixed integer which is not multiple of 3 and the Newman sum over numbers multiple of a fixed integer divisible by 3 is o(1) when the upper limit of summing tends to infinity.We also discuss a connection of our results with a digit conjecture on primes."}
{"category": "Math", "title": "Complexity and cohomology for cut and projection tilings", "abstract": "We consider a subclass of tilings, the tilings obtained by cut and projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent \\alpha in terms of the ranks of certain groups which appear in the construction. We give bounds for \\alpha. These computations apply to some well known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over \\Q. We show that such a link cannot be established for more general tilings, and we present a counter-example in dimension one."}
{"category": "Math", "title": "Proving Touchard's Theorem From Euler's Form", "abstract": "This paper derives Touchard's theorem from Euler's form for odd perfect numbers. It also fine-tunes Euler's form."}
{"category": "Math", "title": "Uniform Eberlein spaces and the finite axiom of choice", "abstract": "We work in set-theory without choice $\\ZF$. Given a closed subset $F$ of $[0,1]^I$ which is a bounded subset of $\\ell^1(I)$ ({\\em resp.} such that $F \\subseteq \\ell^0(I)$), we show that the countable axiom of choice for finite subsets of $I$, ({\\em resp.} the countable axiom of choice $\\ACD$) implies that $F$ is compact. This enhances previous results where $\\ACD$ ({\\em resp.} the axiom of Dependent Choices $\\DC$) was required. Moreover, if $I$ is linearly orderable (for example $I=\\IR$), the closed unit ball of $\\ell^2(I)$ is weakly compact (in $\\ZF$)."}
{"category": "Math", "title": "Riffles, ruffles, and the turning algebra", "abstract": "The rising algebra is a subalgebra of the group algebra of the symmetric group S_n, gotten by lumping together permutations having the same number of rising sequences. This well-known algebra arises naturally when studying riffle shuffles. Here we introduce a number of other subalgebras that arise naturally when studying `ruffles', which are like riffles except that after cutting the deck you turn over the bunch of cards that were on the bottom. This orphaned draft offers no context or motivation, and uses idiosyncratic notation and terminology that `seemed like a good idea at the time'. We're making it available because it has been cited in this form."}
{"category": "Math", "title": "Renormalization of the two-dimensional Lotka--Volterra model", "abstract": "We show that renormalized two-dimensional Lotka--Volterra models near criticality converge to a super-Brownian motion. This is used to establish long-term survival of a rare type for a range of parameter values near the voter model."}
{"category": "Math", "title": "Smooth ergodic theory", "abstract": "This is an expository article/encyclopedia entry explaining the history, techniques, and central results in the field of smooth ergodic theory."}
{"category": "Math", "title": "Extremality and designs in spaces of quadratic forms", "abstract": "A well known theorem of Voronoi caracterizes extreme quadratic forms and Euclidean lattices, that is those which are local maxima for the Hermite function, as perfect and eutactic. This caracterization has been extended in various cases, such that family of lattices, sections of lattices, Humbert forms, etc. Moreover, there is a criterion for extreme lattices, discovered by Venkov, formulated in terms of spherical designs which has been extended in the case of Grassmannians and sections of lattices. In this article, we define a general frame, in which there is a ``Voronoi characterization'', and a ``Venkov criterion'' through an appropriate notion of design. This frame encompasses many interesting situations in which a ``Voronoi characterization'' has been proved. We also discuss the question of extremality relatively to the Epstein zeta function, and we extend to our frame a characterization of final zeta-extremality formulated by Delone and Ryshkov and a criterion in terms of designs found by Coulangeon."}
{"category": "Math", "title": "Continuation of connecting orbits in 3D-ODEs: (II) Cycle-to-cycle connections", "abstract": "In Part I of this paper we discussed new methods for the numerical continuation of point-to-cycle connecting orbits in 3-dimensional autonomous ODE's using projection boundary conditions. In this second part we extend the method to the numerical continuation of cycle-to-cycle connecting orbits. In our approach, the projection boundary conditions near the cycles are formulated using eigenfunctions of the associated adjoint variational equations, avoiding costly and numerically unstable computations of the monodromy matrices. The equations for the eigenfunctions are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find the connecting orbits are discussed in general and illustrated with an example from population dynamics. Complete AUTO demos, which can be easily adapted to any autonomous 3-dimensional ODE system, are freely available."}
{"category": "Math", "title": "On the q-meromorphic Weyl algebra", "abstract": "We introduce a q-analogue MW_q for the meromorphic Weyl algebra, and study the normalization problem and the symmetric powers sym^n(MW_q) for such algebra from a combinatorial viewpoint."}
{"category": "Math", "title": "Bi-Lipschitz geometry of complex surface singularities", "abstract": "We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical."}
{"category": "Math", "title": "The n-homology of representations", "abstract": "This is an expository article, written for a special edition of the journal of the UMA (Union Matematica Argentina) commemorating the mathematician Misha Cotlar. The article gives an introduction to the n-homology groups and surveys some developments, with a particular emphasis on results pertaining to the problem of calculating n-homology groups."}
{"category": "Math", "title": "Gromov's Pinching Constant", "abstract": "In early 80's M.Gromov showed that there exists a constant $\\epsilon$ such that any compact Riemannian manifold $M^n$ with $|K|_{M^n} \\cdot diam^2(M^n) \\leq \\epsilon$ can be finitely covered by a nilmanifold. The present paper illustrates by an explicit example that the pinching constant $\\epsilon$ depends on the dimension $n$ of the manifold, in particular, it decreases with the dimension at least as $\\frac{12}{n^2}.$"}
{"category": "Math", "title": "Singular Chern Classes of Schubert Varieties via Small Resolution", "abstract": "We discuss a method for calculating the Chern-Schwartz-MacPherson (CSM) class of a Schubert variety in the Grassmannian using small resolutions introduced by Zelevinsky. As a consequence, we show how to compute the Chern-Mather class and local Euler obstructions using small resolutions instead of the Nash blowup. The algorithm obtained for CSM classes also allows us to prove new cases of a positivity conjecture of Aluffi and Mihalcea."}
{"category": "Math", "title": "Bent Rectangles", "abstract": "We study generalized regular bent functions using a representation by bent rectangles, that is, special matrices with restrictions on rows and columns. We describe affine transformations of bent rectangles, propose new biaffine and bilinear constructions, study partitions of a vector space into affine planes of the same dimension and use such partitions to build bent rectangles. We illustrate the concept of bent rectangles by examples for the Boolean case."}
{"category": "Math", "title": "Characterization of the critical values of branching random walks on weighted graphs through infinite-type branching processes", "abstract": "We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible."}
{"category": "Math", "title": "Framed Deformation of Galois Representation", "abstract": "We studied framed deformations of two dimensional Galois representation of which the residue representation restrict to decomposition groups are scalars, and established a modular lifting theorem for certain cases. We then proved a family version of the result, and used it to determine the structure of deformation rings over characteristic zero fields. These can be applied to the study of exceptional zero of p-adic L-function."}
{"category": "Math", "title": "A variational theory for monotone vector fields", "abstract": "Monotone vector fields were introduced almost 40 years ago as nonlinear extensions of positive definite linear operators, but also as natural extensions of gradients of convex potentials. These vector fields are not always derived from potentials in the classical sense, and as such they are not always amenable to the standard methods of the calculus of variations. We describe here how the selfdual variational calculus developed recently by the author, provides a variational approach to PDEs and evolution equations driven by maximal monotone operators. To any such a vector field T on a reflexive Banach space X, one can associate a convex selfdual Lagrangian L_T on phase space X x X* that can be seen as a \"potential\" for T, in the sense that the problem of inverting T reduces to minimizing the convex energy L_T. This variational approach to maximal monotone operators allows their theory to be analyzed with the full range of methods --computational or not-- that are available for variational settings. Standard convex analysis (on phase space) can then be used to establish many old and new results concerned with the identification, superposition, and resolution of such vector fields."}
{"category": "Math", "title": "Disk single Hurwitz numbers", "abstract": "It is investigated Hurwitz numbers, that correspond to covering of disk with single non-simple boundary critical value. It is found differential equations, that describe a generating function for these numbers."}
{"category": "Math", "title": "Shannon Multiresolution Analysis on the Heisenberg Group", "abstract": "We present a notion of frame multiresolution analysis on the Heisenberg group, abbreviated by FMRA, and study its properties. Using the irreducible representations of this group, we shall define a sinc-type function which is our starting point for obtaining the scaling function. Further, we shall give a concrete example of a wavelet FMRA on the Heisenberg group which is analogous to the Shannon MRA on $\\RR$."}
{"category": "Math", "title": "An Index Theorem for Toeplitz Operators on the Quarter-Plane", "abstract": "We prove an index theorem for Toeplitz operators on the quarter-plane using the index theory for generalized Toeplitz operators introduced by G. J. Murphy. To prove this index theorem we construct an indicial triple on the tensor product of two C*-algebras provided with indicial triples with general conditions. We show that our results can be extended to some extensions of the theory of Toeplitz operators on the quarter-plane."}
{"category": "Math", "title": "The critical exponent of the Arshon words", "abstract": "Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov (for n = 3), we prove that for all n greater than or equal to 2, the critical exponent of the Arshon word of order $n$ is given by (3n-2)/(2n-2), and this exponent is attained at position 1."}
{"category": "Math", "title": "Quenched large deviations for random walk in a random environment", "abstract": "We take the point of view of a particle performing random walk with bounded jumps on $\\mathbb{Z}^d$ in a stationary and ergodic random environment. We prove the quenched large deviation principle (LDP) for the pair empirical measure of the environment Markov chain. By an appropriate contraction, we deduce the quenched LDP for the mean velocity of the particle and obtain a variational formula for the corresponding rate function. We propose an Ansatz for the minimizer of this formula. When $d=1$, we verify this Ansatz and generalize the nearest-neighbor result of Comets, Gantert and Zeitouni to walks with bounded jumps."}
{"category": "Math", "title": "Stable equivariant abelianization, its properties, and applications", "abstract": "Let $G$ be a finite group. For a based $G$-space $X$ and a Mackey functor $M$, a topological Mackey functor $X\\widetilde\\otimes M$ is constructed, which will be called the stable equivariant abelianization of $X$ with coefficients in $M$. When $X$ is a based $G$-CW complex, $X\\widetilde\\otimes M$ is shown to be an infinite loop space in the sense of $\\mathcal{G}$-spaces. This gives a version of the $RO(G)$-graded equivariant Dold-Thom theorem. Applying a variant of Elmendorf's construction, we get a model for the Eilenberg-Mac Lane spectrum $HM$. The proof uses a structure theorem for Mackey functors and our previous results."}
{"category": "Math", "title": "On the quantum homology algebra of toric Fano manifolds", "abstract": "In this paper we study certain algebraic properties of the quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semi-simplicity of the quantum homology algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily-verified sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non semi-simple quantum homology algebra, and others in which the Calabi quasi-morphism is non-unique."}
{"category": "Math", "title": "On the Inner Curvature of the Second Fundamental Form of a Surface in the Hyperbolic Space", "abstract": "The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form can be satisfied only by extrinsic spheres."}
{"category": "Math", "title": "Nonnegative Curvature on Low Dimensional Cohomogeneity One Manifolds", "abstract": "This paper has been withdrawn by the author due to a critical error in the proof of Theorem A pointed out by Burkhard Wilking."}
{"category": "Math", "title": "Partial Chromatic Polynomials and Diagonally Distinct Sudoku Squares", "abstract": "Sudoku grids can be thought of as graphs where the vertices are the squares of the grid, and edges join vertices in the same row, column, or sub-grid. A Sudoku puzzle corresponds to a partial proper coloring of the Sudoku graph. We provide a new and simpler proof of the theorem which states that the number of completions of partial colorings of a graph is a polynomial in the number of colors (originally due to Herzberg and Murty). Moreover, we construct Sudoku squares of arbitrary size with distinct entries on both diagonals (a similar proof was first published by Keedwell, unknown to the author)."}
{"category": "Math", "title": "An example of a solid von Neumann algebra", "abstract": "We prove that the group-measure-space von Neumann algebra $L^\\infty(T^2) \\rtimes SL(2,Z)$ is solid. The proof uses topological amenability of the action of $SL(2,Z)$ on the Higson corona of $Z^2$."}
{"category": "Math", "title": "Generalised Hermite Constants, Voronoi Theory and Heights on Flag Varieties", "abstract": "This paper explores the study of the general Hermite constant associated to the general linear group and its irreducible representations, as defined by T. Watanabe. To that end, a height, which naturally applies to flag varieties, is built and notions of perfection and eutaxy characterising extremality are introduced. Finally we acquaint some relations (e.g. with Korkine--Zolotareff reduction), upper bounds and computation relative to these constants."}
{"category": "Math", "title": "The Pointwise Estimates of Solutions for Semilinear Dissipative Wave Equation", "abstract": "In this paper we focus on the global-in-time existence and the pointwise estimates of solutions to the initial value problem for the semilinear dissipative wave equation in multi-dimensions. By using the method of Green function combined with the energy estimates, we obtain the pointwise estimates of the solution."}
{"category": "Math", "title": "Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation", "abstract": "Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation."}
{"category": "Math", "title": "Noncanonical Polynomial Representations of Classical Lie Algebras", "abstract": "Using the skew-symmetry of the differential operators and multiplication operators in the canonical representations of finite-dimensional classical Lie algebras, we obtain some noncanonical polynomial representations of the classical Lie algebras. The representation spaces of all polynomials are decomposed into irreducible submodules, which are infinite-dimensional. Bases for the irreducible submodules are constructed. In particular, we obtain some new infinite-dimensional irreducible modules of symplectic Lie algebras that are not of highest weight type."}
{"category": "Math", "title": "On the obstruction to linearizability of 2-order ordinary differential equations", "abstract": "In this paper, we investigate the action of pseudogroup of all point transformations on the natural bundle of equations $y'' = u^0(x,y) + u^1(x,y)y' + u^2(x,y)(y')^2 + u^3(x,y)(y')^3$. We calculate the 1-st nontrivial differential invariant of this action. It is a horizontal differential 2-form with values in some algebra, it is defined on the bundle of 2--jets of sections of the considered bundle. We prove that this form is a unique obstruction to linearizability of these equations by point transformations."}
{"category": "Math", "title": "Probabilistic Interpretation for Systems of Isaacs Equations with Two Reflecting Barriers", "abstract": "In this paper we investigate zero-sum two-player stochastic differential games whose cost functionals are given by doubly controlled reflected backward stochastic differential equations (RBSDEs) with two barriers. For admissible controls which can depend on the whole past and so include, in particular, information occurring before the beginning of the game, the games are interpreted as games of the type \"admissible strategy\" against \"admissible control\", and the associated lower and upper value functions are studied. A priori random, they are shown to be deterministic, and it is proved that they are the unique viscosity solutions of the associated upper and the lower Bellman-Isaacs equations with two barriers, respectively. For the proofs we make full use of the penalization method for RBSDEs with one barrier and RBSDEs with two barriers. For this end we also prove new estimates for RBSDEs with two barriers, which are sharper than those in [18]. Furthermore, we show that the viscosity solution of the Isaacs equation with two reflecting barriers not only can be approximated by the viscosity solutions of penalized Isaacs equations with one barrier, but also directly by the viscosity solutions of penalized Isaacs equations without barrier."}
{"category": "Math", "title": "Zero-sum free sequences with small sum-set", "abstract": "Let A be a zero-sum free subset of Z_n with |A|=k. We compute for k\\le 7 the least possible size of the set of all subset-sums of A."}
{"category": "Math", "title": "Stability results for uniquely determined sets from two directions in discrete tomography", "abstract": "In this paper we prove several new stability results for the reconstruction of binary images from two projections. We consider an original image that is uniquely determined by its projections and possible reconstructions from slightly different projections. We show that for a given difference in the projections, the reconstruction can only be disjoint from the original image if the size of the image is not too large. We also prove an upper bound for the size of the image given the error in the projections and the size of the intersection between the image and the reconstruction."}
{"category": "Math", "title": "Existence of an infinite particle limit of stochastic ranking process", "abstract": "We study a stochastic particle system which models the time evolution of the ranking of books by online bookstores (e.g., Amazon). In this system, particles are lined in a queue. Each particle jumps at random jump times to the top of the queue, and otherwise stays in the queue, being pushed toward the tail every time another particle jumps to the top. In an infinite particle limit, the random motion of each particle between its jumps converges to a deterministic trajectory. (This trajectory is actually observed in the ranking data on web sites.) We prove that the (random) empirical distribution of this particle system converges to a deterministic space-time dependent distribution. A core of the proof is the law of large numbers for {\\it dependent} random variables."}
{"category": "Math", "title": "Injective Spaces via Adjunction", "abstract": "Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relation $\\mathfrak{x}\\to x$ between ultrafilters and points of a topological space $X$ as arrows in $X$. Naturally, this point of view opens the door to the use of concepts and ideas from (enriched) Category Theory for the investigation of (for instance) topological spaces. In this paper we study cocompleteness, adjoint functors and Kan extensions in the context of topological theories. We show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on $\\SET$. This way we obtain enriched versions of known results about injective topological spaces and continuous lattices."}
{"category": "Math", "title": "Remarks on special symplectic connections", "abstract": "The notion of special symplectic connections is closely related to contact parabolic geometries due to the work of M. Cahen and L. Schwachh\\\"ofer. We remind their characterization and reinterpret the result in terms of generalized Weyl connections. The aim of this paper is to provide an alternative and more explicit construction of special symplectic connections of three types from the list. This is done by pulling back an ambient linear connection from the total space of a natural scale bundle over the homogeneous model of the corresponding contact parabolic structure."}
{"category": "Math", "title": "The heat kernel and frequency localized functions on the Heisenberg group", "abstract": "The goal of this paper is to study the action of the heat operator on the Heisenberg group H^d, and in particular to characterize Besov spaces of negative index on H^d in terms of the heat kernel. That characterization can be extended to positive indexes using Bernstein inequalities. As a corollary we obtain a proof of refined Sobolev inequalities in W^{s,p} spaces."}
{"category": "Math", "title": "Optimal transport and Perelman's reduced volume", "abstract": "We show that a certain entropy-like function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman's reduced volume."}
{"category": "Math", "title": "Limit sets and a problem in dynamical systems", "abstract": "We approximate a chain recurrent dynamical system by periodic dynamical systems. This is similar to the well known Bohr theorem on approximation of almost periodic functions by periodic functions."}
{"category": "Math", "title": "Relative Property (T) Actions and Trivial Outer Automorphism Groups", "abstract": "We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S.-Popa \\cite[Def. 4.1]{Pop06}. There are uncountably many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives in particular the first examples of such actions for the free group on $2$ generators."}
{"category": "Math", "title": "Congruences between modular forms given by the divided beta family in homotopy theory", "abstract": "We characterize the 2-line of the p-local Adams-Novikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p > 3. We give a similar characterization of the 1-line, reinterpreting some earlier work of A. Baker and G. Laures. These results are then used to deduce that, for l a prime which generates the p-adic units, the spectrum Q(l) detects the alpha and beta families in the stable stems."}
{"category": "Math", "title": "Uniqueness of fast travelling fronts in reaction-diffusion equations with delay", "abstract": "We consider positive travelling fronts of the time-delayed reaction-diffusion equation with the monostable birth function. Our main result says that for every fixed and sufficiently large velocity c, the positive travelling front is unique (modulo translations). To prove the uniqueness, we introduce a small parameter 1/c and realize the Lyapunov-Schmidt reduction in a scale of Banach spaces."}
{"category": "Math", "title": "Projective spectrum in Banach algebras", "abstract": "For a tuple $A=(A_0, A_1, ..., A_n)$ of elements in a unital Banach algebra ${\\mathcal B}$, its {\\em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0, z_1, ..., z_n]\\in \\pn$ such that $A(z)=z_0A_0+z_1A_1+... +z_nA_n$ is not invertible in ${\\mathcal B}$. The pre-image of $p(A)$ in ${\\cc}^{n+1}$ is denoted by $P(A)$. When ${\\mathcal B}$ is the $k\\times k$ matrix algebra $M_k(\\cc)$, the projective spectrum is a projective hypersurface. In infinite dimensional cases, projective spectrums can be very complicated, but also have some properties similar to that of hypersurfaces. When $A$ is commutative, $P(A)$ is a union of hyperplanes. When ${\\mathcal B}$ is reflexive or is a $C^*$-algebra, the {\\em projective resolvent set} $P^c(A):=\\cc^{n+1}\\setminus P(A)$ is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type ${\\mathcal B}$-valued 1-form $A^{-1}(z)dA(z)$ on $P^c(A)$. As a consequence, we show that if ${\\mathcal B}$ is a $C^*$-algebra with a trace $\\phi$, then $\\phi(A^{-1}(z)dA(z))$ is a nontrivial element in the de Rham cohomology space $H^1_d(P^c(A), \\cc)$."}
{"category": "Math", "title": "Fibred surfaces with general pencils of genus 5", "abstract": "Let $f:S \\fr B$ be a surface fibration with fibres of genus 5. We find a linear relation between the fundamental invariants of the surface. Namely $K_f^2=\\chi_f+N$ where $N$ is the number of trigonal fibres. Our proof is based on the analysis of the relative canonical algebra $\\cal{R}(f)$."}
{"category": "Math", "title": "A practical procedure to find matching priors for frequentist inference", "abstract": "In the manuscript, we present a practical way to find the matching priors proposed by Welch & Peers (1963) and Peers (1965). We investigate the use of saddlepoint approximations combined with matching priors and obtain p-values of the test of an interest parameter in the presence of nuisance parameter. The advantage of our procedure is the flexibility of choosing different initial conditions so that one can adjust the performance of the test. Two examples have been studied, with coverage verified via Monte Carlo simulation. One relates to the ratio of two exponential means, and the other relates the logistic regression model. Particularly, we are interested in small sample settings."}
{"category": "Math", "title": "Blow-ups and resolutions of strong K\\\"ahler with torsion metrics", "abstract": "On a compact complex manifold we study the behaviour of strong K\\\"ahler with torsion (strong KT) structures under small deformations of the complex structure and the problem of extension of a strong KT metric. In this context we obtain the analogous result of Miyaoka extension theorem. Studying the blow-up of a strong KT manifold at a point or along a complex submanifold, we prove that a complex orbifold endowed with a strong KT metric admits a strong KT resolution. In this way we obtain new examples of compact simply-connected strong KT manifolds."}
{"category": "Math", "title": "An involution on the K-theory of bimonoidal categories with anti-involution", "abstract": "We construct a combinatorially defined involution on the algebraic $K$-theory of the ring spectrum associated to a bimonoidal category with anti-involution. Particular examples of such are braided bimonoidal categories. We investigate examples such as algebraic K-theory of connective complex and real topological K-theory and Waldhausen's K-theory of spaces of the form BBG, for abelian groups G. We show that the involution agrees with the classical one for a bimonoidal category associated to a ring and prove that it is not trivial in the above mentioned examples."}
{"category": "Math", "title": "Geodesic manifolds with a transitive subset of smooth biLipschitz maps", "abstract": "This paper is connected with the problem of describing path metric spaces that are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. Let $X = G/H$ be a homogeneous manifold of a Lie group $G$ and let $d$ be a geodesic distance on $X$ inducing the same topology. Suppose there exists a subgroup $G_S$ of $G$ that acts transitively on $X$, such that each element $g \\in G_S$ induces a locally biLipschitz homeomorphism of the metric space $(X,d)$. Then the metric is locally biLipschitz equivalent to a sub-Riemannian metric. Any such metric is defined by a bracket generating $G_S$-invariant sub-bundle of the tangent bundle. The result is a consequence of a more general fact that requires a transitive family of uniformly biLipschitz diffeomorphisms with a control on their differentials. It will be relevant that the group acting transitively on the space is a Lie group and so it is locally compact, since, in general, the whole group of biLipschitz maps, unlikely the isometry group, is not locally compact. Our method also gives an elementary proof of the following fact. Given a Lipschitz sub-bundle of the tangent bundle of a Finsler manifold, then both the class of piecewise differentiable curves tangent to the sub-bundle and the class of Lipschitz curves almost everywhere tangent to the sub-bundle give rise to the same Finsler-Carnot-Carath\\'eodory metric, under the condition that the topologies induced by these distances coincide with the manifold topology."}
{"category": "Math", "title": "Boundedness and convergence for singular integrals of measures separated by Lipschitz graphs", "abstract": "We shall consider the truncated singular integral operators T_{\\mu, K}^{\\epsilon}f(x)=\\int_{\\mathbb{R}^{n}\\setminus B(x,\\epsilon)}K(x-y)f(y)d\\mu y and related maximal operators $T_{\\mu,K}^{\\ast}f(x)=\\underset{\\epsilon >0}{\\sup}| T_{\\mu,K}^{\\epsilon}f(x)|$. We shall prove for a large class of kernels $K$ and measures $\\mu$ and $\\nu$ that if $\\mu$ and $\\nu$ are separated by a Lipschitz graph, then $T_{\\nu,K}^{\\ast}:L^p(\\nu)\\to L^p(\\mu)$ is bounded for $1<p<\\infty$. We shall also show that the truncated operators $T_{\\mu, K}^{\\epsilon}$ converge weakly in some dense subspaces of $L^2(\\mu)$ under mild assumptions for the measures and the kernels."}
{"category": "Math", "title": "Asymptotic Properties of the Maximum Likelihood Estimator for Stochastic Parabolic Equations with Additive Fractional Brownian Motion", "abstract": "A parameter estimation problem is considered for a diagonaliazable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter $H\\geq 1/2$. The objective is to study asymptotic properties of the maximum likelihood estimator as the number of the Fourier coefficients increases. A necessary and sufficient condition for consistency and asymptotic normality is presented in terms of the eigenvalues of the operators in the equation."}
{"category": "Math", "title": "Dynamics of symmetric dynamical systems with delayed switching", "abstract": "We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value so-called event collisions occur. This paper classifies and analyzes event collisions, a special type of discontinuity induced bifurcations, for periodic orbits. Our focus is on event collisions of symmetric periodic orbits in systems with full reflection symmetry, a symmetry that is prevalent in applications. We derive an implicit expression for the Poincare map near the colliding periodic orbit. The Poincare map is piecewise smooth, finite-dimensional, and changes the dimension of its image at the collision. In the second part of the paper we apply this general result to the class of unstable linear single-degree-of-freedom oscillators where we detect and continue numerically collisions of invariant tori. Moreover, we observe that attracting closed invariant polygons emerge at the torus collision."}
{"category": "Math", "title": "Twisted cscK metrics and K\\\"ahler slope stability", "abstract": "We introduce a cohomological obstruction to solving the constant scalar curvature K\\\"ahler (cscK) equation twisted by a semipositive form, appearing in works of Fine and Song-Tian. Geometrically this gives an obstruction for a manifold to be the base of a holomorphic submersion carrying a cscK metric in certain ``adiabatic'' classes. In turn this produces many new examples of general type threefolds with classes which do not admit a cscK representative. When the twist vanishes our obstruction extends the slope stability of Ross-Thomas to effective divisors on a K\\\"ahler manifold. Thus we find examples of non-projective slope unstable manifolds."}
{"category": "Math", "title": "Fibered orbifolds and crystallographic groups", "abstract": "In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group G determines a geometric fibered orbifold structure on the flat orbifold E^n/G, and conversely every geometric fibered orbifold structure on E^n/G is determined by a normal subgroup N of G, which is maximal in its commensurability class of normal subgroups of G. In particular, we prove that E^n/G is a fiber bundle, with totally geodesic fibers, over a b-dimensional torus, where b is the first Betti number of G. Let N be a normal subgroup of G which is maximal in its commensurability class. We study the relationship between the exact sequence 1 -> N -> G -> G/N -> 1 splitting and the corresponding fibration projection having an affine section. If N is torsion-free, we prove that the exact sequence splits if and only if the fibration projection has an affine section. If the generic fiber F = Span(N)/N has an ordinary point that is fixed by every isometry of F, we prove that the exact sequence always splits. Finally, we describe all the geometric fibrations of the orbit spaces of all 2- and 3-dimensional crystallographic groups building on the work of Conway and Thurston."}
{"category": "Math", "title": "The centralizer of an element in an endomorphism ring", "abstract": "We have results about the centralizer."}
{"category": "Math", "title": "Parabolically induced representations of graded Hecke algebras", "abstract": "We study the representation theory of graded Hecke algebras, starting from scratch and focusing on representations that are obtained with induction from a discrete series representation of a parabolic subalgebra. We determine all intertwining operators between such parabolically induced representations, and use them to parametrize the irreducible representations."}
{"category": "Math", "title": "Four-dimensional Osserman metrics of neutral signature", "abstract": "In the algebraic context, we show that null Osserman, spacelike Osserman, and timelike Osserman are equivalent conditions for a model of signature (2,2). We also classify the null Jordan Osserman models of signature (2,2). In the geometric context, we show that a pseudo-Riemannian manifold of signature (2,2) is null Jordan Osserman if and only if either it has constant sectional curvature or it is locally a complex space form."}
{"category": "Math", "title": "A Multilinear Operator for Almost Product Evaluation of Hankel Determinants", "abstract": "In a recent paper we have presented a method to evaluate certain Hankel determinants as almost products; i.e. as a sum of a small number of products. The technique to find the explicit form of the almost product relies on differential-convolution equations and trace calculations. In the trace calculations a number of intermediate nonlinear terms involving determinants occur, but only to cancel out in the end. In this paper, we introduce a class of multilinear operators \\gamma acting on tuples of matrices as an alternative to the trace method. These operators do not produce extraneous nonlinear terms, and can be combined easily with differentiation. The paper is self contained. An example of an almost product evaluation using \\gamma-operators is worked out in detail and tables of the \\gamma-operator values on various forms of matrices are provided. We also present an explicit evaluation of a new class of Hankel determinants and conjectures."}
{"category": "Math", "title": "Weakly irreducible subgroups of $Sp(1,n+1)$", "abstract": "Connected weakly irreducible not irreducible subgroups of $Sp(1,n+1)\\subset SO(4,4n+4)$ that satisfy a certain additional condition are classified. This will be used to classify connected holonomy groups of pseudo-hyper-K\\\"ahlerian manifolds of index 4."}
{"category": "Math", "title": "Asymptotic behavior of permutation records", "abstract": "We study the asymptotic behavior of two statistics defined on the symmetric group S_n when n tends to infinity: the number of elements of S_n having k records, and the number of elements of S_n for which the sum of the positions of their records is k. We use a probabilistic argument to show that the scaled asymptotic behavior of these statistics can be described by remarkably simple functions."}
{"category": "Math", "title": "Hyperdeterminantal point processes", "abstract": "As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between $2M$ points for some integer $M$. The role of matrices is now played by $2M$-dimensional \"hypercubic\" arrays, and the determinant is replaced by a suitable generalization of it to such arrays -- Cayley's first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization."}
{"category": "Math", "title": "On the role of Convexity in Functional and Isoperimetric Inequalities", "abstract": "This is a continuation of our previous work 0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz-Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz'ya and extended by Barthe--Cattiaux--Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, $q$-log-Sobolev inequalities ($q \\in [1,2]$) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry--Ledoux and Bobkov--Zegarlinski. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the $CD(0,\\infty)$ curvature-dimension condition of Bakry--\\'Emery."}
{"category": "Math", "title": "Algebro-Geometric Invariants of Finitely Generated Groups (The Profile of a Representation Variety)", "abstract": "If G is a finitely generated group, and A an algebraic group, then Hom(G,A) is a possibly reducible algebraic variety denoted by R_A(G). Here we define the profile function, P_d(R_A(G)), of the representation variety of G over A to be P_d(R_A(G))=(N_d(R_A(G)),...,N_0(R_A(G))), where N_i(R_A(G)) stands for the number of irreducible components of R_A(G) of dimension i, where 0\\leq i\\leq d, and d=Dim(R_A(G)). We then use this invariant in the study of fg groups and prove various results. In particular, we show that if G an orientable surface group of genus g\\geq 1, then P_d(R_{SL(2,C)}(G))\\neq P_d(R_{PSL(2,C)}(G)). We also show that the same holds for G a torus knot group with presentation <x,y;x^p=y^t> where both p,t are greater than 2, and that the same also holds when G is a the fundamental group of a compact non-orientable surface of genus g\\geq 3. Further, we show that if a group G can be n+1 generated, and presented by <x_1,...,x_n,y ; W=y^p>, where W is a non-trivial word in F_n=<x_1,...,x_n>, and A=PSL(2, C), that then Dim(R_{A}(G)) is equal to Max{3n, Dim(R_{A}(G'))+2 \\} \\leq 3n+1, where G'=<x_1,...,x_n; W=1>. We also give a condition guaranteeing that the resulting algebraic variety is reducible."}
{"category": "Math", "title": "A Note on Approximately Divisible C$^*$-algebras", "abstract": "Let $\\mathcal A$ be a separable, unital, approximately divisible C$^*$-algebra. We show that $\\mathcal A$ is generated by two self-adjoint elements and the topological free entropy dimension of any finite generating set of $\\mathcal A$ is less than or equal to 1. In addition, we show that the similarity degree of $\\mathcal A$ is at most 5. Thus an approximately divisible C$^*$-algebra has an affirmative answer to Kadison's similarity problem."}
{"category": "Math", "title": "Ramification estimates for the hyperbolic Gauss map", "abstract": "We give the best possible upper bound on the number of exceptional values and the totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic constant mean curvature one surfaces in the hyperbolic three-space and some partial results on the Osserman problem for algebraic case. Moreover, we study the value distribution of the hyperbolic Gauss map for complete constant mean curvature one faces in de Sitter three-space."}
{"category": "Math", "title": "Cohen-Macaulay Monomial Ideals of Codimension 2", "abstract": "We give a structure theorem for Cohen Macaulay monomial ideals of codimension 2, and describe all possible relation matrices of such ideals. In case that the ideal has a linear resolution, the relation matrices can be identified with the spanning trees of a connected chordal graph with the property that each distinct pair of maximal cliques of the graph has at most one vertex in common."}
{"category": "Math", "title": "On the Mullineux involution for Ariki-Koike algebras", "abstract": "This note is concerned with a natural generalization of the Mullineux involution for Ariki-Koike algebras. Using a result of Fayers together with previous results by the authors, we give an efficient algorithm for computing this generalized Mullineux involution. Our algorithm notably does not involve the determination of paths in affine crystals."}
{"category": "Math", "title": "Dimension expanders", "abstract": "We show that there exists $k \\in \\bbn$ and $0 < \\e \\in\\bbr$ such that for every field $F$ of characteristic zero and for every $n \\in \\bbn$, there exists explicitly given linear transformations $T_1,..., T_k: F^n \\to F^n$ satisfying the following: For every subspace $W$ of $F^n$ of dimension less or equal $\\frac n2$, $ \\dim(W+\\suml^k_{i=1} T_iW) \\ge (1+\\e) \\dim W$. This answers a question of Avi Wigderson [W]. The case of fields of positive characteristic (and in particular finite fields) is left open."}
{"category": "Math", "title": "The False Dilemma: Bayesian vs. Frequentist", "abstract": "There are two main opposing schools of statistical reasoning, Frequentist and Bayesian approaches. Until recent days, the frequentist or classical approach has dominated the scientific research, but Bayesianism has reappeared with a strong impulse that is starting to change the situation. Recently the controversy about the primacy of one of the two approaches seems to be unfinished at a philosophical level, but scientific practices are giving an increasingly important position to the Bayesian approach. This paper eludes philosophical debate to focus on the pragmatic point of view of scientists' day-to-day practices, in which Bayesian methodology is very useful. Several facts and operational values are described as the core-set for understanding the change."}
{"category": "Math", "title": "Extrema de valeurs propres dans une classe conforme", "abstract": "Survey about extremum of eigenvalues of geometric operators within a conformal class of a compact riemannian manifold."}
{"category": "Math", "title": "Properly discontinuous actions on bounded domains", "abstract": "We give sufficient conditions for the quotient of a free, properly discontinuous action on a bounded domain of holomorphy to be a Stein manifold in terms of Poincar\\'e series or limit sets for orbits. An immediate consequence is that the quotient of any cyclic, free, properly discontinuous action on the unit ball or the bidisc is Stein."}
{"category": "Math", "title": "Quantization of quasi-Lie bialgebras", "abstract": "We construct quantization functors of quasi-Lie bialgebras. We establish a bijection between this set of quantization functors, modulo equivalence and twist equivalence, and the set of quantization functors of Lie bialgebras, modulo equivalence. This is based on the acyclicity of the kernel of the natural morphism from the universal deformation complex of quasi-Lie bialgebras to that of Lie bialgebras. The proof of this acyclicity consists in several steps, ending up in the acyclicity of a complex related to free Lie algebras, namely, the universal version of the Lie algebra cohomology complex of a Lie algebra in its enveloping algebra, viewed as the left regular module. Using the same arguments, we also prove the compatibility of quantization functors of quasi-Lie bialgebras with twists, which allows us to recover our earlier results on compatibility of quantization functors with twists in the case of Lie bialgebras."}
{"category": "Math", "title": "Chemical trees minimizing energy and Hosoya index", "abstract": "The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph's eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. Trees minimizing the energy under various additional conditions have been determined in the past, e.g., trees with a given diameter or trees with a perfect matching. However, it is quite a natural problem to minimize the energy of trees with bounded maximum degree--clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimal energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree."}
{"category": "Math", "title": "Singular integrals on Sierpinski gaskets", "abstract": "We construct a class of singular integral operators associated with homogeneous Calder\\'{o}n-Zygmund standard kernels on $d$-dimensional, $d <1$, Sierpinski gaskets $E_d$. These operators are bounded in $L^2(\\mu_d)$ and their principal values diverge $\\mu_d$ almost everywhere, where $\\mu_d$ is the natural (d-dimensional) measure on $E_d$."}
{"category": "Math", "title": "Directed porosity on conformal iterated function systems and weak convergence of singular integrals", "abstract": "The aim of the present paper is twofold. We study directed porosity in connection with conformal iterated function systems (CIFS) and with singular integrals. We prove that limit sets of finite CIFS are porous in a stronger sense than already known. Furthermore we use directed porosity to establish that truncated singular integral operators, with respect to general Radon measures $\\mu$ and kernels $K$, converge weakly in some dense subspaces of $L^2(\\mu)$ when the support of $\\mu$ belongs to a broad family of sets. This class contains many fractal sets like CIFS's limit sets."}
{"category": "Math", "title": "A special case of the $\\Gamma_{00}$ conjecture", "abstract": "In this paper we prove the $\\Gamma_{00}$ conjecture of van Geemen and van der Geer, under the additional assumption that the matrix of coefficients of the tangent has rank at most 2. This assumption is satisfied by Jacobians, and thus our result gives a characterization of the locus of Jacobians among all principally polarized abelian varieties. The proof is by reduction to the (stronger version of the) characterization of Jacobians by semidegenerate trisecants, i.e. by the existence of lines tangent to the Kummer variety at one point and intersecting it in another, proven by Krichever in the course of his proof of the Welters' trisecant conjecture."}
{"category": "Math", "title": "Partial Translation Algebras for Trees", "abstract": "In arXiv:math/0603621 we introduced the notion of a partial translation $C^*$-algebra for a discrete metric space. Here we demonstrate that several important classical $C^*$-algebras and extensions arise naturally by considering partial translation algebras associated with subspaces of trees."}
{"category": "Math", "title": "Approximation of center-valued Betti-numbers", "abstract": "In this paper we generalize the approximation theorem for L^2-Betti numbers to an approximation theorem for center-valued Betti-numbers."}
{"category": "Math", "title": "Convergence Properties of Kemp's q-Binomial Distribution", "abstract": "We consider Kemp's q-analogue of the binomial distribution. Several convergence results involving the classical binomial, the Heine, the discrete normal, and the Poisson distribution are established. Some of them are q-analogues of classical convergence properties. Besides elementary estimates, we apply Mellin transform asymptotics."}
{"category": "Math", "title": "The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees", "abstract": "A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of fixed genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all rooted maps by Marcus and Schaeffer's bijection. This gives an immediate derivation of the asymptotic number of unicellular maps of given genus, and a simple bijective proof of a formula of Lehman and Walsh on the number of triangulations with one vertex. From the labelled case, we deduce an expression of the asymptotic number of maps of genus g with n edges involving the ISE random measure, and an explicit characterization of the limiting profile and radius of random bipartite quadrangulations of genus g in terms of the ISE."}
{"category": "Math", "title": "Semistablity of syzygy bundles on projective spaces in positive characteristics", "abstract": "In char $k = p >0$, A. Langer proved a strong restriction theorem (in the style of H. Flenner) for semistable sheaves to a very general hypersurface of degree $d$, on certain varieties, with the condition that `char $k > d$'. He remarked that to remove this condition, it is enough to answer either of the following questions affirmatively: {\\it For the syzygy bundle $\\sV_d$ of ${\\mathcal O}(d)$, is $\\sV_d$ semistable for arbitrary $n, d$ and $p = {char} k$?, or is there a good estimate on $\\mu_{max}(\\sV_d^*)$?} Here we prove that (1) the bundle $\\sV_d$ is semistable, for a certain infinite set of integers $d\\geq 0$, and (2) for arbitrary $d$, there is a good enough estimate on $\\mu_{max}(\\sV_d^*)$ in terms of $d$ and $n$. In particular one obtains Langer's theorem, in arbitrary characeristic."}
{"category": "Math", "title": "Statistical performance of support vector machines", "abstract": "The support vector machine (SVM) algorithm is well known to the computer learning community for its very good practical results. The goal of the present paper is to study this algorithm from a statistical perspective, using tools of concentration theory and empirical processes. Our main result builds on the observation made by other authors that the SVM can be viewed as a statistical regularization procedure. From this point of view, it can also be interpreted as a model selection principle using a penalized criterion. It is then possible to adapt general methods related to model selection in this framework to study two important points: (1) what is the minimum penalty and how does it compare to the penalty actually used in the SVM algorithm; (2) is it possible to obtain ``oracle inequalities'' in that setting, for the specific loss function used in the SVM algorithm? We show that the answer to the latter question is positive and provides relevant insight to the former. Our result shows that it is possible to obtain fast rates of convergence for SVMs."}
{"category": "Math", "title": "Gauge theoretical methods in the classification of non-Kaehlerian surfaces", "abstract": "The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with $b_2>0$ has $b_2$ curves. By the results of Kato, Nakamura and Dloussky/Oeljeklaus/Toma, this conjecture (if true) would solve this classification problem completely. We explain a new approach (based on techniques from Donaldson theory) to prove existence of curves on class VII surfaces, and we present recent results obtained using this approach."}
{"category": "Math", "title": "High order relaxed schemes for nonlinear reaction diffusion problems in nonconservative form", "abstract": "Different relaxation approximations to partial differential equations, including conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems, have been recently proposed. The present paper focuses onto diffusive relaxed schemes for the numerical approximation of nonlinear reaction diffusion equations. We choose here a nonstandard relaxation scheme that allow the treatment of diffusion equations in their nonconservative form. A comparison with the traditional approach in the case of conservative equations is also included. High order methods are obtained by coupling ENO and WENO schemes for space discretization with IMEX schemes for time integration, where the implicit part can be explicitly solved at a linear cost. To illustrate the high accuracy and good properties of the proposed numerical schemes, also in the degenerate case, we consider various examples in one and two dimensions: the Fisher-Kolmogoroff equation, the porous-Fisher equation and the porous medium equation with strong absorption. Moreover we show a test on a system of PDEs that describe an ecological model for the dispersal and settling of animal populations."}
{"category": "Math", "title": "Homogenization of variational problems in manifold valued BV-spaces", "abstract": "This paper extends the result of \\cite{BM} on the homogenization of integral functionals with linear growth defined for Sobolev maps taking values in a given manifold. Through a $\\Gamma$-convergence analysis, we identify the homogenized energy in the space of functions of bounded variation. It turns out to be finite for $BV$-maps with values in the manifold. The bulk and Cantor parts of the energy involve the tangential homogenized density introduced in \\cite{BM}, while the jump part involves an homogenized surface density given by a geodesic type problem on the manifold."}
{"category": "Math", "title": "Nonstandard Hulls of Locally Exponential Lie Algebras", "abstract": "We show how to construct the nonstandard hull of certain infinite-dimensional Lie algebras in order to generalize a theorem of Pestov on the enlargeability of Banach-Lie algebras. In the process, we consider a nonstandard smoothness condition on functions between locally convex spaces to ensure that the induced function between the nonstandard hulls is smooth. We also discuss some conditions on a function between locally convex spaces which guarantee that its nonstandard extension maps finite points to finite points."}
{"category": "Math", "title": "Cauchy-Kowaleskaya-Kashiwara theorem with growth conditions", "abstract": "We prove the Cauchy-Kowaleskaya-Kashiwara theorem for holomorphic functions with growth conditions."}
{"category": "Math", "title": "Characteristic polynomials of automorphisms of hyperelliptic curves", "abstract": "Let alpha be an automorphism of a hyperelliptic curve C of genus g, and let alpha' be the automorphism of P^1 induced by alpha. Let n be the order of alpha and let n' be the order of alpha'. We show that the triple (g,n,n') completely determines the characteristic polynomial of the automorphism alpha^* of the Jacobian of C, unless n is even, n=n', and (2g+2)/n is even, in which case there are two possibilities. We give explicit formulas for the characteristic polynomial in all cases."}
{"category": "Math", "title": "Tropical Hurwitz Numbers", "abstract": "Hurwitz numbers count genus g, degree d covers of the projective line with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers."}
{"category": "Math", "title": "Mazur intersection property for Asplund spaces", "abstract": "The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund space of density character $\\omega_1$ has a renorming with the Mazur intersection property. Combined with the previous result of Jim\\' enez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP normability of Asplund spaces of density $\\omega_1$ is undecidable in ZFC."}
{"category": "Math", "title": "Potential level-lowering for GSp(4)", "abstract": "In this preprint, we explore a beautiful idea of Skinner and Wiles in the context of GSp(4) over a totally real field. The main result provides congruences between automorphic forms which are Iwahori-spherical at a certain place w, and forms with a tamely ramified principal series at w. Thus, after base change to a totally real finite solvable extension, one can often lower the level at w. For the proof, we first establish an analogue of the Jacquet-Langlands correspondence, using the stable trace formula. The congruences are then obtained on inner forms, which are compact at infinity mod center, and split at all finite places. The crucial ingredient allowing us to do so, is an important result of Roche on types for principal series representations of split reductive groups."}
{"category": "Math", "title": "Minimal links and a result of Gaeta", "abstract": "If $V$ is an equidimensional codimension $c$ subscheme of an $n$-dimensional projective space, and $V$ is linked to $V'$ by a complete intersection $X$, then we say that $V$ is {\\em minimally linked} to $V'$ if $X$ is a codimension $c$ complete intersection of smallest degree containing $V$. Gaeta showed that if $V$ is any arithmetically Cohen-Macaulay (ACM) subscheme of codimension two then there is a finite sequence of minimal links beginning with $V$ and arriving at a complete intersection. We extend this work in the following ways: 1) In the codimension 2 non-ACM case, we show that for any $n \\geq 3$ there are examples of subschemes that are not minimal in their even liaison class, and cannot be minimally linked in any number of steps to a minimal subscheme. 2) Nevertheless, there are examples of non-ACM liaison classes of curves in projective 3-space where all elements are minimally linked in a finite number of steps to a minimal curve. 3) Extending previous work of the authors with Huneke and Ulrich (about the licci case), we show that also in the non-ACM case in any higher codimension there are non-minimal subschemes that are not minimally linked to a minimal subscheme in the even liaison class. 4) J. Watanabe had shown many years ago that codimension 3 graded Gorenstein ideals of any dimension are licci. Here we show that any such ideal is minimally linked in a finite number of steps to a complete intersection, and that it admits a sequence of strictly decreasing CI-biliaisons down to a complete intersection, extending work of Hartshorne, Sabadini and Schlesinger."}
{"category": "Math", "title": "Prop profile of bi-Hamiltonian structures", "abstract": "Recently S.A. Merkulov established a link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of minimal resolutions of props. In particular he described the prop profile of Poisson geometry. In this paper we define a prop such that representations of its minimal resolution in a vector space V are in a one-to-one correspondence with bi-Hamiltonian structures, i.e. pairs of compatible Poisson structures, on the formal manifold associated to V."}
{"category": "Math", "title": "About the embedding of Moufang loops in alternative algebras", "abstract": "It is proved that any free Moufang loop can be embedded in a loop of invertible elements of some alternative algebra."}
{"category": "Math", "title": "Special cycles on unitary Shimura varieties I. unramified local theory", "abstract": "The supersingular locus in the fiber at p of a Shimura variety attached to a unitary similitude group GU(1,n-1) over Q is uniformized by a formal scheme \\Cal N. In the case when p is inert, we define special cycles Z(x) in \\Cal N, associated to a collection x of m `special homomorphisms' with fundamental matrix T in Herm_m(OK). When m=n and T is nonsingular, we show that the cycle Z(x) is a union of components of the Ekedahl-Oort stratification, and we give a necessary and sufficient conditions, in terms of T, for Z(x) to be irreducible. When Z(x) is zero dimensional -- in which case it reduces to a single point -- we determine the length of the corresponding local ring by using a variant of the theory of quasi-canonical liftings. We show that this length coincides with the derivative of a representation density for hermitian forms."}
{"category": "Math", "title": "The fixed point property via dual space properties", "abstract": "A Banach space has the weak fixed point property if its dual space has a weak$^*$ sequentially compact unit ball and the dual space satisfies the weak$^*$ uniform Kadec-Klee property; and it has the \\fpp if there exists $\\epsilon>0$ such that, for every infinite subset $A$ of the unit sphere of the dual space, $A\\cup (-A)$ fails to be $(2-\\epsilon)$-separated. In particular, $E$-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property."}
{"category": "Math", "title": "Riemann surfaces with boundary and natural triangulations of the Teichmueller space", "abstract": "We compare some natural triangulations of the Teichm\\\"uller space of hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt Scannell-Wolf's proof to show that grafting semi-infinite cylinders at the ends of hyperbolic surfaces with fixed boundary lengths is a homeomorphism. This way, we construct a family of triangulations of the Teichm\\\"uller space of punctures surfaces that interpolates between Penner-Bowditch-Epstein's (using the spine construction) and Harer-Mumford-Thurston's (using Strebel's differentials). Finally, we show (adapting arguments of Dumas) that on a fixed punctured surface, when the triangulation approaches HMT's, the associated Strebel differential is well-approximated by the Schwarzian of the associated projective structure and by the Hopf differential of the collapsing map."}
{"category": "Math", "title": "Some addition to the generalized Riemann-Hilbert problem", "abstract": "We give some additions to the article \"On the generalized Riemann-Hilbert problem with irregular singularities\" by Bolibruch, Malek, Mitschi (math/0410483). In particular, a weak GRH-problem and the GRH-problem for scalar differential equations are discussed."}
{"category": "Math", "title": "Local similarities and the Haagerup property", "abstract": "A new class of groups, the locally finitely determined groups of local similarities on compact ultrametric spaces, is introduced and it is proved that groups in this class have the Haagerup property (that is, they are a-T-menable in the sense of Gromov). The class includes Thompson's groups, which have already been shown to have the Haagerup property by D. S. Farley, as well as many other groups acting on boundaries of trees. A sufficient condition, used in this paper, for the Haagerup property is shown in the appendix by D. S. Farley to be equivalent to the well-known property of having a proper action on a space with walls."}
{"category": "Math", "title": "The Number of Pseudo-Anosov Elements in the Mapping Class Group of a Four-Holed Sphere", "abstract": "We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the identity tends to one as the radius of the ball tends to infinity."}
{"category": "Math", "title": "Geometric spaces from arbitrary convex polytopes", "abstract": "We associate a geometric space to an arbitrary convex polytope. Our construction parallels the construction by D. Cox of a toric variety as a GIT quotient. The spaces that we obtain are endowed with a natural stratification and perfectly mimic the features of toric varieties associated to rational convex polytopes."}
{"category": "Math", "title": "Non-Commutative Partial Matrix Convexity", "abstract": "Let $p$ be a polynomial in the non-commuting variables $(a,x)=(a_1,...,a_{g_a},x_1,...,x_{g_x})$. If $p$ is convex in the variables $x$, then $p$ has degree two in $x$ and moreover, $p$ has the form $p = L + \\Lambda ^T \\Lambda,$ where $L$ has degree at most one in $x$ and $\\Lambda$ is a (column) vector which is linear in $x,$ so that $\\Lambda^T\\Lambda$ is a both sum of squares and homogeneous of degree two. Of course the converse is true also. Further results involving various convexity hypotheses on the $x$ and $a$ variables separately are presented."}
{"category": "Math", "title": "A Complete Classification of Ternary Self-Dual Codes of Length 24", "abstract": "Ternary self-dual codes have been classified for lengths up to 20. At length 24, a classification of only extremal self-dual codes is known. In this paper, we give a complete classification of ternary self-dual codes of length 24 using the classification of 24-dimensional odd unimodular lattices."}
{"category": "Math", "title": "Groebner-Shirshov bases for dialgebras", "abstract": "In this paper, we define the Gr\\\"obner-Shirshov basis for a dialgebra. The Composition-Diamond lemma for dialgebras is given then. As results, we give Gr\\\"obner-Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra. We obtain some normal forms for algebras mentioned the above."}
{"category": "Math", "title": "Groebner-Shirshov bases for extensions of algebras", "abstract": "An algebra $\\cal{R}$ is called an extension of the algebra $M$ by $B$ if $M^2=0$, $M$ is an ideal of $\\cal{R}$ and $\\cal{R}$$/M\\cong B$ as algebras. In this paper, by using the Gr\\\"{o}bner-Shirshov bases, we characterize completely the extensions of $M$ by $B$. An algorithm to find the conditions of an algebra $A$ to be an extension of $M$ by $B$ is obtained."}
{"category": "Math", "title": "Further Hopping with Toads and Frogs", "abstract": "We show the value of positions of the combinatorial game ``Toads and Frogs''. We present new values of starting positions. Moreover, we discuss the values of all positions with exactly one $\\Box, \\regT^{a}\\Box\\Box \\regF^{a}, \\regT^{a} \\Box \\Box \\Box \\regF \\regF \\regF,\\regT^{a}\\Box\\Box \\regF^{b}$, $\\regT^{a}\\Box\\Box\\Box \\regF^{b}$. At the end, we post five new conjectures and discuss the possible future work."}
{"category": "Math", "title": "Groebner-Shirshov bases for Schreier extensions of groups", "abstract": "In this paper, by using the Groebner-Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier extension is obtained. By introducing a special total order, we obtain the structure of the Schreier extension by an HNN group."}
{"category": "Math", "title": "Groebner-Shirshov basis for HNN extensions of groups and for the alternating group", "abstract": "In this paper, we generalize the Shirshov's Composition Lemma by replacing the monomial order for others. By using Groebner-Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained."}
{"category": "Math", "title": "Groebner-Shirshov bases for some one-relator groups", "abstract": "In this paper, we prove that two-generator one-relator groups with depth less than or equal to 3 can be effectively embedded into a tower of HNN-extensions in which each group has the effective standard normal form. We give an example to show how to deal with some general cases for one-relator groups. By using the Magnus method and Composition-Diamond Lemma, we reprove the G. Higman, B. H. Neumann and H. Neumann's embedding theorem."}
{"category": "Math", "title": "Finite Trigonometric Character Sums Via Discrete Fourier Analysis", "abstract": "We prove several old and new theorems about finite sums involving characters and trigonometric functions. These sums can be traced back to theta function identities from Ramanujan's notebooks and were systematically first studied by Berndt and Zaharescu; their proofs involved complex contour integration. We show how to prove most of Berndt-Zaharescu's and some new identities by elementary methods of discrete Fourier Analysis."}
{"category": "Math", "title": "Homological mirror symmetry is T-duality for $\\mathbb P^n$", "abstract": "In this paper, we apply the idea of T-duality to projective spaces. From a connection on a line bundle on $\\mathbb P^n$, a Lagrangian in the mirror Landau-Ginzburg model is constructed. Under this correspondence, the full strong exceptional collection $\\mathcal O_{\\mathbb P^n}(-n-1),...,\\mathcal O_{\\mathbb P^n}(-1)$ is mapped to standard Lagrangians in the sense of \\cite{nz}. Passing to constructible sheaves, we explicitly compute the quiver structure of these Lagrangians, and find that they match the quiver structure of this exceptional collection of $\\mathbb P^n$. In this way, T-duality provides quasi-equivalence of the Fukaya category generated by these Lagrangians and the category of coherent sheaves on $\\mathbb P^n$, which is a kind of homological mirror symmetry."}
{"category": "Math", "title": "Storms prediction : Logistic regression vs random forest for unbalanced data", "abstract": "The aim of this study is to compare two supervised classification methods on a crucial meteorological problem. The data consist of satellite measurements of cloud systems which are to be classified either in convective or non convective systems. Convective cloud systems correspond to lightning and detecting such systems is of main importance for thunderstorm monitoring and warning. Because the problem is highly unbalanced, we consider specific performance criteria and different strategies. This case study can be used in an advanced course of data mining in order to illustrate the use of logistic regression and random forest on a real data set with unbalanced classes."}
{"category": "Math", "title": "Identities between Appell's and hypergeometric functions", "abstract": "Univariate specializations of Appell's hypergeometric functions F1, F2, F3, F4 satisfy ordinary Fuchsian equations of order at most 4. In special cases, these differential equations are of order 2, and could be simple (pullback) transformations of Euler's differential equation for the Gauss hypergeometric function. The paper classifies these cases, and presents corresponding relations between univariate specializations of Appell's functions and univariate hypergeometric functions. The computational aspect and interesting identities are discussed."}
{"category": "Math", "title": "The discrete square peg problem", "abstract": "The square peg problem asks whether every Jordan curve in the plane has four points which form a square. The problem has been resolved (positively) for various classes of curves, but remains open in full generality. We present two new direct proofs for the case of piecewise linear curves."}
{"category": "Math", "title": "Estimating the Number of Components in a Mixture of Multilayer Perceptrons", "abstract": "BIC criterion is widely used by the neural-network community for model selection tasks, although its convergence properties are not always theoretically established. In this paper we will focus on estimating the number of components in a mixture of multilayer perceptrons and proving the convergence of the BIC criterion in this frame. The penalized marginal-likelihood for mixture models and hidden Markov models introduced by Keribin (2000) and, respectively, Gassiat (2002) is extended to mixtures of multilayer perceptrons for which a penalized-likelihood criterion is proposed. We prove its convergence under some hypothesis which involve essentially the bracketing entropy of the generalized score-functions class and illustrate it by some numerical examples."}
{"category": "Math", "title": "On a secondary invariant of the hyperelliptic mapping class group", "abstract": "In this paper, we discuss relations among several invariants of 3-manifolds including Meyer's function, the eta-invariant, the von Neumann rho-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface."}
{"category": "Math", "title": "Small Resolutions and Non-Liftable Calabi-Yau threefolds", "abstract": "We use properties of small resolutions of the ordinary double point in dimension three to construct smooth non-liftable Calabi-Yau threefolds. In particular, we construct a smooth projective Calabi-Yau threefold over $\\F_3$ that does not lift to characteristic zero and a smooth projective Calabi-Yau threefold over $\\F_5$ having an obstructed deformation. We also construct many examples of smooth Calabi-Yau algebraic spaces over $\\F_p$ that do not lift to algebraic spaces in characteristic zero."}
{"category": "Math", "title": "A theoretical comparison of the data augmentation, marginal augmentation and PX-DA algorithms", "abstract": "The data augmentation (DA) algorithm is a widely used Markov chain Monte Carlo (MCMC) algorithm that is based on a Markov transition density of the form $p(x|x')=\\int_{\\mathsf{Y}}f_{X|Y}(x|y)f_{Y|X}(y|x') dy$, where $f_{X|Y}$ and $f_{Y|X}$ are conditional densities. The PX-DA and marginal augmentation algorithms of Liu and Wu [J. Amer. Statist. Assoc. 94 (1999) 1264--1274] and Meng and van Dyk [Biometrika 86 (1999) 301--320] are alternatives to DA that often converge much faster and are only slightly more computationally demanding. The transition densities of these alternative algorithms can be written in the form $p_R(x|x')=\\int_{\\mathsf{Y}}\\int _{\\mathsf{Y}}f_{X|Y}(x|y')R(y,dy')f_{Y|X}(y|x') dy$, where $R$ is a Markov transition function on $\\mathsf{Y}$. We prove that when $R$ satisfies certain conditions, the MCMC algorithm driven by $p_R$ is at least as good as that driven by $p$ in terms of performance in the central limit theorem and in the operator norm sense. These results are brought to bear on a theoretical comparison of the DA, PX-DA and marginal augmentation algorithms. Our focus is on situations where the group structure exploited by Liu and Wu is available. We show that the PX-DA algorithm based on Haar measure is at least as good as any PX-DA algorithm constructed using a proper prior on the group."}
{"category": "Math", "title": "Generalizing Hartogs' Trichotomy Theorem", "abstract": "A celebrated argument of F. Hartogs (1915) deduces the Axiom of Choice from the hypothesis of comparability for any pair of cardinals. We show how each of a sequence of seemingly much weaker hypotheses suffices. Fixing a finite number $k>1$, the Axiom of Choice follows if merely any family of $k$ cardinals contains at least one comparable pair."}
{"category": "Math", "title": "Differential invariants of 2--order ODEs, I", "abstract": "In this paper, we investigate the action of pseudogroup of all point transformations on the natural bundle of equations $y''=a^3(x,y)(y')^3+a^2(x,y)(y')^2+a^1(x,y)y'+a^0(x,y)$. We construct differential invariants of this action and solve the equivalence problem for some classes of these equations in particular for generic equations."}
{"category": "Math", "title": "Second-order asymptotic expansion for a non-synchronous covariation estimator", "abstract": "In this paper, we consider the problem of estimating the covariation of two diffusion processes when observations are subject to non-synchronicity. Building on recent papers \\cite{Hay-Yos03, Hay-Yos04}, we derive second-order asymptotic expansions for the distribution of the Hayashi-Yoshida estimator in a fairly general setup including random sampling schemes and non-anticipative random drifts. The key steps leading to our results are a second-order decomposition of the estimator's distribution in the Gaussian set-up, a stochastic decomposition of the estimator itself and an accurate evaluation of the Malliavin covariance. To give a concrete example, we compute the constants involved in the resulting expansions for the particular case of sampling scheme generated by two independent Poisson processes."}
{"category": "Math", "title": "A rigidity theorem for Moor-bialgebras", "abstract": "We introduce the operad Moor, dual of the operad NAP and the notion of Moor-bialgebras. We warn the reader that the compatibility relation linking the Moor-operation with the Moor-cooperation is not distributive in the sense of Loday. Nevertheless, a rigidity theorem (\\`a la Hopf-Borel) for the category of connected Moor-bialgebras is given. We show also that free permutative algebras can be equipped with a Moor-cooperation whose compatibility with the permutative product looks like the infinitesimal relation."}
{"category": "Math", "title": "Consistency of spectral clustering", "abstract": "Consistency is a key property of all statistical procedures analyzing randomly sampled data. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of the popular family of spectral clustering algorithms, which clusters the data with the help of eigenvectors of graph Laplacian matrices. We develop new methods to establish that, for increasing sample size, those eigenvectors converge to the eigenvectors of certain limit operators. As a result, we can prove that one of the two major classes of spectral clustering (normalized clustering) converges under very general conditions, while the other (unnormalized clustering) is only consistent under strong additional assumptions, which are not always satisfied in real data. We conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering."}
{"category": "Math", "title": "Generalizations of product-free subsets", "abstract": "For any group G of order n, a subset A of G is said to be product-free if there is no solution of the equation ab=c with a,b,c in A. Previous results of Gowers showed that the size of any product-free subset of G is at most n/d^(1/3), where d is the smallest dimension of a nontrivial representation of G. However, this upper bound does not match the best lower bound. We will generalize the upper bound to the case of product-poor subsets A, in which the equation ab=c is allowed to have a few solutions with a,b,c in A. We prove that the upper bound for the size of product-poor subsets matches the best lower bound in many families of groups. We will also generalize the concept of product-free to the case in which we have many subsets of a group, and different constraints about products of the elements in the subsets."}
{"category": "Math", "title": "Asymptotic properties of bridge estimators in sparse high-dimensional regression models", "abstract": "We study the asymptotic properties of bridge estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase to infinity with the sample size. We are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero. We show that under appropriate conditions, bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance. Thus, bridge estimators have an oracle property in the sense of Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348--1360] and Fan and Peng [Ann. Statist. 32 (2004) 928--961]. In general, the oracle property holds only if the number of covariates is smaller than the sample size. However, under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients, we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size."}
{"category": "Math", "title": "A 3-manifold complexity via immersed surfaces", "abstract": "We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on P2-irreducible manifolds. Moreover, for P2-irreducible manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere S3, the projective space RP3 and the lens space L41, which have surface-complexity zero. We will also give estimations of the surface-complexity by means of triangulations, Heegaard splittings, surgery presentations and Matveev complexity."}
{"category": "Math", "title": "Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds", "abstract": "The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of a special type with respect to the considered diffeomorphism."}
{"category": "Math", "title": "About Adaptive Singular Systems with External Delay", "abstract": "This paper is mainly concerned with the robustly stable adaptive control of single-input single-output impulse-free linear time-invariant singular dynamic systems of known order and unknown parameterizations subject to single external point delays. The control law is of pole-placement type and based on input/output measurements and parametrical estimation only. The parametrical estimation incorporates adaptation dead zones to prevent against potential instability caused by disturbances and unmodeled dynamics. The Weierstrass canonical form is investigated in detail to discuss controllability and observability via testable conditions of the given arbitrary state-space realization of the same order."}
{"category": "Math", "title": "A_k singularities of wave fronts", "abstract": "In this paper, we discuss the recognition problem for A_k-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors' previous work \"the geometry of fronts\" for surfaces. The crucial point to prove our criteria for A_k-singularities is to introduce a suitable parametrization of the singularities called the \"k-th KRSUY-coordinates\". Using them, we can directly construct a versal unfolding for a given singularity. As an application, we prove that a given nondegenerate singular point p on a real (resp. complex) hypersurface (as a wave front) in R^{n+1} (resp. C^{n+1}) is differentiably (resp. holomorphically) right-left equivalent to the A_{k+1}-type singular point if and only if the linear projection of the singular set around p into a generic hyperplane R^n (resp. C^n) is right-left equivalent to the A_k-type singular point in R^n (resp. C^{n}). Moreover, we show that the restriction of a C-infinity-map f:R^n --> R^n to its Morin singular set gives a wave front consisting of only A_k-type singularities. Furthermore, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts."}
{"category": "Math", "title": "High-dimensional generalized linear models and the lasso", "abstract": "We consider high-dimensional generalized linear models with Lipschitz loss functions, and prove a nonasymptotic oracle inequality for the empirical risk minimizer with Lasso penalty. The penalty is based on the coefficients in the linear predictor, after normalization with the empirical norm. The examples include logistic regression, density estimation and classification with hinge loss. Least squares regression is also discussed."}
{"category": "Math", "title": "Admissible submonoids of Artin-Tits monoids", "abstract": "We show the analogue of Muhlherr's [Coxeter groups in Coxeter groups, Finite Geom. and Combinatorics, Cambridge Univ. Press (1993), 277-287] for Artin-Tits monoids, and for Artin-Tits groups of spherical type. That is, the submonoid (resp. subgroup) of an Artin-Tits monoid (resp. group of spherical type) induced by an admissible partition of the Coxeter graph is an Artin-Tits monoid (resp. group). This generalizes and unifies the situation of the submonoid (resp. subgroup) of fixed elements of an Artin-Tits monoid (resp. group of spherical type) under the action of graph automorphisms, and the notion of LCM-homomorphisms defined by Crisp in [Injective maps between Artin groups, Geom. Group Theory Down Under, Canberra (1996) 119-137] and generalized by Godelle in [Morphismes injectifs entre groupes d'Artin-Tits, Algebr. Geom. Topol. 2 (2002), 519--536]. We then complete the classification of the admissible partitions for which the Coxeter graphs involved have no infinite label, started by Muhlherr in [Some contributions to the theory of buildings based on the gate property, Dissertation, T\\\"ubingen (1994)]. This leads us to the classification of Crisp's LCM-homomorphisms."}
{"category": "Math", "title": "Weak Finsler Strutures and the Funk Metric", "abstract": "We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we show that these are given by the so-called Funk weak metric. We conclude the paper with a discussion of geodesics, of metric balls and of convexity properties of the Funk weak metric."}
{"category": "Math", "title": "Moves for standard skeleta of 3-manifolds with marked boundary", "abstract": "We prove that the classical set of moves for standard spines of 3-manifolds (i.e. the MP-move and the V-move) does not suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary. We also describe a condition on the 3-manifold with marked boundary that tells whether the generalised set of moves, made up of the MP-move and the L-move, suffices to relate to each other any two standard skeleta of the 3-manifold with marked boundary. For the 3-manifolds with marked boundary that do not fulfil this condition, we give three other moves: the CR-move, the T1-move and the T2-move. The first one is local and, with the MP-move and the L-move, suffices to relate to each other any two standard skeleta of a 3-manifold with marked boundary fulfilling another condition. For the universal case, we prove that the non-local T1-move and T2-move, with the MP-move and the L-move, suffice to relate to each other any two standard skeleta of a generic 3-manifold with marked boundary. As a corollary, we get that disc-replacements suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary."}
{"category": "Math", "title": "Effect of mean on variance function estimation in nonparametric regression", "abstract": "Variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived. We are particularly interested in the effect of the unknown mean on the estimation of the variance function. Our results indicate that, contrary to the common practice, it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth. Instead it is more desirable to use estimators of the mean with minimal bias. On the other hand, when the mean function is very smooth, our numerical results show that the residual-based method performs better, but not substantial better than the first-order-difference-based estimator. In addition our asymptotic results also correct the optimal rate claimed in Hall and Carroll [J. Roy. Statist. Soc. Ser. B 51 (1989) 3--14]."}
{"category": "Math", "title": "On deconvolution with repeated measurements", "abstract": "In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without it. However, if additional data are available, then it is possible to estimate consistently the unknown error density. Data are seldom available directly on the transformation, but repeated, or replicated, measurements increasingly are becoming available. Such data consist of ``intrinsic'' values that are measured several times, with errors that are generally independent. Working in this setting we treat the nonparametric deconvolution problems of density estimation with observation errors, and regression with errors in variables. We show that, even if the number of repeated measurements is quite small, it is possible for modified kernel estimators to achieve the same level of performance they would if the error distribution were known. Indeed, density and regression estimators can be constructed from replicated data so that they have the same first-order properties as conventional estimators in the known-error case, without any replication, but with sample size equal to the sum of the numbers of replicates. Practical methods for constructing estimators with these properties are suggested, involving empirical rules for smoothing-parameter choice."}
{"category": "Math", "title": "On the zeros of functions in the Selberg class", "abstract": "It is proved that under some suitable conditions, the degree two functions in the Selberg class have infinitely many zeros on the critical line."}
{"category": "Math", "title": "Estimation of a semiparametric transformation model", "abstract": "This paper proposes consistent estimators for transformation parameters in semiparametric models. The problem is to find the optimal transformation into the space of models with a predetermined regression structure like additive or multiplicative separability. We give results for the estimation of the transformation when the rest of the model is estimated non- or semi-parametrically and fulfills some consistency conditions. We propose two methods for the estimation of the transformation parameter: maximizing a profile likelihood function or minimizing the mean squared distance from independence. First the problem of identification of such models is discussed. We then state asymptotic results for a general class of nonparametric estimators. Finally, we give some particular examples of nonparametric estimators of transformed separable models. The small sample performance is studied in several simulations."}
{"category": "Math", "title": "Estimating deformations of isotropic Gaussian random fields on the plane", "abstract": "This paper presents a new approach to the estimation of the deformation of an isotropic Gaussian random field on $\\mathbb{R}^2$ based on dense observations of a single realization of the deformed random field. Under this framework we investigate the identification and estimation of deformations. We then present a complete methodological package--from model assumptions to algorithmic recovery of the deformation--for the class of nonstationary processes obtained by deforming isotropic Gaussian random fields."}
{"category": "Math", "title": "Using the Incompressibility Method to obtain Local Lemma results for Ramsey-type Problems", "abstract": "We reveal a connection between the incompressibility method and the Lovasz local lemma in the context of Ramsey theory. We obtain bounds by repeatedly encoding objects of interest and thereby compressing strings. The method is demonstrated on the example of van der Waerden numbers. It applies to lower bounds of Ramsey numbers, large transitive subtournaments and other Ramsey phenomena as well."}
{"category": "Math", "title": "Automorphs of indefinite binary quadratic forms and K3-surfaces with Picard number 2", "abstract": "Every indefinite binary form occurs as the Picard lattice of some K3-surface. The group of its isometries, or automorphs, coincides with the automorphism group of the K3-surface, but only up to finite groups. The classical theory of automorphs for binary forms can then be applied to study these automorphism groups. The result is a precise description of all possible automorphism groups of ``general'' K3's with Picard number two."}
{"category": "Math", "title": "A system of grabbing particles related to Galton-Watson trees", "abstract": "We consider a system of particles with arms that are activated randomly to grab other particles as a toy model for polymerization. We assume that the following two rules are fulfilled: Once a particle has been grabbed then it cannot be grabbed again, and an arm cannot grab a particle that belongs to its own cluster. We are interested in the shape of a typical polymer in the situation when the initial number of monomers is large and the numbers of arms of monomers are given by i.i.d. random variables. Our main result is a limit theorem for the empirical distribution of polymers, where limit is expressed in terms of a Galton-Watson tree."}
{"category": "Math", "title": "Direct image for multiplicative and relative K-theories from transgression of the families index theorem, part 3", "abstract": "This is the final part of the work started in math.DG/0611281 and math.DG/0703916. Here the question of double fibration ois adressed both for relative k-theory and free multiplicative K-theory. In the case of relative and ``nonfree'' multiplicative K-theory, the direct image is proved to be functorial for double submersions."}
{"category": "Math", "title": "Complements and signed digit representations: Analysis of a multi-exponentiation-algorithm of Wu, Lou, Lai and Chang", "abstract": "Wu, Lou, Lai and Chang proposed a multi-exponentiation algorithm using binary complements and the non-adjacent form. The purpose of this paper is to show that neither the analysis of the algorithm given by its original proposers nor that by other authors are correct. In fact it turns out that the complement operation does not have significant influence on the performance of the algorithm and can therefore be omitted."}
{"category": "Math", "title": "Algebraic curves P(x)-Q(y)=0 and functional equations", "abstract": "In this paper we give several conditions implying the irreducibility of the algebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also apply the results obtained to the functional equations P(f)=Q(g) and P(f)=cP(g), where c\\in C. For example, we show that for a generic pair of rational functions P,Q the first equation has no non-constant solutions f,g meromorphic on C whenever (\\deg P-1)(\\deg Q-1) \\geq 2."}
{"category": "Math", "title": "Normalized least-squares estimation in time-varying ARCH models", "abstract": "We investigate the time-varying ARCH (tvARCH) process. It is shown that it can be used to describe the slow decay of the sample autocorrelations of the squared returns often observed in financial time series, which warrants the further study of parameter estimation methods for the model. Since the parameters are changing over time, a successful estimator needs to perform well for small samples. We propose a kernel normalized-least-squares (kernel-NLS) estimator which has a closed form, and thus outperforms the previously proposed kernel quasi-maximum likelihood (kernel-QML) estimator for small samples. The kernel-NLS estimator is simple, works under mild moment assumptions and avoids some of the parameter space restrictions imposed by the kernel-QML estimator. Theoretical evidence shows that the kernel-NLS estimator has the same rate of convergence as the kernel-QML estimator. Due to the kernel-NLS estimator's ease of computation, computationally intensive procedures can be used. A prediction-based cross-validation method is proposed for selecting the bandwidth of the kernel-NLS estimator. Also, we use a residual-based bootstrap scheme to bootstrap the tvARCH process. The bootstrap sample is used to obtain pointwise confidence intervals for the kernel-NLS estimator. It is shown that distributions of the estimator using the bootstrap and the ``true'' tvARCH estimator asymptotically coincide. We illustrate our estimation method on a variety of currency exchange and stock index data for which we obtain both good fits to the data and accurate forecasts."}
{"category": "Math", "title": "Complex and Kaehler structures on compact homogeneous manifolds - their existence, classification and moduli problem", "abstract": "The existence of some complex geometrical structures on a compact manifold such as complex structures, Kaehler (pseudo-Kaehler) structures often impose certain restrictions on its underling topological or differentiable manifold. In this article we survey recent developments in the study of the existence, classification and moduli problems of such structures on compact homogeneous manifolds."}
{"category": "Math", "title": "On the equation P(f)=Q(g), where P,Q are polynomials and f,g are entire functions", "abstract": "In 1922 Ritt described polynomial solutions of the functional equation P(f)=Q(g). In this paper we describe solutions of the equation above in the case when P,Q are polynomials while f,g are allowed to be arbitrary entire functions. In fact, we describe solutions of the more general functional equation s=P(f)=Q(g), where s,f,g are entire functions and P,Q are arbitrary rational functions. Besides, we solve the problem of description of \"strong uniqueness polynomials\" for entire functions."}
{"category": "Math", "title": "Sequential change detection revisited", "abstract": "In sequential change detection, existing performance measures differ significantly in the way they treat the time of change. By modeling this quantity as a random time, we introduce a general framework capable of capturing and better understanding most well-known criteria and also propose new ones. For a specific new criterion that constitutes an extension to Lorden's performance measure, we offer the optimum structure for detecting a change in the constant drift of a Brownian motion and a formula for the corresponding optimum performance."}
{"category": "Math", "title": "Moduli of Flat Conformal Structures of Hyperbolic Type", "abstract": "To each flat conformal structure (FCS) of hyperbolic type in the sense of Kulkarni-Pinkall, we associate, for all $\\theta\\in[(n-1)\\pi/2,n\\pi/2[$ and for all $r>\\opTan(\\theta/n)$ a unique immersed hypersurface $\\Sigma_{r,\\theta}=(M,i_{r,\\theta})$ in $\\Bbb{H}^{n+1}$ of constant $\\theta$-special Lagrangian curvature equal to $r$. We show that these hypersurfaces smoothly approximate the boundary of the canonical hyperbolic end associated to the FCS by Kulkarni and Pinkall and thus obtain results concerning the continuous dependance of the hyperbolic end and of the Kulkarni-Pinkall metric on the flat conformal structure."}
{"category": "Math", "title": "Orbital stability of the black soliton to the Gross-Pitaevskii equation", "abstract": "We establish the orbital stability of the black soliton, or kink solution, $\\v_0(x) = \\th \\big(\\frac{x}{\\sqrt{2}} \\big)$, to the one-dimensional Gross-Pitaevskii equation, with respect to perturbations in the energy space."}
{"category": "Math", "title": "Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets", "abstract": "For every fixed graph $H$ and every fixed $0 < \\alpha < 1$, we show that if a graph $G$ has the property that all subsets of size $\\alpha n$ contain the ``correct'' number of copies of $H$ one would expect to find in the random graph $G(n,p)$ then $G$ behaves like the random graph $G(n,p)$; that is, it is $p$-quasi-random in the sense of Chung, Graham, and Wilson. This solves a conjecture raised by Shapira and solves in a strong sense an open problem of Simonovits and S\\'os."}
{"category": "Math", "title": "Betti numbers of mixed product ideals", "abstract": "We compute the Betti numbers of the resolution of a special class of square-free monomial ideals, the ideals of mixed products. Moreover when these ideals are Cohen-Macaulay we calculate their type."}
{"category": "Math", "title": "Closed-form likelihood expansions for multivariate diffusions", "abstract": "This paper provides closed-form expansions for the log-likelihood function of multivariate diffusions sampled at discrete time intervals. The coefficients of the expansion are calculated explicitly by exploiting the special structure afforded by the diffusion model. Examples of interest in financial statistics and Monte Carlo evidence are included, along with the convergence of the expansion to the true likelihood function."}
{"category": "Math", "title": "Bounds for Bayesian order identification with application to mixtures", "abstract": "The efficiency of two Bayesian order estimators is studied. By using nonparametric techniques, we prove new underestimation and overestimation bounds. The results apply to various models, including mixture models. In this case, the errors are shown to be $O(e^{-an})$ and $O((\\log n)^b/\\sqrt{n})$ ($a,b>0$), respectively."}
{"category": "Math", "title": "Some consequences of reflection on the approachability ideal", "abstract": "We study the approachability ideal I[\\kappa^+] in the context of large cardinals properties of the regular cardinals below a singular \\kappa. As a guiding example consider the approachability ideal I[\\aleph_{\\omega+1}] assuming that \\aleph_\\omega is strong limit. In this case we obtain that club many points in \\aleph_{\\omega+1} of cofinality \\aleph_n for some n>1 are approachable assuming the joint reflection of countable families of stationary subsets of \\aleph_n. This reflection principle holds under Martin's maximum for all n>1 and for each n>1 is equiconsistent with \\aleph_n being weakly compact in L. This characterizes the structure of the approachability ideal I[\\aleph_{\\omega+1}] in models of Martin's maximum."}
{"category": "Math", "title": "Decompositions involving Anick's spaces", "abstract": "Recently Stephen Theriault and I found an elementary construction of Anick's spaces and proved their main properties(arXiv:0710.1024).In this work the fundamental fibration is decomposed. This is useful in studying maps out of Anick's spaces and will be needed in order to determine it's universal properties."}
{"category": "Math", "title": "Renormalization in the H\\'enon family, II: The heteroclinic web", "abstract": "We study highly dissipative H\\'enon maps $$ F_{c,b}: (x,y) \\mapsto (c-x^2-by, x) $$ with zero entropy. They form a region $\\Pi$ in the parameter plane bounded on the left by the curve $W$ of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in $\\Pi$, but there exist infinitely many different topological types of such maps (even away from $W$). We also prove that in the infinitely renormalizable case, the average Jacobian $b_F$ on the attracting Cantor set $\\OO_F$ is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside $\\OO_F$ if and only if there are no heteroclinic tangencies."}
{"category": "Math", "title": "Mean Curvature Flow of Spacelike Graphs", "abstract": "We prove the mean curvature flow of a spacelike graph in $(\\Sigma_1\\times \\Sigma_2, g_1-g_2)$ of a map $f:\\Sigma_1\\to \\Sigma_2$ from a closed Riemannian manifold $(\\Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(\\Sigma_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2\\leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2\\leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and $K_2\\leq -c$, $c>0$ constant, any map $f:\\Sigma_1\\to \\Sigma_2$ is trivially homotopic provided $f^*g_2<\\rho g_1$ where $\\rho=\\min_{\\Sigma_1}K_1/\\sup_{\\Sigma_2}K_2^+\\geq 0$, in case $K_1>0$, and $\\rho=+\\infty$ in case $K_2\\leq 0$. This largely extends some known results for $K_i$ constant and $\\Sigma_2$ compact, obtained using the Riemannian structure of $\\Sigma_1\\times \\Sigma_2$, and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one."}
{"category": "Math", "title": "New Graphs of Finite Mutation Type", "abstract": "To a directed graph without loops and 2-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph G is the set of all isomorphism classes of graphs that can be obtained from G by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. Fomin, Shapiro and Thurston constructed mutation-finite graphs from triangulations of oriented bordered surfaces with marked points. We will call such graphs \"of geometric type\". Besides graphs with 2 vertices, and graphs of geometric type, there are only 9 other \"exceptional\" mutation classes that are known to be finite. In this paper we introduce 2 new exceptional finite mutation classes."}
{"category": "Math", "title": "Abelian solutions of the soliton equations and geometry of abelian varieties", "abstract": "We introduce the notion of abelian solutions of the 2D Toda lattice equations and the bilinear discrete Hirota equation and show that all of them are algebro-geometric."}
{"category": "Math", "title": "The 2-generalized knot group determines the knot", "abstract": "Generalized knot groups $G_n(K)$ were introduced independently by Kelly (1991) and Wada (1992). We prove that $G_2(K)$ determines the unoriented knot type and sketch a proof of the same for $G_n(K)$ for $n>2$."}
{"category": "Math", "title": "The fluctuations in the number of points on a hyperelliptic curve over a finite field", "abstract": "The number of points on a hyperelliptic curve over a field of $q$ elements may be expressed as $q+1+S$ where $S$ is a certain character sum. We study fluctuations of $S$ as the curve varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $S/\\sqrt{q}$ is distributed as the trace of a random $2g\\times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the the limiting distribution of $S$ is that of a sum of $q$ independent trinomial random variables taking the values $\\pm 1$ with probabilities $1/2(1+q^{-1})$ and the value 0 with probability $1/(q+1)$. When both the genus and the finite field grow, we find that $S/\\sqrt{q}$ has a standard Gaussian distribution."}
{"category": "Math", "title": "Gevrey solutions for irregular hypergeometric systems I", "abstract": "We describe the Gevrey solutions at singular points of irregular hypergeometric systems (GKZ systems) associated with affine monomial curves."}
{"category": "Math", "title": "Large tilting modules and representation type", "abstract": "We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsion-free modules proven in [AHT] for tame hereditary algebras."}
{"category": "Math", "title": "On Nichols algebras with standard braiding", "abstract": "The class of standard braided vector spaces, introduced by Andruskiewitsch and the author in \\texttt{arXiv:math/0703924v2} to understand the proof of a theorem of Heckenberger \\cite{H2}, is slightly more general than the class of braided vector spaces of Cartan type. In the present paper, we classify standard braided vector spaces with finite-dimensional Nichols algebra. For any such braided vector space, we give a PBW-basis, a closed formula of the dimension and a presentation by generators and relations of the associated Nichols algebra."}
{"category": "Math", "title": "Twisted Burnside Theory for the Discrete Heisenberg Group and for Wreath Products of Some Groups", "abstract": "The RP-property of Fel'shtyn and Troitsky is proved for wreath products of finitely generated Abelian groups with the group of integers. Such wreath products become the first known example of finitely generated RP-groups being not almost polycyclic."}
{"category": "Math", "title": "q-Analogue of Gauss' Divisibility Theorem", "abstract": "We give a q-analogue of Gauss' divisibility theorem"}
{"category": "Math", "title": "Integrable Isotropic Geometrical Flows and Heisenberg Ferromagnets", "abstract": "Geometrical flows (GF) play an important role in modern mathematics and physics. In this letter we have considered some integrable isotropic GF -- Ricci flows (RF) and mean curvature flows (MCF) -- which are related with integrable Heisenberg ferromagnets. In 2+1 dimensions, these GF have a singularity at $t=t_{0}$."}
{"category": "Math", "title": "A Schur-type addition theorem for primes", "abstract": "Suppose that all primes are colored with k colors. Then there exist monochromatic primes p1, p2, p3 such that p1+p2=p3+1."}
{"category": "Math", "title": "Symmetries of some motivic integrals", "abstract": "We give an explicit formula for the motivic integrals related to the Milnor number over spaces of parametrised arcs on the plane with fixed tangency orders with the axis. These integrals are rational functions of the parameters and the class of the affine line. Using a set of natural recurrence relations between them, we prove an unexpected invariance property with respect to the simultaneous inversion of the parameters and the class of the affine line. We also discuss a generalization of this system of recurrence relations whose solutions are also symmetric and satisfy additional differential equations."}
{"category": "Math", "title": "Concave functions of positive operators, sums and congruences", "abstract": "Subaddivity type matrix inequalities for concave funcions and symetric norms are given."}
{"category": "Math", "title": "Markov Jump Processes Approximating a Nonsymmetric Generalized Diffusion: numerics explained to probabilists", "abstract": "Consider a non-symmetric generalized diffusion $X(\\cdot)$ in ${\\bbR}^d$ determined by the differential operator $A(\\msx)=-\\sum_{ij} \\partial_ia_{ij}(\\msx)\\partial_j +\\sum_i b_i(\\msx)\\partial_i$. In this paper the diffusion process is approximated by Markov jump processes $X_n(\\cdot)$, in homogeneous and isotropic grids $G_n \\subset {\\bbR}^d$, which converge in distribution to the diffusion $X(\\cdot)$. The generators of $X_n(\\cdot)$ are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for $d\\geq3$ can be applied to processes for which the diffusion tensor $\\{a_{ij}(\\msx)\\}_{11}^{dd}$ fulfills an additional condition. The proposed construction offers a simple method for simulation of sample paths of non-symmetric generalized diffusion. Simulations are carried out in terms of jump processes $X_n(\\cdot)$. For $d=2$ the construction can be easily implemented into a computer code."}
{"category": "Math", "title": "Surviving particles for subcritical branching processes in random environment", "abstract": "The asymptotic behavior of a subcritical Branching Process in Random Environment (BPRE) starting with several particles depends on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS). %Descendances of particles for BPRE are not independent. In the (SS+IS) case, the asymptotic probability of survival is proportional to the initial number of particles, and conditionally on the survival of the population, only one initial particle survives $a.s.$ These two properties do not hold in the (WS) case and different asymptotics are established, which require new results on random walks with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition on the survival of particles. This also raises the problem of the dependence of the Yaglom quasistationary distributions on the initial number of particles and the asymptotic behavior of the Q-process associated with a subcritical BPRE."}
{"category": "Math", "title": "Super-potentials for currents on compact Kaehler manifolds and dynamics of automorphisms", "abstract": "We introduce a notion of super-potential (canonical function) associated to positive closed (p,p)-currents on compact Kaehler manifolds and we develop a calculus on such currents. One of the key points in our study is the use of deformations in the space of currents. As an application, we obtain several results on the dynamics of holomorphic automorphisms: regularity and uniqueness of the Green currents. We also get the regularity, the entropy, the ergodicity and the hyperbolicity of the equilibrium measures."}
{"category": "Math", "title": "Sparse Approximate Solution of Partial Differential Equations", "abstract": "A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure is presented that uses linear programming to find a good approximation to the sparse solution on a given refinement level. Then only those parts of the mesh are refined that belong to large expansion coefficients. Error estimates for this procedure are refined and the behavior of the procedure is demonstrated via some simple elliptic model problems."}
{"category": "Math", "title": "On the ring of approximation triples attached to a class of extremal real numbers", "abstract": "We attach a ring of sequences to each number from a certain class of extremal real numbers, and we study the properties of this ring both from an analytic point of view by exhibiting elements with specific behaviors, and also from an algebraic point of view by identifying it with the quotient of a polynomial ring over Q. The link between these points of view relies on combinatorial results of independent interest. We apply this theory to estimate the dimension of a certain space of sequences satisfying prescribed growth constrains."}
{"category": "Math", "title": "Clique percolation", "abstract": "Derenyi, Palla and Vicsek introduced the following dependent percolation model, in the context of finding communities in networks. Starting with a random graph $G$ generated by some rule, form an auxiliary graph $G'$ whose vertices are the $k$-cliques of $G$, in which two vertices are joined if the corresponding cliques share $k-1$ vertices. They considered in particular the case where $G=G(n,p)$, and found heuristically the threshold for a giant component to appear in $G'$. Here we give a rigorous proof of this result, as well as many extensions. The model turns out to be very interesting due to the essential global dependence present in $G'$."}
{"category": "Math", "title": "Generalized uncertainty inequalities", "abstract": "In this paper, Heisenberg-Pauli-Weyl-type uncertainty inequalities are obtained for a pair of positive-self adjoint operators on a Hilbert space, whose spectral projectors satisfy a ``balance condition'' involving certain operator norms. This result is then applied to obtain uncertainty inequalities on Riemannian manifolds, Riemannian symmetric spaces of non-compact type, homogeneous graphs and unimodular Lie groups with sublaplacians."}
{"category": "Math", "title": "Contraction and restriction of positroids in terms of decorated permutations", "abstract": "A positroid is a matroid defined by Postnikov to study the cells in the non-negative part of the Grassmannian. They are in bijection with decorated permutations. We show a way to explain contraction and restriction of positroids in terms of decorated permutations."}
{"category": "Math", "title": "On Gaussian Brunn-Minkowski inequalities", "abstract": "In this paper, we are interested in Gaussian versions of the classical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrard inequality for $m$ Borel or convex sets based on a previous work by Borell. Our method also allows us to have semigroup proofs of the geometric Brascamp-Lieb inequality and of the reverse one which follow exactly the same lines."}
{"category": "Math", "title": "Spiked Models in Wishart Ensemble", "abstract": "The spiked model is an important special case of the Wishart ensemble, and a natural generalization of the white Wishart ensemble. Mathematically, it can be defined on three kinds of variables: the real, the complex and the quaternion. For practical application, we are interested in the limiting distribution of the largest sample eigenvalue. We first give a new proof of the result of Baik, Ben Arous and P\\'{e}ch\\'{e} for the complex spiked model, based on the method of multiple orthogonal polynomials by Bleher and Kuijlaars. Then in the same spirit we present a new result of the rank 1 quaternionic spiked model, proven by combinatorial identities involving quaternionic Zonal polynomials (\\alpha = 1/2 Jack polynomials) and skew orthogonal polynomials. We find a phase transition phenomenon for the limiting distribution in the rank 1 quaternionic spiked model as the spiked population eigenvalue increases, and recognize the seemingly new limiting distribution on the critical point as the limiting distribution of the largest sample eigenvalue in the real white Wishart ensemble. Finally we give conjectures for higher rank quaternionic spiked model and the real spiked model."}
{"category": "Math", "title": "Clique Numbers of Graphs and Irreducible Exact m-Covers of Z", "abstract": "For each m>=1 and k>=2, we construct a graph G=(V,E) with \\omega(G)=m such that max_{1\\leq i\\leq k} \\omega(G[V_i])=m for arbitrary partition V=V_1\\cup...\\cup V_k, where \\omega(G) is the clique number of G and G[V_i] is the induced subgraph of G with the vertex set V_i. Using this result, we show that for each m>=2 there exists an exact m-cover of Z which is not the union of two 1-covers."}
{"category": "Math", "title": "Anti-commutative Groebner-Shirshov basis of a free Lie algebra", "abstract": "One of the natural ways to prove that the Hall words (Philip Hall, 1933) consist of a basis of a free Lie algebra is a direct construction: to start with a linear space spanned by Hall words, to define the Lie product of Hall words, and then to check that the product yields the Lie identities (Marshall Hall, 1950). Here we suggest another way using the Composition-Diamond lemma for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962)."}
{"category": "Math", "title": "Some remarks for the Akivis algebras and the Pre-Lie algebras", "abstract": "In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gr\\\"{o}bner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I.P. Shestakov's result that any Akivis algebra is linear and D. Segal's result that the set of all good words in $X^{**}$ forms a linear basis of the free Pre-Lie algebra $PLie(X)$ generated by the set $X$. For completeness, we give the details of the proof of Shirshov's Composition-Diamond lemma for non-associative algebras."}
{"category": "Math", "title": "Chernoff and Trotter-Kato theorems for locally convex spaces", "abstract": "We develop new approach for studying the abstract Cauchy problem $\\dot{x}=Ax$, $x(0)=x_0\\in D(A)$ for linear operators $A$ defined on a locally convex space $X$. This approach was firstly introduced in the paper \"Chernoff and Trotter type product formulas\" to study the problem for Banach spaces. In this paper we not only generalize the results of the previous paper to more general topological spaces but also get new results for Banach spaces. In particular, we prove the \"local\" extension of Chernoff-Trotter-Kato type theorems. Applying this result, we prove Chernoff, Lie-Trotter and Trotter-Kato theorems for locally convex spaces. Also we find necessary and sufficient conditions for the validity of the Chernoff and Trotter product formulas."}
{"category": "Math", "title": "Composition-Diamond Lemma for Modules", "abstract": "In this paper we give some relationships among the Groebner-Shirshov bases in free associative algebras, free left modules and \"double-free\" left modules (free modules over a free algebra). We give the Chibrikov's Composition-Diamond lemma for modules and show that Kang-Lee's Composition-Diamond lemma follows from this lemma. As applications, we also deal with highest weight module over the Lie algebra $sl_2$, Verma module over a Kac-Moody algebra, Verma module over Lie algebra of coefficients of a free conformal algebra and the universal enveloping module for a Sabinin algebra."}
{"category": "Math", "title": "Structural Dynamics of Various Causes of Migration in Jaipur", "abstract": "Various social causes for migration in Jaipur are studied and statistical hypotheses are made in this paper."}
{"category": "Math", "title": "Linear projections and successive minima", "abstract": "Using linear projections one gets new inequalities for the successive minima of the lattice of sections of an hermitian line bundle on an arithmetic surface."}
{"category": "Math", "title": "Representations of pointed Hopf algebras and their Drinfel'd quantum doubles", "abstract": "We study representations of nilpotent type nontrivial liftings of quantum linear spaces and their Drinfel'd quantum doubles. We construct a family of Verma- type modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules."}
{"category": "Math", "title": "Linear Koszul Duality", "abstract": "In this paper we construct, for F_1 and F_2 subbundles of a vector bundle E, a \"Koszul duality\" equivalence between derived categories of G_m-equivariant coherent (dg-)sheaves on the derived intersection of F_1 and F_2 inside E, and the corresponding derived intersection of the orthogonals of F_1 and F_2 inside the dual vector bundle E^*. We also propose applications to Hecke algebras."}
{"category": "Math", "title": "Computing ODE Symmetries as Abnormal Variational Symmetries", "abstract": "We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [Comput. Methods Appl. Math. 5 (2005), no. 4, pp. 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods."}
{"category": "Math", "title": "Mod-2 Equivalence of the K-theoretic Euler and Signature Classes", "abstract": "This note proves that, as K-theory elements, the symbol classes of the de Rham operator and the signature operator on a closed manifold of even dimension are congruent mod 2. An equivariant generalization is given pertaining to the equivariant Euler characteristic and the multi-signature."}
{"category": "Math", "title": "The Branch Locus for One-Dimensional Pisot Tiling Spaces", "abstract": "If phi is a Pisot substitution of degree d, then the inflation and substitution homeomorphism Phi on the tiling space T_Phi factors via geometric realization onto a d-dimensional solenoid. Under this realization, the collection of Phi-periodic asymptotic tilings corresponds to a finite set that projects onto the branch locus in a d-torus. We prove that if two such tiling spaces are homeomorphic, then the resulting branch loci are the same up to the action of certain affine maps on the torus."}
{"category": "Math", "title": "Stochastic evolution equations in UMD Banach spaces", "abstract": "We discuss existence, uniqueness, and space-time H\\\"older regularity for solutions of the parabolic stochastic evolution equation dU(t) = (AU(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), t\\in [0,\\Tend], U(0) = u_0, where $A$ generates an analytic $C_0$-semigroup on a UMD Banach space $E$ and $W_H$ is a cylindrical Brownian motion with values in a Hilbert space $H$. We prove that if the mappings $F:[0,T]\\times E\\to E$ and $B:[0,T]\\times E\\to \\mathscr{L}(H,E)$ satisfy suitable Lipschitz conditions and $u_0$ is $\\F_0$-measurable and bounded, then this problem has a unique mild solution, which has trajectories in $C^\\l([0,T];\\D((-A)^\\theta)$ provided $\\lambda\\ge 0$ and $\\theta\\ge 0$ satisfy $\\l+\\theta<\\frac12$. Various extensions of this result are given and the results are applied to parabolic stochastic partial differential equations."}
{"category": "Math", "title": "On Birational Transformations of Pairs in the Complex Plane", "abstract": "This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of art and provide some new results on this subject."}
{"category": "Math", "title": "Analysis of Discrete and Hybrid Stochastic Systems by Nonlinear Contraction Theory", "abstract": "We investigate the stability properties of discrete and hybrid stochastic nonlinear dynamical systems. More precisely, we extend the stochastic contraction theorems (which were formulated for continuous systems) to the case of discrete and hybrid resetting systems. In particular, we show that the mean square distance between any two trajectories of a discrete (or hybrid resetting) contracting stochastic system is upper-bounded by a constant after exponential transients. Using these results, we study the synchronization of noisy nonlinear oscillators coupled by discrete noisy interactions."}
{"category": "Math", "title": "Large sets with small doubling modulo p are well covered by an arithmetic progression", "abstract": "We prove that there is a small but fixed positive integer e such that for every prime larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < p is contained in an arithmetic progression of length |2S|-|S|+1. This is the first result of this nature which places no unnecessary restrictions on the size of S."}
{"category": "Math", "title": "Recovery of inhomogeneities and buried obstacles", "abstract": "In this paper we consider the unique determination of inhomogeneities together with possible buried obstacles by scattering measurements. Under the assumption that the buried obstacles have only planar contacts with the inhomogeneities, we prove that one can recover both of them by knowing the associated scattering amplitude at a fixed energy."}
{"category": "Math", "title": "A non-commutative generalization of $k$-Schur functions", "abstract": "We introduce non-commutative analogues of $k$-Schur functions of Lapointe-Lascoux and Morse. We give an explicit formulas for the expansions of non-commutive functions with one and two parameters in terms of these new functions. These results are similar to the conjectures existing in the commutative case."}
{"category": "Math", "title": "The Dynkin diagram cohomology of finite Coxeter groups", "abstract": "Let D be a connected graph. The Dynkin complex CD(A) of a D-algebra A was introduced by the second author in [TL2] to control the deformations of quasi-Coxeter algebra structures on A. In the present paper, we study the cohomology of this complex when A is the group algebra of a Coxeter group W and D is the Dynkin diagram of W. We compute this cohomology when W is finite and prove in particular the rigidity of quasi-Coxeter algebra structures on kW. For an arbitrary W, we compute the top cohomology group and obtain a number of additional partial results when W is affine. Our computations are carried out by filtering CD(A) by the number of vertices of subgraphs of D. The corresponding graded complex turns out to be dual to the sum of the Coxeter complexes of all standard, irreducible parabolic subgroups of W."}
{"category": "Math", "title": "Rosso-Yamane Theorem on PBW basis of $U_q(A_N)$", "abstract": "Let $U_q(A_N)$ be the Drinfeld-Jimbo quantum group of type $A_N$. In this paper, by using Groebner-Shirshov bases, we give a simple (but not short) proof of the Rosso-Yamane Theorem on PBW basis of $U_q(A_N)$."}
{"category": "Math", "title": "Groebner-Shirshov bases for free inverse semigroups", "abstract": "A new construction of a free inverse semigroup was obtained by Poliakova and Schein in 2005. Based on their result, we find a Groebner-Shirshov basis of a free inverse semigroup relative to the deg-lex order of words. In particular, we give the (unique and shortest) Groebner-Shirshov normal forms in the classes of equivalent words of a free inverse semigroup together with the Groebner-Shirshov algorithm to transform any word to its normal form."}
{"category": "Math", "title": "Three-dimensional terminal toric flips", "abstract": "We describe three-dimensional terminal toric flips. We obtain the complete local description of three-dimensional terminal toric flips."}
{"category": "Math", "title": "A log-type moment result for perpetuities and its application to martingales in supercritical branching random walks", "abstract": "Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection, first observed by the second author in [Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114, 27-50.], arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons in [Lyons, R.(1997). A simple path to Biggins' martingale convergence for branching random walk. In Athreya, K.B., Jagers, P. (eds.). Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications, vol. 84, Springer, Berlin, 217-221.]. We also provide a necessary and sufficient condition for uniform integrability of these martingales in the most general situation which particularly means that the classical (LlogL)-condition is not always needed."}
{"category": "Math", "title": "Asymptotic stability of certain sets of associated prime ideals of local cohomology modules", "abstract": "Let $(R,\\m)$ be a Noetherian local ring $I, J$ two ideals of $R$ and $M$ a finitely generated $R-$module. It is first shown that for $k\\geq -1$ the integer $r_k = \\depth_k(I,J^nM/J^{n+1}M)$, it is the length of a maximal $(J^nM/J^{n+1}M)-$sequence in dimension $>k$ in $I$ defined by M. Brodmann and L. T. Nhan \\cite{BN}, becomes for large $n$ independent of $n$. Then we prove in this paper that the sets $\\bigcup_{j\\le r_k}\\Ass_R(H^j_I(J^nM/J^{n+1}M))$ with $k=-1$ or $k=0$, and $\\bigcup_{j\\le r_1}\\Ass_R(H^j_I(J^nM/J^{n+1}M))\\cup\\{\\m\\}$ are stable for large $n$. We also obtain similar results for modules $M/J^nM$."}
{"category": "Math", "title": "Groebner-Shirshov Basis for the Chinese Monoid", "abstract": "In this paper, a Groebner-Shirshov basis for the Chinese monoid is obtained and an algorithm for the normal form of the Chinese monoid is given."}
{"category": "Math", "title": "Normality in group rings", "abstract": "Let $KG$ be the group ring of a group $G$ over a commutative ring $K$ with unity. The rings $KG$ are described for which $xx^\\sigma=x^\\sigma x$ for all $x=\\sum_{g\\in G}\\alpha_gg\\in KG$, where \\quad $x\\mapsto x^\\sigma=~\\sum_{g\\in G}\\alpha_gf(g)\\sigma(g)$\\quad is an involution of $KG$; here $f: G\\to U(K)$ is a homomorphism and $\\sigma$ is an anti-automorphism of order two of $G$."}
{"category": "Math", "title": "Objective priors for the bivariate normal model", "abstract": "Study of the bivariate normal distribution raises the full range of issues involving objective Bayesian inference, including the different types of objective priors (e.g., Jeffreys, invariant, reference, matching), the different modes of inference (e.g., Bayesian, frequentist, fiducial) and the criteria involved in deciding on optimal objective priors (e.g., ease of computation, frequentist performance, marginalization paradoxes). Summary recommendations as to optimal objective priors are made for a variety of inferences involving the bivariate normal distribution. In the course of the investigation, a variety of surprising results were found, including the availability of objective priors that yield exact frequentist inferences for many functions of the bivariate normal parameters, including the correlation coefficient."}
{"category": "Math", "title": "On the 2D Cahn-Hilliard equation with inertial term", "abstract": "P. Galenko et al. proposed a modified Cahn-Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solutions. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor."}
{"category": "Math", "title": "A geometric Newton method for Oja's vector field", "abstract": "Newton's method for solving the matrix equation $F(X)\\equiv AX-XX^TAX=0$ runs up against the fact that its zeros are not isolated. This is due to a symmetry of $F$ by the action of the orthogonal group. We show how differential-geometric techniques can be exploited to remove this symmetry and obtain a ``geometric'' Newton algorithm that finds the zeros of $F$. The geometric Newton method does not suffer from the degeneracy issue that stands in the way of the original Newton method."}
{"category": "Math", "title": "Quadratic distances on probabilities: A unified foundation", "abstract": "This work builds a unified framework for the study of quadratic form distance measures as they are used in assessing the goodness of fit of models. Many important procedures have this structure, but the theory for these methods is dispersed and incomplete. Central to the statistical analysis of these distances is the spectral decomposition of the kernel that generates the distance. We show how this determines the limiting distribution of natural goodness-of-fit tests. Additionally, we develop a new notion, the spectral degrees of freedom of the test, based on this decomposition. The degrees of freedom are easy to compute and estimate, and can be used as a guide in the construction of useful procedures in this class."}
{"category": "Math", "title": "Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux", "abstract": "We introduce a new family of noncommutative analogues of the Hall-Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explicit formula for: the steady state probability of each state in the partially asymmetric exclusion process (PASEP); the polynomial enumerating permutations with a fixed set of weak excedances according to crossings; the polynomial enumerating permutations with a fixed set of descent bottoms according to occurrences of the generalized pattern 2-31."}
{"category": "Math", "title": "Quotient correlation: A sample based alternative to Pearson's correlation", "abstract": "The quotient correlation is defined here as an alternative to Pearson's correlation that is more intuitive and flexible in cases where the tail behavior of data is important. It measures nonlinear dependence where the regular correlation coefficient is generally not applicable. One of its most useful features is a test statistic that has high power when testing nonlinear dependence in cases where the Fisher's $Z$-transformation test may fail to reach a right conclusion. Unlike most asymptotic test statistics, which are either normal or $\\chi^2$, this test statistic has a limiting gamma distribution (henceforth, the gamma test statistic). More than the common usages of correlation, the quotient correlation can easily and intuitively be adjusted to values at tails. This adjustment generates two new concepts--the tail quotient correlation and the tail independence test statistics, which are also gamma statistics. Due to the fact that there is no analogue of the correlation coefficient in extreme value theory, and there does not exist an efficient tail independence test statistic, these two new concepts may open up a new field of study. In addition, an alternative to Spearman's rank correlation, a rank based quotient correlation, is also defined. The advantages of using these new concepts are illustrated with simulated data and a real data analysis of internet traffic."}
{"category": "Math", "title": "Fundamental groups and Diophantine geometry", "abstract": "We give a brief exposition on the uses of non-commutative fundamental groups for the study of Diophantine problems via a non-abelian Albanese map."}
{"category": "Math", "title": "Multiplicity for critical and overcritical equations", "abstract": "On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the classical critical Sobolev exponent, for instance the Yamabe equation. When there is no finite orbits, the multiplicity is obtained for equations with overcritical exponents."}
{"category": "Math", "title": "Controllability of networks of one-dimensional second order p.d.e. - An algebraic approach", "abstract": "We discuss controllability of systems that are initially given by boundary coupled p.d.e. of second order. Those systems may be described by modules over a certain subring R of the ring of Mikusinski operators with compact support. We show that the ring R is a Bezout domain. This property is utilized in order to derive algebraic and trajectory related controllability results."}
{"category": "Math", "title": "Noncommutative Gorenstein Projective Schemes and Gorenstein-Injective Sheaves", "abstract": "We prove that if a positively-graded ring $R$ is Gorenstein and the associated torsion functor has finite cohomological dimension, then the corresponding noncommutative projective scheme ${\\rm Tails}(R)$ is a Gorenstein category in the sense of \\cite{EEG}. Moreover, under this condition, a (right) recollement relating Gorenstein-injective sheaves in ${\\rm Tails}(R)$ and (graded) Gorenstein-injective $R$-modules is given."}
{"category": "Math", "title": "The focusing energy-critical nonlinear Schr\\\"odinger equation in dimensions five and higher", "abstract": "We consider the focusing energy-critical nonlinear Schr\\\"odinger equation $iu_t+\\Delta u = - |u|^{\\frac4{d-2}}u$ in dimensions $d\\geq 5$. We prove that if a maximal-lifespan solution $u:I\\times\\R^d\\to \\C$ obeys $\\sup_{t\\in I}\\|\\nabla u(t)\\|_2<\\|\\nabla W\\|_2$, then it is global and scatters both forward and backward in time. Here $W$ denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state $W$ at some point in time, then the solution is global and scatters. We also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state. Similar results were obtained by Kenig and Merle in \\cite{Evian, kenig-merle} for spherically symmetric initial data and dimensions $d=3,4,5$."}
{"category": "Math", "title": "Submanifolds with ample normal bundles and a conjecture of Hartshorne", "abstract": "The Hartshorne conjecture predicts that two submanifolds X and Y in a projective manifold Z with ample normal bundles meets as soon as dim X + dim Y is at least dim Z. We mostly assume slightly stronger that one of the normal bundles is positive in the sense of Griffiths. We observe that the conjecture holds generically, relate it to question on cones of cycles and verify it in various cases."}
{"category": "Math", "title": "A Sobolev-like inequality for the Dirac operator", "abstract": "In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved."}
{"category": "Math", "title": "Testing for Homogeneity with Kernel Fisher Discriminant Analysis", "abstract": "We propose to investigate test statistics for testing homogeneity in reproducing kernel Hilbert spaces. Asymptotic null distributions under null hypothesis are derived, and consistency against fixed and local alternatives is assessed. Finally, experimental evidence of the performance of the proposed approach on both artificial data and a speaker verification task is provided."}
{"category": "Math", "title": "Pruning a L\\'evy continuum random tree", "abstract": "Given a general critical or sub-critical branching mechanism, we define a pruning procedure of the associated L\\'evy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using L\\'evy snake techniques. We then prove that the resulting sub-tree after pruning is still a L\\'evy continuum random tree. This last result is proved using the exploration process that codes the CRT, a special Markov property and martingale problems for exploration processes. We finally give the joint law under the excursion measure of the lengths of the excursions of the initial exploration process and the pruned one."}
{"category": "Math", "title": "Narayana numbers and Schur-Szego composition", "abstract": "In the present paper we find a new interpretation of Narayana polynomials N_n(x) which are the generating polynomials for the Narayana numbers N_{n,k} counting Dyck paths of length n and with exactly k peaks. Strangely enough Narayana polynomials also occur as limits as n->oo of the sequences of eigenpolynomials of the Schur-Szego composition map sending (n-1)-tuples of polynomials of the form (x+1)^{n-1}(x+a) to their Schur-Szego product, see below. As a corollary we obtain that every N_n(x) has all roots real and non-positive. Additionally, we present an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence {N_n(x)}."}
{"category": "Math", "title": "The compound Poisson distribution and return times in dynamical systems", "abstract": "Previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere Poissonian. Here we study the behaviour of return times at periodic points and show that the limiting distribution is a compound Poissonian distribution. We also derive error terms for the convergence to the limiting distribution. We also prove a very general theorem that can be used to establish compound Poisson distributions in many other settings."}
{"category": "Math", "title": "Estimation of Ambiguity Functions With Limited Spread", "abstract": "This paper proposes a new estimation procedure for the ambiguity function of a non-stationary time series. The stochastic properties of the empirical ambiguity function calculated from a single sample in time are derived. Different thresholding procedures are introduced for the estimation of the ambiguity function. Such estimation methods are suitable if the ambiguity function is only non-negligible in a limited region of the ambiguity plane. The thresholds of the procedures are formally derived for each point in the plane, and methods for the estimation of nuisance parameters that the thresholds depend on are proposed. The estimation method is tested on several signals, and reductions in mean square error when estimating the ambiguity function by factors of over a hundred are obtained. An estimator of the spread of the ambiguity function is proposed."}
{"category": "Math", "title": "On the Spectral Properties of Matrices Associated with Trend Filters", "abstract": "This paper is concerned with the spectral properties of matrices associated with linear filters for the estimation of the underlying trend of a time series. The interest lies in the fact that the eigenvectors can be interpreted as the latent components of any time series that the filter smooths through the corresponding eigenvalues. A difficulty arises because matrices associated with trend filters are finite approximations of Toeplitz operators and therefore very little is known about their eigenstructure, which also depends on the boundary conditions or, equivalently, on the filters for trend estimation at the end of the sample. Assuming reflecting boundary conditions, we derive a time series decomposition in terms of periodic latent components and corresponding smoothing eigenvalues. This decomposition depends on the local polynomial regression estimator chosen for the interior. Otherwise, the eigenvalue distribution is derived with an approximation measured by the size of the perturbation that different boundary conditions apport to the eigenvalues of matrices belonging to algebras with known spectral properties, such as the Circulant or the Cosine. The analytical form of the eigenvectors is then derived with an approximation that involves the extremes only. A further topic investigated in the paper concerns a strategy for a filter design in the time domain. Based on cut-off eigenvalues, new estimators are derived, that are less variable and almost equally biased as the original estimator, based on all the eigenvalues. Empirical examples illustrate the effectiveness of the method."}
{"category": "Math", "title": "Secondary invariants for Frechet algebras and quasihomomorphisms", "abstract": "A Frechet algebra endowed with a multiplicatively convex topology has two types of invariants: homotopy invariants (topological K-theory and periodic cyclic homology) and secondary invariants (multiplicative K-theory and the non-periodic versions of cyclic homology). The aim of this paper is to establish a Riemann-Roch-Grothendieck theorem relating direct images for homotopy and secondary invariants of Frechet m-algebras under finitely summable quasihomomorphisms."}
{"category": "Math", "title": "On a question of Koll\\'ar", "abstract": "We show: If a bounded domain in a Stein space covers a compact complex space, it must be smooth. This give a negative answer to a question of Koll\\'ar. Furthermore, we deduce some related results."}
{"category": "Math", "title": "Lifting and restricting recollement data", "abstract": "We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in \"Parametrizing recollement data\", allows us to show that many well-known recollements of right bounded derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a 'right bounded' derived category of a differential graded(=dg) category to be a recollement of 'right bounded' derived categories of dg categories. In Theorem 2 we consider the particular case in which those dg categories are just ordinary algebras."}
{"category": "Math", "title": "Adaptivity in convolution models with partially known noise distribution", "abstract": "We consider a semiparametric convolution model. We observe random variables having a distribution given by the convolution of some unknown density $f$ and some partially known noise density $g$. In this work, $g$ is assumed exponentially smooth with stable law having unknown self-similarity index $s$. In order to ensure identifiability of the model, we restrict our attention to polynomially smooth, Sobolev-type densities $f$, with smoothness parameter $\\beta$. In this context, we first provide a consistent estimation procedure for $s$. This estimator is then plugged-into three different procedures: estimation of the unknown density $f$, of the functional $\\int f^2$ and goodness-of-fit test of the hypothesis $H_0:f=f_0$, where the alternative $H_1$ is expressed with respect to $\\mathbb {L}_2$-norm (i.e. has the form $\\psi_n^{-2}\\|f-f_0\\|_2^2\\ge \\mathcal{C}$). These procedures are adaptive with respect to both $s$ and $\\beta$ and attain the rates which are known optimal for known values of $s$ and $\\beta$. As a by-product, when the noise density is known and exponentially smooth our testing procedure is optimal adaptive for testing Sobolev-type densities. The estimating procedure of $s$ is illustrated on synthetic data."}
{"category": "Math", "title": "Inverse Indefinite Spectral Problem for Second Order Differential Operator with Complex Periodoc Coefficients", "abstract": "The inverse problem for the Sturm- Liouville operator with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is studied. We give formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem."}
{"category": "Math", "title": "A Hilbert--Mumford criterion for polystability in Kaehler geometry", "abstract": "Consider a Hamiltonian action by biholomorphisms of a compact Lie group $K$ on a Kaehler manifold $X$, with moment map $\\mu:X\\to\\klie^*$. We characterize which orbits of the complexified action of $G=K^{\\CC}$ in $X$ intersect $\\mu^{-1}(0)$ in terms of the maximal weights $\\lim_{t\\to\\infty}\\la\\mu(e^{\\imag ts}\\cdot x),s\\ra$, where $s$ belongs to the Lie algebra of $K$. We do not impose any a priori restriction on the stabilizer of $x$. Assuming some mild growth conditions on the action of $K$ on $X$, we view the maximal weights as defining a maps $\\lambda_x$ from the boundary at infinity of the symmetric space $K\\backslash G$ to $\\RR\\cup\\{\\infty\\}$. We prove that $G\\cdot x$ meets $\\mu^{-1}(0)$ if: (1) $\\lambda_x$ is everywhere nonnegative, (2) any boundary point $y$ such that $\\lambda_x(y)=0$ can be connected with a geodesic in $K\\backslash G$ to another boundary point $y'$ satisfying $\\lambda_x(y')=0$. We also prove that $\\lambda_{g\\cdot x}(y)=\\lambda_x(y\\cdot g)$ for any $g\\in G$ and $y\\in \\partial_{\\infty}(K\\backslash G)$."}
{"category": "Math", "title": "Addendum to \"Ricci-flat holonomy: a Classification\": the case of Spin(10)", "abstract": "This note fills a hole in the author's previous paper ``Ricci-Flat Holonomy: a Classification'', by dealing with irreducible holonomy algebras that are subalgebras or real forms of $\\mbb{C} \\oplus \\mf{spin}(10,\\mbb{C})$. These all turn out to be of Ricci-type."}
{"category": "Math", "title": "Desingularized fiber products of semi-stable elliptic surfaces with vanishing third Betti number", "abstract": "Desingularized fiber products of semi-stable, non-isotrivial jacobian elliptic surfaces with vanishing third Betti number are classified. Such varieties may play a role in the study of supersingular threefolds, of the deformation theory of varieties with trivial canonical bundle, and of the arithmetic degenerations of rigid Calabi-Yau threefolds."}
{"category": "Math", "title": "The functors Wbar and Diag o Nerve are simplicially homotopy equivalent", "abstract": "Given a simplicial group G, there are two known classifying simplicial set constructions, the Kan classifying simplicial set Wbar G and Diag N G, where N denotes the dimensionwise nerve. They are known to be weakly homotopy equivalent. We will show that Wbar G is a strong simplicial deformation retract of Diag N G. In particular, Wbar G and Diag N G are simplicially homotopy equivalent."}
{"category": "Math", "title": "Extensions of the Frobenius to ring of differential operators on polynomial algebra in prime characteristic", "abstract": "Let $K$ be a field of characteristic $p>0$. It is proved that each automorphism $\\s \\in \\Aut_K(\\CDPn)$ of the ring $\\CDPn$ of differential operators on a polynomial algebra $P_n= K[x_1, ..., x_n]$ is {\\em uniquely} determined by the elements $\\s (x_1), ... ,\\s (x_n)$, and the set $\\Frob (\\CDPn)$ of all the extensions of the Frobenius from certain maximal commutative polynomial subalgebras of $\\CDPn$, like $P_n$, is equal to $\\Aut_K(\\CDPn) \\cdot \\CF$ where $\\CF$ is the set of all the extensions of the Frobenius from $P_n$ to $\\CDPn$ that leave invariant the subalgebra of scalar differential operators. The set $\\CF$ is found explicitly, it is large (a typical extension depends on {\\em countably} many independent parameters)."}
{"category": "Math", "title": "The special subgroup of invertible non-commutative rational power series as a metric group", "abstract": "We give an easy proof of Sch\\\"utzenberger's Theorem stating that non-commutative formal power series are rational if and only if they are recognisable. A byproduct of this proof is a natural metric on a subgroup of invertible rational non-commutative power series. We describe a few features of this metric group."}
{"category": "Math", "title": "Regularity of the minimizers in the composite membrane problem in R^2", "abstract": "We study the regularity of minimizers to the composite membrane problem in the plane (ie given a domain omega and a positive number A, smaller than the measure of omega, minimize the first Dirichlet eigenvalue for the Schrodinger operator with potential equal to a fixed multiple of the characteristic function of a subset D of omega, with measure A). We show that for minimizers, the boundary of D is analytic."}
{"category": "Math", "title": "Graph pegging numbers", "abstract": "In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like ``pegging moves'' allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging number (respectively, the optimal pegging number) of a graph is the minimum number of pegs such that for every (respectively, some) distribution of that many pegs on the graph, any vertex can be reached by a sequence of pegging moves. We prove several basic properties of pegging and analyze the pegging number and optimal pegging number of several classes of graphs, including paths, cycles, products with complete graphs, hypercubes, and graphs of small diameter."}
{"category": "Math", "title": "Perturbations of rational Misiurewicz maps", "abstract": "In this paper we investigate the perturbation properties of rational Misiurewicz maps, when the Julia set is the whole sphere (the other case is treated in [1]). In particular, we show that if f is a Misiurewicz map and not a flexible Lattes map, then we can find a hyperbolic map arbitrarily close to f."}
{"category": "Math", "title": "A fractional Poisson equation: existence, regularity and approximations", "abstract": "We consider a stochastic boundary value elliptic problem on a bounded domain $D\\subset \\mathbb{R}^k$, driven by a fractional Brownian field with Hurst parameter $H=(H_1,...,H_k)\\in[{1/2},1[^k$. First we define the stochastic convolution derived from the Green kernel and prove some properties. Using monotonicity methods, we prove existence and uniqueness of solution, along with regularity of the sample paths. Finally, we propose a sequence of lattice approximations and prove its convergence to the solution of the SPDE at a given rate."}
{"category": "Math", "title": "Theories of bundles with additional homotopy conditions", "abstract": "In the present paper we study bundles equipped with extra homotopy conditions, in particular so-called simplicial $n$-bundles. It is shown that (under some condition) the classifying space of 1-bundles is the double coset space of some finite dimensional Lie group. We also establish some relation between our bundles and C*-algebras."}
{"category": "Math", "title": "Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS", "abstract": "Let $d\\geq 4$ and let $u$ be a global solution to the focusing mass-critical nonlinear Schr\\\"odinger equation $iu_t+\\Delta u=-|u|^{\\frac 4d}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state $Q$. We prove that if $u$ does not scatter then, up to phase rotation and scaling, $u$ is the solitary wave $e^{it}Q$. Combining this result with that of Merle \\cite{merle2}, we obtain that in dimensions $d\\geq 4$, the only spherically symmetric minimal-mass blowup solutions are, up to phase rotation and scaling, the pseudo-conformal ground state and the solitary wave."}
{"category": "Math", "title": "Sliced Inverse Moment Regression Using Weighted Chi-Squared Tests for Dimension Reduction", "abstract": "We propose a new method for dimension reduction in regression using the first two inverse moments. We develop corresponding weighted chi-squared tests for the dimension of the regression. The proposed method considers linear combinations of Sliced Inverse Regression (SIR) and the method using a new candidate matrix which is designed to recover the entire inverse second moment subspace. The optimal combination may be selected based on the p-values derived from the dimension tests. Theoretically, the proposed method, as well as Sliced Average Variance Estimate (SAVE), are more capable of recovering the complete central dimension reduction subspace than SIR and Principle Hessian Directions (pHd). Therefore it can substitute for SIR, pHd, SAVE, or any linear combination of them at a theoretical level. Simulation study indicates that the proposed method may have consistently greater power than SIR, pHd, and SAVE."}
{"category": "Math", "title": "On a generalized Sierpinski fractal in RP^n", "abstract": "We associate a fractal in $\\RPn$ to each vector basis of $\\bR^{n+1}$ and we study its measure and asymptotic properties. Then we discuss and study numerically in detail the cases $n=1,2,3$, evaluating in particular their Hausdorff dimension."}
{"category": "Math", "title": "Branching processes in random environment die slowly", "abstract": "Let $Z_{n,}n=0,1,...,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $% f_{0}(s),f_{1}(s),...,$ and let $S_{0}=0,S_{k}=X_{1}+...+X_{k},k\\geq 1,$ be the associated random walk with $X_{i}=\\log f_{i-1}^{\\prime}(1),$ $\\tau (m,n)$ be the left-most point of minimum of $\\left\\{S_{k},k\\geq 0\\right\\} $ on the interval $[m,n],$ and $T=\\min \\left\\{k:Z_{k}=0\\right\\} $. Assuming that the associated random walk satisfies the Doney condition $P(S_{n}>0) \\to \\rho \\in (0,1),n\\to \\infty ,$ we prove (under the quenched approach) conditional limit theorems, as $n\\to \\infty $, for the distribution of $Z_{nt},$ $Z_{\\tau (0,nt)},$ and $Z_{\\tau (nt,n)},$ $t\\in (0,1),$ given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\\tau (0,n)$ with respect to the point $nt.$"}
{"category": "Math", "title": "Convergence of compact Ricci solitons", "abstract": "We show that sequences of compact gradient Ricci solitons converge to complete orbifold gradient solitons, assuming constraints on volume, the $L^{n/2}$-norm of curvature, and the auxiliary constant $C_1$. The strongest results are in dimension 4, where $L^2$ curvature bounds are equivalent to upper bounds on the Euler number. We obtain necessary and sufficient conditions for limits to be compact."}
{"category": "Math", "title": "Remarks on generators and dimensions of triangulated categories", "abstract": "In this paper we prove that the dimension of the bounded derived category of coherent sheaves on a smooth quasi-projective curve is equal to one. We also discuss dimension spectrums of these categories."}
{"category": "Math", "title": "Explicit reduction modulo $p$ of certain crystalline representations", "abstract": "We use the p-adic local Langlands correspondence for GL_2(Q_p) to explicitly compute the reduction modulo p of crystalline representations of small slope, and give applications to modular forms."}
{"category": "Math", "title": "Microlocalization and nonexistence of $C^2$ Levi-flat hypersurfaces in $\\Bbb CP^2", "abstract": "To be prudent, the paper has been withdrawn by the authors, due an error (missing complex conjugate sign) in Equation (2.5). We are very grateful to Marco Brunella for pointed out the error."}
{"category": "Math", "title": "A short, based on the mixed volume, proof of Liggett's theorem on the convolution of ultra-logconcave sequences", "abstract": "R. Pemantle conjectured, and T.M. Liggett proved in 1997, that the convolution of two ultra-logconcave is ultra-logconcave. Liggett's proof is elementary but long. We present here a short proof, based on the mixed volume of convex sets."}
{"category": "Math", "title": "A unified approach to compact symmetric spaces of rank one", "abstract": "A relatively simple algebraic framework is given, in which all the compact symmetric spaces can be described and handled without distinguishing cases. We also give some applications and further results."}
{"category": "Math", "title": "A leave-p-out based estimation of the proportion of null hypotheses", "abstract": "In the multiple testing context, a challenging problem is the estimation of the proportion $\\pi_0$ of true-null hypotheses. A large number of estimators of this quantity rely on identifiability assumptions that either appear to be violated on real data, or may be at least relaxed. Under independence, we propose an estimator $\\hat{\\pi}_0$ based on density estimation using both histograms and cross-validation. Due to the strong connection between the false discovery rate (FDR) and $\\pi_0$, many multiple testing procedures (MTP) designed to control the FDR may be improved by introducing an estimator of $\\pi_0$. We provide an example of such an improvement (plug-in MTP) based on the procedure of Benjamini and Hochberg. Asymptotic optimality results may be derived for both $\\hat{\\pi}_0$ and the resulting plug-in procedure. The latter ensures the desired asymptotic control of the FDR, while it is more powerful than the BH-procedure. Finally, we compare our estimator of $\\pi_0$ with other widespread estimators in a wide range of simulations. We obtain better results than other tested methods in terms of mean square error (MSE) of the proposed estimator. Finally, both asymptotic optimality results and the interest in tightly estimating $\\pi_0$ are confirmed (empirically) by results obtained with the plug-in MTP."}
{"category": "Math", "title": "A note on combinatorial splicing formulas for Heegaard Floer homology", "abstract": "We give a precise description of splicing formulas from a previous paper in terms of knot Floer complex associated with a knot in homology sphere."}
{"category": "Math", "title": "Rewriting Systems in Alternating Knot groups with the Dehn presentation", "abstract": "Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the Dehn presentation of the knot group of a tame, prime, alternating knot, with a regular and common projection has a finite and complete rewriting system. Although there are rules in the rewriting system with left-hand side a generator and which increase the length of the words we show that the system is terminating."}
{"category": "Math", "title": "Rewriting Systems and Embedding of monoids in groups", "abstract": "In this paper, a connection between rewriting systems and embedding of monoids in groups is found. We show that if a group with a positive presentation has a complete rewriting system $\\Re$ that satisfies the condition that each rule in $\\Re$ with positive left-hand side has a positive right-hand side, then the monoid presented by the subset of positive rules from $\\Re$ embeds in the group. As an example, we give a simple proof that right angled Artin monoids embed in the corresponding right angled Artin groups. This is a special case of the well-known result of Paris \\cite{paris} that Artin monoids embed in their groups."}
{"category": "Math", "title": "Quasi-homogeneous linear systems on P2 with base points of multiplicity 7, 8, 9, 10", "abstract": "In the paper we prove Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems on $\\mathbb P^2$ for $m=7$, 8, 9, 10, i.e. systems of curves of given degree passing through points in general position with multiplicities at least $m,...,m,m_0$, where $m=7$, 8, 9, 10, $m_0$ is arbitrary."}
{"category": "Math", "title": "Strong cleanness of matrix rings over commutative rings", "abstract": "Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $\\mathbb{M}_n(R)$ is strongly clean for each integer $n>0$ if $R$ is Henselian and we show that the converse holds if either the residue class field of $R$ is algebraically closed or $R$ is an integrally closed domain or $R$ is a valuation ring. It is also shown that each $R$-algebra which is locally a direct limit of module-finite algebras, is strongly clean if $R$ is a $\\pi$-regular commutative ring."}
{"category": "Math", "title": "Index of transversally elliptic operators", "abstract": "In 1996, Berline and Vergne gave a cohomological formula for the index of a transversally elliptic operator. In this paper we propose a new point of view where the cohomological formulae make use of equivariant Chern characters with generalized coefficients and with compact suppport. This kind of Chern characters was studied by the authors in a previous paper (see arXiv:0801.2822)."}
{"category": "Math", "title": "Real Elements in Spin Groups", "abstract": "Let $F$ be a field of characteristic $\\neq 2$. Let $G$ be an algebraic group defined over $F$. An element $t\\in G(F)$ is called {\\bf real} if there exists $s\\in G(F)$ such that $sts^{-1}=t^{-1}$. A semisimple element $t$ in $GL_n(F), SL_n(F), O(q), SO(q), Sp(2n)$ and the groups of type $G_2$ over $F$ is real if and only if $t=\\tau_1\\tau_2$ where $\\tau_1^2=\\pm 1=\\tau_2^2$ (ref. \\cite{st1,st2}). In this paper we extend this result to the semisimple elements in $Spin$ groups when $\\dim(V)\\equiv 0,1,2 \\imod 4$."}
{"category": "Math", "title": "Classifying foliations", "abstract": "We give a survey of the approaches to classifying foliations, starting with the Haefliger classifying spaces and the various results and examples about the secondary classes of foliations. Various dynamical properties of foliations are introduced and discussed, including expansion rate, local entropy, and orbit growth rates. This leads to a decomposition of the foliated space into Borel or measurable components with these various dynamical types. The dynamical structure is compared with the classification via secondary classes."}
{"category": "Math", "title": "Divergence form operators in Reifenberg flat domains", "abstract": "We study the boundary regularity of solutions of elliptic operators in divergence form with $C^{0,\\alpha}$ coefficients or operators which are small perturbations of the Laplacian in non-smooth domains. We show that, as in the case of the Laplacian, there exists a close relationship between the regularity of the corresponding elliptic measure and the geometry of the domain."}
{"category": "Math", "title": "Reality Properties of Conjugacy Classes in G_2", "abstract": "Let $G$ be an algebraic group over a field $k$. We call $g\\in G(k)$ {\\bf real} if $g$ is conjugate to $g^{-1}$ in $G(k)$. In this paper we study reality for groups of type $G_2$ over fields of characteristic different from 2. Let $G$ be such a group over $k$. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element in $G(k)$ is real if and only if it is a product of two involutions in $G(k)$. Every unipotent element in $G(k)$ is a product of two involutions in $G(k)$. We discuss reality for $G_2$ over special fields and construct examples to show that reality fails for semisimple elements in $G_2$ over $\\Q$ and $\\Q_p$. We show that semisimple elements are real for $G_2$ over $k$ with $cd(k)\\leq 1$. We conclude with examples of nonreal elements in $G_2$ over $k$ finite, with characteristic $k$ not 2 or 3, which are not semisimple or unipotent."}
{"category": "Math", "title": "Reality Properties of Conjugacy Classes in algebraic Groups", "abstract": "Let $G$ be an algebraic group defined over a field $k$. We call $g\\in G$ {\\bf real} if $g$ is conjugate to $g^{-1}$ and $g\\in G(k)$ as {\\bf $k$-real} if $g$ is real in $G(k)$. An element $g\\in G$ is {\\bf strongly real} if $\\exists h\\in G$, $h^{2}=1$ (i.e. $h$ is an {\\bf involution}) such that $hgh^{-1}=g^{-1}$. Clearly, strongly real elements are real and are product of two involutions. Let $G$ be a connected adjoint semisimple group over a perfect field $k$, with -1 in the Weyl group. We prove that any strongly regular $k$-real element in $G(k)$ is strongly $k$-real (i.e. is a product of two involutions in $G(k)$). For classical groups, with some mild exceptions, over an arbitrary field $k$ of characteristic not 2, we prove that $k$-real semisimple elements are strongly $k$-real. We compute an obstruction to reality and prove some results on reality specific to fields $k$ with $cd(k)\\leq 1$. Finally, we prove that in a group $G$ of type $G_2$ over $k$, characteristic of $k$ different from 2 and 3, any real element in $G(k)$ is strongly $k$-real. This extends our results in \\cite{st}, on reality for semisimple and unipotent real elements in groups of type $G_2$."}
{"category": "Math", "title": "Groebner-Shirshov Bases for Lie Algebras: after A. I. Shirshov", "abstract": "In this paper, we review Shirshov's method for free Lie algebras invented by him in 1962 which is now called the Groebner-Shirshov bases theory."}
{"category": "Math", "title": "Cohomology of Oriented Tree Diagram Lie Algebras", "abstract": "Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this paper, we use Hodge Laplacian to study the cohomology of these Lie algebras. The \"total rank conjecture\" and \"$b_2$-conjecture\" for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler-Poincare principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras."}
{"category": "Math", "title": "Chaining Techniques and their Application to Stochastic Flows", "abstract": "We review several competing chaining methods to estimate the supremum, the diameter of the range or the modulus of continuity of a stochastic process in terms of tail bounds of their two-dimensional distributions. Then we show how they can be applied to obtain upper bounds for the growth of bounded sets under the action of a stochastic flow."}
{"category": "Math", "title": "Abelian Ideals and Cohomology of Symplectic Type", "abstract": "For symplectic Lie algebras $\\mathfrak{sp}(2n,\\mathbb{C})$, denote by $\\mathfrak{b}$ and $\\mathfrak{n}$ its Borel subalgebra and maximal nilpotent subalgebra, respectively. We construct a relationship between the abelian ideals of $\\mathfrak{b}$ and the cohomology of $\\mathfrak{n}$ with trivial coefficients. By this relationship, we can enumerate the number of abelian ideals of $\\mathfrak{b}$ with certain dimension via the Poincare polynomials of Weyl groups of type $A_{n-1}$ and $C_n$."}
{"category": "Math", "title": "Singular cobordism categories", "abstract": "Recently Galatius, Madsen, Tillmann and Weiss identified the homotopy type of the classifying space of the cobordism category of embedded d-dimensional manifolds [7] for each positive integer d. Their result lead to a new proof of the generalized standard Mumford conjecture. We extend the main theorem of [7] to the case of cobordism categories of embedded d-dimensional manifolds with prescribed singularities, and explain the relation of singular cobordism categories to the bordism version of the Gromov h-principle."}
{"category": "Math", "title": "k-Wise Independent Random Graphs", "abstract": "We study the k-wise independent relaxation of the usual model G(N,p) of random graphs where, as in this model, N labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. This relaxation can be relevant in modeling phenomena where only k-wise independence is assumed to hold, and is also useful when the relevant graphs are so huge that handling G(N,p) graphs becomes infeasible, and cheaper random-looking distributions (such as k-wise independent ones) must be used instead. Unfortunately, many well-known properties of random graphs in G(N,p) are global, and it is thus not clear if they are guaranteed to hold in the k-wise independent case. We explore the properties of k-wise independent graphs by providing upper-bounds and lower-bounds on the amount of independence, k, required for maintaining the main properties of G(N,p) graphs: connectivity, Hamiltonicity, the connectivity-number, clique-number and chromatic-number and the appearance of fixed subgraphs. Most of these properties are shown to be captured by either constant k or by some k= poly(log(N)) for a wide range of values of p, implying that random looking graphs on N vertices can be generated by a seed of size poly(log(N)). The proofs combine combinatorial, probabilistic and spectral techniques."}
{"category": "Math", "title": "A note on Chern character, loop spaces and derived algebraic geometry", "abstract": "In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere."}
{"category": "Math", "title": "Geometric optics and boundary layers for Nonlinear Schrodinger equations", "abstract": "We justify supercritical geometric optics in small time for the defocusing semiclassical Nonlinear Schrodinger Equation for a large class of non-necessarily homogeneous nonlinearities. The case of a half-space with Neumann boundary condition is also studied."}
{"category": "Math", "title": "Notes on link homology", "abstract": "This article consists of six lectures on the categorification of the Burau representation and on link homology groups which categorify the Jones and the HOMFLY-PT polynomial. The notes are based on the lecture course at the PCMI 2006 summer school in Park City, Utah."}
{"category": "Math", "title": "Maximal integral point sets over $\\mathbb{Z}^2$", "abstract": "Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set $P=\\{p_1,...,p_n\\}\\subset\\mathbb{Z}^2$ a maximal integral point set over $\\mathbb{Z}^2$ if all pairwise distances are integral and every additional point $p_{n+1}$ destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle."}
{"category": "Math", "title": "Maximal integral point sets in affine planes over finite fields", "abstract": "Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $\\mathbb{F}_q^2$ over a finite field $\\mathbb{F}_q$, where the formally defined squared Euclidean distance of every pair of points is a square in $\\mathbb{F}_q$. It turns out that integral point sets over $\\mathbb{F}_q$ can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point sets can be restated as cliques in Paley graphs of square order. In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over $\\mathbb{F}_q$ for $q\\le 47$. Furthermore, we give two series of maximal integral point sets and prove their maximality."}
{"category": "Math", "title": "Integral point sets over finite fields", "abstract": "We consider point sets in the affine plane $\\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean spaces $\\mathbb{E}^m$. We determine their maximal cardinality $\\mathcal{I}(\\mathbb{F}_q,2)$. For arbitrary commutative rings $\\mathcal{R}$ instead of $\\mathbb{F}_q$ or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality."}
{"category": "Math", "title": "Pointwise Trichotomy for Skew-Evolution Semiflows on Banach Spaces", "abstract": "The paper introduces the notion of skew-evolution semiflows and presents the concept of pointwise trichotomy in the case of skew-evolution semiflows on a Banach space. The connection with the classic notion of trichotomy presented by us in a previous paper in 2006 for evolution operators, is also emphasized, as well as some characterizations. The approach of the theory is from uniform point of view. The study can also be extended to systems with control whose state evolution can be described by skew-evolution semiflows."}
{"category": "Math", "title": "Khovanov homology and star-like isotopies", "abstract": "A star-like isotopy for oriented links in 3-space is an isotopy which uses only Reidemeister II moves with opposite orientations and Reidemeister III moves with alternating orientations when checking the strands clockwise (or anticlockwise). We define a link polynomial derived from the Jones polynomial which is, in general, only invariant under star-like isotopies and we categorify it."}
{"category": "Math", "title": "Bounds for the minimum oriented diameter", "abstract": "We consider the problem of finding an orientation with minimum diameter of a connected bridgeless graph. Fomin et. al. discovered a relation between the minimum oriented diameter an the size of a minimal dominating set. We improve their upper bound."}
{"category": "Math", "title": "Bounds for the minimum diameter of integral point sets", "abstract": "Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space diagonals of integer lengths, remains open. More generally an integral point set $\\mathcal{P}$ is a set of $n$ points in the $m$-dimensional Euclidean space $\\mathbb{E}^m$ with pairwise integral distances where the largest occurring distance is called its diameter. From the combinatorial point of view there is a natural interest in the determination of the smallest possible diameter $d(m,n)$ for given parameters $m$ and $n$. We give some new upper bounds for the minimum diameter $d(m,n)$ and some exact values."}
{"category": "Math", "title": "Integral point sets over $\\mathbb{Z}_n^m$", "abstract": "There are many papers studying properties of point sets in the Euclidean space $\\mathbb{E}^m$ or on integer grids $\\mathbb{Z}^m$, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of $\\mathbb{Z} / \\mathbb{Z}n$, and study the properties of the resulting combinatorial structures."}
{"category": "Math", "title": "Universal star products", "abstract": "One defines the notion of universal deformation quantization: given any manifold $M$, any Poisson structure $\\P$ on $M$ and any torsionfree linear connection $\\nabla$ on $M$, a universal deformation quantization associates to this data a star product on $(M,\\P)$ given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor $\\P$, the curvature tensor $R$ and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the appropriate universal Poisson cohomology."}
{"category": "Math", "title": "There are integral heptagons, no three points on a line, no four on a circle", "abstract": "We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erd\\H{o}s."}
{"category": "Math", "title": "Weak approximation of stochastic partial differential equations: the non linear case", "abstract": "We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that as it is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case a semilinear stochastic heat equation driven by a space-time white noise."}
{"category": "Math", "title": "LERF and the Lubotzky-Sarnak conjecture", "abstract": "We prove that every closed hyperbolic 3-manifold has a family of (possibly infinite sheeted) coverings with the property that the Cheeger constants in the family tend to zero. This is used to show that, if in addition the fundamental group of the manifold is LERF, then it satisfies the Lubotzky-Sarnak conjecture."}
{"category": "Math", "title": "On the minimum diameter of plane integral point sets", "abstract": "Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter $d(2,n)$ of a plane integral point set consisting of $n$ points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for $d(2,n)$ achieving the known upper bound $n^{c_2\\log\\log n}$ up to a constant in the exponent. A famous question of Erd\\H{o}s asks for plane integral point sets with no 3 points on a line and no 4 points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by $\\dot{d}(2,n)$. Recently $\\dot{d}(2,7)=22 270$ could be determined via an exhaustive search."}
{"category": "Math", "title": "Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's", "abstract": "We present a general result of transverse nonlinear instability of 1-d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1d model and the transverse perturbation \"have the same sign\". Our result applies to the generalized KP-I equation, the Nonlinear Schr\\\"odinger equation, the generalized Boussinesq system and the Zakharov-Kuznetsov equation and we hope that it may be useful in other contexts."}
{"category": "Math", "title": "Surface subgroups of Kleinian groups with torsion", "abstract": "We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup."}
{"category": "Math", "title": "Enumeration of integral tetrahedra", "abstract": "We determine the numbers of integral tetrahedra with diameter $d$ up to isomorphism for all $d\\le 1000$ via computer enumeration. Therefore we give an algorithm that enumerates the integral tetrahedra with diameter at most $d$ in $O(d^5)$ time and an algorithm that can check the canonicity of a given integral tetrahedron with at most 6 integer comparisons. For the number of isomorphism classes of integral $4\\times 4$ matrices with diameter $d$ fulfilling the triangle inequalities we derive an exact formula."}
{"category": "Math", "title": "Tilting modules arising from ring epimorphisms", "abstract": "Given a ring R, we investigate tilting modules of the form S \\oplus S/R for some injective ring epimorphism R \\to S. In particular, we are interested in tilting modules arising from Schofield's universal localization. For some rings, in this way one obtains a classification of all tilting modules."}
{"category": "Math", "title": "Plurisubharmonicity in a General Geometric Context", "abstract": "Recently the authors have explored new concepts of plurisubharmonicity and pseudoconvexity, with much of the attendant analysis, in the context of calibrated manifolds. Here a much broader extension is made. This development covers a wide variety of geometric situations, including, for example, Lagrangian plurisubhamonicity and convexity. It also applies in a number of non-geometric situations. Results include: fundamental properties of $P^+$-plurisubharmonic functions, plurisubharmonic distributions and regularity, $P^+$-convex domains and $P^+$-convex boundaries, topological restrictions on and construction of such domains, continuity of upper envelopes, and solutions of the Dirichlet problem for related Monge-Ampere-type equations. Many results in this paper have been generalized in recent work of the authors. However, this article covers many cases of geometric interest, and certain convexity assumptions here allow the use of classical analytic methods, making the exposition more accessible."}
{"category": "Math", "title": "On the underestimation of model uncertainty by Bayesian K-nearest neighbors", "abstract": "When using the K-nearest neighbors method, one often ignores uncertainty in the choice of K. To account for such uncertainty, Holmes and Adams (2002) proposed a Bayesian framework for K-nearest neighbors (KNN). Their Bayesian KNN (BKNN) approach uses a pseudo-likelihood function, and standard Markov chain Monte Carlo (MCMC) techniques to draw posterior samples. Holmes and Adams (2002) focused on the performance of BKNN in terms of misclassification error but did not assess its ability to quantify uncertainty. We present some evidence to show that BKNN still significantly underestimates model uncertainty."}
{"category": "Math", "title": "Packing index of subsets in Polish groups", "abstract": "For a subset $A$ of a Polish group $G$, we study the (almost) packing index $\\ind_P(A)$ (resp. $\\Ind_P(A)$) of $A$, equal to the supremum of cardinalities $|S|$ of subsets $S\\subset G$ such that the family of shifts $\\{xA\\}_{x\\in S}$ is (almost) disjoint (in the sense that $|xA\\cap yA|<|A|$ for any distinct points $x,y\\in S$). Subsets $A\\subset G$ with small (almost) packing index are small in a geometric sense. We show that $\\ind_P(A)\\in \\IN\\cup\\{\\aleph_0,\\cc\\}$ for any $\\sigma$-compact subset $A$ of a Polish group. If $A\\subset G$ is Borel, then the packing indices $\\ind_P(A)$ and $\\Ind_P(A)$ cannot take values in the half-interval $[\\sq(\\Pi^1_1),\\cc)$ where $\\sq(\\Pi^1_1)$ is a certain uncountable cardinal that is smaller than $\\cc$ in some models of ZFC. In each non-discrete Polish Abelian group $G$ we construct two closed subsets $A,B\\subset G$ with $\\ind_P(A)=\\ind_P(B)=\\cc$ and $\\Ind_P(A\\cup B)=1$ and then apply this result to show that $G$ contains a nowhere dense Haar null subset $C\\subset G$ with $\\ind_P(C)=\\Ind_P(C)=\\kappa$ for any given cardinal number $\\kappa\\in[4,\\cc]$."}
{"category": "Math", "title": "Symmetric monochromatic subsets in colorings of the Lobachevsky plane", "abstract": "We prove that for each partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset."}
{"category": "Math", "title": "The expansion for the overlap function", "abstract": "In this work, it is proved the complete expansion for the second moment of the overlap function for the Sherrington-Kirkpatrick model. It is a technical result which takes advantage of the cavity method and other induction arguments."}
{"category": "Math", "title": "A little noticed right triangle", "abstract": "Given a right triangle ABC, with the ninety degree angle at A; consider the triangle O1OO2.Where the point O is the midpoint of the hypotenuseBC(and so the center of the triangle ABC's circumcircle), the point O1 being the triangle AOB's circumcenter, and the point O2 being the triangle AOC circumcenter. The triangle O1OO2 is actually a right triangle similar to ABC. In the early part of this work, some geometrical formulas are derived, including some lengths of segments from a nearby trapezoid. After that, the focus changes. We examine the case when ABC is a Pythagorean triangle, and find the precise conditions under which the triangle O1OO2 is also Pythagorean. A numerical table of values is also offered at the end of the paper."}
{"category": "Math", "title": "Groebner-Shirshov Bases: Some New Results", "abstract": "In this survey article, we report some new results of Groebner-Shirshov bases, including new Composition-Diamond lemmas, applications of some known Composition-Diamond lemmas and content of some expository papers."}
{"category": "Math", "title": "Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic--parabolic systems", "abstract": "Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic-parabolic systems including the Navier--Stokes equations of compressible gas- and magnetohydrodynamics, establishing that linear and nonlinear stability are both equivalent to an Evans function, or generalized spectral stability, condition. The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in particular, it has been shown by Costanzino, Humpherys, Nguyen, and Zumbrun to hold for sufficiently large-amplitude layers for isentropic ideal gas dynamics, with general adiabiatic index $\\gamma \\ge 1$. Together with these previous results, our results thus give nonlinear stability of large-amplitude isentropic boundary layers, the first such result for compressive (``shock-type'') layers in other than the nearly-constant case. The analysis, as in the strictly parabolic case, proceeds by derivation of detailed pointwise Green function bounds, with substantial new technical difficulties associated with the more singular, hyperbolic behavior in the high-frequency/short time regime."}
{"category": "Math", "title": "Splitting the spectral flow and the SU(3) Casson invariant for spliced sums", "abstract": "We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus."}
{"category": "Math", "title": "Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation", "abstract": "Given a knot complement X and its p-fold cyclic cover X_p, we identify twisted polynomials associated to 1-dimensional linear representations of the fundamental group of X_p with twisted polynomials associated to related p-dimensional linear representations of the fundamental group of X. This provides a simpler and faster algorithm to compute these twisted polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P(3,7,9,11,15), corresponding to permutations of (7,9,11,15), represent distinct concordance classes."}
{"category": "Math", "title": "Carath\\'eodory, Helly and the others in the max-plus world", "abstract": "Carath\\'eodory's, Helly's and Radon's theorems are three basic results in discrete geometry. Their max-plus counterparts have been proved by various authors. In this paper, more advanced results in discrete geometry are shown to have also their max-plus counterparts: namely, the colorful Carath\\'eodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem -- Sierksma's conjecture --, although still open for the usual convexity, is shown to be true in the max-plus settings."}
{"category": "Math", "title": "Seiberg-Witten Floer homology and symplectic forms on S^1 X M^3", "abstract": "Let M be a closed, connected, orientable 3-manifold. The purpose of this paper is to study the Seiberg-Witten Floer homology of M given that S^1 X M admits a symplectic form. In particular, we prove that M fibers over the circle if M has first Betti number 1 and S^1 X M admits a symplectic form with non-torsion canonical class."}
{"category": "Math", "title": "A bijection on core partitions and a parabolic quotient of the affine symmetric group", "abstract": "Let $\\ell,k$ be fixed positive integers. In an earlier work, the first and third authors established a bijection between $\\ell$-cores with first part equal to $k$ and $(\\ell-1)$-cores with first part less than or equal to $k$. This paper gives several new interpretations of that bijection. The $\\ell$-cores index minimal length coset representatives for $\\widetilde{S_{\\ell}} / S_{\\ell}$ where $\\widetilde{S_{\\ell}}$ denotes the affine symmetric group and $S_{\\ell}$ denotes the finite symmetric group. In this setting, the bijection has a beautiful geometric interpretation in terms of the root lattice of type $A_{\\ell-1}$. We also show that the bijection has a natural description in terms of another correspondence due to Lapointe and Morse."}
{"category": "Math", "title": "A Note on Approximate Liftings", "abstract": "In this paper, we prove approximate lifting results in the C$^{\\ast}$-algebra and von Neumann algebra settings. In the C$^{\\ast}$-algebra setting, we show that two (weakly) semiprojective unital C*-algebras, each generated by $n$ projections, can be glued together with partial isometries to define a larger (weakly) semiprojective algebra. In the von Neumann algebra setting, we prove lifting theorems for trace-preserving *-homomorphisms from abelian von Neumann algebras or hyperfinite von Neumann algebras into ultraproducts. We also extend a classical result of S. Sakai \\cite{sakai} by showing that a tracial ultraproduct of C*-algebras is a von Neumann algebra, which yields a generalization of Lin's theorem \\cite{Lin} on almost commuting selfadjoint operators with respect to $\\Vert\\cdot\\Vert_{p}$ on any unital C*-algebra with trace."}
{"category": "Math", "title": "Topological Stable Rank of Nest Algebras", "abstract": "We establish a general result about extending a right invertible row over a Banach algebra to an invertible matrix. This is applied to the computation of right topological stable rank of a split exact sequence. We also introduce a quantitative measure of stable rank. These results are applied to compute the right (left) topological stable rank for all nest algebras. This value is either 2 or infinity, and rtsr(T(N)) = 2 occurs only when N is of ordinal type less than omega^2 and the dimensions of the atoms grows sufficiently quickly. We introduce general results on `partial matrix algebras' over a Banach algebra. This is used to obtain an inequality akin to Rieffel's formula for matrix algebras over a Banach algebra. This is used to give further insight into the nest case."}
{"category": "Math", "title": "A Height Gap Theorem For Finite Subsets Of GL_d(\\bar{Q}) and Non Amenable Subgroups", "abstract": "We show a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. We introduce a conjugation invariant normalized height $\\hat{h}(F)$ of a finite set of matrices $F$ in $GL_{n}(\\bar{\\Bbb{Q}})$ which is the adelic analog of the minimal displacement on a symmetric space. We then show, making use of theorems of Bilu and Zhang on the equidistribution of Galois orbits of small points, that $\\hat{h}(F)>\\epsilon $ as soon as $F$ generates a non-virtually solvable subgroup of $SL_{n}(\\bar{\\Bbb{Q}}),$ where $\\epsilon =\\epsilon (n)>0$ is an absolute constant."}
{"category": "Math", "title": "Coverage Probability of Wald Interval for Binomial Parameters", "abstract": "In this paper, we develop an exact method for computing the minimum coverage probability of Wald interval for estimation of binomial parameters. Similar approach can be used for other type of confidence intervals."}
{"category": "Math", "title": "Optimal Explicit Binomial Confidence Interval with Guaranteed Coverage Probability", "abstract": "In this paper, we develop an approach for optimizing the explicit binomial confidence interval recently derived by Chen et al. The optimization reduces conservativeness while guaranteeing prescribed coverage probability."}
{"category": "Math", "title": "A Strong Tits Alternative", "abstract": "We show that for every integer $d$, there is a constant $N(d)$ such that if $K$ is any field and $F$ is a finite subset of $GL_d(K)$, which generates a non amenable subgroup, then $F^{N(d)}$ contains two elements, which freely generate a non abelian free subgroup. This improves the original statement of the Tits alternative. It also implies a growth gap and a co-growth gap for non-amenable linear groups, and has consequences about the girth and uniform expansion of small sets in finite subgroups of $GL_d(\\Bbb{F}_q)$ as well as other diophantine properties of non-discrete subgroups of Lie groups."}
{"category": "Math", "title": "Presentations of finite simple groups: a computational approach", "abstract": "All nonabelian finite simple groups of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, have presentations with at most $80 $ relations and bit-length $O(\\log n +\\log q)$. Moreover, $A_n$ and $S_n$ have presentations with 3 generators$,$ 7 relations and bit-length $O(\\log n)$, while $\\SL(n,q)$ has a presentation with 7 generators, $2 5$ relations and bit-length $O(\\log n +\\log q)$."}
{"category": "Math", "title": "On Estimation and Optimization of Mean Values of Bounded Variables", "abstract": "In this paper, we develop a general approach for probabilistic estimation and optimization. An explicit formula and a computational approach are established for controlling the reliability of probabilistic estimation based on a mixed criterion of absolute and relative errors. By employing the Chernoff-Hoeffding bound and the concept of sampling, the minimization of a probabilistic function is transformed into an optimization problem amenable for gradient descendent algorithms."}
{"category": "Math", "title": "The generalized Lefschetz number of homeomorphisms on punctured disks", "abstract": "We compute the generalized Lefschetz number of orientation-preserving self-homeomorphisms of a compact punctured disk, using the fact that homotopy classes of these homeomorphisms can be identified with braids. This result is applied to study Nielsen-Thurston canonical homeomorphisms on a punctured disk. We determine, for a certain class of braids, the rotation number of the corresponding canonical homeomorphisms on the outer boundary circle. Also,it is shown that the canonical homeomorphisms corresponding to some braids are pseudo-Anosov."}
{"category": "Math", "title": "Relative support varieties", "abstract": "We define relative support varieties with respect to some fixed module over a finite dimensional algebra. These varieties share many of the standard properties of classical support varieties. Moreover, when introducing finite generation conditions on cohomology, we show that relative support varieties contain homological information on the modules involved. As an application, we provide a new criterion for a selfinjective algebra to be of wild representation type."}
{"category": "Math", "title": "Lotsize optimization leading to a $p$-median problem with cardinalities", "abstract": "We consider the problem of approximating the branch and size dependent demand of a fashion discounter with many branches by a distributing process being based on the branch delivery restricted to integral multiples of lots from a small set of available lot-types. We propose a formalized model which arises from a practical cooperation with an industry partner. Besides an integer linear programming formulation and a primal heuristic for this problem we also consider a more abstract version which we relate to several other classical optimization problems like the p-median problem, the facility location problem or the matching problem."}
{"category": "Math", "title": "The Top-Dog Index: A New Measurement for the Demand Consistency of the Size Distribution in Pre-Pack Orders for a Fashion Discounter with Many Small Branches", "abstract": "We propose the new Top-Dog-Index, a measure for the branch-dependent historic deviation of the supply data of apparel sizes from the sales data of a fashion discounter. A common approach is to estimate demand for sizes directly from the sales data. This approach may yield information for the demand for sizes if aggregated over all branches and products. However, as we will show in a real-world business case, this direct approach is in general not capable to provide information about each branch's individual demand for sizes: the supply per branch is so small that either the number of sales is statistically too small for a good estimate (early measurement) or there will be too much unsatisfied demand neglected in the sales data (late measurement). Moreover, in our real-world data we could not verify any of the demand distribution assumptions suggested in the literature. Our approach cannot estimate the demand for sizes directly. It can, however, individually measure for each branch the scarcest and the amplest sizes, aggregated over all products. This measurement can iteratively be used to adapt the size distributions in the pre-pack orders for the future. A real-world blind study shows the potential of this distribution free heuristic optimization approach: The gross yield measured in percent of gross value was almost one percentage point higher in the test-group branches than in the control-group branches."}
{"category": "Math", "title": "A lower bound for the principal eigenvalue of the Stokes operator in a random domain", "abstract": "This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound for the Stokes PE. The arguments are based on the method proposed by F. Merkl and M. V. W\\\"{u}trich for localization of the PE of the Schr\\\"{o}dinger operator in a similar setting. Some additional work is needed to circumvent the complications arising from the restriction to divergence-free vector fields of the class of test functions in the variational characterization of the Stokes PE."}
{"category": "Math", "title": "The C1 generic diffeomorphism has trivial centralizer", "abstract": "Answering a question of Smale, we prove that the space of C1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial."}
{"category": "Math", "title": "Smooth Yamabe invariant and surgery", "abstract": "We prove a surgery formula for the smooth Yamabe invariant $\\sigma(M)$ of a compact manifold $M$. Assume that $N$ is obtained from $M$ by surgery of codimension at least 3. We prove the existence of a positive number $\\Lambda_n$, depending only on the dimension $n$ of $M$, such that $$ \\sigma(N) \\geq \\min{\\sigma(M),\\Lambda_n}. $$"}
{"category": "Math", "title": "Sur la g\\'eom\\'etrie systolique des vari\\'et\\'es de Bieberbach", "abstract": "The systole of a compact non simply connected Riemannian manifold is the smallest length of a non-contractible closed curve ; the systolic ratio is the quotient $(\\mathrm{systole})^n/\\mathrm{volume}$. Its supremum on the set of all the riemannian metrics, is known to be finite for a large class of manifolds, including the $K(\\pi,1)$. We study the optimal systolic ratio of compact, 3-dimensional non orientable Bieberbach manifolds, and prove that it cannot be realized by a flat metric."}
{"category": "Math", "title": "Probabilistic computation of wind farm power generation based on wind turbine dynamic modeling", "abstract": "This paper addresses the problem of predicting a wind farm's power generation when no or few statistical data is available. The study is based on a time-series wind speed model and on a simple dynamic model of a DFIG wind turbine including cut-off and cut-in behaviours. The wind turbine is modeled as a stochastic hybrid system with three operation modes. Numerical results, obtained using Monte-Carlo simulations, provide the annual distribution of a wind farm's active power generation. For different numbers of wind turbines, we compare the numerical results obtained using the dynamic model with those obtained considering the wind turbine's steady-state power curve. Simulations show that the wind turbine's dynamics do not need to be considered for analyzing the annual distribution of a wind farm generation."}
{"category": "Math", "title": "Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms", "abstract": "We show that a multiplicative form of Dirichlet's theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R^k which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled by showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in R^{k+1} gets equidistributed in the limit. We use Ratner's theorem on unipotent flows, linearization techniques, and a new observation about intertwined linear dynamics of various SL(m,R)'s contained in SL(k+1,R)."}
{"category": "Math", "title": "Torsion points on elliptic curves over function fields and a theorem of Igusa", "abstract": "If F is a global function field of characteristic p>3, we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a fixed finite set of places of F. We end the paper with an application to torsion points rational over abelian extensions of F."}
{"category": "Math", "title": "Coherent structures and isolated spectrum for Perron-Frobenius cocycles", "abstract": "We present an analysis of one-dimensional models of dynamical systems that possess 'coherent structures'; global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron-Frobenius cocycles. We prove that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron-Frobenius cocycle has at most finitely many isolated points. Moreover, we develop a strengthened version of the Multiplicative Ergodic Theorem for non-invertible matrices and construct an invariant splitting into Oseledets subspaces. We detail examples of cocycles of expanding maps with isolated Lyapunov spectrum and calculate the Oseledets subspaces, which lead to an identification of the underlying coherent structures. Our constructions generalise the notions of almost-invariant and almost-cyclic sets to non-autonomous dynamical systems and provide a new ensemble-based formalism for coherent structures in one-dimensional non-autonomous dynamics."}
{"category": "Math", "title": "Representations of quivers via reflection functors", "abstract": "These are the notes for a course on representations of quivers for second year students in Paderborn in summer 2007. My aim was to provide a basic introduction without using any advanced methods. It turns out that a good knowledge of linear algebra is sufficient for proving Gabriel's theorem. Thus we classify the quivers of finite representation type and study their representations using reflection functors. Further material has been added after giving this course in Bielefeld in summer 2010. This includes a discussion of regular representations and wild phenomena. In particular, two classical examples are covered: representations of the Kronecker quiver and representations of the Klein four group."}
{"category": "Math", "title": "Nonautonomous Kolmogorov parabolic equations with unbounded coefficients", "abstract": "We study a class of elliptic operators $A$ with unbounded coefficients defined in $I\\times\\CR^d$ for some unbounded interval $I\\subset\\CR$. We prove that, for any $s\\in I$, the Cauchy problem $u(s,\\cdot)=f\\in C_b(\\CR^d)$ for the parabolic equation $D_tu=Au$ admits a unique bounded classical solution $u$. This allows to associate an evolution family $\\{G(t,s)\\}$ with $A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for $\\{G(t,s)\\}$ and we study the first properties of the extension of $G(t,s)$ to the $L^p$-spaces with respect to such measures."}
{"category": "Math", "title": "An asymptotic result for Brownian polymers", "abstract": "We consider a model of the shape of a growing polymer introduced by Durrett and Rogers (Probab. Theory Related Fields 92 (1992) 337--349). We prove their conjecture about the asymptotic behavior of the underlying continuous process $X_t$ (corresponding to the location of the end of the polymer at time $t$) for a particular type of repelling interaction function without compact support."}
{"category": "Math", "title": "Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces", "abstract": "Let (E,H,mu) be an abstract Wiener space and let D_V := VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space G. Given a bounded operator B on G, coercive on the closure of the range of V, we consider the realisation of the operator D_V* B D_V in L^p(E,mu) for 1<p<\\infty. Our main result states that the following assertions are equivalent: (1) dom((sqrt(D_V* B D_V)) = dom(D_V) and Meyer's inequalities hold for D_V* B D_V; (2) D_V D_V* B admits a bounded H-infinity calculus on the closure of the range of D_V; (3) dom(sqrt(V*BV)) = dom(V) and Meyer's inequalities hold for V*BV; (4) VV*B admits a bounded H-infinity calculus on the closure of the range of V. Moreover, if these conditions are satisfied, then dom(L) = dom(D_V^2) \\cap dom(D_A). The equivalence of (1)-(4) is a non-symmetric generalisation of the classical Meyer inequalities (which correspond to the case G=H, V=I, B=I). A one-sided version of the main result, giving L^p-boundedness of the associated Riesz transforms in terms of a square function estimate, is also obtained. As an application let -A generate an analytic C_0-contraction semigroup on a Hilbert space H and let -L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A."}
{"category": "Math", "title": "Extreme flatness of normed modules and Arveson-Wittstock type theorems", "abstract": "We show in this paper that a certain class of normed modules over the algebra of all bounded operators on a Hilbert space possesses a homological property which is a kind of a functional-analytic version of the standard algebraic property of flatness. We mean the preservation, under projective tensor multiplication of modules, of the property of a given morphism to be isometric. As an application, we obtain several extension theorems for different types of modules, called Arveson-Wittstock type theorems. These, in their turn, have, as a straight corollary, the `genuine' Arveson-Wittstock Theorem in its non-matricial presentation. We recall that the latter theorem plays the role of a `quantum' version of the classical Hahn-Banach theorem on the extension of bounded linear functionals. It was originally proved by Wittstock (1981), and a crucial preparatory step was done by Arveson (1969)."}
{"category": "Math", "title": "On countably compact 0-simple topological inverse semigroups", "abstract": "We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups."}
{"category": "Math", "title": "On linearly ordered $H$-closed topological semilattices", "abstract": "We give a criterium when a linearly ordered topological semilattice is $H$-closed. We also prove that any linearly ordered $H$-closed topological semilattice is absolutely $H$-closed and we show that every linearly ordered semilattice is a dense subsemilattice of an $H$-closed topological semilattice."}
{"category": "Math", "title": "Semigroup Closures of Finite Rank Symmetric Inverse Semigroups", "abstract": "We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations $\\mathscr{I}_\\lambda^n$ of the rank $\\leqslant n$ is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider."}
{"category": "Math", "title": "Quenched large deviations for multidimensional random walk in random environment: a variational formula", "abstract": "We take the point of view of the particle in a multidimensional nearest neighbor random walk in random environment (RWRE). We prove a quenched large deviation principle and derive a variational formula for the quenched rate function. Most of the previous results in this area rely on the subadditive ergodic theorem. We employ a different technique which is based on a minimax theorem. Large deviation principles for RWRE have been proven for i.i.d. nestling environments subject to a moment condition and for ergodic uniformly elliptic environments. We assume only that the environment is ergodic and the transition probabilities satisfy a moment condition."}
{"category": "Math", "title": "Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. II", "abstract": "We find many conditions equivalent to the model-theoretical property $\\lambda \\stackrel{\\kappa}{\\Rightarrow} \\mu$ introduced in [L1]. Our conditions involve uniformity of ultrafilters, compactness properties of products of topological spaces and the existence of certain infinite matrices."}
{"category": "Math", "title": "On chains in $H$-closed topological pospaces", "abstract": "We study chains in an $H$-closed topological partially ordered space. We give sufficient conditions for a maximal chain $L$ in an $H$-closed topological partially ordered space such that $L$ contains a maximal (minimal) element. Also we give sufficient conditions for a linearly ordered topological partially ordered space to be $H$-closed. We prove that any $H$-closed topological semilattice contains a zero. We show that a linearly ordered $H$-closed topological semilattice is an $H$-closed topological pospace and show that in the general case this is not true. We construct an example an $H$-closed topological pospace with a non-$H$-closed maximal chain and give sufficient conditions that a maximal chain of an $H$-closed topological pospace is an $H$-closed topological pospace."}
{"category": "Math", "title": "Cluster algebras and semipositive symmetrizable matrices", "abstract": "In this paper, we give a description of the skew-symmetrizable matrices and their mutation classes which are determined by the generalized Cartan matrices of affine type."}
{"category": "Math", "title": "Monads and comonads in module categories", "abstract": "Let $A$ be a ring and $\\M_A$ the category of $A$-modules. It is well known in module theory that for any $A $-bimodule $B$, $B$ is an $A$-ring if and only if the functor $-\\otimes_A B: \\M_A\\to \\M_A$ is a monad (or triple). Similarly, an $A $-bimodule $\\C$ is an $A$-coring provided the functor $-\\otimes_A\\C:\\M_A\\to \\M_A$ is a comonad (or cotriple). The related categories of modules (or algebras) of $-\\otimes_A B$ and comodules (or coalgebras) of $-\\otimes_A\\C$ are well studied in the literature. On the other hand, the right adjoint endofunctors $\\Hom_A(B,-)$ and $\\Hom_A(\\C,-)$ are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of $\\Hom_A(B,-)$-comodules is isomorphic to the category of $B$-modules, while the category of $\\Hom_A(\\C,-)$-modules (called $\\C$-contramodules by Eilenberg and Moore) need not be equivalent to the category of $\\C$-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of $\\C$-comodules and $\\Hom_A(\\C,-)$-modules are equivalent provided $\\C$ is a coseparable coring. Furthermore, a bialgebra $H$ over a commutative ring $R$ is a Hopf algebra if and only if $\\Hom_R(H-)$ is a Hopf bimonad on $\\M_R$ and in this case the categories of $H$-Hopf modules and mixed $\\Hom_R(H,-)$-bimodules are both equivalent to $\\M_R$."}
{"category": "Math", "title": "Stabilit\\'{e} du comportement des marches al\\'{e}atoires sur un groupe localement compact", "abstract": "Dans cet article nous d\\'{e}montrons un th\\'{e}or\\`{e}me de stabilit\\'{e} des probabilit\\'{e}s de retour sur un groupe localement compact unimodulaire, s\\'{e}parable et compactement engendr\\'{e}. Nous d\\'{e}montrons que le comportement asymptotique de $F^{*(2n)}(e)$ ne d\\'{e}pend pas de la densit\\'{e} $F$ sous des hypoth\\`{e}ses naturelles. A titre d'exemple nous \\'{e}tablissons que la probabilit\\'{e} de retour sur une large classe de groupes r\\'{e}solubles se comporte comme $\\exp(-n^{1/3})$."}
{"category": "Math", "title": "On Semi-Modular Subalgebras of Lie Algebras Over Fields of Arbitrary Characteristic", "abstract": "This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. It is shown that, in certain circumstances, including for all solvable algebras, for all Lie algebras over algebraically closed fields of characteristic p > 0 that have absolute toral rank less than or equal to 1 or are restricted, and for all Lie algebras having the one-and-a-half generation property, the conditions of modularity and semi-modularity are equivalent, but that the same is not true for all Lie algebras over a perfect field of characteristic three. Semi-modular subalgebras of dimensions one and two are characterised over (perfect, in the case of two-dimensional subalgebras) fields of characteristic different from 2, 3."}
{"category": "Math", "title": "Homotopy exponents for large H-spaces", "abstract": "We show that H-spaces with finitely generated cohomology, as an algebra or as an algebra over the Steenrod algebra, have homotopy exponents at all primes. This provides a positive answer to a question of Stanley."}
{"category": "Math", "title": "On Exponential Stability for Skew-Evolution Semiflows on Banach Spaces", "abstract": "The paper emphasizes the property of stability for skew-evolution semiflows on Banach spaces, defined by means of evolution semiflows and evolution cocycles and which generalize the concept introduced by us in a previous paper. There are presented several general characterizations of this asymptotic property out of which can be deduced well known results of the stability theory. A unified treatment in the uniform and in the nonuniform setting is given. The main results are also formulated in discrete time."}
{"category": "Math", "title": "A Littelmann path model for crystals of Generalized Kac-Moody algebras revisited", "abstract": "A Littelmann path model is constructed for crystals pertaining to a not necessarily symmetrizable Borcherds-Cartan matrix. Here one must overcome several combinatorial problems coming from the imaginary simple roots. The main results are an isomorphism theorem and a character formula of Borcherds-Kac-Weyl type for the crystals. In the symmetrizable case, the isomorphism theorem implies that the crystals constructed by this path model coincide with those of Jeong, Kang, Kashiwara and Shin obtained by taking the limit at q=0 in the quantized enveloping algebra."}
{"category": "Math", "title": "On certain enumeration problems in two-dimensional topology", "abstract": "We announce a solution to several enumeration problems in topology of surfaces. This includes an enumeration of homotopy classes of sections of locally trivial fiber bundles over surfaces and a computation of non-abelian 1-cohomology of surfaces."}
{"category": "Math", "title": "Remarks on a normal subgroup of GA_n", "abstract": "We show that the subgroup generated by locally finite polynomial automorphisms of k^n is normal in GA_n. Also, some properties of normal subgroups of GA_n containing all diagonal automorphisms are given."}
{"category": "Math", "title": "Differential modules on p-adic polyannuli", "abstract": "We consider variational properties of some numerical invariants, measuring convergence of local horizontal sections, associated to differential modules on polyannuli over a nonarchimedean field of characteristic zero. This extends prior work in the one-dimensional case of Christol, Dwork, Robba, Young, et al. Our results do not require positive residue characteristic; thus besides their relevance to the study of Swan conductors for isocrystals, they are germane to the formal classification of flat meromorphic connections on complex manifolds."}
{"category": "Math", "title": "Rationality of the moduli spaces of plane curves of sufficiently large degree", "abstract": "We prove that the moduli space of plane curves of degree d is rational for all sufficiently large d."}
{"category": "Math", "title": "On an identity of Ky Fan", "abstract": "We give several applications of an identity for sums of weakly stationary sequences due to Ky Fan."}
{"category": "Math", "title": "The rationality of the moduli space of curves of genus 3 after P. Katsylo", "abstract": "This article is a survey of P. Katsylo's proof that the moduli space of smooth projective complex curves of genus 3 is rational. We hope to make the argument more comprehensible and transparent by emphasizing the underlying geometry in the proof and its key structural features."}
{"category": "Math", "title": "Geometry of whips and chains", "abstract": "We study the geometry of the inextensible string (the whip) and its discrete approximation (the chain). In the absence of gravity, both motions represent geodesic motions on certain manifolds. We show how the motion of the chain converges to that of a whip, and how the curvature of the chain's configuration space converges to that of the whip's configuration space. Finally we speculate on the analogous approximation of an incompressible fluid by a discrete system."}
{"category": "Math", "title": "The supersingular locus of the Shimura variety of GU(1,n-1) II", "abstract": "We complete the study of the supersingular locus in the fiber at $p$ of a Shimura variety attached to a unitary similitude group $\\GU(1,n-1)$ over $\\QQ$ in the case that $p$ is inert. This was started by the first author in \\cite{Vo_Uni} where complete results were obtained for $n =2,3$. The supersingular locus is uniformized by a formal scheme $\\Ncal$ which is a moduli space of so-called unitary $p$-divisible groups. It depends on the choice of a unitary isocrystal $\\Nbf$. We define a stratification of $\\Ncal$ indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of $\\Nbf$. We show that the combinatorial behaviour of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the reduced irreducible components of $\\Ncal$) are identified with (generalized) Deligne-Lusztig varieties. We show that the Bruhat-Tits stratification is a refinement of the Ekedahl-Oort stratification and also relate the Ekedahl-Oort strata to Deligne-Lusztig varieties. We deduce that the supersingular locus is locally a complete intersection, that its irreducible components and each Ekedahl-Oort stratum in every irreducible component is isomorphic to a Deligne-Lusztig variety, and give formulas for the number of irreducible components of every Ekedahl-Oort stratum of the supersingular locus."}
{"category": "Math", "title": "Rooted induced trees in triangle-free graphs", "abstract": "For a graph $G$, let $t(G)$ denote the maximum number of vertices in an induced subgraph of $G$ that is a tree. Further, for a vertex $v\\in V(G)$, let $t^v(G)$ denote the maximum number of vertices in an induced subgraph of $G$ that is a tree, with the extra condition that the tree must contain $v$. The minimum of $t(G)$ ($t^v(G)$, respectively) over all connected triangle-free graphs $G$ (and vertices $v\\in V(G)$) on $n$ vertices is denoted by $t_3(n)$ ($t_3^v(n)$). Clearly, $t^v(G)\\le t(G)$ for all $v\\in V(G)$. In this note, we solve the extremal problem of maximizing $|G|$ for given $t^v(G)$, given that $G$ is connected and triangle-free. We show that $|G|\\le 1+\\frac{(t_v(G)-1)t_v(G)}{2}$ and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\\ge t_3^v(n)=\\lceil {1/2}(1+\\sqrt{8n-7})\\rceil$, improving a recent result by Fox, Loh and Sudakov."}
{"category": "Math", "title": "Symmetric links and Conway sums: volume and Jones polynomial", "abstract": "We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial."}
{"category": "Math", "title": "Suppression of unbounded gradients in a SDE associated with the Burgers equation", "abstract": "We consider the Langevin equation describing a stochastically perturbed by uniform noise non-viscous Burgers fluid and introduce a deterministic function that corresponds to the mean of the velocity when we keep the value of position fixed. We study interrelations between this function and the solution of the non-perturbed Burgers equation. Especially we are interested in the property of the solution of the latter equation to develop unbounded gradients within a finite time. We study the question how the initial distribution of particles for the Langevin equation influences this blowup phenomenon. It is shown that for a wide class of initial data and initial distributions of particles the unbounded gradients are eliminated. The case of a linear initial velocity is particular. We show that if the initial distribution of particles is uniform, then the mean of the velocity for a given position coincides with the solution of the Burgers equation and in particular does not depend on the constant variance of the stochastic perturbation. Further, for a one space space variable we get the following result: if the decay rate of the power-behaved initial particles distribution at infinity is greater or equal $|x|^{-2},$ then the blowup is suppressed, otherwise, the blowup takes place at the same moment of time as in the case of the non-perturbed Burgers equation."}
{"category": "Math", "title": "Inseparable local uniformization", "abstract": "It is known since the works of Zariski in early 40ies that desingularization of varieties along valuations (called local uniformization of valuations) can be considered as the local part of the desingularization problem. It is still an open problem if local uniformization exists in positive characteristic and dimension larger than three. In this paper, we prove that Zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions, i.e. any valuation can be uniformized by a purely inseparable alteration."}
{"category": "Math", "title": "K3 surfaces with Picard rank 20", "abstract": "We determine all complex K3 surfaces with Picard rank 20 over Q. Here the Neron-Severi group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We then apply our methods to general singular K3 surfaces, i.e. with Neron-Severi group of rank 20, but not necessarily generated by divisors over Q."}
{"category": "Math", "title": "Generalized Navier Boundary Condition and Geometric Conservation Law for surface tension", "abstract": "We consider two-fluid flow problems in an Arbitrary Lagrangian Eulerian (ALE) framework. The purpose of this work is twofold. First, we address the problem of the moving contact line, namely the line common to the two fluids and the wall. Second, we perform a stability analysis in the energy norm for various numerical schemes, taking into account the gravity and surface tension effects. The problem of the moving contact line is treated with the so-called Generalized Navier Boundary Conditions. Owing to these boundary conditions, it is possible to circumvent the incompatibility between the classical no-slip boundary condition and the fact that the contact line of the interface on the wall is actually moving. The energy stability analysis is based in particular on an extension of the Geometry Conservation Law (GCL) concept to the case of moving surfaces. This extension is useful to study the contribution of the surface tension. The theoretical and computational results presented in this paper allow us to propose a strategy which offers a good compromise between efficiency, stability and artificial diffusion."}
{"category": "Math", "title": "Modular polynomials for genus 2", "abstract": "Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them."}
{"category": "Math", "title": "Giroux torsion and twisted coefficients", "abstract": "We explain the effect of applying a full Lutz twist along a pre-Lagrangian torus in a contact 3-manifold, on the contact invariant in Heegaard Floer homology with twisted coefficients."}
{"category": "Math", "title": "The Weyl group of type $A_1$ root systems extended by an abelian group", "abstract": "We investigate the class of root systems $R$ obtained by extending an $A_1$-type irreducible root system by a free abelian group $G$. In this context there is a Weyl group $W$ and a group $U$ with the presentation by conjugation. Both groups are reflection groups with respect to a discrete symmetric space $T$ associated to $R$. We show that the natural homomorphism $U\\to W$ is an isomorphism if and only if an associated subset $T^{ab}\\setminus\\{0\\}$ of $G_2=G/2G$ is 2-independent, i.e. its image under the map $G_2\\to G_2\\otimes G_2, g\\mapsto g\\otimes g$ is linearly independent over the Galois field $F_2$."}
{"category": "Math", "title": "Coarse differentiation and multi-flows in planar graphs", "abstract": "We show that the multi-commodity max-flow/min-cut gap for series-parallel graphs can be as bad as 2, matching a recent upper bound Chakrabarti, Jaffe, Lee, and Vincent for this class, and resolving one side of a conjecture of Gupta, Newman, Rabinovich, and Sinclair. This also improves the largest known gap for planar graphs from 3/2 to 2, yielding the first lower bound that doesn't follow from elementary calculations. Our approach uses the {\\em coarse differentiation} method of Eskin, Fisher, and Whyte in order to lower bound the distortion for embedding a particular family of shortest-path metrics into $L_1$."}
{"category": "Math", "title": "Oscillatory Integral Decay, Sublevel Set Growth, and the Newton Polyhedron", "abstract": "Using some resolution of singularities methods of the author, a generalization of a well-known theorem of Varchenko relating decay of oscillatory integrals to the Newton polyhedron is proven. They are derived from analogous results for sublevel integrals, proven here. Varchenko's theorem requires a certain nondegeneracy condition on the faces of the Newton polyhedron on the phase. In this paper, it is shown that the estimates of Varchenko's theorem also hold for a significant class of phase functions for which this nondegeneracy condition does not hold. Thus in problems where one wants to switch coordinates to a coordinate system where Varchenko's estimates are valid, one has greater flexibility. Some additional estimates are also proven for more degenerate situations, including some too degenerate for the Newton polyhedron to give the optimal decay in the sense of Varchenko."}
{"category": "Math", "title": "Fixed point-free isometric actions of topological groups on Banach spaces", "abstract": "We show that every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on Banach spaces. For separable topological groups, in the above statements it is enough to consider affine actions on one particular Banach space: the unique Banach space envelope of the universal Urysohn metric space, known as the Holmes space. At the same time, we show that Polish groups need not admit topologically proper (in particular, free) affine isometric actions on Banach spaces (nor even on complete metric spaces): this is the case for the unitary group of the separable infinite dimensional Hilbert space with strong operator topology, the infinite symmetric group, etc."}
{"category": "Math", "title": "Adaptive nonparametric estimation in heteroscedastic regression models. Part 2: Asymptotic efficiency", "abstract": "The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper (2007) for estimation of unknown nonparametric regression. We prove that this procedure is asymptotically efficient for a quadratic risk. It means that the asymptotic quadratic risk for this procedure coincides with a sharp lower bound."}
{"category": "Math", "title": "A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs", "abstract": "For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explainthis connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs."}
{"category": "Math", "title": "Comparison of secondary invariants of algebraic K-theory", "abstract": "In this paper we prove that the multiplicative character of A. Connes and M. Karoubi and the determinant invariant of L. G. Brown, J. W. Helton and R. E. Howe agree up to a canonical homomorphism."}
{"category": "Math", "title": "Structural Ramsey theory of metric spaces and topological dynamics of isometry groups", "abstract": "In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces - called ultrahomogeneous - is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the present paper is to explore the different aspects of this connection."}
{"category": "Math", "title": "Holomorphic Representation of Constant Mean Curvature Surfaces in Minkowski Space: Consequences of Non-Compactness in Loop Group Methods", "abstract": "We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space $\\real^{2,1}$. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group $SU_2$ with $SU_{1,1}$. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form $SU_{1,1}$, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in $\\real^{2,1}$. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant."}
{"category": "Math", "title": "Invariance of a Shift-Invariant Space", "abstract": "A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. This paper characterizes those shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated spaces, these spaces are characterized in terms of the generators of the space. As a consequence, it is shown that principal shift-invariant spaces with a compactly supported generator cannot be invariant under any non-integer translations."}
{"category": "Math", "title": "Automorphisms of curves fixing the order two points of the Jacobian", "abstract": "Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism \\sigma that fixes pointwise all the order two points of Pic}^0(X), then we prove that X is hyperelliptic with \\sigma being the unique hyperelliptic involution. As a corollary, if a nontrivial automorphisms \\sigma' of X fixes pointwise all the theta characteristics on X, then X is hyperelliptic with \\sigma' being its hyperelliptic involution."}
{"category": "Math", "title": "Linear Optimization over a Polymatroid with Side Constraints -- Scheduling Queues and Minimizing Submodular Functions", "abstract": "Two seemingly unrelated problems, scheduling a multiclass queueing system and minimizing a submodular function, share a rather deep connection via the polymatroid that is characterized by a submodular set function on the one hand and represents the performance polytope of the queueing system on the other hand. We first develop what we call a {\\it grouping} algorithm that solves the queueing scheduling problem under side constraints, with a computational effort of $O(n^3LP(n))$, $n$ being the number of job classes, and LP(n) being the computational efforts of solving a linear program with no more than $n$ variables and $n$ constraints. The algorithm organizes the job classes into groups, and identifies the optimal policy to be a priority rule across the groups and a randomized rule within each group (to enforce the side constraints). We then apply the grouping algorithm to the submodular function minimization, mapping the latter to a queueing scheduling problem with side constraints. %Each time the algorithm is applied, it identifies a subset; and We show the minimizing subset can be identified by applying the grouping algorithm $n$ times. Hence, this results in a algorithm that minimizes a submodular function with an effort of $O(n^4LP(n))$."}
{"category": "Math", "title": "Automorphisms and Verma modules for Generalized Schr\\\"{o}dinger-Virasoro algebras", "abstract": "Let $\\mathbb{F}$ be a field of characteristic 0, $G$ an additive subgroup of $\\mathbb{F}$, $\\alpha\\in \\mathbb{F}$ satisfying $\\alpha\\notin G, 2\\alpha\\in G$. We define a class of infinite-dimensional Lie algebras which are called generalized Schr\\\"{o}dinger-Virasoro algebras and use $\\mathfrak{gsv}[G,\\alpha]$ to denote the one corresponding to $G$ and $\\alpha$. In this paper the automorphism group and irreducibility of Verma modules for $\\mathfrak{gsv}[G,\\alpha]$ are completely determined."}
{"category": "Math", "title": "Hilbert domains quasi-isometric to normed vector spaces", "abstract": "We prove that a Hilbert domain which is quasi-isometric to a normed vector space is actually a convex polytope."}
{"category": "Math", "title": "Hilbert geometry for convex polygonal domains", "abstract": "We prove in this paper that the Hilbert geometry associated with an open convex polygonal set is Lipschitz equivalent to Euclidean plane."}
{"category": "Math", "title": "On the spectrum of asymptotic slopes", "abstract": "The slopes of maximal subbundles of rank $s$ divided by the degree of the map under various pull backs form a bounded collection of numbers called the $s$-spectrum of the bundle. We study the supremum of the $s$-spectrum and determine it in terms of the Harder Narasimhan filtration of the bundle."}
{"category": "Math", "title": "Counting conics in complete intersections", "abstract": "We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasi-lines."}
{"category": "Math", "title": "Existence and boundedness of solutions for a singular phase field system", "abstract": "This paper is devoted to the mathematical analysis of a thermomechanical model describing phase transitions in terms of the entropy and order structure balance law. We consider a macroscopic description of the phenomenon and make a presentation of the model. Then, the initial and boundary value problem is addressed for the related PDE system, which contains some nonlinear and singular terms with respect to the temperature variable. Existence of the solution is shown along with the boundedness of both phase variable $\\chi$ and absolute temperature $\\theta$. Finally, uniqueness is proved in the framework of a source term depending Lipschitz continuously on $\\theta$."}
{"category": "Math", "title": "Exact conditions for no ruin for the generalised Ornstein-Uhlenbeck process", "abstract": "For a bivariate L\\'evy process $(\\xi_t,\\eta_t)_{t\\geq 0}$ the generalised Ornstein-Uhlenbeck (GOU) process is defined as V_t:=e^{\\xi_t}(z+\\int_0^t e^{-\\xi_{s-}}d\\eta_s), t\\ge0, where $z\\in\\mathbb{R}.$ We define necessary and sufficient conditions under which the infinite horizon ruin probability for the process is zero. These conditions are stated in terms of the canonical characteristics of the L\\'evy process and reveal the effect of the dependence relationship between $\\xi$ and $\\eta.$ We also present technical results which explain the structure of the lower bound of the GOU."}
{"category": "Math", "title": "Finite Dimensional Intuitionistic Fuzzy Normed Linear Space", "abstract": "Following the definition of intuitionistic fuzzy n-norm [ 3 ], we have introduced the definition of intuitionistic fuzzy norm (in short IFN) over a linear space and there after a few results on intuitionistic fuzzy normed linear space and finite dimensional intuitionistic fuzzy normed linear space. Lastly, we have introduced the definitions of intuitionistic fuzzy continuity and sequentially intuitionistic fuzzy continuity and proved that they are equivalen"}
{"category": "Math", "title": "Deformation of Curves with Automorphisms and representations on Riemann-Roch spaces", "abstract": "We study the deformation theory of nonsigular projective curves defined over algebraic closed fields of positive characteristic. We show that under some assumptions the local deformation problem for automorphisms of powerseries can be reduced to a deformation problem for matrix representations. We study both equicharacteristic and mixed deformations in the case of two dimensional representations."}
{"category": "Math", "title": "The Terwilliger Algebra of a Distance-Regular Graph of Negative Type", "abstract": "Let $\\Gamma$ denote a distance-regular graph with diameter $D \\ge 3$. Assume $\\Gamma$ has classical parameters $(D,b,\\alpha,\\beta)$ with $b < -1$. Let $X$ denote the vertex set of $\\Gamma$ and let $A \\in MX$ denote the adjacency matrix of $\\Gamma$. Fix $x \\in X$ and let $A^* \\in MX$ denote the corresponding dual adjacency matrix. Let $T$ denote the subalgebra of $MX$ generated by $A, A^*$. We call $T$ the {\\em Terwilliger algebra} of $\\Gamma$ with respect to $x$. We show that up to isomorphism there exist exactly two irreducible $T$-modules with endpoint 1; their dimensions are $D$ and $2D-2$. For these $T$-modules we display a basis consisting of eigenvectors for $A^*$, and for each basis we give the action of $A$"}
{"category": "Math", "title": "Introducing Ramanujan's Class Polynomials in the Generation of Prime Order Elliptic Curves", "abstract": "In this paper, we propose the use of Ramanujan class of polynomials for the construction of prime order elliptic curves using the CM-method. We compare (theoretically and experimentally) the efficiency of using this new class against the use of the Weber, $M_{D,l}(x)$ and $M_{D,p_1,p_2}(x)$ polynomials and show that they clearly outweigh all of them in the generation of prime order elliptic curves."}
{"category": "Math", "title": "Dedekind Zeta motives for totally real fields", "abstract": "Let $k$ be a totally real number field. For every odd $n\\geq 3$, we construct a Dedekind zeta motive in the category $\\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $\\pi^{n[k:\\Q]}\\zeta^*_k(1-n)$, where $\\zeta^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\\Ext^1_{\\MT(k)} (\\Q(0),\\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculate the groups $\\Ext^1_{\\MT(k)} (\\Q(0),\\Q(n))$. In particular, the special value of the Dedekind zeta function is a determinant of volumes of geodesic hyperbolic simplices defined over $k$."}
{"category": "Math", "title": "On percolation in random graphs with given vertex degrees", "abstract": "We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core."}
{"category": "Math", "title": "On ring class eigenspaces of Mordell-Weil groups of elliptic curves over global function fields", "abstract": "If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional eigenspace relative to a fixed complex ring class character provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is nonzero. This represents the analogue in the function field setting of a theorem for rational elliptic curves due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof employs Kolyvagin-type arguments, and the cohomological machinery is started up by the control on the Galois structure of the torsion of E provided by classical results of Igusa in positive characteristic."}
{"category": "Math", "title": "An inequality between multipoint Seshadri constants", "abstract": "Let X be a projective variety of dimension n and L be a nef divisor on X. Denote by e_d(r;X,L) the d-dimensional Seshadri constant of r very general points in X. We prove that e_d(rs;X,L) >= e_d(r;X,L)e_d(s;P^n,O_{P^n}(1))."}
{"category": "Math", "title": "Poles of regular quaternionic functions", "abstract": "This paper studies the singularities of Cullen-regular functions of one quaternionic variable. The quaternionic Laurent series prove to be Cullen-regular. The singularities of Cullen-regular functions are thus classified as removable, essential or poles. The quaternionic analogues of meromorphic complex functions, called semiregular functions, turn out to be quotients of Cullen-regular functions with respect to an appropriate division operation. This allows a detailed study of the poles and their distribution."}
{"category": "Math", "title": "Geometry of plane sections of the infinite regular skew polyhedron {4,6|4}", "abstract": "The asymptotic behavior of open plane sections of triply periodic surfaces is dictated, for an open dense set of plane directions, by an integer second homology class of the three-torus. The dependence of this homology class on the direction can have a rather rich structure, leading in special cases to a fractal. In this paper we present in detail the results for the skew polyhedron {4,6|4} and in particular we show that in this case a fractal arises and that such a fractal can be generated through an elementary algorithm, which in turn allows us to verify for this case a conjecture of S.P.Novikov that such fractals have zero measure."}
{"category": "Math", "title": "Optimal systems of subalgebras for a nonlinear Black-Scholes equation", "abstract": "This paper has been withdraw by the auhor due to a mistake in classification of the algebra"}
{"category": "Math", "title": "Low Cohomogeneity and Polar Actions on Exceptional Compact Lie Groups", "abstract": "We study isometric Lie group actions on the compact exceptional groups E6, E7, E8, F4 and G2 endowed with a biinvariant metric. We classify polar actions on these groups. We determine all isometric actions of cohomogeneity less than three on E6, E7, F4 and all isometric actions of cohomogeneity less than 20 on E8. Moreover we determine the principal isotropy algebras for all isometric actions on G2."}
{"category": "Math", "title": "Non-algebraic Hyperkaehler manifolds", "abstract": "We study the algebraic dimension a(X) of a compact hyperkaehler manfold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact Kaehler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0,n and 2n are possible. In case of middle dimension, the algebraic reduction is holomorphic Lagrangian. If n = 2, then - without any assumptions - the algebraic dimension only takes the values 0,2 and 4. The paper gives structure results for \"generalised hyperkaehler\" manifolds and studies nef lines bundles."}
{"category": "Math", "title": "Approximating optimization problems over convex functions", "abstract": "Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\\Omega)$, and optimizing functionals arising from some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and functions with positive semidefinite discrete Hessian need not be convex in a discrete sense. Previous work has concentrated on non-local descriptions of convexity, making the number of constraints to grow super-linearly with the number of nodes even in dimension 2, and these descriptions are very difficult to extend to higher dimensions. In this paper we propose a finite difference approximation using positive semidefinite programs and discrete Hessians, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to problems in two and three dimensions."}
{"category": "Math", "title": "Sub-Riemannian geodesics on the 3-D sphere", "abstract": "The unit sphere $\\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system."}
{"category": "Math", "title": "An Artin-Rees Theorem and applications to zero cycles", "abstract": "We prove an Artin-Rees type theorem for algebraic cycles and give an application to zero cycles."}
{"category": "Math", "title": "A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem", "abstract": "We consider a transmission wave equation in two embedded domains in $R^2$, where the speed is $a1 > 0$ in the inner domain and $a2 > 0$ in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strictly convex and $a1 > a2$ . As a consequence of this inequality, uniqueness and Lip- schitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement."}
{"category": "Math", "title": "Vortex-like finite-energy asymptotic profiles for isentropic compressible flows", "abstract": "Bidimensional incompressible viscous flows with well-localised vorticity are well-known to develop vortex structures. The purpose of the present paper is to recover the asymptotic profiles describing these phenomena for homogeneous finite-energy flows as asymptotic profiles for near-equilibrium isentropic compressible flows. This task is performed by extending the sharp description of the asymptotic behaviour of near-equilibrium compressible flows obtained by David Hoff and Kevin Zumbrun to the case of finite-energy vortex-like solutions."}
{"category": "Math", "title": "Constructive solution of a bilinear optimal control problem for a Schr\\\"odinger equation", "abstract": "Often considered in numerical simulations related to the control of quantum systems, the so-called monotonic schemes have not been so far much studied from the functional analysis point of view. Yet, these procedures provide an efficient constructive method for solving a certain class of optimal control problems. This paper aims both at extending the results already available about these algorithms in the finite dimensional case (i.e., the time-discretized case) and at completing those of the continuous case. This paper starts with some results about the regularity of a functional related to a wide class of model in quantum chemistry. Those enable us to extend an inequality due to"}
{"category": "Math", "title": "An inverse problem for Schr\\\"odinger equations with discontinuous main coefficient", "abstract": "This paper concerns the inverse problem of retrieving a stationary potential for the Schr\\\"odinger evolution equation in a bounded domain of RN with Dirichlet data and discontinuous principal coefficient a(x) from a single time-dependent Neumann boundary measurement. We consider that the discontinuity of a is located on a simple closed hyper-surface called the interface, and a is constant in each one of the interior and exterior domains with respect to this interface. We prove uniqueness and lipschitz stability for this inverse problem under certain convexity hypothesis on the geometry of the interior domain and on the sign of the jump of a at the interface. The proof is based on a global Carleman inequality for the Schr\\\"odinger equation with discontinuous coefficients, result also interesting by itself."}
{"category": "Math", "title": "Adaptive sequential estimation for ergodic diffusion processes in quadratic metric. Part 2: Asymptotic efficiency", "abstract": "Asymptotic efficiency is proved for the constructed in part 1 procedure, i.e. Pinsker's constant is found in the asymptotic lower bound for the minimax quadratic risk. It is shown that the asymptotic minimax quadratic risk of the constructed procedure coincides with this constant."}
{"category": "Math", "title": "Adaptive nonparametric estimation in heteroscedastic regression models. Part 1: Sharp non-asymptotic Oracle inequalities", "abstract": "An adaptive nonparametric estimation procedure is constructed for the estimation problem of heteroscedastic regression when the noise variance depends on the unknown regression. A non-asymptotic upper bound for a quadratic risk (an oracle inequality) is constructed."}
{"category": "Math", "title": "The Yamabe problem with singularities", "abstract": "Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\geq 3$. Under some assumptions, we prove that there exists a positive function $\\varphi$ solution of the following Yamabe type equation \\Delta \\varphi+ h\\varphi= \\tilde h \\varphi^{\\frac{n+2}{n-2}} where $h\\in L^p(M)$, $p>n/2$ and $\\tilde h\\in \\mathbb R$. We give the regularity of $\\varphi$ with respect to the value of $p$. Finally, we consider the results in geometry when $g$ is a singular Riemannian metric and $h=\\frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar curvature of $g$."}
{"category": "Math", "title": "Robust control of a bimorph mirror for adaptive optics system", "abstract": "We apply robust control technics to an adaptive optics system including a dynamic model of the deformable mirror. The dynamic model of the mirror is a modification of the usual plate equation. We propose also a state-space approach to model the turbulent phase. A continuous time control of our model is suggested taking into account the frequential behavior of the turbulent phase. An H_\\infty controller is designed in an infinite dimensional setting. Due to the multivariable nature of the control problem involved in adaptive optics systems, a significant improvement is obtained with respect to traditional single input single output methods."}
{"category": "Math", "title": "Amenability of algebras of approximable operators", "abstract": "We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson's space."}
{"category": "Math", "title": "Micro-local analysis in Fourier Lebesgue and modulation spaces. Part I", "abstract": "Let $\\omega ,\\omega_0$ be appropriate weight functions and $q\\in [1,\\infty ]$. We introduce the wave-front set, $\\WF_{\\mathscr FL^q_{(\\omega)}}(f)$ of $f\\in \\mathscr S'$ with respect to weighted Fourier Lebesgue space $\\mathscr FL^q_{(\\omega)}$. We prove that usual mapping properties for pseudo-differential operators $\\op (a)$ with symbols $a$ in $S^{(\\omega _0)}_{\\rho, 0}$ hold for such wave-front sets. Especially we prove \\WF_{\\mathscr FL^q_{(\\omega /\\omega_0)}}(\\op (a)f)\\subseteq \\WF_{\\mathscr FL^q_{(\\omega)}}(f) \\subseteq \\WF_{\\mathscr FL^q_{(\\omega /\\omega_0)}}(\\op (a)f)\\ttbigcup \\Char (a). %% Here $\\Char (a)$ is the set of characteristic points of $a$."}
{"category": "Math", "title": "On Parallel Sections of a Vector Bundle", "abstract": "We consider when a smooth vector bundle endowed with a connection possesses non-trivial, local parallel sections. This is accomplished by means of a derived flag of subsets of the bundle. The procedure is algebraic and rests upon the Frobenius Theorem."}
{"category": "Math", "title": "Push-outs of derivations", "abstract": "Let A be a Banach algebra and let X be a Banach A -bimodule. In studying the bounded Hochschild cohomology groups H^1(A,X) it is often useful to extend a given derivation D: A-> X to a Banach algebra B containing A as an ideal, thereby exploiting (or establishing) hereditary properties. This is usually done using (bounded/unbounded) approximate identities to obtain the extension as a limit of operators b->D(ba)-b.D(a), (a in A) in an appropriate operator topology, the main point in the proof being to show that the limit map is in fact a derivation. In this paper we make clear which part of this approach is analytic and which algebraic by presenting an algebraic scheme that gives derivations in all situations at the cost of enlarging the module. We use our construction to give improvements and shorter proofs of some results from the literature and to give a necessary and sufficient condition that biprojectivity and biflatness are inherited to ideals."}
{"category": "Math", "title": "Biharmonic maps and morphisms from conformal mappings", "abstract": "Inspired by the all-important conformal invariance of harmonic maps on two-dimensional domains, this article studies the relationship between biharmonicity and conformality. We first give a characterization of biharmonic morphisms, analogues of harmonic morphisms investigated by Fuglede and Ishihara, which, in particular, explicits the conditions required for a conformal map in dimension four to preserve biharmonicity and helps producing the first example of a biharmonic morphism which is not a special type of harmonic morphism. Then, we compute the bitension field of horizontally weakly conformal maps, which include conformal mappings. This leads to several examples of proper (i.e. non-harmonic) biharmonic conformal maps, in which dimension four plays a pivotal role. We also construct a family of Riemannian submersions which are proper biharmonic maps."}
{"category": "Math", "title": "Lower bounds of martingale measure densities in the Dalang-Morton-Willinger theorem", "abstract": "For a $d$-dimensional stochastic process $(S_n)_{n=0}^N$ we obtain criteria for the existence of an equivalent martingale measure, whose density $z$, up to a normalizing constant, is bounded from below by a given random variable $f$. We consider the case of one-period model (N=1) under the assumptions $S\\in L^p$; $f,z\\in L^q$, $1/p+1/q=1$, where $p\\in [1,\\infty]$, and the case of $N$-period model for $p=\\infty$. The mentioned criteria are expressed in terms of the conditional distributions of the increments of $S$, as well as in terms of the boundedness from above of an utility function related to some optimal investment problem under the loss constraints. Several examples are presented."}
{"category": "Math", "title": "Collision probabilities in the rarefaction fan of asymmetric exclusion processes", "abstract": "We consider the one-dimensional asymmetric simple exclusion process (ASEP) in which particles jump to the right at rate $p\\in(1/2,1]$ and to the left at rate $1-p$, interacting by exclusion. In the initial state there is a finite region such that to the left of this region all sites are occupied and to the right of it all sites are empty. Under this initial state, the hydrodynamical limit of the process converges to the rarefaction fan of the associated Burgers equation. In particular suppose that the initial state has first-class particles to the left of the origin, second-class particles at sites 0 and 1, and holes to the right of site 1. We show that the probability that the two second-class particles eventually collide is $(1+p)/3p$, where a_collision_ occurs when one of the particles attempts to jump over the other. This also corresponds to the probability that two ASEP processes, started from appropriate initial states and coupled using the so-called \"basic coupling\", eventually reach the same state. We give various other results about the behaviour of second-class particles in the ASEP. In the totally asymmetric case ($p=1$) we explain a further representation in terms of a multi-type particle system, and also use the collision result to derive the probability of coexistence of both clusters in a two-type version of the corner growth model."}
{"category": "Math", "title": "On convex functions and the finite element method", "abstract": "Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\\Omega)$, or some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and an adequate discrete version of the Hessian must be given. In this paper we propose a finite element description of the Hessian, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to optimization problems."}
{"category": "Math", "title": "Large semilattices of breadth three", "abstract": "A 1984 problem of S.Z. Ditor asks whether there exists a lattice of cardinality aleph two, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of ZFC, namely (1) Martin's Axiom restricted to collections of aleph one dense subsets in posets of precaliber aleph one, (2) the existence of a gap-1 morass. In particular, the existence of such a lattice is consistent with ZFC, while the non-existence of such a lattice implies that omega two is inaccessible in the constructible universe. We also prove that for each regular uncountable cardinal $\\kappa$ and each positive integer n, there exists a join-semilattice L with zero, of cardinality $\\kappa^{+n}$ and breadth n+1, in which every principal ideal has less than $\\kappa$ elements."}
{"category": "Math", "title": "Solitary waves for linearly coupled nonlinear Schrodinger equations with inhomogeneous coefficients", "abstract": "Motivated by the study of matter waves in Bose-Einstein condensates and coupled nonlinear optical systems, we study a system of two coupled nonlinear Schrodinger equations with inhomogeneous parameters, including a linear coupling. For that system we prove the existence of two different kinds of homoclinic solutions to the origin describing solitary waves of physical relevance. We use a Krasnoselskii fixed point theorem together with a suitable compactness criterion."}
{"category": "Math", "title": "Multidimensional Version of Lagrange's Problem on Mean Motion", "abstract": "The famous mean motion problem which goes back to Lagrange as follows: to prove that any exponential polynomial with exponents on the imaginary axis has an average speed for the amplitude, whenever the variable moves along a horizontal line. It was completely proved by B. Jessen and H. Tornehave in Acta Math.77, 1945. Here we give its multidimensional version."}
{"category": "Math", "title": "Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes", "abstract": "We prove that for a generic $C^1$-diffeomorphism existence of a homoclinic class with periodic saddles of different indices (dimension of the unstable bundle) implies existence an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of $f$."}
{"category": "Math", "title": "Parabolicity of maximal surfaces in Lorentzian product spaces", "abstract": "In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form $M^2\\times\\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface with non-negative Gaussian curvature and $M^2\\times\\mathbb{R}_1$ is endowed with the Lorentzian product metric $<,>=<,>_M-dt^2$. In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain $\\Omega\\subseteq M$ is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi-Bernstein result for entire maximal graphs in $M^2\\times\\mathbb{R}_1$."}
{"category": "Math", "title": "On Type I singularities of the local axi-symmetric solutions of the Navier-Stokes equations", "abstract": "Local regularity of axially symmetric solutions to the Navier-Stokes equations is studied. It is shown that under certain natural assumptions there are no singularities of Type I."}
{"category": "Math", "title": "Long-time Behavior for a Nonlinear Plate Equation with Thermal Memory", "abstract": "We consider a nonlinear plate equation with thermal memory effects due to non-Fourier heat flux laws. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we use a suitable Lojasiewicz--Simon type inequality to show the convergence of global solutions to single steady states as time goes to infinity under the assumption that the nonlinear term $f$ is real analytic. Moreover, we provide an estimate on the convergence rate."}
{"category": "Math", "title": "Last Passage Percolation in Macroscopically Inhomogeneous Media", "abstract": "In this note we investigate the last passage percolation model in the presence of macroscopic inhomogeneity. We analyze how this affects the scaling limit of the passage time, leading to a variational problem that provides an ODE for the deterministic limiting shape of the maximal path. We obtain a sufficient analytical condition for uniqueness of the solution for the variational problem. Consequences for the totally asymmetric simple exclusion process are discussed."}
{"category": "Math", "title": "A characterization of simplicial polytopes with g_2=1", "abstract": "Kalai proved that the simplicial polytopes with g_2=0 are the stacked polytopes. We characterize the g_2=1 case. Specifically, we prove that every simplicial d-polytope (d>=4) which is prime and with g_2=1 is combinatorially equivalent either to a free sum of two simplices whose dimensions add up to d (each of dimension at least 2), or to a free sum of a polygon with a (d-2)-simplex. Thus, every simplicial d-polytope (d>=4) with g_2=1 is combinatorially equivalent to a polytope obtained by stacking over a polytope as above. Moreover, the above characterization holds for any homology (d-1)-sphere (d>=4) with g_2=1, and our proof takes advantage of working with this larger class of complexes."}
{"category": "Math", "title": "Structure of Ann-Categories", "abstract": "This paper presents the structure conversion by which from an Ann-category $\\A,$ we can obtain its reduced Ann-category of the type $(R,M)$ whose structure is a family of five functions $k=(\\xi,\\eta,\\alpha,\\lambda,\\rho)$. Then we will show that each Ann-category is determined by three invariants: 1. The ring $\\Pi_0(\\A)$ of the isomorphic classes of objects of $\\A$, 2. $\\Pi_0(\\A)$-bimodule $\\Pi_1(\\A) = \\Aut_{\\A}(0),$ 3. The element $ \\bar{k}\\in H^{3}_{M}(\\Pi_0(\\A), \\Pi_1(\\A))$ (the ring cohomology due to MacLane)."}
{"category": "Math", "title": "Topological complexity of basis-conjugating automorphism groups", "abstract": "We compute the topological complexity of Eilenberg-Mac Lane spaces associated to the group of automorphisms of a finitely generated free group which act by conjugation on a given basis, and to certain subgroups."}
{"category": "Math", "title": "Grassmann Geometries and Integrable Systems", "abstract": "We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann submanifolds. An example is given from recent work of the author."}
{"category": "Math", "title": "AHS-structures and affine holonomies", "abstract": "We show that a large class of non-metric, non-symplectic affine holonomies can be realized, uniformly and without case by case considerations, by Weyl connections associated to the natural AHS-structures on certain generalized flag manifolds."}
{"category": "Math", "title": "Independence of Four Projective Criteria for the Weak Invariance Principle", "abstract": "Let $(X_i)_{i\\in\\Z}$ be a regular stationary process for a given filtration. The weak invariance principle holds under the condition $\\sum_{i\\in\\Z}\\|P_0(X_i)\\|_2<\\infty$ (see Hannan (1979)}, Dedecker and Merlev\\`ede (2003), Deddecker, Merlev\\'ede and Voln\\'y (2007)). In this paper, we show that this criterion is independent of other known criteria: the martingale-coboundary decomposition of Gordin (see Gordin (1969, 1973)), the criterion of Dedecker and Rio (see Dedecker and Rio (2000)) and the condition of Maxwell and Woodroofe (see Maxwell and Woodroofe (2000), Peligrade and Utev (2005), Voln\\'y (2006, 2007))."}
{"category": "Math", "title": "An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths", "abstract": "We discover an explicit expansion formula for the powers $s$ of the Euler Product (or Dedekind $\\eta$-function) in terms of hook lengths of partitions, where the exponent $s$ is any complex number. Several classical formulas have been derived for certain integers $s$ by Euler, Jacobi, Klein, Fricke, Atkin, Winquist, Dyson and Macdonald. In particular, Macdonald obtained expansion formulas for the integer exponents $s$ for which there exists a semi-simple Lie algebra of dimension $s$. For the type $A_l^{(a)}$ he has expressed the $(t^2-1)$-st power of the Euler Product as a sum of weighted integer vectors of length $t$ for any integer $t$. Kostant has considered the general case for any positive integer $s$ and obtained further properties. ----- The present paper proposes a new approach. We convert the weighted vectors of length $t$ used by Macdonald in his identity for type $A_l^{(a)}$ to weighted partitions with free parameter $t$, so that a new identity on the latter combinatorial structures can be derived without any restrictions on $t$. The surprise is that the weighted partitions have a very simple form in terms of hook lengths of partitions. As applications of our formula, we find some new identities about hook lengths, including the \"marked hook formula\". We also improve a result due to Kostant. The proof of the Main Theorem is based on Macdonald's identity for $A_l^{(a)}$ and on the properties of a bijection between $t$-cores and integer vectors constructed by Garvan, Kim and Stanton."}
{"category": "Math", "title": "Special Kahler Metrics on Complex Line Bundles and the Geometry of $K3$-Surfaces", "abstract": "We construct metrics with the holonomy group SU(2) on the tangent bundles of weighted complex projective lines and give a geometric description of the moduli space of special Kahler metrics on a K3-surface in the neighborhood of the flat orbifold $T^4/Z_3$."}
{"category": "Math", "title": "Elliptic curve configurations on Fano surfaces", "abstract": "The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces."}
{"category": "Math", "title": "Bipotentials for non monotone multivalued operators: fundamental results and applications", "abstract": "This is a survey of recent results about bipotentials representing multivalued operators. The notion of bipotential is based on an extension of Fenchel's inequality, with several interesting applications related to non associated constitutive laws in non smooth mechanics, such as Coulomb frictional contact or non-associated Drucker-Prager model in plasticity. Relations betweeen bipotentials and Fitzpatrick functions are described. Selfdual lagrangians, introduced and studied by Ghoussoub, can be seen as bipotentials representing maximal monotone operators. We show that bipotentials can represent some monotone but not maximal operators, as well as non monotone operators. Further we describe results concerning the construction of a bipotential which represents a given non monotone operator, by using convex lagrangian covers or bipotential convex covers."}
{"category": "Math", "title": "Maximal Commutative Subalgebras Invariant for CP-Maps: (Counter-)Examples", "abstract": "We solve, mainly by counterexamples, many natural questions regarding maximal commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a von Neumann algebra. In particular, we discuss the structure of the generators of norm continuous semigroups on B(G) leaving a maximal commutative subalgebra invariant and show that there exists Markov CP-semigroups on M_d without invariant maximal commutative subalgebras for any d>2."}
{"category": "Math", "title": "Small deviations of stable processes and entropy of the associated random operators", "abstract": "We investigate the relation between the small deviation problem for a symmetric $\\alpha$-stable random vector in a Banach space and the metric entropy properties of the operator generating it. This generalizes former results due to Li and Linde and to Aurzada. It is shown that this problem is related to the study of the entropy numbers of a certain random operator. In some cases, an interesting gap appears between the entropy of the original operator and that of the random operator generated by it. This phenomenon is studied thoroughly for diagonal operators. Basic ingredients here are techniques related to random partitions of the integers. The main result concerning metric entropy and small deviations allows us to determine or provide new estimates for the small deviation rate for several symmetric $\\alpha$-stable random processes, including unbounded Riemann--Liouville processes, weighted Riemann--Liouville processes and the ($d$-dimensional) $\\alpha$-stable sheet."}
{"category": "Math", "title": "A stochastic fixed point equation for weighted minima and maxima", "abstract": "Given any finite or countable collection of real numbers $T_j,j\\in J$, we find all solutions $F$ to the stochastic fixed point equation \\[W\\stackrel{\\mathrm {d}}{=}\\inf_{j\\in J}T_jW_j,\\] where $W$ and the $W_j,j\\in J$, are independent real-valued random variables with distribution $F$ and $\\stackrel{\\mathrm {d}}{=}$ means equality in distribution. The bulk of the necessary analysis is spent on the case when $|J|\\geq 2$ and all $T_j$ are (strictly) positive. Nontrivial solutions are then concentrated on either the positive or negative half line. In the most interesting (and difficult) situation $T$ has a characteristic exponent $\\alpha$ given by $\\sum_{j\\in J}T_j^{\\alpha}=1$ and the set of solutions depends on the closed multiplicative subgroup of $\\mathbb {R}^{>}=(0,\\infty)$ generated by the $T_j$ which is either $\\{1\\}$, $\\mathbb {R}^{>}$ itself or $r^{\\mathbb {Z}}=\\{r^n\\dvt n\\in \\mathbb {Z}\\}$ for some $r>1$. The first case being trivial, the nontrivial fixed points in the second case are either Weibull distributions or their reciprocal reflections to the negative half line (when represented by random variables), while in the third case further periodic solutions arise. Our analysis builds on the observation that the logarithmic survival function of any fixed point is harmonic with respect to $\\varLambda =\\sum_{j\\geq 1}\\delta_{T_j}$, i.e. $\\varGamma =\\varGamma \\star \\varLambda$, where $\\star$ means multiplicative convolution. This will enable us to apply the powerful Choquet--Deny theorem."}
{"category": "Math", "title": "On Local Models with Special Parahoric Level Structure", "abstract": "We consider the local model of a Shimura variety of PEL type, with the unitary similitudes corresponding to a ramified quadratic extension of $\\mathbb{Q}_p$ as defining group. We examine the cases where the level structure at $p$ is given by a parahoric that is the stabilizer of a selfdual periodic lattice chain and that is special in the sense of Bruhat--Tits theory. We prove that in these cases the special fiber of the local model is irreducible and generically reduced; consequently, the special fiber is reduced and is normal, Frobenius split, and with only rational singularities. In addition, we show that in these cases the local model contains an open subset that is isomorphic to affine space."}
{"category": "Math", "title": "On multifractality and time subordination for continuous functions", "abstract": "We show that if $Z$ is \"homogeneously multifractal\" (in a sense we precisely define), then $Z$ is the composition of a monofractal function $g$ with a time subordinator $f$ (i.e. $f$ is the integral of a positive Borel measure supported by $\\zu$). When the initial function $Z$ is given, the monofractality exponent of the associated function $g$ is uniquely determined. We study in details a classical example of multifractal functions $Z$, for which we exhibit the associated functions $g$ and $f$. This provides new insights into the understanding of multifractal behaviors of functions."}
{"category": "Math", "title": "Multivariate normal approximation using Stein's method and Malliavin calculus", "abstract": "We combine Stein's method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for fields subordinated to a fractional Brownian motion."}
{"category": "Math", "title": "Filtering the fiber of the pinch map", "abstract": "A new type of Hopf invariant is described for the fiber of the pinch map from the mapping cone of a map from A to X onto to the suspension of A; this is then used to study the boundary map in the fibration sequence of Cohen, Moore and Neisendorfer in the case that the mapping cone is an odd dimensional Moore space. The components of the boundary map are then shown to be compatible with Hopf invariants and a filtered splitting of the loops on the fiber is obtained."}
{"category": "Math", "title": "On Probabilistic Parametric Inference", "abstract": "An objective operational theory of probabilistic parametric inference is formulated without invoking the so-called non-informative prior probability distributions."}
{"category": "Math", "title": "Intersection homology with field coefficients: $K$-Witt spaces and $K$-Witt bordism", "abstract": "We construct geometric examples of pseudomanifolds that satisfy the Witt condition for intersection homology Poincare duality with respect to certain fields but not others. We also compute the bordism theory of $K$-Witt spaces for an arbitrary field $K$, extending results of Siegel for $K=Q$."}
{"category": "Math", "title": "Variations on Descents and Inversions in Permutations", "abstract": "We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation sigma = sigma_1sigma_2...sigma_n defined as the set of indices i such that either i is odd and sigma_i > sigma_{i+1}, or i is even and sigma_i < sigma_{i+1}. We show that this statistic is equidistributed with the 3-descent set statistic on permutations sigma = sigma_1sigma_2...sigma_{n+1} with sigma_1 = 1, defined to be the set of indices i such that the triple sigma_i sigma_{i+1} sigma_{i+2} forms an odd permutation of size 3. We then introduce Mahonian inversion statistics corresponding to the two new variations of descents and show that the joint distributions of the resulting descent-inversion pairs are the same. We examine the generating functions involving alternating Eulerian polynomials, defined by analogy with the classical Eulerian polynomials sum_{sigma in S_n} t^{des(sigma)+1} using alternating descents. For the alternating descent set statistic, we define the generating polynomial in two non-commutative variables by analogy with the ab-index of the Boolean algebra B_n, and make observations about it. By looking at the number of alternating inversions in alternating (down-up) permutations, we obtain a new q-analog of the Euler number E_n and show how it emerges in a q-analog of an identity expressing E_n as a weighted sum of Dyck paths."}
{"category": "Math", "title": "Unitarizable minimal principal series of reductive groups", "abstract": "The aim of this paper is to give an exposition of recent progress on the determination of the unitarizable Langlands quotients of minimal principal series for reductive groups over the real or p-adic fields in characteristic 0."}
{"category": "Math", "title": "On the global construction of modules over Fedosov deformation quantization algebra", "abstract": "Let $(M,\\omega)$ be a symplectic manifold, $\\mathcal{D}\\subset TM$ a real polarization on $M$ and $\\wp$ a leaf of $\\mathcal{D}$. We construct a Fedosov-type star-product $\\ast_L$ on $M$ such that $C^\\infty (\\wp)[[h]]$ has a natural structure of left module over the deformed algebra $(C^\\infty (M)[[h]], \\ast_L)$. This generalizes the results of 0708.2626."}
{"category": "Math", "title": "On some crystalline representations of $GL_2(Q_p)$", "abstract": "We show that the universal unitary completion of certain locally algebraic representation of $G:=\\GL_2(\\Qp)$ with $p>2$ is non-zero, topologically irreducible, admissible and corresponds to a 2-dimensional crystalline representation with non-semisimple Frobenius via the $p$-adic Langlands correspondence for $G$."}
{"category": "Math", "title": "Conley index and stable sets for flows on flag bundles", "abstract": "Consider a continuous flow of automorphisms of a G-principal bundle which is chain transitive on its compact Hausdorff base. Here G is a connected noncompact semi-simple Lie group with finite center. The finest Morse decomposition of the induced flows on the associated flag bundles were obtained in previous articles. Here we describe the stable sets of these Morse components and, under an additional assumption, their Conley indices."}
{"category": "Math", "title": "Algebras of Almost Periodic Functions with Bohr-Fourier Spectrum in a Semigroup: Hermite Property and its Applications", "abstract": "It is proved that the unital Banach algebra of almost periodic functions of several variables with Bohr-Fourier spectrum in a given additive semigroup is an Hermite ring. The same property holds for the Wiener algebra of functions that in addition have absolutely convergent Bohr-Fourier series. As applications of the Hermite property of these algebras, we study factorizations of Wiener--Hopf type of rectangular matrix functions and the Toeplitz corona problem in the context of almost periodic functions of several variables."}
{"category": "Math", "title": "Characterization of unitary processes with independent and stationary increments", "abstract": "This is a continuation of the earlier work \\cite{SSS} to characterize stationary unitary increment Gaussian processes. The earlier assumption of uniform continuity is replaced by weak continuity and with a technical assumption on the domain of the generator, unitary equivalence of the processes to the solution of Hudson-Parthasarathy equation is proved."}
{"category": "Math", "title": "The $(L^{1},L^{1})$ bilinear Hardy-Littlewood function and Furstenberg averages", "abstract": "Let $(X,\\mathcal{B}, \\mu, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $\\dis R^*:(f, g) \\in L^1\\times L^1 \\to R^*(f, g)(x) = \\sup_{n} \\frac{f(T^nx)g(T^{2n}x)}{n}.$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy--Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\\in L^1\\times L^1.$ The Furstenberg averages do not converge for all pairs of $(L^{1},L^{1})$ functions, while by a result of J. Bourgain these averages converge for all pairs of $(L^{p},L^{q})$ functions with $\\frac{1}{p}+\\frac{1}{q}\\leq 1.$"}
{"category": "Math", "title": "Nonhomeomorphic conjugates of connected Shimura varieties", "abstract": "We show that conjugation by an automorphism of the complex numbers (as an abstract field) may change the topological fundamental group of a locally symmetric variety over C. As a consequence, we obtain a large class of algebraic varieties defined over number fields with the property that different embeddings of the number field into C give complex varieties with nonisomorphic fundamental groups."}
{"category": "Math", "title": "The Hausdorff dimension of self-affine Sierpinski sponges", "abstract": "We compute the Hausdorff dimension of limit sets generated by 3-dimensional self-affine mappings with diagonal matrices of the form A_{ijk}=Diag(a_{ijk}, b_{ij}, c_{i}), where 0<a_{ijk}\\le b_{ij}\\le c_i<1. By doing so we show that the variational principle for the dimension holds for this class."}
{"category": "Math", "title": "Various considerations on hypergeometric series", "abstract": "E661 in the Enestrom index. This was originally published as \"Variae considerationes circa series hypergeometricas\" (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma function. He looks at the relations between some infinite products and integrals. He takes the logarithm of these infinite products, and expands these using the Euler-Maclaurin summation formula. In section 14, Euler seems to be rederiving some of the results he already proved in the paper. However I do not see how these derivations are different. If any readers think they understand please I would appreciate it if you could email me. I am presently examining Euler's work on analytic number theory. The two main topics I want to understand are the analytic continuation of analytic functions and the connection to divergent series, and the asymptotic behavior of the Gamma function."}
{"category": "Math", "title": "The character tables of centralizers in Weyl Groups of $E_6$, $E_7$, $F_4$, $G_2$", "abstract": "To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_6$, $E_7$, $F_4$, $G_2$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP."}
{"category": "Math", "title": "Stein--Sahi complementary series and their degenerations", "abstract": "The aim of the paper is an introduction to Stein--Sahi complementary series, holomorphic series, and 'unipotent representations'. We also discuss some open problems related to these objects. For the sake of simplicity, we consider only the groups U(n,n)."}
{"category": "Math", "title": "The character tables of centralizers in Weyl Group of $E_8$ I", "abstract": "To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 1-- 28."}
{"category": "Math", "title": "Milnor-Wood inequalities for manifolds locally isometric to a product of hyperbolic planes", "abstract": "This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to many other locally symmetric spaces they do admit flat vector bundle of the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles are determined completely by the sign of the Euler number."}
{"category": "Math", "title": "Intersection of subgroups in free groups and homotopy groups", "abstract": "We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group $\\pi_3$. This generalizes a result of Gutierrez-Ratcliffe who relate the intersection of two subgroups with the computation of $\\pi_2$. Let $K$ be a two-dimensional CW-complex with subcomplexes $K_1,K_2,K_3$ such that $K=K_1\\cup K_2\\cup K_3$ and $K_1\\cap K_2\\cap K_3$ is the 1-skeleton $K^1$ of $K$. We construct a natural homomorphism of $\\pi_1(K)$-modules $$ \\pi_3(K)\\to \\frac{R_1\\cap R_2\\cap R_3}{[R_1,R_2\\cap R_3][R_2,R_3\\cap R_1][R_3,R_1\\cap R_2]}, $$ where $R_i=ker\\{\\pi_1(K^1)\\to \\pi_1(K_i)\\}, i=1,2,3$ and the action of $\\pi_1(K)=F/R_1R_2R_3$ on the right hand abelian group is defined via conjugation in $F$. In certain cases, the defined map is an isomorphism. Finally, we discuss certain applications of the above map to group homology."}
{"category": "Math", "title": "Homotopy types of reduced 2-nilpotent simplicial groups", "abstract": "We classify the homotopy types of reduced 2-nilpotent simplicial groups in terms of the homology an d boundary invariants $b,\\beta$. This contains as special cases results of J.H.C. Whitehead on 1-connected 4-dimensional complexes and of Quillen on reduced 2-nilpotent rational simplicial groups. Moreover it yields for 1-nilpotent (or abelian) simplicial groups a classification due to Dold-Kan. Our result describes a new natural structure of the integral homology of any simply connected space. We also classify the homotopy types of connective spectra in the category of 2-nilpotent simplicial groups. Moreover we compute homotopy groups of spheres in the category of $m$-nilpotent groups for $m=2,3$ and partially for $m=4,5$."}
{"category": "Math", "title": "The character tables of centralizers in Weyl Group of $E_8$ II", "abstract": "To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 29--46."}
{"category": "Math", "title": "The character tables of centralizers in Weyl Group of $E_8$ III", "abstract": "To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 47--64."}
{"category": "Math", "title": "The character tables of centralizers in Weyl Group of $E_8$ IV", "abstract": "To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 65--94."}
{"category": "Math", "title": "The character tables of centralizers in Weyl Group of $E_8$ V", "abstract": "To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_8$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 95--112."}
{"category": "Math", "title": "Rouquier blocks of the cyclotomic Ariki-Koike algebras", "abstract": "The definition of Rouquier for the families of characters of Weyl groups in terms of blocks of the associated Iwahori-Hecke algebra has made possible the generalization of this notion to the complex reflection groups. Here we give an algorithm for the determination of the \"Rouquier blocks\" of the cyclotomic Ariki-Koike algebras."}
{"category": "Math", "title": "A multiplication formula for module subcategories of Ext-symmetry", "abstract": "We define evaluation forms associated to objects in a module subcategory of Ext-symmetry generated by finitely many simple modules over a path algebra with relations and prove a multiplication formula for the product of two evaluation forms. It is analogous to a multiplication formula for the product of two evaluation forms associated to modules over a preprojective algebra given by Geiss, Leclerc and Schr\\\"oer in \\cite{GLS2006}."}
{"category": "Math", "title": "Green formula in Hall algebras and cluster algebras", "abstract": "The objective of the present paper is to give a survey of recent progress on applications of the approaches of Ringel-Hall type algebras to quantum groups and cluster algebras via various forms of Green's formula. In this paper, three forms of Green's formula are highlighted, (1) the original form of Green's formula \\cite{Green}\\cite{RingelGreen}, (2) the degeneration form of Green's formula \\cite{DXX} and (3) the projective form of Green's formula \\cite{XX2007a} i.e. Green formula with a $\\bbc^{*}$-action."}
{"category": "Math", "title": "On a Class of Polynomials with Integer Coefficients", "abstract": "A class P_{n,m,p}(x) of polynomials is defined. The combinatorial meaning of its coefficients is given. Chebyshev polynomials are the special cases of P_{n,m,p}(x). It is first shown that P_{n,m,p}(x) may be expressed in terms of P_{n,0,p}(x). From this we derive that P_{n,2,2}(x) may be obtain in terms of trigonometric functions, from which we obtain some of its important properties. Some questions about orthogonality are also concerned. Furthermore, it is shown that P_{n,2,2}(x) fulfills the same three terms recurrence as Chebyshev polynomials. Some others recurrences for P_{n,m,p}(x) and its coefficients are also obtained. At the end a formula for coefficients of Chebyshev polynomials of the second kind is derived."}
{"category": "Math", "title": "Stationary distributions for diffusions with inert drift", "abstract": "Consider a reflecting diffusion in a domain in $R^d$ that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting process and the value of the drift vector has a product form. Moreover, the first component is the symmetrizing measure on the domain for the reflecting diffusion without inert drift, and the second component has a Gaussian distribution. We also consider processes where the drift is given in terms of the gradient of a potential."}
{"category": "Math", "title": "On The Criteria Of The F5 Algorithm", "abstract": "Faugere's F5 algorithm is one of the fastest known algorithms for the computation of Grobner bases. So far only the F5 Criterion is proved, whereas the second powerful criterion, the Rewritten Criterion, is not understood very well until now. We give a proof of both, the F5 Criterion and the Rewritten Criterion showing their connection to syzygies, i.e. the relations between the S-Polynomials to be investigated by the algorithm. Using the example of a Grobner basis computation stated in Faugere's F5 paper we show how the criteria work, and discuss the possibility of improving the F5 Criterion. An introduction to a SINGULAR implementation of F5 is given in the end."}
{"category": "Math", "title": "Positivity and the canonical basis of tensor products of finite-dimensional irreducible representations of quantum sl(k)", "abstract": "In a categorification of tensor products of fundamental representations of quantum sl(k) via highest weight categories, the indecomposable tilting modules descend to the canonical basis. Since projective functors map tilting modules to tilting modules, the coefficients of the canonical basis of tensor products of finite dimensional, irreducible representations under the action of the Chevalley generators are positive."}
{"category": "Math", "title": "On the representations of integers by the sextenary quadratic form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2 and 7-cores", "abstract": "In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x^2+y^2+z^2+ 7s^2+7t^2+ 7u^2. We establish the following intriguing inequalities 2b(n)>=a_7(n)>=b(n) for n not equal to 0,2,6,16. Here a_7(n) is the number of partitions of n that are 7-cores and b(n) is the number of representations of n+2 by the sextenary form (x ^2+ y ^2+z ^2+ 7s ^2 + 7t ^2+ 7u^2)/8 with x,y,z,s,t and u being odd."}
{"category": "Math", "title": "Modular representations of reductive groups and geometry of affine Grassmannians", "abstract": "By the geometric Satake isomorphism of Mirkovic and Vilonen, decomposition numbers for reductive groups can be interpreted as decomposition numbers for equivariant perverse sheaves on the complex affine Grassmannian of the Langlands dual group. Using a description of the minimal degenerations of the affine Grassmannian obtained by Malkin, Ostrik and Vybornov, we are able to recover geometrically some decomposition numbers for reductive groups. In the other direction, we can use some decomposition numbers for reductive groups to prove geometric results, such as a new proof of non-smoothness results, and a proof that some singularities are not equivalent (a conjecture of Malkin, Ostrik and Vybornov). We also give counterexamples to a conjecture of Mirkovic and Vilonen stating that the stalks of standard perverse sheaves over the integers on the affine Grassmannian are torsion-free, and propose a modified conjecture, excluding bad primes."}
{"category": "Math", "title": "The classification on simple Moufang loops", "abstract": "Let $C(F)$ be a matrix Cayley-Dickson algebra over field $F$. By $M_0(F)$ we denote the loop containing of all elements of algebra $C(F)$ with norm 1. It is shown in this paper that with precision till isomorphism the loops $M_0(F)/<-1>$ they and only they are simple non-associative Moufang loops, where $F$ are subfields of algebraic closed field"}
{"category": "Math", "title": "About the embedding of Moufang loops in alternative algebras II", "abstract": "It is known that with precision till isomorphism that only and only loops $M(F) = M_0(F)/<-1>$, where $M_0(F)$ denotes the loop, consisting from elements of all matrix Cayley-Dickson algebra $C(F)$ with norm 1, and $F$ be a subfield of arbitrary fixed algebraically closed field, are simple non-associative Moufang loops. In this paper it is proved that the simple loops $M(F)$ they and only they are not embedded into a loops of invertible elements of any unitaly alternative algebras if $\\text{char} F \\neq 2$ and $F$ is closed under square root operation. For the remaining Moufang loops such an embedding is possible. Using this embedding it is quite simple to prove the well-known finding: the finite Moufang $p$-loop is centrally nilpotent."}
{"category": "Math", "title": "Stochastic chains with memory of variable length", "abstract": "Stochastic chains with memory of variable length constitute an interesting family of stochastic chains of infinite order on a finite alphabet. The idea is that for each past, only a finite suffix of the past, called context, is enough to predict the next symbol. These models were first introduced in the information theory literature by Rissanen (1983) as a universal tool to perform data compression. Recently, they have been used to model up scientific data in areas as different as biology, linguistics and music. This paper presents a personal introductory guide to this class of models focusing on the algorithm Context and its rate of convergence."}
{"category": "Math", "title": "A symmetric version of Kontsevich graph complex and Leibniz homology", "abstract": "Kontsevich has proven that the Lie homology of the Lie algebra of symplectic vector fields can be computed in terms of the homology of a graph complex. We prove that the Leibniz homology of this Lie algebra can be computed in terms of the homology of a variant of the graph complex endowed with an action of the symmetric groups. The resulting isomorphism is shown to be a Zinbiel-associative bialgebra isomorphism."}
{"category": "Math", "title": "Higher dimensional Hermitian Gray manifolds", "abstract": "The aim of this paper is to describe a large class of Hermitian Gray manifolds."}
{"category": "Math", "title": "Automorphisms of elliptic Poisson algebras", "abstract": "We describe the automorphism groups of elliptic Poisson algebras on polynomial algebras in three variables and give an explicit set of generators and defining relations for this group."}
{"category": "Math", "title": "Examples of relative deformation spaces that are not locally connected", "abstract": "We provide an infinite family of pared manifolds whose relative deformation spaces of hyperbolic structures on these manifolds are not locally connected. This is a natural extension of the recent result of Bromberg that shows the space of Kleinian punctured torus groups is not locally connected."}
{"category": "Math", "title": "Kreps-Yan theorem for Banach ideal spaces", "abstract": "Let $C$ be a closed convex cone in a Banach ideal space $X$ on a measurable space with a $\\sigma$-finite measure. We prove that conditions $C\\cap X_+=\\{0\\}$ and $C\\supset -X_+$ imply the existence of a strictly positive continuous functional on $X$, whose restriction to $C$ is non-positive."}
{"category": "Math", "title": "Continuous Families of Rational Surface Automorphisms with Positive Entropy", "abstract": "We construct k-parameter families of rational surface automorphisms for any k. These are automorphisms of surfaces X, which are constructed from iterated blowups over the projective plane. In certain cases: we are able to determine the exact automorphism group of X, as well as when two of the surfaces X are inequivalent."}
{"category": "Math", "title": "A diagrammatic approach to categorification of quantum groups II", "abstract": "We categorify one-half of the quantum group associated to an arbitrary Cartan datum."}
{"category": "Math", "title": "On regularity in codimension one of irreducible components of module varieties", "abstract": "Let A be a tame quasi-tilted algebra and d the dimension vector of an indecomposable A-module. In the paper we prove that each irreducible component of the variety of A-modules of dimension vector d is regular in codimension one."}
{"category": "Math", "title": "A new proof for the multiplicative property of the boolean cumulants with applications to operator-valued case", "abstract": "The paper presents several combinatorial properties of the boolean cumulants. A corollary is a new proof of the multiplicative property of the boolean cumulant series that can be easily adapted for the case of boolean independence with amalgamation over an algebra."}
{"category": "Math", "title": "Composition-Diamond Lemma for Tensor Product of Free Algebras", "abstract": "In this paper, we establish Composition-Diamond lemma for tensor product $k< X> \\otimes k< Y>$ of two free algebras over a field. As an application, we construct a Groebner-Shirshov basis in $k< X> \\otimes k< Y>$ by lifting a Groebner-Shirshov basis in $k[X] \\otimes k< Y>$, where $k[X]$ is a commutative algebra."}
{"category": "Math", "title": "The One-Dimenshional Inverse Wave Spectral Problem with Discontinuous Wave Speed", "abstract": "The inverse problem for the Sturm- Liouville operator with complex periodic potential and positive discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is studied. We give formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem"}
{"category": "Math", "title": "Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations", "abstract": "In this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls, with a proof that works equally well in a more general equivariant setting. For a non-equidimensional surjective holomorphic map between compact ball quotients, our method applies to show that the set of critical values must be nonempty and of codimension 1. In the equivariant setting the line of arguments extend to holomorphic mappings of maximal rank into the complex projective space or the complex Euclidean space, yielding in the latter case a lower estimate on the dimension of the singular locus of certain holomorphic maps defined by integrating holomorphic 1-forms. In another direction, we extend the nonexistence statement on holomorphic submersions to the case of ball quotients of finite volume, provided that the target complex unit ball is of dimension m>=2, giving in particular a new proof that a local biholomorphism between noncompact m-ball quotients of finite volume must be a covering map whenever m>=2. Finally, combining our results with Hermitian metric rigidity, we show that any holomorphic submersion from a bounded symmetric domain into a complex unit ball equivariant with respect to a lattice must factor through a canonical projection to yield an automorphism of the complex unit ball, provided that either the lattice is cocompact or the ball is of dimension at least 2."}
{"category": "Math", "title": "Hyperbolic lattice-point counting and modular symbols", "abstract": "For a cocompact group $\\G$ of $\\slr$ we fix a real non-zero harmonic 1-form $\\alpha$. We study the asymptotics of the hyperbolic lattice-counting problem for $\\G$ under restrictions imposed by the modular symbols $\\modsym{\\gamma}{\\a}$. We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution."}
{"category": "Math", "title": "$G_2$-Holonomy Metrics Connected with a 3-Sasakian Manifold", "abstract": "We construct complete noncompact Riemannian metrics with $G_2$-holonomy on noncompact orbifolds that are $\\Bbb R^3$-bundles with the twistor space $\\mathcal{Z}$ as a spherical fiber."}
{"category": "Math", "title": "Geometry of neutral metrics in dimesnion four", "abstract": "The purpose of this article is to review some recent results on the geometry of neutral signature metrics in dimension four and their twistor spaces. The following topics are considered: Neutral K\\\"ahler and hyperk\\\"ahler surfaces, Walker metrics, Neutral anti-self-dual 4-manifolds and projective structures, Twistor spaces of neutral metrics."}
{"category": "Math", "title": "Representing elementary semi-algebraic sets by a few polynomial inequalities: A constructive approach", "abstract": "Let P be an elementary closed semi-algebraic set in R^d, i.e., there exist real polynomials p_1,...,p_s such that P= \\{x \\in R^d : p_1(x) \\ge 0, >..., p_s(x) \\ge 0 \\}; in this case p_1,...,p_s are said to represent P. Denote by $n$ the maximal number of the polynomials from \\{p_1,...,p_s\\} that vanish in a point of P. If P is non-empty and bounded, we show that it is possible to construct n+1 polynomials representing P. Furthermore, the number n+1 can be reduced to n in the case when the set of points of P in which n polynomials from \\{p_1,...,p_s\\} vanish is finite. Analogous statements are also obtained for elementary open semi-algebraic sets."}
{"category": "Math", "title": "A General Reciprocity Law on arbitrary Vector Spaces", "abstract": "The paper contained a preliminary version of a general theory of reciprocity laws on vector spaces."}
{"category": "Math", "title": "A classification of mahonian maj-inv statistics", "abstract": "Two well known mahonian statistics on words are the inversion number and the major index. In 1996, Foata and Zeilberger introduced generalizations, parameterized by relations, of these statistics. In this paper, we study the statistics which can be written as a sum of these generalized statistics. This leads to generalizations of some classical results. In particular, we characterize all such statistics which are mahonian."}
{"category": "Math", "title": "Combinatorial representations of Coxeter groups over a field of two elements", "abstract": "Let $W$ denote a simply-laced Coxeter group with $n$ generators. We construct an $n$-dimensional representation $\\phi$ of $W$ over the finite field $F_2$ of two elements. The action of $\\phi(W)$ on $F_2^n$ by left multiplication is corresponding to a combinatorial structure extracted and generalized from Vogan diagrams. In each case W of types A, D and E, we determine the orbits of $F_2^n$ under the action of $\\phi(W)$, and find that the kernel of $\\phi$ is the center $Z(W)$ of $W.$"}
{"category": "Math", "title": "On Walkup's class ${\\cal K}(d)$ and a minimal triangulation of $(S^3 \\times \\rotatebox{90}{\\ltimes} S^1)^{\\#3}$", "abstract": "For $d \\geq 2$, Walkup's class ${\\cal K}(d)$ consists of the $d$-dimensional simplicial complexes all whose vertex-links are stacked $(d-1)$-spheres. Kalai showed that for $d \\geq 4$, all connected members of ${\\cal K}(d)$ are obtained from stacked $d$-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold $X$ with Euler characteristic $\\chi$ satisfies $f_1 \\geq 5f_0 - {15/2} \\chi$, with equality only for $X \\in {\\cal K}(4)$. K\\\"{u}hnel observed that this implies $f_0(f_0 - 11) \\geq -15\\chi$, with equality only for 2-neighborly members of ${\\cal K}(4)$. K\\\"{u}hnel also asked if there is a triangulated 4-manifold with $f_0 = 15$, $\\chi = -4$ (attaining equality in his lower bound). In this paper, guided by Kalai's theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product $S^3 \\times {-2.8mm}_{-} S^1$. Because of K\\\"{u}hnel's inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of $(2d+ 3)$-vertex sphere products $S^{d-1} \\times S^1$ (twisted for $d$ odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai's result."}
{"category": "Math", "title": "Simplicial Descent Categories", "abstract": "Much of the homotopical and homological structure of the categories of chain complexes and topological spaces can be deduced from the existence and properties of the 'simple' functors Tot : {double chain complexes} -> {chain complexes} and geometric realization : {sSets} -> {Top}, or similarly, Tot : {simplicial chain complexes} -> {chain complexes} and | | : {sTop} -> {Top}. The purpose of this thesis is to abstract this situation, and to this end we introduce the notion of '(co)simplicial descent category'. It is inspired by Guillen-Navarros's '(cubical) descent categories'. The key ingredients in a (co)simplicial descent category D are a class E of morphisms in D, called equivalences, and a 'simple' functor s : {(co)simplicial objects in D} -> D. They must satisfy axioms like 'Eilenberg-Zilber', 'exactness' and 'acyclicity'. This notion covers a wide class of examples, as chain complexes, sSets, topological spaces, filtered cochain complexes (where E = filtered quasi-isomorphisms or E = E_2-isomorphisms), commutative differential graded algebras (with s = Navarro's Thom-Whitney simple), DG-modules over a DG-category and mixed Hodge complexes, where s = Deligne's simple. From the simplicial descent structure we obtain homotopical structure on D, as cone and cylinder objects. We use them to i) explicitly describe the morphisms of HoD=D[E^{-1}] similarly to the case of calculus of fractions; ii) endow HoD with a non-additive pre-triangulated structure, that becomes triangulated in the stable additive case. These results use the properties of a 'total functor', which associates to any biaugmented bisimplicial object a simplicial object. It is the simplicial analogue of the total chain complex of a double complex, and it is left adjoint to Illusie's 'decalage' functor."}
{"category": "Math", "title": "A Characterization of Semisimple Plane Polynomial Automorphisms", "abstract": "It is well-known that an element of the linear group ${\\rm GL}_n(\\C)$ is semisimple if and only if its conjugacy class is Zariski closed. The aim of this paper is to show that the same result holds for the group of complex plane polynomial automorphisms."}
{"category": "Math", "title": "Local conditions for global representations of quadratic forms", "abstract": "We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form."}
{"category": "Math", "title": "Non-simple abelian varieties in a family: geometric and analytic approaches", "abstract": "Let $A_t$ be a family of abelian varieties over a number field $k$ parametrized by a rational coordinate $t$, and suppose the generic fiber of $A_t$ is geometrically simple. For example, we may take $A_t$ to be the Jacobian of the hyperelliptic curve $y^2 = f(x)(x-t)$ for some polynomial $f$. We give two upper bounds for the number of $t \\in k$ of height at most $B$ such that the fiber $A_t$ is geometrically non-simple. One bound comes from arithmetic geometry, and shows that there are only finitely many such $t$; but one has very little control over how this finite number varies as $f$ changes. Another bound, from analytic number theory, shows that the number of geometrically non-simple fibers grows quite slowly with $B$; this bound, by contrast with the arithmetic one, is effective, and is uniform in the coefficients of $f$. We hope that the paper, besides proving the particular theorems we address, will serve as a good example of the strengths and weaknesses of the two complementary approaches."}
{"category": "Math", "title": "Subsystems of Fock Need Not Be Fock: Spatial CP-Semigroups", "abstract": "We show that a product subsystem of a time ordered system (that is, a product system of time ordered Fock modules), though type I, need not be isomorphic to a time ordered product system. In that way, we answer an open problem in the classification of CP-semigroups by product systems. We define spatial strongly continuous CP-semigroups on a unital C*-algebra and characterize them as those that have a Christensen-Evans generator."}
{"category": "Math", "title": "On Periodic solutions for a reduction of Benney chain", "abstract": "We study periodic solutions for a quasi-linear system, which is the so called dispersionless Lax reduction of the Benney moments chain. This question naturally arises in search of integrable Hamiltonian systems of the form $ H=p^2/2+u(q,t) $ Our main result classifies completely periodic solutions for 3 by 3 system. We prove that the only periodic solutions have the form of traveling waves, so in particular, the potential $u$ is a function of a linear combination of $t$ and $q$. This result implies that the there are no nontrivial cases of existence of the fourth power integral of motion for $H$: if it exists, then it is equal necessarily to the square of the quadratic one. Our method uses two new general observations. The first is the genuine non-linearity of the maximal and minimal eigenvalues for the system. The second observation uses the compatibility conditions of Gibonns-Tsarev in order to give certain exactness for the system in Riemann invariants. This exactness opens a possibility to apply the Lax analysis of blow up of smooth solutions, which usually does not work for systems of higher order."}
{"category": "Math", "title": "Noncanonical number systems in the integers", "abstract": "The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when we choose a digit set that does not contain 0. We prove that such digit sets exist and we provide infinitely many examples for every base b with |b|\\ge 4, and for b=-2. For the special case b=-2, we give a full characterisation of all valid digit sets."}
{"category": "Math", "title": "Parapuzzle of the Multibrot set and typical dynamics of unimodal maps", "abstract": "We study the parameter space of unicritical polynomials $f_c:z\\mapsto z^d+c$. For complex parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic, or Collet-Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the ``principal nest'' of parapuzzle pieces."}
{"category": "Math", "title": "Refinement Equations and Spline Functions", "abstract": "In this paper, we exploit the relation between the regularity of refinable functions with non-integer dilations and the distribution of powers of a fixed number modulo 1, and show the nonexistence of a non-trivial {\\bf C}^{\\infty} solution of the refinement equation with non-integer dilations. Using this, we extend the results on the refinable splines with non-integer dilations and construct a counterexample to some conjecture concerning the refinable splines with non-integer dilations. Finally, we study the box splines satisfying the refinement equation with non-integer dilation and translations. Our study involves techniques from number theory and harmonic analysis."}
{"category": "Math", "title": "Multipoint Pad\\'e Approximants to Complex Cauchy Transforms with Polar Singularities", "abstract": "We study diagonal multipoint Pad\\'e approximants to sums of a Cauchy transform of a complex measure and a rational function. The measure is assumed to have compact regular support included into the real line and an argument of bounded variation on the support. For interpolation sets whose normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution, we show that the counting measures of poles of the approximants converge to the balayage of that distribution onto the support of the measure, in the weak-star sense, that the approximants themselves converge in capacity to the approximated function outside the support of the measure, and that the poles of the additional rational function attract at least as many poles of the approximants as their multiplicity and not much more."}
{"category": "Math", "title": "Stone-Weierstrass type theorems for large deviations", "abstract": "We give a general version of Bryc's theorem valid on any topological space and with any algebra $\\mathcal{A}$ of real-valued continuous functions separating the points, or any well-separating class. In absence of exponential tightness, and when the underlying space is locally compact regular and $\\mathcal{A}$ constituted by functions vanishing at infinity, we give a sufficient condition on the functional $\\Lambda(\\cdot)_{\\mid \\mathcal{A}}$ to get large deviations with not necessarily tight rate function. We obtain the general variational form of any rate function on a completely regular space; when either exponential tightness holds or the space is locally compact Hausdorff, we get it in terms of any algebra as above. Prohorov-type theorems are generalized to any space, and when it is locally compact regular the exponential tightness can be replaced by a (strictly weaker) condition on $\\Lambda(\\cdot)_{\\mid \\mathcal{A}}$."}
{"category": "Math", "title": "Linearity conditions on the Jacobian ideal and logarithmic--meromorphic comparison for free divisors", "abstract": "In this paper we survey the role of D-module theory in the comparison between logarithmic and meromorphic de Rham complexes of integrable logarithmic connections with respect to free divisors, and we present some new linearity conditions on the Jacobian ideal which arise in this setting."}
{"category": "Math", "title": "A note on Todorov surfaces", "abstract": "Let $S$ be a {\\em Todorov surface}, {\\it i.e.}, a minimal smooth surface of general type with $q=0$ and $p_g=1$ having an involution $i$ such that $S/i$ is birational to a $K3$ surface and such that the bicanonical map of $S$ is composed with $i.$ The main result of this paper is that, if $P$ is the minimal smooth model of $S/i,$ then $P$ is the minimal desingularization of a double cover of $\\mathbb P^2$ ramified over two cubics. Furthermore it is also shown that, given a Todorov surface $S$, it is possible to construct Todorov surfaces $S_j$ with $K^2=1,...,K_S^2-1$ and such that $P$ is also the smooth minimal model of $S_j/i_j,$ where $i_j$ is the involution of $S_j.$ Some examples are also given, namely an example different from the examples presented by Todorov in \\cite{To2}."}
{"category": "Math", "title": "On equations of double planes with $p_g=q=1$", "abstract": "This paper describes how to compute equations of plane models of minimal Du Val double planes of general type with $p_g=q=1$ and $K^2=2,...,8.$ A double plane with $K^2=8$ having bicanonical map not composed with the associated involution is also constructed. The computations are done using the algebra system Magma."}
{"category": "Math", "title": "Counting the Number of Site Swap Juggling Patterns with Respect to Particular Ceilings", "abstract": "Site swap is a mathematical notation used by jugglers to communicate, create and study complex juggling patterns. Determining the number of possible site swap juggling patterns with respect to certain limiting parameters such as number of balls etc., is a problem that has been much studied and solved by many mathematicians. However, when the patterns have a throw height restriction (ceiling) the problem becomes difficult and is in general still open. In this article we derive some formulae for computing the number of possible juggling patterns with respect to certain ceiling types."}
{"category": "Math", "title": "Mean values with cubic characters", "abstract": "We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large-sieve type result for order three (and six) Dirichlet characters."}
{"category": "Math", "title": "Lectures on Lie groups over local fields", "abstract": "These are the lecture notes of a 2-hour mini-course on Lie groups over local fields presented at the \"Workshop on Totally Disconnected Groups, Graphs and Geometry\" at the Heinrich-Fabri-Institut Blaubeuren in May 2007. The goal of the notes is to provide an introduction to p-adic Lie groups and Lie groups over fields of formal Laurent series, with an emphasis on relations to the structure theory of totally disconnected, locally compact groups. In particular, they contain a discussion of the scale, tidy subgroups and contraction groups for automorphisms of Lie groups over local fields. Special attention is paid to the case of Lie groups over local fields of positive characteristic."}
{"category": "Math", "title": "On sections of genus two Lefschetz fibrations", "abstract": "In this note we find new relations in the mapping class group of a genus two surface with n boundary components for n=1,..., 8 which induce a genus two Lefschetz fibration $CP^2#13CP^2bar \\to S^2$ with n disjoint sections. As a consequence, we observe any holomorphic genus 2 Lefschetz fibration without separating singular fibers admits a section."}
{"category": "Math", "title": "Good moduli spaces for Artin stacks", "abstract": "We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford's geometric invariant theory and tame stacks."}
{"category": "Math", "title": "Transitivity of Surface Dynamics Lifted to Abelian Covers", "abstract": "A homeomorphism f of a manifold M is called H_1-transitive if there is a transitive lift of an iterate of f to the universal Abelian cover \\tM. Roughly speaking, this means that f has orbits which repeatedly and densely explore all elements of H_1(M). For a rel pseudo-Anosov map \\phi of a compact surface M we show that the following are equivalent: (a) \\phi is H_1-transitive, (b) the action of \\phi on H_1(M) has spectral radius one, and (c) the lifts of the invariant foliations of \\phi to \\tM have dense leaves. The proof relies on a characterization of transitivity for twisted \\Z^d-extensions of a transitive subshift of finite type."}
{"category": "Math", "title": "On central tendency and dispersion measures for intervals and hypercubes", "abstract": "The uncertainty or the variability of the data may be treated by considering, rather than a single value for each data, the interval of values in which it may fall. This paper studies the derivation of basic description statistics for interval-valued datasets. We propose a geometrical approach in the determination of summary statistics (central tendency and dispersion measures) for interval-valued variables."}
{"category": "Math", "title": "Smooth surfaces with non-simply-connected complements", "abstract": "We give two constructions of surfaces in simply-connected 4-manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author. We also construct, for any group G satisfying some simple conditions, a simply-connected symplectic manifold containing a symplectic surface whose complement has fundamental group G. In each case, we produce infinitely many smoothly inequivalent surfaces that are equivalent up to smooth s-cobordism and hence are topologically equivalent for good groups."}
{"category": "Math", "title": "Dynamics and density evolution in piecewise deterministic growth processes", "abstract": "A new sufficient condition is proved for the existence of stochastic semigroups generated by the sum of two unbounded operators. It is applied to one-dimensional piecewise deterministic Markov processes, where we also discuss the existence of a unique stationary density and give sufficient conditions for asymptotic stability."}
{"category": "Math", "title": "Geometric Structures of Collapsing Riemannian Manifolds I", "abstract": "Let (M^n_i,g_i,p_i) be a sequence of smooth pointed complete n-dimensional Riemannian Manifolds with uniform bounds on the sectional curvatures and let (X,d,p) be a metric space such that (M^n_i,g_i,p_i) -> (X,d,p) in the Gromov-Hausdorff sense. Let O \\subseteq X be the set of points x \\in X such that there exists a neighborhood of x which is isometric to an open set in a Riemannian orbifold and let B = O^c be the complement set. Then we have the sharp estimates dim_Haus(B) \\leq min{n-5, dim_Haus(X)-3}, and further for arbitrary x \\in X we have that x \\in O iff a neighborhood of x has bounded Alexandroff curvature. In particular, if n \\leq 4 then B is empty and (X,d) is a Riemannian orbifold. Our main application is to prove that a collapsed limit of Einstein four manifolds has a smooth Riemannian orbifold structure away from a finite number of points, and that near these points the curvatures has a -dist^{-2} lower bound."}
{"category": "Math", "title": "Number Gossip", "abstract": "This article covers my talk at the Gathering for Gardner 2008, with some additions."}
{"category": "Math", "title": "Isomorphism invariants of restricted enveloping algebras", "abstract": "Let $L$ and $H$ be finite-dimensional restricted Lie algebras over a perfect field $\\F$ such that $u(L)\\cong u(H)$, where $u(L)$ is the restricted enveloping algebra of $L$. We prove that if $L$ is $p$-nilpotent and abelian, then $L\\cong H$. We deduce that if $L$ is abelian and $\\F$ is algebraically closed, then $L\\cong H$. We use these results to prove the main result of this paper stating that if $L$ is $p$-nilpotent, then $L/L'^p+\\gamma_3(L)\\cong H/H'^p+\\gamma_3(H)$."}
{"category": "Math", "title": "Automorphisms of two-dimensional RAAGs and partially symmetric automorphisms of free groups", "abstract": "We compute the virtual cohomological dimension (VCD) of the group of partially symmetric outer automorphisms of a free group. We use this to obtain new upper and lower bounds on the VCD of the outer automorphism group of a two-dimensional right-angled Artin group. In the case of a right-angled group with defining graph a tree, the bounds agree."}
{"category": "Math", "title": "Proving modularity for a given elliptic curve over an imaginary quadratic field", "abstract": "We present an algorithm to determine if the $L$-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor and Berger-Harcos (cf. \\cite{harris-taylor}, \\cite{taylorII} and \\cite{berger-harcos}) we can associate to an automorphic representation a family of compatible $p$-adic representations. Our algorithm is based on Faltings-Serre's method to prove that $p$-adic Galois representations are isomorphic."}
{"category": "Math", "title": "Zeta Functions of Complexes Arising from PGL(3)", "abstract": "In this paper we obtain a closed form expression of the zeta function $Z(X_\\Gamma, u)$ of a finite quotient $X_\\Gamma = \\Gamma \\backslash PGL_3(F)/PGL_3(O_F)$ of the Bruhat-Tits building of $PGL_3$ over a nonarchimedean local field $F$. Analogous to a graph zeta function, $Z(X_\\Gamma, u)$ is a rational function and it satisfies the Riemann hypothesis if and only if $X_\\Gamma$ is a Ramanujan complex."}
{"category": "Math", "title": "Bounds for the loss probability in large loss queueing systems", "abstract": "Let $\\mathcal{G}(\\frak{g}_1,\\frak{g}_2)$ be the class of all probability distribution functions of positive random variables having the given first two moments $\\frak{g}_1$ and $\\frak{g}_2$. Let $G_1(x)$ and $G_2(x)$ be two probability distribution functions of this class satisfying the condition $|G_1(x)-G_2(x)|<\\epsilon$ for some small positive value $\\epsilon$ and let $\\widehat{G}_1(s)$ and, respectively, $\\widehat{G}_2(s)$ denote their Laplace-Stieltjes transforms. For real $\\mu$ satisfying $\\mu\\frak{g}_1>1$ let us denote by $\\gamma_{G_1}$ and $\\gamma_{G_2}$ the least positive roots of the equations $z=\\widehat{G}_1(\\mu-\\mu z)$ and $z=\\widehat{G}_2(\\mu-\\mu z)$ respectively. In the paper, the upper bound for $|\\gamma_{G_1}-\\gamma_{G_2}|$ is derived. This upper bound is then used to find lower and upper bounds for the loss probabilities in different large loss queueing systems."}
{"category": "Math", "title": "Nikol'skii-type inequalities for rearrangement invariant spaces", "abstract": "In this paper we generalize the classical Nikol'skii inequality on the many popular classes pairs of rearrangement invariant (r.i.) spaces and construct some examples in order to show the exactness of our estimations."}
{"category": "Math", "title": "Scaling-sharp dispersive estimates for the Korteweg-de Vries group", "abstract": "We prove weighted estimates on the linear KdV group, which are scaling sharp. This kind of estimates are in the spirit of that used to prove small data scattering for the generalized KdV equations."}
{"category": "Math", "title": "Spectral analysis for one class of second-order indefinite non-self-adjoint differential operator pencil", "abstract": "The inverse problem for the differential operator pencil with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is studied. We give formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem"}
{"category": "Math", "title": "Special invited paper. Large deviations", "abstract": "This paper is based on Wald Lectures given at the annual meeting of the IMS in Minneapolis during August 2005. It is a survey of the theory of large deviations."}
{"category": "Math", "title": "From Permutahedron to Associahedron", "abstract": "For each finite real reflection group $W$, we identify a copy of the type-$W$ simplicial generalised associahedron inside the corresponding simplicial permutahedron. This defines a bijection between the facets of the generalised associahedron and the elements of the type $W$ non-crossing partition lattice which is more tractable than previous such bijections. We show that the simplicial fan determined by this associahedron coincides with the Cambrian fan for $W$."}
{"category": "Math", "title": "A change of variables theorem for the multidimensional Riemann integral", "abstract": "The most general change of variables theorem for the Riemann integral of functions of a single variable has been published in 1961 (by Kestelman). In this theorem, the substitution is made by an `indefinite integral', that is, by a function of the form (t\\mapsto c+\\int_a^tg=:G(t)) where (g) is Riemann integrable on ([a,b]) and (c) is any constant. We prove a multidimensional generalization of this theorem for the case where (G) is injective -- using the fact that the Riemann primitives are the same as those Lipschitz functions which are almost everywhere strongly differentiable in ((a,b)). We prove a generalization of Sard's lemma for Lipschitz functions of several variables that are almost everywhere strongly differentiable, which enables us to keep all our proofs within the framework of the Riemannian theory which was our aim."}
{"category": "Math", "title": "Relative homology and maximal l-orthogonal modules", "abstract": "Let $\\L$ be an artin algebra. Iyama conjectures that the endomorphism ring of any two maximal $l$-orthogonal modules, $M_1$ and $M_2$, are derived equivalent. He proves the conjecture for $l=1$, and for $l>1$ he gives some orthogonality condition on $M_1$ and $M_2$, such that the $\\End_\\L(M_2)^\\op$-$\\End_\\L(M_1)$-bimodule $\\Hom_\\L(M_2,M_1)$ is tilting, which implies that the rings $\\End_\\L(M_2)$ and $\\End_\\L(M_1)$ are derived equivalent (see \\cite{H}). The purpose of this paper is to characterize tilting modules of the form $\\Hom_\\L(M_2,M_1)$ in terms of the relative theories induced by the $\\L$-modules $M_1$ and $M_2$, thus getting a generilization of Iyama's result."}
{"category": "Math", "title": "Continuous crystals and Duistermaat-Heckman measure for Coxeter groups", "abstract": "We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We use a generalization of the Littelmann path model to show the existence of the crystals, and study an associated Duistermaat-Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm-Liouville equations."}
{"category": "Math", "title": "On Lipschitz compactifications of trees", "abstract": "We study the Lipschitz structures on the geodesic compactification of a regular tree, that are preserved by the automorphism group. They are shown to be similar to the compactifications introduced by William Floyd, and a complete description is given."}
{"category": "Math", "title": "2-Primary Anick Fibrations", "abstract": "Cohen conjectured that for r>=2 there is a space T^2n+1(2^r) and a homotopy fibration sequence Loop^2 S^2n+1 --> S^2n-1 --> T^2n+1(2^r) --> Loop S^2n+1 with the property that the left map composed with the double suspension, Loop^2 S^2n+1 --> S^2n-1 --> Loop^2 S^2n+1, is homotopic to the 2^r-power map. We positively resolve this conjecture when r>=3. Several preliminary results are also proved which are of interest in their own right."}
{"category": "Math", "title": "Almost homogeneous manifolds with boundary", "abstract": "Let $\\rho_0$ be an action of a Lie group on a manifold with boundary that is transitive on the interior. We study the set of actions that are topologically conjugate to $\\rho_0$, up to smooth or analytic change of coordinates. We show that in many cases, including the compactifications of negatively curved symmetric spaces, this set is infinite."}
{"category": "Math", "title": "On the permanent of random Bernoulli matrices", "abstract": "We show that the permanent of an $n \\times n$ matrix with iid Bernoulli entries $\\pm 1$ is of magnitude $n^{({1/2}+o(1))n}$ with probability $1-o(1)$. In particular, it is almost surely non-zero."}
{"category": "Math", "title": "Almost-sure Growth Rate of Generalized Random Fibonacci sequences", "abstract": "We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $n\\ge 1$, $F_{n+2} = \\lambda F_{n+1} \\pm F_{n}$ (linear case) and $\\widetilde F_{n+2} = |\\lambda \\widetilde F_{n+1} \\pm \\widetilde F_{n}|$ (non-linear case), where each $\\pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<p\\le 1$). Our main result is that, when $\\lambda$ is of the form $\\lambda_k = 2\\cos (\\pi/k)$ for some integer $k\\ge 3$, the exponential growth of $F_n$ for $0<p\\le 1$, and of $\\widetilde F_{n}$ for $1/k < p\\le 1$, is almost surely positive and given by $$ \\int_0^\\infty \\log x d\\nu_{k, \\rho} (x), $$ where $\\rho$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $\\nu_{k, \\rho}$ is an explicit probability distribution on $\\RR_+$ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<p\\le 1$ in the easier case $\\lambda\\ge 2$. Finally, we study the variations of the exponent as a function of $p$."}
{"category": "Math", "title": "Growth rate for the expected value of a generalized random Fibonacci sequence", "abstract": "A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p the probability of a +), and the recurrence relation is of the form g_n = |\\lambda g_{n-1} +/- g_{n-2} |. When \\lambda >=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When \\lambda = \\lambda_k = 2 cos(\\pi/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-\\lambda_k)/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in a previous paper by the second author."}
{"category": "Math", "title": "Enumeration of ad-nilpotent ideals of parabolic subalgebras for exceptional types", "abstract": "Using GAP4, we determine the number of ad-nilpotent and abelian ideals of a parabolic subalgebra of a simple Lie algebra of exceptional types E, F or G."}
{"category": "Math", "title": "Galois objects and cocycle twisting for locally compact quantum groups", "abstract": "In this article, we investigate the notion of a Galois object for a locally compact quantum group M. Such an object consists of a von Neumann algebra N equipped with an ergodic integrable coaction of M on N, such that the crossed product is a type I factor. We show how to construct from such a coaction a new locally compact quantum group P, which we call the reflection of M along N. By way of application, we prove the following statement: any twisting of a locally compact quantum group by a unitary 2-cocycle is again a locally compact quantum group."}
{"category": "Math", "title": "Bayesian Inference on Mixtures of Distributions", "abstract": "This survey covers state-of-the-art Bayesian techniques for the estimation of mixtures. It complements the earlier Marin, Mengersen and Robert (2005) by studying new types of distributions, the multinomial, latent class and t distributions. It also exhibits closed form solutions for Bayesian inference in some discrete setups. Lastly, it sheds a new light on the computation of Bayes factors via the approximation of Chib (1995)."}
{"category": "Math", "title": "Approximating the marginal likelihood in mixture models", "abstract": "In Chib (1995), a method for approximating marginal densities in a Bayesian setting is proposed, with one proeminent application being the estimation of the number of components in a normal mixture. As pointed out in Neal (1999) and Fruhwirth-Schnatter (2004), the approximation often fails short of providing a proper approximation to the true marginal densities because of the well-known label switching problem (Celeux et al., 2000). While there exist other alternatives to the derivation of approximate marginal densities, we reconsider the original proposal here and show as in Berkhof et al. (2003) and Lee et al. (2008) that it truly approximates the marginal densities once the label switching issue has been solved."}
{"category": "Math", "title": "Around the Gysin triangle I", "abstract": "We define and study Gysin morphisms on mixed motives over a perfect field. Our construction extends the case of closed immersions, already known from results of Voevodsky, to arbitrary projective morphisms. We prove several classical formulas in this context, such as the projection and excess intersection formulas, and some more original ones involving residues. Finally, we give an application of this construction to duality and motive with compact support."}
{"category": "Math", "title": "Extension of bounded root functionals of a system of polynomial equations", "abstract": "The notion of a root functional of a system of polynomials or ideal of polynomials is a generalization of the notion of a root, in particular, for a multiple root. A root functional is a linear functional that is defined on a polynomial ring and annuls the ideal of polynomials. A bounded root functional is a functional that annuls d-th component of the ideal in some filtration in this ideal. The paper consider bounded root functionals and their extension operation for a system of polynomial equation at which the number of equations is equal to the number of unknows. The extension operation has connection with the multivariate Bezoutian construction."}
{"category": "Math", "title": "Virtually fibred Montesinos links of type $\\widetilde{SL_2}$", "abstract": "We find a larger class of virtually fibred classic Montesinos links of type $\\widetilde{SL_2}$, extending a result of Agol, Boyer and Zhang."}
{"category": "Math", "title": "State estimation in quantum homodyne tomography with noisy data", "abstract": "In the framework of noisy quantum homodyne tomography with efficiency parameter $0 < \\eta \\leq 1$, we propose two estimators of a quantum state whose density matrix elements $\\rho_{m,n}$ decrease like $e^{-B(m+n)^{r/ 2}}$, for fixed known $B>0$ and $0<r\\leq 2$. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for $0<\\eta \\leq 1/2$), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in $\\mathbb{L}_2$ risk."}
{"category": "Math", "title": "Differential calculus on a Lie algebroid and Poisson manifolds", "abstract": "A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present in this paper the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds."}
{"category": "Math", "title": "A Little Statistical Mechanics for the Graph Theorist", "abstract": "In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zero-temperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches. This paper is an elementary general audience survey, intended to popularize the area and provide an accessible first point of entry for further exploration."}
{"category": "Math", "title": "First coniveau notch of the Dwork family and its mirror", "abstract": "If $X_{\\lambda}$ is a smooth member of the Dwork family over a perfect field $k$, and $Y_{\\lambda}$ is its mirror variety, then the motives of $X_{\\lambda}$ and $Y_{\\lambda}$ are equal up to motives that are in coniveau $\\geq 1$. If $k$ is a finite field, this provides a motivic explanation for Wan's congruence between the zeta functions of $X_{\\lambda}$ and $Y_{\\lambda}$."}
{"category": "Math", "title": "The Atiyah-Singer index formula for subelliptic operators on contact manifolds, Part I", "abstract": "The Atiyah-Singer index theorem is a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and involves analysis on the Heisenberg group. For a hypoelliptic differential operator in the Heisenberg calculus on a contact manifold we construct a symbol class in the K-theory of a noncommutative C*-algebra of symbols. There is a canonical map from this analytic K-theory group to the deRham cohomology of the manifold, which gives a class to which the Atiyah-Singer formula can be applied. We prove that the index formula holds for these hypoelliptic operators. Our methods derive from Connes' tangent groupoid proof of the index theorem."}
{"category": "Math", "title": "The Atiyah-Singer index formula for subelliptic operators on contact manifolds, Part II", "abstract": "We present a new solution to the index problem for hypoelliptic operators in the Heisenberg calculus on contact manifolds, by constructing the appropriate topological K-theory cocycle for such operators. Its Chern character gives a cohomology class to which the Atiyah-Singer index formula can be applied. Such a K-cocycle has already been constructed by Boutet de Monvel for Toeplitz operators, and, more recently, by Melrose and Epstein for the class of Hermite operators. Our construction applies to general hypoelliptic pseudodifferential operators in the Heisenberg calculus. As in the Hermite Index Formula of Melrose and Epstein, our construction gives a vector bundle automorphism of the symmetric tensors of the contact hyperplane bundle. This automorphism is constructed directly from the invertible Heisenberg symbol of the operator, and is easily computed in the case of differential operators."}
{"category": "Math", "title": "More on counting acyclic digraphs", "abstract": "In this note we derive enumerative formulas for several types of labelled acyclic directed graphs by slight modifications of the familiar recursive formula for simple acyclic digraphs. These considerations are motivated by, and based upon, recent combinatorial results in geometric topology obtained by S.Choi, who established exact correspondences between acyclic digraphs and so-called small covers over hypercubes and related polytopes. In particular, we show that the number of equivalence classes of small covers over the cartesian product of $n$ copies of an $r$-simplex is equal to the number of acyclic $(2^r-1)$-multidigraphs of order $n$. Asymptotics follows easily since the main formula is represented by a simple equation in terms of special generating functions."}
{"category": "Math", "title": "Smoothness of radial solutions to Monge-Ampere equations", "abstract": "We characterize when radial weak solutions to Monge-Ampere equations are smooth. This paper extends previous partial results and also covers Generalized Monge-Ampere equations and infinitely vanishing right hand side."}
{"category": "Math", "title": "Remarks on modules of the ortho-symplectic Lie superalgebras", "abstract": "We examine in detail the Jacobi-Trudi characters over the ortho-symplectic Lie superalgebras spo(2|2m+1) and spo(2n|3). We furthermore relate them to Serganova's notion of Euler characters."}
{"category": "Math", "title": "Some New Monotonicity Formulas and the Singular Set in the Lower Dimensional Obstacle Problem", "abstract": "We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set. Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren's frequency formula, first established by Caffarelli, Salsa, and Silvestre."}
{"category": "Math", "title": "Construction of quantized enveloping algebras by cocycle deformation", "abstract": "By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known construction by (generalized) quantum doubles, but unlike in the known situation, it saves us from difficulties in checking complicated defining relations."}
{"category": "Math", "title": "Statistics, ethics, and probiotica", "abstract": "A randomized clinical trial comparing an experimental new treatment to a standard therapy for a life-threatening medical condition should be stopped early on ethical grounds, in either of the following scenarios: (1) it has become overwhelmingly clear that the new treatment is better than the standard; (2) it has become overwhelmingly clear that the trial is not going to show that the new treatment is any better than the standard. The trial is continued in the third scenario: (3) there is a reasonable chance that the new treatment will finally turn out to be better than the standard, but we aren't sure yet. However, the (blinded) data monitoring committee in the \"PROPATRIA\" trial of an experimental probiotica treatment for patients with acute pancreatitis allowed the trial to continue at the half way interim analysis, in effect because there was still a good chance of proving that the probiotica treatment was very harmful to their patients. The committee did not know whether treatment A was the probiotica treatment or the placebo. In itself this should not have caused a problem, since it could easily have determined the appropriate decision under both scenarios. Were the decisions in the two scenarios different, then the data would have to be de-blinded, in order to determine the appropriate decision. The committee mis-read the output of SPSS, which reports the smaller of two one-sided p-values, without informing the user what it is doing. It seems that about 5 lives were sacrificed to the chance of getting a significant result that the probiotica treatment was bad for the patients in the trial."}
{"category": "Math", "title": "Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion", "abstract": "Let $q\\geq 2$ be a positive integer, $B$ be a fractional Brownian motion with Hurst index $H\\in(0,1)$, $Z$ be an Hermite random variable of index $q$, and $H_q$ denote the Hermite polynomial having degree $q$. For any $n\\geq 1$, set $V_n=\\sum_{k=0}^{n-1} H_q(B_{k+1}-B_k)$. The aim of the current paper is to derive, in the case when the Hurst index verifies $H>1-1/(2q)$, an upper bound for the total variation distance between the laws $\\mathscr{L}(Z_n)$ and $\\mathscr{L}(Z)$, where $Z_n$ stands for the correct renormalization of $V_n$ which converges in distribution towards $Z$. Our results should be compared with those obtained recently by Nourdin and Peccati (2007) in the case when $H<1-1/(2q)$, corresponding to the situation where one has normal approximation."}
{"category": "Math", "title": "Dimensions of Biquadratic Spline Spaces over T-meshes", "abstract": "This paper discusses the dimensions of the spline spaces over T-meshes with lower degree. Two new concepts are proposed: extension of T-meshes and spline spaces with homogeneous boundary conditions. In the dimension analysis, the key strategy is linear space embedding with the operator of mixed partial derivative. The dimension of the original space equals the difference between the dimension of the image space and the rank of the constraints which ensuring any element in the image space has a preimage in the original space. Then the dimension formula and basis function construction of bilinear spline spaces of smoothness order zero over T-meshes are discussed in detail, and a dimension lower bound of biquadratic spline spaces over general T-meshes is provided. Furthermore, using level structure of hierarchical T-meshes, a dimension formula of biquadratic spline space over hierarchical T-meshes are proved. A topological explantation of the dimension formula is shown as well."}
{"category": "Math", "title": "On the Numerical Evaluation of Fredholm Determinants", "abstract": "Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painleve transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nystrom method for the solution of Fredholm equations of the second kind. Using Gauss-Legendre or Clenshaw-Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the two-point correlation functions of the more recently studied Airy and Airy1 processes for the first time."}
{"category": "Math", "title": "Microlocal smoothing effect for the Schr\\\"odinger evolution equation in a Gevrey class", "abstract": "We discuss the microlocal Gevrey smoothing effect for the Schr\\\"odinger equation with variable coefficients via the propagation property of the wave front set of homogenous type. We apply the microlocal exponential estimates in a Gevrey case to prove our result."}
{"category": "Math", "title": "Poisson processes for subsystems of finite type in symbolic dynamics", "abstract": "Let $\\Delta\\subsetneq\\V$ be a proper subset of the vertices $\\V$ of the defining graph of an irreducible and aperiodic shift of finite type $(\\Sigma_{A}^{+},\\S)$. Let $\\Sigma_{\\Delta}$ be the subshift of allowable paths in the graph of $\\Sigma_{A}^{+}$ which only passes through the vertices of $\\Delta$. For a random point $x$ chosen with respect to an equilibrium state $\\mu$ of a H\\\"older potential $\\phi$ on $\\Sigma_{A}^{+}$, let $\\tau_{n}$ be the point process defined as the sum of Dirac point masses at the times $k>0$, suitably rescaled, for which the first $n$-symbols of $\\S^k x$ belong to $\\Delta$. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of $\\phi$ to $\\Sigma_{\\Delta}$ and the parameters of the limit law are explicitly computed."}
{"category": "Math", "title": "On the asymptotic measure of periodic subsystems of finite type in symbolic dynamics", "abstract": "Let $\\Delta\\subsetneq\\V$ be a proper subset of the vertices $\\V$ of the defining graph of an aperiodic shift of finite type $(\\Sigma_{A}^{+},\\S)$. Let $\\Delta_{n}$ be the union of cylinders in $\\Sigma_{A}^{+}$ corresponding to the points $x$ for which the first $n$-symbols of $x$ belong to $\\Delta$ and let $\\mu$ be an equilibrium state of a H\\\"older potential $\\phi$ on $\\Sigma_{A}^{+}$. We know that $\\mu(\\Delta_{n})$ converges to zero as $n$ diverges. We study the asymptotic behaviour of $\\mu(\\Delta_{n})$ and compare it with the pressure of the restriction of $\\phi$ to $\\Sigma_{\\Delta}$. The present paper extends some results in \\cite{CCC} to the case when $\\Sigma_{\\Delta}$ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case."}
{"category": "Math", "title": "Infinitesimal Derived Torelli Theorem for K3 surfaces", "abstract": "We prove that the first order deformations of two smooth projective K3 surfaces are derived equivalent under a Fourier--Mukai transform if and only if there exists a special isometry of the total cohomology groups of the surfaces which preserves the Mukai pairing, an infinitesimal weight-2 decomposition and the orientation of a positive 4-dimensional space. This generalizes the derived version of the Torelli Theorem. Along the way we show the compatibility of the actions on Hochschild homology and singular cohomology of any Fourier--Mukai functor."}
{"category": "Math", "title": "On cohomologically complete intersections", "abstract": "An ideal $I$ of a local Gorenstein ring $(R, \\mathfrak m)$ is called cohomologically complete intersection whenever $H^i_I(R) = 0$ for all $i \\not= \\height I.$ Here $H^i_I(R), i \\in \\mathbb Z,$ denotes the local cohomology of $R$ with respect to $I.$ For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view, in particular in terms of their Bass numbers of $H^c_I(R), c = \\height I.$ As a main result it is shown that the vanishing $H^i_I(R) = 0$ for all $i \\not= c$ is completely encoded in homological properties of $H^c_I(R),$ in particular in its Bass numbers."}
{"category": "Math", "title": "Pressure and Equilibrium States in Ergodic Theory", "abstract": "Our goal is to present the basic results on one-dimensional Gibbs and equilibrium states viewed as special invariant measures on symbolic dynamical systems, and then to describe without technicalities a sample of results they allowed to obtain for certain differentiable dynamical systems. We hope that this contribution will illustrate the symbiotic relationship between ergodic theory and statistical mechanics, and also information theory."}
{"category": "Math", "title": "Painleve IV asymptotics for orthogonal polynomials with respect to a modified Laguerre weight", "abstract": "We study polynomials that are orthogonal with respect to the modified Laguerre weight $z^{-n + \\nu} e^{-Nz} (z-1)^{2b}$ in the limit where $n, N \\to \\infty$ with $N/n \\to 1$ and $\\nu$ is a fixed number in $\\mathbb{R} \\setminus \\mathbb{N}_0$. With the effect of the factor $(z-1)^{2b}$, the local parametrix near the critical point $z =1$ can be constructed in terms of $\\Psi$-functions associated with the Painleve IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painleve IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann-Hilbert problem associated with orthogonal polynomials."}
{"category": "Math", "title": "The allelic partition for coalescent point processes", "abstract": "Assume that individuals alive at time $t$ in some population can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d. The ranked sequence of these branches is called a coalescent point process. We have shown in a previous work that splitting trees are important instances of such populations. Here, individuals are given DNA sequences, and for a sample of $n$ DNA sequences belonging to distinct individuals, we consider the number $S_n$ of polymorphic sites (sites at which at least two sequences differ), and the number $A_n$ of distinct haplotypes (sequences differing at one site at least). It is standard to assume that mutations arrive at constant rate (on germ lines), and never hit the same site on the DNA sequence. We study the mutation pattern associated to coalescent point processes under this assumption. Here, $S_n$ and $A_n$ grow linearly as $n$ grows, with explicit rate. However, when the branch lengths have infinite expectation, $S_n$ grows more rapidly, e.g. as $n \\ln(n)$ for critical birth--death processes. Then, we study the frequency spectrum of the sample, that is, the numbers of polymorphic sites/haplotypes carried by $k$ individuals in the sample. These numbers are shown to grow also linearly with sample size, and we provide simple explicit formulae for mutation frequencies and haplotype frequencies. For critical birth--death processes, mutation frequencies are given by the harmonic series and haplotype frequencies by Fisher logarithmic series."}
{"category": "Math", "title": "On the Index of Congruence Subgroups of Aut(F_n)", "abstract": "For an epimorphism pi of the free group F_n onto a finite group G write Gamma(G,pi) for the group of all automorphisms f of F_n for which pi*f = pi. This is called the standard congruence subgroup of Aut(F_n) associated to G and pi. In the case n = 2 we present formulas for the index of Gamma(G,pi) where G is abelian or dihedral. Moreover, we show that congruence subgroups associated to dihedral groups provide a family of subgroups of arbitrary large index in Aut(F_2) generated by a fixed number of elements. This implies that finite index subgroups of Aut(F_2) cannot be written as free products."}
{"category": "Math", "title": "On Greedy Clique Decompositions and Set Representations of Graphs", "abstract": "In 1994 S. McGuinness showed that any greedy clique decompo- sition of an n-vertex graph has at most $\\lfloor n^2/4 \\rfloor$ cliques (The greedy clique decomposition of a graph, J. Graph Theory 18 (1994) 427-430), where a clique decomposition means a clique partition of the edge set and a greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. This result solved a conjecture by P. Winkler. A multifamily set rep- resentation of a simple graph G is a family of sets, not necessarily distinct, each member of which represents a vertex in G, and the in- tersection of two sets is non-empty if and only if two corresponding vertices in G are adjacent. It is well known that for a graph G, there is a one-to-one correspondence between multifamily set representations and clique coverings of the edge set. Further for a graph one may have a one-to-one correspondence between particular multifamily set rep- resentations with intersection size at most one and clique partitions of the edge set. In this paper, we study for an n-vertex graph the variant of the set representations using a family of distinct sets, including the greedy way to get the corresponding clique partition of the edge set of the graph. Similarly, in this case, we obtain a result that any greedy clique decomposition of an n-vertex graph has at most $\\lfloor n^2/4 \\rfloor$ cliques."}
{"category": "Math", "title": "On the orthogonal polynomials associated with a L\\'evy process", "abstract": "Let $X=\\{X_t, t\\ge0\\}$ be a c\\`{a}dl\\`{a}g L\\'{e}vy process, centered, with moments of all orders. There are two families of orthogonal polynomials associated with $X$. On one hand, the Kailath--Segall formula gives the relationship between the iterated integrals and the variations of order $n$ of $X$, and defines a family of polynomials $P_1(x_1), P_2(x_1,x_2),...$ that are orthogonal with respect to the joint law of the variations of $X$. On the other hand, we can construct a sequence of orthogonal polynomials $p^{\\sigma}_n(x)$ with respect to the measure $\\sigma^2\\delta_0(dx)+x^2 \\nu(dx)$, where $\\sigma^2$ is the variance of the Gaussian part of $X$ and $\\nu$ its L\\'{e}vy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the L\\'{e}vy process proved by Nualart and Schoutens. The main objective of this work is to study the probabilistic properties and the relationship of the two families of polynomials. In particular, the L\\'{e}vy processes such that the associated polynomials $P_n(x_1,...,x_n)$ depend on a fixed number of variables are characterized. Also, we give a sequence of L\\'{e}vy processes that converge in the Skorohod topology to $X$, such that all variations and iterated integrals of the sequence converge to the variations and iterated integrals of $X$."}
{"category": "Math", "title": "Nonstandard limit theorem for infinite variance functionals", "abstract": "We consider functionals of long-range dependent Gaussian sequences with infinite variance and obtain nonstandard limit theorems. When the long-range dependence is strong enough, the limit is a Hermite process, while for weaker long-range dependence, the limit is $\\alpha$-stable L\\'{e}vy motion. For the critical value of the long-range dependence parameter, the limit is a sum of a Hermite process and $\\alpha$-stable L\\'{e}vy motion."}
{"category": "Math", "title": "The Calogero-Moser partition for wreath products", "abstract": "This paper has been withdrawn - the results (with updated proofs) now appear in arXiv:0801.1627."}
{"category": "Math", "title": "A note on Pollard's Theorem", "abstract": "Let $A,B$ be nonempty subsets of a an abelian group $G$. Let $N_i(A,B)$ denote the set of elements of $G$ having $i$ distinct decompositions as a product of an element of $A$ and an element of $B$. We prove that $$ \\sum _{1\\le i \\le t} |N_i (A,B)|\\ge t(|A|+|B|- t-\\alpha+1+w)-w, $$ where $\\alpha $ is the largest size of a coset contained in $AB$ and $w=\\min (\\alpha-1,1)$, with a strict inequality if $\\alpha\\ge 3$ and $t\\ge 2$, or if $\\alpha\\ge 2$ and $t= 2$. This result is a local extension of results by Pollard and Green--Ruzsa and extends also for $t>2$ a recent result of Grynkiewicz, conjectured by Dicks--Ivanov (for non necessarily abelian groups) in connection to the famous Hanna Neumann problem in Group Theory."}
{"category": "Math", "title": "Volume preserving subgroups of A and K and singularities in unimodular geometry", "abstract": "For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this moduli space vanishes for A-equivalence with volume-preserving target diffeomorphisms and A-stable maps f and for K-equivalence with volume-preserving source diffeomorphisms and K-simple maps f. On the other hand, there are A-stable maps f with infinite-dimensional moduli space for A-equivalence with volume-preserving source diffeomorphisms."}
{"category": "Math", "title": "On Pointed Hopf Algebras with Weyl Groups of exceptional type", "abstract": "All -1-type pointed Hopf algebras and central quantum linear spaces with Weyl groups of exceptional type are found. It is proved that every non -1-type pointed Hopf algebra with real $G(H)$ is infinite dimensional and every central quantum linear space over finite group is finite dimensional. It is proved that except a few cases Nichols algebras of reducible Yetter-Drinfeld modules over Weyl groups of exceptional type are infinite dimensional."}
{"category": "Math", "title": "Construction of Local Conservation Laws by Generalized Isometric Embeddings of Vector Bundles", "abstract": "This article uses Cartan-K\\\"ahler theory to construct local conservation laws from covariantly closed vector valued differential forms, objects that can be given, for example, by harmonic maps between two Riemannian manifolds. We apply the article's main result to construct conservation laws for covariant divergence free energy-momentum tensors. We also generalize the local isometric embedding of surfaces in the analytic case by applying the main result to vector bundles of rank two over any surface."}
{"category": "Math", "title": "Quasiflats in CAT(0) complexes", "abstract": "We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical."}
{"category": "Math", "title": "Low and high frequency approximations to eigenvibrations of string with double contrasts", "abstract": "We study eigenvibrations for inhomogeneous string consisting of two parts with strongly contrasting stiffness and mass density. In this work we treat a critical case for the high frequency approximations, namely the case when the order of mass density inhomogeneity is the same as the order of stiffness inhomogeneity, with heavier part being softer. The limit problem for high frequency approximations depends nonlinearly on the spectral parameter. The quantization of the spectral semiaxies is applied in order to get a close approximations of eigenvalues as well as eigenfunctions for the prime problem under perturbation."}
{"category": "Math", "title": "Effectively closed sets of measures and randomness", "abstract": "We show that if a real $x$ is strongly Hausdorff $h$-random, where $h$ is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure $\\mu$ such that the $\\mu$-measure of the basic open cylinders shrinks according to $h$. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for $\\Pi^0_1$-classes applied to closed sets of probability measures. We use the main result to give a new proof of Frostman's Lemma, to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman's Theorem."}
{"category": "Math", "title": "The Non-triviality of the Grope Filtrations of The Knot and Link Concordance Groups", "abstract": "We consider the Grope filtration of the classical knot concordance group that was introduced in a paper of Cochran, Orr and Teichner. Our main result is that successive quotients at each stage in this filtration have infinite rank. We also establish the analogous result for the Grope filtration of the concordance group of string links consisting of more than one component."}
{"category": "Math", "title": "A homotopy-theoretic universal property of Leinster's operad for weak omega-categories", "abstract": "We explain how any cofibrantly generated weak factorisation system on a category may be equipped with a universally and canonically determined choice of cofibrant replacement. We then apply this to the theory of weak omega-categories, showing that the universal and canonical cofibrant replacement of the operad for strict omega-categories is precisely Leinster's operad for weak omega-categories."}
{"category": "Math", "title": "Volume estimates for equiangular hyperbolic Coxeter polyhedra", "abstract": "An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where all dihedral angles are equal to \\pi/n for some fixed integer n at least 2. It is a consequence of Andreev's theorem that either n=3 and the polyhedron has all ideal vertices or that n=2. Volume estimates are given for all equiangular hyperbolic Coxeter polyhedra."}
{"category": "Math", "title": "The core of ideals in arbitrary characteristic", "abstract": "We prove formulas for the core of ideals that apply in arbitrary characteristic."}
{"category": "Math", "title": "On the Loss of Compactness in the Vectorial Heteroclinic Connection Problem", "abstract": "We give an alternative proof of the theorem of Alikakos-Fusco [AF] concerning existence of heteroclinic solutions to a Hamiltonian ODE system on the whole real line which arises in the theory of phase transitions. Our method is variational but differs from the original artificial constraint method of [AF] and establishes existence by analysing the loss of compactness in minimising sequences of the action in the appropriate functional space. Our assumptions are slightly different from those considered previously and also imply a priori estimates for the solution."}
{"category": "Math", "title": "Projective background of the infinitesimal rigidity of frameworks", "abstract": "We present proofs of two classical theorems. The first one, due to Darboux and Sauer, states that infinitesimal rigidity is a projective invariant; the other one establishes relations (infinitesimal Pogorelov maps) between the infinitesimal motions of a Euclidean framework and of its hyperbolic and spherical images. The arguments use the static formulation of infinitesimal rigidity. The duality between statics and kinematics is established through the principles of virtual work. A geometric approach to statics, due essentially to Grassmann, makes both theorems straightforward. Besides, it provides a simple derivation of the formulas both for the Darboux-Sauer correspondence and for the infinitesimal Pogorelov maps."}
{"category": "Math", "title": "When is a Connection a Levi-Civita Connection?", "abstract": "We consider the more general question as to when a connection is a metric connection. There are two aspects to this investigation: first, the determination of the integrability conditions that ensure the existence of a local parallel metric in the neighbourhood of a given point and second, the characterization of the topological obstruction to a globally defined parallel metric."}
{"category": "Math", "title": "Four Properties of Reproducing Kernel Hilbert Spaces", "abstract": "A reproducing kernel Hilbert space (RKHS) has four well-known easily derived properties. Since these properties are usually not emphasized as a simple means of gaining insight into RKHS structure, they are singled out and proved in this brief article."}
{"category": "Math", "title": "Discrete surfaces of constant mean curvature", "abstract": "We propose a unified definition for discrete analogues of constant mean curvature surfaces in spaces of constant curvature as a special case of discrete special isothermic nets. B\\\"acklund transformations and Lawson's correspondence are discussed. It is shown that the definition generalizes previous definitions and a construction for discrete cmc surfaces of revolution in space forms is provided."}
{"category": "Math", "title": "Correlated Link Shadow Fading in Multi-hop Wireless Networks", "abstract": "Accurate representation of the physical layer is required for analysis and simulation of multi-hop networking in sensor, ad hoc, and mesh networks. This paper investigates, models, and analyzes the correlations that exist in shadow fading between links in multi-hop networks. Radio links that are geographically proximate often experience similar environmental shadowing effects and thus have correlated fading. We describe a measurement procedure and campaign to measure a large number of multi-hop networks in an ensemble of environments. The measurements show statistically significant correlations among shadowing experienced on different links in the network, with correlation coefficients up to 0.33. We propose a statistical model for the shadowing correlation between link pairs which shows strong agreement with the measurements, and we compare the new model with an existing shadowing correlation model of Gudmundson (1991). Finally, we analyze multi-hop paths in three and four node networks using both correlated and independent shadowing models and show that independent shadowing models can underestimate the probability of route failure by a factor of two or greater."}
{"category": "Math", "title": "Subgroup theorem for valuated groups and the CSA property", "abstract": "A valuated group with normal forms is a group with an integer-valued length function satisfying some Lyndon's axioms and an additional axiom considered by Hurley. We prove a subgroup theorem for valuated groups with normal forms analogous to Grushko-Neumann's theorem. We study also the CSA property in such groups."}
{"category": "Math", "title": "The Taylor expansion of Ruelle L-function at the origin and the Borel regulator", "abstract": "We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rational multiple of the volume up to a certain contribution from cusps. Moreover if the dimension is three we will identify the leading coefficient. Both of them will be intepreted by the Borel regulator in algebraic K-theory. Also a relation with the L^2-torsion will be discussed."}
{"category": "Math", "title": "Covering shadows with a smaller volume", "abstract": "For each i = 1, ..., n constructions are given for convex bodies K and L in n-dimensional Euclidean space such that each rank i orthogonal projection of K can be translated inside the corresponding projection of L, even though K has strictly larger m-th intrinsic volumes (i.e. V_m(K) > V_m(L)) for all m > i. It is then shown that, for each i = 1, ..., n, there is a class of bodies C{n,i}, called i-cylinder bodies of R^n, such that, if the body L with i-dimensional covering shadows is an i-cylinder body, then K will have smaller n-volume than L. The families C{n,i} are shown to form a strictly increasing chain of subsets C{n,1} < C{n,2} < ... < C{n,n-1} < C{n,n}, where C{n,1} is precisely the collection of centrally symmetric compact convex sets in n-dimensional space, while C{n,n} is the collection of all compact convex sets in n-dimensional space. Members of each family C{n,i} are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes, and members of C{n,i} are shown to satisfy certain geometric inequalities. Related open questions are also posed."}
{"category": "Math", "title": "An Exceptional Representation of Sp(4,F_q)", "abstract": "We describe a folklore construction of an exceptional representation of Sp(4,F_q). This representation has the following remarkable combination of properties, namely it is cuspidal, degenerate and unipotent."}
{"category": "Math", "title": "Structural stability of finite dispersion-relation preserving schemes", "abstract": "The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurance of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to has a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solution of the original continuous equations. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes."}
{"category": "Math", "title": "Convergence Rates of Nonparametric Posterior Distributions", "abstract": "We study the asymptotic behavior of posterior distributions. We present general posterior convergence rate theorems, which extend several results on posterior convergence rates provided by Ghosal and Van der Vaart (2000), Shen and Wasserman (2001) and Walker, Lijor and Prunster (2007). Our main tools are the Hausdorff $\\alpha$-entropy introduced by Xing and Ranneby (2008) and a new notion of prior concentration, which is a slight improvement of the usual prior concentration provided by Ghosal and Van der Vaart (2000). We apply our results to several statistical models."}
{"category": "Math", "title": "Universal recursive formulae for Q-curvatures", "abstract": "We formulate and discuss two conjectures concerning recursive formulae for Branson's $Q$-curvatures. The proposed formulae describe all $Q$-curvatures on manifolds of all even dimensions in terms of respective lower order $Q$-curvatures and lower order GJMS-operators. They are universal in the dimension of the underlying space. The recursive formulae are generated by an algorithm which rests on the theory of residue families. We attempt to resolve the algorithm by formulating a conjectural description of the coefficients in the recursive formulae in terms of interpolation polynomials associated to compositions of natural numbers. We prove that the conjectures cover $Q_4$ and $Q_6$ for general metrics, and $Q_8$ for conformally flat metrics. The result for $Q_8$ is proved here for the first time. Moreover, we display explicit (conjectural) formulae for $Q$-curvatures of order up to 16, and test high order cases for round spheres and Einstein metrics."}
{"category": "Math", "title": "Boosting Algorithms: Regularization, Prediction and Model Fitting", "abstract": "We present a statistical perspective on boosting. Special emphasis is given to estimating potentially complex parametric or nonparametric models, including generalized linear and additive models as well as regression models for survival analysis. Concepts of degrees of freedom and corresponding Akaike or Bayesian information criteria, particularly useful for regularization and variable selection in high-dimensional covariate spaces, are discussed as well. The practical aspects of boosting procedures for fitting statistical models are illustrated by means of the dedicated open-source software package mboost. This package implements functions which can be used for model fitting, prediction and variable selection. It is flexible, allowing for the implementation of new boosting algorithms optimizing user-specified loss functions."}
{"category": "Math", "title": "Comment: Boosting Algorithms: Regularization, Prediction and Model Fitting", "abstract": "The authors are doing the readers of Statistical Science a true service with a well-written and up-to-date overview of boosting that originated with the seminal algorithms of Freund and Schapire. Equally, we are grateful for high-level software that will permit a larger readership to experiment with, or simply apply, boosting-inspired model fitting. The authors show us a world of methodology that illustrates how a fundamental innovation can penetrate every nook and cranny of statistical thinking and practice. They introduce the reader to one particular interpretation of boosting and then give a display of its potential with extensions from classification (where it all started) to least squares, exponential family models, survival analysis, to base-learners other than trees such as smoothing splines, to degrees of freedom and regularization, and to fascinating recent work in model selection. The uninitiated reader will find that the authors did a nice job of presenting a certain coherent and useful interpretation of boosting. The other reader, though, who has watched the business of boosting for a while, may have quibbles with the authors over details of the historic record and, more importantly, over their optimism about the current state of theoretical knowledge. In fact, as much as ``the statistical view'' has proven fruitful, it has also resulted in some ideas about why boosting works that may be misconceived, and in some recommendations that may be misguided. [arXiv:0804.2752]"}
{"category": "Math", "title": "On Oliver's p-group conjecture", "abstract": "Let S be a p-group for an odd prime p. Bob Oliver conjectures that a certain characteristic subgroup X(S) always contains the Thompson subgroup J(S). We obtain a reformulation of the conjecture as a statement about modular representations of p-groups. Using this we verify Oliver's conjecture for groups where S/X(S) has nilpotence class at most two."}
{"category": "Math", "title": "A characterization of domains in $\\mathbf C^2$ with noncompact automorphism group", "abstract": "Let $D$ be a bounded domain in $\\mathbf C^2$ with a non-compact group of holomorphic automorphisms. Model domains for $D$ are obtained under the hypothesis that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type."}
{"category": "Math", "title": "Comment: Boosting Algorithms: Regularization, Prediction and Model Fitting", "abstract": "Comment on ``Boosting Algorithms: Regularization, Prediction and Model Fitting'' [arXiv:0804.2752]"}
{"category": "Math", "title": "A General Correspondence between Averages and Integrals", "abstract": "Recent work has generalized the Furstenberg correspondence between sets of integers and dynamical systems to versions which involve sequences of finite graphs or sequences of $L^\\infty$ functions. We give a unified version of the theorem subsuming all these generalizations."}
{"category": "Math", "title": "Largest Laplacian Eigenvalue and Degree Sequences of Trees", "abstract": "We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization."}
{"category": "Math", "title": "Rejoinder: Boosting Algorithms: Regularization, Prediction and Model Fitting", "abstract": "Rejoinder to ``Boosting Algorithms: Regularization, Prediction and Model Fitting'' [arXiv:0804.2752]"}
{"category": "Math", "title": "Reductions of locally conformal symplectic structures and de Rham cohomology tangent to a foliation", "abstract": "We propose a produre of reduction a locally conformal symplectic structure. This procedure of reduction can be applied to wide class of submanifolds. There are no local obstructions for this procedure. But there are global obstructions. We find a necessary and sufficient condition when this reduction holds in terms of the special kind of de Rham cohomology class (tangent to the characteristic foliation) of the Lee form."}
{"category": "Math", "title": "On certain nonlinear elliptic PDE and quasiconfomal mapps between Euclidean surfaces", "abstract": "We mainly investigate some properties of quasiconformal mappings between smooth 2-dimensional surfaces with boundary in the Euclidean space, satisfying certain partial differential equations (inequalities) concerning Laplacian, and in particular satisfying Laplace equation and show that that these mappings are Lipschitz. Conformal parametrization of such surfaces and the method developed in our paper \\cite{km} have important role in this paper.dan curves and is extended to the case of $C^{2,\\alpha}$ surfaces with smooth and compact boundary."}
{"category": "Math", "title": "Topological Hochschild and cyclic homology for Differential graded categories", "abstract": "We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several non-trivial natural transformations from algebraic K-theory to THH(-). In an intermediate step, we prove that the homotopy theory of dg categories is Quillen equivalent, through a four step zig-zag of Quillen equivalences, to the homotopy theory of Eilenberg-Mac Lane spectral categories. Finally, we show that over the rationals two dg categories are topological equivalent if and only if they are quasi-equivalent."}
{"category": "Math", "title": "On three-parametric Lie groups as quasi-Kaehler manifolds with Killing Norden metric", "abstract": "A 3-parametric family of 6-dimensional quasi-Kaehler manifolds with Norden metric is constructed on a Lie group. This family is characterized geometrically. The condition for such a 6-manifold to be isotropic Kaehler is given."}
{"category": "Math", "title": "Existence and behavior of asymmetric traveling wave solutions to thin film equation", "abstract": "We proved the existence and uniqueness of a traveling wave solution to the thin film equation with a Navier slip condition at the liquid-solid interface. We obtain explicit lower and upper bounds for the solution and an absolute error estimate of approximation of a solution to the thin films equation by the traveling-wave solution."}
{"category": "Math", "title": "Quasi-Kaehler manifolds with a pair of Norden metrics", "abstract": "The basic class of the non-integrable almost complex manifolds with a pair of Norden metrics are considered. The interconnections between corresponding quantities at the transformation between the two Levi-Civita connections are given. A 4-parametric family of 4-dimensional quasi-Kaehler manifolds with Norden metric is characterized with respect to the associated Levi-Civita connection."}
{"category": "Math", "title": "A Lie group as a 4-dimensional quasi-Kaehler manifold with Norden metric", "abstract": "A 4-parametric family of 4-dimensional quasi-Kaehler manifolds with Norden metric is constructed on a Lie group. This family is characterized geometrically. The condition for such a 4-manifold to be isotropic Kaehler is given."}
{"category": "Math", "title": "On the geometry of quasi-Kaehler manifolds with Norden metric", "abstract": "The basic class of the non-integrable almost complex manifolds with Norden metric is considered. Its curvature properties are studied. The isotropic Kaehler type of investigated manifolds is introduced and characterized geometrically."}
{"category": "Math", "title": "Tangent Bundles with Sasaki Metric and Almost Hypercomplex Pseudo-Hermitian Structure", "abstract": "The tangent bundle as a $4n$-manifold is equipped with an almost hypercomplex pseudo-Hermitian structure and it is characterized with respect to the relevant classifications. A number of 8-dimensional examples of the considered type of manifold are received from the known explicit examples in that manner."}
{"category": "Math", "title": "Cyclic Cohomology of the Weyl Algebra", "abstract": "We give an explicit formula for symplectically basic representatives of the cyclic cohomology of the Weyl algebra. This paper can be seen as cyclic addendum to the paper by Feigin, Felder and Shoikhet, where the analogous Hochschild case was treated. As an application, we prove a generalization of a Theorem of Nest and Tsygan concerning the relation of the Todd class and the cyclic cohomology of the differential operators on a complex manifold."}
{"category": "Math", "title": "Some four-dimensional almost hypercomplex pseudo-Hermitian manifolds", "abstract": "In this paper, a lot of examples of four-dimensional manifolds with an almost hypercomplex pseudo-Hermitian structure are constructed in several explicit ways. The received 4-manifolds are characterized by their linear invariants in the known aspects."}
{"category": "Math", "title": "On the equivariant and the non-equivariant main conjecture for imaginary quadratic fields", "abstract": "The Iwasawa main conjecture fields has been an important tool to study the arithmetic of special values of $L$-functions of Hecke characters of imaginary quadratic fields. To obtain the finest possible invariants it is important to know the main conjecture for all prime numbers $p$ and also to have an equivariant version at disposal. In this paper we first prove the main conjecture for imaginary quadratic fields for all prime numbers $p$, improving earlier results by Rubin. From this we deduce the equivariant main conjecture in the case that a certain $\\mu$-invariant vanishes. For prime numbers $p\\nmid 6$ which split in $K$, this is a theorem by a result of Gillard."}
{"category": "Math", "title": "Uniform lamda-adjustment and mu-approximation in Banach spaces", "abstract": "We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform lambda-adjustment which is weaker than perturbations by small gap, operator norm, q-norm, and K2-approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform lambda-adjustment, while other fail. However, uniformly lambda-adjusted subspaces and linear operators retain their (semi--)Fredholm properties in a Banach space which dual is Fr\\'{e}chet-Urysohn in weak* topology. We also introduce another concept of perturbation called uniform mu-approximation which is weaker than perturbations by small gap, norm, and compact convergence, yet stronger than uniform lambda-adjustment. We present Fredholm stability theorems for uniform mu-approximation in arbitrary Banach spaces and a theorem on stability of Riesz kernels and ranges for commuting closed essentially Kato operators. Finally, we define the new concepts of a tuple of subspaces and of a complex of subspaces in Banach spaces, and present stability theorems for index and defect numbers of Fredholm tuples and complexes under uniform lambda-adjustment and uniform mu-approximation."}
{"category": "Math", "title": "Inequalities of Hardy-Sobolev type in Carnot-Carath\\'eodory spaces", "abstract": "We consider various types of Hardy-Sobolev inequalities on a Carnot-Carath\\'eodory space $(\\Om, d)$ associated to a system of smooth vector fields $X=\\{X_1, X_2,...,X_m\\}$ on $\\RR^n$ satisfying the H\\\"ormander's finite rank condition $rank Lie[X_1,...,X_m] \\equiv n$. One of our main concerns is the trace inequality \\int_{\\Om}|\\phi(x)|^{p}V(x)dx\\leq C\\int_{\\Om}|X\\phi|^{p}dx,\\qquad \\phi\\in C^{\\infty}_{0}(\\Om), where $V$ is a general weight, i.e., a nonnegative locally integrable function on $\\Om$, and $1<p<+\\infty$. Under sharp geometric assumptions on the domain $\\Om\\subset \\Rn$ that can be measured equivalently in terms of subelliptic capacities or Hausdorff contents, we establish various forms of Hardy-Sobolev type inequalities."}
{"category": "Math", "title": "Frechet differential of a power series in a Banach algebra", "abstract": "We present two new forms in which the Frechet differential of a power series in a Banach algebra can be expressed in terms of absolutely convergent series involving the commutant $C(T):A\\mapsto [A,T]$.Then we apply the results to the analytic functional calculus in a complex Banach space."}
{"category": "Math", "title": "Note on the Cantor-Bendixson rank of limit groups", "abstract": "We show that the Cantor-Bendixson rank of a limit group is finite as well as that of a limit group of a linear group."}
{"category": "Math", "title": "Monotonicity of Subelliptic Estimates on Rigid Pseudoconvex Domains", "abstract": "This paper presents monotonicity of subelliptic estimates on rigid pseudoconvex domains. As an application of monotonicity, we will show that if a rigid monomial domain is of finite type in the D'Angelo's sense, then the sharp subelliptic estimate of this domain equals the reciprocal of the type."}
{"category": "Math", "title": "Relative Riemann-Zariski spaces", "abstract": "In this paper we study relative Riemann-Zariski spaces attached to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasi-separated schemes factors as a composition of an affine morphism and a proper morphism. (In particular, we obtain a new proof of Nagata's compactification theorem.)"}
{"category": "Math", "title": "Information Preserving Component Analysis: Data Projections for Flow Cytometry Analysis", "abstract": "Flow cytometry is often used to characterize the malignant cells in leukemia and lymphoma patients, traced to the level of the individual cell. Typically, flow cytometric data analysis is performed through a series of 2-dimensional projections onto the axes of the data set. Through the years, clinicians have determined combinations of different fluorescent markers which generate relatively known expression patterns for specific subtypes of leukemia and lymphoma -- cancers of the hematopoietic system. By only viewing a series of 2-dimensional projections, the high-dimensional nature of the data is rarely exploited. In this paper we present a means of determining a low-dimensional projection which maintains the high-dimensional relationships (i.e. information) between differing oncological data sets. By using machine learning techniques, we allow clinicians to visualize data in a low dimension defined by a linear combination of all of the available markers, rather than just 2 at a time. This provides an aid in diagnosing similar forms of cancer, as well as a means for variable selection in exploratory flow cytometric research. We refer to our method as Information Preserving Component Analysis (IPCA)."}
{"category": "Math", "title": "Movable algebraic singularities of second-order ordinary differential equations", "abstract": "Any nonlinear equation of the form y''=\\sum_{n=0}^N a_n(z)y^n has a (generally branched) solution with leading order behaviour proportional to (z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\\ne 0. We consider the subclass of equations for which each possible leading order term of this form corresponds to a one-parameter family of solutions represented near z_0 by a Laurent series in fractional powers of z-z_0. For this class of equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This work generalizes previous results of S. Shimomura. The only other possible kind of movable singularity that might occur is an accumulation point of algebraic singularities that can be reached by analytic continuation along infinitely long paths ending at a finite point in the complex plane. This behaviour cannot occur for constant coefficient equations in the class considered. However, an example of R. A. Smith shows that such singularities do occur in solutions of a simple autonomous second-order differential equation outside the class we consider here."}
{"category": "Math", "title": "The sign of Galois representations attached to automorphic forms for unitary groups", "abstract": "We determine the sign of the polarization of any polarized irreducible factor of a Galois representation attached to a polarized cohomological cuspidal automorphic form of Gl_n of a CM field: it is always +1, as was conjectured by Gross."}
{"category": "Math", "title": "Spinning rough disk moving in a rarefied medium", "abstract": "A spinning rough disk moves through a rarefied medium on the plane. The roughness is formed by small cavities on the disk boundary. The medium is so rare that mutual interaction of particles can be neglected. All collisions of particles with the disk are perfectly elastic; there may happen multiple collisions in the cavities. We calculate the force of resistance acting on the body and examine how it depends on the kind of roughness (shape of the cavities). We show that the nonzero transversal component of the force generally appears, resulting in deflection of the disk trajectory. In several simple cases the trajectory is determined. We compare our results with the ones known in the literature. It is known that there is a transversal force acting on a spinning body (most often a sphere or a cylinder) moving in a rarefied gas, due to nonelastic interaction of gas particles with the body. We propose another mechanism of creating the transversal force, resulting from multiple reflections of particles from the body."}
{"category": "Math", "title": "Analytic Functions of a General Matrix Variable", "abstract": "Recent innovations on the differential calculus for functions of non-commuting variables, begun for a quaternionic variable, are now extended to the case of a general matrix over the complex numbers. The expansion of F(X+Delta) is given to first order in Delta for general matrix variables X and Delta that do not commute with each other."}
{"category": "Math", "title": "Characterization of Compact Subsets of $\\mathcal{A}^p$ with Respect to Weak Topology", "abstract": "In this brief article we characterize the relatively compact subsets of $\\mathcal{A}^p$ for the topology $\\sigma(\\mathcal{A}^p,\\mathcal{R}^q)$ (see below), by the weak compact subsets of $L^p$ . The spaces $\\mathcal{R}^q$ endowed with the weak topology induced by $\\mathcal{A}^p$, was recently employed to create the convex risk theory of random processes. The weak compact sets of $\\mathcal{A}^p$ are important to characterize the so-called Lebesgue property of convex risk measures, to give a complete description of the Makcey topology on $\\mathcal{R}^q$ and for their use in the optimization theory."}
{"category": "Math", "title": "Forcing the Strong Lefschetz and the Maximal Rank Properties", "abstract": "Three basic properties that standard graded artinian $k$-algebras may or may not enjoy are the Weak and Strong Lefschetz Properties and the Maximal Rank Property (respectively WLP, SLP, and MRP). In this paper we will assume that the base field $k$ has characteristic zero. It is known that SLP implies MRP, which in turn implies WLP, but that both implications are strict. However, it surprisingly turned out that the set of Hilbert functions admitting any algebras with WLP coincides with the corresponding set for SLP (and therefore with that for MRP). J. Migliore and the first author, using Green's theorem and a result of Wiebe, characterized the Hilbert functions forcing all algebras to enjoy WLP. The purpose of this note is to prove the corresponding characterizations for both SLP and MRP. Unsurprisingly (or surprisingly??), the two characterizations coincide, but they define a class of Hilbert functions strictly smaller than that determined for WLP. Our methods include the Herzog-Popescu theorem on quotients of $k$-algebras modulo a general form, a result of Wiebe, and gins and stable ideals. At the end, we will also discuss the importance of assuming that the characteristic be zero, and we will exhibit a class of codimension 2 monomial complete intersections for which SLP (but not MRP) fails in positive characteristic."}
{"category": "Math", "title": "Uniform observability of hidden Markov models and filter stability for unstable signals", "abstract": "A hidden Markov model is called observable if distinct initial laws give rise to distinct laws of the observation process. Observability implies stability of the nonlinear filter when the signal process is tight, but this need not be the case when the signal process is unstable. This paper introduces a stronger notion of uniform observability which guarantees stability of the nonlinear filter in the absence of stability assumptions on the signal. By developing certain uniform approximation properties of convolution operators, we subsequently demonstrate that the uniform observability condition is satisfied for various classes of filtering models with white-noise type observations. This includes the case of observable linear Gaussian filtering models, so that standard results on stability of the Kalman--Bucy filter are obtained as a special case."}
{"category": "Math", "title": "Hitting Time Statistics and Extreme Value Theory", "abstract": "We consider discrete time dynamical systems and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa. We apply these results to non-uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws (with no extra condition on the speed of mixing, for example). We also give applications of our theory to higher dimensional examples, for which we also obtain classical extreme value laws and exponential hitting time statistics (for balls). We extend these ideas to the subsequent returns to asymptotically small sets, linking the Poisson statistics of both processes."}
{"category": "Math", "title": "A Groebner basis for the secant ideal of the second hypersimplex", "abstract": "We determine a Groebner basis for the secant ideal of the toric ideal associated to the second hypersimplex, with respect to any circular term order. The Groebner basis of the secant ideal requires polynomials of odd degree up to n. This shows that the circular term order is 2-delightful, resolving a conjecture of Drton, Sturmfels, and the author. The proof uses Groebner degenerations for secant ideals, combinatorial characterizations of the secant ideals of monomial ideals, and the relations between secant ideals and prolongations."}
{"category": "Math", "title": "A Note about proving non-$\\Gamma$ under a finite non-microstates free Fisher information Assumption", "abstract": "We prove that if $X_{1},...,X_{n} (n >1)$ are selfadjoints in a $W^{*}$-probability space with finite non-microstates free Fisher information, then the von Neumann algebra $W^{*}(X_{1},...,X_{n})$ they generate doesn't have property $\\Gamma$ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy."}
{"category": "Math", "title": "On Finitely Generated Models of Theories with at Most Countably Many Nonisomorphic Finitely Generated Models", "abstract": "We study finitely generated models of countable theories, having at most countably many nonisomorphic finitely generated models. We intro- duce a notion of rank of finitely generated models and we prove, when T has at most countably many nonisomorphic finitely generated models, that every finitely generated model has an ordinal rank. This rank is used to give a prop- erty of finitely generated models analogue to the Hopf property of groups and also to give a necessary and sufficient condition for a finitely generated model to be prime of its complete theory. We investigate some properties of limit groups of equationally noetherian groups, in respect to their ranks."}
{"category": "Math", "title": "Minimal volume $k$-point lattice $d$-simplices", "abstract": "We extend the results of Bey, Hen, and Wills (http://arxiv.org/abs/math/0606089). In this paper, we show that, up to equivalence under unimodular transformations, there is exactly one class of $d$-simplices having $k \\ge 1$ interior lattice points and minimal volume $\\frac{1}{d!}(dk+1)$."}
{"category": "Math", "title": "Pairs of Noncrossing Free Dyck Paths and Noncrossing Partitions", "abstract": "Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length $2n$ and noncrossing partitions of $[2n+1]$ with $n+1$ blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm."}
{"category": "Math", "title": "The Conjecture of Nowicki on Weitzenboeck derivations of polynomial algebras", "abstract": "The Weitzenboeck theorem states that the algebra of constants of a linear locally nilpotent derivation of the polynomial algebra K[Z]=K[z_1,...,z_m] in m variables over a field K of characteristic 0 is finitely generated. If m=2n and the Jordan normal form of the derivation consists of Jordan cells of size 2 only, we may assume that K[Z]=K[X,Y] and the derivation sends y_i to x_i and x_i to 0, i=1,...,n. Nowicki conjectured that the algebra of constants of this derivation is generated by x_1,...,x_n and x_iy_j-x_jy_i, i<j. Recently this conjecture was confirmed in the Ph.D. thesis of Khoury, and also by Derksen. In this paper we give an elementary proof of the conjecture of Nowicki. Then we find a very simple system of defining relations of the algebra of constants which corresponds to the reduced Groebner basis of the related ideal with respect to a suitable admissible order, and present an explicit basis of the algebra of constants as a vector space."}
{"category": "Math", "title": "The Aleph-zero or zero dichotomy", "abstract": "This paper proves the existence of a dichotomy which being formally derived from the topological successiveness of w-order leads to the same absurdity of Zeno's Dichotomy II. It also derives a contradictory result from the first Zeno's Dichotomy."}
{"category": "Math", "title": "Recursion and w-order", "abstract": "This paper has been withdrawn by the author due to an error."}
{"category": "Math", "title": "Margin-adaptive model selection in statistical learning", "abstract": "A classical condition for fast learning rates is the margin condition, first introduced by Mammen and Tsybakov. We tackle in this paper the problem of adaptivity to this condition in the context of model selection, in a general learning framework. Actually, we consider a weaker version of this condition that allows one to take into account that learning within a small model can be much easier than within a large one. Requiring this \"strong margin adaptivity\" makes the model selection problem more challenging. We first prove, in a general framework, that some penalization procedures (including local Rademacher complexities) exhibit this adaptivity when the models are nested. Contrary to previous results, this holds with penalties that only depend on the data. Our second main result is that strong margin adaptivity is not always possible when the models are not nested: for every model selection procedure (even a randomized one), there is a problem for which it does not demonstrate strong margin adaptivity."}
{"category": "Math", "title": "Superswapping", "abstract": "Supertask theory is used here to prove a contradictory result which involves the consistency of w-order and the Axiom of Infinity."}
{"category": "Math", "title": "A disturbing supertask", "abstract": "This paper examines the consistency of w-order by means of a supertask that functions as a supertrap for the assumed existence of w-ordered collections, which are simultaneously complete (as is required by the Actual infinity) and uncompletable (because no last element completes them)."}
{"category": "Math", "title": "On the $S_n$-equivariant Euler characteristic of moduli spaces of hyperelliptic curves", "abstract": "The generating function for $S_n$-equivariant Euler characteristics of moduli spaces of pointed hyperelliptic curves for any genus g>1 is calculated. This answer generalizes the known ones for genera 2 and 3 and answers obtained by J. Bergstrom for any genus and n<8 points."}
{"category": "Math", "title": "A construction of spherical designs from finite graphs with the theory of crystal lattice", "abstract": "We want to introduce a construction of spherical designs from finite graphs with the theory of crystal lattice. We start from a finite graph, and we consider standard realization of the crystal lattices as the maximal Abelian covering of the graph. Then, we take the set of vectors which form the crystal lattice. If every vector has the same norm, then we can consider a finite set on Euclidean sphere, and then we get a spherical design. In this paper, we observe the results by numerical calculations. We tried constructing vectors from various finite graphs, strongly regular graphs, distance regular graphs, and so on. We also introduce some facts and conjectures."}
{"category": "Math", "title": "Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data", "abstract": "When outcomes are missing for reasons beyond an investigator's control, there are two different ways to adjust a parameter estimate for covariates that may be related both to the outcome and to missingness. One approach is to model the relationships between the covariates and the outcome and use those relationships to predict the missing values. Another is to model the probabilities of missingness given the covariates and incorporate them into a weighted or stratified estimate. Doubly robust (DR) procedures apply both types of model simultaneously and produce a consistent estimate of the parameter if either of the two models has been correctly specified. In this article, we show that DR estimates can be constructed in many ways. We compare the performance of various DR and non-DR estimates of a population mean in a simulated example where both models are incorrect but neither is grossly misspecified. Methods that use inverse-probabilities as weights, whether they are DR or not, are sensitive to misspecification of the propensity model when some estimated propensities are small. Many DR methods perform better than simple inverse-probability weighting. None of the DR methods we tried, however, improved upon the performance of simple regression-based prediction of the missing values. This study does not represent every missing-data problem that will arise in practice. But it does demonstrate that, in at least some settings, two wrong models are not better than one."}
{"category": "Math", "title": "Comment: Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data", "abstract": "Comment on ``Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data'' [arXiv:0804.2958]"}
{"category": "Math", "title": "Comment: Performance of Double-Robust Estimators When ``Inverse Probability'' Weights Are Highly Variable", "abstract": "Comment on ``Performance of Double-Robust Estimators When ``Inverse Probability'' Weights Are Highly Variable'' [arXiv:0804.2958]"}
{"category": "Math", "title": "Comment: Understanding OR, PS and DR", "abstract": "Comment on ``Understanding OR, PS and DR'' [arXiv:0804.2958]"}
{"category": "Math", "title": "Comment: Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data", "abstract": "Comment on ``Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data'' [arXiv:0804.2958]"}
{"category": "Math", "title": "Rejoinder: Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data", "abstract": "Rejoinder to ``Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data'' [arXiv:0804.2958]"}
{"category": "Math", "title": "On some Moduli spaces of stable vector bundles on cubic and quartic threefolds", "abstract": "We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles."}
{"category": "Math", "title": "Measuring Traffic", "abstract": "A traffic performance measurement system, PeMS, currently functions as a statewide repository for traffic data gathered by thousands of automatic sensors. It has integrated data collection, processing and communications infrastructure with data storage and analytical tools. In this paper, we discuss statistical issues that have emerged as we attempt to process a data stream of 2 GB per day of wildly varying quality. In particular, we focus on detecting sensor malfunction, imputation of missing or bad data, estimation of velocity and forecasting of travel times on freeway networks."}
{"category": "Math", "title": "Splitting type, global sections and Chern classes for torsion free sheaves on P^N", "abstract": "In this paper we compare a torsion free sheaf $\\FF$ on $\\PP^N$ and the free vector bundle $\\oplus_{i=1}^n\\OPN(b_i)$ having same rank and splitting type. We show that the first one has always \"less\" global sections, while it has a higher second Chern class. In both cases bounds for the difference are found in terms of the maximal free subsheaves of $\\FF$. As a consequence we obtain a direct, easy and more general proof of the \"Horrocks' splitting criterion\", also holding for torsion free sheaves, and lower bounds for the Chern classes $c_i(\\FF(t))$ of twists of $\\FF$, only depending on some numerical invariants of $\\FF$. Especially, we prove for rank $n$ torsion free sheaves on $\\PP^N$, whose splitting type has no gap (i.e. $b_i\\geq b_{i+1}\\geq b_i-1$ for every $i=1, ...,n-1$), the following formula for the discriminant: \\[ \\Delta(\\FF):=2nc_2-(n-1)c_1^2\\geq -{1/12}n^2(n^2-1)\\] Finally in the case of rank $n$ reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes $c_3(\\FF(t)), ..., c_n(\\FF(t))$, for the dimension of the cohomology modules $H^i\\FF(t)$ and for the Castelnuovo-Mumford regularity of $\\FF$; these polynomial bounds only depend only on $c_1(\\FF)$, $c_2(\\FF)$, the splitting type of $\\FF$ and $t$."}
{"category": "Math", "title": "Higher order invariants of Levi degenerate hypersurfaces", "abstract": "The first part of this paper considers higher order CR invariants of three dimensional hypersurfaces of finite type. Using a full normal form we give a complete characterization of hypersurfaces with trivial local automorphism group, and analogous results for finite groups. The second part considers hypersurfaces of finite Catlin multitype and the Kohn-Nirenberg phenomenon in higher dimensions. We give a necessary condition for local convexifiability of a class of pseudoconvex hypersurfaces in $\\mathbb C^{n+1}$."}
{"category": "Math", "title": "The Convenient Setting for non-Quasianalytic Denjoy--Carleman Differentiable Mappings", "abstract": "For Denjoy--Carleman differential function classes $C^M$ where the weight sequence $M=(M_k)$ is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is $C^M$ if it maps $C^M$-curves to $C^M$-curves. The category of $C^M$-mappings is cartesian closed in the sense that $C^M(E,C^M(F,G))\\cong C^M(E\\x F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $C^M$-diffeomorphisms is a $C^M$-Lie group but not better."}
{"category": "Math", "title": "The Epic Story of Maximum Likelihood", "abstract": "At a superficial level, the idea of maximum likelihood must be prehistoric: early hunters and gatherers may not have used the words ``method of maximum likelihood'' to describe their choice of where and how to hunt and gather, but it is hard to believe they would have been surprised if their method had been described in those terms. It seems a simple, even unassailable idea: Who would rise to argue in favor of a method of minimum likelihood, or even mediocre likelihood? And yet the mathematical history of the topic shows this ``simple idea'' is really anything but simple. Joseph Louis Lagrange, Daniel Bernoulli, Leonard Euler, Pierre Simon Laplace and Carl Friedrich Gauss are only some of those who explored the topic, not always in ways we would sanction today. In this article, that history is reviewed from back well before Fisher to the time of Lucien Le Cam's dissertation. In the process Fisher's unpublished 1930 characterization of conditions for the consistency and efficiency of maximum likelihood estimates is presented, and the mathematical basis of his three proofs discussed. In particular, Fisher's derivation of the information inequality is seen to be derived from his work on the analysis of variance, and his later approach via estimating functions was derived from Euler's Relation for homogeneous functions. The reaction to Fisher's work is reviewed, and some lessons drawn."}
{"category": "Math", "title": "On the Carleson measure criterion in linear systems theory", "abstract": "In Ho, Russell, and Weiss, a Carleson measure criterion for admissibility of one-dimensional input elements with respect to diagonal semigroups is given. We extend their results from the Hilbert space situation $X=\\ell_2$ and $L^2$--admissibility to the more general situation of $L^p$--admissibility on $\\ell_q$--spaces. In case of analytic diagonal semigroups we present a new result that does not rely on Laplace transform methods. A comparison of both criteria leads to result of $L^p$--admissibility for reciprocal systems in the sense of Curtain."}
{"category": "Math", "title": "Rank four vector bundles without theta divisor over a curve of genus two", "abstract": "We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and compute the degree of the rational theta map."}
{"category": "Math", "title": "Generalized SURE for Exponential Families: Applications to Regularization", "abstract": "Stein's unbiased risk estimate (SURE) was proposed by Stein for the independent, identically distributed (iid) Gaussian model in order to derive estimates that dominate least-squares (LS). In recent years, the SURE criterion has been employed in a variety of denoising problems for choosing regularization parameters that minimize an estimate of the mean-squared error (MSE). However, its use has been limited to the iid case which precludes many important applications. In this paper we begin by deriving a SURE counterpart for general, not necessarily iid distributions from the exponential family. This enables extending the SURE design technique to a much broader class of problems. Based on this generalization we suggest a new method for choosing regularization parameters in penalized LS estimators. We then demonstrate its superior performance over the conventional generalized cross validation approach and the discrepancy method in the context of image deblurring and deconvolution. The SURE technique can also be used to design estimates without predefining their structure. However, allowing for too many free parameters impairs the performance of the resulting estimates. To address this inherent tradeoff we propose a regularized SURE objective. Based on this design criterion, we derive a wavelet denoising strategy that is similar in sprit to the standard soft-threshold approach but can lead to improved MSE performance."}
{"category": "Math", "title": "Geodesics in large planar maps and in the Brownian map", "abstract": "We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of the complement of {x} in S. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps."}
{"category": "Math", "title": "Real Paley-Wiener theorems and local spectral radius formulas", "abstract": "We systematically develop real Paley-Wiener theory for the Fourier transform on R^d for Schwartz functions, L^p-functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley-Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of Fourier type as well. An explanation is also given why the easily applied classical Paley-Wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull, whereas real Paley-Wiener theorems can be used to reconstruct the support precisely, albeit at the cost of combinatorial complexity. We indicate a possible application of real Paley-Wiener theory to partial differential equations in this vein and furthermore we give evidence that a number of real Paley-Wiener results can be expected to have an interpretation as local spectral radius formulas. A comprehensive overview of the literature on real Paley-Wiener theory is included."}
{"category": "Math", "title": "A Proposal of Multigrid Methods for Hermitian Positive Definite Linear Systems enjoying an order relation", "abstract": "Given a multigrid procedure for linear systems with coefficient matrices $A_n$, we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices $B_n$: we assume that both $A_n$ and $B_n$ are positive definite with $A_n\\le \\vartheta B_n$, for some positive $\\vartheta$ independent of $n$. In this context we prove the Two-Grid method optimality. We apply this elementary strategy for designing a multigrid solution for modifications of multilevel structured (Toeplitz, circulants, Hartley, sine ($\\tau$ class) and cosine algebras) linear systems, in which the coefficient matrix is banded in a multilevel sense and Hermitian positive definite. In such a way, several linear systems arising from the approximation of integro-differential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to Two-Grid and multigrid procedures."}
{"category": "Math", "title": "Three Dimensional Corners: A Box Norm Proof", "abstract": "In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero element of G. The Ramsey number of interest is R_3(G) the maximal cardinality of a subset of G^3 that does not contain a three-dimensional corner. Furstenberg and Katznelson have shown R_3(Z_N) is little-o of N^3, and in fact the corresponding result holds in all dimensions, a result that is a far reaching extension of the Szemeredi Theorem. We give a new proof of the finite field version of this fact, a proof that is a common generalization of the Gowers proof of Szemeredi's Theorem for four term progressions, and the result of Shkredov on two-dimensional corners. The principal tool are the Gowers Box Norms."}
{"category": "Math", "title": "Torsion Invariants for Families", "abstract": "We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments."}
{"category": "Math", "title": "On the size of identifying codes in binary hypercubes", "abstract": "We consider identifying codes in binary Hamming spaces F^n, i.e., in binary hypercubes. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks. Let C be a subset of F^n. For any subset X of F^n, denote by I_r(X)=I_r(C;X) the set of elements of C within distance r from at least one x in X. Now C is called an (r,<= l)-identifying code if the sets I_r(X) are distinct for all subsets X of size at most l. We estimate the smallest size of such codes with fixed l and r/n converging to some number rho in (0,1). We further show the existence of such a code of size O(n^{3/2}) for every fixed l and r slightly less than n/2, and give for l=2 an explicit construction of small such codes for r the integer part of n/2-1 (the largest possible value)."}
{"category": "Math", "title": "Torsion dans un produit de courbes elliptiques", "abstract": "Let $A$ be an abelian variety defined over a number field $K$, the number of torsion points rational over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. We formulate a question suggesting the optimal exponent for this bound in terms of the dimension of the Mumford-Tate groups of the abelian subvarieties of $A$; we study the behaviour under product and then give a positive answer to our question when $A$ is the product of elliptic curves."}
{"category": "Math", "title": "The Clustering Coefficient of a Scale-Free Random Graph", "abstract": "We consider a random graph process in which, at each time step, a new vertex is added with m out-neighbours, chosen with probabilities proportional to their degree plus a strictly positive constant. We show that the expectation of the clustering coefficient of the graph process is asymptotically proportional to log n/n. Bollob\\'as and Riordan have previously shown that when the constant is zero, the same expectation is asymptotically proportional to ((log n)^2)/n."}
{"category": "Math", "title": "A Simple Sample Size Formula for Estimating Means of Poisson Random Variables", "abstract": "In this paper, we derive an explicit sample size formula based a mixed criterion of absolute and relative errors for estimating means of Poisson random variables."}
{"category": "Math", "title": "Distance graphs in vector spaces over finite fields, coloring and pseudo-randomness", "abstract": "In this paper we systematically study various properties of the distance graph in ${\\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field ${\\Bbb F}_q$ with $q$ elements. In the process we compute the diameter of distance graphs and show that sufficiently large subsets of $d$-dimensional vector spaces over finite fields contain every possible finite configurations."}
{"category": "Math", "title": "Absolute continuity for some one-dimensional processes", "abstract": "We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with H\\\"{o}lder continuous coefficients. Furthermore, we allow such coefficients to be random and to depend on the whole path of the solution. We also show how it can be extended to some stochastic partial differential equations and to some L\\'{e}vy-driven stochastic differential equations. In the cases under study, the Malliavin calculus cannot be used, because the solution in generally not Malliavin differentiable."}
{"category": "Math", "title": "Galois extensions for coquasi-Hopf algebras", "abstract": "The notions of Galois and cleft extensions are generalized for coquasi-Hopf algebras. It is shown that such an extension over a coquasi-Hopf algebra is cleft if and only if it is Galois and has the normal basis property. A Schneider type theorem is proven for coquasi-Hopf algebras with bijective antipode. As an application, we generalize Schauenburg's bialgebroid construction for coquasi-Hopf algebras."}
{"category": "Math", "title": "Grid Diagrams and Legendrian Lens Space Links", "abstract": "Grid diagrams encode useful geometric information about knots in S^3. In particular, they can be used to combinatorially define the knot Floer homology of a knot K in S^3, and they have a straightforward connection to Legendrian representatives of K in (S^3, \\xi_\\st), where \\xi_\\st is the standard, tight contact structure. The definition of a grid diagram was extended to include a description for links in all lens spaces, resulting in a combinatorial description of the knot Floer homology of a knot K in L(p, q) for all p > 0. In the present article, we explore the connection between lens space grid diagrams and the contact topology of a lens space. Our hope is that an understanding of grid diagrams from this point of view will lead to new approaches to the Berge conjecture, which claims to classify all knots in S^3 upon which surgery yields a lens space."}
{"category": "Math", "title": "Conductor inequalities and criteria for Sobolev-Lorentz two-weight inequalities", "abstract": "In this paper we present integral conductor inequalities connecting the Lorentz p,q-(quasi)norm of a gradient of a function to a one-dimensional integral of the p,q-capacitance of the conductor between two level surfaces of the same function. These inequalities generalize an inequality obtained by the second author in the case of the Sobolev norm. Such conductor inequalities lead to necessary and sufficient conditions for Sobolev-Lorentz type inequalities involving two arbitrary measures."}
{"category": "Math", "title": "Small parts in the Bernoulli sieve", "abstract": "Sampling from a random discrete distribution induced by a `stick-breaking' process is considered. Under a moment condition, it is shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the halfline."}
{"category": "Math", "title": "Beyond Chaos", "abstract": "The first part of this paper defines recursive interactions by means of logistic functions and derives a general result on the way interacting systems evolve in attractors. It also defines the notion of coevolution trajectory and presents a new family of attractors: orbital attractors (including single, irregular, folded, complex and discontinuous orbits). The second part summarizes the results of a first experimental analysis of recursive interactions in both binary and multiple interactions. Among other results, this analysis reveals that interacting systems may easily evolve from chaos to order."}
{"category": "Math", "title": "Degeneracy loci of families of Dirac operators", "abstract": "Generalizing some results from R. Leung's thesis, we compute, in rational cohomology, the Poincare dual of the degeneracy locus of the family of Dirac operators parameterized by the moduli space of projectively anti-self-dual $\\SO(3)$ connections. This is the first step in a program to derive a relation between the Donaldson and spin invariants."}
{"category": "Math", "title": "Integral equalities for functions of unbounded spectral operators in Banach spaces", "abstract": "We investigate a limiting procedure for extending local integral operator equalities to the global ones and to applying it to obtaining generalizations of the Newton-Leibnitz formula for operator-valued maps for a wide class of unbounded operators."}
{"category": "Math", "title": "Determinant computations for some classes of Toeplitz-Hankel matrices", "abstract": "The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of the form $ (a_{i-j} \\pm a_{i+j+1-k})_{i,j=0... N-1} $ with $k$ is fixed. We will show that this example as well as some general classes of operators have expansions that are similar to those that appear in the Strong Szeg\\\"{o} Limit Theorem. We also obtain exact identitities for some of the determinants that are analogous to the one derived independently by Geronimo and Case and by Borodin and Okounkov for finite Toeplitz matrices. These problems were motivated by considering certain statistical quantities that appear in random matrix theory."}
{"category": "Math", "title": "(q,t)-analogues and GL_n(F_q)", "abstract": "We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q,t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonald's ``seventh variation'' of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GL_n(F_q)."}
{"category": "Math", "title": "Refined convergence for the Boolean model", "abstract": "In a previous work, two of the authors proposed a new proof of a well known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process. In this paper, we consider the particular case of the two-dimensional Boolean model where the grains are discs with random radii. We investigate the second-order term in this convergence when the Boolean model and the Poisson line process are coupled on the same probability space. A precise coupling between the Boolean model and the Poisson line process is first established, a result of directional convergence in distribution for the difference of the two sets involved is derived as well."}
{"category": "Math", "title": "A characterization of dimension free concentration in terms of transportation inequalities", "abstract": "The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's $\\T_2$ transportation-cost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincar\\'e inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques."}
{"category": "Math", "title": "Unfolding a Codimension-Two, Discontinuous, Andronov-Hopf Bifurcation", "abstract": "We present an unfolding of the codimension-two scenario of the simultaneous occurrence of a discontinuous bifurcation and an Andronov-Hopf bifurcation in a piecewise-smooth, continuous system of autonomous ordinary differential equations in the plane. We find the Hopf cycle undergoes a grazing bifurcation that may be very shortly followed by a saddle-node bifurcation of the orbit. We derive scaling laws for the bifurcation curves that emanate from the codimension-two bifurcation."}
{"category": "Math", "title": "Teichm\\\"uller Structures and Dual Geometric Gibbs Type Measure Theory for Continuous Potentials", "abstract": "The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to have such a theory in general. However, we develop a dual geometric Gibbs type measure theory for certain continuous potentials in this paper following some ideas and techniques from Teichm\\\"uller theory for Riemann surfaces. Furthermore, we prove that the space of those continuous potentials has a Teichm\\\"uller structure. Moreover, this Teichm\\\"uller structure is a complete structure and is the completion of the space of smooth potentials under this Teichm\\\"uller structure. Thus our dual geometric Gibbs type theory is the completion of the Gibbs measure theory for smooth potentials from the dual geometric point of view."}
{"category": "Math", "title": "Subellipticity of the $\\bar\\partial$-Neumann problem on a weakly $q$-pseudoconvex/concave domain", "abstract": "For a domain $D$ of $\\mathbb{C}^n$ which is weakly $q$-pseudoconvex or $q$-pseudoconcave we give a sufficient condition for subelliptic estimates for the $\\bar{\\partial}$-Neumann problem. The paper extends to domains which are not necessarily pseudoconvex, the results and the techniques of Catlin. MSC: 32D10, 32U05, 32V25"}
{"category": "Math", "title": "Feedback Control and the Arrow of Time", "abstract": "The purpose of this paper is to highlight the central role that the time asymmetry of stability plays in feedback control. We show that this provides a new perspective on the use of doubly-infinite or semi-infinite time axes for signal spaces in control theory. We then focus on the implication of this time asymmetry in modeling uncertainty, regulation and robust control. We point out that modeling uncertainty and the ease of control depend critically on the direction of time. We also discuss the relationship of this control-based time-arrow with the well known arrows of time in physics."}
{"category": "Math", "title": "A unified construction yielding precisely Hilbert and James sequences spaces", "abstract": "Following James' approach, we shall define the Banach space $J(e)$ for each vector $e=(e_1,e_2,...,e_d) \\in \\Bbb{R}^d$ with $ e_1 \\ne 0$. The construction immediately implies that J(1) coincides with the Hilbert space $i_2$ and that $J(1;-1)$ coincides with the celebrated quasireflexive James space $J$. The results of this paper show that, up to an isomorphism, there are only the following two possibilities: (i) either $J(e)$ is isomorphic to $l_2$, if $e_1+e_2+...+e_d\\ne 0$ (ii) or $J(e)$ is isomorphic to $J$. Such a dichotomy also holds for every separable Orlicz sequence space $l_M$."}
{"category": "Math", "title": "Siegel modular forms mod p", "abstract": "We prove a vanishing theorem for one forms on the moduli stack of principally polarized abelian varieties of genus g>1 with level structure N over fields of characteristic p different from two. This is used to compute the Picard groups of these stacks. As an application we obtain results one congruences between integral Siegel modular forms."}
{"category": "Math", "title": "Some applications of the Beta function", "abstract": "This short note deals with some applications of the Beta function"}
{"category": "Math", "title": "Interlaced processes on the circle", "abstract": "When two Markov operators commute, it suggests that we can couple two copies of one of the corresponding processes. We explicitly construct a number of couplings of this type for a commuting family of Markov processes on the set of conjugacy classes of the unitary group, using a dynamical rule inspired by the RSK algorithm. Our motivation for doing this is to develop a parallel programme, on the circle, to some recently discovered connections in random matrix theory between reflected and conditioned systems of particles on the line. One of the Markov chains we consider gives rise to a family of Gibbs measures on `bead configurations' on the infinite cylinder. We show that these measures have determinantal structure and compute the corresponding space-time correlation kernel."}
{"category": "Math", "title": "Ruan's Conjecture on Singular symplectic flops", "abstract": "We prove that the orbifold quantum ring is preserved under singular symplectic flops. Hence we verify Ruan's conjecture for this case."}
{"category": "Math", "title": "Singular symplectic flops and Ruan cohomology", "abstract": "In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient $$ W_r=\\{(x,y,z,t)|xy-z^{2r}+t^2=0 \\}/\\mu_r(a,-a,1,0), r\\geq 1, $$ which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let $X$ and $Y$ be two symplectic orbifolds connected by such a flop. We study orbifold Gromov-Witten invariants of exceptional classes on $X$ and $Y$ and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan."}
{"category": "Math", "title": "Derived Categories and Birational Geometry", "abstract": "A survey article for AMS Summer Institute at Seattle in 2005."}
{"category": "Math", "title": "Finite generation of a canonical ring", "abstract": "The purpose of this note is to review an algebraic proof of the finite generation theorem due to Birkar-Cascini-Hacon-McKernan whose method is based on the Minimal Model Program. A survey article for Current Development in Mathematics 2007."}
{"category": "Math", "title": "Bayesian computation for statistical models with intractable normalizing constants", "abstract": "This paper deals with some computational aspects in the Bayesian analysis of statistical models with intractable normalizing constants. In the presence of intractable normalizing constants in the likelihood function, traditional MCMC methods cannot be applied. We propose an approach to sample from such posterior distributions. The method can be thought as a Bayesian version of the MCMC-MLE approach of Geyer and Thompson (1992). To the best of our knowledge, this is the first general and asymptotically consistent Monte Carlo method for such problems. We illustrate the method with examples from image segmentation and social network modeling. We study as well the asymptotic behavior of the algorithm and obtain a strong law of large numbers for empirical averages."}
{"category": "Math", "title": "A Lost Theorem: Definite Integrals in Asymptotic Setting", "abstract": "We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using Riemann sums. In our axiomatic approach even the proof of the existence of the definite integral (which does use Riemann sums) becomes slightly more elegant than the conventional one. We also discuss an interesting connection between our approach and the history of calculus. The article is written for readers who teach calculus and its applications. It might be accessible to students under a teacher's supervision and suitable for senior projects on calculus, real analysis, or history of mathematics."}
{"category": "Math", "title": "Hartman-Mycielski functor of non-metrizable compacta", "abstract": "We investigate some topological properties of a normal functor $H$ introduced earlier by Radul which is a certain functorial compactification of the Hartman-Mycielski construction $HM$. We show that $H$ is open and find the condition when $HX$ is an absolute retract homeomorphic to the Tychonov cube."}
{"category": "Math", "title": "The Arf-Kervaire invariant of framed manifolds as an obstruction to embeddability", "abstract": "We prove that no $14$-connected (resp. $30$-connected) stably parallelizable manifold $N^{30}$ (resp. $N^{62}$) of dimension $30$ (resp. $62$) with the Arf-Kervaire invariant 1 can be smoothly embedded into $\\mathbb{R}^{36}$ (resp. $\\mathbb{R}^{83}$)."}
{"category": "Math", "title": "Preprojective algebras and cluster algebras", "abstract": "We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups."}
{"category": "Math", "title": "Cram\\'{e}r asymptotics for finite time first passage probabilities of general L\\'{e}vy processes", "abstract": "We derive the exact asymptotics of $P(\\sup_{u\\leq t}X(u) > x)$ if $x$ and $t$ tend to infinity with $x/t$ constant, for a L\\'{e}vy process $X$ that admits exponential moments. The proof is based on a renewal argument and a two-dimensional renewal theorem of H\\\"{o}glund (1990)."}
{"category": "Math", "title": "Plane Jacobian Conjecture for rational polynomials", "abstract": "A non-zero constant Jacobian polynomial maps $F=(P,Q)$ of $\\mathbb{C}^2$ is invertible if $P$ and $Q$ are rational polynomials."}
{"category": "Math", "title": "Harold Jeffreys's Theory of Probability Revisited", "abstract": "Published exactly seventy years ago, Jeffreys's Theory of Probability (1939) has had a unique impact on the Bayesian community and is now considered to be one of the main classics in Bayesian Statistics as well as the initiator of the objective Bayes school. In particular, its advances on the derivation of noninformative priors as well as on the scaling of Bayes factors have had a lasting impact on the field. However, the book reflects the characteristics of the time, especially in terms of mathematical rigor. In this paper we point out the fundamental aspects of this reference work, especially the thorough coverage of testing problems and the construction of both estimation and testing noninformative priors based on functional divergences. Our major aim here is to help modern readers in navigating in this difficult text and in concentrating on passages that are still relevant today."}
{"category": "Math", "title": "Higher Green's functions for modular forms", "abstract": "Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, but instead of the usual equation $\\Delta f=0$ we have equation $\\Delta f = k(1-k) f$. Here $k$ is a positive integer. Properties of these functions are related to the space of modular forms of weight $2k$. In the case when there are no cusp forms of weight $2k$ it was conjectured that the values of the Green function at points of complex multiplication are algebraic multiples of logarithms of algebraic numbers. We show that this conjecture can be proved in any particular case if one constructs a family of elements of certain higher Chow groups on the power of a family of elliptic curves. These families have to satisfy certain properties. A different family of elements of Higher Chow groups is needed for a different point of complex multiplication. We give an example of such families, thereby proving the conjecture for the case when the group is $PSL_2(\\mathbf Z)$, $k=2$ and one of the arguments is $\\sqrt{-1}$."}
{"category": "Math", "title": "A summation by Gencev", "abstract": "Gencev has recently reported a closed form summation for an infinite series involving the harmonic numbers and the central binomial numbers. This note indicates a possible approach to the proof involving the dilogarithm function."}
{"category": "Math", "title": "Representations of unramified U(2,2) over a p-adic field I: representations of non-integral level", "abstract": "Let F_0 be a non-archimedean local field of odd residual characteristic and let G be the unramified unitary group U(2,2) defined over F_0. In this paper, we give a classification of the irreducible smooth representations of G of non-integral level using the Hecke algebraic method developed by Allen Moy for GSp(4)."}
{"category": "Math", "title": "Computable counter-examples to the Brouwer fixed-point theorem", "abstract": "This paper is an overview of results that show the Brouwer fixed-point theorem (BFPT) to be essentially non-constructive and non-computable. The main results, the counter-examples of Orevkov and Baigger, imply that there is no procedure for finding the fixed point in general by giving an example of a computable function which does not fix any computable point. Research in reverse mathematics has shown the BFPT to be equivalent to the weak K\\\"onig lemma in RCA$_0$ (the system of recursive comprehension) and this result is illustrated by relating the weak K\\\"onig lemma directly to the Baigger example."}
{"category": "Math", "title": "Stability of Universal Equivalence of Groups under Free Constructions", "abstract": "In 1971 J. Stallings introduced a generalisation of amalgamated products of groups -- called a pregroup, which is a particular kind of a partial group. He defined the universal group U(P) of a pregroup P to be a universal object (in the sense of category theory) extending the partial operations on P to group operations on U(P). This turns out to be a versatile and convenient generalisation of classical group constructions: HNN-extensions and amalgamated products. In this respect the following general question arises. Which properties of pregroups, or relations between pregroups, carry over to the respective universal groups? In this paper it is proved that universal equivalence of pregroups extends to universal equivalence of universal groups. Applications to free products with amalgamation and HNN-extensions are then described."}
{"category": "Math", "title": "A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators", "abstract": "Let $E$ be a closed set on the unit circle. We find a Blaschke-type condition, optimal in a sense of the order, on the Riesz measure of a subharmonic function $v$ in the unit disk with a certain growth at the direction of $E$. In particular case when $E$ is a finite set, and $v=\\log|f|$ with an analytic function $f$, our result agrees with the recent one by A. Borichev, L. Golinskii and S. Kupin. An application to contractions close to unitary operators in the Hilbert space is given."}
{"category": "Math", "title": "A structural approach to subset-sum problems", "abstract": "We discuss a structural approach to subset-sum problems in additive combinatorics. The core of this approach are Freiman-type structural theorems, many of which will be presented through the paper. These results have applications in various areas, such as number theory, combinatorics and mathematical physics."}
{"category": "Math", "title": "Poisson automorphisms and quiver moduli", "abstract": "A factorization formula for certain automorphisms of a Poisson algebra associated to a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing formulas for Donaldson-Thomas type invariants of M. Kontsevich and Y. Soibelman."}
{"category": "Math", "title": "A tessellation for algebraic surfaces in CP3", "abstract": "We present an explicit algorithm for tessellating the algebraic surfaces (real 4-manifolds) F(n) embedded in CP3 defined by the equation z0^n + z1^n + z2^n + z3^n = 0 in the standard homogeneous coordinates [z0, z1, z2, z3], where n is any positive integer. Note that F(4) in particular is a K3 surface. Our tessellation contains a minimal number of vertices, namely the n-th roots of unity in the six standard projective lines CP1 in CP3, which form an obvious framework for constructing a natural tessellation of F(n). Our tessellation is invariant under the action of the obvious isomorphism group of F(n) induced by permutations and phase multiplications of the coordinates, and the action is transitive on the set of 4-cells. The tessellation is built upon a similar triangulation for the corresponding algebraic curves in CP2."}
{"category": "Math", "title": "Stability Functions", "abstract": "In this article we discuss the role of stability functions in geometric invariant theory and apply stability function techniques to problems in toric geometry. In particular we show how one can use these techniques to recover results of Burns-Guillemin-Uribe and Shiffman-Tate-Zelditch on asymptotic properties of sections of holomorphic line bundles over toric varieties."}
{"category": "Math", "title": "On generators of bounded ratios of minors for totally positive matrices", "abstract": "We provide a method for factoring all bounded ratios of the form $$\\det A(I_1|I_1')\\det A(I_2|I_2')/\\det A(J_1|J_1')\\det A(J_2|J_2')$$ where $A$ is a totally positive matrix, into a product of more elementary ratios each of which is bounded by 1, thus giving a new proof of Skandera's result. The approach we use generalizes the one employed by Fallat et al. in their work on principal minors. We also obtain a new necessary condition for a ratio to be bounded for the case of non-principal minors."}
{"category": "Math", "title": "A generalization of Ostrowski inequality on time scales for k points", "abstract": "In this paper we first generalize the Ostrowski inequality on time scales for k points and then unify corresponding continuous and discrete versions. We also point out some particular Ostrowski type inequalities on time scales as special cases."}
{"category": "Math", "title": "An Ostrowski-Gruss type inequality on time scales", "abstract": "In this paper we derive a new inequality of Ostrowski-Gruss type on time scales and thus unify corresponding continuous and discrete versions. We also apply our result to the quantum calculus case."}
{"category": "Math", "title": "Fundamental classes of negatively curved manifolds cannot be represented by products of manifolds", "abstract": "Not every singular homology class is the push-forward of the fundamental class of some manifold. In the same spirit, one can study the following problem: Which singular homology classes are the push-forward of the fundamental class of a given type of manifolds? In the present article, we show that the fundamental classes of negatively curved manifolds cannot be represented by a non-trivial product of manifolds. This observation sheds some light on the functorial semi-norm on singular homology given by products of compact surfaces."}
{"category": "Math", "title": "A Conversation with Seymour Geisser", "abstract": "Seymour Geisser received his bachelor's degree in Mathematics from the City College of New York in 1950, and his M.A. and Ph.D. degrees in Mathematical Statistics at the University of North Carolina in 1952 and 1955, respectively. He then held positions at the National Bureau of Standards and the National Institute of Mental Health until 1961. From 1961 until 1965, he was Chief of the Biometry Section at the National Institute of Arthritis and Metabolic Diseases, and also held the position of Professorial Lecturer at the George Washington University from 1960 to 1965. From 1965 to 1970, he was the founding Chair of the Department of Statistics at the State University of New York, Buffalo, and in 1971, he became the founding Director of the School of Statistics at the University of Minnesota, remaining in that position until 2001. He held visiting professorships at Iowa State University, 1960; University of Wisconsin, 1964; University of Tel-Aviv (Israel), 1971; University of Waterloo (Canada), 1972; Stanford University, 1976, 1977, 1988; Carnegie Mellon University, 1976; University of the Orange Free State (South Africa), 1978, 1993; Harvard University, 1981; University of Chicago, 1985; University of Warwick (England), 1986; University of Modena (Italy), 1996; and National Chiao Tung University (Taiwan), 1998. He was the Lady Davis Visiting Professor, Hebrew University of Jerusalem, 1991, 1994, 1999, and the Schor Scholar, Merck Research Laboratories, 2002-2003. He was a Fellow of the Institute of Mathematical Statistics and the American Statistical Association."}
{"category": "Math", "title": "A Conversation with Monroe Sirken", "abstract": "Born January 11, 1921 in New York City, Monroe Sirken grew up in a suburb of Pasadena, California. He earned B.A. and M.A. degrees in sociology at UCLA in 1946 and 1947, and a Ph.D. in 1950 in sociology with a minor in mathematics at the University of Washington in 1950 where Professor Z. W. Birnbaum was his mentor and thesis advisor. As a Post-Doctoral Fellow of the Social Science Research Council, Monroe spent 1950--1951 at the Statistics Laboratory, University of California at Berkeley and the Office of the Assistant Director for Research, U.S. Bureau of the Census in Suitland, Maryland. Monroe visited the Census Bureau at a time of great change in the use of sampling and survey methods, and decided to remain. He began his government career there in 1951 as a mathematical statistician, and moved to the National Office of Vital Statistics (NOVS) in 1953 where he was an actuarial mathematician and a mathematical statistician. He has held a variety of research and administrative positions at the National Center for Health Statistics (NCHS) and he was the Associate Director, Research and Methodology and the Director, Office of Research and Methodology until 1996 when he became a senior research scientist, the title he currently holds. Aside from administrative responsibilities, Monroe's major professional interests have been conducting and fostering survey and statistical research responsive to the needs of federal statistics. His interest in the design of rare and sensitive population surveys led to the development of network sampling which improves precision by linking multiple selection units to the same observation units. His interest in fostering research on the cognitive aspects of survey methods led to the establishment of permanent questionnaire design research laboratories, first at NCHS and later at other federal statistical agencies here and abroad."}
{"category": "Math", "title": "Poisson type generators for L^1(R)", "abstract": "We characterize the discrete sets L of the real line such that {f(t-l), l in L} span L^1(R), f being an L^1(R)-function whose Fourier transform behaves like the one of the Poisson function."}
{"category": "Math", "title": "Model spaces results for the Gabor and Wavelet transforms", "abstract": "We prove that the unique Gabor atom with analytical model space is the Gaussian function. We give an analogous result for the wavelet transform. For the general case we give a new approach to study the irregular Gabor and wavelet frames. We improve some results for Gabor atoms in the Feichtinger algebra, and for a special class of wavelets."}
{"category": "Math", "title": "Torus fibrations and localization of index I", "abstract": "We define a local Riemann-Roch number for an open symplectic manifold when a complete integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the number of nonsingular Bohr-Sommerfeld fibers and a contribution of the singular fibers. A key step of the proof is formally explained as a version of Witten's deformation applied to a Hilbert bundle."}
{"category": "Math", "title": "Milnor $K$-group attached to a torus and Birch-Tate conjecture", "abstract": "We formulate (and prove under a certain assumption) a conjecture relating the order of Somekawa's Milnor $K$-group attached to a torus $T$ and the value of the Artin $L$-function attached to the cocharacter group of $T$ (regarded as an Artin representation) at $s=-1$. The case $T=\\G_m$ reduces to the classical Birch-Tate conjecture."}
{"category": "Math", "title": "Lifts of matroid representations over partial fields", "abstract": "There exist several theorems which state that when a matroid is representable over distinct fields F_1,...,F_k, it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. First, parts of Whittle's characterization of representations of ternary matroids follow from our theorem. Second, we prove the following theorem by Vertigan: if a matroid is representable over both GF(4) and GF(5), then it is representable over the real numbers by a matrix such that the absolute value of the determinant of every nonsingular square submatrix is a power of the golden ratio. Third, we give a characterization of the 3-connected matroids having at least two inequivalent representations over GF(5). We show that these are representable over the complex numbers. Additionally we provide an algebraic construction that, for any set of fields F_1,...,F_k, gives the best possible result that can be proven using the Lift Theorem."}
{"category": "Math", "title": "Optimal stopping for L\\'evy processes and affine functions", "abstract": "This paper studies an optimal stopping problem for L\\'evy processes. We give a justification of the form of the Snell envelope using standard results of optimal stopping. We also justify the convexity of the value function, and without a priori restriction to a particular class of stopping times, we deduce that the smallest optimal stopping time is necessarily a hitting time. We propose a method which allows to obtain the optimal threshold. Moreover this method allows to avoid long calculations of the integro-differential operatorused in the usual proofs."}
{"category": "Math", "title": "A characterization of double covers of curves in terms of the ample cone of second symmetric product", "abstract": "We investigate the nef cone spanned by the diagonal and the fibre classes of second symmetric product of a curve of genus $g$. This 2-dimensional nef cone gives a characterization of double covers of curves of genus $\\le \\frac{g-1}{8}$. This is a generalization of a result by Debarre."}
{"category": "Math", "title": "Simulation of stochastic reaction-diffusion processes on unstructured meshes", "abstract": "Stochastic chemical systems with diffusion are modeled with a reaction-diffusion master equation. On a macroscopic level, the governing equation is a reaction-diffusion equation for the averages of the chemical species. On a mesoscopic level, the master equation for a well stirred chemical system is combined with Brownian motion in space to obtain the reaction-diffusion master equation. The space is covered by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the Laplace operator on the macroscale. The resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso- or on the macroscale level. The accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology."}
{"category": "Math", "title": "Principal basis in Cartan subalgebra", "abstract": "Let $\\Lie{g}$ be a simple complex Lie algebra and $\\Lie{h}$ a Cartan subalgebra. In this article we explain how to obtain the principal basis of $\\Lie{h}$ starting form a set of generators $\\{p_1,...,p_r\\}$,$r=\\rank(\\Lie{g})$, of the invariants polynomials $\\Sgg$. For each invariant polynomial $p$, we define a $G$-equivariant map $Dp$ form $\\Lie{g}$ to $\\Lie{g}$. We show that the Gram-Schmidt orthogonalization of the elements $\\{Dp_1(\\rho^\\vee), ... Dp_r(\\rho^\\vee) \\}$ gives the principal basis of $\\Lie{h}$. Similarly the orthogonalization of the elements $\\{Dp_1(\\rho), >... Dp_r(\\rho) \\}$ produces the principal basis of the Cartan subalgebra of $\\lLie{g}$, the Langlands dual of $\\Lie{g}$."}
{"category": "Math", "title": "On the H\\\"ormander multiplier theorem and modulation spaces", "abstract": "It is known that the Sobolev space $L^2_s$ with $s>n/2$ appeared in the H\\\"ormander multiplier theorem can be replaced by the Besov space $B^{2,1}_{n/2}$. On the other hand, the Besov space $B_{n/2}^{2,1}$ is continuously embedded in the modulation space $M^{2,1}_0$. In this paper, we consider the problem whether we can replace $B_{n/2}^{2,1}$ by $M^{2,1}_0$."}
{"category": "Math", "title": "Geometry of Carnot--Carath\\'{e}odory Spaces, Differentiability and Coarea Formula", "abstract": "We give a simple proof of Gromov's Theorem on nilpotentization of vector fields, and exhibit a new method for obtaining quantitative estimates of comparing geometries of two different local Carnot groups in Carnot--Carath\\'{e}odory spaces with $C^{1,\\alpha}$-smooth basis vector fields, $\\alpha\\in[0,1]$. From here we obtain the similar estimates for comparing geometries of a Carnot--Carath\\'{e}odory space and a local Carnot group. These two theorems imply basic results of the theory: Gromov type Local Approximation Theorems, and for $\\alpha>0$ Rashevski\\v{\\i}-Chow Theorem and Ball--Box Theorem, etc. We apply the obtained results for proving $hc$-differentiability of mappings of Carnot--Carath\\'{e}odory spaces with continuous horizontal derivatives. The latter is used in proving the coarea formula for some classes of contact mappings of Carnot--Carath\\'{e}odory spaces."}
{"category": "Math", "title": "The Anti-Symmetric GUE Minor Process", "abstract": "Our study is initiated by a multi-component particle system underlying the tiling of a half hexagon by three species of rhombi. In this particle system species $j$ consists of $\\lfloor j/2 \\rfloor$ particles which are interlaced with neigbouring species. The joint probability density function (PDF) for this particle system is obtained, and is shown in a suitable scaling limit to coincide with the joint eigenvalue PDF for the process formed by the successive minors of anti-symmetric GUE matrices, which in turn we compute from first principles. The correlations for this process are determinantal and we give an explicit formula for the corresponding correlation kernel in terms of Hermite polynomials. Scaling limits of the latter are computed, giving rise to the Airy kernel, extended Airy kernel and bead kernel at the soft edge and in the bulk, as well as a new kernel at the hard edge."}
{"category": "Math", "title": "Entire functions of exponential type, almost periodic in Besicovitch's sense on the real hyperplane", "abstract": "Suppose that an almost periodic in Besicovitch's sense function $f(x)$ of several variables is the restriction to the real hyperplane of an entire function of exponential type $b$. Then its spectrum is contained in the ball of radius $b$ with the center in the origin."}
{"category": "Math", "title": "Cluster algebras of finite type via Coxeter elements and principal minors", "abstract": "We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan-Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors."}
{"category": "Math", "title": "A proof of the Dalang-Morton-Willinger theorem", "abstract": "We give a new proof of the Dalang-Morton-Willinger theorem, relating the no-arbitrage condition in stochastic securities market models to the existence of an equivalent martingale measure with bounded density for a $d$-dimensional stochastic sequence $(S_n)_{n=0}^N$ of stock prices. Roughly speaking, the proof is reduced to the assertion that under the no-arbitrage condition for N=1 and $S\\in L^1$ there exists a strictly positive linear fucntional on $L^1$, which is bounded from above on a special subset of the subspace $K\\subset L^1$ of investor's gains."}
{"category": "Math", "title": "R-boundedness of smooth operator-valued functions", "abstract": "In this paper we study $R$-boundedness of operator families $\\mathcal{T}\\subset \\calL(X,Y)$, where $X$ and $Y$ are Banach spaces. Under cotype and type assumptions on $X$ and $Y$ we give sufficient conditions for $R$-boundedness. In the first part we show that certain integral operator are $R$-bounded. This will be used to obtain $R$-boundedness in the case that $\\mathcal{T}$ is the range of an operator-valued function $T:\\R^d\\to \\calL(X,Y)$ which is in a certain Besov space $B^{d/r}_{r,1}(\\R^d;\\calL(X,Y))$. The results will be applied to obtain $R$-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems."}
{"category": "Math", "title": "Tangencies between holomorphic maps and holomorphic laminations", "abstract": "We prove that the set of leaves of a holomorphic lamination of codimension one that are non-transversal to a germ of a holomorphic map is discrete."}
{"category": "Math", "title": "Periodic quotients of hyperbolic and large groups", "abstract": "Let $G$ be either a non-elementary (word) hyperbolic group or a large group (both in the sense of Gromov). In this paper we describe several approaches for constructing continuous families of periodic quotients of $G$ with various properties. The first three methods work for any non-elementary hyperbolic group, producing three different continua of periodic quotients of $G$. They are based on the results and techniques, that were developed by Ivanov and Olshanskii in order to show that there exists an integer $n$ such that $G/G^n$ is an infinite group of exponent $n$. The fourth approach starts with a large group $G$ and produces a continuum of pairwise non-isomorphic periodic residually finite quotients. Speaking of a particular application, we use each of these methods to give a positive answer to a question of Wiegold from Kourovka Notebook."}
{"category": "Math", "title": "Good Representations and Homogeneous Spaces", "abstract": "Let G be a real or complex linear algebraic reductive group. Let H and F be reductive subgroups. We study the natural H action on G/F. The main theorem of this note shows that generic H orbits are closed. This theorem is then applied to study representations of G, representations of H which are induced from representations of G, and intersections of reductive subgroups of G."}
{"category": "Math", "title": "Non-classical Godeaux Surfaces", "abstract": "A non-classical Godeaux surface is a minimal surface of general type with $\\chi=K^2=1$ but with $h^{01}\\neq0$. We prove that such surfaces fulfill $h^{01}=1$ and they can exist only over fields of positive characteristic at most 5. Like non-classical Enriques surfaces they fall into two classes: the singular and the supersingular ones. We give a complete classification in characteristic 5 and compute their Hodge-, Hodge--Witt- and crystalline cohomology (including torsion). Finally, we give an example of a supersingular Godeaux surface in characteristic 5."}
{"category": "Math", "title": "Classifying Dihedral O(2)-Equivariant Spectra", "abstract": "The category of rational O(2)-equivariant spectra splits as a product of cyclic and dihedral parts. Using the classification of rational G-equivariant spectra for finite groups G, we classify the dihedral part of rational O(2)-equivariant spectra in terms of an algebraic model."}
{"category": "Math", "title": "Isomorphisms preserving invariants", "abstract": "Let $V$ and $W$ be finite dimensional real vector spaces and let $G\\subset\\GL(V)$ and $H\\subset\\GL(W)$ be finite subgroups. Assume for simplicity that the actions contain no reflections. Let $Y$ and $Z$ denote the real algebraic varieties corresponding to $\\R[V]^G$ and $\\R[W]^H$, respectively. If $V$ and $W$ are quasi-isomorphic, i.e., if there is a linear isomorphism $L\\colon V\\to W$ such that $L$ sends $G$-orbits to $H$-orbits and $L\\inv$ sends $H$-orbits to $G$-orbits, then $L$ induces an isomorphism of $Y$ and $Z$. Conversely, suppose that $f\\colon Y\\to Z$ is a germ of a diffeomorphism sending the origin of $Y$ to the origin of $Z$. Then we show that $V$ and $W$ are quasi-isomorphic, This result is closely related to a theorem of Strub \\cite{Strub}, for which we give a new proof. We also give a new proof of a result of \\cite{KrieglLosikMichor03} on lifting of biholomorphisms of quotient spaces."}
{"category": "Math", "title": "A Method of Trend Extraction Using Singular Spectrum Analysis", "abstract": "The paper presents a new method of trend extraction in the framework of the Singular Spectrum Analysis (SSA) approach. This method is easy to use, does not need specification of models of time series and trend, allows to extract trend in the presence of noise and oscillations and has only two parameters (besides basic SSA parameter called window length). One parameter manages scale of the extracted trend and another is a method specific threshold value. We propose procedures for the choice of the parameters. The presented method is evaluated on a simulated time series with a polynomial trend and an oscillating component with unknown period and on the seasonally adjusted monthly data of unemployment level in Alaska for the period 1976/01-2006/09."}
{"category": "Math", "title": "Promotion and cyclic sieving via webs", "abstract": "We show that Sch\\\"utzenberger's promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work of the third author, we show that this action admits the cyclic sieving phenomenon."}
{"category": "Math", "title": "Uncountable Graphs and Invariant Measures on the Set of Universal Countable Graphs", "abstract": "We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $s\\geq 3$) that are invariant with respect to the group of all permutations of the vertices. Such measures can be regarded as random graphs (respectively, random $K_s$-free graphs). The well-known example of Erd\\\"os--R\\'enyi (ER) of the random graph corresponds to the Bernoulli measure on the set of adjacency matrices. For the case of the universal $K_s$-free graphs there were no previously known examples of the invariant measures on the space of such graphs. The main idea of our construction is based on the new notions of {\\it measurable universal}, and {\\it topologically universal} graphs, which are interesting themselves. The realization of the construction can be regarded as two-step randomization for universal measurable graph : {\\it \"randomization in vertices\"} and {\\it \"randomization in edges\"}. For $K_s$-free, $s\\geq 3$ there is only randomization in vertices of the measurable graphs. The completeness of our lists is proved using the important theorem by D. Aldous about $S_{\\infty}$-invariant matrices, which we reformulate in appropriate way."}
{"category": "Math", "title": "An iterative scheme for solving nonlinear equations with monotone operators", "abstract": "An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with monotone operators is proposed and its convergence is proved. A new discrepancy principle is proposed and justified. A priori and a posteriori stopping rules for the DSM algorithm are formulated and justified."}
{"category": "Math", "title": "A DSM proof of surjectivity of monotone nonlinear mappings", "abstract": "We prove that if $F$ is twice Frechet differentiable and coercivity conditions hold, then $F$ is surjective, i.e., the equation $F(u)=h$ is solvable for every $h\\in H$. This is a basic result in the theory of monotone operators. Our aim is to give a simple and short proof of this result based on the Dynamical Systems Method (DSM), developed in the monograph A.G. Ramm, Dynamical systems method, Elsevier, Amsterdam, 2007."}
{"category": "Math", "title": "Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$", "abstract": "Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solutions of the approximating Riemannian PDE and the ensuing $C^{\\infty}$ regularity of the sub-Riemannian minimal surface along its Legendrian foliation."}
{"category": "Math", "title": "Uniqueness and Stability in $\\mathcal E(X,\\omega)$", "abstract": "We prove uniqueness for the Dirichlet problem for the complex Monge-Amp\\`ere equation on compact K\\\"ahler manifolds in the case of measures vanishing on pluripolar sets. As a by-product we generalize Xing's stability theorem."}
{"category": "Math", "title": "Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups $H^n, n>1$", "abstract": "We prove that Lipschitz intrinsic graphs in the Heisenberg groups $H^n$, with $n>1$, which are vanishing viscosity solutions of the minimal surface equation are smooth."}
{"category": "Math", "title": "Random Search Algorithms for the Sparse Null Vector Problem", "abstract": "We consider the following problem: Given a matrix A, find minimal subsets of columns of A with cardinality no larger than a given bound that are linear dependent or nearly so. This problem arises in various forms in optimization, electrical engineering, and statistics. In its full generality, the problem is known to be NP-complete. We present a Monte Carlo method that finds such subsets with high confidence. We also give a deterministic method that is capable of proving that no subsets of linearly dependent columns up to a certain cardinality exist. The performance of both methods is analyzed and illustrated with numerical experiments."}
{"category": "Math", "title": "Voyage by Catamaran: Effecting Semantic Network \"Bricolage\" via Infinite-Dimensional Zero-Divisor Ensembles", "abstract": "Continuing arguments presented [1] or announced [2][3] in \"Complex Systems,\" zero-divisor (ZD) foundations for \"scale-free\" networks (evinced, in particular, in the \"fractality\" of the Internet) are decentralized. Spandrels, quartets of ZD-free or \"hidden\" box-kite-like structures (HBKs) in the 2^(N+1)- ions, are \"exploded\" from (and uniquely linked to) each standard box-kite in the 2^N-ions, N > 3. Any HBK houses, in a \"cowbird's nest,\" exactly one copy of the (ZD-free) octonions, the recursive basis for all ZD ensembles. Each is a potential waystation for alien-ensemble infiltration in the large, or metaphor-like jumps in the small. Cowbirding models what evolutionary biologists [4], and structural mythologist Claude Levi-Strauss before them [5], term bricolage: the opportunistic co-opting of objects designed for one purpose to serve others unrelated to it. Such arguments entail switching focus, from the octahedral box-kite's four triangular \"sails,\" to its trio of square \"catamarans\" and their box-kite-switching \"twist products.\""}
{"category": "Math", "title": "Covering data and higher dimensional global class field theory", "abstract": "For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism \\rho_X: C_X --> \\pi_1^\\ab(X), which is surjective and whose kernel is the connected component of the identity. The (topological) group C_X is explicitly given and built solely out of data attached to points and curves on X. A similar but weaker statement holds for smooth varieties over finite fields. Our results are based on earlier work of G. Wiesend."}
{"category": "Math", "title": "A family of 2-graphs arising from two-dimensional subshifts", "abstract": "Higher-rank graphs (or $k$-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz-Krieger $C^*$-algebras of Robertson and Steger. Here we consider a family of finite 2-graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others. We analyse the $C^*$-algebras of these 2-graphs, find criteria under which they are simple and purely infinite, and compute their $K$-theory. We find examples whose $C^*$-algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic to the $C^*$-algebras of ordinary directed graphs."}
{"category": "Math", "title": "Functional calculus extensions on dual spaces", "abstract": "In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply this result to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one."}
{"category": "Math", "title": "Stable Cohomotopy Seiberg-Witten Invariants of Connected Sums of Four-Manifolds with Positive First Betti Number", "abstract": "We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg-Witten invariant of connected sums of 4-manifolds with positive first Betti number. The non-vanishing theorem enables us to find many new examples of 4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and it also gives a partial, but strong affirmative answer to a conjecture concerning non-vanishing of the invariant. Various new applications of the non-vanishing theorem are also given. For example, we shall introduce variants $\\bar{\\lambda}_k$ of Perelman's $\\bar{\\lambda}$ invariants for real numbers $k$ and compute the values for a large class of 4-manifolds including connected sums of certain K{\\\"{a}}hler surfaces. The non-vanishing theorem is also used to construct the first examples of 4-manifolds with non-zero simplicial volume and satisfying the strict Gromov-Hitchin-Thorpe inequality, but admitting infinitely many distinct smooth structures for which no compatible Einstein metric exists. Moreover, we are able to prove a new result on the existence of exotic smooth structures."}
{"category": "Math", "title": "Pairings of Sheaves of $\\mathcal{A}$-Modules through Bilinear $\\mathcal{A}$-Morphisms", "abstract": "It is proved that for any free $\\mathcal{A}$-modules $\\mathcal{F}$ and $\\mathcal{E}$ of finite rank on some $\\mathbb{C}$-algebraized space $(X, \\mathcal{A})$ a \\textit{degenerate} bilinear $\\mathcal{A}$-morphism $\\Phi: \\mathcal{F}\\times \\mathcal{E}\\longrightarrow \\mathcal{A}$ induces a \\textit{non-degenerate} bilinear $\\mathcal{A}$-morphism $\\bar{\\Phi}: \\mathcal{F}/\\mathcal{E}^\\perp\\times \\mathcal{E}/\\mathcal{F}^\\perp\\longrightarrow \\mathcal{A}$, where $\\mathcal{E}^\\perp$ and $\\mathcal{F}^\\perp$ are the \\textit{orthogonal} sub-$\\mathcal{A}$-modules associated with $\\mathcal{E}$ and $\\mathcal{F}$, respectively. This result generalizes the finite case of the classical result, which states that given two vector spaces $W$ and $V$, paired into a field $k$, the induced vector spaces $W/V^\\perp$ and $V/W^\\perp$ have the same dimension. Some related results are discussed as well."}
{"category": "Math", "title": "Semi-bounded unitary representations of infinite-dimensional Lie groups", "abstract": "In this note we introduce the concept of a semi-bounded unitary representations of an infinite-dimensional Lie group $G$. Semi-boundedness is defined in terms of the corresponding momentum set in the dual $\\g'$ of the Lie algebra $\\g$ of $G$. After dealing with some functional analytic issues concerning certain weak-$*$-locally compact subsets of dual spaces, called semi-equicontinuous, we characterize unitary representations which are bounded in the sense that their momentum set is equicontinuous, we characterize semi-bounded representations of locally convex spaces in terms of spectral measures, and we also describe a method to compute momentum sets of unitary representations of reproducing kernel Hilbert spaces of holomorphic functions."}
{"category": "Math", "title": "Bethe-Sommerfeld conjecture for pseudodifferential perturbation", "abstract": "We consider a periodic pseudodifferential operator $H=(-\\Delta)^l+A$ ($l>0$) in $\\R^d$ which satisfies the following conditions: (i) the symbol of $H$ is smooth in $x$, and (ii) the perturbation $A$ has order smaller than $2l-1$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line."}
{"category": "Math", "title": "Affine quotients of supergroups", "abstract": "In this article we consider sheaf quotients of affine superschemes by affine supergroups that act on them freely. The necessary and sufficient conditions for such quotients to be affine are given. If $G$ is an affine supergroup and $H$ is its normal supersubgroup, then we prove that a dur $K$-sheaf $\\tilde{\\tilde{G/H}}$ is again affine supergroup. Additionally, if $G$ is algebraic, then a $K$-sheaf $\\tilde{G/H}$ is also algebraic supergroup and it coincides with $\\tilde{\\tilde{G/H}}$. In particular, any normal supersubgroup of an affine supergroup is faithfully exact."}
{"category": "Math", "title": "Large p-groups actions with |G| /g^2 > 4/ (p^2-1)^2", "abstract": "Let k be an algebraically closed field of characteristic p>0 and C a connected nonsingular projective curve over k with genus g>1. Let (C,G) be a \"big action\", i.e. a pair (C,G) where G is a p-subgroup of the k-automorphism group of C such that |G|/g > 2p / p-1. We first study finiteness results on the values taken by the quotient |G|/g^2 when (C,G) runs over the big actions satisfying |G|/g^2 >M, for a given positive real M>0. Then, we exhibit a classification and a parametrization of such big actions when M=4/ (p^2-1)^2."}
{"category": "Math", "title": "On the Kernel of the affine Dirac operator", "abstract": "Let L be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form, \\sigma an elliptic automorphism of L leaving the form invariant, and A a \\sigma-invariant reductive subalgebra of L, such that the restriction of the form to A is non-degenerate. Consider the associated twisted affine Lie algebras L^, A^, and let F be the \\sigma-twisted Clifford module over A^ associated to the orthocomplement of A in L. Under suitable hypotheses on\\sigma and A, we provide a general formula for the decomposition of the kernel of the affine Dirac operator, acting on the tensor product of an integrable highest weight L^-module and F, into irreducible A^-submodules. As an application, we derive the decomposition of all level 1 integrable irreducible highest weight modules over orthogonal affine Lie algebras with respect to the affinization of the isotropy subalgebra of an arbitrary symmetric space."}
{"category": "Math", "title": "Arr\\^et optimal pour les processus de Markov forts et les fonctions affines", "abstract": "In this Note we study optimal stopping problems for strong Markov processes and affine functions. We give a justification of the Snell envelope form using standard results of optimal stopping. We also justify the convexity of the value function, and without a priori restriction to a particular class of stopping times, we deduce that the smallest optimal stopping time is necessarily a hitting time."}
{"category": "Math", "title": "Random Walk in deterministically changing environment", "abstract": "We consider a random walk with transition probabilities weakly dependent on an environment with a deterministic, but strongly chaotic, evolution. We prove that for almost all initial conditions of the environment the walk satisfies the CLT."}
{"category": "Math", "title": "A geometric study of Wasserstein spaces: Euclidean spaces", "abstract": "We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) \"exotic\" isometric flow. This contrasts with the case of higher-dimensional Euclidean spaces, where all isometries of the Wasserstein space preserve the shape of measures. We also study the curvature and various ranks of these spaces."}
{"category": "Math", "title": "Hardy type inequality in variable Lebesgue spaces", "abstract": "We prove that in variable exponent spaces $L^{p(\\cdot)}(\\Omega)$, where $p(\\cdot)$ satisfies the log-condition and $\\Omega$ is a bounded domain in $\\mathbf R^n$ with the property that $\\mathbf R^n \\backslash \\bar{\\Omega}$ has the cone property, the validity of the Hardy type inequality $$| 1/\\delta(x)^\\alpha \\int_\\Omega \\phi(y) dy/|x-y|^{n-\\alpha}|_{p(\\cdot)} \\leqq C |\\phi|_{p(\\cdot)}, \\quad 0<\\al<\\min(1,\\frac{n}{p_+})$$, where $\\delta(x)=\\mathrm{dist}(x,\\partial\\Omega)$, is equivalent to a certain property of the domain $\\Om$ expressed in terms of $\\al$ and $\\chi_\\Om$."}
{"category": "Math", "title": "Khinchin theorem for integral points on quadratic varieties", "abstract": "We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often."}
{"category": "Math", "title": "A note on 2-subset-regular self-complementary 3-uniform hypergraphs", "abstract": "We show that a 2-subset-regular self-complementary 3-uniform hypergraph with $n$ vertices exists if and only if $n\\ge 6$ and $n$ is congruent to 2 modulo 4."}
{"category": "Math", "title": "On the classification of links up to finite type", "abstract": "We use an action, of 2l-component string links on l-component string links, defined by the first author and Xiao-Song Lin, to lift the indeterminacy of finite type link invariants. The set of links up to this new indeterminacy is in bijection with the orbit space of the restriction of this action to the stabilizer of the identity. Structure theorems for the sets of links up to C_n-equivalence and Self-C_n-equivalence are also given."}
{"category": "Math", "title": "Blow up and regularity for fractal Burgers equation", "abstract": "The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian $\\alpha < 1/2,$ and global existence as well as analyticity of solution for $\\alpha \\geq 1/2.$ We also prove the existence of solutions with very rough initial data $u_0 \\in L^p,$ $1 < p < \\infty.$ Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation."}
{"category": "Math", "title": "Some consequences of Schanuel's Conjecture", "abstract": "During the Arizona Winter School 2008 (held in Tucson, AZ) we worked on the following problems: a) (Expanding a remark by S. Lang). Define $E_0 = \\overline{\\mathbb{Q}}$ Inductively, for $n \\geq 1$, define $E_n$ as the algebraic closure of the field generated over $E_{n-1}$ by the numbers $\\exp(x)=e^x$, where $x$ ranges over $E_{n-1}$. Let $E$ be the union of $E_n$, $n \\geq 0$. Show that Schanuel's Conjecture implies that the numbers $\\pi, \\log \\pi, \\log \\log \\pi, \\log \\log \\log \\pi, \\ldots $ are algebraically independent over $E$. b) Try to get a (conjectural) generalization involving the field $L$ defined as follows. Define $L_0 = \\overline{\\mathbb{Q}}$. Inductively, for $n \\geq 1$, define $L_n$ as the algebraic closure of the field generated over $L_{n-1}$ by the numbers $y$, where $y$ ranges over the set of complex numbers such that $e^y\\in L_{n-1}$. Let $L$ be the union of $L_n$, $n \\geq 0$. We were able to prove that Schanuel's Conjecture implies $E$ and $L$ are linearly disjoint over $\\overline{\\mathbb{Q}}$."}
{"category": "Math", "title": "Homological symbols and the Quillen conjecture", "abstract": "We formulate a \"correct\" version of the Quillen conjecture on linear group homology for certain arithmetic rings and provide evidence for the new conjecture. In this way we predict that the linear group homology has a direct summand looking like an unstable form of Milnor K-theory and we call this new theory \"homological symbols algebra\". As a byproduct we prove the Quillen conjecture in homological degree two for the rank two and the prime 5."}
{"category": "Math", "title": "Sur les quotients discrets de semi-groupes complexes", "abstract": "Let $X=G/K$ be an irreducible Hermitian symmetric space of the non-compact type and let $S\\in G^\\mbb{C}$ be the associated compression semi-group. Let $\\Gamma$ be a discrete subgroup of $G$. We give a sufficient condition for $\\Gamma\\backslash S$ to be a Stein manifold. Moreover, we show that in general $\\Gamma\\backslash S$ is not Stein, which disproves a conjecture by Achab, Betten and Kr\\\"otz."}
{"category": "Math", "title": "Exponential Dichotomy and Trichotomy for Skew-Evolution Semiflows in Banach Spaces", "abstract": "The paper emphasizes the properties of exponential dichotomy and exponential trichotomy for skew-evolution semiflows in Banach spaces, by means of evolution semiflows and evolution cocycles. The approach is from uniform point of view. Some characterizations which generalize classic results are also provided."}
{"category": "Math", "title": "Log minimal models according to Shokurov", "abstract": "Following Shokurov's ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension d implies that any klt pair of dimension d has a log minimal model or a Mori fibre space. Thus, in particular, any klt pair of dimension 4 has a log minimal model or a Mori fibre space."}
{"category": "Math", "title": "A colimit of classifying spaces", "abstract": "We recall a group-theoretic description of the first non-vanishing homotopy group of a certain (n+1)-ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu's group-theoretic description of the homotopy groups of a 2-sphere."}
{"category": "Math", "title": "Measure conjugacy invariants for actions of countable sofic groups", "abstract": "Sofic groups were defined implicitly by Gromov in [Gr99] and explicitly by Weiss in [We00]. All residually finite groups (and hence every linear group) is sofic. The purpose of this paper is to introduce, for every countable sofic group $G$, a family of measure-conjugacy invariants for measure-preserving $G$-actions on probability spaces. These invariants generalize Kolmogorov-Sinai entropy for actions of amenable groups. They are computed exactly for Bernoulli shifts over $G$, leading to a complete classification of Bernoulli systems up to measure-conjugacy for many groups including all countable linear groups. Recent rigidity results of Y. Kida and S. Popa are utilized to classify Bernoulli shifts over mapping class groups and property T groups up to orbit equivalence and von Neumann equivalence respectively."}
{"category": "Math", "title": "Rigidity of measures invariant under the action of a multiplicative semigroup of polynomial growth on $\\T$", "abstract": "We prove that if a Borel probability measure (\\mu) on (\\T) is invariant under the action of a \"large\" multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then (\\mu) is either Lebesgue or has finite support."}
{"category": "Math", "title": "Tensor categories: A selective guided tour", "abstract": "These are the lecture notes for a short course on tensor categories. The coverage in these notes is relatively non-technical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on k-linear categories with finite dimensional hom-spaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography."}
{"category": "Math", "title": "The equation asinx+bcosx=c and a family of cyclic Heron quadrilaterals", "abstract": "In the beginning of this paper, we present the general solution to the trigonometric equation asinx+bcosx=c. After that, we focus on the case when a^2+b^2=c^2. In this case, the general solution is expressed in terms of the acute angle theta which satisfies tan(theta)=a/b+c .From the right trianle with leglengths a and b, and hypotenuse length c, we construct a cyclic quadrilateral within which the angle theta is illustrated.Then we examine the case when a,b,and c are integers;and we derive or construct a family of cyclic Heron quadrilaterals. These are quadrilaterals with integer sidelengths or edges, integer diagonal lengths, and integer area.Also these quadrilaterals have their four vertices lying on a circle. The said family of quadrilaterals, is a three parameter infinite family."}
{"category": "Math", "title": "An identity for sums of polylogarithm functions", "abstract": "We derive an identity for certain linear combinations of polylogarithm functions with negative exponents, which implies relations for linear combinations of Eulerian numbers. The coefficients of our linear combinations are related to expanding moments of Satake parameters of holomorphic cuspidal newforms in terms of the moments of the corresponding Fourier coefficients, which has applications in analyzing lower order terms in the behavior of zeros of L-functions near the central point."}
{"category": "Math", "title": "On sensitive dependence on initial conditions and existence of physical measure for 3-flows", "abstract": "After reviewing known results on sensitiveness and also on robustness of attractors together with observations on their proofs, we show that for attractors of three-dimensional flows, robust chaotic behavior meaning sensitiveness to initial conditions for the past as well for the future for all nearby flows) is equivalent to the existence of certain hyperbolic structures. These structures, in turn, are associated to the existence of physical measures. In short in low dimensions, robust chaotic behavior for smooth flows ensures the existence of a physical measure."}
{"category": "Math", "title": "Lorenz-like chaotic attractors revised", "abstract": "We describe some recent results on the dynamics of singular-hyperbolic (or Lorenz-like) attractors: attractors in this class are expansive and so sensitive with respect to initial data; they admit a unique physical measure whose support is the whole attractor, which is hyperbolic and the equilibrium state with respect to the center-unstable Jacobian; the hitting time associated to a geometric Lorenz attractor satisfies a logarithm law; the rate of large deviations for the physical measure on the ergodic basin of a geometric Lorenz attractor is exponential."}
{"category": "Math", "title": "Heegaard Floer homology and genus one, one boundary component open books", "abstract": "We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T,\\phi), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of any T^2-bundle over S^1 with first Betti number equal to one, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-filllings of the contact structures compatible with such open books, to narrow down somewhat the class of 3-braid knots with finite concordance order, and to identify all quasi-alternating links with braid index at most 3."}
{"category": "Math", "title": "Consensus Problems in Networks of Agents under Nonlinear Protocols with Directed Interaction Topology", "abstract": "The purpose of this short paper is to provide a theoretical analysis for the consensus problem under nonlinear protocols. A main contribution of this work is to generalize the previous consensus problems under nonlinear protocols for networks with undirected graphs to directed graphs (information flow). Our theoretical result is that if the directed graph is strongly connected and the nonlinear protocol is strictly increasing, then consensus can be realized. Some simple examples are also provided to demonstrate the validity of our theoretical result."}
{"category": "Math", "title": "Virasoro constraints and descendant Hurwitz-Hodge Integrals", "abstract": "Virasoro constraints are applied to degree zero Gromov-Witten theory of weighted projective stacks $\\mathbb{P}(1,N)$ and $\\mathbb{P}(1,1,N)$ to obtain formulas of descendant cyclic Hurwitz-Hodge integrals in higher genera."}
{"category": "Math", "title": "Separating twists and the Magnus representation of the Torelli group", "abstract": "The Magnus representation of the Torelli subgroup of the mapping class group of a surface is a homomorphism r: I_{g,1} -> GL_{2g}(Z[H]). Here H is the first homology group of the surface. This representation is not faithful; in particular, Suzuki previously described precisely when the commutator of two Dehn twists about separating curves is in the kernel of r. Using the trace of the Magnus representation, we apply a new method of showing that two endomorphisms generate a free group to prove that the images of two positive separating multitwists under the Magnus representation either commute or generate a free group, and we characterize when each case occurs."}
{"category": "Math", "title": "New hook length formulas for binary trees", "abstract": "We find two new hook length formulas for binary trees. The particularity of our formulas is that the hook length $h_v$ appears as an exponent."}
{"category": "Math", "title": "On the Log-Concavity of Hilbert Series of Veronese Subrings and Ehrhart Series", "abstract": "For every positive integer $n$, consider the linear operator $\\U_{n}$ on polynomials of degree at most $d$ with integer coefficients defined as follows: if we write $\\frac{h(t)}{(1 - t)^{d + 1}} = \\sum_{m \\geq 0} g(m) t^{m}$, for some polynomial $g(m)$ with rational coefficients, then $\\frac{\\U_{n}h(t)}{(1- t)^{d + 1}} = \\sum_{m \\geq 0} g(nm) t^{m}$. We show that there exists a positive integer $n_{d}$, depending only on $d$, such that if $h(t)$ is a polynomial of degree at most $d$ with nonnegative integer coefficients and $h(0) \\geq 1$, then for $n \\geq n_{d}$, $\\U_{n}h(t)$ has simple, real, strictly negative roots and positive, strictly log concave and strictly unimodal coefficients. Applications are given to Ehrhart $\\delta$-polynomials and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen--MacCauley graded rings."}
{"category": "Math", "title": "Super edge-graceful paths", "abstract": "A graph $G(V,E)$ of order $|V|=p$ and size $|E|=q$ is called super edge-graceful if there is a bijection $f$ from $E$ to $\\{0,\\pm 1,\\pm 2,...,\\pm \\frac{q-1}{2}\\}$ when $q$ is odd and from $E$ to $\\{\\pm 1,\\pm 2,...,\\pm \\frac{q}{2}\\}$ when $q$ is even such that the induced vertex labeling $f^*$ defined by $f^*(x) = \\sum_{xy\\in E(G)}f(xy)$ over all edges $xy$ is a bijection from $V$ to $\\{0,\\pm 1,\\pm 2...,\\pm \\frac{p-1}{2}\\}$ when $p$ is odd and from $V$ to $\\{\\pm 1,\\pm 2,...,\\pm \\frac{p}{2}\\}$ when $p$ is even. \\indent We prove that all paths $P_n$ except $P_2$ and $P_4$ are super edge-graceful."}
{"category": "Math", "title": "The diffeomorphism groups of the real line are pairwise bihomeomorphic", "abstract": "We prove that the group D^r(R) of C^r diffeomorphisms of the real line, endowed with the compact-open and Whitney C^r topologies, is bihomeomorphic to the group H(R) of homeomorphisms of the real line endowed with the compact-open and Whitney topologies. This implies that the diffeomorphism group D^r(R) endowed with the Whitney C^r topology is homeomorphic to the countable box-power of the separable Hilbert space."}
{"category": "Math", "title": "On The Behavior of Subgradient Projections Methods for Convex Feasibility Problems in Euclidean Spaces", "abstract": "We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific self-adapting manner. This strategy leaves enough user-flexibility but gives a mathematical guarantee for the algorithm's behavior in the inconsistent case. We present numerical results of computational experiments that illustrate the computational advantage of the new method."}
{"category": "Math", "title": "Monodromy Filtrations and the Topology of Tropical Varieties", "abstract": "We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical compactifications, and use it to construct normal crossings degenerations of a subvariety X of a torus, under mild hypotheses on X. These degenerations allow us to construct a natural, \"multiplicity-free\" parameterization of Trop(X) by a topological space \\Gamma_X. We give a geometric interpretation of the cohomology of \\Gamma_X in terms of the action of a monodromy operator on the cohomology of X. This gives bounds on the Betti numbers of $\\Gamma_X$ in terms of the Betti numbers of $X$. When $X$ is a sufficiently general complete intersection, this allows us to show that the cohomology of Trop(X) vanishes in degree less than dim(X). In addition, we give a description for the top power of the monodromy operator acting on middle cohomology in terms of the volume pairing on $\\Gamma_X$."}
{"category": "Math", "title": "Symplectic Dirac Operators on Hermitian Symmetric Spaces", "abstract": "We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones."}
{"category": "Math", "title": "Conjugacy Classes of Centralizers in $G_2$", "abstract": "Let $G$ be an algebraic group of type $G_2$ over a field $k$ of characteristic $\\neq 2,3$. In this paper we calculate centralizers of semisimple elements in anisotropic $G_2$. Using these, we show explicitly that there are six conjugacy classes of centralizers in the compact real form of $G_2$."}
{"category": "Math", "title": "Cayley decompositions of lattice polytopes and upper bounds for h^*-polynomials", "abstract": "We give an effective upper bound on the h^*-polynomial of a lattice polytope in terms of its degree and leading coefficient, confirming a conjecture of Batyrev. We deduce this bound as a consequence of a strong Cayley decomposition theorem which says, roughly speaking, that any lattice polytope with a large multiple that has no interior lattice points has a nontrivial decomposition as a Cayley sum of polytopes of smaller dimension. In an appendix, we interpret this result in terms of adjunction theory for toric varieties."}
{"category": "Math", "title": "Phase Transition in the 1d Random Field ising model with long range interaction", "abstract": "We study the one dimensional Ising model with ferromagnetic, long range interaction which decays as |i-j|^{-2+a}, 1/2< a<1, in the presence of an external random filed. we assume that the random field is given by a collection of independent identically distributed random variables, subgaussian with mean zero. We show that for temperature and strength of the randomness (variance) small enough with P=1 with respect to the distribution of the random fields there are at least two distinct extremal Gibbs measures."}
{"category": "Math", "title": "Exact Exponent of Remainder Term of Gelfond's Digit Theorem in Binary Case", "abstract": "We give a simple formula for the exact exponent in the remainder term of the main Gelfond's digit theorem in the binary case."}
{"category": "Math", "title": "Distinct Distances in Graph Drawings", "abstract": "The \\emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in $\\mathcal{O}(\\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distance-number. Moreover, as $\\Delta$ increases the existential lower bound on the distance-number of $\\Delta$-regular graphs tends to $\\Omega(n^{0.864138})$."}
{"category": "Math", "title": "Slicing surfaces and Fourier restriction conjecture", "abstract": "We deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid, the elliptic hyperboloid in $\\mathbb{R}^n$ implies that for the cone in $\\mathbb{R}^{n+1}$. We also prove a new restriction estimate for any surface in $\\mathbb{R}^3$ locally isometric to the plane and of finite type."}
{"category": "Math", "title": "Some results on non-self-adjoint operators, a survey", "abstract": "This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint operators with small random perturbations. In the introduction we also review the notion of pseudo-spectrum and its relation to non-self-adjoint spectral problems."}
{"category": "Math", "title": "Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes", "abstract": "This paper is devoted to the estimation of a vector $\\bm {\\theta}$ parametrizing an energy function of a Gibbs point process, via the maximum pseudolikelihood method. Strong consistency and asymptotic normality results of this estimator depending on a single realization are presented. In the framework of exponential family models, sufficient conditions are expressed in terms of the local energy function and are verified on a wide variety of examples."}
{"category": "Math", "title": "On the Collatz Problem", "abstract": "Taking a new approach towards analyzing the Collatz Problem, or, 3x+1 conjecture. Introducing some new functions, the Collatz-2 and Collatz-3 sequences, as well as deducing results related to Collatz-2 and Collatz-3 sequences."}
{"category": "Math", "title": "Genericity of nondegenerate critical points and Morse geodesic functionals", "abstract": "We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Based on an idea of B. White, we prove an abstract genericity result that employs the infinite dimensional Sard--Smale theorem. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds."}
{"category": "Math", "title": "Homogeneous Metrics with nonnegative curvature", "abstract": "Given compact Lie groups H\\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics."}
{"category": "Math", "title": "Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. III", "abstract": "We generalize the results from \"P. Lipparini, Productive $[\\lambda,\\mu]$-compactness and regular ultrafilters, Topology Proceedings, 21 (1996), 161--171\"; in particular the present results apply to singular cardinals, too."}
{"category": "Math", "title": "A class of matrix-valued polynomials generalizing Jacobi Polynomials", "abstract": "A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a two-step recurrence relation, integral inter-relations, and quasi-orthogonality relations."}
{"category": "Math", "title": "On the $p$-adic distance between a point of finite order and a curve of genus higher or equal to two", "abstract": "Let $A$ be an abelian variety over ${\\bf C}_p$ ($p$ a prime number) and $V\\hookrightarrow A$ a closed subvariety. The conjecture of Tate-Voloch predicts that the $p$-adic distance from a torsion point $T\\not\\in V({\\bf C}_p)$ to the variety $V$ is bounded below by a strictly positive constant. This conjecture is proven by Hrushovski and Scanlon, when $A$ has a model over $\\bar{{\\bf C}}_p$. We give an explicit formula for this constant, in the case where $V$ is a curve embedded into its Jacobian and $V$ has a model over a number field. This explicit formula involves analytic and arakelovian invariants of the curve."}
{"category": "Math", "title": "Ratios of Norms for Polynomials and Connected n-width Problems", "abstract": "Let G be a bounded simply connected domain and E be a regular compact subset of G with connected complement. We investigate the asymptotic behavior of the Kolmogorov k-width, k=k(n), of the set of polynomials of degree at most n having the supremum norm at most 1 on G restricted to E in the space of continuous functions on E."}
{"category": "Math", "title": "Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds - II", "abstract": "Extending the earlier results for analytic curve segments, in this article we describe the asymptotic behaviour of evolution of a finite segment of a C^n-smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n-manifold. In particular, we show that if the curve satisfies certain natural geometric conditions, the pushforward of the parameter measure on the curve under the geodesic flow converges to the normalized canonical Riemannian measure on the tangent bundle in the limit. We also study the limits of geodesic evolution of shrinking segments. We use Ratner's classification of ergodic invariant measures for unipotent flows on homogeneous spaces of SO(n,1), and an observation relating local growth properties of smooth curves and dynamics of linear SL(2,R)-actions."}
{"category": "Math", "title": "Mixed sums of squares and triangular numbers (III)", "abstract": "In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if T_m=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p^2=x^2+8(y^2+z^2) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2T_m (m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4."}
{"category": "Math", "title": "Equivariant sl(n)-link homology", "abstract": "For every positive integer $n$ we construct a bigraded homology theory for links, such that the corresponding invariant of the unknot is closely related to the U(n)-equivariant cohomology ring of $\\mathbb{CP}^{n-1}$; our construction specializes to the Khovanov-Rozansky $sl_n$-homology. We are motivated by the \"universal\" rank two Frobenius extension studied by M. Khovanov in \\cite{Kh3} for $sl_2$-homology."}
{"category": "Math", "title": "Measures with positive Lyapunov exponent and conformal measures in rational dynamics", "abstract": "Ergodic properties of rational maps are studied, generalising the work of F.\\ Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for an ergodic invariant probability measure with positive Lyapunov exponent to be absolutely continuous with respect to a general conformal measure. If they hold, we can construct an induced expanding Markov map with integrable return time which generates the invariant measure."}
{"category": "Math", "title": "Real double coset spaces and their invariants", "abstract": "Let G be a real form of a complex reductive group. Suppose that we are given involutions \\sigma and \\theta of G. Let H=G^\\sigma denote the fixed group of \\sigma and let K=G^\\theta denote the fixed group of \\theta. We are interested in calculating the double coset space H\\backslash G/K. We use moment map and invariant theoretic techniques to calculate the double cosets, especially the ones that are closed. One salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles."}
{"category": "Math", "title": "Graded mapping cone theorem, multisecants and syzygies", "abstract": "Let $X$ be a reduced closed subscheme in $\\mathbb P^n$. As a slight generalization of property $\\textbf{N}_p$ due to Green-Lazarsfeld, we can say that $X$ satisfies property $\\textbf{N}_{2,p}$ scheme-theoretically if there is an ideal $I$ generating the ideal sheaf $\\mathcal I_{X/\\P^n}$ such that $I$ is generated by quadrics and there are only linear syzygies up to $p$-th step (cf. \\cite{EGHP1}, \\cite{EGHP2}, \\cite{V}). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property $\\textbf{N}_{2,p}$(cf. \\cite{CKP}, \\cite{EGHP1}, \\cite{EGHP2} \\cite {KP}). In this case, the Castelnuovo regularity and normality can be obtained by the blowing-up method as $\\reg(X)\\le e+1$ where $e$ is the codimension of a smooth variety $X$ (cf. \\cite{BEL}). On the other hand, projection methods have been very useful and powerful in bounding Castelnuovo regularity, normality and other classical invariants in geometry(cf. \\cite{BE}, \\cite{K}, \\cite{KP}, \\cite{L} \\cite {R}). In this paper, we first prove the graded mapping cone theorem on partial eliminations as a general algebraic tools and give some applications. Then, we bound the length of zero dimensional intersection of $X$ and a linear space $L$ in terms of graded Betti numbers and deduce a relation between $X$ and its projections with respect to the geometry and syzygies in the case of projective schemes satisfying property $\\textbf{N}_{2,p}$ scheme-theoretically. In addition, we give not only interesting information on the regularity of fibers and multiple loci for the case of $\\textbf{N}_{d,p}, d\\ge 2$ but also geometric structures for projections according to moving the center."}
{"category": "Math", "title": "Fourier series on compact symmetric spaces", "abstract": "The Fourier coefficients F(t) of a function f on a compact symmetric space U/K are given by integration of f against matrix coefficients of irreducible representations of U. The coefficients depend on a spectral parameter t, which determines the representation, and they can be represented by elements F(t) in a common Hilbert space H. We obtain a theorem of Paley-Wiener type which describes the size of the support of f by means of the exponential type of a holomorphic H-valued extension of F, provided f is K-finite and of sufficiently small support. The result was obtained previously for K-invariant functions, to which case we reduce."}
{"category": "Math", "title": "Cauchy-Pompeiu type formulas for d-bar on affine algebraic Riemann surfaces and some applications", "abstract": "We have obtained the explicit versions and precisions for the Hodge-Riemann decomposition of formes on affine algebraic curve V. The main application consists in the construction of Faddeev-Green function for Laplacian on V. Basing on this [HM](arXiv:0804.3951 and J.Geom.Anal., 2008,18), we extended from the case X in C to the case of bordered Riemann surface X in V the R.Novikov (1988) scheme for the effective reconstruction of conductivity function sigma on X through Dirichlet-to-Neumann mapping on bX for solutions of d(sigma d^cU)=0. In Sec.4 we give a correction of the paper [HM]."}
{"category": "Math", "title": "Conditions implying the uniqueness of the weak$^*$-topology on certain group algebras", "abstract": "We investigate possible preduals of the measure algebra $M(G)$ of a locally compact group and the Fourier algebra $A(G)$ of a separable compact group. Both of these algebras are canonically dual spaces and the canonical preduals make the multiplication separately weak$^*$-continuous so that these algebras are dual Banach algebras. In this paper we find additional conditions under which the preduals $C_0(G)$ of $M(G)$ and $C^*(G)$ of $A(G)$ are uniquely determined. In both cases we consider a natural coassociative multiplication and show that the canonical predual gives rise to the unique weak$^*$-topology making both the multiplication separately weak$^*$-continuous and the coassociative multiplication weak$^*$-continuous. In particular, dual cohomological properties of these algebras are well defined with this additional structure."}
{"category": "Math", "title": "The ideal structure of reduced crossed products", "abstract": "Let (A,G) be a C*-dynamical system with G discrete. In this paper we investigate the ideal structure of the reduced crossed product C*-algebra and in particular we determine sufficient - and in some cases also necessary - conditions for A to separate the ideals in Ax_rG. When A separates the ideals in Ax_rG, then there is a one-to-one correspondence between the ideals in Ax_rG and the invariant ideals in A. We extend the concept of topological freeness and present a generalization of the Rokhlin property. Exactness properties of (A,G) turns out to be crucial in these investigations."}
{"category": "Math", "title": "On Estimation of Finite Population Proportion", "abstract": "In this paper, we study the classical problem of estimating the proportion of a finite population. First, we consider a fixed sample size method and derive an explicit sample size formula which ensures a mixed criterion of absolute and relative errors. Second, we consider an inverse sampling scheme such that the sampling is continue until the number of units having a certain attribute reaches a threshold value or the whole population is examined. We have established a simple method to determine the threshold so that a prescribed relative precision is guaranteed. Finally, we develop a multistage sampling scheme for constructing fixed-width confidence interval for the proportion of a finite population. Powerful computational techniques are introduced to make it possible that the fixed-width confidence interval ensures prescribed level of coverage probability."}
{"category": "Math", "title": "Hecke group algebras as quotients of affine Hecke algebras at level 0", "abstract": "The Hecke group algebra $HW_0$ of a finite Coxeter group $W_0$, as introduced by the first and last author, is obtained from $W_0$ by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $W_0$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $HW_0$ is the natural quotient of the affine Hecke algebra through its level 0 representation. We further show that the level 0 representation is a calibrated principal series representation for a suitable choice of character, so that the quotient factors (non trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the classical 0-Hecke algebra and that of the affine Hecke algebra at this specialization."}
{"category": "Math", "title": "On parahoric subgroups", "abstract": "We prove some basic facts on parahoric subgroups and on Iwahori-Weyl groups."}
{"category": "Math", "title": "Representations of Higher Rank Graph Algebras", "abstract": "Let $\\Fth$ be a $\\Bk$-graph on a single vertex. We show that every irreducible atomic $*$-representation is the minimal $*$-dilation of a group construction representation. It follows that every atomic representation decomposes as a direct sum or integral of such representations. We characterize periodicity of $\\Fth$ and identify a symmetry subgroup $H_\\theta$ of $\\bZ^\\Bk$. If this has rank $s$, then $\\ca(\\Fth) \\cong \\rC(\\bT^s) \\otimes \\fA$ for some simple C*-algebra $\\fA$."}
{"category": "Math", "title": "Mutation of cluster-tilting objects and potentials", "abstract": "We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras associated with quivers with potentials. We show that cluster-tilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3-CY algebras. The nearly Morita equivalence for 2-CY-tilted algebras is shown to hold for the finite length modules over Jacobian algebras."}
{"category": "Math", "title": "Invariance of Gromov--Witten theory under a simple flop", "abstract": "We show that the generating functions of Gromov--Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended K\\\"ahler moduli space. This is a sequel to [LLW]."}
{"category": "Math", "title": "Interaction cohomology of forward or backward self-similar systems", "abstract": "We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings${\\cal U} $ of the invariant set, where each ${\\cal U}$ consists of (backward)images of the invariant set under elements of finite word length. Inparticular, we give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of finitely generated semigroups of polynomials on the complex plane. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given."}
{"category": "Math", "title": "A deconvolution estimate and localization in spline-type spaces", "abstract": "In this article some explicit estimates on the decay of the convolutive inverse of a sequence are proved. They are derived from the functional calculus for Sobolev algebras. Applications include localization in spline-type spaces and oversampling schemes."}
{"category": "Math", "title": "The concrete theory of numbers : New Mersenne conjectures. Simplicity and other wonderful properties of numbers $L(n) = 2^{2n}\\pm2^n\\pm1$", "abstract": "New Mersenne conjectures. The problems of simplicity, common prime divisors and free from squares of numbers $L(n) = 2^{2n}\\pm2^n\\pm1$ are investigated. Wonderful formulas $gcd $ for numbers $L (n) $ and numbers repunit are proved."}
{"category": "Math", "title": "A Quasi-Newton Approach to Nonsmooth Convex Optimization Problems in Machine Learning", "abstract": "We extend the well-known BFGS quasi-Newton method and its memory-limited variant LBFGS to the optimization of nonsmooth convex objectives. This is done in a rigorous fashion by generalizing three components of BFGS to subdifferentials: the local quadratic model, the identification of a descent direction, and the Wolfe line search conditions. We prove that under some technical conditions, the resulting subBFGS algorithm is globally convergent in objective function value. We apply its memory-limited variant (subLBFGS) to L_2-regularized risk minimization with the binary hinge loss. To extend our algorithm to the multiclass and multilabel settings, we develop a new, efficient, exact line search algorithm. We prove its worst-case time complexity bounds, and show that our line search can also be used to extend a recently developed bundle method to the multiclass and multilabel settings. We also apply the direction-finding component of our algorithm to L_1-regularized risk minimization with logistic loss. In all these contexts our methods perform comparable to or better than specialized state-of-the-art solvers on a number of publicly available datasets. An open source implementation of our algorithms is freely available."}
{"category": "Math", "title": "A binomial-coefficient identity arising from the middle discrete series of SU(2,2)", "abstract": "The aim of this paper is to give an elementary proof of certain identities on binomials and state an answer to Remark 8.2 in Takahiro Hayata, Harutaka Koseki, and Takayuki Oda, Matrix coefficients of the middle discrete series of SU(2,2), J. Funct. Anal. 185 (2001), 297-341."}
{"category": "Math", "title": "A quantitative Mean Ergodic Theorem for uniformly convex Banach spaces", "abstract": "We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of T. Tao of the Mean Ergodic Theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad, Gerhardy and Towsner (arXiv:0706.1512v2 [math.DS]) and T. Tao (arXiv:0707.1117v1 [math.DS])."}
{"category": "Math", "title": "Automorphisms of real rational surfaces and weighted blow-up singularities", "abstract": "Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e=[e_1,...,e_l] be a partition of n. Denote by X^e the set of l-tuples (P_1,...,P_l) of distinct nonsingular curvilinear infinitely near points of X of orders (e_1,...,e_l). We show that the group Aut(X) acts transitively on X^e. This statement generalizes earlier work where the case of the trivial partition e=[1,...,1] was treated under the supplementary condition that X is nonsingular. As an application we classify singular real rational surfaces obtained from nonsingular surfaces by performing weighted blow-ups."}
{"category": "Math", "title": "The real quadrangle of type E6", "abstract": "We give a geometric interpretation of the building associated to the real Lie group E_6(-14) in terms of its 54-dimensional module."}
{"category": "Math", "title": "On the isolated singularities of the solutions of the Gaussian curvature equation for nonnegative curvature", "abstract": "The precise asymptotic behaviour of the solutions to the twodimensional curvature equation $\\Delta u=k(z) e^{2 u}$ with $e^{2 u} \\in L^1$ for bounded nonnegative curvature functions $-k(z)$ near isolated singularities is obtained."}
{"category": "Math", "title": "Functional inequalities derived from the Brunn-Minkowski inequalities for quermassintegrals", "abstract": "We use Brunn-Minkowski inequalities for quermassintegrals to deduce a family of inequalities of Poincar\\'e type on the unit sphere and on the boundary of smooth convex bodies in the $n$-dimensional Euclidean space."}
{"category": "Math", "title": "The Hijazi inequality on conformally parabolic manifolds", "abstract": "We prove the Hijazi inequality, an estimate for Dirac eigenvalues, for complete manifolds of finite volume. Under some additional assumptions on the dimension and the scalar curvature, this inequality is also valid for elements of the essential spectrum. This allows to prove the conformal version of the Hijazi inequality on conformally parabolic manifolds if the spin analog to the Yamabe invariant is positive."}
{"category": "Math", "title": "Remark on the Boundedness of the Cauchy Singular Integral Operator on Variable Lebesgue Spaces with Radial Oscillating Weights", "abstract": "Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result."}
{"category": "Math", "title": "A Bergman kernel proof of the Kawamata subadjunction theorem", "abstract": "The main purpose of the following article is to give a proof of Y. Kawamata's celebrated subadjunction theorem in the spirit of our previous work on Bergman kernels. We will use two main ingredients : an $\\displaystyle L^{2\\over m}$--extension theorem of Ohsawa-Takegoshi type (which is also a new result) and a more complete version of our former results."}
{"category": "Math", "title": "Formality of Cyclic Chains", "abstract": "We prove a conjecture raised by Tsygan, namely the existence of an L-infinity-quasiisomorphism of L-infinity-modules between the cyclic chain complex of smooth functions on a manifold and the differential forms on that manifold. Concretely, we prove that the obvious u-linear extension of Shoikhet's morphism of Hochschild chains solves Tsygan's conjecture."}
{"category": "Math", "title": "Witt vectors. Part 1", "abstract": "This is the first part of a 2 part survey on the functor of the big and p-adic Witt vectors."}
{"category": "Math", "title": "A characterization of quaternionic projective space by the conformal-Killing equation", "abstract": "We prove that a compact quaternionic-K\\\"{a}hler manifold of dimension $4n\\geq 8$ admitting a conformal-Killing 2-form which is not Killing, is isomorphic to the quaternionic projective space, with its standard quaternionic-K\\\"{a}hler structure."}
{"category": "Math", "title": "Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions", "abstract": "The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in addition to an unbounded operator acting on the Y-component of the solution there is still another one acting on the Z-component. With the help of this dual equation we then deduce the duality between approximate controllability and observability. Finally, under the assumption that the unbounded operator acting on the state process of the forward equation is an infinitesimal generator of an exponentially stable semigroup, we show that the generalized Hautus test provides a necessary condition for the approximate controllability. The paper generalizes former results by Buckdahn, Quincampoix and Tessitore (2006) and Goreac (2007) from the finite dimensional to the infinite dimensional case."}
{"category": "Math", "title": "Smooth conjugacy of Anosov diffeomorphisms on higher dimensional tori", "abstract": "Let $L$ be a hyperbolic automorphism of $\\mathbb T^d$, $d\\ge3$. We study the smooth conjugacy problem in a small $C^1$-neighborhood $\\mathcal U$ of $L$. The main result establishes $C^{1+\\nu}$ regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$ with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations. We elaborate on the example of de la Llave of two Anosov systems on $\\mathbb T^4$ with the same constant periodic eigenvalue data that are only H\\\"older conjugate. We show that these examples exhaust all possible ways to perturb $C^{1+\\nu}$ conjugacy class without changing periodic eigenvalue data. Also we generalize these examples to majority of reducible toral automorphisms as well as to certain product diffeomorphisms of $\\mathbb T^4$ $C^1$-close to the original example."}
{"category": "Math", "title": "CMC tori of revolution in $\\mathbb{S}^3$: additional data on the spectra of their Jacobi operators", "abstract": "We prove a theorem about elliptic operators with symmetric potential functions, defined on a function space over a closed loop. The result is similar to a known result for a function space on an interval with Dirichlet boundary conditions. These theorems provide accurate numerical methods for finding the spectra of those operators over either type of function space. As an application, we numerically compute the Morse index of constant mean curvature tori of revolution in the unit 3-sphere $\\mathbb{S}^3$, confirming that every such torus has Morse index at least five, and showing that other known lower bounds for this Morse index are close to optimal."}
{"category": "Math", "title": "Maximum Probability and Relative Entropy Maximization. Bayesian Maximum Probability and Empirical Likelihood", "abstract": "Works, briefly surveyed here, are concerned with two basic methods: Maximum Probability and Bayesian Maximum Probability; as well as with their asymptotic instances: Relative Entropy Maximization and Maximum Non-parametric Likelihood. Parametric and empirical extensions of the latter methods - Empirical Maximum Maximum Entropy and Empirical Likelihood - are also mentioned. The methods are viewed as tools for solving certain ill-posed inverse problems, called Pi-problem, Phi-problem, respectively. Within the two classes of problems, probabilistic justification and interpretation of the respective methods are discussed."}
{"category": "Math", "title": "On the global boundedness of Fourier integral operators", "abstract": "We consider a class of Fourier integral operators, globally defined on $\\mathbb{R}^{d}$, with symbols and phases satisfying product type estimates (the so-called $SG$ or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces $M^p$. The minimal loss of derivatives is shown to be $d|1/2-1/p|$. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on $L^p$ spaces are presented."}
{"category": "Math", "title": "The M/M/1 queue is Bernoulli", "abstract": "The classical output theorem for the M/M/1 queue, due to Burke (1956), states that the departure process from a stationary M/M/1 queue, in equilibrium, has the same law as the arrivals process, that is, it is a Poisson process. In this paper we show that the associated measure-preserving transformation is metrically isomorphic to a two-sided Bernoulli shift. We also discuss some extensions of Burke's theorem where it remains an open problem to determine if, or under what conditions, the analogue of this result holds."}
{"category": "Math", "title": "Highly Degenerate Harmonic Mean Curvature Flow", "abstract": "We study the evolution of a weakly convex surface $\\Sigma_0$ in $\\R^3$ with flat sides by the Harmonic Mean Curvature flow. We establish the short time existence as well as the optimal regularity of the surface and we show that the boundaries of the flat sides evolve by the curve shortening flow. It follows from our results that a weakly convex surface with flat sides of class $C^{k,\\gamma}$, for some $k\\in \\mathbb{N}$ and $0 < \\gamma \\leq 1$, remains in the same class under the flow. This distinguishes this flow from other, previously studied, degenerate parabolic equations, including the porous medium equation and the Gauss curvature flow with flat sides, where the regularity of the solution for $t >0$ does not depend on the regularity of the initial data."}
{"category": "Math", "title": "Quantum Gross Laplacian and Applications", "abstract": "In this paper, we introduce and study a noncommutative extension of the Gross Laplacian, called quantum Gross Laplacian. Then, applying the quantum Gross Laplacian to the particular case where the operator is the multiplication operator, we find a relation between classical and quantum Gross Laplacian. As application, we give explicit solution of linear quantum white noise differential equation. In particular, we give a explicit solution of the quantum Gross heat equation."}
{"category": "Math", "title": "The noisy veto-voter model: a Recursive Distributional Equation on [0,1]", "abstract": "We study a particular example of a recursive distributional equation (RDE) on the unit interval. We identify all invariant distributions, the corresponding \"basins of attraction\" and address the issue of endogeny for the associated tree-indexed problem, making use of an extension of a recent result of Warren."}
{"category": "Math", "title": "Inverse conductivity problem on Riemann surfaces", "abstract": "An electrical potential U on a bordered Riemann surface X with conductivity function sigma>0 satisfies equation d(sigma d^cU)=0. The problem of effective reconstruction of sigma is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R.Novikov (1988) for simply connected X. We apply for this new kernels for dbar on affine algebraic Riemann surfaces constructed in Henkin, arXiv:0804.3761"}
{"category": "Math", "title": "Partial pluricomplex energy and integrability exponents of plurisubharmonic functions", "abstract": "We give a sufficient condition on the Monge-Amp\\`ere mass of a plurisubharmonic function $u$ for $\\exp (- 2 u)$ to be locally integrable. This gives a pluripotential theoretic proof of a theorem by J-P. Demailly."}
{"category": "Math", "title": "The commutative Moufang loops with minimum conditions for subloops I", "abstract": "The structure of the commutative Moufang loops (CML) with minimum condition for subloops is examined. In particular it is proved that such a CML $Q$ is a finite extension of a direct product of a finite number of the quasicyclic groups, lying in the centre of the CML $Q$. It is shown that the minimum conditions for subloops and for normal subloops are equivalent in a CML. Moreover, such CML also characterized by different conditions of finiteness of its multiplication groups."}
{"category": "Math", "title": "The commutative Moufang loops with minimum conditions for subloops II", "abstract": "It is proved that the following conditions are equivalent for an infinite non-associative commutative Moufang loop $Q$: 1) $Q$ satisfies the minimum condition for subloops; 2) if the loop $Q$ contains a centrally solvable subloop of class $s$, then it satisfies the minimum condition for centrally solvable subloops of class $s$; 3) if the loop $Q$ contains a centrally nilpotent subloop of class $n$, then it satisfies the minimum condition for centrally nilpotent subloops of class $n$; 4) $Q$ satisfies the minimum condition for non-invatiant associative subloops. The structure of the commutative Moufang loops, whose infinite non-associative subloops are normal, is examined."}
{"category": "Math", "title": "The commutative Moufang loops with maximum conditions for subloops", "abstract": "It is proved that the maximum condition for subloops in a commutative Moufang loop $Q$ is equivalent with the conditions of finite generating of different subloops of the loop $Q$ and different subgroups of the multiplication group of the loop $Q$. An analogue equivalence is set for the commutative Moufang $ZA$-loops."}
{"category": "Math", "title": "On some criterions of finiteness conditions in commutative Moufang loops", "abstract": "The various finiteness conditions in commutative Moufang loops are characterized using the notions of centralizer of subloops and centralizer of subgroups of its multiplication group."}
{"category": "Math", "title": "JSJ Decompositions of Coxeter Groups", "abstract": "The idea of \"JSJ-decompositions\" for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of 2-sided, incompressible tori that separate a given closed irreducible 3-manifold into pieces with strong topological structure. Sela introduced the idea of JSJ-decompositions for groups, an idea that has flourished in a variety of directions. The general idea is to consider a certain class X of groups and splittings of groups in X by groups in another class Y. E.g. Rips and Sela considered splittings of finitely presented groups by infinite cyclic groups. For an arbitrary group G in X the goal is to produce a unique graph of groups decomposition T of G with edge groups in Y so that T reveals all graph of groups decompositions of G with edge groups in Y. More specifically, if V is a vertex group of T then either there is no Y-group that splits both G and V, or V has a special \"surface group-like\" structure. It is standard to call vertex groups of the second type \"orbifold groups\". For a finitely generated Coxeter system (W,S) we produce a reduced JSJ-decomposition T for splittings of W over virtually abelian subgroups. We show T is unique with each vertex and edge group generated by a subset of S (and so T is \"visual\"). The construction of T is algorithmic. If V, a subset of S, generates an orbifold vertex group of T then V is the disjoint union of K and M, where < M > is virtually abelian, < K > is virtually a closed surface group or virtually free and < V > is the direct product of < M > and < K >."}
{"category": "Math", "title": "On Frattini subloops and normalizers of commutative Moufang loops", "abstract": "Let $L$ be a commutative Moufang loop (CML) with multiplication group $\\frak M$, and let $\\frak F(L)$, $\\frak F(\\frak M)$ be the Frattini subgroup and Frattini subgroup of $L$ and $\\frak M$ respectively. It is proved that $\\frak F(L) = L$ if and only if $\\frak F(\\frak M) = \\frak M$ and is described the structure of this CLM. Constructively it is defined the notion of normalizer for subloops in CML. Using this it is proved that if $\\frak F(L) \\neq L$ then $L$ satisfies the normalizer condition and that any divisible subgroup of $\\frak M$ is an abelian group and serves as a direct factor for $\\frak M$."}
{"category": "Math", "title": "Metaheuristics and Their Hybridization to Solve the Bi-objective Ring Star Problem: a Comparative Study", "abstract": "This paper presents and experiments approaches to solve a new bi-objective routing problem called the ring star problem. It consists of locating a simple cycle through a subset of nodes of a graph while optimizing two kinds of cost. The first objective is the minimization of a ring cost that is related to the length of the cycle. The second one is the minimization of an assignment cost from non-visited nodes to visited ones. In spite of its obvious bi-objective formulation, this problem has always been investigated in a single-objective way. To tackle the bi-objective ring star problem, we first investigate different stand-alone search methods. Then, we propose two cooperative strategies that combines two multiple objective metaheuristics: an elitist evolutionary algorithm and a population-based local search. We apply this new hybrid approaches to well-known benchmark test instances and demonstrate their effectiveness in comparison to non-hybrid algorithms and to state-of-the-art methods."}
{"category": "Math", "title": "Hyperlinear and sofic groups: a brief guide", "abstract": "Relatively recently, two new classes of (discrete, countable) groups have been isolated: hyperlinear groups and sofic groups. They come from different corners of mathematics (operator algebras and symbolic dynamics, respectively), and were introduced independently from each other, but are closely related nevertheless. Hyperlinear groups have their origin in Connes' Embedding Conjecture about von Neumann factors of type $II_1$, while sofic groups, introduced by Gromov, are motivated by Gottschalk Surjunctivity Conjecture (can a shift $A^G$ contain a proper isomorphic copy of itself, where $A$ is a finite discrete space and $G$ is a group?). Groups from both classes can be characterized as subgroups of metric ultraproducts of families of certain metric groups (formed in the same way as ultraproducts of Banach spaces): unitary groups of finite rank lead to hyperlinear groups, symmetric groups of finite rank - to sofic groups. We offer an introductory guide to some of the main concepts, results, and sources of the theory, following Connes, Gromov, Benjamin Weiss, Kirchberg, Ozawa, Radulescu, Elek and Szab\\'o, and others, and discuss open questions which are for the time being perhaps more numerous than the results."}
{"category": "Math", "title": "Limit Theorems for Translation Flows", "abstract": "The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2)."}
{"category": "Math", "title": "True amplitude one-way propagation in heterogeneous media", "abstract": "This paper deals with the numerical analysis of two one-way systems derived from the general complete modeling proposed by M.V. De Hoop. The main goal of this work is to compare two different formulations in which a correcting term allows to improve the amplitude of the numerical solution. It comes out that even if the two systems are equivalent from a theoretical point of view, nothing of the kind is as far as the numerical simulation is concerned. Herein a numerical analysis is performed to show that as long as the propagation medium is smooth, both the models are equivalent but it is no more the case when the medium is associated to a quite strongly discontinuous velocity."}
{"category": "Math", "title": "Growth of Sobolev norms and controllability of Schr\\\"odinger equation", "abstract": "In this paper we obtain a stabilization result for the Schr\\\"odinger equation under generic assumptions on the potential. Then we consider the Schr\\\"odinger equation with a potential which has a random time-dependent amplitude. We show that if the distribution of the amplitude is sufficiently non-degenerate, then any trajectory of system is almost surely non-bounded in Sobolev spaces."}
{"category": "Math", "title": "On the doubled tetrus", "abstract": "The tetrus is a sort of big brother to the tripus, W.P. Thurston's example of a compact hyperbolic 3-manifold with totally geodesic boundary. We describe a sixfold cover of the double of the tetrus, itself a double, which fibers over the circle with fiber a closed surface of genus 19. We also record arithmeticity of the doubles and certain twisted doubles of the tripus and tetrus, and point out some consequences regarding families of covers."}
{"category": "Math", "title": "Maximum likelihood estimation of a multidimensional log-concave density", "abstract": "Let X_1, ..., X_n be independent and identically distributed random vectors with a log-concave (Lebesgue) density f. We first prove that, with probability one, there exists a unique maximum likelihood estimator of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non-constructive, we are able to reformulate the issue of computation in terms of a non-differentiable convex optimisation problem, and thus combine techniques of computational geometry with Shor's r-algorithm to produce a sequence that converges to the maximum likelihood estimate. For the moderate or large sample sizes in our simulations, the maximum likelihood estimator is shown to provide an improvement in performance compared with kernel-based methods, even when we allow the use of a theoretical, optimal fixed bandwidth for the kernel estimator that would not be available in practice. We also present a real data clustering example, which shows that our methodology can be used in conjunction with the Expectation--Maximisation (EM) algorithm to fit finite mixtures of log-concave densities. An R version of the algorithm is available in the package LogConcDEAD -- Log-Concave Density Estimation in Arbitrary Dimensions."}
{"category": "Math", "title": "Numerical Algorithms for Finding Balanced Metrics on Vector Bundles", "abstract": "In \\cite{D3}, Donaldson defines a dynamical system on the space of Fubini-Study metrics on a polarized compact K\\\"ahler manifold. Sano proved that if there exists a balanced metric for the polarization, then this dynamical system always converges to the balanced metric (\\cite{S}). In \\cite{DKLR}, Douglas, et. al., conjecture that the same holds in the case of vector bundles. In this paper, we give an affirmative answer to their conjecture."}
{"category": "Math", "title": "Which Ambient Spaces Admit Isoperimetric Inequalities for Submanifolds?", "abstract": "We give simple conditions on an ambient manifold that are necessary and sufficient for isoperimetric inequalities (for submanifolds) to hold."}
{"category": "Math", "title": "On sublattice determinants in reduced bases", "abstract": "We prove several inequalities on the determinants of sublattices in LLL-reduced bases. They generalize the inequalities on the length of the shortest vector proven by Lenstra, Lenstra, and Lovasz, and show that LLL-reduction finds not only a short vector, but more generally, sublattices with small determinants. We also prove new upper bounds on the product of the norms of the first few vectors."}
{"category": "Math", "title": "Distinguishing Number of Countable Homogeneous Relational Structures", "abstract": "The distinguishing number of a graph $G$ is the smallest positive integer $r$ such that $G$ has a labeling of its vertices with $r$ labels for which there is no non-trivial automorphism of $G$ preserving these labels. Albertson and Collins computed the distinguishing number for various finite graphs, and Imrich, Klav\\v{z}ar and Trofimov computed the distinguishing number of some infinite graphs, showing in particular that the Random Graph has distinguishing number 2. We compute the distinguishing number of various other finite and countable homogeneous structures, including undirected and directed graphs, and posets. We show that this number is in most cases two or infinite, and besides a few exceptions conjecture that this is so for all primitive homogeneous countable structures."}
{"category": "Math", "title": "The problem of the body of revolution of minimal resistance", "abstract": "Newton's problem of the body of minimal aerodynamic resistance is traditionally stated in the class of {\\it convex} axially symmetric bodies with fixed length and width. We state and solve the minimal resistance problem in the wider class of axially symmetric but {\\it generally nonconvex} bodies. The infimum in this problem is not attained. We construct a sequence of bodies minimizing the resistance. This sequence approximates a convex body with smooth front surface, while the surface of approximating bodies becomes more and more complicated. The shape of the resulting convex body and the value of minimal resistance are compared with the corresponding results for Newton's problem and for the problem in the intermediate class of axisymmetric bodies satisfying the {\\it single impact} assumption \\cite{CL1}. In particular, the minimal resistance in our class is smaller than in Newton's problem; the ratio goes to 1/2 as (length)/(width of the body) $\\to 0$, and to 1/4 as (length)/(width) $\\to +\\infty$."}
{"category": "Math", "title": "Infinitely many solution for prescribed curvature problem on $S^N$", "abstract": "We consider the following prescribed scalar curvature problem on $ S^N$ (*)$$\\left\\{\\begin{array}{l} - \\Delta_{S^N} u + \\frac{N(N-2)}{2} u = \\tilde{K} u^{\\frac{N+2}{N-2}} {on} S^N, u >0 \\end{array}\\right. $$ where $ \\tilde{K}$ is positive and rotationally symmetric. We show that if $\\tilde{K}$ has a local maximum point between the poles then equation (*) has {\\bf infinitely many non-radial positive} solutions, whose energy can be made arbitrarily large."}
{"category": "Math", "title": "Infinitely many positive solutions for the nonlinear Shcrodinger equations in $R^N$", "abstract": "We consider the following nonlinear problem in $\\R^N$ $$\\label{eq} - \\Delta u +V(|y|)u=u^{p},\\quad u>0 {in} \\R^N, u \\in H^1(\\R^N) $$ where $V(r)$ is a positive function, $1<p <\\frac{N+2}{N-2}$. We show that if $V(r)$ has the following expansion: There are constants $a>0$, $m>1$, $\\theta>0$, and $V_0>0$, such that \\[ V(r)= V_0+\\frac a {r^m} +O\\bigl(\\frac1{r^{m+\\theta}}\\bigr),\\quad \\text{as $r\\to +\\infty$,} \\] then \\eqref{eq} has {\\bf infinitely many non-radial positive} solutions, whose energy can be made arbitrarily large."}
{"category": "Math", "title": "On the principal symbols of $K_{\\mathbb C}$-invariant differential operators on Hermitian symmetric spaces", "abstract": "Let $(G,K)$ be one of the following classical irreducible Hermitian symmetric pairs of noncompact type: $(SU(p,q), S(U(p) \\times U(q))),(Sp(n,R), U(n))$, or $(SO*(2n), U(n))$. Let $G_{\\mathbb C}$ and $K_{\\mathbb C}$ be complexifications of $G$ and $K$, respectively, and let $P$ be a maximal parabolic subgroup of $G_{\\mathbb C}$ whose Levi subgroup is $K_{\\mathbb C}$. Let $V$ be the holomorphic part of the complexifiaction of the tangent space at the origin of $G/K$. It is well known that the ring of $K_{\\mathbb C}$-invariant differential operators on $V$ has a generating system $\\{\\varGamma_k \\}$ given in terms of determinant or Pfaffian that plays an essential role in the Capelli identities. Our main result of this paper is that determinant or Pfaffian of the ``moment map'' on the holomorphic cotangent bundle of $G_{\\mathbb C}/P$ provides a generating function for the principal symbols of $\\varGamma_k$'s."}
{"category": "Math", "title": "Eigenvalue distributions and Weyl laws for semi-classical non-self-adjoint operators in 2 dimensions", "abstract": "In this note we compare two recent results about the distribution of eigenvalues for semi-classical pseudodifferential operators in two dimensions. For classes of analytic operators A. Melin and the author obtained a complex Bohr-Sommerfeld rule, showing that the eigenvalues are situated on a distorted lattice. On the other hand, with M. Hager we showed in any dimension that Weyl asymptotics holds with probability close to 1 for small random perturbations of the operator. In both cases the eigenvalues distribute to leading order according two smooth densities and we show here that the two densities are in general different."}
{"category": "Math", "title": "On the exceptional locus of the birational projections of normal surface singularity into a plane", "abstract": "Given a normal surface singularity $(X, Q)$ and a birational morphism to a non- singular surface $\\pi : X \\to S$, we investigate the local geometry of the exceptional divisor $L$ of $\\pi$. We prove that the dimension of the tangent space to $L$ at $Q$ equals the number of exceptional components meeting at $Q$. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterization of minimal singularities is obtained in these terms."}
{"category": "Math", "title": "Geometric construction of cluster algebras and cluster categories", "abstract": "In this note we explain how to obtain cluster algebras from triangulations of (punctured) discs following the approach of S. Fomin, M. Shapiro and D. Thurston. Furthermore, we give a description of m-cluster categories via diagonals (arcs) in (punctured) polygons and of m-cluster categories via powers of translation quivers as given in joint work with R. Marsh."}
{"category": "Math", "title": "A family of series representations of the multiparameter fractional Brownian motion", "abstract": "We derive a family of series representations of the multiparameter fractional Brownian motion in the centred ball of radius $R$ in the $N$-dimensional space $\\mathbb{R}^N$. Some known examples of series representations are shown to be the members of the family under consideration."}
{"category": "Math", "title": "A note on standard systems and ultrafilters", "abstract": "Let $(M,\\scott X) \\models \\ACA$ be such that $P_\\scott X$, the collection of all unbounded sets in $\\scott X$, admits a definable complete ultrafilter and let $T$ be a theory extending first order arithmetic coded in $\\scott X$ such that $M$ thinks $T$ is consistent. We prove that there is an end-extension $N \\models T$ of $M$ such that the subsets of $M$ coded in $N$ are precisely those in $\\scott X$. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory $T$ extending first order arithmetic is the standard system of a recursively saturated model of $T$."}
{"category": "Math", "title": "On the geometry of the $B$-connection on quasi-K\\\"ahler manifolds with Norden metric", "abstract": "The $B$-connection on almost complex manifolds with Norden metric is an analogue of the first canonical connection of Lihnerovich in Hermitian geometry. In the present paper it is considered a $B$-connection in the class of the quasi-K\\\"ahler manifold with Norden metric. Some necessary and sufficient conditions are derived for the corresponding curvature tensor to be K\\\"ahlerian. Curvature properties for this connection are obtained. Conditions are given for the considered manifolds to be isotropic-K\\\"ahler."}
{"category": "Math", "title": "Lie groups as 4-dimensional Riemannian or pseudo-Riemannian almost product manifolds with nonintegrable structure", "abstract": "A Lie group as a 4-dimensional pseudo-Riemannian manifold is considered. This manifold is equipped with an almost product structure and a Killing metric in two ways. In the first case Riemannian almost product manifold with nonintegrable structure is obtained, and in the second case - a pseudo-Riemannian one. Each belongs to a 4-parametric family of manifolds, which are characterized geometrically."}
{"category": "Math", "title": "A connection with skew symmetric torsion and K\\\"ahler curvature tensor on quasi-K\\\"ahler manifolds with Norden metric", "abstract": "There is considered a connection with skew symmetric torsion on a quasi-K\\\"ahler manifold with Norden metric. Some necessary and sufficient conditions are derived for the corresponding curvature tensor to be K\\\"ahlerian. In the case when this tensor is K\\\"ahlerian, some relations are obtained between its scalar curvature and the scalar curvature of other curvature tensors. Conditions are given for the considered manifolds to be isotropic-K\\\"ahler."}
{"category": "Math", "title": "Reconstruction of p-disconnected graphs", "abstract": "We prove that Kelly-Ulam conjecture is true for p-disconnected graphs."}
{"category": "Math", "title": "Convex bodies and algebraic equations on affine varieties", "abstract": "Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This is a far reaching generalization of usual theory of Newton polytopes (which is concerned with toric varieties). As applications we give new, simple and transparent proofs of some well-known theorems in both algebraic geometry (e.g. Hodge Index Theorem) and convex geometry (e.g. Alexandrov-Fenchel inequality). Our main tools are classical Hilbert theory on degree of subvarieties of a projective space (in algebraic geometry) and Brunn-Minkowski inequality (in convex geometric)."}
{"category": "Math", "title": "Logarithmic components of the vacant set for random walk on a discrete torus", "abstract": "This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus (Z/NZ)^d up to time uN^d in high dimension d. If u>0 is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments of length c_0 log N for some constant c_0 > 0, and this component occupies a non-degenerate fraction of the total volume as N tends to infinity. Within the same setup, we investigate here the complement of the giant component in the vacant set and show that some components consist of segments of logarithmic size. In particular, this shows that the choice of a sufficiently large constant c_0 > 0 is crucial in the definition of the giant component."}
{"category": "Math", "title": "Hermitian Curvature Flow", "abstract": "We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to K\\\"ahler-Einstein metrics, and are automatically K\\\"ahler-Einstein under certain conditions. Given this, a natural parabolic flow equation arises. We prove short time existence and regularity results for this flow, as well as stability for the flow near K\\\"ahler-Einstein metrics with negative or zero first Chern class."}
{"category": "Math", "title": "Generalization of a criterion for semistable vector bundles", "abstract": "It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that the cohomologies of E\\otimes F vanish. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on $X$ such that the cohomologies of E\\otimes F vanish. We also give an explicit bound for the rank of $F$."}
{"category": "Math", "title": "Gaussian limits for generalized spacings", "abstract": "Nearest neighbor cells in $R^d,d\\in\\mathbb{N}$, are used to define coefficients of divergence ($\\phi$-divergences) between continuous multivariate samples. For large sample sizes, such distances are shown to be asymptotically normal with a variance depending on the underlying point density. In $d=1$, this extends classical central limit theory for sum functions of spacings. The general results yield central limit theorems for logarithmic $k$-spacings, information gain, log-likelihood ratios and the number of pairs of sample points within a fixed distance of each other."}
{"category": "Math", "title": "Optimization of operational aircraft parameters Reducing Noise Emission", "abstract": "The objective of this paper is to develop a model and a minimization method to provide flight path optimums reducing aircraft noise in the vicinity of airports. Optimization algorithm has solved a complex optimal control problem, and generates flight paths minimizing aircraft noise levels. Operational and safety constraints have been considered and their limits satisfied. Results are here presented and discussed."}
{"category": "Math", "title": "The first moment of quadratic Dirichlet L-functions", "abstract": "We obtain an asymptotic formula for the smoothly weighted first moment of primitive quadratic Dirichlet L-functions at the central point, with an error term that is \"square-root\" of the main term. Our approach uses a recursive technique that feeds the result back into itself, successively improving the error term."}
{"category": "Math", "title": "The Cops and Robber game on graphs with forbidden (induced) subgraphs", "abstract": "The two-player, complete information game of Cops and Robber is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph. In this paper, we study the cop number in the classes of graphs defined by forbidding one or more graphs as either subgraphs or induced subgraphs. In the case of a single forbidden graph we completely characterize (for both relations) the graphs which force bounded cop number. En passant, we bound the cop number in terms of tree-width."}
{"category": "Math", "title": "The Other Group of as Galois Extension", "abstract": "Let $k\\subseteq K$ be a finite Galois extension of fields with Galois group $G$. Let $\\mathscr{G}$ be the automorphism $k$-group scheme of $K$. We construct a canonical $k$-subgroup scheme $\\underline{G}\\subset\\mathscr{G}$ with the property that $Spec_k(K)$ is a $k$-torsor for $\\underline{G}$. $\\underline{G}$ is a constant $k$-group if and only if $G$ is abelian, in which case $G=\\underline{G}$."}
{"category": "Math", "title": "Tiling tripartite graphs with 3-colorable graphs", "abstract": "For a fixed integer h>=1, let G be a tripartite graph with N vertices in each vertex class, N divisible by 6h, such that every vertex is adjacent to at least 2N/3+h-1 vertices in each of the other classes. We show that if N is sufficiently large, then G can be tiled perfectly by copies of K_{h,h,h}. This extends the work in [19] and also gives a sufficient condition for tiling by any (fixed) 3-colorable graph. Furthermore, we show that this minimum-degree condition is best possible and provide very tight bounds when N is divisible by h but not by 6h."}
{"category": "Math", "title": "Cohomology Jumping Loci and Relative Malcev Completion", "abstract": "Two standard invariants used to study the fundamental group G of the complement X of a hyperplane arrangement are the Malcev completion of G and the cohomology groups of X with coefficients in rank one local systems. In this paper, we develop a tool that unifies these two approaches. This tool is the Malcev completion S_p of G relative to a homomorphism p from G into (C^*)^N. This is a prosolvable group that is tightly controlled by the cohomology groups of X with coefficients in rank one local systems. The prounipotent radical U_p of the relative completion S_p corresponds to a pronilpotent Lie algebra u_p. We provide an example of a hyperplane complement X for which this algebra is not quadratically presented. In addition, we show that if X is a hyperplane complement and Y is a subtorus of the character torus, then S_p is combinatorially determined for general p in Y. Finally, we show that the relative completion S_p is generally constant over subvarieties of the character torus."}
{"category": "Math", "title": "Locally constant n-operads as higher braided operads", "abstract": "We introduce a category of locally constant $n$-operads which can be considered as the category of higher braided operads. For $n=1,2,\\infty$ the homotopy category of locally constant $n$-operads is equivalent to the homotopy category of classical nonsymmetric, braided and symmetric operads correspondingly."}
{"category": "Math", "title": "Mirabolic Langlands duality and the quantum Calogero-Moser system", "abstract": "We give a generic spectral decomposition of the derived category of twisted D-modules on a moduli stack of mirabolic vector bundles on a curve X in characteristic p: that is, we construct an equivalence with the derived category of quasi-coherent sheaves on a moduli stack of mirabolic local systems on X. This equivalence may be understood as a tamely ramified form of the geometric Langlands equivalence. When X has genus 1, this equivalence generically solves (in the sense of noncommutative geometry) the quantum Calogero-Moser system."}
{"category": "Math", "title": "Syzygies of multiplier ideals on singular varieties", "abstract": "It was recently established by the first two authors that multiplier ideals on a smooth variety satisfy some special syzygetic properties. The purpose of this note is to show how some of these can be extended to the singular setting."}
{"category": "Math", "title": "The Morse Index of Wente Tori", "abstract": "We find various lower and upper bounds for the index of Wente tori that contain a continuous family of planar principal curves. We then prove a result that gives an algorithm for computing the index sharply."}
{"category": "Math", "title": "Mean curvature one surfaces in hyperbolic space, and their relationship to minimal surfaces in Euclidean space", "abstract": "We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results on these surfaces. We include computer graphics of a number of examples."}
{"category": "Math", "title": "Minimal surfaces with planar boundary curves", "abstract": "We consider compact connected minimal surfaces, with a pair of boundary curves (not necessarily convex) in distinct planes, that have least-area amongst all orientable surfaces with the same boundary. When the planes containing these two boundary curves are either parallel or sufficiently close to parallel, and when the boundary curves themselves are sufficiently close to each other, we draw specific conclusions about the geometry and topology of the surfaces. We also strength the following result: Let $M$ be any compact minimal annulus with two planar boundary curves of diameters $d_1$ and $d_2$ in parallel planes $P_1$ and $P_2$; if the distance between $P_1$ and $P_2$ is $h$, then the inequality $h \\leq {3/2}\\max\\{d_1,d_2\\}$ is satisfied. We strength it by removing the assumption that $M$ is an annulus and by showing that the stronger conclusion $h \\leq \\max\\{d_1,d_2\\}$ holds. We also include a similar result for nonminimal constant mean curvature surfaces."}
{"category": "Math", "title": "High-Dimensional Graphical Model Selection Using $\\ell_1$-Regularized Logistic Regression", "abstract": "We consider the problem of estimating the graph structure associated with a discrete Markov random field. We describe a method based on $\\ell_1$-regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an $\\ell_1$-constraint. Our framework applies to the high-dimensional setting, in which both the number of nodes $p$ and maximum neighborhood sizes $d$ are allowed to grow as a function of the number of observations $n$. Our main results provide sufficient conditions on the triple $(n, p, d)$ for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously. Under certain assumptions on the population Fisher information matrix, we prove that consistent neighborhood selection can be obtained for sample sizes $n = \\Omega(d^3 \\log p)$, with the error decaying as $\\order(\\exp(-C n/d^3))$ for some constant $C$. If these same assumptions are imposed directly on the sample matrices, we show that $n = \\Omega(d^2 \\log p)$ samples are sufficient."}
{"category": "Math", "title": "Minimal Surfaces with Catenoid Ends", "abstract": "In this paper, we use the conjugate surface construction to prove the existence of certain non-periodic symmetric immersed minimal surfaces. These surfaces have finite total curvature and embedded catenoid ends, and they have positive genus yet maintain the symmetry of their genus-zero counterparts constructed by Jorge-Meeks and Xu."}
{"category": "Math", "title": "Minimal surfaces in $R^3$ with dihedral symmetry", "abstract": "We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry group the natural $\\bfZ_2$ extension of the dihedral group $D_n$. The surfaces are constructed by proving existence of the conjugate surfaces. We extend this method to cases where the conjugate surface of the fundamental piece is noncompact and is not a graph over a convex plane domain."}
{"category": "Math", "title": "On embeddedness of area-minimizing disks, and an application to constructing complete minimal surfaces", "abstract": "Let $\\alpha$ be a polygonal Jordan curve in $\\bfR^3$. We show that if $\\alpha$ satisfies certain conditions, then the least-area Douglas-Rad\\'{o} disk in $\\bfR^3$ with boundary $\\alpha$ is unique and is a smooth graph. As our conditions on $\\alpha$ are not included amongst previously known conditions for embeddedness, we are enlarging the set of Jordan curves in $\\bfR^3$ which are known to be spanned by an embedded least-area disk. As an application, we consider the conjugate surface construction method for minimal surfaces. With our result we can apply this method to a wider range of complete catenoid-ended minimal surfaces in $\\bfR^3$."}
{"category": "Math", "title": "Sums of Zeros for Certain Special Functions", "abstract": "Key words and phrases: q-Airy function (Ramanujan's entire function); q-Bessel function; Bessel function; Airy function; Riemann zeta function; Dirichlet L-series."}
{"category": "Math", "title": "Constant mean curvature surfaces with two ends in hyperbolic space", "abstract": "We investigate the close relationship between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. Just as in the case of minimal surfaces in Euclidean 3-space, the only complete connected embedded constant mean curvature 1 surfaces with two ends in hyperbolic space are well-understood surfaces of revolution -- the catenoid cousins. In contrast to this, we show that, unlike the case of minimal surfaces in Euclidean 3-space, there do exist complete connected immersed constant mean curvature 1 surfaces with two ends in hyperbolic space that are not surfaces of revolution -- the genus 1 catenoid cousins. The genus 1 catenoid cousins are of interest because they show that, although minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space are intimately related, there are essential differences between these two sets of surfaces (when the surfaces are considered globally). The proof we give of existence of the genus 1 catenoid cousins is a mathematically rigorous verification that the results of a computer experiment are sufficiently accurate to imply existence."}
{"category": "Math", "title": "On the idempotents of Hecke algebras", "abstract": "We give a new construction of primitive idempotents of the Hecke algebras associated with the symmetric groups. The idempotents are found as evaluated products of certain rational functions thus providing a new version of the fusion procedure for the Hecke algebras. We show that the normalization factors which occur in the procedure are related to the Ocneanu--Markov trace of the idempotents."}
{"category": "Math", "title": "Complex and Kaehler structures on compact solvmanifolds", "abstract": "We discuss our recent results on the existence and classification problem of complex and Kaehler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact solvmanifolds (and compact homogeneous manifolds in general)."}
{"category": "Math", "title": "Potentially $K_{m}-G$-graphical Sequences: A Survey", "abstract": "The set of all non-increasing nonnegative integers sequence $\\pi=$ ($d(v_1),$ $d(v_2),$ $...,$ $d(v_n)$) is denoted by $NS_n$. A sequence $\\pi\\in NS_n$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices, and such a graph $G$ is called a realization of $\\pi$. The set of all graphic sequences in $NS_n$ is denoted by $GS_n$. A graphical sequence $\\pi$ is potentially $H$-graphical if there is a realization of $\\pi$ containing $H$ as a subgraph, while $\\pi$ is forcibly $H$-graphical if every realization of $\\pi$ contains $H$ as a subgraph. Let $K_k$ denote a complete graph on $k$ vertices. Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). This paper summarizes briefly some recent results on potentially $K_{m}-G$-graphic sequences and give a useful classification for determining $\\sigma(H,n)$."}
{"category": "Math", "title": "An intrinsic non-triviality of graphs", "abstract": "We say that a graph is intrinsically non-trivial if every spatial embedding of the graph contains a non-trivial spatial subgraph. We prove that an intrinsically non-trivial graph is intrinsically linked, namely every spatial embedding of the graph contains a non-splittable 2-component link. We also show that there exists a graph such that every spatial embedding of the graph contains either a non-splittable 3-component link or an irreducible spatial handcuff graph whose constituent 2-component link is split."}
{"category": "Math", "title": "A state sum invariant for regular isotopy of links having a polynomial number of states", "abstract": "The state sum regular isotopy invariant of links which I introduce in this work is a generalization of the Jones Polynomial. So it distinguishes any pair of links which are distinguishable by Jones'. This new invariant, denoted {\\em VSE-invariant} is strictly stronger than Jones': I detected a pair of links which are not distinguished by Jones' but are distinguished by the new invariant. The full VSE-invariant has $3^n$ states. However, there are useful specializations of it parametrized by an integer k, having $O(n^k)=\\sum_{\\ell=0}^k {n \\choose \\ell} 2^\\ell $ states. The link with more crossings of the pair which was distinguished by the VSE-invariant has 20 crossings. The specialization which is enough to distinguish corresponds to k=2 and has only 801 states, as opposed to the $2^{20} = 1,048,576$ states of the Jones polynomial of the same link. The full VSE-invariant of it has $3^{20} = 3,486,784,401$ states. The VSE-invariant is a good alternative for the Jones polynomial when the number of crossings makes the computation of this polynomial impossible. For instance, for $k=2$ the specialization of the VSE-invariant of a link with $n=500$ crossings can be computed in a few minutes, since it has only $2 n^2+1 = 500,001$ states."}
{"category": "Math", "title": "Products of Toeplitz Operators on a Vector Valued Bergman Space", "abstract": "We give a necessary and a sufficient condition for the boundedness of the Toeplitz product $T_FT_{G^*}$ on the vector valued Bergman space $L_a^2(\\mathbb{C}^n)$, where $F$ and $G$ are matrix symbols with scalar valued Bergman space entries. The results generalize existing results in the scalar valued Bergman space case. We also characterize boundedness and invertibility of Toeplitz products $T_FT_{G^*}$ in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space."}
{"category": "Math", "title": "Twistors, 4-symmetric spaces and integrable systems", "abstract": "An order four automorphism of a Lie algebra gives rise to an integrable system discussed by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space. Specialising to 4-dimensional target, we find that surfaces with holomorphic mean curvature in 4-dimensional spaces with constant sectional or holomorphic sectional curvatures constitute an integrable system as do Hamiltonian stationary Lagrangian surfaces in a 4-dimensional Hermitian symmetric space (this last being a result of Helein-Romon)."}
{"category": "Math", "title": "Entropy and its variational principle for noncompact metric spaces", "abstract": "In the present paper, we introduce a natural extension of AKM-topological entropy for noncompact spaces and prove a variational principle which states that the topological entropy, the supremum of the measure theoretical entropies and the minimum of the metric theoretical entropies always coincide. We apply the variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the classical formula for the entropy of an automorphism of a noncompact Lie group is just an upper bound for its topological entropy."}
{"category": "Math", "title": "Embeddings of four-valent framed graphs into two-surfaces", "abstract": "We study the problem of finding the minimal (maximal) genus for a surface where a given four-valent graph with fixed opposite edge structure can be embedded into. We find several partial relations and give new reformulations in combinatorial and knot theoretic languages."}
{"category": "Math", "title": "Finite generation of Tate cohomology", "abstract": "Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable non-projective kG-module M, we conjecture that if the Tate cohomology $\\HHHH^*(G, M)$ of G with coefficients in M is finitely generated over the Tate cohomology ring $\\HHHH^*(G, k)$, then the support variety V_G(M) of M is equal to the entire maximal ideal spectrum V_G(k). We prove various results which support this conjecture. The converse of this conjecture is established for modules in the connected component of k in the stable Auslander-Reiten quiver for kG, but it is shown to be false in general. It is also shown that all finitely generated kG-modules over a group G have finitely generated Tate cohomology if and only if G has periodic cohomology."}
{"category": "Math", "title": "A hysteresis model for two-dimensional input signals", "abstract": "We formulate and analyze a new model of vector hysteresis for the case of two-dimensional input signals. We prove certain properties of this model and we present the solutions to two identification problems connected with our model."}
{"category": "Math", "title": "Tilting generators via ample line bundles", "abstract": "It is known that a tilting generator on an algebraic variety $X$ gives a derived equivalence between $X$ and a certain non-commutative algebra. In this paper, we explain a method to construct a tilting generator from an ample line bundle, and construct it in several examples."}
{"category": "Math", "title": "Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein--Uhlenbeck processes", "abstract": "Properties of the law $\\mu$ of the integral $\\int_0^{\\infty}c^{-N_{t-}}\\,dY_t$ are studied, where $c>1$ and $\\{(N_t,Y_t),t\\geq0\\}$ is a bivariate L\\'{e}vy process such that $\\{N_t\\}$ and $\\{Y_t\\}$ are Poisson processes with parameters $a$ and $b$, respectively. This is the stationary distribution of some generalized Ornstein--Uhlenbeck process. The law $\\mu$ is parametrized by $c$, $q$ and $r$, where $p=1-q-r$, $q$, and $r$ are the normalized L\\'{e}vy measure of $\\{(N_t,Y_t)\\}$ at the points $(1,0)$, $(0,1)$ and $(1,1)$, respectively. It is shown that, under the condition that $p>0$ and $q>0$, $\\mu_{c,q,r}$ is infinitely divisible if and only if $r\\leq pq$. The infinite divisibility of the symmetrization of $\\mu$ is also characterized. The law $\\mu$ is either continuous-singular or absolutely continuous, unless $r=1$. It is shown that if $c$ is in the set of Pisot--Vijayaraghavan numbers, which includes all integers bigger than 1, then $\\mu$ is continuous-singular under the condition $q>0$. On the other hand, for Lebesgue almost every $c>1$, there are positive constants $C_1$ and $C_2$ such that $\\mu$ is absolutely continuous whenever $q\\geq C_1p\\geq C_2r$. For any $c>1$ there is a positive constant $C_3$ such that $\\mu$ is continuous-singular whenever $q>0$ and $\\max\\{q,r\\}\\leq C_3p$. Here, if $\\{N_t\\}$ and $\\{Y_t\\}$ are independent, then $r=0$ and $q=b/(a+b)$."}
{"category": "Math", "title": "A posteriori error control for discontinuous Galerkin methods for parabolic problems", "abstract": "We derive energy-norm a posteriori error bounds for an Euler time-stepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case, and then move to the fully discrete scheme by introducing the implicit Euler time-stepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To illustrate the method, we apply it to the interior penalty discontinuous Galerkin method, which requires the derivation of novel a posteriori error bounds. For the analysis of the time-dependent problems we use the elliptic reconstruction technique and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it."}
{"category": "Math", "title": "Families of absolutely simple hyperelliptic jacobians", "abstract": "We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field $K$ of characteristic zero if $t \\in K$ and the Galois group of the polynomial $h(x)$ over $K$ is \"very big\" and $deg(h)$ is an even number >8. (The case of odd $deg(h)>3$ follows easily from previous results of the author.)"}
{"category": "Math", "title": "Parameter estimation of high-dimensional linear differential equations", "abstract": "We study the problem of estimating the coefficients in linear ordinary differential equations (ODE's) with a diverging number of variables when the solutions are observed with noise. The solution trajectories are first smoothed with local polynomial regression and the coefficients are estimated with nonconcave penalty proposed by \\cite{fan01}. Under some regularity and sparsity conditions, we show the procedure can correctly identifies nonzero coefficients with probability converging to one and the estimators for nonzero coefficients have the same asymptotic normal distribution as they would have when the zero coefficients are known and the same two-step procedure is used. Our asymptotic results are valid under the misspecified case where linear ODE's are only used as an approximation to nonlinear ODE's, and the estimates will converge to the coefficients of the best approximating linear system. From our results, when the solution trajectories of the ODE's are sufficiently smooth, the parametric $\\sqrt{n}$ rate is achieved even though nonparametric regression estimator is used in the first step of the procedure. The performance of the two-step procedure is illustrated by a simulation study as well as an application to yeast cell-cycle data."}
{"category": "Math", "title": "Yet another generalization of Postnikov's hook length formula for binary trees", "abstract": "We discover another one-parameter generalization of Postnikov's hook length formula for binary trees. The particularity of our formula is that the hook length $h_v$ appears as an exponent. As an application, we derive another simple hook length formula for binary trees when the underlying parameter takes the value 1/2."}
{"category": "Math", "title": "The Frolicher--Kriegl differentiabilities as a particular case of the Bertram--Glockner--Neeb construction", "abstract": "We prove that the order $k$ differentiability classes for $k=0,1,...\\infty$ in the \"arc-generated\" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl can be obtained as particular cases of the general construction by Bertram, Glockner and Neeb leading to $C^k$ differentiabilities from a given $C^0$ concept."}
{"category": "Math", "title": "On the non-existence of exceptional automorphisms on Shimura curves", "abstract": "We study the group of automorphisms of Shimura curves $X_0(D, N)$ attached to an Eichler order of square-free level $N$ in an indefinite rational quaternion algebra of discriminant $D>1$. We prove that, when the genus $g$ of the curve is greater than or equal to 2, $\\Aut (X_0(D, N))$ is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions $W_0(D, N)$ as a subgroup of index 1 or 2. It is conjectured that $\\Aut (X_0(D, N)) = W_0(D, N)$ except for finitely many values of $(D, N)$ and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik-Drinfeld theory on their rigid analytic uniformization at primes $p\\mid D$."}
{"category": "Math", "title": "Detecting phylogenetic relations out from sparse context trees", "abstract": "The goal of this paper is to study the similarity between sequences using a distance between the \\emph{context} trees associated to the sequences. These trees are defined in the framework of \\emph{Sparse Probabilistic Suffix Trees} (SPST), and can be estimated using the SPST algorithm. We implement the Phyl-SPST package to compute the distance between the sparse context trees estimated with the SPST algorithm. The distance takes into account the structure of the trees, and indirectly the transition probabilities. We apply this approach to reconstruct a phylogenetic tree of protein sequences in the globin family of vertebrates. We compare this tree with the one obtained using the well-known PAM distance."}
{"category": "Math", "title": "Supercharacters of the Sylow p-subgroups of the finite symplectic and orthogonal groups", "abstract": "We define and study supercharacters of the classical finite unipotent groups of symplectic and orthogonal types (over any finite field of odd characteristic). We show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. We also describe all irreducible characters of maximum degree in terms of the root system, and show how they can be obtained as constituents of particular supercharacters."}
{"category": "Math", "title": "Structure of wrap groups of quaternion and octonion fiber bundles", "abstract": "This article is devoted to the investigation of structure of wrap groups of connected fiber bundles over the fields of real $\\bf R$, complex $\\bf C$ numbers, the quaternion skew field $\\bf H$ and the octonion algebra $\\bf O$. Iterated wrap groups are studied as well. Their smashed products are constructed."}
{"category": "Math", "title": "Simple polytopes arising from finite graphs", "abstract": "Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${\\cal P}_G$ and the toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is proved that the toric ideals $I_G$ of $G$ possesses a quadratic Gr\\\"obner basis if the edge polytope ${\\cal P}_G$ of $G$ is simple. It is also shown that, for a finite graph $G$, the edge polytope is simple but not a simplex if and only if it is smooth but not a simplex. Moreover, the Ehrhart polynomial and the normalized volume of simple edge polytopes are computed."}
{"category": "Math", "title": "Null structure and almost optimal local well-posedness of the Maxwell-Dirac system", "abstract": "We uncover the full null structure of the Maxwell-Dirac system in Lorenz gauge. This structure, which cannot be seen in the individual component equations, but only when considering the system as a whole, is expressed in terms of tri- and quadrilinear integral forms with cancellations measured by the angles between spatial frequencies. In the 3D case, we prove frequency-localized $L^2$ space-time estimates for these integral forms at the scale invariant regularity up to a logarithmic loss, hence we obtain almost optimal local well-posedness of the system by iteration."}
{"category": "Math", "title": "Anisotropic bilinear $L^2$ estimates related to the 3d wave equation", "abstract": "We first review the $L^2$ bilinear generalizations of the $L^4$ estimate of Strichartz for solutions of the homogeneous 3D wave equation, and give a short proof based solely on an estimate for the volume of intersection of two thickened spheres. We then go on to prove a number of new results, the main theme being how additional, anisotropic Fourier restrictions influence the estimates. Moreover, we prove some refinements which are able to simultaneously detect both concentrations and nonconcentrations in Fourier space."}
{"category": "Math", "title": "A new generalization of Ostrowski type inequality on time scales", "abstract": "In this paper we first extend a generalization of Ostrowski type inequality on time scales for functions whose derivatives are bounded and then unify corresponding continuous and discrete versions. We also point out some particular integral type inequalities on time scales as special cases."}
{"category": "Math", "title": "Large Deviations for Random Spectral Measures and Sum Rules", "abstract": "We prove a Large Deviation Principle for the random spec- tral measure associated to the pair $(H_N; e)$ where $H_N$ is sampled in the GUE(N) and e is a fixed unit vector (and more generally in the $\\beta$- extension of this model). The rate function consists of two parts. The contribution of the absolutely continuous part of the measure is the reversed Kullback information with respect to the semicircle distribution and the contribution of the singular part is connected to the rate function of the extreme eigenvalue in the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but in thoses cases the expression of the rate function is not so explicit."}
{"category": "Math", "title": "The Connectivity Order of Links", "abstract": "We associate at each link a connectivity space which describes its splittability properties. Then, the notion of order for finite connectivity spaces results in the definition of a new numerical invariant for links, their connectivity order. A section of this short paper presents a theorem which asserts that every finite connectivity structure can be realized by a link : the Brunn-Debrunner-Kanenobu Theorem."}
{"category": "Math", "title": "Int\\'egration du contr\\^ole automatique dans la ma\\^itrise statistique des proc\\'ed\\'es", "abstract": "The Statistical Process Control (SPC) and the Automated Process Control (APC) have a common goal: achieve optimal product quality by controlling variations in the process. The work in this paper will present a developed integration methodology of the APC in the SPC which is based on discretization of the transfer functions relating to each component of the process. We proposed on the one hand, a new control rule which is based on a system of first order. In the other hand, we showed how to establish control charts to a process of the type AR (1). Using simulation experiments, we showed that the proposed control rule reduced variability by comparing it with that proposed in literature."}
{"category": "Math", "title": "Some remarks on cabling, contact structures, and complex curves", "abstract": "We determine the relationship between the contact structure induced by a fibered knot, K, in the three-sphere and the contact structures induced by its various cables. Understanding this relationship allows us to classify fibered cable knots which bound a properly embedded complex curve in the four-ball satisfying a genus constraint. This generalizes the well-known classification of links of plane curve singularities."}
{"category": "Math", "title": "Fourier-Laplace transform of a variation of polarized complex Hodge structure, II", "abstract": "We show that the limit, by rescaling, of the `new supersymmetric index' attached to the Fourier-Laplace transform of a polarized variation of Hodge structure on a punctured affine line is equal to the spectral polynomial attached to the same object. We also extend the definition by Deligne of a Hodge filtration on the de Rham cohomology of a exponentially twisted polarized variation of complex Hodge structure and prove a E_1 degeneration property for it."}
{"category": "Math", "title": "Integral structures on $p$-adic Fourier theory", "abstract": "In this article, we give an explicit construction of the $p$-adic Fourier transform by Schneider and Teitelbaum, which allows for the investigation of the integral property. As an application, we give a certain integral basis of the space of $K$-locally analytic functions on the ring of integers $\\mathcal{O}_K$ for any finite extension $K$ of $\\mathbb{Q}_p$, generalizing the basis constructed by Amice for locally analytic functions on $\\mathbb{Z}_p$. We also use our result to prove congruences of Bernoulli-Hurwitz numbers at non-ordinary (i.e. supersingular) primes originally investigated by Katz and Chellali."}
{"category": "Math", "title": "A note on solitary waves solutions of classical wave equations", "abstract": "The goal of this work is to determine whole classes of solitary wave solutions general for wave equations."}
{"category": "Math", "title": "Some more proofs from the Book: solvability and insolvability of equations in radicals", "abstract": "This paper is purely expository. We present short elementary proofs of * the Gauss Theorem on constructibility of regular polygons; * the existence of a cubic equation unsolvable in real radicals; * the existence of a quintic equation unsolvable in complex radicals (Galois Theorem). The statements of these celebrated results are simple and well-known. However, their proofs given in most textbooks rely upon much unmotivated material and are far from being economic. We do not use the terms `Galois group' or even `group'. The paper is accessible for students familiar with polynomials and complex numbers, and could be an interesting easy reading for professional mathematicians. Short English version is followed by an extended Russian version where before presenting the proofs we illustrate the main ideas by sequences of problems with hints or solutions."}
{"category": "Math", "title": "The $C^{\\a}$ regularity of a class hypoelliptic ultraparabolic equations", "abstract": "We obtained the $C^{\\a}$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form ${\\ptl_t u}= \\sum_{i,j=1}^{m_0}X_i(a_{ij}(x,t)X_j u)+X_0 u$. The result is proved by simplifying and generalizing our earlier arguments for the $C^{\\a}$ regularity of homogeneous ultraparabolic equations."}
{"category": "Math", "title": "Improving Coverage Accuracy of Block Bootstrap Confidence Intervals", "abstract": "The block bootstrap confidence interval based on dependent data can outperform the computationally more convenient normal approximation only with non-trivial Studentization which, in the case of complicated statistics, calls for highly specialist treatment. We propose two different approaches to improving the accuracy of the block bootstrap confidence interval under very general conditions. The first calibrates the coverage level by iterating the block bootstrap. The second calculates Studentizing factors directly from block bootstrap series and requires no non-trivial analytic treatment. Both approaches involve two nested levels of block bootstrap resampling and yield high-order accuracy with simple tuning of block lengths at the two resampling levels. A simulation study is reported to provide empirical support for our theory."}
{"category": "Math", "title": "Ergodic Optimal Quadratic Control for an Affine Equation with Stochastic and Stationary Coefficients", "abstract": "We study ergodic quadratic optimal stochastic control problems for an affine state equation with state and control dependent noise and with stochastic coefficients. We assume stationarity of the coefficients and a finite cost condition. We first treat the stationary case and we show that the optimal cost corresponding to this ergodic control problem coincides with the one corresponding to a suitable stationary control problem and we provide a full characterization of the ergodic optimal cost and control."}
{"category": "Math", "title": "Periodic solutions for a class of nonlinear partial differential equations in higher dimension", "abstract": "We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schroedinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases where the bifurcation equation is infinite-dimensional, such as the nonlinear Schroedinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist."}
{"category": "Math", "title": "Noncommutative topological entropy of endomorphisms of Cuntz algebras", "abstract": "Noncommutative topological entropy estimates are obtained for polynomial gauge invariant endomorphisms of Cuntz algebras, generalising known results for the canonical shift endomorphisms. Exact values for the entropy are computed for a class of permutative endomorphisms related to branching function systems introduced and studied by Bratteli, Jorgensen and Kawamura."}
{"category": "Math", "title": "Fractional Brownian flows", "abstract": "We consider stochastic flow on n-dimensional Euclidean space driven by fractional Brownian motion with Hurst parameter H greater than half, and study tangent flow and the growth of the Hausdorff measure of sub-manifolds of the ambient n-dimensional Euclidean space, as they evolve under the flow. The main result is a bound on the rate of (global) growth in terms of the (local) Holder norm of the flow."}
{"category": "Math", "title": "On the Finsler metrics obtained as limits of chessboard structures", "abstract": "We study the geodesics in a planar chessboard structure with two values 1 and $\\beta>1$. The results for a fixed structure allow us to infer the properties of the Finsler metrics obtained, with an homogenization procedure, as limit of oscillating chessboard structures."}
{"category": "Math", "title": "Symmetry and holomorphy of the third-order ordinary differential equation satisfied by the third Painlev\\'e Hamiltonian", "abstract": "We study symmetry and holomorphy of the third-order ordinary differential equation satisfied by the third Painlev\\'e Hamiltonian."}
{"category": "Math", "title": "Op\\'erateurs d'entrelacement et alg\\`ebres de Hecke avec param\\`etres d'un groupe r\\'eductif $p$-adique - le cas des groupes classiques", "abstract": "For $G$ a symplectic or orthogonal $p$-adic group (not necessarily split), or an inner form of a general linear $p$-adic group, we compute the endomorphism algebras of some induced projective generators \\`a la Bernstein of the category of smooth representations of $G$ and show that these algebras are isomorphic to the semi-direct product of a Hecke algebra with parameters by a finite group algebra. Our strategy and parts of our intermediate results apply to a general reductive connected $p$-adic group."}
{"category": "Math", "title": "Families of vector fields which generate the group of diffeomorphisms", "abstract": "Given a compact manifold M, we prove that any bracket generating and invariant under multiplication on smooth functions family of vector fields on M generates the connected component of unit of the group Diff(M)."}
{"category": "Math", "title": "A weak energy identity and the length of necks for a Sacks-Uhlenbeck $\\alpha$-harmonic map sequence", "abstract": "We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks."}
{"category": "Math", "title": "An Equivalence Relation on A Set of Words of Finite Length", "abstract": "In this work, we study several equivalence relations induced from the partitions of the sets of words of finite length. We have results on words over finite fields extending the work of Bacher (2002, Europ. J. Combinatorics, {\\bf 23}, 141-147). Cardinalities of its equivalence classes and explicit relationships between two words are determined. Moreover, we deal with words of finite length over the ring $\\mathbb{Z}/N\\mathbb{Z}$ where $N$ is a positive integer. We have arithmetic results parallel to Bacher's."}
{"category": "Math", "title": "The diamond-alpha Riemann integral and mean value theorems on time scales", "abstract": "We study diamond-alpha integrals on time scales. A diamond-alpha version of Fermat's theorem for stationary points is also proved, as well as Rolle's, Lagrange's, and Cauchy's mean value theorems on time scales."}
{"category": "Math", "title": "On homology classes not representable by products", "abstract": "This paper has been withdrawn, because it is subsumed by the new preprint arXiv:0806.4540 ."}
{"category": "Math", "title": "The isometry group of L^{p}(\\mu,\\X) is SOT-contractible", "abstract": "We will show that if (\\Omega,\\Sigma,\\mu) is an atomless positive measure space, X is a Banach space and 1\\leq p<\\infty, then the group of isometric automorphisms on the Bochner space L^{p}(\\mu,X) is contractible in the strong operator topology. We do not require \\Sigma or X above to be separable."}
{"category": "Math", "title": "The intrinsic asymmetry and inhomogeneity of Teichmuller space", "abstract": "Royden proved that any isometry of Teichmuller space in the Teichmuller metric must be an element of the extended mapping class group M(S). He also proved that the Teichmuller metric is not symmetric at any point. In this paper we give extensions of Royden's theorems from the Teichmuller metric to an arbitrary complete, finite covolume, M(S)-invariant Finsler (e.g. Riemannian) metric on Teichmuller space. In particular this gives a new mechanism behind Royden's original theorem."}
{"category": "Math", "title": "Volume Laws for Boxed Plane Partitions and Area Laws for Ferrers Diagrams", "abstract": "We asymptotically analyse the volume-random variables of general, symmetric and cyclically symmetric plane partitions fitting inside a box. We consider the respective symmetry class equipped with the uniform distribution. We also prove area limit laws for two ensembles of Ferrers diagrams. Most of the limit laws are Gaussian."}
{"category": "Math", "title": "On the Chern number of a filtration", "abstract": "We study the first Hilbert coefficient (after the multiplicity) $e_1$ of a local ring $(A,\\m). $ Under various circumstances, it is also called the {\\bf Chern number} of the local ring $A.$ Starting from the work of D.G. Northcott in the 60's, several results have been proved which give some relationships between the Hilbert coefficients, but always assuming the Cohen-Macaulayness of the basic ring. Recent papers of S. Goto, K. Nishida, A. Corso and W. Vasconcelos pushed the interest toward a more general setting. In this paper we extend an upper bound on $e_1$ proved by S. Huckaba and T. Marley. Thus we get the Cohen-Macaulayness of the ring $A$ as a consequence of the extremal behavior of the integer $e_1.$ The result can be considered a confirm of the general philosophy of the paper of W. Vasconcelos where the Chern number is conjectured to be a measure of the distance from the Cohen-Macaulyness of $A.$ This main result of the paper is a consequence of a nice and perhaps unexpected property of superficial elements. It is essentially a kind of \"Sally machine\" for local rings. In the last section we describe an application of these results, concerning an upper bound on the multiplicity of the Sally module of a good filtration of a module which is not necessarily Cohen-Macaulay. It is an extension to the non Cohen-Macaulay case of a result of Vaz Pinto."}
{"category": "Math", "title": "Existence and regularity of a nonhomogeneous transition matrix under measurability conditions", "abstract": "This paper is about the existence and regularity of the transition probability matrix of a nonhomogeneous continuous-time Markov process with a countable state space. A standard approach to prove the existence of such a transition matrix is to begin with a continuous (in t) and conservative matrix Q(t)=[q_{ij}(t)] of nonhomogeneous transition rates q_{ij}(t), and use it to construct the transition probability matrix. Here we obtain the same result except that the q_{ij}(t) are only required to satisfy a mild measurability condition, and Q(t) may not be conservative. Moreover, the resulting transition matrix is shown to be the minimum transition matrix and, in addition, a necessary and sufficient condition for it to be regular is obtained. These results are crucial in some applications of nonhomogeneous continuous-time Markov processes, such as stochastic optimal control problems and stochastic games, which motivated this work in the first place."}
{"category": "Math", "title": "Minimal free resolution of a finitely generated module over a regular local ring", "abstract": "Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R,\\n)$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $\\n$-stable filtrations ${\\mathbb M} $ of $M $ and to compare the Betti numbers of $M$ with those of the associated graded module $ gr_{\\mathbb M}(M). $ This approach has the advantage that the same module $M$ can be detected by using different filtrations on it. It provides interesting upper bounds for the Betti numbers and we study the modules for which the extremal values are attained. Among others, the Koszul modules have this behavior. As a consequence of the main result, we extend some results by Aramova, Conca, Herzog and Hibi on the rigidity of the resolution of standard graded algebras to the local setting."}
{"category": "Math", "title": "A new characterization of Baire class 1 functions", "abstract": "We give a new characterization of the Baire class 1 functions (defined on an ultrametric space) by proving that they are exactly the pointwise limits of sequences of full functions (which are particularly simple Lipschitz functions). Moreover we highlight the link between the two classical stratifications of the Borel functions by showing that the Baire class functions of some level are exactly those obtained as uniform limits of sequences of Delta functions (of a corresponding level)."}
{"category": "Math", "title": "Non-unitary representations, Baum-Connes morphism and unconditional completions", "abstract": "We show that the Baum-Connes morphism twisted by a non-unitary representation, defined in [GA08], is an isomorphism for a large class of groups satisfying the Baum-Connes conjecture. Such class contains all the real semi-simple Lie groups, all hyperbolic groups and many infinite discret groups having Kazhdan's property (T). We define a tensorisation by a non-unitary finite dimensional representation on the left handside of the Baum-Connes morphism and we show that its analogue in K-theory must be defined on the K-theory of the twisted group algebras introduced in [GA07b]."}
{"category": "Math", "title": "Matrix Cubes Parametrized by Eigenvalues", "abstract": "An elimination problem in semidefinite programming is solved by means of tensor algebra. It concerns families of matrix cube problems whose constraints are the minimum and maximum eigenvalue function on an affine space of symmetric matrices. An LMI representation is given for the convex set of all feasible instances, and its boundary is studied from the perspective of algebraic geometry. This generalizes the earlier work [12] with Parrilo on k-ellipses and k-ellipsoids."}
{"category": "Math", "title": "On an area property of the sum cotA+cotB+cotC in a triangle", "abstract": "Given a triangle ABC, a new triangle A'B'C' can be formed as follows: Draw the perpendicular to the line AB at the pointA; then the perpendicular to the line BC at B, and lastly the perpendicular to the line CA at C.the two triangles ABC and A'B'C' are always similar. In Postulate1 we prove that the ratio E'/E is equal to (cotA+cotB+cotC)^2, which is the main result in this work.Here E' and E stand for the areas of the triangles A'B'C' and ABC respectively. In Postulate 2, we show that the above ratio has minimum value 3, which is attained when ABC and A'B'C' are equilateral triangles.In Postulate 3, we show that if we consider only those pairs of triangles (ABC,A'B'C'), with both ABC and A'B'C' being right triangles, then the minimum value of the above ratio of areas, is 4."}
{"category": "Math", "title": "Infinite sequences in the framework of classical logic", "abstract": "Infinite sequences are considered in the framework of classical logic from a new point of view."}
{"category": "Math", "title": "On the spectra of a Cantor measure", "abstract": "We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in $\\{0, 1, 2, 3\\}$) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis."}
{"category": "Math", "title": "Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields", "abstract": "We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas-Stein exponent. Using incidence theorems, we also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimates for Gauss and Kloosterman sums and their variants play an important role."}
{"category": "Math", "title": "A Geometric Proof to Cantor's Theorem and an Irrationality Measure for Some Cantor's Series", "abstract": "Generalizing a geometric idea due to J. Sondow, we give a geometric proof for the Cantor's Theorem. Moreover, it is given an irrationality measure for some Cantor series."}
{"category": "Math", "title": "On the holomorphic closure dimension of real analytic sets", "abstract": "Given a real analytic (or, more generally, semianalytic) set R in the n-dimensional complex space, there is, for every point p in the closure of R, a unique smallest complex analytic germ X_p that contains the germ R_p. We call the complex dimension of X_p the holomorphic closure dimension of R at p. We show that the holomorphic closure dimension of an irreducible R is constant on the complement of a closed proper analytic subset of R, and discuss the relationship between this dimension and the CR dimension of R."}
{"category": "Math", "title": "Cordes conditions and some alternatives for parabolic equations and discontinuous diffusion", "abstract": "The paper considers parabolic equations in non-divergent form with discontinuous coefficients at higher derivatives. Their investigation is most complicated because, in general, in the case of discontinuous coefficients, the uniqueness of a solution for nonlinear parabolic or elliptic equations can fail, and there is no a priory estimate for partial derivatives of a solution. In this paper, existence and regularity results are obtained under some Cordes type restrictions on the coefficients. The results are applied to diffusion processes."}
{"category": "Math", "title": "Numerical solutions of integrodifferential systems by hybrid of general block-pulse functions and the second Chebyshev polynomials", "abstract": "By applying hybrid functions of general block-pulse functions and the second Chebyshev polynomials,integrodifferential systems are converted into a system of algebraic equations. The approximate solutions of integrodifferential systems are derived. The numerical examples illustrate that the algorithms are valid."}
{"category": "Math", "title": "Long heterochromatic paths in heterochromatic triangle free graphs", "abstract": "In this paper, graphs under consideration are always edge-colored. We consider long heterochromatic paths in heterochromatic triangle free graphs. Two kinds of such graphs are considered, one is complete graphs with Gallai colorings, i.e., heterochromatic triangle free complete graphs; the other is heterochromatic triangle free graphs with $k$-good colorings, i.e., minimum color degree at least $k$. For the heterochromatic triangle free graphs $K_n$, we obtain that for every vertex $v\\in V(K_n)$, $K_n$ has a heterochromatic $v$-path of length at least $d^c(v)$; whereas for the heterochromatic triangle free graphs $G$ we show that if, for any vertex $v\\in V(G)$, $d^c(v)\\geq k\\geq 6$, then $G$ a heterochromatic path of length at least $\\frac{3k}{4}$."}
{"category": "Math", "title": "Assuad-Nagata dimension of nilpotent groups with arbitrary left invariant metrics", "abstract": "Suppose $G$ is a countable, not necessarily finitely generated, group. We show $G$ admits a proper, left-invariant metric $d_G$ such that the Assouad-Nagata dimension of $(G,d_G)$ is infinite, provided the center of $G$ is not locally finite. As a corollary we solve two problems of A.Dranishnikov."}
{"category": "Math", "title": "Indestructible colourings and rainbow Ramsey theorems", "abstract": "We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring $c:[{\\omega_2}]^2\\to 2$ establishing $\\omega_2 \\not\\to [(\\omega_1;{\\omega})]^2_2$ such that some colouring $g:[\\omega_1]^2\\to 2$ can not be embedded into $c$. It is also consistent that $2^{\\omega_1}$ is arbitrarily large, and a function $g$ establishes $2^{\\omega_1} \\not\\to [(\\omega_1,\\omega_2)]^2_{\\omega_1}$ such that there is no uncountable $g$-rainbow subset of $2^{\\omega_1}$. We also show that for each $k\\in {\\omega}$ it is consistent with Martin's Axiom that the negative partition relation $\\omega_1 \\not\\to^* [(\\omega_1;\\omega_1)]_{k-bdd}$ holds."}
{"category": "Math", "title": "Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk", "abstract": "We consider a special case of the Patlak-Keller-Segel system in a disc, which arises in the modelling of chemotaxis phenomena. For a critical value of the total mass, the solutions are known to be global in time but with density becoming unbounded, leading to a phenomenon of mass-concentration in infinite time. We establish the precise grow-up rate and obtain refined asymptotic estimates of the solutions. Unlike in most of the similar, recently studied, grow-up problems, the rate is neither polynomial nor exponential. In fact, the maximum of the density behaves like $e^{\\sqrt{2t}}$ for large time. In particular, our study provides a rigorous proof of a behaviour suggested by Sire and Chavanis [Phys. Rev. E, 2002] on the basis of formal arguments."}
{"category": "Math", "title": "Invariant measures of minimal post-critical sets of logistic maps", "abstract": "We construct logistic maps whose restriction to the omega-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor system corresponding to the restriction of a logistic map to the omega-limit set of its critical point. Furthermore, we show that such a logistic map $f$ can be taken so that each such invariant measure has zero Lyapunov exponent and is an equilibrium state of $f$ for the potential $-\\ln |f'|$."}
{"category": "Math", "title": "Modelling recorded crime: a full search for cointegrated models", "abstract": "A modelgenerator is developed that searches for cointegrated models among a potentially large group of candidate models. The generator employs the first step of the Engle-Granger procedure and orders cointegrated models according to the information criterions AIC and BIC. Assisted by the generator, a cointegrated relation is established between recorded violent crime in the Netherlands, the number of males aged 15-25 years (split into Western and non-Western background) and deflated consumption. In-sample forecasts reveal that the cointegrated model outperforms the best short-run models."}
{"category": "Math", "title": "On the topological essential range and regularity of cocycles over compact and generic systems", "abstract": "We consider the notions of topological essential range and regularity for continuous cocycles over minimal $\\Z$-systems introduced in \\cite{GH} and discuss relations with their generic counterparts. The alternative generic definitions can be given by using the notion of generic Mackey action associated with a cocycle. We further present a description of recurrent cocycles over minimal rotations with values in discrete groups and derive several consequences."}
{"category": "Math", "title": "A compact null set containing a differentiability point of every Lipschitz function", "abstract": "We prove that in a Euclidean space of dimension at least two, there exists a compact set of Lebesgue measure zero such that any real-valued Lipschitz function defined on the space is differentiable at some point in the set. Such a set is constructed explicitly."}
{"category": "Math", "title": "On termination of log flips in dimension four", "abstract": "We prove the termination of 4-fold log flips for klt pairs of Kodaira dimension $\\kappa\\ge 2$."}
{"category": "Math", "title": "Perelman's invariant and collapse via geometric characteristic splittings", "abstract": "Any closed orientable and smooth non-positively curved manifold M is known to admit a geometric characteristic splitting, analogous to the JSJ decomposition in three dimensions. We show that when this splitting consists of pieces which are Seifert fibered or pieces each of whose fundamental group has non-trivial centre, M collapses with bounded curvature and has zero Perelman invariant."}
{"category": "Math", "title": "An Identity of the Symmetry for the Frobenius-Euler polynomials associated with the fermionic p-adic invariant q-integrals on Z_p", "abstract": "The main purpose of this paper is to prove an identity of symmetry for the Frobenius-Euler polynomials."}
{"category": "Math", "title": "On the Kummer construction", "abstract": "We discuss a generalization of Kummer construction which, on the base of an integral representation of a finite group and local resolution of its quotient, produces a higher dimensional variety with trivial canonical class. As an application we compute cohomology of some generalized Kummer varieties."}
{"category": "Math", "title": "Gabor (Super)Frames with Hermite Functions", "abstract": "We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions $H_n$. Let $h= (H_0, H_1, ..., H_n)$ be the vector of the first $n+1$ Hermite functions. We give a complete characterization of all lattices $\\Lambda \\subseteq \\bR ^2$ such that the Gabor system $\\{e^{2\\pi i \\lambda_2 t} \\boh (t-\\lambda_1): \\lambda = (\\lambda_1, \\lambda_2) \\in \\Lambda \\}$ is a frame for $L^2 (\\bR, \\bC ^{n+1})$. As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass $\\sigma $-function, a new type of interpolation problem for entire functions on the Bargmann-Fock space, and structural results about vector-valued Gabor frames."}
{"category": "Math", "title": "Spaces H^1 and BMO on ax+b-groups", "abstract": "Let S be the semidirect product of R^d and R^+ endowed with the Riemannian symmetric space metric and the right Haar measure: this is a Lie group of exponential growth. In this paper we define an Hardy space H^1 and a BMO space in this context. We prove that the functions in BMO satisfy the John-Nirenberg inequality and that BMO may be identified with the dual space of H^1. We then prove that singular integral operators which satisfy a suitable integral Hormander condition are bounded from H^1 to L^1 and from L^{\\infty} to BMO. We also study the real interpolation between H^1, BMO and the L^p spaces."}
{"category": "Math", "title": "The Kodaira dimension of the moduli space of Prym varieties", "abstract": "We study the enumerative geometry of the moduli space R_g of Prym varieties of dimension g-1 (also known as the space of admissible double covers). Our main result is that the compactification of R_g is of general type as soon as g>13. We achieve this by computing the class of two types of cycles on R_g: one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym-Green conjecture on syzygies of Prym-canonical curves. In the appendix we show that even though R_g has non-canonical singularities, pluricanonical forms on R_g extend to any desingularization."}
{"category": "Math", "title": "On Intersection Representations and Clique Partitions of Graphs", "abstract": "A multifamily set representation of a finite simple graph $G$ is a multifamily $\\mathcal{F}$ of sets (not necessarily distinct) for which each set represents a vertex in $G$ and two sets in $\\mathcal{F}$ intersects if and only if the two corresponding vertices are adjacent. For a graph $G$, an \\textit{edge clique covering} (\\textit{edge clique partition}, respectively) $\\mathcal{Q}$ is a set of cliques for which every edge is contained in \\textit{at least} (\\textit{exactly}, respectively) one member of $\\mathcal{Q}$. In 1966, P. Erd\\\"{o}s, A. Goodman, and L. P\\'{o}sa (The representation of a graph by set intersections, \\textit{Canadian J. Math.}, \\textbf{18}, pp.106-112) pointed out that for a graph there is a one-to-one correspondence between multifamily set representations $\\mathcal{F}$ and clique coverings $\\mathcal{Q}$ for the edge set. Furthermore, for a graph one may similarly have a one-to-one correspondence between particular multifamily set representations with intersection size at most one and clique partitions of the edge set. In 1990, S. McGuinness and R. Rees (On the number of distinct minimal clique partitions and clique covers of a line graph, \\textit{Discrete Math.} \\textbf{83} (1990) 49-62.) calculated the number of distinct clique partitions for line graphs. In this paper, we study the set representations of graphs corresponding to edge clique partitions in various senses, namely family representations of \\textit{distinct} sets, antichain representations of \\textit{mutually exclusive} sets, and uniform representations of sets with the \\textit{same cardinality}. Among others, we completely determine the number of distinct family representations and the number of antichain representations of line graphs."}
{"category": "Math", "title": "K_1-injectivity for properly infinite C*-algebras", "abstract": "One of the main tools to classify \\cst-algebras is the study of its projections and its unitaries. It was proved by Cuntz in \\cite{Cu81} that if $A$ is a \\textit{purely infinite} simple \\cst-algebra, then the kernel of the natural map for the unitary group $\\U(A)$ to the $K$-theory group $K_1(A)$ is reduced to the connected component $\\U^0(A)$, i.e. $A$ is \\textit{$K_1$-injective} (see \\S 3). We study in this note a finitely generated \\cst-algebra, the $K_1$-injectivity of which would imply the $K_1$-injectivity of all unital \\textit{properly infinite} \\cst-algebras."}
{"category": "Math", "title": "On Base Partitions And Cover Partitions Of Skew Characters", "abstract": "In this paper we give an easy combinatorial description for the base partition B of a skew character [A], which is the intersection of all partitions alpha whose corresponding character [alpha] appears in [A]. This we use to construct the cover partition C for the ordinary outer product as well as for the Schubert product of two characters and for some skew characters, here the cover partition is the union of all partitions whose corresponding character appears in the product or in the skew character. This gives us also the Durfee size for arbitrary Schubert products."}
{"category": "Math", "title": "Homotopy shadowing", "abstract": "Michael Shub proved in 1969 that the topological conjugacy class of an expanding endomorphism on a compact manifold is determined by its homotopy type. In this article we generalize this result in two directions. In one direction we consider certain expanding maps on metric spaces. In a second direction we consider maps which are hyperbolic with respect to product cone fields on a product manifold. A key step in the proof is to establish a shadowing theorem for pseudo--orbits with some additional homotopy information."}
{"category": "Math", "title": "Eigenvalue Estimates for submanifolds of $N \\times \\mathbb{R}$ with locally bounded mean curvature", "abstract": "We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $ N \\times \\mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} \\leq \\kappa$. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces has positive fundamental tone."}
{"category": "Math", "title": "Remarks on the Dynamic of the Ruelle Operator and invariant differentials", "abstract": "Let $ R $ be a rational map. We are interesting in the dynamic of the Ruelle operator on suitable spaces of differentials. In particular the necessary and sufficient conditions (in terms of convergence of sequences of measures) of existence of invariant conformal structures on $ J(R) $ are obtained."}
{"category": "Math", "title": "Heron isosceles with integral external radii", "abstract": "Each triangle has three exterior or external circles tangential to the three straight lines containing the three sides of the triangle.Among the preliminaries in this paper, is deriving formulas for the radii of the three exterior circles in terms of the triangle's sidelengths. After that we focus on Heron triangles. Heron triangles are known in the literature as triangles with integer sidelengths and integral area.Pythagorean triangles are examples of Heron triangles. In the first part of the paper, we parametrically describe all Heron isosceles triangles. In the second part, we parametrically(in terms of three independent parameters) describe the subfamily which consists of all Heron isosceles triangles which also have integral external radii. We provide numerical examples and tables."}
{"category": "Math", "title": "The End Curve Theorem for normal complex surface singularities", "abstract": "We prove the \"End Curve Theorem,\" which states that a normal surface singularity $(X,o)$ with rational homology sphere link $\\Sigma$ is a splice-quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An \"end-curve function\" is an analytic function $(X,o)\\to (\\C,0)$ whose zero set intersects $\\Sigma$ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A \"splice-quotient singularity\" $(X,o)$ is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in $\\C^t$, where $t$ is the number of leaves in the resolution graph for $(X,o)$, together with an explicit description of the covering transformation group. Among the immediate consequences of the End Curve Theorem are the previously known results: $(X,o)$ is a splice quotient if it is weighted homogeneous (Neumann 1981), or rational or minimally elliptic (Okuma 2005)."}
{"category": "Math", "title": "Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions", "abstract": "A pair of dual frames with almost exponentially localized elements (needlets) are constructed on $\\RR_+^d$ based on Laguerre functions. It is shown that the Triebel-Lizorkin and Besov spaces induced by Laguerre expansions can be characterized in terms of respective sequence spaces that involve the needlet coefficients."}
{"category": "Math", "title": "Digraphs with a fixed number of edges and vertices, having a maximal number of walks of length 2", "abstract": "Inspired by the work of Backelin on non-commutative correspondences to Macaulay's theorem of the growth of the Hilbert series of affine algebras, we study embedding dimension dependant versions of his degree 2 to degree 3 result. In graph-theoretical terms, we study the following question: what is the maximal number of directed walks of length 2 in a digraph with (k) edges and (n) vertices? The problem can also be formulated as follows: maximize (< \\lambda, \\lambda^T >) when (\\lambda) is a partition of (k), contained in an (n \\times n) box. We show that for mild restrictions on (n), optimal digraphs are the ``stars of saturated stars''."}
{"category": "Math", "title": "Brill-Noether-type Theorems with a Movable Ramification Point", "abstract": "The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in r-dimensional projective space. These theorems can be extended to include ramification conditions at fixed general points. This paper deals with the problem of imposing a ramification condition at an unspecified point. We solve the problem completely in dimensions 1 and 2, and provide an existence test and bound the dimension of the family in the general case."}
{"category": "Math", "title": "S_3-covers of schemes", "abstract": "We analyze flat S_3-covers, attempting to create structures parallel to those found in the abelian theory. We use an initial local analysis as a guide in finding a global description."}
{"category": "Math", "title": "Transitive latin bitrades", "abstract": "In this note we give two results. First, if a latin bitrade $(T_1, T_2)$ is primary, thin, separated, and the autotopism group of $T_1$ acts regularly on $T_1$, then $(T_1, T_2)$ may be derived from a group-based construction. Second, if a latin bitrade $(T_1, T_2)$ has genus 0 then the disjoint mate $T_2$ is unique and the autotopism group of $T_1$ is equal to the autotopism group of $T_2$."}
{"category": "Math", "title": "Scalar curvature and holomorphy potentials", "abstract": "A holomorphy potential is a complex valued function whose complex gradient, with respect to some K\\\"ahler metric, is a holomorphic vector field. Given $k$ holomorphic vector fields on a compact complex manifold, form, for a given K\\\"ahler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where $k=0$ and the functional is the square of the $L^2$-norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the \\sk metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above."}
{"category": "Math", "title": "Mass formulas for local Galois representations to wreath products and cross products", "abstract": "Bhargava proved a formula for counting, with certain weights, degree n etale extensions of a local field, or equivalently, local Galois representations to S_n. This formula is motivation for his conjectures about the density of discriminants of S_n-number fields. We prove there are analogous ``mass formulas'' that count local Galois representations to any group that can be formed from symmetric groups by wreath products and cross products, corresponding to counting towers and direct sums of etale extensions. We obtain as a corollary that the above mentioned groups have rational character tables. Our result implies that D_4 has a mass formula for certain weights, but we show that D_4 does not have a mass formula when the local Galois representations to D_4 are weighted in the same way as representations to S_4 are weighted in Bhargava's mass formula."}
{"category": "Math", "title": "$d$-Regular graphs of acyclic chromatic index at least $d+2$", "abstract": "An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \\emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that $a'(G)\\le \\Delta+2$, where $\\Delta =\\Delta(G)$ denotes the maximum degree of the graph. Alon et.al also raised the question whether the complete graphs of even order are the only regular graphs which require $\\Delta+2$ colors to be acyclically edge colored. In this paper, using a simple counting argument we observe not only that this is not true, but infact all d-regular graphs with $2n$ vertices and $d > n$, requires at least $d+2$ colors. We also show that $a'(K_{n,n}) \\ge n+2$, when $n$ is odd using a more non-trivial argument(Here $K_{n,n}$ denotes the complete bipartite graph with $n$ vertices on each side). This lower bound for $K_{n,n}$ can be shown to be tight for some families of complete bipartite graphs and for small values of $n$. We also infer that for every $d,n$ such that $d \\ge 5$, $n \\ge 2d + 3$ and $dn$ even, there exist $d$-regular graphs which require at least $d+2$-colors to be acyclically edge colored."}
{"category": "Math", "title": "Gaussian Processes and Limiting Linear Models", "abstract": "Gaussian processes retain the linear model either as a special case, or in the limit. We show how this relationship can be exploited when the data are at least partially linear. However from the perspective of the Bayesian posterior, the Gaussian processes which encode the linear model either have probability of nearly zero or are otherwise unattainable without the explicit construction of a prior with the limiting linear model in mind. We develop such a prior, and show that its practical benefits extend well beyond the computational and conceptual simplicity of the linear model. For example, linearity can be extracted on a per-dimension basis, or can be combined with treed partition models to yield a highly efficient nonstationary model. Our approach is demonstrated on synthetic and real datasets of varying linearity and dimensionality."}
{"category": "Math", "title": "Braided and coboundary monoidal categories", "abstract": "In this expository paper, we discuss and compare the notions of braided and coboundary monoidal categories. Coboundary monoidal categories are analogues of braided monoidal categories in which the role of the braid group is replaced by the cactus group. We focus on the categories of representations of quantum groups and crystals and explain how while the former is a braided monoidal category, this structure does not pass to the crystal limit. However, the categories of representations of quantum groups of finite type also possess the structure of a coboundary category which does behave well in the crystal limit. We explain this construction and also a recent interpretation of the coboundary structure using quiver varieties. This geometric viewpoint allows one to show that the category of crystals is in fact a coboundary monoidal category for arbitrary symmetrizable Kac-Moody type."}
{"category": "Math", "title": "Introduction to Potential Theory via Applications", "abstract": "We introduce the basic concepts related to subharmonic functions and potentials, mainly for the case of the complex plane and prove the Riesz decomposition theorem. Beyond the elementary facts of the theory we deviate slightly from the usual path of exposition and introduce further concepts alongside with applications. We cover the Dirichlet problem in detail and illustrate the relations between potential theory and probability by considering harmonic measure and its relation to Brownian motion. Furthermore Green's function is introduced and an application to growth of polynomials is given. Equilibrium measures are motivated by their original development in physics and we end with a brief discussion of capacity and its relation to Hausdorff measure. We hope that the reader, who is familiar with the main elements of real analysis, complex analysis, measure theory and some probability theory benefits from these notes. No new results are presented but we hope that the style of presentation enables the reader to understand quickly the basic ideas of potential theory and how it can be used in different contexts. The notes can also be used for a short course on potential theory. Therefore the required prerequisites are described in the Appendix. References are given where expositions and details can be found; roughly speaking, familiarity with the basic foundations of real and complex analysis should suffice to proceed without any background reading."}
{"category": "Math", "title": "Characterizing right-veering homeomorphisms of the punctured torus via the Burau representation", "abstract": "We classify right-veering homeomorphisms of the once-punctured torus using the Burau representation of the 3-strand braid group. We show that reducible and periodic mapping classes in B_3 can be identified as right-veering by consideration of the reduced version of the Burau representation. Given any element beta in B_3, we give a method to quickly determine its action on the generators of the fundamental group of the 3-times punctured disk. This action which determines whether beta is right-veering, left-veering, or neither."}
{"category": "Math", "title": "Extending the Ehresmann-Schein-Nambooripad Theorem", "abstract": "We extend the `join-premorphisms' part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (1991) for the `morphisms' part. However, it is so-called `meet-premorphisms' which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for meet-premorphisms in the case of two-sided restriction semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case."}
{"category": "Math", "title": "Hamiltonicity thresholds in Achlioptas processes", "abstract": "In this paper we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K=K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem has three regimes, depending on the value of K. For K=o(\\log n), the threshold for Hamiltonicity is (1+o(1))n\\log n /(2K), i.e., typically we can construct a Hamilton cycle K times faster that in the usual random graph process. When K=\\omega(\\log n) we can essentially waste almost no edges, and create a Hamilton cycle in n+o(n) rounds with high probability. Finally, in the intermediate regime where K=\\Theta(\\log n), the threshold has order n and we obtain upper and lower bounds that differ by a multiplicative factor of 3."}
{"category": "Math", "title": "Hall-Littlewood Polynomials, Alcove Walks, and Fillings of Young Diagrams", "abstract": "A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. The inversion statistic, which is the more intricate one, suffices for specializing a closely related formula to one for the type A Hall-Littlewood Q-polynomials (spherical functions on p-adic groups). An apparently unrelated development, at the level of arbitrary finite root systems, led to Schwer's formula (rephrased and rederived by Ram) for the Hall-Littlewood P-polynomials of arbitrary type. The latter formula is in terms of so-called alcove walks, which originate in the work of Gaussent-Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, we relate the above developments, by deriving a Haglund-Haiman-Loehr type formula for the Hall-Littlewood P-polynomials of type A from Ram's version of Schwer's formula via a \"compression\" procedure."}
{"category": "Math", "title": "On Combinatorial Formulas for Macdonald Polynomials", "abstract": "A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of so-called alcove walks; these originate in the work of Gaussent-Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. In this paper, we relate the above developments, by explaining how the Ram-Yip formula compresses to a new formula, which is similar to the Haglund-Haiman-Loehr one but contains considerably fewer terms."}
{"category": "Math", "title": "Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis", "abstract": "This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of sl_n, the celebrated Gelfand-Tsetlin basis is the only such basis known. Using the setup of supporting graphs developed by Donnelly, we present a simple combinatorial proof of the Gelfand-Tsetlin formulas based on a rational function identity. Some properties of the Gelfand-Tsetlin basis are derived via an algorithm for solving certain equations on the lattice of semistandard Young tableaux."}
{"category": "Math", "title": "Representations of Aut(A(Gamma)) acting on homogeneous components of A(Gamma) and A(Gamma) dual", "abstract": "In this paper we will study the structure of algebras A(Gamma) associated to two directed, layered graphs Gamma. These are algebras associated with Hasse graphs of n-gons and the algebras Q_n related to pseudoroots of noncommutative polynomials. We will find the filtration preserving automorphism group of these algebras and then we will find the multiplicities of the irreducible representations of Aut(A(Gamma)) acting on the homogeneous components of A(Gamma) and A(Gamma) dual."}
{"category": "Math", "title": "Twisted Fourier-Mukai number of a K3 surface", "abstract": "We give a counting formula for the twisted Fourier-Mukai partners of a projective K3 surface. As an application, we describe all twisted Fourier-Mukai partners of a projective K3 surface of Picard number 1."}
{"category": "Math", "title": "Structural shrinkage of nonparametric spectral estimators for multivariate time series", "abstract": "In this paper we investigate the performance of periodogram based estimators of the spectral density matrix of possibly high-dimensional time series. We suggest and study shrinkage as a remedy against numerical instabilities due to deteriorating condition numbers of (kernel) smoothed periodogram matrices. Moreover, shrinking the empirical eigenvalues in the frequency domain towards one another also improves at the same time the Mean Squared Error (MSE) of these widely used nonparametric spectral estimators. Compared to some existing time domain approaches, restricted to i.i.d. data, in the frequency domain it is necessary to take the size of the smoothing span as \"effective or local sample size\" into account. While B\\\"{o}hm and von Sachs (2007) proposes a multiple of the identity matrix as optimal shrinkage target in the absence of knowledge about the multidimensional structure of the data, here we consider \"structural\" shrinkage. We assume that the spectral structure of the data is induced by underlying factors. However, in contrast to actual factor modelling suffering from the need to choose the number of factors, we suggest a model-free approach. Our final estimator is the asymptotically MSE-optimal linear combination of the smoothed periodogram and the parametric estimator based on an underfitting (and hence deliberately misspecified) factor model. We complete our theoretical considerations by some extensive simulation studies. In the situation of data generated from a higher-order factor model, we compare all four types of involved estimators (including the one of B\\\"{o}hm and von Sachs (2007))."}
{"category": "Math", "title": "Batalin-Vilkovisky coalgebra of string topology", "abstract": "We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed oriented manifold, and a gravity coalgebra structure on the reduced equivariant homology."}
{"category": "Math", "title": "Conjugacy of P-configurations and nonlinear solutions to a certain conditional Cauchy equation", "abstract": "We study the connection between conjugations of a special kind of dynamical systems, called P-configurations, and solutions to homogeneous Cauchy type functional equations. We find that any two regular P-configurations are conjugate by a homeomorphism, but cannot be conjugate by a diffeomorphism. This leads us to the following conclusion (resolving an open problem posed by Paneah): there exist continuous nonlinear solutions to the functional equation: f(t) = f((t+1)/2) + f((t-1)/2), t \\in [-1,1] ."}
{"category": "Math", "title": "The defect of weak approximation for homogeneous spaces. II", "abstract": "Let X be a homogeneous space of a connected linear algebraic group G' over a number field k, containing a k-point x. Assume that the stabilizer of x in G' is connected. Using the notion of a quasi-trivial group, recently introduced by Colliot-Th\\'el\\`ene, we can represent X in the form X=G/H, where G is a quasi-trivial k-group and H is a connected k-subgroup of G. Let S be a finite set of places of k. Applying results of [B2], we compute the defect of weak approximation for X with respect to S in terms of the biggest toric quotient T of H. In particular, we show that if T splits over a metacyclic extension of k, then X has the weak approximation property. We show also that any homogeneous space X with connected stabilizer (without assumptions on T) has the real approximation property."}
{"category": "Math", "title": "Compact complete proper minimal immersions in strictly convex bounded regular domains of R^3", "abstract": "Consider a strictly convex bounded regular domain $C$ of $\\R^3$. For any arbitrary finite topological type we find a compact Riemann surface $\\mathcal{M}$, an open domain $M\\subset \\mathcal{M}$ with the fixed topological type, and a conformal complete proper minimal immersion $X:M\\to C$ which can be extended to a continuous map $X:\\overline{M}\\to \\overline{C}$."}
{"category": "Math", "title": "Incorporating a contrast in the Bayesian formula: What consequences for the MAP estimator and the posterior distribution? Applications in spatial statistics", "abstract": "In order to estimate model parameters and circumvent possible difficulties encountered with the likelihood function, we propose to replace the likelihood in the formula of the posterior distribution by a function depending on a contrast. The properties of the contrast-based (CB) posterior distribution and MAP estimator are studied to understand what the consequences of incorporating a contrast in the Bayesian formula are. We show that the proposed method can be used to make frequentist inference and allows the reduction of analytical calculations to get the limit variance matrix of the estimator. For specific contrasts, the CB--posterior distribution directly approximates the limit distribution of the estimator; the calculation of the limit variance matrix is then avoided. Moreover, for these contrasts, the CB--posterior distribution can also be used to make inference in the Bayesian way. The method is applied to three spatial data sets."}
{"category": "Math", "title": "The abelianization of the level 2 mapping class group", "abstract": "In this paper, we determine the abelianization of the level d mapping class group for d=2 and odd d. We also extend the homomorphism of the Torelli group defined by Heap to a homomorphism of the level 2 mapping class group."}
{"category": "Math", "title": "Hyperbolic graphs of small complexity", "abstract": "In this paper we enumerate and classify the ``simplest'' pairs (M,G) where M is a closed orientable 3-manifold and G is a trivalent graph embedded in M. To enumerate the pairs we use a variation of Matveev's definition of complexity for 3-manifolds, and we consider only (0,1,2)-irreducible pairs, namely pairs (M,G) such that any 2-sphere in M intersecting G transversely in at most 2 points bounds a ball in M either disjoint from G or intersecting G in an unknotted arc. To classify the pairs our main tools are geometric invariants defined using hyperbolic geometry. In most cases, the graph complement admits a unique hyperbolic structure with parabolic meridians; this structure was computed and studied using Heard's program Orb and Goodman's program Snap. We determine all (0,1,2)-irreducible pairs up to complexity 5, allowing disconnected graphs but forbidding components without vertices in complexity 5. The result is a list of 129 pairs, of which 123 are hyperbolic with parabolic meridians. For these pairs we give detailed information on hyperbolic invariants including volumes, symmetry groups and arithmetic invariants. Pictures of all hyperbolic graphs up to complexity 4 are provided. We also include a partial analysis of knots and links. The theoretical framework underlying the paper is twofold, being based on Matveev's theory of spines and on Thurston's idea (later developed by several authors) of constructing hyperbolic structures via triangulations. Many of our results were obtained (or suggested) by computer investigations."}
{"category": "Math", "title": "Cohen-Macaulayness with respect to Serre classes", "abstract": "Let $R$ be a commutative Noetherian ring. The notion of regular sequences with respect to a Serre class of $R$-modules is introduced and some of their essential properties are given. Then in the local case, we explore a theory of Cohen-Macaulayness with respect to Serre classes."}
{"category": "Math", "title": "Borel-amenable Reducibilities for Sets of Reals", "abstract": "We show that if $\\mathcal{F}$ is any \"well-behaved\" subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on $\\pow(\\mathbb{R})$ induced by $\\mathcal{F}$ turns out to look like the Wadge hierarchy (which is the special case where $\\mathcal{F}$ is the set of continuous functions)."}
{"category": "Math", "title": "Intrinsic pseudo-volume forms for logarithmic pairs", "abstract": "We study an adaptation to the logarithmic case of the Kobayashi-Eisenman pseudo-volume form, or rather an adaptation of its variant defined by Claire Voisin, for which she replaces holomorphic maps by holomorphic K-correspondences. We define an intrinsic logarithmic pseudo-volume form \\Phi_{X,D} for every pair (X,D) consisting of a complex manifold X and a normal crossing Weil divisor, the positive part of which is reduced. We then prove that \\Phi_{X,D} is generically non-degenerate when X is projective and K_X+D is ample. This result is analogous to the classical Kobayashi-Ochiai theorem. We also show the vanishing of \\Phi_{X,D} for a large class of log-K-trivial pairs, which is an important step in the direction of the Kobayashi conjecture about infinitesimal measure hyperbolicity in the logarithmic case."}
{"category": "Math", "title": "The spectrum of the random environment and localization of noise", "abstract": "We consider random walk on a mildly random environment on finite transitive d- regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized."}
{"category": "Math", "title": "Relevant First-Order Logic $LP^\\#$ and Curry's Paradox resolution", "abstract": "In 1942 Haskell B.Curry presented what is now called Curry paradox which can be found in a logic independently of its stand on negation.In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry's Paradox and Shaw-Kwei paradox without rejection any contraction postulate is proposed."}
{"category": "Math", "title": "On Einstein Metrics on 4-Manifolds with Finite Cyclic Fundamental Group", "abstract": "The existence or non-existence of Einstein metrics on 4-manifolds with non-trivial fundamental group and the relation with the underlying differential structure are analyzed. For most points $(n,m)$ in a large region of the integer lattice, the manifold $n\\mathbb C \\mathbb P^2\\#m \\overline{\\mathbb C \\mathbb P^2}$ is shown to admit infinitely many inequivalent free actions of finite cyclic groups and there are no Einstein metrics which are invariant under any of these actions. The main tools are Seiberg-Witten theory, cyclic branched coverings of complex surfaces and symplectic surgeries."}
{"category": "Math", "title": "Logarithmic Fourier integrals for the Riemann Zeta Function", "abstract": "We use symmetric Poisson-Schwarz formulas for analytic functions $f$ in the half-plane ${Re}(s)>\\frac12$ with $\\bar{f(\\bar{s})}=f(s)$ in order to derive factorisation theorems for the Riemann zeta function. We prove a variant of the Balazard-Saias-Yor theorem and obtain explicit formulas for functions which are important for the distribution of prime numbers. In contrast to Riemann's classical explicit formula, these representations use integrals along the critical line ${Re}(s)=\\frac12$ and Blaschke zeta zeroes."}
{"category": "Math", "title": "Multiplicative Bundle Gerbes with Connection", "abstract": "Multiplicative bundle gerbes are gerbes over a Lie group which are compatible with the group structure. In this article connections on such bundle gerbes are introduced and studied. It is shown that multiplicative bundle gerbes with connection furnish geometrical constructions of the following objects: smooth central extensions of loop groups, Chern-Simons actions for arbitrary gauge groups, and symmetric bi-branes for WZW models with topological defect lines."}
{"category": "Math", "title": "Computation of Selberg zeta functions on Hecke triangle groups", "abstract": "In this paper, a heuristic method to compute the Selberg zeta function for Hecke triangle groups, $G_q$ ($q>=3$) is described. The algorithm is based on the transfer operator method and an overview of the relevant background is given. We give numerical support for the claim that the method works and can be used to compute the Selberg Zeta function for $G_q$ to any desired precision. We also present some numerical results obtained by implementing the algorithm."}
{"category": "Math", "title": "On the Index and the Order of Quasi-regular Implicit Systems of Differential Equations", "abstract": "This paper is mainly devoted to the study of the differentiation index and the order for quasi-regular implicit ordinary differential algebraic equation (DAE) systems. We give an algebraic definition of the differentiation index and prove a Jacobi-type upper bound for the sum of the order and the differentiation index. Our techniques also enable us to obtain an alternative proof of a combinatorial bound proposed by Jacobi for the order. As a consequence of our approach we deduce an upper bound for the Hilbert-Kolchin regularity and an effective ideal membership test for quasi-regular implicit systems. Finally, we prove a theorem of existence and uniqueness of solutions for implicit differential systems."}
{"category": "Math", "title": "Families of prudent self-avoiding walks", "abstract": "A self-avoiding walk (SAW) on the square lattice is prudent if it never takes a step towards a vertex it has already visited. Prudent walks differ from most classes of SAW that have been counted so far in that they can wind around their starting point. Their enumeration was first addressed by Pr\\'ea in 1997. He defined 4 classes of prudent walks, of increasing generality, and wrote a system of recurrence relations for each of them . However, these relations involve more and more parameters as the generality of the class increases. The first class actually consists of partially directed walks, and its generating function is well-known to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (2005). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even D-finite. The fourth class -- general prudent walks -- is the only isotropic one, and still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-D-finite. We also study the asymptotic properties of these classes of walks, with the (somewhat disappointing) conclusion that their endpoint moves away from the origin at a positive speed. This is confirmed visually by the random generation procedures we have designed."}
{"category": "Math", "title": "A combinatorial proof of the Removal Lemma for Groups", "abstract": "Green [Geometric and Functional Analysis 15 (2005), 340--376] established a version of the Szemer\\'edi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations."}
{"category": "Math", "title": "Dynamics of Functions with an Eventual Negative Schwarzian Derivative", "abstract": "In the study of one dimensional dynamical systems one often assumes that the functions involved have a negative Schwarzian derivative. In this paper we consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. The introduction of this class was motivated by some maps arising in neuroscience."}
{"category": "Math", "title": "Motivic Weight Complexes for Arithmetic Varieties", "abstract": "We associate weight complexes of (homological) motives, and hence Euler characteristics in the Grothendieck group of motives, to arithmetic varieties and Deligne-Mumford stacks; this extends the results in the paper \"Descent, Motives and K-theory\" in volume 478 of Crelle, where a similar result was proved for varieties over a field of characteristic zero. We use K_0-motives with rational coefficients, rather than Chow motives, because we cannot appeal to resolution of singularities, but rather must use de Jong's results. In addition, for varieties over a field we prove a general result on contravariance of weight complexes, in particular showing that any morphism of finite tor-dimension between varieties induces a morphism of weight complexes."}
{"category": "Math", "title": "Arithmetic partial differential equations, II: modular curves", "abstract": "We classify ``arithmetic convection equations'' on modular curves, and describe their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions of the same modular forms; in this sense, our arithmetic convection equations can be seen as \"unifying\" the two types of expansions. The theory can be generalized to one of ``arithmetic heat equations'' on modular curves, but we prove that modular curves do not carry ``arithmetic wave equations.'' Finally, we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation."}
{"category": "Math", "title": "Closedness of the tangent spaces to the orbits of proper actions", "abstract": "In this note we show that for any proper action of a Banach--Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps $\\g \\to T_x(M)$ have closed range for each $x \\in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $M/G$ carries a natural manifold structure."}
{"category": "Math", "title": "The Nagata automorphism is shifted linearizable", "abstract": "A polynomial automorphism $F$ is called {\\em shifted linearizable} if there exists a linear map $L$ such that $LF$ is linearizable. We prove that the Nagata automorphism $N:=(X-Y\\Delta -Z\\Delta^2,Y+Z\\Delta, Z)$ where $\\Delta=XZ+Y^2$ is shifted linearizable. More precisely, defining $L_{(a,b,c)}$ as the diagonal linear map having $a,b,c$ on its diagonal, we prove that if $ac=b^2$, then $L_{(a,b,c)}N$ is linearizable if and only if $bc\\not = 1$. We do this as part of a significantly larger theory: for example, any exponent of a homogeneous locally finite derivation is shifted linearizable. We pose the conjecture that the group generated by the linearizable automorphisms may generate the group of automorphisms, and explain why this is a natural question."}
{"category": "Math", "title": "A right inverse of the divergence for planar H\\\"older-$\\alpha$ domains", "abstract": "If $\\Omega\\subset\\R^n$ is a bounded domain, the existence of solutions ${\\bf u}\\in H^1_0(\\Omega)^n$ of ${div} {\\bf u} = f$ for $f\\in L^2(\\Omega)$ with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular it allows to show the existence of a solution $({\\bf u},p)\\in H^1_0(\\Omega)^n\\times L^2(\\Omega)$, where ${\\bf u}$ is the velocity and $p$ the pressure. It is known that the above mentioned result holds when $\\Omega$ is a Lipschitz domain and that it is not valid for arbitrary H\\\"older-$\\alpha$ domains. In this paper we prove that if $\\Omega$ is a planar simply connected H\\\"older-$\\alpha$ domain, there exist right inverses of the divergence which are continuous in appropriate weighted spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. In our results, the zero boundary condition is replaced by a weaker one. For the particular case of domains with an external cusp of power type, we prove that our weaker boundary condition is equivalent to the standard one. In this case we show the well posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution $({\\bf u},p)\\in H^1_0(\\Omega)^n\\times L^r(\\Omega)$ for some $r<2$ depending on the power of the cusp."}
{"category": "Math", "title": "On the field intersection problem of solvable quintic generic polynomials", "abstract": "We study a general method of the field intersection problem of generic polynomials over an arbitrary field $k$ via formal Tschirnhausen transformation. In the case of solvable quintic, we give an explicit answer to the problem by using multi-resolvent polynomials."}
{"category": "Math", "title": "On Splitting Types, Discriminant Bounds, and Conclusive Tests for the Galois Group", "abstract": "Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types of polynomials modulo primes, and cycle types of the Galois groups of polynomials. One remarkable example is the removal of all artificial constraints from the Kummer-Dedekind Theorem that relates splitting and factorization patterns. Finally, we present an elementary proof that the discriminant of the splitting field of a monic irreducible polynomial with integer coefficients has a computable upper bound in terms of the coefficients. This result, combined with one of Lagarias et al., shows that tests of polynomials for the cycle types of the Galois group are conclusive. In particular, the Galois groups of monic irreducible cubics, quartics, and quintics with integer coefficients can be completely determined in finitely many steps (though not necessarily in one's lifetime)."}
{"category": "Math", "title": "On the deformation chirality of real cubic fourfolds", "abstract": "According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformation classification, that is how to respond to the chirality question: which cubics are not deformation equivalent to their image under a mirror reflection. We provide an arithmetical criterion of chirality, in terms of the eigen-sublattices of the complex conjugation involution in homology, and show how this criterion can be effectively applied taking as examples $M$-cubics (that is those for which the real locus has the richest topology) and $(M-1)$-cubics (the next case with respect to complexity of the real locus). It happens that there is one chiral class of $M$-cubics and three chiral classes of $(M-1)$-cubics, contrary to two achiral classes of $M$-cubics and three achiral classes of $(M-1)$-cubics."}
{"category": "Math", "title": "Eternal solutions and heteroclinic orbits of a semilinear parabolic equation", "abstract": "This dissertation describes the space of heteroclinic orbits for a class of semilinear parabolic equations, focusing primarily on the case where the nonlinearity is a second degree polynomial with variable coefficients. Along the way, a new and elementary proof of existence and uniqueness of solutions is given. Heteroclinic orbits are shown to be characterized by a particular functional being finite. A novel asymptotic-numeric matching scheme is used to uncover delicate bifurcation behavior in the equilibria. The exact nature of this bifurcation behavior leads to a demonstration that the equilibria are degenerate critical points in the sense of Morse. Finally, the space of heteroclinic orbits is shown to have a cell complex structure, which is finite dimensional when the number of equilibria is finite."}
{"category": "Math", "title": "Substochastic semigroups and densities of piecewise deterministic Markov processes", "abstract": "Necessary and sufficient conditions are given for a substochastic semigroup on $L^1$ obtained through the Kato--Voigt perturbation theorem to be either stochastic or strongly stable. We show how such semigroups are related to piecewise deterministic Markov process, provide a probabilistic interpretation of our results, and apply them to fragmentation equations."}
{"category": "Math", "title": "Square-Difference-Free Sets of Size Omega(n^{0.7334...})", "abstract": "A set A is square-difference free (henceforth SDF) if there do not exist x,y\\in A, x\\ne y, such that |x-y| is a square. Let sdf(n) be the size of the largest SDF subset of {1,...,n}. Ruzsa has shown that sdf(n) = \\Omega(n^{0.5(1+ \\log_{65} 7)}) = \\Omega(n^{0.733077...}) We improve on the lower bound by showing sdf(n) = \\Omega(n^{0.5(1+ \\log_{205} 12)})= \\Omega(n^{.7443...}) As a corollary we obtain a new lower bound on the quadratic van der Waerden numbers."}
{"category": "Math", "title": "An analog of the Furstenberg-Katznelson-Weiss theorem on triangles in sets of positive density in finite field geometries", "abstract": "We prove that if the cardinality of a subset of the 2-dimensional vector space over a finite field with $q$ elements is $\\ge \\rho q^2$, with $\\frac{1}{\\sqrt{q}}<<\\rho \\leq 1$, then it contains an isometric copy of $\\ge c\\rho q^3$ triangles."}
{"category": "Math", "title": "Automorphisms of the truth-table degrees are fixed on some cone", "abstract": "Let Dtt denote the set of truth-table degrees. A bijection p from Dtt to Dtt is an automorphism if for all truth-table degrees x and y we have x <=tt y if and only if p(x) <=tt p(y). We say an automorphism p is fixed on some cone if there is a degree b such that for all x >=tt b we have p(x) = x. We first prove that for every 2-generic real X we have X' is not tt below X + 0'. We next prove that for every real X >=tt 0' there is a real Y such that Y + 0' =tt Y' =tt X. Finally, we use this to demonstrate that every automorphism of the truth-table degrees is fixed on some cone."}
{"category": "Math", "title": "Campedelli surfaces with fundamental group of order 8", "abstract": "We prove that an etale cover Y of degree 8 of a Campedelli surface S is a complete intersection of four quadrics in P^6, obtaining as a consequence that Y is the universal cover of S, the covering group G=Gal(Y/S)is the topological fundamental group of S and that G cannot be the dihedral group of order 8. This paper patches up an incomplete manuscript of the third author."}
{"category": "Math", "title": "3--symmetric and 3--decomposable drawings of $K_n$ (extended version)", "abstract": "Even the most superficial glance at the vast majority of crossing-minimal geometric drawings of $K_n$ reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply {\\em 3-symmetric}) . And second, they all are {\\em 3-decomposable}, that is, there is a triangle $T$ enclosing the drawing, and a balanced partition $A, B, C$ of the underlying set of points $P$, such that the orthogonal projections of $P$ onto the sides of $T$ show $A$ between $B$ and $C$ on one side, $B$ between $A$ and $C$ on another side, and $C$ between $A$ and $B$ on the third side. In fact, we conjecture that all optimal drawings are 3-decomposable, and that there are 3-symmetric optimal constructions for all $n$ multiple of 3. In this paper, we show that any 3-decomposable geometric drawing of $K_n$ has at least $0.380029\\binom{n}{4}+\\Theta(n^3)$ crossings. On the other hand, we produce 3-symmetric and 3-decomposable drawings that improve the {\\em general} upper bound for the rectilinear crossing number of $K_n$ to $0.380488\\binom{n}{4}+\\Theta(n^3)$. We also give explicit 3-symmetric and 3-decomposable constructions for $n<100$ that are at least as good as those previously known."}
{"category": "Math", "title": "Optimized Schwarz preconditioning for SEM based magnetohydrodynamics", "abstract": "A recent theoretical result on optimized Schwarz algorithms demonstrated at the algebraic level enables the modification of an existing Schwarz procedure to its optimized counterpart. In this work, it is shown how to modify a bilinear FEM based Schwarz preconditioning strategy originally presented in [Fischer, JCP 133:84 1997] to its optimized version. The latter is employed to precondition the pseudo--Laplacian operator arising from the spectral element discretization of the magnetohydrodynamic equations in Elsasser form."}
{"category": "Math", "title": "A new series of compact minitwistor spaces and Moishezon twistor spaces over them", "abstract": "In recent papers math.DG/0701278 and arXiv:0705.0060, we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called minitwistor spaces). These include the twistor spaces studied in the papers as very special cases. Our source of the result is a series of self-dual metrics with torus action constructed by D. Joyce. Actually, for arbitrary Joyce metrics and U(1)-subgroups of the torus which fixes a torus-invariant 2-sphere, we first determine the associated minitwistor spaces in explicit forms. Next by analyzing the meromorphic maps from the twistor spaces to the minitwistor spaces, we realize projective models of the twistor spaces of all Joyce metrics, as conic bundles over the minitwistor spaces. Then we prove that for any one of these minitwistor spaces, there exist Moishezon twistor spaces with only C*-action whose quotient space is the given minitwistor space. This result generates numerous Moishezon twistor spaces which cannot be found in the literature (including the author's papers), in quite explicit form."}
{"category": "Math", "title": "Affine Deligne-Lusztig varieties in affine flag varieties", "abstract": "This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures concerning their dimensions, and generalizes the superset method."}
{"category": "Math", "title": "Projective models of the twistor spaces of Joyce metrics", "abstract": "We provide a simple algebraic construction of the twistor spaces of arbitrary Joyce's self-dual metrics on the 4-manifold H^2 x T^2 that extend smoothly to nCP^2, the connected sum of complex projective planes. Indeed, we explicitly realize projective models of the twistor spaces of arbitrary Joyce metrics on nCP^2 in a CP^4-bundle over CP^1, and show that they contain the twistor spaces of H^2 x T^2 as dense non-Zariski open subsets. In particular, we see that the last non-compact twistor spaces can be realized in rank-4 vector bundles over CP^1 by quite simple defining equations."}
{"category": "Math", "title": "A Discrete Construction for Gaussian Markov Processes", "abstract": "In the L\\'evy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process $W^{N}$ and to define the Wiener process as the almost sure path-wise limit of $W^{N}$ when $N$ tends to infinity. We generalize such a construction to the class of centered Gaussian Markov processes $X$ which can be written $X_{t} = g(t) \\cdot \\int_{0}^{t} f(t) dW_{t}$ with $f$ and $g$ being continuous functions. We build the finite-dimensional process $X^{N}$ so that it gives an exact representation of the conditional expectation of $X$ with respect to the filtration generated by ${\\lbrace X_{k/2^{N}}\\rbrace}$ for $0 \\leq k \\leq 2^{N}$. Moreover, we prove that the process $X^{N}$ converges in distribution toward $X$."}
{"category": "Math", "title": "On a variance for twins of $k-$free numbers in arithmetic progressions", "abstract": "In this paper, we give a new upper bound of Barban-Davenport-Halberstam type for twins of $k-$free numbers in arithmetic progressions."}
{"category": "Math", "title": "Quantile tomography: using quantiles with multivariate data", "abstract": "The use of quantiles to obtain insights about multivariate data is addressed. It is argued that incisive insights can be obtained by considering directional quantiles, the quantiles of projections. Directional quantile envelopes are proposed as a way to condense this kind of information; it is demonstrated that they are essentially halfspace (Tukey) depth levels sets, coinciding for elliptic distributions (in particular multivariate normal) with density contours. Relevant questions concerning their indexing, the possibility of the reverse retrieval of directional quantile information, invariance with respect to affine transformations, and approximation/asymptotic properties are studied. It is argued that the analysis in terms of directional quantiles and their envelopes offers a straightforward probabilistic interpretation and thus conveys a concrete quantitative meaning; the directional definition can be adapted to elaborate frameworks, like estimation of extreme quantiles and directional quantile regression, the regression of depth contours on covariates. The latter facilitates the construction of multivariate growth charts---the question that motivated all the development."}
{"category": "Math", "title": "A bijective enumeration of labeled trees with given indegree sequence", "abstract": "For a labeled tree on the vertex set $\\set{1,2,\\ldots,n}$, the local direction of each edge $(i\\,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence $\\lambda = 1^{e_1}2^{e_2} \\ldots$ of a tree on the vertex set $\\set{1,2,\\ldots,n}$ is a partition of $n-1$. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Pr\\\"ufer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a $q$-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case."}
{"category": "Math", "title": "Non-commutative A-G mean inequality", "abstract": "In this paper we consider non-commutative analogue for the arithmeticgeometric mean inequality $$a^{r}b^{1-r}+(r-1)b\\geq ra$$ for two positive numbers $a,b$ and $r> 1$. We show that under some assumptions the non-commutative analogue for $a^{r}b^{1-r}$ which satisfies this inequality is unique and equal to $r$-mean. The case $0<r<1$ is also considered. In particular, we give a new characterization of the geometric mean."}
{"category": "Math", "title": "A nonparametric estimator of the spectral density of a continuous-time Gaussian process observed at random times", "abstract": "In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a nonparametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between successive observations. In the case of stationary Gaussian processes, one can compare this estimator with estimators based on the empirical periodogram. Both estimators reach the same optimal rate of convergence, but the estimator based on wavelet analysis converges for a different class of random times. Simulation examples and application to biological data are also provided."}
{"category": "Math", "title": "Existence results for quasilinear elliptic boundary value problems via topological methods", "abstract": "In this paper, existence and localization results of $C^1$-solutions to elliptic Dirichlet boundary value problems are established. The approach is based on the nonlinear alternative of Leray-Schauder."}
{"category": "Math", "title": "Graph braid groups and right-angled Artin groups", "abstract": "We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $\\ge 5$. In order to have the necessity part, graphs are organized into small classes so that one of homological or cohomological characteristics of right-angled Artin groups can be applied. Finally we show that a given graph is planar iff the first homology of its 2-braid group is torsion-free and leave the corresponding statement for $n$-braid groups as a conjecture along with few other conjectures about graphs whose braid groups of index $\\le 4$ are right-angled Artin groups."}
{"category": "Math", "title": "Random walks, arrangements, cell complexes, greedoids, and self-organizing libraries", "abstract": "The starting point is the known fact that some much-studied random walks on permutations, such as the Tsetlin library, arise from walks on real hyperplane arrangements. This paper explores similar walks on complex hyperplane arrangements. This is achieved by involving certain cell complexes naturally associated with the arrangement. In a particular case this leads to walks on libraries with several shelves. We also show that interval greedoids give rise to random walks belonging to the same general family. Members of this family of Markov chains, based on certain semigroups, have the property that all eigenvalues of the transition matrices are non-negative real and given by a simple combinatorial formula. Background material needed for understanding the walks is reviewed in rather great detail."}
{"category": "Math", "title": "The volume conjecture for augmented knotted trivalent graphs", "abstract": "We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume conjecture holds."}
{"category": "Math", "title": "Resonances for a diffusion with small noise", "abstract": "We study resonances for the generator of a diffusion with small noise in $R^d$ :$ L_\\epsilon = -\\epsilon\\Delta + \\nabla F \\cdot \\nabla$, when the potential F grows slowly at infinity (typically as a square root of the norm). The case when F grows fast is well known, and under suitable conditions one can show that there exists a family of exponentially small eigenvalues, related to the wells of F . We show that, for an F with a slow growth, the spectrum is R+, but we can find a family of resonances whose real parts behave as the eigenvalues of the \"quick growth\" case, and whose imaginary parts are small."}
{"category": "Math", "title": "Bilinear Hilbert transforms along curves I. The monomial case", "abstract": "We establish an L^2 \\times L^2 to L^1 estimate for the bilinear Hilbert transform along a curve defined by a monomial. Our proof is closely related to multi-linear oscillatory integrals."}
{"category": "Math", "title": "Generalizations of Han's Hook Length Identities", "abstract": "Han recently discovered new hook length identities for binary trees. In this paper, we extend Han's identities to binomial families of trees. Moreover, we present a bijective proof of one of the identities for the family of ordered trees."}
{"category": "Math", "title": "Kummer surfaces associated to (1,2)-polarized abelian surfaces", "abstract": "The aim of this paper is to describe the geometry of the generic Kummer surface associated to a $(1,2)$-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an elliptic fibration with twelve singular fibers of type $I_2.$"}
{"category": "Math", "title": "Del Pezzo Surfaces of degree 6 over an arbitrary field", "abstract": "We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field $F$. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen $K$-theory of the surface. As a consequence, we obtain an index reduction formula for the function field of the surface."}
{"category": "Math", "title": "A generalised Joyce construction for a family of nonlinear partial differential equations", "abstract": "We explain a simple construction of solutions to a family of PDE's in two dimensions which includes that defining zero scalar curvature Kahler metrics, with two Killing fields, and the affine maximal equation."}
{"category": "Math", "title": "Constant scalar curvature metrics on toric surfaces", "abstract": "This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/"}
{"category": "Math", "title": "Uniform uniform exponential growth of subgroups of the mapping class group", "abstract": "Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any such subgroup explicit free group generators which are \"short\" in any word metric. Besides bounding growth, this allows a bound on the return probability of simple random walks."}
{"category": "Math", "title": "Cullen-regular quaternionic functions in a Fueter operator framework", "abstract": "We show characterizations of the class of Cullen-regular functions in the sense of Gentili-Struppa for any domain $\\Omega$ in terms of the Fueter operator. We then state a Integral Theorem and discuss how it can be used to define a more general version of Cullen-regularity, that does not require the function to be of class $C^{1}$."}
{"category": "Math", "title": "On the calculation of the cohomology of the third finite subset space of spheres", "abstract": "In this paper we provide a computation of the mod 2 cohomology groups of the third finite subset space of the sphere $S^n$ using known results about the cohomology of the symmetric product of spheres."}
{"category": "Math", "title": "Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry", "abstract": "We study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. We work in the general setting of derived algebraic geometry: our basic objects are derived stacks X and their oo-categories QC(X) of quasicoherent sheaves. We show that for a broad class of derived stacks, called perfect stacks, algebraic and geometric operations on their categories of sheaves are compatible. We identify the category of sheaves on a fiber product with the tensor product of the categories of sheaves on the factors. We also identify the category of sheaves on a fiber product with functors between the categories of sheaves on the factors (thus realizing functors as integral transforms, generalizing a theorem of Toen for ordinary schemes). As a first application, for a perfect stack X, consider QC(X) with its usual monoidal tensor product. Then our main results imply the equivalence of the Drinfeld center (or Hochschild cohomology category) of QC(X), the trace (or Hochschild homology category) of QC(X) and the category of sheaves on the loop space of X. More generally, we show that the E_n-center and the E_n-trace (or E_n-Hochschild cohomology and homology categories respectively) of QC(X) are equivalent to the category of sheaves on the space of maps from the n-sphere into X. This directly verifies geometric instances of the categorified Deligne and Kontsevich conjectures on the structure of Hochschild cohomology. As a second application, we use our main results to calculate the Drinfeld center of categories of linear endofunctors of categories of sheaves. This provides concrete applications to the structure of Hecke algebras in geometric representation theory. Finally, we explain how all of the above results can be interpreted in the context of topological field theory."}
{"category": "Math", "title": "Operator-valued dyadic BMO spaces", "abstract": "We consider BMO spaces of operator-valued functions, among them the space of operator-valued functions $B$ which define a bounded paraproduct on $L^2(H)$. We obtain several equivalent formulations of $\\|\\pi_B\\|$ in terms of the norm of the \"sweep\" function of $B$ or of averages of the norms of martingales transforms of $B$ in related spaces. Furthermore, we investigate a connection between John-Nirenberg type inequalities and Carleson-type inequalities via a product formula for paraproducts and deduce sharp dimensional estimates for John-Nirenberg type inequalities."}
{"category": "Math", "title": "Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality", "abstract": "Let $\\alpha$ be a quadratic Poisson bivector on a vector space $V$. Then one can also consider $\\alpha$ as a quadratic Poisson bivector on the vector space $V^*[1]$. Fixed a universal deformation quantization (prediction some weights to all Kontsevich graphs [K97]), we have deformation quantization of the both algebras $S(V^*)$ and $\\Lambda(V)$. These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on $\\alpha$, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the framework of the Tamarkin's theory [T1]."}
{"category": "Math", "title": "Spectral parameterization for the power sums of quantum supermatrix", "abstract": "A parameterization for the power sums of GL(m|n) type quantum (super)matrix is obtained in terms of it's spectral values."}
{"category": "Math", "title": "Normal forms for real quadratic forms", "abstract": "We investigate the non-diagonal normal forms of a quadratic form on R^n, in particular for n=3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of \\R^5."}
{"category": "Math", "title": "On normal contact pairs", "abstract": "We consider manifolds endowed with a contact pair structure. To such a structure are naturally associated two almost complex structures. If they are both integrable, we call the structure a normal contact pair. We generalize the Morimoto's Theorem on product of almost contact manifolds to flat bundles. We construct some examples on Boothby--Wang fibrations over contact-symplectic manifolds. In particular, these results give new methods to construct complex manifolds."}
{"category": "Math", "title": "On the equipartition of energy for critical NLW", "abstract": "We study some qualitative properties of global solutions to the following focusing and defocusing critical $NLW$: \\begin{equation*} \\Box u+ \\lambda u|u|^{2^*-2}=0, \\hbox{} \\lambda\\in {\\mathbf R} \\end{equation*} $$\\hspace{2cm} u(0)=f\\in \\dot H^1({\\mathbf R}^n), \\partial_t u(0)=g\\in L^2({\\mathbf R}^n)$$ on ${\\mathbf R}\\times {\\mathbf R}^n$ for $n\\geq 3$, where $2^*\\equiv \\frac{2n}{n-2}$. We will consider the global solutions of the defocusing $NLW$ whose existence and scattering property is shown in \\cite{shst}, \\cite{sb} and \\cite{bg}, without any restriction on the initial data $(f,g)\\in \\dot H^1({\\mathbf R}^n) \\times L^2({\\mathbf R}^n)$. As well as the solutions constructed in \\cite{pecher} to the focusing $NLW$ for small initial data and to the ones obtained in \\cite{mk}, where a sharp condition on the smallness of the initial data is given. We prove that the solution $u(t, x)$ satisfies a family of identities, that turn out to be a precised version of the classical Morawetz estimates (see \\cite{mor1}). As a by--product we deduce that any global solution to critical $NLW$ belonging to a natural functional space satisfies: $$\\lim_{R\\to \\infty}\\frac 1R \\int_{\\mathbf R} \\int_{|x|<R} |\\nabla_{x} u(t,x)|^2 \\hbox{} dxdt $$ $$=\\lim_{R\\to \\infty} \\frac 1{2R} \\int_{\\mathbf R} \\int_{|x|<R} (|\\nabla_{t,x} u(t,x)|^2 + \\frac{2 \\lambda}{2^*} |u(t,x)|^{2^*}) \\hbox{} dxdt$$ $$=\\int_{{\\mathbf R}^n} (|\\nabla_{t, x} u(0, x)|^2+ \\frac{2 \\lambda}{2^*} |u(0, x)|^{2^*}) \\hbox{} dx.$$"}
{"category": "Math", "title": "Application of topological radicals to calculation of joint spectral radii", "abstract": "It is shown that the joint spectral radius $\\rho(M)$ of a precompact family $M$ of operators on a Banach space $X$ is equal to the maximum of two numbers: the joint spectral radius $\\rho_{e}(M)$ of the image of $M$ in the Calkin algebra and the Berger-Wang radius $r(M)$ defined by the formula \\[ r(M)=\\underset{n\\to\\infty}{\\limsup}(\\sup\\left\\{\\rho(a):a\\in M^{n}\\right\\} ^{1/n}) . \\] Some more general Banach-algebraic results of this kind are also proved. The proofs are based on the study of special radicals on the class of Banach algebras."}
{"category": "Math", "title": "The Auslander-Reiten conjecture for Gorenstein rings", "abstract": "The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this note is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring $R$, then it holds for $R$."}
{"category": "Math", "title": "Evaluation for moments of a ratio with application to regression estimation", "abstract": "Ratios of random variables often appear in probability and statistical applications. We aim to approximate the moments of such ratios under several dependence assumptions. Extending the ideas in Collomb [C. R. Acad. Sci. Paris 285 (1977) 289--292], we propose sharper bounds for the moments of randomly weighted sums and for the $L^p$-deviations from the asymptotic normal law when the central limit theorem holds. We indicate suitable applications in finance and censored data analysis and focus on the applications in the field of functional estimation."}
{"category": "Math", "title": "Topological chaos: what may this mean ?", "abstract": "We confront existing definitions of chaos with the state of the art in topological dynamics. The article does not propose any new definition of chaos but, starting from several topological properties that can be reasonably called chaotic, tries to sketch a theoretical view of chaos. Among the main ideas in this article are the distinction between overall chaos and partial chaos, and the fact that some dynamical properties may be considered more chaotic than others."}
{"category": "Math", "title": "Properties and refinements of the fused lasso", "abstract": "We consider estimating an unknown signal, both blocky and sparse, which is corrupted by additive noise. We study three interrelated least squares procedures and their asymptotic properties. The first procedure is the fused lasso, put forward by Friedman et al. [Ann. Appl. Statist. 1 (2007) 302--332], which we modify into a different estimator, called the fused adaptive lasso, with better properties. The other two estimators we discuss solve least squares problems on sieves; one constrains the maximal $\\ell_1$ norm and the maximal total variation seminorm, and the other restricts the number of blocks and the number of nonzero coordinates of the signal. We derive conditions for the recovery of the true block partition and the true sparsity patterns by the fused lasso and the fused adaptive lasso, and we derive convergence rates for the sieve estimators, explicitly in terms of the constraining parameters."}
{"category": "Math", "title": "Diamond-alpha Integral Inequalities on Time Scales", "abstract": "The theory of the calculus of variations was recently extended to the more general time scales setting, both for delta and nabla integrals. The primary purpose of this paper is to further extend the theory on time scales, by establishing some basic diamond-alpha dynamic integral inequalities. We prove generalized versions of H\\\"{o}lder, Cauchy-Schwarz, Minkowski, and Jensen's inequalities. For the particular case when alpha is equal to one or alpha is equal to zero one gets, respectively, correspondent delta and nabla inequalities. If we further restrict ourselves by fixing the time scale to the real or integer numbers, then we obtain the classical inequalities, whose role in optimal control is well known. By analogy, we trust that the diamond-alpha integral inequalities we prove here will be important in the study of control systems on times scales."}
{"category": "Math", "title": "Derivative loss for Kirchhoff equations with non-Lipschitz nonlinear term", "abstract": "In this paper we consider the Cauchy boundary value problem for the abstract Kirchhoff equation with a continuous nonlinearity m : [0,+\\infty) --> [0,+\\infty). It is well known that a local solution exists provided that the initial data are regular enough. The required regularity depends on the continuity modulus of m. In this paper we present some counterexamples in order to show that the regularity required in the existence results is sharp, at least if we want solutions with the same space regularity of initial data. In these examples we construct indeed local solutions which are regular at t = 0, but exhibit an instantaneous (often infinite) derivative loss in the space variables."}
{"category": "Math", "title": "Logarithms and Square Roots of Real Matrices", "abstract": "In these notes, we consider the problem of finding the logarithm or the square root of a real matrix. It is known that for every real n x n matrix, A, if no real eigenvalue of A is negative or zero, then A has a real logarithm, that is, there is a real matrix, X, such that e^X = A. Furthermore, if the eigenvalues, xi, of X satisfy the property -pi < Im(xi) < pi, then X is unique. It is also known that under the same condition every real n x n matrix, A, has a real square root, that is, there is a real matrix, X, such that X^2 = A. Moreover, if the eigenvalues, rho e^{i theta}, of X satisfy the condition -pi/2 < theta < pi/2, then X is unique. These theorems are the theoretical basis for various numerical methods for exponentiating a matrix or for computing its logarithm using a method known as scaling and squaring (resp. inverse scaling and squaring). Such methods play an important role in the log-Euclidean framework due to Arsigny, Fillard, Pennec and Ayache and its applications to medical imaging. Actually, there is a necessary and sufficient condition for a real matrix to have a real logarithm (or a real square root) but it is fairly subtle as it involves the parity of the number of Jordan blocks associated with negative eigenvalues. As far as I know, with the exception of Higham's recent book, proofs of these results are scattered in the literature and it is not easy to locate them. Moreover, Higham's excellent book assumes a certain level of background in linear algebra that readers interested in the topics of this paper may not possess so we feel that a more elementary presentation might be a valuable supplement to Higham. In these notes, I present a unified exposition of these results and give more direct proofs of some of them using the Real Jordan Form."}
{"category": "Math", "title": "Arithmetic Laplacians", "abstract": "We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one dimensional groups, and analyze their spaces of solutions."}
{"category": "Math", "title": "On multiplicative conditionally free convolution", "abstract": "Using the combinatorics of non-crossing partitions, we construct a conditionally free analogue of the Voiculescu's S-transform. The result is applied to analytical description of conditionally free multiplicative convolution and characterization of infinite divisibility."}
{"category": "Math", "title": "Diamond-alpha Polynomial Series on Time Scales", "abstract": "The objective of this paper is twofold: (i) to survey existing results of generalized polynomials on time scales, covering definitions and properties for both delta and nabla derivatives; (ii) to extend previous results by using the more general notion of diamond-alpha derivative on time scales. We introduce a new notion of combined-polynomial series on a time scale, as a convex linear combination of delta and nabla generalized series. Main results are formulated for homogenous time scales. As an example, we compute diamond-alpha derivatives on time scales for delta and nabla exponential functions."}
{"category": "Math", "title": "Isoperimetric Problems of the Calculus of Variations on Time Scales", "abstract": "We prove a necessary optimality condition for isoperimetric problems on time scales in the space of delta-differentiable functions with rd-continuous derivatives. The results are then applied to Sturm-Liouville eigenvalue problems on time scales."}
{"category": "Math", "title": "An example of a weighted algebra $L_p^w(G)$ on uncountable group", "abstract": "We construct examples of weighted algebras $L_p^w(G)$ with $1<p\\le 2$ on uncountable free groups. For $p>2$ no weighted algebras exist on these groups. From the other side, we prove that an amenable group on which exist weighted algebras with $p>1$ must be sigma-compact."}
{"category": "Math", "title": "Convolution and Cross-Correlation of Ramanujan-Fourier Series", "abstract": "This paper uses the machinery of almost periodic functions to prove that even without uniform convergence the connection between a pair of almost periodic functions and the constants of the associated Fourier series exists for both the convolution and cross-correlation. The general results for two almost periodic functions are narrowed and applied to Ramanujan sums and finally applied to support the specific relation of the Wiener-Khinchin formula for arithemic functions with a Ramanujan-Fourier Series."}
{"category": "Math", "title": "The Bar-Natan skein module of the solid torus and the homology of (n,n) Springer varieties", "abstract": "This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n,n) Springer variety. The results build on Khovanov's work with crossingless matchings and the cohomology of the (n,n) Springer variety. We also give a formula for comultiplication in the Bar-Natan skein module for this specific three-manifold and boundary curve system."}
{"category": "Math", "title": "Rouquier blocks of the cyclotomic Hecke algebras of G(de,e,r)", "abstract": "The \"Rouquier blocks\" of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the \"families of characters\", defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite series, G(de,e,r), thus completing their calculation for all complex reflection groups."}
{"category": "Math", "title": "Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations", "abstract": "Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also included some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of ``projective polyhedron'' based on cones. To the best of our knowledge, this notion of projective polyhedron is new. We also believe that some of our proofs establishing the equivalence of V-polyhedra and H-polyhedra are new."}
{"category": "Math", "title": "A Khasminskii type averaging principle for stochastic reaction-diffusion equations", "abstract": "We prove that an averaging principle holds for a general class of stochastic reaction-diffusion systems, having unbounded multiplicative noise, in any space dimension. We show that the classical Khasminskii approach for systems with a finite number of degrees of freedom can be extended to infinite dimensional systems."}
{"category": "Math", "title": "Averaging principle for a class of stochastic reaction-diffusion equations", "abstract": "We consider the averaging principle for stochastic reaction-diffusion equations. Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, we calculate limiting slow motion. The study of solvability of Kolmogorov equations in Hilbert spaces and the analysis of regularity properties of solutions, allow to generalize the classical approach to finite-dimensional problems of this type in the case of SPDE's."}
{"category": "Math", "title": "Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions", "abstract": "One of the main goals of these notes is to explain how rotations in reals^n are induced by the action of a certain group, Spin(n), on reals^n, in a way that generalizes the action of the unit complex numbers, U(1), on reals^2, and the action of the unit quaternions, SU(2), on reals^3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and reals^n). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of an algebra, Cl_n, the Clifford algebra associated with reals^n. Since the spinor groups are certain well chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. These notes provide a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Clifford algebra Cl_{p, q} associated with a nondegenerate symmetric bilinear form of signature (p, q) and culminating in the beautiful \"8-periodicity theorem\" of Elie Cartan and Raoul Bott (with proofs)."}
{"category": "Math", "title": "Sums of norm spheres are norm shells and lower triangle inequalities are sharp", "abstract": "The statements in the title are explained and proved, as a little exercise in elementary normed vector space theory at the level of Chapter 5 of Dieudonn\\'e's \"Foundations of Mathematical Analysis\". A connection to recent moment bounds for submartingales is sketched."}
{"category": "Math", "title": "On the norm convergence of nonconventional ergodic averages", "abstract": "We offer a generalization of the recent result of Tao (building on earlier results of Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler) that the nonconventional ergodic averages associated to an arbitrary number of commuting probability-preserving transformations always converge to some limit in L^2. We prove the corresponding result for a collection of commuting actions of a larger discrete Abelian group, and gives convergence that is uniform in the start-point of the averages. While Tao's proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result."}
{"category": "Math", "title": "On submanifolds with tamed second fundamental form", "abstract": "We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_{N}\\leq \\kappa \\leq 0$ are proper, (compact if $N$ is compact). In addition, if $N$ is Hadamard then $M$ has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realized as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below."}
{"category": "Math", "title": "Width of homoclinic zone for quadratic maps", "abstract": "We study several families of planar quadratic diffeomorphisms near a Bogdanov-Takens bifurcation. For each family, the associate bifurcation diagram can be deduced from the interpolating flow. However, a zone of chaos confined between two lines of homoclinic bifurcation that are exponentially close to one-another is observed. The goal of this paper is to test numerically an accurate asymptotic expansion for the width of this chaotic zone for different families."}
{"category": "Math", "title": "Nathanson's Heights and the CSS Conjecture for Cayley Graphs", "abstract": "Let $G$ be a finite directed graph, $\\beta(G)$ the minimum size of a subset $X$ of edges such that the graph $G' = (V,E \\smallsetminus X)$ is directed acyclic and $\\gamma(G)$ the number of pairs of nonadjacent vertices in the undirected graph obtained from $G$ by replacing each directed edge with an undirected edge. Chudnovsky, Seymour and Sullivan \\cite{CSS07} proved that if $G$ is triangle-free, then $\\beta(G) \\leq \\gamma(G)$. They conjectured a sharper bound (so called the \"CSS conjecture\") that $\\beta(G) \\leq \\dfrac{\\gamma(G)}{2}$. Nathanson and Sullivan verified this conjecture for the directed Cayley graph $\\Cay(\\bbZ/N\\bbZ, E_A)$ whose vertex set is the additive group $\\bbZ/N\\bbZ$ and whose edge set $E_A$ is determined by $E_A = {(x,x+a) : x \\in \\bbZ/N\\bbZ, a \\in A}$ when $N$ is prime in \\cite{NS07} by introducing \"height\". In this work, we extend the definition of height and the proof of CSS conjecture for $\\Cay(\\bbZ/N\\bbZ, E_A)$ to any positive integer $N$."}
{"category": "Math", "title": "Universal spaces for manifolds equipped with a closed integral k-form", "abstract": "In this note we prove that any integral closed k-form $\\phi ^k$, $k\\ge 3$, on a m-dimensional manifold $M^m$, $m \\ge k$, is the restriction of a universal closed k-form $h^k$ on a universal manifold $U^{d(m,k)}$ as a result of an embedding of $M^m$ to $U^{d(m,k)}$."}
{"category": "Math", "title": "Yang-Mills bar connections over compact K\\\"ahler manifolds", "abstract": "In this note we introduce a Yang-Mills bar equation on complex vector bundles over compact Hermitian manifolds as the Euler-Lagrange equation for a Yang-Mills bar functional. We show the existence of a non-trivial solution of this equation over compact K\\\"ahler manifolds as well as a short time existence of the negative Yang-Mills bar gradient flow. We also show a rigidity of holomorphic connections among a class of Yang-Mills bar connections over compact K\\\"ahler manifolds of positive Ricci curvature."}
{"category": "Math", "title": "Periods and elementary real numbers", "abstract": "The periods, introduced by Kontsevich and Zagier, form a class of complex numbers which contains all algebraic numbers and several transcendental quantities. Little has been known about qualitative properties of periods. In this paper, we compare the periods with hierarchy of real numbers induced from computational complexities. In particular we prove that periods can be effectively approximated by elementary rational Cauchy sequences. As an application, we exhibit a computable real number which is not a period."}
{"category": "Math", "title": "Asymptotic enumeration of constellations and related families of maps on orientable surfaces", "abstract": "We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces of genus g: m-hypermaps and m-constellations. For m=2, they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees. Our proofs rely on the generalisation to orientable surfaces of the Bouttier-Di Francesco-Guitter bijection, and on generating series methods. We show that each of the 2g fondamental cycles of the surface contributes a factor m between the numbers of m-hypermaps and m-constellations -- for example, large maps of genus g with even face degrees are bipartite with probability tending to 1/2^{2g}. A special case of our results implies former conjectures of Gao."}
{"category": "Math", "title": "M\\\"obius transformations and the Poincar\\'e distance in the quaternionic setting", "abstract": "In the space $\\hh$ of quaternions, we investigate the natural, invariant geometry of the open, unit disc $\\Delta_{\\hh}$ and of the open half-space $\\hh^{+}$. These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of M\\\"obius transformations of $\\Delta_{\\hh}$ and $\\hh^{+}$ and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the M\\\"obius transformations, and use it to define the analogous of the Poincar\\'e distances on $\\Delta_{\\hh}$ and $\\hh^{+}$. We easily deduce that there exists no isometry between the quaternionic Poincar\\'e distance of $\\Delta_{\\hh}$ and the Kobayashi distance inherited by $\\Delta_{\\hh}$ as a domain of $\\mathbb{C}^{2}$, in accordance with a direct consequence of the classification of the non compact, rank 1, symmetric spaces."}
{"category": "Math", "title": "Strong cleanness of the $2\\times 2$ matrix ring over a general local ring", "abstract": "A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey \\cite{BDD05a} completely characterized the commutative local rings $R$ for which ${\\mathbb M}_n(R)$ is strongly clean. For a general local ring $R$ and $n>1$, however, it is unknown when the matrix ring ${\\mathbb M}_n(R)$ is strongly clean. Here we completely determine the local rings $R$ for which ${\\mathbb M}_2(R)$ is strongly clean."}
{"category": "Math", "title": "On Smarandache Bryant Schneider group of a Smarandache loop", "abstract": "The concept of Smarandache Bryant Schneider Group of a Smarandache loop is introduced. Relationship(s) between the Bryant Schneider Group and the Smarandache Bryant Schneider Group of an S-loop are discovered and the later is found to be useful in finding Smarandache isotopy-isomorphy condition(s) in S-loops just like the formal is useful in finding isotopy-isomorphy condition(s) in loops. Some properties of the Bryant Schneider Group of a loop are shown to be true for the Smarandache Bryant Schneider Group of a Smarandache loop. Some interesting and useful cardinality formulas are also established for a type of finite Smarandache loop."}
{"category": "Math", "title": "Topological methods in analysis of periodic and chaotic canard-type trajectories", "abstract": "We investigate the role of topological methods in the analysis of canard-type periodic and chaotic trajectories. In the first part of the paper, we apply topological degree to the analysis of multi-dimensional canards. The second part is devoted to an application of a special corollary of the Poincare-Bendixson theorem to the existence of periodic two-dimensional canards."}
{"category": "Math", "title": "A Limit Theorem for Products of Free Unitary Operators", "abstract": "This paper establishes necessary and sufficient conditions for the products of freely independent unitary operators to converge in distribution to the uniform law on the unit circle."}
{"category": "Math", "title": "Alternatives to Pearson's and Spearman's Correlation Coefficients", "abstract": "This article presents several alternatives to Pearson's correlation coefficient and many examples. In the samples where the rank in a discrete variable counts more than the variable values, the mixtures that we propose of Pearson's and Spearman's correlation coefficients give better results."}
{"category": "Math", "title": "Freiman-Ruzsa-type theory for small doubling constant", "abstract": "In this paper, we study the linear structure of sets $A \\subset \\mathbb{F}_2^n$ with doubling constant $\\sigma(A)<2$, where $\\sigma(A):=\\frac{|A+A|}{|A|}$. In particular, we show that $A$ is contained in a small affine subspace. We also show that $A$ can be covered by at most four shifts of some subspace $V$ with $|V|\\leq |A|$. Finally, we classify all binary sets with small doubling constant."}
{"category": "Math", "title": "Gr\\\"unbaum Colorings of Toroidal Triangulations", "abstract": "We prove that if G is a triangulation of the torus and \\chi(G) \\neq 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors."}
{"category": "Math", "title": "Representation Growth of Linear Groups", "abstract": "Let $\\Gamma$ be a group and $r_n(\\Gamma)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function $\\calz_\\Gamma(s) = \\suml^\\infty_{n=1} r_n(\\Gamma)n^{-s}$. When $\\Gamma$ is an arithmetic group satisfying the congruence subgroup property then $\\calz_\\Gamma(s)$ has an ``Euler factorization\". The \"factor at infinity\" is sometimes called the \"Witten zeta function\" counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups $U$ of the associated simple group $G$ over the associated local field $K$. Here we show a surprising dichotomy: if $G(K)$ is compact (i.e. $G$ anisotropic over $K$) the abscissa of convergence goes to 0 when $\\dim G$ goes to infinity, but for isotropic groups it is bounded away from 0. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa."}
{"category": "Math", "title": "The Player's Effect", "abstract": "In a function that takes its inputs from various players, the effect of a player measures the variation he can cause in the expectation of that function. In this paper we prove a tight upper bound on the number of players with large effect, a bound that holds even when the players' inputs are only known to be pairwise independent. We also study the effect of a set of players, and show that there always exists a \"small\" set that, when eliminated, leaves every set with little effect. Finally, we ask whether there always exists a player with positive effect. We answer this question differently in various scenarios, depending on the properties of the function and the distribution of players' inputs. More specifically, we show that if the function is non-monotone or the distribution is only known to be pairwise independent, then it is possible that all players have 0 effect. If the distribution is pairwise independent with minimal support, on the other hand, then there must exist a player with \"large\" effect."}
{"category": "Math", "title": "Collinear triples in permutations", "abstract": "Let $\\alpha:\\mathbb{F}_q\\to\\mathbb{F}_q$ be a permutation and $\\Psi(\\alpha)$ be the number of collinear triples in the graph of $\\alpha$, where $\\mathbb{F}_q$ denotes a finite field of $q$ elements. When $q$ is odd Cooper and Solymosi once proved $\\Psi(\\alpha)\\geq(q-1)/4$ and conjectured the sharp bound should be $\\Psi(\\alpha)\\geq(q-1)/2$. In this note we indicate that the Cooper-Solymosi conjecture is true."}
{"category": "Math", "title": "Some conjectures about q-Fibonacci polynomials", "abstract": "In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials."}
{"category": "Math", "title": "Small volume expansions for elliptic equations", "abstract": "This paper analyzes the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form $\\nabla \\cdot D^\\eps \\nabla u^\\eps=0$ or $(-\\Delta + q^\\eps) u^\\eps=0$ with prescribed Neumann conditions. The theory is well-known when the constitutive parameters in the elliptic equation assume the values of different and smooth functions in the background and inside the inclusions. We generalize the results to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions of the trace of the solution at the domain's boundary up to an order $\\eps^{2d}$, where $d$ is dimension and $\\eps$ is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most $\\eps^{d+1}$ rather than the expected $\\eps^d$. We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables."}
{"category": "Math", "title": "New Proof on some sharp double integral Inequalities of the Hermite-Hadamard Type", "abstract": "In this paper, we derive a new proof on some sharp double integral inequalities of the Hermite-Hadamard type. Our approach is mainly based on well-known Taylor's theorem with the integral remainder."}
{"category": "Math", "title": "Connected components of strata of quadratic differentials over Teichmuller space", "abstract": "In this paper, we study connected components of strata of the space of quadratic differentials lying over $\\T_g$. We use certain general properties of sections of line bundles to put a upper bound on the number of connected components, and a generalized version of the Gauss map as an invariant to put a lower bound on the number of such components. For strata with sufficiently many zeroes of the same order we can state precisely the number of components."}
{"category": "Math", "title": "Motivic Integration on Toric Stacks", "abstract": "We present a decomposition of the space of twisted arcs of a toric stack. As a consequence, we give a combinatorial description of the motivic integral associated to a torus-invariant divisor of a toric stack."}
{"category": "Math", "title": "Denominators in cluster algebras of affine type", "abstract": "The Fomin-Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for the denominators of cluster variables in cluster algebras of affine type. The formulas are in terms of the dimensions of spaces of homomorphisms in the corresponding cluster category, and hold for any choice of initial cluster."}
{"category": "Math", "title": "Uniform Bounds on Pre-Images under Quadratic Dynamical Systems", "abstract": "For any elements b,c of a number field K, let G(b,c) denote the backwards orbit of b under the map f_c: C-->C given by f_c(x)=x^2+c. We prove an upper bound on the number of elements of G(b,c) whose degree over K is at most some constant B. This bound depends only on b, [K:Q], and B, and is valid for all b outside an explicit finite set. We also show that, for any N>3 and any b in K outside a finite set, there are only finitely many pairs of complex numbers (y,c) for which [K(y,c):K]<2^(N-3) and the value of the N-th iterate of f_c(x) at x=y is b. Moreover, the bound 2^(N-3) in this result is optimal."}
{"category": "Math", "title": "The mixing advantage is less than 2", "abstract": "Corresponding to $n$ independent non-negative random variables $X_1,...,X_n$, are values $M_1,...,M_n$, where each $M_i$ is the expected value of the maximum of $n$ independent copies of $X_i$. We obtain an upper bound to the expected value of the maximum of $X_1,...,X_n$ in terms of $M_1,...,M_n$. This inequality is sharp in the sense that the quantity and its bound can be made as close to each other as we want. We also present related comparison results."}
{"category": "Math", "title": "The Elliptic Hypergeometric Functions Associated to the Configuration Space of Points on an Elliptic Curve I : Twisted Cycles", "abstract": "We consider the Euler type integral associated to the configuration space of points on an elliptic curve, which is an analogue of the hypergeometric function associated to the configuration space of points on a projective line. We calculate the {\\it twisted homology group}, with coefficients in the local system associated to a power function $g^{\\alpha}$ of an elliptic function $g$, and the intersection form. Applying these calculations, we describe the {\\it connection matrices} representing the linear isomorphisms induced from analytic continuations of the functions defined by the integrations of $g^{\\alpha}$ over twisted cycles."}
{"category": "Math", "title": "Distance-based clustering of sparsely observed stochastic processes, with applications to online auctions", "abstract": "We propose a distance between two realizations of a random process where for each realization only sparse and irregularly spaced measurements with additional measurement errors are available. Such data occur commonly in longitudinal studies and online trading data. A distance measure then makes it possible to apply distance-based analysis such as classification, clustering and multidimensional scaling for irregularly sampled longitudinal data. Once a suitable distance measure for sparsely sampled longitudinal trajectories has been found, we apply distance-based clustering methods to eBay online auction data. We identify six distinct clusters of bidding patterns. Each of these bidding patterns is found to be associated with a specific chance to obtain the auctioned item at a reasonable price."}
{"category": "Math", "title": "Twisted Poincar\\'e Lemma and Twisted \\v{C}ech-de Rham Isomorphism in case of Projective Line", "abstract": "In this paper, we give a direct proof of the twisted Poincar\\'{e} lemma by using the integrations over regularized paths. This method tells us a concrete description of the \\v{C}ech-de Rham isomorphism."}
{"category": "Math", "title": "Consistency of restricted maximum likelihood estimators of principal components", "abstract": "In this paper we consider two closely related problems : estimation of eigenvalues and eigenfunctions of the covariance kernel of functional data based on (possibly) irregular measurements, and the problem of estimating the eigenvalues and eigenvectors of the covariance matrix for high-dimensional Gaussian vectors. In Peng and Paul (2007), a restricted maximum likelihood (REML) approach has been developed to deal with the first problem. In this paper, we establish consistency and derive rate of convergence of the REML estimator for the functional data case, under appropriate smoothness conditions. Moreover, we prove that when the number of measurements per sample curve is bounded, under squared-error loss, the rate of convergence of the REML estimators of eigenfunctions is near-optimal. In the case of Gaussian vectors, asymptotic consistency and an efficient score representation of the estimators are obtained under the assumption that the effective dimension grows at a rate slower than the sample size. These results are derived through an explicit utilization of the intrinsic geometry of the parameter space, which is non-Euclidean. Moreover, the results derived in this paper suggest an asymptotic equivalence between the inference on functional data with dense measurements and that of the high dimensional Gaussian vectors."}
{"category": "Math", "title": "Stanley decompositions and localization", "abstract": "We study the behavior of Stanley depth under the operation of localization with respect to a variable."}
{"category": "Math", "title": "Spectral gap for the interchange process in a box", "abstract": "We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to $\\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in any graph the spectral gap for the interchange process is the same as the spectral gap for a corresponding continuous-time random walk. Our proof uses a technique that is similar to that used by Handjani and Jungreis, who proved that Aldous's conjecture holds when the graph is a tree."}
{"category": "Math", "title": "Quotient categories, stability conditions, and birational geometry", "abstract": "This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. It turns out that this category has homological dimension c. As an application of this, we will describe the space of stability conditions on its derived category in the case c=1. Moreover, we describe all exact equivalences between these quotient categories in this particular case which is closely related to classification problems in birational geometry."}
{"category": "Math", "title": "Degrees of maps between Grassmann manifolds", "abstract": "Let $f:G_{n,k}\\longrightarrow G_{m,l}$ be any continuous map between any two distinct complex Grassmann manifolds of the same dimension where the target is not the complex projective space. We show that, for any given $k,l$, the degree of $f$ is zero provided that $m,n$ are sufficiently large. If the degree of $f$ is $\\pm 1$, we show that $(m,l)=(n,k)$ and $f$ is a homotopy equivalence. Also, we prove that the image under $f^*$ of elements of a set of algebra generators of $H^*(G_{m,l};\\mathbb{Q})$ is determined upto a sign, $\\pm$, if the degree of $f$ is non-zero. Our proofs cover the case of quaternionic Grassmann manifolds as well."}
{"category": "Math", "title": "Notes on algebra and geometry of polynomial representations", "abstract": "Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following template. Assume that in a neighborhood of a boundary point the semi-algebraic set A can be described by an irreducible polynomial f. Then f is a factor of a certain multiplicity of some of the polynomials p_1,...,p_m. Special cases when A is elementary closed, elementary open, a polygon, or a polytope are considered separately."}
{"category": "Math", "title": "On non-vanishing of cohomologies of generalized Raynaud polarized surfaces", "abstract": "We consider a family of slightly extended version of the Raynaud's surfaces X over the field of positive characteristic with Mumford-Szpiro type polarizations Z, which have Kodaira non-vanishing H^1(X, Z^{-1})\\ne 0. The surfaces are at least normal but smooth under a special condition. We compute the cohomologies H^i(X, Z^n), for intergers i and n, and study their (non-)vanishing. Finally, we give a fairly large family of non Mumford-Szpiro type polarizations Z_{a,b} with Kodaira non-vanishing."}
{"category": "Math", "title": "Zeros of p-adic forms", "abstract": "A variant of Brauer's induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms."}
{"category": "Math", "title": "Steady periodic water waves under nonlinear elastic membranes", "abstract": "This is a study of two-dimensional steady periodic travelling waves on the surface of an infinitely deep irrotational ocean, when the top streamline is in contact with a membrane which has a nonlinear response to stretching and bending, and the pressure in the air above is constant. It is not supposed that the waves have small amplitude. The problem of existence of such waves is addressed using methods from the calculus of variations. The analysis involves the Hilbert transform and a Riemann-Hilbert formulation."}
{"category": "Math", "title": "Ideals generated by submaximal minors", "abstract": "The goal of this paper is to study irreducible families W(b;a) of codimension 4, arithmetically Gorenstein schemes X of P^n defined by the submaximal minors of a t x t matrix A with entries homogeneous forms of degree a_j-b_i. Under some numerical assumption on a_j and b_i we prove that the closure of W(b;a) is an irreducible component of Hilb^{p(x)}(P^n), we show that Hilb^{p(x)}(P^n) is generically smooth along W(b;a) and we compute the dimension of W(b;a) in terms of a_j and b_i. To achieve these results we first prove that X is determined by a regular section of the twisted conormal sheaf I_Y/I^2_Y(s) where s=deg(det(A)) and Y is a codimension 2, arithmetically Cohen-Macaulay scheme of P^n defined by the maximal minors of the matrix obtained deleting a suitable row of A."}
{"category": "Math", "title": "Local time steps for a finite volume scheme", "abstract": "We present a strategy for solving time-dependent problems on grids with local refinements in time using different time steps in different regions of space. We discuss and analyze two conservative approximations based on finite volume with piecewise constant projections and domain decomposition techniques. Next we present an iterative method for solving the composite-grid system that reduces to solution of standard problems with standard time stepping on the coarse and fine grids. At every step of the algorithm, conservativity is ensured. Finally, numerical results illustrate the accuracy of the proposed methods."}
{"category": "Math", "title": "Ideal clones: Solution to a problem of Czedli and Heindorf", "abstract": "Given an infinite set X and an ideal I of subsets of X, the set of all finitary operations on X which map all (powers of) I-small sets to I-small sets is a clone. In a 2001 article, G. Czedli and L. Heindorf asked whether or not for two particular ideals I and J on a countably infinite set X, the corresponding ideal clones were a covering in the lattice of clones. We give an affirmative answer to this question."}
{"category": "Math", "title": "A martingale approach to minimal surfaces", "abstract": "We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in the theory of minimal surfaces, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems."}
{"category": "Math", "title": "Intermittence and nonlinear parabolic stochastic partial differential equations", "abstract": "We consider nonlinear parabolic SPDEs of the form $\\partial_t u=\\sL u + \\sigma(u)\\dot w$, where $\\dot w$ denotes space-time white noise, $\\sigma:\\R\\to\\R$ is [globally] Lipschitz continuous, and $\\sL$ is the $L^2$-generator of a L\\'evy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $\\sigma$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of $\\sL$ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for $\\sL$ in dimension $(1+1)$. When $\\sL=\\kappa\\partial_{xx}$ for $\\kappa>0$, these formulas agree with the earlier results of statistical physics \\cite{Kardar,KrugSpohn,LL63}, and also probability theory \\cite{BC,CM94} in the two exactly-solvable cases where $u_0=\\delta_0$ and $u_0\\equiv 1$."}
{"category": "Math", "title": "The Classification Theorem for Compact Surfaces And A Detour On Fractals", "abstract": "The purpose of these notes is to present a fairly complete proof of the classification Theorem for compact surfaces. Other presentations are often quite informal (see the references in Chapter V) and we have tried to be more rigorous. Our main source of inspiration is the beautiful book on Riemann Surfaces by Ahlfors and Sario. However, Ahlfors and Sario's presentation is very formal and quite compact. As a result, uninitiated readers will probably have a hard time reading this book. Our goal is to help the reader reach the top of the mountain and help him not to get lost or discouraged too early. This is not an easy task! We provide quite a bit of topological background material and the basic facts of algebraic topology needed for understanding how the proof goes, with more than an impressionistic feeling. We hope that these notes will be helpful to readers interested in geometry, and who still believe in the rewards of serious hiking!"}
{"category": "Math", "title": "New congruences for central binomial coefficients", "abstract": "Let p be a prime and let a be a positive integer. In this paper we determine $\\sum_{k=0}^{p^a-1}\\binom{2k}{k+d}/m^k$ and $\\sum_{k=1}^{p-1}\\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p. For example, we show that if $p\\not=2,5$ then $$\\sum_{k=1}^{p-1}(-1)^k\\frac{\\binom{2k}k}k=-5\\frac{F_{p-(\\frac p5)}}p (mod p),$$ where F_n is the n-th Fibonacci number and (-) is the Jacobi symbol. We also prove that if p>3 then $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}k={8/9} p^2B_{p-3} (mod p^3),$$ where B_n denotes the n-th Bernoulli number."}
{"category": "Math", "title": "Hulls and Husks", "abstract": "The aim of this note is to prove an analog of the flattening decomposition theorem for reflexive hulls. The main applications are: the construction of the moduli space of varieties of general type, improved flatness conditions and criteria for simultaneous normalizations."}
{"category": "Math", "title": "Cyclic homology of crossed products", "abstract": "We obtain a mixed complex, simpler that the canonical one, given the Hochschild, cyclic, negative and periodic homology of a crossed product E=A#fH, where H is an arbitrary Hopf algebra and f is a convolution invertible cocycle with values in A. Actually, we work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A which is stable under the action of H, and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. As an application we obtain two spectral sequences converging to the cyclic homology of E relative to K. The first one in the general setting and the second one (which generalizes those previously found by several authors) when f takes its values in K."}
{"category": "Math", "title": "Homotopy of unitaries in simple C*-algebras with tracial rank one", "abstract": "Let $\\epsilon>0$ be a positive number. Is there a number $\\delta>0$ satisfying the following? Given any pair of unitaries $u$ and $v$ in a unital simple $C^*$-algebra $A$ with $[v]=0$ in $K_1(A)$ for which $$ \\|uv-vu\\|<\\dt, $$ there is a continuous path of unitaries $\\{v(t): t\\in [0,1]\\}\\subset A$ such that $$ v(0)=v, v(1)=1 \\and \\|uv(t)-v(t)u\\|<\\epsilon \\forall t\\in [0,1]. $$ An answer is given to this question when $A$ is assumed to be a unital simple $C^*$-algebra with tracial rank no more than one. Let $C$ be a unital separable amenable simple $C^*$-algebra with tracial rank no more than one which also satisfies the UCT. Suppose that $\\phi: C\\to A$ is a unital monomorphism and suppose that $v\\in A$ is a unitary with $[v]=0$ in $K_1(A)$ such that $v$ almost commutes with $\\phi.$ It is shown that there is a continuous path of unitaries $\\{v(t): t\\in [0,1]\\}$ in $A$ with $v(0)=v$ and $v(1)=1$ such that the entire path $v(t)$ almost commutes with $\\phi,$ provided that an induced Bott map vanishes. Other versions of the so-called Basic Homotopy Lemma are also presented."}
{"category": "Math", "title": "Homological and homotopical higher-order filling functions", "abstract": "We construct groups in which FV^3(n) != \\delta^2(n). This construction also leads to groups G_k, k >= 3 for which \\delta^{k}(n) is not subrecursive."}
{"category": "Math", "title": "Rational and algebraic series in combinatorial enumeration", "abstract": "Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally algebraic. This property has a practical interest, since one can usually say a lot on the numbers a(n), but also a combinatorial one: the rational or algebraic nature of the generating function suggests that the objects have a (possibly hidden) structure, similar to the linear structure of words in the rational case, and to the branching structure of trees in the algebraic case. We describe and illustrate this combinatorial intuition, and discuss its validity. While it seems to be satisfactory in the rational case, it is probably incomplete in the algebraic one. We conclude with open questions."}
{"category": "Math", "title": "A characterization of Dirac morphisms", "abstract": "Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields."}
{"category": "Math", "title": "Descendent bounds for effective divisors on the moduli space of curves", "abstract": "The slope of the moduli space of genus g curves is bounded from below by 60/(g+4) via a descendent calculation."}
{"category": "Math", "title": "Limit theorems for additive c-free convolution", "abstract": "In this paper we find necessary and sufficient conditions for the weak convergence of c-free convolution of pairs of measures, where the measures are assumed to be infinitesimal and their support may be unbounded. These results are obtained by complex analytic methods."}
{"category": "Math", "title": "Hardy inequalities for general elliptic operators with improvements", "abstract": "We obtain optimal generalized versions of Hardy inequalities, which as special cases contain Hardy's inequality and Hardy's inequality involving the distance function to the boundary of $ \\Omega$. In addition we obtain neccesary and sufficient conditions to add improvements in the form of non negative potentials."}
{"category": "Math", "title": "Functional moderate deviations for triangular arrays and applications", "abstract": "Motivated by the study of dependent random variables by coupling with independent blocks of variables, we obtain first sufficient conditions for the moderate deviation principle in its functional form for triangular arrays of independent random variables. Under some regularity assumptions our conditions are also necessary in the stationary case. The results are then applied to derive moderate deviation principles for linear processes, kernel estimators of a density and some classes of dependent random variables."}
{"category": "Math", "title": "Embeddings between operator-valued dyadic BMO spaces", "abstract": "We investigate a scale of dyadic operator-valued BMO spaces, corresponding to the different yet equivalent characterizations of dyadic BMO in the scalar case. In the language of operator spaces, we investigate different operator space structures on the scalar dyadic BMO space which arise naturally from the different characterisations of scalar BMO. We also give sharp dimensional growth estimates for the sweep of functions and its bilinear extension in some of those different dyadic BMO spaces."}
{"category": "Math", "title": "Logarithmic decay of hyperbolic equations with arbitrary boundary damping", "abstract": "In this paper, we study the logarithmic stability for the hyperbolic equations by arbitrary boundary observation. Based on Carleman estimate, we first prove an estimate of the resolvent operator of such equation. Then we prove the logarithmic stability estimate for the hyperbolic equations without any assumption on an observation subboundary."}
{"category": "Math", "title": "Curvature extrema and four-vertex theorems for polygons and polyhedra", "abstract": "Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. For smooth curves and polygonal lines in the plane, a formula relating the number of extrema of curvature to the winding numbers of the curves (polygonal lines) and their evolutes is obtained. Also are considered higher-dimensional analogs of the four-vertex theorem for regular and shellable triangulations."}
{"category": "Math", "title": "Classifications of Linear Controlled Systems", "abstract": "This paper is devoted to a study of linear, differential and topological classifications for linear controlled systems governed by ordinary differential equations. The necessary and sufficient conditions for the linear and topological equivalence are given. It is also shown that the differential equivalence is the same as the linear equivalence for the linear controlled systems."}
{"category": "Math", "title": "Computational Geometric Optimal Control of Rigid Bodies", "abstract": "This paper formulates optimal control problems for rigid bodies in a geometric manner and it presents computational procedures based on this geometric formulation for numerically solving these optimal control problems. The dynamics of each rigid body is viewed as evolving on a configuration manifold that is a Lie group. Discrete-time dynamics of each rigid body are developed that evolve on the configuration manifold according to a discrete version of Hamilton's principle so that the computations preserve geometric features of the dynamics and guarantee evolution on the configuration manifold; these discrete-time dynamics are referred to as Lie group variational integrators. Rigid body optimal control problems are formulated as discrete-time optimization problems for discrete Lagrangian/Hamiltonian dynamics, to which standard numerical optimization algorithms can be applied. This general approach is illustrated by presenting results for several different optimal control problems for a single rigid body and for multiple interacting rigid bodies. The computational advantages of the approach, that arise from correctly modeling the geometry, are discussed."}
{"category": "Math", "title": "Groebner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras", "abstract": "In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Groebner-Shirshov bases of free Rota-Baxter algebra, $\\lambda$-differential algebra and $\\lambda$-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to those obtained by Ebrahimi-Fard and Guo, and Guo and Keigher recently by using other methods."}
{"category": "Math", "title": "Einstein solvmanifolds attached to two-step nilradicals", "abstract": "A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite a substantial progress towards the understanding of Einstein nilradicals, there is still a lack of classification results even for some well-studied classes of nilpotent Lie algebras, such as the two-step ones. In this paper, we give a classification of two-step nilpotent Einstein nilradicals in one of the rare cases when the complete set of affine invariants is known: for the two-step nilpotent Lie algebras with the two-dimensional center. Informally speaking, we prove that such a Lie algebra is an Einstein nilradical, if it is defined by a matrix pencil having no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity. We also discuss the connection between the property of a two-step nilpotent Lie algebra and its dual to be an Einstein nilradical."}
{"category": "Math", "title": "On the coordinate ring of spherical conjugacy classes", "abstract": "Let G be a simple algebraic group over an algebraically closed field of characteristic zero and X be a spherical conjugacy class of G. We determine the decomposition of the coordinate ring of X into simple G-modules."}
{"category": "Math", "title": "On the Holonomy of Kaluza-Klein metrics", "abstract": "We investigate Kaluza-Klein metrics with a recurrent light-like vector field over a pseudo-Riemannian manifold."}
{"category": "Math", "title": "Tensor products of type III factor representations of Cuntz-Krieger algebras", "abstract": "We introduced a non-symmetric tensor product of any two states or any two representations of Cuntz-Krieger algebras associated with a certain non-cocommutative comultiplication in previous our work. In this paper, we show that a certain set of KMS states is closed with respect to the tensor product. From this, we obtain formulae of tensor product of type {\\rm III} factor representations of Cuntz-Krieger algebras which is different from results of the tensor product of factors of type {\\rm III}."}
{"category": "Math", "title": "Multidimensional Hahn polynomials, intertwining functions on the symmetric group and Clebsch-Gordan coefficients", "abstract": "We generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klymik, we develop a tree-method approach for those intertwining functions. We also give a group theoretic proof of the relation between Hahn polynomials and Clebesh-Gordan coefficients, given analytically by Koornwinder and by Nikiforov, Smorodinski\\u{i} and Suslov. Such relation is also extended to the multidimensional case."}
{"category": "Math", "title": "A Note on Fuzzy Real and Complex Field", "abstract": "Using the concept of fuzzy field, we have considered the fuzzy field of real and complex numbers and thereafter we have established a few standard results of real and complex numbers with respect to a membership function."}
{"category": "Math", "title": "The Euler characteristic as a polynomial in the Chern classes", "abstract": "In this paper we obtain some explicit expressions for the Euler characteristic of a rank n coherent sheaf F on P^N and of its twists F(t) as polynomials in the Chern classes c_i(F), also giving algorithms for the computation. The employed methods use techniques of umbral calculus involving symmetric functions and Stirling numbers."}
{"category": "Math", "title": "On the Calculation of gl.dim$G^{\\mathbb{N}}(A)$ and gl.dim$\\widetilde{A}$ by Using Gr\\\"obner Bases", "abstract": "Let $A=K< X_1,...,X_n> /< {\\cal G}>$ be a $K$-algebra defined by a finite Gr\\\"obner basis ${\\cal G}$. It is shown how to use the Ufnarovski graph $\\Gamma ({\\bf LM}({\\cal G}))$ and the graph of $n$-chains $\\Gamma_{\\rm C}({\\bf LM}({\\cal G}))$ to calculate gl.dim$G^{\\mathbb{N}}(A)$ and gl.dim$\\widetilde{A}$, where $G^{\\mathbb{N}}(A)$, respectively $\\widetilde{A}$, is the associated $\\mathbb{N}$-graded algebra of $A$, respectively the Rees algebra of $A$ with respect to the $\\mathbb{N}$-filtration $FA$ of $A$ induced by a weight $\\mathbb{N}$-grading filtration of $K< X_1,...,X_n>$."}
{"category": "Math", "title": "Fractional, Maximal and Singular Operators in Variable Exponent Lorentz Spaces", "abstract": "We introduce the Lorentz space $\\mathcal{L}^{p(\\cdot), q(\\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The main goal of the paper is to show that the boundedness of these operators in the spaces $\\mathcal{L}^{p(\\cdot), q(\\cdot)}$ is possible without the local log-condition on the exponents, typical for the variable exponent Lebesgue spaces; instead the exponents $p(s)$ and $q(s)$ should only satisfy decay conditions of log-type as $s\\to 0$ and $s\\to\\infty$. To prove this, we base ourselves on the recent progress in the problem of the validity of Hardy inequalities in variable exponent Lebesgue spaces."}
{"category": "Math", "title": "Higher order cohomology of arithmetic groups", "abstract": "Higher order cohomology of arithmetic groups is expressed in terms of (g,K)-cohomology. Generalizing results of Borel, it is shown that the latter can be computed using functions of (uniform) moderate growth. A higher order versions of Borel's conjecture is stated, asserting that the cohomology can be computed using automorphic forms."}
{"category": "Math", "title": "Principal eigenvalue for random walk among random traps on Z^d", "abstract": "Let $(\\tau_x)_{x \\in \\Z^d}$ be i.i.d. random variables with heavy (polynomial) tails. Given $a \\in [0,1]$, we consider the Markov process defined by the jump rates $\\omega_{x \\to y} = {\\tau_x}^{-(1-a)} {\\tau_y}^a$ between two neighbours $x$ and $y$ in $\\Z^d$. We give the asymptotic behaviour of the principal eigenvalue of the generator of this process, with Dirichlet boundary condition. The prominent feature is a phase transition that occurs at some threshold depending on the dimension."}
{"category": "Math", "title": "A necessary condition for dynamic equivalence", "abstract": "If two control systems on manifolds of the same dimension are dynamic equivalent, we prove that either they are static equivalent --i.e. equivalent via a classical diffeomorphism-- or they are both ruled; for systems of different dimensions, the one of higher dimension must ruled. A ruled system is one whose equations define at each point in the state manifold, a ruled submanifold of the tangent space. Dynamic equivalence is also known as equivalence by endogenous dynamic feedback, or by a Lie-B\\\"acklund transformation when control systems are viewed as underdetermined systems of ordinary differential equations; it is very close to absolute equivalence for Pfaffian systems. It was already known that a differentially flat system must be ruled; this is a particular case of the present result, in which one of the systems is assumed to be \"trivial\" (or linear controllable)."}
{"category": "Math", "title": "Free subalgebras of Lie algebras close to nilpotent", "abstract": "We prove that for every automata algebra of exponential growth, the associated Lie algebra contains a free subalgebra. For n\\geq 1, let L_{n+2} be a Lie algebra with generator set x_1,..., x_{n+2} and the following relations: for k\\leq n, any commutator of length $k$ which consists of fewer than k different symbols from {x_1,...,x_{n+2}} is zero. As an application of this result about automata algebras, we prove that for every n\\geq 1, L_{n+2} contains a free subalgebra. We also prove the similar result about groups defined by commutator relations."}
{"category": "Math", "title": "Covariance of centered distributions on manifold", "abstract": "We define and study a family of distributions with domain complete Riemannian manifold. They are obtained by projection onto a fixed tangent space via the inverse exponential map. This construction is a popular choice in the literature for it makes it easy to generalize well known multivariate Euclidean distributions. However, most of the available solutions use coordinate specific definition that makes them less versatile. %We propose improvements in two directions. We define the distributions of interest in coordinate independent way by utilizing co-variant 2-tensors. Then we study the relation of these distributions to their Euclidean counterparts. In particular, we are interested in relating the covariance to the tensor that controls distribution concentration. We find approximating expression for this relation in general and give more precise formulas in case of manifolds of constant curvature, positive or negative. Results are confirmed by simulation studies of the standard normal distribution on the unit-sphere and hyperbolic plane."}
{"category": "Math", "title": "Lecture by Michael Hopkins: the string orientation of tmf", "abstract": "These are the notes of a lecture held by Michael Hopkins in march 2007, at the Talbot workshop."}
{"category": "Math", "title": "Maximum likelihood estimation in a partially observed stratified regression model with censored data", "abstract": "The stratified proportional intensity model generalizes Cox's proportional intensity model by allowing different groups of the population under study to have distinct baseline intensity functions. In this article, we consider the problem of estimation in this model when the variable indicating the stratum is unobserved for some individuals in the studied sample. In this setting, we construct nonparametric maximum likelihood estimators for the parameters of the stratified model and we establish their consistency and asymptotic normality. Consistent estimators for the limiting variances are also obtained."}
{"category": "Math", "title": "A Microscopic Convexity Principle for Nonlinear Partial Differential Equations", "abstract": "We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form."}
{"category": "Math", "title": "Conformal metrics on $\\R^{2m}$ with constant Q-curvature", "abstract": "We study the conformal metrics on $\\R^{2m}$ with constant Q-curvature $Q$ having finite volume, particularly in the case $Q\\leq 0$. We show that when $Q<0$ such metrics exist in $\\R^{2m}$ if and only if $m>1$. Moreover we study their asymptotic behavior at infinity, in analogy with the case $Q>0$, which we treated in a recent paper. When Q=0, we show that such metrics have the form $e^{2p}g_{\\R^{2m}}$, where $p$ is a polynomial such that $2\\leq \\deg p\\leq 2m-2$ and $\\sup_{\\R^{2m}}p<+\\infty$. In dimension 4, such metrics are exactly the polynomials $p$ of degree 2 with $\\lim_{|x|\\to+\\infty}p(x)=-\\infty$."}
{"category": "Math", "title": "Construction of eternal solutions for a semilinear parabolic equation", "abstract": "Eternal solutions of parabolic equations (those which are defined for all time) are typically rather rare. For example, the heat equation has exactly one eternal solution -- the trivial solution. While solutions to the heat equation exist for all forward time, they cannot be extended backwards in time. Nonlinearities exasperate the situation somewhat, in that solutions may form singularities in both backward and forward time. However, semilinear parabolic equations can also support nontrivial eternal solutions. This article shows how nontrivial eternal solutions can be constructed for a semilinear equation that has at least two distinct equilibrium solutions. The resulting eternal solution is a heteroclinic orbit which connects the two given equilibria."}
{"category": "Math", "title": "Which powers of holomorphic functions are integrable?", "abstract": "We show that every limit of log canonical thresholds of n-variable functions is also a log canonical threshold of an (n-1)-variable function."}
{"category": "Math", "title": "Splitting Polytopes", "abstract": "A split of a polytope $P$ is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of $P$ admits a unique decomposition as a linear combination of weight functions corresponding to the splits of $P$ (with a split prime remainder). This generalizes a result of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite metric spaces. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with any polytope $P$, the split complex of $P$. Complete descriptions of the split complexes of all hypersimplices are obtained. Moreover, it is shown that these complexes arise as subcomplexes of the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)]."}
{"category": "Math", "title": "Square-free discriminants of Frobenius rings", "abstract": "Let $E$ be an elliptic curve over $\\Q$. It is well known that the ring of endomorphisms of $E_p$, the reduction of $E$ modulo a prime $p$ of ordinary reduction, is an order of the quadratic imaginary field $Q(\\pi_p)$ generated by the Frobenius element $\\pi_p$. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of $a_p^2-4p$, the discriminant of the characteristic polynomial of the Frobenius element. In this paper, we are interested in the function $\\pi_{E,r,h}(x)$ counting the number of primes $p$ up to $x$ such that $a_p^2-4p$ is square-free and in the congruence class $r$ modulo $h$. We give in this paper the precise asymptotic for $\\pi_{E,r,h}(x)$ when averaging over elliptic curves defined over the rationals, and we discuss the relation of this result with the Lang-Trotter conjecture, and with some other problems related to the curve modulo $p$."}
{"category": "Math", "title": "Finding and investigating exact spherical codes", "abstract": "In this paper we present the results of computer searches using a variation of an energy minimization algorithm used by Kottwitz for finding good spherical codes. We prove that exact codes exist by representing the inner products between the vectors as algebraic numbers. For selected interesting cases, we include detailed discussion of the configurations. Of particular interest are the 20-point code in $\\mathbb{R}^6$ and the 24-point code in $\\mathbb{R}^7$, which are both the union of two cross polytopes in parallel hyperplanes. Finally, we catalogue all of the codes we have found."}
{"category": "Math", "title": "Asymptotic normality of wavelet estimators of the memory parameter for linear processes", "abstract": "We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi-parametrically using wavelets from a sample $X_1,...,X_n$ of the process. We treat both the log-regression wavelet estimator and the wavelet Whittle estimator. We show that these estimators are asymptotically normal as the sample size $n\\to\\infty$ and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the empirical scalogram for linear processes, conveniently centered and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. In contrast with quadratic forms computed on the Fourier coefficients such as the periodogram, the scalogram involves correlations which do not vanish as the sample size $n\\to\\infty$."}
{"category": "Math", "title": "Central Limit Theorems for arrays of decimated linear processes", "abstract": "Linear processes are defined as a discrete-time convolution between a kernel and an infinite sequence of i.i.d. random variables. We modify this convolution by introducing decimation, that is, by stretching time accordingly. We then establish central limit theorems for arrays of squares of such decimated processes. These theorems are used to obtain the asymptotic behavior of estimators of the spectral density at specific frequencies. Another application, treated elsewhere, concerns the estimation of the long-memory parameter in time-series, using wavelets."}
{"category": "Math", "title": "Notes on character sheaves", "abstract": "In the first section we study a functor of Bezrukavnikov, Finkelberg and Ostrik defined on character sheaves; we compute it in a Grothendieck group taking weights into account. In the second section we enlarge the class of character sheaves to a larger class of simple perverse sheaves which behaves well under tensor product (unlike the character sheaves themselves)."}
{"category": "Math", "title": "On Certain Vanishing Identities For Gromov-Witten Invariants", "abstract": "In this paper, we study some vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in the case that the summation range is large compare to genus. In fact, in such cases, we can obtain a vanishing identity which is stronger than their conjectures. Moreover we will also prove their conjectures in low genus cases."}
{"category": "Math", "title": "Constructing Big Indecomposable modules", "abstract": "Let $R$ be local Noetherian ring of depth at least two. We prove that there are indecomposable $R$-modules which are free on the punctured spectrum of constant, arbitrarily large, rank."}
{"category": "Math", "title": "An exotic sphere with positive sectional curvature", "abstract": "We show that there is a metric on the Gromoll-Meyer sphere with positive sectional curvature."}
{"category": "Math", "title": "Extension Theorems for Paraboloids in the Finite Field Setting", "abstract": "In this paper we study the $L^p-L^r$ boundedness of the extension operators associated with paraboloids in vector spaces over finite fields.In higher even dimensions, we estimate the number of additive quadruples in the subset $E$ of the paraboloids, that is the number of quadruples $(x,y,z,w) \\in E^4$ with $x+y=z+w.$ As a result, in higher even dimensions, we improve upon the standard Tomas-Stein exponents which Mockenhaupt and Tao obtained for the boundedness of extension operators for paraboloids by estimating the decay of the Fourier transform of measures on paraboloids. In particular, we obtain the sharp $L^p-L^4$ bound up to endpoints in higher even dimensions. Moreover, we also study the $L^2-L^r$ estimates.In the case when -1 is not a square number in the underlying finite field, we also study the $L^p-L^r$ bound in higher odd dimensions.The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof."}
{"category": "Math", "title": "Probabilistic proofs of hook length formulas involving trees", "abstract": "Recently, Han discovered two formulas involving binary trees which have the interestig property that hooklengths appear as exponents. The purpose of this note is to give a probabilistic proof of one of Han's formulas. Yang has generalized Han's results to ordered trees. We show how the probabilistic approach can also be used in Yang's setting, as well as for a generalization of Han's formula in terms of certain infinite trees."}
{"category": "Math", "title": "Some Remarks on the Algebra of Bounded Dirichlet Series", "abstract": "We examine the algebra of all Dirichlet Series bounded on the right half plane. We consider the analogue of the Corona theorem in this setting, and show that it is false, i.e. the right half-plane is not dense in the maximal ideal space. We also prove some refinements of the Hille-Bohnenblust theorem, where both probabilistic and deterministic devices are used, and we show how the proof is carried out for each."}
{"category": "Math", "title": "Gradient-like observers for invariant dynamics on a Lie group", "abstract": "This paper proposes a design methodology for non-linear state observers for invariant kinematic systems posed on finite dimensional connected Lie groups, and studies the associated fundamental system structure. The concept of synchrony of two dynamical systems is specialised to systems on Lie groups. For invariant systems this leads to a general factorisation theorem of a nonlinear observer into a synchronous (internal model) term and an innovation term. The synchronous term is fully specified by the system model. We propose a design methodology for the innovation term based on gradient-like terms derived from invariant or non-invariant cost functions. The resulting nonlinear observers have strong (almost) global convergence properties and examples are used to demonstrate the relevance of the proposed approach."}
{"category": "Math", "title": "The Analogue of Dedekind Eta Functions for Calabi-Yau Manifilolds II. (Algebraic, Analytic Discriminants and the Analogue of Baily-Borel Compactification of the Moduli Space of CY Manifolds.)", "abstract": "In this paper we construct the analogue of Dedekind eta-function on the moduli space of polarized CY manifolds. We prove that the L-two norm of eta is the regularized determinants of the Laplacians of the CY metric on (0,1) forms. We construct the analogue of the Baily-Borel Compactification of the moduli space of polarized CY and prove that it has the same properties as the Baily-Borel compactification of the locally symmetric Hermitian spaces. We proved that the compactification constructed in the paper is the minimal."}
{"category": "Math", "title": "A note on non-reduced Picard schemes", "abstract": "The Picard scheme of a smooth curve and a smooth complex variety is reduced. In this note we discuss which classes of surfaces in terms of the Enriques-Kodaira classification can have non-reduced Picard schemes and whether there are restrictions on the characteristic of the ground field. It turns out that non-reduced Picard schemes are uncommon in Kodaira dimension $\\kappa\\leq0$, that this phenomenon can be bounded for $\\kappa=2$ (general type) and that it is as bad as can be in $\\kappa=1$."}
{"category": "Math", "title": "On the Eigenspaces of Lamplighter Random Walks and Percolation Clusters on Graphs", "abstract": "We show that the Plancherel measure of the lamplighter random walk on a graph coincides with the expected spectral measure of the absorbing random walk on the Bernoulli percolation clusters. In the subcritical regime the spectrum is pure point and we construct a complete orthonormal basis of finitely supported eigenfunctions."}
{"category": "Math", "title": "Cut Points and Diffusions in Random Environment", "abstract": "In this article we investigate the asymptotic behavior of a new class of multi-dimensional diffusions in random environment. We introduce cut times in the spirit of the work done by Bolthausen, Sznitman and Zeitouni, see [4], in the discrete setting providing a decoupling effect in the process. This allows us to take advantage of an ergodic structure to derive a strong law of large numbers with possibly vanishing limiting velocity and a central limit theorem under the quenched measure."}
{"category": "Math", "title": "Concentration of measure via approximated Brunn--Minkowski inequalities", "abstract": "We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces."}
{"category": "Math", "title": "Lyapunov control of a quantum particle in a decaying potential", "abstract": "A Lyapunov-based approach for the trajectory generation of an $N$-dimensional Schr{\\\"o}dinger equation in whole $\\RR^N$ is proposed. For the case of a quantum particle in an $N$-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization."}
{"category": "Math", "title": "A uniformly spread measure criterion", "abstract": "We prove that if all shifts of a measure in the Euclidean space are close in a sense to each other, then this measure is close to the Lebesgue one."}
{"category": "Math", "title": "On fake lens spaces with the fundamental group of order a power of 2", "abstract": "We present a classification of fake lens spaces of dimension greater or equal to 5 which have as fundamental group the cyclic group of order N = 2^K, in that we extend the results of Wall and others in the case N=2."}
{"category": "Math", "title": "Behavior near the extinction time in self-similar fragmentations I: the stable case", "abstract": "The stable fragmentation with index of self-similarity $\\alpha \\in [-1/2,0)$ is derived by looking at the masses of the subtrees formed by discarding the parts of a $(1 + \\alpha)^{-1}$--stable continuum random tree below height $t$, for $t \\geq 0$. We give a detailed limiting description of the distribution of such a fragmentation, $(F(t), t \\geq 0)$, as it approaches its time of extinction, $\\zeta$. In particular, we show that $t^{1/\\alpha}F((\\zeta - t)^+)$ converges in distribution as $t \\to 0$ to a non-trivial limit. In order to prove this, we go further and describe the limiting behavior of (a) an excursion of the stable height process (conditioned to have length 1) as it approaches its maximum; (b) the collection of open intervals where the excursion is above a certain level and (c) the ranked sequence of lengths of these intervals. Our principal tool is excursion theory. We also consider the last fragment to disappear and show that, with the same time and space scalings, it has a limiting distribution given in terms of a certain size-biased version of the law of $\\zeta$."}
{"category": "Math", "title": "On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities", "abstract": "Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. Additionally, our analysis of Sobolev type inequalities for two-component mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincar\\'e constant of a two-component mixture may remain bounded as the mixture proportion goes to 0 or 1 while the logarithmic Sobolev constant may surprisingly blow up. This counter-intuitive result is not reducible to support disconnections, and appears as a reminiscence of the variance-entropy comparison on the two-point space. As far as mixtures are concerned, the logarithmic Sobolev inequality is less stable than the Poincar\\'e inequality and the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-component mixtures. This work leads to many open questions."}
{"category": "Math", "title": "Fibonacci Identities and Graph Colorings", "abstract": "We generalize both the Fibonacci and Lucas numbers to the context of graph colorings, and prove some identities involving these numbers. As a corollary we obtain new proofs of some known identities involving Fibonacci numbers such as \\[F_{r+s+t} = F_{r+1}F_{s+1}F_{t+1} + F_r F_s F_t - F_{r-1}F_{s-1}F_{t-1}.\\]"}
{"category": "Math", "title": "One-dimensional Schr\\\"odinger operators with singular periodic potentials", "abstract": "We study the one-dimensional Schr\\\"odinger operators $$ S(q)u:=-u\"+q(x)u,\\quad u\\in \\mathrm{Dom}\\left(S(q)\\right), $$ with $1$-periodic real-valued singular potentials $q(x)\\in H_{\\operatorname{per}}^{-1}(\\mathbb{R},\\mathbb{R})$ on the Hilbert space $L_{2}\\left(\\mathbb{R}\\right)$. We show equivalence of five basic definitions of the operators $S(q)$ and prove that they are self-adjoint. A new proof of continuity of the spectrum of the operators $S(q)$ is found. Endpoints of spectrum gaps are precisely described."}
{"category": "Math", "title": "Geodesics on weighted projective spaces", "abstract": "We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can \"hear\" the weights of a weighted projective space."}
{"category": "Math", "title": "Admissible unitary completions of locally $Q_p$-rational representations of $GL_2(F)$", "abstract": "Let $F$ be a finite extension of $Q_p$, $p>2$. We construct admissible unitary completions of certain representations of $GL_2(F)$ on $L$-vector spaces, where $L$ is a finite extension of $F$. When $F=Q_p$ using the results of Berger, Breuil and Colmez we obtain some results about lifting 2-dimensional mod $p$ representations of the absolute Galois group of $Q_p$ to crystabelline representations with given Hodge-Tate weights."}
{"category": "Math", "title": "Bounds on Fake Weighted Projective Space", "abstract": "A fake weighted projective space X is a Q-factorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (\\lambda_0,...,\\lambda_n). We see how the singularities of P(\\lambda_0,...,\\lambda_n) influence the singularities of X, and how the weights bound the number of possible fake weighted projective spaces for a fixed dimension. Finally, we present an upper bound on the ratios \\lambda_j/\\sum\\lambda_i if we wish X to have only terminal (or canonical) singularities."}
{"category": "Math", "title": "Coalescent processes in subdivided populations subject to recurrent mass extinctions", "abstract": "We investigate the infinitely many demes limit of the genealogy of a sample of individuals from a subdivided population subject to sporadic mass extinction events. By exploiting a separation of timescales property of Wright's island model, we show that as the number of demes tends to infinity the limiting form of the genealogy can be described in terms of the alternation of instantaneous 'scattering' phases dominated by local demographic processes, and extended 'collecting' phases dominated by global processes. When extinction and recolonization events are local, this genealogy is given by Kingman's coalescent and the scattering phase influences only the overall rate of the process. In contrast, if the vacant demes left by a mass extinction event can be recolonized by individuals emerging from a small number of demes, then the limiting genealogy is a colaescent with simultaneous multiple mergers. In this case, the details of the within-deme population dynamics influence not only the overall rate of the coalescent process, but also the statistics of the complex mergers that can occur within sample genealogies. This study gives some insight into the genealogical consequences of mass extinction in structured populations."}
{"category": "Math", "title": "Category theorems for stable operators on Hilbert spaces", "abstract": "We discuss the two closely related, but different concepts of weak and almost weak stability for the powers of a contraction on a separable Hilbert space. Extending Halmos' and Rohlin's theorems in ergodic theory as a model, we show that the set of all weakly stable contractions is of first category while the set of all almost weakly stable contractions is of second category and is residual. Analogous statements for unitary and isometric operators are also proved."}
{"category": "Math", "title": "Spectral analysis of finite dimensional algebras and singularities", "abstract": "We give a summary on spectral techniques for finite dimensional algebras and study its link to singularity theory. In particular, we offer a contribution to the categorification of the Milnor lattice of two-dimensional singularities through triangulated categories naturally associated with a weighted projective line."}
{"category": "Math", "title": "Prym varieties of cyclic coverings", "abstract": "The Prym map of type (g,n,r) associates to every cyclic covering of degree n of a curve of genus g, ramified at a reduced divisor of degree r, the corresponding Prym variety. We show that the corresponding map of moduli spaces is generically finite in most cases. From this we deduce the dimension of the image of the Prym map."}
{"category": "Math", "title": "Width and flow of hypersurfaces by curvature functions", "abstract": "We give a bound on the extinction time for a compact, strictly convex hypersurface in R^{n+1} evolving by a geometric flow where the velocity is given in terms of the curvature. This result generalizes a theorem of Colding and Minicozzi for mean curvature flow solutions to a wider class of flows studied by Ben Andrews. In the proof, we use the concept of the width of a hypersurface, introduced by Colding and Minicozzi. We also extend the result to 2-convex hypersurfaces, using the 2-width."}
{"category": "Math", "title": "Category theorems for stable semigroups", "abstract": "Inspired by the classical category theorems of Halmos and Rohlin for the discrete measure preserving transformations, we prove analogous results in the abstract setting of unitary and isometric C_0-semigroups on a separable Hilbert space. More presicely, we show that the set of all weakly stable unitary groups (isometric semigroups) is of first category, while the set of all almost weakly stable unitary groups (isometric semigroups) is residual for an appropriate topology."}
{"category": "Math", "title": "Selecting universities: personal preference and rankings", "abstract": "Polyhedral geometry can be used to quantitatively assess the dependence of rankings on personal preference, and provides a tool for both students and universities to assess US News and World Report rankings."}
{"category": "Math", "title": "On the weak analogue of the Trotter-Kato theorem", "abstract": "In the Trotter-Kato approximation theorem for C_0-semigroups on Banach spaces, we replace the strong by the weak operator topology and discuss the validity of the relevant implications."}
{"category": "Math", "title": "Some New Random Field Tools for Spatial Analysis", "abstract": "This is a brief review, in relatively non-technical terms, of recent advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric characteristics of excursion sets of random fields. As well as a review of the theory, we provide brief descriptions of some of the more interesting applications."}
{"category": "Math", "title": "Asymptotically Affine and Asymptotically Conformal Circle Endomorphisms", "abstract": "We show that every uniformly asymptotically affine circle endomorphism has a uniformly asymptotically conformal extension."}
{"category": "Math", "title": "Cluster categories for algebras of global dimension 2 and quivers with potential", "abstract": "Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\\leq 2$. We construct a triangulated category $\\Cc_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When $\\Cc_A$ is $\\Hom$-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schr{\\\"o}er and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluster category $\\Cc_{(Q,W)}$ associated to a quiver with potential $(Q,W)$. When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra $\\Jj(Q,W)$."}
{"category": "Math", "title": "Nonlinear dynamics of phase separation in ultra-thin films", "abstract": "We present a long-wavelength approximation to the Navier-Stokes Cahn-Hilliard equations to describe phase separation in thin films. The equations we derive underscore the coupled behaviour of free-surface variations and phase separation. Since we are interested in the long-time behaviour of the phase-separating fluid, we restrict our attention to films that do not rupture. To do this, we introduce a regularising Van der Waals potential. We analyse the resulting fourth-order equations by constructing a solution as the limit of a Galerkin approximation, and obtain existence and regularity results. In our analysis, we find a nonzero lower bound for the height of the film, which precludes the possibility of rupture. The lower bound depends on the parameters of the problem, and we compare this dependence with numerical simulations. We find that while the theoretical lower bound is crucial to the construction of a smooth, unique solution to the PDEs, it is not sufficiently sharp to represent accurately the parametric dependence of the observed dips in free-surface height."}
{"category": "Math", "title": "Weakly and almost weakly stable C_0-semigroups", "abstract": "In this paper we survey results concerning the asymptotic properties of C_0-semigroups on Banach spaces with respect to the weak operator topology. The property \"no eigenvalues of the generator on the imaginary axis\" is equivalent to weak stability for most time values; a phenomenon called \"almost weak stability\". Further, sufficient conditions actually implying weak stability are also given. By several examples we explain weak and almost weak stability and illustrate the fundamental difference between them. Many historical and bibliographical remarks position the material in the literature. We conclude the paper with some open questions and comments."}
{"category": "Math", "title": "Some results on the crystal commutor and affine sl(n) crystals", "abstract": "There are two parts to this work, which are largely independent. The first consists of a series of results concerning the crystal commutor of Henriques and Kamnitzer. We first describe the relationship between the crystal commutor and Drinfeld's unitarized R-matrix. We then give a new definition for the crystal commutor, which makes sense for any symmetrizable Kac-Moody algebra. We show that this new definition agrees with A. Henriques and J. Kamnitzer's definition in the finite type case, but we cannot prove our commutor remains a coboundary structure in other cases. Next, we extend these ideas to give a new formula for the standard R-matrix. In the second part, we define three combinatorial models for affine sl(n) crystals. These are parameterized by partitions, configurations of beads on an \"abacus\", and cylindric plane partitions, respectively. Our models are reducible, but we can identify an irreducible subcrystal corresponding to any dominant integral highest weight. Cylindric plane partitions actually parameterize a basis for an irreducible representation of affine gl(n). This allows us to calculate the partition function for a system of random cylindric plane partitions first studied by A. Borodin. We also observe a form of rank-level duality, originally due to I. Frenkel."}
{"category": "Math", "title": "A nonextension result on the spectral metric", "abstract": "The spectral metric, defined by Schwarz and Oh using Floer-theoretical method, is a bi-invariant metric on the Hamiltonian diffeomorphism group. We show in this note that for certain symplectic manifolds, this metric can not be extended to a bi-invariant metric on the full group of symplectomorphisms. We also study the bounded isometry conjecture of Lalonde and Polterovich in the context of the spectral metric. In particular, we show that the conjecture holds for the torus with all linear symplectic forms."}
{"category": "Math", "title": "On the Markov sequence problem for Jacobi polynomials", "abstract": "We give a simple and entirely elementary proof of Gasper's theorem on the Markov sequence problem for Jacobi polynomials. It is based on the spectral analysis of an operator that arises in the study of a probabilistic model of colliding molecules introduced by Marc Kac. In the process, we obtain some new integral formulas for ratios of Jacobi polynomials that generalize Gasper's product formula and a well known formula of Koornwinder."}
{"category": "Math", "title": "Two Theorems on the structure of Pythagorean triples and some diophantine consequences", "abstract": "Even though four theorems are actually proved in this paper, two are the main ones,Teorems 1 and 3. In Theorem 1 we show that if a and be are odd squarefree positive integers satisfying certain quadratic residue conditions; then there exists no primitive Pythagorean triangle one of whose leglengths is equal to a times an integer square, while the other leglength is equal to b times a perfect square. The family of all such pairs (a,b) is slightly complicated in its description. A subfamily of the said family consists of pairs (a,b), with a being congruent to 1, while b being congruent to 5 modulo8; and also with both a and b being primes, and with a being a quadratic nonresidue ofb(and so by the quadratic reciprocity law, b also being a nonresidue of a). Theorem 3 is similar in nature, but less complicated in its hypothesis. It states that if p and q are primes, both congruent to 1 modulo4, and one of them being a quadratic nonresidue of the other.Then the diophantine equation, p^2x^4 + q^2y^4 = z^2, Has no solutions in positive integers x, y, and z, satisfying (px, qy)=1."}
{"category": "Math", "title": "Uniform rectifiability, Calderon-Zygmund operators with odd kernel, and quasiorthogonality", "abstract": "In this paper we study some questions in connection with uniform rectifiability and the $L^2$ boundedness of Calderon-Zygmund operators. We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the Jones' $\\beta$ numbers. We also use these new coefficients to prove that n-dimensional Calderon-Zygmund operators with odd kernel of type $C^2$ are bounded in $L^2(\\mu)$ if $\\mu$ is an n-dimensional uniformly rectifiable measure."}
{"category": "Math", "title": "Fixed points in non-invariant plane continua", "abstract": "If $f:[a,b]\\to \\mathbb{R}$, with $a<b$, is continuous and such that $a$ and $b$ are mapped in opposite directions by $f$, then $f$ has a fixed point in $I$. Suppose that $f:\\mathbb{C}\\to\\mathbb{C}$ is map and $X$ is a continuum. We extend the above for certain continuous maps of dendrites $X\\to D, X\\subset D$ and for positively oriented maps $f:X\\to \\mathbb{C}, X\\subset \\mathbb{C}$ with the continuum $X$ not necessarily invariant. Then we show that in certain cases a holomorphic map $f:\\mathbb{C}\\to\\mathbb{C}$ must have a fixed point $a$ in a continuum $X$ so that either $a\\in \\mathrm{Int}(X)$ or $f$ exhibits rotation at $a$."}
{"category": "Math", "title": "Eigenvalues of sums of pseudo-Hermitian matrices", "abstract": "We study analogues of classical inequalities for the eigenvalues of sums of pseudo-Hermitian matrices."}
{"category": "Math", "title": "Elliptic Dedekind Domains Revisited", "abstract": "We give an affirmative answer to a 1976 question of M. Rosen: every abelian group is isomorphic to the class group of an elliptic Dedekind domain R. We can choose R to be the integral closure of a PID in a separable quadratic field extension. In particular, this yields new and -- we feel -- simpler proofs of theorems of L. Claborn and C.R. Leedham-Green."}
{"category": "Math", "title": "Exact and non-stiff sampling of highly oscillatory systems: an implicit mass-matrix penalization approach", "abstract": "We propose and analyze an implicit mass-matrix penalization (IMMP) technique which enables efficient and exact sampling of the (Boltzmann/Gibbs) canonical distribution associated to Hamiltonian systems with fast degrees of freedom (fDOFs). The penalty parameters enable arbitrary tuning of the timescale for the selected fDOFs, and the method is interpreted as an interpolation between the exact Hamiltonian dynamics and the dynamics with infinitely slow fDOFs (equivalent to geometrically corrected rigid constraints). This property translates in the associated numerical methods into a tunable trade-off between stability and dynamical modification. The penalization is based on an extended Hamiltonian with artificial constraints associated with each fDOF. By construction, the resulting dynamics is statistically exact with respect to the canonical distribution in position variables. The algorithms can be easily implemented with standard geometric integrators with algebraic constraints given by the expected fDOFs, and has no additional complexity in terms of enforcing the constraint and force evaluations. The method is demonstrated on a high dimensional system with non-convex interactions. Prescribing the macroscopic dynamical timescale, it is shown that the IMMP method increases the time-step stability region with a gain that grows linearly with the size of the system. The latter property, as well as consistency of the macroscopic dynamics of the IMMP method is proved rigorously for linear interactions. Finally, when a large stiffness parameter is introduced, the IMMP method can be tuned to be asymptotically stable, converging towards the heuristically expected Markovian effective dynamics on the slow manifold."}
{"category": "Math", "title": "On a degenerate parabolic equation arising in pricing of Asian options", "abstract": "We study a certain one dimensional, degenerate parabolic partial differential equation with a boundary condition which arises in pricing of Asian options. Due to degeneracy of the partial differential operator and the non-smooth boundary condition, regularity of the generalized solution of such a problem remained unclear. We prove that the generalized solution of the problem is indeed a classical solution."}
{"category": "Math", "title": "A Chebyshev criterion for Abelian integrals", "abstract": "We present a criterion that provides an easy sufficient condition in order that a collection of Abelian integrals has the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced."}
{"category": "Math", "title": "f-Vectors of 3-Manifolds", "abstract": "In 1970, Walkup completely described the set of $f$-vectors for the four 3-manifolds $S^3$, $S^2 twist S^1$, $S^2 \\times S^1$, and $RP^3$. We improve one of Walkup's main restricting inequalities on the set of $f$-vectors of 3-manifolds. As a consequence of a bound by Novik and Swartz, we also derive a new lower bound on the number of vertices that are needed for a combinatorial $d$-manifold in terms of its $\\beta_1$-coefficient, which partially settles a conjecture of K\\\"uhnel. Enumerative results and a search for small triangulations with bistellar flips allow us, in combination with the new bounds, to completely determine the set of $f$-vectors for twenty further 3-manifolds, that is, for the connected sums of sphere bundles $(S^2 \\times S^1)^{# k}$ and twisted sphere bundles $(S^2 twist S^1)^{# k}$, where $k=2,3,4,5,6,7,8,10,11,14$. For many more 3-manifolds of different geometric types we provide small triangulations and a partial description of their set of $f$-vectors. Moreover, we show that the 3-manifold $RP^3 # RP^3$ has (at least) two different minimal $g$-vectors."}
{"category": "Math", "title": "A cellular algebra with certain idempotent decomposition", "abstract": "For a cellular algebra $\\A$ with a cellular basis $\\ZC$, we consider a decomposition of the unit element $1_\\A$ into orthogonal idempotents (not necessary primitive) satisfying some conditions. By using this decomposition, the cellular basis $\\ZC$ can be partitioned into some pieces with good properties. Then by using a certain map $\\a$, we give a coarse partition of $\\ZC$ whose refinement is the original partition. We construct a Levi type subalgebra $\\aA$ of $\\A$ and its quotient algebra $\\oA$, and also construct a parabolic type subalgebra $\\tA$ of $\\A$, which contains $\\aA$ with respect to the map $\\a$. Then, we study the relation of standard modules, simple modules and decomposition numbers among these algebras. Finally, we study the relationship of blocks among these algebras."}
{"category": "Math", "title": "The real analytic Feigenbaum-Coullet-Tresser attractor in the disk", "abstract": "We consider a real analytic diffeomorphism $\\psi_0$ on a n-dimensional disk D, n >= 2, exhibiting a Feigenbaum-Coullet-Tresser (F.C.T.) attractor, being far, in the standard topology of the real analytic diffeomorphism space C(D), from the standard F.C.T. map $\\phi_0$ fixed by the double renormalization. We prove that $\\psi_0$ persists along a codimension-one manifold M \\subset C(D), and that it is the bifurcating map along any one-parameter family in $C(D)$ transversal to M, from diffeomorphisms attracted to sinks, to those which exhibit chaos. The main tool in the proofs is a theorem of Functional Analysis, which we state and prove in this paper, characterizing the existence of codimension one submanifolds in any abstract functional Banach space."}
{"category": "Math", "title": "Essential dimension of Hermitian spaces", "abstract": "Given an hermitian space we compute its essential dimension, Chow motive and prove its incompressibility in certain dimensions."}
{"category": "Math", "title": "Note on 2-rational fields", "abstract": "We compute the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field."}
{"category": "Math", "title": "Geometric Approach to Pontryagin's Maximum Principle", "abstract": "Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self\\textendash{}contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry."}
{"category": "Math", "title": "Autoregressive Process Modeling via the Lasso Procedure", "abstract": "The Lasso is a popular model selection and estimation procedure for linear models that enjoys nice theoretical properties. In this paper, we study the Lasso estimator for fitting autoregressive time series models. We adopt a double asymptotic framework where the maximal lag may increase with the sample size. We derive theoretical results establishing various types of consistency. In particular, we derive conditions under which the Lasso estimator for the autoregressive coefficients is model selection consistent, estimation consistent and prediction consistent. Simulation study results are reported."}
{"category": "Math", "title": "The plane fixed point problem", "abstract": "In this paper we present proofs of basic results, including those developed so far by H. Bell, for the plane fixed point problem. Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. We develop a prime end theory through hyperbolic chords in maximal round balls contained in the complement of a non-separating plane continuum $X$. We define the concept of an {\\em outchannel} for a fixed point free map which carries the boundary of $X$ minimally into itself and prove that such a map has a \\emph{unique} outchannel, and that outchannel must have variation $=-1$. We also extend Bell's linchpin theorem for a foliation of a simply connected domain, by closed convex subsets, to arbitrary domains in the sphere. We introduce the notion of an oriented map of the plane. We show that the perfect oriented maps of the plane coincide with confluent (that is composition of monotone and open) perfect maps of the plane. We obtain a fixed point theorem for positively oriented, perfect maps of the plane. This generalizes results announced by Bell in 1982 (see also \\cite{akis99}). It follows that if $X$ is invariant under an oriented map $f$, then $f$ has a point of period at most two in $X$."}
{"category": "Math", "title": "Direct minimization for calculating invariant subspaces in density functional computations of the electronic structure", "abstract": "We analyse three related preconditioned steepest descent algorithms, which are partially popular in Hartree-Fock and Kohn-Sham theory as well as invariant subspace computations, from the viewpoint of minimization of the corresponding functionals, constrained by orthogonality conditions. We exploit the geometry of the of the admissible manifold, i.e. the invariance with respect to unitary transformations, to reformulate the problem on the Grassmann manifold as the admissible set. We then prove asymptotical linear convergence of the algorithms under the condition that the Hessian of the corresponding Lagrangian is elliptic on the tangent space of the Grassmann manifold at the minimizer."}
{"category": "Math", "title": "Exactness of martingale approximation and the central limit theorem", "abstract": "The article is showing sharpness of central limit theorems of Kipnis and Varadhan, Derriennic and Lin, Maxwell and Woodroofe. In the case of the CLT of Derriennic and Lin (for Markov chains with a normal operator) it is shown that the assumption of normality cannot be relaxed. In the case of the CLT of Maxwell and Woodroofe, the example of Peligrad and Utev is improved in the sense of getting a convergence to different laws."}
{"category": "Math", "title": "Fake weighted projective spaces", "abstract": "We define fake weighted projective spaces as a generalisation of weighted projective spaces. We introduce the notions of fundamental group in codimension 1 and of universal covering in codimension 1. We prove that for every fake weighted projective space its universal cover in codimension 1 is a weighted projective space."}
{"category": "Math", "title": "Large Selmer groups over number fields", "abstract": "Let p be a prime number and M a quadratic number field, M not equal to Q(\\sqrt{p}) if p is congruent to 1 modulo 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D_{2p} and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least p^d."}
{"category": "Math", "title": "Twisted Alexander polynomials detect fibered 3-manifolds", "abstract": "A classical result in knot theory says that the Alexander polynomial of a fibered knot is monic and that its degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3-manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S^1 x N^3 admits a symplectic structure, then N fibers over S^1. In fact we will completely determine the symplectic cone of S^1 x N in terms of the fibered faces of the Thurston norm ball of N."}
{"category": "Math", "title": "Schur-Weyl duality over finite fields", "abstract": "We prove a version of Schur--Weyl duality over finite fields. We prove that for any field $k$, if $k$ has at least $r+1$ elements, then Schur--Weyl duality holds for the $r$th tensor power of a finite dimensional vector space $V$. Moreover, if the dimension of $V$ is at least $r+1$, the natural map $k\\Sym_r \\to End\\_{GL(V)}(V^{\\otimes r})$ is an isomorphism. This isomorphism may fail if $\\dim_k V$ is not strictly larger than $r$."}
{"category": "Math", "title": "The embedded contact homology index revisited", "abstract": "Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the relative grading on ECH; the ECH differential counts holomorphic curves in the symplectization of Y whose relative homology classes have ECH index 1. A known index inequality implies that such curves are (mostly) embedded and satisfy some additional constraints. In this paper we prove four new results about the ECH index. First, we refine the relative grading on ECH to an absolute grading, which associates to each union of Reeb orbits a homotopy class of oriented 2-plane fields on Y. Second, we extend the ECH index inequality to symplectic cobordisms between three-manifolds with Hamiltonian structures, and simplify the proof. Third, we establish general inequalities on the ECH index of unions and multiple covers of holomorphic curves in cobordisms. Finally, we define a new relative filtration on ECH, or any other kind of contact homology of a contact 3-manifold, which is similar to the ECH index and related to the Euler characteristic of holomorphic curves. This does not give new topological invariants except possibly in special situations, but it is a useful computational tool."}
{"category": "Math", "title": "Lattice polytopes cut out by root systems and the Koszul property", "abstract": "We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties."}
{"category": "Math", "title": "Exact generating function for 2-convex polygons", "abstract": "Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their `concavity index', $m$. Such polygons are called \\emph{$m$-convex} polygons and are characterised by having up to $m$ indentations in their perimeter. We first describe how we conjectured the (isotropic) generating function for the case $m=2$ using a numerical procedure based on series expansions. We then proceed to prove this result for the more general case of the full anisotropic generating function, in which steps in the $x$ and $y$ direction are distinguished. In so doing, we develop tools that would allow for the case $m > 2$ to be studied. %In our proof we use a `divide and conquer' approach, factorising 2-convex %polygons by extending a line along the base of its indents. We then use %the inclusion-exclusion principle, the Hadamard product and extensions to %known methods to derive the generating functions for each case."}
{"category": "Math", "title": "Some Topics Related to Bergman Kernel", "abstract": "Actually we will discuss some topics related to Bergman kernel on Cartan- Hartogs domain."}
{"category": "Math", "title": "On higher-power moments of $\\Delta(x)$(II)", "abstract": "Let $\\Delta(x)$ be the error term of the Dirichlet divisor problem. The asymptotic formula of the integral $\\int_1^T\\Delta^k(x)dx$ is established for any integer $3\\leq k\\leq 9$ by an unified method. Similar results are also established for some other well-known error terms in the analytic number theory."}
{"category": "Math", "title": "Two kinds of hook length formulas for complete $m$-ary trees", "abstract": "In this paper, we define two kinds of hook length for internal vertices of complete $m$-ary trees, and deduce their corresponding hook length formulas, which generalize the main results obtained by Du and Liu."}
{"category": "Math", "title": "Bell Polynomials and $k$-generalized Dyck Paths", "abstract": "A {\\em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\\mathbb{Z}\\times\\mathbb{Z}$ consisting of horizontal-steps $(k, 0)$ for a given integer $k\\geq 0$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis. The present paper studies three kinds of statistics on $k$-generalized Dyck paths: \"number of $u$-segments\", \"number of internal $u$-segments\" and \"number of $(u,h)$-segments\". The Lagrange inversion formula is used to represent the generating function for the number of $k$-generalized Dyck paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. Many important special cases are considered leading to several surprising observations. Moreover, enumeration results related to $u$-segments and $(u,h)$-segments are also established, which produce many new combinatorial identities, and specially, two new expressions for Catalan numbers."}
{"category": "Math", "title": "Identities involving Narayana polynomials and Catalan numbers", "abstract": "We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities."}
{"category": "Math", "title": "On the number of combinations without certain separations", "abstract": "In this paper we enumerate the number of ways of selecting $k$ objects from $n$ objects arrayed in a line such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects and provide three different formulas when $m,p\\geq 1$ and $n\\geq pm(k-1)$. Also, we prove that the number of ways of selecting $k$ objects from $n$ objects arrayed in a circle such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects is given by $\\frac{n}{n-pk}\\binom{n-pk}{k}$, where $m,p\\geq 1$ and $n\\geq mpk+1$."}
{"category": "Math", "title": "The Star of David Rule", "abstract": "In this note, a new concept called {\\em $SDR$-matrix} is proposed, which is an infinite lower triangular matrix obeying the generalized rule of David star. Some basic properties of $SDR$-matrices are discussed and two conjectures on $SDR$-matrices are presented, one of which states that if a matrix is a $SDR$-matrix, then so is its matrix inverse (if exists)."}
{"category": "Math", "title": "A simple bijection between binary trees and colored ternary trees", "abstract": "In this short note, we first present a simple bijection between binary trees and colored ternary trees and then derive a new identity related to generalized Catalan numbers."}
{"category": "Math", "title": "Pattern Avoidance in Generalized Non-crossing Trees", "abstract": "In this paper, the problem of pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding patterns of length one and two are obtained. Lagrange inversion formula is used to obtain the explicit formulas for some special cases. Bijection is also established between generalized non-crossing trees with special pattern avoidance and the little Schr\\\"{o}der paths."}
{"category": "Math", "title": "The $\\square_b$ Heat Equation and Multipliers via the Wave Equation", "abstract": "Recently, Nagel and Stein studied the $\\square_b$-heat equation, where $\\square_b$ is the Kohn Laplacian on the boundary of a weakly-pseudoconvex domain of finite type in $\\C^2$. They showed that the Schwartz kernel of $e^{-t\\square_b}$ satisfies good \"off-diagonal\" estimates, while that of $e^{-t\\square_b}-\\pi$ satisfies good \"on-diagonal\" estimates, where $\\pi$ is the Szeg\\\"o projection. We offer a simple proof of these results, which easily generalizes to other, similar situations. Our methods involve adapting the well-known relationship between the heat equation and the finite propagation speed of the wave equation to this situation. In addition, we apply these methods to study multipliers of the form $m\\l(\\square_b\\r)$. In particular, we show that $m\\l(\\square_b\\r)$ is an NIS operator, where $m$ satisfies an appropriate Mihlin-H\\\"ormander condition."}
{"category": "Math", "title": "A note on extension of sliced average variance estimation to multivariate regression", "abstract": "This paper has been withdrawn because the Editor of Electronical Journal of Statistics declined the paper."}
{"category": "Math", "title": "Hierarchical Models, Marginal Polytopes, and Linear Codes", "abstract": "In this paper, we explore a connection between binary hierarchical models, their marginal polytopes and codeword polytopes, the convex hulls of linear codes. The class of linear codes that are realizable by hierarchical models is determined. We classify all full dimensional polytopes with the property that their vertices form a linear code and give an algorithm that determines them."}
{"category": "Math", "title": "Genus two curves with quaternionic multiplication and modular jacobian", "abstract": "We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces $A_f$ with quaternionic multiplication attached to a normalized newform $f$ without complex multiplication. We include an example of $A_f$ with quaternionic multiplication for which we find numerically a curve $C$ whose Jacobian is $A_f$ up to numerical approximation, and we prove that it has quaternionic multiplication and is isogenous to $A_f$."}
{"category": "Math", "title": "Models of some simple modular Lie superalgebras", "abstract": "Models of the exceptional simple modular Lie superalgebras in characteristic $p\\geq 3$, that have appeared in the classification due to Bouarroudj, Grozman and Leites of the Lie superalgebras with indecomposable symmetrizable Cartan matrices, are provided. The models relate these exceptional Lie superalgebras to some low dimensional nonassociative algebraic systems."}
{"category": "Math", "title": "Tropical Resultants for Curves and Stable Intersection", "abstract": "We introduce the notion of resultant of two planar curves in the tropical geometry framework. We prove that the tropicalization of the algebraic resultant can be used to compute the stable intersection of two tropical plane curves. It is shown that, for two generic preimages of the curves to an algebraic framework, their intersection projects exactly onto the stable intersection of the curves. It is also given sufficient conditions for such a generality in terms of the residual coefficients of the algebraic coefficients of defining equations of the curves."}
{"category": "Math", "title": "The Finite Horizon Optimal Multi-Modes Switching Problem: the Viscosity Solution Approach", "abstract": "In this paper we show existence and uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. This problem is in relation with the valuation of firms in a financial market."}
{"category": "Math", "title": "Canonical metrics on Hartogs domains", "abstract": "An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural Kaehler metric $g_F$. This paper contains two results. In the first one we prove that if $g_F$ is an extremal Kaehler metric then $(D_F, g_F)$ is holomorphically isometric to an open subset of the $n$-dimensional complex hyperbolic space. In the second one we prove the same assertion under the assumption that there exists a real holomorphic vector field $X$ on $D_F$ such that $(g_F, X)$ is a Kaehler-Ricci soliton."}
{"category": "Math", "title": "The Fujita Exponent for Semilinear Heat Equations with Quadratically Decaying Potential or in an Exterior Domain", "abstract": "Consider the equation u_t=\\Delta u-Vu +au^p \\text{in} R^n\\times (0,T); u(x,0)=\\phi(x)\\gneq0, \\text{in} R^n, where $p>1$, $n\\ge2$, $T\\in(0,\\infty]$, $V(x)\\sim\\frac\\omega{|x|^2}$ as $|x|\\to\\infty$, for some $\\omega\\neq0$, and $a(x)$ is on the order $|x|^m$ as $|x|\\to\\infty$, for some $m\\in (-\\infty,\\infty)$. A solution to the above equation is called global if $T=\\infty$. Under some additional technical conditions, we calculate a critical exponent $p^*$ such that global solutions exist for $p>p^*$, while for $1<p\\le p^*$, all solutions blow up in finite time. We also show that when $V\\equiv0$, the blow-up/global solution dichotomy for \\eqref{abstract} coincides with that for the corresponding problem in an exterior domain with the Dirichlet boundary condition, including the case in which $p$ is equal to the critical exponent."}
{"category": "Math", "title": "A trace formula for varieties over a discretely valued field", "abstract": "We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety $X$ over a complete discretely valued field $K$ with perfect residue field $k$. If $K$ has characteristic zero, we extend the definition to arbitrary $K$-varieties using Bittner's presentation of the Grothendieck ring and a process of N\\'eron smoothening of pairs of varieties. The motivic Serre invariant can be considered as a measure for the set of unramified points on $X$. Under certain tameness conditions, it admits a cohomological interpretation by means of a trace formula. In the curve case, we use T. Saito's geometric criterion for cohomological tameness to obtain more detailed results. We discuss some applications to Weil-Ch\\^atelet groups, Chow motives, and the structure of the Grothendieck ring."}
{"category": "Math", "title": "Classification of bijections between 321- and 132-avoiding permutations", "abstract": "It is well-known, and was first established by Knuth in 1969, that the number of 321-avoiding permutations is equal to that of 132-avoiding permutations. In the literature one can find many subsequent bijective proofs of this fact. It turns out that some of the published bijections can easily be obtained from others. In this paper we describe all bijections we were able to find in the literature and show how they are related to each other via ``trivial'' bijections. We classify the bijections according to statistics preserved (from a fixed, but large, set of statistics), obtaining substantial extensions of known results. Thus, we give a comprehensive survey and a systematic analysis of these bijections. We also give a recursive description of the algorithmic bijection given by Richards in 1988 (combined with a bijection by Knuth from 1969). This bijection is equivalent to the celebrated bijection of Simion and Schmidt (1985), as well as to the bijection given by Krattenthaler in 2001, and it respects 11 statistics--the largest number of statistics any of the bijections respects."}
{"category": "Math", "title": "Hydrodynamic limit of particle systems with long jumps", "abstract": "We consider some interacting particle processes with long-range dynamics: the zero-range and exclusion processes with long jumps. We prove that the hydrodynamic limit of these processes corresponds to a (possibly non-linear) fractional heat equation. The scaling in this case is superdiffusive. In addition, we discuss a central limit theorem for a tagged particle on the zero-range process and existence and uniqueness of solutions of the Cauchy problem for the fractional heat equation."}
{"category": "Math", "title": "Small deviations of general L\\'{e}vy processes", "abstract": "We study the small deviation problem $\\log\\mathbb{P}(\\sup_{t\\in[0,1]}|X_t|\\leq\\varepsilon)$, as $\\varepsilon\\to0$, for general L\\'{e}vy processes $X$. The techniques enable us to determine the asymptotic rate for general real-valued L\\'{e}vy processes, which we demonstrate with many examples. As a particular consequence, we show that a L\\'{e}vy process with nonvanishing Gaussian component has the same (strong) asymptotic small deviation rate as the corresponding Brownian motion."}
{"category": "Math", "title": "A degree inequality for Lie algebras with a regular Poisson semi-center", "abstract": "For Lie algebras whose Poisson semi-center is a polynomial ring we give a bound for the sum of the degrees of the generating semi-invariants. This bound was previously known in many special cases."}
{"category": "Math", "title": "Variations on a theme of Runge: effective determination of integral points on certain varieties", "abstract": "We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge's theorem valid for higher-dimensional varieties, generalizing a uniform version of Runge's theorem due to Bombieri. We then take up the study of how Runge's method may be expanded by taking advantage of certain coverings. We prove both a result for arbitrary curves and a more explicit result for superelliptic curves. As an application of our method, we solve certain equations involving squares in products of terms in an arithmetic progression."}
{"category": "Math", "title": "A note on the enumeration of directed animals via gas considerations", "abstract": "In the literature, most of the results about the enumeration of directed animals on lattices via gas considerations are obtained by a formal passage to the limit of enumeration of directed animals on cyclical versions of the lattice. Here we provide a new point of view on this phenomenon. Using the gas construction given in [Electron. J. Combin. (2007) 14 R71], we describe the gas process on the cyclical versions of the lattices as a cyclical Markov chain (roughly speaking, Markov chains conditioned to come back to their starting point). Then we introduce a notion of convergence of graphs, such that if $(G_n)\\to G$ then the gas process built on $G_n$ converges in distribution to the gas process on $G$. That gives a general tool to show that gas processes related to animals enumeration are often Markovian on lines extracted from lattices. We provide examples and computations of new generating functions for directed animals with various sources on the triangular lattice, on the $\\mathcal {T}_n$ lattices introduced in [Ann. Comb. 4 (2000) 269--284] and on a generalization of the $\\mathcaligr {L}_n$ lattices introduced in [J. Phys. A 29 (1996) 3357--3365]."}
{"category": "Math", "title": "Canonical triangulations of Dehn fillings", "abstract": "Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori: in a broad range of cases, generic in an appropriate sense, this decomposition can be predicted from that of the unfilled manifold. As an example, we treat all hyperbolic fillings on one cusp of the Whitehead link complement."}
{"category": "Math", "title": "Ideal class groups and torsion in Picard groups of varieties", "abstract": "We give a new general technique for constructing and counting number fields with an ideal class group of nontrivial m-rank. Our results can be viewed as providing a way of specializing the Picard group of a variety V over $\\mathbb{Q}$ to obtain class groups for number fields $\\mathbb{Q}(P)$, $P\\in V(\\Qbar)$, for certain families of points P. In particular, we show how the problem of constructing quadratic number fields with a large-rank ideal class group can be reduced to the problem of finding a hyperelliptic curve with a rational Weierstrass point and a large rational torsion subgroup in its Jacobian. Furthermore, we show how many previous results on constructing large-rank ideal class groups can be fit into our framework and rederived. As an application of our technique, we derive a quantitative version of a theorem of Nakano. This gives the best known general quantitative result on number fields with a large-rank ideal class group."}
{"category": "Math", "title": "The geometry of twisted conjugacy classes in wreath products", "abstract": "We give a geometric proof based on recent work of Eskin, Fisher and Whyte that the lamplighter group $L_n$ has infinitely many twisted conjugacy classes for any automorphism $\\vp$ only when $n$ is divisible by 2 or 3, originally proved by Gon\\c{c}alves and Wong. We determine when the wreath product $G \\wr \\Z$ has this same property for several classes of finite groups $G$, including symmetric groups and some nilpotent groups."}
{"category": "Math", "title": "Seidel's Representation on the Hamiltonian Group of a Cartesian Product", "abstract": "Let $(M,\\omega)$ be a closed symplectic manifold and $\\textup{Ham}(M,\\omega)$ the group of Hamiltonian diffeomorphisms of $(M,\\omega)$. Then the Seidel homomorphism is a map from the fundamental group of $\\textup{Ham}(M,\\omega)$ to the quantum homology ring $QH_*(M;\\Lambda)$. Using this homomorphism we give a sufficient condition for when a nontrivial loop $\\psi$ in $\\textup{Ham}(M,\\omega)$ determines a nontrivial loop $\\psi\\times\\textup{id}_N$ in $\\textup{Ham}(M\\times N,\\omega\\oplus\\eta)$, where $(N,\\eta)$ is a closed symplectic manifold such that $\\pi_2(N)=0$."}
{"category": "Math", "title": "A spanning tree model for the Heegaard Floer homology of a branched double-cover", "abstract": "Given a diagram of a link K in S^3, we write down a Heegaard diagram for the branched-double cover Sigma(K). The generators of the associated Heegaard Floer chain complex correspond to Kauffman states of the link diagram. Using this model we make some computations of the homology \\hat{HF}(Sigma(K)) as a graded group. We also conjecture the existence of a delta-grading on \\hat{HF}(Sigma(K)) analogous to the delta-grading on knot Floer and Khovanov homology."}
{"category": "Math", "title": "Random projection trees for vector quantization", "abstract": "A simple and computationally efficient scheme for tree-structured vector quantization is presented. Unlike previous methods, its quantization error depends only on the intrinsic dimension of the data distribution, rather than the apparent dimension of the space in which the data happen to lie."}
{"category": "Math", "title": "The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications", "abstract": "The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of t-cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function for t-cores. Several applications are derived, including the \"marked hook formula\"."}
{"category": "Math", "title": "Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections", "abstract": "Given an i.i.d. sample from a distribution $F$ on $\\mathbb{R}$ with uniformly continuous density $p_0$, purely data-driven estimators are constructed that efficiently estimate $F$ in sup-norm loss and simultaneously estimate $p_0$ at the best possible rate of convergence over H\\\"older balls, also in sup-norm loss. The estimators are obtained by applying a model selection procedure close to Lepski's method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or $B$-splines. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-type analogs of the inequalities in Koltchinskii [Ann. Statist. 34 (2006) 2593-2656] for the deviation of suprema of empirical processes from their Rademacher symmetrizations."}
{"category": "Math", "title": "Uniform limit theorems for wavelet density estimators", "abstract": "Let $p_n(y)=\\sum_k\\hat{\\alpha}_k\\phi(y-k)+\\sum_{l=0}^{j_n-1}\\sum_k\\hat {\\beta}_{lk}2^{l/2}\\psi(2^ly-k)$ be the linear wavelet density estimator, where $\\phi$, $\\psi$ are a father and a mother wavelet (with compact support), $\\hat{\\alpha}_k$, $\\hat{\\beta}_{lk}$ are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density $p_0$ on $\\mathbb{R}$, and $j_n\\in\\mathbb{Z}$, $j_n\\nearrow\\infty$. Several uniform limit theorems are proved: First, the almost sure rate of convergence of $\\sup_{y\\in\\mathbb{R}}|p_n(y)-Ep_n(y)|$ is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that $\\sup_{y\\in\\mathbb{R}}|p_n(y)-p_0(y)|$ attains the optimal almost sure rate of convergence for estimating $p_0$, if $j_n$ is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of $p_n$, that is, for the stochastic processes $\\sqrt{n}(F_n ^W(s)-F(s))=\\sqrt{n}\\int_{-\\infty}^s(p_n-p_0),s\\in\\mathbb{R}$, are proved; and more generally, uniform central limit theorems for the processes $\\sqrt{n}\\int(p_n-p_0)f$, $f\\in\\mathcal{F}$, for other Donsker classes $\\mathcal{F}$ of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho et al. [Ann. Statist. 24 (1996) 508--539]."}
{"category": "Math", "title": "Cyclic cocycles on deformation quantizations and higher index theorems", "abstract": "We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the $K$-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes--Moscovici and its extension to orbifolds."}
{"category": "Math", "title": "A New Definition of the Steenrod Operations in Algebraic Geometry", "abstract": "The Steenrod operations (mod p) in Chow theory are defined for any prime p for a quasi-projective scheme, without appealing to the results of any domain but Milnor's K-theory. The new definition also gives a direct formula that depends only on the scheme itself. Additionally, basic properties of the operations are proved from the new definition. The idea is based on a construction of M. Rost."}
{"category": "Math", "title": "On existence and asymptotic behaviour of solutions of a fractional integral equation with linear modification of the argument", "abstract": "We study the solvability of a quadratic integral equation of fractional order with linear modification of the argument. This equation is considered in the Banach space of real functions defined, bounded and continuous on an unbounded interval. Moreover, we will obtain some asymptotic characterization of solutions."}
{"category": "Math", "title": "Standard isotrivial fibrations with p_g=q=1. II", "abstract": "A smooth, projective surface $S$ is called a $\\emph{standard isotrivial fibration}$ if there exists a finite group $G$ which acts faithfully on two smooth projective curves $C$ and $F$ so that $S$ is isomorphic to the minimal desingularization of $T:=(C \\times F)/G$. Standard isotrivial fibrations of general type with $p_g=q=1$ have been classified in \\cite{Pol07} under the assumption that $T$ has only Rational Double Points as singularities. In the present paper we extend this result, classifying all cases where $S$ is a minimal model. As a by-product, we provide the first examples of minimal surfaces of general type with $p_g=q=1$, $K_S^2=5$ and Albanese fibration of genus 3. Finally, we show with explicit examples that the case where $S$ is not minimal actually occurs."}
{"category": "Math", "title": "High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities", "abstract": "We define a discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors d-regular measure on H by integrating the discrete curvature according to the product measure. The aim of this work, continued in a subsequent paper, is to estimate multiscale least squares approximations of such measures by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature is comparable to the ``Jones-type flatness''. The latter quantity adds up scaled errors of approximations of a measure by d-planes at different scales and locations, and is commonly used to characterize uniform rectifiability. We thus obtain a characterization of uniform rectifiability by using the Menger-type curvature. In the current paper (part I) we control the continuous Menger-type curvature of an Ahlfors d-regular measure by its Jones-type flatness."}
{"category": "Math", "title": "The Dynamics Theorem for CMC surfaces in R^3", "abstract": "In this paper, we study the space of translational limits T(M) of a surface M properly embedded in R^3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface M' in T(M), the set T(M') in T(M). Among various dynamics type results we prove that surfaces in minimal T-invariant sets of T(M) are chord-arc. We also show that if M has an infinite number of ends, then there exists a nonempty minimal T-invariant set in T(M) consisting entirely of surfaces with planes of Alexandrov symmetry. Finally, when M has a plane of Alexandrov symmetry, we prove the following characterization theorem: M has finite topology if and only if M has a finite number of ends greater than one."}
{"category": "Math", "title": "On d-dimensional d-Semimetrics and Simplex-Type Inequalities for High-Dimensional Sine Functions", "abstract": "We show that high-dimensional analogues of the sine function (more precisely, the d-dimensional polar sine and the d-th root of the d-dimensional hypersine) satisfy a simplex-type inequality in a real pre-Hilbert space H. Adopting the language of Deza and Rosenberg, we say that these d-dimensional sine functions are d-semimetrics. We also establish geometric identities for both the d-dimensional polar sine and the d-dimensional hypersine. We then show that when d=1 the underlying functional equation of the corresponding identity characterizes a generalized sine function. Finally, we show that the d-dimensional polar sine satisfies a relaxed simplex inequality of two controlling terms \"with high probability\"."}
{"category": "Math", "title": "Degree-distribution stability of scale-free networks", "abstract": "Based on the concept and techniques of first-passage probability in Markov chain theory, this letter provides a rigorous proof for the existence of the steady-state degree distribution of the scale-free network generated by the Barabasi-Albert (BA) model, and mathematically re-derives the exact analytic formulas of the distribution. The approach developed here is quite general, applicable to many other scale-free types of complex networks."}
{"category": "Math", "title": "Quivers, long exact sequences and Horn type inequalities II", "abstract": "We study the set of all m-tuples $(\\lambda(1),...,\\lambda(m))$ of possible types of finite abelian p-groups $M_{\\lambda(1)}, ..., M_{\\lambda(m)}$ for which there exists a long exact sequence $M_{\\lambda(1)} \\to ... \\to M_{\\lambda(m)}$. When m=3, we recover Fulton's results on the possible eigenvalues of majorized Hermitian matrices."}
{"category": "Math", "title": "Notes on GIT-fans for quivers", "abstract": "These are notes on the construction of the GIT-fans for quivers without oriented cycles. We follow closely the steps outlined by N. Ressayre in \"The GIT-Equivalence for G-Line Bundles\" (Geometriae Dedicata, Volume 81, Numbers 1-3, 2000). A simple construction of the GIT-fan for normal, affine G-varities has been recently given by Ivan V. Arzhantsev and Juergen Hausen in \"Geometric Invariant Theory via Cox Rings\""}
{"category": "Math", "title": "Braid ordering and the geometry of closed braid", "abstract": "The relationships between braid ordering and the geometry of its closure is studied. We prove that if an essential closed surface $F$ in the complements of closed braid has relatively small genus with respect to the Dehornoy floor of the braid, $F$ is circular-foliated in a sense of Birman-Menasco's Braid foliation theory. As an application of the result, we prove that if Dehornoy floor of braids are larger than three, Nielsen-Thurston classification of braids and the geometry of their closure's complements are in one-to-one correspondence. Using this result, we construct infinitely many hyperbolic knots explicitly from pseudo-Anosov element of mapping class groups."}
{"category": "Math", "title": "On the Ramsey numbers for paths and generalized Jahangir graphs", "abstract": "For given graphs $G$ and $H,$ the \\emph{Ramsey number} $R(G,H)$ is the least natural number $n$ such that for every graph $F$ of order $n$ the following condition holds: either $F$ contains $G$ or the complement of $F$ contains $H.$ In this paper, we determine the Ramsey number of paths versus generalized Jahangir graphs. We also derive the Ramsey number $R(tP_n,H)$, where $H$ is a generalized Jahangir graph $J_{s,m}$ where $s\\geq2$ is even, $m\\geq3$ and $t\\geq1$ is any integer."}
{"category": "Math", "title": "Conditions for stochastic integrability in UMD Banach spaces", "abstract": "A detailed theory of stochastic integration in UMD Banach spaces has been developed recently by the authors. The present paper is aimed at giving various sufficient conditions for stochastic integrability."}
{"category": "Math", "title": "Periodic twisted cohomology and T-duality", "abstract": "The initial motivation of this work was to give a topological interpretation of two-periodic twisted de-Rham cohomology which is generalizable to arbitrary coefficients. To this end we develop a sheaf theory in the context of locally compact topological stacks with emphasis on the construction of the sheaf theory operations in unbounded derived categories, elements of Verdier duality and integration. The main result is the construction of a functorial periodization functor associated to a U(1)-gerbe. As applications we verify the $T$-duality isomorphism in periodic twisted cohomology and in periodic twisted orbispace cohomology."}
{"category": "Math", "title": "The Drinfel'd polynomial of a tridiagonal pair", "abstract": "Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \\to V$ and $A^*:V \\to V$ that satisfy the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\\{V_i\\}{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \\subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \\leq i \\leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\\{V^*_i\\}{i=0}^\\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \\subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \\leq i \\leq \\delta$, where $V^*_{-1}=0$ and $V^*_{\\delta+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \\subseteq W$, $A^* W \\subseteq W$, $W \\neq 0$, $W \\neq V$. We call such a pair a {\\it tridiagonal pair} on $V$. It is known that $d=\\delta$ and for $0 \\leq i \\leq d$ the dimensions of $V_i$, $V_{d-i}$, $V^*_i$, $V^*_{d-i}$ coincide. The pair $A,A^*$ is called {\\it sharp} whenever $\\dim V_0=1$. It is known that if $K$ is algebraically closed then $A,A^*$ is sharp. Assuming $A,A^*$ is sharp, we use the data $\\Phi=(A; \\{V_i\\}{i=0}^d; A^*; \\{V^*_i\\}{i=0}^d)$ to define a polynomial $P$ in one variable and degree at most $d$. We show that $P$ remains invariant if $\\Phi$ is replaced by $(A;\\{V_{d-i}\\}{i=0}^d; A^*; \\{V^*_i\\}{i=0}^d)$ or $(A;\\{V_i\\}{i=0}^d; A^*; \\{V^*_{d-i}\\}{i=0}^d)$ or $(A^*; \\{V^*_i\\}{i=0}^d; A; \\{V_i\\}{i=0}^d)$. We call $P$ the {\\it Drinfel'd polynomial} of $A,A^*$. We explain how $P$ is related to the classical Drinfel'd polynomial from the theory of Lie algebras and quantum groups. We expect that the roots of $P$ will be useful in a future classification of the sharp tridiagonal pairs. We compute the roots of $P$ for the case in which $V_i$ and $V^*_i$ have dimension 1 for $0 \\leq i \\leq d$."}
{"category": "Math", "title": "The Iterative Simplicity of Basic Topological Operations", "abstract": "Semigroups generated by topological operations such as closure, interior or boundary are considered. It is noted that some of these semigroups are in general finite and noncommutative. The problem is formulated whether they are always finite."}
{"category": "Math", "title": "Two combinatorial formulas concerning marked partitions", "abstract": "A partition of degree $n$ is a decomposition $n=i_1+i_2+\\dots+i_q$, where ${i_1,i_2,\\dots,i_q}$ are positive integers called the parts of the partition. Let $\\lambda>0$ be an integer. The partition is said to be a $\\lambda$--partition if $i_{a+1}-i_a\\geqslant \\lambda$ for all $a$ such that $1\\leqslant a<q$. The main result of this note are combinatorial formulas, which express the quantity of $1$-partitions of a given degree in terms of the $\\lambda$--partitions of the same degree, where $\\lambda=2$ or $\\lambda=3$, some special parts of which are marked depending on $\\lambda$. The presented proofs of both formulas are bijective. It is shown that for $\\lambda=3$ the corresponding formula is equivalent to the classical Sylvester identity. The obtained combinatorial formulas as well as their bijective proofs are generalized to the quantities of $1$--partitions, all parts of which are $\\geqslant k$ for any fixed integer $k\\geqslant 1$."}
{"category": "Math", "title": "Modular Reduction in Abstract Polytopes", "abstract": "The paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind: first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in Z[t] (with t=\\tau the golden ratio), to construct new regular 4-polytopes of hyperbolic types {3,5,3} and {5,3,5} with automorphism groups given by finite orthogonal groups."}
{"category": "Math", "title": "Paraconsistent First-Order Logic with restricted modus ponens rule and infinite hierarchy levels of contradiction $LP^\\#_{\\omega}$. Axiomatical system $HST^\\#_{\\omega}$, as paraconsistent generalization of Hrbacek set theory HST", "abstract": "In this paper paraconsistent first-order logic LP^{#}_{\\omega} with restricted modus ponens rule and infinite hierarchy levels of contradiction is proposed. Corresponding paraconsistent set theory KSth^{#}_{\\omega} is discussed.Axiomatical system HST^{#}_{\\omega} as paraconsistent generalization of Hrbacek set theory HST is considered."}
{"category": "Math", "title": "Square-integrable coactions of locally compact quantum groups", "abstract": "We define and study square-integrable coactions of locally compact quantum groups on Hilbert modules, generalising previous work for group actions. As special cases, we consider square-integrable Hilbert space corepresentations and integrable coactions on C*-algebras. Our main result is an equivariant generalisation of Kasparov's Stabilisation Theorem."}
{"category": "Math", "title": "Local product structure for expansive homeomorphisms", "abstract": "Let $f\\colon M\\to M$ be an expansive homeomorphism with dense topologically hyperbolic periodic points, $M$ a compact manifold. Then there is a local product structure in an open and dense subset of $M$. Moreover, if some topologically hyperbolic periodic point has codimension one, then this local product structure is uniform. In particular, we conclude that the homeomorphism is conjugated to a linear Anosov diffeomorphism of a torus."}
{"category": "Math", "title": "Weights of mixed tilting sheaves and geometric Ringel duality", "abstract": "We describe several general methods for calculating weights of mixed tilting sheaves. We introduce a notion called \"non-cancellation property\" which implies a strong uniqueness of mixed tilting sheaves and enables one to calculate their weights effectively. When we have a certain Radon transform, we prove a geometric analogue of Ringel duality which sends tilting objects to projective objects. We apply these methods to (partial) flag varieties and affine (partial) flag varieties and show that the weight polynomials of mixed tilting sheaves on flag and affine flag varieties are essentially given by Kazhdan-Lusztig polynomials. This verifies a mixed geometric analogue of a conjecture by W.Soergel in \\cite{Sg1}."}
{"category": "Math", "title": "Hypergeometric functions and hyperbolic metric", "abstract": "We obtain new inequalities for certain hypergeometric functions. Using these inequalities, we deduce estimates for the hyperbolic metric and the induced distance function on a certain canonical hyperbolic plane domain."}
{"category": "Math", "title": "Structure of Ann-categories", "abstract": "Each Ann-category $\\A$ is equivalent to an Ann-category of the type $(R,M),$ where $M$ is an $R$-bimodule. The family of constraints of $A$ induces a {\\it structure} on $(R,M).$ The main result of the paper is: 1. {\\it There exists a bijection between the set of structures on $(R,M)$ and the group of Mac Lane 3-cocycles $Z^{3}_{MaL}(R, M).$} 2. {\\it There exists a bijection between $C(R,M)$ of congruence classes of Ann-categories whose pre-stick is of the type $(R,M)$ and the Mac Lane cohomology group $H^3_{\\textrm{MaL}}(R,M).$}"}
{"category": "Math", "title": "Agnihotri-Woodward-Belkale polytope and the intersection of Klyachko cones", "abstract": "Agnihotri-Woodward-Belkale polytope $\\Delta$ (resp. Klyachko cone $K$) is the set of solutions of the multiplicative (resp. additive) Horn's problem, i.e., the set of triples of spectra of special unitary (resp. traceless Hermitian) $n\\times n$ matrices satisfying $AB=C$ (resp. $A+B=C$). $K$ is the tangent cone of $\\Delta$ at the origin. The group $G=\\Bbb Z_n \\oplus \\Bbb Z_n$ acts naturally on $\\Delta$. In this note, we report on a computer calculation which shows that $\\Delta$ coincides with the intersection of $gK$, $g\\in G$, for $n\\le 14$ but does not coincide for $n=15$. Our motivation was an attempt to understand how to solve the multiplicative Horn problem in practice for given conjugacy classes in SU(n)."}
{"category": "Math", "title": "On higher-power moments of $\\Delta(x)$(III)", "abstract": "Let $\\Delta(x)$ be the error term of the Dirichlet divisor problem. An asymptotic formula with the error term $O(T^{53/28+\\epsilon})$ is established for the integral $\\int_1^T\\Delta^4(x)dx.$ Similar results are also established for some other well-known error terms in the analytic number theory ."}
{"category": "Math", "title": "The Dolgachev Surface", "abstract": "We prove that the Dolgachev surface E(1)_{2,3} admits a handlebody decomposition without 1- and 3- handles, and we draw the explicit picture of this handlebody. We also locate a \"cork\" inside of E(1)_{2,3}, so that E(1)_{2,3} is obtained from E(1) by twisting along this cork."}
{"category": "Math", "title": "On the fourth power moment of $\\Delta(x)$ and $E(x)$ in short intervals", "abstract": "Let $\\Delta(x)$ and $E(x)$ be error terms of the sum of divisor function and the mean square of the Riemann zeta function, respectively. In this paper their fourth power moments for short intervals of Jutila's type are considered. We get an asymptotic formula for $U$ in some range."}
{"category": "Math", "title": "Small values of the Lusternik-Schnirelmann category for manifolds", "abstract": "We prove that manifolds of Lusternik-Schnirelmann category 2 necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larranaga and Gonzalez-Acuna, by generalizing their result in dimension 3, to all higher dimensions. We also obtain some general results on the relations between the fundamental group of a closed manifold M, the dimension of M, and the Lusternik-Schnirelmann category of M, and relate the latter to the systolic category of M."}
{"category": "Math", "title": "Twisted cohomologies of wrap groups over quaternions and octonions", "abstract": "This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real $\\bf R$, complex $\\bf C$ numbers, the quaternion skew field $\\bf H$ and the octonion algebra $\\bf O$. Cohomologies of wrap groups and their structure are investigated. Sheaves of wrap groups are constructed and studied. Moreover, twisted cohomologies and sheaves over quaternions and octonions are investigated as well."}
{"category": "Math", "title": "Homotopy theory of presheaves of Gamma-spaces", "abstract": "We consider the category of presheaves of Gamma-spaces, or equivalently, of Gamma-objects in simplicial presheaves. Our main result is the construction of stable model structures on this category parametrised by local model structures on simplicial presheaves. If a local model structure on simplicial presheaves is monoidal, the corresponding stable model structure on presheaves of Gamma-spaces is monoidal and satisfies the monoid axiom. This allows us to lift the stable model structures to categories of algebras and modules over commutative algebras."}
{"category": "Math", "title": "A.D. Alexandrov's problem for Busemann non-positively curved spaces", "abstract": "The paper is the last in the cycle devoted to the solution of Alexandrov's problem for non-positively curved spaces. Here we study non-positively curved spaces in the sense of Busemann. We prove that if $X$ is geodesically complete connected at infinity proper Busemann space, then it has the following characterization of isometries. For any bijection $f:X\\to X$, if $f$ and $f^{-1}$ preserve the distance 1, then $f$ is an isometry."}
{"category": "Math", "title": "The action of SL(2) on abelian varieties", "abstract": "We show that the algebraic group SL(2) acts on any polarized abelian variety A through correspondences. As a consequence we recover the action of SL(2) on the Chow group CH(A) (with rational coefficients), and this gives rise to Lefschetz type properties for algebraic cycles on A ."}
{"category": "Math", "title": "A global compact attractor for high-dimensional defocusing non-linear Schr\\\"odinger equations with potential", "abstract": "We study the asymptotic behavior of large data solutions in the energy space $H := H^1(\\R^d)$ in very high dimension $d \\geq 11$ to defocusing Schr\\\"odinger equations $i u_t + \\Delta u = |u|^{p-1} u + Vu$ in $\\R^d$, where $V \\in C^\\infty_0(\\R^d)$ is a real potential (which could contain bound states), and $1+\\frac{4}{d} < p < 1+\\frac{4}{d-2}$ is an exponent which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as $t \\to +\\infty$, these solutions split into a radiation term that evolves according to the linear Schr\\\"odinger equation, and a remainder which converges in $H$ to a compact attractor $K$, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in $H$. The main novelty of this result is that $K$ is a \\emph{global} attractor, being independent of the initial energy of the initial data; in particular, no matter how large the initial data is, all but a bounded amount of energy is radiated away in the limit."}
{"category": "Math", "title": "Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. IV", "abstract": "We extend to singular cardinals the model-theoretical relation $\\lambda \\stackrel{\\kappa}{\\Rightarrow} \\mu$ introduced in P. Lipparini, The compactness spectrum of abstract logics, large cardinals and combinatorial principles, Boll. Unione Matematica Italiana ser. VII, {\\bf 4-B} 875--903 (1990). We extend some results obtained in Part II, finding equivalent conditions involving uniformity of ultrafilters and the existence of certain infinite matrices. Our present definition suggests a new compactness property for abstract logics."}
{"category": "Math", "title": "A continued fraction expansion for a q-tangent function: An elementary proof", "abstract": "We prove a continued fraction expansion for a certain $q$-tangent function that was conjectured by the present writer, then proved by Fulmek, now in a completely elementary way."}
{"category": "Math", "title": "Pleijel's nodal domain theorem for free membranes", "abstract": "We prove an analogue of Pleijel's nodal domain theorem for piecewise analytic planar domains with Neumann boundary conditions. This confirms a conjecture made by Pleijel in 1956. The proof is a combination of Pleijel's original approach and an estimate due to Toth and Zelditch for the number of boundary zeros of Neumann eigenfunctions."}
{"category": "Math", "title": "A finiteness property for preperiodic points of Chebyshev polynomials", "abstract": "Let K be a number field with algebraic closure K-bar, let S be a finite set of places of K containing the archimedean places, and let f be a Chebyshev polynomial. We prove that if a in K-bar is not preperiodic, then there are only finitely many preperiodic points b in K-bar which are S-integral with respect to a."}
{"category": "Math", "title": "Periodic points, linearizing maps, and the dynamical Mordell-Lang problem", "abstract": "We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism f:X --> X. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics."}
{"category": "Math", "title": "A Linear Programming Relaxation and a Heuristic for the Restless Bandit Problem with General Switching Costs", "abstract": "We extend a relaxation technique due to Bertsimas and Nino-Mora for the restless bandit problem to the case where arbitrary costs penalize switching between the bandits. We also construct a one-step lookahead policy using the solution of the relaxation. Computational experiments and a bound for approximate dynamic programming provide some empirical support for the heuristic."}
{"category": "Math", "title": "The weak type $(1,1)$ bounds for the maximal function associated to cubes grow to infinity with the dimension", "abstract": "Let $M_d$ be the centered Hardy-Littlewood maximal function associated to cubes in $\\mathbb{R}^d$ with Lebesgue measure, and let $c_d$ denote the lowest constant appearing in the weak type (1,1) inequality satisfied by $M_d$. We show that $c_d \\to \\infty$ as $d\\to \\infty$, thus answering, for the case of cubes, a long standing open question of E. M. Stein and J. O. Str\\\"{o}mberg."}
{"category": "Math", "title": "Bass-Serre rigidity results in von Neumann algebras", "abstract": "We obtain new Bass-Serre type rigidity results for ${\\rm II_1}$ equivalence relations and their von Neumann algebras, coming from free ergodic actions of free products of groups on the standard probability space. As an application, we show that any non-amenable factor arising as an amalgamated free product of von Neumann algebras $\\mathcal{M}_1 \\ast_B \\mathcal{M}_2$ over an abelian von Neumann algebra $B$, is prime, i.e. cannot be written as a tensor product of diffuse factors. This gives, both in the type ${\\rm II_1}$ and in the type ${\\rm III}$ case, new examples of prime factors."}
{"category": "Math", "title": "Order Statistics and Probabilistic Robust Control", "abstract": "Order statistics theory is applied in this paper to probabilistic robust control theory to compute the minimum sample size needed to come up with a reliable estimate of an uncertain quantity under continuity assumption of the related probability distribution. Also, the concept of distribution-free tolerance intervals is applied to estimate the range of an uncertain quantity and extract the information about its distribution. To overcome the limitations imposed by the continuity assumption in the existing order statistics theory, we have derived a cumulative distribution function of the order statistics without the continuity assumption and developed an inequality showing that this distribution has an upper bound which equals to the corresponding distribution when the continuity assumption is satisfied. By applying this inequality, we investigate the minimum computational effort needed to come up with an reliable estimate for the upper bound (or lower bound) and the range of a quantity. We also give conditions, which are much weaker than the absolute continuity assumption, for the existence of such minimum sample size. Furthermore, the issue of making tradeoff between performance level and risk is addressed and a guideline for making this kind of tradeoff is established. This guideline can be applied in general without continuity assumption."}
{"category": "Math", "title": "Fast Construction of Robustness Degradation Function", "abstract": "We develop a fast algorithm to construct the robustness degradation function, which describes quantitatively the relationship between the proportion of systems guaranteeing the robustness requirement and the radius of the uncertainty set. This function can be applied to predict whether a controller design based on an inexact mathematical model will perform satisfactorily when implemented on the true system."}
{"category": "Math", "title": "Constrained Optimal Synthesis and Robustness Analysis by Randomized Algorithms", "abstract": "In this paper, we consider robust control using randomized algorithms. We extend the existing order statistics distribution theory to the general case in which the distribution of population is not assumed to be continuous and the order statistics is associated with certain constraints. In particular, we derive an inequality on distribution for related order statistics. Moreover, we also propose two different approaches in searching reliable solutions to the robust analysis and optimal synthesis problems under constraints. Furthermore, minimum computational effort is investigated and bounds for sample size are derived."}
{"category": "Math", "title": "On the Binomial Confidence Interval and Probabilistic Robust Control", "abstract": "The Clopper-Pearson confidence interval has ever been documented as an exact approach in some statistics literature. More recently, such approach of interval estimation has been introduced to probabilistic control theory and has been referred as non-conservative in control community. In this note, we clarify the fact that the so-called exact approach is actually conservative. In particular, we derive analytic results demonstrating the extent of conservatism in the context of probabilistic robustness analysis. This investigation encourages seeking better methods of confidence interval construction for robust control purpose."}
{"category": "Math", "title": "Ranks of the Sylow 2-subgroups of the classical groups", "abstract": "We determine the ranks and normal ranks of the Sylow 2-subgroups of the classical groups of odd characteristic."}
{"category": "Math", "title": "Ranks of the Sylow 2-subgroups of the classical Simple groups", "abstract": "We determine the ranks of the Sylow 2-subgroups of the classical simple groups of odd characteristic."}
{"category": "Math", "title": "On Stein's Conjecture on the Polynomial Carleson Operator", "abstract": "We prove that the generalized Carleson operator $C_d$ with polynomial phase function is of strong type $(p,r)$, $1<r<p<\\infty$; this yields a positive answer in the $1<p<2$ case to a conjecture of Stein which asserts that for $1<p<\\infty$ we have that $C_d$ is of strong type $(p,p)$. A key ingredient in this proof is the further extension of the {\\it relational} time-frequency perspective (introduced in \\cite{q}) to the general polynomial phase."}
{"category": "Math", "title": "Algebra in the superextensions of groups: minimal left ideals", "abstract": "We prove that the minimal left ideals of the superextension $\\lambda(Z)$ of the discrete group $Z$ of integers are metrizable topological semigroups, topologically isomorphic to minimal left ideals of the superextension $\\lambda(Z_2)$ of the compact group $Z_2$ of integer 2-adic numbers."}
{"category": "Math", "title": "On isoperimetric inequalities for log-convex measures", "abstract": "This paper has been withdrawn by the author"}
{"category": "Math", "title": "Resonances for non-analytic potentials", "abstract": "We consider semiclassical Schroedinger operators on R^n, with C^\\infty potentials decaying polynomially at infinity. The usual theories of resonances do not apply in such a non-analytic framework. Here, under some additional conditions, we show that resonances are invariantly defined up to any power of their imaginary part. The theory is based on resolvent estimates for families of approximating distorted operators with potentials that are holomorphic in narrow complex sectors around R^n."}
{"category": "Math", "title": "The covariogram determines three-dimensional convex polytopes", "abstract": "The cross covariogram g_{K,L} of two convex sets K, L in R^n is the function which associates to each x in R^n the volume of the intersection of K with L+x. The problem of determining the sets from their covariogram is relevant in stochastic geometry, in probability and it is equivalent to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. The two main results of this paper are that g_{K,K} determines three-dimensional convex polytopes K and that g_{K,L} determines both K and L when K and L are convex polyhedral cones satisfying certain assumptions. These results settle a conjecture of G. Matheron in the class of convex polytopes. Further results regard the known counterexamples in dimension n>=4. We also introduce and study the notion of synisothetic polytopes. This concept is related to the rearrangement of the faces of a convex polytope."}
{"category": "Math", "title": "Unstable attractors in manifolds", "abstract": "Let $M$ be a locally compact metric space endowed with a continuous flow $\\phi : M \\times \\mathbb{R} \\longrightarrow M$. Frequently an attractor $K$ for $\\phi$ exists which is of interest, not only in itself but also the dynamics in its basin of attraction $\\mathcal{A}(K)$. In this paper the class of {\\sl attractors with no external explosions}, which is intermediate between the well known {\\sl stable attractors} and the extremely wild {\\sl unstable attractors}, is studied. We are mainly interested in their cohomological properties, as well as in the strong relations which exist between their shape (in the sense of Borsuk) and the topology of the phase space."}
{"category": "Math", "title": "Bernstein Operators for Extended Chebyshev Systems", "abstract": "Let $U_{n}\\subset C^{n}[ a,b] $ be an extended Chebyshev space of dimension $n+1$. Suppose that $f_{0}\\in U_{n}$ is strictly positive and $% f_{1}\\in U_{n}$ has the property that $f_{1}/f_{0}$ is strictly increasing. We search for conditions ensuring the existence of points $% t_{0},...,t_{n}\\in [ a,b] $ and positive coefficients $\\alpha_{0},...,\\alpha_{n}$ such that for all $f\\in C[ a,b]$, the operator $B_{n}:C[ a,b] \\to U_{n}$ defined by $% B_{n}f=\\sum_{k=0}^{n}f(t_{k}) \\alpha_{k}p_{n,k}$ satisfies $% B_{n}f_{0}=f_{0}$ and $B_{n}f_{1}=f_{1}.$ Here it is assumed that $% p_{n,k},k=0,...,n$, is a Bernstein basis, defined by the property that each $% p_{n,k}$ has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b.$"}
{"category": "Math", "title": "Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces", "abstract": "We study the existence and shape preserving properties of a generalized Bernstein operator $B_{n}$ fixing a strictly positive function $f_{0}$, and a second function $f_{1}$ such that $f_{1}/f_{0}$ is strictly increasing, within the framework of extended Chebyshev spaces $U_{n}$. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator $B_{n}:C[a,b]\\to U_{n}$ with strictly increasing nodes, fixing $f_{0}, f_{1}\\in U_{n}$. If $U_{n}\\subset U_{n + 1}$ and $U_{n + 1}$ has a non-negative Bernstein basis, then there exists a Bernstein operator $B_{n+1}:C[a,b]\\to U_{n+1}$ with strictly increasing nodes, fixing $f_{0}$ and $f_{1}.$ In particular, if $% f_{0},f_{1},...,f_{n}$ is a basis of $U_{n}$ such that the linear span of $% f_{0},..,f_{k}$ is an extended Chebyshev space over $[ a,b] $ for each $k=0,...,n$, then there exists a Bernstein operator $B_{n}$ with increasing nodes fixing $f_{0}$ and $f_{1}.$ The second main result says that under the above assumptions the following inequalities hold B_{n}f\\geq B_{n+1}f\\geq f for all $(f_{0},f_{1})$-convex functions $f\\in C[ a,b] .$ Furthermore, $B_{n}f$ is $(f_{0},f_{1})$-convex for all $(f_{0},f_{1})$% -convex functions $f\\in C[ a,b] .$ In the specific case of exponential polynomials we give alternative proofs of shape preserving properties by computing derivatives of the generalized Bernstein polynomials."}
{"category": "Math", "title": "Bernstein operators for exponential polynomials", "abstract": "Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $\\lambda_{0},...,\\lambda_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex conjugation. If the length of the interval $[ a,b] $ is smaller than $\\pi /M_{n}$, where $M_{n}:=\\max \\left\\{| \\text{Im}% \\lambda_{j}| :j=0,...,n\\right\\} $, then there exists a basis $p_{n,k}$%, $k=0,...n$, of the space $U_{n}$ with the property that each $p_{n,k}$ has a zero of order $k$ at $a$ and a zero of order $n-k$ at $b,$ and each $% p_{n,k}$ is positive on the open interval $(a,b) .$ Under the additional assumption that $\\lambda_{0}$ and $\\lambda_{1}$ are real and distinct, our first main result states that there exist points $% a=t_{0}<t_{1}<...<t_{n}=b$ and positive numbers $\\alpha_{0},..,\\alpha_{n}$%, such that the operator \\begin{equation*} B_{n}f:=\\sum_{k=0}^{n}\\alpha_{k}f(t_{k}) p_{n,k}(x) \\end{equation*} satisfies $B_{n}e^{\\lambda_{j}x}=e^{\\lambda_{j}x}$, for $j=0,1.$ The second main result gives a sufficient condition guaranteeing the uniform convergence of $B_{n}f$ to $f$ for each $f\\in C[ a,b] $."}
{"category": "Math", "title": "Monge-Ampere boundary measures", "abstract": "We study swept-out Monge-Ampere measures of plurisubharmonic functions and boundary values related to these measures."}
{"category": "Math", "title": "Partitions of $\\mathbb{Z}_n$ into Arithmetic Progressions", "abstract": "We introduce the notion of arithmetic progression blocks or AP-blocks of $\\mathbb{Z}_n$, which can be represented as sequences of the form $(x, x+m, x+2m, ..., x+(i-1)m) \\pmod n$. Then we consider the problem of partitioning $\\mathbb{Z}_n$ into AP-blocks for a given difference $m$. We show that subject to a technical condition, the number of partitions of $\\mathbb{Z}_n$ into $m$-AP-blocks of a given type is independent of $m$. When we restrict our attention to blocks of sizes one or two, we are led to a combinatorial interpretation of a formula recently derived by Mansour and Sun as a generalization of the Kaplansky numbers. These numbers have also occurred as the coefficients in Waring's formula for symmetric functions."}
{"category": "Math", "title": "Noncommutative Geometry in the Framework of Differential Graded Categories", "abstract": "In this survey article we discuss a framework of noncommutative geometry with differential graded categories as models for spaces. We outline a construction of the category of noncommutative spaces and also include a discussion on noncommutative motives. We propose a motivic measure with values in a motivic ring. This enables us to introduce certain zeta functions of noncommutative spaces."}
{"category": "Math", "title": "Overall and Pairwise Segregation Tests Based on Nearest Neighbor Contingency Tables", "abstract": "Multivariate interaction between two or more classes (or species) has important consequences in many fields and causes multivariate clustering patterns such as segregation or association. The spatial segregation occurs when members of a class tend to be found near members of the same class (i.e., near conspecifics) while spatial association occurs when members of a class tend to be found near members of the other class or classes. These patterns can be studied using a nearest neighbor contingency table (NNCT). The null hypothesis is randomness in the nearest neighbor (NN) structure, which may result from -- among other patterns -- random labeling (RL) or complete spatial randomness (CSR) of points from two or more classes (which is called the CSR independence, henceforth). In this article, we introduce new versions of overall and cell-specific tests based on NNCTs (i.e., NNCT-tests) and compare them with Dixon's overall and cell-specific tests. These NNCT-tests provide information on the spatial interaction between the classes at small scales (i.e., around the average NN distances between the points). Overall tests are used to detect any deviation from the null case, while the cell-specific tests are post hoc pairwise spatial interaction tests that are applied when the overall test yields a significant result. We analyze the distributional properties of these tests; assess the finite sample performance of the tests by an extensive Monte Carlo simulation study. Furthermore, we show that the new NNCT-tests have better performance in terms of Type I error and power. We also illustrate these NNCT-tests on two real life data sets."}
{"category": "Math", "title": "The probability of exceeding a piecewise deterministic barrier by the heavy-tailed renewal compound process", "abstract": "We analyze the asymptotics of crossing a high piecewise linear barriers by a renewal compound process with the subexponential jumps. The study is motivated by ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions when the initial reserves of both companies tend to infinity."}
{"category": "Math", "title": "On reductions of families of crystalline Galois representations", "abstract": "Let K_{f} be the finite unramified extension of Q_{p} of degree f and E any finite large enough coefficient field containing K_{f}. We construct analytic families of \\'etale (Phi,Gamma)-modules which give rise to families of crystalline E-representations of the absolute Galois group G_{K_{f}} of K_{f}. For any irreducible effective two-dimensional crystalline E-representation of G_{K_{f}} with labeled Hodge-Tate weights {0,-k_{i}}_{{\\tau}_{i}} induced from a crystalline character of G_{K_{2f}}, we construct an infinite family of crystalline E-representations of G_{K_{f}} of the same Hodge-Tate type which contains it. As an application, we compute the semisimplified mod p reductions of the members of each such family."}
{"category": "Math", "title": "The eta invariant in the doubly K\\\"ahlerian conformally compact Einstein case", "abstract": "On a 3-manifold bounding a compact 4-manifold, let a conformal structure be induced from a complete Einstein metric which conformally compactifies to a K\\\"ahler metric. Formulas are derived for the eta invariant of this conformal structure under additional assumptions. One such assumption is that the K\\\"ahler metric admits a special K\\\"ahler-Ricci potential in the sense defined by Derdzinski and Maschler. Another is that the K\\\"ahler metric is part of an ambitoric structure, in the sense defined by Apostolov, Calderbank and Gauduchon, as well as a toric one. The formulas are derived using the Duistermaat-Heckman theorem. This result is closely related to earlier work of Hitchin on the Einstein selfdual case."}
{"category": "Math", "title": "Heat Kernel Analysis on Infinite-Dimensional Heisenberg Groups", "abstract": "We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, $\\{\\nu_t\\}_{t>0},$ are also studied. We show that these heat kernel measures admit: 1) Gaussian like upper bounds, 2) Cameron-Martin type quasi-invariance results, 3) good $L^p$ -- bounds on the corresponding Radon-Nykodim derivatives, 4) integration by parts formulas, and 5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor."}
{"category": "Math", "title": "Fast Universal Algorithms for Robustness Analysis", "abstract": "In this paper, we develop efficient randomized algorithms for estimating probabilistic robustness margin and constructing robustness degradation curve for uncertain dynamic systems. One remarkable feature of these algorithms is their universal applicability to robustness analysis problems with arbitrary robustness requirements and uncertainty bounding set. In contrast to existing probabilistic methods, our approach does not depend on the feasibility of computing deterministic robustness margin. We have developed efficient methods such as probabilistic comparison, probabilistic bisection and backward iteration to facilitate the computation. In particular, confidence interval for binomial random variables has been frequently used in the estimation of probabilistic robustness margin and in the accuracy evaluation of estimating robustness degradation function. Motivated by the importance of fast computing of binomial confidence interval in the context of probabilistic robustness analysis, we have derived an explicit formula for constructing the confidence interval of binomial parameter with guaranteed coverage probability. The formula overcomes the limitation of normal approximation which is asymptotic in nature and thus inevitably introduce unknown errors in applications. Moreover, the formula is extremely simple and very tight in comparison with classic Clopper-Pearson's approach."}
{"category": "Math", "title": "Arithmetical rank of the cyclic and bicyclic graphs", "abstract": "We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex or a path, the arithmetical rank equals the projective dimension."}
{"category": "Math", "title": "Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity", "abstract": "We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard's celebrated theorem that (the boundary of) the Mandelbrot set is connected. For these parameter spaces, a fundamental conjecture is that hyperbolic dynamics is dense. For quadratic polynomials, this would follow from the famous stronger conjecture that the bifurcation locus (or equivalently the Mandelbrot set) is locally connected. It turns out that a formally slightly weaker statement is sufficient, namely that every point in the bifurcation locus is the landing point of a parameter ray. For exponential maps, the bifurcation locus is not locally connected. We describe a different conjecture (triviality of fibers) which naturally generalizes the role that local connectivity has for quadratic or unicritical polynomials."}
{"category": "Math", "title": "Sample Reuse Techniques of Randomized Algorithms for Control under Uncertainty", "abstract": "Sample reuse techniques have significantly reduced the numerical complexity of probabilistic robustness analysis. Existing results show that for a nested collection of hyper-spheres the complexity of the problem of performing $N$ equivalent i.i.d. (identical and independent) experiments for each sphere is absolutely bounded, independent of the number of spheres and depending only on the initial and final radii. In this chapter we elevate sample reuse to a new level of generality and establish that the numerical complexity of performing $N$ equivalent i.i.d. experiments for a chain of sets is absolutely bounded if the sets are nested. Each set does not even have to be connected, as long as the nested property holds. Thus, for example, the result permits the integration of deterministic and probabilistic analysis to eliminate regions from an uncertainty set and reduce even further the complexity of some problems. With a more general view, the result enables the analysis of complex decision problems mixing real-valued and discrete-valued random variables."}
{"category": "Math", "title": "Fast Parallel Frequency Sweeping Algorithms for Robust ${\\cal D}$-Stability Margin", "abstract": "This paper considers the robust ${\\cal D}$-stability margin problem under polynomic structured real parametric uncertainty. Based on the work of De Gaston and Safonov (1988), we have developed techniques such as, a parallel frequency sweeping strategy, different domain splitting schemes, which significantly reduce the computational complexity and guarantee the convergence."}
{"category": "Math", "title": "Parallel Branch and Bound Algorithm for Computing Maximal Structured Singular Value", "abstract": "In this paper, we have developed a parallel branch and bound algorithm which computes the maximal structured singular value $\\mu$ without tightly bounding $\\mu$ for each frequency and thus significantly reduce the computational complexity."}
{"category": "Math", "title": "Universal Cycles of Discrete Functions", "abstract": "A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian; this baseline result is used as the basis of existence proofs for universal cycles (also known as deBruijn cycles or $U$-cycles) of several combinatorial objects. We present new results on the existence of universal cycles of certain classes of functions. These include onto functions, and 1-inequitable sequences on a binary alphabet. In each case the connectedness of the underlying graph is the non-trivial aspect to be established."}
{"category": "Math", "title": "On Abstract Strichartz Estimates and the Strauss Conjecture for Nontrapping Obstacles", "abstract": "The purpose of this paper is to show how local energy decay estimates for certain linear wave equations involving compact perturbations of the standard Laplacian lead to optimal global existence theorems for the corresponding small amplitude nonlinear wave equations with power nonlinearities. To achieve this goal, at least for spatial dimensions $n=3$ and 4, we shall show how the aforementioned linear decay estimates can be combined with \"abstract Strichartz\" estimates for the free wave equation to prove corresponding estimates for the perturbed wave equation when $n\\ge3$. As we shall see, we are only partially successful in the latter endeavor when the dimension is equal to two, and therefore, at present, our applications to nonlinear wave equations in this case are limited."}
{"category": "Math", "title": "Restricted Radon transforms and projections of planar sets", "abstract": "We establish a mixed norm estimate for the Radon transform in the plane when the set of directions has fractional dimension. This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to a conjecture analogous to a well-known conjecture of Furstenberg."}
{"category": "Math", "title": "Geometric and spectral properties of locally tessellating planar graphs", "abstract": "In this article, we derive bounds for values of the global geometry of locally tessellating planar graphs, namely, the Cheeger constant and exponential growth, in terms of combinatorial curvatures. We also discuss spectral implications for the Laplacians."}
{"category": "Math", "title": "Comparison theory and smooth minimal C*-dynamics", "abstract": "We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar's Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimension functions. These results continue to hold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one. Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C*-algebra. This is also used to construct uncountably many non-Morita-equivalent simple separable amenable C*-algebras with the same K-theory and tracial state space, providing a C*-algebraic analogue of McDuff's uncountable family of II_1 factors. We prove in passing that the range of the radius of comparison is exhausted by simple C*-algebras."}
{"category": "Math", "title": "An Asymptotic Formula for the Sequence ||exp(i n h(t))||_A", "abstract": "Given a function f with an absolutely convergent Fourier series, we define the norm of f as ||f||_A = the sum of absolute values of the Fourier coefficients of f. We study the behavior of ||f^n||_A as n goes to infinity, for f of the form exp(ih(t)) where h is a real, odd and twice continuously differentiable function such that h(t + 2\\pi) = h(t) + 2k\\pi for some integer k. We obtain a remarkably simple asymptotic formula for the case when h'' has no zeros in (0,\\pi) and satisfies an additional condition near 0 and near \\pi. Corollaries of our formula are an asymptotic formula due to D.Girard, and a formula on Bessel functions, due to G.Stey."}
{"category": "Math", "title": "Lattice points on the plane ax+by+cz=d and the diophantine system ax+by+cz=d ex+fy+gz=h", "abstract": "The subject matter of this work are the linear, three variable diophantine equation ax+by+cz=d (1), and the diophantine system ax+by+cz=d (2) ex+fy+gz=h with the coefficients a,b,c,d,e,f,g,h being integers. Introductory number theory books, typically contain only a brief outline of how to solve equation (1). Even less or no material is offered on the system (2). The purpose of this work is to fill this gap. After some preliminary, introductory material, which includes the general solution of the two variable linear diophantine equation ax+by=c(material which we use later in the paper); we present a complete and detailed analysis of equation (1). We determine the precise conditions that the coefficients a,b,c,d must satisfy in order for integer solutions to exist. We then derive a two-parameter, parametric description of The solution set. The solution set of (1), if not empty, consists of all integer triples (x,y,z) that satisfy (1). Geometrically, this is interpreted as the set of all lattice points in 3-D space which lie on the plane with equation (1). Similarly, we offer an exhaustive analysis of the system (2) by determining the precise conditions that the coeffients must satisfy in order for integer solutions to exist. We offer seven examples with detailed solutions."}
{"category": "Math", "title": "Computing stability of multi-dimensional travelling waves", "abstract": "We present a numerical method for computing the pure-point spectrum associated with the linear stability of multi-dimensional travelling fronts to parabolic nonlinear systems. Our method is based on the Evans function shooting approach. Transverse to the direction of propagation we project the spectral equations onto a finite Fourier basis. This generates a large, linear, one-dimensional system of equations for the longitudinal Fourier coefficients. We construct the stable and unstable solution subspaces associated with the longitudinal far-field zero boundary conditions, retaining only the information required for matching, by integrating the Riccati equations associated with the underlying Grassmannian manifolds. The Evans function is then the matching condition measuring the linear dependence of the stable and unstable subspaces and thus determines eigenvalues. As a model application, we study the stability of two-dimensional wrinkled front solutions to a cubic autocatalysis model system. We compare our shooting approach with the continuous orthogonalization method of Humpherys and Zumbrun. We then also compare these with standard projection methods that directly project the spectral problem onto a finite multi-dimensional basis satisfying the boundary conditions."}
{"category": "Math", "title": "Approximating the Value Functions of Stochastic Knapsack Problems: A Homogeneous Monge-Amp\\'ere Equation and Its Stochastic Counterparts", "abstract": "Stochastic knapsack problem originally was a versatile model for controls in telecommunication networks. Recently, it draws attentions of revenue management community by serving as a basic model for allocating resources over time. We develop approximation schemes for knapsack problems in this paper, a system of nonlinear but solvable partial differential equations and stochastic partial differential equation are shown to be the limit of the process that following the optimal solution of the stochastic knapsack problem."}
{"category": "Math", "title": "The Minkowski question mark function: explicit series for the dyadic period function and moments", "abstract": "Previously, several natural integral transforms of the Minkowski question mark function F(x) were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and the functional equation, thus encoding intrinsic information about F(x). One of them - the dyadic period function G(z) - was defined as a Stieltjes transform. In this paper we introduce a family of \"distributions\" F_p(x) for Re p>=1, such that F_1(x) is the question mark function and F_2(x) is a discrete distribution with support on x=1. We prove that the generating function of moments of F_p(x) satisfies the three term functional equation. This has an independent interest, though our main concern is the information it provides about F(x). This approach yields the following main result: we prove that the dyadic period function is a sum of infinite series of rational functions with rational coefficients."}
{"category": "Math", "title": "The ABC Theorem for Meromorphic Functions", "abstract": "Using a `height-to-radical' identity, we define the archimedean contribution to the radical, $r_\\arch$, and we give a new proof of the abc theorem for the field of meromorphic functions. The first step of the proof is completely formal and yields that the height is bounded by the radical, $h\\leq r$, where $r=r_\\na+r_\\arch$ is the radical completed with the archimedean contribution. The second step is analytic in nature and uses the lemma on the logarithmic derivative to derive a bound for $r_\\arch$."}
{"category": "Math", "title": "Rank-level duality of conformal blocks of GL_n and SL_n", "abstract": "We generalise the proof by Marian and Oprea of rank-level duality for non-abelian theta functions to the case of sections of line bundles (conformal blocks) over moduli spaces of parabolic vector bundles over a projective smooth curve. We also describe how it implies the rank-level duality between conformal blocks of Sp(2) and Sp(2n)."}
{"category": "Math", "title": "Geometric representation of binary codes and computation of weight enumerators", "abstract": "For every linear binary code $C$, we construct a geometric triangular configuration $\\Delta$ so that the weight enumerator of $C$ is obtained by a simple formula from the weight enumerator of the cycle space of $\\Delta$. The triangular configuration $\\Delta$ thus provides a geometric representation of $C$ which carries its weight enumerator. This is the initial step in the suggestion by M. Loebl, to extend the theory of Pfaffian orientations from graphs to general linear binary codes. Then we carry out also the second step by constructing, for every triangular configuration $\\Delta$, a triangular configuration $\\Delta'$ and a bijection between the cycle space of $\\Delta$ and the set of the perfect matchings of $\\Delta'$."}
{"category": "Math", "title": "On Some Discrete Differential Equations", "abstract": "In this short note, we present few results on the use of the discrete Laplace transform in solving first and second order initial value problems of discrete differential equations."}
{"category": "Math", "title": "Lower Bounds for the Size of Random Maximal H-Free Graphs", "abstract": "We consider the next greedy randomized process for generating maximal H-free graphs: Given a fixed graph H and an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph. Then, construct an n-vertex graph, M_n(H), iteratively as follows. Traverse the permuted edges of the complete n-vertex graph and add each one to the (initially empty) evolving graph M_n(H) - unless its addition creates a copy of H. The result of this process is a maximal H-free graph M_n(H). The basic question we are concerned with in here is: What is the expected number of edges in M_n(H)? We give new lower bounds on the expected number of edges in M_n(H) for the case where H is a regular, strictly 2-balanced graph. In particular, we obtain new lower bounds for Turan numbers of complete balanced bipartite graphs K_{r,r}, for every fixed r > 4. This improves an old lower bound of Erdos and Spencer."}
{"category": "Math", "title": "Combable functions, quasimorphisms, and the central limit theorem", "abstract": "A function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left and right invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include (i) homomorphisms to Z (ii) word length with respect to a finite generating set (iii) most known explicit constructions of quasimorphisms (e.g. the Epstein-Fujiwara counting quasimorphisms) We show that bicombable functions on word-hyperbolic groups satisfy a central limit theorem: if \\bar{\\phi}_n is the value of \\phi on a random element of word length n (in a certain sense), there are E and \\sigma for which there is convergence in the sense of distribution n^{-1/2}(\\bar{\\phi}_n - nE) \\to N(0,\\sigma), where N(0,\\sigma) denotes the normal distribution with standard deviation \\sigma. As a corollary, we show that if S_1 and S_2 are any two finite generating sets for G, there is an algebraic number lambda_{1,2} depending on S_1 and S_2 such that almost every word of length n in the S_1 metric has word length n\\lambda_{1,2} in the S_2 metric, with error of size O(\\sqrt{n})."}
{"category": "Math", "title": "Cut ideals of K4-minor free graphs are generated by quadrics", "abstract": "Cut ideals are used in algebraic statistics to study statistical models defined by graphs. Intuitively, topological restrictions on the graphs should imply structural statements about the corresponding cut ideals. Several theorems and many computer calculations support that. Sturmfels and Sullivant conjectured that the cut ideal is generated by quadrics if and only if the graph is free of K4-minors. Parts of the conjecture has been resolved by Brennan and Chen, and later by Nagel and Petrovic. We prove the full conjecture by introducing a new type of toric fiber product theorem."}
{"category": "Math", "title": "Algebraic Levi-flat hypervarieties in complex projective space", "abstract": "We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in $\\mathbb{C}$ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety."}
{"category": "Math", "title": "Singularities on normal varieties", "abstract": "In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being Q-Gorenstein or the pair being log Q-Gorenstein. The main features of the theory extend to this setting in a natural way."}
{"category": "Math", "title": "Panel Cointegration with Global Stochastic Trends", "abstract": "This paper studies estimation of panel cointegration models with cross-sectional dependence generated by unobserved global stochastic trends. The standard least squares estimator is, in general, inconsistent owing to the spuriousness induced by the unobservable I(1) trends. We propose two iterative procedures that jointly estimate the slope parameters and the stochastic trends. The resulting estimators are referred to respectively as CupBC (continuously-updated and bias-corrected) and the CupFM (continuously-updated and fully-modified) estimators. We establish their consistency and derive their limiting distributions. Both are asymptotically unbiased and asymptotically mixed normal and permit inference to be conducted using standard test statistics. The estimators are also valid when there are mixed stationary and non-stationary factors, as well as when the factors are all stationary."}
{"category": "Math", "title": "Log-Level Comparison Principle for Small Ball Probabilities", "abstract": "We prove a new variant of comparison principle for logarithmic $L_2$-small ball probabilities of Gaussian processes. As an application, we obtain logarithmic small ball asymptotics for some well-known processes with smooth covariances."}
{"category": "Math", "title": "Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains", "abstract": "We prove the existence of a compact random attractor for the stochastic Benjamin-Bona-Mahony Equation defined on an unbounded domain. This random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity."}
{"category": "Math", "title": "Higgs Bundles and Geometric Structures on Surfaces", "abstract": "This paper concerns the relationship between locally homogeneous geometric structures on topological surfaces and the moduli of polystable Higgs bundles on Riemann surfaces, due to Hitchin and Simpson. In particular we discuss the uniformization of Riemann surfaces by hyperbolic geometry from this viewpoint, and survey more recent developments in this theory."}
{"category": "Math", "title": "The cross covariogram of a pair of polygons determines both polygons, with a few exceptions", "abstract": "The cross covariogram g_{K,L} of two convex sets K and L in R^n is the function which associates to each x in R^n the volume of the intersection of K and L+x. Very recently Averkov and Bianchi [AB] have confirmed Matheron's conjecture on the covariogram problem, that asserts that any planar convex body K is determined by the knowledge of g_{K,K}. The problem of determining the sets from their covariogram is relevant in probability, in statistical shape recognition and in the determination of the atomic structure of a quasicrystal from X-ray diffraction images. We prove that when K and L are convex polygons (and also when K and L are planar convex cones) g_{K,L} determines both K and L, up to a described family of exceptions. These results imply that, when K and L are in these classes, the information provided by the cross covariogram is so rich as to determine not only one unknown body, as required by Matheron's conjecture, but two bodies, with a few classified exceptions. These results are also used by Bianchi [Bia] to prove that any convex polytope P in R^3 is determined by g_{P,P}."}
{"category": "Math", "title": "Vector- valued distributions and Hardy's uncertainty principle for operators", "abstract": "In this paper, we generalise Hardy's uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised."}
{"category": "Math", "title": "Weyl groupoids with at most three objects", "abstract": "We adapt the generalization of root systems of the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we completely classify all finite Weyl groupoids with at most three objects. The classification yields that there exist infinitely many standard, but only 9 exceptional cases. Key words: Nichols algebra, reflection, root system, Weyl groupoid"}
{"category": "Math", "title": "Canonical measures and the dynamical systems of Bergman kernels", "abstract": "In this article, we construct the canonical semipositive current or the canonical measure ($=$ the potential of the canonical semipositive current) on a smooth projective variety of nonnegative Kodaira dimension in terms of a dynamical system of Bergman kernels. This current is considered to be a generalization of a K\\\"{a}hler-Einstein metric and coincides the one constructed independently by J. Song and G. Tian. The major difference between their work and the present article is that they found the canonical measure in terms of K\\\"{a}her-Ricci flows, while I found the canonical measure in terms of the dynamical system of Bergman kernels. Hence the present approach can be viewed as the discrete version of the K\\\"{a}hler-Ricci flow."}
{"category": "Math", "title": "Complex structures on quasi-filiform nilpotent Lie algebras", "abstract": "We present the classification of real nilpotent quasi-filiform Lie algebras endowed with a complex structure. A nilpotent Lie algebra g is called quasi-filiform is the nilindex is equal to dim(n)-2. We recall that the filiform case (nilindex =dim(g)-1) has already been studied."}
{"category": "Math", "title": "A finiteness theorem for dual graphs of surface singularities", "abstract": "Consider a fixed connected, finite graph $\\Gamma$ and equip its vertices with weights $p_i$ which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically Gorenstein surface singularity having $\\Gamma$ as the dual graph of the minimal resolution, the weights $p_i$ of the vertices being the arithmetic genera of the corresponding irreducible components. As a consequence we get that if $\\Gamma$ is not a cycle, then there is a finite number of possibilities of self-intersection numbers which one can attach to the vertices which are of valency $\\geq 2$, such that one gets the dual graph of the minimal resolution of a numerically Gorenstein surface singularity. Moreover, we describe precisely the situations when there exists an infinite number of possibilities for the self-intersections of the component corresponding to some fixed vertex of $\\Gamma$."}
{"category": "Math", "title": "On the $L_{q,p}$-cohomology of Riemannian Manifolds with Negative Curvature", "abstract": "We prove a non-vanishing result for the $L_{q,p}$-cohomology of complete simply-connected Riemannian manifolds with pinched negative curvature."}
{"category": "Math", "title": "Technical report: Adaptivity and optimality of the monotone least squares estimator for four different models", "abstract": "In this paper we will consider the estimation of a monotone regression (or density) function in a fixed point by the least squares (Grenander) estimator. We will show that this estimator is fully adaptive, in the sense that the attained rate is given by a functional relation using the underlying function $f_0$, and not by some smoothness parameter, and that this rate is optimal when considering the class of all monotone functions, in the sense that there exists a sequence of alternative monotone functions $f_1$, such that no other estimator can attain a better rate for both $f_0$ and $f_1$. We also show that under mild conditions the estimator attains the same rate in $L^q$ sense, and we give general conditions for which we can calculate a (non-standard) limiting distribution for the estimator."}
{"category": "Math", "title": "Unconditional basic sequences in spaces of large density", "abstract": "We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show, in particular, the relative consistency with GCH of the statement that every Banach space of density $\\aleph_\\omega$ contains an unconditional basic sequence."}
{"category": "Math", "title": "Cell contamination and branching process in random environment with immigration", "abstract": "We consider a branching model for a population of dividing cells infected by parasites. Each cell receives parasites by inheritance from its mother cell and independent contamination from outside the population. Parasites multiply randomly inside the cell and are shared randomly between the two daughter cells when the cell divides. The law of the number of parasites which contaminate a given cell depends only on whether the cell is already infected or not. We determine the asymptotic behavior of the number of parasites in a cell line, which follows a branching process in random environment with state dependent immigration. We then derive a law of large numbers for the asymptotic proportions of cells with a given number of parasites. The main tools are branching processes in random environment and laws of large numbers for Markov tree."}
{"category": "Math", "title": "A origami of genus 2 with a translation", "abstract": "We study an example of a Teichmueller curve in the moduli space of algebraic curves of genus 2 coming from an origami S. It is particular in that its points admit the Klein four group as a subgroup of the automorphism group. We give an explicit description of its points in terms of affine plane curves, we show that the Teichmueller curve is a nonsingular, affine curve of genus 0 and we determine the number of cusps in the boundary of the moduli space."}
{"category": "Math", "title": "Avoidance of Partially Ordered Generalized Patterns of the form $k$-$\\sigma$-$k$", "abstract": "Sergey Kitaev has shown that the exponential generating function for permutations avoiding the generalized pattern $\\sigma$-$k$, where $\\sigma$ is a pattern without dashes and $k$ is one greater than the biggest element in $\\sigma$, is determined by the exponential generating function for permutations avoiding $\\sigma$. We show that this also holds for permutations avoiding all the generalized patterns $\\sigma_1$-$k_1$, $...$, $\\sigma_n$-$k_n$, where $\\sigma_1$, $...$, $\\sigma_n$ are patterns without dashes and $k_i$ is one greater than the biggest element in $\\sigma_i$. Similarly the exponential generating function for permutations avoiding the partially ordered generalized patterns $k_1$-$\\sigma_1$-$k_1$, $...$, $k_n$-$\\sigma_n$-$k_n$ can be determined from the exponential generating function for permutations avoiding the generalized patterns $\\sigma_1$, $...$, $\\sigma_n$, where $\\sigma_1$, $...$, $\\sigma_n$ are patterns without dashes and $k_i$ is one greater than the largest element in $\\sigma_i$. Using this we construct a bijection between bicolored set partitions and permutations avoiding the partially ordered generalized pattern 3-12-3 (that is, permutations avoiding both the patterns 3-12-4 and 4-12-3). By using this method twice, we find a closed formula for the exponential generating function for permutations avoiding the partially ordered generalized pattern 3-121-3. Finally, we give a complete classification of when single partially ordered generalized patterns have the same set of avoiders."}
{"category": "Math", "title": "The holomorphy conjecture for ideals in dimension two", "abstract": "The holomorphy conjecture predicts that the local Igusa zeta function associated to a hypersurface and a character is holomorphic on $\\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this note we propose the holomorphy conjecture for arbitrary subschemes at the level of the topological zeta function and we prove this conjecture for subschemes defined by an ideal that is generated by a finite number of complex polynomials in two variables."}
{"category": "Math", "title": "Poles of the topological zeta function for plane curves and Newton polyhedra", "abstract": "The local topological zeta function is a rational function associated to a germ of a complex holomorphic function. This function can be computed from an embedded resolution of singularities of the germ. For nondegenerate functions it is also possible to compute it from the Newton polyhedron. Both ways give rise to a set of candidate poles of the topological zeta function, containing all poles. For plane curves, Veys showed how to filter the actual poles out of the candidate poles induced by the resolution graph. In this note we show how to determine from the Newton polyhedron of a nondegenerate plane curve which candidate poles are actual poles."}
{"category": "Math", "title": "Effective categoricity of equivalence Structures", "abstract": "An equivalence structure is a set with a single binary relation, satisfying sentences stating that the relation is an equivalence relation. A computable structure A is said to be $\\Delta^0_\\alpha$ categorical if for any computable structure B isomorphic to A there is a $\\Delta^0_\\alpha$ function witnessing that the two are isomorphic. The present paper gives an exact characterization of $\\Delta^0_\\alpha$ equivalence structures where $\\alpha = 1$ or $\\alpha \\geq 3$. Extensive results for $\\alpha = 2$ are also given, and open cases are exhaustively described."}
{"category": "Math", "title": "Effective categoricity of Abelian p-groups", "abstract": "Let p be a fixed prime. An Abelian p-group is an Abelian group (not necessarily finitely generated) in which every element has for its order some power of p. The countable Abelian p-groups are classified by Ulm's theorem, and Khisamiev characterized the Abelian p-groups with computable copies. A computable structure A is said to be $\\Delta^0_\\alpha$ categorical if for any computable structure B isomorphic to A there is a $\\Delta^0_\\alpha$ function witnessing that the two are isomorphic. The present paper seeks to characterize $\\Delta^0_\\alpha$ categoricity for Abelian p-groups, and results of this kind are given for broad classes of Abelian p-groups and values of $\\alpha$. The remaining open cases are exhaustively described."}
{"category": "Math", "title": "Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane", "abstract": "Let N be a complete, simply-connected surface of constant curvature \\kappa \\leq 0. Moreover, suppose that \\Omega and \\tilde{\\Omega} are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from \\Omega to \\tilde{\\Omega} whose graph is a minimal submanifold of N \\times N."}
{"category": "Math", "title": "On the martingale probem associated to the 2D and 3D Stochastic Navier-Stokes equations", "abstract": "We consider the martingale problem associated to the Navier-Stokes in dimension 2 or 3. Existence is well known and it has been recently shown that markovian transition semi group associated to these equations can be constructed. We study the Kolmogorov operator associated to these equations. It can be defined formally as a differential operator on an infinite dimensional Hilbert space. It can be also defined in an abstract way as the infinitesimal generator of the transition semi group. We explicit cores for these abstract operators and identify them with the concrete differential operators on these cores. In dimension 2, the core is explicit and we can use a classical argument to prove uniqueness for the martingale problem. In dimension 3, we are only able to exhibit a core which is defined abstractly and does not allow to prove uniqueness for the martingale problem. Instead, we exhibit a core for a modified Kolmogorov operator which enables us to prove uniqueness for the martingale problem up to the time the solutions are regular."}
{"category": "Math", "title": "On one property of distances in the infinite random quadrangulation", "abstract": "We show that the Schaeffer's tree for an infinite quadrangulation only changes locally when changing the root of the quadrangulation. This follows from one property of distances in the infinite uniform random quadrangulation."}
{"category": "Math", "title": "On universal Lie nilpotent associative algebras", "abstract": "We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, we completely describe such quotient for i=4 (for i<=3 this was done previously by Feigin and Shoikhet). We also study properties of the ideals M_i(A), e.g. when M_i(A)M_j(A) is contained in M_{i+j-1}(A) (by a result of Gupta and Levin, it is always contained in M_{i+j-2}(A))."}
{"category": "Math", "title": "Analytification is the limit of all tropicalizations", "abstract": "We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings."}
{"category": "Math", "title": "Krull dimension for limit groups II: aligning JSJ decompositions", "abstract": "This is the second paper in a sequence on Krull dimension for limit groups, answering a question of Z. Sela. In it we develop the notion of a resolution of a sequence of limit groups and show how to derive resolutions of low complexity from resolutions of high complexity."}
{"category": "Math", "title": "Aggregation of weakly dependent doubly stochastic processes", "abstract": "The aim of this paper is to extend the aggregation convergence results given in (Dacunha-Castelle and Fermin 2005, Dacunha-Castelle and Fermin 2008) to doubly stochastic linear and nonlinear processes with weakly dependent innovations. First, we introduce a weak dependence notion for doubly stochastic processes, based in the weak dependence definition given in (Doukhan and Louhichi 1999), and we exhibe several models satisfying this notion, such as: doubly stochastic Volterra processes and doubly stochastic Bernoulli scheme with weakly dependent innovations. Afterwards we derive a central limit theorem for the partial aggregation sequence considering weakly dependent doubly stochastic processes. Finally, show a new SLLN for the covariance function of the partial aggregation process in the case of doubly stochastic Volterra processes with interactive innovations. Keywords: Aggregation, weak dependence, doubly stochastic processes, Volterra processes, Bernoulli shift, TCL, SLLN."}
{"category": "Math", "title": "A splitting theorem for holomorphic Banach bundles", "abstract": "This paper is motivated by Grothendieck's splitting theorem. In the 1960s, Gohberg generalized this to a class of Banach bundles. We consider a compact complex manifold $X$ and a holomorphic Banach bundle $E \\to X$ that is a compact perturbation of a trivial bundle in a sense recently introduced by Lempert. We prove that $E$ splits into the sum of a finite rank bundle and a trivial bundle, provided $H^{1}(X, \\O)=0$."}
{"category": "Math", "title": "Matrix subadditivity inequalities and block-matrices", "abstract": "Several subadditivity results and conjectures are given for matrices (or operators), block-matrices, concave functions and norms."}
{"category": "Math", "title": "Moving frames on the twistor space of self-dual positive Einstein 4-manifolds", "abstract": "The twistor space of self-dual positive Einstein manifolds naturally admits two 1-parameter families of Riemannian metrics, one is the family of canonical deformation metrics and the other is the family introduced by B. Chow and D. Yang in 1989. The purpose of this paper is to compare these two families. In particular we compare the Ricci tensor and the behavior under the Ricci flow of these families. As an application, we propose a new proof to the fact that a locally irreducible self-dual positive Einstein 4-manifold is isometric to either $S^4$ with a standard metric or $\\PP^2(\\C)$ with a Fubini-Study metric."}
{"category": "Math", "title": "Alternating, pattern-avoiding permutations", "abstract": "We study the problem of counting alternating permutations avoiding collections of permutation patterns including 132. We construct a bijection between the set S_n(132) of 132-avoiding permutations and the set A_{2n + 1}(132) of alternating, 132-avoiding permutations. For every set p_1, ..., p_k of patterns and certain related patterns q_1, ..., q_k, our bijection restricts to a bijection between S_n(132, p_1, ..., p_k), the set of permutations avoiding 132 and the p_i, and A_{2n + 1}(132, q_1, ..., q_k), the set of alternating permutations avoiding 132 and the q_i. This reduces the enumeration of the latter set to that of the former."}
{"category": "Math", "title": "On a set of transformations of Gaussian random functions", "abstract": "We consider a set of one-dimensional transformations of Gaussian random functions. Under natural assumptions we obtain a connection between $L_2$-small ball asymptotics of the transformed function and of the original one. Also the explicit Karhunen -- Lo\\'eve expansion is obtained for a proper class of Gaussian processes."}
{"category": "Math", "title": "Confidence regions for the multinomial parameter with small sample size", "abstract": "Consider the observation of n iid realizations of an experiment with d>1 possible outcomes, which corresponds to a single observation of a multinomial distribution M(n,p) where p is an unknown discrete distribution on {1,...,d}. In many applications, the construction of a confidence region for p when n is small is crucial. This concrete challenging problem has a long history. It is well known that the confidence regions built from asymptotic statistics do not have good coverage when n is small. On the other hand, most available methods providing non-asymptotic regions with controlled coverage are limited to the binomial case d=2. In the present work, we propose a new method valid for any d>1. This method provides confidence regions with controlled coverage and small volume, and consists of the inversion of the \"covering collection\"' associated with level-sets of the likelihood. The behavior when d/n tends to infinity remains an interesting open problem beyond the scope of this work."}
{"category": "Math", "title": "On the Rankin-Selberg problem: higher power moments of the Riesz mean error term", "abstract": "Let $\\Delta_1(x;\\phi)$ be the error term of the first Riesz means of the Rankin-Selberg zeta function. We study the higher power moments of $\\Delta_1(x;\\phi)$ and derive an asymptotic formula for 3-rd, 4-th and 5-th power moments by using Ivi\\'c 's large value arguments."}
{"category": "Math", "title": "On Berry--Esseen bounds for non-instantaneous filters of linear processes", "abstract": "Let $X_n=\\sum_{i=1}^{\\infty}a_i\\epsilon_{n-i}$, where the $\\epsilon_i$ are i.i.d. with mean 0 and at least finite second moment, and the $a_i$ are assumed to satisfy $|a_i|=O(i^{-\\beta})$ with $\\beta >1/2$. When $1/2<\\beta<1$, $X_n$ is usually called a long-range dependent or long-memory process. For a certain class of Borel functions $K(x_1,...,x_{d+1})$, $d\\ge0$, from ${\\mathcal{R}}^{d+1}$ to $\\mathcal{R}$, which includes indicator functions and polynomials, the stationary sequence $K(X_n,X_{n+1},...,X_{n+d})$ is considered. By developing a finite orthogonal expansion of $K(X_n,...,X_{n+d})$, the Berry--Esseen type bounds for the normalized sum $Q_N/\\sqrt{N},Q_N=\\sum_{n=1}^N(K(X_ n,...,X_{n+d})-\\mathrm{E}K(X_n,...,X_{n+d}))$ are obtained when $Q_N/\\sqrt{N}$ obeys the central limit theorem with positive limiting variance."}
{"category": "Math", "title": "Number of Least Area Planes in Gromov Hyperbolic 3-Spaces", "abstract": "We show that for a generic simple closed curve C in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exist a unique least area plane P in X with asymptotic boundary C. This result has interesting topological applications for constructions of canonical 2-dimensional objects in 3-manifolds."}
{"category": "Math", "title": "The dbar steepest descent method for orthogonal polynomials on the real line with varying weights", "abstract": "We obtain Plancherel-Rotach type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form $e^{-NV(x)}$ on the real line, assuming that $V$ has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. Our methodology involves developing a new technique of asymptotic analysis for matrix Riemann-Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior."}
{"category": "Math", "title": "On the symmetry of arithmetical functions in almost all short intervals,IV", "abstract": "We study the arithmetic (real) function, with f 'essentially bounded'. In particular, we obtain non-trivial bounds, through f 'correlations', for the 'Selberg integral' and the 'symmetry integral' of f in almost all short intervals $[x-h,x+h]$, $N\\le x\\le 2N$, beyond the 'classical' level, up to a very high level of distribution (for $h$ not too small). This time we go beyond Large Sieve inequality. Precisely, our method applies Weil bound for Kloosterman sums."}
{"category": "Math", "title": "Structure of the spaces of matrix monotone functions and of matrix convex functions and Jensen's type inequality for operators", "abstract": "Let $n \\in \\N$ and $M_n$ be the algebra of $n \\times n$ matrices. We call a function $f$ matrix monotone of order $n$ or $n$-monotone in short whenever the inequality $f(a) \\leq f(b)$ holds for every pair of selfadjoint matrices $a, b \\in M_n$ such that $a \\leq b$ and all eigenvalues of $a$ and $b$ are contained in $I$. Matrix convex (concave) functions on $I$ are similarily defined. The spaces for $n$-monotone functions and $n$-convex functions are written as $P_n(I)$ and $K_n(I)$. In this note we discuss several assertions at each leven $n$ for which we regard themas the problems of double piling structure of those sequences $\\{P_n(I)\\}_{n\\in\\N}$ and $\\{K_n(I)\\}_{n\\in\\N}$. In order to see clear insight of the aspect of the problems, however, we choose the following three main assertions among them and discuss their mutual dependence: \\begin{enumerate} \\item[(i)] $f(0)\\leq 0$ and $f$ is $n$-convex in $[0,\\alpha)$, \\item[(ii)] For each matrix $a$ with its spectrum in $[0,\\alpha)$ and a contraction $c$ in the matrix algebra $M_n$, \\[ f(c^{\\star}a c)\\leq c^{\\star}f(a)c, \\] \\item[(iii)] The functon $g(t)/t$ is $n$-monotone in $(0,\\alpha)$. \\end{enumerate} In particular, we show that for any $n \\in \\N$ two conditions $(ii)$ and $(iii)$ are equivalent."}
{"category": "Math", "title": "Stochastic analysis on Gaussian space applied to drift estimation", "abstract": "In this paper we consider the nonparametric functional estimation of the drift of Gaussian processes using Paley-Wiener and Karhunen-Lo\\`eve expansions. We construct efficient estimators for the drift of such processes, and prove their minimaxity using Bayes estimators. We also construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and stochastic analysis on Gaussian space, in which superharmonic functionals of the process paths play a particular role. Our results are illustrated by numerical simulations and extend the construction of James-Stein type estimators for Gaussian processes by Berger and Wolper."}
{"category": "Math", "title": "Currents and Flat Chains Associated to Varifolds, with an Application to Mean Curvature Flow", "abstract": "We prove under suitable hypotheses that convergence of integral varifolds implies convergence of associated mod 2 flat chains and subsequential convergence of associated integer-multiplicity rectifiable currents. The convergence results imply restrictions on the kinds of singularities that can occur in mean curvature flow."}
{"category": "Math", "title": "The Kolmogorov operator associated to a Burgers SPDE in spaces of continuous functions", "abstract": "We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation."}
{"category": "Math", "title": "Equiangular tight frames from complex Seidel matrices containing cube roots of unity", "abstract": "We derive easily verifiable conditions which characterize when complex Seidel matrices containing cube roots of unity have exactly two eigenvalues. The existence of such matrices is equivalent to the existence of equiangular tight frames for which the inner product between any two frame vectors is always a common multiple of the cube roots of unity. We also exhibit a relationship between these equiangular tight frames, complex Seidel matrices, and highly regular, directed graphs. We construct examples of such frames with arbitrarily many vectors."}
{"category": "Math", "title": "A symmetry property of some harmonic algebraic curves", "abstract": "The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots $z_i$ of a complex polynomial $P$ lie on the unit circle $\\U$, the points of $\\U$ different from the $z_i$, and such that $\\Arg(P(z))=\\theta$, form a regular $n$-gon, where $n$ is the degree of $P$."}
{"category": "Math", "title": "The growth of a C_0-semigroup characterised by its cogenerator", "abstract": "We characterise contractivity, boundedness and polynomial boundedness for a C_0-semigroup on a Banach space in terms of its cogenerator V (or the Cayley transform of the generator) or its resolvent. In particular, we extend results of Gomilko and Brenner, Thomee and show that polynomial boundedness of a semigroup implies polynomial boundedness of its cogenerator. As is shown by an example, the result is optimal. For analytic semigroups we show that the converse holds, i.e., polynomial boundedness of the cogenerators implies polynomial boundedness of the semigroup. In addition, we show by simple examples in (C^2,\\|\\cdot\\|_p), p \\neq 2, that our results on the characterisation of contractivity are sharp. These examples also show that the famous Foias-Sz.-Nagy theorem on cogenerators of contraction semigroups on Hilbert spaces fails in (C^2,\\|\\cdot\\|_p) for p\\neq 2."}
{"category": "Math", "title": "Operators of Harmonic Analysis in Weighted Spaces with Non-standard Growth", "abstract": "Last years there was increasing an interest to the so called function spaces with non-standard growth, known also as variable exponent Lebesgue spaces. For weighted such spaces on homogeneous spaces, we develop a certain variant of Rubio de Francia's extrapolation theorem. This extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis, such as maximal and singular operators, potential operators, Fourier multipliers, dominants of partial sums of trigonometric Fourier series and others, in weighted Lebesgue spaces with variable exponent. There are also given their vector-valued analogues."}
{"category": "Math", "title": "Codings of separable compact subsets of the first Baire class", "abstract": "Let $X$ be a Polish space and $K$ a separable compact subset of the first Baire class on $X$. For every sequence $\\bs$ dense in $\\kk$, the descriptive set-theoretic properties of the set \\[ \\lbf=\\{L\\in[\\nn]: (f_n)_{n\\in L} \\text{is pointwise convergent}\\} \\] are analyzed. It is shown that if $K$ is not first countable, then $\\lbf$ is $\\PB^1_1$-complete. This can also happen even if $K$ is a pre-metric compactum of degree at most two, in the sense of S. Todorcevic. However, if $K$ is of degree exactly two, then $\\lbf$ is always Borel. A deep result of G. Debs implies that $\\lbf$ contains a Borel cofinal set and this gives a tree-representation of $\\kk$. We show that classical ordinal assignments of Baire-1 functions are actually $\\PB^1_1$-ranks on $\\kk$. We also provide an example of a $\\SB^1_1$ Ramsey-null subset $A$ of $[\\nn]$ for which there does not exist a Borel set $B\\supseteq A$ such that the difference $B\\setminus A$ is Ramsey-null."}
{"category": "Math", "title": "Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces", "abstract": "We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\\cdot)}(\\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result of the boundedness of the Hardy-Littlewood maximal operator in the space $L^{p(\\cdot)}(X,\\varrho)$ over a metric measure space $X$ satisfying the doubling condition. In the case where $X$ is bounded, the weight function satisfies a certain version of a general Muckenhoupt-type condition For a bounded or unbounded $X$ we also consider a class of weights of the form $\\varrho(x)=[1+d(x_0,x)]^{\\bt_\\infty}\\prod_{k=1}^m w_k(d(x,x_k))$, $x_k\\in X$, where the functions $w_k(r)$ have finite upper and lower indices $m(w_k)$ and $M(w_k)$. Some of the results are new even in the case of constant $p$."}
{"category": "Math", "title": "Sample autocovariances of long-memory time series", "abstract": "We find the asymptotic distribution of the sample autocovariances of long-memory processes in cases of finite and infinite fourth moment. Depending on the interplay of assumptions on moments and the intensity of dependence, there are three types of convergence rates and limit distributions. In particular, a normal approximation with the standard rate does not always hold in practically relevant cases."}
{"category": "Math", "title": "On filling families of finite subsets of the Cantor set", "abstract": "Let $\\ee>0$ and $\\fff$ be a family of finite subsets of the Cantor set $\\ccc$. Following D. H. Fremlin, we say that $\\fff$ is $\\ee$-filling over $\\ccc$ if $\\fff$ is hereditary and for every $F\\subseteq\\ccc$ finite there exists $G\\subseteq F$ such that $G\\in\\fff$ and $|G|\\geq\\ee |F|$. We show that if $\\fff$ is $\\ee$-filling over $\\ccc$ and $C$-measurable in $[\\ccc]^{<\\omega}$, then for every $P\\subseteq\\ccc$ perfect there exists $Q\\subseteq P$ perfect with $[Q]^{<\\omega}\\subseteq\\fff$. A similar result for weaker versions of density is also obtained."}
{"category": "Math", "title": "A classification of separable Rosenthal compacta and its applications", "abstract": "The present work consists of three parts. In the first one we determine the prototypes of separable Rosenthal compacta and we provide a classification theorem. The second part concerns an extension of a theorem of S. Todorcevic. The last one is devoted to applications."}
{"category": "Math", "title": "On pairs of definable orthogonal families", "abstract": "We introduce the notion of an M-family of infinite subsets of $\\nn$ which is implicitly contained in the work of A. R. D. Mathias. We study the structure of a pair of orthogonal hereditary families $\\aaa$ and $\\bbb$, where $\\aaa$ is analytic and $\\bbb$ is $C$-measurable and an M-family."}
{"category": "Math", "title": "A strong boundedness result for separable Rosenthal compacta", "abstract": "It is proved that the class of separable Rosenthal compacta on the Cantor set having a uniformly bounded dense sequence of continuous functions, is strongly bounded."}
{"category": "Math", "title": "Definability under duality", "abstract": "It is shown that if $A$ is an analytic class of separable Banach spaces with separable dual, then the set $A^*=\\{Y:\\exists X\\in A \\text{with} Y\\cong X^*\\}$ is analytic. The corresponding result for pre-duals is false."}
{"category": "Math", "title": "On antichains of spreading models of Banach spaces", "abstract": "We show that for every separable Banach space $X$, either $\\spw(X)$ (the set of all spreading models of $X$ generated by weakly-null sequences in $X$, modulo equivalence) is countable, or $\\spw(X)$ contains an antichain of the size of the continuum. This answers a question of S. J. Dilworth, E. Odell and B. Sari."}
{"category": "Math", "title": "Limit distributions for the problem of collecting pairs", "abstract": "Let $N_n=\\{1,2,...,n\\}$. Elements are drawn from the set $N_n$ with replacement, assuming that each element has probability $1/n$ of being drawn. We determine the limiting distributions for the waiting time until the given portion of pairs $jj$, $j\\in N_n$, is sampled. Exact distributions of some related random variables and their characteristics are also obtained."}
{"category": "Math", "title": "Braid ordering and knot genus", "abstract": "The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering."}
{"category": "Math", "title": "On classes of Banach spaces admitting \"small\" universal spaces", "abstract": "We characterize those classes $\\ccc$ of separable Banach spaces admitting a separable universal space $Y$ (that is, a space $Y$ containing, up to isomorphism, all members of $\\ccc$) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class $\\mathrm{NU}$ of non-universal separable Banach spaces is strongly bounded. This settles in the affirmative the main conjecture form \\cite{AD}. Our approach is based, among others, on a construction of $\\llll_\\infty$-spaces, due to J. Bourgain and G. Pisier. As a consequence we show that there exists a family $\\{Y_\\xi:\\xi<\\omega_1\\}$ of separable, non-universal, $\\llll_\\infty$-spaces which uniformly exhausts all separable Banach spaces. A number of other natural classes of separable Banach spaces are shown to be strongly bounded as well."}
{"category": "Math", "title": "On the elicitation of continuous, symmetric, unimodal distributions", "abstract": "In this brief note, we highlight some difficulties that can arise when fitting a continuous, symmetric, unimodal distribution to a set of expert's judgements. A simple analysis shows it is possible to fit a Cauchy distribution to an expert's beliefs when their beliefs actually follow a normal distribution. This example stresses the need for careful distribution fitting and for feedback to the expert about what the fitted distribution implies about their beliefs."}
{"category": "Math", "title": "On unconditionally saturated Banach spaces", "abstract": "We prove a structural property of the class of unconditionally saturated separable Banach spaces. We show, in particular, that for every analytic set $\\aaa$, in the Effros-Borel space of subspaces of $C[0,1]$, of unconditionally saturated separable Banach spaces, there exists an unconditionally saturated Banach space $Y$, with a Schauder basis, that contains isomorphic copies of every space $X$ in the class $\\aaa$."}
{"category": "Math", "title": "On the number of unique expansions in non-integer bases", "abstract": "Let $q > 1$ be a real number and let $m=m(q)$ be the largest integer smaller than $q$. It is well known that each number $x \\in J_q:=[0, \\sum_{i=1}^{\\infty} m q^{-i}]$ can be written as $x=\\sum_{i=1}^{\\infty}{c_i}q^{-i}$ with integer coefficients $0 \\le c_i < q$. If $q$ is a non-integer, then almost every $x \\in J_q$ has continuum many expansions of this form. In this note we consider some properties of the set $\\mathcal{U}_q$ consisting of numbers $x \\in J_q$ having a unique representation of this form. More specifically, we compare the size of the sets $\\mathcal{U}_q$ and $\\mathcal{U}_r$ for values $q$ and $r$ satisfying $1< q < r$ and $m(q)=m(r)$."}
{"category": "Math", "title": "Absolute Chow-Kuenneth decomposition for rational homogeneous bundles and for log homogeneous varieties", "abstract": "In this paper, we investigate Murre's conjecture on the existence of a Chow--Kuenneth decomposition for a rational homogeneous bundle $Z\\to S$ over a smooth variety, defined over complex numbers. Chow-K\\\"unneth decomposition is exhibited for $Z$ whenever $S$ has a Chow--Kuenneth decomposition. The same conclusion holds for a class of log homogeneous varieties, studied by M. Brion."}
{"category": "Math", "title": "Asymptotic Nets and Discrete Affine Surfaces with Indefinite Metric", "abstract": "Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres."}
{"category": "Math", "title": "KW-sections for exceptional type Vinberg's $\\theta$-groups", "abstract": "Let $k$ be an algebraically closed field of characteristic not equal to 2 or 3, let $G$ be an almost simple algebraic group of type $F_4$, $G_2$ or $D_4$ and let $\\theta$ be an automorphism of $G$ of finite order, coprime to the characteristic. In this paper we consider the $\\theta$-group (in the sense of Vinberg) associated to these choices; we classify the positive rank automorphisms and give their Kac diagrams and we describe the little Weyl group in each case. As a result we show that all such $\\theta$-groups have KW-sections, confirming a conjecture of Popov in these cases."}
{"category": "Math", "title": "A 3-Variable Bracket", "abstract": "Kauffman's bracket is an invariant of regular isotopy of knots and links which since its discovery in 1985 it has been used in many different directions: (a) it implies an easy proof of the invariance of (in fact, it is equivalent to) the Jones polynomial; (b) it is the basic ingredient in a completely combinatorial construction for quantum 3-manifold invariants; (c) by its fundamental character it plays an important role in some theories in Physics; it has been used in the context of virtual links; it has connections with many objects other objects in Mathematics and Physics. I show in this note that, surprisingly enough, the same idea that produces the bracket can be slightly modified to produce algebraically stronger regular isotopy and ambient isotopy invariants living in the quotient ring $R/I$, where the ring $R$ and the ideal $I$ are: \\begin{center} $R=\\Z[\\alpha,\\beta,\\delta]$, $I=< p_1, p_2 >$, with $p_1=\\alpha^2 \\delta + 2 \\alpha \\beta \\delta ^2 -\\delta ^2+\\beta ^2 \\delta, p_2=\\alpha \\beta \\delta ^3+\\alpha ^2 \\delta ^2+\\beta ^2 \\delta ^2+\\alpha \\beta \\delta -\\delta.$ \\end{center} It is easy to prove that any pair of links distinguished by the usual bracket is also distinguishable by the new invariant. The contrary is not necessarily true. However, a explicit example of a pair of knots not distinguished by the bracket and distinguished by this new invariant is an open problem."}
{"category": "Math", "title": "Manifolds covered by lines and extremal rays", "abstract": "Let $X$ be a smooth complex projective variety and let $H \\in \\pic(X)$ be an ample line bundle. Assume that $X$ is covered by rational curves with degree one with respect to $H$ and with anticanonical degree greater than or equal to $(\\dim X -1)/2$. We prove that there is a covering family of such curves whose numerical class spans an extremal ray in the cone of curves $\\cone(X)$."}
{"category": "Math", "title": "Estimation of the Brownian dimension of a continuous It\\^{o} process", "abstract": "In this paper, we consider a $d$-dimensional continuous It\\^{o} process which is observed at $n$ regularly spaced times on a given time interval $[0,T]$. This process is driven by a multidimensional Wiener process and our aim is to provide asymptotic statistical procedures which give the minimal dimension of the driving Wiener process, which is between 0 (a pure drift) and $d$. We exhibit several different procedures, all similar to asymptotic testing hypotheses."}
{"category": "Math", "title": "Infinite smooth Lyndon words", "abstract": "In a recent paper, Brlek, Jamet and Paquin showed that some extremal infinite smooth words are also infinite Lyndon words. This result raises a natural question: are they the only ones? If no, what do the infinite smooth words that are also Lyndon words look like? In this paper, we give the answer, proving that the only infinite smooth Lyndon words are $m_{\\{a<b\\}}$, with $a,b$ even, $m_{\\{1<b\\}}$ and $\\Delta^{-1}_1(m_{\\{1<b\\}})$, with $b$ odd, where $m_\\A$ is the minimal infinite smooth word with respect to the lexicographic order over a numerical alphabet $\\A$ and $\\Delta$ is the run-length encoding function."}
{"category": "Math", "title": "Stochastic calculus for convoluted L\\'{e}vy processes", "abstract": "We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation L\\'{e}vy process with a Volterra-type kernel. This class of processes contains, for example, fractional L\\'{e}vy processes as studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an It\\^{o} formula which separates the different contributions from the memory due to the convolution and from the jumps."}
{"category": "Math", "title": "Fully closed maps and non-metrizable higher-dimensional Anderson-Choquet continua", "abstract": "Fedorchuk's fully closed (continuous) maps and resolutions are applied in constructions of non-metrizable higher-dimensional analogues of Anderson, Choquet, and Cook's continua. Certain theorems on dimension-lowering maps are proved for inductive dimensions and fully closed maps from spaces that need not be hereditarily normal, and some examples of continua have non-coinciding dimensions."}
{"category": "Math", "title": "On the Form of Odd Perfect Gaussian Integers", "abstract": "We extend the sum-of-divisors function to the complex plane via the Gaussian integers. Then we prove a modified form of Euler's classification of odd perfect numbers."}
{"category": "Math", "title": "New examples of p-adically rigid automorphic forms", "abstract": "We give examples of cohomological automorphic forms for unitary groups which are $p$-adically rigid."}
{"category": "Math", "title": "Invariant measures for interval maps with critical points and singularities", "abstract": "We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure."}
{"category": "Math", "title": "Asymptotic Behavior of Systems involving Delays: Preliminary Results", "abstract": "This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed) linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics. The proofs are based on a Perron type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtained results are also applied to investigate the global stability of the perturbed equation based on that of its corresponding limiting equation."}
{"category": "Math", "title": "Quadrature formulas for integrals transforms generated by orthogonal polynomials", "abstract": "By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a modified Fourier Transform in the case of the Hermite polynomials, a Bessel Transform in the case of the Laguerre polynomials and to an Appell Transform in the case of the Jacobi polynomials."}
{"category": "Math", "title": "On special values of spinor L-functions of Siegel cusp eigenforms of genus 3", "abstract": "We compute the special values for the spinor L-function L(s,F12) in the critical strip s={12,...,19}, where F12 is the unique (up to a scalar) Siegel cusp form of degree 3 and weight 12, which was constructed by Miyawaki. These values are proportional to the product of Petersson inner products of Ramanujan's Delta by itself and the cusp form of weight 20 for SL(2,Z) by itself by a rational number and some power of Pi. We also verify this result numerically using Dokchitser's ComputeL PARI package. To our knowledge this is the first example of a spinor L-function of Siegel cusp forms of degree 3, when the special values can be computed explicitly."}
{"category": "Math", "title": "Eight Hateful Sequences", "abstract": "In his July 1974 Scientific American column, Martin Gardner mentioned the Handbook of Integer Sequences, which then contained 2372 sequences. Today the On-Line Encyclopedia of Integer Sequences (the OEIS) contains 140000 sequences. This paper discusses eight of them, suggested by the theme of the Eighth Gathering For Gardner: they are all infinite, and all 'ateful in one way or another. Each one is connected with an unsolved problem. The sequences are related to: hateful numbers, Angelini's 1995 puzzle, the persistence of a number, Alekseyev's 123 sequence, the curling number conjecture, Quet's prime-generating recurrence, the traveling salesman's problem, and the Riemann Hypothesis."}
{"category": "Math", "title": "Functoriality in Morse theory on closed manifolds", "abstract": "We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct map on homology. This association does not in general respect composition. We give sufficient conditions under which composition is preserved. As an application we provide a new proof that the cup product as defined in Morse theory on the chain level agrees with the cup product in singular cohomology. In appendices we present a proof (due to Paul Biran) that the unstable manifolds of a Morse-Smale system are the open cells of a CW structure on the underlying manifold, and also we show that the Morse complex of the triple is canonically isomorphic to the cellular complex of the CW structure. This gives a new proof that the Morse complex is actually a complex and that it computes the homology of the manifold."}
{"category": "Math", "title": "Spectral gaps of the one-dimensional Schr\\\"odinger operators with singular periodic potentials", "abstract": "The behaviour of the lengths of spectral gaps $\\{\\gamma_{n}(q)\\}_{n=1}^{\\infty}$ of the Hill-Schr\\\"odinger operators S(q)u=-u''+q(x)u,\\quad u\\in \\mathrm{Dom}(S(q)) with real-valued 1-periodic distributional potentials $q(x)\\in H_{1{-}per}^{-1}(\\mathbb{R})$ is studied. We show that they exhibit the same behaviour as the Fourier coefficients $\\{\\widehat{q}(n)\\}_{n=-\\infty}^{\\infty}$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,\\varphi}$, $s>-1$, $\\varphi\\in \\mathrm{SV}$. The case $q(x)\\in L_{1{-}per}^{2}(\\mathbb{R})$, $s\\in \\mathbb{Z}_{+}$, $\\varphi\\equiv 1$ corresponds to the Marchenko-Ostrovskii Theorem."}
{"category": "Math", "title": "An Improvement on Olson's Constant for Z_p+Z_p", "abstract": "For a prime number p greater than 6000, the Olson's constant for the group Z_p+Z_p is given by Ol(Z_p+Z_p)=p-1+Ol(Z_p)."}
{"category": "Math", "title": "On the adjoint quotient of Chevalley groups over arbitrary base schemes", "abstract": "For a split semisimple Chevalley group scheme G with Lie algebra g over an arbitrary base scheme S, we consider the quotient of g by the adjoint action of G. We study in detail the structure of g over S. Given a maximal torus T with Lie algebra t and associated Weyl group W, we show that the Chevalley morphism t/W -> g/G is an isomorphism except for the group Sp_{2n} over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient g -> g/G commutes, or does not commute, with base change on S."}
{"category": "Math", "title": "Modular forms on noncongruence subgroups and Atkin-Swinnerton-Dyer relations", "abstract": "We give new examples of weight three cusp forms on noncongruence subgroups of SL(2, Z) whose Scholl representation is modular and which satisfy three term Atkin-Swinnerton-Dyer relations."}
{"category": "Math", "title": "Orderings of the rationals and dynamical systems", "abstract": "This paper is devoted to a systematic study of a class of binary trees encoding the structure of rational numbers both from arithmetic and dynamical point of view. The paper is divided into two parts. The first one is a critical review of rather standard topics such as Stern-Brocot and Farey trees and their connections with continued fraction expansion and the question mark function. In the second part we introduce a class of one-dimensional maps which can be used to generate the binary trees in different ways and study their ergodic properties. This also leads us to study some random processes (Markov chains and martingales) arising in a natural way in this context."}
{"category": "Math", "title": "Yoneda representations of flat functors and classifying toposes", "abstract": "In this paper, we first introduce a technique that we call \"Yoneda representation of flat functors\", based on ideas from indexed category theory; then we provide applications of this technique to the theory of classifying toposes. Specifically, we obtain results characterizing the models of a theory classified by a topos of the form Sh(C,J) in terms of the models of a theory classified by the topos [C^op, Set]."}
{"category": "Math", "title": "On the moduli space of Donaldson-Thomas instantons", "abstract": "In alignment with a programme by Donaldson and Thomas [DT], Thomas [Th] constructed a deformation invariant for smooth projective Calabi-Yau threefolds, which is now called the Donaldson-Thomas invariant, from the moduli space of (semi-)stable sheaves by using algebraic geometry techniques. In the same paper [Th], Thomas noted that certain perturbed Hermitian-Einstein equations might possibly produce an analytic theory of the invariant. This article sets up the equations on symplectic 6-manifolds, and gives the local model and structures of the moduli space coming from the equations. We then describe a Hitchin-Kobayashi style correspondence for the equations on compact K\\\"ahler threefolds, which turns out to be a special case of results by Alvarez-Consul and Garcia-Prada [AG]."}
{"category": "Math", "title": "A weak compactness theorem of the Donaldson-Thomas instantons on compact K\\\"ahler threefolds", "abstract": "In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the Donaldson-Thomas invariants. In this article, we consider the equation on compact K\\\"ahler threefolds, and study some of analytic properties of solutions to them, using analytic methods in higher-dimensional Yang-Mills theory developed by Nakajima and Tian with some additional arguments concerning an extra non-linear term coming from the Higgs fields. We prove that a sequence of solutions to the Donaldson-Thomas equation of a unitary vector bundle over a compact K\\\"ahler threefold has a converging subsequence outside a closed subset whose real 2-dimensional Hausdorff measure is finite, provided that the L^2-norms of the Higgs fields are uniformly bounded. We also prove an n/2-compactness theorem of solutions to the equations on compact K\\\"ahler threefolds."}
{"category": "Math", "title": "The Donaldson-Thomas instantons on compact Kahler threefolds and a convergence", "abstract": "The contents of this article are now presented in the appendix of arXiv:0805.2195v2."}
{"category": "Math", "title": "Mass formula for self-orthogonal codes over Z_{p^2}", "abstract": "In this note, we establish a mass formula for self-orthogonal codes over Z_{p^2}, where p is a prime. As a consequence, an alternative proof of the known mass formulas for self-dual codes over Z_{p^2} is obtained. We also establish a mass formula for even quaternary codes, which includes a mass formula for Type II quaternary codes as a special case."}
{"category": "Math", "title": "Augmented GARCH sequences: Dependence structure and asymptotics", "abstract": "The augmented GARCH model is a unification of numerous extensions of the popular and widely used ARCH process. It was introduced by Duan and besides ordinary (linear) GARCH processes, it contains exponential GARCH, power GARCH, threshold GARCH, asymmetric GARCH, etc. In this paper, we study the probabilistic structure of augmented $\\mathrm {GARCH}(1,1)$ sequences and the asymptotic distribution of various functionals of the process occurring in problems of statistical inference. Instead of using the Markov structure of the model and implied mixing properties, we utilize independence properties of perturbed GARCH sequences to directly reduce their asymptotic behavior to the case of independent random variables. This method applies for a very large class of functionals and eliminates the fairly restrictive moment and smoothness conditions assumed in the earlier theory. In particular, we derive functional CLTs for powers of the augmented GARCH variables, derive the error rate in the CLT and obtain asymptotic results for their empirical processes under nearly optimal conditions."}
{"category": "Math", "title": "Density estimation with heteroscedastic error", "abstract": "It is common, in deconvolution problems, to assume that the measurement errors are identically distributed. In many real-life applications, however, this condition is not satisfied and the deconvolution estimators developed for homoscedastic errors become inconsistent. In this paper, we introduce a kernel estimator of a density in the case of heteroscedastic contamination. We establish consistency of the estimator and show that it achieves optimal rates of convergence under quite general conditions. We study the limits of application of the procedure in some extreme situations, where we show that, in some cases, our estimator is consistent, even when the scaling parameter of the error is unbounded. We suggest a modified estimator for the problem where the distribution of the errors is unknown, but replicated observations are available. Finally, an adaptive procedure for selecting the smoothing parameter is proposed and its finite-sample properties are investigated on simulated examples."}
{"category": "Math", "title": "Consistency of the $\\alpha$-trimming of a probability. Applications to central regions", "abstract": "The sequence of $\\alpha$-trimmings of empirical probabilities is shown to converge, in the Painlev\\'{e}--Kuratowski sense, on the class of probability measures endowed with the weak topology, to the $\\alpha$-trimming of the population probability. Such a result is applied to the study of the asymptotic behaviour of central regions based on the trimming of a probability."}
{"category": "Math", "title": "A New Differential Test for Series of Positive Terms", "abstract": "A new differential test for series of positive terms is proved. Let f(x) be a positive continuous function corresponded to a series of positive terms f(k), and g(x) is a derivative of reciprocal of f(x). Then, the convergence and divergence of the series may be determined from a value of fgx for enough large x. The rest may make the limit form, and is universal and complete."}
{"category": "Math", "title": "Analytic perturbations and systematic bias in statistical modeling and inference", "abstract": "In this paper we provide a comprehensive study of statistical inference in linear and allied models which exhibit some analytic perturbations in their design and covariance matrices. We also indicate a few potential applications. In the theory of perturbations of linear operators it has been known for a long time that the so-called ``singular perturbations'' can have a big impact on solutions of equations involving these operators even when their size is small. It appears that so far the question of whether such undesirable phenomena can also occur in statistical models and their solutions has not been formally studied. The models considered in this article arise in the context of nonlinear models where a single parameter accounts for the nonlinearity."}
{"category": "Math", "title": "Pranab Kumar Sen: Life and works", "abstract": "In this article, we describe briefly the highlights and various accomplishments in the personal as well as the academic life of Professor Pranab Kumar Sen."}
{"category": "Math", "title": "Smooth estimation of mean residual life under random censoring", "abstract": "We propose here a smooth estimator of the mean residual life function based on randomly censored data. This is derived by smoothing the product-limit estimator using the Chaubey-Sen technique (Chaubey and Sen (1998)). The resulting estimator does not suffer from boundary bias as is the case with standard kernel smoothing. The asymptotic properties of the estimator are investigated. We establish strong uniform consistency and asymptotic normality. This complements the work of Chaubey and Sen (1999) which considered a similar estimation procedure in the case of complete data. It is seen that the properties are similar, though technically more difficult to prove, to those in the complete data case with appropriate modifications due to censoring."}
{"category": "Math", "title": "Conformal Metrics", "abstract": "This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and conformal differential geometry."}
{"category": "Math", "title": "Order restricted inference for comparing the cumulative incidence of a competing risk over several populations", "abstract": "There is a substantial literature on testing for the equality of the cumulative incidence functions associated with one specific cause in a competing risks setting across several populations against specific or all alternatives. In this paper we propose an asymptotically distribution-free test when the alternative is that the incidence functions are linearly ordered, but not equal. The motivation stems from the fact that in many examples such a linear ordering seems reasonable intuitively and is borne out generally from empirical observations. These tests are more powerful when the ordering is justified. We also provide estimators of the incidence functions under this ordering constraint, derive their asymptotic properties for statistical inference purposes, and show improvements over the unrestricted estimators when the order restriction holds."}
{"category": "Math", "title": "Statistical inference under order restrictions on both rows and columns of a matrix, with an application in toxicology", "abstract": "We present a general methodology for performing statistical inference on the components of a real-valued matrix parameter for which rows and columns are subject to order restrictions. The proposed estimation procedure is based on an iterative algorithm developed by Dykstra and Robertson (1982) for simple order restriction on rows and columns of a matrix. For any order restrictions on rows and columns of a matrix, sufficient conditions are derived for the algorithm to converge in a single application of row and column operations. The new algorithm is applicable to a broad collection of order restrictions. In practice, it is easy to design a study such that the sufficient conditions derived in this paper are satisfied. For instance, the sufficient conditions are satisfied in a balanced design. Using the estimation procedure developed in this article, a bootstrap test for order restrictions on rows and columns of a matrix is proposed. Computer simulations for ordinal data were performed to compare the proposed test with some existing test procedures in terms of size and power. The new methodology is illustrated by applying it to a set of ordinal data obtained from a toxicological study."}
{"category": "Math", "title": "Totally free arrangements of hyperplanes", "abstract": "A central arrangement $\\A$ of hyperplanes in an $\\ell$-dimensional vector space $V$ is said to be {\\it totally free} if a multiarrangement $(\\A, m)$ is free for any multiplicity $ m : \\A\\to \\Z_{> 0}$. It has been known that $\\A$ is totally free whenever $\\ell \\le 2$. In this article, we will prove that there does not exist any totally free arrangement other than the obvious ones, that is, a product of one-dimensional arrangements and two-dimensional ones."}
{"category": "Math", "title": "On the structure of a family of probability generating functions induced by shock models", "abstract": "We explore conditions for a class of functions defined via an integral representation to be a probability generating function of some positive integer valued random variable. Interest in and research on this question is motivated by an apparently surprising connection between a family of classic shock models due to Esary et. al. (1973) and the negatively aging nonparametric notion of ``strongly decreasing failure rate'' (SDFR) introduced by Bhattacharjee (2005). A counterexample shows that there exist probability generating functions with our integral representation which are not discrete SDFR, but when used as shock resistance probabilities can give rise to a SDFR survival distribution in continuous time."}
{"category": "Math", "title": "Adaptive approximate Bayesian computation", "abstract": "Sequential techniques can enhance the efficiency of the approximate Bayesian computation algorithm, as in Sisson et al.'s (2007) partial rejection control version. While this method is based upon the theoretical works of Del Moral et al. (2006), the application to approximate Bayesian computation results in a bias in the approximation to the posterior. An alternative version based on genuine importance sampling arguments bypasses this difficulty, in connection with the population Monte Carlo method of Cappe et al. (2004), and it includes an automatic scaling of the forward kernel. When applied to a population genetics example, it compares favourably with two other versions of the approximate algorithm."}
{"category": "Math", "title": "A Bayesian test for excess zeros in a zero-inflated power series distribution", "abstract": "Power series distributions form a useful subclass of one-parameter discrete exponential families suitable for modeling count data. A zero-inflated power series distribution is a mixture of a power series distribution and a degenerate distribution at zero, with a mixing probability $p$ for the degenerate distribution. This distribution is useful for modeling count data that may have extra zeros. One question is whether the mixture model can be reduced to the power series portion, corresponding to $p=0$, or whether there are so many zeros in the data that zero inflation relative to the pure power series distribution must be included in the model i.e., $p\\geq0$. The problem is difficult partially because $p=0$ is a boundary point. Here, we present a Bayesian test for this problem based on recognizing that the parameter space can be expanded to allow $p$ to be negative. Negative values of $p$ are inconsistent with the interpretation of $p$ as a mixing probability, however, they index distributions that are physically and probabilistically meaningful. We compare our Bayesian solution to two standard frequentist testing procedures and find that using a posterior probability as a test statistic has slightly higher power on the most important ranges of the sample size $n$ and parameter values than the score test and likelihood ratio test in simulations. Our method also performs well on three real data sets."}
{"category": "Math", "title": "Posterior consistency of Dirichlet mixtures of beta densities in estimating positive false discovery rates", "abstract": "In recent years, multiple hypothesis testing has come to the forefront of statistical research, ostensibly in relation to applications in genomics and some other emerging fields. The false discovery rate (FDR) and its variants provide very important notions of errors in this context comparable to the role of error probabilities in classical testing problems. Accurate estimation of positive FDR (pFDR), a variant of the FDR, is essential in assessing and controlling this measure. In a recent paper, the authors proposed a model-based nonparametric Bayesian method of estimation of the pFDR function. In particular, the density of p-values was modeled as a mixture of decreasing beta densities and an appropriate Dirichlet process was considered as a prior on the mixing measure. The resulting procedure was shown to work well in simulations. In this paper, we provide some theoretical results in support of the beta mixture model for the density of p-values, and show that, under appropriate conditions, the resulting posterior is consistent as the number of hypotheses grows to infinity."}
{"category": "Math", "title": "On hereditarily indecomposable compacta and factorization of maps", "abstract": "We prove a general factorization theorem for maps with hereditarily indecomposable fibers and apply it to reprove a theorem of Mackoviak on the existence of universal hereditarily indecomposable continua."}
{"category": "Math", "title": "Robust estimation in finite population sampling", "abstract": "The paper proposes some robust estimators of the finite population mean. Such estimators are particularly suitable in the presence of some outlying observations. Included as special cases of our general result are robust versions of the ratio estimator and the Horvitz-Thompson estimator. The robust estimators are derived on the basis of certain predictive influence functions."}
{"category": "Math", "title": "Estimation of population-level summaries in general semiparametric repeated measures regression models", "abstract": "This paper considers a wide family of semiparametric repeated measures regression models, in which the main interest is on estimating population-level quantities such as mean, variance, probabilities etc. Examples of our framework include generalized linear models for clustered/longitudinal data, among many others. We derive plug-in kernel-based estimators of the population level quantities and derive their asymptotic distribution. An example involving estimation of the survival function of hemoglobin measures in the Kenya hemoglobin study data is presented to demonstrate our methodology."}
{"category": "Math", "title": "Fundamental groups of symmetric sextics. II", "abstract": "We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with the set of inner singularities $2\\bold{A}_8$ or $\\bold{A}_{17}$. We also compute the fundamental groups of a number of other sextics, both of and not of torus type. The groups found are simplest possible, i.e., $\\Bbb{Z}_2*\\Bbb{Z}_3$ and $\\Bbb{Z}_6$, respectively."}
{"category": "Math", "title": "Smoothing-inspired lack-of-fit tests based on ranks", "abstract": "A rank-based test of the null hypothesis that a regressor has no effect on a response variable is proposed and analyzed. This test is identical in structure to the order selection test but with the raw data replaced by ranks. The test is nonparametric in that it is consistent against virtually any smooth alternative, and is completely distribution free for all sample sizes. The asymptotic distribution of the rank-based order selection statistic is obtained and seen to be the same as that of its raw data counterpart. Exact small sample critical values of the test statistic are provided as well. It is shown that the Pitman-Noether efficiency of the proposed rank test compares very favorably with that of the order selection test. In fact, their asymptotic relative efficiency is identical to that of the Wilcoxon signed rank and $t$-tests. An example involving microarray data illustrates the usefulness of the rank test in practice."}
{"category": "Math", "title": "A nonparametric control chart based on the Mann-Whitney statistic", "abstract": "Nonparametric or distribution-free charts can be useful in statistical process control problems when there is limited or lack of knowledge about the underlying process distribution. In this paper, a phase II Shewhart-type chart is considered for location, based on reference data from phase I analysis and the well-known Mann-Whitney statistic. Control limits are computed using Lugannani-Rice-saddlepoint, Edgeworth, and other approximations along with Monte Carlo estimation. The derivations take account of estimation and the dependence from the use of a reference sample. An illustrative numerical example is presented. The in-control performance of the proposed chart is shown to be much superior to the classical Shewhart $\\bar{X}$ chart. Further comparisons on the basis of some percentiles of the out-of-control conditional run length distribution and the unconditional out-of-control ARL show that the proposed chart is almost as good as the Shewhart $\\bar{X}$ chart for the normal distribution, but is more powerful for a heavy-tailed distribution such as the Laplace, or for a skewed distribution such as the Gamma. Interactive software, enabling a complete implementation of the chart, is made available on a website."}
{"category": "Math", "title": "On the length of lemniscates", "abstract": "We show that for a monic polynomial p of degree d, the length of the level set {z: |p(z)|=1} is at most 9.2 d, which improves an earlier estimate due to P. Borwein. For d=2 we show that the extremal level set is the Bernoullis' Lemniscate. One ingredient of our proofs is the fact that for an extremal polynomial this level set is connected."}
{"category": "Math", "title": "Regression rank scores in nonlinear models", "abstract": "Consider the nonlinear regression model $Y_i=g({\\bf x}_i,\\boldmath $\\theta$)+e_i,\\quad i=1,...,n$(1) with ${\\bf x}_i\\in \\mathbb{R}^k,$ $\\boldmath{\\theta}=(\\theta_0,\\theta_1,...,\\theta_p)^{\\prime}\\in \\boldmath $\\Theta$$ (compact in $\\mathbb{R}^{p+1}$), where $g({\\bf x},\\boldmath $\\theta$)=\\theta_0+\\tilde{g}({\\bf x},\\theta_1,...,\\theta_p)$ is continuous, twice differentiable in $\\boldmath $\\theta$$ and monotone in components of $\\boldmath $\\theta$$. Following Gutenbrunner and Jure\\v{c}kov\\'{a} (1992) and Jure\\v{c}kov\\'{a} and Proch\\'{a}zka (1994), we introduce regression rank scores for model (1), and prove their asymptotic properties under some regularity conditions. As an application, we propose some tests in nonlinear regression models with nuisance parameters."}
{"category": "Math", "title": "Chernoff-Savage and Hodges-Lehmann results for Wilks' test of multivariate independence", "abstract": "We extend to rank-based tests of multivariate independence the Chernoff-Savage and Hodges-Lehmann classical univariate results. More precisely, we show that the Taskinen, Kankainen and Oja (2004) normal-score rank test for multivariate independence uniformly dominates -- in the Pitman sense -- the classical Wilks (1935) test, which establishes the Pitman non-admissibility of the latter, and provide, for any fixed space dimensions $p,q$ of the marginals, the lower bound for the asymptotic relative efficiency, still with respect to Wilks' test, of the Wilcoxon version of the same."}
{"category": "Math", "title": "Stanley depth of complete intersection monomial ideals", "abstract": "We compute the Stanley depth of irreducible monomial ideals and we show that the Stanley depth of a monomial complete intersection ideal is the same as the Stanley depth of it's radical. Also, we give some bounds for the Stanley depth of a monomial complete intersection ideal."}
{"category": "Math", "title": "Surgery presentations of coloured knots and of their covering links", "abstract": "We consider knots equipped with a representation of their knot groups onto a dihedral group D_{2n} (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relating to the problem of passing from the initial data of a D_{2n}-coloured knot to a surgery presentation of the corresponding branched covering space and covering link. In particular, we describe effective algorithms for constructing such presentations. A by-product of these investigations is a proof of the conjecture that two D_{2n}-coloured knots are related by a sequence of surgeries along unit-framed unknots in the kernel of the representation if and only if they have the same coloured untying invariant (a Z_{n}-valued algebraic invariant of D_{2n}-coloured knots)."}
{"category": "Math", "title": "Ext^1-quivers for the Witt algebra W(1,1)", "abstract": "Let g be the finite dimensional Witt algebra W(1,1) over an algebraically closed field of characteristic p > 3. It is well known that all simple W(1,1)-modules are finite dimensional. Each simple module admits a character \\chi in g^*. Given such a \\chi one can form the (finite dimensional) reduced enveloping algebra u(g,\\chi). The simple modules for u(g,\\chi) are precisely those simple W(1,1)-modules admitting the character \\chi. In this paper the authors compute Ext^1 between pairs of simple modules for u(g,\\chi)."}
{"category": "Math", "title": "Long-time extinction of solutions of some semilinear parabolic equations", "abstract": "We study the long time behaviour of solutions of semi-linear parabolic equation of the following type $\\partial_t u-\\Delta u+a_0(x)u^q=0$ where $a_0(x) \\geq d_0 \\exp(\\frac{\\omega(|x|)}{|x|^2})$, $d_0>0$, $1>q>0$ and $\\omega$ a positive continuous radial function. We give a Dini-like condition on the function $\\omega$ by two different method which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schr\\\"odinger operators."}
{"category": "Math", "title": "$U$-tests for variance components in one-way random effects models", "abstract": "We consider a test for the hypothesis that the within-treatment variance component in a one-way random effects model is null. This test is based on a decomposition of a $U$-statistic. Its asymptotic null distribution is derived under the mild regularity condition that the second moment of the random effects and the fourth moment of the within-treatment errors are finite. Under the additional assumption that the fourth moment of the random effect is finite, we also derive the distribution of the proposed $U$-test statistic under a sequence of local alternative hypotheses. We report the results of a simulation study conducted to compare the performance of the $U$-test with that of the usual $F$-test. The main conclusions of the simulation study are that (i) under normality or under moderate degrees of imbalance in the design, the $F$-test behaves well when compared to the $U$-test, and (ii) when the distribution of the random effects and within-treatment errors are nonnormal, the $U$-test is preferable even when the number of treatments is small."}
{"category": "Math", "title": "On the Simes inequality and its generalization", "abstract": "The Simes inequality has received considerable attention recently because of its close connection to some important multiple hypothesis testing procedures. We revisit in this article an old result on this inequality to clarify and strengthen it and a recently proposed generalization of it to offer an alternative simpler proof."}
{"category": "Math", "title": "Composition-Diamond lemma for differential algebras", "abstract": "In this paper, we establish the Composition-Diamond lemma for free differential algebras. As applications, we give Groebner-Shirshov bases for free Lie-differential algebra and free commutative-differential algebra, respectively."}
{"category": "Math", "title": "Multiple testing procedures under confounding", "abstract": "While multiple testing procedures have been the focus of much statistical research, an important facet of the problem is how to deal with possible confounding. Procedures have been developed by authors in genetics and statistics. In this chapter, we relate these proposals. We propose two new multiple testing approaches within this framework. The first combines sensitivity analysis methods with false discovery rate estimation procedures. The second involves construction of shrinkage estimators that utilize the mixture model for multiple testing. The procedures are illustrated with applications to a gene expression profiling experiment in prostate cancer."}
{"category": "Math", "title": "New HKT manifolds arising from quaternionic representations", "abstract": "We give a procedure for constructing an $8n$-dimensional HKT Lie algebra starting from a $4n$-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-K\\\"ahler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in $SL(n,\\Bbb H)$ which is not a nilmanifold. We find in addition new compact strong HKT manifolds. We also show that every K\\\"ahler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional K\\\"ahler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8. Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics."}
{"category": "Math", "title": "Stochastic expansions and Hopf algebras", "abstract": "We study solutions to nonlinear stochastic differential systems driven by a multi-dimensional Wiener process. A useful algorithm for strongly simulating such stochastic systems is the Castell--Gaines method, which is based on the exponential Lie series. When the diffusion vector fields commute, it has been proved that at low orders this method is more accurate in the mean-square error than corresponding stochastic Taylor methods. However it has also been shown that when the diffusion vector fields do not commute, this is not true for strong order one methods. Here we prove that when there is no drift, and the diffusion vector fields do not commute, the exponential Lie series is usurped by the sinh-log series. In other words, the mean-square error associated with a numerical method based on the sinh-log series, is always smaller than the corresponding stochastic Taylor error, in fact to all orders. Our proof utilizes the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations. We illustrate the benefits of the proposed series in numerical studies."}
{"category": "Math", "title": "A Fast Algorithm for Stallings' Folding Process", "abstract": "We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stalling's folding process in time O(N log^*(N)), where N is the sum of the word lengths of the given generators of H."}
{"category": "Math", "title": "Compact anti-self-dual orbifolds with torus actions", "abstract": "We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a point at infinity to an asymptotically locally Euclidean (ALE) scalar-flat K\\\"ahler orbifold. We use this observation to classify ALE scalar-flat K\\\"ahler 4-orbifolds whose isometry group contain a 2-torus."}
{"category": "Math", "title": "On the Spectrum of geometric operators on K\\\"ahler manifolds", "abstract": "On a compact K\\\"ahler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after taking these symmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin^c-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved K\\\"ahler manifolds of odd complex dimension."}
{"category": "Math", "title": "Noncommutative geometry and dual coalgebras", "abstract": "In arXiv:math/0606241v2 M. Kontsevich and Y. Soibelman argue that the category of noncommutative (thin) schemes is equivalent to the category of coalgebras. We propose that under this correspondence the affine scheme of a k-algebra A is the dual coalgebra A^o and draw some consequences. In particular, we describe the dual coalgebra A^o of A in terms of the A-infinity structure on the Yoneda-space of all the simple finite dimensional A-representations."}
{"category": "Math", "title": "Probabilistic representation for solutions of an irregular porous media type equation", "abstract": "We consider a porous media type equation over all of $\\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. One of the main analytic ingredients of the proof, is a new result on uniqueness of distributional solutions of a linear PDE on $\\R^1$ with non-continuous coefficients."}
{"category": "Math", "title": "Integer program with bimodular matrix", "abstract": "This paper has been accepted for publication in Discrete Optimization."}
{"category": "Math", "title": "The antipode of a dual quasi-Hopf algebra with nonzero integrals is bijective", "abstract": "For $A$ a Hopf algebra of arbitrary dimension over a field $K$, it is well-known that if $A$ has nonzero integrals, or, in other words, if the coalgebra $A$ is co-Frobenius, then the space of integrals is one-dimensional and the antipode of $A$ is bijective. Bulacu and Caenepeel recently showed that if $H$ is a dual quasi-Hopf algebra with nonzero integrals, then the space of integrals is one-dimensional, and the antipode is injective. In this short note we show that the antipode is bijective."}
{"category": "Math", "title": "Local recognition of reflection graphs on Coxeter groups", "abstract": "We provide local recognition results for the reflection graphs on spherical Coxeter groups. In particular, we study the case $F_4$ which is locally recognizable under additional constraints only. It is then demonstrated in the cases $A_n$ and $F_4$ how these graph theoretical recognition results can be used to characterize the corresponding Coxeter groups in terms of their reflection centralizers. Finally, we outline the connection to the local recognition of Chevalley groups which is particularly important in the classification of the finite simple groups."}
{"category": "Math", "title": "On knots with infinite smooth concordance order", "abstract": "We use the Heegaard Floer obstructions defined by Grigsby, Ruberman, and Strle to show that forty-six of the sixty-seven knots through eleven crossings whose concordance orders were previously unknown have infinite concordance order."}
{"category": "Math", "title": "Eulerian quasisymmetric functions and poset topology", "abstract": "We introduce a family of quasisymmetric functions called {\\em Eulerian quasisymmetric functions}, which have the property of specializing to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising $q$-analog of a classical formula for the exponential generating function of the Eulerian polynomials. This $q$-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain $q$-analogs, $(q,p)$-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts such as in MacMahon's study of multiset derangements, in work of Procesi and Stanley on toric varieties of Coxeter complexes and in Stanley's work on symmetric chromatic polynomials. Here we present yet another occurence in connection with the homology of a poset introduced by Bj\\\"orner and Welker."}
{"category": "Math", "title": "The birational type of the moduli space of even spin curves", "abstract": "We determine the Kodaira dimension of the moduli space of even spin curves for all genera, with one possible exception: The scheme S_g has negative Kodaira dimension for g<8 and it is of general type for g>8. The Kodaira dimension of S_8 is non-negative."}
{"category": "Math", "title": "Minimal triangulations for an infinite family of lens spaces", "abstract": "The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein in earlier work, and, unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its \"minimal layered triangulation.\" This paper proves that for each integer n>1, the minimal layered triangulation of the lens space L(2n,1) is its unique minimal triangulation. More generally, the minimal triangulations (and hence the complexity) are determined for an infinite family of lens spaces containing the lens spaces L(2n,1)."}
{"category": "Math", "title": "Isometric Immersions and Compensated Compactness", "abstract": "A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\\mathcal M}^2$ which can be realized as isometric immersions into $\\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in $\\R^3$. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in $\\R^3$. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a $C^{1,1}$ isometric immersion of the two-dimensional manifold in $\\R^3$ satisfying our prescribed initial conditions. T"}
{"category": "Math", "title": "Studies on the second member of the second Painlev\\'e hierarchy", "abstract": "In this paper, we study the second member of the second Painlev\\'e hierarchy $P_{II}^{(2)}$. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four, and this Hamiltonian system can be considered as a 1-parameter family of coupled Painlev\\'e systems. This Hamiltonian is new. We also show that this system admits extended affine Weyl group symmetry of type $A_1^{(1)}$, and can be recovered by its holomorphy conditions. We also study a fifth-order ordinary differential equation satisfied by this Hamiltonian. After we transform this equation into a system of the first-order ordinary differential equations of polynomial type in dimension five by birational transformations, we give its symmetry and holomorphy conditions."}
{"category": "Math", "title": "An abstract characterization of unital operator spaces", "abstract": "In this article, we give an abstract characterization of the ``identity'' of an operator space $V$ by looking at a quantity $n_{cb}(V,u)$ which is defined in analogue to a well-known quantity in Banach space theory. More precisely, we show that there exists a complete isometry from $V$ to some $\\mathcal{L}(H)$ sending $u$ to ${\\rm id}_H$ if and only if $n_{cb}(V,u) =1$. We will use it to give an abstract characterization of operator systems. Moreover, we will show that if $V$ is a unital operator space and $W$ is a proper complete $M$-ideal, then $V/W$ is also a unital operator space. As a consequece, the quotient of an operator system by a proper complete $M$-ideal is again an operator system. In the appendix, we will also give an abstract characterisation of ``non-unital operator systems'' using an idea arose from the definition of $n_{cb}(V,u)$."}
{"category": "Math", "title": "A pattern mixture model for a paired $2\\times2$ crossover design", "abstract": "When conducting a paired $2\\times2$ crossover design, each subject is paired with another subject with similar characteristics. The pair is then randomized to the same sequence of two treatments. That is, the two subjects receive the first experimental treatment, and then they cross over and receive the other experimental treatment(s). The paired $2\\times2$ crossover design that was used in the Beta Adrenergic Response by GEnotype (BARGE) Study conducted by the National Heart, Lung and Blood Institute's Asthma Clinical Research Network (ACRN) has been described elsewhere. When the data arising from such a design are balanced and complete -- or if at least any missingness that occurs is at random -- general linear mixed-effects model methods can be used to analyze the data. In this paper, we present a method based on a pattern-mixture model for analyzing the data arising from a paired $2\\times2$ crossover design when some of the data are missing in a non-ignorable fashion. Because of its inherent scientific interest, we focus our particular attention on the estimation of the treatment-by-type of subject interaction term. Finally, we illustrate the pattern-mixture model methods described in this paper on the data arising from the BARGE study."}
{"category": "Math", "title": "Projected likelihood contrasts for testing homogeneity in finite mixture models with nuisance parameters", "abstract": "This paper develops a test for homogeneity in finite mixture models where the mixing proportions are known a priori (taken to be 0.5) and a common nuisance parameter is present. Statistical tests based on the notion of Projected Likelihood Contrasts (PLC) are considered. The PLC is a slight modification of the usual likelihood ratio statistic or the Wilk's $\\Lambda$ and is similar in spirit to the Rao's score test. Theoretical investigations have been carried out to understand the large sample statistical properties of these tests. Simulation studies have been carried out to understand the behavior of the null distribution of the PLC statistic in the case of Gaussian mixtures with unknown means (common variance as nuisance parameter) and unknown variances (common mean as nuisance parameter). The results are in conformity with the theoretical results obtained. Power functions of these tests have been evaluated based on simulations from Gaussian mixtures."}
{"category": "Math", "title": "Hook lengths and 3-cores", "abstract": "Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that $A000731(n)=0$ if and only if $A033687(n)=0$. The numbers $A000731(n)$ are given in terms of hook lengths of partitions, while $A033687(n)$ equals the number of 3-core partitions of $n$. Here we prove this conjecture."}
{"category": "Math", "title": "Discovering hook length formulas by expansion technique", "abstract": "We introduce the hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas those for partitions are much more difficult and some of them still remain open conjectures. We also develop a Maple package HookExp for computing the hook length expansion. The paper can be seen as a collection of hook length formulas for partitons and plane trees. All examples are illustrated by HookExp and, for many easy cases, expained by well-known combinatorial arguments."}
{"category": "Math", "title": "Schr\\\"oder Paths and Pattern Avoiding Partitions", "abstract": "In this paper, we show that both 12312-avoiding partitions and 12321-avoiding partitions of the set $[n+1]$ are in one-to-one correspondence with Schr\\\"oder paths of semilength $n$ without peaks at even level. As a consequence, the refined enumeration of 12312-avoiding (resp. 12321-avoiding) partitions according to the number of blocks can be reduced to the enumeration of certain Schr\\\"oder paths according to the number of peaks. Furthermore, we get the enumeration of irreducible 12312-avoiding (resp. 12321-avoiding) partitions, which are closely related to skew Dyck paths."}
{"category": "Math", "title": "The Liouville phenomenon in the deformation problem of coisotropics", "abstract": "The work of Oh and Park ([OP]) on the deformation problem of coisotropic submanifolds opened the possibility of studying a large and interesting class of foliations with some explicit geometric tools. These tools assemble into the structure of an L-infinity algebra on the shifted foliation complex (\\Omega^*[1](\\fol), d_\\fol), which allows a concise description of deformations in terms of a Maurer-Cartan equation. Infinitesimal deformations are given by d_\\fol-closed forms, and the relation between infinitesimal deformations and full deformations can be studied in terms of obstruction classes lying in the foliation cohomology H^*_\\fol. Closely related to the foliation cohomology is Haefliger's group \\Omega^*_c(T/H), an under-appreciated model for the leaf space of a foliation. We make integral use of this group in showing solvability and unsolvability of the obstruction equations. We also show the L-infinity apparatus to be capable of detecting the Liouville/diophantine distinction of KAM theory, and argue for the greater significance of Haefliger's integration-over-leaves map in passing this fine structure to a geometric model for the leaf space."}
{"category": "Math", "title": "Bootstrapping the Grenander estimator", "abstract": "The goal of this paper is to study the bootstrap for the Grenander estimator. The first result is a proof of the inconsistency of the nonparametric bootstrap for the Grenander estimator at a given point. The second result is the development and verification of a bootstrap for the $L_1$ confidence band for the Grenander estimator. As part of this work, kernel estimators are studied as alternatives to the Grenander estimator. We show that when the second derivative of the true density is assumed to be uniformly bounded, there exist kernel estimators with faster convergence rates than the Grenander estimator. We study the implications of this in developing $L_1$ and uniform confidence bands and discuss some open questions."}
{"category": "Math", "title": "Ratio tests for change point detection", "abstract": "We propose new tests to detect a change in the mean of a time series. Like many existing tests, the new ones are based on the CUSUM process. Existing CUSUM tests require an estimator of a scale parameter to make them asymptotically distribution free under the no change null hypothesis. Even if the observations are independent, the estimation of the scale parameter is not simple since the estimator for the scale parameter should be at least consistent under the null as well as under the alternative. The situation is much more complicated in case of dependent data, where the empirical spectral density at 0 is used to scale the CUSUM process. To circumvent these difficulties, new tests are proposed which are ratios of CUSUM functionals. We demonstrate the applicability of our method to detect a change in the mean when the errors are AR(1) and GARCH(1,1) sequences."}
{"category": "Math", "title": "A comparison of the Benjamini-Hochberg procedure with some Bayesian rules for multiple testing", "abstract": "In the spirit of modeling inference for microarrays as multiple testing for sparse mixtures, we present a similar approach to a simplified version of quantitative trait loci (QTL) mapping. Unlike in case of microarrays, where the number of tests usually reaches tens of thousands, the number of tests performed in scans for QTL usually does not exceed several hundreds. However, in typical cases, the sparsity $p$ of significant alternatives for QTL mapping is in the same range as for microarrays. For methodological interest, as well as some related applications, we also consider non-sparse mixtures. Using simulations as well as theoretical observations we study false discovery rate (FDR), power and misclassification probability for the Benjamini-Hochberg (BH) procedure and its modifications, as well as for various parametric and nonparametric Bayes and Parametric Empirical Bayes procedures. Our results confirm the observation of Genovese and Wasserman (2002) that for small p the misclassification error of BH is close to optimal in the sense of attaining the Bayes oracle. This property is shared by some of the considered Bayes testing rules, which in general perform better than BH for large or moderate $p$'s."}
{"category": "Math", "title": "The character table of a split extension of the Heisenberg group $H_1(q)$ by $Sp(2,q)$, $q$ odd", "abstract": "In this paper we determine the full character table of a certain split extension $H_1(q)\\rtimes Sp(2,q)$ of the Heisenberg group $H_1$ by the odd-characteristic symplectic group $Sp(2,q)$."}
{"category": "Math", "title": "On estimating the change point in generalized linear models", "abstract": "Statistical models incorporating change points are common in practice, especially in the area of biomedicine. This approach is appealing in that a specific parameter is introduced to account for the abrupt change in the response variable relating to a particular independent variable of interest. The statistical challenge one encounters is that the likelihood function is not differentiable with respect to this change point parameter. Consequently, the conventional asymptotic properties for the maximum likelihood estimators fail to hold in this situation. In this paper, we propose an estimating procedure for estimating the change point along with other regression coefficients under the generalized linear model framework. We show that the proposed estimators enjoy the conventional asymptotic properties including consistency and normality. Simulation work we conducted suggests that it performs well for the situations considered. We applied the proposed method to a case-control study aimed to examine the relationship between the risk of myocardial infarction and alcohol intake."}
{"category": "Math", "title": "Using statistical smoothing to date medieval manuscripts", "abstract": "We discuss the use of multivariate kernel smoothing methods to date manuscripts dating from the 11th to the 15th centuries, in the English county of Essex. The dataset consists of some 3300 dated and 5000 undated manuscripts, and the former are used as a training sample for imputing dates for the latter. It is assumed that two manuscripts that are ``close'', in a sense that may be defined by a vector of measures of distance for documents, will have close dates. Using this approach, statistical ideas are used to assess ``similarity'', by smoothing among distance measures, and thus to estimate dates for the 5000 undated manuscripts by reference to the dated ones."}
{"category": "Math", "title": "Sequential nonparametrics and semiparametrics: Theory, implementation and applications to clinical trials", "abstract": "One of Pranab K. Sen's major research areas is sequential nonparametrics and semiparametrics and their applications to clinical trials, to which he has made many important contributions. Herein we review a number of these contributions and related developments. We also describe some recent work on nonparametric and semiparametric inference and the associated computational methods in time-sequential clinical trials with survival endpoints."}
{"category": "Math", "title": "Combinatorial cube packings in cube and torus", "abstract": "We consider sequential random packing of cubes $z+[0,1]^n$ with $z\\in \\frac{1}{N}\\ZZ^n$ into the cube $[0,2]^n$ and the torus $\\QuotS{\\RR^n}{2\\ZZ^n}$ as $N\\to\\infty$. In the cube case $[0,2]^n$ as $N\\to\\infty$ the random cube packings thus obtained are reduced to a single cube with probability $1-O(\\frac{1}{N})$. In the torus case the situation is different: for $n\\leq 2$, sequential random cube packing yields cube tilings, but for $n\\geq 3$ with strictly positive probability, one obtains non-extensible cube packings. So, we introduce the notion of combinatorial cube packing, which instead of depending on $N$ depend on some parameters. We use use them to derive an expansion of the packing density in powers of $\\frac{1}{N}$. The explicit computation is done in the cube case. In the torus case, the situation is more complicate and we restrict ourselves to the case $N\\to\\infty$ of strictly positive probability. We prove the following results for torus combinatorial cube packings: We give a general Cartesian product construction. We prove that the number of parameters is at least $\\frac{n(n+1)}{2}$ and we conjecture it to be at most $2^n-1$. We prove that cube packings with at least $2^n-3$ cubes are extensible. We find the minimal number of cubes in non-extensible cube packings for $n$ odd and $n\\leq 6$."}
{"category": "Math", "title": "Rectangle groups", "abstract": "A class of groups is investigated, each of which has a fairly simple presentation . For example the group $R = (a, b, c, d | a^3 = b^3 = c^3 = d^3 = 1, ba^{-1} =dc^{-1}, ca^{-1} = db^{-1}) $ is in the class. Such a group does not have as a homomorphic image any group which is a 2-orbifold group or which is a group of isometries of the reals. However it does have incompatible splittings over subgroups which are not small. This contradicts some ideas I had about universal JSJ decompostions for finitely presented groups over finitely generated subgroups. Such a group also has an unstable action on an R-tree and a cocompact action on a CAT(0) cube complex with finite cyclic point stabilizers, and trivial edge stabilizers."}
{"category": "Math", "title": "Estimating medical costs from a transition model", "abstract": "Nonparametric estimators of the mean total cost have been proposed in a variety of settings. In clinical trials it is generally impractical to follow up patients until all have responded, and therefore censoring of patient outcomes and total cost will occur in practice. We describe a general longitudinal framework in which costs emanate from two streams, during sojourn in health states and in transition from one health state to another. We consider estimation of net present value for expenditures incurred over a finite time horizon from medical cost data that might be incompletely ascertained in some patients. Because patient specific demographic and clinical characteristics would influence total cost, we use a regression model to incorporate covariates. We discuss similarities and differences between our net present value estimator and other widely used estimators of total medical costs. Our model can accommodate heteroscedasticity, skewness and censoring in cost data and provides a flexible approach to analyses of health care cost."}
{"category": "Math", "title": "Correcting for selection bias via cross-validation in the classification of microarray data", "abstract": "There is increasing interest in the use of diagnostic rules based on microarray data. These rules are formed by considering the expression levels of thousands of genes in tissue samples taken on patients of known classification with respect to a number of classes, representing, say, disease status or treatment strategy. As the final versions of these rules are usually based on a small subset of the available genes, there is a selection bias that has to be corrected for in the estimation of the associated error rates. We consider the problem using cross-validation. In particular, we present explicit formulae that are useful in explaining the layers of validation that have to be performed in order to avoid improperly cross-validated estimates."}
{"category": "Math", "title": "Some regular symmetric pairs", "abstract": "In [AG2] we explored the question what symmetric pairs are Gelfand pairs. We introduced the notion of regular symmetric pair and conjectured that all symmetric pairs are regular. This conjecture would imply that many symmetric pairs are Gelfand pairs, and in particular that any connected symmetric pair over C is a Gelfand pair. In this paper we show that the pairs $$(GL(V),O(V)), (GL(V),U(V)), (U(V),O(V)), (O(V \\oplus W),O(V) \\times O(W)), (U(V \\oplus W),U(V) \\times U(W))$$ are regular where V and W are quadratic or hermitian spaces over arbitrary local field of characteristic zero. We deduce from that that the pairs $(GL_n(\\C),O_n(\\C))$ and $(O_{n+m}(\\C),O_n(\\C) \\times O_m(\\C))$ are Gelfand pairs."}
{"category": "Math", "title": "Parity-induced Selmer Growth For Symplectic, Ordinary Families", "abstract": "Let $p$ be an odd prime, and let $K/K_0$ be a quadratic extension of number fields. Denote by $K_\\pm$ the maximal $\\mathbb{Z}_p$-power extensions of $K$ that are Galois over $K_0$, with $K_+$ abelian over $K_0$ and $K_-$ dihedral over $K_0$. In this paper we show that for a Galois representation over $K_0$ satisfying certain hypotheses, if it has odd Selmer rank over $K$ then for one of $K_\\pm$ its Selmer rank over $L$ is bounded below by $[L:K]$ for $L$ ranging over the finite subextensions of $K$ in $K_\\pm$. Our method or proof generalizes a method of Mazur--Rubin, building upon results of Nekov\\'a\\v{r}, and applies to abelian varieties of arbitrary dimension, (self-dual twists of) modular forms of even weight, and (twisted) Hida families."}
{"category": "Math", "title": "Bimodule herds", "abstract": "The notion of a bimodule herd is introduced and studied. A bimodule herd consists of a $B$-$A$ bimodule, its formal dual, called a pen, and a map, called a shepherd, which satisfies untiality and coassociativity conditions. It is shown that every bimodule herd gives rise to a pair of corings and coactions. If, in addition, a bimodule herd is tame i.e. it is faithfully flat and a progenerator, then these corings are associated to entwining structures; the bimodule herd is a Galois comodule of these corings. The notion of a bicomodule coherd is introduced as a formal dualisation of the definition of a bimodule herd. Every bicomodule coherd defines a pair of (non-unital) rings. It is shown that a tame $B$-$A$ bimodule herd defines a bicomodule coherd, and sufficient conditions for the derived rings to be isomorphic to $A$ and $B$ are discussed. The composition of bimodule herds via the tensor product is outlined. The notion of a bimodule herd is illustrated by the example of Galois co-objects of a commutative, faithfully flat Hopf algebra."}
{"category": "Math", "title": "An asymptotically normal test for the selective neutrality hypothesis", "abstract": "An important parameter in the study of population evolution is $\\theta=4N\\nu$, where $N$ is the effective population size and $\\nu$ is the rate of mutation per locus per generation. Therefore, $\\theta$ represents the mean number of mutations per site per generation. There are many estimators of $\\theta$, one of them being the mean number of pairwise nucleotide differences, which we call $\\mathcal{T}_2$. Other estimators are $\\mathcal{T}_1$, based on the number of segregating sites and $\\mathcal{T}_3$, based on the number of singletons. The concept of selective neutrality can be interpreted as a differentiated nucleotide distribution for mutant sites when compared to the overall nucleotide distribution. Tajima (1989) has proposed the so-called Tajima's test of selective neutrality based on $\\mathcal{T}_2-\\mathcal{T}_1$. Its complex empirical behavior (Kiihl, 2005) motivates us to propose a test statistic solely based on $\\mathcal{T}_2$. We are thus able to prove asymptotic normality under different assumptions on the number of sequences and number of sites via $U$-statistics theory."}
{"category": "Math", "title": "Complex structures on tangent and cotangent Lie algebras of dimension six", "abstract": "This paper deals with complex structures on Lie algebras $\\ct_{\\pi} \\hh=\\hh \\ltimes_{\\pi} V$, where $\\pi$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\\ct_{\\pi} \\hh$ for $\\hh$ a three dimensional real Lie algebra. First it was proposed the study of complex structures $J$ satisfying the constrain $J\\hh=V$. Whenever $\\pi$ is the adjoint representation this kind of complex structures are associated to non singular derivations of $\\hh$. This fact derives different kind of applications. Finally an approach to the pseudo K\\\"ahler geometry was done."}
{"category": "Math", "title": "Model selection and sensitivity analysis for sequence pattern models", "abstract": "In this article we propose a maximal a posteriori (MAP) criterion for model selection in the motif discovery problem and investigate conditions under which the MAP asymptotically gives a correct prediction of model size. We also investigate robustness of the MAP to prior specification and provide guidelines for choosing prior hyper-parameters for motif models based on sensitivity considerations."}
{"category": "Math", "title": "Maximal Solutions of Semilinear Elliptic Equations with Locally Integrable Forcing Term", "abstract": "We study the existence of a maximal solution of $-\\Gd u+g(u)=f(x)$ in a domain $\\Gw\\subset \\BBR^N$ with compact boundary, assuming that $f\\in (L^1_{loc}(\\Gw))_+$ and that $g$ is nondecreasing, $g(0)\\geq 0$ and $g$ satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classical $C_{1,2}$ Wiener criterion then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition we discuss the question of uniqueness of large solutions."}
{"category": "Math", "title": "Asymptotic behaviour for the gradient of large solutions to some nonlinear elliptic equations", "abstract": "If $h$ is a nondecreasing real valued function and $0\\leq q\\leq 2$, we analyse the boundary behaviour of the gradient of any solution $u$ of $-\\Delta u+h(u)+\\abs {\\nabla u}^q=f$ in a smooth N-dimensional domain $\\Omega$ with the condition that $u$ tends to infinity when $x$ tends to $\\partial\\Omega$. We give precise expressions of the blow-up which, in particular, point out the fact that the phenomenon occurs essentially in the normal direction to $\\partial\\Omega$. Motivated by the blow--up argument in our proof, we also give in Appendix a symmetry result for some related problems in the half space."}
{"category": "Math", "title": "The rationality of certain moduli spaces of curves of genus 3", "abstract": "We show, for each algebraically closed field, the rationality of the following two moduli spaces: M(3,3) parametrizing pairs (C, \\eta) where C has genus 3 and \\eta is a 3-torsion divisor class, respectively of M(3,<3>) parametrizing pairs (C, <\\eta>) as above and where <\\eta> is the cyclic subgroup of order 3 in Pic_0(C) generated by \\eta."}
{"category": "Math", "title": "Symmetry of large solutions of nonlinear elliptic equations in a ball", "abstract": "Let $g$ be a locally Lipschitz continuous real valued function which satisfies the Keller-Osserman condition and is convex at infinity, then any large solution of $-\\Delta u+g(u)=0$ in a ball is radially symmetric."}
{"category": "Math", "title": "A nonlinear inequality", "abstract": "A quadratic inequality is formulated in the paper. An estimate on the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations."}
{"category": "Math", "title": "The regular algebra of a poset", "abstract": "Let $K$ be a field. We attach to each finite poset $\\mathbb P$ a von Neumann regular $K$-algebra $Q_K(\\mathbb P)$ in a functorial way. We show that the monoid of isomorphism classes of finitely generated projective $Q_K(\\mathbb P)$-modules is the abelian monoid generated by $\\mathbb P$ with the only relations given by $p=p+q$ whenever $q<p$ in $\\mathbb P$. This extends the class of monoids for which there is a positive solution to the realization problem for von Neumann regular rings."}
{"category": "Math", "title": "Ambiguity theory, old and new", "abstract": "This is a introductory survey of some recent developments of \"Galois ideas\" in Arithmetic, Complex Analysis, Transcendental Number Theory and Quantum Field Theory, and of some of their interrelations."}
{"category": "Math", "title": "Galois theory, motives and transcendental numbers", "abstract": "From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn't one associate conjugates and a Galois group to transcendental numbers such as $\\pi$? Beyond, can't one envision an appropriate Galois theory in the field of transcendental number theory? In which role? The aim of this text is to indicate what Grothendieck's theory of motives has to say, at least conjecturally, on these questions."}
{"category": "Math", "title": "Triangulordinary Selmer Groups", "abstract": "Let $p$ be a prime number, and let $K$ be a $p$-adic local field. We study a class of semistable $p$-adic Galois representations of $K$, which we call {\\it triangulordinary} because it includes the ordinary ones yet allows non-\\'etale behavior in the associated $(\\phi,\\Gamma_K)$-modules over the Robba ring. Our main result provides a description of the Bloch--Kato local condition of such representations. We also propose a program, using variational techniques, that would give a definition of the Selmer group along the eigencurve of Coleman--Mazur, including notably its nonordinary locus."}
{"category": "Math", "title": "Nilpotent centralizers and Springer isomorphisms", "abstract": "Let G be a semisimple algebraic group over a field K whose characteristic is very good for G, and let sigma be any G-equivariant isomorphism from the nilpotent variety to the unipotent variety; the map sigma is known as a Springer isomorphism. Let y in G(K), let Y in Lie(G)(K), and write C_y = C_G(y) and C_Y= C_G(Y) for the centralizers. We show that the center of C_y and the center of C_Y are smooth group schemes over K. The existence of a Springer isomorphism is used to treat the crucial cases where y is unipotent and where Y is nilpotent. Now suppose G to be quasisplit, and write C for the centralizer of a rational regular nilpotent element. We obtain a description of the normalizer N_G(C) of C, and we show that the automorphism of Lie(C) determined by the differential of sigma at zero is a scalar multiple of the identity; these results verify observations of J-P. Serre."}
{"category": "Math", "title": "Lines on hypersurfaces", "abstract": "This is a detailed study of the infinitesimal variation of the variety of lines through a point of a low degree hypersurface in pro jective space. The motion is governed by a system of partial differential equations which we describe explicitly."}
{"category": "Math", "title": "The Yang-Mills stratification for surfaces revisited", "abstract": "We revisit Atiyah and Bott's study of Morse theory for the Yang-Mills functional over a Riemann surface, and establish new formulas for the minimum codimension of a (non-semi-stable) stratum. These results yield the exact connectivity of the natural map (C_{min} E)//G(E) --> Map^E (M, BU(n)) from the homotopy orbits of the space of central Yang-Mills connections to the classifying space of the gauge group G(E). All of these results carry over to non-orientable surfaces via Ho and Liu's non-orientable Yang-Mills theory. A somewhat less detailed version of this paper (titled \"On the Yang-Mills stratification for surfaces\") will appear in the Proceedings of the AMS."}
{"category": "Math", "title": "On Two Related Questions of Wilf Concerning Standard Young Tableaux", "abstract": "We consider two questions of Wilf related to Standard Young Tableaux. We provide a partial answer to one question, and that will lead us to a more general answer to the other question. Our answers are purely combinatorial."}
{"category": "Math", "title": "Overcrowding and hole probabilities for random zeros on complex manifolds", "abstract": "We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a holomorphic section of the N-th power of a positive line bundle on a compact Kaehler manifold. In particular, we show that for all $\\delta>0$, the probability that this volume differs by more than $\\delta N$ from its average value is less than $\\exp(-C_{\\delta,U}N^{m+1})$, for some constant $C_{\\delta,U}>0$. As a consequence, the \"hole probability\" that a random section does not vanish in U has an upper bound of the form $\\exp(-C_{U}N^{m+1})$."}
{"category": "Math", "title": "On the homotopy theory of spectral categories", "abstract": "This paper has been withdrawn by the author, due a critical mistake on page 3."}
{"category": "Math", "title": "Separating algebras and finite reflection groups", "abstract": "A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating algebra. This allows us to prove that only groups generated by reflections may have polynomial separating algebras, and only groups generated by bireflections may have complete intersection separating algebras."}
{"category": "Math", "title": "A model category structure on the category of simplicial multicategories", "abstract": "We establish a Quillen model category structure on the category of symmetric simplicial multicategories. This model structure extends the model structure on simplicial categories due to J. Bergner."}
{"category": "Math", "title": "Convergence of nonlocal threshold dynamics approximations to front propagation", "abstract": "In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order $\\alpha \\in (0,2)$ converge to moving fronts. When $\\alpha \\geqq 1$ the resulting interface moves by weighted mean curvature, while for $\\alpha <1$ the normal velocity is nonlocal of ``fractional-type.'' The results easily extend to general nonlocal anisotropic threshold dynamics schemes."}
{"category": "Math", "title": "(GL(2n,C),SP(2n,C)) is a Gelfand Pair", "abstract": "We prove that (GL_{2n}(C),Sp_{2n}(C)) is a Gelfand pair. More precisely, we show that for an irreducible smooth admissible Frechet representation (\\pi,E) of GL_{2n}(C) the space of continuous functionals Hom_{Sp_{2n}(\\cc)}(E,C) is at most one dimensional. For this we show that any distribution on GL_{2n}(C) invariant with respect to the double action Sp_{2n}(C) \\times Sp_{2n}(C) is transposition invariant. Such a result was previously proven for p-adic fields by M. Heumos and S. Rallis."}
{"category": "Math", "title": "On the moduli space of negatively curved metrics of a hyperbolic manifold", "abstract": "We study the moduli space of negatively curved metrics of a hyperbolic manifold."}
{"category": "Math", "title": "On the $k$-free divisor problem", "abstract": "Let $\\Delta^{(k)}(x)$ denote the error term of the $k$-free divisor problem for $k\\geq 2$. In this paper we establish an asymptotic formula of the integral $\\int_1^T|\\Delta^{(k)}(x)|^2dx$ for each $k\\geq 4.$"}
{"category": "Math", "title": "Regularity issues in the problem of fluid structure interaction", "abstract": "We investigate the evolution of rigid bodies in a viscous incompressible fluid. The flow is governed by the 2D Navier-Stokes equations, set in a bounded domain with Dirichlet boundary conditions. The boundaries of the solids and the domain have H\\\"older regularity $C^{1, \\alpha}$, $0 < \\alpha \\le 1$. First, we show the existence and uniqueness of strong solutions up to collision. A key ingredient is a BMO bound on the velocity gradient, which substitutes to the standard $H^2$ estimate for smoother domains. Then, we study the asymptotic behaviour of one $C^{1, \\alpha}$ body falling over a flat surface. We show that collision is possible in finite time if and only if $\\alpha < 1/2$."}
{"category": "Math", "title": "Universal Upper Bound for the Growth of Artin Monoids", "abstract": "In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given."}
{"category": "Math", "title": "Disjointness of representations arising in harmonic analysis on the infinite-dimensional unitary group", "abstract": "We prove pairwise disjointness of representations T_{z,w} of the infinite-dimensional unitary group. These representations provide a natural generalization of the regular representation for the case of \"big\" group U(\\infty). They were introduced and studied by G.Olshanski and A.Borodin. Disjointness of the representations can be reduced to disjointness of certain probability measures on the space of paths in the Gelfand-Tsetlin graph. We prove the latter disjointness using probabilistic and combinatorial methods."}
{"category": "Math", "title": "On the solutions of Knizhnik-Zamolodchikov system", "abstract": "We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of KZ system is rational too. We give the method of constructing the corresponding rational solution. We deduce the asymptotic formulas for the solution of KZ system when $\\rho$ is integer."}
{"category": "Math", "title": "Le probleme de Bogomolov effectif sur les varietes abeliennes", "abstract": "We give a new lower bound for the essential minimum of subvarieties of abelian varieties with small codimension, under a conjecture about ordinary primes in abelian varieties. This lower bound is already known in the toric case since the work of Amoroso and David. It is the best expected, \"up to an epsilon\", in the degree of the subvariety."}
{"category": "Math", "title": "Generalizations of Gronwall-Bihari Inequalities on Time Scales", "abstract": "We establish some nonlinear integral inequalities for functions defined on a time scale. The results extend some previous Gronwall and Bihari type inequalities on time scales. Some examples of time scales for which our results can be applied are provided. An application to the qualitative analysis of a nonlinear dynamic equation is discussed."}
{"category": "Math", "title": "Implicit higher derivatives, and a formula of Comtet and Fiolet", "abstract": "Let F(x,y) be a function of two variables, and suppose y = f(x) satisfies F(x,y)=0 in some range. Then dy/dx = -Fx/Fy, where Fx and Fy denote the partial derivatives of F with respect to x and y. It is natural to seek a general expression for the higher derivatives d^ny/dx^n, in terms of partial derivatives of F, and such an expression was given in 1974 by L. Comtet and M. Fiolet. Their formula, however, contains some errors. In this note, we give a corrected expression. We give a derivation using Lagrange inversion and also an elementary proof by induction. We further correct a minor error in Comtet and Fiolet's expression for the number of terms in their formula."}
{"category": "Math", "title": "Time-preserving structural stability of hyperbolic differential dynamics with noncompact phase spaces", "abstract": "In this paper, we study the structural stability of hyperbolic differential systems on Euclidean spaces using Liao theory."}
{"category": "Math", "title": "A Quasi Curtis-Tits-Phan theorem for the symplectic group", "abstract": "We obtain the symplectic group $\\SP(V)$ as the universal completion of an amalgam of low rank subgroups akin to Levi components. We let $\\SP(V)$ act flag-transitively on the geometry of maximal rank subspaces of $V$. We show that this geometry and its rank $\\ge 3$ residues are simply connected with few exceptions. The main exceptional residue is described in some detail. The amalgamation result is then obtained by applying Tits' lemma. This provides a new way of recognizing the symplectic groups from a small collection of small subgroups."}
{"category": "Math", "title": "Analytic multiplicative cocycles over holomorphic dynamical systems", "abstract": "We prove some properties of analytic multiplicative and sub-multiplicative cocycles. The results allow to construct natural invariant analytic sets associated to complex dynamical systems."}
{"category": "Math", "title": "Whittaker Modules for the Virasoro Algebra", "abstract": "Whittaker modules have been well studied in the setting of complex semisimple Lie algebras. Their definition can easily be generalized to certain other Lie algebras with triangular decomposition, including the Virasoro algebra. We define Whittaker modules for the Virasoro algebra and obtain analogues to several results from the classical setting, including a classification of simple Whittaker modules by central characters and composition series for general Whittaker modules."}
{"category": "Math", "title": "Finite Rank Toeplitz Operators: Some Extensions of D.Luecking's Theorem", "abstract": "The recent theorem by D.Luecking about finite rank Bergman-Toeplitz operators is extended to weights being distributions with compact support and to the spaces of harmonic functions."}
{"category": "Math", "title": "Singular Solutions of Hessian Fully Nonlinear Elliptic Equations", "abstract": "We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+\\epsilon}, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<\\alpha<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial."}
{"category": "Math", "title": "An improved estimate on sums of product sets", "abstract": "In a recent paper \\cite{Gl} A. Glibichuk proved that if $A,B$ are subsets of an arbitrary finite filed $\\F_q$, such that $|A||B|>q$, then $16AB = \\F_q$. We improve this to $10AB = \\F_q.$"}
{"category": "Math", "title": "Recurrence Formulas for Fibonacci Sums", "abstract": "In this article we present a new recurrence formula for a finite sum involving the Fibonacci sequence. Furthermore, we state an algorithm to compute the sum of a power series related to Fibonacci series, without the use of term-by-term differentiation theorem"}
{"category": "Math", "title": "Cops and robbers in random graphs", "abstract": "We consider the pursuit and evasion game on finite, connected, undirected graphs known as cops and robbers. Meyniel conjectured that for every graph on n vertices a rootish number of cops can win the game. We prove that this holds up to a log(n) factor for random graphs G(n,p) if p is not very small, and this is close to be tight unless the graph is very dense. We analyze the area-defending strategy (used by Aigner in case of planar graphs) and show examples where it can not be too efficient."}
{"category": "Math", "title": "Observable invariant measures", "abstract": "For continuous maps on a compact manifold M, particularly for those that do not preserve the Lebesgue measure m, we define the observable invariant probability measures as a generalization of the physical measures. We prove that any continuous map has observable measures, and characterize those that are physical in terms of the observability. We prove that there exist physical measures whose basins cover Lebesgue a.e, if and only if the set of all observable measures is finite or infinite numerable. We define for any continuous map, its generalized attractors using the set of observable invariant measures where there is no physical measure, and prove that any continuous map defines a decomposition of the space in up to infinitely many generalized attractors whose basins cover Lebesgue a.e. We apply the results to the C1 expanding maps f in the circle, proving that the set of observable measures (even if f is not C1 plus Holder, is a subset of the equilibrium states."}
{"category": "Math", "title": "Multi-bump Solutions for a Strongly Indefinite Semilinear Schr\\\"odinger Equation Without Symmetry or convexity Assumptions", "abstract": "In this paper, we study the following semilinear Schr\\\"odinger equation with periodic coefficient: $$-\\triangle u +V(x)u=f(x,u), u\\in H^{1}(\\mathbb{R}^{N}).$$ The functional corresponding to this equation possesses strongly indefinite structure. The nonlinear term $f(x,t)$ satisfies some superlinear growth conditions and need not be odd or increasing strictly in $t$. Using a new variational reduction method and a generalized Morse theory, we proved that this equation has infinitely many geometrically different solutions. Furthermore, if the solutions of this equation under some energy level are isolated, then we can show that this equation has infinitely many $m-$bump solutions for any positive integer $m\\geq 2.$"}
{"category": "Math", "title": "Local solutions in Sobolev spaces with negative indices for the \"good\" Boussinesq equation", "abstract": "We study the local well-posedness of the initial-value problem for the nonlinear \"good\" Boussinesq equation with data in Sobolev spaces \\textit{$H^s$} for negative indices of $s$."}
{"category": "Math", "title": "CW-groups associated with wrap groups", "abstract": "This article is devoted to the investigation of wrap groups of connected fiber bundles. CW-groups associated with wrap groups are studied."}
{"category": "Math", "title": "Intersections of base rings associated to transversal polymatroids", "abstract": "The discrete polymatroids and their base rings are studied recently in many papers (see \\cite{HH}, \\cite{HHV}, \\cite{V1}, \\cite{V2}). It is important to give conditions when the base ring associated to a transversal polymatroid is Gorenstein (see \\cite{HH}). In \\cite{SA} we introduced a class of such base rings. In this paper we note that an intersection of such base rings (introduced in \\cite{SA}) is Gorenstein and give necessary and sufficient conditions for the intersection of two base rings from \\cite{SA} to be still a base ring of a transversal polymatroid. Also, we compute the $a$-invariant of those base rings. The results presented were discovered by extensive computer algebra experiments performed with {\\it{Normaliz}} \\cite{BK}."}
{"category": "Math", "title": "On the continuity of separately continuous bihomomorphisms", "abstract": "Separately continuous bihomomorphisms on a product of convergence or topological groups occur with great frequency. Of course, in general, these need not be jointly continuous. In this paper, we exhibit some results of Banach-Steinhaus type and use these to derive joint continuity from separate continuity. The setting of convergence groups offers two advantages. First, the continuous convergence structure is a powerful tool in many duality arguments. Second, local compactness and first countability, the usual requirements for joint continuity, are available in much greater abundance for convergence groups."}
{"category": "Math", "title": "The local Langlands conjecture for Sp(4)", "abstract": "We show that the local Langlands conjecture for $Sp(2n)$ follows from that for $GSp(2n)$. In particular, we prove the local Langlands conjecture for $Sp(4)$, based on our previous work on the local Langlands conjecture for $GSp(4)$. We also determine the possible sizes of $L$-packets for $Sp(4)$."}
{"category": "Math", "title": "Connes' metric for states in group algebras", "abstract": "In this article we follow the main idea of A. Connes for the construction of a metric in the state space of a C*-algebra. We focus in the reduced algebra of a discrete group $\\Gamma$, and prove some equivalences and relations between two central objects of this category: the word-length growth (connected with the degree of the extension of $\\Gamma$ when the group is an extension of Z by a finite group), and the topological equivalence between the w*-topology and the one introduced with this metric in the state space of $C_r*(\\Gamma)$."}
{"category": "Math", "title": "The quandle of the trefoil knot as the Dehn quandle of the torus", "abstract": "We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space $\\Z \\oplus \\Z$. The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. We also show that the fundamental quandle of the long trefoil knot is isomorphic to the cord quandle on a 2-sphere with a hole and three punctures."}
{"category": "Math", "title": "Symmetry in Data Mining and Analysis: A Unifying View based on Hierarchy", "abstract": "Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical domain of interest. \"Structure\" has long been understood as symmetry which can take many forms with respect to any transformation, including point, translational, rotational, and many others. Beginning with the role of number theory in expressing data, we show how we can naturally proceed to hierarchical structures. We show how this both encapsulates traditional paradigms in data analysis, and also opens up new perspectives towards issues that are on the order of the day, including data mining of massive, high dimensional, heterogeneous data sets. Linkages with other fields are also discussed including computational logic and symbolic dynamics. The structures in data surveyed here are based on hierarchy, represented as p-adic numbers or an ultrametric topology."}
{"category": "Math", "title": "On the distribution of imaginary parts of zeros of the Riemann zeta function, II", "abstract": "We continue our investigation of the distribution of the fractional parts of $a \\gamma$, where $a$ is a fixed non-zero real number and $\\gamma$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to pair correlation functions and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. This is a sequel to the paper math.NT/0405459."}
{"category": "Math", "title": "A short proof of the Khukhro--Makarenko theorem on large characteristic subgroups with laws", "abstract": "We give a short proof and some strengthening of the Khukhro--Makarenko theorem that each group virtually satisfying an outer commutator identity contains a finite-index characteristic subgroup satisfying this identity. An estimate for the index of this characteristic subgroup is obtained."}
{"category": "Math", "title": "Universal Khovanov-Rozansky sl(2) cohomology", "abstract": "We generalize the Khovanov-Rozansky cohomology for n=2 by means of a homogeneous potential that depends on two parameters, to obtain the universal Khovanov-Rozansky sl(2) link cohomology. This theory is equivalent to the universal foam sl(2) link cohomology, after tensoring both theories with appropriate rings."}
{"category": "Math", "title": "The Remarkable Simplicity of Very High Dimensional Data: Application of Model-Based Clustering", "abstract": "An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a hierarchical embedding. Such hierarchical structure can be global in the data set, or local. By quantifying extent or degree of ultrametricity in a data set, we show that ultrametricity becomes pervasive as dimensionality and/or spatial sparsity increases. This leads us to assert that very high dimensional data are of simple structure. We exemplify this finding through a range of simulated data cases. We discuss also application to very high frequency time series segmentation and modeling."}
{"category": "Math", "title": "Invariant tori for commuting Hamiltonian PDEs", "abstract": "We generalize to some PDEs a theorem by Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with $r$ integrals of motion and $n$ degrees of freedom, $r\\leq n$. The result we get ensures the persistence of an $r$-parameter family of $r$-dimensional invariant tori. The parameters belong to a Cantor-like set. The proof is based on the Lyapunof-Schmidt decomposition and on the standard implicit function theorem. Some of the persistent tori are resonant. We also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of coupled beam equations. In the first case we construct 2 dimensional tori, while in the second case we construct 3 dimensional tori."}
{"category": "Math", "title": "Correlation decay and recurrence estimates for some robust nonuniformly hyperbolic maps", "abstract": "We study decay of correlations, the asymptotic distribution of hitting times and fluctuations of the return times for a robust class of multidimensional non-uniformly hyperbolic transformations. Oliveira and Viana [15] proved that there is a unique equilibrium state for a large class of non- uniformly expanding transformations and Holder continuous potentials with small variation. For an open class of potentials with small variation, we prove quasi-compactness of the Ruelle-Perron-Frobenius operator in a space $V_\\theta$ of functions with essential bounded variation that strictly contain Holder continuous observables. We deduce that the equilibrium states have exponential decay of correlations. Furthermore, we prove exponential asymptotic distribu- tion of hitting times and log-normal fluctuations of the return times around the average given by the metric entropy."}
{"category": "Math", "title": "Weil-Petersson geometry of Teichmuller-Coxeter complex and its finite rank property", "abstract": "We construct a Weil-Petersson geodesic completion of Teichmuller space through the formalism of Coxeter complex with the Teichmuller space as its non-linear non-homogeneous fundamental domain. We show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-compact symmetric spaces of semi-simple Lie groups."}
{"category": "Math", "title": "Quantum D-modules, elliptic braid groups, and double affine Hecke algebras", "abstract": "We build representations of the elliptic braid group from the data of a quantum D-module M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid, and also certain geometric constructions of Calaque, Enriquez, and Etingof concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasi-classical limit of U=U_t(sl_N), as t limits 1. In the latter case, we produce representations of the double affine Hecke algebra of type A_{n-1}, for each n."}
{"category": "Math", "title": "A class of simple $C^*$-algebras arising from certain nonsofic subshifts", "abstract": "We present a class of subshifts $Z_N, N = 1,2,...$ whose associated $C^*$-algebras ${\\cal O}_{Z_N}$ are simple, purely infinite and not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The class of the subshifts is the first examples whose associated $C^*$-algebras are not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The subshifts $Z_N$ are coded systems whose languages are context free. We compute the topological entropy for the subshifts and show that KMS-state for gauge action on the associated $C^*$-algebra ${\\cal O}_{Z_N}$ exists if and only if the logarithm of the inverse temperature is the topological entropy for the subshift $Z_N$, and the corresponding KMS-state is unique."}
{"category": "Math", "title": "On the distribution of coefficients of residue polynomials", "abstract": "Using the formalism of polynomials with positive coefficients, the fact that exactly half of all subsets of a finite set have even cardinality can be generalized asymptotically."}
{"category": "Math", "title": "Integral representations for a generalized Hermite linear functional", "abstract": "In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given."}
{"category": "Math", "title": "Applications of Klee's Dehn-Sommerville relations", "abstract": "We use Klee's Dehn-Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai's conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify K\\\"uhnel's conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove K\\\"uhnel's conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee's Dehn-Sommerville relations and strengthen Kalai's result on the number of their edges."}
{"category": "Math", "title": "Fraisse's construction from a topos-theoretic perspective", "abstract": "We present a topos-theoretic interpretation of (a categorical generalization of) Fraisse's construction in model theory, with applications to countably categorical theories."}
{"category": "Math", "title": "The srank Conjecture on Schur's $Q$-Functions", "abstract": "We show that the shifted rank, or srank, of any partition $\\lambda$ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's $Q_{\\lambda}$ function in terms of power sum symmetric functions. This gives an affirmative answer to a conjecture of Clifford. As pointed out by Clifford, the notion of the srank can be naturally extended to a skew partition $\\lambda/\\mu$ as the minimum number of bars among the corresponding skew bar tableaux. While the srank conjecture is not valid for skew partitions, we give an algorithm to compute the srank."}
{"category": "Math", "title": "Logarithmically Improved Criteria for Navier-Stokes Equations", "abstract": "In this paper we prove the logarithmically improved Serrin's criteria to the three-dimensional incompressible Navier-Stokes equations."}
{"category": "Math", "title": "Ramanujan-type supercongruences", "abstract": "We present several supercongruences that may be viewed as $p$-adic analogues of Ramanujan-type series for $1/\\pi$ and $1/\\pi^2$, and prove three of these examples."}
{"category": "Math", "title": "The length of the shortest closed geodesics on a positively curved manifold", "abstract": "We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics."}
{"category": "Math", "title": "Probability theory and its models", "abstract": "This paper argues for the status of formal probability theory as a mathematical, rather than a scientific, theory. David Freedman and Philip Stark's concept of model based probabilities is examined and is used as a bridge between the formal theory and applications."}
{"category": "Math", "title": "Long-time Asymptotics for the NLS equation via dbar methods", "abstract": "We present a new method for obtaining sharp asymptotics of solutions of the defocussing nonlinear Schr\\\"odinger (NLS) equation, based on dbar methods and under essentially minimal regularity assumptions on initial data."}
{"category": "Math", "title": "Dutch book in simple multivariate normal prediction: Another look", "abstract": "In this expository paper we describe a relatively elementary method of establishing the existence of a Dutch book in a simple multivariate normal prediction setting. The method involves deriving a nonstandard predictive distribution that is motivated by invariance. This predictive distribution satisfies an interesting identity which in turn yields an elementary demonstration of the existence of a Dutch book for a variety of possible predictive distributions."}
{"category": "Math", "title": "Continuous families of Hamiltonian torus actions", "abstract": "We determine conditions under which two Hamiltonian torus actions on a symplectic manifold $M$ are homotopic by a family of Hamiltonian torus actions, when $M$ is a toric manifold and when $M$ is a coadjoint orbit."}
{"category": "Math", "title": "Connectivity of the Uniform Random Intersection Graph", "abstract": "A \\emph{uniform random intersection graph} $G(n,m,k)$ is a random graph constructed as follows. Label each of $n$ nodes by a randomly chosen set of $k$ distinct colours taken from some finite set of possible colours of size $m$. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks. Such graphs arise in particular when modelling the network graph of the well known key predistribution technique due to Eschenauer and Gligor. The paper determines the threshold for connectivity of the graph $G(n,m,k)$ when $n\\to \\infty$ with $k$ a function of $n$ such that $k\\geq 2$ and $m=\\lfloor n^\\alpha\\rfloor$ for some fixed positive real number $\\alpha$. In this situation, $G(n,m,k)$ is almost surely connected when \\[ \\liminf k^2n/m\\log n>1, \\] and $G(n,m,k)$ is almost surely disconnected when \\[ \\limsup k^2n/m\\log n<1. \\]"}
{"category": "Math", "title": "Connexions affines et projectives sur les surfaces complexes compactes", "abstract": "We prove that holomorphic normal projective connections on compact complex surfaces are flat. We show that a holomorphic torsion-free affine connection $\\nabla$ on a compact complex surface is locally modelled on a translations-invariant affine connection on $\\C^2$, except if $\\nabla$ is a generic connection on a principal elliptic bundle over a Riemann surface of genus $g \\geq 2$, with odd first Betti number. In the last case, the local Killing Lie algebra is of dimension one, generated by the fundamental vector field of the principal fibration."}
{"category": "Math", "title": "Local Unitary Cocycles of E_0-semigroups", "abstract": "This paper concerns the structure of the group of local unitary cocycles, also called the gauge group, of an E_0-semigroup. The gauge group of a spatial E_0-semigroup has a natural action on the set of units by operator multiplication. Arveson has characterized completely the gauge group of E_0-semigroups of type I, and as a consequence it is known that in this case the gauge group action is transitive. In fact, if the semigroup has index k, then the gauge group action is transitive on the set of k+1-tuples of appropriately normalized independent units. An action of the gauge group having this property is called k+1-fold transitive. We construct examples of E_0-semigroups of type II and index 1 which are not 2-fold transitive. These new examples also illustrate that an E_0-semigroup of type II_k need not be a tensor product of an E_0-semigroup of type II_0 and another of type I_k."}
{"category": "Math", "title": "R\\'eseaux d'induction des repr\\'esentations elliptiques de Lubin-Tate", "abstract": "We study the reduction modulo $l$ of some elliptic representations; for each of these representations, we give a particular lattice naturally obtained by parabolic induction in giving the graph of extensions between its irreducible sub-quotient of its reduction modulo $l$. The principal motivation for this work, is that these lattices appear in the cohomology of Lubin-Tate towers."}
{"category": "Math", "title": "Characteristics of hand and machine-assigned scores to college students' answers to open-ended tasks", "abstract": "Assessment of learning in higher education is a critical concern to policy makers, educators, parents, and students. And, doing so appropriately is likely to require including constructed response tests in the assessment system. We examined whether scoring costs and other concerns with using open-end measures on a large scale (e.g., turnaround time and inter-reader consistency) could be addressed by machine grading the answers. Analyses with 1359 students from 14 colleges found that two human readers agreed highly with each other in the scores they assigned to the answers to three types of open-ended questions. These reader assigned scores also agreed highly with those assigned by a computer. The correlations of the machine-assigned scores with SAT scores, college grades, and other measures were comparable to the correlations of these variables with the hand-assigned scores. Machine scoring did not widen differences in mean scores between racial/ethnic or gender groups. Our findings demonstrated that machine scoring can facilitate the use of open-ended questions in large-scale testing programs by providing a fast, accurate, and economical way to grade responses."}
{"category": "Math", "title": "Generating uniform random vectors in $\\QTR{bf}{Z}_{p}^{k}$: the general case", "abstract": "This paper is about the rate of convergence of the Markov chain $X_{n+1}=AX_{n}+B_{n}$ (mod $p$), where $A$ is an integer matrix with nonzero eigenvalues and ${B_{n}}_{n}$ is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of $Q^{k}$ invariant under $A$. If $|\\lambda_{i}|\\not=1$ for all eigenvalues $\\lambda_{i}$ of $A$, then $n=O((\\ln p)^{2}) $ steps are sufficient and $n=O(\\ln p)$ steps are necessary to have $X_{n}$ sampling from a nearly uniform distribution. Conversely, if $A$ has the eigenvalues $\\lambda_{i}$ that are roots of positive integer numbers, $|\\lambda_{1}|=1$ and $|\\lambda_{i}|>1$ for all $i\\not=1$, then $O(p^{2}) $ steps are necessary and sufficient."}
{"category": "Math", "title": "Plane curves as Pfaffians", "abstract": "Let $C$ be a smooth curve in $\\PP^2$ given by an equation F=0 of degree $d$. In this paper we parametrise all linear pfaffian representations of $F$ by an open subset in the moduli space $M_C(2,K_C)$. We construct an explicit correspondence between pfaffian representations of $C$ and rank 2 vector bundles on $C$ with canonical determinant and no sections."}
{"category": "Math", "title": "Alternative formulas for synthetic dual system estimation in the 2000 census", "abstract": "The U.S. Census Bureau provides an estimate of the true population as a supplement to the basic census numbers. This estimate is constructed from data in a post-censal survey. The overall procedure is referred to as dual system estimation. Dual system estimation is designed to produce revised estimates at all levels of geography, via a synthetic estimation procedure. We design three alternative formulas for dual system estimation and investigate the differences in area estimates produced as a result of using those formulas. The primary target of this exercise is to better understand the nature of the homogeneity assumptions involved in dual system estimation and their consequences when used for the enumeration data that occurs in an actual large scale application like the Census. (Assumptions of this nature are sometimes collectively referred to as the ``synthetic assumption'' for dual system estimation.) The specific focus of our study is the treatment of the category of census counts referred to as imputations in dual system estimation. Our results show the degree to which varying treatment of these imputation counts can result in differences in population estimates for local areas such as states or counties."}
{"category": "Math", "title": "On the history and use of some standard statistical models", "abstract": "This paper tries to tell the story of the general linear model, which saw the light of day 200 years ago, and the assumptions underlying it. We distinguish three principal stages (ignoring earlier more isolated instances). The model was first proposed in the context of astronomical and geodesic observations, where the main source of variation was observational error. This was the main use of the model during the 19th century. In the 1920's it was developed in a new direction by R.A. Fisher whose principal applications were in agriculture and biology. Finally, beginning in the 1930's and 40's it became an important tool for the social sciences. As new areas of applications were added, the assumptions underlying the model tended to become more questionable, and the resulting statistical techniques more prone to misuse."}
{"category": "Math", "title": "Counting the homeless in Los Angeles County", "abstract": "Over the past two decades, a variety of methods have been used to count the homeless in large metropolitan areas. In this paper, we report on an effort to count the homeless in Los Angeles County, one that employed the sampling of census tracts. A number of complications are discussed, including\\^{E} the need to impute homeless counts to areas of \\^{E}the County\\^{E} not sampled. We conclude that, despite their imperfections, estimated counts provided useful and credible information to the stakeholders involved."}
{"category": "Math", "title": "Statistical adjustment for a measure of healthy lifestyle doesn't yield the truth about hormone therapy", "abstract": "The Women's Health Initiative randomized clinical trial of hormone therapy found no benefit of hormones in preventive cardiovascular disease, a finding in striking contrast with a large body of observational research. Understanding whether better methodology and/or statistical adjustment might have prevented the erroneous conclusions of observational research is important. This is a re-analysis of data from a case-control study examining the relationship of postmenopausal hormone therapy and the risks of myocardial infarction (MI) and ischemic stroke in which we reported no overall increase or decrease in the risk of either event. Variables measuring health behavior/lifestyle that are not likely to be causally with the risks of MI and stroke (e.g., sunscreen use) were included in multivariate analysis along with traditional confounders (age, hypertension, diabetes, smoking, body mass index, ethnicity, education, prior coronary heart disease for MI and prior stroke/TIA for stroke) to determine whether adjustment for the health behavior/lifestyle variables could reproduce or bring the results closer to the findings in a large and definitive randomized clinical trial of hormone therapy, the Women's Health Initiative. For both MI and stroke, measures of health behavior/lifestyle were associated with odds ratios (ORs) less than 1.0. Adjustment for traditional cardiovascular disease confounders did not alter the magnitude of the ORs for MI or stroke. Addition of a subset of these variables selected using stepwise regression to the final MI or stroke models along with the traditional cardiovascular disease confounders moved the ORs for estrogen and estrogen/progestin use closer to values observed in the Women Health Initiative clinical trial, but did not reliably reproduce the clinical trial results for these two endpoints."}
{"category": "Math", "title": "Convergence of Bergman measures for high powers of a line bundle", "abstract": "Let $L$ be a holomorphic line bundle on a compact complex manifold $X$ of dimension $n,$ and let $e^{-\\phi}$ be a continuous metric on $L.$ Fixing a measure $d\\mu$ on $X$ gives a sequence of Hilbert spaces consisting of holomorphic sections of tensor powers of $L.$ We prove that the corresponding sequence of scaled Bergman measures converges, in the high tensor power limit, to the equilibrium measure of the pair $(K,\\phi),$ where $K$ is the support of $d\\mu,$ as long as $d\\mu$ is stably Bernstein-Markov with respect to $(K,\\phi).$ Here the Bergman measure denotes $d\\mu$ times the restriction to the diagonal of the pointwise norm of the corresponding orthogonal projection operator. In particular, an extension to higher dimensions is obtained of results concerning random matrices and classical orthogonal polynomials."}
{"category": "Math", "title": "On the Hochschild homology of elliptic Sklyanin algebras", "abstract": "In this paper, we compute the Hochschild homology of elliptic Sklyanin algebras. These algebras are deformations of polynomial algebra with a Poisson bracket called the Sklyanin Poisson bracket."}
{"category": "Math", "title": "Diagonal invariants and the refined multimahonian distribution", "abstract": "Combinatorial aspects of multivariate diagonal invariants of the symmetric group are studied. As a consequence it is proved the existence of a multivariate extension of the classical Robinson-Schensted correspondence. Further byproduct are a pure combinatorial algorithm to describe the irreducible decomposition of the tensor product of two irreducible representations of the symmetric group, and new symmetry results on permutation enumeration with respect to descent sets."}
{"category": "Math", "title": "Cohomology algebra of orbit spaces of free involutions on lens spaces", "abstract": "Let $G$ be a group acting continuously on a space $X$ and let $X/G$ be its orbit space. Determining the topological or cohomological type of the orbit space $X/G$ is a classical problem in the theory of transformation groups. In this paper, we consider this problem for cohomology lens spaces. Let $X$ be a finitistic space having the mod 2 cohomology algebra of the lens space $L_p^{2m-1}(q_1,...,q_m)$. Then we classify completely the possible mod 2 cohomology algebra of orbit spaces of arbitrary free involutions on $X$. We also give examples of spaces realizing the possible cohomology algebras. In the end, we give an application of our results to non-existence of $\\mathbb{Z}_2$-equivariant map $\\mathbb{S}^n \\to X$."}
{"category": "Math", "title": "Diophantine approximation with arithmetic functions, I", "abstract": "We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes."}
{"category": "Math", "title": "Natural Topologies on Colombeau Algebras", "abstract": "We define natural topologies on the Colombeau algebras which are compatible with the algebraic structure. These topologies reduces do Scarpalezos sharp topologies when restricted. with this we take a positive step towards topological methods of solving P.D. Equations in Colombeau algebras. Applications will appear elsewhere."}
{"category": "Math", "title": "On the classification of twisting maps between $K^n$ and $K^m$", "abstract": "We define the notion of admissible pair for an algebra $A$, consisting on a couple $(\\Gamma,R)$, where $\\Gamma$ is a quiver and $R$ a unital, splitted and factorizable representation of $\\Gamma$, and prove that the set of admissible pairs for $A$ is in one to one correspondence with the points of the variety of twisting maps $\\mathcal{T}_A^n:=\\mathcal{T}(K^n,A)$. We describe all these representations in the case $A=K^m$."}
{"category": "Math", "title": "Multivariate data analysis: The French way", "abstract": "This paper presents exploratory techniques for multivariate data, many of them well known to French statisticians and ecologists, but few well understood in North American culture. We present the general framework of duality diagrams which encompasses discriminant analysis, correspondence analysis and principal components, and we show how this framework can be generalized to the regression of graphs on covariates."}
{"category": "Math", "title": "The future of census coverage surveys", "abstract": "A quarter-century of statistical research has shown that census coverage surveys, valuable as they are in offering a report card on each decennial census, do not provide usable estimates of geographical differences in coverage. The determining reason is the large number of ``doubly missing'' people missing both from the census enumeration and from coverage survey estimates. Future coverage surveys should be designed to meet achievable goals, foregoing efforts at spatial specificity. One implication is a sample size no more than about $30,000$, setting free resources for controlling processing errors and investing in coverage improvement. Possible integration of coverage measurement with the American Community Survey would have many benefits and should be given careful consideration."}
{"category": "Math", "title": "Hypergeometric functions over F_p and relations to elliptic curves and modular forms", "abstract": "For primes p congruent to 1 mod 12, we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728/t and values of a particular 2F1-hypergeometric function over F_p. Additionally, we determine a formula for traces of Hecke operators T_k(p) on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's tau-function in terms of hypergeometric functions."}
{"category": "Math", "title": "Quantization of pseudo-differential operators on the torus", "abstract": "Pseudo-differential and Fourier series operators on the n-torus are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on $L^2$ under certain conditions on their phases and amplitudes."}
{"category": "Math", "title": "Moderate deviations for stationary sequences of Hilbert valued bounded random variables", "abstract": "In this paper, we derive the moderate deviation principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are martingale approximations and a new Hoeffding inequality for non adpated sequences of Hilbert-valued random variables. Applications to Cramer-Von Mises statistics, functions of linear processes and stable Markov chains are given."}
{"category": "Math", "title": "Counter example to Strichartz estimates for the wave equation in domains", "abstract": "Let U be a bounded, regular, strictly convex domain of R^2 and consider the wave equation on U with Dirichlet boundary condition. We prove that in such a domain the Strichartz estimates for the wave equation suffer losses when compared to the case U=R^2, at least for a subset of the usual range of indices."}
{"category": "Math", "title": "Howe type duality for metaplectic group acting on symplectic spinor valued forms", "abstract": "Let $\\lambda: \\tilde{G}\\to G$ be the non-trivial double covering of the symplectic group $G=Sp(V,\\omega)$ of the symplectic vector space $(V,\\omega)$ by the metaplectic group $\\tilde{G}=Mp(V,\\omega).$ In this case, $\\lambda$ is also a representation of $\\tilde{G}$ on the vector space $V$ and thus, it gives rise to the representation of $\\tilde{G}$ on the space of exterior forms $\\bigwedge^{\\bullet}V^*$ by taking wedge products. Let $S$ be the minimal globalization of the Harish-Chandra module of the complex Segal-Shale-Weil representation of the metaplectic group $\\tilde{G}.$ We prove that the associative commutant algebra $\\hbox{End}_{\\tilde{G}}(\\bigwedge^{\\bullet}V^*\\otimes S)$ of the metaplectic group $\\tilde{G}$ acting on the $S$-valued exterior forms is generated by certain representation of the super ortho-symplectic Lie algebra $osp(1|2)$ and two distinguished operators. This establishes a Howe type duality between the metaplectic group and the super Lie algebra $\\mathfrak{osp}(1|2).$ Also the space $\\bigwedge^{\\bullet}V^*\\otimes S$ is decomposed wr. to the joint action of $Mp(V,\\omega)$ and $osp(1|2).$"}
{"category": "Math", "title": "Vector partition function and generalized Dahmen-Micchelli spaces", "abstract": "This is the first of two papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory will appear in a subsequent paper."}
{"category": "Math", "title": "The line bundles on moduli stacks of principal bundles on a curve", "abstract": "Let G be an affine reductive algebraic group over an algebraically closed field k. We determine the Picard group of the moduli stacks of principal G-bundles on any smooth projective curve over k."}
{"category": "Math", "title": "Optimal reconstruction systems for erasures and for the q-potential", "abstract": "We introduce the $q$-potential as an extension of the Benedetto-Fickus frame potential, defined on general reconstruction systems and we show that protocols are the minimizers of this potential under certain restrictions. We extend recent results of B.G. Bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst (normalized) reconstruction error is computed with respect to a compatible unitarily invariant norm. We finally describe necessary and sufficient (spectral) conditions, that we call $q$-fundamental inequalities, for the existence of protocols with prescribed properties by relating this problem to Klyachko's and Fulton's theory on sums of hermitian operators."}
{"category": "Math", "title": "Asymptotically linear solutions in H^1 of the 2-d defocusing nonlinear Schroedinger and Hartree equations", "abstract": "In the 2-d setting, given an $H^1$ solution $v(t)$ to the linear Schr\\\"odinger equation $i\\partial_t v +\\Delta v =0$, we prove the existence (but not uniqueness) of an $H^1$ solution $u(t)$ to the defocusing nonlinear Schr\\\"odinger (NLS) equation $i\\partial_t u + \\Delta u -|u|^{p-1}u=0$ for nonlinear powers $2<p<3$ and the existence of an $H^1$ solution $u(t)$ to the defocusing Hartree equation $i\\partial_t u + \\Delta u -(|x|^{-\\gamma}\\star|u|^{2})u=0$ for interaction powers $1<\\gamma<2$, such that $\\|u(t)-v(t)\\|_{H^1} \\to 0$ as $t\\to +\\infty$. This is a partial result toward the existence of well-defined continuous wave operators $H^1 \\to H^1$ for these equations. For NLS in 2-d, such wave operators are known to exist for $p\\geq 3$, while for $p\\leq 2$ it is known that they cannot exist. The Hartree equation in 2-d only makes sense for $0<\\gamma<2$, and it was previously known that wave operators cannot exist for $0<\\gamma\\leq 1$, while no result was previously known in the range $1<\\gamma<2$. Our proof in the case of NLS applies a new estimate of Colliander-Grillakis-Tzirakis (2008) to a strategy devised by Nakanishi (2001). For the Hartree equation, we prove a new correlation estimate following the method of Colliander-Grillakis-Tzirakis (2008)."}
{"category": "Math", "title": "The Space of Symplectic Structures on Closed 4-Manifolds", "abstract": "This is a survey paper on the space of symplectic structures on closed 4-manifolds, for the Proceedings ICCM 2004"}
{"category": "Math", "title": "Wintenberger's Functor for Abelian Extensions", "abstract": "Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$-adic Lie extensions $E/F$, where $F$ is a local field with residue field $k$, and a category whose objects are pairs $(K,A)$, where $K\\cong k((T))$ and $A$ is an abelian $p$-adic Lie subgroup of $\\Aut_k(K)$. In this paper we extend this equivalence to allow $\\Gal(E/F)$ and $A$ to be arbitrary abelian pro-$p$ groups."}
{"category": "Math", "title": "Modified Schmidt games and Diophantine approximation with weights", "abstract": "We show that the sets of weighted badly approximable vectors in $\\Bbb R^n$ are winning sets of certain games, which are modifications of $(\\alpha,\\beta)$-games introduced by W. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension."}
{"category": "Math", "title": "The Relative Symplectic Cone and T^2-Fibrations", "abstract": "In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T^2-fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kaehler surfaces raised by the second author."}
{"category": "Math", "title": "Projective C*-Algebras and Boundary Maps", "abstract": "Both boundary maps in K-theory are expressed in terms of surjections from projective C*-algebras to semiprojective C*-algebras."}
{"category": "Math", "title": "Adjoint functors and tree duality", "abstract": "A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions."}
{"category": "Math", "title": "Generalized BSDE With 2-Reflecting Barriers and Stochastic Quadratic Growth", "abstract": "We study the existence of a solution for a one-dimensional generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under assumptions on the input data which are weaker than that on the current literature. In particular, we construct a maximal solution for such a GRBSDE when the terminal condition \\xi is only F_T-measurable and the driver f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z without assuming any P-integrability conditions. The work is suggested by the interest the results might have in Dynkin game problem and American game option."}
{"category": "Math", "title": "Cocycle Superrigidity for Profinite Actions of Property (T) Groups", "abstract": "Consider a free ergodic measure preserving profinite action $\\Gamma\\curvearrowright X$ (i.e. an inverse limit of actions $\\Gamma\\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $\\Gamma$ (more generally of a group $\\Gamma$ which admits an infinite normal subgroup $\\Gamma_0$ such that the inclusion $\\Gamma_0\\subset\\Gamma$ has relative property (T) and $\\Gamma/\\Gamma_0$ is finitely generated) on a standard probability space $X$. We prove that if $w:\\Gamma\\times X\\to \\Lambda$ is a measurable cocycle with values in a countable group $\\Lambda$, then $w$ is cohomologous to a cocycle $w'$ which factors through the map $\\Gamma\\times X\\to \\Gamma\\times X_n$, for some $n$. As a corollary, we show that any orbit equivalence of $\\Gamma\\curvearrowright X$ with any free ergodic measure preserving action $\\Lambda\\curvearrowright Y$ comes from a (virtual) conjugacy of actions."}
{"category": "Math", "title": "Multiple tests of association with biological annotation metadata", "abstract": "We propose a general and formal statistical framework for multiple tests of association between known fixed features of a genome and unknown parameters of the distribution of variable features of this genome in a population of interest. The known gene-annotation profiles, corresponding to the fixed features of the genome, may concern Gene Ontology (GO) annotation, pathway membership, regulation by particular transcription factors, nucleotide sequences, or protein sequences. The unknown gene-parameter profiles, corresponding to the variable features of the genome, may be, for example, regression coefficients relating possibly censored biological and clinical outcomes to genome-wide transcript levels, DNA copy numbers, and other covariates. A generic question of great interest in current genomic research regards the detection of associations between biological annotation metadata and genome-wide expression measures. This biological question may be translated as the test of multiple hypotheses concerning association measures between gene-annotation profiles and gene-parameter profiles. A general and rigorous formulation of the statistical inference question allows us to apply the multiple hypothesis testing methodology developed in [Multiple Testing Procedures with Applications to Genomics (2008) Springer, New York] and related articles, to control a broad class of Type I error rates, defined as generalized tail probabilities and expected values for arbitrary functions of the numbers of Type I errors and rejected hypotheses. The resampling-based single-step and stepwise multiple testing procedures of [Multiple Testing Procedures with Applications to Genomics (2008) Springer, New York] take into account the joint distribution of the test statistics and provide Type I error control in testing problems involving general data generating distributions (with arbitrary dependence structures among variables), null hypotheses, and test statistics."}
{"category": "Math", "title": "Three months journeying of a Hawaiian monk seal", "abstract": "Hawaiian monk seals (Monachus schauinslandi) are endemic to the Hawaiian Islands and are the most endangered species of marine mammal that lives entirely within the jurisdiction of the United States. The species numbers around 1300 and has been declining owing, among other things, to poor juvenile survival which is evidently related to poor foraging success. Consequently, data have been collected recently on the foraging habitats, movements, and behaviors of monk seals throughout the Northwestern and main Hawaiian Islands. Our work here is directed to exploring a data set located in a relatively shallow offshore submerged bank (Penguin Bank) in our search of a model for a seal's journey. The work ends by fitting a stochastic differential equation (SDE) that mimics some aspects of the behavior of seals by working with location data collected for one seal. The SDE is found by developing a time varying potential function with two points of attraction. The times of location are irregularly spaced and not close together geographically, leading to some difficulties of interpretation. Synthetic plots generated using the model are employed to assess its reasonableness spatially and temporally. One aspect is that the animal stays mainly southwest of Molokai. The work led to the estimation of the lengths and locations of the seal's foraging trips."}
{"category": "Math", "title": "Ces\\`aro means of Jacobi expansions on the parabolic biangle", "abstract": "We study Ces\\`aro $(C,\\delta)$ means for two-variable Jacobi polynomials on the parabolic biangle $B=\\{(x_1,x_2)\\in{\\mathbb R}^2:0\\leq x_1^2\\leq x_2\\leq 1\\}$. Using the product formula derived by Koornwinder & Schwartz for this polynomial system, the Ces\\`aro operator can be interpreted as a convolution operator. We then show that the Ces\\`aro $(C,\\delta)$ means of the orthogonal expansion on the biangle are uniformly bounded if $\\delta>\\alpha+\\beta+1$, $\\alpha-\\frac 12\\geq\\beta\\geq 0$. Furthermore, for $\\delta\\geq\\alpha+2\\beta+\\frac 32$ the means define positive linear operators."}
{"category": "Math", "title": "DNA Probabilities in People v. Prince: When are racial and ethnic statistics relevant?", "abstract": "When a defendant's DNA matches a sample found at a crime scene, how compelling is the match? To answer this question, DNA analysts typically use relative frequencies, random-match probabilities or likelihood ratios. They compute these quantities for the major racial or ethnic groups in the United States, supplying prosecutors with such mind-boggling figures as ``one in nine hundred and fifty sextillion African Americans, one in one hundred and thirty septillion Caucasians, and one in nine hundred and thirty sextillion Hispanics.\" In People v. Prince, a California Court of Appeals rejected this practice on the theory that only the perpetrator's race is relevant to the crime; hence, it is impermissible to introduce statistics about other races. This paper critiques this reasoning. Relying on the concept of likelihood, it presents a logical justification for referring to a range of races and identifies some problems with the one-race-only rule. The paper also notes some ways to express the probative value of a DNA match quantitatively without referring to variations in DNA profile frequencies among races or ethnic groups."}
{"category": "Math", "title": "A-infinity monads and completion", "abstract": "Given an operad A of topological spaces, we consider A-monads in a topological category C . When A is an A-infinity-operad, any A-monad K : C -> C can be thought of as a monad up to coherent homotopies. We define the completion functor with respect to an A-infinity-monad and prove that it is an A-infinity-monad itself."}
{"category": "Math", "title": "Testing earthquake predictions", "abstract": "Statistical tests of earthquake predictions require a null hypothesis to model occasional chance successes. To define and quantify `chance success' is knotty. Some null hypotheses ascribe chance to the Earth: Seismicity is modeled as random. The null distribution of the number of successful predictions -- or any other test statistic -- is taken to be its distribution when the fixed set of predictions is applied to random seismicity. Such tests tacitly assume that the predictions do not depend on the observed seismicity. Conditioning on the predictions in this way sets a low hurdle for statistical significance. Consider this scheme: When an earthquake of magnitude 5.5 or greater occurs anywhere in the world, predict that an earthquake at least as large will occur within 21 days and within an epicentral distance of 50 km. We apply this rule to the Harvard centroid-moment-tensor (CMT) catalog for 2000--2004 to generate a set of predictions. The null hypothesis is that earthquake times are exchangeable conditional on their magnitudes and locations and on the predictions--a common ``nonparametric'' assumption in the literature. We generate random seismicity by permuting the times of events in the CMT catalog. We consider an event successfully predicted only if (i) it is predicted and (ii) there is no larger event within 50 km in the previous 21 days. The $P$-value for the observed success rate is $<0.001$: The method successfully predicts about 5% of earthquakes, far better than `chance,' because the predictor exploits the clustering of earthquakes -- occasional foreshocks -- which the null hypothesis lacks. Rather than condition on the predictions and use a stochastic model for seismicity, it is preferable to treat the observed seismicity as fixed, and to compare the success rate of the predictions to the success rate of simple-minded predictions like those just described. If the proffered predictions do no better than a simple scheme, they have little value."}
{"category": "Math", "title": "Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems", "abstract": "It has been widely realized that Monte Carlo methods (approximation via a sample ensemble) may fail in large scale systems. This work offers some theoretical insight into this phenomenon in the context of the particle filter. We demonstrate that the maximum of the weights associated with the sample ensemble converges to one as both the sample size and the system dimension tends to infinity. Specifically, under fairly weak assumptions, if the ensemble size grows sub-exponentially in the cube root of the system dimension, the convergence holds for a single update step in state-space models with independent and identically distributed kernels. Further, in an important special case, more refined arguments show (and our simulations suggest) that the convergence to unity occurs unless the ensemble grows super-exponentially in the system dimension. The weight singularity is also established in models with more general multivariate likelihoods, e.g. Gaussian and Cauchy. Although presented in the context of atmospheric data assimilation for numerical weather prediction, our results are generally valid for high-dimensional particle filters."}
{"category": "Math", "title": "Semi-continuit\\'e des cellules de Kazhdan-Lusztig", "abstract": "Computations in small Coxeter groups and infinite dihedral groups suggest that Kazhdan-Lusztig cells for unequal parameters obey to some \"semicontinuity\" phenomenon (as the parameter vary). The aim of this paper is to provide a rigorous theoretical background that allows to state some precise conjectures."}
{"category": "Math", "title": "Higher order influence functions and minimax estimation of nonlinear functionals", "abstract": "We present a theory of point and interval estimation for nonlinear functionals in parametric, semi-, and non-parametric models based on higher order influence functions (Robins (2004), Section 9; Li et al. (2004), Tchetgen et al. (2006), Robins et al. (2007)). Higher order influence functions are higher order U-statistics. Our theory extends the first order semiparametric theory of Bickel et al. (1993) and van der Vaart (1991) by incorporating the theory of higher order scores considered by Pfanzagl (1990), Small and McLeish (1994) and Lindsay and Waterman (1996). The theory reproduces many previous results, produces new non-$\\sqrt{n}$ results, and opens up the ability to perform optimal non-$\\sqrt{n}$ inference in complex high dimensional models. We present novel rate-optimal point and interval estimators for various functionals of central importance to biostatistics in settings in which estimation at the expected $\\sqrt{n}$ rate is not possible, owing to the curse of dimensionality. We also show that our higher order influence functions have a multi-robustness property that extends the double robustness property of first order influence functions described by Robins and Rotnitzky (2001) and van der Laan and Robins (2003)."}
{"category": "Math", "title": "Projection pursuit for discrete data", "abstract": "This paper develops projection pursuit for discrete data using the discrete Radon transform. Discrete projection pursuit is presented as an exploratory method for finding informative low dimensional views of data such as binary vectors, rankings, phylogenetic trees or graphs. We show that for most data sets, most projections are close to uniform. Thus, informative summaries are ones deviating from uniformity. Syllabic data from several of Plato's great works is used to illustrate the methods. Along with some basic distribution theory, an automated procedure for computing informative projections is introduced."}
{"category": "Math", "title": "Finitude homotopique et isotopique des structures de contact tendues", "abstract": "Let V be a closed 3-manifold. In this paper we prove that the homotopy classes of plane fields on V that contain tight contact structures are in finite number and that, if V is atoroidal, the isotopy classes of tight contact structures are also in finite number."}
{"category": "Math", "title": "Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions", "abstract": "It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ \\xi_j $ for $ j \\in \\mathbb Z,$ when the ceiling of the walk $ S_n $ is located at $ j.$ The random reward (or scenery) $ \\xi_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion (fBm). This work extends a previous work where the random walk $ S_n$ had independent increments limits."}
{"category": "Math", "title": "Objective Bayesian analysis under sequential experimentation", "abstract": "Objective priors for sequential experiments are considered. Common priors, such as the Jeffreys prior and the reference prior, will typically depend on the stopping rule used for the sequential experiment. New expressions for reference priors are obtained in various contexts, and computational issues involving such priors are considered."}
{"category": "Math", "title": "J. K. Ghosh's contribution to statistics: A brief outline", "abstract": "Professor Jayanta Kumar Ghosh has contributed massively to various areas of Statistics over the last five decades. Here, we survey some of his most important contributions. In roughly chronological order, we discuss his major results in the areas of sequential analysis, foundations, asymptotics, and Bayesian inference. It is seen that he progressed from thinking about data points, to thinking about data summarization, to the limiting cases of data summarization in as they relate to parameter estimation, and then to more general aspects of modeling including prior and model selection."}
{"category": "Math", "title": "Sequential tests and estimates after overrunning based on $p$-value combination", "abstract": "Often in sequential trials additional data become available after a stopping boundary has been reached. A method of incorporating such information from overrunning is developed, based on the ``adding weighted Zs'' method of combining $p$-values. This yields a combined $p$-value for the primary test and a median-unbiased estimate and confidence bounds for the parameter under test. When the amount of overrunning information is proportional to the amount available upon terminating the sequential test, exact inference methods are provided; otherwise, approximate methods are given and evaluated. The context is that of observing a Brownian motion with drift, with either linear stopping boundaries in continuous time or discrete-time group-sequential boundaries. The method is compared with other available methods and is exemplified with data from two sequential clinical trials."}
{"category": "Math", "title": "Classification of complex naturally graded quasi-filiform Zinbiel algebras", "abstract": "In this work the description up to isomorphism of complex naturally graded quasi-filiform Zinbiel algebras is obtained."}
{"category": "Math", "title": "On predictive probability matching priors", "abstract": "We revisit the question of priors that achieve approximate matching of Bayesian and frequentist predictive probabilities. Such priors may be thought of as providing frequentist calibration of Bayesian prediction or simply as devices for producing frequentist prediction regions. Here we analyse the $O(n^{-1})$ term in the expansion of the coverage probability of a Bayesian prediction region, as derived in [Ann. Statist. 28 (2000) 1414--1426]. Unlike the situation for parametric matching, asymptotic predictive matching priors may depend on the level $\\alpha$. We investigate uniformly predictive matching priors (UPMPs); that is, priors for which this $O(n^{-1})$ term is zero for all $\\alpha$. It was shown in [Ann. Statist. 28 (2000) 1414--1426] that, in the case of quantile matching and a scalar parameter, if such a prior exists then it must be Jeffreys' prior. In the present article we investigate UPMPs in the multiparameter case and present some general results about the form, and uniqueness or otherwise, of UPMPs for both quantile and highest predictive density matching."}
{"category": "Math", "title": "A note on the ABC-PRC algorithm of Sissons et al. (2007)", "abstract": "This note describes the results of some tests of the ABC-PRC algorithm of Sissons et al. (PNAS, 2007), and demonstrates with a toy example that the method does not converge on the true posterior distribution."}
{"category": "Math", "title": "Galois actions on Neron models of Jacobians", "abstract": "Let $X$ be a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, and let $K'/K$ be a tame extension. We study extensions of the $G = \\Gal(K'/K)$-action on $ X_{K'} $ to certain regular models of $X_{K'}$ over $R'$, the integral closure of $R$ in $K'$. In particular, we consider the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model, and obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. We apply these results to study a natural filtration of the special fiber of the N\\'eron model of the Jacobian of $X$ by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for $X$ over $R$, and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber types for curves of genus 1 and 2."}
{"category": "Math", "title": "Price dynamics in a strategic model of trade between two regions", "abstract": "This paper develops a strategic model of trade between two regions in which, depending on the relation among output, financial resources and transportation costs, the adjustment of prices towards an equilibrium is studied. We derive conditions on the relations among output and financial resources which produce different types of Nash equilibria. The paths obtained in the process of converging toward a steady state for prices under discrete-time and continuous-time dynamics are derived and compared. It turns out that the results in the two cases differ substantially. Some of the effects of random disturbances on the price dynamics in continuous time are also studied."}
{"category": "Math", "title": "Triply Periodic Minimal Surfaces Bounded by Vertical Symmetry Planes", "abstract": "We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces."}
{"category": "Math", "title": "Variational characterizations of the effective multiplication factor of a nuclear reactor core", "abstract": "We derive some new inf-sup and sup-inf formulae for the so-called effective multiplication factor arising in the study of reactor analysis. We treat in a same formalism the transport equation and the energy-dependent diffusion equation."}
{"category": "Math", "title": "The cone of lower semicontinuous traces on a C*-algebra", "abstract": "The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2-quasitraces on a (non-exact) C*-algebra is considered as well. These results are applied to the study of the Cuntz semigroup. It is shown that if a C*-algebra absorbs the Jiang-Su algebra, then the subsemigroup of its Cuntz semigroup consisting of the purely non-compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2-quasitraces. This yields a computation of the Cuntz semigroup for the following two classes of C*-algebras: C*-algebras that absorb the Jiang-Su algebra and have no non-zero simple subquotients, and simple C*-algebras that absorb the Jiang-Su algebra."}
{"category": "Math", "title": "Derived deformations of Artin stacks", "abstract": "We generalise the techniques of arXiv:0908.1963 to describe derived deformations in simplicial categories. This allows us to consider deformation problems with higher automorphisms, such as chain complexes (which have homotopies) and stacks (which have 2-automorphisms). We also give a general approach for studying deformations of diagrams."}
{"category": "Math", "title": "Rigidity of Quasi-Einstein Metrics", "abstract": "We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some K\\\"ahler quasi-Einstein metrics."}
{"category": "Math", "title": "Asymptotic expansion of the heat kernel for orbifolds", "abstract": "We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the time-zero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbit space of a manifold under the action of a discrete group of isometries), H. Donnelly proved the existence of the heat kernel and constructed the asymptotic expansion for the heat trace. We extend Donnelly's work to the case of general compact orbifolds. Moreover, in both the good case and the general case, we express the heat invariants in a form that clarifies the asymptotic contribution of each part of the singular set of the orbifold. We calculate several terms in the asymptotic expansion explicitly in the case of two-dimensional orbifolds; we use these terms to prove that the spectrum distinguishes elements within various classes of two-dimensional orbifolds."}
{"category": "Math", "title": "Higher order derivative estimates for finite-difference schemes", "abstract": "We give sufficient conditions under which solutions of finite-difference schemes in the space variable for second order possibly degenerate parabolic and elliptic equations admit estimates of spatial derivatives up to any given order independent of the mesh size."}
{"category": "Math", "title": "An elementary proof of a series evaluation in terms of harmonic numbers", "abstract": "An elementary proof of an identity by Lyons, Paule and Riese is given. It is simpler than all the 3 published proofs."}
{"category": "Math", "title": "Smooth analysis of the condition number and the least singular value", "abstract": "Let $\\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $\\a$ is gaussian. Our investigation reveals an interesting fact that the \"core\" matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n} $. This does not occur in Spielman-Teng studies when $\\a$ is gaussian. Consequently, our general estimate involves the norm $\\|M\\|$. In the special case when $\\|M\\|$ is relatively small, this estimate is nearly optimal and extends or refines existing results."}
{"category": "Math", "title": "Hopf algebra extensions of group algebras and Tambara-Yamagami categories", "abstract": "We determine the structure of Hopf algebra extensions of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension $<36$ is necessarily group-theoretical; thus 36 is the smallest possible dimension where a non group-theoretical example occurs."}
{"category": "Math", "title": "Unknotting numbers of diagrams of a given nontrivial knot are unbounded", "abstract": "We show that for any nontrivial knot $K$ and any natural number $n$ there is a diagram $D$ of $K$ such that the unknotting number of $D$ is greater than or equal to $n$. It is well known that twice the unknotting number of $K$ is less than or equal to the crossing number of $K$ minus one. We show that the equality holds only when $K$ is a $(2,p)$-torus knot."}
{"category": "Math", "title": "Thue's Fundamentaltheorem, I: The General Case", "abstract": "In this paper, Thue's Fundamentaltheorem is analysed. We show that it includes, and often strengthens, known effective irrationality measures obtained via the so-called hypergeometric method as well as showing that it can be applied to previously unconsidered families of algebraic numbers. Furthermore, we extend the method to also cover approximation by algebraic numbers in imaginary quadratic number fields."}
{"category": "Math", "title": "A model of hydrodynamic interaction between swimming bacteria", "abstract": "We study the dynamics and interaction of two swimming bacteria, modeled by self-propelled dumbbell-type structures. We focus on alignment dynamics of a coplanar pair of elongated swimmers, which propel themselves either by pushing\" or \"pulling\" both in three- and quasi-two-dimensional geometries of space. We derive asymptotic expressions for the dynamics of the pair, which, complemented by numerical experiments, indicate that the tendency of bacteria to swim in or swim off depends strongly on the position of the propulsion force. In particular, we observe that positioning of the effective propulsion force inside the dumbbell results in qualitative agreement with the dynamics observed in experiments, such as mutual alignment of converging bacteria."}
{"category": "Math", "title": "Accurate Evaluation of Polynomials", "abstract": "For a large class of polynomials, the standard method of polynomial evaluation, Horner's method, can be very inaccurate. The alternative method given here is on average 100 to 1000 times more accurate than Horner's Method. The number of floating point operations is twice that of Horner's method for a single evaluation. For repeated evaluations at nearby points, the number of floating point operations is only doubled for the first evaluation, and is the same as Horner's Method for all following evaluations. This new method is tested with random polynomials."}
{"category": "Math", "title": "Hecke algebras from groups acting on trees and HNN extensions", "abstract": "We study Hecke algebras of groups acting on trees with respect to geometrically defined subgroups. In particular, we consider Hecke algebras of groups of automorphisms of locally finite trees with respect to vertex and edge stabilizers and the stabilizer of an end relative to a vertex stabilizer, assuming that the actions are sufficiently transitive. We focus on identifying the structure of the resulting Hecke algebras, give explicit multiplication tables of the canonical generators and determine whether the Hecke algebra has a universal C*-completion. The paper unifies past algebraic and analytic approaches by focusing on the common geometric thread.The results have implications for the general theory of totally disconnected locally compact groups."}
{"category": "Math", "title": "Data-dependent probability matching priors for empirical and related likelihoods", "abstract": "We consider a general class of empirical-type likelihoods and develop higher order asymptotics with a view to characterizing members thereof that allow the existence of possibly data-dependent probability matching priors ensuring approximate frequentist validity of posterior quantiles. In particular, for the usual empirical likelihood, positive results are obtained. This is in contrast with what happens if only data-free priors are entertained."}
{"category": "Math", "title": "Probability matching priors for some parameters of the bivariate normal distribution", "abstract": "This paper develops some objective priors for certain parameters of the bivariate normal distribution. The parameters considered are the regression coefficient, the generalized variance, and the ratio of the conditional variance of one variable given the other to the marginal variance of the other variable. The criterion used is the asymptotic matching of coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. The paper uses various matching criteria, namely, quantile matching, matching of distribution functions, highest posterior density matching, and matching via inversion of test statistics. One particular prior is found which meets all the matching criteria individually for all the parameters of interest."}
{"category": "Math", "title": "Fuzzy set representation of a prior distribution", "abstract": "In the subjective Bayesian approach uncertainty is described by a prior distribution chosen by the statistician. Fuzzy set theory is another way of representing uncertainty. Here we give a decision theoretic approach which allows a Bayesian to convert their prior distribution into a fuzzy set membership function. This yields a formal relationship between these two different methods of expressing uncertainty."}
{"category": "Math", "title": "Fuzzy sets in nonparametric Bayes regression", "abstract": "A simple Bayesian approach to nonparametric regression is described using fuzzy sets and membership functions. Membership functions are interpreted as likelihood functions for the unknown regression function, so that with the help of a reference prior they can be transformed to prior density functions. The unknown regression function is decomposed into wavelets and a hierarchical Bayesian approach is employed for making inferences on the resulting wavelet coefficients."}
{"category": "Math", "title": "Radical Parametrization of Algebraic Curves by Adjoint Curves", "abstract": "We present algorithms for parametrizing by radicals an irreducible curve, not necessarily plane, when the genus is less o equal to 4 and they are defined over an algebraically closed field of characteristic zero. In addition, we also present an algorithm for parametrizing by radicals any irreducible plane curve of degree $d$ having at least a point of multiplicity $d-r$, with $1\\leq r \\leq 4$ and, as a consequence, every irreducible plane curve of degree $d \\leq 5$ and every irreducible singular plane curve of degree 6."}
{"category": "Math", "title": "Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations", "abstract": "S. Montgomery-Smith provided a one dimensional model for the three dimensional, incompressible Navier-Stokes equations, for which he proved the blow up of solutions associated to a class of large initial data, while the same global existence results as for the Navier-Stokes equations hold for small data. In this note the model is adapted to the case of two and three space dimensions, with the additional feature that the divergence free condition is preserved. It is checked that the family of initial data constructed previously by J.-Y Chemin and I. Gallagher which is arbitrarily large but yet generates a global solution to the Navier-Stokes equations in three space dimensions, actually causes blow up for the toy model -- meaning that the precise structure of the nonlinear term is crucial to understand the dynamics of large solutions to the Navier-Stokes equations."}
{"category": "Math", "title": "The initial value problem for a third-order dispersive flow into compact almost Hermitian manifolds", "abstract": "We present a time-local existence theorem of the initial value problem for a third-order dispersive evolution equation for open curves on compact almost Hermitian manifolds arising in the geometric analysis of vortex filaments. This equation causes the so-called loss of one-derivative since the target manifold is not supposed to be a K\\\"ahler manifold. We overcome this difficulty by using a gauge transformation of a multiplier on the pull-back bundle to eliminate the bad first order terms essentially."}
{"category": "Math", "title": "Objective Bayes testing of Poisson versus inflated Poisson models", "abstract": "The Poisson distribution is often used as a standard model for count data. Quite often, however, such data sets are not well fit by a Poisson model because they have more zeros than are compatible with this model. For these situations, a zero-inflated Poisson (ZIP) distribution is often proposed. This article addresses testing a Poisson versus a ZIP model, using Bayesian methodology based on suitable objective priors. Specific choices of objective priors are justified and their properties investigated. The methodology is extended to include covariates in regression models. Several applications are given."}
{"category": "Math", "title": "Consistent selection via the Lasso for high dimensional approximating regression models", "abstract": "In this article we investigate consistency of selection in regression models via the popular Lasso method. Here we depart from the traditional linear regression assumption and consider approximations of the regression function $f$ with elements of a given dictionary of $M$ functions. The target for consistency is the index set of those functions from this dictionary that realize the most parsimonious approximation to $f$ among all linear combinations belonging to an $L_2$ ball centered at $f$ and of radius $r_{n,M}^2$. In this framework we show that a consistent estimate of this index set can be derived via $\\ell_1$ penalized least squares, with a data dependent penalty and with tuning sequence $r_{n,M}>\\sqrt{\\log(Mn)/n}$, where $n$ is the sample size. Our results hold for any $1\\leq M\\leq n^{\\gamma}$, for any $\\gamma>0$."}
{"category": "Math", "title": "Relative log convergent cohomology and relative rigid cohomology III", "abstract": "In this paper, we prove the generic overconvergence of relative rigid cohomology with coefficient, by using the semistable reduction conjecture for overconvergent $F$-isocrystals (which is recently shown by Kedlaya)."}
{"category": "Math", "title": "Asymptotic optimality of a cross-validatory predictive approach to linear model selection", "abstract": "In this article we study the asymptotic predictive optimality of a model selection criterion based on the cross-validatory predictive density, already available in the literature. For a dependent variable and associated explanatory variables, we consider a class of linear models as approximations to the true regression function. One selects a model among these using the criterion under study and predicts a future replicate of the dependent variable by an optimal predictor under the chosen model. We show that for squared error prediction loss, this scheme of prediction performs asymptotically as well as an oracle, where the oracle here refers to a model selection rule which minimizes this loss if the true regression were known."}
{"category": "Math", "title": "Risk and resampling under model uncertainty", "abstract": "In statistical exercises where there are several candidate models, the traditional approach is to select one model using some data driven criterion and use that model for estimation, testing and other purposes, ignoring the variability of the model selection process. We discuss some problems associated with this approach. An alternative scheme is to use a model-averaged estimator, that is, a weighted average of estimators obtained under different models, as an estimator of a parameter. We show that the risk associated with a Bayesian model-averaged estimator is bounded as a function of the sample size, when parameter values are fixed. We establish conditions which ensure that a model-averaged estimator's distribution can be consistently approximated using the bootstrap. A new, data-adaptive, model averaging scheme is proposed that balances efficiency of estimation without compromising applicability of the bootstrap. This paper illustrates that certain desirable risk and resampling properties of model-averaged estimators are obtainable when parameters are fixed but unknown; this complements several studies on minimaxity and other properties of post-model-selected and model-averaged estimators, where parameters are allowed to vary."}
{"category": "Math", "title": "On a Monge-Amp\\`{e}re type equation in the Cegrell class $\\mathcal{E}_{\\chi}$", "abstract": "In this paper we prove an existence and uniqueness result for a Monge-Amp\\`{e}re type equation in the Cegrell class $\\mathcal{E}_{\\chi}$."}
{"category": "Math", "title": "Remarks on consistency of posterior distributions", "abstract": "In recent years, the literature in the area of Bayesian asymptotics has been rapidly growing. It is increasingly important to understand the concept of posterior consistency and validate specific Bayesian methods, in terms of consistency of posterior distributions. In this paper, we build up some conceptual issues in consistency of posterior distributions, and discuss panoramic views of them by comparing various approaches to posterior consistency that have been investigated in the literature. In addition, we provide interesting results on posterior consistency that deal with non-exponential consistency, improper priors and non i.i.d. (independent but not identically distributed) observations. We describe a few examples for illustrative purposes."}
{"category": "Math", "title": "Voting power and Qualified Majority Voting with a \"no vote\" option", "abstract": "In recent years, enlargement of the European Union has led to increased interest in the allocation of voting weights to member states with hugely differing population numbers. While the eventually agreed voting scheme lacks any strict mathematical basis, the Polish government suggested a voting scheme based on the Penrose definition of voting power, leading to an allocation of voting weights proportional to the square root of the population (the \"Jagiellonian Compromise\"). The Penrose definition of voting power is derived from the citizens' freedom to vote either \"yes\" or \"no\". This paper defines a corresponding voting power based on \"yes\", \"no\" and \"abstain\" options, and it is found that this definition also leads to a square root law, and to the same optimal vote allocation as the Penrose scheme."}
{"category": "Math", "title": "Reproducing kernel Hilbert spaces of Gaussian priors", "abstract": "We review definitions and properties of reproducing kernel Hilbert spaces attached to Gaussian variables and processes, with a view to applications in nonparametric Bayesian statistics using Gaussian priors. The rate of contraction of posterior distributions based on Gaussian priors can be described through a concentration function that is expressed in the reproducing Hilbert space. Absolute continuity of Gaussian measures and concentration inequalities play an important role in understanding and deriving this result. Series expansions of Gaussian variables and transformations of their reproducing kernel Hilbert spaces under linear maps are useful tools to compute the concentration function."}
{"category": "Math", "title": "Self-dual projective toric varieties", "abstract": "Let T be a torus over an algebraically closed field k of characteristic 0, and consider a projective T-module P(V). We determine when a projective toric subvariety X of P(V) is self-dual, in terms of the configuration of weights of V."}
{"category": "Math", "title": "A Bayesian semi-parametric model for small area estimation", "abstract": "In public health management there is a need to produce subnational estimates of health outcomes. Often, however, funds are not available to collect samples large enough to produce traditional survey sample estimates for each subnational area. Although parametric hierarchical methods have been successfully used to derive estimates from small samples, there is a concern that the geographic diversity of the U.S. population may be oversimplified in these models. In this paper, a semi-parametric model is used to describe the geographic variability component of the model. Specifically, we assume Dirichlet process mixtures of normals for county-specific random effects. Results are compared to a parametric model based on the base measure of the Dirichlet process, using binary health outcomes related to mammogram usage."}
{"category": "Math", "title": "A Logical Calculus To Intuitively And Logically Denote Number Systems", "abstract": "Simple continued fractions, base-b expansions, Dedekind cuts and Cauchy sequences are common notations for number systems. In this note, first, it is proven that both simple continued fractions and base-b expansions fail to denote real numbers and thus lack logic; second, it is shown that Dedekind cuts and Cauchy sequences fail to join in algebraical operations and thus lack intuition; third, we construct a logical calculus and deduce numbers to intuitively and logically denote number systems."}
{"category": "Math", "title": "On Perelman's Dilaton", "abstract": "By means of a Kaluza-Klein type argument we show that the Perelman's F-functional is the Einstein-Hilbert action in a space with extra ``phantom'' dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his first famous paper. As a consequence the Ricci flow (modified by a diffeomorphism and a time-dependent factor) is the evolution of the ``real'' part of the metric under a constrained gradient flow of the Einstein-Hilbert gravitational action in higher dimension."}
{"category": "Math", "title": "A hierarchical Bayesian approach for estimating the origin of a mixed population", "abstract": "We propose a hierarchical Bayesian model to estimate the proportional contribution of source populations to a newly founded colony. Samples are derived from the first generation offspring in the colony, but mating may occur preferentially among migrants from the same source population. Genotypes of the newly founded colony and source populations are used to estimate the mixture proportions, and the mixture proportions are related to environmental and demographic factors that might affect the colonizing process. We estimate an assortative mating coefficient, mixture proportions, and regression relationships between environmental factors and the mixture proportions in a single hierarchical model. The first-stage likelihood for genotypes in the newly founded colony is a mixture multinomial distribution reflecting the colonizing process. The environmental and demographic data are incorporated into the model through a hierarchical prior structure. A simulation study is conducted to investigate the performance of the model by using different levels of population divergence and number of genetic markers included in the analysis. We use Markov chain Monte Carlo (MCMC) simulation to conduct inference for the posterior distributions of model parameters. We apply the model to a data set derived from grey seals in the Orkney Islands, Scotland. We compare our model with a similar model previously used to analyze these data. The results from both the simulation and application to real data indicate that our model provides better estimates for the covariate effects."}
{"category": "Math", "title": "Finite element scheme for integro-partial differential equations", "abstract": "We construct a finite element like scheme for fully non-linear integro-partial differential equations arising in optimal control of jump-processes. Special cases of these equations include optimal portfolio and option pricing equations in Finance. The schemes are monotone and robust. We prove that they converge in very general situations, including degenerate equations, multiple dimensions, relatively low regularity of the data, and for most (if not all) types of jump-models used in Finance. In all cases we provide (probably optimal) error bounds. These bounds apply when grids are unstructured and integral terms are very singular, two features that are new or highly unusual in this setting."}
{"category": "Math", "title": "Kendall's tau in high-dimensional genomic parsimony", "abstract": "High-dimensional data models, often with low sample size, abound in many interdisciplinary studies, genomics and large biological systems being most noteworthy. The conventional assumption of multinormality or linearity of regression may not be plausible for such models which are likely to be statistically complex due to a large number of parameters as well as various underlying restraints. As such, parametric approaches may not be very effective. Anything beyond parametrics, albeit, having increased scope and robustness perspectives, may generally be baffled by the low sample size and hence unable to give reasonable margins of errors. Kendall's tau statistic is exploited in this context with emphasis on dimensional rather than sample size asymptotics. The Chen--Stein theorem has been thoroughly appraised in this study. Applications of these findings in some microarray data models are illustrated."}
{"category": "Math", "title": "A Note on Partial List Colorings", "abstract": "Let $G$ be a simple graph with $n$ vertices and list chromatic number $\\chi_\\ell(G)=\\chi_\\ell$. Suppose that $0\\leq t\\leq \\chi_\\ell$ and each vertex of $G$ is assigned a list of $t$ colors. Albertson, Grossman and Haas [1] conjectured that at least $\\frac{tn}{\\chi_\\ell}$ vertices of $G$ can be colored from these lists. In this paper we find some new results in partial list coloring which help us to show that the conjecture is true for at least half of the numbers of the set $\\{1,2,...,\\chi_\\ell(G)-1\\}$. In addition we introduce a new related conjecture and finally we present some results about this conjecture."}
{"category": "Math", "title": "Orthogonalized smoothing for rescaled spike and slab models", "abstract": "Rescaled spike and slab models are a new Bayesian variable selection method for linear regression models. In high dimensional orthogonal settings such models have been shown to possess optimal model selection properties. We review background theory and discuss applications of rescaled spike and slab models to prediction problems involving orthogonal polynomials. We first consider global smoothing and discuss potential weaknesses. Some of these deficiencies are remedied by using local regression. The local regression approach relies on an intimate connection between local weighted regression and weighted generalized ridge regression. An important implication is that one can trace the effective degrees of freedom of a curve as a way to visualize and classify curvature. Several motivating examples are presented."}
{"category": "Math", "title": "Nonparametric statistics on manifolds with applications to shape spaces", "abstract": "This article presents certain recent methodologies and some new results for the statistical analysis of probability distributions on manifolds. An important example considered in some detail here is the 2-D shape space of k-ads, comprising all configurations of $k$ planar landmarks ($k>2$)-modulo translation, scaling and rotation."}
{"category": "Math", "title": "An ensemble approach to improved prediction from multitype data", "abstract": "We have developed a strategy for the analysis of newly available binary data to improve outcome predictions based on existing data (binary or non-binary). Our strategy involves two modeling approaches for the newly available data, one combining binary covariate selection via LASSO with logistic regression and one based on logic trees. The results of these models are then compared to the results of a model based on existing data with the objective of combining model results to achieve the most accurate predictions. The combination of model predictions is aided by the use of support vector machines to identify subspaces of the covariate space in which specific models lead to successful predictions. We demonstrate our approach in the analysis of single nucleotide polymorphism (SNP) data and traditional clinical risk factors for the prediction of coronary heart disease."}
{"category": "Math", "title": "Sharp failure rates for the bootstrap particle filter in high dimensions", "abstract": "We prove that the maximum of the sample importance weights in a high-dimensional Gaussian particle filter converges to unity unless the ensemble size grows exponentially in the system dimension. Our work is motivated by and parallels the derivations of Bengtsson, Bickel and Li (2007); however, we weaken their assumptions on the eigenvalues of the covariance matrix of the prior distribution and establish rigorously their strong conjecture on when weight collapse occurs. Specifically, we remove the assumption that the nonzero eigenvalues are bounded away from zero, which, although the dimension of the involved vectors grow to infinity, essentially permits the effective system dimension to be bounded. Moreover, with some restrictions on the rate of growth of the maximum eigenvalue, we relax their assumption that the eigenvalues are bounded from above, allowing the system to be dominated by a single mode."}
{"category": "Math", "title": "Handle moves in contact surgery diagrams", "abstract": "We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the respective classifications of long and loose Legendrian knots."}
{"category": "Math", "title": "Hochschild and ordinary cohomology rings of small categories", "abstract": "Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C|=BC. We study the ring homomorphism HH*(kC) --> H*(|C|,k) and prove it is split surjective. This generalizes the well-known results for groups and posets. Based on this result, we construct a seven-dimensional category algebra whose Hochschild cohomology ring modulo nilpotents is not finitely generated, against a conjecture of Snashall and Solberg."}
{"category": "Math", "title": "Proving the existence of the $n$th root by induction", "abstract": "In this paper we prove by induction on $n$ that any positive real number has $n$th root."}
{"category": "Math", "title": "On a certain generalization of the Balog-Szemeredi-Gowers Theorem", "abstract": "In this note, we prove a certain hypergraph generalization of the Balog-Szemeredi-Gowers Theorem. Our result shares some features in common with a similar such generalizsation due to Sudakov, Szemeredi and Vu, though the conclusion of our result is somewhat different."}
{"category": "Math", "title": "An Introduction to Smooth Infinitesimal Analysis", "abstract": "An exposition of smooth infinitesimal analysis, which is a way to do calculus with nilsquare infinitesimals, is given."}
{"category": "Math", "title": "Interior Cauchy-Schauder estimates for the heat flow in Carnot-Caratheodory spaces", "abstract": "The purpose of this paper is to establish some basic interior estimates of Cauchy-Schauder type for the heat flow associated with a system of smooth vector fields satisfying Hormander's finite rank condition"}
{"category": "Math", "title": "Characterizing indecomposable plane continua from their complements", "abstract": "We show that a plane continuum X is indecomposable iff X has a sequence (U_n) of not necessarily distinct complementary domains satisfying what we call the double-pass condition: If one draws an open arc A_n in each U_n whose ends limit into the boundary of U_n, one can choose components of U_n minus A_n whose boundaries intersected with the continuum (which we call shadows) converge to the continuum."}
{"category": "Math", "title": "Any counterexample to Makienko's conjecture is an indecomposable continuum", "abstract": "Makienko's conjecture, a proposed addition to Sullivan's dictionary, can be stated as follows: The Julia set of a rational function R has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko's conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational functions whose Julia set is an indecomposable continuum."}
{"category": "Math", "title": "Existence and stability of noncharacteristic boundary-layers for the compressible Navier-Stokes and viscous MHD equations", "abstract": "For a general class of hyperbolic-parabolic systems including the compressible Navier-Stokes and compressible MHD equations, we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical Navier-Stokes boundary conditions. Our first main result, using the abstract framework established by the authors in the companion work \\cite{GMWZ6}, is to show that existence and stability of arbitrary amplitude exact boundary-layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an Evans function (uniform Evans stability). Whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with \"residual\" boundary conditions. Our second is to show that uniform Evans stability for small-amplitude layers is equivalent to Evans stability of the limiting constant layer, which in turn can be checked by a linear-algebraic computation. Finally, for a class of symmetric-dissipative systems including the physical examples mentioned above, we carry out energy estimates showing that constant (and thus small-amplitude) layers always satisfy uniform Evans stability. This yields existence of small-amplitude multi-dimensional boundary layers for the compressible Navier-Stokes and MHD equations. For both equations these appear to be the first such results in the compressible case."}
{"category": "Math", "title": "Natural Equivariant Dirac Operators", "abstract": "We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of the equivariant index. We also show that the components of the representation-valued equivariant index coincide with those of an elliptic operator constructed from the original data."}
{"category": "Math", "title": "Slices of motivic Landweber spectra", "abstract": "We show that the Conjecture of Voevodsky concerning slices of the algebraic cobordism spectrum MGL implies a general statement about the slices of motivic Landweber spectra. In particular it confirms the possible approach suggested by Voevodsky for the computation of the slices of the homotopy algebraic K-theory spectrum KGL via a Conner-Floyd isomorphism complementing Levine's unconditional proof of these slices over perfect fields. A similar result, and Voevodsky's conjecture over fields of char. 0, are also announced by Hopkins-Morel."}
{"category": "Math", "title": "Solubility of Fermat equations", "abstract": "The arithmetic of ternary diagonal equation is considered for degree d >1, with the outcome that the set of coefficients for which the equation admits a non-zero integer solution is shown to have density zero."}
{"category": "Math", "title": "On what Ontology Is and not-Is", "abstract": "In this paper I study the connection between logic and metaphysics in Plato's participation theory, from the structural properties of the latter. Although Plato was the first ever to formulate the contradiction principle explicitly (in the Phaedo), the logic underlying his system appears to be paraconsistent. This confirms an earlier suggestion by G. Priest. Its technical characteristics and the textual evidence supporting this interpretation are both studied in detail."}
{"category": "Math", "title": "An $L_\\infty$ algebra structure on polyvector fields", "abstract": "It is well-known that the Kontsevich formality [K97] for Hochschild cochains of the polynomial algebra $A=S(V^*)$ fails if the vector space $V$ is infinite-dimensional. In the present paper, we study the corresponding obstructions. We construct an $L_\\infty$ structure on polyvector fields on $V$ having the even degree Taylor components, with the degree 2 component given by the Schouten-Nijenhuis bracket, but having as well higher non-vanishing Taylor components. We prove that this $L_\\infty$ algebra is quasi-isomorphic to the corresponding Hochschild cochain complex. We prove that our $L_\\infty$ algebra is $L_\\infty$ quasi-isomorphic to the Lie algebra of polyvector fields on $V$ with the Schouten-Nijenhuis bracket, if $V$ is finite-dimensional."}
{"category": "Math", "title": "On minimal extensions of rings", "abstract": "Given two rings $R \\subseteq S$, $S$ is said to be a minimal ring extension of $R$ if $R$ is a maximal subring of $S$. In this article, we study minimal extensions of an arbitrary ring $R$, with particular focus on those possessing nonzero ideals that intersect $R$ trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs & Shapiro, and Ferrand & Olivier on commutative minimal extensions."}
{"category": "Math", "title": "Proper holomorphic mapppings between Reinhardt domains in $\\mathbb C^2$", "abstract": "We describe all possibilities of existence of non-elementary proper holomorphic maps between non-hyperbolic Reinhardt domains in $\\mathbb C^2$ and the corresponding pairs of domains."}
{"category": "Math", "title": "Bernstein measures on convex polytopes", "abstract": "We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functions which is a generalization of the asymptotic expansion of the Bernstein polynomials on the standard $m$-simplex obtained by Abel-Ivan and H\\\"{o}rmander. These are different from the Bergman-Bernstein approximations over Delzant polytopes recently introduced by Zelditch. We discuss relations between Bernstein approximations defined in this paper and Zelditch's Bergman-Bernstein approximations."}
{"category": "Math", "title": "Cross Curvature Flow on Locally Homogeneous Three-manifolds (II)", "abstract": "In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cross curvature flow on locally homogeneous 3-manifolds. We show that, typically, the positive cross curvature flow on locally homogeneous 3-manifold produce an Heisenberg type sub-Riemannian geometry."}
{"category": "Math", "title": "Differential Harnack Estimates for Backward Heat Equations with Potentials under the Ricci Flow", "abstract": "In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow."}
{"category": "Math", "title": "A First Sight Towards Primitively Generated Connected Braided Bialgebras", "abstract": "The main aim of this paper is to investigate the structure of primitively generated connected braided bialgebras $A$ with respect to the braided vector space $P$ consisting of their primitive elements. When the Nichols algebra of $P$ is obtained dividing out the tensor algebra $T(P) $ by the two-sided ideal generated by its primitive elements of degree at least two, we show that $A$ can be recovered as a sort of universal enveloping algebra of $P$. One of the main applications of our construction is the description, in terms of universal enveloping algebras, of connected braided bialgebras whose associated graded coalgebra is a quadratic algebra."}
{"category": "Math", "title": "Estimation in models driven by fractional Brownian motion", "abstract": "Let $\\{b_H(t),t\\in\\mathbb{R}\\}$ be the fractional Brownian motion with parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of the type \\[X(t)=c+\\int_0^t\\sigma\\bigl(X(u)\\bigr)\\mathrm {d}b_H(u)+\\int _0^t\\mu\\bigl(X(u)\\bigr)\\mathrm {d}u.\\] In different particular models where $\\sigma(x)=\\sigma$ or $\\sigma(x)=\\sigma x$ and $\\mu(x)=\\mu$ or $\\mu(x)=\\mu x$, we propose a central limit theorem for estimators of $H$ and of $\\sigma$ based on regression methods. Then we give tests of the hypothesis on $\\sigma$ for these models. We also consider functional estimation on $\\sigma(\\cdot)$ in the above more general models based in the asymptotic behavior of functionals of the 2nd-order increments of the fBm."}
{"category": "Math", "title": "On Asymptotic Stability of Solitary Waves in Discrete Schr\\\"odinger Equation Coupled to Nonlinear Oscillator", "abstract": "The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schr\\\"odinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group. For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schr\\\"odinger equation. The proofs use the strategy of Buslaev-Perelman: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc."}
{"category": "Math", "title": "Cohomology of Courant algebroids with split base", "abstract": "We study the (standard) cohomology $H^\\bullet_{st}(E)$ of a Courant algebroid $E$. We prove that if $E$ is transitive, the standard cohomology coincides with the naive cohomology $H_{naive}^\\bullet(E)$ as conjectured by Stienon and Xu. For a general Courant algebroid we define a spectral sequence converging to its standard cohomology. If $E$ is with split base, we prove that there exists a natural transgression homomorphism $T_3$ (with image in $H^3_{naive}(E)$) which, together with the naive cohomology, gives all $H^\\bullet_{st}(E)$. For generalized exact Courant algebroids, we give an explicit formula for $T_3$ depending only on the \\v{S}evera characteristic clas of $E$."}
{"category": "Math", "title": "The least singular value of a random square matrix is O(n^{-1/2})", "abstract": "Let A be a matrix whose entries are real i.i.d. centered random variables with unit variance and suitable moment assumptions. Then the smallest singular value of A is of order n^{-1/2} with high probability. The lower estimate of this type was proved recently by the authors; in this note we establish the matching upper estimate."}
{"category": "Math", "title": "A note on James spaces and superstrictly singular operators", "abstract": "An elementary lemma is used in order to show that the natural inclusion $J_p\\to J_q$ of James spaces is superstrictly singular for $p<q$. As a consequence, it is shown that an operator without nontrivial invariant subspaces constructed by Charles Read is superstrictly singular."}
{"category": "Math", "title": "On Upper-Confidence Bound Policies for Non-Stationary Bandit Problems", "abstract": "Multi-armed bandit problems are considered as a paradigm of the trade-off between exploring the environment to find profitable actions and exploiting what is already known. In the stationary case, the distributions of the rewards do not change in time, Upper-Confidence Bound (UCB) policies have been shown to be rate optimal. A challenging variant of the MABP is the non-stationary bandit problem where the gambler must decide which arm to play while facing the possibility of a changing environment. In this paper, we consider the situation where the distributions of rewards remain constant over epochs and change at unknown time instants. We analyze two algorithms: the discounted UCB and the sliding-window UCB. We establish for these two algorithms an upper-bound for the expected regret by upper-bounding the expectation of the number of times a suboptimal arm is played. For that purpose, we derive a Hoeffding type inequality for self normalized deviations with a random number of summands. We establish a lower-bound for the regret in presence of abrupt changes in the arms reward distributions. We show that the discounted UCB and the sliding-window UCB both match the lower-bound up to a logarithmic factor."}
{"category": "Math", "title": "Vitesse de convergence dans le th\\'{e}or\\`{e}me limite central pour des cha\\^{i}nes de Markov fortement ergodiques", "abstract": "Let $Q$ be a transition probability on a measurable space $E$ which admits an invariant probability measure, let $(X_n)_n$ be a Markov chain associated to $Q$, and let $\\xi$ be a real-valued measurable function on $E$, and $S_n=\\sum _{k=1}^n\\xi(X_k)$. Under functional hypotheses on the action of $Q$ and the Fourier kernels $Q(t)$, we investigate the rate of convergence in the central limit theorem for the sequence $(\\frac{S_n}{\\sqrt{n}})_n$. According to the hypotheses, we prove that the rate is, either $\\mathrm{O}(n^{-{\\tau}/{2}})$ for all $\\tau<1$, or $\\mathrm{O}(n^{-{1}/{2}})$. We apply the spectral Nagaev's method which is improved by using a perturbation theorem of Keller and Liverani, and a majoration of $|\\mathbb{E}[\\mathrm{e}^{\\mat hrm{i}t{S_n}/{\\sqrt{n}}}]-\\mathrm{e}^{{-t^2}/{2}}|$ obtained by a method of martingale difference reduction. When $E$ is not compact or $\\xi$ is not bounded, the conditions required here on $Q(t)$ (in substance, some moment conditions on $\\xi$) are weaker than the ones usually imposed when the standard perturbation theorem is used in the spectral method. For example, in the case of $V$-geometric ergodic chains or Lipschitz iterative models, the rate of convergence in the c.l.t. is $\\mathrm{O}(n^{-{1}/{2}})$ under a third moment condition on $\\xi$."}
{"category": "Math", "title": "Fesenko reciprocity map", "abstract": "In recent papers, Fesenko has defined the non-abelian local reciprocity map for every totally-ramified arithmetically profinite ($APF$) Galois extension of a given local field $K$ by extending the works of Hazewinkel and Neukirch-Iwasawa. The theory of Fesenko extends the previous non-abelian generalizations of local class field theory given by Koch-de Shalit and by A. Gurevich. In this paper, which is research-expository in nature, we give a detailed account of Fesenko's work including all the skipped proofs."}
{"category": "Math", "title": "Some results on the second Gaussian map for curves", "abstract": "We study the second Gaussian map for a curve X of genus g, in relation with the second fundamental form of the period map. We exhibit a class of infinitely many curves with surjective second Gaussian map. We compute its rank on the hyperelliptic and trigonal locus and show global generation of its image for X not hyperelliptic nor trigonal."}
{"category": "Math", "title": "Siegel metric and curvature of the moduli space of curves", "abstract": "We study the curvature of the moduli space M_g of curves of genus g with the Siegel metric induced by the period map. We give an explicit formula for the holomorphic sectional curvature of M_g along a Schiffer variation at a point P on the curve X, in terms of the holomorphic sectional curvature of A_g and the second Gaussian map. Finally we extend the Kaehler form of the Siegel metric as a closed current on the Deligne-Mumford compatification of M_g and we determine its cohomology class as a multiple of the first Chern class of the Hodge bundle."}
{"category": "Math", "title": "Generalized Fesenko reciprocity map", "abstract": "In this paper, which is the natural continuation and generalization of Fesenko's non-abelian reciprocity map, we extend the theory of Fesenko to infinite $APF$-Galois extensions $L$ over a local field $K$, with finite residue-class field $\\kappa_K$ of $q=p^f$ elements, satisfying $\\pmb{\\mu}_p(K^{sep})\\subset K$ and $K\\subset L\\subset K_{\\phi^d}$ where the residue-class degree $[\\kappa_L:\\kappa_K]=d$. More precisely, for such extensions $L/K$, fixing a Lubin-Tate splitting $\\phi$ over $K$, we construct a 1-cocycle, \\pmb{\\Phi}_{L/K}^{(\\phi)}:\\text{Gal}(L/K)\\to K^\\times/N_{L_0/K}L_0^\\times\\times U_{\\widetilde{\\mathbb X}(L/K)}^\\diamond /Y_{L/L_0}, where $L_0=L\\cap K^{nr}$, and study its functorial and ramification-theoretic properties. The case $d=1$ recovers the theory of Fesenko."}
{"category": "Math", "title": "L-R-smash biproducts, double biproducts and a braided category of Yetter-Drinfeld-Long bimodules", "abstract": "Let H be a bialgebra and D an H-bimodule algebra H-bicomodule coalgebra. We find sufficient conditions on D for the L-R-smash product algebra and coalgebra structures on D\\otimes H to form a bialgebra (in this case we say that (H, D) is an L-R-admissible pair), called L-R-smash biproduct. The Radford biproduct is a particular case, and so is, up to isomorphism, a double biproduct with trivial pairing. We construct a prebraided monoidal category {\\cal LR}(H), whose objects are H-bimodules H-bicomodules M endowed with left-left and right-right Yetter-Drinfeld module as well as left-right and right-left Long module structures over H, with the property that, if (H, D) is an L-R-admissible pair, then D is a bialgebra in {\\cal LR}(H)."}
{"category": "Math", "title": "A Solution to the Monotonicity Problem for Unimodal Families", "abstract": "In this note we consider a collection C of one parameter families of unimodal maps of [0,1]. Each family in the collection has the form uf where u is in [0,1]. Denoting the kneading sequence of uf by K(uf), we will prove that for each member of C, the map u->K(uf) is monotone. It then follows that for each member of C the map u -> h(uf) is monotone, where h(uf) is the topological entropy of uf. For interest, uf(x)=4ux(1-x) and uf(x)=usin(pi x) are shown to belong to C."}
{"category": "Math", "title": "Quasi-elementary H-Azumaya algebras arising from generalized (anti) Yetter-Drinfeld modules", "abstract": "Let H be a Hopf algebra with bijective antipode, let \\alpha, \\beta be two Hopf algebra automorphisms of H and M a finite dimensional (\\alpha, \\beta )-Yetter-Drinfeld module. We prove that End(M) endowed with certain structures becomes an H-Azumaya algebra, and the set of H-Azumaya algebras of this type is a subgroup of BQ(k, H), the Brauer group of H."}
{"category": "Math", "title": "On the Milnor fibers of cyclic quotient singularities", "abstract": "The oriented link of the cyclic quotient singularity $\\mathcal{X}_{p,q}$ is orientation-preserving diffeomorphic to the lens space $L(p,q)$ and carries the standard contact structure $\\xi_{st}$. Lisca classified the Stein fillings of $(L(p,q), \\xi_{st})$ up to diffeomorphisms and conjectured that they correspond bijectively through an {\\it explicit} map to the Milnor fibers associated with the irreducible components (all of them being smoothing components) of the reduced miniversal space of deformations of $\\mathcal{X}_{p,q}$. We prove this conjecture using the smoothing equations given by Christophersen and Stevens. Moreover, based on a different description of the Milnor fibers given by de Jong and van Straten, we also canonically identify these fibers with Lisca's fillings. Using these and a newly introduced additional structure - the order - associated with lens spaces, we prove that the above Milnor fibers are pairwise non-diffeomorphic (by diffeomorphisms which preserve the orientation and order). This also implies that de Jong and van Straten parametrize in the same way the components of the reduced miniversal space of deformations as Christophersen and Stevens."}
{"category": "Math", "title": "Comparison between criteria leading to the weak invariance principle", "abstract": "The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR 188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields 110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincar\\'{e} Probab. Statist. 36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab. 28 (2000) later improved upon by Peligrad and Utev in Ann. Probab. 33 (2005). We prove that in every ergodic dynamical system with positive entropy, if we consider two of these criteria, we can find a function in $\\mathbb{L}^2$ satisfying the first but not the second."}
{"category": "Math", "title": "Synchronization in Networks of Identical Linear Systems", "abstract": "The paper investigates the synchronization of a network of identical linear state-space models under a possibly time-varying and directed interconnection structure. The main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected. The result can be interpreted as a generalization of classical consensus algorithms. Stronger conditions are shown to be sufficient but to some extent, also necessary to ensure synchronization with the diffusive static output coupling often considered in the literature."}
{"category": "Math", "title": "Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0", "abstract": "Let $K$ be a one-variable function field over a field of constants of characteristic 0. Let $R$ be a holomorphy subring of $K$, not equal to $K$. We prove the following undecidability results for $R$: If $K$ is recursive, then Hilbert's Tenth Problem is undecidable in $R$. In general, there exist $x_1,...,x_n \\in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\\Q(x_1,...,x_n)$ has solutions in $R$."}
{"category": "Math", "title": "Extended affine Lie algebras and other generalizations of affine Lie algebras - a survey", "abstract": "This is a survey on extended affine Lie algebras and related types of Lie algebras, which generalize affine Lie algebras."}
{"category": "Math", "title": "Leading coefficients of the Kazhdan-Lusztig polynomials for an Affine Weyl group of type $\\widetilde{B_2}$", "abstract": "In this paper we compute the leading coefficients $\\mu (u,w)$ of the Kazhdan--Lusztig polynomials $P_{u,w}$ for an affine Weyl group of type $\\tilde{B}_2$. By using the \\textbf{a}-function of a Coxeter group defined by Lusztig (see [L1, \\S2]), we compute most $\\mu (u,w)$ explicitly. With part of these values $\\mu (u,w)$, we show that a conjecture of Lusztig on distinguished involutions is true for an affine Weyl group of type $\\tilde{B}_2$. We also show that the conjectural formula in [L3, (12)] needs a modification."}
{"category": "Math", "title": "Singular value decomposition of large random matrices (for two-way classification of microarrays)", "abstract": "Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to Wigner-noise is investigated.It is proved that such an m\\times n matrix almost surely has a constant number of large singular values (of order \\sqrt{mn}), while the rest of the singular values are of order \\sqrt{m+n} as m,n\\to\\infty. Concentration results of Alon et al. for the eigenvalues of large symmetric random matrices are adapted to the rectangular case, and on this basis, almost sure results for the singular values as well as for the corresponding isotropic subspaces are proved. An algorithm, applicable to two-way classification of microarrays, is also given that finds the underlying block structure."}
{"category": "Math", "title": "Locally Toroidal Polytopes and Modular Linear Groups", "abstract": "When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d>1, one obtains a finite group G^d which is often the automorphism group of an abstract regular polytope. Building on earlier work in the case that d is an odd prime, we here develop methods to handle composite moduli and completely describe the corresponding modular polytopes when G is of spherical or Euclidean type. Using a modular variant of the quotient criterion, we then describe the locally toroidal polytopes provided by our construction, most of which are new."}
{"category": "Math", "title": "Counting numerical sets with no small atoms", "abstract": "A numerical set $S$ with Frobenius number $g$ is a set of integers with $\\min(S) = 0$ and $\\max(\\Zbb - S)=g$, and its atom monoid is $A(S) = \\setpres{n \\in \\Zbb}{$n+s \\in S$ for all $s \\in S$}$. Let $\\gamma_g$ be the number of numerical sets $S$ having $A(S) = \\set{0} \\cup (g,\\infty)$ divided by the total number of numerical sets with Frobenius number $g$. We show that the sequence $\\set{\\gamma_g}$ is decreasing and converges to a number $\\gamma_\\infty \\approx .4844$ (with accuracy to within $.0050$). We also examine the singularities of the generating function for $\\set{\\gamma_g}$. Parallel results are obtained for the ratio $\\gsymm{g}$ of the number of symmetric numerical sets $S$ with $A(S) = \\set{0} \\cup (g,\\infty)$ by the number of symmetric numerical sets with Frobenius number $g$. These results yield information regarding the asymptotic behavior of the number of finite additive 2-bases."}
{"category": "Math", "title": "Effective log Iitaka fibrations for surfaces and threefolds", "abstract": "We prove an analogue of Fujino and Mori's ``bounding the denominators'' in the log canonical bundle formula (see also Prokhorov and Shokurov) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair $(X,\\Delta)$ of Kodaira codimension one and dimension at most three such that the coefficients of $\\Delta$ are in a DCC set $\\mathcal{A}$, there is a natural number $N$ that depends only on $\\mathcal{A}$ for which the round down of $\\N(K_X+\\Delta)$ induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type."}
{"category": "Math", "title": "Riemannian manifolds with geometric structures", "abstract": "Some geometric structures with associated Riemannian metrics have been considered in the book."}
{"category": "Math", "title": "Extremely Non-symmetric, Non-multiplicative, Non-commutative Operator Spaces", "abstract": "Motivated by importance of operator spaces contained in the set of all scalar multiples of isometries ($MI$-spaces) in a separable Hilbert space for $C^*$-algebras and E-semigroups we exhibit more properties of such spaces. For example, if an $MI$-space contains an isometry with shift part of finite multiplicity, then it is one-dimensional. We propose a simple model of a unilateral shift of arbitrary multiplicity and show that each separable subspace of a Hilbert space is the range of a shift. Also, we show that $MI$-spaces are non-symmetric, very unfriendly to multiplication, and prove a Commutator Identity which elucidates the extreme non-commutativity of these spaces."}
{"category": "Math", "title": "Hyper-atoms and the critical pair Theory", "abstract": "We introduce the notion of a hyper-atom. One of the main results of this paper is the $\\frac{2|G|}3$--Theorem: Let $S$ be a finite generating subset of an abelian group $G$ of order $\\ge 2$. Let $T$ be a finite subset of $G$ such that $2\\le |S|\\le |T|$, $S+T$ is aperiodic, $0\\in S\\cap T$ and $$ \\frac{2|G|+2}3\\ge |S+T|= |S|+|T|-1.$$ Let $H$ be a hyper-atom of $S$. Then $S$ and $T$ are $H$--quasi-periodic. Moreover $\\phi(S)$ and $\\phi(T)$ are arithmetic progressions with the same difference, where $\\phi :G\\mapsto G/H$ denotes the canonical morphism. This result implies easily the traditional critical pair Theory and its basic stone: Kemperman's Structure Theorem."}
{"category": "Math", "title": "Deformation-obstruction theory for complexes via Atiyah and Kodaira--Spencer classes", "abstract": "We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira--Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds."}
{"category": "Math", "title": "New cubature formulae and hyperinterpolation in three variables", "abstract": "A new algebraic cubature formula of degree $2n+1$ for the product Chebyshev measure in the $d$-cube with $\\approx n^d/2^{d-1}$ nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree $n$ in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpolant provides a new Clenshaw-Curtis type cubature formula in the 3-cube."}
{"category": "Math", "title": "Poisson Geometry of Directed Networks in a Disk", "abstract": "We investigate Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if the space of edge weights is equipped with a representative of a 6-parameter family of universal quadratic Poisson brackets and the Grasmannian is viewed as a Poisson homogeneous space of the general linear group equipped with an appropriately chosen R-matrix Poisson-Lie structure. We also prove that Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure."}
{"category": "Math", "title": "Weighted Vogan diagrams associated to real nilpotent orbits", "abstract": "We associate to each nilpotent orbit of a real semisimple Lie algebra $g_o$ a weighted Vogan diagram, that is a Dynkin diagram with an involution of the diagram, a subset of painted nodes and a weight for each node. Every nilpotent element of $g_o$ is noticed in some subalgebra of $g_o$. In this paper we characterize the weighted Vogan diagrams associated to orbits of noticed nilpotent elements."}
{"category": "Math", "title": "Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends", "abstract": "Suppose M is a noncompact connected oriented C^infty n-manifold and omega is a positive volume form on M. Let D^+(M) denote the group of orientation preserving diffeomorphisms of M endowed with the compact-open C^infty topology and D(M; omega) denote the subgroup of omega-preserving diffeomorphisms of M. In this paper we propose a unified approach for realization of mass transfer toward ends by diffeomorphisms of M. This argument together with Moser's theorem enables us to deduce two selection theorems for the groups D^+(M) and D(M; omega). The first one is the extension of Moser's theorem to noncompact manifolds, that is, the existence of sections for the orbit maps under the action of D^+(M) on the space of volume forms. This implies that D(M; omega) is a strong deformation retract of the group D^+(M; E^omega_M) consisting of h in D^+(M) which preserves the set E^omega_M of omega-finite ends of M. The second one is related to the mass flow toward ends under volume-preserving diffeomorphisms of M. Let D_{E_M}(M; omega) denote the subgroup consisting of all h in D(M; omega) which fix the ends E_M of M. S.R.Alpern and V.S.Prasad introduced the topological vector space S(M; omega) of end charges of M and the end charge homomorphism c^omega : D_{E_M}(M; omega) to S(M; omega), which measures the mass flow toward ends induced by each h in D_{E_M}(M; omega). We show that the homomorphism c^omega has a continuous section. This induces the factorization D_{E_M}(M; omega) cong ker c^omega times S(M; omega) and it implies that ker c^omega is a strong deformation retract of D_{E_M}(M; omega)."}
{"category": "Math", "title": "Generalized improper integral definition for infinite limit", "abstract": "A generalization of the definition of a one-dimensional improper integral with an infinite limit is presented. The new definition extends the range of valid integrals to include integrals which were previously considered to not be integrable. This definition is shown to preserve linearity and uniqueness. Integrals which are valid under the conventional definition have the same value under the new definition. Criteria for interchanging the order of integration and differentiation, and for interchanging the order of integration with a second integration, are determined. Examples are provided. A restriction on changing the variable of integration using integration by substitution with the new definition is demonstrated."}
{"category": "Math", "title": "Homology rigidity of Grassmannians", "abstract": "Applying the theory of Gr\\\"{o}bner basis to the Schubert presentation of the cohomology of Grassmannians, we extend the homology rigidity results known for the classical Grassmannians to the exceptional cases."}
{"category": "Math", "title": "Averages of ratios of characteristic polynomials in circular beta-ensembles and super-Jack polynomials", "abstract": "We study the averages of ratios of characteristic polynomials over circular $\\beta$-ensembles, where $\\beta$ is a positive real number. Using Jack polynomial theory, we obtain three expressions for ratio averages. Two of them are given as sums of super-Jack polynomials and another one is given by a hyperdeterminant. As applications, we give dual relations for ratio averages between $\\beta$ and $4/\\beta$."}
{"category": "Math", "title": "Embedding property of $J$-holomorphic curves in Calabi-Yau manifolds for generic $J$", "abstract": "In this paper, we prove that for a generic choice of tame (or compatible) almost complex structures $J$ on a symplectic manifold $(M^{2n},\\omega)$ with $n \\geq 3$ and with its first Chern class $c_1(M,\\omega) = 0$, all somewhere injective $J$-holomorphic maps from any closed smooth Riemann surface into $M$ are \\emph{embedded}. We derive this result as a consequence of the general optimal 1-jet evaluation transversality result of $J$-holomorphic maps in general symplectic manifolds that we also prove in this paper."}
{"category": "Math", "title": "On adaptive Bayesian inference", "abstract": "We study the rate of Bayesian consistency for hierarchical priors consisting of prior weights on a model index set and a prior on a density model for each choice of model index. Ghosal, Lember and Van der Vaart [2] have obtained general in-probability theorems on the rate of convergence of the resulting posterior distributions. We extend their results to almost sure assertions. As an application we study log spline densities with a finite number of models and obtain that the Bayes procedure achieves the optimal minimax rate $n^{-\\gamma/(2\\gamma+1)}$ of convergence if the true density of the observations belongs to the H\\\"{o}lder space $C^{\\gamma}[0,1]$. This strengthens a result in [1; 2]. We also study consistency of posterior distributions of the model index and give conditions ensuring that the posterior distributions concentrate their masses near the index of the best model."}
{"category": "Math", "title": "Explicit error bounds for lazy reversible Markov Chain Monte Carlo", "abstract": "We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte Carlo methods, such as the Metropolis algorithm. The problem is to compute the expectation (or integral) of f with respect to a measure which can be given by a density with respect to another measure. A straight simulation of the desired distribution by a random number generator is in general not possible. Thus it is reasonable to use Markov chain sampling with a burn-in. We study such an algorithm and extend the analysis of Lovasz and Simonovits (1993) to obtain an explicit error bound."}
{"category": "Math", "title": "Computationally Efficient Nonparametric Importance Sampling", "abstract": "The variance reduction established by importance sampling strongly depends on the choice of the importance sampling distribution. A good choice is often hard to achieve especially for high-dimensional integration problems. Nonparametric estimation of the optimal importance sampling distribution (known as nonparametric importance sampling) is a reasonable alternative to parametric approaches.In this article nonparametric variants of both the self-normalized and the unnormalized importance sampling estimator are proposed and investigated. A common critique on nonparametric importance sampling is the increased computational burden compared to parametric methods. We solve this problem to a large degree by utilizing the linear blend frequency polygon estimator instead of a kernel estimator. Mean square error convergence properties are investigated leading to recommendations for the efficient application of nonparametric importance sampling. Particularly, we show that nonparametric importance sampling asymptotically attains optimal importance sampling variance. The efficiency of nonparametric importance sampling algorithms heavily relies on the computational efficiency of the employed nonparametric estimator. The linear blend frequency polygon outperforms kernel estimators in terms of certain criteria such as efficient sampling and evaluation. Furthermore, it is compatible with the inversion method for sample generation. This allows to combine our algorithms with other variance reduction techniques such as stratified sampling. Empirical evidence for the usefulness of the suggested algorithms is obtained by means of three benchmark integration problems. As an application we estimate the distribution of the queue length of a spam filter queueing system based on real data."}
{"category": "Math", "title": "Gene profiling for determining pluripotent genes in a time course microarray experiment", "abstract": "In microarray experiments, it is often of interest to identify genes which have a pre-specified gene expression profile with respect to time. Methods available in the literature are, however, typically not stringent enough in identifying such genes, particularly when the profile requires equivalence of gene expression levels at certain time points. In this paper, the authors introduce a new methodology, called gene profiling, that uses simultaneous differential and equivalent gene expression level testing to rank genes according to a pre-specified gene expression profile. Gene profiling treats the vector of true gene expression levels as a linear combination of appropriate vectors, i.e., vectors that give the required criteria for the profile. This gene-profile model is fitted to the data and the resultant parameter estimates are summarized in a single test statistic that is then used to rank the genes. The theoretical underpinnings of gene profiling (equivalence testing, intersection-union tests) are discussed in this paper, and the gene profiling methodology is applied to our motivating stem cell experiment."}
{"category": "Math", "title": "Marginal Likelihood Integrals for Mixtures of Independence Models", "abstract": "Inference in Bayesian statistics involves the evaluation of marginal likelihood integrals. We present algebraic algorithms for computing such integrals exactly for discrete data of small sample size. Our methods apply to both uniform priors and Dirichlet priors. The underlying statistical models are mixtures of independent distributions, or, in geometric language, secant varieties of Segre-Veronese varieties."}
{"category": "Math", "title": "Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy", "abstract": "We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach constant equilibrium state in the Lp-norm at a rate O(t^(-m/2(1-1/p))), as t tends to $\\infty$, for p in [min (m,2),+ \\infty]. Moreover, we can show that we can approximate, with a faster order of convergence, theconservative part of the solution in terms of the linearized hyperbolic operator for m >= 2, and by a parabolic equation in the spirit of Chapman-Enskog expansion. The main tool is given by a detailed analysis of the Green function for the linearized problem."}
{"category": "Math", "title": "Simultaneous realization of function space structures in transitive Banach spaces", "abstract": "This paper has been withdrawn, since it contains a corollary with impossible consequences and the source of an error is currently unknown."}
{"category": "Math", "title": "Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. II", "abstract": "Let C be a curve over a non-singular base variety S. We study algebraic cycles on the symmetric powers C^[n] and on the Jacobian J. The Chow homology of C^[*], the sum of all C^[n], is a ring using the Pontryagin product. We prove that this ring is isomorphic to CH(J)[t]<u>, the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over the Chow ring CH(J). We give two such isomorphisms that over a general base are different. Further we give some precise results on how CH(J) sits embedded in CH(C^[*]) and we give an explicit geometric description of how the derivations with regard to t and u act. Our results give rise to a new grading on the Chow ring of the Jacobian. After tensoring with Q the associated descending filtration coincides with the one coming from Beauville's decomposition. The grading we obtain is in general different from Beauville's. Finally we give a version of our main result for tautological classes, and we show how our methods give a very simple and geometric proof of some relations obtained by Herbaut and van der Geer-Kouvidakis."}
{"category": "Math", "title": "Riemannian products which are conformally equivalent to Einstein metrics", "abstract": "Necessary and sufficient conditions for a Riemannian product to be conformally equivalent to an Einstein manifold are given. Such spaces which are complete are characterized."}
{"category": "Math", "title": "On Caccetta-Haggkvist Conjecture", "abstract": "We show that we cannot avoid the existence of at least one directed circuit of length less than or equal to (n/r) in a digraph on n vertices with out-degree greater than or equal to r. This is well-known Caccetta-Haggkvist problem."}
{"category": "Math", "title": "Spectral properties of general advection operators and weighted translation semigroups", "abstract": "We investigate the spectral properties of a class of weighted shift semigroups associated to abstract transport equations with a Lipschitz--continuous vector field with no--reentry boundary conditions. We illustrate our results with various examples taken from collisionless kinetic theory."}
{"category": "Math", "title": "Sym\\'etrie des grandes solutions d'\\'equations elliptiques semi lin\\'eaire", "abstract": "We prove that, if $g$ is a continuous asymptotically convex function, any solution $u$ of $-\\Delta u+g(u)=0$ in a ball B which tends to infinity on $\\partial B$ is radially symmetric."}
{"category": "Math", "title": "Likelihood for generally coarsened observations from multi-state or counting process models", "abstract": "We consider first the mixed discrete-continuous scheme of observation in multistate models; this is a classical pattern in epidemiology because very often clinical status is assessed at discrete visit times while times of death or other events are observed exactly. A heuristic likelihood can be written for such models, at least for Markov models; however, a formal proof is not easy and has not been given yet. We present a general class of possibly non-Markov multistate models which can be represented naturally as multivariate counting processes. We give a rigorous derivation of the likelihood based on applying Jacod's formula for the full likelihood and taking conditional expectation for the observed likelihood. A local description of the likelihood allows us to extend the result to a more general coarsening observation scheme proposed by Commenges & G\\'egout-Petit. The approach is illustrated by considering models for dementia, institutionalization and death."}
{"category": "Math", "title": "Diffusion versus absorption in semilinear parabolic equations", "abstract": "We study the limit, when $k\\to\\infty$, of the solutions $u=u_{k}$ of (E) $\\prt_{t}u-\\Delta u+ h(t)u^q=0$ in $\\BBR^N\\ti (0,\\infty)$, $u_{k}(.,0)=k\\delta_{0}$, with $q>1$, $h(t)>0$. If $h(t)=e^{-\\gw(t)/t}$ where $\\gw>0$ satisfies to $\\int_{0}^1\\sqrt{\\gw(t)}t^{-1}dt<\\infty$, the limit function $u_{\\infty}$ is a solution of (E) with a single singularity at $(0,0)$, while if $\\gw(t)\\equiv 1$, $u_{\\infty}$ is the maximal solution of (E). We examine similar questions for equations such as $\\prt_{t}u-\\Gd u^m+ h(t)u^q=0$ with $m>1$ and $\\prt_{t}u-\\Gd u+ h(t)e^{u}=0$."}
{"category": "Math", "title": "Capacitary representations of positive solutions of semilinear parabolic equations", "abstract": "We give a global bilateral estimate on the maximal solution $\\bar u_F$ of $ \\prt_tu-\\Delta u+u^q=0$ in $\\BBR^N\\times (0,\\infty)$, $q>1$, $N\\geq 1$, which vanishes at $t=0 $ on the complement of a closed subset $F\\subset \\BBR^N$. This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2/q,q'}$. We deduce from this estimate that $\\bar {u}_F$ is $\\sigma$-moderate in Dynkin's sense."}
{"category": "Math", "title": "Boundary singularities of solutions of N-harmonic equations with absorption", "abstract": "We study the boundary behaviour of solutions $u$ of $-\\Delta_{N}u+ |u|^{q-1}u=0$ in a bounded smooth domain $\\Omega\\subset\\mathbb R^{N}$ subject to the boundary condition $u=0$ except at one point, in the range $q>N-1$. We prove that if $q\\geq 2N-1$ such a $u$ is identically zero, while, if $N-1<q<2N-1$, $u$ inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed."}
{"category": "Math", "title": "A more intuitive definition of limit", "abstract": "Limit can be defined by two axioms: 1. Strict inequality between limits implies, ultimately, strict inequality between functions. 2. For constant functions limit is trivial. How can basic results on convergence be derived from these axioms? In this paper we propose two answers: a) at the most elementary level- add two more axioms, b) at somewhat higher level, do it in three steps, and, in our forthcoming paper \"Axiomatic definition of limit\", a third answer- c) do it neater - in an abstract framework, where only order relations are present."}
{"category": "Math", "title": "The singularities of the principal component of the Hilbert scheme of points", "abstract": "We show that the principal component of the Hilbert scheme of 9 points in C^8 is not Cohen-Macaulay."}
{"category": "Math", "title": "Morita Theory For Derived Categories: A Bicategorical Perspective", "abstract": "We present a bicategorical perspective on derived Morita theory for rings, DG algebras, and spectra. This perspective draws a connection between Morita theory and the bicategorical Yoneda Lemma, yielding a conceptual unification of Morita theory in derived and bicategorical contexts. This is motivated by study of Rickard's theorem for derived equivalences of rings and of Morita theory for ring spectra, which we present in Sections 2 and 4. Along the way, we gain an understanding of the barriers to Morita theory for DG algebras and give a conceptual explanation for the counterexample of Dugger and Shipley."}
{"category": "Math", "title": "Partial Crossed Product of a group G vs Crossed Product of S(G)", "abstract": "In this work we present a new definition to the Partial Crossed Product by actions of inverse semigroups in a C^*-algebra, without using the covariant representations as Sieben did in [5]. Also we present an isomorphism between the partial crossed products by partial actions of a group G and the partial crossed product by actions of S(G), an inverse semigroup associated to G introduced by Exel in [2]."}
{"category": "Math", "title": "Gradient estimates for $u_t=\\Delta F(u)$ on manifolds and some Liouville-type theorems", "abstract": "In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $$u_t=\\Delta F(u),$$ with $F'(u) > 0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): $$u_t=\\Delta (u^p),\\qquad p>0,$$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of $p$ than that for Aronson-B\\'enilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yaus celebrated Liouville theorem for positive harmonic functions."}
{"category": "Math", "title": "In\\'egalit\\'es de Poincar\\'e cin\\'etiques", "abstract": "In this note we prove Poincar\\'e type inequalities for a family of kinetic equations. We apply this inequality to the variational solution of a linear kinetic model."}
{"category": "Math", "title": "Amenability properties of the centres of group algebras", "abstract": "Let G be a locally compact group, and ZL1(G) be the centre of its group algebra. We show that when $G$ is compact ZL1(G) is not amenable when G is either nonabelian and connected, or is a product of infinitely many finite nonabelian groups. We also, study, for some non-compact groups G, some conditions which imply amenability and hyper-Tauberian property, for ZL1(G)."}
{"category": "Math", "title": "The tropical analogue of polar cones", "abstract": "We study the max-plus or tropical analogue of the notion of polar: the polar of a cone represents the set of linear inequalities satisfied by its elements. We establish an analogue of the bipolar theorem, which characterizes all the inequalities satisfied by the elements of a tropical convex cone. We derive this characterization from a new separation theorem. We also establish variants of these results concerning systems of linear equalities."}
{"category": "Math", "title": "Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer", "abstract": "We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic."}
{"category": "Math", "title": "A boundary value problem for minimal Lagrangian graphs", "abstract": "Let \\Omega and \\tilde{\\Omega} be uniformly convex domains in \\mathbb{R}^n with smooth boundary. We show that there exists a diffeomorphism f: \\Omega \\to \\tilde{\\Omega} such that the graph \\Sigma = \\{(x,f(x)): x \\in \\Omega\\} is a minimal Lagrangian submanifold of \\mathbb{R}^n \\times \\mathbb{R}^n."}
{"category": "Math", "title": "Spectral Curves for Almost-Complex Tori in $ S ^6 $", "abstract": "To each non-isotropic almost-complex immersion of a 2-torus into $ S ^ 6 $ we associate an algebraic curve, called the spectral curve, and a linear flow in the intersection of two Prym varieties on this spectral curve. We show that generically the spectral curve is smooth and compute the dimension of the moduli space of such curves and of the torus in which the eigenline bundles lie."}
{"category": "Math", "title": "Embedded Associated Primes of Powers of Square-free Monomial Ideals", "abstract": "An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all natural numbers t. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that I^t has embedded primes is bigger than beta_1, where beta_1 is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If in addition I fails to have the packing property, then embedded primes of I^t do occur when t=beta_1 +1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornu\\'ejols."}
{"category": "Math", "title": "Multiplicative functional for reflected Brownian motion via deterministic ODE", "abstract": "We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory."}
{"category": "Math", "title": "Bijective Proofs of Identities from Colored Binary Trees", "abstract": "This note provide bijective proofs of two combinatorial identities involving generalized Catalan number $C_{m,5}(n)={m\\over 5n+m}{5n+m\\choose n}$ recently proposed by Sun."}
{"category": "Math", "title": "Pullback Attractors for the Non-autonomous FitzHugh-Nagumo System on Unbounded Domains", "abstract": "The existence of a pullback attractor is established for the singularly perturbed FitzHugh-Nagumo system defined on the entire space $R^n$ when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero."}
{"category": "Math", "title": "Constructing mean curvature 1 surfaces in $H^3$ with irregular ends", "abstract": "With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space $H^3$, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface has an essential singularity, resp. at most a pole, there.) There are some known surfaces with irregular ends, but they are all either reducible or of infinite total curvature. (The surface is reducible if and only if the monodromy of the secondary Gauss map can be simultaneously diagonalized.) Up to now there have been no known complete irreducible CMC 1 surfaces in $H^3$ with finite total curvature and irregular ends. The purpose of this paper is to construct countably many 1-parameter families of genus zero CMC 1 surfaces with irregular ends and finite total curvature, which have either dihedral or Platonic symmetries. For all the examples we produce, we show that they have finite total curvature and irregular ends. For the examples with dihedral symmetry and the simplest example with tetrahedral symmetry, we show irreducibility. Moreover, we construct a genus one CMC 1 surface with four irregular ends, which is the first known example with positive genus whose ends are all irregular."}
{"category": "Math", "title": "Birational geometry and localisation of categories", "abstract": "This version corrects a wrong proof of Proposition 6.3.2 and simplifies the exposition in Section 6."}
{"category": "Math", "title": "On q-deformed gl(l+1)-Whittaker function III", "abstract": "We identify q-deformed gl(l+1)-Whittaker functions with a specialization of Macdonald polynomials. This provides a representation of q-deformed gl(l+1)-Whittaker functions in terms of Demazure characters of affine Lie algebra \\hat{gl(l+1)}. We also define a system of dual Hamiltonians for q-deformed gl(l+1)-Toda chains and give a new integral representation for q-deformed gl(l+1)-Whittaker functions. Finally an expression of q-deformed gl(l+1)-Whittaker function as a matrix element of a quantum torus algebra is derived."}
{"category": "Math", "title": "Contractions of low-dimensional nilpotent Jordan algebras", "abstract": "In this paper we classify the laws of three-dimensional and four-dimensional nilpotent Jordan algebras over the field of complex numbers. We describe the irreducible components of their algebraic varieties and extend contractions and deformations among them. In particular, we prove that J2 and J3 are irreducible and that J4 is the union of the Zariski closures of two rigid Jordan algebras."}
{"category": "Math", "title": "The Solvability and Subellipticity of Systems of Pseudodifferential Operators", "abstract": "The paper studies the local solvability and subellipticity for square systems of principal type. These are the systems for which the principal symbol vanishes of first order on its kernel. For systems of principal type having constant characteristics, local solvability is equivalent to condition (PSI) on the eigenvalues, see arXiv:0801.4043. This is a condition on the sign changes of the imaginary part of the eigenvalue along the oriented bicharacteristics of the real part. In the generic case when the principal symbol does not have constant characteristics, condition (PSI) is not sufficient, not invariant and in general not well defined. Instead we study systems which are quasi-symmetrizable, these systems have natural invariance properties and are of principal type. We prove that quasi-symmetrizable systems are locally solvable. We also study the subellipticity of quasi-symmetrizable systems in the case when principal symbol vanishes of finite order along the bicharacteristics. In order to prove subellipticity, we assume that the principal symbol has the approximation property, which implies that there are no transversal bicharacteristics."}
{"category": "Math", "title": "Period problems for mean curvature one surfaces in $H^3$ (with application to surfaces of low total curvature)", "abstract": "We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of \"total curvature\" -- one is the total absolute curvature, which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature, which is the total absolute curvature of the dual CMC 1 surface. Here we discuss results on both notions (proven in two other papers by the authors), and we introduce new results as well."}
{"category": "Math", "title": "Some linear and nonlinear integral inequalities on time scales in two independent variables", "abstract": "We establish some linear and nonlinear integral inequalities of Gronwall-Bellman-Bihari type for functions with two independent variables on general time scales. The results are illustrated with examples, obtained by fixing the time scales to concrete ones. An estimation result for the solution of a partial delta dynamic equation is given as an application."}
{"category": "Math", "title": "Logarithmic bounds on Sobolev norms for time-dependent linear Schr\\\"odinger equations", "abstract": "We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schr\\\"odinger equations is at most logarithmic in time for any (fixed) potential which is analytic (or Gevrey). Recently it was proven in [N] that almost surely the Sobolev norms are unbounded, which indicates that the log is almost surely necessary. In [W], the author showed that the Sobolev norms remain bounded for an explicit time periodic potential. This is in the exceptional set in the sense of [N]. The present paper together with [N, W] give a rather complete picture of time dependent linear Schr\\\"odinger equations on the circle."}
{"category": "Math", "title": "An Intrinsic Impulse Observability Criterion for Descriptor System", "abstract": "Analyzing the order of unobservable impulse in descriptor system leads to a new testing criterion for impulse observability, both the statement and the proof of which use only the original system data."}
{"category": "Math", "title": "On the generic and typical ranks of 3-tensors", "abstract": "We study the generic and typical ranks of 3-tensors of dimension l x m x n using results from matrices and algebraic geometry. We state a conjecture about the exact values of the generic rank of 3-tensors over the complex numbers, which is verified numerically for l,m,n not greater than 14. We also discuss the typical ranks over the real numbers, and give an example of an infinite family of 3-tensors of the form l=m, n=(m-1)^2+1, m=3,4,..., which have at least two typical ranks."}
{"category": "Math", "title": "Maximal solutions of equation u = uq in arbitrary domains", "abstract": "We prove bilateral capacitary estimates for the maximal solution $U_F$ of $-\\Delta u+u^q=0$ in the complement of an arbitrary closed set $F\\subset\\mathbb R^N$, involving the Bessel capacity $C_{2,q'}$, for $q$ in the supercritical range $q\\geq q_{c}:=N/(N-2)$. We derive a pointwise necessary and sufficient condition, via a Wiener type criterion, in order that $U_F(x)\\to\\infty$ as $x\\to y$ for given $y\\in\\prt F$. Finally we prove a general uniqueness result for large solutions."}
{"category": "Math", "title": "Semigroups of valuations on local rings, II", "abstract": "Given a noetherian local domain $R$ and a valuation $\\nu$ of its field of fractions which is non negative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values lying in certain parts of the value group $\\Gamma$ of $\\nu$. We show that this growth condition imposes restrictions on the semigroups $\\nu(R\\setminus \\{0\\})$ for noetherian $R$ which are stronger that those resulting from the previous paper \\cite{C2} of the first author. Given an ordered embedding $\\Gamma\\subset ({\\mathbf R}^h)_{\\hbox{\\rm lex}}$, where $h$ is the rank of $\\nu$, we also study the shape in ${\\mathbf R}^h$ of the parts of $\\Gamma$ which appear naturally in this study. We give examples which show that this shape can be quite wild in a way which does not depend on the embedding and suggest that it is a good indicator of the complexity of the semigroup $\\nu(R\\setminus \\{0\\})$."}
{"category": "Math", "title": "The balance between diffusion and absorption in semilinear parabolic equations", "abstract": "Let $h:[0,\\infty)\\mapsto [0,\\infty)$ be continuous and nondecreasing, $h(t)>0$ if $t>0$, and $m,q$ be positive real numbers. We investigate the behavior when $k\\to\\infty$ of the fundamental solutions $u=u_{k}$ of $\\prt_{t} u-\\Delta u^m+h(t)u^q=0$ in $\\Omega\\ti (0,T)$ satisfying $u_{k}(x,0)=k\\delta_0$. The main question is wether the limit is still a solution of the above equation with an isolated singularity at $(0,0)$, or a solution of the associated ordinary differential equation $ u'+h(t)u^q=0$ which blows-up at $t=0$."}
{"category": "Math", "title": "The precise boundary trace of solutions of a class of supercritical nonlinear equations", "abstract": "We construct and study the properties of the precise boundary trace of positive solutions of $-\\Delta u+u^q=0$ in a smooth bounded domain of $\\mathbb R^N$, in the supercritical case $q\\geq q_c=(N+1)/(N-1)$"}
{"category": "Math", "title": "Isonemal Prefabrics with Perpendicular Axes of Symmetry", "abstract": "This paper continues the refinement of Richard Roth's taxonomy of isonemal weaving designs through types 11--32 of the 39 in order to solve three problems for those designs: which designs exist in various sizes, which prefabrics can be doubled and remain isonemal, and which can be halved and remain isonemal."}
{"category": "Math", "title": "On the nontrivial projection problem", "abstract": "The Nontrivial Projection Problem asks whether every finite-dimensional normed space of dimension greater than one admits a well-bounded projection of non-trivial rank and corank or, equivalently, whether every centrally symmetric convex body (of arbitrary dimension greater than one) is approximately affinely equivalent to a direct product of two bodies of non-trivial dimension. We show that this is true \"up to a logarithmic factor.\""}
{"category": "Math", "title": "Approximating with Gaussians", "abstract": "Linear combinations of translations of a single Gaussian, e^{-x^2}, are shown to be dense in L^2(R). Two algorithms for determining the coefficients for the approximations are given, using orthogonal Hermite functions and least squares. Taking the Fourier transform of this result shows low-frequency trigonometric series are dense in L^2 with Gaussian weight function."}
{"category": "Math", "title": "Estimation of Large Precision Matrices Through Block Penalization", "abstract": "This paper focuses on exploring the sparsity of the inverse covariance matrix $\\bSigma^{-1}$, or the precision matrix. We form blocks of parameters based on each off-diagonal band of the Cholesky factor from its modified Cholesky decomposition, and penalize each block of parameters using the $L_2$-norm instead of individual elements. We develop a one-step estimator, and prove an oracle property which consists of a notion of block sign-consistency and asymptotic normality. In particular, provided the initial estimator of the Cholesky factor is good enough and the true Cholesky has finite number of non-zero off-diagonal bands, oracle property holds for the one-step estimator even if $p_n \\gg n$, and can even be as large as $\\log p_n = o(n)$, where the data $\\y$ has mean zero and tail probability $P(|y_j| > x) \\leq K\\exp(-Cx^d)$, $d > 0$, and $p_n$ is the number of variables. We also prove an operator norm convergence result, showing the cost of dimensionality is just $\\log p_n$. The advantage of this method over banding by Bickel and Levina (2008) or nested LASSO by Levina \\emph{et al.} (2007) is that it allows for elimination of weaker signals that precede stronger ones in the Cholesky factor. A method for obtaining an initial estimator for the Cholesky factor is discussed, and a gradient projection algorithm is developed for calculating the one-step estimate. Simulation results are in favor of the newly proposed method and a set of real data is analyzed using the new procedure and the banding method."}
{"category": "Math", "title": "Hopf Algebroids", "abstract": "This is a preprint version of a chapter for Handbook of Algebra."}
{"category": "Math", "title": "Null controllability for the parabolic equation with a complex principal part", "abstract": "This paper is addressed to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators $(\\a+i\\b)\\pa_t+\\sum\\limits_{j,k=1}^n\\pa_k(a^{jk}\\pa_j)$ (with real functions $\\a$ and $\\b$), by which we develop a universal approach, based on global Carleman estimate, to deduce not only the desired explicit observability estimate for the linearized complex Ginzburg-Landau equation, but also all the known controllability/observability results for the parabolic, hyperbolic, Schr\\\"odinger and plate equations that are derived via Carleman estimates."}
{"category": "Math", "title": "Gelfand pairs on the Heisenberg group and Schwartz functions", "abstract": "Let $\\Hn$ be the $(2n+1)$-dimensional Heisenberg group and $K$ a compact group of automorphisms of $\\Hn$ such that $(K\\ltimes \\Hn,K)$ is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of $K$-invariant Schwartz functions on $\\Hn$ and the space of Schwartz function on a closed subset of $\\R^s$ homeomorphic to the Gelfand spectrum of the Banach algebra of $K$-invariant integrable functions on $\\Hn$."}
{"category": "Math", "title": "Convergence in the Boundary Layer for Nonhomogeneous Linear Singularly Perturbed Systems", "abstract": "Convergence of the solutions of nonhomogeneous linear singularly perturbed systems to that of the corresponding reduced singular system on the half-line [0, $\\infty $) is considered. To include the situation on a neighborhood of initial instant, a boundary layer, a distributional approach to convergence is adopted. An explicit analytical expression for the limit as a distribution is proved."}
{"category": "Math", "title": "On fixed points and uniformly convex spaces", "abstract": "The purpose of this note is to present two elementary, but useful, facts concerning actions on uniformly convex spaces. We demonstrate how each of them can be used in an alternative proof of the triviality of the first $L_p$-cohomology of higher rank simple Lie groups, proved in [BFGM]."}
{"category": "Math", "title": "Eigenfunction concentration for polygonal billiards", "abstract": "In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in \\cite{M1}. There, the methods developed in Burq-Zworski \\cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$ \\int_U |u|^2 \\geq c \\int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$."}
{"category": "Math", "title": "Sharp capacitary estimates for rings in metric spaces", "abstract": "We establish sharp estimates for the $p$-capacity of metric rings with unrelated radii in metric measure spaces equipped with a doubling measure and supporting a Poincar\\'e inequality. These estimates play an essential role in the study of the local behavior of \\p-harmonic Green's functions."}
{"category": "Math", "title": "Missing observation analysis for matrix-variate time series data", "abstract": "Bayesian inference is developed for matrix-variate dynamic linear models (MV-DLMs), in order to allow missing observation analysis, of any sub-vector or sub-matrix of the observation time series matrix. We propose modifications of the inverted Wishart and matrix $t$ distributions, replacing the scalar degrees of freedom by a diagonal matrix of degrees of freedom. The MV-DLM is then re-defined and modifications of the updating algorithm for missing observations are suggested."}
{"category": "Math", "title": "Bi-isometries and commutant lifting", "abstract": "A new functional model for pairs of commuting isometries is described. Intertwining operators between such models are then studied in order to approach the classification of invariant subspaces of such pairs."}
{"category": "Math", "title": "The Use of Labeled Cortical Distance Maps for Quantization and Analysis of Anatomical Morphometry of Brain Tissues", "abstract": "Anatomical shape differences in cortical structures in the brain can be associated with various neuropsychiatric and neuro-developmental diseases or disorders. Labeled Cortical Distance Map (LCDM), can be a powerful tool to quantize such morphometric differences. In this article, we investigate various issues regarding the analysis of LCDM distances in relation to morphometry. The length of the LCDM distance vector provides the number of voxels (approximately a multiple of volume (in mm^3)); median, mode, range, and variance of LCDM distances are all suggestive of size, thickness, and shape differences. However these measures provide a crude summary based on LCDM distances which may convey much more information about the tissue in question. To utilize more of this information, we pool (merge) the LCDM distances from subjects in the same group or condition. The statistical methodology we employ require normality and within and between sample independence. We demonstrate that the violation of these assumptions have mild influence on the tests. We specify the types of alternatives the parametric and nonparametric tests are more sensitive for. We also show that the pooled LCDM distances provide powerful results for group differences in distribution, left-right morphometric asymmetry of the tissues, and variation of LCDM distances. As an illustrative example, we use gray matter (GM) tissue of ventral medial prefrontal cortices (VMPFCs) from subjects with major depressive disorder, subjects at high risk, and control subjects. We find significant evidence that VMPFCs of subjects with depressive disorders are different in shape compared to those of normal subjects."}
{"category": "Math", "title": "Cohen-Macaulay clutters with combinatorial optimization properties and parallelizations of normal edge ideals", "abstract": "Let C be a uniform clutter and let I=I(C) be its edge ideal. We prove that if C satisfies the packing property (resp. max-flow min-cut property), then there is a uniform Cohen-Macaulay clutter C1 satisfying the packing property (resp. max-flow min-cut property) such that C is a minor of C1. For arbitrary edge ideals of clutters we prove that the normality property is closed under parallelizations. Then we show some applications to edge ideals and clutters which are related to a conjecture of Conforti and Cornu\\'ejols and to max-flow min-cut problems."}
{"category": "Math", "title": "Homogeneous Polynomials with Isomorphic Milnor Algebras", "abstract": "In Theorem 3.2 we show that two homogeneous polynomials $f$ and $g$ having isomorphic Milnor algebras are right-equivalent."}
{"category": "Math", "title": "Dynamics of meromorphic maps with small topological degree II: Energy and invariant measure", "abstract": "We continue our study of the dynamics of meromorphic mappings with small topological degree on a compact K\\\"ahler surface $X$. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and admits a natural geometric description. Our hypotheses are always satisfied when $X$ has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of $\\mathbf{C}^2$. They are new even in the birational case. We also exhibit families of mappings where our assumptions are generically satisfied and show that if counterexamples exist, the corresponding measure must give mass to a pluripolar set."}
{"category": "Math", "title": "Differentiable Rigidity under Ricci curvature lower bound", "abstract": "In this article we prove a differentiable rigidity result. Let $(Y, g)$ and $(X, g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n\\geqslant 3$) and $f:Y\\to X$ be a continuous map of degree $1$. We furthermore assume that the metric $g_0$ is real hyperbolic and denote by $d$ the diameter of $(X,g_0)$. We show that there exists a number $\\varepsilon:=\\varepsilon (n, d)>0$ such that if the Ricci curvature of the metric $g$ is bounded below by $-n(n-1)$ and its volume satisfies $\\vol_g (Y)\\leqslant (1+\\varepsilon) \\vol_{g_0} (X)$ then the manifolds are diffeomorphic. The proof relies on Cheeger-Colding's theory of limits of Riemannian manifolds under lower Ricci curvature bound."}
{"category": "Math", "title": "Algebraic Legendrian Varieties", "abstract": "Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional properties. The most remarkable case is the Legendrian subvarieties of projective space and prior to the author's research only few smooth examples of these were known. The first series of results of this thesis is related to the automorphism group of any Legendrian subvariety in any projective contact manifold. The connected component of this group (under suitable minor assumptions) is completely determined by the sections of the distinguished line bundle on the contact manifold vanishing on the Legendrian variety. Moreover its action preserves the contact structure. The second series of results is devoted to finding new examples of smooth Legendrian subvarieties of projective space. The contribution of this thesis is in three steps: First we find an example of a smooth toric surface. Next we find a smooth quasihomogeneous Fano 8-fold that admits a Legendrian embedding. Finally, we realise that both of these are special cases of a very general construction: a general hyperplane section of a smooth Legendrian variety, after a suitable projection, is a smooth Legendrian variety of smaller dimension. By applying this result to known examples and decomposable Legendrian varieties, we construct infinitely many new examples in every dimension, with various Picard rank, canonical degree, Kodaira dimension and other invariants."}
{"category": "Math", "title": "Generalized Chinese restaurant construction of exchangeable Gibbs partitions and related results", "abstract": "By resorting to sequential constructions of exchangeable random partitions (Pitman, 2006), and exploiting some known facts about generalized Stirling numbers, we derive a generalized Chinese restaurant process construction of exchangeable Gibbs partitions of type $\\alpha$ (Gnedin and Pitman, 2006). Our construction represents the natural theoretical probabilistic framework in which to embed some recent results about a Bayesian nonparametric treatment of estimation problems arising in genetic experiment under Gibbs, species sampling, models priors."}
{"category": "Math", "title": "On the error term in Weyl's law for the Heisenberg manifolds", "abstract": "For a fixed integer $l\\geq 1$, let $R(t)$ denote the error term in the Weyl's law of a $(2l+1)$-dimensional Heisenberg manifold with the metric $g_l.$ In this paper we shall prove the asymptotic formula of the $k$-th power moment for any integers $3\\leq k\\leq 9.$ We shall also prove that the function $t^{-(l-1/4)}R(t)$ has a distribution function."}
{"category": "Math", "title": "A vanishing theorem for log canonical pairs", "abstract": "Using inversion of adjunction, we deduce from Nadel's theorem a vanishing property for ideals sheaves on projective varieties, a special case of which recovers a result due to Bertram--Ein--Lazarsfeld. This enables us to generalize to a large class of projective schemes certain bounds on Castelnuovo--Mumford regularity previously obtained by Bertram--Ein--Lazarsfeld in the smooth case and by Chardin--Ulrich for locally complete intersection varieties with rational singularities. Our results are tested on several examples."}
{"category": "Math", "title": "Gamma convergence of an energy functional related to the fractional Laplacian", "abstract": "We prove a Gamma-convergence result for an energy functional related to some fractional powers of the Laplacian operator, with two singular perturbations (one in the interior and one on the boundary)."}
{"category": "Math", "title": "Excursions away from a regular point for one-dimensional symmetric Levy processes without Gaussian part", "abstract": "The characteristic measure of excursions away from a regular point is studied for a class of symmetric L\\'evy processes without Gaussian part. It is proved that the harmonic transform of the killed process enjoys Feller property. The result is applied to prove extremeness of the excursion measure and to prove several sample path behaviors of the excursion and the $ h $-path processes."}
{"category": "Math", "title": "Asymptotic stability of ground states in 2D nonlinear Schr\\\"odinger equation including subcritical cases", "abstract": "We consider a class of nonlinear Schr\\\"odinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that \"shadows\" the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations."}
{"category": "Math", "title": "When is the Hawking mass monotone under Geometric Flows", "abstract": "In this paper, we study the relation of the monotonicity of Hawking Mass and geometric flow problems. We show that along the Hamilton-DeTurck flow with bounded curvature coupled with the modified mean curvature flow, the Hawking mass of the hypersphere with a sufficiently large radius in Schwarzschild spaces is monotone non-decreasing."}
{"category": "Math", "title": "The Poincar\\'e series of a local Gorenstein ring of multiplicity up to 10 is rational", "abstract": "Let $R$ be a local, Gorenstein ring with algebraically closed residue field $k$ of characteristic 0 and let $P_R(z):=\\sum_{p=0}^{\\infty}\\dim_k(\\tor_p^R(k,k))z^p$ be its Poincar\\'e series. We compute $P_R$ when $R$ belongs to a particular class defined in the introduction, proving its rationality. As a by--product we prove the rationality of $P_R$ for all local, Gorenstein rings of multiplicity at most 10."}
{"category": "Math", "title": "Local spectral radius formulas on compact Lie groups", "abstract": "We determine the local spectrum of a central element of the complexified universal enveloping algebra of a compact connected Lie group at a smooth function as an element of L^p(G). Based on this result we establish a corresponding local spectral radius formula."}
{"category": "Math", "title": "Truncated Wiener-Hopf operators with Fisher Hartwig singularities", "abstract": "We derive the asymptotic behavior of determinants of truncated Wiener-Hopf operators generated by symbols having Fisher-Hartwig singularities. This task is achieved thanks to an asymptotic resolution of the Riemann-Hilbert problem associated to some generalized sine kernel. As a byproduct, we give yet another derivation of the asymptotic behavior of Toeplitz determinants having Fisher-Hartwig singularities. The Riemann-Hilbert problem approach to these asymptotics yields a systematic although quickly cumbersome way to compute their sub-leading asymptotics."}
{"category": "Math", "title": "Inference for Multivariate Normal Mixtures", "abstract": "Multivariate normal mixtures provide a flexible model for high-dimensional data. They are widely used in statistical genetics, statistical finance, and other disciplines. Due to the unboundedness of the likelihood function, classical likelihood-based methods, which may have nice practical properties, are inconsistent. In this paper, we recommend a penalized likelihood method for estimating the mixing distribution. We show that the maximum penalized likelihood estimator is strongly consistent when the number of components has a known upper bound. We also explore a convenient EM-algorithm for computing the maximum penalized likelihood estimator. Extensive simulations are conducted to explore the effectiveness and the practical limitations of both the new method and the ratified maximum likelihood estimators. Guidelines are provided based on the simulation results."}
{"category": "Math", "title": "A set-valued framework for birth-and-growth process", "abstract": "We propose a set-valued framework for the well-posedness of birth-and-growth process. Our birth-and-growth model is rigorously defined as a suitable combination, involving Minkowski sum and Aumann integral, of two very general set-valued processes representing nucleation and growth respectively. The simplicity of the used geometrical approach leads us to avoid problems arising by an analytical definition of the front growth such as boundary regularities. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, it is not local, i.e. for a fixed time instant, growth is the same at each space point."}
{"category": "Math", "title": "Extrinsic symplectic symmetric spaces", "abstract": "We define the notion of extrinsic symplectic symmetric spaces and exhibit some of their properties. We construct large families of examples and show how they fit in the perspective of a complete classification of these manifolds. We also build a natural star-quantization on a class of examples."}
{"category": "Math", "title": "Optimal local H\\\"{o}lder index for density states of superprocesses with $(1+\\beta)$-branching mechanism", "abstract": "For $0<\\alpha\\leq2$, a super-$\\alpha$-stable motion $X$ in $\\mathsf{R}^d$ with branching of index $1+\\beta\\in(1,2)$ is considered. Fix arbitrary $t>0$. If $d<\\alpha/\\beta$, a dichotomy for the density function of the measure $X_t$ holds: the density function is locally H\\\"{o}lder continuous if $d=1$ and $\\alpha>1+\\beta$ but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local H\\\"{o}lder index."}
{"category": "Math", "title": "Binary Additive Problems Involving Naturals with Binary Decompositions of a Special Kind", "abstract": "Let $h$ and $l$ be integers such that $0\\le h\\le 2$, $0\\le l\\le 4$. We obtain asymptotic formulas for the numbers of solutions of the equations $n-3m=h$, $n-5m=l$ in positive integers $m$ and $n$ of a special kind, $m\\le X$."}
{"category": "Math", "title": "Estimate of a Trigonometrical Sum Involving Naturals with Binary Decompositions of a Special Kind", "abstract": "Let $\\mathbb{N}_0$ be a class of natural numbers whose binary decompositions has even number of 1. We estimate of the sum $\\sum\\limits_{n\\in \\mathbf{N}_0,n\\le X}\\exp(2\\pi i \\alpha n^2)$."}
{"category": "Math", "title": "On Fano threefolds with canonical Gorenstein singularities", "abstract": "We classify three-dimensional Fano varieties with canonical Gorenstein singularities of degree bigger than 64."}
{"category": "Math", "title": "On Morita equivalence for simple Generalized Weyl algebras", "abstract": "We give a necessary condition for Morita equivalence of simple Generalized Weyl algebras of classical type. We propose a reformulation of Hodges' result, which describes Morita equivalences in case the polynomial defining the Generalized Weyl algebra has degree 2, in terms of isomorphisms of quantum tori, inspired by similar considerations in noncommutative differential geometry. We study how far this link can be generalized for $n\\ge 3$."}
{"category": "Math", "title": "Global controllability and stabilization for the nonlinear Schrodinger equation on an interval", "abstract": "We prove global internal controllability in large time for the nonlinear Schrodinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result on L2. We also get a regularity result about the control if the data are assumed smoother."}
{"category": "Math", "title": "Nonstandard Models and Optimization", "abstract": "This is an overview of a few possibilities that are open by model theory in applied mathematics. Most attention is paid to the present state and frontiers of the Cauchy method of majorants, approximation of operator equations with finite-dimensional analogs, and the Lagrange multiplier principle in multiobjective decision making."}
{"category": "Math", "title": "A Few Results about the Geometry of Model Averages", "abstract": "Given a collection of computational models that all estimate values of the same natural process, we compare the performance of the average of the collection to the individual member whose estimates are nearest a given set of observations. Performance is the ability of a model, or average, to reproduce a sequence of observations of the process. We identify a condition that determines if a single model performs better than the average. That result also yields a necessary condition for when the average performs better than any individual model. We also give sharp bounds for the performance of the average on a given interval. Since the observation interval is fixed, performance is evaluated in a vector space, and we can add intuition to our results by explaining them geometrically. We conclude with some comments on directions statistical tests of performance might take."}
{"category": "Math", "title": "On distribution and almost convergence of bounded sequences", "abstract": "In this paper, we give the concepts of properly distributed and simply distributed sequences, and prove that they are almost convergent. Basing on these, we review the work of Feng and Li [Feng, B. Q. and Li, J. L., Some estimations of Banach limits, J. Math. Anal. Appl. 323(2006) No. 1, 481-496. MR2262220 46B45 (46A45).], which is shown to be a special case of our generalized theory."}
{"category": "Math", "title": "The volume flux group and nonpositive curvature", "abstract": "We show that every closed nonpositively curved manifold with non-trivial volume flux group has zero minimal volume, and admits a finite covering with circle actions whose orbits are homologically essential. This proves a conjecture of Kedra-Kotschick-Morita for this class of manifolds."}
{"category": "Math", "title": "Projectivity of analytic Hilbert and Kaehler quotients", "abstract": "We investigate algebraicity properties of quotients of complex spaces by complex reductive Lie groups G. We obtain a projectivity result for compact momentum map quotients of algebraic G-varieties. Furthermore, we prove equivariant versions of Kodaira's Embedding Theorem and Chow's Theorem relative to an analytic Hilbert quotient. Combining these results we derive an equivariant algebraisation theorem for complex spaces with projective quotient."}
{"category": "Math", "title": "Generic ordinarity for semi-stable fibration", "abstract": "In this paper, we proved that, for a semi-stable fibration of a proper smooth surface to a proper smooth curve over a field of positive characteristic, if the generic fiber is ordinary, then the semi-positivity theorem holds. As an application, we constrcuted a counterexample to Parshin's conjecture on the Miyaoka-Yau inequality."}
{"category": "Math", "title": "Unirational Surfaces on the Noether Line", "abstract": "We show that among simply connected surfaces of general type unirationality is a common feature, even when fixing the positive characteristic or numerical invariants. To do so, we construct unirational Horikawa surfaces in abundance."}
{"category": "Math", "title": "Low pole order frames on vertical jets of the universal hypersurface", "abstract": "Of the two techniques introduced by Bloch, Green-Griffiths and developed by Siu, Demailly to establish Kobayashi hyperbolicity of generic high degree complex algebraic hypersurfaces X in P^(n+1), the second one, initiated by Clemens, Ein, Voisin and developed by Siu, Paun, Rousseau consists in constructing meromorphic frames on the space of the so-called vertical k-jets J_vert^k (X_univ) of the universal hypersurface X_univ parametrizing all X in P^(n+1) of degree d. In 2004, Siu announced that there exists a constant c_n such that the twisting of the tangent bundle to J_vert^n (X_univ) by O (c_n) is globally generated (frame property). The present article provides c_n = (n^2 + 5n) / 2, recovering c_2 = 7 (Paun), c_3 = 12 (Rousseau). Applications to effective degree estimates for algebraic degeneracy or hyperbolicity are expected, especially in dimension n = 4, granted that the Demailly-Semple algebra of jet polynomials invariant under reparametrization and under a certain unipotent action is, for n = k = 4, generated by 16 fundamental bi-invariants enjoying 41 groebnerized syzygies."}
{"category": "Math", "title": "Semi-stable fibration of generic p-rank 0", "abstract": "In this paper, we proved that, for a semi-stable fibration of a proper smooth surface to a proper smooth curve over a filed of positive characteristic, if the p-rank of the generic fiber is 0, then the base change of the fibration by a sufficiently many iterative Frobenius morphism of the base curve violates the semi-positivity theorem. As an application, we suggest a statement on a distribution of p-ranks of reductions for a certain non-closed point in the moduli space over a number field."}
{"category": "Math", "title": "The Relation between Approximation in Distribution and Shadowing in Molecular Dynamics", "abstract": "Molecular dynamics refers to the computer simulation of a material at the atomic level. An open problem in numerical analysis is to explain the apparent reliability of molecular dynamics simulations. The difficulty is that individual trajectories computed in molecular dynamics are accurate for only short time intervals, whereas apparently reliable information can be extracted from very long-time simulations. It has been conjectured that long molecular dynamics trajectories have low-dimensional statistical features that accurately approximate those of the original system. Another conjecture is that numerical trajectories satisfy the shadowing property: that they are close over long time intervals to exact trajectories but with different initial conditions. We prove that these two views are actually equivalent to each other, after we suitably modify the concept of shadowing. A key ingredient of our result is a general theorem that allows us to take random elements of a metric space that are close in distribution and embed them in the same probability space so that they are close in a strong sense. This result is similar to the Strassen-Dudley Theorem except that a mapping is provided between the two random elements. Our results on shadowing are motivated by molecular dynamics but apply to the approximation of any dynamical system when initial conditions are selected according to a probability measure."}
{"category": "Math", "title": "Embedding groups of class two and prime exponent in capable and non-capable groups", "abstract": "We show that if $G$ is any $p$-group of class at most two and exponent $p$, then there exist groups $G_1$ and $G_2$ of class two and exponent $p$ that contain $G$, neither of which can be expressed as a central product, and with $G_1$ capable and $G_2$ not capable. We provide upper bounds for ${\\rm rank}(G_i^{\\rm ab})$ in terms of ${\\rm rank}(G^{\\rm ab})$ in each case."}
{"category": "Math", "title": "Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives", "abstract": "This paper is devoted to the study of some coherent sheaves on non reduced curves that can be locally embedded in smooth surfaces. If Y is such a curve then there is a filtration by subschemes C_i such that C_1 is the reduced curve associated to Y, and that for every P in C, if z is an equation of C_1 in the local ring of Y at P, then (z^i) is the ideal of C_i. A coherent sheaf on Y is called torsion free if it does not have any non zero subsheaf with finite support. We prove that torsion free sheaves are reflexive. We study then the quasi locally free sheaves, i.e. sheaves which are locally isomorphic to direct sums of the structure sheaves of the C_i. We define an invariant for these sheaves, the complete type, and prove the irreducibility of the set of sheaves of given complete type. We study the generic quasi locally free sheaves, with applications to the moduli spaces of stable sheaves on Y."}
{"category": "Math", "title": "Scattering in flatland: Efficient representations via wave atoms", "abstract": "This paper presents a numerical compression strategy for the boundary integral equation of acoustic scattering in two dimensions. These equations have oscillatory kernels that we represent in a basis of wave atoms, and compress by thresholding the small coefficients to zero. This phenomenon was perhaps first observed in 1993 by Bradie, Coifman, and Grossman, in the context of local Fourier bases \\cite{BCG}. Their results have since then been extended in various ways. The purpose of this paper is to bridge a theoretical gap and prove that a well-chosen fixed expansion, the nonstandard wave atom form, provides a compression of the acoustic single and double layer potentials with wave number $k$ as $O(k)$-by-$O(k)$ matrices with $O(k^{1+1/\\infty})$ nonnegligible entries, with a constant that depends on the relative $\\ell_2$ accuracy $\\eps$ in an acceptable way. The argument assumes smooth, separated, and not necessarily convex scatterers in two dimensions. The essential features of wave atoms that enable to write this result as a theorem is a sharp time-frequency localization that wavelet packets do not obey, and a parabolic scaling wavelength $\\sim$ (essential diameter)${}^2$. Numerical experiments support the estimate and show that this wave atom representation may be of interest for applications where the same scattering problem needs to be solved for many boundary conditions, for example, the computation of radar cross sections."}
{"category": "Math", "title": "Continuation of root functionals of a system of polynomial equations and the reduction of polynomials modulo its ideal", "abstract": "The notion of a root functional of polynomials is a generalization of the notion of a root for a multiple root. A root functional is a linear functional that is defined on a polynomial ring and annuls the ideal of a system of polynomials. A bounded root functional is a functional that annuls d-th component of the ideal in some filtration in this ideal. It was constructed the operation of continuation of root functionals and the operation of reduction of polynomials modulo the ideal on the basis of the extension operation for bounded root functionals when the number of polynomials is equal to the number of variables and the ideal of polynomials is 0-dimensional. The extension operation has connection with the multivariate Bezoutian construction."}
{"category": "Math", "title": "Positivity for toric vector bundles", "abstract": "We show that an equivariant vector bundle on a complete toric variety is nef or ample if and only if its restriction to every invariant curve is nef or ample, respectively. Furthermore, we show that nef toric vector bundles have a nonvanishing global section at every point, and deduce that the underlying vector bundle is trivial if and only if its restriction to every invariant curve is trivial. We apply our methods and results to study, in particular, the vector bundles M_L that arise as the kernel of the evaluation map on sections of L, when L is an ample line bundle. We give examples of twists of such bundles that are ample but not globally generated."}
{"category": "Math", "title": "Realization of a certain class of semi-groups as value semi-groups of valuations", "abstract": "Given a well-ordered semi-group $\\Gamma$ with a minimal system of generators of ordinal type at most $\\omega n$ and of rational rank $r$, which satisfies a positivity and increasing condition, we construct a zero-dimensional valuation centered on the ring of polynomials with $r$ variables such that the semi-group of the values of the polynomial ring is equal to $\\Gamma$. The construction uses a generalization of Favre and Jonsson's version of MacLane's sequence of key-polynomials."}
{"category": "Math", "title": "Truth as value and duty: lessons of mathematics", "abstract": "I discuss some connotations of mathematical notion of \"truth\" in the context of humanistic discourse"}
{"category": "Math", "title": "Eigenfunctions of Dirac operators at the threshold energies", "abstract": "We show that the eigenspaces of the Dirac operator $H=\\alpha\\cdot (D - A(x)) + m \\beta $ at the threshold energies $\\pm m$ are coincide with the direct sum of the zero space and the kernel of the Weyl-Dirac operator $\\sigma\\cdot (D - A(x))$. Based on this result, we describe the asymptotic limits of the eigenfunctions of the Dirac operator corresponding to these threshold energies. Also, we discuss the set of vector potentials for which the kernels of $H\\mp m$ are non-trivial, i.e. ${Ker}(H\\mp m) \\not = \\{0 \\}$."}
{"category": "Math", "title": "On Multidimensional Pythagorean Numbers", "abstract": "To represent positive integers by regular patterns on a plane or in three-dimensional space may be traced back to the Pythagoreans. The aim of the present article is to explore the possibility of extending the representation framework for integers to spaces with more than three dimensions. Thus, taking up a definition of polygonal numbers given by Diophantus and by Nicomachus, and generalizing the Pythagorean concept of gnomon, one is led through quite elementary means to a single, unified definition of multidimensional number formations henceforth called hypersolids."}
{"category": "Math", "title": "Twisted diagrams and homotopy sheaves", "abstract": "Twisted diagrams are \"diagrams\" with components in different categories. Structure maps are defined using auxiliary data which consists of functors relating the various categories to each other. Prime examples of the construction are spectra (in the sense of homotopy theory) and quasi-coherent sheaves on schemes. We develop the basic theory of twisted diagrams, and establish various model structures (which are well-known in special cases). We also introduce a notion of homotopy sheaves, a collection of local data which is compatible up to weak equivalence, and study basic properties of such objects. These objects occur in nature; for example, the notion of an Omega-spectrum fits into this framework. The main purpose of the paper is to provide a convenient reference for model structures on twisted diagrams, and for the language of sheaves and homotopy sheaves as defined here."}
{"category": "Math", "title": "Crossed product of groups. Applications", "abstract": "We survey the extensions of a group by a group using crossed products instead of exact sequences of groups. The approach has various advantages, one of them being that the crossed product is an universal object. Several new applications are given, a general Schreier type theorem is proved and a few open problems are left."}
{"category": "Math", "title": "On the equi-normalizable deformations of singularities of complex plane curves", "abstract": "We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $\\delta$ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A natural invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in equi-normalizable families. We consider in details deformations of ordinary multiple points, the deformations of a singularity into the collections of ordinary multiple points and deformations of the type $x^p+y^{pk}$ into the collections of $A_k$'s."}
{"category": "Math", "title": "Generalized Stirling permutations, families of increasing trees and urn models", "abstract": "Bona [2007+] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [1978]. Recently, Janson [2008+] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of Bona and Janson, and relate them with certain families of generalized plane recursive trees, and also $(k+1)$-ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) Polya urn models using various methods."}
{"category": "Math", "title": "On the derived category of a regular toric scheme", "abstract": "Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting \"twisted\" diagram of modules satisfies a certain gluing condition, stating that the data is compatible with restriction to smaller open sets. In case X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category D(X) of quasi-coherent sheaves on X can be obtained from a category of twisted diagrams which do not necessarily satisfy any gluing condition by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we given an explicit construction of a finite set of weak generators for the derived category. For example, if X is projective n-space then D(X) is generated by n+1 successive twists of the structure sheaf; the present paper gives a new homotopy-theoretic proof of this classical result. The approach taken uses the language of model categories, and the machinery of Bousfield-Hirschhorn colocalisation. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets agree on intersections up to quasi-isomorphism only. In a second step it is shown that the homotopy category of homotopy sheaves is the derived category of X."}
{"category": "Math", "title": "Semidirect product decomposition of Coxeter groups", "abstract": "Let $(W,S)$ be a Coxeter system, let $S=I \\dot{\\cup} J$ be a partition of $S$ such that no element of $I$ is conjugate to an element of $J$, let $\\widetilde{J}$ be the set of $W_I$-conjugates of elements of $J$ and let $\\widetilde{W}$ be the subgroup of $W$ generated by $\\widetilde{J}$. We show that $W=\\widetilde{W} \\rtimes W_I$ and that $(\\widetilde{W},\\widetilde{J})$ is a Coxeter system."}
{"category": "Math", "title": "On the uniqueness of the infinite cluster of the vacant set of random interlacements", "abstract": "We consider the model of random interlacements on $\\mathbb{Z}^d$ introduced in Sznitman [Vacant set of random interlacements and percolation (2007) preprint]. For this model, we prove the uniqueness of the infinite component of the vacant set. As a consequence, we derive the continuity in $u$ of the probability that the origin belongs to the infinite component of the vacant set at level $u$ in the supercritical phase $u<u_*$."}
{"category": "Math", "title": "A theorem of Tits type for compact Kahler manifolds", "abstract": "We prove a theorem of Tits type about automorphism groups for compact Kahler manifolds, which has been conjectured in the paper [KOZ]."}
{"category": "Math", "title": "Mixed Volume Techniques for Embeddings of Laman Graphs", "abstract": "Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the viewpoint of Bernstein's Theorem. The focus of the paper is to provide the methods to study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints. While in most cases the resulting bounds are weaker than the best known bounds on the number of embeddings, for some classes of graphs the bounds are tight."}
{"category": "Math", "title": "Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations", "abstract": "We carry on the study of Fourier integral operators of H{\\\"o}rmander's type acting on the spaces $(\\mathcal{F}L^p)_{comp}$, $1\\leq p\\leq\\infty$, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank $r$ of the Hessian of the phase $\\Phi(x,\\eta)$ with respect to the space variables $x$. Indeed, we show that operators of order $m=-r|1/2-1/p|$ are bounded on $(\\mathcal{F}L^p)_{comp}$, if the mapping $x\\longmapsto\\nabla_x\\Phi(x,\\eta)$ is constant on the fibers, of codimension $r$, of an affine fibration."}
{"category": "Math", "title": "Reduced measures associated to parabolic problems", "abstract": "We study the existence and the properties of the reduced measures for the parabolic equations $\\partial_tu-\\Delta u+g(u)=0$ in $\\Omega\\times (0,\\infty)$ subject to the conditions ($P$): $u=0$ on $\\partial\\Omega\\times (0,\\infty)$, $u(x,0)=\\mu$ and ($P'$): $u=\\mu'$ on $\\partial\\Omega\\times (0,\\infty)$, $u(x,0)=0$ where $\\mu$ and $\\mu'$ are positive Radon measures and $g$ a continuous nondecreasing function"}
{"category": "Math", "title": "On rational normal curves in projective space", "abstract": "In this paper we consider a generalization of a well known result by Veronese about rational normal curves. More precisely, given a collection of linear spaces in $\\PP^n$ we study the existence of rational normal curves intersecting each component of the configuration maximally. We introduce different methods to show existence and non-existence of such curves. We also show how to apply these techniques to the study of defectivity of Segre-Veronese varieties."}
{"category": "Math", "title": "Convergence of Point Processes with Weakly Dependent Points", "abstract": "For each $n \\geq 1$, let $\\{X_{j,n}\\}_{1 \\leq j \\leq n}$ be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_n=\\sum_{j=1}^{n}\\delta_{X_{j,n}}$ to an infinitely divisible point process. From the point process convergence, we obtain the convergence in distribution of the partial sum sequence $S_n=\\sum_{j=1}^{n}X_{j,n}$ to an infinitely divisible random variable, whose L\\'{e}vy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model."}
{"category": "Math", "title": "Stability of projective Poincare and Picard bundles", "abstract": "Let $X$ be an irreducible smooth projective curve of genus $g\\ge3$ defined over the complex numbers and let ${\\mathcal M}_\\xi$ denote the moduli space of stable vector bundles on $X$ of rank $n$ and determinant $\\xi$, where $\\xi$ is a fixed line bundle of degree $d$. If $n$ and $d$ have a common divisor, there is no universal vector bundle on $X\\times {\\mathcal M}_\\xi$. We prove that there is a projective bundle on $X\\times {\\mathcal M}_\\xi$ with the property that its restriction to $X\\times\\{E\\}$ is isomorphic to $P(E)$ for all $E\\in\\mathcal{M}_\\xi$ and that this bundle (called the projective Poincar\\'e bundle) is stable with respect to any polarization; moreover its restriction to $\\{x\\}\\times\\mathcal{M}_\\xi$ is also stable for any $x\\in X$. We prove also stability results for bundles induced from the projective Poincar\\'e bundle by homomorphisms $\\text{PGL}(n)\\to H$ for any reductive $H$. We show further that there is a projective Picard bundle on a certain open subset $\\mathcal{M}'$ of $\\mathcal{M}_\\xi$ for any $d>n(g-1)$ and that this bundle is also stable. We obtain new results on the stability of the Picard bundle even when $n$ and $d$ are coprime."}
{"category": "Math", "title": "Poisson Cloning Model for Random Graphs", "abstract": "In the random graph $G(n,p)$ with $pn$ bounded, the degrees of the vertices are almost i.i.d Poisson random variables with mean $\\gl:= p(n-1)$. Motivated by this fact, we introduce the Poisson cloning model $G_{PC} (n,p)$ for random graphs in which the degrees are i.i.d Poisson random variables with mean $\\gl$. Then, we first establish a theorem that shows the new model is equivalent to the classical model $G(n,p)$ in an asymptotic sense. Next, we introduce a useful algorithm, called the cut-off line algorithm, to generate the random graph $G_{PC} (n,p)$. The Poisson cloning model $G_{PC}(n,p)$ equipped with the cut-off line algorithm enables us to very precisely analyze the sizes of the largest component and the $t$-core of $G(n,p)$. This new approach to the problems yields not only elegant proofs but also improved bounds that are essentially best possible. We also consider the Poisson cloning models for random hypergraphs and random $k$-SAT problems. Then, the $t$-core problem for random hypergraphs and the pure literal algorithm for random $k$-SAT problems are analyzed."}
{"category": "Math", "title": "La th\\'eorie des invariants des formes quadratiques ternaires revisit\\'ee", "abstract": "The simultaneous invariants of 2, 3, 4 and 5 ternary quadratic forms under the group $\\SL(3, {\\Bbb C})$ were given by several authors (P. Gordan, C. Ciamberlini, H.W. Turnbull, J.A Todd), utilizing the symbolic method. Using the Jordan algebra structure of the space of ternary quadratic forms, we give these invariants explicitely."}
{"category": "Math", "title": "Algebras with involution that become hyperbolic over the fonction field of a conic", "abstract": "We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra $Q$. We classify these algebras in degree~4 and give an example of such a division algebra with orthogonal involution of degree~8 that does not contain $Q$ with its canonical involution, even though it contains $Q$ and is totally decomposable into a tensor product of quaternion algebras with involution."}
{"category": "Math", "title": "Cohomologically hyperbolic endomorphisms of complex manifolds", "abstract": "We show that if a compact Kahler manifold X admits a cohomologically hyperbolic surjective endomorphism then its Kodaira dimension is non-positive. This gives an affirmative answer to a conjecture of Guedj in the holomorphic case. The main part of the paper is to determine the geometric structure and the fundamental groups (up to finite index) for those X of dimension 3."}
{"category": "Math", "title": "On the path density of a gradient field", "abstract": "We consider the problem of reliably finding filaments in point clouds. Realistic data sets often have numerous filaments of various sizes and shapes. Statistical techniques exist for finding one (or a few) filaments but these methods do not handle noisy data sets with many filaments. Other methods can be found in the astronomy literature but they do not have rigorous statistical guarantees. We propose the following method. Starting at each data point we construct the steepest ascent path along a kernel density estimator. We locate filaments by finding regions where these paths are highly concentrated. Formally, we define the density of these paths and we construct a consistent estimator of this path density."}
{"category": "Math", "title": "Stokes Theorem for Lipschitz forms on a smooth manifold", "abstract": "Stokes theorem holds for Lipschitz forms on a smooth manifold."}
{"category": "Math", "title": "The loci of abelian varieties with points of high multiplicity on the theta divisor", "abstract": "We study the loci of principally polarized abelian varieties with points of high multiplicity on the theta divisor. Using the heat equation and degeneration techniques, we relate these loci and their closures to each other, as well as to the singular set of the universal theta divisor. We obtain bounds on the dimensions of these loci and relations among their dimensions, and make further conjectures about their structure."}
{"category": "Math", "title": "L^2-Invariants of Finite Aspherical CW-Complexes", "abstract": "Let $X$ be a finite aspherical CW-complex whose fundamental group $\\pi_1(X)$ possesses a subnormal series $\\pi_1(X) \\rhd G_m \\rhd ... \\rhd G_0$ with a non-trivial elementary amenable group $G_0$. We investigate the $L^2$-invariants of the universal covering of such a CW-complex $X$. We show that the Novikov-Shubin invariants $\\alpha_n({\\tilde X})$ are positive. We further prove that the $L^2$-torsion $\\rho^{(2)}({\\tilde X})$ vanishes if $\\pi_1(X)$ has semi-integral determinant."}
{"category": "Math", "title": "Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces", "abstract": "We study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone."}
{"category": "Math", "title": "Phase space distribution of Gabor expansions", "abstract": "We present an example of a complete and minimal Gabor system consisting of time-frequency shifts of a Gaussian, localized at the coordinate axes in the time-frequency plane (phase space). Asymptotically, the number of time-frequency shifts contained in a disk centered at the origin is only 2/pi times the number of points from the von Neumann lattice found in the same disk. Requiring a certain regular distribution in phase space, we show that our system has minimal density among all complete and minimal systems of time-frequency shifts of a Gaussian."}
{"category": "Math", "title": "The scattering matrix and associated formulas in Hamiltonian mechanics", "abstract": "We survey the basic notions of scattering theory in Hamiltonian mechanics with a particular attention to the analogies with scattering theory in quantum mechanics. We discuss the scattering symplectomorphism, which is analogous to the scattering matrix. We prove identities which relate the Calabi invariant of the scattering symplectomorphism to the total time delay and the regularised phase space volume. These identities are analogous to the Birman-Krein formula and the Eisenbud-Wigner formula in quantum scattering theory."}
{"category": "Math", "title": "On a generalization of Christoffel words: epichristoffel words", "abstract": "Sturmian sequences are well-known as the ones having minimal complexity over a 2-letter alphabet. They are also the balanced sequences over a 2-letter alphabet and the sequences describing discrete lines. They are famous and have been extensively studied since the 18th century. One of the {extensions} of these sequences over a $k$-letter alphabet, with $k\\geq 3$, are the episturmian sequences, which generalizes a construction of Sturmian sequences using the palindromic closure operation. There exists a finite version of the Sturmian sequences called the Christoffel words. They are known since the works of Christoffel and have interested many mathematicians. In this paper, we introduce a generalization of Christoffel words for an alphabet with 3 letters or more, using the episturmian morphisms. We call them the {\\it epichristoffel words}. We define this new class of finite words and show how some of the properties of the Christoffel words can be generalized naturally or not for this class."}
{"category": "Math", "title": "Baxter permutations and plane bipolar orientations", "abstract": "We present a simple bijection between Baxter permutations of size $n$ and plane bipolar orientations with n edges. This bijection translates several classical parameters of permutations (number of ascents, right-to-left maxima, left-to-right minima...) into natural parameters of plane bipolar orientations (number of vertices, degree of the sink, degree of the source...), and has remarkable symmetry properties. By specializing it to Baxter permutations avoiding the pattern 2413, we obtain a bijection with non-separable planar maps. A further specialization yields a bijection between permutations avoiding 2413 and 3142 and series-parallel maps."}
{"category": "Math", "title": "A degree bound for globally generated vector bundles", "abstract": "We obtain a sharp bound on the degree of a globally generated vector bundle over a reduced irreducible projective variety defined over an algebraically closed field of characteristic zero. As an application, we obtain a Del Pezzo-Bertini type theorem on varieties of minimal degree for subvarieties of Grassmannians."}
{"category": "Math", "title": "Four manifolds with two-positive Ricci curvature", "abstract": "This paper has been withdrawn by the author, due to an error in the proof of Theorem 3.8."}
{"category": "Math", "title": "Free fields in skew fields", "abstract": "Building on the work of K. Chiba (J. Algebra 263 (2003), 75-87), we present sufficient conditions for the completion of a division ring with respect to the metric defined by a discrete valuation function to contain a free field, i.e. the universal field of fractions of a free associative algebra. Several applications to division rings generated by torsion-free nilpotent groups, skew Laurent series and related division rings are discussed."}
{"category": "Math", "title": "Transverse conformal Killing forms and a Gallot-Meyer Theorem for foliations", "abstract": "We study transverse conformal Killing forms on foliations and prove a Gallot-Meyer theorem for foliations. Moreover, we show that on a foliation with $C$-positive normal curvature, if there is a closed basic 1-form $\\phi$ such that $\\Delta_B\\phi=qC\\phi$, then the foliation is transversally isometric to the quotient of a $q$-sphere."}
{"category": "Math", "title": "Crossed products and cleft extensions for coquasi-Hopf algebras", "abstract": "The notion of crossed product by a coquasi-bialgebra H is introduced and studied. The resulting crossed product is an algebra in the monoidal category of right H-comodules. We give an interpretation of the crossed product as an action of a monoidal category. In particular, necessary and sufficient conditions for two crossed products to be equivalent are provided. Then, two structure theorems for coquasi Hopf modules are given. First, these are relative Hopf modules over the crossed product. Second, the category of coquasi-Hopf modules is trivial, namely equivalent to the category of modules over the starting associative algebra. In connection the crossed product, we recall the notion of a cleft extension over a coquasi-Hopf algebra. A Morita context of Hom spaces is constructed in order to explain these extensions, which are shown to be equivalent with crossed product with invertible cocycle. At the end, we give a complete description of all cleft extensions by the non-trivial coquasi-Hopf algebras of dimension two and three."}
{"category": "Math", "title": "An Orthogonal Test of the L-Functions Ratios Conjecture", "abstract": "We test the predictions of the L-functions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N --> oo through the primes or N=1 and k --> oo. We study the main and lower order terms in the 1-level density. We provide evidence for the Ratios Conjecture by computing and confirming its predictions up to a power savings in the family's cardinality, at least for test functions whose Fourier transforms are supported in (-2, 2). We do this both for the weighted and unweighted 1-level density (where in the weighted case we use the Petersson weights), thus showing that either formulation may be used. These two 1-level densities differ by a term of size 1 / log(k^2 N). Finally, we show that there is another way of extending the sums arising in the Ratios Conjecture, leading to a different answer (although the answer is such a lower order term that it is hopeless to observe which is correct)."}
{"category": "Math", "title": "Best subspace tensor approximations", "abstract": "In many applications such as data compression, imaging or genomic data analysis, it is important to approximate a given tensor by a tensor that is sparsely representable. For matrices, i.e. 2-tensors, such a representation can be obtained via the singular value decomposition which allows to compute the best rank $k$ approximations. For $t$-tensors with $t>2$ many generalizations of the singular value decomposition have been proposed to obtain low tensor rank decompositions. In this paper we will present a different approach which is based on best subspace approximations, which present an alternative generalization of the singular value decomposition to tensors."}
{"category": "Math", "title": "Unitary Braid Representations with Finite Image", "abstract": "We characterize unitary representations of braid groups $B_n$ of degree linear in $n$ and finite images of such representations of degree exponential in $n$."}
{"category": "Math", "title": "Chains of distributions, hierarchical Bayesian models and Benford's Law", "abstract": "Kossovsky recently conjectured that the distribution of leading digits of a chain of probability distributions converges to Benford's law as the length of the chain grows. We prove his conjecture in many cases, and provide an interpretation in terms of products of independent random variables and a central limit theorem. An interesting consequence is that in hierarchical Bayesian models priors tend to satisfy Benford's Law as the number of levels of the hierarchy increases, which allows us to develop some simple tests (based on Benford's law) to test proposed models. We give explicit formulas for the error terms as sums of Mellin transforms, which converges extremely rapidly as the number of terms in the chain grows. We may interpret our results as showing that certain Markov chain Monte Carlo processes are rapidly mixing to Benford's law."}
{"category": "Math", "title": "Some Bounds for ramification of $p^n$-torsion semi-stable representations", "abstract": "Let p be an odd prime, K a finite extension of Q_p, G=Gal(\\bar K/K) the Galois group and e=e(K/Q_p) the ramification index. Suppose T is a p^n torsion representation such that T is isomorphic to a quotient of two G-stable Z_p-lattices in a semi-stable representation with Hodge-Tate weights in {0,...,r}. We prove that there exists a constant \\mu explicitly depending on n, e and r such that the upper numbering ramification group G^{(\\mu)} acts on T trivially."}
{"category": "Math", "title": "Surface Homeomorphisms That Do Not Extend to Any Handlebody and the Johnson Filtration", "abstract": "We prove the existence of homeomorphisms of a closed, orientable surface of genus 3 or greater that do not extend to any handlebody bounded by the surface. We show that such homeomorphisms exist arbitrarily deep in the Johnson filtration of the mapping class group. The second and third terms of the Johnson filtration are the well-known Torelli group and Johnson subgroup, respectively. Richard Hain has obtained very similar results by different methods."}
{"category": "Math", "title": "Monotone Linear Relations: Maximality and Fitzpatrick Functions", "abstract": "We analyze and characterize maximal monotonicity of linear relations (set-valued operators with linear graphs). An important tool in our study are Fitzpatrick functions. The results obtained partially extend work on linear and at most single-valued operators by Phelps and Simons and by Bauschke, Borwein and Wang. Furthermore, a description of skew linear relations in terms of the Fitzpatrick family is obtained. We also answer one of Simons problems by showing that if a maximal monotone operator has a convex graph, then this graph must actually be affine."}
{"category": "Math", "title": "Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility", "abstract": "In this paper we characterize, in two different ways, the Newton polygons which are jacobian Newton polygons of a branch. These characterizations give in particular combinatorial criteria of irreducibility for complex series in two variables and necessary conditions which a complex curve has to satisfy in order to be the discriminant of a complex plane branch."}
{"category": "Math", "title": "Automorphisms of Coxeter groups and Lusztig's conjectures for Hecke algebras with unequal parameters", "abstract": "Let $(W,S)$ be a Coxeter system, let $G$ be a finite solvable group of automorphisms of $(W,S)$ and let $\\varphi$ be a weight function which is invariant under $G$. Let $\\varphi_G$ denote the weight function on $W^G$ obtained by restriction from $\\varphi$. The aim of this paper is to compare the ${\\mathbf{a}}$-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated to $(W,\\varphi)$ and to $(W^G,\\varphi_G)$, provided that Lusztig's Conjectures hold."}
{"category": "Math", "title": "On pluricanonical systems of algebraic varieties of general type", "abstract": "We extend Kollar's technique to look for an explicit function h(n) with phi_m birational onto its image for all integers $m\\geq h(n)$ and for all n-dimensional nonsingular projective varieties of general type."}
{"category": "Math", "title": "Adjoint bi-continuous semigroups and semigroups on the space of measures", "abstract": "For a given bi-continuous semigroup T on a Banach space X we define its adjoint on an appropriate closed subspace X^o of the norm dual X'. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology (X^o,X). An application is the following: For K a Polish space we consider operator semigroups on the space C(K) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(K) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(K) are precisely those that are adjoints of a bi-continuous semigroups on C(K). We also prove that the class of bi-continuous semigroups on C(K) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if K is not Polish space this is not the case."}
{"category": "Math", "title": "On Shintani's ray class invariant for totally real number fields", "abstract": "We introduce a ray class invariant $X(C)$ for a totally real field, following Shintani's work in the real quadratic case. We prove a factorization formula $X=X_1... X_n$ where each $X_i=X_i(C)$ corresponds to a real place. Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices. Finally, we describe the behavior of $X_i(C)$ when the signature of $C$ at a real place is changed. This last result is also interpreted into an interesting behavior of the derivative $L'(0,\\chi)$ of $L$-functions."}
{"category": "Math", "title": "Remarks on lines and minimal rational curves", "abstract": "We determine all of lines in the moduli space $M$ of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section."}
{"category": "Math", "title": "Exact Edgeworth expansion for a L\\'{e}vy process", "abstract": "The one dimensional distribution of a L\\'{e}vy process is not known in general even though its characteristic function is given by the famous L\\'{e}vy-Khinchine theorem. This article gives an exact series representation for the one dimensional distribution of a L\\'{e}vy process satisfying certain moment conditions. Moreover, this work clarifies an old result by Cram\\'{e}r on Edgeworth expansions for the distribution function of a L\\'{e}vy process."}
{"category": "Math", "title": "Universal L^p improving for averages along polynomial curves in low dimensions", "abstract": "We prove sharp $L^p-L^q$ estimates for averaging operators along general polynomial curves in two and three dimensions. These operators are translation-invariant, given by convolution with the so-called affine arclength measure of the curve and we obtain universal bounds over the class of curves given by polynomials of bounded degree. Our method relies on a geometric inequality for general vector polynomials together with a combinatorial argument due to M. Christ. Almost sharp Lorentz space estimates are obtained as well."}
{"category": "Math", "title": "On subexponentiality of the L\\'evy measure of the diffusion inverse local time; with applications to penalizations", "abstract": "For a recurrent linear diffusion on $\\R_+$ we study the asymptotics of the distribution of its local time at 0 as the time parameter tends to infinity. Under the assumption that the L\\'evy measure of the inverse local time is subexponential this distribution behaves asymtotically as a multiple of the L\\'evy measure. Using spectral representations we find the exact value of the multiple. For this we also need a result on the asymptotic behavior of the convolution of a subexponential distribution and an arbitrary distribution on $\\R_+.$ The exact knowledge of the asymptotic behavior of the distribution of the local time allows us to analyze the process derived via a penalization procedure with the local time. This result generalizes the penalizations obtained in Roynette, Vallois and Yor \\cite{rvyV} for Bessel processes."}
{"category": "Math", "title": "Configuration spaces of rings and wickets", "abstract": "The main result in this paper is that the space of all smooth links in Euclidean 3-space isotopic to the trivial link of n components has the same homotopy type as its finite-dimensional subspace consisting of configurations of n unlinked Euclidean circles (the \"rings\" in the title). There is also an analogous result for spaces of arcs in upper half-space, with circles replaced by semicircles (the \"wickets\" in the title). A key part of the proofs is a procedure for greatly reducing the complexity of tangled configurations of rings and wickets. This leads to simple methods for computing presentations for the fundamental groups of these spaces of rings and wickets as well as various interesting subspaces. The wicket spaces are also shown to be K(G,1)'s."}
{"category": "Math", "title": "Potential Polynomials and Motzkin Paths", "abstract": "A {\\em Motzkin path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\\mathbb{Z}\\times\\mathbb{Z}$ consisting of horizontal-steps $(1, 0)$, up-steps $(1,1)$, and down-steps $(1,-1)$, which never passes below the x-axis. A {\\em $u$-segment {\\rm (resp.} $h$-segment {\\rm)}} of a Motzkin path is a maximum sequence of consecutive up-steps ({\\rm resp.} horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: \"number of $u$-segments\" and \"number of $h$-segments\". The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided."}
{"category": "Math", "title": "Adaptive design and analysis of supercomputer experiments", "abstract": "Computer experiments are often performed to allow modeling of a response surface of a physical experiment that can be too costly or difficult to run except using a simulator. Running the experiment over a dense grid can be prohibitively expensive, yet running over a sparse design chosen in advance can result in obtaining insufficient information in parts of the space, particularly when the surface calls for a nonstationary model. We propose an approach that automatically explores the space while simultaneously fitting the response surface, using predictive uncertainty to guide subsequent experimental runs. The newly developed Bayesian treed Gaussian process is used as the surrogate model, and a fully Bayesian approach allows explicit measures of uncertainty. We develop an adaptive sequential design framework to cope with an asynchronous, random, agent--based supercomputing environment, by using a hybrid approach that melds optimal strategies from the statistics literature with flexible strategies from the active learning literature. The merits of this approach are borne out in several examples, including the motivating computational fluid dynamics simulation of a rocket booster."}
{"category": "Math", "title": "Analytic approximation of matrix functions in $L^p$", "abstract": "We consider the problem of approximation of matrix functions of class $L^p$ on the unit circle by matrix functions analytic in the unit disk in the norm of $L^p$, $2\\le p<\\be$. For an $m\\times n$ matrix function $\\Phi$ in $L^p$, we consider the Hankel operator $H_\\Phi:H^q(C^n)\\to H^2_-(C^m)$, $1/p+1/q=1/2$. It turns out that the space of $m\\times n$ matrix functions in $L^p$ splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If $\\Phi$ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of $H_\\Phi$. For weird matrix functions, to obtain the distance formula, we consider Hankel operators defined on spaces of matrix functions. We also describe the set of $p$-badly approximable matrix functions in terms of special factorizations and give a parametrization formula for all best analytic approximants in the norm of $L^p$. Finally, we introduce the notion of $p$-superoptimal approximation and prove the uniqueness of a $p$-superoptimal approximant for rational matrix functions."}
{"category": "Math", "title": "Differentiability of functions of contractions", "abstract": "In this paper we study differentiability properties of the map $T\\mapsto\\phi(T)$, where $\\phi$ is a given function in the disk-algebra and $T$ ranges over the set of contractions on Hilbert space. We obtain sharp conditions (in terms of Besov spaces) for differentiability and existence of higher derivatives. We also find explicit formulae for directional derivatives (and higher derivatives) in terms of double (and multiple) operator integrals with respect to semi-spectral measures."}
{"category": "Math", "title": "Conditioning on an extreme component: Model consistency with regular variation on cones", "abstract": "Multivariate extreme value theory assumes a multivariate domain of attraction condition for the distribution of a random vector. This necessitates that each component satisfies a marginal domain of attraction condition. An approximation of the joint distribution of a random vector obtained by conditioning on one of the components being extreme was developed by Heffernan and Tawn [12] and further studied by Heffernan and Resnick [11]. These papers left unresolved the consistency of different models obtained by conditioning on different components being extreme and we here provide clarification of this issue. We also clarify the relationship between these conditional distributions, multivariate extreme value theory and standard regular variation on cones of the form $[0,\\infty]\\times(0,\\infty]$."}
{"category": "Math", "title": "Characteristic varieties and logarithmic differential 1-forms", "abstract": "We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety $M$, see Theorem (3.1) and Corollaries (3.2) and (4.2). A logarithmic resonance variety is also considered in Proposition (4.5). As an application, we determine the first characteristic variety of the configuration space of $n$ distinct labeled points on an elliptic curve, see Proposition (5.1). Finally, for a logarithmic one form $\\alpha$ on $M$ we investigate the relation between the resonance degree of $\\alpha$ and the codimension of the zero set of $\\alpha$ on a good compactification of $M$, see Corollary (1.1). This question was inspired by the recent work by D. Cohen, G. Denham, M. Falk and A. Varchenko."}
{"category": "Math", "title": "A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling", "abstract": "We introduce a mixed discontinuous/continuous finite element pair for ocean modelling, with continuous quadratic pressure/layer depth and discontinuous velocity. We investigate the finite element pair applied to the linear shallow-water equations on an f-plane. The element pair has the property that all geostrophically balanced states which strongly satisfy the boundary conditions have discrete divergence equal to exactly zero and hence are exactly steady states of the discretised equations. This means that the finite element pair has excellent geostrophic balance properties. We illustrate these properties using numerical tests and provide convergence calculations which show that the discretisation has quadratic errors, indicating that the element pair is stable."}
{"category": "Math", "title": "A cell complex structure for the space of heteroclines for a semilinear parabolic equation", "abstract": "It is well known that for many semilinear parabolic equations there is a global attractor which has a cell complex structure with finite dimensional cells. Additionally, many semilinear parabolic equations have equilibria with finite dimensional unstable manifolds. In this article, these results are unified to show that for a specific parabolic equation on an unbounded domain, the space of heteroclinic orbits has a cell complex structure with finite dimensional cells. The result depends crucially on the choice of spatial dimension and the degree of the nonlinearity in the parabolic equation, and thereby requires some delicate treatment."}
{"category": "Math", "title": "Cobordism invariance of the family index", "abstract": "We give a K-theory proof of the invariance under cobordism of the family index. We consider elliptic pseudodifferential families on a continuous fibre bundle with smooth fibres over a compact base space B, and define a notion of cobordant families using K^1-groups on fibrations with boundary. We show that the index of two such families is the same using properties of the push-forward map in K-theory to reduce it to families on B x R^n."}
{"category": "Math", "title": "Khovanov homology of the 2-cable detects the unknot", "abstract": "We prove that the Khovanov homology of the 2-cable detects the unknot. A corollary is that Khovanov's categorification of the 2-colored Jones polynomial detects the unknot."}
{"category": "Math", "title": "Flipping and stabilizing Heegaard splittings", "abstract": "We show that the number of stabilizations needed to interchange the handlebodies of a Heegaard splitting of a closed 3-manifold by an isotopy is bounded below by the smaller of twice its genus or half its Hempel distance. This is a combinatorial version of a proof by Hass, Thompson and Thurston of a similar theorem, but with an explicit bound in terms of distance. We also show that in a 3-manifold with boundary, the stable genus of a Heegaard splitting and a boundary stabilization of itself is bounded below by the same value."}
{"category": "Math", "title": "Does Khovanov homology detect the unknot?", "abstract": "We determine a wide class of knots, which includes unknotting number one knots, within which Khovanov homology detects the unknot. A corollary is that the Khovanov homology of many satellite knots, including the Whitehead double, detects the unknot."}
{"category": "Math", "title": "On motivic cohomology with Z/l coefficients", "abstract": "In this paper we give a proof of the Bloch-Kato conjecture relating motivic cohomology and etale cohomology. It is a corrected version of the paper with the same title which posted earlier."}
{"category": "Math", "title": "Motives over simplicial schemes", "abstract": "In this paper we define the triangulated category of motives over a simplicial scheme. The morphisms between the Tate objects in this category compute the motivic cohomology of the underlying scheme. In the last section we consider the special case of \"embedded\" simplicial schemes, which correspond to the subsheaves of the constant sheaf and naturally appear in the descent problems for motivic cohomology such as the Bloch-Kato conjecture."}
{"category": "Math", "title": "Motivic Eilenberg-Maclane spaces", "abstract": "This paper is the second one in a series of papers about operations in motivic cohomology. Here we show that in the context of smooth schemes over a field of characteristic zero all the bi-stable operations can be obtained in the usual way from the motivic reduced powers and the Bockstein homomorphism."}
{"category": "Math", "title": "Simplicial radditive functors", "abstract": "The simplicial extension of any functor from Sets to Sets which commutes with directed colimits takes weak equivalences to weak equivalences. The goal of the present paper is construct a framework which can be used to proof results of this kind for a wide class of closed model categories and functors between those categories."}
{"category": "Math", "title": "Lectures on motivic cohomology 2000/2001 (written by Pierre Deligne)", "abstract": "These lecture notes cover four topics. There is a proof of the fact that the functors represented by the motivic Eilenberg-Maclane spaces on the motivic homotopy category coincide with the motivic cohomology defined in terms of the motivic complexes. There is a description of the equivariant motivic homotopy category for a finite flat group scheme (over a noetherian base) together with a new characterization of A^1-equivalences. There is a part where we introduce a class of sheaves called solid sheaves. Finally there is a part where we study functors of the form X -> X/G and X -> X^W and show that they preserve equivalences between term-wise ind-solid simplicial sheaves."}
{"category": "Math", "title": "Intersections of Apartments", "abstract": "We show that, if a building is endowed with its complete system of apartments, and if each panel is contained in at least four chambers, then the intersection of two apartments can be any convex subcomplex contained in an apartment. This combinatorial result is particularly interesting for lower dimensional convex subcomplexes of apartments, where we definitely need the assumption on the four chambers per panel in the building. The corresponding statement is not true anymore for arbitrary systems of apartments, and counter-examples for infinite convex subcomplexes exist for any type of buildings. However, when we restrict to finite convex subcomplexes, the above remains true for arbitrary systems of apartments if and only if every finite subset of chambers of the standard Coxeter complex is contained in the convex hull of two chambers."}
{"category": "Math", "title": "On some equivalent definitions of $\\rho$- Carleson measures on the unit ball", "abstract": "We give in this paper some equivalent definitions of the so called $\\rho$-Carleson measures when $\\rho(t)=(\\log(4/t))^p(\\log\\log(e^4/t))^q$, $0\\le p,q<\\infty$. As applications, we characterize the pointwise multipliers on $LMOA(\\mathbb S^n)$ and from this space to $BMOA(\\mathbb S^n)$. Boundedness of the Ces\\`aro type integral operators on $LMOA(\\mathbb S^n)$ and from $LMOA(\\mathbb S^n)$ to $BMOA(\\mathbb S^n)$ is considered as well."}
{"category": "Math", "title": "Toric ideals of normalized graph algebras", "abstract": "A graph-theoretic method, simpler than existing ones, is used to characterize the minimal set of monomial generators for the integral closure of any algebra of polynomials generated by quadratic monomials. The toric ideal of relations between these generators is generated by a set of binomials, defined graphically. The spectra of the original algebra and of its integral closure turn out to be canonically homeomorphic."}
{"category": "Math", "title": "Existence, Regularity, and Properties of Generalized Apparent Horizons", "abstract": "We prove a conjecture of Tom Ilmanen's and Hubert Bray's regarding the existence of the outermost generalized apparent horizon in an initial data set and that it is outer area minimizing."}
{"category": "Math", "title": "Stanley depth of complete intersection monomial ideals and upper-discrete partitions", "abstract": "Let $I$ be an $m$-generated complete intersection monomial ideal in $S=K[x_1,...,x_n]$. We show that the Stanley depth of $I$ is $n-\\floor{\\frac{m}{2}}$. We also study the upper-discrete structure for monomial ideals and prove that if $I$ is a squarefree monomial ideal minimally generated by 3 elements, then the Stanley depth of $I$ is $n-1$."}
{"category": "Math", "title": "Second order numerical scheme for motion of polygonal curves with constant area speed", "abstract": "We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of general constant area speed motion of polygons. Accuracy and efficiency of our numerical scheme are checked through numerical simulations for several polygonal motions such as motion by curvature and area-preserving advected flow etc."}
{"category": "Math", "title": "Queueing systems with pre-scheduled random arrivals", "abstract": "We consider a point process $i+\\xi_i$, where $i\\in \\bZ$ and the $\\xi_{i}$'s are i.i.d. random variables with variance $\\sigma^{2}$. This process, with a suitable rescaling of the distribution of $\\xi_i$'s, converges to the Poisson process in total variation for large $\\sigma$. We then study a simple queueing system with our process as arrival process, and we provide a complete analytical description of the system. Although the arrival process is very similar to the Poisson process, due to negative autocorrelation the resulting queue is very different from the Poisson case. We found interesting connections of this model with the statistical mechanics of Fermi particles. This model is motivated by air traffic systems."}
{"category": "Math", "title": "Deformations and automorphisms: a framework for globalizing local tangent and obstruction spaces", "abstract": "Building on Schlessinger's work, we define a framework for studying geometric deformation problems which allows us to systematize the relationship between the local and global tangent and obstruction spaces of a deformation problem. Starting from Schlessinger's functors of Artin rings, we proceed in two steps: we replace functors to sets by categories fibered in groupoids, allowing us to keep track of automorphisms, and we work with deformation problems naturally associated to a scheme X, and which naturally localize on X, so that we can formalize the local behavior. The first step is already carried out by Rim in the context of his homogeneous groupoids, but we develop the theory substantially further. In this setting, many statements known for a range of specific deformation problems can be proved in full generality, under very general stack-like hypotheses."}
{"category": "Math", "title": "Micro-local analysis in Fourier Lebesgue and modulation spaces. Part II", "abstract": "We consider different types of (local) products $f_1 f_2$ in Fourier Lebesgue spaces. Furthermore, we prove the existence of such products for other distributions satisfying appropriate wave-front properties. We also consider semi-linear equations of the form $$ \\qquad P(x,D)f = G(x,J_k f), $$ with appropriate polynomials $P $ and $G$. If the solution locally belongs to appropriate weighted Fourier Lebesgue space ${\\mathscr F}L^q_{(\\omega)} (\\rr d)$ and $P$ is non-characteristic at $(x_0,\\xi_0),$ then we prove that $(x_0,\\xi_0)\\not \\in WF_{{\\mathscr F}L^q_{(\\widetilde {\\omega})}} (f)$, where $\\widetilde{\\omega}$ depends on $\\omega$, $P$ and $G$."}
{"category": "Math", "title": "Problem-based learning and teacher training in mathematics", "abstract": "Problem-based learning (PBL) is a constructivist learner-centered instructional approach based on the analysis, resolution and discussion of a given problem. It can be applied to any subject, indeed it is especially useful for the teaching of mathematics. When compared to \"traditional\" teaching, the PBL approach requires increased responsibility for the teachers (in addition to the presentation of mathematical knowledge, they need to engage students in gathering information and using their knowledge to solve given problems). It thus become crucial that the future teachers become aware of its effectiveness. One of the main obstacle to this awareness lies usually on the fact that future teachers did not find this methodology in their own pre-service training. In this paper we will describe the attempt to introduce PBL in University courses so to have future maths teacher \"experience mathematics\" themselves."}
{"category": "Math", "title": "Postnikov towers, k-invariants and obstruction theory for DG categories", "abstract": "By inspiring ourselves in Drinfeld's DG quotient, we develop Postnikov towers, k-invariants and an obstruction theory for dg categories. As an application, we obtain the following `rigidification' theorem: let A be a homologically connective dg category and F:B -> H0(A) a dg functor to its homotopy category. If the family of obstruction classes vanishes, then a lift for F exists."}
{"category": "Math", "title": "Meromorphic almost rigid geometric structures", "abstract": "We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures."}
{"category": "Math", "title": "Algebraic continued fractions in F_q((T^{-1})) and recurrent sequences in F_q", "abstract": "There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called hyperquadratic, can sometimes be made explicit. Here we describe this expansion for a wide family of hyperquadratic power series in odd characteristic. This leads to consider interesting recurrent sequences in the finite base field when it is not a prime field."}
{"category": "Math", "title": "Involutions on surfaces with $p_g=q=1$", "abstract": "In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S/i$ is a non-ruled surface and such that the bicanonical map of $S$ is not composed with $i.$ A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type with $p_g=q=1$ and $K^2=7$ having birational bicanonical map is obtained."}
{"category": "Math", "title": "Q-factorial Gorenstein toric Fano varieties with large Picard number", "abstract": "Q-factorial Gorenstein toric Fano varieties X of dimension d with Picard number rho(X) correspond to simplicial reflexive d-polytopes with rho(X)+d vertices. Casagrande showed that any simplicial reflexive d-polytope has at most 3d vertices, if d is even, respectively, 3d-1, if d is odd. Moreover, it is known that equality for d even implies uniqueness up to unimodular equivalence. In this paper we completely classify all simplicial reflexive d-polytopes having 3d-1 vertices, corresponding to d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number 2d-1. For d even, there exist three such varieties, with two being singular, while for d odd (d > 1) there exist precisely two, both being nonsingular toric fiber bundles over the projective line. This generalizes recent work of the second author."}
{"category": "Math", "title": "Asymptotic Properties of an Estimator of the Drift Coefficients of Multidimensional Ornstein-Uhlenbeck Processes that are not Necessarily Stable", "abstract": "In this paper, we investigate the consistency and asymptotic efficiency of an estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are not necessarily stable. We consider all the cases. (1) The eigenvalues of $F$ are in the right half space (i.e., eigenvalues with positive real parts). In this case the process grows exponentially fast. (2) The eigenvalues of $F$ are on the left half space (i.e., the eigenvalues with negative or zero real parts). The process where all eigenvalues of $F$ have negative real parts is called a stable process and has a unique invariant (i.e., stationary) distribution. In this case the process does not grow. When the eigenvalues of $F$ have zero real parts (i.e., the case of zero eigenvalues and purely imaginary eigenvalues) the process grows polynomially fast. Considering (1) and (2) separately, we first show that an estimator, $\\hat{F}$, of $F$ is consistent. We then combine them to present results for the general Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic efficiency of the estimator."}
{"category": "Math", "title": "From the hyperbolic 24-cell to the cuboctahedron", "abstract": "We describe a family of 4-dimensional hyperbolic orbifolds, constructed by deforming an infinite volume orbifold obtained from the ideal, hyperbolic 24-cell by removing two walls. This family provides an infinite number of infinitesimally rigid, infinite covolume, geometrically finite discrete subgroups of the isometry group of hyperbolic 4-space. It also leads to finite covolume Coxeter groups which are the homomorphic image of the group of reflections in the hyperbolic 24-cell. The examples are constructed very explicitly, both from an algebraic and a geometric point of view. The method used can be viewed as a 4-dimensional, but infinite volume, analog of 3-dimensional hyperbolic Dehn filling."}
{"category": "Math", "title": "Two dimensional adelic analysis and cuspidal automorphic representations of GL(2)", "abstract": "Two dimensional adelic objects were introduced by I. Fesenko in his study of the Hasse zeta function associated to a regular model $\\mathcal E$ of the elliptic curve $E$. The Hasse-Weil $L$-function $L(E,s)$ of $E$ appears in the denominator of the Hasse zeta function of $\\mathcal E$. The two dimensional adelic analysis predicts that the integrand $h$ of the boundary term of the two dimensional zeta integral attached to $\\mathcal E$ is mean-periodic. The mean-periodicity of $h$ implies the meromorphic continuation and the functional equation of $L(E,s)$. On the other hand, if $E$ is modular, several nice analytic properties of $L(E,s)$, in particular the analytic continuation and the functional equation, are obtained by the theory of the cuspical automorphic representation of GL(2) over the ordinary ring of adele (one dimensional adelic object). In this article we try to relate the theory of two dimensional adelic object to the theory of cuspidal automorphic representation of GL(2) over the one dimensional adelic object, under the assumption that $E$ is modular. Roughly speaking, they are dual each other."}
{"category": "Math", "title": "On Upper Bounds for the Tail Distribution of Geometric Sums of Subexponential Random Variables", "abstract": "The approach used by Kalashnikov and Tsitsiashvili for constructing upper bounds for the tail distribution of a geometric sum with subexponential summands is reconsidered. By expressing the problem in a more probabilistic light, several improvements and one correction are made, which enables the constructed bound to be significantly tighter. Several examples are given, showing how to implement the theoretical result."}
{"category": "Math", "title": "Optimal conditions for $L^\\infty$-regularity and a priori estimates for elliptic systems, I: two components", "abstract": "In this paper we present a new bootstrap procedure for elliptic systems with two unknown functions. Combining with the $L^p$-$L^q$-estimates, it yields the optimal $L^\\infty$-regularity conditions for the three well-known types of weak solutions: $H_0^1$-solutions, $L^1$-solutions and $L^1_\\delta$-solutions. Thanks to the linear theory in $L^p_\\delta(\\Omega)$, it also yields the optimal conditions for a priori estimates for $L^1_\\delta$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems."}
{"category": "Math", "title": "Optimal conditions for $L^\\infty$-regularity and a priori estimates for elliptic systems, II: $n(\\geq 3)$ components", "abstract": "In this paper, we present a bootstrap procedure for general elliptic systems with $n(\\geq 3)$ components. Combining with the $L^p$-$L^q$-estimates, it yields the optimal $L^\\infty$-regularity conditions for the three well-known types of weak solutions: $H_0^1$-solutions, $L^1$-solutions and $L^1_\\delta$-solutions. Thanks to the linear theory in $L^p_\\delta(\\Omega)$, it also yields the optimal conditions for a priori estimates for $L^1_\\delta$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems."}
{"category": "Math", "title": "Convex Bodies Associated to Linear Series", "abstract": "In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens the door to a number of extensions. The purpose of this paper is to initiate a systematic development of the theory, and to give a number of applications and examples."}
{"category": "Math", "title": "Prym-Tyurin varieties via Hecke algebras", "abstract": "Let $G$ denote a finite group and $\\pi: Z \\to Y$ a Galois covering of smooth projective curves with Galois group $G$. For every subgroup $H$ of $G$ there is a canonical action of the corresponding Hecke algebra $\\mathbb{Q}[H \\backslash G/H]$ on the Jacobian of the curve $X = Z/H$. To each rational irreducible representation $\\mathcal{W}$ of $G$ we associate an idempotent in the Hecke algebra, which induces a correspondence of the curve $X$ and thus an abelian subvariety $P$ of the Jacobian $JX$. We give sufficient conditions on $\\mathcal{W}$, $H$, and the action of $G$ on $Z$, which imply $P$ to be a Prym-Tyurin variety. We obtain many new families of Prym-Tyurin varieties of arbitrary exponent in this way."}
{"category": "Math", "title": "Local symplectic algebra of quasi-homogeneous curves", "abstract": "We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of an analytic curve is a finite dimensional vector space. We also show that the action of local diffeomorphisms preserving the curve on this vector space is determined by the infinitesimal action of liftable vector fields. We apply these results to obtain the complete symplectic classification of curves with the semigroups (3,4,5), (3,5,7), (3,7,8)."}
{"category": "Math", "title": "On a conjecture of Kottwitz and Rapoport", "abstract": "We prove a conjecture of Kottwitz and Rapoport which implies a converse to Mazur's Inequality for all split and quasi-split (connected) reductive groups. These results are related to the non-emptiness of certain affine Deligne-Lusztig varieties."}
{"category": "Math", "title": "Unstable motivic homotopy categories in Nisnevich and cdh-topologies", "abstract": "The motivic homotopy categories can be defined with respect to different topologies and different underlying categories of schemes. For a number of reasons (mainly because of the Gluing Theorem) the motivic homotopy category built out of smooth schemes with respect to the Nisnevich topology plays a distinguished role but in some cases it is very desirable to be able to work with all schemes instead of the smooth ones. In this paper we prove that, under the resolution of singularities assumption, the unstable motivic homotopy category of all schemes over a field with respect to the cdh-topology is almost equivalent to the unstable motivic homotopy category of smooth schemes over the same field with respect to the Nisnevich topology. In order to do it we show that the standard cd-topologies on the category of Noetherian schemes, including the cdh-topology, satisfy certain conditions which allows one to use the generalized version of the Brown-Gersten approach to the homotopy theory of simplicial sheaves."}
{"category": "Math", "title": "Homotopy theory of simplicial presheaves in completely decomposable topologies", "abstract": "There are two approaches to the homotopy theory of simplicial (pre-)sheaves. One developed by Joyal and Jardine works for all sites but produces a model structure which is not finitely generated even in the case of sheaves on a Noetherian topological space. The other one developed by Brown and Gersten gives a nice model structure for sheaves on a Noetherian space of finite dimension but does not extend to all sites. In this paper we define a class of sites for which a generalized version of the Brown-Gersten approach works."}
{"category": "Math", "title": "Distance Expanding Random Mappings, Thermodynamic Formalism, Gibbs Measures, and Fractal Geometry", "abstract": "In this paper we define distance expanding random dynamical systems. We develop the appropriate thermodynamic formalism of such systems. We obtain in particular the existence and uniqueness of invariant Gibbs states, the appropriate pressure function and exponentially fast convergence of iterates of Perron-Frobenius operators resulting, in particular, in an exponential decay of correlations. We also obtain the formula for the derivative of the expected value of the pressure function. Next, we define and investigate in detail conformal random expanding repellers. Applying the developed machinery of thermodynamic formalism we prove a version of Bowen's formula which identifies the Hausdorff dimension $h$ of almost all fibers with the only zero of expected value of the pressure function. We then turn to more refined fractal properties by, firstly, showing that the multifractal formalism of the Gibbs states is valid and, secondly, that the $h$--Hausdorff measure vanishes while the corresponding packing measure is infinite provided the system is not quasi-deterministic."}
{"category": "Math", "title": "The maximal number of singular points on log del Pezzo surfaces", "abstract": "We prove that a del Pezzo surface with Picard number one has at most four singular points."}
{"category": "Math", "title": "Mean Curvature Motion of Graphs with Constant Contact Angle and Moving Boundaries", "abstract": "We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\\mathbb{R}^n$, intersecting $\\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary, and derive a continuation criterion based on the second fundamental form. If the initial graph is concave, we show this is preserved, and that the solution exists only for finite time. This corresponds to a symmetric version of triple junction motion of hypersurfaces by mean curvature, with constant angles at the junction."}
{"category": "Math", "title": "A Mathematical Approach to the Plato's Problem", "abstract": "Maybe the first inverse problem presented in the history of the occidental thought is described in the book Republic, written by Plato. The problem is posed in the Book VII in a text known as the Allegory of the Cave. That text motivated us to formulate a simple mathematical model that simulates, in a sense, the situation of the persons described in that problem."}
{"category": "Math", "title": "A new old class of maximal monotone operators", "abstract": "In a recent paper in Journal of Convex Analysis the authors studied, in non-reflexive Banach spaces, a class of maximal monotone operators, characterized by the existence of a function in Fitzpatrick's family of the operator which conjugate is above the duality product. This property was used to prove that such operators satisfies a restricted version of Brondsted-Rockafellar property. In this work we will prove that if a single Fitzpatrick function of a maximal monotone operator has a conjugate above the duality product, then all Fitzpatrick function of the operator have a conjugate above the duality product. As a consequence, the family of maximal monotone operators with this property is just the class NI, previously defined and studied by Simons. We will also prove that an auxiliary condition used by the authors to prove the restricted Brondsted-Rockafellar property is equivalent to the assumption of the conjugate of the Fitzpatrick function to majorize the duality product."}
{"category": "Math", "title": "Maximum likelihood estimation of cloud height from multi-angle satellite imagery", "abstract": "We develop a new estimation technique for recovering depth-of-field from multiple stereo images. Depth-of-field is estimated by determining the shift in image location resulting from different camera viewpoints. When this shift is not divisible by pixel width, the multiple stereo images can be combined to form a super-resolution image. By modeling this super-resolution image as a realization of a random field, one can view the recovery of depth as a likelihood estimation problem. We apply these modeling techniques to the recovery of cloud height from multiple viewing angles provided by the MISR instrument on the Terra Satellite. Our efforts are focused on a two layer cloud ensemble where both layers are relatively planar, the bottom layer is optically thick and textured, and the top layer is optically thin. Our results demonstrate that with relative ease, we get comparable estimates to the M2 stereo matcher which is the same algorithm used in the current MISR standard product (details can be found in [IEEE Transactions on Geoscience and Remote Sensing 40 (2002) 1547--1559]). Moreover, our techniques provide the possibility of modeling all of the MISR data in a unified way for cloud height estimation. Research is underway to extend this framework for fast, quality global estimates of cloud height."}
{"category": "Math", "title": "Maximal monotone operators with a unique extension to the bidual", "abstract": "We present a new sufficient condition under which a maximal monotone operator $T:X\\tos X^*$ admits a unique maximal monotone extension to the bidual $\\widetilde T:X^{**} \\rightrightarrows X^*$. For non-linear operators this condition is equivalent to uniqueness of the extension. The class of maximal monotone operators which satisfy this new condition includes class of Gossez type D maximal monotone operators, previously defined and studied by J.-P. Gossez, and all maximal monotone operators of this new class satisfies a restricted version of Brondsted-Rockafellar condition. The central tool in our approach is the $\\mathcal{S}$-function defined and studied by Burachik and Svaiter in 2000 \\cite{BuSvSet02}(submission date, July 2000). For a generic operator, this function is the supremum of all convex lower semicontinuous functions which are majorized by the duality product in the graph of the operator. We also prove in this work that if the graph of a maximal monotone operator is convex, then this graph is an affine linear subspace."}
{"category": "Math", "title": "On the surjectivity properties of perturbations of maximal monotone operators in non-reflexive Banach spaces", "abstract": "We are concerned with surjectivity of perturbations of maximal monotone operators in non-reflexive Banach spaces. While in a reflexive setting, a classical surjectivity result due to Rockafellar gives a necessary and sufficient condition to maximal monotonicity, in a non-reflexive space we characterize maximality using a ``enlarged'' version of the duality mapping, introduced previously by Gossez."}
{"category": "Math", "title": "Bicategory of entwinings", "abstract": "We define a bicategory in which the 0-cells are the entwinings over variable rings. The 1-cells are triples of a bimodule and two maps of bimodules which satisfy an additional hexagon, two pentagons and two (co)unit triangles; and the 2-cells are the maps of bimodules satisfying two simple compatibilities. The operation of getting the \"composed coring\" from a given entwining, is promoted here to a canonical morphism of bicategories from a bicategory of entwinings to the Street's bicategory of corings."}
{"category": "Math", "title": "On the first passage time for Brownian motion subordinated by a Levy process", "abstract": "This paper considers the class of L\\'evy processes that can be written as a Brownian motion time changed by an independent L\\'evy subordinator. Examples in this class include the variance gamma model, the normal inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that standard first passage time is the almost sure limit of iterations of first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are lead to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms."}
{"category": "Math", "title": "Regularity and Segre-Veronese embeddings", "abstract": "This paper studies the regularity of certain coherent sheaves that arise naturally from Segre-Veronese embeddings of a product of projective spaces. We give an explicit formula for the regularity of these sheaves and show that their regularity is subadditive. We then apply our results to study the Tate resolutions of these sheaves."}
{"category": "Math", "title": "The perverse filtration and the Lefschetz Hyperplane Theorem", "abstract": "We describe the perverse filtration in cohomology using the Lefschetz Hyperplane Theorem."}
{"category": "Math", "title": "Reversible linear differential equations", "abstract": "Let $\\nabla$ be a meromorphic connection on a vector bundle over a compact Riemann surface $\\Gamma$. An automorphism $\\sigma:\\Gamma\\to\\Gamma$ is called a symmetry of $\\nabla$ if the pull-back bundle and the pull-back connection can be identified with $\\nabla$. We study the symmetries by means of what we call the Fano Group; and, under the hypothesis that $\\nabla$ has a unimodular reductive Galois group, we relate the differential Galois group, the Fano group and the symmetries by means of an exact sequence."}
{"category": "Math", "title": "On the sl(2) foam cohomology computations", "abstract": "We show how to use Bar-Natan's `divide and conquer' approach to computations to efficiently compute the universal sl(2) dotted foam cohomology groups, even for big knots and links. We also describe a purely topological version of the sl(2) foam theory, in the sense that no dots are needed on foams."}
{"category": "Math", "title": "Labeled Trees and Localized Automorphisms of the Cuntz Algebras", "abstract": "We initiate a detailed and systematic study of automorphisms of the Cuntz algebras $\\O_n$ which preserve both the diagonal and the core $UHF$-subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of ${\\rm Aut}(\\O_n)$ and leads to numerous new examples. In particular, we completely classify all such automorphisms of ${\\mathcal O}_2$ for the permutation unitaries in $\\otimes^4 M_2$. We show that the subgroup of ${\\rm Out}(\\O_2)$ generated by these automorphisms contains a copy of the infinite dihedral group ${\\mathbb Z} \\rtimes {\\mathbb Z}_2$."}
{"category": "Math", "title": "Computing the Jones index of quadratic permutation endomorphisms of O_2", "abstract": "We compute the index of the inclusions of type $III_{1/2}$ factors arising from endomorphisms of the Cuntz algebra ${\\mathcal O}_2$ associated to the rank-two permutation matrices."}
{"category": "Math", "title": "A note on closed isometric embeddings", "abstract": "This note is about a little extension of Nash's embedding theorem in the case of complete manifolds."}
{"category": "Math", "title": "Numerical Computations for Backward Doubly SDEs and SPDEs", "abstract": "In this paper we present two numerical schemes of approximating solutions of backward doubly stochastic differential equations (BDSDEs for short). We give a method to discretize a BDSDE. And we also give the proof of the convergence of these two kinds of solutions for BDSDEs respectively. We give a sample of computation of BDSDEs."}
{"category": "Math", "title": "Global regularity of wave maps III. Large energy from $\\R^{1+2}$ to hyperbolic spaces", "abstract": "We show that wave maps $\\phi$ from two-dimensional Minkowski space $\\R^{1+2}$ to hyperbolic spaces $\\H^m$ are globally smooth in time if the initial data is smooth, conditionally on some reasonable claims concerning the local theory of such wave maps, as well as the self-similar and travelling (or stationary solutions); we will address these claims in the sequels \\cite{tao:heatwave2}, \\cite{tao:heatwave3}, \\cite{tao:heatwave4} to this paper. Following recent work in critical dispersive equations, the strategy is to reduce matters to the study of an \\emph{almost periodic} maximal Cauchy development in the energy class. We then repeatedly analyse the stress-energy tensor of this development (as in \\cite{tao:forges}) to extract either a self-similar, travelling, or degenerate non-trivial energy class solution to the wave maps equation. We will then rule out such solutions in the sequels to this paper, establishing the desired global regularity result for wave maps."}
{"category": "Math", "title": "Backward SDEs with constrained jumps and quasi-variational inequalities", "abstract": "We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems. As a direct consequence, this suggests a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs."}
{"category": "Math", "title": "Minimizing the Number of Tiles in a Tiled Rectangle", "abstract": "In this paper, we prove that if a finite number of rectangles, every of which has at least one integer side, perfectly tile a big rectangle then there exists a strategy which reduces the number of these tiles (rectangles) without violating the condition on the borders of the tiles. Consequently this strategy leads to yet another solution to the famous rectangle tiling theorem."}
{"category": "Math", "title": "Averages of Euler products, distribution of singular series and the ubiquity of Poisson distribution", "abstract": "We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the $k$-tuple conjecture and more general problems of polynomial representation of primes. We show that the singular series for the $k$-tuple conjecture have a limiting distribution when taken over $k$-tuples with (distinct) entries of growing size, and observe that its moments have a curious symmetry property. We also give conditional arguments that would imply that the number of twin primes (or more general polynomial prime patterns) in suitable short intervals are asymptotically Poisson distributed."}
{"category": "Math", "title": "Les deux quadrangulations infinies uniformes ont m\\^eme loi", "abstract": "We prove that the uniform infinite random quadrangulations introduced respectively by Chassaing-Durhuus and Krikun have the same distribution."}
{"category": "Math", "title": "A point counting algorithm using cohomology with compact support", "abstract": "We describe an algorithm to count the number of rational points of an hyperelliptic curve defined over a finite field of odd characteristic which is based upon the computation of the action of the Frobenius morphism on a basis of the Monsky-Washnitzer cohomology with compact support. This algorithm follows the vein of a systematic exploration of potential applications of cohomology theories to point counting. Our algorithm decomposes in two steps. A first step which consists in the computation of a basis of the cohomology and then a second step to obtain a representation of the Frobenius morphism. We achieve a $\\tilde{O}(g^4 n^{3})$ worst case time complexity and $O(g^3 n^3)$ memory complexity where $g$ is the genus of the curve and $n$ is the absolute degree of its base field. We give a detailed complexity analysis of the algorithm as well as a proof of correctness."}
{"category": "Math", "title": "Poisson (co)homology of truncated polynomial algebras in two variables", "abstract": "We study the Poisson (co)homology of the algebra of truncated polynomials in two variables viewed as the semi-classical limit of a quantum complete intersection studied by Bergh and Erdmann. We show in particular that the Poisson cohomology ring of such a Poisson algebra is isomorphic to the Hochschild cohomology ring of the corresponding quantum complete intersection."}
{"category": "Math", "title": "Restricted centrosymmetric permutations", "abstract": "The paper was withdrawn because of its significant overlap with a paper appeared recently."}
{"category": "Math", "title": "Denseness of certain smooth L\\'evy functionals in $\\DD_{1,2}$", "abstract": "The Malliavin derivative for a L\\'evy process $(X_t)$ can be defined on the space $\\DD_{1,2}$ using a chaos expansion or in the case of a pure jump process also via an increment quotient operator \\cite{sole-utzet-vives}. In this paper we define the Malliavin derivative operator $\\D$ on the class $\\mathcal{S}$ of smooth random variables $f(X_{t_1}, ..., X_{t_n}),$ where $f$ is a smooth function with compact support. We show that the closure of $L_2(\\Om) \\supseteq \\mathcal{S} \\stackrel{\\D}{\\to} L_2(\\m\\otimes \\mass)$ yields to the space $\\DD_{1,2}.$ As an application we conclude that Lipschitz functions map from $\\DD_{1,2}$ into $\\DD_{1,2}.$"}
{"category": "Math", "title": "The crossing number of composite knots", "abstract": "It is a very old conjecture that the crossing number of knots is additive under connected sum. In other words, if K#K' is the connected sum of knots K and K', then does the equality c(K#K') = c(K) + c(K') hold? We prove that c(K#K') is at most c(K) + c(K') and at least (c(K) + c(K'))/152."}
{"category": "Math", "title": "Subspaces with a common complement in a Banach space", "abstract": "We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement coincide with those pairs which are isomorphic to a pair of graphs of bounded linear operators between two other Banach spaces. (2) The pairs of subspaces of a Banach space X with a common complement coincide with those pairs for which there exists an involution S on X exchanging the two subspaces, such that I+S is bounded from below on their union. Moreover we show that, in a separable Hilbert space, the only pairs of subspaces with a common complement are those which are either equivalently positioned or not completely asymptotic to one another. We also obtain characterizations for the existence of a common complement for subspaces with closed sum."}
{"category": "Math", "title": "Generalizations of the Lax-Milgram theorem", "abstract": "We prove a linear and a nonlinear generalization of the Lax-Milgram theorem. In particular we give sufficient conditions for a real-valued function defined on the product of a reflexive Banach space and a normed space to represent all bounded linear functionals of the latter. We also give two applications to singular differential equations."}
{"category": "Math", "title": "Astala's Conjecture on Distortion of Hausdorff Measures under Quasiconformal Maps in the Plane", "abstract": "Let E be a compact set in the plane, g be a K-quasiconformal map, and let 0<t<2. Then H^t (E) = 0 implies H^{t'} (g E) = 0, for t'=[2Kt]/[2+(K-1)t]. This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala in his celebrated paper on area distortion for quasiconformal maps and answers in the positive a Conjecture of K. Astala in op. cit."}
{"category": "Math", "title": "Cohomological tautness for Riemannian foliations", "abstract": "In this paper we present some new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterization to a class of foliations which includes the foliated strata of any singular Riemannian foliation of a closed manifold."}
{"category": "Math", "title": "On the varieties of representations and characters of a family of one-relator subgroups. Their irreducible components", "abstract": "Let us consider the group $G = < x,y \\mid x^m = y^n>$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ and the character variety $X(G)$ in $SL(2,\\C)$ of the group $G$,obtaining by elementary methods an explicit primary decomposition of the ideal corresponding to $X(G)$ in the coordinates $X=t_x$, $Y=t_y$ and $Z=t_{xy}$. As an easy consequence, a formula for computing the number of irreducible components of $X(G)$ as a function of $m$ and $n$ is given. We provide a combinatorial description of $X(G)$ and we prove that in most cases it is possible to recover $(m,n)$ from the combinatorial structure of $X(G)$. Finally we compute the number of irreducible components of $R(G)$ and study the behavior of the projection $t:R(G)\\longrightarrow X(G)$."}
{"category": "Math", "title": "Hilbert space structure and positive operators", "abstract": "Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the non-symmetric case."}
{"category": "Math", "title": "The Schur Algorithm in Terms of System Realization", "abstract": "The main goal of this paper is to demonstrate the usefulness of certain ideas from System Theory in the study of problems from complex analysis. With this paper, we also aim to encourage analysts, who might not be familiar with System Theory, colligations or operator models to take a closer look at these topics. For this reason, we present a short introduction to the necessary background. The method of system realizations of analytic functions often provides new insights into and interpretations of results relating to the objects under consideration. In this paper we will use a well-studied topic from classical analysis as an example. More precisely, we will look at the classical Schur algorithm from the perspective of System Theory. We will confine our considerations to rational inner functions. This will allow us to avoid questions involving limits and will enable us to concentrate on the algebraic aspects of the problem at hand. Given a non-negative integer (n), we describe all system realizations of a given rational inner function of degree (n) in terms of an appropriately constructed equivalence relation in the set of all unitary ((n + 1) \\times (n + 1) )-matrices. The concept of Redheffer coupling of colligations gives us the possibility to choose a particular representative from each equivalence class. The Schur algorithm for a rational inner function is, consequently, described in terms of the state space representation."}
{"category": "Math", "title": "Strichartz estimates for the wave equation on manifolds with boundary", "abstract": "We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcricital case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions."}
{"category": "Math", "title": "Affine interval exchange maps with a wandering interval", "abstract": "For almost all interval exchange maps T_0, with combinatorics of genus g>=2, we construct affine interval exchange maps T which are semi-conjugate to T_0 and have a wandering interval."}
{"category": "Math", "title": "A first-countable non-remainder of H", "abstract": "We give a (consistent) example of a first-countable continuum that is not a remainder of the real line."}
{"category": "Math", "title": "Lattice Homomorphisms between Sobolev Spaces", "abstract": "We show that every vector lattice homomorphism $T$ between Sobolev spaces can be represented by a composition and a multiplication, that is, $T$ is of the form $Tu(x)=u(h(x))g(x)$ for quasi every/almost every $x$ and all $u$."}
{"category": "Math", "title": "Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold", "abstract": "We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=\\Delta (u^{m}/m)={\\rm div} (u^{m-1}\\nabla u)$ posed for $x\\in\\RR^d$, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \\cite{BBDGV} since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace-Beltrami operator of a suitable Riemannian Manifold $(\\RR^d,{\\bf g})$, with a metric ${\\bf g}$ which is conformal to the standard $\\RR^d$ metric. Studying the pointwise heat kernel behaviour allows to prove {suitable Gagliardo-Nirenberg} inequalities associated to the generator. Such inequalities in turn allow to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker--Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of $m$."}
{"category": "Math", "title": "Sobolev spaces with respect to measures in curves and zeros of Sobolev orthogonal polynomials", "abstract": "In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces."}
{"category": "Math", "title": "Distortions of the Helicoid", "abstract": "Colding and Minicozzi have shown that an embedded minimal disk $0\\in\\Sigma\\subset B_R$ in $\\Real^3$ with large curvature at 0 looks like a helicoid on the scale of $R$. Near 0, this can be sharpened: on the scale of $|A|^{-1}(0)$, $\\Sigma$ is close, in a Lipschitz sense, to a piece of a helicoid. We use surfaces constructed by Colding and Minicozzi to see this description cannot hold on the scale $R$."}
{"category": "Math", "title": "Limits of families of Brieskorn lattices and compactified classifying spaces", "abstract": "We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Brieskorn lattices and prove that its pure polarized part carries a natural hermitian structure and that the induced distance makes it into a complete metric space."}
{"category": "Math", "title": "Prym-Tyurin varieties using self-products of groups", "abstract": "Given Prym-Tyurin varieties of exponent $q$ with respect to a finite group $G$, a subgroup $H$ and a set of rational irreducible representations of $G$ satisfying some additional properties, we construct a Prym-Tyurin variety of exponent $[G:H]q$ in a natural way. We study an example of this result, starting from the dihedral group $\\mathbf{D}_p$ for any odd prime $p$. This generalizes the construction of arXiv:math/0412103v2[math.AG] for $p=3$. Finally, we compute the isogeny decomposition of the Jacobian of the curve underlying the above mentioned example."}
{"category": "Math", "title": "Products of Jacobians as Prym-Tyurin varieties", "abstract": "Let $X_1, ..., X_m$ denote smooth projective curves of genus $g_i \\geq 2$ over an algebraically closed field of characteristic 0 and let $n$ denote any integer at least equal to $1+\\max_{i=1}^m g_i$. We show that the product $JX_1 \\times ... \\times JX_m$ of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent $n^{m-1}$. This exponent is considerably smaller than the exponent of the structure of a Prym-Tyurin variety known to exist for an arbitrary principally polarized abelian variety. Moreover it is given by explicit correspondences."}
{"category": "Math", "title": "Relatively spectral morphisms and applications to K-theory", "abstract": "Spectral morphisms between Banach algebras are useful for comparing their K-theory and their \"noncommutative dimensions\" as expressed by various notions of stable ranks. In practice, one often encounters situations where the spectral information is only known over a dense subalgebra. We investigate such relatively spectral morphisms. We prove a relative version of the Density Theorem regarding isomorphism in K-theory. We also solve Swan's problem for the connected stable rank, in fact for an entire hierarchy of higher connected stable ranks that we introduce."}
{"category": "Math", "title": "The Equivalence of Two Graph Polynomials and a Symmetric Function", "abstract": "The U-polynomial, the polychromate and the symmetric function generalization of the Tutte polynomial due to Stanley are known to be equivalent in the sense that the coefficients of any one of them can be obtained as a function of the coefficients of any other. The definition of each of these functions suggests a natural way in which to generalize them which also captures Tutte's universal V-functions as a specialization. We show that the equivalence remains true for the extended functions thus answering a question raised by Dominic Welsh."}
{"category": "Math", "title": "Projective Reeds-Shepp car on $S^2$ with quadratic cost", "abstract": "Fix two points $x,\\bar{x}\\in S^2$ and two directions (without orientation) $\\eta,\\bar\\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$ J[\\gamma]=\\int_0^T g_{\\gamma(t)}(\\dot\\gamma(t),\\dot\\gamma(t))+ K^2_{\\gamma(t)}g_{\\gamma(t)}(\\dot\\gamma(t),\\dot\\gamma(t)) ~dt$$ along all smooth curves starting from $x$ with direction $\\eta$ and ending in $\\bar{x}$ with direction $\\bar\\eta$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_\\gamma$ is the corresponding geodesic curvature. The interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1). We compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology."}
{"category": "Math", "title": "Fusion rules for quantum reflection groups", "abstract": "We find the fusion rules for the quantum analogues of the complex reflection groups $H_n^s=\\mathbb Z_s\\wr S_n$. The irreducible representations can be indexed by the elements of the free monoid $\\mathbb N^{*s}$, and their tensor products are given by formulae which remind the Clebsch-Gordan rules (which appear at $s=1$)."}
{"category": "Math", "title": "Enveloping Actions for Partial Hopf Actions", "abstract": "Motivated by partial group actions on unital algebras, in this article we extend many results obtained by Exel and Dokuchaev to the context of partial actions of Hopf algebras, according to Caenepeel and Jansen. First, we generalize the theorem about the existence of an enveloping action, also known as the globalization theorem. Second, we construct a Morita context between the partial smash product and the smash product related to the enveloping action. Third, we dualize the globalization theorem to partial coactions and finally, we define partial representations of Hopf algebras and show some results relating partial actions and partial representations."}
{"category": "Math", "title": "Stratified Subcartesian Spaces", "abstract": "We show that, if the family \\cal{O} of orbits of all vector fields on a subcartesian space P is locally finite and each orbit in \\cal{O} is locally closed, then \\cal{O} defines a smooth Whitney A stratification of P. We also show that the stratification by orbit type of the space M/G of orbits of a proper action of a Lie group G on a smooth manifold M is given by orbits of the family of all vector fields on M/G."}
{"category": "Math", "title": "Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor", "abstract": "It is known that the eigenvalues of selfadjoint elements a,b,c with a+b+c=0 in the factor R^omega (ultrapower of the hyperfinite II1 factor) are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. A matricial (`complete') form of this result is equivalent to an embedding question formulated by Connes."}
{"category": "Math", "title": "Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations", "abstract": "We investigate qualitative properties of local solutions $u(t,x)\\ge 0$ to the fast diffusion equation, $\\partial_t u =\\Delta (u^m)/m$ with $m<1$, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form $[0,T]\\times\\RR^d$. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low $m$ in the so-called very fast diffusion range, precisely for all $m\\le m_c=(d-2)/d.$ The boundedness statements are true even for $m\\le 0$, while the positivity ones cannot be true in that range."}
{"category": "Math", "title": "New topological recursion relations", "abstract": "Simple boundary expressions for the k-th power of the cotangent line class on the moduli space of stable 1-pointed genus g curves are found for k >= 2g. The method is by virtual localization on the moduli space of maps to the projective line. As a consequence, nontrivial tautological classes in the kernel of the push-forward map associated to the irreducible boundary divisor of the moduli space of stable g+1 curves are constructed. The geometry of genus g+1 curves then provides universal equations in genus g Gromov-Witten theory. As an application, we prove all the Gromov-Witten identities conjectured recently by K. Liu and H. Xu."}
{"category": "Math", "title": "Loebl-Komlos-Sos Conjecture: dense case", "abstract": "We prove a version of the Loebl-Komlos-Sos Conjecture for dense graphs. For each q>0 there exists a number $n_0\\in \\mathbb{N}$ such that for any n>n_0 and k>qn the following holds: if G be a graph of order n with at least n/2 vertices of degree at least k, then any tree of order k+1 is a subgraph of G."}
{"category": "Math", "title": "Associativity of the Commutator Operation in Groups", "abstract": "The study of associativity of the commutator operation in groups goes back to some work of Levi in 1942. In the 1960's Richard J. Thompson created a group F whose elements are representatives of the generalized associative law for an arbitrary binary operation. In 2006, Geoghegan and Guzman proved that a group G is solvable if and only if the commutator operation in G eventually satisfies ALL instances of the associative law, and also showed that many non-solvable groups do not satisfy any instance of the generalized associative law. We will address the question: Is there a non-solvable group which satisfies SOME instance of the generalized associative law? For finite groups, we prove that the answer is no."}
{"category": "Math", "title": "Pairs of periodic orbits with fixed homology difference", "abstract": "We obtain an asymptotic formula for the number of pairs of closed orbits of a weak-mixing transitive Anosov flow whose homology classes have a fixed difference."}
{"category": "Math", "title": "On the non-persistence of Hamiltonian identity cycles", "abstract": "We study the leading term of the holonomy map of a perturbed plane polynomial Hamiltonian foliation. The non-vanishing of this term implies the non-persistence of the corresponding Hamiltonian identity cycle. We prove that this does happen for generic perturbations and cycles, as well for cycles which are commutators in Hamiltonian foliations of degree two. Our approach relies on the Chen's theory of iterated path integrals which we briefly resume."}
{"category": "Math", "title": "Algebraic Graph Theory (a short course for postgraduate students and researchers)", "abstract": "This submission has been withdrawn by arXiv administration."}
{"category": "Math", "title": "Isoperimetry and symmetrization for logarithmic Sobolev inequalities", "abstract": "Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts."}
{"category": "Math", "title": "Successors of Singular Cardinals and Coloring Theorems II", "abstract": "We investigate negative square-brackets partition relations at successors of singular cardinals of countable cofinality. Along the way we prove some club-guessing results."}
{"category": "Math", "title": "Fuzzy signed measure", "abstract": "we will define a fuzzy signed measure on $\\sigma$-algebras, as well as positive and negative sets. Herein, we will show that the Fuzzy Hahn Decomposition Theorem, which is a generalization of the classical Hahn Decomposition Theorem, decompose any space X into a positive set A and a negative set B such that A+B=X and the signed measure of $A \\wedge B $ is 0."}
{"category": "Math", "title": "Einstein solvmanifolds and nilsolitons", "abstract": "The purpose of the present expository paper is to give an account of the recent progress and present status of the classification of solvable Lie groups admitting an Einstein left invariant Riemannian metric, the only known examples so far of noncompact Einstein homogeneous manifolds. The problem turns to be equivalent to the classification of Ricci soliton left invariant metrics on nilpotent Lie groups."}
{"category": "Math", "title": "On the field algebra construction", "abstract": "A pure algebraic variant of John Roberts' field algebra construction is presented and applied to bialgebroid Galois extensions and certain generalized fusion categories."}
{"category": "Math", "title": "The Riemann Hypothesis for Function Fields over a Finite Field", "abstract": "We discuss Enrico Bombieri's proof of the Riemann hypothesis for curves over a finite field. Reformulated, it states that the number of points on a curve $\\C$ defined over the finite field $\\F_q$ is of the order $q+O(\\sqrt{q})$. The first proof was given by Andr\\'e Weil in 1942. This proof uses the intersection of divisors on $\\C\\times\\C$, making the application to the original Riemann hypothesis so far unsuccessful, because $\\spec\\Z\\times\\spec\\Z=\\spec\\Z$ is one-dimensional. A new method of proof was found in 1969 by S. A. Stepanov. This method was greatly simplified and generalized by Bombieri in 1973. Bombieri's method uses functions on $\\C\\times\\C$, again precluding a direct translation to a proof of the original Riemann hypothesis. However, the two coordinates on $\\C\\times\\C$ have different roles, one coordinate playing the geometric role of the variable of a polynomial, and the other coordinate the arithmetic role of the coefficients of this polynomial. The Frobenius automorphism of $\\C$ acts on the geometric coordinate of $\\C\\times\\C$. In the last section, we make some suggestions how Nevanlinna theory could provide a model of $\\spec\\Z\\times\\spec\\Z$ that is two-dimensional and carries an action of Frobenius on the geometric coordinate."}
{"category": "Math", "title": "Graph norms and Sidorenko's conjecture", "abstract": "Let $H$ and $G$ be two finite graphs. Define $h_H(G)$ to be the number of homomorphisms from $H$ to $G$. The function $h_H(\\cdot)$ extends in a natural way to a function from the set of symmetric matrices to $\\mathbb{R}$ such that for $A_G$, the adjacency matrix of a graph $G$, we have $h_H(A_G)=h_H(G)$. Let $m$ be the number of edges of $H$. It is easy to see that when $H$ is the cycle of length $2n$, then $h_H(\\cdot)^{1/m}$ is the $2n$-th Schatten-von Neumann norm. We investigate a question of Lov\\'{a}sz that asks for a characterization of graphs $H$ for which the function $h_H(\\cdot)^{1/m}$ is a norm. We prove that $h_H(\\cdot)^{1/m}$ is a norm if and only if a H\\\"{o}lder type inequality holds for $H$. We use this inequality to prove both positive and negative results, showing that $h_H(\\cdot)^{1/m}$ is a norm for certain classes of graphs, and giving some necessary conditions on the structure of $H$ when $h_H(\\cdot)^{1/m}$ is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact for such graphs we can prove statements that are much stronger than the assertion of Sidorenko's conjecture. We also investigate the $h_H(\\cdot)^{1/m}$ norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the $2n$-th Schatten-von Neumann norms."}
{"category": "Math", "title": "Symmetrization and sharp Sobolev inequalities in metric spaces", "abstract": "We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for H\\\"{o}rmander vector fields."}
{"category": "Math", "title": "Descriptive set theoretic methods applied to strictly singular and strictly cosingular operators", "abstract": "The class of strictly singular operators originating from the dual of a separable Banach space is written as an increasing union of $\\omega_1$ subclasses which are defined using the Schreier sets. A question of J. Diestel, of whether a similar result can be stated for strictly cosingular operators, is studied."}
{"category": "Math", "title": "A new approach to the Ramsey-type games and the Gowers dichotomy in F-spaces", "abstract": "We give a new approach to the Ramsey-type results of Gowers on block bases in Banach spaces and apply our results to prove the Gowers dichotomy in F-spaces."}
{"category": "Math", "title": "Sur l'existence d'une cat\\'egorie ayant une matrice strictement positive donn\\'ee", "abstract": "The Leinster matrix corresponding to a finite category has entries counting the number of morphisms between objects. A first question is to know which positive integer matrices come from at least one finite category. Here, that question will be answered when the entries are strictly positive."}
{"category": "Math", "title": "Generating functions of stable pair invariants via wall-crossings in derived categories", "abstract": "The notion of limit stability on Calabi-Yau 3-folds is introduced by the author to construct an approximation of Bridgeland-Douglas stability conditions at the large volume limit. It has also turned out that the wall-crossing phenomena of limit stable objects seem relevant to the rationality conjecture of the generating functions of Pandharipande-Thomas invariants. In this article, we shall make it clear how wall-crossing formula of the counting invariants of limit stable objects solves the above conjecture."}
{"category": "Math", "title": "Infinity inner products on A-infinity algebras", "abstract": "We give a self contained introduction to A$_\\infty$-algebras, A$_\\infty$-bimodules and maps between them. The case of A$_\\infty$-bimodule-map between $A$ and its dual space $A^{*}$, which we call $\\infty$-inner-product, will be investigated in detail. In particular, we describe the graph complex associated to $\\infty$-inner-product. In a later paper, we show how $\\infty$-inner-products can be used to model the string topology BV-algebra on the free loop space of a Poincar\\'e duality space."}
{"category": "Math", "title": "Elliptic systems and material interpenetration", "abstract": "We classify the second order, linear, two by two systems for which the two fundamental theorems for planar harmonic mappings, the Rado'-Kneser-Choquet Theorem and the H. Lewy Theorem, hold. They are those which, up to a linear change of variable, can be written in diagonal form with the same operator on both diagonal blocks. In particular, we prove that the aforementioned Theorems cannot be extended to solutions of either the Lame' system of elasticity, or of elliptic systems in diagonal form, even with just slightly different operators for the two components."}
{"category": "Math", "title": "Gelfand-Tsetlin algebras and cohomology rings of Laumon spaces", "abstract": "Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL_n. We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand-Tsetlin subalgebra of U(gl_n), and formulate a conjectural answer for the small quantum cohomology rings in terms of certain commutative shift of argument subalgebras of U(gl_n)."}
{"category": "Math", "title": "An abstract Coifman-Rochberg-Weiss commutator theorem", "abstract": "We formulate and prove a version of the celebrated Coifman-Rochberg-Weiss commutator theorem for the real method of interpolation"}
{"category": "Math", "title": "On the equations for universal torsors over del Pezzo surfaces", "abstract": "We show that every split del Pezzo surface of degree d=5,4,3 or 2 has a universal torsor which is a dense open subset of the intersection of 6-d dilatations of the affine cone over the corresponding generalized Grassmannian G/P. Here a dilatation is the linear transformation by an element of the 'diagonal' torus. This gives a concise description of the quadratic equations of universal torsors obtained by Popov and Derenthal. Any (possibly, non-split) del Pezzo surface with a rational point has a universal torsor which embeds into the same homogeneous space as a split surface of the same degree. The proof uses a recent result of Ph. Gille and Raghunathan."}
{"category": "Math", "title": "Structure Theorem for Riemannian surfaces with arbitrary curvature", "abstract": "In this paper we prove that any Riemannian surface, with no restriction of curvature at all, can be decomposed into blocks belonging just to some of these types: generalized Y-pieces, generalized funnels and halfplanes."}
{"category": "Math", "title": "Asymptotically tight bounds on subset sums", "abstract": "For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\\in B}a:B\\subset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and Samal. We improve this bound to the asymptotically best possible result |Sigma(A)|>= (1/4-o(1))|A|^{2}. We also study a related problem in which A is any subset of Z_{n} with all elements of A coprime to n; it has recently been shown, by Vu, that if such a set A has the property Sigma(A) is not Z_{n} then |A|=O(sqrt{n}). This bound was improved to |A|<= 8sqrt{n} by De Vos, Goddyn, Mohar and Samal, we further improve the bound to the asymptotically best possible result |A|<= (2+o(1))sqrt{n}."}
{"category": "Math", "title": "A real variable characterization of Gromov hyperbolicity of flute surfaces", "abstract": "In this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This theorem has several consequences; in particular, it allows to deduce a result about stability of hyperbolicity, even though the original surface and the modified one are not quasi-isometric."}
{"category": "Math", "title": "Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics", "abstract": "We obtain explicit and simple conditions which in many cases allow one decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows to deduce the non-hyperbolicity of any periodic Denjoy domain."}
{"category": "Math", "title": "Random Attractors for the Stochastic FitzHugh-Nagumo System on Unbounded Domains", "abstract": "The existence of a random attractor for the stochastic FitzHugh-Nagumo system defined on an unbounded domain is established. The pullback asymptotic compactness of the stochastic system is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique."}
{"category": "Math", "title": "New demonstrations about the resolution of numbers into squares", "abstract": "Translated from the Latin original \"Novae demonstrationes circa resolutionem numerorum in quadrata\" (1774). E445 in the Enestrom index. See Chapter III, section XI of Weil's \"Number theory: an approach through history\". Also, a very clear proof of the four squares theorem based on Euler's is Theorem 370 in Hardy and Wright, \"An introduction to the theory of numbers\", fifth ed. It uses Theorem 87 in Hardy and Wright, but otherwise does not assume anything else from their book. I translated most of the paper and checked those details a few months ago, but only finished last few parts now. If anything isn't clear please email me."}
{"category": "Math", "title": "On iterating operators and on generalized periodic orbits", "abstract": "We try to define the more general form of iterative processes in which the Pomeau-Manneville and the Feigenbaum scenario may occur along with their specific scaling properties. Doing this we need to generalize other basic concepts. Thus, what we call a periodic carousel is a generalization of what is usually called a periodic orbit."}
{"category": "Math", "title": "Differentiability of stochastic flow of reflected Brownian motions", "abstract": "We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation."}
{"category": "Math", "title": "On the degenerated Arnold-Givental conjecture", "abstract": "We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, \\omega, \\tau)$ with nonempty and compact real part $L={\\rm Fix}(\\tau)$. For given $\\Lambda\\in (0, +\\infty]$ and $m\\in\\N\\cup\\{0\\}$ we show the equivalence of the following two claims: (i) $\\sharp(L\\cap\\phi^H_1(L))\\ge m$ for any Hamiltonian function $H\\in C_0^\\infty([0, 1]\\times M)$ with Hofer's norm $\\|H\\|<\\Lambda$; (ii) $\\sharp {\\cal P}(H,\\tau)\\ge m$ for every $H\\in C^\\infty_0(\\R/\\Z\\times M)$ satisfying $H(t,x)=H(-t,\\tau(x))\\;\\forall (t,x)\\in\\mathbb{R}\\times M$ and with Hofer's norm $\\|H\\|<2\\Lambda$, where ${\\cal P}(H, \\tau)$ is the set of all $1$-periodic solutions of $\\dot{x}(t)=X_{H}(t,x(t))$ satisfying $x(-t)=\\tau(x(t))\\;\\forall t\\in\\R$ (which are also called brake orbits sometimes). Suppose that $(M, \\omega)$ is geometrical bounded for some $J\\in{\\cal J}(M,\\omega)$ with $\\tau^\\ast J=-J$ and has a rationality index $r_\\omega>0$ or $r_\\omega=+\\infty$. Using Hofer's method we prove that if the Hamiltonian $H$ in (ii) above has Hofer's norm $\\|H\\|<r_\\omega$ then $\\sharp(L\\cap\\phi^H_1(L))\\ge\\sharp {\\cal P}_0(H,\\tau)\\ge {\\rm Cuplength}_{\\F}(L)$ for $\\F=\\Z_2$, and further for $\\F=\\Z$ if $L$ is orientable, where ${\\cal P}_0(H,\\tau)$ consists of all contractible solutions in ${\\cal P}(H,\\tau)$."}
{"category": "Math", "title": "Symbolic computation of moments of sampling distributions", "abstract": "By means of the notion of umbrae indexed by multisets, a general method to express estimators and their products in terms of power sums is derived. A connection between the notion of multiset and integer partition leads immediately to a way to speed up the procedures. Comparisons of computational times with known procedures show how this approach turns out to be more efficient in eliminating much unnecessary computation."}
{"category": "Math", "title": "Lasso-type recovery of sparse representations for high-dimensional data", "abstract": "The Lasso is an attractive technique for regularization and variable selection for high-dimensional data, where the number of predictor variables $p_n$ is potentially much larger than the number of samples $n$. However, it was recently discovered that the sparsity pattern of the Lasso estimator can only be asymptotically identical to the true sparsity pattern if the design matrix satisfies the so-called irrepresentable condition. The latter condition can easily be violated in the presence of highly correlated variables. Here we examine the behavior of the Lasso estimators if the irrepresentable condition is relaxed. Even though the Lasso cannot recover the correct sparsity pattern, we show that the estimator is still consistent in the $\\ell_2$-norm sense for fixed designs under conditions on (a) the number $s_n$ of nonzero components of the vector $\\beta_n$ and (b) the minimal singular values of design matrices that are induced by selecting small subsets of variables. Furthermore, a rate of convergence result is obtained on the $\\ell_2$ error with an appropriate choice of the smoothing parameter. The rate is shown to be optimal under the condition of bounded maximal and minimal sparse eigenvalues. Our results imply that, with high probability, all important variables are selected. The set of selected variables is a meaningful reduction on the original set of variables. Finally, our results are illustrated with the detection of closely adjacent frequencies, a problem encountered in astrophysics."}
{"category": "Math", "title": "Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory", "abstract": "We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the invertible case and is, to our knowledge, one among not very many instances in which a natural invariant measure for a non-invertible dynamical system is well-understood."}
{"category": "Math", "title": "Fun With Fourier Series", "abstract": "By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. We construct several series whose sums remain unchanged when the $n^{th}$ term is multiplied by $\\sin(n)/n$. One example is this classic series for $\\pi/4$: \\[ \\frac{\\pi}{4} = 1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\dots = 1 \\cdot \\frac{\\sin(1)}{1} - \\frac{1}{3} \\cdot \\frac{\\sin(3)}{3} + \\frac{1}{5} \\cdot \\frac{\\sin(5)}{5} - \\frac{1}{7} \\cdot \\frac{\\sin(7)}{7} + \\dots . \\] Another example is \\[ \\sum_{n=1}^{\\infty} \\frac{\\sin(n)}{n} = \\sum_{n=1}^{\\infty} \\left(\\frac{\\sin(n)}{n}\\right)^2 = \\frac{\\pi-1}{2}. \\] This paper also discusses an included Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions."}
{"category": "Math", "title": "Some properties of non-positively curved lattices", "abstract": "We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity statements are obtained."}
{"category": "Math", "title": "Connected components of partition preserving diffeomorphisms", "abstract": "Let $f:\\mathbb{R}^2 \\to \\mathbb{R}$ be a real homogeneous polynomial and $S(f)$ be the group of diffeomorphisms $h:\\mathbb{R}^2 \\to \\mathbb{R}^2$ preserving $f$, i.e. $f \\circ h = f$. Denote by $S(f,r)$, $(0\\leq r \\leq \\infty)$, the identity path component of $S(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $S(f,\\infty) = \\cdots = S(f,1)$ for all such $f$ and that $S(f,1) \\not= S(f,0)$ if and only if $f$ is a product of at least two distinct irreducible over $\\mathbb{R}$ quadratic forms."}
{"category": "Math", "title": "Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^*$-modules", "abstract": "In this note we show that an unbounded regular operator $t$ on Hilbert $C^*$-modules over an arbitrary $C^*$ algebra $ \\mathcal{A}$ has polar decomposition if and only if the closures of the ranges of $t$ and $|t|$ are orthogonally complemented, if and only if the operators $t$ and $t^*$ have unbounded regular generalized inverses. For a given $C^*$-algebra $ \\mathcal{A}$ any densely defined $\\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules has polar decomposition, if and only if any densely defined $\\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules has generalized inverse, if and only if $\\mathcal A$ is a $C^*$-algebra of compact operators."}
{"category": "Math", "title": "Finite Linear Quotients of $\\B_3$ of Low Dimension", "abstract": "We study the problem of deciding whether or not the image of an irreducible representation of the braid group $\\B_3$ of degree $\\leq 5$ has finite image if we are only given the eigenvalues of a generator. We provide a partial algorithm that determines when the images are finite or infinite in all but finitely many cases, and use these results to study examples coming from quantum groups. Our technique uses two classification theorems and the computational group theory package GAP."}
{"category": "Math", "title": "About Factorial Sums", "abstract": "Certain new inequalities for the sums of factorials are presented."}
{"category": "Math", "title": "Weight Multiplicity Polynomials of multi-variable Weyl Modules", "abstract": "This paper is based on the observation that dimension of weight spaces of multi-variable Weyl modules depends polynomially on the highest weight (Conjecture 1). We support this conjecture by various explicit answers for up to three variable cases and discuss the underlying combinatorics."}
{"category": "Math", "title": "Motives and representability of algebraic cycles on threefolds over a field", "abstract": "We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the corresponding intermediate Jacobian J^2(X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibered by surfaces with algebraic H^2. This gives another new examples of three-dimensional varieties whose motives are finite-dimensional."}
{"category": "Math", "title": "On The Functorialrily Of Stratified Desingularizations", "abstract": "This article is devoted to the study of smooth desingularization, which are customary employed in the definition of De Rham Intersection Cohomology with differential forms. In this paper we work with the category of Thom-Mather simple spaces. We construct a functor which sends each Thom-Mather simple space into a smooth manifold called its primary unfolding. Hence we prove that the primary unfoldings are unique up Thom-Mather isomorphisms."}
{"category": "Math", "title": "A quotient stack related to the Weyl algebra", "abstract": "Let A denote the ring of differential operators on the affine line with its two usual generators t and d/dt given degrees +1 and -1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each integer point. Then the category of graded A-modules is equivalent to the category of quasi-coherent sheaves on X. Version 2: corrected typos and deleted appendix at referee's suggestion."}
{"category": "Math", "title": "On the concentration of the chromatic number of random graphs", "abstract": "Let 0<p<1 be fixed. Shamir and Spencer proved in the 1980s that the chromatic number of a random graph in G(n,p) is concentrated in an interval of length about n^{1/2}. In this explanatory note, we give a proof of a result due due Noga Alon, showing that the chromatic number is concentrated in an interval of length about n^{1/2}/log n."}
{"category": "Math", "title": "Bounded Berezin-Toeplitz operators on the Segal-Bargmann space", "abstract": "We discuss the boundedness of Berezin-Toeplitz operators on a generalized Segal-Bargmann space (Fock space) over the complex $n$-space. This space is characterized by the image of a global Bargmann-type transform introduced by Sj\\\"ostrand. We also obtain the deformation estimates of the composition of Berezin-Toeplitz operators whose symbols and their derivatives up to order three are in the Wiener algebra of Sj\\\"ostrand. Our method of proofs is based on the pseudodifferential calculus and the heat flow determined by the phase function of the Bargmann transform."}
{"category": "Math", "title": "Modular representations and branching rules for wreath Hecke algebras", "abstract": "We introduce a generalization of degenerate affine Hecke algebra, called wreath Hecke algebra, associated to an arbitrary finite group G. The simple modules of the wreath Hecke algebra and of its associated cyclotomic algebras are classified over an algebraically closed field of any characteristic p. The modular branching rules for these algebras are obtained, and when p does not divide the order of G, they are further identified with crystal graphs of integrable modules for quantum affine algebras. The key is to establish an equivalence between a module category of the (cyclotomic) wreath Hecke algebra and its suitable counterpart for the degenerate affine Hecke algebra."}
{"category": "Math", "title": "End-point Estimates and Multi-parameter Paraproducts on Higher Dimensional Tori", "abstract": "Analogues of multi-paramter multiplier operators on R^d are defined on the torus T^d. It is shown that these operators satisfy the classical Coifman-Meyer theorem. In addition, L log L and L (log L)^n end-point estimates are proved."}
{"category": "Math", "title": "The Grothendieck and Picard groups of a complete toric DM stack", "abstract": "We compute the Grothendieck and Picard groups of a complete smooth toric Deligne-Mumford stack by using a suitable category of graded modules over a polynomial ring."}
{"category": "Math", "title": "C-totally real warped product submanifolds", "abstract": "We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequalities involving the Laplacian of the warping function and the squared mean curvature of $C$-totally real warped product submanifolds of $(\\kappa ,\\mu ) $-space forms, Sasakian space forms and non-Sasakian $(\\kappa ,\\mu) $-manifolds. Then we obtain obstructions to the existence of minimal isometric immersions of $C$-totally real warped product submanifolds in $(\\kappa ,\\mu) $-space forms, non-Sasakian $(\\kappa ,\\mu) $-manifolds and Sasakian space forms. In the last, we obtain an example of a warped product $C$-totally real submanifold of a non-Sasakian $(\\kappa ,\\mu) $-manifold, which satisfies the equality case of the basic inequality."}
{"category": "Math", "title": "On Kazhdan-Lusztig cells in type B", "abstract": "We prove that, for any choice of parameters, the Kazhdan-Lusztig cells of a Weyl group of type $B$ are unions of combinatorial cells (defined using the domino insertion algorithm)."}
{"category": "Math", "title": "On indices of 1-forms on determinantal singularities", "abstract": "We consider 1-forms on, so called, essentially isolated determinantal singularities (a natural generalization of isolated ones), show how to define analogues of the Poincar\\'e--Hopf index for them, and describe relations between these indices and the radial index. For isolated determinantal singularities, we discuss properties of the homological index of a holomorphic 1-form and its relation with the Poincar\\'e--Hopf index."}
{"category": "Math", "title": "Minimal graphs in $\\mathbb{R}^{4}$ with bounded Jacobians", "abstract": "We obtain a Bernstein type result for entire two dimensional minimal graphs in $\\mathbb{R}^{4}$, which extends a previous one due to L. Ni. Moreover, we provide a characterization for complex analytic curves."}
{"category": "Math", "title": "On the Weyl law for Toeplitz operators", "abstract": "A Weyl law for Toeplitz operators was proved by Boutet de Monvel and Guillemin for general Toeplitz structures. In the setting of positive line bundles, we revisit this theme in light of local asymptotic techniques based on the microlocal theory of the Szego kernel. By pairing this approach with classical arguments used to estimate the spectral function of a pseudodifferential operator, we first establish a local Weyl law (that is, a pointwise estimate on the spectral function of the Toeplitz operator)."}
{"category": "Math", "title": "The Poincare series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface", "abstract": "For a simple complete ideal $\\wp$ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincar\\'e series $P_{\\wp}$, that gathers in an unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to $\\wp$. This paper is devoted to prove that $P_{\\wp}$ is a rational function giving an explicit expression for it."}
{"category": "Math", "title": "Extended Formulations for Packing and Partitioning Orbitopes", "abstract": "We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in (Kaibel and Pfetsch, 2008). These polytopes are the convex hulls of all 0/1-matrices with lexicographically sorted columns and at most, resp. exactly, one 1-entry per row. They are important objects for symmetry reduction in certain integer programs. Using the extended formulations, we also derive a rather simple proof of the fact that basically shifted-column inequalities suffice in order to describe those orbitopes linearly."}
{"category": "Math", "title": "On Hamilton Decompositions", "abstract": "P. J. Kelly conjectured in 1968 that every diregular tournament on (2n+1) points can be decomposed in directed Hamilton circuits [1]. We define so called leading diregular tournament on (2n+1) points and show that it can be decomposed in directed Hamilton circuits when (2n+1) is a prime number. When (2n+1) is not a prime number this method does not work and we will need to devise some another method. We also propose a general method to find Hamilton decomposition of certain tournament for all sizes."}
{"category": "Math", "title": "Susceptibility in subcritical random graphs", "abstract": "We study the evolution of the susceptibility in the subcritical random graph $G(n,p)$ as $n$ tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal."}
{"category": "Math", "title": "On totally real Hilbert-Speiser Fields of type C_p", "abstract": "Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers O_L is free as an O_K[G]-module. Let C_p denote the cyclic group of prime order p. We show that if p >= 7 (or p=5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilbert-Speiser of type C_p."}
{"category": "Math", "title": "On ergodic behavior of $p$-adic dynamical systems", "abstract": "Monomial mappings, $x\\mapsto x^n$, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an anologous result for monomial dynamical systems over $p-$adic numbers. The process is, however, not straightforward. The result will depend on the natural number $n$. Moreover, in the $p-$adic case we never have ergodicity on the unit circle, but on the circles around the point 1."}
{"category": "Math", "title": "The quantized walled Brauer algebra and mixed tensor space", "abstract": "In this paper we investigate a multi-parameter deformation $\\mathfrak{B}_{r,s}^n(a,\\lambda,\\delta)$ of the walled Brauer algebra which was previously introduced by Leduc (\\cite{leduc}). We construct an integral basis of $\\mathfrak{B}_{r,s}^n(a,\\lambda,\\delta)$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $\\mathfrak{B}_{r,s}^n(q)= \\mathfrak{B}_{r,s}^n(q^{-1}-q,q^n,[n]_q)$ on mixed tensor space and prove that the kernel is free over the ground ring $R$ of rank independent of $R$. As an application, we prove one side of Schur--Weyl duality for mixed tensor space: the image of $\\mathfrak{B}_{r,s}^n(q)$ in the $R$-endomorphism ring of mixed tensor space is, for all choices of $R$ and the parameter $q$, the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra $\\mathbf{U}$ of the general linear Lie algebra $\\mathfrak{gl}_n$ on mixed tensor space. Thus, the $\\mathbf{U}$-invariants in the ring of $R$-linear endomorphisms of mixed tensor space are generated by the action of $\\mathfrak{B}_{r,s}^n(q)$."}
{"category": "Math", "title": "On the Hochschild (co)homology of Quantum Homogeneous Spaces", "abstract": "The recent result of Brown and Zhang establishing Poincare duality in the Hochschild (co)homology of a large class of Hopf algebras is extended to right coideal subalgebras over which the Hopf algebra is faithfully flat, and applied to the standard Podles quantum 2-sphere."}
{"category": "Math", "title": "On the Linearization of the First and Second Painleve' Equations", "abstract": "We found Fuchs--Garnier pairs in 3X3 matrices for the first and second Painleve' equations which are linear in the spectral parameter. As an application of our pairs for the second Painleve' equation we use the generalized Laplace transform to derive an invertible integral transformation relating two its Fuchs--Garnier pairs in 2X2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and the one found by Harnad, Tracy, and Widom. Together with the certain other transformations it allows us to relate all known 2X2 matrix Fuchs--Garnier pairs for the second Painleve' equation with the original Garnier pair."}
{"category": "Math", "title": "Motivic Landweber Exactness", "abstract": "We prove a motivic version of Landweber's exact functor theorem from topology. The main result is that the assignment given by a Landweber-type formula using the MGL-homology of a motivic spectrum defines a homology theory on the stable motivic homotopy category and is representable by a Tate-like (or cellular) spectrum. Using the universal coefficient spectral sequence of Dugger-Isaksen we deduce formulas for operations of motivic Landweber spectra of a certain type including homotopy algebraic K-theory. Finally we construct a Chern character as a map between motivic spectra."}
{"category": "Math", "title": "Local uniformization and free boundary regularity of minimal singular surfaces", "abstract": "In continuing the study of harmonic mapping from 2-dimensional Riemannian simplicial complexes in order to construct minimal surfaces with singularity, we obtain an a-priori regularity result concerning the real analyticity of the free boundary curve. The free boundary is the singular set along which three disk-type minimal surfaces meet. Here the configuration of the singular minimal surface is obtained by a minimization of a weighted energy functional, in the spirit of J.Douglas' approach to the Plateau Problem. Using the free boundary regularity of the harmonic map, we construct a local uniformization of the singular surface as a parameterization of a neighborhood of a point on the free boundary by the singular tangent cone. In addition, applications of the local uniformization are discussed in relation to H.Lewy's real analytic extension of minimal surfaces."}
{"category": "Math", "title": "Ordered k-flaw Preferences Sets", "abstract": "In this paper, we focus on ordered $k$-flaw preference sets. Let $\\mathcal{OP}_{n,\\geq k}$ denote the set of ordered preference sets of length $n$ with at least $k$ flaws and $\\mathcal{S}_{n,k}=\\{(x_1,...,x_{n-k})\\mid x_1+x_2+... +x_{n-k}=n+k, x_i\\in\\mathbb{N}\\}$. We obtain a bijection from the sets $\\mathcal{OP}_{n,\\geq k}$ to $\\mathcal{S}_{n,k}$. Let $\\mathcal{OP}_{n,k}$ denote the set of ordered preference sets of length $n$ with exactly $k$ flaws. An $(n,k)$-\\emph{flaw path} is a lattice path starting at $(0,0)$ and ending at $(2n,0)$ with only two kinds of steps--rise step: $U=(1,1)$ and fall step: $D=(1,-1)$ lying on the line $y = -k$ and touching this line. Let $\\mathcal{D}_{n,k}$ denote the set of $(n, k)$-flaw paths. Also we establish a bijection between the sets $\\mathcal{OP}_{n,k}$ and $\\mathcal{D}_{n,k}$. Let $op_{n,\\geq k,\\leq l}^m$ $(op_{n, k, =l}^m)$ denote the number of preference sets $\\alpha=(a_1,...,a_n)$ with at least $k$ (exact) flaws and leading term $m$ satisfying $a_i\\leq l$ for any $i$ $(\\max\\{a_i\\mid 1\\leq i\\leq n\\}=l)$, respectively. With the benefit of these bijections, we obtain the explicit formulas for $op_{n,\\geq k,\\leq l}^m$. Furthermore, we give the explicit formulas for $op_{n, k, =l}^m$. We derive some recurrence relations of the sequence formed by ordered $k$-flaw preference sets of length $n$ with leading term $m$. Using these recurrence relations, we obtain the generating functions of some corresponding $k$-flaw preference sets."}
{"category": "Math", "title": "Fast winning strategies in Avoider-Enforcer games", "abstract": "In numerous positional games the identity of the winner is easily determined. In this case one of the more interesting questions is not {\\em who} wins but rather {\\em how fast} can one win. These type of problems were studied earlier for Maker-Breaker games; here we initiate their study for unbiased Avoider-Enforcer games played on the edge set of the complete graph $K_n$ on $n$ vertices. For several games that are known to be an Enforcer's win, we estimate quite precisely the minimum number of moves Enforcer has to play in order to win. We consider the non-planarity game, the connectivity game and the non-bipartite game."}
{"category": "Math", "title": "k-flaw Preference Sets", "abstract": "In this paper, let $\\mathcal{P}_{n;\\leq s;k}^l$ denote a set of $k$-flaw preference sets $(a_1,...,a_n)$ with $n$ parking spaces satisfying that $1\\leq a_i\\leq s$ for any $i$ and $a_1=l$ and $p_{n;\\leq s;k}^l=|\\mathcal{P}_{n;\\leq s;k}^l|$. We use a combinatorial approach to the enumeration of $k$-flaw preference sets by their leading terms. The approach relies on bijections between the $k$-flaw preference sets and labeled rooted forests. Some bijective results between certain sets of $k$-flaw preference sets of distinct leading terms are also given. We derive some formulas and recurrence relations for the sequences $p_{n;\\leq s;k}^l$ and give the generating functions for these sequences."}
{"category": "Math", "title": "Representations of logmodular algebras", "abstract": "We study the question of whether or not contractive representations of logmodular algebras are completely contractive. We prove that a 2-contractive representation of a logmodular algebra extends to a positive map on the enveloping C*-algebra, which we show generalizes a result of Foias and Suciu on uniform logmodular algebras. Our proof uses non-commutative operator space generalizations of classical results on 2-summing maps and semispectral measures. We establish some matrix factorization results for uniform logmodular algebras"}
{"category": "Math", "title": "Symmetry and monotonicity of least energy solutions", "abstract": "We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable."}
{"category": "Math", "title": "Classifications of special double-coverings associated to a non-orientable surface", "abstract": "This paper investigates some actions \"\\`a la Johnson\" on the set, denoted by ${\\cal E}$, of Spin-structures which are interpreted as special double-coverings of a trivial $S^1-$fibration over a non-orientable surface $N_{g+1}$. The group acting is first a group of orthogonal isomorphisms assoiciated to $N_{g+1}$. A second approach is to consider the subspace of ${\\cal E}$ (with $2^{g}$ elements) coming from special double-coverings of $S^1\\times F_g$, where $F_g$ is the orientation covering of $N_{g+1}$. The group acting now is a subgroup of the group of symplectic isomorphisms associated to $F_{g}$. In both situations, we obtain results on the number of orbits and the number of elements in each orbit. Except in one case, these results do not depend on any necessary choices. We compare both previous classifications to a third one: weak-equivalence of coverings"}
{"category": "Math", "title": "On the closedness of approximation spectra", "abstract": "Generalizing Cusick's theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant."}
{"category": "Math", "title": "The Gauss higher relative class number problem", "abstract": "Assuming the 2-adic Iwasawa main conjecture, we find all CM fields with higher relative class number at most 16: there are at least 31 and at most 34 such fields, and exactly one is not abelian."}
{"category": "Math", "title": "Scalar Extension of Abelian and Tannakian Categories", "abstract": "We introduce and develop the notion of scalar extension for abelian categories. Given a field extension F'/F, to every F-linear abelian category A satisfying a suitable finiteness condition we associate an F'-linear abelian category A' and an exact F-linear functor t: A --> A'. This functor is universal among F-linear right exact functors with target an F'-linear abelian category. We discuss various basic properties of this concept, among others compatibilities with multilinear endofunctors such as tensor products, and the permanence of favourable properties of the functors and categories involved. We obtain the notion of scalar extension for Tannakian categories, which allows us to deduce consequences for the algebraic monodromy groups of Tannakian categories."}
{"category": "Math", "title": "Representation of Finite Abelian Group Elements by Subsequence Sums", "abstract": "Let $G\\cong C_{n_1}\\oplus ... \\oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|...|n_r$. A conjecture of Hamidoune says that if $W=w_1... w_n$ is a sequence of integers, all but at most one relatively prime to $|G|$, and $S$ is a sequence over $G$ with $|S|\\geq |W|+|G|-1\\geq |G|+1$, the maximum multiplicity of $S$ at most $|W|$, and $\\sigma(W)\\equiv 0\\mod |G|$, then there exists a nontrivial subgroup $H$ such that every element $g\\in H$ can be represented as a weighted subsequence sum of the form $g=\\sum_{i=1}^{n}w_is_i$, with $s_1... s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the counterexamples for large $|W|\\geq {1/2}|G|$. A theorem of Gao, generalizing an older result of Olson, says that if $G$ is a finite abelian group, and $S$ is a sequence over $G$ with $|S|\\geq |G|+D(G)-1$, then either every element of $G$ can be represented as a $|G|$-term subsequence sum from $S$, or there exists a coset $g+H$ such that all but at most $|G/H|-2$ terms of $S$ are from $g+H$. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis $|S|\\geq |G|+D(G)-1$ can be relaxed to $|S|\\geq |G|+d^*(G)$, where $d^*(G)=\\Sum_{i=1}^{r}(n_i-1)$. We also use this method to derive a variation on Hamidoune's conjecture valid when at least $d^*(G)$ of the $w_i$ are relatively prime to $|G|$."}
{"category": "Math", "title": "Acceleration Operators in the Value Iteration Algorithms for Average Reward Markov Decision Processes", "abstract": "One of the most widely used methods for solving average cost MDP problems is the value iteration method. This method, however, is often computationally impractical and restricted in size of solvable MDP problems. We propose acceleration operators that improve the performance of the value iteration for average reward MDP models. These operators are based on two important properties of Markovian operator: contraction mapping and monotonicity. It is well known that the classical relative value iteration methods for average cost criteria MDP do not involve the max-norm contraction or monotonicity property. To overcome this difficulty we propose to combine acceleration operators with variants of value iteration for stochastic shortest path problems associated average reward problems."}
{"category": "Math", "title": "The cyclic cycle complex of a surface", "abstract": "A contractible simplicial complex is constructed that parametrizes different ways of representing a fixed one-dimensional homology class in a closed orientable surface by isotopy classes of systems of disjoint oriented simple closed curves. This is a variant on an earlier construction of Bestvina-Bux-Margalit."}
{"category": "Math", "title": "Double Categories in Mathematical Physics", "abstract": "Expansion of the categorical point of view on many areas of the mathematics and mathematical physics will cause to deeper understanding of genuine features of these problems. New applications of categorical methods are connected with new additional structures on categories. One of such structures, the double category, is considered in this article. The double category structure is defined as generalization of the bicategory structure. It is shown that double categories exist in the topological and ordinary quantum field theories, and for dynamical systems with inputs and outputs. Morphisms of all these double categories are not maps of sets."}
{"category": "Math", "title": "The principal eigenvalue of the $\\infty$-Laplacian with the Neumann boundary condition", "abstract": "We prove the existence of a principal eigenvalue associated to the $\\infty$-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem."}
{"category": "Math", "title": "Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)", "abstract": "The key condition A3w of Ma, Trudinger and Wang for regularity of optimal transportation maps is implied by the nonnegativity of a pseudo-Riemannian curvature -- which we call cross-curvature -- induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature nonnegativity is preserved for products of two manifolds; (2) both A3w and cross-curvature nonnegativity are inherited by Riemannian submersions; and (3) the $n$-dimensional round sphere satisfies cross-curvature nonnegativity. From these results, a large new class of Riemannian manifolds satisfying cross-curvature nonnegativity (thus A3w) is obtained, including many whose sectional curvature is far from constant. All known obstructions to the regularity of optimal maps are absent from these manifolds, making them a class for which it is natural to conjecture that regularity holds. This conjecture is confirmed for complex projective space CP^n."}
{"category": "Math", "title": "The growth exponent for planar loop-erased random walk", "abstract": "We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any two-dimensional discrete lattice."}
{"category": "Math", "title": "Free Analysis Questions II: The Grassmannian Completion and The Series Expansions at the Origin", "abstract": "The fully matricial generalization in part I, of the difference quotient derivation on holomorphic functions, in which ${\\mathbb C}$ is replaced by a Banach algebra $B$, is extended from the affine case to a Grassmannian completion. The infinitesimal bialgebra duality, the duality transform generalizing the Stieltjes transform and the spectral theory with non-commuting scalars all extend to this completion. The series expansions of fully matricial analytic functions are characterized, providing a new way to generate fully matricial functions."}
{"category": "Math", "title": "Density fluctuations for a zero-range process on the percolation cluster", "abstract": "We prove that the density fluctuations for a zero-range process evolving on the supercritical percolation cluster are given by a generalized Ornstein-Uhlenbeck process in the space of distributions $\\mc S'(\\bb R^d)$."}
{"category": "Math", "title": "Coarse graining, fractional moments and the critical slope of random copolymers", "abstract": "For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarse-graining procedure, combined with upper bounds on the fractional moments of the partition function."}
{"category": "Math", "title": "Sasakian Geometry: The Recent Work of Krzysztof Galicki", "abstract": "This is a mainly expository article honoring my recently deceased friend and collaborator Krzysztof Galicki who died after a tragic hiking accident. I give a review of our recent work in Sasakian geometry. A few new results are also presented."}
{"category": "Math", "title": "New Proofs of the Green-Tao-Ziegler Dense Model Theorem: An Exposition", "abstract": "Green, Tao and Ziegler prove ``Dense Model Theorems'' of the following form: if R is a (possibly very sparse) pseudorandom subset of set X, and D is a dense subset of R, then D may be modeled by a set M whose density inside X is approximately the same as the density of D in R. More generally, they show that a function that is majorized by a pseudorandom measure can be written as a sum of a bounded function having the same expectation plus a function that is ``indistinguishable from zero.'' This theorem plays a key role in the proof of the Green-Tao Theorem that the primes contain arbitrarily long arithmetic progressions. In this note, we present a new proof of the Green-Tao-Ziegler Dense Model Theorem, which was discovered independently by ourselves and Gowers. We refer to our full paper for variants of the result with connections and applications to computational complexity theory, and to Gowers' paper for applications of the proof technique to ``decomposition, ``structure,'' and ``transference'' theorems in arithmetic and extremal combinatorics (as well as a broader survey of such theorems)."}
{"category": "Math", "title": "Hopf-cyclic homology with contramodule coefficients", "abstract": "A new class of coefficients for the Hopf-cyclic homology of module algebras and coalgebras is introduced. These coefficients, termed stable anti-Yetter-Drinfeld contramodules, are both modules and contramodules of a Hopf algebra that satisfy certain compatibility conditions."}
{"category": "Math", "title": "Subdivisions and transgressive chains", "abstract": "Combinatorial transgressions are secondary invariants of a space admitting triangulations. They arise from subdivisions and are analogous to transgressive forms such as those arising in Chern-Weil theory. Unlike combinatorial characteristic classes, combinatorial transgressions have not been previously studied. First, this article characterizes transgressions that are path-independent of subdivision sequence. The result is obtained by using a cohomology on posets that is shown to be equivalent to higher derived functors of the inverse (or projective) limit over the opposite poset. Second, a canonical local formula is demonstrated for a particular combinatorial transgression: namely, that relative the difference of Poincar\\'{e} duals to the Euler class."}
{"category": "Math", "title": "Convexity properties of the condition number", "abstract": "We define in the space of n by m matrices of rank n, n less or equal than m, the condition Riemannian structure as follows: For a given matrix A the tangent space of A is equipped with the Hermitian inner product obtained by multiplying the usual Frobenius inner product by the inverse of the square of the smallest singular value of A denoted sigma_n(A). When this smallest singular value has multiplicity 1, the function A -> log (sigma_n(A)^(-2)) is a convex function with respect to the condition Riemannian structure that is t -> log (sigma_n(A(t))^(-2)) is convex, in the usual sense for any geodesic A(t). In a more abstract setting, a function alpha defined on a Riemannian manifold (M,<,>) is said to be self-convex when log alpha (gamma(t)) is convex for any geodesic in (M,<,>). Necessary and sufficient conditions for self-convexity are given when alpha is C^2. When alpha(x) = d(x,N)^(-2) where d(x,N) is the distance from x to a C^2 submanifold N of R^j we prove that alpha is self-convex when restricted to the largest open set of points x where there is a unique closest point in N to x. We also show, using this more general notion, that the square of the condition number ||A|||_F / sigma_n(A) is self-convex in projective space and the solution variety."}
{"category": "Math", "title": "Total curvature and isotopy of graphs in $R^3$", "abstract": "Knot theory is the study of isotopy classes of embeddings of the circle $S^1$ into a 3-manifold, specifically $R^3$. The F\\'ary-Milnor Theorem says that any curve in $R^3$ of total curvature less than $4\\pi$ is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topological type of graphs $\\Gamma$, what limitations on the isotopy class of $\\Gamma$ are implied by a bound on total curvature? What does ``total curvature\" mean for a graph? We define a natural notion of net total curvature of a graph $\\Gamma$ in $R^3$, and prove that if $\\Gamma$ is homeomorphic to the $\\theta$-graph, then the net total curvature of $\\Gamma$ \\geq 3\\pi$; and if it is $< 4\\pi$, then $\\Gamma$ is isotopic in $R^3$ to a planar $\\theta$-graph. Further, the net total curvature $= 3\\pi$ only when $\\Gamma$ is a convex plane curve plus a chord. We begin our discussion with piecewise smooth graphs, and extend all these results to continuous graphs in the final section. In particular, we show that continuous graphs of finite total curvature are isotopic to polygonal graphs."}
{"category": "Math", "title": "Kummer structures", "abstract": "Suppose we take an abelian group G and quotient it by the action of negation. What structure does the quotient K inherit from the group structure of G? We describe this structure (which we call the Kummer of G) in terms of a map from the set of unordered pairs of elements of K to itself. We propose some axioms that hold for such structures, and show that every structure satisfying those axioms either is the Kummer of a unique group, or comes from one other construction, the quotient of a 2-torsion group by an involution."}
{"category": "Math", "title": "The finite precision computation and the nonconvergence of difference scheme", "abstract": "The authors show that the round-off error can break the consistency which is the premise of using the difference equation to replace the original differential equations. We therefore proposed a theoretical approach to investigate this effect, and found that the difference scheme can not guarantee the convergence of the actual compute result to the analytical one. A conservation scheme experiment is applied to solve a simple linear differential equation satisfing the LAX equivalence theorem in a finite precision computer. The result of this experiment is not convergent when time step-size decreases trend to zero, which proves that even the stable scheme can't guarantee the numerical convergence in finite precision computer. Further the relative convergence concept is introduced."}
{"category": "Math", "title": "Some Enumerations for Parking Functions", "abstract": "In this paper, let $\\mathcal{P}_{n,n+k;\\leq n+k}$ (resp. $\\mathcal{P}_{n;\\leq s}$) denote the set of parking functions $\\alpha=(a_1,...,a_n)$ of length $n$ with $n+k$ (respe. $n$)parking spaces satisfying $1\\leq a_i\\leq n+k$ (resp. $1\\leq a_i\\leq s$) for all $i$. Let $p_{n,n+k;\\leq n+k}=|\\mathcal{P}_{n,n+k;\\leq n+k}|$ and $p_{n;\\leq s}=|\\mathcal{P}_{n;\\leq s}|$. Let $\\mathcal{P}_{n;\\leq s}^l$ denote the set of parking functions $\\alpha=(a_1,...,a_n)\\in\\mathcal{P}_{n;\\leq s}$ such that $a_1=l$ and $p_{n;\\leq s}^l=|\\mathcal{P}_{n;\\leq s}^l|$. We derive some formulas and recurrence relations for the sequences $p_{n,n+k;\\leq n+k}$, $p_{n;\\leq s}$ and $p_{n;\\leq s}^l$ and give the generating functions for these sequences. We also study the asymptotic behavior for these sequences."}
{"category": "Math", "title": "The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems", "abstract": "In this paper, the Conley conjecture, which were recently proved by Franks and Handel \\cite{FrHa} (for surfaces of positive genus), Hingston \\cite{Hi} (for tori) and Ginzburg \\cite{Gi} (for closed symplectically aspherical manifolds), is proved for $C^1$-Hamiltonian systems on the cotangent bundle of a $C^3$-smooth compact manifold $M$ without boundary, of a time 1-periodic $C^2$-smooth Hamiltonian $H:\\R\\times T^\\ast M\\to\\R$ which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If $H$ also satisfies $H(-t,q, -p)=H(t,q, p)$ for any $(t,q, p)\\in\\R\\times T^\\ast M$, it is shown that the time-one map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of $T^\\ast M$. If $M$ is $C^5$-smooth and $\\dim M>1$, $H$ is of $C^4$ class and independent of time $t$, then for any $\\tau>0$ the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of $\\tau$ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of $H$, $L:\\R\\times TM\\to\\R$, which is proved to be strongly convex and to have quadratic growth in the velocities yet."}
{"category": "Math", "title": "Subequivalence Relations and Positive-Definite Functions", "abstract": "We study a positive-definite function associated to a measure-preserving equivalence relation on a standard probability space and use it to measure quantitatively the proximity of subequivalence relations. This is combined with a recent co-inducing construction of Epstein to produce new kinds of mixing actions of an arbitrary infinite discrete group and it is also used to show that orbit equivalence of free, measure preserving, mixing actions of non-amenable groups is unclassifiable in a strong sense. Finally, in the case of property (T) groups we discuss connections with invariant percolation on Cayley graphs and the calculation of costs."}
{"category": "Math", "title": "Enumerations of Permutations by Circular Descent Sets", "abstract": "The circular descent of a permutation $\\sigma$ is a set $\\{\\sigma(i)\\mid \\sigma(i)>\\sigma(i+1)\\}$. In this paper, we focus on the enumerations of permutations by the circular descent set. Let $cdes_n(S)$ be the number of permutations of length $n$ which have the circular descent set $S$. We derive the explicit formula for $cdes_n(S)$. We describe a class of generating binary trees $T_k $ with weights. We find that the number of permutations in the set $CDES_n(S)$ corresponds to the weights of $T_k$. As a application of the main results in this paper, we also give the enumeration of permutation tableaux according to their shape."}
{"category": "Math", "title": "Circular Peaks and Hilbert Series", "abstract": "The circular peak set of a permutation $\\sigma$ is the set $\\{\\sigma(i)\\mid \\sigma(i-1)<\\sigma(i)>\\sigma(i+1)\\}$. Let $\\mathcal{P}_n$ be the set of all the subset $S\\subseteq [n]$ such that there exists a permutation $\\sigma$ which has the circular set $S$. We can make the set $\\mathcal{P}_n$ into a poset $\\mathscr{P}_n$ by defining $S\\preceq T$ if $S\\subseteq T$ as sets. In this paper, we prove that the poset $\\mathscr{P}_n$ is a simplicial complex on the vertex set $[3,n]$. We study the $f$-vector, the $f$-polynomial, the reduced Euler characteristic, the M$\\ddot{o}$bius function, the $h$-vector and the $h$-polynomial of $\\mathscr{P}_n$. We also derive the zeta polynomial of $\\mathscr{P}_n$ and give the formula for the number of the chains in $\\mathscr{P}_n$. By the poset $\\mathscr{P}_n$, we define two algebras $\\mathcal{A}_{\\mathscr{P}_n}$ and $\\mathcal{B}_{\\mathscr{P}_n}$. We consider the Hilbert polynomials and the Hilbert series of the algebra $\\mathcal{A}_{\\mathscr{P}_n}$ and $\\mathcal{B}_{\\mathscr{P}_n}$."}
{"category": "Math", "title": "Enumerations for Permutations by Circular Peak Sets", "abstract": "The circular peak set of a permutation $\\sigma$ is the set $\\{\\sigma(i)\\mid \\sigma(i-1)<\\sigma(i)>\\sigma(i+1)\\}$. In this paper, we focus on the enumeration problems for permutations by circular peak sets. Let $cp_n(S)$ denote the number of the permutations of order $n$ which have the circular peak set $S$. For the case with $|S|=0,1,2$, we derive the explicit formulas for $cp_n(S)$. We also obtain some recurrence relations for the sequence $cp_n(S)$ and give the formula for $cp_n(S)$ in the general case."}
{"category": "Math", "title": "Diffusion determines the manifold", "abstract": "We prove under a weak smoothness condition that two Riemannian manifold are isomorphic if and only there exists an order isomorphism which intertwines with the Dirichlet type heat semigroups on the manifolds."}
{"category": "Math", "title": "Degree estimate for commutators", "abstract": "Let K<X> be a free associative algebra over a field K of characteristic 0 and let each of the noncommuting polynomials f,g generate its centralizer in K<X>. Assume that the leading homogeneous components of f and g are algebraically dependent with degrees which do not divide each other. We give a counterexample to the recent conjecture of Jie-Tai Yu that deg([f,g])=deg(fg-gf) > min{deg(f),deg(g)}. Our example satisfies deg(g)/2 < deg([f,g]) < deg(g) < deg(f) and deg([f,g]) can be made as close to deg(g)/2 as we want. We obtain also a counterexample to another related conjecture of Makar-Limanov and Jie-Tai Yu stated in terms of Malcev - Neumann formal power series. These counterexamples are found using the description of the free algebra K<X> considered as a bimodule of K[u] where u is a monomial which is not a power of another monomial and then solving the equation [u^m,s]=[u^n,r] with unknowns r,s in K<X>."}
{"category": "Math", "title": "Generalized parking functions, descent numbers, and chain polytopes of ribbon posets", "abstract": "We consider the inversion enumerator I_n(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking functions. Substituting q = -1 into this generalized polynomial produces the number of permutations with a certain descent set. In the classical case, this result implies the formula I_n(-1) = E_n, the number of alternating permutations. We give a combinatorial proof of these formulas based on the involution principle. We also give a geometric interpretation of these identities in terms of volumes of generalized chain polytopes of ribbon posets. The volume of such a polytope is given by a sum over generalized parking functions, which is similar to an expression for the volume of the parking function polytope of Pitman and Stanley."}
{"category": "Math", "title": "Conditions for existence and smoothness of the distribution density for an Ornstein-Uhlenbeck process with Levy noise", "abstract": "Conditions are given, sufficient for the distribution of an Ornstein-Uhlenbeck process with L\\'evy noise to be absolutely continuous or to possess a smooth density. For the processes with non-degenerate drift coefficient, these conditions are a necessary ones. A multidimensional analogue for the non-degeneracy condition on the drift coefficient is introduced."}
{"category": "Math", "title": "Stability of the LCD Model", "abstract": "In this paper, first-passage probability of Markov chains is used to get a strict proof of the existence of degree distribution of the LCD model presented by Bollobas (Random Structures and Algorithms 18(2001)). Also, a precise expression of degree distribution is presented."}
{"category": "Math", "title": "Uniform estimates for paraproducts and related multilinear multipliers", "abstract": "In this paper, we prove some uniform estimates between Lebesgue and Hardy spaces for operators closely related to the multilinear paraproducts on R^d. We are looking for uniformity with respect to parameters, which allow us to disturb the geometry and the metric on $R^d$."}
{"category": "Math", "title": "A general asymptotic decay lemma for elliptic problems", "abstract": "We prove a general asymptotic decay lemma which is applicable in various contexts. As an example, the general theorem is shown to give lower growth estimates for entire and exterior solutions of the minimal surface equation."}
{"category": "Math", "title": "Real Homotopy Theory of Semi-Algebraic Sets", "abstract": "We complete the details of a theory outlined by Kontsevich and Soibelman that associates to a semi-algebraic set a certain graded commutative differential algebra of \"semi-algebraic differential forms\" in a functorial way. This algebra encodes the real homotopy type of the semi-algebraic set in the spirit of the DeRham algebra of differential forms on a smooth manifold. Its development is needed for Kontsevich's proof of the formality of the little cubes operad."}
{"category": "Math", "title": "Groebner bases for the polynomial ring with infinite variables and their applications", "abstract": "We develop the theory of Gr\\\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division algorithm."}
{"category": "Math", "title": "Lower bounds for the normalized height and non-dense subsets of varieties in an abelian variety", "abstract": "This work is the third part of a series of papers. In the first two we consider curves and varieties in a power of an elliptic curve. Here we deal with subvarieties of an abelian variety in general. Let V be an irreducible variety of dimension d embedded in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. For V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d+1 translated by the points close to a subgroup G of finite rank are non Zariski-dense in V. If G has rank zero, it is sufficient to assume that V is weak-transverse. The notion of closeness is defined using a height function."}
{"category": "Math", "title": "On Sums of Conditionally Independent Subexponential Random Variables", "abstract": "The asymptotic tail behaviour of sums of independent subexponential random variables is well understood, one of the main characteristics being the principle of the single big jump. We study the case of dependent subexponential random variables, for both deterministic and random sums, using a fresh approach, by considering conditional independence structures on the random variables. We seek sufficient conditions for the results of the theory with independent random variables still to hold. For a subexponential distribution, we introduce the concept of a boundary class of functions, which we hope will be a useful tool in studying many aspects of subexponential random variables. The examples we give in the paper demonstrate a variety of effects owing to the dependence, and are also interesting in their own right."}
{"category": "Math", "title": "Stable solutions of $-\\Delta u = f(u)$ in $\\R^N$", "abstract": "The paper proves Liouville-type results for stable solutions of semilinear elliptic PDEs with convex nonlinearity, posed on the entire Euclidean space. Extensions to solutions which are stable outside a compact set are also presented."}
{"category": "Math", "title": "The Dirichlet problem for the minimal surface equation -with possible infinite boundary data- over domains in a Riemannian surface", "abstract": "In this paper, we study existence and uniqueness of solutions to Jenkins-Serrin type problems on domains in a Riemannian surface. In the case of unbounded domains, the study is focused on the hyperbolic plane."}
{"category": "Math", "title": "Examples of quantum commutants", "abstract": "We describe the notion of a quantum family of maps of a quantum space and that of a quantum commutant of such a family. Quantum commutants are quantum semigroups defined by a certain universal property. We give a few examples of these objects acting on a classical $n$-point space and on the quantum space underlying the algebra of two by two matrices. We show that some of the resulting quantum semigroups are not compact quantum groups. The proof of one result touches on an interesting problem of the theory of compact quantum groups."}
{"category": "Math", "title": "Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices", "abstract": "Using the variational method, it is shown that the set of all strong peak functions in a closed algebra $A$ of $C_b(K)$ is dense if and only if the set of all strong peak points is a norming subset of $A$. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space $X$ is a norming subset of $\\mathcal{P}({}^n X)$, then the set of all strongly norm attaining elements in $\\mathcal{P}({}^n X)$ is dense. In particular, the set of all points at which the norm of $\\mathcal{P}({}^n X)$ is Fr\\'echet differentiable is a dense $G_\\delta$ subset. In the last part, using Reisner's graph theoretic-approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space $X$ with an absolute norm, its polynomial numerical indices are one if and only if $X$ is isometric to $\\ell_\\infty^n$. Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm."}
{"category": "Math", "title": "On the binomial convolution of arithmetical functions", "abstract": "Let $n=\\prod_p p^{\\nu_p(n)}$ denote the canonical factorization of $n\\in \\N$. The binomial convolution of arithmetical functions $f$ and $g$ is defined as $(f\\circ g)(n)=\\sum_{d\\mid n} (\\prod_p \\binom{\\nu_p(n)}{\\nu_p(d)}) f(d)g(n/d),$ where $\\binom{a}{b}$ is the binomial coefficient. We provide properties of the binomial convolution. We study the $\\C$-algebra $({\\cal A},+,\\circ,\\C)$, characterizations of completely multiplicative functions, Selberg multiplicative functions, exponential Dirichlet series, exponential generating functions and a generalized binomial convolution leading to various M\\\"obius-type inversion formulas. Throughout the paper we compare our results with those of the Dirichlet convolution *. Our main result is that $({\\cal A},+,\\circ,\\C)$ is isomorphic to $({\\cal A},+,*,\\C)$. We also obtain a \"multiplicative\" version of the multinomial theorem."}
{"category": "Math", "title": "Line bundles on spectral curves and the generalised Legendre transform construction of hyperkaehler metrics", "abstract": "An analogue of the correspondence between GL(k)-conjugacy classes of matricial polynomials and line bundles is given for K-conjugacy classes, where K is one of the following: maximal parabolic, maximal torus, GL(k-1) embedded diagonally. The generalised Legendre transform construction of hyperkaehler metrics is studied further, showing that many known hyperkaehler metrics (including the ones on coadjoint orbits) arise in this way, and giving a large class of new (pseudo-)hyperkaehler metrics, analogous to monopole metrics."}
{"category": "Math", "title": "Tromino Tiling Deficient Cubes of Any Side Length", "abstract": "We show that three dimensional cubes of any size can be tiled with trominoes and, when necessary, one or two singletons in any positions. Cubes of side length a multiple of three can always be tiled with trominoes (known), cubes of side length congruent to 1 mod 3 can always be tiled with an arbitrary single cube and trominoes, and cubes of side length congruent to 2 mod 3 can always be tiled with two single cubes in arbitrary locations and trominoes."}
{"category": "Math", "title": "On the definition of L2-Betti numbers of equivalence relations", "abstract": "We show that the L2-Betti numbers of equivalence relations defined by R. Sauer coincide with those defined by D. Gaboriau."}
{"category": "Math", "title": "Thresholding Projection Estimators in Functional Linear Models", "abstract": "We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove these estimators are minimax and rates of convergence are given for some particular cases."}
{"category": "Math", "title": "Nonparametric Partial Importance Sampling for Financial Derivative Pricing", "abstract": "Importance sampling is a promising variance reduction technique for Monte Carlo simulation based derivative pricing. Existing importance sampling methods are based on a parametric choice of the proposal. This article proposes an algorithm that estimates the optimal proposal nonparametrically using a multivariate frequency polygon estimator. In contrast to parametric methods, nonparametric estimation allows for close approximation of the optimal proposal. Standard nonparametric importance sampling is inefficient for high-dimensional problems. We solve this issue by applying the procedure to a low-dimensional subspace, which is identified through principal component analysis and the concept of the effective dimension. The mean square error properties of the algorithm are investigated and its asymptotic optimality is shown. Quasi-Monte Carlo is used for further improvement of the method. It is easy to implement, particularly it does not require any analytical computation, and it is computationally very efficient. We demonstrate through path-dependent and multi-asset option pricing problems that the algorithm leads to significant efficiency gains compared to other algorithms in the literature."}
{"category": "Math", "title": "Asymptotic Harmonic Analysis on the Space of Square Complex Matrices", "abstract": "In this paper, we determine the spherical functions of positive type on the inductive limit space of square complex matrices."}
{"category": "Math", "title": "Some special functions identities arising from commuting operators", "abstract": "Commuting is an important property in many cases of investigation of properties of operators as well as in various applications, especially in quantum physics. Using the observation that the generalized weighted differential operator of order $k$ and the weighted Hardy-type operator commute we derive a number of new and interesting identities involving some functions of mathematical physics."}
{"category": "Math", "title": "Finite Generation of Algebras Associated to Powers of Ideals", "abstract": "We study generalized symbolic powers and form ideals of powers of ideals and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated."}
{"category": "Math", "title": "Tensor-triangulated categories and dualities", "abstract": "In a triangulated symmetric monoidal closed category, there are natural dualities induced by the internal Hom. Given a monoidal functor f^* between two such catgories and adjoint couples (f^*,f_*) and (f_*,f^!), we prove the necessary commutative diagrams for f^* and f_* to respect certain dualities, for a projection formula to hold between them (as duality preserving functors) and for classical base change and composition formulas to hold when such duality preserving functors are composed. This framework is for example useful to define push-forwards for Witt groups."}
{"category": "Math", "title": "Push-forwards for Witt groups of schemes", "abstract": "We define push-forwards for Witt groups of schemes along proper morphisms, using Grothendieck duality theory. This article is an application of results of the authors on tensor-triangulated closed categories to such structures on some derived categories of schemes together with classical derived functors."}
{"category": "Math", "title": "Sampling Spatially Correlated Clutter", "abstract": "Correlated ${\\cal G}$ distributions can be used to describe the clutter seen in images obtained with coherent illumination, as is the case of B-scan ultrasound, laser, sonar and synthetic aperture radar (SAR) imagery. These distributions are derived using the square root of the generalized inverse Gaussian distribution for the amplitude backscatter within the multiplicative model. A two-parameters particular case of the amplitude ${\\mathcal G}$ distribution, called ${\\mathcal G}_{A}^{0}$, constitutes a modeling improvement with respect to the widespread ${\\mathcal K}_{A}$ distribution when fitting urban, forested and deforested areas in remote sensing data. This article deals with the modeling and the simulation of correlated ${\\mathcal G}_{A}^{0}$-distributed random fields. It is accomplished by means of the Inverse Transform method, applied to Gaussian random fields with spatial correlation. The main feature of this approach is its generality, since it allows the introduction of negative correlation values in the resulting process, necessary for the proper explanation of the shadowing effect in many SAR images."}
{"category": "Math", "title": "Properties of cut ideals associated to ring graphs", "abstract": "A cut ideal of a graph records the relations among the cuts of the graph. These toric ideals have been introduced by Sturmfels and Sullivant who also posed the problem of relating their properties to the combinatorial structure of the graph. We study the cut ideals of the family of ring graphs, which includes trees and cycles. We show that they have quadratic Gr\\\"obner bases and that their coordinate rings are Koszul, Hilbertian, and Cohen-Macaulay, but not Gorenstein in general."}
{"category": "Math", "title": "Jumping numbers of a unibranch curve on a smooth surface", "abstract": "A formula for the jumping numbers of a curve unibranch at a singular point is established. The jumping numbers are expressed in terms of the Enriques diagram of the log resolution of the singularity, or equivalently in terms of the canonical set of generators of the semigroup of the curve at the singular point."}
{"category": "Math", "title": "Local-global principles for embedding of fields with involution into simple algebras with involution", "abstract": "In this paper we prove local-global principles for embedding of fields with involution into central simple algebras with involution over a global field. These should be of interest in study of classical groups over global fields. We deduce from our results that in a group of type D_n, n>4 even, two weakly commensurable Zariski-dense S-arithmetic subgroups are actually commensurable. A consequence of this result is that given an absolutely simple algebraic K-group G of type D_n, n>4 even, K a number field, any K-form G' of G having the same set of isomorphism classes of maximal K-tori as G, is necessarily K-isomorphic to G. These results lead to results about isolength and isospectral compact hyperbolic spaces of dimension 2n-1 with n even."}
{"category": "Math", "title": "Genuine deformations of submanifolds II:the conformal case", "abstract": "We extend to the conformal realm the concept of genuine deformations of submanifolds, introduced by Dajczer and the first author for the isometric case. Analogously to that case, we call a conformal deformation of a submanifold $M^n$ genuine if no open subset of $M^n$ can be included as a submanifold of a higher dimensional conformally deformable submanifold in such a way that the conformal deformation of the former is induced by a conformal deformation of the latter. We describe the geometric structure of a submanifold that admits a genuine conformal deformation and give several applications showing the unifying character of this concept."}
{"category": "Math", "title": "Functoriality for the su(3) Khovanov homology", "abstract": "We prove that Morrison and Nieh's categorification of the su(3) quantum knot invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified su(2) theory, which was not functorial as originally defined. We use methods of Bar-Natan to construct explicit chain maps for each variation of the third Reidemeister move. Then, to show functoriality, we modify arguments used by Clark, Morrison, and Walker to show that induced chain maps are invariant under Carter and Saito's movie moves."}
{"category": "Math", "title": "On common divisors of multinomial coefficients", "abstract": "Erd\\H{o}s and Szekeres showed in 1978 that for any four positive integers satisfying m_1+m_2 = n_1+n_2, the two binomial coefficients (m_1+m_2)!/m_1! m_2! and (n_1+n_2)!/n_1! n_2! have a common divisor >1. The analogous statement for families of k k-nomial coefficients (k>1) was conjectured in 1997 by David Wasserman. Erd\\H{o}s and Szekeres remark that if m_1, m_2, n_1, n_2 as above are all >1, there is probably a lower bound on the common divisor in question which goes to infinity as a function of m_1+m_2. Such a bound is here obtained. Results are proved that narrow the class of possible counterexamples to Wasserman's conjecture. On the other hand, several plausible generalizations of that conjecture are shown to be false."}
{"category": "Math", "title": "Local additive estimation", "abstract": "Additive models are popular in high--dimensional regression problems because of flexibility in model building and optimality in additive function estimation. Moreover, they do not suffer from the so-called {\\it curse of dimensionality} generally arising in nonparametric regression setting. Less known is the model bias incurring from the restriction to the additive class of models. We introduce a new class of estimators that reduces additive model bias and at the same time preserves some stability of the additive estimator. This estimator is shown to partially relieve the dimensionality problem as well. The new estimator is constructed by localizing the assumption of additivity and thus named {\\it local additive estimator}. Implementation can be easily made with any standard software for additive regression. For detailed analysis we explicitly use the smooth backfitting estimator by Mammen, Linton and Nielsen (1999)."}
{"category": "Math", "title": "The existence problem for Steiner networks in strictly convex domains", "abstract": "We consider the existence problem for `Steiner networks' (trivalent graphs with 120 degree angles at each junction) in strictly convex domains, with `Neumann' boundary conditions (orthogonal intersection with the domain boundary.) For each of the three possible combinatorial possibilities, sufficient conditions on the domain are derived for existence; in addition, in each case explicit examples of nonexistence are given."}
{"category": "Math", "title": "The Antinomy of the Liar and Provability", "abstract": "This work evidences that a sentence cannot be denominated by P and written as P IS NOT TRUE. It demonstrates that in a system in which Q denominates the sentence Q IS NOT PROVABLE it is not provable that Q is true and not provable."}
{"category": "Math", "title": "Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras", "abstract": "Let $C$ and $A$ be two unital separable amenable simple C*-algebras with tracial rank no more than one. Suppose that $C$ satisfies the Universal Coefficient Theorem and suppose that $\\phi_1, \\phi_2: C\\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\\{u_t: t\\in [0, \\infty)\\}$ of $A$ such that $$ \\lim_{t\\to\\infty}u_t^*\\phi_1(c)u_t=\\phi_2(c)\\tforal c\\in C $$ if and only if $[\\phi_1]=[\\phi_2]$ in $KK(C,A),$ $\\phi_1^{\\ddag}=\\phi_2^{\\ddag},$ $(\\phi_1)_T=(\\phi_2)_T$ and a rotation related map $\\bar{R}_{\\phi_1,\\phi_2}$ associated with $\\phi_1$ and $\\phi_2$ is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class ${\\cal A}$ of unital separable simple amenable \\CA s which is strictly larger than the class of separable \\CA s whose tracial rank are zero or one. The class contains all unital simple ASH-algebras whose state spaces of $K_0$ are the same as the tracial state spaces as well as the simple inductive limits of dimension drop circle algebras. Moreover it contains some unital simple ASH-algebras whose $K_0$-groups are not Riesz. One consequence of the main result is that all unital simple AH-algebras which are ${\\cal Z}$-stable are isomorphic to ones with no dimension growth."}
{"category": "Math", "title": "On \"small geodesics\" and free loop spaces", "abstract": "A topological group is constructed which is homotopy equivalent to the pointed loop space of a path-connected Riemannian manifold $M$ and which is given in terms of \"composable small geodesics\" on $M$. This model is analogous to J. Milnor's free group construction \\cite{Milnor} which provides a model for the pointed loop space of a connected simplicial complex. Related function spaces are constructed from \"composable small geodesics\" which provide models for the free loop space of $M$ as well as the space of continuous maps from a surface to $M$."}
{"category": "Math", "title": "On the Sum-Product Problem on Elliptic Curves", "abstract": "Let $\\E$ be an ordinary elliptic curve over a finite field $\\F_{q}$ of $q$ elements and $x(Q)$ denote the $x$-coordinate of a point $Q = (x(Q),y(Q))$ on $\\E$. Given an $\\F_q$-rational point $P$ of order $T$, we show that for any subsets $\\cA, \\cB$ of the unit group of the residue ring modulo $T$, at least one of the sets $$ \\{x(aP) + x(bP) : a \\in \\cA, b \\in \\cB\\} \\quad\\text{and}\\quad \\{x(abP) : a \\in \\cA, b \\in \\cB\\} $$ is large. This question is motivated by a series of recent results on the sum-product problem over finite fields and other algebraic structures."}
{"category": "Math", "title": "Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci Hamiltonian", "abstract": "We consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently small. As a consequence, for these values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of the Fibonacci Hamiltonian all coincide and are smooth functions of the coupling constant."}
{"category": "Math", "title": "Linearization of the inverse conductivity problem", "abstract": "A positive function (conductivity) on the edges of a graph induces the Dirichlet-to- Neumann map between boundary values of harmonic functions. The inverse conductivity problem is to find the conductivity from the Dirichlet-to-Neumann map. We will show that the map from logarithm of conductivity to the certain logarithms of the determinants of the submatrices of the Dirichlet-to-Neumann map is linear(!) and so the solution of the inverse problem is reduced to solution of the system of linear equations that arise from disjoint paths in the graph. We will make a calculation for a simple tensor product lattice graph and conjecture that it generalizes to planar and three dimensional graphs and also to the continuous case. Depending on the graph the algorithm resembles or not the layer-stripping."}
{"category": "Math", "title": "The Square of the Dirichlet-to-Neumann map equals minus Laplacian", "abstract": "The Dirichlet-to-Neumann maps connect boundary values of harmonic functions. It is an amazing fact that the square of the non-local Dirichlet-to-Neumann map for the uniform conductivity 1 on the unit disc equals minus the local(!) Laplace operator on the boundary circle. To establish a new connection between discrete and continuous Dirichlet-to-Neumann maps and for the approximations I construct a finite and an infinite graphs which Dirichlet-to-Neumann map have the same property: \\Lambda^2(1) = - d^2/d \\theta^2. The construction gives a new continued fraction identity. It is interesting to consider the geometric and probabilistic (trajectories of the random walk) consequences of this localizing identity unifying discrete and continuous equations for potentials."}
{"category": "Math", "title": "Spectrum of analytic continuation", "abstract": "I will show that operator of analytic (harmonic) continuation on a lattice graph has a positive spectrum. I use a theorem about positivity of eigenvalues of totally positive matrices. I conjecture that by approximation the similar result holds in continuous case on a plane."}
{"category": "Math", "title": "The generalized Cassels-Tate dual exact sequence for 1-motives", "abstract": "We establish a generalized Cassels-Tate dual exact sequence for 1-motives over global fields. We thereby extend the main theorem of [4] from abelian varieties to arbitrary 1-motives."}
{"category": "Math", "title": "Computing a Generating Set of Arithmetic Kleinian Groups", "abstract": "The goal of this paper is to demonstrate the use of techniques from hyperbolic geometry to compute generating sets of certain subgroups of $SL^+(2,\\mathbb{C})$; specifically, $SO^+(Q,\\mathbb{Z})$ for $Q$ some integral quadratic form of signature $(3,1)$ that does not represent 0. The algorithm is illustrated for the form $Q_7=x_1^2+x_2^2+x_3-7x^4$, and explicit generating matrices are found."}
{"category": "Math", "title": "(2+2)-free posets, ascent sequences and pattern avoiding permutations", "abstract": "We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2+2)-free posets and a certain class of involutions (or chord diagrams), already appeared in the literature, but were apparently not known to be equinumerous. We present a direct bijection between them. The third class is a family of permutations defined in terms of a new type of pattern. An attractive property of these patterns is that, like classical patterns, they are closed under the action of $D_8$, the symmetry group of the square. The fourth class is formed by certain integer sequences, called ascent sequences, which have a simple recursive structure and are shown to encode (2+2)-free posets and permutations. Our bijections preserve numerous statistics. We determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for the class of chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3{\\bar 1}52{\\bar 4}$ and use this to enumerate those permutations, thereby settling a conjecture of Pudwell."}
{"category": "Math", "title": "Simply branched covers of an elliptic curve and the moduli space of curves", "abstract": "Consider genus g curves that admit degree d covers to an elliptic curve simply branched at 2g-2 points. Vary a branch point and the locus of such covers forms a one-parameter family W. We investigate the geometry of W by using admissible covers to study its slope, genus and components. The results can also be applied to study slopes of effective divisors on the moduli space of genus g curves."}
{"category": "Math", "title": "The equivariant Euler characteristic of real Coxeter toric varieties", "abstract": "Let $W$ be a Weyl group, and let $\\CT_W$ be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of $W$, and its weight lattice. The real locus $\\CT_W(\\R)$ is a smooth, connected, compact manifold with a $W$-action. We give a formula for the equivariant Euler characteristic of $\\CT_W(\\R)$ as a generalised character of $W$. In type $A_{n-1}$ for $n$ odd, one obtains a generalised character of $\\Sym_n$ whose degree is (up to sign) the $n^{\\text{th}}$ Euler number."}
{"category": "Math", "title": "Automorphisms of polynomial algebras and Dirichlet series", "abstract": "Let GF(q)[x,y] be the polynomial algebra in two variables over the finite field GF(q) with q elements. We give an exact formula and the asymptotics for the number p(n) of automorphisms (f,g) of GF(q)[x,y] such that max{deg(f),deg(g)}=n. We describe also the Dirichlet series generating function p(1)/1^s+p(2)/2^s+p(3)/3^s+.... The same results hold for the automorphisms of the free associative algebra GF(q)<x,y>. We have also obtained analogues for free algebras with two generators in Nielsen - Schreier varieties of algebras."}
{"category": "Math", "title": "A Temperley-Lieb analogue for the BMW algebra", "abstract": "The Temperley-Lieb algebra may be thought of as a quotient of the Hecke algebra of type A, acting on tensor space as the commutant of the usual action of quantum sl(2) on the n-th tensor power of the 2-dimensional irreducible module. We define and study a quotient of the Birman-Wenzl-Murakami algebra, which plays an analogous role for the 3-dimensional representation of quantum sl(2). In the course of the discussion we prove some general results about the radical of a cellular algebra, which may be of independent interest."}
{"category": "Math", "title": "Infinite-dimensionality of the Automorphism Groups of Homogeneous Stein Manifolds", "abstract": "We show that the group of holomorphic automorphisms of a Stein manifold X of dimension greater than 1 is infinite-dimensional, provided X is a homogeneous space of a holomorphic action of a complex Lie group."}
{"category": "Math", "title": "Staggered t-structures on toric varieties", "abstract": "Achar has recently introduced a family of t-structures on the derived category of equivariant coherent sheaves on a $G$-scheme, generalizing the perverse coherent t-structures of Bezrukavnikov and Deligne. They are called \\emph{staggered} t-structures, and their main point of interest so far is that they are more often self-dual. In this paper we investigate these t-structures on the $T$-equivariant derived category of a toric variety."}
{"category": "Math", "title": "Non-cuspidality outside the middle degree of l-adic cohomology of the Lubin-Tate tower", "abstract": "In this article, we consider the representations of the general linear group over a non-archimedean local field obtained from the vanishing cycle cohomology of the Lubin-Tate tower. We give an easy and direct proof of the fact that no supercuspidal representation appears as a subquotient of such representations unless they are obtained from the cohomology of the middle degree. Our proof is purely local and does not require Shimura varieties."}
{"category": "Math", "title": "To sample or not to sample: Self-triggered control for nonlinear systems", "abstract": "Feedback control laws have been traditionally implemented in a periodic fashion on digital hardware. Although periodicity simplifies the analysis of the mismatch between the control design and its digital implementation, it also leads to conservative usage of resources such as CPU utilization in the case of embedded control. We present a novel technique that abandons the periodicity assumption by using the current state of the plant to decide the next time instant in which the state should be measured, the control law computed, and the actuators updated. This technique, termed self-triggered control, is developed for two classes of nonlinear control systems, namely, state-dependent homogeneous systems and polynomial systems. The wide applicability of the proposed results is illustrated in two well known physical examples: a jet engine compressor and the rigid body."}
{"category": "Math", "title": "Track billiards", "abstract": "We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the segments and the arcs, the billiards considered have non-zero Lyapunov exponents almost everywhere. These results are then extended to a similar class of of 3-dimensional billiards. Finally, we find that for some subclasses of track billiards, the mechanism generating hyperbolicity is not the defocusing one that requires every infinitesimal beam of parallel rays to defocus after every reflection off of the focusing boundary."}
{"category": "Math", "title": "On lattices of maximal index two", "abstract": "The maximal index of a Euclidean lattice L of dimension n is the maximal index of the sub-lattices of L spanned by n independent minimal vectors of L. In this paper, we prove that a perfect lattice of maximal index two not provided by a cross-section has dimension at most 5."}
{"category": "Math", "title": "Multifractal analysis of weak Gibbs measures for non-uniformly expanding C^1 maps", "abstract": "We consider the local dimension spectrum of a weak Gibbs measure on a C^1 non-uniformly hyperbolic system of Manneville- Pomeau type. We present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems."}
{"category": "Math", "title": "Justification of an asymptotic expansion at infinity", "abstract": "A family of asymptotic solutions at infinity for the system of ordinary differential equations is considered. Existence of exact solutions which have these asymptotics is proved."}
{"category": "Math", "title": "High dimensional gaussian classification", "abstract": "High dimensional data analysis is known to be as a challenging problem. In this article, we give a theoretical analysis of high dimensional classification of Gaussian data which relies on a geometrical analysis of the error measure. It links a problem of classification with a problem of nonparametric regression. We give an algorithm designed for high dimensional data which appears straightforward in the light of our theoretical work, together with the thresholding estimation theory. We finally attempt to give a general treatment of the problem that can be extended to frameworks other than gaussian."}
{"category": "Math", "title": "Chow Stability of Curves of Genus 4 in P^3", "abstract": "In the paper, we study the GIT construction of the moduli space of Chow semistable curves of genus 4 in P^3. By using the GIT method developed by Mumford and a deformation theoretic argument, we give a modular description of this moduli space. We classify Chow stable or Chow semistable curves when they are irreducible or nonreduced. Then we work out the case when a curve has two components. Our classification provides some clues to understand the birational map from the moduli space of stable curves of genus 4 to the moduli space of Chow semistable curves of genus 4 in P^3."}
{"category": "Math", "title": "Quasi-compactness of transfer operators for contact Anosov flows", "abstract": "For any $C^r$ contact Anosov flow with $r\\ge 3$, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all $C^r$ functions, such that the transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. Further we give explicit bounds on the essential spectral radii of the extensions in terms of the differentiability r and the hyperbolicity exponents of the flow."}
{"category": "Math", "title": "On astheno-Kaehler metrics", "abstract": "A Hermitian metric on a complex manifold of complex dimension $n$ is called {\\em astheno-K\\\"ahler} if its fundamental $2$-form $F$ satisfies the condition $\\partial \\overline \\partial F^{n - 2} =0$. If $n =3$, then the metric is {\\em strong KT}, i.e. $F$ is $\\partial \\overline \\partial$-closed. By using blow-ups and the twist construction, we construct simply-connected astheno-K\\\"ahler manifolds of complex dimension $n > 3$. Moreover, we construct a family of astheno-K\\\"ahler (non strong KT) $2$-step nilmanifolds of complex dimension $4$ and we study deformations of strong KT structures on nilmanifolds of complex dimension $3$. Finally, we study the relation between astheno-K\\\"ahler condition and (locally) conformally balanced one and we provide examples of locally conformally balanced astheno-K\\\"ahler metrics on $\\T^2$-bundles over (non-K\\\"ahler) homogeneous complex surfaces."}
{"category": "Math", "title": "Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures", "abstract": "In this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let $\\b$ denote the inverse temperature and let $\\L_\\b \\subset \\Z^2$ be a square box with periodic boundary conditions such that $\\lim_{\\b\\to\\infty}|\\L_\\b|=\\infty$. We run the dynamics on $\\L_\\b$ starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles)are small. For large $\\b$, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single \\emph{critical droplet} somewhere in $\\L_\\b$. Using potential-theoretic methods, we compute the \\emph{average nucleation time} (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as $\\b\\to\\infty$. It turns out that this time grows as $Ke^{\\Gamma\\b}/|\\L_\\b|$ for Glauber dynamics and $K\\b e^{\\Gamma\\b}/|\\L_\\b|$ for Kawasaki dynamics, where $\\Gamma$ is the local canonical, respectively, grand-canonical energy to create a critical droplet and $K$ is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on $|\\L_\\b|$). The fact that the average nucleation time is inversely proportional to $|\\L_\\b|$ is referred to as \\emph{homogeneous nucleation}, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition $\\L_\\b$."}
{"category": "Math", "title": "\\\"Uber Pro-p-Fundamentalgruppen markierter arithmetischer Kurven", "abstract": "Let k be a global field, p an odd prime number different from char(k) and S, T disjoint, finite sets of primes of k. Let G_S^T(k)(p)=Gal(k_S^T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and completely split at T. We prove the existence of a finite set of primes S_0, which can be chosen disjoint from any given set M of Dirichlet density zero, such that the cohomology of G_{S\\cup S_0}^T(k)(p) coincides with the etale cohomology of the associated marked arithmetic curve. In particular, cd G_{S\\cup S_0}^T(k)(p)=2. Furthermore, we can choose S_0 in such a way that k_{S\\cup S_0}^T(p) realizes the maximal p-extension k_\\p(p) of the local field k_\\p for all \\p\\in S\\cup S_0, the cup-product H^1(G_{S\\cup S_0}^T(k)(p),\\F_p) \\otimes H^1(G_{S\\cup S_0}^T(k)(p),\\F_p) --> H^2(G_{S\\cup S_0}^T(k)(p),\\F_p) is surjective and the decomposition groups of the primes in S establish a free product inside G_{S\\cup S_0}^T(k)(p). This generalizes previous work of the author where similar results were shown in the case T=\\emptyset under the restrictive assumption p\\nmid Cl(k) and \\zeta_p\\notin k."}
{"category": "Math", "title": "On Triple Veronese Embeddings of $\\PP_n$ in the Grassmannians", "abstract": "We classify all the embeddings of $\\mathbb{P}_n$ in a Grassmannian $Gr(1,N)$ such that the composition with Pl\\\"{u}cker embedding is given by a linear system of cubics on $\\mathbb{P}_n$. As a corollary in the direction of the Hartshorne conjecture, we prove that every vector bundle giving such an embedding, splits if $n\\geq 3$."}
{"category": "Math", "title": "Upper bounds for the moments of zeta prime rho", "abstract": "Assuming the Riemann Hypothesis, we obtain an upper bound for the 2k-th moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\\zeta(s)$ for every positive integer k. Our bounds are nearly as sharp as the conjectured asymptotic formulae for these moments. The proof is based upon a recent method of K. Soundararajan that provides analogous bounds for continuous moments of the Riemann zeta-function as well as for moments L-functions at the central point, averaged over families."}
{"category": "Math", "title": "Power reductivity over an arbitrary base", "abstract": "Our starting point is Mumford's conjecture, on representations of Chevalley groups over fields, as it is phrased in the preface of \"Geometric Invariant Theory\". After extending the conjecture appropriately, we show that it holds over an arbitrary commutative base ring. We thus obtain the first fundamental theorem of invariant theory (often referred to as Hilbert's fourteenth problem) over an arbitrary Noetherian ring. We also prove results on the Grosshans graded deformation of an algebra in the same generality. We end with tentative finiteness results for rational cohomology over the integers."}
{"category": "Math", "title": "On slowdown and speedup of transient random walks in random environment", "abstract": "We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^\\kappa)$ from the origin, $\\kappa\\in(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{\\nu_0})$ from the origin, $\\nu_0\\in (0,\\kappa)$), and speedup (at time $n$, the particle is at a distance of order $n^{\\nu_1}$ from the origin, $\\nu_1\\in (\\kappa,1)$), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time $n$, the particle is located around $(-n^\\nu)$, thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well."}
{"category": "Math", "title": "Mixed succession rules: the commutative case", "abstract": "We begin a systematic study of the enumerative combinatorics of mixed succession rules, which are succession rules such that, in the associated generating tree, the nodes are allowed to produce their sons at several different levels according to different production rules. Here we deal with a specific case, namely that of two different production rules whose rule operators commute. In this situation, we are able to give a general formula expressing the sequence associated with the mixed succession rules in terms of the sequences associated with the component production rules. We end by providing some examples illustrating our approach."}
{"category": "Math", "title": "Recurrence relations for powers of q-Fibonacci polynomials", "abstract": "We derive some q-analogs of Euler-Cassini-type identities and of recurrence formulas for powers of Fibonacci polynomials."}
{"category": "Math", "title": "A class of non homogeneous self interacting random processes with applications to learning in games and vertex-reinforced random walks", "abstract": "Using an approximation by a set-valued dynamical system, this paper studies a class of non Markovian and non homogeneous stochastic processes on a finite state space. It provides an unified approach to simulated annealing type processes. It permits to study new models of vertex reinforced random walks and new models of learning in games including Markovian fictitious play."}
{"category": "Math", "title": "A generalization of Watts's Theorem: Right exact functors on module categories", "abstract": "Watts's Theorem says that a right exact functor F:Mod R-->Mod S that commutes with direct sums is isomorphic to -\\otimes_R B where B is the R-S-bimodule FR. The main result in this paper is the following: if A is a cocomplete abelian category and F:Mod R --> A is a right exact functor commuting with direct sums, then F is isomorphic to - \\otimes_R B where B is a suitable R-module in A, i.e., a pair (B,f) consisting of an object B in A and a ring homomorphism f:R --> Hom_A(B,B). Part of the point is to give meaning to the notation -\\otimes_R B. That is done in the paper by Artin and Zhang on Abstract Hilbert Schemes. The present paper is a natural extension of some of the ideas in the first part of their paper."}
{"category": "Math", "title": "Quasimaps, straightening laws, and quantum cohomology for the Lagrangian Grassmannian", "abstract": "The Drinfel'd Lagrangian Grassmannian compactifies the space of algebraic maps of fixed degree from the projective line into the Lagrangian Grassmannian. It has a natural projective embedding arising from the canonical embedding of the Lagrangian Grassmannian. We show that the defining ideal of any Schubert subvariety of the Drinfel'd Lagrangian Grassmannian is generated by polynomials which give a straightening law on an ordered set. Consequentially, any such subvariety is Cohen-Macaulay and Koszul. The Hilbert function is computed from the straightening law, leading to a new derivation of certain intersection numbers in the quantum cohomology ring of the Lagrangian Grassmannian."}
{"category": "Math", "title": "$p$-adic Hurwitz numbers", "abstract": "We introduce stable tropical curves and use these to count covers of the $p$-adic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the covers."}
{"category": "Math", "title": "A new (?) continued fraction expansion for the reciprocal of a $q$-series", "abstract": "We prove a continued fraction expansion for the reciprocal of a certain $q$-series. All the specialists in the world are asked whether it is new or not."}
{"category": "Math", "title": "Maple Symbolic Computation in the Calculus of Variations", "abstract": "It is the aim of this work to identify and illustrate the potential and weaknesses of the computer algebra system Maple in the area of the Calculus of Variations: a classical area of mathematics that studies the methods for finding maximum and minimum values of functionals of integral type. Variational problems are usually solved with the help of the necessary optimality conditions of Euler-Lagrange, which are, in general, nonlinear and difficult second order differential equations to be solved. We show how the computer algebra system Maple can be useful in the determination and resolution of these equations. We will also present the solution to the celebrated brachistochrone problem from the point of view of the calculus of variations and the Maple system, and a reformulation of the classical problem obtained by restricting the class of admissible functions."}
{"category": "Math", "title": "Noncommutative Dunkl operators and braided Cherednik algebras", "abstract": "We introduce braided Dunkl operators that are acting on a q-polynomial algebra and q-commute. Generalizing the approach of Etingof and Ginzburg, we explain the q-commutation phenomenon by constructing braided Cherednik algebras for which the above operators form a representation. We classify all braided Cherednik algebras using the theory of braided doubles developed in our previous paper. Besides ordinary rational Cherednik algebras, our classification gives new algebras attached to an infinite family of subgroups of even elements in complex reflection groups, so that the corresponding braided Dunkl operators pairwise anti-commute. We explicitly compute these new operators in terms of braided partial derivatives and divided differences."}
{"category": "Math", "title": "Elliptic Littlewood identities", "abstract": "We prove analogues for elliptic interpolation functions of Macdonald's version of the Littlewood identity for (skew) Macdonald polynomials, in the process developing an interpretation of general elliptic \"hypergeometric\" sums as skew interpolation functions. One such analogue has an interpretation as a \"vanishing integral\", generalizing a result of arXiv:math/0606204; the structure of this analogue gives sufficient insight to enable us to conjecture elliptic versions of most of the other vanishing integrals of arXiv:math/0606204 as well. We are thus led to formulate ten conjectures, each of which can be viewed as a multivariate quadratic transformation, and can be proved in a number of special cases."}
{"category": "Math", "title": "A note on Artin-Markov normal form theorem for braid groups", "abstract": "In a recent paper by L. A. Bokut, V. V. Chaynikov and K. P. Shum in 2007, Braid group $B_n$ is represented by Artin-Burau's relations. For such a representation, it is told that all other compositions can be checked in the same way. In this note, we support this claim and check all compositions."}
{"category": "Math", "title": "The Gram matrix of a Temperley-Lieb algebra is similar to the matrix of chromatic joins", "abstract": "In this paper we show that the matrix of chromatic joins and the Gram matrix of the Temperley-Lieb algebra are similar (after rescaling), with the change of basis given by diagonal matrices."}
{"category": "Math", "title": "Characterization of Vibrating Plates by Bi-Laplacian Eigenvalue Problems", "abstract": "In this paper we derive boundary integral identities for the bi-Laplacian eigenvalue problems under Dirichlet, Navier and simply-supported boundary conditions. By using these identities, we first obtain the uniqueness criteria for the solutions of the bi-Laplacian eigenvalue problems, and then prove that each eigenvalue of the problem with simply-supported boundary condition increases strictly with Poisson's ratio, thereby showing that each natural frequency of a simply-supported vibrating plate increases strictly with Poisson's ratio. In addition, we obtain boundary integral representations for the strain energies of the vibrating plates under the three boundary conditions."}
{"category": "Math", "title": "On randomly placed arcs on the circle", "abstract": "We completely describe in terms of Hausdorff measures the size of the set of points of the circle that are covered infinitely often by a sequence of random arcs with given lengths. We also show that this set is a set with large intersection."}
{"category": "Math", "title": "Weighted grassmannians and stable hyperplane arrangements", "abstract": "We give a common generalization of (1) Hassett's weighted stable curves, and (2) Hacking-Keel-Tevelev's stable hyperplane arrangements."}
{"category": "Math", "title": "Tridiagonal pairs and the q-tetrahedron algebra", "abstract": "In this paper we further develop the connection between tridiagonal pairs and the q-tetrahedron algebra $\\boxtimes_q$. Let V denote a finite dimensional vector space over an algebraically closed field and let A, A^* denote a tridiagonal pair on V. For $0 \\leq i \\leq d$ let $\\theta_i$ (resp. $\\theta^*_i$) denote a standard ordering of the eigenvalues of A (resp. A^*). Fix a nonzero scalar q which is not a root of unity. T. Ito and P. Terwilliger have shown that when $\\theta_i = q^{2i-d}$ and $\\theta^*_i = q^{d-2i}$ there exists an irreducible $\\boxtimes_q$-module structure on V such that the $\\boxtimes_q$ generators x_{01}, x_{23} act as A, A^* respectively. In this paper we examine the case in which there exists a nonzero scalar c in K such that $\\theta_i = q^{2i-d}$ and $\\theta^*_i = q^{2i-d} + c q^{d-2i}$. In this case we associate to A,A^* a polynomial P and prove the following equivalence. The following are equivalent: (i) There exists a $\\boxtimes_q$-module structure on V such that x_{01} acts as A and x_{30} + cx_{23} acts as A^*, where x_{01}, x_{30}, x_{23} are standard generators for $\\boxtimes_q$. (ii) P(q^{2d-2} (q-q^{-1})^{-2}) \\neq 0. Suppose (i),(ii) hold. Then the $\\boxtimes_q$-module structure on V is unique and irreducible."}
{"category": "Math", "title": "Involutions of 3-dimensional handlebodies", "abstract": "We study the orientation preserving involutions of the orientable 3-dimensional handlebody $H_g$, for any genus $g$. A complete classification of such involutions is given in terms of their fixed points."}
{"category": "Math", "title": "Homotopy Type of the Boolean Complex of a Coxeter System", "abstract": "In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank n, the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group. of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group."}
{"category": "Math", "title": "Computation of Topological Entropy via $\\phi$-expansion, an Inverse Problem for the Dynamical Systems $\\beta x+\\alpha \\mod 1$", "abstract": "We give an algorithm, based on the $\\phi$-expansion of Parry, in order to compute the topological entropy of a class of shift spaces. The idea is the solve an inverse problem for the dynamical systems $\\beta x+\\alpha \\mod1$.The first part is an exposition of the $\\phi$-expansion applied to piecewise monotone dynamical systems. We formulate for the validity of the $\\phi$-expansion, necessary and sufficient conditions, which are different from those in Parry's paper."}
{"category": "Math", "title": "Reflected Solutions of Backward Doubly Stochastic Differential Equations", "abstract": "We study reflected solutions of one-dimensional backward doubly stochastic differential equations (BDSDEs in short). The \"reflected\" keeps the solution above a given stochastic process. We get the uniqueness and existence by penalization. For the existence of backward stochastic integral, our proof is different from [KKPPQ] slightly. We also obtain a comparison theorem for reflected BDSDEs."}
{"category": "Math", "title": "Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers", "abstract": "Let $P$ be a probability distribution on $\\mathbb{R}^d$ (equipped with an Euclidean norm $|\\cdot|$). Let $ r> 0 $ and let $(\\alpha_n)_{n \\geq1}$ be an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(\\alpha_n)_{n \\geq1}$ defined for every $n \\geq1$ by $\\rho(\\alpha_n) = \\max{|a|, a \\in\\alpha_n}$. When $\\card(\\supp(P))$ is infinite, the maximal radius sequence goes to $\\sup{|x|, x \\in\\operatorname{supp}(P)}$ as $n$ goes to infinity. We then give the exact rate of convergence for two classes of distributions with unbounded support: distributions with hyper-exponential tails and distributions with polynomial tails. In the one-dimensional setting, a sharp rate and constant are provided for distributions with hyper-exponential tails."}
{"category": "Math", "title": "Calculus on Lie algebroids, Lie groupoids and Poisson manifolds", "abstract": "We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds, whose links with Lie algebroids are very close."}
{"category": "Math", "title": "Cycles Of Given Length In Oriented Graphs", "abstract": "We show that for each \\ell\\geq 4 every sufficiently large oriented graph G with \\delta^+(G), \\delta^-(G) \\geq \\lfloor |G|/3 \\rfloor +1 contains an \\ell-cycle. This is best possible for all those \\ell\\geq 4 which are not divisible by 3. Surprisingly, for some other values of \\ell, an \\ell-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an \\ell-cycle (with \\ell \\geq 4 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity and consider \\ell-cycles in general digraphs."}
{"category": "Math", "title": "Prime pairs and Zeta's zeros", "abstract": "There is extensive numerical support for the prime-pair conjecture (PPC) of Hardy and Littlewood (1923) on the asymptotic behavior of pi_{2r}(x), the number of prime pairs (p,p+2r) with p not exceeding x. However, it is still not known whether there are infinitely many prime pairs with given even difference! Using a strong hypothesis on (weighted) equidistribution of primes in arithmetic progressions, Goldston, Pintz and Yildirim have shown (2007) that there are infinitely many pairs of primes differing by at most sixteen. The present author uses a Tauberian approach to derive that the PPC is equivalent to specific boundary behavior of certain functions involving zeta's complex zeros. Under Riemann's Hypothesis and on the real axis, these functions resemble pair-correlation expressions. A speculative extension of Montgomery's classical work (1973) would imply that there must be an abundance of prime pairs."}
{"category": "Math", "title": "On the factor set of code loops", "abstract": "A Code loop on a binary linear code that is doubly even with a factor set is shown to be a central loop, conjugacy closed loop, Burn loop and extra loop. General forms of the identities that define the factor set of a code are deduced."}
{"category": "Math", "title": "Optimal Convergence Rates for Tikhonov Regularization in Besov Scales", "abstract": "In this paper we deal with linear inverse problems and convergence rates for Tikhonov regularization. We consider regularization in a scale of Banach spaces, namely the scale of Besov spaces. We show that regularization in Banach scales differs from regularization in Hilbert scales in the sense that it is possible that stronger source conditions may lead to weaker convergence rates and vive versa. Moreover, we present optimal source conditions for regularization in Besov scales."}
{"category": "Math", "title": "Existence and symmetry of least energy solutions for a class of quasi-linear elliptic equations", "abstract": "For a general class of autonomous quasi-linear elliptic equations on R^n we prove the existence of a least energy solution and show that all least energy solutions do not change sign and are radially symmetric up to a translation in R^n."}
{"category": "Math", "title": "Location and phase segregation of ground and excited states for 2D Gross-Pitaevskii systems", "abstract": "We consider a system of Gross-Pitaevskii equations in R^2 modelling a mixture of two Bose-Einstein condensates with repulsive interaction. We aim to study the qualitative behaviour of ground and excited state solutions. We allow two different harmonic and off-centered trapping potentials and study the spatial patterns of the solutions within the Thomas-Fermi approximation as well as phase segregation phenomena within the large-interaction regime."}
{"category": "Math", "title": "Asymptotic behavior of a thermoviscoelastic plate with memory effects", "abstract": "We consider a coupled linear system describing a thermoviscoelastic plate with hereditary effects. The system consists of a hyperbolic integrodifferential equation, governing the temperature, which is linearly coupled with the partial differential equation ruling the evolution of the vertical deflection, presenting a convolution term accounting for memory effects. It is also assumed that the thermal power contains a memory term characterized by a relaxation kernel. We prove that the system is exponentially stable and we obtain a closeness estimate between the system with memory effects and the corresponding memory-free limiting system, as the kernels fade in a suitable sense."}
{"category": "Math", "title": "The action of a nilpotent group on its horofunction boundary has finite orbits", "abstract": "We study the action of a nilpotent group G with finite generating set S on its horofunction boundary. We show that there is one finite orbit associated to each facet of the polytope obtained by projecting S into the infinite component of the abelianisation of G. We also prove that these are the only finite orbits of Busemann points. To finish off, we examine in detail the Heisenberg group with its usual generators."}
{"category": "Math", "title": "On the long term spatial segregation for a competition-diffusion system", "abstract": "We investigate the long term behavior for a class of competition-diffusion systems of Lotka-Volterra type for two competing species in the case of low regularity assumptions on the data. Due to the coupling that we consider the system cannot be reduced to a single equation yielding uniform estimates with respect to the inter-specific competition rate parameter. Moreover, in the particular but meaningful case of initial data with disjoint support and Dirichlet boundary data which are time-independent, we prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially segregated state satisfying some variational inequalities."}
{"category": "Math", "title": "Some combinatorics related to central binomial coefficients: Grand-Dyck paths, coloured noncrossing partitions and signed pattern avoiding permutations", "abstract": "We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus showing, in particular, that it is isomorphic to the structure induced by the (strong) Bruhat order on a certain set of signed pattern avoiding permutations."}
{"category": "Math", "title": "Twisted exterior derivatives for universal enveloping algebras I", "abstract": "Consider any representation $\\phi$ of a finite-dimensional Lie algebra $g$ by derivations of the completed symmetric algebra $\\hat{S}(g^*)$ of its dual. Consider the tensor product of $\\hat{S}(g^*)$ and the exterior algebra $\\Lambda(g)$. We show that the representation $\\phi$ extends canonically to the representation $\\tilde\\phi$ of that tensor product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncommutative differentials and coordinates are formal power series in partial derivatives. The square of the corresponding exterior derivative is zero like in the classical case, but the Leibniz rule is deformed."}
{"category": "Math", "title": "Supersequences, rearrangements of sequences, and the spectrum of bases in additive number theory", "abstract": "The set A = {a_n} of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be represented as the sum of h elements of A. If a_n ~ alpha n^h for some real number alpha > 0, then alpha is called an additive eigenvalue of order h. The additive spectrum of order h is the set N(h) consisting of all additive eigenvalues of order h. It is proved that there is a positive number eta_h <= 1/h! such that N(h) = (0, eta_h) or N(h) = (0, eta_h]. The proof uses results about the construction of supersequences of sequences with prescribed asymptotic growth, and also about the asymptotics of rearrangements of infinite sequences. For example, it is proved that there does not exist a strictly increasing sequence of integers B = {b_n} such that b_n ~ 2^n and B contains a subsequence {b_{n_k}} such that b_{n_k} ~ 3^k."}
{"category": "Math", "title": "Trivial centralizers for codimension-one attractors", "abstract": "We show that if $\\Lambda$ is a codimension-one hyperbolic attractor for a $C^r$ diffeomorphism $f$, where $2\\leq r\\leq \\infty$, and $f$ is not Anosov, then there is a neighborhood $\\mathcal{U}$ of $f$ in $\\mathrm{Diff}^r(M)$ and an open and dense set $\\mathcal{V}$ of $\\mathcal{U}$ such that any $g\\in\\mathcal{V}$ has a trivial centralizer on the basin of attraction for $\\Lambda$."}
{"category": "Math", "title": "The stochastic Hamilton-Jacobi equation", "abstract": "We extend some aspects of the Hamilton-Jacobi theory to the category of stochastic Hamiltonian dynamical systems. More specifically, we show that the stochastic action satisfies the Hamilton-Jacobi equation when, as in the classical situation, it is written as a function of the configuration space using a regular Lagrangian submanifold. Additionally, we will use a variation of the Hamilton-Jacobi equation to characterize the generating functions of one-parameter groups of symplectomorphisms that allow to rewrite a given stochastic Hamiltonian system in a form whose solutions are very easy to find; this result recovers in the stochastic context the classical solution method by reduction to the equilibrium of a Hamiltonian system."}
{"category": "Math", "title": "Ueber die Geometrie der alten Aegypter", "abstract": "Lecture given before the Royal Academy Vienna that summarizes the state of knowledge about the mathematics of the ancient Egyptians, up to 1884. Contains all relevant references to classical Greek texts, and the 'latest' archeology results. Later published as booklet. A list with completed bibliographic references is appended. ----- Vortrag vor der k.u.k. Akademie der Wissenschaften in Wien, 1884, ueber den damaligen Wissensstand zur Mathematik der Aegypter, mit Referenzen zu den klassischen griechischen Texten und damalige 'neue' Erkenntnisse der Archaeologie. Spaeter als Heftchen gedruckt. Hinzugefiegt wurden komplette bibliografische Referenzen fuer diejenigen, die im Text unvollstaendig angegeben wurden."}
{"category": "Math", "title": "Does Ten Have a Friend?", "abstract": "Any positive integer $n$ other than 10 with abundancy index 9/5 must be a square with at least 6 distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime power factorization of $n$."}
{"category": "Math", "title": "Competition between Discrete Random Variables, with Applications to Occupancy Problems", "abstract": "Consider $n$ players whose \"scores\" are independent and identically distributed values $\\{X_i\\}_{i=1}^n$ from some discrete distribution $F$. We pay special attention to the cases where (i) $F$ is geometric with parameter $p\\to0$ and (ii) $F$ is uniform on $\\{1,2,...,N\\}$; the latter case clearly corresponds to the classical occupancy problem. The quantities of interest to us are, first, the $U$-statistic $W$ which counts the number of \"ties\" between pairs $i,j$; second, the univariate statistic $Y_r$, which counts the number of strict $r$-way ties between contestants, i.e., episodes of the form ${X_i}_1={X_i}_2=...={X_i}_r$; $X_j\\ne {X_i}_1;j\\ne i_1,i_2,...,i_r$; and, last but not least, the multivariate vector $Z_{AB}=(Y_A,Y_{A+1},...,Y_B)$. We provide Poisson approximations for the distributions of $W$, $Y_r$ and $Z_{AB}$ under some general conditions. New results on the joint distribution of cell counts in the occupancy problem are derived as a corollary."}
{"category": "Math", "title": "Rigidity at the boundary for conformal structures and other Cartan geometries", "abstract": "In this paper, we consider the problem of building a conformal boundary, embedding a pseudo-Riamnnian manifold as an open subset of a bigger one. We get first results about conformal maximality. We also show that in dimension $\\geq 3$, there are rigidity properties for the topological boundary of such a conformal embedding. We get results of the same kind about general Cartan geometries."}
{"category": "Math", "title": "Negative Entropy, Zero temperature and stationary Markov Chains on the interval", "abstract": "We analyze some properties of maximizing stationary Markov probabilities on the Bernoulli space $[0,1]^\\mathbb{N}$, More precisely, we consider ergodic optimization for a continuous potential $A$, where $A: [0,1]^\\mathbb{N}\\to \\mathbb{R}$ which depends only on the two first coordinates. We are interested in finding stationary Markov probabilities $\\mu_\\infty$ on $ [0,1]^\\mathbb{N}$ that maximize the value $ \\int A d \\mu,$ among all stationary Markov probabilities $\\mu$ on $[0,1]^\\mathbb{N}$. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities $\\mu_\\beta$ which weakly converges to $\\mu_\\infty$. The probabilities $\\mu_\\beta$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure. Under the hypothesis of $A$ being $C^2$ and the twist condition, that is, $\\frac{\\partial^2 A}{\\partial_x \\partial_y} (x,y) \\neq 0$, for all $(x,y) \\in [0,1]^2$, we show the graph property."}
{"category": "Math", "title": "Hyperbolic Groups Which Fiber in Infinitely Many Ways", "abstract": "We construct examples of free-by-cyclic hyperbolic groups which fiber in infinitely many ways over Z. The construction involves adding a specialized square 2-cell to a non-positively curved, squared 2-complex defined by labeled oriented graphs. The fundamental groups of the resulting complexes are hyperbolic, free-by-cyclic and can be mapped onto Z in infinitely many ways."}
{"category": "Math", "title": "Gorenstein rings through face rings of manifolds", "abstract": "The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the sphere $g$-conjecture implies all enumerative consequences of its far reaching generalization (due to Kalai) to manifolds. A special case of Kalai's manifold $g$-conjecture is established for homology manifolds that have a codimension-two face whose link contains many vertices."}
{"category": "Math", "title": "Equivalence and Disintegration Theorems for Fell Bundles and their C*-algebras", "abstract": "We study the C*-algebras of Fell bundles. In particular, we prove the analogue of Renault's disintegration theorem for groupoids. As in the groupoid case, this result is the key step in proving a deep equivalence theorem for the C*-algebras of Fell bundles."}
{"category": "Math", "title": "Survival time of random walk in random environment among soft obstacles", "abstract": "We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d.\\ random field of obstacles. When the particle hits an obstacle, it disappears with a positive probability. We obtain quenched and annealed bounds on the tails of the survival time in the general $d$-dimensional case. We then consider a simplified one-dimensional model (where transition probabilities and obstacles are independent and the RWRE only moves to neighbour sites), and obtain finer results for the tail of the survival time. In addition, we study also the \"mixed\" probability measures (quenched with respect to the obstacles and annealed with respect to the transition probabilities and vice-versa) and give results for tails of the survival time with respect to these probability measures. Further, we apply the same methods to obtain bounds for the tails of hitting times of Branching Random Walks in Random Environment (BRWRE)."}
{"category": "Math", "title": "Frobenius Amplitude, Ultraproducts, and Vanishing on Singular Spaces", "abstract": "When X is a singular complex projective variety, we prove a Kodaira type vanishing theorem generalizing results of Navarro Aznar and others. This is proved by extending the Deligne-Illusie decomposition to the Du Bois complex. We also give a new definition of Frobenius amplitude (introduced in math.AG/0202129) using ultraproducts."}
{"category": "Math", "title": "Spectral analysis of transport equations with bounce-back boundary conditions", "abstract": "We investigate the spectral properties of the time-dependent linear transport equation with bounce-back boundary conditions. A fine analysis of the spectrum of the streaming operator is given and the explicit expression of the strongly continuous streaming semigroup is derived. Next, making use of a recent result from M. Sbihi \\cite{Sbihi}, we prove, via a compactness argument, that the essential spectrum of the transport semigroup and that of the streaming semigroup coincide on all $L^p$-spaces with $1<p<\\infty$."}
{"category": "Math", "title": "Wave Equations on Lorentzian Manifolds and Quantization", "abstract": "This book provides a detailed introduction to linear wave equations on Lorentzian manifolds (for vector-bundle valued fields). After a collection of preliminary material in the first chapter one finds in the second chapter the construction of local fundamental solutions together with their Hadamard expansion. The third chapter establishes the existence and uniqueness of global fundamental solutions on globally hyperbolic spacetimes and discusses Green's operators and well-posedness of the Cauchy problem. The last chapter is devoted to field quantization in the sense of algebraic quantum field theory. The necessary basics on C*-algebras and CCR-representations are developed in full detail. The text provides a self-contained introduction to these topics addressed to graduate students in mathematics and physics. At the same time it is intended as a reference for researchers in global analysis, general relativity, and quantum field theory."}
{"category": "Math", "title": "The group of order preserving automorphisms of the ring of differential operators on Laurent polynomial algebra in prime characteristic", "abstract": "Let $K$ be a field of characteristic $p>0$. It is proved that the group $\\Aut_{ord}(\\CD (L_n))$ of order preserving automorphisms of the ring $\\CD (L_n)$ of differential operators on a Laurent polynomial algebra $L_n:= K[x_1^{\\pm 1}, ..., x_n^{\\pm 1}]$ is isomorphic to a skew direct product of groups $\\Zp^n \\rtimes \\Aut_K(L_n)$ where $\\Zp$ is the ring of $p$-adic integers. Moreover, the group $\\Aut_{ord}(\\CD (L_n))$ is found explicitly. Similarly, $\\Aut_{ord}(\\CDPn)\\simeq \\Aut_K(P_n)$ where $P_n: =K[x_1, ..., x_n]$ is a polynomial algebra."}
{"category": "Math", "title": "An upper bound on the multiplicative energy", "abstract": "We prove that the sumset or the productset of any finite set of real numbers, $A,$ is at least $|A|^{4/3-\\epsilon},$ improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, $E(A,A).$"}
{"category": "Math", "title": "Irregular connections and Kac-Moody root systems", "abstract": "Some moduli spaces of irregular connections on the trivial bundle over the Riemann sphere will be identified with Nakajima quiver varieties. In particular this enables us to associate a Kac-Moody root system to such connections (yielding many isomorphisms between such moduli spaces, via the reflection functors for the corresponding Weyl group). The possibility of 'reading' a quiver in different ways also yields numerous isomorphisms between such moduli spaces, often between spaces of connections on different rank bundles and with different polar divisors. Finally some results of Crawley-Boevey on the existence of stable connections will be extended to this more general context."}
{"category": "Math", "title": "Knot homology groups from instantons", "abstract": "For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities."}
{"category": "Math", "title": "Homological stability of non-orientable mapping class groups with marked points", "abstract": "Wahl recently proved that the homology of the non-orientable mapping class group stabilizes as the genus increases. In this short note we analyse the situation where the underlying non-orientable surfaces have marked points."}
{"category": "Math", "title": "On Robustness Properties of Beta Encoders and Golden Ratio Encoders", "abstract": "The beta-encoder was recently proposed as a quantization scheme for analog-to-digital conversion; in contrast to classical binary quantization, in which each analog sample x in [-1,1] is mapped to the first N bits of its base-2 expansion, beta-encoders replace each sample x with its expansion in a base beta satisfying 1 < beta < 2. This expansion is non-unique when 1 < beta < 2, and the beta-encoder exploits this redundancy to correct inevitable errors made by the quantizer component of its circuit design. The multiplier element of the beta-encoder will also be imprecise; effectively, the true value beta at any time can only be specified to within an interval [ beta_{low}, beta_{high} ]. This problem was addressed by the golden ratio encoder, a close relative of the beta-encoder that does not require a precise multiplier. However, the golden ratio encoder is susceptible to integrator leak in the delay elements of its hardware design, and this has the same effect of changing beta to an unknown value. In this paper, we present a method whereby exponentially precise approximations to the value of beta in both golden ratio encoders and beta encoders can be recovered amidst imprecise circuit components from the truncated beta-expansions of a \"test\" number x_{test} in [-1,1], and its negative counterpart, -x_{test}. That is, beta-encoders and golden ratio encoders are robust with respect to unavoidable analog component imperfections that change the base beta needed for reconstruction."}
{"category": "Math", "title": "Unmixed bipartite graphs and sublattices of the Boolean lattices", "abstract": "The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs."}
{"category": "Math", "title": "Incremental Stochastic Subgradient Algorithms for Convex Optimization", "abstract": "In this paper we study the effect of stochastic errors on two constrained incremental sub-gradient algorithms. We view the incremental sub-gradient algorithms as decentralized network optimization algorithms as applied to minimize a sum of functions, when each component function is known only to a particular agent of a distributed network. We first study the standard cyclic incremental sub-gradient algorithm in which the agents form a ring structure and pass the iterate in a cycle. We consider the method with stochastic errors in the sub-gradient evaluations and provide sufficient conditions on the moments of the stochastic errors that guarantee almost sure convergence when a diminishing step-size is used. We also obtain almost sure bounds on the algorithm's performance when a constant step-size is used. We then consider \\ram{the} Markov randomized incremental subgradient method, which is a non-cyclic version of the incremental algorithm where the sequence of computing agents is modeled as a time non-homogeneous Markov chain. Such a model is appropriate for mobile networks, as the network topology changes across time in these networks. We establish the convergence results and error bounds for the Markov randomized method in the presence of stochastic errors for diminishing and constant step-sizes, respectively."}
{"category": "Math", "title": "Groebner-Shirshov basis for the braid semigroup", "abstract": "We found Groebner-Shirshov basis for the braid semigroup $B^+_{n+1}$. It gives a new algorithm for the solution of the word problem for the braid semigroup and so for the braid group."}
{"category": "Math", "title": "On the KP I transonic limit of two-dimensional Gross-Pitaevskii travelling waves", "abstract": "We provide a rigorous mathematical derivation of the convergence in the long-wave transonic limit of the minimizing travelling waves for the two-dimensional Gross-Pitaevskii equation towards ground states for the Kadomtsev-Petviashvili equation (KP I)."}
{"category": "Math", "title": "Groebner-Shirshov basis for the braid group in the Birman-Ko-Lee-Garside generators", "abstract": "In this paper, we obtain Groebner-Shirshov (non-commutative Gr\\\"obner) bases for the braid groups in the Birman-Ko-Lee generators enriched by new ``Garside word\" $\\delta$. It gives a new algorithm for getting the Birman-Ko-Lee Normal Form in the braid groups, and thus a new algorithm for solving the word problem in these groups."}
{"category": "Math", "title": "Markov and Artin normal form theorem for braid groups", "abstract": "In this paper we will present the results of Artin--Markov on braid groups by using the Groebner--Shirshov basis. As a consequence we can reobtain the normal form of Artin--Markov--Ivanovsky as an easy corollary."}
{"category": "Math", "title": "Groebner-Shirshov basis for the braid group in the Artin-Garside generators", "abstract": "In this paper, we give a Groebner-Shirshov basis of the braid group $B_{n+1}$ in the Artin--Garside generators. As results, we obtain a new algorithm for getting the Garside normal form, and a new proof that the braid semigroup $B^+{n+1}$ is the subsemigroup in $B_{n+1}$."}
{"category": "Math", "title": "Multivariate Splines and Polytopes", "abstract": "In this paper, we use multivariate splines to investigate the volume of polytopes. We first present an explicit formula for the multivariate truncated power, which can be considered as a dual version of the famous Brion's formula for the volume of polytopes. We also prove that the integration of polynomials over polytopes can be dealt with by the multivariate truncated power. Moreover, we show that the volume of the cube slicing can be considered as the maximum value of the box spline. Based on this connection, we give a simple proof for Good's conjecture, which has been settled by probability methods."}
{"category": "Math", "title": "Linear Stability of Equilibrium Points in the Generalized Photogravitational Chermnykh's Problem", "abstract": "The equilibrium points and their linear stability has been discussed in the generalized photogravitational Chermnykh's problem. The bigger primary is being considered as a source of radiation and small primary as an oblate spheroid. The effect of radiation pressure has been discussed numerically. The collinear points are linearly unstable and triangular points are stable in the sense of Lyapunov stability provided $\\mu< \\mu_{Routh}=0.0385201$. The effect of gravitational potential from the belt is also examined. The mathematical properties of this system are different from the classical restricted three body problem."}
{"category": "Math", "title": "Limit theorems for sample eigenvalues in a generalized spiked population model", "abstract": "In the spiked population model introduced by Johnstone (2001),the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to quantify the effect of the perturbation caused by the spike eigenvalues. Baik and Silverstein (2006) establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. In a recent work (Bai and Yao, 2008), we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we extend this theory to a {\\em generalized} spiked population model where the base population covariance matrix is arbitrary, instead of the identity matrix as in Johnstone's case. New mathematical tools are introduced for establishing the almost sure convergence of the sample eigenvalues generated by the spikes."}
{"category": "Math", "title": "Local Aronson-B\\'enilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds", "abstract": "In this work we derive local gradient and Laplacian estimates of the Aronson-B\\'enilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discover some new entropy formulae for these equations."}
{"category": "Math", "title": "Abel-Jacobi maps for hypersurfaces and non commutative Calabi-Yau's", "abstract": "It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed p-form with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y of degree n. We provide several definitions of this form - via the Abel-Jacobi map, via Hochschild homology, and via the linkage class, and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show that the Fano scheme is birational to a certain moduli space of sheaves on a p-dimensional Calabi--Yau variety X arising naturally in the context of homological projective duality, and that the constructed form is induced by the holomorphic volume form on X. This remains true for a general non Pfaffian hypersurface but the dual Calabi-Yau becomes non commutative."}
{"category": "Math", "title": "Associated primes of monomial ideals and odd holes in graphs", "abstract": "Let $G$ be a finite simple graph with edge ideal $I(G)$. Let $J(G)$ denote the Alexander dual of $I(G)$. We show that a description of all induced cycles of odd length in $G$ is encoded in the associated primes of $J(G)^2$. This result forms the basis for a method to detect odd induced cycles of a graph via ideal operations, e.g., intersections, products and colon operations. Moreover, we get a simple algebraic criterion for determining whether a graph is perfect. We also show how to determine the existence of odd holes in a graph from the value of the arithmetic degree of $J(G)^2$."}
{"category": "Math", "title": "Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs", "abstract": "The problem of computing the smallest fixed point of an order-preserving map arises in the study of zero-sum positive stochastic games. It also arises in static analysis of programs by abstract interpretation. In this context, the discount rate may be negative. We characterize the minimality of a fixed point in terms of the nonlinear spectral radius of a certain semidifferential. We apply this characterization to design a policy iteration algorithm, which applies to the case of finite state and action spaces. The algorithm returns a locally minimal fixed point, which turns out to be globally minimal when the discount rate is nonnegative."}
{"category": "Math", "title": "Tangential interpolation in weighted vector-valued H^p spaces, with applications", "abstract": "In this paper, norm estimates are obtained for the problem of minimal-norm tangential interpolation by vector-valued analytic functions in weighted H^p spaces, expressed in terms of the Carleson constants of related scalar measures. Applications are given to the notion of p-controllability properties of linear semigroup systems and controllability by functions in certain Sobolev spaces."}
{"category": "Math", "title": "Supertropical algebra", "abstract": "We develop the algebraic polynomial theory for \"supertropical algebra,\" as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of \"ghost elements,\" which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geometric approach to tropical geometry, viewing tropical varieties as sets of roots of (supertropical) polynomials, leading to an analog of the Hilbert Nullstellensatz. Particular attention is paid to factorization of polynomials. In one indeterminate, any polynomial can be factored into linear and quadratic factors, and unique factorization holds in a certain sense. On the other hand, the failure of unique factorization in several indeterminates is explained by geometric phenomena described in the paper."}
{"category": "Math", "title": "Asymptotics of posteriors for binary branching processes", "abstract": "We compute the posterior distributions of the initial population and parameter of binary branching processes, in the limit of a large number of generations. We compare this Bayesian procedure with a more na\\\"ive one, based on hitting times of some random walks. In both cases, central limit theorems are available, with explicit variances."}
{"category": "Math", "title": "Completions, Reversals, and Duality for Tropical Varieties", "abstract": "The algebraic foundation of tropical polynomial algebra provides the framework for the geometric construction of the supplement and the reversal of tropical varieties, thereby inducing a duality of reduced tropical varieties; for classes of tropical hypersurfaces the corresponding point symmetry is obtained for their Newton polytopes and lattice polytopes."}
{"category": "Math", "title": "Supertropical matrix algebra", "abstract": "The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows: * The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible. * There exists an adjoint matrix $\\adj{A}$ such that the matrix $A \\adj{A}$ behaves much like the identity matrix (times $|A|$). * Every matrix $A$ is a supertropical root of its Hamilton-Cayley polynomial $f_A$. If these roots are distinct, then $A$ is conjugate (in a certain supertropical sense) to a diagonal matrix. * The tropical determinant of a matrix $A$ is a ghost iff the rows of $A$ are tropically dependent, iff the columns of $A$ are tropically dependent. * Every root of $f_A$ is a \"supertropical\" eigenvalue of $A$ (appropriately defined), and has a tangible supertropical eigenvector."}
{"category": "Math", "title": "Feynman graphs, rooted trees, and Ringel-Hall algebras", "abstract": "We construct symmetric monoidal categories $\\LRF, \\FD$ of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of $\\LRF, \\FD$, $\\HH_{\\LRF}, \\HH_{\\FD}$ are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman graphs. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras."}
{"category": "Math", "title": "Incompressible flow in porous media with fractional diffusion", "abstract": "In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with $\\alpha\\in (1,2]$, we obtain the existence of the global attractor for the solutions in the space $H^s$ for any $s > (N/2)+1-\\alpha$."}
{"category": "Math", "title": "Four-free groups and hyperbolic geometry", "abstract": "We give new information about the geometry of closed, orientable hyperbolic 3-manifolds with 4-free fundamental group. As an application we show that such a manifold has volume greater than 3.44. This is in turn used to show that if M is a closed orientable hyperbolic 3-manifold such that vol M < 3.44, then H_1(M;Z/2Z) has dimension at most 7."}
{"category": "Math", "title": "Note on bipartite graph tilings", "abstract": "Let s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our result extends the work of Zhao, who determined the minimum degree threshold which guarantees that a bipartite graph has a K_{s,s}-factor."}
{"category": "Math", "title": "Paperfolding sequences, paperfolding curves and local isomorphism", "abstract": "For each integer n, an n-folding curve is obtained by folding n times a strip of paper in two, possibly up or down, and unfolding it with right angles. Generalizing the usual notion of infinite folding curve, we define complete folding curves as the curves without endpoint which are unions of increasing sequences of n-folding curves for n integer. We prove that there exists a standard way to extend any complete folding curve into a covering of the plane by disjoint such curves, which satisfies the local isomorphism property introduced to investigate aperiodic tiling systems. This covering contains at most six curves."}
{"category": "Math", "title": "Pre-torsors and Galois comodules over mixed distributive laws", "abstract": "We study comodule functors for comonads arising from mixed distributive laws. Their Galois property is reformulated in terms of a (so-called) regular arrow in Street's bicategory of comonads. Between categories possessing equalizers, we introduce the notion of a regular adjunction. An equivalence is proven between the category of pre-torsors over two regular adjunctions $(N_A,R_A)$ and $(N_B,R_B)$ on one hand, and the category of regular comonad arrows $(R_A,\\xi)$ from some equalizer preserving comonad ${\\mathbb C}$ to $N_BR_B$ on the other. This generalizes a known relationship between pre-torsors over equal commutative rings and Galois objects of coalgebras.Developing a bi-Galois theory of comonads, we show that a pre-torsor over regular adjunctions determines also a second (equalizer preserving) comonad ${\\mathbb D}$ and a co-regular comonad arrow from ${\\mathbb D}$ to $N_A R_A$, such that the comodule categories of ${\\mathbb C}$ and ${\\mathbb D}$ are equivalent."}
{"category": "Math", "title": "Painlev\\'e VI equations with algebraic solutions and family of curves", "abstract": "In families of Painlev\\'e VI differential equations having common algebraic solutions we classify all the members which come from geometry, i.e. the corresponding linear differential equations which are Picard-Fuchs associated to families of algebraic varieties. In our case, we have one family with zero dimensional fibers and all others are families of curves. We use the classification of families of elliptic curves with four singular fibers done by Herfurtner in 1992 and generalize the results of Doran in 2001 and Ben Hamed and Gavrilov in 2005."}
{"category": "Math", "title": "On the number of extreme measures with fixed marginals", "abstract": "In this paper we give an improved upper bound, as compared to the one given in [3] for the number of extreme points of the convex set of all G-invariant probability measures on X*Y with given marginals of full support."}
{"category": "Math", "title": "Monotone loop models and rational resonance", "abstract": "Let $T_{n,m}=\\mathbb Z_n\\times\\mathbb Z_m$, and define a random mapping $\\phi\\colon T_{n,m}\\to T_{n,m}$ by $\\phi(x,y)=(x+1,y)$ or $(x,y+1)$ independently over $x$ and $y$ and with equal probability. We study the orbit structure of such ``quenched random walks'' $\\phi$ in the limit $m,n\\to\\infty$, and show how it depends sensitively on the ratio $m/n$. For $m/n$ near a rational $p/q$, we show that there are likely to be on the order of $\\sqrt{n}$ cycles, each of length O(n), whereas for $m/n$ far from any rational with small denominator, there are a bounded number of cycles, and for typical $m/n$ each cycle has length on the order of $n^{4/3}$."}
{"category": "Math", "title": "Degenerate and star colorings of graphs on surfaces", "abstract": "We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of Fertin et al.: If $G$ is a graph of maximum degree $\\Delta$, then $G$ admits a degenerate star coloring using $O(\\Delta^{3/2})$ colors. We use this result to prove that every graph of genus $g$ admits a degenerate star coloring with $O(g^{3/5})$ colors. It is also shown that these results are sharp up to a logarithmic factor."}
{"category": "Math", "title": "Noether-Lefschetz theorem with base locus", "abstract": "We compute the class groups of very general normal surfaces in complex projective three-space containing an arbitrary base locus $Z$, thereby extending the classic Noether-Lefschetz theorem (the case when $Z$ is empty). Our method is an adaptation of Griffiths and Harris' degeneration proof, simplified by a cohomology and base change argument. We give applications to computing Picard groups, which generalize several known results."}
{"category": "Math", "title": "O-segments on topological measure spaces", "abstract": "Let $X$ be a topological space and $\\mu$ be a nonatomic finite measure on a $\\sigma$-algebra $\\Sigma$ containing the Borel $\\sigma$-algebra of $X$. We say $\\mu$ is weakly outer regular, if for every $A \\in \\Sigma$ and $\\epsilon>0$, there exists an open set $O$ such that $\\mu(A \\backslash O)=0$ and $\\mu(O \\backslash A)<\\epsilon$. The main result of this paper is to show that if $f,g \\in L^1(X,\\Sigma, \\mu)$ with $\\int_X f d\\mu=\\int_X g d\\mu=1$, then there exists an increasing family of open sets $u(t)$, $t\\in [0,1]$, such that $u(0)=\\emptyset$, $u(1)=X$, and $\\int_{u(t)} f d\\mu=\\int_{u(t)} g d\\mu=t$ for all $t\\in [0,1]$. We also study a similar problem for a finite collection of integrable functions on general finite and $\\sigma$-finite nonatomic measure spaces."}
{"category": "Math", "title": "Geometric decompositions and local bases for spaces of finite element differential forms", "abstract": "We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields, frequently referred to as Raviart-Thomas, Brezzi-Douglas-Marini, and Nedelec spaces. In the present paper, we derive geometric decompositions of these spaces which lead directly to explicit local bases for them, generalizing the Bernstein basis for ordinary Lagrange finite elements. The approach applies to both families of finite element spaces, for arbitrary polynomial degree, arbitrary order of the differential forms, and an arbitrary simplicial triangulation in any number of space dimensions. A prominent role in the construction is played by the notion of a consistent family of extension operators, which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis."}
{"category": "Math", "title": "Dirac structures, nonholonomic systems and reduction", "abstract": "The reduction of nonholonomic systems is formulated in terms of Dirac reduction. An optimal reduction method for a class of nonholonomic systems is formulated. Several examples are studied in detail."}
{"category": "Math", "title": "Semigroup inequalities, stochastic domination, Hardy's inequality, and strong ergodicity", "abstract": "There is a mathematical error in the first version of this paper. A new corrected version will be posted when the error is fixed, possibly with a modified title."}
{"category": "Math", "title": "Pseudo-radial solutions of semi-linear elliptic equations on symmetric domains", "abstract": "In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a dynamical system. In particular, we give a full description of the set of pseudo-radial solutions of equations of the form $\\Delta u = \\pm a^2(|x|) u|u|^{q-1}$, with $q>0$, $q\\neq 1$. We also study such equations over spherical or hyperbolic symmetric domains."}
{"category": "Math", "title": "Stability of Tails and 4-Canonical Models", "abstract": "We show that the GIT quotients of suitable loci in the Hilbert and Chow schemes of 4-canonically embedded curves of genus $g\\ge 3$ are the moduli space $\\bar{M}_g^{\\text{ps}}$ of pseudo-stable curves constructed by Schubert in \\cite{Schubert} using Chow varieties and 3-canonical models. The only new ingredient needed in the Hilbert scheme variant is a more careful analysis of the stability with respect to a certain 1-ps $\\lambda$ of the $m^{\\text{th}}$ Hilbert points of curves $X$ with elliptic tails. We compute the exact weight with which $\\lambda$ acts, and not just the leading term in $m$ of this weight. A similar analysis of stability of curves with rational cuspidal tails allows us to determine the stable and semistable 4-canonical Chow loci. Although here the geometry of the quotient is more complicated because there are strictly semi-stable orbits, we are able to again identify it as $\\bar{M}_g^{\\text{ps}}$. Our computations yield, as byproducts, examples of both $m$-Hilbert unstable and $m$-Hilbert stable $X$ that are Chow strictly semi-stable."}
{"category": "Math", "title": "Matricial formulae for partitions", "abstract": "The exponential of the triangular matrix whose entries in the diagonal at distance $n$ from the principal diagonal are all equal to the sum of the inverse of the divisors of $n$ is the triangular matrix whose entries in the diagonal at distance $n$ from the principal diagonal are all equal to the number of partitions of $n$. A similar result is true for any pair of sequences satisfying a special recurrence."}
{"category": "Math", "title": "Maximal plurisubharmonic models", "abstract": "An analytic pair of dimension n and center V is a pair (V, M) where M is a complex manifold of (complex) dimension n and V is a closed totally real analytic submanifold of dimension n. To an analytic pair (V, M) we associate the class of the functions u from M to a positive bounded interval which are plurisubharmonic in M and such that u(p) = 0 for each p in V. If the class admits a maximal function u, the triple (V, M, u) is said to be a maximal plurisubharmonic model. After defining a pseudo-metric E(V,M) on the center V of an analytic pair (V, M) we prove (see Theorem 4.1, Theorem 5.1) that maximal plurisubharmonic models provide a natural generalization of the Monge-Ampere models introduced by Lempert and Szoke in [16]."}
{"category": "Math", "title": "Multifractal Formalism and Inequality involving Packing Dimension", "abstract": "This article fits in many studies of multifractal analysis of measure. We took as a starting point the work of F. Ben Nasr in \" Calculs de dimension de packing \" to give a new inequality involving $Dim(\\bar{X}^{\\alpha})$ which would be, in certain cases, finer than the inequality established by L. Olsen in \" A multifractal formalism \" . Besides we elaborated an application of our result which gives a better inequality involving $Dim(\\bar{X}^{\\alpha})$."}
{"category": "Math", "title": "Generalized Jacobi operators in Krein spaces", "abstract": "A special class of generalized Jacobi operators which are self-adjoint in Krein spaces is presented. A description of the resolvent set of such operators in terms of solutions of the corresponding recurrence relations is given. In particular, special attention is paid to the periodic generalized Jacobi operators. Finally, the spectral properties of generalized Jacobi operators are applied to prove convergence results for Pad\\'e approximants."}
{"category": "Math", "title": "Comparison between shape optimization and volumic level set approximation for geometrical functionals", "abstract": "We propose to differentiate a general curvature functional with two different approaches. In the first one we compute the derivative with the tools of shape optimization and in the second one we compute the derivative of a volumic approximation of the functional with respect to a level set function. We show that the two previous approaches give the same result."}
{"category": "Math", "title": "Computing expected transition events in reducible Markov chains", "abstract": "We present a closed-form, computable expression for the expected number of times any transition event occurs during the transient phase of a reducible Markov chain. Examples of events include time to absorption, number of visits to a state, traversals of a particular transition, loops from a state to itself, and arrivals to a state from a particular subset of states. We give an analogous expression for time-average events, which describe the steady-state behavior of reducible chains as well as the long-term behavior of irreducible chains."}
{"category": "Math", "title": "Stabilizing Randomly Switched Systems", "abstract": "This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunov-like functions, and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control design"}
{"category": "Math", "title": "Sur la convergence des s\\'eries trigonom\\'etriques qui servent \\`a repr\\'esenter une fonction arbitraire entre des limites donn\\'ees", "abstract": "Dirichlet proves the general convergence of Fourier series, after pointing out errors in an earlier attempt by Cauchy. We transcribed from Crelle's Journal (1829) with numerous typographical corrections, and added a completed bibliography. Dirichlet prouve la convergence g\\'en\\'erale de la s\\'eries de Fourier, apr\\`es avoir montr\\'e des erreurs dans un essai par Cauchy. Nous avons transcrit de Crelle's journal (1829) avec de nombreuses corrections typographiques, et avons ajout\\'e une bibliographie compl\\`ete."}
{"category": "Math", "title": "Cores of Geometric Graphs", "abstract": "Cameron and Kazanidis have recently shown that rank-3 graphs are either cores or have complete cores, and they asked whether this holds for all strongly regular graphs. We prove that this is true for the point graphs and line graphs of generalized quadrangles and that when the number of points is sufficiently large, it is also true for the block graphs of Steiner systems and orthogonal arrays."}
{"category": "Math", "title": "Regular Maximal Monotone Multifunctions and Enlargements", "abstract": "In this note we use recent results concerning the sum theorem for maximal monotone multifunctions in general Banach spaces to find new characterizations and properties of regular maximal monotone multifunctions and then use these to describe the domain of certain enlargements."}
{"category": "Math", "title": "Parabolic principal Higgs bundles", "abstract": "In ``Ramified G-bundles as parabolic bundles'' (J. Ramanujan Math. Soc. (2003) Vol. 18) with Balaji and Nagaraj we introduced the ramified principal bundles. The aim here is to define the Higgs bundles in the ramified context."}
{"category": "Math", "title": "Insufficiency of the Brauer-Manin obstruction applied to etale covers", "abstract": "Let k be any global field of characteristic not 2. We construct a k-variety X such that X(k) is empty, but for which the emptiness cannot be explained by the Brauer-Manin obstruction or even by the Brauer-Manin obstruction applied to finite etale covers."}
{"category": "Math", "title": "Embeddings and chains of free groups", "abstract": "We build two non-abelian CSA-groups in which maximal abelian subgroups are conjugate and divisible."}
{"category": "Math", "title": "Localization theory for triangulated categories", "abstract": "These notes provide an introduction to the theory of localization for triangulated categories. Localization is a machinery to formally invert morphisms in a category. We explain this formalism in some detail and we show how it is applied to triangulated categories."}
{"category": "Math", "title": "Volume Growth and Curvature Decay of Complete Positively Curved K\\\"{a}hler Manifolds", "abstract": "This paper constructs a class of complete K\\\"{a}hler metrics of positive holomorphic sectional curvature on ${\\bf C}^n$ and finds that the constructed metrics satisfy the following properties: As the geodesic distance $\\rho\\to\\infty,$ the volume of geodesic balls grows like $O(\\rho^{\\frac{2(\\beta+1)n}{\\beta+2}})$ and the Riemannian scalar curvature decays like $O(\\rho^{-\\frac{2(\\beta+1)}{\\beta+2}}),$ where $\\beta\\geq 0.$"}
{"category": "Math", "title": "Cesaro averages of Euler-like functions", "abstract": "By Euler-like function we mean a function defined on the positive integers and associating to $n$ the product, over all primes $p$ dividing $n$, of 1 plus (or minus) the inverse of $p$ to the power $s$. We calculate the limit of the Cesaro mean of these functions."}
{"category": "Math", "title": "On Discrete Subgroups of automorphism of $P^2_C$", "abstract": "We study the geometry and dynamics of discrete subgroups $\\Gamma$ of $\\PSL(3,\\mathbb{C})$ with an open invariant set $\\Omega \\subset \\PC^2$ where the action is properly discontinuous and the quotient $\\Omega/\\Gamma$ contains a connected component whicis compact. We call such groups {\\it quasi-cocompact}. In this case $\\Omega/\\Gamma$ is a compact complex projective orbifold and $\\Omega$ is a {\\it divisible set}. Our first theorem refines classical work by Kobayashi-Ochiai and others about complex surfaces with a projective structure: We prove that every such group is either virtually affine or complex hyperbolic. We then classify the divisible sets that appear in this way, the corresponding quasi-cocompact groups and the orbifolds $\\Omega/\\Gamma$. We also prove that excluding a few exceptional cases, the Kulkarni region of discontinuity coincides with the equicontinuity region and is the largest open invariant set where the action is properly discontinuous."}
{"category": "Math", "title": "Asymptotics of Plancherel measures for $GL(n,q)$", "abstract": "We introduce the Plancherel measure on the set of partition collections, which parameterize irreducible representations of order n general linear group over a finite field. We prove that as n goes to infinity, the random partitions from the random collections converge in finite dimencional distribution to independent random variables. We give explicit formulas for the corresponding limit distributions."}
{"category": "Math", "title": "Classes of 3-regular graphs that are (7, 2)-edge-choosable", "abstract": "A graph is (7, 2)-edge-choosable if, for every assignment of lists of size 7 to the edges, it is possible to choose two colors for each edge from its list so that no color is chosen for two incident edges. We show that every 3-edge-colorable graph is (7, 2)-edge-choosable and also that many non-3-edge-colorable 3-regular graphs are (7, 2)-edge-choosable."}
{"category": "Math", "title": "Seshadri constants and surfaces of minimal degree", "abstract": "In \"Seshadri fibrations of algebraic surfaces\" [arXiv:0709.2592v1] we showed that if the multiple point Seshadri constants of an ample line bundle on a smooth projective surface in very general points satisfy certain inequality then the surface is fibred by curves computing these constants. Here we characterize the border case of polarized surfaces whose Seshadri constants in general points fulfill the equality instead of inequality and which are not fibred by Seshadri curves. It turns out that these surfaces are the projective plane and surfaces of minimal degree."}
{"category": "Math", "title": "Asymptotic Behavior of Solutions of a Free Boundary Problem Modelling the Growth of Tumors with Stokes Equations", "abstract": "We study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The fluid velocity satisfies Stokes equations with a source determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. It is easy to prove that this problem has a unique radially symmetric stationary solution. By using a functional approach, we prove that there exists a threshold value $\\gamma_*>0$ for the surface tension coefficient $\\gamma$, such that in the case $\\gamma>\\gamma_*$ this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable."}
{"category": "Math", "title": "Analysis of the Effect of Speed Limit Increases on Accident-Injury Severities", "abstract": "The influence of speed limits on roadway safety has been a subject of continuous debate in the State of Indiana and nationwide. In Indiana, highway-related accidents result in about 900 fatalities and forty thousand injuries annually and place an incredible social and economic burden on the state. Still, speed limits posted on highways and other roads are routinely exceeded as individual drivers try to balance safety, mobility (speed), and the risks and penalties associated with law enforcement efforts. The speed-limit/safety issue has been a matter of considerable concern in Indiana since the state raised its speed limits on rural interstates and selected multilane highways on July 1, 2005. In this paper, the influence of the posted speed limit on the severity of vehicle accidents is studied using Indiana accident data from 2004 (the year before speed limits were raised) and 2006 (the year after speed limits were raised on rural interstates and some multi-lane non-interstate routes). Statistical models of the injury severity of different types of accidents on various roadway classes were estimated. The results of the model estimations showed that, for the speed limit ranges currently used, speed limits did not have a statistically significant effect on the severity of accidents on interstate highways. However, for some non-interstate highways, higher speed limits were found to be associated with higher accident severities - suggesting that future speed limit changes, on non-interstate highways in particular, need to be carefully assessed on a case-by-case basis."}
{"category": "Math", "title": "Differentiable Categories, gerbes and G-structures", "abstract": "The theories of strings and $D$-branes have motivated the development of non Abelian cohomology techniques in differential geometry, on the purpose to find a geometric interpretation of characteristic classes. The spaces studied here, like orbifolds are not often smooth. In classical differential geometry, non smooth spaces appear also naturally, for example in the theory of foliations, the space of leaves can be an orbifold with singularities. The scheme to study these structures is identical: classical tools used in differential geometry, like connections, curvature are adapted. The purpose of this paper is to present the notion of differential category which unifies all these points of view. This enables us to provide a geometric interpretation of 5-characteristic classes, and to interpret classical problems which appear in the theory of $G$-structures by using gerbes."}
{"category": "Math", "title": "Asymptotic Stability of Stationary Solutions of a Free Boundary Problem Modeling the Growth of Tumors with Fluid Tissues", "abstract": "This paper aims at proving asymptotic stability of the radial stationary solution of a free boundary problem modeling the growth of nonnecrotic tumors with fluid-like tissues. In a previous paper we considered the case where the nutrient concentration $\\sigma$ satisfies the stationary diffusion equation $\\Delta\\sigma=f(\\sigma)$, and proved that there exists a threshold value $\\gamma_*>0$ for the surface tension coefficient $\\gamma$, such that the radial stationary solution is asymptotically stable in case $\\gamma>\\gamma_*$, while unstable in case $\\gamma<\\gamma_*$. In this paper we extend this result to the case where $\\sigma$ satisfies the non-stationary diffusion equation $\\epsln\\partial_t\\sigma=\\Delta\\sigma-f(\\sigma)$. We prove that for the same threshold value $\\gamma_*$ as above, for every $\\gamma>\\gamma_*$ there is a corresponding constant $\\epsln_0(\\gamma)>0$ such that for any $0<\\epsln<\\epsln_0(\\gamma)$ the radial stationary solution is asymptotically stable with respect to small enough non-radial perturbations, while for $0<\\gamma<\\gamma_*$ and $\\epsln$ sufficiently small it is unstable under non-radial perturbations."}
{"category": "Math", "title": "Isotriviality is equivalent to potential good reduction for endomorphisms of ${\\mathbb P}^N$ over function fields", "abstract": "Let $K=k(C)$ be the function field of a complete nonsingular curve $C$ over an arbitrary field $k$. The main result of this paper states that a morphism $\\phi:{\\mathbb P}^N_K\\to{\\mathbb P}^N_K$ is isotrivial if and only if it has potential good reduction at all places $v$ of $K$; this generalizes results of Benedetto for polynomial maps on ${\\mathbb P}^1_K$ and Baker for arbitrary rational maps on ${\\mathbb P}^1_K$. We offer two proofs: the first uses algebraic geometry and geometric invariant theory, and it is new even in the case N=1. The second proof uses non-archimedean analysis and dynamics, and it more directly generalizes the proofs of Benedetto and Baker. We will also give two applications. The first states that an endomorphism of ${\\mathbb P}^N_K$ of degree at least two is isotrivial if and only if it has an isotrivial iterate. The second gives a dynamical criterion for whether (after base change) a locally free coherent sheaf ${\\mathcal E}$ of rank $N+1$ on $C$ decomposes as a direct sum ${\\mathcal L}\\oplus...\\oplus{\\mathcal L}$ of $N+1$ copies of the same invertible sheaf ${\\mathcal L}$."}
{"category": "Math", "title": "Unranking permutations in transposition order and linear time", "abstract": "An algorithm is presented for unranking permutations in transposition order: Given a seed s\\in N, the algorithm produces a permutation P(s) that differs from the permutation P(s+1) by the transposition of two elements."}
{"category": "Math", "title": "Global well-posedness for the $L^2$ critical Hartree equation on $\\bbr^n$, $n\\ge 3$", "abstract": "We consider the initial value problem for the L^2-critical defocusing Hartree equation in R^n, n \\ge 3. We show that the problem is globally well posed in H^s(R^n) when 1>s> \\frac{2(n-2)}{3n-4}$. We use the \"I-method\" combined with a local in time Morawetz estimate for the smoothed out solution."}
{"category": "Math", "title": "On the non-existence of some Steiner $t$-$(v,k)$ trades of certain volumes", "abstract": "Mahmoodian and Soltankhah $\\cite{MMS}$ conjectured that there does not exist any $t$-$(v,k)$ trade of volume $s_{i}< s <s_{i+1}$, where $s_{i}=2^{t+1}-2^{t-i}, i=0,1,..., t-1$. Also they showed that the conjecture is true for $i=0$. In this paper we prove the correctness of this conjecture for Steiner trades."}
{"category": "Math", "title": "Constructing regular graphs with smallest defining number", "abstract": "In a given graph $G$, a set $S$ of vertices with an assignment of colors is a {\\sf defining set of the vertex coloring of $G$}, if there exists a unique extension of the colors of $S$ to a $\\Cchi(G)$-coloring of the vertices of $G$. A defining set with minimum cardinality is called a {\\sf smallest defining set} (of vertex coloring) and its cardinality, the {\\sf defining number}, is denoted by $d(G, \\Cchi)$. Let $ d(n, r, \\Cchi = k)$ be the smallest defining number of all $r$-regular $k$-chromatic graphs with $n$ vertices. Mahmoodian et. al \\cite{rkgraph} proved that, for a given $k$ and for all $n \\geq 3k$, if $r \\geq 2(k-1)$ then $d(n, r, \\Cchi = k)=k-1$. In this paper we show that for a given $k$ and for all $n < 3k$ and $r\\geq 2(k-1)$, $d(n, r, \\Cchi=k)=k-1$."}
{"category": "Math", "title": "Some Divisibility Properties in Ring of Polynomials over a Unique Factorization Domain", "abstract": "Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is possible to prove that under certain conditions, the domain of coefficients must have infinitely many prime elements. We give alternative characterizations for D-rings and present various examples."}
{"category": "Math", "title": "A remark on the rational cohomology of $\\bar{S}_{1,n}$", "abstract": "We focus on the rational cohomology of Cornalba's moduli space of spin curves of genus 1 with $n$ marked points. In particular, we show that both its first and its third cohomology group vanish and the second cohomology group is generated by boundary classes."}
{"category": "Math", "title": "New Upper Bound for the Edge Folkman Number Fe(3,5;13)", "abstract": "In this paper we prove that the edge Folkman number Fe(3,5;13) is not greater than 21."}
{"category": "Math", "title": "Equivalent Definitions for Uniform Exponential Trichotomy of Evolution Operators in Banach Spaces", "abstract": "The aim of this paper is to give necessary and sufficient conditions for the uniform exponential trichotomy property of nonlinear evolution operators in Banach spaces. The obtained results are generalizations for infinite-dimensional case of some well-known results of Elaydi and Hajek on exponential trichotomy of differential systems."}
{"category": "Math", "title": "Nonuniform Behaviors for Skew-Evolution Semiflows in Banach Spaces", "abstract": "The paper emphasizes some asymptotic behaviors for skew-evolution semiflows in Banach spaces. These are defined by means of evolution semiflows and evolution cocycles. Some characterizations which generalize classical results are also provided. The approach is from nonuniform point of view."}
{"category": "Math", "title": "The Relative Capacity", "abstract": "The purpose of this article is to introduce the relative $p$-capacity $\\Cap_{p,\\Omega}$ with respect to an open set $\\Omega$ in $\\IR^N$. It is a Choquet capacity on the closure of $\\Omega$ and extends the classical $p$-capacity $\\Cap_p$ in the sense that $\\Cap_{p,\\Omega}=\\Cap_p$ if $\\Omega=\\IR^N$. The importance of the relative $p$-capacity stems from the fact that a large class of Sobolev functions defined on a 'bad domain' admit a trace on the boundary $\\partial\\Omega$ which is then unique up to $\\Cap_{p,\\Omega}$-polar set. As an application we prove a characterization of $W^{1,p}_0(\\Omega)$ for open sets $\\Omega\\subset\\IR^N$."}
{"category": "Math", "title": "Quasi-co-Frobenius corings as Galois comodules", "abstract": "We compare several quasi-Frobenius-type properties for corings that appeared recently in literature and provide several new characterizations for each of these properties. By applying the theory of Galois comodules with a firm coinvariant ring, we can characterize a locally quasi-Frobenius (quasi-co-Frobenius) coring as a locally projective generator in its category of comodules."}
{"category": "Math", "title": "Smooth Structures and Einstein Metrics on $CP^2#5,6,7\\bar{CP^2}$", "abstract": "We show that each of the topological 4-manifolds $CP^2#k\\bar{CP^2}, for $k = 6, 7$ admits a smooth structure which has an Einstein metric of scalar curvature $s > 0$, a smooth structure which has an Einstein metric with $s < 0$ and infinitely many non-diffeomorphic smooth structures which do not admit Einstein metrics. We show that there are infinitely many manifolds homeomorphic non-diffeomorphic to $CP^2#5\\bar{CP^2}$ which do not admit an Einstein metric. We also exhibit new higher dimensional examples of manifolds carrying Einstein metrics of both positive and negative scalar curvature. The main ingredients are recent constructions of exotic symplectic or complex manifolds with small topological numbers."}
{"category": "Math", "title": "An extension problem for convex functions", "abstract": "We give a statement on extension with estimates of convex functions defined on a linear subspace, inspired by similar extension results concerning metrics on positive line bundles"}
{"category": "Math", "title": "On invariants for Legendrian knots", "abstract": "Suppose that L is a null--homologous Legendrian knot in a contact 3--manifold. We determine the connection between the sutured invariant of the complement of L and the Legendrian invariant defined by Lisca, Ozsvath, Stipsicz and Szabo. In particular, we derive a vanishing theorem for the Legendrian invariant in the presence of Giroux torsion in the complement of the knot, and reprove several known properties of the Legendrian invariant from this perspective."}
{"category": "Math", "title": "On the extendability of elliptic surfaces of rank two and higher", "abstract": "We study threefolds X in a projective space having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings of Weierstrass fibrations of any rank, under which every such threefold must be a cone."}
{"category": "Math", "title": "Generalized regularity and solution concepts for differential equations", "abstract": "As the title ``Generalized regularity and solution concepts for differential equations'' suggests, the main topic of my thesis is the investigation of generalized solution concepts for differential equations, in particular first order hyperbolic partial differential equations with real-valued, non-smooth coefficients and their characteristic system of ordinary differential equations. In Colombeau theory there have been developed existence results that yield solutions for ordinary and partial differential equations beyond the scope of classical approaches. Nevertheless this comes at the price of sacrificing regularity (in general a Colombeau solution may even lack a distributional shadow). It is prevailing in the Colombeau setting that the question of mere existence of solutions is much easier to answer than to determine their regularity properties (i.e. if a distributional shadow exists and how regular it is). In order order to address these regularity question and encouraged by the fact that the solution of a (homogeneous) first order partial differential equation can be written as a pullback of the initial condition by the characteristic backward flow, a main topic of my thesis deals with the microlocal analysis of pullbacks of c-bounded Colombeau generalized functions. Another topic is the comparsion of Colombeau techniques for solving ordinary and partial differential equations to other generalized solution concepts, which has led to a joint article with G\\\"unther H\\\"ormann. A useful tool for this purpose is the concept of a generalized graph, which has been developed in the thesis."}
{"category": "Math", "title": "A counter example on nontangential convergence for oscillatory integrals", "abstract": "Consider the solution of the time-dependent Schr{\\\"o}dinger equation with initial data $f$. It is shown in \\cite{artikel} that there exists $f$ in the Sobolev space $H^s(\\RR), s=n/2$ such that tangential convergence can not be widened to convergence regions. In this paper we show that the corresponding result holds when $-\\Delta_x$ is replaced by an operator $\\phi(D)$, with special conditions on $\\phi$."}
{"category": "Math", "title": "Sharp bounds for symmetric and asymmetric Diophantine approximation", "abstract": "In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method and give both sharp upper and lower bounds. We also calculate the asymptotic frequency that these bounds occur."}
{"category": "Math", "title": "Generalized Thomas hyperplane sections and relations between vanishing cycles", "abstract": "As is remarked by B. Totaro, R. Thomas essentially proved that the Hodge conjecture is inductively equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a given primitive Hodge class does not vanish. We study the relations between the vanishing cycles in the cohomology of a general fiber, and show that each relation between the vanishing cycles of type (0,0) with unipotent monodromy around a singular hyperplane section defines a primitive Hodge class such that this singular hyperplane section is a generalized Thomas hyperplane section if and only if the pairing between a given primitive Hodge class and some of the constructed primitive Hodge classes does not vanish. This is a generalization of a construction by P. Griffiths."}
{"category": "Math", "title": "Analysis of Metric Distances and Volumes of Hippocampi Indicates Different Morphometric Changes over Time in Dementia of Alzheimer Type and Nondemented Subjects", "abstract": "In this article, we analyze the morphometry of hippocampus in subjects with very mild dementia of Alzheimer's type (DAT) and nondemented controls and how it changes over a two-year period. Morphometric differences with respect to a template hippocampus were measured by the metric distance obtained from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) algorithm which was previously used to calculate dense one-to-one correspondence vector fields between the shapes. LDDMM assigns metric distances on the space of anatomical images thereby allowing for the direct comparison and quantization of morphometric changes. We use various statistical methods to compare the metric distances in a cross-sectional and longitudinal manner. At baseline, the metric distances for demented subjects are found not to be significantly different from those for nondemented subjects. At follow-up, the metric distances for demented subjects were significantly larger compared to nondemented subjects. The metric distances for demented subjects increased significantly from baseline to follow-up but not for nondemented subjects. We also use the metric distances in logistic regression for diagnostic discrimination of subjects. We compare metric distances with the volumes and obtain similar results. In classification, the model that uses volume, metric distance, and volume loss over time together performs better in detecting DAT. Thus, metric distances with respect to a template computed via LDDMM can be a powerful tool in detecting differences in shape in cross-sectional as well as longitudinal studies."}
{"category": "Math", "title": "On the opposite of the category of rings", "abstract": "We define a faithful contravariant functor NCSpec from the category of rings to the category of ringed spaces, and show that if R is a commutative ring then NCSpec(R) may be viewed as a completion of Spec(R) in an appropriate sense. We then explain how the spaces NCSpec(R) may be glued, and study quasicoherent sheaves on them. As an example, we compute the category of quasicoherent sheaves on a space constructed from a skew-polynomial ring R by an analogue of the Proj construction."}
{"category": "Math", "title": "On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries", "abstract": "We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution to a convex optimization problem. We show that if Sigma(R, C) is sufficiently large, then a random matrix D in Sigma(R, C) sampled from the uniform probability measure in Sigma(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions."}
{"category": "Math", "title": "Polytopal complexes: maps, chain complexes and... necklaces", "abstract": "The notion of polytopal map between two polytopal complexes is defined. Surprisingly, this definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the associated chain complexes. Finally, we use this new tool to give the first combinatorial proof of the splitting necklace theorem of Alon. The paper ends with open questions, such as the existence of Sperner's lemma for a polytopal complex or the existence of a cubical approximation theorem."}
{"category": "Math", "title": "Electromagnetism and geometry", "abstract": "This work is an introduction to modern mathematical physics. We begin with Maxwell laws and vector calculus, pass next to consider the action and the Feynman integral in quantum mechanics, next relativity and differential geometry to formulate the electromagnetic laws in intrinsic form. Next we see gravitation to study the covariant derivative. We end with the electromagnetic bundle U(1). It contains the know-how + the know-why. The text is written in Spanish. 340 pps."}
{"category": "Math", "title": "The M\\\"{o}bius Function of a Restricted Composition Poset", "abstract": "We study a poset of compositions restricted by part size under a partial ordering introduced by Bj\\\"{o}rner and Stanley. We show that our composition poset $C_{d+1}$ is isomorphic to the poset of words $A_d^*$. This allows us to use techniques developed by Bj\\\"{o}rner to study the M\\\"{o}bius function of $C_{d+1}$. We use counting arguments and shellability as avenues for proving that the M\\\"{o}bius function is $\\mu(u,w)=(-1)^{|u|+|w|}{w\\choose u}_{dn}$, where ${w\\choose u}_{dn}$ is the number of $d$-normal embeddings of $u$ in $w$. We then prove that the formal power series whose coefficients are given by the zeta and the M\\\"{o}bius functions are both rational. Following in the footsteps of Bj\\\"{o}rner and Reutenauer and Bj\\\"{o}rner and Sagan, we rely on definitions to prove rationality in one case, and in another case we use finite-state automata."}
{"category": "Math", "title": "Local inverses of shift maps along orbits of flows", "abstract": "Let $M$ be a smooth manifold and $F$ be a vector field on $M$. My article [\"Smooth shifts along trajectories of flows\", Topol. Appl. 130 (2003) 183-204, arXiv:math/0106199] concerning the homotopy types of the group of diffeomorphisms preserving orbits of $F$ contains two errors. They imply that the principal statement of that paper holds under additional assumptions on $F$. Unfortunately this result was essentially used in another paper of mine [\"Homotopy types of stabilizers and orbits of Morse functions on surfaces\" Ann. Glob. Anal. Geom., 29 no. 3, (2006) 241-285, arXiv:math/0310067]. The aim of this article is to expose the results of the first paper in a right way, extend them to a larger class of flows with degenerate singularities, and show that the results of the second paper remain true."}
{"category": "Math", "title": "Homology representations arising from the half cube", "abstract": "We construct a CW decomposition $C_n$ of the $n$-dimensional half cube in a manner compatible with its structure as a polytope. For each $3 \\leq k \\leq n$, the complex $C_n$ has a subcomplex $C_{n, k}$, which coincides with the clique complex of the half cube graph if $k = 4$. The homology of $C_{n, k}$ is concentrated in degree $k-1$ and furthermore, the $(k-1)$-st Betti number of $C_{n, k}$ is equal to the $(k-2)$-nd Betti number of the complement of the $k$-equal real hyperplane arrangement. These Betti numbers, which also appear in theoretical computer science, numerical analysis and engineering, are the coefficients of a certain Pascal-like triangle (Sloane's sequence A119258). The Coxeter groups of type $D_n$ act naturally on the complexes $C_{n, k}$, and thus on the associated homology groups."}
{"category": "Math", "title": "Relative Chow-Kunneth decompositions for conic bundles and Prym varieties", "abstract": "We construct a relative Chow-Kunneth decomposition for a conic bundle over a surface such that the middle projector gives the Prym variety of the associated double covering of the discriminant of the conic bundle. This gives a refinement (up to an isogeny) of Beauville's theorem on the relation between the intermediate Jacobian of the conic bundle and the Prym variety of the double covering."}
{"category": "Math", "title": "On a Theorem on sums of the form 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m) and a result linking Fermat with Mersenne numbers", "abstract": "In his book \"250 Problems in Elementary Number Theory\", W.Sierpinski shows that the numbers 1+2^(2^n)+2^(2^n+1) are divisible by 21; for n=1,2,.... In this paper, we prove a similar but more general result.Consider the natural numbers of the form I(n.m)= 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m).In Theorem 1 we prove that for every odd integer N greater than 1, there exist infinitely many natural numbers n and m such that the integers I(n.m) are divisible by N. We give an explicit construction of the numbers n and m, for a given N. As an example, when N=31, and with n=4k and m=94+124i, the numbers I(n,m) are divisible by 31. A similar example is offered for N=(31)(7)=217. In Theorem 2, we prove a result pertaining to Mersenne numbers.There are also three Corollaries in this work, one of which deals with Fermat numbers."}
{"category": "Math", "title": "The Christoffel-Darboux Kernel", "abstract": "A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results."}
{"category": "Math", "title": "Highest weight categories arising from Khovanov's diagram algebra I: cellularity", "abstract": "This is the first of four articles studying some slight generalisations H(n,m) of Khovanov's diagram algebra, as well as quasi-hereditary covers K(n,m) of these algebras in the sense of Rouquier, and certain infinite dimensional limiting versions. In this article we prove that H(n,m) is a cellular symmetric algebra and that K(n,m) is a cellular quasi-hereditary algebra. In subsequent articles, we relate these algebras to level two blocks of degenerate cyclotomic Hecke algebras, parabolic category O and the general linear supergroup, respectively."}
{"category": "Math", "title": "Manin's conjecture on a nonsingular quartic del Pezzo surface", "abstract": "Given a nonsingular quartic del Pezzo surface, a conjecture of Manin predicts the density of rational points on the open subset of the surface formed by deleting the lines. We prove that this prediction is of the correct order of magnitude for a particular surface."}
{"category": "Math", "title": "A basis of bideterminants for the coordinate ring of the orthogonal group", "abstract": "We give a basis of bideterminants for the coordinate ring K[O(n)] of the orthogonal group O(n,K), where K is an infinite field of characteristic not 2. The bideterminants are indexed by pairs of Young tableaux which are O(n)-standard in the sense of King-Welsh. We also give an explicit filtration of K[O(n)] as an O(n,K)-bimodule, whose factors are isormorphic to the tensor product of orthogonal analogues of left and right Schur modules."}
{"category": "Math", "title": "Model category extensions of the Pirashvili-S{\\l}omi\\'{n}ska theorems", "abstract": "We describe the class of semi-stable model categories, which generalize the equivalence of finite products and coproducts in abelian and stable model categories, and use this to establish Morita equivalences among categories of functors. We provide a construction of pairs of small categories--known as conjugate pairs--whose associated categories of diagrams are Quillen equivalent in the semi-stable setting. We frame our development in the context of Morita theory, following Slominska's work on similar questions for categories of functors enriched over and taking values in R-modules."}
{"category": "Math", "title": "It\\^o's formula for the $L_{p}$-norm of stochastic $W^{1}_{p}$-valued processes", "abstract": "We prove It\\^o's formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form."}
{"category": "Math", "title": "Schur Positivity and the $q$-Log-convexity of the Narayana Polynomials", "abstract": "Using Schur positivity and the principal specialization of Schur functions, we provide a proof of a recent conjecture of Liu and Wang on the $q$-log-convexity of the Narayana polynomials, and a proof of the second conjecture that the Narayana transformation preserves the log-convexity. Based on a formula of Br\\\"and$\\mathrm{\\acute{e}}$n which expresses the $q$-Narayana numbers as the specializations of Schur functions, we derive several symmetric function identities using the Littlewood-Richardson rule for the product of Schur functions, and obtain the strong $q$-log-convexity of the Narayana polynomials and the strong $q$-log-concavity of the $q$-Narayana numbers."}
{"category": "Math", "title": "Transcendence of Power Series for Some Number Theoretic Functions", "abstract": "We give a new proof of Fatou's theorem: {\\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely multiplicative function from $\\mathbb{N}$ to $\\{-1,1\\}$, the series $\\sum_{n=1}^\\infty f(n)z^n$ is transcendental over $\\mathbb{Z}[z]$; in particular, $\\sum_{n=1}^\\infty \\lambda(n)z^n$ is transcendental, where $\\lambda$ is Liouville's function. The transcendence of $\\sum_{n=1}^\\infty \\mu(n)z^n$ is also proved."}
{"category": "Math", "title": "On the Structure of the Fusion Ideal", "abstract": "We prove that there is a finite level-independent bound on the number of relations defining the fusion ring of positive energy representations of the loop group of a simple, simply connected Lie group. As an illustration, we compute the fusion ring of $G_2$ at all levels."}
{"category": "Math", "title": "On the comparison of positive elements of a C*-algebra by lower semicontinuous traces", "abstract": "It is shown in this paper that two positive elements of a C*-algebra agree on all lower semicontinuous traces if and only if they are equivalent in the sense of Cuntz and Pedersen. A similar result is also obtained in the more general case where the two elements are comparable by their values on the lower semicontinuous traces. This result is used to give a characterization of the functions on the cone of lower semicontinuous traces of a stable C*-algebra that arise from positive elements of the algebra."}
{"category": "Math", "title": "On a certain asymptotic relationship involving $\\vartheta(t) - \\lfloor t \\rfloor$ and $t^{1/2}$", "abstract": "Let $\\lfloor t \\rfloor$ denote the greatest positive integer less than or equal to a given positive real number $t$ and $\\vartheta(t)$ the Chebyshev $\\vartheta$-function. In this paper, we prove a certain asymptotic relationship involving $\\vartheta(t) - \\lfloor t \\rfloor $ and $t^{1/2}$."}
{"category": "Math", "title": "Comparing 2-handle additions to a genus 2 boundary component", "abstract": "We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus two boundary component of a compact, orientable 3-manifold. These results give a near complete solution to a conjecture of Scharlemann and provide evidence for a conjecture of Scharlemann and Wu. The proofs make use of a new theorem concerning the effects of attaching a 2-handle to a suture in the boundary of a sutured manifold."}
{"category": "Math", "title": "SiZer for Censored Density and Hazard Estimation", "abstract": "The SiZer method is extended to nonparametric hazard estimation and also to censored density and hazard estimation. The new method allows quick, visual statistical inference about the important issue of statistically significant increases and decreases in the smooth curve estimate. This extension has required the opening of a new avenue of research on the interface between statistical inference and scale space."}
{"category": "Math", "title": "On a six-parameter generalized Burr XII distribution", "abstract": "In this paper, we derive a probability density function that generalizes the Burr XII distribution. The cumulative distribution function and the $n^{th}$ moment of the generalized distribution are obtained while the distribution of some order statistics of the distribution are established. A theorem that relate the new distribution to another statistical distribution is established."}
{"category": "Math", "title": "On type III generalized half logistic distribution", "abstract": "It is well known that generalized models is attracting the attention of researchers in recent times because of their flexibilities. Particularly, the logistic model has been generalized and applied by many authors while the half logistic distribution has not recieve much attention in term of its generalization. In this paper, we considered a generalized form of half logistic model called type III generalized half logistic distribution. We obtained its probability density function, the cumulative distribution function, the $n^{th}$ moment, the median, the mode, the 100$p$-percentage points and the order statistics of the generalized distribution are established."}
{"category": "Math", "title": "Distinguished principal series representations for GLn over a p-adic field", "abstract": "In the following article, we give a description of the distingushed irreducible principal series representations of the general linear group over a p-adic field in terms of inducing datum. This provides a counter-example to a conjecture of Jacquet about distinction (Conjecture 1 in U.K Anandavardhanan, \"Distinguished non-Archimedean representations \", Proc. Hyderabad Conference on Algebra and Number Theory, 2005, 183-192)."}
{"category": "Math", "title": "A positive solution to a conjecture of A. Katok for diffeomorphism case", "abstract": "In this paper, using Pesin theory and Liao theory, we give a positive solution to a conjecture of A. Katok."}
{"category": "Math", "title": "A Tamed 3D Navier-Stokes Equation in Domains", "abstract": "In this paper, we analyze a tamed 3D Navier-Stokes equation in uniform $C^2$-domains (not necessarily bounded), which obeys the scaling invariance principle, and prove the existence and uniqueness of strong solutions to this tamed equation. In particular, if there exists a bounded solution to the classical 3D Navier-Stokes equation, then this solution satisfies our tamed equation. Moreover, the existence of a global attractor for the tamed equation in bounded domains is also proved. As simple applications, some well known results for the classical Navier-Stokes equations in unbounded domains are covered."}
{"category": "Math", "title": "The Hochschild cohomology ring of the standard Podles quantum sphere", "abstract": "The cup and cap product in twisted Hochschild (co)homology is computed for the standard quantum 2-sphere and used to construct a cyclic 2-cocycle that represents the fundamental Hochschild class."}
{"category": "Math", "title": "Some Critical Issues for the \"Equation-Free\" Approach to Multiscale Modeling", "abstract": "The \"equation-free'' approach has been proposed in recent years as a general framework for developing multiscale methods to efficiently capture the macroscale behavior of a system using only the microscale models. In this paper, we take a close look at some of the algorithms proposed under the \"equation-free'' umbrella, the projective integrators and the patch dynamics. We discuss some very simple examples in the context of the \"equation-free'' approach. These examples seem to indicate that while its general philosophy is quite attractive and indeed similar to many other approaches in concurrent multiscale modeling, there are severe limitations to the specific implementation proposed by the equation-free approach."}
{"category": "Math", "title": "Bayesian outlier detection in Capital Asset Pricing Model", "abstract": "We propose a novel Bayesian optimisation procedure for outlier detection in the Capital Asset Pricing Model. We use a parametric product partition model to robustly estimate the systematic risk of an asset. We assume that the returns follow independent normal distributions and we impose a partition structure on the parameters of interest. The partition structure imposed on the parameters induces a corresponding clustering of the returns. We identify via an optimisation procedure the partition that best separates standard observations from the atypical ones. The methodology is illustrated with reference to a real data set, for which we also provide a microeconomic interpretation of the detected outliers."}
{"category": "Math", "title": "Geodesically Complete Lorentzian Metrics on Some Homogeneous 3 Manifolds", "abstract": "In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with non compact (local) isotropy group, those that are geodesically complete."}
{"category": "Math", "title": "Counterpropagating Two-Soliton Solutions in the FPU Lattice", "abstract": "We study the interaction of small amplitude, long wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbor interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intemediate values of time these solutions describe the interaction of two counterpropagating pulses. These solutions are stable with respect to perturbations in $\\ell^2$ and asymptotically stable with respect to perturbations which decay exponentially at spatial $\\pm \\infty$.}"}
{"category": "Math", "title": "H\\\"older Regularity of Two-Dimensional Almost-Minimal Sets in $\\R^n$", "abstract": "We give a different and probably more elementary proof of a good part of Jean Taylor's regularity theorem for Almgren almost-minimal sets of dimension 2 in $\\R^3$. We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor's result to almost-minimal sets of dimension 2 in $\\R^n$, and give the expected characterization of the closed sets $E$ of dimension 2 in $\\R^3$ that are minimal, in the sense that $H^2(E\\setminus F) \\leq H^2(F\\setminus E)$ for every closed set $F$ such that there is a bounded set $B$ so that $F=E$ out of $B$ and $F$ separates points of $\\R^3 \\setminus B$ that $E$ separates."}
{"category": "Math", "title": "Haar Shifts, Commutators, and Hankel Operators", "abstract": "Hankel operators lie at the junction of analytic and real-variables. We will explore this junction, from the point of view of Haar shifts and commutators. An decomposition of the commutator [H,b] into paraproducts is presented."}
{"category": "Math", "title": "Confidence Sets Based on Penalized Maximum Likelihood Estimators in Gaussian Regression", "abstract": "Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and it is shown that symmetric intervals are the shortest. The length of the shortest intervals based on the hard-thresholding estimator is larger than the length of the shortest interval based on the adaptive LASSO, which is larger than the length of the shortest interval based on the LASSO, which in turn is larger than the standard interval based on the maximum likelihood estimator. In the case where the penalized estimators are tuned to possess the `sparsity property', the intervals based on these estimators are larger than the standard interval by an order of magnitude. Furthermore, a simple asymptotic confidence interval construction in the `sparse' case, that also applies to the smoothly clipped absolute deviation estimator, is discussed. The results for the known-variance case are shown to carry over to the unknown-variance case in an appropriate asymptotic sense."}
{"category": "Math", "title": "Explicit solutions of division problems for matrices of polynomials", "abstract": "We present integral representations of solutions to division problems involving matrices of polynomials in several complex variables. We also find estimates of the polynomial degree of the solutions by means of careful degree estimates of the so-called Hefer forms which are components of the representations."}
{"category": "Math", "title": "Mean value one of prime-pair constants", "abstract": "For k greater than 1 and r different from 0, let pi^k_{2r}(x) denote the number of prime pairs (p,p^k+2r) with p not exceeding (large) x. By the Bateman-Horn conjecture, the function pi^k_{2r}(x) should be asymptotic to (2/k)C^k_{2r}li_2(x), with certain specific constants C^k_{2r}. Heuristic arguments lead to the conjecture that these constants have mean value one, just like the Hardy-Littlewood constants C_{2r} for prime pairs (p,p+2r). The conjecture is supported by extensive numerical work."}
{"category": "Math", "title": "Group actions and Helly's theorem", "abstract": "We describe a connection between the combinatorics of generators for certain groups and the combinatorics of Helly's 1913 theorem on convex sets. We use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces. These results are encode d in a property that we introduce called ``property $\\FA_r$'', which reduces to Serre's property $\\FA$ when $r=1$. The method applies to $S$-arithmetic groups in higher $\\Q$-rank, to simplex reflection groups (including some non-arithmetic ones), and to higher rank Chevalley groups over polynomial and other rings (for example $\\SL_n(\\Z[x_1,..., x_d]), n>2$)."}
{"category": "Math", "title": "Transcendence of the Gaussian Liouville number and relatives", "abstract": "{\\em The Liouville number}, denoted $l$, is defined by $$l:=0.100101011101101111100...,$$ where the $n$th bit is given by ${1/2}(1+\\gl(n))$; here $\\gl$ is the Liouville function for the parity of prime divisors of $n$. Presumably the Liouville number is transcendental, though at present, a proof is unattainable. Similarly, define {\\em the Gaussian Liouville number} by $$\\gamma:=0.110110011100100111011...$$ where the $n$th bit reflects the parity of the number of rational Gaussian primes dividing $n$, 1 for even and 0 for odd. In this paper, we prove that the Gaussian Liouville number and its relatives are transcendental. One such relative is the number $$\\sum_{k=0}^\\infty\\frac{2^{3^k}}{2^{3^k2}+2^{3^k}+1}=0.101100101101100100101...,$$ where the $n$th bit is determined by the parity of the number of prime divisors that are equivalent to 2 modulo 3. We use methods similar to that of Dekking's proof of the transcendence of the Thue--Morse number \\cite{Dek1} as well as a theorem of Mahler's \\cite{Mahl1}. (For completeness we provide proofs of all needed results.) This method involves proving the transcendence of formal power series arising as generating functions of completely multiplicative functions."}
{"category": "Math", "title": "Gauss-Manin connection and t-adic geometry", "abstract": "We show that the de Rham cohomology of any separated and smooth rigid variety over a field of Laurent series of characteristic zero carries a natural formal meromorphic connection, which we call the Gauss-Manin connection. We compare it with the Gauss-Manin connection of a proper and smooth variety over a curve, and with the Gauss-Manin connection of the Milnor fibration at an isolated complex hypersurface singularity."}
{"category": "Math", "title": "Multiplicative Properties of the Slice Filtration", "abstract": "We show that the slice filtration introduced by Voevodsky is compatible in a suitable sense with the symmetric monoidal structure in the category of motivic symmetric T-spectra constructed by Jardine. It follows from this compatibility that the zero slice of the sphere spectrum s_{0}(1) is a ring spectrum and that for every symmetric T-spectrum X, and every integer n; the n-slice s_{n}(X) is a module over s_{0}(1). In particular, if the base scheme is a field of characteristic zero, we have that all the slices s_{n}(X) are big motives in the sense of Voevodsky. We also get as a corollary that the smash product of symmetric T-spectra induces pairings in the motivic Atiyah-Hirzebruch spectral sequence."}
{"category": "Math", "title": "Schottky groups cannot act on $\\mathbb{P}^{2n}_{\\mathbb{C}}$ as subgroups of $PSL(2n+1,\\Bbb{C})$", "abstract": "In this paper we look at a special type of discrete subgroups of $PSL_{n+1}(\\Bbb{C})$ called Schottky groups. We develop some basic properties of these groups and their limit set when $n > 1$, and we prove that Schottky groups only occur in odd dimensions, {\\it i.e.}, they cannot be realized as subgroups of $PSL_{2n+1}(\\Bbb{C})$."}
{"category": "Math", "title": "Automorphisms of $P_8$ singularities and the complex crystallographic groups", "abstract": "The paper completes the study of symmetries of parabolic function singularities with relation to complex crystallographic groups that was started in \\cite{GM,X9}. We classify smoothable automorphisms of $P_8$ singularities which split the kernel of the intersection form on the second homology. For such automorphisms, the monodromy groups acting on the duals to the eigenspaces with degenerate intersection form are then identified as some of complex affine reflection groups tabled in \\cite{P}."}
{"category": "Math", "title": "Linearizing a certain family of nonlinear differential equations", "abstract": "We show how to reduce the problem of solving members of a certain family of nonlinear differential equations to that of solving some corresponding linear differential equations."}
{"category": "Math", "title": "Linear Fractional Stable Sheets: wavelet expansion and sample path properties", "abstract": "In this paper we give a detailed description of the random wavelet series representation of real-valued linear fractional stable sheet introduced in Ayache, Roueff and Xiao (2007). By using this representation, in the case where the sample paths are continuous, an anisotropic uniform and quasi-optimal modulus of continuity of these paths is obtained as well as an upper bound for their behavior at infinity and around the coordinate axes. The Hausdorff dimensions of the range and graph of these stable random fields are then derived."}
{"category": "Math", "title": "Noncommutative geometry as a functor", "abstract": "In this note the noncommutative geometry is interpreted as a functor, whose range is a family of the operator algebras. Some examples are given and a program is sketched."}
{"category": "Math", "title": "On the transverse invariant for bindings of open books", "abstract": "Consider a transverse knot which is the binding of an open book for the ambient contact manifold. In this paper, we show that the transverse invariants defined by Lisca, Ozsvath, Stipsicz, and Szabo (LOSS) are nonvanishing for such transverse knots. This is true regardless of whether or not the ambient contact structure is tight. We also prove a vanishing theorem for LOSS's Legendrian and transverse invariants. As a corollary, we show that if (T,\\pi) is an open book with connected binding, then the complement of T has no Giroux torsion."}
{"category": "Math", "title": "Eigenvalue multiplicity and volume growth", "abstract": "Let $G$ be a finite group with symmetric generating set $S$, and let $c = \\max_{R > 0} |B(2R)|/|B(R)|$ be the doubling constant of the corresponding Cayley graph, where $B(R)$ denotes an $R$-ball in the word-metric with respect to $S$. We show that the multiplicity of the $k$th eigenvalue of the Laplacian on the Cayley graph of $G$ is bounded by a function of only $c$ and $k$. More specifically, the multiplicity is at most $\\exp((\\log c)(\\log c + \\log k))$. Similarly, if $X$ is a compact, $n$-dimensional Riemannian manifold with non-negative Ricci curvature, then the multiplicity of the $k$th eigenvalue of the Laplace-Beltrami operator on $X$ is at most $\\exp(n^2 + n log k)$. The first result (for $k=2$) yields the following group-theoretic application. There exists a normal subgroup $N$ of $G$, with $[G : N] \\leq \\alpha(c)$, and such that $N$ admits a homomorphism onto the cyclic group $Z_M$, where $M \\geq |G|^{\\delta(c)}$ and $\\alpha(c), \\delta(c) > 0$ are explicit functions depending only on $c$. This is the finitary analog of a theorem of Gromov which states that every infinite group of polynomial growth has a subgroup of finite index which admits a homomorphism onto the integers. This addresses a question of Trevisan, and is proved by scaling down Kleiner's proof of Gromov's theorem. In particular, we replace the space of harmonic functions of fixed polynomial growth by the second eigenspace of the Laplacian on the Cayley graph of $G$."}
{"category": "Math", "title": "On coproducts in varieties, quasivarieties and prevarieties", "abstract": "If the free algebra F on one generator in a variety V of algebras (in the sense of universal algebra) has a subalgebra free on two generators, must it also have a subalgebra free on three generators? In general, no; but yes if F generates the variety V. Generalizing the argument, it is shown that if we are given an algebra and subalgebras, A_0\\supseteq ... \\supseteq A_n, in a prevariety (SP-closed class of algebras) P such that A_n generates P, and also subalgebras B_i\\subseteq A_{i-1} (0<i\\leq n) such that for each i>0 the subalgebra of A_{i-1} generated by A_i and B_i is their coproduct in P, then the subalgebra of A generated by B_1, ..., B_n is the coproduct in P of these algebras. Some further results on coproducts are noted: If P satisfies the amalgamation property, one has the stronger \"transitivity\" statement: if A has a finite family of subalgebras (B_i)_{i\\in I} such that the subalgebra of A generated by the B_i is their coproduct, and each B_i has a finite family of subalgebras (C_{ij})_{j\\in J_i} with the same property, then the subalgebra of A generated by all the C_{ij} is their coproduct. For P a residually small prevariety or an arbitrary quasivariety, relationships are proved between the least number of algebras needed to generate P as a prevariety or quasivariety, and behavior of the coproduct operation in P. It is shown by example that for B a subgroup of the group S = Sym(\\Omega) of all permutations of an infinite set \\Omega, the group S need not have a subgroup isomorphic over B to the coproduct with amalgamation S \\cP_B S. But under weak additional hypotheses on B, the question remains open."}
{"category": "Math", "title": "Threshold solutions for the focusing 3d cubic Schroedinger equation", "abstract": "We study the focusing 3d cubic NLS equation with H^1 data at the mass-energy threshold, namely, when M[u_0]E[u_0] = M[Q]E[Q]. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) is classified when M[u_0]E[u_0] < M[Q]E[Q]. In this paper, we first exhibit 3 special solutions: e^{it}Q and Q^+, Q^-; here Q is the ground state, and Q^+, Q^- exponentially approach the ground state solution in the positive time direction, meanwhile Q^+ having finite time blow up and Q^- scattering in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to \\dot{H}^{1/2} symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schroedinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation."}
{"category": "Math", "title": "Classification of compact ancient solutions to the curve shortening flow", "abstract": "We consider an embedded convex ancient solution $\\Gamma_t$ to the curve shortening flow in $\\mathbb{R}^2$. We prove that there are only two possibilities: the family $\\Gamma_t$ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II ancient solution to the curve shortening flow. We also give a necessary and sufficient curvature condition for an embedded, closed ancient solution to the curve shortening flow to be convex."}
{"category": "Math", "title": "The harmonic mean curvature flow of nonconvex surfaces in $\\mathbb{R}^3$", "abstract": "We consider a compact, star-shaped, mean convex hypersurface $\\Sigma^2\\subset \\mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \\cite{An1}). We also prove that in the case we have a surface of revolution which is star-shaped and mean convex, a smooth solution always exists up to some finite time $T < \\infty$ at which the flow shrinks to a point asymptotically spherically."}
{"category": "Math", "title": "Gregory Trees, The Continuum, And Martin's Axiom", "abstract": "We continue the investigation of Gregory trees and the Cantor Tree Property carried out by Hart and Kunen. We produce models of MA with the Continuum arbitrarily large in which there are Gregory trees, and in which there are no Gregory trees."}
{"category": "Math", "title": "Permutation equivalent maximal irreducible Goppa codes", "abstract": "We consider the problem of finding the number of permutation non-equivalent classical irreducible maximal Goppa codes having fixed parameters q, n and r from a group theory point of view."}
{"category": "Math", "title": "Formes automorphes et theoremes de Riemann-Roch arithmetiques", "abstract": "Nous construisons trois familles de formes automorphes au moyen du theoreme de Riemann-Roch arithmetique et de la formule de Lefschetz arithmetique. Deux de ces familles ont deja ete construites par Yoshikawa et notre construction met en lumiere leur origine arithmetique. ----- We construct three families of automorphic forms following the arithmetic Riemann-Roch theorem and the arithmetic Lefschetz formula. Two of these families were already constructed by Yoshikawa and our construction illuminates their arithmetic origin."}
{"category": "Math", "title": "Correction to \"Simplicial monoids and Segal categories\"", "abstract": "In this note we make a minor correction to the paper ``Simplicial monoids and Segal categories.\""}
{"category": "Math", "title": "On 2-partitionable clutters and the MFMC property", "abstract": "We introduce 2-partitionable clutters as the simplest case of the class of $k$-partitionable clutters and study some of their combinatorial properties. In particular, we study properties of the rank of the incidence matrix of these clutters and properties of their minors. A well known conjecture of Conforti and Cornu\\'ejols \\cite{ConfortiCornuejols,cornu-book} states: That all the clutters with the packing property have the max-flow min-cut property, i.e. are mengerian. Among the general classes of clutters known to verify the conjecture are: balanced clutters (Fulkerson, Hoffman and Oppenheim \\cite{FulkersonHoffmanOppenheim}), binary clutters (Seymour \\cite{Seymour}) and dyadic clutters (Cornu\\'ejols, Guenin and Margot \\cite{CornuejolsGueninMargot}). We find a new infinite family of 2-partitionable clutters, that verifies the conjecture. On the other hand we are interested in studying the normality of the Rees algebra associated to a clutter and possible relations with the Conforti and Cornu\\'ejols conjecture. In fact this conjecture is equivalent to an algebraic statement about the normality of the Rees algebra \\cite{rocky}."}
{"category": "Math", "title": "Some reducible Specht modules for Iwahori--Hecke algebras of type $A$ with $q=-1$", "abstract": "The reducibility of the Specht modules for the Iwahori--Hecke algebras in type $A$ is still open in the case where the defining parameter $q$ equals -1. We prove the reducibility of a large class of Specht modules for these algebras."}
{"category": "Math", "title": "A non-crossing standard monomial theory", "abstract": "The second author has introduced non-crossing tableaux, objects whose non-nesting analogues are semi-standard Young tableaux. We relate non-crossing tableaux to Gelfand-Tsetlin patterns and develop the non-crossing analogue of standard monomial theory. Leclerc and Zelevinsky's weakly separated sets are special cases of non-crossing tableaux, and we suggest that non-crossing tableaux may help illuminate the theory of weakly separated sets."}
{"category": "Math", "title": "Analytic families of holomorphic iterated function systems", "abstract": "This paper deals with analytic families of holomorphic iterated function systems. Using real analyticity of the pressure function (which we prove), we establish a classification theorem for analytic families of holomorphic iterated function systems which depend continuously on a parameter when the space of holomorphic iterated function systems is endowed with the \"$\\lambda$-topology\". This classification theorem allows us to generalize some geometric results from the paper (\"Lambda-Topology vs. Pointwise Topology\", to appear in Ergod. Th. & Dynam. Sys.) of the authors, and gives us a better and clearer understanding of the global structure of the space of conformal IFSs."}
{"category": "Math", "title": "Well-posedness for compressible Euler equations with physical vacuum singularity", "abstract": "An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristics coincide and have unbounded derivative. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local in time well-posedness of one-dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity."}
{"category": "Math", "title": "Energy-efficient motion camouflage in three dimensions", "abstract": "Recent observations suggest that one insect may camouflage its own motion whilst tracking another. Here we present a strategy by which one agent can efficiently track and intercept another agent, whilst camouflaging its own motion and minimizing its energy consumption"}
{"category": "Math", "title": "Planar Algebra of the Subgroup-Subfactor", "abstract": "We give an identification between the planar algebra of the subgroup-subfactor $R \\rtimes H \\subset R \\rtimes G$ and the $G$-invariant planar subalgebra of the planar algebra of the bipartite graph $\\star_n$, where $n = [G : H]$. The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant, of $R \\rtimes H \\subset R \\rtimes G$ in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor $R^G \\subset R^H$ and the $G$-invariant planar subalgebra of the planar algebra of the `flip' of $\\star_n $."}
{"category": "Math", "title": "On isomorphism classes and invariants of low dimensional complex filiform Leibniz algebras (part 2)", "abstract": "The paper is an implementation in low dimensional cases of the classification method presented before by Rakhimov and Bekbaev. We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from the naturally graded non-Lie filiform Leibniz algebras. A hypothetic formula for the adapted number of isomorphism classes is given."}
{"category": "Math", "title": "Coefficients of squares of Newman polynomials", "abstract": "We show that there are polynomials $p_N$ of arbitrarily large degree $N$, with coefficients equal to 0 or 1 (Newman polynomials), such that $$ \\liminf_{N \\to \\infty} N \\Linf{p_N^2} \\bigl / p_N^2(1) < 1, $$ where $\\Linf{q}$ denotes the maximum coefficient of the polynomial $q$ and which, at the same time, are sparse: $p_N(1)/N \\to 0$. This disproves a conjecture of Yu \\cite{yu}. We build on some previous results of Berenhaut and Saidak \\cite{berenhaut-saidak} and Dubickas \\cite{dubickas} whose examples lacked the sparsity. This sparsity we create from these examples by randomization."}
{"category": "Math", "title": "Cohomology algebra of the orbit space of free circle group actions on lens spaces", "abstract": "Suppose that G=S^1 acts freely on a finitistic space X whose mod p cohomology ring isomorphic to that of a lens space L^{2m-1}(p;q_1,...,q_m). In this paper, we determine the mod p cohomology ring of the orbit space X/G. If the characteristic class \\alpha\\belongs H^2(X/G;Z_p) of the S^1-bundle S--> X--> X/G is nonzero, then mod p ndex of the action is deined to be the largest integer n such that \\alpha^n is nonzero. We also show that the mod p index of a free action of S^1 on a lens space L^(2m-1)(p;q_1,...,q_m) is p-1, provided that \\alpha is nonzero."}
{"category": "Math", "title": "Quotients of Banach spaces with the Daugavet property", "abstract": "We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak$^*$ analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of $L_1[0,1]$ over an $\\ell_1$-subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pelczynski in the negative."}
{"category": "Math", "title": "An algorithm for weighted fractional matroid matching", "abstract": "Let M be a matroid on ground set E. A subset l of E is called a `line' when its rank equals 1 or 2. Given a set L of lines, a `fractional matching' in (M,L) is a nonnegative vector x indexed by the lines in L, that satisfies a system of linear constraints, one for each flat of M. Fractional matchings were introduced by Vande Vate, who showed that the set of fractional matchings is a half-integer relaxation of the matroid matching polytope. It was shown by Chang et al. that a maximum size fractional matching can be found in polynomial time. In this paper we give a polynomial time algorithm to find for any given weights on the lines in L, a maximum weight fractional matching."}
{"category": "Math", "title": "Distal actions and shifted convolution property", "abstract": "A locally compact group $G$ is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure $\\mu$ on $G$, either $\\sup_{x\\in G} \\mu ^n (Cx) \\ra 0$ for all compact subsets $C$ of $G$, or there exist $x\\in G$ and a compact subgroup $K$ normalised by $x$ such that $\\mu^nx^{-n} \\ra \\omega_K$, the Haar measure on $K$. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors under compact groups invariant under the action and for factors under the connected component of identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (point-wise distality). This has some interesting corollaries to distality of certain actions and Choquet Deny measures which actually motivated SCP and point-wise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures."}
{"category": "Math", "title": "Alexandrov curvature of Kaehler curves", "abstract": "We study the intrinsic geometry of a one-dimensional complex space provided with a Kaehler metric in the sense of Grauert. We show that if K is an upper bound for the Gaussian curvature on the regular locus, then the intrinsic metric has curvature at most K in the sense of Alexandrov."}
{"category": "Math", "title": "Groupoid representations and modules over the convolution algebras", "abstract": "The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an etale Lie groupoid and the category of modules over its convolution algebra that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita bicategory of etale Lie groupoids and the given correspondence represents a natural equivalence between them."}
{"category": "Math", "title": "Index and nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces", "abstract": "The aim of this work is to extend the results of S. Nayatani about the index and the nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces for values of the genus bigger than 37. That allows us to state that these minimal surfaces are non degenerate for all the values of the genus in the sense of the definition of J. Perez and A. Ros."}
{"category": "Math", "title": "Stochastic equations with delay: optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations", "abstract": "We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process $X$ with values in a space of continuous functions $\\mathbf C$, with generator $\\mathcal L$. We then consider a backward stochastic differential equation depending on $X$, with unknown processes $(Y,Z)$, and we study properties of the resulting system, in particular we identify the process $Z$ as a deterministic functional of $X$. We next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space $\\mathbf C$ driven by $\\mathcal L$, and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms the process $X$."}
{"category": "Math", "title": "Darboux transforms and spectral curves of Hamiltonian stationary Lagrangian tori", "abstract": "The multiplier spectral curve of a conformal torus in the 4-sphere is essentially, see arXiv:0712.2311, given by all Darboux transforms of the conformal torus. In the particular case when the conformal immersion is a Hamiltonian stationary torus in Euclidean 4-space, the left normal of the immersion is harmonic, hence we can associate a second Riemann surface: the eigenline spectral curve of the left normal, as defined by Hitchin. We show that the multiplier spectral curve of a Hamiltonian stationary torus and the eigenline spectral curve of its left normal are biholomorphic Riemann surfaces of genus zero. Moreover, we prove that all Darboux transforms, which arise from generic points on the spectral curve, are Hamiltonian stationary whereas we also provide examples of Darboux transforms which are not even Lagrangian."}
{"category": "Math", "title": "Base-point-free pencils on triple covers of smooth curves", "abstract": "Let $X$ be a smooth algebraic curve. Suppose that there exists a triple covering $f : X \\to Y$ where $Y$ is a smooth algebraic curve. In this paper, we investigate the existence of morphisms from $X$ to the projective line $\\mathbf{P}^1$ which do not factor through the covering $f$. For this purpose, we generalize the classical results of Maroni concerning base-point-free pencils on trigonal curves to the case of triple covers of arbitrary smooth irrational curves."}
{"category": "Math", "title": "Spiraling spectra of geodesic lines in negatively curved manifolds", "abstract": "Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lines around these objects. We prove analogs of the theorems of Dirichlet, Hall and Cusick in this context. As a consequence, we obtain Diophantine approximation results of real numbers, complex numbers, or elements of the Heisenberg group by irrational quadratic ones."}
{"category": "Math", "title": "On pro-p fundamental groups of marked arithmetic curves", "abstract": "Let k be a global field, p an odd prime number different from char(k) and S, T disjoint, finite sets of primes of k. Let G_S^T(k)(p)=Gal(k_S^T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and completely split at T. We prove the existence of a finite set of primes S_0, which can be chosen disjoint from any given set M of Dirichlet density zero, such that the cohomology of G_{S\\cup S_0}^T(k)(p) coincides with the etale cohomology of the associated marked arithmetic curve. In particular, cd G_{S\\cup S_0}^T(k)(p)=2. Furthermore, we can choose S_0 in such a way that k_{S\\cup S_0}^T(p) realizes the maximal p-extension k_\\p(p) of the local field k_\\p for all \\p\\in S\\cup S_0, the cup-product H^1(G_{S\\cup S_0}^T(k)(p),\\F_p) \\otimes H^1(G_{S\\cup S_0}^T(k)(p),\\F_p) --> H^2(G_{S\\cup S_0}^T(k)(p),\\F_p) is surjective and the decomposition groups of the primes in S establish a free product inside G_{S\\cup S_0}^T(k)(p). This generalizes previous work of the author where similar results were shown in the case T=\\emptyset under the restrictive assumption p\\nmid Cl(k) and \\zeta_p\\notin k."}
{"category": "Math", "title": "Making research on symmetric functions using MuPAD-Combinat", "abstract": "We present an overview of the implementation of symmetric functions in MuPAD-Combinat. We also explain how to interface C++ programs with MuPAD in order to make efficient research during concrete work sessions."}
{"category": "Math", "title": "Mock Jacobi forms in basic hypergeometric series", "abstract": "We show that some $q$-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of $q$. And we prove that certain linear sums of $q$-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition."}
{"category": "Math", "title": "Equality of multiplicity free skew characters", "abstract": "In this paper we show that two skew diagrams lambda/mu and alpha/beta can represent the same multiplicity free skew character [lambda/mu]=[alpha/beta] only in the the trivial cases when lambda/mu and alpha/beta are the same up to translation or rotation or if lambda=alpha is a staircase partition lambda=(l,l-1,...,2,1) and lambda/mu and alpha/beta are conjugate of each other."}
{"category": "Math", "title": "A family of transversely nonsimple knots", "abstract": "We apply knot Floer homology to exhibit an infinite family of transversely nonsimple prime knots starting with $10_{132}$. We also discuss the combinatorial relationship between grid diagrams, braids, and Legendrian and transverse knots in standard contact $\\mathbb{R}^3$."}
{"category": "Math", "title": "Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations", "abstract": "In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems."}
{"category": "Math", "title": "Extrapolation of Threshold-Limited Null Measurement Frequencies", "abstract": "The total measurable level of a pathogen is due to many sources, which produce a variety of pulses, overlapping in time, that rise suddenly and then decay. What is measured is the level of the total contribution of the sources at a given time. But since we are only capable of measuring the total level above some threshold $x_0$, we would like to predict the distribution below this level. Our principal model assumption is that of the asymptotic exponential decay of all pulses. We show that this implies a power law distribution for the frequencies of low amplitude observations. As a consequence, there is a simple extrapolation procedure for carrying the data to the region below $x_0$."}
{"category": "Math", "title": "The Stochastic Heat Equation Driven by a Gaussian Noise: germ Markov Property", "abstract": "Let $u=\\{u(t,x);t \\in [0,T], x \\in {\\mathbb{R}}^{d}\\}$ be the process solution of the stochastic heat equation $u_{t}=\\Delta u+ \\dot F, u(0,\\cdot)=0$ driven by a Gaussian noise $\\dot F$, which is white in time and has spatial covariance induced by the kernel $f$. In this paper we prove that the process $u$ is locally germ Markov, if $f$ is the Bessel kernel of order $\\alpha=2k,k \\in \\bN_{+}$, or $f$ is the Riesz kernel of order $\\alpha=4k,k \\in \\bN_{+}$."}
{"category": "Math", "title": "Minimal surfaces in circle bundles over Riemann surfaces", "abstract": "For a compact 3-manifold $M$ which is a circle bundle over a compact Riemann surface $\\Sigma$ with even Euler number $e(M)$, and with a Riemannian metric compatible with the bundle projection, there exists a compact minimal surface $S$ in $M$. $S$ is embedded and is a section of the restriction of the bundle to the complement of a finite number of points in $\\Sigma$."}
{"category": "Math", "title": "The mixing time evolution of Glauber dynamics for the mean-field Ising model", "abstract": "We consider Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime ($\\beta < 1$) has order $n\\log n$, whereas the mixing-time in the case $\\beta > 1$ is exponential in $n$. Recently, Levin, Luczak and Peres proved that for any fixed $\\beta < 1$ there is cutoff at time $[2(1-\\beta)]^{-1} n\\log n$ with a window of order $n$, whereas the mixing-time at the critical temperature $\\beta=1$ is $\\Theta(n^{3/2})$. It is natural to ask how the mixing-time transitions from $\\Theta(n\\log n)$ to $\\Theta(n^{3/2})$ and finally to $\\exp(\\Theta(n))$. That is, how does the mixing-time behave when $\\beta=\\beta(n)$ is allowed to tend to 1 as $n\\to\\infty$. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point $\\beta_c=1$. In particular, we find a scaling window of order $1/\\sqrt{n}$ around the critical temperature. In the high temperature regime, $\\beta = 1 - \\delta$ for some $0 < \\delta < 1$ so that $\\delta^2 n \\to\\infty$ with $n$, the mixing-time has order $(n/\\delta)\\log(\\delta^2 n)$, and exhibits cutoff with constant 1/2 and window size $n/\\delta$. In the critical window, $\\beta = 1\\pm \\delta$ where $\\delta^2 n$ is O(1), there is no cutoff, and the mixing-time has order $n^{3/2}$. At low temperature, $\\beta = 1 + \\delta$ for $\\delta > 0$ with $\\delta^2 n \\to\\infty$ and $\\delta=o(1)$, there is no cutoff, and the mixing time has order $(n/\\delta)\\exp(({3/4}+o(1))\\delta^2 n)$."}
{"category": "Math", "title": "Bitwist 3-manifolds", "abstract": "Our earlier twisted-face-pairing construction showed how to modify an arbitrary orientation-reversing face-pairing on a faceted 3-ball in a mechanical way so that the quotient is automatically a closed, orientable 3-manifold. The modifications were, in fact, parametrized by a finite set of positive integers, arbitrarily chosen, one integer for each edge class of the original face-pairing. This allowed us to find very simple face-pairing descriptions of many, though presumably not all, 3-manifolds. Here we show how to modify the construction to allow negative parameters, as well as positive parameters, in the twisted-face-pairing construction. We call the modified construction the bitwist construction. We prove that all closed connected orientable 3-manifolds are bitwist manifolds. As with the twist construction, we analyze and describe the Heegaard splitting naturally associated with a bitwist description of a manifold."}
{"category": "Math", "title": "The Kohn Algorithm on Denjoy-Carleman Classes", "abstract": "The equivalence of the Kohn finite ideal type and the D'Angelo finite type with the subellipticity of the $\\bar\\partial$-Neumann problem is extended to pseudoconvex domains in $C^n$ whose defining function is in a Denjoy-Carleman quasianalytic class closed under differentiation. The proof involves algebraic geometry over a ring of germs of Denjoy-Carleman quasianalytic functions that is not known to be Noetherian and that is intermediate between the ring of germs of real-analytic functions and the ring of germs of smooth functions. It is also shown that this type of ring of germs of Denjoy-Carleman functions satisfies the $\\sqrt{acc}$ property, one of the strongest properties a non-Noetherian ring could possess."}
{"category": "Math", "title": "Analogue of Sylvester-Cayley formula for invariants of ternary form", "abstract": "The number $\\nu_d(n)$ of linearly independed homogeneous invariants of degree $n$ for the ternary form of degree $d$ is calculated."}
{"category": "Math", "title": "A characterisation of the Hoffman-Wohlgemuth surfaces in terms of their symmetries", "abstract": "For an embedded singly periodic minimal surface M with genus bigger than or equal to 4 and annular ends, some weak symmetry hypotheses imply its congruence with one of the Hoffman-Wohlgemuth examples. We give a very geometrical proof of this fact, along which they come out many valuable clues for the understanding of these surfaces."}
{"category": "Math", "title": "On divergence form SPDEs with VMO coefficients", "abstract": "We present several results on solvability in Sobolev spaces $W^{1}_{p}$ of SPDEs in divergence form in the whole space."}
{"category": "Math", "title": "On Picture (2+1)-TQFTs", "abstract": "The goal of the paper is an exposition of the simplest $(2+1)$-TQFTs in a sense following a pictorial approach. In the end, we fell short on details in the later sections where new results are stated and proofs are outlined. Comments are welcome and should be sent to the 4th author."}
{"category": "Math", "title": "A conjecture on the forms of the roots of equations", "abstract": "E30 in the Enestrom index. Translated from the Latin original \"De formis radicum aequationum cuiusque ordinis coniectatio\" (1733). For an equation of degree n, Euler wants to define a \"resolvent equation\" of degree n-1 whose roots are related to the roots of the original equation. Thus by solving the resolvent we can solve the original equation. In sections 2 to 7 he works this out for quadratic, cubic and biquadratic equations. Apparently he gives a new method for solving the quartic in section 5. Then in section 8 Euler says that he wants to try the same approach for solving the quintic equation and general nth degree equations. In the rest of the paper Euler tries to figure out in what cases resolvents will work. Two references I found useful were Chapter 14, p.p. 106-113 of C. Edward Sandifer, \"The Early Mathematics of Leonhard Euler\", published 2007 by The Mathematical Association of America and Olaf Neumann, \"Cyclotomy: from Euler through Vandermonde to Gauss\", p.p. 323-362 in the collection \"Leonhard Euler: Life, Work and Legacy\" edited by Bradley and Sandifer, 2007. Stacy Langton has given a lot of details about Euler's work on the theory of equations, and also some advice on the translation; of course any mistakes are my own. If Langton ends up writing anything about Euler' and the theory of equations I would highly recommend reading it."}
{"category": "Math", "title": "Fibers of Generic Projections", "abstract": "Let X be a smooth projective variety of dimension n in P^r. We study the fibers of a general linear projection pi: X --> P^{n+c}, with c > 0. When n is small it is classical that the degree of any fiber is bounded by n/c+1, but this fails for n >> 0. We describe a new invariant of the fiber that agrees with the degree in many cases and is always bounded by n/c+1. This implies, for example, that if we write a fiber as the disjoint union of schemes Y' and Y'' such that Y' is the union of the locally complete intersection components of Y, then deg Y'+deg Y''_red <= n/c+1 and this formula can be strengthened a little further. Our method also gives a sharp bound on the subvariety of P^r swept out by the l-secant lines of X for any positive integer l, and we discuss a corresponding bound for highly secant linear spaces of higher dimension. These results extend Ziv Ran's \"Dimension+2 Secant Lemma\"."}
{"category": "Math", "title": "Algebraic density property of homogeneous spaces", "abstract": "Let $X$ be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that $X$ is equipped with several non-degenerate fixed point free $SL_2$-actions satisfying some mild additional assumption. Then we show that the Lie algebra generated by completely integrable algebraic vector fields on $X$ coincides with the set of all algebraic vector fields. In particular, we show that apart from a few exceptions this fact is true for any homogeneous space of form $G/R$ where $G$ is a linear algebraic group and $R$ is its proper reductive subgroup."}
{"category": "Math", "title": "Semitoric integrable systems on symplectic 4-manifolds", "abstract": "Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce."}
{"category": "Math", "title": "A Dedekind Finite Borel Set", "abstract": "In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if B is a G-delta-sigma set, then either B is countable or B contains a perfect subset. Second, we prove that if the real line is the countable union of countable sets, then there exists an F-sigma-delta set which is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite set of reals which is F-sigma-delta."}
{"category": "Math", "title": "On divergence form SPDEs with VMO coefficients in a half space", "abstract": "We extend several known results on solvability in the Sobolev spaces $W^{1}_{p}$, $p\\in[2,\\infty)$, of SPDEs in divergence form in $\\bR^{d}_{+}$ to equations having coefficients which are discontinuous in the space variable."}
{"category": "Math", "title": "Curvature flows and CMC hypersurfaces", "abstract": "We give an overview of the existence and regularity results for curvature flows and how these flows can be used to solve some problems in geometry and physics."}
{"category": "Math", "title": "On the question of ergodicity for minimal group actions on the circle", "abstract": "This work is devoted to the study of minimal, smooth actions of finitely generated groups on the circle. We provide a sufficient condition for such an action to be ergodic (with respect to the Lebesgue measure), and we illustrate this condition by studying two relevant examples. Under an analogous hypothesis, we also deal with the problem of the zero Lebesgue measure for exceptional minimal sets. This hypothesis leads to many other interesting conclusions, mainly concerning the stationary and conformal measures. Moreover, several questions are left open. The methods work as well for codimension-one foliations, though the results for this case are not explicitly stated."}
{"category": "Math", "title": "The moduli space of flat SU(2)-bundles over a nonorientable surface", "abstract": "We study the topology of the moduli space of flat SU(2)-bundles over a nonorientable surface X. This moduli space may be identified with the space of homomorphisms Hom(\\pi_1(X),SU(2)) modulo conjugation by SU(2). In particular, we compute the (rational) equivariant cohomology ring of Hom(\\pi_1(X),SU(2)) and use this to compute the ordinary cohomology groups of the quotient Hom(\\pi_1(X),SU(2))/SU(2). A key property is that the conjugation action is equivariantly formal."}
{"category": "Math", "title": "Projections of Singular Vectors of Verma Modules over Rank 2 Kac-Moody Lie Algebras", "abstract": "We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of $sl_{2}$ (Theorem 3). The formula is derived from a more general but less explicit formula due to Feigin, Fuchs and Malikov [Funct. Anal. Appl. 20 (1986), no. 2, 103-113]. In the simpler case of $A_{1}^{1}$ the formula was obtained in [Fuchs D., Funct. Anal. Appl. 23 (1989), no. 2, 154-156]."}
{"category": "Math", "title": "Simple SL(n)-Modules with Normal Closures of Maximal Torus Orbits", "abstract": "Let $T$ be the subgroup of diagonal matrices in the group SL(n). The aim of this paper is to find all finite-dimensional simple rational SL(n)-modules $V$ with the following property: for each point $v\\in V$ the closure $\\bar{Tv}$ of its $T$-orbit is a normal affine variety. Moreover, for any SL(n)-module without this property a $T$-orbit with non-normal closure is constructed. The proof is purely combinatorial: it deals with the set of weights of simple SL(n)-modules. The saturation property is checked for each subset in the set of weights."}
{"category": "Math", "title": "Words Maps and Spectra of Random Graph Lifts", "abstract": "We begin with a new analysis of formal words. Let w be a formal word in letters g_1,...,g_k. The word map associated with w maps the permutations s_1,...,s_k in S_n to the permutation obtained by replacing for each i, every occurrence of g_i in w by s_i. We investigate the random variable X_w^n that counts the fixed points in this permutation when the s_i are selected uniformly at random. A major ingredient of our work is a new categorization of words which considerably extends the dichotomy of primitive vs. imprimitive words. We establish some results and make a few conjectures about the relation between the expectation E(X_w^n) and this new categorization. This analysis contributes deeply to our study of the spectra of random lifts of graphs. Let G be a connected graph, and let the infinite tree T be its universal cover space. If L and R are the spectral radii of G and T respectively, then, as shown by J. Friedman, for almost every n-lift H of G, all \"new\" eigenvalues of H are < O(L^(1/2)R^(1/2)). We improve this upper bound to O(L^(1/3)R^(2/3)), and our aforementioned conjectures suggest a possible approach to proving an upper bound of O(R). This is a generalization of the problem of bounding the second eigenvalue in a random 2d-regular graph. As an aside, we obtain a new conceptual and relatively simple proof of a theorem of A. Nica, which determines, for every fixed w, the limit distribution (as n \\to \\infty) of X_w^n. A surprising aspect of this theorem is that the answer depends only on the largest integer d so that w=u^d for some word u."}
{"category": "Math", "title": "Examples of limits of Frobenius (type) structures: the singularity case", "abstract": "We give examples of families of Frobenius type structures on the punctured plane and we study their limits at the boundary. We then discuss the existence of a limit Frobenius manifold. We also give an example of a logarithmic Frobenius manifold."}
{"category": "Math", "title": "Zeta functions, heat kernels and spectral asymptotics on degenerating families of discrete tori", "abstract": "By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with generating set given by choosing a generator for each cyclic factor. In this article we study the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First we show that the sequence of heat kernels corresponding to the degenerating family converges, after re-scaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz type zeta functions."}
{"category": "Math", "title": "Shadows and intersections: stability and new proofs", "abstract": "We give a short new proof of a version of the Kruskal-Katona theorem due to Lov\\'asz. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lov\\'asz's theorem that answers a question of Frankl and Tokushige."}
{"category": "Math", "title": "Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces", "abstract": "In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective 4-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case."}
{"category": "Math", "title": "Triangle packings and 1-factors in oriented graphs", "abstract": "An oriented graph is a directed graph which can be obtained from a simple undirected graph by orienting its edges. In this paper we show that any oriented graph G on n vertices with minimum indegree and outdegree at least (1/2-o(1))n contains a packing of cyclic triangles covering all but at most 3 vertices. This almost answers a question of Cuckler and Yuster and is best possible, since for n = 3 mod 18 there is a tournament with no perfect triangle packing and with all indegrees and outdegrees (n-1)/2 or (n-1)/2 \\pm 1. Under the same hypotheses, we also show that one can embed any prescribed almost 1-factor, i.e. for any sequence n_1,...,n_t with n_1+...+n_t < n-O(1) we can find a vertex-disjoint collection of directed cycles with lengths n_1,...,n_t. In addition, under quite general conditions on the n_i we can remove the O(1) additive error and find a prescribed 1-factor."}
{"category": "Math", "title": "Unique mixing of the shift on the C*--algebras generated by the q--canonical commutation relations", "abstract": "The shift on the C^*--algebras generated by the Fock representation of the q--commutation relations has the strong ergodic property of unique mixing, when |q|<1."}
{"category": "Math", "title": "Moduli Spaces of Semistable Sheaves on Singular Genus One Curves", "abstract": "We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle $E_N$ of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves $\\mathcal{O}(-1)$ supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank $r$ is isomorphic to the $r$-th symmetric product of the rational curve with one node."}
{"category": "Math", "title": "A commuting derivations theorem on UFDs", "abstract": "Let $A$ be the polynomial ring over $k$ (a field of characteristic zero) in $n+1$ variables. The commuting derivations conjecture states that $n$ commuting locally nilpotent derivations on $A$, linearly independent over $A$, must satisfy $A^{D_1,...,D_m}=k[f]$ where $f$ is a coordinate. The conjecture can be formulated as stating that a $(G_m)^n$-action on $k^{n+1}$ must have invariant ring $k[f]$ where $f$ is a coordinate. In this paper we prove a statement (theorem \\ref{CDH2}) where we assume less on $A$ ($A$ is a {\\sc UFD} over $k$ of transcendence degree $n+1$ satisfying $A^*=k$) and prove less ($A/(f-\\alpha)$ is a polynomial ring for all but finitely many $\\alpha$). Under certain additional conditions (the $D_i$ are linearly independent modulo $(f-\\alpha)$ for each $\\alpha\\in k$) we prove that $A$ is a polynomial ring itself and $f$ is a coordinate. This statement is proven even more generally by replacing ``free unipotent action of dimension $n$'' for ``$G_a^n$-action''. We make links with the (Abhyankar-)Sataye conjecture and give a new equivalent formulation of the Sataye conjecture."}
{"category": "Math", "title": "Stochastic calculus for symmetric Markov processes", "abstract": "Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It\\^{o} formula for Dirichlet processes is obtained."}
{"category": "Math", "title": "Classification of quasi-trigonometric solutions of the classical Yang-Baxter equation", "abstract": "It was proved by Montaner and Zelmanov that up to classical twisting Lie bialgebra structures on $\\mathfrak{g}[u]$ fall into four classes. Here $\\mathfrak{g}$ is a simple complex finite-dimensional Lie algebra. It turns out that classical twists within one of these four classes are in a one-to-one correspondence with the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. In this paper we give a complete list of the quasi-trigonometric solutions in terms of sub-diagrams of the certain Dynkin diagrams related to $\\mathfrak{g}$. We also explain how to quantize the corresponding Lie bialgebra structures."}
{"category": "Math", "title": "Local structure of Brill-Noether strata in the moduli space of flat stable bundles", "abstract": "We study the Brill-Noether stratification of the coarse moduli space of locally free stable and flat sheaves of a compact Kahler manifold, proving that these strata have quadratic algebraic singularities."}
{"category": "Math", "title": "A Method of Classifying Simple Laced Root Systems", "abstract": "A root system in which all roots have same norm is known as a simply laced root system. We present a simple method of classifying all simply laced root systems."}
{"category": "Math", "title": "Switching Game of Backward Stochastic Differential Equations and Associated System of Obliquely Reflected Backward Stochastic Differential Equations", "abstract": "This paper is concerned with the switching game of a one-dimensional backward stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the penalization method, the monotone convergence, and the a priori estimates."}
{"category": "Math", "title": "On the distance between separatrices for the discretized pendulum equation", "abstract": "We consider the discretization q(t+\\epsilon)+q(t-\\epsilon)-2q(t)=\\epsilon^{2}\\sin\\big(q(t)\\big), $\\epsilon>0$ a small parameter, of the pendulum equation $ q '' = \\sin (q) $; in system form, we have the discretization q(t+\\epsilon)-q(t)=\\epsilon p(t+\\epsilon), p(t+\\epsilon)-p(t)=\\epsilon\\sin\\big(q(t)\\big). of the system q'=p, p'=\\sin(q). The latter system of ordinary differential equations has two saddle points at $A=(0,0)$, $B=(2\\pi, 0)$ and near both, there exist stable and unstable manifolds. It also admits a heteroclinic orbit connecting the stationary points $B$ and $A$ parametrised by $q_0(t)=4\\arctan\\big(e^{-t}\\big)$ and which contains the stable manifold of this system at $A$ as well as its unstable manifold at $B$. We prove that the stable manifold of the point $A$ and the unstable manifold of the point $B$ do not coincide for the discretization. More precisely, we show that the vertical distance between these two manifolds is exponentially small but not zero and in particular we give an asymptotic estimate of this distance. For this purpose we use a method adapted from the article of Sch\\\"afke-Volkmer \\cite{SV} using formal series and accurate estimates of the coefficients. Our result is similar to that of Lazutkin et. al. \\cite{LS}; our method of proof, however, is quite different."}
{"category": "Math", "title": "Values of the t-invariant for small Seifert manifolds", "abstract": "The t-invariant can be considered as the Turaev-Viro invariant of order 5 computed for integer colors only. We compute all values of the t-invariant for Seifert manifolds with base sphere and three singular fibers. As a result we show that the manifolds parameters modulo five define the value of the t-invariant. Partially we show that there are 12 distinct values of the t-invariant for these manifolds. Some examples show that the t-invariant for these manifolds is not defined by the first homology group."}
{"category": "Math", "title": "A note on the span of Hadamard products of vectors", "abstract": "We give a new proof of Theorem 6 in [L. Qiu and X. Zhan, On the span of Hadamard products of vectors, Linear Algebra Appl. 422 (2007) 304--307]."}
{"category": "Math", "title": "Distinguishing Primitive Permutation Groups", "abstract": "Let $G$ be a permutation group acting on a set $V$. A partition $\\pi$ of $V$ is distinguishing if the only element of $G$ that fixes each cell of $\\pi$ is the identity. The distinguishing number of $G$ is the minimum number of cells in a distinguishing partition. We prove that if $G$ is a primitive permutation group and $|V|\\ge336$, its distinguishing number is two."}
{"category": "Math", "title": "$C^{1+\\alpha}$-Regularity for Two-Dimensional Almost-Minimal Sets in $\\R^n$", "abstract": "We give a new proof and a partial generalization of Jean Taylor's result [Ta] that says that Almgren almost-minimal sets of dimension 2 in $\\R^3$ are locally $C^{1+\\alpha}$-equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in [D3] and an extension of Reifenberg's parameterization theorem [DDT]. The key idea is still that if $X$ is the cone over an arc of small Lipschitz graph in the unit sphere, but $X$ is not contained in a disk, we can use the graph of a harmonic function to deform $X$ and diminish substantially its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in $\\R^n$, but in this setting our final regularity result on $E$ may depend on the list of minimal cones obtained as blow-up limits of $E$ at a point."}
{"category": "Math", "title": "Closure properties of solutions to heat inequalities", "abstract": "We prove that if $u_1,u_2 : (0,\\infty) \\times \\R^d \\to (0,\\infty)$ are sufficiently well-behaved solutions to certain heat inequalities on $\\R^d$ then the function $u: (0,\\infty) \\times \\R^d \\to (0,\\infty)$ given by $u^{1/p}=u_1^{1/p_1} * u_2^{1/p_2}$ also satisfies a heat inequality of a similar type provided $\\tfrac{1}{p_1} + \\tfrac{1}{p_2} = 1 + \\tfrac{1}{p}$. On iterating, this result leads to an analogous statement concerning $n$-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp $n$-fold Young convolution inequality and its reverse form."}
{"category": "Math", "title": "Factorizations of EP operators", "abstract": "In this paper we characterize EP operators through the existence of different types of factorizations. Our results extend to EP operators existing characterizations for EP matrices and give new characterizations both for EP matrices and EP operators."}
{"category": "Math", "title": "Tools for working with multiplier Hopf algebras", "abstract": "Let $(A,\\Delta)$ be a multiplier Hopf algebra. In general, the underlying algebra $A$ need not have an identity and the coproduct $\\Delta$ does not map $A$ into $A\\otimes A$ but rather into its multiplier algebra $M(A\\otimes A)$. In this paper, we study {\\it some tools} that are frequently used when dealing with such multiplier Hopf algebras and that are typical for working with algebras without identity in this context. The {\\it basic ingredient} is a unital left $A$-module $X$. And the basic construction is that of extending the module by looking at linear maps $\\rho:A\\to X$ satisfying $\\rho(aa')=a\\rho(a')$ where $a,a'\\in A$. We write the module action as multiplication. Of course, when $x\\in X$, and when $\\rho(a)=ax$, we get such a linear map. And if $A$ has an identity, all linear maps $\\rho$ have this form for $x=\\rho(1)$. However, the point is that in the case of a non-unital algebra, the space of such maps is in general strictly bigger than $X$ itself. We get an {\\it extended module}, denoted by $\\bar X$ (for reasons that will be explained in the paper). We study all sorts of more complicated situations where such extended modules occur and we illustrate all of this with {\\it several examples}, from very simple ones to more complex ones where iterated extensions come into play. We refer to cases that appear in the literature. We use this basic idea of extending modules to explain, in a more rigorous way, the so-called {\\it covering technique}, which is needed when using {\\it Sweedler notations} for coproducts and coactions. Again, we give many examples and refer to the existing literature where this technique is applied."}
{"category": "Math", "title": "The Limiting Distributions of the Coefficients of the q-Derangement Numbers", "abstract": "We show that the distribution of the coefficients of the q-derangement numbers is asymptotically normal. We also show that this property holds for the q-derangement numbers of type B."}
{"category": "Math", "title": "Log canonical thresholds of smooth Fano threefolds. With an appendix by Jean-Pierre Demailly", "abstract": "We compute global log canonical thresholds of some smooth Fano threefolds."}
{"category": "Math", "title": "The index of a vector field on an orbifold with boundary", "abstract": "A Poincar\\'{e}-Hopf theorem in the spirit of Pugh is proven for compact orbifolds with boundary. The theorem relates the index sum of a smooth vector field in generic contact with the boundary orbifold to the Euler-Satake characteristic of the orbifold and a boundary term. The boundary term is expressed as a sum of Euler characteristics of tangency and exit-region orbifolds. As a corollary, we express the index sum of the vector field induced on the inertia orbifold to the Euler characteristics of the associated underlying topological spaces."}
{"category": "Math", "title": "On the excedance sets of colored permutations", "abstract": "We define the excedence set and the excedance word on $G_{r,n}$, generalizing a work of Ehrenborg and Steingrimsson and use the inclusion-exclusion principle to calculate the number of colored permutations having a prescribed excedance word. We show some symmetric properties as Log concavity and unimodality of a specific sequence of excedance words."}
{"category": "Math", "title": "Bloch-Kato exponential maps for local fields with imperfect residue fields", "abstract": "In this paper, we generalise the construction of the Bloch-Kato exponential map to complete discrete valuation fields of mixed characteristic (0,p) whose residue fields have a finite p-basis. As an application we prove an explicit reciprocity law, extending a result of Cherbonnier and Colmez in the classical case. This result relies on the calculation of the Galois cohomology of a p-adic representation V in terms of its (phi,G)-module."}
{"category": "Math", "title": "Affine algebraic groups with periodic components", "abstract": "A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finite number of fixed points. It is also discussed which connected groups have finite extensions with periodic components. The results are applied to the study of the normalizer of a maximal torus in a simple algebraic group."}
{"category": "Math", "title": "The vanishing-off subgroup", "abstract": "In this paper, we define the vanishing-off subgroup of a nonabelian group. We study the structure of the quotient of this subgroup and a central series obtained from this subgroup."}
{"category": "Math", "title": "Twisted Alexander polynomials, symplectic 4-manifolds and surfaces of minimal complexity", "abstract": "Let M be a 4-manifold which admits a free circle action. We use twisted Alexander polynomials to study the existence of symplectic structures and the minimal complexity of surfaces in M. The results on the existence of symplectic structures summarize previous results of the authors in [FV08a,FV08,FV07]. The results on surfaces of minimal complexity are new."}
{"category": "Math", "title": "Freyd's generating hypothesis with almost split sequences", "abstract": "Freyd's generating hypothesis for the stable module category of a non-trivial finite group G is the statement that a map between finitely generated kG-modules that belongs to the thick subcategory generated by k factors through a projective if the induced map on Tate cohomology is trivial. In this paper we show that Freyd's generating hypothesis fails for kG when the Sylow p-subgroup of G has order at least 4 using almost split sequences. By combining this with our earlier work, we obtain a complete answer to Freyd's generating hypothesis for the stable module category of a finite group. We also derive some consequences of the generating hypothesis."}
{"category": "Math", "title": "On knot Floer homology and cabling II", "abstract": "We continue our study of the knot Floer homology invariants of cable knots. For large |n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p,pn+1) cable of a knot, K, are isomorphic to those of K. This result allows us to obtain information about the behavior of the Ozsvath-Szabo concordance invariant under cabling, which has geometric consequences for the cabling operation. Applications considered include quasipositivity in the braid group, the knot theory of complex curves, smooth concordance, and lens space (or, more generally, L-space) surgeries."}
{"category": "Math", "title": "From particle to kinetic and hydrodynamic descriptions of flocking", "abstract": "We discuss the Cucker-Smale's (C-S) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasov-type kinetic model for the C-S particle model and prove it exhibits time-asymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the C-S Vlasov-type kinetic model and prove flocking behavior \\emph{without} closure of higher moments."}
{"category": "Math", "title": "Asymptotic analysis of a boundary-value problem with the nonlinear boundary multiphase interactions in a perforated domain", "abstract": "We consider a boundary-value problem for the second order elliptic differential operator with rapidly oscillating coefficients in a domain $\\Omega_{\\epsilon}$ that is $\\epsilon-$periodically perforated by small holes. The holes are divided into two $\\epsilon-$periodical sets depending on the boundary interaction at their surfaces. Therefore, two different nonlinear Robin boundary conditions $\\sigma_\\epsilon (u_\\epsilon) + \\epsilon \\kappa_{m} (u_\\epsilon) = \\epsilon g^{(m)}_\\epsilon, m=1, 2,$ are given on the corresponding boundaries of the small holes. The asymptotic analysis of this problem is made as $\\epsilon\\to0,$ namely the convergence theorem both for the solution and for the energy integral is proved without using extension operators, the asymptotic approximations both for the solution and for the energy integral are constructed and the corresponding error estimates are obtained."}
{"category": "Math", "title": "Explicit Constructions of the non-Abelian $p^3$-Extensions Over $\\QQ$", "abstract": "Let $p$ be an odd prime. Let $F/k$ be a cyclic extension of degree $p$ and of characteristic different from $p$. The explicit constructions of the non-abelian $p^{3}$-extensions over $k$, are induced by certain elements in ${F(\\mu_{p})}^{*}$. In this paper we let $k=\\QQ$ and present sufficient conditions for these elements to be suitable for the constructions. Polynomials for the non-abelian groups of order 27 over $\\QQ$ are constructed."}
{"category": "Math", "title": "Improved Likelihood Inference in Birnbaum-Saunders Regressions", "abstract": "The Birnbaum-Saunders regression model is commonly used in reliability studies. We address the issue of performing inference in this class of models when the number of observations is small. We show that the likelihood ratio test tends to be liberal when the sample size is small, and we obtain a correction factor which reduces the size distortion of the test. The correction makes the error rate of he test vanish faster as the sample size increases. The numerical results show that the modified test is more reliable in finite samples than the usual likelihood ratio test. We also present an empirical application."}
{"category": "Math", "title": "Invariant measure for the continual Cartan subgroup", "abstract": "We construct and study the one-parameter semigroup of $\\sigma$-finite measures ${\\cal L}^{\\theta}$, $\\theta>0$, on the space of Schwartz distributions that have an infinite-dimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup of diagonal positive matrices of the group $SL(n,R)$. The parameter $\\theta$ is the degree of homogeneity with respect to homotheties of the space, we prove uniqueness theorem for measures with given degree of homogeneity, and call the measure with degree of homogeneity equal to one the infinite-dimensional Lebesgue measure $\\cal L$. The structure of these measures is very closely related to the so-called Poisson--Dirichlet measures $PD(\\theta)$, and to the well-known gamma process. The nontrivial properties of the Lebesgue measure are related to the superstructure of the measure PD(1), which is called the conic Poisson--Dirichlet measure -- $CPD$. This is the most interesting $\\sigma$-finite measure on the set of positive convergent monotonic real series."}
{"category": "Math", "title": "The oriented swap process", "abstract": "Particles labelled $1,...,n$ are initially arranged in increasing order. Subsequently, each pair of neighboring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behavior of this process as $n\\to\\infty$. We prove that the space--time trajectories of individual particles converge (when suitably scaled) to a certain family of random curves with two points of non-differentiability, and that the permutation matrix at a given time converges to a certain deterministic measure with absolutely continuous and singular parts. The absorbing state (where all particles are in decreasing order) is reached at time $(2+o(1))n$. The finishing times of individual particles converge to deterministic limits, with fluctuations asymptotically governed by the Tracy--Widom distribution."}
{"category": "Math", "title": "Tree-valued resampling dynamics: Martingale Problems and applications", "abstract": "The measure-valued Fleming-Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the \"individuals\" in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies. We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with the Gromov-weak topology serves as state space for tree-valued processes. We use well-posed martingale problems to construct the tree-valued resampling dynamics of the evolving genealogies for both the finite population Moran model and the infinite population Fleming-Viot diffusion. We show that sufficient information about any ultra-metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled \"individuals\". We give explicit formulas for the evolution of the Laplace transform of the distribution of finite subtrees under the tree-valued Fleming-Viot dynamics."}
{"category": "Math", "title": "Cut and singular loci up to codimension 3", "abstract": "We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension $n-2$ is well known. We go further in this direction by giving a clasification of all points up to a set of Hausdorff dimension $n-3$."}
{"category": "Math", "title": "Two interacting Hopf algebras of trees", "abstract": "Hopf algebra structures on rooted trees are by now a well-studied object, especially in the context of combinatorics. In this work we consider a Hopf algebra H by introducing a coproduct on a (commutative) algebra of rooted forests, considering each tree of the forest (which must contain at least one edge) as a Feynman-like graph without loops. The primitive part of the graded dual is endowed with a pre-Lie product defined in terms of insertion of a tree inside another. We establish a surprising link between the Hopf algebra H obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted trees by means of a natural H-bicomodule structure on the latter. This enables us to recover recent results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua."}
{"category": "Math", "title": "A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4", "abstract": "We prove a change of variable formula for the 2D fractional Brownian motion of index H bigger of equal to 1/4. For H strictly bigger than 1/4, our formula coincides with that obtained by using the rough paths theory. For H=1/4 (the more interesting case), there is an additional term that is a classical Wiener integral against an independent standard Brownian motion."}
{"category": "Math", "title": "The initial meadows", "abstract": "A \\emph{meadow} is a commutative ring with an inverse operator satisfying $0^{-1}=0$. We determine the initial algebra of the meadows of characteristic 0 and show that its word problem is decidable."}
{"category": "Math", "title": "Interface evolution: the Hele-Shaw and Muskat problems", "abstract": "We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy's law. The free boundary is given by the discontinuity among the densities and viscosities of the fluids. This physical scenario is known as the two dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition."}
{"category": "Math", "title": "Classification of strict wonderful varieties", "abstract": "In the setting of strict wonderful varieties we answer positively to Luna's conjecture, saying that wonderful varieties are classified by combinatorial objects, the so-called spherical systems. In particular, we prove that strict wonderful varieties are mostly obtained from symmetric spaces, spherical nilpotent orbits or model spaces. To make the paper self-contained as much as possible, we shall gather some known results on these families and more generally on wonderful varieties."}
{"category": "Math", "title": "Moduli spaces of rank 2 ACM bundles on prime Fano threefolds", "abstract": "Given a smooth non-hyperelliptic prime Fano threefold X, we prove the existence of all rank 2 ACM vector bundles on X by deformation of semistable sheaves. We show that these bundles move in generically smooth components of the corresponding moduli space. We give applications to pfaffian representations of quartic threefolds in P^4 and cubic hypersurfaces of a smooth quadric of P^5."}
{"category": "Math", "title": "Homogeneous para-K\\\"ahler Einstein manifolds", "abstract": "A para-K\\\"ahler manifold can be defined as a pseudo-Riemannian manifold $(M,g)$ with a parallel skew-symmetric para-complex structures $K$, i.e. a parallel field of skew-symmetric endomorphisms with $ K^2 = \\mathrm{Id} $ or, equivalently, as a symplectic manifold $(M,\\omega)$ with a bi-Lagrangian structure $L^\\pm$, i.e. two complementary integrable Lagrangian distributions. A homogeneous manifold $M = G/H$ of a semisimple Lie group $G$ admits an invariant para-K\\\"ahler structure $(g,K)$ if and only if it is a covering of the adjoint orbit $\\mathrm{Ad}_Gh$ of a semisimple element $h.$ We give a description of all invariant para-K\\\"ahler structures $(g,K)$ on a such homogeneous manifold. Using a para-complex analogue of basic formulas of K\\\"ahler geometry, we prove that any invariant para-complex structure $K$ on $M = G/H$ defines a unique para-K\\\"ahler Einstein structure $(g,K)$ with given non-zero scalar curvature. An explicit formula for the Einstein metric $g$ is given. A survey of recent results on para-complex geometry is included."}
{"category": "Math", "title": "Holomorphic self-maps of the disk intertwining two linear fractional maps", "abstract": "We characterize (in almost all cases) the holomorphic self-maps of the unit disk that intertwine two given linear fractional self-maps of the disk. The proofs are based on iteration and a careful analysis of the Denjoy-Wolff points. In particular, we characterize the maps that commute with a given linear fractional map (in the cases that are not already known) and, as an application, determine all \"roots\" of such maps in the sense of iteration (if any). This yields as a byproduct a short proof of a recent theorem on the embedding of a linear fractional transformation into a semigroup of holomorphic self-maps of the disk."}
{"category": "Math", "title": "The univalent Bloch-Landau constant, harmonic symmetry and conformal glueing", "abstract": "By modifying a domain first suggested by Ruth Goodman in 1935 and by exploiting the explicit solution by Fedorov of the Poly\\'a-Chebotarev problem in the case of four symmetrically placed points, an improved upper bound for the univalent Bloch-Landau constant is obtained. The domain that leads to this improved bound takes the form of a disk from which some arcs are removed in such a way that the resulting simply connected domain is harmonically symmetric in each arc with respect to the origin. The existence of domains of this type is established, using techniques from conformal welding, and some general properties of harmonically symmetric arcs in this setting are established."}
{"category": "Math", "title": "Random ideal triangulations and the Weil-Petersson distance between finite degree covers of punctured Riemann surfaces", "abstract": "We prove that any two finite-area non-compact hyperbolic Riemann surfaces S and T have finite covers that are arbitrarily close in the normalized Weil-Petersson metric, where we normalize by dividing the square of the metric by the area of the surface. In the case where T is the modular surface this reduces to showing that S has a finite cover with a proper ideal triangulation where most of the shear coordinates are small; we will construct such a cover out of a random collection of immersed ideal triangles in S."}
{"category": "Math", "title": "Erdelyi-Kober integrals on the cone of positive definite matrices and Radon transforms on Grassmann manifolds", "abstract": "We introduce bi-parametric fractional integrals of the Erdelyi-Kober type that generalize known Garding-Gindikin constructions associated to the cone of positive definite matrices. It is proved that the Radon transform, which maps a zonal function on the Grassmann manifold $G_{n,m}$ of $m$-dimensional linear subspaces of $R^n$ into a function on the similar manifold $G_{n,k}$, $ 1\\leq m<k \\leq n-1$, is represented as analytic continuation of the corresponding Erdelyi-Kober integral. This result shows that different Grinberg-Rubin's formulas for such transforms [GR] have, in fact, a common structure."}
{"category": "Math", "title": "Bounds for codes and designs in complex subspaces", "abstract": "We introduce the concepts of complex Grassmannian codes and designs. Let G(m,n) denote the set of m-dimensional subspaces of C^n: then a code is a finite subset of G(m,n) in which few distances occur, while a design is a finite subset of G(m,n) that polynomially approximates the entire set. Using Delsarte's linear programming techniques, we find upper bounds for the size of a code and lower bounds for the size of a design, and we show that association schemes can occur when the bounds are tight. These results are motivated by the bounds for real subspaces recently found by Bachoc, Coulangeon and Nebe, and the bounds generalize those of Delsarte, Goethals and Seidel for codes and designs on the complex unit sphere."}
{"category": "Math", "title": "Special points of the Brownian net", "abstract": "The Brownian net, which has recently been introduced by Sun and Swart [SS08], and independently by Newman, Ravishankar and Schertzer [NRS08], generalizes the Brownian web by allowing branching. In this paper, we study the structure of the Brownian net in more detail. In particular, we give an almost sure classification of each point in $R^2$ according to the configuration of the Brownian net paths entering and leaving the point. Along the way, we establish various other structural properties of the Brownian net."}
{"category": "Math", "title": "Rapid factorization of structured matrices via randomized sampling", "abstract": "Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an extension of such techniques to a wider class of matrices that are not themselves rank-deficient, but have off-diagonal blocks that are. Such matrices arise frequently in numerical analysis and signal processing, and there exist several methods for rapidly performing algebraic operations (matrix-vector multiplications, matrix factorizations, matrix inversion, \\textit{etc}) on them once low-rank approximations to all off-diagonal blocks have been constructed. The paper demonstrates that if such a matrix can be applied to a vector in O(N) time, where the matrix is of size $N\\times N$, and if individual entries of the matrix can be computed rapidly, then in many cases, the task of constructing approximate low-rank factorizations for all off-diagonal blocks can be performed in $O(N k^{2})$ time, where $k$ is an upper bound for the numerical rank of the off-diagonal blocks."}
{"category": "Math", "title": "Delay Analysis for Max Weight Opportunistic Scheduling in Wireless Systems", "abstract": "We consider the delay properties of max-weight opportunistic scheduling in a multi-user ON/OFF wireless system, such as a multi-user downlink or uplink. It is well known that max-weight scheduling stabilizes the network (and hence yields maximum throughput) whenever input rates are inside the network capacity region. We show that when arrival and channel processes are independent, average delay of the max-weight policy is order-optimal, in the sense that it does not grow with the number of network links. While recent queue-grouping algorithms are known to also yield order-optimal delay, this is the first such result for the simpler class of max-weight policies. We then consider multi-rate transmission models and show that average delay in this case typically does increase with the network size due to queues containing a small number of \"residual\" packets."}
{"category": "Math", "title": "Rational vertex operator algebras are finitely generated", "abstract": "It is proved that any vertex operator algebra for which the image of the Virasoro element in Zhu's algebra is algebraic over complex numbers is finitely generated. In particular, any vertex operator algebra with a finite dimensional Zhu's algebra is finitely generated. As a result, any rational vertex operator algebra is finitely generated."}
{"category": "Math", "title": "Formal deformations of Poisson structures in low dimensions", "abstract": "In this paper, we study formal deformations of Poisson structures, especially for three families of Poisson varieties in dimensions two and three. For these families of Poisson structures, using an explicit basis of the second Poisson cohomology space, we solve the deformation equations at each step and obtain a large family of formal deformations for each Poisson structure which we consider. With the help of an explicit formula, we show that this family contains, modulo equivalence, all possible formal deformations. We show moreover that, when the Poisson structure is generic, all members of the family are non-equivalent."}
{"category": "Math", "title": "A sharp uniform bound for the distribution of sums of Bernoulli trials", "abstract": "In this note we establish a uniform bound for the distribution of a sum $S_n=X_1+\\cdots+X_n$ of independent non-homogeneous Bernoulli trials. Specifically, we prove that $\\sigma_n \\mathbb{P}(S_n\\!=\\!j)\\leq\\eta$ where $\\sigma_n$ denotes the standard deviation of $S_n$ and $\\eta$ is a universal constant. We compute the best possible constant $\\eta\\sim 0.4688$ and we show that the bound also holds for limits of sums and differences of Bernoullis, including the Poisson laws which constitute the worst case and attain the bound. We also investigate the optimal bounds for $n$ and $j$ fixed. An application to estimate the rate of convergence of Mann's fixed point iterations is presented."}
{"category": "Math", "title": "A mapping from the unitary to doubly stochastic matrices and symbols on a finite set", "abstract": "We prove that the mapping from the unitary to the doubly stochastic matrices that maps a unitary matrix (u_{kl}) to the doubly stochastic matrix (|u_{kl}|^2) is a submersion for almost all unitary matrices. The proof uses the framework of operator symbols on a finite set. We give detailed proofs of the results announced in our earlier preprint."}
{"category": "Math", "title": "Cubical cospans and higher cobordisms (Cospans in algebraic topology, III)", "abstract": "After two papers on weak cubical categories and {\\it collarable} cospans, respectively, we put things together and construct a {\\it weak} cubical category of cubical {\\it collared} cospans of topological spaces. We also build a second structure, called a {\\it quasi} cubical category, formed of arbitrary cubical cospans concatenated by homotopy pushouts. This structure, simpler but weaker, has {\\it lax} identities. It contains a similar framework for cobordisms of manifolds with corners and could therefore be the basis to extend the study of TQFT's of Part II to higher cubical degree."}
{"category": "Math", "title": "On the zeros of the Riemann zeta function", "abstract": "In 2008 I thought I found a proof of the Riemann Hypothesis, but there was an error. In the Spring 2020 I believed to have fixed the error, but it cannot be fixed. I describe here where the error was. It took me several days to find the error in a careful checking before a possible submission to a payable review offered by one leading journal. There were three simple lemmas and one simple theorem, all were correct, yet there was an error: what Lemma 2 proved was not exactly what Lemma 3 needed. So, it was the connection of the lemmas. This paper came out empty, but I have found a different proof of the Riemann Hypothesis and it seems so far correct. In the discussion at the end of this paper I raise a matter that I think is of importance to the review process in mathematics."}
{"category": "Math", "title": "On a non-combinatorial definition of Stirling numbers", "abstract": "In Combinatorics Stirling numbers may be defined in several ways. One such definition is given in [1], where an extensive consideration of Stirling numbers is presented. In this paper an alternative definition of Stirling numbers of both kind is given. Namely, Stirling numbers of the first kind appear in the closed formula for the n-th derivative of ln x. In the same way Stirling numbers of the second kind appear in the formula for the n-th derivative of f(e^x), where f(x) is an arbitrary smooth real function. This facts allow us to define Stirling numbers within the frame of differential calculus. These definitions may be interesting because arbitrary functions appear in them. Choosing suitable function we may obtain different properties of Stirling numbers by the use of derivatives only. Using simple properties of derivatives we obtain here three important properties of Stirling numbers. First are so called two terms recurrence relations, from which one can easily derive the combinatorial meaning of Stirling numbers. Next we obtain expansion formulas of powers into falling factorials, and vise versa. These expansions usually serve as the definitions of Stirling numbers, as in [1]. Finally, we obtain the exponential generating functions for Stirling and Bell numbers. As a by product the closed formulas for the $n$-th derivative of the functions f(e^x) and f(ln x) are obtained."}
{"category": "Math", "title": "Rootless pairs of $EE_8$-lattices", "abstract": "We describe a classification of pairs $M, N$ of lattices isometric to $EE_8:=\\sqrt 2 E_8$ such that the lattice $M + N$ is integral and rootless and such that the dihedral group associated to them has order at most 12. It turns out that most of these pairs may be embedded in the Leech lattice. Complete proofs will appear in another article. This theory of integral lattices has connections to vertex operator algebra theory and moonshine."}
{"category": "Math", "title": "Morrey Spaces and Fractional Integral Operators", "abstract": "The present paper is devoted to the boundedness of fractional integral operators in Morrey spaces defined on quasimetric measure spaces. In particular, Sobolev, trace and weighted inequalities with power weights for potential operators are established. In the case when measure satisfies the doubling condition the derived conditions are simultaneously necessary and sufficient for appropriate inequalities."}
{"category": "Math", "title": "Fun with $\\F_1$", "abstract": "We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost--Connes naturally arises by extension of scalars from the \"field with one element\" to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that \"field\", while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra."}
{"category": "Math", "title": "Asymptotic series for the splitting of separatrices near a Hamiltonian bifurcation", "abstract": "This is a proof of an asymptotic formula which describes exponentially small splitting of separatrices in a generic analytic family of area-preserving maps near a Hamiltonian saddle-centre bifurcation. As a particular case and in combination with an earlier work on a Stokes constant for the H\\'enon map (Gelfreich, Sauzin (2001)), it implies exponentially small transversality of separatrices in the area-preserving H\\'enon family when the multiplicator of the fixed point is close to one."}
{"category": "Math", "title": "On the shock wave spectrum for isentropic gas dynamics with capillarity", "abstract": "We consider the stability problem for shock layers in Slemrod's model of an isentropic gas with capillarity. We show that these traveling waves are monotone in the weak capillarity case, and become highly oscillatory as the capillarity strength increases. Using a spectral energy estimate we prove that small-amplitude monotone shocks are spectrally stable. We also show, through the use of a novel spectral energy estimate, that monotone shocks have no unstable real spectrum regardless of amplitude; this implies that any instabilities of these monotone traveling waves, if they exist, must occur through a Hopf-like bifurcation, where one or more conjugate pairs of eigenvalues cross the imaginary axis. We then conduct a systematic numerical Evans function study, which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths. In particular, we show that the transition from monotone to non-monotone profiles does not appear to trigger any instabilities."}
{"category": "Math", "title": "Compactly supported cohomology of buildings", "abstract": "We compute the compactly supported cohomology of the standard realization of any locally finite building."}
{"category": "Math", "title": "Canonical RNA pseudoknot structures", "abstract": "In this paper we study $k$-noncrossing, canonical RNA pseudoknot structures with minimum arc-length $\\ge 4$. Let ${\\sf T}_{k,\\sigma}^{[4]} (n)$ denote the number of these structures. We derive exact enumeration results by computing the generating function ${\\bf T}_{k,\\sigma}^{[4]}(z)= \\sum_n{\\sf T}_{k,\\sigma}^{[4]}(n)z^n$ and derive the asymptotic formulas ${\\sf T}_{k,3}^{[4]}(n)^{}\\sim c_k n^{-(k-1)^2-\\frac{k-1}{2}} (\\gamma_{k,3}^{[4]})^{-n}$ for $k=3,...,9$. In particular we have for $k=3$, ${\\sf T}_{3,3}^{[4]}(n)^{}\\sim c_3 n^{-5} 2.0348^n$. Our results prove that the set of biophysically relevant RNA pseudoknot structures is surprisingly small and suggest a new structure class as target for prediction algorithms."}
{"category": "Math", "title": "Sharp logarithmic Sobolev inequalities on gradient solitons and applications", "abstract": "We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. The geometric invariant appearing in the sharp lower bound is shown to be nonnegative. We also characterize the expanders when such invariant is zero. In the proof various useful volume growth estimates are also established for gradient shrinking and expanding solitons. In particular, we prove that the {\\it asymptotic volume ratio} of any gradient shrinking soliton with nonnegative Ricci curvature must be zero."}
{"category": "Math", "title": "Dominating Sets in Plane Triangulations", "abstract": "In 1996, Matheson and Tarjan conjectured that any n-vertex triangulation with n sufficiently large has a dominating set of size at most n/4. We prove this for graphs of maximum degree 6."}
{"category": "Math", "title": "Developing Bayesian Information Entropy-based Techniques for Spatially Explicit Model Assessment", "abstract": "The aim of this paper is to explore and develop advanced spatial Bayesian assessment methods and techniques for land use modeling. The paper provides a comprehensive guide for assessing additional informational entropy value of model predictions at the spatially explicit domain of knowledge, and proposes a few alternative metrics and indicators for extracting higher-order information dynamics from simulation tournaments. A seven-county study area in South-Eastern Wisconsin (SEWI) has been used to simulate and assess the accuracy of historical land use changes (1963-1990) using artificial neural network simulations of the Land Transformation Model (LTM). The use of the analysis and the performance of the metrics helps: (a) understand and learn how well the model runs fits to different combinations of presence and absence of transitions in a landscape, not simply how well the model fits our given data; (b) derive (estimate) a theoretical accuracy that we would expect a model to assess under the presence of incomplete information and measurement; (c) understand the spatially explicit role and patterns of uncertainty in simulations and model estimations, by comparing results across simulation runs; (d) compare the significance or estimation contribution of transitional presence and absence (change versus no change) to model performance, and the contribution of the spatial drivers and variables to the explanatory value of our model; and (e) compare measurements of informational uncertainty at different scales of spatial resolution."}
{"category": "Math", "title": "Relations between invasion percolation and critical percolation in two dimensions", "abstract": "We study invasion percolation in two dimensions. We compare connectivity properties of the origin's invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. To exhibit similarities, we show that for any $k\\geq1$, the $k$-point function of the first so-called pond has the same asymptotic behavior as the probability that $k$ points are in the critical cluster of the origin. More prominent, though, are the differences. We show that there are infinitely many ponds that contain many large disjoint $p_c$-open clusters. Further, for $k>1$, we compute the exact decay rate of the distribution of the radius of the $k$th pond and see that it differs from that of the radius of the critical cluster of the origin. We finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular."}
{"category": "Math", "title": "Estimation of conditional laws given an extreme component", "abstract": "Let $(X,Y)$ be a bivariate random vector. The estimation of a probability of the form $P(Y\\leq y \\mid X >t) $ is challenging when $t$ is large, and a fruitful approach consists in studying, if it exists, the limiting conditional distribution of the random vector $(X,Y)$, suitably normalized, given that $X$ is large. There already exists a wide literature on bivariate models for which this limiting distribution exists. In this paper, a statistical analysis of this problem is done. Estimators of the limiting distribution (which is assumed to exist) and the normalizing functions are provided, as well as an estimator of the conditional quantile function when the conditioning event is extreme. Consistency of the estimators is proved and a functional central limit theorem for the estimator of the limiting distribution is obtained. The small sample behavior of the estimator of the conditional quantile function is illustrated through simulations."}
{"category": "Math", "title": "Sobolev $W^1_p$-spaces on closed subsets of $R^n$", "abstract": "For each $p>n$ we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the Sobolev space $W_p^1(R^n)$ to an arbitrary closed subset of $R^n$."}
{"category": "Math", "title": "Well-posedness of the water-wave problem with surface tension", "abstract": "In this paper, we prove the local well-posedness of the water wave problem with surface tension in the case of finite depth by working in the Eulerian setting. For the flat bottom, as surface tension tends to zero, the solution of the water wave problem with surface tension converges to the solution of the water wave problem without surface tension."}
{"category": "Math", "title": "Universal structures in some mean field spin glasses, and an application", "abstract": "We discuss a spin glass reminiscent of the Random Energy Model, which allows in particular to recast the Parisi minimization into a more classical Gibbs variational principle, thereby shedding some light on the physical meaning of the order parameter of the Parisi theory. As an application, we study the impact of an extensive cavity field on Derrida's REM: Despite its simplicity, this model displays some interesting features such as ultrametricity and chaos in temperature."}
{"category": "Math", "title": "On the simply connectedness of non-negatively curved K\\\"ahler manifolds and applications", "abstract": "We study complete noncompact long time solutions $(M, g(t))$ to the K\\\"ahler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. $ R_\\ijb \\ge cRg_\\ijb$ at $(p,t)$ for all $t$ for some $c>0$, then there always exists a local gradient K\\\"ahler Ricci soliton limit around $p$ after possibly rescaling $g(t)$ along some sequence $t_i \\to \\infty$. We will show as an immediate corollary that the injectivity radius of $g(t)$ along $t_i$ is uniformly bounded from below along $t_i$, and thus $M$ must in fact be simply connected. Additional results concerning the uniformization of $M$ and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci for $(M, g(t))$. Combining our results with Cao's splitting for K\\\"ahler Ricci flow \\cite{Cao04} and techniques of Ni-Tam \\cite{NiTam03}, we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point $p \\in M$, then $M$ has a special holomorphic fiber bundle structure. We will treat a special cases, complete K\\\"ahler manifolds with non-negative holomorphic bisectional and average quadratic curvature decay as well as the case of steady gradient K\\\"ahler Ricci solitons."}
{"category": "Math", "title": "Survey on the Burnside ring of compact Lie groups", "abstract": "The definition and basic properties of the Burnside ring of compact Lie groups are presented, with emphasis on the analogy with the construction of the Burnside ring of finite groups."}
{"category": "Math", "title": "On Poisson quasi-Nijenhuis Lie algebroids", "abstract": "We propose a definition of Poisson quasi-Nijenhuis Lie algebroids as a natural generalization of Poisson quasi-Nijenhuis manifolds and show that any such Lie algebroid has an associated quasi-Lie bialgebroid. Therefore, also an associated Courant algebroid is obtained. We introduce the notion of a morphism of quasi-Lie bialgebroids and of the induced Courant algebroids morphism and provide some examples of Courant algebroid morphisms. Finally, we use paired operators to deform doubles of Lie and quasi-Lie bialgebroids and find an application to generalized complex geometry."}
{"category": "Math", "title": "Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations", "abstract": "We study the fully nonlinear elliptic equation $F(D^2u,Du,u,x) = f$ in a smooth bounded domain $\\Omega$, under the assumption the nonlinearity $F$ is uniformly elliptic and positively homogeneous. Recently, it has been shown that such operators have two principal \"half\" eigenvalues, and that the corresponding Dirichlet problem possesses solutions, if both of the principal eigenvalues are positive. In this paper, we prove the existence of solutions of the Dirichlet problem if both principal eigenvalues are negative, provided the \"second\" eigenvalue is positive, and generalize the anti-maximum principle of Cl\\'{e}ment and Peletier to homogeneous, fully nonlinear operators."}
{"category": "Math", "title": "Howe duality and Kostant Homology Formula for infinite-dimensional Lie superalgebras", "abstract": "Using Howe duality we compute explicitly Kostant-type homology groups for a wide class of representations of the infinite-dimensional Lie superalgebra $\\hat{\\frak{gl}}_{\\infty|\\infty}$ and its classical subalgebras at positive integral levels. We also obtain Kostant-type homology formulas for the Lie algebra $ widehat{\\frak{gl}}_\\infty$ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for these representations."}
{"category": "Math", "title": "On sums of figurate numbers by using techniques of poset representation theory", "abstract": "We use representations and differentiation algorithms of posets, in order to obtain results concerning unsolved problems on figurate numbers. In particular, we present criteria for natural numbers which are the sum of three octahedral numbers, three polygonal numbers of positive rank or four cubes with two of them equal. Some identities of the Rogers-Ramanujan type involving this class of numbers are also obtained."}
{"category": "Math", "title": "Pell's equation without irrational numbers", "abstract": "We solve Pell's equation in a simple way without continued fractions or irrational numbers, and relate the algorithm to the Stern Brocot tree."}
{"category": "Math", "title": "The q-WZ Method for Infinite Series", "abstract": "Motivated by the telescoping proofs of two identities of Andrews and Warnaar, we find that infinite q-shifted factorials can be incorporated into the implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and Mu to prove nonterminating basic hypergeometric series identities. This observation enables us to extend the q-WZ method to identities on infinite series. As examples, we will give the q-WZ pairs for some classical identities such as the q-Gauss sum, the $_6\\phi_5$ sum, Ramanujan's $_1\\psi_1$ sum and Bailey's $_6\\psi_6$ sum."}
{"category": "Math", "title": "Neuberg cubics over finite fields", "abstract": "The framework of universal geometry allows us to consider metrical properties of affine views of elliptic curves, even over finite fields. We show how the Neuberg cubic of triangle geometry extends to the finite field situation and provides interesting potential invariants for elliptic curves, focussing on an explicit example over $\\mathbb{F}_{23}$. We also prove that tangent conics for a Weierstrass cubic are identical or disjoint."}
{"category": "Math", "title": "The sum-product phenomenon in arbitrary rings", "abstract": "The \\emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \\cdot A$ unless it is in some sense \"close\" to a finite subring of $R$. This phenomenon has been analysed intensively for various specific rings, notably the reals $\\R$ and cyclic groups $\\Z/q\\Z$. In this paper we consider the problem in arbitrary rings $R$, which need not be commutative or contain a multiplicative identity. We obtain rigorous formulations of the sum-product phenomenon in such rings in the case when $A$ encounters few zero-divisors of $R$. As applications we recover (and generalise) several sum-product theorems already in the literature."}
{"category": "Math", "title": "Central limit theorems for eigenvalues in a spiked population model", "abstract": "In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms."}
{"category": "Math", "title": "Quenched law of large numbers for branching Brownian motion in a random medium", "abstract": "We study a spatial branching model, where the underlying motion is $d$-dimensional ($d\\ge1$) Brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all $d\\ge1$. We also show that the branching Brownian motion with mild obstacles spreads less quickly than ordinary branching Brownian motion by giving an upper estimate on its speed. When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the quenched local growth that is independent of the Poissonian intensity. More general offspring distributions (beyond the dyadic one considered in the main theorems) as well as mild obstacle models for superprocesses are also discussed."}
{"category": "Math", "title": "Geometric torsions and an Atiyah-style topological field theory", "abstract": "The construction of invariants of three-dimensional manifolds with a triangulated boundary, proposed earlier by the author for the case when the boundary consists of not more than one connected component, is generalized to any number of components. These invariants are based on the torsion of acyclic complexes of geometric origin. The relevant tool for studying our invariants turns out to be F.A. Berezin's calculus of anti-commuting variables; in particular, they are used in the formulation of the main theorem of the paper, concerning the composition of invariants under a gluing of manifolds. We show that the theory obeys a natural modification of M. Atiyah's axioms for anti-commuting variables."}
{"category": "Math", "title": "Homogenization of a singular random one-dimensional PDE", "abstract": "This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order term. We show that under a proper choice of the scaling factor for the said zero order terms, the family of solutions of the studied problem converges in law, and describe the limit process. It should be noted that the limit dynamics remain random."}
{"category": "Math", "title": "Bundles of C*-categories, II: C*-dynamical systems and Dixmier-Douady invariants", "abstract": "We introduce a cohomological invariant arising from a class in nonabelian cohomology. This invariant generalizes the Dixmier-Douady class and encodes the obstruction to a C*-algebra bundle being the fixed-point algebra of a gauge action. As an application, the duality breaking for group bundles vs. tensor C*-categories with non-simple unit is discussed in the setting of Nistor-Troitsky gauge-equivariant K-theory: there is a map assigning a nonabelian gerbe to a tensor category, and \"triviality\" of the gerbe is equivalent to the existence of a dual group bundle. At the C*-algebraic level, this corresponds to studying C*-algebra bundles with fibre a fixed-point algebra of the Cuntz algebra and in this case our invariant describes the obstruction to finding an embedding into the Cuntz-Pimsner algebra of a vector bundle."}
{"category": "Math", "title": "The quenched invariance principle for random walks in random environments admitting a bounded cycle representation", "abstract": "We derive a quenched invariance principle for random walks in random environments whose transition probabilities are defined in terms of weighted cycles of bounded length. To this end, we adapt the proof for random walks among random conductances by Sidoravicius and Sznitman (Probab. Theory Related Fields 129 (2004) 219--244) to the non-reversible setting."}
{"category": "Math", "title": "Torelli theorem for the moduli spaces of pairs", "abstract": "Let X be a smooth projective curve of genus at least two over the complex numbers. A pair (E,\\phi) over X consists of an algebraic vector bundle E over X and a holomorphic section \\phi of E. There is a concept of stability for pairs which depends on a real parameter \\tau. Here we prove that the third cohomology groups of the moduli spaces of \\tau-stable pairs with fixed determinant and rank at least two are polarised pure Hodge structures, and they are isomorphic to H^1(X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover that the third cohomology group of the moduli space of stable bundles of rank at least two and fixed determinant is a polarised pure Hodge structure, which is isomorphic to H^1(X). We also prove Torelli theorems for the corresponding moduli spaces of pairs and bundles with non-fixed determinant."}
{"category": "Math", "title": "Approximate Hermite quasi-interpolation", "abstract": "In this paper we derive approximate quasi-interpolants when the values of a function $u$ and of some of its derivatives are prescribed at the points of a uniform grid. As a byproduct of these formulas we obtain very simple approximants which provide high order approximations for solutions to elliptic differential equations with constant coefficients."}
{"category": "Math", "title": "Subelliptic Li-Yau estimates on three dimensional model spaces", "abstract": "We describe three elementary models in three dimensional subelliptic geometry which correspond to the three models of the Riemannian geometry (spheres, Euclidean spaces and Hyperbolic spaces) which are respectively the SU(2), Heisenberg and SL(2) groups. On those models, we prove parabolic Li-Yau inequalities on positive solutions of the heat equation. We use for that the $\\Gamma_{2}$ techniques that we adapt to those elementary model spaces. The important feature developed here is that although the usual notion of Ricci curvature is meaningless (or more precisely leads to bounds of the form $-\\infty$ for the Ricci curvature), we describe a parameter $\\rho$ which plays the same role as the lower bound on the Ricci curvature, and from which one deduces the same kind of results as one does in Riemannian geometry, like heat kernel upper bounds, Sobolev inequalities and diameter estimates."}
{"category": "Math", "title": "On the Small Deviation Problem for Some Iterated Processes", "abstract": "We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for $n$-iterated Brownian motions and, more generally, for the iteration of $n$ fractional Brownian motions. We also give a new and correct proof of some results in E. Nane, Laws of the iterated logarithm for $\\alpha$-time Brownian motion, Electron. J. Probab. 11 (2006), no. 18, 434--459."}
{"category": "Math", "title": "On a class of optimal stopping problems for diffusions with discontinuous coefficients", "abstract": "In this paper, we introduce a modification of the free boundary problem related to optimal stopping problems for diffusion processes. This modification allows the application of this PDE method in cases where the usual regularity assumptions on the coefficients and on the gain function are not satisfied. We apply this method to the optimal stopping of integral functionals with exponential discount of the form $E_x\\int_0^{\\tau}e^{-\\lambda s}f(X_s) ds$, $\\lambda\\ge0$ for one-dimensional diffusions $X$. We prove a general verification theorem which justifies the modified version of the free boundary problem. In the case of no drift and discount, the free boundary problem allows to give a complete and explicit discussion of the stopping problem."}
{"category": "Math", "title": "The Weil-Steinberg character of finite classical groups", "abstract": "We compute the irreducible constitutents of the product of the Weil character and the Steinberg character in those finite classical groups for which a Weil character is defined, namely the symplectic, unitary and general linear groups. It turns out that this product is multiplicity free for the symplectic and general unitary groups, but not for the general linear groups. As an application we show that the restriction of the Steinberg character of such a group to the subgroup stabilizing a vector in the natural module is multiplicity free. The proof of this result for the unitary groups uses an observation of Brunat, published as an appendix to our paper. As our \"Weil character\" for the symplectic groups in even characteristic we use the 2-modular Brauer character of the generalized spinor representation. Its product with the Steinberg character is the Brauer character of a projective module. We also determine its indecomposable direct summands."}
{"category": "Math", "title": "A Local-Global Criterion for Dynamics on P^1", "abstract": "Let K be a number field or a function field, let F:P^1 --> P^1 be a rational map of degree at least two defined over K, let P be a point in P^1(K) having infinite F-orbit, and let Z be a finite subset of Z. We prove a local-global criterion for the intersection of the F-orbit of P and the finite set Z. This is a special case of a dynamical Brauer-Manin criterion suggested by Hsia and Silverman."}
{"category": "Math", "title": "A unified framework for utility maximization problems: An Orlicz space approach", "abstract": "We consider a stochastic financial incomplete market where the price processes are described by a vector-valued semimartingale that is possibly nonlocally bounded. We face the classical problem of utility maximization from terminal wealth, with utility functions that are finite-valued over $(a,\\infty)$, $a\\in\\lbrack-\\infty,\\infty)$, and satisfy weak regularity assumptions. We adopt a class of trading strategies that allows for stochastic integrals that are not necessarily bounded from below. The embedding of the utility maximization problem in Orlicz spaces permits us to formulate the problem in a unified way for both the cases $a\\in\\mathbb{R}$ and $a=-\\infty$. By duality methods, we prove the existence of solutions to the primal and dual problems and show that a singular component in the pricing functionals may also occur with utility functions finite on the entire real line."}
{"category": "Math", "title": "Connected subgroups of SO(2,n) acting irreducibly on $\\R^{2,n}$", "abstract": "We classify all connected subgroups of SO(2,n) that act irreducibly on $\\R^{2,n}$. Apart from $SO_0(2,n)$ itself these are $U(1,n/2)$, $SU(1,n/2)$, if $n$ even, $S^1\\cdot SO(1,n/2)$ if $n$ even and $n\\ge 2$, and $SO_0(1,2)$ for $n=3$. Our proof is based on the Karpelevich Theorem and uses the classification of totally geodesic submanifolds of complex hyperbolic space and of the Lie ball. As an application we obtain a list of possible irreducible holonomy groups of Lorentzian conformal structures, namely $SO_0(2,n)$, SU(1,n), and $SO_0(1,2)$."}
{"category": "Math", "title": "Group schemes of period p>2", "abstract": "For a prime number p>2, we give a direct proof of Breuil's classification of killed by p finite flat group schemes over the valuation ring of a p-adic field with perfect residue field. As application we prove that the Galois modules of geometric points of such group schemes and of their characteristic p analogues coming from Faltings's strict modules can be identified via the Fontaine-Wintenberger field-of-norms functor."}
{"category": "Math", "title": "Explicit representation of membership in polynomial ideals", "abstract": "We introduce a new division formula on projective space which provides explicit solutions to various polynomial division problems with sharp degree estimates. We consider simple examples as the classical Macaulay theorem as well as a quite recent result by Hickel, related to the effective Nullstellensatz. We also obtain a related result that generalizes Max Noether's classical AF+BG theorem."}
{"category": "Math", "title": "The combinatorics of k-marked Durfee symbols", "abstract": "Andrews recently introduced k-marked Durfee symbols which are connected to moments of Dyson's rank. By these connections, Andrews deduced their generating functions and some combinatorial properties and left their purely combinatorial proofs as open problems. The primary goal of this article is to provide combinatorial proofs in answer to Andrews' request. We obtain a relation between k-marked Durfee symbols and Durfee symbols by constructing bijections, and all identities on k-marked Durfee symbols given by Andrews could follow from this relation. In a similar manner, we also prove the identities due to Andrews on k-marked odd Durfee symbols combinatorially, which resemble ordinary k-marked Durfee symbols with a modified subscript and with odd numbers as entries."}
{"category": "Math", "title": "Canonical toric Fano threefolds", "abstract": "An inductive approach to classifying toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688 such varieties."}
{"category": "Math", "title": "Simplicity of vacuum modules over affine Lie superalgebras", "abstract": "We prove an explicit condition on the level $k$ for the irreducibility of a vacuum module $V^{k}$ over a (non-twisted) affine Lie superalgebra, which was conjectured by M. Gorelik and V.G. Kac. An immediate consequence of this work is the simplicity conditions for the corresponding minimal W-algebras obtained via quantum reduction, in all cases except when the level $k$ is a non-negative integer."}
{"category": "Math", "title": "$H_D$-Quantum Vertex Algebras", "abstract": "We discuss a class of quantum vertex algebras where not only the commutativity of the vertex algebra is broken by a braiding map $S^{(\\tau)}$, but also the translation covariance is broken by a translation map $S^{(\\gamma)}$. The new class of quantum vertex operators satisfy a Braided Jacobi Identity containing both the braiding and the translation maps."}
{"category": "Math", "title": "An operator equality involving a continuous field of operators and its norm inequalities", "abstract": "Let ${\\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a probability measure $P$ and let $(A_t)_{t\\in T}$ be a continuous field of operators in ${\\mathfrak A}$ such that the function $t \\mapsto A_t$ is norm continuous on $T$ and the function $t \\mapsto \\|A_t\\|$ is integrable. Then the following equality including Bouchner integrals holds \\begin{eqnarray}\\label{oi} \\int_T|A_t - \\int_TA_s{\\rm d}P|^2 {\\rm d}P=\\int_T|A_t|^2{\\rm d}P - |\\int_TA_t{\\rm d}P|^2 . \\end{eqnarray} This equality is related both to the notion of variance in statistics and to a characterization of inner product spaces. With this operator equality, we present some uniform norm and Schatten $p$-norm inequalities."}
{"category": "Math", "title": "An operator algebraic proof of Agler's factorization theorem", "abstract": "We give a short direct proof of Agler's factorization theorem that uses the abstract characterization of operator algebras. the key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional information about these factorizations in the case of polynomials."}
{"category": "Math", "title": "Bounded Ratios of Products of Principal Minors of Positive Definite Matrices", "abstract": "Considered is the multiplicative semigroup of ratios of products of principal minors bounded over all positive definite matrices. A long history of literature identifies various elements of this semigroup, all of which lie in a sub-semigroup generated by Hadamard-Fischer inequalities. Via cone-theoretic techniques and the patterns of nullity among positive semidefinite matrices, a semigroup containing all bounded ratios is given. This allows the complete determination of the semigroup of bounded ratios for 4-by-4 positive definite matrices, whose 46 generators include ratios not implied by Hadamard-Fischer and ratios not bounded by 1. For n > 4 it is shown that the containment of semigroups is strict, but a generalization of nullity patterns, of which one example is given, is conjectured to provide a finite determination of all bounded ratios."}
{"category": "Math", "title": "Manifold Learning: The Price of Normalization", "abstract": "We analyze the performance of a class of manifold-learning algorithms that find their output by minimizing a quadratic form under some normalization constraints. This class consists of Locally Linear Embedding (LLE), Laplacian Eigenmap, Local Tangent Space Alignment (LTSA), Hessian Eigenmaps (HLLE), and Diffusion maps. We present and prove conditions on the manifold that are necessary for the success of the algorithms. Both the finite sample case and the limit case are analyzed. We show that there are simple manifolds in which the necessary conditions are violated, and hence the algorithms cannot recover the underlying manifolds. Finally, we present numerical results that demonstrate our claims."}
{"category": "Math", "title": "Local Procrustes for Manifold Embedding: A Measure of Embedding Quality and Embedding Algorithms", "abstract": "We present the Procrustes measure, a novel measure based on Procrustes rotation that enables quantitative comparison of the output of manifold-based embedding algorithms (such as LLE (Roweis and Saul, 2000) and Isomap (Tenenbaum et al, 2000)). The measure also serves as a natural tool when choosing dimension-reduction parameters. We also present two novel dimension-reduction techniques that attempt to minimize the suggested measure, and compare the results of these techniques to the results of existing algorithms. Finally, we suggest a simple iterative method that can be used to improve the output of existing algorithms."}
{"category": "Math", "title": "Real Regulators on Self-Products of K3 Surfaces", "abstract": "Based on a novel application of an archimedean type pairing to the geometry and deformation theory of $K3$ surfaces, we construct a regulator indecomposable $K_1$-class on a self-product of a $K3$ surface. In the Appendix, we explain how this pairing is a special instance of a general pairing on precycles in the equivalence relation defining Bloch's higher Chow groups."}
{"category": "Math", "title": "Symmetric polynomials, p-norm inequalities, and certain functionals related to majorization", "abstract": "We study a variant of the majorization relation. In particular we consider inequalities involving some Schur-concave symmetric polynomials related to the multinomial expansion. We also discuss how these topics were motivated by conjectures on sharp Lp inequalities between complex exponential sums conjectured by Hardy and Littlewood (still open problems), and why the usual majorization relation does not hold in that context."}
{"category": "Math", "title": "Canonical integral structures on the de Rham cohomology of curves", "abstract": "For a smooth and proper curve X over the fraction field K of a discrete valuation ring R, we explain (under very mild hypotheses) how to equip the de Rham cohomology H^1_{dR}(X/K) with a canonical integral structure: i.e. an R-lattice which is functorial in finite (generically etale) K-morphisms of X and which is preserved by the cup-product auto-duality on H^1_{dR}(X/K). Our construction of this lattice uses a certain class of normal proper models of X and relative dualizing sheaves. We show that our lattice naturally contains the lattice furnished by the (truncated) de Rham complex of a regular proper R-model of X and that the index for this inclusion of lattices is a numerical invariant of X (we call it the de Rham conductor). Using work of Bloch and Liu-Saito, we prove that the de Rham conductor of X is bounded above by the Artin conductor, and bounded below by the Efficient conductor. We then study how the position of our canonical lattice inside the de Rham cohomology of X is affected by finite extension of scalars."}
{"category": "Math", "title": "Pieri-Type Formulas for the Nonsymmetric Macdonald Polynomials", "abstract": "In symmetric Macdonald polynomial theory the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function and the Macdonald polynomial. In this paper we give the nonsymmetric analogues for the cases r=1 and r=n-1. We do this by first deducing the the decomposition for the product of any nonsymmetric Macdonald polynomial with a linear function in terms of nonsymmetric Macdonald polynomials. As a corollary of finding the branching coefficients of the product of the first elementary function with a nonsymmetric Macdonald polynomial we evaluate the generalised binomial coefficients associated with the nonsymmetric Macdonald polynomials for |u|=|v|+1."}
{"category": "Math", "title": "A construction of Einstein-Weyl spaces via LeBrun-Mason type twistor correspondence", "abstract": "We construct infinitely many Einstein-Weyl structures on $S^2 \\times R$ of signature (-++) which is sufficiently close to the model case of constant curvature, and whose space-like geodesics are all closed. Such structures are obtained from small perturbations of the diagonal of $CP^1 \\times CP^1$ using the method of LeBrun-Mason type twistor theory. The geometry of constructed Einstein-Weyl space is well understood from the configuration of holomorphic disks. We also review Einstein-Weyl structures and their properties in the former half of this article."}
{"category": "Math", "title": "Note on geodesic rays tamed by simple test configurations", "abstract": "In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration."}
{"category": "Math", "title": "A Theorem of Beurling and H\\\"ormander on Damek-Ricci Spaces", "abstract": "This paper has been withdrawn by the authors, due to the requirement of the Journal where a modified version will be published."}
{"category": "Math", "title": "A Morita theorem for dual operator algebras", "abstract": "We prove that two dual operator algebras are weak$^*$ Morita equivalent if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak$^*$-continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak$^*$ Morita equivalence bimodule. We also develop the theory of the $W^*$-dilation, which connects the non-selfadjoint dual operator algebra with the $W^*$-algebraic framework. In the case of weak$^*$ Morita equivalence, this $W^*$-dilation is a $W^*$-module over a von Neumann algebra generated by the non-selfadjoint dual operator algebra. The theory of the $W^*$-dilation is a key part of the proof of our main theorem."}
{"category": "Math", "title": "Adaptation of the generic PDE's results to the notion of prevalence", "abstract": "Many generic results have been proved, especially concerning the qualitative behaviour of solutions of partial differential equations. Recently, a new notion of \"almost always\", the prevalence, has been developped for vectorial spaces. This notion is interesting since, for example, prevalence sets are equivalent to the full Lebesgue measure sets in finite dimensional spaces. The purpose of this article is to adapt the generic PDE's results to the notion of prevalence. In particular, we consider the cases where Sard-Smale theorems or arguments of analytic perturbations of the parameters are used."}
{"category": "Math", "title": "Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle", "abstract": "In this paper, we show that, for scalar reaction-diffusion equations on the circle S1, the property of hyperbolicity of all equilibria and periodic orbits is generic with respect to the non-linearity . In other words, we prove that in an appropriate functional space of nonlinear terms in the equation, the set of functions, for which all equilibria and periodic orbits are hyperbolic, is a countable intersection of open dense sets. The main tools in the proof are the property of the lap number and the Sard-Smale theorem."}
{"category": "Math", "title": "A property of C_p[0,1]", "abstract": "We prove that for every finite dimensional compact metric space $X$ there is an open continuous linear surjection from $C_p[0,1]$ onto $C_p(X)$. The proof makes use of embeddings introduced by Kolmogorov and Sternfeld in connection with Hilbert's 13th problem."}
{"category": "Math", "title": "Conditionally identically distributed species sampling sequences", "abstract": "Conditional identity in distribution (Berti et al. (2004)) is a new type of dependence for random variables, which generalizes the well-known notion of exchangeability. In this paper, a class of random sequences, called Generalized Species Sampling Sequences, is defined and a condition to have conditional identity in distribution is given. Moreover, a class of generalized species sampling sequences that are conditionally identically distributed is introduced and studied: the Generalized Ottawa sequences (GOS). This class contains a '`randomly reinforced'' version of the P\\'olya urn and of the Blackwell-MacQueen urn scheme. For the empirical means and the predictive means of a GOS, we prove two convergence results toward suitable mixtures of Gaussian distributions. The first one is in the sense of stable convergence and the second one in the sense of almost sure conditional convergence. In the last part of the paper we study the length of the partition induced by a GOS at time $n$, i.e. the random number of distinct values of a GOS until time $n$. Under suitable conditions, we prove a strong law of large numbers and a central limit theorem in the sense of stable convergence. All the given results in the paper are accompanied by some examples."}
{"category": "Math", "title": "Symbolic dynamics for the geodesic flow on locally symmetric orbifolds of rank one", "abstract": "We present a strategy for a geometric construction of cross sections for the geodesic flow on locally symmetric orbifolds of rank one. We work it out in detail for $\\Gamma\\backslash H$, where $H$ is the upper half plane and $\\Gamma=\\PGamma_0(p)$, $p$ prime. Its associated discrete dynamical system naturally induces a symbolic dynamics on $\\R$. The transfer operator produced from this symbolic dynamics has a particularly simple structure."}
{"category": "Math", "title": "${L^p}$-variations for multifractal fractional random walks", "abstract": "A multifractal random walk (MRW) is defined by a Brownian motion subordinated by a class of continuous multifractal random measures $M[0,t], 0\\le t\\le1$. In this paper we obtain an extension of this process, referred to as multifractal fractional random walk (MFRW), by considering the limit in distribution of a sequence of conditionally Gaussian processes. These conditional processes are defined as integrals with respect to fractional Brownian motion and convergence is seen to hold under certain conditions relating the self-similarity (Hurst) exponent of the fBm to the parameters defining the multifractal random measure $M$. As a result, a larger class of models is obtained, whose fine scale (scaling) structure is then analyzed in terms of the empirical structure functions. Implications for the analysis and inference of multifractal exponents from data, namely, confidence intervals, are also provided."}
{"category": "Math", "title": "A mixed singular/switching control problem for a dividend policy with reversible technology investment", "abstract": "We consider a mixed stochastic control problem that arises in Mathematical Finance literature with the study of interactions between dividend policy and investment. This problem combines features of both optimal switching and singular control. We prove that our mixed problem can be decoupled in two pure optimal stopping and singular control problems. Furthermore, we describe the form of the optimal strategy by means of viscosity solution techniques and smooth-fit properties on the corresponding system of variational inequalities. Our results are of a quasi-explicit nature. From a financial viewpoint, we characterize situations where a firm manager decides optimally to postpone dividend distribution in order to invest in a reversible growth opportunity corresponding to a modern technology. In this paper a reversible opportunity means that the firm may disinvest from the modern technology and return back to its old technology by receiving some gain compensation. The results of our analysis take qualitatively different forms depending on the parameters values."}
{"category": "Math", "title": "Variance bounding Markov chains", "abstract": "We introduce a new property of Markov chains, called variance bounding. We prove that, for reversible chains at least, variance bounding is weaker than, but closely related to, geometric ergodicity. Furthermore, variance bounding is equivalent to the existence of usual central limit theorems for all $L^2$ functionals. Also, variance bounding (unlike geometric ergodicity) is preserved under the Peskun order. We close with some applications to Metropolis--Hastings algorithms."}
{"category": "Math", "title": "$EE_8$-lattices and dihedral groups", "abstract": "We classify integral rootless lattices which are sums of pairs of $EE_8$-lattices (lattices isometric to $\\sqrt 2$ times the $E_8$-lattice) and which define dihedral groups of orders less than or equal to 12. Most of these may be seen in the Leech lattice. Our classification may help understand Miyamoto involutions on lattice type vertex operator algebras and give a context for the dihedral groups which occur in the Glauberman-Norton moonshine theory."}
{"category": "Math", "title": "Finitely summable Fredholm modules over higher rank groups and lattices", "abstract": "We give a complete classification (up to smooth homotopy) of finitely summable Fredholm representations (Fredholm modules) over higher rank groups and lattices. Our results are a direct consequence of work of Bader, Furman, Gelander and Monod on generalisations of Kazhdan's property T for locally compact groups."}
{"category": "Math", "title": "Stochastic Impulse Control of Non-Markovian Processes", "abstract": "We consider a class of stochastic impulse control problems of general stochastic processes i.e. not necessarily Markovian. Under fairly general conditions we establish existence of an optimal impulse control. We also prove existence of combined optimal stochastic and impulse control of a fairly general class of diffusions with random coefficients. Unlike, in the Markovian framework, we cannot apply quasi-variational inequalities techniques. We rather derive the main results using techniques involving reflected BSDEs and the Snell envelope."}
{"category": "Math", "title": "Central limit theorem for signal-to-interference ratio of reduced rank linear receiver", "abstract": "Let $\\mathbf{s}_k=\\frac{1}{\\sqrt{N}}(v_{1k},...,v_{Nk})^T,$ with $\\{v_{ik},i,k=1,...\\}$ independent and identically distributed complex random variables. Write $\\mathbf{S}_k=(\\mathbf{s}_1,...,\\mathbf {s}_{k-1},\\mathbf{s}_{k+1},... ,\\mathbf{s}_K),$ $\\mathbf{P}_k=\\operatorname {diag}(p_1,...,p_{k-1},p_{k+1},...,p_K)$, $\\mathbf{R}_k=(\\mathbf{S}_k\\mathbf{P}_k\\mathbf{S}_k^*+\\sigma ^2\\mathbf{I})$ and $\\mathbf{A}_{km}=[\\mathbf{s}_k,\\mathbf{R}_k\\mathbf{s}_k,... ,\\mathbf{R}_k^{m-1}\\mathbf{s}_k]$. Define $\\beta_{km}=p_k\\mathbf{s}_k^*\\mathbf{A}_{km}(\\mathbf {A}_{km}^*\\times\\ mathbf{R}_k\\mathbf{A}_{km})^{-1}\\mathbf{A}_{km}^*\\mathbf{s}_k$, referred to as the signal-to-interference ratio (SIR) of user $k$ under the multistage Wiener (MSW) receiver in a wireless communication system. It is proved that the output SIR under the MSW and the mutual information statistic under the matched filter (MF) are both asymptotic Gaussian when $N/K\\to c>0$. Moreover, we provide a central limit theorem for linear spectral statistics of eigenvalues and eigenvectors of sample covariance matrices, which is a supplement of Theorem 2 in Bai, Miao and Pan [Ann. Probab. 35 (2007) 1532--1572]. And we also improve Theorem 1.1 in Bai and Silverstein [Ann. Probab. 32 (2004) 553--605]."}
{"category": "Math", "title": "Stable length in stable groups", "abstract": "We show that the stable commutator length vanishes for certain groups defined as infinite unions of smaller groups. The argument uses a group-theoretic analogue of the Mazur swindle, and goes back to the works of Anderson, Fisher, and Mather on homeomorphism groups."}
{"category": "Math", "title": "Regular Steinhaus graphs of odd degree", "abstract": "A Steinhaus matrix is a binary square matrix of size $n$ which is symmetric, with diagonal of zeros, and whose upper-triangular coefficients satisfy $a_{i,j}=a_{i-1,j-1}+a_{i-1,j}$ for all $2\\leq i<j\\leq n$. Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states that even Steinhaus graphs, i.e. those with all vertex degrees even, have doubly-symmetric Steinhaus matrices. In 1979 Dymacek conjectured that the complete graph on two vertices $K_2$ is the only regular Steinhaus graph of odd degree. Using Dymacek's theorem, we prove that if $(a_{i,j})_{1\\leq i,j\\leq n}$ is a Steinhaus matrix associated with a regular Steinhaus graph of odd degree then its sub-matrix $(a_{i,j})_{2\\leq i,j\\leq n-1}$ is a multi-symmetric matrix, that is a doubly-symmetric matrix where each row of its upper-triangular part is a symmetric sequence. We prove that the multi-symmetric Steinhaus matrices of size $n$ whose Steinhaus graphs are regular modulo 4, i.e. where all vertex degrees are equal modulo 4, only depend on $\\lceil \\frac{n}{24}\\rceil$ parameters for all even numbers $n$, and on $\\lceil \\frac{n}{30}\\rceil$ parameters in the odd case. This result permits us to verify the Dymacek's conjecture up to 1500 vertices in the odd case."}
{"category": "Math", "title": "Stationary max-stable fields associated to negative definite functions", "abstract": "Let $W_i,i\\in{\\mathbb{N}}$, be independent copies of a zero-mean Gaussian process $\\{W(t),t\\in{\\mathbb{R}}^d\\}$ with stationary increments and variance $\\sigma^2(t)$. Independently of $W_i$, let $\\sum_{i=1}^{\\infty}\\delta_{U_i}$ be a Poisson point process on the real line with intensity $e^{-y} dy$. We show that the law of the random family of functions $\\{V_i(\\cdot),i\\in{\\mathbb{N}}\\}$, where $V_i(t)=U_i+W_i(t)-\\sigma^2(t)/2$, is translation invariant. In particular, the process $\\eta(t)=\\bigvee_{i=1}^{\\infty}V_i(t)$ is a stationary max-stable process with standard Gumbel margins. The process $\\eta$ arises as a limit of a suitably normalized and rescaled pointwise maximum of $n$ i.i.d. stationary Gaussian processes as $n\\to\\infty$ if and only if $W$ is a (nonisotropic) fractional Brownian motion on ${\\mathbb{R}}^d$. Under suitable conditions on $W$, the process $\\eta$ has a mixed moving maxima representation."}
{"category": "Math", "title": "Another Correction. Error estimates for Binomial approximations of game options", "abstract": "The Annals of Applied Probability 16 (2006) 984--1033 [URL: http://projecteuclid.org/euclid.aoap/1151592257]"}
{"category": "Math", "title": "Sorting a Permutation by block moves", "abstract": "We prove a lower and an upper bound on the number of block moves necessary to sort a permutation. We put our results in contrast with existing results on sorting by block transpositions, and raise some open questions."}
{"category": "Math", "title": "Free products, Orbit Equivalence and Measure Equivalence Rigidity", "abstract": "We study the analogue in orbit equivalence of free product decomposition and free indecomposability for countable groups. We introduce the (orbit equivalence invariant) notion of freely indecomposable ({\\FI}) standard probability measure preserving equivalence relations and establish a criterion to check it, namely non-hyperfiniteness and vanishing of the first $L^2$-Betti number. We obtain Bass-Serre rigidity results, \\textit{i.e.} forms of uniqueness in free product decompositions of equivalence relations with ({\\FI}) components. The main features of our work are weak algebraic assumptions and no ergodicity hypothesis for the components. We deduce, for instance, that a measure equivalence between two free products of non-amenable groups with vanishing first $\\ell^2$-Betti numbers is induced by measure equivalences of the components. We also deduce new classification results in Orbit Equivalence and II$_1$ factors."}
{"category": "Math", "title": "Polynomial least squares fitting in the Bernstein basis", "abstract": "The problem of polynomial regression in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix A of the overdetermined system to be solved in the least squares sense is then a rectangular Bernstein-Vandermonde matrix. In order to use the method based on the QR decomposition of A, the first stage consists of computing the bidiagonal decomposition of the coefficient matrix A. Starting from that bidiagonal decomposition, an algorithm for obtaining the QR decomposition of A is the applied. Finally, a triangular system is solved by using the bidiagonal decomposition of the R-factor of A. Some numerical experiments showing the behavior of this approach are included."}
{"category": "Math", "title": "A tight closure approach to a result of G. Faltings", "abstract": "Using a result of M. Hochster and C. Huneke on $F$-rational rings a criterion for complete intersection rings of characteristic $p>0$ is presented. As an application, we give a completely different proof for an algebraic result of G. Faltings that was used by Taylor and Wiles in \\cite{TW} for a simplification of the proof of the minimal deformation problem."}
{"category": "Math", "title": "Strict abnormal extremals in nonholonomic and kinematic control systems", "abstract": "In optimal control problems, there exist different kinds of extremals, that is, curves candidates to be solution: abnormal, normal and strictly abnormal. The key point for this classification is how those extremals depend on the cost function. We focus on control systems such as nonholonomic control mechanical systems and the associated kinematic systems as long as they are equivalent. With all this in mind, first we study conditions to relate an optimal control problem for the mechanical system with another one for the associated kinematic system. Then, Pontryagin's Maximum Principle will be used to connect the abnormal extremals of both optimal control problems. An example is given to glimpse what the abnormal solutions for kinematic systems become when they are considered as extremals to the optimal control problem for the corresponding nonholonomic mechanical systems."}
{"category": "Math", "title": "Torsional rigidity of submanifolds with controlled geometry", "abstract": "We prove explicit upper and lower bounds for the torsional rigidity of extrinsic domains of submanifolds P^m with controlled radial mean curvature in ambient Riemannian manifolds N^n with a pole p and with sectional curvatures bounded from above and from below, respectively. These bounds are given in terms of the torsional rigidities of corresponding Schwarz symmetrizations of the domains in warped product model spaces. Our main results are obtained using methods from previously established isoperimetric inequalities, as found in e.g. [MP4] and [MP5]. As in [MP4] we also characterize the geometry of those situations in which the bounds for the torsional rigidity are actually attained and study the behavior at infinity of the so-called geometric average of the mean exit time for Brownian motion."}
{"category": "Math", "title": "Doubly periodic textile patterns", "abstract": "Knitted and woven textile structures are examples of doubly periodic structures in a thickened plane made out of intertwining strands of yarn. Factoring out the group of translation symmetries of such a structure gives rise to a link diagram in a thickened torus. Such a diagram on a standard torus is converted into a classical link by including two auxiliary components which form the cores of the complementary solid tori. The resulting link, called a kernel for the structure, is determined by a choice of generators u and v for the group of symmetries. A normalised form of the multi-variable Alexander polynomial of a kernel is used to provide polynomial invariants of the original structure which are essentially independent of the choice of generators. It gives immediate information about the existence of closed curves and other topological features in the original textile structure. Because of its natural algebraic properties under coverings we can recover the polynomial for kernels based on a proper subgroup from the polynomial derived from the full symmetry group of the structure. This enables two structures to be compared at similar scales, even when one has a much smaller minimal repeating cell than the other. Examples of simple traditional structures are given, and their Alexander data polynomials are presented to illustrate the techniques and results."}
{"category": "Math", "title": "String topology on Gorenstein spaces", "abstract": "The purpose of this paper is to describe a general and simple setting for defining $(g,p+q)$-string operations on a Poincar\\'e duality space and more generally on a Gorenstein space. Gorenstein spaces include Poincar\\'e duality spaces as well as classifying spaces or homotopy quotients of connected Lie groups. Our presentation implies directly the homotopy invariance of each $(g,p+q)$-string operation as well as it leads to explicit computations."}
{"category": "Math", "title": "Supervised functional classification: A theoretical remark and some comparisons", "abstract": "The problem of supervised classification (or discrimination) with functional data is considered, with a special interest on the popular k-nearest neighbors (k-NN) classifier. First, relying on a recent result by Cerou and Guyader (2006), we prove the consistency of the k-NN classifier for functional data whose distribution belongs to a broad family of Gaussian processes with triangular covariance functions. Second, on a more practical side, we check the behavior of the k-NN method when compared with a few other functional classifiers. This is carried out through a small simulation study and the analysis of several real functional data sets. While no global \"uniform\" winner emerges from such comparisons, the overall performance of the k-NN method, together with its sound intuitive motivation and relative simplicity, suggests that it could represent a reasonable benchmark for the classification problem with functional data."}
{"category": "Math", "title": "2-step nilpotent Lie groups arising from semisimple modules", "abstract": "Every finite dimensional real representation of a compact real semisimple Lie algebra determines a metric 2-step nilpotent Lie algebra and a corresponding simply connected metric 2-step nilpotent Lie group N. We study the differential geometry of N using representation theory of the complexified complex semisimple Lie algebra."}
{"category": "Math", "title": "Set Linear Algebra and Set Fuzzy Linear Algebra", "abstract": "In this book, the authors define the new notion of set vector spaces which is the most generalized form of vector spaces. Set vector spaces make use of the least number of algebraic operations, therefore, even a non-mathematician is comfortable working with it. It is with the passage of time, that we can think of set linear algebras as a paradigm shift from linear algebras. Here, the authors have also given the fuzzy parallels of these new classes of set linear algebras. This book is divided into seven chapters. The first chapter briefly recalls some of the basic concepts in order to make this book self-contained. Chapter two introduces the notion of set vector spaces which is the most generalized concept of vector spaces. Set vector spaces lends itself to define new classes of vector spaces like semigroup vector spaces and group vector spaces. These are also generalization of vector spaces. The fuzzy analogue of these concepts are given in Chapter three. In Chapter four, set vector spaces are generalized to biset bivector spaces and not set vector spaces. This is done taking into account the advanced information technology age in which we live. As mathematicians, we have to realize that our computer-dominated world needs special types of sets and algebraic structures. Set n-vector spaces and their generalizations are carried out in Chapter five. Fuzzy n-set vector spaces are introduced in the sixth chapter. The seventh chapter suggests more than three hundred problems."}
{"category": "Math", "title": "A note on K\\\"ahler-Ricci soliton", "abstract": "In this note we provide a proof of the following: Any compact KRS with positive bisectional curvature is biholomorphic to the complex projective space. As a corollary, we obtain an alternative proof of the Frankel conjecture by using the K\\\"ahler-Ricci flow."}
{"category": "Math", "title": "A direct proof of Z-stability for AH algebras of bounded topological dimension", "abstract": "We prove that a unital simple approximately homogeneous (AH) C*-algebra with no dimension growth absorbs the Jiang-Su algebra tensorially without appealing to the classification theory of these algebras. Our main result continues to hold under the slightly weaker hypothesis of exponentially slow dimension growth."}
{"category": "Math", "title": "Generating sequences and Poincar\\'e series for a finite set of plane divisorial valuations", "abstract": "Let $V$ be a finite set of divisorial valuations centered at a 2-dimensional regular local ring $R$. In this paper we study its structure by means of the semigroup of values, $S_V$, and the multi-index graded algebra defined by $V$, $\\gr_V R$. We prove that $S_V$ is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in $V$, the approximation of a reduced plane curve singularity $C$ by families of sets $V^{(k)}$ of divisorial valuations, and the relationship between the value semigroup of $C$ and the semigroups of the sets $V^{(k)}$, allow us to obtain the (finite) minimal generating sequences for $C$ as well as for $V$. We also analyze the structure of the homogeneous components of $\\gr_V R$. The study of their dimensions allows us to relate the Poincar\\'e series for $V$ and for a general curve $C$ of $V$. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincar\\'e series of $V$. Moreover, the Poincar\\'e series of $C$ could be seen as the limit of the series of $V^{(k)}$, $k\\ge 0$."}
{"category": "Math", "title": "Maximizing Sum Rates in Gaussian Interference-limited Channels", "abstract": "We study the problem of maximizing sum rates in a Gaussian interference-limited channel that models multiuser communication in a CDMA wireless network or DSL cable binder. Using tools from nonnegative irreducible matrix theory, in particular the Perron-Frobenius Theorem and the Friedland-Karlin inequalities, we provide insights into the structural property of optimal power allocation strategies that maximize sum rates. Our approach is similar to the treatment of linear models in mathematical economies, where interference is viewed in the context of competition. We show that this maximum problem can be restated as a maximization problem of a convex function on a closed convex set. We suggest three algorithms to find the exact and approximate values of the optimal sum rates. In particular, our algorithms exploit the eigenspace of specially crafted nonnegative {\\it interference matrices}, which, with the use of standard optimization tools, can provide useful upper bounds and feasible solutions to the nonconvex problem."}
{"category": "Math", "title": "Rational convexity of non generic immersed lagrangian submanifolds", "abstract": "We prove that an immersed lagrangian submanifold in $\\C^n$ with quadratic self-tangencies is rationally convex. This generalizes former results for the embedded and the immersed transversal cases."}
{"category": "Math", "title": "Thompson's Group F and Uniformly Finite Homology", "abstract": "This paper demonstrates the uniformly finite homology developed by Block and Weinberger and its relationship to amenable spaces via applications to the Cayley graph of Thompson's Group F. In particular, a certain class of subgraph of F is shown to be non-amenable. This shows that if F is amenable, these subsets (which include every finitely generated submonoid of the positive monoid of F) must necessarily have measure zero."}
{"category": "Math", "title": "Detecting automorphic orbits in free groups", "abstract": "We present an effective algorithm for detecting automorphic orbits in free groups, as well as a number of algorithmic improvements of train tracks for free group automorphisms."}
{"category": "Math", "title": "Statistical Behaviour of the Leaves of Riccati Foliations", "abstract": "We introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension 1 and hyperbolic, corresponding to the unique complete metric of curvature -1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivisation of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation r:pi_1(S) -> GL(n,C). We give conditions under which the foliated geodesic flow has a generic repellor-attractor statistical dynamics. That is, there are measures m- and m+ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to m- and for positive time to m+ (i.e. m+ is the unique SRB-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behavior of the leaves of the foliation."}
{"category": "Math", "title": "Optimal designs for mixed models in experiments based on ordered units", "abstract": "We consider experiments for comparing treatments using units that are ordered linearly over time or space within blocks. In addition to the block effect, we assume that a trend effect influences the response. The latter is modeled as a smooth component plus a random term that captures departures from the smooth trend. The model is flexible enough to cover a variety of situations; for instance, most of the effects may be either random or fixed. The information matrix for a design will be a function of several variance parameters. While data will shed light on the values of these parameters, at the design stage, they are unlikely to be known, so we suggest a maximin approach, in which a minimal information matrix is maximized. We derive maximin universally optimal designs and study their robustness. These designs are based on semibalanced arrays. Special cases correspond to results available in the literature."}
{"category": "Math", "title": "Symplectic topology of SU(2)-representation varieties and link homology, I: Symplectic braid action and the first Chern class", "abstract": "There are some similarities between cohomology of SU(2)-representation varieties of the fundamental group of some link complements and the Khovanov homology of the links. We start here a program to explain a possible source of these similarities. We introduce a symplectic manifold ${\\mathscr M}$ with an action of the braid group $B_{2n}$ preserving the symplectic structure. The action allows to associate a Lagrangian submanifold of ${\\mathscr M}$ to every braid. The representation variety of a link can then be described as the intersection of such Lagrangian submanifolds, given a braid presentation of the link. We expect this to go some way in explaining the similarities mentioned above."}
{"category": "Math", "title": "Composite quantile regression and the oracle Model Selection Theory", "abstract": "Coefficient estimation and variable selection in multiple linear regression is routinely done in the (penalized) least squares (LS) framework. The concept of model selection oracle introduced by Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348--1360] characterizes the optimal behavior of a model selection procedure. However, the least-squares oracle theory breaks down if the error variance is infinite. In the current paper we propose a new regression method called composite quantile regression (CQR). We show that the oracle model selection theory using the CQR oracle works beautifully even when the error variance is infinite. We develop a new oracular procedure to achieve the optimal properties of the CQR oracle. When the error variance is finite, CQR still enjoys great advantages in terms of estimation efficiency. We show that the relative efficiency of CQR compared to the least squares is greater than 70% regardless the error distribution. Moreover, CQR could be much more efficient and sometimes arbitrarily more efficient than the least squares. The same conclusions hold when comparing a CQR-oracular estimator with a LS-oracular estimator."}
{"category": "Math", "title": "Lower order terms for the one-level densities of symmetric power $L$-functions in the level aspect", "abstract": "In a previous paper, the authors determined, among other things, the main terms for the one-level densities for low-lying zeros of symmetric power L-functions in the level aspect. In this paper, the lower order terms of these one-level densities are found. The combinatorial difficulties, which should arise in such context, are drastically reduced thanks to Chebyshev polynomials, which are the characters of the irreducible representations of SU(2). %"}
{"category": "Math", "title": "Adaptive estimation of and oracle inequalities for probability densities and characteristic functions", "abstract": "The theory of adaptive estimation and oracle inequalities for the case of Gaussian-shift--finite-interval experiments has made significant progress in recent years. In particular, sharp-minimax adaptive estimators and exact exponential-type oracle inequalities have been suggested for a vast set of functions including analytic and Sobolev with any positive index as well as for Efromovich--Pinsker and Stein blockwise-shrinkage estimators. Is it possible to obtain similar results for a more interesting applied problem of density estimation and/or the dual problem of characteristic function estimation? The answer is ``yes.'' In particular, the obtained results include exact exponential-type oracle inequalities which allow to consider, for the first time in the literature, a simultaneous sharp-minimax estimation of Sobolev densities with any positive index (not necessarily larger than 1/2), infinitely differentiable densities (including analytic, entire and stable), as well as of not absolutely integrable characteristic functions. The same adaptive estimator is also rate minimax over a familiar class of distributions with bounded spectrum where the density and the characteristic function can be estimated with the parametric rate."}
{"category": "Math", "title": "Vari\\'et\\'es rationnellement connexes sur un corps alg\\'ebriquement clos", "abstract": "These are lectures notes on rationally connected varieties, written for the \"Etats de la Recherche\" of the French Mathematical Society held in Strasbourg (May 2008). We focus on geometric aspects. These notes have been written in order that a wide audience can easily read them, except maybe the last section, a bit more technical, where we give the proof of Shokurov rational connectedness conjecture following Hacon and McKernan. ----- Ce sont les notes d'un mini-cours sur les vari\\'et\\'es rationnellement connexes, \\'ecrit pour les Etats de la Recherche de la Soci\\'et\\'e Math\\'ematique de France (Strasbourg, 2008). On met l'accent sur les aspects g\\'eom\\'etriques. Ce cours est r\\'edig\\'e dans l'espoir de s'adresser \\`a un public large, \\`a l'exception peut-\\^etre du \\S 7, o\\`u nous donnons les grandes lignes de la preuve de la conjecture de connexit\\'e rationnelle de Shokurov par Hacon et McKernan, plus technique et o\\`u les pr\\'erequis sont un peu plus importants."}
{"category": "Math", "title": "Admissible predictive density estimation", "abstract": "Let $X|\\mu\\sim N_p(\\mu,v_xI)$ and $Y|\\mu\\sim N_p(\\mu,v_yI)$ be independent $p$-dimensional multivariate normal vectors with common unknown mean $\\mu$. Based on observing $X=x$, we consider the problem of estimating the true predictive density $p(y|\\mu)$ of $Y$ under expected Kullback--Leibler loss. Our focus here is the characterization of admissible procedures for this problem. We show that the class of all generalized Bayes rules is a complete class, and that the easily interpretable conditions of Brown and Hwang [Statistical Decision Theory and Related Topics (1982) III 205--230] are sufficient for a formal Bayes rule to be admissible."}
{"category": "Math", "title": "$L^2-$interpolation with error and size of spectra", "abstract": "Given a compact set $S$ and a uniformly discrete sequence $\\La$, we show that \"approximate interpolation\" of delta--functions on $\\La$ by a bounded sequence of $L^2-$functions with spectra in $S$ implies an estimate on measure of $S$ through the density of $\\La$."}
{"category": "Math", "title": "Kostant's problem and parabolic subgroups", "abstract": "Let $\\frak g$ be a finite dimensional complex semi-simple Lie algebra with Weyl group $W$ and simple reflections $S$. For $I\\subseteq S$ let $\\frak g_I$ be the corresponding semi-simple subalgebra of $\\frak g$. Denote by $W_I$ the Weyl group of $\\frak g_I$ and let $w_o$ and $w^I_o$ be the longest elements of $W$ and $W_I$, respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight $\\frak g_I$-module $L_I(x)$ of highest weight $x\\cdot 0$, $x\\in W_I$, as the answer for the simple highest weight $\\frak g$-module $L(x w^I_o w_o)$ of highest weight $(x w^I_o w_o)\\cdot 0$. We also give a new description of the unique quasi-simple quotient of the Verma module $\\Delta(e)$ with the same annihilator as $L(y)$, $y\\in W$."}
{"category": "Math", "title": "Wave equation and multiplier estimates on Damek-Ricci spaces", "abstract": "Let S be a Damek-Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators associated with L. This generalizes previous results proved by D. Mueller and C. Thiele on ax+b-groups. We also prove pointwise estimates of the gradient of these convolution kernels. As a corollary we reprove basic multiplier estimates from previous papers of W. Hebisch and T. Steger and M. Vallarino by different methods. Finally we derive Sobolev estimates for the solution to the wave equation associated with L."}
{"category": "Math", "title": "A Point is Normal for Almost All Maps $\\beta x + \\alpha \\mod 1$ or Generalized $\\beta$-Maps", "abstract": "We consider the map $T_{\\alpha,\\beta}(x):= \\beta x + \\alpha \\mod 1$, which admits a unique probability measure of maximal entropy $\\mu_{\\alpha,\\beta}$. For $x \\in [0,1]$, we show that the orbit of $x$ is $\\mu_{\\alpha,\\beta}$-normal for almost all $(\\alpha,\\beta)\\in[0,1)\\times(1,\\infty)$ (Lebesgue measure). Nevertheless we construct analytic curves in $[0,1)\\times(1,\\infty)$ along them the orbit of $x=0$ is at most at one point $\\mu_{\\alpha,\\beta}$-normal. These curves are disjoint and they fill the set $[0,1)\\times(1,\\infty)$. We also study the generalized $\\beta$-maps (in particular the tent map). We show that the critical orbit $x=1$ is normal with respect to the measure of maximal entropy for almost all $\\beta$."}
{"category": "Math", "title": "A Note on Dominant Contractions of Jordan Algebras", "abstract": "In the paper we consider two positive contractions $T,S:L^{1}(A,\\tau)\\longrightarrow L^{1}(A,\\tau)$ such that $T\\leq S$, here $(A,\\t)$ is a semi-finite $JBW$-algebra. If there is an $n_{0}\\in\\mathbb{N}$ such that $\\|S^{n_{0}}-T^{n_{0}}\\|<1$. Then we prove that $\\|S^{n}-T^{n}\\|<1$ holds for every $n\\geq n_{0}.$"}
{"category": "Math", "title": "Parametric bootstrap approximation to the distribution of EBLUP and related prediction intervals in linear mixed models", "abstract": "Empirical best linear unbiased prediction (EBLUP) method uses a linear mixed model in combining information from different sources of information. This method is particularly useful in small area problems. The variability of an EBLUP is traditionally measured by the mean squared prediction error (MSPE), and interval estimates are generally constructed using estimates of the MSPE. Such methods have shortcomings like under-coverage or over-coverage, excessive length and lack of interpretability. We propose a parametric bootstrap approach to estimate the entire distribution of a suitably centered and scaled EBLUP. The bootstrap histogram is highly accurate, and differs from the true EBLUP distribution by only $O(d^3n^{-3/2})$, where $d$ is the number of parameters and $n$ the number of observations. This result is used to obtain highly accurate prediction intervals. Simulation results demonstrate the superiority of this method over existing techniques of constructing prediction intervals in linear mixed models."}
{"category": "Math", "title": "On the ergodicity of the adaptive Metropolis algorithm on unbounded domains", "abstract": "This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen [Bernoulli 7 (2001) 223--242] for target distributions with a noncompact support. The conditions ensuring a strong law of large numbers require that the tails of the target density decay super-exponentially and have regular contours. The result is based on the ergodicity of an auxiliary process that is sequentially constrained to feasible adaptation sets, independent estimates of the growth rate of the AM chain and the corresponding geometric drift constants. The ergodicity result of the constrained process is obtained through a modification of the approach due to Andrieu and Moulines [Ann. Appl. Probab. 16 (2006) 1462--1505]."}
{"category": "Math", "title": "Additivity of Heegaard genera of bounded surface sums", "abstract": "Let $M$ be a surface sum of 3-manifolds $M_1$ and $M_2$ along a bounded connected surface $F$ and $\\partial_i$ be the component of $\\partial M_i$ containing $F$. If $M_i$ has a high distance Heegaard splitting, then any minimal Heegaard splitting of $M$ is the amalgamation of those of $M^1, M^2$ and $M^*$, where $M^i=M_i\\setminus\\partial_i\\times I$, and $M^{*}=\\partial_1\\times I\\cup_{F} \\partial_2\\times I$. Furthermore, once both $\\partial_i\\setminus F$ are connected, then $g(M) = Min\\bigl\\{g(M_1)+g(M_2), \\alpha\\bigr\\}$, where $\\alpha = g(M_1) + g(M_2) + 1/2(2\\chi(F) + 2 - \\chi(\\partial_1) - \\chi(\\partial_2)) - Max\\bigl\\{g(\\partial_1), g(\\partial_2)\\bigl\\}$; in particular $g(M)=g(M_1)+g(M_2)$ if and only if $\\chi(F)\\geq 1/2Max\\bigl\\{\\chi(\\partial_1), \\chi(\\partial_2)\\bigr\\}.$ The proofs rely on Scharlemann-Tomova's theorem."}
{"category": "Math", "title": "q,t-Fuss-Catalan numbers for complex reflection groups", "abstract": "In type A, the q,t-Fuss -Catalan numbers can be defined as a bigraded Hilbert series of a module associated to the symmetric group S_n. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in q and t. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress."}
{"category": "Math", "title": "Data-driven Sobolev tests of uniformity on compact Riemannian manifolds", "abstract": "Data-driven versions of Sobolev tests of uniformity on compact Riemannian manifolds are proposed. These tests are invariant under isometries and are consistent against all alternatives. The large-sample asymptotic null distributions are given."}
{"category": "Math", "title": "Construction of A Lattice on the completion space of an algebra and an isomorphism to its Caratheodory Extension", "abstract": "In this paper, we will show how the Caratheodory Extension process is intimately related to the metric completion process. In particular, it will be shown how one is able to construct a lattice on the completion and to obtain an isomorphism to its Caratheodory Extension."}
{"category": "Math", "title": "New Techniques for Empirical Process of Dependent Data", "abstract": "We present a new technique for proving empirical process invariance principle for stationary processes $(X_n)_{n\\geq 0}$. The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions $(f(X_n))_{n\\geq 0}$, not containing the indicator functions. Our approach can be applied to Markov chains and dynamical systems, using spectral properties of the transfer operator. Our proof consists of a novel application of chaining techniques."}
{"category": "Math", "title": "Decomposition rank and Z-stability", "abstract": "We show that separable, simple, unital C*-algebras with finite decomposition rank absorb the Jiang-Su algebra Z tensorially. This has a number of consequences for Elliott's program to classify nuclear C*-algebras by their K-theory data. In particular, it completes the classification of C*-algebras associated to uniquely ergodic, smooth, minimal dynamical systems by their ordered K-groups."}
{"category": "Math", "title": "Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space", "abstract": "Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $\\Delta u = f(u)$ on hyperbolic space $\\HH^n$ must reduce to functions of one variable provided they admit asymptotic boundary values on the infinite boundary of $\\HH^n$ which are invariant under a cohomogeneity one subgroup of the group of isometries of $\\HH^n$. We also prove existence of these one-dimensional solutions."}
{"category": "Math", "title": "Branchfolds and rational conifolds", "abstract": "We extend the concept of orbifold to that of branchfold, in order to allow any cone singularities with rational angles, and show why branchfolds naturally fit in the theory of branched coverings. Then, we obtain a geometric goodness theorem for branchfolds and apply it to prove that a conifold can be endowed with branchfold structure if and only if it has locally finite holonomy."}
{"category": "Math", "title": "Remarks on an 1828 theorem of Clausen", "abstract": "We list some explicit calculations related to a theorem of Clausen originally published in 1828, more commonly known as the result that describes the linear third order differential equation satisfied by the squares and the product of any two solutions of a linear second order differential equation in the real domain. The case of the cube of a solution dates to Appell, 1880. Although not commonly known and perhaps even new to some extent we show that the fourth and fifth powers of such solutions actually satisfy a linear differential equation of order five and six respectively provided the coefficients are sufficiently smooth. Indeed, it is the case that given any solution of a linear second order equation its m-th power satisfies an effectively computable linear differential equation of order m+1."}
{"category": "Math", "title": "Optimal rank-based tests for homogeneity of scatter", "abstract": "We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in $m$ elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly robust against heavy-tailed distributions (validity robustness). Nevertheless, they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all (efficiency robustness). In particular, their normal-score version outperforms traditional Gaussian likelihood ratio tests and their pseudo-Gaussian robustifications under a very broad range of non-Gaussian densities including, for instance, all multivariate Student and power-exponential distributions."}
{"category": "Math", "title": "Quasi-factors for infinite-measure preserving transformations", "abstract": "This paper is a study of Glasner's definition of quasi-factors in the setting of infinite-measure preserving system. The existence of a system with zero Krengel entropy and a quasi-factor with positive entropy is obtained. On the other hand, relative zero-entropy for conservative systems implies relative zero-entropy of any quasi-factor with respect to its natural projection onto the factor. This extends (and is based upon) results of Glasner, Thouvenot and Weiss. Following and extending Glasner and Weiss, we also prove that any conservative measure preserving system with positive entropy in the sense of Danilenko and Rudolph admits any probability preserving system with positive entropy as a factor. Some applications and connections with Poisson-suspensions are presented."}
{"category": "Math", "title": "Analytic pro-p groups of small dimensions", "abstract": "According to Lazard, every p-adic Lie group contains an open pro-p subgroup which is saturable. This can be regarded as the starting point of p-adic Lie theory, as one can naturally associate to every saturable pro-p group G a Lie lattice L(G) over the p-adic integers. Essential features of saturable pro-p groups include that they are torsion-free and p-adic analytic. In the present paper we prove a converse result in small dimensions: every torsion-free p-adic analytic pro-p group of dimension less than p is saturable. This leads to useful consequences and interesting questions. For instance, we give an effective classification of 3-dimensional soluble torsion-free p-adic analytic pro-p groups for p > 3. Our approach via Lie theory is comparable with the use of Lazard's correspondence in the classification of finite p-groups of small order."}
{"category": "Math", "title": "Multivariate spacings based on data depth: I. Construction of nonparametric multivariate tolerance regions", "abstract": "This paper introduces and studies multivariate spacings. The spacings are developed using the order statistics derived from data depth. Specifically, the spacing between two consecutive order statistics is the region which bridges the two order statistics, in the sense that the region contains all the points whose depth values fall between the depth values of the two consecutive order statistics. These multivariate spacings can be viewed as a data-driven realization of the so-called ``statistically equivalent blocks.'' These spacings assume a form of center-outward layers of ``shells'' (``rings'' in the two-dimensional case), where the shapes of the shells follow closely the underlying probabilistic geometry. The properties and applications of these spacings are studied. In particular, the spacings are used to construct tolerance regions. The construction of tolerance regions is nonparametric and completely data driven, and the resulting tolerance region reflects the true geometry of the underlying distribution. This is different from most existing approaches which require that the shape of the tolerance region be specified in advance. The proposed tolerance regions are shown to meet the prescribed specifications, in terms of $\\beta$-content and $\\beta$-expectation. They are also asymptotically minimal under elliptical distributions. Finally, a simulation and comparison study on the proposed tolerance regions is presented."}
{"category": "Math", "title": "A general trimming approach to robust Cluster Analysis", "abstract": "We introduce a new method for performing clustering with the aim of fitting clusters with different scatters and weights. It is designed by allowing to handle a proportion $\\alpha$ of contaminating data to guarantee the robustness of the method. As a characteristic feature, restrictions on the ratio between the maximum and the minimum eigenvalues of the groups scatter matrices are introduced. This makes the problem to be well defined and guarantees the consistency of the sample solutions to the population ones. The method covers a wide range of clustering approaches depending on the strength of the chosen restrictions. Our proposal includes an algorithm for approximately solving the sample problem."}
{"category": "Math", "title": "Energy Critical NLS in two space dimensions", "abstract": "We investigate the initial value problem for a defocusing nonlinear Schr\\\"odingerequation with exponential nonlinearity. We identify subcritical, critical and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case."}
{"category": "Math", "title": "A fourth moment inequality for functionals of stationary processes", "abstract": "In this paper, a fourth moment bound for partial sums of functional of strongly ergodic Markov chain is established. This type of inequality plays an important role in the study of empirical process invariance principle. This one is specially adapted to the technique of Dehling, Durieu and Voln\\'y (2008). The same moment bound can be proved for dynamical system whose transfer operator has some spectral properties. Examples of applications are given."}
{"category": "Math", "title": "Adaptive goodness-of-fit tests based on signed ranks", "abstract": "Within the nonparametric regression model with unknown regression function $l$ and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis $l=0$ against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and it is uniformly consistent against alternatives whose sup-norm tends to zero at the fastest possible rate. The test is shown to be asymptotically optimal in two senses: It is rate-optimal adaptive against H\\\"{o}lder classes. Furthermore, its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup-norm loss is close to 1 in case of homoscedastic Gaussian errors within a broad range of H\\\"{o}lder classes simultaneously."}
{"category": "Math", "title": "Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets", "abstract": "Let P be an hyperplane in R^N, and denote by dH the Hausdorff distance. We show that for all positive radius r < 1 there is an epsilon > 0, such that if K is a Reifenberg-flat set in B(0; 1), a ball in R^N, that contains the origin, with d_H(K; P) <epsilon, and if u is an energy minimizing function in B(0; 1)\\K with restricted values on @B(0; 1)\\K, then the normalized energy of u in B(0; r)\\K is bounded by the normalized energy of u in B(0; 1)\\K. We also prove the same result in R^3 when K is a epsilon-minimal set, that is a generalization of Reifenberg-flat sets with minimal cones of type Y and T. Moreover, the result is still true for a further generalization of sets called (eps; eps_0)-minimal. This article is a preliminary study for a forthcoming paper where a regularity result for the singular set of the Mumford-Shah functional close to minimal cones in R^3 is proved by the same author."}
{"category": "Math", "title": "Regularity of the singular set for Mumford-Shah minimizers in R^3 near a minimal cone", "abstract": "We show that if (u;K) is a minimizer of the Mumford-Shah functional in an open set of R^3, and if x, K and r > 0 are such that K is close enough to a minimal cone of type P (a plane), Y (three half planes meeting with 120 degrees angles) or T (cone over a regular tetrahedron centered at the origin) in terms of Hausdorff distance in B(x; r), then K is C^1,alpha equivalent to the minimal cone in B(x; cr) where c < 1 is an universal constant."}
{"category": "Math", "title": "Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves", "abstract": "We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, where they are vulnerable to faster index calculus attacks. We provide explicit formulae for isogenies with kernel isomorphic to $(\\ZZ/2\\ZZ)^3$ (over an algebraic closure of the base field) for any hyperelliptic genus 3 curve over a field of characteristic not 2 or 3. These isogenies are rational for a positive fraction of all hyperelliptic genus 3 curves defined over a finite field of characteristic $p > 3$. Subject to reasonable assumptions, our constructions give an explicit and efficient reduction of instances of the DLP from hyperelliptic to non-hyperelliptic Jacobians for around 18.57% of all hyperelliptic genus 3 curves over a given finite field. We conclude with a discussion on extending these ideas to isogenies with more general kernels. A condensed version of this work appeared in the proceedings of the EUROCRYPT 2008 conference."}
{"category": "Math", "title": "Invertible and nilpotent matrices over antirings", "abstract": "In this paper we characterize invertible matrices over an arbitrary commutative antiring S and find the structure of GL_n (S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent $n \\times n$ matrix over an entire antiring can be written as a sum of $\\lceil \\log_2 n \\rceil$ square-zero matrices and also find the necessary number of square-zero summands for an arbitrary trace-zero matrix to be expressible as their sum."}
{"category": "Math", "title": "Deligne-Lusztig restriction of a Gelfand-Graev module", "abstract": "Using Deodhar's decomposition of a double Schubert cell, we study the regular representations of finite groups of Lie type arising in the cohomology of Deligne-Lusztig varieties associated to tori. We deduce that the Deligne-Lusztig restriction of a Gelfand-Graev module is a shifted Gelfand-Graev module."}
{"category": "Math", "title": "Stochastic control up to a hitting time: optimality and rolling-horizon implementation", "abstract": "We present a dynamic programming-based solution to a stochastic optimal control problem up to a hitting time for a discrete-time Markov control process. Firstly, we determine an optimal control policy to steer the process toward a compact target set while simultaneously minimizing an expected discounted cost. We then provide a rolling-horizon strategy for approximating the optimal policy, together with quantitative characterization of its sub-optimality with respect to the optimal policy. Finally, we address related issues of asymptotic discount-optimality of the value-iteration policy. Both the state and action spaces are assumed to be Polish."}
{"category": "Math", "title": "Asymptotic Uncorrelation for Mexican Needlets", "abstract": "We recall Mexican needlets from [5]. We derive an estimate for certain types of Legendre series, which we apply to the statistical properties of Mexican needlets. More precisely, we shall show that, under isotropy and Gaussianity assumptions, the Mexican needlet coefficients of a random field on the sphere are asymptotically uncorrelated, as the frequency parameter goes to infinity. This property is important in the analysis of cosmic microwave background radiation."}
{"category": "Math", "title": "Corks, Plugs and exotic structures", "abstract": "We discuss corks, and introduce new objects which we call plugs. Though plugs are fundamentally different objects, they also detect exotic smooth structures in 4-manifolds like corks. We discuss relation between corks, plugs and rational blow-downs. We show how to detect corks and plugs inside of some exotic manifolds. Furthermore, we construct knotted corks and plugs."}
{"category": "Math", "title": "Nonparametric estimation for an autoregressive model", "abstract": "The paper deals with the nonparametric estimation problem at a given fixed point for an autoregressive model with unknown distributed noise. Kernel estimate modifications are proposed. Asymptotic minimax and efficiency properties for proposed estimators are shown."}
{"category": "Math", "title": "Two-sided localizations of bimodules", "abstract": "We extend to bimodules Schelter's localization of a ring with respect to Gabriel filters of left and right ideals. Our two-sided localization of bimodules provides an endofunctor on a convenient bicategory of rings with filters of ideals. This is used to study the Picard group of a ring relative to a filter of ideals."}
{"category": "Math", "title": "Squaring rectangles for dumbbells", "abstract": "The theorem on squaring a rectangle from a tiling of a quadrilateral (Schramm and Cannon-Floyd-Parry) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectangle-squaring as an approximation to the classical Riemamnn mapping."}
{"category": "Math", "title": "Uniconvergence theorems for Sturm--Liouville operators with potentials from Sobolev space $W_2^{-1}[0,\\pi]$", "abstract": "We consider a Sturm--Liouville $Ly=-y''+q(x)y$ in space $L_2[0,\\pi]$ with potential from Sobolev space $W_2^{-1}[0,\\pi]$. Moreover, we assume, that $q=u'$, where $u\\in L_2[0,\\pi]$. We consider Direchlet boundary conditions $y(0)=y(\\pi)=0$, although we can treat a boundary conditions of Sturm type. It is known, that operators of such class have a discrete spectr with only accumulation point $+\\infty$ and the system $\\{y_k\\}_1^\\infty$ of eigen and associated functions is a Riesz basis in $L_2[0,\\pi]$. Moreover, this basis is a Hilbert--Schmidt perturbation of the basis $\\{sin(kx)\\}_1^\\infty$. In this paper we prove the uniconvergence theorem: for any element $f\\in L_2[0,\\pi]$ the sequence $P_nf-S_nf\\to0$ as $n\\to\\infty$ in $C[0,\\pi]$ (here $P_n$ and $S_n$ are the Riesz projectors to $\\{y_k\\}_1^n$ and $\\{\\sin(kt)\\}_1^n$ respectively)."}
{"category": "Math", "title": "Rates of contraction of posterior distributions based on Gaussian process priors", "abstract": "We derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on Gaussian processes. The rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel Hilbert space of the Gaussian process and the small ball probabilities of the Gaussian process. We determine these quantities for a range of examples of Gaussian priors and in several statistical settings. For instance, we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a (fractionally integrated) Brownian motion. We also consider regression with Gaussian errors and smooth classification under a logistic or probit link function combined with various priors."}
{"category": "Math", "title": "Codimension one generic homoclinic classes with interior", "abstract": "We study generic diffeomorphisms with a homoclinc class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive."}
{"category": "Math", "title": "A Filtration of (q,t)-Catalan numbers", "abstract": "We define new generalizations of (q,t)-Catalan numbers applying nabla operator on k-Schur functions indexed by column partitions. In some special cases, we give a combinatorial interpretation of these numbers using configurations of Dyck paths. In some other special cases, we also interpret these new polynomial as Hilbert series of explicit sub-modules of the alternating diagonal harmonics built using differential operators."}
{"category": "Math", "title": "Isoperimetry and Symmetrization for Sobolev spaces on metric spaces", "abstract": "Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, as well as metric versions of the P\\'{o}% lya-Szeg\\\"{o} and Faber-Krahn principles."}
{"category": "Math", "title": "Syzygies of the secant variety of a curve", "abstract": "We show that the secant variety of a linearly normal smooth curve of degree at least 2g+3 is arithmetically Cohen-Macaulay, and we use this information to study the graded Betti numbers of the secant variety."}
{"category": "Math", "title": "Higher Order Birkhoff Averages", "abstract": "There are well-known examples of dynamical systems for which the Birkhoff averages with respect to a given observable along some or all of the orbits do not converge. It has been suggested that such orbits could be classified using higher order averages. In the case of a bounded observable, we show that a classical result of G.H. Hardy implies that if the Birkhoff averages do not converge, then neither do the higher order averages. If the Birkhoff averages do not converge then we may denote by $[\\alpha_k,\\beta_k]$ the limit set of the $k$-th order averages. The sequence of intervals thus generated is nested: $[\\alpha_{k+1},\\beta_{k+1}] \\subset [\\alpha_k,\\beta_k]$. We can thus make a distinction among nonconvergent Birkhoff averages; either: B1. $\\cap_{k=1}^\\infty [\\alpha_k,\\beta_k]$ is a point $B_\\infty$, or, B2. $\\cap_{k=1}^\\infty [\\alpha_k,\\beta_k]$ is a non-trivial interval $[\\alpha_\\infty,\\beta_\\infty]$. We give characterizations of the types B1 and B2 in terms of how slowly they oscillate and we give examples that exhibit both behaviours B1 and B2 in the context of full shifts on finite symbols and \"Bowen's example\". For finite full shifts, we show that the set of orbits with type B2 behaviour has full topological entropy."}
{"category": "Math", "title": "The algebraic concordance order of a knot", "abstract": "Levine defined the rational algebraic knot concordance group and proved that each nontrivial element is of order two, of order four, or of infinite order. The determination of the order of an element depends on a p-adic analysis for all primes p. Here we develop effective means to determine the order of any element that is in the image of the integral algebraic concordance group by restricting the set of primes that need to be considered and by finding simple tests that often avoid p-adic considerations. The paper includes an outline of how the results apply to give the determination of the algebraic orders of all 2,977 prime knots of 12 or fewer crossings. The paper also includes a short expository account of the necessary background in p-adic numbers and Witt groups of bilinear forms."}
{"category": "Math", "title": "Graphs of bounded degree and the $p$-harmonic boundary", "abstract": "Let $p$ be a real number greater than one and let $G$ be a connected graph of bounded degree. In this paper we introduce the $p$-harmonic boundary of $G$. We use this boundary to characterize the graphs $G$ for which the constant functions are the only $p$-harmonic functions on $G$. It is shown that any continuous function on the $p$-harmonic boundary of $G$ can be extended to a function that is $p$-harmonic on $G$. Some properties of this boundary that are preserved under rough-isometries are also given. Now let $\\Gamma$ be a finitely generated group. As an application of our results we characterize the vanishing of the first reduced $\\ell^p$-cohomology of $\\Gamma$ in terms of the cardinality of its $p$-harmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on $\\Gamma$, the $p$-harmonic boundary of $\\Gamma$ with the first reduced $\\ell^p$-cohomology of $\\Gamma$."}
{"category": "Math", "title": "A limit-method for solving period problems on minimal surfaces", "abstract": "We introduce a new technique to solve period problems on minimal surfaces called limit-method. If a family of surfaces has Weierstrass-data converging to the data of a known example, and this presents a transversal solution of periods, then the original family contains a sub-family with closed periods."}
{"category": "Math", "title": "Triply periodic minimal surfaces which converge to the Hoffman-Wohlgemuth example", "abstract": "We get a continuous one-parameter new family of embedded minimal surfaces, of which the period problems are two-dimensional. Moreover, one proves that it has Scherk second surface and Hoffman-Wohlgemuth example as limit-members."}
{"category": "Math", "title": "On a question of Erdos and Ulam", "abstract": "Ulam asked in 1945 if there is an everywhere dense \\emph{rational set}, i.e. a point set in the plane with all its pairwise distances rational. Erd\\H os conjectured that if a set $S$ has a dense rational subset, then $S$ should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erd\\H os's conjecture for algebraic curves, by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set."}
{"category": "Math", "title": "Elliptic and parabolic second-order PDEs with growing coefficients", "abstract": "We consider a second-order parabolic equation in $\\bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally H\\\"older continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the $L_{\\infty}$-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients."}
{"category": "Math", "title": "The Bass and Topological Stable Ranks of $H^\\infty_\\R(\\D)$ and $A_\\R(\\D)$", "abstract": "In this note we prove that the Bass stable rank of $H^\\infty_\\R(\\D)$ is two. This establishes the validity of a conjecture by S. Treil. We accomplish this in two different ways, one by giving a direct proof, and the other, by first showing that the topological stable rank of $H^\\infty_\\R(\\D)$ is two. We apply these results to give new proofs of results by R. Rupp and A. Sasane stating that the Bass stable rank of $A_\\R(\\D)$ is two and the topological stable rank of $A_\\R(\\D)$ is two, settling a conjecture by the second author. We also present a $\\bar\\partial$-free proof of the second author's characterization of the reducible pairs in $A_\\R(\\D)$."}
{"category": "Math", "title": "Classification of singular Q-homology planes. I. Structure and singularities", "abstract": "A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient singularity then it is a quotient of an affine cone over a projective curve by an action of a finite group respecting the set of lines through the vertex. In particular, it is contractible, has negative Kodaira dimension and only one singular point. We describe minimal normal completions of such planes."}
{"category": "Math", "title": "Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal", "abstract": "We give a necessary and sufficient condition for an MV polytope $P$ in a highest weight crystal to lie in an arbitrary fixed Demazure crystal (resp., opposite Demazure crystal), in terms of the lengths of edges along a path through the 1-skeleton of $P$ corresponding to a reduced word for the longest element of the Weyl group $W$. % Also, we give an explicit description as a pseudo-Weyl polytope for extremal MV polytopes in a highest weight crystal. % Finally, by combining the results above, we obtain a polytopal condition for an MV polytope $P$ to lie in an arbitrary fixed opposite Demazure crystal."}
{"category": "Math", "title": "Classification and realizations of type III factor representations of Cuntz-Krieger algebras associated with quasi-free states", "abstract": "We completely classify type III factor representations of Cuntz-Krieger algebras associated with quasi-free states up to unitary equivalence. Furthermore, we realize these representations on concrete Hilbert spaces without using GNS construction. Free groups and their type ${\\rm II}_{1}$ factor representations are used in these realizations."}
{"category": "Math", "title": "Laws of the iterated logarithm for a class of iterated processes", "abstract": "Let $X=\\{X(t), t\\geq 0\\}$ be a Brownian motion or a spectrally negative stable process of index $1<\\a<2$. Let $E=\\{E(t),t\\geq 0\\}$ be the hitting time of a stable subordinator of index $0<\\beta<1$ independent of $X$. We use a connection between $X(E(t))$ and the stable subordinator of index $\\beta/\\a$ to derive information on the path behavior of $X(E_t)$. This is an extension of the connection of iterated Brownian motion and (1/4)-stable subordinator due to Bertoin \\cite{bertoin}. Using this connection, we obtain various laws of the iterated logarithm for $X(E(t))$. In particular, we establish law of the iterated logarithm for local time Brownian motion, $X(L(t))$, where $X$ is a Brownian motion (the case $\\a=2$) and $L(t)$ is the local time at zero of a stable process $Y$ of index $1<\\gamma\\leq 2$ independent of $X$. In this case $E(\\rho t)=L(t)$ with $\\beta=1-1/\\gamma$ for some constant $\\rho>0$. This establishes the lower bound in the law of the iterated logarithm which we could not prove with the techniques of our paper \\cite{MNX}. We also obtain exact small ball probability for $X(E_t)$ using ideas from \\cite{aurzada}."}
{"category": "Math", "title": "On the uniqueness of promotion operators on tensor products of type A crystals", "abstract": "The affine Dynkin diagram of type $A_n^{(1)}$ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type $A_n$ crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type $A_n$ crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara's conjecture that all `good' affine crystals are tensor products of Kirillov-Reshetikhin crystals."}
{"category": "Math", "title": "A comparison of formulations and solution methods for the Minimum-Envy Location Problem. Additional results", "abstract": "We consider a discrete facility location problem with a new form of equity criterion. The model discussed in the paper analyzes the case where demand points only have strict preference order on the sites where the plants can be located. The goal is to find the location of the facilities minimizing the total envy felt by the entire set of demand points. We define this new total envy criterion and provide several integer linear programming formulations that reflect and model this approach. Extensive computational tests are reported, showing the potentials and limits of each formulation on several types of instances."}
{"category": "Math", "title": "On De Giorgi Conjecture in Dimension $N \\geq 9$", "abstract": "A celebrated conjecture due to De Giorgi states that any bounded solution of the equation $\\Delta u + (1-u^2) u = 0 \\hbox{in} \\R^N $ with $\\pp_{y_N}u >0$ must be such that its level sets $\\{u=\\la\\}$ are all hyperplanes, {\\em \\bf at least} for dimension $N\\le 8$. A counterexample for $N\\ge 9$ has long been believed to exist. Based on a minimal graph $\\Gamma$ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in $\\R^N$, $N\\ge 9$, we prove that for any small $\\alpha >0$ there is a bounded solution $u_\\alpha(y)$ with $\\pp_{y_N}u_\\alpha >0$, which resembles $ \\tanh (\\frac t{\\sqrt{2}}) $, where $t=t(y)$ denotes a choice of signed distance to the blown-up minimal graph $\\Gamma_\\alpha := \\alpha^{-1}\\Gamma$. This solution constitutes a counterexample to De Giorgi conjecture for $N\\ge 9$."}
{"category": "Math", "title": "Energy scattering for 2D critical wave equation", "abstract": "We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schr\\\"odinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities can not control the nonlinear term uniformly on each time interval, but with constants depending on how much the solution is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and then to implement Bourgain's induction argument. We show the same result for the \"subcritical\" nonlinear Schr\\\"odinger equation."}
{"category": "Math", "title": "Computation of a Feynman integral", "abstract": "A Feymnan integral is computed exactly using LLL"}
{"category": "Math", "title": "Asymptotics of the number of partitions into p-cores and some trigonometric sums", "abstract": "An asymptotic formula for the number of partitions into p-cores is derived. As a byproduct some integer valued trigonometric sums are found"}
{"category": "Math", "title": "Lower Bounds for Boxicity", "abstract": "An axis-parallel $b$-dimensional box is a Cartesian product $R_1\\times R_2\\times...\\times R_b$ where $R_i$ is a closed interval of the form $[a_i,b_i]$ on the real line. For a graph $G$, its \\emph{boxicity} box(G) is the minimum dimension $b$, such that $G$ is representable as the intersection graph of boxes in $b$-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below: (1) The boxicity of a graph on $n$ vertices with no universal vertices and minimum degree $\\delta$ is at least $n/2(n-\\delta-1)$. (2) Consider the $\\mathcal{G}(n,p)$ model of random graphs. Let $ p \\le 1- \\frac{40 \\log n}{n^2}$. Then, for $G \\in \\mathcal{G}(n,p)$, almost surely $box(G)=\\Omega(np(1-p))$. On setting $p=1/2$ we immediately infer that almost all graphs have boxicity $\\Omega(n)$. (3) Spectral lower bounds for the boxicity of $k$-regular graphs. (4) The boxicity of random $k$-regular graphs on $n$ vertices (where $k$ is fixed) is $\\Omega(k/\\log k)$. (5) There exists a positive constant$c$ such that almost all balanced bipartite graphs on $2n$ vertices with exactly $m$ edges have boxicity at least $c m/n$, for $ m\\le c' n^2/3$ for any positive constant $c' < 1$."}
{"category": "Math", "title": "Directionally Convex Ordering of Random Measures, Shot Noise Fields and Some Applications to Wireless Communications", "abstract": "Directionally convex ($dcx$) ordering is a tool for comparison of dependence structure of random vectors that also takes into account the variability of the marginal distributions. When extended to random fields it concerns comparison of all finite dimensional distributions. Viewing locally finite measures as non-negative fields of measure-values indexed by the bounded Borel subsets of the space, in this paper we formulate and study the $dcx$ ordering of random measures on locally compact spaces. We show that the $dcx$ order is preserved under some of the natural operations considered on random measures and point processes, such as deterministic displacement of points, independent superposition and thinning as well as independent, identically distributed marking. Further operations such as position dependent marking and displacement of points though do not preserve the $dcx$ order on all point processes, are shown to preserve the order on Cox point processes. We also examine the impact of $dcx$ order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions, pair correlation functions as well as examples seem to indicate that point processes higher in $dcx$ order cluster more. As the main result, we show that non-negative integral shot-noise fields with respect to $dcx$ ordered random measures inherit this ordering from the measures. Numerous applications of this result are shown, in particular to comparison of various Cox processes and some performance measures of wireless networks, in both of which shot-noise fields appear as key ingredients. We also mention a few pertinent open questions."}
{"category": "Math", "title": "Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in $R^{2}$ with non--zero mean", "abstract": "Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with $EG_{1}G_{2}\\neq 0$. Let $c_{1},c_{2}\\in R^{1}$. A necessary and sufficient condition for $G=((G_{1}+c_{1}\\alpha)^{2},(G_{2}+c_{2}\\alpha)^{2})$ to be infinitely divisible for all $\\alpha\\in R^{1}$ is that \\[ \\Ga_{i,i}\\geq \\frac{c_{i}}{c_{j}}\\Ga_{i,j}>0\\qquad\\forall 1\\le i\\ne j\\le 2.\\] In this paper we show that when this does not hold there exists an $0<\\alpha_{0}<\\ff $ such that $G=((G_{1}+c_{1}\\alpha)^{2},(G_{2}+c_{2}\\alpha)^{2})$ is infinitely divisible for all $|\\alpha|\\leq \\alpha_{0}$ but not for any $|\\al|>\\al_{0}$."}
{"category": "Math", "title": "Strongly Fillable Contact Manifolds and J-holomorphic Foliations", "abstract": "We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of the 3-torus similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of the 3-torus are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured J-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on the cotangent bundle of the 2-torus is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result for contact manifolds with positive Giroux torsion."}
{"category": "Math", "title": "Renewal series and square-root boundaries for Bessel processes", "abstract": "We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian motion, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor."}
{"category": "Math", "title": "Holomorphic Functions of Exponential Type and Duality for Stein Groups with Algebraic Connected Component of Identity", "abstract": "We suggest a generalization of Pontryagin duality from the category of commutative Stein groups to the category of (not necessarily commutative) Stein groups with algebraic connected component of identity. In contrast to the other similar generalizations, in our approach the enveloping category consists of Hopf algebras (in a proper symmetrical monoidal category)."}
{"category": "Math", "title": "Shephard-Todd-Chevalley Theorem for skew polynomial rings", "abstract": "We prove the following generalization of the classical Shephard-Todd-Chevalley Theorem. Let $G$ be a finite group of graded algebra automorphisms of a skew polynomial ring $A:=k_{p_{ij}}[x_1,...,x_n]$. Then the fixed subring $A^G$ has finite global dimension if and only if $G$ is generated by quasi-reflections. In this case the fixed subring $A^G$ is isomorphic a skew polynomial ring with possibly different $p_{ij}$'s. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings."}
{"category": "Math", "title": "Hydrodynamic limit of gradient exclusion processes with conductances", "abstract": "Fix a strictly increasing right continuous with left limits function $W: \\bb R \\to \\bb R$ and a smooth function $\\Phi : [l,r] \\to \\bb R$, defined on some interval $[l,r]$ of $\\bb R$, such that $0<b \\le \\Phi'\\le b^{-1}$. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes, with conductances given by $W$, is described by the weak solutions of the non-linear differential equation $\\partial_t \\rho = (d/dx)(d/dW) \\Phi(\\rho)$. We derive some properties of the operator $(d/dx)(d/dW)$ and prove uniqueness of weak solutions of the previous non-linear differential equation."}
{"category": "Math", "title": "Sparse Regularization with $l^q$ Penalty Term", "abstract": "We consider the stable approximation of sparse solutions to non-linear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate $O(\\sqrt{\\delta})$ of the regularized solutions in dependence of the noise level $\\delta$. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to $O(\\delta)$."}
{"category": "Math", "title": "Knot Group Epimorphisms, II", "abstract": "We consider the relations $\\ge$ and $\\ge_p$ on the collection of all knots, where $k \\ge k'$ (respectively, $k \\ge_p k'$) if there exists an epimorphism $\\pi k \\to \\pi k'$ of knot groups (respectively, preserving peripheral systems). When $k$ is a torus knot, the relations coincide and $k'$ must also be a torus knot; we determine the knots $k'$ that can occur. If $k$ is a 2-bridge knot and $k \\ge_p k'$, then $k'$ is a 2-bridge knot with determinant a proper divisor of the determinant of $k$; only finitely many knots $k'$ are possible."}
{"category": "Math", "title": "Linear precision for toric surface patches", "abstract": "We classify the homogeneous polynomials in three variables whose toric polar linear system defines a Cremona transformation. This classification also includes, as a proper subset, the classification of toric surface patches from geometric modeling which have linear precision. Besides the well-known tensor product patches and B\\'ezier triangles, we identify a family of toric patches with trapezoidal shape, each of which has linear precision. B\\'ezier triangles and tensor product patches are special cases of trapezoidal patches."}
{"category": "Math", "title": "New Stability Conditions for Linear Differential Equations with Several Delays", "abstract": "New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation $$ \\dot{x}(t)+\\sum_{k=1}^m a_k(t)x(h_k(t))=0 $$ with measurable delays and coefficients. These results are compared to known stability tests."}
{"category": "Math", "title": "Non-perturbative approach to random walk in markovian environment", "abstract": "We prove an averaged CLT for a random walk in a dynamical environment where the states of the environment at different sites are independent Markov chains."}
{"category": "Math", "title": "Rational Witt classes of pretzel knots", "abstract": "In his pioneering work from 1969, Jerry Levine introduced a complete set of invariants of algebraic concordance of knots. The evaluation of these invariants requires a factorization of the Alexander polynomial of the knot, and is therefore in practice often hard to realize. We thus propose the study of an alternative set of invariants of algebraic concordance - the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the rational Witt classes of all pretzel knots. We give many examples and provide applications to obstructing sliceness for pretzel knots. We also obtain explicit formulae for the determinants and signatures of all pretzel knots. This article is dedicated to Jerry Levine and his lasting mathematical legacy; on the occasion of the conference \"Fifty years since Milnor and Fox\" held at Brandeis University on June 2-5, 2008."}
{"category": "Math", "title": "Maximum Likelihood Drift Estimation for Multiscale Diffusions", "abstract": "We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarse-grained equation for the slow variables can be rigorously derived, which we refer to as averaging and homogenization problems. We ask whether, given data from the slow variable in the fast/slow system, we can correctly estimate parameters in the drift of the coarse-grained equation for the slow variable, using maximum likelihood. We show that, whereas the maximum likelihood estimator is asymptotically unbiased for the averaging problem, for the homogenization problem maximum likelihood fails unless we subsample the data at an appropriate rate. An explicit formula for the asymptotic error in the log likelihood function is presented. Our theory is applied to two simple examples from molecular dynamics."}
{"category": "Math", "title": "Gale duality and Koszul duality", "abstract": "Given an affine hyperplane arrangement with some additional structure, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O."}
{"category": "Math", "title": "The Bloch-Okounkov correlation functions, a classical half-integral case", "abstract": "Bloch and Okounkov's correlation function on the infinite wedge space has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and certain character functions of $\\hgl_\\infty$-modules of level one. Recent works have calculated these character functions for higher levels for $\\hgl_\\infty$ and its Lie subalgebras of classical type. Here we obtain these functions for the subalgebra of type $D$ of half-integral levels and as a byproduct, obtain $q$-dimension formulas for integral modules of type $D$ at half-integral level."}
{"category": "Math", "title": "The cycle-convergence of restarted GMRES for normal matrices is sublinear", "abstract": "We prove that the cycle-convergence of the restarted GMRES applied to a system of linear equations with a normal coefficient matrix is sublinear."}
{"category": "Math", "title": "Envelope Algebras of Partial Actions as Groupoid C*-Algebras", "abstract": "We describe the envelope C*-algebra associated to a partial action of a countable discrete group on a locally compact space as a groupoid C*-algebra (more precisely as a C*-algebra from an equivalence relation) and we use our approach to show that, for a large class of partial actions of Z on the Cantor set, the envelope C*-algebra is an AF-algebra. We also completely characterize partial actions of a countable discrete group on a compact space such that the envelope action acts in a Hausdorff space."}
{"category": "Math", "title": "Sobre o papel dos Departamentos de Matem\\'atica na vida e desenvolvimento da comunidade", "abstract": "The objective of this article is to stimulate discussions in mathematical society about the role of mathematical departments in the life of the community. University community is the center of knowledge and promotes the intellectual development. However, this is questioned today because of its reduced participation on the global learning process as there are many other components."}
{"category": "Math", "title": "Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology", "abstract": "We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence."}
{"category": "Math", "title": "BART: Bayesian additive regression trees", "abstract": "We develop a Bayesian \"sum-of-trees\" model where each tree is constrained by a regularization prior to be a weak learner, and fitting and inference are accomplished via an iterative Bayesian backfitting MCMC algorithm that generates samples from a posterior. Effectively, BART is a nonparametric Bayesian regression approach which uses dimensionally adaptive random basis elements. Motivated by ensemble methods in general, and boosting algorithms in particular, BART is defined by a statistical model: a prior and a likelihood. This approach enables full posterior inference including point and interval estimates of the unknown regression function as well as the marginal effects of potential predictors. By keeping track of predictor inclusion frequencies, BART can also be used for model-free variable selection. BART's many features are illustrated with a bake-off against competing methods on 42 different data sets, with a simulation experiment and on a drug discovery classification problem."}
{"category": "Math", "title": "The Classification of Exceptional CDQL Webs on Compact Complex Surfaces", "abstract": "Codimension one webs are configurations of finitely many codimension one foliations in general position. Much of the classical theory evolved around the concept of abelian relation: a functional relation among the first integrals of the foliations defining the web reminiscent of Abel's addition theorem in classical algebraic geometry. The abelian relations of a given web form a finite dimensional vector space with dimension (the rank of the web) bounded by Castelnuovo number p(n,k) where n is the dimension of the ambient space and k is the number of foliations defining the web. A fundamental problem in web geometry is the classification of exceptional webs, that is, webs of maximal rank not equivalent to the dual of a projective curve. Recently, J.-M. Trepreau proved that there are no exceptional k-webs for n>2 and k > 2n-1. In dimension two there are examples of exceptional k-webs for arbitrary k and the classification problem is wide open. In this paper, we classify the exceptional Completely Decomposable Quasi-Linear (CDQL) webs globally defined on compact complex surfaces. By definition, the CDQL (k+1)-webs are formed by the superposition of k linear foliations and one non-linear foliation. For instance, we show that up to projective transformations there are exactly four countable families and thirteen sporadic exceptional CDQL webs on the projective plane."}
{"category": "Math", "title": "Jack polynomials and the coinvariant ring of $G(r,p,n)$", "abstract": "We study the coinvariant ring of the complex reflection group $G(r,p,n)$ as a module for the corresponding rational Cherednik algebra $\\HH$ and its generalized graded affine Hecke subalgebra $\\mathcal{H}$. We construct a basis consisting of non-symmetric Jack polynomials, and using this basis decompose the coinvariant ring into irreducible modules for $\\mathcal{H}$. The basis consists of certain non-symmetric Jack polynomials, whose leading terms are the ``descent monomials'' for $G(r,p,n)$ recently studied by Adin, Brenti, and Roichman and Bagno and Biagoli. The irreducible $\\mathcal{H}$-submodules of the coinvariant ring are their ``colored descent representations''."}
{"category": "Math", "title": "Mean representation number of integers as the sum of primes", "abstract": "Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best possible."}
{"category": "Math", "title": "F-injective singularities are Du Bois", "abstract": "In this paper, we prove that singularities of $F$-injective type are Du Bois. This extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of Frobenius in positive characteristic."}
{"category": "Math", "title": "Zariski-van Kampen method and transcendental lattices of certain singular K3 surfaces", "abstract": "We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular K3 surfaces associated with an arithmetic Zariski pair of maximizing sextics of type $A_{10}+A_{9}$ that are defined over a real quadratic field and are conjugate to each other over the field of rational numbers."}
{"category": "Math", "title": "The Volume of a Local Nodal Domain", "abstract": "Let M either be a closed real analytic Riemannian manifold or a closed smooth Riemannian surface. We estimate from below the volume of a nodal domain component in an arbitrary ball provided that this component enters the ball deeply enough."}
{"category": "Math", "title": "Minimal Surfaces in the Four-Dimensional Euclidean Space", "abstract": "We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for strongly regular minimal surfaces of general type in terms of their invariants. We introduce canonical parameters on strongly regular minimal surfaces of general type and prove that any such a surface is determined up to a motion by two invariant functions satisfying a system of two natural partial differential equations. On any minimal surface of the basic class of non strongly regular minimal surfaces we define canonical parameters and prove that any such a surface is determined up to a motion by two invariant functions of one variable satisfying a system of two natural ordinary differential equations. We find a geometric description of this class of non strongly regular minimal surfaces."}
{"category": "Math", "title": "Cyclic orders on the quantum grassmannian", "abstract": "The quantum grassmannian is known to be a graded quantum algebra with a straightening law when the poset of generating quantum minors is endowed with the standard partial ordering. In this paper it is shown that this result remains true when the ordering is subjected to cyclic shifts. The method involves proving that noncommutative dehomogenisation is possible at any consecutive quantum minor."}
{"category": "Math", "title": "Fibr\\'es vectoriels de rang deux sur $\\P^2$ provenant d'un rev\\^etement double", "abstract": "Since Schwarzenberger and his celebrated paper called \"Vector bundles on the projective plane\" we know that any rank two vector bundle on $\\P^2$ is a direct image of a line bundle on a double covering of the plane. This theorem suggests to study the rank two vector bundles according to the branch curve of the covering which they come from. Thus, in the first part we prove that, given a double covering ramified over an irreducible curve $C_{2r}$ with degree $2r$, the jumping lines of fixed order (order depending on $r$ and on the parity of the rank two vector bundle) of the direct images vector bundles are necessarely $r$-tangent to $C_{2r}$. In the second part we concentrate on the case $r=2$. Then we give a list of vector bundles for which the jumping lines are exactly the bitangent lines to the branch quartic."}
{"category": "Math", "title": "Approximation of Holder continuous homeomorphisms by piecewise affine homeomorphisms", "abstract": "This paper is concerned with the problem of approximating a homeomorphism by piecewise affine homeomorphisms. The main result is as follows: every homeomorphism from a planar domain with a polygonal boundary to R^2 that is globally Holder continuous of exponent \\alpha, and whose inverse is also globally Holder continuous of exponent \\alpha can be approximated in the Holder norm of exponent \\beta by piecewise affine homeomorphisms, for some \\beta that only depends on \\alpha. The proof is constructive. We adapt the proof of simplicial approximation in the supremum norm, and measure the side lengths and angles of the triangulation over which the approximating homeomorphism is piecewise affine. The approximation in the supremum norm, and a control on the minimum angle and on the ratio between the maximum and minimum side lengths of the triangulation suffice to obtain approximation in the Holder norm."}
{"category": "Math", "title": "Nonparametric adaptive estimation for pure jump L\\'evy processes", "abstract": "This paper is concerned with nonparametric estimation of the L\\'evy density of a pure jump L\\'evy process. The sample path is observed at $n$ discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the ${\\mathbb L}^2$-risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under additional assumptions on the L\\'evy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed."}
{"category": "Math", "title": "Low frequency estimates for long range perturbations in divergence form", "abstract": "We prove low frequency estimates for the boundary values of the resolvent of long range perturbations of the flat Laplacian in divergence form."}
{"category": "Math", "title": "Well-posedness of the spatially homogeneous Landau equation for soft potentials", "abstract": "We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderatley soft potentials, we assume additionnally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka."}
{"category": "Math", "title": "Chow motives without projectivity", "abstract": "In paper 0704.4003, Bondarko recently defined the notion of weight structure, and proved that the category $\\DgM$ of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander, is canonically equipped with such a structure. Building on this result, and under a condition on the weights avoided by the boundary motive, we describe a method to construct intrinsically in $\\DgM$ a motivic version of interior cohomology of smooth, but possibly non-projective schemes. In a sequel to this work, this method will be applied to Shimura varieties."}
{"category": "Math", "title": "Log-concavity and q-Log-convexity Conjectures on the Longest Increasing Subsequences of Permutations", "abstract": "Let $P_{n,k}$ be the number of permutations $\\pi$ on [n]={1, 2,..., n} such that the length of the longest increasing subsequences of $\\pi$ equals k, and let $M_{2n, k}$ be the number of matchings on [2n] with crossing number k. Define $P_n(x)= \\sum_k P_{n,k}x^k$ and $M_{2n}(x)=\\sum_{k} M_{2n,k}x^k$. We propose some conjectures on the log-concavity and q-log-convexity of the polynomials $P_n(x)$ and $M_{2n}(x)$. We also introduce the notions of $\\infty$-q-log-convexity and $\\infty$-q-log-concavity, and the notion of higher order log-concavity with respect to $\\infty$-q-log-convex or $\\infty$-q-log-concavity. A conjecture on the $\\infty$-q-log-convexity of the Boros-Moll polynomials is presented. It seems that $M_{2n}(x)$ are log-concave of any order with respect to $\\infty$-q-log-convexity."}
{"category": "Math", "title": "Calculating Effective Diffusivities in the Limit of Vanishing Molecular Diffusion", "abstract": "In this paper we study the problem of the numerical calculation (by Monte Carlo Methods) of the effective diffusivity for a particle moving in a periodic divergent-free velocity filed, in the limit of vanishing molecular diffusion. In this limit traditional numerical methods typically fail, since they do not represent accurately the geometry of the underlying deterministic dynamics. We propose a stochastic splitting method that takes into account the volume preserving property of the equations motion in the absence of noise, and when inertial effects can be neglected. An extension of the method is then proposed for the cases where the noise has a non trivial time-correlation structure and when inertial effects cannot be neglected. Modified equations are used to perform backward error analysis. The new stochastic geometric integrators are shown to outperform standard Euler-based integrators. Various asymptotic limits of physical interest are investigated by means of numerical experiments, using the new integrators."}
{"category": "Math", "title": "Decay estimates for variable coefficient wave equations in exterior domains", "abstract": "In this article we consider variable coefficient, time dependent wave equations in exterior domains. We prove localized energy estimates if the domain is star-shaped and global in time Strichartz estimates if the domain is strictly convex."}
{"category": "Math", "title": "N-th root", "abstract": "The quaternion equation X^n=A is solved for any integer number n > 1. A is a given quaternion with komplex numbers as its elements. We use the isomorphism between quaternions and (4,4)-matrices to solve this equation."}
{"category": "Math", "title": "Overpseudoprimes, Mersenne Numbers and Wieferich primes", "abstract": "We introduce a new class of pseudoprimes-so called \"overpseudoprimes\" which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime iff value of h(n) is invariant of all divisors d>1 of n. In particular, we prove that all composite Mersenne numbers 2^p-1,where p is prime, and squares of Wieferich primes are overpseudoprimes. We give also a generalization of the results on arbitrary base a>1 and prove that every overpseudoprime is strong pseudoprime of the same base."}
{"category": "Math", "title": "Norm Varieties and the Chain Lemma (after Markus Rost)", "abstract": "The goal of this paper is to present proofs of two results of Markus Rost: the Chain Lemma and the Norm Principle. These are the final steps needed to complete the publishable verification of the Bloch-Kato conjecture, that the norm residue maps are isomorphisms between Milnor K-theory $K_n^M(k)/p$ and etale cohomology $H^n(k,\\mu_p^n)$ for every prime p, every n and every field k containing 1/p. Our proofs of these two results are based on Rost's 1998 preprints, his web site and Rost's lectures at the Institute for Advanced Study in 1999-2000 and 2005."}
{"category": "Math", "title": "M\\'etodos de Multiresoluci\\'on y su Aplicaci\\'on a un Modelo de Ingenier\\'ia", "abstract": "The main objective of this dissertation is to present an adaptation of some finite volume methods used in the resolution of problems arising in sedimentation processes of flocculated suspensions (or sedimentation with compression). This adaptation is based on the utilization of multiresolution techniques, originally designed to reduce the computational cost incurred in solving using high resolution schemes in the numerical solution of hyperbolic systems of conservation laws."}
{"category": "Math", "title": "Multiresolution Schemes and its Application to Sedimentation Models", "abstract": "A numerical method is presented to obtain approximate solutions to problems arising from sedimentation models. These processes are widely utilized in minery for recovering water from suspensions coming out of flotation processes. The main idea is to apply a multiresolution method to the existing schemes developed by B\\\"urger et al. [2, 3, 4] and to observe the good performance of the multiresolution strategy when applied to these kind of problems. We obtain high rates of memory compression without affecting the quality of the solution."}
{"category": "Math", "title": "Toeplitz Corona Theorems for the Polydisk and the Unit Ball", "abstract": "The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar concerning the Corona problem for the polydisk and the unit ball in $\\mathbb{C}^n$."}
{"category": "Math", "title": "Minimal supporting subtrees for the free energy of polymers on disordered trees", "abstract": "We consider a model of directed polymers on a regular tree with a disorder given by independent, identically distributed weights attached to the vertices. For suitable weight distributions this model undergoes a phase transition with respect to its localization behaviour. We show that, for high temperatures, the free energy is supported by a random tree of positive exponential growth rate, which is strictly smaller than that of the full tree. The growth rate of the minimal supporting subtree is decreasing to zero as the temperature decreases to the critical value. At the critical value and all lower temperatures, a single polymer suffices to support the free energy. Our proofs rely on elegant martingale methods adapted from the theory of branching random walks."}
{"category": "Math", "title": "Boolean 2-designs and the embedding of a 2-design in a group", "abstract": "We try to embed a t-design in a finite commutative group in such a way that the sum of the k points of a block is zero. We can compute the number of blocks of the boolean 2-design having all the non zero vectors of $(Z_2)^n$ as the set of points and the k-subsets of elements the sum of which is zero as blocks."}
{"category": "Math", "title": "Stabilization of Three-Dimensional Collective Motion", "abstract": "This paper proposes a methodology to stabilize relative equilibria in a model of identical, steered particles moving in three-dimensional Euclidean space. Exploiting the Lie group structure of the resulting dynamical system, the stabilization problem is reduced to a consensus problem on the Lie algebra. The resulting equilibria correspond to parallel, circular and helical formations. We first derive the stabilizing control laws in the presence of all-to-all communication. Providing each agent with a consensus estimator, we then extend the results to a general setting that allows for unidirectional and time-varying communication topologies."}
{"category": "Math", "title": "Log minimal model program for the moduli space of stable curves: The first flip", "abstract": "We give a geometric invariant theory (GIT) construction of the log canonical model $\\bar M_g(\\alpha)$ of the pairs $(\\bar M_g, \\alpha \\delta)$ for $\\alpha \\in (7/10 - \\epsilon, 7/10]$ for small $\\epsilon \\in \\mathbb Q_+$. We show that $\\bar M_g(7/10)$ is isomorphic to the GIT quotient of the Chow variety bicanonical curves; $\\bar M_g(7/10-\\epsilon)$ is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions, e.g., they do not contain elliptic bridges. We show that there is a small contraction $\\Psi: \\bar M_g(7/10+\\epsilon) \\to \\bar M_g(7/10)$ that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction $\\Psi^+ : \\bar M_g(7/10-\\epsilon) \\to \\bar M_g(7/10)$ that is the Mori flip of $\\Psi$."}
{"category": "Math", "title": "Shape derivative of potential energy and energy release rate in fracture mechanics", "abstract": "We study general mathematical framework for variation of potential energy with respect to domain deformation. It enables rigorous derivation of the integral formulas for the energy release rate in crack problems. Applying a technique of the shape sensitivity analysis, we formulate the shape derivative of potential energy as a variational problem with a parameter. Key tools of our abstract theory are a new parameter variational principle and the classical implicit function theorem in Banach spaces."}
{"category": "Math", "title": "On The Existence of Globally Solvable Vector Fields in Smooth Manifolds", "abstract": "Let $(\\mathrm{M}, \\omega_{0})$ be a connected paracompact smooth oriented manifold. We establish a necessary and sufficient conditions on Lie subalgebra $\\mathfrak{a}$ of $\\mathrm{T M}$ such that its orbits becomes diffeomorphic to an open convex set of $\\mathbb{R}^{n}$, where $n$ is the orbit dimension."}
{"category": "Math", "title": "Conjugate Generators of Knot and Link Groups", "abstract": "This note shows that if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic. This includes the figure-eight knot and Whitehead link groups. Similarly, if two conjugate elements generate the trefoil knot group, then the elements are peripheral."}
{"category": "Math", "title": "$\\zeta$-phenomenology", "abstract": "It is well known that Euler experimentally discovered the functional equation of the Riemann zeta function. Indeed he detected the fundamental $s\\mapsto 1-s$ invariance of $\\zeta(s)$ by looking only at special values. In particular, via this functional equation, the permutation group on two letters, $S_2\\simeq\\Z/(2)$, is realized as a group of symmetries of $\\zeta(s)$. In this paper, we use the theory of special-values of our characteristic $p$ zeta functions to experimentally detect a natural symmetry group $S_{(q)}$ for these functions of cardinality ${\\mathfrak c}=2^{\\aleph_0}$ (where $\\mathfrak c$ is the cardinality of the continuum); $S_{(q)}$ is a realization of the permutation group on $\\{0,1,2...\\}$ as homeomorphisms of $\\Zp$ stabilizing both the nonpositive and nonnegative integers. We present a number of distinct instances in which $S_{(q)}$ acts (or appears to act) as symmetries of our functions. In particular, we present a natural, but highly mysterious, action of $S_{(q)}$ on a large subset of the domain of our functions that appears to stabilize zeta-zeroes. As of this writing, we do not yet know an overarching formalism that unifies these examples; however, it would seem that this formalism will involve an interplay between the 1-unit group $U_1$ -- playing the role of a \"gauge group\" -- and $S_{(q)}$. Furthermore, we show that $S_{(q)}$ may be naturally realized as an automorphism group of the convolution algebras of characteristic $p$ valued measures."}
{"category": "Math", "title": "The role of binomial type sequences in determination identities for Bell polynomials", "abstract": "Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and binomial type sequences in first part, and, we generalize the results obtained in [4] in second part."}
{"category": "Math", "title": "Rationality, irrationality, and Wilf equivalence in generalized factor order", "abstract": "Let $P$ be a partially ordered set and consider the free monoid $P^*$ of all words over $P$. If $w,w'\\in P^*$ then $w'$ is a factor of $w$ if there are words $u,v$ with $w=uw'v$. Define generalized factor order on $P^*$ by letting $u\\le w$ if there is a factor $w'$ of $w$ having the same length as $u$ such that $u\\le w'$, where the comparison of $u$ and $w'$ is done componentwise using the partial order in $P$. One obtains ordinary factor order by insisting that $u=w'$ or, equivalently, by taking $P$ to be an antichain. Given $u\\in P^*$, we prove that the language $\\cF(u)=\\{w : w\\ge u\\}$ is accepted by a finite state automaton. If $P$ is finite then it follows that the generating function $F(u)=\\sum_{w\\ge u} w$ is rational. This is an analogue of a theorem of Bj\\\"orner and Sagan for generalized subword order. We also consider $P=\\bbP$, the positive integers with the usual total order, so that $P^*$ is the set of compositions. In this case one obtains a weight generating function $F(u;t,x)$ by substituting $tx^n$ each time $n\\in\\bbP$ appears in $F(u)$. We show that this generating function is also rational by using the transfer-matrix method. Words $u,v$ are said to be Wilf equivalent if $F(u;t,x)=F(v;t,x)$ and we prove various Wilf equivalences combinatorially. Bj\\\"orner found a recursive formula for the M\\\"obius function of ordinary factor order on $P^*$. It follows that one always has $\\mu(u,w)=0,\\pm1$. Using the Pumping Lemma we show that the generating function $M(u)=\\sum_{w\\ge u} |\\mu(u,w)| w$ can be irrational."}
{"category": "Math", "title": "Highest weight categories arising from Khovanov's diagram algebra II: Koszulity", "abstract": "This is the second of a series of four articles studying various generalisations of Khovanov's diagram algebra. In this article we develop the general theory of Khovanov's diagrammatically defined \"projective functors\" in our setting. As an application, we give a direct proof of the fact that the quasi-hereditary covers of generalised Khovanov algebras are Koszul."}
{"category": "Math", "title": "The ancient Greeks present: Rational Trigonometry", "abstract": "Pythagoras' theorem, the area of a triangle as one half the base times the height, and Heron's formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, using quadrance not distance. This leads to a simpler and more elegant trigonometry, in which angle is replaced by spread, and which extends to arbitrary fields and more general quadratic forms."}
{"category": "Math", "title": "The twisted Floer homology of torus bundles", "abstract": "Given a torus bundle $Y$ over the circle and a cohomology class $[\\omega]\\in H^2(Y;\\mathbb{Z})$ which evaluates nontrivially on the fiber, we compute the Heegaard Floer homology of $Y$ with twisted coefficients in the universal Novikov ring."}
{"category": "Math", "title": "Hedlund-Metrics and the Stable Norm", "abstract": "The real homology of a compact Riemannian manifold $M$ is naturally endowed with the stable norm. The stable norm on $H_1(M,\\mathbb{R})$ arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space $H_1(M,\\mathbb{R})$ are stable norms of a Riemannian metric on $M$. If the dimension of $M$ is at least three, I. Babenko and F. Balacheff proved in \\cite{baba} that every polyhedral norm ball in $H_1(M,\\mathbb{R})$, whose vertices are rational with respect to the lattice of integer classes in $H_1(M,\\mathbb{R})$, is the stable norm ball of a Riemannian metric on $M$. This metric can even be chosen to be conformally equivalent to any given metric. The proof in \\cite{baba} uses singular Riemannian metrics on polyhedra which are finally smoothed. Here we present an alternative construction of such metrics which remains in the geometric framework of smooth Riemannian metrics."}
{"category": "Math", "title": "White-noise-aided Control", "abstract": "The issue of white-noise-aided control is considered and its availability is proved. And a noise-aiding way is developed to stabilize perturbed systems to be input-to-state stable (ISS) with respect to (w.r.t.) perturbations. To illustrate its effectiveness, the white-noise-aided control of a parameter perturbed chaotic Chen system is given as an example. And numerically, it shows that, comparing to the un-noise-aided case, noise-aided control can not only shorten the control's transient process but also save its cost. These are also demonstrated by various aiding noises such as common (symmetric) noise and non-common (independent or asymmetric) noise, where common noise is found to be the most efficient in enhancing the control."}
{"category": "Math", "title": "Unbounded representations of $q$-deformation of Cuntz algebra", "abstract": "We study a deformation of the Cuntz-Toeplitz $C^*$-algebra determined by the relations $a_i^*a_i=1+q a_ia_i^*, a_i^*a_j=0$. We define well-behaved unbounded *-representations of the *-algebra defined by relations above and classify all such irreducible representations up to unitary equivalence."}
{"category": "Math", "title": "Riemannian Geometry of Lie Algebroids", "abstract": "We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the integrability of Riemannian Lie algebroids."}
{"category": "Math", "title": "Matrix valued polynomials generated by the scalar-type Rodrigues' formulas", "abstract": "The properties of matrix valued polynomials generated by the scalar-type Rodrigues' formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are established, as well as, in the commutative case, the second order equations for which these polynomials are eigenfunctions and the corresponding eigenvalues, and ladder operators. The conjecture of Duran and Grunbaum that if the weights are self-adjoint and positive semidefinite then they are necessarily of scalar type is proved for Q(x)=x and Q(x)=x^2-1 in dimension two, and for any dimension under genericity assumptions. Commutative classes of quasi-orthogonal polynomials are found, which satisfy all the properties usually associated to orthogonal polynomials."}
{"category": "Math", "title": "Cohomology of GKM Fiber Bundles", "abstract": "The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties."}
{"category": "Math", "title": "Relaxed optimality conditions for mu-differentiable functions", "abstract": "We prove some fundamental properties of mu-differentiable functions. A new notion of local minimizer and maximizer is introduced and several extremum conditions are formulated using the language of nonstandard analysis."}
{"category": "Math", "title": "$C^*$-algebras associated with algebraic correspondences on the Riemann sphere", "abstract": "Let $p(z,w)$ be a polynomial in two variables. We call the solution of the algebraic equation $p(z,w) = 0$ the algebraic correspondence. We regard it as the graph of the multivalued function $z \\mapsto w$ defined implicitly by $p(z,w) = 0$. Algebraic correspondences on the Riemann sphere $\\hat{\\mathbb C}$ give a generalization of dynamical systems of Klein groups and rational functions. We introduce $C^*$-algebras associated with algebraic correspondences on the Riemann sphere. We show that if an algebraic correspondence is free and expansive on a closed $p$-invariant subset $J$ of $\\hat{\\mathbb C}$, then the associated $C^*$-algebra ${\\mathcal O}_p(J)$ is simple and purely infinite."}
{"category": "Math", "title": "Time-dependent Schr\\\"odinger perturbations of transition densities", "abstract": "We construct the fundamental solution of $\\partial_t-\\Delta_y- q(t,y)$, for functions $q$ with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces. We also discuss specific applications to Schr\\\"odinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution corresponding to the operator."}
{"category": "Math", "title": "A mu-differentiable Lagrange multiplier rule", "abstract": "We present some properties of the gradient of a mu-differentiable function. The Method of Lagrange Multipliers for mu-differentiable functions is then exemplified."}
{"category": "Math", "title": "Normal numbers from Steinhaus viewpoint", "abstract": "In this paper we recall a non-standard construction of the Borel sigma-algebra B in [0,1] and construct a family of measures (in particular, Lebesgue measure) in B by a completely non-topological method. This approach, that goes back to Steinhaus, in 1923, is now used to introduce natural generalizations of the concept of normal numbers and explore their intrinsic probabilistic properties. We show that, in virtually all the cases, almost all real number in [0,1] is normal (with respect to this generalized concept). This procedure highlights some apparently hidden but interesting features of the Borel sigma-algebra and Lebesgue measure in [0,1]."}
{"category": "Math", "title": "The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector", "abstract": "We classify irreducible $\\sigma$-twisted modules for the N=1 super triplet vertex operator superalgebra $\\mathcal{SW}(m)$ introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of $\\sigma$-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the $SL(2,\\mathbb{Z})$-closure of the space spanned by irreducible characters, irreducible supercharacters and $\\sigma$-twisted irreducible characters is $(9m+3)$-dimensional. We present strong evidence that this is also the (full) space of generalized characters for $\\mathcal{SW}(m)$. We are also able to relate irreducible $\\mathcal{SW}(m)$ characters to characters for the triplet vertex algebra $\\mathcal{W}(2m+1)$, studied in [Adamovic D., Milas A., Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857]."}
{"category": "Math", "title": "Stochastic relations of random variables and processes", "abstract": "This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov processes, it suffices that the generators of the processes preserve some, not necessarily reflexive or transitive, subrelation of the order relation. The main contributions of the paper are: a functional characterization of stochastic relations, necessary and sufficient conditions for the preservation of stochastic relations, and an algorithm for finding subrelations preserved by probability kernels. The theory is illustrated with applications to hidden Markov processes, population processes, and queueing systems."}
{"category": "Math", "title": "Espace des modules de certains polyedres projectifs miroirs", "abstract": "A projective mirror polyhedron is a projective polyhedron endowed with reflections across its faces. We construct an explicit diffeomorphism between the moduli space of a mirror projective polyhedron with fixed dihedral angles in $(0,\\frac{\\pi}{2}]$, and the union of $n$ copies of $\\R^d$, when the polyhedron has the combinatorics of an \\emph{ecimahedron}, an infinite class of combinatorial polyhedra we introduce here. Moreover, the integers $n$ and $d$ can be computed explicitly in terms of the combinatorics and the fixed dihedral angles."}
{"category": "Math", "title": "Secure two-dimensional tori are flat", "abstract": "A riemannian manifold is secure if the geodesics between any pair of points in the manifold can be blocked by a finite number of point obstacles. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. The conjecture claims, in particular, that a riemannian torus of any dimension is secure if and only if it is flat. We prove this for two-dimensional tori."}
{"category": "Math", "title": "Realisation of cycles by aspherical manifolds", "abstract": "We develop a new purely combinatorial approach to N. Steenrod's problem on realisation of cycles. We prove that every n-dimensional homology class of every topological space can be realised with some multiplicity by an image of a finite-fold covering over the manifold M^n, where M^n is the isospectral manifold of real symmetric tridiagonal (n=1)x(n+1) matrices. In particular, every homology class of every arcwise connected topological space can be realised by a continuous image of an aspherical manifold."}
{"category": "Math", "title": "Carries, Shuffling and An Amazing Matrix", "abstract": "The number of ``carries'' when $n$ random integers are added forms a Markov chain [23]. We show that this Markov chain has the same transition matrix as the descent process when a deck of $n$ cards is repeatedly riffle shuffled. This gives new results for the statistics of carries and shuffling."}
{"category": "Math", "title": "Improved Bounds on the Sizes of S.P Numbers", "abstract": "A number which is S.P in base r is a positive integer which is equal to the sum of its base-r digits multiplied by the product of its base-r digits. These numbers have been studied extensively in The Mathematical Gazette. Recently, Shah Ali obtained the first effective bound on the sizes of S.P numbers. Modifying Shah Ali's method, we obtain an improved bound on the number of digits in a base-r S.P number. Our bound is the first sharp bound found for the case r=2."}
{"category": "Math", "title": "The structural equations of Cartan and the wave solution Einstein's equation", "abstract": "Work consists of introduction, two chapters, conclusion and four applications. In this work is examined the condition, with which the wave space metrics of Riemann- Cartan is the solution of Einstein equation in the void. Geometric structures were for this purpose studied on the differentiated variety: connectedness, curvature and twisting connectedness. For this was used this analytical apparatus of differential geometry as the calculation of exterior forms. In the first chapter the following concepts are examined: - variety; - vectors and 1- forms on the varieties; - metric tensor, connectedness and the covariant derivative; - form with the values in the vector spaces; - the structural equations of Cartan. The second chapter is dedicated to the presence of condition, during which wave certificate is the solution of Einstein equation in the void. Using the first and second structural equations of Cartan for the variety without twisting of that allotted by wave certificate, they were calculated: the 1- forms of connectedness and the coefficients of connectedness, and also Riemann tensor, Ricci tensor and scalar of curvature. In appendix 1 the derivation of formula for the coefficients of connected-ness in the space with the twisting is given. In appendix 2 the calculation of the components of the 1- forms of connectedness is carried out. In appendix 3 the calculations of the components of the 2- forms of curvature are presented. In appendix 4 the components of Ricci tensor are calculated."}
{"category": "Math", "title": "Equivariant structure constants for ordinary and weighted projective space", "abstract": "We compute the integral torus-equivariant cohomology ring for weighted projective space for two different torus actions by embedding the cohomology in a sum of polynomial rings $\\oplus_{i=0}^n \\Z[t_1, t_2,..., t_n]$. One torus action gives a result complementing that of Bahri, Franz, and Ray. For the other torus action, each basis class for weighted projective space is a multiple of the basis class for ordinary projective space; we identify each multiple explicitly. We also give a simple formula for the structure constants of the equivariant cohomology ring of ordinary projective space in terms of the basis of Schubert classes, as a sequence of divided difference operators applied to a specific polynomial."}
{"category": "Math", "title": "Characteristic classes for Riemannian foliations", "abstract": "The purpose of this paper is to both survey and offer some new results on the non-triviality of the characteristic classes of Riemannian foliations. We give examples where the primary Pontrjagin classes are all linearly independent. The independence of the secondary classes is also discussed, along with their total variation. Finally, we give a negative solution of a conjecture that the map of classifying spaces $\\FRGq \\to \\FGq$ is trivial for codimension $q > 1$."}
{"category": "Math", "title": "Hypergeometric formulas for lattice sums and Mahler measures", "abstract": "We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd's conjectured identities between Mahler measures and special values of $L$-series of elliptic curves."}
{"category": "Math", "title": "Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class", "abstract": "Using the harmonic map heat flow, we construct an energy class for wave maps $\\phi$ from two-dimensional Minkowski space $\\R^{1+2}$ to hyperbolic spaces $\\H^m$, and then show (conditionally on a large data well-posedness claim for such wave maps) that no stationary, travelling, self-similar, or degenerate wave maps exist in this energy class. These results form three of the five claims required in our earlier paper (arXiv:0805.4666) to prove global regularity for such wave maps. (The conditional claim of large data well-posedness is one of the remaining claims required in that paper.)"}
{"category": "Math", "title": "A Graph-Theoretic Approach to a Partial Order of Knots and Links", "abstract": "We say that a link $L_1$ is an s-major of a link $L_2$ if any diagram of $L_1$ can be transformed into a diagram of $L_2$ by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime alternating links. We determine this partial order for all prime alternating knots and links with the crossing number less than or equal to six. The proofs are given by graph-theoretic methods."}
{"category": "Math", "title": "Stabilization under shared communication with message losses and its limitations", "abstract": "We consider a synthesis problem for a remotely controlled linear system where the communication is constrained because of the shared and unreliable nature of the channel. Modeling the constraints by a periodic transmission scheme and random message losses, we present an H-infinity design framework and study the limitations in the communication required for stabilization."}
{"category": "Math", "title": "A Quiver Construction of Symmetric Crystals", "abstract": "In the recent papers with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type conjectures for the affine Hecke algebras of type $B$. Namely, we conjectured that certain composition multiplicities and branching rules for the affine Hecke algebras of type $B$ are described by using the lower global basis of symmetric crystals of $V_\\theta(\\lambda)$. In this paper, we prove the existence of crystal bases and global bases of $V_\\theta(0)$ for any symmetric quantized Kac-Moody algebra by using a geometry of quivers (with a Dynkin diagram involution). This is analogous to George Lusztig's geometric construction of $U_v^-$ and its lower global basis."}
{"category": "Math", "title": "Chromogeometry", "abstract": "Chromogeometry brings together Euclidean geometry (called blue) and two relativistic geometries (called red and green), in a surprising three-fold symmetry. We show how the red and green `Euler lines' and `nine-point circles' of a triangle interact with the usual blue ones, and how the three orthocenters form an associated triangle with interesting collinearities. This is developed in the framework of rational trigonometry using quadrance and spread instead of distance and angle. The former are more suitable for relativistic geometries."}
{"category": "Math", "title": "Divisor and Totient Functions Estimates", "abstract": "New unconditional estimates of the divisor and totient functions are contributed to the literature. These results are consistent with the Riemann hypothesis and seem to solve the Nicolas inequality for all sufficiently large integers."}
{"category": "Math", "title": "A noncommutative extended de Finetti theorem", "abstract": "The extended de Finetti theorem characterizes exchangeable infinite random sequences as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem. In contrast to the classical result of Ryll-Nadzewski, exchangeability turns out to be stronger than spreadability for infinite noncommutative random sequences. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of commuting squares and K\\\"ummerer's generalized Bernoulli shifts. Our main result is applicable to classical probability, quantum probability, in particular free probability, braid group representations and Jones subfactors."}
{"category": "Math", "title": "On Lehner's `free' noncommutative analogue of De Finetti's theorem", "abstract": "Inspired by Lehner's results on exchangeability systems we define `weak conditional freeness' and `conditional freeness' for stationary processes in an operator algebraic framework of noncommutative probability. We show that these two properties are equivalent and thus the process embeds into a von Neumann algebraic amalgamated free product over the fixed point algebra of the stationary process."}
{"category": "Math", "title": "A reduction method for noncommutative $L_p$-spaces and applications", "abstract": "We consider the reduction of problems on general noncommutative $L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative $L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent general noncommutative martingale/ergodic inequalities to those in the tracial case."}
{"category": "Math", "title": "White Noise Calculus and Hamiltonian of a Quantum Stochastic Process", "abstract": "A white noise quantum stochastic calculus is developped using classical measure theory as mathematical tool. Wick's and Ito's theorems have been established. The simplest quantum stochastic differential equation has been solved, unicity and the conditions for unitarity have been proven. The Hamiltonian of the associated one parameter strongly continuous group has been calculated explicitely."}
{"category": "Math", "title": "Recurrence Relations for Strongly q-Log-Convex Polynomials", "abstract": "We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity."}
{"category": "Math", "title": "Inverse zero-sum problems and algebraic invariants", "abstract": "In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds true for finite cyclic groups, finite Abelian p-groups and for finite Abelian groups of rank two. Also, the results obtained here enable us to improve, via the resolution of a linear integer program, a result of W. Gao and A. Geroldinger concerning the minimal number of elements with maximal order in a long zero-sumfree sequence of a finite Abelian group of rank two."}
{"category": "Math", "title": "The Slice Algorithm For Irreducible Decomposition of Monomial Ideals", "abstract": "Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice."}
{"category": "Math", "title": "Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions", "abstract": "We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products $\\Gamma\\wr\\SG_n$ and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon's multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B."}
{"category": "Math", "title": "On the total curvature of semialgebraic graphs", "abstract": "We define the total curvature of a semialgebraic embedding of a graph in the 3-dimensional Euclidean space. We prove that it satisfies a Chern-Lashof type inequality and we describe when the equality holds. We also prove a generalization of a classical result of Fary and Milnor stating that certain graphs cannot be knotted if they are not too curved. The techniques employed are Morse theoretic."}
{"category": "Math", "title": "Estimation of a diffusion model with trends taking in account the extremes. Application to temperature in France", "abstract": "We built a model of the daily temperature based on a diffusion process and address to extreme values not taken into account in the literature on this kind of models. We first study, using non parametric tools, the trends on mean and variance. In a second step we estimate a stationary model first non parametrically and then using likelihood methods. Extreme values are taken into account in the estimation of model and to obtain a definitive estimation we use in a specific framework extreme theory for diffusions. A test of suitable model by simulation is done."}
{"category": "Math", "title": "A matrix interpolation between classical and free max operations: I. The univariate case", "abstract": "Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings."}
{"category": "Math", "title": "Quantum Group of Orientation preserving Riemannian Isometries", "abstract": "We formulate a quantum group analogue of the group of orinetation-preserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly $R$-twisted in the sense of a paper of one of the authors, and of compact type) spectral triple. The main advantage of this formulation, which is directly in terms of the Dirac operator, is that it does not need the existence of any `good ' Laplacian as in our previous works on quantum isometry groups. Several interesting examples, including those coming from Rieffel-type deformation as well as the equivariant spectral triples on $SU_\\mu(2)$ and $S^2_{\\mu 0}$ are dicussed."}
{"category": "Math", "title": "Noncommutative independence from the braid group $B_\\infty$", "abstract": "We introduce `braidability' as a new symmetry for (infinite) sequences of noncommutative random variables related to representations of the braid group $B_\\infty$. It provides an extension of exchangeability which is tied to the symmetric group $S_\\infty$. Our key result is that braidability implies spreadability and thus conditional independence, according to the noncommutative extended de Finetti theorem (of C. K\\\"{o}stler). This endows the braid groups $B_n$ with a new intrinsic (quantum) probabilistic interpretation. We underline this interpretation by a braided extension of the Hewitt-Savage Zero-One Law. Furthermore we use the concept of product representations of endomorphisms (of R. Gohm) with respect to certain Galois type towers of fixed point algebras to show that braidability produces triangular towers of commuting squares and noncommutative Bernoulli shifts. As a specific case we study the left regular representation of $B_\\infty$ and the irreducible subfactor with infinite Jones index in the non-hyperfinite $II_1$-factor $L(B_\\infty)$ related to it. Our investigations reveal a new presentation of the braid group $B_\\infty$, the `square root of free generator presentation' $F_\\infty^{1/2}$. These new generators give rise to braidability while the squares of them yield a free family. Hence our results provide another facet of the strong connection between subfactors and free probability theory and we speculate about braidability as an extension of (amalgamated) freeness on the combinatorial level."}
{"category": "Math", "title": "On the Briancon-Skoda theorem on a singular variety", "abstract": "Let $Z$ be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform Briancon-Skoda theorem for the local ring $\\mathcal{O}_Z$; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results."}
{"category": "Math", "title": "Optimal time and space regularity for solutions of degenerate differential equations", "abstract": "We derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time."}
{"category": "Math", "title": "Integral representation of renormalized self-intersection local times", "abstract": "In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the $d$% -dimensional fractional Brownian motion with Hurst parameter $H\\in (0,1)$. As a consequence, we derive the existence of some exponential moments for this random variable."}
{"category": "Math", "title": "On arithmetic partitions of Z_n", "abstract": "Generalizing a classical problem in enumerative combinatorics, Mansour and Sun counted the number of subsets of $\\Z_n$ without certain separations. Chen, Wang, and Zhang then studied the problem of partitioning $\\Z_n$ into arithmetical progressions of a given type under some technical conditions. In this paper, we improve on their main theorems by applying a convolution formula for cyclic multinomial coefficients due to Raney-Mohanty."}
{"category": "Math", "title": "Principal bundles, quasi-abelian varieties and structure of algebraic groups", "abstract": "We classify principal bundles over anti-affine schemes with affine and commutative structural group. We show that this yields the classification of quasi-abelian varieties over a field k (i.e., group k-schemes with no non constant global functions). The interest of this result is given by the fact that the classification of smooth group k-schemes is reduced to the classification of quasi-abelian varieties and of certain affine group schemes."}
{"category": "Math", "title": "Distinguished Orbits of Reductive Groups", "abstract": "We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group $G$ acting linearly and rationally on a real vector space $V$. $G$ can be viewed as the real points of a complex reductive group $G^\\mathbb C$ which acts on $V^\\mathbb C := V \\otimes \\mathbb C$. Borel-Harish-Chandra show that $G^\\mathbb C \\cdot v \\cap V$ is a finite union of $G$-orbits; moreover, $G^\\mathbb C \\cdot v$ is closed if and only if $G\\cdot v$ is closed. We show that the same result holds not just for closed orbits but for the so-called distinguished orbits. An orbit is called distinguished if it contains a critical point of the norm squared of the moment map on projective space. Our main result compares the complex and real settings to show $G\\cdot v$ is distinguished if and only if $G^\\mathbb C \\cdot v$ is distinguished. In addition, we show that if an orbit is distinguished, then under the negative gradient flow of the norm squared of the moment map the entire $G$-orbit collapses to a single $K$-orbit. This result holds in both the complex and real settings. We finish with some applications to the study of the left-invariant geometry of Lie groups."}
{"category": "Math", "title": "On the difference between solutions of discrete tomography problems", "abstract": "We consider the problem of reconstructing binary images from their horizontal and vertical projections. We present a condition that the projections must necessarily satisfy when there exist two disjoint reconstructions from those projections. More generally, we derive an upper bound on the symmetric difference of two reconstructions from the same projections. We also consider two reconstructions from two different sets of projections and prove an upper bound on the symmetric difference in this case."}
{"category": "Math", "title": "Ricci-flat K\\\"ahler metrics on crepant resolutions of K\\\"ahler cones", "abstract": "We prove that a crepant resolution of a Ricci-flat K\\\"ahler cone X admits a complete Ricci-flat K\\\"ahler metric asymptotic to the cone metric in every K\\\"ahler class in H^2_c(Y,R). This result contains as a subcase the existence of ALE Ricci-flat K\\\"ahler metrics on crepant resolutions of X=C^n /G, where G is a finite subgroup of SL(n,C). We consider the case in which X is toric. A result of A. Futaki, H. Ono, and G. Wang guarantees the existence of a Ricci-flat K\\\"ahler cone metric if X is Gorenstein. We use toric geometry to construct crepant resolutions."}
{"category": "Math", "title": "An invariant of embeddings of 3-manifolds in 6-manifolds and Milnor's triple linking number", "abstract": "We give a simple axiomatic definition of a rational-valued invariant s(W,V,e) of triples (W,V,e), where W is a (smooth, oriented, closed) 6-manifold and V is a 3-submanifold of W, and where e is a second rational cohomology class of the complement of V satisfying a certain condition. The definition is stated in terms of cobordisms of such triples and the signature of 4-manifolds. When W = S^6 and V is a smoothly embedded 3-sphere, and when e/2 is the Poincare dual of a Seifert surface of V, the invariant coincides with -8 times Haefliger's embedding invariant of (S^6,V). Our definition recovers a more general invariant due to Takase, and contains a new definition for Milnor's triple linking number of algebraically split 3-component links in R^3 that is close to the one given by the perturbative series expansion of the Chern-Simons theory of links in R^3."}
{"category": "Math", "title": "A sharp bound for the reconstruction of partitions", "abstract": "Answering a question of Cameron, Pretzel and Siemons proved that every integer partition of $n\\ge 2(k+3)(k+1)$ can be reconstructed from its set of $k$-deletions. We describe a new reconstruction algorithm that lowers this bound to $n\\ge k^2+2k$ and present examples showing that this bound is best possible."}
{"category": "Math", "title": "Cohomology and Support Varieties for Lie Superalgebras of Type W(n)", "abstract": "Boe, Kujawa and Nakano recently investigated relative cohomology for classical Lie superalgebras and developed a theory of support varieties. The dimensions of these support varieties give a geometric interpretation of the combinatorial notions of defect and atypicality due to Kac, Wakimoto, and Serganova. In this paper we calculate the cohomology ring of the Cartan type Lie superalgebra W(n) relative to the degree zero component W(n)_{0} and show that this ring is a finitely generated polynomial ring. This allows one to define support varieties for finite dimensional W(n)-supermodules which are completely reducible over W(n)_{0}. We calculate the support varieties of all simple supermodules in this category. Remarkably our computations coincide with the prior notion of atypicality for Cartan type superalgebras due to Serganova. We also present new results on the realizability of support varieties which hold for both classical and Cartan type superalgebras."}
{"category": "Math", "title": "Regularity conditions for arbitrary Leavitt path algebras", "abstract": "We show that if $E$ is an arbitrary acyclic graph then the Leavitt path algebra $L_K(E)$ is locally $K$-matricial; that is, $L_K(E)$ is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field $K$. As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph $E$: (1) $L_K(E)$ is von Neumann regular. (2) $L_K(E)$ is $\\pi$-regular. (3) $E$ is acyclic. (4) $L_K(E)$ is locally $K$-matricial. (5) $L_K(E)$ is strongly $\\pi$-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions."}
{"category": "Math", "title": "Improved testing inference in mixed linear models", "abstract": "Mixed linear models are commonly used in repeated measures studies. They account for the dependence amongst observations obtained from the same experimental unit. Oftentimes, the number of observations is small, and it is thus important to use inference strategies that incorporate small sample corrections. In this paper, we develop modified versions of the likelihood ratio test for fixed effects inference in mixed linear models. In particular, we derive a Bartlett correction to such a test and also to a test obtained from a modified profile likelihood function. Our results generalize those in Zucker et al. (Journal of the Royal Statistical Society B, 2000, 62, 827-838) by allowing the parameter of interest to be vector-valued. Additionally, our Bartlett corrections allow for random effects nonlinear covariance matrix structure. We report numerical evidence which shows that the proposed tests display superior finite sample behavior relative to the standard likelihood ratio test. An application is also presented and discussed."}
{"category": "Math", "title": "Survey on geometric group theory", "abstract": "This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research. The sections are: classical examples, basics about quasiisometry,properties and invariants of groups invariant under quasiisometry, rigidity, hyperbolic spaces and CAT(k)-spaces, the boundary of a hyperbolic space, hyperbolic groups, CAT(0)-groups, classifying spaces for proper actions, measurable group theory, some open problems."}
{"category": "Math", "title": "Observations regarding compactness in the $\\overline{\\partial}$-Neumann problem", "abstract": "We show that compactness of the $\\overline{\\partial}$-Neumann operator is independent of the metric, and we give a new proof of this independence for subellipticity. We define an abstract obstruction to compactness, namely the common zero set of all the compactness multipliers, and we identify this subset of the boundary for convex domains in $\\mathbb{C}^{n}$ and for complete Hartogs domains in $\\mathbb{C}^{2}$."}
{"category": "Math", "title": "Convexity in semi-algebraic geometry and polynomial optimization", "abstract": "We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide a certificate of convexity if a sufficient (and almost necessary) condition is satified. This condition can be checked numerically and also provides a new condition for K to have semidefinite representation. For this we use (and extend) some of recent results from the author and Helton and Nie. Finally, we show that when restricting to a certain class of convex polynomials, the celebrated Jensen's inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures."}
{"category": "Math", "title": "Prescription of Q-curvature on closed Riemannian manifolds", "abstract": "In this paper it is hown that given any smooth, positive function f on a closed, smooth manifold of dimension greater than four and with positive Paneitz invariant, there exists a metric on M such that $Q_g$ = f."}
{"category": "Math", "title": "Connected sums of closed Riemannian manifolds and fourth order conformal invariants", "abstract": "In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities change, analogous to how the Yamabe constants and the Yamabe invariants do, under the connected sum operations."}
{"category": "Math", "title": "A structure theorem of Dirac-harmonic maps between spheres", "abstract": "For an arbitrary Dirac-harmonic map $(\\phi,\\psi)$ between compact oriented Riemannian surfaces, we shall study the zeros of $|\\psi|$. With the aid of Bochner-type formulas, we explore the relationship between the order of the zeros of $|\\psi|$ and the genus of $M$ and $N$. On the basis, we could clarify all of nontrivial Dirac-harmonic maps from $S^2$ to $S^2$."}
{"category": "Math", "title": "Multiple fibers of del Pezzo fibrations", "abstract": "We prove that a terminal three-dimensional del Pezzo fibration has no fibers of multiplicity $\\ge 6$. We also obtain a rough classification possible configurations of singular points on multiple fibers and give some examples."}
{"category": "Math", "title": "Gap conjecture for 3-dimensional canonical thresholds", "abstract": "We prove that the interval $(5/6, 1)$ contains no 3-dimensional canonical thresholds."}
{"category": "Math", "title": "A Generalization of Fulton-MacPherson Configuration Spaces", "abstract": "Presented is a wonderful compactification of n distinct labeled points in X away from D, where X is a nonsingular variety and D is a nonsingular proper subvariety. When D is empty, it is the Fulton-MacPherson configuration space."}
{"category": "Math", "title": "Cohomogeneity one disk bundles with normal homogeneous collars", "abstract": "We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in $\\{2,3,4,6,8\\}$. Moreover, we give a complete classification of such bundles of rank 6 and 8, and a partial classification for rank 3."}
{"category": "Math", "title": "A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics", "abstract": "Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection $D$ is introduced such that the structure of these manifolds is parallel with respect to D. Of special interest is the class of the locally conformally equivalent manifolds of the manifolds with covariantly constant almost complex structures and the case when the torsion of D is D-parallel. Curvature properties of these manifolds are studied. An example of 4-dimensional manifolds in the considered basic class is constructed and characterized."}
{"category": "Math", "title": "Approximation of conformal mappings by circle patterns", "abstract": "A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in $(0,\\pi)$. Two sequences of circle patterns are employed to approximate a given conformal map $g$ and its first derivative. For the domain of $g$ we use embedded circle patterns where all circles have the same radius decreasing to 0 and which have uniformly bounded intersection angles. The image circle patterns have the same combinatorics and intersection angles and are determined from boundary conditions (radii or angles) according to the values of $g'$ ($|g'|$ or $\\arg g'$). For quasicrystallic circle patterns the convergence result is strengthened to $C^\\infty$-convergence on compact subsets."}
{"category": "Math", "title": "On Pseudo Algebraically Closed Extensions of Fields", "abstract": "The notion of `Pseudo Algebraically Closed (PAC) extensions' is a generalization of the classical notion of PAC fields. It was originally motivated by Hilbert's tenth problem, and recently had new applications. In this work we develop a basic machinery to study PAC extensions. This machinery is based on a generalization of embedding problems to field extensions. The main goal is to prove that the Galois closure of any proper separable algebraic PAC extension is its separable closure. This vastly generalizes earlier works of Jarden-Razon, Jarden, and Jarden and the author. This also leads to a classification of all finite PAC extensions which in turn proves the `bottom conjecture' for finitely generated infinite fields. The secondary goal of this work is to unify proofs of known results about PAC extensions and to establish new basic properties of PAC extensions, e.g.\\ transitiveness of PAC extensions."}
{"category": "Math", "title": "On reduction integer programs to knapsack problem", "abstract": "Let $A$ be an integral nonnegative $m\\times n$ matrix, $b$ be an integral nonnegative vector. It is suggested new method for reduction of integer program $\\max \\{cx| Ax=b, x\\ge 0, x\\in\\mathbf{Z}^n\\}$ to knapsack problem $\\max \\{c'x| fAx=fb, x\\ge 0, x\\in\\mathbf{Z}^n\\}$."}
{"category": "Math", "title": "Almost Paracontact Manifolds", "abstract": "In this paper eleven basic classes of almost paracontact manifolds are introduced and some examples are constructed."}
{"category": "Math", "title": "G\\'eom\\'etrie et cognition; l'exemple du continu", "abstract": "In this paper I propose the idea to establish a clear distinction between the foundations of truth and the foundations of meaning in Mathematics. I explore on the most basic example, the mathematical line, the possibility that the foundations of its meaning are provided by a protomathematical object resulting from the identification by our perceptual system of the visual line and the vestibular line, a point of view suggested by recent results of neurophysiology."}
{"category": "Math", "title": "Special Examples of Diffusions in Random Environment", "abstract": "In this note we present some examples of diffusions in random environment whose asymptotic behavior is rather surprising. We construct a family of diffusions that are small perturbations of Brownian motion with non-vanishing expected local drift under the static measure of the environment but where the ballistic behavior is lost. As slight modifications of this collection of diffusions we also provide examples with ballistic behavior where the non-vanishing limiting velocity points to a direction opposite to the expected local drift under the static measure."}
{"category": "Math", "title": "A prolongation-projection algorithm for computing the finite real variety of an ideal", "abstract": "We provide a real algebraic symbolic-numeric algorithm for computing the real variety $V_R(I)$ of an ideal $I$, assuming it is finite while $V_C(I)$ may not be. Our approach uses sets of linear functionals on $R[X]$, vanishing on a given set of polynomials generating $I$ and their prolongations up to a given degree, as well as on polynomials of the real radical ideal of $I$, obtained from the kernel of a suitably defined moment matrix assumed to be positive semidefinite and of maximum rank. We formulate a condition on the dimensions of projections of these sets of linear functionals, which serves as stopping criterion for our algorithm. This algorithm, based on standard numerical linear algebra routines and semidefinite optimization, combines techniques from previous work of the authors together with an existing algorithm for the complex variety. This results in a unified methodology for the real and complex cases."}
{"category": "Math", "title": "Small overlap monoids II: automatic structures and normal forms", "abstract": "We show that any finite monoid or semigroup presentation satisfying the small overlap condition C(4) has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms can be computed in linear time. We also deduce that C(4) monoids and semigroups are rational (in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in the sense of Duncan and Gilman). From this it follows that C(4) monoids satisfy analogues of Kleene's theorem, and admit decision algorithms for the rational subset and finitely generated submonoid membership problems. We also prove some automata-theoretic results which may be of independent interest."}
{"category": "Math", "title": "Combinatorics of line arrangements and characteristic varieties", "abstract": "This paper has been withdrawn by the author due to a sheaf-theoretic error, in the end of the proof of the main theorem."}
{"category": "Math", "title": "Crossed products by twisted partial actions and graded algebras", "abstract": "For a twisted partial action \\Theta of a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A X_\\Theta G is proved to be associative. Given a G-graded k-algebra B = \\oplus_{g\\in G}\\B_g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B_1 X_\\Theta G for some twisted partial action of G on B_1. The equality B_g\\B_{g^{-1}}B_g = \\B_g for all g\\in G is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G."}
{"category": "Math", "title": "On the mean square of the error term for the two-dimensional divisor problem (I)", "abstract": "Suppose $a$ and $b$ are two fixed positive integers such that $(a,b)=1.$ In this paper we shall establish an asymptotic formula for the mean square of the error term $\\Delta_{a,b}(x)$ of the general two-dimensional divisor problem."}
{"category": "Math", "title": "The topological cyclic Deligne conjecture", "abstract": "Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to the framed little 2-discs operad."}
{"category": "Math", "title": "What does a random contingency table look like?", "abstract": "Let R=(r_1, ..., r_m) and C=(c_1, ..., c_n) be positive integer vectors such that r_1 +... + r_m=c_1 +... + c_n. We consider the set Sigma(R, C) of non-negative mxn integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D in Sigma(R,C) is close with high probability to a particular matrix (\"typical table'') Z defined as follows. We let g(x)=(x+1) ln(x+1)-x ln x for non-negative x and let g(X)=sum_ij g(x_ij) for a non-negative matrix X=(x_ij). Then g(X) is strictly concave and attains its maximum on the polytope of non-negative mxn matrices X with row sums R and column sums C at a unique point, which we call the typical table Z."}
{"category": "Math", "title": "Computing the Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials", "abstract": "Let $\\R$ be a real closed field, $ {\\mathcal Q} \\subset \\R[Y_1,...,Y_\\ell,X_1,...,X_k], $ with $ \\deg_{Y}(Q) \\leq 2, \\deg_{X}(Q) \\leq d, Q \\in {\\mathcal Q}, #({\\mathcal Q})=m$, and $ {\\mathcal P} \\subset \\R[X_1,...,X_k] $ with $\\deg_{X}(P) \\leq d, P \\in {\\mathcal P}, #({\\mathcal P})=s$. Let $S \\subset \\R^{\\ell+k}$ be a semi-algebraic set defined by a Boolean formula without negations, with atoms $P=0, P \\geq 0, P \\leq 0, P \\in {\\mathcal P} \\cup {\\mathcal Q}$. We describe an algorithm for computing the the Betti numbers of $S$. The complexity of the algorithm is bounded by $(\\ell s m d)^{2^{O(m+k)}}$. The complexity of the algorithm interpolates between the doubly exponential time bounds for the known algorithms in the general case, and the polynomial complexity in case of semi-algebraic sets defined by few quadratic inequalities known previously. Moreover, for fixed $m$ and $k$ this algorithm has polynomial time complexity in the remaining parameters."}
{"category": "Math", "title": "Harmonic measures versus quasiconformal measures for hyperbolic groups", "abstract": "We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as \\qc measures on the boundary of the group."}
{"category": "Math", "title": "Nested iterative algorithms for convex constrained image recovery problems", "abstract": "The objective of this paper is to develop methods for solving image recovery problems subject to constraints on the solution. More precisely, we will be interested in problems which can be formulated as the minimization over a closed convex constraint set of the sum of two convex functions f and g, where f may be non-smooth and g is differentiable with a Lipschitz-continuous gradient. To reach this goal, we derive two types of algorithms that combine forward-backward and Douglas-Rachford iterations. The weak convergence of the proposed algorithms is proved. In the case when the Lipschitz-continuity property of the gradient of g is not satisfied, we also show that, under some assumptions, it remains possible to apply these methods to the considered optimization problem by making use of a quadratic extension technique. The effectiveness of the algorithms is demonstrated for two wavelet-based image restoration problems involving a signal-dependent Gaussian noise and a Poisson noise, respectively."}
{"category": "Math", "title": "On elliptic modular foliations", "abstract": "In this article we consider the three parameter family of elliptic curves $E_t: y^2-4(x-t_1)^3+t_2(x-t_1)+t_3=0, t\\in\\C^3$ and study the modular holomorphic foliation $\\F_{\\omega}$ in $\\C^3$ whose leaves are constant locus of the integration of a 1-form $\\omega$ over topological cycles of $E_t$. Using the Gauss-Manin connection of the family $E_t$, we show that $\\F_{\\omega}$ is an algebraic foliation. In the case $\\omega=\\frac{xdx}{y}$, we prove that a transcendent leaf of $\\F_{\\omega}$ contains at most one point with algebraic coordinates and the leaves of $\\F_{\\omega}$ corresponding to the zeros of integrals, never cross such a point. Using the generalized period map associated to the family $E_t$, we find a uniformization of $\\F_{\\omega}$ in $T$, where $T\\subset \\C^3$ is the locus of parameters $t$ for which $E_t$ is smooth. We find also a real first integral of $\\F_\\omega$ restricted to $T$ and show that $\\F_{\\omega}$ is given by the Ramanujan relations between the Eisenstein series."}
{"category": "Math", "title": "$q$-Partition Algebra Combinatorics", "abstract": "We compute the dimension $d_{n,r}(q) = \\dim(\\IR_q^r)$ of the defining module $\\IR_q^r$ for the $q$-partition algebra. This module comes from $r$-iterations of Harish-Chandra restriction and induction on $\\GL_n(\\FF_q)$. This dimension is a polynomial in $q$ that specializes as $d_{n,r}(1) = n^r$ and $d_{n,r}(0) = B(r)$, the $r$th Bell number. We compute $d_{n,r}(q)$ in two ways. The first is purely combinatorial. We show that $d_{n,r}(q) = \\sum_\\lambda f^\\lambda(q) m_r^\\lambda$, where $f^\\lambda(q)$ is the $q$-hook number and $m_r^\\lambda$ is the number of $r$-vacillating tableaux. Using a Schensted bijection, we write this as a sum over integer sequences which, when $q$-counted by inverse major index, gives $d_{n,r}(q)$. The second way is algebraic. We find a basis of $\\IR_q^r$ that is indexed by $n$-restricted $q$-set partitions of $\\{1,..., r\\}$, and we show that there are $d_{n,r}(q)$ of these."}
{"category": "Math", "title": "A note on palindromic $\\delta$-vectors for certain rational polytopes", "abstract": "Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi's Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart $\\delta$-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact."}
{"category": "Math", "title": "Sums of squares and orthogonal integral vectors", "abstract": "Two vectors in $\\BZ^3$ are called \\emph{twins} if they are orthogonal and have the same length. The paper describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers $M$ with the property that each integral vector with length $\\sqrt{M}$ has a twin are called twin-complete. They are completely characterized modulo a famous conjecture in number theory. The main tool is the decomposition theory of Hurwitz integral quaternions. Throughout the paper we made a concerted effort to keep the exposition as elementary as possible."}
{"category": "Math", "title": "On primitive Dirichlet characters and the Riemann hypothesis", "abstract": "For any natural number $n$, let $X'_n$ be the set of primitive Dirichlet characters modulo $n$. We show that if the Riemann hypothesis is true, then the inequality $|X'_{2n_k}|\\le C_2 e^{-\\gamma} \\phi(2n_k)/\\log\\log(2n_k)$ holds for all $k\\ge 1$, where $n_k$ is the product of the first $k$ primes, $\\gamma$ is the Euler-Mascheroni constant, $C_2$ is the twin prime constant, and $\\phi(n)$ is the Euler function. On the other hand, if the Riemann hypothesis is false, then there are infinitely many $k$ for which the same inequality holds and infinitely many $k$ for which it fails to hold."}
{"category": "Math", "title": "Quasisymmetric conjugacy between quadratic dynamics and iterated function systems", "abstract": "We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the \"overlap set\" $\\Ok$ is finite, and which are \"invertible\" on the attractor $A$, the sense that there is a continuous surjection $q: A\\to A$ whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the sense that $q$ is not a local homeomorphism precisely at $\\Ok$. We suppose also that there is a rational function $p$ with the Julia set $J$ such that $(A,q)$ and $(J,p)$ are conjugate. We prove that if $A$ has bounded turning and $p$ has no parabolic cycles, then the conjugacy is quasisymmetric. This result is applied to some specific examples including an uncountable family. Our main focus is on the family of IFS $\\{\\lambda z,\\lambda z+1\\}$ where $\\lambda$ is a complex parameter in the unit disk, such that its attractor $A_\\lam$ is a dendrite, which happens whenever $\\Ok$ is a singleton. C. Bandt observed that a simple modification of such an IFS (without changing the attractor) is invertible and gives rise to a quadratic-like map $q_\\lam$ on $A_\\lam$. If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map $p_c(z)=z^2+c$, with the Julia set $J_c$ such that $(A_\\lam,q_\\lam)$ and $(J_c,p_c)$ are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case."}
{"category": "Math", "title": "The Planar Rook Algebra and Pascal's Triangle", "abstract": "We study the combinatorial representation theory of the ``planar rook algebra\" $P_n$. This algebra has a basis consisting of planar rook diagrams and multiplication given by diagram concatenation. For each integer $0 \\le k \\le n$, we construct natural representations $V^n_k$ which form a complete set of non-isomorphic, irreducible $P_n$-representations. We explicitly decompose the regular representation of $P_n$ into a direct sum of irreducible modules. We compute the Bratteli diagram for the tower of algebras $P_0 \\subseteq P_1 \\subseteq P_2 \\subseteq ...$ and show that this Bratteli diagram is Pascal's triangle. In fact, we show that many of the binomial identities, both additive and multiplicative, have interpretations in terms of the representation theory of the planar rook algebra."}
{"category": "Math", "title": "Short note on additive sequences and on recursive processes", "abstract": "Simple methods permit to generalize the concepts of iteration and of recursive processes. We shall see briefly on several examples what these methods generate. In additive sequences, we shall encounter not only the golden or the silver ratio, but a dense set of ratio limits that corresponds to an infinity of conceivable recursive additive rules. We shall show that some of these limits have nice properties. Identities involving Fibonacci and Lucas sequences will be viewed as special cases of more general identities. We shall show that some properties of the Pascal Triangle belong also to other similar objects. In Dynamical Systems and Chaos Theory we shall encounter weird orbits, whose order is higher than the number of its distinct elements and, beyond the chaos point, a rather unexpected belated convergence to 0, after a pseudo chaotic behaviour during as many terms as one may wish. Time and again, we shall find here the Feigenbaum constant. In Formal Grammars we shall see that recursive rules applied to concatenation are sometimes equivalent to formal grammars although generally more restrictive."}
{"category": "Math", "title": "Localization of algebras over coloured operads", "abstract": "We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. It is principally the treatment of module spectra and their maps that led us to the use of coloured operads (also called enriched multicategories) in this context."}
{"category": "Math", "title": "On localization in Kronecker's diophantine theorem", "abstract": "We study the localization problem appearing in Kronecker's diophantine theorem. We introduce a probabilistic approach allowing to extend for general $\\Q$-linearly independent sequences a result of T\\'uran concerning the sequence $ (\\log p_\\ell)$, $p_\\ell$ being the $\\ell$-th prime."}
{"category": "Math", "title": "Metric connections in projective differential geometry", "abstract": "We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure."}
{"category": "Math", "title": "Gradient estimates for a nonlinear parabolic equation under Ricci flow", "abstract": "Let $(M,g(t))$, $0\\le t\\le T$, be a n-dimensional complete noncompact manifold, $n\\ge 2$, with bounded curvatures and metric $g(t)$ evolving by the Ricci flow $\\frac{\\partial g_{ij}}{\\partial t}=-2R_{ij}$. We will extend the result of L. Ma and Y. Yang and prove a local gradient estimate for positive solutions of the nonlinear parabolic equation $\\frac{\\1 u}{\\1 t}=\\Delta u-au\\log u-qu$ where $a\\in\\R$ is a constant and $q$ is a smooth function on $M\\times [0,T]$."}
{"category": "Math", "title": "Recognizing indecomposable subcontinua of surfaces from their complements", "abstract": "We prove two theorems which allow one to recognize indecomposable subcontinua of closed surfaces without boundary. If $X$ is a subcontinuum of a closed surface $S$, we call the components of $S \\setminus X$ the complementary domains of $X$. We prove that a continuum is either indecomposable or the union of two indecomposable continua whenever it has a sequence of distinct complementary domains whose boundaries limit to the continuum in the Hausdorff metric. We define a slightly stronger condition on the complementary domains of a continuum, called the double-pass condition, which we conjecture is equivalent to indecomposability of the continuum. We prove that this is so for continua which are not the boundary of one of their complementary domains."}
{"category": "Math", "title": "Relations on Generalized Degree Sequences", "abstract": "We study degree sequences for simplicial posets and polyhedral complexes, generalizing the well-studied graphical degree sequences. Here we extend the more common generalization of vertex-to-facet degree sequences by considering arbitrary face-to-flag degree sequences. In particular, these may be viewed as natural refinements of the flag f-vector of the poset. We investigate properties and relations of these generalized degree sequences, proving linear relations between flag degree sequences in terms of the composition of rank jumps of the flag. As a corollary, we recover an f-vector inequality on simplicial posets first shown by Stanley."}
{"category": "Math", "title": "Isometry types of profinite groups", "abstract": "Let T be a rooted tree and Iso(T) be the group of isometries of T. Using model-theoretic tools we study closed subgroups G of Iso(T) with respect to the number of conjugacy classes of Iso(T) having representatives in G."}
{"category": "Math", "title": "Infinite sequence of fixed point free pseudo-Anosov homeomorphisms", "abstract": "We construct infinite sequences of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of abelian differentials. The existence of such sequences show that all pseudo-Anosov homeomorphisms fixing orientable measured foliations cannot be obtained by the Rauzy-Veech induction strategy."}
{"category": "Math", "title": "On Unitary Representations of GL2n Distinguished by the Symplectic Group", "abstract": "We provide a family of representations of GL(2n) over a p-adic field that admit a non-vanishing linear functional invariant under the symplectic group (i.e. representations that are Sp(2n)- distinguished). While our result generalizes a result of M. Heumos and S. Rallis our methods, unlike their purely local technique, re- lies on the theory of automorphic forms. The results of this paper together with later works by the authors imply that the family of representations studied in this paper contains all irreducible, unitary representations of the general linear group that are distin- guished by the symplectic group."}
{"category": "Math", "title": "Index Theory for Coverings", "abstract": "We establish the basics of the analysis of operators on coverings of manifolds with cylindrical ends with a group of deck transformations $\\Gamma$. We prove the $\\Gamma$-analogue of the Atiyah-Patodi-Singer formula for Dirac operators on such coverings."}
{"category": "Math", "title": "Invariant tensors and cellular categories", "abstract": "Let U be the quantised enveloping algebra associated to a Cartan matrix of finite type. Let W be the tensor product of a finite list of highest weight representations of U. Then the centraliser algebra of W has a basis called the dual canonical basis which gives an integral form. We show that this integral form is cellular by using results due to Lusztig."}
{"category": "Math", "title": "About the maximal rank of 3-tensors over the real and the complex number field", "abstract": "High dimensional array data, tensor data, is becoming important in recent days. Then maximal rank of tensors is important in theory and applications. In this paper we consider the maximal rank of 3 tensors. It can be attacked from various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and Atkinson-Lloyd(1980). They treated the problem in the complex field, and we will present various bounds over the real field by proving several lemmas and propositions, which is real counterparts of their results."}
{"category": "Math", "title": "Herleitung von Skalarprodukten aus Symmetrieprinzipien", "abstract": "This is an attempt to model ambient space as a three-dimensional real affine space with a distinguished group of automorphisms containing the translations and acting freely and transitively on pairs consisting of a half-plane together with a half-line on its boundary. From there the existence of an invariant scalar product is deduced, which then also implies Pythagoras theorem in a quite precise form. This is in contrast to the usual procedure to model ambient space by asking for a distinguished scalar product and using Pythagoras theorem as known from high school to connect with reality."}
{"category": "Math", "title": "Mackey functors and bisets", "abstract": "For any finite group G, we define a bivariant functor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way."}
{"category": "Math", "title": "Lower bound for the remainder in the prime-pair conjecture", "abstract": "For any positive integer r, let pi_{2r}(x) denote the number of prime pairs (p, p+2r) with p not exceeding (large) x. According to the prime-pair conjecture of Hardy and Littlewood, pi_{2r}(x) should be asymptotic to 2C_{2r}li_2(x) with an explicit positive constant C_{2r}. A heuristic argument indicates that the remainder e_{2r}(x) in this approximation cannot be of lower order than x^beta, where beta is the supremum of the real parts of zeta's zeros. The argument also suggests an approximation for pi_{2r}(x) similar to one of Riemann for pi(x)."}
{"category": "Math", "title": "On the number of components of a complete intersection of real quadrics", "abstract": "Our main results concern complete intersections of three real quadrics. We prove that the maximal number $B^0_2(N)$ of connected components that a regular complete intersection of three real quadrics in $\\Bbb{P}^N$ can have differs at most by one from the maximal number of ovals of the submaximal depth $[(N-1)/2]$ of a real plane projective curve of degree $d=N+1$. As a consequence, we obtain a lower bound \\smash{$\\frac14 N^2+O(N)$} and an upper bound \\smash{$\\frac38 N^2+O(N)$} for $B^0_2(N)$."}
{"category": "Math", "title": "Global well-posedness issues for the inviscid Boussinesq system with Yudovich's type data", "abstract": "The present paper is dedicated to the study of the global existence for the inviscid two-dimensional Boussinesq system. We focus on finite energy data with bounded vorticity and we find out that, under quite a natural additional assumption on the initial temperature, there exists a global unique solution. None smallness conditions are imposed on the data. The global existence issues for infinite energy initial velocity, and for the B\\'enard system are also discussed."}
{"category": "Math", "title": "The Leray and Fujita-Kato theorems for the Boussinesq system with partial viscosity", "abstract": "We are concerned with the so-called Boussinesq equations with partial viscosity. These equations consist of the ordinary incompressible Navier-Stokes equations with a forcing term which is transported {\\it with no dissipation} by the velocity field. Such equations are simplified models for geophysics (in which case the forcing term is proportional either to the temperature, or to the salinity or to the density). In the present paper, we show that the standard theorems for incompressible Navier-Stokes equations may be extended to Boussinesq system despite the fact that there is no dissipation or decay at large time for the forcing term. More precisely, we state the global existence of finite energy weak solutions in any dimension, and global well-posedness in dimension $N\\geq3$ for small data. In the two-dimensional case, the finite energy global solutions are shown to be unique for any data in $L^2(\\R^2).$"}
{"category": "Math", "title": "Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces", "abstract": "This paper is devoted to the study of the Cauchy problem for the Boussinesq system with partial viscosity in dimension $N\\geq3.$ First we prove a global existence result for data in Lorentz spaces satisfying a smallness condition which is at the scaling of the equations. Second, we get a uniqueness result in Besov spaces with {\\it negative} indices of regularity (despite the fact that there is no smoothing effect on the temperature). The proof relies on a priori estimates with loss of regularity for the nonstationary Stokes system with convection. As a corollary, we obtain a global existence and uniqueness result for small data in Lorentz spaces."}
{"category": "Math", "title": "Hochschild two-cocycles and the good triple $(As,Hoch,Mag^\\infty)$", "abstract": "Hochschild two-cocycles play an important role in the deformation \\`a la Gerstenhaber of associative algebras. The aim of this paper is to introduce the category of Hoch-algebras whose objects are associative algebras equipped with an extra magmatic operation \\succ verifying the Hochschild two-cocycle relation: (x \\succ y)*z+ (x*y)\\succ z= x\\succ (y*z)+ x*(y\\succ z). The free Hoch-algebra over a K-vector space is given in terms of planar rooted trees and the triples of operads (As,Hoch, Mag^\\infty) endowed with the infinitesimal relations are shown to be good. We then obtain an equivalence of categories between connected infinitesimal Hoch-bialgebras and Mag^\\infty-algebras."}
{"category": "Math", "title": "On the spectral flow for paths of essentially hyperbolic bounded operators on Banach spaces", "abstract": "We give a definition of the spectral flow for paths of bounded essentially hyperbolic operators on a Banach space. The spectral flow induces a group homomorphism on the fundamental group of every connected component of the space of essentially hyperbolic operators. We prove that this homomorphism completes the exact homotopy sequence of a Serre fibration. This allows us to characterise its kernel and image and to produce examples of spaces where it is not injective or not surjective, unlike what happens for Hilbert spaces. For a large class of paths, namely the essentially splitting, the spectral flow of $ A $ coincides with $ -\\ind(F_A) $, the Fredholm index of the differential operator $ F_A (u) = u' - A u $."}
{"category": "Math", "title": "Formality of cyclic cochains", "abstract": "We prove Kontsevich's cyclic formality conjecture."}
{"category": "Math", "title": "Finding the sum of any series from a given general term", "abstract": "Translation from the Latin original, \"Inventio summae cuiusque seriei ex dato termino generali\" (1735). E47 in the Enestrom index. In this paper Euler derives the Euler-Maclaurin summation formula, by expressing y(x-1) with the Taylor expansion of y about x. In sections 21 to 23 Euler uses the formula to find expressions for the sums of the nth powers of the first x integers. He gives the general formula for this, and works it out explicitly up to n=16. In sections 25 to 28 he applies the summation formula to getting approximations to partial sums of the harmonic series, and in sections 29 to 30 to partial sums of the reciprocals of the odd positive integers. In sections 31 to 32, Euler gets an approximation to zeta(2); in section 33, approximations for zeta(3) and zeta(4). I found David Pengelley's paper \"Dances between continuous and discrete: Euler's summation formula\", in the MAA's \"Euler at 300: An Appreciation\", edited by Robert E. Bradley, Lawrence A. D'Antonio, and C. Edward Sandifer, very helpful and I recommend it if you want to understand the summation formula better."}
{"category": "Math", "title": "On the fundamental solution of an elliptic equation in nondivergence form", "abstract": "We consider the existence and asymptotics for the fundamental solution of an elliptic operator in nondivergence form, ${\\mathcal L}(x,\\del_x)=a_{ij}(x)\\del_i\\del_i$, for $n\\geq 3$. We assume that the coefficients have modulus of continuity satisfying the square Dini condition. For fixed $y$, we construct a solution of ${\\mathcal L}Z_y(x)=0$ for $0<|x-y|<\\e$ with explicit leading order term which is $O(|x-y|^{2-n}e^{I(x,y)})$ as $x\\to y$, where $I(x,y)$ is given by an integral and plays an important role for the fundamental solution: if $I(x,y)$ approaches a finite limit as $x\\to y$, then we can solve ${\\mathcal L}(x,\\del_x)F(x,y)=\\de(x-y)$, and $F(x,y)$ is asymptotic as $x\\to y$ to the fundamental solution for the constant coefficient operator ${\\mathcal L}(y,\\del_x)$. On the other hand, if $I(x,y)\\to -\\infty$ as $x\\to y$ then the solution $Z_y(x)$ violates the \"extended maximum principle\" of Gilbarg & Serrin \\cite{GS} and is a distributional solution of ${\\mathcal L}(x,\\del_x)Z_y(x)=0$ for $|x-y|<\\e$ although $Z_y$ is not even bounded as $x\\to y$."}
{"category": "Math", "title": "Topological complexity of configuration spaces", "abstract": "The topological complexity TC(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X, viewed as configuration space of a mechanical system. In this paper we complete the computation of the topological complexity of the configuration space of n distinct points in Euclidean m-space for all m>1$ and n>1; the answer was previously known in the cases m=2 and m odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity."}
{"category": "Math", "title": "Motion planning in spaces with small fundamental group", "abstract": "We establish sharp upper bounds for the topological complexity of motion planning problem in spaces with small fundamental group, i.e. when it is finite of small order or has small cohomological dimension."}
{"category": "Math", "title": "High-dimensional additive modeling", "abstract": "We propose a new sparsity-smoothness penalty for high-dimensional generalized additive models. The combination of sparsity and smoothness is crucial for mathematical theory as well as performance for finite-sample data. We present a computationally efficient algorithm, with provable numerical convergence properties, for optimizing the penalized likelihood. Furthermore, we provide oracle results which yield asymptotic optimality of our estimator for high dimensional but sparse additive models. Finally, an adaptive version of our sparsity-smoothness penalized approach yields large additional performance gains."}
{"category": "Math", "title": "Theoretical properties of Cook's PFC dimension reduction algorithm for linear regression", "abstract": "We analyse the properties of the Principal Fitted Components (PFC) algorithm proposed by Cook. We derive theoretical properties of the resulting estimators, including sufficient conditions under which they are $\\sqrt{n}$-consistent, and explain some of the simulation results given in Cook's paper. We use techniques from random matrix theory and perturbation theory. We argue that, under Cook's model at least, the PFC algorithm should outperform the Principal Components algorithm."}
{"category": "Math", "title": "Irrevocable Multi-Armed Bandit Policies", "abstract": "This paper considers the multi-armed bandit problem with multiple simultaneous arm pulls. We develop a new `irrevocable' heuristic for this problem. In particular, we do not allow recourse to arms that were pulled at some point in the past but then discarded. This irrevocable property is highly desirable from a practical perspective. As a consequence of this property, our heuristic entails a minimum amount of `exploration'. At the same time, we find that the price of irrevocability is limited for a broad useful class of bandits we characterize precisely. This class includes one of the most common applications of the bandit model, namely, bandits whose arms are `coins' of unknown biases. Computational experiments with a generative family of large scale problems within this class indicate losses of up to 5 to 10% relative to an upper bound on the performance of an optimal policy with no restrictions on exploration. We also provide a worst-case theoretical analysis that shows that for this class of bandit problems, the price of irrevocability is uniformly bounded: our heuristic earns expected rewards that are always within a factor of 1/8 of an optimal policy with no restrictions on exploration. In addition to being an indicator of robustness across all parameter regimes, this analysis sheds light on the structural properties that afford a low price of irrevocability."}
{"category": "Math", "title": "On some block ciphers and imprimitive groups", "abstract": "The group generated by the round functions of a block ciphers is a widely investigated problem. We identify a large class of block ciphers for which such group is easily guaranteed to be primitive. Our class includes the AES and the SERPENT."}
{"category": "Math", "title": "On the Allen-Cahn equation in the Grushin plane: a monotone entire solution that is not one-dimensional", "abstract": "We consider solutions of the Allen-Cahn equation in the whole Grushin plane and we show that if they are monotone in the vertical direction, then they are stable and they satisfy a good energy estimate. However, they are not necessarily one-dimensional, as a counter-example shows."}
{"category": "Math", "title": "Optimal oracle inequalities for model selection", "abstract": "Model selection is often performed by empirical risk minimization. The quality of selection in a given situation can be assessed by risk bounds, which require assumptions both on the margin and the tails of the losses used. Starting with examples from the 3 basic estimation problems, regression, classification and density estimation, we formulate risk bounds for empirical risk minimization under successively weakening conditions and prove them at a very general level, for general margin and power tail behavior of the excess losses."}
{"category": "Math", "title": "Noncommutative Cartan sub-algebras of C*-algebras", "abstract": "J. Renault has recently found a generalization of the caracterization of C*-diagonals obtained by A. Kumjian in the eighties, which in turn is a C*-algebraic version of J. Feldman and C. Moore's well known Theorem on Cartan subalgebras of von Neumann algebras. Here we propose to give a version of Renault's result in which the Cartan subalgebra is not necessarily commutative [sic]. Instead of describing a Cartan pair as a twisted groupoid C*-algebra we use N. Sieben's notion of Fell bundles over inverse semigroups which we believe should be thought of as \"twisted etale groupoids with noncommutative unit space\". En passant we prove a theorem on uniqueness of conditional expectations."}
{"category": "Math", "title": "Smooth self-similar blow-up profiles for the wave map equation", "abstract": "Consider the equivariant wave map equation from Minkowski space to a rotationnally symmetric manifold which has an equator (example: the sphere). In dimension 3, this article gives a necessary and sufficient condition for the existence of a smooth self-similar blow up profile. More generally, we study the relation between 1. the minimizing properties of the equator map for the (elliptic) Dirichlet energy and 2. the existence of a smooth blow-up profile for the (hyperbolic) wave map problem. Several applications of this approach are described."}
{"category": "Math", "title": "Geometrically Constructed Bases for Homology of Non-Crossing Partition Lattices", "abstract": "For any finite, real reflection group $W$, we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Bj\\\"{o}rner and Wachs in \\cite{BW} using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by $W$."}
{"category": "Math", "title": "Dimension of automorphisms with fixed degree for polynomial algebras", "abstract": "Let $K[x,y]$ be the polynomial algebra in two variables over an algebraically closed field $K$. We generalize to the case of any characteristic the result of Furter that over a field of characteristic zero the set of automorphisms $(f,g)$ of $K[x,y]$ such that $\\max\\{\\text{deg}(f),\\text{deg}(g)\\}=n\\geq 2$ is constructible with dimension $n+6$. The same result holds for the automorphisms of the free associative algebra $K< x,y>$. We have also obtained analogues for free algebras with two generators in Nielsen -- Schreier varieties of algebras."}
{"category": "Math", "title": "Finite energy scattering for the Lorentz-Maxwell equation", "abstract": "In the case where the charge of the particle is small compared to its mass, we describe the asymptotics of the Lorentz-Maxwell equation for any finite-energy data. As time goes to infinity, we prove that the speed of the particle converges to a certain limit, whereas the electromagnetic field can be decomposed into a soliton plus a free solution of the Maxwell equation. It is the first instance of a scattering result for general finite energy data in a field-particle equation."}
{"category": "Math", "title": "First integrals of linear differential systems", "abstract": "We investigate the problem of the existence of first integrals for multidimensional and ordinary linear differential systems with constant coefficients. The spectral method of the first integrals basis construction for these systems of linear differential equations is developed."}
{"category": "Math", "title": "Non-simple purely infinite rings", "abstract": "In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-)unital $K$-algebra having enough nonzero idempotents (for example, for a von Neumann regular algebra) its tensor product over $K$ with many nonsimple Leavitt path algebras is purely infinite."}
{"category": "Math", "title": "Orbifolds as stacks?", "abstract": "The first goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps has to be thought of as a 2-category. Compare this with the classical definition of Satake and Thurston of orbifolds as a 1-category of sets with extra structure and/or with the \"modern\" definition of orbifolds as proper etale Lie groupoids up to Morita equivalence. The second goal is to describe two complementary ways of thinking of orbifolds as a 2-category: 1. the weak 2-category of foliation Lie groupoids, bibundles and equivariant maps between bibundles and 2. the strict 2-category of Deligne-Mumford stacks over the category of smooth manifolds."}
{"category": "Math", "title": "Asymptotique des nombres de Betti des vari\\'et\\'es arithm\\'etiques", "abstract": "We study the question of the growth of Betti numbers of certain arithmetic varieties in tower of congruence coverings. In fact, our results are about Siegel varieties and varieties associated to orthogonal groups. We explain how a theorem of Waldspurger can be used to obtain lower and upper bound. Our results are in the direction of conjectures made by Sarnak and Xue."}
{"category": "Math", "title": "Computing invariants and semi-invariants by means of Frobenius Lie algebras", "abstract": "Let U(L) be the enveloping algebra of a finite dimensional Lie algebra L over a field k of characteristic zero, Z(U(L)) its center and Sz(U(L)) its semicenter. A sufficient condition is given in order for Sz(U(L)) to be a polynomial algebra over k. Surprisingly, this condition holds for many Lie algebras, especially among those for which the radical is nilpotent, in which case Sz(U(L))=Z(U(L)). In particular, it allows the explicit description of Z(U(L)) for more than half of all complex, indecomposable nilpotent Lie algebras of dimension at most 7."}
{"category": "Math", "title": "Solving period problems for minimal surfaces with the support function", "abstract": "In this paper we show how to bypass the usual difficulties in the analysis of elliptic integrals that arise when solving period problems for minimal surfaces. The method consists of replacing period problems with ordinary Sturm-Liouville problems involving the support function. We give a practical application by proving existence of the sheared Scherk-Karcher family of surfaces numerically described by Wei. Moreover, we show that this family is continuous, and both of its limit-members are the singly periodic genus-one helicoid."}
{"category": "Math", "title": "Jet and prolongation spaces", "abstract": "The notion of prolongation of an algebraic variety is developed in an abstract setting that generalises the difference and (Hasse) differential contexts. An interpolating map that compares the prolongation spaces with algebraic jet spaces is introduced and studied."}
{"category": "Math", "title": "Some examples of compact composition operators on $H^2$", "abstract": "To appear in J. Functional Analysis"}
{"category": "Math", "title": "A criterion of weak compactness for operators on subspaces of Orlicz spaces", "abstract": "To appear in J. Funct. Spaces and Appl."}
{"category": "Math", "title": "Compact composition operators on $H^2$ and Hardy-Orlicz spaces", "abstract": "We compare the compactness of composition operators on $H^2$ and on Orlicz-Hardy spaces $H^\\Psi$. We show in particular that exists an Orlicz function $\\Psi$ such that $H^{3+\\eps} \\subseteq H^\\Psi \\subseteq H^3$ for every $\\eps >0$, and a composition operator $C_\\phi$ which is compact on $H^3$ and on $H^{3+\\eps}$, but not compact on $H^\\Psi$."}
{"category": "Math", "title": "More Constructions for Tur\\'an's (3, 4)-Conjecture", "abstract": "For Tur\\'an's (3, 4)-conjecture, in the case of n = 3k+1 vertices, (.5)6^{k-1} non-isomorphic complexes are constructed that attain the conjecture. In the case of n = 3k+2 vertices, 6^{k-1} non-isomorphic complexes are constructed that attain the conjecture."}
{"category": "Math", "title": "The Charney-Davis conjecture for certain subdivisions of spheres", "abstract": "Notions of sesquiconstructible complexes and odd iterated stellar subdivisions are introduced, and some of their basic properties are verified. The Charney-Davis conjecture is then proven for odd iterated stellar subdivisions of sesquiconstructible balls and spheres."}
{"category": "Math", "title": "A strong antidiamond principle compatible with CH", "abstract": "A strong antidiamond principle (*c) is shown to be consistent with CH. This principle can be stated as a \"P-ideal dichotomy\": every P-ideal on omega-1 (i.e. an ideal that is sigma-directed under inclusion modulo finite) either has a closed unbounded subset of omega-1 locally inside of it, or else has a stationary subset of omega-1 orthogonal to it. We rely on Shelah's theory of parameterized properness for NNR iterations, and make a contribution to the theory with a method of constructing the properness parameter simultaneously with the iteration. Our handling of the application of the NNR iteration theory involves definability of forcing notions in third order arithmetic, analogous to Souslin forcing in second order arithmetic."}
{"category": "Math", "title": "Online data processing: comparison of Bayesian regularized particle filters", "abstract": "The aim of this paper is to compare three regularized particle filters in an online data processing context. We carry out the comparison in terms of hidden states filtering and parameters estimation, considering a Bayesian paradigm and a univariate stochastic volatility model. We discuss the use of an improper prior distribution in the initialization of the filtering procedure and show that the regularized Auxiliary Particle Filter (APF) outperforms the regularized Sequential Importance Sampling (SIS) and the regularized Sampling Importance Resampling (SIR)."}
{"category": "Math", "title": "Convex functions on Grassmannian manifolds and Lawson-Osserman problem", "abstract": "We derive estimates of the Hessian of two smooth functions defined on Grassmannian manifold. Based on it, we can derive curvature estimates for minimal submanifolds in Euclidean space via Gauss map. In this way, the result for Bernstein type theorem done by Jost and the first author could be improved."}
{"category": "Math", "title": "Classification of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III", "abstract": "We classify a certain class of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III. Our main technical tools are the structural analysis of type III factors and the theory of canonical extension of endomorphisms introduced by Izumi."}
{"category": "Math", "title": "The notion of $\\psi$-weak dependence and its applications to bootstrapping time series", "abstract": "We give an introduction to a notion of weak dependence which is more general than mixing and allows to treat for example processes driven by discrete innovations as they appear with time series bootstrap. As a typical example, we analyze autoregressive processes and their bootstrap analogues in detail and show how weak dependence can be easily derived from a contraction property of the process. Furthermore, we provide an overview of classes of processes possessing the property of weak dependence and describe important probabilistic results under such an assumption."}
{"category": "Math", "title": "On the vanishing of Selmer groups for elliptic curves over ring class fields", "abstract": "Let E be a rational elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let H be the ring class field of K of conductor c prime to ND with Galois group G over K. Fix a complex character \\chi of G. Our main result is that if the special value of the \\chi-twisted L-function of E/K is non-zero then the tensor product (with respect to \\chi) of the p-Selmer group of E/H with W over Z[G] is 0 for all but finitely many primes p, where W is a suitable finite extension of Z_p containing the values of \\chi. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a \\chi-twisted version of the Birch and Swinnerton-Dyer conjecture for E over H (Bertolini and Darmon) and of the vanishing of the p-Selmer group of E/K for almost all p (Kolyvagin) in the case of analytic rank zero."}
{"category": "Math", "title": "Complex Spaces and Nonstandard Schemes", "abstract": "We apply methods of nonstandard mathematics in order to regard analytic geometry in a very different way. For example, complex spaces are seen to be the \"standard part\" of certain algebraic nonstandard schemes. We construct a category of such schemes, sitting in between usual algebraic schemes (over the complex numbers) and that of complex spaces. We clarify the structure of prime ideals in a Stein algebra, coming from nonstandard points and show in particular that ANY maximal and minimal prime ideal in a Stein algebra is the vanishing ideal of a nonstandard point. Other applications of our point of view are given for differential forms (a la Leibniz), generic points (as nonstandard ones), meromorphic functions, hyperbolicity. The essential tools taken from nonstandard mathematics and adapted for our purposes, are summarized in the appendix."}
{"category": "Math", "title": "Dynamic programming for infinite horizon boundary control problems of PDE's with age structure", "abstract": "We develop the dynamic programming approach for a family of infinite horizon boundary control problems with linear state equation and convex cost. We prove that the value function of the problem is the unique regular solution of the associated stationary Hamilton--Jacobi--Bellman equation and use this to prove existence and uniqueness of feedback controls. The idea of studying this kind of problem comes from economic applications, in particular from models of optimal investment with vintage capital. Such family of problems has already been studied in the finite horizon case by Faggian. The infinite horizon case is more difficult to treat and it is more interesting from the point of view of economic applications, where what mainly matters is the behavior of optimal trajectories and controls in the long run. The study of infinite horizon is here performed through a nontrivial limiting procedure from the corresponding finite horizon problem."}
{"category": "Math", "title": "Entropy of meromorphic maps and dynamics of birational maps", "abstract": "We study the dynamics of meromorphic maps for a compact Kaehler manifold X. More precisely, we give a simple criterion that allows us to produce a measure of maximal entropy. We can apply this result to bound the Lyapunov exponents. Then, we study the particular case of a family of generic birational maps of P^k for which we construct the Green currents and the equilibrium measure. We use for that the theory of super-potentials. We show that the measure is mixing and gives no mass to pluripolar sets. Using the criterion we get that the measure is of maximal entropy. It implies finally that the measure is hyperbolic."}
{"category": "Math", "title": "Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen", "abstract": "Pranab K. Sen has contributed extensively to many areas of Statistics including order statistics, nonparametrics, robust inference, sequential methods, asymptotics, biostatistics, clinical trials, bioenvironmental studies and bioinformatics. His long list of over 600 publications and 22 books and volumes along with numerous citations during the past 5 decades bear testimony to his work. All three of us have had the good fortune of being associated with him in different capacities. He has given professional and personal advice on many occasions to all of us, and we feel that our lives have certainly been enriched by our association with him. He has been over the years a friend, philosopher and a guide to us, and still continues to be one! While parametric statistical inference remains ever so popular, semi-parametric, Bayesian and nonparametric inferential methods have attracted great attention from numerous applied scientists because of their weaker assumptions, which make them naturally robust and so more appropriate in real-life applications. This clearly signals for ``beyond parametrics'' approaches which include nonparametrics, semi-parametrics, Bayes methods and many others. Motivated by this feature, and his drive in the ``beyond parametrics'' area, we thought that it will be only appropriate for a volume in honor of Pranab Kumar Sen to focus on this aspect of statistical inference and its applications. With this in mind, we have put together this volume in order to (i) review some of the recent developments in this direction, (ii) focus on some new methodologies and highlight their applications, and (iii) suggest some interesting open problems and possible new directions for further research."}
{"category": "Math", "title": "Proof of Ira Gessel's Lattice Path Conjecture", "abstract": "We present a computer-aided, yet fully rigorous, proof of Ira Gessel's tantalizingly simply-stated conjecture that the number of ways of walking $2n$ steps in the region $x+y \\geq 0, y \\geq 0$ of the square-lattice with unit steps in the east, west, north, and south directions, that start and end at the origin, equals $16^n\\frac{(5/6)_n(1/2)_n}{(5/3)_n(2)_n}$ ."}
{"category": "Math", "title": "Deformation of central charges, vertex operator algebras whose Griess algebras are Jordan algebras", "abstract": "If a vertex operator algebra $V=\\oplus_{n=0}^{\\infty}V_n$ satisfies $\\dim V_0=1, V_1=0$, then $V_2$ has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set $Sym_d(\\C)$ of symmetric matrices of degree $d$ becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In this paper, we construct vertex operator algebras with central charge $c$ and its Griess algebra is isomorphic to $Sym_d(\\C)$ for any complex number $c$."}
{"category": "Math", "title": "The doubly periodic Scherk-Costa surfaces", "abstract": "We present a new family of embedded doubly periodic minimal surfaces, of which the symmetry group does not coincide with any other example known before."}
{"category": "Math", "title": "The Quasi-Holonomic Ansatz and Restricted Lattice Walks", "abstract": "The great enumerator Germain Kreweras empirically discovered this intriguing fact, and then needed lots of pages[K], and lots of human ingenuity, to prove it. Other great enumerators, for example, Heinrich Niederhausen[N], Ira Gessel[G1], and Mireille Bousquet-M\\'elou[B], found other ingenious, ``simpler'' proofs. Yet none of them is as simple as ours! Our proof (with the generous help of our faithful computers) is ``ugly'' in the traditional sense, since it would be painful for a lowly human to follow all the steps. But according to our humble aesthetic taste, this proof is much more elegant, since it is (conceptually) one-line. So what if that line is rather long (a huge partial-recurrence equation satisfied by the general counting function), it occupies less storage than a very low-resolution photograph."}
{"category": "Math", "title": "Heat-flow monotonicity related to the Hausdorff--Young inequality", "abstract": "It is known that if $q$ is an even integer then the $L^q(\\mathbb{R}^d)$ norm of the Fourier transform of a superposition of translates of a fixed gaussian is monotone increasing as their centres \"simultaneously slide\" to the origin. We provide explicit examples to show that this monotonicity property fails dramatically if $q > 2$ is not an even integer. These results are equivalent, upon rescaling, to similar statements involving solutions to heat equations. Such considerations are natural given the celebrated theorem of Beckner concerning the gaussian extremisability of the Hausdorff--Young inequality."}
{"category": "Math", "title": "Sums and Differences of Three k-th Powers", "abstract": "Let k>2 be a fixed integer exponent and let \\theta > 9/10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3 k-th powers, using integers of size at most B, in O(B^{\\theta}N^{1/10}) ways, providing that N << B^{3/13}. The significance of this is that we may take \\theta strictly less than 1. We also prove the estimate O(B^{10/k}), (subject to N << B) which is better for large k. The results extend to representations by an arbitrary fixed nonsingular ternary from. However ``non-trivial'' must then be suitably defined. Consideration of the singular form x^{k-1}y-z^k allows us to establish an asymptotic formula for (k-1)-free values of p^k+c, when p runs over primes, answering a problem raised by Hooley."}
{"category": "Math", "title": "The Ratio Monotonicity of the Boros-Moll Polynomials", "abstract": "In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this paper, we show that the Boros-Moll polynomials possess the ratio monotone property which implies the log-concavity and the spiral property. We conclude with a conjecture which is stronger than Moll's conjecture on the $\\infty$-log-concavity."}
{"category": "Math", "title": "On cotilting cotorsion and torsion pairs", "abstract": "In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the existence of a strong relationship between the $\\Sigma$-pure injectivity of the cotilting module and the property of the induced cotorsion pair to be of finite type. In particular for cotilting modules of injective dimension at most 1, or for noetherian rings, the two notions are equivalent. On the other hand we prove that a torsion pair is cogenerated by a $\\Sigma$-pure injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a $\\Sigma$-pure injective cotilting module of injective dimension at most 1 is necessarily coherent. Finally, for noetherian rings, we characterize cotilting torsion pairs induced by $\\Sigma$-pure injective cotilting modules."}
{"category": "Math", "title": "Invariants associated with linear ordinary differential equations", "abstract": "We apply a novel method for the equivalence group and its infinitesimal generators to the investigation of invariants of linear ordinary differential equations. First, a comparative study of this method is illustrated by an example. Next, the method is used to obtain the invariants of low order linear ordinary differential equations, and the structure invariance group for an arbitrary order of these equations. Other properties of these equations are also discussed, including the exact number of their invariants."}
{"category": "Math", "title": "On submanifolds with recurrent second fundamental form in spaces of constant curvature", "abstract": "The complete local classification and geometric description of n-dimensional submanifolds F with recurrent nonparallel second fundamental form in the spaces of constant curvature M(c) are obtained in this article."}
{"category": "Math", "title": "The triangle-free process", "abstract": "Consider the following stochastic graph process. We begin with the empty graph on n vertices and add edges one at a time, where each edge is chosen uniformly at random from the collection of potential edges that do not form triangles when added to the graph. The process terminates at a maximal traingle-free graph. Here we analyze the triangle-free process, determining the likely order of magnitude of the number of edges in the final graph. As a corollary we show that the triangle-free process is very likely to produce a Ramsey R(3,t) graph; that is, our analysis of the triangle-free process gives a new proof of the lower bound on R(3,t) previously established by Jeong Han Kim. The techniques introduced extend to the K_4-free process thereby establishing a small improvement in the best known lower bound on the Ramsey number R(4,t)."}
{"category": "Math", "title": "Nice inducing schemes and the thermodynamics of rational maps", "abstract": "We consider the thermodynamic formalism of a complex rational map $f$ of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter $t$ we study the (non-)existence of equilibrium states of $f$ for the potential $-t \\ln |f'|$, and the analytic dependence on $t$ of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism of a rational map that is \"expanding away from critical points\" and that has arbitrarily small \"nice sets\" with some additional properties. Our results apply in particular to non-renormalizable polynomials without indifferent periodic points, infinitely renormalizable quadratic polynomials with a priori bounds, real quadratic polynomials, topological Collet-Eckmann rational maps, and to backward contracting rational maps. As an application, for these maps we describe the dimension spectrum of Lyapunov exponents, and of pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results."}
{"category": "Math", "title": "On logarithmic extension of overconvergent isocrystals", "abstract": "In this paper, we establish a criterion for an overconvergent isocrystal on a smooth variety over a field of characteristic $p>0$ to extend logarithmically to its smooth compactification whose complement is a strict normal crossing divisor. This is a generalization of a result of Kedlaya, who treated the case of unipotent monodromy. Our result is regarded as a $p$-adic analogue of the theory of canonical extension of regular singular integrable connections on smooth varieties of characteristic 0."}
{"category": "Math", "title": "Parabolic, hyperbolic and elliptic Poincar\\'e series", "abstract": "Following Petersson, we study the parabolic, hyperbolic and elliptic expansions of holomorphic cusp forms and the associated Poincar\\'e series. We show how these ideas extend to the space of second-order cusp forms."}
{"category": "Math", "title": "The Semigroup of Betti Diagrams", "abstract": "The recent proof of the Boij-Soederberg conjectures reveals new structure about Betti diagrams of modules, giving a complete description of the cone of Betti diagrams. We begin to expand on this new structure by investigating the semigroup of Betti diagrams. We prove that this semigroup is finitely generated, and we answer several other fundamental questions about this semigroup."}
{"category": "Math", "title": "Normal Families of Bicomplex Holomorphic Functions", "abstract": "In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples of bicomplex Julia sets are also given for the specific slice $\\bold{j}=0$."}
{"category": "Math", "title": "Column Subset Selection, Matrix Factorization, and Eigenvalue Optimization", "abstract": "Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called rank-revealing {\\sf QR}, which seeks a well-conditioned collection of columns that spans the (numerical) range of the matrix. The functional analysis literature contains another strand of work on column selection whose algorithmic implications have not been explored. In particular, a celebrated result of Bourgain and Tzafriri demonstrates that each matrix with normalized columns contains a large column submatrix that is exceptionally well conditioned. Unfortunately, standard proofs of this result cannot be regarded as algorithmic. This paper presents a randomized, polynomial-time algorithm that produces the submatrix promised by Bourgain and Tzafriri. The method involves random sampling of columns, followed by a matrix factorization that exposes the well-conditioned subset of columns. This factorization, which is due to Grothendieck, is regarded as a central tool in modern functional analysis. The primary novelty in this work is an algorithm, based on eigenvalue minimization, for constructing the Grothendieck factorization. These ideas also result in a novel approximation algorithm for the $(\\infty, 1)$ norm of a matrix, which is generally {\\sf NP}-hard to compute exactly. As an added bonus, this work reveals a surprising connection between matrix factorization and the famous {\\sc maxcut} semidefinite program."}
{"category": "Math", "title": "Andr\\'e-Quillen cohomology of algebras over an operad", "abstract": "We study the Andr\\'e-Quillen cohomology with coefficients of an algebra over an operad. Using resolutions of algebras coming from Koszul duality theory, we make this cohomology theory explicit and we give a Lie theoretic interpretation. For which operads is the associated Andr\\'e-Quillen cohomology equal to an Ext-functor ? We give several criterion, based on the cotangent complex, to characterize this property. We apply it to homotopy algebras, which gives a new homotopy stable property for algebras over cofibrant operads."}
{"category": "Math", "title": "Some New Examples of Non-K\\\"ahler Ricci Solitons", "abstract": "We produce non-K\\\"ahler complete steady gradient Ricci solitons generalising those constructed by Bryant and Ivey."}
{"category": "Math", "title": "Summing the curious series of Kempner and Irwin", "abstract": "In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ... where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. The actual sum is about 22.92068. In 1916, Irwin proved, among other things, that the sum of 1/n where n has at most a finite number of 9's is also a convergent series. We show how to compute sums of Irwins' series to high precision. For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9, is about 23.04428 70807 47848 31968. Another example: the sum of 1/n where n has exactly 100 zeros is about 10 ln(10) + 1.00745 x 10^-197 ~ 23.02585; note that the first, and largest, term in this series is the tiny 1/googol. Finally, we discuss a class of related series whose summation algorithm has not yet been developed."}
{"category": "Math", "title": "Iitaka conjecture $C_{n,m}$ in dimension six", "abstract": "We prove that the Iitaka conjecture $C_{n,m}$ for algebraic fibre spaces holds up to dimension 6, that is, when $n\\le 6$."}
{"category": "Math", "title": "Nonabelian free group actions: Markov processes, the Abramov-Rohlin formula and Yuzvinskii's formula", "abstract": "This paper introduces Markov chains and processes over nonabelian free groups and semigroups. We prove a formula for the $f$-invariant of a Markov chain over a free group in terms of transition matrices that parallels the classical formula for the entropy a Markov chain. Applications include free group analogues of the Abramov-Rohlin formula for skew-product actions and Yuzvinskii's addition formula for algebraic actions."}
{"category": "Math", "title": "Split reductions of simple abelian varieties", "abstract": "Consider an absolutely simple abelian variety X over a number field K. If the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then the reduction X_p is absolutely simple for almost all p. Conversely, if the absolute endomorphism ring of X is noncommutative, then X_p is reducible for p in a set of positive density."}
{"category": "Math", "title": "On the generalization of Gurland distribution", "abstract": "In the present paper a generalization of Gurland distribution [3] is obtained as a beta mixture of the generalized Poisson distribution (GPD) of Consul and Jain [2]. The first two moments of the distribution and a recurrence relation among probabilities are obtained. The present distribution is supposed to be more general in nature and wider in scope."}
{"category": "Math", "title": "On endomorphism rings and dimensions of local cohomology modules", "abstract": "Let $(R,\\mathfrak m)$ denote an $n$-dimensional complete local Gorenstein ring. For an ideal $I$ of $R$ let $H^i_I(R), i \\in \\mathbb Z,$ denote the local cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i \\not= c = \\height I,$ then the endomorphism ring of $H^c_I(R)$ is isomorphic to $R$ (cf. \\cite{HSt} and \\cite{HS}). Here we prove that this is true if and only if $H^i_I(R) = 0, i = n, n -1$ provided $c \\geq 2$ and $R/I$ has an isolated singularity resp. if $I$ is set-theoretically a complete intersection in codimension at most one. Moreover, there is a vanishing result of $H^i_I(R)$ for all $i > m, m$ a given integer, resp. an estimate of the dimension of $H^i_I(R).$"}
{"category": "Math", "title": "Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations", "abstract": "In this paper we study the following non-autonomous stochastic evolution equation on a UMD Banach space $E$ with type 2, {equation}\\label{eq:SEab}\\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), \\quad t\\in [0,T], U(0) & = u_0. {aligned}. {equation} Here $(A(t))_{t\\in [0,T]}$ are unbounded operators with domains $(D(A(t)))_{t\\in [0,T]}$ which may be time dependent. We assume that $(A(t))_{t\\in [0,T]}$ satisfies the conditions of Acquistapace and Terreni. The functions $F$ and $B$ are nonlinear functions defined on certain interpolation spaces and $u_0\\in E$ is the initial value. $W_H$ is a cylindrical Brownian motion on a separable Hilbert space $H$. Under Lipschitz and linear growth conditions we show that there exists a unique mild solution of \\eqref{eq:SEab}. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapie\\'n, and Zabczyk, to obtain space-time regularity results for the solution $U$ of \\eqref{eq:SEab}. For Hilbert spaces $E$ we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second order stochastic partial differential equation which has been studied by Sanz-Sol\\'e and Vuillermot. This leads to several improvements of their result."}
{"category": "Math", "title": "Probability and Statistics: Essays in Honor of David A. Freedman", "abstract": "This volume is our tribute to David A. Freedman, whom we regard as one of the great statisticians of our time. He received his B.Sc. degree from McGill University and his Ph.D. from Princeton, and joined the Department of Statistics of the University of California, Berkeley, in 1962, where, apart from sabbaticals, he has been ever since. In a career of over 45 years, David has made many fine contributions to probability and statistical theory, and to the application of statistics. His early research was on Markov chains and martingales, and two topics with which he has had a lifelong fascination: exchangeability and De Finetti's theorem, and the consistency of Bayes estimates. His asymptotic theory for the bootstrap was also highly influential. David was elected to the American Academy of Arts and Sciences in 1991, and in 2003 he received the John J. Carty Award for the Advancement of Science from the U.S. National Academy of Sciences. In addition to his purely academic research, David has extensive experience as a consultant, including working for the Carnegie Commission, the City of San Francisco, and the Federal Reserve, as well as several Departments of the U.S. Government--Energy, Treasury, Justice, and Commerce. He has testified as an expert witness on statistics in a number of law cases, including Piva v. Xerox (employment discrimination), Garza v. County of Los Angeles (voting rights), and New York v. Department of Commerce (census adjustment). Lastly, he is an exceptionally good writer and teacher, and his many books and review articles are arguably his most important contribution to our subject."}
{"category": "Math", "title": "Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh", "abstract": "Jayanta Kumar Ghosh is one of the most extraordinary professors in the field of Statistics. His research in numerous areas, especially asymptotics, has been groundbreaking, influential throughout the world, and widely recognized through awards and other honors. His leadership in Statistics as Director of the Indian Statistical Institute and President of the International Statistical Institute, among other eminent positions, has been likewise outstanding. In recognition of Jayanta's enormous impact, this volume is an effort to honor him by drawing together contributions to the main areas in which he has worked and continues to work. The papers naturally fall into five categories. First, sequential estimation was Jayanta's starting point. Thus, beginning with that topic, there are two papers, one classical by Hall and Ding leading to a variant on p-values, and one Bayesian by Berger and Sun extending reference priors to stopping time problems. Second, there are five papers in the general area of prior specification. Much of Jayanta's earlier work involved group families as does Sweeting's paper here for instance. There are also two papers dwelling on the link between fuzzy sets and priors, by Meeden and by Delampady and Angers. Equally daring is the work by Mukerjee with data dependent priors and the pleasing confluence of several prior selection criteria found by Ghosh, Santra and Kim. Jayanta himself studied a variety of prior selection criteria including probability matching priors and reference priors."}
{"category": "Math", "title": "M-curves of degree 9 with three nests", "abstract": "The first part of Hilbert's sixteenth problem deals with the classification of the isotopy types realizable by real plane algebraic curves of a given degree $m$. For $m = 9$, the classification of the $M$-curves is still wide open. Let $C_9$ be an $M$-curve of degree 9 and $O$ be a non-empty oval of $C_9$. If $O$ contains in its interior $\\alpha$ ovals that are all empty, we say that $O$ together with these $\\alpha$ ovals forms a nest. The present paper deals with the $M$-curves with three nests. Let $\\alpha_i, i = 1, 2, 3$ be the numbers of empty ovals in each nest. We prove that at least one of the $\\alpha_i$ is odd. This is a step towards a conjecture of A. Korchagin, claiming that at least two of the $\\alpha_i$ should be odd."}
{"category": "Math", "title": "Numerical simulation of BSDEs using empirical regression methods: theory and practice", "abstract": "This article deals with the numerical resolution of backward stochastic differential equations. Firstly, we consider a rather general case where the filtration is generated by a Brownian motion and a Poisson random measure. We provide a simulation algorithm based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. We state fully explicit error bounds. Secondly, restricting to the case of a Brownian filtration, we consider reflected BSDEs and adapt the previous algorithm to that situation. The complexity of the algorithm is very competitive and allows us to treat numerical results in dimension 10."}
{"category": "Math", "title": "Trees of definable sets over the p-adics", "abstract": "To a definable subset of Z_p^n (or to a scheme of finite type over Z_p) one can associate a tree in a natural way. It is known that the corresponding Poincare series P(X) = \\sum_i N_i X^i is rational, where N_i is the number of nodes of the tree at depth i. This suggests that the trees themselves are far from arbitrary. We state a conjectural, purely combinatorial description of the class of possible trees and provide some evidence for it. We verify that any tree in our class indeed arises from a definable set, and we prove that the tree of a definable set (or of a scheme) lies in our class in three special cases: under weak smoothness assumptions, for definable subsets of Z_p^2, and for one-dimensional sets."}
{"category": "Math", "title": "On generating relative and absolute invariants of linear differential equations", "abstract": "A general expression for a relative invariant of a linear ordinary differential equations is given in terms of the fundamental semi-invariant and an absolute invariant. This result is used to established a number of properties of relative invariants, and it is explicitly shown how to generate fundamental sets of relative and absolute invariants of all orders for the general linear equation. Explicit constructions are made for the linear ODE of order five. The approach used for the explicit determination of invariants is based on an infinitesimal method."}
{"category": "Math", "title": "On Positive Integers Represented as Arithmetic Series", "abstract": "The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In particular, we shall be concerned with the representation of positive integers as arithmetic series of the simplest kind, i.e., either as sums of successive odd positive numbers, or as sums of successive even positive numbers (both treated as Problem 1), or as sums of consecutive positive integers (treated as Problem 2)."}
{"category": "Math", "title": "Simultaneous packing and covering in sequence spaces", "abstract": "We adapt a construction of Klee (1981) to find a packing of unit balls in $\\ell_p$ ($1\\leq p<\\infty$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1/p}$ covers the whole space. We show that the value $2^{1-1/p}$ is optimal."}
{"category": "Math", "title": "Bootstrap percolation in three dimensions", "abstract": "By bootstrap percolation we mean the following deterministic process on a graph $G$. Given a set $A$ of vertices \"infected\" at time 0, new vertices are subsequently infected, at each time step, if they have at least $r\\in\\mathbb{N}$ previously infected neighbors. When the set $A$ is chosen at random, the main aim is to determine the critical probability $p_c(G,r)$ at which percolation (infection of the entire graph) becomes likely to occur. This bootstrap process has been extensively studied on the $d$-dimensional grid $[n]^d$: with $2\\leq r\\leq d$ fixed, it was proved by Cerf and Cirillo (for $d=r=3$), and by Cerf and Manzo (in general), that \\[p_c([n]^d,r)=\\Theta\\biggl(\\frac{1}{\\log_{(r-1)}n}\\biggr)^{d-r+1},\\] where $\\log_{(r)}$ is an $r$-times iterated logarithm. However, the exact threshold function is only known in the case $d=r=2$, where it was shown by Holroyd to be $(1+o(1))\\frac{\\pi^2}{18\\log n}$. In this paper we shall determine the exact threshold in the crucial case $d=r=3$, and lay the groundwork for solving the problem for all fixed $d$ and $r$."}
{"category": "Math", "title": "Confinement of matroid representations to subsets of partial fields", "abstract": "Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P' of P. We say that B confines M to P' if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled P'-matrix. We show that, under some conditions on the partial fields, on M, and on B, verifying whether B confines M to P' amounts to a finite check. A corollary of this result is Whittle's Stabilizer Theorem. A combination of the Confinement Theorem and the Lift Theorem from arXiv:0804.3263 leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields. We also use a combination of the Confinement Theorem and the Lift Theorem to prove a characterization, in terms of representability over partial fields, of the 3-connected matroids that have k inequivalent representations over GF(5), for k = 1, ..., 6. Additionally we give, for a fixed matroid M, an algebraic construction of a partial field P_M and a representation A over P_M such that every representation of M over a partial field P is equal to f(A) for some homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al."}
{"category": "Math", "title": "Seshadri constants on the self-product of an elliptic curve", "abstract": "The purpose of this paper is to study Seshadri constants on the self-product $E\\times E$ of an elliptic curve $E$. We provide explicit formulas for computing the Seshadri constants of all ample line bundles on the surfaces considered. As an application, we obtain a good picture of the behaviour of the Seshadri function on the nef cone."}
{"category": "Math", "title": "Gap in Nonlinear Equivalence for Numerical Methods for PDEs", "abstract": "For a large class of nonlinear evolution PDEs, and more generally, of nonlinear semigroups, as well as their approximating numerical methods, two rather natural stability type convergence conditions are given, one being necessary, while the other is sufficient. The gap between these two stability conditions is analyzed, thus leading to a general nonlinear equivalence between stability and convergence."}
{"category": "Math", "title": "The Poincare series of divisorial valuations in the plane defines the topology of the set of divisors", "abstract": "To a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincare series of this filtration turnes out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincare series of the corresponding multi-index filtration similar to the one associated to plane germs. Here we show that the Poincare series of a set of divisorial valuations on the ring of germs of functions of two variables defines \"the topology of the set of the divisors\" in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which proves that the Alexander polynomial is equivalent to the embedded topology."}
{"category": "Math", "title": "Generators of simple Lie algebras II", "abstract": "This paper is a continuation of earlier work on generators of simple Lie algebras in arbitrary characteristic (see arXiv:0708.1711). We show that, in contrast to classical Lie algebras, simple graded Lie algebras of Cartan type S,H or K never enjoy the 'one-and-a-half generation' property. The methods rely on a study of centralisers in Cartan type Lie algebras."}
{"category": "Math", "title": "Minimax State Estimation for a Dynamic System Described by a Differential-Algebraic Equation", "abstract": "In this report we address the linear state estimation problem: to estimate a linear transformation $\\ell(\\varphi)$ of the state $\\varphi$ through an algorithm $\\widehat{\\ell(\\varphi)}$ operating on measurements $y$, where $L\\varphi=f,y=H\\varphi+\\eta$. We study the estimation problem in terms of the minimax estimation framework: to find a linear algorithm $\\widehat{\\widehat{\\ell(\\varphi)}}$ that minimizes the worst case error $\\sup_{\\varphi,\\eta}d(\\ell(\\varphi),\\widehat{\\ell(\\varphi)}) $. A key feature of the presented estimation approach is to fix a class of linear operators $L$, $H$; given any pair $L,H$ from that class we describe a class $\\mathcal L$ of all solution operators $\\ell$ such that the worst case error is finite. We formulate a duality theorem (like Kalman duality principle) that is the estimation problem is equal to the optimal control problem if $G$ is convex bounded subset of the corresponding Hilbert space, $L$ is a closed linear mapping. We obtain optimal estimations as solutions of the linear operator equations if $G$ is an ellipsoid. Then we apply this to the state estimation for the linear differential-algebraic equations (DAE). The minimax observer for DAE is represented in the form of the minimax filter. For discrete time DAEs we present the online minimax estimator."}
{"category": "Math", "title": "Aronszajn Compacta", "abstract": "We consider a class of compacta X such that the maps from X onto metric compacta define an Aronszajn tree of closed subsets of X."}
{"category": "Math", "title": "The Brauer algebra and the symplectic Schur algebra", "abstract": "Let k be an algebraically closed field of characteristic p>0, let m,r be integers with m\\ge1, r\\ge0 and m\\ge r and let S_0(2m,r) be the symplectic Schur algebra over k as introduced by the first author. We introduce the symplectic Schur functor, derive some basic properties of it and relate this to work of Hartmann and Paget. We do the same for the orthogonal Schur algebra. We give a modified Jantzen sum formula and a block result for the symplectic Schur algebra under the assumption that r and the residue of 2m mod p are small relative to p. From this we deduce a block result for the orthogonal Schur algebra under similar assumptions. Finally, we deduce from the previous results a new proof of the geometric description of the blocks of the Brauer algebra in characteristic 0 as obtained by Cox, De Visscher and Martin."}
{"category": "Math", "title": "Cauchy problem for viscous rotating shallow water equations", "abstract": "We consider the Cacuhy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modelling of motions for shallow water with free surface in a rotating sub-domain. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without coriolis forces, the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments."}
{"category": "Math", "title": "Families of equivariant differential operators and anti de Sitter spaces", "abstract": "We prove existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces."}
{"category": "Math", "title": "Ergodic actions of S_\\mu U(2) on C*-algebras from II_1 subfactors", "abstract": "To a proper inclusion N\\subset M of II_1 factors of finite Jones index [M:N], we associate an ergodic C*-action of the quantum group S_\\mu U(2). The deformation parameter is determined by -1<\\mu<0 and [M:N]=|\\mu+\\mu^{-1}|. The higher relative commutants can be identified with the spectral spaces of the tensor powers of the defining representation of the quantum group. This ergodic action may be thought of as a virtual subgroup of S_\\mu U(2) in the sense of Mackey arising from the tensor category generated by M regarded as a bimodule over N. \\mu is negative as M is a real bimodule."}
{"category": "Math", "title": "The second iterate for the Navier-Stokes equation", "abstract": "We consider the iterative resolution scheme for the Navier-Stokes equation, and focus on the second iterate, more precisely on the map from the initial data to the second iterate at a given time t. We investigate boundedness properties of this bilinear operator. This new approach yields very interesting results: a new perspective on Koch-Tataru solutions; a first step towards weak strong uniqueness for Koch-Tataru solutions; and finally an instability result in $B^{-1}_{\\infty,q}$, for q>2."}
{"category": "Math", "title": "Poset Resolutions of Monomial Ideals", "abstract": "We introduce the class of lattice-linear monomial ideals and use the LCM-lattice to give an explicit construction for their minimal free resolution. The class of lattice-linear ideals includes (among others) the class of monomial ideals with linear free resolution and the class of Scarf monomial ideals. Our main tool is a new construction by Tchernev that produces from a map of posets $\\eta:P\\lra\\mbb{N}^n$ a sequence of multigraded modules and maps."}
{"category": "Math", "title": "A Note on Algebraic Multigrid Methods for the Discrete Weighted Laplacian", "abstract": "In recent contributions, algebraic multigrid methods have been designed and studied from the viewpoint of the spectral complementarity. In this note we focus our efforts on specific applications and, more precisely, on large linear systems arising from the approximation of weighted Laplacian with various boundary conditions. We adapt the multigrid idea to this specific setting and we present and critically discuss a wide numerical experimentation showing the potentiality of the considered approach."}
{"category": "Math", "title": "Worst Case to Average Case Reductions for Polynomials", "abstract": "A degree-$d$ polynomial $p$ in $n$ variables over a field $\\F$ is {\\em equidistributed} if it takes on each of its $|\\F|$ values close to equally often, and {\\em biased} otherwise. We say that $p$ has a {\\em low rank} if it can be expressed as a bounded combination of polynomials of lower degree. Green and Tao [gt07] have shown that bias imply low rank over large fields (i.e. for the case $d < |\\F|$). They have also conjectured that bias imply low rank over general fields. In this work we affirmatively answer their conjecture. Using this result we obtain a general worst case to average case reductions for polynomials. That is, we show that a polynomial that can be {\\em approximated} by few polynomials of bounded degree, can be {\\em computed} by few polynomials of bounded degree. We derive some relations between our results to the construction of pseudorandom generators, and to the question of testing concise representations."}
{"category": "Math", "title": "On ill-posedness for the one-dimensional periodic cubic Schrodinger equation", "abstract": "We prove the ill-posedness in $ H^s(\\T) $, $s<0$, of the periodic cubic Schr\\\"odinger equation in the sense that the flow-map is not continuous from $H^s(\\T) $ into itself for any fixed $ t\\neq 0 $. This result is slightly stronger than the one obtained by Christ-Colliander-Tao where the discontinuity of the solution map is established. Moreover our proof is different and clarifies the ill-posedness phenomena. Our approach relies on a new result on the behavior of the associated flow-map with respect to the weak topology of $ L^2(\\T) $."}
{"category": "Math", "title": "Fundamental classes not representable by products", "abstract": "We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of non-compact type. For closed manifolds from certain classes, say non-positively curved ones, or certain surface bundles over surfaces, we show that they do admit maps of non-zero degree from non-trivial products if and only if they are virtually diffeomorphic to products."}
{"category": "Math", "title": "Note on a theorem of Bousfield and Friedlander", "abstract": "We examine the proof of a classical localization theorem of Bousfield and Friedlander and we remove the assumption that the underlying model category be right proper. The key to the argument is a lemma about factoring in morphisms in the arrow category of a model category."}
{"category": "Math", "title": "Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations", "abstract": "We prove the existence of a compact, finite dimensional, global attractor for a coupled PDE system comprising a nonlinearly damped semilinear wave equation and a nonlinear system of thermoelastic plate equations, without any mechanical (viscous or structural) dissipation in the plate component. The plate dynamics is modelled following Berger's approach; we investigate both cases when rotational inertia is included into the model and when it is not. A major part in the proof is played by an estimate--known as stabilizability estimate--which shows that the difference of any two trajectories can be exponentially stabilized to zero, modulo a compact perturbation. In particular, this inequality yields bounds for the attractor's fractal dimension which are independent of two key parameters, namely $\\gamma$ and $\\kappa$, the former related to the presence of rotational inertia in the plate model and the latter to the coupling terms. Finally, we show the upper semi-continuity of the attractor with respect to these parameters."}
{"category": "Math", "title": "Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts", "abstract": "We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time $\\tau_\\alpha$ from a wedge with apex at the origin and interior half-angle $\\alpha$ by a non-homogeneous random walk on the square lattice with mean drift at $x$ of magnitude $O(1/|x|)$ as $|x| \\to \\infty$. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors (see arXiv:0910.1772) stated that $\\tau_\\alpha < \\infty$ a.s. for any $\\alpha$ (while for a stronger drift field $\\tau_\\alpha$ is infinite with positive probability). Here we study the more difficult problem of the existence and non-existence of moments $E[\\tau_\\alpha^s]$, $s>0$. Assuming (in common with much of the literature) a uniform bound on the walk's increments, we show that for $\\alpha < \\pi/2$ there exists $s_0 \\in (0,\\infty)$ such that $E[\\tau_\\alpha^s]$ is finite for $s < s_0$ but infinite for $s > s_0$; under specific assumptions on the drift field we show that we can attain $E[\\tau_\\alpha^s] = \\infty$ for any $s > 1/2$. We show that for $\\alpha \\leq \\pi$ there is a phase transition between drifts of magnitude $O(1/|x|)$ (the critical regime) and $o(1/|x|)$ (the subcritical regime). In the subcritical regime we obtain a non-homogeneous random walk analogue of a theorem for Brownian motion due to Spitzer, under considerably weaker conditions than those previously given (including work by Varopoulos) that assumed zero drift."}
{"category": "Math", "title": "Springer theory via the Hitchin fibration", "abstract": "In this paper, we translate the Springer theory of Weyl group representations into the language of symplectic topology. Given a semisimple complex group G, we describe a Lagrangian brane in the cotangent bundle of the adjoint quotient g/G that produces the perverse sheaves of Springer theory. The main technical tool is an analysis of the Fourier transform for constructible sheaves from the perspective of the Fukaya category. Our results can be viewed as a toy model of the quantization of Hitchin fibers in the Geometric Langlands program."}
{"category": "Math", "title": "Global attractor and asymptotic smoothing effects for the weakly damped cubic Schr\\\"odinger equation in $L^2(\\T)$", "abstract": "We prove that the weakly damped cubic Schr\\\"odinger flow in $L^2(\\T)$ provides a dynamical system that possesses a global attractor. The proof relies on a sharp study of the behavior of the associated flow-map with respect to the weak $ L^2(\\T) $-convergence inspired by a previous work of the author. Combining the compactness in $ L^2(\\T) $ of the attractor with the approach developed by Goubet, we show that the attractor is actually a compact set of $ H^2(\\T) $. This asymptotic smoothing effect is optimal in view of the regularity of the steady states."}
{"category": "Math", "title": "Consecutive integers in high-multiplicity sumsets", "abstract": "Sharpening (a particular case of) a result of Szemeredi and Vu and extending earlier results of Sarkozy and ourselves, we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers of length, comparable with the lengths of the set summands. A corollary of our main result is as follows. Let $k,l\\ge 1$ and $n\\ge 3$ be integers, and suppose that $A_1,...,A_k\\subset[0,l]$ are integer sets of size at least $n$, none of which is contained in an arithmetic progression with difference greater than 1. If $k\\ge 2\\lceil(l-1)/(n-2)\\rceil$, then the sumset $A_1+...+A_k$ contains a block of consecutive integers of length $k(n-1)$."}
{"category": "Math", "title": "Generating Simplicial Complexes", "abstract": "In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on simplicial complexes. For expansive systems remarkable properties are observed. Known examples are revisited and new examples are presented."}
{"category": "Math", "title": "Rational Symplectic Field Theory for Legendrian knots", "abstract": "We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology."}
{"category": "Math", "title": "An Analogue of the Gallai-Edmonds Structure Theorem for Nonzero Roots of the Matching Polynomial", "abstract": "Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing \"deficiency\" with multiplicity of $\\theta$ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, which describes how the matching structure of a graph changes upon deletion of a single vertex. An analogue of Gallai's Lemma follows. Together these two results imply an analogue of the Gallai-Edmonds Structure Theorem. Consequently, the matching polynomial of a vertex transitive graph has simple roots."}
{"category": "Math", "title": "Algebraic cycles and motivic iterated integrals II", "abstract": "This is a sequel to our previous paper (joint with Furusho). It will give a more natural framework for constructing elements in the Hopf algebra of framed mixed Tate motives according to Bloch and Kriz. This framework allows us to extend our previous results to interpret all multiple zeta values (including the divergent ones) and the multiple polylogarithms in one variable as elements of this Hopf algebra. It implies that the pro-unipotent completion of the torsor of paths on projective line minus three points, is a mixed Tate motive in the sense of Bloch-Kriz. Also It allows us to interpret the multiple logarithm as an element of this Hopf algebra as long as the products of consecutive arguments are not 1."}
{"category": "Math", "title": "Predicting Regional Classification of Levantine Ivory Sculptures: A Machine Learning Approach", "abstract": "Art historians and archaeologists have long grappled with the regional classification of ancient Near Eastern ivory carvings. Based on the visual similarity of sculptures, individuals within these fields have proposed object assemblages linked to hypothesized regional production centers. Using quantitative rather than visual methods, we here approach this classification task by exploiting computational methods from machine learning currently used with success in a variety of statistical problems in science and engineering. We first construct a prediction function using 66 categorical features as inputs and regional style as output. The model assigns regional style group (RSG), with 98 percent prediction accuracy. We then rank these features by their mutual information with RSG, quantifying single-feature predictive power. Using the highest- ranking features in combination with nomographic visualization, we have found previously unknown relationships that may aid in the regional classification of these ivories and their interpretation in art historical context."}
{"category": "Math", "title": "Norm one idempotent cb-multipliers with applications to the Fourier algebra in the cb-multiplier norm", "abstract": "For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{Mcb}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \\subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{Mcb}(G)$ with an approximate identity bounded by 1, and we characterize those $G$ for which $A_{Mcb}(G)$ is 1-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)"}
{"category": "Math", "title": "Discrete spectrum and Weyl's asymptotic formula for incomplete manifolds", "abstract": "Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With certain restrictions on the metric, we establish that the spectrum is discrete and satisfies Weyl's asymptotic formula."}
{"category": "Math", "title": "Uniform Local Existence for Inhomogeneous Rotating Fluid Equations", "abstract": "We investigate the equations of anisotropic incompressible viscous fluids in $\\R^3$, rotating around an inhomogeneous vector $B(t, x_1, x_2)$. We prove the global existence of strong solutions in suitable anisotropic Sobolev spaces for small initial data, as well as uniformlocal existence result with respect to the Rossby number in the same functional spaces under the additional assumption that $B=B(t,x_1)$ or $B=B(t,x_2)$. We also obtain the propagation of the isotropic Sobolev regularity using a new refined product law."}
{"category": "Math", "title": "Lower bounds for index of Wente tori", "abstract": "We show numerically that any of the constant mean curvature tori first found by Wente must have index at least eight."}
{"category": "Math", "title": "On the index of constant mean curvature 1 surfaces in hyperbolic space", "abstract": "We show that the index of a constant mean curvature 1 surface in hyperbolic 3-space is completely determined by the compact Riemann surface and secondary Gauss map that represent it in Bryant's Weierstrass representation. We give three applications of this observation. Firstly, it allows us to explicitly compute the index of the catenoid cousins and some other examples. Secondly, it allows us to be able to apply a method similar to that of Choe (using Killing vector fields on minimal surfaces in Euclidean 3-space) to our case as well, resulting in lower bounds of index for other examples. And thirdly, it allows us to give a more direct proof of the result by do Carmo and Silveira that if a constant mean curvature 1 surface in hyperbolic 3-space has finite total curvature, then it has finite index. Finally, we show that for any constant mean curvature 1 surface in hyperbolic 3-space that has been constructed via a correspondence to a minimal surface in Euclidean 3-space, we can take advantage of this correspondence to find a lower bound for its index."}
{"category": "Math", "title": "Power sums of Hecke's eigenvalues and application", "abstract": "We sharpen some estimates of Rankin on power sums of Hecke eigenvalues, by using Kim & Shahidi's recent results on higher order symmetric powers. As an application, we improve Kohnen, Lau & Shparlinski's lower bound for the number of Hecke eigenvalues of same signs."}
{"category": "Math", "title": "Ehrhart Theory for Lawrence Polytopes and Orbifold Cohomology of Hypertoric Varieties", "abstract": "We establish a connection between the orbifold cohomology of hypertoric varieties and the Ehrhart theory of Lawrence polytopes. More specifically, we show that the dimensions of the orbifold cohomology groups of a hypertoric variety are equal to the coefficients of the Ehrhart $\\delta$-polynomial of the associated Lawrence polytope. As a consequence, we deduce a formula for the Ehrhart $\\delta$-polynomial of a Lawrence polytope and use the injective part of the Hard Lefschetz Theorem for hypertoric varieties to deduce some inequalities between the coefficients of the $\\delta$-polynomial."}
{"category": "Math", "title": "Limit Surfaces of Riemann Examples", "abstract": "The family of embedded, singly periodic minimal surfaces of Riemann have as limit-surfaces the helicoid, the catenoid, a single plane, or an infinite set of equally-spaced parallel planes."}
{"category": "Math", "title": "Nonsingular Ricci flow on a noncompact manifold in dimension three", "abstract": "We consider the Ricci flow $\\frac{\\partial}{\\partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rm\\geq 0, |Rm(p)|\\to 0, ~as ~d(o,p)\\to 0.$ We prove that the Ricci flow on such a manifold is nonsingular in any finite time."}
{"category": "Math", "title": "Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities", "abstract": "We study AAK-type meromorphic approximants to functions $F$, where $F$ is a sum of a rational function $R$ and a Cauchy transform of a complex measure $\\lambda$ with compact regular support included in $(-1,1)$, whose argument has bounded variation on the support. The approximation is understood in $L^p$-norm of the unit circle, $p\\geq2$. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of $\\lambda$ relative to the unit disk, that the approximants themselves converge in capacity to $F$, and that the poles of $R$ attract at least as many poles of the approximants as their multiplicity and not much more."}
{"category": "Math", "title": "On the Degree Sequence and its Critical Phenomenon of an Evolving Random Graph Process", "abstract": "In this paper we focus on the problem of the degree sequence for the following random graph process. At any time-step $t$, one of the following three substeps is executed: with probability $\\alpha_1$, a new vertex $x_t$ and $m$ edges incident with $x_t$ are added; or, with probability $\\alpha-\\alpha_1$, $m$ edges are added; or finally, with probability $1-\\a$, $m$ random edges are deleted. Note that in any case edges are added in the manner of preferential attachment. we prove that there exists a critical point $\\alpha_c$ satisfying: 1) if $\\alpha_1<\\alpha_c$, then the model has power law degree sequence; 2) if $\\alpha_1>\\alpha_c$, then the model has exponential degree sequence; and 3) if $\\alpha_1=\\alpha_c$, then the model has a degree sequence lying between the above two cases."}
{"category": "Math", "title": "Remnant inequalities and doubly-twisted conjugacy in free groups", "abstract": "We give two results for computing doubly-twisted conjugacy relations in free groups with respect to homomorphisms $\\phi$ and $\\psi$ such that certain remnant words from $\\phi$ are longer than the images of generators under $\\psi$. Our first result is a remnant inequality condition which implies that two words $u$ and $v$ are not doubly-twisted conjugate. Further we show that if $\\psi$ is given and $\\phi$, $u$, and $v$ are chosen at random, then the probability that $u$ and $v$ are not doubly-twisted conjugate is 1. In the particular case of singly-twisted conjugacy, this means that if $\\phi$, $u$, and $v$ are chosen at random, then $u$ and $v$ are not in the same singly-twisted conjugacy class with probability 1. Our second result generalizes Kim's \"bounded solution length\". We give an algorithm for deciding doubly-twisted conjugacy relations in the case where $\\phi$ and $\\psi$ satisfy a similar remnant inequality. In the particular case of singly-twisted conjugacy, our algorithm suffices to decide any twisted conjugacy relation if $\\phi$ has remnant words of length at least 2. As a consequence of our generic properties we give an elementary proof of a recent result of Martino, Turner, and Ventura, that computes the densities of injective and surjective homomorphisms from one free group to another. We further compute the expected value of the density of the image of a homomorphism."}
{"category": "Math", "title": "Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)", "abstract": "We construct families of manifolds that have pairs of genus $g$ Heegaard splittings that must be stabilized roughly $g$ times to become equivalent. We also show that when two unstabilized, boundary-unstabilized Heegaard splittings are amalgamated by a \"sufficiently complicated\" map, the resulting splitting is unstabilized. As a corollary, we produce a manifold that has distance one Heegaard splittings of arbitrarily high genus. Finally, we show that in a 3-manifold formed by a sufficiently complicated gluing, a low genus, unstabilized Heegaard splitting can be expressed in a unique way as an amalgamation over the gluing surface."}
{"category": "Math", "title": "Band-Dominated Fredholm Operators and a Question of Rabinovich, Roch and Silbermann", "abstract": "Withdrawn due to a likely error with the homeomorphism at line (4). Old abstract: In the monograph 'Limit Operators and their Applications in Operator Theory', the authors define the operator spectrum of a band-dominated operator T and prove that T is Fredholm if and only if all of the operators in its operator spectrum are invertible with uniformly bounded inverses. They also ask whether the uniform boundedness condition can in fact be dispensed with. In this note we answer this question affirmatively."}
{"category": "Math", "title": "Graph polynomials and their applications II: Interrelations and interpretations", "abstract": "This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial information it contains. The polynomials we discuss here are not generally specializations of the Tutte polynomial, but they are each in some way related to the Tutte polynomial, and often to one another. We emphasize these interrelations and explore how an understanding of one polynomial can guide research into others. We also discuss multivariable generalizations of some of these polynomials and the theory facilitated by this. We conclude with two examples, one from biology and one from physics, that illustrate the applicability of graph polynomials in other fields. This is the second chapter of a two chapter series, and concludes Graph Polynomials and Their Applications I: The Tutte Polynomial, arXiv:0803.3079"}
{"category": "Math", "title": "Functions with isolated singularities on surfaces", "abstract": "Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1, and f:M-->P be a smooth mapping. In a previous series of papers for the case when f is a Morse map the author calculated the homotopy types of stabilizers and orbits of f with respect to the right action of the diffeomorphisms group of M. The present paper extends those calculations to a large class of maps M-->P with degenerate singularities satisfying certain set of axioms."}
{"category": "Math", "title": "Kahler manifolds with quasi-constant holomorphic curvature", "abstract": "The aim of this paper is to classify compact Kahler manifolds with quasi-constant holomorphic sectional curvature."}
{"category": "Math", "title": "Retarded integral inequalities of Gronwall-Bihari type", "abstract": "We establish two nonlinear retarded integral inequalities. Bounds on the solution of some retarded equations are then obtained."}
{"category": "Math", "title": "Some Examples of Blackadar and Kirchberg's MF Algebras", "abstract": "In the paper, we provide some examples of MF algebras by considering minimal or maximal tensor products of MF algebras and crossed products of MF algebras by finite groups or an integer group. We also present some examples of C$^*$-algebras, whose BDF extension semigroup is not group. These examples include, for example, $ C_r^*(F_n)\\otimes_{max} C^*(F_n) $, $ C_r^*(F_n)\\otimes_{min} C^*(F_n) $, $C_r^*(H_1\\ast H_2)$ with $2\\le |H_1|<\\infty$ and $3\\le |H_2|<\\infty$ where $|H_1|$, $|H_2|$ are the orders of thr groups $H1$, or $H_2$ respectively, and many others."}
{"category": "Math", "title": "Promotion and Evacuation", "abstract": "Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schutzenberger. This paper surveys the basic properties of these two operations and discusses some generalizations."}
{"category": "Math", "title": "First Order Conditions for Semidefinite Representations of Convex Sets Defined by Rational or Singular Polynomials", "abstract": "A set is called semidefinite representable or semidefinite programming (SDP) representable if it can be represented as the projection of a higher dimensional set which is represented by some Linear Matrix Inequality (LMI). This paper discuss the semidefinite representability conditions for convex sets of the form S_D(f) = {x \\in D: f(x) >= 0}. Here D={x\\in R^n: g_1(x) >= 0, ..., g_m(x) >= 0} is a convex domain defined by some \"nice\" concave polynomials g_i(x) (they satisfy certain concavity certificates), and f(x) is a polynomial or rational function. When f(x) is concave over \\mc{D}, we prove that S_D(f) has some explicit semidefinite representations under certain conditions called preordering concavity or q-module concavity, which are based on the Positivstellensatz certificates for the first order concavity criteria. When f(x) is a polynomial or rational function having singularities on the boundary of S_D(f), a perspective transformation is introduced to find some explicit semidefinite representations for S_D(f) under certain conditions. In the particular case n=2, if the Laurent expansion of f(x) around one singular point has only two consecutive homogeneous parts, we show that S_D(f) always admits an explicitly constructible semidefinite representation."}
{"category": "Math", "title": "A version of smooth K-theory adapted to the total Chern class", "abstract": "A version of smooth K-theory is constructed, which is adapted to the total Chern class instead of the Chern character (contrarily to previous theories). Some total Chern class morphism from this K-theory to Cheeger-Simons differential characters is constructed. This answers a question raised by U. Bunke."}
{"category": "Math", "title": "On the strength of Hausdorff's gap condition", "abstract": "Hausdorff's gap condition was satisfied by his original 1936 construction of an (omega-1,omega-1) gap in P(N)/Fin. We solve an open problem in determining whether Hausdorff's condition is actually stronger than the more modern indestructibility condition, by constructing an indestructible (omega-1,omega-1) gap not equivalent to any gap satisfying Hausdorff's condition, from uncountably many random reals."}
{"category": "Math", "title": "Reasonable non--Radon--Nikodym ideals", "abstract": "For any abelian Polish sigma-compact group H there exist a sigma-ideal Z over N and a Borel Z-approximate homomorphism f : H --> H^N which is not Z-approximable by a continuous true homomorphism g : H --> H^N."}
{"category": "Math", "title": "About the almost everywhere convergence of the spectral expansions of functions from $L_1^\\a(S^N)$", "abstract": "In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace-Beltrami operator on the unit sphere. The sufficient conditions for summability is obtained. The more general properties and representation by the eigenfunctions of the Laplace-Beltrami operator of the Liouville space $L_1^\\a$ is used. For the orders of Riesz means, which greater than critical index $\\frac{N-1}{2}$ we proved the positive results on summability of Fourier-Laplace series. Note that when order $\\alpha$ of Riesz means is less than critical index then for establish of the almost everywhere convergence requests to use other methods form proving negative results. We have constructed different method of summability of Laplace series, which based on spectral expansions property of self-adjoint Laplace-Beltrami operator on the unit sphere."}
{"category": "Math", "title": "Maximum principle for viscosity solutions on Riemannian manifolds", "abstract": "In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different method, we generalize maximum principles of Omori and Yau to a viscosity version."}
{"category": "Math", "title": "Analytic Disks and the Projective Hull", "abstract": "Let X be a complex manifold and c a simple closed curve in X. We address the question: What conditions on c ensure the existence of a 1-dimensional complex subvariety V with boundary c in X. When X = C^n, an answer to this question involves the polynomial hull of gamma. When X = P^n, complex projective space, the projective hull hat{c} of c comes into play. One always has V contained in hat{c}, and for analytic curves they conjecturally coincide. In this paper we establish an approximate analogue of this idea which holds without the analyticity of c. We characterize points in hat{c} as those which lie on a sequence of analytic disks whose boundaries converge down to c. This is in the spirit of work of Poletsky and of Larusson-Sigurdsson, whose work is essential here. The results are applied to construct a remarkable example of a closed curve c in P^2, which is real analytic at all but one point, and for which the closure of hat{c} is W \\cup L where L is a projective line and W is an analytic (non-algebraic) subvariety of P^2 - L. Furthermore, hat{c} itself is the union of W with only two points on L."}
{"category": "Math", "title": "\"Iff\" is not expressible in independence-friendly logic", "abstract": "Ordinary first-order logic has the property that two formulas \\phi and \\psi have the same meaning in a structure if and only if the formula ``\\phi iff \\psi'' is true in the structure. We prove that independence-friendly logic does not have this property."}
{"category": "Math", "title": "Automata generating free products of groups of order 2", "abstract": "We construct a family of automata with n states, n>3, acting on a rooted binary tree that generate the free products of cyclic groups of order 2."}
{"category": "Math", "title": "Computing j-multiplicity", "abstract": "The j-multiplicity is an invariant that can be defined for any ideal in a Noetherian local ring $(R, m)$. It is equal to the Hilbert-Samuel multiplicity if the ideal is $m$-primary. In this paper we explore the computability of the j-multiplicity, giving another proof for the length formula and the additive formula."}
{"category": "Math", "title": "Selberg's zeta functions for congruence subgroups of modular groups in SL(2,R) and SL(2,C)", "abstract": "It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to any congruence subgroup of the modular groups in SL(2,R) and SL(2,C)."}
{"category": "Math", "title": "$n$-level density of the low-lying zeros of quadratic Dirichlet $L$-functions", "abstract": "The Density Conjecture of Katz and Sarnak associates a classical compact group to each reasonable family of $L$-functions. Under the assumption of the Generalized Riemann Hypothesis, Rubinstein computed the $n$-level density of low-lying zeros for the family of quadratic Dirichlet $L$-functions in the case that the Fourier transform $\\hat{f}(u)$ of any test function $f$ is supported in the region $\\sum^n_{j=1}u_j < 1$ and showed that the result agrees with the Density Conjecture. In this paper, we improve Rubinstein's result on computing the $n$-level density for the Fourier transform $\\hat{f}(u)$ being supported in the region $\\sum^n_{j=1}u_j < 2$."}
{"category": "Math", "title": "Graph polynomials and Tutte-Grothendieck invariants: an application of elementary finite Fourier analysis", "abstract": "This paper is based on a series of talks given at the Patejdlovka Enumeration Workshop held in the Czech Republic in November 2007. The topics covered are as follows. The graph polynomial, Tutte-Grothendieck invariants, an overview of relevant elementary finite Fourier analysis, the Tutte polynomial of a graph as a Hamming weight enumerator of its set of tensions (or flows), and a description of a family of polynomials containing the graph polynomial which yield Tutte-Grothendieck invariants in a similar way."}
{"category": "Math", "title": "Weighted pluricomplex energy", "abstract": "We study the complex Monge-Ampre operator on the classes of finite pluricomplex energy $\\mathcal{E}_\\chi (\\Omega)$ in the general case ($\\chi(0)=0$ i.e. the total Monge-Ampre mass may be infinite). We establish an interpretation of these classes in terms of the speed of decrease of the capacity of sublevel sets and give a complete description of the range of the operator $(dd^c \\cdot)^n$ on the classes $\\mathcal{E}\\chi(\\Omega).$"}
{"category": "Math", "title": "The PBW Filtration, Demazure Modules and Toroidal Current Algebras", "abstract": "Let $L$ be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra $\\hat{\\mathfrak g}$. The $m$-th space $F_m$ of the PBW filtration on $L$ is a linear span of vectors of the form $x_1... x_lv_0$, where $l\\le m$, $x_i\\in \\hat{\\mathfrak g}$ and $v_0$ is a highest weight vector of $L$. In this paper we give two descriptions of the associated graded space $L^{\\rm gr}$ with respect to the PBW filtration. The \"top-down\" description deals with a structure of $L^{\\rm gr}$ as a representation of the abelianized algebra of generating operators. We prove that the ideal of relations is generated by the coefficients of the squared field $e_\\theta(z)^2$, which corresponds to the longest root $\\theta$. The \"bottom-up\" description deals with the structure of $L^{\\rm gr}$ as a representation of the current algebra ${\\mathfrak g}\\otimes {\\mathbb C}[t]$. We prove that each quotient $F_m/F_{m-1}$ can be filtered by graded deformations of the tensor products of $m$ copies of ${\\mathfrak g}$."}
{"category": "Math", "title": "Products of Functions in Hardy and Lipschitz or Bmo Spaces", "abstract": "We define as a distribution the product of a function (or distribution) h in some Hardy space Hp with a function b in the dual space of Hp. Moreover, we prove that the product bxh may be written as the sum of an integrable function with a distribution that belongs to some Hardy-Orlicz space, or to the same Hardy space Hp, depending on the values of p."}
{"category": "Math", "title": "New estimates and tests of independence in some copula models", "abstract": "We introduce new estimates and tests of independence in copula models with unknown margins using $\\phi$-divergences and the duality technique. The asymptotic laws of the estimates and the test statistics are established both when the parameter is an interior or a boundary value of the parameter space. Simulation results show that the choice of $\\chi^2$-divergence has good properties in terms of efficiency-robustness."}
{"category": "Math", "title": "Algorithm for computing local Bernstein-Sato ideals", "abstract": "Given $p$ polynomials of $n$ variables over a field $k$ of characteristic 0 and a point $a \\in k^n$, we propose an algorithm computing the local Bernstein-Sato ideal at $a$. Moreover with the same algorithm we compute a constructible stratification of $k^n$ such that the local Bernstein-Sato ideal is constant along each stratum. Finally, we present non-trivial examples computed with our algorithm."}
{"category": "Math", "title": "A Satake type theorem for Super Automorphic forms", "abstract": "Aim of this article is a Satake type theorem for super automorphic forms on a complex bounded symmetric super domain B of rank 1 with respect to a lattice. This theorem - roughly speaking - says that for large weight k and all p from 1 to infinity (both including) a super automorphic form on B is a super cusp form if and only if its p-norm with respect to a certain measure on the quotient of B is finite. And so in particular all these Lp-spaces coincide! We will give a proof of this theorem using an unbounded realization of B and Fourier decomposition at the cusps of the quotient mapped to infinity via a partial Cayley transform."}
{"category": "Math", "title": "Self-similar Focusing in Porous Media: An Explicit Calculation", "abstract": "We consider a porous mediaum flow in which the gas is initially distributed in the exterior of an empty region (a hole) and study the final stage of the hole-filling process. Hole-filling is asymptotically described by a self-similar solution which depends on a constant determined by the initial configuration. In general, this constant must be found numerically.. Here we give an example of a one-dimensional symmetric flow where the constant is obtained explicitly."}
{"category": "Math", "title": "Geometry of syzygies via Poncelet varieties", "abstract": "We consider the Grassmannian $\\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({\\P^1},\\O_{\\P^1}(n))$. We define $\\frak{X}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on $\\P^1$ whose basis realize a fixed number of polynomial relations of fixed degree, say a fixed number of syzygies of a certain degree. The first result of this paper is the computation of the dimension of $\\frak{X}_{k,r,d}$. In the second part we make a link between $\\frak{X}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the Poncelet varieties."}
{"category": "Math", "title": "Instability results for an elliptic equation on compact Riemannian manifolds with non-negative Ricci curvature", "abstract": "We prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smooth N-dimensional Riemannian manifold without boundary with non-negative Ricci curvature. Conversely for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative we prove existence of nonconstant local minimizers for the same class of functionals."}
{"category": "Math", "title": "Smoothing Effects for Navier-Stokes Equations", "abstract": "We prove some smoothing effects for the 3-D Navier-Stokes equations for initial data belonging to the critical Sobolev space $H^{1/2}(\\R^3)$. Asymptotic behavior of the global solution when the time goes to infinity is studied. We also obtain a new energy estimate. Other results in this direction and with different methods can be found in \\cite{C4}."}
{"category": "Math", "title": "Asymptotic Analysis of MHD Systems", "abstract": "In this paper, we study the convergence of strong solutions of a Magneto-Hydro-Dynamic system. On the torus ${\\mathbb{T}}^3$, the proof is based on Schochet's methods, whereas in the case of the whole space ${\\bf \\mathbb{R}^3}$, we use Strichartz's type estimates and a product law's $2D\\times3D$."}
{"category": "Math", "title": "The Problem with the Linpack Benchmark Matrix Generator", "abstract": "We characterize the matrix sizes for which the Linpack Benchmark matrix generator constructs a matrix with identical columns."}
{"category": "Math", "title": "Diffusive stability of oscillations in reaction-diffusion systems", "abstract": "We study nonlinear stability of spatially homogeneous oscillations in reaction-diffusion systems. Assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase, we prove that spatially localized perturbations decay algebraically with the diffusive rate t^{-n/2} in space dimension n. We also compute the leading order term in the asymptotic expansion of the solution, and show that it corresponds to a spatially localized modulation of the phase. Our approach is based on a normal form transformation in the kinetics ODE which partially decouples the phase equation, at the expense of making the whole system quasilinear. Stability is then obtained by a global fixed point argument in temporally weighted Sobolev spaces."}
{"category": "Math", "title": "Constructing arithmetic subgroups of unipotent groups", "abstract": "Let G be a unipotent algebraic subgroup of some GL_m(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G \\cap GL_m(Z). This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists."}
{"category": "Math", "title": "Discrete Tracy-Widom Operators", "abstract": "Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. This paper considers discrete Tracy-Widom operators, and gives sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation."}
{"category": "Math", "title": "Automorphisms and derivations of Borel subalgebras and their nilradicals in Kac-Moody algebras", "abstract": "In this paper, we determine derivations of Borel subalgebras and their derived subalgebras called nilradicals, in Kac-Moody algebras (and contragredient Lie algebras) over any field of characteristic 0; and we also determine automorphisms of those subalgebras in symmetrizable Kac-Moody algebras. The results solve a conjecture posed by R. V. Moody about 30 years ago which generalizes a result by B. Kostant and which is discussed by A. Fialowski using Lie algebra cohomology in case of affine type."}
{"category": "Math", "title": "Matrix factorizations and intertwiners of the fundamental representations of quantum group U_q (sl_n)", "abstract": "We want to construct a homological link invariant whose Euler characteristic is MOY polynomial as Khovanov and Rozansky constructed a categorification of HOMFLY polynomial. The present paper gives the first step to construct a categorification of MOY polynomial. For the essential colored planar diagrams with additional data which is a sequence naturally induced by coloring, we define matrix factorizations, and then we define a matrix factorization for planar diagram obtained by gluing the essential colored planar diagrams as tensor product of the matrix factorizations for the essential planar diagrams. Moreover, we show that some matrix factorizations deribed from tensor product of the essential matrix factorizations have homotopy equivalences corresponding to MOY relations."}
{"category": "Math", "title": "Injectives in residuated algebras", "abstract": "Injectives in several classes of structures associated with logic are characterized. Among the classes considered are residuated lattices, MTL-algebras, IMTL-algebras, BL-algebras, NM-algebras and bounded hoops."}
{"category": "Math", "title": "Examples of finite $p$-divisible sets of MHS", "abstract": "We provide some examples which give evidence to the conjectures contained in my paper \"Finiteness of $p$-Divisible Sets of Multiple Harmonic Sums\" (math.NT/0303043). All the main theoretical results can be found in that paper."}
{"category": "Math", "title": "Pavelka-style completeness in expansions of \\L ukasiewicz logic", "abstract": "An algebraic setting for the validity of Pavelka style completeness for some natural expansions of \\L ukasiewicz logic by new connectives and rational constants is given. This algebraic approach is based on the fact that the standard MV-algebra on the real segment $[0, 1]$ is an injective MV-algebra. In particular the logics associated with MV-algebras with product and with divisible MV-algebras are considered."}
{"category": "Math", "title": "On the boundary of the moduli spaces of log Hodge structures: triviality of the torsor", "abstract": "In this paper we will study the moduli spaces of log Hodge structures introduced by Kato-Usui. This moduli space is a partial compactification of a discrete quotient of a period domain. We treat the following 2 cases: (A) the case where the period domain is Hermitian symmetric, (B) the case where the Hodge structures are of the mirror quintic type. Especially we study a property of the torsor."}
{"category": "Math", "title": "Bounds for solution of linear diophantine equations", "abstract": "Given linear diophantine equation Ax=b, rank A=m. Let d be the maximum of absolute values of the mxm minors of the matrix (A | b). It is shown that if M={x : Ax=b, x nonnegative and integer} is nonempty, then there exists x=(x1,...,xn) in M, such that xi does not exceed d (i=1,2,..,n)."}
{"category": "Math", "title": "On Galois representations and Hilbert-Siegel modular forms", "abstract": "In this paper, we associate Galois representations to globally generic cuspidal automorphic representations on GSp(4), over a totally real field F, which are Steinberg at some finite place. This association is compatible with the local Langlands correspondence for GSp(4) studied recently in a preprint of Gan and Takeda. As a corollary, we relate the rank of the monodromy operator at p to the dimensions of the parahoric fixed spaces at p. The Galois representations are constructed by first passing to GL(4) over a CM extension, then applying the book of Harris-Taylor plus a refinement due to Taylor-Yoshida, and finally descending to F by a delicate patching argument. This is a variation of the techniques used by Blasius and Rogawski in order to attach motives to Hilbert modular forms."}
{"category": "Math", "title": "Weakly almost periodic functionals on the measure algebra", "abstract": "It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C$^*$-algebra. This implies that the weakly almost periodic functionals on $M(G)$, the measure algebra of a locally compact group $G$, is a C$^*$-subalgebra of $M(G)^* = C_0(G)^{**}$. The proof builds upon a factorisation result, due to Young and Kaiser, for weakly compact module maps. The main technique is to adapt some of the theory of corepresentations to the setting of general reflexive Banach spaces."}
{"category": "Math", "title": "Growth of intersection numbers for free group automorphisms", "abstract": "For a fully irreducible automorphism \\phi of the free group F_k we compute the asymptotics of the intersection number n \\mapsto i(T,T'\\phi^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the Guirardel core for the trees T and T'\\phi^n for n large."}
{"category": "Math", "title": "Geometry of configuration spaces of tensegrities", "abstract": "Consider a graph G with n vertices. In this paper we study geometric conditions for an n-tuple of points in R^d to admit a tensegrity with underlying graph G. We introduce and investigate a natural stratification, depending on G, of the configuration space of all n-tuples in R^d. In particular we find surgeries on graphs that give relations between different strata. Based on numerous examples we give a description of geometric conditions defining the strata for plane tensegrities, we conjecture that the list of such conditions is sufficient to describe any stratum. We conclude the paper with particular examples of strata for tensegrities in the plane with a small number of vertices."}
{"category": "Math", "title": "Singular Points of Real Quintic Curves Via Computer Algebra", "abstract": "There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for reducible real quintic curves. The classification of real singular points for irreducible real quintic curves is originally due to Golubina and Tai. There are 28 types of singular points for irreducible complex quintic curves and 33 types of singular points for reducible complex quintic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, a major component of the proof consists of a sequence of large symbolic computations that can be done nicely using Maple."}
{"category": "Math", "title": "Zeros of partial sums of the Riemann zeta-function", "abstract": "We investigate the distribution of the zeros of partial sums of the Riemann zeta-function, sum_{n\\leq X}n^{-s}, estimating the number of zeros up to height T, the number of zeros to the right of a given vertical line, and other aspects of their horizontal distribution."}
{"category": "Math", "title": "Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure", "abstract": "Two types of Gaussian processes, namely the Gaussian field with generalized Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy covariance (GSGCC) are considered. Some of the basic properties and the asymptotic properties of the spectral densities of these random fields are studied. The associated self-similar random fields obtained by applying the Lamperti transformation to GFGCC and GSGCC are studied."}
{"category": "Math", "title": "Deformations of linear Poisson orbifolds", "abstract": "Let $\\Gamma$ be a finite group acting faithfully and linearly on a vector space $V$. Let $T(V)$ ($S(V)$) be the tensor (symmetric) algebra associated to $V$ which has a natural $\\Gamma$ action. We study generalized quadratic relations on the tensor algebra $T(V)\\rtimes \\Gamma$. We prove that the quotient algebras of $T(V)\\rtimes \\Gamma$ by such relations satisfy PBW property. Such quotient algebras can be viewed as quantizations of linear or constant Poisson structures on $S(V)\\rtimes \\Gamma$, and are natural generalizations of symplectic reflection algebras."}
{"category": "Math", "title": "Equidistribution of Fekete points on complex manifolds", "abstract": "We prove the several variable version of the classical equidistribution theorem for Fekete points of a compact subset of the complex plane, which settles a well-known conjecture in pluri-potential theory. The result is obtained as a special case of a general equidistribution theorem for Fekete points in the setting of a given holomorphic line bundle over a compact complex manifold. The proof builds on our recent work \"Capacities and weighted volumes for line bundles\"."}
{"category": "Math", "title": "Nielsen theory, Floer homology and a generalisation of the Poincare-Birkhoff theorem", "abstract": "The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltotic symplectic invariant. A generalisation of the Poincare- Birkhoff fixed point theorem and Arnold conjecture is proposed."}
{"category": "Math", "title": "A construction of symmetric linear functions of the restricted quantum group \\overline{U}_q (sl_2)", "abstract": "In this paper we construct all the primitive idempotents of the restricted quantum group $\\overline{U}_q (sl_2)$ and also determine the multiplication rules among a basis given by the action of generators of $\\bar{U}_q (sl_2)$ to the idempotents. By using this result we construct a basis of the space of symmetric linear functions of $\\overline{U}_q (sl_2)$ and determine the decomposition of the integral of the dual of $\\overline{U}_q (sl_2)$ twisted by the balancing element to the basis of the space of symmetric linear functions."}
{"category": "Math", "title": "Frequentist and Bayesian measures of confidence via multiscale bootstrap for testing three regions", "abstract": "A new computation method of frequentist $p$-values and Bayesian posterior probabilities based on the bootstrap probability is discussed for the multivariate normal model with unknown expectation parameter vector. The null hypothesis is represented as an arbitrary-shaped region. We introduce new parametric models for the scaling-law of bootstrap probability so that the multiscale bootstrap method, which was designed for one-sided test, can also computes confidence measures of two-sided test, extending applicability to a wider class of hypotheses. Parameter estimation is improved by the two-step multiscale bootstrap and also by including higher-order terms. Model selection is important not only as a motivating application of our method, but also as an essential ingredient in the method. A compromise between frequentist and Bayesian is attempted by showing that the Bayesian posterior probability with an noninformative prior is interpreted as a frequentist $p$-value of ``zero-sided'' test."}
{"category": "Math", "title": "Discontinuous Superprocesses with Dependent Spatial Motion", "abstract": "We construct a class of discontinuous superprocesses with dependent spatial motion and general branching mechanism. The process arises as the weak limit of critical interacting-branching particle systems where the spatial motions of the particles are not independent. The main work is to solve the martingale problem. When we turn to the uniqueness of the process, we generalize the localization method introduced by [D.W. Stroock, Diffusion processes associated with Levy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32(1975) 209--244] to the measure-valued context. As for existence, we use particle system approximation and a perturbation method. This work generalizes the model introduced in [D.A. Dawson, Z. Li, H. Wang, Superprocesses with dependent spatial motion and general branching densities, Electron. J. Probab. 6(2001), no.25, 33 pp. (electronic)] where quadratic branching mechanism was considered. We also investigate some properties of the process."}
{"category": "Math", "title": "Asymptotic formulas for partial sums of class numbers of indefinite binary quadratic forms", "abstract": "Sarnak obtained the asymptotic formula of the sum of the class numbers of indefinite binary quadratic forms from the prime geodesic theorem for the modular group. In the present paper, we show several asymptotic formulas of partial sums of the class numbers by using the prime geodesic theorems for the congruence subgroups of the modular group."}
{"category": "Math", "title": "Equivariant differential characters and symplectic reduction", "abstract": "We describe equivariant differential characters (classifying equivariant circle bundles with connections), their prequantization, and reduction."}
{"category": "Math", "title": "Some Remarks on the Braided Thompson Group BV", "abstract": "Matthew Brin and Patrick Dehornoy independently discovered a braided version BV of Thompson's group V. In this paper, we discuss some properties of BV that might make the group interesting for group based cryptography. In particular, we show that BV does not admit a non-trivial linear representation."}
{"category": "Math", "title": "Concentration inequalities for $s$-concave measures of dilations of Borel sets and applications", "abstract": "We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in $\\mathbb{R}^n$ by a $s$-concave probability. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Gu\\'edon. Applying our inequality to the level sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary $s$-concave probability."}
{"category": "Math", "title": "A homogeneous Gibbons-Hawking ansatz and Blaschke products", "abstract": "A homogeneous Gibbons-Hawking ansatz is described, leading to 4-dimensional hyperkahler metrics with homotheties. In combination with Blaschke products on the unit disc in the complex plane, this ansatz allows one to construct infinite-dimensional families of such hyperkahler metrics that are, in a suitable sense, complete. Our construction also gives rise to incomplete metrics on 3-dimensional contact manifolds that induce complete Carnot-Caratheodory distances."}
{"category": "Math", "title": "A proof of the Riemann hypothesis", "abstract": "This paper has been withdrawn by the author, due to a mistake on page 29."}
{"category": "Math", "title": "Sets of non-differentiability for conjugacies between expanding interval maps", "abstract": "We study differentiability of topological conjugacies between expanding piecewise $C^{1+\\epsilon}$ interval maps. If these conjugacies are not $C^1$, then they have zero derivative almost everywhere. We obtain the result that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Using multifractal analysis and thermodynamic formalism, we show that this Hausdorff dimension is explicitly determined by the Lyapunov spectrum. Moreover, we show that these results give rise to a \"rigidity dichotomy\" for the type of conjugacies under consideration."}
{"category": "Math", "title": "Weyl groupoids of rank two and continued fractions", "abstract": "A relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two is found. This allows to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for the entries of the Cartan matrices. Key words: Cartan matrix, continued fraction, Nichols algebra, Weyl groupoid"}
{"category": "Math", "title": "A note on the Ricci flow on noncompact manifolds", "abstract": "Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\\to 0$ as $x\\to \\infty$. Then the Ricci flow with initial data $(M^3,g_0)$ has a long time solution. This extends a recent result of Ma and Zhu. We also have a higher dimensional version, and we reprove a K$\\ddot{a}$hler analogy due to Chau, Tam and Yu."}
{"category": "Math", "title": "Isochronicity conditions for some real polynomial systems", "abstract": "This paper focuses on isochronicity of linear center perturbed by a polynomial. Isochronicity of a linear center perturbed by a degree four and degree five polynomials is studied, several new isochronous centers are found. For homogeneous isochronous perturbations, a first integral and a linearizing change of coordinates are presented. Moreover, a family of Abel polynomial systems is also considered. By investigations until degree 10 we prove the existence of a unique isochronous center. These results are established using a computer implementation based on Urabe theorem."}
{"category": "Math", "title": "An elementary proof of the Briancon-Skoda theorem", "abstract": "We give a new elementary proof of the Brian\\c{c}on-Skoda theorem, which states that for an $m$-generated ideal $\\mathfrak{a}$ in the ring of germs of analytic functions at $0\\in \\C^n$, the $\\nu$:th power of its integral closure is contained in $\\mathfrak{a}$, where $\\nu = \\min(m,n)$."}
{"category": "Math", "title": "Nonhomogeneous analytic families of trees", "abstract": "We consider a dichotomy for analytic families of trees stating that either there is a colouring of the nodes for which all but finitely many levels of every tree are nonhomogeneous, or else the family contains an uncountable antichain. This dichotomy implies that every nontrivial Souslin poset satisfying the countable chain condition adds a splitting real. We then reduce the dichotomy to a conjecture of Sperner Theory. This conjecture is concerning the asymptotic behaviour of the product of the sizes of the m-shades of pairs of cross-t-intersecting families."}
{"category": "Math", "title": "Sur Les Suites D'Interpolation Pour Les Espaces De Bergman a Poids Dans la Boule De $\\mathbb{C}^n$}", "abstract": "Let $A$ be a sequence of points of $\\mathbb{B}^n$ the unit ball in $\\mathbb{C}^n.$ In terms of interpolating vectorial function (or Amar's function)[1], we give a necessary condition on $A$ to be interpolating for the weighted Bergman space $B^p_\\alpha (\\mathbb{B}^n). In the particular case of Hardy space $H^p (\\mathbb{B}^2)$, this condition is sufficient no optimal. In the main theorem proof, we resolve Gleason's problem (vectorial form) in $B^p_\\alpha (\\mathbb{B}^n)$"}
{"category": "Math", "title": "On the unitarization of linear representations of primitive partially ordered sets", "abstract": "We describe all weights which are appropriated for the unitarization of linear representations of primitive partially ordered sets of finite type."}
{"category": "Math", "title": "Classifying Camina groups: a theorem of Dark and Scoppola", "abstract": "Recall that a group $G$ is a Camina group if every nonlinear irreducible character of $G$ vanishes on $G \\setminus G'$. Dark and Scoppola classified the Camina groups that can occur. We present a different proof of this classification using Theorem 2, which strengthens a result of Isaacs on Camina pairs. Theorem 2 is of independent interest."}
{"category": "Math", "title": "A conjecture on homotopy groups of spheres, details on the algebra of higher cohomology operations", "abstract": "The theory of secondary chomology operations leads to a conjecture concerning the algebra of higher cohomology operations in general. This conjecture is discussed here in detail and its connection with homotopy groups of spheres and the Adams spectral sequence is described."}
{"category": "Math", "title": "Properties of Design-Based Functional Principal Components Analysis", "abstract": "This work aims at performing Functional Principal Components Analysis (FPCA) with Horvitz-Thompson estimators when the observations are curves collected with survey sampling techniques. One important motivation for this study is that FPCA is a dimension reduction tool which is the first step to develop model assisted approaches that can take auxiliary information into account. FPCA relies on the estimation of the eigenelements of the covariance operator which can be seen as nonlinear functionals. Adapting to our functional context the linearization technique based on the influence function developed by Deville (1999), we prove that these estimators are asymptotically design unbiased and consistent. Under mild assumptions, asymptotic variances are derived for the FPCA' estimators and consistent estimators of them are proposed. Our approach is illustrated with a simulation study and we check the good properties of the proposed estimators of the eigenelements as well as their variance estimators obtained with the linearization approach."}
{"category": "Math", "title": "Une minoration du minimum essentiel sur les varietes abeliennes", "abstract": "We extend to the general codimension a lower bound for the essential minimum on abelian varieties found in a previous work, under a conjecture about ordinary primes in abelian varieties. This lower bound is the best expected, ``up to an epsilon'', in the degree of the subvariety. Following a strategy of Amoroso, we change the transcendance phasis in order to simplify the zero lemma and its combinatorics. The last argument is a descent procedure on varieties, which is far more intricate in the abelian setting since there is not, in general, a lifting of the Frobenius in characteristic zero."}
{"category": "Math", "title": "The mapping class group cannot be realized by homeomorphisms", "abstract": "Let $M$ be a closed surface. By $\\Homeo(M)$ we denote the group of orientation preserving homeomorphisms of $M$ and let $\\MC(M)$ denote the Mapping class group. In this paper we complete the proof of the conjecture of Thurston that says that for any closed surface $M$ of genus $\\g \\ge 2$, there is no homomorphic section $\\E:\\MC(M) \\to \\Homeo(M)$ of the standard projection map $\\Proj:\\Homeo(M) \\to \\MC(M)$."}
{"category": "Math", "title": "Covers of surfaces with fixed branch locus", "abstract": "Given a connected smooth projective surface X over the complex numbers, together with a simple normal crossings divisor D on it, we study finite normal covers Y of X that are unramified outside D. Given moreover a fibration of X onto a curve C, we prove that the `height' of Y over C is bounded linearly in terms of the degree of Y over X. We indicate how an arithmetic analogue of this result, if true, can be auxiliary in proving the existence of a polynomial time algorithm that computes the mod-l Galois representations associated to a given smooth projective geometrically connected surface over the rational numbers. A precise conjecture is formulated."}
{"category": "Math", "title": "Singular Points of Reducible Sextic Curves", "abstract": "There are 106 individual types of singular points for reducible complex sextic curves."}
{"category": "Math", "title": "Characterizing two-timescale nonlinear dynamics using finite-time Lyapunov exponents and vectors", "abstract": "Finite-time Lyapunov exponents and vectors are used to define and diagnose boundary-layer type, two-timescale behavior in the tangent linear dynamics and to determine the associated manifold structure in the flow of a finite-dimensional nonlinear autonomous dynamical system. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for a state space region. The slow-fast splitting is defined using finite-time Lyapunov exponents and vectors, guided by the asymptotic theory of partially hyperbolic sets, with important modifications for the finite-time case; for example, finite-time Lyapunov analysis relies more heavily on the Lyapunov vectors due to their relatively fast convergence compared to that of the corresponding exponents. The splitting is used to locate points on normally hyperbolic center manifolds. Determining manifolds from tangent bundle structure is more generally applicable than approaches, such as the singular perturbation method, that require special normal forms or other a priori knowledge. The use, features, and accuracy of the approach are illustrated via several detailed examples."}
{"category": "Math", "title": "Selfdual representations of division algebras and Weil groups: A contrast", "abstract": "Selfdual representations of any group fall into two classes when they are irreducible: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the irreducible representations \\sigma of the Weil group of a local field k of dimension n with the irreducible representations \\pi of the invertible elements of a division algebra D over k of index n, takes selfdual representations to selfdual representations. In this paper we use global methods to study how the Langlands correspondence behaves relative to this distinction among selfdual representations. We prove in particular that for n even, \\sigma is symplectic if and only if \\pi is orthogonal. Our results treat more generally the case of GL_m(B), for B a division algebra over k of index r, and n=mr."}
{"category": "Math", "title": "A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure", "abstract": "We characterize two-weight inequalities for certain maximal truncations of the Hilbert transform in terms of testing conditions on simpler functions. For 1<p<2 and two positive Borel measures u, v on R, we assume that u is doubling, and we consider maximal truncations T_# of the Hilbert transform. The norm estimate || T(f u) ||_{L^p(v)} < C || f ||_{L^p(u)} is characterized in terms of an A_p condition on the weights and two testing conditions. The first is the norm condition above, but the function f varies over bounded functions supported on a cube. The second is a dual weak-type condition, for arbitrary functions. This result should be compared to the result of Nazarov, Treil and Volberg, arXiv:math/0702758. Additional results are obtained for 2<p<\\infty, and for the weak type inequality."}
{"category": "Math", "title": "Discrete Symbol Calculus", "abstract": "This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phase-space, i.e., functions of space $x$ and frequency $\\xi$. The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow to write fast-converging, non-asymptotic expansions in adequate systems of rational Chebyshev functions or hierarchical splines. The classical results of closedness of such symbol classes under multiplication, inversion and taking the square root translate into practical iterative algorithms for realizing these operations directly in the proposed expansions. Because symbol-based numerical methods handle operators and not functions, their complexity depends on the desired resolution $N$ very weakly, typically only through $\\log N$ factors. We present three applications to computational problems related to wave propagation: 1) preconditioning the Helmholtz equation, 2) decomposing wavefields into one-way components and 3) depth-stepping in reflection seismology."}
{"category": "Math", "title": "Random systems of polynomial equations. The expected number of roots under smooth analysis", "abstract": "We consider random systems of equations over the reals, with $m$ equations and $m$ unknowns $P_i(t)+X_i(t)=0$, $t\\in\\mathbb{R}^m$, $i=1,...,m$, where the $P_i$'s are non-random polynomials having degrees $d_i$'s (the \"signal\") and the $X_i$'s (the \"noise\") are independent real-valued Gaussian centered random polynomial fields defined on $\\mathbb{R}^m$, with a probability law satisfying some invariance properties. For each $i$, $P_i$ and $X_i$ have degree $d_i$. The problem is the behavior of the number of roots for large $m$. We prove that under specified conditions on the relation signal over noise, which imply that in a certain sense this relation is neither too large nor too small, it follows that the quotient between the expected value of the number of roots of the perturbed system and the expected value corresponding to the centered system (i.e., $P_i$ identically zero for all $i=1,...,m$), tends to zero geometrically fast as $m$ tends to infinity. In particular, this means that the behavior of this expected value is governed by the noise part."}
{"category": "Math", "title": "Tridiagonal pairs of $q$-Racah type", "abstract": "Let $K$ denote an algebraically closed field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \\to V$ and $A^*:V \\to V$ that satisfy the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\\lbrace V_i\\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \\subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \\leq i \\leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\\lbrace V^*_i\\rbrace_{i=0}^\\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \\subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \\leq i \\leq \\delta$, where $V^*_{-1}=0$ and $V^*_{\\delta+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \\subseteq W$, $A^* W \\subseteq W$, $W \\neq 0$, $W \\neq V$. We call such a pair a {\\it tridiagonal pair} on $V$. It is known that $d=\\delta$. For $0 \\leq i \\leq d$ let $\\theta_i$ (resp. $\\theta^*_i$) denote the eigenvalue of $A$ (resp. $A^*$) associated with $V_i$ (resp. $V^*_i$). The pair $A,A^*$ is said to have {\\it $q$-Racah type} whenever $\\theta_i = a + b q^{2i-d}+ c q^{d-2i}$ and $\\theta^*_i = a^* + b^*q^{2i-d}+c^*q^{d-2i}$ for $0 \\leq i \\leq d$, where $q, a,b,c,a^*,b^*,c^*$ are scalars in $K$ with $q,b,c,b^*,c^*$ nonzero and $q^2 \\not\\in \\lbrace 1,-1\\rbrace$. This type is the most general one. We classify up to isomorphism the tridiagonal pairs over $K$ that have $q$-Racah type. Our proof involves the representation theory of the quantum affine algebra $U_q(\\widehat{\\mathfrak{sl}}_2)$."}
{"category": "Math", "title": "Stratification of Unfoldings of Corank 1 Singularities", "abstract": "In the study of equisingularity of families of mappings Gaffney introduced the crucial notion of excellent unfoldings. This definition essentially says that the family can be stratified so that there are no strata of dimension 1 other than the parameter axis for the family. Consider a family of corank 1 multi-germs with source dimension less than target. In this paper it is shown how image Milnor numbers can ensure some of the conditions involved in being excellent. The methods used can also be successfully applied to cases where the double point set is a curve. In order to prove the results the rational cohomology description of the disentanglement of a corank 1 multi-germ is given for the first time. Then, using a simple generalization of the Marar-Mond Theorem on the multiple point space of such maps, this description is applied to give conditions which imply the upper semi-continuity of the image Milnor number. From this the main results follow."}
{"category": "Math", "title": "The symmetries of the 2phi1", "abstract": "We show that the only symmetries of the 2phi1 within a large class of possible transformations are Heine's transformations. The class of transformations considered consists of equation of the form 2phi1(a,b;c;q,z)= f(a,b,c,z) 2phi1(L(a,b,c,q,z)), where f is a q-hypergeometric term and L a linear operator on the logarithms of the parameters. We moreover prove some results on q-difference equations satisfied by 2phi1, which are used to prove the main result."}
{"category": "Math", "title": "Melnikov theory to all orders and Puiseux series for subharmonic solutions", "abstract": "We study the problem of subharmonic bifurcations for analytic systems in the plane with perturbations depending periodically on time, in the case in which we only assume that the subharmonic Melnikov function has at least one zero. If the order of zero is odd, then there is always at least one subharmonic solution, whereas if the order is even in general other conditions have to be assumed to guarantee the existence of subharmonic solutions. Even when such solutions exist, in general they are not analytic in the perturbation parameter. We show that they are analytic in a fractional power of the perturbation parameter. To obtain a fully constructive algorithm which allows us not only to prove existence but also to obtain bounds on the radius of analyticity and to approximate the solutions within any fixed accuracy, we need further assumptions. The method we use to construct the solution -- when this is possible -- is based on a combination of the Newton-Puiseux algorithm and the tree formalism. This leads to a graphical representation of the solution in terms of diagrams. Finally, if the subharmonic Melnikov function is identically zero, we show that it is possible to introduce higher order generalisations, for which the same kind of analysis can be carried out."}
{"category": "Math", "title": "Effective Construction of a Positive Operator which does not admit Triangular Factorization", "abstract": "We have constructed a concrete example of a non-factorable positive operator. As a result, for the well-known problems (Ringrose, Kadison and Singer problems) we replace existence theorems by concrete examples."}
{"category": "Math", "title": "Hypercontact structures and Floer homology", "abstract": "We introduce a new Floer theory associated to a pair consisting of a Cartan hypercontact 3-manifold M and a hyperkaehler manifold X."}
{"category": "Math", "title": "Optimal consumption policies in illiquid markets", "abstract": "We investigate optimal consumption policies in the liquidity risk model introduced in Pham and Tankov (2007). Our main result is to derive smoothness results for the value functions of the portfolio/consumption choice problem. As an important consequence, we can prove the existence of the optimal control (portfolio/consumption strategy) which we characterize both in feedback form in terms of the derivatives of the value functions and as the solution of a second-order ODE. Finally, numerical illustrations of the behavior of optimal consumption strategies between two trading dates are given."}
{"category": "Math", "title": "Harmonic symmetrization of convex sets and of Finsler structures, with applications to Hilbert geometry", "abstract": "David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain $\\Omega$ in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of the boundary of $\\Omega$ and refers to a theorem by Busemann and Mayer that produces the norm of a tangent vector from the distance function. In this paper, we develop a new approach for the study of the Hilbert metric where no differentiability is assumed. The approach exhibits the Hilbert metric on a domain as a symmetrization of a natural weak metric, known as the Funk metric. The Funk metric is described as a tautological weak Finsler metric, in which the unit ball at each tangent space is naturally identified with the domain $\\Omega$ itself. The Hilbert metric is then identified with the reversible tautological weak Finsler structure on $\\Omega$, and the unit ball at each point is described as the harmonic symmetrization of the unit ball of the Funk metric. Properties of the Hilbert metric then follow from general properties of harmonic symmetrizations of weak Finsler structures."}
{"category": "Math", "title": "Harish-Chandra bimodules for quantized Slodowy slices", "abstract": "The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizaions of the Poisson algebra of polynomial functions on the Slodowy slice. In this paper, we define and study Harish-Chandra bimodules over Premet's algebras. We apply the technique of Harish-Chandra bimodules to prove a conjecture of Premet concerning primitive ideals, and to construct `noncommutative resolutions' of Slodowy slices via translation functors."}
{"category": "Math", "title": "Cayley-Dicksonia Revisited", "abstract": "In the theory of the hypercomplex, the laws governing the algebra are based on units that are naturally associated with an orthogonal vector space, a requirement that is far from mandatory in many algebraic formulations arising in the context of the reals or the complex numbers.In this article the complementing view is held, in that the laws of hypercomplex algebra are recast in terms of quite generally posited units. Proceeding in this manner, a generalized form of the Cayley-Dickson process is examined. The representations given are regular bimodular; the resulting matrices are standard except they are allowed nonstandard multiplication for noncommutative matrix elements."}
{"category": "Math", "title": "Confluence of singularities of differential equation: a Lie algebra contraction approach", "abstract": "We investigate here the confluence of singularities of Mathieu differential equation by means of the Lie algebra contraction of the Lie algebra of the motion group M(2) on the Heisenberg Lie algebra H(3). A similar approach for the Lam\\'e equation in terms of the Lie algebra contraction of $SO_0(2,1)$ on the Lie algebra of the motion group M(2) is outlined."}
{"category": "Math", "title": "Rigidity Theorems For Lagrangian Submanifolds of $C^n$ and $CP^n$ With Conformal Maslov Form", "abstract": "In this paper, we obtain a rigidity theorem for Lagrangian submanifolds of $C^n$ and $CP^n$ with conformal Maslov form."}
{"category": "Math", "title": "Isometric Lattice Homomorphisms between Sobolev Spaces", "abstract": "Given bounded domains $\\Omega_1$ and $\\Omega_2$ in $\\mathds{R}^N$ and an isometry $T$ from $W^{1,p}(\\Omega_1)$ to $W^{1,p}(\\Omega_2)$, we give sufficient conditions ensuring that $T$ corresponds to a rigid motion of the space, i.e., $Tu = \\pm (u \\circ \\xi)$ for an isometry $\\xi$, and that the domains are congruent. More general versions of the involved results are obtained along the way."}
{"category": "Math", "title": "Berezin transform in polynomial Bergman spaces", "abstract": "We study the reproducing kernel for weighted polynomial Bergman spaces and consider applications to the Berezin transform. Some of our results have applications in random matrix theory, a topic which we discuss in a separate (companion) paper."}
{"category": "Math", "title": "Small exotic rational surfaces without 1- and 3-handles", "abstract": "We give new rational blowdown constructions of exotic CP^2#n(-CP^2) (5\\leq n\\leq 9) without using elliptic fibrations. We also show that our 4-manifolds admit handle decompositions without 1- and 3-handles, for 7\\leq n\\leq 9. A strategy for rational blowdown constructions of exotic CP^2#n(-CP^2) (1\\leq n\\leq 4) is also proposed."}
{"category": "Math", "title": "Fluctuations of eigenvalues of random normal matrices", "abstract": "In this note, we prove Gaussian field convergence of fluctuations of eigenvalues of random normal matrices in the interior of a quantum droplet."}
{"category": "Math", "title": "Some Combinatorial Properties of Hook Lengths, Contents, and Parts of Partitions", "abstract": "This paper proves a generalization of a conjecture of Guoniu Han, inspired originally by an identity of Nekrasov and Okounkov. The main result states that certain sums over partitions p of n, involving symmetric functions of the squares of the hook lengths of p, are polynomial functions of n. A similar result is obtained for symmetric functions of the contents and shifted parts of n."}
{"category": "Math", "title": "A bilinear form relating two Leonard systems", "abstract": "Let $\\Phi$, $\\Phi'$ be Leonard systems over a field $\\mathbb{K}$, and $V$, $V'$ the vector spaces underlying $\\Phi$, $\\Phi'$, respectively. In this paper, we introduce and discuss a balanced bilinear form on $V\\times V'$. Such a form naturally arises in the study of $Q$-polynomial distance-regular graphs. We characterize a balanced bilinear form from several points of view."}
{"category": "Math", "title": "Faces of the scl norm ball", "abstract": "Let F be the fundamental group of S, where S is a compact, connected, oriented surface with negative Euler characteristic and nonempty boundary. (1) The projective class of the chain \\partial S in B_1(F) intersects the interior of a codimension one face of the unit ball in the stable commutator length pseudo-norm. (2) The unique homogeneous quasimorphism on F dual to this face (up to scale and elements of H^1) is the rotation quasimorphism associated to the action of F on the ideal boundary of the hyperbolic plane, coming from a hyperbolic structure on S. These facts follow from the fact that every homologically trivial 1-chain in S rationally cobounds an immersed surface with a sufficiently large multiple of the boundary. This is true even if S has no boundary."}
{"category": "Math", "title": "Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension", "abstract": "We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whose leaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the $L^1$ contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate the computational efficiency together with the convergence properties."}
{"category": "Math", "title": "Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux", "abstract": "A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist--Osher approximation for the flux and explicit time--stepping. An adaptivemultiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier--thickener model illustrate the efficiency of this method."}
{"category": "Math", "title": "Surgery on a knot in (Surface x I)", "abstract": "Suppose F is a compact orientable surface, K is a knot in F x I, and N is the 3-manifold obtained by some non-trivial surgery on K. If F x {0} compresses in N, then there is an annulus in F x I with one end K and the other end an essential simple closed curve in F x {0}. Moreover, the end of the annulus at K determines the surgery slope. An application: suppose M is a compact orientable 3-manifold that fibers over the circle. If surgery on a knot K in M yields a reducible manifold, then either: the projection of K to S^1 has non-trivial winding number; or K lies in a ball; or K lies in a fiber; or K is a cabled knot."}
{"category": "Math", "title": "Decoding generalised hyperoctahedral groups and asymptotic analysis of correctible error patterns", "abstract": "We demonstrate a majority-logic decoding algorithm for decoding the generalised hyperoctahedral group $C_m \\wr S_n$ when thought of as an error-correcting code. We also find the complexity of this decoding algorithm and compare it with that of another, more general, algorithm. Finally, we enumerate the number of error patterns exceeding the correction capability that can be successfully decoded by this algorithm, and analyse this asymptotically."}
{"category": "Math", "title": "The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence", "abstract": "One of the most widely used methods for eigenvalue computation is the $QR$ iteration with Wilkinson's shift: here the shift $s$ is the eigenvalue of the bottom $2\\times 2$ principal minor closest to the corner entry. It has been a long-standing conjecture that the rate of convergence of the algorithm is cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_X$ be the $3 \\times 3$ matrix having only two nonzero entries $(T_X)_{12} = (T_X)_{21} = 1$ and let $T_L$ be the set of real, symmetric tridiagonal matrices with the same spectrum as $T_X$. There exists a neighborhood $U \\subset T_L$ of $T_X$ which is invariant under Wilkinson's shift strategy with the following properties. For $T_0 \\in U$, the sequence of iterates $(T_k)$ exhibits either strictly quadratic or strictly cubic convergence to zero of the entry $(T_k)_{23}$. In fact, quadratic convergence occurs exactly when $\\lim T_k = T_X$. Let $X$ be the union of such quadratically convergent sequences $(T_k)$: the set $X$ has Hausdorff dimension 1 and is a union of disjoint arcs $X^\\sigma$ meeting at $T_X$, where $\\sigma$ ranges over a Cantor set."}
{"category": "Math", "title": "Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus", "abstract": "We establish a connection between the strong solution to the spatially periodic Navier-Stokes equations and a solution to a system of forward-backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct representations of the strong solution to the Navier-Stokes equations in terms of diffusion processes."}
{"category": "Math", "title": "La forme asymptotique du processus de contact en environnement al\\'eatoire", "abstract": "The aim of this article is to prove asymptotic shape theorems for the contact process in stationary random environment. These theorems generalize known results for the classical contact process. In particular, if H_t denotes the set of already occupied sites at time t, we show that for almost every environment, when the contact process survives, the set H_t/t almost surely converges to a compact set that only depends on the law of the environment. To this aim, we prove a new almost subadditive ergodic theorem."}
{"category": "Math", "title": "Unchained polygons and the N-body problem", "abstract": "The simplest solutions of the N-body problem --symmetric relative equilibria-- are shown to be organizing centers from which stem some recently studied classes of periodic solutions. We focus on the relative equilibrium of the equal-mass regular N-gon, assumed horizontal, and study the families of Lyapunov quasi-periodic solutions bifurcating from them in the vertical direction. The proof of the local existence of such solutions relies on the fact that the restriction to the corresponding directions of the quadratic part of the energy is positive definite. We then discuss the possibility of continuing the families globally as action minimizers under symmetry constraints by using the fact that, in rotating frames where they become periodic, these solutions are highly symmetric. The paradigmatic examples are the \"Eight\" families for an odd number of bodies and the \"Hip-Hop\" families for an even number. We argue that it is precisely for these two families that global minimization may be used. We also study the relation with the regular N-gon, of the so-called \"chain\" choreographies (see C. Sim\\'o, New families of Solutions in N-Body Problems, Progr. Math. 201, 2001): here, only a local minimization property is true (except for N=3) and moreover the parity plays a deciding role, in particular through the value of the angular momentum."}
{"category": "Math", "title": "Analogue of Sylvester-Cayley formula for invariants of $n$-ary form", "abstract": "The number $\\nu_{n,d}(k)$ of linearly independed homogeneous invariants of degree $k$ for the $n$-ary form of degree $d$ is calculated."}
{"category": "Math", "title": "A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology", "abstract": "This work deals with the numerical solution of the monodomain and bidomain models of electrical activity of myocardial tissue. The bidomain model is a system consisting of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time-dependent ODE modeling the evolution of the so-called gating variable. In the simpler sub-case of the monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple models for the membrane and ionic currents are considered, the Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical examples demonstrates thatthese methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, an optimalthreshold for discarding non-significant information in the multiresolution representation of the solution is derived, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed-up, memory compression, and errors in different norms."}
{"category": "Math", "title": "A generalized Major index statistic", "abstract": "Inspired by the $k$-inversion statistic for LLT polynomials, we define a $k$-inversion number and $k$-descent set for words. Using these, we define a new statistic on words, called the $k$-major index, that interpolates between the major index and inversion number. We give a bijective proof that the $k$-major index is equidistributed with the major index, generalizing a classical result of Foata and rediscovering a result of Kadell. Inspired by recent work of Haglund and Stevens, we give a partial extension of these definitions and constructions to standard Young tableaux. Finally, we give an application to Macdonald polynomials made possible through connections with LLT polynomials."}
{"category": "Math", "title": "Higher-order Analogues of the Slice Genus of a Knot", "abstract": "For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann $\\rho$-invariants, which we call higher-order signatures. The higher-order genera offer a refinement of the Grope filtration of the knot concordance group."}
{"category": "Math", "title": "The rank of a hypergeometric system", "abstract": "The holonomic rank of the A-hypergeometric system M_A(\\beta) is the degree of the toric ideal I_A for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of nongeneric parameters to construct a combinatorial formula for the rank jump of M_A(\\beta). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of M_A(\\beta) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space. These results hold for generalized A-hypergeometric systems as well, where the semigroup ring of A is replaced by a nontrivial weakly toric module M contained in \\CC[\\ZZ A]. We also provide a direct proof of the result of M. Saito and W. Traves regarding the isomorphism classes of M_A(\\beta)."}
{"category": "Math", "title": "The Postage Stamp Problem and Essential Subsets in Integer Bases", "abstract": "Plagne recently determined the asymptotic behavior of the function E(h), which counts the maximum possible number of essential elements in an additive basis for N of order h. Here we extend his investigations by studying asymptotic behavior of the function E(h,k), which counts the maximum possible number of essential subsets of size k, in a basis of order h. For a fixed k and with h going to infinity, we show that E(h,k) = \\Theta_{k} ([h^{k}/\\log h]^{1/(k+1)}). The determination of a more precise asymptotic formula is shown to depend on the solution of the well-known \"postage stamp problem\" in finite cyclic groups. On the other hand, with h fixed and k going to infinity, we show that E(h,k) \\sim (h-1) {\\log k \\over \\log \\log k}."}
{"category": "Math", "title": "Quaternionic contact normal coordinates", "abstract": "This paper constructs a family of coordinate systems about a point on a quaternionic contact manifold, called quaternionic contact pseudohermitian normal coordinates. Once defined, conformal variations of the quaternionic contact structure induce changes on the coordinates which are studied in an effort to simplify the torsion and curvature at the center point."}
{"category": "Math", "title": "Computing points of small height for cubic polynomials", "abstract": "Let f in Q[z] be a polynomial of degree d at least two. The associated canonical height \\hat{h}_f is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of f. Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on d. A related conjecture claims that at non-preperiodic rational points, \\hat{h}_f is bounded below by a positive constant (depending only on d) times some kind of height of f itself. In this paper, we provide support for these conjectures in the case d=3 by computing the set of small height points for several billion cubic polynomials."}
{"category": "Math", "title": "Complete intersection Approximation, Dual Filtrations and Applications", "abstract": "We give a two step method to study certain questions regarding associated graded module of a Cohen-Macaulay (CM) module $M$ w.r.t an $\\mathfrak{m}$-primary ideal $\\mathfrak{a}$ in a complete Noetherian local ring $(A,\\mathfrak{m})$. The first step, we call it complete intersection approximation, enables us to reduce to the case when both $A$, $ G_\\mathfrak{a}(A) = \\bigoplus_{n \\geq 0} \\mathfrak{a}^n/\\mathfrak{a}^{n+1} $ are complete intersections and $M$ is a maximal CM $A$-module. The second step consists of analyzing the classical filtration $\\{Hom_A(M,\\mathfrak{a}^n) \\}_{\\mathbb{Z}}$ of the dual $Hom_A(M,A)$. We give many applications of this point of view. For instance let $(A,\\mathfrak{m})$ be equicharacteristic and CM. Let $a(G_\\mathfrak{a}(A))$ be the $a$-invariant of $G_\\mathfrak{a}(A)$. We prove: 1. $a(G_\\mathfrak{a}(A)) = -\\dim A$ iff $\\mathfrak{a}$ is generated by a regular sequence. 2. If $\\mathfrak{a}$ is integrally closed and $a(G_\\mathfrak{a}(A)) = -\\dim A + 1$ then $\\mathfrak{a}$ has minimal multiplicity. We extend to modules a result of Ooishi relating symmetry of $h$-vectors. As another application we prove a conjecture of Itoh, if $A$ is a CM local ring and $\\mathfrak{a}$ is a normal ideal with $e_3^\\mathfrak{a}(A) = 0$ then $G_\\mathfrak{a}(A)$ is CM."}
{"category": "Math", "title": "CR manifolds admitting a CR contraction", "abstract": "We classify the germs of $\\mathcal{C}^\\infty$ CR manifolds that admit a smooth CR contraction. We show that such a CR manifold is embedded into $\\CC^n$ as a real hypersurface defined by a polynomial defining function consisting of monomials whose degrees are completely determined by the extended resonances of the contraction. Furthermore the contraction map extends to a holomorphic contraction that coincides in fact with its polynomial normal form. Consequently, several results concerning Complex and CR geometry are derived."}
{"category": "Math", "title": "Combinatorial computation of the motivic Poincare series", "abstract": "We give the explicit algorithm computing the motivic generalization of the Poincare series of the plane curve singularity introduced by A. Campillo, F. Delgado and S. Gusein-Zade. It is done in terms of the embedded resolution of the curve. The result is a rational function depending of the parameter q, at q=1 it coincides with the Alexander polynomial of the corresponding link. For irreducible curves we relate this invariant to the Heegard-Floer knot homologies constructed by P. Ozsvath and Z. Szabo. Many explicit examples are considered."}
{"category": "Math", "title": "Curves without automorphisms and integral invariants of Calabi-Yau three-folds", "abstract": "This paper has been withdrawn by the author, due a crucial mistake in proof of lemma 4.2."}
{"category": "Math", "title": "Faltings heights of CM cycles and derivatives of L-functions", "abstract": "We study the Faltings height pairing of arithmetic Heegner divisors and CM cycles on Shimura varieties associated to orthogonal groups. We compute the Archimedian contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a Rankin L-function. We prove the conjecture in certain cases where the Shimura variety has dimension 0, 1, or 2. In particular, we obtain a new proof of the Gross-Zagier formula."}
{"category": "Math", "title": "On a surprising relation between rectangular and square free convolutions", "abstract": "Debbah and Ryan have recently proved a result about the limit empirical singular distribution of the sum of two rectangular random matrices whose dimensions tend to infinity. In this paper, we reformulate it in terms of the rectangular free convolution introduced in a previous paper and then we give a new, shorter, proof of this result under weaker hypothesis: we do not suppose the \\pro measure in question in this result to be compactly supported anymore. At last, we discuss the inclusion of this result in the family of relations between rectangular and square random matrices."}
{"category": "Math", "title": "On the finite cyclicity of open period annuli", "abstract": "Let $\\Pi$ be an open, relatively compact period annulus of real analytic vector field $X_0$ on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from $\\Pi$ under a given multi-parameter analytic deformation $X_\\lambda$ of $X_0$ is finite, provided that $X_0$ is either Hamiltonian, or generic Darbouxian vector field."}
{"category": "Math", "title": "Irreducibility of the Hilbert-Blumenthal moduli spaces with parahoric level structure", "abstract": "We determine the number of irreducible components of the reduction mod p of any Hilbert-Blumenthal moduli space with a parahoric level structure, where p is unramified in the totally real field."}
{"category": "Math", "title": "Singular solutions to the Loewner equation", "abstract": "We consider the L\\\"owner differential equation generating univalent self-maps of the unit disk (or of the upper half-plane). If the solution to this equation represents a one-slit map, then the driving term is a continuous function. The reverse statement is not true in general as a famous Kufarev's example shows. We address the following main problem: to find a criterium for the L\\\"owner equation to generate one-slit solutions. New examples of non-slit solutions to the L\\\"owner equation are presented. Properties of singular slit solutions are revealed."}
{"category": "Math", "title": "Asymptotic analysis for bifurcating autoregressive processes via a martingale approach", "abstract": "We study the asymptotic behavior of the least squares estimators of the unknown parameters of bifurcating autoregressive processes. Under very weak assumptions on the driven noise of the process, namely conditional pair-wise independence and suitable moment conditions, we establish the almost sure convergence of our estimators together with the quadratic strong law and the central limit theorem. All our analysis relies on non-standard asymptotic results for martingales."}
{"category": "Math", "title": "Ikehara-type theorem involving boundedness", "abstract": "Consider any Dirichlet series sum a_n/n^z with nonnegative coefficients a_n and finite sum function f(z)=f(x+iy) when x is greater than 1. Denoting the partial sum a_1+...+a_N by s_N, the paper gives the following necessary and sufficient condition in order that (s_N)/N remain bounded as N goes to infinity. For x tending to 1 from above, the quotient q(x+iy)=f(x+iy)/(x+iy) must converge to a pseudomeasure q(1+iy), the distributional Fourier transform of a bounded function. The paper also gives an optimal estimate for (s_N)/N under the \"real condition\" that (1-x)f(x) remain bounded as x tends to 1 from above."}
{"category": "Math", "title": "On exponentials of exponential generating series", "abstract": "Identifying the algebra of exponential generating series with the shuffle algebra of formal power series, one can define an exponential map ${\\mathop{exp}}_!:X\\mathbb K[[X]]\\longrightarrow 1+X\\mathbb K[[X]]$ for the associated Lie group formed by exponential generating series with constant coefficient 1 over an arbitrary field $\\mathbb K$. The main result of this paper states that the map ${\\mathop{exp}}_!$ (and its inverse map ${\\mathop{log}}_!$) induces a bijection between rational, respectively algebraic, series in $X\\mathbb K [[X]]$ and $1+X\\mathbb K[[X]]$ if the field $\\mathbb K$ is a subfield of the algebraically closed field $\\bar{\\mathbb F}_p$ of characteristic $p$."}
{"category": "Math", "title": "Solution of large linear systems with embedded network structure for a non-homogeneous network flow programming problem", "abstract": "In the paper we consider the linear underdetermined system of a special type. Systems of this type appear in non-homogeneous network flow programming problems in the form of systems of constraints and can be characterized as systems with a large sparse submatrix representing the embedded network structure. We develop a direct method for finding solutions of the system. The algorithm is based on the theoretic-graph specificities for the structure of the support and properties of the basis of a solution space of a homogeneous system. One of the key steps is decomposition of the system. A simple example is regarded at the end of the paper."}
{"category": "Math", "title": "A new family of Markov branching trees: the alpha-gamma model", "abstract": "We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this paper, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour."}
{"category": "Math", "title": "On the symmetry of b-functions of linear free divisors", "abstract": "We introduce the concept of a prehomogeneous determinant as a nonreduced version of a linear free divisor. Both are special cases of prehomogeneous vector spaces. We show that the roots of the $b$-function are symmetric about -1 for reductive prehomogeneous determinants and for regular special linear free divisors. For general prehomogeneous determinants, we describe conditions under which this symmetry still holds. Combined with Kashiwara's theorem on the roots of b-functions, our symmetry result shows that -1 is the only integer root of the b-function. This gives a positive answer to a problem posed by Castro-Jimenez and Ucha-Enriquez in the above cases. We study the condition of (strong) Euler homogeneity in terms of the action of the stabilizers on the normal spaces. As an application of our results, we show that the logarithmic comparison theorem holds for Koszul free reductive linear free divisors exactly if they are (strongly) Euler homogeneous."}
{"category": "Math", "title": "Differential Harnack Estimates for Time-dependent Heat Equations with Potentials", "abstract": "In this paper, we prove a differential Harnack inequality for positive solutions of time-dependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the time-dependent heat equation."}
{"category": "Math", "title": "Equisingularity and The Euler Characteristic of a Milnor Fibre", "abstract": "We study the Euler characteristic of the Milnor fibre of a hypersurface singularity. This invariant is given in terms of the Euler characteristic of a fibre in between the original singularity and its Milnor fibre and in terms of the Euler characteristics associated to strata of the in-between fibre. From this we can deduce a result of Massey and Siersma regarding singularities with a one-dimensional critical locus. The result is also applied to the study of equisingularity. The famous Brian\\c{c}on-Speder-Teissier result states that a family of isolated hypersurface singularities is equisingular if and only if its $\\mu ^*$-sequence is constant. We show that if a similar sequence for a family of corank 1 complex analytic mappings from n-space to (n+1)-space is constant, then the image of the family of mappings is equisingular. For families of corank 1 maps from 3-space to 4-space we show that the converse is true also."}
{"category": "Math", "title": "Iteratively re-weighted least squares minimization for sparse recovery", "abstract": "We analyze an Iteratively Re-weighted Least Squares (IRLS) algorithm for promoting l1-minimization in sparse and compressible vector recovery. We prove its convergence and we estimate its local rate. We show how the algorithm can be modified in order to promote lt-minimization for t<1, and how this modification produces superlinear rates of convergence."}
{"category": "Math", "title": "Three-dimensional compact manifolds and the Poincare conjecture", "abstract": "The aim of the work is to prove the following main theorem. Theorem. Let M3 be a three-dimensional, connected, simple-connected, closed, compact, smooth manifold. Tnen the manifold M3 is diffeomorphic to the three-dimensional sphere."}
{"category": "Math", "title": "On Locally Conformally Flat Gradient Shrinking Ricci Solitons", "abstract": "In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence of this identity, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below."}
{"category": "Math", "title": "Orthogonal systems in vector spaces over finite fields", "abstract": "We prove that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors."}
{"category": "Math", "title": "Steady compressible Oseen flow with slip boundary conditions", "abstract": "We prove the existence of solution in a class H^2(\\Omega) x H^1(\\Omega) to steady compressible Oseen system with slip boundary conditions in a two dimensional, convex domain with the boundary of class H^{5/2}. The method is to regularize a weak solution obtained via the Galerkin method. The problem of regularization is reduced to a problem of solvability of a certain transport equation by application of the Helmholtz decomposition. The method works under additional assumption on the geometry of the boundary."}
{"category": "Math", "title": "Semisimple Hopf algebras and their depth two Hopf subalgebras", "abstract": "We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field $k$ is algebraically closed of characteristic zero). This means on the one hand that a Hopf subalgebra is normal when inducing (then restricting) modules several times as opposed to one time creates no new simple constituents. This point of view was taken in the paper http://arxiv.org/abs/math/0409346 which established a normality result in case H and K are finite group algebras. On the other hand this means that K is normal in H when H | K is a Galois extension with respect to action of generalized bialgebras such as bialgebroids, weak Hopf algebras or Hopf algebroids. The generalized Galois picture of depth two is the point of view we take here: after showing the centralizer R is separable algebra via Hopf invariant theory, we compute that the depth two semisimple Hopf algebra pair H | K is free Frobenius extension with Markov trace satisfying all hypotheses considered in Kadison and Nikshych, Frobenius Extensions and Weak Hopf Algebras, J.Algebra 244 (2001). By the main theorem in that paper it is then a Galois extension with action of semisimple weak Hopf algebra (also regular and possessing Haar integral). Then the Galois canonical isomorphism (via coring theory) restricted to integral induces algebra homomorphism from Hopf algebra into weak Hopf algebra with kernel HK^+ = K^+H."}
{"category": "Math", "title": "Estimates on the Probability of Outliers for Real Random Bargmann-Fock functions", "abstract": "In this paper we consider the distribution of the zeros of a real random Bargmann-Fock function of one or more variables. For these random functions we prove estimates for two types of families of events, both of which are large deviations from the mean. First, we prove that the probability there are no zeros in $[-r,r]^m\\subset\\R^m$ decays at least exponentially in terms of $r^m$. For this event we also prove a lower bound on the order of decay, which we do not expect to be sharp. Secondly, we compute the order of decay for the probability of families of events where the volume of the complex zero set is either much larger or much smaller then expected."}
{"category": "Math", "title": "The Freiheitssatz and the automorphisms of free right-symmetric algebras", "abstract": "We prove the Freiheitssatz for right-symmetric algebras and the decidability of the word problem for right-symmetric algebras with a single defining relation. We also prove that two generated subalgebras of free right-symmetric algebras are free and automorphisms of two generated free right-symmetric algebras are tame."}
{"category": "Math", "title": "Distinguished Torsion, Curvature and Deflection Tensors in the Multi-Time Hamilton Geometry", "abstract": "The aim of this paper is to present the main geometrical objects on the dual 1-jet bundle $J^{1*}(\\cal{T},M)$ (this is the polymomentum phase space of the De Donder-Weyl covariant Hamiltonian formulation of field theory) that characterize our approach of multi-time Hamilton geometry. In this direction, we firstly introduce the geometrical concept of a nonlinear connection $N$ on the dual 1-jet space $J^{1*}(\\cal{T},M)$. Then, starting with a given $N$-linear connection $D$ on $J^{1*}(\\cal{T},M)$, we describe the adapted components of the torsion, curvature and deflection distinguished tensors attached to the $N$-linear connection $D$."}
{"category": "Math", "title": "Galois covers of the open p-adic disc", "abstract": "This paper investigates Galois branched covers of the open $p$-adic disc and their reductions to characteristic $p$. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive a criterion for good reduction in the abelian case, and give an arithmetic reformulation of the local Oort Conjecture concerning the liftability of cyclic covers of germs of curves."}
{"category": "Math", "title": "L-functions for holomorphic forms on GSp(4) x GL(2) and their special values", "abstract": "We provide an explicit integral representation for L-functions of pairs (F,g) where F is a holomorphic genus 2 Siegel newform and g a holomorphic elliptic newform, both of squarefree levels and of equal weights. When F,g have level one, this was earlier known by the work of Furusawa. The extension is not straightforward. Our methods involve precise double-coset and volume computations as well as an explicit formula for the Bessel model for GSp(4) in the Steinberg case; the latter is possibly of independent interest. We apply our integral representation to prove an algebraicity result for a critical special value of L(s, F \\times g). This is in the spirit of known results on critical values of triple product L-functions, also of degree 8, though there are significant differences."}
{"category": "Math", "title": "Dynamic Model of Smoothing Problem in Water Power Systems", "abstract": "In this paper the problem of optimal performance of a power system is considered. The problem is posed in various aspects within the frames of the theory of optimal control of stores. Mathematical models are presented by means of the recurrent equations of a dynamic programming. In the general case a new method is presented which avoids the \"curse of dimensionality.\""}
{"category": "Math", "title": "Algebra of formal vector fields on the line and Buchstaber's conjecture", "abstract": "Let L_1 denotes the Lie algebra of formal vector fields on the line which vanish at the origin together with their first derivatives. Buchstaber and Shokurov have shown that the universal enveloping algebra U(L_1) is isomorphic to the tensor product of the Landweber-Novikov algebra S in complex cobordism theory by reals. The cohomology H*(L_1) has trivial multiplication. Buchstaber conjectured that H*(L_1) is generated with respect to non-trivial Massey products by H^1(L_1). Feigin, Fuchs and Retakh found representation of H*(L_1) by trivial Massey products. In the present article we prove that H*(L_1) is generated with respect to non-trivial Massey products by two elements from H^1(L_1)."}
{"category": "Math", "title": "The quaternionic Cullen-regular product for a larger class of functions", "abstract": "We introduce the regular product for Cullen-regular quaternionic functions in a manner that does not depend upon a representation in power series but upon another, weaker kind of representation. The special case when the functions are represented as quaternionic power series is studied. We show that the regular ring of quaternionic power series is a subring of the regular associative ring of real-analytical Hyperholomorphic functions."}
{"category": "Math", "title": "A diagrammatic approach to Hopf monads", "abstract": "Given a Hopf algebra in a symmetric monoidal category with duals, the category of modules inherits the structure of a monoidal category with duals. If the notion of algebra is replaced with that of monad on a monoidal category with duals then Bruguieres and Virelizier showed when the category of modules inherits this structure of being monoidal with duals, and this gave rise to what they called a Hopf monad. In this paper it is shown that there are good diagrammatic descriptions of dinatural transformations which allows the three-dimensional, object-free nature of their constructions to become apparent."}
{"category": "Math", "title": "A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation", "abstract": "We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen `exchangeability' (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables, we prove that invariance of their joint distribution under quantum permutations is equivalent to the fact that the random variables are identically distributed and free with respect to the conditional expectation onto their tail algebra."}
{"category": "Math", "title": "Representations of SO(3) and angular polyspectra", "abstract": "We characterize the angular polyspectra, of arbitrary order, associated with isotropic fields defined on the sphere S^2. Our techniques rely heavily on group representation theory, and specifically on the properties of Wigner matrices and Clebsch-Gordan coefficients. The findings of the present paper constitute a basis upon which one can build formal procedures for the statistical analysis and the probabilistic modelization of the Cosmic Microwave Background radiation, which is currently a crucial topic of investigation in cosmology. We also outline an application to random data compression and \"simulation\" of Clebsch-Gordan coefficients."}
{"category": "Math", "title": "Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type", "abstract": "Generalized Reynolds ideals are ideals of the center of a symmetric algebra over a field of positive characteristic. They have been shown by the second author to be invariant under derived equivalences. In this paper we determine the generalized Reynolds ideals of algebras of dihedral and semidihedral type (as defined by Erdmann), in characteristic 2. In this way we solve some open problems about scalars occurring in the derived equivalence classification of these algebras."}
{"category": "Math", "title": "Stacks in canonical RNA pseudoknot structures", "abstract": "In this paper we study the distribution of stacks in $k$-noncrossing, $\\tau$-canonical RNA pseudoknot structures ($<k,\\tau> $-structures). An RNA structure is called $k$-noncrossing if it has no more than $k-1$ mutually crossing arcs and $\\tau$-canonical if each arc is contained in a stack of length at least $\\tau$. Based on the ordinary generating function of $<k,\\tau>$-structures \\cite{Reidys:08ma} we derive the bivariate generating function ${\\bf T}_{k,\\tau}(x,u)=\\sum_{n \\geq 0} \\sum_{0\\leq t \\leq \\frac{n}{2}} {\\sf T}_{k, \\tau}^{} (n,t) u^t x^n$, where ${\\sf T}_{k,\\tau}(n,t)$ is the number of $<k,\\tau>$-structures having exactly $t$ stacks and study its singularities. We show that for a certain parametrization of the variable $u$, ${\\bf T}_{k,\\tau}(x,u)$ has a unique, dominant singularity. The particular shift of this singularity parametrized by $u$ implies a central limit theorem for the distribution of stack-numbers. Our results are of importance for understanding the ``language'' of minimum-free energy RNA pseudoknot structures, generated by computer folding algorithms."}
{"category": "Math", "title": "Root systems and Weyl groupoids for Nichols algebras", "abstract": "Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The obtained combinatorial structure fits perfectly into an existing framework of generalized root systems associated to a family of Cartan matrices, and provides novel insight into Nichols algebras. We demonstrate the power of our construction with new results on Nichols algebras over finite non-abelian simple groups and symmetric groups. Key words: Hopf algebra, quantum group, root system, Weyl group"}
{"category": "Math", "title": "Elliptic fibrations on cubic surfaces", "abstract": "We classify elliptic fibrations birational to a nonsingular, minimal cubic surface over a field of characteristic zero. Our proof is adapted to provide computational techniques for the analysis of such fibrations, and we describe an implementation of this analysis in computer algebra."}
{"category": "Math", "title": "Transfer of Gorenstein dimensions along ring homomorphisms", "abstract": "A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved for the Gorenstein flat and the Gorenstein projective dimensions; here we give a solution for the Gorenstein injective dimension. Moreover, we establish two formulas for the Gorenstein injective dimension of modules in terms of the depth invariant; they extend formulas for the injective dimension due to Bass and Chouinard."}
{"category": "Math", "title": "On the geometry of the slice of trace--free SL(2,C)-characters of a knot group", "abstract": "Let K be a knot in an integral homology 3-sphere and let B denote the 2-fold branched cover of the integral homology sphere branched along K. We construct a map from the slice of characters with trace free along meridians in the SL(2, C)-character variety of the knot exterior to the SL(2, C)-character variety of 2-fold branched cover B. When this map is surjective, it describes the slice as the 2-fold branched cover over the SL(2, C)-character variety of B with branched locus given by the abelian characters, whose preimage is precisely the set of metabelian characters. We show that each of metabelian character can be represented as the character of a binary dihedral representation of the knot group. This map is shown to be surjective for all 2-bridge knots and all pretzel knots of type (p, q, r). An extension of this framework to n-fold branched covers is also described."}
{"category": "Math", "title": "Case-deletion importance sampling estimators: Central limit theorems and related results", "abstract": "Case-deleted analysis is a popular method for evaluating the influence of a subset of cases on inference. The use of Monte Carlo estimation strategies in complicated Bayesian settings leads naturally to the use of importance sampling techniques to assess the divergence between full-data and case-deleted posteriors and to provide estimates under the case-deleted posteriors. However, the dependability of the importance sampling estimators depends critically on the variability of the case-deleted weights. We provide theoretical results concerning the assessment of the dependability of case-deleted importance sampling estimators in several Bayesian models. In particular, these results allow us to establish whether or not the estimators satisfy a central limit theorem. Because the conditions we derive are of a simple analytical nature, the assessment of the dependability of the estimators can be verified routinely before estimation is performed. We illustrate the use of the results in several examples."}
{"category": "Math", "title": "Modeling solutions with jumps for rate-independent systems on metric spaces", "abstract": "Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given."}
{"category": "Math", "title": "Detection of cellular aging in a Galton-Watson process", "abstract": "We consider the bifurcating Markov chain model introduced by Guyon to detect cellular aging from cell lineage. To take into account the possibility for a cell to die, we use an underlying Galton-Watson process to describe the evolution of the cell lineage. We give in this more general framework a weak law of large number, an invariance principle and thus fluctuation results for the average over one generation or up to one generation. We also prove the fluctuations over each generation are independent. Then we present the natural modifications of the tests given by Guyon in cellular aging detection within the particular case of the auto-regressive model."}
{"category": "Math", "title": "P-spectrum and collapsing of connected sums, calculus of the limit", "abstract": "The goal of the paper is to calculate the limit sectrum of the Hodge-Laplace operator under the perturbation of collapse of one part of a connected sum. This gives some new results concerning the 'conformal spectrum' on differential forms."}
{"category": "Math", "title": "Unconstrained Recursive Importance Sampling", "abstract": "We propose an unconstrained stochastic approximation method of finding the optimal measure change (in an a priori parametric family) for Monte Carlo simulations. We consider different parametric families based on the Girsanov theorem and the Esscher transform (or exponential-tilting). In a multidimensional Gaussian framework, Arouna uses a projected Robbins-Monro procedure to select the parameter minimizing the variance. In our approach, the parameter (scalar or process) is selected by a classical Robbins-Monro procedure without projection or truncation. To obtain this unconstrained algorithm we intensively use the regularity of the density of the law without assume smoothness of the payoff. We prove the convergence for a large class of multidimensional distributions and diffusion processes. We illustrate the effectiveness of our algorithm via pricing a Basket payoff under a multidimensional NIG distribution, and pricing a barrier options in different markets."}
{"category": "Math", "title": "Localisable moving average stable and multistable processes", "abstract": "We study a particular class of moving average processes which possess a property called localisability. This means that, at any given point, they admit a ``tangent process'', in a suitable sense. We give general conditions on the kernel g defining the moving average which ensures that the process is localisable and we characterize the nature of the associated tangent processes. Examples include the reverse Ornstein-Uhlenbeck process and the multistable reverse Ornstein-Uhlenbeck process. In the latter case, the tangent process is, at each time t, a L\\'evy stable motion with stability index possibly varying with t. We also consider the problem of path synthesis, for which we give both theoretical results and numerical simulations."}
{"category": "Math", "title": "The concordance genus of a knot, II", "abstract": "The concordance genus of a knot K is the minimum three-genus among all knots concordant to K. For prime knots of 10 or fewer crossings there have been three knots for which the concordance genus was unknown. Those three cases are now resolved. Two of the cases are settled using invariants of Levine's algebraic concordance group. The last case depends on the use of twisted Alexander polynomials, viewed as Casson-Gordon invariants."}
{"category": "Math", "title": "The modified complex Busemann-Petty problem on sections of convex bodies", "abstract": "Since the answer to the complex Busemann-Petty problem is negative in most dimensions, it is natural to ask what conditions on the (n-1)-dimensional volumes of the central sections of complex convex bodies with complex hyperplanes allow to compare the n-dimensional volumes. In this article we give necessary conditions on the section function in order to obtain an affirmative answer in all dimensions."}
{"category": "Math", "title": "A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space", "abstract": "We consider the task of computing an approximate minimizer of the sum of a smooth and non-smooth convex functional, respectively, in Banach space. Motivated by the classical forward-backward splitting method for the subgradients in Hilbert space, we propose a generalization which involves the iterative solution of simpler subproblems. Descent and convergence properties of this new algorithm are studied. Furthermore, the results are applied to the minimization of Tikhonov-functionals associated with linear inverse problems and semi-norm penalization in Banach spaces. With the help of Bregman-Taylor-distance estimates, rates of convergence for the forward-backward splitting procedure are obtained. Examples which demonstrate the applicability are given, in particular, a generalization of the iterative soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as well as total-variation based image restoration in higher dimensions are presented."}
{"category": "Math", "title": "The complex Busemann-Petty problem for arbitrary measures", "abstract": "The complex Busemann-Petty problem asks whether origin symmetric convex bodies in C^n with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n\\leq 3 and negative if n\\geq 4. In this article we show that the answer remains the same if the volume is replaced by an \"almost\" arbitrary measure. This result is the complex analogue of Zvavitch's generalization to arbitrary measures of the original real Busemann-Petty problem."}
{"category": "Math", "title": "Comparing and interpolating distributions on manifold", "abstract": "We are interested in comparing probability distributions defined on Riemannian manifold. The traditional approach to study a distribution relies on locating its mean point and finding the dispersion about that point. On a general manifold however, even if two distributions are sufficiently concentrated and have unique means, a comparison of their covariances is not possible due to the difference in local parametrizations. To circumvent the problem we associate a covariance field with each distribution and compare them at common points by applying a similarity invariant function on their representing matrices. In this way we are able to define distances between distributions. We also propose new approach for interpolating discrete distributions and derive some criteria that assure consistent results. Finally, we illustrate with some experimental results on the unit 2-sphere."}
{"category": "Math", "title": "Zeros of Dirichlet series with periodic coefficients", "abstract": "Let $a=(a_n)_{n\\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic continuation of $\\sum_{n\\ge 1} a_n/n^s$, and $N_a(\\sigma_1, \\sigma_2, T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle $\\sigma_1 < \\Re s < \\sigma_2$, $|\\Im s | \\le T$. We extend previous results of Laurin\\v{c}ikas, Kaczorowski, Kulas, and Steuding, by showing that if $F_a(s)$ is not of the form $P(s) L_{\\chi} (s)$, where $P(s)$ is a Dirichlet polynomial and $L_{\\chi}(s)$ a Dirichlet L-function, then there exists an $\\eta=\\eta(a)>0$ such that for all $1/2 < \\sigma_1 < \\sigma_2 < 1+\\eta$, we have $c_1 T \\le N_a(\\sigma_1, \\sigma_2, T) \\le c_2 T$ for sufficiently large $T$, and suitable positive constants $c_1$ and $c_2$ depending on $a$, $\\sigma_1$, and $\\sigma_2$."}
{"category": "Math", "title": "From Black-Scholes and Dupire formulae to last passage times of local martingales. Part B : The finite time horizon", "abstract": "These notes are the second half of the contents of the course given by the second author at the Bachelier Seminar (8-15-22 February 2008) at IHP. They also correspond to topics studied by the first author for her Ph.D.thesis."}
{"category": "Math", "title": "Cech homology for shape recognition in the presence of occlusions", "abstract": "In Computer Vision the ability to recognize objects in the presence of occlusions is a necessary requirement for any shape representation method. In this paper we investigate how the size function of a shape changes when a portion of the shape is occluded by another shape. More precisely, considering a set $X=A\\cup B$ and a measuring function $\\phi$ on $X$, we establish a condition so that $\\ell_{(X,\\phi)=\\ell_{(A,\\phi|_A)}+\\ell_{(B,\\phi|_B)}-\\ell_{(A\\cap B,\\phi|_{A\\cap B})}$. The main tool we use is the Mayer-Vietoris sequence of \\v{C}ech homology groups. This result allows us to prove that size functions are able to detect partial matching between shapes by showing a common subset of cornerpoints."}
{"category": "Math", "title": "Algebraic Independence in SL(3,C) Character Varieties of Free Groups", "abstract": "Let X be the moduli space of SL(3,C) representations of a free group of rank r. In this paper we describe maximal algebraically independent subsets of certain minimal sets of coordinate functions on X. These subsets locally parametrize the moduli space."}
{"category": "Math", "title": "A Comparison Theorem for Gromov-Witten Invariants in the Symplectic Category", "abstract": "In this paper we exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare the virtual fundamental classes of stable maps to a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. This extends versions of a statement well-known in the algebraic category to the symplectic category, where it appears to be less familiar. The latter's inherent flexibility then leads to a confirmation of Pandharipande's Gopakumar-Vafa prediction for GW-invariants of Fano classes in 6-dimensional symplectic manifolds. In a forthcoming paper, we use a similar approach to relative Gromov-Witten invariants and the absolute/relative correspondence in genus~0."}
{"category": "Math", "title": "Classification of irreducible modules of the vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of an arbitrary rank", "abstract": "In this paper, we first classify all irreducible modules of the vertex algebra $V_L^+$ when $L$ is a negative definite even lattice of arbitrary rank. In particular, we show that any irreducible $V_L^+$-module is isomorphic to a submodule of an irreducible twisted $V_L$-module. We then extend this result to a vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of finite rank."}
{"category": "Math", "title": "A threshold phenomenon for random independent sets in the discrete hypercube", "abstract": "Let $I$ be an independent set drawn from the discrete $d$-dimensional hypercube $Q_d=\\{0,1\\}^d$ according to the hard-core distribution with parameter $\\lambda>0$ (that is, the distribution in which each independent set $I$ is chosen with probability proportional to $\\lambda^{|I|}$). We show a sharp transition around $\\lambda=1$ in the appearance of $I$: for $\\lambda>1$, $\\min\\{|I \\cap {\\cal E}|, |I \\cap {\\cal O}|\\}=0$ asymptotically almost surely, where ${\\cal E}$ and ${\\cal O}$ are the bipartition classes of $Q_d$, whereas for $\\lambda<1$, $\\min\\{|I \\cap {\\cal E}|, |I \\cap {\\cal O}|\\}$ is asymptotically almost surely exponential in $d$. The transition occurs in an interval whose length is of order $1/d$. A key step in the proof is an estimation of $Z_\\lambda(Q_d)$, the sum over independent sets in $Q_d$ with each set $I$ given weight $\\lambda^{|I|}$ (a.k.a. the hard-core partition function). We obtain the asymptotics of $Z_\\lambda(Q_d)$ for $\\lambda>\\sqrt{2}-1$, and nearly matching upper and lower bounds for $\\lambda \\leq \\sqrt{2}-1$, extending work of Korshunov and Sapozhenko. These bounds allow us to read off some very specific information about the structure of an independent set drawn according to the hard-core distribution. We also derive a long-range influence result. For all fixed $\\lambda>0$, if $I$ is chosen from the independent sets of $Q_d$ according to the hard-core distribution with parameter $\\lambda$, conditioned on a particular $v \\in {\\cal E}$ being in $I$, then the probability that another vertex $w$ is in $I$ is $o(1)$ for $w \\in {\\cal O}$ but $\\Omega(1)$ for $w \\in {\\cal E}$."}
{"category": "Math", "title": "A class of locally conformally flat 4-manifolds", "abstract": "We construct infinite families of non-simply connected locally conformally flat (LCF) 4-manifolds realizing rich topological types. These manifolds have strictly negative scalar curvature and the underlying topological 4-manifolds do not admit any Einstein metrics. Such 4-manifolds are of particular interest as examples of Bach-flat but non-Einstein spaces in the non-simply connected case. Besides that the underlying smooth manifolds are examples of spaces that admit open book decomposition in dimension 4."}
{"category": "Math", "title": "Symmetry of Quadratic Homogeneous Differential Systems", "abstract": "In this paper, the symmetry group of a differential system of n quadratic homogeneous first order ODEs of n variables is studied. For this purpose, we consider the action of both point and contact transformations to signify the corresponding Lie algebras. We also find the independent differential invariants of these actions."}
{"category": "Math", "title": "Symmetries of 2nd order ODE: y'' + G(x)y' + H(x)y = 0", "abstract": "This paper is devoted to study the Lie algebra of linear symmetries of a homogenous 2nd order ODE, by the method of Kushner, Lychagin and Robstov."}
{"category": "Math", "title": "The local Calderon problem and the determination at the boundary of the conductivity", "abstract": "We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $\\Omega\\subset\\mathbb{R}^{n}$ when the so--called Dirichlet-to-Neumann map is locally given on a non empty portion $\\Gamma$ of the boundary $\\partial\\Omega$. We extend results of uniqueness and stability at the boundary, obtained by the same authors in SIAM J. Math. Anal. 33 (2001), no. 1, 153--171, where the Dirichlet-to-Neumann map was given on all of $\\partial\\Omega$ instead. We also obtain a pointwise stability result at the boundary among the class of conductivities which are continuous at some point $y\\in\\Gamma$. Our arguments also apply when the local Neumann-to-Dirichlet map is available."}
{"category": "Math", "title": "Approximation of subharmonic functions in the unit disk", "abstract": "Let u be a subharmonic function in D={|z|<1}. There exist an absolute constant C and an analytic function f in D such that \\int_D |u(z)-log|f(z)|| dm(z)<C where m denotes the plane Lebesgue measure. We also consider uniform approximation."}
{"category": "Math", "title": "Functoriality and the Inverse Galois problem II: groups of type B_n and G_2", "abstract": "For every finite field F and every positive integer r, there exists a finite extension F' of F such that either SO(2r+1,F') or its simple derived group can be realized as a Galois group over Q. If the characteristic of F is 3 or 5 (mod 8), then we can guarantee that the derived group of SO(2r+1,F') can be realized. Likewise, for every finite field F, there exists a finite extension F' of F such that the finite simple group G_2(F') can be realized a Galois group over Q. The proof uses automorphic forms to construct Galois representations which cut out Galois extensions of the desired type."}
{"category": "Math", "title": "Quantifying Residual Finiteness", "abstract": "We introduce the concept of quantifying the extent to which a finitely generated group is residually finite. The quantification is carried out for some examples including free groups, the first Grigorchuk group, finitely generated nilpotent groups, and certain arithmetic groups such as $SL_n(\\mathbb{Z})$. In the context of finite nilpotent quotients, we find a new characterization of nilpotent groups."}
{"category": "Math", "title": "Tensor products and Correlation Estimates with applications to Nonlinear Schr\\\"odinger equations", "abstract": "We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the original approach of Lin and Strauss but applied to tensor products of solutions. We then demonstrate the proof using commutator vector operators acting on the conservation laws of the equation. This method can be generalized to obtain correlation estimates in all dimensions. In one dimension we use the Gauss-Weierstrass summability method acting on the conservation laws. We then apply the 2d estimate to nonlinear Schr\\\"odinger equations and derive a direct proof of Nakanishi's $H^1$ scattering result for every $L^{2}$-supercritical nonlinearity. We also prove scattering below the energy space for a certain class of $L^{2}$-supercritical equations."}
{"category": "Math", "title": "History of the formulas and algorithms for pi", "abstract": "Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals. Some expressions which are (amazingly) related to pi have been evaluated. In addition, a continual battle has been waged just to break the records computing digits of this number; records have been set using rapidly converging series, ultra fast algorithms and really surprising ones, calculating isolated digits. The development of powerful computers has played a fundamental role in these achievements of calculus."}
{"category": "Math", "title": "Conformally K\\\"ahler base metrics for Einstein warped products", "abstract": "A Riemannian metric $\\wht{g}$ with Ricci curvature $\\wht{\\ri}$ is called nontrivial quasi-Einstein, in the sense of Case, Shu and Wei, if it satisfies $(-a/f)\\wht{\\nab} df+\\wht{\\ri}=\\lambda \\wht{g}$, for a smooth nonconstant function $f$ and constants $\\lambda$ and $a>0$. If $a$ is a positive integer, by a result of Kim and Kim, such a metric forms a base for certain warped Einstein metrics. On a manifold $M$ of real dimension at least six, let $(g,\\t)$ be a pair consisting of a K\\\"ahler metric $g$ which is locally K\\\"ahler irreducible, and a nonconstant Killing potential $\\t$. Suppose the metric $\\wht{g}=g/\\t^2$ is nontrivial \\bee on $M\\setminus\\t^{-1}(0)$, and the associated function $f$ is locally a function of $\\t$. Then $(g,\\t)$ is an \\sk\\ pair, a notion defined by Derdzinski and Maschler. This implies that $M$ is biholomorphic to an open set in the total space of a $CP^1$ bundle whose base manifold admits a K\\\"ahler-Einstein metric. If $M$ is additionally compact, it is a total space of such a bundle or complex projective space. Also, the function $f$ is affine in $\\t^{-1}$ with nonzero constants. Conversely, in all even dimensions $n\\geq 4$, there exist \\sk pairs $(g,\\t)$ and corresponding nonzero constants $K$ and $L$ for which $g/\\t^2$ is nontrivial quasi-Einstein with $f=K\\t^{-1}+L$. Additionally, a result of Case, Shu and Wei on the K\\\"ahler reducibility of nontrivial K\\\"ahler \\bers is reproduced in dimension at least six in a more explicit form."}
{"category": "Math", "title": "Ill-posedness of the Navier-Stokes equations in a critical space in 3D", "abstract": "We prove that the Cauchy problem for the three dimensional Navier-Stokes equations is ill posed in $\\dot{B}^{-1,\\infty}_{\\infty}$ in the sense that a ``norm inflation'' happens in finite time. More precisely, we show that initial data in the Schwartz class $\\mathcal{S}$ that are arbitrarily small in $\\dot{B}^{-1, \\infty}_{\\infty}$ can produce solutions arbitrarily large in $\\dot{B}^{-1, \\infty}_{\\infty}$ after an arbitrarily short time. Such a result implies that the solution map itself is discontinuous in $\\dot{B}^{-1, \\infty}_{\\infty}$ at the origin."}
{"category": "Math", "title": "The Coin Exchange Problem and the Structure of Cube Tilings", "abstract": "Let k_1,...,k_d be positive integers, and D be a subset of [k_1]x...x[k_d], whose complement can be decomposed into disjoint sets of the form {x_1}x...x{x_{s-1}}x[k_s]x{x_{s+1}}x...x{x_d}. We conjecture that the number of elements of D can be represented as a linear combination of the numbers k_1,..., k_d with non-negative integer coefficients. A connexion of this conjecture with the structure of periodical cube tilings is revealed."}
{"category": "Math", "title": "Polytopes with mass linear functions, part I", "abstract": "We analyze mass linear functions $H$ on simple polytopes $\\De$, where a mass linear function is an affine function on $\\De$ whose value on the center of mass depends linearly on the positions of the supporting hyperplanes. We show that certain types of symmetries of $\\De$ give rise to nonconstant mass linear functions on $\\De$. These are called inessential; the others are essential. We also show that most polytopes do not admit any nonconstant mass linear functions. Our main result shows that there is only one family of smooth polytopes of dimension $\\leq 3$ which admit essential mass linear functions. These results have geometric implications. Fix a symplectic toric manifold $(M,\\om,T,\\Phi)$ with moment polytope $\\De = \\Phi(M)$; let $\\Symp(M,\\om)$ be its group of symplectomorphisms. Any linear function $H$ on $\\De$ generates a Hamiltonian $\\R$ action on $M$ whose closure is a subtorus $T_H$ of $T$. We show that if the map $\\pi_1(T_H)\\to \\pi_1(\\Symp(M,\\om))$ has finite image, then $H$ is mass linear. Therefore, in most cases the induced map $\\pi_1(T) \\to \\pi_1(\\Symp(M,\\om))$ is an injection. We also show that this map does not have finite image unless $M$ is a product of projective spaces. Moreover, the inessential $H$ correspond to elements in the kernel of the map $\\pi_1(T)\\to \\Isom(M)$, where the Kahler isometry group $\\Isom(M)\\subset \\Symp(M,\\om)$ consists of elements that also preserve the natural compatible complex structure on $M$. Therefore if $\\De$ supports no nonconstant essential mass linear $H$, the map $\\pi_1(\\Isom(M))\\to pi_1(\\Symp(M,\\om)$ is injective."}
{"category": "Math", "title": "Simple modules over factorpowers", "abstract": "In this paper we study complex representations of the factorpower $\\fp(G,M)$ of a finite group $G$ acting on a finite set $M$. This includes the finite monoid $\\FP$, which can be seen as a kind of a ``balanced'' generalization of the symmetric group $S_n$ inside the semigroup of all binary relations. We describe all irreducible representations of $\\fp(G,M)$ and relate them to irreducible representations of certain inverse semigroups. In particular, irreducible representations of $\\FP$ are related to irreducible representations of the maximal factorizable submonoid of the dual symmetric inverse monoid. We also show that in the latter cases irreducible representations lead to an interesting combinatorial problem in the representation theory of $S_n$, which, in particular, is related to Foulkes' conjecture. Finally, we show that all simple $\\fp(G,M)$-modules are unitarizable and that tensor products of simple $\\fp(G,M)$-modules are completely reducible."}
{"category": "Math", "title": "Cyclic and finite surgeries on Montesinos knots", "abstract": "We give a complete classification of the Dehn surgeries on Montesinos knots which yield manifolds with cyclic or finite fundamental groups."}
{"category": "Math", "title": "Canonical Nonlinear Connections in the Multi-Time Hamilton Geometry", "abstract": "In this paper we study some geometrical objects (d-tensors, multi-time semisprays of polymomenta and nonlinear connections) on the dual 1-jet vector bundle $J^{1*}(\\cal{T}, M)\\to \\cal{T}\\times M$. Some geometrical formulas, which connect the last two geometrical objects, are also derived. Finally, a canonical nonlinear connection produced by a Kronecker $h$-regular multi-time Hamiltonian is given."}
{"category": "Math", "title": "On a $p$--Laplace equation with multiple critical nonlinearities", "abstract": "Using the Mountain--Pass Theorem of Ambrosetti and Rabinowitz we prove that $-\\Delta_p u-\\mu|x|^{-p}{u^{p-1}}=|x|^{-s}{u^{\\crits-1}}+u^{\\crit-1}$ admits a positive weak solution in $\\rn$ of class $\\dunp\\cap C^1(\\rn\\setminus\\{0\\})$, whenever $\\mu<\\mu_1$, and $\\mu_1=[(n-p)/p]^p$. The technique is based on the existence of extremals of some Hardy--Sobolev type embeddings of independent interest. We also show that if $u\\in\\dunp$ is a weak solution in $\\rn$ of $-\\Delta_p u-\\mu|x|^{-p}{|u|^{p-2}u}=|x|^{-s}{|u|^{\\crits-2}u}+|u|^{q-2}u$, then $u\\equiv0$ when either $1<q<\\crit$, or $q>\\crit$ and $u$ is also of class $L^\\infty_\\text{\\scriptsize{loc}}(\\rn\\setminus\\{0\\})$."}
{"category": "Math", "title": "On the symmetry of ascents and descents over 01-fillings of moon polyominoes", "abstract": "The purpose of this short paper is to put recent results on the symmetry of the joint distribution of the numbers of crossings and nestings of two edges over matchings, set partitions and linked partitions, in the larger context of the enumeration of increasing and decreasing chains of length 2 in fillings of moon polyominoes."}
{"category": "Math", "title": "Generating the mapping class group of a punctured surface by involutions", "abstract": "Let $\\Sigma_{g,b}$ denote a closed orientable surface of genus $g$ with $b$ punctures and let $\\rm Mod(\\Sigma_{\\textit{g,b}})$ denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, $\\rm Mod(\\Sigma_{\\textit{g,b}})$ is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate $\\rm Mod(\\Sigma_{\\textit{g,b}})$. Brendle and Farb [BF] gave an answer in the case of $g\\geq 3, b=0$ and $g\\geq 4, b=1$, by describing a generating set consisting of 6 involutions. Kassabov showed that for every $b$ $\\rm Mod(\\Sigma_{\\textit{g,b}})$ can be generated by 4 involutions if $g\\geq 8$, 5 involutions if $g\\geq 6$ and 6 involutions if $g\\geq 4$. We proved that for every $b$ $\\rm Mod(\\Sigma_{\\textit{g,b}})$ can be generated by 4 involutions if $g\\geq 7$ and 5 involutions if $g\\geq 5$."}
{"category": "Math", "title": "Multiorder, Kleene stars and cyclic projectors in the geometry of max cones", "abstract": "This paper summarizes results on some topics in the max-plus convex geometry, mainly concerning the role of multiorder, Kleene stars and cyclic projectors, and relates them to some topics in max algebra. The multiorder principle leads to max-plus analogues of some statements in the finite-dimensional convex geometry and is related to the set covering conditions in max algebra. Kleene stars are fundamental for max algebra, as they accumulate the weights of optimal paths and describe the eigenspace of a matrix. On the other hand, the approach of tropical convexity decomposes a finitely generated semimodule into a number of convex regions, and these regions are column spans of uniquely defined Kleene stars. Another recent geometric result, that several semimodules with zero intersection can be separated from each other by max-plus halfspaces, leads to investigation of specific nonlinear operators called cyclic projectors. These nonlinear operators can be used to find a solution to homogeneous multi-sided systems of max-linear equations. The results are presented in the setting of max cones, i.e., semimodules over the max-times semiring."}
{"category": "Math", "title": "Second-order elliptic equations with variably partially VMO coefficients", "abstract": "The solvability in $W^{2}_{p}(\\bR^{d})$ spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the ball. This generalizes to a very large extent the case of equations with continuous or VMO coefficients."}
{"category": "Math", "title": "On the number of minimal surfaces with a given boundary", "abstract": "We generalize the following result of White: Suppose $N$ is a compact, strictly convex domain in $\\RR^3$ with smooth boundary. Let $\\Sigma$ be a compact 2-manifold with boundary. Then a generic smooth curve $\\Gamma\\cong \\partial\\Sigma$ in $\\partial N$ bounds an odd or even number of embedded minimal surfaces diffeomorphic to $\\Sigma$ according to whether $\\Sigma$ is or is not a union of disks. First, we prove that the parity theorem holds for any compact riemannian 3-manifold $N$ such that $N$ is strictly mean convex, $N$ is homeomorphic to a ball, $\\partial N$ is smooth, and $N$ contains no closed minimal surfaces. We then further relax the hypotheses by allowing $N$ to be weakly mean convex and to have piecewise smooth boundary. We extend the parity theorem yet further by showing that, under an additional hypothesis, it remains true for minimal surfaces with prescribed symmetries. The parity theorems are used in an essential way to prove the existence of embedded genus-$g$ helicoids in $\\SS^2\\times \\RR$. We give a very brief outline of this application. (The full argument will appear elsewhere.)"}
{"category": "Math", "title": "On pseudo-Riemannian Lie algebras: a class of new Lie-admissible algebras", "abstract": "M. Boucetta introduced the notion of pseudo-Riemannian Lie algebra in [2] when he studied the line Poisson structure on the dual of a Lie algebra. In this paper, we redefine pseudo-Riemannian Lie algebra, which, in essence, is a class of new Lie admissible algebras and prove that all pseudo-Riemannian Lie algebras are solvable. Using our main result and method, we prove some of M. Boucetta's results in [2, 3] in a simple and new way, then we give an explicit construction of Riemann-Lie algebras and a classification of pseudo-Riemannian Lie algebras of dimension 2 and 3."}
{"category": "Math", "title": "On $G$--modular functor", "abstract": "In this paper, we extend the notion of modular functor and fusion category to what we called $G$ equivariant modular functor and $G$ equivariant fusion category, where $G$ is a finite group, and establish a correspondence between between these notions."}
{"category": "Math", "title": "Quantizations of Character Varieties and Quantum Knot Invariants", "abstract": "Let G be a simple complex algebraic group and g its Lie algebra. We show that the g-Witten-Reshetikhin-Turaev quantum invariants determine a deformation-quantization, C_q[X_G(torus)], of the coordinate ring of the G-character variety of the torus. We prove that this deformation is in the direction of the Goldman's bracket. Furthermore, we show that every knot K defines an ideal I_K in C_q[X_G(torus)]. We conjecture that the homomorphism C_q[X_G(torus)] -> C[X_G(torus)], q -> 1, maps I_K to the ideal whose radical is the kernel of the map C[X_G(torus)] -> C[X_G(S^3 K)]. This conjecture is related to AJ-conjecture for sl(2,\\C). The results of this paper are inspired by the theory of q-holonomic relations between quantum invariants of Garoufalidis and Le. Along the way, we disprove Conjecture 2 in Le's \"The Colored Jones and the A-polynomial of Two-Bridge knots\"."}
{"category": "Math", "title": "Componentwise condition numbers of random sparse matrices", "abstract": "We prove an O(log n) bound for the expected value of the logarithm of the componentwise (and, a fortiori, the mixed) condition number of a random sparse n x n matrix. As a consequence, small bounds on the average loss of accuracy for triangular linear systems follow."}
{"category": "Math", "title": "Poincar\\'e recurrence for observations", "abstract": "A high dimensional dynamical system is often studied by experimentalists through the measurement of a relatively low number of different quantities, called an observation. Following this idea and in the continuity of Boshernitzan's work, for a measure preserving system, we study Poincar\\'e recurrence for the observation. The link between the return time for the observation and the Hausdorff dimension of the image of the invariant measure is considered. We prove that when the decay of correlations is super polynomial, the recurrence rates for the observations and the pointwise dimensions relatively to the push-forward are equal."}
{"category": "Math", "title": "An end-to-end-construction for singly periodic minimal surfaces", "abstract": "We show the existence of various families of properly embedded singly periodic minimal surfaces in R^3 with finite arbitrary genus and Scherk type ends in the quotient. The proof of our results is based on the gluing of small perturbations of pieces of already known minimal surfaces."}
{"category": "Math", "title": "On automorphism groups of some types of generic distributions", "abstract": "To certain types of generic distributions (subbundles in a tangent bundle) one can associate canonical Cartan connections. Many of these constructions fall into the class of parabolic geometries. The aim of this article is to show how strong restrictions on the possibles sizes of automorphism groups of such distributions can be deduced from the existence of canonical Cartan connections. This needs no information on how the Cartan connections are actually constructed and only very basic information on their properties. In particular, we discuss the examples of generic distributions of rank two in dimension five, rank three in dimension six, and rank four in dimension seven."}
{"category": "Math", "title": "On different notions of tameness in arithmetic geometry", "abstract": "The notion of a tamely ramified covering is canonical only for curves. Several notions of tameness for coverings of higher dimensional schemes have been used in the literature. We show that all these definitions are essentially equivalent. Furthermore, we prove finiteness theorems for the tame fundamental groups of arithmetic schemes."}
{"category": "Math", "title": "Varieties with generically nef tangent bundles", "abstract": "We study the geometry of projective manifolds whose tangent bundles are nef on sufficiently general curves (i.e. the tangent bundle is generically nef) and show that manifolds whose anticanonical bundles are semi-ample have this property. Furthermore we introduce a notion of sufficient nefness and investigate the relation with manifolds whose anticanonical bundles are nef."}
{"category": "Math", "title": "On the Oppenheim's \"factorisatio numerorum\" function", "abstract": "Let $f(n)$ denote the number of distinct unordered factorisations of the natural number $n$ into factors larger than 1.In this paper, we address some aspects of the function $f(n)$."}
{"category": "Math", "title": "A Spanning Set for the space of Super Cusp forms", "abstract": "Aim of this article is the construction of a spanning set for the space of super cusp forms on a complex bounded symmetric super domain B of rank 1 with respect to a lattice. The main ingredients are a generalization of the Anosov closing lemma for partially hyperbolic diffeomorphisms and an unbounded realization of B, in particular Fourier decomposition at the cusps mapped to infinity via a partial Cayley transformation. The elements of the spanning set are in finite-to-one correspondence with closed geodesics, the number of elements corresponding to a geodesic growing linearly with its length."}
{"category": "Math", "title": "Hopf decomposition and horospheric limit sets", "abstract": "By looking at the relationship between the recurrence properties of a countable group action with a quasi-invariant measure and the structure of its ergodic components we establish a simple general description of the Hopf decomposition of the action into the conservative and the dissipative parts in terms of the Radon--Nikodym derivatives of the action. As an application we prove that the conservative part of the boundary action of a discrete group of isometries of a Gromov hyperbolic space with respect to any invariant quasi-conformal stream coincides (mod 0) with the big horospheric limit set of the group."}
{"category": "Math", "title": "Minimal surfaces and harmonic diffeomorphisms from the complex plane onto a Hadamard surface", "abstract": "We construct harmonic diffeomorphisms from the complex plane $C$ onto any Hadamard surface $M$ whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in $M\\times R$ over domains of $M$ bounded by ideal geodesic polygons and show the existence of a sequence of minimal graphs over polygonal domains converging to an entire minimal graph in $M\\times R$ with the conformal structure of $C$."}
{"category": "Math", "title": "Bayesian Analysis of Marginal Log-Linear Graphical Models for Three Way Contingency Tables", "abstract": "This paper deals with the Bayesian analysis of graphical models of marginal independence for three way contingency tables. We use a marginal log-linear parametrization, under which the model is defined through suitable zero-constraints on the interaction parameters calculated within marginal distributions. We undertake a comprehensive Bayesian analysis of these models, involving suitable choices of prior distributions, estimation, model determination, as well as the allied computational issues. The methodology is illustrated with reference to two real data sets."}
{"category": "Math", "title": "Nonstandard model categories and homotopy theory", "abstract": "In order to apply nonstandard methods to questions of algebraic geometry we continue our investigation from \"Enlargements of categories\" (Theory Appl. Categ. 14 (2005), No. 16, 357--398) and show how important homotopical constructions behave under enlargements."}
{"category": "Math", "title": "Etale and motivic cohomology and ultraproducts of schemes", "abstract": "This paper is a continuation of the authors article \"Enlargements of schemes\" (Log. Anal.1 (2007), no. 1, 1-60) We mainly study the behaviour of etale cohomology, algebraic cycles and motives under ultraproducts respectively enlargements. The main motivation for that is to find methods to transfer statements about etale cohomology and algebraic cycles from characteristic zero to positive characteristic and vice versa. We give one application to the independence of $l$ of Betti numbers in etale cohomology and applications to the complexity of algebraic cycles."}
{"category": "Math", "title": "Diagonalisation schemes and applications", "abstract": "These notes develop aspects of perturbation theory of matrices related to so-called diagonalisation schemes. Primary focus is on constructive tools to derive asymptotic expansions for small/large parameters of eigenvalues and eigenprojections of families of matrices depending upon real/complex parameters. Applications of the schemes in different frameworks --including hyperbolic polynomials, asymptotic integration of ordinary differential equations, diagonalisation within symbolic hierarchies and pseudo-differential decoupling of hyperbolic-parabolic coupled systems-- are also discussed and references to further applications given."}
{"category": "Math", "title": "Cauchon diagrams for quantized enveloping algebras", "abstract": "Let $\\mathfrak{g}$ be a finite dimensional complex simple Lie algebra, $\\mathbb{K}$ a commutative field and $q$ a nonzero element of $\\mathbb{K}$ which is not a root of unity. To each reduced decomposition of the longest element $w_0$ of the Weyl group $W$ corresponds a PBW basis of the quantised enveloping algebra $\\mathcal{U}_q^+(\\mathfrak{g})$, and one can apply the theory of deleting-derivation to this iterated Ore extension. In particular, for each decomposition of $w_0$, this theory constructs a bijection between the set of prime ideals in $\\mathcal{U}_q^+(\\mathfrak{g})$ that are invariant under a natural torus action and certain combinatorial objects called Cauchon diagrams. In this paper, we give an algorithmic description of these Cauchon diagrams when the chosen reduced decomposition of $w_0$ corresponds to a good ordering (in the sense of Lusztig \\cite{Lu2}) of the set of positive roots. This algorithmic description is based on the constraints that are coming from Lusztig's admissible planes \\cite{Lu2}: each admissible plane leads to a set of constraints that a diagram has to satisfy to be Cauchon. Moreover, we explicitely describe the set of Cauchon diagrams for explicit reduced decomposition of $w_0$ in each possible type. In any case, we check that the number of Cauchon diagrams is always equal to the cardinal of $W$. In \\cite{CM}, we use these results to prove that Cauchon diagrams correspond canonically to the positive subexpressions of $w_0$. So the results of this paper also give an algorithmic description of the positive subexpressions of any reduced decomposition of $w_0$ corresponding to a good ordering."}
{"category": "Math", "title": "Matrix random products with singular harmonic measure", "abstract": "Any Zariski dense countable subgroup of $SL(d,R)$ is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in $R^d$ in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and Lyapunov exponents going to infinity."}
{"category": "Math", "title": "Finite dimensional representations of W-algebras", "abstract": "W-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete classification of finite dimensional irreducible modules for W-algebras. Also we study a relation between Harish-Chandra bimodules and bimodules over $W$-algebras."}
{"category": "Math", "title": "Gaussian Multiplicative Chaos revisited", "abstract": "In this article, we extend the theory of multiplicative chaos for positive definite functions in Rd of the form f(x) = 2 ln+ T|x|+ g(x) where g is a continuous and bounded function. The construction is simpler and more general than the one defined by Kahane in 1985. As main application, we give a rigorous mathematical meaning to the Kolmogorov-Obukhov model of energy dissipation in a turbulent flow."}
{"category": "Math", "title": "The homotopy type of the space of symplectic balls in rational ruled 4-manifolds", "abstract": "Let M:=(M^{4},\\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_{\\omega}(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \\subset \\R^4 of radius r and of capacity c:= \\pi r^2 into (M,\\om). By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity \\ccrit \\in (0,w_{M}] such that, for all c\\in(0,\\ccrit), the embedding space Emb_{\\omega}(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \\SFr(M). We also know that the homotopy type of Emb_{\\omega}(B^{4}(c),M) changes when c reaches \\ccrit and that it remains constant for all c \\in [\\ccrit,w_{M}). In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of \\Emb_{\\omega}(B^{4}(c),M) in the remaining case c \\in [\\ccrit,w_{M}). In particular, we show that it does not have the homotopy type of a finite CW-complex."}
{"category": "Math", "title": "The Word and Geodesic Problems in Free Solvable Groups", "abstract": "We study the computational complexity of the Word Problem (WP) in free solvable groups $S_{r,d}$, where $r \\geq 2$ is the rank and $d \\geq 2$ is the solvability class of the group. It is known that the Magnus embedding of $S_{r,d}$ into matrices provides a polynomial time decision algorithm for WP in a fixed group $S_{r,d}$. Unfortunately, the degree of the polynomial grows together with $d$, so the uniform algorithm is not polynomial in $d$. In this paper we show that WP has time complexity $O(r n \\log_2 n)$ in $S_{r,2}$, and $O(n^3 r d)$ in $S_{r,d}$ for $d \\geq 3$. However, it turns out, that a seemingly close problem of computing the geodesic length of elements in $S_{r,2}$ is $NP$-complete. We prove also that one can compute Fox derivatives of elements from $S_{r,d}$ in time $O(n^3 r d)$, in particular one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as, on a relatively new geometric ideas, in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs."}
{"category": "Math", "title": "KPZ formula for log-infinitely divisible multifractal random measures", "abstract": "We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in \\cite{bacry} . If M is a non degenerate multifractal measure with associated metric $\\rho(x,y)=M([x,y])$ and structure function $\\zet a$, we show that we have the following relation between the (Euclidian) Hausdorff dimension ${\\rm dim}_H$ of a measurable set K and the Hausdorff dimension ${\\rm dim}_H^{\\rho}$ with respect to \\rho of the same set: $\\zeta({\\rm dim}_H^{\\rho}(K))={\\r m dim}_H(K)$. Our results can be extended to higher dimensions in the log normal case: inspired by quantum gravity in dime nsion 2, we consider the 2 dimensional case."}
{"category": "Math", "title": "Hydrodynamic turbulence and intermittent random fields", "abstract": "In this article, we construct two families of nonsymmetrical multifractal fields. One of these families is used for the modelization of the velocity field of turbulent flows."}
{"category": "Math", "title": "$AC(\\sigma)$ operators", "abstract": "In this paper we present a new extension of the theory of well-bounded operators to cover operators with complex spectrum. In previous work a new concept of the class of absolutely continuous functions on a nonempty compact subset $\\sigma$ of the plane, denoted $AC(\\sigma)$, was introduced. An $\\AC(\\sigma)$ operator is one which admits a functional calculus for this algebra of functions. The class of $AC(\\sigma)$ operators includes all of the well-bounded operators and trigonometrically well-bounded operators, as well as all scalar-type spectral operators, but is strictly smaller than Berkson and Gillespie's class of $AC$ operators. This paper develops the spectral properties of $AC(\\sigma)$ operators and surveys some of the problems which remain in extending results from the theory of well-bounded operators."}
{"category": "Math", "title": "Compact $AC(\\sigma)$ operators", "abstract": "All compact $AC(\\sigma)$ operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact $AC(\\sigma)$ into real and imaginary parts."}
{"category": "Math", "title": "Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems", "abstract": "Let M denote the space of probability measures on R^D endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in M was introduced by Ambrosio, Gigli and Savare'. In this paper we develop a calculus for the corresponding class of differential forms on M. In particular we prove an analogue of Green's theorem for 1-forms and show that the corresponding first cohomology group, in the sense of de Rham, vanishes. For D=2d we then define a symplectic distribution on M in terms of this calculus, thus obtaining a rigorous framework for the notion of Hamiltonian systems as introduced by Ambrosio and Gangbo. Throughout the paper we emphasize the geometric viewpoint and the role played by certain diffeomorphism groups of R^D."}
{"category": "Math", "title": "Global regularity and probabilistic schemes for free boundary surfaces of multivariate American derivatives and their Greeks", "abstract": "In a rather general setting of multivariate stochastic volatility market models we derive global iterative probabilistic schemes for computing the free boundary and its Greeks for a generic class of American derivative models using front-fixing methods. Convergence is closely linked to a proof of global regularity of the free boundary surface."}
{"category": "Math", "title": "Discrete time nonlinear filters with informative observations are stable", "abstract": "The nonlinear filter associated with the discrete time signal-observation model $(X_k,Y_k)$ is known to forget its initial condition as $k\\to\\infty$ regardless of the observation structure when the signal possesses sufficiently strong ergodic properties. Conversely, it stands to reason that if the observations are sufficiently informative, then the nonlinear filter should forget its initial condition regardless of any properties of the signal. We show that for observations of additive type $Y_k=h(X_k)+\\xi_k$ with invertible observation function $h$ (under mild regularity assumptions on $h$ and on the distribution of the noise $\\xi_k$), the filter is indeed stable in a weak sense without any assumptions at all on the signal process. If the signal satisfies a uniform continuity assumption, weak stability can be strengthened to stability in total variation."}
{"category": "Math", "title": "A simple method for generating rational triangles", "abstract": "In the early part of the paper, various geometrical formulas are derived. Then, at some point in the paper, the concept of a Pythagorean rational is introduced. A Pythagorean rational is a rational number which is the ratio of two integers which are the leglengths of a Pythagorean triangle. Using the idea of Pythagorean rationals, we generate two families of rational triangles. We define a rational triangle to be a triangle with rational sidelengths and area."}
{"category": "Math", "title": "Crystalline representations of G_Qp^a with coefficients", "abstract": "This paper studies crystalline representations of G_K with coefficients of any dimension, where K is the unramified extension of Q_p of degree a. We prove a theorem of Fontaine-Laffaille type when \\sigma-invariant Hodge-Tate weight less than p-1, which establishes the bijection between Galois stable lattices in crystalline representations and strongly divisible \\phi-lattice. In generalizing Breuil's work, we classify all reducible and irreducible crystalline representations of G_K of dimensional 2, then describe their mod p reductions. We generalize some results (of Deligne, Fontaine-Serre, and Edixhoven) to representations arising from Hilbert modular forms when \\sigma-invariant Hodge-Tate weight less than p-1."}
{"category": "Math", "title": "Interpretation of the Arithmetic in certain groups of piecewise affine permutations of an interval", "abstract": "The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group $F$ and some of its generalizations interpret the Arithmetic without parameters."}
{"category": "Math", "title": "Non-enlargeable operators and self-cancelling operators", "abstract": "The epsilon-enlargement of a maximal monotone operator is a construct similar to the Br{\\o}ndsted and Rocakfellar epsilon-subdifferential enlargement of the subdifferential. Like the epsilon-subdifferential, the epsilon-enlargement of a maximal monotone operator has practical and theoretical applications. In a recent paper in Journal of Convex Analysis Burachik and Iusem studied conditions under which a maximal monotone operator is non-enlargeable, that is, its epsilon-enlargement coincides with the operator. Burachik and Iusem studied these non-enlargeable operators in reflexive Banach spaces, assuming the interior of the domain of the operator to be nonempty. In the present work, we remove the assumption on the domain of non-enlargeable operators and also present partial results for the non-reflexive case."}
{"category": "Math", "title": "Principal components analysis for sparsely observed correlated functional data using a kernel smoothing approach", "abstract": "In this paper, we consider the problem of estimating the covariance kernel and its eigenvalues and eigenfunctions from sparse, irregularly observed, noise corrupted and (possibly) correlated functional data. We present a method based on pre-smoothing of individual sample curves through an appropriate kernel. We show that the naive empirical covariance of the pre-smoothed sample curves gives highly biased estimator of the covariance kernel along its diagonal. We attend to this problem by estimating the diagonal and off-diagonal parts of the covariance kernel separately. We then present a practical and efficient method for choosing the bandwidth for the kernel by using an approximation to the leave-one-curve-out cross validation score. We prove that under standard regularity conditions on the covariance kernel and assuming i.i.d. samples, the risk of our estimator, under $L^2$ loss, achieves the optimal nonparametric rate when the number of measurements per curve is bounded. We also show that even when the sample curves are correlated in such a way that the noiseless data has a separable covariance structure, the proposed method is still consistent and we quantify the role of this correlation in the risk of the estimator."}
{"category": "Math", "title": "Generalization of the Apollonius Circles", "abstract": "The three Apollonius circles of a triangle, each passing through a triangle vertex, the corresponding vertex of the cevian triangle of the incenter and the corresponding vertex of the circumcevian triangle of the symmedian point, are coaxal. Similarly defined three circles remain coaxal, when the circumcevian triangle is defined with respect to any point on the triangle circumconic through the incenter and symmedian point. Inversion in the incircle of the reference triangle carries these three coaxal circles into coaxal circles, each passing through a vertex of the inverted triangle and centered on the opposite sideline, at the intersection of the orthotransversal with respect to a point on the Euler line of the inverted triangle. A similar circumconic exists in a more general configuration, when the cevian triangle is defined with respect to an arbitrary point, passing through this arbitrary point and isogonal conjugate of its complement."}
{"category": "Math", "title": "Boole's formula as a consequence of Lagrange's Interpolating Polynomial theorem", "abstract": "We present a slightly more general version of Boole's additive formula for factorials as a simple consequence of Lagrange's Interpolating Polynomial theorem."}
{"category": "Math", "title": "Cohomology of normic systems and fake Z_p extensions", "abstract": "We set up a general framework to study Tate cohomology groups of Galois modules along $\\mathbb{Z}_p$-extensions of number fields. Under suitable assumptions on the Galois modules, we establish the existence of a five-term exact sequence in a certain quotient category whose objects are simultaneously direct and inverse systems, subject to some compatibility. The exact sequence allows one, in particular, to control the behaviour of the Tate cohomology groups of the units along $\\mathbb{Z}_p$-extensions. As an application, we study the growth of class numbers along what we call \"fake $\\mathbb{Z}_p$-extensions of dihedral type\". This study relies on a previous work, where we established a class number formula for dihedral extensions in terms of the cohomology groups of the units."}
{"category": "Math", "title": "The coarse classification of countable abelian groups", "abstract": "We prove that two countable locally finite-by-abelian groups G,H endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely-generated or both are infinitely generated. On the other hand, we show that each countable group G that coarsely embeds into a countable abelian group is locally nilpotent-by-finite. Moreover, the group G is locally abelian-by-finite if and only if G is undistorted in the sense that G can be written as the union of countably many finitely generated subgroups G_n such that each G_n is undistorted in G_{n+1} (which means that the identity inclusion from G_n to G_{n+1} is a quasi-isometric embedding with respect to word metrics)."}
{"category": "Math", "title": "Test elements, retracts and automorphic orbits", "abstract": "Let $A_2$ be a free associative or polynomial algebra of rank two over a field $K$ of characteristic zero. Based on the degree estimate of Makar-Limanov and J.-T.Yu, we prove: 1) An element $p \\in A_2$ is a test element if $p$ does not belong to any proper retract of $A_2$; 2) Every endomorphism preserving the automorphic orbit of a nonconstant element of $A_2$ is an automorphism."}
{"category": "Math", "title": "A characterization of quiver algebras based on double derivations", "abstract": "Let k a characteristic zero field. We give a characterization for the finite quiver k-algebras, based on double derivations. More precisely, we prove that if an associative and unitary k-algebra have a family of double derivations satisfying suitable conditions, then it is (canonically isomorphic to) a quiver algebra. This is the non-commutative version of a result of D. Wright."}
{"category": "Math", "title": "The Grothendieck-Katz Conjecture for certain locally symmetric varieties", "abstract": "Using Margulis's results on lattices in semisimple Lie groups, we prove the Grothendieck-Katz $p$-Curvature Conjecture for certain locally symmetric varieties, including the moduli space of abelian varieties ${\\cal A}_g$ when $g > 1.$"}
{"category": "Math", "title": "The Chow group of zero-cycles on certain Ch{\\^a}telet surfaces over local fields", "abstract": "We compute the Chow group of zero-cycles on certain Ch{\\^a}telet surfaces over local fields."}
{"category": "Math", "title": "Absolute norms of p-primary units", "abstract": "We prove a local analogue of a theorem of J. Martinet about the absolute norm of the relative discriminant ideal of an extension of number fields. The result can be seen as a statement about 2-primary units. We also prove a similar statement about the absolute norms of p-primary units, for all primes p."}
{"category": "Math", "title": "On the moments of the Riemann zeta-function in short intervals", "abstract": "Assuming the Riemann Hypothesis it is proved that, for fixed $k>0$ and $H = T^\\theta$ with fixed $0<\\theta \\le 1$, $$ \\int_T^{T+H}|\\zeta(1/2+it)|^{2k} dt \\ll H(\\log T)^{k^2(1+O(1/\\log_3T))}, $$ where $\\log_jT = \\log(\\log_{j-1}T)$. The proof is based on the recent method of K. Soundararajan for counting the occurrence of large values of $\\log|\\zeta(1/2+it)|$, who proved that $$ \\int_0^{T}|\\zeta(1/2+it)|^{2k} dt \\ll_\\epsilon T(\\log T)^{k^2+\\epsilon}. $$"}
{"category": "Math", "title": "An essential relation between Einstein metrics, volume entropy, and exotic smooth structures", "abstract": "We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positive minimal volume entropy. 2) They satisfy a strict version of the Gromov-Hitchin-Thorpe inequality, with a minimal volume entropy term. 3) They nevertheless admit infinitely many distinct smooth structures for which no compatible Einstein metric exists."}
{"category": "Math", "title": "On generic properties of finitely presented monoids and semigroups", "abstract": "We study the generic properties of finitely presented monoids and semigroups. We show that for positive integers a > 1, k and m, the generic a-generator k-relation monoid and semigroup presentation (defined in any of several definite statistical senses) satisfy the small overlap condition C(m). It follows that the generic monoid is torsion-free and J-trivial and, by a recent result of the author, admits a linear time solution to its word problem and a regular language of unique normal forms for its elements. Moreover, the uniform word problem for finitely presented monoids is generically solvable in time linear in the word lengths and quadratic in the presentation size. We also prove some technical results about generic sets which may be of independent interest."}
{"category": "Math", "title": "A cocycle on the group of symplectic diffeomorphisms", "abstract": "We define a cocycle on the group of symplectic diffeomorphisms of a symplectic manifold and investigate its properties. The main applications are concerned with symplectic actions of discrete groups. For example, we give an alternative proof of the Polterovich theorem about the distortion of cyclic subgroups in finitely generated groups of Hamiltonian diffeomorphisms."}
{"category": "Math", "title": "Spatial homogenization in a stochastic network with mobility", "abstract": "A stochastic model for a mobile network is studied. Users enter the network, and then perform independent Markovian routes between nodes where they receive service according to the Processor-Sharing policy. Once their service requirement is satisfied, they leave the system. The stability region is identified via a fluid limit approach, and strongly relies on a \"spatial homogenization\" property: at the fluid level, customers are instantaneously distributed across the network according to the stationary distribution of their Markovian dynamics and stay distributed as such as long as the network is not empty. In the unstable regime, spatial homogenization almost surely holds asymptotically as time goes to infinity (on the normal scale), telling how the system fills up. One of the technical achievements of the paper is the construction of a family of martingales associated to the multidimensional process of interest, which makes it possible to get crucial estimates for certain exit times."}
{"category": "Math", "title": "Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes", "abstract": "We consider the class of all the Hermite processes $(Z_{t}^{(q,H)})_{t\\in \\lbrack 0,1]}$ of order $q\\in \\mathbf{N}^{\\ast}$ and with Hurst parameter $% H\\in (\\frac{1}{2},1)$. The process $Z^{(q,H)}$ is $H$-selfsimilar, it has stationary increments and it exhibits long-range dependence identical to that of fractional Brownian motion (fBm). For $q=1$, $Z^{(1,H)}$ is fBm, which is Gaussian; for $q=2$, $Z^{(2,H)}$ is the Rosenblatt process, which lives in the second Wiener chaos; for any $q>2$, $Z^{(q,H)}$ is a process in the $q$th Wiener chaos. We study the variations of $Z^{(q,H)}$ for any $q$, by using multiple Wiener -It\\^{o} stochastic integrals and Malliavin calculus. We prove a reproduction property for this class of processes in the sense that the terms appearing in the chaotic decomposition of their variations give rise to other Hermite processes of different orders and with different Hurst parameters. We apply our results to construct a strongly consistent estimator for the self-similarity parameter $H$ from discrete observations of $Z^{(q,H)}$; the asymptotics of this estimator, after appropriate normalization, are proved to be distributed like a Rosenblatt random variable (value at time $1$ of a Rosenblatt process).with self-similarity parameter $1+2(H-1)/q$."}
{"category": "Math", "title": "The Rank of the Endomorphism Monoid of a Partition", "abstract": "The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open question."}
{"category": "Math", "title": "ULD-Lattices and Delta-Bonds", "abstract": "We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colorings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on Delta-bonds with invariant circular flow-difference. This instance generalizes several previously studied lattice structures, in particular, c-orientations (Propp), alpha-orientations of planar graphs (Felsner, resp. de Mendez) and planar flows (Khuller, Naor and Klein). The characterization also applies to other instances, e.g. to chip-firing games."}
{"category": "Math", "title": "Negative volatility for a 2-dimensional square root SDE", "abstract": "In affine term structure models the short rate is modelled as an affine transformation of a multi-dimensional square root process. Sufficient conditions to avoid negative volatility factors are the multivariate Feller conditions. We will prove their necessity for a 2-dimensional square root SDE with one volatility factor by presenting a methodology based on measure transformations and solving linear systems of ordinary differential equations."}
{"category": "Math", "title": "The supersingular locus in Siegel modular varieties with Iwahori level structure", "abstract": "We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl-Oort stratification on the former, the Kottwitz-Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case $g$ is even."}
{"category": "Math", "title": "Reducing almost Lagrangian structures and almost CR geometries to partially integrable structures", "abstract": "This paper demostrates a method for analysing almost CR geometries $(H,J)$, by uniquley defining a partially integrable structure $(H,K)$ from the same data. Thus two almost CR geometries $(H,J)$ and $(H',J')$ are equivalent if and and only if they generate isomorphic induced partially integrable CR geometries $(H,K)$ and $(H',K')$, and if the set of CR morphisms between these spaces contains an element that maps $J$ to $J'$. Similar results hold for almost Lagrangian structures."}
{"category": "Math", "title": "Algebras over Cobar(coFrob)", "abstract": "We show that a square zero, degree one element in W(V), the Weyl algebra on a vector space V, is equivalent to providing V with the structure of an algebra over the properad Cobar(coFrob), the properad arising from the cobar construction applied to the cofrobenius coproperad."}
{"category": "Math", "title": "Meromorphic functions with several essential singularities", "abstract": "A two-parameter characteristic of functions meromorphic on annuli is introduced and an extension of the Nevanlinna value distribution theory for such functions is proposed."}
{"category": "Math", "title": "Pivoting in Linear Complementarity: Two Polynomial-Time Cases", "abstract": "We study the behavior of simple principal pivoting methods for the P-matrix linear complementarity problem (P-LCP). We solve an open problem of Morris by showing that Murty's least-index pivot rule (under any fixed index order) leads to a quadratic number of iterations on Morris's highly cyclic P-LCP examples. We then show that on K-matrix LCP instances, all pivot rules require only a linear number of iterations. As the main tool, we employ unique-sink orientations of cubes, a useful combinatorial abstraction of the P-LCP."}
{"category": "Math", "title": "$\\sigma$-continuity and related forcings", "abstract": "The Steprans forcing notion arises as a quotient of Borel sets modulo the ideal of $\\sigma$-continuity of a certain Borel not $\\sigma$-continuous function. We give a characterization of this forcing in the language of trees and using this characterization we establish such properties of the forcing as fusion and continuous reading of names. Although the latter property is usually implied by the fact that the associated ideal is generated by closed sets, we show it is not the case with Steprans forcing. We also establish a connection between Steprans forcing and Miller forcing thus giving a new description of the latter. Eventually, we exhibit a variety of forcing notions which do not have continuous reading of names in any presentation."}
{"category": "Math", "title": "Firmly nonexpansive and Kirszbraun-Valentine extensions: a constructive approach via monotone operator theory", "abstract": "Utilizing our recent proximal-average based results on the constructive extension of monotone operators, we provide a novel approach to the celebrated Kirszbraun-Valentine Theorem and to the extension of firmly nonexpansive mappings."}
{"category": "Math", "title": "Global regularity for some classes of large solutions to the Navier-Stokes equations", "abstract": "In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in the last paper is that it varies slowly in one direction, though in some sense it is ``well prepared'' (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of that last paper to an ``ill prepared'' situation (the norm blows up as the small parameter goes to zero).The proof uses the special structure of the nonlinear term of the equation."}
{"category": "Math", "title": "Semiparametric curve alignment and shift density estimation for biological data", "abstract": "Assume that we observe a large number of curves, all of them with identical, although unknown, shape, but with a different random shift. The objective is to estimate the individual time shifts and their distribution. Such an objective appears in several biological applications like neuroscience or ECG signal processing, in which the estimation of the distribution of the elapsed time between repetitive pulses with a possibly low signal-noise ratio, and without a knowledge of the pulse shape is of interest. We suggest an M-estimator leading to a three-stage algorithm: we split our data set in blocks, on which the estimation of the shifts is done by minimizing a cost criterion based on a functional of the periodogram; the estimated shifts are then plugged into a standard density estimator. We show that under mild regularity assumptions the density estimate converges weakly to the true shift distribution. The theory is applied both to simulations and to alignment of real ECG signals. The estimator of the shift distribution performs well, even in the case of low signal-to-noise ratio, and is shown to outperform the standard methods for curve alignment."}
{"category": "Math", "title": "Holomorphic transforms with application to affine processes", "abstract": "In a rather general setting of It\\^o-L\\'evy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine It\\^o-L\\'evy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform."}
{"category": "Math", "title": "On real moduli spaces over M-curves", "abstract": "Let $F$ be a genus $g$ curve and $\\sigma: F \\to F$ a real structure with the maximal possible number of fixed circles. We study the real moduli space $\\N' = \\Fix (\\sigma^{#})$ where $\\sigma^{#}: \\N \\to \\N$ is the induced real structure on the moduli space $\\N$ of stable holomorphic bundles of rank 2 over $F$ with fixed non-trivial determinant. In particular, we calculate $H^* (\\N',\\mathbb Z)$ in the case of $g = 2$, generalizing Thaddeus' approach to computing $H^* (\\N,\\mathbb Z)$."}
{"category": "Math", "title": "Cohomology and Immersed Curves", "abstract": "We introduce a new cohomology-theoretic method for classifying generic immersed curves in closed compact surfaces by using Gauss codes. This subsumes a result of J.S. Carter on classifying immersed curves in oriented compact surfaces, and provides a criterion for when an immersion is two-colorable. We note an application to twisted virtual link theory."}
{"category": "Math", "title": "Column basis reduction, and decomposable knapsack problems", "abstract": "We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b' <= Ax <= b, x \\in Z^n with b' <= AUy <= b, y \\in Z^n, where U is a unimodular matrix computed via basis reduction, to make the columns of $AU$ short and nearly orthogonal. The reformulation is called rangespace reformulation. It is motivated by the reformulation technique proposed for equality constrained IPs by Aardal, Hurkens and Lenstra. We also study a family of IP instances, called decomposable knapsack problems (DKPs). DKPs generalize the instances proposed by Jeroslow, Chvatal and Todd, Avis, Aardal and Lenstra, and Cornuejols et al. DKPs are knapsack problems with a constraint vector of the form $pM + r, $ with $p >0$ and $r$ integral vectors, and $M$ a large integer. If the parameters are suitably chosen in DKPs, we prove 1) hardness results for these problems, when branch-and-bound branching on individual variables is applied; 2) that they are easy, if one branches on the constraint $px$ instead; and 3) that branching on the last few variables in either the rangespace- or the AHL-reformulations is equivalent to branching on $px$ in the original problem. We also provide recipes to generate such instances. Our computational study confirms that the behavior of the studied instances in practice is as predicted by the theoretical results."}
{"category": "Math", "title": "Local behavior of p-harmonic Green's functions in metric spaces", "abstract": "We describe the behavior of p-harmonic Green's functions near a singularity in metric measure spaces equipped with a doubling measure and supporting a Poincar\\'e inequality."}
{"category": "Math", "title": "The geometry of unitary 2-representations of finite groups and their 2-characters", "abstract": "Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and their 2-characters. We show how the basic ideas of geometric quantization are `categorified' in this context: just as representations of groups correspond to equivariant line bundles, 2-representations of groups correspond to equivariant gerbes. We also show how the 2-character of a 2-representation can be made functorial with respect to morphisms of 2-representations. Under the geometric correspondence, the 2-character of a 2-representation corresponds to the geometric character of its associated equivariant gerbe. This enables us to show that the complexified 2-character is a unitarily fully faithful functor from the complexified homotopy category of unitary 2-representations to the category of unitary equivariant vector bundles over the group."}
{"category": "Math", "title": "Surgery obstructions from Khovanov homology", "abstract": "For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links."}
{"category": "Math", "title": "A multimodular algorithm for computing Bernoulli numbers", "abstract": "We describe an algorithm for computing Bernoulli numbers. Using a parallel implementation, we have computed B(k) for k = 10^8, a new record. Our method is to compute B(k) modulo p for many small primes p, and then reconstruct B(k) via the Chinese Remainder Theorem. The asymptotic time complexity is O(k^2 log(k)^(2+epsilon)), matching that of existing algorithms that exploit the relationship between B(k) and the Riemann zeta function. Our implementation is significantly faster than several existing implementations of the zeta-function method."}
{"category": "Math", "title": "Nonsymmetric interpolation Macdonald polynomials and g_n basic hypergeometric series", "abstract": "The Knop-Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type g_n. Our main results include a new q-binomial theorem, new q-Gauss sum, and several transformation formulae for g_n series."}
{"category": "Math", "title": "Second structure relation for $q$-semiclassical polynomials of the Hahn Tableau", "abstract": "The q-classical orthogonal polynomials of the q-Hahn Tableau are characterized from their orthogonality condition and by a first and a second structure relation. Unfortunately, for the q-semiclassical orthogonal polynomials (a generalization of the classical ones) we find only in the literature the first structure relation. In this paper, a second structure relation is deduced. In particular, by means of a general finite-type relation between a q-semiclassical polynomial sequence and the sequence of its q-differences such a structure relation is obtained."}
{"category": "Math", "title": "Thermoacoustic tomography with detectors on an open curve: an efficient reconstruction algorithm", "abstract": "Practical applications of thermoacoustic tomography require numerical inversion of the spherical mean Radon transform with the centers of integration spheres occupying an open surface. Solution of this problem is needed (both in 2-D and 3-D) because frequently the region of interest cannot be completely surrounded by the detectors, as it happens, for example, in breast imaging. We present an efficient numerical algorithm for solving this problem in 2-D (similar methods are applicable in the 3-D case). Our method is based on the numerical approximation of plane waves by certain single layer potentials related to the acquisition geometry. After the densities of these potentials have been precomputed, each subsequent image reconstruction has the complexity of the regular filtration backprojection algorithm for the classical Radon transform. The peformance of the method is demonstrated in several numerical examples: one can see that the algorithm produces very accurate reconstructions if the data are accurate and sufficiently well sampled, on the other hand, it is sufficiently stable with respect to noise in the data."}
{"category": "Math", "title": "Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems", "abstract": "We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology. Solutions of these equations in these applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree. By a thresholding procedure, namely the elimination of leaves that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen optimally, in the sense that the total error of the adaptive scheme is of the same slope as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled."}
{"category": "Math", "title": "Some sufficient conditions of a given series with rational terms converging to an irrational number or a transcdental number", "abstract": "In this paper, we propose various sufficient conditions to determine if a given real number is an irrational number or a transcendental number and also apply these conditions to some interesting examples, particularly,one of them comes from complex analytic dynamics"}
{"category": "Math", "title": "Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface", "abstract": "Given an oriented Riemannian surface $(\\Sigma, g)$, its tangent bundle $T\\Sigma$ enjoys a natural pseudo-K\\\"{a}hler structure, that is the combination of a complex structure $\\J$, a pseudo-metric $\\G$ with neutral signature and a symplectic structure $\\Om$. We give a local classification of those surfaces of $T\\Sigma$ which are both Lagrangian with respect to $\\Om$ and minimal with respect to $\\G$. We first show that if $g$ is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in $\\R^3$ or $\\R^3_1$ induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in $T\\S^2$ or $T \\H^2$ respectively. We relate the area of the congruence to a second-order functional $\\mathcal{F}=\\int \\sqrt{H^2-K} dA$ on the original surface."}
{"category": "Math", "title": "Notes on De Jong's period=index theorem for central simple algebras over fields of transcendence degree two", "abstract": "These are notes on de Jong's proof of the period=index theorem over fields of transcendence degree two. They are actually about the simplified proof sketched by de Jong in the last section of his paper. These notes were meant as support for my lectures at the summer school \"Central Simple Algebras over Function Fields\" at the Universitat Konstanz between August, 26 and September, 1 2007 (other lectures on this subject were given by Philippe Gille, Andrew Kresch, Max Lieblich, Tamas Szamuely and Jan Van Geel). No originality is intended (except perhaps a little in the proof of the Artin splitting theorem). Various sources on which the material is based are indicated in the notes. The reader should be warned that these notes have not been updated to reflect developments in the subject which occurred after the end of the summerschool."}
{"category": "Math", "title": "The Regular C*-algebra of an Integral Domain", "abstract": "To each integral domain R with finite quotients we associate a purely infinite simple C*-algebra in a very natural way. Its stabilization can be identified with the crossed product of the algebra of continuous functions on the \"finite adele space\" corresponding to R by the action of the ax+b-group over the quotient field Q(R). We study the relationship to generalized Bost-Connes systems and deduce for them a description as universal C*-algebras with the help of our construction."}
{"category": "Math", "title": "A uniqueness result for Kirchhoff equations with non-Lipschitz nonlinear term", "abstract": "We consider the second order Cauchy problem $$u''+\\m{u}Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\\infty)\\to[0,+\\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that $u_{0}$ and $u_{1}$ are regular enough, depending on the continuity modulus of $m$. It is also well known that the solution is unique when $m$ is locally Lipschitz continuous. In this paper we prove that if either $<Au_{0},u_{1}>\\neq 0$, or $|A^{1/2}u_{1}|^{2}\\neq\\m{u_{0}}|Au_{0}|^{2}$, then the local solution is unique even if $m$ is not Lipschitz continuous."}
{"category": "Math", "title": "Almost Sure Stabilization for Adaptive Controls of Regime-switching LQ Systems with A Hidden Markov Chain", "abstract": "This work is devoted to the almost sure stabilization of adaptive control systems that involve an unknown Markov chain. The control system displays continuous dynamics represented by differential equations and discrete events given by a hidden Markov chain. Different from previous work on stabilization of adaptive controlled systems with a hidden Markov chain, where average criteria were considered, this work focuses on the almost sure stabilization or sample path stabilization of the underlying processes. Under simple conditions, it is shown that as long as the feedback controls have linear growth in the continuous component, the resulting process is regular. Moreover, by appropriate choice of the Lyapunov functions, it is shown that the adaptive system is stabilizable almost surely. As a by-product, it is also established that the controlled process is positive recurrent."}
{"category": "Math", "title": "A Generalized Mixed Zero-sum Stochastic Differential Game and Double Barrier Reflected BSDEs with Quadratic Growth Coefficient", "abstract": "This article is dedicated to the study of mixed zero-sum two-player stochastic differential games in the situation when the player's cost functionals are modeled by doubly controlled reflected backward stochastic equations with two barriers whose coefficients have quadratic growth in Z. This is a generalization of the risk-sensitive payoffs. We show that the lower and the upper value function associated with this stochastic differential game with reflection are deterministic and they are also the unique viscosity solutions for two Isaacs equations with obstacles."}
{"category": "Math", "title": "The Weil--Petersson geometry of the moduli space of Riemann surfaces", "abstract": "In [4], Z. Huang showed that in the thick part of the moduli space $\\mathcal{M}_g$ of compact Riemann surfaces of genus $g$, the sectional curvature of the Weil--Petersson metric is bounded below by a constant depending on injectivity radius, but independent of the genus $g$. In this article, we prove this result by a different method. We also show that the same result holds for Ricci curvature. For the universal Teichm\\\"uller space equipped with Hilbert structure induced by Weil--Petersson metric, we prove that its sectional curvature is bounded below by a universal constant."}
{"category": "Math", "title": "On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology", "abstract": "Let K in S^3 be a knot, and let \\widetilde{K} denote the preimage of K inside its double branched cover, \\Sigma(K). We prove, for each integer n > 1, the existence of a spectral sequence from Khovanov's categorification of the reduced n-colored Jones polynomial of the mirror of K to the knot Floer homology of (\\Sigma(K),\\widetilde{K}) (when n odd) and to (S^3, K # K) (when n even). A corollary of our result is that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1."}
{"category": "Math", "title": "Two Generalizations of Tensor Products, Beyond Vector Spaces", "abstract": "Two successive generalizations of the usual tensor products are given. One can be constructed for arbitrary binary operations, and not only for semigroups, groups or vector spaces. The second one, still more general, is constructed for arbitrary generators on sets."}
{"category": "Math", "title": "Mean curvature flow via convex functions on Grassmannian manifolds", "abstract": "Using the convex functions in Grassmannian manifolds we can carry out interior estimates for mean curvature flow of higher codimension. In this way some of the results of Ecker-Huisken can be generalized to higher codimension"}
{"category": "Math", "title": "A Multidimensional Central Sets Theorem", "abstract": "The Theorems of Hindman and van der Waerden belong to the classical theorems of partition Ramsey Theory. The Central Sets Theorem is a strong simultaneous extension of both theorems that applies to general commutative semigroups. We give a common extension of the Central Sets Theorem and Ramsey's Theorem."}
{"category": "Math", "title": "Strong characterizing sequences of countable groups", "abstract": "Andr\\'as Bir\\'o and Vera S\\'os prove that for any subgroup $G$ of $\\T$ generated freely by finitely many generators there is a sequence $A\\subset \\N$ such that for all $\\beta \\in \\T$ we have ($\\|.\\|$ denotes the distance to the nearest integer) $$\\beta\\in G \\Rightarrow \\sum_{n\\in A} \\| n \\beta\\| < \\infty,\\quad \\quad \\quad \\beta\\notin G \\Rightarrow \\limsup_{n\\in A, n \\to \\infty} \\|n \\beta\\| > 0. $$ We extend this result to arbitrary countable subgroups of $\\T$. We also show that not only the sum of norms but the sum of arbitrary small powers of these norms can be kept small. Our proof combines ideas from the above article with new methods, involving a filter characterization of subgroups of $\\T$."}
{"category": "Math", "title": "A variant of the Hales-Jewett Theorem", "abstract": "It was shown by V. Bergelson that any set B with positive upper multiplicative density contains nicely intertwined arithmetic and geometric progressions: For each positive integer k there exist integers a,b,d such that $ {b(a+id)^j:i,j \\in\\nhat k}\\subset B. $ In particular one cell of each finite partition of the positive integers contains such configurations. We prove a Hales-Jewett type extension of this partition theorem."}
{"category": "Math", "title": "Duality for Borel measurable cost functions", "abstract": "We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if $c:X\\times Y\\to [0,\\infty)$ is an arbitrary Borel measurable cost function on the product of Polish spaces $X,Y$. In the course of the proof we show how to relate a non - optimal transport plan to the optimal transport costs via a ``subsidy'' function and how to identify the dual optimizer. We also provide some examples showing the limitations of the duality relations."}
{"category": "Math", "title": "Fixed point theory and trace for bicategories", "abstract": "The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is the identification of these different invariants. We define a generalization of the trace in symmetric monoidal categories to a trace in bicategories with shadows. We show the invariants used in the converse of the Lefschetz fixed point theorem are examples of this trace and that the functoriality of the trace provides some of the necessary identifications. The methods used here do not use simplicial techniques and so generalize readily to other contexts."}
{"category": "Math", "title": "Coupled Hamiltonian systems with extended affine Weyl group symmetry of type $D_3^{(2)}$", "abstract": "We find a two-parameter family of ordinary differential systems in dimension five with the affine Weyl group symmetry of type $D_3^{(2)}$. We show its symmetry and holomorphy conditions. This is the second example which gave higher order Painlev\\'e type systems of type $D_{3}^{(2)}$. By obtaining its first integrals of polynomial type, we can obtain a two-parameter family of coupled Hamiltonian systems in dimension four with the polynomial Hamiltonian."}
{"category": "Math", "title": "Blocks and families for cyclotomic Hecke algebras", "abstract": "The families of characters, defined by Lusztig for Weyl groups, play an important role in the representation theory of finite reductive groups. The definition of Rouquier for the families of characters in terms of blocks of the Hecke algebra has made possible the generalization of this notion to the case of complex reflection groups. In this article, we study the Rouquier blocks of the cyclotomic Hecke algebras and we show that they depend on numerical data determined by the generic Hecke algebra. Moreover, we provide the algorithm and the results of their determination for all exceptional complex reflection groups."}
{"category": "Math", "title": "Correction to ``Knotted Hamiltonian cycles in spatial embeddings of complete graphs\"", "abstract": "We state and prove a correct version of a theorem presented in an earlier paper."}
{"category": "Math", "title": "Brill-Noether theory of binary curves", "abstract": "Final version, to appear in Mathematical Research Letters."}
{"category": "Math", "title": "Approximation of the Semigroup generated by the Robin Laplacian in terms of the Gaussian Semigroup", "abstract": "For smooth bounded open sets in euclidean space, we construct corresponding contractive linear extension operators for the space of continuous functions which preserve regularity of functions in the domain of the Robin Laplacian. We also prove a Trotter-like approximation for the semigroup generated by the Laplacian subject to Robin boundary conditions in terms of these extension operators. The limiting case of Dirichlet boundary conditions is treated separately."}
{"category": "Math", "title": "On Weyl resolutions associated to Frobenius twists", "abstract": "We construct Weyl resolutions associated to certain Frobenius twists of divided powers. Using these and other related complexes we obtain the Weyl filtration dimension of the Schur algebras S(2,r), a result due to A. Parker."}
{"category": "Math", "title": "On the Kauffman bracket skein module of surgery on a (2,2b) torus link", "abstract": "We show that the Kauffman bracket skein modules of certain manifolds obtained from integral surgery on a (2,2b) torus link are finitely generated, and list the generators for select examples."}
{"category": "Math", "title": "Regular polynomial interpolation and approximation of global solutions of linear partial differential equations", "abstract": "We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the 'limit' of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics."}
{"category": "Math", "title": "The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions", "abstract": "This paper is concerned with the existence and the regularity of global solutions to the linear wave equation associated with two-point type boundary conditions. We also investigate the decay properties of the global solutions to this problem by the construction of a suitable Lyapunov functional."}
{"category": "Math", "title": "Ergodic BSDEs and related PDEs with Neumann boundary conditions", "abstract": "We study a new class of ergodic backward stochastic differential equations (EBSDEs for short) which is linked with semi-linear Neumann type boundary value problems related to ergodic phenomenas. The particularity of these problems is that the ergodic constant appears in Neumann boundary conditions. We study the existence and uniqueness of solutions to EBSDEs and the link with partial differential equations. Then we apply these results to optimal ergodic control problems."}
{"category": "Math", "title": "Depth-zero base change for ramified U(2,1)", "abstract": "We give an explicit description of L-packets and quadratic base change for depth-zero representations of ramified unitary groups in two and three variables. We show that this base change lifting is compatible with a certain lifting of families of representations of finite groups. We conjecture that such a compatibility is valid in much greater generality."}
{"category": "Math", "title": "On convergence of solutions to equilibria for quasilinear parabolic problems", "abstract": "We show convergence of solutions to equilibria for quasilinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which is normally hyperbolic. Our results do not depend on the presence of an appropriate Lyapunov functional as in the \\L ojasiewicz-Simon approach, but are of local nature."}
{"category": "Math", "title": "Bounding Betti numbers of bipartite graph ideals", "abstract": "We prove a conjectured lower bound of Nagel and Reiner on Betti numbers of edge ideals of bipartite graphs."}
{"category": "Math", "title": "Scaling of Saddle-Node Bifurcations: Degeneracies and Rapid Quantitative Changes", "abstract": "The scaling of the time delay near a \"bottleneck\" of a generic saddle-node bifurcation is well-known to be given by an inverse square-root law. We extend the analysis to several non-generic cases for smooth vector fields. We proceed to investigate $C^0$ vector fields. Our main result is a new phenomenon in two-parameter families having a saddle-node bifurcation upon changing the first parameter. We find distinct scalings for different values of the second parameter ranging from power laws with exponents in (0,1) to scalings given by O(1). We illustrate this rapid quantitative change of the scaling law by a an overdamped pendulum with varying length."}
{"category": "Math", "title": "Iterated Point-Line Configurations Grow Doubly-Exponentially", "abstract": "Begin with a set of four points in the real plane in general position. Add to this collection the intersection of all lines through pairs of these points. Iterate. Ismailescu and Radoi\\v{c}i\\'{c} (2003) showed that the limiting set is dense in the plane. We give doubly exponential upper and lower bounds on the number of points at each stage. The proof employs a variant of the Szemer\\'edi-Trotter Theorem and an analysis of the ``minimum degree'' of the growing configuration."}
{"category": "Math", "title": "Gradings on the Kac superalgebra", "abstract": "We classify the gradings on the Kac superalgebra up to equivalence."}
{"category": "Math", "title": "Some Ramsey results for the n-cube", "abstract": "In this note we establish a Ramsey-type result for certain subsets of the $n$-dimensional cube. This can then be applied to obtain reasonable bounds on various related structures, such as (partial) Hales-Jewett lines for alphabets of sized 3 and 4, Hilbert cubes in sets of real numbers with small sumsets, \"corner\" in the integer lattice in the plane, and 3-term geometric progressions in integers."}
{"category": "Math", "title": "Large-Sample Confidence Intervals for the Treatment Difference in a Two-Period Crossover Trial, Utilizing Prior Information", "abstract": "Consider a two-treatment, two-period crossover trial, with responses that are continuous random variables. We find a large-sample frequentist 1-alpha confidence interval for the treatment difference that utilizes the uncertain prior information that there is no differential carryover effect."}
{"category": "Math", "title": "A splitting theorem for equifocal submanifolds with non-flat section", "abstract": "In this paper, I prove a splitting theorem for equifocal submanifolds with non-flat section in a simply connected symmetric space of compact type. Also, by using the splitting theorem, I prove that the sections of equifocal submanifolds with non-flat section in an irreducible simply connected symmetric space of compact type are isometric to a sphere or a real projective space."}
{"category": "Math", "title": "On the Completeness of Gradient Ricci Solitons", "abstract": "A gradient Ricci soliton is a triple $(M,g,f)$ satisfying $R_{ij}+\\nabla_i\\nabla_j f=\\lambda g_{ij}$ for some real number $\\lambda$. In this paper, we will show that the completeness of the metric $g$ implies that of the vector field $\\nabla f$."}
{"category": "Math", "title": "Gradient Shrinking Solitons with Vanishing Weyl Tensor", "abstract": "In this paper, we will give a local version of the Hamilton-Ivey type pinching estimate of the gradient shrinking soliton with vanishing Weyl tensor, and then give a complete classification on gradient shrinking solitons with vanishing Weyl tensor."}
{"category": "Math", "title": "Lie properties of crossed products", "abstract": "Let $F^\\lambda_{\\sigma} [G]$ be a crossed product of a group $G$ and the field $F$. We study the Lie properties of $F^\\lambda_{\\sigma} [G]$ in order to obtain a characterization of those crossed products which are upper (lower) Lie nilpotent and Lie $(n,m)$-Engel."}
{"category": "Math", "title": "Uniqueness of signed measures solving the continuity equation for Osgood vector fields", "abstract": "Nonnegative measure-valued solutions of the continuity equation are uniquely determined by their initial condition, if the characteristic ODE associated to the velocity field has a unique solution. In this paper we give a partial extension of this result to signed measure-valued solutions, under a quantitative two-sided Osgood condition on the velocity field. Our results extend those obtained for log-Lipschitz vector fields by Bahouri and Chemin."}
{"category": "Math", "title": "On the Conley decomposition of Mather sets", "abstract": "In the context of Mather's theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian."}
{"category": "Math", "title": "Evolution Families and the Loewner Equation I: The Unit Disc", "abstract": "In this paper we introduce a general version of the Loewner differential equation which allows us to present a new and unified treatment of both the radial equation introduced in 1923 by K. Loewner and the chordal equation introduced in 2000 by O. Schramm. In particular, we prove that evolution families in the unit disc are in one to one correspondence with solutions to this new type of Loewner equations. Also, we give a Berkson-Porta type formula for non-autonomous weak holomorphic vector fields which generate such Loewner differential equations and study in detail geometric and dynamical properties of evolution families."}
{"category": "Math", "title": "The homogeneous slice theorem for the complete complexification of a proper complex equifocal submanifold", "abstract": "We investigate the homogeneity of certain kind of slices of the complete complexification of a proper complex equifocal submanifold in a symmetric space of non-compact type."}
{"category": "Math", "title": "Hamiltonian Systems: Stability and Instability Theory", "abstract": "This is a short survey on Nekhoroshev theory, KAM theory, and Arnold's diffusion."}
{"category": "Math", "title": "A generic property of families of Lagrangian systems", "abstract": "We prove that a generic lagrangian has finitely many minimizing measures for every cohomology class."}
{"category": "Math", "title": "Perturbation d'un hamiltonien partiellement hyperbolique", "abstract": "Arnold's diffusion in quasi integrable hamiltonian systems occurs in exponentially large time. We study an initially hyperbolic system which admits diffusion in polynomial time."}
{"category": "Math", "title": "The complexifications of pseudo-Riemannian manifolds and anti-Kaehler geometry", "abstract": "In this paper, we first define the complexification of a real analytic map between real analytic Koszul manifolds and show that the complexified map is the holomorphic extension of the original map. Next we define an anti-Kaehler metric compatible with the adapted complex structure on the complexification of a real analytic pseudo-Riemannian manifold. In particular, for a pseudo-Riemannian homogeneous space, we define another complexification and a (complete) anti-Kaehler metric on the complexification. One of main purposes of this paper is to find the interesting relation between these two complexifications (equipped with the anti-Kaehler metrics) of a pseudo-Riemannian homogeneous space. Another of main purposes of this paper is to show that almost all principal orbits of some isometric action on the first complexification (equipped with the anti-Kaehler metric) of a semi-simple pseudo-Riemannian symmetric space are curvature-adapted isoparametric submanifolds with flat section in the sense of this paper."}
{"category": "Math", "title": "Hermann type actions on a pseudo-Riemannian symmetric space", "abstract": "We first investigate the geometry of orbits of the isotropy action on a semi-simple pseudo-Riemannian symmetric space by investigating the complexified action. Next we investigate the geometry of the orbits of Hermann type actions on the symmetric spaces. By considering two special Hermann type actions on the symmetric space, we recognize an interesting structure of the symmetric space."}
{"category": "Math", "title": "Homogeneity of proper complex equifocal submanifolds", "abstract": "In this paper, we show that an irreducible proper complex equifocal submanifold of codimension greater than one in a symmetric space of non-compact type. The proof is performed by showing the homogeneity of the lift of the complexification of the original submanifold to some infinite dimensional anti-Kahelerian space."}
{"category": "Math", "title": "On equifocal submanifolds with non-flat section in symmetric spaces of rank two", "abstract": "In this paper, we show that there exists no equifocal submanifold with non-flat section in four irreducible simply connected symmetric spaces of compact type and rank two. Also, we show a fact for the sections of equifocal submanifolds with non-flat section in other irreducible simply connected symmetric spaces of compact type and rank two."}
{"category": "Math", "title": "Examples of a complex hyperpolar actions without singular orbit", "abstract": "The notion of a complex hyperpolar action on a symmetric space of non-compact type has recently been introduced as counterpart of a hyperpolar action on a symmetric space of compact type. In this paper, we construct examples of a complex hyperpolar action without singular orbit and investigate the geometry of the orbits of the examples."}
{"category": "Math", "title": "Ergodic actions of mapping class groups on moduli spaces of representations of non-orientable surfaces", "abstract": "The purpose of this paper is to study the action of the mapping class group on the moduli space of representations of the fundamental group of a non-orientable surface into SU(2). The action is shown to be ergodic with respect to a natural measure. This measure is defined using the push-forward measure associated to a map defined by the presentation of the surface group. given by the pushforward of Haar measure. This result is an extension of earlier results of Goldman for orientable surfaces."}
{"category": "Math", "title": "Finding Short Cycles in an Embedded Graph in Polynomial Time", "abstract": "Let ${\\cal{C}}_1$ be the set of fundamental cycles of breadth-first-search trees in a graph $G$ and ${\\cal{C}}_2$ the set of the sums of two cycles in ${\\cal{C}}_1$. Then we show that $(1) {\\cal{C}}={\\cal{C}}_1\\bigcup{\\cal{C}}_2$ contains a shortest $\\Pi$-twosided cycle in a $\\Pi$-embedded graph $G$;$(2)$ $\\cal{C}$ contains all the possible shortest even cycles in a graph $G$;$(3)$ If a shortest cycle in a graph $G$ is an odd cycle, then $\\cal{C}$ contains all the shortest odd cycles in $G$. This implies the existence of a polynomially bounded algorithm to find a shortest $\\Pi-$twosided cycle in an embedded graph and thus solves an open problem of B.Mohar and C.Thomassen[2,pp112]"}
{"category": "Math", "title": "Piatetski-Shapiro's phenomenon and related problems", "abstract": "This Ph.D. thesis, prepared under the supervision of Prof. Alexander Olevskii, is concerned with some problems in two areas of Fourier Analysis: uniqueness theory of trigonometric expansions, and the theory of translation invariant subspaces in function spaces. Our main result in the first area extends to $\\ell_q$ spaces ($q > 2$) a deep phenomenon found by Piatetski-Shapiro in 1954 for the space $c_0$. The approach we developed also enabled us to get a result in the second mentioned area, which a priori does not look connected with the first one. The result (maybe, a bit surprising) is: one cannot characterize the functions in $\\ell_p(\\Z)$ or $L^p(\\R)$, $1 < p < 2$, whose translates span the whole space, by the zero set of their Fourier transform. This should be contrasted against the classical Wiener theorems related to the cases $p=1,2$."}
{"category": "Math", "title": "The centipede is determined by its Laplacian spectrum", "abstract": "A centipede is a graph obtained by appending a pendant vertex to each vertex of degree 2 of a path. In this paper we prove that the centipede is determined by its Laplacian spectrum."}
{"category": "Math", "title": "Continuous dependence results for Non-linear Neumann type boundary value problems", "abstract": "We obtain estimates on the continuous dependence on the coefficient for second order non-linear degenerate Neumann type boundary value problems. Our results extend previous work of Cockburn et.al., Jakobsen-Karlsen, and Gripenberg to problems with more general boundary conditions and domains. A new feature here is that we account for the dependence on the boundary conditions. As one application of our continuous dependence results, we derive for the first time the rate of convergence for the vanishing viscosity method for such problems. We also derive new explicit continuous dependence on the coefficients results for problems involving Bellman-Isaacs equations and certain quasilinear equation."}
{"category": "Math", "title": "Sur la Caracterisation des Retractes Holomorphes a l'aide de la Metrique Infinitesimale de Kobayashi", "abstract": "Dans cet article, on donne une caracterisation des retractes holomorphes a l'aide de la metrique infinitesimale de Kobayashi ----- This article gives a characterization of holomorphic retractions using Kobayashi's infinitesimal metric."}
{"category": "Math", "title": "Schwarzenberger bundles of arbitary rank on the projective space", "abstract": "We introduce a generalized notion of Schwarzenberger bundle on the projective space. Associated to this more general definition, we give an ad-hoc notion of jumping subspaces of a Steiner bundle on ${\\Bbb P^n}$ (which in rank $n$ coincides with the notion of unstable hyperplane introduced by Vall\\`es, Ancona and Ottaviani). For the set of jumping hyperplanes, we find a sharp bound for its dimension. We also classify those Steiner bundles whose set of jumping hyperplanes have maximal dimension and prove that they are generalized Schwarzenberger bundles."}
{"category": "Math", "title": "The lazy homology of a Hopf algebra", "abstract": "To any Hopf algebra H we associate two commutative Hopf algebras, which we call the first and second lazy homology Hopf algebras of H. These algebras are related to the lazy cohomology groups based on the so-called lazy cocycles of H by universal coefficient theorems. When H is a group algebra, then its lazy homology can be expressed in terms of the 1- and 2-homology of the group. When H is a cosemisimple Hopf algebra over an algebraically closed field of characteristic zero, then its first lazy homology is the Hopf algebra of the universal abelian grading group of the category of corepresentations of H. We also compute the lazy homology of the Sweedler algebra."}
{"category": "Math", "title": "Fundamental Cycles and Graph Embeddings", "abstract": "In this paper we present a new Good Characterization of maximum genus of a graph which makes a common generalization of the works of Xuong, Liu, and Fu et al. Based on this, we find a new polynomially bounded algorithm to find the maximum genus of a graph."}
{"category": "Math", "title": "Centers of F-purity", "abstract": "In this paper, we study a positive characteristic analogue of the centers of log canonicity of a pair $(R, \\Delta)$. We call these analogues centers of $F$-purity. We prove positive characteristic analogues of subadjunction-like results, prove new stronger subadjunction-like results, and in some cases, lift these new results to characteristic zero. Using a generalization of centers of $F$-purity which we call uniformly $F$-compatible ideals, we give a characterization of the test ideal (which unifies several previous characterizations). Finally, in the case that $\\Delta = 0$, we show that uniformly $F$-compatible ideals coincide with the annihilators of the $\\mathcal{F}(E_R(k))$-submodules of $E_R(k)$ as defined by Smith and Lyubeznik."}
{"category": "Math", "title": "Verifying the Congruence Conjecture for Rubin-Stark Elements", "abstract": "The `Congruence Conjecture' was developed by the second author in a previous paper. It provides a conjectural explicit reciprocity law for a certain element associated to an abelian extension of a totally real number field whose existence is predicted by earlier conjectures of Rubin and Stark. The first aim of the present paper is to design and apply techniques to investigate the Congruence Conjecture numerically. We then present complete verifications of the conjecture in 48 varied cases with real quadratic base fields."}
{"category": "Math", "title": "Vector valued reproducing kernel Hilbert spaces and universality", "abstract": "This paper is devoted to the study of vector valued reproducing kernel Hilbert spaces. We focus on two aspects: vector valued feature maps and universal kernels. In particular we characterize the structure of translation invariant kernels on abelian groups and we relate it to the universality problem."}
{"category": "Math", "title": "A note on two-positive Ricci curvature and positive Einstein curvature", "abstract": "In this short note we discuss some recent results about two-positive Ricci curvature and their applications to positive Einstein curvature."}
{"category": "Math", "title": "Quasi-K\\\"ahler manifolds with trivial Chern Holonomy", "abstract": "In this paper we study almost complex manifolds admitting a quasi-K\\\"ahler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-K\\\"ahler Chern-flat structure can be tamed by a symplectic form if and only if the ambient space is isomorphic to a flat torus."}
{"category": "Math", "title": "Classes of Monomial Ideals", "abstract": "In this thesis, we focus on the study of some classes of monomial ideals, namely lexsegment ideals and monomial ideals with linear quotients."}
{"category": "Math", "title": "Percolation of arbitrary words in one dimension", "abstract": "We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in others."}
{"category": "Math", "title": "The Hexatangle", "abstract": "We are interested in knowing what type of manifolds are obtained by doing Dehn surgery on closed pure 3-braids in the 3-sphere. In particular, we want to determine when we get the 3-sphere by surgery on such a link. We consider links which are small closed pure 3-braids; these are the closure of 3-braids of the form $({\\sigma_1}^{2e_1})({\\sigma_2}^{2f_1})(\\sigma_2\\sigma_1\\sigma_2)^{2e}$, where $\\sigma_1$, $\\sigma_2$ are the generators of the 3-braid group and $e_1$, $f_1$, $e$ are integers. We study Dehn surgeries on these links, and determine exactly which ones admit an integral surgery producing the 3-sphere. This is equivalent to determining the surgeries of some type on a certain six component link $L$ that produce $S^3$. The link $L$ is strongly invertible and its exterior double branch covers a certain configuration of arcs and spheres, which we call the Hexatangle. Our problem is equivalent to determine which fillings of the spheres by integral tangles produce the trivial knot, which is what we explicitly solve. This hexatangle is a generalization of the Pentangle, which is studied by Gordon and Luecke."}
{"category": "Math", "title": "A modified logarithmic Sobolev inequality for the Hamming cube and some applications", "abstract": "The logarithmic Sobolev inequality for the Hamming cube {0,1}^n states that for any real-valued function f on the cube holds E(f,f) \\ge 2 Ent(f^2), where E(f,f) is the appropriate Dirichlet form (also known as \"sum of influences\"). We show that the constant C = 2 at the right hand side of this inequality can be replaced by a function C(rho) depending on rho = Ent(f^2) / (n Ef^2). The function C is an increasing convex function taking [0,log 2] to [2, 2/log 2]. We present some applications of this modified inequality. In particular, it is used to obtain a discrete version of the Faber-Krahn inequality for small subsets of the Hamming cube, answering a question of Friedman and Tillich. We introduce, following the approach of Friedman and Tillich, the notion of a fractional edge-boundary size of a subset of {0,1}^n, and show Hamming balls of radius at most n/2 - O(n^{3/4}) to be sets with (asymptotically) the smallest fractional edge-boundary for their size."}
{"category": "Math", "title": "The Eyring-Kramers law for potentials with nonquadratic saddles", "abstract": "The Eyring-Kramers law describes the mean transition time of an overdamped Brownian particle between local minima in a potential landscape. In the weak-noise limit, the transition time is to leading order exponential in the potential difference to overcome. This exponential is corrected by a prefactor which depends on the principal curvatures of the potential at the starting minimum and at the highest saddle crossed by an optimal transition path. The Eyring-Kramers law, however, does not hold whenever one of these principal curvatures vanishes, since it would predict a vanishing or infinite transition time. We derive the correct prefactor up to multiplicative errors that tend to one in the zero-noise limit. As an illustration, we discuss the case of a symmetric pitchfork bifurcation, in which the prefactor can be expressed in terms of modified Bessel functions, as well as bifurcations with two vanishing eigenvalues. The corresponding transition times are studied in a full neighbourhood of the bifurcation point. These results extend work by Bovier, Eckhoff, Gayrard and Klein, who rigorously analysed the case of quadratic saddles, using methods from potential theory."}
{"category": "Math", "title": "Young measures, Cartesian maps, and polyconvexity", "abstract": "We consider the variational problem consisting of minimizing a polyconvex integrand for maps between manifolds. We offer a simple and direct proof of the existence of a minimizing map. The proof is based on Young measures."}
{"category": "Math", "title": "Convergence of the law of the Environment Seen by the Particle for Directed Polymers in Random Media in the $L^2$ region", "abstract": "We consider the model of Directed Polymers in an i.i.d. gaussian or bounded Environment in the $L^2$ region. We prove the convergence of the law of the environment seen by the particle. As a main technical step, we establish a lower tail concentration inequality for the partition function for bounded environments. Our proof is based on arguments developed by Talagrand in the context of the Hopfield Model. This improves in some sense a concentration inequality obtained by Carmona and Hu for gaussian environments. We use this and a Local Limit Theorem to prove the $L^1$ convergence of the density of the law of the environment seen by the particle with respect to the product measure."}
{"category": "Math", "title": "Floer homology in disc bundles and symplectically twisted geodesic flows", "abstract": "We show that if K: P \\to R is an autonomous Hamiltonian on a symplectic manifold (P,\\Omega) which attains 0 as a Morse-Bott nondegenerate minimum along a symplectic submanifold M, and if c_1(TP)|_M vanishes in real cohomology, then the Hamiltonian flow of K has contractible periodic orbits with bounded period on all sufficiently small energy levels. As a special case, if the geodesic flow on the cotangent bundle of M is twisted by a symplectic magnetic field form, then the resulting flow has contractible periodic orbits on all low energy levels. These results were proven by Ginzburg and G\\\"urel when \\Omega|_M is spherically rational, and our proof builds on their work; the argument involves constructing and carefully analyzing at the chain level a version of filtered Floer homology in the symplectic normal disc bundle to M."}
{"category": "Math", "title": "Investigation of solutions of boundary value problems for a composite type equation with non-local boundary conditions", "abstract": "Since the order of elliptic type model equation (Laplace equation) is two [1], [2], then it is natural the order of composite type model equation must be [3] [4] [5] three. At each point of the domain under consideration these equations have both real and complex characteristics. Notice that a boundary value problem for a composite type equation of second order first appeared in the paper [6]. The method for investigating the Fredholm property of boundary value problems is distinctive and belongs to one of the authors of the present paper."}
{"category": "Math", "title": "Convenient Categories of Smooth Spaces", "abstract": "A \"Chen space\" is a set X equipped with a collection of \"plots\" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's \"diffeological spaces\" share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of \"concrete sheaves on a concrete site\". As a result, the categories of such spaces are locally cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use."}
{"category": "Math", "title": "Evolution Families and the Loewner Equation II: complex hyperbolic manifolds", "abstract": "We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds."}
{"category": "Math", "title": "The GIT-stability of Polarised Varieties via discrepancy", "abstract": "We prove that various GIT semistabilities of polarized varieties imply semi-log-canonicity."}
{"category": "Math", "title": "Sur la classification de quelques phi-modules simples", "abstract": "This note is an appendix to a preprint by E. Hellmann. We give a complete classification of simple objects of the category of vector spaces D over K = Fpbar((u)) equipped with an endomorphism phi whose image generates D and that are semi-linear with respect to the ring morphism sending u to u^b (b > 1 is an integer) and acting on elements of k through a fixed automorphism. Some of these phi-modules are involved in the classification of finite flat group schemes over ring of integers of p-adic fields."}
{"category": "Math", "title": "The Mobius function is strongly orthogonal to nilsequences", "abstract": "We show that the Mobius function mu(n) is strongly asymptotically orthogonal to any polynomial nilsequence n -> F(g(n)L). Here, G is a simply-connected nilpotent Lie group with a discrete and cocompact subgroup L (so G/L is a nilmanifold), g : Z -> G is a polynomial sequence and F: G/L -> R is a Lipschitz function. More precisely, we show that the inner product of mu(n) with F(g(n)L) over {1,...,N} is bounded by 1/log^A N, for all A > 0. In particular, this implies the Mobius and Nilsequence conjecture MN(s) from our earlier paper \"Linear equations in primes\" for every positive integer s. This is one of two major ingredients in our programme, outlined in that paper, to establish a large number of cases of the generalised Hardy-Littlewood conjecture, which predicts how often a collection \\psi_1,...,\\psi_t : Z^d -> Z of linear forms all take prime values. The proof is a relatively quick application of the results in our recent companion paper on the distribution of polynomial orbits on nilmanifolds. We give some applications of our main theorem. We show, for example, that the Mobius function is uncorrelated with any bracket polynomial. We also obtain a result about the distribution of nilsequences n -> a^nxL as n ranges only over the primes."}
{"category": "Math", "title": "Calabi-Yau components in general type hypersurfaces", "abstract": "For a one-parameter family of general type hypersurfaces with bases of holomorphic n-forms, we construct open covers using tropical geometry. We show that after normalization, each holomorphic n-form is approximately supported on a unique open component and such a pair approximates a Calabi-Yau hypersurface together with its holomorphic n-form as the parameter becomes large. We also show that the Lagrangian fibers in the fibration constructed by Mikhalkin are asymptotically special Lagrangian. As the holomorphic n-form plays an important role in mirror symmetry for Calabi-Yau manifolds, our results is a step toward understanding mirror symmetry for general type manifolds."}
{"category": "Math", "title": "Selfimprovemvent of the inequality between arithmetic and geometric means", "abstract": "We present a refinement, by selfimprovement, of the arithmetic geometric inequality."}
{"category": "Math", "title": "On Entanglement and Separability", "abstract": "We present a necessary and sufficient condition to determine the entanglement status of an arbitrary N-qubit quantum state (may be pure or mixed) represented by the density matrix, (Rho)N. We develop a new approach and a new criterion for the problem of deciding entanglement status. Further, we develop as an important application of entanglement a new quantum protocol for superluminal communication of classical information in terms of a desired ordered sequence of classical bits. We then show that this new quantum protocol for superluminal communication of classical information can be used in the quantum teleportation protocol [12] for achieving superluminal quantum teleportation."}
{"category": "Math", "title": "A class of Sasakian 5-manifolds", "abstract": "We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group $H_{2n + 1}$. Furthermore, we classify Sasakian Lie algebras of dimension 5 and determine which of them carry a Sasakian $\\alpha$-Einstein structure. We show that a 5-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either $H_5$ or a semidirect product $\\R \\ltimes (H_3 \\times \\R)$. In particular, the compact quotient is an $S^1$-bundle over a 4-dimensional K\\\"ahler solvmanifold."}
{"category": "Math", "title": "Hook lengths and shifted parts of partitions", "abstract": "Some conjectures on partition hook lengths, recently stated by the author, have been proved and generalized by Stanley, who also needed a formula by Andrews, Goulden and Jackson on symmetric functions to complete his derivation. Another identity on symmetric functions can be used instead. The purpose of this note is to prove it."}
{"category": "Math", "title": "Determine the spacial term of a two-dimensional heat source", "abstract": "We consider the problem of determining a pair of functions $(u,f)$ satisfying the heat equation $u_t -\\Delta u =\\varphi(t)f (x,y)$, where $(x,y)\\in \\Omega=(0,1)\\times (0,1)$ and the function $\\varphi$ is given. The problem is ill-posed. Under a slight condition on $\\varphi$, we show that the solution is determined uniquely from some boundary data and the initial temperature. Using the interpolation method and the truncated Fourier series, we construct a regularized solution of the source term $f$ from non-smooth data. The error estimate and numerical experiments are given."}
{"category": "Math", "title": "Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2", "abstract": "Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in S^2xS^2 and H^2xH^2 are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of S^2xR and H^2xR is established. Using that, surfaces with vanishing Hopf differentials (in particular spheres with parallel mean curvature vector) are classified and a rigidity result for constant mean curvature surfaces of S^2xR and H^2xR is proved."}
{"category": "Math", "title": "Stochastic 2D hydrodynamical type systems: Well posedness and large deviations", "abstract": "We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and 2D magnetic B\\'enard problem and also some shell models of turbulence. We first prove the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by weak convergence method."}
{"category": "Math", "title": "Relative Chern characters for nilpotent ideals", "abstract": "If I is a nilpotent ideal in a $\\mathbb{Q}$-algebra $A$, Goodwillie defined two isomorphisms from $K_*(A,I)$ to negative cyclic homology, $HN_*(A,I)$. One is the relative version of the absolute Chern character, and the other is defined using rational homotopy theory. The question of whether they agree was implicit in Goodwillie's 1986 Annals paper. In this paper, we show that the two isomorphisms agree. Here are three applications. 1.Cathelineau proved that the rational homotopy character is compatible with the $\\lambda$-filtration. It follows that the relative Chern character is also compatible with this filtration for nilpotent ideals. 2.This agreement, together with Cathelineau's result, was used by the authors and Haesemeyer to show that the absolute Chern character, from $K(A)$ to $HN(A)$, is compatible with the $\\lambda$-filtration for every commutative $\\mathbb{Q}$-algebra. This is the main result of Infinitesimal cohomology and the Chern character to negative cyclic homology, arXiv:math/0703133v1. 3.This agreement can be used to strengthen Ginot's results in \"Formules explicites pour le charactere de Chern en $K$-th\\'eorie alg\\'ebrique\", Ann. Inst. Fourier 54 (2004)."}
{"category": "Math", "title": "Determine the source term of a two-dimensional heat equation", "abstract": "Let $\\Omega$ be a two-dimensional heat conduction body. We consider the problem of determining the heat source $F(x,t)=\\varphi(t)f(x,y)$ with $\\varphi$ be given inexactly and $f$ be unknown. The problem is nonlinear and ill-posed. By a specific form of Fourier transforms, we shall show that the heat source is determined uniquely by the minimum boundary condition and the temperature distribution in $\\Omega$ at the initial time $t=0$ and at the final time $t=1$. Using the methods of Tikhonov's regularization and truncated integration, we construct the regularized solutions. Numerical part is given."}
{"category": "Math", "title": "On paraquaternionic submersions between paraquaternionic K\\\"ahler manifolds", "abstract": "In this paper we deal with some properties of a class of semi-Riemannian submersions between manifolds endowed with paraquaternionic structures, proving a result of non-existence of paraquaternionic submersions between paraquaternionic K\\\"ahler non locally hyper paraK\\\"ahler manifolds. Then we examine, as an example, the canonical projection of the tangent bundle, endowed with the Sasaki metric, of an almost paraquaternionic Hermitian manifold."}
{"category": "Math", "title": "The best constant in a fractional Hardy inequality", "abstract": "We prove an optimal Hardy inequality for the fractional Laplacian on the half-space."}
{"category": "Math", "title": "Factorization structures with a 2-dimensional factor", "abstract": "We introduce the notion of quantum duplicates of an (associative, unital) algebra, motivated by the problem of constructing toy-models for quantizations of certain configuration spaces in quantum mechanics. The proposed (algebraic) model relies on the classification of factorization structures with a two-dimensional factor. In the present paper, main properties of this particular kind of structures are determined, and we present a complete description of quantum duplicates of finite set algebras. As an application, we obtain a classification (up to isomorphism) of all the algebras of dimension 4 (over an arbitrary field) that can be factorized as a product of two factors."}
{"category": "Math", "title": "Hamiltonian degree sequences in digraphs", "abstract": "We show that for each \\eta>0 every digraph G of sufficiently large order n is Hamiltonian if its out- and indegree sequences d^+_1\\le ... \\le d^+_n and d^- _1 \\le ... \\le d^-_n satisfy (i) d^+_i \\geq i+ \\eta n or d^-_{n-i- \\eta n} \\geq n-i and (ii) d^-_i \\geq i+ \\eta n or d^+_{n-i- \\eta n} \\geq n-i for all i < n/2. This gives an approximate solution to a problem of Nash-Williams concerning a digraph analogue of Chv\\'atal's theorem. In fact, we prove the stronger result that such digraphs G are pancyclic."}
{"category": "Math", "title": "Alg\\`ebres et cog\\`ebres de Gerstenhaber et cohomologie de Chevalley-Harrison", "abstract": "The fundamental example of Gerstenhaber algebra is the space $T_{poly}({\\mathbb R}^d)$ of polyvector fields on $\\mathbb{R}^d$, equipped with the wedge product and the Schouten bracket. In this paper, we explicitely describe what is the enveloping $G_\\infty$ algebra of a Gerstenhaber algebra $\\mathcal{G}$. This structure gives us a definition of the Chevalley-Harrison cohomology operator for $\\mathcal{G}$. We finally show the nontriviality of a Chevalley-Harrison cohomology group for a natural Gerstenhaber subalgebra in $T_{poly}({\\mathbb R}^d)$."}
{"category": "Math", "title": "A rooted-trees q-series lifting a one-parameter family of Lie idempotents", "abstract": "We define and study a series indexed by rooted trees and with coefficients in Q(q). We show that it is related to a family of Lie idempotents. We prove that this series is a q-deformation of a more classical series and that some of its coefficients are Carlitz q-Bernoulli numbers."}
{"category": "Math", "title": "Content Algebras Over Commutative Rings With Zero-Divisors", "abstract": "Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = \\cap \\lbrace I \\colon I \\text{is an ideal of} R \\text{and} x \\in IM \\rbrace $. $M$ is said to be a content $R$-module if $x \\in c(x)M $, for all $x \\in M$. $B$ is called a content $R$-algebra, if it is a faithfully flat and content $R$-module and it satisfies the Dedekind-Mertens content formula. In this article, we prove some new results for content modules and algebras by using ideal theoretic methods."}
{"category": "Math", "title": "Biharmonic maps between doubly warped product manifolds", "abstract": "In this paper biharmonic maps between doubly warped product manifolds are studied. We show that the inclusion maps of Riemannian manifolds $B$ and $F$ into the doubly warped product $_{f}B\\times_{b}F$ can not be proper biharmonic maps. Also we analyze the conditions for the biharmonicity of projections $_{f}B\\times_{b}F\\to B$ and $_{f}B\\times_{b}F\\to F$, respectively. Some characterizations for non-harmonic biharmonic maps are given by using product of harmonic maps and warping metric. Specially, in the case of $f=1$, the results for warped product in \\cite{Balmus-mont} are obtained."}
{"category": "Math", "title": "An algebraic formula for the intersection number of a polynomial immersion", "abstract": "There is presented an algorithm for computing the topological degree for a large class of polynomial mappings. As an application there is given an effective algebraic formula for the intersection number of a polynomial immersion M --> R^2m, where M is an m-dimensional algebraic manifold."}
{"category": "Math", "title": "Lp-Solutions for Reected Backward Stochastic Differential Equations", "abstract": "This paper deals with the problem of existence and uniqueness of a solution for a backward stochastic differential equation (BSDE for short) with one reflecting barrier in the case when the terminal value, the generator and the obstacle process are Lp-integrable with p in ]1,2[. To construct the solution we use two methods: penalization and Snell envelope. As an application we broaden the class of functions for which the related obstacle partial differential equation problem has a unique viscosity solution."}
{"category": "Math", "title": "Linear family of Lie brackets on the space of matrices $Mat(n\\times m,\\K)$ and Ado's Theorem", "abstract": "In this paper we classify a linear family of Lie brackets on the space of rectangular matrices $Mat(n\\times m,\\K)$ and we give an analogue of the Ado's Theorem. We give also a similar classification on the algebra of the square matrices $Mat(n, \\K)$ and as a consequence, we prove that we can't built a faithful representation of the $(2n+1)$-dimensional Heisenberg Lie algebra $\\mathfrak{H}_n$ in a vector space $V$ with $\\dim V\\leq n+1$. Finally, we prove that in the case of the algebra of square matrices $Mat(n,\\K)$, the corresponding Lie algebras structures are a contraction of the canonical Lie algebra $\\mathfrak{gl}(n,\\K)$."}
{"category": "Math", "title": "Les (a,b)-alg\\`ebres \\`a homotopie pr\\`es", "abstract": "We study in this article the concepts of algebra up to homotopy for a structure defined by two operations $ \\pt $ and $[, ]$. Having determined the structure of $ G_\\infty $ algebras and $ P_\\infty $ algebras, we generalize this construction and we define a structure of $ (a, b)$-algebra up to homotopy. Given a structure of commutative and differential graded Lie algebra for two shifts degree given by $a$ and $b$, we will give an explicit construction of the associate algebra up to homotopy"}
{"category": "Math", "title": "A symplectic Gysin sequence", "abstract": "We use the theory of pseudo-holomorphic quilts to establish a counterpart, in symplectic Floer homology, to the Gysin sequence for the homology of a sphere-bundle. In a motivating class of examples, this \"symplectic Gysin sequence\" is precisely analogous to an exact sequence describing the behaviour of Seiberg-Witten monopole Floer homology for 3-manifolds under connected sum."}
{"category": "Math", "title": "Brown representability does not come for free", "abstract": "We exhibit a triangulated category T having both products and coproducts, and a triangulated subcategory S of T which is both localizing and colocalizing, for which neither a Bousfield localization nor a colocalization exists. It follows that neither the category S nor its dual satisfy Brown representability. Our example involves an abelian category whose derived category does not have small Hom-sets."}
{"category": "Math", "title": "Teichmuller geometry of moduli space, II: M(S) seen from far away", "abstract": "We construct a metric simplicial complex which is an almost isometric model of the moduli space M(S) of Riemann surfaces. We then use this model to compute the \"tangent cone at infinity\" of M(S): it is the topological cone on the quotient of the complex of curves C(S) by the mapping class group of S, endowed with an explicitly described metric. The main ingredient is Minsky's product regions theorem."}
{"category": "Math", "title": "On quasiinvariants of $S_n$ of hook shape", "abstract": "Chalykh, Veselov and Feigin introduced the notions of quasiinvariants for Coxeter groups, which is a generalization of invariants. In [2], Bandlow and Musiker showed that for the symmetric group $S_n$ of order $n$, the space of quasiinvariants has a decomposition indexed by standard tableaux. They gave a description of basis for the components indexed by standard tableaux of shape $(n-1,1)$. In this paper, we generalize their results to a description of basis for the components indexed by standard tableaux of arbitrary hook shape."}
{"category": "Math", "title": "Classification of Almost Quarter-Pinched Manifolds", "abstract": "We show that if a simply connected manifold is almost quarter pinched then it is diffeomorphic to a CROSS (a compact rank one symmetric space) or a sphere."}
{"category": "Math", "title": "Initial logarithmic Lie algebras of hypersurface singularities", "abstract": "We introduce a Lie algebra of initial terms of logarithmic vector fields along a hypersurface singularity. Extending the formal structure theorem in [GS06, Thm. 5.4], we show that the completely reducible part of its linear projection lifts formally to a linear Lie algebra of logarithmic vector fields. For quasihomogeneous singularities, we prove convergence of this linearization. We relate our construction to the work of Hauser and M\"uller [M\"ul86, HM89] on Levi subgroups of automorphism groups of singularities, which proves convergence even for algebraic singularities. Based on the initial Lie algebra, we introduce a notion of reductive hypersurface singularity and show that any reductive free divisor is linear. As an application, we describe a lower bound for the dimension of hypersurface singularities in terms of the semisimple part of their initial Lie algebra."}
{"category": "Math", "title": "On the Gaussian q-Distribution", "abstract": "We present a study of the Gaussian q-measure introduced by Diaz and Teruel from a probabilistic and from a combinatorial viewpoint. A main motivation for the introduction of the Gaussian q-measure is that its moments are exactly the q-analogues of the double factorial numbers. We show that the Gaussian q-measure interpolates between the uniform measure on the interval [-1,1] and the Gaussian measure on the real line."}
{"category": "Math", "title": "A PL-manifold of nonnegative curvature homeomorphic to $S^2 \\times S^2$ is a direct metric product (Preliminary version)", "abstract": "Let $M^4$ be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, $S^2 \\times S^2$. We prove that $M$ is a direct metric product of two spheres endowed with some polyhedral metrics. In other words, $M$ is a direct metric product of the surfaces of two convex polyhedra in $\\mathbb{R}^3$. The background for the question is the following. The classical H.Hopf's hypothesis states: for any Riemannian metric on $S^2 \\times S^2$ of nonnegative sectional curvature the curvature cannot be strictly positive at all points. There is no quick answer to this question: it is known that a Riemannian metric on $S^2 \\times S^2$ of nonnegative sectional curvature need not be a product metric. However, M.Gromov has pointed out that the condition of nonnegative curvature in the PL-case appears to be stronger than nonnegative sectional curvature of Riemannian manifolds and analogous to some condition on the curvature operator. So the motivation for the question addressed in this text is to settle the PL-version of the Hopf's hypothesis."}
{"category": "Math", "title": "Energy image density property and the lent particle method for Poisson measures", "abstract": "We introduce a new approach to absolute continuity of laws of Poisson functionals. It is based on the {\\it energy image density} property for Dirichlet forms and on what we call {\\it the lent particle method} which consists in adding a particle and taking it back after some calculation."}
{"category": "Math", "title": "On the Real Multidimensional Rational K-Moment Problem", "abstract": "We present a solution to the real multidimensional rational K-moment problem, where K is defined by finitely many polynomial inequalities. More precisely, let S be a finite set of real polynomials in X=(X_1,...,X_n) such that the corresponding basic closed semialgebraic set K_S is nonempty. Let E=D^{-1}R[X] be a localization of the real polynomial algebra, and T_S^E the preordering on E generated by S. We show that every linear functional L on E that is nonnegative on T_S^E is represented by a positive measure on a certain subset of K_S, provided D contains an element that grows fast enough on K_S."}
{"category": "Math", "title": "Diophantine sets of polynomials over number fields", "abstract": "Let R be a recursive subring of a number field. We show that recursively enumerable sets are diophantine for the polynomial ring R[Z]."}
{"category": "Math", "title": "An Upper Estimate for the Overpseudoprime Counting Function", "abstract": "We prove that the number of overpseudoprimes to base 2 not exceeding x does not exceed x^(3/4)(1+o(1))."}
{"category": "Math", "title": "Exploring Parameter Spaces in Dynamical Systems", "abstract": "The parameter space of dynamical systems arising in applications is often found to be high-dimensional and difficult to explore. We construct a fast algorithm to numerically analyze \"quantitative features\" of dynamical systems depending on parameters. Using a classical problem from mathematical ecology as an example, we demonstrate how to apply the algorithm to investigate the amplitude of a limit cycle depending on seven parameters. We stress the practical value of the algorithm but we also provide a rigorous error analysis to justify the overall strategy. Our approach turns out to be particularly useful in the case of comparing experimental data to a model defined by differential equations and to investigate whether the equations can approximate the modeled system."}
{"category": "Math", "title": "A \"typical\" contraction is unitary", "abstract": "We show that (for the weak operator topology) the set of unitary operators on a separable infinite-dimensional Hilbert space is residual in the set of all contractions. The analogous result holds for isometries and the strong operator topology as well. These results are applied to the problem of embedding operators into strongly continuous semigroups."}
{"category": "Math", "title": "Markov Stochastic Operators of Heredity", "abstract": "Applying non-ergodic quadratic stochastic operator the continual family of weak ergodic non-homogeneous Markov chains is constructed."}
{"category": "Math", "title": "Compression of Finite Group Actions and Covariant Dimension, II", "abstract": "Let $G$ be a finite group and $\\phi : V\\to W$ an equivariant morphism of finite dimensional $G$-modules. We say that $\\phi$ is faithful if $G$ acts faithfully on $\\phi(V)$. The covariant dimension of $G$ is the minimum of the dimension of $\\bar{\\phi(V)}$ taken over all faithful $\\phi$. In \\cite{KS07} we investigated covariant dimension and were able to determine it in many cases. Our techniques largely depended upon finding homogeneous faithful covariants. After publication of \\cite{KS07}, the junior author of this article pointed out several gaps in our proofs. Fortunately, this inspired us to find better techniques, involving multihomogeneous covariants, which have enabled us to extend and complete the results, simplify the proofs and fill the gaps of \\cite{KS07}."}
{"category": "Math", "title": "Methods in Industrial Biotechnology for Chemical Engineers", "abstract": "In keeping with the definition that biotechnology is really no more than a name given to a set of techniques and processes, the authors apply some set of fuzzy techniques to chemical industry problems such as finding the proper proportion of raw mix to control pollution, to study flow rates, to find out the better quality of products. We use fuzzy control theory, fuzzy neural networks, fuzzy relational equations, genetic algorithms to these problems for solutions. When the solution to the problem can have certain concepts or attributes as indeterminate, the only model that can tackle such a situation is the neutrosophic model. The authors have also used these models in this book to study the use of biotechnology in chemical industries. This book has six chapters. First chapter gives a brief description of biotechnology. Second chapter deals will proper proportion of mix of raw materials in cement industries to minimize pollution using fuzzy control theory. Chapter three gives the method of determination of temperature set point for crude oil in oil refineries. Chapter four studies the flow rates in chemical industries using fuzzy neutral networks. Chapter five gives the method of minimization of waste gas flow in chemical industries using fuzzy linear programming. The final chapter suggests when in these studies indeterminancy is an attribute or concept involved, the notion of neutrosophic methods can be adopted."}
{"category": "Math", "title": "Multiloop Lie algebras and the construction of extended affine Lie algebras", "abstract": "It is known that a multiloop Lie algebra, which constructed using multiloop realization, can be a Lie torus if the given multiloop Lie algebra satisfies several conditions, and it is also known that a family of extended affine Lie algebras (EALAs) are obtained from a Lie torus. In many cases, however, even if a given multiloop Lie algebra does not satisfy these conditions, we can also construct a family of EALAs from it. In this paper, we study this construction, and prove that two families of EALAs constructed from two multiloop Lie algebras coincide up to isomorphisms as EALAs if and only if two multiloop Lie algebras are \"support-isomorphic\". Also, we give a necessary and sufficient condition for two multiloop Lie algebras to be support-isomorphic."}
{"category": "Math", "title": "Growth in SL_3(Z/pZ)", "abstract": "Let G=SL_3(Z/pZ), p a prime. Let A be a set of generators of G. Then A grows under the group operation. To be precise: denote by |S| the number of elements of a finite set S. Assume |A| < |G|^{1-\\epsilon} for some \\epsilon>0. Then |A\\cdot A\\cdot A|>|A|^{1+\\delta}, where \\delta>0 depends only on \\epsilon. We also study subsets A\\subset G that do not generate G. Other results on growth and generation follow."}
{"category": "Math", "title": "Spaces of closed subgroups of locally compact groups", "abstract": "The set $\\Cal C(G)$ of closed subgroups of a locally compact group $G$ has a natural topology which makes it a compact space. This topology has been defined in various contexts by Vietoris, Chabauty, Fell, Thurston, Gromov, Grigorchuk, and many others. The purpose of the talk was to describe the space $\\Cal C(G)$ first for a few elementary examples, then for $G$ the complex plane, in which case $\\Cal C(G)$ is a 4--sphere (a result of Hubbard and Pourezza), and finally for the 3--dimensional Heisenberg group $H$, in which case $\\Cal C(H)$ is a 6--dimensional singular space recently investigated by Martin Bridson, Victor Kleptsyn and the author \\cite{BrHK}. These are slightly expanded notes prepared for a talk given at several places: the Kortrijk workshop on {\\it Discrete Groups and Geometric Structures, with Applications III,} May 26--30, 2008; the {\\it Tripode 14,} \\'Ecole Normale Sup\\'erieure de Lyon, June 13, 2008; and seminars at the EPFL, Lausanne, and in the Universit\\'e de Rennes 1. The notes do not contain any other result than those in \\cite{BrHK}, and are not intended for publication."}
{"category": "Math", "title": "Geometries with intransitive equivalence relation", "abstract": "One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry cannot be axiomatized, in general. It is obtained as a result of deformation of the proper Euclidean geometry. Class of physical geometries is more powerful, than the class of axiomatized geometries. The physical geometry admits one to describe such geometric properties as discreteness, granularity and limited divisibility. These properties are important in application to the space-time. They admits one to explain the discrimination properties of the space-time, which generate discrete parameters of elementary particles. Mathematical formalism of a physical geometry is very simple. The physical geometry is formulated in geometrical terms (in terms of points and world function) without a use of means of description (coordinate system, space dimension, manifold, etc.)."}
{"category": "Math", "title": "Bloch-Wigner theorem over rings with many units", "abstract": "The purpose of this article is to provide a version of Bloch-Wigner theorem over the class of rings with many units."}
{"category": "Math", "title": "Sparse random graphs with clustering", "abstract": "In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non-Poisson) multi-type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with power-law degree sequences with a wide range of degree exponents and clustering coefficients."}
{"category": "Math", "title": "Towards the generalized Shapiro and Shapiro conjecture", "abstract": "We find a new, asymptotically better, bound $g\\le\\frac14d^2+O(d)$ on the genus of a curve that may violate the generalized total reality conjecture. The bound covers all known cases except $g=0$ (the original conjecture)."}
{"category": "Math", "title": "Balanced split sets and Hamilton-Jacobi equations", "abstract": "We study the singular locus of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of $u$. In a previous paper, we proved that the singular locus is what we call a balanced split locus. In this paper, we find and classify all balanced split sets, identifying the cases where the only balanced split locus is the singular locus, and the cases where this doesn't hold. This clarifies the relationship between viscosity solutions and the more classical approach of characteristics and shocks."}
{"category": "Math", "title": "Approximation of subharmonic functions", "abstract": "In certain classes of subharmonic functions u on C distinguished in terms of lower bounds for the Riesz measure of u, a sharp estimate is obtained for the rate of approximation by functions of the form log |f(z)|, where f is an entire function. The results complement and generalize those recently obtained by Yu. Lyubarskii and Eu. Malinnikova."}
{"category": "Math", "title": "An extremal problem for a class of entire functions of exponential type", "abstract": "We find the exact upper estimate for the upper density of zeros of entire functions of exponential type whose indicator diagram is contained in a given interval."}
{"category": "Math", "title": "About Some Quadratic Scalar Curvatures and the $h_{4}$ Yamabe Equation", "abstract": "This is a paper based on a talk given at the conference on Conformal Geometry which held at Roscoff in France in the 2008 summer. We study some aspects of the equation arising from the problem of the existence on a given closed Riemannian manifold of dimension at leat 4, of a conformal metric with constant $h_4$ curvature. We establish a simple formula relating the second Gauss-Bonnet curvature $h_4$ to the $\\sigma_2$ curvature and we study some positivity properties of these two quadratic curvatures. We use different quadratic curvatures to characterize space forms, Einstein metrics and conformally flat metrics. In the appendix we introduce natural generalizations of Newton transformations, the corresponding Newton identities are used to obtain Avez type formulas for all the Gauss-Bonnet curvatures."}
{"category": "Math", "title": "Cycle Space Constructions for Exhaustions of Flag Domains", "abstract": "In the study of complex flag manifolds, flag domains and their cycle spaces, a key point is the fact that the cycle space $\\mathcal M_D$ of a flag domain $D$ is a Stein manifold. That fact has a long history. The earliest approach relied on construction of a strictly plurisubharmonic function on $\\mathcal M_D$, starting with a $q$--convex exhaustion function on $D$, where $q$ is the dimension of a particular maximal compact subvariety of $D$ (we use the normalization that 0--convex means Stein). Construction of that exhaustion function on $D$ required that $D$ be measurable. In that case the exhaustion on $D$ was transferred to $\\mathcal M_D$, using a special case of a method due to Barlet. Here we do the opposite: we use an incidence method to construct a canonical plurisubharmonic exhaustion function on $\\mathcal M_D$ and use it in turn to construct a canonical $q$--convex exhaustion function on $D$. This promises to have strong consequences for cohomology vanishing theorems and the construction of admissible representations of real reductive Lie groups."}
{"category": "Math", "title": "Problems in Additive Number Theory, III: Thematic Seminars at the Centre de Recerca Matematica", "abstract": "This is a survey of open problems in different parts of combinatorial and additive number theory. The paper is based on lectures at the Centre de Recerca Matematica in Barcelona on January 23 and January 25, 2008."}
{"category": "Math", "title": "Reflected Backward Stochastic Differential Equations with Continuous Coefficient and L^2 Barriers", "abstract": "In this paper we study reflected backward stochastic differential equations with a continuous, linear growth coefficient and two barriers which belong to L^2. We prove that there exists at least by penalization method."}
{"category": "Math", "title": "Reflected Backward Stochastic Differential Equations Driven by L\\'{e}vy Process", "abstract": "In this paper, we deal with a class of reflected backward stochastic differential equations associated to the subdifferential operator of a lower semi-continuous convex function driven by Teugels martingales associated with L\\'{e}vy process. We obtain the existence and uniqueness of solutions to these equations by means of the penalization method. As its application, we give a probabilistic interpretation for the solutions of a class of partial differential-integral inclusions."}
{"category": "Math", "title": "Nonexistence of arithmetic fake compact hermitian symmetric spaces of types other than An", "abstract": "We show that there are no arithmetic fake compact hermitian symmetric spaces of type other than An for n>4."}
{"category": "Math", "title": "Zeros of p-adic forms", "abstract": "We show that all $p$-adic quintic forms in at least $n>4562911$ variables have a non-trivial zero. We also derive new result concerning systems of cubic and quadratic forms."}
{"category": "Math", "title": "Ruled Lagrangian Submanifolds of the 6-Sphere", "abstract": "This article sets out to serve a dual purpose. On the one hand, we give an explicit description of the Lagrangian submanifolds of the nearly Kaehler 6-sphere which are ruled by circles of constant radius using Weierstrass formulae. On the other, we recognise all previous known examples of these Lagrangians as being ruled by such circles. Therefore, we describe all families of Lagrangians in the 6-sphere whose second fundamental form satisfies natural pointwise conditions: so-called second order families."}
{"category": "Math", "title": "The asymptotically optimal estimating equation for longitudinal data. Strong Consistency", "abstract": "In this article, we introduce a conditional marginal model for longitudinal data, in which the residuals form a martingale difference sequence. This model allows us to consider a rich class of estimating equations, which contains several estimating equations proposed in the literature. A particular sequence of estimating equations in this class contains a random matrix $\\mathcal{R}_{i-1}^*(\\beta)$, as a replacement for the ``true'' conditional correlation matrix of the $i$-th individual. Using the approach of [12], we identify some sufficient conditions under which this particular sequence of equations is asymptotically optimal (in our class). In the second part of the article, we identify a second set of conditions, under which we prove the existence and strong consistency of a sequence of estimators of $\\beta$, defined as roots of estimation equations which are martingale transforms (in particular, roots of the sequence of asymptotically optimal equations)."}
{"category": "Math", "title": "A Bernstein-Von Mises Theorem for discrete probability distributions", "abstract": "We investigate the asymptotic normality of the posterior distribution in the discrete setting, when model dimension increases with sample size. We consider a probability mass function $\\theta_0$ on $\\mathbbm{N}\\setminus \\{0\\}$ and a sequence of truncation levels $(k_n)_n$ satisfying $k_n^3\\leq n\\inf_{i\\leq k_n}\\theta_0(i).$ Let $\\hat{\\theta}$ denote the maximum likelihood estimate of $(\\theta_0(i))_{i\\leq k_n}$ and let $\\Delta_n(\\theta_0)$ denote the $k_n$-dimensional vector which $i$-th coordinate is defined by \\sqrt{n} (\\hat{\\theta}_n(i)-\\theta_0(i)) for $1\\leq i\\leq k_n.$ We check that under mild conditions on $\\theta_0$ and on the sequence of prior probabilities on the $k_n$-dimensional simplices, after centering and rescaling, the variation distance between the posterior distribution recentered around $\\hat{\\theta}_n$ and rescaled by $\\sqrt{n}$ and the $k_n$-dimensional Gaussian distribution $\\mathcal{N}(\\Delta_n(\\theta_0),I^{-1}(\\theta_0))$ converges in probability to $0.$ This theorem can be used to prove the asymptotic normality of Bayesian estimators of Shannon and R\\'{e}nyi entropies. The proofs are based on concentration inequalities for centered and non-centered Chi-square (Pearson) statistics. The latter allow to establish posterior concentration rates with respect to Fisher distance rather than with respect to the Hellinger distance as it is commonplace in non-parametric Bayesian statistics."}
{"category": "Math", "title": "The 8-universality Criterion is Unique", "abstract": "Using the methods developed for the proof that the 2-universality criterion is unique, we partially characterize criteria for the n-universality of positive-definite integer-matrix quadratic forms. We then obtain the uniqueness of Oh's 8-universality criterion as an application of our characterization results."}
{"category": "Math", "title": "The Stein hull", "abstract": "We are interested in the statistical linear inverse problem $Y=Af+\\epsilon\\xi$, where $A$ denotes a compact operator and $\\epsilon\\xi$ a stochastic noise. In a first time, we investigate the link between some threshold estimators and the risk hull point of view introduced in (5). The penalized blockwise Stein's rule plays a central role in this study. In particular, this estimator may be considered as a risk hull minimization method, provided the penalty is well-chosen. Using this perspective, we study the properties of the threshold and propose an admissible range for the penalty leading to accurate results. We eventually propose a penalty close to the lower bound of this range. The risk hull point of view provides interesting tools for the construction of adaptive estimators. It sheds light on the processes governing the behavior of linear estimators. The variability of the problem may be indeed quite large and should be carefully controlled."}
{"category": "Math", "title": "Generalized fractional Ornstein-Uhlenbeck processes", "abstract": "We introduce an extended version of the fractional Ornstein-Uhlenbeck (FOU) process where the integrand is replaced by the exponential of an independent L\\'evy process. We call the process the generalized fractional Ornstein-Uhlenbeck (GFOU) process. Alternatively, the process can be constructed from a generalized Ornstein-Uhlenbeck (GOU) process using an independent fractional Brownian motion (FBM) as integrator. We show that the GFOU process is well-defined by checking the existence of the integral included in the process, and investigate its properties. It is proved that the process has a stationary version and exhibits long memory. We also find that the process satisfies a certain stochastic differential equation. Our underlying intention is to introduce long memory into the GOU process which has short memory without losing the possibility of jumps. Note that both FOU and GOU processes have found application in a variety of fields as useful alternatives to the Ornstein-Uhlenbeck (OU) process."}
{"category": "Math", "title": "The large sieve, monodromy and zeta functions of algebraic curves, II: independence of the zeros", "abstract": "Using the sieve for Frobenius, we show that, in a certain sense, the roots of the L-functions of \"most\" algebraic curves over finite fields do not satisfy any non-trivial (linear or multiplicative) rational dependency relations. This can be seen as an analogue of conjectures of linear independence among ordinates of zeros of L-functions over number fields. As a corollary, we find, for \"most\" pairs of distinct algebraic curves over a finite field, the limiting distribution of the (suitably normalized) difference between the number of rational points over extensions of the ground field. The method of proof also emphasizes the relevance of Random Matrix models for this type of arithmetic questions. We also describe an alternative approach, which relies on Serre's theory of Frobenius tori, and we give a number of examples."}
{"category": "Math", "title": "The Irregular Set for Maps with the Specification Property has Full Topological Pressure", "abstract": "Let $(X,d)$ be a compact metric space, $f:X \\mapsto X$ be a continuous map with the specification property, and $\\varphi: X \\mapsto \\IR$ a continuous function. We consider the set of points for which the Birkhoff average of $\\varphi$ does not exist (which we call the irregular set for $\\varphi$) and show that this set is either empty or carries full topological pressure (in the sense of Pesin and Pitskel). We formulate various equivalent natural conditions on $\\varphi$ that completely describe when the latter situation holds and give examples of interesting systems to which our results apply but were not previously known. As an application, we show that for a suspension flow over a continuous map with specification, the irregular set carries full topological entropy."}
{"category": "Math", "title": "A Thermodynamic Definition of Topological Pressure for Non-Compact Sets", "abstract": "We give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle, leading to an alternative definition of an equilibrium state. We study the properties of this new quantity and compare it with existing notions of topological pressure. We are particularly interested in the situation when the ambient metric space is assumed to be compact. We motivate the naturality of our definition by applying it to some interesting examples, including the level sets of the pointwise Lyapunov exponent for the Manneville-Pomeau family of maps."}
{"category": "Math", "title": "Comultiplication rules for the double Schur functions and Cauchy identities", "abstract": "The double Schur functions form a distinguished basis of the ring \\Lambda(x||a) which is a multiparameter generalization of the ring of symmetric functions \\Lambda(x). The canonical comultiplication on \\Lambda(x) is extended to \\Lambda(x||a) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood-Richardson coefficients in two different ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood-Richardson coefficients provide a multiplication rule for the dual Schur functions."}
{"category": "Math", "title": "Spectral flow is the integral of one forms on the Banach manifold of self adjoint Fredholm operators", "abstract": "One may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 Vancouver ICM address of I.M. Singer. Our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self-adjoint bounded Breuer-Fredholm operators in a semi-finite von Neumann algebra. These formulae have a geometric interpretation which derives from the proof. Namely we define a family of Banach submanifolds of all bounded self-adjoint Breuer-Fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path contained in the submanifold calculates the spectral flow along this path. We emphasise that our methods do not give a single globally defined one form on the self adjoint Breuer- Fredholms whose integral along all paths is spectral flow rather, as the choice of the plural `forms' in the title suggests, we need a family of such one forms in order to confirm Singer's idea. The original context for this result concerned paths of unbounded self-adjoint Fredholm operators. We therefore prove analogous formulae for spectral flow in the unbounded case as well. The proof is a synthesis of key contributions by previous authors, whom we acknowledge in detail in the introduction, combined with an additional important recent advance in the differential calculus of functions of non-commuting operators."}
{"category": "Math", "title": "Three-dimensional polyhedra can be described by three polynomial inequalities", "abstract": "Bosse et al. conjectured that for every natural number $d \\ge 2$ and every $d$-dimensional polytope $P$ in $\\real^d$ there exist $d$ polynomials $p_0(x),...,p_{d-1}(x)$ satisfying $P=\\{x \\in \\mathbb{R}^d : p_0(x) \\ge 0, >..., p_{d-1}(x) \\ge 0 \\}.$ We show that for dimensions $d \\le 3$ even every $d$-dimensional polyhedron can be described by $d$ polynomial inequalities. The proof of our result is constructive."}
{"category": "Math", "title": "Tits indices over semilocal rings", "abstract": "We give a simplified proof of Tits' classification of semisimple algebraic groups that remains valid over semilocal rings. In particular, we provide explicit necessary and sufficient conditions that anisotropic groups of a given type appear as anisotropic kernels of semisimple groups of a given Tits index. We also give a new proof of the existence of all indices of exceptional inner type using the notion of canonical dimension of projective homogeneous varieties."}
{"category": "Math", "title": "Wilf conjecture", "abstract": "Let S(n,k) be the Stirling number of the second kind. Wilf conjectured that the alternating sum of S(n,k) for k from 0 to n is not zero for all n>2. In this paper, we prove that Wilf conjecture is true except at most one number with the properties of weighted Motzkin number."}
{"category": "Math", "title": "Uniform-in-bandwidth consistency for kernel-type estimators of Shannon's entropy", "abstract": "We establish uniform-in-bandwidth consistency for kernel-type estimators of the differential entropy. We consider two kernel-type estimators of Shannon's entropy. As a consequence, an asymptotic 100% confidence interval of entropy is provided."}
{"category": "Math", "title": "Whittaker limits of difference spherical functions", "abstract": "We introduce the (global) q-Whittaker function as the limit at t=0 of the q,t-spherical function extending the symmetric Macdonald polynomials to arbitrary eigenvalues. The construction heavily depends on the technique of the q-Gaussians developed by the author (and Stokman in the non-reduced case). In this approach, the q-Whittaker function is given by a series convergent everywhere, a kind of generating function for multi-dimensional q-Hermite polynomials (closely related to the level 1 Demazure characters). One of the applications is a q-version of the Shintani- Casselman- Shalika formula, which appeared directly connected with q-Mehta- Macdonald identities in terms of the Jackson integrals. This formula generalizes that of type A due to Gerasimov et al. to arbitrary reduced root systems. At the end of the paper, we obtain a q,t-counterpart of the Harish-Chandra asymptotic formula for the spherical functions, including the Whittaker limit."}
{"category": "Math", "title": "Homogenization of the Signorini boundary-value problem in a thick plane junction", "abstract": "We consider a mixed boundary-value problem for the Poisson equation in a plane thick junction $\\Omega_{\\varepsilon}$ which is the union of a domain $\\Omega_0$ and a large number of $\\varepsilon$-periodically situated thin rods. The nonuniform Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is made as $\\varepsilon \\to 0$, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove the convergence theorem and show that the nonuniform Signorini conditions are transformed (as $\\varepsilon \\to 0$) in the limiting variational inequalities in the region that is filled up by the thin rods in the limit passage. The existence and uniqueness of the solution to this non-standard limit problem is established. The convergence of the energy integrals is proved as well."}
{"category": "Math", "title": "Consistency of a needlet spectral estimator on the sphere", "abstract": "The angular power spectrum of a stationary random field on the sphere is estimated from the needlet coefficients of a single realization, observed with increasingly fine resolution. The estimator we consider is similar to the one recently used in practice by (Fa\\\"{y} et al. 2008) to estimate the power spectrum of the Cosmic Microwave Background. The consistency of the estimator, in the asymptotics of high frequencies, is proved for a model with a stationary Gaussian field corrupted by heteroscedastic noise and missing data."}
{"category": "Math", "title": "Spherical Averages on Regular and Semiregular Graphs", "abstract": "In 1966, P. Guenther proved the following result: Given a continuous function f on a compact surface M of constant curvature -1 and its periodic lift g to the universal covering, the hyperbolic plane, then the averages of the lift g over increasing spheres converge to the average of the function f over the surface M. In this article, we prove similar results for functions on the vertices and edges of regular and semiregular graphs, with special emphasis on the convergence rate. We also consider averages over more general sets like arcs, tubes and horocycles."}
{"category": "Math", "title": "Stanley depth of monomial ideals in three variables", "abstract": "We show that $\\depth(S/I)=0$ if and only if $\\sdepth(S/I)=0$, where $I\\subset S=K[x_1,...,x_n]$ is a monomial ideal. We give an algorithm to compute the Stanley depth of $S/I$, where $I\\subset S=K[x_1,x_2,x_3]$ is a monomial ideal. Also, we prove that a monomial ideal $I\\subset K[x_1,x_2,x_3]$ minimally generated by three monomials has $\\sdepth(I)=2$."}
{"category": "Math", "title": "The Normalized Ricci Flow on Four-Manifolds and Exotic Smooth Structures", "abstract": "In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \\in [0, \\infty)$ with uniformly bounded sectional curvature. In dimension 4, there exist many compact topological manifolds admitting distinct smooth structures, i.e., exotic smooth structures. Interestingly, in this article, the difference between existence and non-existence of non-singular solutions to the normalized Ricci flow on 4-manifolds turns out to strictly depend on the choice of smooth structure. In fact, we shall prove that, for every natural number $\\ell$, there exists a compact topological 4-manifold $X_{\\ell}$ which admits smooth structures for which non-singular solutions of the normalized Ricci flow exist, but also admits smooth structures for which no non-singular solution of the normalized Ricci flow exists. Hence, in dimension 4, smooth structures become definite obstructions to the existence of non-singular solutions to the normalized Ricci flow."}
{"category": "Math", "title": "Canonical tilting modules over shod algebras are regular in codimension one", "abstract": "We show that for a class of modules over shod algebras, including the canonical tilting modules, the closures of the corresponding orbits in module varieties are regular in codimension one."}
{"category": "Math", "title": "Schmidt games and Markov partitions", "abstract": "Let T be a C^2-expanding self-map of a compact, connected, smooth, Riemannian manifold M. We correct a minor gap in the proof of a theorem from the literature: the set of points whose forward orbits are nondense has full Hausdorff dimension. Our correction allows us to strengthen the theorem. Combining the correction with Schmidt games, we generalize the theorem in dimension one: given a point x in M, the set of points whose forward orbit closures miss x is a winning set."}
{"category": "Math", "title": "Splittings of monomial ideals", "abstract": "We provide some new conditions under which the graded Betti numbers of a monomial ideal can be computed in terms of the graded Betti numbers of smaller ideals, thus complementing Eliahou and Kervaire's splitting approach. As applications, we show that edge ideals of graphs are splittable, and we provide an iterative method for computing the Betti numbers of the cover ideals of Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which one can find particular splittings of monomial ideals and raise questions about ideals whose resolutions are characteristic-dependent."}
{"category": "Math", "title": "Quiver representations in toric geometry", "abstract": "This article is based on my lecture notes from summer schools at the Universities of Utah (June 2007) and Warwick (September 2007). We provide an introduction to explicit methods in the study of moduli spaces of quiver representations and derived categories arising in toric geometry. The first main goal is to present the noncommutative geometric approach to semiprojective toric varieties via quivers. To achieve this, we use geometric invariant theory to construct both semiprojective toric varieties and moduli spaces of quiver representations. The second main goal builds on the first by presenting an introduction to explicit methods in derived categories of coherent sheaves in toric geometry. We recall the notion of tilting bundles with examples, and describe the McKay correspondence as a derived equivalence in some detail following Bridgeland, King and Reid. We also describe extensions of their result beyond the $G$-Hilbert scheme to other fine moduli spaces of bound quiver representations."}
{"category": "Math", "title": "Windings of planar random walks and averaged Dehn function", "abstract": "We prove a sharp estimate on the expected value of the integral of the index of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc."}
{"category": "Math", "title": "Well-posedness for the generalized Benjamin-Ono equations with arbitrary large initial data in the critical space", "abstract": "We prove that the generalized Benjamin-Ono equations $\\partial_tu+\\mathcal{H}\\partial_x^2u\\pm u^k\\partial_xu=0$, $k\\geq 4$ are locally well-posed in the scaling invariant spaces $\\dot{H}^{s_k}(\\R)$ where $s_k=1/2-1/k$. Our results also hold in the non-homogeneous spaces $H^{s_k}(\\R)$. In the case $k=3$, local well-posedness is obtained in $H^{s}(\\R)$, $s>1/3$."}
{"category": "Math", "title": "On the vanishing of the Rokhlin invariant", "abstract": "It is a natural consequence of fundamental properties of the Casson invariant that the Rokhlin invariant of an amphichiral integral homology 3-sphere M vanishes. In this paper, we give a new direct proof of this vanishing property. For such an M, we construct a manifold pair (Y,Q) of dimensions 6 and 3 equipped with some additional structure (6-dimensional spin e-manifold), such that Q = M \\cup M \\cup (-M) and (Y,Q) \\cong (-Y,-Q). We prove that (Y,Q) bounds a 7-dimensional spin e-manifold (Z,X) by studying the cobordism group of 6-dimensional spin e-manifolds and the Z/2-actions on the two--point configuration space of M minus one point. For any such (Z,X), the signature of X vanishes, and this implies the vanishing of the Rokhlin invariant. The idea of the construction of (Y,Q) comes from the definition of the Kontsevich-Kuperberg-Thurston invariant for rational homology 3-spheres."}
{"category": "Math", "title": "Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints", "abstract": "We consider the well-known following shape optimization problem: $$\\lambda_1(\\Omega^*)=\\min_{\\stackrel{|\\Omega|=a} {\\Omega\\subset{D}}} \\lambda_1(\\Omega), $$ where $\\lambda_1$ denotes the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary condition, and $D$ is an open bounded set (a box). It is well-known that the solution of this problem is the ball of volume $a$ if such a ball exists in the box $D$ (Faber-Krahn's theorem). In this paper, we prove regularity properties of the boundary of the optimal shapes $\\Omega^*$ in any case and in any dimension. Full regularity is obtained in dimension 2."}
{"category": "Math", "title": "Note on the Deodhar decomposition of a double Schubert cell", "abstract": "We show that for an algebraic reductive group $G$, the partition of a double Schubert cell in the flag variety $G/B$ defined by Deodhar, and coming from a Bialynicki-Birula decomposition, is not a stratification in general. We give a counterexample for a group of type B$_n$, where the closure of some specific cell of dimension $2n$ has a non-trivial intersection with a cell of dimension $3n-3$"}
{"category": "Math", "title": "Twistor structures, tt*-geometry and singularity theory", "abstract": "We give an overview on the tt*-geometry defined for isolated hypersurface singularities and tame functions via Brieskorn lattices. We discuss nilpotent orbits in this context, as well as classifying spaces of Brieskorn lattices and (limits of) period maps."}
{"category": "Math", "title": "Differential forms on locally convex spaces and the Stokes formula", "abstract": "We prove a version of the Stokes formula for differential forms on locally convex spaces. The main tool used for proving this formula is the surface layer theorem proved in another paper by the author. Moreover, for differential forms of a Sobolev-type class relative to a differentiable measure, we compute the operator adjoint to the exterior differential in terms of standard operations of calculus of differential forms and the logarithmic derivative. Previously, this connection was established under essentially stronger assumptions."}
{"category": "Math", "title": "On a constrained 2-D Navier-Stokes equation", "abstract": "The planar Navier-Stokes equation exhibits, in absence of external forces, a trivial asymptotics in time. Nevertheless the appearence of coherent structures suggests non-trivial intermediate asymptotics which should be explained in terms of the equation itself. Motivated by the separation of the different time scales observed in the dynamics of the Navier-Stokes equation, we study the well-posedness and asymptotic behaviour of a constrained equation which neglects the variation of the energy and moment of inertia."}
{"category": "Math", "title": "An exact Ramsey principle for block sequences", "abstract": "We prove an exact, i.e., formulated without $\\Delta$-expansions, Ramsey principle for infinite block sequences in vector spaces over countable fields, where the two sides of the dichotomic principle are represented by respectively winning strategies in Gowers' block sequence game and winning strategies in the infinite asymptotic game. This allows us to recover Gowers' dichotomy theorem for block sequences in normed vector spaces by a simple application of the basic determinacy theorem for infinite asymptotic games."}
{"category": "Math", "title": "$n$-Subspaces in linear and unitary spaces", "abstract": "We study a relation between brick $n$-tuples of subspaces of a finite dimensional linear space, and irreducible $n$-tuples of subspaces of a finite dimensional Hilbert (unitary) space such that a linear combination, with positive coefficients, of orthogonal projections onto these subspaces equals the identity operator. We prove that brick systems of one-dimensional subspaces and the systems obtained from them by applying the Coxeter functors (in particular, all brick triples and quadruples of subspaces) can be unitarized. For each brick triple and quadruple of subspaces, we describe sets of characters that admit a unitarization."}
{"category": "Math", "title": "On Sun's conjecture concerning disjoint cosets", "abstract": "In 2004, Zhi-Wei Sun posed the following conjecture: If a_1G_1,...,a_kG_k (k>1) are finitely many pairwise disjoint left cosets in a group G with all the indices [G:G_i] finite, then for some 1\\le i<j\\le k, the greatest common divisor of [G:G_i] and [G:G_j] is at least k. In this paper, we confirm Sun's conjecture for k=3,4."}
{"category": "Math", "title": "Nonoscillation and Stability of the Second Order Ordinary Differential Equations with a Damping Term", "abstract": "In this paper we consider the linear ordinary equation of the second order $$ L x(t)\\equiv \\ddot{x}(t) +a(t)\\dot{x}(t)+b(t)x(t)=f(t), \\eqno{(1)} $$ and the corresponding homogeneous equation $$ \\ddot{x}(t) +a(t)\\dot{x}(t)+b(t)x(t)=0. \\eqno{(2)} $$ Note that $[\\alpha ,\\beta ]$ is called a nonoscillation interval if every nontrivial solution has at most one zero on this interval. Many investigations which seem to have no connection such as differential inequalities, the Polia-Mammana decomposition (i.e. representation of the operator $L$ in the form of products of the first order differential operators), unique solvability of the interpolation problems, kernels oscillation, separation of zeros, zones of Lyapunov's stability and some others have a certain common basis - nonoscillation. Presumably Sturm was the first to consider the two problems which naturally appear here: to develop corollaries of nonoscillation and to find methods to check nonoscillation. In this paper we obtain several tests for nonoscillation on the semiaxis and apply them to propose new results on asymptotic properties and the exponential stability of the second order equation (2). Using the Floquet representations and upper and lower estimates of nonoscillation intervals of oscillatory solutions we deduce results on the exponential and Lyapunov's stability and instability of equation (2)."}
{"category": "Math", "title": "Hausdorff dimension for ergodic measures of interval exchange transformations", "abstract": "I show that there exist minimal interval exchange transformations with an ergodic measure whose Hausdorff dimension is arbitrarily small, even 0. I will also show that in particular cases one can bound the Hausdorff dimension between $\\frac 1 {2r+4}$ and $\\frac 1 r$ for any r greater than 1."}
{"category": "Math", "title": "Aspects of area formulas by way of Luzin, Rad\\'o, and Reichelderfer on metric measure spaces", "abstract": "We consider some measure-theoretic properties of functions belonging to a Sobolev-type class on metric measure spaces that admit a Poincar\\'e inequality and are equipped with a doubling measure. The properties we have selected to study are those that are related to area formulas."}
{"category": "Math", "title": "Comparison of cobordism theories", "abstract": "Relying on results of Hopkins-Morel, we show that, for $X$ a quasi-projective variety over a field of characteristic zero, the canonical map $\\Omega_n(X)\\to MGL_{2n,n}'(X)$ is an isomorphism. Here $\\Omega_*(X)$ is the theory of algebraic cobordism defined by Levine-Morel, and $MGL_{*,*}'$ is the Borel-Moore homology version of the theory of algebraic cobordism defined via the algebraic Thom complex in the Morel-Voevodsky motivic stable homotopy category."}
{"category": "Math", "title": "Nemirovski's Inequalities Revisited", "abstract": "An important tool for statistical research are moment inequalities for sums of independent random vectors. Nemirovski and coworkers (1983, 2000) derived one particular type of such inequalities: For certain Banach spaces $(\\B,\\|\\cdot\\|)$ there exists a constant $K = K(\\B,\\|\\cdot\\|)$ such that for arbitrary independent and centered random vectors $X_1, X_2, ..., X_n \\in \\B$, their sum $S_n$ satisfies the inequality $ E \\|S_n \\|^2 \\le K \\sum_{i=1}^n E \\|X_i\\|^2$. We present and compare three different approaches to obtain such inequalities: Nemirovski's results are based on deterministic inequalities for norms. Another possible vehicle are type and cotype inequalities, a tool from probability theory on Banach spaces. Finally, we use a truncation argument plus Bernstein's inequality to obtain another version of the moment inequality above. Interestingly, all three approaches have their own merits."}
{"category": "Math", "title": "Mathematical Analysis of a Kinetic Model for Cell Movement in Network Tissues", "abstract": "Mesenchymal motion describes the movement of cells in biological tissues formed by fiber networks. An important example is the migration of tumor cells through collagen networks during the process of metastasis formation. We investigate the mesenchymal motion model proposed by T. Hillen (J. Math. Biol. 53:585-616, 2006) in higher dimensions. We formulate the problem as an evolution equation in a Banach space of measure-valued functions and use methods from semigroup theory to show the global existence of classical solutions. We investigate steady states of the model and show that patterns of network type exist as steady states. For the case of constant fiber distribution, we find an explicit solution and we prove the convergence to the parabolic limit."}
{"category": "Math", "title": "Oriented cohomology, Borel-Moore homology and algebraic cobordism", "abstract": "We examine various versions of oriented cohomology and Borel-Moore homology theories in algebraic geometry and put these two together in the setting of an \"oriented duality theory\", a generalization of Bloch-Ogus twisted duality theory. This combines and exends work of Panin and Mocanasu. We apply this to give a Borel-Moore homology version $MGL'_{*,*}$ of Voevodsky's $MGL^{*,*}$-theory, and a natural map $\\vartheta:\\Omega_*\\to MGL'_{2*,*}$, where $\\Omega_*$ is the algebraic cobordism theory defined by Levine-Morel. We conjecture that $\\vartheta$ is an isomorphism and describe a program for proving this conjecture."}
{"category": "Math", "title": "Rate of Escape of Random Walks on Regular Languages and Free Products by Amalgamation of Finite Groups", "abstract": "We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques."}
{"category": "Math", "title": "Smooth motives", "abstract": "Following ideas of Bondarko, we construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme $S$ generated by the motives of smooth projective $S$-schemes, assuming that $S$ is itself smooth over a perfect field."}
{"category": "Math", "title": "Structure theorems of mixable shuffle algebras and free commutative Rota-Baxter algebras", "abstract": "We study the ring theoretical structures of mixable shuffle algebras and their associated free commutative Rota-Baxter algebras. For this study we utilize the connection of the mixable shuffle algebras with the overlapping shuffle algebra of Hazewinkel, quasi-shuffle algebras of Hoffman and quasi-symmetric functions. This connection allows us to apply methods and results on shuffle products and Lyndon words on ordered sets. As a result, we obtain structure theorems for a large class of mixable shuffle algebras and free commutative Rota-Baxter algebras with various coefficient rings."}
{"category": "Math", "title": "Classes on compactifications of the moduli space of curves through solutions to the quantum master equation", "abstract": "In this paper we describe a construction which produces classes in a compactification of the moduli space of curves. This construction extends a construction of Kontsevich which produces classes in the open moduli space from the initial data of a cyclic A-infinity algebra. The initial data for our construction is what we call a `quantum A-infinity algebra', which arises as a type of deformation of a cyclic A-infinity algebra. The deformation theory for these structures is described explicitly. We construct a family of examples of quantum A-infinity algebras which extend a family of cyclic A-infinity algebras, introduced by Kontsevich, which are known to produce all the Miller-Morita-Mumford classes using his construction."}
{"category": "Math", "title": "Ice formation in the Arctic during summer: false-bottoms", "abstract": "The only source of ice formation in the Arctic during summer is a layer of ice called false-bottoms between an under-ice melt pond and the underlying ocean. Of interest is to give a mathematical model in order to determine the simultaneous growth and ablation of false-bottoms, which is governed by both of heat fluxes and salt fluxes. In one dimension, this problem may be considered mathematically as a two-phase Stefan problem with two free boundaries. Our main result is to prove the existence and uniqueness of the solution from the initial condition."}
{"category": "Math", "title": "Two Dimensional Density Estimation using Smooth Invertible Transformations", "abstract": "We investigate the problem of estimating a smooth invertible transformation f when observing independent samples X_1, ..., X_n ~ P \\circ f, where P is a known measure. We focus on the two dimensional case where P and f are defined on R^2. We present a flexible class of smooth invertible transformations in two dimensions with variational equations for optimizing over the classes, then study the problem of estimating the transformation f by penalized maximum likelihood estimation. We apply our methodology to the case when P \\circ f has a density with respect to Lebesgue measure on R^2 and demonstrate improvements over kernel density estimation on three examples."}
{"category": "Math", "title": "Counterexamples in Cake-Cutting", "abstract": "This article contains counterexamples to theorems and claims in Brams, Jones and Klamler's article \"Better Ways to Cut a Cake\" in the December 2006 Notices of the American Mathematical Society."}
{"category": "Math", "title": "Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber", "abstract": "We show that the natural morphism $\\phi:\\pi_1(X_{\\eta},x_{\\eta})\\to \\pi_1(X,x)_{\\eta}$ between the fundamental group scheme of the generic fiber $X_{\\eta}$ of a scheme $X$ over a connected Dedekind scheme and the generic fiber of the fundamental group scheme of $X$ is always faithfully flat. As an application we give a necessary and sufficient condition for a finite, dominated pointed $G$-torsor over $X_{\\eta}$ to be extended over $X$. We finally provide examples where $\\phi:\\pi_1(X_{\\eta},x_{\\eta})\\to \\pi_1(X,x)_{\\eta}$ is an isomorphism.."}
{"category": "Math", "title": "Adapted coordinates in two dimensions and a proof of Puiseux's theorem", "abstract": "A method for finding Puiseux series goes back to Isaac Newton, which gives the terms of Puiseux series through an infinite recursive process; an additional argument is then used to show that the resulting Puiseux series are convergent. This paper provides an argument based on Newton's method and some ideas from resolution of singularities that gives a quick proof of both the existence and convergence of Puiseux series. It is then shown that similar ideas can be used to give a short proof of the existence of smooth adapted coordinates for oscillatory integrals in two dimensions, a result first proved in the real-analytic case by Varchenko [V] and then recently for the general smooth case by Ikromov-Muller [IM]. The arguments of this paper are entirely elementary."}
{"category": "Math", "title": "A cone theorem for nef curves", "abstract": "Following ideas of V. Batyrev, we prove an analogue of the Cone Theorem for the closed cone of nef curves: an enlargement of the cone of nef curves is the closure of the sum of a K_X-non-negative portion and countably many K_X-negative coextremal rays. An example shows that this enlargement is necessary. We also describe the relationship between K_X-negative faces of this cone and the possible outcomes of the minimal model program."}
{"category": "Math", "title": "A note on K$\\ddot{a}$hler manifolds with almost nonnegative bisectional curvature", "abstract": "In this note we prove the following result: There is a positive constant $\\epsilon(n,\\Lambda)$ such that if $M^n$ is a simply connected compact K$\\ddot{a}$hler manifold with sectional curvature bounded from above by $\\Lambda$, diameter bounded from above by 1, and with holomorphic bisectional curvature $H \\geq -\\epsilon(n,\\Lambda)$, then $M^n$ is diffeomorphic to the product $M_1\\times ... \\times M_k$, where each $M_i$ is either a complex projective space or an irreducible K$\\ddot{a}$hler-Hermitian symmetric space of rank $\\geq 2$. This resolves a conjecture of F. Fang under the additional upper bound restrictions on sectional curvature and diameter."}
{"category": "Math", "title": "The energy space for the Gross-Pitaevskii equation with magnetic field", "abstract": "We study the energy space for the Gross-Pitaevskii equation with magnetic field and non-vanishing conditions at infinity. We provide necessary and sufficient conditions on the magnetic field for which the energy space is non-empty."}
{"category": "Math", "title": "An output-sensitive algorithm for multi-parametric LCPs with sufficient matrices", "abstract": "This paper considers the multi-parametric linear complementarity problem (pLCP) with sufficient matrices. The main result is an algorithm to find a polyhedral decomposition of the set of feasible parameters and to construct a piecewise affine function that maps each feasible parameter to a solution of the associated LCP in such a way that the function is affine over each cell of the decomposition. The algorithm is output-sensive in the sense that its time complexity is polynomial in the size of the input and linear in the size of the output, when the problem is non-degenerate. We give a lexicographic perturbation technique to resolve degeneracy as well. Unlike for the non-parametric case, the resolution turns out to be nontrivial, and in particular, it involves linear programming (LP) duality and multi-objective LP."}
{"category": "Math", "title": "Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation", "abstract": "We consider the semiclassical limit of the Hartree equation with a data causing a focusing at a point. We study the asymptotic behavior of phase function associated with the WKB approximation near the caustic when a nonlinearity is supercritical. In this case, it is known that a phase shift occurs in a neighborhood of focusing time in the case of focusing cubic nonlinear Schr\\\"odinger equation. Thanks to the smoothness of the nonlocal nonlinearities, we justify the WKB-type approximation of the solution for a data which is larger than in the previous results and is not necessarily well-prepared. We also show by an analysis of the limit hydrodynamical equaiton that, however, this WKB-type approximation breaks down before reaching the focal point: Nonlinear effects lead to the formation of singularity of the leading term of the phase function."}
{"category": "Math", "title": "On Fano manifolds with a birational contraction sending a divisor to a curve", "abstract": "Let X be a smooth Fano variety of dimension at least 4. We show that if X has an elementary birational contraction sending a divisor to a curve, then the Picard number of X is smaller or equal to 5."}
{"category": "Math", "title": "Floer homology and existence of incompressible tori in homology spheres", "abstract": "We show that if a prime homology sphere has the same Floer homology as the standard three-sphere, it does not contain any incompressible tori."}
{"category": "Math", "title": "Local rigidity of quasi-regular varieties", "abstract": "For a $G$-variety $X$ with an open orbit, we define its boundary $\\partial X$ as the complement of the open orbit. The action sheaf $S_X$ is the subsheaf of the tangent sheaf made of vector fields tangent to $\\partial X$. We prove, for a large family of smooth spherical varieties, the vanishing of the cohomology groups $H^i(X,S_X)$ for $i>0$, extending results of F. Bien and M. Brion. We apply these results to study the local rigidity of the smooth projective varieties with Picard number one classified in a previous paper of the first author."}
{"category": "Math", "title": "Process of \"Primoverization\" of Numbers of the Form a^n-1", "abstract": "We call an integer N>1 primover to base a if it either prime or overpseudoprime to base a. We prove, in particular, that every Fermat number is primover to base 2. We also indicate a simple process of receiving of primover divisors of numbers of the form a^n-1."}
{"category": "Math", "title": "On the stratified vector bundles", "abstract": "The stratified vector bundles on a smooth variety defined over an algebraically closed field $k$ form a neutral Tannakian category over $k$. We investigate the affine group--scheme corresponding to this neutral Tannakian category."}
{"category": "Math", "title": "Computing faithful representations for nilpotent Lie algebras", "abstract": "We describe three methods to determine a faithful representation of small dimension for a finite-dimensional nilpotent Lie algebra over an arbitrary field. We apply our methods in finding bounds for the smallest dimension $\\mu(\\Lg)$ of a faithful $\\Lg$-module for some nilpotent Lie algebras $\\Lg$. In particular, we describe an infinite family of filiform nilpotent Lie algebras $\\Lf_n$ of dimension $n$ over $\\Q$ and conjecture that $\\mu(\\Lf_n) > n+1$. Experiments with our algorithms suggest that $\\mu(\\Lf_n)$ is polynomial in $n$."}
{"category": "Math", "title": "Large Deviations of the Front in a one dimensional model of $X+Y \\to 2X$", "abstract": "We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction $X+Y \\to 2X$ on the integer lattice in which $Y$ particles do not move while $X$ particles move as independent simple continuous time random walks of total jump rate $2$. For a wide class of initial conditions, we prove that a large deviations principle holds and we show that the zero set of the rate function is the interval $[0,v]$, where $v$ is the velocity of the front given by the law of large numbers. We also give more precise estimates for the rate of decay of the slowdown probabilities. Our results indicate a gapless property of the generator of the process as seen from the front, as it happens in the context of nonlinear diffusion equations describing the propagation of a pulled front into an unstable state."}
{"category": "Math", "title": "Bounds for Integral j -Invariants and Cartan Structures on Elliptic Curves", "abstract": "We bound the j -invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that the modular curve Xsplit (p3) has no non-trivial rational point if p is a sufficiently large prime number. Assuming the GRH, one can replace p3 by p2 ."}
{"category": "Math", "title": "Algebraic string bracket as a Poisson bracket", "abstract": "In this paper we construct a Lie algebra representation of the algebraic string bracket on negative cyclic cohomology of an associative algebra with appropriate duality. This is a generalized algebraic version of the main theorem of [AZ] which extends Goldman's results using string topology operations.The main result can be applied to the de Rham complex of a smooth manifold as well as the Dolbeault resolution of the endomorphisms of a holomorphic bundle on a Calabi-Yau manifold."}
{"category": "Math", "title": "The growth rate of an entire function and the Hausdorff dimension of its Julia set", "abstract": "Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract."}
{"category": "Math", "title": "The height of random binary unlabelled trees", "abstract": "This extended abstract is dedicated to the analysis of the height of non-plane unlabelled rooted binary trees. The height of such a tree chosen uniformly among those of size $n$ is proved to have a limiting theta distribution, both in a central and local sense. Moderate as well as large deviations estimates are also derived. The proofs rely on the analysis (in the complex plane) of generating functions associated with trees of bounded height."}
{"category": "Math", "title": "Transitivity of codimension one Anosov actions of R^k on closed manifolds", "abstract": "In this paper, we define codimension one Anosov actions of $\\RR^k, k\\geq 2,$ on a closed connected orientable manifold $M$. We prove that if the ambient manifold has dimension greater than $k+2$, then the action is topologically transitive. This generalizes a result of Verjovsky for codimension one Anosov flows."}
{"category": "Math", "title": "Complexity of comparing monomials and two improvements of the BM-algorithm", "abstract": "We give a new algorithm for merging sorted lists of monomials. Together with a projection technique we obtain a new complexity bound for the BM-algorithm."}
{"category": "Math", "title": "The type of the base ring associated to a transversal polymatroid", "abstract": "In this paper we determine the facets of the polyhedral cone generated by the exponent set of the monomials defining the base ring associated to some transversal polymatroid. We need the description of these facets to find the canonical module of the base ring which is expressed in terms of the relative interior of the cone. This would allow us to compute the $a$-invariant of those base rings. The results presented were discovered by extensive computer algebra experiments performed with {\\it{Normaliz}} \\cite{BK}."}
{"category": "Math", "title": "Boundedness of Schroedinger type propagators on modulation spaces", "abstract": "It is known that Fourier integral operators arising when solving Schr\\\"odinger-type operators are bounded on the modulation spaces $\\cM^{p,q}$, for $1\\leq p=q\\leq\\infty$, provided their symbols belong to the Sj\\\"ostrand class $M^{\\infty,1}$. However, they generally fail to be bounded on $\\cM^{p,q}$ for $p\\not=q$. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on $\\cM^{p,q}$ for $p\\not=q$, and between $\\cM^{p,q}\\to\\cM^{q,p}$, $1\\leq q< p\\leq\\infty$. We also study similar problems for operators acting on Wiener amalgam spaces, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle."}
{"category": "Math", "title": "Strictly singular non-compact diagonal operators on HI spaces", "abstract": "We construct a Hereditarily Indecomposable Banach space $\\eqs_d$ with a Schauder basis \\seq{e}{n} on which there exist strictly singular non-compact diagonal operators. Moreover, the space $\\mc{L}_{\\diag}(\\eqs_d)$ of diagonal operators with respect to the basis \\seq{e}{n} contains an isomorphic copy of $\\ell_{\\infty}(\\N)$."}
{"category": "Math", "title": "Saturated extensions, the attractors method and Hereditarily James Tree Space", "abstract": "In the present work we provide a variety of examples of HI Banach spaces containing no reflexive subspace and we study the structure of their duals as well as the spaces of their linear bounded operators. Our approach is based on saturated extensions of ground sets and the method of attractors."}
{"category": "Math", "title": "On pointed Hopf algebras associated with the symmetric groups", "abstract": "It is an important open problem whether the dimension of the Nichols algebra B(O,\\rho) is finite when O is the class of the transpositions and \\rho is the sign representation, with m>= 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs (O,\\rho) might give rise to finite-dimensional Nichols algebras."}
{"category": "Math", "title": "On rich lines in grids", "abstract": "In this paper we show that if one has a grid A x B, where A and B are sets of n real numbers, then there can be only very few ``rich'' lines in certain quite small families. Indeed, we show that if the family has lines taking on n^epsilon distinct slopes, and where each line is parallel to n^epsilon others (so, at least n^(2 epsilon) lines in total), then at least one of these lines must fail to be ``rich''. This result immediately implies non-trivial sum-product inequalities; though, our proof makes use of the Szemeredi-Trotter inequality, which Elekes used in his argument for lower bounds on |C+C| + |C.C|."}
{"category": "Math", "title": "Contact structures, sutured Floer homology and TQFT", "abstract": "We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of one sutured manifold (M',\\Gamma') into a larger sutured manifold (M,\\Gamma), together with a contact structure on M-M'. As an application of this gluing map, we produce a (1+1)-dimensional TQFT by dimensional reduction and study its properties."}
{"category": "Math", "title": "Positive mass theorem for the Paneitz-Branson operator", "abstract": "We prove that under suitable assumptions, the constant term in the Green function of the Paneitz-Branson operator on a compact Riemannian manifold $(M,g)$ is positive unless $(M,g)$ is conformally diffeomophic to the standard sphere. The proof is inspired by the positive mass theorem on spin manifolds by Ammann-Humbert."}
{"category": "Math", "title": "On the minimal speed of traveling waves for a non-local delayed reaction-diffusion equation", "abstract": "In this note, we give constructive upper and lower bounds for the minimal speed of propagation of traveling waves for non-local delayed reaction-diffusion equation."}
{"category": "Math", "title": "Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory", "abstract": "In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters."}
{"category": "Math", "title": "Secant varieties to osculating varieties of Veronese embeddings of $\\mathbb{P}^n$", "abstract": "A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of $V_{n,d}$ (the $d$-uple embedding of $\\mathbb{P}^n$) have the expected dimension, with few known exceptions. We study here the same problem for $T_{n,d}$, the tangential variety to $V_{n,d}$, and prove a conjecture, which is the analogous of Alexander-Hirschowitz theorem, for $n\\leq 9$. Moreover. we prove that it holds for any $n,d$ if it holds for $d=3$. Then we generalize to the case of $O_{k,n,d}$, the $k$-osculating variety to $V_{n,d}$, proving, for $n=2$, a conjecture that relates the defectivity of $\\sigma_s(O_{k,n,d})$ to the Hilbert function of certain sets of fat points in $\\mathbb{P}^n$."}
{"category": "Math", "title": "Automata and cells in affine Weyl groups", "abstract": "Let W~ be an affine Weyl group, and let C be a left, right, or two-sided Kazhdan--Lusztig cell in W~. Let Reduced (C) be the set of all reduced expressions of elements of C, regarded as a formal language in the sense of the theory of computation. We show that Reduced (C) is a regular language. Hence the reduced expressions of the elements in any Kazhdan--Lusztig cell can be enumerated by a finite state automaton."}
{"category": "Math", "title": "Differential operators, shifted parts, and hook lengths", "abstract": "We discuss Sekiguchi-type differential operators, their eigenvalues, and a generalization of Andrews-Goulden-Jackson formula. These will be applied to extract explicit formulae involving shifted partitions and hook lengths."}
{"category": "Math", "title": "Strong A-infinity weights, Besov and Sobolev capacities in metric measure spaces", "abstract": "This article studies strong A-infinity weights in Ahlfors Q-regular and geodesic metric spaces satisfying a weak (1,s)-Poincare inequality for some 1<s<=Q, where Q is finite. It is shown that whenever max(1,Q-1)<s<=Q, a function u yields a strong A-infinity weight of the form w=exp(Qu) if u has a minimal s-weak upper gradient with sufficiently small Morrey norm. Similarly, it is proved that if 1<Q<p for some finite p, then w=exp(Qu) is a strong A-infinity weight whenever u has sufficiently small Besov p-seminorm."}
{"category": "Math", "title": "The relative hyperbolicity of one-relator relative presentations", "abstract": "We prove that if $G$ is a free-torsion group and $w(t)$ is a word in the alphabet $G \\sqcup \\{t^{\\pm 1}\\}$ with exponent sum one, then the group $<G,t|(w(t))^k = 1>$, where $k \\geq 2$, is relatively hyperbolic with respect to $G$."}
{"category": "Math", "title": "Symplectic Floer homology of area-preserving surface diffeomorphisms", "abstract": "The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel's HF_*(h) for h any orientation-preserving mapping class."}
{"category": "Math", "title": "Differential Tannakian Categories", "abstract": "We define a differential Tannakian category and show that under a natural assumption it has a fibre functor. If in addition this category is neutral, that is, the target category for the fibre functor are finite dimensional vector spaces over the base field, then it is equivalent to the category of representations of a (pro-)linear differential algebraic group. Our treatment of the problem is via differential Hopf algebras and Deligne's fibre functor construction."}
{"category": "Math", "title": "The fine intersection problem for Steiner triple systems", "abstract": "The intersection of two Steiner triple systems (X,A) and (X,B) is the set A intersect B. The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m,n) such that there exist two Steiner triple systems of order v whose intersection has n blocks over m points. We show that for v = 1 or 3 (mod 6), |I(v)| = Omega(v^3), where previous results only imply that |I(v)| = Omega(v^2)."}
{"category": "Math", "title": "Approximate volume and integration for basic semi-algebraic sets", "abstract": "Given a basic compact semi-algebraic set $\\K\\subset\\R^n$, we introduce a methodology that generates a sequence converging to the volume of $\\K$. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on $\\K$ can be approximated as closely as desired, and so permits to approximate the integral on $\\K$ of any given polynomial; extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed."}
{"category": "Math", "title": "Polar transform of Spacelike isothermic surfaces in 4-dimensional Lorentzian space forms", "abstract": "The conformal geometry of spacelike surfaces in 4-dimensional Lorentzian space forms has been studied by the authors in a previous paper, where the so-called polar transform was introduced. Here it is shown that this transform preserves spacelike conformal isothermic surfaces. We relate this new transform with the known transforms (Darboux transform and spectral transform) of isothermic surfaces by establishing the permutability theorems."}
{"category": "Math", "title": "Partially specified prior", "abstract": "This note introduces the concept of a partially specified prior distribution for certain post hoc inference problems, where a finite population is sampled once in order to make a decision on the presence or complete absence of some attribute. If the decision is made to accept complete absence, a probability statement may be required that the population is indeed free of the attribute. A partially specified prior is shown to be advantageous in making such statements realistic and useful."}
{"category": "Math", "title": "Simultaneous estimation of the mean and the variance in heteroscedastic Gaussian regression", "abstract": "Let $Y$ be a Gaussian vector of $\\mathbb{R}^n$ of mean $s$ and diagonal covariance matrix $\\Gamma$. Our aim is to estimate both $s$ and the entries $\\sigma_i=\\Gamma_{i,i}$, for $i=1,...,n$, on the basis of the observation of two independent copies of $Y$. Our approach is free of any prior assumption on $s$ but requires that we know some upper bound $\\gamma$ on the ratio $\\max_i\\sigma_i/\\min_i\\sigma_i$. For example, the choice $\\gamma=1$ corresponds to the homoscedastic case where the components of $Y$ are assumed to have common (unknown) variance. In the opposite, the choice $\\gamma>1$ corresponds to the heteroscedastic case where the variances of the components of $Y$ are allowed to vary within some range. Our estimation strategy is based on model selection. We consider a family $\\{S_m\\times\\Sigma_m, m\\in\\mathcal{M}\\}$ of parameter sets where $S_m$ and $\\Sigma_m$ are linear spaces. To each $m\\in\\mathcal{M}$, we associate a pair of estimators $(\\hat{s}_m,\\hat{\\sigma}_m)$ of $(s,\\sigma)$ with values in $S_m\\times\\Sigma_m$. Then we design a model selection procedure in view of selecting some $\\hat{m}$ among $\\mathcal{M}$ in such a way that the Kullback risk of $(\\hat{s}_{\\hat{m}},\\hat{\\sigma}_{\\hat{m}})$ is as close as possible to the minimum of the Kullback risks among the family of estimators $\\{(\\hat{s}_m,\\hat{\\sigma}_m), m\\in\\mathcal{M}\\}$. Then we derive uniform rates of convergence for the estimator $(\\hat{s}_{\\hat{m}},\\hat{\\sigma}_{\\hat{m}})$ over H\\\"{o}lderian balls. Finally, we carry out a simulation study in order to illustrate the performances of our estimators in practice."}
{"category": "Math", "title": "A generalized logarithmic module and duality of Coxeter multiarrangements", "abstract": "We introduce a new definition of a generalized logarithmic module of multiarrangements by uniting those of the logarithmic derivation and the differential modules. This module is realized as a logarithmic derivation module of an arrangement of hyperplanes with a multiplicity consisting of both positive and negative integers. We consider several properties of this module including Saito's criterion and reflexivity. As applications, we prove a shift isomorphism and duality of some Coxeter multiarrangements by using the primitive derivation."}
{"category": "Math", "title": "Statistical tools used to identify scientific misconduct in mobile phone research (REFLEX program)", "abstract": "A severe case of scientific misconduct was discovered in a paper from 2005 allegedly showing harmful effects (DNA breakage) of non-thermal mobile phone electromagnetic field exposure on human and rat cells. Here we describe the way how the fraudulent data were identified. The low variations of the reported biological data are shown to be below theoretical lower limits (multinomial distributions). Another reason for doubts was highly significant non-equal distributions of last digits, a known hint towards data fabrication. The Medical University Vienna, where the research was conducted, was informed about these findings and came to the conclusion that the data in this and another, related paper by the same group were fabricated, and that both papers should be retracted."}
{"category": "Math", "title": "Comparing two samples by penalized logistic regression", "abstract": "Inference based on the penalized density ratio model is proposed and studied. The model under consideration is specified by assuming that the log--likelihood function of two unknown densities is of some parametric form. The model has been extended to cover multiple samples problems while its theoretical properties have been investigated using large sample theory. A main application of the density ratio model is testing whether two, or more, distributions are equal. We extend these results by arguing that the penalized maximum empirical likelihood estimator has less mean square error than that of the ordinary maximum likelihood estimator, especially for small samples. In fact, penalization resolves any existence problems of estimators and a modified Wald type test statistic can be employed for testing equality of the two distributions. A limited simulation study supports further the theory."}
{"category": "Math", "title": "A note on grid transfer operators for multigrid methods", "abstract": "The Local Fourier analysis (LFA) is a classic tool to prove convergence theorems for multigrid methods (MGMs). In particular, we are interested in optimality that is a convergence speed independent of the size of the involved matrices. For elliptic partial differential equations (PDEs), a well known optimality result requires that the sum of the orders of the grid transfer operators is not lower than the order of the PDE to solve. Analogously, when dealing with MGMs for Toeplitz matrices in the literature an optimality condition on the position and on the order of the zeros of the symbols of the grid transfer operators has been found. In this work we show that in the case of elliptic PDEs with constant coefficients, the two different approaches lead to an equivalent condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA, which allows to deal not only with differential problems but also for instance with integral problems. The equivalence of the two approaches gives the possibility of using grid transfer operators with different orders also for MGMs for Toeplitz matrices. We give also a class of grid transfer operators related to the B-spline's refinement equation and we study their geometric properties. This analysis suggests further links between wavelets and multigrid methods. A numerical experimentation confirms the correctness of the proposed analysis."}
{"category": "Math", "title": "On the homotopy type of the Deligne-Mumford compactification", "abstract": "An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne-Mumford compactification. We give an integral refinement: the classifying space of the Charney-Lee category actually has the same homotopy type as the moduli stack of stable curves, and the etale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney-Lee category."}
{"category": "Math", "title": "Spatio-temporal Functional Regression on Paleo-ecological Data", "abstract": "The influence of climate on biodiversity is an important ecological question. Various theories try to link climate change to allelic richness and therefore to predict the impact of global warming on genetic diversity. We model the relationship between genetic diversity in the European beech forests and curves of temperature and precipitation reconstructed from pollen databases. Our model links the genetic measure to the climate curves through a linear functional regression. The interaction in climate variables is assumed to be bilinear. Since the data are georeferenced, our methodology accounts for the spatial dependence among the observations. The practical issues of these extensions are discussed."}
{"category": "Math", "title": "Algebraic versus topological triangulated categories", "abstract": "The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called `algebraic' because they originate from abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the source categories are usually very `non-additive' before passing to homotopy classes of morphisms. Because of their origin I refer to these examples as `topological triangulated categories'. In these extended talk notes I explain some systematic differences between these two kinds of triangulated categories. There are certain properties -- defined entirely in terms of the triangulated structure -- which hold in all algebraic examples, but which fail in some topological ones. These differences are all torsion phenomena, and rationally there is no difference between algebraic and topological triangulated categories."}
{"category": "Math", "title": "Calculus of Variations on Time Scales with Nabla Derivatives", "abstract": "We prove a necessary optimality condition of Euler-Lagrange type for variational problems on time scales involving nabla derivatives of higher-order. The proof is done using a new and more general fundamental lemma of the calculus of variations on time scales."}
{"category": "Math", "title": "A decomposition result for the Haar distribution on the orthogonal group", "abstract": "Let H be a Haar distributed random matrix on the group of pxp real orthogonal matrices. Partition H into four blocks: (1) the (1,1) element, (2)the rest of the first row, (3) the rest of the first column, and (4)the remaining (p-1)x(p-1) matrix. The marginal distribution of (1) is well known. In this paper, we give the conditional distribution of (2) and (3) given (1), and the conditional distribution of (4) given (1), (2), (3). This conditional specification uniquely determines the Haar distribution. The two conditional distributions involve well known probability distributions namely, the uniform distribution on the unit sphere in p-1 dimensional space and the Haar distribution on (p-2)x(p-2) orthogonal matrices. Our results show how to construct the Haar distribution on pxp orthogonal matrices from the Haar distribution on (p-2)x(p-2) orthogonal matrices coupled with the uniform distribution on the unit sphere in p-1 dimensions."}
{"category": "Math", "title": "The Jones polynomial and the planar algebra of alternating links", "abstract": "It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module, we describe an alternating condition on elements of this skein module, show that it is satisfied by the Jones invariant of the single crossing tangles, and prove that it is preserved by appropriately \"alternating\" planar algebra compositions. Hence, this condition is satisfied by the Jones polynomial of all alternating tangles. Finally, in the case of 0-tangles, that is links, our condition is equivalent to simple alternation of the coefficients of the Jones polynomial."}
{"category": "Math", "title": "Quenched large deviation principle for words in a letter sequence", "abstract": "When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the \\emph{annealed} large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the \\emph{quenched} LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an \\emph{algebraic} tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. The previous version (arXiv:0807.2611v2) appeared in Probab. Theory Relat. Fields 148, no. 3/4 (2010), 403--456. Meanwhile, it has turned out that the original proof of the representation of the rate function is flawed when the mean word length is infinite. We add an erratum in which we fix the flaw in the proof. Along the way we derive new representations of the rate function that are interesting in their own right. A key ingredient in the proof is the observation that if the rate function in the annealed large deviation principle is finite at a stationary word process, then the letters in the tail of the long words in this process are typical."}
{"category": "Math", "title": "A Proximal Decomposition Method for Solving Convex Variational Inverse Problems", "abstract": "A broad range of inverse problems can be abstracted into the problem of minimizing the sum of several convex functions in a Hilbert space. We propose a proximal decomposition algorithm for solving this problem with an arbitrary number of nonsmooth functions and establish its convergence. The algorithm fully decomposes the problem in that it involves each function individually via its own proximity operator. A significant improvement over the methods currently in use in the area of inverse problems is that it is not limited to two nonsmooth functions. Numerical applications to signal and image processing problems are demonstrated."}
{"category": "Math", "title": "Character sheaves on disconnected groups, X", "abstract": "We classify the unipotent character sheaves on a fixed connected component of a reductive algebraic group under a mild hypothesis on the characteristic of the ground field."}
{"category": "Math", "title": "Newton polygons of higher order in algebraic number theory", "abstract": "We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by \\O{}. Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and integral bases of number fields."}
{"category": "Math", "title": "Instant Evaluation and Demystification of zeta(n),L(n,chi) that Euler,Ramanujan Missed - II", "abstract": "For Hurwitz zeta function, we obtain power series expression in second variable for its higher order derivatives (with respect to first variable) at non-positive integer arguments and consequently obtain rapidly decreasing series expression for Riemann zeta fuction at positive odd integer arguments. Further, we obtain corresponding results for Dirichlet L-series. We also a unified proof of various classical identities involving Riemann zeta values."}
{"category": "Math", "title": "Level set approach for fractional mean curvature flows", "abstract": "This paper is concerned with the study of a geometric flow whose law involves a singular integral operator. This operator is used to define a non-local mean curvature of a set. Moreover the associated flow appears in two important applications: dislocation dynamics and phase field theory for fractional reaction-diffusion equations. It is defined by using the level set method. The main results of this paper are: on one hand, the proper level set formulation of the geometric flow; on the other hand, stability and comparison results for the geometric equation associated with the flow."}
{"category": "Math", "title": "Divisibility by 2 and 3 of certain Stirling numbers", "abstract": "The numbers e_p(k,n) defined as min(nu_p(S(k,j)j!): j >= n) appear frequently in algebraic topology. Here S(k,j) is the Stirling number of the second kind, and nu_p(-) the exponent of p. The author and Sun proved that if L is sufficiently large, then e_p((p-1)p^L + n -1, n) >= n-1+nu_p([n/p]!). In this paper, we determine the set of integers n for which equality holds in this inequality when p=2 and 3. The condition is roughly that, in the base-p expansion of n, the sum of two consecutive digits must always be less than p."}
{"category": "Math", "title": "Through a Glass Darkly", "abstract": "We consider the question of how mathematicians view themselves and how non-mathematicians view us. What is our role in society? Is it effective? Is it rewarding? How could it be improved? This paper will be part of a forthcoming volume on this circle of questions."}
{"category": "Math", "title": "A categorification of the quantum sl(N)-link polynomials using foams", "abstract": "In this thesis we define and study a categorification of the sl(N)-link polynomial using foams, for N\\geq 3. For N=3 we define the universal sl(3)-link homology, using foams, which depends on three parameters and show that it is functorial, up to scalars, with respect to link cobordisms. Our theory is integral. We show that tensoring it with Q yields a theory which is equivalent to the rational universal Khovanov-Rozansky sl(3)-link homology. For N\\geq 4 we construct a rational theory categorifying the sl(N)-link polynomial using foams. Our theory is functorial, up to scalars, with respect to link cobordisms. To evaluate closed foams we use the Kapustin-Li formula. We show that for any link our homology is isomorphic to the Khovanov-Rozansky homology. We conjecture that the theory is integral and we compute the conjectured integral sl(N)-link homology for the (2,m)-torus links and show that it has torsion of order N."}
{"category": "Math", "title": "Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schr\\\"odinger equation", "abstract": "We consider the focusing mass-critical NLS $iu_t + \\Delta u = - |u|^{4/d} u$ in high dimensions $d \\geq 4$, with initial data $u(0) = u_0$ having finite mass $M(u_0) = \\int_{\\R^d} |u_0(x)|^2 dx < \\infty$. It is well known that this problem admits unique (but not global) strong solutions in the Strichartz class $C^0_{t,\\loc} L^2_x \\cap L^2_{t,\\loc} L^{2d/(d-2)}_x$, and also admits global (but not unique) weak solutions in $L^\\infty_t L^2_x$. In this paper we introduce an intermediate class of solution, which we call a \\emph{semi-Strichartz class solution}, for which one does have global existence and uniqueness in dimensions $d \\geq 4$. In dimensions $d \\geq 5$ and assuming spherical symmetry, we also show the equivalence of the Strichartz class and the strong solution class (and also of the semi-Strichartz class and the semi-strong solution class), thus establishing ``unconditional'' uniqueness results in the strong and semi-strong classes. With these assumptions we also characterise these solutions in terms of the continuity properties of the mass function $t \\mapsto M(u(t))$."}
{"category": "Math", "title": "A cabling formula for the colored Jones polynomial", "abstract": "We prove an explicit cabling formula for the colored Jones polynomial. As an application we prove the volume conjecture for all zero volume knots and links, i.e. all knots and links that are obtained from the unknot by repeated cabling and connected sum."}
{"category": "Math", "title": "q-Euler Numbers and Polynomials Associated with Basic Zeta Functions", "abstract": "In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Finally we woll treat some identities of the q-extension of the euler numbers by using fermionic p-adic q-integration on Z_p."}
{"category": "Math", "title": "The Signature of the Chern Coefficients of Local Rings", "abstract": "This paper considers the following conjecture: If $R$ is an unmixed, equidimensional local ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal $J$ generated by a system of parameters, the Chern coefficient $e_1(J)< 0$ is equivalent to $R$ being non Cohen-Macaulay. The conjecture is established if $R$ is a homomorphic image of a Gorenstein ring, and for all universally catenary integral domains containing fields. Criteria for the detection of Cohen-Macaulayness in equi-generated graded modules are derived."}
{"category": "Math", "title": "On kaleidoscopic pseudo-randomness of finite Euclidean graphs", "abstract": "In this paper we study the kaleidoscopic pseudo-randomness of finite Euclidean graphs using probabilistic methods. Roughly speaking, we show that sufficiently large subsets of d-dimensional vector spaces over finite fields contain every possible finite configurations."}
{"category": "Math", "title": "On the number of orthogonal systems in vector spaces over finite fields", "abstract": "Iosevich and Senger (2008) showed that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors. In this note, we provide a graph theoretic proof of this result."}
{"category": "Math", "title": "Explicit tough Ramsey graphs", "abstract": "A graph G is t-tough if any induced subgraph of it with x > 1 connected components is obtained from G by deleting at least tx vertices. Chvatal conjectured that there exists an absolute constant t_0 so that every t_0-tough graph is pancyclic. This conjecture was disproved by Bauer, van den Heuvel and Schmeichel by constructing a t_0-tough triangle-free graph for every real t_0. For each finite field F_q with q odd, we consider graphs associated to the finite Euclidean plane and the finite upper half plane over F_q. These graphs have received serious attention as they have been shown to be Ramanujan (or asymptotically Ramanujan) for large q. We will show that for infinitely many q, these graphs provide further counterexamples to Chvatal's conjecture. They also provide a good constructive lower bound for the Ramsey number R(3,k)."}
{"category": "Math", "title": "On the volume functional of compact manifolds with boundary with constant scalar curvature", "abstract": "We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and ''small'' hyperbolic and spherical balls in dimensions 3 to 5, the standard space form metrics are indeed saddle points for the volume functional."}
{"category": "Math", "title": "Unitary isomorphism of Fock spaces of bosons and fermions arising from a representation of the Cuntz algebra $\\co{2}$", "abstract": "Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. According to branching laws associated with these descriptions, a certain representation of the Cuntz algebra $\\co{2}$ induces Fock representations ${\\cal H}_{B}$ and ${\\cal H}_{F}$ of bosons and fermions simultaneously. From this, a unitary operator $U$ from ${\\cal H}_{B}$ to ${\\cal H}_{F}$ is obtained. We show the explicit formula of the action of $U$ on the standard basis of ${\\cal H}_{B}$. It is shown that $U$ preserves the particle number of ${\\cal H}_{B}$ and ${\\cal H}_{F}$."}
{"category": "Math", "title": "Optimal lower bounds on the maximal p-negative type of finite metric spaces", "abstract": "This article derives lower bounds on the supremal (strict) p-negative type of finite metric spaces using purely elementary techniques. The bounds depend only on the cardinality and the (scaled) diameter of the underlying finite metric space. Examples show that these lower bounds can easily be best possible under clearly delineated circumstances. We further point out that the entire theory holds (more generally) for finite semi-metric spaces without modification and wherein the lower bounds are always optimal."}
{"category": "Math", "title": "Latin trades in groups defined on planar triangulations", "abstract": "For a finite triangulation of the plane with faces properly coloured white and black, let A be the abelian group constructed by labelling the vertices with commuting indeterminates and adding relations which say that the labels around each white triangle add to the identity. We show that A has free rank exactly two. Let A* be the torsion subgroup of A, and B* the corresponding group for the black triangles. We show that A* and B* have the same order, and conjecture that they are isomorphic. For each spherical latin trade W, we show there is a unique disjoint mate B such that (W,B) is a connected and separated bitrade. The bitrade (W,B) is associated with a two-colourable planar triangulation and we show that W can be embedded in A*, thereby proving a conjecture due to Cavenagh and Drapal. The proof involves constructing a (0,1) presentation matrix whose permanent and determinant agree up to sign. The Smith Normal Form of this matrix determines A*, so there is an efficient algorithm to construct the embedding. Contrasting with the spherical case, for each genus g>0 we construct a latin trade which is not embeddable in any group and another that is embeddable in a cyclic group. We construct a sequence of spherical latin trades which cannot be embedded in any family of abelian groups whose torsion ranks are bounded. Also, we show that any trade that can be embedded in a finitely generated abelian group can be embedded in a finite abelian group. As a corollary, no trade can be embedded in a free abelian group."}
{"category": "Math", "title": "The polynomial representation of the double affine Hecke algebra of type $(C^\\vee_n, C_n)$ for specialized parameters", "abstract": "In this paper, we study the polynomial representation of the double affine Hecke algebra of type $(C^\\vee_n, C_n)$ for specialized parameters. Inductively and combinatorially, we give a linear basis of the representation in terms of linear combinations of non-symmetric Koornwinder polynomials. The basis consists of generalized eigenfunctions with respect to $q$-Dunkl-Cherednik operators $\\hat{Y}_i$, and it gives a way to cancel out poles of non-symmetric Koornwinder polynomials. We examine irreducibility and $Y$-semisimplicity of the representation for the specialized parameters. For some cases, we give a characterization of the subrepresentations by vanishing conditions for Laurent polynomials."}
{"category": "Math", "title": "Penalized estimate of the number of states in Gaussian linear AR with Markov regime", "abstract": "We deal with the estimation of the regime number in a linear Gaussian autoregressive process with a Markov regime (AR-MR). The problem of estimating the number of regimes in this type of series is that of determining the number of states in the hidden Markov chain controlling the process. We propose a method based on penalized maximum likelihood estimation and establish its strong consistency (almost sure) without assuming previous bounds on the number of states."}
{"category": "Math", "title": "Lower bounds for posterior rates with Gaussian process priors", "abstract": "Upper bounds for rates of convergence of posterior distributions associated to Gaussian process priors are obtained by van der Vaart and van Zanten in [14] and expressed in terms of a concentration function involving the Reproducing Kernel Hilbert Space of the Gaussian prior. Here lower-bound counterparts are obtained. As a corollary, we obtain the precise rate of convergence of posteriors for Gaussian priors in various settings. Additionally, we extend the upper-bound results of [14] about Riemann-Liouville priors to a continuous family of parameters."}
{"category": "Math", "title": "Nonvanishing vector fields on orbifolds", "abstract": "We introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold $Q$. Motivated by the inertia orbifold, the space of multi-sectors, and the generalized orbifold Euler characteristics, we construct for each finitely generated group $\\Gamma$ an orbifold called the space of $\\Gamma$-sectors of $Q$. The obstruction occurs as the Euler-Satake characteristics of the $\\Gamma$-sectors for an appropriate choice of $\\Gamma$; in the case that $Q$ is oriented, this obstruction is expressed as a cohomology class, the $\\Gamma$-Euler-Satake class. We also acquire a complete obstruction in the case that $Q$ is compact with boundary and in the case that $Q$ is an open suborbifold of a closed orbifold."}
{"category": "Math", "title": "On asymptotics of exchangeable coalescents with multiple collisions", "abstract": "We study the number of collisions $X_n$ of an exchangeable coalescent with multiple collisions ($\\Lambda$-coalescent) which starts with $n$ particles and is driven by rates determined by a finite characteristic measure $\\nu({\\rm d}x)=x^{-2}\\Lambda({\\rm d}x)$. Via a coupling technique we derive limiting laws of $X_n$, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The possible limiting laws of $X_n$ include normal, stable with index $1\\le\\alpha<2$ and Mittag-Leffler distributions. The results apply, in particular, to the case when $\\nu$ is a beta$(a-2,b)$ distribution with parameters $a>2$ and $b>0$. The approach taken allows to derive asymptotics of three other functionals of the coalescent, the absorption time, the length of an external branch chosen at random from the $n$ external branches, and the number of collision events that occur before the randomly selected external branch coalesces with one of its neighbours."}
{"category": "Math", "title": "Distinguished representations and exceptional poles of the Asai-L-function", "abstract": "In this article, we proove that it is equivalent for a generic irreducible representation of GL(n,K), with K a p-adic field, to be distinguished, and for its Rankin-Selberg Asai L-function to have an exceptional pole at zero. This extends a criterion of Kable claiming that a discrete series representation is distinguished if and only if its Asai L-function has a pole at zero. As an application we compute by local methods Asai L-functions of ordinary representations of GL(2,K), in particular we give a formula for Asai L-functions of principal series representations of GL(2,K)."}
{"category": "Math", "title": "Annealed vs Quenched Critical Points for a Random Walk Pinning Model", "abstract": "We study a random walk pinning model, where conditioned on a simple random walk Y on Z^d acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with Hamiltonian -L_t(X,Y), where L_t(X,Y) is the collision local time between X and Y up to time t. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with Brownian noise, and the directed polymer model. It falls in the same framework as the pinning and copolymer models, and exhibits a localization-delocalization transition as the inverse temperature \\beta varies. We show that in dimensions d=1,2, the annealed and quenched critical values of \\beta are both 0, while in dimensions d\\geq 4, the quenched critical value of \\beta is strictly larger than the annealed critical value (which is positive). This implies the existence of certain intermediate regimes for the parabolic Anderson model with Brownian noise and the directed polymer model. For d\\geq 5, the same result has recently been established by Birkner, Greven and den Hollander via a quenched large deviation principle. Our proof is based on a fractional moment method used recently by Derrida, Giacomin, Lacoin and Toninelli to establish the non-coincidence of annealed and quenched critical points for the pinning model in the disorder-relevant regime. The critical case d=3 remains open."}
{"category": "Math", "title": "ABC likelihood-freee methods for model choice in Gibbs random fields", "abstract": "Gibbs random fields (GRF) are polymorphous statistical models that can be used to analyse different types of dependence, in particular for spatially correlated data. However, when those models are faced with the challenge of selecting a dependence structure from many, the use of standard model choice methods is hampered by the unavailability of the normalising constant in the Gibbs likelihood. In particular, from a Bayesian perspective, the computation of the posterior probabilities of the models under competition requires special likelihood-free simulation techniques like the Approximate Bayesian Computation (ABC) algorithm that is intensively used in population genetics. We show in this paper how to implement an ABC algorithm geared towards model choice in the general setting of Gibbs random fields, demonstrating in particular that there exists a sufficient statistic across models. The accuracy of the approximation to the posterior probabilities can be further improved by importance sampling on the distribution of the models. The practical aspects of the method are detailed through two applications, the test of an iid Bernoulli model versus a first-order Markov chain, and the choice of a folding structure for two proteins."}
{"category": "Math", "title": "Minimax state estimation for linear discrete-time differential-algebraic equations", "abstract": "This paper presents a state estimation approach for an uncertain linear equation with a non-invertible operator in Hilbert space. The approach addresses linear equations with uncertain deterministic input and noise in the measurements, which belong to a given convex closed bounded set. A new notion of a minimax observable subspace is introduced. By means of the presented approach, new equations describing the dynamics of a minimax recursive estimator for discrete-time non-causal differential-algebraic equations (DAEs) are presented. For the case of regular DAEs it is proved that the estimator's equation coincides with the equation describing the seminal Kalman filter. The properties of the estimator are illustrated by a numerical example."}
{"category": "Math", "title": "Codistances of 3-spherical buildings", "abstract": "We show that a 3-spherical building in which each rank 2 residue is connected far away from a chamber, and each rank 3 residue is simply 2-connected far away from a chamber, admits a twinning (i.e., is one half of a twin building) as soon as it admits a codistance, i.e., a twinning with a single chamber."}
{"category": "Math", "title": "On the connectedness of Deligne-Lusztig varieties", "abstract": "We give a criterion which determines when a union of one-dimensional Deligne-Lusztig varieties has a connected closure. We also obtain a new, short proof of the connectedness criterion for Deligne-Lusztig varieties due to Lusztig."}
{"category": "Math", "title": "Weak-strong uniqueness for the isentropic compressible Navier-Stokes system", "abstract": "We prove weak-strong uniqueness results for the isentropic compressible Navier-Stokes system on the torus. In other words, we give conditions on a strong solution so that it is unique in a class of weak solutions. Known weak-strong uniqueness results are improved. Classical uniqueness results for this equation follow naturally."}
{"category": "Math", "title": "Zariski decomposition of b-divisors", "abstract": "Based on a recent work by Thomas Bauer reproving the existence of Zariski decompositions for surfaces, we construct a b-divisorial analogue of Zariski decomposition in all dimensions."}
{"category": "Math", "title": "Boundaries for Banach spaces determine weak compactness", "abstract": "A boundary for a Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. Godefroy's Boundary Problem asks whether nevertheless both topologies have the same bounded compact sets. This paper contains the answer in the positive."}
{"category": "Math", "title": "Phase Transition on The Degree Sequence of a Mixed Random Graph Process", "abstract": "This paper focuses on the problem of the degree sequence for a mixed random graph process which continuously combines the {\\it classical} model and the BA model. Note that the number of step added edges for the mixed model is random and non-uniformly bounded. By developing a comparing argument, phase transition on the degree distributions of the mixed model is revealed: while the {\\it pure} classical model possesses a {\\it exponential} degree sequence, the {\\it pure} BA model and the mixed model possess {\\it power law} degree sequences. As an application of the methodology, phase transition on the degree sequence of {\\it another} mixed model with {\\it hard copying} is also studied, especially, in the power law region, the inverse power can take any value greater than 1."}
{"category": "Math", "title": "Play Ground for Victor's Magic Squares", "abstract": "This article presents a new development of magic squares with a simple set up."}
{"category": "Math", "title": "Permutation classes of every growth rate above 2.48188", "abstract": "We prove that there are permutation classes (hereditary properties of permutations) of every growth rate (Stanley-Wilf limit) at least \\lambda \\approx 2.48187, the unique real root of x^5-2x^4-2x^2-2x-1, thereby establishing a conjecture of Albert and Linton."}
{"category": "Math", "title": "The Degree Sequence of a Scale-Free Random Graph Process with Hard Copying", "abstract": "In this paper we consider a simple model of random graph process with {\\it hard} copying as follows: At each time step $t$, with probability $0<\\alpha\\leq 1$ a new vertex $v_t$ is added and $m$ edges incident with $v_t$ are added in the manner of {\\it preferential attachment}; or with probability $1-\\alpha$ an existing vertex is copied uniformly at random. In this way, while a vertex with large degree is copied, the number of added edges is its degree and thus the number of added edges is not upper bounded. We prove that, in the case of $\\alpha$ being large enough, the model possesses a mean degree sequence as $ d_{k}\\sim Ck^{-(1+2\\alpha)}$, where $d_k$ is the limit mean proportion of vertices of degree $k$."}
{"category": "Math", "title": "Piecewise linear parametrization of canonical bases", "abstract": "We extend the known piecewise linear parametrization of the canonical basis of the plus part of an enveloping algebra of type ADE to the nonsimplylaced case."}
{"category": "Math", "title": "Approximation properties and entropy estimates for crossed products by actions of amenable discrete quantum groups", "abstract": "We construct explicit approximating nets for crossed products of C*-algebras by actions of discrete quantum groups. This implies that C*-algebraic approximation properties such as nuclearity, exactness or completely bounded approximation property are preserved by taking crossed products by actions of amenable discrete quantum groups. We also show that the noncommutative topological entropy of a transformation commuting with the quantum group action does not change when we pass to the canonical extension to the crossed product. Both these results are extended to twisted crossed products via a stabilisation trick."}
{"category": "Math", "title": "Method of Moments Estimation of Ornstein-Uhlenbeck Processes Driven by General L\\'{e}vy Process", "abstract": "Ornstein-Uhlenbeck processes driven by general L\\'{e}vy process are considered in this paper. We derive strongly consistent estimators for the moments of the underlying L\\'{e}vy process and for the mean reverting parameter of the Ornstein-Uhlenbeck process. Moreover, we prove that the estimators are asymptotically normal. Finally, we test the empirical performance of our estimators in a simulation study and we fit the model to real data."}
{"category": "Math", "title": "Uneven Splitting of Ham Sandwiches", "abstract": "Let m_1,...,m_n be continuous probability measures on R^n and a_1,...,a_n in [0,1]. When does there exist an oriented hyperplane H such that the positive half-space H^+ has m_i(H^+)=a_i for all i in [n]? It is well known that such a hyperplane does not exist in general. The famous ham sandwich theorem states that if a_i=1/2 for all i, then such a hyperplane always exists. In this paper we give sufficient criteria for the existence of H for general a_i in [0,1]. Let f_1,...,f_n:S^{n-1}->R^n denote auxiliary functions with the property that for all i the unique hyperplane H_i with normal v that contains the point f_i(v) has m_i(H_i^+)=a_i. Our main result is that if Im(f_1),...,Im(f_n) are bounded and can be separated by hyperplanes, then there exists a hyperplane H with m_i(H^+)=a_i for all i. This gives rise to several corollaries, for instance if the supports of m_1,...,m_n are bounded and can be separated by hyperplanes, then H exists for any choice of a_1,...,a_n in [0,1]. We also obtain results that can be applied if the supports of m_1,...,m_n overlap."}
{"category": "Math", "title": "Stable flatness of nonarchimedean hyperenveloping algebras", "abstract": "Let L be a p-adic local field and g a finite dimensional Lie algebra over L. We show that its hyperenveloping algebra F(g) is a stably flat completion of its universal enveloping algebra. As a consequence the relative cohomology for the locally convex algebra F(g) coincides with the underlying Lie algebra cohomology. Final version. Some minor items corrected. Appeared in Journal of Algebra (2010)."}
{"category": "Math", "title": "On a Furstenberg-Katznelson-Weiss type theorem over finite fields", "abstract": "Using Fourier analysis, Covert, Hart, Iosevich and Uriarte-Tuero (2008) showed that if the cardinality of a subset of the 2-dimensional vector space over a finite field with q elements is >= rq^2, with q^{-1/2} << r <= 1 then it contains an isometric copy of >= crq^3 triangles. In this note, we give a graph theoretic proof of this result."}
{"category": "Math", "title": "Thin position and planar surfaces for graphs in the 3-sphere", "abstract": "We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful planar surface."}
{"category": "Math", "title": "Bounding the stable genera of Heegaard splittings from below", "abstract": "We describe for each postive integer $k$ a 3-manifold with Heegaard surfaces of genus $2k$ and $2k-1$ such that any common stabilization of these two surfaces has genus at least $3k-1$. We also show that for every positive $n$, there is a 3-manifold that has $n$ pairwise non-isotopic Heegaard splittings of the same genus all of which are stabilized."}
{"category": "Math", "title": "Heegaard surfaces and the distance of amalgamation", "abstract": "Let $M_1$ and $M_2$ be orientable irreducible 3--manifolds with connected boundary and suppose $\\partial M_1\\cong\\partial M_2$. Let $M$ be a closed 3--manifold obtained by gluing $M_1$ to $M_2$ along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then $M$ is not homeomorphic to $S^3$ and all small-genus Heegaard splittings of $M$ are standard in a certain sense. In particular, $g(M)=g(M_1)+g(M_2)-g(\\partial M_i)$, where $g(M)$ denotes the Heegaard genus of $M$. This theorem is also true for certain manifolds with multiple boundary components."}
{"category": "Math", "title": "Algorithms and Classification in Groups of Piecewise-Linear Homeomorphisms", "abstract": "This manuscript represents the author's PhD dissertation thesis.The first part studies decision problems in Thompson's groups F,T,V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson's group F and suitable larger groups of piecewise-linear homeomorphisms of the unit interval. We determine algorithms to compute roots and centralizers in these groups and to detect periodic points and their behavior by looking at a particular diagram associated to an element. In the second part, we describe the structure of subgroups of the group of all homeomorphisms of the unit circle, with the additional requirement that they contain no non-abelian free subgroup. It is shown that in this setting the rotation number map is a group homomorphism. We give a classification of such subgroups as subgroups of certain wreath products and we show that such subgroups can exist by building examples. Similar techniques are then used to compute centralizers in these groups."}
{"category": "Math", "title": "Polynomial functors and trees", "abstract": "We explore the relationship between polynomial functors and (rooted) trees. In the first part we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category $\\Omega$ of Moerdijk and Weiss; its main properties are described in terms of some factorisation systems. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. In the second part we describe polynomial endofunctors and monads as structures built from trees, characterising the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a certain projectivity condition, which serves also to characterise polynomial endofunctors and monads among (coloured) collections and operads."}
{"category": "Math", "title": "Generalized Long-Moody representations of braid groups", "abstract": "Long and Moody gave a method of constructing representations of the braid group B_n. We discuss some ways to generalize their construction. One of these gives representations of subgroups of B_n, including the Gassner representation of the pure braid group as a special case. Another gives representations of the Hecke algebra."}
{"category": "Math", "title": "A K-theoretic L2-index theorem for families", "abstract": "This paper has been withdrawn due to a error in Theorem 3.1."}
{"category": "Math", "title": "Jumping sequences", "abstract": "An integer sequence a(n) is called a jump sequence if a(1)=1 and 1<=a(n)<n for n>=2. Such a sequence has the property that a^k(n)=a(a(...(a(n))...)) goes to 1 in finitely many steps and we call the pattern (n,a(n),a^2(n),...,a^k(n)=1) a jumping pattern from n down to 1. In this paper we look at jumping sequences which are weight minimizing with respect to various weight functions (where a weight w(i,j) is given to each jump from j down to i). Our main result is to show that if w(i,j)=(i+j)/i^2 then the cost minimizing jump sequence has the property that the number m satisfies m=a^q(p) for arbitrary q and some p (depending on q) if and only if m is a Pell number."}
{"category": "Math", "title": "Linearly repetitive Delone systems have a finite number of non periodic Delone system factors", "abstract": "We prove linearly repetitive Delone systems have finitely many Delone system factors up to conjugacy. This result is also applicable to linearly repetitive tiling systems."}
{"category": "Math", "title": "Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation", "abstract": "We consider the energy-critical semilinear focusing wave equation in dimension $N=3,4,5$. An explicit solution $W$ of this equation is known. By the work of C. Kenig and F. Merle, any solution of initial condition $(u_0,u_1)$ such that $E(u_0,u_1)<E(W,0)$ and $\\|\\nabla u_0\\|_{L^2}<\\|\\nabla W\\|_{L^2}$ is defined globally and has finite $L^{\\frac{2(N+1)}{N-2}}_{t,x}$-norm, which implies that it scatters. In this note, we show that the supremum of the $L^{\\frac{2(N+1)}{N-2}}_{t,x}$-norm taken on all scattering solutions at a certain level of energy below $E(W,0)$ blows-up logarithmically as this level approaches the critical value $E(W,0)$. We also give a similar result in the case of the radial energy-critical focusing semilinear Schr\\\"odinger equation. The proofs rely on the compactness argument of C. Kenig and F. Merle, on a classification result, due to the authors, at the energy level $E(W,0)$, and on the analysis of the linearized equation around $W$."}
{"category": "Math", "title": "Depth Two Hopf Subalgebras of Semisimple Hopf algebras", "abstract": "Let H be a finite dimensional semisimple Hopf algebra over an algebraically closed field of characteristic zero. In this note we give a short proof of the fact that a Hopf subalgebra of H is a depth two subalgebra if and only if it is normal Hopf subalgebra."}
{"category": "Math", "title": "A Family of Multistep Methods with Zero Phase-Lag and Derivatives for the Numerical Integration of Oscillatory ODEs", "abstract": "In this paper we develop a family of three 8-step methods, optimized for the numerical integration of oscillatory ordinary differential equations. We have nullified the phase-lag of the methods and the first r derivatives, where r=1,2,3. We show that with this new technique, the method gains efficiency with each derivative of the phase-lag nullified. This is the case for the integration of both the Schrodinger equation and the N-body problem. A local truncation error analysis is performed, which, for the case of the Schrodinger equation, also shows the connection of the error and the energy, revealing the importance of the zero phase-lag derivatives. Also the stability analysis shows that the methods with more derivatives vanished, have a bigger interval of periodicity."}
{"category": "Math", "title": "Comportement asymptotique des hauteurs des points de Heegner", "abstract": "The leading order term for the average, over quadratic discriminants satisfying the so-called Heegner condition, of the Neron-Tate height of Heegner points on a rational elliptic curve E has been determined in [12]. In addition, the second order term has been conjectured. In this paper, we prove that this conjectured second order term is the right one; this yields a power saving in the remainder term. Cancellations of Fourier coefficients of GL(2)-cusp forms in arithmetic progressions lie in the core of the proof."}
{"category": "Math", "title": "Learning from Experts: A Survey", "abstract": "The survey is concerned with the issue of information transmission from experts to non-experts. Two main approaches to the use of experts can be traced. According to the game-theoretic approach expertise is a case of asymmetric information between the expert, who is the better informed agent, and the non-expert, who is either a decision-maker or an evaluator of the expert's performance. According to the Bayesian decision-theoretic approach the expert is the agent who announces his probabilistic opinion, and the non-expert has to incorporate that opinion into his beliefs in a consistent way, despite his poor understanding of the expert's substantive knowledge. The two approaches ground the relationships between experts and non-experts on so different premises that their results are very poorly connected."}
{"category": "Math", "title": "Codimension one subgroups and boundaries of hyperbolic groups", "abstract": "We construct hyperbolic groups with the following properties: The boundary of the group has big dimension, it is separated by a Cantor set and the group does not split. This shows that Bowditch's theorem that characterizes splittings of hyperbolic groups over 2-ended groups in terms of the boundary can not be extended to splittings over more complicated subgroups."}
{"category": "Math", "title": "On Blocking Numbers of Surfaces", "abstract": "The blocking number of a manifold is the minimal number of points needed to block out lights between any two given points in the manifold. It has been conjectured that if the blocking number of a manifold is finite, then the manifold must be flat. In this paper we prove that this is true for 2-dimensional manifolds with non-trivial fundamental groups."}
{"category": "Math", "title": "Phase Lag Sensitivity Analysis for Numerical Integration", "abstract": "In the field of numerical integration, methods specially tuned on oscillating functions, are of great practical importance. Such methods are needed in various branches of natural sciences, particularly in physics, since a lot of physical phenomena exhibit a pronounced oscillatory behavior. Among others, probably the most important tool used to construct efficient methods for oscillatory problems is the exponential (trigonometric) fitting. The basic characteristic of these methods is that their phase lag vanishes at a predefined frequency. In this work, we introduce a new tool which improves the behavior of exponentially fitted numerical methods. The new technique is based on the vanishing of the first derivatives of the phase lag function at the fitted frequency. It is proved in the text that these methods present improved characteristics in oscillatory problems."}
{"category": "Math", "title": "On the commutant of $C(X)$ in $C^*$-crossed products by $\\mathbb{Z}$ and their representations", "abstract": "For the $C^*$-crossed product $C^*(\\Sigma)$ associated with an arbitrary topological dynamical system $\\Sigma = (X, \\sigma)$, we provide a detailed analysis of the commutant, in $C^* (\\Sigma)$, of $C(X)$ and the commutant of the image of $C(X)$ under an arbitrary Hilbert space representation $\\tilde{\\pi}$ of $C^* (\\Sigma)$. In particular, we give a concrete description of these commutants, and also determine their spectra. We show that, regardless of the system $\\Sigma$, the commutant of $C(X)$ has non-zero intersection with every non-zero, not necessarily closed or self-adjoint, ideal of $C^* (\\Sigma)$. We also show that the corresponding statement holds true for the commutant of $\\tilde{\\pi}(C(X))$ under the assumption that a certain family of pure states of $\\tilde{\\pi}(C^* (\\Sigma))$ is total. Furthermore we establish that, if $C(X) \\subsetneq C(X)'$, there exist both a $C^*$-subalgebra properly between $C(X)$ and $C(X)'$ which has the aforementioned intersection property, and such a $C^*$-subalgebra which does not have this property. We also discuss existence of a projection of norm one from $C^*(\\Sigma)$ onto the commutant of $C(X)$."}
{"category": "Math", "title": "A New Family of Multistep Methods with Improved Phase Lag Characteristics for the Integration of Orbital Problems", "abstract": "In this work we introduce a new family of ten-step linear multistep methods for the integration of orbital problems. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. The efficiency of the new family of methods is proved via error analysis and numerical applications."}
{"category": "Math", "title": "Limit theorems for some adaptive MCMC algorithms with subgeometric kernels", "abstract": "This paper deals with the ergodicity and the existence of a strong law of large numbers for adaptive Markov Chain Monte Carlo. We show that a diminishing adaptation assumption together with a drift condition for positive recurrence is enough to imply ergodicity. Strengthening the drift condition to a polynomial drift condition yields a strong law of large numbers for possibly unbounded functions. These results broaden considerably the class of adaptive MCMC algorithms for which rigorous analysis is now possible. As an example, we give a detailed analysis of the Adaptive Metropolis Algorithm of Haario et al. (2001) when the target distribution is sub-exponential in the tails."}
{"category": "Math", "title": "An estimate from below for the Buffon needle probability of the four-corner Cantor set", "abstract": "Let $\\Cant_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $\\K_n = \\Cant_n \\times \\Cant_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $\\K_n$ is essentially the average length of the projections of $\\K_n$, also known as the Favard length of $\\K_n$. A classical theorem of Besicovitch implies that the Favard length of $\\K_n$ tends to zero. It is still an open problem to determine its exact rate of decay. Until recently, the only explicit upper bound was $\\exp(- c\\log_* n)$, due to Peres and Solomyak. ($\\log_* n$ is the number of times one needs to take log to obtain a number less than 1 starting from $n$). In Nazarov-Peres-Volberg paper (arxiv:math 0801.2942) the power estimate from above was obtained. The exponent in this paper was less than 1/6 but could have been slightly improved. On the other hand, a simple estimate shows that from below we have the estimate $\\frac{c}{n}$. Here we apply the idea from papers of Nets Katz (MRL (1996), 527-536) and Bateman-Katz (arxiv:math/0609187v1 2006) to show that the estimate from below can be in fact improved to $c \\frac{\\log n}{n}$. This is in drastic difference from the case of {\\em random} Cantor sets studied by Peres and Solomyak in Pacific J. Math. 204 (2002), 473-496."}
{"category": "Math", "title": "Classical Theory of Fourier Series:Demystified and Generalised", "abstract": "For a Riemann integrable function on an interval and for a point therein,we define 'Fourier series at the point on the interval' and bring out how and when the function element becomes expressible as Fourier series.In this process,we also generalise the theory by bringing in such concepts as finite Fourier series,right/left hand Fourier series.We also sum up subseries corresponding to terms in an arithmetic progression,of the basic Fourier series."}
{"category": "Math", "title": "Gorenstein Modules of Finite Length", "abstract": "In Commutative Algebra structure results on minimal free resolutions of Gorenstein modules are of classical interest. We define Gorenstein modules of finite length over the weighted polynomial ring via symmetric matrices in divided powers. We show that their graded minimal free resolution is selfdual in a strong sense. Applications include a proof of the dependence of the monoid of Betti tables of Cohen-Macaulay modules on the characteristic of the base field. Moreover we give a new proof of the failure of the generalization of Green's Conjecture to characteristic 2 in the case of general curves of genus $2^n -1$."}
{"category": "Math", "title": "The stochastic approximation method for the estimation of a multivariate probability density", "abstract": "We apply the stochastic approximation method to construct a large class of recursive kernel estimators of a probability density, including the one introduced by Hall and Patil (1994). We study the properties of these estimators and compare them with Rosenblatt's nonrecursive estimator. It turns out that, for pointwise estimation, it is preferable to use the nonrecursive Rosenblatt's kernel estimator rather than any recursive estimator. A contrario, for estimation by confidence intervals, it is better to use a recursive estimator rather than Rosenblatt's estimator."}
{"category": "Math", "title": "Remarks on cycle classes of sections of the arithmetic fundamental group", "abstract": "Given a smooth and separated K(pi,1) variety X over a field k, we associate a \"cycle class\" in etale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of X to the absolute Galois group of k. We discuss the algebraicity of this class in the case of curves over p-adic fields, and deduce in particular a new proof of Stix's theorem according to which the index of a curve X over a p-adic field k must be a power of p as soon as the natural map from the arithmetic fundamental group of X to the absolute Galois group of k admits a section. Finally, an etale adaptation of Beilinson's geometrization of the pronilpotent completion of the topological fundamental group allows us to lift this cycle class in suitable cohomology groups."}
{"category": "Math", "title": "Local cohomology: Associated primes, artinianness and asymptotic behaviour", "abstract": "Let $R$ be a noetherian ring, $\\fa$ an ideal of $R$, $M$ an $R$--module and $n$ a non-negative integer. In this paper we first will study the finiteness properties of the kernel and the cokernel of the natural map $f:\\Ext^n_{R}(R/\\fa,M)\\lo \\Hom_{R}(R/\\fa,\\lc^{n}_{\\fa}(M))$. Then we will get some corollaries about the associated primes and artinianness of local cohomology modules. Finally we will study the asymptotic behaviour of the kernel and the cokernel of this natural map in the graded case."}
{"category": "Math", "title": "On the Riesz and Baez-Duarte criteria for the Riemann Hypothesis", "abstract": "We investigate the relation between the Riesz and the B{\\'a}ez-Duarte criterion for the Riemann Hypothesis. In particular we present the relation between the function $R(x)$ appearing in the Riesz criterion and the sequence $c_k$ appearing in the B{\\'a}ez-Duarte formulation. It is shown that $R(x)$ can be expressed by $c_k$, and, vice versa, the sequence $c_k$ can be obtained from the values of $R(x)$ at integer arguments. Also, we give some relations involving $c_k$ and $R(x)$, and value of the alternating sum of $c_k$."}
{"category": "Math", "title": "On the finiteness theorem for rational maps on a variety of general type", "abstract": "The dominant rational maps of finite degree from a fixed variety to varieties of general type, up to birational isomorphisms, form a finite set. This has been known as the Iitaka-Severi conjecture, and is nowdays an established result, in virtue of some recent advances in the theory of pluricanonical maps. We study the question of finding some effective estimate for the finite number of maps, and to this aim we provide some update and refinement of the classical treatment of the subject."}
{"category": "Math", "title": "The Littlewood--Paley--Rubio de Francia property of a Banach space for the case of equal intervals", "abstract": "Let $X$ be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of $X$-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents $p$ greater or equal than 2 if and only if the space $X$ is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series."}
{"category": "Math", "title": "Weak Expectations and the Injective Envelope", "abstract": "Given a unital C*-subalgebra of B(H), we study the set of all possible images of its injective envelope that are contained in B(H) and their position relative to the double commutant of the algebra in order to obtain more information about the existence or non-existence of weak expectations. We study the subset of B(H) that is the intersection of all possible images of the injective envelope and show that it is simultaneously a reflexive cover and a new type of order completion of the algebra."}
{"category": "Math", "title": "Stochastic domination for the last passage percolation model", "abstract": "A competition model on $\\mathbb{Z}_+^{2}$ governed by directed last passage percolation is considered. A stochastic domination argument between subtrees of the last passage percolation is put forward."}
{"category": "Math", "title": "Classification of 64-element finite semifields", "abstract": "A finite semifield $D$ is a finite nonassociative ring with identity such that the set $D^*=D\\setminus\\{0\\}$ is closed under the product. In this paper we obtain a computer-assisted description of all 64-element finite semifields, which completes the classification of finite semifields of order 125 or less."}
{"category": "Math", "title": "Some conjectures on addition and multiplication of complex (real) numbers", "abstract": "We discuss conjectures related to the following two conjectures: (1) for each complex numbers x_1,...,x_n there exist rationals y_1,...,y_n \\in [-2^{n-1},2^{n-1}] such that \\forall i \\in {1,...,n} (x_i=1 \\Rightarrow y_i=1) \\forall i,j,k \\in {1,...,n} (x_i+x_j=x_k \\Rightarrow y_i+y_j=y_k) (2) for each complex (real) numbers x_1,...,x_n there exist complex (real) numbers y_1,...,y_n such that \\forall i \\in {1,...,n} |y_i| \\leq 2^{2^{n-2}} \\forall i \\in {1,...,n} (x_i=1 \\Rightarrow y_i=1) \\forall i,j,k \\in {1,...,n} (x_i+x_j=x_k \\Rightarrow y_i+y_j=y_k) \\forall i,j,k \\in {1,...,n} (x_i \\cdot x_j=x_k \\Rightarrow y_i \\cdot y_j=y_k)"}
{"category": "Math", "title": "Super Linear Algebra", "abstract": "In this book, the authors introduce the notion of Super linear algebra and super vector spaces using the definition of super matrices defined by Horst (1963). This book expects the readers to be well-versed in linear algebra. Many theorems on super linear algebra and its properties are proved. Some theorems are left as exercises for the reader. These new class of super linear algebras which can be thought of as a set of linear algebras, following a stipulated condition, will find applications in several fields using computers. The authors feel that such a paradigm shift is essential in this computerized world. Some other structures ought to replace linear algebras which are over a century old. Super linear algebras that use super matrices can store data not only in a block but in multiple blocks so it is certainly more powerful than the usual matrices. This book has 3 chapters. Chapter one introduces the notion of super vector spaces and enumerates a number of properties. Chapter two defines the notion of super linear algebra, super inner product spaces and super bilinear forms. Several interesting properties are derived. The main application of these new structures in Markov chains and Leontief economic models are also given in this chapter. The final chapter suggests 161 problems mainly to make the reader understand this new concept and apply them."}
{"category": "Math", "title": "On the first Stiefel-Whitney class of moduli space for real rational stable curves in the projective space", "abstract": "Moduli space of genus zero stable maps to the projective three-space naturally carries a real structure such that the fixed locus is a moduli space for real rational spatial curves with real marked points. The latter is a normal projective real variety. The singular locus being in codimension at least two, a first Stiefel-Whitney class is well defined. In this paper, we determine a representative for the first Stiefel-Whitney class of such real space when the evaluation map is generically finite. This can be done by means of Poincar\\'e duals of boundary divisors."}
{"category": "Math", "title": "On the Global Attractor of Delay Differential Equations with Unimodal Feedback", "abstract": "We give bounds for the global attractor of the delay differential equation $x'(t) =-\\mu x(t)+f(x(t-\\tau))$, where $f$ is unimodal and has negative Schwarzian derivative. If $f$ and $\\mu$ satisfy certain condition, then, regardless of the delay, all solutions enter the domain where f is monotone decreasing and the powerful results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions. In this situation we determine the sharpest interval that contains the global attractor for any delay. In the absence of that condition, improving earlier results, we show that if the d5A5Aelay is sufficiently small, then all solution enter the domain where $f'$ is negative. Our theorems then are illustrated by numerical examples using Nicholson's blowflies equation and the Mackey-Glass equation."}
{"category": "Math", "title": "The Spectrum and the Spectral Type of the Off-Diagonal Fibonacci Operator", "abstract": "We consider Jacobi matrices with zero diagonal and off-diagonals given by elements of the hull of the Fibonacci sequence and show that the spectrum has zero Lebesgue measure and all spectral measures are purely singular continuous. In addition, if the two hopping parameters are distinct but sufficiently close to each other, we show that the spectrum is a dynamically defined Cantor set, which has a variety of consequences for its local and global fractal dimension."}
{"category": "Math", "title": "Osserman and conformally Osserman manifolds with warped and twisted product structure", "abstract": "We characterize Osserman and conformally Osserman Riemannian manifolds with the local structure of a warped product. By means of this approach we analyze the twisted product structure and obtain, as a consequence, that the only Osserman manifolds which can be written as a twisted product are those of constant curvature."}
{"category": "Math", "title": "Minimal atlases of closed contact manifolds", "abstract": "We study the minimal number C(M,\\xi) of contact charts that one needs to cover a closed connected contact manifold (M,\\xi). Our basic result is C(M,\\xi) \\le \\dim M + 1. We compute C(M,\\xi) for all closed connected contact 3-manifolds: C (M,\\xi) = 2 if M = S^3 and \\xi is tight, 3 if M = S^3 and \\xi is overtwisted or if M = #_k (S^2 \\times S^1), 4 otherwise. We also show that on every sphere S^{2n+1} there exists a contact structure with C(S^{2n+1},\\xi) \\ge 3."}
{"category": "Math", "title": "Closed String TCFT for Hermitian Calabi-Yau Elliptic Spaces", "abstract": "We describe an explicit action of the prop of the chains on the moduli space of Riemann surfaces on the Hochschild complex of a Calabi-Yau elliptic space. One example of such an elliptic space extends the known string topology operations, for all compact simply-connected manifolds, to a collection indexed by the de Rham currents on the moduli space. Another example pertains to the B-model at all genera."}
{"category": "Math", "title": "Deformations of Symmetric Simple Modular Lie (Super)Algebras", "abstract": "We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank $<9$, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated."}
{"category": "Math", "title": "Fermionic realization of toroidal Lie algebras of classical types", "abstract": "We use fermionic operators to construct toroidal Lie algebras of classical types, including in particular that of symplectic affine algebras, which is first realized by fermions."}
{"category": "Math", "title": "On 3-manifolds with locally-standard (Z_2)^3-actions", "abstract": "As a generalization of Davis-Januszkiewicz theory, there is an essential link between locally standard $(\\Z_2)^n$-actions (or $T^n$-actions) actions and nice manifolds with corners, so that a class of nicely behaved equivariant cut-and-paste operations on locally standard actions can be carried out in step on nice manifolds with corners. Based upon this, we investigate what kinds of closed manifolds admit locally standard $(\\Z_2)^n$-actions; especially for the 3-dimensional case. Suppose $M$ is an orientable closed connected 3-manifold. When $H_1(M;\\Z_2)=0$, it is shown that $M$ admits a locally standard $(\\Z_2)^3$-action if and only if $M$ is homeomorphic to a connected sum of 8 copies of some $\\Z_2$-homology sphere $N$, and if further assuming $M$ is irreducible, then $M$ must be homeomorphic to $S^3$. In addition, the argument is extended to rational homology 3-sphere $M$ with $H_1(M;\\Z_2) \\cong \\Z_2$ and an additional assumption that the $(\\Z_2)^3$-action has a fixed point."}
{"category": "Math", "title": "L'Algebre Tropicale Comme Algebre De la Caracteristique 1 : Algebre Lineaire Sur Les Semi-Corps Idempotents", "abstract": "We define a formal framework for the study of algebras of type Max-plus, Min-Plus, tropical algebras, and more generally algebras over a commutative idempotent semi-field. This work is motivated by the increasingly diversified use of these algebras which occur also in control theory, automata theory as well as in algebraic geometry, and in more specific ways in other parts of mathematics such as the theory of monoids. In this first article, we expecially re-examine linear algebra over idempotent semi-fields: the most delicate, but undoubtedly the most interesting point is the notion of a singular point seen as a generalization of the notion of zero. We thus rediscover many notions of regularity already introduced for matrices, and this permits us to define further notions, new in this context, such as that of the kernel of a linear form, and to apply duality to obtain a good notion of tropical dimension of a submodule."}
{"category": "Math", "title": "Algorithms for Representation Theory of Real Reductive Groups", "abstract": "We give an algorithm for computing the irreducible admissible representations of a real reductive group with regular integral infinitesimal character. This algorithm has been implemented on a computer, as part of the Atlas of Lie Groups and Representations project."}
{"category": "Math", "title": "Guide to the Atlas Software: Computational Representation Theory of Real Reductive Groups", "abstract": "This is an introduction to the Atlas of Lie Groups and Representations software, for computing representation and structure theory of real reductive groups. The user is led through the basic commands of the software, via numerous examples. See Algorithms for Representation Theory of Real Reductive Groups, arXiv number 0807.3093, for a more complete mathematical description of the algorithm"}
{"category": "Math", "title": "The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups", "abstract": "In this paper we search for conditions on a countably compact (pseudo-compact) topological semigroup under which: (i) each maximal subgroup $H(e)$ in $S$ is a (closed) topological subgroup in $S$; (ii) the Clifford part $H(S)$(i.e. the union of all maximal subgroups) of the semigroup $S$ is a closed subset in $S$; (iii) the inversion $\\operatorname{inv}\\colon H(S)\\to H(S)$ is continuous; and (iv) the projection $\\pi\\colon H(S)\\to E(S)$, $\\pi\\colon x\\longmapsto xx^{-1}$, onto the subset of idempotents $E(S)$ of $S$, is continuous."}
{"category": "Math", "title": "On the splitting problem for selections", "abstract": "We investigate when does the Repov\\v{s}-Semenov Splitting problem for selections have an affirmative solution for continuous set-valued mappings in finite-dimensional Banach spaces. We prove that this happens when images of set-valued mappings or even their graphs are P-sets (in the sense of Balashov) or strictly convex sets. We also consider an example which shows that there is no affirmative solution of this problem even in the simplest case in $\\R^{3}$. We also obtain affirmative solution of the Approximate splitting problem for Lipschitz continuous selections in the Hilbert space."}
{"category": "Math", "title": "Broue's Abelian Defect Group Conjecture for the Tits Group", "abstract": "In this paper we prove that Broue's abelian defect group conjecture holds for the Tits group $^2F_4(2)'$. Also we prove that under certain conditions we are able to lift derived equivalences and use this to prove Broue's conjecture for the group $^2F_4(2)$."}
{"category": "Math", "title": "On the Inviscid Burgers Equation and the Axiom of Choice", "abstract": "The article considers the Choice Axiom."}
{"category": "Math", "title": "Relative critical exponents, non-vanishing and metrics with minimal singularities", "abstract": "In this article we prove a non-vanishing statement, as well as several properties of metrics with minimal singularities of adjoint bundles. Our arguments involve many ideas from Y.-T. Siu's analytic proof of the finite generation of the canonical ring. An important technical tool is the notion of relative critical exponent of two closed positive currents with respect to a measure."}
{"category": "Math", "title": "Functional inequalities for heavy tails distributions and application to isoperimetry", "abstract": "This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincar\\'e and weak Cheeger, weighted Poincar\\'e and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures on $\\R^n$ we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve many previous results."}
{"category": "Math", "title": "A note on state space representations of locally stationary wavelet time series", "abstract": "In this note we show that the locally stationary wavelet process can be decomposed into a sum of signals, each of which following a moving average process with time-varying parameters. We then show that such moving average processes are equivalent to state space models with stochastic design components. Using a simple simulation step, we propose a heuristic method of estimating the above state space models and then we apply the methodology to foreign exchange rates data."}
{"category": "Math", "title": "Stability for t-intersecting families of permutations", "abstract": "A family of permutations (\\mathcal{A} \\subset S_{n}) is said to be (t)-\\textit{intersecting} if any two permutations in (\\mathcal{A}) agree on at least (t) points, i.e. for any (\\sigma, \\pi \\in \\mathcal{A}), (|\\{i \\in [n]: \\sigma(i)=\\pi(i)\\}| \\geq t). It was recently proved by Friedgut, Pilpel and the author that for (n) sufficiently large depending on (t), a (t)-intersecting family (\\mathcal{A} \\subset S_{n}) has size at most ((n-t)!), with equality only if (\\mathcal{A}) is a coset of the stabilizer of (t) points (or `(t)-coset' for short), proving a conjecture of Deza and Frankl. Here, we first obtain a rough stability result for (t)-intersecting families of permutations, namely that for any (t \\in \\mathbb{N}) and any positive constant (c), if (\\mathcal{A} \\subset S_{n}) is a (t)-intersecting family of permutations of size at least (c(n-t)!), then there exists a (t)-coset containing all but at most a (O(1/n))-fraction of (\\mathcal{A}). We use this to prove an exact stability result: for (n) sufficiently large depending on (t), if (\\mathcal{A} \\subset S_{n}) is a (t)-intersecting family which is not contained within a (t)-coset, then (\\mathcal{A}) is at most as large as the family \\mathcal{D} & = & \\{\\sigma \\in S_{n}: \\sigma(i)=i \\forall i \\leq t, \\sigma(j)=j \\textrm{for some} j > t+1\\} && \\cup \\{(1 t+1),(2 t+1),...,(t t+1)\\} which has size ((1-1/e+o(1))(n-t)!). Moreover, if (\\mathcal{A}) is the same size as (\\mathcal{D}) then it must be a `double translate' of (\\mathcal{D}), meaning that there exist (\\pi,\\tau \\in S_{n}) such that (\\mathcal{A}=\\pi \\mathcal{D} \\tau). We also obtain an analogous result for (t)-intersecting families in the alternating group (A_{n})."}
{"category": "Math", "title": "Cutting Cakes Correctly", "abstract": "Without additional hypotheses, Proposition 7.1 in Brams and Taylor's book \"Fair Division\" (Cambridge University Press, 1996) is false, as are several related Pareto-optimality theorems of Brams, Jones and Klamler in their 2006 cake-cutting paper."}
{"category": "Math", "title": "A Proof of the Cameron-Ku conjecture", "abstract": "A family of permutations A \\subset S_n is said to be intersecting if any two permutations in A agree at some point, i.e. for any \\sigma, \\pi \\in A, there is some i such that \\sigma(i)=\\pi(i). Deza and Frankl showed that for such a family, |A| <= (n-1)!. Cameron and Ku showed that if equality holds then A = {\\sigma \\in S_{n}: \\sigma(i)=j} for some i and j. They conjectured a `stability' version of this result, namely that there exists a constant c < 1 such that if A \\subset S_{n} is an intersecting family of size at least c(n-1)!, then there exist i and j such that every permutation in A maps i to j (we call such a family `centred'). They also made the stronger `Hilton-Milner' type conjecture that for n \\geq 6, if A \\subset S_{n} is a non-centred intersecting family, then A cannot be larger than the family C = {\\sigma \\in S_{n}: \\sigma(1)=1, \\sigma(i)=i \\textrm{for some} i > 2} \\cup {(12)}, which has size (1-1/e+o(1))(n-1)!. We prove the stability conjecture, and also the Hilton-Milner type conjecture for n sufficiently large. Our proof makes use of the classical representation theory of S_{n}. One of our key tools will be an extremal result on cross-intersecting families of permutations, namely that for n \\geq 4, if A,B \\subset S_{n} are cross-intersecting, then |A||B| \\leq ((n-1)!)^{2}. This was a conjecture of Leader; it was recently proved for n sufficiently large by Friedgut, Pilpel and the author."}
{"category": "Math", "title": "Intersections of several disks of the Riemann sphere as K-spectral sets", "abstract": "We prove that if $n$ closed disks $D_1, D_2, ..., D_n$, of the Riemann sphere are spectral sets for a bounded linear operator $A$ on a Hilbert space, then their intersection $D_1\\cap D_2...\\cap D_n$ is a complete $K$-spectral set for $A$, with $K\\leq n+n(n-1)/\\sqrt3$. When $n=2$ and the intersection $X_1\\cap X_2$ is an annulus, this result gives a positive answer to a question of A.L. Shields (1974)."}
{"category": "Math", "title": "Weight Reduction for Mod l Bianchi Modular Forms", "abstract": "Let K be an imaginary quadratic field with class number one and ring of integers O. We prove that mod l, a system of Hecke eigenvalues occurring in the first cohomology group of some congruence subgroup Gamma of SL(2,O) can be realized in the first cohomology with trivial coefficients after increasing the level of Gamma by l."}
{"category": "Math", "title": "Instant Evaluation and Demystification of zeta(n),L(n,chi) that Euler,Ramanujan Missed III", "abstract": "We show that for a non-positive value of the first variable,Hurwitz zeta function becomes a polynomial in the second variable. We show this, using 'integration approach', instead of 'power series approach', which we had resorted to, in our earlier paper with the same title. This, in particular, explains why Riemann zeta function at positive even integer arguments, can be evaluated and why it cannot be evaluated explicitly at positive odd integer arguments."}
{"category": "Math", "title": "Principle of detailed balance and convergence assessment of Markov Chain Monte Carlo methods and simulated annealing", "abstract": "Markov Chain Monte Carlo (MCMC) methods are employed to sample from a given distribution of interest, whenever either the distribution does not exist in closed form, or, if it does, no efficient method to simulate an independent sample from it is available. Although a wealth of diagnostic tools for convergence assessment of MCMC methods have been proposed in the last two decades, the search for a dependable and easy to implement tool is ongoing. We present in this article a criterion based on the principle of detailed balance which provides a qualitative assessment of the convergence of a given chain. The criterion is based on the behaviour of a one-dimensional statistic, whose asymptotic distribution under the assumption of stationarity is derived; our results apply under weak conditions and have the advantage of being completely intuitive. We implement this criterion as a stopping rule for simulated annealing in the problem of finding maximum likelihood estimators for parameters of a 20-component mixture model. We also apply it to the problem of sampling from a 10-dimensional funnel distribution via slice sampling and the Metropolis-Hastings algorithm. Furthermore, based on this convergence criterion we define a measure of efficiency of one algorithm versus another."}
{"category": "Math", "title": "Super-rigidity and finiteness of embedded $J$-holomorphic curves on Calabi-Yau threefolds", "abstract": "The paper contains a fundamental defect in its framework of using the gauge action to study the rigidity problem. As a result, the calculations leading to the main formula is also incorrect."}
{"category": "Math", "title": "A topological pinching for the injectivity radius of a compact surface in S^3 and in H^3", "abstract": "It is given a topological pinching for the injectivity radius of a compact embedded surface either in the sphere or in the hyperbolic space"}
{"category": "Math", "title": "Algebraic relations among periods and logarithms of rank 2 Drinfeld modules", "abstract": "For any rank 2 Drinfeld module rho defined over an algebraic function field, we consider its period matrix P, which is analogous to the period matrix of an elliptic curve defined over a number field. Suppose that the characteristic of F_q is odd and rho is without complex multiplication. We show that the transcendence degree of the field generated by the entries of P over F_q(theta) is 4. As a consequence, we show also the algebraic independence of Drinfeld logarithms of algebraic functions which are linearly independent over F_q(theta)."}
{"category": "Math", "title": "A comparative review of recent researches in geometry", "abstract": "Felix Klein's so-called Erlangen Program was published in 1872 as professoral dissertation. It proposed a new solution to the problem how to classify and characterize geometries on the basis of projective geometry and group theory. The given translation was made in 1892 by Dr. M. W. Haskell and transcribed by N. C. Rughoonauth. We replaced bibliographical data in text and footnotes with pointers to a complete bibliography section."}
{"category": "Math", "title": "A geometric degree formula for $A$-discriminants and Euler obstructions of toric varieties", "abstract": "We give explicit formulas for the dimensions and the degrees of $A$-discriminant varieties introduced by Gelfand-Kapranov-Zelevinsky. Our formulas can be applied also to the case where the $A$-discriminant varieties are higher-codimensional and their degrees are described by the geometry of the configurations $A$. Moreover combinatorial formulas for the Euler obstructions of general (not necessarily normal) toric varieties will be also given."}
{"category": "Math", "title": "The pattern of genetic hitchhiking under recurrent mutation", "abstract": "Genetic hitchhiking describes evolution at a neutral locus that is linked to a selected locus. If a beneficial allele rises to fixation at the selected locus, a characteristic polymorphism pattern (so-called selective sweep) emerges at the neutral locus. The classical model assumes that fixation of the beneficial allele occurs from a single copy of this allele that arises by mutation. However, recent theory (Pennings and Hermisson, 2006a; Pennings and Hermisson, 2006b) has shown that recurrent beneficial mutation at biologically realistic rates can lead to markedly different polymorphism patterns, so called soft selective sweeps. We extend an approach that has recently been developed for the classical hitchhiking model (Schweinsbergand Durrett, 2005; Etheridge, Pfaffelhuber, Wakolbinger, 2006) to study the recurrent mutation scenario. We show that the genealogy at the neutral locus can be approximated (to leading orders in the selection strength) by a marked Yule process with immigration. Using this formalism, we derive an improved analytical approximation for the expected heterozygosity at the neutral locus at the time of fixation of the beneficial allele."}
{"category": "Math", "title": "Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces", "abstract": "In the present investigation we are linking noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. Therefore we are in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to exploit the deeper properties of Gabor frames. Furthermore we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) that the Morita-Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the {\\it existence} of good multi-window Gabor frames for modulation spaces with Gabor atoms in Feichtinger's algebra and in Schwartz space."}
{"category": "Math", "title": "Permutations Which Make Transitive Groups Primitive", "abstract": "In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating groups, there are infinitely many primitive groups with one primitive generator each. These primitive groups are certain Mathieu groups, certain projective general and projective special linear groups, and certain subgroups of some affine special linear groups."}
{"category": "Math", "title": "On uniqueness of large solutions of nonlinear parabolic equations in nonsmooth domains", "abstract": "We study the existence and uniqueness of the positive solutions of the problem (P): $\\partial_tu-\\Delta u+u^q=0$ ($q>1$) in $\\Omega\\times (0,\\infty)$, $u=\\infty$ on $\\partial\\Omega\\times (0,\\infty)$ and $u(.,0)\\in L^1(\\Omega)$, when $\\Omega$ is a bounded domain in $\\mathbb R^N$. We construct a maximal solution, prove that this maximal solution is a large solution whenever $q<N/(N-2)$ and it is unique if $\\partial\\Omega=\\partial\\bar\\Omega^c$. If $\\partial\\Omega$ has the local graph property, we prove that there exists at most one solution to problem (P)"}
{"category": "Math", "title": "The moduli space of 3-dimensional associative algebras", "abstract": "In this paper, we give a classification of the 3-dimensional associative algebras over the complex numbers, including a construction of the moduli space, using versal deformations to determine how the space is glued together."}
{"category": "Math", "title": "Un th\\'eor\\`eme de la masse positive pour le probl\\`eme de Yamabe en dimension paire", "abstract": "Let $(M,g)$ be a compact conformally flat manifold of dimension $n\\geq4$ with positive scalar curvature. According to a positive mass theorem by Schoen and Yau, the constant term in the development of the Green function of the conformal Laplacian is positive if $(M,g)$ is not conformally equivalent to the sphere. On spin manifolds, there is an elementary proof of this fact by Ammann and Humbert, based on a proof of Witten. Using differential forms instead of spinors, we give an elementary proof on even dimensional manifolds, without any other topological assumption."}
{"category": "Math", "title": "Miniversal deformations of dialgebras", "abstract": "We develop the theory of versal deformations of dialgebras and describe a method for constructing a miniversal deformation of a dialgebra."}
{"category": "Math", "title": "On the localized phase of a copolymer in an emulsion: subcritical percolation regime", "abstract": "The present paper is a continuation of \\cite{dHP07b}. The object of interest is a two-dimensional model of a directed copolymer, consisting of a random concatenation of hydrophobic and hydrophilic monomers, immersed in an emulsion, consisting of large blocks of oil and water arranged in a percolation-type fashion. The copolymer interacts with the emulsion through an interaction Hamiltonian that favors matches and disfavors mismatches between the monomers and the solvents, in such a way that the interaction with the oil is stronger than with the water. The model has two regimes, supercritical and subcritical, depending on whether the oil blocks percolate or not. In \\cite{dHP07b} we focussed on the supercritical regime and obtained a complete description of the phase diagram, which consists of two phases separated by a single critical curve. In the present paper we focus on the subcritical regime and show that the phase diagram consists of four phases separated by three critical curves meeting in two tricritical points."}
{"category": "Math", "title": "On a conjecture of Ira Gessel", "abstract": "Let F(m; n1, n2) denote the number of lattice walks from (0,0) to (n1,n2), always staying in the first quadrant {(n_1,n_2); n1 >= 0, n2 >= 0} and having exactly m steps, each of which belongs to the set {E=(1,0), W=(-1,0), NE=(1,1), SW=(-1,-1)}. Ira Gessel conjectured that F(2n; 0, 0) = 16^n (1/2)_n (5/6)_n / ((2)_n (5/3)_n) where (a)_n is the Pochhammer symbol. We pose similar conjectures for some other values of (n1,n2), and give closed-form formulas for F(n1; n1, n2) when n1 >= n2 as well as for F(2n2 - n1; n1, n2) when n1 <= n2. In the main part of the paper, we derive a functional equation satisfied by the generating function of F(m; n1, n2), use the kernel method to turn it into an infinite lower-triangular system of linear equations satisfied by the values of F(m; n1, 0) and F(m; 0, n2) + F(m; 0, n2 - 1), and express these values explicitly as determinants of lower-Hessenberg matrices with unit superdiagonals whose non-zero entries are products of two binomial coefficients."}
{"category": "Math", "title": "Concentration of maps and group action", "abstract": "In this paper, from the viewpoint of the concentration theory of maps, we study a compact group and a L\\'{e}vy group action to a large class of metric spaces, such as R-trees, doubling spaces, metric graphs, and Hadamard manifolds."}
{"category": "Math", "title": "Brill-Noether theory for moduli spaces of sheaves on algebraic varieties", "abstract": "Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be an ample line bundle on $X$. Let $M_{X,H}(r;c_1, ..., c_{s})$ be the moduli space of $H$-stable vector bundles $E$ on $X$ of rank $r$ and Chern classes $c_i(E)=c_i$ for $i=1, ..., s:=min\\{r,n\\}$. We define the Brill-Noether filtration on $M_{X,H}(r;c_1, ..., c_{s})$ as $W_{H}^{k}(r;c_1,..., c_{s})= \\{E \\in M_{X,H}(r;c_1, ..., c_{s}) | h^0(X,E) \\geq k \\}$ and we realize $W_{H}^{k}(r;c_1,..., c_{s})$ as the $k$th determinantal variety of a morphism of vector bundles on $M_{X,H}(r;c_1, ..., c_{s})$, provided $H^i(E)=0$ for $i \\geq 2$ and $E \\in M_{X,H}(r;c_1, ..., c_{s})$. We also compute the expected dimension of $W_{H}^{k}(r;c_1,..., c_{s})$. Very surprisingly we will see that the Brill-Noether stratification allow us to compare moduli spaces of vector bundles on Hirzebruch surfaces stables with respect to different polarizations. We will also study the Brill-Noether loci of the moduli space of instanton bundles and we will see that they have the expected dimension."}
{"category": "Math", "title": "List Colouring Squares of Planar Graphs", "abstract": "In 1977, Wegner conjectured that the chromatic number of the square of every planar graph $G$ with maximum degree $\\Delta\\ge8$ is at most $\\bigl\\lfloor\\frac32\\Delta\\bigr\\rfloor+1$. We show that it is at most $\\frac32 \\Delta (1+o(1))$ (where the $o(1)$ is as $\\Delta\\to+\\infty$), and indeed that this is true for the list chromatic number and for more general classes of graphs."}
{"category": "Math", "title": "A Connection on Manifolds with a Nilpotent Structure", "abstract": "All the connections, pure toward the nilpotent structure, are found. Examples of manifolds, for which the curvature tensor is pure or hybrid, are given. For a manifold of B-type a necessary and sufficient condition for purity of the curvature tensor is proved. It is verified that the conformal change of the metric of a B-manifold does not retain its purity."}
{"category": "Math", "title": "Closed 1-forms in topology and dynamics", "abstract": "This article surveys recent progress of results in topology and dynamics based on techniques of closed one-forms. Our approach allows us to draw conclusions about properties of flows by studying homotopical and cohomological features of manifolds. More specifically we describe a Lusternik - Schnirelmann type theory for closed one-forms, the focusing effect for flows and the theory of Lyapunov one-forms. We also discuss recent results about cohomological treatment of the invariants cat(X, \\xi) and cat^1(X, \\xi) and their explicit computation in certain examples."}
{"category": "Math", "title": "A diagrammatic approach to categorification of quantum groups III", "abstract": "We categorify the idempotented form of quantum sl(n)."}
{"category": "Math", "title": "Diamond representations of rank two semisimple Lie algebras", "abstract": "The present work is a part of a larger program to construct explicit combinatorial models for the (indecomposable) regular representation of the nilpotent factor $N$ in the Iwasawa decomposition of a semi-simple Lie algebra $\\mathfrak g$, using the restrictions to $N$ of the simple finite dimensional modules of $\\mathfrak g$. Such a description is given in \\cite{[ABW]}, for the cas $\\mathfrak g=\\mathfrak{sl}(n)$. Here, we give the analog for the rank 2 semi simple Lie algebras (of type $A_1\\times A_1$, $A_2$, $C_2$ and $G_2$). The algebra $\\mathbb C[N]$ of polynomial functions on $N$ is a quotient, called reduced shape algebra of the shape algebra for $\\mathfrak g$. Basis for the shape algebra are known, for instance the so called semi standard Young tableaux (see \\cite{[ADLMPPrW]}). We select among the semi standard tableaux, the so called quasi standard ones which define a kind basis for the reduced shape algebra."}
{"category": "Math", "title": "Positive Polynomials and Sequential Closures of Quadratic Modules", "abstract": "Let S be a basic closed semi-algebraic set in R^n and P the corresponding preordering in R[X_1,...,X_n]. We examine for which polynomials f there exist identities f+\\ep q \\in P for all \\ep>0. These are precisely the elements of the sequential closure of P with respect to the finest locally convex topology. We solve the open problem whether this equals the double dual cone of P, by providing a counterexample. We then prove a theorem that allows to obtain identities for polynomials as above, by looking at a family of fibre-preorderings, constructed from bounded polynomials. These fibre-preorderings are easier to deal with than the original preordering in general. For a large class of examples we are thus able to show that either every polynomial f that is nonnegative on S admits such representations, or at least the polynomials from the double dual cone of P do. The results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings."}
{"category": "Math", "title": "Realisation of measured dynamics as uniquely ergodic minimal homeomorphisms on manifolds", "abstract": "We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss relative version of Jewett-Krieger theorem: any extension between two ergodic systems is isomorphic to a skew-product on Cantor sets."}
{"category": "Math", "title": "Reversibility of Some Chordal SLE$(\\kappa;\\rho)$ Traces", "abstract": "We prove that, for $\\kappa\\in(0,4)$ and $\\rho\\ge (\\kappa-4)/2$, the chordal SLE$(\\kappa;\\rho)$ trace started from $(0;0^+)$ or $(0;0^-)$ satisfies the reversibility property. And we obtain the equation for the reversal of the chordal SLE$(\\kappa;\\rho)$ trace started from $(0;b_0)$, where $b_0>0$."}
{"category": "Math", "title": "On local geometry of rank 3 distributions with 6-dimensional square", "abstract": "We solve the equivalence problem for rank 3 completely nonholonomic vector distributions with 6-dimensional square on a smooth manifold of arbitrary dimension n under very mild genericity conditions. The main idea is to consider the projectivization of the annihilator of a given 3-dimensional distribution. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. The problem of equivalence of distributions can be essentially reduced to the differential geometry of such curves. The class of all 3-distributions under consideration is split into a finite number of subclasses according to the Young diagram of their flags. The local geometry of distributions can be recovered from the properties of the symmetry group of so-called flat curves of flags associated with this Young diagram. In each subclass we describe the flat distribution and construct a canonical frame for any other distribution. It turns out that for n>6 in the most nontrivial case the symmetry algebra of the flat distribution can be described in terms of rational normal curves (their secants and tangential developables) in projective spaces and its dimension grows exponentially with respect to n."}
{"category": "Math", "title": "Convergence of symmetric Markov chains on $\\Z^d$", "abstract": "For each $n$ let $Y^n_t$ be a continuous time symmetric Markov chain with state space $n^{-1} \\Z^d$. A condition in terms of the conductances is given for the convergence of the $Y^n_t$ to a symmetric Markov process $Y_t$ on $\\R^d$. We have weak convergence of $\\{Y^n_t: t\\leq t_0\\}$ for every $t_0$ and every starting point. The limit process $Y$ has a continuous part and may also have jumps."}
{"category": "Math", "title": "Composition operators on the space of Couchy-Stiltjes transforms", "abstract": "In this note is given a new proof of the norm estimate of J. Cima and A. Matheson."}
{"category": "Math", "title": "Jordan decomposition and dynamics on flag manifolds", "abstract": "Let $\\g$ be a semisimple Lie algebra and $G = \\Int(\\g)$. In this article, we relate the Jordan decomposition of $X \\in \\g$ (or $g \\in G$) with the dynamics induced on generalized flag manifolds by the right invariant continuous-time flow generated by $X$ (or the discrete-time flow generated by $g$). We characterize the recurrent set and the finest Morse decomposition (including its stable sets) of these flows and show that its entropy always vanishes. We characterize the structurally stable ones and compute the Conley index of the attractor Morse component. When the nilpotent part of $X$ is trivial, we compute the Conley indexes of all Morse components. Finally, we consider the dynamical aspects of linear differential equations with periodic coefficients in $\\g$, which can be regarded as an extension of the dynamics generated by an element $X \\in \\g$. In this context, we generalize Floquet theory and extend the previous results to this case."}
{"category": "Math", "title": "Geometric description of the connecting homomorphism for Witt groups", "abstract": "We give a geometric setup in which the connecting homomorphism in the localization long exact sequence for Witt groups decomposes as the pull-back to the exceptional fiber of a suitable blow-up followed by a push-forward."}
{"category": "Math", "title": "A note on finite abelian gerbes over toric Deligne-Mumford stacks", "abstract": "Any toric Deligne-Mumford stack is a $\\mu$-gerbe over the underlying toric orbifold for a finite abelian group $\\mu$. In this paper we give a sufficient condition so that certain kinds of gerbes over a toric Deligne-Mumford stack are again toric Deligne-Mumford stacks."}
{"category": "Math", "title": "The Analysis of Rotated Vector Field for the Pendulum", "abstract": "The pendulum, in the presence of linear dissipation and a constant torque, is a non-integrable, nonlinear differential equation. In this paper, using the idea of rotated vector fields, derives the relation between the applied force $\\beta$ and the periodic solution, and a conclusion that the critical value of $\\beta$ is a fixed one in the over damping situation. These results are of practical significance in the study of charge-density waves in physics."}
{"category": "Math", "title": "Conjugation spaces and edges of compatible torus actions", "abstract": "Duistermaat introduced the concept of ``real locus'' of a Hamiltonian manifold. In that and in others' subsequent works, it has been shown that many of the techniques developed in the symplectic category can be used to study real loci, so long as the coefficient ring is restricted to the integers modulo 2. It turns out that these results seem not necessarily to depend on the ambient symplectic structure, but rather to be topological in nature. This observation prompts the definition of ``conjugation space'' in a paper of the two authors with V. Puppe. Our main theorem in this paper gives a simple criterion for recognizing when a topological space is a conjugation space."}
{"category": "Math", "title": "Unique representation domains, II", "abstract": "Given a star operation * of finite type, we call a domain R a *-unique representation domain (*-URD) if each *-invertible *-ideal of R can be uniquely expressed as a *-product of pairwise *-comaximal ideals with prime radical. When * is the t-operation we call the *-URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah and Brewer-Heinzer, we give conditions for a *-ideal to be a unique *-product of pairwise *-comaximal ideals with prime radical and characterize *-URDs. We show that the class of URDs includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t-spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D+XD_S[X] construction."}
{"category": "Math", "title": "Witt groups of Grassmann varieties", "abstract": "We compute the total Witt groups of (split) Grassmann varieties, over any regular base X. The answer is a free module over the total Witt ring of X. We provide an explicit basis for this free module, which is indexed by a special class of Young diagrams, that we call even Young diagrams."}
{"category": "Math", "title": "Remarks on shrinking target properties", "abstract": "This paper defines and describes a few (related) notions of shrinking target property. We show that simultaneous expanding circle maps have a certain shrinking target property, but that circle homeomorphisms and isometries of complete, separable metric spaces do not have any shrinking target property."}
{"category": "Math", "title": "w-Divisoriality in Polynomial Rings", "abstract": "We extend the Bass-Matlis characterization of local Noetherian divisorial domains to the non-Noetherian case. This result is then used to study the following question: If a domain D is w-divisorial, that is, if each w-ideal of D is divisorial, then is D[X] automatically w-divisorial? We show that the answer is yes if D is either integrally closed or Mori."}
{"category": "Math", "title": "On Nonlinear Gauge Theories", "abstract": "In this note, we study non-linear gauge theories for principal bundles, where the structure group is replaced by a Lie groupoid. We follow the approach of Moerdijk-Mrcun and establish its relation with the existing physics literature. In particular, we derive a new formula for the gauge transformation which closely resembles and generalizes the classical formulas found in Yang Mills gauge theories."}
{"category": "Math", "title": "On Parametrization of Compact Wavelet Matrices", "abstract": "It is given an efficient complete parametrization of wavelet matrices of rank $m$, genus $g+1$, and degree $g$, which are naturally identified with corresponding polynomial paraunitary matrix-functions. The parametrization depends on Wiener-Hopf factorization of unitary matrix-functions with constant determinant given in the unit circle. This method allows us to construct in real time the coefficients of wavelet matrices from the above class."}
{"category": "Math", "title": "Existence, covolumes and infinite generation of lattices for Davis complexes", "abstract": "Let $\\Sigma$ be the Davis complex for a Coxeter system (W,S). The automorphism group G of $\\Sigma$ is naturally a locally compact group, and a simple combinatorial condition due to Haglund--Paulin determines when G is nondiscrete. The Coxeter group W may be regarded as a uniform lattice in G. We show that many such G also admit a nonuniform lattice $\\Gamma$, and an infinite family of uniform lattices with covolumes converging to that of $\\Gamma$. It follows that the set of covolumes of lattices in G is nondiscrete. We also show that the nonuniform lattice $\\Gamma$ is not finitely generated. Examples of $\\Sigma$ to which our results apply include buildings and non-buildings, and many complexes of dimension greater than 2. To prove these results, we introduce a new tool, that of \"group actions on complexes of groups\", and use this to construct our lattices as fundamental groups of complexes of groups with universal cover $\\Sigma$."}
{"category": "Math", "title": "Space-Time Current Process for Independent Random Walks in One Dimension", "abstract": "In a system made up of independent random walks, fluctuations of order $n^{1/4}$ from the hydrodynamic limit come from particle current across characteristics. We show that a two-parameter space-time particle current process converges to a two-parameter Gaussian process. These Gaussian processes also appear as the limit for the one-dimensional random average process. The final section of this paper looks at large deviations of the current process."}
{"category": "Math", "title": "On representations of Bol algebras", "abstract": "In this paper, we introduce the notion of representation of Bol algebra. We prove an analogue of the classical Engel's theorem and the ex- tension of Ado-Iwasawa theorem for Bol Algebras. We study the category of representations of Bol algebras and show that it is a tensor category. In the case of regular representations of Bol algebras, we give a complete classification of them for all two-dimensional Bol algebras."}
{"category": "Math", "title": "The topology of moduli spaces of free group representations", "abstract": "For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topological methods, we show that the SL(3,C)-character variety of a rank 2 free group is homotopic to an 8 sphere and the SL(2,C)-character variety of a rank 3 free group is homotopic to a 6 sphere."}
{"category": "Math", "title": "Boundary of the Rauzy fractal sets in $\\RR \\times \\CC$ generated by $P(x)=x^4-x^3-x^2-x-1$", "abstract": "We study the boundary of the 3-dimensional Rauzy fractal ${\\mathcal E} \\subset \\RR \\times \\CC$ generated by the polynomial $P(x) = x^4-x^3-x^2-x-1$. The finite automaton characterizing the boundary of ${\\mathcal E}$ is given explicitly. As a consequence we prove that the set ${\\mathcal E}$ has 18 neighborhoods where 6 of them intersect the central tile ${\\mathcal E}$ in a point. Our construction shows that the boundary is generated by an iterated function system starting with 2 compact sets."}
{"category": "Math", "title": "A characterization of substitutive sequences using return words", "abstract": "We prove that a sequence is primitive substitutive if and only if the set of its derived sequences is finite; we defined these sequences here."}
{"category": "Math", "title": "A version of Lomonosov's theorem for collections of positive operators", "abstract": "It is known that for every Banach space X and every proper WOT-closed subalgebra A of L(X), if A contains a compact operator then it is not transitive. That is, there exist non-zero x in X and f in X* such that f(Tx)=0 for all T in A. In the case of algebras of adjoint operators on a dual Banach space, V.Lomonosov extended this as follows: without having a compact operator in the algebra, |f(Tx)| is less than or equal to the essential norm of the pre-adjoint operator T_* for all T in A. In this paper, we prove a similar extension (in case of adjoint operators) of a result of R.Drnovsek. Namely, we prove that if C is a collection of positive adjoint operators on a Banach lattice X satisfying certain conditions, then there exist non-zero positive x in X and f in X* such that f(Tx) is less than or equal to the essential norm of T_* for all T in C."}
{"category": "Math", "title": "Desingularization of G_2 manifolds with isolated conical singularities", "abstract": "We present a method to desingularize a compact G_2 manifold with isolated conical singularities by cutting out a neighbourhood of each singular point and glueing in an asymptotically conical G_2 manifold. Controlling the error on the overlap glueing region enables us to use a result of Joyce to conclude that the resulting compact smooth 7-manifold admits a torsion-free G_2 structure, with full G_2 holonomy. There are topological obstructions for this procedure to work, which arise from the degree 3 and degree 4 cohomology of the asymptotically conical G_2 manifolds which are glued in at each conical singularity. When a certain necessary topological condition on the manifold with isolated conical singularities is satisfied, we can introduce correction terms to the glueing procedure to ensure that it still works. In the case of degree 4 obstructions, these correction terms are trivial to construct, but in the case of degree 3 obstructions we need to solve an elliptic equation on a non-compact manifold. For this we use the Lockhart-McOwen theory of weighted Sobolev spaces on manifolds with ends. This theory is also used to obtain a good asymptotic expansion of the G_2 structure on an asymptotically conical G_2 manifold under an appropriate gauge-fixing condition, which is required to make the glueing procedure work."}
{"category": "Math", "title": "Parallel Approximation and Integer Programming Reformulation", "abstract": "We show that in a knapsack feasibility problem an integral vector $p$, which is short, and near parallel to the constraint vector gives a branching direction with small integer width. We use this result to analyze two computationally efficient reformulation techniques on low density knapsack problems. Both reformulations have a constraint matrix with columns reduced in the sense of Lenstra, Lenstra, and Lov\\'asz. We prove an upper bound on the integer width along the last variable, which becomes 1, when the density is sufficiently small. In the proof we extract from the transformation matrices a vector which is near parallel to the constraint vector $a.$ The near parallel vector is a good branching direction in the original knapsack problem, and this transfers to the last variable in the reformulations."}
{"category": "Math", "title": "Equivariant CW-Complexes and the Orbit Category", "abstract": "We give a general framework for studying G-CW complexes via the orbit category. As an application we show that the symmetric group G=S_5 admits a finite G-CW complex X homotopy equivalent to a sphere, with cyclic isotropy subgroups."}
{"category": "Math", "title": "Inclusions between parabolic geometries", "abstract": "Some of the well known Fefferman like constructions of parabolic geometries end up with a new structure on the same manifold. In this paper, we classify all such cases with the help of the classical Onishchik's lists \\cite{onish1} and we treat in detail the only new series of inclusions providing the spinorial structures on the manifolds with generic free distributions. Our technique relies on the cohomological understanding of the canonical normal Cartan connections for parabolic geometries and the classical computations with exterior forms. Apart of the complete discussion of the distributions from the geometrical point of view and the new functorial construction of the inclusion into the spinorial geometry, we also discuss the normality problem of the resulting spinorial connections. In particular, there is a non--trivial subclass of distributions providing normal spinorial connections directly by the construction."}
{"category": "Math", "title": "Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 1 standard modules for $\\tilde{\\mathfrak sl}(\\ell+1,\\C)$", "abstract": "Let $\\tilde{\\mathfrak g}$ be an affine Lie algebra of type $A_\\ell^{(1)}$. Suppose we're given a $\\mathbb Z$-gradation of the corresponding simple finite-dimensional Lie algebra ${\\mathfrak g}={\\mathfrak g}_{-1}\\oplus{\\mathfrak g}_0 \\oplus {\\mathfrak g}_1$; then we also have the induced $\\mathbb Z$-gradation of the affine Lie algebra $$\\tilde{\\mathfrak g}=\\tilde{\\mathfrak g}_{-1} \\oplus \\tilde{\\mathfrak g}_0 \\oplus \\tilde{\\mathfrak g}_1.$$ Let $L(\\Lambda)$ be a standard module of level 1. Feigin-Stoyanovsky's type subspace $W(\\Lambda)$ is the $\\tilde{\\mathfrak g}_1$-submodule of $L(\\Lambda)$ generated by the highest-weight vector $v_\\Lambda$, $$W(\\Lambda)=U(\\tilde{\\mathfrak g}_1)\\cdot v_\\Lambda\\subset L(\\Lambda).$$ We find a combinatorial basis of $W(\\Lambda)$ given in terms of difference and initial conditions. Linear independence of the generating set is proved inductively by using coefficients of intertwining operators. A basis of $L(\\Lambda)$ is obtained as an ``inductive limit'' of the basis of $W(\\Lambda)$."}
{"category": "Math", "title": "Discrete piecewise linear functions", "abstract": "The concept of permutograph is introduced and properties of integral functions on permutographs are established. The central result characterizes the class of integral functions that are representable as lattice polynomials. This result is used to establish lattice polynomial representations of piecewise linear functions on convex domains and continuous selectors on linear orders."}
{"category": "Math", "title": "The geodesic problem in quasimetric spaces", "abstract": "In this article, we study the geodesic problem in a generalized metric space, in which the distance function satisfies a relaxed triangle inequality $d(x,y)\\leq \\sigma (d(x,z)+d(z,y))$ for some constant $\\sigma \\geq 1$, rather than the usual triangle inequality. Such a space is called a quasimetric space. We show that many well-known results in metric spaces (e.g. Ascoli-Arzel\\`{a} theorem) still hold in quasimetric spaces. Moreover, we explore conditions under which a quasimetric will induce an intrinsic metric. As an example, we introduce a family of quasimetrics on the space of atomic probability measures. The associated intrinsic metrics induced by these quasimetrics coincide with the $d_{\\alpha}$ metric studied early in the study of branching structures arisen in ramified optimal transportation. An optimal transport path between two atomic probability measures typically has a \"tree shaped\" branching structure. Here, we show that these optimal transport paths turn out to be geodesics in these intrinsic metric spaces."}
{"category": "Math", "title": "The converse to Curtiss' theorem for one-sided moment generating functions", "abstract": "A set of necessary and sufficient conditions for a sequence of moment generating functions to converge to a moment generating function on an interval (a,b) not necessarily containing 0, is given. The result is derived using recent results by Mukherjea, et al. (2006) and Chareka (2007)."}
{"category": "Math", "title": "Schroedinger flow into almost Hermitian manifolds", "abstract": "We present a short-time existence theorem of solutions to the initial value problem for Schroedinger maps of a closed Riemannian manifold to a compact almost Hermitian manifold. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and essentially eliminate the loss of one derivative from the partial differential equation of the Schroedinger map."}
{"category": "Math", "title": "What Did Fisher Mean by An Estimate?", "abstract": "Fisher's Method of Maximum Likelihood is shown to be a procedure for the construction of likelihood intervals or regions, instead of a procedure of point estimation. Based on Fisher's articles and books it is justified that by estimation Fisher meant the construction of likelihood intervals or regions from appropriate likelihood function and that an estimate is a statistic, that is, a function from a sample space to a parameter space such that the likelihood function obtained from the sampling distribution of the statistic at the observed value of the statistic is used to construct likelihood intervals or regions. Thus Problem of Estimation is how to choose the 'best' estimate. Fisher's solution for the problem of estimation is Maximum Likelihood Estimate (MLE). Fisher's Theory of Statistical Estimation is a chain of ideas used to justify MLE as the solution of the problem of estimation. The construction of confidence intervals by the delta method from the asymptotic normal distribution of MLE is based on Fisher's ideas, but is against his 'logic of statistical inference'. Instead the construction of confidence intervals from the profile likelihood function of a given interest function of the parameter vector is considered as a solution more in line with Fisher's 'ideology'. A new method of calculation of profile likelihood-based confidence intervals for general smooth interest functions in general statistical models is considered."}
{"category": "Math", "title": "Low regularity global well-posedness for the two-dimensional Zakharov system", "abstract": "The two-dimensional Zakharov system is shown to have a unique global solution for data without finite energy if the L^2 - norm of the Schr\\\"odinger part is small enough. The proof uses a refined I-method originally initiated by Colliander, Keel, Staffilani, Takaoka and Tao. A polynomial growth bound for the solution is also given."}
{"category": "Math", "title": "Monodromy and isotopy of monotone Lagrangian tori", "abstract": "We define new Hamiltonian isotopy invariants for a monotone Lagrangian torus embedded in a symplectic 4-manifold. We show that, in the standard symplectic 4-space, these invariants distinguish a monotone Clifford torus from a Chekanov torus."}
{"category": "Math", "title": "A generalization of Cobham's Theorem", "abstract": "If a non-periodic sequence $X$ is the image by a morphism of a fixed point of both a primitive substitution $\\sigma$ and a primitive substitution $\\tau$, then the dominant eigenvalues of the matrices of $\\sigma$ and of $\\tau$ are multiplicatively dependent. This is the way we propose to generalize Cobham's Theorem."}
{"category": "Math", "title": "Building Hyper Dirichlet Processes for Graphical Models", "abstract": "Graphical models are used to describe the conditional independence relations in multivariate data. They have been used for a variety of problems, including log-linear models (Liu and Massam, 2006), network analysis (Holland and Leinhardt, 1981; Strauss and Ikeda, 1990; Wasserman and Pattison, 1996; Pattison and Wasserman, 1999; Robins et al., 1999);, graphical Gaussian models (Roverato and Whittaker, 1998; Giudici and Green, 1999; Marrelec and Benali, 2006), and genetics (Dobra et al., 2004). A distribution that satisfies the conditional independence structure of a graph is Markov. A graphical model is a family of distributions that is restricted to be Markov with respect to a certain graph. In a Bayesian problem, one may specify a prior over the graphical model. Such a prior is called a hyper Markov law if the random marginals also satisfy the independence constraints. Previous work in this area includes (Dempster, 1972; Dawid and Lauritzen, 1993; Giudici and Green, 1999; Letac and Massam, 2007). We explore graphical models based on a non-parametric family of distributions, developed from Dirichlet processes."}
{"category": "Math", "title": "Almost Vanishing Polynomials for Sets of Limited Precision Points", "abstract": "Let X be a set of s points whose coordinates are known with only limited From the numerical point of view, given a set X of s real points whose coordinates are known with only limited precision, each set X* of real points whose elements differ from those of X of a quantity less than the data uncertainty can be considered equivalent to X. We present an algorithm that, given X and a tolerance Tol on the data error, computes a set G of polynomials such that each element of G \"almost vanishing\" at X and at all its equivalent sets X*. Even if G is not, in the general case, a basis of the vanishing ideal I(X), we show that, differently from the basis of I(X) that can be greatly influenced by the data uncertainty, G can determine a geometrical configuration simultaneously characterizing the set X and all its equivalent sets X*."}
{"category": "Math", "title": "A Berger type normal holonomy theorem for complex submanifolds", "abstract": "We prove a Berger type theorem for the normal holonomy group (i.e., the holonomy group of the normal connection) of a full complete complex submanifold of the complex projective space. Namely, if the normal holonomy does not act transitively, then the submanifold is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of the complex Euclidean space the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the projective case) and basic facts of complex submanifolds."}
{"category": "Math", "title": "Elastic-Net Regularization in Learning Theory", "abstract": "Within the framework of statistical learning theory we analyze in detail the so-called elastic-net regularization scheme proposed by Zou and Hastie for the selection of groups of correlated variables. To investigate on the statistical properties of this scheme and in particular on its consistency properties, we set up a suitable mathematical framework. Our setting is random-design regression where we allow the response variable to be vector-valued and we consider prediction functions which are linear combination of elements ({\\em features}) in an infinite-dimensional dictionary. Under the assumption that the regression function admits a sparse representation on the dictionary, we prove that there exists a particular ``{\\em elastic-net representation}'' of the regression function such that, if the number of data increases, the elastic-net estimator is consistent not only for prediction but also for variable/feature selection. Our results include finite-sample bounds and an adaptive scheme to select the regularization parameter. Moreover, using convex analysis tools, we derive an iterative thresholding algorithm for computing the elastic-net solution which is different from the optimization procedure originally proposed by Zou and Hastie"}
{"category": "Math", "title": "Limit theorem for random walk in weakly dependent random scenery", "abstract": "Let $S=(S_k)_{k\\geq 0}$ be a random walk on $\\mathbb{Z}$ and $\\xi=(\\xi_{i})_{i\\in\\mathbb{Z}}$ a stationary random sequence of centered random variables, independent of $S$. We consider a random walk in random scenery that is the sequence of random variables $(\\Sigma_n)_{n\\geq 0}$ where $$\\Sigma_n=\\sum_{k=0}^n \\xi_{S_k}, n\\in\\mathbb{N}.$$ Under a weak dependence assumption on the scenery $\\xi$ we prove a functional limit theorem generalizing Kesten and Spitzer's theorem (1979)."}
{"category": "Math", "title": "Dichotomy results for delay differential equations with negative Schwarzian", "abstract": "We gain further insight into the use of the Schwarzian derivative to obtain new results for a family of functional differential equations including the famous Wright's equation and the Mackey-Glass type delay differential equations. We present some dichotomy results, which allow us to get easily computable bounds of the global attractor. We also discuss related conjectures, and formulate new open problems."}
{"category": "Math", "title": "A simple proof that any additive basis has only finitely many essential subsets", "abstract": "Let $A$ be an additive basis. We call ``essential subset'' of $A$ any finite subset $P$ of $A$ such that $A \\setminus P$ is not an additive basis and that $P$ is minimal (for the inclusion order) to have this property. A recent theorem due to B. Deschamps and the author states that any additive basis has only finitely many essential subsets (see ``Essentialit\\'e dans les bases additives, J. Number Theory, 123 (2007), p. 170-192''). The aim of this note is to give a simple proof of this theorem."}
{"category": "Math", "title": "A stochastic SIR model with contact-tracing: large population limits and statistical inference", "abstract": "A stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease is studied. Precisely, individuals identified as infected may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The population evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size roughly speaking, becomes large. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relation of the model to the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out."}
{"category": "Math", "title": "Nonparametric estimation of the characteristic triplet of a discretely observed L\\'evy process", "abstract": "Given a discrete time sample $X_1,... X_n$ from a L\\'evy process $X=(X_t)_{t\\geq 0}$ of a finite jump activity, we study the problem of nonparametric estimation of the characteristic triplet $(\\gamma,\\sigma^2,\\rho)$ corresponding to the process $X.$ Based on Fourier inversion and kernel smoothing, we propose estimators of $\\gamma,\\sigma^2$ and $\\rho$ and study their asymptotic behaviour. The obtained results include derivation of upper bounds on the mean square error of the estimators of $\\gamma$ and $\\sigma^2$ and an upper bound on the mean integrated square error of an estimator of $\\rho.$"}
{"category": "Math", "title": "Inference with Discriminative Posterior", "abstract": "We study Bayesian discriminative inference given a model family $p(c,\\x, \\theta)$ that is assumed to contain all our prior information but still known to be incorrect. This falls in between \"standard\" Bayesian generative modeling and Bayesian regression, where the margin $p(\\x,\\theta)$ is known to be uninformative about $p(c|\\x,\\theta)$. We give an axiomatic proof that discriminative posterior is consistent for conditional inference; using the discriminative posterior is standard practice in classical Bayesian regression, but we show that it is theoretically justified for model families of joint densities as well. A practical benefit compared to Bayesian regression is that the standard methods of handling missing values in generative modeling can be extended into discriminative inference, which is useful if the amount of data is small. Compared to standard generative modeling, discriminative posterior results in better conditional inference if the model family is incorrect. If the model family contains also the true model, the discriminative posterior gives the same result as standard Bayesian generative modeling. Practical computation is done with Markov chain Monte Carlo."}
{"category": "Math", "title": "On the Existence of Symplectic Resolutions of Symplectic Reductions", "abstract": "We compute the symplectic reductions for the action of Sp_2n on several copies of C^2n and for all coregular representations of Sl_2. If it exists we give at least one symplectic resolution for each example. In the case Sl_2 acting on sl_2+C^2 we obtain an explicit description of Fu's and Namikawa's example of two non-equivalent symplectic resolutions connected by a Mukai flop."}
{"category": "Math", "title": "Frequency locking in the injection-locked frequency divider equation", "abstract": "We consider a model for the injection-locked frequency divider, and study analytically the locking onto rational multiples of the driving frequency. We provide explicit formulae for the width of the plateaux appearing in the devil's staircase structure of the lockings, and in particular show that the largest plateaux correspond to even integer values for the ratio of the frequency of the driving signal to the frequency of the output signal. Our results prove the experimental and numerical results available in the literature."}
{"category": "Math", "title": "Preconditioned HSS Method for Finite Element Approximations of Convection-Diffusion Equations", "abstract": "A two-step preconditioned iterative method based on the Hermitian/Skew-Hermitian splitting is applied to the solution of nonsymmetric linear systems arising from the Finite Element approximation of convection-diffusion equations. The theoretical spectral analysis focuses on the case of matrix sequences related to FE approximations on uniform structured meshes, by referring to spectral tools derived from Toeplitz theory. In such a setting, if the problem is coercive, and the diffusive and convective coefficients are regular enough, then the proposed preconditioned matrix sequence shows a strong clustering at unity, i.e., a superlinear preconditioning sequence is obtained. Under the same assumptions, the optimality of the PHSS method is proved and some numerical experiments confirm the theoretical results. Tests on unstructured meshes are also presented, showing the some convergence behavior."}
{"category": "Math", "title": "The Garman-Klass volatility estimator revisited", "abstract": "The Garman-Klass unbiased estimator of the variance per unit time of a zero-drift Brownian Motion B, based on the usual financial data that reports for time windows of equal length the open (OPEN), minimum (MIN), maximum (MAX) and close (CLOSE) values, is quadratic in the statistic S1=(CLOSE-OPEN, OPEN-MIN, MAX-OPEN). This estimator, with efficiency 7.4 with respect to the classical estimator (CLOSE-OPEN)^2, is widely believed to be of minimal variance. The current report disproves this belief by exhibiting an unbiased estimator with slightly but strictly higher efficiency 7.7322. The essence of the improvement lies in the observation that the data should be compressed to the statistic S2 defined on W(t)= B(0)+[B(t)-B(0)]sign[(B(1)-B(0)] as S1 was defined on the Brownian path B(t). The best S2-based quadratic unbiased estimator is presented explicitly. The Cramer-Rao upper bound for the efficiency of unbiased estimators, corresponding to the efficiency of large-sample Maximum Likelihood estimators, is 8.471. This bound cannot be attained because the distribution is not of exponential type. Regression-fitted quadratic functions of S2 (with mean 1) markedly out-perform those of S1 when applied to random walks with heavy-tail-distributed increments. Performance is empirically studied in terms of the tail parameter."}
{"category": "Math", "title": "Billiards in Nearly Isosceles Triangles", "abstract": "We prove that any sufficiently small perturbation of an isosceles triangle has a periodic billiard path. Our proof involves the analysis of certain infinite families of Fourier series that arise in connection with triangular billiards, and reveals some self-similarity phenomena in irrational triangular billiards. Our analysis illustrates the surprising fact that billiards on a triangle near a Veech triangle is extremely complicated even though Billiards on a Veech triangle is very well understood."}
{"category": "Math", "title": "Modules with cosupport and injective functors", "abstract": "Several authors have studied the filtered colimit closure lim(B) of a class B of finitely presented modules. Lenzing called lim(B) the category of modules with support in B, and proved that it is equivalent to the category of flat objects in the functor category (B^{op},Ab). In this paper, we study the category (Mod-R)^B of modules with cosupport in B. We show that (Mod-R)^B is equivalent to the category of injective objects in (B,Ab), and thus recover a classical result by Jensen-Lenzing on pure injective modules. Works of Angeleri-Hugel, Enochs, Krause, Rada, and Saorin make it easy to discuss covering and enveloping properties of (Mod-R)^B, and furthermore we compare the naturally associated notions of B-coherence and B-noetherianness. Finally, we prove a number of stability results for lim(B) and (Mod-R)^B. Our applications include a generalization of a result by Gruson-Jensen and Enochs on pure injective envelopes of flat modules."}
{"category": "Math", "title": "Modular Galois covers associated to symplectic resolutions of singularities", "abstract": "Let Y be a normal projective variety and p a morphism from X to Y, which is a projective holomorphic symplectic resolution. Namikawa proved that the Kuranishi deformation spaces Def(X) and Def(Y) are both smooth, of the same dimension, and p induces a finite branched cover f from Def(X) to Def(Y). We prove that f is Galois. We proceed to calculate the Galois group G, when X is simply connected, and its holomorphic symplectic structure is unique, up to a scalar factor. The singularity of Y is generically of ADE-type, along every codimension 2 irreducible component B of the singular locus, by Namikawa's work. The modular Galois group G is the product of Weyl groups of finite type, indexed by such irreducible components B. Each Weyl group factor W_B is that of a Dynkin diagram, obtained as a quotient of the Dynkin diagram of the singularity-type of B, by a group of Dynkin diagram automorphisms. Finally we consider generalizations of the above set-up, where Y is affine symplectic, or a Calabi-Yau threefold with a curve of ADE-singularities. We prove that the morphism f from Def(X) to Def(Y) is a Galois cover of its image. This explains the analogy between the above results and related work of Nakajima, on quiver varieties, and of Szendroi on enhanced gauge symmetries for Calabi-Yau threefolds."}
{"category": "Math", "title": "On the adjustment coefficient, drawdowns and Lundberg-type bounds for random walk", "abstract": "Consider a random walk whose (light-tailed) increments have positive mean. Lower and upper bounds are provided for the expected maximal value of the random walk until it experiences a given drawdown d. These bounds, related to the Calmar ratio in Finance, are of the form (exp{alpha d}-1)/alpha and (K exp{alpha d}-1)/alpha for some K>1, in terms of the adjustment coefficient alpha (E[exp{-alpha X}]=1) of the insurance risk literature. Its inverse 1/alpha has been recently derived by Aumann and Serrano as an index of riskiness of the random variable X. This article also complements the Lundberg exponential stochastic upper bound and the Cramer-Lundberg approximation for the expected minimum of the random walk, with an exponential stochastic lower bound. The tail probability bounds are of the form C exp{-alpha x} and exp{-alpha x} respectively, for some 1/K < C < 1. Our treatment of the problem involves Skorokhod embeddings of random walks in Martingales, especially via the Azema-Yor and Dubins stopping times, adapted from standard Brownian Motion to exponential Martingales."}
{"category": "Math", "title": "X-rays of currents and projections of forms", "abstract": "We introduce and study a new Radon-like transform that averages projected differential p-forms in R^n over affine (n-k)-planes. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms. Our transform differs from the one in Gelfand-Graev-Shapiro. Moreover, if it can be extended to a somewhat larger space of p-forms, our inversion formula will allow the synthesis of any rapidly-decaying smooth p-form on R^n as a (continuous) superposition of pullbacks from p-forms on k-dimensional subspaces. In turn, such synthesis implies an explicit formula (which we derive) for reconstructing compactly supported currents in R^n (e.g., compact oriented k-dimensional subvarieties) from their oriented projections onto k-planes."}
{"category": "Math", "title": "Hyperpolar homogeneous foliations on symmetric spaces of noncompact type", "abstract": "A foliation on a Riemannian manifold is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold that intersects each leaf of the foliation orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space of noncompact type."}
{"category": "Math", "title": "A curious result related to Kempner's series", "abstract": "It is well known since A. J. Kempner's work that the series of the reciprocals of the positive integers whose the decimal representation does not contain any digit 9, is convergent. This result was extended by F. Irwin and others to deal with the series of the reciprocals of the positive integers whose the decimal representation contains only a limited quantity of each digit of a given nonempty set of digits. Actually, such series are known to be all convergent. Here, letting $S^{(r)}$ $(r \\in \\mathbb{N})$ denote the series of the reciprocal of the positive integers whose the decimal representation contains the digit 9 exactly $r$ times, the impressive obtained result is that $S^{(r)}$ tends to $10 \\log{10}$ as $r$ tends to infinity!"}
{"category": "Math", "title": "On the support of the free Lie algebra: the Sch\\\"utzenberger problems", "abstract": "M.-P. Sch\\\"utzenberger asked to determine the support of the free Lie algebra ${\\mathcal L}_{{\\mathbb Z}_{m}}(A)$ on a finite alphabet $A$ over the ring ${\\mathbb Z}_{m}$ of integers $\\bmod m$ and all the corresponding pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We study these problems using the adjoint endomorphism $l^{*}$ of the left normed Lie bracketing $l$ of ${\\mathcal L}_{{\\mathbb Z}_{m}}(A)$. Calculating $l^{*}(w)$ via all factors of a given word $w$ of fixed length and the shuffle product, we recover the result of Duchamp and Thibon $(1989)$ for the support of the free Lie ring in a much more natural way. We rephrase these problems, for words of length $n$, in terms of the action of the left normed multi-linear Lie bracketing $l_{n}$ of ${\\mathcal L}_{{\\mathbb Z}_{m}}(A)$ - viewed as an element of the group ring of the symmetric group ${\\mathcal S}_{n}$ - on $\\lambda$-tabloids, where $\\lambda$ is a partition of $n$. For words $w$ in two letters, represented by a subset $I$ of $[n] = \\{1, 2, ..., n \\}$, this leads us to the {\\em Pascal descent polynomial} $p_{n}(I)$, a particular commutative multi-linear polynomial which equals to a signed binomial coefficient when $|I| = 1$ and allows us to obtain a sufficient condition on $n$ and $I$ in order that $w$ lies in ${\\mathcal L}_{{\\mathbb Z}_{m}}(A)$. We also have a particular conjecture for twin and anti-twin words for the free Lie ring and show that it is enough to be checked for $|A| = 2$."}
{"category": "Math", "title": "Testing the Nullspace Property using Semidefinite Programming", "abstract": "Recent results in compressed sensing show that, under certain conditions, the sparsest solution to an underdetermined set of linear equations can be recovered by solving a linear program. These results either rely on computing sparse eigenvalues of the design matrix or on properties of its nullspace. So far, no tractable algorithm is known to test these conditions and most current results rely on asymptotic properties of random matrices. Given a matrix A, we use semidefinite relaxation techniques to test the nullspace property on A and show on some numerical examples that these relaxation bounds can prove perfect recovery of sparse solutions with relatively high cardinality."}
{"category": "Math", "title": "Integral Representaion for L-functions for GSp(4)xGL(2)", "abstract": "Let pi be a cuspidal, automorphic representation of GSp(4) attached to a Siegel modular form of degree 2. We refine the method of Furusawa to obtain an integral representation for the degree-8 L-function L(s,pi x tau), where tau runs through certain cuspidal, automorphic representation of GL(2). Our calculations include the case of square-free level for the p-adic components of tau, and a wide class of archimedean types including Maass forms. As an application we obtain a special value result for L(s,pi x tau)."}
{"category": "Math", "title": "A Kinetic Model for Grain Growth", "abstract": "We provide a well-posedness analysis of a kinetic model for grain growth introduced by Fradkov which is based on the von Neumann-Mullins law. The model consists of an infinite number of transport equations with a tri-diagonal coupling modelling topological changes in the grain configuration. Self-consistency of this kinetic model is achieved by introducing a coupling weight which leads to a nonlinear and nonlocal system of equations. We prove existence of solutions by approximation with finite dimensional systems. Key ingredients in passing to the limit are suitable super-solutions, a bound from below on the total mass, and a tightness estimate which ensures that no mass is transported to infinity in finite time."}
{"category": "Math", "title": "On the extension of twisted holomorphic sections of singular hermitian line bundles", "abstract": "We prove a sharp Ohsawa-Takegoshi-Manivel type extension result for twisted holomorphic sections of singular hermitian line bundles over almost Stein manifolds. We establish as corollaries some extension results for pluri-twisted holomorphic sections of singular hermitian line bundles over projective manifolds."}
{"category": "Math", "title": "Asymptotic normality of the deconvolution kernel density estimator under the vanishing error variance", "abstract": "Let $X_1,...,X_n$ be i.i.d. observations, where $X_i=Y_i+\\sigma_n Z_i$ and the $Y$'s and $Z$'s are independent. Assume that the $Y$'s are unobservable and that they have the density $f$ and also that the $Z$'s have a known density $k.$ Furthermore, let $\\sigma_n$ depend on $n$ and let $\\sigma_n\\to 0$ as $n\\to\\infty.$ We consider the deconvolution problem, i.e. the problem of estimation of the density $f$ based on the sample $X_1,...,X_n.$ A popular estimator of $f$ in this setting is the deconvolution kernel density estimator. We derive its asymptotic normality under two different assumptions on the relation between the sequence $\\sigma_n$ and the sequence of bandwidths $h_n.$ We also consider several simulation examples which illustrate different types of asymptotics corresponding to the derived theoretical results and which show that there exist situations where models with $\\sigma_n\\to 0$ have to be preferred to the models with fixed $\\sigma.$"}
{"category": "Math", "title": "Recognition of generalized network matrices", "abstract": "In this PhD thesis, we deal with binet matrices, an extension of network matrices. The main result of this thesis is the following. A rational matrix A of size n times m can be tested for being binet in time O(n^6 m). If A is binet, our algorithm outputs a nonsingular matrix B and a matrix N such that [B N] is the node-edge incidence matrix of a bidirected graph (of full row rank) and A=B^{-1} N. Furthermore, we provide some results about Camion bases. For a matrix M of size n times m', we present a new characterization of Camion bases of M, whenever M is the node-edge incidence matrix of a connected digraph (with one row removed). Then, a general characterization of Camion bases as well as a recognition procedure which runs in O(n^2m') are given. An algorithm which finds a Camion basis is also presented. For totally unimodular matrices, it is proven to run in time O((nm)^2) where m=m'-n. The last result concerns specific network matrices. We give a characterization of nonnegative {r,s}-noncorelated network matrices, where r and s are two given row indexes. It also results a polynomial recognition algorithm for these matrices."}
{"category": "Math", "title": "A geometric interpretation of Stanley's monotonicity theorem", "abstract": "We present a new geometric proof of Stanley's monotonicity theorem for lattice polytopes, using an interpretation of $\\delta$-polynomials of lattice polytopes in terms of orbifold Chow rings."}
{"category": "Math", "title": "Topological obstructions to embedding of a matrix algebra bundle into a trivial one", "abstract": "In the present paper we describe topological obstructions to embedding of a (complex) matrix algebra bundle into a trivial one under some additional arithmetic condition on their dimensions. We explain a relation between this problem and some principal bundles with structure groupoid. Finally, we briefly discuss a relation of our results to the twisted K-theory."}
{"category": "Math", "title": "Remarks on the McKay Conjecture", "abstract": "The McKay Conjecture (MC) asserts the existence of a bijection between the (inequivalent) complex irreducible representations of degree coprime to $p$ ($p$ a prime) of a finite group $G$ and those of the subgroup $N$, the normalizer of Sylow $p$-subgroup. In this paper we observe that MC implies the existence of analogous bijections involving various pairs of algebras, including certain crossed products, and that MC is \\emph{equivalent} to the analogous statement for (twisted) quantum doubles. Using standard conjectures in orbifold conformal field theory, MC is \\emph{equivalent} to parallel statements about holomorphic orbifolds $V^G, V^N$. There is a uniform formulation of MC covering these different situations which involves quantum dimensions of objects in pairs of ribbon fusion categories."}
{"category": "Math", "title": "Effective completeness for real computation", "abstract": "The main result of this paper, as previously presented to arxiv, was incorrect. See the full text for details and for reference to the remaining results."}
{"category": "Math", "title": "On the expected diameter of an L2-bounded martingale", "abstract": "It is shown that the ratio between the expected diameter of an L2-bounded martingale and the standard deviation of its last term cannot exceed sqrt(3). Moreover, a one-parameter family of stopping times on standard Brownian Motion is exhibited, for which the sqrt(3) upper bound is attained. These stopping times, one for each cost-rate c, are optimal when the payoff for stopping at time t is the diameter D(t) obtained up to time t minus the hitherto accumulated cost c t. A quantity related to diameter, maximal drawdown (or rise), is introduced and its expectation is shown to be bounded by sqrt(2) times the standard deviation of the last term of the martingale. These results complement the Dubins and Schwarz respective bounds 1 and sqrt(2) for the ratios between the expected maximum and maximal absolute value of the martingale and the standard deviation of its last term. Dynamic programming (gambling theory) methods are used for the proof of optimality."}
{"category": "Math", "title": "Variational Particle Schemes for the Porous Medium Equation and for the System of Isentropic Euler Equations", "abstract": "Both the porous medium equation and the system of isentropic Euler equations can be considered as steepest descents on suitable manifolds of probability measures in the framework of optimal transport theory. By discretizing these variational characterizations instead of the partial differential equations themselves, we obtain new schemes with remarkable stability properties. We show that they capture successfully the nonlinear features of the flows, such as shocks and rarefaction waves for the isentropic Euler equations. We also show how to design higher order methods for these problems in the optimal transport setting using backward differentiation formula (BDF) multi-step methods or diagonally implicit Runge-Kutta methods."}
{"category": "Math", "title": "Linear relations between polynomial orbits", "abstract": "We study the orbits of a polynomial f in C[X], namely the sets {e,f(e),f(f(e)),...} with e in C. We prove that if nonlinear complex polynomials f and g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C^d with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell--Lang conjecture."}
{"category": "Math", "title": "On Ritt's polynomial decomposition theorems", "abstract": "Ritt studied the functional decomposition of a univariate complex polynomial f into prime (indecomposable) polynomials, f = u_1 o u_2 o ... o u_r. His main achievement was a procedure for obtaining any decomposition of f from any other by repeatedly applying certain transformations. However, Ritt's results provide no control on the number of times one must apply the basic transformations, which makes his procedure unsuitable for many theoretical and algorithmic applications. We solve this problem by giving a new description of the collection of all decompositions of a polynomial. Our results have been used by Ghioca, Tucker and Zieve (arXiv:0807.3576) to describe the polynomials f,g having orbits with infinite intersection; they have also been used by Medvedev and Scanlon to describe the affine curves invariant under a coordinatewise polynomial action."}
{"category": "Math", "title": "Zero patterns and unitary similarity", "abstract": "A subspace of the space, L(n), of traceless complex $n\\times n$ matrices can be specified by requiring that the entries at some positions $(i,j)$ be zero. The set, $I$, of these positions is a (zero) pattern and the corresponding subspace of L(n) is denoted by $L_I(n)$. A pattern $I$ is universal if every matrix in L(n) is unitarily similar to some matrix in $L_I(n)$. The problem of describing the universal patterns is raised, solved in full for $n\\le3$, and partial results obtained for $n=4$. Two infinite families of universal patterns are constructed. They give two analogues of Schur's triangularization theorem."}
{"category": "Math", "title": "On a Conjecture of Harvey and Lawson", "abstract": "We consider complex projective space P^{n} and a smooth closed curve gamma in P^{n}. Harvey and Lawson have defined the notion of the projective hull \\hat{K} of a compact subset K in P^n. This concept is an analogue of the polynomial hull of compact subsets of C^{n}. In the present note we study the relation between the following two properties of the curve gamma: (1) \\hat{gamma} - gamma is a one-dimensional complex analytic subvariety of P^{n} - gamma, and (2) There exists a Stein subdomain of P^{n} which contains the projective hull \\hat{gamma} of gamma."}
{"category": "Math", "title": "Alcove walks, buildings, symmetric functions and representations", "abstract": "For a complex simple Lie algebra, the dimension $K_{\\lambda\\mu}$ of the $\\mu$ weight space of a finite dimensional representation of highest weight $\\lambda$ is the same as the number of Littelmann paths of type $\\lambda$ and weight $\\mu$. In this paper we give an explicit construction of a path of type $\\lambda$ and weight $\\mu$ whenever $K_{\\lambda\\mu}\\ne 0$. This construction has additional consequences, it produces an explicit point in the building which chamber retracts to $\\lambda$ and sector retracts to $\\mu$, and an explicit point of the affine Grassmannian in the corresponding Mirkovi\\'c-Vilonen intersection. In an appendix we discuss the connection between retractions in buildings and alcove walks."}
{"category": "Math", "title": "Rank-Crank type PDE's and non-holomorphic Jacobi forms", "abstract": "In this paper we show how Rank-Crank type PDE's (first found by Atkin and Garvan) occur naturally in the framework of non-holomorphic Jacobiforms and find an infinite family of such differential equations. As an application we show an infinite family of congruences for odd Durfee symbols, a partition statistic introduced by George Andrews."}
{"category": "Math", "title": "Logarithmic Stable Maps", "abstract": "We introduce the notion of a logarithmic stable map from a minimal log prestable curve to a log twisted semi-stable variety of form $xy=0$. We study the compactification of the moduli spaces of such maps and provide a perfect obstruction theory, applicable to the moduli spaces of (un)ramified stable maps and stable relative maps. As an application, we obtain a modular desingularization of the main component of Kontsevich's moduli space of elliptic stable maps to a projective space."}
{"category": "Math", "title": "Existence of traveling waves for a nonlocal monostable equation: an abstract approach", "abstract": "We consider a nonlocal analogue of the Fisher-KPP equation. We do not assume that the Borel-measure for the convolution is absolutely continuous. In order to show the main result, we modify a recursive method for abstract monotone discrete dynamical systems by Weinberger. We note that the monotone semiflow generated by the equation does not have compactness with respect to the compact-open topology. At the end, we propose a discrete model that describes the measurement process."}
{"category": "Math", "title": "Substitutional dynamical systems, Bratteli diagrams and dimension groups", "abstract": "The present paper explores substitution minimal systems and their relation to stationary Bratteli diagrams and stationary dimension groups. The constructions involved are algorithmic and explicit, and render an effective method to compute an invariant of (ordered) $K$-theoretic nature for these systems. This new invariant is independent of spectral invariants which have previously been extensively studied. Before we state the main results we give some background."}
{"category": "Math", "title": "Hamiltonian monodromy via geometric quantization and theta functions", "abstract": "In this paper, Hamiltonian monodromy is studied from the point of view of geometric quantization abd theta functions, and various differential geometric aspects thereof are dealt with, all related to holonomies of suitable flat connections."}
{"category": "Math", "title": "A Sparse-Sparse Iteration for Computing a Sparse Incomplete Factorization of the Inverse of an SPD Matrix", "abstract": "In this paper, a method via sparse-sparse iteration for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix is proposed. The resulting factorized sparse approximate inverse is used as a preconditioner for solving symmetric positive definite linear systems of equations by using the preconditioned conjugate gradient algorithm. Some numerical experiments on test matrices from the Harwell-Boeing collection for comparing the numerical performance of the presented method with one available well-known algorithm are also given."}
{"category": "Math", "title": "Free interpolation of families of Cauchy-Stiltjes integrals and their multipliers", "abstract": "The present paper contains a generalization of some interpolation theorems of S. A. Vinogradov."}
{"category": "Math", "title": "Analytic Mappings Between LB-spaces and Applications in Infinite-Dimensional Lie Theory", "abstract": "We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: (a) groups of germs of analytic diffeomorphisms around a compact set in a Banach space and (b) unions of ascending sequences of Banach Lie groups."}
{"category": "Math", "title": "An\\'alisis de distancias temporales y espaciales entre el Lugar de La Mancha y cuatro puntos de referencia", "abstract": "The identity of the famous place of La Mancha appearing at the Quijote is an unknown with a history almost as long as that of the famous book by Miguel de Cervantes. This work analyzes data obtained from a Geographic Information System and compares the results with those of the previous works. Three different variables with two possible values each are considered: time or space data, 3 or 4 reference points, and the commonly used distances to the place of La Mancha or a set of recently proposed ones. The village in the Campo de Montiel which is closest to be the place of La Mancha happens to be Carrizosa or Villanueva de los Infantes, depending on the configuration, with the latter being the solution for the configuration in which the relative errors are the smallest and the second candidate village is furthest from the first. ----- La identidad del famoso lugar de la Mancha que aparece en El Quijote es una inc\\'ognita con una historia casi tan larga como la publicaci\\'on de la famosa obra de Miguel de Cervantes. Este trabajo analiza datos obtenidos mediante un Sistema de Informaci\\'on Geogr\\'afica y compara los resultados con los de los trabajos anteriores. Se consideran tres variables diferentes con dos posibles valores cada una: datos temporales o espaciales, 3 \\'o 4 puntos de referencia y las distancias al lugar de La Mancha usadas habitualmente o unas recientemente introducidas. La localidad del Campo de Montiel m\\'as cercana a ser el lugar de La Mancha resulta ser Carrizosa o Villanueva de los Infantes, dependiendo de la configuraci\\'on, siendo \\'esta \\'ultima la soluci\\'on para la configuraci\\'on en que los errores relativos son los m\\'as peque\\~nos y la segunda localidad candidata est\\'a m\\'as lejos de la primera."}
{"category": "Math", "title": "Homotopy Type of Disentanglements of Multi-germs", "abstract": "For a complex analytic map f from n-space to p-space with n<p and with an isolated instability at the origin, the disentanglement of f is a local stabilization of f that is analogous to the Milnor fibre for functions. For mono-germs it is known that the disentanglement is a wedge of spheres of possibly varying dimensions. In this paper we give a condition that allows us to deduce that the same is true for a large class of multi-germs."}
{"category": "Math", "title": "Eigenvectors of random graphs: Nodal domains", "abstract": "We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our main focus in this paper is on the nodal domains associated with the different eigenfunctions. In the analogous realm of Laplacians of Riemannian manifolds, nodal domains have been the subject of intensive research for well over a hundred years. Graphical nodal domains turn out to have interesting and unexpected properties. Our main theorem asserts that there is a constant c such that for almost every graph G, each eigenfunction of G has at most two large nodal domains, and in addition at most c exceptional vertices outside these primary domains. We also discuss variations of these questions and briefly report on some numerical experiments which, in particular, suggest that almost surely there are just two nodal domains and no exceptional vertices."}
{"category": "Math", "title": "A bootstrap method for estimating bias and variance in statistical multispecies models using highly disparate data sets", "abstract": "Statistical multispecies models of multiarea marine ecosystems use a variety of data sources to estimate parameters using composite or weighted likelihood functions with associated weighting issues and questions on how to obtain variance estimates. Regardless of the method used to obtain point estimates, a method is needed for variance estimation. A bootstrap technique is introduced for the evaluation of uncertainty in such models, taking into account inherent spatial and temporal correlations in the data sets thus avoiding many model--specification issues, which are commonly transferred as assumptions from a likelihood estimation procedure into Hessian--based variance estimation procedures. The technique is demonstrated on a real data set and used to look for estimation bias and the effects of different aggregation levels in population dynamics models."}
{"category": "Math", "title": "Universality of the limit shape of convex lattice polygonal lines", "abstract": "Let ${\\varPi}_n$ be the set of convex polygonal lines $\\varGamma$ with vertices on $\\mathbb {Z}_+^2$ and fixed endpoints $0=(0,0)$ and $n=(n_1,n_2)$. We are concerned with the limit shape, as $n\\to\\infty$, of \"typical\" $\\varGamma\\in {\\varPi}_n$ with respect to a parametric family of probability measures $\\{P_n^r,0<r<\\infty\\}$ on ${\\varPi}_n$, including the uniform distribution ($r=1$) for which the limit shape was found in the early 1990s independently by A. M. Vershik, I. B\\'ar\\'any and Ya. G. Sinai. We show that, in fact, the limit shape is universal in the class $\\{P^r_n\\}$, even though $P^r_n$ ($r\\ne1$) and $P^1_n$ are asymptotically singular. Measures $P^r_n$ are constructed, following Sinai's approach, as conditional distributions $Q_z^r(\\cdot |{\\varPi}_n)$, where $Q_z^r$ are suitable product measures on the space ${\\varPi}=\\bigcup_n{\\varPi}_n$, depending on an auxiliary \"free\" parameter $z=(z_1,z_2)$. The transition from $({\\varPi},Q_z^r)$ to $({\\varPi}_n,P_n^r)$ is based on the asymptotics of the probability $Q_z^r({\\varPi}_n)$, furnished by a certain two-dimensional local limit theorem. The proofs involve subtle analytical tools including the M\\\"obius inversion formula and properties of zeroes of the Riemann zeta function."}
{"category": "Math", "title": "On a certain relation between Legendre's conjecture and Bertrand's postulate", "abstract": "We prove a certain relation between Legendre's conjecture and Bertrand's postulate in terms of a certain transformation of Legendre's function phi. We show a certain property of a prime."}
{"category": "Math", "title": "Disjoint minimal graphs", "abstract": "We prove that the number s(n) of disjoint minimal graphs supported on domains in R^n is bounded by e(n+1)^2. In the two-dimensional case we show that s(2) is at most three (the conjectured number is two)."}
{"category": "Math", "title": "On families of subsets with a forbidden subposet", "abstract": "Let $\\F\\subset 2^{[n]}$ be a family of subsets of $\\{1,2,..., n\\}$. For any poset $H$, we say $\\F$ is $H$-free if $\\F$ does not contain any subposet isomorphic to $H$. Katona and others have investigated the behavior of $\\La(n,H)$, which denotes the maximum size of $H$-free families $\\F\\subset 2^{[n]}$. Here we use a new approach, which is to apply methods from extremal graph theory and probability theory to identify new classes of posets $H$, for which $\\La(n,H)$ can be determined asymptotically as $n\\to\\infty$ for various posets $H$, including two-end-forks, up-down trees, and cycles $C_{4k}$ on two levels."}
{"category": "Math", "title": "From subfactor planar algebras to subfactors", "abstract": "We present a purely planar algebraic proof of the main result of a paper of Guionnet-Jones-Shlaykhtenko which constructs an extremal subfactor from a subfactor planar algebra whose standard invariant is given by that planar algebra."}
{"category": "Math", "title": "K3 surfaces with non-symplectic automorphisms of 2-power order", "abstract": "This paper concerns complex algebraic K3 surfaces with an automorphism which acts trivially on the Neron-Severi group. Complementing a result by Vorontsov and Kondo, we determine those K3 surfaces where the order of the automorphism is a 2-power and equals the rank of the transcendental lattice. We also study the arithmetic of these K3 surfaces and comment on mirror symmetry"}
{"category": "Math", "title": "Moreira's Theorem on the arithmetic sum of dynamically defined Cantor sets", "abstract": "We present a complete proof of a theorem of C.G. Moreira. Under mild checkable conditions, the theorem asserts that the Hausdorff dimension of the arithmetic sum of two dynamically defined Cantor subsets of the real line, equals either the sum of the dimensions or 1, whichever is smaller."}
{"category": "Math", "title": "Favard, Baxter, Geronimus, Rakhmanov, Szeg\\\"o and the strong Szeg\\\"o theorems for orthogonal trigonometric polynomials", "abstract": "In this paper, we obtain some analogs of Favard, Baxter, Geronimus, Rakhmanov, Szeg\\\"o and the strong Szeg\\\"o theorems appeared in the theory of orthogonal polynomials on the unit circle (OPUC) for orthogonal trigonometric polynomials (OTP). The key tool is the mutual representation theorem for OPUC and OTP."}
{"category": "Math", "title": "Data spectroscopy: Eigenspaces of convolution operators and clustering", "abstract": "This paper focuses on obtaining clustering information about a distribution from its i.i.d. samples. We develop theoretical results to understand and use clustering information contained in the eigenvectors of data adjacency matrices based on a radial kernel function with a sufficiently fast tail decay. In particular, we provide population analyses to gain insights into which eigenvectors should be used and when the clustering information for the distribution can be recovered from the sample. We learn that a fixed number of top eigenvectors might at the same time contain redundant clustering information and miss relevant clustering information. We use this insight to design the data spectroscopic clustering (DaSpec) algorithm that utilizes properly selected eigenvectors to determine the number of clusters automatically and to group the data accordingly. Our findings extend the intuitions underlying existing spectral techniques such as spectral clustering and Kernel Principal Components Analysis, and provide new understanding into their usability and modes of failure. Simulation studies and experiments on real-world data are conducted to show the potential of our algorithm. In particular, DaSpec is found to handle unbalanced groups and recover clusters of different shapes better than the competing methods."}
{"category": "Math", "title": "Numerical simulation of optimal transport paths", "abstract": "This article provides numerical simulation of an optimal transport path from a single source to an atomic measure of equal total mass. We first construct an initial transport path, and then modify the path as much as possible by using both local and global minimization algorithms."}
{"category": "Math", "title": "Profinite properties of graph manifolds", "abstract": "Let $M$ be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of $\\pi_1(M)$ is efficient with respect to the JSJ decomposition of $M$. We go on to prove that $\\pi_1(M)$ is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if $M$ is a graph manifold then $\\pi_1(M)$ is conjugacy separable."}
{"category": "Math", "title": "A path following algorithm for Sparse Pseudo-Likelihood Inverse Covariance Estimation (SPLICE)", "abstract": "Given n observations of a p-dimensional random vector, the covariance matrix and its inverse (precision matrix) are needed in a wide range of applications. Sample covariance (e.g. its eigenstructure) can misbehave when p is comparable to the sample size n. Regularization is often used to mitigate the problem. In this paper, we proposed an l1-norm penalized pseudo-likelihood estimate for the inverse covariance matrix. This estimate is sparse due to the l1-norm penalty, and we term this method SPLICE. Its regularization path can be computed via an algorithm based on the homotopy/LARS-Lasso algorithm. Simulation studies are carried out for various inverse covariance structures for p=15 and n=20, 1000. We compare SPLICE with the l1-norm penalized likelihood estimate and a l1-norm penalized Cholesky decomposition based method. SPLICE gives the best overall performance in terms of three metrics on the precision matrix and ROC curve for model selection. Moreover, our simulation results demonstrate that the SPLICE estimates are positive-definite for most of the regularization path even though the restriction is not enforced."}
{"category": "Math", "title": "Odd-graceful labelings of trees of diameter 5", "abstract": "A difference vertex labeling of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy the weight |f(x)-f(y)|. A difference vertex labeling f of a graph G of size n is odd-graceful if f is an injection from V(G) to {0,1,...,2n-1} such that the induced weights are {1,3,...,2n-1}. We show here that any forest whose components are caterpillars is odd-graceful. We also show that every tree of diameter up to five is odd-graceful."}
{"category": "Math", "title": "Non-commutative quadrics", "abstract": "In this paper we describe non-commutative versions of $\\PP^1\\times \\PP^1$. These contain the class of non-commutative deformations of $\\PP^1\\times \\PP^1$."}
{"category": "Math", "title": "Generalized Cantor manifolds and homogeneity", "abstract": "A classical theorem of Alexandroff states that every $n$-dimensional compactum $X$ contains an $n$-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds, and $V^n$-continua, and prove corresponding versions of the above theorem. We apply our results to show that each homogeneous metrizable continuum which is not in a given class $\\mathcal C$ is a strong Cantor manifold (or at least a Cantor manifold) with respect to $\\mathcal C$. Here, the class $\\mathcal C$ is one of four classes that are defined in terms of dimension-like invariants. A class of spaces having bases of neighborhoods satisfying certain special conditions is also considered."}
{"category": "Math", "title": "On a generalized Jones conjecture", "abstract": "We solve the Jones conjecture, which states that the exponent sum in a minimal braid representation of a knot in S^3 is a knot invariant, by proving a generalized version of the original one. We apply contact geometry to study this problem in knot theory."}
{"category": "Math", "title": "On the unipotent characters of the Ree groups of type G_2", "abstract": "This note is concerned with the unipotent characters of the Ree groups of type G_2. We determine the roots of unity associated by Lusztig and Digne-Michel to unipotent characters of ^2G_2(3^{2n+1}) and we prove that the Fourier matrix of ^2G_2(3^{2n+1}) defined by Geck and Malle satisfies a conjecture of Digne-Michel. Our main tool is the Shintani descent of Ree groups of type G_2."}
{"category": "Math", "title": "Dynamic Data Compression with Distortion Constraints for Wireless Transmission over a Fading Channel", "abstract": "We consider a wireless node that randomly receives data from different sensor units. The arriving data must be compressed, stored, and transmitted over a wireless link, where both the compression and transmission operations consume power. Specifically, the controller must choose from one of multiple compression options every timeslot. Each option requires a different amount of power and has different compression ratio properties. Further, the wireless link has potentially time-varying channels, and transmission rates depend on current channel states and transmission power allocations. We design a dynamic algorithm for joint compression and transmission, and prove that it comes arbitrarily close to minimizing average power expenditure, with an explicit tradeoff in average delay. Our approach uses stochastic network optimization together with a concept of place holder bits to provide efficient energy-delay performance. The algorithm is simple to implement and does not require knowledge of probability distributions for packet arrivals or channel states. Extensions that treat distortion constraints are also considered."}
{"category": "Math", "title": "Adiabatic limit, Bismut-Freed connection, and the real analytic torsion form", "abstract": "For a complex flat vector bundle over a fibered manifold, we consider the 1-parameter family of certain deformed sub-signature operators introduced by Ma-Zhang. We compute the adiabatic limit of the Bismut-Freed connection associated to this family and show that the Bismut-Lott analytic torsion form shows up naturally under this procedure."}
{"category": "Math", "title": "On derivation of Euler-Lagrange Equations for incompressible energy-minimizers", "abstract": "We prove that any distribution $q$ satisfying the equation $\\nabla q=\\div{\\bf f}$ for some tensor ${\\bf f}=(f^i_j), f^i_j\\in h^r(U)$ ($1\\leq r<\\infty$) -the {\\it local Hardy space}, $q$ is in $h^r$, and is locally represented by the sum of singular integrals of $f^i_j$ with Calder\\'on-Zygmund kernel. As a consequence, we prove the existence and the local representation of the hydrostatic pressure $p$ (modulo constant) associated with incompressible elastic energy-minimizing deformation ${\\bf u}$ satisfying $|\\nabla {\\bf u}|^2, |{\\rm cof}\\nabla{\\bf u}|^2\\in h^1$. We also derive the system of Euler-Lagrange equations for incompressible local minimizers ${\\bf u}$ that are in the space $K^{1,3}_{\\rm loc}$; partially resolving a long standing problem. For H\\\"older continuous pressure $p$, we obtain partial regularity of area-preserving minimizers."}
{"category": "Math", "title": "Small exotic Stein manifolds", "abstract": "It is known that the only Stein filling of the standard contact structure on S^3 is B^4. In this paper, we construct simply connected exotic compact Stein 4-manifold pairs for any Betti number $b_2 \\geq 1$; we do this by enlarging corks and plugs."}
{"category": "Math", "title": "Reflection principle and Ocone martingales", "abstract": "Let $M =(M_t)_{t\\geq 0}$ be any continuous real-valued stochastic process. We prove that if there exists a sequence $(a_n)_{n\\geq 1}$ of real numbers which converges to 0 and such that $M$ satisfies the reflection property at all levels $a_n$ and $2a_n$ with $n\\geq 1$, then $M$ is an Ocone local martingale with respect to its natural filtration. We state the subsequent open question: is this result still true when the property only holds at levels $a_n$? Then we prove that the later question is equivalent to the fact that for Brownian motion, the $\\sigma$-field of the invariant events by all reflections at levels $a_n$, $n\\ge1$ is trivial. We establish similar results for skip free $\\mathbb{Z}$-valued processes and use them for the proof in continuous time, via a discretisation in space."}
{"category": "Math", "title": "Hopf images and inner faithful representations", "abstract": "We develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image."}
{"category": "Math", "title": "Biseparating maps between Lipschitz function spaces", "abstract": "For complete metric spaces $X$ and $Y$, a description of linear biseparating maps between spaces of vector-valued Lipschitz functions defined on $X$ and $Y$ is provided. In particular it is proved that $X$ and $Y$ are bi-Lipschitz homeomorphic, and the automatic continuity of such maps is derived in some cases. Besides, these results are used to characterize the separating bijections between scalar-valued Lipschitz function spaces when $Y$ is compact."}
{"category": "Math", "title": "Accuracy and Robustness of Clustering Algorithms for Small-Size Applications in Bioinformatics", "abstract": "The performance (accuracy and robustness) of several clustering algorithms is studied for linearly dependent random variables in the presence of noise. It turns out that the error percentage quickly increases when the number of observations is less than the number of variables. This situation is common situation in experiments with DNA microarrays. Moreover, an {\\it a posteriori} criterion to choose between two discordant clustering algorithm is presented."}
{"category": "Math", "title": "Derived equivalence classification of m-cluster tilted algebras of type A", "abstract": "We use the maximal faces of the $m$-cluster complex of type A to describe the m-cluster tilted algebras of type A as quivers with relations. We then classify connected components of m-cluster tilted algebras of type A up to derived equivalence using tilting complexes directly related to the combinatorics of the m-cluster complex of type A. This generalizes a result of Buan and Vatne."}
{"category": "Math", "title": "On the artificial compressibility method for the Navier Stokes Fourier system", "abstract": "This paper deals with the approximation of the weak solutions of the incompressible Navier Stokes Fourier system. In particular it extends the artificial compressibility method for the Leray weak solutions of the Navier Stokes equation, used by Temam, in the case of a bounded domain and later in the case of the whole space. By exploiting the wave equation structure of the pressure of the approximating system the convergence of the approximating sequences is achieved by means of dispersive estimate of Strichartz type. It will be proved that the projection of the approximating velocity fields on the divergence free vectors is relatively compact and converges to a weak solution of the incompressible Navier Stokes Fourier system."}
{"category": "Math", "title": "Quasi-convex density and determining subgroups of compact abelian groups", "abstract": "For an abelian topological group G let G^* denote the dual group of all continuous characters endowed with the compact open topology. Given a closed subset X of an infinite compact abelian group G such that w(X) < w(G) and an open neighbourhood U of 0 in the circle group, we show that the set of all characters which send X into U has the same size as G^*. (Here, w(G) denotes the weight of G.) A subgroup D of G determines G if the restriction homomorphism G^* --> D^* is an isomorphism between G^* and D^*. We prove that w(G) = min {|D|: D is a subgroup of G that determines G} for every infinite compact abelian group G. In particular, an infinite compact abelian group determined by a countable subgroup is metrizable. This gives a negative answer to questions of Comfort, Hernandez, Macario, Raczkowski and Trigos-Arrieta. As an application, we furnish a short elementary proof of the result from [13] that a compact abelian group G is metrizable provided that every dense subgroup of G determines G."}
{"category": "Math", "title": "Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller--Segel type models", "abstract": "In this paper we deal with diffusive relaxation limits of nonlinear systems of Euler type modeling chemotactic movement of cells toward Keller--Segel type systems. The approximating systems are either hyperbolic--parabolic or hyperbolic--elliptic. They all feature a nonlinear pressure term arising from a \\emph{volume filling effect} which takes into account the fact that cells do not interpenetrate. The main convergence result relies on compensated compactness tools and is obtained for large initial data under suitable assumptions on the approximating solutions. In order to justify such assumptions, we also prove an existence result for initial data which are small perturbation of a constant state. Such result is proven via classical Friedrichs's symmetrization and linearization. In order to simplify the coverage, we restrict to the two--dimensional case with periodical boundary conditions."}
{"category": "Math", "title": "Groupoid actions as quantale modules", "abstract": "For an arbitrary localic etale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of etendues."}
{"category": "Math", "title": "On shrinking targets for Z^m actions on tori", "abstract": "Let A be an n by m matrix with real entries. Consider the set Bad_A of x \\in [0,1)^n for which there exists a constant c(x)>0 such that for any q \\in Z^m the distance between x and the point {Aq} is at least c(x) |q|^{-m/n}. It is shown that the intersection of Bad_A with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new."}
{"category": "Math", "title": "On the strong approximation and functional limit laws for the increments of the non-overlapping k-spacings processes", "abstract": "The first aim of the present paper, is to establish strong approximations of the uniform non-overlapping k-spacings process extending the results of Aly et al. (1984). Our methods rely on the invariance principle in Mason and van Zwet (1987). The second goal, is to generalize the Dindar (1997) results for the increments of the spacings quantile process to the uniforme non-overlapping k-spacings quantile process. We apply the last result to characterize the limit laws of functionals of the increments k-spacings quantile process."}
{"category": "Math", "title": "Blackbox computation of $A_\\infty$-algebras", "abstract": "Kadeishvili's proof of the minimality theorem induces an algorithm for the inductive computation of an $A_\\infty$-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete $A_\\infty$-algebra structure after a finite amount of computational work."}
{"category": "Math", "title": "Ideals with an assigned initial ideal", "abstract": "The stratum St(J,<) (the homogeneous stratum Sth(J,<) respectively) of a monomial ideal J in a polynomial ring R is the family of all (homogeneous) ideals of R whose initial ideal with respect to the term order < is J. St(J,<) and Sth(J,<) have a natural structure of affine schemes. Moreover they are homogeneous w.r.t. a non-standard grading called level. This property allows us to draw consequences that are interesting from both a theoretical and a computational point of view. For instance a smooth stratum is always isomorphic to an affine space (Corollary 3.6). As applications, in Sec. 5 we prove that strata and homogeneous strata w.r.t. any term ordering < of every saturated Lex-segment ideal J are smooth. For Sth(J,Lex) we also give a formula for the dimension. In the same way in Sec. 6 we consider any ideal R in k[x0,..., xn] generated by a saturated RevLex-segment ideal in k[x,y,z]. We also prove that Sth(R,RevLex) is smooth and give a formula for its dimension."}
{"category": "Math", "title": "Birational geometry of singular Fano varieties", "abstract": "We prove divisorial canonicity of Fano hypersurfaces and double spaces of general position with elementary singularities."}
{"category": "Math", "title": "Hopf cyclic cohomology in braided monoidal categories", "abstract": "We extend the formalism of Hopf cyclic cohomology to the context of braided categories. For a Hopf algebra in a braided monoidal abelian category we introduce the notion of stable anti-Yetter-Drinfeld module. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution in the sense of Connes and Moscovici. When the braiding is symmetric the full formalism of Hopf cyclic cohomology with coefficients can be extended to our categorical setting."}
{"category": "Math", "title": "Alcove geometry and a translation principle for the Brauer algebra", "abstract": "There are similarities between algebraic Lie theory and a geometric description of the blocks of the Brauer algebra in characteristic zero. Motivated by this, we study the alcove geometry of a certain reflection group action. We provide analogues of translation functors for a tower of recollement, and use these to construct Morita equivalences between blocks containing weights in the same facet. Moreover, we show that the determination of decomposition numbers for the Brauer algebra in characteristic zero can be reduced to a study of the block containing the weight 0. We define parabolic Kazhdan-Lusztig polynomials for the Brauer algebra and show in certain low rank examples that they determine standard module decomposition numbers and filtrations."}
{"category": "Math", "title": "Rationality of the vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of arbitrary rank", "abstract": "In this paper we prove that the vertex algebra $V_L^+$ is rational if $L$ is a negative definite even lattice of finite rank, or if $L$ is a non-degenerate even lattice of a finite rank that is neither positive definite nor negative definite. In particular, for such even lattices $L$, we show that the Zhu algebras of the vertex algebras $V_L^+$ are semisimple. This extends the result of Abe which establishes the rationality of $V_L^+$ when $L$ is a positive definite even lattice of finite rank."}
{"category": "Math", "title": "Conditional density estimation in a censored single-index regression model", "abstract": "Under a single-index regression assumption, we introduce a new semiparametric procedure to estimate a conditional density of a censored response. The regression model can be seen as a generalization of Cox regression model and also as a profitable tool to perform dimension reduction under censoring. This technique extends the results of Delecroix et al. (2003). We derive consistency and asymptotic normality of our estimator of the index parameter by proving its asymptotic equivalence with the (uncomputable) maximum likelihood estimator, using martingales results for counting processes and arguments of empirical processes theory. Furthermore, we provide a new adaptive procedure which allows us both to chose the smoothing parameter involved in our approach and to circumvent the weak performances of Kaplan-Meier estimator (1958) in the right-tail of the distribution. Through a simulation study, we study the behavior of our estimator for small samples."}
{"category": "Math", "title": "Local convergence analysis of inexact Newton-like methods under majorant condition", "abstract": "We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, which relaxes the Lipschitz continuity of the derivative, and the nonlinear operator under consideration. It also allow us to obtain some important special cases"}
{"category": "Math", "title": "A transference principle for general groups and functional calculus on UMD spaces", "abstract": "We prove a transference principle for general (i.e., not necessarily bounded) strongly continuous groups on Banach spaces. If the Banach space has the UMD property, the transference principle leads to estimates for the functional calculus of the group generator. In the Hilbert space case, the results cover classical theorems of McIntosh and Boyadzhiev-de Laubenfels; in the UMD case they are analogues of classical results by Hieber and Pruess. By using functional calculus methods, consequences for sectorial operators are derived. For instance it is proved, that every generator of a cosine function on a UMD space has bounded H-infinity calculus on sectors."}
{"category": "Math", "title": "Nonlinear optimization for matroid intersection and extensions", "abstract": "We address optimization of nonlinear functions of the form $f(Wx)$, where $f:\\R^d\\to \\R$ is a nonlinear function, $W$ is a $d\\times n$ matrix, and feasible $x$ are in some large finite set $F$ of integer points in $\\R^n$. One motivation is multi-objective discrete optimization, where $f$ trades off the linear functions given by the rows of $W$. Another motivation is to extend known results about polynomial-time linear optimization over discrete structures to nonlinear optimization. We assume that the convex hull of $F$ is well-described by linear inequalities. For example, the set of characteristic vectors of common bases of a pair of matroids on a common ground set. When $F$ is well described, $f$ is convex (or even quasiconvex), and $W$ has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic algorithm for maximization. When $F$ is well described, $f$ is a norm, and binary-encoded $W$ is nonnegative, we give an efficient deterministic constant-approximation algorithm for maximization. When $F$ is well described, $f$ is ``ray concave'' and non-decreasing, and $W$ has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic constant-approximation algorithm for minimization. When $F$ is the set of characteristic vectors of common bases of a pair of vectorial matroids on a common ground set, $f$ is arbitrary, and $W$ has a fixed number of rows and is unary encoded, we give an efficient randomized algorithm for optimization."}
{"category": "Math", "title": "Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins", "abstract": "Advances in nanotechnology have allowed scientists to study biological processes on an unprecedented nanoscale molecule-by-molecule basis, opening the door to addressing many important biological problems. A phenomenon observed in recent nanoscale single-molecule biophysics experiments is subdiffusion, which largely departs from the classical Brownian diffusion theory. In this paper, by incorporating fractional Gaussian noise into the generalized Langevin equation, we formulate a model to describe subdiffusion. We conduct a detailed analysis of the model, including (i) a spectral analysis of the stochastic integro-differential equations introduced in the model and (ii) a microscopic derivation of the model from a system of interacting particles. In addition to its analytical tractability and clear physical underpinning, the model is capable of explaining data collected in fluorescence studies on single protein molecules. Excellent agreement between the model prediction and the single-molecule experimental data is seen."}
{"category": "Math", "title": "Etale groupoids as germ groupoids and their base extensions", "abstract": "We introduce the notion of wide representation of an inverse semigroup and prove that with a suitably defined topology there is a space of germs of such a representation which has the structure of an etale groupoid. This gives an elegant description of Paterson's universal groupoid and of the translation groupoid of Skandalis, Tu, and Yu. In addition we characterize the inverse semigroups that arise from groupoids, leading to a precise bijection between the class of etale groupoids and the class of complete and infinitely distributive inverse monoids equipped with suitable representations, and we explain the sense in which quantales and localic groupoids carry a generalization of this correspondence."}
{"category": "Math", "title": "Essentially Reductive Weighted Shift Hilbert Modules", "abstract": "We discuss the relation between questions regarding the essential normality of finitely generated essentially spherical isometries and some results and conjectures of Arveson and Guo-Wang on the closure of homogeneous ideals in the m-shift space. We establish a general results for the case of two tuples and ideals with one dimensional zero variety. Further, we show how to reduce the analogous question for quasi-homogeneous ideals, to those results for homogeneous ones. Finally, we show that the essential reductivity of positive regular Hilbert modules is directly related to a generalization of the Arveson problem."}
{"category": "Math", "title": "On the behaviors of solution near possible blow-up time in the incompressible Euler and related equations", "abstract": "We study behaviors of scalar quantities near the possible blow-up time, which is made of smooth solutions of the Euler equations, Navier-Stokes equations and the surface quasi-geostrophic equations. Integrating the dynamical equations of the scaling invariant norms, we derive the possible blow-up behaviors of the above quantities, from which we obtain new type of blow-up criteria and some necessary conditions for the blow-up."}
{"category": "Math", "title": "Explicit Construction of a Robust Family of Compact Inertial Manifolds", "abstract": "A construction of a robust family of compact inertial manifolds is presented. The result aims to complete an analysis of certain types of attracting sets for a class of dissipative infinite dimensional dynamical systems. Application to a hyperbolically relaxed Chaffee-Infante reaction diffusion equation is also discussed."}
{"category": "Math", "title": "Note on the growth of Area functions", "abstract": "A new proof of the inequalities of D. J. Hallenbeck for the Area functions of multipliers of fractional Cauchy transforms is given."}
{"category": "Math", "title": "A hyperelliptic Hodge integral", "abstract": "We calculate the hyperelliptic Hodge integral lambda_g lambda_{g-1} / (1 - psi) for use in arXiv:math/0702219. The proof uses the WDVV equations for the genus zero Gromov--Witten invariants of P(1,1,2)."}
{"category": "Math", "title": "Eigenfunctions of the Laplacian and associated Ruelle operator", "abstract": "Let $\\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\\'e disk $\\DD$ and $\\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\\Delta$, equivariant by $\\Gamma$ with real eigenvalue $\\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution $\\dd_{f,s}$ (the Helgason distribution) which is equivariant by $\\Gamma$ and supported at infinity $\\partial\\DD=\\SS^1$. The geodesic flow on the compact surface $\\DD/\\Gamma$ is conjugate to a suspension over a natural extension of a piecewise analytic map $T:\\SS^1\\to\\SS^1$, the so-called Bowen-Series transformation. Let $\\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\\ln |T'|$. M. Pollicott showed that $\\dd_{f,s}$ is an eigenfunction of the dual operator $\\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $\\psi_{f,s}$ of $\\ll_s$ for the eigenvalue 1, given by an integral formula \\[ \\psi_{f,s} (\\xi)=\\int \\frac{J(\\xi,\\eta)}{|\\xi-\\eta|^{2s}} \\dd_{f,s} (d\\eta), \\] \\noindent where $J(\\xi,\\eta)$ is a $\\{0,1\\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface $\\DD/\\Gamma$."}
{"category": "Math", "title": "Tukey classes of ultrafilters on omega", "abstract": "Motivated by a question of Isbell, we show that Jensen's Diamond Principle implies there is a non-P-point ultrafilter U on omega such that U, whether ordered by reverse inclusion or reverse inclusion mod finite, is not Tukey equivalent to the finite sets of reals ordered by inclusion. We also show that, for every regular infinite kappa not greater than 2^{aleph_0}, if MA_{sigma-centered} holds, then some ultrafilter U on omega, ordered by reverse inclusion mod finite, is Tukey equivalent to the sets of reals of size less than kappa, ordered by inclusion. We also prove two negative ZFC results about the possible Tukey classes of ultrafilters on omega."}
{"category": "Math", "title": "Proper actions on corank-one reductive homogeneous spaces", "abstract": "Let k be a local field and G the set of k-points of a connected semisimple algebraic k-group of rank one. We describe all torsion-free discrete subgroups of G\\times G acting properly discontinuously on G by left and right multiplication. To this end, we prove a general result on the Cartan projection of discrete groups acting properly discontinuously on corank-one reductive homogeneous spaces over k."}
{"category": "Math", "title": "The Calabi flow on toric Fano surface", "abstract": "We prove the longtime existence and convergence of the Calabi flow on toric Fano surfaces in a large family of Kahler classes where the class has positive extremal Hamiltonian potential and the initial Calabi energy is bounded by some constant. This is an extension of our previous work. We use the toric condition in a more essential way to rule out bubbles."}
{"category": "Math", "title": "How to sharpen a tridiagonal pair", "abstract": "Let $\\F$ denote a field and let $V$ denote a vector space over $\\F$ with finite positive dimension. We consider a pair of linear transformations $A:V \\to V$ and $A^*:V \\to V$ that satisfy the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\\lbrace V_i\\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \\subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \\leq i \\leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\\lbrace V^*_i\\rbrace_{i=0}^\\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \\subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \\leq i \\leq \\delta$, where $V^*_{-1}=0$ and $V^*_{\\delta+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \\subseteq W$, $A^* W \\subseteq W$, $W \\neq 0$, $W \\neq V$. We call such a pair a {\\it tridiagonal pair} on $V$. It is known that $d=\\delta$, and for $0 \\leq i \\leq d$ the dimensions of $V_i, V^*_i, V_{d-i}, V^*_{d-i}$ coincide. Denote this common dimension by $\\rho_i$ and call $A,A^*$ {\\it sharp} whenever $\\rho_0=1$. Let $T$ denote the $\\F$-subalgebra of ${\\rm End}_\\F(V)$ generated by $A,A^*$. We show: (i) the center $Z(T)$ is a field whose dimension over $\\F$ is $\\rho_0$; (ii) the field $Z(T)$ is isomorphic to each of $E_0TE_0$, $E_dTE_d$, $E^*_0TE^*_0$, $E^*_dTE^*_d$, where $E_i$ (resp. $E^*_i$) is the primitive idempotent of $A$ (resp. $A^*$) associated with $V_i$ (resp. $V^*_i$); (iii) with respect to the $Z(T)$-vector space $V$ the pair $A,A^*$ is a sharp tridiagonal pair."}
{"category": "Math", "title": "Multi-center clinical trials: Randomization and ancillary statistics", "abstract": "The purpose of this paper is to investigate and develop methods for analysis of multi-center randomized clinical trials which only rely on the randomization process as a basis of inference. Our motivation is prompted by the fact that most current statistical procedures used in the analysis of randomized multi-center studies are model based. The randomization feature of the trials is usually ignored. An important characteristic of model based analysis is that it is straightforward to model covariates. Nevertheless, in nearly all model based analyses, the effects due to different centers and, in general, the design of the clinical trials are ignored. An alternative to a model based analysis is to have analyses guided by the design of the trial. Our development of design based methods allows the incorporation of centers as well as other features of the trial design. The methods make use of conditioning on the ancillary statistics in the sample space generated by the randomization process. We have investigated the power of the methods and have found that, in the presence of center variation, there is a significant increase in power. The methods have been extended to group sequential trials with similar increases in power."}
{"category": "Math", "title": "Should the democrats move to the left on economic policy?", "abstract": "Could John Kerry have gained votes in the 2004 Presidential election by more clearly distinguishing himself from George Bush on economic policy? At first thought, the logic of political preferences would suggest not: the Republicans are to the right of most Americans on economic policy, and so in a one-dimensional space with party positions measured with no error, the optimal strategy for the Democrats would be to stand infinitesimally to the left of the Republicans. The median voter theorem suggests that each party should keep its policy positions just barely distinguishable from the opposition. In a multidimensional setting, however, or when voters vary in their perceptions of the parties' positions, a party can benefit from putting some daylight between itself and the other party on an issue where it has a public-opinion advantage (such as economic policy for the Democrats). We set up a plausible theoretical model in which the Democrats could achieve a net gain in votes by moving to the left on economic policy, given the parties' positions on a range of issue dimensions. We then evaluate this model based on survey data on voters' perceptions of their own positions and those of the candidates in 2004. Under our model, it turns out to be optimal for the Democrats to move slightly to the right but staying clearly to the left of the Republicans' current position on economic issues."}
{"category": "Math", "title": "Conservative statistical post-election audits", "abstract": "There are many sources of error in counting votes: the apparent winner might not be the rightful winner. Hand tallies of the votes in a random sample of precincts can be used to test the hypothesis that a full manual recount would find a different outcome. This paper develops a conservative sequential test based on the vote-counting errors found in a hand tally of a simple or stratified random sample of precincts. The procedure includes a natural escalation: If the hypothesis that the apparent outcome is incorrect is not rejected at stage $s$, more precincts are audited. Eventually, either the hypothesis is rejected--and the apparent outcome is confirmed--or all precincts have been audited and the true outcome is known. The test uses a priori bounds on the overstatement of the margin that could result from error in each precinct. Such bounds can be derived from the reported counts in each precinct and upper bounds on the number of votes cast in each precinct. The test allows errors in different precincts to be treated differently to reflect voting technology or precinct sizes. It is not optimal, but it is conservative: the chance of erroneously confirming the outcome of a contest if a full manual recount would show a different outcome is no larger than the nominal significance level. The approach also gives a conservative $P$-value for the hypothesis that a full manual recount would find a different outcome, given the errors found in a fixed size sample. This is illustrated with two contests from November, 2006: the U.S. Senate race in Minnesota and a school board race for the Sausalito Marin City School District in California, a small contest in which voters could vote for up to three candidates."}
{"category": "Math", "title": "$p$-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas", "abstract": "Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\\geq 5$ and has complex multiplication by the full ring of integers $\\mathcal{O}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove $p$-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function."}
{"category": "Math", "title": "The Kronecker limit formulas via the distribution relation", "abstract": "In this paper, we give a proof of the classical Kronecker limit formulas using the distribution relation of the Eisenstein-Kronecker series. Using a similar idea, we then prove $p$-adic analogues of the Kronecker limit formulas for the $p$-adic Eisenstein-Kronecker functions defined in our previous paper."}
{"category": "Math", "title": "Prediction of multivariate responses with a select number of principal components", "abstract": "This paper proposes a new method and algorithm for predicting multivariate responses in a regression setting. Research into classification of High Dimension Low Sample Size (HDLSS) data, in particular microarray data, has made considerable advances, but regression prediction for high-dimensional data with continuous responses has had less attention. Recently Bair et al (2006) proposed an efficient prediction method based on supervised principal component regression (PCR). Motivated by the fact that a larger number of principal components results in better regression performance, this paper extends the method of Bair et al in several ways: a comprehensive variable ranking is combined with a selection of the best number of components for PCR, and the new method further extends to regression with multivariate responses. The new method is particularly suited to HDLSS problems. Applications to simulated and real data demonstrate the performance of the new method. Comparisons with Bair et al (2006) show that for high-dimensional data in particular the new ranking results in a smaller number of predictors and smaller errors."}
{"category": "Math", "title": "Discussion of: Treelets--An adaptive multi-Scale basis for sparse unordered data", "abstract": "Discussion of \"Treelets--An adaptive multi-Scale basis for sparse unordered data\" [arXiv:0707.0481]"}
{"category": "Math", "title": "Discussion of: Treelets--An adaptive multi-scale basis for sparse unordered data", "abstract": "Discussion of \"Treelets--An adaptive multi-scale basis for sparse unordered data\" [arXiv:0707.0481]"}
{"category": "Math", "title": "Discussion of: Treelets--An adaptive multi-scale basis for sparse unordered data", "abstract": "We congratulate Lee, Nadler and Wasserman (henceforth LNW) on a very interesting paper on new methodology and supporting theory [arXiv:0707.0481]. Treelets seem to tackle two important problems of modern data analysis at once. For datasets with many variables, treelets give powerful predictions even if variables are highly correlated and redundant. Maybe more importantly, interpretation of the results is intuitive. Useful insights about relevant groups of variables can be gained. Our comments and questions include: (i) Could the success of treelets be replicated by a combination of hierarchical clustering and PCA? (ii) When choosing a suitable basis, treelets seem to be largely an unsupervised method. Could the results be even more interpretable and powerful if treelets would take into account some supervised response variable? (iii) Interpretability of the result hinges on the sparsity of the final basis. Do we expect that the selected groups of variables will always be sufficiently small to be amenable for interpretation?"}
{"category": "Math", "title": "Discussion of: Treelets--An adaptive multi-scale basis for sparse unordered data", "abstract": "Discussion of \"Treelets--An adaptive multi-scale basis for sparse unordered data\" [arXiv:0707.0481]"}
{"category": "Math", "title": "Global $L^{p}$ estimates for degenerate Ornstein-Uhlenbeck operators", "abstract": "We consider a class of degenerate Ornstein-Uhlenbeck operators in $\\mathbb{R}^{N}$, of the kind \\[ \\mathcal{A}\\equiv\\sum_{i,j=1}^{p_{0}}a_{ij}\\partial_{x_{i}x_{j}}^{2} +\\sum_{i,j=1}^{N}b_{ij}x_{i}\\partial_{x_{j}}% \\] where $(a_{ij}) ,(b_{ij}) $ are constant matrices, $(a_{ij}) $ is symmetric positive definite on $\\mathbb{R} ^{p_{0}}$ ($p_{0}\\leq N$), and $(b_{ij}) $ is such that $\\mathcal{A}$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\\infty$) of the kind:% \\[ \\Vert \\partial_{x_{i}x_{j}}^{2}u\\Vert_{L^{p}(\\mathbb{R}% ^{N})}\\leq c\\{\\Vert \\mathcal{A}u\\Vert_{L^{p}(\\mathbb{R}^{N})}+\\Vert u\\Vert_{L^{p}(\\mathbb{R}% ^{N})}\\} \\text{for}i,j=1,2,...,p_{0}% \\] and corresponding weak (1,1) estimates. This result seems to be the first case of global estimates, in Lebesgue $L^{p}$ spaces, for complete H\\\"{o}rmander's operators $ \\sum X_{i}^{2}+X_{0},$ proved in absence of a structure of homogeneous group. We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:% \\[ \\Vert \\partial_{x_{i}x_{j}}^{2}u\\Vert_{L^{p}(S)}\\leq c\\Vert Lu\\Vert_{L^{p}(S)}% \\] for any $u\\in C_{0}^{\\infty}(S) ,$ where $S$ is the strip $\\mathbb{R}^{N}\\times[ -1,1] $ and $L$ is the Kolmogorov-Fokker-Planck operator $\\mathcal{A}-\\partial_{t}.$"}
{"category": "Math", "title": "Discussion of: Treelets--An adaptive multi-scale basis for sparse unordered data", "abstract": "This is a discussion of paper \"Treelets--An adaptive multi-scale basis for sparse unordered data\" [arXiv:0707.0481] by Ann B. Lee, Boaz Nadler and Larry Wasserman. In this paper the authors defined a new type of dimension reduction algorithm, namely, the treelet algorithm. The treelet method has the merit of being completely data driven, and its decomposition is easier to interpret as compared to PCR. It is suitable in some certain situations, but it also has its own limitations. I will discuss both the strength and the weakness of this method when applied to microarray data analysis."}
{"category": "Math", "title": "Discussion of: Treelets--An adaptive multi-scale basis for sparse unordered data", "abstract": "We would like to congratulate Lee, Nadler and Wasserman on their contribution to clustering and data reduction methods for high $p$ and low $n$ situations. A composite of clustering and traditional principal components analysis, treelets is an innovative method for multi-resolution analysis of unordered data. It is an improvement over traditional PCA and an important contribution to clustering methodology. Their paper [arXiv:0707.0481] presents theory and supporting applications addressing the two main goals of the treelet method: (1) Uncover the underlying structure of the data and (2) Data reduction prior to statistical learning methods. We will organize our discussion into two main parts to address their methodology in terms of each of these two goals. We will present and discuss treelets in terms of a clustering algorithm and an improvement over traditional PCA. We will also discuss the applicability of treelets to more general data, in particular, the application of treelets to microarray data."}
{"category": "Math", "title": "Rejoinder of: Treelets--An adaptive multi-scale basis for spare unordered data", "abstract": "Rejoinder of \"Treelets--An adaptive multi-scale basis for spare unordered data\" [arXiv:0707.0481]"}
{"category": "Math", "title": "Homology cylinders and sutured manifolds for homologically fibered knots", "abstract": "Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured manifolds associated with a special class of knots which we call {\\it homologically fibered knots}. Then we use invariants of homology cylinders to give applications to knot theory such as fibering obstructions, Reidemeister torsions and handle numbers of homologically fibered knots."}
{"category": "Math", "title": "A short proof and a generalization of the BKR-inequality", "abstract": "There is a serious mistake in the proof."}
{"category": "Math", "title": "New identities for the Glasser transform and their applications", "abstract": "In the present paper the authors show that an iteration of the $\\mathscr{L}_{2}$-transform by itself is a constant multiple of the Glasser transform. Using this iteration identity, a Parseval-Goldstein type theorem for $\\mathscr{L}_{2}$-transform and the Glasser transform is given. By making use of these results a number of new Parseval-Goldstein type identities are obtained for these and many other well-known integral transforms. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results presented here."}
{"category": "Math", "title": "A duality theorem for Dieudonne displays", "abstract": "We show that the Zink equivalence between p-divisible groups and Dieudonne displays over a complete local ring with perfect residue field of characteristic p is compatible with duality. The proof relies on a new explicit formula for the p-divisible group associated to a Dieudonne display."}
{"category": "Math", "title": "Long-time behavior of stochastically perturbed neuronal networks", "abstract": "Our investigation is specially motivated by the stochastic version of a common model of potential spread in a dendritic tree. We do not assume the noise in the junction points to be Markovian. In fact, we allow for long-range dependence in time of the stochastic perturbation. This leads to an abstract formulation in terms of a stochastic diffusion with dynamic boundary conditions, featuring fractional Brownian motion. We prove results on existence, uniqueness and asymptotics of weak and strong solutions to such a stochastic differential equation."}
{"category": "Math", "title": "Polynomial selections and separation by polynomials", "abstract": "Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex functions of higher order. Another application is some Hyers-Ulam-stability-type result."}
{"category": "Math", "title": "Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields", "abstract": "We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory requirements of the algorithm appear to be very good: for a given prime number p, it computes the p-valuation of the discriminant and the factorization of p in a number field of degree 1000 in a few seconds, in a personal computer."}
{"category": "Math", "title": "Bas du spectre de surfaces hyperboliques de volume infini", "abstract": "This article presents some methods to control the bottom of the spectrum of the Laplacian $\\lambda_0$ on hyperbolic surfaces with infinite volume. Our first result bounds the $\\lambda_0$ of a geometrically finite surface in terms of the geometry of its convex core. We then focus on infinite type periodic hyperbolic surfaces built by gluing copies of a geometrically finite surface with boundary according to the plan of an infinite graph. We control the $\\lambda_0$ of the so-obtained infinite surfaces by constants coming from spectral properties of the building brick and combinatorial datae of the graph. We then use these methods to control the $\\lambda_0$ of two other kind of infinite type hyperbolic surfaces : those admitting a splitting into bounded pieces, and some riemannian coverings."}
{"category": "Math", "title": "Forecasting time series of inhomogeneous Poisson processes with application to call center workforce management", "abstract": "We consider forecasting the latent rate profiles of a time series of inhomogeneous Poisson processes. The work is motivated by operations management of queueing systems, in particular, telephone call centers, where accurate forecasting of call arrival rates is a crucial primitive for efficient staffing of such centers. Our forecasting approach utilizes dimension reduction through a factor analysis of Poisson variables, followed by time series modeling of factor score series. Time series forecasts of factor scores are combined with factor loadings to yield forecasts of future Poisson rate profiles. Penalized Poisson regressions on factor loadings guided by time series forecasts of factor scores are used to generate dynamic within-process rate updating. Methods are also developed to obtain distributional forecasts. Our methods are illustrated using simulation and real data. The empirical results demonstrate how forecasting and dynamic updating of call arrival rates can affect the accuracy of call center staffing."}
{"category": "Math", "title": "Optimal weighting for false discovery rate control", "abstract": "How to weigh the Benjamini-Hochberg procedure? In the context of multiple hypothesis testing, we propose a new step-wise procedure that controls the false discovery rate (FDR) and we prove it to be more powerful than any weighted Benjamini-Hochberg procedure. Both finite-sample and asymptotic results are presented. Moreover, we illustrate good performance of our procedure in simulations and a genomics application. This work is particularly useful in the case of heterogeneous $p$-value distributions."}
{"category": "Math", "title": "The cylinder over the Koras-Russell cubic threefold has a trivial Makar-Limanov invariant", "abstract": "We show that the Makar-Limanov invariant of the cylinder over the Koras-Russell cubic affine threefold is trivial. This means that regular functions which are invariant under all algebraic actions of the additive group on this variety are constants."}
{"category": "Math", "title": "Estimating a difference between Kullback-Leibler risks by a normalized difference of AIC", "abstract": "AIC is commonly used for model selection but the precise value of AIC has no direct interpretation. We are interested in quantifying a difference of risks between two models. This may be useful for both an explanatory point of view or for prediction, where a simpler model may be preferred if it does nearly as well as a more complex model. The difference of risks can be interpreted by linking the risks with relative errors in the computation of probabilities and looking at the values obtained for simple models. A scale of values going from negligible to large is proposed. We propose a normalization of a difference of Akaike criteria for estimating the difference of expected Kullback-Leibler risks between maximum likelihood estimators of the distribution in two different models. The variability of this statistic can be estimated. Thus, an interval can be constructed which contains the true difference of expected Kullback-Leibler risks with a pre-specified probability. A simulation study shows that the method works and it is illustrated on two examples. The first is a study of the relationship between body-mass index and depression in elderly people. The second is the choice between models of HIV dynamics, where one model makes the distinction between activated CD4+ T lymphocytes and the other does not."}
{"category": "Math", "title": "Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation", "abstract": "In this paper, we prove the nonlinear orbital stability of the stationary traveling wave of the one-dimensional Gross-Pitaevskii equation by using Zakharov-Shabat's inverse scattering method."}
{"category": "Math", "title": "On spatial extremes: with application to a rainfall problem", "abstract": "We consider daily rainfall observations at 32 stations in the province of North Holland (the Netherlands) during 30 years. Let $T$ be the total rainfall in this area on one day. An important question is: what is the amount of rainfall $T$ that is exceeded once in 100 years? This is clearly a problem belonging to extreme value theory. Also, it is a genuinely spatial problem. Recently, a theory of extremes of continuous stochastic processes has been developed. Using the ideas of that theory and much computer power (simulations), we have been able to come up with a reasonable answer to the question above."}
{"category": "Math", "title": "Cohomology of the toroidal compactification of A_3", "abstract": "We prove that the cohomology groups with rational coefficients of the Voronoi compactification of the moduli space of abelian threefolds coincide with the Chow groups of that space, as determined by Van der Geer."}
{"category": "Math", "title": "On the non-generic representation theory of the symplectic blob algebra", "abstract": "This paper reports some advances in the study of the symplectic blob algebra. We find a presentation for this algebra. We find a minimal poset for this as a quasi-hereditary algebra. We discuss how to reduce the number of parameters defining the algebra from 6 to 4 (or even 3) without loss of representation theoretic generality. We then find some non-semisimple specialisations by calculating Gram determinants for certain cell modules (or standard modules) using the good parametrisation defined. We finish by considering some quotients of specialisations of the symplectic blob algebra which are isomorphic to Temperley--Lieb algebras of type $A$."}
{"category": "Math", "title": "Support-type properties of convex functions of higher order and Hadamard-type inequalities", "abstract": "It is well-known that every convex function admits an affine support at every interior point of a domain. Convex functions of higher order (precisely of an odd order) have a similar property: they are supported by the polynomials of degree no greater than the order of convexity. In this paper the attaching method is developed. It is applied to obtain the general result Theorem 2, from which the mentioned above support theorem and some related properties of convex functions of higher (both odd and even) order are derived. They are applied to obtain some known and new Hadamard-type inequalities between the quadrature operators and the integral approximated by them. It is also shown that the error bounds of quadrature rules follow by inequalities of this kind."}
{"category": "Math", "title": "On a remark by Y. Namikawa", "abstract": "The aim of the present paper is on the one hand to produce examples supporting the conclusion of Y. Namikawa in Remark 2.8 of \\cite{N} and improving considerations of Example 1.11 of the same paper. On the other hand, it is intended to give a geometric interpretation of the rigidity properties of some trees of exceptional rational curves, as observed by Namikawa, which can be obtained by factorizing small resolutions through nodal threefolds."}
{"category": "Math", "title": "A study of pre-validation", "abstract": "Given a predictor of outcome derived from a high-dimensional dataset, pre-validation is a useful technique for comparing it to competing predictors on the same dataset. For microarray data, it allows one to compare a newly derived predictor for disease outcome to standard clinical predictors on the same dataset. We study pre-validation analytically to determine if the inferences drawn from it are valid. We show that while pre-validation generally works well, the straightforward \"one degree of freedom\" analytical test from pre-validation can be biased and we propose a permutation test to remedy this problem. In simulation studies, we show that the permutation test has the nominal level and achieves roughly the same power as the analytical test."}
{"category": "Math", "title": "A block decomposition of finite-dimensional representations of twisted loop algebras", "abstract": "In this paper we consider the category of F^\\sigma of finite-dimensional representations of a twisted loop algebra corresponding to a finite-dimensional Lie algebra with non-trivial diagram automorphism. Although F^\\sigma is not semisimple, it can be written as a sum of indecomposable subcategories (the blocks of the category). To describe these summands, we introduce the twisted spectral characters for the twisted loop algebra. These are certain equivalence classes of the spectral characters defined by Chari and Moura for an untwisted loop algebra, which were used to provide a description of the blocks of finite--dimensional representations of the untwisted loop algebra. Here we adapt this decomposition to parametrize and describe the blocks of F^\\sigma, via the twisted spectral characters."}
{"category": "Math", "title": "Relative injectivity as cocompleteness for a class of distributors", "abstract": "Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:0804.0326 [math.CV]] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadicity of the category of injective spaces and left adjoints over $\\mathsf{Set}$. In this paper we generalise these results, studying cocompleteness with respect to a given class of distributors. We show in particular that the description of several semantic domains presented in [M. Escard\\'o and B. Flagg, Semantic domains, injective spaces and monads, Electronic Notes in Theoretical Computer Science 20 (1999)] can be translated into the $\\mathsf{V}$-enriched setting."}
{"category": "Math", "title": "Some properties of generalized higher-order convexity", "abstract": "The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved."}
{"category": "Math", "title": "Hermite-Hadamard-type inequalities in the approximate integration", "abstract": "We give a slight extension of the Hermite-Hadamard inequality on simplices and we use it to establish error bounds of the operators connected with the approximate integration."}
{"category": "Math", "title": "On some extremalities in the approximate integration", "abstract": "Some extremalities for quadrature operators are proved for convex functions of higher order. Such results are known in the numerical analysis, however they are often proved under suitable differentiability assumptions. In our considerations we do not use any other assumptions apart from higher order convexity itself. The obtained inequalities refine the inequalities of Hadamard type. They are applied to give error bounds of quadrature operators under the assumptions weaker from the commonly used."}
{"category": "Math", "title": "The planar algebra of group-type subfactors", "abstract": "If $G$ is a countable, discrete group generated by two finite subgroups $H$ and $K$ and $P$ is a II$_1$ factor with an outer G-action, one can construct the group-type subfactor $P^H \\subset P \\rtimes K$ introduced in \\cite{BH}. This construction was used in \\cite{BH} to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra (in the sense of Jones \\cite{J2}) of this subfactor and prove that any subfactor with an abstract planar algebra of \"group type\" arises from such a subfactor. The action of Jones' planar operad is determined explicitly."}
{"category": "Math", "title": "Braid Group Representations arising from the Yang Baxter Equation", "abstract": "This paper aims to determine the images of the braid group under representations afforded by the Yang Baxter equation when the solution is a nontrivial $4 \\times 4$ matrix. Making the assumption that all the eigenvalues of the Yang Baxter solution are roots of unity, leads to the conclusion that all the images are finite. Using results of Turaev, we have also identified cases in which one would get a link invariant. Finally, by observing the group algebra generated by the image of the braid group sometimes factor through known algebras, in certain instances we can identify the invariant as particular specializations of a known invariant."}
{"category": "Math", "title": "Une suite de matrices sym\\'etriques en rapport avec la fonction de Mertens", "abstract": "In this paper we explore a class of equivalence relations over $\\N^\\ast$ from which is constructed a sequence of symetric matrices related to the Mertens function. From numerical experimentations we suggest a conjecture, about the growth of the quadratic norm of these matrices, which implies the Riemann hypothesis. This suggests that matrix analysis methods may play a more important part in this classical and difficult problem."}
{"category": "Math", "title": "An orthogonal approach to the subfactor of a planar algebra", "abstract": "By changing to an orthogonal basis, we give a short proof that the subfactor of the graded algebra of a planar algebra reproduces the planar algebra."}
{"category": "Math", "title": "Integral stability of Calder\\'on inverse conductivity problem in the plane", "abstract": "It is proved that, in two dimensions, the Calder\\'on inverse conductivity problem in Lipschitz domains is stable in the $L^p$ sense when the conductivities are uniformly bounded in any fractional Sobolev space $W^{\\alpha,p}$ $\\alpha>0, 1<p<\\infty$."}
{"category": "Math", "title": "The representation theory of cyclotomic BMW algebras", "abstract": "In this paper, we go on Rui-Xu's work on cyclotomic Birman-Wenzl algebras $\\W_{r, n}$ in \\cite{RX}. In particular, we use the representation theory of cellular algebras in \\cite{GL} to classify the irreducible $\\W_{r, n}$-modules for all positive integers $r$ and $n$. By constructing cell filtrations for all cell modules of $\\W_{r, n}$, we compute the discriminants associated to all cell modules for $\\W_{r, n} $. Via such discriminats together with induction and restriction functors given in section~5, we determine explicitly when $\\W_{r, n}$ is semisimple over a field. This generalizes our previous result on Birman-Wenzl algebras in \\cite{RS1}."}
{"category": "Math", "title": "On affine selections of set-valued functions", "abstract": "The main result states that every convex set-valued function defined on a real interval with compact values in a locally convex space, admits an affine selection. In the case if the target space is a real line and the values are closed real intervals, we can replace the convexity assumption by the more general condition."}
{"category": "Math", "title": "Marked tubes and the graph multiplihedron", "abstract": "Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of the multiphihedron, but features of this polytope appear in works related to quilted disks, bordered Riemann surfaces, and operadic structures. Certain examples of graph multiplihedra are related to Minkowski sums of simplices and cubes and others to the permutohedron."}
{"category": "Math", "title": "Analytic relations on a dynamical orbit", "abstract": "Let $(K,|\\cdot|)$ be a complete discretely valued field and $f:{\\mathbb B}_1(K,1) \\to {\\mathbb B}_1(K,1)$ a nonconstant analytic map from the unit back to itself. We assume that 0 is an attracting fixed point of $f$. Let $a \\in K$ with $\\lim_{n \\to \\infty} f^n(a) = 0$ and consider the orbit ${\\mathcal O}_f(a) := \\{f^n(a) : n \\in {\\mathbb N} \\}$. We show that if 0 is a \\emph{superattracting} fixed point, then every irreducible analytic subvariety of ${\\mathbb B}_n(K,1)$ meeting ${\\mathcal O}_f(a)^n$ in an analytically Zariski dense set is defined by equations of the form $x_i = b$ and $x_j = f^\\ell(x_k)$. When 0 is an attracting, non-superattracting point, we show that all analytic relations come from algebraic tori."}
{"category": "Math", "title": "Combinatorial cell complexes and Poincare duality", "abstract": "We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces, combinatorial cell complexes (or c.c.c). We define orientability, homology and cohomology of c.c.c's and develop enough algebraic topology in this setting to prove the Poincare duality theorem for a c.c.c satisfying suitable regularity conditions. The definitions and proofs are completely finitary and combinatorial in nature."}
{"category": "Math", "title": "Hopf algebras and the logarithm of the S-transform in free probability", "abstract": "Let k be a positive integer and let G_k denote the set of non-commutative k-variable distributions \\mu such that \\mu (X_1) = ... = \\mu (X_k) = 1. G_k is a group under the operation of free multiplicative convolution. We identify G_k as the group of characters of a certain Hopf algebra Y_k. Then, by using the log map from characters to infinitesimal characters of Y_k, we introduce a transform LS_{\\mu} for distributions \\mu in G_k. The main property of the LS-transform is that it linearizes commuting products in G_k. For \\mu in G_k, the transform LS_{\\mu} is a power series in k non-commuting indeterminates; its coefficients can be computed from the coefficients of the R-transform of \\mu by using summations over chains in the lattices NC(n) of non-crossing partitions. In the particular case k=1 one has that Y_1 is naturally isomorphic to the Hopf algebra Sym of symmetric functions, and that the LS-transform is very closely related to the logarithm of the S-transform of Voiculescu, by the formula LS(z) = - z log S(z). In this case the group G_1 can be identified as the group of characters of Sym, in such a way that the S-transform, its reciprocal 1/S and its logarithm log S relate in a natural sense to the sequences of complete, elementary and respectively power sum symmetric functions."}
{"category": "Math", "title": "The canonical shrinking soliton associated to a Ricci flow", "abstract": "To every Ricci flow on a manifold M over a time interval I, we associate a shrinking Ricci soliton on the space-time M x I. We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author, and McCann and the second author; we briefly survey the link between these subjects."}
{"category": "Math", "title": "Jacobi-Bernoulli cohomology and deformations of schemes and maps", "abstract": "We introduce a notion of Jacobi-Bernoulli cohomology associated to a semi-simplicial Lie algebra (SELA). For an algebraic scheme $X$ over $\\C$, we construct a tangent SELA $\\t_X$ and show that the Jacobi-Bernoulli cohomology of $\\t_X$ is related to infinitesimal deformations of $X$."}
{"category": "Math", "title": "Aggregation of Risks and Asymptotic independence", "abstract": "We study the tail behavior of the distribution of the sum of asymptotically independent risks whose marginal distributions belong to the maximal domain of attraction of the Gumbel distribution. We impose conditions on the distribution of the risks $(X,Y)$ such that $P(X + Y > x) \\sim (const)P (X > x)$. With the further assumption of non-negativity of the risks, the result is extended to more than two risks. We note a sufficient condition for a distribution to belong to both the maximal domain of attraction of the Gumbel distribution and the subexponential class. We provide examples of distributions which satisfy our assumptions. The examples include cases where the marginal distributions of $X$ and $Y$ are subexponential and also cases where they are not. In addition, the asymptotic behavior of linear combinations of such risks with positive coefficients is explored leading to an approximate solution of an optimization problem which is applied to portfolio design."}
{"category": "Math", "title": "The weight filtration for real algebraic varieties", "abstract": "Using the work of Guillen and Navarro Aznar we associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on Borel-Moore homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties."}
{"category": "Math", "title": "Matzoh ball soup in spaces of constant curvature", "abstract": "In this paper, we generalize Magnanini-Sakaguchi's result [MS3] from Euclidean space to spaces of constant curvature. More precisely, we show that if a conductor satisfying the exterior geodesic sphere condition in the space of constant curvature has initial temperature 0 and its boundary is kept at temperature 1 (at all times), if the thermal conductivity of the conductor is inverse of its metric, and if the conductor contains a proper sub-domain, satisfying the interior geodesic cone condition and having constant boundary temperature at each given time, then the conductor must be a geodesic ball. Moreover, we show similar result for the wave equations and the Schr\\\"{o}dinger equations in spaces of constant curvature."}
{"category": "Math", "title": "QR-Adjustment for Clustering Tests Based on Nearest Neighbor Contingency Tables", "abstract": "The spatial interaction between two or more classes of points may cause spatial clustering patterns such as segregation or association, which can be tested using a nearest neighbor contingency table (NNCT). A NNCT is constructed using the frequencies of class types of points in nearest neighbor (NN) pairs. For the NNCT-tests, the null pattern is either complete spatial randomness (CSR) of the points from two or more classes (called CSR independence) or random labeling (RL). The distributions of the NNCT-test statistics depend on the number of reflexive NNs (denoted by $R$) and the number of shared NNs (denoted by $Q$), both of which depend on the allocation of the points. Hence $Q$ and $R$ are fixed quantities under RL, but random variables under CSR independence. Using their observed values in NNCT analysis makes the distributions of the NNCT-test statistics conditional on $Q$ and $R$ under CSR independence. In this article, I use the empirically estimated expected values of $Q$ and $R$ under CSR independence pattern to remove the conditioning of NNCT-tests (such a correction is called the \\emph{QR-adjustment}, henceforth). I present a Monte Carlo simulation study to compare the conditional NNCT-tests and QR-adjusted tests under CSR independence and segregation and association alternatives. I demonstrate that QR-adjustment does not significantly improve the empirical size estimates under CSR independence and power estimates under segregation or association alternatives. For illustrative purposes, I apply the conditional and empirically corrected tests on two example data sets."}
{"category": "Math", "title": "Large scale behavior of semiflexible heteropolymers", "abstract": "We consider a general discrete model for heterogeneous semiflexible polymer chains. Both the thermal noise and the inhomogeneous character of the chain (the disorder) are modeled in terms of random rotations. We focus on the quenched regime, i.e., the analysis is performed for a given realization of the disorder. Semiflexible models differ substantially from random walks on short scales, but on large scales a Brownian behavior emerges. By exploiting techniques from tensor analysis and non-commutative Fourier analysis, we establish the Brownian character of the model on large scale and we obtain an expression for the diffusion constant. We moreover give conditions yielding quantitative mixing properties."}
{"category": "Math", "title": "Analytic subdivision invariants", "abstract": "This paper introduces an inner product on chain complexes of finite simplicial complexes that is well-adapted to the harmonic study of subdivisions. Its definition utilizes a decomposition of the chain spaces that suggests a sequence of subdivision invariants which we show do not all vanish for non-trivial subdivisions. We exhibit a combinatorial lower bound for these invariants and provide an effective algorithm for their computation. Unfortunately, these invariants cannot distinguish every subdivision nor do they necessarily increase over successive subdivisions."}
{"category": "Math", "title": "On the Use of Nearest Neighbor Contingency Tables for Testing Spatial Segregation", "abstract": "For two or more classes (or types) of points, nearest neighbor contingency tables (NNCTs) are constructed using nearest neighbor (NN) frequencies and are used in testing spatial segregation of the classes. Pielou's test of independence, Dixon's cell-specific, class-specific, and overall tests are the tests based on NNCTs (i.e., they are NNCT-tests). These tests are designed and intended for use under the null pattern of random labeling (RL) of completely mapped data. However, it has been shown that Pielou's test is not appropriate for testing segregation against the RL pattern while Dixon's tests are. In this article, we compare Pielou's and Dixon's NNCT-tests; introduce the one-sided versions of Pielou's test; extend the use of NNCT-tests for testing complete spatial randomness (CSR) of points from two or more classes (which is called \\emph{CSR independence}, henceforth). We assess the finite sample performance of the tests by an extensive Monte Carlo simulation study and demonstrate that Dixon's tests are also appropriate for testing CSR independence; but Pielou's test and the corresponding one-sided versions are liberal for testing CSR independence or RL. Furthermore, we show that Pielou's tests are only appropriate when the NNCT is based on a random sample of (base, NN) pairs. We also prove the consistency of the tests under their appropriate null hypotheses. Moreover, we investigate the edge (or boundary) effects on the NNCT-tests and compare the buffer zone and toroidal edge correction methods for these tests. We illustrate the tests on a real life and an artificial data set."}
{"category": "Math", "title": "Introduction to Normed *-Algebras and their Representations, 6th ed", "abstract": "This book treats: - spectral theory of Banach *-algebras, - basic representation theory of normed *-algebras, - spectral theory of representations of commutative *-algebras. A novel feature of the book is the construction of the enveloping C*-algebra of a general normed *-algebra."}
{"category": "Math", "title": "Powers of Ideals and Fibers of Morphisms", "abstract": "Let X\\subset PP^n be a projective scheme over a field, and let phi:X --> Y be a finite morphism. Our main result is a formula in terms of global data for the maximum of the Castelnuovo-Mumford regularity of the fibers of \\phi, considered as subschemes of \\PP^n. From an algebraic point of view, our formula is related to the theorem of Cutkosky-Herzog-Trung and Kodiyalam showing that for any homogeneous ideal I in a standard graded algebra S, the regularity of I^t can be written as dt+\\epsilon for some non-negative integers d, \\epsilon, and all large t. In the special case where I contains a power of S_+ and is generated by forms of a single degree, our formula gives an interpretation of \\epsilon: it is one less than the maximum regularity of a fiber of the morphism associated to I. These formulas have strong consequences for ideals generated by generic forms."}
{"category": "Math", "title": "Cluster algebras and Grassmannians of type G2", "abstract": "We prove a conjecture of Geiss, Leclerc and Schr\\\"{o}er, producing cluster algebra structures on multi-homogeneous coordinate ring of partial flag varieties, for the case $G_2$. As a consequence we sharpen the known fact that coordinate ring of the double Bruhat cell $G^{e,w_0}$ is an upper cluster algebra, by proving that it is a cluster algebra."}
{"category": "Math", "title": "Concrete Duality for Strict Infinity Categories", "abstract": "An elementary theory of strict $\\infty $-categories with application to concrete duality is given. All known famous dualities (Gelfand-Naimark, Pontryagin, Stone, etc.) are so-called natural. A criterion of existence of such a duality for higher categories is formulated. New examples are presented."}
{"category": "Math", "title": "Cohomological rigidity of real Bott manifolds", "abstract": "A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a real toric manifold and admits a flat riemannian metric invariant under the natural action of an elementary abelian 2-group. We also prove that the converse is true, namely a real toric manifold which admits a flat riemannian metric invariant under the action of an elementary abelian 2-group is a real Bott manifold."}
{"category": "Math", "title": "On a class of II_1 factors with at most one Cartan subalgebra II", "abstract": "This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II_1. We provide more examples of II_1 factors having either zero, one or several Cartan subalgebras. We also prove a rigidity result for some group measure space II_1 factors."}
{"category": "Math", "title": "C^0-rigidity of the double Poisson bracket", "abstract": "The paper is devoted to function theory on symplectic manifolds. We study a natural class of functionals involving the double Poisson brackets from the viewpoint of their robustness properties with respect to small perturbations in the uniform norm. We observe an hierarchy of such robustness properties. The methods involve Hofer's geometry on the symplectic side and Landau-Hadamard-Kolmogorov inequalities on the function-theoretic side."}
{"category": "Math", "title": "Generating families and Legendrian contact homology in the standard contact space", "abstract": "We show that if a Legendrian knot in standard contact ${\\bb R}^3$ possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting we show Sabloff's duality result for LCH may be viewed as Alexander duality. In addition, we provide an explicit construction of a generating family for a front diagram with graded normal ruling and give a new approach to augmentation $\\Rightarrow$ normal ruling."}
{"category": "Math", "title": "The $\\Lambda$-coalescent speed of coming down from infinity", "abstract": "Consider a $\\Lambda$-coalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number $N_t$ of blocks at any positive time $t>0$). We exhibit a deterministic function $v:(0,\\infty)\\to(0,\\infty)$ such that $N_t/v(t)\\to1$, almost surely, and in $L^p$ for any $p\\geq1$, as $t\\to0$. Our approach relies on a novel martingale technique."}
{"category": "Math", "title": "Quantum Isometry Groups of 0- Dimensional Manifolds", "abstract": "Quantum isometry groups of spectral triples associated with approximately finite-dimensional C*-algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli diagrams. This is used to determine explicitly the quantum isometry group of the natural spectral triple on the algebra of continuous functions on the middlethird Cantor set. It is also shown that the quantum symmetry groups of finite graphs or metric spaces coincide with the quantum isometry groups of the corresponding classical objects equipped with natural Laplacians."}
{"category": "Math", "title": "Necessary and sufficient optimality conditions for relaxed and strict control problems of backward systems", "abstract": "We consider a stochastic control problem where the set of strict (classical) controls is not necessarily convex, and the system is governed by a nonlinear backward stochastic differential equation. By introducing a new approach, we establish necessary as well as sufficient conditions of optimality for two models. The first concerns the relaxed controls, who are measure-valued processes. The second is a particular case of the first and relates to strict control problems."}
{"category": "Math", "title": "Algorithms for computing multiplier ideals", "abstract": "We give algorithms for computing multiplier ideals using Gr\\\"obner bases in Weyl algebras. The algorithms are based on a newly introduced notion which is a variant of Budur--Musta\\c{t}\\v{a}--Saito's (generalized) Bernstein--Sato polynomial. We present several examples computed by our algorithms."}
{"category": "Math", "title": "A-graded methods for monomial ideals", "abstract": "We use \\ZZ^d-gradings to study d-dimensional monomial ideals. The Koszul functor is employed to interpret the quasidegrees of local cohomology in terms of the geometry of distractions and to explicitly compute the multiplicities of exponents. These multigraded techniques originate from the study of hypergeometric systems of differential equations."}
{"category": "Math", "title": "Singularities in positive characteristic, stratification and simplification of the singular locus", "abstract": "We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a form of simplification of the singularities, also known as a reduction to the monomial case."}
{"category": "Math", "title": "Complete description of derivations on $\\tau$-compact operators for type I von Neumann algebras", "abstract": "Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $\\tau,$ let $S_0(M, \\tau)$ be the algebra of all $\\tau$-compact operators affiliated with $M.$ We give a complete description of all derivations on the algebra $S_0(M, \\tau).$ In particular, we prove that if $M$ is of type I$_\\infty$ then every derivation on $S_0(M, \\tau)$ is spatial."}
{"category": "Math", "title": "On multiplicative congruences", "abstract": "Let $\\epsilon$ be a fixed positive quantity, $m$ be a large integer, $x_j$ denote integer variables. We prove that for any positive integers $N_1,N_2,N_3$ with $N_1N_2N_3>m^{1+\\epsilon},$ the set $$ \\{x_1x_2x_3 \\pmod m: \\quad x_j\\in [1,N_j] \\} $$ contains almost all the residue classes modulo $m$ (i.e., its cardinality is equal to $m+o(m)$). We further show that if $m$ is cubefree, then for any positive integers $N_1,N_2,N_3,N_4$ with $N_1N_2N_3N_4>m^{1+\\epsilon},$ the set $$ \\{x_1x_2x_3x_4 \\pmod m: \\quad x_j\\in [1,N_j] \\} $$ also contains almost all the residue classes modulo $m.$ Let $p$ be a large prime parameter and let $p>N>p^{63/76+\\epsilon}.$ We prove that for any nonzero integer constant $k$ and any integer $\\lambda\\not\\equiv 0\\pmod p$ the congruence $$ p_1p_2(p_3+k)\\equiv \\lambda\\pmod p $$ admits $(1+o(1))\\pi(N)^3/p$ solutions in prime numbers $p_1, p_2, p_3\\le N.$"}
{"category": "Math", "title": "A Brief History of Future Set Theory", "abstract": "Mathematicians still use Naive Set Theory when generating sets without danger of producing any contradiction. Therefore their working method can be considered as a consistent inference system with an experience of over 100 years. My conjecture is that this method works well because mathematicians use only those predicates to form sets, which yield closed hereditary consistent predicate extensions. And for every open formula they use in the process of constructing of a certain (special) set (bottom up), we can always find an \"almost-closed\" formula (i.e. a parameter-free formula with only the free variable \"x\") which yields the same certain (special) set as predicate extension as constructed in the bottom up process before. Therefore the use of predicates with free parameters in the Comprehension Scheme does not cause any difficulties and can be \"lifted\" by meta-mathematical considerations. KEYWORDS: naive set theory, Quine, new foundation, NF, universal sets, comprehension schema, predicate extension, philosophy of set theory, Zermelo, Fraenkel, ZF, complement. CT is the Class-Theoretical frame: logic + \"=\" + Church Schema + Extensionality Axiom."}
{"category": "Math", "title": "The Notion \"Pathology\" in Set Theory", "abstract": "When we study the paradoxes of set theory we find out that there are mainly 2 types: the pathologies and the antinomies. These 2 notions are made precise and compared with the somehow inductively definable concept \"abnormal\". (See my paper \"Naive Axiomatic Mengenlehre for Experiments\" in arXiv.) In the following 5 Patho Theses are discussed in order to formalize this notion of pathology. This allows us to define formally the property \"Hereditary-non-Pathological\" for well-formed formulas. With this property the system NACT* of Naive Axiomatic Class Theory is constructed, which has a \"unique maximal\" universe (in a special sense)."}
{"category": "Math", "title": "Naive Axiomatic Class Theory: A Solution for the Antinomies of Naive Mengenlehre", "abstract": "Since the axioms in (Consi-CoS) are not recursively enumerable, NACT* is no axiom system in the classical sense . Therefore we construct a series of partial systems which form a recursive axiom system too. Starting with the \"dichotomic\" systems NACT# and its variant NACT#4, we are going on to the \"disjunctive\" systems NACT+ and NACT+4, and eventually to NACT+Strat. After that we discuss the medium classes of these systems. Finally we present the inconsistent NSA-systems based on Not-SelfApplicability and explain their help for computational set theory."}
{"category": "Math", "title": "Naive Axiomatic Mengenlehre for Experiments", "abstract": "The main goal of \"Naive Axiomatic Mengenlehre\" (NAM) is to find a more or less adequately explicit criterion that precisely formalizes the intuitive notion of a \"normal set\". NAM is mainly a construction procedure for building several formal systems NAMix, each of which can turn out to be an adequate codification of the contentual naive set theory. (\"i\" is a natural number which enumerates the used \"normality\" condition, and \"x\" is a letter which points to the variants of the used axioms.) Parallel to NAM, the Naive Axiomatic Class Theory NACT is constructed as a system of systems too."}
{"category": "Math", "title": "Estimates of the norms of the Toeplitz operators of H determined by rational inner functions", "abstract": "An elementary method is given for estimates of the norms of the Toeplitz operators, determined by rational inner functions"}
{"category": "Math", "title": "Generalized ABC theorems for non-Archimedean entire functions of several variables in arbitrary characteristic", "abstract": "We prove generalized ABC theorems for vanishing sums of non-Archimedean entire functions of several variables in arbitrary characteristic."}
{"category": "Math", "title": "Topological classification of generalized Bott towers", "abstract": "If $B$ is a toric manifold and $E$ is a Whitney sum of complex line bundles over $B$, then the projectivization $P(E)$ of $E$ is again a toric manifold. Starting with $B$ as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a generalized Bott tower. We prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces. This gives a supporting evidence to what we call cohomological rigidity problem for toric manifolds \"Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic?\" We provide two more results which support the cohomological rigidity problem."}
{"category": "Math", "title": "Penalising symmetric stable L\\'evy paths", "abstract": "Limit theorems for the normalized laws with respect to two kinds of weight functionals are studied for any symmetric stable L\\'evy process of index $ 1 < \\alpha \\le 2 $. The first kind is a function of the local time at the origin, and the second kind is the exponential of an occupation time integral. Special emphasis is put on the role played by a stable L\\'evy counterpart of the universal $ \\sigma $-finite measure, found in [9] and [10], which unifies the corresponding limit theorems in the Brownian setup for which $ \\alpha =2 $."}
{"category": "Math", "title": "On the spectrum and Lyapunov exponent of limit periodic Schrodinger operators", "abstract": "We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schr\\\"odinger operator has a positive Lyapunov exponent for all energies and a spectrum of zero Lebesgue measure. No example with those properties was previously known, even in the larger class of ergodic potentials. We also conclude that the generic limit periodic potential has a spectrum of zero Lebesgue measure."}
{"category": "Math", "title": "Infinitely generated free nilpotent groups: completeness of the automorphism groups", "abstract": "Baumslag conjectured in the 1970s that the automorphism tower of a finitely generated free nilpotent group must be very short. Let F_{n,c} denote a free nilpotent group of finite rank n at least two and of nilpotency class c at least two. In 1976 Dyer and Formanek proved that the automorphism group of F_{n,2} is even complete (and hence the height of the aumorphism tower of F_{n,2} is two) provided that n is not three; in the case when n=3, the height of the automorphism tower of F_{n,2} is three. The author proved in 2001 that the automorphism group of any infinitely generated free nilpotent of class two is complete. In his Ph. D. thesis (2003) Kassabov found an upper bound u(n,c) (a natural number) for the height of the automorphism tower of F_{n,c} in terms of n and c, thereby finally proving Baumslag's conjecture. By analyzing the function u(n,c), one can conclude that if c is small compared to n, then the height of the automorphism tower of F_{n,c} is at most three. The main result of the present paper states that the automorphism group of any infinitely generated free nilpotent group of nilpotency class at least two is complete. Thus the automorphism tower of any free nilpotent group terminates after finitely many steps."}
{"category": "Math", "title": "The automorphism groups of relatively free groups of infinite rank", "abstract": "A survey article that presents some recent algebraic and model-theoretic results on the automorphism groups of relatively free groups of infinite rank. The topics include topological aspects, generating sets, descripition of automorpisms and expressive power of the first-order theories."}
{"category": "Math", "title": "Finitely presented groups and homotopy of presentations of triangular algebras", "abstract": "Given any finitely presented group G we find a triangular algebra such that has two presentations, one with fundamental group G and another with trivial group. Thus proving that given a collection G1,...,Gn of finitely presented groups there exist a triangular algebra A such that all Gi appear as fundamental group of some presentation of A, extending one of the result in arXiv:math/0405127v3 [math.RA]"}
{"category": "Math", "title": "Multivariable spectral multipliers and quasielliptic operators", "abstract": "We study multivariable spectral multipliers $F(L_1,L_2)$ acting on Cartesian product of ambient spaces of two self-adjoint operators $L_1$ and $L_2$. We prove that if $F$ satisfies H\\\"ormander type differentiability condition then the operator $F(L_1,L_2)$ is of Calder\\'on-Zygmund type. We apply obtained results to the analysis of quasielliptic operators acting on product of some fractal spaces. The existence and surprising properties of quasielliptic operators have been recently observed in works of Bockelman, Drenning and Strichartz. This paper demonstrates that Riesz type operators corresponding to quasielliptic operators are continuous on $L^p$ spaces."}
{"category": "Math", "title": "A note on the Bruhat decomposition of semisimple Lie groups", "abstract": "Let a split element of a connected semisimple Lie group act on one of its flag manifolds. We prove that each connected set of fixed points of this action is itself a flag manifold. With this we can obtain the generalized Bruhat decomposition of a semisimple Lie group by entirely dynamical arguments."}
{"category": "Math", "title": "Mock Tridiagonal Systems", "abstract": "We introduce the notion of a {\\it mock tridiagonal system}. This is a generalization of a tridiagonal system in which the irreducibility assumption is replaced by a certain non-vanishing condition. We show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications. This paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism."}
{"category": "Math", "title": "Projectivity of modules over Fourier algebras", "abstract": "In this paper we will study the homological properties of various natural modules associated to the Fourier algebra of a locally compact group. In particular, we will focus on the question of identifying when such modules will be projective in the category of operator spaces. We will show that projectivity often implies that the underlying group is discrete and give evidence to show that amenability also plays an important role."}
{"category": "Math", "title": "Some combinatorial properties of flag simplicial pseudomanifolds and spheres", "abstract": "A simplicial complex $\\Delta$ is called flag if all minimal nonfaces of $\\Delta$ have at most two elements. The following are proved: First, if $\\Delta$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $\\Delta$ (i) is $(2d-2)$-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the $d$-dimensional cross-polytope. Second, the $h$-vector of a flag simplicial homology sphere $\\Delta$ of dimension $d-1$ is minimized when $\\Delta$ is the boundary complex of the $d$-dimensional cross-polytope."}
{"category": "Math", "title": "Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities", "abstract": "We prove weak type inequalities for a large class of noncommutative square functions. In conjunction with BMO type estimates, interpolation and duality, we will obtain the corresponding equivalences in the whole Lp scale. The main novelty of our approach relies on a row/column valued theory for noncommutative martingale transforms and operator-valued Calderon-Zygmund operators. This seems to be new in the noncommutative setting and might be regarded as a first step towards a vector-valued noncommutative theory. Some examples and applications are also explored."}
{"category": "Math", "title": "Hermitian vector bundles and extension groups on arithmetic schemes. II. The arithmetic Atiyah extension", "abstract": "In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element -- the arithmetic Atiyah class -- in a suitable arithmetic extension group. If $\\overline{E}$ is a hermitian vector bundle on an arithmetic scheme $X$, its arithmetic Atiyah class is an obstruction to the algebraicity of the unitary connection on the vector bundle $E_\\C$ over the complex manifold $X(\\C)$ that is compatible with its holomorphic structure. We develop basic properties of the arithmetic Atiyah class and study its vanishing in the case of hermitian line bundles. This may be translated into a concrete problem of diophantine geometry, concerning rational points of the universal vector extension of the Picard variety of $X$. We investigate this problem, which was already considered and solved in some cases by Bertrand, by using a classical transcendence result of Schneider-Lang, and we derive a finiteness result. We also consider a geometric analog of our arithmetic situation, namely a smooth, projective variety $X$ which is fibered on a curve $C$ defined over some field $k$ of characteristic zero. To any line bundle $L$ over $X$ is attached its relative Atiyah class ${\\rm at}_{X/C}L$. We describe precisely when this class vanishes. In particular, when the fixed part of the relative Picard variety of $X$ over $C$ is trivial, this holds only when the restriction of $L$ to the generic fiber $X_K$ of $X$ over $C$ is a torsion line bundle."}
{"category": "Math", "title": "Spectral gap global solutions for degenerate Kirchhoff equations", "abstract": "We consider the second order Cauchy problem $$u''+m(|A^{1/2}u|^2)Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\\infty)\\to[0,+\\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that $u_{0}$ and $u_{1}$ are regular enough, depending on the continuity modulus of $m$, and on the strict/weak hyperbolicity of the equation. We prove that for such initial data $(u_{0},u_{1})$ there exist two pairs of initial data $(\\overline{u}_{0},\\overline{u}_{1})$, $(\\widehat{u}_{0},\\widehat{u}_{1})$ for which the solution is global, and such that $u_{0}=\\overline{u}_{0}+\\widehat{u}_{0}$, $u_{1}=\\overline{u}_{1}+\\widehat{u}_{1}$. This is a byproduct of a global existence result for initial data with a suitable spectral gap, which extends previous results obtained in the strictly hyperbolic case with a smooth nonlinearity $m$."}
{"category": "Math", "title": "Multilinear H\\\"older-type inequalities on Lorentz sequence spaces", "abstract": "We establish H\\\"older type inequalities for Lorentz sequence spaces and their duals. In order to achieve these and some related inequalities, we study diagonal multilinear forms in general sequence spaces, and obtain estimates for their norms. We also consider norms of multilinear forms in different Banach multilinear ideals."}
{"category": "Math", "title": "Blow-up estimates at horizontal points and applications", "abstract": "Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathe'odory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure at all horizontal points of $C^{1,1}$ smooth submanifolds. As an application, we establish an integral formula to compute the spherical Hausdorff measure of any $C^{1,1}$ submanifold. Our technique also shows that $C^2$ smooth submanifolds everywhere admit an intrinsic blow-up and that the limit set is an intrinsically homogeneous algebraic variety."}
{"category": "Math", "title": "Stability of Quadratic Modules", "abstract": "A finitely generated quadratic module or preordering in the real polynomial ring is called stable, if it admits a certain degree bound on the sums of squares in the representation of polynomials. Stability, first defined explicitly by Powers and Scheiderer, is a very useful property. It often implies that the quadratic module is closed; furthermore it helps settling the Moment Problem, solves the Membership Problem for quadratic modules and allows applications of methods from optimization to represent nonnegative polynomials. We provide sufficient conditions for finitely generated quadratic modules in real polynomial rings of several variables to be stable. These conditions can be checked easily. For a certain class of semi-algebraic sets, we obtain that the nonexistence of bounded polynomials implies stability of every corresponding quadratic module. As stability often implies the non-solvability of the Moment Problem, this complements Schmuedgen's result which uses bounded polynomials to check the solvability of the Moment Problem by dimensional induction. We also use stability to generalize a result on the Invariant Moment Problem by Cimpric, Kuhlmann and Scheiderer."}
{"category": "Math", "title": "Viscosity solutions for systems of parabolic variational inequalities", "abstract": "In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:\\[\\{[c]{l}\\dfrac{\\partial u}{\\partial t}(t,x)+\\mathcal{L}_tu(t,x)+f(t,x,u(t,x))\\in\\partial\\phi (u(t,x)),\\quad t\\in[0,T),x\\in\\mathbb{R}^d, u(T,x)=h(x),\\quad x\\in\\mathbb{R}^d,\\] where $\\partial\\phi$ is the subdifferential operator of the proper convex lower semicontinuous function $\\phi:\\mathbb{R}^k\\to (-\\infty,+\\infty]$ and $\\mathcal{L}_t$ is a second differential operator given by $\\mathcal{L}_tv_i(x)={1/2}\\operatorname {Tr}[\\sigma(t,x)\\sigma^*(t,x)\\mathrm{D}^2v_i(x)]+< b(t,x),\\nabla v_i(x)>$, $i\\in\\bar{1,k}$. We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution $u:[0,T]\\times\\mathbb{R}^d\\to\\mathbb{R}^k$ of the above parabolic variational inequality."}
{"category": "Math", "title": "Coordinated motion design on Lie groups", "abstract": "The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics. It thereby shows that coordination can be studied in a systematic way once the Lie group geometry of the configuration space is well characterized. This allows among others to retrieve control laws in the literature for particular examples. A link with Brockett's double bracket flows is also made. The concepts are illustrated on SO(3), SE(2) and SE(3)."}
{"category": "Math", "title": "On quasiconformal maps with identity boundary values", "abstract": "Quasiconformal homeomorphisms of the unit ball $B^n$ of ${\\mathbb R}^n, n \\ge 3,$ onto itself with identity boundary values are studied. A spatial analogue of Teichm\\\"uller's theorem is proved."}
{"category": "Math", "title": "Low-rank optimization for semidefinite convex problems", "abstract": "We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of $X$. It is thus very effective for solving programs that have a low rank solution. The factorization $X=Y Y^T$ evokes a reformulation of the original problem as an optimization on a particular quotient manifold. The present paper discusses the geometry of that manifold and derives a second order optimization method. It furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem. The efficiency of the proposed algorithm is illustrated on two applications: the maximal cut of a graph and the sparse principal component analysis problem."}
{"category": "Math", "title": "Linearly recurrent subshifts have a finite number of non-periodic subshift factors", "abstract": "A minimal subshift $(X,T)$ is linearly recurrent if there exists a constant $K$ so that for each clopen set $U$ generated by a finite word $u$ the return time to $U$, with respect to $T$, is bounded by $K|u|$. We prove that given a linearly recurrent subshift $(X,T)$ the set of its non-periodic subshift factors is finite up to isomorphism. We also give a constructive characterization of these subshifts."}
{"category": "Math", "title": "Measure of submanifolds in the Engel group", "abstract": "We find all intrinsic measures of $C^{1,1}$ smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding $d$-dimensional spherical Hausdorff measure restricted to the submanifold. The integer $d$ is the degree of the submanifold. These results follow from a different approach to negligibility, based on a blow-up technique."}
{"category": "Math", "title": "Construction of Minimal Bracketing Covers for Rectangles", "abstract": "We construct explicit $\\delta$-bracketing covers with minimal cardinality for the set system of (anchored) rectangles in the two dimensional unit cube. More precisely, the cardinality of these $\\delta$-bracketing covers are bounded from above by $\\delta^{-2} + o(\\delta^{-2})$. A lower bound for the cardinality of arbitrary $\\delta$-bracketing covers for $d$-dimensional anchored boxes from [M. Gnewuch, Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy, J. Complexity 24 (2008) 154-172] implies the lower bound $\\delta^{-2}+O(\\delta^{-1})$ in dimension $d=2$, showing that our constructed covers are (essentially) optimal. We study also other $\\delta$-bracketing covers for the set system of rectangles, deduce the coefficient of the most significant term $\\delta^{-2}$ in the asymptotic expansion of their cardinality, and compute their cardinality for explicit values of $\\delta$."}
{"category": "Math", "title": "Boundary regularity via Uhlenbeck-Rivi\\`ere decomposition", "abstract": "We prove that weak solutions of systems with skew-symmetric structure, which possess a continuous boundary trace, have to be continuous up to the boundary. This applies, e.g., to the H-surface system with bounded H and thus extends an earlier result by P. Strzelecki and proves the natural counterpart of a conjecture by E. Heinz. Methodically, we use estimates below natural exponents of integrability and a recent decomposition result by T.Rivi\\`ere."}
{"category": "Math", "title": "Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank", "abstract": "This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute."}
{"category": "Math", "title": "Approximate Multipartite Version of the Hajnal--Szemer\\'edi Theorem", "abstract": "Let $q$ be a positve integer, and $G$ be a $q$-partite simple graph on $qn$ vertices, with $n$ vertices in each vertex class. Let $\\delta={k_q \\over k_q+1}$, where $k_q=q+O(\\log{q})$. If each vertex of $G$ is adjacent to at least $\\delta n$ vertices in each of the other vertex classes, $q$ is bounded and $n$ is large enough, then $G$ has a $K_q$-factor."}
{"category": "Math", "title": "Rank functions on rooted tree quivers", "abstract": "The free abelian group R(Q) on the set of indecomposable representations of a quiver Q, over a field K, has a ring structure where the multiplication is given by the tensor product. We show that if Q is a rooted tree (an oriented tree with a unique sink), then the ring $R(Q)_{red}$ is a finitely generated $\\Z$-module (here $R(Q)_{red}$ is the ring R(Q) modulo the ideal of all nilpotent elements). We will describe the ring $R(Q)_{red}$ explicitly, by studying functors from the category Rep(Q) of representations of Q over K to the category of finite dimensional K-vector spaces. We also present an open problem for future direction."}
{"category": "Math", "title": "Smooth metrics on jet bundles and applications", "abstract": "Following a suggestion made by J.-P. Demailly, for each $k\\ge 1$, we endow, by an induction process, the $k$-th (anti)tautological line bundle $\\mathcal O_{X_k}(1)$ of an arbitrary complex directed manifold $(X,V)$ with a natural smooth hermitian metric. Then, we compute recursively the Chern curvature form for this metric, and we show that it depends (asymptotically -- in a sense to be specified later) only on the curvature of $V$ and on the structure of the fibration $X_k\\to X$. When $X$ is a surface and $V=T_X$, we give explicit formulae to write down the above curvature as a product of matrices. As an application, we obtain a new proof of the existence of global invariant jet differentials vanishing on an ample divisor, for $X$ a minimal surface of general type whose Chern classes satisfy certain inequalities, without using a strong vanishing theorem of Bogomolov."}
{"category": "Math", "title": "Properties of special classes of analytic functions which are multipliers of the Cauchy-Stieltjes integrals", "abstract": "Special classes of analytic functions, denoting by K and J arc considered in this paper."}
{"category": "Math", "title": "Note on the multipliers of Cauchy integrals of logarithmic potentials", "abstract": "The present note contains a generalization of a theorem of Hallenbeck and Samotij for the multipliers of Cauchy integrals of logarithmic potentials."}
{"category": "Math", "title": "Bers slices are Zariski dense", "abstract": "We prove that every Bers slice of quasi-Fuchsian space is Zariski dense in the character variety."}
{"category": "Math", "title": "On the Symmetric Homology of Algebras", "abstract": "Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using derived functors and the symmetric bar construction of Fiedorowicz. The symmetric homology of group rings is related to stable homotopy theory. Two chain complexes are constructed that compute symmetric homology, as well as two spectral sequences. In the setup of the second spectral sequence, a complex isomorphic to the suspension of the cycle-free chessboard complex of Vrecica and Zivaljevic appears. Homology operations are defined on the symmetric homology groups over Z/p, p a prime. Finally, an explicit partial resolution is presented, permitting the computation of the zeroth and first symmetric homology groups of finite-dimensional algebras."}
{"category": "Math", "title": "Factorization in generalized Calogero-Moser spaces", "abstract": "Using a recent construction of Bezrukavnikov and Etingof we prove that there is a factorization of the Etingof-Ginzburg sheaf on the generalized Calogero-Moser space associated to a complex reflection group. In the case W = S_n, this confirms a conjecture of Etingof and Ginzburg."}
{"category": "Math", "title": "A Bialgebraic Approach to Automata and Formal Language Theory", "abstract": "A bialgebra is a structure which is simultaneously an algebra and a coalgebra, such that the algebraic and coalgebraic parts are \"compatible\". Bialgebras are normally studied over a field or commutative ring. In this paper, we show how to apply the defining diagrams of algebras, coalgebras, and bialgebras to categories of semimodules and semimodule homomorphisms over a commutative semiring. We then show that formal language theory and the theory of bialgebras have essentially undergone \"convergent evolution\", with the same constructions appearing in both contexts. For example, formal languages correspond to elements of dual algebras of coalgebras, automata are \"pointed representation objects\" of algebras, automaton morphisms are instances of linear intertwiners, and a construction from the theory of bialgebras shows how to run two automata in parallel. We also show how to associate an automaton with an arbitrary algebra, which in the classical case yields the automaton whose states are formal languages and whose transitions are given by language differentiation."}
{"category": "Math", "title": "Functorial reconstruction theorems for stacks", "abstract": "We study the circumstances under which one can reconstruct a stack from its associated functor of isomorphism classes. This is possible surprisingly often: we show that many of the standard examples of moduli stacks are determined by their functors. Our methods seem to exhibit new anabelian-type phenomena, in the form of structures in the category of schemes that encode automorphism data in groupoids."}
{"category": "Math", "title": "Bounds for the Transition Density of Time-Homogeneous Diffusion Processes", "abstract": "The paper presents new simple sharp bounds for transition density functions for time-homogeneous diffusions processes. The bounds are obtained under mild conditions on the drift and diffusion coefficients, extending and substantially improving previous results in the literature which were limited to drifts satisfying a linear growth condition. They lead to an asymptotic expression for the transition density as the transition time approaches zero. While the focus is on the one-dimensional case, an extension to multiple dimensions is discussed. Results are illustrated by numerical examples."}
{"category": "Math", "title": "A third order dispersive flow for closed curves into almost Hermitian manifolds", "abstract": "We discuss a short-time existence theorem of solutions to the initial value problem for a third order dispersive flow for closed curves into a compact almost Hermitian manifold. Our equations geometrically generalize a physical model describing the motion of vortex filament. The classical energy method cannot work for this problem since the almost complex structure of the target manifold is not supposed to be parallel with respect to the Levi-Civita connection. In other words, a loss of one derivative arises from the covariant derivative of the almost complex structure. To overcome this difficulty, we introduce a bounded pseudodifferential operator acting on sections of the pullback bundle, and eliminate the loss of one derivative from the partial differential equation of the dispersive flow."}
{"category": "Math", "title": "On the cohomological equation of magnetic flows", "abstract": "We consider a magnetic flow without conjugate points on a closed manifold $M$ with generating vector field $\\G$. Let $h\\in C^{\\infty}(M)$ and let $\\theta$ be a smooth 1-form on $M$. We show that the cohomological equation \\[\\G(u)=h\\circ \\pi+\\theta\\] has a solution $u\\in C^{\\infty}(SM)$ only if $h=0$ and $\\theta$ is closed. This result was proved in \\cite{DP2} under the assumption that the flow of $\\G$ is Anosov."}
{"category": "Math", "title": "On the $M_t/M_t/K_t+M_t$ queue in heavy traffic", "abstract": "The focus of this paper is on the asymptotics of large-time numbers of customers in time-periodic Markovian many-server queues with customer abandonment in heavy traffic. Limit theorems are obtained for the periodic number-of-customers processes under the fluid and diffusion scalings. Other results concern limits for general time-dependent queues and for time-homogeneous queues in steady state."}
{"category": "Math", "title": "Counting perfect colourings of plane regular tilings", "abstract": "A first step in investigating colour symmetries of periodic and nonperiodic patterns is determining the number of colours which allow perfect colourings of the pattern under consideration. A perfect colouring is one where each symmetry of the uncoloured pattern induces a global permutation of the colours. Two cases are distinguished: Either perfect colourings with respect to all symmetries, or with respect to orientation preserving symmetries only (no reflections). For the important class of colourings of regular tilings (and some Laves tilings) of the Euclidean or hyperbolic plane, this mainly combinatorial question is addressed here using group theoretical methods."}
{"category": "Math", "title": "Hyperbolic conservation laws on manifolds. Error estimate for finite volume schemes", "abstract": "Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h^(1/4) at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch's theory which was originally developed in the Euclidian setting. We extent the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties."}
{"category": "Math", "title": "Hidden Markov models for the assessment of chromosomal alterations using high-throughput SNP arrays", "abstract": "Chromosomal DNA is characterized by variation between individuals at the level of entire chromosomes (e.g., aneuploidy in which the chromosome copy number is altered), segmental changes (including insertions, deletions, inversions, and translocations), and changes to small genomic regions (including single nucleotide polymorphisms). A variety of alterations that occur in chromosomal DNA, many of which can be detected using high density single nucleotide polymorphism (SNP) microarrays, are linked to normal variation as well as disease and are therefore of particular interest. These include changes in copy number (deletions and duplications) and genotype (e.g., the occurrence of regions of homozygosity). Hidden Markov models (HMM) are particularly useful for detecting such alterations, modeling the spatial dependence between neighboring SNPs. Here, we improve previous approaches that utilize HMM frameworks for inference in high throughput SNP arrays by integrating copy number, genotype calls, and the corresponding measures of uncertainty when available. Using simulated and experimental data, we, in particular, demonstrate how confidence scores control smoothing in a probabilistic framework. Software for fitting HMMs to SNP array data is available in the R package VanillaICE."}
{"category": "Math", "title": "Round about Theta. Part I Prehistory", "abstract": "There is a huge amount of work on different kinds of theta functions, the theta correspondence, cohomology classes coming from special Schwartz classes via theta distribution, and much more. The aim of this text is to try to find joint construction principles while often leaving aside relevant but cumbersome details."}
{"category": "Math", "title": "Non-diffusive large time behaviour for a degenerate viscous Hamilton-Jacobi equation", "abstract": "The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem $\\partial_t u = \\Delta_p u + |\\nabla u|^q$ when the initial data converge to zero at infinity. Sufficient conditions on the exponents $p>2$ and $q>1$ are given that guarantee that the diffusion becomes negligible for large times and the $L^\\infty$-norm of $u(t)$ converges to a positive value as $t\\to\\infty$."}
{"category": "Math", "title": "Unsupervised empirical Bayesian multiple testing with external covariates", "abstract": "In an empirical Bayesian setting, we provide a new multiple testing method, useful when an additional covariate is available, that influences the probability of each null hypothesis being true. We measure the posterior significance of each test conditionally on the covariate and the data, leading to greater power. Using covariate-based prior information in an unsupervised fashion, we produce a list of significant hypotheses which differs in length and order from the list obtained by methods not taking covariate-information into account. Covariate-modulated posterior probabilities of each null hypothesis are estimated using a fast approximate algorithm. The new method is applied to expression quantitative trait loci (eQTL) data."}
{"category": "Math", "title": "Modules of finite homological dimension with respect to a semidualizing module", "abstract": "We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify special cases of a question of Takahashi and White."}
{"category": "Math", "title": "Gamma shape mixtures for heavy-tailed distributions", "abstract": "An important question in health services research is the estimation of the proportion of medical expenditures that exceed a given threshold. Typically, medical expenditures present highly skewed, heavy tailed distributions, for which (a) simple variable transformations are insufficient to achieve a tractable low-dimensional parametric form and (b) nonparametric methods are not efficient in estimating exceedance probabilities for large thresholds. Motivated by this context, in this paper we propose a general Bayesian approach for the estimation of tail probabilities of heavy-tailed distributions, based on a mixture of gamma distributions in which the mixing occurs over the shape parameter. This family provides a flexible and novel approach for modeling heavy-tailed distributions, it is computationally efficient, and it only requires to specify a prior distribution for a single parameter. By carrying out simulation studies, we compare our approach with commonly used methods, such as the log-normal model and nonparametric alternatives. We found that the mixture-gamma model significantly improves predictive performance in estimating tail probabilities, compared to these alternatives. We also applied our method to the Medical Current Beneficiary Survey (MCBS), for which we estimate the probability of exceeding a given hospitalization cost for smoking attributable diseases. We have implemented the method in the open source GSM package, available from the Comprehensive R Archive Network."}
{"category": "Math", "title": "A Strong Law of Large Numbers for Strongly Mixing Processes", "abstract": "We prove a strong law of large numbers for a class of strongly mixing processes. Our result rests on recent advances in understanding of concentration of measure. It is simple to apply and gives finite-sample (as opposed to asymptotic) bounds, with readily computable rate constants. In particular, this makes it suitable for analysis of inhomogeneous Markov processes. We demonstrate how it can be applied to establish an almost-sure convergence result for a class of models that includes as a special case a class of adaptive Markov chain Monte Carlo algorithms."}
{"category": "Math", "title": "Quantitative magnetic resonance image analysis via the EM algorithm with stochastic variation", "abstract": "Quantitative Magnetic Resonance Imaging (qMRI) provides researchers insight into pathological and physiological alterations of living tissue, with the help of which researchers hope to predict (local) therapeutic efficacy early and determine optimal treatment schedule. However, the analysis of qMRI has been limited to ad-hoc heuristic methods. Our research provides a powerful statistical framework for image analysis and sheds light on future localized adaptive treatment regimes tailored to the individual's response. We assume in an imperfect world we only observe a blurred and noisy version of the underlying pathological/physiological changes via qMRI, due to measurement errors or unpredictable influences. We use a hidden Markov random field to model the spatial dependence in the data and develop a maximum likelihood approach via the Expectation--Maximization algorithm with stochastic variation. An important improvement over previous work is the assessment of variability in parameter estimation, which is the valid basis for statistical inference. More importantly, we focus on the expected changes rather than image segmentation. Our research has shown that the approach is powerful in both simulation studies and on a real dataset, while quite robust in the presence of some model assumption violations."}
{"category": "Math", "title": "How to Compute a Puiseux Expansion", "abstract": "In this paper, an explanation of the Newton-Peiseux algorithm is given. This explanation is supplemented with well-worked and explained examples of how to use the algorithm to find fractional power series expansions for all branches of a polynomial at the origin."}
{"category": "Math", "title": "The Generalized Effros-Hahn Conjecture for Groupoids", "abstract": "The generalized Effros-Hahn conjecture for groupoid C*-algebras says that, if G is amenable, then every primitive ideal of the groupoid C*-algebra C*(G) is induced from a stability group. We prove that the conjecture is valid for all second countable amenable locally compact Hausdorff groupoids. Our results are a sharpening of previous work of Jean Renault and depend significantly on his results."}
{"category": "Math", "title": "The smooth representations of GL_2(O)", "abstract": "We present some unpublished results of Kutzko together with results of Hill, giving a classification of the smooth (complex) representations of $\\mathrm{GL}_{2}(\\mathcal{O})$, where $\\mathcal{O}$ is the ring of integers in a local field with finite residue field."}
{"category": "Math", "title": "A note on the Jordan decomposition", "abstract": "In this article we prove that the elliptic, hyperbolic and nilpotent (or unipotent) additive (or multiplicative) Jordan components of an endomorphism $X$ (or an isomorphism $g$) of a finite dimensional vector space are given by polynomials in $X$ (or in $g$). By using this, we provide a simple proof that, for an element $X$ of a linear semisimple Lie algebra $\\g$ (or $g$ of a linear semisimple connected Lie group $G$), its three Jordan components lie again in the algebra (in the group). This was previously unknown for linear Lie groups other then $\\Int(\\g)$. This implies that, for this class of algebras and groups, the usual linear Jordan decomposition coincides with the abstract Jordan decomposition."}
{"category": "Math", "title": "Covariance fields", "abstract": "We introduce and study covariance fields of distributions on a Riemannian manifold. At each point on the manifold, covariance is defined to be a symmetric and positive definite (2,0)-tensor. Its product with the metric tensor specifies a linear operator on the respected tangent space. Collectively, these operators form a covariance operator field. We show that, in most circumstances, covariance fields are continuous. We also solve the inverse problem: recovering distribution from a covariance field. Surprisingly, this is not possible on Euclidean spaces. On non-Euclidean manifolds however, covariance fields are true distribution representations."}
{"category": "Math", "title": "Some sharp Hardy inequalities on spherically symmetric domains", "abstract": "We prove some sharp Hardy inequalities for domains with a spherical symmetry. In particular, we prove an inequality for domains of the unit $n$-dimensional sphere with a point singularity, and an inequality for functions defined on the half-space $\\R_+^{n+1}$} vanishing on the hyperplane $\\{x_{n+1}=0\\}$, with singularity along the $x_{n+1}$-axis. The proofs rely on a one-dimensional Hardy inequality involving a weight function related to the volume element on the sphere, as well as on symmetrization arguments. The one-dimensional inequality is derived in a general form."}
{"category": "Math", "title": "Reduction mod $\\ell$ of Theta Series of Level $\\ell^n$", "abstract": "It is proved that the theta series of an even lattice whose level is a power of a prime $\\ell$ is congruent modulo $\\ell$ to an elliptic modular form of level~1. The proof uses arithmetic and algebraic properties of lattices rather than methods from the theory of modular forms. The methods presented here may therefore be especially pleasing to those working in the theory of quadratic forms, and they admit generalizations to more general types of theta series as they occur e.g. in the theory of Siegel or Hilbert modular forms."}
{"category": "Math", "title": "Irreducibility criterion for the set of two matrices", "abstract": "We give the criterion for the irreducibility, the Schur irreducibility and the indecomposability of the set of two $n\\times n$ matrices $\\Lambda_n$ and $A_n$ in terms of the subalgebra associated with the \"support\" of the matrix $A_n$, where $\\Lambda_n$ is a diagonal matrix with different non zeros eigenvalues and $A_n$ is an arbitrary one. The list of all maximal subalgebras of the algebra ${\\rm Mat}(n,{\\mathbb C})$ and the list of the corresponding invariant subspaces connected with these two matrices is also given. The properties of the corresponding subalgebras are expressed in terms of the graphs associated with the support of the second matrix. For arbitrary $n$ we describe all minimal subsets of the elementary matrices $E_{km}$ that generate the algebra ${\\rm Mat}(n,{\\mathbb C})$."}
{"category": "Math", "title": "Rescaled weighted random balls models and stable self-similar random fields", "abstract": "We consider weighted random balls in $\\real^d$ distributed according to a random Poisson measure with heavy-tailed intensity and study the asymptotic behaviour of the total weight of some configurations in $\\real^d$. This procedure amounts to be very rich and several regimes appear in the limit, depending on the intensity of the balls, the zooming factor, the tail parameters of the radii and of the weights. Statistical properties of the limit fields are also evidenced, such as isotropy, self-similarity or dependence. One regime is of particular interest and yields $\\alpha$-stable stationary isotropic self-similar generalized random fields which recovers Takenaka fields, Telecom process or fractional Brownian motion."}
{"category": "Math", "title": "Large scale geometry of commutator subgroups", "abstract": "Let G be a finitely presented group, and G' its commutator subgroup. Let C be the Cayley graph of G' with all commutators in G as generators. Then C is large scale simply connected. Furthermore, if G is a torsion-free nonelementary word-hyperbolic group, C is one-ended. Hence (in this case), the asymptotic dimension of C is at least 2."}
{"category": "Math", "title": "A generalization of Gabriel's Galois covering functors and derived equivalences", "abstract": "Let $G$ be a group acting on a category $\\mathcal{C}$. We give a definition for a functor $F\\colon \\mathcal{C} \\to \\mathcal{C}'$ to be a $G$-covering and three constructions of the orbit category $\\mathcal{C}/G$, which generalizes the notion of a Galois covering of locally finite-dimensional categories with group $G$ whose action on $\\mathcal{C}$ is free and locally bonded defined by Gabriel. Here $\\mathcal{C}/G$ is defined for any category $\\mathcal{C}$ and we do not require that the action of $G$ is free or locally bounded. We show that a $G$-covering is a universal \"$G$-invariant\" functor and is essentially given by the canonical functor $\\mathcal{C} \\to \\mathcal{C}/G$. By using this we improve a covering technique for derived equivalence. Also we prove theorems describing the relationships between smash product construction and the orbit category construction by Cibils and Marcos (2006) without the assumption that the $G$-action is free. The orbit category construction by a cyclic group generated by an auto-equivalence modulo natural isomorphisms (e.g., the construction of cluster categories) is justified by a notion of the \"colimit orbit category\". In addition, we give a presentation of the orbit category of a category with a monoid action by a quiver with relations, which enables us to calculate many examples."}
{"category": "Math", "title": "Strangely dispersed minimal sets in the quasiperiodically forced Arnold map", "abstract": "We study quasiperiodically forced circle endomorphisms, homotopic to the identity, and show that under suitable conditions these exhibit uncountably many minimal sets with a complicated structure, to which we refer to as `strangely dispersed'. Along the way, we generalise some well-known results about circle endomorphisms to the uniquely ergodically forced case. Namely, all rotation numbers in the rotation interval of a uniquely ergodically forced circle endomorphism are realised on minimal sets, and if the rotation interval has non-empty interior then the topological entropy is strictly positive. The results apply in particular to the quasiperiodically forced Arnold circle map, which serves as a paradigm example."}
{"category": "Math", "title": "Eventually Expanding Maps", "abstract": "In this paper we show that the piecewise linear map f(x) = px for x in [0,1/p], and sx-s/p for x in (1/p,1], p > 1, 0 < s < 1 which has an expanding, onto branch and a contracting branch is eventually piecewise expanding and exact."}
{"category": "Math", "title": "Incompressible surfaces, hyperbolic volume, Heegaard genus and homology", "abstract": "We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 4, and if the image of the cup product map in H^2(M;Z_2) has dimension at most 1, then M has volume greater than 3.08. The proofs of these geometric results involve new topological results relating the Heegaard genus of a closed Haken manifold M to the Euler characteristic of the kishkes (i.e guts) of the complement of an incompressible surface in M."}
{"category": "Math", "title": "On an Auxiliary Function for Log-Density Estimation", "abstract": "In this note we provide explicit expressions and expansions for a special function which appears in nonparametric estimation of log-densities. This function returns the integral of a log-linear function on a simplex of arbitrary dimension. In particular it is used in the R-package \"LogCondDEAD\" by Cule et al. (2007)."}
{"category": "Math", "title": "The Colombeau Quaternion Algebra", "abstract": "We introduce the Colombeau Quaternio Algebra and study its algebraic structure. We also study the dense ideal, dense in the algebraic sense, of the algebra of Colombeau generalized numbers and use this show the existence of a maximal ting of quotions which is Von Neumann regular. Recall that it is already known that then algebra of COlombeau generalized numbers is not Von Neumann regular. We also use the study of the dense ideals to give a criteria for a generalized holomorphic function to satisfy the identity theorem. Aragona-Fernadez-Juriaans showed that a generalized holomorphic function has a power series. This is one of the ingredients use to prove the identity theorem."}
{"category": "Math", "title": "On a conjecture of Laugesen and Morpurgo", "abstract": "A well known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal element of the Neumann heat kernel of the unit ball in $\\mathbb{R}^{n}$ ($n\\geq1$) is a radially increasing function. In this paper, we use probabilistic arguments to settle this conjecture, and, as an application, we derive a new proof of the Hot Spots conjecture of J. Rauch in the case of the unit disk."}
{"category": "Math", "title": "The GBG-Rank and t-Cores I. Counting and 4-Cores", "abstract": "Let r_j(\\pi,s) denote the number of cells, colored j, in the s-residue diagram of partition \\pi. The GBG-rank of \\pi mod s is defined as r_0+r_1*w_s+r_2*w_s^2+...+r_(s-1)*w_s^(s-1), where w_s=exp(2*\\Pi*I/s). We will prove that for (s,t)=1, v(s,t) <= binomial(s+t,s)/(s+t), where v(s,t) denotes a number of distinct values that GBG-rank mod s of t-core may assume. The above inequality becomes an equality when s is prime or when s is composite and t<=2p_s, where p_s is a smallest prime divisor of s. We will show that the generating functions for 4-cores with the prescribed values of GBG-rank mod 3 are all eta-products."}
{"category": "Math", "title": "Maxwell strata in sub-Riemannian problem on the group of motions of a plane", "abstract": "The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parametrized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained."}
{"category": "Math", "title": "Manin's conjecture for a cubic surface with D_5 singularity", "abstract": "The Manin conjecture is established for a split singular cubic surface in P^3, with singularity type D_5."}
{"category": "Math", "title": "Morse theory of the moment map for representations of quivers", "abstract": "The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-H\\\"older filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts."}
{"category": "Math", "title": "Conformal actions of nilpotent groups on pseudo-Riemannian manifolds", "abstract": "We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p,q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of nilpotence of H is at most 2p+1, assuming p <= q; further, if this maximal degree is attained, then M is conformally equivalent to the universal type-(p,q), compact, conformally flat space, up to finite covers. The proofs make use of the canonical Cartan geometry associated to a pseudo-Riemannian conformal structure."}
{"category": "Math", "title": "Mod-Gaussian convergence: new limit theorems in probability and number theory", "abstract": "We introduce a new type of convergence in probability theory, which we call ``mod-Gaussian convergence''. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of $L$-functions over function fields in the Katz-Sarnak framework. A similar phenomenon of ``mod-Poisson convergence'' turns out to also appear in the classical Erd\\H{o}s-K\\'ac Theorem."}
{"category": "Math", "title": "Generalizing the Croke-Kleiner Construction", "abstract": "It is well known that every word hyperbolic group has a well-defined visual boundary. An example of C. Croke and B. Kleiner shows that the same cannot be said for CAT(0) groups. All boundaries of a CAT(0) group are, however, shape equivalent, as observed by M. Bestvina and R. Geoghegan. Bestvina has asked if they also satisfy the stronger condition of being cell-like equivalent. This article describes a construction which will produce CAT(0) groups with multiple boundaries. These groups have very complicated boundaries in high dimensions. It is our hope that their study may provide insight into Bestvina's question."}
{"category": "Math", "title": "Les groupes de triangles $(2,p,q)$ sont d\\'etermine\\'es par leur spectre des longueurs", "abstract": "We describe the length spectra of triangle groups $(2,p,q)$ before showing that the length spectra characterizes the isometry class of such a group."}
{"category": "Math", "title": "Prime regular Hopf Algebras of GK-dimension One", "abstract": "This paper constitutes the first part of a program to classify all affine prime regular Hopf algebras $H$ of Gelfand-Kirillov dimension one over an algebraically closed field of characteristic zero. We prove a number of properties of such an algebra, list some classes of examples, and then prove that - when the PI-degree of $H$ is prime - our list contains all such algebras."}
{"category": "Math", "title": "Certain aperiodic automorphisms of unital simple projectionless C*-algebras", "abstract": "Let $G$ be an inductive limit of finite cyclic groups and let $A$ be a unital simple projectionless C*-algebra with $K_1(A) \\cong G$ and with a unique tracial state, as constructed based on dimension drop algebras by Jiang and Su. First, we show that any two aperiodic elements in $\\Aut(A)/\\WInn(A)$ are conjugate, where $\\WInn(A)$ means the subgroup of $\\Aut(A)$ consisting of automorphisms which are inner in the tracial representation. In the second part of this paper, we consider a class of unital simple C*-algebras with a unique tracial state which contains the class of unital simple AT-algebras of real rank zero with a unique tracial state. This class is closed under inductive limits and under crossed products by actions of $\\Z$ with the Rohlin property. Let $A$ be a TAF-algebra in this class. We show that for any automorphism $\\alpha$ of $A$ there exists an automorphism $\\widetilde{\\alpha}$ of $A$ with the Rohlin property such that $\\widetilde{\\alpha}$ and $\\alpha$ are asymptotically unitarily equivalent. In its proof we use an aperiodic automorphism of the Jiang-Su algebra."}
{"category": "Math", "title": "Jet schemes, invariant chiral differential operators, and Howe duality", "abstract": "This paper has been withdrawn due to an error in the proof of Theorem 5.3."}
{"category": "Math", "title": "Center and representations of infinitesimal Hecke algebras of sl_2", "abstract": "In this paper, we compute the center of the infinitesimal Hecke algebras Hz associated to sl_2 ; then using nontriviality of the center, we study representations of these algebras in the framework of the BGG category O. We also discuss central elements in infinitesimal Hecke algebras over gl(n) and sp(2n) for all n. We end by proving an analogue of the theorem of Duflo for Hz."}
{"category": "Math", "title": "On the colored Jones polynomials of certain cable of the torus knots", "abstract": "In this paper, we study the asymptotic behavior of the colored Jones polynomials evaluated at roots of unity for a special class of knots. We show that certain limit is zero as predicted by the volume conjecture."}
{"category": "Math", "title": "Wild cyclic-by-tame extensions", "abstract": "Suppose G is a semi-direct product of the form Z/p^n \\rtimes Z/m where p is prime and m is relatively prime to p. Suppose K is a local field of characteristic p > 0. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration whose dimension depends only on the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p^3."}
{"category": "Math", "title": "Incompressible one-sided surfaces in filled link spaces", "abstract": "When a Dehn filled link manifold contains a geometrically incompressible one-sided surface, it is shown there is a unique boundary incompressible position that the surface can take in the link space. The proof uses a version of the sweep-out technique from two-sided Heegaard splitting theory. When applied to one-sided Heegaard splittings, this result can be used to complete the classification of one-sided splittings of (2p, q) fillings of Figure 8 knot space: determining that fillings with |2p/q|<3 have two non-isotopic geometrically incompressible one-sided splitting surfaces."}
{"category": "Math", "title": "Peak reduction and finite presentations for automorphism groups of right-angled Artin groups", "abstract": "We generalize the peak-reduction algorithm (Whitehead's theorem) for free groups to a theorem about a general right-angled Artin group A_Gamma. As an application, we find a finite presentation for the automorphism group Aut A_Gamma that generalizes McCool's presentation for the automorphism group of a finite rank free group. We also give consider a stronger generalization of peak-reduction, giving a counterexample and proving a special case."}
{"category": "Math", "title": "Toric cohomological rigidity of simple convex polytopes", "abstract": "A simple convex polytope $P$ is \\emph{cohomologically rigid} if its combinatorial structure is determined by the cohomology ring of a quasitoric manifold over $P$. Not every $P$ has this property, but some important polytopes such as simplices or cubes are known to be cohomologically rigid. In this article we investigate the cohomological rigidity of polytopes and establish it for several new classes of polytopes including products of simplices. Cohomological rigidity of $P$ is related to the \\emph{bigraded Betti numbers} of its \\emph{Stanley--Reisner ring}, another important invariants coming from combinatorial commutative algebra."}
{"category": "Math", "title": "Symplectic structures on right-angled Artin groups: between the mapping class group and the symplectic group", "abstract": "We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we call mapping class groups over graphs, are indexed over labeled simplicial graphs with 2g vertices. The mapping class group over the graph Gamma is defined to be a subgroup of the automorphism group of the right-angled Artin group A_Gamma of Gamma. We also prove that the kernel of the map Aut A_Gamma to Aut H_1(A_Gamma) is finitely generated, generalizing a theorem of Magnus."}
{"category": "Math", "title": "Matrix representations for toric parametrizations", "abstract": "In this paper we show that a surface in P^3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over P^2 established in [BJ03] and [BC05]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case T = P^1 x P^1 in detail and give numerous examples."}
{"category": "Math", "title": "H2-optimal approximation of MIMO linear dynamical systems", "abstract": "We consider the problem of approximating a multiple-input multiple-output (MIMO) $p\\times m$ rational transfer function $H(s)$ of high degree by another $p\\times m$ rational transfer function $\\hat H(s)$ of much smaller degree, so that the ${\\cal H}_2$ norm of the approximation error is minimized. We characterize the stationary points of the ${\\cal H}_2$ norm of the approximation error by tangential interpolation conditions and also extend these results to the discrete-time case. We analyze whether it is reasonable to assume that lower-order models can always be approximated arbitrarily closely by imposing only first-order interpolation conditions. Finally, we analyze the ${\\cal H}_2$ norm of the approximation error for a simple case in order to illustrate the complexity of the minimization problem."}
{"category": "Math", "title": "Transformations of algebraic Gauss hypergeometric functions", "abstract": "A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pull-back of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are hypergeometric equations with tetrahedral, octahedral or icosahedral monodromy groups. We give an algorithm for computing Klein's pull-back coverings in these cases, based on certain explicit expressions (Darboux evaluations) of algebraic hypergeometric functions. The explicit expressions can be computed using contiguous relations and a data base of simplest Darboux evaluations (covering the Schwarz table). Klein's pull-back transformations also induce algebraic transformations between hypergeometric solutions and a standard hypergeometric function with the same finite monodromy group."}
{"category": "Math", "title": "Singular cotangent model", "abstract": "Any singular level of a completely integrable system (c.i.s.) with non-degenerate singularities has a singular affine structure. We shall show how to construct a simple c.i.s. around the level, having the above affine structure. The cotangent budle of the desingularised level is used to perform the construction, and the c.i.s. obtained looks like the simplest one associated to the affine structure. This method of construction is used to provide several examples of c.i.s. with different kinds of non-degenerate singularities."}
{"category": "Math", "title": "Torus quotients of homogeneous spaces-minimal dimensional Schubert Variety admitting semi-stable points", "abstract": "In this paper, for any simple, simply connected algebraic group $G$ of type $B_n,C_n$ or $D_n$ and for any maximal parabolic subgroup $P$ of $G$, we describe all minimal dimensional Schubert varieties in $G/P$ admitting semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$. In this paper, we also describe, for any semi-simple simply connected algebraic group $G$ and for any Borel subgroup $B$ of $G$, all Coxeter elements $\\tau$ for which the Schubert variety $X(\\tau)$ admits a semistable point for the action of the torus $T$ with respect to a non-trivial line bundle on $G/B$."}
{"category": "Math", "title": "Sur le corps des modules de certaines vari\\'et\\'es", "abstract": "To every covering of curves, we associate several varieties having the same field of moduli and same fields of definition. We deduce examples of curves having Q (the field of rationals) as field of moduli, that admit models over any completion of Q but no model over Q."}
{"category": "Math", "title": "When are dual Cayley automaton semigroups finite?", "abstract": "In this note we prove that, for a finite semigroup $S$, the dual Cayley automaton semigroup $\\mathbf{C^{\\ast}}(S)$ is finite if and only if $S$ is $\\mathcal{H}$-trivial and has no non-trivial right zero subsemigroups."}
{"category": "Math", "title": "Concentration of the ratio between the geometric and arithmetic means", "abstract": "We explore the concentration properties of the ratio between the geometric mean and the arithmetic mean, showing that for certain sequences of weights one does obtain concentration, around a value that depends on the sequence."}
{"category": "Math", "title": "Mock Theta Functions", "abstract": "In this Ph.D. thesis, written under the direction of D.B. Zagier and R.W. Bruggeman, we study the mock theta functions, that were introduced by Ramanujan. We show how they can be interpreted in the theory of (real-analytic) modular forms. In Chapter 1 we give results for Lerch sums (also called Appell functions, or generalized Lambert series). In Chapter 2 we consider indefinite theta functions of type (r-1,1). Chapter 3 deals with Fourier coefficients of meromorphic Jacobi forms. In Chapter 4 we use the results from Chapter 2 to give explicit results for 8 of the 10 fifth order mock theta functions and all 3 seventh order functions, that were originally defined by Ramanujan. The result is that we can find a correction term, which is a period integral of a weight 3/2 unary theta functions, such that if we add it to the mock theta function, we get a weight 1/2 real-analytic modular form, which is annihilated by the hyperbolic Laplacian."}
{"category": "Math", "title": "Identities for the Hankel transform and their applications", "abstract": "In the present paper the authors show that iterations of the Hankel transform with $\\mathscr{K}_{\\nu}$-transform is a constant multiple of the Widder transform. Using these iteration identities, several Parseval-Goldstein type theorems for these transforms are given. By making use of these results a number of new Goldstein type exchange identities are obtained for these and the Laplace transform. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results presented here."}
{"category": "Math", "title": "Paradigms-Shift in Set Theory", "abstract": "In this article the author claims that there is a paradigm shift from ZFC to NFUM and further to NACT - due to philosophical reasons, not mathematical ones. The goal is to construct systems where every \"Not-Properclass\" is a set! With help of Non-Monotonic Logic, the consistent systems NACT-MoonW, NACT*W, and NACT-SunW are producing \"largest possible universes\" of sets. Using self-evident philosophical principles, three approximations are suggested: NACT+NFUM-closed (to NACT*), NACT&ZFC4+(GCH) (also to NACT*) and NACT-NFUM (to NACT-Sun). Also the system NACT[&ZFC4-closed]+(FCA) is considered. NFUM = NFU with (AC) and measurable properclass Ord. NFUM-closed is NFUM where the set-constituting formulas Ai in the set operator need not only to be stratified but also to be made parameter-free (i.e., have only x as free variable over which the set is comprehended). In other formulas, free variables are allowed. Keywords: Set Theory, ZFC, Naive Set Theory, Predicate-Extension, Church Schema of Comprehension, NF, NFUM, Stratification, Universal Sets, Eliminative Class Theory, NBG, NACT (= Naive Axiomatic Class Theory), NACT-Sun, NACT-Moon, NACT-Star, NACT+NFUM-closed, ZFCK, NACT+NFUM, (GCH), (FCA) [= Finite or Countable Anti-Thesis], NACT+ZFC4+(GCH), NACT[+ZFC4-[closed]+(FCA)."}
{"category": "Math", "title": "Poincar\\'e series and monodromy of the simple and unimodal boundary singularities", "abstract": "A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V. I. Arnold and V. I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincar\\'e series of the coordinate algebra of the ambient singularity."}
{"category": "Math", "title": "An instance of umbral methods in representation theory: the parking function module", "abstract": "We test the umbral methods introduced by Rota and Taylor within the theory of representation of symmetric group. We define a simple bijection between the set of all parking functions of length $n$ and the set of all noncrossing partitions of $\\{1,2,...,n\\}$. Then we give an umbral expression of the Frobenius characteristic of the parking function module introduced by Haiman that allows an explicit relation between this symmetric function and the volume polynomial of Pitman and Stanley."}
{"category": "Math", "title": "Groupoid sheaves as quantale sheaves", "abstract": "Several notions of sheaf on various types of quantale have been proposed and studied in the last twenty five years. It is fairly standard that for an involutive quantale Q satisfying mild algebraic properties the sheaves on Q can be defined to be the idempotent self-adjoint Q-valued matrices. These can be thought of as Q-valued equivalence relations, and, accordingly, the morphisms of sheaves are the Q-valued functional relations. Few concrete examples of such sheaves are known, however, and in this paper we provide a new one by showing that the category of equivariant sheaves on a localic etale groupoid G (the classifying topos of G) is equivalent to the category of sheaves on its involutive quantale O(G). As a means towards this end we begin by replacing the category of matrix sheaves on Q by an equivalent category of complete Hilbert Q-modules, and we approach the envisaged example where Q is an inverse quantal frame O(G) by placing it in the wider context of stably supported quantales, on one hand, and in the wider context of a module theoretic description of arbitrary actions of \\'etale groupoids, both of which may be interesting in their own right."}
{"category": "Math", "title": "On interpretations of bounded arithmetic and bounded set theory", "abstract": "In a recent paper, Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic. THEOREM: The first-order theories of Peano arithmetic and ZF with the axiom of infinity negated are bi-interpretable: that is, they are mutually interpretable with interpretations that are inverse to each other. In this note, I describe a theory of sets that stands in the same relation to the bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of arithmetic in set theory. Instead, I am forced to produce a different interpretation."}
{"category": "Math", "title": "The V/S test of long-range dependence in random fields", "abstract": "Recently, Giraitis et al. (2003, [10]) proposed the $V/S$ statistic for testing long memory in random sequences. We generalize this statistic to the setting of random fields. The null hypothesis is concerned with short memory random fields while the alternative contains a very large family of long memory random fields. Contrary to most of the previous works dealing with long-range dependence, no assumption is made about the isotropy of the strong dependence. Some simulations are presented in order to assess the power of the test according to the kind of long memory in presence."}
{"category": "Math", "title": "Complex vector fields and hypoelliptic partial differential operators", "abstract": "We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for $CR$ manifolds and H\\\"ormander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of $(0,1)$ vector fields satisfies a subelliptic estimate. v2: minor revision, to appear in Ann. Inst. Fourier"}
{"category": "Math", "title": "Construction of weakly CUD sequences for MCMC sampling", "abstract": "In Markov chain Monte Carlo (MCMC) sampling considerable thought goes into constructing random transitions. But those transitions are almost always driven by a simulated IID sequence. Recently it has been shown that replacing an IID sequence by a weakly completely uniformly distributed (WCUD) sequence leads to consistent estimation in finite state spaces. Unfortunately, few WCUD sequences are known. This paper gives general methods for proving that a sequence is WCUD, shows that some specific sequences are WCUD, and shows that certain operations on WCUD sequences yield new WCUD sequences. A numerical example on a 42 dimensional continuous Gibbs sampler found that some WCUD inputs sequences produced variance reductions ranging from tens to hundreds for posterior means of the parameters, compared to IID inputs."}
{"category": "Math", "title": "Adaptive complexity regularization for linear inverse problems", "abstract": "We tackle the problem of building adaptive estimation procedures for ill-posed inverse problems. For general regularization methods depending on tuning parameters, we construct a penalized method that selects the optimal smoothing sequence without prior knowledge of the regularity of the function to be estimated. We provide for such estimators oracle inequalities and optimal rates of convergence. This penalized approach is applied to Tikhonov regularization and to regularization by projection."}
{"category": "Math", "title": "Functional principal components analysis via penalized rank one approximation", "abstract": "Two existing approaches to functional principal components analysis (FPCA) are due to Rice and Silverman (1991) and Silverman (1996), both based on maximizing variance but introducing penalization in different ways. In this article we propose an alternative approach to FPCA using penalized rank one approximation to the data matrix. Our contributions are four-fold: (1) by considering invariance under scale transformation of the measurements, the new formulation sheds light on how regularization should be performed for FPCA and suggests an efficient power algorithm for computation; (2) it naturally incorporates spline smoothing of discretized functional data; (3) the connection with smoothing splines also facilitates construction of cross-validation or generalized cross-validation criteria for smoothing parameter selection that allows efficient computation; (4) different smoothing parameters are permitted for different FPCs. The methodology is illustrated with a real data example and a simulation."}
{"category": "Math", "title": "Hierarchical pinning model with site disorder: Disorder is marginally relevant", "abstract": "We study a hierarchical disordered pinning model with site disorder for which, like in the bond disordered case [6, 9], there exists a value of a parameter b (enters in the definition of the hierarchical lattice) that separates an irrelevant disorder regime and a relevant disorder regime. We show that for such a value of b the critical point of the disordered system is different from the critical point of the annealed version of the model. The proof goes beyond the technique used in [9] and it takes explicitly advantage of the inhomogeneous character of the Green function of the model."}
{"category": "Math", "title": "Rank and crank moments for overpartitions", "abstract": "We study two types of crank moments and two types of rank moments for overpartitions. We show that the crank moments and their derivatives, along with certain linear combinations of the rank moments and their derivatives, can be written in terms of quasimodular forms. We then use this fact to prove exact relations involving the moments as well as congruence properties modulo 3, 5, and 7 for some combinatorial functions which may be expressed in terms of the second moments. Finally, we establish a congruence modulo 3 involving one such combinatorial function and the Hurwitz class number H(n)."}
{"category": "Math", "title": "FDR control for multiple hypothesis testing on composite nulls", "abstract": "Multiple hypothesis testing often involves composite nulls, i.e., nulls that are associated with two or more distributions. In many cases, it is reasonable to assume that there is a prior distribution on the distributions despite it is unknown. When the number of distributions under true nulls is finite, we show that under the above assumption, the false discover rate (FDR) can be controlled using $p$-values computed under constraints imposed by the empirical distribution of the observations. Comparing to FDR control using $p$-values defined as maximum significance level over all null distributions, the proposed FDR control can have substantially more power."}
{"category": "Math", "title": "Dihedral Gauss hypergeometric functions", "abstract": "Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the argument variable. The paper presents general elementary expressions of these dihedral hypergeometric functions, involving finite bivariate sums expressible as terminating Appell's F2 or F3 series. Additionally, trigonometric expressions for the dihedral functions are presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z) are considered."}
{"category": "Math", "title": "Knots, sutures and excision", "abstract": "We develop monopole and instanton Floer homology groups for balanced sutured manifolds. Applications include a new proof of Property P for knots."}
{"category": "Math", "title": "Complex cobordisms and singular manifolds arising from Chern classes", "abstract": "This paper deals with the question of J.Morava on existence of canonical complex cobordism class of singular submanifold. We present several solutions of this question for $X_r(\\xi)$ -- the set of points where $\\dim\\xi-r+1$ generic sections of a complex vector bundle $\\xi$ are linearly dependent. The corresponding complex cobordism classes $Q_r(\\xi)$ and $P_r(\\xi)$ tend to have many nice properties, such as deformed sum formula, but they don't coincide with Chern classes $c_r^U(\\xi)$. They also have relation to the theory of $IH$-small resolutions."}
{"category": "Math", "title": "Random matrices: Universality of ESDs and the circular law", "abstract": "Given an $n \\times n$ complex matrix $A$, let $$\\mu_{A}(x,y):= \\frac{1}{n} |\\{1\\le i \\le n, \\Re \\lambda_i \\le x, \\Im \\lambda_i \\le y\\}|$$ be the empirical spectral distribution (ESD) of its eigenvalues $\\lambda_i \\in \\BBC, i=1, ... n$. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD $\\mu_{\\frac{1}{\\sqrt{n}} A_n}$ of a random matrix $A_n = (a_{ij})_{1 \\leq i,j \\leq n}$ where the random variables $a_{ij} - \\E(a_{ij})$ are iid copies of a fixed random variable $x$ with unit variance. We prove a \\emph{universality principle} for such ensembles, namely that the limit distribution in question is {\\it independent} of the actual choice of $x$. In particular, in order to compute this distribution, one can assume that $x$ is real of complex gaussian. As a related result, we show how laws for this ESD follow from laws for the \\emph{singular} value distribution of $\\frac{1}{\\sqrt{n}} A_n - zI$ for complex $z$. As a corollary we establish the Circular Law conjecture (in both strong and weak forms), that asserts that $\\mu_{\\frac{1}{\\sqrt{n}} A_n}$ converges to the uniform measure on the unit disk when the $a_{ij}$ have zero mean."}
{"category": "Math", "title": "A Proof of Green's Conjecture Regarding the Removal Properties of Sets of Linear Equations", "abstract": "A system of \\ell linear equations in p unknowns Mx=b is said to have the removal property if every set S \\subseteq {1,...,n} which contains o(n^{p-\\ell}) solutions of Mx=b can be turned into a set S' containing no solution of Mx=b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homogenous linear equation always has the removal property and conjectured that every set of homogenous linear equations has the removal property. We confirm Green's conjecture by showing that every set of linear equations (even non-homogenous) has the removal property."}
{"category": "Math", "title": "Random networks with sublinear preferential attachment: Degree evolutions", "abstract": "We define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sublinear function of their degree. We first give a strong limit law for the empirical degree distribution, and then have a closer look at the temporal evolution of the degrees of individual vertices, which we describe in terms of large and moderate deviation principles. Using these results, we expose an interesting phase transition: in cases of strong preference of large degrees, eventually a single vertex emerges forever as vertex of maximal degree, whereas in cases of weak preference, the vertex of maximal degree is changing infinitely often. Loosely speaking, the transition between the two phases occurs in the case when a new edge is attached to an existing vertex with a probability proportional to the root of its current degree."}
{"category": "Math", "title": "An Ore-type theorem for perfect packings in graphs", "abstract": "We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H, we determine, asymptotically, the Ore-type degree condition which ensures that a graph G has a perfect H-packing. More precisely, let \\delta_{\\rm Ore} (H,n) be the smallest number k such that every graph G whose order n is divisible by |H| and with d(x)+d(y)\\geq k for all non-adjacent x \\not = y \\in V(G) contains a perfect H-packing. We determine \\lim_{n\\to \\infty} \\delta_{\\rm Ore} (H,n)/n."}
{"category": "Math", "title": "The cubic fourth-order Schrodinger equation", "abstract": "We investigate the cubic defocusing fourth order Schr\\\"odinger equation $iu_t + \\Delta^2u + |u|^2u=0$ in arbitrary space dimension $\\mathbb{R}^n$ for arbitrary $H^2$ initial data. We prove that the equation is globally well-posed when $n \\le 8$ and ill-posed when $n \\ge 9$, with the additional important information that scattering holds true when $5 \\le n \\le 8$."}
{"category": "Math", "title": "The Burnside Ring-Valued Morse Formula for Vector Fields on Manifolds with Boundary", "abstract": "Let G be a compact Lie group and A(G) its Burnside Ring. For a compact smooth n-dimensional G-manifold X equipped with a generic G-invariant vector field v, we prove an equivariant analog of the Morse formula Ind^G(v) = \\sum_{k = 0}^{n} (-1)^k \\chi^G(\\d_k^+X) which takes its values in A(G). Here Ind^G(v) denotes the equivariant index of the field v, {\\d_k^+X\\} the v-induced Morse stratification (see [M]) of the boundary \\d X, and \\chi^G(\\d_k^+X) the class of the (n - k)-manifold \\d_k^+X in $A(G)$. We examine some applications of this formula to the equivariant real algebraic fields v in compact domains X \\subset \\R^n defined via a generic polynomial inequality. Next, we link the above formula with the equivariant degrees of certain Gauss maps. This link is an equivariant generalization of Gottlieb's formulas."}
{"category": "Math", "title": "ZFK := ZFC with a Complement, or: Hegel and the Synto-Set-Theory", "abstract": "What is the slightest modification of ZF to add a complement-axiom? The answer in my Ph.D. thesis 1971 was ZF'': Zermelo-Fraenkel with replacement for only well-founded domains and an omega-axiom. In 1974, Alonzo Church published a similar system, as did Urs Oswald in 1976. In his 1976 Ph.D. thesis, E. Mitchell also designed, a system very similar to ZF''. In this article we argue that ZF'' is the slightest modification of ZF among all these systems and that its successor ZFK is the simplest set theory with a universal set at all."}
{"category": "Math", "title": "Higher Cardinals are only a Convention", "abstract": "Zermelo's Axiom of Separation is: Exist x: Forall y: (y in x <==> y in a & E(y)) with definite(E) and parameter a. Thoralf Skolem suggested to characterize the terminus \"definite\" by \"the property E should be representable by a FOL formula\". But that is trivial. \"definite\" must mean more. The author claims that \"definite\" means \"in accordance with the theory of definitions of logic\". In this case the theorem of Cantor is no longer a theorem, but a undecidable sentence, and has to be established explicitly as axiom. This is not done by the community, but it is made a silent assumption that we can drop the appendix \"definite(E)\" from the axiom of separation at all. But this is a convention (even when it is silent) and it is nothing else than an axiom."}
{"category": "Math", "title": "Persistence of laminations", "abstract": "We present a modern proof of some extensions of the celebrated Hirsch-Pugh-Shub theorem on persistence of normally hyperbolic compact laminations. Our extensions consist of allowing the dynamics to be an endomorphism, of considering the complex analytic case and of allowing the laminations to be non compact. To study the analytic case, we use the formalism of deformations of complex structures. We present various persistent complex laminations which appear in dynamics of several complex variables: Henon maps, fibered holomorphic maps... In order to proof the persistence theorems, we construct a laminar structure on the stable and unstable of the normally hyperbolic laminations."}
{"category": "Math", "title": "Serre's uniformity problem in the split Cartan case", "abstract": "We prove that there exists an integer p_0 such that X_split(p)(Q) is made of cusps and CM-points for any prime p>p_0. Equivalently, for any non-CM elliptic curve E over Q and any prime p>p_0 the image of the Galois representation induced by the Galois action on the p-division points of E is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an old question of Serre."}
{"category": "Math", "title": "A family of non-isomorphism results", "abstract": "We give a short argument showing that if $m, n \\in {1, 2, ...} \\cup {\\omega}$, then the groups mV and nV are not isomorphic. This answers a question of Brin."}
{"category": "Math", "title": "A Quillen model category structure on some categories of comonoids", "abstract": "We prove that for certain monoidal (Quillen) model categories, the category of comonoids therein also admits a model structure."}
{"category": "Math", "title": "Nonhyperbolicity of invariant measures on maximal attractor", "abstract": "The article states that for every compact manifold M of dimension 4 or higher there is an area U in a set of smooth diffeomorphisms over M such that every map f from U has local maximal partially hyperbolic attractor and nonatomic ergodic invariant measure on it where one of Lyapunov exponents vanish. The result is first proved for skew products over horseshoe and then transferred onto smooth diffeomorphisms."}
{"category": "Math", "title": "Analysis on some infinite modules, inner projection, and applications", "abstract": "A projective scheme $X$ is called `quadratic' if $X$ is scheme-theoretically cut out by homogeneous equations of degree 2. Furthermore, we say $X$ satisfies `property $\\textbf{N}_{2,p}$' if it is quadratic and the quadratic ideal has only linear syzygies up to first $p$-th steps. In the present paper, we compare the linear syzygies of the inner projections with those of $X$ and obtain a theorem on `embedded linear syzygies' as one of our main results. This is the natural projection-analogue of `restricting linear syzygies' in the linear section case, \\cite{EGHP1}. As an immediate corollary, we show that the inner projections of $X$ satisfy property $\\textbf{N}_{2,p-1}$ for any reduced scheme $X$ with property $\\textbf{N}_{2,p}$. Moreover, we also obtain the neccessary lower bound $(\\codim X)\\cdot p -\\frac{p(p-1)}{2}$, which is sharp, on the number of quadrics vanishing on $X$ in order to satisfy $\\textbf{N}_{2,p}$ and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme $X$. These results admit an interesting `syzygetic' rigidity theorem on property $\\textbf{N}_{2,p}$ which leads the classifications of extremal and next to extremal cases. For these results we develope the elimination mapping cone theorem for infinitely generated graded modules and improve the partial elimination ideal theory initiated by M. Green. This new method allows us to treat a wider class of projective schemes which can not be covered by the Koszul cohomology techniques, because these are not projectively normal in general."}
{"category": "Math", "title": "C*-Algebra Relations", "abstract": "We investigate relations on elements in C*-algebras, including *-polynomial relations, order relations and all relations that correspond to universal C*-algebras. We call these C*-relations and define them axiomatically. Within these are the compact C*-relations, which are those that determine universal C*-algebras, and we introduce the more flexible concept of a closed C*-relation. In the case of a finite set of generators, we show that closed C*-relations correspond to the zero-sets of elements in a free sigma -C*-algebra. This provides a solid link between two of the previous theories on relations in C*-algebras. Applications to lifting problems are briefly considered in the last section."}
{"category": "Math", "title": "W-Symmetry of the Adelic Grassmannian", "abstract": "We give a geometric construction of the W_{1+infty} vertex algebra as the infinitesimal form of a factorization structure on an adelic Grassmannian. This gives a concise interpretation of the higher symmetries and Backlund-Darboux transformations for the KP hierarchy and its multicomponent extensions in terms of a version of \"W_{1+infty}-geometry\": the geometry of D-bundles on smooth curves, or equivalently torsion-free sheaves on cuspidal curves."}
{"category": "Math", "title": "Positivity in power series rings", "abstract": "We extend and generalize the results of Scheiderer (2006) on the representation of polynomials nonnegative on two-dimensional basic closed semialgebraic sets. Our extension covers some situations where the defining polynomials do not satisfy the transversality condition. Such situations arise naturally when one considers semialgebraic sets invariant under finite group actions."}
{"category": "Math", "title": "A new method for fast computing unbiased estimators of cumulants", "abstract": "We propose new algorithms for generating $k$-statistics, multivariate $k$-statistics, polykays and multivariate polykays. The resulting computational times are very fast compared with procedures existing in the literature. Such speeding up is obtained by means of a symbolic method arising from the classical umbral calculus. The classical umbral calculus is a light syntax that involves only elementary rules to managing sequences of numbers or polynomials. The cornerstone of the procedures here introduced is the connection between cumulants of a random variable and a suitable compound Poisson random variable. Such a connection holds also for multivariate random variables."}
{"category": "Math", "title": "Maximal quadratic modules on *-rings", "abstract": "We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to $\\ast$-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime $\\ast$-ideal, that every maximal proper quadratic module in a Noetherian $\\ast$-ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schm\\\" udgen's Strict Positivstellensatz for the Weyl algebra: Let $c$ be an element of the Weyl algebra $\\mathcal{W}(d)$ which is not negative semidefinite in the Schr\\\" odinger representation. It is shown that under some conditions there exists an integer $k$ and elements $r_1,...,r_k \\in \\mathcal{W}(d)$ such that $\\sum_{j=1}^k r_j c r_j^\\ast$ is a finite sum of hermitian squares. This result is not a proper generalization however because we don't have the bound $k \\le d$."}
{"category": "Math", "title": "A gap property for the growth of closed 3-manifold groups", "abstract": "We provide a lower bound for the uniform exponential growth rate of closed nonflat nonpositively curved 3-manifold groups. A detailed study of the uniform exponential growth rate of closed 3-manifold groups is also presented."}
{"category": "Math", "title": "Formally real involutions on central simple algebras", "abstract": "An involution $#$ on an associative ring $R$ is \\textit{formally real} if a sum of nonzero elements of the form $r^# r$ where $r \\in R$ is nonzero. Suppose that $R$ is a central simple algebra (i.e. $R=M_n(D)$ for some integer $n$ and central division algebra $D$) and $#$ is an involution on $R$ of the form $r^# = a^{-1} r^\\ast a$, where $\\ast$ is some transpose involution on $R$ and $a$ is an invertible matrix such that $a^\\ast=\\pm a$. In section 1 we characterize formal reality of $#$ in terms of $a$ and $\\ast|_D$. In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on $D=(K/F,\\Phi)$ that extend to a formally real involution on the split algebra $D \\otimes_F K \\cong M_n(K)$. Every such involution is formally real but we show that there exist formally real involutions on $D$ which are not of this form. In particular, there exists a formally real involution $#$ for which the hermitian trace form $x \\mapsto \\tr(x^#x)$ is not positive semidefinite."}
{"category": "Math", "title": "A representation theorem for archimedean quadratic modules on *-rings", "abstract": "We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings, \\cite[Theorem 5]{jacobi}. We show that this theorem is a consequence of the Gelfand-Naimark representation theorem for commutative $C^\\ast$-algebras. A noncommutative version of Gelfand-Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution."}
{"category": "Math", "title": "Approximately bisimilar symbolic models for incrementally stable switched systems", "abstract": "Switched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. This paper describes one particular approach to address these different objectives based on the construction of approximately equivalent (bisimilar) symbolic models for switched systems. The main contribution of this paper consists in showing that under standard assumptions ensuring incremental stability of a switched system (i.e. existence of a common Lyapunov function, or multiple Lyapunov functions with dwell time), it is possible to construct a finite symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori. To support the computational merits of the proposed approach, we use symbolic models to synthesize controllers for two examples of switched systems, including the boost DC-DC converter."}
{"category": "Math", "title": "Counting Abelian Squares", "abstract": "An abelian square is a string of length 2n where the last n symbols form a permutation of the first n symbols. In this note we count the number of abelian squares and give an asymptotic estimate of this quantity."}
{"category": "Math", "title": "On one-sided primitivity of Banach algebras", "abstract": "Let $S$ be the semigroup with identity, generated by $x$ and $y$, subject to $y$ being invertible and $yx=xy^2$. We study two Banach algebra completions of the semigroup algebra $\\mathbb{C}S$. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an alternative proof that $\\mathbb{C}S$ is left-primitive but not right-primitive. We show further that, in contrast to the completions, every irreducible right module for $\\mathbb{C}S$ is finite dimensional and hence that $\\mathbb{C}S$ has a separating family of such modules."}
{"category": "Math", "title": "Stein's method and normal approximation of Poisson functionals", "abstract": "We combine Stein's method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry-Ess\\'een bounds in Central Limit Theorems (CLTs) involving multiple Wiener-It\\^o integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein-Uhlenbeck L\\'evy processes."}
{"category": "Math", "title": "On a variant of the large sieve", "abstract": "We introduce a variant of the large sieve and give an example of its use in a sieving problem. Take the interval [N] = {1,...,N} and, for each odd prime p <= N^{1/2}, remove or ``sieve out'' by all n whose reduction mod p lies in some interval I_p of Z/pZ of length (p-1)/2. Let A be the set that remains: then |A| << N^{1/3 + o(1)}, a bound which improves slightly on the bound of |A| << N^{1/2} which results from applying the large sieve in its usual form. This is a very, very weak result in the direction of a question of Helfgott and Venkatesh, who suggested that nothing like equality can occur in applications of the large sieve unless the unsieved set is essentially the set of values of a polynomial (e.g. A is the set of squares). Assuming the ``exponent pairs conjecture'' (which is deep, as it implies a host of classical questions including the Lindel\\\"of hypothesis, Gauss circle problem and Dirichlet divisor problem) the bound can be improved to |A| << N^{o(1)}. This raises the worry that even reasonably simple sieve problems are connected to issues of which we have little understanding at the present time."}
{"category": "Math", "title": "Cohomological dimension, self-linking, and systolic geometry", "abstract": "Given a closed manifold M, we prove the upper bound of (n+d)/2 for the length of a product of systoles that can form a curvature-free lower bound for the total volume of M, in the spirit of M. Gromov's systolic inequalities. Here n is the dimension of M, while d is the is the cohomological dimension of its fundamental group. We apply this upper bound to show that, in the case of a 4-manifold, the Lusternik--Schnirelmann category is an upper bound for such length. Furthermore we prove a systolic inequality on a manifold M with b_1(M)=2 in the presence of a nontrivial self-linking class of the typical fiber of its Abel--Jacobi map to the 2-torus."}
{"category": "Math", "title": "Finiteness properties of formal local cohomology modules and Cohen-Macaulayness", "abstract": "Let $\\fa$ be an ideal of a local ring $(R,\\fm)$ and $M$ a finitely generated $R$-module. We investigate the structure of the formal local cohomology modules ${\\vpl}_nH^i_{\\fm}(M/\\fa^n M)$, $i\\geq 0$. We prove several results concerning finiteness properties of formal local cohomology modules which indicate that these modules behave very similar to local cohomology modules. Among other things, we prove that if $\\dim R\\leq 2$ or either $\\fa$ is principal or $\\dim R/\\fa\\leq 1$, then $\\Tor_j^R(R/\\fa,{\\vpl}_nH^i_{\\fm}(M/\\fa^n M))$ is Artinian for all $i$ and $j$. Also, we examine the notion $\\fgrade(\\fa,M)$, the formal grade of $M$ with respect to $\\fa$ (i.e. the least integer $i$ such that ${\\vpl}_nH^i_{\\fm}(M/\\fa^n M) \\neq 0$). As applications, we establish a criterion for Cohen-Macaulayness of $M$, and also we provide an upper bound for cohomological dimension of $M$ with respect to $\\fa$."}
{"category": "Math", "title": "Geometric characterization of flat groups of automorphisms", "abstract": "If H is a flat group of automorphisms of finite rank n of a totally disconnected, locally compact group G, then each orbit of H in the metric space B(G) of compact, open subgroups of G is quasi-isometric to n-dimensional euclidean space. In this note we prove the following partial converse: Assume that G is a totally disconnected, locally compact group such that B(G) is a proper metric space and let H be a group of automorphisms of G such that some (equivalently every) orbit of H in B(G) is quasi-isometric to n-dimensional euclidean space, then H has a finite index subgroup which is flat of rank n. We can draw this conclusion under weaker assumptions. We also single out a naturally defined flat subgroup of such groups of automorphisms."}
{"category": "Math", "title": "Lattice Gauge Field Theory and Prismatic Sets", "abstract": "We study prismatics sets analogously to simplical sets except that realization involves prisms, i.e., products of simplices rather than just simplices. Particular examples are the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S and in particular the latter we use to study lattice gauge theory in the sense of Phillips and Stone. Thus for a Lie group G and a set of parallel transport functions defining the transition over faces of the simplices, we define a classifying map from the prismatic star to a prismatic version of the classifying space of G. In turn this defines a G-bundle over the prismatic star."}
{"category": "Math", "title": "The structure of strong linear preservers of gw-majorization on $ \\mathbf{M}_{{n,m}", "abstract": "Let M_{n,m} be the set of all n by m matrices with entries in F, where F is the field of real or complex numbers. A matrix R in M_{n} with the property Re=e, is said to be a g-row stochastic (generalized row stochastic) matrix. Let A,B in M_{n,m}, so B is said to be gw-majorized by A if there exists an n by n g-row stochastic matrix R such that B=RA. In this paper we characterize all linear operators that strongly preserve gw-majorization on M_{n,m} and all linear operators that strongly preserve matrix majorization on M_{n} ."}
{"category": "Math", "title": "Moment bounds for non-linear functionals of the periodogram", "abstract": "In this paper, we prove the validity of the Edgeworth expansion of the Discrete Fourier transforms of some linear time series. This result is applied to approach moments of non linear functionals of the periodogram. As an illustration, we give an expression of the mean square error of the Geweke and Porter-Hudak estimator of the long memory parameter. We prove that this estimator is rate optimal, extending the result of Giraitis, Robinson, Samarov (1997) from Gaussian to linear processes."}
{"category": "Math", "title": "On a theorem of Shkredov", "abstract": "We show that if A is a finite subset of an abelian group with additive energy at least c|A|^3 then there is a subset L of A with |L|=O(c^{-1}\\log |A|) such that |A \\cap Span(L)| >> c^{1/3}|A|."}
{"category": "Math", "title": "Roth's theorem in Z_4^n", "abstract": "We show that if A is a subset of Z_4^n containing no three-term arithmetic progression in which all the elements are distinct then |A|=o(4^n/n)."}
{"category": "Math", "title": "Chowla's cosine problem", "abstract": "Suppose that G is a discrete abelian group and A is a finite symmetric subset of G. We show two main results. i) Either there is a set H of O(log^c|A|) subgroups of G with |A \\triangle \\bigcup H| = o(|A|), or there is a character X on G such that -wh{1_A}(X) >> log^c|A|. ii) If G is finite and |A|>> |G| then either there is a subgroup H of G such that |A \\triangle H| = o(|A|), or there is a character X on G such that -wh{1_A}(X)>> |A|^c."}
{"category": "Math", "title": "Popular difference sets", "abstract": "We provide further explanation of the significance of a construction in a recent paper of Wolf [Israel J. Math. 179 (2010), 253-278] in the context of the problem of finding large subspaces in sumsets."}
{"category": "Math", "title": "Delocalized Betti numbers and Morse type inequalities", "abstract": "In this paper we state and prove Morse type inequalities for Morse functions as well as for closed differential 1-forms. These inequalities involve delocalized Betti numbers. As an immediate consequence, we prove the vanishing of delocalized Betti numbers of manifolds fibering over the circle."}
{"category": "Math", "title": "Formality of the homotopy calculus algebra of Hochschild (co)chains", "abstract": "The Kontsevich-Soibelman solution of the cyclic version of Deligne's conjecture and the formality of the operad of little discs on a cylinder provide us with a natural homotopy calculus structure on the pair (C^*(A), C_*(A)) ``Hochschild cochains + Hochschild chains'' of an associative algebra A. We show that for an arbitrary smooth algebraic variety X with the structure sheaf O_X the sheaf (C^*(O_X), C_*(O_X)) of homotopy calculi is formal. This result was announced in paper [29] by the second and the third author."}
{"category": "Math", "title": "The supremum of autoconvolutions, with applications to additive number theory", "abstract": "We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if f(x) is in L^1 and L^2, is nonnegative, and is supported on an interval of length I, then the supremum of the convolution f*f is at least 0.631 \\| f \\|_1^2 / I. This improves the previous bound of 0.591389 \\| f \\|_1^2 / I. Consequently, we improve the known bounds on several related number-theoretic problems. For a subset A of {1,2, ..., n}, let g be the maximum multiplicity of any element of the multiset {a+b: a,b in A}. Our main corollary is the inequality gn>0.631|A|^2, which holds uniformly for all g, n, and A."}
{"category": "Math", "title": "Morse inequalities for manifolds with boundary", "abstract": "The aim of this paper is to provide a proof for a version of Morse inequality for manifolds with boundary. Our main results are certainly known to the experts on Morse theory, nevertheless it seems necessary to write down a complete proof for it. Our proof is analytic and is based on J. Roe's account of Witten's approach to Morse Theory."}
{"category": "Math", "title": "Quantum Functor $Mor$", "abstract": "Let $Top_c$ be the category of compact spaces and continuous maps and $Top_f\\subset Top_c$ be the full subcategory of finite spaces. Consider the covariant functor $Mor:Top_f^{op}\\times Top_c\\to Top_c$ that associates any pair $(X,Y)$ with the space of all morphisms from $X$ to $Y$. In this paper, we describe a non commutative version of $Mor$. More pricelessly, we define a functor $\\mathfrak{M}\\mathfrak{o}\\mathfrak{r}$, that takes any pair $(B,C)$ of a finitely generated unital C*-algebra $B$ and a finite dimensional C*-algebra $C$ to the quantum family of all morphism from $B$ to $C$."}
{"category": "Math", "title": "BMOA estimates and radial growth of $B_{\\phi}$ functions", "abstract": "BMO estimates and the radial growth of Bloch functions have been studied by B. Korenblum [3]. The present paper contains some natural generalizations of these results."}
{"category": "Math", "title": "On some generalizations of Newton non degeneracy for hypersurface singularities", "abstract": "We introduce two generalizations of Newton-non-degenerate (Nnd) singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called topologically Newton-non-degenerate (tNnd) if the local embedded topological singularity type can be restored from a collection of Newton diagrams (for some coordinate choices). A singularity that is not tNnd is called essentially Newton-degenerate. For plane curves we give an explicit characterization of tNnd singularities; for hypersurfaces we provide several examples. Next, we treat the question: whether Nnd or tNnd is a property of singularity types or of particular representatives. Namely, is the non-degeneracy preserved in an equi-singular family? This fact is proved for curves. For hypersurfaces we give an example of a Nnd hypersurface whose equi-singular deformation consists of essentially Newton-degenerate hypersurfaces. Finally, we define the directionally Newton-non-degenerate germs, a subclass of tNnd germs. For such singularities the classical formulas for the Milnor number and the zeta function of the Nnd hypersurface are generalized."}
{"category": "Math", "title": "The Sigma Invariants of Thompson's Group F", "abstract": "Thompson's group F is the group of all increasing dyadic piecewise linear homeomorphisms of the closed unit interval. We compute Sigma^m(F) and Sigma^m(F;Z), the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F, and we show that Sigma^m(F) = Sigma^m(F;Z). As an application, we show that, for every m, F has subgroups of type F_{m-1} which are not of type F_{m}."}
{"category": "Math", "title": "A note on strong Jordan separation", "abstract": "We provide a strengthening of Jordan separation, to the setting of maps from a compact topological space X into a sphere, where the source space X is not necessarily a codimension one sphere, and the map is not necessarily injective."}
{"category": "Math", "title": "Sigma Invariants of Direct Products of Groups", "abstract": "The Product Conjecture for the homological Bieri-Neumann-Strebel-Renz invariants is proved over a field. Under certain hypotheses the Product Conjecture is shown to also hold over Z, even though D. Schuetz has recently shown that the Conjecture is false in general over Z. Our version over Z is applied in a joint paper with D. Kochloukova to derive new information about subgroups of Thompson's group F, namely that F has subgroups F_m which are not of type F_{m+1}."}
{"category": "Math", "title": "Geometries enumeratives complexe, reelle et tropicale", "abstract": "This text is an introduction to algebraic enumerative geometry and to applications of tropical geometry to classical geometry, based on a course given during the X-UPS mathematical days, 2008 May 14th and 15th. The aim of this text is to be understandable by a first year master student."}
{"category": "Math", "title": "Homotopy classification of maps into homogeneous spaces", "abstract": "We give an alternative to Postnikov's homotopy classification of maps from 3-dimensional CW-complexes to homogeneous spaces G/H of Lie groups. It describes homotopy classes in terms of lifts to the group G and is suitable for extending the notion of homotopy to Sobolev maps. This is required for applications to variational problems of mathematical physics."}
{"category": "Math", "title": "Link invariants from finite racks", "abstract": "We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles."}
{"category": "Math", "title": "Sums of squares and moment problems in equivariant situations", "abstract": "We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group $G$ over $\\R$ acting on an affine $\\R$-variety $V$, we consider the induced dual action on the coordinate ring $\\R[V]$ and on the linear dual space of $\\R[V]$. In this setting, given an invariant closed semialgebraic subset $K$ of $V(\\R)$, we study the problem of representation of invariant nonnegative polynomials on $K$ by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on $\\R[V]$ by invariant measures supported on $K$. To this end, we analyse the relation between quadratic modules of $\\R[V]$ and associated quadratic modules of the (finitely generated) subring $\\R[V]^G$ of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional $K$-moment problem. Most of our results are specific to the case where the group $G(\\R)$ is compact."}
{"category": "Math", "title": "An anticipating It\\^o formula for L\\'evy processes", "abstract": "In this paper, we use the Malliavin calculus techniques to obtain an anticipative version of the change of variable formula for L\\'evy processes. Here the coefficients are in the domain of the anihilation (gradient) operator in the \"future sense\", which includes the family of all adapted and square-integrable processes. This domain was introduced on the Wiener space by Al\\`os and Nualart."}
{"category": "Math", "title": "Representations of Lie Superalgebras in Prime Characteristic I", "abstract": "We initiate the representation theory of restricted Lie superalgebras over an algebraically closed field of characteristic p>2. A superalgebra generalization of the celebrated Kac-Weisfeiler Conjecture is formulated, which exhibits a mixture of p-power and 2-power divisibilities of dimensions of modules. We establish the Conjecture for basic classical Lie superalgebras."}
{"category": "Math", "title": "A note on the invariant subspace problem relative to a type ${\\rm II}_1$ factor", "abstract": "Let $\\M$ be a type ${\\rm II}_1$ factor with a faithful normal tracial state $\\tau$ and let $\\M^\\omega$ be the ultrapower algebra of $\\M$. In this paper, we prove that for every operator $T\\in \\M^\\omega$, there is a family of projections $\\{P_t\\}_{0\\leq t\\leq 1}$ in $\\M^\\omega$ such that $TP_t=P_tTP_t$, $P_s\\leq P_t$ if $s\\leq t$, and $\\tau_\\omega(P_t)=t$. Let $\\mathfrak{M}=\\{Z \\in \\M: \\text{there is a family of projections} \\{P_t\\}_{0\\leq t\\leq 1} \\text{in} \\M \\text{such that} ZP_t=P_tZP_t, P_s\\leq P_t \\text{if} s\\leq t, \\text{and} \\tau(P_t)=t\\}$. As an application we show that for every operator $T\\in \\M$ and $\\epsilon>0$, there is an operator $S\\in \\mathfrak{M}$ such that $\\|S\\|\\leq \\|T\\|$ and $\\|S-T\\|_2<\\epsilon$. We also show that $\\prod_n^\\omega M_n(\\cc)$ is not $\\ast$-isomorphic to the ultrapower algebra of the hyperfinite type ${\\rm II}_1$ factor."}
{"category": "Math", "title": "Birth and death in discrete Morse theory", "abstract": "Suppose $M$ is a finite simplicial complex and that for $0=t_0,t_1,...,t_r=1$ we have a discrete Morse function $F_{t_i}:M\\to \\zr$. In this paper, we study the births and deaths of critical cells for the functions $F_{t_i}$ and present an algorithm for pairing the cells that occur in adjacent slices. We first study the case where the triangulation of $M$ is the same for each $t_i$, and then generalize to the case where the triangulations may differ. This has potential applications in data imaging, where one has function values at a sample of points in some region in space at several different times or at different levels in an object."}
{"category": "Math", "title": "Classifying subcategories of modules over a commutative noetherian ring", "abstract": "Let R be a quotient ring of a commutative coherent regular ring by a finitely generated ideal. Hovey gave a bijection between the set of coherent subcategories of the category of finitely presented R-modules and the set of thick subcategories of the derived category of perfect R-complexes. Using this isomorphism, he proved that every coherent subcategory of finitely presented R-modules is a Serre subcategory. In this paper, it is proved that this holds whenever R is a commutative noetherian ring. This paper also yields a module version of the bijection between the set of localizing subcategories of the derived category of R-modules and the set of subsets of Spec R which was given by Neeman."}
{"category": "Math", "title": "On tabulating virtual strings", "abstract": "A virtual string can be defined as a closed curve on a surface modulo certain equivalence relations. Turaev defined several invariants of virtual strings which we use to produce a table of virtual strings up to 4 crossings. We discuss progress in extending the tabulation to 5 crossings. We also provide a counter-example to a statement of Kadokami."}
{"category": "Math", "title": "On varieties of almost minimal degree I: Secant loci of rational normal scrolls", "abstract": "To complete the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let $\\tilde X \\subset {\\mathbb P}^{r+1}_K$ be a variety of minimal degree and of codimension at least 2, and consider $X_p = \\pi_p (\\tilde X) \\subset {\\mathbb P}^r_K$ where $p \\in {\\mathbb P}^{r+1}_K \\backslash \\tilde X$. By \\cite{B-Sche}, it turns out that the cohomological and local properties of $X_p$ are governed by the secant locus $\\Sigma_p (\\tilde X)$ of $\\tilde X$ with respect to $p$. Along these lines, the present paper is devoted to give a geometric description of the secant stratification of $\\tilde X$, that is of the decomposition of ${\\mathbb P}^{r+1}_K$ via the types of secant loci. We show that there are exactly six possibilities for the secant locus $\\Sigma_p (\\tilde X)$, and we precisely describe each stratum of the secant stratification of $\\tilde X$, each of which turns out to be a quasi-projective variety. As an application, we obtain the classification of all non-normal Del Pezzo varieties by providing a complete list of pairs $(\\tilde X, p)$ where $\\tilde X \\subset {\\mathbb P}^{r+1}_K$ is a variety of minimal degree, $p$ is a closed point in $\\mathbb P^{r+1}_K \\setminus \\tilde X$ and $X_p \\subset {\\mathbb P}^r _K$ is a Del Pezzo variety."}
{"category": "Math", "title": "Group Invariant Entanglements in Generalized Tensor Products", "abstract": "The group invariance of entanglement is obtained within a very general and simple setup of the latter, given by a recently introduced considerably extended concept of tensor products. This general approach to entanglement - unlike the usual one given in the particular setup of tensor products of vector spaces - turns out not to need any specific algebraic structure. The resulting advantage is that, entanglement being in fact defined by a negation, its presence in a general setup increases the chances of its manifestations, thus also its availability as a resource."}
{"category": "Math", "title": "The Fundamental Theorem of Algebra made effective: an elementary real-algebraic proof via Sturm chains", "abstract": "Sturm's theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any real polynomial. In his residue calculus (1831/37) Cauchy extended Sturm's method to count and locate the complex roots of any complex polynomial. For holomorphic functions Cauchy's index is based on contour integration, but in the special case of polynomials it can effectively be calculated via Sturm chains using euclidean division as in the real case. In this way we provide an algebraic proof of Cauchy's theorem for polynomials over any real closed field. As our main tool, we formalize Gauss' geometric notion of winding number (1799) in the real-algebraic setting, from which we derive a real-algebraic proof of the Fundamental Theorem of Algebra. The proof is elementary inasmuch as it uses only the intermediate value theorem and arithmetic of real polynomials. It can thus be formulated in the first-order language of real closed fields. Moreover, the proof is constructive and immediately translates to an algebraic root-finding algorithm."}
{"category": "Math", "title": "Yang-Baxter deformations and rack cohomology", "abstract": "Every rack $Q$ provides a set-theoretic solution $c_Q$ of the Yang-Baxter equation. This article examines the deformation theory of $c_Q$ within the space of Yang-Baxter operators over a ring $\\A$, a problem initiated by Freyd and Yetter in 1989. As our main result we classify deformations in the modular case, which had previously been left in suspense, and establish that every deformation of $c_Q$ is gauge-equivalent to a quasi-diagonal one. Stated informally, in a quasi-diagonal deformation only behaviourally equivalent elements interact. In the extreme case, where all elements of $Q$ are behaviourally distinct, Yang-Baxter cohomology thus collapses to its diagonal part, which we identify with rack cohomology. The latter has been intensively studied in recent years and, in the modular case, is known to produce non-trivial and topologically interesting Yang-Baxter deformations."}
{"category": "Math", "title": "Similarity versus Coincidence Rotations of Lattices", "abstract": "The groups of similarity and coincidence rotations of an arbitrary lattice L in d-dimensional Euclidean space are considered. It is shown that the group of similarity rotations contains the coincidence rotations as a normal subgroup. Furthermore, the structure of the corresponding factor group is examined. If the dimension d is a prime number, this factor group is an elementary Abelian d-group. Moreover, if L is a rational lattice, the factor group is trivial (d odd) or an elementary Abelian 2-group (d even)."}
{"category": "Math", "title": "Global and touchdown behaviour of the generalized MEMS device equation", "abstract": "We prove the local and global existence of solutions of the generalized micro-electromechanical system (MEMS) equation $u_t =\\Delta u+\\lambda f(x)/g(u)$, $u<1$, in $\\Omega\\times (0,\\infty)$, $u(x,t)=0$ on $\\partial\\Omega\\times (0,\\infty)$, $u(x,0)=u_0$ in $\\Omega$, where $\\Omega\\subset\\Bbb{R}^n$ is a bounded domain, $\\lambda >0$ is a constant, $0\\le f\\in C^{\\alpha}(\\overline{\\Omega})$, $f\\not\\equiv 0$, for some constant $0<\\alpha<1$, $0<g\\in C^2((-\\infty,1))$ such that $g'(s)\\le 0$ for any $s<1$ and $u_0\\in L^1(\\Omega)$ with $u_0\\le a<1$ for some constant $a$. We prove that there exists a constant $\\lambda^{\\ast}=\\lambda^{\\ast}(\\Omega, f,g)>0$ such that the associated stationary problem has a solution for any $0\\le\\lambda<\\lambda^*$ and has no solution for any $\\lambda>\\lambda^*$. We obtain comparison theorems for the generalized MEMS equation. Under a mild assumption on the initial value we prove the convergence of global solutions to the solution of the corresponding stationary elliptic equation as $t\\to\\infty$ for any $0\\le\\lambda<\\lambda^*$. We also obtain various conditions for the existence of a touchdown time $T>0$ for the solution $u$. That is a time $T>0$ such that $\\lim_{t\\nearrow T}\\sup_{\\Omega}u(\\cdot,t)=1$."}
{"category": "Math", "title": "Complete Linear Series on a Hyperelliptic Curve", "abstract": "In this paper we study complete linear series on a hyperelliptic curve $C$ of arithmetic genus $g$. Let $A$ be the unique line bundle on $C$ such that $|A|$ is a $g^1_2$, and let $\\mathcal{L}$ be a line bundle on $C$ of degree $d$. Then $\\mathcal{L}$ can be factorized as $\\mathcal{L} = A^m \\otimes B$ where $m$ is the largest integer satisfying $H^0 (C,\\mathcal{L} \\otimes A^{-m}) \\neq 0$. Let $b = {deg}(B)$. We say that \\textit{the factorization type of} $\\mathcal{L}$ is $(m,b)$. Our main results in this paper assert that $(m,b)$ gives a precise answer for many natural questions about $\\mathcal{L}$."}
{"category": "Math", "title": "Foliations and Global Inversion", "abstract": "We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism $f: M \\to\\mathbb{R}^n$ is bijective if and only if $H_{n-1}(M)=0$ and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometric constructions involving foliations and tools from intersection theory. This topological result generalizes in finite dimensions the classical analytic theorem of Hadamard-Plastock, including its recent improvement by Nollet-Xavier. The main theorem also relates to a conjecture of the aforementioned authors, involving the well known Jacobian Conjecture in algebraic geometry."}
{"category": "Math", "title": "Defining the Mean of a Real-Valued Function on an Arbitrary Metric Space", "abstract": "We show how a metric space induces a linear functional (a \"mean\") on real-valued functions with domains in that metric space. This immediately induces a \"relative\" measure on a collection of subsets of the underlying set."}
{"category": "Math", "title": "Existence and asymptotics of solutions of the Debye-Nernst-Planck system in R^2", "abstract": "In this paper we investigate a system describing electrically charged particles in the whole space R^2. Our main goal is to describe large time behavior of solutions which start their evolution from initial data of small size. This is achieved using radially symmetric self-similar solutions."}
{"category": "Math", "title": "Rational points on cubic hypersurfaces that split off a form", "abstract": "Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms that share no common variables. ----- This paper features an appendix \"Groupe de Brauer non ramifi\\'e des hypersurfaces cubiques singuli\\`eres (d'apr\\`es P. Salberger)\", by J.-L. Colliot-Th\\'l\\`ene."}
{"category": "Math", "title": "Cohomology of SL(2,C) character varieties of surface groups and the action of the Torelli group", "abstract": "We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat PSL(2,C) connections. The non-trivial part consists of the even alternating products of degree two Prym representations, so that the kernel of the action is precisely the Prym-Torelli group. We compute the Betti numbers of the ordinary cohomology of the moduli space of flat SL(2,C) connections. Using results of Cappell-Lee-Miller we show that the Prym-Torelli group, which acts trivially on equivariant cohomology, acts non-trivially on ordinary cohomology."}
{"category": "Math", "title": "Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles", "abstract": "We consider finite families of SL(2,R) matrices whose products display uniform exponential growth. These form open subsets of (SL(2,R))^N, and we study their components, boundary, and complement. We also consider the more general situation where the allowed products of matrices satisfy a Markovian rule."}
{"category": "Math", "title": "Theorem of completeness for a Dirac-type operator with generalized $\\lambda$-depending boundary conditions", "abstract": "A completeness theorem is proved involving a system of integro-differential equations with some $\\lambda$-depending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established."}
{"category": "Math", "title": "A period map for generalized deformations", "abstract": "For every compact Kaehler manifold we give a canonical extension of Griffith's period map to generalized deformations, intended as solutions of Maurer-Cartan equation in the algebra of polyvector fields. Our construction involves the notion of Cartan homotopy and a canonical L-infinity structure on mapping cones of morphisms of differential graded Lie algebras."}
{"category": "Math", "title": "Ergodic averages with deterministic weights", "abstract": "The purpose of this paper is to study ergodic averages with deterministic weights. More precisely we study the convergence of the ergodic averages of the type $\\frac{1}{N} \\sum_{k=0}^{N-1} \\theta (k) f \\circ T^{u_k}$ where $\\theta = (\\theta (k) ; k\\in \\NN)$ is a bounded sequence and $u = (u_k ; k\\in \\NN)$ a strictly increasing sequence of integers such that for some $\\delta<1$ $$ S_N (\\theta, u) := \\sup_{\\alpha \\in \\pRR} | \\sum_{k=0}^{N-1} \\theta (k) \\exp (2i\\pi \\alpha u_k) | = O (N^{\\delta}) \\leqno{({\\cal H}_1)} $$ i.e., there exists a constant $C$ such that $S_N (\\theta, u) \\leq C N^{\\delta} $. We define $\\delta (\\theta, u)$ to be the infimum of the $\\delta $ satisfying $\\H_1$ for $\\theta $ and $u$."}
{"category": "Math", "title": "H^1 and BMO for certain nondoubling metric measure spaces", "abstract": "Suppose that (M,d,m) is an unbounded metric measure space, which possesses two geometric properties, called \"isoperimetric property\" and \"approximate midpoint property\", and that the measure m is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure m is not globally doubling. In this paper we define an atomic Hardy space H1(m), where atoms are supported only on \"small balls\", and a corresponding space BMO(m) of functions of bounded mean oscillation, where the control is only on the oscillation over small balls. We prove that BMO(m) is the dual of H1(m) and that an inequality of John-Nirenberg type on small balls holds for functions in BMO(m). Furthermore, we show that the Lp(m) spaces are intermediate spaces between H1(m) and BMO(m), and we develop a theory of singular integral operators acting on function spaces on M. Finally, we show that our theory is strong enough to give H1(m)-L1(m) and L1(m)-BMO(m) estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded on L1(m) and on L\\infty(m)."}
{"category": "Math", "title": "Structure of derivations on various algebras of measurable operators for type I von Neumann algebras", "abstract": "Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $\\tau$ on $M$ let $S(M, \\tau)$ (resp. $S_0(M, \\tau)$) be the algebra of all $\\tau$-measurable (resp. $\\tau$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_\\infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,\\tau)$) is inner, and each derivation on $S_0(M, \\tau)$ is spatial and implemented by an element from $S(M, \\tau).$"}
{"category": "Math", "title": "Multi-Harnack smoothings of real plane branches", "abstract": "We introduce a new method for the construction of smoothings of a real plane branch $(C, 0)$ by using Viro Patchworking method. Since real plane branches are Newton degenerated in general, we cannot apply Viro Patchworking method directly. Instead we apply the Patchworking method for certain Newton non degenerate curve singularities with several branches. These singularities appear as a result of iterating deformations of the strict transforms of the branch at certain infinitely near points of the toric embedded resolution of singularities of $(C,0)$. We characterize the $M$-smoothings obtained by this method by the local data. In particular, we analyze the class of multi-Harnack smoothings, those smoothings arising in a sequence $M$-smoothings of the strict transforms of (C,0) which are in maximal position with respect to the coordinate lines. We prove that there is a unique the topological type of multi-Harnack smoothings, which is determined by the complex equisingularity type of the branch. This result is a local version of a recent Theorem of Mikhalkin."}
{"category": "Math", "title": "Approximate Roots, Toric Resolutions and Deformations of a Plane Branch", "abstract": "We analyze the expansions in terms of the approximate roots of a Weierstrass polynomial $f$ defining a plane branch $(C,0)$, in the light of the toric embedded resolution of the branch. This leads to the definition of a class of (non equisingular) deformations of a plane branch $(C,0)$ supported on certain monomials in the approximate roots of $f$. As a consequence we find out a Kouchnirenko type formula for the Milnor number $(C,0)$. Our results provide a geometrical approach to Abhyankar's straight line conditions and its consequences. As an application we give an equisingularity criterion for a family of plane curves to be equisingular to a plane branch and we express it algorithmically."}
{"category": "Math", "title": "Moduli space of CR-projective complex foliated tori", "abstract": "We study the moduli space of CR-projective complex foliated tori. We describe it in terms of isotropic subspaces of Grassmannian and we show that it is a normal complex analytic space."}
{"category": "Math", "title": "Modular compactifications of M_{1,n}", "abstract": "We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne-Mumford stability. For every pair of integers 0<m<n, we prove that the moduli problem of n-pointed m-stable curves of arithmetic genus one is representable by a proper irreducible Deligne-Mumford stack. We also consider weighted variants of these stability conditions, and construct the corresponding moduli stacks. In forthcoming work, we will prove that these stacks have projective coarse moduli and use the resulting spaces to give a complete description of the log minimal model program for M_{1,n}."}
{"category": "Math", "title": "Cubature formula and interpolation on the cubic domain", "abstract": "Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in \\cite{LSX}. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on $[-1,1]^2$, as well as new results on $[-1,1]^3$. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on $n^3/4 +\\CO(n^2)$ nodes of a cubature formula on $[-1,1]^3$."}
{"category": "Math", "title": "Relative Chern character, boundaries and index formulae", "abstract": "For three classes of elliptic pseudodifferential operators on a compact manifold with boundary which have `geometric K-theory', namely the `transmission algebra' introduced by Boutet de Monvel, the `zero algebra' introduced by Mazzeo and the `scattering algebra' from [MR95k:58168] we give explicit formulae for the Chern character of the index bundle in terms of the symbols (including normal operators at the boundary) of a Fredholm family of fibre operators. This involves appropriate descriptions, in each case, of the cohomology with compact supports in the interior of the total space of a vector bundle over a manifold with boundary in which the Chern character, mapping from the corresponding realization of K-theory, naturally takes values."}
{"category": "Math", "title": "A converse to the Whitehead Theorem", "abstract": "We show that finite-dimensional Lie algebras over a field of characteristic zero such that their high-degree cohomology in any finite-dimensional non-trivial irreducible module vanishes, are, essentially, direct sums of semisimple and nilpotent algebras."}
{"category": "Math", "title": "Damped wave equations with dynamic boundary conditions", "abstract": "We discuss several classes of linear second order initial-boundary value problems, where damping terms appear in the main wave equation as well as in the dynamic boundary condition. We investigate their well-posedness and describe some qualitative properties of their solutions, including boundedness, stability, or almost periodicity. In particular, we are able to characterize the analyticity of certain $C_0$-semigroups associated to such problems. Applications to several problems on domains and networks are shown."}
{"category": "Math", "title": "The second homology group of current Lie algebras", "abstract": "This is an old paper put here for archeological purposes. We derive a general formula expressing the second homology of a Lie algebra of the form L\\otimes A with coefficients in the trivial module through homology of $L$, cyclic homology of $A$, and other invariants of $L$ and $A$. This is achieved by using the Hopf formula expressing the second homology of a Lie algebra in terms of its presentation. We also derive a similar formula for the associated Lie algebra of the tensor product of two associative algebras."}
{"category": "Math", "title": "Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials", "abstract": "We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie on the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-side estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also discuss convergence rates of the method as well as illustrate our results with various numerical experiments."}
{"category": "Math", "title": "A variant of Tao's method with application to restricted sumsets", "abstract": "In this paper, we develop Terence Tao's harmonic analysis method and apply it to restricted sumsets. The well known Cauchy-Davenport theorem asserts that if $A$ and $B$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|A+B|\\ge min{p,|A|+|B|-1}$, where $A+B={a+b: a\\in A, b\\in B}$. In 2005, Terence Tao gave a harmonic analysis proof of the Cauchy-Davenport theorem, by applying a new form of the uncertainty principle on Fourier transform. We modify Tao's method so that it can be used to prove the following extension of the Erdos-Heilbronn conjecture: If $A,B,S$ are nonempty subsets of $Z/pZ$ with $p$ a prime, then $|{a+b: a\\in A, b\\in B, a-b not\\in S}|\\ge min {p,|A|+|B|-2|S|-1}$."}
{"category": "Math", "title": "Motivic double shuffle", "abstract": "The goal of this article is to give an elementary proof of the double shuffle relations directly for the Goncharov and Manin motivic multiple zeta values. The shuffle relation is straightforward, but for the stuffle we use a modification of a method first introduced by P. Cartier for the purpose of proving stuffle for the real multiple zeta values via integrals and blow-up sequences."}
{"category": "Math", "title": "Contraction in $L^1$ and large time behavior for a system arising in chemical reactions and molecular motors", "abstract": "We prove a contraction in $L^1$ property for the solutions of a nonlinear reaction--diffusion system whose special cases include intercellular transport as well as reversible chemical reactions. Assuming the existence of stationary solutions we show that the solutions stabilize as $t$ tends to infinity. Moreover, in the special case of linear reaction terms, we prove the existence and the uniqueness (up to a multiplicative constant) of the stationary solution."}
{"category": "Math", "title": "Numerical approximation of a reaction-diffusion system with fast reversible reaction", "abstract": "We consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction. We deduce from a priori estimates that the approximate solution converges to the weak solution of the reaction-diffusion problem and satisfies estimates which do not depend on the chemical kinetics factor. It follows that the solution converges to the solution of a nonlinear diffusion problem, as the size of the volume elements and the time steps converge to zero while the kinetic rate tends to infinity."}
{"category": "Math", "title": "Classifications of Cohen-Macaulay modules - The base ring associated to a transversal polymatroid", "abstract": "In this thesis, we focus on the study of the base rings associated to some transversal polymatroids. A transversal polymatroid is a special kind of discrete polymatroid. Discrete polymatroids were introduced by Herzog and Hibi \\cite{HH} in 2002."}
{"category": "Math", "title": "Maps, sheaves, and K3 surfaces", "abstract": "The conjectural equivalence of curve counting on Calabi-Yau 3-folds via stable maps and stable pairs is discussed. By considering Calabi-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3 surfaces. New results and conjectures (with D. Maulik) about descendent integration on K3 surfaces are announced. The recent proof of the Yau-Zaslow conjecture is surveyed. The paper accompanies my lecture at the Clay research conference in Cambridge, MA in May 2008."}
{"category": "Math", "title": "The f-invariant and index theory", "abstract": "In this paper we prove a tertiary index theorem which relates a spectral geometric and a homotopy theoretic invariant of an almost complex manifold with framed boundary. It is derived from the index theoretic and homotopy theoretic versions of a complex elliptic genus and interestingly related with the structure of the stable homotopy groups of spheres."}
{"category": "Math", "title": "Maximal operators on variable Lebesgue spaces with weights related to oscillations of Carleson curves", "abstract": "We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights $\\varphi_{t,\\gamma}(\\tau)=|(\\tau-t)^\\gamma|$, where $\\gamma$ is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point $t$ and $\\gamma$ is not real, then $\\varphi_{t,\\gamma}$ is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko."}
{"category": "Math", "title": "Complexifications of Morse functions and the directed Donaldson-Fukaya category", "abstract": "Let N be a closed four dimensional manifold which admits a self-indexing Morse function f with only 3 critical values 0,2,4, and a unique maximum and minimum. Let g be a Riemannian metric on N such that (f,g) is Morse-Smale. We construct from (N,f,g) a certain six dimensional exact symplectic manifold M, together with some exact Lagrangian spheres V_4, V_2^j, V_0 in M, j=1,...,k. These spheres correspond to the critical points x_4, x_2^j, x_0 of f, where the subscript indicates the Morse index. (In a companion paper we explain how (M, V_4,{V_2^j},V_0) is a model for the regular fiber and vanishing spheres of the complexification of f, viewed as a Lefschetz fibration on the disk cotangent bundle D(T^*N).) Our main result is a computation of the Lagrangian Floer homology groups HF(V_4,V_2^j), HF(V_2^j,V_0), HF(V_4,V_0) and the triangle product mu_2: HF(V_4,V_2^j) \\otimes HF(V_2^j,V_0) --> HF(V_4,V_0). The outcome is that the directed Donaldson-Fukaya category of (M,V_4,{V_2^j},V_0) is isomorphic to the flow category of (N,f,g)."}
{"category": "Math", "title": "General solutions to equation $axb^*-bx^*a^*=c$ in rings with involution", "abstract": "In [Q. Xu et al., The solutions to some operator equations, Linear Algebra Appl.(2008), doi:10.1016/j.laa.2008.05.034], Xu et al. provided the necessary and sufficient conditions for the existence of a solution to the equation $AXB^*-BX^*A^*=C$ in the general setting of the adjointable operators between Hilbert $C^*$-modules. Based on the generalized inverses, they also obtained the general expression of the solution in the solvable case. In this paper, we generalize their work in the more general setting of ring $R$ with involution * and reobtain results for rectangular matrices and operators between Hilbert $C^*$-modules by embedding the \"rectangles\" into rings of square matrices or rings of operators acting on the same space."}
{"category": "Math", "title": "Trivial extensions defined by Prufer conditions", "abstract": "This paper deals with well-known extensions of the Prufer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zerodivisors subject to various Prufer conditions. The new examples give further evidence for the validity of Bazzoni-Glaz conjecture on the weak dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials."}
{"category": "Math", "title": "Typical elements in free groups are in different doubly-twisted conjugacy classes", "abstract": "We give an easily checkable algebraic condition which implies that two elements of a finitely generated free group are members of distinct doubly-twisted conjugacy classes with respect to a pair of homomorphisms. We further show that this criterion is satisfied with probability 1 when the homomorphisms and elements are chosen at random."}
{"category": "Math", "title": "The moduli of curves of genus 6 and K3 surfaces", "abstract": "We prove that the coarse moduli space of curves of genus 6 is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized K3 surfaces."}
{"category": "Math", "title": "Ellipses Inscribed in Parallelograms", "abstract": "We prove that there exists a unique ellipse of minimal eccentricity, E_{I}, inscribed in a parallelogram, D. We also prove that the smallest nonnegative angle between equal conjugate diameters of E_{I} equals the smallest nonnegative angle between the diagonals of D. We also prove that if E_{M} is the unique ellipse inscribed in a rectangle, R, which is tangent at the midpoints of the sides of R, then E_{M} is the unique ellipse of minimal eccentricity, maximal area, and maximal arc length inscribed in R. Let D be any convex quadrilateral. In previous papers, the author proved that there is a unique ellipse of minimal eccentricity, E_{I}, inscribed in D, and a unique ellipse, E_{O}, of minimal eccentricity circumscribed about D. We defined D to be bielliptic if E_{I} and E_{O} have the same eccentricity. In this paper we show that a parallelogram, D, is bielliptic if and only if the square of the length of one of the diagonals of D equals twice the square of the length of one of the sides of D."}
{"category": "Math", "title": "C*-algebras associated to shift spaces", "abstract": "These are some notes I wrote for the summer school \"Symbolic dynamics and homeomorphisms of the Cantor set\" at the University of Copenhagen, 23 - 27 June 2008. The notes contain the definition of the C*-algebra associated to a shift space and some basic facts about these. The notes furthermore contain a proof of the fact that the C*-algebra associated to a shift space is a one-sided conjugacy invariant, and a proof of the fact that the Morita equivalence class of the C*-algebra associated to a shift space is a two-sided conjugacy and a flow invariant. The notes also contain a section (without proofs) about the K-theory of C*-algebras associated to shift spaces. The notes are written for people without a background in operator algebra and contains a short appendix about C*-algebras, Morita equivalence and K-theory of C*-algebras."}
{"category": "Math", "title": "Perelman's reduced volume and a gap theorem for the Ricci flow", "abstract": "In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of Anderson's result for Ricci-flat manifolds. As a corollary, a gap theorem for gradient shrinking Ricci solitons is also obtained."}
{"category": "Math", "title": "Exceptional (Z/2Z) x (Z/2Z)-symmetric spaces", "abstract": "The notion of (Z/2Z) x (Z/2Z)-symmetric spaces is a generalization of classical symmetric spaces, where the group Z/2Z is replaced by (Z/2Z) x (Z/2Z). In this article, a classification is given of the (Z/2Z) x (Z/2Z)-symmetric spaces G/K where G is an exceptional compact Lie group or Spin(8), complementing recent results of Bahturin and Goze. Our results are equivalent to a classification of (Z/2Z) x (Z/2Z)-gradings on the exceptional simple Lie algebras e6, e7, e8, f4, g2 and so(8)."}
{"category": "Math", "title": "Notes on periodic elements of Garside groups", "abstract": "Let $G$ be a Garside group with Garside element $\\Delta$. An element $g$ in $G$ is said to be \\emph{periodic} if some power of $g$ lies in the cyclic group generated by $\\Delta$. This paper shows the following. (i) The periodicity of an element does not depend on the choice of a particular Garside structure if and only if the center of $G$ is cyclic. (ii) If $g^k=\\Delta^{ka}$ for some nonzero integer $k$, then $g$ is conjugate to $\\Delta^a$. (iii) Every finite subgroup of the quotient group $G/<\\Delta^m>$ is cyclic, where $\\Delta^m$ is the minimal positive central power of $\\Delta$."}
{"category": "Math", "title": "On a local characterization of pseudoconvex domains", "abstract": "Pseudoconvexity of a domain in $\\Bbb C^n$ is described in terms of the existence of a locally defined plurisubharmonic/holomorphic function near any boundary point that is unbounded at the point."}
{"category": "Math", "title": "Smooth Structures and Normalized Ricci Flows on Non-Simply Connected Four-Manifolds", "abstract": "A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on 4-manifolds with non-trivial fundamental group and the relation with the smooth structures. For example, we prove that, for any finite cyclic group ${\\mathbb Z}_{d}$, where $d>1$, there exists a compact topological 4-manifold $X$ with fundamental group ${\\mathbb Z}_{d}$, which admits at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admits infinitely many distinct smooth structures for which {\\it no} non-singular solution of the normalized Ricci flow exists. Related non-existence results on non-singular solutions are also proved. Among others, we show that there are no non-singular $\\ZZ_d-$equivariant solutions to the normalized Ricci flow on appropriate connected sums of $\\bcp ^2$s and $\\cpb $s ($d>1$)."}
{"category": "Math", "title": "A generalization of the Widder potential transform and applications", "abstract": "In the present paper the authors consider the $\\mathcal{P}_{\\nu,2}$-transform as a generalization of the Widder potential transform and the Glasser transform. The $\\mathcal{P}_{\\nu,2}$-transform is obtained as an iteration of the the $\\mathcal{L}_{2}$-transform with itself. Many identities involving these transforms are given. By making use of these identities, a number of new Parseval-Goldstein type identities are obtained for these and many other well-known integral transforms. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results presented here."}
{"category": "Math", "title": "Double shuffle relation for associators", "abstract": "It is proved that Drinfel'd's pentagon equation implies the generalized double shuffle relation. As a corollary, an embedding from the Grothendieck-Teichm\\\"uller group $GRT_1$ into Racinet's double shuffle group $DMR_0$ is obtained, which settles the project of Deligne-Terasoma. It is also proved that the gamma factorization formula follows from the generalized double shuffle relation."}
{"category": "Math", "title": "Non-standard approximations of the Ito-map", "abstract": "The Wong-Zakai theorem asserts that ODEs driven by \"reasonable\" (e.g. piecewise linear) approximations of Brownian motion converge to the corresponding Stratonovich stochastic differential equation. With the aid of rough path analysis, we study \"non-reasonable\" approximations and go beyond a well-known criterion of [Ikeda--Watanabe, North Holland 1989] in the sense that our result applies to perturbations on all levels, exhibiting additional drift terms involving any iterated Lie brackets of the driving vector fields. In particular, this applies to the approximations by McShane ('72) and Sussmann ('91). Our approach is not restricted to Brownian driving signals. At last, these ideas can be used to prove optimality of certain rough path estimates."}
{"category": "Math", "title": "Threefolds of order one in the six-quadric", "abstract": "Consider the smooth quadric Q_6 in P^7. The middle homology group H_6(Q_6,Z) is two-dimensional with a basis given by two classes of linear subspaces. We classify all threefolds of bidegree (1,p) inside Q_6."}
{"category": "Math", "title": "Quantized dual graded graphs", "abstract": "We study quantized dual graded graphs, which are graphs equipped with linear operators satisfying the relation DU - qUD = rI. We construct examples based upon: the Fibonacci poset, permutations, standard Young tableau, and plane binary trees."}
{"category": "Math", "title": "On implicit ODEs with hexagonal web of solutions", "abstract": "Solutions of an implicit ODE form a web. Already for cubic ODEs the 3-web of solutions has a nontrivial local invariant, namely the curvature form. Thus any local classification of implicit ODEs necessarily has functional moduli if no restriction on the class of ODEs is imposed. Here the most symmetric case of hexagonal 3-web of solutions is discussed, i.e. the curvature is supposed to vanish identically. A finite list of normal forms is established under some natural regularity assumptions. Geometrical meaning of these assumptions is that the surface, defined by ODE in the space of 1-jets, is smooth as well as the criminant, which is the critical set of this surface projection to the plane."}
{"category": "Math", "title": "Livsic theorem for matrix cocycles", "abstract": "We prove the Liv\\v{s}ic Theorem for arbitrary $GL(m,\\mathbb R)$ cocycles. We consider a hyperbolic dynamical system $f : X \\to X$ and a H\\\"older continuous function $A: X \\to GL(m,\\mathbb R)$. We show that if $A$ has trivial periodic data, i.e. $A(f^{n-1} p) ... A(fp) A(p) = Id$ for each periodic point $p=f^n p$, then there exists a H\\\"older continuous function $C: X \\to GL(m,\\mathbb R)$ satisfying $A (x) = C(f x) C(x) ^{-1}$ for all $x \\in X$. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on growth of products along orbits for an arbitrary H\\\"older function $A$."}
{"category": "Math", "title": "The Kuratowski covering conjecture for graphs of order < 10 for the nonorientable surfaces of genus 3 and 4", "abstract": "Kuratowski proved that a finite graph embeds in the plane if it does not contain a subdivision of either K_5 or K_{3,3}, called Kuratowski subgraphs. A conjectured generalization of this result to all nonorientable surfaces says that a finite minimal forbidden subgraph for the nonorientable surface of genus g can be written as the union of g+1 Kuratowski subgraphs such that the union of each pair of these fails to embed in the projective plane, the union of each triple of these fails to embed in the Klein bottle if g >= 2, and the union of each triple of these fails to embed in the torus if g >= 3. We show that this conjecture is true for all minimal forbidden subgraphs of order < 10 for the nonorientable surfaces of genus 3 and 4."}
{"category": "Math", "title": "Some notable properties of the standard oncology phase I design", "abstract": "We identify three properties of the standard oncology phase I trial design or 3 + 3 design. We show that the standard design implicitly uses isotonic regression to estimate a maximum tolerated dose. We next illustrate the relationship between the standard design and a Bayesian design proposed by Ji et al. (2007). A slight modification to this Bayesian design, under a particular model specification, would assign treatments in a manner identical to the standard design. We finally present calculations revealing the behavior of the standard design in a worst case scenario and compare its behavior with other 3 + 3-like designs."}
{"category": "Math", "title": "Graph Powers and Graph Homomorphisms", "abstract": "In this paper we investigate some basic properties of fractional powers. In this regard, we show that for any rational number $1\\leq {2r+1\\over 2s+1}< og(G)$, $G^{{2r+1\\over 2s+1}}\\longrightarrow H$ if and only if $G\\longrightarrow H^{-{2s+1\\over 2r+1}}.$ Also, for two rational numbers ${2r+1\\over 2s+1} < {2p+1\\over 2q+1}$ and a non-bipartite graph $G$, we show that $G^{2r+1\\over 2s+1} < G^{2p+1\\over 2q+1}$. In the sequel, we introduce an equivalent definition for circular chromatic number of graphs in terms of fractional powers. We also present a sufficient condition for equality of chromatic number and circular chromatic number."}
{"category": "Math", "title": "About the asymptotic formula for spectral function of the Laplace-Beltrami operator on sphere", "abstract": "In this work we established asymptotical behavior for Riesz means of the spectral function of the Laplace operator on unit sphere."}
{"category": "Math", "title": "Global regularity of wave maps V. Large data local wellposedness and perturbation theory in the energy class", "abstract": "Using the harmonic map heat flow and the function spaces of Tataru and the author, we establish a large data local well-posedness result in the energy class for wave maps from two-dimensional Minkowski space $\\R^{1+2}$ to hyperbolic spaces $\\H^m$. This is one of the five claims required in an earlier paper in this series to prove global regularity for such wave maps."}
{"category": "Math", "title": "A note on localizations of mapping spaces", "abstract": "We show that if A is a simply connected, finite, pointed CW-complex then the mapping spaces Map(A, -) are preserved by the localization functors only if A has the rational homotopy type of a wedge of spheres of a fixed dimension."}
{"category": "Math", "title": "Characterization of SL(2,q) by its non-commuting graph", "abstract": "Let $G$ be a non-abelian group and $Z(G)$ be its center. The non-commuting graph $\\mathcal{A}_G$ of $G$ is the graph whose vertex set is $G\\backslash Z(G)$ and two vertices are joined by an edge if they do not commute. Let $\\mathrm{SL}(2,q)$ be the special linear group of degree 2 over the finite field of order $q$. In this paper we prove that if $G$ is a group such that $\\mathcal{A}_G\\cong \\mathcal{A}_{\\mathrm{SL}(2,q)}$ for some prime power $q\\geq 2$, then $G\\cong \\mathrm{SL}(2,q)$."}
{"category": "Math", "title": "Discrete Asymptotic Behaviors for Skew-Evolution Semiflows on Banach Spaces", "abstract": "The paper emphasizes asymptotic behaviors, as stability, instability, dichotomy and trichotomy for skew-evolution semiflows, defined by means of evolution semiflows and evolution cocycles and which can be considered generalizations for evolution operators and skew-product semiflows. The definition are given in continuous time, but the unified treatment for the characterization of the studied properties in the nonuniform case is given in discrete time. The property of trichotomy, introduced in finite dimension by S. Elaydi and O. Hajek in 1988 as a natural generalization for the dichotomy of linear time-varying differential systems, was studied by us in continuous time and from uniform point of view and in discrete time and from nonuniform point of view but for a particular case of one-parameter semiflows."}
{"category": "Math", "title": "The Chow ring of relative Fulton-MacPherson space", "abstract": "Suppose that X is a nonsingular variety and D is a nonsingular proper subvariety. Configuration spaces of distinct and non-distinct n points in X away from D were constructed by the author and B. Kim in arXiv:0806.3819, by using the method of wonderful compactification. In this paper, we give an explicit presentation of Chow motives and Chow rings of these configuration spaces."}
{"category": "Math", "title": "An Overview of Mixture Models", "abstract": "This paper has been withdrawn. With the advancement of statistical theory and computing power, data sets are providing a greater amount of insight into the problems of today. Statisticians have an ever increasing number of tools to attack these problems, some of which can be implemented in the area of mixture modeling. There is a great deal of literature on mixture models and this work attempts to provide a general overview of the subject, including the discussion of relevant issues and algorithms. The reader can hope to gain a broad understanding of concepts in mixture modeling and find the references cited within as a valuable resource for the next stage of their research."}
{"category": "Math", "title": "Lantern relations and rational blowdowns", "abstract": "We discuss a connection between the lantern relation in mapping class groups and the rational blowing down process for 4-manifolds. More precisely, if we change a positive relator in Dehn twist generators of the mapping class group by using a lantern relation, the corresponding Lefschetz fibration changes into its rational blowdown along a copy of the configuration C_2. We exhibit examples of such rational blowdowns of Lefschetz fibrations whose blowup is homeomorphic but not diffeomorphic to the original fibration."}
{"category": "Math", "title": "Hard Lefschetz actions in Riemannian geometry with special holonomy", "abstract": "It is known that the hard Lefschetz action, together with K\\\"ahler identities for K\\\"ahler (resp. hyperk\\\"ahler) manifolds, determines a $\\mathfrak{su}(1,1)_{sup}$ (resp. $\\mathfrak{sp}(1,1)_{sup}$) Lie superalgebra action on differential forms. In this paper, we explain the geometric origin of this action, and we also generalize it to manifolds with other holonomy groups. For semi-flat Calabi-Yau (resp. hyperk\\\"ahler) manifolds, these symmetries can be enlarged to a $\\mathfrak{so}(2,2)_{sup}$ (resp. $\\mathfrak{su}(2,2)_{sup}$) action."}
{"category": "Math", "title": "Coverings, composites and cables of virtual strings", "abstract": "A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string or virtual strings which produce another virtual string, namely covering, composition and cabling. In particular we study virtual strings unchanged by the covering operation. We also show how the based matrix of a composite virtual string is related to the based matrices of its components, correcting a result by Turaev. Finally we investigate what happens under cabling to some invariants defined by Turaev."}
{"category": "Math", "title": "Non-commutative Castelnuovo-Mumford Regularity and AS-regular Algebras", "abstract": "Let $A$ be a connected graded $k$-algebra with a balanced dualizing complex. We prove that $A$ is a Koszul AS-regular algebra if and only if that the Castelnuovo-Mumford regularity and the Ext-regularity coincide for all finitely generated $A$-modules. This can be viewed as a non-commutative version of \\cite[Theorem 1.3]{ro}. By using Castelnuovo-Mumford regularity, we prove that any Koszul standard AS-Gorenstein algebra is AS-regular. As a preparation to prove the main result, we also prove the following statements are equivalent: (1) $A$ is AS-Gorenstein; (2) $A$ has finite left injective dimension; (3) the dualizing complex has finite left projective dimension. This generalizes \\cite[Corollary 5.9]{mori}."}
{"category": "Math", "title": "Vacca-type series for values of the generalized-Euler-constant function and its derivative", "abstract": "We generalize well-known Catalan-type integrals for Euler's constant to values of the generalized-Euler-constant function and its derivatives. Using generating functions appeared in these integral representations we give new Vacca and Ramanujan-type series for values of the generalized-Euler-constant function and Addison-type series for values of the generalized-Euler-constant function and its derivative. As a consequence, we get base $B$ rational series for $\\log\\frac{4}{\\pi},$ $\\frac{G}{\\pi}$ (where $G$ is Catalan's constant), $\\frac{\\zeta'(2)}{\\pi^2}$ and also for logarithms of Somos's and Glaisher-Kinkelin's constants."}
{"category": "Math", "title": "A modified lookdown construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks", "abstract": "Let $\\Lambda$ be a finite measure on the unit interval. A $\\Lambda$-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions ($\\Lambda$-coalescent) in analogy to the duality known for the classical Fleming Viot process and Kingman's coalescent, where $\\Lambda$ is the Dirac measure in 0. We explicitly construct a dual process of the coalescent with simultaneous multiple collisions ($\\Xi$-coalescent) with mutation, the $\\Xi$-Fleming-Viot process with mutation, and provide a representation based on the empirical measure of an exchangeable particle system along the lines of Donnelly and Kurtz (1999). We establish pathwise convergence of the approximating systems to the limiting $\\Xi$-Fleming-Viot process with mutation. An alternative construction of the semigroup based on the Hille-Yosida theorem is provided and various types of duality of the processes are discussed. In the last part of the paper a population is considered which undergoes recurrent bottlenecks. In this scenario, non-trivial $\\Xi$-Fleming-Viot processes naturally arise as limiting models."}
{"category": "Math", "title": "Estimates for differential operators of vector analysis involving $L^1$-norm", "abstract": "New Hardy and Sobolev type inequalities involving $L^1$-norms of scalar and vector-valued functions in $\\Bbb{R}^n$ are obtained. The work is related to some problems stated in the recent paper by Bourgain and Brezis"}
{"category": "Math", "title": "Gromov Compactness in Hoelder Spaces and Minimal Connections on Jet Bundles", "abstract": "The goal of this work is to establish a proof of the Gromov convergence in Hoelder spaces for curves with a totally real boundary condition following the original geometric idea of Gromov. We use a local reflection principle in neighbourhoods of the totally real submanifold as developed by Ivashkovich and Shevchishin and existence results for special connections on spaces of jet bundles to obtain higher regularity and Gromov-Schwarz estimates along the boundary."}
{"category": "Math", "title": "Refined Analytic Torsion on Manifolds with Boundary", "abstract": "The refined analytic torsion, defined by M. Braverman and T. Kappeler on closed manifolds, can be viewed as a refinement of the Ray-Singer torsion, since it is a canonical choice of an element with Ray-Singer norm one, in case of unitary representations. The complex phase of the refinement is given by the rho-invariant of the odd-signature operator. Unfortunately there seems to be no canonical way to extend the construction of Braverman and Kappeler to compact manifolds with boundary. In particular a gluing formula seems to be out of reach. We propose a different refinement of analytic torsion, similar to Braverman and Kappeler, which does apply to compact manifolds with and without boundary. In a subsequent publication we establish a gluing formula for our construction, which in fact can also be viewed as a gluing law for the original definition of refined analytic torsion by Braverman and Kappeler."}
{"category": "Math", "title": "Counting squarefree discriminants of trinomials under abc", "abstract": "For an odd positive integer $n\\ge 5$, assuming the truth of the $abc$ conjecture, we show that for a positive proportion of pairs $(a,b)$ of integers the trinomials of the form $t^n+at+b (a,b\\in \\mathbb Z)$ are irreducible and their discriminants are squarefree."}
{"category": "Math", "title": "The Hausdorff dimension of the double points on the Brownian frontier", "abstract": "The frontier of a planar Brownian motion is the boundary of the unbounded component of the complement of its range. In this paper we find the Hausdorff dimension of the set of double points on the frontier."}
{"category": "Math", "title": "Strong laws for balanced triangular urns", "abstract": "Consider an urn model whose replacement matrix is triangular, has all entries nonnegative and the row sums are all equal to one. We obtain the strong laws for the counts of balls corresponding to each color. The scalings for these laws depend on the diagonal elements of a rearranged replacement matrix. We use the strong laws obtained to study further behavior of certain three color urn models."}
{"category": "Math", "title": "On the geometry of the f-invariant", "abstract": "The f-invariant is a higher version of the e-invariant that takes values in the divided congruences between modular forms; it can be formulated as an elliptic genus of manifolds with corners of codimension two. In this thesis, we develop a geometrical interpretation of the f-invariant in terms of index theory, thereby providing an analytical link between the stable homotopy groups of the spheres and the arithmetic of modular forms. In particular, we are able to establish a formula that allows us to compute the f-invariant from a single face. Furthermore, we apply our results to the situation of cartesian products and principal circle bundles, performing explicit calculations."}
{"category": "Math", "title": "Maximally homogeneous para-CR manifolds of semisimple type", "abstract": "An almost para-CR structure on a manifold $M$ is given by a distribution $HM \\subset TM$ together with a field $K \\in \\Gamma({\\rm End}(HM))$ of involutive endomorphisms of $HM$. If $K$ satisfies an integrability condition, then $(HM,K)$ is called a para-CR structure. The notion of maximally homogeneous para-CR structure of a semisimple type is given. A classification of such maximally homogeneous para-CR structures is given in terms of appropriate gradations of real semisimple Lie algebras."}
{"category": "Math", "title": "Smoothings of schemes with non-isolated singularities", "abstract": "In this paper we study the deformation and Q-Gorenstein deformation theory of schemes with non-isolated singularities. We obtain obstruction spaces for the existence of deformations and also for local deformations to exist globally. Finally we obtain explicit criteria in order for a pure and reduced scheme of finite type over a field $k$ to have smoothings and Q-Gorenstein smoothings."}
{"category": "Math", "title": "Analytic Torsion of a Bounded Generalized Cone", "abstract": "Torsion invariants for manifolds which are not simply connected were introduced by K. Reidemeister and generalized to higher dimensions by W. Franz. The Reidemeister torsion, was the first invariant of manifolds which was not a homotopy invariant. The analytic counterpart of the combinatorial Reidemeister torsion was introduced by D. B. Ray and I. M. Singer in form of a weighted product of zeta-regularized determinants of Laplace operators on differential forms. The celebrated Cheeger-Mueller Theorem, established independently by J. Cheeger and W. Mueller, proved equality between the analytic Ray-Singer torsion and the combinatorial Reidemeister torsion for any smooth closed manifold with an orthogonal representation of its fundamental group. Motivated by the vision of a Cheeger-Mueller type result on manifolds with conical singularities, we compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of M. Spreafico and using the symmetry of the de Rham complex, as established by M. Lesch."}
{"category": "Math", "title": "Gluing Formula for Refined Analytic Torsion", "abstract": "In the previous article \"Refined Analytic Torsion on Manifolds with Boundary\" we have presented a construction of refined analytic torsion in the spirit of Braverman and Kappeler, which does apply to compact manifolds with and without boundary. We now derive a gluing formula for our construction, which can be viewed as a gluing law for the original definition of refined analytic torsion by Braverman and Kappeler. A gluing formula allows to compute the torsion invariant by cutting the manifold into elementary pieces and performing computations on each component. Certainly, the general fact of existence of such gluing formulas is remarkable from the analytic point of view, since the secondary spectral invariants are uppermost non-local."}
{"category": "Math", "title": "Formality of the little N-disks operad", "abstract": "We develop the details of Kontsevich's proof of the formality of little N-disks operad over the field of real numbers. Formality holds in the category of operads of chain complexes and also in some sense in the category of commutative differential graded algebras, which is the category encoding \"real\" homotopy theory. We also prove a relative version of the formality for the inclusion of the little m-disks operad in the little N-disks operad for N>=2m+1."}
{"category": "Math", "title": "On the Danilov-Gizatullin Isomorphism Theorem", "abstract": "A Danilov-Gizatullin surface is a normal affine surface V, which is a complement to an ample section S in a Hirzebruch surface of index d. By a surprising result due to Danilov and Gizatullin, V depends only on the self-intersection number of S and neither on d nor on S. In this note we provide a new and simple proof of this Isomorphism Theorem."}
{"category": "Math", "title": "Sums of squares on reducible real curves", "abstract": "We ask whether every polynomial function that is non-negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring. Scheiderer has classified all irreducible curves for which this is the case. For reducible curves, we show how the answer depends on the configuration of the irreducible components and give complete necessary and sufficient conditions. We also prove partial results in the more general case of finitely generated preorderings and discuss applications to the moment problem for semialgebraic sets."}
{"category": "Math", "title": "Fourier Series Of the Derivatives of Hurwitz and Lerch Zeta Functions", "abstract": "As a function of second variable, we identify the Fourier series of Hurwitz zeta function and its derivatives on the unit interval. Consequently, we obtain results based on the formula for Fourier coefficients and also on Parseval's theorem. We do likewise in the case of Lerch's zeta function and its derivatives."}
{"category": "Math", "title": "Spectral duality for a class of unbounded operators", "abstract": "We establish a spectral duality for certain unbounded operators in Hilbert space. The class of operators includes discrete graph Laplacians arising from infinite weighted graphs. The problem in this context is to establish a practical approximation of infinite models with suitable sequences of finite models which in turn allow (relatively) easy computations. Let $X$ be an infinite set and let $\\H$ be a Hilbert space of functions on $X$ with inner product $\\ip{\\cdot}{\\cdot}=\\ip{\\cdot}{\\cdot}_{\\H}$. We will be assuming that the Dirac masses $\\delta_x$, for $x\\in X$, are contained in $\\H$. And we then define an associated operator $\\Delta$ in $\\H$ given by $$(\\Delta v)(x):=\\ip{\\delta_x}{v}_{\\H}.$$ Similarly, for every finite subset $F\\subset X$, we get an operator $\\Delta_F$. If $F_1\\subset F_2\\subset...$ is an ascending sequence of finite subsets such that $\\cup_{k\\in\\bn}F_k=X$, we are interested in the following two problems: (a) obtaining an approximation formula $$\\lim_{k\\to\\infty}\\Delta_{F_k}=\\Delta;$$ and (b) establish a computational spectral analysis for the truncated operators $\\Delta_F$ in (a)."}
{"category": "Math", "title": "A Multivariate Fast Discrete Walsh Transform with an Application to Function Interpolation", "abstract": "For high dimensional problems, such as approximation and integration, one cannot afford to sample on a grid because of the curse of dimensionality. An attractive alternative is to sample on a low discrepancy set, such as an integration lattice or a digital net. This article introduces a multivariate fast discrete Walsh transform for data sampled on a digital net that requires only $O(N \\log N)$ operations, where $N$ is the number of data points. This algorithm and its inverse are digital analogs of multivariate fast Fourier transforms. This fast discrete Walsh transform and its inverse may be used to approximate the Walsh coefficients of a function and then construct a spline interpolant of the function. This interpolant may then be used to estimate the function's effective dimension, an important concept in the theory of numerical multivariate integration. Numerical results for various functions are presented."}
{"category": "Math", "title": "O carater de Chern-Connes para C$^*$-sistemas dinamicos calculado em algumas algebras de operadores pseudodiferenciais", "abstract": "Given a C$^*$-dynamical system $(A, G, \\alpha)$ one defines a homomorphism, called the Chern-Connes character, that take an element in $K_0(A) \\oplus K_1(A)$, the K-theory groups of the C$^*$-algebra $A$, and maps it into $H_{\\mathbb{R}}^*(G)$, the real deRham cohomology ring of $G$. We explictly compute this homomorphism for the examples $(\\overline{\\Psi_{cl}^0(S^1)}, S^1, \\alpha)$ and $(\\overline{\\Psi_{cl}^0(S^2)}, SO(3), \\alpha)$, where $\\overline{\\Psi_{cl}^0(M)}$ denotes the C$^*$-algebra generated by the classical pseudodifferential operators of zero order in the manifold $M$ and $\\alpha$ the action of conjugation by the regular representation (translations)."}
{"category": "Math", "title": "Series Jackson networks and non-crossing probabilities", "abstract": "This paper studies the queue length process in series Jackson networks with external input to the first station. We show that its Markov transition probabilities can be written as a finite sum of non-crossing probabilities, so that questions on time-dependent queueing behavior are translated to questions on non-crossing probabilities. This makes previous work on non-crossing probabilities relevant to queueing systems and allows new queueing results to be established. To illustrate the latter, we prove that the relaxation time (i.e., the reciprocal of the `spectral gap') of a positive recurrent system equals the relaxation time of an M/M/1 queue with the same arrival and service rates as the network's bottleneck station. This resolves a conjecture of Blanc, which he proved for two queues in series."}
{"category": "Math", "title": "L^1 Ergodic Theorems for Random Group Averages", "abstract": "This is an earlier, but more general, version of \"An L^1 Ergodic Theorem for Sparse Random Subsequences\". We prove an L^1 ergodic theorem for averages defined by independent random selector variables, in a setting of general measure-preserving group actions. A far more readable version of this paper is in the works."}
{"category": "Math", "title": "Partial choice functions for families of finite sets", "abstract": "Let m>2 be an integer. We show that ZF + \"For every integer n, Every countable family of non-empty sets of cardinality at most n has an infinite partial choice function\" is not strong enough to prove that every countable set of m-element sets has a choice function. In the case where m=p is prime, to obtain the independence result we make use of a permutation model in which the set of atoms has the structure of a vector space over the field of p elements. When m is non-prime, a suitable permutation model is built from the models used in the prime cases."}
{"category": "Math", "title": "On the union stabilization of two Heegaard splittings", "abstract": "Let two Heegaard splittings $V_1 \\cup W_1$ and $V_2 \\cup W_2$ of a 3-manifold $M$ be given. We consider the union stabilization $M=V \\cup W$ which is a common stabilization of $V_1 \\cup W_1$ and $V_2 \\cup W_2$ having the property that $V=V_1 \\cup V_2$. We show that any two Heegaard splittings of a 3-manifold have a union stabilization. We also give some examples with numerical bounds on the minimal genus of union stabilization. On the other hand, we give an example of a candidate for which the minimal genus of union stabilization is strictly larger than the usual stable genus -- the minimal genus of common stabilization."}
{"category": "Math", "title": "Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity", "abstract": "For odd dimensional Poincar\\'e-Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k<\\ndemi$) which are $C^m$ (with $m\\in\\nn$) on the conformal compactification $\\bar{X}$ of $X$. This is infinite dimensional for small $m$ but it becomes finite dimensional if $m$ is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology $H^k(\\bar{X},\\pl\\bar{X})$ and the kernel of the Branson-Gover \\cite{BG} differential operators $(L_k,G_k)$ on the conformal infinity $(\\pl\\bar{X},[h_0])$. In a second time we relate the set of $C^{n-2k+1}(\\Lambda^k(\\bar{X}))$ forms in the kernel of $d+\\delta_g$ to the conformal harmonics on the boundary in the sense of \\cite{BG}, providing some sort of long exact sequence adapted to this setting. This study also provides another construction of Branson-Gover differential operators, including a parallel construction of the generalization of $Q$ curvature for forms."}
{"category": "Math", "title": "A Proof of George Andrews' and Dave Robbins' q-TSPP Conjecture (modulo a finite amount of routine calculations)", "abstract": "In the historic conference Combinatoire Enumerative[LL] wonderfully organized by Gilbert Labelle and Pierre Leroux there were many stimulating lectures, including a very interesting one by Pierre Leroux himself, who talked about his joint work with Xavier Viennot[LV], on solving differential equations combinatorially! During the problem session of that very same colloque, chaired by Pierre Leroux, Richard Stanley raised some intriguing problems about the enumeration of plane partitions, that he later expanded into a fascinating article[Sta1]. Most of these problems concerned the enumeration of symmetry classes of plane partitions, that were discussed in more detail in another article of Stanley[Sta2]. All of the conjectures in the latter article have since been proved (see Dave Bressoud's modern classic[B]), except one, that, so far, resisted the efforts of the greatest minds in enumerative combinatorics. It concerns the proof of an explicit formula for the q-enumeration of totally symmetric plane partitions, conjectured independently by George Andrews and Dave Robbins([Sta2],[Sta1](conj. 7), [B](conj. 13)). In this tribute to Pierre Leroux, we describe how to prove that last stronghold."}
{"category": "Math", "title": "Microarrays, Empirical Bayes and the Two-Groups Model", "abstract": "The classic frequentist theory of hypothesis testing developed by Neyman, Pearson and Fisher has a claim to being the twentieth century's most influential piece of applied mathematics. Something new is happening in the twenty-first century: high-throughput devices, such as microarrays, routinely require simultaneous hypothesis tests for thousands of individual cases, not at all what the classical theory had in mind. In these situations empirical Bayes information begins to force itself upon frequentists and Bayesians alike. The two-groups model is a simple Bayesian construction that facilitates empirical Bayes analysis. This article concerns the interplay of Bayesian and frequentist ideas in the two-groups setting, with particular attention focused on Benjamini and Hochberg's False Discovery Rate method. Topics include the choice and meaning of the null hypothesis in large-scale testing situations, power considerations, the limitations of permutation methods, significance testing for groups of cases (such as pathways in microarray studies), correlation effects, multiple confidence intervals and Bayesian competitors to the two-groups model."}
{"category": "Math", "title": "The dealternating number and the alternation number of a closed 3-braid", "abstract": "We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation number) and any braid index."}
{"category": "Math", "title": "Comment: Microarrays, Empirical Bayes and the Two-Groups Model", "abstract": "Comment on ``Microarrays, Empirical Bayes and the Two-Groups Model'' [arXiv:0808.0572]"}
{"category": "Math", "title": "Comment: Microarrays, Empirical Bayes and the Two-Group Model", "abstract": "Comment on ``Microarrays, Empirical Bayes and the Two-Group Model'' [arXiv:0808.0572]"}
{"category": "Math", "title": "Comment: Microarrays, Empirical Bayes and the Two-Groups Model", "abstract": "Brad Efron's paper [arXiv:0808.0572] has inspired a return to the ideas behind Bayes, frequency and empirical Bayes. The latter preferably would not be limited to exchangeable models for the data and hyperparameters. Parallels are revealed between microarray analyses and profiling of hospitals, with advances suggesting more decision modeling for gene identification also. Then good multilevel and empirical Bayes models for random effects should be sought when regression toward the mean is anticipated."}
{"category": "Math", "title": "Comment: Microarrays, Empirical Bayes and the Two-Groups Model", "abstract": "Comment on ``Microarrays, Empirical Bayes and the Two-Groups Model'' [arXiv:0808.0572]"}
{"category": "Math", "title": "Limit laws of entrance times for low complexity Cantor minimal systems", "abstract": "This paper is devoted to the study of limit laws of entrance times to cylinder sets for Cantor minimal systems of zero entropy using their representation by means of ordered Bratteli diagrams. We study in detail substitution subshifts and we prove these limit laws are piecewise linear functions. The same kind of results is obtained for classical low complexity systems given by non stationary ordered Bratteli diagrams."}
{"category": "Math", "title": "Rejoinder: Microarrays, Empirical Bayes and the Two-Groups Model", "abstract": "Rejoinder to ``Microarrays, Empirical Bayes and the Two-Groups Model'' [arXiv:0808.0572]"}
{"category": "Math", "title": "A note on the Capelli identities for symmetric pairs of Hermitian type", "abstract": "We get several identities of differential operators in determinantal form. These identities are non-commutative versions of the formula of Cauchy-Binet or Laplace expansions of determinants, and if we take principal symbols, they are reduced to such classical formulas. These identities are naturally arising from the generators of the rings of invariant differential operators over symmetric spaces, and have strong resemblance to the classical Capelli identities. Thus we call those identities the Capelli identities for symmetric pairs."}
{"category": "Math", "title": "Proofs On Arnold Conjectures", "abstract": "In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture."}
{"category": "Math", "title": "The 2005 Neyman Lecture: Dynamic Indeterminism in Science", "abstract": "Jerzy Neyman's life history and some of his contributions to applied statistics are reviewed. In a 1960 article he wrote: ``Currently in the period of dynamic indeterminism in science, there is hardly a serious piece of research which, if treated realistically, does not involve operations on stochastic processes. The time has arrived for the theory of stochastic processes to become an item of usual equipment of every applied statistician.'' The emphasis in this article is on stochastic processes and on stochastic process data analysis. A number of data sets and corresponding substantive questions are addressed. The data sets concern sardine depletion, blowfly dynamics, weather modification, elk movement and seal journeying. Three of the examples are from Neyman's work and four from the author's joint work with collaborators."}
{"category": "Math", "title": "On the Shortest Identity in Finite Simple Groups of Lie Type", "abstract": "We prove that the length of the shortest identity in a finite simple group of Lie type of rank $r$ defined over $\\mathbb{F}_q$, is bounded (from above and below) by explicit polynomials in $q$ and $r$."}
{"category": "Math", "title": "The third homology of the special linear group of a field", "abstract": "We prove that for any infinite field homology stability for the third integral homology of the special linear groups $SL(n,F)$ begins at $n=3$. When $n=2$ the cokernel of the map from the third homology of $SL(2,F)$ to the third homology of $SL(3,F)$ is naturally isomorphic to the square of Milnor $K_3$. We discuss applications to the indecomposable $K_3$ of the field and to Milnor-Witt K-theory."}
{"category": "Math", "title": "Comment: The 2005 Neyman Lecture: Dynamic Indeterminism in Science", "abstract": "Comment on ``The 2005 Neyman Lecture: Dynamic Indeterminism in Science'' [arXiv:0808.0620]"}
{"category": "Math", "title": "Comment: The 2005 Neyman Lecture: Dynamic Indeterminism in Science", "abstract": "Comment on ``The 2005 Neyman Lecture: Dynamic Indeterminism in Science'' [arXiv:0808.0620]"}
{"category": "Math", "title": "Rejoinder: The 2005 Neyman Lecture: Dynamic Indeterminism in Science", "abstract": "Rejoinder to ``The 2005 Neyman Lecture: Dynamic Indeterminism in Science'' [arXiv:0808.0620]"}
{"category": "Math", "title": "Criteria equivalent to the Riemann Hypothesis", "abstract": "We give a brief overview of a few criteria equivalent to the Riemann Hypothesis. Next we concentrate on the Riesz and B{\\'a}ez-Duarte criteria. We proof that they are equivalent and we provide some computer data to support them. It is not compressed to six pages version of the talk delivered by M.W. during the XXVII Workshop on Geometrical Methods in Physics, 28 June -- 6 July, 2008, Bia{\\l}owie{\\.z}a, Poland."}
{"category": "Math", "title": "Molecules in Coorbit Spaces and Boundedness of Operators", "abstract": "We study the notion of molecules in coorbit spaces. The main result states that if an operator, originally defined on an appropriate space of test functions, maps atoms to molecules, then it can be extended to a bounded operator on coorbit spaces. For time-frequency molecules we recover some boundedness results on modulation spaces, for time-scale molecules we obtain the boundedness on homogenous Besov spaces."}
{"category": "Math", "title": "Verbal Autopsy Methods with Multiple Causes of Death", "abstract": "Verbal autopsy procedures are widely used for estimating cause-specific mortality in areas without medical death certification. Data on symptoms reported by caregivers along with the cause of death are collected from a medical facility, and the cause-of-death distribution is estimated in the population where only symptom data are available. Current approaches analyze only one cause at a time, involve assumptions judged difficult or impossible to satisfy, and require expensive, time-consuming, or unreliable physician reviews, expert algorithms, or parametric statistical models. By generalizing current approaches to analyze multiple causes, we show how most of the difficult assumptions underlying existing methods can be dropped. These generalizations also make physician review, expert algorithms and parametric statistical assumptions unnecessary. With theoretical results, and empirical analyses in data from China and Tanzania, we illustrate the accuracy of this approach. While no method of analyzing verbal autopsy data, including the more computationally intensive approach offered here, can give accurate estimates in all circumstances, the procedure offered is conceptually simpler, less expensive, more general, as or more replicable, and easier to use in practice than existing approaches. We also show how our focus on estimating aggregate proportions, which are the quantities of primary interest in verbal autopsy studies, may also greatly reduce the assumptions necessary for, and thus improve the performance of, many individual classifiers in this and other areas. As a companion to this paper, we also offer easy-to-use software that implements the methods discussed herein."}
{"category": "Math", "title": "On the formal grade of finitely generated modules over local rings", "abstract": "Let \\fa be an ideal of a local ring (R,\\fm) and M a finitely generated R-module. This paper concerns the notion \\fgrade(\\fa,M), the formal grade of M with respect to \\fa (i.e. the least integer i such that {\\vpl}_nH^i_{\\fm}(M/\\fa^n M)\\neq 0). We show that \\fgrade(\\fa,M)\\geq \\depth M-\\cd_{\\fa}(M), and as a result, we establish a new characterization of Cohen-Macaulay modules. As an application of this characterization, we show that if M is Cohen-Macaulay and L a pure submodule of M with the same support as M, then \\fgrade(\\fa,L)=\\fgrade(\\fa,M). Also, we give a generalization of the Hochster-Eagon result on Cohen-Macaulayness of invariant rings."}
{"category": "Math", "title": "High-Breakdown Robust Multivariate Methods", "abstract": "When applying a statistical method in practice it often occurs that some observations deviate from the usual assumptions. However, many classical methods are sensitive to outliers. The goal of robust statistics is to develop methods that are robust against the possibility that one or several unannounced outliers may occur anywhere in the data. These methods then allow to detect outlying observations by their residuals from a robust fit. We focus on high-breakdown methods, which can deal with a substantial fraction of outliers in the data. We give an overview of recent high-breakdown robust methods for multivariate settings such as covariance estimation, multiple and multivariate regression, discriminant analysis, principal components and multivariate calibration."}
{"category": "Math", "title": "Geometry of minimal energy Yang-Mills connections", "abstract": "We study the converse to the statement that instantons are minimizers of the Yang--Mills energy in four dimensions. We show that given an energy minimizing connection, A, the curvature of A takes values in a subbundle of the adjoint bundle which decomposes as a sum of instantons."}
{"category": "Math", "title": "Arithmetic differential operators on Z_p", "abstract": "We prove that a function f from Z_p to itself is analytic if and only if it can be represented as f(x)=F(x, dx, ..., d^r x) where dx=(x-x^p)/p is the Fermat quotient operator and F is a restricted power series with coefficients in Z_p."}
{"category": "Math", "title": "Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves", "abstract": "We give examples over arbitrary fields of rings of invariants that are not finitely generated. The group involved can be as small as three copies of the additive group, as in Mukai's examples over the complex numbers. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell-Weil rank. Our work suggests a generalization of the Morrison-Kawamata cone conjecture from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in dimension 2 in the case of minimal rational elliptic surfaces."}
{"category": "Math", "title": "Fourier transform and middle convolution for irregular D-modules", "abstract": "S.Block and H.Esnault constructed the local Fourier transform for D-modules. We present a different approach to the local Fourier transform, which makes its properties almost tautological. We apply the local Fourier transform to compute the local version of Katz's middle convolution."}
{"category": "Math", "title": "Support union recovery in high-dimensional multivariate regression", "abstract": "In multivariate regression, a $K$-dimensional response vector is regressed upon a common set of $p$ covariates, with a matrix $B^*\\in\\mathbb{R}^{p\\times K}$ of regression coefficients. We study the behavior of the multivariate group Lasso, in which block regularization based on the $\\ell_1/\\ell_2$ norm is used for support union recovery, or recovery of the set of $s$ rows for which $B^*$ is nonzero. Under high-dimensional scaling, we show that the multivariate group Lasso exhibits a threshold for the recovery of the exact row pattern with high probability over the random design and noise that is specified by the sample complexity parameter $\\theta(n,p,s):=n/[2\\psi(B^*)\\log(p-s)]$. Here $n$ is the sample size, and $\\psi(B^*)$ is a sparsity-overlap function measuring a combination of the sparsities and overlaps of the $K$-regression coefficient vectors that constitute the model. We prove that the multivariate group Lasso succeeds for problem sequences $(n,p,s)$ such that $\\theta(n,p,s)$ exceeds a critical level $\\theta_u$, and fails for sequences such that $\\theta(n,p,s)$ lies below a critical level $\\theta_{\\ell}$. For the special case of the standard Gaussian ensemble, we show that $\\theta_{\\ell}=\\theta_u$ so that the characterization is sharp. The sparsity-overlap function $\\psi(B^*)$ reveals that, if the design is uncorrelated on the active rows, $\\ell_1/\\ell_2$ regularization for multivariate regression never harms performance relative to an ordinary Lasso approach and can yield substantial improvements in sample complexity (up to a factor of $K$) when the coefficient vectors are suitably orthogonal. For more general designs, it is possible for the ordinary Lasso to outperform the multivariate group Lasso. We complement our analysis with simulations that demonstrate the sharpness of our theoretical results, even for relatively small problems."}
{"category": "Math", "title": "New estimates for the length of the Erdos-Herzog-Piranian lemniscate", "abstract": "Let p(z) be a monic polynomial of degree n. Consider the lemniscate L={z:|p(z)|=1}. It has been conjectured that L has the largest length when p(z)=z^n-1. We show that the length of L attains a local maximum at this polynomial and prove the asymptotically sharp bound |L|<2n+o(n)."}
{"category": "Math", "title": "Global geometry under isotropic Brownian flows", "abstract": "We consider global geometric properties of a codimension one manifold embedded in Euclidean space, as it evolves under an isotropic and volume preserving Brownian flow of diffeomorphisms. In particular, we obtain expressions describing the expected rate of growth of the Lipschitz-Killing curvatures, or intrinsic volumes, of the manifold under the flow. These results shed new light on some of the intriguing growth properties of flows from a global perspective, rather than the local perspective, on which there is a much larger literature."}
{"category": "Math", "title": "On the ring structure of spark characters", "abstract": "We give a new description of the ring structure on the differential characters of a smooth manifold via the smooth hyperspark complex. We show the explicit product formula, and as an application, calculate the product for differential characters of the unit circle. Applying the presentation of spark classes by smooth hypersparks, we give an explicit construction of the isomorphism between groups of spark classes and the $(p,p)$ part of smooth Deligne cohomology groups associated to a smooth manifold. We then give a new direct proof that this is an isomorphism of ring structures."}
{"category": "Math", "title": "Boolean formulae, hypergraphs and combinatorial topology", "abstract": "With a view toward studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial complex, called the theta complex, associated to any hypergraph, which is the Alexander dual of the more well-known independence complex. In particular, the set of satisfiable formulae in k-conjunctive normal form with less than or equal to n variables has the homotopy type of Theta(Cube(n,n-k)), where Cube(n,n-k) is a hypergraph associated to the (n-k)-skeleton of an n-cube. We make partial progress in calculating the homotopy type of theta for these cubical hypergraphs, and we also give calculations and examples for other hypergraphs as well. Indeed studying the theta complex of hypergraphs is an interesting problem in its own right."}
{"category": "Math", "title": "Logarithm laws and shrinking target properties", "abstract": "We survey some of the recent developments in the study of logarithm laws and shrinking target properties for various families of dynamical systems. We discuss connections to geometry, diophantine approximation, and probability theory."}
{"category": "Math", "title": "Rigid irregular connections on P^1", "abstract": "N.Katz's middle convolution algorithm provides a description of rigid connections on the projective line with regular singularities. We extend the algorithm by adding the Fourier transform to it. The extended algorithm provides a description of rigid connections with arbitrary singularities."}
{"category": "Math", "title": "Positive Dehn Twist Expression for a $\\mathbb{Z}_3$ action on $\\Sigma_g$", "abstract": "A positive Dehn twist product for a $\\mathbb{Z}_3$ action with $g+2$ fixed points on the 2-dimensional closed, compact, oriented surface $\\Sigma_g$ is presented. The homeomorphism invariants of the resulting symplectic 4-manifolds are computed."}
{"category": "Math", "title": "On the Convergence of Optimal Measures", "abstract": "Using recent results of Berman and Boucksom we show that for a non-pluripolar compact set K in C^d and an admissible weight function w=e^{-\\phi} any sequence of so-called optimal measures converges weak-* to the equilibrium measure \\mu_{K,\\phi} of (weighted) Pluripotential Theory for K,\\phi."}
{"category": "Math", "title": "Skein theory for the D_{2n} planar algebras", "abstract": "We give a combinatorial description of the ``$D_{2n}$ planar algebra,'' by generators and relations. We explain how the generator interacts with the Temperley-Lieb braiding. This shows the previously known braiding on the even part extends to a `braiding up to sign' on the entire planar algebra. We give a direct proof that our relations are consistent (using this `braiding up to sign'), give a complete description of the associated tensor category and principal graph, and show that the planar algebra is positive definite. These facts allow us to identify our combinatorial construction with the standard invariant of the subfactor $D_{2n}$."}
{"category": "Math", "title": "Elementary Techniques for Erdos-Ko-Rado-like Theorems", "abstract": "The well-known Erdos-Ko-Rado Theorem states that if F is a family of k-element subsets of {1,2,...,n} (n>2k-1) such that every pair of elements in F has a nonempty intersection, then |F| is at most $\\binom{n-1}{k-1}$. The theorem also provides necessary and sufficient conditions for attaining the maximum. We present elementary methods for deriving generalizations of the Erdos-Ko-Rado Theorem on several classes of combinatorial objects. We also extend our results to systems under Hamming intersection."}
{"category": "Math", "title": "A Conversation with Peter Huber", "abstract": "Peter J. Huber was born on March 25, 1934, in Wohlen, a small town in the Swiss countryside. He obtained a diploma in mathematics in 1958 and a Ph.D. in mathematics in 1961, both from ETH Zurich. His thesis was in pure mathematics, but he then decided to go into statistics. He spent 1961--1963 as a postdoc at the statistics department in Berkeley where he wrote his first and most famous paper on robust statistics, ``Robust Estimation of a Location Parameter.'' After a position as a visiting professor at Cornell University, he became a full professor at ETH Zurich. He worked at ETH until 1978, interspersed by visiting positions at Cornell, Yale, Princeton and Harvard. After leaving ETH, he held professor positions at Harvard University 1978--1988, at MIT 1988--1992, and finally at the University of Bayreuth from 1992 until his retirement in 1999. He now lives in Klosters, a village in the Grisons in the Swiss Alps. Peter Huber has published four books and over 70 papers on statistics and data analysis. In addition, he has written more than a dozen papers and two books on Babylonian mathematics, astronomy and history. In 1972, he delivered the Wald lectures. He is a fellow of the IMS, of the American Association for the Advancement of Science, and of the American Academy of Arts and Sciences. In 1988 he received a Humboldt Award and in 1994 an honorary doctorate from the University of Neuch\\^{a}tel. In addition to his fundamental results in robust statistics, Peter Huber made important contributions to computational statistics, strategies in data analysis, and applications of statistics in fields such as crystallography, EEGs, and human growth curves."}
{"category": "Math", "title": "LLE with low-dimensional neighborhood representation", "abstract": "The local linear embedding algorithm (LLE) is a non-linear dimension-reducing technique, widely used due to its computational simplicity and intuitive approach. LLE first linearly reconstructs each input point from its nearest neighbors and then preserves these neighborhood relations in the low-dimensional embedding. We show that the reconstruction weights computed by LLE capture the high-dimensional structure of the neighborhoods, and not the low-dimensional manifold structure. Consequently, the weight vectors are highly sensitive to noise. Moreover, this causes LLE to converge to a linear projection of the input, as opposed to its non-linear embedding goal. To overcome both of these problems, we propose to compute the weight vectors using a low-dimensional neighborhood representation. We prove theoretically that this straightforward and computationally simple modification of LLE reduces LLE's sensitivity to noise. This modification also removes the need for regularization when the number of neighbors is larger than the dimension of the input. We present numerical examples demonstrating both the perturbation and linear projection problems, and the improved outputs using the low-dimensional neighborhood representation."}
{"category": "Math", "title": "The Early Statistical Years: 1947--1967 A Conversation with Howard Raiffa", "abstract": "Howard Raiffa earned his bachelor's degree in mathematics, his master's degree in statistics and his Ph.D. in mathematics at the University of Michigan. Since 1957, Raiffa has been a member of the faculty at Harvard University, where he is now the Frank P. Ramsey Chair in Managerial Economics (Emeritus) in the Graduate School of Business Administration and the Kennedy School of Government. A pioneer in the creation of the field known as decision analysis, his research interests span statistical decision theory, game theory, behavioral decision theory, risk analysis and negotiation analysis. Raiffa has supervised more than 90 doctoral dissertations and written 11 books. His new book is Negotiation Analysis: The Science and Art of Collaborative Decision Making. Another book, Smart Choices, co-authored with his former doctoral students John Hammond and Ralph Keeney, was the CPR (formerly known as the Center for Public Resources) Institute for Dispute Resolution Book of the Year in 1998. Raiffa helped to create the International Institute for Applied Systems Analysis and he later became its first Director, serving in that capacity from 1972 to 1975. His many honors and awards include the Distinguished Contribution Award from the Society of Risk Analysis; the Frank P. Ramsey Medal for outstanding contributions to the field of decision analysis from the Operations Research Society of America; and the Melamed Prize from the University of Chicago Business School for The Art and Science of Negotiation. He earned a Gold Medal from the International Association for Conflict Management and a Lifetime Achievement Award from the CPR Institute for Dispute Resolution. He holds honorary doctor's degrees from Carnegie Mellon University, the University of Michigan, Northwestern University, Ben Gurion University of the Negev and Harvard University. The latter was awarded in 2002."}
{"category": "Math", "title": "Growth rate and extinction rate of a reaction diffusion equation with a singular nonlinearity", "abstract": "We prove the growth rate of global solutions of the equation $u_t=\\Delta u-u^{-\\nu}$ in $\\R^n\\times (0,\\infty)$, $u(x,0)=u_0>0$ in $\\R^n$, where $\\nu>0$ is a constant. More precisely for any $0<u_0\\in C(\\R^n)$ satisfying $A_1(1+|x|^2)^{\\alpha_1}\\le u_0\\le A_2(1+|x|^2)^{\\alpha_2}$ in $\\R^n$ for some constants $1/(1+\\nu)\\le\\alpha_1<1$, $\\alpha_2\\ge\\alpha_1$ and $A_2\\ge A_1= (2\\alpha_1(1-\\3)(n+2\\alpha_1-2))^{-1/(1+\\nu)}$ where $0<\\3<1$ is a constant, the global solution $u$ exists and satisfies $A_1(1+|x|^2+b_1t)^{\\alpha_1}\\le u(x,t)\\le A_2(1+|x|^2+b_2t)^{\\alpha_2}$ in $\\R^n\\times (0,\\infty)$ where $b_1=2(n+2\\alpha_1-2)\\3$ and $b_2=2n$ if $0<\\alpha_2\\le 1$ and $b_2=2(n+2\\alpha_2-2)$ if $\\alpha_2>1$. We also find various conditions on the initial value for the solution to extinct in a finite time and obtain the corresponding decay rate of the solution near the extinction time."}
{"category": "Math", "title": "Chevalley Supergroups", "abstract": "In the framework of algebraic supergeometry, we give a construction of the scheme-theoretic supergeometric analogue of Chevalley groups, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. In particular both basic (even exceptional) and strange cases are covered. This provides a unified approach to most of the algebraic supergroups considered so far in literature, and an effective method to construct new ones. As an intermediate step, we prove an existence theorem for Chevalley bases of simple classical Lie superalgebras and a PBW-like theorem for their associated Kostant superalgebras."}
{"category": "Math", "title": "Relative Yamabe invariant and c-concordant metrics", "abstract": "We show a surgery formula for the relative Yamabe invariant and give applications to the study of concordance classes of metrics."}
{"category": "Math", "title": "On positivity in algebras of tempered generalized functions", "abstract": "An explicit counterexample shows that contrary to the situation in the special Colombeau algebra, positivity and invertibility cannot be characterized pointwise in algebras of tempered generalized functions. Further a point value characterization of the latter is refined."}
{"category": "Math", "title": "Knotted holomorphic discs in C^2", "abstract": "We construct knotted proper holomorphic embeddings of the unit disc in C^2."}
{"category": "Math", "title": "Backward stochastic variational inequalities with locally bounded generators", "abstract": "The paper deals with the existence and uniqueness of the solution of the backward stochastic variational inequality: \\begin{equation} \\left\\{\\begin{array} {l}-dY_{t}+\\partial \\varphi(Y_{t})dt \\ni F(t,Y_{t},Z_{t})dt-Z_{t}dB_{t},\\;0\\leq t<T \\\\ Y_{T}=\\eta, \\end{array} \\right.\\end{equation} where $F$ satisfies a local boundedness condition."}
{"category": "Math", "title": "Stochastic approach for a multivalued Dirichlet-Neumann problem", "abstract": "We prove the existence and uniqueness of a viscosity solution of the parabolic variational inequality with a nonlinear multivalued Neumann-Dirichlet boundary condition:% {equation*} \\{{array}{r} \\dfrac{\\partial u(t,x)}{\\partial t}-\\mathcal{L}_{t}u(t,x) {+}{% \\partial \\phi}\\big(u(t,x)\\big)\\ni f\\big(t,x,u(t,x),(\\nabla u\\sigma)(t,x)\\big), t>0, x\\in \\mathcal{D},\\medskip \\multicolumn{1}{l}{\\dfrac{\\partial u(t,x)}{\\partial n}+{\\partial \\psi}\\big(% u(t,x)\\big)\\ni g\\big(t,x,u(t,x)\\big), t>0, x\\in Bd(\\mathcal{D}%),\\multicolumn{1}{l}{u(0,x)=h(x), x\\in \\bar{\\mathcal{D}},}% {array}%. {equation*}% where $\\partial \\phi $ and $\\partial \\psi $ are subdifferentials operators and $\\mathcal{L}_{t}$ is a second differential operator. The result is obtained by a Feynman-Ka\\c{c} representation formula starting from the backward stochastic variational inequality:% {equation*} \\{{array}{l} dY_{t}{+}F(t,Y_{t},Z_{t}) dt{+}G(t,Y_{t}) dA_{t}\\in \\partial \\phi (Y_{t}) dt{+}\\partial \\psi (Y_{t}) dA_{t}{+}Z_{t}dW_{t}, 0\\leq t\\leq T,\\medskip \\ Y_{T}=\\xi .% {array}%. {equation*}"}
{"category": "Math", "title": "Iterated Riesz Commutators: A Simple Proof of Boundedness", "abstract": "We give a simple proof of L^p boundedness of iterated commutators of Riesz transforms and a product BMO function. We use a representation of the Riesz transforms by means of simple dyadic operators - dyadic shifts - which in turn reduces the estimate quickly to paraproduct estimates."}
{"category": "Math", "title": "Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for infinite matrices", "abstract": "In this paper we extend work of Kawamura, see kawamura, for Cuntz-Krieger algebras O_A for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of O_A. We use these representations to describe the Perron-Frobenius operator, associated to an nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples."}
{"category": "Math", "title": "A Novel Proof of the Heine-Borel Theorem", "abstract": "Every beginning real analysis student learns the classic Heine-Borel theorem, that the interval [0,1] is compact. In this article, we present a proof of this result that doesn't involve the standard techniques such as constructing a sequence and appealing to the completeness of the reals. We put a metric on the space of infinite binary sequences and prove that compactness of this space follows from a simple combinatorial lemma. The Heine-Borel theorem is an immediate corollary."}
{"category": "Math", "title": "Proof of the Caratheodory Conjecture", "abstract": "A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in ${\\mathbb E}^3$ must be greater than one. In this paper we prove this for $C^{3+\\alpha}$-smooth surfaces. The Conjecture is first reformulated in terms of complex points on a Lagrangian surface in $TS^2$, viewed as the space of oriented geodesics in ${\\mathbb E}^3$. Here complex and Lagrangian refer to the canonical neutral Kaehler structure on $TS^2$. We then prove that the existence of a closed convex surface with only one umbilic point implies the existence of a totally real Lagrangian hemisphere in $TS^2$, to which it is not possible to attach the edge of a holomorphic disc. The main step in the proof is to establish the existence of a holomorphic disc with edge contained on any given totally real Lagrangian hemisphere. To construct the holomorphic disc we utilize mean curvature flow with respect to the neutral metric. Long-time existence of this flow is proven by a priori estimates and we show that the flowing disc is asymptotically holomorphic. Existence of a holomorphic disc is then deduced from Schauder estimates."}
{"category": "Math", "title": "Divergences Test Statistics for Discretely Observed Diffusion Processes", "abstract": "In this paper we propose the use of $\\phi$-divergences as test statistics to verify simple hypotheses about a one-dimensional parametric diffusion process $\\de X_t = b(X_t, \\theta)\\de t + \\sigma(X_t, \\theta)\\de W_t$, from discrete observations $\\{X_{t_i}, i=0, ..., n\\}$ with $t_i = i\\Delta_n$, $i=0, 1, >..., n$, under the asymptotic scheme $\\Delta_n\\to0$, $n\\Delta_n\\to\\infty$ and $n\\Delta_n^2\\to 0$. The class of $\\phi$-divergences is wide and includes several special members like Kullback-Leibler, R\\'enyi, power and $\\alpha$-divergences. We derive the asymptotic distribution of the test statistics based on $\\phi$-divergences. The limiting law takes different forms depending on the regularity of $\\phi$. These convergence differ from the classical results for independent and identically distributed random variables. Numerical analysis is used to show the small sample properties of the test statistics in terms of estimated level and power of the test."}
{"category": "Math", "title": "Reduction of Almost Poisson brackets and Hamiltonization of the Chaplygin Sphere", "abstract": "We construct different almost Poisson brackets for nonholonomic systems than those existing in the literature and study their reduction. Such brackets are built by considering non-canonical two-forms on the cotangent bundle of configuration space and then carrying out a projection onto the constraint space that encodes the Lagrange-D'Alembert principle. We justify the need for this type of brackets by working out the reduction of the celebrated Chaplygin sphere rolling problem. Our construction provides a geometric explanation of the Hamiltonization of the problem given by A. V. Borisov and I. S. Mamaev."}
{"category": "Math", "title": "Orthogonal bundles, theta characteristics and the symplectic strange duality", "abstract": "A basis for the space of generalized theta functions of level one for the spin groups, parameterized by the theta characteristics (the even theta characteristcs for the odd spin groups) on a curve, is shown to be projectively flat over the moduli space of curves (for Hitchin's connection). The symplectic strange duality conjecture, conjectured by Beauville is shown to hold for all curves of genus at least two, by using Abe's proof of the conjecture for generic curves, and the above monodromy result."}
{"category": "Math", "title": "Constant-length substitutions and countable scrambled sets", "abstract": "In this paper we provide examples of topological dynamical systems having either finite or countable scrambled sets. In particular we study conditions for the existence of Li-Yorke, asymptotic and distal pairs in constant--length substitution dynamical systems. Starting from a circle rotation we also construct a dynamical system having Li--Yorke pairs, none of which is recurrent."}
{"category": "Math", "title": "Corrigendum and addendum to: Linearly recurrent subshifts have a finite number of non-periodic factors", "abstract": "We prove that a subshift $(X,T)$ is linearly recurrent if and only if it is a primitive and proper $S$-adic subshift. This corrects Proposition 6 in F. Durand ({\\it Ergod. Th. & Dynam. Sys. {\\bf 20}} (2000), 1061--1078)."}
{"category": "Math", "title": "Some metrics on Teichm\\\"uller spaces of surfaces of infinite type", "abstract": "Unlike the case of surfaces of topologically finite type, there are several different Teichm\\\"uller spaces that are associated to a surface of topological infinite type. These Teichm\\\"uller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, given the choice of a point of view (hyperbolic or conformal), on the choice of a distance function on Teichm\\\"uller space. Examples of distance functions that appear naturally in the hyperbolic setting are the length spectrum distance and the bi-Lipschitz distance, and there are other useful distance functions. The Teichm\\\"uller spaces also depend on the choice of a basepoint. The aim of this paper is to present some examples, results and questions on the Teichm\\\"uller theory of surfaces of infinite topological type that do not appear in the setting the Teichm\\\"uller theory of surfaces of finite type. In particular, we point out relations and differences between the various Teichm\\\"uller spaces associated to a given surface of topological infinite type."}
{"category": "Math", "title": "Robust 4-manifolds and robust embeddings", "abstract": "A link in the 3-sphere is homotopically trivial, according to Milnor, if its components bound disjoint maps of disks in the 4-ball. This paper concerns the question of what spaces give rise to the same class of homotopically trivial links when used in place of disks in an analogous definition. We show that there are 4-manifolds for which this property depends on their embedding in the 4-ball. This work is motivated by the A-B slice problem, a reformulation of the 4-dimensional topological surgery conjecture. As a corollary this provides a new, secondary, obstruction in the A-B slice problem for a certain class of decompositions of D^4."}
{"category": "Math", "title": "Microsupport of tempered solutions of D-Modules associated to smooth morphisms", "abstract": "Let $f:X\\to Y$ be a smooth morphism of complex analytic manifolds and let $F$ be an $\\mathbb{R}$-constructible complex on $Y$. Let $\\cal{M}$ be a coherent $\\shd_X$-module. We prove that the microsupport of the solution complex of $\\shm$ in the tempered holomorphic functions $t \\shh \\text{om} (f^{-1} F, \\sho_X)$, is contained in the 1-characteristic variety of $\\cal{M}$ associated to $f$, and that the microsupport of the solution complex in the tempered microfunctions $t\\mu hom(f^{-1}F, \\sho_X)$ is contained in the 1-microcharacteristic variety of the microlocalized of $\\shm$ along $T^*Y\\times_Y X$. This applies in particular to the complex of solutions of $\\shm$ in the sheaf of distributions holomorphic in the fibers of an arbitrary smooth morphism."}
{"category": "Math", "title": "Laplacian spectrum for the nilpotent Kac-Moody Lie algebras", "abstract": "We prove that the maximal nilpotent subalgebra of a Kac-Moody Lie algebra has an (essentially unique) Euclidean metric with respect to which the Laplace operator in the chain complex is scalar on each component of a given degree. Moreover, both the Lie algebra structure and the metric are uniquely determined by this property."}
{"category": "Math", "title": "On some interpolation theorems for the multipliers of the Cauchy- Stiltjes type integrals", "abstract": "In this paper we prove some interpolation theorems for the multipliers of the Cauchy- Stiltjes type integrals"}
{"category": "Math", "title": "Langlands Functoriality Conjecture", "abstract": "Functoriality conjecture is one of the central subjects of the present-day mathematics. Functoriality is the profound problem formulated by Robert Langlands in the late 1960s in order to establish nonabelian class field theory. In this expository paper, we describe the Langlands-Shahidi method, the local and global Langlands conjectures and the converse theorems which are powerful tools for the establishment of functoriality of some important cases, and survey the interesting results related to functoriality conjecture."}
{"category": "Math", "title": "Proof of Han's Hook Expansion Conjecture", "abstract": "We prove a conjecture by Guo-Niu Han which interpolates between two known hook expansion formulas."}
{"category": "Math", "title": "Triangles of Baumslag-Solitar Groups", "abstract": "Our main result is that many triangles of Baumslag-Solitar groups collapse to finite groups, generalizing a famous example of Hirsch. A triangle of Baumslag-Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of class<4. We obtain a lot of information about finite quotients, even when we cannot determine developability."}
{"category": "Math", "title": "Corrigendum to \"Knot Floer homology detects fibred knots\"", "abstract": "We correct a mistake on the citation of JSJ theory in \\cite{Ni}. Some arguments in \\cite{Ni} are also slightly modified accordingly."}
{"category": "Math", "title": "Contributions to Seymour's Second Neighborhood Conjecture", "abstract": "Let D be a simple digraph without loops or digons. For any v in V(D) let N_1(v) be the set of all nodes at out-distance 1 from v and let N_2(v) be the set of all nodes at out-distance 2. We provide sufficient conditions under which there must exist some v in V(D) such that |N_1(v)| is less than or equal to |N_2(v)|, as well as examine properties of a minimal graph which does not have such a node. We show that if one such graph exists, then there exist infinitely many strongly-connected graphs having no such vertex."}
{"category": "Math", "title": "Unitary orbits in a full matrix algebra", "abstract": "The Hilbert manifold $\\Sigma$ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of unitary orbits $\\Omega\\subset \\Sigma$ is studied from the topological and metric viewpoints: we seek for conditions that ensure the existence of a smooth local structure for the set $\\Omega$, and we study the convexity of this set for the geodesic structures that arise when we give $\\Sigma$ two Riemannian metrics."}
{"category": "Math", "title": "On the Euler Numbers and its Applications", "abstract": "Recently the new q-Euler numbers are defined. In this paper we derive the the Kummer type congruence related to q-Euler numbers and we introduce some interesting formulae related to these q-Euler numbers."}
{"category": "Math", "title": "The sparsity and bias of the Lasso selection in high-dimensional linear regression", "abstract": "Meinshausen and Buhlmann [Ann. Statist. 34 (2006) 1436--1462] showed that, for neighborhood selection in Gaussian graphical models, under a neighborhood stability condition, the LASSO is consistent, even when the number of variables is of greater order than the sample size. Zhao and Yu [(2006) J. Machine Learning Research 7 2541--2567] formalized the neighborhood stability condition in the context of linear regression as a strong irrepresentable condition. That paper showed that under this condition, the LASSO selects exactly the set of nonzero regression coefficients, provided that these coefficients are bounded away from zero at a certain rate. In this paper, the regression coefficients outside an ideal model are assumed to be small, but not necessarily zero. Under a sparse Riesz condition on the correlation of design variables, we prove that the LASSO selects a model of the correct order of dimensionality, controls the bias of the selected model at a level determined by the contributions of small regression coefficients and threshold bias, and selects all coefficients of greater order than the bias of the selected model. Moreover, as a consequence of this rate consistency of the LASSO in model selection, it is proved that the sum of error squares for the mean response and the $\\ell_{\\alpha}$-loss for the regression coefficients converge at the best possible rates under the given conditions. An interesting aspect of our results is that the logarithm of the number of variables can be of the same order as the sample size for certain random dependent designs."}
{"category": "Math", "title": "Statistics of extremes by oracle estimation", "abstract": "We use the fitted Pareto law to construct an accompanying approximation of the excess distribution function. A selection rule of the location of the excess distribution function is proposed based on a stagewise lack-of-fit testing procedure. Our main result is an oracle type inequality for the Kullback--Leibler loss."}
{"category": "Math", "title": "Dimension reduction based on constrained canonical correlation and variable filtering", "abstract": "The ``curse of dimensionality'' has remained a challenge for high-dimensional data analysis in statistics. The sliced inverse regression (SIR) and canonical correlation (CANCOR) methods aim to reduce the dimensionality of data by replacing the explanatory variables with a small number of composite directions without losing much information. However, the estimated composite directions generally involve all of the variables, making their interpretation difficult. To simplify the direction estimates, Ni, Cook and Tsai [Biometrika 92 (2005) 242--247] proposed the shrinkage sliced inverse regression (SSIR) based on SIR. In this paper, we propose the constrained canonical correlation ($C^3$) method based on CANCOR, followed by a simple variable filtering method. As a result, each composite direction consists of a subset of the variables for interpretability as well as predictive power. The proposed method aims to identify simple structures without sacrificing the desirable properties of the unconstrained CANCOR estimates. The simulation studies demonstrate the performance advantage of the proposed $C^3$ method over the SSIR method. We also use the proposed method in two examples for illustration."}
{"category": "Math", "title": "Fence methods for mixed model selection", "abstract": "Many model search strategies involve trading off model fit with model complexity in a penalized goodness of fit measure. Asymptotic properties for these types of procedures in settings like linear regression and ARMA time series have been studied, but these do not naturally extend to nonstandard situations such as mixed effects models, where simple definition of the sample size is not meaningful. This paper introduces a new class of strategies, known as fence methods, for mixed model selection, which includes linear and generalized linear mixed models. The idea involves a procedure to isolate a subgroup of what are known as correct models (of which the optimal model is a member). This is accomplished by constructing a statistical fence, or barrier, to carefully eliminate incorrect models. Once the fence is constructed, the optimal model is selected from among those within the fence according to a criterion which can be made flexible. In addition, we propose two variations of the fence. The first is a stepwise procedure to handle situations of many predictors; the second is an adaptive approach for choosing a tuning constant. We give sufficient conditions for consistency of fence and its variations, a desirable property for a good model selection procedure. The methods are illustrated through simulation studies and real data analysis."}
{"category": "Math", "title": "Derived Kodaira Spencer map, Cosection lemma, and semiregularity", "abstract": "The cosection lemma proved by J. Li and Y.H. Kiem said the intrinsic normal cone lies inside the kernel of any cosection of the obstruction sheaf when the moduli has a perfect obstruction theory. With a definition of higher tangent vectors of a scheme at a point, and a construction of the derived Kodaira Spencer map by K. Behrend and B. Fantechi, we prove a derived version of cosection lemma without perfect obstruction theory condition. As an application we give a short proof of the Kodaira's Principle \\textit{ambient cohomology annihilates obstruction} (semiregularity), assuming the existence of locall universal family."}
{"category": "Math", "title": "Semiparametric detection of significant activation for brain fMRI", "abstract": "Functional magnetic resonance imaging (fMRI) aims to locate activated regions in human brains when specific tasks are performed. The conventional tool for analyzing fMRI data applies some variant of the linear model, which is restrictive in modeling assumptions. To yield more accurate prediction of the time-course behavior of neuronal responses, the semiparametric inference for the underlying hemodynamic response function is developed to identify significantly activated voxels. Under mild regularity conditions, we demonstrate that a class of the proposed semiparametric test statistics, based on the local linear estimation technique, follow $\\chi^2$ distributions under null hypotheses for a number of useful hypotheses. Furthermore, the asymptotic power functions of the constructed tests are derived under the fixed and contiguous alternatives. Simulation evaluations and real fMRI data application suggest that the semiparametric inference procedure provides more efficient detection of activated brain areas than the popular imaging analysis tools AFNI and FSL."}
{"category": "Math", "title": "Normal Forms for Semilinear Quantum Harmonic Oscillators", "abstract": "We consider the semilinear harmonic oscillator $$i\\psi_t=(-\\Delta +\\va{x}^{2} +M)\\psi +\\partial_2 g(\\psi,\\bar \\psi), \\quad x\\in \\R^d, t\\in \\R$$ where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple (in particular bounded) dynamics when $d\\geq 2$. As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions."}
{"category": "Math", "title": "On finite groups acting on acyclic low-dimensional manifolds", "abstract": "We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogue remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups A_5 and A_6, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds."}
{"category": "Math", "title": "Happy places or happy people? A multi-level modelling approach to the analysis of happiness and well-being", "abstract": "This paper aims to enhance our understanding of substantive questions regarding self-reported happiness and well-being through the specification and use of multi-level models. To date, there have been numerous quantitative research studies of the happiness of individuals, based on single-level regression models, where typically a happiness index is related to a set of explanatory variables. There are also several single-level studies comparing aggregate happiness levels between countries. Nevertheless, there have been very few studies that attempt to simultaneously take into account variations in happiness and well-being at several different levels, such as individual, household, and area. Here, multilevel models are used with data from the British Household Panel Survey to assess the nature and extent of variations in happiness and well-being to determine the relative importance of the area (district, region), household and individual characteristics on these outcomes. Moreover, having taken into account the characteristics at these different levels in the multilevel models, the paper shows how it is possible to identify any areas that are associated with especially positive or negative feelings of happiness and well-being."}
{"category": "Math", "title": "Confidence bands in nonparametric time series regression", "abstract": "We consider nonparametric estimation of mean regression and conditional variance (or volatility) functions in nonlinear stochastic regression models. Simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct. The imposed dependence structure allows applications in many linear and nonlinear auto-regressive processes. The results are applied to the S&P 500 Index data."}
{"category": "Math", "title": "One-step sparse estimates in nonconcave penalized likelihood models", "abstract": "Fan and Li propose a family of variable selection methods via penalized likelihood using concave penalty functions. The nonconcave penalized likelihood estimators enjoy the oracle properties, but maximizing the penalized likelihood function is computationally challenging, because the objective function is nondifferentiable and nonconcave. In this article, we propose a new unified algorithm based on the local linear approximation (LLA) for maximizing the penalized likelihood for a broad class of concave penalty functions. Convergence and other theoretical properties of the LLA algorithm are established. A distinguished feature of the LLA algorithm is that at each LLA step, the LLA estimator can naturally adopt a sparse representation. Thus, we suggest using the one-step LLA estimator from the LLA algorithm as the final estimates. Statistically, we show that if the regularization parameter is appropriately chosen, the one-step LLA estimates enjoy the oracle properties with good initial estimators. Computationally, the one-step LLA estimation methods dramatically reduce the computational cost in maximizing the nonconcave penalized likelihood. We conduct some Monte Carlo simulation to assess the finite sample performance of the one-step sparse estimation methods. The results are very encouraging."}
{"category": "Math", "title": "Discussion: One-step sparse estimates in nonconcave penalized likelihood models", "abstract": "Discussion of ``One-step sparse estimates in nonconcave penalized likelihood models'' [arXiv:0808.1012]"}
{"category": "Math", "title": "Discussion: One-step sparse estimates in nonconcave penalized likelihood models: Who cares if it is a white cat or a black Cat?", "abstract": "Discussion of ``One-step sparse estimates in nonconcave penalized likelihood models'' [arXiv:0808.1012]"}
{"category": "Math", "title": "Discussion: One-step sparse estimates in nonconcave penalized likelihood models", "abstract": "Discussion of ``One-step sparse estimates in nonconcave penalized likelihood models'' [arXiv:0808.1012]"}
{"category": "Math", "title": "Rejoinder: One-step sparse estimates in nonconcave penalized likelihood models", "abstract": "We would like to take this opportunity to thank the discussants for their thoughtful comments and encouragements on our work [arXiv:0808.1012]. The discussants raised a number of issues from theoretical as well as computational perspectives. Our rejoinder will try to provide some insights into these issues and address specific questions asked by the discussants."}
{"category": "Math", "title": "A Banach space determined by the Weil height", "abstract": "The absolute logarithmic Weil height is well defined on the group of units of the algebraic closure of the rational numbers, modulo roots of unity, and induces a metric topology on this group. We show that the completion of this metric space is a Banach space over the field of real numbers. We further show that this Banach space is isometrically isomorphic to a co-dimension one subspace of L1 of a certain totally disconnected, locally compact space, equipped with a certain measure satisfying an invariance property with respect to the absolute Galois group."}
{"category": "Math", "title": "About the geometry of almost para-quaternionic manifolds", "abstract": "We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the canonical almost complex structure of the twistor space and to the integrability of the canonical almost para-complex structure of the reflector space of (M,P). We show that (M, P) has plenty of locally defined, compatible, complex and para-complex structures, provided that P is para-quaternionic."}
{"category": "Math", "title": "The Laurent norm", "abstract": "We generalize a semi-norm for the Alexander polynomial of a connected, compact, oriented 3-manifold on its first cohomology group to a semi-norm for an arbitrary Laurent polynomial f on the dual vector space to the space of exponents of f. We determine a decomposition formula for this Laurent norm; an expression for the Laurent norm for f in terms of the Laurent norms for each of the irreducible factors of f. For an n-variable polynomial f, we introduce a space of m \\leq n essential variables which determine the reduced Laurent norm unit ball; a convex polyhedron of the same dimension m as the Newton polyhedron of f. In the space spanned by the essential variables, the Laurent semi-norm for polynomials with at least two terms is shown to be a norm."}
{"category": "Math", "title": "Arithmetic of K3 surfaces", "abstract": "We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry."}
{"category": "Math", "title": "Infinite log-concavity: developments and conjectures", "abstract": "Given a sequence (a_k) = a_0, a_1, a_2,... of real numbers, define a new sequence L(a_k) = (b_k) where b_k = a_k^2 - a_{k-1} a_{k+1}. So (a_k) is log-concave if and only if (b_k) is a nonnegative sequence. Call (a_k) \"infinitely log-concave\" if L^i(a_k) is nonnegative for all i >= 1. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the nth row for all n <= 1450. We also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of infinite log-concavity. We investigate related questions about the columns of Pascal's triangle, q-analogues, symmetric functions, real-rooted polynomials, and Toeplitz matrices. In addition, we offer several conjectures."}
{"category": "Math", "title": "Alexander and Thurston norms of graph links", "abstract": "We show that the Alexander and Thurston norms are the same for all irreducible Eisenbud-Neumann graph links in homology 3-spheres. These are the links obtained by splicing Seifert links in homology 3-spheres together along tori. By combining this result with previous results, we prove that the two norms coincide for all links in S^3 if either of the following two conditions are met; the link is a graph link, so that the JSJ decomposition of its complement in S^3 is made up of pieces which are all Seifert-fibered, or the link is alternating and not a (2,n)-torus link, so that the JSJ decomposition of its complement in S^3 is made up of pieces which are all hyperbolic. We use the E-N obstructions to fibrations for graph links together with the Thurston cone theorem on link fibrations to deduce that every facet of the reduced Thurston norm unit ball of a graph link is a fibered facet."}
{"category": "Math", "title": "An intermediate regime for exit phenomena driven by non-Gaussian Levy noises", "abstract": "A dynamical system driven by non-Gaussian L\\'evy noises of small intensity is considered. The first exit time of solution orbits from a bounded neighborhood of an attracting equilibrium state is estimated. For a class of non-Gaussian L\\'evy noises, it is shown that the mean exit time is asymptotically faster than exponential (the well-known Gaussian Brownian noise case) but slower than polynomial (the stable L\\'evy noise case), in terms of the reciprocal of the small noise intensity."}
{"category": "Math", "title": "The homotopy fixed point spectra of profinite Galois extensions", "abstract": "Let E be a k-local profinite G-Galois extension of an E_infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points."}
{"category": "Math", "title": "On finite and elementary generation of SL_2(R)", "abstract": "Motivated by a question of A. Rapinchuk concerning general reductive groups, we are investigating the following question: Given a finitely generated integral domain $R$ with field of fractions $F$, is there a \\emph{finitely generated subgroup} $\\Gamma$ of $SL_2(F)$ containing $SL_2(R)$? We shall show in this paper that the answer to this question is negative for any polynomial ring $R$ of the form $R = R_0[s,t]$, where $R_0$ is a finitely generated integral domain with infinitely many (non--associate) prime elements. The proof applies Bass--Serre theory and reduces to analyzing which elements of $SL_2(R)$ can be generated by elementary matrices with entries in a given finitely generated $R$--subalgbra of $F$. Using Bass--Serre theory, we can also exhibit new classes of rings which do not have the $GE_2$ property introduced by P.M. Cohn."}
{"category": "Math", "title": "General Matrix-Valued Inhomogeneous Linear Stochastic Differential Equations and Applications", "abstract": "The expressions of solutions for general $n\\times m$ matrix-valued inhomogeneous linear stochastic differential equations are derived. This generalizes a result of Jaschke (2003) for scalar inhomogeneous linear stochastic differential equations. As an application, some $\\R^n$ vector-valued inhomogeneous nonlinear stochastic differential equations are reduced to random differential equations, facilitating pathwise study of the solutions."}
{"category": "Math", "title": "Local Galois Symbols on E x E", "abstract": "This article is the first part of a two-part work on the Albanese kernel T_F(E x E), for an elliptic curve E over F. The main result furnishes information, for any odd prime p, about the kernel and image of the Galois symbol map from T_F(E \\times E)/p to the Galois cohomology group H^2(F, E[p] (x) E[p]), for F a p-adic field and E/F ordinary, without requiring that the p-torsion points are F-rational. A key step is to show that the image is zero when the Galois module E[p] is non-semisimple. The forthcoming second part will deal with global questions."}
{"category": "Math", "title": "Universal bounds and semiclassical estimates for eigenvalues of abstract Schroedinger operators", "abstract": "We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These include a family of differential inequalities for generalized Riesz means and theorems stating that arithmetic means of lambda_k^p for p <= 3 are universally bounded from above by multiples of the geometric mean of the lambda_k. For Schroedinger operators and the Dirichlet Laplacian these bounds are Weyl-sharp, i.e., saturated by the standard semiclassical estimates for lambda_k at large k."}
{"category": "Math", "title": "A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components", "abstract": "Tutte has described in the book \"Connectivity in graphs\" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general grammar expressing any family of graphs (with some stability conditions) in terms of the 3-connected subfamily. A key ingredient we use is an extension of the so-called dissymmetry theorem, which yields negative signs in the grammar. As a main application we recover in a purely combinatorial way the analytic expression found by Gim\\'enez and Noy for the series counting labelled planar graphs (such an expression is crucial to do asymptotic enumeration and to obtain limit laws of various parameters on random planar graphs). Besides the grammar, an important ingredient of our method is a recent bijective construction of planar maps by Bouttier, Di Francesco and Guitter."}
{"category": "Math", "title": "Biflatness and biprojectivity of the Fourier algebra", "abstract": "We show that the biflatness - in the sense of A. Ya. Helemskii - of the Fourier algebra $A(G)$ of a locally compact group $G$ forces $G$ to either have an abelian subgroup of finite index or to be non-amenable without containing $F_2$, the free group in two generators, as a closed subgroup. An analogous dichotomy is obtained for biprojectivity."}
{"category": "Math", "title": "Relations among conditional probabilities", "abstract": "We describe a Groebner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events. They may be specialized to the partially-observed random variable case, the purely conditional case, and other special cases. We also investigate the connection to generalized permutohedra and describe a conditional probability simplex."}
{"category": "Math", "title": "Enumeration of $(k,2)$-noncrossing partitions", "abstract": "A set partition is said to be $(k,d)$-noncrossing if it avoids the pattern $12... k12... d$. We find an explicit formula for the ordinary generating function of the number of $(k,d)$-noncrossing partitions of $\\{1,2,...,n\\}$ when $d=1,2$."}
{"category": "Math", "title": "A trivial formalization of the theory of grossone", "abstract": "A trivial formalization is given for the informal reasonings presented in a series of papers by Ya.D.Sergeyev on a positional numeral system with an infinitely large base, grossone; the system which is groundlessly opposed by its originator to the classical nonstandard analysis."}
{"category": "Math", "title": "Heat flow on Finsler manifolds", "abstract": "We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space $\\mathcal{P}_2(M)$ of probability measures on $M$ for the relative entropy. Both approaches depend on the choice of a measure $m$ on $M$ and then lead to the same nonlinear evolution semigroup. We prove $\\mathcal{C}^{1,\\alpha}$-regularity for solutions to the (nonlinear) heat equation on the Finsler space $(M,F,m)$. Typically, solutions to the heat equation will not be $\\mathcal{C}^2$. Moreover, we derive pointwise comparison results a la Cheeger-Yau and integrated upper Gaussian estimates a la Davies."}
{"category": "Math", "title": "Rank of 3-tensors with 2 slices and Kronecker canonical forms", "abstract": "Tensor type data are becoming important recently in various application fields. We determine a rank of a tensor T so that A+T is diagonalizable for a given 3-tensor A with 2 slices over the complex and real number field."}
{"category": "Math", "title": "Frame and wavelet systems on the sphere", "abstract": "In this paper we formulate a weighted version of minimum problem (1.4) on the sphere and we show that, for $K\\le L$, if $\\set{\\phi_k}^K_{k=1}$ consists of the spherical functions with degree less than $N$ we can localize the points $(\\xi_1,...,\\xi_L)$ on the sphere so that the solution of this problem is the simplest possible. This localization is connected to the discrete orthogonality of the spherical functions which was proved in [3]. Using these points we construct a frame system and a wavelet system on the sphere and we study the properties of these systems. For $K>L$ a similar construction was made in paper [4], but in that case the solution of the minimum problem (1.4) is not as efficient as it is in our case. The analogue of Fej\\'er and de la Val\\'ee-Poussin summation methods introduced in [3] can be expressed by the frame system introduced in this paper."}
{"category": "Math", "title": "The maximal number of exceptional Dehn surgeries", "abstract": "We prove two conjectures of C. Gordon. We show that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10, and that the maximal intersection number between exceptional slopes is 8. The proof uses a combination of new geometric techniques and a rigorous computer-assisted calculation."}
{"category": "Math", "title": "Manifolds with weighted Poincar\\'e inequality and uniqueness of minimal hypersurfaces", "abstract": "In this paper, we obtain results on rigidity of complete Riemannian manifolds with weighted Poincar\\'e inequality. As an application, we prove that if $M$ is a complete $\\frac{n-2}{n}$-stable minimal hypersurface in $\\mathbb{R}^{n+1}$ with $n\\geq 3$ and has bounded norm of the second fundamental form, then $M$ must either have only one end or be a catenoid."}
{"category": "Math", "title": "On approximability by embeddings of cycles in the plane", "abstract": "We obtain a criterion for approximability by embeddings of piecewise linear maps of a circle to the plane, analogous to the one proved by Minc for maps of a segment to the plane. Theorem. Let S be a triangulation of a circle with s vertices. Let f be a simplicial map of the graph S to the plane. The map f is approximable by embeddings if and only if for each i=0,...,s the i-th derivative of the map f (defined by Minc) neither contains transversal self-intersections nor is the standard winding of degree greater than 1. We deduce from the Minc result the completeness of the van Kampen obstruction to approximability by embeddings of piecewise linear maps of a segment to the plane. We also generalize these criteria to simplicial maps of a graph without vertices of degree >3 to a circle."}
{"category": "Math", "title": "Interpolation sequences for the Bernstein algebra", "abstract": "We give a description, in analytic and geometric terms, of the interpolation sequences for the algebra of entire functions of exponential type which are bounded on the real line."}
{"category": "Math", "title": "Smoothing effects of dispersive equations on real rank one symmetric spaces", "abstract": "In this article we prove time-global smoothing effects of dispersive pseudodifferential equations with constant coefficient radially symmetric symbols on real rank one symmetric spaces of noncompact type. We also discuss gain of regularities according to decay rates of initial values for the Schroedinger evolution equation. We introduce some isometric operators and reduce the arguments to the well-known Euclidean case. In our proof, Helgason's Fourier transform and the Radon transform as an elliptic Fourier integral operator play crucial roles."}
{"category": "Math", "title": "Embedding products of graphs into Euclidean spaces", "abstract": "For any collection of graphs we find the minimal dimension d such that the product of these graphs is embeddable into the d-dimensional Euclidean space. In particular, we prove that the n-th powers of the Kuratowsky graphs are not embeddable into the 2n-dimensional Euclidean space. This is a solution of a problem of Menger from 1929. The idea of the proof is the reduction to a problem from so-called Ramsey link theory: we show that any embedding of L into the (2n-1)-dimensional sphere, where L is the join of n copies of a 4-point set, has a pair of linked (n-1)-dimensional spheres."}
{"category": "Math", "title": "Balanced Hermitian metrics from SU(2)-structures", "abstract": "We study the intrinsic geometrical structure of hypersurfaces in 6-manifolds carrying a balanced Hermitian SU(3)-structure, which we call {\\em balanced} SU(2)-{\\em structures}. We provide conditions which imply that such a 5-manifold can be isometrically embedded as a hypersurface in a manifold with a balanced SU(3)-structure. We show that any 5-dimensional compact nilmanifold has an invariant balanced SU(2)-structure as well as new examples of balanced Hermitian SU(3)-metrics constructed from balanced SU(2)-structures. Moreover, for $n=3,4$, we present examples of compact manifolds, endowed with a balanced SU(n)-structure, such that the corresponding Bismut connection has holonomy equal to SU(n)."}
{"category": "Math", "title": "Equidistribution of the Fekete points on the sphere", "abstract": "The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere. The way we proceed is by showing their connection with other array of points, the Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been studied recently."}
{"category": "Math", "title": "The cohomology of lattices in SL(2,C)", "abstract": "This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H^1(G,E_n), where Gamma is a lattice in SL(2,C) and E_n is one of the standard self-dual modules. In the case Gamma = SL(2,O) for the ring of integers O in an imaginary quadratic number field, we make the theory of lifting explicit and obtain lower bounds linear in n. We have accumulated a large amount of experimental data in this case, as well as for some geometrically constructed and mostly non-arithmetic groups. The computations for SL(2,O) lead us to discover two instances with non-lifted classes in the cohomology. We also derive an upper bound of size O(n^2 / log n) for any fixed lattice Gamma in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data."}
{"category": "Math", "title": "Abrahamse's interpolation theorem and Fuchsian groups", "abstract": "We generalize Abrahamse's interpolation theorem from the setting of a multiply connected domain to that of a more general Riemann surface. Our main result provides the scalar-valued interpolation theorem for the fixed-point subalgebra of $H^\\infty$ associated to the action of a Fuchsian group. We rely on two results from a paper of Forelli. This allows us to prove the interpolation result using duality techniques that parallel Sarason's approach to the interpolation problem for $H^\\infty$. In this process we prove a more general distance formula, very much like Nehari's theorem, and obtain relations between the kernel function for the character automorphic Hardy spaces and the Szeg\\\"o kernel for the disk. Finally, we examine our interpolation results in the context of the two simplest examples of Fuchsian groups acting on the disk."}
{"category": "Math", "title": "On the Pontryagin-Steenrod-Wu theorem", "abstract": "This paper is on homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f of an (n+1)-manifold into the n-sphere define the degree deg f to be the class dual to f^*[S^n], where [S^n] is the fundamental class. We present a short and direct proof of the following specific case of the Pontryagin-Steenrod-Wu theorem: Theorem. Let M be a connected orientable closed smooth (n+1)-manifold, n>2. Then the map deg:\\pi^n(M)\\to H_1(M;Z) is 1-to-1 (i.e., bijective), if the product w_2(M) x r_2 H_2(M;Z) is nonzero, where r_2 is the mod2 reduction; 2-to-1 (i.e., each element of H_1(M;Z) has exactly 2 preimages) - otherwise. The proof is based on the Pontryagin-Thom construction and a geometric definition of the Stiefel-Whitney classes w_2(M)."}
{"category": "Math", "title": "On the mean square of the error term for the two-dimensional divisor problems(II)", "abstract": "Let $\\Delta(a,b;x)$ denote the error term of the general two-dimensional divisor problem. In this paper we shall study the relation between the discrete mean value $\\sum_{n\\leq T}\\Delta^2(a,b;n)$ and the continuous mean value $\\int_1^T\\Delta^2(a,b;x)dx$."}
{"category": "Math", "title": "A short proof of the Twelve points theorem", "abstract": "We present a short elementary proof of the following Twelve Points Theorem: Let M be a convex polygon with vertices at the lattice points, containing a single lattice point in its interior. Denote by m (resp. m*) the number of lattice points in the boundary of M (resp. in the boundary of the dual polygon). Then m+m*=12."}
{"category": "Math", "title": "Teichm\\\"uller's problem in space", "abstract": "Quasiconformal homeomorphisms of the whole space Rn, onto itself normalized at one or two points are studied. In particular, the stability theory, the case when the maximal dilatation tends to 1, is in the focus. Our main result provides a spatial analogue of a classical result due to Teichm\\\"uller. Unlike Teichm\\\"uller's result, our bounds are explicit. Explicit bounds are based on two sharp well-known distortion results: the quasiconformal Schwarz lemma and the bound for linear dilatation. Moreover, Bernoulli type inequalities and asymptotically sharp bounds for special functions involving complete elliptic integrals are applied to simplify the computations. Finally, we discuss the behavior of the quasihyperbolic metric under quasiconformal maps and prove a sharp result for quasiconformal maps of R^n \\ {0} onto itself."}
{"category": "Math", "title": "On the incidence-prevalence relation and length-biased sampling", "abstract": "For many diseases, logistic and other constraints often render large incidence studies difficult, if not impossible, to carry out. This becomes a drawback, particularly when a new incidence study is needed each time the disease incidence rate is investigated in a different population. However, by carrying out a prevalent cohort study with follow-up it is possible to estimate the incidence rate if it is constant. In this paper we derive the maximum likelihood estimator (MLE) of the overall incidence rate, $\\lambda$, as well as age-specific incidence rates, by exploiting the well known epidemiologic relationship, prevalence = incidence $\\times$ mean duration ($P = \\lambda \\times \\mu$). We establish the asymptotic distributions of the MLEs, provide approximate confidence intervals for the parameters, and point out that the MLE of $\\lambda$ is asymptotically most efficient. Moreover, the MLE of $\\lambda$ is the natural estimator obtained by substituting the marginal maximum likelihood estimators for P and $\\mu$, respectively, in the expression $P = \\lambda \\times \\mu$. Our work is related to that of Keiding (1991, 2006), who, using a Markov process model, proposed estimators for the incidence rate from a prevalent cohort study \\emph{without} follow-up, under three different scenarios. However, each scenario requires assumptions that are both disease specific and depend on the availability of epidemiologic data at the population level. With follow-up, we are able to remove these restrictions, and our results apply in a wide range of circumstances. We apply our methods to data collected as part of the Canadian Study of Health and Ageing to estimate the incidence rate of dementia amongst elderly Canadians."}
{"category": "Math", "title": "Similar Sublattices and Coincidence Rotations of the Root Lattice A4 and its Dual", "abstract": "A natural way to describe the Penrose tiling employs the projection method on the basis of the root lattice A4 or its dual. Properties of these lattices are thus related to properties of the Penrose tiling. Moreover, the root lattice A4 appears in various other contexts such as sphere packings, efficient coding schemes and lattice quantizers. Here, the lattice A4 is considered within the icosian ring, whose rich arithmetic structure leads to parametrisations of the similar sublattices and the coincidence rotations of A4 and its dual lattice. These parametrisations, both in terms of a single icosian, imply an index formula for the corresponding sublattices. The results are encapsulated in Dirichlet series generating functions. For every index, they provide the number of distinct similar sublattices as well as the number of coincidence rotations of A4 and its dual."}
{"category": "Math", "title": "An algorithm for the unit group of the Burnside ring of a finite group", "abstract": "In this note we present an algorithm for the construction of the unit group of the Burnside ring $\\Omega(G)$ of a finite group $G$ from a list of representatives of the conjugacy classes of subgroups of G."}
{"category": "Math", "title": "Dynamics Groups of Asynchronous Cellular Automata", "abstract": "We say that a finite asynchronous cellular automaton (or more generally, any sequential dynamical system) is pi-independent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the dynamics group. We have previously shown that exactly 104 of the possible 256 cellular automaton rules are pi-independent. In this article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information about permissible dynamics as a function of update sequence and, as such, connect discrete dynamical systems, group theory, and algebraic combinatorics in a new and interesting way. We conclude with a discussion of numerous open problems and directions for future research."}
{"category": "Math", "title": "Homology and finiteness properties of SL_2(Z[t,t^{-1}])", "abstract": "We show that the group $H_2(\\slzti;\\zz)$ is not finitely generated, answering a question mentioned by Bux and Wortman in \\cite{bux}."}
{"category": "Math", "title": "Coverings of Laura Algebras: the Standard Case", "abstract": "In this paper, we study the covering theory of laura algebras. We prove that if a connected laura algebra is standard (that is, it is not quasi-tilted of canonical type and its connecting components are standard), then this algebra has nice Galois coverings associated to the coverings of the connecting component. As a consequence, we show that the first Hochschild cohomology group of a standard laura algebra vanishes if and only if it has no proper Galois coverings."}
{"category": "Math", "title": "A Hecke Correspondence Theorem for Automorphic Integrals with Symmetric Rational Period Functions on the Hecke Groups", "abstract": "Marvin Knopp developed the theory of automorphic integrals, which generalize automorphic forms; each automorphic integral has an additional period function in its automorphic relation. The period functions satisfy relations that arise from the underlying group relations. Knopp showed that entire automorphic integrals with rational period functions satisfy a Hecke correspondence theorem, provided the rational period functions have poles only at 0 or infinity. For other automorphic integrals each corresponding Dirichlet series has a functional equation with a remainder term that arises from the nonzero poles of the rational period function. In this paper we prove a Hecke correspondence theorem for a class of automorphic integrals with rational period functions on the Hecke groups. We restrict our attention to automorphic integrals of weight that is twice an odd integer and to rational period functions that satisfy a symmetry property we call \"Hecke-symmetry.\" We explicate the relationship between the structure of the rational period functions and the corresponding remainder terms. Each remainder term satisfies two relations (the second of which is new in this paper) corresponding to the two relations for the rational period function."}
{"category": "Math", "title": "Extensions of positive definite functions on amenable groups", "abstract": "Let $S$ be a subset of a amenable group $G$ such that $e\\in S$ and $S^{-1}=S$. The main result of the paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as a corollary. New applications are also presented."}
{"category": "Math", "title": "Obstructions to the Existence and Squeezing of Lagrangian Cobordisms", "abstract": "Capacities that provide both qualitative and quantitative obstructions to the existence of a Lagrangian cobordism between two $(n-1)$-dimensional submanifolds in parallel hyperplanes of $\\mathbb{R}^{2n}$ are defined using the theory of generating families. Qualitatively, these capacities show that, for example, in $\\mathbb R^4$ there is no Lagrangian cobordism between two $\\infty$-shaped curves with a negative crossing when the lower end is \"smaller\". Quantitatively, when the boundary of a Lagrangian ball lies in a hyperplane of $\\mathbb{R}^{2n}$, the capacity of the boundary gives a restriction on the size of a rectangular cylinder into which the Lagrangian ball can be squeezed."}
{"category": "Math", "title": "Linear systems and determinantal random point fields", "abstract": "Tracy and Widom showed that fundamentally important kernels in random matrix theory arise from differential equations with rational coefficients. More generally, this paper considers symmetric Hamiltonian systems abd determines the properties of kernels that arise from them. The inverse spectral problem for self-adjoint Hankel operators gives a sufficient condition for a self-adjoint operator to be the Hankel operator on $L^2(0, \\infty)$ from a linear system in continuous time; thus this paper expresses certain kernels as squares of Hankel operators. For a suitable linear system $(-A,B,C)$ with one dimensional input and output spaces, there exists a Hankel operator $\\Gamma$ with kernel $\\phi_{(x)}(s+t)=Ce^{-(2x+s+t)A}B$ such that $\\det (I+(z-1)\\Gamma\\Gamma^\\dagger)$ is the generating function of a determinantal random point field."}
{"category": "Math", "title": "A Partial Ordering on Slices of Planar Lagrangians", "abstract": "A collection of simple closed curves in $\\rr^3$ is called a negative slice if it is the intersection of a flat-at-infinity planar Lagrangian surface and $\\{y_2 = a \\}$ for some $a < 0$. Examples and non-examples of negative slices are given. Embedded Lagrange cobordisms define a relation on slices and in some (and perhaps all) cases this relation defines a partial order. The set of slices is a commutative monoid and the additive structure has an interesting relationship with the ordering relation."}
{"category": "Math", "title": "Quantum metric spaces of quantum maps", "abstract": "We show that any quantum family of maps from a non commutative space to a compact quantum metric space has a canonical quantum semi metric structure."}
{"category": "Math", "title": "Optimal control of a stochastic network driven by a fractional Brownian motion input", "abstract": "We consider a stochastic control model driven by a fractional Brownian motion. This model is a formal approximation to a queueing network with an on-off input process. We study stochastic control problems associated with the long-run average cost, the infinite horizon discounted cost, and the finite horizon cost. In addition, we find a solution to a constrained minimization problem as an application of our solution to the long-run average cost problem. We also establish Abelian limit relationships among the value functions of the above control problems."}
{"category": "Math", "title": "Stability of multi-dimensional viscous shocks for symmetric systems with variable multiplicities", "abstract": "We establish long-time stability of multi-dimensional viscous shocks of a general class of symmetric hyperbolic--parabolic systems with variable multiplicities, notably including the equations of compressible magnetohydrodynamics (MHD) in dimensions $d\\ge 2$. This extends the existing result established by K. Zumbrun for systems with characteristics of constant multiplicity to the ones with variable multiplicity, yielding the first such a stability result for (fast) MHD shocks. At the same time, we are able to drop a technical assumption on structure of the so--called glancing set that was necessarily used in previous analyses. The key idea to the improvements is to introduce a new simple argument for obtaining a $L^1\\to L^p$ resolvent bound in low--frequency regimes by employing the recent construction of degenerate Kreiss' symmetrizers by O. Gu\\`es, G. M\\'etivier, M. Williams, and K. Zumbrun. Thus, at the low-frequency resolvent bound level, our analysis gives an alternative to the earlier pointwise Green's function approach of K. Zumbrun. High--frequency solution operator bounds have been previously established entirely by nonlinear energy estimates."}
{"category": "Math", "title": "Universal Cycles of Restricted Classes of Words", "abstract": "It is well known that Universal Cycles of $k$-letter words on an $n$-letter alphabet exist for all $k$ and $n$. In this paper, we prove that Universal Cycles exist for restricted classes of words, including: non-bijections, equitable words (under suitable restrictions), ranked permutations, and \"passwords\"."}
{"category": "Math", "title": "Periodic resolutions and self-injective algebras of finite type", "abstract": "We say that an algebra A is periodic if it has a periodic projective resolution as an (A,A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B --> A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba's classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi-Yau dimensions."}
{"category": "Math", "title": "The rationality of the moduli space of genus four curves endowed with an order three subgroup of their Jacobian", "abstract": "Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and on a theorem of Dolgachev, the rationality of the moduli space R mentioned in the title is proved. Let M be the moduli space of 6 points in the plane, modulo the natural involution induced by double-six configurations on cubic surfaces. It is proved that R is birational to a tower of locally trivial projective bundles ending onto M. The rationality of R then follows from Dolgachev's theorem that M is rational."}
{"category": "Math", "title": "Cohomology algebra of the orbit space of some free actions on spaces of cohomology type (a, b)", "abstract": "Let X be a finitistic space with non-trivial cohomology groups H^in(X;Z)=Z with generators v_i, where i = 0, 1, 2, 3. We say that X has cohomology type (a, b) if v_1^2 = av_2 and v_1v_2 = bv_3 . In this note, we determine the mod 2 cohomology ring of the orbit space X/G of a free action of G = Z_2 on X, where both a and b are even. In this case, we observed that there is no equivariant map S^m --> X for m > 3n, where S^m has the antipodal action. Moreover, it is shown that G can not act freely on space X which is of cohomology type (a, b) where a is odd and b is even. We also obtain the mod 2 cohomology ring of the orbit space X/G of free action of G = S^1 on the space X of type (0, b)."}
{"category": "Math", "title": "Presentations of Semigroup Algebras of Weighted Trees", "abstract": "We find presentations for subalgebras of invariants of the coordinate algebras of binary symmetric models of phylogenetic trees studied by Buczynska and Wisniewski in \\cite{BW}. These algebras arise as toric degenerations of rings of global sections of weight varieties of the Grassmanian of two planes associated to the Pl\\\"ucker embedding, and as toric degenerations of rings of invariants of Cox-Nagata rings."}
{"category": "Math", "title": "Addendum to: \"Knots, sutures and excision\"", "abstract": "We observe that the main theorem in \\cite{KMsuture} immediately implies its analogue for closed 3--manifolds."}
{"category": "Math", "title": "Schubert polynomials and Arakelov theory of symplectic flag varieties", "abstract": "Let X be the flag variety of the symplectic group. We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of X. We use these polynomials to describe the arithmetic Schubert calculus on X. Moreover, we give a method to compute the natural arithmetic Chern numbers on X, and show that they are all rational numbers."}
{"category": "Math", "title": "Semistar dimension of polynomial rings and Pr\\\"{u}fer-like domains", "abstract": "Let $D$ be an integral domain and $\\star$ a semistar operation stable and of finite type on it. In this paper we define the semistar dimension (inequality) formula and discover their relations with $\\star$-universally catenarian domains and $\\star$-stably strong S-domains. As an application we give new characterizations of $\\star$-quasi-Pr\\\"{u}fer domains and UM$t$ domains in terms of dimension inequality formula (and the notions of universally catenarian domain, stably strong S-domain, strong S-domain, and Jaffard domains). We also extend Arnold's formula to the setting of semistar operations."}
{"category": "Math", "title": "Biharmonic space-like hypersurfaces in pseudo-Riemannian space", "abstract": "We classify the space-like biharmonic surfaces in 3-dimension pseudo-Riemannian space form, and construct explicit examples of proper biharmonic hypersurfaces in general ADS space."}
{"category": "Math", "title": "Unramified representations of reductive groups over finite rings", "abstract": "Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic p, extending the construction of Deligne and Lusztig of representations of reductive groups over finite fields. We generalize Lusztig's results to reductive groups over arbitrary finite local rings. This generalization uses the Greenberg functor and the theory of group schemes over Artinian local rings."}
{"category": "Math", "title": "Comparison results for conjugate and focal points in semi-Riemannian geometry via Maslov index", "abstract": "We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold."}
{"category": "Math", "title": "Compact multipliers on spaces of analytic functions", "abstract": "In the paper compact multiplier operators on Banach spaces of analytic functions on the unit disk with the range in Banach sequence lattices are studied. If the domain space $X$ is such that $H_\\infty\\hookrightarrow X\\hookrightarrow H_1$, necessary and sufficient conditions for compactness are presented. Moreover, the calculation of the Hausdorff measure of noncompactness for diagonal operators between Banach sequence lattices is applied to obtaining the characterization of compact multipliers in case the domain space $X$ satisfies $H_\\infty\\hookrightarrow X\\hookrightarrow H_2$."}
{"category": "Math", "title": "Asymptotic equivariant index of Toeplitz operators and relative index of CR structures", "abstract": "Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures."}
{"category": "Math", "title": "On surfaces of general type with maximal Albanese dimension", "abstract": "Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic curves on any surface of general type with two linearly independent regular one forms."}
{"category": "Math", "title": "Hardy spaces of operator-valued analytic functions", "abstract": "We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of $\\mathbb{C}.$ In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operator-valued $\\mathrm{H}^1$-BMOA duality theorem is proved. Finally, by the $\\mathrm{H}^1$-BMOA duality we present the Lusin area integral and Littlewood-Paley $g$-function characterizations of the operator-valued analytic Hardy space."}
{"category": "Math", "title": "Algebraic topology from geometric viewpoint", "abstract": "This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and invariants, characteristic classes). Thus main ideas of algebraic topology are presented with minimal technicalities. Familiarity of a reader with basic notions of topology (such as 2-dimensional manifolds and vector fields) is desirable, although definitions are given at the beginning. The book is accessible to undergraduates and could also be an interesting easy reading for professional mathematicians."}
{"category": "Math", "title": "Coupled Painlev\\'e VI systems in dimension four with affine Weyl group symmetry of type $E_6^{(2)}$", "abstract": "We find a four-parameter family of coupled Painlev\\'e VI systems in dimension four with affine Weyl group symmetry of type $E_6^{(2)}$. This is the first example which gave higher order Painlev\\'e type systems of type $E_{6}^{(2)}$. We study its symmetry and holomorphy conditions."}
{"category": "Math", "title": "A Counter Example To the Hodge Conjecture", "abstract": "In this paper, we give a simple counter example to the famous Hodge conjecture."}
{"category": "Math", "title": "Circle correspondence $C^*$-algebras", "abstract": "We investigate Cuntz-Pimsner $C^*$-algebras associated with certain correspondences of the unit circle $\\mathbb{T}$. We analyze these $C^*$-algebras by analogy with irrational rotation algebras $A_\\theta$ and Cuntz algebras $\\mathcal{O}_n$. We construct a Rieffel type projection, study the fixed point algebras of certain actions of finite groups, and calculate the entropy of a certain endomorphism. We also study the induced map of the dual action of the gauge action on $K$-groups."}
{"category": "Math", "title": "Weighted Semigroup Algebras as Dual Banach algebras", "abstract": "In this paper, among other things, we study those conditions under which the weighted semigroup algebra $\\ell^1(S,\\omega)$ is a dual Banach algebra with respect to predual $c_0(S)$. Some useful examples, illustrating the results, are also included."}
{"category": "Math", "title": "There are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime", "abstract": "Dirichlet's proof of infinitely many primes in arithmetic progressions was published in 1837, introduced L-series for the first time, and it is said to have started rigorous analytic number theory. Dirichlet uses Euler's earlier work on the zeta function and the distribution of primes. He first proves a simpler case before going to full generality. The paper was translated from German by R. Stephan and given a reference section."}
{"category": "Math", "title": "New Tests of Spatial Segregation Based on Nearest Neighbor Contingency Tables", "abstract": "The spatial clustering of points from two or more classes (or species) has important implications in many fields and may cause the spatial patterns of segregation and association, which are two major types of spatial interaction between the classes. The null patterns we consider are random labeling (RL) and complete spatial randomness (CSR) of points from two or more classes, which is called CSR independence. The segregation and association patterns can be studied using a nearest neighbor contingency table (NNCT) which is constructed using the frequencies of nearest neighbor (NN) types in a contingency table. Among NNCT-tests Pielou's test is liberal the null pattern but Dixon's test has the desired significance level under the RL pattern. We propose three new multivariate clustering tests based on NNCTs. We compare the finite sample performance of these new tests with Pielou's and Dixon's tests and Cuzick & Edward's k-NN tests in terms of empirical size under the null cases and empirical power under various segregation and association alternatives and provide guidelines for using the tests in practice. We demonstrate that the newly proposed NNCT-tests perform relatively well compared to their competitors and illustrate the tests using three example data sets. Furthermore, we compare the NNCT-tests with the second-order methods using these examples."}
{"category": "Math", "title": "Frames and Oversampling Formulas for Band Limited Functions", "abstract": "In this article we obtain families of frames for the space B_\\omega of functions with band in [-\\omega,\\omega] by using the theory of shift-invariant spaces. Our results are based on the Gramian analysis of A. Ron and Z. Shen and a variant, due to Bownik, of their characterization of families of functions whose shifts form frames or Riesz bases. We give necessary and sufficient conditions for the translates of a finite number of functions (generators) to be a frame or a Riesz basis for B_\\omega. We also give explicit formulas for the dual generators and we apply them to Hilbert transform sampling and derivative sampling. Finally, we provide numerical experiments which support the theory."}
{"category": "Math", "title": "Model selection for density estimation with L2-loss", "abstract": "We consider here estimation of an unknown probability density s belonging to L2(mu) where mu is a probability measure. We have at hand n i.i.d. observations with density s and use the squared L2-norm as our loss function. The purpose of this paper is to provide an abstract but completely general method for estimating s by model selection, allowing to handle arbitrary families of finite-dimensional (possibly non-linear) models and any density s belonging to L2(mu). We shall, in particular, consider the cases of unbounded densities and bounded densities with unknown bound and investigate how the L-infinity-norm of s may influence the risk. We shall also provide applications to adaptive estimation and aggregation of preliminary estimators. Although of a purely theoretical nature, our method leads to results that cannot presently be reached by more concrete methods."}
{"category": "Math", "title": "Maslov class rigidity for Lagrangian submanifolds via Hofer's geometry", "abstract": "In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit a metric of negative sectional curvature. Such Lagrangian submanifolds exist in every symplectic manifold of dimension greater than six or equal to four. The proof utilizes the relations between closed geodesics on the Lagrangian, the periodic orbits of geometric Hamiltonian flows supported near the Lagrangian, and the length minimizing properties of these flows with respect to the negative Hofer length functional."}
{"category": "Math", "title": "Determining sets, resolving sets, and the exchange property", "abstract": "A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U. A subset W is called a resolving set if every vertex in G is uniquely determined by its distances to the vertices of W. Determining (resolving) sets are said to have the exchange property in G if whenever S and R are minimal determining (resolving) sets for G and r\\in R, then there exists s\\in S so that S-\\{s\\}\\cup \\{r\\} is a minimal determining (resolving) set. This work examines graph families in which these sets do, or do not, have the exchange property. This paper shows that neither determining sets nor resolving sets have the exchange property in all graphs, but that both have the exchange property in trees. It also gives an infinite graph family (n-wheels where n\\geq 8) in which determining sets have the exchange property but resolving sets do not. Further, this paper provides necessary and sufficient conditions for determining sets to have the exchange property in an outerplanar graph."}
{"category": "Math", "title": "Cerny's conjecture, synchronizing automata, group representation theory", "abstract": "Let us say that a Cayley graph $\\Gamma$ of a group $G$ of order $n$ is a Cerny Cayley graph if every synchronizing automaton containing $\\Gamma$ as a subgraph with the same vertex set admits a synchronizing word of length at most $(n-1)^2$. In this paper we use the representation theory of groups over the rational numbers to obtain a number of new infinite families of {\\v{C}}ern{\\'y} Cayley graphs."}
{"category": "Math", "title": "The cyclic sliding operation in Garside groups", "abstract": "We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoretical properties. We show, in particular, that if a super summit element has conjugates which are 'rigid' (that is, which have a certain particularly simple structure), then the optimal way of obtaining such a rigid conjugate through conjugation by positive elements is given by iterated cyclic sliding."}
{"category": "Math", "title": "Derivatives of Knots and Second-order Signatures", "abstract": "We define a set of \"second-order\" L^(2)-signature invariants for any algebraically slice knot. These obstruct a knot's being a slice knot and generalize Casson-Gordon invariants, which we consider to be \"first-order signatures\". As one application we prove: If K is a genus one slice knot then, on any genus one Seifert surface, there exists a homologically essential simple closed curve of self-linking zero, which has vanishing zero-th order signature and a vanishing first-order signature. This extends theorems of Cooper and Gilmer. We introduce a geometric notion, that of a derivative of a knot with respect to a metabolizer. We also introduce a new equivalence relation, generalizing homology cobordism, called null-bordism."}
{"category": "Math", "title": "Asymptotic upper bounds on the shades of t-intersecting families", "abstract": "We examine the m-shades of t-intersecting families of k-subsets of [n], and conjecture on the optimal upper bound on their cardinalities. This conjecture extends Frankl's General Conjecture that was proven true by Ahlswede-Khachatrian. From this we deduce the precise asymptotic upper bounds on the cardinalities of m-shades of t(m)-intersecting families of k(m)-subsets of [2m], as m -> infinity. A generalization to cross-t-intersecting families is also considered."}
{"category": "Math", "title": "L-functions for GSp(4)xGL(2)in the case of high GL(2) conductor", "abstract": "Furusawa has given an integral representation for the degree 8 L-function of GSp(4) x GL(2) and has carried out the unramified calculation. The local p-adic zeta integrals were calculated in our earlier work under the assumption that the GSp(4) representation \\pi is unramified and the GL(2) representation \\tau has conductor p. In the present work we generalize to the case where the GL(2) representation has arbitrarily high conductor. The result is that the zeta integral represents the local Euler factor L(s,\\pi \\times \\tau) in all cases. As a global application we obtain a special value result for a GSp(4) x GL(2) global L-function coming from classical holomorphic cusp forms with arbitrarily high level for the elliptic modular form."}
{"category": "Math", "title": "Markov switching models: an application to roadway safety", "abstract": "In this research, two-state Markov switching models are proposed to study accident frequencies and severities. These models assume that there are two unobserved states of roadway safety, and that roadway entities (e.g., roadway segments) can switch between these states over time. The states are distinct, in the sense that in the different states accident frequencies or severities are generated by separate processes (e.g., Poisson, negative binomial, multinomial logit). Bayesian inference methods and Markov Chain Monte Carlo (MCMC) simulations are used for estimation of Markov switching models. To demonstrate the applicability of the approach, we conduct the following three studies. In the first study, two-state Markov switching count data models are considered as an alternative to zero-inflated models for annual accident frequencies, in order to account for preponderance of zeros typically observed in accident frequency data. In the second study, two-state Markov switching Poisson model and two-state Markov switching negative binomial model are estimated using weekly accident frequencies on selected Indiana interstate highway segments over a five-year time period. In the third study, two-state Markov switching multinomial logit models are estimated for severity outcomes of accidents occurring on Indiana roads over a four-year time period. One of the most important results found in each of the three studies, is that in each case the estimated Markov switching models are strongly favored by roadway safety data and result in a superior statistical fit, as compared to the corresponding standard (non-switching) models."}
{"category": "Math", "title": "Locally adaptive estimation of evolutionary wavelet spectra", "abstract": "We introduce a wavelet-based model of local stationarity. This model enlarges the class of locally stationary wavelet processes and contains processes whose spectral density function may change very suddenly in time. A notion of time-varying wavelet spectrum is uniquely defined as a wavelet-type transform of the autocovariance function with respect to so-called autocorrelation wavelets. This leads to a natural representation of the autocovariance which is localized on scales. We propose a pointwise adaptive estimator of the time-varying spectrum. The behavior of the estimator studied in homogeneous and inhomogeneous regions of the wavelet spectrum."}
{"category": "Math", "title": "A stochastic differential game for the inhomogeneous $\\infty$-Laplace equation", "abstract": "Given a bounded $\\mathcaligr{C}^2$ domain $G\\subset{\\mathbb{R}}^m$, functions $g\\in\\mathcaligr{C}(\\partial G,{\\mathbb{R}})$ and $h\\in\\mathcaligr {C}(\\bar{G},{\\mathbb{R}}\\setminus\\{0\\})$, let $u$ denote the unique viscosity solution to the equation $-2\\Delta_{\\infty}u=h$ in $G$ with boundary data $g$. We provide a representation for $u$ as the value of a two-player zero-sum stochastic differential game."}
{"category": "Math", "title": "A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras", "abstract": "Let $\\lie g$ be a simple Lie algebra and let $\\bs^{\\lie g}$ be the locally finite part of the algebra of invariants $(_\\bc\\bv\\otimes S(\\lie g))^{\\lie g}$ where $\\bv$ is the direct sum of all simple finite-dimensional modules for $\\lie g$ and $S(\\lie g)$ is the symmetric algebra of $\\lie g$. Given an integral weight $\\xi$, let $\\Psi=\\Psi(\\xi)$ be the subset of roots which have maximal scalar product with $\\xi$. Given a dominant integral weight $\\lambda$ and $\\xi$ such that $\\Psi$ is a subset of the positive roots we construct a finite-dimensional subalgebra $\\bs^{\\lie g}_\\Psi(\\le_\\Psi\\lambda)$ of $\\bs^{\\lie g}$ and prove that the algebra is Koszul of global dimension at most the cardinality of $\\Psi$. Using this we then construct naturally an infinite-dimensional Koszul algebra of global dimension equal to the cardinality of $\\Psi$. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras."}
{"category": "Math", "title": "Heegner points and Eisenstein series", "abstract": "We give an alternative computation of the twisted second moment of critival values of class group $L$-functions attached to an imaginary quadratic field. The method avoids long calculations and yields the expected polynomial growth in the $s$-parameter for the remaining term."}
{"category": "Math", "title": "Looking for Groebner Basis Theory for (Almost) Skew 2-Nomial Algebras", "abstract": "In this paper, we introduce (almost) skew 2-nomial algebras and look for a one-sided or two-sided Gr\\\"obner basis theory for such algebras at a modest level. That is, we establish the existence of a skew multiplicative $K$-basis for every skew 2-nomial algebra, and we explore the existence of a (left, right, or two-sided) monomial ordering for an (almost) skew 2-nomial algebra. As distinct from commonly recognized algebras holding a Gr\\\"obner basis theory (such as algebras of the solvable type [K-RW] and some of their homomorphic images), a subclass of skew 2-nomial algebras that have a left Gr\\\"obner basis theory but may not necessarily have a two-sided Gr\\\"obner basis theory, respectively a subclass of skew 2-nomial algebras that have a right Gr\\\"obner basis theory but may not necessarily have a two-sided Gr\\\"obner basis theory, are determined such that numerous quantum binomial algebras (which provide binomial solutions to the Yang-baxter equation [Laf], [G-I2]) are involved."}
{"category": "Math", "title": "Apery, Bessel, Calabi-Yau and Verrill", "abstract": "A differential equation related to the moments of Bessel functions is shown to have a solution at infinity with coefficients being squares of binomial coefficients."}
{"category": "Math", "title": "On hyperbolic cohomology classes", "abstract": "We study hyperbolic cohomology classes in the general context of simplicial complexes and prove homological invariance statements for them. We relate the existence of hyperbolic cohomology classes to the non-amenability of the fundamental group. In degree two we clarify the relation between hyperbolic and atoroidal classes, leading to an application to symplectically atoroidal manifolds."}
{"category": "Math", "title": "Graded annihilators and tight closure test ideals", "abstract": "Let $R$ be a commutative Noetherian local ring of prime characteristic $p$. The main purposes of this paper are to show that if the injective envelope $E$ of the simple $R$-module has a structure as a torsion-free left module over the Frobenius skew polynomial ring over $R$, then $R$ has a tight closure test element (for modules) and is $F$-pure, and to relate the test ideal of $R$ to the smallest '$E$-special' ideal of $R$ of positive height. A byproduct is an analogue of a result of Janet Cowden Vassilev: she showed, in the case where $R$ is an $F$-pure homomorphic image of an $F$-finite regular local ring, that there exists a strictly ascending chain $0 = \\tau_0 \\subset \\tau_1 \\subset ... \\subset \\tau_t = R$ of radical ideals of $R$ such that, for each $i = 0, ..., t-1$, the reduced local ring $R/\\tau_i$ is $F$-pure and its test ideal (has positive height and) is exactly $\\tau_{i+1}/\\tau_i$. This paper presents an analogous result in the case where $R$ is complete (but not necessarily $F$-finite) and $E$ has a structure as a torsion-free left module over the Frobenius skew polynomial ring. Whereas Cowden Vassilev's results were based on R. Fedder's criterion for $F$-purity, the arguments in this paper are based on the author's work on graded annihilators of left modules over the Frobenius skew polynomial ring."}
{"category": "Math", "title": "On Jordan type inequalities for hyperbolic functions", "abstract": "This paper deals with some inequalities for trigonometric and hyperbolic functions such as the Jordan inequality and its generalizations. In particular, lower and upper bounds for functions such as (sin x)/x and x/(sinh x) are proved."}
{"category": "Math", "title": "Circular Law Theorem for Random Markov Matrices", "abstract": "Consider an nxn random matrix X with i.i.d. nonnegative entries with bounded density, mean m, and finite positive variance sigma^2. Let M be the nxn random Markov matrix with i.i.d. rows obtained from X by dividing each row of X by its sum. In particular, when X11 follows an exponential law, then M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Our main result states that with probability one, the counting probability measure of the complex spectrum of n^(1/2)M converges weakly as n tends to infinity to the uniform law on the centered disk of radius sigma/m. The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent."}
{"category": "Math", "title": "New results on the least common multiple of consecutive integers", "abstract": "When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions $g_k$ $(k \\in \\mathbb{N})$, defined by $g_k(n) := \\frac{n (n + 1) ... (n + k)}{\\lcm(n, n + 1, >..., n + k)}$ $(\\forall n \\in \\mathbb{N} \\setminus \\{0\\})$. He proved that $g_k$ $(k \\in \\mathbb{N})$ is periodic and $k!$ is a period of $g_k$. He raised the open problem consisting to determine the smallest positive period $P_k$ of $g_k$. Very recently, S. Hong and Y. Yang have improved the period $k!$ of $g_k$ to $\\lcm(1, 2, ..., k)$. In addition, they have conjectured that $P_k$ is always a multiple of the positive integer $\\frac{\\lcm(1, 2, >..., k, k + 1)}{k + 1}$. An immediate consequence of this conjecture states that if $(k + 1)$ is prime then the exact period of $g_k$ is precisely equal to $\\lcm(1, 2, ..., k)$. In this paper, we first prove the conjecture of S. Hong and Y. Yang and then we give the exact value of $P_k$ $(k \\in \\mathbb{N})$. We deduce, as a corollary, that $P_k$ is equal to the part of $\\lcm(1, 2, ..., k)$ not divisible by some prime."}
{"category": "Math", "title": "Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise", "abstract": "We prove existence, uniqueness and Lipschitz dependence on the initial datum for mild solutions of stochastic partial differential equations with Lipschitz coefficients driven by Wiener and Poisson noise. Under additional assumptions, we prove Gateaux and Frechet differentiability of solutions with respect to the initial datum. As an application, we obtain gradient estimates for the resolvent associated to the mild solution. Finally, we prove the strong Feller property of the associated semigroup."}
{"category": "Math", "title": "The power of multifolds: Folding the algebraic closure of the rational numbers", "abstract": "It is well known that the usual Huzita-Hatori axioms for origami enable angle trisection but not angle quintisection. Using the concept of a multifold, Lang has achieved quintisection but not arbitrary algebraic numbers. We define the n-parameter multifold and show how to use one-parameter multifolds to obtain the algebraic closure of the rational numbers."}
{"category": "Math", "title": "Constructive Gelfand duality for C*-algebras", "abstract": "We present a constructive proof of Gelfand duality for C*-algebras by reducing the problem to Gelfand duality for real C*-algebras."}
{"category": "Math", "title": "De Morgan classifying toposes", "abstract": "We present a general method for deciding whether a Grothendieck topos satisfies De Morgan's law (resp. the law of excluded middle) or not; applications to the theory of classifying toposes follow. Specifically, we obtain a syntactic characterization of the class of geometric theories whose classifying toposes satisfy De Morgan's law (resp. are Boolean), as well as model-theoretic criteria for theories whose classifying toposes arise as localizations of a given presheaf topos."}
{"category": "Math", "title": "Integrals and Valuations", "abstract": "We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure."}
{"category": "Math", "title": "Bounding sup-norms of cusp forms of large level", "abstract": "Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $\\chi$ and Laplacian eigenvalue $\\lambda\\geq 1/4$. It is shown that $\\| f \\|_{\\infty} \\ll_{\\lambda} N^{-1/37}$, from which the hybrid bound $\\|f \\|_{\\infty} \\ll \\lambda^{1/4} (N\\lambda)^{-\\delta}$ (for some $\\delta > 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k."}
{"category": "Math", "title": "Eigenvalue Asymptotics in a Twisted Waveguide", "abstract": "We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence."}
{"category": "Math", "title": "On the complexity group of stable curves", "abstract": "In this paper, we study combinatorial properties of stable curves. To the dual graph of any nodal curve, it is naturally associated a group, which is the group of components of the N\\'eron model of the generalized Jacobian of the curve. We study the order of this group, called the complexity. In particular, we provide a partial characterization of the stable curves having maximal complexity, and we provide an upper bound, depending only on the genus $g$ of the curve, on the maximal complexity of stable curves; this bound is asymptotically sharp for $g\\gg 0$. Eventually, we state some conjectures on the behavior of stable curves with maximal complexity, and prove partial results in this direction."}
{"category": "Math", "title": "Integration with Functions of a Quaternionic Variable", "abstract": "Recent innovations in the differential calculus for functions of non-commuting variables, beginning with a quaternionic variable, are now extended to consider some integration."}
{"category": "Math", "title": "Smoothable del Pezzo surfaces with quotient singularities", "abstract": "We give a complete classification of del Pezzo surfaces with quotient singularities and Picard rank 1 which admit a Q-Gorenstein smoothing. There are 14 infinite families of toric examples. The surfaces in each family correspond to solutions of a Markov-type equation. The remaining surfaces are obtained as deformations of the toric surfaces or belong to a finite list of sporadic surfaces."}
{"category": "Math", "title": "On SYZ mirror transformations", "abstract": "In this expository paper, we discuss how Fourier-Mukai-type transformations, which we call SYZ mirror transformations, can be applied to provide a geometric understanding of the mirror symmetry phenomena for semi-flat Calabi-Yau manifolds and toric Fano manifolds. We also speculate the possible applications of these transformations to other more general settings."}
{"category": "Math", "title": "Spacings and pair correlations for finite Bernoulli convolutions", "abstract": "We consider finite Bernoulli convolutions with a parameter $1/2 < r < 1$ supported on a discrete point set, generically of size $2^N$. These sequences are uniformly distributed with respect to the infinite Bernoulli convolution measure $\\nu_r$, as $N$ tends to infinity. Numerical evidence suggests that for a generic $r$, the distribution of spacings between appropriately rescaled points is Poissonian. We obtain some partial results in this direction; for instance, we show that, on average, the pair correlations do not exhibit attraction or repulsion in the limit. On the other hand, for certain algebraic $r$ the behavior is totally different."}
{"category": "Math", "title": "Link invariants from finite Coxeter racks", "abstract": "We study Coxeter racks over $\\mathbb{Z}_n$ and the knot and link invariants they define. We exploit the module structure of these racks to enhance the rack counting invariants and give examples showing that these enhanced invariants are stronger than the unenhanced rack counting invariants."}
{"category": "Math", "title": "Homological Dimensions in Cotorsion Pairs", "abstract": "Two classes $\\mathcal A$ and $\\mathcal B$ of modules over a ring $R$ are said to form a cotorsion pair $(\\mathcal A, \\mathcal B)$ if $\\mathcal A={\\rm Ker Ext}^1_R(-,\\mathcal B)$ and $\\mathcal B={\\rm Ker Ext}^1_R(\\mathcal A,-)$. We investigate relative homological dimensions in cotorsion pairs. This can be applied to study the big and the little finitistic dimension of $R$. We show that $\\Findim R<\\infty$ if and only if the following dimensions are finite for some cotorsion pair $(\\mathcal A, \\mathcal B)$ in $\\mathrm{Mod} R$: the relative projective dimension of $\\A$ with respect to itself, and the $\\mathcal A$-resolution dimension of the category $\\mathcal P$ of all $R$-modules of finite projective dimension. Moreover, we obtain an analogous result for $\\findim R$, and we characterize when $\\Findim R=\\findim R.$"}
{"category": "Math", "title": "A Generalized Publication Bias Model", "abstract": "Scargle (2000) has discussed Rosenthal and Rubin's (1978) \"fail-safe number\" (FSN) method for estimating the number of unpublished studies in meta-analysis. He concluded that this FSN cannot possibly be correct because a central assumption the authors used conflicts with the very definition of publication bias. While this point has been made by others before (Elsahoff, 1978; Darlington, 1980; Thomas, 1985, Iyengar & Greenhouse, 1988), Scargle showed, by way of a simple 2-parameter model, how far off Rosenthal & Rubin' s estimate can be in practice. However, his results relied on the assumption that the decision variable is normally distributed with zero mean. In this case the ratio of unpublished to published papers is large only in a tiny region of the parameter plane. Building on these results, we now show that (1) Replacing densities with probability masses greatly simplifies Scargle's derivations and permits an explicit statement of the relation between the probability alpha of Type I errors and the step-size beta; (2) This result does not require any distribution assumptions; (3) The distinction between 1-sided and 2-sided rejection regions becomes immaterial; (4) This distribution-free approach leads to an immediate generalization to partitions involving more than two intervals, and thus covers more general selection functions."}
{"category": "Math", "title": "On the average growth exponent for beta-expansions", "abstract": "Let $\\be\\in(1,2)$. Each $x\\in I_\\be:=[0,\\frac{1}{\\be-1}]$ can be represented in the form \\[ x=\\sum_{k=1}^\\infty a_k\\be^{-k}, \\] where $a_k\\in\\{0,1\\}$ for all $k$ (a $\\be$-expansion of $x$). It was shown in \\cite{S} that a.e. $x\\in I_\\be$ has a continuum of distinct $\\be$-expansions. In this paper we show that for a generic $x$, this continuum has one and the same growth rate, i.e., the general $\\be$-expansions exhibit an ergodic behaviour. When $\\be<\\frac{1+\\sqrt5}2$, we show that the set of $\\be$-expansions grows exponentially for every $x\\in(0,\\frac{1}{\\be-1})$. Special attention is paid to the case $\\be=\\frac{1+\\sqrt5}2$, for which we explicitly compute the average growth exponent and apply this result to evaluating the local dimension of the corresponding Bernoulli convolution at a Lebesgue-generic $x$."}
{"category": "Math", "title": "The sharp energy-capacity inequality", "abstract": "Using the Oh-Schwarz spectral invariants and some arguments of Frauenfelder, Ginzburg, and Schlenk, we show that the \\pi_1-sensitive Hofer-Zehnder capacity of any subset of a closed symplectic manifold is less than or equal to its displacement energy. This estimate is sharp, and implies some new extensions of the Non-Squeezing Theorem."}
{"category": "Math", "title": "A Tale of Three Kernels", "abstract": "We consider the Bergman, Szeg\\H{o}, and Poisson Szeg\\H{o} kernels on a domain in $\\CC$ or $\\CC^n$. Various properties and relationships are established. The paper has both an expository component and a research component."}
{"category": "Math", "title": "Manin's conjecture for quartic del Pezzo surfaces with a conic fibration", "abstract": "An asymptotic formula is established for the number of rational points of bounded height on a non-singular quartic del Pezzo surface with a conic bundle structure."}
{"category": "Math", "title": "Continuity Points of Typical Bounded Functions", "abstract": "Kostyrko and Salat showed that if a linear space of bounded functions has an element that is discontinuous almost everywhere, then a typical element in the space is discontinuous almost everywhere. We give a topological analogue of this theorem and provide some examples."}
{"category": "Math", "title": "Coupled Painlev\\'e systems with affine Weyl group symmetry of types $A_7^{(2)},A_5^{(2)}$ and $D_4^{(3)}$", "abstract": "We find a four-parameter family of coupled Painlev\\'e VI systems in dimension four with affine Weyl group symmetry of type $A_7^{(2)}$. This is the first example which gave higher-order Painlev\\'e equations of type $A_{2l+5}^{(2)}$. We then give an explicit description of a confluence process from this system to a 3-parameter family of coupled Painlev\\'e V and III systems in dimension four with $W(A_5^{(2)})$-symmetry. For a degenerate system of $A_5^{(2)}$ system, we also find a two-parameter family of ordinary differential systems in dimension four with affine Weyl group symmetry of type $D_4^{(3)}$. This is the first example which gave higher-order Painlev\\'e equations of type $D_4^{(3)}$. We show that for each system, we give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new."}
{"category": "Math", "title": "The nonlinear Schr\\\"odinger equations with combined nonlinearities of power-type and Hartree", "abstract": "This paper is devoted to a comprehensive study of the nonlinear Schr\\\"odinger equations with combined nonlinearities of the power-type and Hartree-type in any dimension n\\ge3. With some structural conditions, a nearly whole picture of the interactions of these nonlinearities in the energy space is given. The method is based on the Morawetz estimates and perturbation principles."}
{"category": "Math", "title": "Quasi-commutative algebras", "abstract": "We characterise algebras commutative with respect to a Yang-Baxter operator (quasi-commutative algebras) in terms of certain cosimplicial complexes. In some cases this characterisation allows the classification of all possible quasi-commutative structures."}
{"category": "Math", "title": "Purity of level m stratifications", "abstract": "Let $k$ be a field of characteristic $p>0$. Let $D_m$ be a $\\BT_m$ over $k$ (i.e., an $m$-truncated Barsotti--Tate group over $k$). Let $S$ be a\\break $k$-scheme and let $X$ be a $\\BT_m$ over $S$. Let $S_{D_m}(X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\\ge 5$, we show that $S_{D_m}(X)$ is pure in $S$ i.e., the immersion $S_{D_m}(X) \\hookrightarrow S$ is affine. For $p\\in\\{2,3\\}$, we prove purity if $D_m$ satisfies a certain property depending only on its $p$-torsion $D_m[p]$. For $p\\ge 5$, we apply the developed techniques to show that all level $m$ stratifications associated to Shimura varieties of Hodge type are pure."}
{"category": "Math", "title": "Generalized Cheeger-Gromoll Metrics and the Hopf map", "abstract": "We show, using two different approaches, that there exists a family of Riemannian metrics on the tangent bundle of a two-sphere, which induces metrics of constant curvature on its unit tangent bundle. In other words, given such a metric on the tangent bundle of a two-sphere, the Hopf map is identified with a Riemannian submersion from the universal covering space of the unit tangent bundle onto the two-sphere. A hyperbolic counterpart dealing with the tangent bundle of a hyperbolic plane is also presented."}
{"category": "Math", "title": "Superfilters, Ramsey theory, and van der Waerden's Theorem", "abstract": "Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its variant for ultrafilters on the natural numbers. We use them to confirm a conjecture of Ko\\v{c}inac and Di Maio, which is a generalization of a Ramsey theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindman's 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman-Rado-Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers."}
{"category": "Math", "title": "How incomputable is the separable Hahn-Banach theorem?", "abstract": "We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak Konig's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function Sep and a natural notion of reducibility for multi-valued functions, we obtain a computational counterpart of the subsystem of second order arithmetic WKL_0. We study analogies and differences between WKL_0 and the class of Sep-computable multi-valued functions. Extending work of Brattka, we show that a natural multi-valued function associated with the Hahn-Banach Extension Theorem is Sep-complete."}
{"category": "Math", "title": "Combinatorics of labelling in higher dimensional automata", "abstract": "The main idea for interpreting concurrent processes as labelled precubical sets is that a given set of n actions running concurrently must be assembled to a labelled n-cube, in exactly one way. The main ingredient is the non-functorial construction called labelled directed coskeleton. It is defined as a subobject of the labelled coskeleton, the latter coinciding in the unlabelled case with the right adjoint to the truncation functor. This non-functorial construction is necessary since the labelled coskeleton functor of the category of labelled precubical sets does not fulfil the above requirement. We prove in this paper that it is possible to force the labelled coskeleton functor to be well-behaved by working with labelled transverse symmetric precubical sets. Moreover, we prove that this solution is the only one. A transverse symmetric precubical set is a precubical set equipped with symmetry maps and with a new kind of degeneracy map called transverse degeneracy. Finally, we also prove that the two settings are equivalent from a directed algebraic topological viewpoint. To illustrate, a new semantics of CCS, equivalent to the old one, is given."}
{"category": "Math", "title": "Extremal distributions for tail probabilities of sums of iid random variables on [0,1]", "abstract": "Two old conjectures from problem sections, one of which from SIAM Review, concern the question of finding distributions that maximize P(Sn <= t), where Sn is the sum of i.i.d. random variables X1, ..., Xn on the interval [0,1], satisfying E[X1]=m. In this paper a Lagrange multiplier technique is applied to this problem, yielding necessary conditions for distributions to be extremal, for arbitrary n. For n=2, a complete solution is derived from them: extremal distributions are discrete and have one of the following supports, depending on m and t: {0,t}, {t-1,1}, {t/2,1}, or {0,t,1}. These results suffice to refute both conjectures. However, acquired insight naturally leads to a revised conjecture: that extremal distributions always have at most three support points and belong to a (for each n, specified) finite collection of two and three point distributions."}
{"category": "Math", "title": "Regularisation and the Mullineux map", "abstract": "We classify the pairs of conjugate partitions whose regularisations are images of each other under the Mullineux map. This classification proves a conjecture of Lyle, answering a question of Bessenrodt, Olsson and Xu."}
{"category": "Math", "title": "On idempotent states on quantum groups", "abstract": "Idempotent states on a compact quantum group are shown to yield group-like projections in the multiplier algebra of the dual discrete quantum group. This allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the Haar state on a finite quantum hypergroup. A natural order structure on the set of idempotent states is also studied and some examples discussed."}
{"category": "Math", "title": "Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology", "abstract": "The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link."}
{"category": "Math", "title": "Describing all bi-orderings on Thompson's group F", "abstract": "We describe all possible ways of bi-ordering Thompson group F: its space of bi-orderings is made up of eight isolated points and four canonical copies of the Cantor set."}
{"category": "Math", "title": "Extended flux maps on surfaces and the contracted Johnson homomorphism", "abstract": "On a closed symplectic surface Sigma of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group Symp(Sigma) to the cohomology group H^1(Sigma;R) that extends the flux homomorphism). This construction uses the topology of the Jacobian of the surface and a correction factor related to the Johnson homomorphism. For surfaces of genus three or more, we give another new construction of an extended flux map using hyperbolic geometry."}
{"category": "Math", "title": "Conservative discrete time-invariant systems and block operator CMV matrices", "abstract": "It is well known that an operator-valued function $\\Theta$ from the Schur class ${\\bf S}(\\mathfrak M,\\mathfrak N)$, where $\\mathfrak M$ and $\\mathfrak N$ are separable Hilbert spaces, can be realized as the transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function $\\Theta$ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so called \"Schur parameters\". In this paper we construct simple conservative realizations using the Schur parameters only. It turns out that the unitary operators corresponding to the systems take the form of five diagonal block operator matrices, which are the analogs of Cantero--Moral--Vel\\'azquez (CMV) matrices appeared recently in the theory of scalar orthogonal polynomials on the unit circle. We obtain new models given by truncated block operator CMV matrices for an arbitrary completely non-unitary contraction. It is shown that the minimal unitary dilations of a contraction in a Hilbert space and the minimal Naimark dilations of a semi-spectral operator measure on the unit circle can also be expressed by means of block operator CMV matrices."}
{"category": "Math", "title": "Analytic continuation in mapping spaces", "abstract": "We consider a Stein manifold $M$ of dimension $\\geq 2$ and a compact subset $K\\subset M$ such that $M'=M\\backslash K$ is connected. Let $S$ be a compact differential manifold, and let $M_S$, resp. $M'_S$ stand for the complex manifold of maps $S\\to M$, resp. $S\\to M'$, of some specified regularity, that are homotopic to constant. We prove that any holomorphic function on $M'_S$ continues analytically to $M_S$ (perhaps as a multivalued function)."}
{"category": "Math", "title": "Loose Hamilton cycles in hypergraphs", "abstract": "We prove that any k-uniform hypergraph on n vertices with minimum degree at least n/(2(k-1))+o(n) contains a loose Hamilton cycle. The proof strategy is similar to that used by K\\\"uhn and Osthus for the 3-uniform case. Though some additional difficulties arise in the k-uniform case, our argument here is considerably simplified by applying the recent hypergraph blow-up lemma of Keevash."}
{"category": "Math", "title": "On the G_2 bundle of a Riemannian 4-manifold", "abstract": "We study the natural G_2 structure on the unit tangent sphere bundle SM of any given orientable Riemannian 4-manifold M, as it was discovered in \\cite{AlbSal}. A name is proposed for the space. We work in the context of metric connections, or so called geometry with torsion, and describe the components of the torsion of the connection which imply certain equations of the G_2 structure. This article is devoted to finding the G_2-torsion tensors which classify our structure according to the theory in \\cite{FerGray}."}
{"category": "Math", "title": "Coherent sheaves and cohesive sheaves", "abstract": "We consider coherent and cohesive sheaves of $\\cO$--modules over open sets $\\Omega\\subset\\bC^n$. We prove that coherent sheaves, and certain other sheaves derived from them, are cohesive; and conversely, certain sheaves derived from cohesive sheaves are coherent. An important tool in all this, also proved here, is that the sheaf of Banach space valued holomorphic germs is flat."}
{"category": "Math", "title": "Generalized Polar Decompositions for Closed Operators in Hilbert Spaces and Some Applications", "abstract": "We study generalized polar decompositions of densely defined, closed linear operators in Hilbert spaces and provide some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators."}
{"category": "Math", "title": "D-bar Spark Theory and Deligne Cohomology", "abstract": "We study the Harvey-Lawson spark characters of level p on complex manifolds. Presenting Deligne cohomology classes by sparks of level $p$, we give an explicit analytic product formula for Deligne cohomology. We also define refined Chern classes in Deligne cohomology for holomorphic vector bundles over complex manifolds. Applications to algebraic cycles are given. A Bott-type vanishing theorem in Deligne cohomology for holomorphic foliations is established. A general construction of Nadel-type invariants is given together with a new proof of Nadel's conjecture."}
{"category": "Math", "title": "Weakly turbulent solutions for the cubic defocusing nonlinear Schr\\\"odinger equation", "abstract": "We consider the cubic defocusing nonlinear Schr\\\"odinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This weakly turbulent behavior is quantified by the growth of higher Sobolev norms: given any delta << 1, K >>1, s >1, we construct smooth initial data u_0 with ||u_0||_{{H}^s} < delta, so that the corresponding time evolution u satisfies ||u(T)||_{{H}^s} > K at some time T. This growth occurs despite the Hamiltonian's bound on ||u(t)||_{\\dot{H}^1} and despite the conservation of the quantity ||u(t)||_{L^2}. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution's frequency support that simplifies the system of ODE's describing each Fourier mode's evolution. The second is a construction of solutions to these simpler systems of ODE's which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems."}
{"category": "Math", "title": "Central Simple Algebras with Involution: A Geometric Approach", "abstract": "Let $k$ be an algebraically closed base field of characteristic zero. The category equivalence between central simple algebras and irreducible, generically free $PGL_n$-varieties is extended to the context of central simple algebras with involution. The associated variety of a central simple algebra with involution comes with an action of the semidirect product $P_{n,\\tau}:=PGL_n\\rtimes<\\tau>$, where $\\tau$ is the automorphism of $\\PGLn$ given by $\\tau(h)=(h^{-1})^transpose$. Basic properties of an involution are described in terms of the action of $P_{n,\\tau}$ on the associated variety, and in particular in terms of the stabilizer in general position for this action."}
{"category": "Math", "title": "On the chain-level intersection pairing for PL pseudomanifolds", "abstract": "James McClure recently showed that the domain for the intersection pairing of PL chains on a PL manifold $M$ is a subcomplex of $C_*(M)\\otimes C_*(M)$ that is quasi-isomorphic to $C_*(M)\\otimes C_*(M)$ and, more generally, that the intersection pairing endows $C_*(M)$ with the structure of a partially-defined commutative DGA. We generalize this theorem to intersection pairings of PL intersection chains on PL stratified pseudomanifolds and demonstrate the existence of a partial restricted commutative DGA structure. This structure is shown to generalize the iteration of the Goresky-MacPherson intersection product. As an application, we construct an explicit \"roof\" representation of the intersection homology pairing in the derived category of sheaves and verify that this sheaf theoretic pairing agrees with that arising from the geometric Goresky-MacPherson intersection pairing."}
{"category": "Math", "title": "Intersection homology Kunneth theorems", "abstract": "Cohen, Goresky and Ji showed that there is a Kunneth theorem relating the intersection homology groups $I^{\\bar p}H_*(X\\times Y)$ to $I^{\\bar p}H_*(X)$ and $I^{\\bar p}H_*(Y)$, provided that the perversity $\\bar p$ satisfies rather strict conditions. We consider biperversities and prove that there is a K\\\"unneth theorem relating $I^{\\bar p,\\bar q}H_*(X\\times Y)$ to $I^{\\bar p}H_*(X)$ and $I^{\\bar q}H_*(Y)$ for all choices of $\\bar p$ and $\\bar q$. Furthermore, we prove that the Kunneth theorem still holds when the biperversity $p,q$ is \"loosened\" a little, and using this we recover the Kunneth theorem of Cohen-Goresky-Ji."}
{"category": "Math", "title": "On the K-theory of Toric Stack Bundles", "abstract": "Simplicial toric stack bundles are smooth Deligne-Mumford stacks over smooth varieties with fibre a toric Deligne-Mumford stack. We compute the Grothendieck $K$-theory of simplicial toric stack bundles and study the Chern character homomorphism."}
{"category": "Math", "title": "On Hecke Eigenvalues at Piatetski-Shapiro Primes", "abstract": "Let $\\lambda(n)$ be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant $C > 0$ depending on the cusp form and every fixed $c$ in the range $1 < c < 8/7$, the mean value of $\\lambda(p)$ is $\\ll \\exp (-C \\sqrt{\\log N})$ as p runs over all (Piatetski-Shapiro) primes of the form $[n^c]$ with a natural number $n \\leq N$."}
{"category": "Math", "title": "A relative version of Kummer theory", "abstract": "Let $E/F$ be a cyclic Galois extension of degree $p^l$ with Galois group $G$. It is shown that the Galois module structure of both sides of the Kummer pairing (for Kummer extensions of $E$) are the same. In other words, we show that the Kummer duality holds in the level of finitely generated $G$-modules."}
{"category": "Math", "title": "Injective and non-injective realizations with symmetry", "abstract": "In this paper, we introduce a natural classification of bar and joint frameworks that possess symmetry. This classification establishes the mathematical foundation for extending a variety of results in rigidity, as well as infinitesimal or static rigidity, to frameworks that are realized with certain symmetries and whose joints may or may not be embedded injectively in the space. In particular, we introduce a symmetry-adapted notion of `generic' frameworks with respect to this classification and show that `almost all' realizations in a given symmetry class are generic and all generic realizations in this class share the same infinitesimal rigidity properties. Within this classification we also clarify under what conditions group representation theory techniques can be applied to further analyze the rigidity properties of a (not necessarily injective) symmetric realization."}
{"category": "Math", "title": "Quantum Statistical Mechanics of $\\mathbb{Q}$-lattices and noncommutative geometry", "abstract": "After recalling some basic notions of quantum statistical mechanics, we explain the Bost-Connes system that relates the structure of the maximal abelian extension of $\\mathbb{Q}$ to the space of \\kms states of a \\cs-dynamical system. Afterwards, we study briefly the Connes-Marcolli $\\text{GL}_2$-system as a generalization of the former system."}
{"category": "Math", "title": "Nonpositively curved metric in the positive cone of a finite von Neumann algebra", "abstract": "In this paper we study the metric geometry of the space $\\Sigma$ of positive invertible elements of a von Neumann algebra ${\\mathcal A}$ with a finite, normal and faithful tracial state $\\tau$. The trace induces an incomplete Riemannian metric $<x,y>_a=\\tau (ya^{-1}xa^{-1})$, and though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the $n\\times n$ matrices when they are regarded as a symmetric space. For instance we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an intrinsic (algebraic) characterization of the geodesically convex submanifolds $M$ of $\\Sigma$, and under suitable hypothesis we prove a factorization theorem for elements in the algebra that resembles the Iwasawa decomposition for matrices. This factorization is obtained \\textit{via} a nonlinear orthogonal projection $\\Pi_M:\\Sigma\\to M$, a map which turns out to be contractive for the geodesic distance."}
{"category": "Math", "title": "Indecomposable $PD_3$-complexes", "abstract": "We show that if $X$ is an indecomposable $PD_3$-complex and $\\pi_1(X) is the fundamental group of a reduced finite graph of finite groups but is not virtually cyclic then $X$ is orientable, the underlying graph is a tree, all the edge groups are $Z/2Z$ and all but at most one of the vertex groups is dihedral of order $2m$ with $m$ odd. Every such group is realized by some $PD_3$-complex. We also propose a strategy for tackling the question of whether every $PD_3$-complex has a finite covering space which is homotopy equivalent to a closed orientable 3-manifold."}
{"category": "Math", "title": "Some extensions of the class of convex bodies", "abstract": "We introduce and study a new class of $\\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, and find applications to geometric tomography. We introduce the notion of a circular projection and investigate the problem of determination of $\\eps$-convex bodies by their projection-type images. The results generalize corresponding stability theorems by H. Groemer."}
{"category": "Math", "title": "On the number of allelic types for samples taken from exchangeable coalescents with mutation", "abstract": "Let $K_n$ denote the number of types of a sample of size $n$ taken from an exchangeable coalescent process ($\\Xi$-coalescent) with mutation. A distributional recursion for the sequence $(K_n)_{n\\in{\\mathbb N}}$ is derived. If the coalescent does not have proper frequencies, i.e., if the characterizing measure $\\Xi$ on the infinite simplex $\\Delta$ does not have mass at zero and satisfies $\\int_\\Delta |x|\\Xi(dx)/(x,x)<\\infty$, where $|x|:=\\sum_{i=1}^\\infty x_i$ and $(x,x):=\\sum_{i=1}^\\infty x_i^2$ for $x=(x_1,x_2,...)\\in\\Delta$, then $K_n/n$ converges weakly as $n\\to\\infty$ to a limiting variable $K$ which is characterized by an exponential integral of the subordinator associated with the coalescent process. For so-called simple measures $\\Xi$ satisfying $\\int_\\Delta\\Xi(dx)/(x,x)<\\infty$ we characterize the distribution of $K$ via a fixed-point equation."}
{"category": "Math", "title": "A classification of smooth embeddings of 4-manifolds in 7-space, II", "abstract": "Let N be a closed, connected, smooth 4-manifold with H_1(N;Z)=0. Our main result is the following classification of the set E^7(N) of smooth embeddings N->R^7 up to smooth isotopy. Haefliger proved that the set E^7(S^4) with the connected sum operation is a group isomorphic to Z_{12}. This group acts on E^7(N) by embedded connected sum. Boechat and Haefliger constructed an invariant BH:E^7(N)->H_2(N;Z) which is injective on the orbit space of this action; they also described im(BH). We determine the orbits of the action: for u in im(BH) the number of elements in BH^{-1}(u) is GCD(u/2,12) if u is divisible by 2, or is GCD(u,3) if u is not divisible by 2. The proof is based on a new approach using modified surgery as developed by Kreck."}
{"category": "Math", "title": "Conflations of Probability Distributions", "abstract": "The conflation of a finite number of probability distributions P_1,..., P_n is a consolidation of those distributions into a single probability distribution Q=Q(P_1,..., P_n), where intuitively Q is the conditional distribution of independent random variables X_1,..., X_n with distributions P_1,..., P_n, respectively, given that X_1= ... =X_n. Thus, in large classes of distributions the conflation is the distribution determined by the normalized product of the probability density or probability mass functions. Q is shown to be the unique probability distribution that minimizes the loss of Shannon Information in consolidating the combined information from P_1,..., P_n into a single distribution Q, and also to be the optimal consolidation of the distributions with respect to two minimax likelihood-ratio criteria. When P_1,..., P_n are Gaussian, Q is Gaussian with mean the classical weighted-mean-squares reciprocal of variances. A version of the classical convolution theorem holds for conflations of a large class of a.c. measures."}
{"category": "Math", "title": "Shellability and the strong gcd-condition", "abstract": "Shellability is a well-known combinatorial criterion for verifying that a simplicial complex is Cohen-Macaulay. Another notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, a criterion on simplicial complexes reminiscent of shellability, called the strong gcd-condition, was shown to imply Golodness of the associated Stanley-Reisner ring. The two algebraic notions were tied together by Herzog, Reiner and Welker who showed that if the Alexander dual of a complex is sequentially Cohen-Macaulay then the complex itself is Golod. In this paper, we present a combinatorial companion of this result, namely that if the Alexander dual of a complex is (non-pure) shellable then the complex itself satisfies the strong gcd-condition. Moreover, we show that all implications just mentioned are strict in general but that they are equivalences for flag complexes."}
{"category": "Math", "title": "Spherical conjugacy classes and Bruhat decomposition", "abstract": "Let G be a connected, reductive algebraic group over an algebraically closed field of characteristic zero or good and odd. We characterize the spherical conjugacy classes of G as those intersecting only Bruhat cells corresponding to involutions in the Weyl group of G."}
{"category": "Math", "title": "Sheaves of ordered spaces and interval theories", "abstract": "We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category turns out to be a certain full subcategory of a topos of sheaves over a simpler site. A precise characterisation of this subcategory is provided. The ambient topos makes available some general homotopical machinery."}
{"category": "Math", "title": "Relations between the leading terms of a polynomial automorphism", "abstract": "Let $I$ be the ideal of relations between the leading terms of the polynomials defining an automorphism of $K^n$. In this paper, we prove the existence of a locally nilpotent derivation which preserves $I$. Moreover, if $I$ is principal, i.e. $I=(R)$, we compute an upper bound for $\\deg_2(R)$ for some degree function $\\deg_2$ defined by the automorphism. As applications, we determine all the principal ideals of relations for automorphisms of $K^3$ and deduce two elementary proofs of the Jung-van der Kulk Theorem about the tameness of automorphisms of $K^{2}$."}
{"category": "Math", "title": "Fourier analysis, linear programming, and densities of distance avoiding sets in R^n", "abstract": "In this paper we derive new upper bounds for the densities of measurable sets in R^n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2,..., 24. This gives new lower bounds for the measurable chromatic number in dimensions 3,..., 24. We apply it to get a new, short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss and Bourgain and Falconer about sets avoiding many distances."}
{"category": "Math", "title": "Subgroups of free idempotent generated semigroups need not be free", "abstract": "We use topological methods to study the maximal subgroups of the free idempotent generated semigroup on a biordered set. We use these to give an example of a free idempotent generated semigroup with maximal subgroup isomorphic to the free abelian group of rank 2. This is the first example of a non-free subgroup of a free idempotent generated semigroup."}
{"category": "Math", "title": "Note on the Euler Numbers and Polynomials", "abstract": "In this paper we investigate the properties of the Euler functions. By using the Fourier transform for the Euler function, we derive the interesting formula related to the infinite series. Finally we give some interesting identities between the Euler numbers and the second kind stirling numbers."}
{"category": "Math", "title": "Fibered nonlinearities for $p(x)$-Laplace equations", "abstract": "In $\\R^m\\times\\R^{n-m}$, endowed with coordinates $X=(x,y)$, we consider the PDE $$ -{\\rm div} \\big(\\alpha(\\x) |\\nabla u(\\X)|^{p(x)-2}\\nabla u(\\X)\\big)=f(x,u(\\X)).$$ We prove a geometric inequality and a symmetry result."}
{"category": "Math", "title": "Primitive Collections and Toric Varieties", "abstract": "This paper studies Batyrev's notion of primitive collection. We use primitive collections to characterize the nef cone of a quasi-projective toric variety whose fan has convex support, a result stated without proof by Batyrev in the smooth projective case. When the fan is non-simplicial, we modify the definition of primitive collection and explain how our definition relates to primitive collections of simplicial subdivisons. The paper ends with some open problems."}
{"category": "Math", "title": "Intrinsic geometry of oriented congruences in three dimensions", "abstract": "Starting from the classical notion of an oriented congruence (i.e. a foliation by oriented curves) in $R^3$, we abstract the notion of an oriented congruence structure. This is a 3-dimensional CR manifold $(M,H, J)$ with a preferred splitting of the tangent space $TM=V\\oplus H$. We find all local invariants of such structures using Cartan's equivalence method refining Cartan's classification of 3-dimensional CR structures. We use these invariants and perform Fefferman like constructions, to obtain interesting Lorentzian metrics in four dimensions, which include explicit Ricci-flat and Einstein metrics, as well as not conformally Einstein Bach-flat metrics."}
{"category": "Math", "title": "Questions about determinants and polynomials", "abstract": "We discuss several conjectures about the real-rootedness of polynomials whose coefficients are determinants of coefficients of a real-rooted polynomial. We also consider some questions about matrices generalizing totally positive matrices, namely totally stable matrices and totally upper matrices."}
{"category": "Math", "title": "Weak amenability of Fourier algebras on compact groups", "abstract": "We give for a compact group G, a full characterisation of when its Fourier algebra A(G) is weakly amenable: when the connected component of the identity G_e is abelian. This condition is also equivalent to the hyper-Tauberian property for A(G), and to having the anti-diagonal D^v={(s,s^{-1}):s is in G} being a set of spectral synthesis for A(GXG). We show the relationship between amenability and weak amenability of A(G), and (operator) amenability and (operator) weak amenability of A_D(G), an algebra defined by the authors in arXiv:0705.4277. We close by extending our results to some classes of non-compact, locally compact groups, including small invariant neighbourhood groups and maximally weakly almost periodic groups."}
{"category": "Math", "title": "Varieties with Definable Factor Congruences", "abstract": "We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are \"definable\" in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0&1 and Boolean Factor Congruences. We also obtain an explicit first order definition of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner it is preserved by taking direct products and direct factors. The main tool is the use of \"central elements,\" which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice."}
{"category": "Math", "title": "Lower Bounds for Dimensions of Sums of Sets", "abstract": "We study lower bounds for the Minkowski and Hausdorff dimensions of the algebraic sum E+K of two subsets E and K of d-dimensional Euclidean space."}
{"category": "Math", "title": "Squishing dimers on the hexagon lattice", "abstract": "We describe an operation on dimer configurations on the hexagon lattice, called \"squishing\", and use this operation to explain some of the properties of dimer generating functions."}
{"category": "Math", "title": "Weighted interlace polynomials", "abstract": "The interlace polynomials introduced by Arratia, Bollobas and Sorkin extend to invariants of graphs with vertex weights, and these weighted interlace polynomials have several novel properties. One novel property is a version of the fundamental three-term formula q(G)=q(G-a)+q(G^{ab}-b)+((x-1)^{2}-1)q(G^{ab}-a-b) that lacks the last term. It follows that interlace polynomial computations can be represented by binary trees rather than mixed binary-ternary trees. Binary computation trees provide a description of $q(G)$ that is analogous to the activities description of the Tutte polynomial. If $G$ is a tree or forest then these \"algorithmic activities\" are associated with a certain kind of independent set in $G$. Three other novel properties are weighted pendant-twin reductions, which involve removing certain kinds of vertices from a graph and adjusting the weights of the remaining vertices in such a way that the interlace polynomials are unchanged. These reductions allow for smaller computation trees as they eliminate some branches. If a graph can be completely analyzed using pendant-twin reductions then its interlace polynomial can be calculated in polynomial time. An intuitively pleasing property is that graphs which can be constructed through graph substitutions have vertex-weighted interlace polynomials which can be obtained through algebraic substitutions."}
{"category": "Math", "title": "Turbulence, representations, and trace-preserving actions", "abstract": "We establish criteria for turbulence in certain spaces of C*-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X,\\mu) and on the hyperfinite II_1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving flows on (X,\\mu) is generically turbulent."}
{"category": "Math", "title": "The number of 2x2 integer matrices having a prescribed integer eigenvalue", "abstract": "Random matrices arise in many mathematical contexts, and it is natural to ask about the properties that such matrices satisfy. If we choose a matrix with integer entries at random, for example, what is the probability that it will have a particular integer as an eigenvalue, or an integer eigenvalue at all? If we choose a matrix with real entries at random, what is the probability that it will have a real eigenvalue in a particular interval? The purpose of this paper is to resolve these questions, once they are made suitably precise, in the setting of 2x2 matrices."}
{"category": "Math", "title": "Connections on non-abelian Gerbes and their Holonomy", "abstract": "We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the role of a band, or structure 2-group. Upon choosing certain examples of Lie 2-groups, our axiomatic framework reproduces in a systematical way several known concepts of gerbes with connection: non-abelian differential cocycles, Breen-Messing gerbes, abelian and non-abelian bundle gerbes. These relationships convey a well-defined notion of surface holonomy from our axiomatic framework to each of these concrete models. Till now, holonomy was only known for abelian gerbes; our approach reproduces that known concept and extends it to non-abelian gerbes. Several new features of surface holonomy are exposed under its extension to non-abelian gerbes; for example, it carries an action of the mapping class group of the surface."}
{"category": "Math", "title": "Existence of minimal models for varieties of log general type II", "abstract": "We prove the existence of pl-flips."}
{"category": "Math", "title": "New coins from old, smoothly", "abstract": "Given a (known) function $f:[0,1] \\to (0,1)$, we consider the problem of simulating a coin with probability of heads $f(p)$ by tossing a coin with unknown heads probability $p$, as well as a fair coin, $N$ times each, where $N$ may be random. The work of Keane and O'Brien (1994) implies that such a simulation scheme with the probability $\\P_p(N<\\infty)$ equal to 1 exists iff $f$ is continuous. Nacu and Peres (2005) proved that $f$ is real analytic in an open set $S \\subset (0,1)$ iff such a simulation scheme exists with the probability $\\P_p(N>n)$ decaying exponentially in $n$ for every $p \\in S$. We prove that for $\\alpha>0$ non-integer, $f$ is in the space $C^\\alpha [0,1]$ if and only if a simulation scheme as above exists with $\\P_p(N>n) \\le C (\\Delta_n(p))^\\alpha$, where $\\Delta_n(x)\\eqbd \\max \\{\\sqrt{x(1-x)/n},1/n \\}$. The key to the proof is a new result in approximation theory: Let $\\B_n$ be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree $n$. We show that a function $f:[0,1] \\to (0,1)$ is in $C^\\alpha [0,1]$ if and only if $f$ has a series representation $\\sum_{n=1}^\\infty F_n$ with $F_n \\in \\B_n$ and $\\sum_{k>n} F_k(x) \\le C(\\Delta_n(x))^\\alpha$ for all $ x \\in [0,1]$ and $n \\ge 1$. We also provide a counterexample to a theorem stated without proof by Lorentz (1963), who claimed that if some $\\phi_n \\in \\B_n$ satisfy $|f(x)-\\phi_n(x)| \\le C (\\Delta_n(x))^\\alpha$ for all $ x \\in [0,1]$ and $n \\ge 1$, then $f \\in C^\\alpha [0,1]$."}
{"category": "Math", "title": "Action integrals and infinitesimal characters", "abstract": "Let $G$ be a reductive Lie group and ${\\mathcal O}$ the coadjoint orbit of a hyperbolic element of ${\\frak g}^*$. By $\\pi$ is denoted the unitary irreducible representation of $G$ associated with ${\\mathcal O}$ by the orbit method. We give geometric interpretations in terms of concepts related to ${\\mathcal O}$ of the constant $\\pi(g)$, for $g\\in Z(G)$. We also offer a description of the invariant $\\pi(g)$ in terms of action integrals and Berry phases. In the spirit of the orbit method we interpret geometrically the infinitesimal character of the differential representation of $\\pi$."}
{"category": "Math", "title": "Nonstandard Mathematics and New Zeta and L-Functions", "abstract": "This Ph.D. thesis, prepared under the supervision of Professor Ivan Fesenko, defines new zeta functions in a nonstandard setting and their analytical properties are developed. Further, p-adic interpolation is presented within a nonstandard setting which enables the concept of interpolating with respect to two, or more, distinct primes to be defined. The final part of the dissertation examines the work of M. J. Shai Haran and makes initial attempts of viewing it from a nonstandard perspective."}
{"category": "Math", "title": "De Morgan's law and the theory of fields", "abstract": "We show that the classifying topos for the theory of fields does not satisfy De Morgan's law, and we identify its largest dense De Morgan subtopos as the classifying topos for the theory of fields of nonzero characteristic which are algebraic over their prime fields."}
{"category": "Math", "title": "Calculating intersection numbers on moduli spaces of pointed curves", "abstract": "This is a note on calculating intersection numbers on moduli spaces of curves. A codimension 3 relation among tautological classes on the moduli space of genus 4 curves is given."}
{"category": "Math", "title": "Conformally Invariant Operators via Curved Casimirs: Examples", "abstract": "We discuss a general scheme for a construction of linear conformally invariant differential operators from curved Casimir operators; we then explicitly carry this out for several examples. Apart from demonstrating the efficacy of the approach via curved Casimirs, this shows that this method applies both in regular and in singular infinitesimal character, and also that it can be used to construct standard as well as non--standard operators. The examples treated include conformally invariant operators with leading term, in one case, a square of the Laplacian, and in another case, a cube of the Laplacian."}
{"category": "Math", "title": "d-collapsibility is NP-complete for d greater or equal to 4", "abstract": "A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing (collapsing) a face of dimension at most d-1 that is contained in a unique maximal face. We prove that the algorithmic question whether a given simplicial complex is d-collapsible is NP-complete for d greater or equal to 4 and polynomial time solvable for d at most 2. As an intermediate step, we prove that d-collapsibility can be recognized by the greedy algorithm for d at most 2, but the greedy algorithm does not work for d greater or equal 3. A simplicial complex is d-representable if it is the nerve of a collection of convex sets in R^d. The main motivation for studying d-collapsible complexes is that every d-representable complex is d-collapsible. We also observe that known results imply that analogical algorithmic question for d-representable complexes is NP-hard for d greater or equal to 2."}
{"category": "Math", "title": "On visualisation scaling, subeigenvectors and Kleene stars in max algebra", "abstract": "The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is strict visualisation scaling, which means finding, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X^{-1}AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max-algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry."}
{"category": "Math", "title": "Torelli theorem for the moduli space of framed bundles", "abstract": "Let X be an irreducible smooth complex projective curve of genus g>2, and let x be a fixed point. A framed bundle is a pair (E,\\phi), where E is a vector bundle over X, of rank r and degree d, and \\phi:E_x\\to C^r is a non-zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter \\tau>0, which gives rise to the moduli space of \\tau-semistable framed bundles M^\\tau. We prove a Torelli theorem for M^\\tau, for \\tau>0 small enough, meaning, the isomorphism class of the one-pointed curve (X,x), and also the integer r, are uniquely determined by the isomorphism class of the variety M^\\tau."}
{"category": "Math", "title": "Reconsidering the asymptotic null distribution of likelihood ratio tests for genetic linkage in multivariate variance components models", "abstract": "Accurate knowledge of the null distribution of hypothesis tests is important for valid application of the tests. In previous papers and software, the asymptotic null distribution of likelihood ratio tests for detecting genetic linkage in multivariate variance components models has been stated to be a mixture of chi-square distributions with binomial mixing probabilities. Here we show, by simulation and by theoretical arguments based on the geometry of the parameter space, that all aspects of the previously stated asymptotic null distribution are incorrect--both the binomial mixing probabilities and the chi-square components. Correcting the null distribution gives more conservative critical values than previously stated, yielding P values that can easily be ten times larger. The true mixing probabilities give the highest probability to the case where all variance parameters are estimated positive, and the mixing components show severe departures from chi-square distributions. Thus, the asymptotic null distribution has complex features that raise challenges for the assessment of significance of multivariate linkage findings. We propose a method to generate an asymptotic null distribution that is much faster than other empirical methods such as gene-dropping, enabling us to obtain P values with higher precision more efficiently."}
{"category": "Math", "title": "An application of Jacquet-Langlands correspondence to transfer operators for geodesic flows on Riemann surfaces", "abstract": "In the paper as a new application of the Jacquet-Langlands correspondence we connect the transfer operators for different cofinite Fuchsian groups by comparing the corresponding Selberg zeta functions."}
{"category": "Math", "title": "The B\\\"{a}cklund transforms of Peterson's deformations of quadrics", "abstract": "In trying to provide explicit deformations of quadrics the starting point of our investigation is to use Bianchi's link between real deformations of totally real regions of real paraboloids and various totally real forms of the sine-Gordon equation coupled with Bianchi's simple observation that the vacuum soliton of these totally real forms of the sine-Gordon equation provides precisely Peterson's deformations of such quadrics in order to derive explicit B\\\"{a}cklund transforms of Peterson's deformations of quadrics. Based also on Bianchi's approach of the B\\\"{a}cklund transformation for quadrics via common conjugate systems and in analogy to the solitons of the sine-Gordon equation corresponding at the level of the geometric picture to the solitons of the pseudo-sphere we propose a model for the solitons of quadrics."}
{"category": "Math", "title": "On secant loci and simple linear projections of some projective varieties", "abstract": "In this paper, we study how simple linear projections of some projective varieties behave when the projection center runs through the ambient space. More precisely, let $X \\subset \\P^r$ be a projective variety satisfying Green-Lazarsfeld's property $N_p$ for some $p \\geq 2$, $q \\in \\P^r$ a closed point outside of $X$, and $X_q := \\pi_q (X) \\subset \\P^{r-1}$ the projected image of $X$ from $q$. First, it is shown that the secant locus $\\Sigma_q (X)$ of $X$ with respect to $q$, i.e. the set of all points on $X$ spanning secant lines passing through $q$, is either empty or a quadric in a subspace of $\\P^r$. This implies that the finite morphism $\\pi_q : X \\to X_q$ is birational. Our main result is that cohomological and local properties of $X_q$ are precisely determined by $\\Sigma_q (X)$. To complete this result, the next step should be to classify all possible secant loci and to decompose the ambient space via the classification of secant loci. We obtain such a decomposition for Veronese embeddings and Segre embeddings. Also as an application of the main result, we study cohomological properties of low degree varieties."}
{"category": "Math", "title": "Bianchi's B\\\"{a}cklund transformation for higher dimensional quadrics", "abstract": "We provide a generalization of Bianchi's B\\\"{a}cklund transformation from 2-dimensional quadrics to higher dimensional quadrics. The starting point of our investigation is the higher dimensional (infinitesimal) version of Bianchi's main four theorems on the theory of deformations of quadrics and Bianchi's treatment of the B\\\"{a}cklund transformation for diagonal paraboloids via conjugate systems."}
{"category": "Math", "title": "Stably diffeomorphic manifolds and l_{2q+1}(Z[\\pi])", "abstract": "The monoids l_{2q+1}(Z[\\pi]) detect s-cobordisms amongst certain bordisms between stably diffeomorphic 2q-dimensional manifolds and generalise the Wall simple surgery obstruction groups, L_{2q+1}^s(Z[\\pi]) \\subset l_{2q+1}(Z[\\pi]). In this paper we give exact sequences which completely describe l_{2q+1}(Z[\\pi]) as a set and which we use to compute its Grothendieck group. As a consequence we deduce cancellation results for stably diffeomorphic manifolds with polycyclic-by-finite fundamental group."}
{"category": "Math", "title": "Perfect, strongly eutactic lattices are periodic extreme", "abstract": "We introduce a parameter space for periodic point sets, given as unions of $m$ translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension $d\\leq 8$ and $d=24$."}
{"category": "Math", "title": "Scattering configuration spaces", "abstract": "For a compact manifold with boundary $X$ we introduce the $n$-fold scattering stretched product $X^n_{\\text{sc}}$ which is a compact manifold with corners for each $n,$ coinciding with the previously known cases for $n=2,3.$ It is constructed by iterated blow up of boundary faces and boundary faces of multi-diagonals in $X^n.$ The resulting space is shown to map smoothly, by a b-fibration, covering the usual projection, to the lower stretched products. It is anticipated that this manifold with corners, or at least its combinatorial structure, is a universal model for phenomena on asymptotically flat manifolds in which particle clusters emerge at infinity. In particular this is the case for magnetic monopoles on $\\mathbb{R}^3$ in which case these spaces are closely related to compactifications of the moduli spaces with the boundary faces mapping to lower charge idealized moduli spaces."}
{"category": "Math", "title": "Regular induced subgraphs of a random graph", "abstract": "An old problem of Erd\\H{o}s, Fajtlowicz and Staton asks for the order of a largest induced regular subgraph that can be found in every graph on n vertices. Motivated by this problem, we consider the order of such a subgraph in a typical graph on n vertices, i.e., in a binomial random graph G(n,1/2). We prove that with high probability a largest induced regular subgraph of G(n,1/2) has about n^{2/3} vertices."}
{"category": "Math", "title": "On asymptotic stability of standing waves of discrete Schr\\\"odinger equation in $\\Bbb Z$", "abstract": "We prove an analogue of a classical asymptotic stability result of standing waves of the Schr\\\"odinger equation originating in work by Soffer and Weinstein. Specifically, our result is a transposition on the lattice Z of a result by Mizumachi and it involves a discrete Schr\\\"odinger operator H. The decay rates on the potential are less stringent than in Mizumachi, since we require for the potential $q\\in \\ell ^{1,1}$. We also prove $|e^{itH}(n,m)|\\le C < t > ^{-1/3}$ for a fixed $C$ requiring, in analogy to Goldberg and Schlag only $q\\in \\ell ^{1,1}$ if $H$ has no resonances and $q\\in \\ell ^{1,2}$ if it has resonances. In this way we ease the hypotheses on H contained in Pelinovsky and Stefanov, which have a similar dispersion estimate."}
{"category": "Math", "title": "Efficient computation of resonance varieties via Grassmannians", "abstract": "Associated to the cohomology ring A of the complement X(A) of a hyperplane arrangement A in complex m-space are the resonance varieties R^k(A). The most studied of these is R^1(A), which is the union of the tangent cones at the origin to the characteristic varieties of the fundamental group of X. R^1(A) may be described in terms of Fitting ideals, or as the locus where a certain Ext module is supported. Both these descriptions give obvious algorithms for computation. In this note, we show that interpreting R^1(A) as the locus of decomposable two-tensors in the Orlik-Solomon ideal leads to a description of R^1(A) as the intersection of a Grassmannian with a linear space, determined by the quadratic generators of the Orlik-Solomon ideal. This method is much faster than previous alternatives."}
{"category": "Math", "title": "Eigenvalues Estimates for the p-Laplace Operator on Manifolds", "abstract": "We obtain geometric estimates for the first eigenvalue and the fundamental tone of the p-laplacian on manifolds in terms of admissible vector fields. Also, we defined a new spectral invariant and we show its relation with the geometry of the manifold."}
{"category": "Math", "title": "Piecewise polynomials on polyhedral complexes", "abstract": "For a d-dimensional polyhedral complex P, the dimension of the space of piecewise polynomial functions (splines) on P of smoothness r and degree k is given, for k sufficiently large, by a polynomial f(P,r,k) of degree d. When d=2 and P is simplicial, Alfeld and Schumaker determined a formula for all three coefficients of f. However, in the polyhedral case, no formula is known. Using localization techniques and specialized dual graphs associated to codimension--2 linear spaces, we obtain the first three coefficients of f(P,r,k), giving a complete answer when d=2."}
{"category": "Math", "title": "Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras", "abstract": "We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki's categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial Z-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic."}
{"category": "Math", "title": "Inductive Analysis on Singular Minimal Hypersurfaces", "abstract": "The present paper describes a way to relate Martin boundaries on spaces of varying topology. This enables us to approach some detailed inductive analysis of the eigenfunctions of conformal Laplacians on minimal hypersurfaces near their singularities. This can directly be used resp. translated to understand the way how such minimal hypersurfaces inherit positive scalar curvature from their ambience resp. how to smooth singular minimal hypersurfaces to regular hypersurfaces with positive mean curvature."}
{"category": "Math", "title": "Weak* continuous states on Banach algebras", "abstract": "We prove that if a unital Banach algebra $A$ is the dual of a Banach space $\\pd{A}$, then the set of weak* continuous states is weak* dense in the set of all states on $A$. Further, weak* continuous states linearly span $\\pd{A}$."}
{"category": "Math", "title": "Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme", "abstract": "We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere's tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here \"equatorial periodic solutions\", analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct \"confined solutions\", which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test-cases are presented."}
{"category": "Math", "title": "Almost filling laminations and the connectivity of ending lamination space", "abstract": "We show that if S is a finite type orientable surface of negative Euler characteristic which is not the 3-holed sphere, 4-holed sphere or 1-holed torus, then the ending lamination space of S is connected, locally path connected and cyclic."}
{"category": "Math", "title": "Abelian ideals with given dimension in Borel subalgebras", "abstract": "A well-known Peterson's theorem says that the number of abelian ideals in a Borel subalgebra of a rank-$r$ finite dimensional simple Lie algebra is exactly $2^r$. In this paper, we determine the dimensional distribution of abelian ideals in a Borel subalgebra of finite dimensional simple Lie algebras, which is a refinement of the Peterson's theorem capturing more Lie algebra invariants."}
{"category": "Math", "title": "Connectivity Properties of Horospheres in Euclidean Buildings and Applications to Finiteness Properties of Discrete Groups", "abstract": "Let G(O_S) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O_S) established in an earlier paper is sharp in this case. The geometric analysis underlying our result determines the conectivity properties of horospheres in thick Euclidean buildings."}
{"category": "Math", "title": "Density estimates and concentration inequalities with Malliavin calculus", "abstract": "We show how to use the Malliavin calculus to obtain density estimates of the law of general centered random variables. In particular, under a non-degeneracy condition, we prove and use a new formula for the density of a random variable which is measurable and differentiable with respect to a given isonormal Gaussian process. Among other results, we apply our techniques to bound the density of the maximum of a general Gaussian process from above and below; several new results ensue, including improvements on the so-called Borell-Sudakov inequality. We then explain what can be done when one is only interested in or capable of deriving concentration inequalities, i.e. tail bounds from above or below but not necessarily both simultaneously."}
{"category": "Math", "title": "All exceptional surgeries on alternating knots are integral surgeries", "abstract": "We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries."}
{"category": "Math", "title": "The flipping puzzle on a graph", "abstract": "Let $S$ be a connected graph which contains an induced path of $n-1$ vertices, where $n$ is the order of $S.$ We consider a puzzle on $S$. A configuration of the puzzle is simply an $n$-dimensional column vector over $\\{0, 1\\}$ with coordinates of the vector indexed by the vertex set $S$. For each configuration $u$ with a coordinate $u_s=1$, there exists a move that sends $u$ to the new configuration which flips the entries of the coordinates adjacent to $s$ in $u.$ We completely determine if one configuration can move to another in a sequence of finite steps."}
{"category": "Math", "title": "The Maskit embedding of the twice punctured torus", "abstract": "The Maskit embedding M of a surface \\Sigma is the space of geometrically finite groups on the boundary of quasifuchsian space for which the `top' end is homeomorphic to \\Sigma, while the `bottom' end consists of two triply punctured spheres, the remains of \\Sigma when two fixed disjoint curves have been pinched. As such representations vary in the character variety, the conformal structure on the top side varies over the Teichm\\\"uller space T(\\Sigma). We investigate M when \\Sigma is a twice punctured torus, using the method of pleating rays. Fix a projective measure class [\\mu] supported on closed curves on \\Sigma. The pleating ray P_[\\mu] consists of those groups in M for which the bending measure of the top component of the convex hull boundary of the associated 3-manifold is in [\\mu]. It is known that P is a real 1-submanifold of M. Our main result is a formula for the asymptotic direction of P in M as the bending measure tends to zero, in terms of natural parameters for the 2-complex dimensional representation space R and the Dehn-Thurston coordinates of the support curves to [\\mu] relative to the pinched curves on the bottom side. This leads to a method of locating M in R."}
{"category": "Math", "title": "Adjoint vector fields and differential operators on representation spaces", "abstract": "Let $G$ be a semisimple algebraic group with Lie algebra $\\g$. In 1979, J. Dixmier proved that any vector field annihilating all $G$-invariant polynomials on $\\g$ lies in the $\\bbk[\\g]$-module generated by the \"adjoint vector fields\", i.e., vector fields $\\varsigma$ of the form $\\varsigma(y)(x)=[x,y]$, $x,y\\in\\g$. A substantial generalisation of Dixmier's theorem was found by Levasseur and Stafford. They explicitly described the centraliser of $\\bbk[\\g]^G$ in the algebra of differential operators on $\\g$. On the level of vector fields, their result reduces to Dixmier's theorem. The purpose of this paper is to explore similar problems in the general context of affine algebraic groups and their rational representations."}
{"category": "Math", "title": "Two-dimensional models of type theory", "abstract": "We describe a non-extensional variant of Martin-L\\\"of type theory which we call two-dimensional type theory, and equip it with a sound and complete semantics valued in 2-categories."}
{"category": "Math", "title": "High-dimensional Gaussian model selection on a Gaussian design", "abstract": "We consider the problem of estimating the conditional mean of a real Gaussian variable $\\nolinebreak Y=\\sum_{i=1}^p\\nolinebreak\\theta_iX_i+\\nolinebreak \\epsilon$ where the vector of the covariates $(X_i)_{1\\leq i\\leq p}$ follows a joint Gaussian distribution. This issue often occurs when one aims at estimating the graph or the distribution of a Gaussian graphical model. We introduce a general model selection procedure which is based on the minimization of a penalized least-squares type criterion. It handles a variety of problems such as ordered and complete variable selection, allows to incorporate some prior knowledge on the model and applies when the number of covariates $p$ is larger than the number of observations $n$. Moreover, it is shown to achieve a non-asymptotic oracle inequality independently of the correlation structure of the covariates. We also exhibit various minimax rates of estimation in the considered framework and hence derive adaptiveness properties of our procedure."}
{"category": "Math", "title": "Parallel calibrations and minimal submanifolds", "abstract": "Given a parallel calibration $\\phi \\in \\Omega^p(M)$ on a Riemannian manifold $M$, I prove that the $\\phi$--critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the $\\phi$--critical submanifolds are precisely the integral manifolds of a $\\mathscr{C}^\\infty(M)$--linear subspace $\\sP \\subset \\Omega^p(M)$. In particular, the calibrated submanifolds are necessarily integral submanifolds of the system. (Examples of parallel calibrations include the special Lagrangian calibration on Calabi-Yau manifolds, (co)associative calibrations on $G_2$--manifolds, and the Cayley calibration on $\\tSpin(7)$--manifolds.)"}
{"category": "Math", "title": "Local energy estimates for the finite element method on sharply varying grids", "abstract": "Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global \"pollution\" term that measures the influence of solution quality from outside the domain of interest and is heuristically of higher order. However, the original analysis of Nitsche and Schatz is restricted to quasi-uniform grids. We present local a priori energy estimates that are valid on shape regular grids, an assumption which allows for highly graded meshes and which much more closely matches the typical practical situation. Our chief technical innovation is an improved superapproximation result."}
{"category": "Math", "title": "Tangent Cone of Numerical Semigroup Rings of Embedding Dimension Three", "abstract": "In this paper, we give new characterizations of the Buchsbaum and Cohen-Macaulay properties of the tangent cone $\\gr_\\frakm(R)$, where $(R,\\frakm)$ is a numerical semigroup ring of embedding dimension 3. In particular, we confirm the conjectures raised by Sapko on the Buchsbaumness of $\\gr_\\frakm(R)$."}
{"category": "Math", "title": "\\'Etale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields", "abstract": "We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang-Weil inequality. Moreover, we prove the Lang-Weil inequality for affine as well as projective varieties with an explicit description and a bound for the constant appearing therein. We also prove a conjecture of Lang and Weil concerning the Picard varieties and \\'etale cohomology spaces of projective varieties. The general inequality for complete intersections may be viewed as a more precise version of the estimates given by Hooley and Katz. The proof is primarily based on a suitable generalization of the Weak Lefschetz Theorem to singular varieties together with some Bertini-type arguments and the Grothendieck-Lefschetz Trace Formula. We also describe some auxiliary results concerning the \\'etale cohomology spaces and Betti numbers of projective varieties over finite fields and a conjecture along with some partial results concerning the number of points of projective algebraic sets over finite fields."}
{"category": "Math", "title": "The Continuous Graph FFT", "abstract": "The discrete Fourier transform and the FFT algorithm are extended from the circle to continuous graphs with equal edge lengths."}
{"category": "Math", "title": "The local recognition of reflection graphs of spherical Coxeter groups", "abstract": "Based on the third author's thesis in this article we complete the local recognition of commuting reflection graphs of spherical Coxeter groups arising from irreducible crystallographic root systems."}
{"category": "Math", "title": "The Katok-Spatzier Conjecture and Generalized Symmetries", "abstract": "Within the smooth category, an intertwining is exhibited between the global rigidity of irreducible higher rank abelian Anosov actions on the n-torus and the classification of equilibrium-free flows on the n-torus that possess nontrivial generalized symmetries."}
{"category": "Math", "title": "Pinned Repetitions in Symbolic Flows: Preliminary Results", "abstract": "We consider symbolic flows over finite alphabets and study certain kinds of repetitions in these sequences. Positive and negative results for the existence of such repetitions are given for codings of interval exchange transformations and codings of quadratic polynomials."}
{"category": "Math", "title": "Multidimensional Chebyshev Systems - just a definition", "abstract": "We provide a definition of Multidimensional Chebyshev Systems of order N which is satisfied by the solutions of a wide class of elliptic equations of order 2N. This definition generalizes a very large class of Extended Complete Chebyshev systems in the one-dimensional case. This is the first of a series of papers in this area, which solves the longstanding problem of finding a satisfactory multidimensional generalization of the classical Chebyshev systems introduced already by A. Markov more than hundered years ago, and studied later by S. Bernstein and M. Krein."}
{"category": "Math", "title": "On isoperimetric profiles of algebras", "abstract": "Isoperimetric profile in algebras was first introduced by Gromov. We study the behavior of the isoperimetric profile under various ring theoretic constructions and its relation with the Gelfand-Kirillov dimension."}
{"category": "Math", "title": "Relatively Open Gromov-Witten Invariants for Symplectic Manifolds of Lower Dimensions", "abstract": "Let $(X,\\omega)$ be a compact symplectic manifold, $L$ be a Lagrangian submanifold and $V$ be a codimension 2 symplectic submanifold of $X$, we consider the pseudoholomorphic maps from a Riemann surface with boundary $(\\Sigma,\\partial\\Sigma)$ to the pair $(X,L)$ satisfying Lagrangian boundary conditions and intersecting $V$. In some special cases, for instance, under the semi-positivity condition, we study the stable moduli space of such open pseudoholomorphic maps involving the intersection data. If $L\\cap V=\\emptyset$, we study the problem of orientability of the moduli space. Moreover, assume that there exists an anti-symplectic involution $\\phi$ on $X$ such that $L$ is the fixed point set of $\\phi$ and $V$ is $\\phi$-anti-invariant, then we define the so-called \"relatively open\" invariants for the tuple $(X,\\omega,V,\\phi)$ if $L$ is orientable and dim$X\\le 6$. If $L$ is nonorientable, we define such invariants under the condition that dim$X\\le4$ and some additional restrictions on the number of marked points on each boundary component of the domain."}
{"category": "Math", "title": "The distribution of the irreducibles in an algebraic number field", "abstract": "We study the distribution of principal ideals generated by irreducible elements in an algebraic number field."}
{"category": "Math", "title": "Sum of squares of degrees in a graph", "abstract": "Let $\\G(v,e)$ be the set of all simple graphs with $v$ vertices and $e$ edges and let $P_2(G)=\\sum d_i^2$ denote the sum of the squares of the degrees, $d_1, >..., d_v$, of the vertices of $G$. It is known that the maximum value of $P_2(G)$ for $G \\in \\G(v,e)$ occurs at one or both of two special graphs in $\\G(v,e)$--the \\qs graph or the \\qc graph. For each pair $(v,e)$, we determine which of these two graphs has the larger value of $P_2(G)$. We also determine all pairs $(v,e)$ for which the values of $P_2(G)$ are the same for the \\qs and the \\qc graph. In addition to the \\qs and \\qc graphs, we find all other graphs in $\\G(v,e)$ for which the maximum value of $P_2(G)$ is attained. Density questions posed by previous authors are examined."}
{"category": "Math", "title": "Implicit-Explicit Variational Integration of Highly Oscillatory Problems", "abstract": "In this paper, we derive a variational integrator for certain highly oscillatory problems in mechanics. To do this, we take a new approach to the splitting of fast and slow potential forces: rather than splitting these forces at the level of the differential equations or the Hamiltonian, we split the two potentials with respect to the Lagrangian action integral. By using a different quadrature rule to approximate the contribution of each potential to the action, we arrive at a geometric integrator that is implicit in the fast force and explicit in the slow force. This can allow for significantly longer time steps to be taken (compared to standard explicit methods, such as St\\\"ormer/Verlet) at the cost of only a linear solve rather than a full nonlinear solve. We also analyze the stability of this method, in particular proving that it eliminates the linear resonance instabilities that can arise with explicit multiple-time-stepping methods. Next, we perform some numerical experiments, studying the behavior of this integrator for two test problems: a system of coupled linear oscillators, for which we compare against the resonance behavior of the r-RESPA method; and slow energy exchange in the Fermi--Pasta--Ulam problem, which couples fast linear oscillators with slow nonlinear oscillators. Finally, we prove that this integrator accurately preserves the slow energy exchange between the fast oscillatory components, which explains the numerical behavior observed for the Fermi--Pasta--Ulam problem."}
{"category": "Math", "title": "Persistent Clustering and a Theorem of J. Kleinberg", "abstract": "We construct a framework for studying clustering algorithms, which includes two key ideas: persistence and functoriality. The first encodes the idea that the output of a clustering scheme should carry a multiresolution structure, the second the idea that one should be able to compare the results of clustering algorithms as one varies the data set, for example by adding points or by applying functions to it. We show that within this framework, one can prove a theorem analogous to one of J. Kleinberg, in which one obtains an existence and uniqueness theorem instead of a non-existence result. We explore further properties of this unique scheme, stability and convergence are established."}
{"category": "Math", "title": "Compact symmetric spaces, triangular factorization, and Cayley coordinates", "abstract": "Let U/K represent a connected, compact symmetric space, where theta is an involution of U that fixes K, phi: U/K to U is the geodesic Cartan embedding, and G is the complexification of U. We investigate the intersection of phi(U/K) with the Bruhat decomposition of G corresponding to a theta-stable triangular, or LDU, factorization of the Lie algebra of G. When g in phi(U/K) is generic, the corresponding factorization g=ld(g)u is unique, where l in N^-, d(g) in H, and u in N^+. In this paper we present an explicit formula for d in Cayley coordinates, compute it in several types of symmetric spaces, and use it to identify representatives of the connected components of the generic part of phi(U/K). This formula calculates a moment map for a torus action on the highest dimensional symplectic leaves of the Evens-Lu Poisson structure on U/K."}
{"category": "Math", "title": "Explicit multidimensional Ingham--Beurling type estimates", "abstract": "Recently a new proof was given for Beurling's Ingham type theorem on one-dimensional nonharmonic Fourier series, providing explicit constants. We improve this result by applying a short elementary method instead of the previous complex analytical approach. Our proof equally works in the multidimensional case."}
{"category": "Math", "title": "Cayley automaton semigroups", "abstract": "In this paper we characterize when a Cayley automaton semigroup is a group, is trivial, is finite, is free, is a left zero semigroup, or is a right zero semigroup."}
{"category": "Math", "title": "Uniform bounds for point cohomology of $\\ell^1({\\mathbb Z}_+)$ and related algebras", "abstract": "It is well-known that the point cohomology of the convolution algebra $\\ell^1({\\mathbb Z}_+)$ vanishes in degrees 2 and above. We sharpen this result by obtaining splitting maps whose norms are bounded independently of the choice of point module. Our construction is a by-product of new estimates on projectivity constants of maximal ideals in $\\ell^1({\\mathbb Z}_+)$. Analogous results are obtained for some other $L^1$-algebras which arise from `rank one' subsemigroups of ${\\mathbb R}_+$."}
{"category": "Math", "title": "Superefficiency from the Vantage Point of Computability", "abstract": "In 1952 Lucien Le Cam announced his celebrated result that, for regular univariate statistical models, sets of points of superefficiency have Lebesgue measure zero. After reviewing the turbulent history of early studies of superefficiency, I suggest using the notion of computability as a tool for exploring the phenomenon of superefficiency. It turns out that only computable parameter points can be points of superefficiency for computable estimators. This algorithmic version of Le Cam's result implies, in particular, that sets of points of superefficiency not only have Lebesgue measure zero but are even countable."}
{"category": "Math", "title": "Deducing the multidimensional Szemeredi Theorem from an infinitary removal lemma", "abstract": "We offer a new proof of the Furstenberg-Katznelson multiple recurrence theorem for several commuting probability-preserving transformations T_1, T_2, >..., T_d: \\bbZ\\curvearrowright (X,\\S,\\mu), and so, via the Furstenberg correspondence principle introduced in, a new proof of the multi-dimensional Szemeredi Theorem. We bypass the careful manipulation of certain towers of factors of a probability-preserving system that underlies the Furstenberg-Katznelson analysis, instead modifying an approach recently developed for the study of convergence of nonconventional ergodic averages to pass to a large extension of our original system in which this analysis greatly simplifies. The proof is then completed using an adaptation of arguments developed by Tao for his study of an infinitary analog of the hypergraph removal lemma. In a sense, this addresses the difficulty, highlighted by Tao, of establishing a direct connection between his infinitary, probabilistic approach to the hypergraph removal lemma and the infinitary, ergodic-theoretic approach to Szemeredi's Theorem set in motion by Furstenberg."}
{"category": "Math", "title": "On the geometry of a class of invariant measures and a problem of Aldous", "abstract": "In his 1985 survey of notions of exchangeability, Aldous introduced a form of exchangeability corresponding to the symmetries of the infinite discrete cube, and asked whether these exchangeable probability measures enjoy a representation theorem similar to those for exchangeable sequences, arrays and set-indexed families. In this note we to prove that, whereas the known representation theorems for different classes of partially exchangeable probability measure imply that the compact convex set of such measures is a Bauer simplex (that is, its subset of extreme points is closed), in the case of cube-exchangeability it is a copy of the Poulsen simplex (in which the extreme points are dense). This follows from the arguments used by Glasner and Weiss' for their characterization of property (T) in terms of the geometry of the simplex of invariant measures for associated generalized Bernoulli actions. The emergence of this Poulsen simplex suggests that, if a representation theorem for these processes is available at all, it must take a very different form from the case of set-indexed exchangeable families."}
{"category": "Math", "title": "Descent Construction for Gspin Groups: Main Results and Applications", "abstract": "The purpose of this note is to announce an extension of the descent method of Ginzburg, Rallis and Soudry to the setting of essentially self dual representations. This extension of the descent construction provides a complement to recent work of Asgari and Shahidi on the generic transfer for general Spin groups as well as to the work of Asgari and Raghuram on cuspidality of the exterior square lift for representations of GL4. Complete proofs of the results announced in the present note will appear in our forthcoming articles."}
{"category": "Math", "title": "Lagrangian Grassmannian in Infinite Dimension", "abstract": "Given a complex structure $J$ on a real (finite or infinite dimensional) Hilbert space $H$, we study the geometry of the Lagrangian Grassmannian $\\Lambda(H)$ of $H$, i.e. the set of closed linear subspaces $L\\subset H$ such that $$J(L)=L^\\perp.$$ The complex unitary group $U(H_J)$, consisting of the elements of the orthogonal group of $H$ which are complex linear for the given complex structure, acts transitively on $\\Lambda(H)$ and induces a natural linear connection in $\\Lambda(H)$. It is shown that any pair of Lagrangian subspaces can be joined by a geodesic of this connection. A Finsler metric can also be introduced, if one regards subspaces $L$ as projections $p_L$ (=the orthogonal projection onto $L$) or symmetries $\\e_L=2p_L-I$, namely measuring tangent vectors with the operator norm. We show that for this metric the Hopf-Rinow theorem is valid in $\\Lambda(H)$: a geodesic joining a pair of Lagrangian subspaces can be chosen to be of minimal length. We extend these results to the classical Banach-Lie groups of Schatten."}
{"category": "Math", "title": "Finsler geometry and actions of the p-Schatten unitary groups", "abstract": "Let $p$ be an even positive integer and $U_p(H)$ be the Banach-Lie group of unitary operators $u$ which verify that $u-1$ belongs to the $p$-Schatten ideal $B_p(H)$. Let ${\\cal O}$ be a smooth manifold on which $U_p(H)$ acts transitively and smoothly. Then one can endow ${\\cal O}$ with a natural Finsler metric in terms of the $p$-Schatten norm and the action of $U_p(H)$. Our main result establishes that for any pair of given initial conditions $$ x\\in {\\cal O}\\hbox{and} X\\in (T{\\cal O})_x $$ there exists a curve $\\delta(t)=e^{tz}\\cdot x$ in ${\\cal O}$, with $z$ a skew-hermitian element in the $p$-Schatten class such that $$ \\delta(0)=x \\hbox{and} \\dot{\\delta}(0)=X, $$ which remains minimal as long as $t\\|z\\|_p\\le \\pi/4$. Moreover, $\\delta$ is unique with these properties. We also show that the metric space $({\\cal O},d)$ ($d=$ rectifiable distance) is complete. In the process we establish minimality results in the groups $U_p(H)$, and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit $$ {\\cal O}=\\{uAu^*: u\\in U_p(H)\\} $$ of a self-adjoint operator $A\\in B(H)$."}
{"category": "Math", "title": "Norm Inequalities in Operator Ideals", "abstract": "In this paper we introduce a new technique for proving norm inequalities in operator ideals with an unitarily invariant norm. Among the well known inequalities which can be proved with this technique are the L\\\"owner-Heinz inequality, inequalities relating various operator means and the Corach-Porta-Recht inequality. We prove two general inequalities and from them we derive several inequalities by specialization, many of them new. We also show how some inequalities, known to be valid for matrices or bounded operators, can be extended with this technique to normed ideals in $C^*$-algebras, in particular to the noncommutative $L^p$-spaces of a semi-finite von Neumann algebra."}
{"category": "Math", "title": "On conformal biharmonic immersions", "abstract": "This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter family of non-minimal conformal biharmonic immersions of cylinder into $\\r^3$ and some examples of conformal biharmonic immersions of 4-dimensional Euclidean space into sphere and hyperbolic space thus provide many simple examples of proper biharmonic maps with rich geometric meanings. These suggest that there are abundant proper biharmonic maps in the family of conformal immersions. We also explore the relationship between biharmonicity and holomorphicity of conformal immersions of surfaces."}
{"category": "Math", "title": "Duality Relation for the Hilbert Series of Almost Symmetric Numerical Semigroups", "abstract": "We derive the duality relation for the Hilbert series H(d^m;z) of almost symmetric numerical semigroup S(d^m) combining it with its dual H(d^m;z^{-1}). On this basis we establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps F_j, generators d_i and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd indices separately. We apply the duality relation to the simple case of the almost symmetric semigroups of maximal embedding dimension, and give the necessary and efficient conditions for minimal set d^m to generate such semigroups."}
{"category": "Math", "title": "Markov paths, loops and fields", "abstract": "This is an extended version of a series of lectures given in St Flour. It includes a discussion of relations between the occupation field of Markov loops with the corresponding free field."}
{"category": "Math", "title": "Boundaries of systolic groups", "abstract": "For all systolic groups we construct boundaries which are EZ--structures. This implies the Novikov conjecture for torsion--free systolic groups. The boundary is constructed via a system of distinguished geodesics in a systolic complex, which we prove to have coarsely similar properties to geodesics in CAT(0) spaces."}
{"category": "Math", "title": "Boolean complexes for Ferrers graphs", "abstract": "In this paper we provide an explicit formula for calculating the boolean number of a Ferrers graph. By previous work of the last two authors, this determines the homotopy type of the boolean complex of the graph. Specializing to staircase shapes, we show that the boolean numbers of the associated Ferrers graphs are the Genocchi numbers of the second kind, and obtain a relation between the Legendre-Stirling numbers and the Genocchi numbers of the second kind. In another application, we compute the boolean number of a complete bipartite graph, corresponding to a rectangular Ferrers shape, which is expressed in terms of the Stirling numbers of the second kind. Finally, we analyze the complexity of calculating the boolean number of a Ferrers graph using these results and show that it is a significant improvement over calculating by edge recursion."}
{"category": "Math", "title": "The arithmetic-geometric scaling spectrum for continued fractions", "abstract": "To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued fraction digits exceeding a given number which tends to infinity."}
{"category": "Math", "title": "Gamma-invariant ideals in Iwasawa algebras", "abstract": "Let kG be the completed group algebra of a uniform pro-p group G with coefficients in a field k of characteristic p. We study right ideals I in kG that are invariant under the action of another uniform pro-p group Gamma. We prove that if I is non-zero then an irreducible component of the characteristic support of kG/I must be contained in a certain finite union of rational linear subspaces of Spec gr kG. The minimal codimension of these subspaces gives a lower bound on the homological height of I in terms of the action of a certain Lie algebra on G/G^p. If we take Gamma to be G acting on itself by conjugation, then Gamma-invariant right ideals of kG are precisely the two-sided ideals of kG, and we obtain a non-trivial lower bound on the homological height of a possible non-zero two-sided ideal. For example, when G is open in SL_n(\\Zp) this lower bound equals 2n - 2. This gives a significant improvement of the results of Ardakov, Wei and Zhang on reflexive ideals in Iwasawa algebras."}
{"category": "Math", "title": "A new secant method for unconstrained optimization", "abstract": "We present a gradient-based algorithm for unconstrained minimization derived from iterated linear change of basis. The new method is equivalent to linear conjugate gradient in the case of a quadratic objective function. In the case of exact line search it is a secant method. In practice, it performs comparably to BFGS and DFP and is sometimes more robust."}
{"category": "Math", "title": "Fitting Martingales To Given Marginals", "abstract": "We consider the problem of finding a real valued martingale fitting specified marginal distributions. For this to be possible, the marginals must be increasing in the convex order and have constant mean. We show that, under the extra condition that they are weakly continuous, the marginals can always be fitted in a unique way by a martingale which lies in a particular class of strong Markov processes. It is also shown that the map that this gives from the sets of marginal distributions to the martingale measures is continuous. Furthermore, we prove that it is the unique continuous method of fitting martingale measures to the marginal distributions."}
{"category": "Math", "title": "The Penrose Transform for Complex Projective Space", "abstract": "Various complexes of differential operators are constructed on complex projective space via the Penrose transform, which also computes their cohomology."}
{"category": "Math", "title": "Distortion elements in $Diff^\\infty(R/Z)$", "abstract": "We consider the group of smooth diffeomorphisms of the circle. We show that any recurrent $f$ (in the sense that $\\{f^n\\}_{n \\in Z}$ is not discrete) is in fact a distortion element (in the sense that its iterates can be written as short compositions involving finitely many smooth diffeomorphisms). Thus rotations are distortion elements."}
{"category": "Math", "title": "Khovanov homology, open books, and tight contact structures", "abstract": "We define the reduced Khovanov homology of an open book (S,h), and we identify a distinguished \"contact element\" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,h). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology of an open book which parallels the comultiplication on Heegaard Floer homology defined previously by the first author."}
{"category": "Math", "title": "Decomposable Principal Component Analysis", "abstract": "We consider principal component analysis (PCA) in decomposable Gaussian graphical models. We exploit the prior information in these models in order to distribute its computation. For this purpose, we reformulate the problem in the sparse inverse covariance (concentration) domain and solve the global eigenvalue problem using a sequence of local eigenvalue problems in each of the cliques of the decomposable graph. We demonstrate the application of our methodology in the context of decentralized anomaly detection in the Abilene backbone network. Based on the topology of the network, we propose an approximate statistical graphical model and distribute the computation of PCA."}
{"category": "Math", "title": "Le probl\\`eme des diviseurs pour des formes binaires de degr\\'e 4", "abstract": "We study the average order of the divisor function, as it ranges over the values of binary quartic forms that are reducible over the rationals."}
{"category": "Math", "title": "A Rejoinder to Mackintosh and some Remarks on the Concept of General Intelligence", "abstract": "In 2000 Nicholas J. Mackintosh (2000) published an article in \"Nature\" referring to the concept of general intelligence (\"g\") claiming that there is clear empirical evidence for the existence of the g factor and psychologists are \"united in their support of g\". Surprisingly, his view remained yet unchallenged although this issue is by no means as clear-cut as Mackintosh argues. Let us therefore attempt to clarify some common but unfortunately major misconceptions about g, which Mackintosh, following Jensen's (1998) precedent, recounted in his \"Nature\" article. The bottom line is that Spearman's g does not exist, that this has been known and acknowledged by leading scholars (Guttman, 1992; Thurstone, 1947) of factor analysis for decades so that the task of objectively defining human intelligence remains unfinished."}
{"category": "Math", "title": "A spline interpretation of Eulerian numbers", "abstract": "In this paper, we explore the interrelationship between Eulerian numbers and B splines. Specifically, using B splines, we give the explicit formulas of the refined Eulerian numbers, and descents polynomials. Moreover, we prove that the coefficients of descent polynomials $D_d^n(t)$ are log-concave. This paper also provides a new approach to study Eulerian numbers and descent polynomials."}
{"category": "Math", "title": "Cluster-tilted algebras and their intermediate coverings", "abstract": "We construct the intermediate coverings of cluster-tilted algebras by defining the generalized cluster categories. These generalized cluster categories are Calabi-Yau triangulated categories with fraction CY-dimension and have also cluster tilting objects (subcategories). Furthermore we study the representations of these intermediate coverings of cluster-tilted algebras."}
{"category": "Math", "title": "Complete minimal surfaces in R3 with a prescribed coordinate function", "abstract": "In this paper we construct complete simply connected minimal surfaces with a prescribed coordinate function. Moreover, we prove that these surfaces are dense in the space of all minimal surfaces with this coordinate function (with the topology of the smooth convergence on compact sets)."}
{"category": "Math", "title": "Strong Spectral Gaps for Compact Quotients of Products of $\\PSL(2,\\bbR)$", "abstract": "The existence of a strong spectral gap for quotients $\\Gamma\\bs G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan-Selberg Conjectures. If $G$ has no compact factors then for general lattices a strong spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the strong spectral gap for an irreducible co-compact lattice $\\Gamma$ in $G=\\PSL(2,\\bbR)^d$ for $d\\geq 2$ which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting."}
{"category": "Math", "title": "On finite-index extensions of subgroups of free groups", "abstract": "We study the lattice of finite-index extensions of a given finitely generated subgroup $H$ of a free group $F$. This lattice is finite and we give a combinatorial characterization of its greatest element, which is the commensurator of $H$. This characterization leads to a fast algorithm to compute the commensurator, which is based on a standard algorithm from automata theory. We also give a sub-exponential and super-polynomial upper bound for the number of finite-index extensions of $H$, and we give a language-theoretic characterization of the lattice of finite-index subgroups of $H$. Finally, we give a polynomial time algorithm to compute the malnormal closure of $H$."}
{"category": "Math", "title": "How to Draw Tropical Planes", "abstract": "The tropical Grassmannian parameterizes tropicalizations of linear spaces, while the Dressian parameterizes all planes in $\\TP^{n-1}$. We study these parameter spaces and we compute them explicitly for $n \\leq 7$. Planes are identified with matroid subdivisions and with arrangements of trees. These representations are used to draw pictures."}
{"category": "Math", "title": "Weighted Hardy and singular operators in Morrey spaces", "abstract": "We study the weighted boundedness of the Cauchy singular integral operator $S_\\Gm$ in Morrey spaces $L^{p,\\lambda}(\\Gm)$ on curves satisfying the arc-chord condition, for a class of \"radial type\" almost monotonic weights. The non-weighted boundedness is shown to hold on an arbitrary Carleson curve. We show that the weighted boundedness is reduced to the boundedness of weighted Hardy operators in Morrey spaces $L^{p,\\lambda}(0,\\ell), \\ell>0$. We find conditions for weighted Hardy operators to be bounded in Morrey spaces. To cover the case of curves we also extend the boundedness of the Hardy-Littlewood maximal operator in Morrey spaces, known in the Euclidean setting, to the case of Carleson curves. Key words and phrases: Morrey space, singular operator, Hardy operator, Hardy-Littlewood maximal operator, weighted estimate."}
{"category": "Math", "title": "Fat-Tailed Distributions and Levy Processes", "abstract": "The notion that natural disasters can be controlled is, of course, farcical; history is permeated with examples of countless failed attempts at this pointless task; it is synonymous with trying to build a perpetual motion machine. Nonetheless, there are ways to reduce their impact on human communities, particularly by looking away from the normal hypothesis."}
{"category": "Math", "title": "Rankin's method and Jacobi forms of several variables", "abstract": "Following Rankin's method, D. Zagier computed the $n$-th Rankin-Cohen bracket of a modular form $g$ of weight $k_1$ with the Eisenstein series of weight $k_2$ and then computed the inner product of this Rankin-Cohen bracket with a cusp form $f$ of weight $k = k_1+k_2+2n$ and showed that this inner product gives, upto a constant, the special value of the Rankin-Selberg convolution of $f$ and $g$. This result was generalized to Jacobi forms of degree 1 by Y. Choie and W. Kohnen. In this paper, we generalize this result to Jacobi forms defined over ${\\mathcal H} \\times {\\mathbb C}^{(g, 1)}$."}
{"category": "Math", "title": "Tropical R maps and Affine Geometric Crystals", "abstract": "By modifying the method in [KNO], certain affine geometric crystals are realized in affinization of the fundamental representation $W(\\varpi_1)_l$ and the tropical R maps for the affine geometric crystals are described explicitly. We also define prehomogeneous geometric crystals and show that for a positive geometric crystal, the connectedness of the corresponding ultra-discretized crystal is the sufficient condition for prehomogeneity of the positive geometric crystal. Moreover, the uniqueness of tropical R maps is discussed."}
{"category": "Math", "title": "Embedding operators into strongly continuous semigroups", "abstract": "We study linear operators $T$ on Banach spaces for which there exists a $C_0$-semigroup $(T(t))_{t\\geq 0}$ such that $T=T(1)$. We present a necessary condition in terms of the spectral value 0 and give classes of examples where this can or cannot be achieved."}
{"category": "Math", "title": "Graphs, Frobenius functionals, and the classical Yang-Baxter equation", "abstract": "A Lie algebra is Frobenius if it admits a linear functional F such that the Kirillov form F([x,y]) is non-degenerate. If g is the m-th maximal parabolic subalgebra P(n,m) of sl(n) this occurs precisely when (n,m) = 1. We define a \"cyclic\" functional F on P(n,m) and prove it is non-degenerate using properties of certain graphs associated to F. These graphs also provide in some cases readily computable associated solutions of the classical Yang-Baxter equation. We also define a local ring associated to each connected loopless graph from which we show that the graph can be reconstructed. Finally, we examine the seaweed Lie algebras of Dergachev and Kirillov from our perspective."}
{"category": "Math", "title": "Maximal regularity for parabolic partial differential equations on manifolds with cylindrical ends", "abstract": "We give a short, simple proof of maximal regularity for linear parabolic evolution equations on manifolds with cylindrical ends by making use of pseudodifferential parametrices and the concept of R-boundedness for the resolvent."}
{"category": "Math", "title": "Simple Cohen-Macaulay Codimension 2 Singularities", "abstract": "In this article, we provide a complete list of simple Cohen-Macaulay codimension 2 singularities together with a list of adjacencies which is complete in the case of fat point and space curve singularities."}
{"category": "Math", "title": "Rigidity and uniruling for Lagrangian submanifolds", "abstract": "This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$."}
{"category": "Math", "title": "Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality", "abstract": "We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis inequality for surfaces. The proof is based on the theory of Generic Vanishing and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative algebra. Along the way we give a positive answer, in the setting of K\\\"ahler manifolds, to a question of Green-Lazarsfeld on the vanishing of higher direct images of Poincar\\'e bundles. We indicate generalizations to arbitrary Fourier-Mukai transforms."}
{"category": "Math", "title": "Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation", "abstract": "We give new proofs of two basic results in number theory: The law of quadratic reciprocity and the sign of the Gauss sum. We show that these results are encoded in the relation between the discrete Fourier transform and the action of the Weyl element in the Weil representation modulo p,q and pq."}
{"category": "Math", "title": "Multivariable operator-valued Nevanlinna-Pick interpolation: a survey", "abstract": "The theory of Nevanlinna-Pick and Carath\\'eodory-Fej\\'er interpolation for matrix- and operator-valued Schur class functions on the unit disk is now well established. Recent work has produced extensions of the theory to a variety of multivariable settings, including the ball and the polydisk (both commutative and noncommutative versions), as well as a time-varying analogue. Largely independent of this is the recent Nevanlinna-Pick interpolation theorem by P.S. Muhly and B. Solel for an abstract Hardy algebra set in the context of a Fock space built from a $W^*$-correspondence E over a $W^{*}$-algebra $\\cA$ and a *-representation $\\sigma$ of $\\cA$. In this review we provide an exposition of the Muhly-Solel interpolation theory accessible to operator theorists, and explain more fully the connections with the already existing interpolation literature. The abstract point evaluation first introduced by Muhly-Solel leads to a tensor-product type functional calculus in the main examples. A second kind of point-evaluation for the $W^*$-correspondence Hardy algebra, also introduced by Muhly and Solel, is here further investigated, and a Nevanlinna-Pick theorem in this setting is proved. It turns out that, when specified for examples, this alternative point-evaluation leads to an operator-argument functional calculus and corresponding Nevanlinna-Pick interpolation. We also discuss briefly several Nevanlinna-Pick interpolation results for Schur classes that do not fit into the Muhly-Solel $W^*$-correspondence formalism."}
{"category": "Math", "title": "Operator-Valued $L_{q}\\to L_{p}$ Fourier Multipliers", "abstract": "This paper is withdrawn due to some gaps in the proof"}
{"category": "Math", "title": "On Exceptional Collections on Grassmannians", "abstract": "The author was informed that the result in the original version had been obtained earlier by K. Ueda (arXiv:math/0503355 [math.AG]). The paper is retracted."}
{"category": "Math", "title": "SO(3) quantum invariants are dense", "abstract": "We show that when $r \\geq 5$ is prime, the SO(3) Witten-Reshetikhin-Turaev quantum invariants for three-manifolds at the level $r$ form a dense set in the complex plane. This confirms a conjecture of Larsen and Wang."}
{"category": "Math", "title": "Algebraic points of small height missing a union of varieties", "abstract": "Let $K$ be a number field, $\\overline{\\mathbb Q}$, or the field of rational functions on a smooth projective curve over a perfect field, and let $V$ be a subspace of $K^N$, $N \\geq 2$. Let $Z_K$ be a union of varieties defined over $K$ such that $V \\nsubseteq Z_K$. We prove the existence of a point of small height in $V \\setminus Z_K$, providing an explicit upper bound on the height of such a point in terms of the height of $V$ and the degree of a hypersurface containing $Z_K$, where dependence on both is optimal. This generalizes and improves upon the previous results of the author. As a part of our argument, we provide a basic extension of the function field version of Siegel's lemma of J. Thunder to an inequality with inhomogeneous heights. As a corollary of the method, we derive an explicit lower bound for the number of algebraic integers of bounded height in a fixed number field."}
{"category": "Math", "title": "Relaxed commutant lifting: existence of a unique solution", "abstract": "In this paper we present necessary and sufficient conditions for the existence of a unique solution to the relaxed commutant lifting problem. The obtained conditions are more complicated than those for the classical commutant lifting setting, and earlier obtained sufficient conditions turn out not to be necessary conditions. It is also shown that these conditions simplify in certain special cases."}
{"category": "Math", "title": "Inequalities for analytic functions with the derivative in H1", "abstract": "It is proved an inequality - integrated analogue of the Hardy inequality and as application simplified proof of the theorem of S. A. Vinogradov for the bounded Toeplitz operators on the space of functions analytic and bounded in the unit disc is given."}
{"category": "Math", "title": "Commuting-Square Subfactors and Central Sequences", "abstract": "Let $M_0 \\subset M_1$ be a finite-index infinite-depth hyperfinite $II_1$ subfactor and $\\omega$ a free ultrafilter of the natural numbers. We show that if this subfactor is constructed from a commuting square then the central sequence inclusion $M_0^{\\omega} \\cap M_1' \\subset (M_0)_{\\omega}$ has infinite Pimnser-Popa index. We will also demonstrate this result for certain infinite-depth hyperfinite subfactors coming from groups."}
{"category": "Math", "title": "Improved lower bound on the size of Kakeya sets over finite fields", "abstract": "In a recent breakthrough, Dvir showed that every Kakeya set in $\\F^n$ must be of cardinality at least $c_n |\\F|^n$ where $c_n \\approx 1/n!$. We improve this lower bound to $\\beta^n |\\F|^n$ for a constant $\\beta > 0$. This pins down the growth of the leading constant to the right form as a function of $n$."}
{"category": "Math", "title": "The Conjugacy Problem in the Grigorchuk Group is polynomial time decidable", "abstract": "In this paper we prove that the Conjugacy Problem in the Grigorchuk group $\\Gamma$ has polynomial time complexity."}
{"category": "Math", "title": "An observation about submatrices", "abstract": "Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer less than or equal to n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of M of order k. The proof uses results about random walks on symmetric groups and concentration of measure. In a similar way, we also show that almost all k x n submatrices of M have almost the same distribution of singular values."}
{"category": "Math", "title": "Unification theorems in algebraic geometry", "abstract": "In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along the way we lay down foundations of algebraic geometry over arbitrary algebraic structures."}
{"category": "Math", "title": "A Geometrical Approach to Hilbert-Schmidt Operators", "abstract": "We give a Riemannian structure to the set $\\Sigma$ of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of $\\Sigma$ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold $\\Sigma$ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into $\\Sigma$. We give an intrinsic algebraic characterization of convex closed submanifolds $M$. We study the group of isometries of such submanifolds: we prove that $G_M$, the Banach-Lie group generated by $M$, acts isometrically and transitively on $M$. Moreover, $G_M$ admits a polar decomposition relative to $M$, namely $G_M\\simeq M\\times K$ as Hilbert manifolds (here $K$ is the isotropy of $p=1$ for the action $I_g: p\\mapsto gpg^*$), and also $G_M/K\\simeq M$ so $M$ is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds $M$. These decompositions are obtained \\textit{via} a nonlinear but analytic orthogonal projection $\\Pi_M:\\Sigma\\to M$, a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism $NM\\simeq\\Sigma$ (here $NM$ stands for the normal bundle of a convex closed submanifold $M$). Writing down the factorizations for fixed ${\\rm e}^a$, we obtain ${\\rm e}^a={\\rm e}^x{\\rm e}^v{\\rm e}^x$ with ${\\rm e}^x\\in M$ and $v$ orthogonal to $M$ at $p=1$. As a corollary we obtain decompositions for the full group of invertible elements $G\\simeq M\\times \\exp(T_1M^{\\perp})\\times K$."}
{"category": "Math", "title": "Hopf-Rinow Theorem in the Sato Grassmannian", "abstract": "Let $U_2({\\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\\cal H}$ which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit $$\\{upu^*: u\\in U_2({\\cal H})\\},$$ of an infinite projection $p$ in ${\\cal H}$. This orbit coincides with the connected component of $p$ in the Hilbert-Schmidt restricted Grassmannian $Gr_{res}(p)$ (also known in the literature as the Sato Grassmannian) corresponding to the polarization ${\\cal H}=p({\\cal H})\\oplus p({\\cal H})^\\perp$. It is known that the components of $Gr_{res}(p)$ are differentiable manifolds. Here we give a simple proof of the fact that $Gr_{res}^0(p)$ is a smooth submanifold of the affine Hilbert space $p+{\\cal B}_2({\\cal H})$, where ${\\cal B}_2({\\cal H})$ denotes the space of Hilbert-Schmidt operators of ${\\cal H}$. We prove that the geodesics of the natural connection, which are of the form $\\gamma(t)=e^{tz}pe^{-tz}$, for $z$ a $p$-codiagonal anti-hermitic element of ${\\cal B}_2({\\cal H})$, have minimal length provided that $\\|z\\|\\le \\pi/2$. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points $p_1,p_2\\in Gr_{res}^0(p)$ are joined by a minimal geodesic. If moreover $\\|p_1-p_2\\|<1$, the minimal geodesic is unique. Finally, we replace the 2-norm by the $k$-Schatten norm ($k>2$), and prove that the geodesics are also minimal for these norms, up to a critical value of $t$, which is estimated also in terms of the usual operator norm. In the process, minimality results in the $k$-norms are also obtained for the group $U_2({\\cal H})$."}
{"category": "Math", "title": "Weak Riemannian manifolds from finite index subfactors", "abstract": "Let $N\\subset M$ be a finite Jones' index inclusion of II$_1$ factors, and denote by $U_N\\subset U_M$ their unitary groups. In this paper we study the homogeneous space $U_M/U_N$, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit $$ {\\cal O}(p) =\\{u p u^*: u\\in U_M\\} $$ of the Jones projection $p$ of the inclusion. We endow ${\\cal O}(p) $ with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete), therefore ${\\cal O}(p)$ is a weak Riemannian manifold. We show that ${\\cal O}(p)$ enjoys certain properties similar to classic Hilbert-Riemann manifolds. Among them, metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point $p_1$ of ${\\cal O}(p)$, there is a ball $\\{q\\in {\\cal O}(p):\\|q-p_1\\|<r\\}$ (of uniform radius $r$) of the usual norm of $M$, such that any point $p_2$ in the ball is joined to $p_1$ by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion ${\\cal O}(p)\\subset {\\cal P}(M_1)$, where the last set denotes the Grassmann manifold of the von Neumann algebra generated by $M$ and $p$."}
{"category": "Math", "title": "On Integral Operators with Operator Valued Kernels", "abstract": "Here Lq-Lp boundedness of integral operator with operator-valued kernels is studied and the main result is applied to convolution operators. Using these results Besov space regularity for Fourier multiplier operator is established."}
{"category": "Math", "title": "Ekedahl-Oort strata and Kottwitz-Rapoport strata", "abstract": "We study the moduli space A_g of g-dimensional principally polarized abelian varieties in positive characteristic, and its variant A_I with Iwahori level structure. Both supersingular Ekedahl-Oort strata and supersingular Kottwitz-Rapoport strata are isomorphic to disjoint unions of Deligne-Lusztig varieties (see [Hoeve 2008] and [Goertz, Yu 2008], resp.). Here we compare these isomorphisms. We also give an explicit description of Kottwitz-Rapoport strata contained in the supersingular locus in the general parahoric case. Finally, we show that every Ekedahl-Oort stratum is isomorphic to a parahoric Kottwitz-Rapoport stratum."}
{"category": "Math", "title": "Vector partition functions and index of transversally elliptic operators", "abstract": "Let G be a torus acting linearly on a complex vector space M, and let X be the list of weights of G in M. We determine the equivariant K-theory of the open subset of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the lattice of weights of G, satisfying the cocircuit difference equations associated to X, introduced by Dahmen--Micchelli in the context of the theory of splines in order to study vector partition functions. This allows us to determine the range of the index map from G-transversally elliptic operators on M to generalized functions on G and to prove that the index map is an isomorphism on the image. This is a setting studied by Atiyah-Singer which is in a sense universal for index computations."}
{"category": "Math", "title": "A bicategorical version of Masuoka's theorem. Applications to bimodules over functor categories and to firm bimodules", "abstract": "We give a bicategorical version of the main result of A. Masuoka ({Corings and invertible bimodules,} {\\em Tsukuba J. Math.} \\textbf{13} (1989), 353--362) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings $R \\subseteq S$, the relative Picard group $Pic(S/R)$ is isomorphic to the Amitsur 1--cohomology group $H^1(S/R,U)$ with coefficients in the units functor $U$."}
{"category": "Math", "title": "Galois sections for abelianized fundamental groups", "abstract": "Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the \\'etale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the \\'etale fundamental group `with abelianized geometric part' onto the Galois group. We give a criterion for the existence of sections in arbitrary dimension and over arbitrary perfect fields, and then study the case of curves over local and global fields more closely. We also point out the relation to the elementary obstruction of Colliot-Th\\'el\\`ene and Sansuc."}
{"category": "Math", "title": "Equivalence of robust stabilization and robust performance via feedback", "abstract": "One approach to robust control for linear plants with structured uncertainty as well as for linear parameter-varying (LPV) plants (where the controller has on-line access to the varying plant parameters) is through linear-fractional-transformation (LFT) models. Control issues to be addressed by controller design in this formalism include robust stability and robust performance. Here robust performance is defined as the achievement of a uniform specified $L^{2}$-gain tolerance for a disturbance-to-error map combined with robust stability. By setting the disturbance and error channels equal to zero, it is clear that any criterion for robust performance also produces a criterion for robust stability. Counter-intuitively, as a consequence of the so-called Main Loop Theorem, application of a result on robust stability to a feedback configuration with an artificial full-block uncertainty operator added in feedback connection between the error and disturbance signals produces a result on robust performance. The main result here is that this performance-to-stabilization reduction principle must be handled with care for the case of dynamic feedback compensation: casual application of this principle leads to the solution of a physically uninteresting problem, where the controller is assumed to have access to the states in the artificially-added feedback loop. Application of the principle using a known more refined dynamic-control robust stability criterion, where the user is allowed to specify controller partial-state dimensions, leads to correct robust-performance results. These latter results involve rank conditions in addition to Linear Matrix Inequality (LMI) conditions."}
{"category": "Math", "title": "Stickelberger elements and Kolyvagin systems", "abstract": "In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements, utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. We show how to apply this construction to prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and deduce the main conjecture of Iwasawa theory for totally real fields (for totally odd characters). Although the main results of this paper have already been established by Wiles, our approach provides another example (which slightly differs from the case of Stark elements) on how to study Kolyvagin systems of core rank r > 1 (in the sense of Mazur and Rubin). Also, by making use of the 'rigidity' of the collection of Kolyvagin systems, we establish a link between the Stickelberger elements and the Rubin-Stark elements."}
{"category": "Math", "title": "Degree formula for connective K-theory", "abstract": "We apply the degree formula for connective $K$-theory to study rational contractions of algebraic varieties. Examples include rationally connected varieties and complete intersections."}
{"category": "Math", "title": "Universal Malliavin Calculus in Fock and L\\'{e}vy-It\\^{o} Spaces", "abstract": "We review and extend Lindsay's work on abstract gradient and divergence operators in Fock space over a general complex Hilbert space. Precise expressions for the domains are given, the $L^2$-equivalence of norms is proved and an abstract version of the It\\^{o}-Skorohod isometry is established. We then outline a new proof of It\\^{o}'s chaos expansion of complex L\\'{e}vy-It\\^{o} space in terms of multiple Wiener-L\\'{e}vy integrals based on Brownian motion and a compensated Poisson random measure. The duality transform now identifies L\\'{e}vy-It\\^{o} space as a Fock space. We can then easily obtain key properties of the gradient and divergence of a general L\\'{e}vy process. In particular we establish maximal domains of these operators and obtain the It\\^{o}-Skorohod isometry on its maximal domain."}
{"category": "Math", "title": "Quantum groups and quantization of Weyl group symmetries of Painlev\\'e systems", "abstract": "We shall construct the quantized q-analogues of the birational Weyl group actions arising from nilpotent Poisson algebras, which are conceptual generalizations, proposed by Noumi and Yamada, of the B\\\"acklund transformations for Painlev\\'e equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra of arbitrary symmetrizable Kac-Moody type. Then non-integral powers of the image of the Chevalley generators generate the quantized q-analogue of the birational Weyl group action. Using the same method, we shall reconstruct the quantized B\\\"acklund transformations of q-Painlev\\'e equations constructed by Hasegawa. We shall also prove that any subquotient integral domain of a quantized universal enveloping algebra of finite or affine type is an Ore domain."}
{"category": "Math", "title": "On Bogovski\\u{\\i} and regularized Poincar\\'e integral operators for de Rham complexes on Lipschitz domains", "abstract": "We study integral operators related to a regularized version of the classical Poincar\\'e path integral and the adjoint class generalizing Bogovski\\u{\\i}'s integral operator, acting on differential forms in $R^n$. We prove that these operators are pseudodifferential operators of order -1. The Poincar\\'e-type operators map polynomials to polynomials and can have applications in finite element analysis. For a domain starlike with respect to a ball, the special support properties of the operators imply regularity for the de Rham complex without boundary conditions (using Poincar\\'e-type operators) and with full Dirichlet boundary conditions (using Bogovski\\u{\\i}-type operators). For bounded Lipschitz domains, the same regularity results hold, and in addition we show that the cohomology spaces can always be represented by $C^\\infty$ functions."}
{"category": "Math", "title": "Explicit double shuffle relations and a generalization of Euler's decomposition formula", "abstract": "We give an explicit formula for the shuffle relation in a general double shuffle framework that specializes to double shuffle relations of multiple zeta values and multiple polylogarithms. As an application, we generalize the well-known decomposition formula of Euler that expresses the product of two Riemann zeta values as a sum of double zeta values to a formula that expresses the product of two multiple polylogarithm values as a sum of other multiple polylogarithm values."}
{"category": "Math", "title": "Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices", "abstract": "We give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is, functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted minimum and maximum functions, and present their characterizations, accordingly. Moreover, we discuss normal form representations of these functions."}
{"category": "Math", "title": "Liberation of orthogonal Lie groups", "abstract": "We show that under suitable assumptions, we have a one-to-one correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: $O_n,S_n,H_n,B_n,S_n',B_n'$. We investigate the representation theory aspects of the correspondence, with the result that for $O_n,S_n,H_n,B_n$, this is compatible with the Bercovici-Pata bijection. Finally, we discuss some more general classification problems in the compact orthogonal case, notably with the construction of a new quantum group."}
{"category": "Math", "title": "A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand", "abstract": "We give a new representation of fractional Brownian motion with Hurst parameter H<=1/2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Levy."}
{"category": "Math", "title": "The Laguerre polynomials preserve real-rootedness", "abstract": "The linear transformation that sends $x^n$ to the n'th Laguerre polynomial preserves real-rootedness."}
{"category": "Math", "title": "Operator-valued Fourier multipliers in Besov spaces and its applications", "abstract": "The present paper, is devoted to investigation of operator--valued Fourier multiplier theorems from $B_{q_{1},r}^{s}$ to $B_{q_{2},r}^{s}$, optimal embedding of Besov spaces, the separability and positivity of differential operators. Here, we show that these differential operators generate analytic semigroup."}
{"category": "Math", "title": "Some remarks on the Stanley's depth for multigraded modules", "abstract": "We show that the Stanley's conjecture holds for any multigraded $S$-module $M$ with $\\sdepth(M)=0$, where $S=K[x_1,...,x_n]$. Also, we give some bounds for the Stanley depth of the powers of the maximal irrelevant ideal in $S$."}
{"category": "Math", "title": "Invariant conformal metrics on S^n", "abstract": "In this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on the sphere which are invariant by a $k-$parameter subgroup of conformal diffeomorphisms of the sphere, giving a bound on its maximum dimension. Moreover, we classify conformal metrics on the sphere whose eigenvalues of the Shouten tensor are all constant (we call them \\emph{isoparametric conformal metrics}), and we use a classification result for radial conformal metrics which are solution of some $\\sigma _k -$Yamabe type problem for obtaining existence of rotational spheres and Delaunay-type hypersurfaces for some classes of Weingarten hypersurfaces in $\\h ^{n+1}$."}
{"category": "Math", "title": "Communication-optimal parallel and sequential QR and LU factorizations", "abstract": "We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform, and just as stable as Householder QR. We prove optimality by extending known lower bounds on communication bandwidth for sequential and parallel matrix multiplication to provide latency lower bounds, and show these bounds apply to the LU and QR decompositions. We not only show that our QR algorithms attain these lower bounds (up to polylogarithmic factors), but that existing LAPACK and ScaLAPACK algorithms perform asymptotically more communication. We also point out recent LU algorithms in the literature that attain at least some of these lower bounds."}
{"category": "Math", "title": "Conformal dimension: Cantor sets and moduli", "abstract": "In this paper we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension 1 thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede \\cite{Fug}. It implies in particular that there are many sets $E\\subset\\mathbb{R}$ of zero length such that $X\\times Y$ is minimal for conformal dimension for every compact $Y$."}
{"category": "Math", "title": "Integer Points in Backward Orbits", "abstract": "A theorem of J. Silverman states that a forward orbit of a rational map $\\phi(z)$ on $\\mathbb P^1(K)$ contains finitely many $S$-integers in the number field $K$ when $(\\phi\\circ\\phi)(z)$ is not a polynomial. We state an analogous conjecture for the backward orbits using a general $S$-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map $\\phi(z)=z^d$, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for $z^n-\\beta$ when $\\beta\\not =0$ is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for $\\phi^n(z)-\\beta$ is bounded independently of $n$."}
{"category": "Math", "title": "Discretely ordered groups", "abstract": "We consider group orders and right-orders which are discrete, meaning there is a least element which is greater than the identity. We note that free groups cannot be given discrete orders, although they do have right-orders which are discrete. More generally, we give necessary and sufficient conditions that a given orderable group can be endowed with a discrete order. In particular, every orderable group G embeds in a discretely orderable group. We also consider conditions on right-orderable groups to be discretely right-orderable. Finally, we discuss a number of illustrative examples involving discrete orderability, including the Artin braid groups and Bergman's non-locally-indicable right orderable groups."}
{"category": "Math", "title": "A simple estimation of the maximal rank of tensors with two slices by row and column operations, symmetrization and induction", "abstract": "The determination of the maximal ranks of a set of a given type of tensors is a basic problem both in theory and application. In statistical applications, the maximal rank is related to the number of necessary parameters to be built in a tensor model. Based on this classical theorem by Bosch we will show the tight bound for 2 x n x n tensors by simple row and column operations, symmetrization and mathematical induction, which has been given by several authors based on eigenvalue theories."}
{"category": "Math", "title": "Complex manifolds with generating tangent bundles", "abstract": "Let $M$ be a close complex manifold and $TM$ its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then $M$ is a complex homogeneous manifold. Our proof depends on the complex version of Chow-Rashevskii theorem in Carnot-Caratheodory spaces."}
{"category": "Math", "title": "A dozen integrals: Russell-style", "abstract": "On June 15, 1876, the Proceedings of the Royal Society of London published a paper by Mr. W. H. L. Russell entitled \"On certain integrals\". The paper starts with \"The following are certain integrals which will, I hope, be found interesting\". The rest of the paper is simply a list of 12 integrals. We provide a list of 12 integrals that we hope would be interesting."}
{"category": "Math", "title": "A construction of Frobenius manifolds with logarithmic poles and applications", "abstract": "A construction theorem for Frobenius manifolds with logarithmic poles is established. This is a generalization of a theorem of Hertling and Manin. As an application we prove a generalization of the reconstruction theorem of Kontsevich and Manin for projective smooth varieties with convergent Gromov-Witten potential. A second application is a construction of Frobenius manifolds out of a variation of polarized Hodge structures which degenerates along a normal crossing divisor when certain generation conditions are fulfilled."}
{"category": "Math", "title": "Posterior Convergence and Model Estimation in Bayesian Change-point Problems", "abstract": "We study the posterior distribution of the Bayesian multiple change-point regression problem when the number and the locations of the change-points are unknown. While it is relatively easy to apply the general theory to obtain the $O(1/\\sqrt{n})$ rate up to some logarithmic factor, showing the exact parametric rate of convergence of the posterior distribution requires additional work and assumptions. Additionally, we demonstrate the asymptotic normality of the segment levels under these assumptions. For inferences on the number of change-points, we show that the Bayesian approach can produce a consistent posterior estimate. Finally, we argue that the point-wise posterior convergence property as demonstrated might have bad finite sample performance in that consistent posterior for model selection necessarily implies the maximal squared risk will be asymptotically larger than the optimal $O(1/\\sqrt{n})$ rate. This is the Bayesian version of the same phenomenon that has been noted and studied by other authors."}
{"category": "Math", "title": "Formal topology and constructive mathematics: the Gelfand and Stone-Yosida representation theorems", "abstract": "We present a constructive proof of the Stone-Yosida representation theorem for Riesz spaces motivated by considerations from formal topology. This theorem is used to derive a representation theorem for f-algebras. In turn, this theorem implies the Gelfand representation theorem for C*-algebras of operators on Hilbert spaces as formulated by Bishop and Bridges. Our proof is shorter, clearer, and we avoid the use of approximate eigenvalues."}
{"category": "Math", "title": "Invariant manifolds for analytic dynamical systems over ultrametric fields", "abstract": "We give an exposition of the theory of invariant manifolds around a fixed point, in the case of time-discrete, analytic dynamical systems over a complete ultrametric field K. Typically, we consider an analytic manifold M modelled on an ultrametric Banach space over K, an analytic self-map f of M, and a fixed point p of f. Under suitable conditions on the tangent map of f at p, we construct a centre-stable manifold, a centre manifold, respectively, an r-stable manifold around p, for a given positive real number r not exceeding 1. The invariant manifolds are useful in the theory of Lie groups over local fields, where they allow results to be extended to the case of positive characteristic which previously were only available in characteristic zero (i.e., for p-adic Lie groups)."}
{"category": "Math", "title": "Recognizing trace graphs of closed braids", "abstract": "To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetraherdal moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral moves in polynomial time with respect to the braid length."}
{"category": "Math", "title": "Cusp areas of Farey manifolds and applications to knot theory", "abstract": "This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to explicit bounds on the volume of Dehn fillings of these manifolds, for example sharp bounds on volumes of hyperbolic closed 3-braids in terms of the Schreier normal form of the associated braid word. Finally, these results are applied to derive relations between the Jones polynomial and the volume of hyperbolic knots, and to disprove a related conjecture."}
{"category": "Math", "title": "Smoothing noisy spectroscopic data with many-knot spline method", "abstract": "In this paper, we present the development of a many-knot spline method derived to remove the statistical noise in the spectroscopic data. This method is an expansion of the B-spline method. Compared to the B-spline method, the many-knot spline method is significantly faster."}
{"category": "Math", "title": "Perturbation method for determining the group of invariance of hierarchical models", "abstract": "We propose a perturbation method for determining the (largest) group of invariance of a toric ideal defined in Aoki and Takemura [2008a]. In the perturbation method, we investigate how a generic element in the row space of the configuration defining a toric ideal is mapped by a permutation of the indeterminates. Compared to the proof in Aoki and Takemura [2008a] which was based on stabilizers of a subset of indeterminates, the perturbation method gives a much simpler proof of the group of invariance. In particular, we determine the group of invariance for a general hierarchical model of contingency tables in statistics, under the assumption that the numbers of the levels of the factors are generic. We prove that it is a wreath product indexed by a poset related to the intersection poset of the maximal interaction effects of the model."}
{"category": "Math", "title": "Multiplicity one theorem for (GL(n+1,R),GL(n,R))", "abstract": "Let F be either R or C. Consider the standard embedding GL(n,F)<GL(n+1,F) and the action of GL(n,F) on GL(n+1,F) by conjugation. In this paper we show that any GL(n,F)-invariant distribution on GL(n+1,F) is invariant with respect to transposition. We show that this implies that for any irreducible admissible smooth Frechet representations $\\pi$ of GL(n+1,F) and $\\tau$ of GL(n,F), $$dim Hom_{GL(n,F)}(\\pi,\\tau) \\leq 1.$$ For p-adic fields those results were proven in [AGRS]."}
{"category": "Math", "title": "Efficient rare-event simulation for the maximum of heavy-tailed random walks", "abstract": "Let $(X_n:n\\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean. Set $S_0=0$ and define $S_n=X_1+... +X_n$. We propose an importance sampling algorithm to estimate the tail of $M=\\max \\{S_n:n\\geq 0\\}$ that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmund's (1979) algorithm. The rigorous analysis of efficiency requires new Lyapunov-type inequalities that can be useful in the study of more general importance sampling algorithms."}
{"category": "Math", "title": "Two orbits: When is one in the closure of the other?", "abstract": "Let $G$ be a connected linear algebraic group, let $V$ be a finite dimensional algebraic $G$-module, and let $\\mathcal O_1$, $\\mathcal O_2$ be two $G$-orbits in $V$. We describe a constructive way to find out whether $\\mathcal O_1$ lies in the closure of $\\mathcal O_2$ or not."}
{"category": "Math", "title": "A Kind of Compact Quantum Semigroups", "abstract": "We show that the quantum family of all maps from a finite space to a finite dimensional compact quantum semigroup has a canonical quantum semigroup structure."}
{"category": "Math", "title": "Unifying Practical Uncertainty Representations: I. Generalized P-Boxes", "abstract": "There exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Ferson's p-boxes and Neumaier's clouds. Both for theoretical and practical considerations, it is very useful to know whether one representation is equivalent to or can be approximated by other ones. In this paper, we define a generalized form of usual p-boxes. These generalized p-boxes have interesting connections with other previously known representations. In particular, we show that they are equivalent to pairs of possibility distributions, and that they are special kinds of random sets. They are also the missing link between p-boxes and clouds, which are the topic of the second part of this study."}
{"category": "Math", "title": "An iterative method for numerical integration of rational functions", "abstract": "The rational Landen transformations are used to produce a highly efficient numerical method for the integration of rational functions."}
{"category": "Math", "title": "An invitation to toric degenerations", "abstract": "This is an expository paper which explores the ideas of the authors' paper \"From Affine Geometry to Complex Geometry\", arXiv:0709.2290. We explain the basic ideas of the latter paper by going through a large number of concrete, increasingly complicated examples."}
{"category": "Math", "title": "A class of logarithmic integrals", "abstract": "We present a systematic study of integrals over [0,1] where the integrand is of the form Q(x) log log 1/x. Here Q is a rational function."}
{"category": "Math", "title": "The integrals in Gradshteyn and Ryzhik. Part 11: The incomplete beta function", "abstract": "The table of Gradshteyn and Ryzhik contains some integrals that can be expressed in terms of the incomplete beta function. We describe some elementary properties of this function and use them to check some of the formulas in the mentioned table."}
{"category": "Math", "title": "Exterior algebras and two conjectures on finite abelian groups", "abstract": "Let G be a finite abelian group with |G|>1. Let a_1,...,a_k be k distinct elements of G and let b_1,...,b_k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that there is a permutation $\\pi$ on {1,...,k} such that a_1b_{\\pi(1)},...,a_kb_{\\pi(k)} are distinct, provided that any other prime divisor of |G| (if there is any) is greater than k!. This in particular confirms the Dasgupta-Karolyi-Serra-Szegedy conjecture for abelian p-groups. We also pose a new conjecture involving determinants and characters, and show that its validity implies Snevily's conjecture for abelian groups of odd order. Our methods involve exterior algebras and characters."}
{"category": "Math", "title": "Exponential sums nondegenerate relative to a lattice", "abstract": "Our previous theorems on exponential sums often did not apply or did not give sharp results when certain powers of a variable appearing in the polynomial were divisible by p. We remedy that defect in this paper by systematically applying \"p-power reduction,\" making it possible to strengthen and extend our earlier results."}
{"category": "Math", "title": "On almost universal mixed sums of squares and triangular numbers", "abstract": "In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form $x^2+y^2+10z^2$, equivalently the form $2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $T_z$ denotes the triangular number $z(z+1)/2$. Given positive integers $a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ax^2+by^2+cT_z$ represents sufficiently large integers and establish similar results for the forms $ax^2+bT_y+cT_z$ and $aT_x+bT_y+cT_z$. Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form $2ax^2+y^2+z^2$ if and only if all prime divisors of $a$ are congruent to 1 modulo 4. (ii) The form $ax^2+y^2+T_z$ is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of $a$ is congruent to 1 or 3 modulo 8. (iii) $ax^2+T_y+T_z$ is almost universal if and only if all odd prime divisors of $a$ are congruent to 1 modulo 4. (iv) When $v_2(a)\\not=3$, the form $aT_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $a$ are congruent to 1 modulo 4 and $v_2(a)\\not=5,7,...$, where $v_2(a)$ is the 2-adic order of $a$."}
{"category": "Math", "title": "On the (ir)rationality of Kontsevich weights", "abstract": "We compute the weight of a Kontsevich graph in deformation quantization. Up to rationals, the result is Zeta(3)^2/Pi^6."}
{"category": "Math", "title": "Algebraic Values of Transcendental Functions at Algebraic Points", "abstract": "In this paper, the authors will prove that any subset of $\\overline{\\QQ}$ can be the exceptional set of some transcendental entire function. Furthermore, we could generalize this theorem to a much more general version and present a unified proof."}
{"category": "Math", "title": "Unifying Practical Uncertainty Representations: II. Clouds", "abstract": "There exist many simple tools for jointly capturing variability and incomplete information by means of uncertainty representations. Among them are random sets, possibility distributions, probability intervals, and the more recent Ferson's p-boxes and Neumaier's clouds, both defined by pairs of possibility distributions. In the companion paper, we have extensively studied a generalized form of p-box and situated it with respect to other models . This paper focuses on the links between clouds and other representations. Generalized p-boxes are shown to be clouds with comonotonic distributions. In general, clouds cannot always be represented by random sets, in fact not even by 2-monotone (convex) capacities."}
{"category": "Math", "title": "Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces", "abstract": "We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term--the top one--with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata-Viehweg vanishing to bear."}
{"category": "Math", "title": "Strongly t-logarithmic t-generating sets: Geometric properties of some soluble groups", "abstract": "We introduce the concept of a strongly t-logarithmic t-generating set for a Z[t,t^{-1}]-module, which enables us to prove that a large class of soluble groups are not almost convex. We also prove some results about dead-end depth."}
{"category": "Math", "title": "Breuil's classification of $p$-divisible groups over regular local rings of arbitrary dimension", "abstract": "Let $k$ be a perfect field of characteristic $p \\geq 3$. We classify $p$-divisible groups over regular local rings of the form $W(k)[[t_1,...,t_r,u]]/(u^e+pb_{e-1}u^{e-1}+...+pb_1u+pb_0)$, where $b_0,...,b_{e-1}\\in W(k)[[t_1,...,t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin."}
{"category": "Math", "title": "Locally Connected HL Compacta", "abstract": "It is consistent with MA plus not CH that there is a locally connected hereditarily Lindelof compact space which is not metrizable."}
{"category": "Math", "title": "A remark on Khovanov homology and two-fold branched covers", "abstract": "Examples of knots and links distinguished by the total rank of their Khovanov homology but sharing the same two-fold branched cover are given. As a result, Khovanov homology does not yield an invariant of two-fold branched covers."}
{"category": "Math", "title": "On satellites in semi-abelian categories: Homology without projectives", "abstract": "Working in a semi-abelian context, we use Janelidze's theory of generalised satellites to study universal properties of the Everaert long exact homology sequence. This results in a new definition of homology which does not depend on the existence of projective objects. We explore the relations with other notions of homology, and thus prove a version of the higher Hopf formulae. We also work out some examples."}
{"category": "Math", "title": "Curvature structure of self-dual 4-manifolds", "abstract": "We show the existence of a modified Cliff(1,1) structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds."}
{"category": "Math", "title": "SPM Bulletin 25", "abstract": "Contents: 1. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, I; 2. Frechet-Urysohn fans in free topological groups; 3. Packing index of subsets in Polish groups; 4. Symmetric monochromatic subsets in colorings of the Lobachevsky plane; 5. Structural Ramsey theory of metric spaces and topological dynamics of isometry groups; 6. Distinguishing Number of Countable Homogeneous Relational Structures; 7. Indestructible colourings and rainbow Ramsey theorems; 8. Products of Borel subgroups; 9. Selection theorems and treeability; 10. Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces, IV; 11. A property of Cp[0, 1]; 12. A Dedekind Finite Borel Set; 13. Aronszajn Compacta; 14. A strong antidiamond principle compatible with CH; 15. On the strength of Hausdorff's gap condition; 16. Nonhomogeneous analytic families of trees; 17. Reasonable non-Radon-Nikodym ideals; 18. Continuity and related forcing; 19. An exact Ramsey principle for block sequences; 20. Baire reflection; 21. Tukey classes of ultrafilters on; 22. Countably determined compact abelian groups; 23. A topological reflection principle equivalent to Shelah's Strong Hypothesis; 24. Superfilters, Ramsey theory, and van der Waerden's Theorem."}
{"category": "Math", "title": "Notes on the Heegaard-Floer Link Surgery Spectral Sequence", "abstract": "This largely technical paper is divided into two parts: part I: An account of P. Ozsvath and Z. Szabo's construction of the link surgery spectral sequence. There are no new results here, but this part slightly modifies and expands their proof and is included as an aid to part II. part II: Some modifications of the spectral sequence suitable for knot Floer homology; an exposition of the invariance of the spectral sequence, under a suitable equivalence, from alterations of the underlying bouquet; and an account of the morphisms induced on the spectral sequences by suitable four dimensional cobordisms and how they depend upon the four manifold. This is the author's effort to investigate a question at the end of the introduction to P. Ozsvath's and Z. Szabo's paper \"On the Heegaard-Floer homology of double branched covers.\""}
{"category": "Math", "title": "More bijective Catalan combinatorics on permutations and on signed permutations", "abstract": "In this paper, we construct bijections between Dyck paths, noncrossing partitions, and 231-avoiding permutations, which send the area statistic on Dyck paths to the inversion number on noncrossing partitions and on 231-avoiding permutations. This bijection has the additional property that it simultaneously sends the major index on Dyck paths to the sum of the major index and the inverse major index on noncrossing partitions and on 231-avoiding permutations, respectively. Moreover, we provide generalizations of these constructions to the group of signed permutations."}
{"category": "Math", "title": "Quandle-like Structures From Groups", "abstract": "We define a class of quandle-like structures called pseudoquandles and analyze some of their algebraic properties."}
{"category": "Math", "title": "Real AlphaBeta-Geometries", "abstract": "By a real alphabeta-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of alpha-planes and an integrable distribution of beta-planes. We obtain a local characterization of the metric when at least one of the distributions is parallel (i.e., is a Walker geometry) and the three-dimensional distribution spanned by the alpha- and beta-distributions is integrable. The case when both distributions are parallel, which has been called two-sided Walker geometry, is obtained as a special case. We also consider real \\alpha\\beta-geometries for which the corresponding spinors are both multiple Weyl principal spinors."}
{"category": "Math", "title": "Symmetric Functions and Caps", "abstract": "Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the \"moments\" F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a product of two matrices, ultimately yielding a polynomial in q=p^d. The first matrix is independent of n while the second makes no mention of finite fields. However, the complexity of calculating each grows with m. The main tools here are the Schur-Weyl duality theorem, and some elementary properties of symmetric functions. This problem is closely to the study of maximal caps."}
{"category": "Math", "title": "The Haar system in the preduals of hyperfinite factors", "abstract": "We shall present examples of Schauder bases in the preduals to the hyperfinite factors of types $\\hbox{II}_1$, $\\hbox{II}_\\infty$, $\\hbox{III}_\\lambda$, $0 < \\lambda \\leq 1$. In the semifinite (respectively, purely infinite) setting, these systems form Schauder bases in any associated separable symmetric space of measurable operators (respectively, in any non-commutative $L^p$-space)."}
{"category": "Math", "title": "Knots yielding homeomorphic lens spaces by Dehn surgery", "abstract": "We show that there exist infinitely many pairs of distinct knots in the 3-sphere such that each pair can yield homeomorphic lens spaces by the same Dehn surgery. Moreover, each knot of the pair can be chosen to be a torus knot, a satellite knot or a hyperbolic knot, except that both cannot be satellite knots simultaneously. This exception is shown to be unavoidable by the classical theory of binary quadratic forms."}
{"category": "Math", "title": "Unbounded Fredholm modules and double operator integrals", "abstract": "In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability constraints. These can be formulated in two ways, either for spectral triples or for bounded Fredholm modules. We study the relationship between these by proving various properties of the map on unbounded self adjoint operators $D$ given by $f(D)=D(1+D^2)^{-1/2}$. In particular we prove commutator estimates which are needed for the bounded case. In fact our methods work in the setting of semifinite noncommutative geometry where one has $D$ as an unbounded self adjoint linear operator affiliated with a semi-finite von Neumann algebra $\\aM$. More precisely we show that for a pair $D,D_0$ of such operators with $D-D_0$ a bounded self-adjoint linear operator from $\\aM$ and $ ({\\bf 1}+D_0^2)^{-1/2}\\in \\sE$, where $\\sE$ is a noncommutative symmetric space associated with $\\aM$, then $$ \\Vert f(D) - f (D_0) \\Vert_{\\sE} \\leq C\\cdot \\Vert D-D_0\\Vert_{\\aM}. $$ This result is further used to show continuous differentiability of the mapping between an odd $\\sE$-summable spectral triple and its bounded counterpart."}
{"category": "Math", "title": "Non- (quantum) differentiable $C^1$-functions in the spaces with trivial Boyd indices", "abstract": "If E is a separable symmetric sequence space with trivial Boyd indices and $\\cC^E$ is the corresponding ideal of compact operators, then there exists a $C^1$-function $f_E$, a self-adjoint element $W\\in \\cC^E$ and a densely defined closed symmetric derivation $\\delta$ on $\\cC^E$ such that $W \\in Dom \\delta$, but $f_E(W) \\notin Dom \\delta$."}
{"category": "Math", "title": "Simple homotopy invariance of higher signatures", "abstract": "We prove that the higher signature for any close oriented manifold is a simple-homotopy invariant."}
{"category": "Math", "title": "Bayesian nonparametric estimators derived from conditional Gibbs structures", "abstract": "We consider discrete nonparametric priors which induce Gibbs-type exchangeable random partitions and investigate their posterior behavior in detail. In particular, we deduce conditional distributions and the corresponding Bayesian nonparametric estimators, which can be readily exploited for predicting various features of additional samples. The results provide useful tools for genomic applications where prediction of future outcomes is required."}
{"category": "Math", "title": "Central limit theorem for a many-server queue with random service rates", "abstract": "Given a random variable $N$ with values in ${\\mathbb{N}}$, and $N$ i.i.d. positive random variables $\\{\\mu_k\\}$, we consider a queue with renewal arrivals and $N$ exponential servers, where server $k$ serves at rate $\\mu_k$, under two work conserving routing schemes. In the first, the service rates $\\{\\mu_k\\}$ need not be known to the router, and each customer to arrive at a time when some servers are idle is routed to the server that has been idle for the longest time (or otherwise it is queued). In the second, the service rates are known to the router, and a customer that arrives to find idle servers is routed to the one whose service rate is greatest. In the many-server heavy traffic regime of Halfin and Whitt, the process that represents the number of customers in the system is shown to converge to a one-dimensional diffusion with a random drift coefficient, where the law of the drift depends on the routing scheme. A related result is also provided for nonrandom environments."}
{"category": "Math", "title": "Stein's method for discrete Gibbs measures", "abstract": "Stein's method provides a way of bounding the distance of a probability distribution to a target distribution $\\mu$. Here we develop Stein's method for the class of discrete Gibbs measures with a density $e^V$, where $V$ is the energy function. Using size bias couplings, we treat an example of Gibbs convergence for strongly correlated random variables due to Chayes and Klein [Helv. Phys. Acta 67 (1994) 30--42]. We obtain estimates of the approximation to a grand-canonical Gibbs ensemble. As side results, we slightly improve on the Barbour, Holst and Janson [Poisson Approximation (1992)] bounds for Poisson approximation to the sum of independent indicators, and in the case of the geometric distribution we derive better nonuniform Stein bounds than Brown and Xia [Ann. Probab. 29 (2001) 1373--1403]."}
{"category": "Math", "title": "Stability of the slow manifold in the primitive equations", "abstract": "We show that, under reasonably mild hypotheses, the solution of the forced--dissipative rotating primitive equations of the ocean loses most of its fast, inertia--gravity, component in the small Rossby number limit as $t\\to\\infty$. At leading order, the solution approaches what is known as \"geostrophic balance\" even under ageostrophic, slowly time-dependent forcing. Higher-order results can be obtained if one further assumes that the forcing is time-independent and sufficiently smooth. If the forcing lies in some Gevrey space, the solution will be exponentially close to a finite-dimensional \"slow manifold\" after some time."}
{"category": "Math", "title": "A concrete co-existential map that is not confluent", "abstract": "We give a concrete example of a co-existential map between continua that is not confluent."}
{"category": "Math", "title": "Ordinary differential system in dinension six with affine Weyl group symmetry of type $D_4^{(2)}$", "abstract": "We find a three-parameter family of ordinary differential systems in dimension six with affine Weyl group symmetry of type $D_4^{(2)}$. This is the second example which gave higher order Painlev\\'e type systems of type $D_{4}^{(2)}$. We show that we give its symmetry and holomorphy conditions. These symmetries, holomorphy conditions and invariant divisors are new."}
{"category": "Math", "title": "Hyperbolic conservation laws on manifolds with limited regularity", "abstract": "We introduce a formulation of the initial and boundary value problem for nonlinear hyperbolic conservation laws posed on a differential manifold endowed with a volume form, possibly with a boundary; in particular, this includes the important case of Lorentzian manifolds. Only limited regularity is assumed on the geometry of the manifold. For this problem, we establish the existence and uniqueness of an L1 semi-group of weak solutions satisfying suitable entropy and boundary conditions."}
{"category": "Math", "title": "A Short History of Markov Chain Monte Carlo: Subjective Recollections from Incomplete Data", "abstract": "We attempt to trace the history and development of Markov chain Monte Carlo (MCMC) from its early inception in the late 1940s through its use today. We see how the earlier stages of Monte Carlo (MC, not MCMC) research have led to the algorithms currently in use. More importantly, we see how the development of this methodology has not only changed our solutions to problems, but has changed the way we think about problems."}
{"category": "Math", "title": "On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase", "abstract": "A uniformly random graph on $n$ vertices with a fixed degree sequence, obeying a $\\gamma$ subpower law, is studied. It is shown that, for $\\gamma>3$, in a subcritical phase with high probability the largest component size does not exceed $n^{1/\\gamma+\\varepsilon_n}$, $\\varepsilon_n=O(\\ln\\ln n/\\ln n)$, $1/\\gamma$ being the best power for this random graph. This is similar to the best possible $n^{1/(\\gamma-1)}$ bound for a different model of the random graph, one with independent vertex degrees, conjectured by Durrett, and proved recently by Janson."}
{"category": "Math", "title": "The density of integral points on hypersurfaces of degree at least four", "abstract": "Let $f$ be a polynomial of degree at least four with integer-valued coefficients. We establish new bounds for the density of integer solutions to the equation $f=0$, using an iterated version of Heath-Browns $q$-analogue of van der Corput's method of exponential sums."}
{"category": "Math", "title": "Krull dimension of solvable groups", "abstract": "In this paper we prove that free solvable groups have finite Krull dimension. In fact, this is true for much wider class of solvable groups, termed rigid groups. Along the way we study the algebraic structure of the limit solvable groups (fully residually free solvable groups)."}
{"category": "Math", "title": "Morita Equivalence of Brandt Semigroup Algebras", "abstract": "We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $\\ell(B(I,G))$ and $\\ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, we show that if $G$ is an amenable group, then for a wide class of Banach $\\ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(\\ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant."}
{"category": "Math", "title": "A binary tree representation for the 2-adic valuation of a sequence arising from a rational integral", "abstract": "We present a binary tree that describes the 2-adic valuation of a sequence of coefficients arising from the evaluation of a rational integral."}
{"category": "Math", "title": "Parseval frames for ICC groups", "abstract": "We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of norm $\\frac{1}{\\sqrt{N}}$, the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by an Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left regular representation of a free group has this property."}
{"category": "Math", "title": "Probability and Fourier duality for affine iterated function systems", "abstract": "Let $d$ be a positive integer, and let $\\mu$ be a finite measure on $\\br^d$. In this paper we ask when it is possible to find a subset $\\Lambda$ in $\\br^d$ such that the corresponding complex exponential functions $e_\\lambda$ indexed by $\\Lambda$ are orthogonal and total in $L^2(\\mu)$. If this happens, we say that $(\\mu, \\Lambda)$ is a spectral pair. This is a Fourier duality, and the $x$-variable for the $L^2(\\mu)$-functions is one side in the duality, while the points in $\\Lambda$ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures $\\mu$ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in $\\br^d$; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems."}
{"category": "Math", "title": "Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems", "abstract": "We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of given dimension n having m singular points. As a function of n,m, this bound turns out to be double exponential in the precise sense explained in the paper. As a corollary, we obtain a solution of the so called restricted infinitesimal Hilbert 16th problem, an explicit upper bound for the number of isolated zeros of Abelian integrals which is polynomially growing as the Hamiltonian tends to the degeneracy locus. This improves the exponential bounds recently established by A. Glutsyuk and Yu. Ilyashenko."}
{"category": "Math", "title": "Hecke-Clifford algebras and spin Hecke algebras III: the trigonometric type", "abstract": "The notion of trigonometric spin double affine Hecke algebras (tsDaHa) and trigonometric double affine Hecke-Clifford algebras (tDaHCa) associated to classical Weyl groups are introduced. The PBW basis property is established. An algebra isomorphism relating tDaHCa to tsDaHa is obtained."}
{"category": "Math", "title": "On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem", "abstract": "We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group."}
{"category": "Math", "title": "Models of expansions of N with no end extensions", "abstract": "We deal with models of Peano arithmetic (specifically with a question of Ali Enayat). The methods are from creature forcing. We find an expansion of N such that its theory has models with no (elementary) end extensions. In fact there is a Borel uncountable set of subsets of N such that expanding N by any uncountably many of them suffice. Also we find arithmetically closed A with no definably closed ultrafilter on it."}
{"category": "Math", "title": "Continuous Bounded Cohomology of Topological Semigroups", "abstract": "In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices."}
{"category": "Math", "title": "On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues", "abstract": "We prove that the isoperimetric inequality due to Hersch-Payne-Schiffer for the n-th nonzero Steklov eigenvalue of a bounded simply-connected planar domain is sharp for all n=1,2,... The equality is attained in the limit by a sequence of simply-connected domains degenerating to the disjoint union of n identical disks. We give a new proof of this inequality for n=2 and show that it is strict in this case. Related results are also obtained for the product of two consecutive Steklov eigenvalues."}
{"category": "Math", "title": "Beilinson-Hodge cycles on semiabelian varieties", "abstract": "Beilinson conjectured that all rational cycles of type (q,q) on the qth cohomology of a smooth complex algebraic variety should come from motivic cohomology. The purpose of this note is to prove this when the variety is a semiabelian variety or a product of curves. The proof is based on the study of invariants under the Mumford-Tate group."}
{"category": "Math", "title": "Nilpotent orbits in classical Lie algebras over finite fields of characteristic 2 and the Springer correspondence", "abstract": "Let $G$ be an adjoint algebraic group of type $B$, $C$, or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of $G$. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained."}
{"category": "Math", "title": "Representations of $asl_2$", "abstract": "We study representations of the simple Lie antialgebra $asl_2$ introduced by Ovsienko. We show that representations of $asl_2$ are closely related to the famous ghost Casimir element of the universal enveloping algebra $osp(1|2)$. We prove that $asl_2$ has no non-trivial finite-dimensional representations; we define and classify some particular infinite-dimensional representations that we call weighted representations."}
{"category": "Math", "title": "Nilpotent orbits in the dual of classical Lie algebras in characteristic 2 and the Springer correspondence", "abstract": "Let $G$ be a simply connected algebraic group of type $B,C$ or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the dual vector space of the Lie algebra of $G$. In particular, we classify the nilpotent orbits in the duals of symplectic and orthogonal Lie algebras over algebraically closed or finite fields of characteristic 2."}
{"category": "Math", "title": "Up-and-Down and the Percentile-Finding Problem", "abstract": "Up-and-Down (U&D) is a popular sequential design for estimating threshold percentiles in binary experiments. However, U&D application practices have stagnated, and significant gaps in understanding its properties persist. The first part of my work aims to fill gaps in U&D theory. New results concerning stationary distribution properties are proven. A second focus of this study is nonparametric U&D estimation. An improvement to isotonic regression called \"centered isotonic regression\" (CIR), and a new averaging estimator called \"auto-detect\" are introduced and their properties studied. Bayesian percentile-finding designs, most notably the continual reassessment method (CRM) developed for Phase I clinical trials, are also studied. In general, CRM convergence depends upon random run-time conditions -- meaning that convergence is not always assured. Small-sample behavior is studied as well. It is shown that CRM is quite sensitive to outlier sub-sequences of thresholds, resulting in highly variable small-sample behavior between runs under identical conditions. Nonparametric CRM variants exhibit a similar sensitivity. Ideas to combine the advantages of U&D and Bayesian designs are examined. A new approach is developed, using a hybrid framework, that evaluates the evidence for overriding the U&D allocation with a Bayesian one."}
{"category": "Math", "title": "A Colding-Minicozzi Stability inequality and its applications", "abstract": "We consider operators $L$ acting on functions on a Riemannian surface, $\\Sigma$, of the form $L = \\Delta + V +a K.$ Here $\\Delta$ is the Laplacian of $\\Sigma$, $V$ a non-negative potential on $\\Sigma$, K the Gaussian curvature and $a$ is a non-negative constant. Such operators $L$ arise as the stability operator of $\\Sigma$ immersed in a Riemannian 3-manifold with constant mean curvature (for particular choices of $V$ and $a$). We assume L is nonpositive acting on functions compactly supported on $\\Sigma$ and we obtain results in the spirit of some theorems of Ficher-Colbrie-Schoen, Colding-Minicozzi, and Castillon. We extend these theorems to $a \\leq 1/4$. We obtain results on the conformal type of $\\Sigma$ and a distance (to the boundary) lemma."}
{"category": "Math", "title": "Non-structure in lambda^{++} using instances of WGCH", "abstract": "We try to redo, improve and continue the non-structure parts in some works on a.e.c., which uses weak diamond, in lambda^+ and lambda^{++} getting better and more results and do what is necessary for the book on a.e.c. Comparing with math.LO/9805146 we make the context closer to the examples, hence hopefully improve transparency, though losing some generality. Toward this we work also on the positive theory, i.e. structure side of \"low frameworks\" like almost good lambda-frames."}
{"category": "Math", "title": "On the concentration and the convergence rate with a moment condition in first passage percolation", "abstract": "We consider the first passage percolation model on the ${\\bf Z}^d$ lattice. In this model, we assign independently to each edge $e$ a non-negative passage time $t(e)$ with a common distribution $F$. Let $a_{0,n}$ be the passage time from the origin to $(n,0,..., 0)$. Under the exponential tail assumption, Kesten (1993) and Talagrand (1995) investigated the concentration of $a_{0,n}$ from its mean using different methods. With this concentration and the exponential tail assumption, Alexander gave an estimate for the convergence rate for ${\\bf E} a_{0,n}$. In this paper, focusing on a moment condition, we reinvestigate the concentration and the convergence rate for $a_{0,n}$ using a special martingale structure."}
{"category": "Math", "title": "Categoricity and solvability of A.E.C., quite highly", "abstract": "We investigate in ZFC what can be the family of large enough cardinals mu in which an a.e.c. K is categorical or even just solvable. We show that for not few cardinals lambda<mu there is a superlimit model in K_lambda. Moreover, our main result is that we can find a good lambda-frame s categorical in lambda such that K_s subseteq K_lambda. We then show how to use [Sh:705] to get categoricity in every large enough cardinality if K has cases of mu-amalgamation for enough mu and 2^mu<2^{mu^{+1}} <... < 2^{mu^{+n}}... for enough mu."}
{"category": "Math", "title": "Twisted cyclic theory, equivariant KK theory and KMS States", "abstract": "Recently, examples of an index theory for KMS states of circle actions were discovered, \\cite{CPR2,CRT}. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C*-algebra A to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in \\cite{CPR2,CRT} as special cases. Next we use the Araki-Woods III_\\lambda representations of the Fermion algebra to show that there are examples which are not Cuntz-Krieger systems."}
{"category": "Math", "title": "Skew-product representations of multidimensional Dunkl Markov processes", "abstract": "In this paper we obtain skew-product representations of the multidimensional Dunkl processes which generalize the skew-product decomposition in dimension 1 obtained in L. Gallardo and M. Yor. Some remarkable properties of the Dunkl martingales. S\\'{e}minaire de Probabilit\\'{e}s XXXIX, 2006. We also study the radial part of the Dunkl process, i.e. the projection of the Dunkl process on a Weyl chamber."}
{"category": "Math", "title": "Exponential Lower Bounds for Quasimodes of Semiclassical Schr\\\"{o}dinger Operators", "abstract": "We prove quantitative unique continuation results for the semiclassical Schrodinger operator on smooth, compact domains. These take the form of exponentially decreasing (in h) local L^{2} lower bounds for exponentially precise quasimodes. We also show that these lower bounds are sharp in h, and that, moreover, the hypothesized quasimode accuracy is also sharp."}
{"category": "Math", "title": "Limit laws for the energy of a charged polymer", "abstract": "In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy \\[H_n=\\sum_{1\\le j<k\\le n}\\omega_j\\omega_k1_{\\{S_j=S_k\\}}\\] of the polymer $\\{S_1,...,S_n\\}$ equipped with random electrical charges $\\{\\omega_1,...,\\omega_n\\}$. Our approach is based on comparison of the moments between $H_n$ and the self-intersection local time \\[Q_n=\\sum_{1\\le j<k\\le n}1_{\\{S_j=S_k\\}}\\] run by the $d$-dimensional random walk $\\{S_k\\}$. As partially needed for our main objective and partially motivated by their independent interest, the central limit theorems and exponential integrability for $Q_n$ are also investigated in the case $d\\ge3$."}
{"category": "Math", "title": "Explicit and Almost Explicit Spectral Calculations for Diffusion Operators", "abstract": "The diffusion operator $$ H_D=-\\frac12\\frac d{dx}a\\frac d{dx}-b\\frac d{dx}=-\\frac12\\exp(-2B)\\frac d{dx}a\\exp(2B)\\frac d{dx}, $$ where $B(x)=\\int_0^x\\frac ba(y)dy$, defined either on $R^+=(0,\\infty)$ with the Dirichlet boundary condition at $x=0$, or on $R$, can be realized as a self-adjoint operator with respect to the density $\\exp(2Q(x))dx$. The operator is unitarily equivalent to the Schr\\\"odinger-type operator $H_S=-\\frac12\\frac d{dx}a\\frac d{dx}+V_{b,a}$, where $V_{b,a}=\\frac12(\\frac{b^2}a+b')$. We obtain an explicit criterion for the existence of a compact resolvent and explicit formulas up to the multiplicative constant 4 for the infimum of the spectrum and for the infimum of the essential spectrum for these operators. We give some applications which show in particular how $\\inf\\sigma(H_D)$ scales when $a=\\nu a_0$ and $b=\\gamma b_0$, where $\\nu$ and $\\gamma$ are parameters, and $a_0$ and $b_0$ are chosen from certain classes of functions. We also give applications to self-adjoint, multi-dimensional diffusion operators."}
{"category": "Math", "title": "On mean central limit theorems for stationary sequences", "abstract": "In this paper, we give estimates of the minimal ${\\mathbb{L}}^1$ distance between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary sequences satisfying projective criteria in the style of Gordin or weak dependence conditions."}
{"category": "Math", "title": "Global well-posedness of incompressible flow in porous media with critical diffusion in Besov spaces", "abstract": "In this paper we study the model of heat transfer in a porous medium with a critical diffusion. We obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in Besov spaces $\\dot B^{3/p}_{p,1}(\\mathbb{R}^3)$ with $1\\le p\\le\\infty$ by the method of modulus of continuity and Fourier localization technique."}
{"category": "Math", "title": "On the topological entropy of families of braids", "abstract": "A method for computing the topological entropy of each braid in an infinite family, making use of Dynnikov's coordinates on the boundary of Teichm\\\"uller space, is described. The method is illustrated on two two-parameter families of braids."}
{"category": "Math", "title": "Joint continuity of the local times of fractional Brownian sheets", "abstract": "Let $B^H=\\{B^H(t),t\\in{{\\mathbb{R}}_+^N}\\}$ be an $(N,d)$-fractional Brownian sheet with index $H=(H_1,...,H_N)\\in(0,1)^N$ defined by $B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\\in {\\mathbb{R}}_+^N),$ where $B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<\\sum_{\\ell=1}^NH_{\\ell}^{-1}$, then the local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global H\\\"{o}lder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$."}
{"category": "Math", "title": "Minimal average degree aberration and the state polytope for experimental designs", "abstract": "For a particular experimental design, there is interest in finding which polynomial models can be identified in the usual regression set up. The algebraic methods based on Groebner bases provide a systematic way of doing this. The algebraic method does not in general produce all estimable models but it can be shown that it yields models which have minimal average degree in a well-defined sense and in both a weighted and unweighted version. This provides an alternative measure to that based on \"aberration\" and moreover is applicable to any experimental design. A simple algorithm is given and bounds are derived for the criteria, which may be used to give asymptotic Nyquist-like estimability rates as model and sample sizes increase."}
{"category": "Math", "title": "A quantized Tits-Kantor-Koecher algebra", "abstract": "We propose a quantum analogue of a Tits-Kantor-Koecher algebra with a Jordan torus as an coordinated algebra by looking at the vertex operator construction over a Fock space."}
{"category": "Math", "title": "Evaluations of the twisted Alexander polynomials of 2-bridge knots at $\\pm 1$", "abstract": "Let $H(p)$ be the set of 2-bridge knots $K$ whose group $G$ is mapped onto a non-trivial free product, $Z/2 * Z/p$, $p$ being odd. Then there is an algebraic integer $s_0$ such that for any $K$ in $H(p)$, $G$ has a parabolic representation $\\rho$ into $SL(2, Z[s_0]) \\subset SL(2,C)$. Let $\\Delta(t)$ be the twisted Alexander polynomial associated to $\\rho$. Then we prove that for any $K$ in $H(p)$, $\\Delta(1)=-2s_0^{-1}$ and $\\Delta(-1)=-2s_0^{-1}\\mu^2$, where $s_0^{-1}, \\mu \\in Z[s_0]$. The number $\\mu$ can be recursively evaluated."}
{"category": "Math", "title": "Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions", "abstract": "Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach to the problem of non-parametric kernel drift estimation in the one-dimensional recurrent case. As a particular example we obtain the rate of convergence of the Nadaraya--Watson estimator in the case of a locally H\\\"{o}lder-continuous drift."}
{"category": "Math", "title": "A theory of hierarchical consequence and conditionals", "abstract": "We introduce A-ranked preferential structures and combine them with an accessibility relation. This framework allows us to formalize contrary to duty obligations. Representation results are proved."}
{"category": "Math", "title": "Roadmap for preferential logics", "abstract": "We give an overview of logical and semantical rules for nonmonotonic and related logics."}
{"category": "Math", "title": "Reactive preferential structures and nonmonotonic consequence", "abstract": "We introduce information bearing systems (IBRS) as an abstraction of many logical systems. We define a general semantics for IBRS, and show that IBRS generalize in a natural way preferential semantics and solve open representation problems."}
{"category": "Math", "title": "Cumulativity without closure of the domain under finite unions", "abstract": "For nonmonotonic logics, Cumulativity is an important rule. We show here that Cumulativity fans out into an infinity of different conditions, if the domain is not closed under finite unions."}
{"category": "Math", "title": "Decoration invariants for horseshoe braids", "abstract": "The Decoration Conjecture describes the structure of the set of braid types of Smale's horseshoe map ordered by forcing, providing information about the order in which periodic orbits can appear when a horseshoe is created. A proof of this conjecture is given for the class of so-called lone decorations, and it is explained how to calculate associated braid conjugacy invariants which provide additional information about forcing for horseshoe braids."}
{"category": "Math", "title": "Isomonodromic deformatiion with an irregular singularity and hyperelliptic curve", "abstract": "In this paper, we extend the result of Kitaev and Korotkin to the case where a monodromy-preserving deformation has an irregular singularity. For the monodromy-preserving deformation, we obtain the $\\tau$-function whose deformation parameters are the positions of regular singularities and the parameter $t$ of an irregular singularity. Furthermore, the $\\tau$-function is expressed by the hyperelliptic $\\Theta$ function moving the argument $\\z$ and the period $\\B,$ where $t$ and the positions of regular singularities move $z$ and $\\B,$ respectively."}
{"category": "Math", "title": "Global Smooth Effects and Well-Posedness for the Derivative Nonlinear Schr\\\"odinger Equation with Small Rough Data", "abstract": "\\rm We obtain the global smooth effects for the solutions of the linear Schr\\\"odinger equation in anisotropic Lebesgue spaces. Applying these estimates, we study the Cauchy problem for the generalized elliptical and non-elliptical derivative nonlinear Schr\\\"odinger equations (DNLS) and get the global well posedness of solutions with small data in modulation spaces $M^{3/2}_{2,1}(\\mathbb{R}^n)$. Noticing that $H^{\\tilde{s}} \\subset M^s_{2,1}$ $(\\tilde{s}-s>n/2)$ is an optimal inclusion, we have shown the global well posedness of DNLS with a class of very rough data."}
{"category": "Math", "title": "Identities for hyperelliptic P-functions of genus one, two and three in covariant form", "abstract": "We give a covariant treatment of the quadratic differential identities satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of genera 1, 2 and 3."}
{"category": "Math", "title": "Some properties of multivariate measures of concordance", "abstract": "We explore the consequences of a set of axioms which extend Scarsini's axioms for bivariate measures of concordance to the multivariate case and exhibit the following results: (1) A method of extending measures of concordance from the bivariate case to arbitrarily high dimensions. (2) A formula expressing the measure of concordance of the random vectors $(\\pm X_1,...,\\pm X_n)$ in terms of the measures of concordance of the \"marginal\" random vectors $(X_{i_1},...,X_{i_k})$. (3) A method of expressing the measure of concordance of an odd-dimensional copula in terms of the measures of concordance of its even-dimensional marginals. (4) A family of relations which exist between the measures of concordance of the marginals of a given copula."}
{"category": "Math", "title": "Top homology of hypergraph matching complexes, $p$-cycle complexes and Quillen complexes of symmetric groups", "abstract": "We investigate the representation of a symmetric group $S_n$ on the homology of its Quillen complex at a prime $p$. For homology groups in small codimension, we derive an explicit formula for this representation in terms of the representations of symmetric groups on homology groups of $p$-uniform hypergraph matching complexes. We conjecture an explicit formula for the representation of $S_n$ on the top homology group of the corresponding hypergraph matching complex when $n \\equiv 1 \\bmod p$. Our conjecture follows from work of Bouc when $p=2$, and we prove the conjecture when $p=3$."}
{"category": "Math", "title": "Some relative stable categories are compactly generated", "abstract": "Let G be a finite group. The stable module category of G has been applied extensively in group representation theory. In particular, it has been used to great effect that it is a triangulated category which is compactly generated. Let H be a subgroup of G. It is possible to define a stable module category of G relative to H. It too is a triangulated category, but no non-trivial examples have been known where this relative stable category was compactly generated. We show here that the relative stable category is compactly generated if the group algebra of H has finite representation type. In characteristic p, this is equivalent to the Sylow p-subgroups of H being cyclic."}
{"category": "Math", "title": "Saturated subfields and invariants of finite groups", "abstract": "Every subfield $\\kk(\\phi)$ of the field of rational functions $\\kk(x_1,...,x_n)$ is contained in a unique maximal subfield of the form $\\kk(\\psi)$. The element $\\psi$ is called generative for the element $\\phi$. A subfield of $\\kk(x_1,...,x_n)$ is called saturated if it contains a generative element of each its element. We study the saturation property for subfields of invariants $\\kk(x_1,...,x_n)^G$, where $G$ is a finite group of automorphisms of the field $\\kk(x_1,...,x_n)$."}
{"category": "Math", "title": "Multiple solutions for the $p-$laplace operator with critical growth", "abstract": "In this note we show the existence of at least three nontrivial solutions to the following quasilinear elliptic equation $-\\Delta_p u = |u|^{p^*-2}u + \\lambda f(x,u)$ in a smooth bounded domain $\\Omega$ of $\\R^N$ with homogeneous Dirichlet boundary conditions on $\\partial\\Omega$, where $p^*=Np/(N-p)$ is the critical Sobolev exponent and $\\Delta_p u =div(|\\nabla u|^{p-2}\\nabla u)$ is the $p-$laplacian. The proof is based on variational arguments and the classical concentrated compactness method."}
{"category": "Math", "title": "On slim double Lie groupoids", "abstract": "We prove that every slim double Lie groupoid with proper core action is completely determined by a factorization of a certain canonically defined \"diagonal\" Lie groupoid."}
{"category": "Math", "title": "On independent sets in purely atomic probability spaces with geometric distribution", "abstract": "We are interested in constructing concrete independent events in purely atomic probability spaces with geometric distribution. Among other facts we prove that there are uncountable many sequences of independent events."}
{"category": "Math", "title": "Algebraic characterization of the isometries of the hyperbolic 5-space", "abstract": "Using the representation of the isometries as 2x2 invertible matrices over the division algebra $\\H$ of quaternions, we give an algebraic characterization of the dynamical types of the orientation-preserving isometries of the hyperbolic 5-space. We also determine the conjugacy classes and the conjugacy classes of centralizers or the z-classes in $GL(2, \\H)$."}
{"category": "Math", "title": "The Annihilating-Ideal Graph of Commutative Rings I", "abstract": "Let $R$ be a commutative ring with ${\\Bbb{A}}(R)$ its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the {\\it annihilating-ideal graph} of $R$, denoted by ${\\Bbb{AG}}(R)$. It is the (undirected) graph with vertices ${\\Bbb{A}}(R)^*:={\\Bbb{A}}(R)\\setminus\\{(0)\\}$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. First, we study some finiteness conditions of ${\\Bbb{AG}}(R)$. For instance, it is shown that if $R$ is not a domain, then ${\\Bbb{AG}}(R)$ has ACC (resp., DCC) on vertices if and only if $R$ is Noetherian (resp., Artinian). Moreover, the set of vertices of ${\\Bbb{AG}}(R)$ and the set of nonzero proper ideals of $R$ have the same cardinality when $R$ is either an Artinian or a decomposable ring. This yields for a ring $R$, ${\\Bbb{AG}}(R)$ has $n$ vertices $(n\\geq 1)$ if and only if $R$ has only $n$ nonzero proper ideals. Next, we study the connectivity of ${\\Bbb{AG}}(R)$. It is shown that ${\\Bbb{AG}}(R)$ is a connected graph and $diam(\\Bbb{AG})(R)\\leq 3$ and if ${\\Bbb{AG}}(R)$ contains a cycle, then $gr({\\Bbb{AG}}(R))\\leq 4$. Also, rings $R$ for which the graph ${\\Bbb{AG}}(R)$ is complete or star, are characterized, as well as rings $R$ for which every vertex of ${\\Bbb{AG}}(R)$ is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs."}
{"category": "Math", "title": "The Annihilating-Ideal Graph of Commutative Rings II", "abstract": "In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in Part I (see [5]). Let $R$ be a commutative ring with ${\\Bbb{A}}(R)$ its set of ideals with nonzero annihilator and $Z(R)$ its set of zero divisors. The annihilating-ideal graph of $R$ is defined as the (undirected) graph ${\\Bbb{AG}}(R)$ that its vertices are $\\Bbb{A}(R)^* =\\Bbb{A}(R)\\hspace{-1mm}\\setminus\\{(0)\\}$ in which for every distinct vertices $I$ and $J$, $I\\hspace{-0.6mm}-\\hspace{-1.7mm}-\\hspace{-1.7mm}-\\hspace{-0.5mm}J$ is an edge if and only if $IJ=(0)$. First, we study the diameter of ${\\Bbb{AG}}(R)$. A complete characterization for the possible diameter is given exclusively in terms of the ideals of $R$ when either $R$ is a Noetherian ring or $Z(R)$ is not an ideal of $R$. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either $\\chi({\\Bbb{AG}}(R))\\leq 2$ or $R$ is reduced and $\\chi({\\Bbb{AG}}(R))\\leq \\infty$. Also it is shown that for each reduced ring $R$, $\\chi(\\Bbb{AG}(R))= cl(\\Bbb{AG}(R))$. Moreover, if $\\chi(\\Bbb{AG}(R))$ is finite, then $R$ has a finite number of minimal primes, and if $n$ is this number, then $\\chi(\\Bbb{AG}(R))= cl(\\Bbb{AG}(R))= n$. Finally, we show that for a Noetherian ring $R$, $cl(\\Bbb{AG}(R))$ is finite if and only if for every ideal $I$ of $R$ with $I^2=(0)$, $I$ has finite number of $R$-submodules."}
{"category": "Math", "title": "Nonparametric goodness-of fit testing in quantum homodyne tomography with noisy data", "abstract": "In the framework of quantum optics, we study the problem of goodness-of-fit testing in a severely ill-posed inverse problem. A novel testing procedure is introduced and its rates of convergence are investigated under various smoothness assumptions. The procedure is derived from a projection-type estimator, where the projection is done in $\\mathbb{L}_2$ distance on some suitably chosen pattern functions. The proposed methodology is illustrated with simulated data sets."}
{"category": "Math", "title": "Uniqueness of pairings in Hopf-cyclic cohomology", "abstract": "We show that all pairings defined in the literature extending Connes-Moscovici characteristic map in Hopf-cyclic cohomology are isomorphic as natural transformations of derived double functors."}
{"category": "Math", "title": "The Stadium Theorem", "abstract": "A proof of a curious planar embedding theorem."}
{"category": "Math", "title": "Maximizing orbits for higher dimensional convex billiards", "abstract": "The main result of this paper is, that for convex billiards in higher dimensions, in contrast with 2D case, for every point on the boundary and for every $n$ there always exist billiard trajectories developing conjugate points at the $n$-th collision with the boundary. We shall explain that this is a consequence of the following variational property of the billiard orbits in higher dimension. If a segment of an orbit is locally maximizing, then it can not pass too close to the boundary. This fact follows from the second variation formula for the Length functional. It turns out that this formula behaves differently with respect to \"longitudinal\" and \"transversal\" variations."}
{"category": "Math", "title": "Baric structures on triangulated categories and coherent sheaves", "abstract": "We introduce the notion of a \"baric structure\" on a triangulated category, as an abstraction of S. Morel's weight truncation formalism for mixed l-adic sheaves. We study these structures on the derived category D_G(X) of G-equivariant coherent sheaves on a G-scheme X. Our main result shows how to endow this derived category with a family of nontrivial baric structures when G acts on X with finitely many orbits. We also describe a general construction for producing a new t-structure on a triangulated category equipped with given t- and baric structures, and we prove that the staggered t-structures on D_G(X) introduced by the first author arise in this way."}
{"category": "Math", "title": "Purity and decomposition theorems for staggered sheaves", "abstract": "Two major results in the theory of l-adic mixed constructible sheaves are the purity theorem (every simple perverse sheaf is pure) and the decomposition theorem (every pure object in the derived category is a direct sum of shifts of simple perverse sheaves). In this paper, we prove analogues of these results for coherent sheaves. Specificially, we work with staggered sheaves, which form the heart of a certain t-structure on the derived category of equivariant coherent sheaves. We prove, under some reasonable hypotheses, that every simple staggered sheaf is pure, and that every pure complex of coherent sheaves is a direct sum of shifts of simple staggered sheaves."}
{"category": "Math", "title": "Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds", "abstract": "A manifold (M,I,J,K) is called hypercomplex if I,J,K are complex structures satisfying quaternionic relations. A quaternionic Hermitian metric is called HKT (hyperkaehler with torsion) if $Id\\omega_I = Jd \\omega_J=Kd\\omega_K$, where $\\omega_I,\\omega_J, \\omega_K$ are Hermitian forms associated with I, J, K. A Hermitian metric $\\omega$ on a complex manifold is called balanced if $d^*\\omega=0$. We show that balanced HKT metrics are precisely the quaternionic Calabi-Yau metrics defined in terms of the quaternionic Monge-Ampere equation. In particular, a balanced HKT-metric is unique in its cohomology class, and it always exists if the quaternionic Calabi-Yau theorem is true. We investigate the cohomological properties of strong HKT metrics (the quaternionic Hermitian metrics, satisfying, in addition to the HKT condition, the relation $dd^c \\omega=0$), and show that the space of strong HKT metrics is finite-dimensional. Using Howe's duality for representations of Sp(n), we prove a hyperkaehler version of Hodge-Riemann bilinear relations. We use it to show that a manifold admitting a balanced HKT-metric never admits a strong HKT-metric, if $\\dim_\\R M \\geq 12$."}
{"category": "Math", "title": "Open book decompositions and stable Hamiltonian structures", "abstract": "We show that every open book decomposition of a contact 3-manifold can be represented (up to isotopy) by a smooth R-invariant family of pseudoholomorphic curves on its symplectization with respect to a suitable stable Hamiltonian structure. In the planar case, this family survives small perturbations, and thus gives a concrete construction of a stable finite energy foliation that has been used in various applications to planar contact manifolds, including the Weinstein conjecture and equivalence of strong and Stein fillability."}
{"category": "Math", "title": "Entire curves, Integral sets and Principal bundles", "abstract": "We compare the behaviour of entire curves and integral sets, in particular in relation to locally trivial fiber bundles, algebraic groups and finite ramified covers over semi-abelian varieties."}
{"category": "Math", "title": "Chen--Ruan cohomology of some moduli spaces", "abstract": "Let X be a compact connected Riemann surface of genus at least two. We compute the Chen--Ruan cohomology ring of the moduli space of stable PSL(2, C)--bundles of nontrivial second Stiefel--Whitney class over X."}
{"category": "Math", "title": "Concentration of 1-Lipschitz maps into an infinite dimensional $\\ell^p$-ball with $\\ell^q$-distance function", "abstract": "In this paper, we study the L\\'{e}vy-Milman concentration phenomenon of 1-Lipschitz maps into infinite dimensional metric spaces. Our main theorem asserts that the concentration to an infinite dimensional $\\ell^p$-ball with the $\\ell^q$-distance function for $1\\leq p<q\\leq +\\infty$ is equivalent to the concentration to the real line."}
{"category": "Math", "title": "Moduli of Continuity of Quasiregular Mappings", "abstract": "This thesis consists of Chapters 1 and 2. The main results are contained in the two preprints and two published papers, listed below. Chapter 1 deals with conformal invariants in the euclidean space Rn; n >= 2; and their interrelation. In particular, conformally invariant metrics and balls of the respective metric spaces are studied. Another theme in Chapter 1 is the study of quasiconformal maps with identity boundary values in two diferent cases, the unit ball and the whole space minus two points. These results are based on the two preprints: R. Klen, V. Manojlovic and M. Vuorinen: Distortion of two point normalized quasiconformal mappings, arXiv:0808.1219[math.CV], 13 pp., V. Manojlovic and M. Vuorinen: On quasiconformal maps with identity boundary values, arXiv:0807.4418[math.CV], 16 pp. Chapter 2 deals with harmonic quasiregular maps. Topics studied are: Preservation of modulus of continuity, in particular Lipschitz continuity, from the boundary to the interior of domain in case of harmonic quasiregular maps and quasiisometry property of harmonic quasiconformal maps. Chapter 2 is based mainly on the two published papers: M. Arsenovic, V. Kojic and M. Mateljevic: On Lipschitz continuity of harmonic quasiregular maps on the unit ball in Rn., Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 1, 315-318. V. Kojic and M. Pavlovic: Subharmonicity of jfjp for quasiregular harmonic functions, with applications, J. Math. Anal. Appl. 342 (2008) 742-746"}
{"category": "Math", "title": "Maximal Inequalities in Bilateral Grand Lebesque Spaces Over Unbounded Measure", "abstract": "In this paper non-asymptotic exact rearrangement invariant norm estimates are derived for the maximum distribution of the family elements of some rearrangement invariant (r.i.) space over unbounded measure in the entropy terms and in the terms of generic chaining. We consider some applications in the martingale theory and in the theory of Fourier series."}
{"category": "Math", "title": "Support of Borelian Measures in Separable Banach Spaces", "abstract": "We prove in this article that every Borelian measure, for example, the distribution of a random variable, in separable Banach space has a support which is compact embedded Banach subspace; and prove that if the norm of the random variable belongs to some exponential Orlicz space, then the new subspace can be choose such that the norm of this variable in the new space also belongs to other exponential Orlicz space."}
{"category": "Math", "title": "Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon Nikodym property", "abstract": "We prove the differentiability of Lipschitz maps X-->V, where X is a complete metric measure space satisfying a doubling condition and a Poincar\\'e inequality, and V is a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direction of tangent vectors to suitable rectifiable curves."}
{"category": "Math", "title": "Champs alg\\'ebriques et foncteur de Picard", "abstract": "This text is my thesis, defended in June 2007, in the status it was at this time. The most important results are contained in the article \"Foncteur de Picard d'un champ alg\\'ebrique\" to appear in \"Mathematische Annalen\" (see the preprint arXiv:0711.4545). In the article, some results have been added, and some previous results have been strengthened. However, the proofs of the results contained in the appendix (concerning the smooth-\\'etale cohomology on an algebraic stack) have been removed. The thesis is only put on the ArXiv to provide a more lasting reference than my webpage for these proofs."}
{"category": "Math", "title": "Ratliff-Rush Filtration, regularity and depth of Higher Associated graded modules: Part II", "abstract": "Let $(A,\\m)$ be a Noetherian local ring, let $M$ be a finitely generated \\CM $A$-module of dimension $r \\geq 2$ and let $I$ be an ideal of definition for $M$. Set $L^I(M) = \\bigoplus_{n\\geq 0}M/I^{n+1}M$. In part one of this paper we showed that $L^I(M)$ is a module over $\\R$, the Rees algebra of $I$ and we gave many applications of $L^I(M)$ to study the associated graded module, $G_I(M)$. In this paper we give many further applications of our technique; most notable is a reformulation of a classical result due to Narita in terms of the Ratliff-Rush filtration. This reformulation can be extended to all dimensions $\\geq 2$."}
{"category": "Math", "title": "Positivity of relative canonical bundles for families of canonically polarized manifolds", "abstract": "Given an effectively parameterized family of canonically polarized manifolds the Kaehler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. For degenerating families we obtain a singular hermitian metric. Main application is an analytic proof of the quasi-projectivity of the moduli space of canonically polarized manifolds. Further applications in arXiv:1002.4858v2."}
{"category": "Math", "title": "Coupled vortex equations and Moduli: Deformation theoretic Approach and Kaehler Geometry", "abstract": "We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kaehler manifold X. These solutions are known to be related to polystable triples via a Kobayashi-Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kaehler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kaehler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi--positive. It is shown that in the case where X is a smooth complex projective variety, the Kaehler form is the Chern form of a Quillen metric on a certain determinant line bundle."}
{"category": "Math", "title": "On behavior of solvable ideals of Lie algebras under outer derivations", "abstract": "Let $L$ be a finite dimensional Lie algebra over a field $F$. It is well known that the solvable radical $S(L)$ of the algebra $L$ is a characteristic ideal of $L$ if $\\char F=0$ and there are counterexamples to this statement in case $\\char F=p>0$. We prove that the sum $S(L)$ of all solvable ideals of a Lie algebra $L$ (not necessarily finite dimensional) is a characteristic ideal of $L$ in the following cases: 1) $\\char F=0;$ 2) $S(L)$ is solvable and its derived length is less than $\\log_{2}p.$ Some estimations (in characteristic 0) for the derived length of ideals $I+D(I)+... +D^{k}(I)$ are obtained where $I$ is a solvable ideal of $L$ and $D\\in Der(L).$"}
{"category": "Math", "title": "Proof of a dynamical Bogomolov conjecture for lines under polynomial actions", "abstract": "We prove a dynamical version of the Bogomolov conjecture in the special case of lines in affine space A^m under the action of a map (f_1,...,f_m) where each f_i is a polynomial in Q-bar[X] of the same degree."}
{"category": "Math", "title": "The dynamical Mordell-Lang problem for etale maps", "abstract": "We prove a dynamical version of the Mordell-Lang conjecture for etale endomorphisms of quasiprojective varieties. We use p-adic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. As special cases of our result we obtain a new proof of the classical Mordell-Lang conjecture for cyclic subgroups of a semiabelian variety, and we also answer positively a question of Keeler/Rogalski/Stafford for critically dense sequences of closed points of a Noetherian integral scheme."}
{"category": "Math", "title": "Homological interpretation of extensions and biextensions of 1-motives", "abstract": "Let k be a separably closed field. Let K_i=[A_i \\to B_i] (for i=1,2,3) be three 1-motives defined over k. We define the geometrical notions of extension of K_1 by K_3 and of biextension of (K_1,K_2) by K_3. We then compute the homological interpretation of these new geometrical notions: namely, the group Biext^0(K_1,K_2;K_3) of automorphisms of any biextension of (K_1,K_2) by K_3 is canonically isomorphic to the cohomology group Ext^0(K_1 \\otimes K_2,K_3), and the group Biext^1(K_1,K_2;K_3) of isomorphism classes of biextensions of (K_1,K_2) by K_3 is canonically isomorphic to the cohomology group Ext^1(K_1 \\otimes K_2,K_3)."}
{"category": "Math", "title": "Surfaces with p_g=q=1, K^2=8 and nonbirtional bicanonical map", "abstract": "We prove that if the bicanonical map of a minimal surface of general type S with p_{g}=q=1 and K^2=8 is non birational, then it is a double cover onto a rational surface. An application of this theorem is the complete classification of minimal surfaces of general type with p_{g}=q=1, K^2=8 and nonbirational bicanonical map."}
{"category": "Math", "title": "A note on $l^p$ norms of weighted mean matrices", "abstract": "We present some results concerning the $l^p$ norms of weighted mean matrices. These results can be regarded as analogues to a result of Bennett concerning weighted Carleman's inequalities."}
{"category": "Math", "title": "On homogenization of space-time dependent and degenerate random flows II", "abstract": "We study the long time behavior (homogenization) of a diffusion in random medium with time and space dependent coefficients. The diffusion coefficient may degenerate. In Stochastic Process. Appl. (2007) (to appear), an invariance principle is proved for the critical rescaling of the diffusion. Here, we generalize this approach to diffusions whose space-time scaling differs from the critical one."}
{"category": "Math", "title": "Siciak-Zahariuta extremal functions, analytic discs and polynomial hulls", "abstract": "We prove two disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in terms of analytic discs. Similar results in previous work of ours required the subsets to be connected."}
{"category": "Math", "title": "Ergodic actions of compact quantum groups from solutions of the conjugate equations", "abstract": "We use a tensor C*-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a *-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal C*-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the C*-algebra in question. The quantum groups involved are A_u(Q) and B_u(Q)."}
{"category": "Math", "title": "The Bi-Musquash Conjecture", "abstract": "D.R. Woodall's definition of a 'musquash' is extended to that of a 'bi-musquash' and the existence/uniqueness of these is discussed."}
{"category": "Math", "title": "Derived categories of cubic fourfolds", "abstract": "We discuss the structure of the derived category of coherent sheaves on cubic fourfolds of three types: Pfaffian cubics, cubics containing a plane and singular cubics, and discuss its relation to the rationality of these cubics."}
{"category": "Math", "title": "Seidel-Smith Cohomology for Tangles", "abstract": "We generalize the \"symplectic Khovanov cohomology\" of Seidel and Smith to tangles using the notion of symplectic valued topological field theory introduced by Wehrheim and Woodward."}
{"category": "Math", "title": "A bracket polynomial for graphs", "abstract": "A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeister equivalence to an equivalence relation on looped graphs. The graph bracket polynomial can be defined recursively using the same pivot and local complementation operations used to define the interlace polynomial, and it gives rise to a graph Jones polynomial that is invariant under the graph Reidemeister moves."}
{"category": "Math", "title": "On a doubly nonlinear diffusion model of chemotaxis with prevention of overcrowding", "abstract": "This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two-sided fashion, including an extra nonlinearity represented by a $p$-Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed-point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local H\\\"older regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what extent the properties of the classical Keller-Segel chemotaxis models are preserved in a more general setting. Some numerical examples illustrate the model."}
{"category": "Math", "title": "LRSA: A new computational method for analyzing time course microarray data", "abstract": "Motivation: Time course data obtained from biological samples subject to specific treatments can be very useful for revealing complex and novel biological phenomena. Although an increasing number of time course microarray datasets becomes available, most of them contain few biological replicates and time points. So far there are few computational methods that can effectively reveal differentially expressed genes and their patterns in such data. Results: We have proposed a new two-step nonparametric statistical procedure, LRSA, to reveal differentially expressed genes and their expression trends in temporal microarray data. We have also employed external controls as a surrogate to estimate false discovery rates and thus to guide the discovery of differentially expressed genes. Our results showed that LRSA reveals substantially more differentially expressed genes and have much lower than two other methods, STEM and ANOVA, in both real data and the simulated data. Our computational results are confirmed using real-time PCRs. Contact: wuw2@upmc.edu"}
{"category": "Math", "title": "Call option prices based on Bessel processes", "abstract": "As a complement to some recent work by Pal and Protter, \"Strict local martingales, bubbles, and no early exercise\", we show that the call option prices associated with the Bessel strict local martingales are integrable over time, and we discuss the probability densities obtained thus."}
{"category": "Math", "title": "The SL(2)-type and Base Change", "abstract": "The SL(2)-type of any smooth, irreducible and unitarizable representation of GL(n) over a p-adic field was defined by Venkatesh. We provide a natural way to extend the definition to all smooth and irreducible representations. For unitarizable representations we show that the SL(2)-type of a representation is preserved under base change with respect to any finite extension. The Klyachko model of a smooth, irreducible and unitarizable representation \\pi of GL(n) depends only on the SL(2)-type of \\pi. As a consequence we observe that the Klyachko model of \\pi and of its base-change are of the same type."}
{"category": "Math", "title": "Complete Constant Mean Curvature surfaces and Bernstein type Theorems in $\\mathbb{M}^2\\times \\mathbb{R}$", "abstract": "In this paper we study constant mean curvature surfaces $\\Sigma$ in a product space, $\\mathbb{M}^2\\times \\mathbb{R}$, where $\\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function $\\nu = \\meta{N}{\\partial_t}$ does not change sign on $\\Sigma$. We classify these surfaces according to the infimum $c(\\Sigma)$ of the Gaussian curvature of the projection of $\\Sigma$. When $H \\neq 0$ and $c(\\Sigma)\\geq 0$, then $\\Sigma $ is a cylinder over a complete curve with curvature 2H. If H=0 and $c(\\Sigma) \\geq 0$, then $\\Sigma$ must be a vertical plane or $\\Sigma$ is a slice $\\mathbb{M}^2 \\times {t}$, or $\\mathbb{M}^2 \\equiv \\mathbb{R}^2$ with the flat metric and $\\Sigma$ is a tilted plane (after possibly passing to a covering space). When $c(\\Sigma)<0$ and $H>\\sqrt{-c(\\Sigma)} /2$, then $\\Sigma$ is a vertical cylinder over a complete curve of $\\mathbb{M}^2$ of constant geodesic curvature $2H$. This result is optimal. We also prove a non-existence result concerning complete multi-graphs in $\\mathbb{M}^2\\times \\mathbb{R}$, when $c(\\mathbb{M}^2)<0$."}
{"category": "Math", "title": "On the Cayley semigroup of a finite aperiodic semigroup", "abstract": "Let $S$ be a finite semigroup. In this paper we introduce the functions $\\phi_s:S^* \\to S^*$, first defined by Rhodes, given by $\\phi_s([a_1,a_2 ,...,a_n]) = [sa_1,sa_1a_2,..., sa_1a_2 ... a_n]$. We show that if $S$ is a finite aperiodic semigroup, then the semigroup generated by the functions $\\{\\phi_s\\}_{s \\in S}$ is finite and aperiodic."}
{"category": "Math", "title": "A New Invariant Metric and Applications", "abstract": "We construct a new invariant metric for a compact subgroup of the automorphism group of a domain in complex space. Applications are provided."}
{"category": "Math", "title": "The Range of a Class of Classifiable Separable Simple Amenable C*-Algebras", "abstract": "We study the range of a classifiable class ${\\cal A}$ of unital separable simple amenable $C^*$-algebras which satisfy the Universal Coefficient Theorem. The class ${\\cal A}$ contains all unital simple AH-algebras. We show that all unital simple inductive limits of dimension drop circle $C^*$-algebras are also in the class. This unifies some of the previous known classification results for unital simple amenable $C^*$-algebras. We also show that there are many other $C^*$-algebras in the class. We prove that, for any partially ordered, simple weakly unperforated rationally Riesz group $G_0$ with order unit $u,$ any countable abelian group $G_1,$ any metrizable Choquet simplex $S,$ and any surjective affine continuous map $r: S\\to S_u(G_0)$ (where $S_u(G_0)$ is the state space of $G_0$) which preserves extremal points, there exists one and only one (up to isomorphism) unital separable simple amenable $C^*$-algebra $A$ in the classifiable class ${\\cal A}$ such that $$ ((K_0(A), K_0(A)_+, [1_A]), K_1(A), T(A), \\lambda_A)=((G_0, (G_0)_+, u), G_1,S, r)."}
{"category": "Math", "title": "The base change fundamental lemma for central elements in parahoric Hecke algebras", "abstract": "Clozel and Labesse proved the base change fundamental lemma for spherical Hecke algebras attached to an unramified group over a p-adic field. This paper proves an analogous fundamental lemma for centers of parahoric Hecke algebras attached to the same class of groups. This provides an ingredient needed for the author's program to study Shimura varieties with parahoric level structure at p."}
{"category": "Math", "title": "Combinatorial bases for multilinear parts of free algebras with double compatible brackets", "abstract": "Let X be an ordered alphabet. Lie_2(n) (and P_2(n) respectively) are the multilinear parts of the free Lie algebra (and the free Poisson algebra respectively) on X with a pair of compatible Lie brackets. In this paper, we prove the dimension formulas for these two algebras conjectured by B. Feigin by constructing bases for Lie_2(n) (and P_2(n)) from combinatorial objects. We also define a complementary space Eil_2(n) to Lie_2(n), give a pairing between Lie_2(n) and Eil_2(n), and show that the pairing is perfect."}
{"category": "Math", "title": "Riesz basis for strongly continuous groups", "abstract": "Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none of the conditions can be weakened. However, if the eigenvalues (counted with multiplicity) can be grouped into subsets of at most $K$ elements, and the distance between the groups is (uniformly) bounded away from zero, then the spectral projections associated to the groups form a Riesz family. This implies that if in every range of the spectral projection we construct an orthonormal basis, then the union of these bases is a Riesz basis in the Hilbert space."}
{"category": "Math", "title": "Tropical enumerative invariants of F_0 and F_2", "abstract": "There is an equation relating numbers of curves on F_0 satisfying incidence conditions and numbers of curves on F_2 satisfying incidence conditions. The purpose of this paper is to give a tropical proof of this equation in the case of rational curves. We use induction on the degree and two Kontsevich-type formulas for curves on F_0 and on F_2. The formula for F_2 was not known before and is proved using tropical geometry."}
{"category": "Math", "title": "Scattering for small energy solutions of NLS with periodic potential in 1D", "abstract": "We prove scattering for small solutions to of nonlinear Schroedinger equations in 1D with a space periodic potential"}
{"category": "Math", "title": "A central limit theorem for the rescaled L\\'evy area of two-dimensional fractional Brownian motion with Hurst index $H<1/4$", "abstract": "Let $B=(B^{(1)},B^{(2)})$ be a two-dimensional fractional Brownian motion with Hurst index $\\alpha\\in (0,1/4)$. Using an analytic approximation $B(\\eta)$ of $B$ introduced in \\cite{Unt08}, we prove that the rescaled L\\'evy area process $(s,t)\\to \\eta^{\\half(1-4\\alpha)}\\int_s^t dB_{t_1}^{(1)}(\\eta) \\int_s^{t_1} dB_{t_2}^{(2)}(\\eta)$ converges in law to $W_t-W_s$ where $W$ is a Brownian motion independent from $B$. The method relies on a very general scheme of analysis of singularities of analytic functions, applied to the moments of finite-dimensional distributions of the L\\'evy area."}
{"category": "Math", "title": "On sequential Monte Carlo, partial rejection control and approximate Bayesian computation", "abstract": "We present a sequential Monte Carlo sampler variant of the partial rejection control algorithm, and show that this variant can be considered as a sequential Monte Carlo sampler with a modified mutation kernel. We prove that the new sampler can reduce the variance of the incremental importance weights when compared with standard sequential Monte Carlo samplers. We provide a study of theoretical properties of the new algorithm, and make connections with some existing algorithms. Finally, the sampler is adapted for application under the challenging \"likelihood free,\" approximate Bayesian computation modelling framework, where we demonstrate superior performance over existing likelihood-free samplers."}
{"category": "Math", "title": "Generalized mean curvature flow in Carnot groups", "abstract": "In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting by Evans-Spruck and Chen-Giga-Goto. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow."}
{"category": "Math", "title": "Discreteness Criteria and the Hyperbolic Geometry of Palindroms", "abstract": "We consider non-elementary representations of two generator free groups in $PSL(2,\\mathbb{C})$, not necessarily discrete or free, $G = < A, B >$. A word in $A$ and $B$, $W(A,B)$, is a palindrome if it reads the same forwards and backwards. A word in a free group is {\\sl primitive} if it is part of a minimal generating set. Primitive elements of the free group on two generators can be identified with the positive rational numbers. We study the geometry of palindromes and the action of $G$ in $\\HH^3$ whether or not $G$ is discrete. We show that there is a {\\sl core geodesic} $\\L$ in the convex hull of the limit set of $G$ and use it to prove three results: the first is that there are well defined maps from the non-negative rationals and from the primitive elements to $\\L$; the second is that $G$ is geometrically finite if and only if the axis of every non-parabolic palindromic word in $G$ intersects $\\L$ in a compact interval; the third is a description of the relation of the pleating locus of the convex hull boundary to the core geodesic and to palindromic elements."}
{"category": "Math", "title": "Periodic orbits of linear endomorphisms on the 2-torus and its lattices", "abstract": "Counting periodic orbits of endomorphisms on the 2-torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The situation on the lattices, up to local conjugacy, is completely determined by the determinant, the trace and a third invariant of the matrix defining the toral endomorphism."}
{"category": "Math", "title": "K-processes, scaling limit and aging for the trap model in the complete graph", "abstract": "We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform distribution. We show how these processes arise, in a particular instance, as scaling limits of the trap model in the complete graph, and subsequently derive aging results for those models in this context."}
{"category": "Math", "title": "Tail asymptotics for a random sign Lindley recursion", "abstract": "We investigate the tail behaviour of the steady state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queuing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems."}
{"category": "Math", "title": "Differential systems with Fuchsian linear part: correction and linearization, normal forms and multiple orthogonal polynomials", "abstract": "Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable) and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable. More generally, normal forms are found. The corrections and the normal forms are found constructively. Expansions in multiple orthogonal polynomials and their generalization to matrix-valued polynomials are instrumental to these constructions."}
{"category": "Math", "title": "A Reconstruction Algorithm for Photoacoustic Imaging based on the Nonuniform FFT", "abstract": "Fourier reconstruction algorithms significantly outperform conventional back-projection algorithms in terms of computation time. In photoacoustic imaging, these methods require interpolation in the Fourier space domain, which creates artifacts in reconstructed images. We propose a novel reconstruction algorithm that applies the one-dimensional nonuniform fast Fourier transform to photoacoustic imaging. It is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of Fourier reconstruction."}
{"category": "Math", "title": "A Chevalley's theorem in class C^r", "abstract": "Let W be a finite reflection group acting orthogonally on R^n, P be the Chevalley polynomial mapping determined by an integrity basis of the algebra of W-invariant polynomials, and h be the highest degree of the coordinate polynomials in P.There exists a linear mapping from (C^r(R^n))^W to C^[r/h](R^n), f\\mapsto F such that f=F \\circ P, continuous for the natural Fr\\'echet topologies. A general counterexample shows that this result is the best possible. The proof uses techniques of division by linear forms and a study of compensation phenomenons. An extension to P^{-1}(R^n) of invariant formally holomorphic regular fields is needed."}
{"category": "Math", "title": "Edge percolation on a random regular graph of low degree", "abstract": "Consider a uniformly random regular graph of a fixed degree $d\\ge3$, with $n$ vertices. Suppose that each edge is open (closed), with probability $p(q=1-p)$, respectively. In 2004 Alon, Benjamini and Stacey proved that $p^*=(d-1)^{-1}$ is the threshold probability for emergence of a giant component in the subgraph formed by the open edges. In this paper we show that the transition window around $p^*$ has width roughly of order $n^{-1/3}$. More precisely, suppose that $p=p(n)$ is such that $\\omega:=n^{1/3}|p-p^*|\\to\\infty$. If $p<p^*$, then with high probability (whp) the largest component has $O((p-p^*)^{-2}\\log n)$ vertices. If $p>p^*$, and $\\log\\omega\\gg\\log\\log n$, then whp the largest component has about $n(1-(p\\pi+q)^d)\\asymp n(p-p^*)$ vertices, and the second largest component is of size $(p-p^*)^{-2}(\\log n)^{1+o(1)}$, at most, where $\\pi=(p\\pi+q)^{d-1},\\pi\\in(0,1)$. If $\\omega$ is merely polylogarithmic in $n$, then whp the largest component contains $n^{2/3+o(1)}$ vertices."}
{"category": "Math", "title": "Topologie sur l'ensemble des parties positives d'un r\\'eseau", "abstract": "We define a notion of {\\it positive part} of a lattice $\\Lambda$ and we endow the set of such positive parts with a topology. We then study some properties of this topology, by comparing it with the one of $V^*/\\RM_{> 0}$, where $V^*$ is the dual vector space of $\\RM \\otimes_\\ZM \\Lambda$."}
{"category": "Math", "title": "The universal Cannon--Thurston maps and the boundary of the curve complex", "abstract": "In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent--Leininger--Schleimer and Mitra, we construct a universal Cannon--Thurston map from a subset of the circle at infinity for the closed surface group onto the boundary of the curve complex of the once-punctured surface. Using the techniques we have developed, we also show that the boundary of this curve complex is locally path-connected."}
{"category": "Math", "title": "Semicontinuity properties of Kazhdan-Lusztig cells", "abstract": "Computations in small Coxeter groups or dihedral groups suggest that the partition into Kazhdan-Lusztig cells with unequal parameters should obey to some semicontinuity phenomenon (as the parameters vary). The aim of this paper is to provide a rigorous theoretical background for supporting this intuition that will allow to state several precise conjectures."}
{"category": "Math", "title": "The universal Kummer congruences", "abstract": "Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number ${{\\hat B_n}\\over n}$ when $n$ is divisible by $p-1$. Using these we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers ${{\\hat B_n}\\over n}$ when $n$ is divisible by $p-1$."}
{"category": "Math", "title": "Applications of computational invariant theory to Kobayashi hyperbolicity and to Green-Griffiths algebraic degeneracy", "abstract": "A major unsolved problem (according to Demailly 1997) towards the Kobayashi hyperbolicity conjecture in optimal degree is to understand jet differentials of germs of holomorphic discs that are invariant under any reparametrization of the source. The underlying group action is not reductive, but we provide a complete algorithm to generate all invariants, in arbitrary dimension n and for jets of arbitrary order k. Two main new situations are studied in great details. For jets of order 4 in dimension 4, we establish that the algebra of Demailly-Semple invariants is generated by 2835 polynomials, while the algebra of bi-invariants is generated by 16 mutually independent polynomials sharing 41 groebnerized syzygies. Nonconstant entire holomorphic curves valued in an algebraic 3-fold (resp. 4-fold) X^3 in P^4 (C) (resp. X^4 in P^5(C)) of degree d satisfy global differential equations as soon as d > 71 (resp. d > 258). A useful asymptotic formula for the Euler-Poincare characteristic of Schur bundles in terms of Giambelli's determinants is derived. For jets of order 5 in dimension 2, we establish that the algebra of Demailly-Semple invariants is generated by 56 polynomials, while the algebra of bi-invariants is generated by 17 mutually independent polynomials sharing 105 groebnerized syzygies."}
{"category": "Math", "title": "Some 6-dimensional Hamiltonian S^1 manifolds", "abstract": "In an earlier paper we explained how to convert the problem of symplectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into \\CP^2 by using a new way to desingularize orbifold blow ups Z of the weighted projective space \\CP^2_{1,m,n}. We now use a related method to construct symplectomorphisms of these spaces Z. This allows us to construct some well known Fano 3-folds (including the Mukai--Umemura 3-fold) in purely symplectic terms, using a classification by Tolman of a particular class of Hamiltonian S^1-manifolds. We also show that these manifolds are uniquely determined by their fixed point data up to equivariant symplectomorphism. As part of this argument we show that the symplectomorphism group of a certain weighted blow up of a weighted projective plane is connected."}
{"category": "Math", "title": "Infinite divisibility of Smith matrices", "abstract": "Given an arithmetical function $f$, by $f(a, b)$ and $f[a, b]$ we denote the function $f$ evaluated at the greatest common divisor $(a, b)$ of positive integers $a$ and $b$ and evaluated at the least common multiple $[a, b]$ respectively. A positive semi-definite matrix $A=(a_{ij})$ with $a_{ij}\\ge 0$ for all $i$ and $j$ is called infinitely divisible if the fractional Hadamard power $A^{\\circ r}=(a_{ij}^r)$ is positive semi-definite for every nonnegative real number $r$. Let $S=\\{x_1, ..., x_n\\}$ be a set of $n$ distinct positive integers. In this paper, we show that if $f$ is a multiplicative function such that $(f*\\mu)(d)\\ge 0$ whenever $d|x$ for any $x\\in S$, then the $n\\times n$ matrices $(f(x_i, x_j))$, $(\\frac{1}{f[x_i, x_j]})$ and $(\\frac{f(x_i, x_j)}{f[x_i, x_j]})$ are infinitely divisible. Finally we extend these results to the Dirichlet convolution case which produces infinitely many examples of infinitely divisible matrices. Our results extend the results obtained previously by Bourque, Ligh, Bhatia, Hong, Lee, Lindqvist and Seip."}
{"category": "Math", "title": "Alg\\`ebres enveloppantes \\`a homotopie pr\\`es, homologie et cohomologie", "abstract": "We present an unified construction for algebras and modules homologies and cohomologies, in the case of associative, commuttaive, Lie and Gerstenhaber algebras. We make a distinction between the linear part of the construction of algebras and cogebras, characterized by the symmetries of the defining relations and the structure itself which appears as a differential on these algebras and cogebras."}
{"category": "Math", "title": "Formal and Informal Model Selection with Incomplete Data", "abstract": "Model selection and assessment with incomplete data pose challenges in addition to the ones encountered with complete data. There are two main reasons for this. First, many models describe characteristics of the complete data, in spite of the fact that only an incomplete subset is observed. Direct comparison between model and data is then less than straightforward. Second, many commonly used models are more sensitive to assumptions than in the complete-data situation and some of their properties vanish when they are fitted to incomplete, unbalanced data. These and other issues are brought forward using two key examples, one of a continuous and one of a categorical nature. We argue that model assessment ought to consist of two parts: (i) assessment of a model's fit to the observed data and (ii) assessment of the sensitivity of inferences to unverifiable assumptions, that is, to how a model described the unobserved data given the observed ones."}
{"category": "Math", "title": "Discrete Affine Surfaces based on Quadrangular Meshes", "abstract": "In this paper we are interested in defining affine structures on discrete quadrangular surfaces of the affine three-space. We introduce, in a constructive way, two classes of such surfaces, called respectively indefinite and definite surfaces. The underlying meshes for indefinite surfaces are asymptotic nets satisfying a non-degeneracy condition, while the underlying meshes for definite surfaces are non-degenerate conjugate nets satisfying a certain natural condition. In both cases we associate to any of these nets several discrete affine invariant quantities: a metric, a normal and a co-normal vector fields, and a mean curvature. Moreover, we derive structural and compatibility equations which are shown to be necessary and sufficient conditions for the existence of a discrete quadrangular surface with a given affine structure."}
{"category": "Math", "title": "Painl\\'eve III and a singular linear statistics in Hermitian random matrix ensembles I", "abstract": "In this paper, we study a certain linear statistics of the unitary Laguerre ensembles, motivated in part by an integrable quantum field theory at finite temperature. It transpires that this is equivalent to the characterization of a sequence of polynomials orthogonal with respect to the weight $$ w(x)=w(x,s):=x^{\\al}\\rme^{-x}\\rme^{-s/x}, \\quad 0\\leq x<\\infty, \\al>0, s>0, $$ namely, the determination of the associated Hankel determinant and recurrence coefficients. Here $w(x,s)$ is the Laguerre weight $x^{\\al}\\:\\rme^{-x}$ 'perturbed' by a multiplicative factor $\\rme^{-s/x},$ which induces an infinitely strong zero at the origin. For polynomials orthogonal on the unit circle, a particular example where there are explicit formulas, the weight of which has infinitely strong zeros, was investigated by Pollazcek and Szeg\\\"o many years ago. Such weights are said to be 'singular' or irregular due to the violation of the Szeg\\\"o condition. In our problem, the linear statistics is a sum of the reciprocal of positive random variables $\\{x_j:j=1,..,,n\\};$ $\\sum_{j=1}^{n}1/x_j.$ We show that the moment generating function, or the Laplace transform of the probability density function of this linear statistics is expressed as the ratio of Hankel determinants and as an integral of the combination of a particular third Painlev\\'e function."}
{"category": "Math", "title": "Vector space bases associated to vanishing ideals of points", "abstract": "We discuss four different constructions of vector space bases associated to vanishing ideals of points. We show how to compute normal forms with respect to these bases and give new complexity bounds. As an application, we improve the computational algebra approach to the reverse engineering of gene regulatory networks."}
{"category": "Math", "title": "Exceptional Times for the Dynamical Discrete Web", "abstract": "The dynamical discrete web (DyDW),introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical time parameter \\tau. The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed \\tau. In this paper, we study the existence of exceptional (random) values of \\tau where the paths of the web do not behave like usual random walks and the Hausdorff dimension of the set of exceptional such \\tau. Our results are motivated by those about exceptional times for dynamical percolation in high dimension by H\\\"{a}ggstrom, Peres and Steif, and in dimension two by Schramm and Steif. The exceptional behavior of the walks in the DyDW is rather different from the situation for the dynamical random walks of Benjamini, H\\\"{a}ggstrom, Peres and Steif. For example, we prove that the walk from the origin S^\\tau_0 violates the law of the iterated logarithm (LIL) on a set of \\tau of Hausdorff dimension one. We also discuss how these and other results extend to the dynamical Brownian web, the natural scaling limit of the DyDW."}
{"category": "Math", "title": "On the dimension of the Hilbert scheme of curves", "abstract": "Consider a component of the Hilbert scheme whose general point corresponds to a degree d genus g smooth irreducible and nondegenerate curve in a projective variety X. We give lower bounds for the dimension of such a component when X is P^3, P^4 or a smooth quadric threefold in P^4 respectively. Those bounds make sense from the asymptotic viewpoint if we fix d and let g vary. Some examples are constructed using determinantal varieties to show the sharpness of the bounds for d and g in a certain range. The results can also be applied to study rigid curves."}
{"category": "Math", "title": "On Universal Cycles of Labeled Graphs", "abstract": "A universal cycle is a compact listing of a class of combinatorial objects. In this paper, we prove the existence of universal cycles of classes of labeled graphs, including simple graphs, trees, graphs with m edges, graphs with loops, graphs with multiple edges (with up to m duplications of each edge), directed graphs, hypergraphs, and k-uniform hypergraphs."}
{"category": "Math", "title": "Circular Digraph Walks, k-Balanced Strings, Lattice Paths and Chebychev Polynomials", "abstract": "We count the number of walks of length n on a k-node circular digraph that cover all k nodes in two ways. The first way illustrates the transfer-matrix method. The second involves counting various classes of height-restricted lattice paths. We observe that the results also count so-called k-balanced strings of length n, generalizing a 1996 Putnam problem."}
{"category": "Math", "title": "The action of Hecke operators on hypergeometric functions", "abstract": "We study the action of Hecke operators on the set of hypergeometric functions. We show that the spectrum of these operators is the set of powers n^a and that polylogarithms play a dominant role in the study of the corresponding eigenfunctions. As a corollary, we obtain a characterization of completely multiplicative hypergeometric coefficients."}
{"category": "Math", "title": "Butterflies I: morphisms of 2-group stacks", "abstract": "Weak morphisms of non-abelian complexes of length 2, or crossed modules, are morphisms of the associated 2-group stacks, or gr-stacks. We present a full description of the weak morphisms in terms of diagrams we call butterflies. We give a complete description of the resulting bicategory of crossed modules, which we show is fibered and biequivalent to the 2-stack of 2-group stacks. As a consequence we obtain a complete characterization of the non-abelian derived category of complexes of length 2. Deligne's analogous theorem in the case of Picard stacks and abelian sheaves becomes an immediate corollary. Commutativity laws on 2-group stacks are also analyzed in terms of butterflies, yielding new characterizations of braided, symmetric, and Picard 2-group stacks. Furthermore, the description of a weak morphism in terms of the corresponding butterfly diagram allows us to obtain a long exact sequence in non-abelian cohomology, removing a preexisting fibration condition on the coefficients short exact sequence."}
{"category": "Math", "title": "Large deviations for infinite dimensional stochastic dynamical systems", "abstract": "The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process."}
{"category": "Math", "title": "Floer homology for negative line bundles and Reeb chords in pre-quantization spaces", "abstract": "In this article we prove existence of Reeb orbits for Bohr-Sommerfeld Legendrians in certain pre-quantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibre-wise quadratic Hamiltonian functions on negative line bundles."}
{"category": "Math", "title": "Finite type invariants of words and Arnold's invariants", "abstract": "We define a new finite type invariant for stably homeomorphic class of curves on compact oriented surfaces without boundaries and extend to a regular homotopy invariant for spherical curves."}
{"category": "Math", "title": "Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties", "abstract": "Given a p-form defined on the smooth locus of a normal variety, and a resolution of singularities, we study the problem of extending the pull-back of the p-form over the exceptional set of the desingularization. For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along the exceptional set. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the well-known Bogomolov-Sommese vanishing theorem to log canonical threefold pairs follows."}
{"category": "Math", "title": "Reversibility of chordal SLE", "abstract": "We prove that the chordal SLE$_{\\kappa}$ trace is reversible for $\\kappa\\in(0,4]$."}
{"category": "Math", "title": "Some Applications of the Isoperimetric Inequality for Integral Varifolds", "abstract": "In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calder\\'on's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds."}
{"category": "Math", "title": "Associative algebra deformations of the Connes-Moscovici's Hopf algebra $\\mathcal{H}_1$", "abstract": "We compute the second Hochschild cohomology space $HH^2(\\mathcal{H}_1)$ of Connes-Moscovici's Hopf algebra $\\mathcal{H}_1$, giving the infinitesimal deformations (up to equivalence) of the associative structure. $HH^2(\\mathcal{H}_1)$ is shown to be one dimensional, and thus Connes-Moscovici's formal deformation of $\\mathcal{H}_1$ using Rankin-Cohen brackets is unique up to equivalence."}
{"category": "Math", "title": "Martingale approach to stochastic differential games of control and stopping", "abstract": "We develop a martingale approach for studying continuous-time stochastic differential games of control and stopping, in a non-Markovian framework and with the control affecting only the drift term of the state-process. Under appropriate conditions, we show that the game has a value and construct a saddle pair of optimal control and stopping strategies. Crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities."}
{"category": "Math", "title": "A Sobolev Poincar\\'e type inequality for integral varifolds", "abstract": "In this work a local inequality is provided which bounds the distance of an integral varifold from a multivalued plane (height) by its tilt and mean curvature. The bounds obtained for the exponents of the Lebesgue spaces involved are shown to be sharp."}
{"category": "Math", "title": "Second order rectifiability of integral varifolds of locally bounded first variation", "abstract": "In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the collection agrees with the mean curvature of the varifold almost everywhere with respect to the varifold."}
{"category": "Math", "title": "Pseudocyclic association schemes and strongly regular graphs", "abstract": "Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence matrix of a symmetric design and the all-ones matrix. Amorphous pseudocyclic association schemes are examples of such association schemes whose associated symmetric design is trivial. We present several non-amorphous examples, which are either cyclotomic association schemes, or their fusion schemes. Special properties of symmetric designs guarantee the existence of further fusions, and the two known non-amorphous association schemes of class 4 discovered by van Dam and by the authors, are recovered in this way. We also give another pseudocyclic non-amorphous association scheme of class 7 on GF(2^{21}), and a new pseudocyclic amorphous association scheme of class 5 on GF(2^{12})."}
{"category": "Math", "title": "Triangulated structures induced by simplicial descent categories", "abstract": "The present paper is devoted to study the homotopy category associated with a simplicial descent category (D,s,E) (arXiv:0808.3684v2). We prove that the class E of equivalences has a calculus of left fractions over a quotient category of D modulo homotopy. We study the fiber/cofiber sequences induced by a (co)simplicial descent structure. Examples of such fiber/cofiber sequences are deduced for (commutative) differential graded algebras, simplicial sets or topological spaces. We prove that the homotopy category of a stable simplicial descent category is triangulated. In addition, these triangulated structures may be extended to the homotopy categories of diagram categories of D. As a corollary, we obtain the triangulated structures on: (filtered) derived categories of abelian categories, the derived category of DG-modules over a DG-category, the stable derived category of fibrant spectra and the localized category of mixed Hodge complexes."}
{"category": "Math", "title": "(Co)Simplicial Descent Categories", "abstract": "In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD --> D, which we call simple functor. For instance, the Bousfield-Kan homotopy colimit in a Quillen simplicial model category is an example of simple functor. As a remarkable example outside the setting of Quillen models we include Deligne simple of mixed Hodge complexes. We prove here that the simple functor induces an equivalence on the corresponding localized categories. We also describe a natural structure of Brown category of cofibrant objects on sD. We use these facts to produce cofiber sequences on the localized category of D by E, which give rise to a natural Verdier triangulated structure in the stable case."}
{"category": "Math", "title": "Asymptotic behaviour of thermoviscoelastic Berger plate", "abstract": "System of partial differential equations with a convolution terms and non-local nonlinearity describing oscillations of plate due to Berger approach and with accounting for thermal regime in terms of Coleman-Gurtin and Gurtin-Pipkin law and fading memory of material is considered. The equation is transformed into a dynamical system in a suitable Hilbert space which asymptotic behaviour is analysed. Existence of the compact global attractor in this dynamical system and some of its properties are proved in this article. Main tool in analysis of asymptotic behaviour is stabilizability inequality."}
{"category": "Math", "title": "Asymptotics of randomly stopped sums in the presence of heavy tails", "abstract": "We study conditions under which $P(S_\\tau>x)\\sim P(M_\\tau>x)\\sim E\\tau P(\\xi_1>x)$ as $x\\to\\infty$, where $S_\\tau$ is a sum $\\xi_1+...+\\xi_\\tau$ of random size $\\tau$ and $M_\\tau$ is a maximum of partial sums $M_\\tau=\\max_{n\\le\\tau}S_n$. Here $\\xi_n$, $n=1$, 2, ..., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case where $\\tau$ is independent of the summands; also, in a particular situation, we deal with a stopping time. Also we consider the case where $E\\xi>0$ and where the tail of $\\tau$ is comparable with or heavier than that of $\\xi$, and obtain the asymptotics $P(S_\\tau>x) \\sim E\\tau P(\\xi_1>x)+P(\\tau>x/E\\xi)$ as $x\\to\\infty$. This case is of a primary interest in the branching processes. In addition, we obtain new uniform (in all $x$ and $n$) upper bounds for the ratio $P(S_n>x)/P(\\xi_1>x)$ which substantially improve Kesten's bound in the subclass ${\\mathcal S}^*$ of subexponential distributions."}
{"category": "Math", "title": "A hyperbolic Out(F_n)-complex", "abstract": "For any finite collection $f_i$ of fully irreducible automorphisms of the free group $F_n$ we construct a connected $\\delta$-hyperbolic $Out(F_n)$-complex in which each $f_i$ has positive translation length."}
{"category": "Math", "title": "Survival of contact processes on the hierarchical group", "abstract": "We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals."}
{"category": "Math", "title": "The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum", "abstract": "In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction $A_B$ of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces $\\Sc$ and $\\tilde{\\Sc}$ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples -- one involving a Hain-L\\\"{u}st type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line -- which together indicate that the abstract results are probably best possible."}
{"category": "Math", "title": "Geometric Approach to the Weil-Petersson Symplectic Form", "abstract": "In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on the Teichmueller spaces of compact Riemann surfaces in a purely geometric way. The method can also be applied to situations like moduli spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available."}
{"category": "Math", "title": "Some questions about $\\mathcal G$-bundles on curves", "abstract": "We define the notion of a parahoric group scheme $\\mathcal G$ over a smooth projective curve, and formulate four conjectures on the structure of the stack of $\\mathcal G$-bundles, which generalize to this case well-known results on $G$-bundles with $G$ a constant reductive group. The conjectures concern the set of connected components, the uniformization by affine flag varieties of twisted loop groups, the Picard groups, and the space of global sections of a dominant line bundle. Since a first version of this paper was circulated, Heinloth [arXiv:0711.4450] has proved a good part of these conjectures."}
{"category": "Math", "title": "Necessary and sufficient conditions for the existence of the q-optimal measure", "abstract": "This paper presents the general form and essential properties of the q-optimal measure following the approach of Delbaen and Schachermayer (1996) and proves its existence under mild conditions. Most importantly, it states a necessary and sufficient condition for a candidate measure to be the q-optimal measure in the case even of signed measures. Finally, an updated characterization of the q-optimal measure for continuous asset price processes is presented in the light of the counterexample appearing in Cerny and Kallsen (2006) concerning Hobson's (2004) approach."}
{"category": "Math", "title": "Hilbert-Chow morphism for non commutative Hilbert schemes and moduli spaces of linear representations", "abstract": "Let $k$ be a commutative ring and let $R$ be a commutative $k-$algebra. The aim of this paper is to define and discuss some connection morphisms between schemes associated to the representation theory of a (non necessarily commutative) $R-$algebra $A. $ We focus on the scheme $\\ran//\\GL_n$ of the $n-$dimensional representations of $A, $ on the Hilbert scheme $\\Hilb_A^n$ parameterizing the left ideals of codimension $n$ of $A$ and on the affine scheme Spec $\\Gamma_R^n(A)^{ab} $ of the abelianization of the divided powers of order $n$ over $A. $ We give a generalization of the Grothendieck-Deligne norm map from $\\Hilb_A^n$ to Spec $\\Gamma_R^n(A)^{ab} $ which specializes to the Hilbert Chow morphism on the geometric points when $A$ is commutative and $k$ is an algebraically closed field. Describing the Hilbert scheme as the base of a principal bundle we shall factor this map through the moduli space $\\ran//\\GL_n$ giving a nice description of this Hilbert-Chow morphism, and consequently proving that it is projective."}
{"category": "Math", "title": "Occupation times of subcritical branching immigration system with Markov motions", "abstract": "We consider a branching system consisting of particles moving according to a Markov family in $\\Rd$ and undergoing subcritical branching with a constant rate $V>0$. New particles immigrate to the system according to homogeneous space-time Poisson random field. The process of the fluctuations of the rescaled occupation time is studied with very mild assumptions on the Markov family. In this general setting a functional central limit theorem is proved. The subcriticality of the branching law is crucial for the limit behaviour and in a sense overwhelms the properties of the particles' motion. It is for this reason that the limit is the same for all dimensions and can be obtained for a wide class of Markov processes. Another consequence is the form of the limit - $\\SP$-valued Wiener process with a simple temporal structure and a complicated spatial one. This behaviour contrasts sharply with the case of critical branching systems."}
{"category": "Math", "title": "Hypergraph Ramsey numbers", "abstract": "The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for r_k(s,n) for k \\geq 3 and s fixed. In particular, we show that r_3(s,n) \\leq 2^{n^{s-2}\\log n}, which improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper bound of Erdos and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there are constants c_1,c_2>0 such that r_3(s,n) \\geq 2^{c_1 sn \\log (n/s)} for all 4 \\leq s \\leq c_2n. When s is a constant, it gives the first superexponential lower bound for r_3(s,n), answering an open question posed by Erdos and Hajnal in 1972. Next, we consider the 3-color Ramsey number r_3(n,n,n), which is the minimum N such that every 3-coloring of the triples of an N-element set contains a monochromatic set of size n. Improving another old result of Erdos and Hajnal, we show that r_3(n,n,n) \\geq 2^{n^{c \\log n}}. Finally, we make some progress on related hypergraph Ramsey-type problems."}
{"category": "Math", "title": "The isocohomological property, higher Dehn functions, and relatively hyperbolic groups", "abstract": "The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type $HF^\\infty$, i.e. that has a classifying space with the homotopy type of a cellular complex with finitely many cells in each dimension, we show that the isocohomological property is equivalent to the universal cover of the classifying space satisfying polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable (respectively $HF^\\infty$ and isocohomological), then we show that the group itself has these respective properties too. Combining with the results of Connes-Moscovici and Dru{\\c{t}}u-Sapir we conclude that a group satisfies the Novikov conjecture if it is relatively hyperbolic to subgroups that are of property RD, of type $HF^\\infty$ and isocohomological."}
{"category": "Math", "title": "The commutant of L(H) in its ultrapower may or may not be trivial", "abstract": "Kirchberg asked in 2004 whether the commutant of L(H)$ in its (norm) ultrapower is trivial. Assuming the Continnuum Hypothesis, we prove that the answer depends on the choice of the ultrafilter."}
{"category": "Math", "title": "KBSM of the product of a disk with two holes and S^{1}", "abstract": "We introduce diagrams and Reidemeister moves for links in FxS^{1}, where F is an orientable surface. Using these diagrams we compute (in a new way) the Kauffman Bracket Skein Modules (KBSM) for D^{2}xS^{1} and AxS^{1}, where D^{2} is a disk and A is an annulus. Moreover, we also find the KBSM for the F_{0,3}xS^{1}, where F_{0,3} denotes a disk with two holes, and thus show that the module is free."}
{"category": "Math", "title": "Equilibrium policies when preferences are time inconsistent", "abstract": "This paper characterizes differentiable and subgame Markov perfect equilibria in a continuous time intertemporal decision problem with non-constant discounting. Capturing the idea of non commitment by letting the commitment period being infinitesimally small, we characterize the equilibrium strategies by a value function, which must satisfy a certain equation. The equilibrium equation is reminiscent of the classical Hamilton-Jacobi-Bellman equation of optimal control, but with a non-local term leading to differences in qualitative behavior. As an application, we formulate an overlapping generations Ramsey model where the government maximizes a utilitarian welfare function defined as the discounted sum of successive generations' lifetime utilities. When the social discount rate is different from the private discount rate, the optimal command allocation is time inconsistent and we retain subgame perfection as a principle of intergenerational equity. Existence of multiple subgame perfect equilibria is established. The multiplicity is due to the successive governments' inability to coordinate their beliefs and we single out one of them as (locally) renegotiation-proof. Decentralization can be achieved with both age and time dependent lump sum transfers and, long term distorting capital interest income taxes/subsidy."}
{"category": "Math", "title": "Symplectic fillings of links of quotient surface singularities", "abstract": "We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity."}
{"category": "Math", "title": "A New Hypoelliptic Operator on Almost CR Manifolds", "abstract": "The aim of this paper is to present the construction, out of the Kohn-Rossi complex, of a new hypoelliptic operator $Q_{L}$ on almost CR manifolds equipped with a real structure. The operator acts on all (p,q)-forms, but when restricted to (p,0)-forms and (p,n)-forms it is a sum of squares up to sign factor and lower order terms. Therefore, only a finite type condition condition is needed to have hypoellipticity on those forms. However, outside these forms $Q_{L}$ may fail to be hypoelliptic, as it is shown in the example of the Heisenberg group. We also look at the Fredholm properties of $Q_{L}$ and show that the corresponding Fredholm index is zero."}
{"category": "Math", "title": "Homotopy types of topological stacks", "abstract": "We define the notion of {\\em classifying space} of a topological stack and show that every topological stack \\X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain paracompactness condition on \\X, we show that X is actually well-defined up to homotopy equivalence. These results are formulated in terms of functors from the category of topological stacks to the (weak) homotopy category of topological spaces. We prove similar results for (small) diagrams of topological stacks."}
{"category": "Math", "title": "Meromorphic functions with linearly distributed values and Julia sets of rational functions", "abstract": "If the preimage of a four-point set under a meromorphic function belongs to the real line, then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if the Julia set of a rational function is contained in a smooth curve then it is contained in a circle."}
{"category": "Math", "title": "Some Characterizations of Domination", "abstract": "We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets $\\Sigma$ in $GL(d,\\mathbb{R})$ with the property that any cocycle with values in $\\Sigma$ has a dominated splitting. We characterize these sets in terms of existence of invariant multicones, thus extending a 2-dimensional result by Avila, Bochi, and Yoccoz. We give an example showing how these multicones can fail to have convexity properties."}
{"category": "Math", "title": "On the Koszul Betti numbers in positive characteristic", "abstract": "We have observed a gap in one of the arguments in the main theorem. We choose to withdraw the paper until we rectify the gap."}
{"category": "Math", "title": "Algorithmic enumeration of ideal classes for quaternion orders", "abstract": "We provide algorithms to count and enumerate representatives of the (right) ideal classes of an Eichler order in a quaternion algebra defined over a number field. We analyze the run time of these algorithms and consider several related problems, including the computation of two-sided ideal classes, isomorphism classes of orders, connecting ideals for orders, and ideal principalization. We conclude by giving the complete list of definite Eichler orders with class number at most 2."}
{"category": "Math", "title": "Minimal hypersurfaces in $\\HH^n \\times \\R$, total curvature and index", "abstract": "In this paper, we consider minimal hypersurfaces in the product space $\\mathbb{H}^n \\times \\mathbb{R}$. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the corresponding curvature goes to zero uniformly at infinity. We show that surfaces with finite total intrinsic curvature have finite index. The converse statement is not true as shown by our examples which also serve as useful barriers."}
{"category": "Math", "title": "On the Pfister Number of Quadratic Forms", "abstract": "The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of -1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n-2 terms and not less. The number of 2-fold Pfister forms needed to express a quadratic form of dimension 6 with trivial discriminant is determined in various cases."}
{"category": "Math", "title": "Tetrahedra on deformed spheres and integral group cohomology", "abstract": "We show that for every injective continuous map f: S^2 --> R^3 there are four distinct points in the image of f such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four edges are also of equal length. This result represents a partial result for the topological Borsuk problem for R^3. Our proof of the geometrical claim, via Fadell-Husseini index theory, provides an instance where arguments based on group cohomology with integer coefficients yield results that cannot be accessed using only field coefficients."}
{"category": "Math", "title": "Directed polymer in random environment and last passage percolation", "abstract": "The sequence of random probability measures $\\nu_n$ that gives a path of length $n$, $\\unsur{n}$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment. Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn."}
{"category": "Math", "title": "Gibbs Sampling, Exponential Families and Orthogonal Polynomials", "abstract": "We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions."}
{"category": "Math", "title": "Comment: Gibbs Sampling, Exponential Families and Orthogonal Polynomials", "abstract": "Comment on ``Gibbs Sampling, Exponential Families and Orthogonal Polynomials'' [arXiv:0808.3852]"}
{"category": "Math", "title": "Comment: Gibbs Sampling, Exponential Families, and Orthogonal Polynomials", "abstract": "Comment on ``Gibbs Sampling, Exponential Families, and Orthogonal Polynomials'' [arXiv:0808.3852]"}
{"category": "Math", "title": "Covariate Balance in Simple, Stratified and Clustered Comparative Studies", "abstract": "In randomized experiments, treatment and control groups should be roughly the same--balanced--in their distributions of pretreatment variables. But how nearly so? Can descriptive comparisons meaningfully be paired with significance tests? If so, should there be several such tests, one for each pretreatment variable, or should there be a single, omnibus test? Could such a test be engineered to give easily computed $p$-values that are reliable in samples of moderate size, or would simulation be needed for reliable calibration? What new concerns are introduced by random assignment of clusters? Which tests of balance would be optimal? To address these questions, Fisher's randomization inference is applied to the question of balance. Its application suggests the reversal of published conclusions about two studies, one clinical and the other a field experiment in political participation."}
{"category": "Math", "title": "Comment: Lancaster Probabilities and Gibbs Sampling", "abstract": "Comment on ``Lancaster Probabilities and Gibbs Sampling'' [arXiv:0808.3852]"}
{"category": "Math", "title": "Comment: On Random Scan Gibbs Samplers", "abstract": "Comment on ``On Random Scan Gibbs Samplers'' [arXiv:0808.3852]"}
{"category": "Math", "title": "Parallel in Time Simulation of Multiscale Stochastic Chemical Kinetics", "abstract": "A version of the time-parallel algorithm parareal is analyzed and applied to stochastic models in chemical kinetics. A fast predictor at the macroscopic scale (evaluated in serial) is available in the form of the usual reaction rate equations. A stochastic simulation algorithm is used to obtain an exact realization of the process at the mesoscopic scale (in parallel). The underlying stochastic description is a jump process driven by the Poisson measure. A convergence result in this arguably difficult setting is established suggesting that a homogenization of the solution is advantageous. We devise a simple but highly general such technique. Three numerical experiments on models representative to the field of computational systems biology illustrate the method. For non-stiff problems, it is shown that the method is able to quickly converge even when stochastic effects are present. For stiff problems we are instead able to obtain fast convergence to a homogenized solution. Overall, the method builds an attractive bridge between on the one hand, macroscopic deterministic scales and, on the other hand, mesoscopic stochastic ones. This construction is clearly possible to apply also to stochastic models within other fields."}
{"category": "Math", "title": "Rejoinder: Gibbs Sampling, Exponential Families and Orthogonal Polynomials", "abstract": "We are thankful to the discussants for their hard, interesting work. The main purpose of our paper was to give reasonably sharp rates of convergence for some simple examples of the Gibbs sampler. We chose examples from expository accounts where direct use of available techniques gave practically useless answers. Careful treatment of these simple examples grew into bivariate modeling and Lancaster families. Since bounding rates of convergence is our primary focus, let us begin there. [arXiv:0808.3852]"}
{"category": "Math", "title": "O-minimal homotopy and generalized (co)homology", "abstract": "This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspond (up to an isomorphism) to the known topological generalized homology and cohomology theories on pointed CW-complexes."}
{"category": "Math", "title": "Embeddings of C*-surfaces into weighted projective spaces", "abstract": "Let V be a normal affine surface which admits a C*- and a C+-action. In this note we show that in many cases V can be embedded as a principal Zariski open subset into a hypersurface of a weighted projective space. In particular, we recover a result of D. Daigle and P. Russell."}
{"category": "Math", "title": "Higher Rank Wavelets", "abstract": "A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in $L^2(\\bR^d)$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable."}
{"category": "Math", "title": "A chain morphism for Adams operations on rational algebraic K-theory", "abstract": "For any regular noetherian scheme X and every k>0, we define a chain morphism between two chain complexes whose homology with rational coefficients is isomorphic to the algebraic K-groups of X tensored by the field of rational numbers. It is shown that these morphisms induce in homology the Adams operations defined by Gillet and Soule or the ones defined by Grayson."}
{"category": "Math", "title": "Quasi-analytische Zerlegungen", "abstract": "The leaves in singular holomorphic foliation theory are examples of quasi-analytic layers. In the first part of our publication we are concerned with a theory of these subjects. A quasi-analytic decomposition of a complex manifold is a decomposition into pairwise disjoint connected quasi-analytic layers. These are holomorphic foliations in the sense of P. Stefan and K. Spallek. A very different but more usual conception of holomorphic foliations is develloped by P. Baum and R. Bott. It is based on holomorphic sheaf theory. In the second part we study the relation between quasi-analytic decompositions and singular holomorphic foliations in the sense of Baum and Bott."}
{"category": "Math", "title": "Randomization Does Not Justify Logistic Regression", "abstract": "The logit model is often used to analyze experimental data. However, randomization does not justify the model, so the usual estimators can be inconsistent. A consistent estimator is proposed. Neyman's non-parametric setup is used as a benchmark. In this setup, each subject has two potential responses, one if treated and the other if untreated; only one of the two responses can be observed. Beside the mathematics, there are simulation results, a brief review of the literature, and some recommendations for practice."}
{"category": "Math", "title": "The modular branching rule for affine Hecke algebras of type A", "abstract": "For the affine Hecke algebra of type A at roots of unity, we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule. The latter generalizes work by Vazirani."}
{"category": "Math", "title": "A sectional characterization of the Dade group", "abstract": "Let $k$ be a field of characteristic $p$, let $P$ be a finite $p$- group, where $p$ is an odd prime, and let $D(P)$ be the Dade group of endo-permutation $kP$-modules. It is known that $D(P)$ is detected via deflation--restriction by the family of all sections of $P$ which are elementary abelian of rank $\\leq2$. In this paper, we improve this result by characterizing $D(P)$ as the limit (with respect to deflation--restriction maps and conjugation maps) of all groups $D(T/S)$ where $T/S$ runs through all sections of $P$ which are either elementary abelian of rank $\\leq3$ or extraspecial of order $p^3$ and exponent $p$."}
{"category": "Math", "title": "On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions", "abstract": "n this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko-Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given."}
{"category": "Math", "title": "Generators and defining relations for ring of differential operators on smooth affine algebraic variety in prime characteristic", "abstract": "For ring of differential operators on smooth affine algebraic variety over perfect field of prime characteristic a set of algebra generators and a set of defining relations are found explicitly."}
{"category": "Math", "title": "Stability of roots of polynomials under linear combinations of derivatives", "abstract": "Let $T=\\alpha_0 I + \\alpha_1 D + ...+\\alpha_n D^n$, where $D$ is the differentiation operator and $\\alpha_0\\not= 0$, and let $f$ be a square-free polynomial with large minimum root separation. We prove that the roots of $Tf$ are close to the roots of $f$ translated by $-\\alpha_1/\\alpha_0$."}
{"category": "Math", "title": "Anti-Perfect Morse Stratification", "abstract": "For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivariantly perfect case). We also introduce a weaker condition of local equivariant antiperfection. We prove that the Morse stratification of the Yang-Mills functional on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface is locally equivariantly Q-antiperfect when the rank n=2,3; we propose that it is actually equivariantly Q-antiperfect when n=2,3. Our proposal yields formulas of G-equivariant Poincare series of the representation variety of flat G-connections for the nonorientable surface where G=U(2), SU(2), U(3), SU(3). Our rank 2 formulas agree with formulas proved by T. Baird in arXiv:0806.1975. Baird verified our conjectural rank 3 formulas when the nonorientable surface is the real projective plane or the Klein bottle (arXiv:0901.1604); he proved our conjectural U(3) formula for any closed nonorientable surfaces by establishing equivariant Q-antiperfection in this case (arXiv:0902.4581)."}
{"category": "Math", "title": "Least Squares Methods for Equidistant Tree Reconstruction", "abstract": "UPGMA is a heuristic method identifying the least squares equidistant phylogenetic tree given empirical distance data among $n$ taxa. We study this classic algorithm using the geometry of the space of all equidistant trees with $n$ leaves, also known as the Bergman complex of the graphical matroid for the complete graph $K_n$. We show that UPGMA performs an orthogonal projection of the data onto a maximal cell of the Bergman complex. We also show that the equidistant tree with the least (Euclidean) distance from the data is obtained from such an orthogonal projection, but not necessarily given by UPGMA. Using this geometric information we give an extension of the UPGMA algorithm. We also present a branch and bound method for finding the best equidistant tree. Finally, we prove that there are distance data among $n$ taxa which project to at least $(n-1)!$ equidistant trees."}
{"category": "Math", "title": "Lagrangian Quantum Homology", "abstract": "The present paper is mainly a survey of our work arXiv:0708.4221 and arXiv:0808.2440 but it also contains the announcement of some new results. Its main purpose is to present an accessible introduction to a technique allowing efficient calculations in Lagrangian Floer theory."}
{"category": "Math", "title": "Viability for stochastic differential equations driven by fractional Brownian motion", "abstract": "In this paper we prove a viability result for multidimensional, time dependent, stochastic differential equations driven by fractional Brownian motion with Hurst parameter1/2 < H < 1, using pathwise approach. The sufficient condition is also an alternative global existence result for the fractional differential equations with restrictions on the state."}
{"category": "Math", "title": "The Scalar Curvature Equation on $S^3$", "abstract": "We give existence results for solutions of the prescribed scalar curvature equation on $S^3$, when the curvature function is a positive Morse function and satisfies an index-count condition."}
{"category": "Math", "title": "Fourier transform of function on locally compact Abelian groups taking value in Banach spaces", "abstract": "We consider Fourier transform of vector-valued functions on a locally compact group $G$, which take value in a Banach space $X$, and are square-integrable in Bochner sense. If $G$ is a finite group then Fourier transform is a bounded operator. If $G$ is an infinite group then Fourier transform $F: L_2(G,X)\\to L_2(\\widehat G,X)$ is a bounded operator if and only if Banach space $X$ is isomorphic to a Hilbert one."}
{"category": "Math", "title": "A priori Holder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps", "abstract": "In this paper, we consider the following type of non-local (pseudo-differential) operators $\\LL $ on $\\R^d$: $$ \\LL u(x) =\\frac12 \\sum_{i, j=1}^d \\frac{\\partial}{\\partial x_i} (a_{ij}(x) \\frac{\\partial}{\\partial x_j}) + \\lim_{\\eps \\downarrow 0} \\int_{\\{y\\in \\R^d: |y-x|>\\eps\\}} (u(y)-u(x)) J(x, y) dy, $$ where $A(x)=(a_{ij}(x))_{1\\leq i, j\\leq d}$ is a measurable $d\\times d$ matrix-valued function on $\\R^d$ that is uniform elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on $\\R^d\\times \\R^d$ satisfying certain conditions. Corresponding to $\\LL$ is a symmetric strong Markov process $X$ on $\\R^d$ that has both the diffusion component and pure jump component. We establish a priori H\\\"older estimate for bounded parabolic functions of $\\LL$ and parabolic Harnack principle for positive parabolic functions of $\\LL$. Moreover, two-sided sharp heat kernel estimates are derived for such operator $\\LL$ and jump-diffusion $X$. In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on $\\R^d$. To establish these results, we employ methods from both probability theory and analysis."}
{"category": "Math", "title": "The H\\\"older continuity of a class of 3-dimension ultraparabolic equations", "abstract": "We obtained the $C^\\alpha$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form $${\\partial_t u} = \\partial_x(a(x,y,t)\\partial_x u) +b_0(x,y,t)\\partial_x u+b(x,y,t)\\partial_y u,$$ which generalized our recent results on KFP equations."}
{"category": "Math", "title": "Fundamental Dominations in Graphs", "abstract": "Nine variations of the concept of domination in a simple graph are identified as fundamental domination concepts, and a unified approach is introduced for studying them. For each variation, the minimum cardinality of a subset of dominating elements is the corresponding fundamental domination number. It is observed that, for each nontrivial connected graph, at most five of these nine numbers can be different, and inequalities between these five numbers are given. Finally, these fundamental dominations are interpreted in terms of the total graph of the given graph, a concept introduced by the second author in 1965. It is argued that the very first domination concept, defined by O. Ore in 1962 and under a different name by C. Berge in 1958, deserves to be called the most fundamental of graph dominations."}
{"category": "Math", "title": "The Center of Mass for Spatial Branching Processes and an Application for Self-Interaction", "abstract": "In this paper we prove that the center of mass of a supercritical branching-Brownian motion, or that of a supercritical super-Brownian motion tends to a limiting position almost surely, which, in a sense complements a result of Tribe on the final behavior of a critical super-Brownian motion. This is shown to be true also for a model where branching Brownian motion is modified by attraction/repulsion between particles. We then put this observation together with the description of the interacting system as viewed from its center of mass, and get the following asymptotic behavior: the system asymptotically becomes a branching Ornstein Uhlenbeck process (inward for attraction and outward for repulsion), but the origin is shifted to a random point which has normal distribution, and the Ornstein Uhlenbeck particles are not independent but constitute a system with a degree of freedom which is less by their number by precisely one."}
{"category": "Math", "title": "Regular projections of graphs with at most three double points", "abstract": "A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free if it contains neither a Hopf link nor a trefoil knot."}
{"category": "Math", "title": "On a Construction of L. Hua for Positive Reproducing Kernels", "abstract": "We study a positive reproducing kernel for holomorphic functions on a domain in a complex space. The technique is based on an idea of L. Hua. Applications are provided. These ideas were developed in another context (quantization of K\\\"{a}hler manifolds) by Berezin."}
{"category": "Math", "title": "Nakajima's remark on Henn's proof", "abstract": "We fill up a gap in Henn's proof concerning large automorphism groups of function fields of degree 1 over an algebraically closed field of positive characteristic."}
{"category": "Math", "title": "Hybrid data regression modelling in measurement", "abstract": "Measurement involves the determination of quantitative estimates of physical quantities from experiment, along with estimates of their associated uncertainties. Herewith an experimental system model is the key to extracting information from the experimental data. The measurement information obtained depends directly on the quality of the model. With this concern novel regression modelling techniques have been fashioned by data integration from computer-simulation and physical designed experiments. These techniques have allowed attaining the advanced level of model completeness, parsimony, and precision via approximation of the exact unknown model by mathematical product of available theoretical and appropriate empirical functions. The purpose of this approximation is to represent adequately the true model on the considered region of factor space with all advantages of theoretical modelling. This allows a further focus on the measurement science of issue. Pneumatic gauge hybrid data candidate model building, solving and validation reviled that such adequate models permit to attain minimum discrepancy from empirical evidence."}
{"category": "Math", "title": "Karl Pearson's Theoretical Errors and the Advances They Inspired", "abstract": "Karl Pearson played an enormous role in determining the content and organization of statistical research in his day, through his research, his teaching, his establishment of laboratories, and his initiation of a vast publishing program. His technical contributions had initially and continue today to have a profound impact upon the work of both applied and theoretical statisticians, partly through their inadequately acknowledged influence upon Ronald A. Fisher. Particular attention is drawn to two of Pearson's major errors that nonetheless have left a positive and lasting impression upon the statistical world."}
{"category": "Math", "title": "Sur l'homologie des groupes orthogonaux et symplectiques \\`a coefficients tordus", "abstract": "We compute the stable homology of orthogonal and symplectic groups over a finite field k with coefficients coming from an usual endofunctor F of k-vector spaces (exterior, symmetric, divided powers...), that is, for all natural integer i, we compute the colimits of the vector spaces $H_i(O_{n,n}(k) ; F(k^{2n}))$ and $H_i(Sp_{2n}(k) ; F(k^{2n}))$. In this situation, the stabilization is a classical result of Charney. We give a formal framework to connect stable homology of some families of groups and homology of suitable small categories thanks to a spectral sequence which collapses in several cases. By our purely algebraic methods (i.e. without stable K-theory) we obtain again results of Betley for stable homology of linear groups and symmetric groups. For orthogonal and symplectic groups over a field we prove a categorical result for vector spaces equipped with quadratic or alternating forms and use powerful cancellation results known in homology of functors (Suslin, Scorichenko, Djament) to deduce a spectacular simplification of the second sheet of our general spectral sequence. When we consider the orthogonal and symplectic groups over a finite field and we take coefficients with values in vector spaces over the same field, we can compute the second sheet of the spectral sequence thanks to classical results: homological cancellation with trivial coefficients (Quillen, Fiedorowicz-Priddy) and calculation of torsion groups between usual functors (Franjou-Friedlander-Scorichenko-Suslin, Chalupnik)."}
{"category": "Math", "title": "Closed Magnetic Geodesics on $S^2$", "abstract": "We give existence results for simple closed curves with prescribed geodesic curvature on $S^{2}$, which correspond to periodic orbits of a charge in a magnetic field."}
{"category": "Math", "title": "Statistical models, likelihood, penalized likelihood and hierarchical likelihood", "abstract": "We give an overview of statistical models and likelihood, together with two of its variants: penalized and hierarchical likelihood. The Kullback-Leibler divergence is referred to repeatedly, for defining the misspecification risk of a model, for grounding the likelihood and the likelihood crossvalidation which can be used for choosing weights in penalized likelihood. Families of penalized likelihood and sieves estimators are shown to be equivalent. The similarity of these likelihood with a posteriori distributions in a Bayesian approach is considered."}
{"category": "Math", "title": "A Conversation with Jayaram Sethuraman", "abstract": "Jayaram Sethuraman was born in the town of Hubli in Bombay Province (now Karnataka State) on October 3, 1937. His early years were spent in Hubli and in 1950 his family moved to Madras (now renamed Chennai). He graduated from Madras University in 1957 with a B.Sc. (Hons) degree in statistics and he earned his M.A. degree in statistics from Madras University in 1958. He earned a Ph.D. in statistics from the Indian Statistical Institute in 1962. Before returning to ISI in 1965 as an Associate Professor, he was a Research Associate at the University of North Carolina 1962--1963, at Michigan State University in 1963--1964 and at Stanford University 1964--1965. After three years at the ISI, Sethuraman moved to Florida State University in 1968 as Full Professor. During his career at FSU, he made sojourns as Visiting Professor to the University of Michigan, 1974--1975, the ISI in fall 1977, as a Visiting Professor and Acting Head, ISI Bangalore Center, 1979--1980. He was a senior ASA/NSF/NIST Fellow 1994--1995 and a Fulbright Senior Researcher at ISI Bangalore 1995--1996. Although Sethuraman officially retired on January 31, 2004 and was named Professor Emeritus at FSU, he continues to be extremely active. He participates in all activities in the Department of Statistics and holds a Courtesy Professor appointment in the Department of Religion. He held an appointment as Professor, University of Pittsburgh in the fall of 2004, and was a Fulbright Senior Lecturer at the Indian Statistical Institute of Technology, Chennai, 2005."}
{"category": "Math", "title": "The Unitary Implementation of a Measured Quantum Groupoid action", "abstract": "Mimicking the von Neumann version of Kustermans and Vaes' locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In a former article, the author had introduced the notion of actions, crossed-product, dual actions of a measured quantum groupoid: a biduality theorem for actions had been proved. This article continues that program : we prove the existence of a standard implementation for an action, and a bidulaity theorem for weights. We generalize this way results which were proved, for locally compact quantum groups by S. Vaes, and for measured groupoids by T. Yamanouchi."}
{"category": "Math", "title": "Honest variable selection in linear and logistic regression models via $\\ell_1$ and $\\ell_1+\\ell_2$ penalization", "abstract": "This paper investigates correct variable selection in finite samples via $\\ell_1$ and $\\ell_1+\\ell_2$ type penalization schemes. The asymptotic consistency of variable selection immediately follows from this analysis. We focus on logistic and linear regression models. The following questions are central to our paper: given a level of confidence $1-\\delta$, under which assumptions on the design matrix, for which strength of the signal and for what values of the tuning parameters can we identify the true model at the given level of confidence? Formally, if $\\widehat{I}$ is an estimate of the true variable set $I^*$, we study conditions under which $\\mathbb{P}(\\widehat{I}=I^*)\\geq 1-\\delta$, for a given sample size $n$, number of parameters $M$ and confidence $1-\\delta$. We show that in identifiable models, both methods can recover coefficients of size $\\frac{1}{\\sqrt{n}}$, up to small multiplicative constants and logarithmic factors in $M$ and $\\frac{1}{\\delta}$. The advantage of the $\\ell_1+\\ell_2$ penalization over the $\\ell_1$ is minor for the variable selection problem, for the models we consider here. Whereas the former estimates are unique, and become more stable for highly correlated data matrices as one increases the tuning parameter of the $\\ell_2$ part, too large an increase in this parameter value may preclude variable selection."}
{"category": "Math", "title": "Measured quantum groupoids with a central basis", "abstract": "Mimicking the von Neumann version of Kustermans and Vaes' locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In this article, we suppose that the basis of the measured quantum groupoid is central; in that case, we prove that a specific sub-${\\bf C}^*$ algebra is invariant under all the data of the measured quantum groupoid; moreover, this sub-${\\bf C}^*$-algebra is a continuous field of ${\\bf C}^*$-algebras; when the basis is central in both the measured quantum groupoid and its dual, we get that the measured quantum groupoid is a continuous field of locally compact quantum groups. On the other hand, using this sub-${\\bf C}^*$-algebra, we prove that any abelian measured quantum groupoid comes from a locally compact groupoid."}
{"category": "Math", "title": "Sparse sampling: Spatial design for monitoring stream networks", "abstract": "Spatial designs for monitoring stream networks, especially ephemeral systems, are typically non-standard, `sparse' and can be very complex, reflecting the complexity of the ecosystem being monitored, the scale of the population, and the competing multiple monitoring objectives. The main purpose of this paper is to present a review of approaches to spatial design to enable informed decisions to be made about developing practical and optimal spatial designs for future monitoring of streams."}
{"category": "Math", "title": "Fonctions L d'Artin et nombre de Tamagawa motiviques", "abstract": "In the first part of this text, we define motivic Artin L-fonctions via a motivic Euler product, and show that they coincide with the analogous functions introduced by Dhillon and Minac. In the second part, we define under some assumptions a motivic Tamagawa number and show that it specializes to the Tamagawa number introduced by Peyre in the context of Manin's conjectures about rational points of bounded height on Fano varieties."}
{"category": "Math", "title": "Any sub-Riemannian Metric has Points of Smoothness", "abstract": "We prove the result stated in the title; it is equivalent to the existence of a regular point of the sub-Riemannian exponential mapping. We also prove that the metric is analytic on an open everywhere dense subset in the case of a complete real-analytic sub-Riemannian manifold."}
{"category": "Math", "title": "The diameter of sparse random graphs", "abstract": "In this paper we study the diameter of the random graph $G(n,p)$, i.e., the the largest finite distance between two vertices, for a wide range of functions $p=p(n)$. For $p=\\la/n$ with $\\la>1$ constant, we give a simple proof of an essentially best possible result, with an $O_p(1)$ additive correction term. Using similar techniques, we establish 2-point concentration in the case that $np\\to\\infty$. For $p=(1+\\epsilon)/n$ with $\\epsilon\\to 0$, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an $O_p(1/\\epsilon)$ additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph $G(n,p)$ to an accuracy of the order of its standard deviation (or better), for all functions $p=p(n)$. Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2-core and the trees attached to it."}
{"category": "Math", "title": "On a question of Goss", "abstract": "In this note we answer the question raised by D. Goss in [Applications of non-Archimedean integration to the $L$-series of $\\tau$-sheaves, {\\em J. Number Theory,} 110 (2005), no. 1, 83--113] by proving that the group of locally analytic endomorphisms on the 1-units of a locally compact field of characteristic $p>0$ is isomorphic to the $p$-adic integers."}
{"category": "Math", "title": "Factoriality of complete intersection threefolds", "abstract": "Let X be a complete intersection of two hypersurfaces F_n and F_k in the projective space P^5 of degree n and k respectively with n >= k, such that the singularities of X are nodal and F_k is smooth. We prove that if the threefold X has at most (n+k-2)(n-1)-1 singular points, then it is factorial."}
{"category": "Math", "title": "Split Orders and Convex Polytopes in Buildings", "abstract": "As part of his work to develop an explicit trace formula for Hecke operators on congruence subgroups of $SL_2(\\Z)$, Hijikata defines and characterizes the notion of a split order in $M_2(k)$, where $k$ is a local field. In this paper, we generalize the notion of a split order to $M_n(k)$ for $n>2$ and give a natural geometric characterization in terms of the affine building for $SL_n(k)$. In particular, we show that there is a one-to-one correspondence between split orders in $M_n(k)$ and a collection of convex polytopes in apartments of the building such that the split order is the intersection of all the maximal orders representing the vertices in the polytope. This generalizes the geometric interpretation in the $n=2$ case in which split orders correspond to geodesics in the tree for $SL_2(k)$ with the split order given as the intersection of the endpoints of the geodesic."}
{"category": "Math", "title": "PEL moduli spaces without $\\mathbb C$-valued points", "abstract": "We give several new moduli interpretations of the fibers of certain Shimura varieties over several prime numbers. As a consequence (of our theorem 9.1) one obtains that for every prescribed odd prime characteristic $p$ every bounded symmetric domain possesses quotients by arithmetic groups whose models have good reduction at a prime divisor of $p$."}
{"category": "Math", "title": "A Thomason Model Structure on the Category of Small n-fold Categories", "abstract": "We construct a cofibrantly generated Quillen model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. This is an n-fold analogue to Thomason's Quillen model structure on Cat. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, we completely prove that the unit and counit of the adjunction between simplicial sets and n-fold categories are natural weak equivalences."}
{"category": "Math", "title": "A note on cyclic semiregular subgroups of some 2-transitive permutation groups", "abstract": "We determine the semi-regular subgroups of the 2-transitive permutation groups PGL(2,n), PSL(2,n), PGU(3,n), PSU(3,n), Sz(n) and Ree(n) with n a suitable power of a prime number p."}
{"category": "Math", "title": "Length Four Polynomial Automorphisms", "abstract": "We study the structure of length four polynomial automorphisms of $R[X,Y]$ when $R$ is a UFD. The results from this study are used to prove that if $\\text{SL}_m(R[X_1,X_2,..., X_n]) = \\text{E}_m(R[X_1,X_2,..., X_n])$ for all $n, m \\ge 0$ then all length four polynomial automorphisms of $R[X,Y]$ that are conjugates are stably tame."}
{"category": "Math", "title": "Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r)", "abstract": "In this paper we present systematically analysis on the smallest eigenvalue of matrices associated with completely even functions (mod $r$). We obtain several theorems on the asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod $r$). In particular, we get information on the asymptotic behavior of the smallest eigenvalue of the famous Smith matrices. Finally some examples are given to demonstrate the main results."}
{"category": "Math", "title": "Inverse Limits of Uniform Covering Maps", "abstract": "In ``Rips complexes and covers in the uniform category'' the authors define, following James, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper notes that the universal generalized uniform covering map is uniformly equivalent to the inverse limit of uniform covering maps and is therefore approximated by uniform covering maps. A characterization of generalized uniform covering maps that are approximated by uniform covering maps is provided as well as a characterization of generalized uniform covering maps that are uniformly equivalent to the inverse limit of uniform covering maps. Inverse limits of group actions that induce generalized uniform covering maps are also treated."}
{"category": "Math", "title": "Poisson quasi-Nijenhuis structures with background", "abstract": "We define the Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that to any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid is associated such a structure. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background constitutes, with its dual, a quasi-Lie bialgebroid. We also prove that any pair $(\\pi,\\omega)$ of a Poisson bivector and a 2-form induces a Poisson quasi-Nijenhuis structure with background and we observe that particular cases correspond to already known compatibilities between $\\pi$ and $\\omega$."}
{"category": "Math", "title": "Bianchi-B\\\"{a}cklund transforms and dressing actions, revisited", "abstract": "We characterize Bianchi-B\\\"{a}cklund transformations of surfaces of positive constant Gauss curvature in terms of dressing actions of the simplest type on the space of harmonic maps."}
{"category": "Math", "title": "Asymptotic and descent formulas for weighted orbital integrals", "abstract": "We rewrite Arthur's asymptotic formula for weighted orbital integrals on real groups with the aid of a residue calculus and extend the resulting formula to the Schwartz space. Then we extract the available information about the coefficients in the decomposition of the Fourier transforms of Arthur's invariant distributions I_M(\\gamma) in terms of standard solutions of the pertinent holonomic system of differential equations. This allows us to determine some of those coefficients explicitly. Finally, we prove descent formulas for those differential equations, for their standard solutions and for the aforementioned coefficients, which reduce each of them to the case that \\gamma is elliptic in M."}
{"category": "Math", "title": "On the compactification of concave ends", "abstract": "Let X be a complex manifold of dimension 2, which admits a strictly plurisubharmonic function r which is proper as a function with values in the intervall ]inf r, sup r[. We prove that the concave end of X can be compactified, if and only if, the first cohomology of X is Hausdorff."}
{"category": "Math", "title": "Can you hear the shape of a Beatty sequence?", "abstract": "Let K(x_1,...,x_d) be a polynomial. If you are not given the real numbers \\alpha_1, \\alpha_2, ...,\\alpha_d, but are given the polynomial K and the sequence a_n=K(\\floor{n\\alpha_1},\\floor{n\\alpha_2},...,\\floor{n\\alpha_d}), can you deduce the values of \\alpha_i? Not, it turns out, in general. But with additional irrationality hypotheses and certain polynomials, it is possible. We also consider the problem of deducing \\alpha_i from the integer sequence with nested flooring (\\floor{\\floor{... \\floor{\\floor{n\\alpha_1}\\alpha_2}... \\alpha_{d-1}}\\alpha_d})_{n=1}^\\infty."}
{"category": "Math", "title": "Jamming as Information: a Geometric Approach", "abstract": "In this paper I discuss the kinds of information that can be extracted by our enemy if our jamming is too precise. I show geometric solutions for reconstructing linear routes given certain information about them, such as the shortest distance to a point or the times of entering and exiting a circle."}
{"category": "Math", "title": "On cuspidal sections of algebraic fundamental groups", "abstract": "Rational points in the boundary of a hyperbolic curve over a field with sufficiently nontrivial Kummer theory are the source for an abundance of sections of the fundamental group exact sequence. We follow and refine Nakamura's approach towards these boundary sections. For example, we obtain a weak anabelian theorem for hyperbolic genus 0 curves over quite general fields including for example the maximal abelian extension of the rational numbers."}
{"category": "Math", "title": "Second symmetric powers of chain complexes", "abstract": "We investigate Buchbaum and Eisenbud's construction of the second symmetric power S^2_R(X) of a chain complex X of modules over a commutative ring R. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construction to prove the following version of a result of Avramov, Buchweitz, and Sega: Let R \\to S be a module-finite ring homomorphism such that R is noetherian and local, and such that 2 is a unit in R. Let X be a complex of finite rank free S-modules such that X_n = 0 for each n < 0. If \\cup_n Ass_R(H_n(X \\otimes_S X)) \\subseteq Ass(R) and if X_P \\simeq S_P for each P \\in Ass(R), then X \\simeq S."}
{"category": "Math", "title": "Comodules for some simple $\\mathcal O$-forms of $\\mathbb G_m$", "abstract": "This paper gives a rather concrete description of the category Rep(G) for certain flat commutative affine group schemes G over a discrete valuation ring such that the general fiber of G is the multiplicative group."}
{"category": "Math", "title": "On modular forms for some noncongruence subgroups of SL_2(Z) II", "abstract": "In this paper we show two classes of noncongruence subgroups satisfy the so-called unbounded denominator property. In particular, we establish our conjecture in [KL08] which says that every type II noncongruence character group of Gamma^0(11) satisfies the unbounded denominator property."}
{"category": "Math", "title": "A spectral method for elliptic equations: the Dirichlet problem", "abstract": "An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate polynomials as the approximants. For a smooth boundary and smooth problem parameter functions, the method is proven to converge faster than any power of 1/n with n the degree of the approximate Galerkin solution. Examples in two and three variables are given as numerical illustrations. Empirically, the condition number of the associated linear system increases like O(N), with N the order of the linear system."}
{"category": "Math", "title": "The density of Lawrence-Krammer and non-conjugate braid representations of links", "abstract": "We use some Lie group theory and Budney's unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on any non-minimal number of (and at least 4) strands."}
{"category": "Math", "title": "Weil-Petersson geometry for families of hyperbolic conical Riemann Surfaces", "abstract": "We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1."}
{"category": "Math", "title": "Weak convergence of the regularization path in penalized M-estimation", "abstract": "We consider an estimator $\\hbbeta_n(t)$ defined as the element $\\bphi\\in\\bPhi$ minimizing a contrast process $\\pencontrast(\\bphi, t)$ for each t. We give some general results for deriving the weak convergence of $\\sqrt{n}(\\hbbeta_n-\\bbeta)$ in the space of bounded functions, where, for each t, $\\bbeta(t)$ is the $\\bphi\\in\\bPhi$ minimizing the limit of $\\pencontrast(\\bphi, t)$ as $n\\to\\infty$. These results are applied in the context of penalized M-estimation, that is, when $\\pencontrast(\\bphi, t)=M_n(\\bphi)+ t J_n(\\bphi)$, where $M_n$ is a usual contrast process and $J_n$ a penalty such as the $\\ell^1$ norm or the squared $\\ell^2$ norm. The function $\\hbbeta_n$ is then called a \\emph{regularization path}. For instance we show that the central limit theorem established for the lasso estimator in Knight and Fu (2000) continues to hold in a functional sense for the regularization path. Other examples include various possible contrast processes for $M_n$ such as those considered in Pollard (1985)."}
{"category": "Math", "title": "The $\\ell$-adic Dualizing Complex on an Excellent Surface with Rational Singularities", "abstract": "In this article, we show that if $X$ is an excellent surface with rational singularities, the constant sheaf $\\mathbb{Q}_{\\ell}$ is a dualizing complex. In coefficient $\\mathbb{Z}_{\\ell}$, we also prove that the obstruction for $\\mathbb{Z}_{\\ell}$ to become a dualizing complex lying on the divisor class groups at singular points. As applications, we study the perverse sheaves and the weights of $\\ell$-adic cohomology groups on such surfaces."}
{"category": "Math", "title": "Group algebras of finite groups as Lie algebras", "abstract": "We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of $G$."}
{"category": "Math", "title": "Matrices related to Dirichlet series", "abstract": "We attach a certain $n \\times n$ matrix $A_n$ to the Dirichlet series $L(s)=\\sum_{k=1}^{\\infty}a_k k^{-s}$. We study the determinant, characteristic polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of $A_n$ can be understood as a weighted sum of the first $n$ coefficients of the Dirichlet series $L(s)^{-1}$. We give an interpretation of the partial sum of a Dirichlet series as a product of eigenvalues. In a special case, the determinant of $A_n$ is the sum of the M\\\"obius function. We disprove a conjecture of Barrett and Jarvis regarding the eigenvalues of $A_n$."}
{"category": "Math", "title": "Generalized asymptotic Euler's relation for certain families of polytopes", "abstract": "According to Euler's relation any polytope P has as many faces of even dimension as it has faces of odd dimension. As a generalization of this fact one can compare the number of faces whose dimension is congruent to i modulo m with the number of all faces of P for some positive integer m and for some 0 < i < m+1. We show some classes of polytopes for which the above proportion is asymptotically equal to 1/m."}
{"category": "Math", "title": "Dualizing complex of a toric face ring", "abstract": "A \"toric face ring\", which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Roemer and their coauthors recently. In this paper, under the \"normality\" assumption, we describe a dualizing complex of a toric face ring $R$ in a very concise way. Since $R$ is not a graded ring in general, the proof is not straightforward. We also develop the squarefree module theory over $R$, and show that the Buchsbaum property and the Gorenstein* property of $R$ are topological properties of its associated cell complex."}
{"category": "Math", "title": "Twisting of the Quantum double and the Weyl algebra", "abstract": "Quantum double construction, originally due to Drinfeld and has been since generalized even to the operator algebra framework, is naturally associated with a certain (quasitriangular) $R$-matrix ${\\mathcal R}$. It turns out that ${\\mathcal R}$ determines a twisting of the comultiplication on the quantum double. It then suggests a twisting of the algebra structure on the dual of the quantum double. For $D(G)$, the $C^*$-algebraic quantum double of an ordinary group $G$, the \"twisted $\\hat{D(G)}$\" turns out to be the Weyl algebra $C_0(G)\\times_{\\tau}G$, which is in turn isomorphic to ${\\mathcal K}(L^2(G))$. This is the $C^*$-algebraic counterpart to an earlier (finite-dimensional) result by Lu. It is not so easy technically to extend this program to the general locally compact quantum group case, but we propose here some possible approaches, using the notion of the (generalized) Fourier transform."}
{"category": "Math", "title": "Some families of supersingular Artin-Schreier curves in characteristic > 2", "abstract": "In this short paper we prove that the following two 1-dimensional families of Artin-Schreier curves are supersingular: y^7 - y = x^5 + c.x^2 over F_7 y^5 - y = x^7 + c.x over F_5 (for some parameter c). Our method is developed upon the p-adic Dwork's method."}
{"category": "Math", "title": "Finite Sets and Counting", "abstract": "We start by presenting a theory of finite sets using the approach which is essentially that taken by Whitehead and Russell in Principia Mathematica}, and which does not involve the natural numbers (or any other infinite set). This theory is then applied to prove results about structures which, like the natural numbers, satisfy the principle of mathematical induction, but do not necessarily satisfy the remaining Peano axioms."}
{"category": "Math", "title": "Bodies of zero resistance and bodies invisible in one direction", "abstract": "We consider a body in a parallel flow of non-interacting particles. The interaction of particles with the body is perfectly elastic. We introduce the notions of a body of zero resistance, a body that leaves no trace, and an invisible body, and prove that all such bodies do exist."}
{"category": "Math", "title": "On the noncommutative Donaldson-Thomas invariants arising from brane tilings", "abstract": "Given a brane tiling, that is a bipartite graph on a torus, we can associate with it a quiver potential and a quiver potential algebra. Under certain consistency conditions on a brane tiling, we prove a formula for the Donaldson-Thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra. We relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling. We prove that the same consistency conditions imply that the quiver potential algebra is a 3-Calabi-Yau algebra. We also formulate a rationality conjecture for the generating functions of the Donaldson-Thomas type invariants."}
{"category": "Math", "title": "Nonsmoothable group actions on spin 4-manifolds", "abstract": "We show that every closed, simply connected, spin topological 4-manifold except $S^4$ and $S^2\\times S^2$ admits a homologically trivial, pseudofree, locally linear action of $\\mathbb{Z}_p$ for any sufficiently large prime number $p$ which is nonsmoothable for any possible smooth structure."}
{"category": "Math", "title": "The Reverse Ultra Log-Concavity of the Boros-Moll Polynomials", "abstract": "We prove the reverse ultra log-concavity of the Boros-Moll polynomials. We further establish an inequality which implies the log-concavity of the sequence $\\{i!d_i(m)\\}$ for any $m\\geq 2$, where $d_i(m)$ are the coefficients of the Boros-Moll polynomials $P_m(a)$. This inequality also leads to the fact that in the asymptotic sense, the Boros-Moll sequences are just on the borderline between ultra log-concavity and reverse ultra log-concavity. We propose two conjectures on the log-concavity and reverse ultra log-concavity of the sequence $\\{d_{i-1}(m) d_{i+1}(m)/d_i(m)^2\\}$ for $m\\geq 2$."}
{"category": "Math", "title": "Representation zeta functions of wreath products with finite groups", "abstract": "Let G be a group which has for all n a finite number r_n(G) of irreducible complex linear representations of dimension n. Let $\\zeta(G,s) = \\sum_{n=1}^{\\infty} r_n(G) n^{-s}$ be its representation zeta function. First, in case G is a permutational wreath product of H with a permutation group Q acting on a finite set X, we establish a formula for $\\zeta(G,s)$ in terms of the zeta functions of H and of subgroups of Q, and of the Moebius function associated with the lattice of partitions of X in orbits under subgroups of Q. Then, we consider groups W(Q,k) which are k-fold iterated wreath products of Q, and several related infinite groups W(Q), including the profinite group, a locally finite group, and several finitely generated groups, which are all isomorphic to a wreath product of themselves with Q. Under convenient hypotheses (in particular Q should be perfect), we show that r_n(W(Q)) is finite for all n, and we establish that the Dirichlet series $\\zeta(W(Q),s)$ has a finite and positive abscissa of convergence s_0. Moreover, the function $\\zeta(W(Q),s)$ satisfies a remarkable functional equation involving $\\zeta(W(Q),es)$ for e=1,...,|X|. As a consequence of this, we exhibit some properties of the function, in particular that $\\zeta(W(Q),s)$ has a singularity at s_0, a finite value at s_0, and a Puiseux expansion around s_0. We finally report some numerical computations for Q the simple groups of order 60 and 168."}
{"category": "Math", "title": "Hodge polynomials of some moduli spaces of Coherent Systems", "abstract": "When $k<n$, we study the coherent systems that come from a BGN extension in which the quotient bundle is strictly semistable. In this case we describe a stratification of the moduli space of coherent systems. We also describe the strata as complements of determinantal varieties and we prove that these are irreducible and smooth. These descriptions allow us to compute the Hodge polynomials of this moduli space in some cases. In particular, we give explicit computations for the cases in which $(n,d,k)=(3,d,1)$ and $d$ is even, obtaining from them the usual Poincar\\'e polynomials."}
{"category": "Math", "title": "High-Dimensional Menger-Type Curvatures-Part II: d-Separation and a Menagerie of Curvatures", "abstract": "This is the second of two papers wherein we estimate multiscale least squares approximations of certain measures by Menger-type curvatures. More specifically, we study an arbitrary d-regular measure on a real separable Hilbert space. The main result of the paper bounds the least squares error of approximation at any ball by an average of the discrete Menger-type curvature over certain simplices in in the ball. A consequent result bounds the Jones-type flatness by an integral of the discrete curvature over all simplices. The preceding paper provided the opposite inequalities. Furthermore, we demonstrate some other discrete curvatures for characterizing uniform rectifiability and additional continuous curvatures for characterizing special instances of the (p, q)-geometric property. We also show that a curvature suggested by Leger (Annals of Math, 149(3), p. 831-869, 1999) does not fit within our framework."}
{"category": "Math", "title": "The Weinstein conjecture for stable Hamiltonian structures", "abstract": "We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3-manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y. We prove that if Y is not a T^2-bundle over S^1, then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3-manifold with a contact form such that all Reeb orbits are nondegenerate and elliptic, then Y is a lens space. Related arguments show that if Y is a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate, and if Y is not a lens space, then there exist at least three distinct embedded Reeb orbits."}
{"category": "Math", "title": "The t-stability number of a random graph", "abstract": "Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n grows, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression for the expected number of t-stable sets of order k. We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most k. Using the above results, we also obtain asymptotic bounds on the t-improper chromatic number of a random graph (this is the generalisation of the chromatic number, where we partition of the vertex set of the graph into t-stable sets)."}
{"category": "Math", "title": "The Adjoint L-function of SU(2,1)", "abstract": "We modify Ginzburg's construction for the Adjoint L function of GL(3) (unfolding and unramified computations only) to accomodate quasisplit unitary groups."}
{"category": "Math", "title": "Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions", "abstract": "We hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many times around the unique minimum of the function."}
{"category": "Math", "title": "Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schr\\\"odinger equation", "abstract": "In this paper, we first show that there exists a maximizer for the non-endpoint Strichartz inequalities for the Schr\\\"odinger equation in all dimensions based on the recent linear profile decomposition results. We then present a new proof of the linear profile decomposition for the Schr\\\"oindger equation with initial data in the homogeneous Sobolev space; as a consequence, there exists a maximizer for the Sobolev-Strichartz inequality."}
{"category": "Math", "title": "Optimal Betti numbers of forest ideals", "abstract": "We prove a tight lower bound on the Betti numbers of tree and forest ideals and a tight upper bound on certain graded Betti numbers of squarefree monomial ideals."}
{"category": "Math", "title": "The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality", "abstract": "In this paper, we establish the linear profile decomposition for the Airy equation with complex or real initial data in $L^2$, respectively. As an application, we obtain a dichotomy result on the existence of maximizers for the symmetric Airy-Strichartz inequality."}
{"category": "Math", "title": "Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence", "abstract": "We prove that the classes of graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence. This result answers the long-standing open question of whether every Exel-Laca algebra is Morita equivalent to a graph algebra. Given an ultragraph G we construct a directed graph E such that C*(G) is isomorphic to a full corner of C*(E). As applications, we characterize real rank zero for ultragraph algebras and describe quotients of ultragraph algebras by gauge-invariant ideals."}
{"category": "Math", "title": "Some Hecke Algebra Products and Corresponding Random Walks", "abstract": "Let $\\bm{i}=1+q+...+q^{i-1}$. For certain sequences $(r_1,...,r_l)$ of positive integers, we show that in the Hecke algebra $\\mathscr{H}_n(q)$ of the symmetric group $\\mathfrak{S}_n$, the product $(1+\\bm{r_1}T_{r_1})... (1+\\bm{r_l}T_{r_l})$ has a simple explicit expansion in terms of the standard basis $\\{T_w\\}$. An interpretation is given in terms of random walks on $\\mathfrak{S}_n$."}
{"category": "Math", "title": "Diophantine Exponents of Affine Subspaces: The Simultaneous Approximation Case", "abstract": "We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of $\\R^n$ and their nondegenerate submanifolds."}
{"category": "Math", "title": "Functions measuring smoothness and the constants in Jackson--Stechkin theorem", "abstract": "This paper is devoted to the equivalence of two type direct theorems in Approximation Theory: a) for smooth functions (Favard's estimates). b) for arbitrary continuous function (Jackson--Stechkin estimates). Specifically, we will show that Jackson--Stechkin inequality with optimal respect to the order of smoothness constants follows from Favard's inequality."}
{"category": "Math", "title": "Local Fourier transform and epsilon factors", "abstract": "Laumon introduced the local Fourier transform for $\\ell$-adic Galois representations of local fields, of equal characteristic $p$ different from $\\ell$, as a powerful tool to study the Fourier-Deligne transform of $\\ell$-adic sheaves over the affine line. In this article, we compute explicitly the local Fourier transform of monomial representations satisfying a certain ramification condition, and deduce Laumon's formula relating the epsilon factor to the determinant of the local Fourier transform under the same condition."}
{"category": "Math", "title": "Unknotting sequences for torus knots", "abstract": "The unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this note we characterize quasipositive knots for which the genus bound is sharp: the slice genus of a quasipositive knot equals its unknotting number, if and only if the given knot appears in an unknotting sequence of a torus knot."}
{"category": "Math", "title": "Subelliptic Estimates for Quadratic Differential Operators", "abstract": "We prove global subelliptic estimates for quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. In a previous joint work with M. Hitrik, we pointed out the existence of a particular linear subvector space in the phase space intrinsically associated to their Weyl symbols, called singular space, which rules spectral properties of non-elliptic quadratic operators. The purpose of the present paper is to prove that quadratic operators whose singular spaces are reduced to zero, are subelliptic with a loss of \"derivatives\" depending directly on particular algebraic properties of the Hamilton maps of their Weyl symbols. More generally, when singular spaces are symplectic spaces, we prove that quadratic operators are subelliptic in any direction of the symplectic orthogonal complements of their singular spaces."}
{"category": "Math", "title": "The 1,2-coloured HOMFLY-PT link homology", "abstract": "In this paper we define the 1,2-coloured HOMFLY-PT link homology and prove that it is a link invariant. We conjecture that this homology categorifies the coloured HOMFLY-PT polynomial for links whose components are labelled 1 or 2."}
{"category": "Math", "title": "Quadratic maps between modules", "abstract": "We introduce a notion of $R$-quadratic maps between modules over a commutative ring $R$ which generalizes several classical notions arising in linear algebra and group theory. On a given module $M$ such maps are represented by $R$-linear maps on a certain module $P^2_R(M)$. The structure of this module is described in term of the symmetric tensor square $Sym^2_R(M)$, the degree 2 component $\\Gamma^2_R(M)$ of the divided power algebra over $M$, and the ideal $I_2$ of $R$ generated by the elements $r^2-r$, $r\\in R$. The latter is shown to represent quadratic derivations on $R$ which arise in the theory of modules over square rings. This allows to extend the classical notion of nilpotent $R$-group of class 2 with coefficients in a 2-binomial ring $R$ to any ring $R$. We provide a functorial presentation of $I_2$ and several exact sequences embedding the modules $P^2_R(M)$ and $\\Gamma^2_R(M)$."}
{"category": "Math", "title": "The structure of the nilpotent cone, the Kazhdan-Lusztig map and algebraic group analogues of the Slodowy slices", "abstract": "We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic group G with Lie algebra g. These slices are associated to (the conjugacy classes of) elements s of the Weyl group W of g. For such slices we prove an analogue of the Kostant cross-section theorem for the action of a unipotent group."}
{"category": "Math", "title": "A necessary and sufficient condition for the invertibility of adapted perturbations of identity on the Wiener space", "abstract": "Let $(W,H,\\mu)$ be the classical Wiener space, assume that $U=I_W+u$ is an adapted perturbation of identity satisfying the Girsanov identity. Then, $U$ is invertible if and only if the kinetic energy of $u$ is equal to the relative entropy of the measure induced with the action of $U$ on the Wiener measure $\\mu$, in other words $U$ is invertible if and only if $$ \\half \\int_W|u|_H^2d\\mu=\\int_W \\frac{dU\\mu}{d\\mu}\\log\\frac{dU\\mu}{d\\mu}d\\mu . $$"}
{"category": "Math", "title": "The Semisimplicity Conjecture for A-Motives", "abstract": "We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Akio Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G_p(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G_p(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V_p(M). The second states that the connected component of G_p(M) is reductive if M is semisimple and has a separable endomorphism algebra."}
{"category": "Math", "title": "Derived categories of Fano threefolds", "abstract": "We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the Appendix we discuss how the ring of algebraic cycles of a smooth projective variety is related to the Grothendieck group of its derived category."}
{"category": "Math", "title": "L'Algebre Tropicale Comme Algebre De la Caracteristique 1 : Polynomes Rationnels Et Fonctions Polynomiales", "abstract": "We continue, in this second article, the study of the the algebraic tools which play a role in tropical algebra. We especially examine here the polynomial algebras over idempotent semi-fields. this work is motivated by the development of tropical geometry which appears to be the algebraic geometry of tropical algebra. In fact, the most interesting object is the image of a polynomial algebra in its semi-field of fractions. We can thus obtain, over good semi-fields, the analog of classical correpondences between polynomials, and varieties of zeros... For example, we show that the algebras of polynomial functions over a tropical curves associated to a polynomial P, is, as in classical algebraic geometry, the quotient of the polynomial algebra by the ideal generated by P."}
{"category": "Math", "title": "Ideals in Parabolic Subalgebras of Simple Lie Algebras", "abstract": "We study ad-nilpotent ideals of a parabolic subalgebra of a simple Lie algebra. Any such ideal determines an antichain in a set of positive roots of the simple Lie algebra. We give a necessary and sufficient condition for an antichain to determine an ad-nilpotent ideal of the parabolic. We write down all such antichains for the classical simple Lie algebras and in particular recover the results of D. Peterson. In section 2 of the paper we study the unique ideal in a parabolic which is irreducible as a module for the reductive part and give several equivalent statements that are satisfied by the corresponding subset of roots."}
{"category": "Math", "title": "Optimization problem for extremals of the trace inequality in domains with holes", "abstract": "We study the Sobolev trace constant for functions defined in a bounded domain $\\O$ that vanish in the subset $A.$ We find a formula for the first variation of the Sobolev trace with respect to hole. As a consequence of this formula, we prove that when $\\O$ is a centered ball, the symmetric hole is critical when we consider deformation that preserve volume but is not optimal for some case."}
{"category": "Math", "title": "Pathwise uniqueness for stochastic heat equations with H\\\"older continuous coefficients: the white noise case", "abstract": "We prove pathwise uniqueness for solutions of parabolic stochastic pde's with multiplicative white noise if the coefficient is H\\\"older continuous of index $\\gamma>3/4$. The method of proof is an infinite-dimensional version of the Yamada-Watanabe argument for ordinary stochastic differential equations."}
{"category": "Math", "title": "Quantum Bases in Uq(g)", "abstract": "This paper is devoted to analize inside the infinitely many possible bases of Uq(g), same that can be considered \"more equal then others\". The element of selection has been a privileged relation with the bialgebra. A new parameter z' has been found that determines the commutation relations, independent from the z=log(q) that defines Uq(g). Both z and z' are necessary to fix the relations between the basic set and its coproducts. Three cases are particularly relevant: the analytical set with z'=z, the Lie set with Lie-like commutation relations (for z'=0) and the canonical/crystal basis with z' infinity. To simplify the exposition, we discuss in details the easy generalizable case of Uq(su(2))."}
{"category": "Math", "title": "Chern classes, K-theory and Landweber exactness over nonregular base schemes", "abstract": "The purpose of this paper is twofold. First, we use the motivic Landweber exact functor theorem to deduce that the Bott inverted infinite projective space is homotopy algebraic $K$-theory. The argument is considerably shorther than any other known proofs and serves well as an illustration of the effectiveness of Landweber exactness. Second, we dispense with the regularity assumption on the base scheme which is often implicitly required in the notion of oriented motivic ring spectra. The latter allows us to verify the motivic Landweber exact functor theorem and the universal property of the algebraic cobordism spectrum for every noetherian base scheme of finite Krull dimension."}
{"category": "Math", "title": "Linearizing non-linear inverse problems and an application to inverse backscattering", "abstract": "We propose an abstract approach to prove local uniqueness and conditional H\\\"older stability to non-linear inverse problems by linearization. The main condition is that, in addition to the injectivity of the linearization $A$, we need a stability estimate for $A$ as well. That condition is satisfied in particular, if $A^*A$ is an elliptic pseudo-differential operator. We apply this scheme to show uniqueness and H\\\"older stability for the inverse backscattering problem for the acoustic equation near a constant sound speed."}
{"category": "Math", "title": "The longest minimum-weight path in a complete graph", "abstract": "We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about \\alpha^* \\log n$ edges where \\alpha^* ~ 3.5911 is the unique solution of the equation $alpha log(alpha) - \\alpha =1. This answers a question posed by Janson (1999)."}
{"category": "Math", "title": "Structural stability of attractor-repellor endomorphisms with singularities", "abstract": "We prove a theorem on structural stability of smooth attractor-repellor endomorphisms of compact manifolds, with singularities. By attractor-repellor, we mean that the non-wandering set of the dynamics $f$ is the disjoint union of a repulsive compact subset with a hyperbolic attractor on which $f$ acts bijectively. The statement of this result is both infinitesimal and dynamical. Up to our knowledge, this is the first in this hybrid direction. Our results generalize also a Mather's theorem in singularity theory which states that infinitesimal stability implies structural stability for composed mappings, to the larger category of laminations."}
{"category": "Math", "title": "Competing risks within shock models", "abstract": "We consider a competing risks model, in which system failures are due to one out of two mutually exclusive causes, formulated within the framework of shock models driven by bivariate Poisson process. We obtain the failure densities and the survival functions as well as other related quantities under three different schemes. Namely, system failures are assumed to occur at the first instant in which a random constant threshold is reached by (a) the sum of received shocks, (b) the minimum of shocks, (c) the maximum of shocks."}
{"category": "Math", "title": "The Hodge--Poincar\\'e polynomial of the moduli spaces of stable vector bundles over an algebraic curve", "abstract": "Let X be a nonsingular complex projective variety that is acted on by a reductive group $G$ and such that $X^{ss} \\neq X_{(0)}^{s}\\neq \\emptyset$. We give formulae for the Hodge--Poincar\\'e series of the quotient $X_{(0)}^s/G$. We use these computations to obtain the corresponding formulae for the Hodge--Poincar\\'e polynomial of the moduli space of properly stable vector bundles when the rank and the degree are not coprime. We compute explicitly the case in which the rank equals 2 and the degree is even."}
{"category": "Math", "title": "Modelling Richardson orbits for SO_N via Delta-filtered modules", "abstract": "We study the Delta-filtered modules for the Auslander algebra of k[T]/T^n\\rtimes C_2 where C_2 is the cyclic group of order two. The motivation for this is the bijection between parabolic orbits in the nilradical of a parabolic subgroup of SL_n and certain Delta-filtered modules for the Auslander algebra of k[T]/T^n as found by Hille and Roehrle and Bruestle et al. Under this bijection, the Richardson orbit (i.e. the dense orbit) corresponds to the Delta-filtered module without self-extensions. It has remained an open problem to describe such a correspondence for other classical groups. In this paper, we establish the Auslander algebra of k[T]/T^n\\rtimes C_2 as the right candidate for the orthogonal groups. In particular, for any parabolic subgroup of an orthogonal group we construct a map from parabolic orbits to Delta-filtered modules and show that in the case of the Richardson orbit, the result has no self-extensions. One of the consequences of our work is that we are able to describe the extensions between special classes of Delta-filtered modules. In particular, we show that these extensions can grow arbitrarily large."}
{"category": "Math", "title": "Tropical Algebraic Geometry in Maple, a preprocessing algorithm for finding common factors to multivariate polynomials with approximate coefficients", "abstract": "Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic-numeric computing. Taking a tropical view on this problem leads to efficient preprocessing techniques, applying polyhedral methods on the exact exponents with numerical techniques on the approximate coefficients. With Maple we will illustrate our use of tropical algebraic geometry."}
{"category": "Math", "title": "Reversible-equivariant systems and matricial equations", "abstract": "This paper uses tools in group theory and symbolic computing to give a classification of the representations of finite groups with order lower than 9 that can be derived from the study of local reversible-equivariant vector fields in $\\rn{4}$. The results are obtained by solving algebraically matricial equations. In particular, we exhibit the involutions used in a local study of reversible-equivariant vector fields. Based on such approach we present, for each element in this class, a simplified Belitiskii normal form."}
{"category": "Math", "title": "Affine buildings for dihedral groups", "abstract": "We construct rank 2 thick nondiscrete affine buildings associated with an arbitrary finite dihedral group."}
{"category": "Math", "title": "On Bounded Approximate Identities and Existence of Dense Ideals in Real Locally C*- and Locally JB-Algebras", "abstract": "It has been established by Inoue that a complex locally C*-algebra with a dense ideal posesses a bounded approximate identity which belonges to that ideal. It has been shown by Fritzsche that if a unital complex locally C*-algebra has an unbounded element then it also has a dense one-sided ideal. In the present paper we obtain analogues of the aforementioned results of Inoue and Fritzsche for real locally C*-algebras (projective limits of projective families of real C*-algebras), and for locally JB-algebras (projective limits of projective families of JB-algebras)."}
{"category": "Math", "title": "Fluctuations of the quenched mean of a planar random walk in an i.i.d. random environment with forbidden direction", "abstract": "We consider an i.i.d. random environment with a strong form of transience on the two dimensional integer lattice. Namely, the walk always moves forward in the y-direction. We prove a functional CLT for the quenched expected position of the random walk indexed by its level crossing times. We begin with a variation of the Martingale Central Limit Theorem. The main part of the paper checks the conditions of the theorem for our problem."}
{"category": "Math", "title": "Bellman Function and the $H^1-BMO$ Duality", "abstract": "A Bellman function approach to Fefferman's $H^1-BMO$ duality theorem is presented. One Bellman-type argument is used to handle two different one-dimensional cases, dyadic and continuous. An explicit estimate for the constant of embedding $BMO\\subset (H^1)^*$ is given in the dyadic case. The same Bellman function is then used to establish a multi-dimensional analog."}
{"category": "Math", "title": "Arithmetic aspects of self-similar groups", "abstract": "We prove that an irreducible lattice in a semisimple algebraic group is virtually isomorphic to an arithmetic lattice if and only if it admits a faithful self-similar action on a rooted tree of finite valency."}
{"category": "Math", "title": "Quadrivariate existence theorems and strong representability", "abstract": "In this paper, we give conditions under which we can compute the conjugate of a convex function on the product of two Frechet spaces defined in terms of another convex function on the product of two (possibly different) Frechet spaces. We use this result to give simple proofs of some (both old and new) results for Banach spaces, and deduce some (both old and new) stability results for strongly representable multifunctions. We take as our starting point a result on closed convex cones in the product of two Frechet spaces."}
{"category": "Math", "title": "Slope filtrations in families", "abstract": "This paper concerns arithmetic families of $\\varphi$-modules over reduced affinoid spaces. For such a family, we first prove that the slope polygons is lower semicontinuous around any rigid point. If the slope polygons are locally constant around a rigid point, we further prove that around this point, the family has a global slope filtration after base change to some extended Robba ring."}
{"category": "Math", "title": "Dimensional characteristics of invariant measures for circle diffeomorphisms", "abstract": "We consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number $\\tau$ there exists a $C^\\infty$ circle diffeomorphism with rotation number $\\tau$ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value $0\\le \\beta \\le 1$."}
{"category": "Math", "title": "On the deformation theory of pair (X, E)", "abstract": "Huybrechts and Thomas recently constructed relative obstruction theory of objects of the derived category of coherent sheaves over smooth projective family. In this paper, we use this construction to obtain the absolute deformation-obstruction theory of the pair (X, E), with X smooth projective scheme and E perfect complex, and show that the obstruction theories for E, (X,E), and X fit into exact triangle as derived objects on the moduli space."}
{"category": "Math", "title": "Quantitative Riemann existence theorem over a number field", "abstract": "Given a covering of the projective line with ramifications defined over a number field, we define a plain model of the algebraic curve realizing the Riemann existence theorem for this covering, and bound explicitly the defining equation of this curve and its definition field."}
{"category": "Math", "title": "Minimum-volume hyperbolic 3-manifolds", "abstract": "We enumerate the small-volume manifolds that can be obtained by Dehn filling on Mom-2 and Mom-3 manifolds as defined by Gabai, Meyerhoff, and the author. In so doing we complete the proof that the Weeks manifold is the minimum-volume compact hyperbolic 3-manifold, as well as enumerating the 10 smallest one-cusped hyperbolic 3-manifolds."}
{"category": "Math", "title": "Central Forests in Trees", "abstract": "A new 2-parameter family of central structures in trees, called central forests, is introduced. Minieka's $m$-center problem and McMorris's and Reid's central-$k$-tree can be seen as special cases of central forests in trees. A central forest is defined as a forest $F$ of $m$ subtrees of a tree $T$, where each subtree has $k$ nodes, which minimizes the maximum distance between nodes not in $F$ and those in $F$. An $O(n(m+k))$ algorithm to construct such a central forest in trees is presented, where $n$ is the number of nodes in the tree. The algorithm either returns with a central forest, or with the largest $k$ for which a central forest of $m$ subtrees is possible. Some of the elementary properties of central forests are also studied."}
{"category": "Math", "title": "Geometry and arithmetic of verbal dynamical systems on simple groups", "abstract": "We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of Engel-like sequences of words in two variables which characterize finite solvable groups. They also give rise to some new phenomena and concepts in the arithmetic of dynamical systems."}
{"category": "Math", "title": "Special scrolls whose base curve has general moduli", "abstract": "In this paper we study the Hilbert scheme of smooth, linearly normal, special scrolls under suitable assumptions on degree, genus and speciality."}
{"category": "Math", "title": "Multipliers of periodic orbits in spaces of rational maps", "abstract": "Given a polynomial or a rational map f we associate to it a space of maps. We introduce local coordinates in this space, which are essentially the set of critical values of the map. Then we consider an arbitrary periodic orbit of f with multiplier \\rho\\not=1 as a function of the local coordinates, and establish a simple connection between the dynamical plane of f and the function \\rho in the space associated to f. The proof is based on the theory of quasiconformal deformations of rational maps. As a corollary, we show that multipliers of non-repelling periodic orbits are also local coordinates in the space."}
{"category": "Math", "title": "Error calculus and regularity of Poisson functionals : the lent particle method", "abstract": "We propose a new method to apply the Lipschitz functional calculus of local Dirichlet forms to Poisson random measures."}
{"category": "Math", "title": "Reflection groups acting on their hyperplanes", "abstract": "After having established elementary results on the relationship between a finite complex (pseudo-)reflection group W < GL(V) and its reflection arrangement A, we prove that the action of W on A is canonically related with other natural representations of W, through a `periodic' family of representations of its braid group. We also prove that, when W is irreducible, then the squares of defining linear forms for A span the quadratic forms on V, which imply |A| >= n(n+1)/2 for n = dim V, and relate the W-equivariance of the corresponding map with the period of our family."}
{"category": "Math", "title": "Multiplicative Diophantine Exponents of Hyperplanes", "abstract": "We study multiplicative Diophantine approximation property of vectors and compute Diophantine exponents of hyperplanes via dynamics."}
{"category": "Math", "title": "A flexible Bayesian method for adaptive measurement in psychophysics", "abstract": "In psychophysical experiments time and the limited goodwill of participants is usually a major constraint. This has been the main motivation behind the early development of adaptive methods for the measurements of psychometric thresholds. More recently methods have been developed to measure whole psychometric functions in an adaptive way. Here we describe a Bayesian method to measure adaptively any aspect of a psychophysical function, taking inspiration from Kontsevich and Tyler's optimal Bayesian measurement method. Our method is implemented in a complete and easy-to-use MATLAB package."}
{"category": "Math", "title": "Uniform convergence for convexification of dominated pointwise convergent continuous functions", "abstract": "The Lebesgue dominated convergence theorem of the measure theory implies that the Riemann integral of a bounded sequence of continuous functions over the interval [ 0,1] pointwise converging to zero, also converges to zero. The validity of this result is independent of measure theory, on the other hand, this result together with only elementary functional analysis, can generate measure theory itself. The mentioned result was also known before the appearance of measure theory, but the original proof was very complicated. For this reason this result, when presented in teaching, is generally obtained based on measure theory. Later, Eberlein gave an elementary, but still relatively complicated proof, and there were other simpler proofs but burdened with complicated concepts, like measure theory. In this paper we give a short and elementary proof even for the following strenghened form of the mentioned result: a bounded sequence of continuous functions defined on a compact topological space K pointwise converging to zero, has a suitable convexification converging also uniformly to zero on $K,$ thus, e.g., the original sequence converges weakly to zero in C(K). This fact can also be used in the proof of the Krein-Smulian theorem. The usual proof beyond the simple tools of the functional analysis, uses heavy embedding theorems and the Riesz' representation theorem with the whole apparatus of measure theory. Our main result, however, reduces the cited proof to a form in which we need abstract tools only, namely the Hahn-Banach separation theorem and Alaoglu's theorem, without Riesz' representation or any statement of measure theory."}
{"category": "Math", "title": "Supersingular representations of GL_2(Q_p) and (phi,Gamma)-modules", "abstract": "The purpose of this note is to give a direct proof of the fact that if one applies Colmez' functor to a two dimensional irreducible F_p^bar-representation of Gal(Q_p^bar/Q_p), one gets the restriction to the Borel subgroup of GL_2(Q_p) of a supersingular representation."}
{"category": "Math", "title": "The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings", "abstract": "Let G be a profinite group which is topologically finitely generated, p a prime number and d an integer. We show that the functor from rigid analytic spaces over Q_p to sets, which associates to a rigid space Y the set of continuous d-dimensional pseudocharacters G -> O(Y), is representable by a quasi-Stein rigid analytic space X, and we study its general properties. Our main tool is a theory of \"determinants\" extending the one of pseudocharacters but which works over an arbitrary base ring; an independent aim of this paper is to expose the main facts of this theory. The moduli space X is constructed as the generic fiber of the moduli formal scheme of continuous formal determinants on G of dimension d. As an application to number theory, this provides a framework to study the generic fibers of pseudodeformation rings (e.g. of Galois representations), especially in the \"residually reducible\" case, and including when p <= d."}
{"category": "Math", "title": "An extended existence result for quadratic BSDEs with jumps with application to the utility maximization problem", "abstract": "In this study, we consider the exponential utility maximization problem in the context of a jump-diffusion model. To solve the problem, we rely on the dynamic programming principle and we derive from it a quadratic BSDE with jumps. Since this quadratic BSDE is driven both by a Wiener process and by a Poisson random measure having a Levy measure with infinite mass, our main task consists in establishing a new existence result for the specific BSDE introduced."}
{"category": "Math", "title": "Crossed Product of Cyclic Groups", "abstract": "All crossed products of two cyclic groups are explicitly described using generators and relations. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given."}
{"category": "Math", "title": "Conelike soap films spanning tetrahedra", "abstract": "In this paper we provide the first examples of non-flat soap films proven to span tetrahedra. These are members of a continuous two parameter family of soap films with tetrahedral boundaries. Of particular interest is a two parameter subfamily where each spanning soap film has the property that two minimal surfaces meet along an edge of the boundary at an angle greater than 120 degrees."}
{"category": "Math", "title": "On eigenfunctions corresponding to a small resurgent eigenvalue", "abstract": "This article is devoted to some foundational questions of resurgent analysis as applied to the Schr\\\"odinger equation in one dimension."}
{"category": "Math", "title": "Efficiency of the maximum partial likelihood estimator for nested case control sampling", "abstract": "In making inference on the relation between failure and exposure histories in the Cox semiparametric model, the maximum partial likelihood estimator (MPLE) of the finite dimensional odds parameter, and the Breslow estimator of the baseline survival function, are known to achieve full efficiency when data is available for all time on all cohort members, even when the covariates are time dependent. When cohort sizes become too large for the collection of complete data, sampling schemes such as nested case control sampling must be used and, under various models, there exist estimators based on the same information as the MPLE having smaller asymptotic variance. Though the MPLE is therefore not efficient under sampling in general, it approaches efficiency in highly stratified situations, or instances where the covariate values are increasingly less dependent upon the past, when the covariate distribution, not depending on the real parameter of interest, is unknown and there is no censoring. In particular, in such situations, when using the nested case control sampling design, both the MPLE and the Breslow estimator of the baseline survival function achieve the information lower bound both in the distributional and the minimax senses in the limit as the number of cohort members tends to infinity."}
{"category": "Math", "title": "Intrinsically Linked Graphs in Projective Space", "abstract": "We examine graphs that contain a non-trivial link in every embedding into real projective space, using a weaker notion of unlink than was used by Flapan, et al. We call such graphs intrinsically linked in projective space. We fully characterize such graphs with connectivity 0,1 and 2. We also show that only one Petersen-family graph is intrinsically linked in projective space and prove that K7 minus any two edges is also minor-minimal intrinsically linked. In all, 594 graphs are shown to be minor-minimal intrinsically linked in projective space."}
{"category": "Math", "title": "Isometry groups of non-positively curved spaces: structure theory", "abstract": "We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further developments on non-positively curved lattices are exposed in a companion paper: \"Isometry groups of non-positively curved spaces: discrete subgroups\"."}
{"category": "Math", "title": "New Formulas and Methods for Interpolation, Numerical Differentiation and Numerical Integration", "abstract": "We present a new formula for divided difference and few new schemes of divided difference tables in this paper. Through this, we derive new interpolation, numerical differentiation and numerical integration formulas with arbitrary order of accuracy for evanly and unevanly spaced data. First, we study the new interpolation formula which generalizes both Newton's and Lagrange's interpolation formula with the new divided difference table for unevanly spaced points and using this; we derive other interpolation formulas, in terms of differences for evanly spaced data. Second, we study two new different methods of numerical differentiation for both evanly and unevanly spaced points without differentiating the interpolating polynomials or the use of operators. Third, we derive new numerical integration formulas using new differentiation formulas and Taylor formula for both evanly and unevanly spaced data. Basic computer algorithms for few new formulas are given. In Comparison to former polynomial interpolation, numerical differentiation and numerical integration formuals, these new formulas have some featured advantages for approximating functional values, numerical derivatives of higher order and approximate integral values for evanly and unevanly spaced data."}
{"category": "Math", "title": "The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)", "abstract": "We show that there is a homotopy cofiber sequence of spectra relating Carlsson's deformation K-theory of a group G to its \"deformation representation ring,\" analogous to the Bott periodicity sequence relating connective K-theory to ordinary homology. We then apply this to study simultaneous similarity of unitary matrices."}
{"category": "Math", "title": "Notes on Sela's work: Limit groups and Makanin-Razborov diagrams", "abstract": "This is the first in a planned series of papers giving an alternate approach to Zlil Sela's work on the Tarski problems. The present paper is an exposition of work of Kharlampovich-Myasnikov and Sela giving a parametrization of Hom(G,F) where G is a finitely generated group and F is a non-abelian free group."}
{"category": "Math", "title": "Rank one isometries of buildings and quasi-morphisms of Kac-Moody groups", "abstract": "Given an irreducible non-spherical non-affine (possibly non-proper) building $X$, we give sufficient conditions for a group $G < \\Aut(X)$ to admit an infinite-dimensional space of non-trivial quasi-morphisms. The result applies to all irreducible (non-spherical and non-affine) Kac-Moody groups over integral domains. In particular, we obtain finitely presented simple groups of infinite commutator width, thereby answering a question of Valerii G. Bardakov from the Kourovka notebook. Independently of these considerations, we also include a discussion of rank one isometries of proper CAT(0) spaces from a rigidity viewpoint. In an appendix, we show that any homogeneous quasi-morphism of a locally compact group with integer values is continuous."}
{"category": "Math", "title": "Integration in algebraically closed valued fields", "abstract": "The first two steps of the construction of motivic integration in the fundamental work of Hrushovski and Kazhdan have been presented in arXiv:1006.2467v1. In this paper we present the final third step. As in arXiv:1006.2467v1, we limit our attention to the theory of algebraically closed valued fields of pure characteristic 0 expanded by a (VF, \\Gamma)-generated substructure S in the language L_{RV}. A canonical description of the kernel of the lifting map is obtained."}
{"category": "Math", "title": "Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded Rings", "abstract": "The Goto number of a parameter ideal Q in a Noetherian local ring (R,m) is the largest integer q such that Q : m^q is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x^{a_1}, x^{a_2},..., x_{a_{\\nu}}]] are characterized using the semigroup of R. This helps in computing them for classes of numerical semigroup rings, as well as on a case-by-case basis. The minimal Goto number of R and its connection to other invariants is explored. Necessary and sufficient conditions for the associated graded rings of R and R/x^{a_1}R to be Gorenstein are also given, again using the semigroup of R."}
{"category": "Math", "title": "Average Continuous Control of Piecewise Deterministic Markov Processes", "abstract": "This paper deals with the long run average continuous control problem of piecewise deterministic Markov processes (PDMP's) taking values in a general Borel space and with compact action space depending on the state variable. The control variable acts on the jump rate and transition measure of the PDMP, and the running and boundary costs are assumed to be positive but not necessarily bounded. Our first main result is to obtain an optimality equation for the long run average cost in terms of a discrete-time optimality equation related to the embedded Markov chain given by the post-jump location of the PDMP. Our second main result guarantees the existence of a feedback measurable selector for the discrete-time optimality equation by establishing a connection between this equation and an integro-differential equation. Our final main result is to obtain some sufficient conditions for the existence of a solution for a discrete-time optimality inequality and an ordinary optimal feedback control for the long run average cost using the so-called vanishing discount approach."}
{"category": "Math", "title": "Bessel models for lowest weight representations of GSp(4,R)", "abstract": "We prove uniqueness and give precise criteria for existence of split and non-split Bessel models for a class of lowest and highest weight representations of the groups GSp(4,R) and Sp(4,R) including all holomorphic and anti-holomorphic discrete series representations. Explicit formulas for the resulting Bessel functions are obtained by solving systems of differential equations. The formulas are applied to derive an integral representation for a global $L$-function on GSp(4)xGL(2) involving a vector-valued Siegel modular form of degree 2."}
{"category": "Math", "title": "The asymptotic behavior of degenerate oscillatory integrals in two dimensions", "abstract": "A theorem of Varchenko gives the order of decay of the leading term of the asymptotic expansion of a degenerate oscillatory integral with real-analytic phase in two dimensions. His theorem expresses this order of decay in a simple geometric way in terms of its Newton polygon once one is in certain coordinate systems called adapted coordinate systems. In this paper, we give explicit formulas that not only provide the order of decay of the leading term, but also the coefficient of this term. There are three rather different formulas corresponding to three different types of Newton polygon. Analogous results for sublevel integrals are proven, as are analogues for the more general case of smooth phase. The formulas require one to be in certain \"superadapted\" coordinates. These are a type of adapted coordinate system which we show exists for any smooth phase."}
{"category": "Math", "title": "Pattern avoidance in binary trees", "abstract": "This paper considers the enumeration of trees avoiding a contiguous pattern. We provide an algorithm for computing the generating function that counts n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting mechanism, we study the analogue of Wilf equivalence in which two tree patterns are equivalent if the respective n-leaf trees that avoid them are equinumerous. We investigate the equivalence classes combinatorially. Toward establishing bijective proofs of tree pattern equivalence, we develop a general method of restructuring trees that conjecturally succeeds to produce an explicit bijection for each pair of equivalent tree patterns."}
{"category": "Math", "title": "On Q-conic bundles, III", "abstract": "A Q-conic bundle germ is a proper morphism from a threefold with only terminal singularities to the germ $(Z \\ni o)$ of a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. Building upon our previous paper [math/0603736], we prove the existence of a Du Val anti-canonical member under the assumption that the central fiber is irreducible."}
{"category": "Math", "title": "From Data to the p-Adic or Ultrametric Model", "abstract": "We model anomaly and change in data by embedding the data in an ultrametric space. Taking our initial data as cross-tabulation counts (or other input data formats), Correspondence Analysis allows us to endow the information space with a Euclidean metric. We then model anomaly or change by an induced ultrametric. The induced ultrametric that we are particularly interested in takes a sequential - e.g. temporal - ordering of the data into account. We apply this work to the flow of narrative expressed in the film script of the Casablanca movie; and to the evolution between 1988 and 2004 of the Colombian social conflict and violence."}
{"category": "Math", "title": "Gluing endo-permutation modules", "abstract": "In this paper, I show that if $p$ is an odd prime, and if $P$ is a finite $p$-group, then there exists an exact sequence of abelian groups $$0\\to T(P)\\to D(P)\\to\\lproj{P}\\to H^1\\big(\\apdeux(P),\\Z\\big)^{(P)},$$ where $D(P)$ is the Dade group of $P$ and $T(P)$ is the subgroup of endo-trivial modules. Here $\\lproj{P}$ is the group of sequences of compatible elements in the Dade groups $D\\big(N_P(Q)/Q\\big)$ for non trivial subgroups $Q$ of $P$. The poset $\\apdeux(P)$ is the set of elementary abelian subgroups of rank at least 2 of $P$, ordered by inclusion. The group $H^1\\big(\\apdeux(P),\\Z\\big)^{(P)}$ is the subgroup of $H^1\\big(\\apdeux(P),\\Z\\big)$ consisting of classes of $P$-invariant 1-cocycles. Here $\\lproj{P}$ is the group of sequences of compatible elements in the Dade groups $D\\big(N_P(Q)/Q\\big)$ for non trivial subgroups $Q$ of $P$. The poset $\\apdeux(P)$ is the set of elementary abelian subgroups of rank at least 2 of $P$, ordered by inclusion. The group $H^1\\big(\\apdeux(P),\\Z\\big)^{(P)}$ is the subgroup of $H^1\\big(\\apdeux(P),\\Z\\big)$ consisting of classes of $P$-invariant 1-cocycles. A key result to prove that the above sequence is exact is a characterization of elements of $2D(P)$ by sequences of integers, indexed by sections $(T,S)$ of $P$ such that $T/S\\cong (\\Z/p\\Z)^2$, fulfilling certain conditions associated to subquotients of $P$ which are either elementary abelian of rank~3, or extraspecial of order $p^3$ and exponent $p$."}
{"category": "Math", "title": "Direct limits, multiresolution analyses, and wavelets", "abstract": "A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. Our results apply to the classical situation involving dilation matrices on $L^2(\\R^n)$, the wavelets on fractals studied by Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids."}
{"category": "Math", "title": "The 5-local homotopy of $eo_4$", "abstract": "We compute the 5-local cohomology of a 5-local analogue of the Weierstrass Hopf algebroid used to compute $tmf$ homology. We compute the Adams-Novikov differentials in the cohomology, giving the homotopy, V(0)-homology, and V(1)-homology of the putative spectrum $eo_4$. We also link this computation to the homotopy of the higher real $K$-theory spectrum $EO_4$."}
{"category": "Math", "title": "Diffeomorphisms Holder conjugate to Anosov diffeomorphisms", "abstract": "We show by means of a counterexample that a $C^{1+Lip}$ diffeomorphism Holder conjugate to an Anosov diffeomorphism is not necessarily Anosov. The counterexample can bear higher smoothness up to $C^3$. Also we include a result from the 2006 Ph.D. thesis of T. Fisher: a $C^{1+Lip}$ diffeomorphism Holder conjugate to an Anosov diffeomorphism is Anosov itself provided that Holder exponents of the conjugacy and its inverse are sufficiently large."}
{"category": "Math", "title": "Non-Gatherable Triples for Non-Affine Root Systems", "abstract": "This paper contains a complete description of minimal non-gatherable triangle triples in the lambda-sequences for the classical root systems, $F_4$ and $E_6$. Such sequences are associated with reduced decompositions (words) in affine and non-affine Weyl groups. The existence of the non-gatherable triples is a combinatorial obstacle for using the technique of intertwiners for an explicit description of the irreducible representations of the (double) affine Hecke algebras, complementary to their algebraic-geometric theory."}
{"category": "Math", "title": "Some Stably Tame Polynomial Automorphisms", "abstract": "We study the structure of length three polynomial automorphisms of $R[X,Y]$ when $R$ is a UFD. These results are used to prove that if $\\text{SL}_m(R[X_1,X_2,..., X_n]) = \\text{E}_m(R[X_1,X_2,..., X_n])$ for all $n,\\ge 0$ and for all $m \\ge 3$ then all length three polynomial automorphisms of $R[X,Y]$ are stably tame."}
{"category": "Math", "title": "Boolean Algebras and Logic", "abstract": "In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone's representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof of completeness theorem in propositional logic will be given using Stone's theorem from Boolean algebra."}
{"category": "Math", "title": "The nonmeasurability of Bernstein sets and related topics", "abstract": "In this article, we shall explore the constructions of Bernstein sets, and prove that every Bernstein set is nonmeasurable and doesn't have the property of Baire. We shall also prove that Bernstein sets don't have the perfect set property."}
{"category": "Math", "title": "Hurwitz's Freeness Property", "abstract": "The groupoid attached to the action of PSL(2,Z) on the irrational reals by linear fractional transformations is free."}
{"category": "Math", "title": "Smooth words and Chebyshev polynomials", "abstract": "A word $\\sigma=\\sigma_1...\\sigma_n$ over the alphabet $[k]=\\{1,2,...,k\\}$ is said to be {\\em smooth} if there are no two adjacent letters with difference greater than 1. A word $\\sigma$ is said to be {\\em smooth cyclic} if it is a smooth word and in addition satisfies $|\\sigma_n-\\sigma_1|\\le 1$. We find the explicit generating functions for the number of smooth words and cyclic smooth words in $[k]^n$, in terms of {\\it Chebyshev polynomials of the second kind}. Additionally, we find explicit formula for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate smooth necklaces, which are cyclic smooth words that are not equivalent up to rotation."}
{"category": "Math", "title": "Homogeneous Representations of Khovanov-Lauda Algebras", "abstract": "We construct irreducible graded representations of simply laced Khovanov-Lauda algebras which are concentrated in one degree. The underlying combinatorics of skew shapes and standard tableaux corresponding to arbitrary simply laced types has been developed previously by Peterson, Proctor and Stembridge. In particular, the Peterson-Proctor hook formula gives dimensions of the homogeneous irreducible modules corresponding to straight shapes."}
{"category": "Math", "title": "On sharp embeddings of Besov and Triebel-Lizorkin spaces in the subcritical case", "abstract": "We discuss the growth envelopes of Fourier-analytically defined Besov and Triebel-Lizorkin spaces $B^s_{p,q}(\\R^n)$ and $F^s_{p,q}(\\R^n)$ for $s=\\sigma_p=n\\max(\\frac 1p-1,0)$. These results may be also reformulated as optimal embeddings into the scale of Lorentz spaces $L_{p,q}(\\R^n)$. We close several open problems outlined already by H. Triebel in [H. Triebel, The structure of functions, Birkh\\\"auser, Basel, 2001.] and explicitly formulated by D. D. Haroske in [D. D. Haroske, Envelopes and sharp embeddings of function spaces, Chapman & Hall / CRC, Boca Raton, 2007.]."}
{"category": "Math", "title": "Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator", "abstract": "Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator $H_\\epsilon = -\\partial_x^2 + x^2 + i\\epsilon^{-1}f(x)$ on $L^2(R)$, where $f$ is a real-valued function and $\\epsilon > 0$ a small parameter. We define $\\Sigma(\\epsilon)$ as the infimum of the real part of the spectrum of $H_\\epsilon$, and $\\Psi(\\epsilon)^{-1}$ as the supremum of the norm of the resolvent of $H_\\epsilon$ along the imaginary axis. Under appropriate conditions on $f$, we show that both quantities $\\Sigma(\\epsilon)$, $\\Psi(\\epsilon)$ go to infinity as $\\epsilon \\to 0$, and we give precise estimates of the growth rate of $\\Psi(\\epsilon)$. We also provide an example where $\\Sigma(\\epsilon)$ is much larger than $\\Psi(\\epsilon)$ if $\\epsilon$ is small. Our main results are established using variational \"hypocoercive\" methods, localization techniques and semiclassical subelliptic estimates."}
{"category": "Math", "title": "Higgs bundles and surface group representations in the real symplectic group", "abstract": "In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, i.e. representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper arXiv:0909.4487 [math.DG] to identify the space of representations with the moduli space of polystable Sp(2n,R)-Higgs bundles. A key step is provided by the discovery of new discrete invariants of maximal representations. These new invariants arise from an identification, in the maximal case, of the moduli space of Sp(2n,R)-Higgs bundles with a moduli space of twisted Higgs bundles for the group GL(n,R)."}
{"category": "Math", "title": "Low dimensional strongly perfect lattices. III: Dual strongly perfect lattices of dimension 14", "abstract": "The extremal 3-modular lattice $[\\pm G_2(3)]_{14}$ with automorphism group $C_2 \\times G_2(\\F_3) $ is the unique dual strongly perfect lattice of dimension 14."}
{"category": "Math", "title": "Relation between powers of factors and recurrence function characterizing Sturmian words", "abstract": "In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved by Carpi and de Luca, and Damanik and Lenz."}
{"category": "Math", "title": "On the Fourier transform of the symmetric decreasing rearrangements", "abstract": "Inspired by work of Montgomery on Fourier series and Donoho-Strak in signal processing, we investigate two families of rearrangement inequalities for the Fourier transform. More precisely, we show that the $L^2$ behavior of a Fourier transform of a function over a small set is controlled by the $L^2$ behavior of the Fourier transform of its symmetric decreasing rearrangement. In the $L^1$ case, the same is true if we further assume that the function has a support of finite measure. As a byproduct, we also give a simple proof and an extension of a result of Lieb about the smoothness of a rearrangement. Finally, a straightforward application to solutions of the free Shr\\\"odinger equation is given."}
{"category": "Math", "title": "Probabilistic solution of the American options", "abstract": "The existence and uniqueness of probabilistic solutions of variational inequalities for the general American options are proved under the hypothesis of hypoellipticity of the infinitesimal generator of the underlying diffusion process which represents the risky assets of the stock market with which the option is created. The main tool is an extension of the It\\^o formula which is valid for the tempered distributions on $\\R^d$ and for nondegenerate It\\^o processes in the sense of the Malliavin calculus."}
{"category": "Math", "title": "Two applications of twisted Floer homology", "abstract": "Given an irreducible closed 3--manifold $Y$, we show that its twisted Heegaard Floer homology determines whether $Y$ is a torus bundle over the circle. Another result we will prove is, if $K$ is a genus 1 null-homologous knot in an $L$--space, and the 0--surgery on $K$ is fibered, then $K$ itself is fibered. These two results are the missing cases of earlier results due to the second author."}
{"category": "Math", "title": "Mean Curvature Motion of Triple Junctions of Graphs in Two Dimensions", "abstract": "We consider a system of three surfaces, graphs over a bounded domain in ${\\mathbb R}^2$, intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to $2\\pi/3$.) For the corresponding two-dimensional parabolic free boundary problem we prove short-time existence of classical solutions (in parabolic H\\\"{o}lder spaces), for sufficiently regular initial data satisfying a compatibility condition."}
{"category": "Math", "title": "Generic Hopf Galois extensions", "abstract": "In previous joint work with Eli Aljadeff we attached a generic Hopf Galois extension A(H,c) to each twisted algebra H(c) obtained from a Hopf algebra H by twisting its product with the help of a cocycle c. The algebra A(H,c) is a flat deformation of H(c) over a \"big\" central subalgebra B(H,c) and can be viewed as the noncommutative analogue of a versal torsor in the sense of Serre. After surveying the results on A(H,c) obtained with Aljadeff, we establish three new results: we present a systematic method to construct elements of the commutative algebra B(H,c), we show that a certain important integrality condition is satisfied by all finite-dimensional Hopf algebras generated by grouplike and skew-primitive elements, and we compute B(H,c) in the case where H is the Hopf algebra of a cyclic group."}
{"category": "Math", "title": "Equivalent inequalities", "abstract": "Equivalencies of many basic elementary inequalities are given"}
{"category": "Math", "title": "Extremal rational elliptic threefolds", "abstract": "We classify nets of quadrics in P^3 which give rise to elliptic fibrations of Mordell-Weil rank zero."}
{"category": "Math", "title": "The 4-string Braid group $B_4$ has property RD and exponential mesoscopic rank", "abstract": "We prove that the braid group $B_4$ on 4 strings, as well as its central quotient $B_4/< z>$, have the property RD of Haagerup-Jolissaint. It follows that the automorphism group $\\Aut(F_2)$ of the free group $F_2$ on 2 generators has property RD. We also prove that the braid group $B_4$ is a group of intermediate rank (of dimension 3). Namely, we show that both $B_4$ and its central quotient have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats."}
{"category": "Math", "title": "Smooth Affine Surfaces with Non-Unique C*-Actions", "abstract": "In this paper we complete the classification of effective C*-actions on smooth affine surfaces up to conjugation in the full automorphism group and up to inversion of C*. If a smooth affine surface V admits more than one C*-action then it is known to be Gizatullin i.e., it can be completed by a linear chain of smooth rational curves. In our previous paper we gave a sufficient condition, in terms of the Dolgachev- Pinkham-Demazure (or DPD) presentation, for the uniqueness of a C*-action on a Gizatullin surface. In the present paper we show that this condition is also necessary, at least in the smooth case. In fact, if the uniqueness fails for a smooth Gizatullin surface V which is neither toric nor Danilov-Gizatullin, then V admits a continuous family of pairwise non-conjugated C*-actions depending on one or two parameters. We give an explicit description of all such surfaces and their C*-actions in terms of DPD presentations. We also show that for every k > 0 one can find a Danilov- Gizatullin surface V (n) of index n = n(k) with a family of pairwise non-conjugate C+-actions depending on k parameters."}
{"category": "Math", "title": "Accentuate the negative", "abstract": "A survey of mean inequalities with real weights is given."}
{"category": "Math", "title": "Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps", "abstract": "We investigate limit theorems for Birkhoff sums of locally H\\\"older functions under the iteration of Gibbs-Markov maps. Aaronson and Denker have given sufficient conditions to have limit theorems in this setting. We show that these conditions are also necessary: there is no exotic limit theorem for Gibbs-Markov maps. Our proofs, valid under very weak regularity assumptions, involve weak perturbation theory and interpolation spaces. For L^2 observables, we also obtain necessary and sufficient conditions to control the speed of convergence in the central limit theorem."}
{"category": "Math", "title": "An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions", "abstract": "We construct a uniformly expanding map of the interval, preserving Lebesgue measure, such that the corresponding transfer operator admits a spectral gap on the space of Lipschitz functions, but does not act continuously on the space of bounded variation functions."}
{"category": "Math", "title": "An Alternating l1 approach to the compressed sensing problem", "abstract": "Compressed sensing is a new methodology for constructing sensors which allow sparse signals to be efficiently recovered using only a small number of observations. The recovery problem can often be stated as the one of finding the solution of an underdetermined system of linear equations with the smallest possible support. The most studied relaxation of this hard combinatorial problem is the $l_1$-relaxation consisting of searching for solutions with smallest $l_1$-norm. In this short note, based on the ideas of Lagrangian duality, we introduce an alternating $l_1$ relaxation for the recovery problem enjoying higher recovery rates in practice than the plain $l_1$ relaxation and the recent reweighted $l_1$ method of Cand\\`es, Wakin and Boyd."}
{"category": "Math", "title": "Commutative quotients of finite W-algebras", "abstract": "Let U(g,e) be the finite W-algebra associated with a nilpotent element e in a simple Lie algebra g and assume that e is induced from a nilpotent element e_0 in a Levi subalgebra l of g. We show that if the finite W-algebra U(l,e_0) has a 1-dimensional representation, then so does U(g,e). For g classical (and in may other cases), we compute the Krull dimension of the largest commutative quotient of U(g,e). Some applications to representation theory of modular counterparts of g are given."}
{"category": "Math", "title": "Occupation times of branching systems with initial inhomogeneous Poisson states and related superprocesses", "abstract": "The $(d,\\alpha,\\beta,\\gamma)$-branching particle system consists of particles moving in $R^d$ according to a symmetric $\\alpha$-stable L\\'evy process $(0<\\alpha\\leq 2)$, splitting with a critical $(1+\\beta)$-branching law $(0<\\beta\\leq 1)$, and starting from an inhomogeneous Poisson random measure with intensity measure $\\mu_\\gamma(dx)=dx/(1+|x|^\\gamma), \\gamma\\geq 0$. By means of time rescaling $T$ and Poisson intensity measure $H_T\\mu_\\gamma$, occupation time fluctuation limits for the system as $T\\to\\infty$ have been obtained in two special cases: Lebesgue measure ($\\gamma=0$, the homogeneous case), and finite measures $(\\gamma>d)$. In some cases $H_T\\equiv 1$ and in others $H_T\\to\\infty$ as $T\\to\\infty$ (high density systems). The limit processes are quite different for Lebesgue and for finite measures. Therefore the question arises of what kinds of limits can be obtained for Poisson intensity measures that are intermediate between Lebesgue measure and finite measures. In this paper the measures $\\mu_\\gamma, \\gamma\\in (0,d]$, are used for investigating this question. Occupation time fluctuation limits are obtained which interpolate in some way between the two previous extreme cases. The limit processes depend on different arrangements of the parameters $d,\\alpha,\\beta,\\gamma$. Related results for the corresponding $(d,\\alpha,\\beta,\\gamma)$-superprocess are also given."}
{"category": "Math", "title": "Compressive Wave Computation", "abstract": "This paper considers large-scale simulations of wave propagation phenomena. We argue that it is possible to accurately compute a wavefield by decomposing it onto a largely incomplete set of eigenfunctions of the Helmholtz operator, chosen at random, and that this provides a natural way of parallelizing wave simulations for memory-intensive applications. This paper shows that L1-Helmholtz recovery makes sense for wave computation, and identifies a regime in which it is provably effective: the one-dimensional wave equation with coefficients of small bounded variation. Under suitable assumptions we show that the number of eigenfunctions needed to evolve a sparse wavefield defined on N points, accurately with very high probability, is bounded by C log(N) log(log(N)), where C is related to the desired accuracy and can be made to grow at a much slower rate than N when the solution is sparse. The PDE estimates that underlie this result are new to the authors' knowledge and may be of independent mathematical interest; they include an L1 estimate for the wave equation, an estimate of extension of eigenfunctions, and a bound for eigenvalue gaps in Sturm-Liouville problems. Numerical examples are presented in one spatial dimension and show that as few as 10 percents of all eigenfunctions can suffice for accurate results. Finally, we argue that the compressive viewpoint suggests a competitive parallel algorithm for an adjoint-state inversion method in reflection seismology."}
{"category": "Math", "title": "Homological dimensions of K\"othe algebras", "abstract": "Given a metrizable K\"othe algebra $\\lambda(P)$, we compute the global dimension, the weak global dimension, the bidimension, and the weak bidimension of $\\lambda(P)$ in terms of the K\"othe set $P$."}
{"category": "Math", "title": "Refined estimates for some basic random walks on the symmetric and alternating groups", "abstract": "We give refined estimates for the discrete time and continuous time versions of some basic random walks on the symmetric and alternating groups $S_n$ and $A_n$. We consider the following models: random transposition, transpose top with random, random insertion, and walks generated by the uniform measure on a conjugacy class. In the case of random walks on $S_n$ and $A_n$ generated by the uniform measure on a conjugacy class, we show that in continuous time the $\\ell^2$-cuttoff has a lower bound of $(n/2)\\log n$. This result, along with the results of M\\\"uller, Schlage-Puchta and Roichman, demonstrates that the continuous time version of these walks may take much longer to reach stationarity than its discrete time counterpart."}
{"category": "Math", "title": "Coloured quiver mutation for higher cluster categories", "abstract": "We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting in higher cluster categories and especially an algorithm to determine the Gabriel quivers of tilting objects in such categories."}
{"category": "Math", "title": "Hopf algebras of dimension 2p^2", "abstract": "Let H be a non-semisimple Hopf algebra of dimension 2p^2 over an algebraically closed field of characteristic zero, where p is an odd prime. We prove that H or H^* is pointed, which completes the classification for Hopf algebras of these dimensions."}
{"category": "Math", "title": "Long cycles in graphs through fragments", "abstract": "Four basic Dirac-type sufficient conditions for a graph $G$ to be hamiltonian are known involving order $n$, minimum degree $\\delta$, connectivity $\\kappa$ and independence number $\\alpha$ of $G$: (1) $\\delta \\geq n/2$ (Dirac); (2) $\\kappa \\geq 2$ and $\\delta \\geq (n+\\kappa)/3$ (by the author); (3) $\\kappa \\geq 2$ and $\\delta \\geq \\max\\lbrace (n+2)/3,\\alpha \\rbrace$ (Nash-Williams); (4) $\\kappa \\geq 3$ and $\\delta \\geq \\max\\lbrace (n+2\\kappa)/4,\\alpha \\rbrace$ (by the author). In this paper we prove the reverse version of (4) concerning the circumference $c$ of $G$ and completing the list of reverse versions of (1)-(4): (R1) if $\\kappa \\geq 2$, then $c\\geq\\min\\lbrace n,2\\delta\\rbrace$ (Dirac); (R2) if $\\kappa \\geq 3$, then $c\\geq\\min\\lbrace n,3\\delta -\\kappa\\rbrace$ (by the author); (R3) if $\\kappa\\geq 3$ and $\\delta\\geq \\alpha$, then $c\\geq\\min\\lbrace n,3\\delta-3\\rbrace$ (Voss and Zuluaga); (R4) if $\\kappa\\geq 4$ and $\\delta\\geq \\alpha$, then $c\\geq\\min\\lbrace n,4\\delta-2\\kappa\\rbrace$. To prove (R4), we present four more general results centered around a lower bound $c\\geq 4\\delta-2\\kappa$ under four alternative conditions in terms of fragments. A subset $X$ of $V(G)$ is called a fragment of $G$ if $N(X)$ is a minimum cut-set and $V(G)-(X\\cup N(X))\\neq\\emptyset$."}
{"category": "Math", "title": "A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators", "abstract": "This paper is concerned with the fast computation of Fourier integral operators of the general form $\\int_{\\R^d} e^{2\\pi\\i \\Phi(x,k)} f(k) d k$, where $k$ is a frequency variable, $\\Phi(x,k)$ is a phase function obeying a standard homogeneity condition, and $f$ is a given input. This is of interest for such fundamental computations are connected with the problem of finding numerical solutions to wave equations, and also frequently arise in many applications including reflection seismology, curvilinear tomography and others. In two dimensions, when the input and output are sampled on $N \\times N$ Cartesian grids, a direct evaluation requires $O(N^4)$ operations, which is often times prohibitively expensive. This paper introduces a novel algorithm running in $O(N^2 \\log N)$ time, i. e. with near-optimal computational complexity, and whose overall structure follows that of the butterfly algorithm [Michielssen and Boag, IEEE Trans Antennas Propagat 44 (1996), 1086-1093]. Underlying this algorithm is a mathematical insight concerning the restriction of the kernel $e^{2\\pi\\i \\Phi(x,k)}$ to subsets of the time and frequency domains. Whenever these subsets obey a simple geometric condition, the restricted kernel has approximately low-rank; we propose constructing such low-rank approximations using a special interpolation scheme, which prefactors the oscillatory component, interpolates the remaining nonoscillatory part and, lastly, remodulates the outcome. A byproduct of this scheme is that the whole algorithm is highly efficient in terms of memory requirement. Numerical results demonstrate the performance and illustrate the empirical properties of this algorithm."}
{"category": "Math", "title": "Polynomial treewidth forces a large grid-like-minor", "abstract": "Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\\ell\\times\\ell$ grid minor is exponential in $\\ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \\emph{grid-like-minor of order} $\\ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{\\ell}$-minor. For example, the rows and columns of the $\\ell\\times\\ell$ grid are a grid-like-minor of order $\\ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $c\\ell^4\\sqrt{\\log\\ell}$ has a grid-like-minor of order $\\ell$. As an application of this result, we prove that the cartesian product $G\\square K_2$ contains a $K_{\\ell}$-minor whenever $G$ has treewidth at least $c\\ell^4\\sqrt{\\log\\ell}$."}
{"category": "Math", "title": "Three geometric applications of quandle homology", "abstract": "In this paper we describe three geometric applications of quandle homology. We show that it gives obstructions to tangle embeddings, provides the lower bound for the 4-move distance between links, and can be used in determining periodicity of links."}
{"category": "Math", "title": "Distance Geometry in Quasihypermetric Spaces. I", "abstract": "Let $(X, d)$ be a compact metric space and let $\\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \\colon \\mathcal{M}(X) \\to \\R$ by \\[I(\\mu) = \\int_X \\int_X d(x,y) d\\mu(x) d\\mu(y),\\] and set $M(X) = \\sup I(\\mu)$, where $\\mu$ ranges over the collection of signed measures in $\\mathcal{M}(X)$ of total mass 1. The metric space $(X, d)$ is quasihypermetric if for all $n \\in \\N$, all $\\alpha_1, ..., \\alpha_n \\in \\R$ satisfying $\\sum_{i=1}^n \\alpha_i = 0$ and all $x_1, ..., x_n \\in X$, one has $\\sum_{i,j=1}^n \\alpha_i \\alpha_j d(x_i, x_j) \\leq 0$. Without the quasihypermetric property $M(X)$ is infinite, while with the property a natural semi-inner product structure becomes available on $\\mathcal{M}_0(X)$, the subspace of $\\mathcal{M}(X)$ of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of $(X, d)$, the semi-inner product space structure of $\\mathcal{M}_0(X)$ and the Banach space $C(X)$ of continuous real-valued functions on $X$; conditions equivalent to the quasihypermetric property; the topological properties of $\\mathcal{M}_0(X)$ with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-$*$ topology and the measure-norm topology on $\\mathcal{M}_0(X)$; and the functional-analytic properties of $\\mathcal{M}_0(X)$ as a semi-inner product space, including the question of its completeness. A later paper [Peter Nickolas and Reinhard Wolf, Distance Geometry in Quasihypermetric Spaces. II] will apply the work of this paper to a detailed analysis of the constant $M(X)$."}
{"category": "Math", "title": "Derangement Polynomials and Excedances of Type B", "abstract": "Adopting the definition of excedances of type B due to Brenti, we give a type B analogue of the q-derangement polynomials. The connection between q-derangement polynomials and Eulerian polynomials naturally extends to the type B case. Based on this relation, we derive some basic properties of the q-derangement polynomials of type B, including the generating function formula, the Sturm sequence property, and the asymptotic normal distribution. We also show that the q-derangement polynomials are almost symmetric in the sense that the coefficients possess the spiral property."}
{"category": "Math", "title": "Liouville type of theorems for the Euler and the Navier-Stokes equations", "abstract": "We prove Liouville type of theorems for weak solutions of the Navier-Stokes and the Euler equations. In particular, if the pressure satisfies $ p\\in L^1 (0,T; L^1 (\\Bbb R^N))$ with $\\int_{\\Bbb R^N} p(x,t)dx \\geq 0$, then the corresponding velocity should be trivial, namely $v=0$ on $\\Bbb R^N \\times (0,T)$. In particular, this is the case when $p\\in L^1 (0,T; \\mathcal{H}^1 (\\Bbb R^N))$, where $\\mathcal{H}^1 (\\Bbb R^N)$ the Hardy space. On the other hand, we have equipartition of energy over each component, if $p\\in L^1 (0,T; L^1 (\\Bbb R^N))$ with $\\int_{\\Bbb R^N} p(x,t)dx <0$. Similar results hold also for the magnetohydrodynamic equations."}
{"category": "Math", "title": "Distance Geometry in Quasihypermetric Spaces. II", "abstract": "Let $(X, d)$ be a compact metric space and let $\\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \\colon \\mathcal{M}(X) \\to \\R$ by \\[ I(\\mu) = \\int_X \\int_X d(x,y) d\\mu(x) d\\mu(y), \\] and set $M(X) = \\sup I(\\mu)$, where $\\mu$ ranges over the collection of signed measures in $\\mathcal{M}(X)$ of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $\\mathcal{M}(X)$ when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum defining $M(X)$, sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the finiteness of $M(X)$."}
{"category": "Math", "title": "Distance Geometry in Quasihypermetric Spaces. III", "abstract": "Let $(X, d)$ be a compact metric space and let $\\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \\colon \\mathcal{M}(X) \\to \\R$ by \\[ I(\\mu) = \\int_X \\int_X d(x,y) d\\mu(x) d\\mu(y), \\] and set $M(X) = \\sup I(\\mu)$, where $\\mu$ ranges over the collection of signed measures in $\\mathcal{M}(X)$ of total mass 1. This paper, with two earlier papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and II], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $\\mathcal{M}(X)$ when equipped with a natural semi-inner product. Specifically, this paper explores links between the properties of $M(X)$ and metric embeddings of $X$, and the properties of $M(X)$ when $X$ is a finite metric space."}
{"category": "Math", "title": "Group-case commutative association schemes and their character tables", "abstract": "Leading towards the classification of primitive commutative association schemes as the ultimate goal, Bannai and some of his school have been trying to * identify the major sources of (primitive) commutative association schemes, * collect known group-case primitive commutative association schemes, and * compute their character tables over the last twenty years. The construction of their character tables are important first step for a systematic study of such association schemes and towards the classification of those schemes. In this talk, we briefly survey the progress made in this direction of research, and list some open problems."}
{"category": "Math", "title": "A note on k[z]-automorphisms in two variables", "abstract": "We prove that for a polynomial $f\\in k[x,y,z]$ equivalent are: (1)$f$ is a $k[z]$-coordinate of $k[z][x,y]$, and (2) $k[x,y,z]/(f)\\cong k^{[2]}$ and $f(x,y,a)$ is a coordinate in $k[x,y]$ for some $a\\in k$. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate $f\\in k[x,y,z]$ which is also a $k(z)$-coordinate, is a $k[z]$-coordinate. We discuss a method for constructing automorphisms of $k[x,y,z]$, and observe that the Nagata automorphism occurs naturally as the first non-trivial automorphism obtained by this method - essentially linking Nagata with a non-tame $R$-automorphism of $R[x]$, where $R=k[z]/(z^2)$."}
{"category": "Math", "title": "Dynamique transverse de la lamination de Ghys-Kenyon", "abstract": "Using an aperiodic and repetitive subtree of the Cayley graph of the free Abelian group with two generators, described by Kenyon, Ghys has constructed an example of minimal Riemann surface lamination having both Euclidean and hyperbolic leaves. We prove that the transverse dynamics of this lamination is represented (in a measurable way) by a 2-adic odometer. In fact, we can describe its topological transverse dynamics, and prove that the Ghys-Kenyon lamination is affable."}
{"category": "Math", "title": "On hypercomplex pseudo-Hermitian manifolds", "abstract": "The class of the hypercomplex pseudo-Hermitian manifolds is considered. The flatness of the considered manifolds with the 3 parallel complex structures is proved. Conformal transformations of the metrics are introduced. The conformal invariance and the conformal equivalence of the basic types manifolds are studied. A known example is characterized in relation to the obtained results."}
{"category": "Math", "title": "Neighborliness of Marginal Polytopes", "abstract": "A neighborliness property of marginal polytopes of hierarchical models, depending on the cardinality of the smallest non-face of the underlying simplicial complex, is shown. The case of binary variables is studied explicitly, then the general case is reduced to the binary case. A Markov basis for binary hierarchical models whose simplicial complexes is the complement of an interval is given."}
{"category": "Math", "title": "Convergence of eigenvalues for a highly non-self-adjoint differential operator", "abstract": "In this paper we study a family of operators dependent on a small parameter $\\epsilon > 0$, which arise in a problem in fluid mechanics. We show that the spectra of these operators converge to N as $\\epsilon \\to 0$, even though, for fixed $\\epsilon > 0$, the eigenvalue asymptotics are quadratic."}
{"category": "Math", "title": "Even universal binary Hermitian lattices over imaginary quadratic fields", "abstract": "A positive definite even Hermitian lattice is called \\emph{even universal} if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields $\\Q{-m}$ for all positive square-free integers $m$ and we list optimal criterions on even universality of Hermitian lattices over $\\Q{-m}$ which admits even universal binary Hermitian lattices."}
{"category": "Math", "title": "Positive Legendrian regular homotopies", "abstract": "In contrast with what happens for Legendrian embeddings, there always exist positive loops of Legendrian immersions."}
{"category": "Math", "title": "Repr\\'esentations d\\'eterminantales effectives des polyn\\^omes univari\\'es par les matrices fl\\`eches", "abstract": "We first show the existence of an effective determinantal representation for any univariate polynomial with real coefficients. Then, we more precisely establish that any univariate polynomial with real coefficients has an effective determinantal representation with signature (r+s,s) if and only if it has at least r real roots with multiplicity. The effective determinantal representations we construct used arrow matrices."}
{"category": "Math", "title": "Characteristic Classes on Grassmann Manifolds", "abstract": "In this paper, we use characteristic classes of the canonical vector bundles and the Poincar\\' {e} dualality to study the structure of the real homology and cohomology groups of oriented Grassmann manifold $G(k, n)$. Show that for $k=2$ or $n\\leq 8$, the cohomology groups $H^*(G(k,n),{\\bf R})$ are generated by the first Pontrjagin class, the Euler classes of the canonical vector bundles. In these cases, the Poincar\\' {e} dualality: $H^q(G(k,n),{\\bf R}) \\to H_{k(n-k)-q}(G(k,n),{\\bf R})$ can be given explicitly."}
{"category": "Math", "title": "The notion of convexity and concavity on Wiener space", "abstract": "We define, in the frame of an abstract Wiener space, the notions of convexity and of concavity for the equivalence classes of random variables. As application we show that some important inequalities of the finite dimensional case have their natural counterparts in this setting."}
{"category": "Math", "title": "Large Deviations of Vector-valued Martingales in 2-Smooth Normed Spaces", "abstract": "We derive exponential bounds on probabilities of large deviations for \"light tail\" martingales taking values in finite-dimensional normed spaces. Our primary emphasis is on the case where the bounds are dimension-independent or nearly so. We demonstrate that this is the case when the norm on the space can be approximated, within an absolute constant factor, by a norm which is differentiable on the unit sphere with a Lipschitz continuous gradient. We also present various examples of spaces possessing the latter property."}
{"category": "Math", "title": "Nonparametric Denoising of Signals with Unknown Local Structure, I: Oracle Inequalities", "abstract": "We consider the problem of pointwise estimation of multi-dimensional signals $s$, from noisy observations $(y_\\tau)$ on the regular grid $\\bZd$. Our focus is on the adaptive estimation in the case when the signal can be well recovered using a (hypothetical) linear filter, which can depend on the unknown signal itself. The basic setting of the problem we address here can be summarized as follows: suppose that the signal $s$ is \"well-filtered\", i.e. there exists an adapted time-invariant linear filter $q^*_T$ with the coefficients which vanish outside the \"cube\" $\\{0,..., T\\}^d$ which recovers $s_0$ from observations with small mean-squared error. We suppose that we do not know the filter $q^*$, although, we do know that such a filter exists. We give partial answers to the following questions: -- is it possible to construct an adaptive estimator of the value $s_0$, which relies upon observations and recovers $s_0$ with basically the same estimation error as the unknown filter $q^*_T$? -- how rich is the family of well-filtered (in the above sense) signals? We show that the answer to the first question is affirmative and provide a numerically efficient construction of a nonlinear adaptive filter. Further, we establish a simple calculus of \"well-filtered\" signals, and show that their family is quite large: it contains, for instance, sampled smooth signals, sampled modulated smooth signals and sampled harmonic functions."}
{"category": "Math", "title": "Solving variational inequalities with Stochastic Mirror-Prox algorithm", "abstract": "In this paper we consider iterative methods for stochastic variational inequalities (s.v.i.) with monotone operators. Our basic assumption is that the operator possesses both smooth and nonsmooth components. Further, only noisy observations of the problem data are available. We develop a novel Stochastic Mirror-Prox (SMP) algorithm for solving s.v.i. and show that with the convenient stepsize strategy it attains the optimal rates of convergence with respect to the problem parameters. We apply the SMP algorithm to Stochastic composite minimization and describe particular applications to Stochastic Semidefinite Feasability problem and Eigenvalue minimization."}
{"category": "Math", "title": "Symmetric topological complexity of projective and lens spaces", "abstract": "For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even torsion lens spaces and complex projective spaces are discussed."}
{"category": "Math", "title": "Prehomogeneous Affine Representations and Flat Pseudo-Riemannian Manifolds", "abstract": "The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes and characters of prehomogeneous affine representations. We give applications concerning flat affine, as well as Pseudo-Riemannian and symplectic affine flat manifolds."}
{"category": "Math", "title": "Self-adjoint curl operators", "abstract": "We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain $D$. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl) equipped with a symplectic pairing arising from the $\\wedge$-product of 1-forms on $\\partial D$. Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extension. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed."}
{"category": "Math", "title": "Deformation Spaces for Affine Crystallographic Groups", "abstract": "We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study the properties of the action of the homotopy mapping class groups on deformation spaces. In our context these groups are arithmetic groups, and we construct examples of flat affine manifolds where every finite group of mapping classes admits a fixed point on the deformation space. We also show that the existence of fixed points on the deformation space is equivalent to the realisation of finite groups of homotopy equivalences by finite groups of affine diffeomorphisms. Extending ideas of Auslander we relate the deformation spaces of affine space forms with solvable fundamental group to deformation spaces of manifolds with nilpotent fundamental group. We give applications concerning the classification problem for affine space forms."}
{"category": "Math", "title": "Modular forms and K3 surfaces", "abstract": "For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over QQ associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author."}
{"category": "Math", "title": "Cohomology of quantum groups: An analog of Kostant's Theorem", "abstract": "We prove the analog of Kostant's Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostant's cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra $\\mathfrak{g} = \\mathfrak{sl}(n)$. We also show that Kostant's formula holds when $q$ is specialized to an $\\ell$-th root of unity for odd $\\ell \\ge h-1$ (where $h$ is the Coxeter number of $\\mathfrak{g}$) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an extension of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic."}
{"category": "Math", "title": "mu-constancy does not imply constant bi-Lipschitz type", "abstract": "We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\\{(x,y,z)\\in\\C^3 | x^5+z^{15}+y^7z+txy^6=0 \\}$ of normal complex surface germs; we show the germ $(X_0, 0)$ is not bi-Lipschitz homeomorphic with respect to the inner metric to the germ $(X_t,0)$ for $t\\ne 0$."}
{"category": "Math", "title": "Certain free products of graph operator algebras", "abstract": "We develop a notion of a generalized Cuntz-Krieger family of projections and partial isometries where the range of the partial isometries need not have trivial intersection. We associate to these generalized Cuntz-Krieger families a directed graph, with a coloring function on the edge set. We call such a directed graph an edge-colored directed graph. We then study the $C^*$-algebras and the non-selfadjoint operator algebras associated to edge-colored directed graphs. These algebras arise as free products of directed graph algebras with amalgamation. We then determine the $C^*$-envelopes for a large class of the non-selfadjoint algebras. Finally, we relate properties of the edge-colored directed graphs to properties of the associated $C^*$-algebra, including simplicity and nuclearity. Using the free product description of these algebras we investigate the $K$-theory of these algebras."}
{"category": "Math", "title": "Small-time expansions for the transition distributions of L\\'evy processes", "abstract": "Let $X$ be a L\\'evy process with absolutely continuous L\\'evy measure $\\nu$. Small time polynomial expansions of order $n$ in $t$ are obtained for the tails $P(X_{t}\\geq{}y)$ of the process, assuming smoothness conditions on the L\\'evy density away from the origin. By imposing additional regularity conditions on the transition density $p_{t}$ of $X_{t}$, an explicit expression for the remainder of the approximation is also given. As a byproduct, polynomial expansions of order $n$ in $t$ are derived for the transition densities of the process. The conditions imposed on $p_{t}$ require that its derivatives remain uniformly bounded away from the origin, as $t\\to{}0$; such conditions are shown to be satisfied for symmetric stable L\\'evy processes as well as for other related L\\'evy processes of relevance in mathematical finance. The expansions seem to correct asymptotics previously reported in the literature."}
{"category": "Math", "title": "Non-Walker Self-Dual Neutral Einstein Four-Manifolds of Petrov Type III", "abstract": "The local structure of the manifolds named in the title is described. Although curvature homogeneous, they are not, in general, locally homogeneous. Not all of them are Ricci-flat, which answers an existence question about type III Jordan-Osserman metrics, raised by Diaz-Ramos, Garcia-Rio and Vazquez-Lorenzo (2006)."}
{"category": "Math", "title": "A Morse theoretic description of string topology", "abstract": "Let M be a closed, oriented, n-dimensional manifold. In this paper we give a Morse theoretic description of the string topology operations introduced by Chas and Sullivan, and extended by the first author, Jones, Godin, and others. We do this by studying maps from surfaces with cylindrical ends to M, such that on the cylinders, they satisfy the gradient flow equation of a Morse function on the loop space, LM. We then give Morse theoretic descriptions of related constructions, such as the Thom and Euler classes of a vector bundle, as well as the shriek, or unkehr homomorphism."}
{"category": "Math", "title": "Coniveau 2 complete intersections and effective cones", "abstract": "The goal of this paper is first of all to propose a strategy to attack the generalized Hodge conjecture for coniveau 2 complete intersections, and secondly to state a conjecture concerning the cones of effective cycle classes in intermediate dimensions. Our main results show that the generalized Hodge conjecture for coniveau 2 complete intersections would follow from a particular case of this effectiveness conjecture."}
{"category": "Math", "title": "$\\LE$-diagrams and totally positive bases inside the nonnegative Grassmannian", "abstract": "There is a cell decomposition of the nonnegative Grassmannian. For each cell, totally positive bases(TP-bases) is defined as the minimal set of Pl\\\"ucker variables such that all other nonzero Pl\\\"ucker variables in the cell can be expressed in those variables in a subtraction-free rational function. This is the generalization of the TP-bases defined for nonnegative part of $GL_k$ defined in \\cite{FZ5}. For each cell, we have a $\\LE$-diagram and a natural way to label the dots inside the diagram with Pl\\\"ucker variables. Those set of Pl\\\"ucker variables form a TP-bases of the cell. Using mutations coming from 3-term Pl\\\"ucker relation, we conjecture that they can be mutated to a special set of Pl\\\"ucker variable $\\S$. All other nonzero Pl\\\"ucker variables in the cell will be expressed as a subtraction-free Laurent polynomial in variables of $\\S$. We define TP-diagrams to express the transformation procedure in terms of moves on a diagram. We will prove the conjecture for certain class of cells called weakly-connected cells. Then we will study the connection with cluster algebras through lattice-path-matroid cells."}
{"category": "Math", "title": "The Seventeen Elements of Pythagorean Triangles", "abstract": "This is an exhaustive study of the seventeen elements of Pythagorean triangles, from the point of view of when such an element is an irrational number, a rational number, or an integer. For each of these 17 elements,precice conditions for their integrality,rationality, or irrationality; are given. These 17 elements are:The radius of the incircle, the radius of the circumscribed circle,the three radii of the three exterior circles tangential to the three lines containing the triangles'sides, the lengths of the three heights, the lenghts of the three internal angle bisectors, the lengthe of the three external angle bisectors, and the lengths of the three medians."}
{"category": "Math", "title": "The holomorphic Gauss Parametrization", "abstract": "We give a local parametric description of all holomorphic hypersurfaces in complex Euclidean and projective spaces with constant index of relative nullity, together with applications. This is a complex analogue to the parametrization for real hypersurfaces in Euclidean space known as the Gauss parametrization."}
{"category": "Math", "title": "Birational rigidity of Fano varieties and field extensions", "abstract": "The aim of this note is to settle some foundational questions about the behavior of birational rigidity in extensions of algebraically closed fields."}
{"category": "Math", "title": "Nonabelian harmonic analysis and functional equations on compact groups", "abstract": "Making use of nonabelian harmonic analysis and representation theory, we solve the functional equation $$f_1(xy)+f_2(yx)+f_3(xy^{-1})+f_4(y^{-1}x)=f_5(x)f_6(y)$$ on arbitrary compact groups. The structure of its general solution is completely described. Consequently, several special cases of the above equation, in particular, the Wilson equation and the d'Alembert long equation, are solved on compact groups."}
{"category": "Math", "title": "Multi-product splitting and Runge-Kutta-Nystrom integrators", "abstract": "The splitting of $\\e^{h(A+B)}$ into a single product of $\\e^{h A}$ and $\\e^{hB}$ results in symplectic integrators when $A$ and $B$ are classical Lie operators. However, at high orders, a single product splitting, with exponentially growing number of operators, is very difficult to derive. This work shows that, if the splitting is generalized to a sum of products, then a simple choice of the basis product reduces the problem to that of extrapolation, with analytically known coefficients and only quadratically growing number of operators. When a multi-product splitting is applied to classical Hamiltonian systems, the resulting algorithm is no longer symplectic but is of the Runge-Kutta-Nystr\\\"om (RKN) type. Multi-product splitting, in conjunction with a special force-reduction process,explains why at orders $p=4$ and 6, RKN integrators only need $p-1$ force evaluations."}
{"category": "Math", "title": "Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets", "abstract": "We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d=6. This implies that for all pairs (d,n) with n-d \\leq 6 the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n-d \\leq 6. We show this result by showing this bound for a more general structure -- so-called matroid polytopes -- by reduction to a small number of satisfiability problems."}
{"category": "Math", "title": "Convexit\\'e holomorphe du rev\\^etement de Malcev d'apr\\`es S. Leroy", "abstract": "In these notes, we present the strategy employed by S. Leroy to attack the nilpotent case of the Shafarevich conjecture. The projective case was treated in a paper of L. Katzarkov but Leroy's proof is largely independant and works in the K\\\"ahler setting as well. It is entirely based on the higher Albanese manifolds constructed by R. Hain."}
{"category": "Math", "title": "Pseudodifferential operator calculus for generalized Q-rank 1 locally symmetric spaces, I", "abstract": "This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of fibre bundles. The class of operators we consider on such a space includes those arising naturally from metrics which degenerate to various orders at the boundary, in directions given by the tower of fibrations. As well as Q-rank 1 locally symmetric spaces, examples include Ricci-flat metrics on the complement of a divisor in a smooth variety constructed by Tian and Yau. In this first part of the calculus construction, parametrices are found for \"fully elliptic differential \\bfa-operators\", which are uniformly elliptic operators on these manifolds that satisfy an additional invertibility condition at infinity. In the second part we will consider operators that do not satisfy this condition."}
{"category": "Math", "title": "Triangulations of the sphere and degenerations of K3 surfaces", "abstract": "W. Thurston proved that to a triangulation of the sphere of non-negative combinatorial curvature, one can associate an element in a certain lattice over the Eisenstein integers such that its orbit is a complete invariant of the triangulation. In this paper, we show that this association can be obtained naturally by using Type III degenerations of K3 surfaces."}
{"category": "Math", "title": "Finite index subgroups of fully residually free groups", "abstract": "Using graph-theoretic techniques for f.g. subgroups of $F^{\\mathbb{Z}[t]}$ we provide a criterion for a f.g. subgroup of a f.g. fully residually free group to be of finite index. Moreover, we show that this criterion can be checked effectively. Also we obtain an analogue of Greenberg-Stallings Theorem for f.g. fully residually free groups, and prove that a f.g. non-abelian subgroup of a f.g. fully residually free group is of finite index in its commensurator."}
{"category": "Math", "title": "The modified K\\\"ahler-Ricci flow and solitons", "abstract": "We investigate the K\\\"ahler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a K\\\"ahler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy."}
{"category": "Math", "title": "On principal fibrations associated with one algebra", "abstract": "In this paper we study two types of fibrations associated with a 3-dimensional unital associative irreducible algebra and their basic properties. We investigate trivial principal fibrations of degenerate semi-Euclidean sphere and their semi-conformal and projective models. We use Norden normalization method for constructing second model."}
{"category": "Math", "title": "Solving the conjugacy problem in Garside groups by cyclic sliding", "abstract": "We present a solution to the conjugacy decision problem and the conjugacy search problem in Garside groups, which is theoretically simpler than the usual one, with no loss of efficiency. This is done by replacing the well known cycling and decycling operations by a new one, called cyclic sliding, which appears to be a more natural choice. We give an analysis of the complexity of our algorithm in terms of fundamental operations with simple elements, so our analysis is valid for every Garside group. This paper intends to be self-contained, not requiring any previous knowledge of prior algorithms, and includes all the details for the algorithm to be implemented on a computer."}
{"category": "Math", "title": "Connected Components of Hurwitz Schemes and Malle's Conjecture", "abstract": "Let Z(X) be the number of degree-d extensions of F_q(t) with bounded discriminant and some specified Galois group. The problem of computing Z(X) can be related to a problem of counting F_q-rational points on certain Hurwitz spaces. Ellenberg and Venkatesh used this idea to develop a heuristic for the asymptotic behavior of Z'(X), the number of -geometrically connected- extensions, and showed that this agrees with the conjectures of Malle for function fields. We extend Ellenberg-Venkatesh's argument to handle the more complicated case of covers of P^1 which may not be geometrically connected, and show thatthe resulting heuristic suggests a natural modification to Malle's conjecture which avoids the counterexamples, due to Kl\\\"uners, to the original conjecture."}
{"category": "Math", "title": "Counting Multisections in Conic Bundles over a Curve defined over F_q", "abstract": "For a given conic bundle X over a curve C defined over F_q, we count irreducible branch covers of C in X of degree d and height e>>1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field F_q (C)."}
{"category": "Math", "title": "Splitting of Sharply 2-Transitive Groups of Characteristic 3", "abstract": "We give a group theoretic proof of the splitting of sharply 2-transitive groups of characteristic 3."}
{"category": "Math", "title": "On NP complete problems I", "abstract": "We study the quadratic residue problem known as an NP complete problem by way of the prime number and show that a nondeterministic polynomial process does not belong to the class P because of a random distribution of solutions for the quadratic residue problem."}
{"category": "Math", "title": "Fonction constante et d\\'eriv\\'ee nulle : un r\\'esultat si trivial..", "abstract": "We study various proofs of the caracterization of constant functions, more precisely of the theorem: a derivable function, defined on a real interval, is constant if, and only if, its derivative is null. Our aim is to study the relationships of these proofs with the mathematical curriculum of secondary schools and the begining of undergraduate studies in France, from various point of views (epistemological, historical, didactical)."}
{"category": "Math", "title": "Least Squares and Shrinkage Estimation under Bimonotonicity Constraints", "abstract": "In this paper we describe active set type algorithms for minimization of a smooth function under general order constraints, an important case being functions on the set of bimonotone r-by-s matrices. These algorithms can be used, for instance, to estimate a bimonotone regression function via least squares or (a smooth approximation of) least absolute deviations. Another application is shrinkage estimation in image denoising or, more generally, regression problems with two ordinal factors after representing the data in a suitable basis which is indexed by pairs (i,j) in {1,...,r}x{1,...,s}. Various numerical examples illustrate our methods."}
{"category": "Math", "title": "Rational representations of $GL_2$", "abstract": "Let $F$ be an algebraically closed field of characteristic $p$. We fashion an infinite dimensional basic algebra $\\underleftarrow{\\mathcal{C}}_p(F)$, with a transparent combinatorial structure, which we expect to control the rational representation theory of $GL_2(F)$."}
{"category": "Math", "title": "Sudden extinction of a critical branching process in random environment", "abstract": "Let $T$ be the extinction moment of a critical branching process $Z=(Z_{n},n\\geq 0) $ in a random environment specified by iid probability generating functions. We study the asymptotic behavior of the probability of extinction of the process $Z$ at moment $n\\to \\infty$, and show that if the logarithm of the (random) expectation of the offspring number belongs to the domain of attraction of a non-gaussian stable law then the extinction occurs owing to very unfavorable environment forcing the process, having at moment $T-1$ exponentially large population, to die out. We also give an interpretation of the obtained results in terms of random walks in random environment."}
{"category": "Math", "title": "Homotopy, homology, and $GL_2$", "abstract": "We define weak 2-categories of finite dimensional algebras with bimodules, along with collections of operators $\\mathbb{O}_{(c,x)}$ on these 2-categories. We prove that special examples $\\mathbb{O}_p$ of these operators control all homological aspects of the rational representation theory of the algebraic group $GL_2$, over a field of positive characteristic. We prove that when $x$ is a Rickard tilting complex, the operators $\\mathbb{O}_{(c,x)}$ honour derived equivalences, in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight $\\mathbb{Z}_+$-gradings on Schur algebras $S(2,r)$, and the existence of braid group actions on the derived categories of blocks of these Schur algebras."}
{"category": "Math", "title": "Universal classes for algebraic groups", "abstract": "We exhibit cocycles representing certain classes in the rational cohomology of of the general linear group with coefficients in the divided powers of a Frobenius twist of the adjoint representation. These classes' existence was anticipated by van der Kallen, and they intervene in the proof that reductive linear algebraic groups have finitely generated cohomology algebras."}
{"category": "Math", "title": "A phase transition for non-intersecting Brownian motions, and the Painleve II equation", "abstract": "We consider n non-intersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of 'large separation' between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between them, as one would intuitively expect. We give a rigorous proof using the Riemann-Hilbert formalism. In the case of 'critical separation' between the endpoints we are led to a model Riemann-Hilbert problem associated to the Hastings-McLeod solution of the Painleve II equation. We show that the Painleve II equation also appears in the large n asymptotics of the recurrence coefficients of the multiple Hermite polynomials that are associated with the Riemann-Hilbert problem."}
{"category": "Math", "title": "Bifunctor cohomology and Cohomological finite generation for reductive groups", "abstract": "Let G be a reductive linear algebraic group over a field k. Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. Invariant theory tells that the ring of invariants A^G=H^0(G,A) is finitely generated. We show that in fact the full cohomology ring H^*(G,A) is finitely generated. The proof is based on the strict polynomial bifunctor cohomology classes constructed by the junior author. We also continue the study of bifunctor cohomology of the divided powers of a Frobenius twist of the adjoint representation."}
{"category": "Math", "title": "Kobayashi geodesics in A_g", "abstract": "We consider Kobayashi geodesics in the moduli space of abelian varieties A_g that is, algebraic curves that are totally geodesic submanifolds for the Kobayashi metric. We show that Kobayashi geodesics can be characterized as those curves whose logarithmic tangent bundle splits as a subbundle of the logarithmic tangent bundle of A_g. Both Shimura curves and Teichmueller curves are examples of Kobayashi geodesics, but there are other examples. We show moreover that non-compact Kobayashi geodesics always map to the locus of real multiplication and that the Q-irreducibility of the induced variation of Hodge structures implies that they are defined over a number field."}
{"category": "Math", "title": "A simulation study comparing likelihood and non-likelihood approaches in analyzing overdispersed count data", "abstract": "Overdispersed count data are modelled with likelihood and non-likelihood approaches. Likelihood approaches include the Poisson mixtures with three distributions, the gamma, the lognormal, and the inverse Gaussian distributions. Non-likelihood approaches include the robust sandwich estimator and quasilikelihood. In this simulation study, overdispersed count data were simulated under the Poisson mixtures with the gamma, the lognormal and the inverse Gaussian distributions, then analyzed with the five likelihood and non-likelihood approaches. Our results indicated that 1) when the count data are mildly overdispersed, there are virtually no differences in type I error rate, standard error of the main effect, and empirical power among the five methods; 2) when the count data are very overdispersed, none of these five approaches is robust to model misspecification as evaluated by type I error rate, standard error of the main effect, and empirical power. This simulation study raises caution on using non-likelihood method for analyzing very overdispered count data because of likely higher type I error and inappropriate power levels. Unlike non-likelihood approaches, likelihood approaches allow for statistical tests based on likelihood ratios and for checking model fit and provide basis for power and sample size calculations. When likelihood approaches are used, we suggest comparing likelihood values to select the appropriate parametric method for analyzing very overdispersed count data."}
{"category": "Math", "title": "On the frequentist coverage of Bayesian credible intervals for lower bounded means", "abstract": "For estimating a lower bounded location or mean parameter for a symmetric and logconcave density, we investigate the frequentist performance of the $100(1-\\alpha)%$ Bayesian HPD credible set associated with priors which are truncations of flat priors onto the restricted parameter space. Various new properties are obtained. Namely, we identify precisely where the minimum coverage is obtained and we show that this minimum coverage is bounded between $1-\\frac{3\\alpha}{2}$ and $1-\\frac{3\\alpha}{2}+\\frac{\\alpha^2}{1+\\alpha}$; with the lower bound $1-\\frac{3\\alpha}{2}$ improving (for $\\alpha \\leq 1/3$) on the previously established ([9]; [8]) lower bound $\\frac{1-\\alpha}{1+\\alpha}$. Several illustrative examples are given."}
{"category": "Math", "title": "On a $p$-adic extension of the Jacquet-Langlands correspondence to weight 1", "abstract": "We consider a novel version of the classical Jacquet-Langlands {correspondence}, explore a $p$-adic extension of the Jacquet-Langlands correspondence, and as an explicit example we find an overconvergent automorphic form of weight~1 which corresponds to a classical modular form of weight~1, using both experimental and theoretical methods."}
{"category": "Math", "title": "Terwilliger Algebras of Wreath Powers of One-Class Association Schemes", "abstract": "In this paper, we study the subconstituent algebras, also called as Terwilliger algebras, of association schemes that are obtained as the wreath product of one-class association schemes $K_n=H(1, n)$ for $n\\ge 2$. We find that the $d$-class association scheme $K_{n_{1}}\\wr K_{n_{2}} \\wr ... \\wr K_{n_{d}}$ formed by taking the wreath product of $K_{n_{i}}$ has the triple-regularity property. We determine the dimension of the Terwilliger algebra for the association scheme $K_{n_{1}}\\wr K_{n_{2}}\\wr ... \\wr K_ {n_{d}}$. We give a description of the structure of the Terwilliger algebra for the wreath power $(K_n)^{\\wr d}$ for $n \\geq 2$ by studying its irreducible modules. In particular, we show that the Terwilliger algebra of $(K_n)^{\\wr d}$ is isomorphic to $M_{d+1}(\\mathbb{C})\\oplus M_1(\\mathbb{C})^{\\oplus \\frac12d(d+1)}$ for $n\\ge3$, and $M_{d+1}(\\mathbb{C})\\oplus M_1(\\mathbb{C})^{\\oplus \\frac12d(d-1)}$ for $n=2$."}
{"category": "Math", "title": "On the Schrodinger equation outside strictly convex obstacles", "abstract": "We prove sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary. We deduce sharp (classical) Strichartz estimates for the Schrodinger equation outside a strictly convex obstacle, local existence for the H^1-critical (quintic) Schrodinger equation and scattering for the sub-critical Schrodinger equation in 3D."}
{"category": "Math", "title": "The Julia sets of basic uniCremer polynomials of arbitrary degree", "abstract": "Let $P$ be a polynomial of degree $d$ with a Cremer point $p$ and no repelling or parabolic periodic bi-accessible points. We show that there are two types of such Julia sets $J_P$. The \\emph{red dwarf} $J_P$ are nowhere connected im kleinen and such that the intersection of all impressions of external angles is a continuum containing $p$ and the orbits of all critical images. The \\emph{solar} $J_P$ are such that every angle with dense orbit has a degenerate impression disjoint from other impressions and $J_P$ is connected im kleinen at its landing point. We study bi-accessible points and locally connected models of $J_P$ and show that such sets $J_P$ appear through polynomial-like maps for generic polynomials with Cremer points."}
{"category": "Math", "title": "Localized factorizations of integers", "abstract": "We determine the order of magnitude of H^{(k+1)}(x,\\vec{y},2\\vec{y}), the number of integers up to x that are divisible by a product d_1...d_k with y_i<d_i\\le 2y_i, when the numbers \\log y_1,...,\\log y_k have the same order of magnitude and k\\ge 2. This generalizes a result by K. Ford when k=1. As a corollary of these bounds, we determine the number of elements up to multiplicative constants that appear in a (k+1)-dimensional multiplication table as well as how many distinct sums of k+1 Farey fractions there are modulo 1."}
{"category": "Math", "title": "Multifractal analysis for multimodal maps", "abstract": "Given a multimodal interval map $f:I \\to I$ and a H\\\"older potential $\\phi:I \\to \\mathbb{R}$, we study the dimension spectrum for equilibrium states of $\\phi$. The main tool here is inducing schemes, used to overcome the presence of critical points. The key issue is to show that enough points are `seen' by a class of inducing schemes. We also compute the Lyapunov spectrum. We obtain the strongest results when $f$ is a Collet-Eckmann map, but our analysis also holds for maps satisfying much weaker growth conditions."}
{"category": "Math", "title": "Lifting of Characters for Nonlinear Simply Laced Groups", "abstract": "One aspect of the Langlands program for linear groups is lifting of characters, which relates virtual representations on a group $G$ with those on an endoscopic group for $G$. The goal of this paper is to extend this theory to nonlinear two-fold covers of real groups in the simply laced case. Suppose $\\tG$ is a two-fold cover of a real reductive group $G$. The main result is that there is an operation, denoted $\\Lift_G^{\\tG}$, taking a stable virtual character of $G$ to 0 or a virtual genuine character of $\\tG$, and $\\Lift_G^{\\tG}(\\Theta_\\pi)$ may be explicitly computed if $\\pi$ is a stable sum of standard modules."}
{"category": "Math", "title": "Discrete Fourier analysis on fundamental domain of $A_d$ lattice and on simplex in $d$-variables", "abstract": "A discrete Fourier analysis on the fundamental domain $\\Omega_d$ of the $d$-dimensional lattice of type $A_d$ is studied, where $\\Omega_2$ is the regular hexagon and $\\Omega_3$ is the rhombic dodecahedron, and analogous results on $d$-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\\log n)^d$. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas."}
{"category": "Math", "title": "Concordance invariants from higher order covers", "abstract": "We generalize the Manolescu-Owens smooth concordance invariant delta(K) of knots K in the 3-sphere to invariants delta_{p^n}(K) obtained by considering covers of order p^n, with p prime. Our main result shows that for any odd prime p, the direct sum of delta_{p^n} as n ranges through the natural numbers, yields a homomorphism of infinite rank from the smooth concordance group to Z^\\infty. We also show that unlike delta, these new invariants typically are not multiples of the knot signature, even for alternating knots. A significant portion of the article is devoted to exploring examples."}
{"category": "Math", "title": "Well-posedness for the 1D Zakharov-Rubenchik system", "abstract": "Local and global well-posedness results are established for the initial value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving Ill-posedness results otherwise. The global results allow us to study the norm growth of solutions corresponding to the Schrodinger equation term. We use ideas recently introduced to study the classical Zakharov systems."}
{"category": "Math", "title": "The topology of Birkhoff varieties", "abstract": "Our main theorem is that the inclusion of a Birkhoff variety in the affine Grassmannian is a homotopy equivalence. We also construct analogues of tubular neighborhoods for Birkhoff and Schubert varieties. We include some observations on torus-equivariant cohomology."}
{"category": "Math", "title": "Tropical descendant Gromov-Witten invariants", "abstract": "We define tropical Psi-classes on the moduli space of rational tropical curves in R^2 and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psi- and evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin's lattice path algorithm and counts rational plane tropical curves satisfying certain Psi- and evaluation conditions."}
{"category": "Math", "title": "Measurability of optimal transportation and strong coupling of martingale measures", "abstract": "We consider the optimal mass transportation problem in $\\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with specified covariance measure. This is done by pushing forward one martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures."}
{"category": "Math", "title": "An example concerning the Theory of Levels for codimension-one foliations", "abstract": "We give an example of a codimension-one foliation which is transversely of class C^1 and which does not satisfy the \"Local Minimal Set\" property."}
{"category": "Math", "title": "beta-family congruences and the f-invariant", "abstract": "In previous work, the authors have each introduced methods for studying the 2-line of the p-local Adams-Novikov spectral sequence in terms of the arithmetic of modular forms. We give the precise relationship between the congruences of modular forms introduced by the first author with the Q-spectrum and the f-invariant of the second author. This relationship enables us to refine the target group of the f-invariant in a way which makes it more manageable for computations."}
{"category": "Math", "title": "On the generalised Ritt problem as a computational problem", "abstract": "The Ritt problem asks if there is an algorithm that tells whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In particular, we show that it is equivalent to testing if a differential polynomial is a zero divisor modulo a radical differential ideal. The technique used in the proof of equivalence yields algorithms for computing a canonical decomposition of a radical differential ideal into prime components and a canonical generating set of a radical differential ideal. Both proposed representations of a radical differential ideal are independent of the given set of generators and can be made independent of the ranking."}
{"category": "Math", "title": "Continuous extension of arithmetic volumes", "abstract": "This paper is the sequel of the paper \"Continuity of volumes on arithmetic varieties\", in which we established the arithmetic volume function of smooth hermitian Q-invertible sheaves and proved its continuity. The continuity of the volume function has a lot of applications as treated in the paper as above. In this paper, we would like to consider its continuous extension over R."}
{"category": "Math", "title": "On adaptive stratification", "abstract": "This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the strata, which should be ideally fitted to thesubsets where the functions to integrate is nearly constant, and on the allocation of the number of samples within each strata. When the dimension is large and the function to integrate is complex, finding such partitions and allocating the sample is a highly non-trivial problem. In this work, we investigate a novel method to improve the efficiency of the estimator \"on the fly\", by jointly sampling and adapting the strata and the allocation within the strata. The accuracy of estimators when this method is used is examined in detail, in the so-called asymptotic regime (i.e. when both the number of samples and the number of strata are large). We illustrate the use of the method for the computation of the price of path-dependent options in models with both constant and stochastic volatility. The use of this adaptive technique yields variance reduction by factors sometimes larger than 1000 compared to classical Monte Carlo estimators."}
{"category": "Math", "title": "On the equivariant cohomology of subvarieties of a B-regular variety", "abstract": "By a $B$-regular variety, we mean a smooth projective variety over $C$ admitting an algebraic action of the upper triangular Borel subgroup $B \\subset SL_2(C)$ such that the unipotent radical in $B$ has a unique fixed point. A result of M. Brion and the first author describes the equivariant cohomology algebra (over $C$) of a $B$-regular variety $X$ as the coordinate ring of a remarkable affine curve in $X \\times P^1$. The main result of this paper uses this fact to classify the $B$-invariant subvarieties $Y$ of a $B$-regular variety $X$ for which the restriction map $i_Y:H^*(X) \\to H^*(Y)$ is surjective."}
{"category": "Math", "title": "Vertex Degree of Random Intersection Graph", "abstract": "A random intersection graph is constructed by independently assigning a subset of a given set of objects $W,$ to each vertex of the vertex set $V$ of a simple graph $G.$ There is an edge between two vertices of $V,$ iff their respective subsets(in $W$,) have at least one common element. The strong threshold for the connectivity between any two arbitrary vertices of vertex set $V,$ is derived. Also we determine the almost sure probability bounds for the vertex degree of a typical vertex of graph $G.$"}
{"category": "Math", "title": "A Strong threshold for the size of random caps to cover a sphere", "abstract": "In this article, we consider `$N$'spherical caps of area $4\\pi p$ were uniformly distributed over the surface of a unit sphere. We are giving the strong threshold function for the size of random caps to cover the surface of a unit sphere. We have shown that for large $N,$ if $\\frac{Np}{\\log\\:N} > 1/2$ the surface of sphere is completely covered by the $N$ caps almost surely, and if $\\frac{Np}{\\log\\:N} \\leq 1/2$ a partition of the surface of sphere is remains uncovered by the $N$ caps almost surely."}
{"category": "Math", "title": "Number of Edges in Random Intersection Graph on Surface of a Sphere", "abstract": "In this article, we consider `$N$'spherical caps of area $4\\pi p$ were uniformly distributed over the surface of a unit sphere. We study the random intersection graph $G_N$ constructed by these caps. We prove that for $p = \\frac{c}{N^{\\al}},\\:c >0$ and $\\al >2,$ the number of edges in graph $G_N$ follow the Poisson distribution. Also we derive the strong law results for the number of isolated vertices in $G_N$: for $p = \\frac{c}{N^{\\al}},\\:c >0$ for $\\al < 1,$ there is no isolated vertex in $G_N$ almost surely i.e., there are atleast $N/2$ edges in $G_N$ and for $\\al >3,$ every vertex in $G_N$ is isolated i.e., there is no edge in edge set $\\cE_N.$"}
{"category": "Math", "title": "Bialgebra structures of 2-associative algebras", "abstract": "This work is devoted to study new bialgebra structures related to 2-associative algebras. A 2-associative algebra is a vector space equipped with two associative multiplications. We discuss the notions of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras. The first structure was revealed by J.-L. Loday and M. Ronco in an analogue of a Cartier-Milnor-Moore theorem, the second was suggested by Loday and the third is a variation of the second one. The main results of this paper are the construction of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras starting from an associative algebra and the classification of these structures in low dimensions."}
{"category": "Math", "title": "$L^p-L^q$ estimates on the solutions to $u_{tt}-u_{x_1x_1}=\\triangle u_t$", "abstract": "This paper focuses the study on the $L^p-L^q$ estimates on the solutions to an asymmetric wave equation with dissipation which arises in the study for the magneto-hydrodynamics by using the method of Green function."}
{"category": "Math", "title": "On degeneration of surface in Fitting compactification of moduli of stable vector bundles", "abstract": "The new compactification of moduli scheme of Gieseker-stable vector bundles with the given Hilbert polynomial on a smooth projective polarized surface (S;H), over the field k = \\bar k of zero characteristic, is constructed in previous papers of the author. Families of locally free sheaves on the surface S are completed by the locally free sheaves on the schemes which are certain modifications of S. We obtain the class of modified surfaces to appear in the construction. Keywords: moduli space, semistable coherent sheaves, blowup algebra, algebraic surface."}
{"category": "Math", "title": "Combinatorial differential geometry and ideal Bianchi-Ricci identities", "abstract": "We apply the graph complex method to vector fields depending naturally on a set of vector fields and a linear symmetric connection. We characterize all possible systems of generators for such vector-field valued operators including the classical ones given by normal tensors and covariant derivatives. We also describe the size of the space of such operators and prove the existence of an `ideal' basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi-Ricci identities without the correction terms."}
{"category": "Math", "title": "Betti numbers of transversal monomial ideals", "abstract": "In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of $t$-minors of a generic pluri-circulant matrix is a transversal monomial ideal . Using a Gr\\\"obner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervair resolution, the Betti numbers of this initial ideal is computed and it is proved that, for some significant values of $t$, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper, naturally arise in the study of generic singularities of algebraic varieties."}
{"category": "Math", "title": "Global Well-posedness of the 1D Dirac-Klein-Gordon system in Sobolev spaces of negative index", "abstract": "We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in the proof is the theory of almost conservation law and I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. Our proof also relies on the null structure in the system, and bilinear spacetime estimates of Klainerman-Machedon type."}
{"category": "Math", "title": "On the essential spectrum of Nadirashvili-Martin-Morales minimal surfaces", "abstract": "We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of $\\mathbb{R}^{3}$ is discrete. This gives a positive answer to a question of Yau."}
{"category": "Math", "title": "On the connected components of the conjugacy class of projectors on $ \\ell_p\\oplus\\ell_q $", "abstract": "We characterize the projectors $ P $ on a Banach space $ E $ having the property of being connected to all the others projectors obtained as a conjugation of $ P $. Using this characterization we show an example of Banach space where the conjugacy class of a projector splits into several path-connected components, and describe the conjugacy classes of projectors onto subspaces of $ \\ell_p\\oplus\\ell_q $ with $ p\\neq q $."}
{"category": "Math", "title": "The group of symmetries of the Tower of Hanoi graph", "abstract": "I prove that the group of symmetries of the Tower of Hanoi graph with k pegs and n disks, denoted H_n^k, is isomorphic to the group of permutations of k elements, S_k, for all k greater than or equal to 3 and positive n."}
{"category": "Math", "title": "Left invariant complex structures on U(2) and SU(2)xSU(2) revisited", "abstract": "We compute the torsion-free linear maps from the Lie algebra su(2) into itself, deduce a new determination of the integrable complex structures and their equivalence classes under the action of the automorphism group for u(2) and su(2)xsu(2), and prove that in both cases the set of complex structures is a differentiable manifold. u(2)x u(2), su(2)^N and u(2)^N are also considered. Extensions of complex structures from u(2) to su(2)xsu(2) are studied, local holomorphic charts given, and attention is paid to what representations of u(2) we can get from a substitute to the regular representation on a space of holomorphic functions for the complex structure."}
{"category": "Math", "title": "Survey on Affine Spheres", "abstract": "We give a survey of the theory of affine spheres, emphasizing the convex cases and relationsships to Monge-Ampere equations and geometric structures on manifolds."}
{"category": "Math", "title": "Representation theory of mv-algebras", "abstract": "In this paper we develop a general representation theory for mv-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of mv-algebras and mv-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. \\emph{We prove that any mv-algebra is isomorphic to the mv-algebra of all global sections of a sheaf of mv-chains on a compact topological space}. This result is intimately related to McNaughton's theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton's. On spite of the language utilized in this abstract, we wrote this paper in a way that, we hope, could be read without much acquaintance with either sheaf theory or mv-algebra theory."}
{"category": "Math", "title": "Monotone images of Cremer Julia sets", "abstract": "We show that if $P$ is a quadratic polynomial with a fixed Cremer point and Julia set $J$, then for any monotone map $\\ph:J\\to A$ from $J$ onto a locally connected continuum $A$, $A$ is a single point."}
{"category": "Math", "title": "K-theoretic exceptional collections at roots of unity", "abstract": "Using cyclotomic specializations of the equivariant $K$-theory with respect to a torus action we derive congruences for discrete invariants of exceptional objects in derived categories of coherent sheaves on a class of varieties that includes Grassmannians and smooth quadrics. For example, we prove that if $X={\\Bbb P}^{n_1-1}\\times...\\times{\\Bbb P}^{n_k-1}$, where $n_i$'s are powers of a fixed prime number $p$, then the rank of an exceptional object on $X$ is congruent to $\\pm 1$ modulo $p$."}
{"category": "Math", "title": "Compactification for essentially finite-type maps", "abstract": "We show that any separated essentially finite-type map $f$ of noetherian schemes globally factors as $f = hi$ where $i$ is an injective localization map and $h$ a separated finite-type map. In particular, via Nagata's compactification theorem, $h$ can be chosen to be proper. We apply these results to Grothendieck duality. We also obtain other factorization results and provide essentialized versions of many general results such as Zariski's Main Theorem, Chow's Lemma, and blow-up descriptions of birational maps."}
{"category": "Math", "title": "Proving a manifold to be hyperbolic once it has been approximated to be so", "abstract": "The computer program SnapPea can approximate whether or not a three manifold whose boundary consists of tori has a complete hyperbolic structure, but it can not prove conclusively that this is so. This article provides a method for proving that such a manifold has a complete hyperbolic structure based on the approximations of SNAP, a program that includes the functionality of SnapPea plus other features. The approximation is done by triangulating the manifold, identifying consistency and completeness equations with respect to this triangulation, and then trying to solve the system of equations using Newton's Method. This produces an approximate, not actual solution. The method developed here uses Kantorovich's theorem to prove that an actual solution exists, thereby assuring that the manifold has a complete hyperbolic structure. Using this, we can definitively prove that every manifold in the SnapPea cusped census has a complete hyperbolic structure."}
{"category": "Math", "title": "Linear algebra meets Lie algebra: the Kostant-Wallach theory", "abstract": "In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra."}
{"category": "Math", "title": "Schatten-von Neumann properties in the Weyl calculus", "abstract": "Let $\\Op_t(a)$, for $t\\in \\mathbf R$, be the pseudo-differential operator $$ f(x) \\mapsto (2\\pi)^{-n}\\iint a((1-t)x+ty,\\xi)f(y)e^{i\\scal {x-y}\\xi} dyd\\xi $$ and let $\\mathscr I_p$ be the set of Schatten-von Neumann operators of order $p\\in [1,\\infty ]$ on $L^2$. We are especially concerned with the Weyl case (i.{}e. when $t=1/2$). We prove that if $m$ and $g$ are appropriate metrics and weight functions respectively, $h_g$ is the Planck's function, $h_g^{k/2}m\\in L^p$ for some $k\\ge 0$ and $a\\in S(m,g)$, then $\\Op_t(a)\\in \\mathscr I_p$, iff $a\\in L^p$. Consequently, if $0\\le \\delta <\\rho \\le 1$ and $a\\in S^r_{\\rho ,\\delta}$, then $\\Op_t(a)$ is bounded on $L^2$, iff $a\\in L^\\infty$."}
{"category": "Math", "title": "Large deviations for random walk in a random environment", "abstract": "In this work, we study the large deviation properties of random walk in a random environment on $\\mathbb{Z}^d$ with $d\\geq1$. We start with the quenched case, take the point of view of the particle, and prove the large deviation principle (LDP) for the pair empirical measure of the environment Markov chain. By an appropriate contraction, we deduce the quenched LDP for the mean velocity of the particle and obtain a variational formula for the corresponding rate function $I_q$. We propose an Ansatz for the minimizer of this formula. This Ansatz is easily verified when $d=1$. In his 2003 paper, Varadhan proves the averaged LDP for the mean velocity and gives a variational formula for the corresponding rate function $I_a$. Under the non-nestling assumption (resp. Kalikow's condition), we show that $I_a$ is strictly convex and analytic on a non-empty open set $\\mathcal{A}$, and that the true velocity $\\xi_o$ is an element (resp. in the closure) of $\\mathcal{A}$. We then identify the minimizer of Varadhan's variational formula at any $\\xi\\in\\mathcal{A}$. For walks in high dimension, we believe that $I_a$ and $I_q$ agree on a set with non-empty interior. We prove this for space-time walks when the dimension is at least 3+1. In the latter case, we show that the cheapest way to condition the asymptotic mean velocity of the particle to be equal to any $\\xi$ close to $\\xi_o$ is to tilt the transition kernel of the environment Markov chain via a Doob $h$-transform."}
{"category": "Math", "title": "On the notion of Cohen-Macaulayness for non Noetherian rings", "abstract": "There exist many characterizations of Noetherian Cohen-Macaulay rings in the literature. These characterizations do not remain equivalent if we drop the Noetherian assumption. The aim of this paper is to provide some comparisons between some of these characterizations in non Noetherian case. Toward solving a conjecture posed by Glaz, we give a generalization of the Hochster-Eagon result on Cohen-Macaulayness of invariant rings, in the context of non Noetherian rings."}
{"category": "Math", "title": "Multicomplexes and spectral sequences", "abstract": "In this note we present some algebraic examples of multicomplexes whose differentials differ from those in the spectral sequences associated to the multicomplexes. The motivation for constructing examples showing the algebraic distinction between a multicomplex and its associated spectral sequence comes from the author's work on Morse-Bott homology with A. Banyaga."}
{"category": "Math", "title": "The ring of projective invariants of eight points on the line via representation theory", "abstract": "The ring of projective invariants of eight ordered points on the line is a quotient of the polynomial ring on V, where V is a fourteen-dimensional representation of S_8, by an ideal I_8, so the modular fivefold (P^1)^8 // GL(2) is Proj(Sym* (V)/I_8). We show that there is a unique cubic hypersurface S in PV whose equation s is skew-invariant, and that the singular locus of S is the modular fivefold. In particular, over Z[1/3], the modular fivefold is cut out by the 14 partial derivatives of s. Better: these equations generate I_8. In characteristic 3, the cubic s is needed to generate the ideal. The existence of such a cubic was predicted by Dolgachev. Over Q, we recover the 14 quadrics found by computer calculation by Koike, and our approach yields a conceptual representation-theoretic description of the presentation. Additionally we find the graded Betti numbers of a minimal free resolution in any characteristic. The proof over Q is by pure thought, using Lie theory and commutative algebra. Over Z, the assistance of a computer was necessary. This result will be used as the base case describing the equations of the moduli space of an arbitrary number of points on P^1, with arbitrary weighting, in a later paper, completing the program of our previous paper (Duke Math. J.). The modular fivefold, and corresponding ring, are known to have a number of special incarnations, due to Deligne-Mostow, Kondo, and Freitag-Salvati Manni, for example as ball quotients or ring of modular forms respectively."}
{"category": "Math", "title": "Smooth dependence on parameters of solution of cohomology equations over Anosov systems and applications to cohomology equations on diffeomorphism groups", "abstract": "We consider the dependence on parameters of the solutions of cohomology equations over Anosov diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for the solutions of equations taking value in diffeomorphism groups. These results are motivated by applications to rigidity theory, dynamical systems, and geometry. In particular, in the context of diffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism of a compact manifold $M$. Suppose that $\\eta \\in C^{\\reg}(M,\\Diff^r(N))$ for a compact manifold $N$, $k,r \\in \\N$, $r \\geq 1$, and $0 < \\alpha \\leq \\Lip$. We show that if there exists a $\\varphi\\in C^{\\reg}(M,\\Diff^1(N))$ solving \\begin{equation*} \\varphi_{f(x)} = \\eta_x \\circ \\varphi_x \\end{equation*} then in fact $\\varphi \\in C^{\\reg}(M,\\Diff^r(N))$."}
{"category": "Math", "title": "The de Rham comparison theorem for Deligne-Mumford stacks", "abstract": "The de Rham comparison theorem for varieties, first proved by Faltings, gives the de Rham cohomology of a variety in terms of its p-adic etale cohomology. We extend this theorem to proper, smooth Deligne-Mumford stacks. Two approaches are given, which both in the end reduce the problem to the already known comparison theorem for varieties. The first approach employs the formalism of Weil cohomologies. Unfortunately, this does not result in a complete proof of the comparison theorem, as the author was unable to prove the required compatibility between intersections and cup products. Nevertheless, the author thought the results that are obtained and the method that is suggested interesting enough to include them. The second approach uses simplicial methods and is based on versions of the comparison theorem by Kisin and Tsuji. The latter approach does result in a complete proof of the extended comparison theorem."}
{"category": "Math", "title": "About the choice of a basis in Kedlaya's algorithm", "abstract": "Kedlaya's algorithm (Kedlaya, J. Ramanujan Math. Soc 16, 2001) can be used to count the points of arbitrary hyperelliptic curves over finite fields of characteristic p, where p is an odd prime. The algorithm uses the cohomology of a p-adic lift of the curve. The Frobenius morphism of the curve induces an automorphism of this cohomological space. The key step of the algorithm is to determine this automorphism with a sufficiently high p-adic precision: it is given in the form of a matrix with respect to a certain basis. Edixhoven has found a basis that has the property that the coefficients of the matrix are p-adically integral. This allows a smaller required precision, because a (semi-linear) power of this matrix must be computed up to some given precision. This text describes Edixhoven's basis and provides a proof of the fact that the basis is suitable."}
{"category": "Math", "title": "Obstructing Sliceness in a Family of Montesinos Knots", "abstract": "Using Gauge theoretical techniques employed by Lisca for 2-bridge knots and by Greene-Jabuka for 3-stranded pretzel knots, we show that no member of the family of Montesinos knots M(0;[m_1+1,n_1+2],[m_2+1,n_2+2],q), with certain restrictions on m_i, n_i, and q, can be (smoothly) slice. Our techniques use Donaldson's diagonalization theorem and the fact that the 2-fold covers of Montisinos knots bound plumbing 4-manifolds, many of which are negative definite. Some of our examples include knots with signature 0 and square determinant."}
{"category": "Math", "title": "Weak Approximation for General Degree Two del Pezzo Surfaces", "abstract": "We address weak approximation for certain del Pezzo surfaces defined over the function field of a curve. We study the rational connectivity of the smooth locus of degree two del Pezzo surfaces with two A1 singularities in order to prove weak approximation for degree two del Pezzo surfaces with square-free discriminant."}
{"category": "Math", "title": "Necessary and Sufficient Conditions for Success of the Nuclear Norm Heuristic for Rank Minimization", "abstract": "Minimizing the rank of a matrix subject to constraints is a challenging problem that arises in many applications in control theory, machine learning, and discrete geometry. This class of optimization problems, known as rank minimization, is NP-HARD, and for most practical problems there are no efficient algorithms that yield exact solutions. A popular heuristic algorithm replaces the rank function with the nuclear norm--equal to the sum of the singular values--of the decision variable. In this paper, we provide a necessary and sufficient condition that quantifies when this heuristic successfully finds the minimum rank solution of a linear constraint set. We additionally provide a probability distribution over instances of the affine rank minimization problem such that instances sampled from this distribution satisfy our conditions for success with overwhelming probability provided the number of constraints is appropriately large. Finally, we give empirical evidence that these probabilistic bounds provide accurate predictions of the heuristic's performance in non-asymptotic scenarios."}
{"category": "Math", "title": "Constructing geometrically infinite groups on boundaries of deformation spaces", "abstract": "Consider a geometrically finite Kleinian group $G$ without parabolic or elliptic elements, with its Kleinian manifold $M=(\\H^3\\cup \\Omega_G)/G$. Suppose that for each boundary component of $M$, either a maximal and connected measured lamination in the Masur domain or a marked conformal structure is given. In this setting, we shall prove that there is an algebraic limit $\\Gamma$ of quasi-conformal deformations of $G$ such that there is a homeomorphism $h$ from $\\mathrm{Int} M$ to $\\H^3/\\Gamma$ compatible with the natural isomorphism from $G$ to $\\Gamma$, the given laminations are unrealisable in $\\H^3/\\Gamma$, and the given conformal structures are pushed forward by $h$ to those of $\\H^3/\\Gamma$. Based on this theorem and its proof, in the subsequent paper, the Bers-Thurston conjecture, saying that every finitely generated Kleinian group is an algebraic limit of quasi-conformal deformations of minimally parabolic geometrically finite group, is proved using recent solutions of Marden's conjecture by Agol, Calegari-Gabai, and the ending lamination conjecture by Minsky collaborating with Brock, Canary and Masur."}
{"category": "Math", "title": "Combinatorics and topology of straightening maps I: compactness and bijectivity", "abstract": "We study the parameter space structure of degree $d \\ge 3$ one complex variable polynomials as dynamical systems acting on $\\C$. We introduce and study {\\it straightening maps}. These maps are a natural higher degree generalization of the ones introduced by Douady and Hubbard to prove the existence of small copies of the Mandelbrot set inside itself. We establish that straightening maps are always injective and that their image contains all the corresponding hyperbolic systems. Also, we characterize straightening maps with compact domain. Moreover, we give two classes of bijective straightening maps. The first produces an infinite collection of embedded copies of the $(d-1)$-fold product of the Mandelbrot set in the connectedness locus of degree $d \\ge 3$. The second produces an infinite collection of full families of quadratic connected filled Julia sets in the cubic connectedness locus, such that each filled Julia set is quasiconformally embedded."}
{"category": "Math", "title": "Appell Polynomials and Their Zero Attractors", "abstract": "A polynomial family $\\{p_n(x)\\}$ is Appell if it is given by $\\frac{e^{xt}}{g(t)} = \\sum_{n=0}^\\infty p_n(x)t^n$ or, equivalently, $p_n'(x) = p_{n-1}(x)$. If $g(t)$ is an entire function, $g(0)\\neq 0$, with at least one zero, the asymptotics of linearly scaled polynomials $\\{p_n(nx)\\}$ are described by means of finitely zeros of $g$, including those of minimal modulus. As a consequence, we determine the limiting behavior of their zeros as well as their density. The techniques and results extend our earlier work on Euler polynomials."}
{"category": "Math", "title": "The Atiyah--Segal completion theorem in twisted K-theory", "abstract": "In this note we prove the analogue of the Atiyah-Segal completion theorem for equivariant twisted K-theory in the setting of an arbitrary compact Lie group G and an arbitrary twisting of the usually considered type. The theorem generalizes a result by C. Dwyer, who has proven the theorem for finite G and twistings of a more restricted type. While versions of the general result have been known to experts, to our knowledge no proof appears in the current literature. Our goal is to fill in this gap. The proof we give proceeds in two stages. We first prove the theorem in the case of a twisting arising from a graded central extension of G, following the Adams-Haeberly-Jackowski-May proof of the classical Atiyah-Segal completion theorem. After establishing that the theorem holds for this special class of twistings, we then deduce the general theorem by a Mayer-Vietoris argument."}
{"category": "Math", "title": "MultiResolution Anomaly Detection Method for Long Range Dependent Time Series", "abstract": "Driven by network intrusion detection, we propose a MultiResolution Anomaly Detection (MRAD) method, which effectively utilizes the multiscale properties of Internet features and network anomalies. In this paper, several theoretical properties of the MRAD method are explored. A major new result is the mathematical formulation of the notion that a two-scaled MRAD method has larger power than the average power of the detection method based on the given two scales. Test threshold is also developed. Comparisons between MRAD method and other classical outlier detectors in time series are reported as well."}
{"category": "Math", "title": "Strong uniqueness for a class of singular SDEs for catalytic branching diffusions", "abstract": "A new result for the strong uniqueness for catalytic branching diffusions is established, which improves the work of Dawson, D.A.; Fleischmann, K.; Xiong, J.[Strong uniqueness for cyclically symbiotic branching diffusions. Statist. Probab. Lett. 73, no. 3, 251--257 (2005)]."}
{"category": "Math", "title": "Submanifolds with Biharmonic Gauss Map", "abstract": "We generalize the Ruh-Vilms problem by characterizing the submanifolds in Euclidean spaces with proper biharmonic Gauss map and we construct examples of such hypersurfaces."}
{"category": "Math", "title": "Semistar-Krull and Valuative Dimension of Integral Domains", "abstract": "Given a stable semistar operation of finite type $\\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $\\star[X]$ on the polynomial ring $D[X]$, such that, if $n:=\\star$-$\\dim(D)$, then $n+1\\leq \\star[X]\\text{-}\\dim(D[X])\\leq 2n+1$. We also establish that if $D$ is a $\\star$-Noetherian domain or is a Pr\\\"{u}fer $\\star$-multiplication domain, then $\\star[X]\\text{-}\\dim(D[X])=\\star\\text{-}\\dim(D)+1$. Moreover we define the semistar valuative dimension of the domain $D$, denoted by $\\star$-$\\dim_v(D)$, to be the maximal rank of the $\\star$-valuation overrings of $D$. We show that $\\star$-$\\dim_v(D)=n$ if and only if $\\star[X_1,...,X_n]$-$\\dim_v(D[X_1,...,X_n])=2n$, and that if $\\star$-$\\dim_v(D)<\\infty$ then $\\star[X]$-$\\dim_v(D[X])=\\star$-$\\dim_v(D)+1$. In general $\\star$-$\\dim(D)\\leq\\star$-$\\dim_v(D)$ and equality holds if $D$ is a $\\star$-Noetherian domain or is a Pr\\\"{u}fer $\\star$-multiplication domain. We define the $\\star$-Jaffard domains as domains $D$ such that $\\star$-$\\dim(D)<\\infty$ and $\\star$-$\\dim(D)=\\star$-$\\dim_v(D)$. As an application, $\\star$-quasi-Pr\\\"{u}fer domains are characterized as domains $D$ such that each $(\\star,\\star')$-linked overring $T$ of $D$, is a $\\star'$-Jaffard domain, where $\\star'$ is a stable semistar operation of finite type on $T$. As a consequence of this result we obtain that a Krull domain $D$, must be a $w_D$-Jaffard domain."}
{"category": "Math", "title": "Minimal Prime Ideals and Semistar Operations", "abstract": "Let $R$ be a commutative integral domain and let $\\star$ be a semistar operation of finite type on $R$, and $I$ be a quasi-$\\star$-ideal of $R$. We show that, if every minimal prime ideal of $I$ is the radical of a $\\star$-finite ideal, then the set $\\Min(I)$ of minimal prime ideals over $I$ is finite."}
{"category": "Math", "title": "Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems", "abstract": "In this paper we discuss the question of how to decide when a general chemical reaction system is incapable of admitting multiple equilibria, regardless of parameter values such as reaction rate constants, and regardless of the type of chemical kinetics, such as mass-action kinetics, Michaelis-Menten kinetics, etc. Our results relate previously described linear algebraic and graph-theoretic conditions for injectivity of chemical reaction systems. After developing a translation between the two formalisms, we show that a graph-theoretic test developed earlier in the context of systems with mass action kinetics, can be applied to reaction systems with arbitrary kinetics. The test, which is easy to implement algorithmically, and can often be decided without the need for any computation, rules out the possibility of multiple equilibria for the systems in question."}
{"category": "Math", "title": "Well-posedness of the transport equation by stochastic perturbation", "abstract": "We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type."}
{"category": "Math", "title": "A class of iterative methods for solving nonlinear operator equations", "abstract": "The article deals with gradient-like iterative methods for solving nonlinear operator equations on Hilbert and Banach spaces. The authors formulate a general principle of studying such methods. This principle allows to formulate simple conditions of convergence of the method under consideration, to estimate the rate of this convergence and to give effective a priori and aposteriori error estimates in terms of a scalar function that is constructed on the base of estimates for properties of invertibility and smoothness of linearizations of the left-hand side of the equations under study. The principle is applicable for analysis of such classical methods as method of minimal residuals, method of steepest descent, method of minimal errors and others. The main results are obtained for operator equations on Hilbert spaces and Banach spaces with a special property, that is called Bynum property."}
{"category": "Math", "title": "Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2", "abstract": "The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my previous papers cited in the present paper as [K1] and [K2], and for the space SU(3)/SO(3) in [K3], Section 6, we now solve the classification for the remaining irreducible Riemannian symmetric spaces of rank 2 and compact type: SU(6)/Sp(3), SO(10)/U(5), E6/(U(1)*Spin(10)), E6/F4, G2/SO(4), SU(3), Sp(2) and G2. Similarly as for the spaces already investigated in the earlier papers, it turns out that for many of the spaces investigated here, the earlier classification of the maximal totally geodesic submanifolds of Riemannian symmetric spaces by Chen and Nagano ([CN], Paragraph 9) is incomplete. In particular, in the spaces Sp(2), G2/SO(4) and G2, there exist maximal totally geodesic submanifolds, isometric to 2- or 3-dimensional spheres, which have a \"skew\" position in the ambient space in the sense that their geodesic diameter is strictly larger than the geodesic diameter of the ambient space. They are all missing from [CN]."}
{"category": "Math", "title": "A complete lift for semisprays", "abstract": "In this paper, we define a complete lift for semisprays. If $S$ is a semispray on a manifold $M$, its complete lift is a new semispray $S^c$ on $TM$. The motivation for this lift is two-fold: First, geodesics for $S^c$ correspond to the Jacobi fields for $S$, and second, this complete lift generalizes and unifies previously known complete lifts for Riemannian metrics, affine connections, and regular Lagrangians. When $S$ is a spray, we prove that the projective geometry of $S^c$ uniquely determines $S$. We also study how symmetries and constants of motions for $S$ lift into symmetries and constants of motions for $S^c$."}
{"category": "Math", "title": "Reliability analysis of semicoherent systems through their lattice polynomial descriptions", "abstract": "A semicoherent system can be described by its structure function or, equivalently, by a lattice polynomial function expressing the system lifetime in terms of the component lifetimes. In this paper we point out the parallelism between the two descriptions and use the natural connection of lattice polynomial functions and relevant random events to collect exact formulas for the system reliability. We also discuss the equivalence between calculating the reliability of semicoherent systems and calculating the distribution function of a lattice polynomial function of random variables."}
{"category": "Math", "title": "The warping degree of a knot diagram", "abstract": "For an oriented knot diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain the monotone diagram from D in the usual way. We show that d(D) + d(-D) + 1 is less than or equal to the crossing number of D. Moreover the equality holds if and only if D is an alternating diagram."}
{"category": "Math", "title": "Remarques sur un article r'ecent de B. Poonen", "abstract": "B. Poonen recently produced smooth threefolds over a number field which do not have a rational point but have no Brauer-Manin obstruction even after descent to a finite 'etale cover. In this note I show that the varieties he produces have zero-cycles of degree 1. ----- B. Poonen a r'ecemment exhib'e des exemples de vari'et'es projectives et lisses de dimension 3 sur un corps de nombres qui n'ont pas de point rationnel et pour lesquelles il n'y a pas d'obstruction de Brauer-Manin apr`es rev^etement fini 'etale. Dans cette note, je montre que les vari'et'es qu'il construit poss`edent des z'ero-cycles de degr'e 1."}
{"category": "Math", "title": "On the flux of pseudo-Anosov homeomorphisms", "abstract": "We exhibit a pseudo-Anosov homeomorphism of a surface S which acts trivially on the first homology group of S and whose flux is non zero"}
{"category": "Math", "title": "Proteus mirabilis swarm-colony development with drift", "abstract": "We prove a global existence result for a model describing the swarming phenomenon of the bacterium Proteus mirabilis. The model consists of an ordinary differential equation coupled with an age-structured equation involving nonlinear degenerate diffusion and an additional drift term."}
{"category": "Math", "title": "General Existence Results for Reflected BSDE and BSDE", "abstract": "In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator $fds + gdA_s$ is continuous with general growth with respect to the variable $y$ and stochastic quadratic growth with respect to the variable $z$. We deal with the case of a bounded terminal condition $\\xi$ and a bounded barrier $L$ as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in \\cite{EH}. The work is suggested by the interest the results might have in finance, control and game theory."}
{"category": "Math", "title": "Hyperbolicity of geometric orbifolds", "abstract": "We study complex hyperbolicity in the setting of geometric orbifolds introduced by F. Campana. Generalizing classical methods to this context, we obtain degeneracy statements for entire curves with ramification in situations where no Second Main Theorem is known from value distribution theory."}
{"category": "Math", "title": "Kuelshammer ideals and the scalar problem for blocks with dihedral defect groups", "abstract": "In by now classical work, K. Erdmann classified blocks of finite groups with dihedral defect groups (and more generally algebras of dihedral type) up to Morita equivalence. In the explicit description by quivers and relations of such algebras with two simple modules, several subtle problems about scalars occurring in relations remained unresolved. In particular, for the dihedral case it is a longstanding open question whether blocks of finite groups can occur for both possible scalars 0 and 1. In this article, using Kuelshammer ideals (a.k.a. generalized Reynolds ideals), we provide the first examples of blocks where the scalar is 1, thus answering the above question to the affirmative. Our examples are the principal blocks of PGL_2(F_q), the projective general linear group of 2x2-matrices with entries in the finite field F_q, where q=p^n\\equiv \\pm 1 mod 8, with p an odd prime number."}
{"category": "Math", "title": "Some notes on trees and paths", "abstract": "These notes cover background material on trees which are used in the paper `On uniqueness of the signature of a path of variation and the reduced path group'."}
{"category": "Math", "title": "Desperately seeking mathematical truth", "abstract": "This article discusses epistemological problems in the philosophy of mathematics and issues concerning the reliability of the mathematical literature."}
{"category": "Math", "title": "Euclidean components for a class of self-injective algebras", "abstract": "We determine the length of composition series of projective modules of G-transitive algebras with an Auslander-Reiten component of Euclidean tree class. Furthermore we show that modules with certain length of composition series are periodic. We apply these results to G-transitive blocks of the Universal enveloping of restricted p-Lie algebras and prove that G-transitive principal blocks only allow components with Euclidean tree class if p=2. Finally we deduce conditions for a smash product of a local basic algebra with a commutative semi-simple group algebra to have components with Euclidean tree class, depending on the components of the Auslander-Reiten quiver of the basic algebra. We also develop some properties of the bilinear form dim Hom_A(-,-) for the representation ring G(A) of any finite-dimensional algebra A."}
{"category": "Math", "title": "Vari'et'es presque rationnelles, leurs points rationnels et leurs d'eg'en'erescences", "abstract": "This survey, which contains very few proofs, addresses the general question: Over a given type of field, is there a natural class of varieties which automatically have a rational point? Fields under consideration here include: finite fields, p-adic fields, function fields in one or two variables over an algebraically closed field. Classical answers are given by the Chevalley-Warning theorem and by Tsen's theorem. More general answers were provided by a theorem of Graber, Harris and Starr and by a theorem of Esnault. The latter results apply to rationally connected varieties. We discuss these varieties from various angles : weak approximation, R-equivalence, Chow group of zero-cycles. Ongoing work on `rationally simply connected' varieties over function fields in two variables is also mentioned. A common thread in this report is the study of the special fibre of a scheme over a discrete valuation ring: if the generic fibre has a simple geometry, what does it imply for the special fibre?"}
{"category": "Math", "title": "Integral models of unitary representations of current groups with values in semidirect products", "abstract": "We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup $P_0$ and a one-parameter group ${\\mathbb R {}}^*_+=\\{r:r>0\\}$ of automorphisms of $P_0$. This construction is determined by a a faithful unitary representation of $P_0$ (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as $r\\to 0$. We apply this construction to the groups of currents of the maximal parabolic subgroups of the groups of motions of the $n$-dimensional real and complex Lobachevsky spaces. The obtained representations of the groups of parabolic currents can be uniquely extended to the groups of currents with values in the semisimple groups O(n,1) and U(n,1). This gives a new description of the representations of the groups of currents of these groups constructed in the 70s and realized in the Fock space. The key role in our construction is played by the so-called special representation of the parabolic subgroup $P$ and the remarkable $\\sigma$-finite measure (Lebesgue measure) $\\mathcal L$ in the space of distributions."}
{"category": "Math", "title": "Degeneracy and decomposability in abelian crossed products", "abstract": "In this paper we study the relationship between degeneracy and decomposability in abelian crossed products. In particular we construct an indecomposable abelian crossed product division algebra of exponent $p$ and index $p^2$ for $p$ an odd prime. The algebra we construct is generic in the sense of Amitsur and Saltman and has the property that its underlying abelian crossed product is a decomposable division algebra defined by a non-degenerate matrix. This algebra gives an example of an indecomposable generic abelian crossed product which is shown to be indecomposable without using torsion in the Chow group of the corresponding Severi-Brauer variety as was needed in [Karpenko, Codimension 2 cycles on Severi-Brauer varieites (1998)] and [McKinnie, Indecomposable $p$-algebras and Galois subfields in generic abelian crossed products (2008)]. It also gives an example of a Brauer class which is in Tignol's Dec group with respect to one abelian maximal subfield, but not in the Dec group with respect to another."}
{"category": "Math", "title": "Symmetry-preserving observers for some water tank problems: theory and application to a shallow water model", "abstract": "In this paper we consider a tank containing fluid and we want to estimate the horizontal currents when the fluid surface height is measured. The fluid motion is described by shallow water equations in two horizontal dimensions. We build a simple non-linear observer which takes advantage of the symmetries of fluid dynamics laws. As a result its structure is based on convolutions with smooth isotropic kernels, and the observer is remarkably robust to noise. We prove the convergence of the observer around a steady-state. In numerical applications local exponential convergence is expected. The observer is also applied to the problem of predicting the ocean circulation. Realistic simulations illustrate the relevance of the approach compared with some standard oceanography techniques."}
{"category": "Math", "title": "The Zeta Functions of Complexes from $\\PGL(3)$: a Representation-theoretic Approach", "abstract": "The zeta function attached to a finite complex $X_\\Gamma$ arising from the Bruhat-Tits building for $\\PGL_3(F)$ was studied in \\cite{KL}, where a closed form expression was obtained by a combinatorial argument. This identity can be rephrased using operators on vertices, edges, and directed chambers of $X_\\Gamma$. In this paper we reprove the zeta identity from a different aspect by analyzing the eigenvalues of these operators using representation theory. As a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of the eigenvalues of the operators on vertices, edges, and directed chambers, respectively."}
{"category": "Math", "title": "On the unfolding of simple closed curves", "abstract": "I show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve. This result is obtained by approximating the curve with polygons and invoking the result of Connelly, Demaine, and Rote that such a motion exists for polygons. I also formulate a generalization of their program, thereby making steps toward a fully continuous proof of the result. To facilitate this, I generalize two of the primary tools used in their program: the Farkas Lemma of linear programming to Banach spaces and the Maxwell-Cremona Theorem of rigidity theory to apply to stresses represented by measures on the plane."}
{"category": "Math", "title": "On a family of tridiagonal matrices", "abstract": "We show that certain integral positive definite symmetric tridiagonal matrices of determinant $n$ are in one to one correspondence with elements of $(\\mathbb Z/n\\mathbb Z)^*$. We study some properties of this correspondence. In a somewhat unrelated second part we discuss a construction which associates a sequence of integral polytopes to every integral symmetric matrix."}
{"category": "Math", "title": "An Application of Topological Multiple Recurrence to Tiling", "abstract": "We show that given any tiling of Euclidean space, any geometric patterns of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated version of the pattern. The rather simple proof uses Furstenberg's topological multiple recurrence theorem."}
{"category": "Math", "title": "Multivariate Jacobi and Laguerre polynomials, infinite-dimensional extensions, and their probabilistic connections with multivariate Hahn and Meixner polynomials", "abstract": "Multivariate versions of classical orthogonal polynomials such as Jacobi, Hahn, Laguerre and Meixner are reviewed and their connection explored by adopting a probabilistic approach. Hahn and Meixner polynomials are interpreted as posterior mixtures of Jacobi and Laguerre polynomials, respectively. By using known properties of gamma point processes and related transformations, a new infinite-dimensional version of Jacobi polynomials is constructed with respect to the size-biased version of the Poisson--Dirichlet weight measure and to the law of the gamma point process from which it is derived."}
{"category": "Math", "title": "Isoparametric and Dupin Hypersurfaces", "abstract": "A hypersurface $M^{n-1}$ in a real space-form ${\\bf R}^n$, $S^n$ or $H^n$ is isoparametric if it has constant principal curvatures. For ${\\bf R}^n$ and $H^n$, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Elie Cartan showed in a series of four papers in 1938-1940, the subject is much deeper and more complex for hypersurfaces in the sphere $S^n$. A hypersurface $M^{n-1}$ in a real space-form is proper Dupin if the number $g$ of distinct principal curvatures is constant on $M^{n-1}$, and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important generalization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field."}
{"category": "Math", "title": "The Relative Burnside Kernel - The Elementary Abelian Case", "abstract": "We give a conjectural description for the kernel of the map assigning to each finite $\\mathbb Z_p$-free $G\\times\\mathbb Z_p$-set its rational permutation module where G is a finite p-group. We prove that this conjecture is true when G is an elementary abelian p-group or a cyclic p-group."}
{"category": "Math", "title": "Topological Cohen-Macaulay criteria for monomial ideals", "abstract": "Scattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen-Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial ideals. The purpose of this survey is to gather the developments into one location, with self-contained proofs, including direct combinatorial topological connections between them."}
{"category": "Math", "title": "The spherical Hecke algebra for affine Kac-Moody groups I", "abstract": "We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for these algebras, relating it to integrable representations of the Langlands dual affine Kac-Moody group. In the next publication we shall use these results to define and study the notion of Hecke eigenfunction for the group $G_{\\aff}$"}
{"category": "Math", "title": "On Algebraic Solutions to Painleve VI", "abstract": "We announce some results which might bring a new insight into the classification of algebraic solutions to the sixth Painleve equation. The main results consist of the rationality of parameters, trigonometric Diophantine conditions, and what the author calls the Tetrahedral Theorem regarding the absence of algebraic solutions in certain situations. The method is based on fruitful interactions between the moduli theoretical formulation of Painleve VI and dynamics on character varieties via the Riemann-Hilbert correspondence."}
{"category": "Math", "title": "A Simple Proof On Poincar\\'e Conjecture", "abstract": "We give a simple proof on the Poincar\\'e's conjecture which states that every compact smooth $3-$manifold which is homotopically equivalent to $S^3$ is diffeomorphic to $S^3$."}
{"category": "Math", "title": "Dynamics of Connected Rigid Bodies in a Perfect Fluid", "abstract": "This paper presents an analytical model and a geometric numerical integrator for a system of rigid bodies connected by ball joints, immersed in an irrotational and incompressible fluid. The rigid bodies can translate and rotate in three-dimensional space, and each joint has three rotational degrees of freedom. This model characterizes the qualitative behavior of three-dimensional fish locomotion. A geometric numerical integrator, refereed to as a Lie group variational integrator, preserves Hamiltonian structures of the presented model and its Lie group configuration manifold. These properties are illustrated by a numerical simulation for a system of three connected rigid bodies."}
{"category": "Math", "title": "On satellites in arbitrary categories", "abstract": "We generalize the definition of satellites with respect to presheaves (and copresheaves) with trace in the sense of Inassaridze; a presheaf with trace is replaced by a graph with a pair of diagrams defined on it. We show that the right satellite functor is left adjoint to the left satellite functor, and that a functor having a right (left) adjoint preserves right (left) satellites. In particular cases the construction of satellites is given."}
{"category": "Math", "title": "A characteristic number of bundles determined by mass linear pairs", "abstract": "Let $\\Delta$ be a Delzant polytope in ${\\mathbb R}^n$ and ${\\bf b}\\in{\\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$ determined by the pair $(\\Delta, {\\bf b})$. We prove the equivalence between the fact that $(\\Delta, {\\bf b})$ is a mass linear pair (D. McDuff, S. Tolman, {\\em Polytopes with mass linear functions, part I.} {\\tt arXiv:0807.0900 [math.SG]}) and the vanishing of a characteristic number of $E$ in the following cases: When $\\Delta$ is a $\\Delta_{n-1}$ bundle over $\\Delta_1$; when $\\Delta$ is the polytope associated with the one point blow up of ${\\mathbb C}P^n$; and when $\\Delta$ is the polytope associated with a Hirzebruch surface."}
{"category": "Math", "title": "Global well-posedness and scattering for the fourth order nonlinear Schr\\\"{o}dinger equations with small data", "abstract": "For $n\\geq 3$, we study the Cauchy problem for the fourth order nonlinear Schr\\\"{o}dinger equations, for which the existence of the scattering operators and the global well-posedness of solutions with small data in Besov spaces $B^{s}_{2,1}(\\R^n)$ are obtained. In one spatial dimension, we get the global well-posedness result with small data in the critical homogeneous Besov spaces $\\dot{B}^{s}_{2,1}$. As a by-product, the existence of the scattering operators with small data is also obtained. In order to show these results, the global version of the estimates for the maximal functions and the local smoothing effects on the fourth order Schr\\\"{o}dinger semi-groups are established."}
{"category": "Math", "title": "A degenerate kernel method for eigenvalue problems of a class of non-compact operators", "abstract": "We consider the eigenvalue problem of certain kind of non-compact linear operators given as the sum of a multiplication and a kernel operator. A degenerate kernel method is used to approximate isolated eigenvalues. It is shown that entries of the corresponding matrix of this method can be evaluated exactly. The convergence of the method is proved; it is proved that the convergence rate is $O(h)$. By some numerical examples, we confirm the results."}
{"category": "Math", "title": "Q-fundamental surfaces in lens spaces", "abstract": "We determine all the Q-fundamental surfaces in $(p,1)$-lens spaces and $(p,2)$-lens spaces with respect to natural triangulations with $p$ tetrahedra. For general $(p,q)$-lens spaces, we give an upper bound for elements of vectors which represent Q-fundamental surfaces with no quadrilateral normal disks disjoint from the core circles of lens spaces. We also give some examples of non-orientable closed surfaces which are Q-fundamental surfaces with such quadrilateral normal disks."}
{"category": "Math", "title": "Tight Lagrangian surfaces in $S^2 \\times S^2$", "abstract": "We determine all tight Lagrangian surfaces in $S^2 \\times S^2$. In particular, globally tight Lagrangian surfaces in $S^2 \\times S^2$ are nothing but real forms."}
{"category": "Math", "title": "Recursive formulas for Welschinger invariants of the projective plane", "abstract": "Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we prove a Caporaso-Harris type formula which allows one to compute Welschinger invariants for configurations of points with any number of complex conjugated points."}
{"category": "Math", "title": "Inverse problem for a parabolic system with two components by measurements of one component", "abstract": "We consider a $2\\times 2$ system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse problems of determining some or all of the coefficients by observations in an arbitrary subdomain over a time interval of only one component and data of two components at a fixed positive time $\\theta$ over the whole spatial domain. The main results are Lipschitz stability estimates for the inverse problems. For the Lipschitz stability, we have to assume some non-degeneracy condition at $\\theta$ for the two components and for it, we can approximately control the two components of the $2 \\times 2$ system by inputs to only one component. Such approximate controllability is proved also by our new Carleman estimate. Finally we establish a Carleman estimate for a $3\\times 3$ system for parabolic equations with coupling of zeroth-order terms by one component to show the corresponding approximate controllability with a control to one component."}
{"category": "Math", "title": "On the Equicontinuity Region of Discrete Subgroups of PU(1,n)", "abstract": "Let $ G $ be a discrete subgroup of PU(1,n). Then $ G $ acts on $\\mathbb {P}^n_\\mathbb C$ preserving the unit ball $\\mathbb {H}^n_\\mathbb {C}$, where it acts by isometries with respect to the Bergman metric. In this work we determine the equicontinuty region $Eq(G)$ of $G$ in $\\mathbb P^n_{\\mathbb C}$: It is the complement of the union of all complex projective hyperplanes in $\\mathbb {P}^n_{\\mathbb C}$ which are tangent to $\\partial \\mathbb {H}^n_\\mathbb {C}$ at points in the Chen-Greenberg limit set $\\Lambda_{CG}(G )$, a closed $G$-invariant subset of $\\partial \\mathbb {H}^n_\\mathbb {C}$, which is minimal for non-elementary groups. We also prove that the action on $Eq(G)$ is discontinuous."}
{"category": "Math", "title": "On Factorization of a Perturbation of a J-selfadjoint Operator Arising in Fluid Dynamics", "abstract": "We prove that some perturbation of a J-selfadjoint second order differential operator admits factorization and use this new representation of the operator to prove compactness of its resolvent and to find its domain."}
{"category": "Math", "title": "Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows", "abstract": "Robust and accurate schemes are designed to simulate the coupling between subsurface and overland flows. The coupling conditions at the interface enforce the continuity of both the normal flux and the pressure. Richards' equation governing the subsurface flow is discretized using a Backward Differentiation Formula and a symmetric interior penalty Discontinuous Galerkin method. The kinematic wave equation governing the overland flow is discretized using a Godunov scheme. Both schemes individually are mass conservative and can be used within single-step or multi-step coupling algorithms that ensure overall mass conservation owing to a specific design of the interface fluxes in the multi-step case. Numerical results are presented to illustrate the performances of the proposed algorithms."}
{"category": "Math", "title": "Cayley Graph Expanders and Groups of Finite Width", "abstract": "We present new infinite families of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our first family defines a tower of coverings (with covering indices equals 2) and our second family is given as Cayley graphs of finite groups with very short presentations with only 2 generators and 4 relations. Both families are based on particular finite quotients of a group G of infinite upper triangular matrices over the ring M(3,F2). We present explicit vector space bases for the finite abelian quotients of the lower exponent-2 groups of G by upper triangular subgroups and prove a particular 3-periodicity of these quotients. The pro-2 completion of the group G satisfies the Golod-Shafarevich inequality $|R| \\geq (|X|^2)/4$, it is infinite, not p-adic analytic, contains a free nonabelian subgroup, but not a free pro-p group. We also conjecture that the group G has finite width 3 and finite average width 8/3."}
{"category": "Math", "title": "Cyclic generators for irreducible representations of affine Hecke algebras", "abstract": "We give a detailed account of a combinatorial construction, due to Cherednik, of cyclic generators for irreducible modules of the affine Hecke algebra of the general linear group with generic parameter q."}
{"category": "Math", "title": "On the quasi-hereditary property for staggered sheaves", "abstract": "Let G be an algebraic group over an algebraically closed field, acting on a variety X with finitely many orbits. \"Staggered sheaves\" are certain complexes of G-equivariant coherent sheaves on X that seem to possess many remarkable properties. In this paper, we construct \"standard\" and \"costandard\" objects in the category of staggered sheaves, and we prove that that category has enough projectives and injectives."}
{"category": "Math", "title": "Cyclotomy and analytic geometry over F_1", "abstract": "Geometry over non--existent \"field with one element\" $F_1$ conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in at least two related but different guises. In this paper I analyze the crucial role of roots of unity in this geometry and propose a version of the notion of \"analytic functions\" over $F_1$. The paper combines a focused survey with some new constructions. In new version, several local additions and changes are made, references added."}
{"category": "Math", "title": "Stabilization in $H^\\infty_{\\mathbb{R}}(\\mathbb{D})$", "abstract": "In this paper we prove the following theorem: Suppose that $f_1,f_2\\in H^\\infty_\\R(\\D)$, with $\\norm{f_1}_\\infty,\\norm{f_2}_{\\infty}\\leq 1$, with $$ \\inf_{z\\in\\D}(\\abs{f_1(z)}+\\abs{f_2(z)})=\\delta>0. $$ Assume for some $\\epsilon>0$ and small, $f_1$ is positive on the set of $x\\in(-1,1)$ where $\\abs{f_2(x)}<\\epsilon$ for some $\\epsilon>0$ sufficiently small. Then there exists $g_1, g_1^{-1}, g_2\\in H^\\infty_\\R(\\D)$ with $$ \\norm{g_1}_\\infty,\\norm{g_2}_\\infty,\\norm{g_1^{-1}}_\\infty\\leq C(\\delta,\\epsilon) $$ and $$ f_1(z)g_1(z)+f_2(z)g_2(z)=1\\quad\\forall z\\in\\D. $$"}
{"category": "Math", "title": "Square free words as products of commutators", "abstract": "Elements of the commutator subgroup of a free group can be presented as values of canonical forms, called Wicks forms. We show that, starting from sufficiently high genus g, there is a sequence of words w(g) which can be presented by f(g) distinct Wicks forms, where f(g)>g!. Moreover we may choose these words w(g) to be square free."}
{"category": "Math", "title": "On unimodality problems in Pascal's triangle", "abstract": "Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We also propose two more general conjectures."}
{"category": "Math", "title": "On \"Regular Landsberg metrics are always Berwald\" by Z. I. Szabo", "abstract": "A gap in the proof of the finsler \"unicorn\" conjecture in the paper \"Regular Landsberg metrics are always Berwald\" by Z. I. Szabo is pointed out"}
{"category": "Math", "title": "Microlocal condition for non-displaceablility", "abstract": "We formulate a sufficient condition for non-displaceability (by Hamiltonian symplectomorphisms which are identity outside of a compact) of a pair of subsets in a cotangent bundle. This condition is based on micro-local analysis of sheaves on manifolds by Kashiwara-Schapira. This condition is used to prove that the real projective space and the Clifford torus inside the complex projective space are mutually non-displaceable"}
{"category": "Math", "title": "Proof of a conjecture on unimodality", "abstract": "Let $P(x)$ be a polynomial of degree $m$, with nonnegative and non-decreasing coefficients. We settle the conjecture that for any positive real number $d$, the coefficients of $P(x+d)$ form a unimodal sequence, of which the special case $d$ being a positive integer has already been asserted in a previous work. Further, we explore the location of modes of $P(x+d)$ and present some sufficient conditions on $m$ and $d$ for which $P(x+d)$ has the unique mode $\\lceil{m-d\\over d+1}\\rceil$."}
{"category": "Math", "title": "The Auslander-Bridger formula and the Gorenstein property for coherent rings", "abstract": "The concept of Gorenstein dimension, defined by Auslander and Bridger for finitely generated modules over a Noetherian ring, is studied in the context of finitely presented modules over a coherent ring. A generalization of the Auslander-Bridger formula is established and is used as a cornerstone in the development of a theory of coherent Gorenstein rings."}
{"category": "Math", "title": "Maximum Principles for Vectorial Approximate Minimizers of Nonconvex Functionals", "abstract": "We establish Maximum Principles which apply to vectorial approximate minimizers of the general integral functional of Calculus of Variations. Our main result is a version of the Convex Hull Property. The primary advance compared to results already existing in the literature is that we have dropped the quasiconvexity assumption of the integrand in the gradient term. The lack of weak Lower Semicontinuity is compensated by introducing a nonlinear convergence technique, based on the approximation of the projection onto a convex set by reflections and on the invariance of the integrand in the gradient term under the Orthogonal Group. Maximum Principles are implied for the relaxed solution in the case of non-existence of minimizers and for minimizing solutions of the Euler-Lagrange system of PDE."}
{"category": "Math", "title": "A Simple Proof of a Conjecture of Simion", "abstract": "Simion had a unimodality conjecture concerning the number of lattice paths in a rectangular grid with the Ferrers diagram of a partition removed. Hildebrand recently showed the stronger result that these numbers are log concave. Here we present a simple proof of Hildebrand's result."}
{"category": "Math", "title": "Unique resonant normal forms for area preserving maps at an elliptic fixed point", "abstract": "We construct a resonant normal form for an area-preserving map near a generic resonant elliptic fixed point. The normal form is obtained by a simplification of a formal interpolating Hamiltonian. The resonant normal form is unique and therefore provides the formal local classification for area-preserving maps with the elliptic fixed point. The total number of formal invariants is infinite. We consider the cases of weak (of order $n\\ge5$) and strong (of order $n=3,4$) resonances. We also construct unique normal forms for analytic families of area-preserving maps. We note that our constructions involve non-linear grading functions."}
{"category": "Math", "title": "On the First Eigenvalue of Bipartite Graphs", "abstract": "In this paper we study the maximum value of the largest eigenvalue for simple bipartite graphs, where the number of edges is given and the number of vertices on each side of the bipartition is given. We state a conjectured solution, which is an analog of the Brualdi- Hoffman conjecture for general graphs, and prove the conjecture in some special cases."}
{"category": "Math", "title": "Torus actions on small blow ups of CP^2", "abstract": "A manifold obtained by k simultaneous symplectic blow-ups of CP^2 of equal sizes epsilon (where the size of CP^1 in CP^2 is one) admits an effective two-dimensional torus action if k <= 3. We show that it does not admit such an action if k >=4 and epsilon <= 1/(3k 2^{2k}). For the proof, we correspond between the geometry of a symplectic toric four-manifold and the combinatorics of its moment map image. We also use techniques from the theory of J-holomorphic curves."}
{"category": "Math", "title": "Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre", "abstract": "In this survey article we give an overview of the developments on the congruence subgroup and the metaplectic problems after the work of Bass, Milnor and Serre."}
{"category": "Math", "title": "Rank one Z^d actions and directional entropy", "abstract": "We study the directional entropy of rank one Z^d actions. We show that if the sequence of towers generating the action are rectangular in shape, then there is always a direction along which the directional entropy is zero. If the rectangles satisfy a \"sub-exponential growth\" condition, then we show that the directional entropy is zero in every direction."}
{"category": "Math", "title": "Weak subconvexity for central values of $L$-functions", "abstract": "We describe a general method to obtain weak subconvexity bounds for many classes of $L$-functions. This has applications to a conjecture of Rudnick and Sarnak for the mass equidistribution of Hecke eigenforms (see arxiv.org:math/0809.1636)."}
{"category": "Math", "title": "Mass equidistribution for Hecke eigenforms", "abstract": "We prove a conjecture of Rudnick and Sarnak on the mass equidistribution of Hecke eigenforms. This builds upon independent work of the authors see arxiv.org:math/0809.1640 and arxiv.org:math/0809.1635."}
{"category": "Math", "title": "Sieving for mass equidistribution", "abstract": "We approach the holomorphic analogue to the Quantum Unique Ergodicity conjecture through an application of the Large Sieve. We deal with shifted convolution sums as in ([Ho], arXiv:0809.1669), with various simplifications in our analysis due to the knowledge of the Ramanujan-Petersson conjecture in this holomorphic case."}
{"category": "Math", "title": "New Iterative Methods for Interpolation, Numerical Differentiation and Numerical Integration", "abstract": "Through introducing a new iterative formula for divided differnce using Neville's and Aitken's algorithms,we study new iterative methods for interpolation,numerical differentiation and numerical integration formulas with arbitrary order of accuracy for evanly or unevanly spaced data. Basic computer algorithms for new methods are given"}
{"category": "Math", "title": "Eigenvalues of harmonic almost submersions", "abstract": "Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues yield characterizations of harmonicity, totally geodesic maps and biconformal changes of metric preserving harmonicity. A Schwarz lemma for pseudo harmonic morphisms is proved, using the dilatation of the eigenvalues and, in dimension five, a Bochner technique method, involving the Laplacian of the difference of the eigenvalues, gives conditions forcing pseudo harmonic morphisms to be harmonic morphisms."}
{"category": "Math", "title": "On the Cauchy problem for the debar operator", "abstract": "We present new results concerning the solvability, of lack thereof, in the Cauchy problem for the debar operator, with initial values assigned on a weakly pseudoconvex hypersurface, and provide illustrative examples."}
{"category": "Math", "title": "A Sieve Method for Shifted Convolution Sums", "abstract": "We study the average size of shifted convolution summation terms related to the problem of Quantum Unique Ergodicity on ${\\rm SL}_2 (\\mathbbm{Z})\\backslash \\mathbbm{H}$. Establishing an upper-bound sieve method for handling such sums, we achieve an unconditional result which suggests that the average size of the summation terms should be sufficient in application to Quantum Unique Ergodicity. In other words, cancellations among the summation terms, although welcomed, may not be required. Furthermore, the sieve method may be applied to shifted sums of other multiplicative functions with similar results under suitable conditions."}
{"category": "Math", "title": "Unusual Geodesics in generalizations of Thompson's Group F", "abstract": "We prove that seesaw words exist in Thompson's Group F(N) for N=2,3,4,... with respect to the standard finite generating set X. A seesaw word w with swing k has only geodesic representatives ending in g^k or g^{-k} (for given g\\in X) and at least one geodesic representative of each type. The existence of seesaw words with arbitrarily large swing guarantees that F(N) is neither synchronously combable nor has a regular language of geodesics. Additionally, we prove that dead ends (or k--pockets) exist in F(N) with respect to X and all have depth 2. A dead end w is a word for which no geodesic path in the Cayley graph \\Gamma which passes through w can continue past w, and the depth of w is the minimal m\\in\\mathbb{N} such that a path of length m+1 exists beginning at w and leaving B_{|w|}. We represent elements of F(N) by tree-pair diagrams so that we can use Fordham's metric. This paper generalizes results by Cleary and Taback, who proved the case N=2."}
{"category": "Math", "title": "The Infrastructure of a Global Field of Arbitrary Unit Rank", "abstract": "In this paper, we show a general way to interpret the infrastructure of a global field of arbitrary unit rank. This interpretation generalizes the prior concepts of the giant step operation and f-representations, and makes it possible to relate the infrastructure to the (Arakelov) divisor class group of the global field. In the case of global function fields, we present results that establish that effective implementation of the presented methods is indeed possible, and we show how Shanks' baby-step giant-step method can be generalized to this situation."}
{"category": "Math", "title": "The essential dimension of the normalizer of a maximal torus in the projective linear group", "abstract": "Let p be a prime, k be a field of characteristic different from p containing a primitive p-th root of unity and N be the normalizer of the maximal torus in the projective linear group PGLn. We compute the exact value of the essential dimension ed(N;p) of N at p for every n."}
{"category": "Math", "title": "New examples of $c_0$-saturated Banach spaces II", "abstract": "For every Banach space $Z$ with a shrinking unconditional basis satisfying upper $p$-estimates for some $p > 1$, an isomorphically polyhedral Banach space is constructed having an unconditional basis and admitting a quotient isomorphic to $Z$."}
{"category": "Math", "title": "On the decomposition numbers of the Hecke algebra of type $D_n$ when $n$ is even", "abstract": "Let $n\\geq 4$ be an even integer. Let $K$ be a field with $\\cha K\\neq 2$ and $q$ an invertible element in $K$ such that $\\prod_{i=1}^{n-1}(1+q^i)\\neq 0$. In this paper, we study the decomposition numbers over $K$ of the Iwahori--Hecke algebra $\\HH_q(D_n)$ of type $D_n$. We obtain some equalities which relate its decomposition numbers with certain Schur elements and the decomposition numbers of various Iwahori--Hecke algebras of type $A$ with the same parameter $q$. When $\\cha K=0$, this completely determine all of its decomposition numbers. The main tools we used are the Morita equivalence theorem established in \\cite{Hu1} and certain twining character formulae of Weyl modules over a tensor product of two $q$-Schur algebras."}
{"category": "Math", "title": "Completely multiplicative functions taking values in $\\{-1,1\\}$", "abstract": "Define {\\em the Liouville function for $A$}, a subset of the primes $P$, by $\\lambda_{A}(n) =(-1)^{\\Omega_A(n)}$ where $\\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville function, $A$ is the set of all primes. Denote $$L_A(n):=\\sum_{k\\leq n}\\lambda_A(n)\\quad{and}\\quad R_A:=\\lim_{n\\to\\infty}\\frac{L_A(n)}{n}.$$ We show that for every $\\alpha\\in[0,1]$ there is an $A\\subset P$ such that $R_A=\\alpha$. Given certain restrictions on $A$, asymptotic estimates for $\\sum_{k\\leq n}\\lambda_A(k)$ are also given. With further restrictions, more can be said. For {\\em character--like functions} $\\lambda_p$ ($\\lambda_p$ agrees with a Dirichlet character $\\chi$ when $\\chi(n)\\neq 0$) exact values and asymptotics are given; in particular $$\\quad\\sum_{k\\leq n}\\lambda_p(k)\\ll \\log n.$$ Within the course of discussion, the ratio $\\phi(n)/\\sigma(n)$ is considered."}
{"category": "Math", "title": "Elliptic complexes and generalized Poincar\\'e inequalities", "abstract": "We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincar\\'e inequality holds. We show that the constant rank condition is sufficient. The concept of the Moore-Penrose generalized inverse of a matrix comes into play."}
{"category": "Math", "title": "Exceptional Loci on $\\bar M_{0,n}$ and Hypergraph Curves", "abstract": "We give a myriad of examples of extremal divisors, rigid curves, and birational morphisms with unexpected properties for the Grothendieck--Knudsen moduli space $\\bar M_{0,n}$ of stable rational curves. The basic tool is an isomorphism between $M_{0,n}$ and the Brill--Noether locus of a very special reducible curve corresponding to a hypergraph."}
{"category": "Math", "title": "Non-orientable fundamental surfaces in lens spaces", "abstract": "We give a concrete example of an infinite sequence of $(p_n, q_n)$-lens spaces $L(p_n, q_n)$ with natural triangulations $T(p_n, q_n)$ with $p_n$ taterahedra such that $L(p_n, q_n)$ contains a certain non-orientable closed surface which is fundamental with respect to $T(p_n, q_n)$ and of minimal crosscap number among all closed non-orientable surfaces in $L(p_n, q_n)$ and has $n-2$ parallel sheets of normal disks of a quadrilateral type disjoint from the pair of core circles of $L(p_n, q_n)$. Actually, we can set $p_0=0, q_0=1, p_{k+1}=3p_k+2q_k$ and $q_{k+1}=p_k+q_k$."}
{"category": "Math", "title": "Secant Varieties of (P ^1) X .... X (P ^1) (n-times) are NOT Defective for n \\geq 5", "abstract": "Let V_n be the Segre embedding of (P^1) x ... X (P^1) (n times). We prove that the higher secant varieties, \\sigma_s(V_n), always have the expected dimension, except for \\sigma_3(V_4), which is of dimension 1 less than expected."}
{"category": "Math", "title": "Arithmetic Progressions in Abundance by Combinatorial Tools", "abstract": "Using the algebraic structure of the Stone-Cech compactification of the integers, Furstenberg and Glasner proved that for arbitrary k, every piecewise syndetic set contains a piecewise syndetic set of k-term arithmetic progressions. We present a purely combinatorial argument which allows to derive this result directly from van der Waerden's Theorem."}
{"category": "Math", "title": "Duality in spaces of finite linear combinations of atoms", "abstract": "In this note we describe the dual and the completion of the space of finite linear combinations of $(p,\\infty)$-atoms, $0<p\\leq 1$ on ${\\mathbb R}^n$. As an application, we show an extension result for operators uniformly bounded on $(p,\\infty)$-atoms, $0<p < 1$, whose analogue for $p=1$ is known to be false. Let $0 < p <1$ and let $T$ be a linear operator defined on the space of finite linear combinations of $(p,\\infty)$-atoms, $0<p < 1 $, which takes values in a Banach space $B$. If $T$ is uniformly bounded on $(p,\\infty)$-atoms, then $T$ extends to a bounded operator from $H^p({\\mathbb R}^n)$ into $B$."}
{"category": "Math", "title": "Holomorphic self-maps of singular rational surfaces", "abstract": "We give a new proof of the classification of normal singular surface germs admitting non-invertible holomorphic self-maps and due to J. Wahl. We then draw an analogy between the birational classification of singular holomorphic foliations on surfaces, and the dynamics of holomorphic maps. Following this analogy, we introduce the notion of minimal holomorphic model for holomorphic maps. We give sufficient conditions which ensure the uniqueness of such a model."}
{"category": "Math", "title": "Simplicial simple-homotopy of flag complexes in terms of graphs", "abstract": "A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type of a graph and show in particular that two finite graphs have the same s-homotopy type if, and only if, the two flag complexes determined by these graphs have the same simplicial simple-homotopy type (Theorem 2.10, part 1). This result is closely related to similar results established by Barmak and Minian (Adv. in Math., 218 (2008), 87-104) in the framework of posets and we give the relation between the two approaches (theorems 3.5 and 3.7). We conclude with a question about the relation between the s-homotopy and the graph homotopy defined by Chen, Yau and Yeh (Discrete Math., 241(2001), 153-170)."}
{"category": "Math", "title": "Convexity bounds for L-functions", "abstract": "We give a sharp convexity estimate for L-functions which have a functional equation and an Euler product."}
{"category": "Math", "title": "A note concerning a tidying procedure and contraction groups in non-metrizable, totally disconnected groups", "abstract": "In a 2004 article, Udo Baumgartner and George Willis used ideas from the structure theory of totally disconnected, locally compact groups to achieve a better understanding of the contraction group U_f associated with an automorphism f of such a group G, assuming that G is metrizable. (Recall that U_f consists of all group elements x such that f^n(x) tends to the identity element as n tends to infinity). Recently, Wojciech Jaworski showed that the main technical tool of the latter article remains valid in the non-metrizable case. He asserted without proof that, therefore, all results from that article remain valid. However, metrizability enters the arguments at a second point. In this note, we resolve this difficulty, by providing an affirmative answer to a question posed by Willis in 2004."}
{"category": "Math", "title": "Abelian categories in dimension 2", "abstract": "The goal of this thesis is to define a 2-dimensional version of abelian categories, where symmetric 2-groups play the role that abelian groups played in 1-dimensional algebra. Abelian and 2-abelian groupoid enriched categories are defined and it is proved that homology can be developed in them, including the existence of a long exact sequence of homology corresponding to an extension of chain complexes. This generalises known results for symmetric 2-groups. The examples include, in addition to symmetric 2-groups, the 2-modules on a 2-ring, which form a 2-abelian groupoid enriched category. Moreover, internal groupoids, functors and natural transformations in an abelian category C (in particular, Baez-Crans 2-vector spaces) form a 2-abelian groupoid enriched category if and only if the axiom of choice holds in C."}
{"category": "Math", "title": "The extension of Buckley-Feuring solutions for non-polynomial fuzzy partial differential equations", "abstract": "This paper presents the natural extension of Buckley-Feuring method proposed in \\cite{BuckleyFeuring99} for solving fuzzy partial differential equations (FPDE) in a non-polynomial relation, such as the operator $\\varphi(D_{x_1}, D_{x_2})$, which maps to the quotient between both partials. The new assumptions and conditions proceedings from this consideration are given in this document."}
{"category": "Math", "title": "Compatible associative products and trees", "abstract": "We compute dimensions of graded components for free algebras with two compatible associative products, and give a combinatorial interpretation of these algebras in terms of planar rooted trees."}
{"category": "Math", "title": "A Regularized Method for Selecting Nested Groups of Relevant Genes from Microarray Data", "abstract": "Gene expression analysis aims at identifying the genes able to accurately predict biological parameters like, for example, disease subtyping or progression. While accurate prediction can be achieved by means of many different techniques, gene identification, due to gene correlation and the limited number of available samples, is a much more elusive problem. Small changes in the expression values often produce different gene lists, and solutions which are both sparse and stable are difficult to obtain. We propose a two-stage regularization method able to learn linear models characterized by a high prediction performance. By varying a suitable parameter these linear models allow to trade sparsity for the inclusion of correlated genes and to produce gene lists which are almost perfectly nested. Experimental results on synthetic and microarray data confirm the interesting properties of the proposed method and its potential as a starting point for further biological investigations"}
{"category": "Math", "title": "3-Dimensional Lattice Polytopes Without Interior Lattice Points", "abstract": "A theorem of Howe states that every 3-dimensional lattice polytope $P$ whose only lattice points are its vertices, is a Cayley polytope, i.e. $P$ is the convex hull of two lattice polygons with distance one. We want to generalize this result by classifying 3-dimensional lattice polytopes without interior lattice points. The main result will be, that they are up to finite many exceptions either Cayley polytopes or there is a projection, which maps the polytope to the double unimodular 2-simplex. To every such polytope we associate a smooth projective surface of genus 0."}
{"category": "Math", "title": "Special symmetries of Banach spaces isomorphic to Hilbert spaces", "abstract": "In this paper Hilbert spaces are characterized among Banach spaces in terms of transitivity with respect to nicely behaved subgroups of the isometry group. For example, the following result is typical here: If X is a real Banach space isomorphic to a Hilbert space and convex-transitive with respect to the isometric finite-dimensional perturbations of the identity, then X is already isometric to a Hilbert space."}
{"category": "Math", "title": "Solutions of some nonlinear parabolic equations with initial blow-up", "abstract": "We study the existence and uniqueness of solutions of $\\partial_tu-\\Delta u+u^q=0$ ($q>1$) in $\\Omega\\times (0,\\infty)$ where $\\Omega\\subset\\mathbb R^N$ is a domain with a compact boundary, subject to the conditions $u=f\\geq 0$ on $\\partial\\Omega\\times (0,\\infty)$ and the initial condition $\\lim_{t\\to 0}u(x,t)=\\infty$. By means of Brezis' theory of maximal monotone operators in Hilbert spaces, we construct a minimal solution when $f=0$, whatever is the regularity of the boundary of the domain. When $\\partial\\Omega$ satisfies the parabolic Wiener criterion and $f$ is continuous, we construct a maximal solution and prove that it is the unique solution which blows-up at $t=0$."}
{"category": "Math", "title": "New examples of $c_0$-saturated Banach spaces", "abstract": "For every $ 1 < p < \\infty $ an isomorphically polyhedral Banach space $E_p$ is constructed having an unconditional basis and admitting a quotient isomorphic to $\\ell_p$. It is also shown that $E_p$ is not isomorphic to a subspace of a $C(K)$ space for every countable and compact metric space $K$."}
{"category": "Math", "title": "A Markov model for the spread of Hepatitis C", "abstract": "We propose a Markov model for the spread of Hepatitis C virus (HCV) among drug users who use injections. We then proceed to an asymptotic analysis (large initial population) and show that the Markov process is close to the solution of a non linear autonomous differential system. We prove both a law of large numbers and functional central limit theorem to precise the speed of convergence towards the limiting system. The deterministic system itself converges, as time goes to infinity, to an equilibrium point. This corroborates the empirical observations about the prevalence of HCV."}
{"category": "Math", "title": "On semidefinite representations of plane quartics", "abstract": "This note focuses on the problem of representing convex sets as projections of the cone of positive semidefinite matrices, in the particular case of sets generated by bivariate polynomials of degree four. Conditions are given for the convex hull of a plane quartic to be exactly semidefinite representable with at most 12 lifting variables. If the quartic is rationally parametrizable, an exact semidefinite representation with 2 lifting variables can be obtained. Various numerical examples illustrate the techniques and suggest further research directions."}
{"category": "Math", "title": "Diametral Pairs of Linear Extensions", "abstract": "Given a finite poset P, we consider pairs of linear extensions of P with maximal distance, where the distance between two linear extensions L_1, L_2 is the number of pairs of elements of P appearing in different orders in L_1 and L_2. A diametral pair maximizes the distance among all pairs of linear extensions of P. Felsner and Reuter defined the linear extension diameter of P as the distance between a diametral pair of linear extensions. We show that computing the linear extension diameter is NP-complete in general, but can be solved in polynomial time for posets of width 3. Felsner and Reuter conjectured that, in every diametral pair, at least one of the linear extensions reverses a critical pair. We construct a counterexample to this conjecture. On the other hand, we show that a slightly stronger property holds for many classes of posets: We call a poset \"diametrally reversing\" if, in every diametral pair, both linear extensions reverse a critical pair. Among other results we show that interval orders and 3-layer posets are diametrally reversing. From the latter it follows that almost all posets are diametrally reversing."}
{"category": "Math", "title": "Convergence rates for an optimally controlled Ginzburg-Landau equation", "abstract": "An optimal control problem related to the probability of transition between stable states for a thermally driven Ginzburg-Landau equation is considered. The value function for the optimal control problem with a spatial discretization is shown to converge quadratically to the value function for the original problem. This is done by using that the value functions solve similar Hamilton-Jacobi equations, the equation for the original problem being defined on an infinite dimensional Hilbert space. Time discretization is performed using the Symplectic Euler method. Imposing a reasonable condition this method is shown to be convergent of order one in time, with a constant independent of the spatial discretization."}
{"category": "Math", "title": "Asymptotic tail properties of the distributions in the class of dispersion models", "abstract": "The class of dispersion models introduced by J{\\o}rgensen (1997b) covers many known distributions such as the normal, Student t, gamma, inverse Gaussian, hyperbola, von-Mises, among others. We study the small dispersion asymptotic (J{\\o}rgensen, 1987b) behavior of the probability density functions of dispersion models which satisfy the uniformly convergent saddlepoint approximation. Our results extend those obtained by Finner et al. (2008)."}
{"category": "Math", "title": "Geometric structure of NLS evolution", "abstract": "We clarify the relation between the Hamiltonian and Lagrangian approaches to nonlinear evolution equations, focusing specifically on the nonlinear Schroedinger equation. In particular, we explain the least action principle and the Noether theorem in this context."}
{"category": "Math", "title": "Schroedinger Operators on Regular Metric Trees with Long Range Potentials: Weak Coupling Behavior", "abstract": "Consider a regular $d$-dimensional metric tree $\\Gamma$ with root $o$. Define the Schroedinger operator $-\\Delta - V$, where $V$ is a non-negative, symmetric potential, on $\\Gamma$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-\\gamma}$ at infinity, where $1 < \\gamma \\leq d \\leq 2, \\gamma \\neq 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-\\Delta - V$. In other words, we will describe the asymptotical behavior of $\\inf \\sigma(-\\Delta - \\alpha V)$ as $\\alpha \\to 0+$"}
{"category": "Math", "title": "A Removal Lemma for Systems of Linear Equations over Finite Fields", "abstract": "We prove a removal lemma for systems of linear equations over finite fields: let $X_1,...,X_m$ be subsets of the finite field $\\F_q$ and let $A$ be a $(k\\times m)$ matrix with coefficients in $\\F_q$ and rank $k$; if the linear system $Ax=b$ has $o(q^{m-k})$ solutions with $x_i\\in X_i$, then we can destroy all these solutions by deleting $o(q)$ elements from each $X_i$. This extends a result of Green [Geometric and Functional Analysis 15(2) (2005), 340--376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma."}
{"category": "Math", "title": "Some remarks on bounded earthquakes", "abstract": "We first show that an earthquake of a geometrically infinite hyperbolic surface induces an asymptotically conformal change in the hyperbolic metric if and only if the measured lamination associated with the earthquake is asymptotically trivial on the surface. Then we show that the contraction along earthquake paths is continuous in the Teichm\\\"uller space of any hyperbolic surface. Finally, we show that if a measured lamination vanishes while approaching infinity at the rate higher than the distance to the boundary then it must be trivial."}
{"category": "Math", "title": "The distribution of the zeroes of random trigonometric polynomials", "abstract": "We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\\to\\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\\frac{2}{\\sqrt{3}}\\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that $\\frac{Z_{N}-\\E Z_{N}}{\\sqrt{cN}}$ converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals."}
{"category": "Math", "title": "The minimal degree of plane models of double covers of smooth curves", "abstract": "If $X$ is a smooth curve such that the minimal degree of its plane models is not too small compared with its genus, then $X$ has been known to be a double cover of another smooth curve $Y$ under some mild condition on the genera. However there are no results yet for the minimal degrees of plane models of double covers except some special cases. In this paper, we give upper and lower bounds for the minimal degree of plane models of the double cover $X$ in terms of the gonality of the base curve $Y$ and the genera of $X$ and $Y$. In particular, the upper bound equals to the lower bound in case $Y$ is hyperelliptic. We give an example of a double cover which has plane models of degree equal to the lower bound."}
{"category": "Math", "title": "Divisor Problem and an Analogue of Euler's Summation Formula", "abstract": "By simple elementary method,we obtain with ease,a highly simple expression for the remainder term of the divisor problem and use it to obtain an Euler-Maclaurin analogue of summation involving divisor function.We also obtain a relation connecting the remainder term of the divisor problem and the remainders of approximate functional equations for Riemann zeta function and its square,for positive values of arguments."}
{"category": "Math", "title": "The distribution of Pearson residuals in generalized linear models", "abstract": "In general, the distribution of residuals cannot be obtained explicitly. We give an asymptotic formula for the density of Pearson residuals in continuous generalized linear models corrected to order $n^{-1}$, where $n$ is the sample size. We define corrected Pearson residuals for these models that, to this order of approximation, have exactly the same distribution of the true Pearson residuals. Applications for important generalized linear models are provided and simulation results for a gamma model illustrate the usefulness of the corrected Pearson residuals."}
{"category": "Math", "title": "Explicit expressions for moments of the beta Weibull distribution", "abstract": "The beta Weibull distribution was introduced by Famoye et al. (2005) and studied by these authors. However, they do not give explicit expressions for the moments. We now derive explicit closed form expressions for the cumulative distribution function and for the moments of this distribution. We also give an asymptotic expansion for the moment generating function. Further, we discuss maximum likelihood estimation and provide formulae for the elements of the Fisher information matrix. We also demonstrate the usefulness of this distribution on a real data set."}
{"category": "Math", "title": "On the invariant measure of the random difference equation $X_n=A_n X_{n-1}+ B_n$ in the critical case", "abstract": "We consider the autoregressive model on $\\R^d$ defined by the following stochastic recursion $X_n = A_n X_{n-1}+B_n$, where $\\{(B_n,A_n)\\}$ are i.i.d. random variables valued in $\\R^d\\times \\R^+$. The critical case, when $\\E\\big[\\log A_1\\big]=0$, was studied by Babillot, Bougeorol and Elie, who proved that there exists a unique invariant Radon measure $\\nu$ for the Markov chain $\\{X_n \\}$. In the present paper we prove that the weak limit of properly dilated measure $\\nu$ exists and defines a homogeneous measure on $\\R^d\\setminus \\{0\\}$."}
{"category": "Math", "title": "Covering by discrete and closed discrete sets", "abstract": "Say that a cardinal number $\\kappa$ is \\emph{small} relative to the space $X$ if $\\kappa <\\Delta(X)$, where $\\Delta(X)$ is the least cardinality of a non-empty open set in $X$. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire $\\sigma$-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces."}
{"category": "Math", "title": "Some results for beta Fr\\'echet distribution", "abstract": "Nadarajah and Gupta (2004) introduced the beta Fr\\'echet (BF) distribution, which is a generalization of the exponentiated Fr\\'echet (EF) and Fr\\'echet distributions, and obtained the probability density and cumulative distribution functions. However, they do not investigated its moments and the order statistics. In this paper the BF density function and the density function of the order statistics are expressed as linear combinations of Fr\\'echet density functions. This is important to obtain some mathematical properties of the BF distribution in terms of the corresponding properties of the Fr\\'echet distribution. We derive explicit expansions for the ordinary moments and L-moments and obtain the order statistics and their moments. We also discuss maximum likelihood estimation and calculate the information matrix which was not known. The information matrix is easily numerically determined. Two applications to real data sets are given to illustrate the potentiality of this distribution."}
{"category": "Math", "title": "Improved estimators for a general class of beta regression models", "abstract": "In this paper we consider an extension of the beta regression model proposed by Ferrari and Cribari-Neto (2004). We extend their model in two different ways, first, we let the regression structure be nonlinear, second, we allow a regression structure for the precision parameter, moreover, this regression structure may also be nonlinear. Generally, the beta regression is useful to situations where the response is restricted to the standard unit interval and the regression structure involves regressors and unknown parameters. We derive general formulae for second-order biases of the maximum likelihood estimators and use them to define bias-corrected estimators. Our formulae generalizes the results obtained by Ospina et al. (2006), and are easily implemented by means of supplementary weighted linear regressions. We also compare these bias-corrected estimators with three different estimators which are also bias-free to the second-order, one analytical and the other two based on bootstrap methods. These estimators are compared by simulation. We present an empirical application."}
{"category": "Math", "title": "Algebraic structures on the topology of moduli spaces of curves and maps", "abstract": "We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov--Witten theory and integrable systems."}
{"category": "Math", "title": "Dissecting brick into bars", "abstract": "An $N$-dimensional parallelepiped will be called a bar if and only if there are no more than $k$ different numbers among the lengths of its sides (the definition of bar depends on $k$). We prove that a parallelepiped can be dissected into finite number of bars iff the lengths of sides of the parallelepiped span a linear space of dimension no more than $k$ over $\\QQ$. This extends and generalizes a well-known theorem of Max Dehn about partition of rectangles into squares. Several other results about dissections of parallelepipeds are obtained."}
{"category": "Math", "title": "Purity at the end", "abstract": "We consider smooth completion of algebraic manifolds. Having some information about its singular completions or about completions of its images we prove purity of cohohomology of the set at infinity. We deduce also some topological properties. The work is based on the study of perverse direct images for algebraic maps."}
{"category": "Math", "title": "The Beta Generalized Exponential Distribution", "abstract": "We introduce the beta generalized exponential distribution that includes the beta exponential and generalized exponential distributions as special cases. We provide a comprehensive mathematical treatment of this distribution. We derive the moment generating function and the $r$th moment thus generalizing some results in the literature. Expressions for the density, moment generating function and $r$th moment of the order statistics also are obtained. We discuss estimation of the parameters by maximum likelihood and provide the information matrix. We observe in one application to real data set that this model is quite flexible and can be used quite effectively in analyzing positive data in place of the beta exponential and generalized exponential distributions."}
{"category": "Math", "title": "When is tight closure determined by the test ideal?", "abstract": "We characterize the rings in which the equality $(\\tau I:\\tau)= I^*$ holds for every ideal $I \\subset R$. Under certain assumptions, these rings must be either weakly F-regular or one-dimensional."}
{"category": "Math", "title": "A Generalization of the Exponential-Poisson Distribution", "abstract": "The two-parameter distribution known as exponential-Poisson (EP) distribution, which has decreasing failure rate, was introduced by Kus (2007). In this paper we generalize the EP distribution and show that the failure rate of the new distribution can be decreasing or increasing. The failure rate can also be upside-down bathtub shaped. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival and failure rate functions; we also obtain the density of the $i$th order statistic. We derive the $r$th raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher's information matrix. Furthermore, expressions for the R\\'enyi and Shannon entropies are given and estimation of the stress-strength parameter is discussed. Applications using two real data sets are presented."}
{"category": "Math", "title": "Compactness of Hankel operators and analytic discs in the boundary of pseudoconvex domains", "abstract": "Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let f be a function smooth up to the boundary on a smooth bounded pseudoconvex domain D in C^n. We show that, if D is convex or the Levi form of the boundary of D is of rank at least n-2, then compactness of the Hankel operator H_f implies that f is holomorphic \"along\" analytic discs in the boundary. Furthermore, when D is convex in C^2 we show that the condition on f is necessary and sufficient for compactness of H_f"}
{"category": "Math", "title": "Global well-posedness and inviscid limit for the modified Korteweg-de Vries-Burgers equation", "abstract": "Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation $u_t+u_{xxx}+\\epsilon |\\partial_x|^{2\\alpha}u=2(u^{3})_x, u(0)=\\phi$, where $0<\\epsilon,\\alpha\\leq 1$ and $u$ is a real-valued function, we show that it is uniformly globally well-posed in $H^s (s\\geq1)$ for all $\\epsilon \\in (0,1]$. Moreover, we prove that for any $s\\geq 1$ and $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the MKdV equation if $\\epsilon$ tends to 0."}
{"category": "Math", "title": "The Gauss-Dirichlet Orbit Number", "abstract": "Dirichlet computed in some particular cases the number of equivalence classes of representations of a nonzero integer by a representative system for the integral binary quadratic forms of a given discriminant. We complete this computation."}
{"category": "Math", "title": "Gradings on symmetric composition algebras", "abstract": "The group gradings on the symmetric composition algebras over arbitrary fields are classified. Applications of this result to gradings on the exceptional simple Lie algebras are considered too."}
{"category": "Math", "title": "On parallel submanifolds of symmetric spaces", "abstract": "Due to its length and several inaccuracies, this article is no longer suggested for reading. Moreover, in the meantime certain results presented herein could be improved by the author in a non-trivial fashion. Instead, the reader is referred to the following two articles: The extrinsic holonomy Lie algebra of a parallel submanifold (arXiv:0904.2611), Extrinsic homogeneity of parallel submanifolds (arXiv:0904.2636)."}
{"category": "Math", "title": "Gelfand transforms of SO(3)-invariant Schwartz functions on the free group N_{3,2}", "abstract": "The spectrum of a Gelfand pair $(K\\ltimes N, K)$, where $N$ is a nilpotent group, can be embedded in a Euclidean space. We prove that in general, the Schwartz functions on the spectrum are the Gelfand transforms of Schwartz $K$-invariant functions on $N$. We also show the converse in the case of the Gelfand pair $(SO(3)\\ltimes N_{3,2}, SO(3))$, where $N_{3,2}$ is the free two-step nilpotent Lie group with three generators. This extends recent results for the Heisenberg group."}
{"category": "Math", "title": "The Classification of Special Cohen-Macaulay Modules", "abstract": "In this paper we completely classify all the special Cohen-Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our result relies on realizing the specials as those CM modules whose first Ext group vanishes against the ring R, thus reducing the problem to combinatorics on the AR quiver; such possible AR quivers were classified by Auslander and Reiten. We also give some general homological properties of the special CM modules and their corresponding reconstruction algebras."}
{"category": "Math", "title": "Keller's Conjecture on the Existence of Columns in Cube Tilings of R^n", "abstract": "It is shown that if n<7, then each tiling of R^n by translates of the unit cube [0,1)^n contains a column; that is, a family of the form {[0,1)^n+(s+ke_i): k \\in Z}, where s \\in R^n, e_i is an element of the standard basis of R^n and Z is the set of integers."}
{"category": "Math", "title": "The GL(2) McKay Correspondence", "abstract": "In this paper we show that for any affine complete rational surface singularity there is a correspondence between the dual graph of the minimal resolution and the quiver of the endomorphism ring of the special CM modules. We thus call such an algebra the reconstruction algebra. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2,C), it means that the endomorphism ring of the special CM C[[x,y]]^G modules can be used to build the dual graph of the minimal resolution of C^2/G, extending McKay's observation for finite subgroups of SL(2,C) to all finite subgroups of GL(2,C)."}
{"category": "Math", "title": "Intersection Graphs of Pseudosegments: Chordal Graphs", "abstract": "We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. For positive we have a construction which shows that all chordal graphs that can be represented as intersection graph of subpaths on a tree are pseudosegment intersection graphs. We then study the limits of representability. We describe a family of intersection graphs of substars of a star which is not representable as intersection graph of pseudosegments. The degree of the substars in this example, however, has to get large. A more intricate analysis involving a Ramsey argument shows that even in the class of intersection graphs of substars of degree three of a star there are graphs that are not representable as intersection graph of pseudosegments. Motivated by representability questions for chordal graphs we consider how many combinatorially different k-segments, i.e., curves crossing k distinct lines, an arrangement of n pseudolines can host. We show that for fixed k this number is in O(n^2). This result is based on a k-zone theorem for arrangements of pseudolines that should be of independent interest."}
{"category": "Math", "title": "General Lp affine isoperimetric inequalities", "abstract": "Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann--Petty centroid inequality."}
{"category": "Math", "title": "On a generalization of the support problem of Erdos and its analogues for abelian varieties and K-theory", "abstract": "In this paper we consider certain local-global principles for Mordell-Weil type groups over number fields like S-units, abelian varieties and algebraic K-theory groups"}
{"category": "Math", "title": "g-Natural metrics of constant sectional curvature", "abstract": "We prove that the tangent bundle endowed with a g-natural metrics has constant sectional curvature if and only if it is flat, and then we give a characterization of flat g-natural metrics on tangent bundles."}
{"category": "Math", "title": "The homoclinic and heteroclinic C*-algebras of a generalized one-dimensional solenoid", "abstract": "It is shown that the heteroclinic and homoclinic algebras of a generalized one-dimensional solenoid are simple AH-algebras of real rank zero with no dimension growth and a unique trace. In the orientable case they are AT-algebras, and in the non-orientable case they have two-torsion both in K_1 and K_0."}
{"category": "Math", "title": "Rough Volterra equations 1: the algebraic integration setting", "abstract": "We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with H\\\"older exponent greater than 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coefficient. In case of a driving signal with H\\\"older exponent in (1/3,1/2], we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coefficient H>1/3."}
{"category": "Math", "title": "Lecture notes: Semidefinite programs and harmonic analysis", "abstract": "Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands."}
{"category": "Math", "title": "Non-degenerate quadratic laminations", "abstract": "We give a combinatorial criterion for a critical diameter to be compatible with a non-degenerate quadratic lamination."}
{"category": "Math", "title": "Structure on the set of closure operations of a commutative ring", "abstract": "We investigate the algebraic structure on the set of closure operations of a ring. We show the set of closure operations is not a monoid under composition for a discrete valuation ring. Even the set of semiprime operations over a DVR is not a monoid; however, it is the union of two monoids, one being the left but not right act of the other. We also determine all semiprime operations over the ring $K[[t^2, t^3]]$."}
{"category": "Math", "title": "The automorphism group of the free group of rank two is a CAT(0) group", "abstract": "We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above."}
{"category": "Math", "title": "Instability of Standing Waves to the Inhomogeneous Nonlinear Schr\\\"odinger Equation with Harmonic Potential", "abstract": "We study the instability of standing-wave solutions $e^{i\\omega t}\\phi_{\\omega}(x)$ to the inhomogeneous nonlinear Schr\\\"{o}dinger equation $$i\\phi_t=-\\triangle\\phi+|x|^2\\phi-|x|^b|\\phi|^{p-1}\\phi, \\qquad \\in\\mathbb{R}^N, $$ where $ b > 0 $ and $ \\phi_{\\omega} $ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $\\omega $ of wave and the power of nonlinearity $p $ for any fixed $ b > 0. $"}
{"category": "Math", "title": "Exact Series Reconstruction in Photoacoustic Tomography with Circular Integrating Detectors", "abstract": "A method for photoacoustic tomography is presented that uses circular integrals of the acoustic wave for the reconstruction of a three-dimensional image. Image reconstruction is a two-step process: In the first step data from a stack of circular integrating are used to reconstruct the circular projection of the source distribution. In the second step the inverse circular Radon transform is applied. In this article we establish inversion formulas for the first step, which involves an inverse problem for the axially symmetric wave equation. Numerical results are presented that show the validity and robustness of the resulting algorithm."}
{"category": "Math", "title": "Smooth solutions of quasianalytic or ultraholomorphic equations", "abstract": "In the first part of this work, we consider a polynomial $ \\phi(x,y)=y^d+a_1(x)y^{d-1}+...+a_d(x) $ whose coefficients $ a_j $ belong to a Denjoy-Carleman quasianalytic local ring $ \\mathcal{E}_1(M) $. Assuming that $ \\mathcal{E}_1(M) $ is stable under derivation, we show that if $ h $ is a germ of $ C^\\infty $ function such that $ \\phi(x,h(x))=0 $, then $ h $ belongs to $ \\mathcal{E}_1(M) $. This extends a well-known fact about real-analytic functions. We also show that the result fails in general for non-quasianalytic ultradifferentiable local rings. In the second part of the paper, we study a similar problem in the framework of ultraholomorphic functions on sectors of the Riemann surface of the logarithm. We obtain a result that includes suitable non-quasianalytic situations."}
{"category": "Math", "title": "Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations", "abstract": "We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of $N$-dimensional lattice differential equations with unidirectional coupling. This class of lattice equations includes some spatial discretizations for hyperbolic conservation laws with a source term as well as a subclass of monotone systems. We obtain a variational characterization of the critical wave speed above which monotone traveling wave solutions are guaranteed to exist. We also discuss non-monotone waves, and the coexistence of monotone and non-monotone waves."}
{"category": "Math", "title": "Schmidt's game, Badly Approximable Linear Forms and Fractals", "abstract": "We prove that for every two natural numbers M and N, if Tau is a Borel, finite, absolutely friendly measure on a compact set K of R^MN, then the intersection of K and BA(M,N) is a winning set in Schmidt's game sense played on K, where BA(M,N) is the set of badly approximable M\\times N matrices. As an immediate consequence we have the following application. If K is the attractor of an irreducible finite family of contracting similarity maps of R^(M\\times N) satisfying the open set condition, (the Cantor ternary set, Koch's curve and Sierpinski's gasket to name a few examples), the dimK=dimK\\capBA(M,N)."}
{"category": "Math", "title": "On the finite generation of a family of Ext modules", "abstract": "Let $Q$ be a Noetherian ring with finite Krull dimension and let $\\mathbf{f}= f_1,... f_c$ be a regular sequence in $Q$. Set $A = Q/(\\mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $\\projdim_Q M$ finite. Set $\\R = \\bigoplus_{n\\geq 0}I^n$, the Rees-Algebra of $I$. Let $N = \\bigoplus_{j \\geq 0}N_j$ be a finitely generated graded $\\R$-module. We show that \\[\\bigoplus_{j\\geq 0}\\bigoplus_{i\\geq 0} \\Ext^{i}_{A}(M,N_j) \\] is a finitely generated bi-graded module over $\\Sc = \\R[t_1,...,t_c]$. We give two applications of this result to local complete intersection rings."}
{"category": "Math", "title": "Comparing operadic theories of $n$-category", "abstract": "We give a framework for comparing on the one hand theories of n-categories that are weakly enriched operadically, and on the other hand n-categories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (Cheng-Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We will show how to take a theory of n-categories of the former kind and produce a globular operad whose algebras are the n-categories we started with. We first provide a generalisation of Trimble's original theory that allows for the use of other parametrising operads in a very general way, via the notion of categories weakly enriched in V where the weakness is parametrised by an operad P in the category V. We define weak n-categories by iterating the weak enrichment construction using a series of parametrising operads P_i. We then show how to construct from such a theory an n-dimensional globular operad for each $n \\geq 0$ whose algebras are precisely the n-categories we constructed by iterated weak enrichment, and we show that the resulting globular operad is contractible precisely when the operads P_i are contractible. We then show how the globular operad associated with Trimble's topological definition is related to the globular operad used by Batanin to define fundamental n-groupoids of spaces."}
{"category": "Math", "title": "Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra", "abstract": "We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three dimensional maximum angle condition and the regular vertex property, for tetrahedra. Our techniques are different from those used in previous papers on the subject and the results obtained are more general in several aspects. First, intermediate regularity is allowed, that is, for the Raviart-Thomas interpolation of degree $k\\ge 0$, we prove error estimates of order $j+1$ when the vector field being approximated has components in $W^{j+1,p}$, for triangles or tetrahedra, where $0\\le j \\le k$ and $1\\le p \\le\\infty$. These results are new even in the two dimensional case. Indeed, the estimate was known only in the case $j=k$. On the other hand, in the three dimensional case, results under the maximum angle condition were known only for $k=0$."}
{"category": "Math", "title": "Character Average of Second and Fourth Powers of Dirichlet L-Series at Unity", "abstract": "We obtain by simple elementary method, the character average of fourth power of Dirichlet L-series at unity.This result seems to be new.We also obtain character average of second power of Dirichlet L-series at unity,using the power series in second variable of Hurwitz zeta function at unity."}
{"category": "Math", "title": "Singularities of Affine Schubert Varieties", "abstract": "This paper studies the singularities of affine Schubert varieties in the affine Grassmannian (of type $\\mathrm{A}^{(1)}_\\ell$). For two classes of affine Schubert varieties, we determine the singular loci; and for one class, we also determine explicitly the tangent spaces at singular points. For a general affine Schubert variety, we give partial results on the singular locus."}
{"category": "Math", "title": "Some remarks on Betti numbers of random polygon spaces", "abstract": "Polygon spaces like $M_\\ell=\\{(u_1,...,u_n)\\in S^1\\times... S^1 ;\\ \\sum_{i=1}^n l_iu_i=0\\}/SO(2)$ or they three dimensional analogues $N_\\ell$ play an important r\\^ole in geometry and topology, and are also of interest in robotics where the $l_i$ model the lengths of robot arms. When $n$ is large, one can assume that each $l_i$ is a positive real valued random variable, leading to a random manifold. The complexity of such manifolds can be approached by computing Betti numbers, the Euler characteristics, or the related Poincar\\'e polynomial. We study the average values of Betti numbers of dimension $p_n$ when $p_n\\to\\infty$ as $n\\to\\infty$. We also focus on the limiting mean Poincar\\'e polynomial, in two and three dimensions. We show that in two dimensions, the mean total Betti number behaves as the total Betti number associated with the equilateral manifold where $l_i\\equiv \\bar l$. In three dimensions, these two quantities are not any more asymptotically equivalent. We also provide asymptotics for the Poincar\\'e polynomials"}
{"category": "Math", "title": "Wahl's conjecture for a minuscule G/P", "abstract": "We show that Wahl's conjecture holds in all characteristics for a minuscule G/P."}
{"category": "Math", "title": "Organizing Volumes of Right-Angled Hyperbolic Polyhedra", "abstract": "This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angled objects, until it is a disjoint union of L\\\"obell polyhedra, a class of polyhedra which generalizes the dodecahedron. Furthermore, these combinatorial operations are shown to have geometric realizations which are volume decreasing. This allows for an organization of the volumes of right-angled hyperbolic polyhedra and allows, in particular, the determination of the polyhedra with smallest and second-smallest volumes."}
{"category": "Math", "title": "Remarks on Automorphisms of $\\mathbb{C}^* \\times \\mathbb{C}^*$ and their basins", "abstract": "We study basins of attraction of automorphisms of $\\CC^2$ tangent to the identity that fix both axes. Our main result is that, if a well known conjecture about automorphisms of $\\CC^*\\times\\CC^*$ holds, then there are no basins of attraction associated to the non-degenerate characteristic directions (in the sense of Hakim), and therefore we cannot find a Fatou-Bieberbach domain that does not intersect both axis with this method."}
{"category": "Math", "title": "Iterated function systems, moments, and transformations of infinite matrices", "abstract": "We study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Our main object of study is the infinite matrix which encodes all the moment data of a Borel measure on R^d or C. To encode the salient features of a given IFS into precise moment data, we establish an interdependence between IFS equilibrium measures, the encoding of the sequence of moments of these measures into operators, and a new correspondence between the IFS moments and this family of operators in Hilbert space. For a given IFS, our aim is to establish a functorial correspondence in such a way that the geometric transformations of the IFS turn into transformations of moment matrices, or rather transformations of the operators that are associated with them. We first examine the classical existence problem for moments, culminating in a new proof of the existence of a Borel measure on R or C with a specified list of moments. Next, we consider moment problems associated with affine and non-affine IFSs. Our main goal is to determine conditions under which an intertwining relation is satisfied by the moment matrix of an equilibrium measure of an IFS. Finally, using the famous Hilbert matrix as our prototypical example, we study boundedness and spectral properties of moment matrices viewed as Kato-Friedrichs operators on weighted l^2 spaces."}
{"category": "Math", "title": "Numerical solution of a certain type of integral equations on the real half-line", "abstract": "We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the convergence and the rate of convergence of our method. The discretization results in an infinite-dimensional nonlinear system, and we also prove results on the approximation of the solution of the infinite-dimensional system by solution of finite truncations."}
{"category": "Math", "title": "Generalized Whittaker functions for degenerate principal series of $GL(4,\\R)$", "abstract": "We give a characterization of a generalized Whittaker model of a degenerate principal series representation of $GL(n,\\R)$ as the kernel of some differential operators. By this characterization, we investigate some examples on $GL(4,\\R)$. We obtain the dimensions of the generalized Whittaker models and give their basis in terms of hypergeometric functions of one and two variables. We show the multiplicity one of the generalized Whittaker models by using the theory of hypergeometric functions."}
{"category": "Math", "title": "The volume of a differentiable stack", "abstract": "We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the groupoid. Sections of a square root of this line bundle constitute an \"intrinsic Hilbert space\" of the stack."}
{"category": "Math", "title": "Containment in (s,t)-core Partitions", "abstract": "We introduce the idea of (s,t)-closure and delta-sets and show that (s,t)-closed beta-sets which are contained set-wise in (s,t)-closed delta-sets are also contained partition-wise. This implies the maximal (s,t)-core partition theorem of Olsson and Stanton. Along the way, we reprove the (s,s+1)-core case, originally proved by Olsson and Stanton, using new techniques, and several lemmas and propositions regarding the containment of partitions are presented as well. Finally, we study how these apply to partitions which are T-core."}
{"category": "Math", "title": "Extending Cantor Paradox", "abstract": "The inconsistencies involved in the foundation of set theory were invariably caused by infinity and self-reference; and only with the opportune axiomatic restrictions could them be obviated. Throughout history, both concepts have proved to be an exhaustible source of paradoxes and contradictions. It seems therefore legitimate to pose some questions concerning their formal consistency. This is just the objective of this paper. Starting from an extension of Cantor's paradox that suggests the inconsistency of the actual infinity, the paper makes a short review of its controversial history and proposes a new way of criticism based on w-order. Self-reference is also examined from a critique perspective which includes syntactic and semantic considerations. The critique affects the formal sentence involved in Godel's first incompleteness theorem and its ordinary language interpretation."}
{"category": "Math", "title": "Constructing the Primitive Roots of Prime Powers", "abstract": "We use only addition and multiplication to construct the primitive roots of $p^{k+1}$ from the primitive roots of $p^{k}$, where $p$ is an odd prime and $k$ is at least 2."}
{"category": "Math", "title": "Combinatorics and geometry of power ideals", "abstract": "We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We prove that their Hilbert series are determined by the combinatorics of A, and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of the resulting fat point ideals and zonotopal Cox rings. Our work unifies and generalizes results due to Dahmen-Micchelli, Holtz-Ron, Postnikov-Shapiro-Shapiro, and Sturmfels-Xu, among others. It also settles a conjecture of Holtz-Ron on the spline interpolation of functions on the interior lattice points of a zonotope."}
{"category": "Math", "title": "Compactness properties of operator multipliers", "abstract": "We continue the study of multidimensional operator multipliers initiated in [arXiv:math/0701645]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C*-algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C*-algebra of compact operators in terms of tensor products, generalising results of Saar."}
{"category": "Math", "title": "Moving curves and Seshadri constants", "abstract": "We study families of curves covering a projective surface and give lower bounds on the self-intersection of the members of such families, improving results of Ein-Lazarsfeld and Xu. We apply the obtained inequalities to get new insights on Seshadri constants and geometry of surfaces."}
{"category": "Math", "title": "On the descending central sequence of absolute Galois groups", "abstract": "Let $p$ be an odd prime number and $F$ a field containing a primitive $p$th root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group $G_F$ of $F$. Namely, the third subgroup $G_F^{(3)}$ in the descending $p$-central sequence of $G_F$ is the intersection of all open normal subgroups $N$ such that $G_F/N$ is 1, $\\mathbb{Z}/p^2$, or the modular group $M_{p^3}$ of order $p^3$."}
{"category": "Math", "title": "The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point", "abstract": "In this paper we describe the moduli space of germs of generic families of analytic diffeomorphisms which unfold a parabolic fixed point of codimension 1. In [MRR] (and also [R]), it was shown that the Ecalle-Voronin modulus can be unfolded to give a complete modulus for such germs. The modulus is defined on a ramified sector in the canonical perturbation parameter $\\eps$. As in the case of the Ecalle-Voronin modulus, the modulus is defined up to a linear scaling depending only on $\\eps$. Here, we characterize the moduli space for such unfoldings by finding the compatibility conditions on the modulus which are necessary and sufficient for realization as the modulus of an unfolding. The compatibility condition is obtained by considering the region of sectorial overlap in $\\eps$-space. This lies in the Glutsyuk sector where the two fixed points are hyperbolic and connected by the orbits of the diffeomorphism. In this region we have two representatives of the modulus which describe the same dynamics. We identify the necessary compatibility condition between these two representatives by comparing them both with their common Glutsyuk modulus. The compatibility condition implies the existence of a linear scaling for which the modulus is 1/2-summable in $\\eps$, whose direction of non-summability coincides with the direction of real multipliers at the fixed points. Conversely, we show that the compatibility condition (which implies the summability property) is sufficient to realize the modulus as coming from an analytic unfolding, thus giving a complete description of the space of moduli."}
{"category": "Math", "title": "Concentration-compactness phenomena in the higher order Liouville's equation", "abstract": "We investigate different concentration-compactness phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in $R^{2m}$, then that of a closed manifold and, finally, the particular case of the sphere $S^{2m}$. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in $R^{2m}$, concentration phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have compactness."}
{"category": "Math", "title": "Classification of real Bott manifolds", "abstract": "A real Bott manifold is the total space of a sequence of $\\R P^1$ bundles starting with a point, where each $\\R P^1$ bundle is projectivization of a Whitney sum of two real line bundles. A real Bott manifold is a real toric manifold which admits a flat riemannian metric. An upper triangular $(0,1)$ matrix with zero diagonal entries uniquely determines such a sequence of $\\R P^1$ bundles but different matrices may produce diffeomorphic real Bott manifolds. In this paper we determine when two such matrices produce diffeomorphic real Bott manifolds. The argument also proves that any graded ring isomorphism between the cohomology rings of real Bott manifolds with $\\Z/2$ coefficients is induced by an affine diffeomorphism between the real Bott manifolds. In particular, this implies the main theorem of \\cite{ka-ma08} which asserts that two real Bott manifolds are diffeomorphic if and only of their cohomology rings with $\\Z/2$ coefficients are isomorphic as graded rings. We also prove that the decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors."}
{"category": "Math", "title": "Division polynomials for twisted Edwards curves", "abstract": "This paper presents division polynomials for twisted Edwards curves. Their chief property is that they characterise the $n$-torsion points of a given twisted Edwards curve. We also present results concerning the coefficients of these polynomials, which may aid computation."}
{"category": "Math", "title": "Degenerations of pre-Lie algebras", "abstract": "We consider the variety of pre-Lie algebra structures on a given n-dimensional vector space. The group GL_n(K) acts on it, and we study the closure of the orbits with respect to the Zariski topology. This leads to the definition of pre-Lie algebra degenerations. We give fundamental results on such degenerations, including invariants and necessary degeneration criteria. We demonstrate the close relationship to Lie algebra degenerations. Finally we classify all orbit closures in the variety of complex 2-dimensional pre-Lie algebras."}
{"category": "Math", "title": "The Continuous Skolem-Pisot Problem: On the Complexity of Reachability for Linear Ordinary Differential Equations", "abstract": "We study decidability and complexity questions related to a continuous analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of a linear recurrent sequence. In particular, we show that the continuous version of the nonnegativity problem is NP-hard in general and we show that the presence of a zero is decidable for several subcases, including instances of depth two or less, although the decidability in general is left open. The problems may also be stated as reachability problems related to real zeros of exponential polynomials or solutions to initial value problems of linear differential equations, which are interesting problems in their own right."}
{"category": "Math", "title": "Arithmetical ranks of Stanley-Reisner ideals of simplicial complexes with a cone", "abstract": "When a cone is added to a simplicial complex $\\Delta$ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley-Reisner ideals of the original simplicial complex and the new simplicial complex $\\Delta'$. In particular, we show that the arithmetical rank of the Stanley-Reisner ideal of $\\Delta'$ equals the projective dimension of the Stanley-Reisner ring of $\\Delta'$ if the corresponding equality holds for $\\Delta$."}
{"category": "Math", "title": "Limit law of the local time for Brox's diffusion", "abstract": "We consider Brox's model: a one-dimensional diffusion in a Brownian potential W. We show that the normalized local time process (L(t;m_(log t) + x)=t; x \\in R), where m_(log t) is the bottom of the deepest valley reached by the process before time t, behaves asymptotically like a process which only depends on W. As a consequence, we get the weak convergence of the local time to a functional of two independent three-dimensional Bessel processes and thus the limit law of the supremum of the normalized local time. These results are discussed and compared to the discrete time and space case which same questions have been solved recently by N. Gantert, Y. Peres and Z. Shi."}
{"category": "Math", "title": "Cohomological non-rigidity of generalized real Bott manifolds of height 2", "abstract": "We investigate when two generalized real Bott manifolds of height 2 have isomorphic cohomology rings with Z/2 coefficients and also when they are diffeomorphic. It turns out that cohomology rings with Z/2 coefficients do not distinguish those manifolds up to diffeomorphism in general. This gives a counterexample to the cohomological rigidity problem for real toric manifolds posed in \\cite{ka-ma08}. We also prove that generalized real Bott manifolds of height 2 are diffeomorphic if they are homotopy equivalent."}
{"category": "Math", "title": "A free action of a finite group on 3-sphere equivalent to a linear action", "abstract": "In this paper, by use of techniques associated to Cobordism theory and Morse theory, we give a proof of Space-Form-Conjecture, i.e. a free action of a finite group on 3-manifold is equivalent to a linear action."}
{"category": "Math", "title": "Continuous LERW Started from Interior Points", "abstract": "We use the whole-plane Loewner equation to define a family of continuous LERW in finitely connected domains that are started from interior points. These continuous LERW satisfy conformal invariance, preserve some continuous local martingales, and are the scaling limits of the corresponding discrete LERW on the discrete approximation of the domains."}
{"category": "Math", "title": "The Complexity of the Evolution of Graph Labelings", "abstract": "We study the {\\sc Graph Relabeling Problem}--given an undirected, connected, simple graph $G = (V,E)$, two labelings $L$ and $L'$ of $G$, and label {\\em flip} or {\\em mutation} functions determine the complexity of transforming or evolving the labeling $L$ into $L'$\\@. The transformation of $L$ into $L'$ can be viewed as an evolutionary process governed by the types of flips or mutations allowed. The number of applications of the function is the duration of the evolutionary period. The labels may reside on the vertices or the edges. We prove that vertex and edge relabelings have closely related computational complexities. Upper and lower bounds on the number of mutations required to evolve one labeling into another in a general graph are given. Exact bounds for the number of mutations required to evolve paths and stars are given. This corresponds to computing the exact distance between two vertices in the corresponding {\\em Cayley graph}. We finally explore both vertex and edge relabeling with {\\em privileged labels}, and resolve some open problems by providing precise characterizations of when these problems are solvable. Many of our results include algorithms for solving the problems, and in all cases the algorithms are polynomial-time. The problems studied have applications in areas such as bioinformatics, networks, and VLSI."}
{"category": "Math", "title": "Small groups of finite Morley rank with involutions", "abstract": "We consider groups of finite Morley rank with solvable local subgroups of even and mixed types. We also consider miscellaneous aspects of small groups of finite Morley rank of odd type."}
{"category": "Math", "title": "Chern-Weil homomorphism in twisted equivariant cohomology", "abstract": "We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have a cohomology theory, the coefficients of the twisted equivariant cohomology must be taken in the completed polynomial algebra over the dual Lie algebra of $G$. We recall the relation between the equivariant cohomology of exact Courant algebroids and the twisted equivariant cohomology, and we show how to endow with a generalized complex structure the finite dimensional approximations of the Borel construction $M\\times_G EG_k$, whenever the generalized complex manifold $M$ possesses a Hamiltonian $G$ action."}
{"category": "Math", "title": "Some remarks on a result of Jensen and tilting modules for $\\SL_3(k)$ and $q$-$\\GL_3(k)$", "abstract": "This paper reviews a result of Jensen on characters of some tilting modules for $\\SL_3(k)$, where $k$ has characteristic at least five and fills in some gaps in the proof of this result. We then apply the result to finding some decomposition numbers for three part partitions for the symmetric group and the Hecke algebra. We review what is known for characteristic two and three. The quantum case is also considered: analogous results hold for the mixed quantum group where $q$ is an $l$th root of unity with $l$ at least three and thus also hold for the associated Hecke algebra."}
{"category": "Math", "title": "Reciprocity and rationality for the greedy normal form of a Coxeter group", "abstract": "We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series, and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian. As corollaries we obtain many of the known rationality and reciprocity results for the growth series of Coxeter groups as well as some new ones."}
{"category": "Math", "title": "Partitions of trees and ACA'", "abstract": "We show that a version of Ramsey's theorem for trees for arbitrary exponents is equivalent to the subsystem ACA' of reverse mathematics."}
{"category": "Math", "title": "Index, eta and rho-invariants on foliated bundles", "abstract": "We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator $D_m$ on the maximal Connes-Skandalis Hilbert module and explain how the functional calculus of $D_m$ encodes both the leafwise calculus and the monodromy calculus in the corresponding von Neumann algebras. When the foliation is endowed with a holonomy invariant transverse measure, we explain the compatibility of various traces and determinants. We extend Atiyah's index theorem on Galois coverings to these foliations. We define a foliated rho-invariant and investigate its stability properties for the signature operator. Finally, we establish the foliated homotopy invariance of such a signature rho-invariant under a Baum-Connes assumption, thus extending to the foliated context results proved by Neumann, Mathai, Weinberger and Keswani on Galois coverings."}
{"category": "Math", "title": "A randomized algorithm for principal component analysis", "abstract": "Principal component analysis (PCA) requires the computation of a low-rank approximation to a matrix containing the data being analyzed. In many applications of PCA, the best possible accuracy of any rank-deficient approximation is at most a few digits (measured in the spectral norm, relative to the spectral norm of the matrix being approximated). In such circumstances, efficient algorithms have not come with guarantees of good accuracy, unless one or both dimensions of the matrix being approximated are small. We describe an efficient algorithm for the low-rank approximation of matrices that produces accuracy very close to the best possible, for matrices of arbitrary sizes. We illustrate our theoretical results via several numerical examples."}
{"category": "Math", "title": "Operad profiles of Nijenhuis structures", "abstract": "Recently S. Merkulov established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as corresponding to representations of the cobar construction on the Koszul dual of a certain quadratic operad. In this paper we prove, using the PBW-basis method of E. Hoffbeck, that the operad governing Nijenhuis structures is Koszul, thereby showing that Nijenhuis structures correspond to representations of the minimal resolution of this operad. We also construct an operad such that representations of its minimal resolution in a vector space V are in one-to-one correspondence with pairs of compatible Nijenhuis structures on the formal manifold associated to V."}
{"category": "Math", "title": "Solvability of Rado systems in D-sets", "abstract": "Rado's Theorem characterizes the systems of homogenous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations. Furstenberg and Weiss proved that solutions to those systems can in fact be found in every central set. (Since one cell of any finite partition is central, this generalizes Rado's Theorem.) We show that the same holds true for the larger class of $D$-sets. Moreover we will see that the conclusion of Furstenberg's Central Sets Theorem is true for all sets in this class."}
{"category": "Math", "title": "New lower bounds for the number of blocks in balanced incomplete block designs", "abstract": "Bose proved the inequality $b\\geq v+r-1$ for resolvable balanced incomplete block designs (RBIBDs) and Kageyama improved it for RBIBDs which are not affine resolvable. In this note we prove a new lower bound on the number of blocks $b$ that holds for all BIBDs. We further prove that for a significantly large number of BIBDs our bound is tighter than the bounds given by the inequalities of Bose and Kageyama."}
{"category": "Math", "title": "A Local Characterization of Combinatorial Multihedrality in Tilings", "abstract": "A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local characterization of combinatorially multihedral tilings in terms of centered coronas. This generalizes the Local Theorem for Monotypic Tilings, established in an earlier paper, which characterizes the case of combinatorial tile-transitivity."}
{"category": "Math", "title": "High Degree Diophantine Equation c^q=a^p+b^p", "abstract": "The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction, a result is proven that two finite numbers is with unequal logarithms in a corresponding module, and is applied to solving a kind of high degree diophantine equation."}
{"category": "Math", "title": "Asymptotic Unconditionality", "abstract": "We show that a separable real Banach space embeds almost isometrically in a space $Y$ with a shrinking 1-unconditional basis if and only if $\\lim_{n \\to \\infty} \\|x^* + x_n^*\\| = \\lim_{n \\to \\infty} \\|x^* - x_n^*\\|$ whenever $x^* \\in X^*$, $(x_n^*)$ is a weak$^*$-null sequence and both limits exist. If $X$ is reflexive then $Y$ can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng."}
{"category": "Math", "title": "A positively curved manifold homeomorphic to T$_1$S$^4$", "abstract": "We construct a new 7-dimensional manifold with positive sectional curvature which is 2-connected with \\pi_3=\\Z_2 and admits an isometric group action with one dimensional quotient."}
{"category": "Math", "title": "The Model Of Paths For Generalized Kac-Moody Algebras", "abstract": "This submission has been withdrawn by arXiv administrators because of inappropriate authorship claims."}
{"category": "Math", "title": "Virtual retractions, conjugacy separability and omnipotence", "abstract": "We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of Stebe that infinite-order elements of Fuchsian groups (of the first type) are conjugacy distinguished, and a proof that surface groups are omnipotent."}
{"category": "Math", "title": "Normal generation of line bundles on multiple coverings", "abstract": "Any line bundle $\\cl $ on a smooth curve $C$ of genus $g$ with $\\deg \\cl \\ge 2g+1$ is normally generated, i.e., $\\varphi_\\cl (C)\\subseteq \\mathbb P H^0 (C,\\cl)$ is projectively normal. However, it has known that more various line bundles of degree $d$ failing to be normally generated appear on multiple coverings of genus $g$ as $d$ becomes smaller than $2g+1$. Thus, investigating the normal generation of line bundles on multiple coverings can be an effective approach to the normal generation. In this paper, we obtain conditions for line bundles on multiple coverings being normally generated or not, respectively."}
{"category": "Math", "title": "Maximal operator for pseudo-differential operators with homogeneous symbols", "abstract": "The aim of the present paper is to obtain a Sj\\\"{o}lin-type maximal estimate for pseudo-differential operators with homogeneous symbols. The crux of the proof is to obtain a phase decomposition formula which does not involve the time traslation. The proof is somehow parallel to the paper by Pramanik and Terwilleger (P. Malabika and E. Terwilleger, A weak $L^2$ estimate for a maximal dyadic sum operator on $R^n$, Illinois J. Math, {\\bf 47} (2003), no. 3, 775--813). In the present paper, we mainly concentrate on our new phase decomposition formula and the results in the Cotlar type estimate, which are different from the ones by Pramanik and Terwilleger."}
{"category": "Math", "title": "Induced nilpotent orbits and birational geometry", "abstract": "In general, a nilpotent orbit closure in a complex simple Lie algebra \\g, does not have a crepant resolution. But, it always has a Q-factorial terminalization by the minimal model program. According to B. Fu, a nilpotent orbit closure has a crepant resolution only when it is a Richardson orbit, and the resolution is obtained as a Springer map for it. In this paper, we shall generalize this result to Q-factorial terminalizations when \\g$ is classical. Here, the induced orbits play an important role instead of Richardson orbits."}
{"category": "Math", "title": "Frame-type families of translates", "abstract": "We construct a uniformly discrete, and even sparse, sequence of real numbers $\\Lambda=\\{\\lambda_n\\}$ and a function g in $L^2(R)$, such that for every q>2, every function f in $L^2(R)$ can be approximated with arbitrary small error by a linear combination $\\sum c_n g(t-\\lambda_n)$ with an $l_q$ estimate of the coefficients: \\|\\{c_n\\}\\|_{l_q}\\leq C(q)\\|f\\|. This can not be done for q=2, according to [2]."}
{"category": "Math", "title": "The co-universal C*-algebra of a row-finite graph", "abstract": "Let E be a row-finite directed graph. We prove that there exists a C*-algebra C*_{min}(E) with the following co-universal property: given any C*-algebra B generated by a Toeplitz-Cuntz-Krieger E-family in which all the vertex projections are nonzero, there is a canonical homomorphism from B onto C*_{min}(E). We also identify when a homomorphism from B to C*_{min}(E) obtained from the co-universal property is injective. When every loop in E has an entrance, C*_{min}(E) coincides with the graph C*-algebra C*(E), but in general, C*_{min}(E) is a quotient of C*(E). We investigate the properties of C*_{min}(E) with emphasis on the utility of co-universality as the defining property of the algebra."}
{"category": "Math", "title": "Infinite paths and cliques in random graphs", "abstract": "We study some percolation problems on the complete graph over $\\mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability, such as independency, is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory."}
{"category": "Math", "title": "Two-dimensional Blaschke Products: Degree growth and ergodic consequences", "abstract": "We study the dynamics of Blaschke products in two dimensions, particularly the rates of growth of the degrees of iterates and the corresponding implications for the ergodic properties of the map."}
{"category": "Math", "title": "A constrained Nevanlinna-Pick interpolation problem for matrix-valued functions", "abstract": "Recent results of Davidson-Paulsen-Raghupathi-Singh give necessary and sufficient conditions for the existence of a solution to the Nevanlinna-Pick interpolation problem on the unit disk with the additional restriction that the interpolant should have the value of its derivative at the origin equal to zero. This concrete mild generalization of the classical problem is prototypical of a number of other generalized Nevanlinna-Pick interpolation problems which have appeared in the literature (for example, on a finitely-connected planar domain or on the polydisk). We extend the results of Davidson-Paulsen-Raghupathi-Singh to the setting where the interpolant is allowed to be matrix-valued and elaborate further on the analogy with the theory of Nevanlinna-Pick interpolation on a finitely-connected planar domain."}
{"category": "Math", "title": "Toward an inductive description of singularities of pairs", "abstract": "Motivated by Shokurov's ACC Conjecture for log canonical thresholds, we propose an inductive point of view on singularities of pairs, in the case when the ambient variety is smooth. Our main result characterizes the log canonicity of a pair in dimension N+1 by the log canonicity of another pair (not effective, in general) in dimension N."}
{"category": "Math", "title": "Affine algebraic monoids as endomorphisms' monoids of finite-dimensional algebras", "abstract": "In this note we prove that any affine algebraic monoid can be obtained as the endomorphisms' monoid of a finite-dimensional (nonassociative) algebra."}
{"category": "Math", "title": "Entire scalar curvature flow and hypersurfaces of constant scalar curvature in Minkowski space", "abstract": "We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates."}
{"category": "Math", "title": "Measuring Singularity of Generalized Minimizers for Control-Affine Problems", "abstract": "An open question contributed by Yu. Orlov to a recently published volume \"Unsolved Problems in Mathematical Systems and Control Theory\", V.D. Blondel, A. Megretski (eds), Princeton Univ. Press, 2004, concerns regularization of optimal control-affine problems. These noncoercive problems in general admit 'cheap (generalized) controls' as minimizers; it has been questioned whether and under what conditions infima of the regularized problems converge to the infimum of the original problem. Starting with a study of this question we show by simple functional-theoretic reasoning that it admits, in general, positive answer. This answer does not depend on commutativity/noncommtativity of controlled vector fields. It depends instead on presence or absence of a Lavrentiev gap. We set an alternative question of measuring \"singularity\" of minimizing sequences for control-affine optimal control problems by so-called degree of singularity. It is shown that, in the particular case of singular linear-quadratic problems, this degree is tightly related to the \"order of singularity\" of the problem. We formulate a similar question for nonlinear control-affine problem and establish partial results. Some conjectures and open questions are formulated."}
{"category": "Math", "title": "Mapping stacks of topological stacks", "abstract": "We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We prove an invariance theorem which shows that the weak homotopy type of the mapping stack Map(Y,X) does not change if we replace X by its classifying space, provided that Y is paracompact topological space. As an example, we describe the loop stack of the classifying stack BG of a topological group G in terms of twisted loop groups of G."}
{"category": "Math", "title": "Permutations with Kazhdan-Lusztig polynomial P_{id,w}(q) = 1 + q^h", "abstract": "Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial P_{id,w}(q)=1+q^h for some h."}
{"category": "Math", "title": "Testing Linear-Invariant Non-Linear Properties", "abstract": "We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for \"triangle freeness\": a function $f:\\cube^{n}\\to\\cube$ satisfies this property if $f(x),f(y),f(x+y)$ do not all equal 1, for any pair $x,y\\in\\cube^{n}$. Here we extend this test to a more systematic study of testing for linear-invariant non-linear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by $k$ points $v_{1},...,v_{k}\\in\\cube^{k}$ and $f:\\cube^{n}\\to\\cube$ satisfies the property that if for all linear maps $L:\\cube^{k}\\to\\cube^{n}$ it is the case that $f(L(v_{1})),...,f(L(v_{k}))$ do not all equal 1. We show that this property is testable if the underlying matroid specified by $v_{1},...,v_{k}$ is a graphic matroid. This extends Green's result to an infinite class of new properties. Our techniques extend those of Green and in particular we establish a link between the notion of \"1-complexity linear systems\" of Green and Tao, and graphic matroids, to derive the results."}
{"category": "Math", "title": "Homotopy Inner Products for Cyclic Operads", "abstract": "We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\\widehat{\\mathcal O}$, which describes modules over $\\mathcal O$ with invariant inner products. We show that $\\widehat{\\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\\widehat{\\mathcal O}$ in terms of derivations and module maps. As an application we construct a homotopy inner product over the commutative operad on the cochains of any Poincar\\'e duality space."}
{"category": "Math", "title": "Exotic automorphisms of the Schouten algebra of polyvector fields", "abstract": "Using a new compactification of the (braid) configuration space of n points in the upper half plane we construct a family of exotic Lie-infinity automorphisms of the Schouten algebra of polyvector fields on an affine space depending on a Kontsevich type propagator."}
{"category": "Math", "title": "Convergence of diagonal Pad\\'e approximants for a class of definitizable functions", "abstract": "Convergence of diagonal Pad\\'e approximants is studied for a class of functions which admit the integral representation $ {\\mathfrak F}(\\lambda)=r_1(\\lambda)\\int_{-1}^1\\frac{td\\sigma(t)}{t-\\lambda}+r_2(\\lambda), $ where $\\sigma$ is a finite nonnegative measure on $[-1,1]$, $r_1$, $r_2$ are real rational functions bounded at $\\infty$, and $r_1$ is nonnegative for real $\\lambda$. Sufficient conditions for the convergence of a subsequence of diagonal Pad\\'e approximants of $ {\\mathfrak F}$ on $\\dR\\setminus[-1,1]$ are found. Moreover, in the case when $r_1\\equiv 1$, $r_2\\equiv 0$ and $\\sigma$ has a gap $(\\alpha,\\beta)$ containing 0, it turns out that this subsequence converges in the gap. The proofs are based on the operator representation of diagonal Pad\\'e approximants of $ {\\mathfrak F}$ in terms of the so-called generalized Jacobi matrix associated with the asymptotic expansion of $ {\\mathfrak F}$ at infinity."}
{"category": "Math", "title": "Number theoretic techniques in the theory of Lie groups and differential geometry", "abstract": "The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of \"weak commensurability\" of two Zariski-dense subgroups in a semi-simple Lie group. We have shown that for arithmetic subgroups, weak commensurability has surprisingly strong consequences. Our proofs make use of p-adic techniques and results from algebraic and transcendental number theory."}
{"category": "Math", "title": "Estimation of a probability with optimum guaranteed confidence in inverse binomial sampling", "abstract": "Sequential estimation of a probability $p$ by means of inverse binomial sampling is considered. For $\\mu_1,\\mu_2>1$ given, the accuracy of an estimator $\\hat{p}$ is measured by the confidence level $P[p/\\mu_2\\leq\\hat{p}\\leq p\\mu_1]$. The confidence levels $c_0$ that can be guaranteed for $p$ unknown, that is, such that $P[p/\\mu_2\\leq \\hat{p}\\leq p\\mu_1]\\geq c_0$ for all $p\\in(0,1)$, are investigated. It is shown that within the general class of randomized or non-randomized estimators based on inverse binomial sampling, there is a maximum $c_0$ that can be guaranteed for arbitrary $p$. A non-randomized estimator is given that achieves this maximum guaranteed confidence under mild conditions on $\\mu_1$, $\\mu_2$."}
{"category": "Math", "title": "A brief note on the spectrum of the basic Dirac operator", "abstract": "In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation $(M,\\mathcal{F})$ with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O'Neill tensor and the first eigenvalue of the Dirac operator on $M$. We discuss examples and also define a new version of the basic Laplacian whose spectrum does not depend on the choice of bundle-like metric."}
{"category": "Math", "title": "Implementing Communication-Optimal Parallel and Sequential QR Factorizations", "abstract": "We present parallel and sequential dense QR factorization algorithms for tall and skinny matrices and general rectangular matrices that both minimize communication, and are as stable as Householder QR. The sequential and parallel algorithms for tall and skinny matrices lead to significant speedups in practice over some of the existing algorithms, including LAPACK and ScaLAPACK, for example up to 6.7x over ScaLAPACK. The parallel algorithm for general rectangular matrices is estimated to show significant speedups over ScaLAPACK, up to 22x over ScaLAPACK."}
{"category": "Math", "title": "Normal automorphisms of relatively hyperbolic groups", "abstract": "An automorphism $\\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group $G$, $Inn(G)$ has finite index in the subgroup $Aut_n(G)$ of normal automorphisms. If, in addition, $G$ is non-elementary and has no non-trivial finite normal subgroups, then $Aut_n(G)=Inn(G)$. As an application, we show that $Out(G)$ is residually finite for every finitely generated residually finite group $G$ with more than one end."}
{"category": "Math", "title": "Shelling Coxeter-like Complexes and Sorting on Trees", "abstract": "In their work on `Coxeter-like complexes', Babson and Reiner introduced a simplicial complex $\\Delta_T$ associated to each tree $T$ on $n$ nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that $\\Delta_T$ is $(n-b-1)$-connected when the tree has $b$ leaves. We provide a shelling for the $(n-b)$-skeleton of $\\Delta_T$, thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree $T$ which imply shellability of $\\Delta_T$, and we construct such inversion functions for a large enough class of trees to deduce the aforementioned conjecture and also recover the shellability of chessboard complexes $M_{m,n}$ with $n \\ge 2m-1$. We also prove that the existence or nonexistence of an inversion function for a fixed tree governs which networks with a tree structure admit greedy sorting algorithms by inversion elimination and provide an inversion function for trees where each vertex has capacity at least its degree minus one."}
{"category": "Math", "title": "Chern classes of blow-ups", "abstract": "We extend the classical formula of Porteous for blowing-up Chern classes to the case of blow-ups of possibly singular varieties along regularly embedded centers. The proof of this generalization is perhaps conceptually simpler than the standard argument for the nonsingular case, involving Riemann-Roch without denominators. The new approach relies on the explicit computation of an ideal, and a mild generalization of the well-known formula for the normal bundle of a proper transform. We also discuss alternative, very short proofs of the standard formula in some cases: an approach relying on the theory of Chern-Schwartz-MacPherson classes (working in characteristic 0), and an argument reducing the formula to a straightforward computation of Chern classes for sheaves of differential 1-forms with logarithmic poles (when the center of the blow-up is a complete intersection)."}
{"category": "Math", "title": "On Coxeter Diagrams of complex reflection groups", "abstract": "We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\\cE = \\ZZ[e^{2 \\pi i/3}]$: there are only four such lattices, namely, the $\\cE$-lattices whose real forms are $A_2$, $D_4$, $E_6$ and $E_8$. Next, we address the issue of characterizing the diagrams for unitary reflection groups, a question that was raised by Brou\\'{e}, Malle and Rouquier. To this end, we describe an algorithm which, given a unitary reflection group $G$, picks out a set of complex reflections. The algorithm is based on an analogy with Weyl groups. If $G$ is a Weyl group, the algorithm immediately yields a set of simple roots. Experimentally we observe that if $G$ is primitive and $G$ has a set of roots whose $\\ZZ$--span is a discrete subset of the ambient vector space, then the algorithm selects a minimal generating set for $G$. The group $G$ has a presentation on these generators such that if we forget that the generators have finite order then we get a (Coxeter-like) presentation of the corresponding braid group. For some groups, such as $G_{33}$ and $G_{34}$, new diagrams are obtained. For $G_{34}$, our new diagram extends to an \"affine diagram\" with $\\ZZ/7\\ZZ$ symmetry."}
{"category": "Math", "title": "Szasz Analytic Functions and Noncompact K\\\"{a}hler Toric Manifolds", "abstract": "We show that the classical Szasz analytic function $S_N(f)(x)$ is obtained by applying the pseudo-differential operator $f(N^{-1}D_{\\theta})$ to the Bergman kernels for the Bargmann-Fock space. The expression generalizes immediately to any smooth polarized noncompact complete toric \\kahler manifold, defining the generalized Szasz analytic function $S_{h^N}(f)(x)$. About $S_{h^N}(f)(x)$, we prove that it admits complete asymptotics and there exists a universal scaling limit. % We also consider some dilation operator composed with $S_{h^N}(f)(x)$ and we give an estimate about this composition. As an example, we will further compute $S_{h^N}(f)(x)$ for the Bergman metric on the unit ball."}
{"category": "Math", "title": "Wreath Product Symmetric Functions", "abstract": "We systematically study wreath product Schur functions and give a combinatorial construction using colored partitions and tableaux. The Pieri rule and the Littlewood-Richardson rule are studied. We also discuss the connection with representations of generalized symmetric groups."}
{"category": "Math", "title": "A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions", "abstract": "Stents are medical devices designed to modify blood flow in aneurysm sacs, in order to prevent their rupture. Some of them can be considered as a locally periodic rough boundary. In order to approximate blood flow in arteries and vessels of the cardio-vascular system containing stents, we use multi-scale techniques to construct boundary layers and wall laws. Simplifying the flow we turn to consider a 2-dimensional Poisson problem that conserves essential features related to the rough boundary. Then, we investigate convergence of boundary layer approximations and the corresponding wall laws in the case of Neumann type boundary conditions at the inlet and outlet parts of the domain. The difficulty comes from the fact that correctors, for the boundary layers near the rough surface, may introduce error terms on the other portions of the boundary. In order to correct these spurious oscillations, we introduce a vertical boundary layer. Trough a careful study of its behavior, we prove rigorously decay estimates. We then construct complete boundary layers that respect the macroscopic boundary conditions. We also derive error estimates in terms of the roughness size epsilon either for the full boundary layer approximation and for the corresponding averaged wall law."}
{"category": "Math", "title": "Fractional diffusion limit for collisional kinetic equations", "abstract": "This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation."}
{"category": "Math", "title": "Two new Probability inequalities and Concentration Results", "abstract": "Concentration results and probabilistic analysis for combinatorial problems like the TSP, MWST, graph coloring have received much attention, but generally, for i.i.d. samples (i.i.d. points in the unit square for the TSP, for example). Here, we prove two probability inequalities which generalize and strengthen Martingale inequalities. The inequalities provide the tools to deal with more general heavy-tailed and inhomogeneous distributions for combinatorial problems. We prove a wide range of applications - in addition to the TSP, MWST, graph coloring, we also prove more general results than known previously for concentration in bin-packing, sub-graph counts, Johnson-Lindenstrauss random projection theorem. It is hoped that the strength of the inequalities will serve many more purposes."}
{"category": "Math", "title": "Continuity of the radius of convergence of differential equations on $p$-adic analytic curves", "abstract": "This paper deals with connections on $p$-adic analytic curves, in the sense of Berkovich. The curves must be compact but the connections are allowed to have a finite number of meromorphic singularities on them. For any choice of a semistable formal model of the curve, we define an intrinsic notion of normalized radius of convergence as a function on the curve, with values in $(0,1]$. For a sufficiently refined choice of the semistable model, we prove continuity and logarithmic concavity of that function. We characterize \\emph{Robba connections}, that is connections whose sheaf of solutions is constant on any open disk contained in the curve."}
{"category": "Math", "title": "The Polya-Tchebotarov problem", "abstract": "We describe the solutions to the problem of identifying the continuum in the complex plane that minimizes the logarithmic capacity among all the continuum that contain a prefixed finite set of points. This description can be implemented numerically and this can be used to improve the estimates on the Bloch-Landau constant and other related problems as the maximal expected lifetime of the Brownian motion on domains of inner radius one or the principal eigenvalue for the Laplace operator on such domains."}
{"category": "Math", "title": "Asymptotics of generalized Hadwiger numbers", "abstract": "We give asymptotic estimates for the number of non-overlapping homothetic copies of some centrally symmetric oval $B$ which have a common point with a 2-dimensional domain $F$ having rectifiable boundary, extending previous work of the L.Fejes-Toth, K.Borockzy Jr., D.G.Larman, S.Sezgin, C.Zong and the authors. The asymptotics compute the length of the boundary $\\partial F$ in the Minkowski metric determined by $B$. The core of the proof consists of a method for sliding convex beads along curves with positive reach in the Minkowski plane. We also prove that level sets are rectifiable subsets, extending a theorem of Erd\\\"os, Oleksiv and Pesin for the Euclidean space to the Minkowski space."}
{"category": "Math", "title": "Hardness and Algorithms for Rainbow Connection", "abstract": "An edge-colored graph $G$ is {\\em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {\\em rainbow connection} of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In the first result of this paper we prove that computing $rc(G)$ is NP-Hard solving an open problem from \\cite{Ca-Yu}. In fact, we prove that it is already NP-Complete to decide if $rc(G)=2$, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every $\\epsilon >0$, a connected graph with minimum degree at least $\\epsilon n$ has {\\em bounded} rainbow connection, where the bound depends only on $\\epsilon$, and a corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented."}
{"category": "Math", "title": "Coherence for Modalities", "abstract": "Positive modalities in systems in the vicinity of S4 and S5 are investigated in terms of categorial proof theory. Coherence and maximality results are demonstrated, and connections with mixed distributive laws and Frobenius algebras are exhibited."}
{"category": "Math", "title": "Ordinals in Frobenius Monads", "abstract": "This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for self-adjunctions (adjunctions where an endofunctor is adjoint to itself), ordinals in epsilon zero play a prominent role. The paper ends by considering how the notion of Frobenius algebra induces the collapse of the hierarchy of ordinals in epsilon zero, and by raising the question of the exact categorial abstraction of the notion of Frobenius algebra."}
{"category": "Math", "title": "Irrationality proof of a $q$-extension of $\\zeta(2)$ using little $q$-Jacobi polynomials", "abstract": "We show how one can use Hermite-Pad\\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\\zeta_q(2)$. These numbers are $q$-analogues of the well known $\\zeta(2)$. Here $q=\\frac{1}{p}$, with $p$ an integer greater than one. These approximants are good enough to show the irrationality of $\\zeta_q(2)$ and they allow us to calculate an upper bound for its measure of irrationality: $\\mu(\\zeta_q(2))\\leq 10\\pi^2/(5\\pi^2-24) \\approx 3.8936$. This is sharper than the upper bound given by Zudilin (\\textit{On the irrationality measure for a $q$-analogue of $\\zeta(2)$}, Mat. Sb. \\textbf{193} (2002), no. 8, 49--70)."}
{"category": "Math", "title": "Exponential functionals of Brownian motion and class-one Whittaker functions", "abstract": "We consider exponential functionals of a multi-dimensional Brownian motion with drift, defined via a collection of linear functionals. We give a characterization of the Laplace transform of their joint law as the unique bounded solution, up to a constant factor, to a Schrodinger-type partial differential equation. We derive a similar equation for the probability density. We then characterize all diffusion processes which can be interpreted as having the law of the Brownian motion with drift conditioned on the law of its exponential functionals. In the case where the family of linear functionals is a set of simple roots, the Laplace transform of the joint law of the corresponding exponential functionals can be expressed in terms of a (class-one) Whittaker function associated with the corresponding root system. In this setting, we establish some basic properties of the corresponding diffusion processes."}
{"category": "Math", "title": "Integral and isocapacitary inequalities", "abstract": "It is shown by a counterexample that isocapacitary and isoperimetric constants of a multi-dimensional Euclidean domain starshaped with respect to a ball are not equivalent. Sharp integral inequalities involving the harmonic capacity which imply Faber-Krahn property of the fundamental Dirichlet-Laplace eigenvalue are obtained. Necessary and sufficient conditions ensuring integral inequalities between a difference seminorm and the $L_p$-norm of the gradient are found."}
{"category": "Math", "title": "Boundedness of the gradient of a solution to the Neumann-Laplace problem in a convex domain", "abstract": "It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given."}
{"category": "Math", "title": "A sequence to compute the Brauer group of certain quasi-triangular Hopf algebras", "abstract": "A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra $B$ in a braided monoidal category $\\C$, and under certain assumptions on the braiding (fulfilled if $\\C$ is symmetric), we construct a sequence for the Brauer group $\\BM(\\C;B)$ of $B$-module algebras, generalizing Beattie's one. It allows one to prove that $\\BM(\\C;B) \\cong \\Br(\\C) \\times \\Gal(\\C;B),$ where $\\Br(\\C)$ is the Brauer group of $\\C$ and $\\Gal(\\C;B)$ the group of $B$-Galois objects. We also show that $\\BM(\\C;B)$ contains a subgroup isomorphic to $\\Br(\\C) \\times \\Hc(\\C;B,I),$ where $\\Hc(\\C;B,I)$ is the second Sweedler cohomology group of $B$ with values in the unit object $I$ of $\\C$. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct $B \\times H$, where $H$ is a usual Hopf algebra over a field $K$, the Hopf subalgebra generated by the quasi-triangular structure $\\R$ is contained in $H$ and $B$ is a Hopf algebra in the category ${}_H\\M$ of left $H$-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that $\\BM(K,H,\\R) \\times \\Hc({}_H\\M;B,K)$ is a subgroup of the Brauer group $\\BM(K,B \\times H,\\R),$ confirming the suspicion that a certain cohomology group of $B \\times H$ (second lazy cohomology group was conjectured) embeds into $\\BM(K,B \\times H,\\R).$ New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence."}
{"category": "Math", "title": "Quartic double solids with ordinary singularities", "abstract": "We study the mixed Hodge structure on the third homology group of a threefold which is the double cover of projective three-space ramified over a quartic surface with a double conic. We deal with the Torelli problem for such threefolds."}
{"category": "Math", "title": "Growth conditions and uniqueness of the Cauchy problem for the evolutionary infinity Laplacian", "abstract": "We study the Cauchy problem for the parabolic infinity Laplace equation. We prove a new comparison principle and obtain uniqueness of viscosity solutions in the class of functions with a polinomial growth at infinity, improving previous results obtained assuming a linear growth."}
{"category": "Math", "title": "Newton Binomial Formulas in Schubert Calculus", "abstract": "We prove Newton's binomial formulas for Schubert Calculus to determine numbers of base point free linear series on the projective line with prescribed ramification divisor supported at given distinct points."}
{"category": "Math", "title": "Locally semisimple and maximal subalgebras of the finitary Lie algebras $gl(\\infty)$, $sl(\\infty)$, $so(\\infty)$, and $sp(\\infty)$", "abstract": "We describe all locally semisimple subalgebras and all maximal subalgebras of the finitary Lie algebras $\\gl(\\infty), \\sl(\\infty), \\so(\\infty)$, and $\\sp(\\infty)$. For simple finite--dimensional Lie algebras these classes of subalgebras have been described in the classical works of A. Malcev and E. Dynkin. Key words (2000 MSC): 17B05 and 17B65."}
{"category": "Math", "title": "Spanning Trees in Grid Graphs", "abstract": "Building on work by Desjarlais, Molina, Faase, and others, a general method is obtained for counting the number of spanning trees of graphs that are a product of an arbitrary graph and either a path or a cycle, of which grid graphs are a subclass. Results are obtained pertaining to recurrences obtained in this manner, and numerous new integer sequences are found."}
{"category": "Math", "title": "On the mean curvature of Nash isometric embeddings", "abstract": "J. Nash proved that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B_{\\mathbb{R}^{N}}(1) of the Euclidean space R^N. However, the geometry of M appears, to some extent, imposing restrictions on the mean curvature vector of the embedding."}
{"category": "Math", "title": "Uniqueness and Non-uniqueness in inverse radiative transfer", "abstract": "We characterize the non-uniqueness in the inverse problem for the stationary transport model, in which the absorption \"a\" and the scattering coefficient \"k\" are to be recovered from the albedo operator. We show that \"gauge equivalent\" pairs (a,k) yield the same albedo operator, and that we can recover uniquely the class of gauge equivalent pairs. We apply this result to show unique determination of the media when the absorption \"a\" depends on the line of travel through each point while scattering \"k\" obeys a symmetry property. Previously known results concerned directional independent absorption."}
{"category": "Math", "title": "Exercises in the birational geometry of algebraic varieties", "abstract": "This a collection of about 100 exercises. It could be used as a supplement to the book Koll\\'ar--Mori: Birational geometry of algebraic varieties."}
{"category": "Math", "title": "Stability of noncharacteristic boundary layers in the standing shock limit", "abstract": "We investigate one- and multi-dimensional stability of noncharacteristic boundary layers in the limit approaching a standing planar shock wave $\\bar U(x_1)$, $x_1>0$, obtaining necessary conditions of (i) weak stability of the limiting shock, (ii) weak stability of the constant layer $u\\equiv U_-:=\\lim_{z\\to -\\infty} \\bar U(z)$, and (iii) nonnegativity of a modified Lopatinski determinant similar to that of the inviscid shock case. For Lax 1-shocks, we obtain equally simple sufficient conditions; for $p$-shocks, $p>1$, the situation appears to be more complicated. Using these results, we determine stability of certain isentropic and full gas dynamical boundary-layers, generalizing earlier work of Serre--Zumbrun and Costanzino--Humphreys--Nguyen--Zumbrun."}
{"category": "Math", "title": "On cluster algebras arising from unpunctured surfaces II", "abstract": "We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincar\\'e characteristic of quiver Grassmannians in Dynkin type $A$ and affine Dynkin type $\\tilde A$."}
{"category": "Math", "title": "Proximal Point Method for a Special Class of Nonconvex Functions on Hadamard Manifolds", "abstract": "In this paper we present the proximal point method for a special class of nonconvex function on a Hadamard manifold. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, it is proved that each accumulation point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained."}
{"category": "Math", "title": "Chow groups of K3 surfaces and spherical objects", "abstract": "We show that for a K3 surface X the finitely generated subring R(X) of the Chow ring introduced by Beauville and Voisin is preserved under derived equivalences. This is proved by analyzing Chern characters of spherical bundles. As for a K3 surface X defined over a number field all spherical bundles on the associated complex K3 surface are defined over $\\bar\\QQ$, this is compatible with the Bloch-Beilinson conjecture. Besides the work of Beauville and Voisin, Lazarfeld's result on Brill-Noether theory for curves in K3 surfaces and the deformation theory developed with Macri and Stellari in arXiv:0710.1645 are central for the discussion."}
{"category": "Math", "title": "Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips", "abstract": "We prove a monotonicity result at specific points for the Horizontal Perimeter for a class of surfaces in the Heisenberg group."}
{"category": "Math", "title": "The Radio Number of Gear Graphs", "abstract": "Let $d(u,v)$ denote the distance between two distinct vertices of a connected graph $G$, and $\\diam(G)$ be the diameter of $G$. A radio labeling $c$ of $G$ is an assignment of positive integers to the vertices of $G$ satisfying $d(u,v)+|c(u)-c(v)|\\geq \\diam(G) + 1.$ The maximum integer in the range of the labeling is its span. The radio number of $G$, $rn(G)$, is the minimum possible span. The family of gear graphs of order $n$, $G_n$, consists of planar graphs with $2n+1$ vertices and $3n$ edges. We prove that the radio number of the $n$-gear is $4n+2$."}
{"category": "Math", "title": "Quantifying the cost of simultaneous non-parametric approximation of several samples", "abstract": "We consider the standard non-parametric regression model with Gaussian errors but where the data consist of different samples. The question to be answered is whether the samples can be adequately represented by the same regression function. To do this we define for each sample a universal, honest and non-asymptotic confidence region for the regression function. Any subset of the samples can be represented by the same function if and only if the intersection of the corresponding confidence regions is non-empty. If the empirical supports of the samples are disjoint then the intersection of the confidence regions is always non--empty and a negative answer can only be obtained by placing shape or quantitative smoothness conditions on the joint approximation. Alternatively a simplest joint approximation function can be calculated which gives a measure of the cost of the joint approximation, for example, the number of extra peaks required."}
{"category": "Math", "title": "The Symmetry Preserving Removal Lemma", "abstract": "In this note we observe that in the hyper-graph removal lemma the edge removal can be done in a way that the symmetries of the original hyper-graph remain preserved. As an application we prove the following generalization of Szemer\\'edi's Theorem on arithmetic progressions. If in an Abelian group $A$ there are sets $S_1,S_2...,S_t$ such that the number of arithmetic progressions $x_1,x_2,...,x_t$ with $x_i\\in S_i$ is $o(|A|^2)$ then we can shrink each $S_i$ by $o(|A|)$ elements such that the new sets don't have such a diagonal arithmetic progression."}
{"category": "Math", "title": "Dualization of the Hopf algebra of secondary cohomology operations and the Adams spectral sequence", "abstract": "We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulae for the computation of the E_3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres."}
{"category": "Math", "title": "On Verifiable Sufficient Conditions for Sparse Signal Recovery via $\\ell_1$ Minimization", "abstract": "We propose novel necessary and sufficient conditions for a sensing matrix to be \"$s$-good\" - to allow for exact $\\ell_1$-recovery of sparse signals with $s$ nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect $\\ell_1$-recovery (nonzero measurement noise, nearly $s$-sparse signal, near-optimal solution of the optimization problem yielding the $\\ell_1$-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse $\\ell_1$-recovery and to efficiently computable upper bounds on those $s$ for which a given sensing matrix is $s$-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like Restricted Isometry or Restricted Eigenvalue properties."}
{"category": "Math", "title": "Logarithmic Sobolev inequalities: regularizing effect of L\\'evy operators and asymptotic convergence in the L\\'evy-Fokker-Planck equation", "abstract": "In this paper we study some applications of the L\\'evy logarithmic Sobolev inequality to the study of the regularity of the solution of the fractal heat equation, i. e. the heat equation where the Laplacian is replaced with the fractional Laplacian. It is also used to the study of the asymptotic behaviour of the L\\'evy-Ornstein-Uhlenbeck process."}
{"category": "Math", "title": "Balance laws with integrable unbounded sources", "abstract": "We consider the Cauchy problem for a $n\\times n$ strictly hyperbolic system of balance laws $$ \\{{array}{c} u_t+f(u)_x=g(x,u), x \\in \\mathbb{R}, t>0 u(0,.)=u_o \\in L^1 \\cap BV(\\mathbb{R}; \\mathbb{R}^n), | \\lambda_i(u)| \\geq c > 0 {for all} i\\in \\{1,...,n\\}, \\|g(x,\\cdot)\\|_{\\mathbf{C}^2}\\leq \\tilde M(x) \\in L1, {array}. $$ each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that the $\\mathbf{L}^1$ norm of $\\|g(x,\\cdot)\\|_{\\mathbf{C}^1}$ and $\\|u_o\\|_{BV(\\reali)}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [1] to unbounded (in $L^\\infty$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup."}
{"category": "Math", "title": "Geometric aspects of transversal Killing spinors on Riemannian flows", "abstract": "We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those Riemannian flows $M$ carrying non-trivial solutions to that equation in case $M$ is a local Riemannian product, a Sasakian manifold or 3-dimensional."}
{"category": "Math", "title": "Proportional fairness and its relationship with multi-class queueing networks", "abstract": "We consider multi-class single-server queueing networks that have a product form stationary distribution. A new limit result proves a sequence of such networks converges weakly to a stochastic flow level model. The stochastic flow level model found is insensitive. A large deviation principle for the stationary distribution of these multi-class queueing networks is also found. Its rate function has a dual form that coincides with proportional fairness. We then give the first rigorous proof that the stationary throughput of a multi-class single-server queueing network converges to a proportionally fair allocation. This work combines classical queueing networks with more recent work on stochastic flow level models and proportional fairness. One could view these seemingly different models as the same system described at different levels of granularity: a microscopic, queueing level description; a macroscopic, flow level description and a teleological, optimization description."}
{"category": "Math", "title": "Representation of operators in the time-frequency domain and generalized Gabor multipliers", "abstract": "Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator's best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator's best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows."}
{"category": "Math", "title": "The Weibull-Geometric distribution", "abstract": "In this paper we introduce, for the first time, the Weibull-Geometric distribution which generalizes the exponential-geometric distribution proposed by Adamidis and Loukas (1998). The hazard function of the last distribution is monotone decreasing but the hazard function of the new distribution can take more general forms. Unlike the Weibull distribution, the proposed distribution is useful for modeling unimodal failure rates. We derive the cumulative distribution and hazard functions, the density of the order statistics and calculate expressions for its moments and for the moments of the order statistics. We give expressions for the R\\'enyi and Shannon entropies. The maximum likelihood estimation procedure is discussed and an algorithm EM (Dempster et al., 1977; McLachlan and Krishnan, 1997) is provided for estimating the parameters. We obtain the information matrix and discuss inference. Applications to real data sets are given to show the flexibility and potentiality of the proposed distribution."}
{"category": "Math", "title": "On the dimension of invariant measures of endomorphisms of $\\mathbb{CP}^k$", "abstract": "Let $f$ be an endomorphism of $\\mathbb{CP}^k$ and $\\nu$ be an $f$-invariant measure with positive Lyapunov exponents $(\\lambda_1,\\...,\\lambda_k)$. We prove a lower bound for the pointwise dimension of $\\nu$ in terms of the degree of $f$, the exponents of $\\nu$ and the entropy of $\\nu$. In particular our result can be applied for the maximal entropy measure $\\mu$. When $k=2$, it implies that the Hausdorff dimension of $\\mu$ is estimated by $\\dim_{\\cal H} \\mu \\geq {\\log d \\over \\lambda_1} + {\\log d \\over \\lambda_2}$, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the $\\nu$-generic inverse branches of $f^n$ in $\\mathbb{CP}^k$. Our tools are a volume growth estimate for the bounded holomorphic polydiscs in $\\mathbb{CP}^k$ and a normalization theorem for the $\\nu$-generic inverse branches of $f^n$."}
{"category": "Math", "title": "On some new invariants for strong shift equivalence for shifts of finite type", "abstract": "We introduce a new computable invariant for strong shift equivalence of shifts of finite type. The invariant is based on an invariant introduced by Trow, Boyle, and Marcus, but has the advantage of being readily computable. We summarize briefly a large-scale numerical experiment aimed at deciding strong shift equivalence for shifts of finite type given by irreducible $2\\times 2$-matrices with entry sum less than 25, and give examples illustrating to power of the new invariant, i.e., examples where the new invariant can disprove strong shift equivalence whereas the other invariants that we use can not."}
{"category": "Math", "title": "Quantum theta functions and Gabor frames for modulation spaces", "abstract": "Representations of the celebrated Heisenberg commutation relations in quantum mechanics and their exponentiated versions form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we try to bridge the two communities, represented by the two co--authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis."}
{"category": "Math", "title": "Criteria for strong and weak random attractors", "abstract": "The theory of random attractors has different notions of attraction, amongst them pullback attraction and weak attraction. We investigate necessary and sufficient conditions for the existence of pullback attractors as well as of weak attractors."}
{"category": "Math", "title": "The Ricci flow does not preserve the set of Zoll metrics", "abstract": "The question of whether or not the set of Zoll metrics on the 2-sphere is connected is still open. Here we show that a naive application of the Ricci flow is not sufficient to answer this problem."}
{"category": "Math", "title": "Slicely Countably Determined Banach spaces", "abstract": "We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $\\ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $\\ell_1$ and that operators which do not fix copies of $\\ell_1$ on a space with the alternative Daugavet property satisfy the alternative Daugavet equation."}
{"category": "Math", "title": "Harmonic maps and Kaluza-Klein metrics on spheres", "abstract": "This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for surfaces and vector fields of constant norm, and existence is proved on two-tori. Classifications are given for conformal, quadratic and Killing vector fields on spheres. Finally, the class of metric considered on the tangent bundle is enlarged, permitting new vector fields to become harmonic."}
{"category": "Math", "title": "Remarques sur les spineurs de Killing transversaux", "abstract": "We show necessary conditions for the existence of transversal Killing spinors on a spin manifold endowed with a Riemannian flow."}
{"category": "Math", "title": "p(x)-Harmonic functions with unbounded exponent in a subdomain", "abstract": "We study the Dirichlet problem $-\\div(|\\nabla u|^{p(x)-2} \\nabla u) =0 $ in $\\Omega$, with $u=f$ on $\\partial \\Omega$ and $p(x) = \\infty$ in $D$, a subdomain of the reference domain $\\Omega$. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as $n \\to \\infty$ of the solutions $u_n$ to the corresponding problem when $p_n(x) =p(x) \\wedge n$, in particular, with $p_n = n$ in $D$. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, $\\infty$-harmonic within $D$. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of $\\Omega$."}
{"category": "Math", "title": "Modified action and differential operators on the 3-D sub-Riemannian sphere", "abstract": "Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere $\\mathbb S^3$. Our method is based on the Hamiltonian approach, where the corresponding Hamitonian system is solved with mixed boundary conditions. A closed form of the modified action is given. It is a sub-Riemannian invariant and plays the role of a distance on $\\mathbb S^3$."}
{"category": "Math", "title": "McKay correspondence for the Poincar\\'e series of Kleinian and Fuchsian singularities", "abstract": "We give a conceptual proof that the Poincar\\'e series of the coordinate algebra of a Kleinian singularity and of a Fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two Coxeter elements. These Coxeter elements are interpreted geometrically, using triangulated categories and spherical twist functors."}
{"category": "Math", "title": "Dynamics of vertex-reinforced random walks", "abstract": "We generalize a result from Volkov [Ann. Probab. 29 (2001) 66--91] and prove that, on a large class of locally finite connected graphs of bounded degree $(G,\\sim)$ and symmetric reinforcement matrices $a=(a_{i,j})_{i,j\\in G}$, the vertex-reinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete $d$-partite subgraph with possible loops plus its outer boundary. We first show that, in general, any stable equilibrium of a linear symmetric replicator dynamics with positive payoffs on a graph $G$ satisfies the property that its support is a complete $d$-partite subgraph of $G$ with possible loops, for some $d\\ge1$. This result is used here for the study of VRRWs, but also applies to other contexts such as evolutionary models in population genetics and game theory. Next we generalize the result of Pemantle [Probab. Theory Related Fields 92 (1992) 117--136] and Bena\\\"{{\\i}}m [Ann. Probab. 25 (1997) 361--392] relating the asymptotic behavior of the VRRW to replicator dynamics. This enables us to conclude that, given any neighborhood of a strictly stable equilibrium with support $S$, the following event occurs with positive probability: the walk localizes on $S\\cup\\partial S$ (where $\\partial S$ is the outer boundary of $S$) and the density of occupation of the VRRW converges, with polynomial rate, to a strictly stable equilibrium in this neighborhood."}
{"category": "Math", "title": "Ruan's conjecture and integral structures in quantum cohomology", "abstract": "This is an expository article on the recent studies of Ruan's crepant resolution/flop conjecture and its possible relations to the K-theory integral structure in quantum cohomology."}
{"category": "Math", "title": "$L^p$ continuity of wave operators in $\\Bbb Z$", "abstract": "We recover for discrete Schr\\\"odinger operators on the lattice Z, stronger analogues of the results by Weder, Galtabiar and Yajima and by D'Ancona and Fanelli on R."}
{"category": "Math", "title": "Foliations with degenerate Gauss maps on $\\mathbb P^4$", "abstract": "We obtain a classification of codimension one holomorphic foliations on $\\mathbb P^4$ with degenerate Gauss maps."}
{"category": "Math", "title": "On the $q$-Bessel Fourier transform", "abstract": "In this work, we are interested by the $q$-Bessel Fourier transform with a new approach. Many important results of this $q$-integral transform are proved with a new constructive demonstrations and we establish in particular the associated $q$-Fourier-Neumen expansion which involves the $q$-little Jacobi polynomials."}
{"category": "Math", "title": "On Elkies subgroups of l-torsion points in elliptic curves defined over a finite field", "abstract": "As a subproduct of the Schoof-Elkies-Atkin algorithm to count points on elliptic curves defined over finite fields of characteristic p, there exists an algorithm that computes, for l an Elkies prime, l-torsion points in an extension of degree l-1 at cost O(l max(l, \\log q)^2) bit operations in the favorable case where l < p/2. We combine in this work a fast algorithm for computing isogenies due to Bostan, Morain, Salvy and Schost with the p-adic approach followed by Joux and Lercier to get for the first time an algorithm valid without any limitation on l and p but of similar complexity."}
{"category": "Math", "title": "Bounds on the frequency of 1 in the Kolakoski word", "abstract": "We use a method of Goulden and Jackson to bound freq_1(K), the limiting frequency of 1 in the Kolakoski word K. We prove that |freq_1(K) - 1/2| <= 17/762, assuming the limit exists, and establish the semi-rigorous bound |freq_1(K) - 1/2| <= 1/46."}
{"category": "Math", "title": "A topological approach to induction theorems in Springer theory", "abstract": "We give a self-contained account of a construction due to Rossmann which lifts Springer's action of a Weyl group on the cohomology of a Springer fiber to an action on its homotopy type. We use this construction to produce a generalization of an \"induction theorem\" of Alvis and Lusztig, which relates the Springer representations attached to a reductive group to those attached to a Levi subgroup. Our generalization applies to more general centralizers and to representations of Weyl groups on mod p cohomology."}
{"category": "Math", "title": "Smooth and weak synthesis of the anti-diagonal in Fourier algebras of Lie groups", "abstract": "Let $G$ be a Lie group of dimension $n$, and let $A(G)$ be the Fourier algebra of $G$. We show that the anti-diagonal $\\check{\\Delta}_G=\\{(g,g^{-1})\\in G\\times G \\mid g\\in G\\}$ is both a set of local smooth synthesis and a set of local weak synthesis of degree at most $[\\frac{n}{2}]+1$ for $A(G\\times G)$. We achieve this by using the concept of the cone property in \\cite{ludwig-turowska}. For compact $G$, we give an alternative approach to demonstrate the preceding results by applying the ideas developed in \\cite{forrest-samei-spronk}. We also present similar results for sets of the form $HK$, where both $H$ and $K$ are subgroups of $G\\times G\\times G\\times G$ of diagonal forms. Our results very much depend on both the geometric and the algebraic structure of these sets."}
{"category": "Math", "title": "Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves", "abstract": "We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has odd relative dimension at least 11, then CH^{i}(X) is finitely generated for i<=4."}
{"category": "Math", "title": "Hodge realizations of 1-motives and the derived Albanese", "abstract": "We prove that the embedding of the derived category of 1-motives into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor $LAlb$, commute with the Hodge realization."}
{"category": "Math", "title": "Second cohomology groups for algebraic groups and their Frobenius kernels", "abstract": "Let $G$ be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic $p > 0$. Let $T$ be a maximal split torus in $G$, $B \\supset T$ be a Borel subgroup of $G$ and $U$ its unipotent radical. Let $F: G \\rightarrow G$ be the Frobenius morphism. For $r \\geq 1$ define the Frobenius kernel, $G_r$, to be the kernel of $F$ iterated with itself $r$ times. Define $U_r$ (respectively $B_r$) to be the kernel of the Frobenius map restricted to $U$ (respectively $B$). Let $X(T)$ be the integral weight lattice and $X(T)_+$ be the dominant integral weights. The computations of particular importance are $\\h^2(U_1,k)$, $\\h^2(B_r,\\la)$ for $\\la \\in X(T)$, $\\h^2(G_r,H^0(\\la))$ for $\\la \\in X(T)_+$, and $\\h^2(B,\\la)$ for $\\la \\in X(T)$. The above cohomology groups for the case when the field has characteristic 2 one computed in this paper. These computations complete the picture started by Bendel, Nakano, and Pillen for $p \\geq 3$ \\cite{BNP2}."}
{"category": "Math", "title": "The relative commutant of separable C*-algebras of real rank zero", "abstract": "We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C*-algebra in its ultrapower depend on the choice of the ultrafilter?"}
{"category": "Math", "title": "Free brace algebras are free prelie algebras", "abstract": "Let g be a free brace algebra. This structure implies that g is also a prelie algebra and a Lie algebra. It is already known that g is a free Lie algebra. We prove here that g is also a free prelie algebra, using a description of g with the help of planar rooted trees, a permutative product, and anipulations on the Poincar\\'e-Hilbert series of g."}
{"category": "Math", "title": "On a computer-aided approach to the computation of Abelian integrals", "abstract": "An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four."}
{"category": "Math", "title": "A new approach to the family of singularities $Re(x+iy)^m$", "abstract": "Assume that $m\\ge 2$ and let $l$ be a nonnegative integer with $l\\ge m-4$. We give an alternative proof of the fact that any smooth function defined locally around $(0,0)\\in \\mathbb{R}^2$ with the Taylor power series at $(0,0)$ beginning with $$Re(x+iy)^m+0+...+0$$ ($l$ zeros) is diffeomorphically equivalent to $Re(x+iy)^m$ at $(0,0)$. For $m\\ge 5$ and $C\\ne 0$ we show that the function $$Re(x+iy)^m+C(x^2+y^2)^{m-2}$$ is not diffeomorphically equivalent to $Re(x+iy)^m$ at $(0,0)$."}
{"category": "Math", "title": "Basic properties of nonsmooth Hormander's vector fields and Poincare's inequality", "abstract": "We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption that the coefficients belong to C^{r-1,1}, Poincare's inequality. By known results, these facts also imply a Sobolev embedding. All these tools allow to draw some consequences about second order differential operators modeled on these nonsmooth Hormander's vector fields."}
{"category": "Math", "title": "An analogue of the Szemeredi Regularity Lemma for bounded degree graphs", "abstract": "We show that a sufficiently large graph of bounded degree can be decomposed into quasi-homogeneous pieces. The result can be viewed as a \"finitarization\" of the classical Farrell-Varadarajan Ergodic Decomposition Theorem."}
{"category": "Math", "title": "La droite de Berkovich sur Z", "abstract": "We study here the Berkovich line over the ring of integers of a number field. It is a natural object which contains complex and non-Archimedean analytic spaces associated to each place. We prove that this line satisfies good topological and algebraic properties and exhibit a few examples of Stein spaces that lie in it. We derive applications to the study of convergent arithmetic power series: choice of zeroes and poles, noetherianity of global rings and inverse Galois problem. Typical examples of such power series are given by analytic functions on the open complex unit disk whose Taylor development in 0 has integer coefficients."}
{"category": "Math", "title": "The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent", "abstract": "The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The work is motivated by two applications: an existence result for the problem of maximizing the rate of exponential decay of a damped membrane and an approximate controllability result for the bilinear Schr\\\"odinger equation."}
{"category": "Math", "title": "The representation type of Ariki-Koike algebras and cyclotomic q-Schur algebras", "abstract": "We give a necessary and sufficient condition on parameters for Ariki-Koike algebras (resp. cyclotomic q-Schur algebras) to be of finite representation type."}
{"category": "Math", "title": "Transfer and Chern classes for extraspecial p-groups", "abstract": "In the cohomology ring of an extraspecial p-group, the subring generated by Chern classes and transfers is studied. This subring is strictly larger than the Chern subring, but still not the whole cohomology ring, even modulo nilradical. A formula is obtained relating Chern classes to transfers."}
{"category": "Math", "title": "Thom series of contact singularities", "abstract": "Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of contact singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes, and with iterated residue identities are revealed."}
{"category": "Math", "title": "On the notion of geometry over $\\F_1$", "abstract": "We refine the notion of variety over the \"field with one element\" developed by C. Soul\\'e by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. Tits. We then solve an open question of C. Soul\\'e by proving, using results of J. Tits and C. Chevalley, that Chevalley group schemes are examples of varieties over a quadratic extension of the above \"field\"."}
{"category": "Math", "title": "Stability Selection", "abstract": "Estimation of structure, such as in variable selection, graphical modelling or cluster analysis is notoriously difficult, especially for high-dimensional data. We introduce stability selection. It is based on subsampling in combination with (high-dimensional) selection algorithms. As such, the method is extremely general and has a very wide range of applicability. Stability selection provides finite sample control for some error rates of false discoveries and hence a transparent principle to choose a proper amount of regularisation for structure estimation. Variable selection and structure estimation improve markedly for a range of selection methods if stability selection is applied. We prove for randomised Lasso that stability selection will be variable selection consistent even if the necessary conditions needed for consistency of the original Lasso method are violated. We demonstrate stability selection for variable selection and Gaussian graphical modelling, using real and simulated data."}
{"category": "Math", "title": "Entropy and Poincar\\'e recurrence from a geometrical viewpoint", "abstract": "We study Poincar\\'e recurrence from a purely geometrical viewpoint. We prove that the metric entropy is given by the exponential growth rate of return times to dynamical balls. This is the geometrical counterpart of Ornstein-Weiss theorem. Moreover, we show that minimal return times to dynamical balls grow linearly with respect to its length. Finally, some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures are given."}
{"category": "Math", "title": "On $v$--domains and star operations", "abstract": "Let $\\ast$ be a star operation on an integral domain $D$. Let $\\f(D)$ be the set of all nonzero finitely generated fractional ideals of $D$. Call $D$ a $\\ast$--Pr\\\"ufer (respectively, $(\\ast, v)$--Pr\\\"ufer) domain if $(FF^{-1})^{\\ast}=D$ (respectively, $(F^vF^{-1})^{\\ast}=D$) for all $F\\in \\f(D)$. We establish that $\\ast$--Pr\\\"ufer domains (and $(\\ast, v)$--Pr\\\"ufer domains) for various star operations $\\ast $ span a major portion of the known generalizations of Pr\\\"{u}fer domains inside the class of $v$--domains. We also use Theorem 6.6 of the Larsen and McCarthy book [Multiplicative Theory of Ideals, Academic Press, New York--London, 1971], which gives several equivalent conditions for an integral domain to be a Pr\\\"ufer domain, as a model, and we show which statements of that theorem on Pr\\\"ufer domains can be generalized in a natural way and proved for $\\ast$--Pr\\\"ufer domains, and which cannot be. We also show that in a $\\ast $--Pr\\\"ufer domain, each pair of $\\ast $-invertible $\\ast $-ideals admits a GCD in the set of $\\ast $-invertible $\\ast $-ideals, obtaining a remarkable generalization of a property holding for the \"classical\" class of Pr\\\"ufer $v$--multiplication domains. We also link $D$ being $\\ast $--Pr\\\"ufer (or $(\\ast, v)$--Pr\\\"ufer) with the group Inv$^{\\ast}(D)$ of $\\ast $-invertible $\\ast $-ideals (under $\\ast$-multiplication) being lattice-ordered."}
{"category": "Math", "title": "Strong Law of Large Numbers for Fragmentation Processes", "abstract": "In the spirit of a classical results for Crump-Mode-Jagers processes, we prove a strong law of large numbers for homogenous fragmentation processes. Specifically, for self-similar fragmentation processes, including homogenous processes, we prove the almost sure convergence of an empirical measure associated with the stopping line corresponding to first fragments of size strictly smaller than $\\eta$ for $1\\geq \\eta >0$."}
{"category": "Math", "title": "New Discretization of Complex Analysis: The Euclidean and Hyperbolic Planes", "abstract": "Few years ago we developed jointly with I.Dynnikov new discretization of complex analysis (DCA) based on the two-dimensional manifolds with colored black/white triangulation. Especially deep results were obtained for the Euclidean plane with equilateral triangle lattice. In the present work we develop a DCA theory for the analogs of equilateral triangle lattice in Hyperbolic plane. Some specific very interesting \"dynamical phenomena\" appear in this case solving most fundamental boundary problems. Mike Boyle from the University of Maryland helped to use here the methods of symbolic dynamics"}
{"category": "Math", "title": "Prime numbers in logarithmic intervals", "abstract": "Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\\leq X$ is a prime number and $h=\\odi{X}$. Then we will apply this to prove that for every $\\lambda>1/2$ there exists a positive proportion of primes $p\\leq X$ such that the interval $(p,p+ \\lambda\\log X]$ contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers $\\lambda>1$ with the property that there is a positive proportion of integers $m\\leq X$ such that the interval $(m,m+ \\lambda\\log X]$ contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers $m\\leq X$ such that the interval $(m,m+ \\lambda\\log X]$ contains at least a prime number. The last application of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes $p\\leq X$ such that the interval $(p,p+ \\lambda\\log X]$ contains no primes."}
{"category": "Math", "title": "Poisson Limit for Associated Random Fields", "abstract": "We prove that under an easily verifiable set of conditions a sequence of associated random fields converges under rescaling to the Poisson Point Process and give a couple of examples."}
{"category": "Math", "title": "Thermodynamic Limit for Large Random Trees", "abstract": "We consider Gibbs distributions on finite random plane trees with bounded branching. We show that as the order of the tree grows to infinity, the distribution of any finite neighborhood of the root of the tree converges to a limit. We compute the limiting distribution explicitly and study its properties. We introduce an infinite random tree consistent with these limiting distributions and show that it satisfies a certain form of the Markov property. We also study the growth of this tree and prove several limit theorems including a diffusion approximation."}
{"category": "Math", "title": "Coloring Simple Hypergraphs", "abstract": "Fix an integer $k \\ge 3$. A $k$-uniform hypergraph is simple if every two edges share at most one vertex. We prove that there is a constant $c$ depending only on $k$ such that every simple $k$-uniform hypergraph $H$ with maximum degree $\\D$ has chromatic number satisfying $$\\chi(H) <c (\\frac{\\D}{\\log \\D})^{\\frac{1}{k-1}}.$$ This implies a classical result of Ajtai-Koml\\'os-Pintz-Spencer-Szemer\\'edi and its strengthening due to Duke-Lefmann-R\\\"odl. The result is sharp apart from the constant $c$."}
{"category": "Math", "title": "Presenting the cohomology of a Schubert variety", "abstract": "We extend the short presentation due to [Borel '53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its Schubert varieties. Our result is stated in a root-system uniform manner by introducing the essential set of a Coxeter group element, generalizing and giving a new characterization of [Fulton '92]'s definition for permutations. Further refinements are obtained in type A."}
{"category": "Math", "title": "The Lawson-Yau Formula and its generalization", "abstract": "The Euler characteristic of Chow varieties of algebraic cycles of a given degree in complex projective spaces was computed by Blaine Lawson and Stephen Yau by using holomorphic symmetries of cycles spaces. In this paper we compute this in a direct and elementary way and generalize this formula to the l-adic Euler-Poincare characteristic for Chow varieties over any algebraically closed field. Moreover, the Euler characteristic for Chow varieties with certain group action is calculated. In particular, we calculate the Euler characteristic of the space of right quaternionic cycles of a given dimension and degree in complex projective spaces."}
{"category": "Math", "title": "Uniform estimates for order statistics and Orlicz functions", "abstract": "We establish uniform estimates for order statistics of sequences of independent identically distributed random variables with log-concave distribution in terms of Orlicz norms associated with the distribution function of the random variables."}
{"category": "Math", "title": "Derived categories of small toric Calabi-Yau 3-folds and counting invariants", "abstract": "We first construct a derived equivalence between a small crepant resolution of an affine toric Calabi-Yau 3-fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wall-crossing formula for the generating function of the counting invariants of perverse coherent systems. As an application we provide certain equations on Donaldson-Thomas, Pandeharipande-Thomas and Szendroi's invariants. Finally, we show that moduli spaces associated with a quiver given by successive mutations are realized as the moduli spaces associated the original quiver by changing the stability conditions."}
{"category": "Math", "title": "Multidimensional Dilation Operators, Boyd and Shimogaki Indices of Bilateral Weight Grand Lebesque Spaces", "abstract": "In this paper we compute the norm of dilation operators, multidimensional Boyd`s and Shimogaki`s indices in the Bilateral Grand Lebesgue Spaces and consider some applications."}
{"category": "Math", "title": "Partial duality and Bollobas and Riordan's ribbon graph polynomial", "abstract": "Recently S. Chmutov introduced a generalization of the dual of a ribbon (or embedded) graph and proved a relation between Bollobas and Riordan's ribbon graph polynomial of a ribbon graph and its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the relation via the homfly polynomial of a knot."}
{"category": "Math", "title": "Stable solutions of elliptic equations on Riemannian manifolds", "abstract": "This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincar\\'e inequality which allows to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold."}
{"category": "Math", "title": "Weakly group-theoretical and solvable fusion categories", "abstract": "We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups - weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq^2, where p,q,r are distinct primes."}
{"category": "Math", "title": "Simultaneous similarity and triangularization of sets of 2 by 2 matrices", "abstract": "Let $\\mathcal{A}=(A_{1},...,A_{n},...)$ be a finite or infinite sequence of $2\\times2$ matrices with entries in an integral domain. We show that, except for a very special case, $\\mathcal{A}$ is (simultaneously) triangularizable if and only if all pairs $(A_{j},A_{k})$ are triangularizable, for $1\\leq j,k\\leq\\infty$. We also provide a simple numerical criterion for triangularization. Using constructive methods in invariant theory, we define a map (with the minimal number of invariants) that distinguishes simultaneous similarity classes for non-commutative sequences over a field of characteristic $\\neq2$. We also describe canonical forms for sequences of $2\\times2$ matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants."}
{"category": "Math", "title": "Partial Data for the Calderon Problem in Two Dimensions", "abstract": "We show in two dimensions that measuring Dirichlet data for the conductivity equation on an open subset of the boundary and, roughly speaking, Neumann data in slightly larger set than the complement uniquely determines the conductivity on a simply connected domain. The proof is reduced to show a similar result for the Schr\\\"odinger equation. Using Carleman estimates with degenerate weights we construct appropriate complex geometrical optics solutions to prove the results."}
{"category": "Math", "title": "Integrally Closed Ideals on Log Terminal Surfaces are Multiplier Ideals", "abstract": "We show that all integrally closed ideals on log terminal surfaces are multiplier ideals by extending an existing proof for smooth surfaces."}
{"category": "Math", "title": "Groups with the same cohomology as their pro-$p$ completions", "abstract": "For any prime $p$ and group $G$, denote the pro-$p$ completion of $G$ by $\\hat{G}^p$. Let $\\mathcal{C}$ be the class of all groups $G$ such that, for each natural number $n$ and prime number $p$, $H^n(\\hat{G^p},\\mathbb Z/p)\\cong H^n(G, \\mathbb Z/p)$, where $\\mathbb Z/p$ is viewed as a discrete, trivial $\\hat{G}^p$-module. In this article we identify certain kinds of groups that lie in $\\mathcal{C}$. In particular, we show that right-angled Artin groups are in $\\mathcal{C}$ and that this class also contains some special types of free products with amalgamation."}
{"category": "Math", "title": "Commutators on $\\ell_1$", "abstract": "The main result is that the commutators on $\\ell_1$ are the operators not of the form $\\lambda I + K$ with $\\lambda\\neq 0$ and $K$ compact. We generalize Apostol's technique (1972, Rev. Roum. Math. Appl. 17, 1513 - 1534) to obtain this result and use this generalization to obtain partial results about the commutators on spaces $\\X$ which can be represented as $\\displaystyle \\X\\simeq (\\bigoplus_{i=0}^{\\infty} \\X)_{p}$ for some $1\\leq p<\\infty$ or $p=0$. In particular, it is shown that every compact operator on $L_1$ is a commutator. A characterization of the commutators on $\\ell_{p_1}\\oplus\\ell_{p_2}\\oplus...\\oplus\\ell_{p_n}$ is given. We also show that strictly singular operators on $\\ell_{\\infty}$ are commutators."}
{"category": "Math", "title": "Numerical Solution of Multiple Nonlinear Volterra Integral Equations", "abstract": "We analyze a discretization method for solving nonlinear integral equations that contain multiple integrals. These equations include integral equations with a Volterra series, instead of a single integral term, on one side of the equation. We prove existence and uniqueness of solutions, and convergence and estimates of the order of convergence for the numerical methods of solution."}
{"category": "Math", "title": "Toeplitz-composition C*-algebras for certain finite Blaschke products", "abstract": "Let R be a finite Blaschke product of degree at least two with R(0)=0. Then there exists a relation between the associated composition operator C_R on the Hardy space and the C*-algebra associated with the complex dynamical system on the Julia set of R. We study the C*-algebra generated by both the composition operator C_R and the Toeplitz operator T_z to show that the quotient algebra by the ideal of the compact operators is isomorphic to the C*-algebra associated with the complex dynamical system, which is simple and purely infinite."}
{"category": "Math", "title": "On rigid Hirzebruch genera", "abstract": "The classical multiplicative (Hirzebruch) genera of manifolds have the wonderful property which is called rigidity. Rigidity of a genus h means that if a compact connected Lie group G acts on a manifold X, then the equivariant genus h^G(X) is independent on G, i.e. h^G(X)=h(X). In this paper we are considering the rigidity problem for complex manifolds. In particular, we are proving that a genus is rigid if and only if it is a generalized Todd genus."}
{"category": "Math", "title": "Some Notes on Standard Borel and Related Spaces", "abstract": "These notes give an elementary approach to parts of the theory of standard Borel and analytic spaces."}
{"category": "Math", "title": "Desingularization of some moduli scheme of stable sheaves on a surface", "abstract": "Let $X$ be a nonsingular projective surface over an algebraically closed field with characteristic zero, and $H_-$ and $H_+$ ample line bundles on $X$ separated by only one wall of type $(c_1,c_2)$. Suppose the moduli scheme $M(H_-)$ of rank-two $H_-$-stable sheaves with Chern classes $(c_1,c_2)$ is non-singular. We shall construct a desingularization of $M(H_+)$ by using $M(H_-)$. As an application, we consider whether singularities of $M(H_+)$ are terminal or not when $X$ is ruled or elliptic."}
{"category": "Math", "title": "Lifting Galois representations over arbitrary number fields", "abstract": "It is proved that every two-dimensional residual Galois representation of the absolute Galois group of an arbitrary number field lifts to a characteristic zero $p$-adic representation, if local lifting problems at places above $p$ are unobstructed."}
{"category": "Math", "title": "Bandlet Image Estimation with Model Selection", "abstract": "To estimate geometrically regular images in the white noise model and obtain an adaptive near asymptotic minimaxity result, we consider a model selection based bandlet estimator. This bandlet estimator combines the best basis selection behaviour of the model selection and the approximation properties of the bandlet dictionary. We derive its near asymptotic minimaxity for geometrically regular images as an example of model selection with general dictionary of orthogonal bases. This paper is thus also a self contained tutorial on model selection with orthogonal bases dictionary."}
{"category": "Math", "title": "On the geometry of biharmonic submanifolds in Sasakian space forms", "abstract": "We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in Sasakian space forms. Finally, using the Boothby-Wang fibration, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces."}
{"category": "Math", "title": "The vector-valued non-homogeneous Tb theorem", "abstract": "The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces of functions with values in a Banach space with the unconditionality property of martingale differences (UMD). The new proof deals directly with all Lebesgue exponents p in the range 1<p<infinity, and relies on delicate estimates for the non-homogenous \"Haar\" functions, as well as McConnell's (1989) decoupling inequality for tangent martingale differences."}
{"category": "Math", "title": "Degree-one maps, surgery and four-manifolds", "abstract": "We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold $M$ to a closed, oriented 3-manifold $N$ if and only if $M$ can be obtained from $N$ by surgery about a link in $N$ each of whose components is an unknot. We use this to interpret the existence of degree-one maps between closed 3-manifolds in terms of smooth 4-manifolds. More precisely, we show that there is a degree-one map from $M$ to $N$ if and only if there is a smooth embedding of $M$ in $W=(N\\times I)#_n \\bar{\\C P^2}#_m {\\C P^2}$, for some $m\\geq 0$, $n\\geq 0$ which separates the boundary components of $W$. This is motivated by the relation to topological field theories, in particular the invariants of Ozsvath and Szabo."}
{"category": "Math", "title": "How to Construct a Dirac Operator in Infinite Dimensions", "abstract": "We define the notion of a co-Riemannian structure and show how it can be used to define the Dirac operator on an appropriate infinite dimensional manifold. In particular, this approach works for the smooth loop space of a so-called string manifold."}
{"category": "Math", "title": "A Proof On Arnold-Chekanov Conjecture", "abstract": "In this article, we give a proof on the Arnold-Chekanov Lagrangian intersection conjecture on the cotangent bundles and its generalizations."}
{"category": "Math", "title": "T-entropy and Variational principle for the spectral radius of weighted shift operators", "abstract": "In this paper we introduce a new functional invariant of discrete time dynamical systems -- the so-called t-entropy. The main result is that this t-entropy is the Legendre dual functional to the logarithm of the spectral radius of the weighted shift operator on $L^1(X,m)$ generated by the dynamical system. This result is called the Variational principle and is similar to the classical variational principle for the topological pressure."}
{"category": "Math", "title": "Incompressibility and Least-Area surfaces", "abstract": "We show that if $F$ is a smooth, closed, orientable surface embedded in a closed, orientable 3-manifold $M$ such that for each Riemannian metric $g$ on $M$, $F$ is isotopic to a least-area surface $F(g)$, then $F$ is incompressible."}
{"category": "Math", "title": "The co-Riemannian Structure of Smooth Loop Spaces", "abstract": "We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator."}
{"category": "Math", "title": "Algebraic and Geometric intersection numbers for free groups", "abstract": "We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of $S^2\\times S^1$."}
{"category": "Math", "title": "T-entropy and Variational Principle for the spectral radius of transfer and weighted shift operators", "abstract": "The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or it is a topological Markov chain. As the main summands these principles contain the integrals over invariant measures and the Kolmogorov--Sinai entropy. In the article we derive the Variational Principle for an arbitrary dynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov--Sinai entropy but a new invariant of entropy type -- the t-entropy."}
{"category": "Math", "title": "Steiner t-designs for large t", "abstract": "One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist."}
{"category": "Math", "title": "An orthogonality relation for multivariable Bessel polynomials", "abstract": "In a recent paper we introduced a multivariable generalisation of the Bessel polynomials, depending on one extra parameter, and related to the so-called hyperbolic Sutherland model with external Morse potential. In this paper we obtain a corresponding multivariable generalisation of a well-known orthogonality relation for the (one-variable) Bessel polynomials due to Krall and Frink."}
{"category": "Math", "title": "On the norm of the Beurling-Ahlfors operator in several dimensions", "abstract": "The Lp operator norm of the generalized Beurling-Ahlfors transformation in n variables is at most (n/2+1)(p-1) for p>2. This improves on earlier results in all dimensions n>2. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms."}
{"category": "Math", "title": "Anosov Automorphisms of Nilpotent Lie Algebras", "abstract": "Each matrix A in GL_n(Z) naturally defines an automorphism f of the free r-step nilpotent Lie algebra on n generators. We study the relationship between the matrix A and the eigenvalues and rational invariant subspaces for f. We give applications to the study of Anosov automorphisms."}
{"category": "Math", "title": "On Quasitoric Orbifolds", "abstract": "Quasitoric spaces were introduced by Davis and Januskiewicz in their 1991 Duke paper. There they extensively studied topological invariants of quasitoric manifolds. These manifolds are generalizations or topological counterparts of nonsingular projective toric varieties. In this article we study structures and invariants of quasitoric orbifolds. In particular, we discuss equivalent definitions and determine the orbifold fundamental group, rational homology groups and cohomology ring of a quasitoric orbifold. We determine whether any quasitoric orbifold can be the quotient of a smooth manifold by a finite group action or not. We prove existence of stable almost complex structure and describe the Chen-Ruan cohomology groups of an almost complex quasitoric orbifold."}
{"category": "Math", "title": "Classifying smooth lattice polytopes via toric fibrations", "abstract": "We define Q-normal lattice polytopes. Natural examples of such polytopes are Cayley sums of strictly combinatorially equivalent lattice polytopes, which correspond to particularly nice toric fibrations, namely toric projective bundles. In a recent paper Batyrev and Nill have suggested that there should be a bound, N(d), such that every lattice polytope of degree d and dimension at least N(d) decomposes as a Cayley sum. We give a sharp answer to this question for smooth Q-normal polytopes. We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a Cayley sum of strictly combinatorially equivalent polytopes if n is greater than or equal to 2d+1. The proof relies on the study of the nef value morphism associated to the corresponding toric embedding."}
{"category": "Math", "title": "A time-variant norm constrained interpolation problem arising from relaxed commutant lifting", "abstract": "A time-variant analogue of an interpolation problem equivalent to the relaxed commutant lifting problem is introduced and studied. In a somewhat less general form the problem already appears in the analysis of the set of all solutions to the three chain completion problem. The interpolants are upper triangular operator matrices of which the columns induce contractive operators. The set of all solutions of the problem is described explicitly. The results presented are time-variant analogues of the main theorems in [A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to the relaxed commutant lifting problem, Acta Sci. Math. (Szeged) 72 (2006), 299--318]."}
{"category": "Math", "title": "Milnor fibers over singular toric varieties and nearby cycle sheaves", "abstract": "We propose a new sheaf-theoretical method for the calculation of the monodromy zeta functions of Milnor fibrations. As an application, classical formulas of Kushnirenko and Varchenko etc. concerning polynomials on $\\CC^n$ will be generalized to polynomial functions on any toric variety."}
{"category": "Math", "title": "Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves", "abstract": "By using sheaf-theoretical methods such as constructible sheaves, we generalize the formula of Libgober-Sperber concerning the zeta functions of monodromy at infinity of polynomial maps into various directions. In particular, some formulas for the zeta functions of global monodromy along the fibers of bifurcation points of polynomial maps will be obtained."}
{"category": "Math", "title": "Stochastic analysis of Bernoulli processes", "abstract": "These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time."}
{"category": "Math", "title": "Economical toric spines via Cheeger's Inequality", "abstract": "Let $G_{\\infty}=(C_m^d)_{\\infty}$ denote the graph whose set of vertices is $\\{1,..., m\\}^d$, where two distinct vertices are adjacent iff they are either equal or adjacent in $C_m$ in each coordinate. Let $G_{1}=(C_m^d)_1$ denote the graph on the same set of vertices in which two vertices are adjacent iff they are adjacent in one coordinate in $C_m$ and equal in all others. Both graphs can be viewed as graphs of the $d$-dimensional torus. We prove that one can delete $O(\\sqrt d m^{d-1})$ vertices of $G_1$ so that no topologically nontrivial cycles remain. This improves an $O(d^{\\log_2 (3/2)}m^{d-1})$ estimate of Bollob\\'as, Kindler, Leader and O'Donnell. We also give a short proof of a result implicit in a recent paper of Raz: one can delete an $O(\\sqrt d/m)$ fraction of the edges of $G_{\\infty}$ so that no topologically nontrivial cycles remain in this graph. Our technique also yields a short proof of a recent result of Kindler, O'Donnell, Rao and Wigderson; there is a subset of the continuous $d$-dimensional torus of surface area $O(\\sqrt d)$ that intersects all nontrivial cycles. All proofs are based on the same general idea: the consideration of random shifts of a body with small boundary and no- nontrivial cycles, whose existence is proved by applying the isoperimetric inequality of Cheeger or its vertex or edge discrete analogues."}
{"category": "Math", "title": "Compact operators that commute with a contraction", "abstract": "Let $T$ be a $C_0$--contraction on a separable Hilbert space. We assume that $I_H-T^*T$ is compact. For a function $f$ holomorphic in the unit disk $\\DD$ and continuous on $\\bar\\DD$, we show that $f(T)$ is compact if and only if $f$ vanishes on $\\sigma (T)\\cap \\TT$, where $\\sigma (T)$ is the spectrum of $T$ and $\\TT$ the unit circle. If $f$ is just a bounded holomorphic function on $\\DD$ we prove that $f(T)$ is compact if and only if $\\lim_{n\\to \\infty} T^nf(T) =0$."}
{"category": "Math", "title": "Formal deformations and their categorical general fibre", "abstract": "We study the general fibre of a formal deformation over the formal disk of a projective variety from the view point of abelian and derived categories. The abelian category of coherent sheaves of the general fibre is constructed directly from the formal deformation and is shown to be linear over the field of Laurent series. The various candidates for the derived category of the general fibre are compared. If the variety is a surface with trivial canonical bundle, we show that the derived category of the general fibre is again a linear triangulated category with a Serre functor given by the square of the shift functor."}
{"category": "Math", "title": "Matrix Corepresentations for SL_q(N) and SU_q(N)", "abstract": "We give an algorithm for computing matrix corepresentations for special linear and special unitary quantum groups using a combinatorial re-indexing of basis elements."}
{"category": "Math", "title": "Extension of twisted Hodge metrics for K\\\"ahler morphisms", "abstract": "Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X, and consider direct image sheaves F = R^qf_*(K_{X/Y} \\otimes E) for q \\geq 0. In our previous paper, we discussed the Nakano semi-positivity of F with respect to the so-called Hodge metric, when the map f is smooth. In this paper, we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(F) ``over Y-Z'' as a singular Hermitian metric with semi-positive curvature ``over Y''. As a particular consequence, if Y is projective, R^qf_*(K_{X/Y} \\otimes E) is weakly positive over Y-Z in the sense of Viehweg."}
{"category": "Math", "title": "Remarks on the extension of twisted Hodge metrics", "abstract": "This is a technical introduction to the paper \"Extension of twisted Hodge metrics for Kahler morphisms\" by the authors."}
{"category": "Math", "title": "A converse to the Grace--Walsh--Szeg\\H{o} theorem", "abstract": "We prove that the symmetrizer of a permutation group preserves stability of a polynomial if and only if the group is orbit homogeneous. A consequence is that the hypothesis of permutation invariance in the Grace-Walsh-Szeg\\H{o} Coincidence Theorem cannot be relaxed. In the process we obtain a new characterization of the \\emph{Grace-like polynomials} introduced by D. Ruelle, and prove that the class of such polynomials can be endowed with a natural multiplication."}
{"category": "Math", "title": "On $\\mathcal{OL}_\\infty$ structure of nuclear, quasidiagonal C*-algebras", "abstract": "We continue the study of $\\mathcal{OL}_\\infty$ structure of nuclear $C^*$-algebras initiated by Junge, Ozawa and Ruan. In particular, we prove if $\\mathcal{OL}_\\infty(A)<1.005,$ then $A$ has a separating family of irreducible, stably finite representations. As an application we give examples of nuclear, quasidiagonal $C^*$-algebras $A$ with $\\mathcal{OL}_\\infty(A)>1.$"}
{"category": "Math", "title": "Schr\\\"odinger Operators Defined by Interval Exchange Transformations", "abstract": "We discuss discrete one-dimensional Schr\\\"odinger operators whose potentials are generated by an invertible ergodic transformation of a compact metric space and a continuous real-valued sampling function. We pay particular attention to the case where the transformation is a minimal interval exchange transformation. Results about the spectrum and the spectral type of these operators are established. In particular, we provide the first examples of transformations for which the associated Schr\\\"odinger operators have purely singular spectrum for every non-constant continuous sampling function."}
{"category": "Math", "title": "A note on minors determined by clones of semilattices", "abstract": "The C-minor partial orders determined by the clones generated by a semilattice operation (and possibly the constant operations corresponding to its identity or zero elements) are shown to satisfy the descending chain condition."}
{"category": "Math", "title": "On an elliptic Kirchhoff-type problem depending on two parameters", "abstract": "In this paper, we consider the Dirichlet problem associated to an elliptic Kirchhoff-type equation depending on two parameters. Under rather general and natural assumptions, we prove that, for certain values of the parameters, the problem has at least three solutions."}
{"category": "Math", "title": "Compactness for manifolds and integral currents with bounded diameter and volume", "abstract": "By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class or oriented $k$-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio-Kirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which we replace uniform compactness of the sequence with uniform bounds on volume and diameter."}
{"category": "Math", "title": "Uniqueness for solutions of the two-phase Stefan problem with signed measures as data", "abstract": "We show uniqueness of solutions to the two-phase Stefan problem which have signed measures as initial data."}
{"category": "Math", "title": "KP hierarchy for Hodge integrals", "abstract": "Starting from the ELSV formula, we derive a number of new equations on the generating functions for Hodge integrals over the moduli space of complex curves. This gives a new simple and uniform treatment of certain known results on Hodge integrals like Witten's conjecture, Virasoro constrains, Faber's lambda_g conjecture etc. Among other results we show that a properly arranged generating function for Hodge integrals satisfies the equations of the KP hierarchy."}
{"category": "Math", "title": "About Bernoulli's Numbers", "abstract": "In this article we present a simple proof of Borevich-Shafarevich's method to compute the sum of the first n natural numbers of the same power. We also prove several properties of Bernoulli's numbers."}
{"category": "Math", "title": "Continuous Measures on Homogenous Spaces", "abstract": "In this paper we generalize Wiener's characterization of continuous measures to compact homogenous manifolds. In particular, we give necessary and sufficient conditions on probability measures on compact semisimple Lie groups and nilmanifolds to be continuous. The methods use only simple properties of heat kernels."}
{"category": "Math", "title": "Duality questions for operators, spectrum and measures", "abstract": "We explore spectral duality in the context of measures in $\\br^n$, starting with partial differential operators and Fuglede's question (1974) about the relationship between orthogonal bases of complex exponentials in $L^2(\\Omega)$ and tiling properties of $\\Omega$, then continuing with affine iterated function systems. We review results in the literature from 1974 up to the present, and we relate them to a general framework for spectral duality for pairs of Borel measures in $\\br^n$, formulated first by Jorgensen and Pedersen."}
{"category": "Math", "title": "Isometries and spectra of multiplication operators on the Bloch space", "abstract": "In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identify the isometries and spectra of a particular class of weighted composition operators on the Bloch space."}
{"category": "Math", "title": "The Gotzmann Coefficients of Hilbert Functions", "abstract": "In this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the Hilbert function of the general hyperplane section of a variety (Green's Hyperplane Restriction Theorem). These geometric consequences improve some results in this direction first given by Green and extend others by Bigatti, Geramita, and Migliore. Other applications of our detailed investigation of how the Hilbert polynomial is written as a sum of binomials, are to conditions that must be satisfied by a polynomial if it is to be the Hilbert polynomial of a non-degenerate integral subscheme of $\\mathbb P^n$ (a problem posed by R. Stanley). We also give some new restrictions on the Hilbert function of a zero dimensional reduced scheme with the Uniform Position Property."}
{"category": "Math", "title": "Controlled coarse homology and isoperimetric inequalities", "abstract": "We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and show that on any group at most linear control is needed for this class to vanish. The latter is a homological version of the classical Burnside problem for infinite groups, with a positive solution. As applications we characterize existence of primitives of the volume form with prescribed growth and show that coarse homology classes obstruct weighted Poincare inequalities."}
{"category": "Math", "title": "$H^1$ and dyadic $H^1$", "abstract": "In this paper we give a simple proof of the fact that the average over all dyadic lattices of the dyadic $H^1$-norm of a function gives an equivalent $H^1$-norm. The proof we present works for both one-parameter and multi-parameter Hardy spaces. The results of such type are known. The first result (for one-parameter Hardy spces) belongs to Burgess Davis (1980). Also, by duality, such results are equivalent to the \"BMO from dyadic BMO\" statements proved by Garnett-Jones(1982} for one parameter case, and by Pipher-Ward (2008) for two-parameter case. While the paper generalizes these results to the multi-parameter setting, this is not its main goal. The purpose of the paper is to present an approach leading to a simple proof, which works in both one-parameter and multi-parameter cases. The main idea of treating square function as a Calderon--Zygmind operator is a commonplace in harmonic analysis; the main observation, on which the paper is based, is that one can treat the random dyadic square function this way. After that, all is proved by using the standard and well-known results about Calderon--Zygmind operators in the Hilbert-space-valued setting. As an added bonus, we get a simple proof of the (equivalent by duality) inclusion $\\text{BMO}\\subset \\text{BMO}_d$, $H^1_d \\subset H^1$ in the multi-parameter case. Note, that unlike the one-parameter case, the inclusions in the general situation are far from trivial."}
{"category": "Math", "title": "On the spectral sequence from Khovanov homology to Heegaard Floer homology", "abstract": "Ozsvath and Szabo show that there is a spectral sequence whose E^2 term is the reduced Khovanov homology of L, and which converges to the Heegaard Floer homology of the (orientation reversed) branched double cover of S^3 along L. We prove that the E^k term of this spectral sequence is an invariant of the link L for all k >= 2. If L is a transverse link in the standard tight contact structure on S^3, then we show that Plamenevskaya's transverse invariant psi(L) gives rise to a transverse invariant, psi^k(L), in the E^k term for each k >= 2. We use this fact to compute each term in the spectral sequences associated to the torus knots T(3,4) and T(3,5)."}
{"category": "Math", "title": "Residuation of Linear Series and The Effective Cone of C_d", "abstract": "We obtain new information about divisors on the $d-$th symmetric power $C_{d}$ of a general curve $C$ of genus $g \\geq 4.$ This includes a complete description of the effective cone of $C_{g-1}$ and a partial computation of the volume function on one of its non-nef subcones, as well as new bounds for the effective and movable cones of $C_{d}$ in the range $\\frac{g+1}{2} \\leq d \\leq g-2.$ We also obtain, for each $g \\geq 5,$ a divisor on $C_{g-1}$ with non-equidimensional stable base locus. For a general hyperelliptic curve $C$ of genus $g,$ we obtain a complete description of the effective cone of $C_{d}$ for $2 \\leq d \\leq g$ and an integral divisor on $C_{g-1}$ which has non-integral volume whenever $g$ is not a power of 2."}
{"category": "Math", "title": "On Hyperelliptic Abelian Functions of Genus 3", "abstract": "The affine ring A of the affine Jacobian variety of a hyperelliptic curve of genus 3 is studied as a D-module. The conjecture on the minimal D-free resolution previously proposed is proved in this case. As a by-product a linear basis of A is explicitly constructed in terms of derivatives of Klein's hyperelliptic pe functions."}
{"category": "Math", "title": "Hilbert schemes for quantum planes are projective", "abstract": "We show that Hilbert schemes for quantum planes are projective."}
{"category": "Math", "title": "Nonuniform hyperbolicity for C^1-generic diffeomorphisms", "abstract": "We study the ergodic theory of non-conservative C^1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C^1-generic diffeomorphisms are nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set L of any C^1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set L. In addition, confirming a claim made by R. Man\\'e in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin's Stable Manifold Theorem, even if the diffeomorphism is only C^1."}
{"category": "Math", "title": "Estimates for singular integrals along surfaces of revolution", "abstract": "We prove certain $L^p$ estimates ($1<p<\\infty$) for non-isotropic singular integrals along surfaces of revolution. As an application we obtain $L^p$ boundedness of the singular integrals under a sharp size condition on their kernels."}
{"category": "Math", "title": "Quadratic pencil of difference equations: Jost solutions, spectrum, and principal vectors", "abstract": "In this paper, a quadratic pencil of Schr\\\"odinger type difference operator $L_{\\lambda}$ is taken under investigation to give a general perspective on the spectral analysis of non-selfadjoint difference equations of second order. Introducing Jost-type solutions, structural and quantitative properties of spectrum of the operator $L_{\\lambda}$ are analyzed and hence, a discrete analog of the theory in Degasperis, (\\emph{J.Math.Phys}. 11: 551--567, 1970) and Bairamov et. al, (\\emph{Quaest. Math.} 26: 15--30, 2003) is developed. In addition, several analogies are established between difference and $q$-difference cases. Finally, the principal vectors of $L_{\\lambda}$ are introduced to lay a groundwork for the spectral expansion. Mathematics Subject Classification (2000): 39A10, 39A12, 39A13"}
{"category": "Math", "title": "Unary Automatic Graphs: An Algorithmic Perspective", "abstract": "This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by finite automata over unary alphabet). We investigate algorithmic properties of such unfolded graphs given their finite presentations. In particular, we ask whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial-time algorithms for each of these questions. For a fixed input graph, the algorithm for the first question is in constant time and the second question is decided using an automaton that recognizes the reachability relation in a uniform way. Hence, we improve on previous work, in which non-elementary or non-uniform algorithms were found."}
{"category": "Math", "title": "Inversion of noisy Radon transform by SVD based needlet", "abstract": "A linear method for inverting noisy observations of the Radon transform is developed based on decomposition systems (needlets) with rapidly decaying elements induced by the Radon transform SVD basis. Upper bounds of the risk of the estimator are established in $L^p$ ($1\\le p\\le \\infty$) norms for functions with Besov space smoothness. A practical implementation of the method is given and several examples are discussed."}
{"category": "Math", "title": "A Wong-Rosay type theorem for proper holomorphic self-maps", "abstract": "We show that the only proper-holomorphic self-maps of bounded domains in C^k whose dynamics escape to a strictly pseudoconvex point of the boundary are automorphisms of the euclidean ball. This is a Wong-Rosay type result for a sequence of maps whose degrees are a priori unbounded."}
{"category": "Math", "title": "On an extension of the notion of Reedy category", "abstract": "We extend the classical notion of a Reedy category so as to allow non-trivial automorphisms. Our extension includes many important examples occuring in topology such as Segal's category Gamma, or the total category of a crossed simplicial group such as Connes' cyclic category Lambda. For any generalized Reedy category R and any cofibrantly generated model category E, the functor category E^R is shown to carry a canonical model structure of Reedy type."}
{"category": "Math", "title": "The dual Hilbert-Samuel function of a Maximal Cohen-Macaulay module", "abstract": "Let $R$ be a Cohen-Macaulay local ring with a canonical module $\\omega_R$. Let $I$ be an $\\m$-primary ideal of $R$ and $M$, a maximal Cohen-Macaulay $R$-module. We call the function $n\\longmapsto \\ell (\\Hom_R(M,{\\omega_R}/{I^{n+1} \\omega_R}))$ the dual Hilbert-Samuel function of $M$ with respect to $I$. By a result of Theodorescu this function is a polynomial function. We study its first two normalized coefficients."}
{"category": "Math", "title": "Lyapunov spectrum for rational maps", "abstract": "We study the dimension spectrum of Lyapunov exponents for rational maps on the Riemann sphere."}
{"category": "Math", "title": "The colorful Helly theorem and colorful resolutions of ideals", "abstract": "We demonstrate that the topological Helly theorem and the algebraic Auslander-Buchsbaum may be viewed as different versions of the same phenomenon. Using this correspondence we show how the colorful Helly theorem of I.Barany and its generalizations by G.Kalai and R.Meshulam translates to the algebraic side. Our main results are algebraic generalizations of these translations, which in particular gives a syzygetic version of Hellys theorem."}
{"category": "Math", "title": "On phase segregation in nonlocal two-particle Hartree systems", "abstract": "We prove the phase segregation phenomenon to occur in the ground state solutions of an interacting system of two self-coupled repulsive Hartree equations for large nonlinear and nonlocal interactions. A self-consistent numerical investigation visualizes the approach to this segregated regime."}
{"category": "Math", "title": "Strongly closed subgroups of finite groups", "abstract": "This paper gives a complete classification of the finite groups that contain a strongly closed p-subgroup for p any prime."}
{"category": "Math", "title": "The equivariant K-theory of toric varieties", "abstract": "This paper contains two results concerning the equivariant K-theory of toric varieties. The first is a formula for the equivariant K-groups of an arbitrary affine toric variety, generalizing the known formula for smooth ones. In fact, this result is established in a more general context, involving the K-theory of graded projective modules. The second result is a new proof of a theorem due to Vezzosi and Vistoli concerning the equivariant K-theory of smooth (not necessarily affine) toric varieties."}
{"category": "Math", "title": "Resolvent estimates for operators belonging to exponential classes", "abstract": "For $a,\\alpha>0$ let $E(a,\\alpha)$ be the set of all compact operators $A$ on a separable Hilbert space such that $s_n(A)=O(\\exp(-an^\\alpha))$, where $s_n(A)$ denotes the $n$-th singular number of $A$. We provide upper bounds for the norm of the resolvent $(zI-A)^{-1}$ of $A$ in terms of a quantity describing the departure from normality of $A$ and the distance of $z$ to the spectrum of $A$. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in $E(a,\\alpha)$."}
{"category": "Math", "title": "On extensions of covariantly finite subcategories", "abstract": "In \\cite{GT}, Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is still covariantly finite. We give an counterexample to show that Gentle-Todorov's theorem may fail in arbitrary abelian categories; we also prove that a triangulated version of Gentle-Todorov's theorem holds; we make applications of Gentle-Todorov's theorem to obtain short proofs to a classical result by Ringel and a recent result by Krause and Solberg."}
{"category": "Math", "title": "Initial-Boundary Value Problems for Parabolic Equations", "abstract": "We prove new existence and uniqueness results for weak solutions to non-homogeneous initial-boundary value problems for parabolic equations modeled on the evolution of the p-Laplacian."}
{"category": "Math", "title": "Tannaka duality for proper Lie groupoids", "abstract": "The main contribution of this thesis is a Tannaka duality theorem for proper Lie groupoids. This result is obtained by replacing the category of smooth vector bundles over the base manifold of a Lie groupoid with a larger category, the category of smooth Euclidean fields, and by considering smooth actions of Lie groupoids on smooth Euclidean fields. The notion of smooth Euclidean field that is introduced here is the smooth, finite dimensional analogue of the familiar notion of continuous Hilbert field. In the second part of the thesis, ordinary smooth representations of Lie groupoids on smooth vector bundles are systematically studied from the point of view of Tannaka duality, and various results are obtained in this direction."}
{"category": "Math", "title": "Christoffel words and Markoff triples", "abstract": "Markoff triples are parametrized uniquely by Christoffel words."}
{"category": "Math", "title": "Structure computation and discrete logarithms in finite abelian p-groups", "abstract": "We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig-Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without using a relation matrix. The problem of computing a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian group G is addressed, yielding a Monte Carlo algorithm to compute the structure of G using O(|G|^0.5) group operations. These results also improve generic algorithms for extracting pth roots in G."}
{"category": "Math", "title": "G-biliaison of ladder Pfaffian varieties", "abstract": "The ideals generated by pfaffians of mixed size contained in a subladder of a skew-symmetric matrix of indeterminates define arithmetically Cohen-Macaulay, projectively normal, reduced and irreducible projective varieties. We show that these varieties belong to the G-biliaison class of a complete intersection. In particular, they are glicci."}
{"category": "Math", "title": "Quotients of products of curves, new surfaces with $p_g=0$ and their fundamental groups", "abstract": "The first main purpose of this paper is to contribute to the existing knowledge about the complex projective surfaces $S$ of general type with $p_g(S) = 0$ and their moduli spaces, constructing 19 new families of such surfaces with hitherto unknown fundamental groups. We also provide a table containing all the known such surfaces with K^2 <=7. Our second main purpose is to describe in greater generality the fundamental groups of smooth projective varieties which occur as the minimal resolutions of the quotient of a product of curves by the action of a finite group. We classify, in the two dimensional case, all the surfaces with q=p_g = 0 obtained as the minimal resolution of such a quotient, having rational double points as singularities. We show that all these surfaces give evidence to the Bloch conjecture."}
{"category": "Math", "title": "Sub-exponentially localized kernels and frames induced by orthogonal expansions", "abstract": "The aim of this paper is to construct sup-exponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions in Jacobi polynomials, spherical harmonics, orthogonal polynomials on the ball and simplex, and Hermite and Laguerre functions."}
{"category": "Math", "title": "Model Theoretic Complexity of Automatic Structures", "abstract": "We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic well- founded partial order is bounded by \\omega^\\omega ; 2) The ordinal heights of automatic well-founded relations are unbounded below the first non-computable ordinal; 3) For any computable ordinal there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structures of Scott rank the first non-computable ordinal and its successor; 4) For any computable ordinal, there is an automatic successor tree of Cantor-Bendixson rank that ordinal."}
{"category": "Math", "title": "Three Lectures on Automatic Structures", "abstract": "This paper grew out of three tutorial lectures on automatic structures given by the first author at the Logic Colloquium 2007. We discuss variants of automatic structures related to several models of computation: word automata, tree automata, Buchi automata, and Rabin automata. Word automata process finite strings, tree automata process finite labeled trees, Buchi automata process infinite strings, and Rabin automata process infinite binary labeled trees. Automatic structures are mathematical objects which can be represented by (word, tree, Buchi, or Rabin) automata. The study of properties of automatic structures is a relatively new and very active area of research."}
{"category": "Math", "title": "On the isometric composition operators on the Bloch space in $\\mathbb{C}^n$", "abstract": "Let $\\varphi$ be a holomorphic self-map of a bounded homogeneous domain $D$ in $\\mathbb{C}^n$. In this work, we show that the composition operator $C_\\varphi: f\\mapsto f\\circ \\varphi$ is bounded on the Bloch space $\\mathcal{B}$ of the domain and provide estimates on its operator norm. We also give a sufficient condition for $\\varphi$ to induce an isometry on $\\cal{B}$. This condition allows us to construct non-trivial examples of isometric composition operators in the case when $D$ has the unit disk as a factor. We then obtain some necessary conditions for $C_\\varphi$ to be an isometry on $\\cal{B}$ when $D$ is a Cartan classical domain. Finally, we give the complete description of the spectrum of the isometric composition operators in the case of the unit disk and for a wide class of symbols on the polydisk."}
{"category": "Math", "title": "Superstable groups acting on trees", "abstract": "We study superstable groups acting on trees. We prove that an action of an $\\omega$-stable group on a simplicial tree is trivial. This shows that an HNN-extension or a nontrivial free product with amalgamation is not $\\omega$-stable. It is also shown that if $G$ is a superstable group acting nontrivially on a $\\Lambda$-tree, where $\\Lambda=\\mathbb Z$ or $\\Lambda=\\mathbb R$, and if $G$ is either $\\alpha$-connected and $\\Lambda=\\mathbb Z$, or if the action is irreducible, then $G$ interprets a simple group having a nontrivial action on a $\\Lambda$-tree. In particular if $G$ is superstable and splits as $G=G_1*_AG_2$, with the index of $A$ in $G_1$ different from 2, then $G$ interprets a simple superstable non $\\omega$-stable group. We will deal with \"minimal\" superstable groups of finite Lascar rank acting nontrivially on $\\Lambda$-trees, where $\\Lambda=\\mathbb Z$ or $\\Lambda=\\mathbb R$. We show that such groups $G$ have definable subgroups $H_1 \\lhd H_2 \\lhd G$, $H_2$ is of finite index in $G$, such that if $H_1$ is not nilpotent-by-finite then any action of $H_1$ on a $\\Lambda$-tree is trivial, and $H_2/H_1$ is either soluble or simple and acts nontrivially on a $\\Lambda$-tree. We are interested particularly in the case where $H_2/H_1$ is simple and we show that $H_2/H_1$ has some properties similar to those of bad groups."}
{"category": "Math", "title": "Hodge spectrum of hyperplane arrangements", "abstract": "In this article there are two main results. The first result gives a formula, in terms of a log resolution, for the graded pieces of the Hodge filtration on the cohomology of a unitary local system of rank one on the complement of an arbitrary divisor in a smooth projective complex variety. The second result is an application of the first. We give a combinatorial formula for the spectrum of a hyperplane arrangement. M. Saito recently proved that the spectrum of a hyperplane arrangement depends only on combinatorics. However, a combinatorial formula was missing. The formula is achieved by a different method."}
{"category": "Math", "title": "Scaling limit for a drainage network model", "abstract": "We consider the two dimensional version of a drainage network model introduced by Gangopadhyay, Roy and Sarkar, and show that the appropriately rescaled family of its paths converges in distribution to the Brownian web. We do so by verifying the convergence criteria proposed by Fontes, Isopi, Newman and Ravishankar."}
{"category": "Math", "title": "Boundedness on inhomogeneous Lipschitz spaces of fractional integrals, singular integrals and hypersingular integrals associated to non-doubling measures", "abstract": "In the context of a finite measure metric space whose measure satisfies a growth condition, we prove \"T1\" type necessary and sufficient conditions for the boundedness of fractional integrals, singular integrals, and hypersingular integrals on inhomogeneous Lipschitz spaces. We also indicate how the results can be extended to the case of infinite measure. Finally we show applications to Real and Complex Analysis."}
{"category": "Math", "title": "A note on gaps", "abstract": "Let $p_{k}$ denote the $k$-th prime and $d(p_{k}) = p_{k} - p_{k - 1}$, the difference between consecutive primes. We denote by $N_{\\epsilon}(x)$ the number of primes $\\leq x$ which satisfy the inequality $d(p_{k}) \\leq (\\log p_{k})^{2 + \\epsilon}$, where $\\epsilon > 0$ is arbitrary and fixed, and by $\\pi(x)$ the number of primes less than or equal to $x$. In this paper, we first prove a theorem that $\\lim_{x \\to \\infty} N_{\\epsilon}(x)/\\pi(x) = 1$. A corollary to the proof of the theorem concerning gaps between consecutive squarefree numbers is stated."}
{"category": "Math", "title": "Angles as probabilities", "abstract": "We use a probabilistic interpretation of solid angles to generalize the well-known fact that the inner angles of a triangle sum to 180 degrees. For the 3-dimensional case, we show that the sum of the solid inner vertex angles of a tetrahedron T, divided by 2*pi, gives the probability that an orthogonal projection of T onto a random 2-plane is a triangle. More generally, it is shown that the sum of the (solid) inner vertex angles of an n-simplex S, normalized by the area of the unit (n-1)-hemisphere, gives the probability that an orthogonal projection of S onto a random hyperplane is an (n-1)-simplex. Applications to more general polytopes are treated briefly, as is the related Perles-Shephard proof of the classical Gram-Euler relations."}
{"category": "Math", "title": "Multitransgression and regulators", "abstract": "In a parallel way to the work of Wang, we define higher order characteristic classes associated with the Chern character, generalizing the work of Bott-Chern and Gillet-Soul\\'e on secondary characteristic classes. Our formalism is simplicial and the computations are easier. As a consequence, we obtain the comparison of Borel and Beilinson regulators and an explicit formula for the real single-valued function associated with the Grasmannian polylogarithm."}
{"category": "Math", "title": "Convergence of symmetric trap models in the hypercube", "abstract": "We consider symmetric trap models in the d-dimensional hypercube whose ordered mean waiting times, seen as weights of a measure in the natural numbers, converge to a finite measure as d diverges, and show that the models suitably represented converge to a K process as d diverges. We then apply this result to get K processes as the scaling limits of the REM-like trap model and the Random Hopping Times dynamics for the Random Energy Model in the hypercube in time scales corresponding to the ergodic regime for these dynamics."}
{"category": "Math", "title": "Averaged large deviations for random walk in a random environment", "abstract": "In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on $\\mathbb{Z}^d$ with $d\\geq1$, and gives a variational formula for the corresponding rate function $I_a$. Under Sznitman's transience condition (T), we show that $I_a$ is strictly convex and analytic on a non-empty open set $\\mathcal{A}$, and that the true velocity of the particle is an element (resp. in the boundary) of $\\mathcal{A}$ when the walk is non-nestling (resp. nestling). We then identify the unique minimizer of Varadhan's variational formula at any velocity in $\\mathcal{A}$."}
{"category": "Math", "title": "Expressions for Catalan Kronecker Products", "abstract": "We give some elementary manifestly positive formulae for the Kronecker products s_(d,d) * s_(d+k,d-k) and s_(d,d) * s_(2d-k,1^k). These formulae demonstrate some fundamental properties of the Kronecker coefficients, and we use them to deduce a number of enumerative and combinatorial results."}
{"category": "Math", "title": "Double Hall algebras and derived equivalences", "abstract": "We show that the reduced Drinfeld double of the Ringel-Hall algebra of a hereditary category is invariant under derived equivalences. By associating an explicit isomorphism to a given derived equivalence, we also extend the results of [BS1], [BS2], [SVdB], and [XY]."}
{"category": "Math", "title": "Global rough solutions to the cubic nonlinear Boussinesq equation", "abstract": "We prove that the initial value problem (IVP) for the cubic defocusing nonlinear Boussinesq equation $u_{tt}-u_{xx}+u_{xxxx}-(|u|^2u)_{xx}=0$ on the real line is globally well-posed in $H^{s}(\\R)$ provided $2/3<s<1$."}
{"category": "Math", "title": "Dynamics of Asymptotically Hyperbolic Manifolds", "abstract": "We prove a dynamical wave trace formula for asymptotically hyperbolic (n+1) dimensional manifolds with negative (but not necessarily constant) sectional curvatures which equates the renormalized wave trace to the lengths of closed geodesics. A corollary of this dynamical trace formula is a dynamical resonance-wave trace formula for compact perturbations of convex co-compact hyperbolic manifolds which we use to prove a growth estimate for the length spectrum counting function. We next define a dynamical zeta function and prove its analyticity in a half plane. In our main result, we produce a prime orbit theorem for the geodesic flow. This is the first such result for manifolds which have neither constant curvature nor finite volume. As a corollary to the prime orbit theorem, using our dynamical resonance-wave trace formula, we show that the existence of pure point spectrum for the Laplacian on negatively curved compact perturbations of convex co-compact hyperbolic manifolds is related to the dynamics of the geodesic flow."}
{"category": "Math", "title": "On inequivalent factorizations of a cycle", "abstract": "We introduce a bijection between inequivalent minimal factorizations of the n-cycle (1 2 ... n) into a product of smaller cycles of given length, on one side, and trees of a certain structure on the other. We use this bijection to count the factorizations with a given number of different commuting factors that can appear in the first and in the last positions, a problem which has found applications in physics. We also provide a necessary and sufficient condition for a set of cycles to be arrangeable into a product evaluating to (1 2 ... n)."}
{"category": "Math", "title": "Pythagorean Partition-Regularity and Ordered Triple Systems with the Sum Property", "abstract": "Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is even open for two colors. A natural strategy is to show that some small nonbipartite triple systems cannot be realized as a family of Pythagorean triples. It suffices to consider partial triple systems (PTS's), and it is therefore natural to consider the Fano plane, the smallest nonbipartite PTS. We show that the Pythagorean triples do not contain any Fano plane. In fact, our main result is that a much larger family of \"ordered\" triple systems (viz. those with a certain \"sum property\") do not contain any Steiner triple system (STS)."}
{"category": "Math", "title": "Theta Bodies for Polynomial Ideals", "abstract": "Inspired by a question of Lov\\'asz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lov\\'asz's theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre's relaxations for real solutions to ideals, and that they can be computed explicitly using combinatorial moment matrices. Theta bodies provide a new canonical set of semidefinite relaxations for the max cut problem. For vanishing ideals of finite point sets, we give several equivalent characterizations of when the first theta body equals the convex hull of the points. We also determine the structure of the first theta body for all ideals."}
{"category": "Math", "title": "Elliptic curves with maximal Galois action on their torsion points", "abstract": "Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, \\rho_E : Gal(\\bar{k}/k) \\to GL_2(\\hat{Z}). For a fixed number field k, we describe the image of \\rho_E for a \"random\" elliptic curve E over k. In particular, if k\\neq Q is linearly disjoint from the cyclotomic extension of Q, then \\rho_E will be surjective for \"most\" elliptic curves over k."}
{"category": "Math", "title": "Stein Block Thresholding For Image Denoising", "abstract": "In this paper, we investigate the minimax properties of Stein block thresholding in any dimension $d$ with a particular emphasis on $d=2$. Towards this goal, we consider a frame coefficient space over which minimaxity is proved. The choice of this space is inspired by the characterization provided in \\cite{BorupNielsen} of family of smoothness spaces on $\\mathbb{R}^d$, a subclass of so-called decomposition spaces \\cite{Feichtinger}. These smoothness spaces cover the classical case of Besov spaces, as well as smoothness spaces corresponding to curvelet-type constructions. Our main theoretical result investigates the minimax rates over these decomposition spaces, and shows that our block estimator can achieve the optimal minimax rate, or is at least nearly-minimax (up to a $\\log$ factor) in the least favorable situation. Another contribution is that the minimax rates given here are stated for a general noise sequence model in the transform coefficient domain beyond the usual i.i.d. Gaussian case. The choice of the threshold parameter is theoretically discussed and its optimal value is stated for some noise models such as the (non-necessarily i.i.d.) Gaussian case. We provide a simple, fast and a practical procedure. We also report a comprehensive simulation study to support our theoretical findings. The practical performance of our Stein block denoising compares very favorably to the BLS-GSM state-of-the art denoising algorithm on a large set of test images. A toolbox is made available for download on the Internet to reproduce the results discussed in this paper."}
{"category": "Math", "title": "Generalized Calogero-Moser systems from rational Cherednik algebras", "abstract": "We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero-Moser type which we explicitly specify. In the case of classical Coxeter groups we also obtain generalized Calogero-Moser systems with added quadratic potential."}
{"category": "Math", "title": "A Kurosh-Type Theorem for Type III Factors", "abstract": "We prove a generalization of N. Ozawa's Kurosh-type theorem to the setting of free products of semiexact II_1 factors with respect to arbitrary (non-tracial) faithful normal states. We are thus able to distinguish certain resulting type III factors. For example, if M = LF_n \\otimes LF_m and {\\phi_i} is any sequence of faithful normal states on M, then the l-various (M,\\phi_1) * ... * (M,\\phi_l) are all mutually non-isomorphic."}
{"category": "Math", "title": "Perfect simulation and finitary coding for multicolor systems with interactions of infinite range", "abstract": "We consider a particle system on $Z^d$ with finite state space and interactions of infinite range. Assuming that the rate of change is continuous and decays sufficiently fast, we introduce a perfect simulation algorithm for the stationary process. The algorithm follows from a representation of the multicolor system as a finitary coding from a sequence of independent uniform random variables. This implies that the process is exponentially ergodic. The basic tool we use is a representation of the infinite range change rates as a mixture of finite range change rates."}
{"category": "Math", "title": "Growth of rank 1 valuation semigroups", "abstract": "We consider the question of which semigroups can occur as the semigroup $S_R(\\nu)$ of positive values of a rank 1 valuation dominating a Noetherian local ring $R$. We give a number of bounds of polynomial type on the growth of $\\phi(n)=S_R(\\nu)\\cap (0,n)$ for $n\\in\\NN$, starting with the upper bound of $P_R(n)$, where $P_R(n)$ is the Hilbert function of $R$. This bound is generalized to an extremely general bound for arbitrary rank valuations in the paper \"Semigroups of valuations on local rings, II\", by Cutkosky and Teissier, arXiv:0805.3788. This bound is already enough to give simple examples of rank 1 well ordered semigroups which are not the value semigroup $S_R(\\nu)$ of a valuation dominating a Noetherian local ring. In the case of rank 1, it is possible to give more precise estimates of $\\phi(n)$, which we prove in this paper. We also give examples showing that many different rates of growth are possible for $\\phi(n)$ on a regular local ring of dimension 2, such as $n({\\alpha}$ for any rational $\\alpha$ with $1\\le\\alpha\\le 2$, and $n{log}(n)$."}
{"category": "Math", "title": "Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps", "abstract": "Resonance tongues are mode-locking regions of parameter space in which stable periodic solutions occur; they commonly occur, for example, near Neimark-Sacker bifurcations. For piecewise-smooth, continuous maps these tongues typically have a distinctive lens-chain (or sausage) shape in two-parameter bifurcation diagrams. We give a symbolic description of a class of \"rotational\" periodic solutions that display lens-chain structures for a general $N$-dimensional map. We then unfold the codimension-two, shrinking point bifurcation, where the tongues have zero width. A number of codimension-one bifurcation curves emanate from shrinking points and we determine those that form tongue boundaries."}
{"category": "Math", "title": "On the linear wave regime of the Gross-Pitaevskii equation", "abstract": "We study a long wave-length asymptotics for the Gross-Pitaevskii equation corresponding to perturbation of a constant state of modulus one. We exhibit lower bounds on the first occurence of possible zeros (vortices) and compare the solutions with the corresponding solutions to the linear wave equation or variants. The results rely on the use of the Madelung transform, which yields the hydrodynamical form of the Gross-Pitaevskii equation, as well as of an augmented system."}
{"category": "Math", "title": "Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I. Generalizations of the Capelli and Turnbull identities", "abstract": "We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull's Capelli-type identities for symmetric and antisymmetric matrices."}
{"category": "Math", "title": "Bifurcation from a normally degenerate manifold", "abstract": "Local bifurcation theory typically deals with the response of a degenerate but isolated equilibrium state or periodic orbit of a dynamical system to perturbations controlled by one or more independent parameters, and characteristically uses tools from singularity theory. There are many situations, however, in which the equilibrium state or periodic orbit is not isolated but belongs to a manifold $S$ of such states, typically as a result of continuous symmetries in the problem. In this case the bifurcation analysis requires a combination of local and global methods, and is most tractable in the case of normal nondegeneracy, that is when the degeneracy is only along $S$ itself and the system is nondegenerate in directions normal to $S$. In this paper we consider the consequences of relaxing normal nondegeneracy, which can generically occur within 1-parameter families of such systems. We pay particular attention to the simplest but important case where $\\dim S=1$ and where the normal degeneracy occurs with corank 1. Our main focus is on uniform degeneracy along $S$, although we also consider aspects of the branching structure for solutions when the degeneracy varies at different places on $S$. The tools are those of singularity theory adapted to global topology of $S$, which allow us to explain the bifurcation geometry in natural way. In particular, we extend and give a clear geometric setting for earlier analytical results of Hale and Taboas."}
{"category": "Math", "title": "Optimal L$^1$-bounds for submartingales", "abstract": "The optimal function $f$ satisfying $$ \\mathbb{E} |\\sum_{1}^n X_i | \\ge f(\\mathrbb{E}|X_1|,...,\\mathbb{E}|X_n|) $$ for every martingale $(X_1,X_1+X_2, ...,\\sum_{i=1}^n X_i)$ is shown to be given by $$ f(a) = \\max \\Big\\{a_k-\\sum_{i=1}^{k-1} a_i\\Big\\}_{k=1}^n \\cup \\Big\\{\\frac {a_k}2\\Big\\}_{k=3}^n $$ for $a\\in{[0,\\infty[}^n_{}$. A similar result is obtained for submartingales $(0,X_1,X_1+X_2,..., \\sum_{i=1}^n X_i)$. The optimality proofs use a convex-analytic comparison lemma of independent interest."}
{"category": "Math", "title": "Large time behavior of differential equations with drifted periodic coefficients modeling Carbon storage in soil", "abstract": "This paper is concerned with the linear ODE in the form $y'(t)=\\lambda\\rho(t)y(t)+b(t)$, $\\lambda <0$ which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function $\\rho(t)$, a linear drift in the coefficient $b(t)$ involves a linear drift for the solution of this ODE. In the second part, we extend the previous results to a classical heat non-homogeneous equation. The connection with an analytic semi-group associated to the ODE equation is considered in the third part. Numerical examples are given."}
{"category": "Math", "title": "Comparing Absolute and relative Gromov--Witten invariants", "abstract": "This note compares the usual (absolute) Gromov-Witten invariants of a symplectic manifold with the invariants that count the curves relative to a (symplectic) divisor D. We give explicit examples where these invariants differ even though it seems at first that they should agree, for example when counting genus zero curves in a class \\be such that \\be\\cdot D=0. The main tool is the decomposition formula in the form developed by A. Li--Ruan."}
{"category": "Math", "title": "Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations", "abstract": "We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on $\\mathscr{B}(G)$ ($G$ a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The proof is by general abstract nonsense (taken from Arveson's classification of $E_0$-semigroups on $\\mathscr{B}(H)$ by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy. All other results that make similar statements (especially, [Mem. Amer. Math. Soc. 240 (2016), vi+126 pages, arXiv:0901.1798]) for more general $C^*$-algebras) have been proved later by suitable adaptations of the methods exposed here. (They use Hilbert module techniques, which we carefully avoid here in order to make the result available without any appeal to Hilbert modules.)"}
{"category": "Math", "title": "On the sum of the squared multiplicities of the distances in a point set over finite fields", "abstract": "We study a finite analog of a conjecture of Erd\\\"os on the sum of the squared multiplicities of the distances determined by an $n$-element point set. Our result is based on an estimate of the number of hinges in spectral graphs."}
{"category": "Math", "title": "On the K-theory of truncated polynomial algebras over the integers", "abstract": "We show that the K_{2i}(Z[x]/(x^m),(x)) is finite of order (mi)!(i!)^{m-2} and that K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1. This is accomplished by showing that the equivariant homotopy groups of the topological Hochschild spectrum THH(Z) are finite, in odd degrees, and free abelian, in even degrees, and by evaluating their orders and ranks, respectively."}
{"category": "Math", "title": "An inequality for Kruskal-Macaulay functions", "abstract": "Given integers $k\\geq1$ and $n\\geq0$, there is a unique way of writing $n$ as $n=\\binom{n_{k}}{k}+\\binom{n_{k-1}}{k-1}+...+\\binom{n_{1}}{1}$ so that $0\\leq n_{1}<...<n_{k-1}<n_{k}$. Using this representation, the \\emph{Kruskal-Macaulay function of}$n$ is defined as $\\partial^{k}(n) =\\binom{n_{k}-1}{k-1}+\\binom{n_{k-1}-1}{k-2}+...+\\binom{n_{1}-1}% {0}.$ We show that if $a\\geq0$ and $a<\\partial^{k+1}(n) $, then $\\partial^{k}(a) +\\partial^{k+1}(n-a) \\geq \\partial^{k+1}(n) .$ As a corollary, we obtain a short proof of Macaulay's Theorem. Other previously known results are obtained as direct consequences."}
{"category": "Math", "title": "Solutions to 3-dimensional Navier-Stokes equations for incompressible fluid", "abstract": "This article is an updated version of the article that was published in the Electronic Journal of Differential Equations on 10. July 2010. Two footnotes have been added. One corrects a minor error not influencing the proof, the second is only a clarifying text to the existing proof. A discussion how the published article solves the Clay Millennium Prize problem on the Navier-Stokes equations is added, the critizism against the published article is answered and a discussion how the Clay problem statement should be corrected is included. The article gives explicit solutions to the space-periodic Navier-Stokes problem with non-periodic pressure. These type of solutions are not unique and by using such solutions one can construct a periodic, smooth, divergence-free initial vector field allowing a space-periodic and time-bounded external force such that there exists a smooth solution to the 3-dimensional Navier-Stokes equations for incompressible fluid with those initial conditions, but the solution cannot be continued to the whole space."}
{"category": "Math", "title": "Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups", "abstract": "For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups."}
{"category": "Math", "title": "Symmetries in Images on Ancient Seals", "abstract": "We discuss the presence of symmetries in images engraved on ancient seals, in particular on stamp seals. Used to stamp decorations, to secure the containers from tampering and for owner's identification, we can find seals that can be dated from Neolithic times. Earliest seals were engraved with lines, dots and spirals. Nevertheless, these very ancient stamp seals, in the small circular or ovoid space of their bases, possess bilateral and rotational symmetries. The shape of the base seems to determine the symmetries of images engraved on it. We will also discuss what could be the meaning of antisymmetry and broken symmetry for images on seals."}
{"category": "Math", "title": "Infinitesimal rigidity of a compact hyperbolic 4-orbifold with totally geodesic boundary", "abstract": "Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell."}
{"category": "Math", "title": "Surgery in codimension 3 and the Browder--Livesay invariants", "abstract": "The inertia subgroup $I_n(\\pi)$ of a surgery obstruction group $L_n(\\pi)$ is generated by elements which act trivially on the set of homotopy triangulations $\\Cal S(X)$ for some closed topological manifold $X^{n-1}$ with $\\pi_1(X)=\\pi$. This group is a subgroup of the group $C_n(\\pi)$ which consists of the elements which can be realized by normal maps of closed manifolds. In all known cases these groups coincide and the computation of them is one of the basic problems of surgery theory. The computation of the group $C_n(\\pi)$ is equivalent to the computation the image of the assembly map $A:H_{n}(B\\pi, \\bold L_{\\bullet})\\to L_{n}(\\pi)$. Every Browder-Livesay filtration of the manifold $X$ provides a collection of Browder-Livesay invariants which are the forbidden invariants in the closed manifold surgery problem. In the present paper we describe all possible forbidden invariants which can give a Browder-Livesay filtration for computing the inertia subgroup. Our approach is a natural generalization of the approach of Hambleton and Kharshiladze. More precisely, we prove that a Browder-Livesay filtration of a given manifold can give the following forbidden invariants for an element $x\\in L_n(\\pi_1(X))$ to belong to the subgroup $I_n(\\pi)$: the nontrivial Browder-Livesay invariants in codimensions 0, 1, 2 and a nontrivial class of obstructions of a restriction of a normal map to a submanifold in codimension 3."}
{"category": "Math", "title": "Abelian categories versus abelian 2-categories", "abstract": "Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. He posses also a question whether any abelian category comes in this way. We will give a rather trivial solution of this problem in the case when a given abelian category has enough projective or injective objects."}
{"category": "Math", "title": "A class of Solvable Lie algebras", "abstract": "All finite-dimensional indecomposable solvable Lie algebras g, having the filiform Lie algebra Q_(2m+1) as the nilradical, are studied and classified. It turns out that the dimension of g is at most dimQ_(2m+1)+2."}
{"category": "Math", "title": "Knizhnik-Zamolodchikov bundles are topologically trivial", "abstract": "We prove that the vector bundles at the core of the Knizhnik-Zamolodchikov and quantum constructions of braid groups representations are topologically trivial bundles. We provide partial generalizations of this result to generalized braid groups. A crucial intermediate result is that the representation ring of the symmetric group on n letters is generated by the alternating powers of its natural n-dimensional representation."}
{"category": "Math", "title": "Isothermic hypersurfaces in R^{n+1}", "abstract": "A diagonal metric sum_{i=1}^n g_{ii} dx_i^2 is termed Guichard_k if sum_{i=1}^{n-k}g_{ii}-sum_{i=n-k+1}^n g_{ii}=0. A hypersurface in R^{n+1} is isothermic_k if it admits line of curvature co-ordinates such that its induced metric is Guichard_k. Isothermic_1 surfaces in R^3 are the classical isothermic surfaces in R^3. Both isothermic_k hypersurfaces in R^{n+1} and Guichard_k orthogonal co-ordinate systems on R^n are invariant under conformal transformations. A sequence of n isothermic_k hypersurfaces in R^{n+1} (Guichard_k orthogonal co-ordinate systems on R^n resp.) is called a Combescure sequence if the consecutive hypersurfaces (orthogonal co-ordinate systems resp.) are related by Combescure transformations. We give a correspondence between Combescure sequences of Guichard_k orthogonal co-ordinate systems on R^n and solutions of the O(2n-k,k)/O(n)xO(n-k,k)-system, and a correspondence between Combescure sequences of isothermic_k hypersurfaces in R^{n+1} and solutions of the O(2n+1-k,k)/O(n+1)xO(n-k,k)-system, both being integrable systems. Methods from soliton theory can therefore be used to construct Christoffel, Ribaucour, and Lie transforms, and to describe the moduli spaces of these geometric objects and their loop group symmetries."}
{"category": "Math", "title": "Symplectic Reconstruction of Data for Heat and Wave Equations", "abstract": "This report concerns the inverse problem of estimating a spacially dependent coefficient of a partial differential equation from observations of the solution at the boundary. Such a problem can be formulated as an optimal control problem with the coefficient as the control variable and the solution as state variable. The heat or the wave equation is here considered as state equation. It is well known that such inverse problems are ill-posed and need to be regularized. The powerful Hamilton-Jacobi theory is used to construct a simple and general method where the first step is to analytically regularize the Hamiltonian; next its Hamiltonian system, a system of nonlinear partial differential equations, is solved with the Newton method and a sparse Jacobian."}
{"category": "Math", "title": "On congruences mod ${\\mathfrak p}^m$ between eigenforms and their attached Galois representations", "abstract": "Given a prime $p$ and cusp forms $f_1$ and $f_2$ on some $\\Gamma_1(N)$ that are eigenforms outside $Np$ and have coefficients in the ring of integers of some number field $K$, we consider the problem of deciding whether $f_1$ and $f_2$ have the same eigenvalues mod ${\\mathfrak p}^m$ (where ${\\mathfrak p}$ is a fixed prime of $K$ over $p$) for Hecke operators $T_{\\ell}$ at all primes $\\ell\\nmid Np$. When the weights of the forms are equal the problem is easily solved via an easy generalization of a theorem of Sturm. Thus, the main challenge in the analysis is the case where the forms have different weights. Here, we prove a number of necessary and sufficient conditions for the existence of congruences mod ${\\mathfrak p}^m$ in the above sense. The prime motivation for this study is the connection to modular mod ${\\mathfrak p}^m$ Galois representations, and we also explain this connection."}
{"category": "Math", "title": "Handlebody-preserving finite group actions on Haken manifolds with Heegaard genus two", "abstract": "Let $M$ be a closed orientable 3-manifold with a genus two Heegaard splitting $(V_1, V_2; F)$ and a non-trivial JSJ-decomposition, where all components of the intersection of the JSJ-tori and $V_i$ are not $\\partial$-parallel in $V_i$ for $i=1,2$. If $G$ is a finite group of orientation-preserving diffeomorphisms acting on $M$ which preserves each handlebody of the Heegaard splitting and each piece of the JSJ-decomposition of $M$, then $G\\cong \\mathbb{Z}_2$ or $\\mathbb{D}_2$ if $V_j\\cap(\\cup T_i)$ consists of at most two disks or at most two annuli."}
{"category": "Math", "title": "Banach Spaces of Bounded Szlenk Index II", "abstract": "For every $\\alpha<\\omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $\\omega^{\\alpha\\omega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $\\omega^{\\alpha\\omega}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates."}
{"category": "Math", "title": "W-algebras related to parafermion algebras", "abstract": "We study a W-algebra of central charge 2(k-1)/(k+2) with k a positive integer greater than 1"}
{"category": "Math", "title": "A dispersive bound for three-dimensional Schroedinger operators with zero energy eigenvalues", "abstract": "We prove a dispersive estimate for the evolution of Schroedinger operators $H = -\\Delta + V(x)$ in ${\\mathbb R}^3$. The potential is allowed to be a complex-valued function belonging to $L^p(\\R^3)\\cap L^q(\\R^3)$, $p < \\frac32 < q$, so that $H$ need not be self-adjoint or even symmetric. Some additional spectral conditions are imposed, namely that no resonances of $H$ exist anywhere within the interval $[0,\\infty)$ and that eigenfunctions at zero (including generalized eigenfunctions) decay rapidly enough to be integrable."}
{"category": "Math", "title": "Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations", "abstract": "Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called matrix pencil methods, the general ideas used as well as the proofs differ considerably. Moreover, the available theory hardly reveals the relations between the different methods. In this work, a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem: the polynomial two-parameter eigenvalue problem, of which the quadratic two-parameter eigenvalue problem is a special case. This framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods. We also recognize a few new matrix pencil variants to determine DDE stability. Finally, the recognition of the new types of eigenvalue problem opens a door to efficient computation of DDE stability."}
{"category": "Math", "title": "Painlev\\'e V and a Pollaczek-Jacobi type orthogonal polynomials", "abstract": "We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=\\rex^{-t/x}\\:x^{\\al}(1-x)^{\\bt},\\quad t\\geq 0, $$ defined for $x\\in[0,1].$ If $t=0,$ this reduces to a shifted Jacobi weight. Our ladder operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients. For $t>0,$ the factor $\\rex^{-t/x}$ induces an infinitely strong zero at $x=0.$ With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a system of difference equations. These, when suitably combined with a pair of Toda-like equations derived from the orthogonality principle, show that the auxiliary quantities are a particular Painlev\\'e V and/or allied functions. It is also shown that the logarithmic derivative of the Hankel determinant, $$ D_n(t):=\\det(\\int_{0}^{1} x^{i+j} \\:\\rex^{-t/x}\\:x^{\\al}(1-x)^{\\bt}dx)_{i,j=0}^{n-1}, $$ satisfies the Jimbo-Miwa-Okamoto $\\sigma-$form of the Painlev\\'e V and that the same quantity satisfies a second order non-linear difference equation which we believe to be new."}
{"category": "Math", "title": "Some extensions of the class of $k$-convex bodies", "abstract": "We study relations of some classes of $k$-convex, $k$-visible bodies in Euclidean spaces. We introduce and study \\textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular, \\textrm{$k$-circular convex} and \\textrm{$k$-circular visible} ones. Investigation of these bodies more general than $k$-convex and $k$-visible ones allows us to generalize some classical results of geometric tomography and find their new applications."}
{"category": "Math", "title": "The group of isometries of a Banach space and duality", "abstract": "We construct an example of a real Banach space whose group of surjective isometries has no uniformly continuous one-parameter semigroups, but the group of surjective isometries of its dual contains infinitely many of them. Other examples concerning numerical index, hermitian operators and dissipative operators are also shown."}
{"category": "Math", "title": "A Jacobi algorithm for distributed model predictive control of dynamically coupled systems", "abstract": "In this paper we introduce an iterative Jacobi algorithm for solving distributed model predictive control (DMPC) problems, with linear coupled dynamics and convex coupled constraints. The algorithm guarantees stability and persistent feasibility, and we provide a localized procedure for constructing an initial feasible solution by constraint tightening. Moreover, we show that the solution of the iterative process converges to the centralized MPC solution. The proposed iterative approach involves solving local optimization problems consisting of only few subsystems, depending on the choice of the designer and the sparsity of dynamical and constraint couplings. The gain in the overall computational load compared to the centralized problem is balanced by the increased communication requirements. This makes our approach more applicable to situations where the number of subsystems is large, the coupling is sparse, and local communication is relatively fast and cheap. A numerical example illustrates the effects of the local problem size, and the number of iterations on convergence to the centralized solution."}
{"category": "Math", "title": "Raccord sur les espaces de Berkovich", "abstract": "Let $X$ be a Berkovich space over a valued field. We prove that every finite group is a Galois group over $\\Ms(B)(T)$, where $\\Ms(B)$ is the field of meromorphic functions over a part $B$ of $X$ satisfying some conditions. This gives a new geometric proof that every finite group is a Galois group over $K(T)$, where $K$ is a complete valued field with non-trivial valuation. Then we switch to Berkovich spaces over ${\\bf Z}$ and use a similar strategy to give a new proof of the following theorem by D. Harbater: every finite group is a Galois group over a field of convergent arithmetic power series. We believe our proof to be more geometric and elementary that the original one. We have included the necessary background on Berkovich spaces over ${\\bf Z}$."}
{"category": "Math", "title": "Decompositions of the tensor products of irreducible sl(2)-modules in characteristic 3", "abstract": "We completely describe the decompositions (into indecomposable submodules) of the tensor products of irreducible sl(2)-modules in characteristic 3. The answer resembles analogous decompositions for the Lie superalgebra sl(1|1)."}
{"category": "Math", "title": "A hypergraph regularity method for generalised Turan problems", "abstract": "We describe a method that we believe may be foundational for a comprehensive theory of generalised Turan problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that any 3-graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2-q+1 and n/2, but for PG_2(4) we find the somewhat surprising phenomenon that the threshold is less than (1/2-c)n for some small c>0. We conclude by setting out a program for future developments of this method to tackle other problems."}
{"category": "Math", "title": "RCF1: Theories of PR Maps and Partial PR Maps", "abstract": "We give to the categorical theory PR of Primitive Recursion a logically simple, algebraic presentation, via equations between maps, plus one genuine Horner type schema, namely Freyd's uniqueness of the initialised iterated. Free Variables are introduced - formally - as another names for projections. Predicates \\chi: A -> 2 admit interpretation as (formal) Objects {A|\\chi} of a surrounding Theory PRA = PR + (abstr) : schema (abstr) formalises this predicate abstraction into additional Objects. Categorical Theory P\\hat{R}_A \\sqsupset PR_A \\sqsupset PR then is the Theory of formally partial PR-maps, having Theory PR_A embedded. This Theory P\\hat{R}_A bears the structure of a (still) diagonal monoidal category. It is equivalent to \"the\" categorical theory of \\mu-recursion (and of while loops), viewed as partial PR maps. So the present approach to partial maps sheds new light on Church's Thesis, \"embedded\" into a Free-Variables, formally variable-free (categorical) framework."}
{"category": "Math", "title": "School evasion: A hard reality", "abstract": "The present work has as objective to show the profile of students who abandoned the studies in a High School, located in Sao Joao de Meriti city, municipal district of Rio de Janeiro state, by means of statistical analysis. The presented indices portray an undesirable reality with almost 20% school evasion, beyond showing that more the half of the students not standing in adequate series. Keywords: School evasion, High Schools, Educational Statistics."}
{"category": "Math", "title": "Sports scheduling for not all pairs of teams", "abstract": "We consider the following sports scheduling problem. Consider $2n$ teams in a sport league. Each pair of teams must play exactly one match in $2n-1$ days. That is, $n$ games are held simultaneously in a day. We want to make a schedule which has $n(2n-1)$ games for $2n-1$ days. When we make a schedule, the schedule must satisfy a constraint according to the HAP table, which designates a home game or an away game for each team and each date. Two teams cannot play against each other unless one team is assigned to a home game and the other team is assigned to an away game. Recently, D. Briskorn proposed a necessary condition for a HAP table to have a proper schedule. And he proposed a conjecture that such a condition is also sufficient. That is, if a solution to the linear inequalities exists, they must have an integral solution. In this paper, we rewrite his conjecture by using perfect matchings. We consider a monoid in the affine space generated by perfect matchings. In terms of the Hilbert basis of such a monoid, the problem is naturally generalized to a scheduling problem for not all pairs of teams described by a regular graph. In this paper, we show a regular graph such that the corresponding linear inequalities have a solution but do not have any integral solution. Moreover we discuss for which regular graphs the statement generalizing the conjecture holds."}
{"category": "Math", "title": "Parking functions and vertex operators", "abstract": "We introduce several associative algebras and series of vector spaces associated to these algebras. Using lattice vertex operators, we obtain dimension and character formulae for these spaces. In particular, we a series of representations of symmetric groups which turn out to be isomorphic to parking function modules. We also construct series of vector spaces whose dimensions are Catalan numbers and Fuss--Catalan numbers respectively. Conjecturally, these spaces are related to spaces of global sections of vector bundles on (zero fibres of) Hilbert schemes and representations of rational Cherednik algebras."}
{"category": "Math", "title": "In whose mind is Mathematics an \"a priori cognition\"?", "abstract": "According to the philosopher Kant, Mathematics is an \"a priori cognition\". Kant's assumption, together with the unsolvability of Hilbert's 10th problem, implies an astonishing result."}
{"category": "Math", "title": "Grassmannian Estimation", "abstract": "This paper discusses the family of distributions on the Grassmannian of the linear span of r central gaussian vectors parametrized by the covariance matrix. Our main result is an existence and uniqueness criterion for the maximum likelihood estimate of a sample."}
{"category": "Math", "title": "Understanding Weil-Petersson curvature", "abstract": "A brief history of the investigation of the Weil-Petersson curvature and a summary of Teichm\\\"{u}ller theory are provided. A report is presented on the program to describe an intrinsic geometry with the Weil-Petersson metric and geodesic-length functions. Formulas for the metric, covariant derivative and formulas for the curvature tensor are presented. A discussion of methods is included. Recent and new applications are sketched, including results from the work of Liu-Sun-Yau, an examination of the Yamada model metric and a description of Jacobi fields along geodesics to the boundary."}
{"category": "Math", "title": "Some limit theorems for rescaled Wick powers", "abstract": "We establish the strong L2(P)-convergence of properly rescaled Wick powers as the power index tends to infinity. The explicit representation of such limit will also provide the convergence in distribution to normal and log-normal random variables. The proofs rely on some estimates for the L2(P)-norm of Wick products and on the properties of second quantization operators."}
{"category": "Math", "title": "Finitely presented residually free groups", "abstract": "We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all $n\\in\\mathbb{N}$, a residually free group is of type ${\\rm{FP}}_n$ if and only if it is of type ${\\rm{F}}_n$. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\\rm{FP}}_\\infty$ nor of Stallings-Bieri type. The template for these examples leads to a more constructive characterization of finitely presented residually free groups up to commensurability. We show that the class of finitely presented residually free groups is recursively enumerable and present a reduction of the isomorphism problem. A new algorithm is described which, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. The (multiple) conjugacy and membership problems for finitely presented subgroups of residually free groups are solved."}
{"category": "Math", "title": "Surgical distance between lens spaces", "abstract": "It is well-known that any pair of closed orientable 3-manifolds are related by a finite sequence of Dehn surgeries on knots. Furthermore Kawauchi showed that such knots can be taken to be hyperbolic. In this article, we consider the minimal length of such sequences connecting a pair of 3-manifolds, in particular, a pair of lens spaces."}
{"category": "Math", "title": "A note on certain Kronecker coefficients", "abstract": "We prove an explicit formula for the tensor product with itself of an irreducible complex representation of the symmetric group defined by a rectangle of height two. We also describe part of the decomposition for the tensor product of representations defined by rectangles of heights two and four. Our results are deduced, through Schur-Weyl duality, from the observation that certain actions on triple tensor products of vector spaces, are multiplicity free."}
{"category": "Math", "title": "Chirplet approximation of band-limited, real signals made easy", "abstract": "In this paper we present algorithms for approximating real band-limited signals by multiple Gaussian Chirps. These algorithms do not rely on matching pursuit ideas. They are hierarchial and, at each stage, the number of terms in a given approximation depends only on the number of positive-valued maxima and negative-valued minima of a signed amplitude function characterizing part of the signal. Like the algorithms used in \\cite{gre2} and unlike previous methods, our chirplet approximations require neither a complete dictionary of chirps nor complicated multi-dimensional searches to obtain suitable choices of chirp parameters."}
{"category": "Math", "title": "New examples of finitely presented groups with strong fixed point properties", "abstract": "The aim of this note is to give an easy example of a finitely presented group that cannot act without a fix point on a CAT(0) space of finite dimension. Such an example has been recently constructed by Arjantseva et al., using other techniques"}
{"category": "Math", "title": "Cremona transformations and diffeomorphisms of surfaces", "abstract": "We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result says that if X is rational, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the group of self-diffeomorphisms of X."}
{"category": "Math", "title": "Weighted graphs defining facets: a connection between stable set and linear ordering polytopes", "abstract": "A graph is alpha-critical if its stability number increases whenever an edge is removed from its edge set. The class of alpha-critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a remarkable result of Lov\\'asz (1978) is the finite basis theorem for alpha-critical graphs of a fixed defect. The class of alpha-critical graphs is also of interest for at least two topics of polyhedral studies. First, Chv\\'atal (1975) shows that each alpha-critical graph induces a rank inequality which is facet-defining for its stable set polytope. Investigating a weighted generalization, Lipt\\'ak and Lov\\'asz (2000, 2001) introduce critical facet-graphs (which again produce facet-defining inequalities for their stable set polytopes) and they establish a finite basis theorem. Second, Koppen (1995) describes a construction that delivers from any alpha-critical graph a facet-defining inequality for the linear ordering polytope. Doignon, Fiorini and Joret (2006) handle the weighted case and thus define facet-defining graphs. Here we investigate relationships between the two weighted generalizations of alpha-critical graphs. We show that facet-defining graphs (for the linear ordering polytope) are obtainable from 1-critical facet-graphs (linked with stable set polytopes). We then use this connection to derive various results on facet-defining graphs, the most prominent one being derived from Lipt\\'ak and Lov\\'asz's finite basis theorem for critical facet-graphs. At the end of the paper we offer an alternative proof of Lov\\'asz's finite basis theorem for alpha-critical graphs."}
{"category": "Math", "title": "Near universal cycles for subsets exist", "abstract": "Let S be a cyclic n-ary sequence. We say that S is a {\\it universal cycle} ((n,k)-Ucycle) for k-subsets of [n] if every such subset appears exactly once contiguously in S, and is a Ucycle packing if every such subset appears at most once. Few examples of Ucycles are known to exist, so the relaxation to packings merits investigation. A family {S_n} of (n,k)-Ucycle packings for fixed k is a near-Ucycle if the length of S_n is $(1-o(1))\\binom{n}{k}$. In this paper we prove that near-(n,k)-Ucycles exist for all k."}
{"category": "Math", "title": "Characterizations of lattice surfaces", "abstract": "We answer a question of Vorobets by showing that the lattice property for flat surfaces is equivalent to the existence of a positive lower bound for the areas of affine triangles. We show that the set of affine equivalence classes of lattice surfaces with a fixed positive lower bound for the areas of triangles is finite and we obtain explicit bounds on its cardinality. We deduce several other characterizations of the lattice property."}
{"category": "Math", "title": "Twisted geometric Satake equivalence", "abstract": "We generalize the classical Satake equivalence as follows. Let k be an algebraically closed field, set O=k[[t]] and F=k((t)). For an almost simple algebraic group G we classify central extensions of G(F) by the multiplicative group. Any such extension E splits canonically over G(O). Consider the category of G(O)-biinvariant perverse sheaves on E with a given Gm-monodromy . We show that this is a tensor category, which is tensor equivalent to the category of representations of a reductive group. We compute the root datum of this group."}
{"category": "Math", "title": "Frobenius map for quintic threefolds", "abstract": "We calculate the matrix of the Frobenius map on the middle dimensional cohomology of the one parameter family that is related by mirror symmetry to the family of all quintic threefolds."}
{"category": "Math", "title": "Universality in Complex Wishart ensembles: The 2 cut case", "abstract": "We studied the universality of Wishart ensembles whose covariance matrix has 2 distinct eigenvalues. We studied the asymptotic limit when the number of both eigenvalues goes to infinity and obtained universality results. In this case, the limiting eigenvalue distribution can be supported on 1 or 2 disjoint intervals. We obtained a necessary and sufficient condition on the parameters such that the limiting distribution is supported on 2 disjoint intervals and have computed the eigenvalue density in the limit. Furthermore, by using Riemann-Hilbert analysis, we have shown that under proper rescaling of the eigenvalues, the limiting correlation kernel is given by the sine kernel and the Airy kernel in the bulk and the edge of the spectrum respectively. As a consequence, the behavior of the largest eigenvalue in this model is described by the Tracy-Widom distribution."}
{"category": "Math", "title": "Polyfolds And A General Fredholm Theory", "abstract": "We survey a very general (nonlinear) Fredholm theory for a new class of ambient spaces, called polyfolds. This theory is being currently developed jointly with K. Wysocki and E. Zehnder. The basic feature of these new spaces is that in general they may have locally varying dimensions. These new spaces are needed for a functional analytic treatment of nonlinear problems involving analytic limiting behavior. This theory is applicable to Gromov-Witten and Floer Theory as well as Symplectic Field Theory."}
{"category": "Math", "title": "Locally connected models for Julia sets", "abstract": "Let $P$ be a polynomial with a connected Julia set $J$. We use continuum theory to show that it admits a \\emph{finest monotone map $\\ph$ onto a locally connected continuum $J_{\\sim_P}$}, i.e. a monotone map $\\ph:J\\to J_{\\sim_P}$ such that for any other monotone map $\\psi:J\\to J'$ there exists a monotone map $h$ with $\\psi=h\\circ \\ph$. Then we extend $\\ph$ onto the complex plane $\\C$ (keeping the same notation) and show that $\\ph$ monotonically semiconjugates $P|_{\\C}$ to a \\emph{topological polynomial $g:\\C\\to \\C$}. If $P$ does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi's fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a criterion and a useful sufficient condition for the map $\\ph$ not to collapse $J$ into a point."}
{"category": "Math", "title": "An analogue of the Narasimhan-Seshadri theorem and some applications", "abstract": "We prove an analogue in higher dimensions of the classical Narasimhan-Seshadri theorem for strongly stable vector bundles of degree 0 on a smooth projective variety $X$ with a fixed ample line bundle $\\Theta$. As applications, over fields of characteristic zero, we give a new proof of the main theorem in a recent paper of Balaji and Koll\\'ar and derive an effective version of this theorem; over uncountable fields of positive characteristics, if $G$ is a simple and simply connected algebraic group and the characteristic of the field is bigger than the Coxeter index of $G$, we prove the existence of strongly stable principal $G$ bundles on smooth projective surfaces whose holonomy group is the whole of $G$."}
{"category": "Math", "title": "A note on stability conditions for planar switched systems", "abstract": "This paper is concerned with the stability problem for the planar linear switched system $\\dot x(t)=u(t)A_1x(t)+(1-u(t))A_2x(t)$, where the real matrices $A_1,A_2\\in \\R^{2\\times 2}$ are Hurwitz and $u(\\cdot) [0,\\infty[\\to\\{0,1\\}$ is a measurable function. We give coordinate-invariant necessary and sufficient conditions on $A_1$ and $A_2$ under which the system is asymptotically stable for arbitrary switching functions $u(\\cdot)$. The new conditions unify those given in previous papers and are simpler to be verified since we are reduced to study 4 cases instead of 20. Most of the cases are analyzed in terms of the function $\\Gamma(A_1,A_2)={1/2}(\\tr(A_1) \\tr(A_2)- \\tr(A_1A_2))$."}
{"category": "Math", "title": "Test for reality of algebraic functions", "abstract": "In this paper is proved that a complex algebraic function on complexification of a real algebraic curve is equivalent to real algebraic function, if and only if the divisor of preimage of critical values is stable under the involution of complex conjugation."}
{"category": "Math", "title": "Optimal Risk Sharing under Distorted Probabilities", "abstract": "We study optimal risk sharing among $n$ agents endowed with distortion risk measures. Our model includes market frictions that can either represent linear transaction costs or risk premia charged by a clearing house for the agents. Risk sharing under third-party constraints is also considered. We obtain an explicit formula for Pareto optimal allocations. In particular, we find that a stop-loss or deductible risk sharing is optimal in the case of two agents and several common distortion functions. This extends recent result of Jouini et al. (2006) to the problem with unbounded risks and market frictions."}
{"category": "Math", "title": "Survival tree and meld to predict long term survival in liver transplantation waiting list", "abstract": "Background: Many authors have described MELD as a predictor of short-term mortality in the liver transplantation waiting list. However MELD score accuracy to predict long term mortality has not been statistically evaluated. Objective: The aim of this study is to analyze the MELD score as well as other variables as a predictor of long-term mortality using a new model: the Survival Tree analysis. Study Design and Setting: The variables obtained at the time of liver transplantation list enrollment and considered in this study are: sex, age, blood type, body mass index, etiology of liver disease, hepatocellular carcinoma, waiting time for transplant and MELD. Mortality on the waiting list is the outcome. Exclusion, transplantation or still in the transplantation list at the end of the study are censored data. Results: The graphical representation of the survival trees showed that the most statistically significant cut off is related to MELD score at point 16. Conclusion: The results are compatible with the cut off point of MELD indicated in the clinical literature."}
{"category": "Math", "title": "A criterion on instability of rotating cylindrical surfaces", "abstract": "We consider a column of a rotating stationary surface in Euclidean space. We obtain a value $l_0>0$ in such way that if the length $l$ of column satisfies $l>l_0$, then the surface is instable. This extends, in some sense, previous results due to Plateau and Rayleigh for columns of surfaces with constant mean curvature."}
{"category": "Math", "title": "A comparison result for radial solutions of the mean curvature equation", "abstract": "We establish two comparison results between the solutions of a class of mean curvature equations and pieces of arcs of circles that satisfy the same Neumann boundary condition. Finally we present a number of examples where our estimates can be applied, some of them have a physical motivation."}
{"category": "Math", "title": "Artinianness of local cohomology modules", "abstract": "Let $A$ be a noetherian ring, $\\fa$ an ideal of $A$, and $M$ an $A$--module. Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case."}
{"category": "Math", "title": "Boolean Factor Congruences and Property (*)", "abstract": "A variety V has Boolean factor congruences (BFC) if the set of factor congruences of every algebra in V is a distributive sublattice of its congruence lattice; this property holds in rings with unit and in every variety which has a semilattice operation. BFC has a prominent role in the study of uniqueness of direct product representations of algebras, since it is a strengthening of the refinement property. We provide an explicit Mal'cev condition for BFC. With the aid of this condition, it is shown that BFC is equivalent to a variant of the definability property (*), an open problem in R. Willard's work (\"Varieties Having Boolean Factor Congruences,\" J. Algebra, 132 (1990))."}
{"category": "Math", "title": "Minimizers of Convex Functionals Arising in Random Surfaces", "abstract": "We investigate regularity of minimizers in two dimensions for certain classes of non-smooth convex functionals. In particular our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of Kenyon, Okounkov and others"}
{"category": "Math", "title": "Stationary rotating surfaces in Euclidean space", "abstract": "A stationary rotating surface is a compact surface in Euclidean space whose mean curvature $H$ at each point $x$ satisfies $2H(x)=a r^2+b$, where $r$ is the distance from $x$ to a fixed straight-line $L$, and $a$ and $b$ are constants. These surfaces are solutions of a variational problem that describes the shape of a drop of incompressible fluid in equilibrium by the action of surface tension when it rotates about $L$ with constant angular velocity. The effect of gravity is neglected. In this paper we study the geometric configurations of such surfaces, focusing the relationship between the geometry of the surface and the one of its boundary. As special cases, we will consider two families of such surfaces: axisymmetric surfaces and embedded surfaces with planar boundary."}
{"category": "Math", "title": "Parabolic Weingarten surfaces in hyperbolic space", "abstract": "A surface in hyperbolic space $\\h^3$ invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of $\\h^3$ that satisfy a linear Weingarten relation of the form $a\\kappa_1+b\\kappa_2=c$ or $aH+bK=c$, where $a,b,c\\in \\r$ and, as usual, $\\kappa_i$ are the principal curvatures, $H$ is the mean curvature and $K$ is de Gaussian curvature. We classify all parabolic linear Weingarten surfaces in hyperbolic space."}
{"category": "Math", "title": "Factor Congruences in Semilattices", "abstract": "We characterize factor congruences in semilattices by using generalized notions of order ideal and of direct sum of ideals. When the semilattice has a minimum (maximum) element, these generalized ideals turn into ordinary (dual) ideals."}
{"category": "Math", "title": "Free Araki-Woods factors and Connes' bicentralizer problem", "abstract": "We show that for any free Araki-Woods factor $\\mathcal{M} = \\Gamma(H_\\R, U_t)\"$ of type ${\\rm III_1}$, the bicentralizer of the free quasi-free state $\\varphi_U$ is trivial. Using Haagerup's Theorem, it follows that there always exists a faithful normal state $\\psi$ on $\\mathcal{M}$ such that $(\\mathcal{M}^\\psi)' \\cap \\mathcal{M} = \\C$."}
{"category": "Math", "title": "Local minimizers and low energy paths in a model of material microstructure with a surface energy term", "abstract": "A family of integral functionals F which, in a simplified way, model material microstructure occupying a two-dimensional domain D and which take account of surface energy and a variable well-depth is studied. It is shown that there is a critical well-depth, whose scaling with the surface energy density and domain dimensions is given, below which the state u=0 is the global minimizer of a typical f in the class F. It is also shown that u=0 is a strict local minimizer of f in the sense that if a non-zero v is admissible and either its L2 norm or the meaure of the subset of D where |v_{y}| exceeds 1 is sufficiently small (with quantitative bounds given in terms of the parameters appearing in the energy functional f) then f(v) > f(0). Low energy paths between u=0 and the global minimizer (in the case of a sufficiently large well-depth) are given such that the cost of introducing small regions where |v_{y}| is larger 1 (analogous to nucleation of martensite in austenite) into the domain D can be made arbitrarily small."}
{"category": "Math", "title": "The Total Gauss Curvature of a Three-Manifold Immersed in r 4", "abstract": "This paper has been removed. It is already a well known result."}
{"category": "Math", "title": "Multiplicity results for the assigned Gauss curvature problem in R2", "abstract": "To study the problem of the assigned Gauss curvature with conical singularities on Riemanian manifolds, we consider the Liouville equation with a single Dirac measure on the two-dimensional sphere. By a stereographic projection, we reduce the problem to a Liouville equation on the euclidean plane. We prove new multiplicity results for bounded radial solutions, which improve on earlier results of C.-S. Lin and his collaborators. Based on numerical computations, we also present various conjectures on the number of unbounded solutions. Using symmetries, some multiplicity results for non radial solutions are also stated."}
{"category": "Math", "title": "Properties of Koszul homology modules", "abstract": "We investigate various module-theoretic properties of Koszul homology under mild conditions. These include their depth, $S_2$-property and their Bass numbers"}
{"category": "Math", "title": "A note on ccc forcings", "abstract": "The aim of this short note is to communicate a simple solution to the problem posed in [1] as Question 7.2.7: is it true that for every ccc $\\sigma$-ideal I any I-positive Borel set contains modulo I an I-positive closed set?"}
{"category": "Math", "title": "A class of unbiased location invariant Hill-type estimators for heavy tailed distributions", "abstract": "Based on the methods provided in Caeiro and Gomes (2002) and Fraga Alves (2001), a new class of location invariant Hill-type estimators is derived in this paper. Its asymptotic distributional representation and asymptotic normality are presented, and the optimal choice of sample fraction by Mean Squared Error is also discussed for some special cases. Finally comparison studies are provided for some familiar models by Monte Carlo simulations."}
{"category": "Math", "title": "On spectral minimal partitions II, the case of the rectangle", "abstract": "In continuation of \\cite{HHOT}, we discuss the question of spectral minimal 3-partitions for the rectangle $]-\\frac a2,\\frac a2[\\times ] -\\frac b2,\\frac b2[ $, with $0< a\\leq b$. It has been observed in \\cite{HHOT} that when $0<\\frac ab < \\sqrt{\\frac 38}$ the minimal 3-partition is obtained by the three nodal domains of the third eigenfunction corresponding to the three rectangles $]-\\frac a2,\\frac a2[\\times ] -\\frac b2,-\\frac b6[$, $]-\\frac a2,\\frac a2[\\times ] -\\frac b6,\\frac b6[$ and $]-\\frac a2,\\frac a2[\\times ] \\frac b6, \\frac b2[$. We will describe a possible mechanism of transition for increasing $\\frac ab$ between these nodal minimal 3-partitions and non nodal minimal 3-partitions at the value $ \\sqrt{\\frac 38}$ and discuss the existence of symmetric candidates for giving minimal 3-partitions when $ \\sqrt{\\frac 38}<\\frac ab \\leq 1$. Numerical analysis leads very naturally to nice questions of isospectrality which are solved by introducing Aharonov-Bohm Hamiltonians or by going on the double covering of the punctured rectangle."}
{"category": "Math", "title": "RCF2: Evaluation and Consistency", "abstract": "We construct here an iterative evaluation of all PR map codes: progress of this iteration is measured by descending complexity within \"Ordinal\" O := N[\\omega] of polynomials in one indeterminate, ordered lexicographically. Non-infinit descent of such iterations is added as a mild additional axiom schema (\\pi_O) to Theory PR_A = PR+(abstr) of Primitive Recursion with predicate abstraction, out of forgoing part RCF 1. This then gives (correct) \"on\"-termination of iterative evaluation of argumented deduction trees as well, for theories PR_A+(\\pi_O). By means of this constructive evaluation the Main Theorem is proved, on Termination-conditioned (Inner) Soundness for such theories, Ordinal O extending N[\\omega]. As a consequence we get Self-Consistency for these theories, namely derivation of its own free-variable Consistency formula. As to expect from classical setting, Self-Consistency gives (unconditioned) Objective Soundness. Termination-Conditioned Soundness holds \"already\" for PR_A, but it turns out that at least present derivation of Consistency from this conditioned Soundness depends on schema (\\pi_O) of non-infinit descent in Ordinal O := \\N[\\omega]."}
{"category": "Math", "title": "On the geometry of the normal bundle with a metric of Cheeger-Gromoll type", "abstract": "We investigate the geometry of a normal bundle equipped with a $(p,q)$-metric, i.e., Riemannian metric of Cheeger-Gromoll type, to the submanifold of a Riemannian manifold. We derive all natural object as the Levi-Civita connection, curvature tensor, sectional and scalar curvature. We prove that under some natural conditions the sectional curvature of this bundle may be bounded from below by given arbitrary large positive constant. Next we investigate $(p,q)$-metrics from the complex geometry point of view. We show when the normal bundle can by equipped with a structure of almost Hermitian, almost K\\\"ahlerian, conformally almost K\\\"ahlerian or K\\\"ahlerian manifold."}
{"category": "Math", "title": "Vector bundles on Sasakian manifolds", "abstract": "We investigate the analog of holomorphic vector bundles in the context of Sasakian manifolds."}
{"category": "Math", "title": "A globally accelerated numerical method for optical tomography with continuous wave source", "abstract": "A new numerical method for an inverse problem for an elliptic equation with unknown potential is proposed. In this problem the point source is running along a straight line and the source-dependent Dirichlet boundary condition is measured as the data for the inverse problem. A rigorous convergence analysis shows that this method converges globally, provided that the so-called tail function is approximated well. This approximation is verified in numerical experiments, so as the global convergence. Applications to medical imaging, imaging of targets on battlefields and to electrical impedance tomography are discussed."}
{"category": "Math", "title": "Maximal monotonicity, conjugation and the duality product in non-reflexive Banach spaces", "abstract": "Maximal monotone operators on a Banach space into its dual can be represented by convex functions bounded below by the duality product. It is natural to ask under which conditions a convex function represents a maximal monotone operator. A satisfactory answer, in the context of reflexive Banach spaces, has been obtained some years ago. Recently, a partial result on non-reflexive Banach spaces was obtained. In this work we study some others conditions which guarantee that a convex function represents a maximal monotone operator in non-reflexive Banach spaces."}
{"category": "Math", "title": "Multilevel Discretized Random Field Models with \"Spin\" Correlations for the Simulation of Environmental Spatial Data", "abstract": "A problem of practical significance is the analysis of large, spatially distributed data sets. The problem is more challenging for variables that follow non-Gaussian distributions. We show that the spatial correlations between variables can be captured by interactions between \"spins\". The spins represent multilevel discretizations of the initial field with respect to a number of pre-defined thresholds. The spatial dependence between the \"spins\" is imposed by means of short-range interactions. We present two approaches, inspired by the Ising and Potts models, that generate conditional simulations from samples with missing data. The simulations of the \"spin system\" are forced to respect locally the sample values and the system statistics globally. We compare the two approaches in terms of their ability to reproduce the sample statistical properties, to predict data at unsampled locations, as well as in terms of their computational complexity. We discuss the impact of relevant simulation parameters, such as the domain size, the number of discretization levels, and the initial conditions."}
{"category": "Math", "title": "Stein Spaces Characterized by their Endomorphisms", "abstract": "Finite dimensional Stein spaces admitting a proper holomorphic embedding of the complex line are characterized, among all complex spaces, by their holomorphic endomorphism semigroup in the sense that any semigroup isomorphism induces either a biholomorphic or an antibiholomorphic map between them."}
{"category": "Math", "title": "A remark on the structure of the Busemann representative of a polyconvex function", "abstract": "Let W be a polyconvex function defined on the 2 x 2 real matrices. The Busemann representative f, say, of W is the largest possible convex representative of W. Writing L for the set of affine functions on R^{5} such that a(A, det A) is less than or equal to W(A) for all 2 x 2 real matrices A, f can then be expressed as f(X) = sup {a(X): a lies in L}. This short note proves the surprising result that f is in general strictly larger than the `natural' convex representative g(X) = sup {a(X): a lies in L and a(A, det A)=W(A) for some A}."}
{"category": "Math", "title": "A lower bound for the error term in Weyl's law for certain Heisenberg manifolds", "abstract": "This article provides an Omega-result for the remainder term in Weyl's law for the spectral counting function of certain (2l+1)-dimensional Heisenberg manifolds."}
{"category": "Math", "title": "On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions", "abstract": "We extend the result of D. Phillips (On one-homogeneous solutions to elliptic systems in two dimensions. C. R. Math. Acad. Sci. Paris 335 (2002), no. 1, 39-42) by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondly, Phillips's original result is shown to apply to one-homogeneous solutions belonging to the Sobolev space H^{1}, from which his treatment of Lipschitz solutions follows as a special case. A singular one-homogeneous solution to an elliptic system violating the hypotheses of the main theorem is constructed using a variational method which has links to nonlinear elasticity."}
{"category": "Math", "title": "Spin(7) instantons and the Hodge Conjecture for certain abelian four-folds: a modest proposal", "abstract": "The Hodge Conjecture is equivalent to a statement about conditions under which a complex vector bundle on a smooth complex projective variety admits a holomorphic structure. I advertise a class of abelian four-folds due to Mumford where this approach could be tested. I construct explicit smooth vector bundles - which can in fact be constructed in terms of of smooth line bundles - whose Chern characters are given Hodge classes. An instanton connection on these vector bundles would endow them with a holomorphic structure and thus prove that these classes are algebraic. I use complex multiplication to exhibit Cayley cycles representing the given Hodge classes. I find alternate complex structures with respect to which the given bundles are holomorphic, and close with a suggestion (due to G. Tian) as to how this may possibly be put to use."}
{"category": "Math", "title": "Dominance, Intimidation, and `Choking' on the PGA Tour", "abstract": "Extending the work of Connolly and Rendleman (2008), we document the dominance of Tiger Woods during the 1998-2001 PGA Tour seasons. We show that by playing \"average,\" Woods could have won some tournaments and placed no worse than fourth in the tournaments in which he participated in year 2000, his best on the PGA Tour. No other PGA Tour player in our sample could have come close to such a feat. We also are able to quantify the intimidation factor associated with playing with Woods. On average, players who were paired with Woods during the 1998-2001 period scored 0.462 strokes per round worse than normal. Although we find that Woods' presence in a tournament may have had a small, but statistically significant adverse impact on the entire field, this effect was swamped by the apparent intimidation factor associated with having to play with Tiger side-by-side. We also demonstrate that Phil Mickelson's performance in major golf championships over the 1998-2001 period was not nearly as bad as was frequently mentioned in the golf press. Although Mickelson won no majors during this period, he played sufficiently well to have won one or two majors under normal circumstances. Moreover, his overall performance in majors, relative to his estimated skill level, was comparable to that of Tiger Woods, who won five of 16 major golf championships during our four-year sample period. Thus, the general characterization of Woods as golf's dominant player over the 1998-2001 period was accurate, but the frequent characterization of Phil Mickelson choking in majors was not."}
{"category": "Math", "title": "Semistable principal Higgs bundles", "abstract": "We give a Miyaoka-type semistability criterion for principal Higgs G-bundles E on complex projective manifolds of any dimension, i.e., we prove that E is semistable and the second Chern class of its adjoint bundle vanishes if and only if certain line bundles, obtained from the characters of the parabolic subgroups of G, are numerically effective. We also give alternative characterizations in terms of a notion of numerical effectiveness of Higgs vector bundles we have recently introduced."}
{"category": "Math", "title": "Inhomogeneous Diophantine approximation on curves and Hausdorff dimension", "abstract": "The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in $R^n$ akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the fundamental homogeneous theorems of R.C. Baker (1978), Dodson, Dickinson (2000) and Beresnevich, Bernik, Kleinbock, Margulis (2002). In the case of planar curves, the complete Hausdorff dimension theory is developed"}
{"category": "Math", "title": "A variational principle for topological pressure for certain non-compact sets", "abstract": "Let $(X,d)$ be a compact metric space, $f:X \\mapsto X$ be a continuous map with the specification property, and $\\varphi: X \\mapsto \\IR$ be a continuous function. We prove a variational principle for topological pressure (in the sense of Pesin and Pitskel) for non-compact sets of the form \\[ \\{x \\in X : \\lim_{n \\ra \\infty} \\frac{1}{n} \\sum_{i = 0}^{n-1} \\varphi (f^i (x)) = \\alpha \\}. \\] Analogous results were previously known for topological entropy. As an application, we prove multifractal analysis results for the entropy spectrum of a suspension flow over a continuous map with specification and the dimension spectrum of certain non-uniformly expanding interval maps."}
{"category": "Math", "title": "Bockstein theorem for nilpotent groups", "abstract": "We extend the definition of Bockstein basis $\\sigma(G)$ to nilpotent groups $G$. A metrizable space $X$ is called a {\\it Bockstein space} if $\\dim_G(X) = \\sup\\{\\dim_H(X) | H\\in \\sigma(G)\\}$ for all Abelian groups $G$. Bockstein First Theorem says that all compact spaces are Bockstein spaces. Here are the main results of the paper: Let $X$ be a Bockstein space. If $G$ is nilpotent, then $\\dim_G(X) \\leq 1$ if and only if $\\sup\\{\\dim_H(X) | H\\in\\sigma(G)\\}\\leq 1$. $X$ is a Bockstein space if and only if $\\dim_{\\Z_{(l)}} (X) = \\dim_{\\hat{Z}_{(l)}}(X)$ for all subsets $l$ of prime numbers."}
{"category": "Math", "title": "Remarks on Kahler Ricci Flow", "abstract": "We study some estimates along the Kahler Ricci flow on Fano manifolds. Using these estimates, we show the convergence of Kahler Ricci flow directly if the $\\alpha$-invariant of the canonical class is greater than $\\frac{n}{n+1}$. Applying these convergence theorems, we can give a flow proof of Calabi conjecture on such Fano manifolds. In particular, the existence of Kahler Einstein metrics on a lot of Fano surfaces can be proved by flow method. Note that this geometric conclusion (based on the same assumption) was established earlier via elliptic method by G. Tian. However, a new proof based on Kahler Ricci flow should be still interesting in its own right."}
{"category": "Math", "title": "On the Christoffel-Darboux kernel for random Hermitian matrices with external source", "abstract": "Bleher and Kuijlaars, and Daems and Kuijlaars showed that the correlation functions of the eigenvalues of a random matrix from unitary ensemble with external source can be expressed in terms of the Christoffel-Darboux kernel for multiple orthogonal polynomials. We obtain a representation of this Christoffel-Darboux kernel in terms of the usual orthogonal polynomials."}
{"category": "Math", "title": "Geometric idealizers", "abstract": "Let X be a projective variety, $\\sigma$ an automorphism of X, L a $\\sigma$-ample invertible sheaf on X, and Z a closed subscheme of X. Inside the twisted homogeneous coordinate ring $B = B(X, L, \\sigma)$, let I be the right ideal of sections vanishing at Z. We study the subring R = k + I of B. Under mild conditions on Z and $\\sigma$, R is the idealizer of I in B: the maximal subring of B in which I is a two-sided ideal. We give geometric conditions on Z and $\\sigma$ that determine the algebraic properties of R, and show that if Z and $\\sigma$ are sufficiently general, in a sense we make precise, then R is left and right noetherian, has finite left and right cohomological dimension, is strongly right noetherian but not strongly left noetherian, and satisfies right $\\chi_d$ (where d = \\codim Z) but fails left $\\chi_1$. We also give an example of a right noetherian ring with infinite right cohomological dimension, partially answering a question of Stafford and Van den Bergh. This generalizes results of Rogalski in the case that Z is a point in $\\mathbb{P}^d$."}
{"category": "Math", "title": "Symmetric homogeneous diophantine equations of odd degree", "abstract": "We find a parametric solution of an arbitrary symmetric homogeneous diophantine equation of 5th degree in 6 variables using two primitive solutions. We then generalize this approach to symmetric forms of any odd degree by proving the following results. (1) Every symmetric form of odd degree $n\\ge 5$ in $6 \\cdot 2^{n-5}$ variables has a rational parametric solution depending on $2n-8$ parameters. (2) Let $F(x_1, ..., x_N)$ be a symmetric form of odd degree $n\\ge 5$ in $N=6 \\cdot 2^{n-4}$ variables, and let $q$ be any rational number. Then the equation $F(x_i)=q$ has a rational parametric solution depending on $2n-6$ parameters. The latter result can be viewed as a solution of a problem of Waring type for this class of forms."}
{"category": "Math", "title": "Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds", "abstract": "We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Marino, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting."}
{"category": "Math", "title": "Sur la conjecture de Zagier pour n=4", "abstract": "We express a general 4-hyperlogarithm as a linear combination of 4-hyperlogarithms in two variables. We reduce the Zagier's conjecture for $n=4$ to a combinatorial statement. We give a short survey of the strategy of Goncharov and Zagier for reducing the Zagier's conjecture for general $n$ to combinatorial relations between hyperlogarithms. Such a survey is missing in the literature."}
{"category": "Math", "title": "Mean position of a particle submitted to a potential barrier", "abstract": "A one-dimensional Klein-Gordon problem, which is a physical model for a quantum particle submitted to a potential barrier, is studied numerically : using a variational formulation and a Newmark numerical method, we compute the mean position and standard deviation of the particle as well as their time evolution."}
{"category": "Math", "title": "Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves", "abstract": "We define complexes of vector bundles on products of moduli spaces of framed rank r torsion-free sheaves on the complex projective plane. The top non-vanishing Chern classes of the cohomology of these complexes yield actions of the r-colored Heisenberg and Clifford algebras on the equivariant cohomology of the moduli spaces. In this way we obtain a geometric realization of the boson-fermion correspondence and related vertex operators."}
{"category": "Math", "title": "Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors", "abstract": "Arbitrary operator A on a Banach space X which is the generator of C_0-group with certain growth condition at infinity is considered. The relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse theorems on connection between the degree of smoothness of vector $x\\in X$ with respect to operator A, the rate of convergence to zero of the best approximation of x by exponential type entire vectors for operator A, and the k-module of continuity are established. Also, a generalization of the Bernstein-type inequality is obtained. The results allow to obtain Bernstein-type inequalities in weighted L_p spaces."}
{"category": "Math", "title": "A Lefschetz hyperplane theorem for Mori dream spaces", "abstract": "Let X be a smooth Mori dream space of dimension at least 4. We show that, if X satisfies a suitable GIT condition which we call \"small unstable locus\", then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Neron-Severi spaces of X and Y, and under this identification every Mori chamber of Y is a union of some Mori chambers of X, and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking \"Mori dream hypersurfaces\" of an ambient Mori dream space, provided that it satisfies the GIT condition. To facilitate this, we then show that the GIT condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles, and in the case when X is toric, we show that the condition is equivalent to the fan of X being 2-neighborly."}
{"category": "Math", "title": "Differential Calculus and Integration of Generalized Functions over Membranes", "abstract": "In this paper we continue the development of the differential calculus started by Aragona-Ferandez-Juriaans. Guided by the topology introduced recently by those authors we introduce the notion of membranes and extend the definition of integrals given in [2] to integrals defined on membranes. We use this to prove a generalized version of teh Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. We also show that the generalized transport equation can be solved giving an explicit solution"}
{"category": "Math", "title": "Completion of the Proof of the Geometrization Conjecture", "abstract": "This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the approach indicated by Perelman in his preprints on the subject. This approach is to study the collapsed part of the manifold as time goes to infinity in a Ricci flow with surgery. The main technique for this study is the theory of Alexandrov spaces. This theory gives local models for the collapsed part of the manifold. These local models can be glued together to prove that the collapsed part of the manifold is a graph manifold with incompressible boundary. From this and previous results, geometrization follows easily."}
{"category": "Math", "title": "On the regularity of maximal operators", "abstract": "We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\\mathbb{R}) \\times W^{1,q}(\\mathbb{R}) \\to W^{1,r}(\\mathbb{R})$ with $1 <p,q < \\infty$ and $r\\geq 1$, boundedly and continuously. The same result holds on $\\mathbb{R}^n$ when $r>1$. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions."}
{"category": "Math", "title": "Improved Sequential Stopping Rule for Monte Carlo Simulation", "abstract": "This paper presents an improved result on the negative-binomial Monte Carlo technique analyzed in a previous paper for the estimation of an unknown probability p. Specifically, the confidence level associated to a relative interval [p/\\mu_2, p\\mu_1], with \\mu_1, \\mu_2 > 1, is proved to exceed its asymptotic value for a broader range of intervals than that given in the referred paper, and for any value of p. This extends the applicability of the estimator, relaxing the conditions that guarantee a given confidence level."}
{"category": "Math", "title": "Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials", "abstract": "We obtain optimal trigonometric polynomials of a given degree $N$ that majorize, minorize and approximate in $L^1(\\mathbb{R}/\\mathbb{Z})$ the Bernoulli periodic functions. These are the periodic analogues of two works of F. Littmann that generalize a paper of J. Vaaler. As applications we provide the corresponding Erd\\\"{o}s-Tur\\'{a}n-type inequalities, approximations to other periodic functions and bounds for certain Hermitian forms."}
{"category": "Math", "title": "Some extremal functions in Fourier analysis, II", "abstract": "We obtain extremal majorants and minorants of exponential type for a class of even functions on $\\R$ which includes $\\log |x|$ and $|x|^\\alpha$, where $-1 < \\alpha < 1$. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev inequalities. A further application provides an Erd\\\"{o}s-Tur\\'{a}n-type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials."}
{"category": "Math", "title": "Some extremal functions in Fourier analysis, III", "abstract": "We obtain the best approximation in $L^1(\\R)$, by entire functions of exponential type, for a class of even functions that includes $e^{-\\lambda|x|}$, where $\\lambda >0$, $\\log |x|$ and $|x|^{\\alpha}$, where $-1 < \\alpha < 1$. We also give periodic versions of these results where the approximating functions are trigonometric polynomials of bounded degree."}
{"category": "Math", "title": "A sharp inequality for the Strichartz norm", "abstract": "Let $u:\\R \\times \\R^n \\to \\C$ be the solution of the linear Schr\\\"odinger equation $iu_t + \\Delta u =0$ with initial data $u(0,x) = f(x)$. In the first part of this paper we obtain a sharp inequality for the Strichartz norm $\\|u(t,x)\\|_{L^{2k}_tL^{2k}_x(\\R \\times\\R^n)}$, where $k\\in \\Z$, $k \\geq 2$ and $(n,k) \\neq (1,2)$, that admits only Gaussian maximizers. As corollaries we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi and Hundertmark - Zharnitsky) and also sharp forms of some Sobolev-Strichartz inequalities. In the second part of the paper we express Foschi's sharp inequalities for the Schr\\\"odinger and wave equations in the broader setting of sharp restriction/extension estimates for the paraboloid and the cone."}
{"category": "Math", "title": "Deformations of glued G_2-manifolds", "abstract": "We study how a gluing construction, which produces compact manifolds with holonomy G_2 from matching pairs of asymptotically cylindrical G_2-manifolds, behaves under deformations. We show that the gluing construction defines a smooth map from a moduli space of gluing data to the moduli space of torsion-free G_2-structures on the glued manifold, and that this is a local diffeomorphism. We use this to partially compactify the moduli space of torsion-free G_2-structures, including it as the interior of a topological manifold with boundary. The boundary points are equivalence classes of matching pairs of torsion-free asymptotically cylindrical G_2-structures."}
{"category": "Math", "title": "Foliations for Quasi-Fuchsian 3-Manifolds", "abstract": "In this paper, we prove that if a quasi-Fuchsian 3-manifold contains a minimal surface whose principle curvature is less than 1, then it admits a foliation such that each leaf is a surface of constant mean curvature. The key method that we use here is volume preserving mean curvature flow."}
{"category": "Math", "title": "Torsion points of abelian varieties with values in infinite extensions over a p-adic field", "abstract": "Let $A$ be an abelian variety over a $p$-adic field $K$ and $L$ an algebraic infinite extension over $K$. We consider the finiteness of the torsion part of the group of rational points $A(L)$ under some assumptions. In 1975, Hideo Imai proved that such a group is finite if $A$ has good reduction and $L$ is the cyclotomic $\\mathbb{Z}_p$-extension of $K$. In this talk, first we show a generalization of Imai's result in the case where $A$ has ordinary good reduction. Next we give some finiteness results when $A$ is an elliptic curve and $L$ is the field generated by the $p$-power torsion of an elliptic curve."}
{"category": "Math", "title": "On a Cuntz-Krieger functor", "abstract": "We construct a covariant functor from the topological torus bundles to the so-called Cuntz-Krieger algebras; the functor maps homeomorphic bundles into the stably isomorphic Cuntz-Krieger algebras. It is shown, that the K-theory of the Cuntz-Krieger algebra encodes torsion of the first homology group of the bundle. We illustrate the result by examples."}
{"category": "Math", "title": "Space of K\\\"ahler metrics (IV)--On the lower bound of the K-energy", "abstract": "We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given K\\\"ahler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic line consists of cscK metrics only. We also prove that for simple test configuration where the central fibre has a cscK metric, the K energy functionals in the nearby fibre must also have a uniform lower bound in its underlying K\\\"ahler class."}
{"category": "Math", "title": "Asymptotic Behavior of Multidimensional Scalar Relaxation Shocks", "abstract": "We establish pointwise bounds for the Green function and consequent linearized stability for multidimensional planar relaxation shocks of general relaxation systems whose equilibrium model is scalar, under the necessary assumption of spectral stability. Moreover, we obtain nonlinear $L^{2}$ asymptotic behavior/sharp decay rate of perturbed weak shocks of general simultaneously symmetrizable relaxation systems, under small $L^{1}\\cap H^{[d/2]+3}$ perturbations with first moment in the normal direction to the front."}
{"category": "Math", "title": "On base station localization for state estimation over lossy networks", "abstract": "We consider a state estimation problem where observations are made by multiple sensors. These observations are communicated over a lossy wireless network to a central base station that computes estimates via a Kalman filter. The goal is to determine the optimal location of the base station under a certain class of packet loss probability models. It is shown in the two sensor case that the base station is optimally located at one of the sensor locations. Empirical evidence suggests that the result holds in some generality."}
{"category": "Math", "title": "On the rank of elliptic curves", "abstract": "The paper proves that the Birch and Swinnerton-Dyer conjecture is false."}
{"category": "Math", "title": "Property $(T)$ for noncommutative universal lattices", "abstract": "We establish a new spectral criterion for Kazhdan's property $(T)$ which is applicable to a large class of discrete groups defined by generators and relations. As the main application, we prove property $(T)$ for the groups $EL_n(R)$, where $n\\geq 3$ and $R$ is an arbitrary finitely generated associative ring. We also strengthen some of the results on property $(T)$ for Kac-Moody groups from a paper of Dymara and Januszkiewicz (Invent. Math 150 (2002))."}
{"category": "Math", "title": "Group actions on median spaces", "abstract": "We investigate the geometry of median metric spaces. The group-theoretic applications are towards Kazhdan's property (T) and Haagerup's property."}
{"category": "Math", "title": "From Pet to Split", "abstract": "Various forms of the polynomial ergodic theorem (PET) which attracted substantial attention in ergodic theory study the limits of expressions having the form $1/N\\sum_{n=1}^NT^{q_1(n)}f_1... T^{q_\\ell (n)}f_\\ell$ where $T$ is a weakly mixing measure preserving transformation, $f_i$'s are bounded measurable functions and $q_i$'s are polynomials taking on integer values on the integers. Motivated partially by these results we obtain a central limit theorem for expressions of the form $1/\\sqrt{N}\\sum_{n=1}^N (X_1(q_1(n))X_2(q_2(n))... X_\\ell(q_\\ell(n))-a_1a_2... a_\\ell)$ (sum-product limit theorem--SPLIT) where $X_i$'s are fast $\\alpha$-mixing bounded stationary processes, $a_j=EX_j(0)$ and $q_i$'s are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when $q_i$'s are polynomials of growing degrees. This result can be applied to the case when $X_i(n)=T^nf_i$ where $T$ is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well, as to the case when $X_i(n)=f_i(\\xi_n)$ where $\\xi_n$ is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure."}
{"category": "Math", "title": "Use of abstract Hardy spaces, Real interpolation and Applications to bilinear operators", "abstract": "This paper can be considered as the sequel of [6], where the authors have proposed an abstract construction of Hardy spaces H^1. They shew an interpolation result for these Hardy spaces with the Lebesgue spaces. Here we describe a more precise result using the real interpolation theory and we clarify the use of Hardy spaces. Then with the help of the bilinear interpolation theory, we then give applications to study bilinear operators on Lebesgue spaces. These ideas permit us to study singular operators with singularities similar to those of bilinear Calderon-Zygmund operators in a far more abstract framework as in the euclidean case."}
{"category": "Math", "title": "The cellular structure of the classifying spaces of finite groups", "abstract": "In this paper we complete the description of the $B\\mathbb{Z} /p$-cellularization of the classifying spaces of all finite groups, for all primes $p$. The techniques are based in a careful analysis of the $p$-fusion structure of the groups involved -with special attention to their strongly closed subgroups- and Chach\\'olski's description of the $A$-cellular approximation."}
{"category": "Math", "title": "A Generalized Composition of Quadratic Forms based on Quadratic Pairs", "abstract": "For quadratic spaces which represent 1 there is a characterization of hermitian compositions in the language of algebras-with-involutions using the even Clifford algebra. We extend this notion to define a generalized composition based on quadratic pairs and determine the degrees of minimal compositions for any given quadratic pair."}
{"category": "Math", "title": "Hilbert transforms and the Cauchy integral in euclidean space", "abstract": "We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert transforms."}
{"category": "Math", "title": "Duality for a Cohen-Macaulay local ring", "abstract": "Let $(R,\\fm)$ be a Cohen-Macaulay local ring. If $R$ has a canonical module, then there are some interesting results about duality for this situation. In this paper, we show that one can indeed obtain similar these results in the case $R$ has not a canonical module. Also, we give some characterizations of complete big Cohen-Macaulay modules of finite injective dimension and by using it some characterizations of Gorenstein modules over the $\\fm$-adic completion of $R$ are obtained."}
{"category": "Math", "title": "Unknotting genus one knots", "abstract": "For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is exactly one way to unknot it by means of a crossing change. In the case of the figure-eight knot, we prove that there are precisely two unknotting crossing changes. The proof uses sutured manifold theory and an analysis of the arc complex of the once-punctured torus."}
{"category": "Math", "title": "The Continuum is Countable: Infinity is Unique", "abstract": "Since the theory developed by Georg Cantor, mathematicians have taken a sharp interest in the sizes of infinite sets. We know that the set of integers is infinitely countable and that its cardinality is Aleph0. Cantor proved in 1891 with the diagonal argument that the set of real numbers is uncountable and that there cannot be any bijection between integers and real numbers. Cantor states in particular the Continuum Hypothesis. In this paper, I show that the cardinality of the set of real numbers is the same as the set of integers. I show also that there is only one dimension for infinite sets, Aleph."}
{"category": "Math", "title": "Inverse moments of univariate discrete distributions via the Poisson expansion", "abstract": "In this note we present a series expansion of inverse moments of a non-negative discrete random variate in terms of its factorial cumulants, based on the Poisson-Charlier expansion of a discrete distribution. We apply the general method to the positive binomial distribution and obtain a convergent series for its inverse moments with an error residual that is uniformly bounded on the entire interval 0<=p<=1."}
{"category": "Math", "title": "Hamiltonian submanifolds of regular polytopes", "abstract": "We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called {\\it super-neighborly triangulations}) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the $d$-dimensional cross polytope. These are the \"regular cases\" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of 7 copies of $S^2 \\times S^2$. By this example all regular cases of $n$ vertices with $n < 20$ or, equivalently, all cases of regular $d$-polytopes with $d\\leq 9$ are now decided."}
{"category": "Math", "title": "On strong n-perfect rings", "abstract": "In this paper we introduce the notion of \"strong $n$-perfect rings\" which is in some way a generalization of the notion of \"$n$-perfect rings\". We are mainly concerned with those class of rings in the context of pullbacks. Also we exhibit a class of $n$-perfect rings that are not strong $n$-perfect rings. Finally, we establish the transfer of this notion to the direct product notions."}
{"category": "Math", "title": "Coincidences for multiple summing mappings", "abstract": "In this note we prove new coincidence results for multiple summing mappings, related to the cotypes of the Banach spaces involved."}
{"category": "Math", "title": "Some irreducible representations of the braid group B_n of dimension greater than n", "abstract": "For any n>3, we give a family of finite dimensional irreducible representations of the braid group B_n. Moreover, we give a subfamily parametrized by 0<m<n of dimension the combinatoric number (n,m). The representation obtained in the case m=1 is equivalent to the Standard representation."}
{"category": "Math", "title": "On the homogeneity of global minimizers for the Mumford-Shah functional when K is a smooth cone", "abstract": "We show that if $(u,K)$ is a global minimizer for the Mumford-Shah functional in $R^N$, and if K is a smooth enough cone, then (modulo constants) u is a homogenous function of degree 1/2. We deduce some applications in $R^3$ as for instance that an angular sector cannot be the singular set of a global minimizer, that if $K$ is a half-plane then $u$ is the corresponding cracktip function of two variables, or that if K is a cone that meets $S^2$ with an union of $C^1$ curvilinear convex polygones, then it is a $P$, $Y$ or $T$."}
{"category": "Math", "title": "A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains", "abstract": "We consider the following problem in one-dimensional diffusion-limited aggregation (DLA). At time $t$, we have an \"aggregate\" consisting of $\\Bbb{Z}\\cap[0,R(t)]$ [with $R(t)$ a positive integer]. We also have $N(i,t)$ particles at $i$, $i>R(t)$. All these particles perform independent continuous-time symmetric simple random walks until the first time $t'>t$ at which some particle tries to jump from $R(t)+1$ to $R(t)$. The aggregate is then increased to the integers in $[0,R(t')]=[0,R(t)+1]$ [so that $R(t')=R(t)+1$] and all particles which were at $R(t)+1$ at time $t'{-}$ are removed from the system. The problem is to determine how fast $R(t)$ grows as a function of $t$ if we start at time 0 with $R(0)=0$ and the $N(i,0)$ i.i.d. Poisson variables with mean $\\mu>0$. It is shown that if $\\mu<1$, then $R(t)$ is of order $\\sqrt{t}$, in a sense which is made precise. It is conjectured that $R(t)$ will grow linearly in $t$ if $\\mu$ is large enough."}
{"category": "Math", "title": "On Prime Ideals of Noetherian Skew Power Series Rings", "abstract": "We study prime ideals in skew power series rings $T:=R[[y;\\tau,\\delta]]$, for suitably conditioned right noetherian complete semilocal rings $R$, automorphisms $\\tau$ of $R$, and $\\tau$-derivations $\\delta$ of $R$. These rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern \"Cutting Down\" and \"Lying Over.\" In particular, under the additional assumption that $\\delta = \\tau - id$ (a basic feature of the Iwasawa-theoretic context), we prove: If $I$ is an ideal of $R$, then there exists a prime ideal $P$ of $S$ contracting to $I$ if and only if $I$ is a $\\delta$-stable $\\tau$-prime ideal of $R$. Our approach essentially depends on two key ingredients: First, the algebras considered are zariskian (in the sense of Li and Van Oystaeyen), and so the ideals are all topologically closed. Second, topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings."}
{"category": "Math", "title": "Non-linear regression models for Approximate Bayesian Computation", "abstract": "Approximate Bayesian inference on the basis of summary statistics is well-suited to complex problems for which the likelihood is either mathematically or computationally intractable. However the methods that use rejection suffer from the curse of dimensionality when the number of summary statistics is increased. Here we propose a machine-learning approach to the estimation of the posterior density by introducing two innovations. The new method fits a nonlinear conditional heteroscedastic regression of the parameter on the summary statistics, and then adaptively improves estimation using importance sampling. The new algorithm is compared to the state-of-the-art approximate Bayesian methods, and achieves considerable reduction of the computational burden in two examples of inference in statistical genetics and in a queueing model."}
{"category": "Math", "title": "Sampling from Dirichlet populations: estimating the number of species", "abstract": "Consider the random Dirichlet partition of the interval into $n$ fragments with parameter $\\theta >0$. We recall the unordered Ewens sampling formulae from finite Dirichlet partitions. As this is a key variable for estimation purposes, focus is on the number of distinct visited species in the sampling process. These are illustrated in specific cases. We use these preliminary statistical results on frequencies distribution to address the following sampling problem: what is the estimated number of species when sampling is from Dirichlet populations? The obtained results are in accordance with the ones found in sampling theory from random proportions with Poisson-Dirichlet distribution. To conclude with, we apply the different estimators suggested to two different sets of real data."}
{"category": "Math", "title": "Eigenvalue distribution for non-self-adjoint operators on compact manifolds with small multiplicative random perturbations", "abstract": "In this work we extend a previous work about the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint differential operators with small multiplicative random perturbations, by treating the case of operators on compact manifolds"}
{"category": "Math", "title": "Approximate zero-one laws and sharpness of the percolation transition in a class of models including two-dimensional Ising percolation", "abstract": "One of the most well-known classical results for site percolation on the square lattice is the equation $p_c+p_c^*=1$. In words, this equation means that for all values $\\neq p_c$ of the parameter $p$, the following holds: either a.s. there is an infinite open cluster or a.s. there is an infinite closed \"star\" cluster. This result is closely related to the percolation transition being sharp: below $p_c$, the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distribution with an exponential tail. The analog of this result has been proven by Higuchi in 1993 for two-dimensional Ising percolation (at fixed inverse temperature $\\beta<\\beta_c$) with external field $h$, the parameter of the model. Using sharp-threshold results (approximate zero-one laws) and a modification of an RSW-like result by Bollob\\'{a}s and Riordan, we show that these results hold for a large class of percolation models where the vertex values can be \"nicely\" represented (in a sense which will be defined precisely) by i.i.d. random variables. We point out that the ordinary percolation model obviously belongs to this class and we also show that the Ising model mentioned above belongs to it."}
{"category": "Math", "title": "Coinductive properties of Lipschitz functions on streams", "abstract": "A simple hierarchical structure is imposed on the set of Lipschitz functions on streams (i.e. sequences over a fixed alphabet set) under the standard metric. We prove that sets of non-expanding and contractive functions are closed under a certain coiterative construction. The closure property is used to construct new final stream coalgebras over finite alphabets. For an example, we show that the 2-adic extension of the Collatz function and certain variants yield final bitstream coalgebras."}
{"category": "Math", "title": "Algebraic colimit calculations in homotopy theory using fibred and cofibred categories", "abstract": "Higher Homotopy van Kampen Theorems allow the computation as colimits of certain homotopical invariants of glued spaces. One corollary is to describe homotopical excision in critical dimensions in terms of induced modules and crossed modules over groupoids. This paper shows how fibred and cofibred categories give an overall context for discussing and computing such constructions, allowing one result to cover many cases. A useful general result is that the inclusion of a fibre of a fibred category preserves connected colimits. The main homotopical application are to pairs of spaces with several base points, but we also describe briefly the situation for triads."}
{"category": "Math", "title": "Tests for zero-inflation and overdispersion", "abstract": "We propose a new methodology to detect zero-inflation and overdispersion based on the comparison of the expected sample extremes among convexly ordered distributions. The method is very flexible and includes tests for the proportion of structural zeros in zero-inflated models, tests to distinguish between two ordered parametric families and a new general test to detect overdispersion. The performance of the proposed tests is evaluated via some simulation studies. For the well-known fetal lamb data, we conclude that the zero-inflated Poisson model should be rejected against other more disperse models, but we cannot reject the negative binomial model."}
{"category": "Math", "title": "Existence and dynamic properties of a parabolic nonlocal MEMS equation", "abstract": "Let $\\Omega\\subset\\mathbb{R}^n$ be a $C^2$ bounded domain and $\\chi>0$ be a constant. We will prove the existence of constants $\\lambda_N\\ge\\lambda_N^{\\ast}\\ge\\lambda^{\\ast}(1+\\chi\\int_{\\Omega}\\frac{dx}{1-w_{\\ast}})^2$ for the nonlocal MEMS equation $-\\Delta v=\\lam/(1-v)^2(1+\\chi\\int_{\\Omega}1/(1-v)dx)^2$ in $\\Omega$, $v=0$ on $\\1\\Omega$, such that a solution exists for any $0\\le\\lambda<\\lambda_N^{\\ast}$ and no solution exists for any $\\lambda>\\lambda_N$ where $\\lambda^{\\ast}$ is the pull-in voltage and $w_{\\ast}$ is the limit of the minimal solution of $-\\Delta v=\\lam/(1-v)^2$ in $\\Omega$ with $v=0$ on $\\1\\Omega$ as $\\lambda\\nearrow \\lambda^{\\ast}$. We will prove the existence, uniqueness and asymptotic behaviour of the global solution of the corresponding parabolic nonlocal MEMS equation under various boundedness conditions on $\\lambda$. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when $\\lambda$ is large."}
{"category": "Math", "title": "Semiclassical states for weakly coupled nonlinear Schr\\\"odinger systems", "abstract": "We consider systems of weakly coupled Schr\\\"odinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate."}
{"category": "Math", "title": "The number of occurrences of a fixed spread among n directions in vector spaces over finite fields", "abstract": "We study a finite analog of a problem of Erdos, Hickerson and Pach on the maximum number of occurrences of a fixed angle among n directions in three-dimensional spaces."}
{"category": "Math", "title": "An elementary illustrated introduction to simplicial sets", "abstract": "This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to students familiar with just the fundamentals of algebraic topology."}
{"category": "Math", "title": "Bounded harmonic functions for the Heckman--Opdam Laplacian", "abstract": "We describe the set of bounded harmonic functions for the Heckman--Opdam Laplacian, when the multiplicity function is larger than 1/2. We prove that this set is a vector space of dimension the cardinality of the Weyl group. We give some consequences in terms of the associated hypergeometric functions."}
{"category": "Math", "title": "Expected Coalescence Time for a Nonuniform Allocation Process", "abstract": "We give an asymptotic expression for the expected coalescence time for a non-uniform balls-into-boxes allocation model. Connections to coalescent processes in population biology and computer science are discussed."}
{"category": "Math", "title": "Set Theoretic Defining Equations of the Variety of Principal Minors of Symmetric Matrices", "abstract": "The variety of principal minors of $n\\times n$ symmetric matrices, denoted $Z_{n}$, is invariant under the action of a group $G\\subset \\GL(2^{n})$ isomorphic to $\\G$. We describe an irreducible $G$-module of degree $4$ polynomials constructed from Cayley's $2 \\times 2 \\times 2$ hyperdeterminant and show that it cuts out $Z_{n}$ set-theoretically. This solves the set-theoretic version of a conjecture of Holtz and Sturmfels. Standard techniques from representation theory and geometry are explored and developed for the proof of the conjecture and may be of use for studying similar $G$-varieties."}
{"category": "Math", "title": "Nonlinear Digital Post-Processing to Mitigate Jitter in Sampling", "abstract": "This paper describes several new algorithms for estimating the parameters of a periodic bandlimited signal from samples corrupted by jitter (timing noise) and additive noise. Both classical (non-random) and Bayesian formulations are considered: an Expectation-Maximization (EM) algorithm is developed to compute the maximum likelihood (ML) estimator for the classical estimation framework, and two Gibbs samplers are proposed to approximate the Bayes least squares (BLS) estimate for parameters independently distributed according to a uniform prior. Simulations are performed to demonstrate the significant performance improvement achievable using these algorithms as compared to linear estimators. The ML estimator is also compared to the Cramer-Rao lower bound to determine the range of jitter for which the estimator is approximately efficient. These simulations provide evidence that the nonlinear algorithms derived here can tolerate 1.4-2 times more jitter than linear estimators, reducing on-chip ADC power consumption by 50-75 percent."}
{"category": "Math", "title": "A geometric space without conjugate points", "abstract": "From a spray space $S$ on a manifold $M$ we construct a new geometric space $P$ of larger dimension with the following properties: 1. Geodesics in $P$ are in one-to-one correspondence with parallel Jacobi fields of $M$. 2. $P$ is complete if and only if $S$ is complete. 3. If two geodesics in $P$ meet at one point, the geodesics coincide on their common domain, and $P$ has no conjugate points. 4. There exists a submersion $\\pi\\colon P \\to M$ that maps geodesics in $P$ into geodesics on $M$. Space $P$ is constructed by first taking two complete lifts of spray $S$. This will give a spray $S^{cc}$ on the second iterated tangent bundle $TTM$. Then space $P$ is obtained by restricting tangent vectors of geodesics for $S^{cc}$ onto a suitable $(2\\dim M+2)$-dimensional submanifold of $TTTM$. Due to the last restriction, space $P$ is not a spray space. However, the construction shows that conjugate points can be removed if we add dimensions and relax assumptions on the geometric structure."}
{"category": "Math", "title": "When do two planted graphs have the same cotransversal matroid?", "abstract": "Cotransversal matroids are a family of matroids that arise from planted graphs. We prove that two planted graphs give the same cotransversal matroid if and only if they can be obtained from each other by a series of local moves."}
{"category": "Math", "title": "Heavy-traffic limits for waiting times in many-server queues with abandonment", "abstract": "We establish heavy-traffic stochastic-process limits for waiting times in many-server queues with customer abandonment. If the system is asymptotically critically loaded, as in the quality-and-efficiency-driven (QED) regime, then a bounding argument shows that the abandonment does not affect waiting-time processes. If instead the system is overloaded, as in the efficiency-driven (ED) regime, following Mandelbaum et al. [Proceedings of the Thirty-Seventh Annual Allerton Conference on Communication, Control and Computing (1999) 1095--1104], we treat customer abandonment by studying the limiting behavior of the queueing models with arrivals turned off at some time $t$. Then, the waiting time of an infinitely patient customer arriving at time $t$ is the additional time it takes for the queue to empty. To prove stochastic-process limits for virtual waiting times, we establish a two-parameter version of Puhalskii's invariance principle for first passage times. That, in turn, involves proving that two-parameter versions of the composition and inverse mappings appropriately preserve convergence."}
{"category": "Math", "title": "Finite surgeries on three-tangle pretzel knots", "abstract": "We classify Dehn surgeries on (p,q,r) pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are (-2,3,7) and (-2,3,9). Agol and Lackenby's 6-theorem reduces the argument to knots with small indices p,q,r. We treat these using the Culler-Shalen norm of the SL(2,C)-character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety."}
{"category": "Math", "title": "Testing composite hypotheses via convex duality", "abstract": "We study the problem of testing composite hypotheses versus composite alternatives, using a convex duality approach. In contrast to classical results obtained by Krafft and Witting (Z. Wahrsch. Verw. Gebiete 7 (1967) 289--302), where sufficient optimality conditions are derived via Lagrange duality, we obtain necessary and sufficient optimality conditions via Fenchel duality under compactness assumptions. This approach also differs from the methodology developed in Cvitani\\'{c} and Karatzas (Bernoulli 7 (2001) 79--97)."}
{"category": "Math", "title": "Deformation quantization modules II. Hochschild class", "abstract": "This paper is the continuation of arXiv:0802.1245. We construct the Hochschild class for coherent modules over a deformation quantization algebroid on a complex Poisson manifold. We also define the convolution of Hochschild homologies, and prove that the Hochschild class of the convolution of two coherent modules is the convolution of their Hochschild classes. We study with some details the case of symplectic deformations."}
{"category": "Math", "title": "Topology of the octonionic flag manifold", "abstract": "The octonionic flag manifold $Fl(\\mathbb{O})$ is the space of all pairs in $\\mathbb{O}P^2\\times \\mathbb{O}P^2$ (where $\\mathbb{O}P^2$ denotes the octonionic projective plane) which satisfy a certain \"incidence\" relation. It comes equipped with the projections $\\pi_1,\\pi_2 : Fl(\\mathbb{O})\\to \\mathbb{O}P^2$, which are $\\mathbb{O}P^1$ bundles, as well as with an action of the group $Spin(8)$. The first two results of this paper give Borel type descriptions of the usual, respectively $Spin(8)$-equivariant cohomology of $Fl(\\mathbb{O})$ in terms of $\\pi_1$ and $\\pi_2$ (actually the Euler classes of the tangent spaces to the fibers of $\\pi_1$, respectively $\\pi_2$, which are rank 8 vector bundles on $Fl(\\mathbb{O})$). Then we obtain a Goresky-Kottwitz-MacPherson type description of the ring $H^*_{Spin(8)}(Fl(\\mathbb{O}))$. Finally, we consider the $Spin(8)$-equivariant $K$-theory ring of $Fl(\\mathbb{O})$ and obtain a Goresky-Kottwitz-MacPherson type description of this ring."}
{"category": "Math", "title": "Non-Standard Analysis, Multiplication of Schwartz Distributions and Delta-Like Solution of Hopf's Equation", "abstract": "We construct an algebra of generalized functions $^*\\mathcal{E}(\\mathbb{R}^d)$. We also construct an embedding of the space of Schwartz distributions $\\mathcal{D}^\\prime(\\mathbb{R}^d)$ into $^*\\mathcal{E}(\\mathbb{R}^d)$ and thus present a solution of the problem of multiplication of Schwartz distributions which improves J.F. Colombeau's solution. As an application we prove the existence of a weak delta-like solution in ${^*\\mathcal{E}(\\mathbb{R}^d)}$ of the Hopf equation. This solution does not have a counterpart in the classical theory of partial differential equations. Our result improves a similar result by M. Radyna obtained in the framework of perturbation theory."}
{"category": "Math", "title": "Critical points between varieties generated by subspace lattices of vector spaces", "abstract": "We denote by Conc(A) the semilattice of compact congruences of an algebra A. Given a variety V of algebras, we denote by Conc(V) the class of all semilattices isomorphic to Conc(A) for some A in V. Given varieties V1 and V2 varieties of algebras, the critical point of V1 under V2, denote by crit(V1;V2) is the smalest cardinality of a semilattice in Conc(V1) but not in Conc(V2). Given a finitely generated variety V of modular lattices, we obtain an integer l, depending of V, such that crit(V;Var(Sub F^n)) is at least aleph_2 for any n > 1 and any field F. In a second part, we prove that crit(Var(Mn);Var(Sub F^3))=aleph_2, for any finite field F and any integer n such that 1+card F< n. Similarly crit(Var(Sub F^3);Var(Sub K^3))=aleph_2, for all finite fields F and K such that card F>card K."}
{"category": "Math", "title": "The Pythagorean Tree: A New Species", "abstract": "In 1967 the Dutch mathemetician F.J.M. Barning described an infinite, planar, ternary tree*. Seven years later, A. Hall independently discovered the same tree. Both used the method of uni-modular matrices to transform one triple to another. A number of rediscoveries have occurred more recently. In this article we announce the discovery of an entirely different ternary tree, and show how it relates to the one found by Barning and Hall."}
{"category": "Math", "title": "Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras", "abstract": "We define a graded twisted-coassociative coproduct on the tensor algebra $TW$ of any $\\Z^n$-graded vector space $W$. If $W$ is the desuspension space $\\da V$ of a graded vector space $V$, the coderivations (resp. quadratic ``degree 1'' codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences $\\zp_s$, $s\\ge 1$, of $s$-linear maps on $V$ (resp. $\\Z^n$-graded Loday structures on $V$, sequences that we call Loday infinity structures on $V$). We prove a minimal model theorem for Loday infinity algebras, investigate Loday infinity morphisms, and observe that the $\\op{Lod}_{\\infty}$ category contains the $\\op{L}_{\\infty}$ category as a subcategory. Moreover, the graded Lie bracket of coderivations gives rise to a graded Lie ``stem'' bracket on the cochain spaces of graded Loday, Loday infinity, and $2n$-ary graded Loday algebras (the latter extend the corresponding Lie algebras in the sense of Michor and Vinogradov). These algebraic structures have square zero with respect to the stem bracket, so that we obtain natural cohomological theories that have good properties with respect to formal deformations. The stem bracket restricts to the graded Nijenhuis-Richardson and--up to isomorphism--to the Grabowski-Marmo brackets (the last bracket extends the Schouten-Nijenhuis bracket to the space of graded antisymmetric first order polydifferential operators), and it encodes, beyond the already mentioned cohomologies, those of graded Lie, graded Poisson, graded Jacobi, Lie infinity, as well as that of $2n$-ary graded Lie algebras."}
{"category": "Math", "title": "Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory", "abstract": "We explain the precise relationship between two module-theoretic descriptions of sheaves on an involutive quantale, namely the description via so-called Hilbert structures on modules and that via so-called principally generated modules. For a principally generated module satisfying a suitable symmetry condition we observe the existence of a canonical Hilbert structure. We prove that, when working over a modular quantal frame, a module bears a Hilbert structure if and only if it is principally generated and symmetric, in which case its Hilbert structure is necessarily the canonical one. We indicate applications to sheaves on locales, on quantal frames and even on sites."}
{"category": "Math", "title": "Symmetric ladders and G-biliaison", "abstract": "We study the family of ideals generated by minors of mixed size contained in a ladder of a symmetric matrix from the point of view of liaison theory. We prove that they can be obtained from ideals of linear forms by ascending G-biliaison. In particular, they are glicci."}
{"category": "Math", "title": "Iterating the hessian: a dynamical system on the moduli space of elliptic curves and dessins d'enfants", "abstract": "Each elliptic curve can be embedded uniquely in the projective plane, up to projective equivalence. The hessian curve of the embedding is generically a new elliptic curve, whose isomorphism type depends only on that of the initial elliptic curve. One gets like this a rational map from the moduli space of elliptic curves to itself. We call it the hessian dynamical system. We compute it in terms of the $j$-invariant of elliptic curves. We deduce that, seen as a map from a projective line to itself, it has 3 critical values, which correspond to the point at infinity of the moduli space and to the two elliptic curves with special symmetries. Moreover, it sends the set of critical values into itself, which shows that all its iterates have the same set of critical values. One gets like this a sequence of dessins d'enfants. We describe an algorithm allowing to construct this sequence."}
{"category": "Math", "title": "Q-modules are Q-suplattices", "abstract": "It is well known that the internal suplattices in the topos of sheaves on a locale are precisely the modules on that locale. Using enriched category theory and a lemma on KZ doctrines we prove (the generalization of) this fact in the case of ordered sheaves on a small quantaloid. Comparing module-equivalence with sheaf-equivalence for quantaloids and using the notion of centre of a quantaloid, we refine a result of F. Borceux and E. Vitale."}
{"category": "Math", "title": "MAPLE subroutines for computing Milnor and Tyurina numbers of hypersurface singularities with application to Arnol'd adjacencies", "abstract": "In the present paper MAPLE subroutines computing Milnor and Tyurina numbers of an isolated algebraic hypersurface singularity are presented and described in detail. They represents examples, and perhaps the first ones, of a MAPLE implementation of local monomial ordering. As an application, the last section is devoted to writing down equations of algebraic stratifications of Kuranishi spaces of simple Arnol'd singularities: they geometrically represents, by means of inclusions of algebraic subsets, the partial ordering on classes of simple singularities induced by the adjacency relation."}
{"category": "Math", "title": "On isometric dilations of product systems of C*-correspondences and applications to families of contractions associated to higher-rank graphs", "abstract": "Let E be a product system of C*-correspondences over N^r. Some sufficient conditions for the existence of a not necessarily regular isometric dilation of a completely contractive representation of E are established and difference between regular and *-regular dilations discussed. It is in particular shown that a minimal isometric dilation is *-regular if and only if it is doubly commuting. The case of product systems associated with higher-rank graphs is analysed in detail."}
{"category": "Math", "title": "Convergence to stable laws for a class of multidimensional stochastic recursions", "abstract": "We consider a Markov chain $\\{X_n\\}_{n=0}^\\8$ on $\\R^d$ defined by the stochastic recursion $X_{n}=M_n X_{n-1}+Q_n$, where $(Q_n,M_n)$ are i.i.d. random variables taking values in the affine group $H=\\R^d\\rtimes {\\rm GL}(\\R^d)$. Assume that $M_n$ takes values in the similarity group of $\\R^d$, and the Markov chain has a unique stationary measure $\\nu$, which has unbounded support. We denote by $|M_n|$ the expansion coefficient of $M_n$ and we assume $\\E |M|^\\a=1$ for some positive $\\a$. We show that the partial sums $S_n=\\sum_{k=0}^n X_k$, properly normalized, converge to a normal law ($\\a\\ge 2$) or to an infinitely divisible law, which is stable in a natural sense ($\\a<2$). These laws are fully nondegenerate, if $\\nu$ is not supported on an affine hyperplane. Under a natural hypothesis, we prove also a local limit theorem for the sums $S_n$. If $\\a\\le 2$, proofs are based on the homogeneity at infinity of $\\nu$ and on a detailed spectral analysis of a family of Fourier operators $P_v$ considered as perturbations of the transition operator $P$ of the chain $\\{X_n \\}$. The characteristic function of the limit law has a simple expression in terms of moments of $\\nu$ ($\\a > 2$) or of the tails of $\\nu$ and of stationary measure for an associated Markov operator ($\\a\\le 2$). We extend the results to the situation where $M_n$ is a random generalized similarity."}
{"category": "Math", "title": "Artin t-Motifs", "abstract": "We show that analytically trivial t-motifs satisfy a Tannakian duality, without restrictions on the base field, save for that it be of generic characteristic. We show that the group of components of the t-motivic Galois group coincides with the absolute Galois group of the base field."}
{"category": "Math", "title": "Topology of moduli spaces of tropical curves with marked points", "abstract": "In this paper we study topology of moduli spaces of tropical curves of genus $g$ with $n$ marked points. We view the moduli spaces as being imbedded in a larger space, which we call the {\\it moduli space of metric graphs with $n$ marked points.} We describe the shrinking bridges strong deformation retraction, which leads to a substantial simplification of all these moduli spaces. In the rest of the paper, that reduction is used to analyze the case of genus 1. The corresponding moduli space is presented as a quotient space of a torus with respect to the conjugation ${\\mathbb Z}_2$-action; and furthermore, as a homotopy colimit over a simple diagram. The latter allows us to compute all Betti numbers of that moduli space with coefficients in ${\\mathbb Z}_2$."}
{"category": "Math", "title": "Transparent connections over negatively curved surfaces", "abstract": "Let $(M,g)$ be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over $M$ is said to be transparent if its parallel transport along the closed geodesics of $g$ is the identity. We study the space of such connections modulo gauge and we prove a classification result in terms of the solutions of certain PDE that arises naturally in the problem. We also show a local uniqueness result for the trivial connection and that there is a transparent SU(2)-connection associated to each meromorphic function on $M$."}
{"category": "Math", "title": "Moduli spaces of metric graphs of genus 1 with marks on vertices", "abstract": "In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces $MG_{1,n}^v$, which parametrize the isometry classes of metric graphs of genus 1 with $n$ marks on vertices are homotopy equivalent to the spaces $TM_{1,n}$, which are the moduli spaces of tropical curves of genus 1 with $n$ marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus."}
{"category": "Math", "title": "Operator theoretic methods for the eigenvalue counting function in spectral gaps", "abstract": "Using the notion of spectral flow, we suggest a simple approach to various asymptotic problems involving eigenvalues in the gaps of the essential spectrum of self-adjoint operators. Our approach uses some elements of the spectral shift function theory. Using this approach, we provide generalisations and streamlined proofs of two results in this area already existing in the literature. We also give a new proof of the generalised Birman-Schwinger principle."}
{"category": "Math", "title": "Moduli spaces of tropical curves of higher genus with marked points and homotopy colimits", "abstract": "The main characters of this paper are the moduli spaces $TM_{g,n}$ of rational tropical curves of genus $g$ with $n$ marked points, with $g\\geq 2$. We reduce the study of the homotopy type of these spaces to the analysis of compact spaces $X_{g,n}$, which in turn possess natural representations as a homotopy colimits of diagrams of topological spaces over combinatorially defined generalized simplicial complexes $\\Delta_g$, with the latter being interesting on their own right. We use these homotopy colimit representations to describe a CW complex decomposition for each $X_{g,n}$. Furthermore, we use these developments, coupled with some standard tools for working with homotopy colimits, to perform an in-depth analysis of special cases of genus 2 and 3, gaining a complete understanding of the moduli spaces $X_{2,0}$, $X_{2,1}$, $X_{2,2}$, and $X_{3,0}$, as well as a partial understanding of other cases, resulting in several open questions and in further conjectures."}
{"category": "Math", "title": "Extensions of profinite duality groups", "abstract": "We show that the class of profinite duality groups is closed under group extensions provided that the kernel satisfies some finiteness condition. This extends earlier results of Pletch and of Wingberg."}
{"category": "Math", "title": "Law of the Iterated Logarithm for the random walk on the infinite percolation cluster", "abstract": "We show that random walks on the infinite supercritical percolation clusters in Z^d satisfy the usual Law of the Iterated Logarithm. The proof combines Barlow's Gaussian heat kernel estimates and the ergodicity of the random walk on the environment viewed from the random walker as derived by Berger and Biskup."}
{"category": "Math", "title": "Representing an element in F_q[t] as the sum of two irreducibles", "abstract": "A monic polynomial in F_q[t] of degree n over a finite field F_q of odd characteristic can be written as the sum of two irreducible monic elements in F_q[t] of degrees n and n-1 if q is larger than a bound depending only on n. The main tool is a sufficient condition for simultaneous primality of two polynomials in one variable x with coefficients in F_q[t]."}
{"category": "Math", "title": "Automorphic orbits in free groups: words versus subgroups", "abstract": "We show that the following problems are decidable in a rank 2 free group F_2: does a given finitely generated subgroup H contain primitive elements? and does H meet the orbit of a given word u under the action of G, the group of automorphisms of F_2? Moreover, decidability subsists if we allow H to be a rational subset of F_2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, we show the decidability of the following weaker problem: given a finitely generated subgroup H, a word u and an integer k, does H contain the image of u by some k-almost bounded automorphism? An automorphism is k-almost bounded if at most one of the letters has an image of length greater than k."}
{"category": "Math", "title": "Small counts in the infinite occupancy scheme", "abstract": "The paper is concerned with the classical occupancy scheme with infinitely many boxes, in which $n$ balls are thrown independently into boxes $1,2,...$, with probability $p_j$ of hitting the box $j$, where $p_1\\geq p_2\\geq...>0$ and $\\sum_{j=1}^\\infty p_j=1$. We establish joint normal approximation as $n\\to\\infty$ for the numbers of boxes containing $r_1,r_2,...,r_m$ balls, standardized in the natural way, assuming only that the variances of these counts all tend to infinity. The proof of this approximation is based on a de-Poissonization lemma. We then review sufficient conditions for the variances to tend to infinity. Typically, the normal approximation does not mean convergence. We show that the convergence of the full vector of $r$-counts only holds under a condition of regular variation, thus giving a complete characterization of possible limit correlation structures."}
{"category": "Math", "title": "Correction. SDEs with oblique reflections on nonsmooth domains", "abstract": "Correction to The Annals of Probability 21 (1993) 554--580 [http://projecteuclid.org/euclid.aop/1176989415]"}
{"category": "Math", "title": "The edge-flipping group of a graph", "abstract": "Let $X=(V,E)$ be a finite simple connected graph with $n$ vertices and $m$ edges. A configuration is an assignment of one of two colors, black or white, to each edge of $X.$ A move applied to a configuration is to select a black edge $\\epsilon\\in E$ and change the colors of all adjacent edges of $\\epsilon.$ Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping puzzle on $X,$ and it corresponds to a group action. This group is called the edge-flipping group $\\mathbf{W}_E(X)$ of $X.$ This paper shows that if $X$ has at least three vertices, $\\mathbf{W}_E(X)$ is isomorphic to a semidirect product of $(\\mathbb{Z}/2\\mathbb{Z})^k$ and the symmetric group $S_n$ of degree $n,$ where $k=(n-1)(m-n+1)$ if $n$ is odd, $k=(n-2)(m-n+1)$ if $n$ is even, and $\\mathbb{Z}$ is the additive group of integers."}
{"category": "Math", "title": "The Modultional Instability for a Generalized KdV Equation", "abstract": "We study the spectral stability of a family of periodic standing wave solutions to the generalized KdV (g-KdV) in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to perturbations of long wavelength. A study of the normal form of the characteristic polynomial of the monodromy map (the periodic Evan's function) in a neighborhood of the origin in the spectral plane leads to two different instability indices. The first index counts modulo 2 the total number of periodic eigenvalues on the real axis. This index is a generalization of the one which governs the stability of the solitary wave. The second index provides a necessary and sufficient condition for the existence of a long-wavelength instability. This index is essentially the quantity calculated by Haragus and Kapitula in the small amplitude limit. Both of these quantities can be expressed in terms of the map between the constants of integration for the ordinary differential equation defining the traveling waves and the conserved quantities of the partial differential equation. These two indices together provide a good deal of information about the spectrum of the linearized operator. We sketch the connection of this calculation to a study of the linearized operator - in particular we perform a perturbation calculation in terms of the Floquet parameter. This suggests geometric interpretations attached to the vanishing of the modulational instability index previously mentioned."}
{"category": "Math", "title": "On ergodic transformations that are both weakly mixing and uniformly rigid", "abstract": "We examine some of the properties of uniformly rigid transformations, and analyze the compatibility of uniform rigidity and (measurable) weak mixing along with some of their asymptotic convergence properties. We show that on Cantor space, there does not exist a finite measure-preserving, totally ergodic, uniformly rigid transformation. We briefly discuss general group actions and show that (measurable) weak mixing and uniform rigidity can coexist in a more general setting."}
{"category": "Math", "title": "Generator coalgebras are not necessarily quasi-coFrobenius", "abstract": "We study the problem of whether a coalgebra that generates its category of left (right) comodules is left (right) quasi-coFrobenius or not. We prove it does not hold in general, by giving a method of constructing counterexamples. This gives a negative answer to a question stated in \\cite{kn:coalgen}. We also prove it is true for monomial pointed coalgebras and we characterize the quivers $Q$ such that $\\Bbbk Q$ admits a monomial subcoalgebra that is left (right) quasi-coFrobenius."}
{"category": "Math", "title": "Real and strongly real classes in finite linear groups", "abstract": "We classify the real and strongly real conjugacy classes in $GL_n(q)$, $SL_n(q)$, $PGL_n(q)$, $PSL_n(q)$, and all quasi-simple covers of $PSL_n(q)$. In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes."}
{"category": "Math", "title": "A Tannaka Theorem for Proper Lie Groupoids", "abstract": "By replacing the category of smooth vector bundles over a manifold with the category of what we call smooth Euclidean fields, which is a proper enlargement of the former, and by considering smooth actions of Lie groupoids on smooth Euclidean fields, we are able to prove a Tannaka duality theorem for proper Lie groupoids. The notion of smooth Euclidean field we introduce here is the smooth, finite dimensional analogue of the usual notion of continuous Hilbert field."}
{"category": "Math", "title": "Essential cohomology for elementary abelian p-groups", "abstract": "For an odd prime p the cohomology ring of an elementary abelian p-group is polynomial tensor exterior. We show that the ideal of essential classes is the Steenrod closure of the class generating the top exterior power. As a module over the polynomial algebra, the essential ideal is free on the set of Mui invariants."}
{"category": "Math", "title": "Conformality of a differential with respect to Cheeger-Gromoll type metrics", "abstract": "We investigate conformality of the differential of a mapping between Riemannian manifolds if the tangent bundles are equipped with a generalized metric of Cheeger-Gromoll type."}
{"category": "Math", "title": "Hyperinvariant subspace for weighted composition operator on $L^p([0,1]^d)$", "abstract": "The main result of this paper is the existence of a hyperinvariant subspace of weighted composition operator $Tf=vf\\circ\\tau$ on $L^p([0,1]^d)$, ($1 \\leq p \\leq \\infty$) when the weight $v$ is in the class of ``generalized polynomials'' and the composition map is a bijective ergodic transform satisfying a given discrepancy. The work is based on the construction of a functional calculus initiated by Wermer and generalized by Davie."}
{"category": "Math", "title": "Boundary slopes and the numbers of positive/negative crossings for Montesinos knots", "abstract": "We show that a finite numerical boundary slope of an essential surface in the exterior of a Montesinos knot is bounded above and below in terms of the numbers of positive/negative crossings of a specific minimal diagram of the knot."}
{"category": "Math", "title": "Multifractal analysis for conformal graph directed Markov systems", "abstract": "We derive the multifractal analysis of the conformal measure (or equivalently, the invariant measure) associated to a family of weights imposed upon a (multi-dimensional) graph directed Markov system (GDMS) using balls as the filtration. This analysis is done over a large subset of the limit set. In particular, it coincides with the limit set when the GDMS under scrutiny satisfies a boundary separation condition. It also applies to more general situations such as real or complex continued fractions."}
{"category": "Math", "title": "Notes on automorphisms of ultrapowers of II_1 factors", "abstract": "In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II_1 factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is aleph_0-locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *-homomorphisms from a separable nuclear C*-algebra into an ultrapower of a II_1 factor, equality of the induced traces implies unitary equivalence. All statements are proved using operator algebraic techniques, but in the last section of the paper we indicate how the underlying principle is related to theorems of Henson's positive bounded logic."}
{"category": "Math", "title": "Integer Cech Cohomology of Icosahedral Projection Tilings", "abstract": "The integer Cech cohomology of canonical projection tilings of dimension three and codimension three is derived. These formulae are then evaluated for several icosahedral tilings known from the literature. Rather surprisingly, the cohomologies of all these tilings turn out to have torsion. This is the case even for the Danzer tiling, which is, in some sense, the simplest of all icosahedral tilings. This result is in contrast to the case of two-dimensional canonical projection tilings, where many examples without torsion are known."}
{"category": "Math", "title": "Algebraic (2,2)-transformation groups", "abstract": "This paper contains the more significant part of the article with the same title that will appear in the Volume 12 of Journal of Group Theory (2009). In this paper we determine all algebraic transformation groups $G$, defined over an algebraically closed field $\\sf k$, which operate transitively, but not primitively, on a variety $\\Omega$, provided the following conditions are fulfilled. We ask that the (non-effective) action of $G$ on the variety of blocks is sharply 2-transitive, as well as the action on a block $\\Delta$ of the normalizer $G_\\Delta$. Also we require sharp transitivity on pairs $(X,Y)$ of independent points of $\\Omega$, i.e. points contained in different blocks."}
{"category": "Math", "title": "Bounded Cohomology and $l_1$-Homology of Three-Manifolds", "abstract": "In this paper we define, for each aspherical orientable 3-manifold $M$ endowed with a \\emph{torus splitting} $\\c{T}$, a 2-dimensional fundamental $l_1$-class $[M]^{\\c{T}}$ whose $l_1$-norm has similar properties as the Gromov simplicial volume of $M$ (additivity under torus splittings and isometry under finite covering maps). Next, we use the Gromov simplicial volume of $M$ and the $l_1$-norm of $[M]^{\\c{T}}$ to give a complete characterization of those nonzero degree maps $f\\co M\\to N$ which are homotopic to a ${\\rm deg}(f)$-covering map. As an application we characterize those degree one maps $f\\co M\\to N$ which are homotopic to a homeomorphism in terms of bounded cohomology classes."}
{"category": "Math", "title": "The Fitzpatrick function - a bridge between convex analysis and multivalued stochastic differential equations", "abstract": "Using the Fitzpatrick function, we characterize the solutions for different classes of deterministic and stochastic differential equations driven by maximal monotone operators (or in particular subdifferential operators) as the minimum point of a suitably chosen convex lower semicontinuous function. Such technique provides a new approach for the existence of the solutions for the considered equations."}
{"category": "Math", "title": "The arithmetic of trees", "abstract": "The arithmetic of the natural numbers can be extended to arithmetic operations on planar binary trees. This gives rise to a non-commutative arithmetic theory. In this exposition, we describe this arithmetree, first defined by Loday, and investigate prime trees."}
{"category": "Math", "title": "Pattern Rigidity in Hyperbolic Spaces: Duality and PD Subgroups", "abstract": "For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space $\\Hyp^n$ of real dimension $n$, $n \\geq 3$. Let $H_i \\subset G_i$ be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of $H_i$ is a codimension one topological sphere. 2) limit set of $H_i$ is an even dimensional topological sphere. 3) $H_i$ is a codimension one duality group. This generalizes (1). In particular, if $n = 3$, $H_i$ could be any freely indecomposable subgroup of $G_i$. 4) $H_i$ is an odd-dimensional Poincare Duality group $PD(2k+1)$. This generalizes (2). We prove pattern rigidity for such pairs extending work of Schwartz who proved pattern rigidity when $H_i$ is cyclic. All this generalizes to quasiconvex subgroups of uniform lattices in rank one symmetric spaces satisfying one of the conditions (1)-(4), as well as certain special subgroups with disconnected limit sets. In particular, pattern rigidity holds for all quasiconvex subgroups of hyperbolic 3-manifolds that are not virtually free. Combining this with a result of Mosher-Sageev-Whyte, we get quasi-isometric rigidity results for graphs of groups where the vertex groups are uniform lattices in rank one symmetric spaces and edge groups are of any of the above types."}
{"category": "Math", "title": "Nonholonomic Lorentzian geometry on some $\\mathbb H$-type groups", "abstract": "We consider examples of the $\\mathbb H$-type groups with the natural horizontal distribution generated by the commutation relations of the group. In the contrast with the previous studies we furnish the horizontal distribution with the Lorentzian metric, which is nondegenerate metric of index 1 instead of a positive definite quadratic form. The causal character is defined. We study the reachable set by timelike future directed curves. The parametric equations of geodesics are obtained."}
{"category": "Math", "title": "Langlands duality for representations of quantum groups", "abstract": "We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an \"interpolating quantum group\" depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups."}
{"category": "Math", "title": "Cartan Invariants of Symmetric Groups and Iwahori-Hecke Algebras", "abstract": "K\\\"{u}lshammer, Olsson and Robinson conjectured that a certain set of numbers determined the invariant factors of the $\\ell$-Cartan matrix for $S_n$ (equivalently, the invariant factors of the Cartan matrix for the Iwahori-Hecke algebra $\\mathcal{H}_n(q)$, where $q$ is a primitive $\\ell$th root of unity). We call these invariant factors Cartan invariants. In a previous paper, the second author calculated these Cartan invariants when $\\ell=p^r$, $p$ prime, and $r\\leq p$ and went on to conjecture that the formulae should hold for all $r$. Another result was obtained, which is surprising and counterintuitive from a block theoretic point of view. Namely, given the prime decomposition $\\ell=p_1^{r_1}... p_k^{r_k}$, the Cartan matrix of an $\\ell$-block of $S_n$ is a product of Cartan matrices associated to $p_i^{r_i}$-blocks of $S_n$. In particular, the invariant factors of the Cartan matrix associated to an $\\ell$-block of $S_n$ can be recovered from the Cartan matrices associated to the $p_i^{r_i}$-blocks. In this paper, we formulate an explicit combinatorial determination of the Cartan invariants of $S_n$--not only for the full Cartan matrix, \\emph{but for an individual block}. We collect evidence for this conjecture, by showing that the formulae predict the correct determinant of the $\\ell$-Cartan matrix. We then go on to show that Hill's conjecture implies the conjecture of KOR."}
{"category": "Math", "title": "Generation and syzygies of the first secant variety", "abstract": "Under certain effective positivity conditions, we show that the secant variety to a smooth variety satisfies $N_{3,p}$. For smooth curves, we provide the best possible effective bound on the degree $d$ of the embedding, $d\\geq 2g+3+p$."}
{"category": "Math", "title": "The second rational homology group of the moduli space of curves with level structures", "abstract": "Let $\\Gamma$ be a finite-index subgroup of the mapping class group of a closed genus $g$ surface that contains the Torelli group. For instance, $\\Gamma$ can be the level $L$ subgroup or the spin mapping class group. We show that $H_2(\\Gamma;\\Q) \\cong \\Q$ for $g \\geq 5$. A corollary of this is that the rational Picard groups of the associated finite covers of the moduli space of curves are equal to $\\Q$. We also prove analogous results for surface with punctures and boundary components."}
{"category": "Math", "title": "Noncommutative Symmetric Functions VII: Free Quasi-Symmetric Functions Revisited", "abstract": "We prove a Cauchy identity for free quasi-symmetric functions and apply it to the study of various bases. A free Weyl formula and a generalization of the splitting formula are also discussed."}
{"category": "Math", "title": "Inversion of some series of free quasi-symmetric functions", "abstract": "We give a combinatorial formula for the inverses of the alternating sums of free quasi-symmetric functions of the form F_{\\omega(I)} where I runs over compositions with parts in a prescribed set C. This proves in particular three special cases (no restriction, even parts, and all parts equal to 2) which were conjectured by B. C. V. Ung in [Proc. FPSAC'98, Toronto]."}
{"category": "Math", "title": "Applications of patching to quadratic forms and central simple algebras", "abstract": "This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the u-invariant of p-adic function fields, for p odd. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations."}
{"category": "Math", "title": "Dominated polynomials on infinite dimensional spaces", "abstract": "The aim of this paper is to prove a stronger version of a conjecture on the existence of non-dominated scalar-valued m-homogeneous polynomials (m>=3) on arbitrary infinite dimensional Banach spaces."}
{"category": "Math", "title": "Characterization of the Oblique Projector $U(VU)^+V$ with Application to Constrained Least Squares", "abstract": "We provide a full characterization of the oblique projector $U(VU)^+V$ in the general case where the range of $U$ and the null space of $V$ are not complementary subspaces. We discuss the new result in the context of constrained least squares minimization."}
{"category": "Math", "title": "Two Essays on the Archimedean versus Non-Archimedean Debate", "abstract": "Part 1 : For more than two millennia, ever since Euclid's geometry, the so called Archimedean Axiom has been accepted without sufficiently explicit awareness of that fact. The effect has been a severe restriction of our views of space-time, a restriction which above all affects Physics. Here it is argued that, ever since the invention of Calculus by Newton, we may actually have {\\it empirical evidence} that time, and thus space as well, are not Archimedean. Part 2 : There is an insufficient awareness about the {\\it rich and complex} structure of various totally ordered scalar fields obtained through the {\\it ultrapower} construction. This rich and complex structure comes from the presence of {\\it infinitesimals} in such fields, presence which leads to the fact that such fields are {\\it non-Archimedean}. Here, with the concept of {\\it walkable world}, which has highly intuitive and pragmatic geometric meaning, the mentioned rich and complex structure is illustrated. The issues presented have relevance for what are usually called the \"infinities in physics\"."}
{"category": "Math", "title": "Local smoothing effects for the water-wave problem with surface tension", "abstract": "This paper has been merged to 0908.3255."}
{"category": "Math", "title": "Global Dimension of Polynomial Rings in Partially Commuting Variables", "abstract": "For any free partially commutative monoid $M(E,I)$, we compute the global dimension of the category of $M(E,I)$-objects in an Abelian category with exact coproducts. As a corollary, we generalize Hilbert's Syzygy Theorem to polynomial rings in partially commuting variables."}
{"category": "Math", "title": "Equidistribution of Dilations of Polynomial Curves in Nilmanifolds", "abstract": "In this paper we study the asymptotic behaviour under dilations of probability measures supported on polynomial curves in nilmanifolds. We prove, under some mild conditions, effective equidistribution of such measures to the Haar measure. We also formulate a mean ergodic theorem for $\\R^n$-representations on Hilbert spaces, restricted to a moving phase of low dimension. Furthermore, we bound the necessary dilation of a given smooth curve in $ \\R^n$ so that the canonical projection onto $ \\T^n $ is $ \\eps$-dense."}
{"category": "Math", "title": "Rescaled Lotka-Volterra Models Converge to Super Stable Processes", "abstract": "Recently, it has been shown that stochastic spatial Lotka-Volterra models when suitably rescaled can converge to a super Brownian motion. We show that the limit process could be a super stable process if the kernel of the underlying motion is in the domain of attraction of a stable law. The corresponding results in Brownian setting were proved by Cox and Perkins (2005, 2008). As applications of the convergence theorems, some new results on the asymptotics of the voter model started from single 1 at the origin are obtained which improve the results by Bramson and Griffeath (1980)."}
{"category": "Math", "title": "Calogero-Moser Spaces over Algebraic Curves", "abstract": "In these notes, we give a survey of the main results of [BC] and [BW]. Our aim is to generalize the geometric classification of (one-sided) ideals of the first Weyl algebra $ A_1(C) $ (see [BW1, BW2]) to the ring $ D(X) $ of differential operators on an arbitrary complex smooth affine curve X. We approach this problem in two steps: first, we classify the ideals of D(X) up to stable isomorphism, in terms of the Picard group of X; then, we refine this classification by describing each stable isomorphism class as a disjoint union of (certain quotients of) generalized Calogero-Moser spaces C_n(X, I). The latter are defined as representation varieties of deformed preprojective algebras over a one-point extension of the ring of regular functions on X by the line bundle I. As in the classical case, C_n(X, I) turn out to be smooth irreducible varieties of dimension 2n."}
{"category": "Math", "title": "Compositions of consistent systems of rank one discrete valuation rings", "abstract": "Let V be a rank one discrete valuation ring (DVR) on a field F and let L/F be a finite separable algebraic field extension with [L:F] = m. The integral closure of V in L is a Dedekind domain that encodes the following invariants: (i) the number of extensions of V to a valuation ring W on L, (ii) the residue degree of each W over V, and (iii) the ramification degree of each W over V. Given a finite set of DVRs on F, an m-consistent system is a family of sets enumerating what is theoretically possible for the above invariants of each V in the set. The m-consistent system is realizable if there exists a finite separable extension field L/F that gives for each V the listed invariants. We investigate the realizability of m-consistent systems."}
{"category": "Math", "title": "Fundamental Theorem of Calculus", "abstract": "A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been explained. Various classical examples of this theorem, such as the Green's and Stokes' theorem are discussed, as well as the new theory of monogenic functions, which generalizes the concept of an analytic function of a complex variable to higher dimensions."}
{"category": "Math", "title": "More non semigroup Lie gradings", "abstract": "This note is devoted to the construction of two very easy examples, of respective dimensions 4 and 6, of graded Lie algebras whose grading is not given by a semigroup, the latter one being a semisimple algebra. It is shown that 4 is the minimal possible dimension."}
{"category": "Math", "title": "Infinite rate mutually catalytic branching", "abstract": "Consider the mutually catalytic branching process with finite branching rate $\\gamma$. We show that as $\\gamma\\to\\infty$, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in terms of a planar Brownian motion from which we infer a path property of the process. This is the first paper in a series or three, wherein we also construct an interacting version of this process and study its long-time behavior."}
{"category": "Math", "title": "The Natural Logarithm on Time Scales", "abstract": "We define an appropriate logarithm function on time scales and present its main properties. This gives answer to a question posed by M. Bohner in [J. Difference Equ. Appl. {\\bf 11} (2005), no. 15, 1305--1306]."}
{"category": "Math", "title": "On the Brown--Shields conjecture for cyclicity in the Dirichlet space", "abstract": "Let $\\cD$ be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function $f\\in\\cD$ to be {\\em cyclic}, i.e. for $\\{pf: p\\text{a polynomial}\\}$ to be dense in $\\cD$. This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in $\\cD$ iff it is outer and its zero set (defined appropriately) is of capacity zero."}
{"category": "Math", "title": "Topology of Manifolds with Asymptotically Nonnegative Ricci Curvature", "abstract": "In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional curvature decay at most quadratically is diffeomorphic to a Euclidean n-space R^n under some conditions on the density of rays starting from the base point p or on the volume growth of geodesic balls in M."}
{"category": "Math", "title": "A note on smoothness and differential bases in positive characteristic", "abstract": "Let $u:A\\to B$ be a morphism of noetherian local rings. We obtain smoothness criteria for algebras with differential bases, in the case of rings containing a field of characteristic $p>0.$ We also give smoothness criteria for reduced morphisms."}
{"category": "Math", "title": "Boundary non-crossings of Brownian pillow", "abstract": "Let B_0(s,t) be a Brownian pillow with continuous sample paths, and let h,u:[0,1]^2\\to R be two measurable functions. In this paper we derive upper and lower bounds for the boundary non-crossing probability \\psi(u;h):=P{B_0(s,t)+h(s,t) \\le u(s,t), \\forall s,t\\in [0,1]}. Further we investigate the asymptotic behaviour of $\\psi(u;\\gamma h)$ with $\\gamma$ tending to infinity, and solve a related minimisation problem."}
{"category": "Math", "title": "Stable string operations are trivial", "abstract": "We show that in closed string topology and in open-closed string topology with one $D$-brane, higher genus stable string operations are trivial. This is a consequence of Harer's stability theorem and related stability results on the homology of mapping class groups of surfaces with boundaries. In fact, this vanishing result is a special case of a general result which applies to all homological conformal field theories with a property that in the associated topological quantum field theories, the string operations associated to genus one cobordisms with one or two boundaries vanish. In closed string topology, the base manifold can be either finite dimensional, or infinite dimensional with finite dimensional cohomology for its based loop space. The above vanishing result is based on the triviality of string operations associated to homology classes of mapping class groups which are in the image of stabilizing maps."}
{"category": "Math", "title": "SubRiemannian geometry on the sphere $\\mathbb{S}^3$", "abstract": "The present paper starts with an introduction to quaternions and then defines the 3-dimmensional sphere as the set of quaternions of length one. The quaternion group induces on $\\mathbb{S}^3$ a structure of noncommutative Lie group. This group is compact and the results obtained in this case are very different than those obtained in the case of the Heisenberg group, which is a noncompact Lie group. Like in the Heisenberg group case, we introduce a nonintegrable distribution on the sphere and a metric on it using two of the noncommutative left invariant vector fields. This way $\\mathbb{S}^3$ becomes a subRiemannian manifold. It is known that the group $SU(2) $ is isomorphic with the sphere $\\mathbb{S}^3$ and represents an example of subRiemannian manifold where the elements are matrices. The main issue here is to study the connectivity by horizontal curves and its geodesics on this manifold. In this paper, we are using Lagrangian method to study the connectivity theorem on ${\\mathbb S}^3$ by horizontal curves with minimal arc-length. We show that for any two points in ${\\mathbb S}^3$, there exists such a geodesic joining these two points."}
{"category": "Math", "title": "Three examples of the relation between rigid-analytic and algebraic deformation parameters", "abstract": "We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk."}
{"category": "Math", "title": "The Minkowski problem for the torsional rigidity", "abstract": "We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of the $n$-dimensional Euclidean space. For the existence part we apply the variational method introduced by Jerison for analogous problems concerning other variational functionals. Uniqueness follows from the Brunn--Minkowski inequality for the torsional rigidity and corresponding equality conditions."}
{"category": "Math", "title": "English translation of chapter 9 of the book \"Adams spectral sequence and stable homotopy groups of spheres\" by Jijkun Lin (In Chinese)(Sciences Press, Beijing 2007)", "abstract": "This paper is an English translation of chapter nine of the book \"Adams spectral sequence and stable homotopy groups of spheres\" by Jinkun Lin (in Chinese)(Sciences Press, Beijing 2007). In this paper, a sequence of new indecomposable families in the stable homotopy groups of spheres such as h_0h_n,h_nb_n,h_0h_nh_m,h_0(h_mb_{n-1}-b_{m-1}h_n) families was detected. The proof of all the detections was stated in this paper in based on several published papers by the author."}
{"category": "Math", "title": "Random block matrices and matrix orthogonal polynomials", "abstract": "In this paper we consider random block matrices, which generalize the general beta ensembles, which were recently investigated by Dumitriu and Edelmann (2002, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density, which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature."}
{"category": "Math", "title": "Optimal length estimates for stable CMC surfaces in 3-space forms", "abstract": "In this paper, we study stable constant mean curvature $H$ surfaces in $\\R^3$. We prove that, in such a surface, the distance from a point to the boundary is less that $\\pi/(2H)$. This upper-bound is optimal and is extended to stable constant mean curvature surfaces in space forms."}
{"category": "Math", "title": "Explicit constructions of infinite families of MSTD sets", "abstract": "We explicitly construct infinite families of MSTD (more sums than differences) sets. There are enough of these sets to prove that there exists a constant C such that at least C / r^4 of the 2^r subsets of {1,...,r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/2^{r/2} for some polynomial f(r)). We conclude by generalizing our method to compare linear forms epsilon_1 A + ... + epsilon_n A with epsilon_i in {-1,1}."}
{"category": "Math", "title": "POCP: a package for polynomial optimal control problems", "abstract": "POCP is a new Matlab package running jointly with GloptiPoly 3 and, optionally, YALMIP. It is aimed at nonlinear optimal control problems for which all the problem data are polynomial, and provides an approximation of the optimal value as well as some control policy. Thanks to a user-friendly interface, POCP reformulates such control problems as generalized problems of moments, in turn converted by GloptiPoly 3 into a hierarchy of semidefinite programming problems whose associated sequence of optimal values converges to the optimal value of the polynomial optimal control problem. In this paper we describe the basic features of POCP and illustrate them with some numerical examples."}
{"category": "Math", "title": "Applications of Automata and Graphs: Labeling Operators in Hilbert Space II", "abstract": "We introduced a family of infinite graphs directly associated with a class of von Neumann automaton model A_{G}. These are finite state models used in symbolic dynamics: stimuli models and in control theory. In the context of groupoid von Neumann algebras, and an associated fractal group, we prove a classification theorem for representations of automata."}
{"category": "Math", "title": "Solving the 100 Swiss Francs Problem", "abstract": "Sturmfels offered 100 Swiss Francs in 2005 to a conjecture, which deals with a special case of the maximum likelihood estimation for a latent class model. This paper confirms the conjecture positively."}
{"category": "Math", "title": "Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant", "abstract": "Let S be a compact connected oriented surface with one boundary component, and let P be the fundamental group of S. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of S, whose k-th term consists of the self-homeomorphisms of S that act trivially at the level of the k-th nilpotent quotient of P. Morita defined a homomorphism from the k-th term of the Johnson filtration to the third homology group of the k-th nilpotent quotient of P. In this paper, we replace groups by their Malcev Lie algebras and we study the \"infinitesimal\" version of the k-th Morita homomorphism, which is shown to correspond to the original version by a canonical isomorphism. We provide a diagrammatic description of the k-th infinitesimal Morita homomorphism and, given an expansion of the free group P that is \"symplectic\" in some sense, we show how to compute it from Kawazumi's \"total Johnson map\". Besides, we give a topological interpretation of the full tree-reduction of the LMO homomorphism, which is a diagrammatic representation of the Torelli group derived from the Le-Murakami-Ohtsuki invariant of 3-manifolds. More precisely, a symplectic expansion of P is constructed from the LMO invariant, and it is shown that the tree-level of the LMO homomorphism is equivalent to the total Johnson map induced by this specific expansion. It follows that the k-th infinitesimal Morita homomorphism coincides with the degree [k,2k[ part of the tree-reduction of the LMO homomorphism. Our results also apply to the monoid of homology cylinders over S."}
{"category": "Math", "title": "Bounds on Sobolev norms for the nonlinear Schr\\\"odinger equation on general tori", "abstract": "We prove Strichartz estimates on general flat d-torus for arbitrary d. Using these estimates, we prove local wellposedness for the cubic nonlinear Schr\\\"odinger equations in appropriate Sobolev spaces. In dimensions 2 and 3, we prove polynomial bounds on the possible growth of Sobolev norms of smooth solutions."}
{"category": "Math", "title": "Sensitivity for Smoluchowski equation", "abstract": "This article investigates the question of sensitivity of the solutions of Smoluchowski equation on R_+^* with respect to parameters \\lambda in the interaction kernel K^lambda. It is proved that the solution is a C^1 function of (t,lambda) with values in a good space of measures under the hypotheses K^{lambda}(x,y) \\leq phi(x)phi(y), for some sub-linear function phi, a (4+epsilon)-moment assumption on the initial condition, and that the derivative is a solution, in a suitable sense, of a linearized equation."}
{"category": "Math", "title": "Series expansions in Fr\\'echet spaces and their duals; construction of Fr\\'echet frames", "abstract": "Frames for Fr\\'echet spaces $X_F$ with respect to Fr\\'echet sequence spaces $\\Theta_F$ are studied and conditions, implying series expansions in $X_F$ and $X_F^*$, are determined. If $\\{g_i\\}$ is a $\\Theta_0$-frame for $X_0$, we construct a sequence $\\{X_s\\}_{s\\in {\\mathbb N}_0}$, $X_s\\subset X_{s-1}$, $s\\in {\\mathbb N}$, for given $\\Theta_F$, respectively a sequence $\\{\\Theta_s\\}_{s\\in {\\mathbb N}_0}$, $\\Theta_s\\subset \\Theta_{s-1}$, $s\\in {\\mathbb N}$, for given $X_F$, so that $\\{g_i\\}$ is a pre-$F$-frame (or $F$-frame) for $X_F$ with respect to $\\Theta_F$ under different assumptions given on $X_0$, $\\Theta_0$, $\\Theta_F$ or $X_0$, $\\Theta_0$, $X_F$."}
{"category": "Math", "title": "Completeness of determinantal Hamiltonian flows on the matrix affine Poisson space", "abstract": "The matrix affine Poisson space (M_{m,n}, pi_{m,n}) is the space of complex rectangular matrices equipped with a canonical quadratic Poisson structure which in the square case m=n reduces to the standard Poisson structure on GL_n(C). We prove that the Hamiltonian flows of all minors are complete. As a corollary we obtain that all Kogan-Zelevinsky integrable systems on M_{n,n} are complete and thus induce (analytic) Hamiltonian actions of C^{n(n-1)/2} on (M_{n,n}, pi_{n,n}) (as well as on GL_n(C) and on SL_n(C)). We define Gelfand-Zeitlin integrable systems on (M_{n,n}, pi_{n,n}) from chains of Poisson projections and prove that their flows are also complete. This is an analog for the quadratic Poisson structure pi_{n,n} of the recent result of Kostant and Wallach [KW] that the flows of the complexified classical Gelfand-Zeitlin integrable systems are complete."}
{"category": "Math", "title": "Regularization of almost complex structures and and gluing holomorphic discs to tori", "abstract": "We prove a result on removing singularities of almost complex structures pulled back by a non-diffeomorphic map. As an application we prove the existence of global J-holomorphic discs with boundaries attached to real tori."}
{"category": "Math", "title": "Point classification of 2nd order ODEs: Tresse classification revisited and beyond", "abstract": "In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the 2nd order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the equivalence problem."}
{"category": "Math", "title": "Smooth supersaturated models", "abstract": "In areas such as kernel smoothing and non-parametric regression there is emphasis on smooth interpolation and smooth statistical models. Splines are known to have optimal smoothness properties in one and higher dimensions. It is shown, with special attention to polynomial models, that smooth interpolators can be constructed by first extending the monomial basis and then minimising a measure of smoothness with respect to the free parameters in the extended basis. Algebraic methods are a help in choosing the extended basis which can also be found as a saturated basis for an extended experimental design with dummy design points. One can get arbitrarily close to optimal smoothing for any dimension and over any region, giving a simple alternative models of spline type. The relationship to splines is shown in one and two dimensions. A case study is given which includes benchmarking against kriging methods."}
{"category": "Math", "title": "Algebraic K-theory of toric hypersurfaces", "abstract": "We construct classes in the motivic cohomology of certain 1-parameter families of Calabi-Yau hypersurfaces in toric Fano n-folds, with applications to local mirror symmetry (growth of genus 0 instanton numbers) and inhomogeneous Picard-Fuchs equations. In the case where the family is classically modular the classes are related to Belinson's Eisenstein symbol; the Abel-Jacobi map (or rational regulator) is computed in this paper for both kinds of cycles. For the \"modular toric\" families where the cycles essentially coincide, we obtain a motivic (and computationally effective) explanation of a phenomenon observed by Villegas, Stienstra, and Bertin."}
{"category": "Math", "title": "Elements and cyclic subgroups of finite order of the Cremona group", "abstract": "We give the classification of elements - respectively cyclic subgroups - of finite order of the Cremona group, up to conjugation. Natural parametrisations of conjugacy classes, related to fixed curves of positive genus, are provided."}
{"category": "Math", "title": "Deconvolving oscillatory transients with a Kalman filter", "abstract": "This paper describes a method to filter oscillatory transients from measurements of a time series which were at least an order of magnitude larger than the signal to be measured. Based on a Kalman filter, it has an optimality property and a natural scaling parameter that allows to tune it to high resolution or low noise."}
{"category": "Math", "title": "Multistage Estimation of Bounded-Variable Means", "abstract": "In this paper, we develop a multistage approach for estimating the mean of a bounded variable. We first focus on the multistage estimation of a binomial parameter and then generalize the estimation methods to the case of general bounded random variables. A fundamental connection between a binomial parameter and the mean of a bounded variable is established. Our multistage estimation methods rigorously guarantee prescribed levels of precision and confidence."}
{"category": "Math", "title": "Brownian couplings, convexity, and shy-ness", "abstract": "Benjamini, Burdzy and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying a positive distance away from each other for all time. Among other results, they showed no shy couplings could exist for reflected Brownian motions in C^2 bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al. result (a) to all bounded convex domains (whether planar and smooth or not) whose boundaries contain no line segments, (b) to all bounded convex planar domains regardless of further conditions on the boundary."}
{"category": "Math", "title": "Directed graphs without short cycles", "abstract": "For a directed graph $G$ without loops or parallel edges, let $\\beta(G)$ denote the size of the smallest feedback arc set, i.e., the smallest subset $X \\subset E(G)$ such that $G \\sm X$ has no directed cycles. Let $\\gamma(G)$ be the number of unordered pairs of vertices of $G$ which are not adjacent. We prove that every directed graph whose shortest directed cycle has length at least $r \\ge 4$ satisfies $\\beta(G) \\le c\\gamma(G)/r^2$, where $c$ is an absolute constant. This is tight up to the constant factor and extends a result of Chudnovsky, Seymour, and Sullivan. This result can be also used to answer a question of Yuster concerning almost given length cycles in digraphs. We show that for any fixed $0 < \\theta < 1/2$ and sufficiently large $n$, if $G$ is a digraph with $n$ vertices and $\\beta(G) \\ge \\theta n^2$, then for any $0 \\le m \\le \\theta n-o(n)$ it contains a directed cycle whose length is between $m$ and $m+6 \\theta^{-1/2}$. Moreover, there is a constant $C$ such that either $G$ contains directed cycles of every length between $C$ and $\\theta n-o(n)$ or it is close to a digraph $G'$ with a simple structure: every strong component of $G'$ is periodic. These results are also tight up to the constant factors."}
{"category": "Math", "title": "Almost all one-relator groups with at least three generators are residually finite", "abstract": "We prove that with probability tending to 1, a 1-relator group with at least 3 generators and relator of length n is residually finite, virtually residually (finite p)-group for all sufficiently large p, and coherent. The proof uses both combinatorial group theory and non-trivial results about Brownian motions, bridges and excursions in R^k."}
{"category": "Math", "title": "Tropical Convex Hull Computations", "abstract": "This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial ideals, subdivisions of products of simplices, matroid theory, finite metric spaces, and the tropical Grassmannians. The relationship between these topics is explained via one running example throughout the whole paper. The final section explains how the new version 2.9.4 of the software system polymake can be used to compute with tropical polytopes."}
{"category": "Math", "title": "Brauer pairs of Camina p-groups of nilpotence class 2", "abstract": "In this paper, we find a condition that characterizes when two Camina $p$-groups of nilpotence class 2 form a Brauer pair."}
{"category": "Math", "title": "A new algorithm for the recursion of multisums with improved universal denominator", "abstract": "The purpose of the paper is to introduce two new algorithms. The first one computes a linear recursion for proper hypergeometric multisums, by treating one summation variable at a time, and provides rational certificates along the way. A key part in the search of a linear recursion is an improved universal denominator algorithm that constructs all rational solutions $x(n)$ of the equation $$ \\frac{a_m(n)}{b_m(n)}x(n+m)+...+\\frac{a_0(n)}{b_0(n)}x(n)= c(n),$$ where $a_i(n), b_i(n), c(n)$ are polynomials. Our algorithm improves Abramov's universal denominator."}
{"category": "Math", "title": "Perturbed Floer Homology of some fibered three manifolds", "abstract": "In this paper, we write down a special Heegaard diagram for a given product three manifold $\\Sigma_g\\times S^1$. We use the diagram to compute its perturbed Heegaard Floer homology."}
{"category": "Math", "title": "Central limit theorem for linear eigenvalue statistics of random matrices with independent entries", "abstract": "We consider $n\\times n$ real symmetric and Hermitian Wigner random matrices $n^{-1/2}W$ with independent (modulo symmetry condition) entries and the (null) sample covariance matrices $n^{-1}X^*X$ with independent entries of $m\\times n$ matrix $X$. Assuming first that the 4th cumulant (excess) $\\kappa_4$ of entries of $W$ and $X$ is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as $n\\to\\infty$, $m\\to\\infty$, $m/n\\to c\\in[0,\\infty)$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class $\\mathbf{C}^5$). This is done by using a simple ``interpolation trick'' from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially $\\mathbb{C}^5$ test function. Here the variance of statistics contains an additional term proportional to $\\kappa_4$. The proofs of all limit theorems follow essentially the same scheme."}
{"category": "Math", "title": "Deformation of Sasakian metrics", "abstract": "Deformations of the Reeb flow of a Sasakian manifold as transversely K\\\"ahler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely K\\\"ahler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics."}
{"category": "Math", "title": "Rigidity of the \\'Alvarez classes of Riemannian foliations with nilpotent structure Lie algebras", "abstract": "We show that if the structure algebra of a Riemannian foliation F on a closed manifold M is nilpotent, then the integral of the \\'Alvarez class of (M,F) along every closed path is the exponential of an algebraic number. By this result and the continuity of the \\'Alvarez class under deformations shown in arXiv:1009.1098v2, we prove that the \\'Alvarez class and the geometrically tautness of Riemannian foliations on a closed manifold M are invariant under deformation, if the fundamental group of M has polynomial growth."}
{"category": "Math", "title": "Indefinite extrinsic symmetric spaces I", "abstract": "We find a one-to-one correspondence between full extrinsic symmetric spaces in (possibly degenerate) inner product spaces and certain algebraic objects called (weak) extrinsic symmetric triples. In particular, this yields a description of arbitrary extrinsic symmetric spaces in pseudo-Euclidean spaces by corresponding infinitesimal objects."}
{"category": "Math", "title": "A local greedy algorithm and higher order extensions for global numerical continuation of analytically varying subspaces", "abstract": "We present a family of numerical implementations of Kato's ODE propagating global bases of analytically varying invariant subspaces, of which the first-order version is a surprising simple \"greedy algorithm\" that is both stable and easy to program and the second-order version a relaxation of a first-order scheme of Brin and Zumbrun. The method has application to numerical Evans function computations used to assess stability of traveling-wave solutions of time-evolutionary PDE."}
{"category": "Math", "title": "The t-improper chromatic number of random graphs", "abstract": "We consider the $t$-improper chromatic number of the Erd{\\H o}s-R{\\'e}nyi random graph $G(n,p)$. The t-improper chromatic number $\\chi^t(G)$ of $G$ is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$, then this is the usual notion of proper colouring. When the edge probability $p$ is constant, we provide a detailed description of the asymptotic behaviour of $\\chi^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$."}
{"category": "Math", "title": "Spaces with a Finite Family of Basic Functions", "abstract": "A space X is finite dimensional, locally compact and separable metrizable if and only if X has a finite basic family: continuous functions Phi_1,...,Phi_n of X to the reals, R, such that for all continuous f from X to R there are g_1,..., g_n in C(R) satisfying f(x)=g_1(Phi_1(x))+g_2(Phi_2(x))+...+g_n(Phi_n(x)) for all x in X. This give the complete solution to four problems on basic functions posed by Sternfeld, as well as questions posed by Hattori and others."}
{"category": "Math", "title": "Counting the Closed Subgroups of Profinite Groups", "abstract": "The sets of closed and closed-normal subgroups of a profinite group carry a natural profinite topology. Through a combination of algebraic and topological methods the size of these subgroup spaces is calculated, and the spaces partially classified up to homeomorphism."}
{"category": "Math", "title": "Classifying Spaces of Subgroups of Profinite Groups", "abstract": "The set of all closed subgroups of a profinite carries a natural profinite topology. This space of subgroups can be classified up to homeomorphism in many cases, and tight bounds placed on its complexity as expressed by its scattered height."}
{"category": "Math", "title": "The Erd\\\"os-Falconer distance problem on the unit sphere in vector spaces over finite fields", "abstract": "art, Iosevich, Koh and Rudnev (2007) show, using Fourier analysis method, that the finite Erd\\\"os-Falconer distance conjecture holds for subsets of the unit sphere in $\\mathbbm{F}_q^d$. In this note, we give a graph theoretic proof of this result."}
{"category": "Math", "title": "Large deviations for the leaves in some random trees", "abstract": "Large deviation principles and related results are given for a class of Markov chains associated to the \"leaves\" in random recursive trees and preferential attachment random graphs, as well as the \"cherries\" in Yule trees. In particular, the method of proof, combining analytic and Dupuis-Ellis type path arguments, allows for an explicit computation of the large deviation pressure."}
{"category": "Math", "title": "Polynomial Translation Weingarten Surfaces in 3-dimensional Euclidean space", "abstract": "In this paper we will classify those translation surfaces in E3 involving polynomials which are Weingarten surfaces. We analyze Weingarten translation surfaces satisfying 2aH + bK = 0. We study also other types of translation surfaces, involving power functions, for which the second Gaussian curvature vanishes."}
{"category": "Math", "title": "On the curvature of biquotients", "abstract": "As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of non-negatively curved manifolds which contain either a point or an open dense set of points at which all 2-planes have positive curvature. We study infinite families of biquotients defined by Eschenburg and Bazaikin from this viewpoint, together with torus quotients of $S^3 \\x S^3$."}
{"category": "Math", "title": "On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D", "abstract": "We consider a nonlinear Schroedinger equation with a finite bands periodic potential in R . We assume the existence of an orbitally stable family of ground states. We prove that under appropriate hypotheses the ground states are asymptotically stable."}
{"category": "Math", "title": "Perfect Derived Categories of Positively Graded DG Algebras", "abstract": "We investigate the perfect derived category dgPer(A) of a positively graded differential graded (dg) algebra A whose degree zero part is a dg subalgebra and semisimple as a ring. We introduce an equivalent subcategory of dgPer(A) whose objects are easy to describe, define a t-structure on dgPer(A) and study its heart. We show that dgPer(A) is a Krull-Remak-Schmidt category. Then we consider the heart in the case that A is a Koszul ring with differential zero satisfying some finiteness conditions."}
{"category": "Math", "title": "Heat-flow monotonicity of Strichartz norms", "abstract": "Most notably we prove that for $d=1,2$ the classical Strichartz norm $$\\|e^{i s\\Delta}f\\|_{L^{2+4/d}_{s,x}(\\mathbb{R}\\times\\mathbb{R}^d)}$$ associated to the free Schr\\\"{o}dinger equation is nondecreasing as the initial datum $f$ evolves under a certain quadratic heat-flow."}
{"category": "Math", "title": "Equivariant Sheaves on Flag Varieties", "abstract": "We show that the Borel-equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of dg modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves. This proves a conjecture of Soergel and Lunts in the case of flag varieties."}
{"category": "Math", "title": "On the construction of A-infinity structures", "abstract": "We relate a construction of Kadeishvili's establishing an A-infinity-structure on the homology of a differential graded algebra or more generally of an A-infinity algebra with certain constructions of Chen and Gugenheim. Thereafter we establish the links of these constructions with subsequent developments."}
{"category": "Math", "title": "Jump transformations and an embedding of ${\\cal O}_{\\infty}$ into ${\\cal O}_{2}$", "abstract": "A measurable map $T$ on a measure space induces a representation $\\Pi_{T}$ of a Cuntz algebra ${\\cal O}_{N}$ when $T$ satisfies a certain condition. For such two maps $\\tau$ and $\\sigma$ and representations $\\Pi_{\\tau}$ and $\\Pi_{\\sigma}$ associated with them, we show that $\\Pi_{\\tau}$ is the restriction of $\\Pi_{\\sigma}$ when $\\tau$ is a jump transformation of $\\sigma$. Especially, the Gauss map $\\tau_1$ and the Farey map $\\sigma_1$ induce representations $\\Pi_{\\tau_1}$ of ${\\cal O}_{\\infty}$ and that $\\Pi_{\\sigma_1}$ of ${\\cal O}_{2}$, respectively, and $\\Pi_{\\tau_1}=\\Pi_{\\sigma_1}|_{{\\cal O}_{\\infty}}$ with respect to a certain embedding of ${\\cal O}_{\\infty}$ into ${\\cal O}_{2}$."}
{"category": "Math", "title": "A variational multiscale Newton-Schur approach for the incompressible Navier-Stokes equations", "abstract": "In the following paper, we present a consistent Newton-Schur solution approach for variational multiscale formulations of the time-dependent Navier-Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three-dimensional problems, and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non-symmetric matrices. In addition to the quadratic convergence characteristics of a Newton-Raphson based scheme, the Newton-Schur approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two-level approach to stabilizing the incompressible Navier-Stokes equations based on a coarse and fine-scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three-dimensional problems for Reynolds number up to 1000 including steady and time-dependent flows."}
{"category": "Math", "title": "$W$-graph versions of tensoring with the $\\S_n$ defining representation", "abstract": "We further develop the theory of inducing $W$-graphs worked out by Howlett and Yin in \\cite{HY1}, \\cite{HY2}, focusing on the case $W = \\S_n$. Our main application is to give two $W$-graph versions of tensoring with the $\\S_n$ defining representation $V$, one being $\\H \\tsr_{\\H_J} -$ for $\\H, \\H_J$ the Hecke algebras of $\\S_n, \\S_{n-1}$ and the other $(\\pH \\tsr_{\\H} -)_1$, where $\\pH$ is a subalgebra of the extended affine Hecke algebra and the subscript signifies taking the degree 1 part. We look at the corresponding $W$-graph versions of the projection $V \\tsr V \\tsr - \\to S^2 V \\tsr -$. This does not send canonical basis elements to canonical basis elements, but we show that it approximates doing so as the Hecke algebra parameter $\\u \\to 0$. We make this approximation combinatorially explicit by determining it on cells. Also of interest is a combinatorial conjecture stating the restriction of $\\H$ to $\\H_J$ is \"weakly multiplicity-free\" for $|J| = n-1$, and a partial determination of the map $\\H \\tsr_{\\H_J} \\H \\xrightarrow{\\counit} \\H$ on canonical basis elements, where $\\counit$ is the counit of adjunction."}
{"category": "Math", "title": "Reduction of the Number of Quantifiers in Real Analysis through Infinitesimals (Master Thesis, Mathematics Department, California Polytechnic State University, San Luis Obispo)", "abstract": "We construct the non-standard complex (and real) numbers using the ultrapower method in the spirit of Cauchy's construction of the real numbers. We show that the non-standard complex numbers are a non-archimedean, algebraically closed field, and that the non-standard real numbers are a totally ordered, real-closed, non-archimedean field. We explore the various types of non-standard numbers, and develop the non-standard completeness results (Saturation Principle, Supremum Completeness of Bounded Internal Sets, etc) for $\\starr$. We give non-standard characterizations for such usual topological objects as open, closed, bounded, and compact sets in terms of monads. We also consider such traditional topics of real analysis as limits, continuity, uniform continuity, convergence, uniform convergence, etc. in a non-standard setting. In both topology and real analysis we reduce (and in some cases eliminate) the number of quantifiers in the non-standard setting."}
{"category": "Math", "title": "Stochastic solutions of a class of Higher order Cauchy problems in $\\rd$", "abstract": "We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Allouba and Zheng \\cite{allouba1}, Baeumer, Meerschaert and Nane \\cite{bmn-07}, Meerschaert, Nane and Vellaisamy \\cite{MNV}, and Nane \\cite{nane-h}. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index $0<\\beta <1$, or by the absolute value of a symmetric $\\alpha$-stable process with $0<\\alpha\\leq 2$, independent of the Markov process. In some special cases we represent the solutions by running composition of $k$ independent Brownian motions, called $k$-iterated Brownian motion for an integer $k\\geq 2$. We make use of a connection between fractional-time diffusions and higher order partial differential equations established first by Allouba and Zheng \\cite{allouba1} and later extended in several directions by Baeumer, Meerschaert and Nane \\cite{bmn-07}."}
{"category": "Math", "title": "Q-curvature flow with indefinite nonlinearity", "abstract": "In this note, we study Q-curvature flow on $S^4$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^4$ has a solution provided the prescribed Q-curvature $f$ has its positive part, which possesses non-degenerate critical points such that $\\Delta_{S^4} f\\not=0$ at the saddle points and an extra condition such as a nontrivial degree counting condition."}
{"category": "Math", "title": "Secant varieties of Segre-Veronese varieties P^m x P^n embedded by O(1,2)", "abstract": "Let $X_{m,n}$ be the Segre-Veronese variety $\\mathbb{P}^m \\times \\mathbb{P}^n$ embedded by the morphism given by $\\mathcal{O}(1,2)$. In this paper, we provide two functions $\\underline{s}(m,n)\\le \\bar{s}(m,n)$ such that the $s^{\\mathrm{th}}$ secant variety of $X_{m,n}$ has the expected dimension if $s \\leq \\underline{s}(m,n)$ or $ \\bar{s}(m,n) \\leq s$. We also present a conjecturally complete list of defective secant varieties of such Segre-Veronese varieties."}
{"category": "Math", "title": "The Back and Forth Nudging algorithm for data assimilation problems: theoretical results on transport equations", "abstract": "In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (B\\\"urgers' equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), the convergence of the algorithm holds under observability conditions. Convergence can also be proven for viscous linear transport equations under some strong hypothesis, but not for viscous Burgers' equation. Moreover, the convergence rate is always exponential in time. We also notice that the forward and backward system of equations is well posed when no nudging term is considered."}
{"category": "Math", "title": "Folding 3-noncrossing RNA pseudoknot structures", "abstract": "In this paper we present a selfcontained analysis and description of the novel {\\it ab initio} folding algorithm {\\sf cross}, which generates the minimum free energy (mfe), 3-noncrossing, $\\sigma$-canonical RNA structure. Here an RNA structure is 3-noncrossing if it does not contain more than three mutually crossing arcs and $\\sigma$-canonical, if each of its stacks has size greater or equal than $\\sigma$. Our notion of mfe-structure is based on a specific concept of pseudoknots and respective loop-based energy parameters. The algorithm decomposes into three parts: the first is the inductive construction of motifs and shadows, the second is the generation of the skeleta-trees rooted in irreducible shadows and the third is the saturation of skeleta via context dependent dynamic programming routines."}
{"category": "Math", "title": "Monopole Floer homology for rational homology 3-spheres", "abstract": "We give a new construction of monopole Floer homology for spin-c rational homology 3-spheres. As applications we define two invariants of certain smooth compact 4-manifolds with b_1=1 and b^+=0."}
{"category": "Math", "title": "Estimation of Higher Order Moments for Compound Models of Clutter by Mellin Transform", "abstract": "The compound models of clutter statistics are found suitable to describe the nonstationary nature of radar backscattering from high-resolution observations. In this letter, we show that the properties of Mellin transform can be utilized to generate higher order moments of simple and compound models of clutter statistics in a compact manner"}
{"category": "Math", "title": "Groups acting on manifolds: around the Zimmer program", "abstract": "This paper is a survey on the {\\em Zimmer program}. In it's broadest form, this program seeks an understanding of actions of large groups on compact manifolds. The goals of this survey are $(1)$ to put in context the original questions and conjectures of Zimmer and Gromov that motivated the program, $(2)$ to indicate the current state of the art on as many of these conjectures and questions as possible and $(3)$ to indicate a wide variety of open problems and directions of research."}
{"category": "Math", "title": "Concentration of measure and mixing for Markov chains", "abstract": "We consider Markovian models on graphs with local dynamics. We show that, under suitable conditions, such Markov chains exhibit both rapid convergence to equilibrium and strong concentration of measure in the stationary distribution. We illustrate our results with applications to some known chains from computer science and statistical mechanics."}
{"category": "Math", "title": "On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems", "abstract": "This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\\pi/j,$ $j\\in\\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\\'{n}, 2005]."}
{"category": "Math", "title": "The Kodaira Dimension of Lefschetz Fibrations", "abstract": "In this note, we verify that the complex Kodaira dimension $\\kappa^h$ equals the symplectic Kodaira dimension $\\kappa^s$ for smooth 4-manifolds with complex and symplectic structures. We also calculate the Kodaira dimension for many Lefschetz fibrations."}
{"category": "Math", "title": "The cohomological equation for partially hyperbolic diffeomorphisms", "abstract": "We establish a theory for the existence and regularity of solutions to the cohomological equation over an accessible, partially hyperbolic diffeomorphism. As a by-product of our techniques, we show that for $r>1$, any $C^r$ homogeneous, locally compact submanifold of a $C^r$ manifold is in fact a $C^r$ submanifold."}
{"category": "Math", "title": "An Information Geometric Framework for Dimensionality Reduction", "abstract": "This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of learning tasks such as classification, clustering, and visualization, these methods have focused primarily on Riemannian manifolds in Euclidean space. While sufficient for many applications, there are many high-dimensional signals which have no straightforward and meaningful Euclidean representation. In these cases, signals may be more appropriately represented as a realization of some distribution lying on a statistical manifold, or a manifold of probability density functions (PDFs). We present a framework for dimensionality reduction that uses information geometry for both statistical manifold reconstruction as well as dimensionality reduction in the data domain."}
{"category": "Math", "title": "Noncommutative ball maps", "abstract": "In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call \"NC ball maps\". We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, \"NC ball maps\" are very simple, in contrast to the classical result of D'Angelo on such analytic maps over C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables."}
{"category": "Math", "title": "Ahlfors-David regular sets and bilipschitz maps", "abstract": "Given two Ahlfors-David regular sets in metric spaces, we study the question whether one of them has a subset bilipschitz equivalent with the other."}
{"category": "Math", "title": "Square-Free Rings And Their Automorphism Group", "abstract": "Finite-dimensional square-free algebras have been completely characterized by Anderson and D'Ambrosia as certain twisted semigroup algebras over a square-free semigroup S with coefficients in a field K. D'Ambrosia extended the definition of square-free to artinian rings with unity and showed every square-free ring has an associated division ring D and square-free semigroup S. We show a square-free ring can be characterized as a twisted semigroup ring over a square-free semigroup S with coefficients in a division ring D. Also, to each square-free ring there exists a short exact sequence connecting the outer automorphisms of a square-free ring to certain cohomology groups related to S and D."}
{"category": "Math", "title": "Random Heegaard splittings", "abstract": "A random Heegaard splitting is a 3-manifold obtained by using a random walk of length n on the mapping class group as the gluing map between two handlebodies. We show that the joint distribution of random walks of length n and their inverses is asymptotically independent, and converges to the product of the harmonic and reflected harmonic measures defined by the random walk. We use this to show that the translation length on the curve complex of a random walk grows linearly in the length of the walk, and similarly, that distance in the curve complex between the disc sets of a random Heegaard splitting grows linearly in n. In particular, this implies that a random Heegaard splitting is hyperbolic with asymptotic probability one."}
{"category": "Math", "title": "Absolutely simple Prymians of trigonal curves", "abstract": "Using Galois Theory, we construct explicitly absolutely simple (principally polarized) Prym varieties that are not isomorphic to jacobians of curves even if we ignore the polarizations. Our approach is based on the previous papers math/0610138 [math.AG] and math/0605028 [math.AG] ."}
{"category": "Math", "title": "On the cohomology of hyperkahler quotients", "abstract": "This paper gives a partial desingularisation construction for hyperk\\\"ahler quotients and a criterion for the surjectivity of an analogue of the Kirwan map to the cohomology of hyperk\\\"ahler quotients. This criterion is applied to some linear actions on hyperk\\\"ahler vector spaces."}
{"category": "Math", "title": "On the nonexistence of Einstein metric on 4-manifolds", "abstract": "By using the gluing formulae of the Seiberg-Witten invariant, we show the nonexistence of Einstein metric on manifolds obtained from a 4-manifold with nontrivial Seiberg-Witten invariant by performing sufficiently many connected sums or appropriate surgeries along circles or homologically trivial 2-spheres with closed oriented 4-manifolds with negative definite intersection form."}
{"category": "Math", "title": "Spaces of algebraic and continuous maps between real algebraic varieties", "abstract": "We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a ` minimum degree\\rq filtration by finite dimensional subspaces and it is natural to expect that the homotopy types of the terms of the filtration approximate closer and closer the homotopy type of the space of continuous mappings as the degree increases. We prove this and compute the lower bounds of this approximation for ` even\\rq components of these spaces (more precisely, we prove a very similar and closely related result, and state this one as a conjecture)."}
{"category": "Math", "title": "Cluster tilting for higher Auslander algebras", "abstract": "The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The $n$-Auslander-Reiten translation functor $\\tau_n$ plays an important role in the study of $n$-cluster tilting subcategories. We study the category $\\MM_n$ of preinjective-like modules obtained by applying $\\tau_n$ to injective modules repeatedly. We call a finite dimensional algebra $\\Lambda$ \\emph{$n$-complete} if $\\MM_n=\\add M$ for an $n$-cluster tilting object $M$. Our main result asserts that the endomorphism algebra $\\End_\\Lambda(M)$ is $(n+1)$-complete. This gives an inductive construction of $n$-complete algebras. For example, any representation-finite hereditary algebra $\\Lambda^{(1)}$ is 1-complete. Hence the Auslander algebra $\\Lambda^{(2)}$ of $\\Lambda^{(1)}$ is 2-complete. Moreover, for any $n\\ge1$, we have an $n$-complete algebra $\\Lambda^{(n)}$ which has an $n$-cluster tilting object $M^{(n)}$ such that $\\Lambda^{(n+1)}=\\End_{\\Lambda^{(n)}}(M^{(n)})$. We give the presentation of $\\Lambda^{(n)}$ by a quiver with relations. We apply our results to construct $n$-cluster tilting subcategories of derived categories of $n$-complete algebras."}
{"category": "Math", "title": "The signature of the Ricci curvature of left invariant Riemannian metrics on 4-dimensional Lie groups", "abstract": "In this paper, we present the classification of all possible signatures of the Ricci curvature of left-invariant Riemannian metrics on 4-dimensional Lie groups and discuss some related questions."}
{"category": "Math", "title": "A martingale-transform goodness-of-fit test for the form of the conditional variance", "abstract": "In the common nonparametric regression model the problem of testing for a specific parametric form of the variance function is considered. Recently Dette and Hetzler (2008) proposed a test statistic, which is based on an empirical process of pseudo residuals. The process converges weakly to a Gaussian process with a complicated covariance kernel depending on the data generating process. In the present paper we consider a standardized version of this process and propose a martingale transform to obtain asymptotically distribution free tests for the corresponding Kolmogorov-Smirnov and Cram\\'{e}r-von-Mises functionals. The finite sample properties of the proposed tests are investigated by means of a simulation study."}
{"category": "Math", "title": "A new multiple Dirichlet series induced by a higher order form", "abstract": "A multiple Dirichlet series in two variables is constructed as a Mellin transform of a higher order Eisenstein series. It is shown to extend to a meromorphic function and satisfy two independent functional equations."}
{"category": "Math", "title": "Cohomological dimension of Laumon 1-motives up to isogenies", "abstract": "We prove that the category of Laumon 1-motives up isogenies over a field of characteristic zero is of cohomological dimension $\\le 1$. As a consequence this implies the same result for the category of formal Hodge structures of level $\\le 1$ (over $\\mathbb{Q}$)."}
{"category": "Math", "title": "Multi-valued hyperelliptic continued fractions of generalized Halphen type", "abstract": "We introduce and study higher genera generalizations of the Halphen theory of continued fractions. The basic notion we start with is hyperelliptic Haplhen (HH) element $$\\frac{\\sqrt{X_{2g+2}}-\\sqrt{Y_{2g+2}}}{x-y},$$ depending on parameter $y$, where $X_{2g+2}$ is a polynomial of degree $2g+2$ and $Y_{2g+2}=X_{2g+2}(y)$. We study regular and irregular HH elements. their continued fraction development and some basic properties of such development: even and odd symmetry and periodicity."}
{"category": "Math", "title": "On curvature-adapted and proper complex equifocal submaniflds", "abstract": "In this paper, we investigate a curvature-adapted and proper complex equifocal submanifold in a symmetric space of non-compact type. The class of these submanifolds contains principal orbits of Hermann type actions as homogeneous examples. In future, the results in this paper will be used to give a submanifold geometrical characterization of principal orbits of Hermann type actions."}
{"category": "Math", "title": "On the existence of polynomial-time algorithms to the subset sum problem", "abstract": "This paper proves that there does not exist a polynomial-time algorithm to the the subset sum problem. As this problem is in NP, the result implies that the class P of problems admitting polynomial-time algorithms does not equal the class NP of problems admitting nondeterministic polynomial-time algorithms."}
{"category": "Math", "title": "A note on random orthogonal polynomials on a compact interval", "abstract": "We consider a uniform distribution on the set $\\mathcal{M}_k$ of moments of order $k \\in \\mathbb{N}$ corresponding to probability measures on the interval $[0,1]$. To each (random) vector of moments in $\\mathcal{M}_{2n-1}$ we consider the corresponding uniquely determined monic (random) orthogonal polynomial of degree $n$ and study the asymptotic properties of its roots if $n \\to \\infty$."}
{"category": "Math", "title": "Testing for a constant coefficient of variation in nonparametric regression", "abstract": "In this paper we propose a new test for the hypothesis of a constant coefficient of variation in the common nonparametric regression model. The test is based on an estimate of the $L^2$-distance between the square of the regression function and variance function. We prove asymptotic normality of a standardized estimate of this distance under the null hypothesis and fixed alternatives and the finite sample properties of a corresponding bootstrap test are investigated by means of a simulation study. The results are applicable to stationary processes with the common mixing conditions and are used to construct tests for ARCH assumptions in financial time series."}
{"category": "Math", "title": "Optimal experimental designs for inverse quadratic regression models", "abstract": "In this paper optimal experimental designs for inverse quadratic regression models are determined. We consider two different parameterizations of the model and investigate local optimal designs with respect to the $c$-, $D$- and $E$-criteria, which reflect various aspects of the precision of the maximum likelihood estimator for the parameters in inverse quadratic regression models. In particular it is demonstrated that for a sufficiently large design space geometric allocation rules are optimal with respect to many optimality criteria. Moreover, in numerous cases the designs with respect to the different criteria are supported at the same points. Finally, the efficiencies of different optimal designs with respect to various optimality criteria are studied, and the efficiency of some commonly used designs are investigated."}
{"category": "Math", "title": "Higher homotopy of groups definable in o-minimal structures", "abstract": "It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their universal cover are contractible."}
{"category": "Math", "title": "Stochastic vortex method for forced three-dimensional Navier--Stokes equations and pathwise convergence rate", "abstract": "We develop a McKean-Vlasov interpretation of Navier-Stokes equations with external force field in the whole space, by associating with local mild $L^p$-solutions of the 3d-vortex equation a generalized nonlinear diffusion with random space-time birth that probabilistically describes creation of rotation in the fluid due to nonconservativeness of the force. We establish a local well-posedness result for this process and a stochastic representation formula for the vorticity in terms of a vector-weighted version of its law after its birth instant. Then we introduce a stochastic system of 3d vortices with mollified interaction and random space-time births, and prove the propagation of chaos property, with the nonlinear process as limit, at an explicit pathwise convergence rate. Convergence rates for stochastic approximation schemes of the velocity and the vorticity fields are also obtained. We thus extend and refine previous results on the probabilistic interpretation and stochastic approximation methods for the nonforced equation, generalizing also a recently introduced random space-time-birth particle method for the 2d-Navier-Stokes equation with force."}
{"category": "Math", "title": "The bicompletion of the Hausdorff quasi-uniformity", "abstract": "We study conditions under which the Hausdorff quasi-uniformity ${\\mathcal U}_H$ of a quasi-uniform space $(X,{\\mathcal U})$ on the set ${\\mathcal P}_0(X)$ of the nonempty subsets of $X$ is bicomplete. Indeed we present an explicit method to construct the bicompletion of the $T_0$-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It is used to find a characterization of those quasi-uniform $T_0$-spaces $(X,{\\mathcal U})$ for which the Hausdorff quasi-uniformity $\\widetilde{{\\mathcal U}}_H$ of their bicompletion $(\\widetilde{X},{\\widetilde{\\mathcal U}})$ on ${\\mathcal P}_0(\\widetilde{X})$ is bicomplete."}
{"category": "Math", "title": "Partial hyperbolicity far from homoclinic bifurcations", "abstract": "We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in arXiv:math/0605387."}
{"category": "Math", "title": "Quantum Pieri rules for isotropic Grassmannians", "abstract": "We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations."}
{"category": "Math", "title": "Solvability of elliptic systems with square integrable boundary data", "abstract": "We consider second order elliptic divergence form systems with complex measurable coefficients $A$ that are independent of the transversal coordinate, and prove that the set of $A$ for which the boundary value problem with $L_2$ Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when $A$ is either Hermitean, block or constant. Our methods apply to more general systems of PDEs and as an example we prove perturbation results for boundary value problems for differential forms."}
{"category": "Math", "title": "RCF3: Map-Code Interpretation via Closure", "abstract": "For a (minimal) Arithmetical theory with higher Order Objects, i.e. a (minimal) Cartesian closed arithmetical theory -- coming as such with the corresponding closed evaluation -- we interprete here map codes, out of [A,B] say,into these maps \"themselves\", coming as elements (\"names\") within hom-Objects B^A. The interpretation (family) uses a Chain of Universal Objects U_n, one for each Order stratum with respect to \"higher\" Order of the Objects. Combined with closed, axiomatic evaluation, this interpretation family gives code-self-evaluation. Via the usual diagonal argument, Antinomie RICHARD then can be formalised within minimal higher Order (Cartesian closed) arithmetical theory, and yields this way inconsistency for all of its extensions, in particular for set theories as ZF, of the Elementary Theory of (higher Order) Topoi with Natural Numbers Object as considered by FREYD, as well as already for the Theory of Cartesian Closed Categories with NNO considered by LAMBEK and SCOTT."}
{"category": "Math", "title": "Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups", "abstract": "We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure $\\mu$ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the \"Lie algebra\" of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the $L^{2}(\\nu)$-closure of holomorphic polynomials by their values on the Cameron-Martin subgroup."}
{"category": "Math", "title": "Un th\\'eor\\`eme \\`a la \"Thom-Sebastiani\" pour les int\\'egrales-fibres", "abstract": "The aim of this article is to prove a Thom-Sebastiani theorem for the asymptotics of the fiber-integrals. This means that we describe the asymptotics of the fiber-integrals of the function $f \\oplus g : (x,y) \\to f(x) + g(y)$ \\ on $(\\mathbb{C}^p\\times \\mathbb{C}^q, (0,0))$ in term of the asymptotics of the fiber-integrals of the holomorphic germs $f : (\\mathbb{C}^p,0) \\to (\\mathbb{C},0)$ and $g : (\\mathbb{C}^q,0) \\to (\\mathbb{C},0)$. This reduces to compute the asymptotics of a convolution $\\Phi_*\\Psi$ from the asymptotics of $\\Phi$ and $\\Psi$ modulo smooth terms. To obtain a precise theorem, giving the non vanishing of expected singular terms in the asymptotic expansion of $f\\oplus g$, we have to compute the constants coming from the convolution process. We show that they are given by rational fractions of Gamma factors. This enable us to show that these constants do not vanish."}
{"category": "Math", "title": "Global Existence Results for the Anisotropic Boussinesq System in Dimension Two", "abstract": "We study the global existence issue for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. We first examine the case where the Navier-Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transportdiffusion equation with diffusion in the horizontal direction only. For the both systems and for arbitrarily large data, we construct global weak solutions `a la Leray. Next, we state global wellposedness results for more regular data. Our results strongly rely on the fact that the diffusion occurs in a direction perpendicular to the buoyancy force."}
{"category": "Math", "title": "A class of optimal stopping problems for Markov processes", "abstract": "Our purpose is to study a particular class of optimal stopping problems for Markov processes. We justify the value function convexity and we deduce that there exists a boundary function such that the smallest optimal stopping time is the first time when the Markov process passes over the boundary depending on time. Moreover, we propose a method to find the optimal boundary function."}
{"category": "Math", "title": "Asymptotic behavior of solutions to Schr\\\"odinger equations near an isolated singularity of the electromagnetic potential", "abstract": "Asymptotics of solutions to Schroedinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order -1."}
{"category": "Math", "title": "Extreme Value Distributions for some classes of Non-Uniformly Partially Hyperbolic Dynamical Systems", "abstract": "In this note, we obtain verifiable sufficient conditions for the extreme value distribution for a certain class of skew product extensions of non-uniformly hyperbolic base maps. We show that these conditions, formulated in terms of the decay of correlations on the product system and the measure of rapidly returning points on the base, lead to a distribution for the maximum of $\\Phi(p) = -\\log(d(p, p_0))$ that is of the first type. In particular, we establish the Type I distribution for $S^1$ extensions of piecewise $C^2$ uniformly expanding maps of the interval, non-uniformly expanding maps of the interval modeled by a Young Tower, and a skew product extension of a uniformly expanding map with a curve of neutral points."}
{"category": "Math", "title": "Quadratic Twists of Elliptic Curves with Small Selmer Rank", "abstract": "Given an elliptic curve E over the rational with no rational 2-torsion points, we prove the existence of a quadratic twist of E for which the 2-Selmer rank is less than or equal to 1. By the author's earlier result, we establish a lower bound on the number of D's for which the twists E(D) have 2-Selmer rank <= 1. We include in the introduction our (brief) opinion about why it is supposed to be hard to push our technique to make the Selmer group trivial."}
{"category": "Math", "title": "Inversion Formulas for the Dunkl Intertwining Operator and Its Dual on Spaces of Functions and Distributions", "abstract": "In this paper we prove inversion formulas for the Dunkl intertwining operator $V_k$ and for its dual ${}^tV_k$ and we deduce the expression of the representing distributions of the inverse operators $V_k^{-1}$ and ${}^tV_k^{-1}$, and we give some applications."}
{"category": "Math", "title": "A globally convergent matricial algorithm for multivariate spectral estimation", "abstract": "In this paper, we first describe a matricial Newton-type algorithm designed to solve the multivariable spectrum approximation problem. We then prove its global convergence. Finally, we apply this approximation procedure to multivariate spectral estimation, and test its effectiveness through simulation. Simulation shows that, in the case of short observation records, this method may provide a valid alternative to standard multivariable identification techniques such as MATLAB's PEM and MATLAB's N4SID."}
{"category": "Math", "title": "Non annulation des fonctions $L$ des formes modulaires de Hilbert en le point central", "abstract": "Birch and Swinnerton-Dyer conjecture allows for sharp estimates on the rank of certain abelian varieties defined over $ \\Q$. in the case of the jacobian of the modular curves, this problem is equivalent to the estimation of the order of vanishing at 1/2 of $L$-functions of classical modular forms, and was treated, without assuming the Riemann hypothesis, by Kowalski, Michel and VanderKam. The purpose of this paper is to extend this approach in the case of an arbitrary totally real field, which necessitates an appeal of Jacquet-Langlands' theory and the adelization of the problem. To show that the $L$-function (resp. its derivative) of a positive density of forms does not vanish at 1/2, we follow Selberg's method of mollified moments (Iwaniec, Sarnak, Kowalski, Michel and VanderKam among others applied it successfully in the case of classical modular forms). We generalize the Petersson formula, and use it to estimate the first two harmonic moments, this then allows us to match the same unconditional densities as the ones proved over $\\Q$ by Kowalski, Michel and VanderKam. In this setting, there is an additional term, coming from old forms, to control. Finally we convert our estimates for the harmonic moments into ones for the natural moments."}
{"category": "Math", "title": "Identifiability of parameters in latent structure models with many observed variables", "abstract": "While hidden class models of various types arise in many statistical applications, it is often difficult to establish the identifiability of their parameters. Focusing on models in which there is some structure of independence of some of the observed variables conditioned on hidden ones, we demonstrate a general approach for establishing identifiability utilizing algebraic arguments. A theorem of J. Kruskal for a simple latent-class model with finite state space lies at the core of our results, though we apply it to a diverse set of models. These include mixtures of both finite and nonparametric product distributions, hidden Markov models and random graph mixture models, and lead to a number of new results and improvements to old ones. In the parametric setting, this approach indicates that for such models, the classical definition of identifiability is typically too strong. Instead generic identifiability holds, which implies that the set of nonidentifiable parameters has measure zero, so that parameter inference is still meaningful. In particular, this sheds light on the properties of finite mixtures of Bernoulli products, which have been used for decades despite being known to have nonidentifiable parameters. In the nonparametric setting, we again obtain identifiability only when certain restrictions are placed on the distributions that are mixed, but we explicitly describe the conditions."}
{"category": "Math", "title": "The Existence of Soliton Metrics for Nilpotent Lie Groups", "abstract": "We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N,g) satisfy the matrix equation Uv = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac-Moody algebras to analyze the solution spaces for such linear systems. We use these methods to find infinitely many new examples of nilmanifolds with soliton metrics. We give a sufficient condition for a sum of soliton metric nilpotent Lie algebra structures to be soliton, and we use this criterion to show that soliton metrics exist on every naturally graded filiform metric Lie algebra."}
{"category": "Math", "title": "The Generalized Bloch Conjecture for the quotient of certain Calabi-Yau varieties", "abstract": "In this paper, the generalized Bloch Conjecture on zero cycles for the quotient of certain complete intersections with trivial canonical bundle is proved to hold. As an application of Bloch-Srinivas method on the decomposition of the diagonal, we compute the rational coefficient Lawson homology for 1-cycles and codimension two cycles for these quotient varieties. The (Generalized) Hodge Conjecture is proved to hold for codimension two cycles (and hence also for 2-cycles) on these quotient varieties."}
{"category": "Math", "title": "On rack polynomials", "abstract": "We study rack polynomials and the link invariants they define. We show that constant action racks are classified by their generalized rack polynomials and show that $ns^at^a$-quandles are not classified by their generalized quandle polynomials. We use subrack polynomials to define enhanced rack counting invariants, generalizing the quandle polynomial invariants."}
{"category": "Math", "title": "Products of straight spaces", "abstract": "A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight if it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X x Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds: (a) both X and Y are precompact; (b) both X and Y are locally connected; (c) one of the spaces is both precompact and locally connected. In particular, when X satisfies (c), the product X x Z is straight for every straight space Z. Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces."}
{"category": "Math", "title": "Metric groups attached to biextensions", "abstract": "Let $G$ be a connnected, unipotent, perfect group scheme over an algebraically closed field of characteristic p > 0. V. Drinfeld has defined a certain metric group associated to biextensions of $G \\times G$ by the discrete group $Q_p/Z_p$. We prove a certain conjecture of Drinfeld regarding the class of this metric group in the Witt group."}
{"category": "Math", "title": "Lie coalgebras and rational homotopy theory II: Hopf invariants", "abstract": "We give a new solution of the \"homotopy periods\" problem, as highlighted by Sullivan, which places explicit geometrically meaningful formulae first dating back to Whitehead in the context of Quillen's formalism for rational homotopy theory and Koszul-Moore duality. Geometrically, we show that homotopy groups are rationally given by \"generalized linking/intersection invariants\" of cochain data. Moreover, we give a method for determining when two maps from $S^n$ to $X$ are homotopic after allowing for multiplication by some integer. For applications, we investigate wedges of spheres and homogeneous spaces (where homotopy is given by classical linking numbers), and configuration spaces (where homotopy is given by generalized linking numbers); also we propose a generalization of the Hopf invariant one question."}
{"category": "Math", "title": "Reeb vector fields and open book decompositions", "abstract": "We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux. We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjecture. In the case of an open book with pseudo-Anosov monodromy, suppose the boundary of a page of the open book is connected and the fractional Dehn twist coefficient $c={k\\over n}$, where $n$ is the number of prongs along the boundary. If $k\\geq 2$, then there is a well-defined linearized contact homology group. If $k\\geq 3$, then the linearized contact homology is exponentially growing with respect to the action, and every Reeb vector field of the corresponding contact structure admits an infinite number of simple periodic orbits."}
{"category": "Math", "title": "Stationary Solutions of SPDEs and Infinite Horizon BDSDEs with Non-Lipschitz Coefficients", "abstract": "We prove a general theorem that the $L_{\\rho}^2({\\mathbb{R}^{d}};{\\mathbb{R}^{1}})\\otimes L_{\\rho}^2({\\mathbb{R}^{d}};{\\mathbb{R}^{d}})$ valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the $L_{\\rho}^2({\\mathbb{R}^{d}};{\\mathbb{R}^{1}})\\otimes L_{\\rho}^2({\\mathbb{R}^{d}};{\\mathbb{R}^{d}})$ valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result."}
{"category": "Math", "title": "A convolution estimate for two-dimensional hypersurfaces", "abstract": "Given three transversal and sufficiently regular hypersurfaces in R^3 it follows from work of Bennett-Carbery-Wright that the convolution of two L^2 functions supported of the first and second hypersurface, respectively, can be restricted to an L^2 function on the third hypersurface, which can be considered as a nonlinear version of the Loomis-Whitney inequality. We generalize this result to a class of C^{1,beta} hypersurfaces in R^3, under scaleable assumptions. The resulting uniform L^2 estimate has applications to nonlinear dispersive equations."}
{"category": "Math", "title": "Examples of buildings constructed via covering spaces", "abstract": "Covering space theory is used to construct new examples of buildings."}
{"category": "Math", "title": "Solutions of Backward Stochastic Differential Equations on Markov Chains", "abstract": "We consider backward stochastic differential equations (BSDEs) related to finite state, continuous time Markov chains. We show that appropriate solutions exist for arbitrary terminal conditions, and are unique up to sets of measure zero. We do not require the generating functions to be monotonic, instead using only an appropriate Lipschitz continuity condition."}
{"category": "Math", "title": "On Q-factorial terminalizations of nilpotent orbits", "abstract": "In a recent preprint, Y. Namikawa proposed a conjecture on Q-factorial terminalizations and their birational geometry of nilpotent orbits. He proved his conjecture for classical simple Lie algebras. In this note, we prove his conjecture for exceptional simple Lie algebras. For the birational geometry, contrary to the classical case, two new types of Mukai flops appear."}
{"category": "Math", "title": "Weighted sum formula for multiple zeta values", "abstract": "The sum formula is a basic identity of multiple zeta values that expresses a Riemann zeta value as a homogeneous sum of multiple zeta values of a given dimension. This formula was already known to Euler in the dimension two case, conjectured in the early 1990s for higher dimensions and then proved by Granville and Zagier independently. Recently a weighted form of Euler's formula was obtained by Ohno and Zudilin. We generalize it to a weighted sum formula for multiple zeta values of all dimensions."}
{"category": "Math", "title": "Classification of Local Singularities on Torus Curves of Type (2,5)", "abstract": "In this paper, we consider curves of degree 10 of torus type (2,5), C : f_5(x, y)^2 + f_2(x, y)^5 = 0. Assume that f_2(0, 0) = f_5(0, 0) = 0. Then O = (0, 0) is a singular point of C which is called an inner singularity. In this paper, we give a topological classification of singularities of (C,O)."}
{"category": "Math", "title": "Quantum Traces in Quantum Teichm\\\"uller Theory", "abstract": "We prove that for the torus with one hole and p greater than or equal to 1 punctures and the sphere with four holes there is a family of quantum trace functions in the quantum Teichm\\\"uller space, analog to the non-quantum trace functions in Teichm\\\"uller space, satisfying the properties proposed by Chekhov and Fock in their paper \"Observables in 3D Gravity and Geodesic Algebras.\""}
{"category": "Math", "title": "Proof of Riemann's zeta-hypothesis", "abstract": "Make an exponential transformation in the integral formulation of Riemann's zeta-function zeta(s) for Re(s) > 0. Separately, in addition make the substitution s -> 1 - s and then transform back to s again using the functional equation. Using residue calculus, we can in this way get two alternative, equivalent series expansions for zeta(s) of order N, both valid inside the \"critical strip\", i e for 0 < Re(s) < 1. Together, these two expansions embody important characteristics of the zeta-function in this range, and their detailed behavior as N tends to infinity can be used to prove Riemann's zeta-hypothesis that the nontrivial zeros of the zeta-function must all have real part 1/2. In addition to the preprint, the arXiv file also contains a discussion of some forty Frequently Asked Questions from readers. Further questions not adequately dealt with in the existing FAQ are welcome."}
{"category": "Math", "title": "Galois coverings of weakly shod algebras", "abstract": "We investigate the Galois coverings of weakly shod algebras. For a weakly shod algebra not quasi-tilted of canonical type, we establish a correspondence between its Galois coverings and the Galois coverings of its connecting component. As a consequence, we show that a weakly shod algebra is simply connected if and only if its first Hochschild cohomology group vanishes."}
{"category": "Math", "title": "Root polytopes and growth series of root lattices", "abstract": "The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root lattices A_n, C_n, and D_n, and compute their f-and h-vectors. This leads us to recover formulae for the growth series of these root lattices, which were first conjectured by Conway-Mallows-Sloane and Baake-Grimm and proved by Conway-Sloane and Bacher-de la Harpe-Venkov."}
{"category": "Math", "title": "On exceptional collections on some log Del Pezzo surfaces with one singular point", "abstract": "In this paper we construct a full exceptional collection of sheaves on some family of log Del Pezzo surfaces, viewed as a smooth stack. These surfaces can be embedded as a quasi-smooth hypersurfaces into a certain weighted projective space, and they are not toric."}
{"category": "Math", "title": "An algebraic approach to the set of intervals", "abstract": "This paper is devoted to a new approach of the arithmetic of intervals. We present the set of intervals as a normed vector space. We define also a four-dimensional associative algebra whose product gives the product of intervals in any cases. This approach allows to give a notion of divisibility and in some cases an euclidian division. We introduce differential calculus and give some applications."}
{"category": "Math", "title": "On Polyharmonic Interpolation", "abstract": "In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from $C^{\\infty}$ or analytic functions in the ball $B_{R}.$ We prove two main results on the interpolation of $C^{\\infty}$ or analytic functions $f$ in the ball $B_{R}$ by polyharmonic functions $h$ of a given order of polyharmonicity $p.$"}
{"category": "Math", "title": "Semiorthogonal decompositions of derived categories of equivariant coherent sheaves", "abstract": "Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of bounded derived category of G-equivariant coherent sheaves on X into components, equivalent to derived categories of twisted representations of the group. If the group is finite or reductive over the algebraically closed field of zero characteristic, this gives a full exceptional collection in the derived equivariant category. We apply our results to particular varieties such as projective spaces, quadrics, Grassmanians and Del Pezzo surfaces."}
{"category": "Math", "title": "A generalization of the Neron models of Green, Griffiths and Kerr", "abstract": "We generalize a construction of the Neron model for a family of intermediate Jacobians due to Green, Griffiths and Kerr by using the theory of mixed Hodge modules. It is a topological group defined over any partial compactification of the base space, and it `graphs' admissible normal functions. Moreover, there is a stratification of the partial compactification such that the restriction over each stratum is a complex Lie group over the stratum."}
{"category": "Math", "title": "An isomorphism between the fusion algebras of $V_L^+$ and type $D^{(1)}$ level 2", "abstract": "The fusion algebra of the vertex operator algebra $V_L^+$ for a rank 1 even lattice $L$ is explicitly shown to be isomorphic to the fusion algebra of the Kac-Moody algebra of type $D^{(1)}$ at level 2 in almost all cases."}
{"category": "Math", "title": "Anisotropic Navier-Stokes equations in a bounded cylindrical domain", "abstract": "We study the global and local existence and uniqueness of solutions to the Navier-Stokes equations with anisotropic viscosity in a bounded cylindrical domain $Q=\\Omega\\times (0,1)$, where $\\Omega$ is a star-shaped domain in $R^2$. In this paper, we consider the case of homogeneous Dirichlet boundary conditions on the lateral boundary and vanishing normal trace on the top and the bottom."}
{"category": "Math", "title": "Semi-Competing Risks on A Trivariate Weibull Survival Model", "abstract": "A setting of a trivairate survival function using semi-competing risks concept is proposed. The Stanford Heart Transplant data is reanalyzed using a trivariate Weibull distribution model with the proposed survival function."}
{"category": "Math", "title": "Competing Risks Analysis on Times to Commit Crimes", "abstract": "A trivariate Weibull survival model using competing risks concept is applied on studying recidivism of committing 3 types of crimes - sex, violent and others. The assumption of independence of time to commit each type of crimes is relaxed so that the association of the time to recidivism between any two types of crimes can be evaluated. We found that the correlation of time to recidivism between sex crimes and violent crimes are more correlated than other pairs. Probability of experiencing a charged arrest of other crimes is greater than a charged arrest of violent crimes followed by a charged arrest of sex crimes for an individual after release."}
{"category": "Math", "title": "Algebraic group actions on noncommutative spectra", "abstract": "Let G be an affine algebraic group and let R be an associative algebra with a rational action of G by algebra automorphisms. We study the induced G-action on the spectrum Spec R of all prime ideals of R, viewed as a topological space with the Jacobson-Zariski topology, and on the subspace consisting of all rational ideals of R. Here, a prime ideal P of R is said to be rational if the extended centroid of R/P is equal to the base field. The main themes of the article are local closedness of G-orbits in Spec R and the so-called G-stratification of Spec R. This stratification plays a central role in the recent investigation of algebraic quantum groups, in particular in the work of Goodearl and Letzter. We describe the G-strata in terms of certain commutative spectra. Our principal results are based on prior work of Moeglin & Rentschler and Vonessen. We generalize the theory arbitrary associative algebras while also simplifying some of the earlier proofs."}
{"category": "Math", "title": "Visibility of ideal classes", "abstract": "Cremona, Mazur, and others have studied what they call visibility of elements of Shafarevich-Tate groups of elliptic curves. The analogue for an abelian number field $K$ is capitulation of ideal classes of $K$ in the minimal cyclotomic field containing $K$. We develop a new method to study capitulation and use it and classical methods to compute data with the hope of gaining insight into the elliptic curve case. For example, the numerical data for number fields suggests that visibility of nontrivial Shafarevich-Tate elements might be much more common for elliptic curves of positive rank than for curves of rank 0."}
{"category": "Math", "title": "Families of rationally simply connected varieties over surfaces and torsors for semisimple groups", "abstract": "Under suitable hypotheses, we prove that a form of a projective homogeneous variety $G/P$ defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebro-geometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre's Conjecture II in Galois cohomology for function fields over an algebraically closed field."}
{"category": "Math", "title": "Exact solution of two classes of prudent polygons", "abstract": "Prudent walks are self-avoiding walks on the square lattice which never step into the direction of an already occupied vertex. We study the closed version of these walks, called prudent polygons, where the last vertex is adjacent to the first one. More precisely, we give the half-perimeter generating functions of two subclasses of prudent polygons, which turn out to be algebraic and non-D-finite, respectively."}
{"category": "Math", "title": "Dimension and singularity theory for local rings of finite embedding dimension", "abstract": "In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally, variants of the homological theorems are shown to hold in equal characteristic. This theory is then applied to Noetherian local rings in order to get: (i) over a Cohen-Macaulay local ring, uniform bounds on the Betti numbers of a Cohen-Macaulay module in terms of dimension and multiplicity, and similar bounds for the Bass numbers of a finitely generated module; (ii) a characterization for being respectively analytically unramified, analytically irreducible, unmixed, quasi-unmixed, normal, Cohen-Macaulay, pseudo-rational, or weakly F-regular in terms of certain uniform arithmetic behavior; (iii) in mixed characteristic, the Improved New Intersection Theorem when the residual characteristic or ramification index is large with respect to dimension (and some other numerical invariants)."}
{"category": "Math", "title": "Inductive Limits of Subhomogeneous $C^*$-algebras with Hausdorff Spectrum", "abstract": "We consider unital simple inductive limits of generalized dimension drop C*-algebras They are so-called ASH-algebras and include all unital simple AH-algebras and all dimension drop $C^*$-algebras. Suppose that $A$ is one of these C*-algebras. We show that $A\\otimes Q$ has tracial rank no more than one, where $Q$ is the rational UHF-algebra. As a consequence, we obtain the following classification result: Let $A$ and $B$ be two unital simple inductive limits of generalized dimension drop algebras with no dimension growth. Then $A\\cong B$ if and only if they have the same Elliott invariant."}
{"category": "Math", "title": "CLT Variance Associated with Baxendale's SDE", "abstract": "Simple analysis of the leftmost eigenvalue of Ince's equation (a boundary value problem with periodicity) resolves an open issue surrounding a stochastic Lyapunov exponent. Numerical verification is also provided."}
{"category": "Math", "title": "Sweeping Algebraic Curves for Singular Solutions", "abstract": "Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well understood, but these methods no longer work for general types of singularities. In order not to miss any singular solutions along a path we propose to monitor the determinant of the Jacobian matrix. We examine the operation range of deflation and relate the effectiveness of deflation to the winding number. Computational experiments on systems coming from different application fields are presented."}
{"category": "Math", "title": "Dynamics of the heat semigroup on symmetric spaces", "abstract": "The aim of this paper is to show that the dynamics of $L^p$ heat semigroups ($p>2$) on a symmetric space of non-compact type is very different from the dynamics of the $L^p$ heat semigroups if $p\\leq 2$. To see this, it is shown that certain shifts of the $L^p$ heat semigroups have a chaotic behavior if $p>2$ and that such a behavior is not possible in the cases $p\\leq 2$. These results are compared with the corresponding situation for euclidean spaces and symmetric spaces of compact type where such a behavior is not possible."}
{"category": "Math", "title": "Linear Monotone Subspaces of Locally Convex Spaces", "abstract": "The main focus of this paper is to study multi-valued linear monotone operators in the contexts of locally convex spaces via the use of their Fitzpatrick and Penot functions. Notions such as maximal monotonicity, uniqueness, negative-infimum, and (dual-) representability are studied and criteria are provided."}
{"category": "Math", "title": "An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum", "abstract": "We construct an orientable holomorphic quadratic differential on a Riemann surface of genus 4 whose SL(2,R)-orbit is closed and has a highly degenerate Kontsevich - Zorich spectrum. This example is related to a previous similar construction in genus 3 by the first author."}
{"category": "Math", "title": "Deformations of algebroid stacks", "abstract": "In this paper we consider deformations of an algebroid stack on an etale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe."}
{"category": "Math", "title": "Fusion subcategories of representation categories of twisted quantum doubles of finite groups", "abstract": "We describe all fusion subcategories of the representation category of a twisted quantum double of a finite group. In view of the fact that every group-theoretical braided fusion category can be embedded into a representation category of a twisted quantum double of a finite group, this gives a complete description of all group-theoretical braided fusion categories. We describe the lattice and give formulas for some invariants of the fusion subcategories of representation category of a twisted quantum double of a finite group. We also give a characterization of group-theoretical braided fusion categories as equivariantizations of pointed categories."}
{"category": "Math", "title": "Regular and Biregular module algebras", "abstract": "Motivated by the study of von Neumann regular skew groups as carried out by Alfaro, Ara and del Rio in 1995 we investigate regular and biregular Hopf module algebras. If $A$ is an algebra with an action by an affine Hopf algebra $H$, then any $H$-stable left ideal of $A$ is a direct summand if and only if $A^H$ is regular and the invariance functor $(-)^H$ induces an equivalence of $A^H$-Mod to the Wisbauer category of $A$ as $A# H$-module. Analogously we show a similar statement for the biregularity of $A$ relative to $H$ where $A^H$ is replaced by $R=Z(A)\\cap A^H$ using the module theory of $A$ as a module over $A\\otimes A^{op} \\bowtie H$ the envelopping Hopf algebroid of $A$ and $H$. We show that every two-sided $H$-stable ideal of $A$ is generated by a central $H$-invariant idempotent if and only if $R$ is regular and $A_m$ is $H$-simple for all maximal ideals $m$ of $R$. Further sufficient conditions are given for $A# H$ and $A^H$ to be regular."}
{"category": "Math", "title": "When $\\delta$-semiperfect rings are semiperfect", "abstract": "Zhou defined $\\delta$-semiperfect rings as a proper generalization of semiperfect rings. The purpose of this paper is to discuss relative notions of supplemented modules and to show that the semiperfect rings are precisely the semilocal rings which are $\\delta$-supplemented. Module theoretic version of our results are obtained."}
{"category": "Math", "title": "The automorphism group of accessible groups", "abstract": "In this article, we study the outer automorphism group of a group G decomposed as a finite graph of group with finite edge groups and finitely generated vertex groups with at most one end. We show that Out(G) is essentially obtained by taking extensions of relative automorphism groups of vertex groups, groups of Dehn twists and groups of automorphisms of free products. We apply this description and obtain a criterion for Out(G) to be finitely presented, as well as a necessary and sufficient condition for Out(G) to be finite. Consequences for hyperbolic groups are discussed."}
{"category": "Math", "title": "Estimating Speed and Damping in the Stochastic Wave Equation", "abstract": "A parameter estimation problem is considered for a one-dimensional stochastic wave equation driven by additive space-time Gaussian white noise. The estimator is of spectral type and utilizes a finite number of the spatial Fourier coefficients of the solution. The asymptotic properties of the estimator are studied as the number of the Fourier coefficients increases, while the observation time and the noise intensity are fixed."}
{"category": "Math", "title": "Invariants, Kronecker Products, and Combinatorics of Some Remarkable Diophantine Systems (Extended Version)", "abstract": "This work lies across three areas (in the title) of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link consists of a single formal power series with a multifaced interpretation. The deeper exploration of this link yielded results as well as methods for solving some numerical problems in each of these separate areas."}
{"category": "Math", "title": "Fourier transforms of spherical distributions on compact symmetric spaces", "abstract": "In our previous articles \"A local Paley-Wiener theorem for compact symmetric spaces\", Adv. Math. 218 (2008), 202--215, and \"Fourier series on compact symmetric spaces\" (submitted) we studied Fourier series on a compact symmetric space M=U/K. In particular, we proved a Paley-Wiener type theorem for the smooth functions on M, which have sufficiently small support and are K-invariant, respectively K-finite. In this article we extend those results to K-invariant distributions on M. We show that the Fourier transform of a distribution, which is supported in a sufficiently small ball around the base point, extends to a holomorphic function of exponential type. We describe the image of the Fourier transform in the space of holomorphic functions. We characterize the singular support of a distribution in terms of its Fourier transform. Finally, we use the Paley-Wiener theorem to characterize the distributions of small support, which are in the range of a given invariant differential operator."}
{"category": "Math", "title": "Lie algebroid structures on double vector bundles and representation theory of Lie algebroids", "abstract": "A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids."}
{"category": "Math", "title": "Moduli of bundles on exotic del Pezzo orders", "abstract": "Maximal orders of rank 4 on the projective plane, ramified on a smooth plane quartic are examples of exotic del Pezzo orders. We compute the possible Chern classes for line bundles on such orders and show the moduli space of line bundles with minimal second Chern class is either a point or a smooth genus two curve."}
{"category": "Math", "title": "The Cyclotomic Birman-Murakami-Wenzl Algebras", "abstract": "----- Please see the pdf file for the actual abstract and important remarks, which could not be put here due to the arXiv length restrictions. ----- This thesis presents a study of the cyclotomic BMW (Birman-Murakami-Wenzl) algebras, introduced by Haring-Oldenburg as a generalization of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. They are shown to be free of rank k^n (2n-1)!! and to have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. Furthermore, the cyclotomic BMW algebras are proven to be cellular, in the sense of Graham and Lehrer. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007."}
{"category": "Math", "title": "Kergin Approximation in Banach Spaces", "abstract": "We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges."}
{"category": "Math", "title": "Uniqueness for the vortex-wave system when the vorticity is constant near the point vortex", "abstract": "We prove uniqueness for the vortex-wave system with a single point vortex introduced by Marchioro and Pulvirenti in the case where the vorticity is initially constant near the point vortex. Our method relies on the Eulerian approach for this problem and in particular on the formulation in terms of the velocity."}
{"category": "Math", "title": "The half-twist for U_q(g) representations", "abstract": "We introduce the notion of a half-ribbon Hopf algebra, which is a ribbon Hopf algebra along with a distinguished element $t$ corresponding to twisting a ribbon by 180 degrees (the half-twist). We show that U_q(g) is a (topological) half-ribbon Hopf algebra, but only if one uses a modified ribbon element. We then discuss some consequences of using this modified ribbon element."}
{"category": "Math", "title": "Using global invariant manifolds to understand metastability in Burgers equation with small viscosity", "abstract": "The large-time behavior of solutions to Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family of solutions known as diffusive N-waves before finally converging to a stable self-similar diffusion wave. More precisely, it is shown that in terms of similarity, or scaling, variables in an algebraically weighted $L^2$ space, the self-similar diffusion waves correspond to a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus, metastability corresponds to a fast transient in which solutions approach this \"metastable\" manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally, convergence to the self-similar diffusion wave."}
{"category": "Math", "title": "A half-twist type formula for the R-matrix of a symmetrizable Kac-Moody algebra", "abstract": "Kirillov-Reshetikhin and Levendorskii-Soibelman developed a formula for the universal R-matrix of the form R=(X^{-1} \\otimes X^{-1}) \\Delta(X). The action of X on a representation V permutes weight spaces according to the longest element in the Weyl group, so is only defined in finite type. We give a similar formula which is valid for the quantized universal enveloping algebra of any symmetrizable Kac-Moody algebra. This is done by replacing the action of X on V with an endomorphism that preserves weight spaces, but which is bar-linear instead of linear."}
{"category": "Math", "title": "Bundles on non-proper schemes: representability", "abstract": "Let X be a proper scheme over a field k which satisfies Serre's condition S2 and G a reductive group over k. We prove that the functor of principal G-bundles defined away from a non-fixed closed subset in X of codimension at least 3, is an algebraic stack in the sense of Artin."}
{"category": "Math", "title": "On the pre-image of a point under an isogeny", "abstract": "Given a rational point on a curve in a rational isogeny class, a natural question concerns the field of definition of its pre-images. The multiplication by m endomorphism is a powerful and much-used tool. The pre-images for this map are found by factorizing a monic polynomial of degree m^2. For m = 2, Everest and King gave examples where the existence of a quadratic factor coincided with the existence of a rational pre-image via a 2-isogeny. Nelson Stephens asked if this always happens and the question is answered in the affirmative. It is also shown that the analogue for m = 3 can only be false when there exists a rational point of order three and a small number of counterexamples are found. The results are proven over any field with characteristic not two or three."}
{"category": "Math", "title": "A Ring Isomorphism and corresponding Pseudoinverses", "abstract": "This paper studies the set of $n\\times n$ matrices for which all row and column sums equal zero. By representing these matrices in a lower dimensional space, it is shown that this set is closed under addition and multiplication, and furthermore is isomorphic to the set of arbitrary $(n-1)\\times (n-1)$ matrices. The Moore-Penrose pseudoinverse corresponds with the true inverse, (when it exists), in this lower dimension and an explicit representation of this pseudoinverse in terms of the lower dimensional space is given. This analysis is then extended to non-square matrices with all row or all column sums equal to zero."}
{"category": "Math", "title": "Modeling the creative process of the mind by prime numbers and a simple proof of the Riemann Hypothesis", "abstract": "Numbers (positive integers) are the most fundamental creatures of the human mind and the foundation to the scientific understanding of nature. Some mathematicians have suspected a link between prime numbers and secrets of creation. Understanding creativity may help resolve the deepest mysteries of primes. The algorithm that programs the mind and makes the mind creative must be sufficient for the mind to create primes. I found that primes are directly linked to the creation algorithm of the mind. The essence of primes is the duality of uniqueness and uniformity together with the creation algorithm of the mind. The creative process of the mind is lawfully determined but the outcome is unpredictable. The mathematical equivalent or model of this process is the creation of primes. Primes have the inherent property of unpredictability but can be generated by the creation algorithm of the mind, termed the Prime Law, via a fully deterministic lawful process. This new understanding of the essence of primes can deduce some of the best-known properties of primes, including the Riemann Hypothesis (RH). Understanding human creativity is obviously the most fundamental of all scientific enquiries. That this understanding can directly lead to a solution to the RH, widely considered the most important unsolved problem in mathematics, shows a deep connection between creativity and mathematics."}
{"category": "Math", "title": "C*-Algebras over Topological Spaces: Filtrated K-Theory", "abstract": "We define the filtrated K-theory of a C*-algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated K-theory. For finite spaces with totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe a space with four points and two C*-algebras over this space in the bootstrap class that have isomorphic filtrated K-theory but are not KK(X)-equivalent. For this particular space, we enrich filtrated K-theory by another K-theory functor, so that there is again a Universal Coefficient Theorem. Thus the enriched filtrated K-theory is a complete invariant for purely infinite, stable C*-algebras with this particular spectrum and belonging to the appropriate bootstrap class."}
{"category": "Math", "title": "Concentration inequalities for Markov processes via coupling", "abstract": "We obtain moment and Gaussian bounds for general Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling time exists, then we obtain a variance inequality. If a moment of order 1+epsilon of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order 1+epsilon is finite uniformly in the starting point of the coupling, we obtain a Gaussian bound. We illustrate the general results with house of cards processes, in which both uniform and non-uniform behavior of moments of the coupling time can occur."}
{"category": "Math", "title": "Lipschitz behavior of the robust regularization", "abstract": "To minimize or upper-bound the value of a function \"robustly\", we might instead minimize or upper-bound the \"epsilon-robust regularization\", defined as the map from a point to the maximum value of the function within an epsilon-radius. This regularization may be easy to compute: convex quadratics lead to semidefinite-representable regularizations, for example, and the spectral radius of a matrix leads to pseudospectral computations. For favorable classes of functions, we show that the robust regularization is Lipschitz around any given point, for all small epsilon > 0, even if the original function is nonlipschitz (like the spectral radius). One such favorable class consists of the semi-algebraic functions. Such functions have graphs that are finite unions of sets defined by finitely-many polynomial inequalities, and are commonly encountered in applications."}
{"category": "Math", "title": "Distributions that are both log-symmetric and R-symmetric", "abstract": "Two concepts of symmetry for the distributions of positive random variables $Y$ are log-symmetry (symmetry of the distribution of $\\log Y$) and R-symmetry [7]. In this paper, we characterise the distributions that have both properties, which we call doubly symmetric. It turns out that doubly symmetric distributions constitute a subset of those distributions that are moment-equivalent to the lognormal distribution. They include the lognormal, some members of the Berg/Askey class of distributions, and a number of others for which we give an explicit construction (based on work of A.J. Pakes) and note some properties; Stieltjes classes, however, are not doubly symmetric."}
{"category": "Math", "title": "Degenerating curves and surfaces: first results", "abstract": "Let $\\mathcal S\\to\\mathbb A^1$ be a smooth family of surfaces whose general fibre is a smooth surface of $\\mathbb P^3$ and whose special fibre has two smooth components, intersecting transversally along a smooth curve $R$. We consider the Universal Severi-Enriques variety $\\mathcal V$ on $\\mathcal S\\to\\mathbb A^1$. The general fibre of $\\mathcal V$ is the variety of curves on $\\mathcal S_t$ in the linear system $|\\mathcal O_{\\mathcal S_t}(n)|$ with $k$ cusps and $\\delta$ nodes as singularities. Our problem is to find all irreducible components of the special fibre of $\\mathcal V$. In this paper, we consider only the cases $(k,\\delta)=(0,1)$ and $(k,\\delta)=(1,0)$. In particular, we determine all singular curves on the special fibre of $\\mathcal S$ which, counted with the right multiplicity, are a limit of 1-cuspidal curves on the general fibre of $\\mathcal S$."}
{"category": "Math", "title": "Principal noncommutative torus bundles", "abstract": "In this paper we study continuous bundles of C*-algebras which are non-commutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally trivial with respect to a suitable bundle version of bivariant K-theory (denoted RKK-theory) due to Kasparov. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal torus-bundles with H-flux, as studied by Mathai and Rosenberg which possess \"classical\" T-duals."}
{"category": "Math", "title": "On doubling inequalities for elliptic systems", "abstract": "We prove doubling inequalities for solutions of elliptic systems with an iterated Laplacian as diagonal principal part and for solutions of the Lame' system of isotropic linearized elasticity. These inequalities depend on global properties of the solutions."}
{"category": "Math", "title": "Compactification de varietes de Siegel aux places de mauvaise reduction", "abstract": "We construct arithmetic toroidal compactifications of the moduli stack of principally polarized abelian varieties with parahoric level structure. To this end, we extend the methods of Faltings and Chai to a case of bad reduction. ----- Nous construisons des compactifications toroidales arithm\\'etiques du champ de modules des vari\\'et\\'es ab\\'eliennes principalement polaris\\'ees munies d'une structure de niveau parahorique. Pour ce faire, nous \\'etendons la m\\'ethode de Faltings et Chai \\`a un cas de mauvaise r\\'eduction."}
{"category": "Math", "title": "Fibrations with noncommutative fibers", "abstract": "We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C*-algebra bundles. We then derive an analogue of the Leray-Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature."}
{"category": "Math", "title": "Varieties swept out by grassmannians of lines", "abstract": "We classify complex projective varieties of dimension $2r \\geq 8$ swept out by a family of codimension two grassmannians of lines $\\mathbb{G}(1,r)$. They are either fibrations onto normal surfaces such that the general fibers are isomorphic to $\\G(1,r)$ or the grassmannian $\\mathbb{G}(1,r+1)$. The cases $r=2$ and $r=3$ are also considered in the more general context of varieties swept out by codimension two linear spaces or quadrics."}
{"category": "Math", "title": "Smoothing nodal Calabi-Yau n-folds", "abstract": "Let X be an n-dimensional Calabi-Yau with ordinary double points, where n is odd. Friedman showed that for n=3 the existence of a smoothing of X implies a specific type of relation between homology classes on a resolution of X. (The converse is also true, due to work of Friedman, Kawamata and Tian.) We sketch a more topological proof of this result, and then extend it to higher dimensions. For n>3 the \"Yukawa product\" on the middle dimensional (co)homology plays an unexpected role. We also discuss a converse, proving it for nodal Calabi-Yau hypersurfaces in projective space."}
{"category": "Math", "title": "Bounds on the volume entropy and simplicial volume in Ricci curvature $L^p$ bounded from below", "abstract": "Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $\\pi:\\bar{M}\\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of $f\\o\\pi$ on the geodesic balls of $\\bar{M}$ are comparable to the mean of $f$ on $M$. Combined with logarithmic volume estimates, this implies bounds on several topological invariants (volume entropy, simplicial volume, first Betti number and presentations of the fundamental group) in Ricci curvature $L^p$-bounded from below."}
{"category": "Math", "title": "Infinitesimal Variations of Hodge Structure at Infinity", "abstract": "By analyzing the local and infinitesimal behavior of degenerating polarized variations of Hodge structure the notion of infinitesimal variation of Hodge structure at infinity is introduced. It is shown that all such structures can be integrated to polarized variations of Hodge structure and that, conversely, all are limits of infinitesimal variations of Hodge structure (IVHS) at finite points. As an illustration of the rich information encoded in this new structure, some instances of the maximal dimension problem for this type of infinitesimal variation are presented and contrasted with the \"classical\" case of IVHS at finite points."}
{"category": "Math", "title": "The Hermitian Laplace Operator on Nearly K\\\"ahler Manifolds", "abstract": "The moduli space NK of infinitesimal deformations of a nearly K\\\"ahler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6-dimensional homogeneous nearly K\\\"ahler manifolds. It turns out that the nearly K\\\"ahler structure is rigid except for the flag manifold F(1,2)=SU_3/T^2, which carries an 8-dimensional moduli space of infinitesimal nearly K\\\"ahler deformations, modeled on the Lie algebra su_3 of the isometry group."}
{"category": "Math", "title": "Maximal totally complex submanifolds of $\\mathbb{H}\\mathbb{P}^n$: homogeneity and normal holonomy", "abstract": "We prove that a maximal totally complex submanifold $N^{2n}$ of the quaternionic projective space $\\mathbb{H}\\mathbb{P}^n$ ($n\\geq 2$) is a parallel submanifold, provided one of the following conditions is satisfied: (1) $N$ is the orbit of a compact Lie group of isometries, (2) the restricted normal holonomy is a proper subgroup of ${\\rm U}(n)$."}
{"category": "Math", "title": "Euler characteristic and quadrilaterals of normal surfaces", "abstract": "Let $M$ be a compact 3-manifold with a triangulation $\\tau$. We give an inequality relating the Euler characteristic of a surface $F$ normally embedded in $M$ with the number of normal quadrilaterals in $F$. This gives a relation between a topological invariant of the surface and a quantity derived from its combinatorial description. Secondly, we obtain an inequality relating the number of normal triangles and normal quadrilaterals of $F$, that depends on the maximum number of tetrahedrons that share a vertex in $\\tau$."}
{"category": "Math", "title": "Simultaneous Stabilization in $A_\\mathbb{R}(\\mathbb{D})$", "abstract": "In this note we study the problem of simultaneous stabilization for the algebra $A_\\R(\\D)$. Invertible pairs $(f_j,g_j)$, $j=1,..., n$, in a commutative unital algebra are called \\textit{simultaneously stabilizable} if there exists a pair $(\\alpha,\\beta)$ of elements such that $\\alpha f_j+\\beta g_j$ is invertible in this algebra for $j=1,..., n$. For $n=2$, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_\\R(\\D)$ has stable rank two, we are faced here with a different situation. When $n=2$, necessary and sufficient conditions are given so that we have simultaneous stability in $A_\\R(\\D)$. For $n\\geq 3$ we show that under these conditions simultaneous stabilization is not possible and further connect this result to the question of which pairs $(f,g)$ in $A_\\R(\\D)^2$ are totally reducible; that is, for which pairs do there exist two units $u$ and $v$ in $A_\\R(\\D)$ such that $uf+vg=1$."}
{"category": "Math", "title": "Periodic perturbations with delay of autonomous differential equations on manifolds", "abstract": "We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincar\\'e $T$-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index."}
{"category": "Math", "title": "Finite groups of units and their composition factors in the integral group rings of the groups $\\text{PSL}(2,q)$", "abstract": "Let $G$ denote the projective special linear group $\\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\\mathbb{Z}G$ of $G$ is isomorphic to a subgroup of $G$. Furthermore, it is shown that a composition factor of a finite subgroup of $V(\\mathbb{Z}G)$ is isomorphic to a subgroup of $G$."}
{"category": "Math", "title": "Incompressibility and normal minimal surfaces", "abstract": "In this paper we describe a procedure for refining the given triangulation of a 3-manifold that scales the PL-metric according to a given weight function while creating no new normal surfaces. It is known that an incompressible surface $F$ in a triangulated 3-manifold $M$ is isotopic to a normal surface that is of minimal PL-area in the isotopy class of $F$. Using the above scaling refinement we prove the converse. If $F$ is a surface in a closed 3-manifold $M$ such that for any triangulation $\\tau$ of $M$, $F$ is isotopic to a $\\tau$-normal surface $F(\\tau)$ that is of minimal PL-area in its isotopy class, then we show that $F$ is incompressible."}
{"category": "Math", "title": "Global attractors for doubly nonlinear evolution equations with non-monotone perturbations", "abstract": "This paper proposes an abstract theory concerned with dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semi-group approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation whose potential is divided into a convex part and a non-convex part as well as a semilinear parabolic equation with a nonlinear term involving gradients."}
{"category": "Math", "title": "Skew Invariant Theory of Symplectic Groups, Pluri-Hodge Groups and 3-Manifold Invariants", "abstract": "This article deals with a number of topics which are, somewhat surprisingly, related. Firstly, the fundamental theorem of skew invariant theory for the symplectic group giving the generators and relations of symplectic invariants is established. The relations are the so called P_n relations which appear in the study of certain 3-manifold invariants. Next a class of cohomology groups are introduced, called Pluri-Hodge groups (somewhat in keeping with the notion of pluri-canonical groups). These are Dolbeault groups on a complex manifold X with values in tensor powers of sheaves of holomorphic forms of various degrees. By Riemann-Roch one shows that knowledge of the Pluri-Hodge groups gives precise formulae for all Chern numbers of the manifold. When X is holomorphic symplectic the Pluri-Hodge groups form representations of Sp(g) where g counts the number of tensor products. Still with X holomorphic symplectic, the Pluri-Hodge groups are vectors in the vector space H_g(X) of states in the topological field theory of Rozansky and Witten for a 3-manifold with boundary a Riemann surface of genus g. A formulation of the Murakami-Ohtsuki invariants, due to Sawon, allows us to show that quotients of the spaces of trivalent graphs B_g carry representations of the symplectic group. There is a weight system W_X: B_g -> H_g(X) and we show that W_X preserves the various symplectic group actions."}
{"category": "Math", "title": "On real analytic Banach manifolds", "abstract": "Let $X$ be a real Banach space with an unconditional basis (e.g., $X=\\ell_2$ Hilbert space), $\\Omega\\subset X$ open, $M\\subset\\Omega$ a closed split real analytic Banach submanifold of $\\Omega$, $E\\to M$ a real analytic Banach vector bundle, and ${\\Cal A}^E\\to M$ the sheaf of germs of real analytic sections of $E\\to M$. We show that the sheaf cohomology groups $H^q(M,{\\Cal A}^E)$ vanish for all $q\\ge1$, and there is a real analytic retraction $r:U\\to M$ from an open set $U$ with $M\\subset U\\subset\\Omega$ such that $r(x)=x$ for all $x\\in M$. Some applications are also given, e.g., we show that any infinite dimensional real analytic Hilbert submanifold of separable affine or projective Hilbert space is real analytically parallelizable."}
{"category": "Math", "title": "$L^p$ spectral theory and heat dynamics of locally symmetric spaces", "abstract": "In this paper we first derive several results concerning the $L^p$ spectrum of arithmetic locally symmetric spaces whose $\\Q$-rank equals one. In particular, we show that there is an open subset of $\\C$ consisting of eigenvalues of the $L^p$ Laplacian if $p <2$ and that corresponding eigenfunctions are given by certain Eisenstein series. On the other hand, if $p>2$ there is at most a discrete set of real eigenvalues of the $L^p$ Laplacian. These results are used in the second part of this paper in order to show that the dynamics of the $L^p$ heat semigroups for $p<2$ is very different from the dynamics of the $L^p$ heat semigroups if $p\\geq 2$."}
{"category": "Math", "title": "The root closure of a ring of mixed characteristic", "abstract": "We define a closure operation for rings of mixed characteristic and verify that the closure is a ring. We then show that this closure produces a ring with good properties with respect to its Fontaine ring and give an example to show that rings that are not closed in this sense do not satisfy these properties."}
{"category": "Math", "title": "On a continuous time game with incomplete information", "abstract": "For zero-sum two-player continuous-time games with integral payoff and incomplete information on one side, one shows that the optimal strategy of the informed player can be computed through an auxiliary optimization problem over some martingale measures. One also characterizes the optimal martingale measures and compute it explicitely in several examples."}
{"category": "Math", "title": "A note on Sierpi\\'{n}ski problem related to triangular numbers", "abstract": "In this note we show that the system of equations t_{x}+t_{y}=t_{p},\\quad t_{y}+t_{z}=t_{q},\\quad t_{x}+t_{z}=t_{r}, where $t_{x}=x(x+1)/2$ is a triangular number, has infinitely many solutions in integers. Moreover we show that this system has rational three-parametric solution. Using this result we show that the system t_{x}+t_{y}=t_{p},\\quad t_{y}+t_{z}=t_{q},\\quad t_{x}+t_{z}=t_{r},\\quad t_{x}+t_{y}+t_{z}=t_{s} has infinitely many rational two-parametric solutions."}
{"category": "Math", "title": "Differential operators on an affine curve: ideal classes and Picard groups", "abstract": "Let X be a smooth complex affine curve, and let R be the space of right ideal classes in the ring D of differential operators on X. We introduce and study a fibration \\gamma : R \\to Pic(X). We relate this fibration to the corresponding one in the classical limit, and derive an integer invariant $ n $ which indexes the decomposition of the fibres of \\gamma into Calogero-Moser spaces (see [BC]). We also study the action of the group Pic(D) on our fibration; and we explain how to define \\gamma in the framework of the Grassmannian description of R due to Cannings and Holland."}
{"category": "Math", "title": "Rational points on certain quintic hypersurfaces", "abstract": "Let $f(x)=x^5+ax^3+bx^2+cx \\in \\Z[x]$ and consider the hypersurface of degree five given by the equation \\cal{V}_{f}: f(p)+f(q)=f(r)+f(s). Under the assumption $b\\neq 0$ we show that there exists $\\Q$-unirational elliptic surface contained in $\\cal{V}_{f}$. If $b=0, a<0$ and $-a\\not\\equiv 2,18,34 \\pmod {48}$ then there exists $\\Q$-rational surface contained in $\\cal{V}_{f}$. Moreover, we prove that for each $f$ of degree five there exists $\\Q(i)$-rational surface contained in $\\cal{V}_{f}$."}
{"category": "Math", "title": "Reciprocity laws \\`a la Iwasawa-Wiles", "abstract": "This paper is a brief survey on explicit reciprocity laws of Artin-Hasse-Iwasawa-Wiles type for the Kummer pairing on local fields."}
{"category": "Math", "title": "Tensor products of recurrent hypercyclic semigroups", "abstract": "We study tensor products of strongly continuous semigroups on Banach spaces that satisfy the hypercyclicity criterion, the recurrent hypercyclicity criterion or are chaotic."}
{"category": "Math", "title": "Integer Knapsacks: Average Behavior of the Frobenius Numbers", "abstract": "The main result of the paper shows that the asymptotic growth of the Frobenius number in average is significantly slower than the growth of the maximum Frobenius number."}
{"category": "Math", "title": "Groupoid normalizers of tensor products", "abstract": "We consider an inclusion $B\\subseteq M$ of finite von Neumann algebras satisfying $B'\\cap M\\subseteq B$. A partial isometry $v\\in M$ is called a groupoid normalizer if $vBv^*, v^*Bv\\subseteq B$. Given two such inclusions $B_i\\subseteq M_i$, $i=1,2$, we find approximations to the groupoid normalizers of $B_1 \\vnotimes B_2$ in $M_1\\vnotimes M_2$, from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis $B_i'\\cap M_i\\subseteq B_i$, $i=1,2$. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries $v\\in M$ satisfying $vBv^*\\subseteq B$ and $v^*v, vv^*\\in B$."}
{"category": "Math", "title": "Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms", "abstract": "We consider nonlinear hyperbolic conservation laws, posed on a differential (n+1)-manifold with boundary referred to as a spacetime, and in which the \"flux\" is defined as a flux field of n-forms depending on a parameter (the unknown variable). We introduce a formulation of the initial and boundary value problem which is geometric in nature and is more natural than the vector field approach recently developed for Riemannian manifolds. Our main assumption on the manifold and the flux field is a global hyperbolicity condition, which provides a global time-orientation as is standard in Lorentzian geometry and general relativity. Assuming that the manifold admits a foliation by compact slices, we establish the existence of a semi-group of entropy solutions. Moreover, given any two hypersurfaces with one lying in the future of the other, we establish a \"contraction\" property which compares two entropy solutions, in a (geometrically natural) distance equivalent to the L1 distance. To carry out the proofs, we rely on a new version of the finite volume method, which only requires the knowledge of the given n-volume form structure on the (n+1)-manifold and involves the {\\sl total flux} across faces of the elements of the triangulations, only, rather than the product of a numerical flux times the measure of that face."}
{"category": "Math", "title": "Limiting distributions and large deviations for random walks in random environments", "abstract": "This thesis concerns the study of random walks in random environments (RWRE). Since there are two levels of randomness for random walks in random environments, there are two different distributions for the random walk that can be studied. The quenched distribution is the law of the random walk conditioned on a given environment. The annealed distribution is the quenched law averaged over all environments. The main results of the thesis fall into two categories: quenched limiting distributions for one-dimensional, transient RWRE and annealed large deviations for multidimensional RWRE. The analysis of the quenched distributions for transient, one-dimensional RWRE falls into two separate cases. First, when an annealed central limit theorem holds, we prove that a quenched central limit theorem also holds but with a random (depending on the environment) centering. In contrast, when the annealed limit distribution is not Gaussian, we prove that there is no quenched limiting distribution for the RWRE. Moreover, we show that for almost every environment, there exist two random (depending on the environment) sequences of times, along which random walk has different quenched limiting distributions. While an annealed large deviation principle for multidimensional RWRE was known previously, very little qualitative information was available about the annealed large deviation rate function. We prove that if the law on environments is non-nestling, then the annealed large deviation rate function is analytic in a neighborhood of its unique zero (which is the limiting velocity of the RWRE)."}
{"category": "Math", "title": "The asymptotic behavior of least pseudo-Anosov dilatations", "abstract": "For a surface $S$ with $n$ marked points and fixed genus $g\\geq2$, we prove that the logarithm of the minimal dilatation of a pseudo-Anosov homeomorphism of $S$ is on the order of $(\\log n)/n$. This is in contrast with the cases of genus zero or one where the order is $1/n$."}
{"category": "Math", "title": "Affine and toric hyperplane arrangements", "abstract": "We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions."}
{"category": "Math", "title": "Stability of holonomicity over quasi-projective varieties", "abstract": "Let $\\V$ be a mixed characteristic complete discrete valuation ring with perfect residue field $k$. We solve Berthelot's conjectures on the stability of the holonomicity over smooth projective formal $\\V$-schemes. Then we build a category of complexes of arithmetic $\\D$-modules over quasi-projective $k$-varieties with bounded, $F$-holonomic cohomology. We get its stability under Grothendieck's six operations."}
{"category": "Math", "title": "On representable graphs, semi-transitive orientations, and the representation numbers", "abstract": "A graph $G=(V,E)$ is representable if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\\in E$ for each $x\\neq y$. If $W$ is $k$-uniform (each letter of $W$ occurs exactly $k$ times in it) then $G$ is called $k$-representable. It was shown that a graph is representable if and only if it is $k$-representable for some $k$. Minimum $k$ for which a representable graph $G$ is $k$-representable is called its representation number. In this paper we give a characterization of representable graphs in terms of orientations. Namely, we show that a graph is representable if and only if it admits an orientation into a so-called \\emph{semi-transitive digraph}. This allows us to prove a number of results about representable graphs, not the least that 3-colorable graphs are representable. We also prove that the representation number of a graph on $n$ nodes is at most $n$, from which one concludes that the recognition problem for representable graphs is in NP. This bound is tight up to a constant factor, as we present a graph whose representation number is $n/2$. We also answer several questions, in particular, on representability of the Petersen graph and local permutation representability."}
{"category": "Math", "title": "Vertex decomposable graphs and obstructions to shellability", "abstract": "Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability."}
{"category": "Math", "title": "On boundedness, existence and uniqueness of strong solutions of the Navier-Stokes Equations in 3 dimensions", "abstract": "In this paper we consider the Navier-Stokes Equations in 3 dimensions in the vorticity formulation in the absence of the external forces. We derive upper bounds on L_{infinity} norm of omega and use them together with the Local Existence and Uniqueness results to show Global Existence and Uniqueness of the solution provided that at t=0, L_{infinity} norm of omega is finite, or L_4 norm of omega is finite."}
{"category": "Math", "title": "Smooth varieties up to A^1-homotopy and algebraic h-cobordisms", "abstract": "We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A^1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A^1-homotopy type. Arithmetic considerations involving the sheaf of A^1-connected components lead us to introduce several different notions of connectedness in A^1-homotopy theory. We provide concrete links between these notions, connectedness of points by chains of affine lines, and various rationality properties of algebraic varieties (e.g., rational connectedness). We introduce the notion of an A^1-h-cobordism, an algebro-geometric analog of the topological notion of h-cobordism, and use it as a tool to produce non-trivial A^1-weak equivalences of smooth proper varieties. Also, we give explicit computations of refined A^1-homotopy invariants, such as the A^1-fundamental sheaf of groups, for some A^1-connected varieties. We observe that the A^1-fundamental sheaf of groups plays a central yet mysterious role in the structure of A^1-h-cobordisms. As a consequence of these observations, we completely solve the classification problem for rational smooth proper surfaces over an algebraically closed field: while there exist arbitrary dimensional moduli of such surfaces, there are only countably many A^1-homotopy types, each uniquely determined by the isomorphism class of its A^1-fundamental sheaf of groups."}
{"category": "Math", "title": "On Banach Spaces containing $l_p$ or $c_0$", "abstract": "We use the Gowers block Ramsey theorem to characterize Banach spaces containing isomorphs of $\\ell_p$ (for some $1 \\leq p < \\infty$) or $c_0$."}
{"category": "Math", "title": "An equivariant index formula for almost-CR manifolds", "abstract": "We consider a consider the case of a compact manifold M, together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients J(E,X) whose properties we describe. When E is equipped with a complex structure, we define a class of symbol mappings in terms of the resulting almost-CR structure that are H-transversally elliptic whenever the action of H is transverse to E. We determine a formula for the H-equivariant index of such symbols that involves only J(E,X) and standard equivariant characteristic classes. This formula generalizes the formula given in arXiv:0712.2431 for the case of a contact manifold."}
{"category": "Math", "title": "A chain complex and Quadrilaterals for normal surfaces", "abstract": "We interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give a proof of an (unpublished) observation independently given by Casson and Rubinstein saying that quadrilaterals determine a normal surface up to vertex linking spheres. We also characterise the quadrilateral coordinates that correspond to a normal surface in a (possibly ideal) triangulation."}
{"category": "Math", "title": "Non-cyclic graph associated with a group", "abstract": "We associate a graph $\\mathcal{C}_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\\{x\\in G | < x,y> \\text{is cyclic for all} y\\in G\\}$ is called the cyclicizer of $G$, and join two vertices if they do not generate a cyclic subgroup. For a simple graph $\\Gamma$, $w(\\Gamma)$ denotes the clique number of $\\Gamma$, which is the maximum size (if it exists) of a complete subgraph of $\\Gamma$. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group $G$ is solvable whenever $w(\\mathcal{C}_G)<31$ and the equality for a non-solvable group $G$ holds if and only if $G/Cyc(G)\\cong A_5$ or $S_5$."}
{"category": "Math", "title": "Interacting multi-class transmissions in large stochastic networks", "abstract": "The mean-field limit of a Markovian model describing the interaction of several classes of permanent connections in a network is analyzed. Each of the connections has a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes of the route. Since several classes of connections going through the nodes of the network are considered, an original mean-field result in a multi-class context is established. It is shown that, as the number of connections goes to infinity, the behavior of the different classes of connections can be represented by the solution of an unusual nonlinear stochastic differential equation depending not only on the sample paths of the process, but also on its distribution. Existence and uniqueness results for the solutions of these equations are derived. Properties of their invariant distributions are investigated and it is shown that, under some natural assumptions, they are determined by the solutions of a fixed-point equation in a finite-dimensional space."}
{"category": "Math", "title": "Finitely presented algebras and groups defined by permutation relations", "abstract": "The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\\sigma (a)} a_{\\sigma (2)} ... a_{\\sigma (n)}$, where $\\sigma$ runs through a subset $H$ of the symmetric group $\\Sym_{n}$ of degree $n$, is introduced. The emphasis is on the case of a cyclic subgroup $H$ of $\\Sym_{n}$ of order $n$. A normal form of elements of the algebra is obtained. It is shown that the underlying monoid, defined by the same (monoid) presentation, has a group of fractions and this group is described. Properties of the algebra are derived. In particular, it follows that the algebra is a semiprimitive domain. Problems concerning the groups and algebras defined by arbitrary subgroups $H$ of $\\Sym_{n}$ are proposed."}
{"category": "Math", "title": "Arithmetic infinite Grassmannians and the induced central extensions", "abstract": "The construction of families of Sato Grassmannians, their determinant line bundles and the extensions induced by them are given. The base scheme is an arbitrary scheme."}
{"category": "Math", "title": "Commutative rings in which every finitely generated ideal is quasi-projective", "abstract": "This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prufer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky's theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz's related conjecture on Gaussian rings. We also establish an analogue of Bazzoni-Glaz results on the transfer of Prufer conditions between a ring and its total ring of quotients. Section 4 examines various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prufer rings."}
{"category": "Math", "title": "Robust Asymptotic Stabilization of Nonlinear Systems with Non-Hyperbolic Zero Dynamics", "abstract": "In this paper we present a general tool to handle the presence of zero dynamics which are asymptotically but not locally exponentially stable in problems of robust nonlinear stabilization by output feedback. We show how it is possible to design locally Lipschitz stabilizers under conditions which only rely upon a partial detectability assumption on the controlled plant, by obtaining a robust stabilizing paradigm which is not based on design of observers and separation principles. The main design idea comes from recent achievements in the field of output regulation and specifically in the design of nonlinear internal models."}
{"category": "Math", "title": "On o-minimal homotopy groups", "abstract": "We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are also semialgebraically homotopic. This result together with the study of semialgebraic homotopy done by H. Delfs and M. Knebusch allows us to develop an o-minimal homotopy theory. In particular, we obtain o-minimal versions of the Hurewicz theorems and the Whitehead theorem."}
{"category": "Math", "title": "Isometric action of SL(2,R) on homogeneous spaces", "abstract": "We investigate the SL(2,R) invariant geodesic curves with the as- sociated invariant distance function in parabolic geometry. Parabolic geom- etry naturally occurs in the study of SL(2,R) and is placed in between the elliptic and the hyperbolic (also known as the Lobachevsky half-plane and 2- dimensional Minkowski half-plane space-time) geometries. Initially we attempt to use standard methods of finding geodesics but they lead to degeneracy in this setup. Instead, by studying closely the two related elliptic and hyperbolic geometries we discover a unified approach to a more exotic and less obvious parabolic case. With aid of common invariants we describe the possible dis- tance functions that turn out to have some unexpected, interesting properties."}
{"category": "Math", "title": "On the field intersection problem of generic polynomials: a survey", "abstract": "Let $k$ be a field of characteristic $\\neq 2$. We survey a general method of the field intersection problem of generic polynomials via formal Tschirnhausen transformation. We announce some of our recent results of cubic, quartic and quintic cases the details of which are to appear elsewhere. In this note, we give an explicit answer to the problem in the cases of cubic and dihedral quintic by using multi-resolvent polynomials."}
{"category": "Math", "title": "A Lindemann-Weierstrass theorem for semiabelian varieties over function fields", "abstract": "We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of Q-linearly independent algebraic numbers are algebraically independent) for commutative algebraic groups G without unipotent quotients, over function fields. We concentrate on solutions to the the differential algebraic relations satisfied by exp from LG to G."}
{"category": "Math", "title": "Pointwise estimates for the Bergman kernel of the weighted Fock space", "abstract": "We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^2(e^{-2\\phi})$ where $\\phi$ is a subharmonic function with $\\Delta \\phi$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $\\Delta \\phi$."}
{"category": "Math", "title": "Decision problems and profinite completions of groups", "abstract": "We consider pairs of finitely presented, residually finite groups $P\\hookrightarrow\\G$ for which the induced map of profinite completions $\\hat P\\to \\hat\\G$ is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not $P$ is isomorphic to $\\G$. We construct pairs for which the conjugacy problem in $\\G$ can be solved in quadratic time but the conjugacy problem in $P$ is unsolvable. Let $\\mathcal J$ be the class of super-perfect groups that have a compact classifying space and no proper subgroups of finite index. We prove that there does not exist an algorithm that, given a finite presentation of a group $\\G$ and a guarantee that $\\G\\in\\mathcal J$, can determine whether or not $\\G\\cong\\{1\\}$. We construct a finitely presented acyclic group $\\H$ and an integer $k$ such that there is no algorithm that can determine which $k$-generator subgroups of $\\H$ are perfect."}
{"category": "Math", "title": "Passage-time moments and hybrid zones for the exclusion-voter model", "abstract": "We study the non-equilibrium dynamics of a one-dimensional interacting particle system that is a mixture of the voter model and the exclusion process. With the process started from a finite perturbation of the ground state Heaviside configuration consisting of 1's to the left of the origin and 0's elsewhere, we study the relaxation time $\\tau$, that is, the first hitting time of the ground state configuration (up to translation). We give conditions for $\\tau$ to be finite and for certain moments of $\\tau$ to be finite or infinite, and prove a result that approaches a conjecture of Belitsky et al. (Bernoulli 7 (2001) 119--144). Ours are the first non-existence-of-moments results for $\\tau$ for the mixture model. Moreover, we give almost sure asymptotics for the evolution of the size of the hybrid (disordered) region. Most of our results pertain to the discrete-time setting, but several transfer to continuous-time. As well as the mixture process, some of our results also cover pure exclusion. We state several significant open problems."}
{"category": "Math", "title": "On a symmetric space attached to polyzeta values", "abstract": "Quickly convergent series are given to compute polyzeta numbers. The formula involves an intricate combination of (generalized) polylogarithms at 1/2. However, the combinatorics has a very simple geometric interpretation: it corresponds with the square map on some symmetric space P."}
{"category": "Math", "title": "Quantum $\\mathrm{SO}(3)$ groups and quantum group actions on $M_2$", "abstract": "Answering a question of Shuzhou Wang we give a description of quantum $\\SO(3)$ groups of Podle\\'s as universal objects. We use this result to give a complete classification of all continuous compact quantum group actions on $M_2$."}
{"category": "Math", "title": "Direct factors of profinite completions and decidability", "abstract": "We consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the profinite completion of $G$. We explain why $A$ need not be a direct factor of a subgroup of finite index in $G$; indeed $G$ need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether $A$ is a direct factor of a subgroup of finite index in $G$."}
{"category": "Math", "title": "On the relation between cluster and classical tilting", "abstract": "Let D be a triangulated category with a cluster tilting subcategory U. The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to support tilting subcategories of D/U, and that, in turn, support tilting subcategories of D/U can be lifted uniquely to maximal 1-orthogonal subcategories of D."}
{"category": "Math", "title": "Fixed point loci of moduli spaces of sheaves on toric varieties", "abstract": "Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves of any dimension on an arbitrary nonsingular toric variety $X$. Using geometric invariant theory (GIT), this allows us to construct explicit moduli spaces of pure equivariant sheaves on $X$ corepresenting natural moduli functors (similar to work of Payne in the case of equivariant vector bundles). The action of the algebraic torus on $X$ lifts to the moduli space of all Gieseker stable sheaves on $X$ and we express its fixed point locus explicitly in terms of moduli spaces of pure equivariant sheaves on $X$. One of the problems arising is to find an equivariant line bundle on the side of the GIT problem, which precisely recovers Gieseker stability. In the case of torsion free equivariant sheaves, we can always construct such equivariant line bundles. As a by-product, we get a combinatorial description of the fixed point locus of the moduli space of $\\mu$-stable reflexive sheaves on $X$. As an application, we show in a sequel how these methods can be used to compute generating functions of Euler characteristics of moduli spaces of $\\mu$-stable torsion free sheaves on nonsingular complete toric surfaces."}
{"category": "Math", "title": "Continuity of ring *-homomorphisms between C*-algebras", "abstract": "The purpose of this short note is to prove that if $A$ and $B$ are unital C*-algebras and $\\phi : A \\to B$ is a unital *-preserving ring homomorphism, then $\\phi$ is contractive; i.e., $\\| \\phi (a) \\| \\leq \\| a \\|$ for all $a \\in A$. (Note that we do not assume $\\phi$ is linear.) We use this result to deduce a number of corollaries as well as characterize the form of such unital *-preserving ring homomorphisms. (This note may be of interest to C*-algebraists as well as algebraists who study noncommutative rings and algebras. It is meant to be accessible to a general mathematician and does not require any prior knowledge of C*-algebras.)"}
{"category": "Math", "title": "A New Bijection Between Forests and Parking Functions", "abstract": "In 1980, G. Kreweras gave a recursive bijection between forests and parking functions. In this paper we construct a nonrecursive bijection from forests onto parking functions, which answers a question raised by R. Stanley. As a by-product, we obtain a bijective proof of Gessel and Seo's formula for lucky statistic on parking functions."}
{"category": "Math", "title": "Estimating the Parameters of Binomial and Poisson Distributions via Multistage Sampling", "abstract": "In this paper, we have developed a new class of sampling schemes for estimating parameters of binomial and Poisson distributions. Without any information of the unknown parameters, our sampling schemes rigorously guarantee prescribed levels of precision and confidence."}
{"category": "Math", "title": "Combinatorial Hopf algebras", "abstract": "We define a \"combinatorial Hopf algebra\" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). The choice of such an isomorphism implies the existence a finer algebraic structure on the Hopf algebra and on the indecomposables (resp. the primitives). For instance a cofree-cocommutative right-sided combinatorial Hopf algebra is completely determined by its primitive part which is a pre-Lie algebra. The key example is the Connes-Kreimer Hopf algebra. The study of all these combinatorial Hopf algebra types gives rise to several good triples of operads. It involves the operads: dendriform, pre-Lie, brace, and variations of them."}
{"category": "Math", "title": "Obstacle problem for SPDE with nonlinear Neumann boundary condition via reflected generalized backward doubly SDEs", "abstract": "This paper is intended to give a probabilistic representation for stochastic viscosity solution of semi-linear reflected stochastic partial differential equations with nonlinear Neumann boundary condition. We use it connection with reflected generalized backward doubly stochastic differential equation."}
{"category": "Math", "title": "Extended Zeilberger's Algorithm for Identities on Bernoulli and Euler Polynomials", "abstract": "We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals."}
{"category": "Math", "title": "Irreducible Lie-Yamaguti algebras", "abstract": "Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their Lie inner derivation algebra are the algebraic counterpart of the isotropy irreducible homogeneous spaces. These systems will be shown to split into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types will be classified and most of them will be shown to be related to a Generalized Tits Construction of Lie algebras."}
{"category": "Math", "title": "A map from Lawson homology to Deligne Cohomology", "abstract": "A natural map from Lawson homology to Deligne cohomology groups for smooth complex projective varieties is constructed by using the Harvey-Lawson spark complexes. We also compare this to Abel-Jacobi type constructions by others."}
{"category": "Math", "title": "Polynomial maps over $p$-adics and residual properties of mapping tori of group endomorphisms", "abstract": "We continue our study of residual properties of mapping tori of free group endomorphisms. In this paper, we prove that each of these groups are virtually residually (finite $p$)-groups for all but finitely many primes$p$. The method involves further studies of polynomial maps over finite fields and $p$-adic completions of number fields."}
{"category": "Math", "title": "Free holomorphic automorphisms of the unit ball of $B(H)^n$", "abstract": "The theory of characteristic functions for row contractions is used to determine the group $Aut(B(H)^n_1)$ of all free holomorphic automorphisms of the unit ball of $B(H)^n$. We show that the noncommutative Poisson transform commutes with the action of the automorphism group $Aut(B(H)^n_1)$. This leads to a characterization of the unitarily implemented automorphisms of the Cuntz-Toeplitz algebra $C^*(S_1,..., S_n)$, which leave invariant the noncommutative disc algebra $\\cA_n$. This result provides new insight into Voiculescu's group of automorphisms of the Cuntz-Toeplitz algebra and reveals new connections with noncommutative multivariable operator theory, especially, the theory of characteristic functions for row contractions and the noncommutative Poisson transforms. We study the isometric dilations and the characteristic functions of row contractions under the action of the automorphism group $Aut(B(H)^n_1)$. This enables us to obtain some results concerning the behavior of the curvature and the Euler characteristic of a row contraction under $Aut(B(H)^n_1)$. We prove a maximum principle for free holomorphic functions on the noncommutative ball $[B(H)^n]_1$ and provide some extensions of the classical Schwarz lemma to our noncommutative setting."}
{"category": "Math", "title": "Linear extension of the Erdos-Heilbronn conjecture", "abstract": "The famous Erdos-Heilbronn conjecture plays an important role in the development of additive combinatorics. In 2007 Z. W. Sun made the following further conjecture (which is the linear extension of the Erdos-Heilbronn conjecture): For any finite subset A of a field F and nonzero elements $a_1,...,a_n$ of F, the set {a_1x_1+...+a_nx_n: x_1,....,x_n are distinct elements of A} has cardinality at least min{p(F)-delta, n(|A|-n)+1}, where the additive order p(F) of the multiplicative identity of F is different from n+1, and delta=0,1 takes the value 1 if and only if n=2 and $a_1+a_2=0$. In this paper we prove this conjecture of Sun when $p(F)\\geq n(3n-5)/2$. We also obtain a sharp lower bound for the cardinality of the restricted sumset {x_1+...+x_n: x_1\\in A_1,...,x_n\\in A_n, and P(x_1,...,x_n)\\not=0}, where $A_1,...,A_n$ are finite subsets of a field F and $P(x_1,...,x_n)$ is a general polynomial over F."}
{"category": "Math", "title": "The Algebra of Formal Twisted Pseudodifferential Symbols and a Noncommutative Residue", "abstract": "We extend the Adler-Manin trace on the algebra of pseudodifferential symbols to a twisted setting."}
{"category": "Math", "title": "Exact solutions to Waring's problem for finite fields", "abstract": "The Waring function $g(k,q)$ measures the difficulty of Waring's problem for $k$th powers in the field of $q$ elements. Its calculation seems to be difficult, and many partial results have been published, notably upper bounds for certain regions of the $k$-$q$-plane. In this paper, we compute the exact value of $g(k,q)$ for two infinite families of exponent-field pairs. In these, $k$ is large compared to $q$. We use a new method of proof that is mainly combinatorial in nature."}
{"category": "Math", "title": "On some Fano--Enriques threefolds", "abstract": "We give a classification of Fano threefolds $X$ with canonical Gorenstein singularities such that $X$ possess a regular involution, which acts freely on some smooth surface in $|-K_X|$, and the linear system $|-K_X|$ gives a morphism which is not an embedding. From this classification one gets, in particular, a description of some natural class of Fano--Enriques threefolds."}
{"category": "Math", "title": "Nonlinear stability of time-periodic viscous shocks", "abstract": "In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green's distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude."}
{"category": "Math", "title": "An overview of abelian varieties in homotopy theory", "abstract": "This is an expository article on the theory of formal group laws in homotopy theory, with the goal of leading to the connection with higher-dimensional abelian varieties and automorphic forms. These are roughly based on a talk at the conference \"New Topological Contexts for Galois Theory and Algebraic Geometry.\""}
{"category": "Math", "title": "Drawing disconnected graphs on the Klein bottle", "abstract": "We prove that two disjoint graphs must always be drawn separately on the Klein bottle, in order to minimize the crossing number of the whole drawing."}
{"category": "Math", "title": "Coxeter group actions on 4F3(1) hypergeometric series", "abstract": "We investigate a certain linear combination $K(\\vec{x})=K(a;b,c,d;e,f,g)$ of two Saalschutzian hypergeometric series of type ${_4}F_3(1)$. We first show that $K(a;b,c,d;e,f,g)$ is invariant under the action of a certain matrix group $G_K$, isomorphic to the symmetric group $S_6$, acting on the affine hyperplane $V=\\{(a,b,c,d,e,f,g)\\in\\Bbb C^7\\colon e+f+g-a-b-c-d=1\\}$. We further develop an algebra of three-term relations for $K(a;b,c,d;e,f,g)$. We show that, for any three elements $\\mu_1,\\mu_2,\\mu_3$ of a certain matrix group $M_K$, isomorphic to the Coxeter group $W(D_6)$ (of order 23040), and containing the above group $G_K$, there is a relation among $K(\\mu_1\\vec{x})$, $K(\\mu_2\\vec{x})$, and $K(\\mu_3\\vec{x})$, provided no two of the $\\mu_j$'s are in the same right coset of $G_K$ in $M_K$. The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in $a,b,c,d,e,f,g$. The set of $({|M_K|/|G_K|\\atop 3})=({32\\atop 3})=4960$ resulting three-term relations may further be partitioned into five subsets, according to the Hamming type of the triple $(\\mu_1,\\mu_2,\\mu_3) $ in question. This Hamming type is defined in terms of Hamming distance between the $\\mu_j$'s, which in turn is defined in terms of the expression of the $\\mu_j$'s as words in the Coxeter group generators. Each three-term relation of a given Hamming type may be transformed into any other of the same type by a change of variable. An explicit example of each of the five types of three-term relations is provided."}
{"category": "Math", "title": "Boundary estimates for positive solutions to second order elliptic equations", "abstract": "Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which guarantee the Hopf-Oleinik type estimates and the boundary Lipschitz estimates for solutions. These conditions are sharp even for harmonic functions."}
{"category": "Math", "title": "Finite simple groups with narrow prime spectrum", "abstract": "We find the nonabelian finite simple groups with order prime divisors not exceeding 1000. More generally, we determine the sets of nonabelian finite simple groups whose maximal order prime divisor is a fixed prime less than 1000. Our results are based on calculations in the computer algebra system GAP."}
{"category": "Math", "title": "Intersection exponents for biased random walks on discrete cylinders", "abstract": "We prove existence of intersection exponents xi(k,lambda) for biased random walks on d-dimensional half-infinite discrete cylinders, and show that, as functions of lambda, these exponents are real analytic. As part of the argument, we prove convergence to stationarity of a time-inhomogeneous Markov chain on half-infinite random paths. Furthermore, we show this convergence takes place at exponential rate, an estimate obtained via a coupling of weighted half-infinite paths."}
{"category": "Math", "title": "A note on Kaehler-Ricci flow", "abstract": "Let $g(t)$ with $t\\in [0,T)$ be a complete solution to the Kaehler-Ricci flow: $\\frac{d}{dt}g_{i\\bar j}=-R_{i\\bar j}$ where $T$ may be $\\infty$. In this article, we show that the curvatures of $g(t)$ is uniformly bounded if the solution $g(t)$ is uniformly equivalet. This result is stronger than the main result in \\v{S}e\\v{s}um \\cite{sesum} within the category of K\\\"ahler-Ricci flow."}
{"category": "Math", "title": "A geometric proof of the classification of complex vector cross product", "abstract": "In this article, we give a geometric proof of the classification of complex vector cross product due to Lee-Leung."}
{"category": "Math", "title": "Negativity of Perelman's Li-Yau-Hamilton type expression", "abstract": "Chau-Tam-Yu has proved the non-positivity of Perelman's new Li-Yau-Hamilton type expression on noncompact manifolds. In this article, we further prove that $v$ is negative if the Ricci flow is not end up with an Euclidean space."}
{"category": "Math", "title": "Some Estimates of Fundamental Solution on noncompact manifolds with time-dependent metrics", "abstract": "In this article, we obtain some further estimates of fundamental solutions comparing to the result of Chau-Tam-Yu and give some applications of the estimates on asymptotic behaviors of fundamental solutions."}
{"category": "Math", "title": "Almost Sure Convergence of Extreme Order Statistics", "abstract": "Let $M_n^{(k)}$ denote the $k$th largest maximum of a sample $(X_1,X_2,...,X_n)$ from parent $X$ with continuous distribution. Assume there exist normalizing constants $a_n>0$, $b_n\\in \\mathbb{R}$ and a nondegenerate distribution $G$ such that $a_n^{-1}(M_n^{(1)}-b_n)\\stackrel{w}{\\to}G$. Then for fixed $k\\in \\mathbb{N}$, the almost sure convergence of \\[\\frac{1}{D_N}\\sum_{n=k}^Nd_n\\mathbb{I}\\{M_n^{(1)}\\le a_nx_1+b_n,M_n^{(2)}\\le a_nx_2+b_n,...,M_n^{(k)}\\le a_nx_k+b_n\\}\\] is derived if the positive weight sequence $(d_n)$ with $D_N=\\sum_{n=1}^Nd_n$ satisfies conditions provided by H\\\"{o}rmann."}
{"category": "Math", "title": "Cobordism categories of manifolds with corners", "abstract": "In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if {Cob}_{d,<k>} is the category whose objets are a fixed dimension d, with corners of codimension less than or equal to k, then we identify the homotopy type of the classifying space B{Cob}_{d,<k>} as the zero space of a homotopy colimit of certain diagram of Thom spectra. We also identify the homotopy type of the corresponding cobordism category when extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss, and the proofs are an adaptation of the their methods. As an application we describe the homotopy type of the category of open and closed strings with a background space X, as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez and Hanbury."}
{"category": "Math", "title": "Differentiability of Banach Spaces via Constructible Sets", "abstract": "the main goal of this paper is to prove that any Banach space X, that every dual ball in X** is weak* -separable, or every weak* -closed convex subset in X** is weak* -separable, or every norm-closed convex set in X* is constructible, admits an equivalent Frechet differentiable norm."}
{"category": "Math", "title": "Convexity of Chebyshev Sets Through Differentiability of Distance Function", "abstract": "In this paper, we study a part of approximation theory that presents the conditions under which a \\Ceby\\sev set in a Banach space is convex. To do so, we use Gateaux differentiability of the distance function."}
{"category": "Math", "title": "Expansive homoclinic classes", "abstract": "We prove that for $C^1$ generic diffeomorphisms, every expansive homoclinic class is hyperbolic."}
{"category": "Math", "title": "On certain permutation groups and sums of two squares", "abstract": "We consider the question of existence of ramified covers over P_1 matching certain prescribed ramification conditions. This problem has already been faced in a number of papers, but we discuss alternative approaches for an existence proof, involving elliptic curves and universal ramified covers with signature. We also relate the geometric problem with finite permutation groups and with the Fermat-Euler Theorem on the representation of a prime as a sum of two squares."}
{"category": "Math", "title": "Information inequalities and a dependent Central Limit Theorem", "abstract": "We adapt arguments concerning information-theoretic convergence in the Central Limit Theorem to the case of dependent random variables under Rosenblatt mixing conditions. The key is to work with random variables perturbed by the addition of a normal random variable, giving us good control of the joint density and the mixing coefficient. We strengthen results of Takano and of Carlen and Soffer to provide entropy-theoretic, not weak convergence."}
{"category": "Math", "title": "On the Combinatorics of the Boros-Moll Polynomials", "abstract": "The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan's Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin and Rota along with an extension of Foata's bijection between Meixner endofunctions and bi-colored permutations, we find a combinatorial proof of the positivity. In fact, from our combinatorial argument one sees that it is essentially the binomial theorem that makes it possible to reduce the double sum to a single sum."}
{"category": "Math", "title": "On quantum semigroup actions on finite quantum spaces", "abstract": "We show that a continuous action of a quantum semigroup $\\mathcal{S}$ on a finite quantum space (finite dimensional $\\mathrm{C}^*$-algebra) preserving a faithful state comes from a continuous action of the quantum Bohr compactification $\\mathfrak{b}\\mathcal{S}$ of $\\mathcal{S}$. Using the classification of continuous compact quantum group actions on $M_2$ we give a complete description of all continuous quantum semigroup actions on this quantum space preserving a faithful state."}
{"category": "Math", "title": "Translated Poisson approximation for Markov chains", "abstract": "The paper is concerned with approximating the distribution of a sum W of n integer valued random variables Y_i, whose distributions depend on the state of an underlying Markov chain X. The approximation is in terms of a translated Poisson distribution, with mean and variance chosen to be close to those of W, and the error is measured with respect to the total variation norm. Error bounds comparable to those found for normal approximation with respect to the weaker Kolmogorov distance are established, provided that the distribution of the sum of the Y_i's between the successive visits of X to a reference state is aperiodic. Without this assumption, approximation in total variation cannot be expected to be good."}
{"category": "Math", "title": "On meromorphic extendibility", "abstract": "Let D be a bounded domain in the complex plane whose boundary bD consists of finitely many pairwise disjoint real analytic simple closed curves. Let f be an integrable function on bD. In the paper we show how to compute the candidates for poles of a meromorphic extension of f through D and thus reduce the question of meromorphic extendibility to the question of holomorphic extendibility. Let A(D) be the algebra of all continuous functions on the closure of D which are holomorphic on D. For continuous functions f on bD we obtain a characterization of meromorphic extendibility in terms of the argument principle: f extends meromorphically through D if and only if there is a nonnegative integer N such that the winding number of Pf+Q along bD is bounded below by -N for all P, Q in A(D) such that Pf+Q has no zero on bD. If this is the case then the meromorphic extension of f has at most N poles in D, counting multiplicity."}
{"category": "Math", "title": "Rough Sets Determined by Quasiorders", "abstract": "In this paper, the ordered set of rough sets determined by a quasiorder relation $R$ is investigated. We prove that this ordered set is a complete, completely distributive lattice. We show that on this lattice can be defined three different kinds of complementation operations, and we describe its completely join-irreducible elements. We also characterize the case in which this lattice is a Stone lattice. Our results generalize some results of J. Pomykala and J. A. Pomykala (1988) and M. Gehrke and E. Walker (1992) in case $R$ is an equivalence."}
{"category": "Math", "title": "Geometric structure in the principal series of the p-adic group G_2", "abstract": "In the representation theory of reductive $p$-adic groups $G$, the issue of reducibility of induced representations is an issue of great intricacy. It is our contention, expressed as a conjecture in [3], that there exists a simple geometric structure underlying this intricate theory. We will illustrate here the conjecture with some detailed computations in the principal series of $G_2$. A feature of this article is the role played by cocharacters $h_c$ attached to two-sided cells $c$ in certain extended affine Weyl groups. The quotient varieties which occur in the Bernstein programme are replaced by extended quotients. We form the disjoint union $A(G)$ of all these extended quotient varieties. We conjecture that, after a simple algebraic deformation, the space $A(G)$ is a model of the smooth dual $Irr(G)$. In this respect, our programme is a conjectural refinement of the Bernstein programme. The algebraic deformation is controlled by cocharacters $h_c$, one for each two-sided cell $c$ in certain extended affine Weyl groups. The cocharacters themselves appear to be closely related to Langlands parameters."}
{"category": "Math", "title": "Noncommutative hyperbolic geometry on the unit ball of $B(H)^n$", "abstract": "In this paper we introduce a hyperbolic distance $\\delta$ on the noncommutative open ball $[B(H)^n]_1$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, which is a noncommutative extension of the Poincare-Bergman metric on the open unit ball of $C^n$. We prove that $\\delta$ is invariant under the action of the group $Aut([B(H)^n]_1)$ of all free holomorphic automorphisms of $[B(\\cH)^n]_1$, and show that the $\\delta$-topology and the usual operator norm topology coincide on $[B(H)^n]_1$. Moreover, we prove that $[B(H)^n]_1$ is a complete metric space with respect to the hyperbolic metric and obtained an explicit formula for $\\delta$ in terms of the reconstruction operator. A Schwarz-Pick lemma for bounded free holomorphic functions on $[B(H)^n]_1$, with respect to the hyperbolic metric, is also obtained."}
{"category": "Math", "title": "Elliptic Equations Involving Meausres", "abstract": "We present the moste recent results dealing with the theory of semilinear elliptic equations with measures data"}
{"category": "Math", "title": "From persistent random walks to the telegraph noise", "abstract": "We study a family of memory-based persistent random walks and we prove weak convergences after space-time rescaling. The limit processes are not only Brownian motions with drift. We have obtained a continuous but non-Markov process $(Z_t)$ which can be easely expressed in terms of a counting process $(N_t)$. In a particular case the counting process is a Poisson process, and $(Z_t)$ permits to represent the solution of the telegraph equation. We study in detail the Markov process $((Z_t,N_t); t\\ge 0)$."}
{"category": "Math", "title": "Reducibility of quasiperiodic cocycles in linear Lie groups", "abstract": "Let G be a linear Lie group. We define the G-reducibility of a continuous or discrete cocycle modulo N. We show that a G-valued continuous or discrete cocycle which is GL(n,C)-reducible is in fact G-reducible modulo 2 if G=GL(n,R),SL(n,R),Sp(n,R) or O(n) and modulo 1 if G=U(n)."}
{"category": "Math", "title": "Disques J-holomorphes contenus dans une hypersurface", "abstract": "We study germs of J-Holomorphic curves contained in $M$, a real analytic hypersurface of an symplectic manifold of dimension 4. We show, under topological hypothesis on $M$, that if $M$ is compact then $M$ is of finite type and so there is no germs of $J$-holomorphic curves on $M$(with $J$ adapted with the symplectic form). In $\\Bbb{C}^2$ with the standard complex structure, this is a classical result of Diederich-Fornaess."}
{"category": "Math", "title": "The p-Laplace heat equation with a source term : self-similar solutions revisited", "abstract": "We study the self-similar solutions of any sign of the equation u_{t}-div(|&#8711;u|^{p-2}&#8711;u)=|u|^{q-1}u, in R^{N}, where p,q>1. We extend the results of Haraux-Weissler obtained for p=2 to the case q>p-1>0. In particular we study the existence of slow or fast decaying solutions. For given t>0, the fast solutions u(t,.) have a compact support in R^{N} when p>2, and |x|^{p/(2-p)}u(t,x) is bounded at infinity when p<2. We describe the behaviour for large |x| of all the solutions. According to the position of q with respect to the first critical exponent p-1+p/N and the critical Sobolev exponent q^{&#8727;}, we study the existence of positive solutions, or the number of the zeros of u(t,.). We prove that any solution u(t,.) is oscillatory when p<2 and q is closed to 1."}
{"category": "Math", "title": "Hyperbolic geometry on the unit ball of $B(H)^n$ and dilation theory", "abstract": "In this paper we continue our investigation concerning the hyperbolic geometry on the noncommutative ball $[B(H)^n]_1^-$, where $B(H)$ is the algebra of all bounded linear operators on a Hilbert space $H$, and its implications to noncommutative function theory. The central object is an intertwining operator $L_{B,A}$ of the minimal isometric dilations of $A, B\\in [B(H)^n]_1^-$, which establishes a strong connection between noncommutative hyperbolic geometry on $[B(H)^n]_1^-$ and multivariable dilation theory. The goal of this paper is to study the operator $L_{B,A}$ and its connections to the hyperbolic metric $\\delta$ on the Harnack parts of $[B(H)^n]_1^-$. We study the geometric structure of the operator $L_{B,A}$ and obtain new characterizations for the Harnack domination (resp. equivalence) in $[B(H)^n]_1^-$. We express $\\|L_{B,A}\\|$ in terms of the reconstruction operators $R_A$ and $R_B$, and obtain a Schwartz-Pick lemma for contractive free holomorphic functions on $[B(H)^n]_1$ with respect to the intertwining operator $L_{B,A}$. As a consequence, we deduce a Schwartz-Pick lemma for operator-valued multipliers of the Drury-Arveson space, with respect to the hyperbolic metric."}
{"category": "Math", "title": "Quantum Isometry groups of the Podles Spheres", "abstract": "For $\\mu \\in (0,1), c> 0,$ we identify the quantum group $SO_\\mu(3)$ as the universal object in the category of compact quantum groups acting by `orientation and volume preserving isometries' in the sense of \\cite{goswami2} on the natural spectral triple on the Podles sphere $S^2_{\\mu, c}$ constructed by Dabrowski, D'Andrea, Landi and Wagner in \\cite{{Dabrowski_et_al}}."}
{"category": "Math", "title": "Brownian motion and the parabolicity of minimal graphs", "abstract": "We prove that minimal graphs (other than planes) are parabolic in the sense that any bounded harmonic function is determined by its boundary values. The proof relies on using the coupling introduced in the author's earlier paper \"A martingale approach to minimal surfaces\" to show that Brownian motion on such a minimal graph almost surely strikes the boundary in finite time."}
{"category": "Math", "title": "Bordered Heegaard Floer homology: Invariance and pairing", "abstract": "We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A-infinity module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A-infinity tensor product of the type D module of one piece and the type A module from the other piece is HF^ of the glued manifold. As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for HF^. We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling."}
{"category": "Math", "title": "Regular functions on covers of nilpotent coadjoint orbits", "abstract": "In this paper I explore the relationship between regular functions associated to local systems on nilpotent orbits and unipotent representations in the complex groups."}
{"category": "Math", "title": "A recursive presentation for Mihailova's subgroup", "abstract": "We give an explicit recursive presentation for Mihailova's subgroup $M(H)$ of $F_n \\times F_n$ corresponding to a finite, concise and Peiffer aspherical presentation $H=< x_1,..., x_n \\mid R_1,..., R_m>$. This partially answers a question of R.I. Grigorchuk, [8, Problem 4.14]. As a corollary, we construct a finitely generated recursively presented orbit undecidable subgroup of $Aut(F_3)$."}
{"category": "Math", "title": "Slicing planar grid diagrams: a gentle introduction to bordered Heegaard Floer homology", "abstract": "We describe some of the algebra underlying the decomposition of planar grid diagrams. This provides a useful toy model for an extension of Heegaard Floer homology to 3-manifolds with parametrized boundary. This paper is meant to serve as a gentle introduction to the subject, and does not itself have immediate topological applications."}
{"category": "Math", "title": "Birational aspects of the geometry of M_g", "abstract": "We discuss topics on the geometry of the moduli space of curves. We present a short proof of the Harris-Mumford theorem on the Kodaira dimension of the moduli space which replaces the computations on the stack of admissible covers by a simple study of tautological Koszul bundles on M_g. We also present a streamlined self-contained account of Verra's recent work on the unirationality of M_g. Finally, we discuss a proof that the moduli space of curves of genus 22 is of general type. Written for Surveys in Differential Geometry."}
{"category": "Math", "title": "On the fundamental group of II_1 factors and equivalence relations arising from group actions", "abstract": "Given a countable group G, we consider the sets S_factor(G), S_eqrel(G), of subgroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental group of the associated II_1 factor, respectively orbit equivalence relation, equals F. We prove that if G is the free product of Z and infinitely many copies of a non-trivial group \\Gamma, then S_factor(G) and S_eqrel(G) contain R_+ itself, all of its countable subgroups, as well as uncountable subgroups whose log can have any Hausdorff dimension in the interval (0,1). We then prove that if G=\\Gamma*\\Lambda, with \\Gamma, \\Lambda finitely generated ICC groups, one of which has property (T), then S_factor(G)=S_eqrel(G)={1}. We also show that there exist II_1 factors M such that the fundamental group of M is R_+, but the associated II_\\infty factor M tensor B(l^2) admits no continuous trace scaling action of R_+."}
{"category": "Math", "title": "Statistics of incomplete quotients of continued fractions of quadratic irrationalities", "abstract": "V.I. Arnold has experimentally established that the limit of the statistics of incomplete quotients of partial continued fractions of quadratic irrationalities coincides with the Gauss--Kuz'min statistics. Below we briefly prove this fact for roots of the equation $r x^2+p x=q$ with fixed $p$ and $r$ ($r>0$), and with random $q$, $q\\le R$, $R\\to \\infty$. In Section 3 we estimate the sum of incomplete quotients of the period. According to the obtained bound, prior to the passage to the limit, incomplete quotients in average are logarithmically small. We also upper estimate the proportion of the \"red\" numbers among those representable as a sum of two squares."}
{"category": "Math", "title": "A negative mass theorem for surfaces of positive genus", "abstract": "We define the \"sum of squares of the wavelengths\" of a Riemannian surface (M,g) to be the regularized trace of the inverse of the Laplacian. We normalize by scaling and adding a constant, to obtain a \"mass\", which is scale invariant and vanishes at the round sphere. This is an anlaog for closed surfaces of the ADM mass from general relativity. We show that if M has positive genus then on each conformal class, the mass attains a negative minimum. For the minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality."}
{"category": "Math", "title": "BCOV theory via Givental group action on cohomological field theories", "abstract": "In a previous paper (arXiv:0704.1001), Losev, me, and Shneiberg constructed a full descendant potential associated to an arbitrary cyclic Hodge dGBV algebra. This contruction extended the construction of Barannikov and Kontsevich of solution of the WDVV equation, based on the earlier paper of Bershadsky, Cecotti, Ooguri, and Vafa. In the present paper, we give an interpretation of this full descendant potential in terms of Givental group action on cohomological field theories. In particular, the fact that it satisfies all tautological equations becomes a trivial observation."}
{"category": "Math", "title": "A primer on Seshadri constants", "abstract": "Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced by Demailly. The original idea of using them towards a proof of the Fujita conjecture failed but they quickly became a subject of intensive study quite in their own right. Lazarsfeld's book \"Positivity in Algebraic Geometry\" contains a whole chapter devoted to local positivity and serves as a very enjoyable introduction to Seshadri constants. Since this book has appeared, the subject witnessed quite a bit of development. It is the aim of these notes to give an account of recent progress as well as to discuss many open questions and provide some examples."}
{"category": "Math", "title": "On the structure of Goulden-Jackson-Vakil formula", "abstract": "We study the structure of the Goulden-Jackson-Vakil formula that relates Hurwitz numbers to some conjectural \"intersection numbers\" on a conjectural family of varieties $X_{g,n}$ of dimension $4g-3+n$. We give explicit formulas for the properly arranged generating function for these \"intersection numbers\", and prove that it satisfies Hirota equations. This generalizes and substantially simplifies our earlier results with Zvonkine in arXiv:math/0602457."}
{"category": "Math", "title": "A naive parametrization for the vortex-sheet problem", "abstract": "We consider the dynamics of a vortex sheet that evolves by the Birkhoff-Rott equations. The fluid evolution is understood as a weak solution of the incompressible Euler equations where the vorticity is given by a delta function on a curve multiplied by an amplitude. The solutions we study are with finite energy, which implies zero mean amplitude. In this context we choose a parametrization for the motion of the vortex sheet for which the equation is well-posed for analytic initial data. For the equation of the amplitude we show ill-posedness for non-analytic initial data."}
{"category": "Math", "title": "A note on Elkin's improvement of Behrend's construction", "abstract": "We provide a short proof of a recent result of Elkin in which large subsets of the integers 1 up to N free of 3-term progressions are constructed."}
{"category": "Math", "title": "Morley sequences in dependent theories", "abstract": "We characterize nonforking (Morley) sequences in dependent theories in terms of a generalization of Poizat's special sequences and show that average types of Morley sequences are stationary over their domains. We characterize generically stable types in terms of the structure of the \"eventual\" type. We then study basic properties of \"strict Morley sequences\", based on Shelah's notion of strict nonforking. In particular we prove \"Kim's lemma\" for such sequences, and a weak version of local character."}
{"category": "Math", "title": "Unstable classes of metric structures", "abstract": "We prove a strong non-structure theorem for a class of metric structures with an unstable pair of formulae. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies several versions of stability. This is the first step in the direction of the investigation of weak categoricity and weak stability of metric classes."}
{"category": "Math", "title": "Soliton dynamics for CNLS systems with potentials", "abstract": "The soliton dynamics in the semiclassical limit for a weakly coupled nonlinear focusing Schr\\\"odinger systems in presence of a nonconstant potential is studied by taking as initial data some rescaled ground state solutions of an associate elliptic system."}
{"category": "Math", "title": "A General Fredholm Theory III: Fredholm Functors and Polyfolds", "abstract": "We describe a very general (nonlinear) Fredholm theory for a new class of ambient spaces, called polyfolds. The basic feature of these new spaces is that in general they may have locally varying dimensions. These new spaces are needed for a functional analytic treatment of nonlinear problems involving analytic limiting behavior. This theory is applicable to Gromov-Witten and Floer Theory as well as Symplectic Field Theory."}
{"category": "Math", "title": "Variational and Geometric Structures of Discrete Dirac Mechanics", "abstract": "In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew's triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange-Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange-d'Alembert-Pontryagin and Hamilton-d'Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators."}
{"category": "Math", "title": "Observer design for invariant systems with homogeneous observations", "abstract": "This paper considers the design of nonlinear observers for invariant systems posed on finite-dimensional connected Lie groups with measurements generated by a transitive group action on an associated homogeneous space. We consider the case where the group action has the opposite invariance to the system invariance and show that the group kinematics project to a minimal realisation of the systems observable dynamics on the homogeneous output space. The observer design problem is approached by designing an observer for the projected output dynamics and then lifting to the Lie-group. A structural decomposition theorem for observers of the projected system is provided along with characterisation of the invariance properties of the associated observer error dynamics. We propose an observer design based on a gradient-like construction that leads to strong (almost) global convergence properties of canonical error dynamics on the homogeneous output space. The observer dynamics are lifted to the group in a natural manner and the resulting gradient-like error dynamics of the observer on the Lie-group converge almost globally to the unobservable subgroup of the system, the stabiliser of the group action."}
{"category": "Math", "title": "Class-Specific Tests of Spatial Segregation Based on Nearest Neighbor Contingency Tables", "abstract": "The spatial interaction between two or more classes (or species) has important consequences in many fields and might cause multivariate clustering patterns such as segregation or association. The spatial pattern of segregation occurs when members of a class tend to be found near members of the same class (i.e., conspecifics), while association occurs when members of a class tend to be found near members of the other class or classes. These patterns can be tested using a nearest neighbor contingency table (NNCT). The null hypothesis is randomness in the nearest neighbor (NN) structure, which may result from -- among other patterns -- random labeling (RL) or complete spatial randomness (CSR) of points from two or more classes (which is called the CSR independence, henceforth). In this article, we consider Dixon's class-specific tests of segregation and introduce a new class-specific test, which is a new decomposition of Dixon's overall chi-square segregation statistic. We demonstrate that the tests we consider provide information on different aspects of the spatial interaction between the classes and they are conditional under the CSR independence pattern, but not under the RL pattern. We analyze the distributional properties and prove the consistency of these tests; compare the empirical significant levels (Type I error rates) and empirical power estimates of the tests using Monte Carlo simulations. We demonstrate that the new class-specific tests also have comparable performance with the currently available tests based on NNCTs in terms of Type I error and power estimates. For illustrative purposes, we use three example data sets. We also provide guidelines for using these tests."}
{"category": "Math", "title": "Modeling the Dialectic", "abstract": "Three formal first-order finite dialectical schemes are investigated. It is shown that schemes 1 and 2 have significantly different finite models. Further, an infinite natural number model for schemes 1, 2, 3 is constructed, and it is shown that scheme 3 has no finite model."}
{"category": "Math", "title": "Generators and relations for wreath products", "abstract": "Generators and defining relations for wreath products of groups are given. Under some condition (conormality of the generators) they are minimal. In particular, it is just the case for the Sylow subgroups of the symmetric groups."}
{"category": "Math", "title": "Convexity of Chebyshev Sets in Hilbert Spaces", "abstract": "The aim of this paper is state of conditions that ensure the convexity of a Chebyshev sets in Hilbert spaces ."}
{"category": "Math", "title": "A Review on Some Geometric Results of the Smulian s Theorem on Frechet Differentiability of Norms", "abstract": "In this paper, we prove the Smulian s theorem on Frechet differentiability of norm,and present some of its geometric results concerning the Gateaux and Frechet differentiability of norm and properties of the allied space and its dual such as reflexivity and strict convexity."}
{"category": "Math", "title": "Relaxed Lyapunov Criteria for Robust Global Stabilization of Nonlinear Systems", "abstract": "The notion of the relaxed Robust Control Lyapunov Function (relaxed RCLF) is introduced and is exploited for the design of robust feedback stabilizers for nonlinear systems. Particularly, it is shown for systems with input constraints that relaxed RCLFs can be easily obtained, while RCLFs are not available. Moreover, it is shown that the use of relaxed RCLFs usually results to different feedback designs from the ones obtained by the use of the standard RCLF methodology. Using the relaxed RCLFs feedback design methodology, a simple controller that guarantees robust global stabilization of a perturbed chemostat model is provided."}
{"category": "Math", "title": "Badly approximable numbers related to the Littlewood conjecture", "abstract": "By means of Peres-Schlag's method we prove the existence of real numbers $\\alpha, \\beta$ such that $$ \\liminf_{q\\to \\infty} (q\\log^2 q)||\\alpha q|| ||\\beta q|| > 0."}
{"category": "Math", "title": "Contributions to Khovanov Homology", "abstract": "Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We prove that Khovanov's chain complex retracts to a subcomplex, whose generators are related to spanning trees of the Tait graph, and we exploit this result to investigate the structure of Khovanov homology for alternating knots. Further, we extend Rasmussen's invariant to links. Finally, we generalize Khovanov's categorifications of the colored Jones polynomial, and study conditions under which our categorifications are functorial with respect to colored framed link cobordisms. In this context, we develop a theory of Carter--Saito movie moves for framed link cobordisms."}
{"category": "Math", "title": "On the theory of $q$-complete spaces", "abstract": "In this article, we show that if $X$ is a Stein space of dimension $n$ and, if $D$ is a locally $q$-complete open set in $X$, then $D$ is $q$-complete. This gives, in particular, a positive answer to the local Steinness problem, which is one of the most classical problem in complex analytic geometry, namely, if $D$ is a locally Stein open set in a Stein space $X$, then $D$ is Stein"}
{"category": "Math", "title": "Cuspidal representations of reductive groups", "abstract": "This paper proves the existence of cuspidal automorphic forms for a reductive group, invariant under an automorphism of finite order. The techniques used are a local analysis of orbital integrals and the Arthur-Selberg trace formula."}
{"category": "Math", "title": "On some representations of degenerate affine Hecke algebras of type $BC_{n}$", "abstract": "Let G=U(p,q) and K=U(p)xU(q). In arXiv:0801.1530, the authors construct a functor from the category of Harish-Chandra modules for the pair (G,K) to the category of representations of the degenerate affine Hecke algebra of type B_n. In this paper, we study the image of the principal series modules of G under this functor. We describe the dAHA module structure of the image by constructing a family of common eigenvectors of {y_i}. In the rank 1 case, we write down an explicit formula for the generator y_{1} in terms of central elements of U(g) and calculate the eigenvalue of y_{1} by using the central characters."}
{"category": "Math", "title": "Character sheaves on unipotent groups in positive characteristic: foundations", "abstract": "In this article we formulate and prove the main theorems of the theory of character sheaves on unipotent groups over an algebraically closed field of characteristic p>0. In particular, we show that every admissible pair for such a group G gives rise to an L-packet of character sheaves on G, and that, conversely, every L-packet of character sheaves on G arises from a (non-unique) admissible pair. In the appendices we discuss two abstract category theory patterns related to the study of character sheaves. The first appendix sketches a theory of duality for monoidal categories, which generalizes the notion of a rigid monoidal category and is close in spirit to the Grothendieck-Verdier duality theory. In the second one we use a topological field theory approach to define the canonical braided monoidal structure and twist on the equivariant derived category of constructible sheaves on an algebraic group; moreover, we show that this category carries an action of the surface operad. The third appendix proves that the \"naive\" definition of the equivariant constructible derived category with respect to a unipotent algebraic group is equivalent to the \"correct\" one."}
{"category": "Math", "title": "Asymptotic energy of graphs", "abstract": "The energy of a simple graph $G$ arising in chemical physics, denoted by $\\mathcal E(G)$, is defined as the sum of the absolute values of eigenvalues of $G$. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice systems. In general for a type of lattice in statistical physics, to compute the asymptotic energy with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are different tasks with different hardness. In this paper, we show that if $\\{G_n\\}$ is a sequence of finite simple graphs with bounded average degree and $\\{G_n'\\}$ a sequence of spanning subgraphs of $\\{G_n\\}$ such that almost all vertices of $G_n$ and $G_n'$ have the same degrees, then $G_n$ and $G_n'$ have the same asymptotic energy. Thus, for each type of lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions, we have the same asymptotic energy. As applications, we obtain the asymptotic formulae of energies per vertex of the triangular, $3^3.4^2$, and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions simultaneously."}
{"category": "Math", "title": "An Alternative Definition of the Completion of Metric Spaces", "abstract": "In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be generalized to arbitrary categories."}
{"category": "Math", "title": "Robust Stabilization of Nonlinear Systems by Quantized and Ternary Control", "abstract": "Results on the problem of stabilizing a nonlinear continuous-time system by a finite number of control or measurement values are presented. The basic tool is a discontinuous version of the so-called semi-global backstepping lemma. We derive robust practical stabilizability results by quantized and ternary controllers and apply them to some significant control problems."}
{"category": "Math", "title": "Rational homotopy theory and differential graded category", "abstract": "We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan's theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions."}
{"category": "Math", "title": "Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mapping", "abstract": "The emphasis of this course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. We choose to show their effectiveness and to describe the theory for two large families of maps. The first chapter deals with holomorphic endomorphisms of the projective space P^k. We establish the first properties and give several constructions for the Green currents and the equilibrium measure \\mu. The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems and establish ergodic properties of \\mu: K-mixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. Finally, we study the entropy, the Lyapounov exponents and the dimension of \\mu. The second chapter develops the theory of polynomial-like maps in higher dimension. We introduce the dynamical degrees and construct the equilibrium measure \\mu of maximal entropy. Then, under a natural assumption, we prove equidistribution properties of points and various statistical properties of the measure \\mu. The assumption is stable under small pertubations on the map. We also study the dimension of \\mu, the Lyapounov exponents and their variation. Our aim is to get a self-contained text that requires only a minimal background. In order to help the reader, an appendix gives the basics on p.s.h. functions, positive closed currents and super-potentials on projective spaces. Some exercises are proposed and an extensive bibliography is given."}
{"category": "Math", "title": "Superconnections and Index Theory", "abstract": "We investigate index theory in the context of Dirac operators coupled to superconnections. In particular, we prove a local index theorem for such operators, and for families of such operators. We investigate eta-invariants and prove an APS-theorem, and construct a geometric determinant line bundle for families of such operators, computing its curvature and holonomy in terms of familiar index theoretic quantities."}
{"category": "Math", "title": "The Newton stratification on deformations of local G-shtukas", "abstract": "Bounded local G-shtukas are function field analogs for p-divisible groups with extra structure. We describe their deformations and moduli spaces. The latter are analogous to Rapoport-Zink spaces for p-divisible groups. The underlying schemes of these moduli spaces are affine Deligne-Lusztig varieties. For basic Newton polygons the closed Newton stratum in the universal deformation of a local G-shtuka is isomorphic to the completion of a corresponding affine Deligne-Lusztig variety in that point. This yields bounds on the dimension and proves equidimensionality of the basic affine Deligne-Lusztig varieties."}
{"category": "Math", "title": "Some properties of (C,E,P)-algebras : Overgneration and 0-order estimates", "abstract": "We give a new definition of the so-called overgenerated rings, which are the usual tool used to define the asymptotic structure of a (C,E,P)-algebra, written as a factor space M_{(A,E,P)}/N_{(I_{A},E,P)}. With this new definition and in the particular case of E=C^{&#8734;}, we show that a moderate element i.e. in M_{(A,E,P)} is negligible if and only if it satisfies the C&#8304;-order estimate for the ideal N_{(I_{A},E,P)}."}
{"category": "Math", "title": "Multivariate Determinateness", "abstract": "The uniqueness question of the multivariate moment problem is studied by different methods: Hilbert space operators, complex function theory, polynomial approximation, disintegration, integral geometry. Most of the known results in the multi-dimensional case are reviewed and reproved, and a number of new determinacy criteria are developed."}
{"category": "Math", "title": "Structural properties of acyclic heaps of pieces with Kazhdan--Lusztig theory", "abstract": "We introduce the notions of boundary vertex, linear equivalence and effective boundary vertex in the context of Viennot's heaps of pieces. We prove that in the heap of a fully commutative element in a star reducible Coxeter group, every boundary vertex is linearly equivalent to an effective boundary vertex. Using this result, we establish Property W (in the sense of math.QA/0509363) for star reducible Coxeter groups; this corrects a mistake in the latter paper."}
{"category": "Math", "title": "Permanence criteria for semi-free profinite groups", "abstract": "We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups."}
{"category": "Math", "title": "A Basic Elementary Extension of the Duchet-Meyniel Theorem", "abstract": "The Conjecture of Hadwiger implies that the Hadwiger number $h$ times the independence number $\\alpha$ of a graph is at least the number of vertices $n$ of the graph. In 1982 Duchet and Meyniel proved a weak version of the inequality, replacing the independence number $\\alpha$ by $2\\alpha-1$, that is, $$(2\\alpha-1)\\cdot h \\geq n.$$ In 2005 Kawarabayashi, Plummer and the second author published an improvement of the theorem, replacing $2\\alpha - 1$ by $2\\alpha - 3/2$ when $\\alpha$ is at least 3. Since then a further improvement by Kawarabayashi and Song has been obtained, replacing $2\\alpha - 1$ by $2\\alpha - 2$ when $\\alpha$ is at least 3. In this paper a basic elementary extension of the Theorem of Duchet and Meyniel is presented. This may be of help to avoid dealing with basic cases when looking for more substantial improvements. The main unsolved problem (due to Seymour) is to improve, even just slightly, the theorem of Duchet and Meyniel in the case when the independence number $\\alpha$ is equal to 2. The case $\\alpha = 2$ of Hadwiger's Conjecture was first pointed out by Mader as an interesting special case."}
{"category": "Math", "title": "Representations of finite special linear groups in non-defining characteristic", "abstract": "We determine precisely the number of irreducible summands of an irreducible cross characteristic representation of $GL_{n}(q)$ on restriction to $SL_{n}(q)$. Combined with a recent result of C. Bonnafe, this yields a canonical labeling for irreducible $\\ell$-modular representations of $SL_{n}(q)$, where $(\\ell,q)=1$. As an application, we classify for the first time complex representations of $SL_{n}(q)$ whose reductions modulo $\\ell$ are irreducible."}
{"category": "Math", "title": "On the Topology of Kac-Moody groups", "abstract": "We study the topology of spaces related to Kac-Moody groups. Given a split Kac-Moody group over the complex numbers, let K denote the unitary form with maximal torus T having normalizer N(T). In this article we study the cohomology of the flag manifold K/T, as a module over the Nil-Hecke ring, as well as the (co)homology of K as a Hopf algebra. In particular, if F is a field of positive characteristic, we show that H_*(K,F) is a finitely generated algebra, and that H^*(K,F) is finitely generated only if K is a compact Lie group . We also study the stable homotopy type of the classifying space BK and show that it is a retract of the classifying space BN(T). We illustrate our results with the example of rank two Kac-Moody groups."}
{"category": "Math", "title": "Symmetric powers and a problem of Kollar and Larsen", "abstract": "We prove a conjecture of Kollar and Larsen on Zariski closed subgroups of $GL(V)$ which act irreducibly on some symmetric power $Sym^{k}(V)$ with $k \\geq 4$. This conjecture has interesting implications, in particular on the holonomy group of a stable vector bundle on a smooth projective variety, as shown by the recent work of Balaji and Kollar."}
{"category": "Math", "title": "Hall-Higman type theorems for semisimple elements of finite classical groups", "abstract": "We prove an analogue of the celebrated Hall-Higman theorem, which gives a lower bound for the degree of the minimal polynomial of any semisimple element of prime power order $p^{a}$ of a finite classical group in any nontrivial irreducible cross characteristic representation. With a few explicit exceptions, this degree is at least $p^{a-1}(p-1)$."}
{"category": "Math", "title": "Holomorphic projection and duality for domains in complex projective space", "abstract": "We show that the efficiency of a natural pairing between certain projectively invariant Hardy spaces on dual strongly C-linearly convex real hypersurfaces in complex projective space is measured by the norm of the corresponding Leray transform."}
{"category": "Math", "title": "A surgery triangle for lattice cohomology", "abstract": "Lattice cohomology, defined by N\\'emethi in (arXiv:0709.0841), is an invariant of negative definite plumbed 3-manifolds which conjecturally computes the Heegaard Floer homology HF^+. We prove a surgery exact triangle for the lattice cohomology analogous to the one for HF^+. This is a step towards comparing these two invariants."}
{"category": "Math", "title": "Bias-Variance Tradeoffs: Novel Applications", "abstract": "We present several applications of the bias-variance decomposition, beginning with straightforward Monte Carlo estimation of integrals, but progressing to the more complex problem of Monte Carlo Optimization (MCO), which involves finding a set of parameters that optimize a parameterized integral. We present the similarity of this application to that of Parametric Learning (PL). Algorithms in this field use a particular interpretation of the bias-variance trade to improve performance. This interpretation also applies to MCO, and should therefore improve performance. We verify that this is indeed the case for a particular MCO problem related to adaptive importance sampling."}
{"category": "Math", "title": "The structure of the exponent set for finite cyclic groups", "abstract": "We survey properties of the set of possible exponents of subsets of $\\Z_n$ (equivalently, exponents of primitive circulant digraphs on $n$ vertices). Let $E_n$ denote this exponent set. We point out that $E_n$ contains the positive integers up to $\\sqrt{n}$, the `large' exponents $\\lfloor \\frac{n}{3} \\rfloor +1, \\lfloor \\frac{n}{2} \\rfloor, n-1$, and for even $n \\ge 4$, the additional value $\\frac{n}{2}-1$. It is easy to see that no exponent in $[\\frac{n}{2}+1,n-2]$ is possible, and Wang and Meng have shown that no exponent in $[\\lfloor \\frac{n}{3}\\rfloor +2,\\frac{n}{2}-2]$ is possible. Extending this result, we show that the interval $[\\lfloor \\frac{n}{4} \\rfloor +3, \\lfloor \\frac{n}{3} \\rfloor -2]$ is another gap in the exponent set $E_n$. In particular, $11 \\not\\in E_{35}$ and this gap is nonempty for all $n \\ge 57$. A conjecture is made about further gaps in $E_n$ for large $n$."}
{"category": "Math", "title": "Asymptotic evaluation of a function defined by power series", "abstract": "We present an asymptotic evaluation unitary formula for large argument values existing for defined class of functions. The asymptotic evaluation is obtained using only power series expansion coefficients of a function, what is a new result in power series analytic continuation theory."}
{"category": "Math", "title": "On weighted mean matrices whose $l^p$ norms are determined on decreasing sequences", "abstract": "We give a condition on weighted mean matrices so that their $l^p$ norms are determined on decreasing sequences when the condition is satisfied. We apply our result to give a proof of a conjecture of Bennett and discuss some related results."}
{"category": "Math", "title": "HIV with contact-tracing: a case study in Approximate Bayesian Computation", "abstract": "Missing data is a recurrent issue in epidemiology where the infection process may be partially observed. Approximate Bayesian Computation, an alternative to data imputation methods such as Markov Chain Monte Carlo integration, is proposed for making inference in epidemiological models. It is a likelihood-free method that relies exclusively on numerical simulations. ABC consists in computing a distance between simulated and observed summary statistics and weighting the simulations according to this distance. We propose an original extension of ABC to path-valued summary statistics, corresponding to the cumulated number of detections as a function of time. For a standard compartmental model with Suceptible, Infectious and Recovered individuals (SIR), we show that the posterior distributions obtained with ABC and MCMC are similar. In a refined SIR model well-suited to the HIV contact-tracing data in Cuba, we perform a comparison between ABC with full and binned detection times. For the Cuban data, we evaluate the efficiency of the detection system and predict the evolution of the HIV-AIDS disease. In particular, the percentage of undetected infectious individuals is found to be of the order of 40%."}
{"category": "Math", "title": "Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient", "abstract": "Thanks to a change of unknown we compare two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\\mathbb{R}^{N}.$ The first one, of the form $-\\Delta_{p}u=\\beta(u)| \\nabla u| ^{p}+\\lambda f(x),$ where $\\beta$ is nonnegative, involves a gradient term with natural growth. The second one, of the form $-\\Delta_{p}v=\\lambda f(x)(1+g(v))^{p-1}$ where $g$ is nondecreasing, presents a source term of order 0. The correlation gives new results of existence, nonexistence and multiplicity for the two problems."}
{"category": "Math", "title": "Large Scale Variational Inference and Experimental Design for Sparse Generalized Linear Models", "abstract": "Many problems of low-level computer vision and image processing, such as denoising, deconvolution, tomographic reconstruction or super-resolution, can be addressed by maximizing the posterior distribution of a sparse linear model (SLM). We show how higher-order Bayesian decision-making problems, such as optimizing image acquisition in magnetic resonance scanners, can be addressed by querying the SLM posterior covariance, unrelated to the density's mode. We propose a scalable algorithmic framework, with which SLM posteriors over full, high-resolution images can be approximated for the first time, solving a variational optimization problem which is convex iff posterior mode finding is convex. These methods successfully drive the optimization of sampling trajectories for real-world magnetic resonance imaging through Bayesian experimental design, which has not been attempted before. Our methodology provides new insight into similarities and differences between sparse reconstruction and approximate Bayesian inference, and has important implications for compressive sensing of real-world images."}
{"category": "Math", "title": "A Note on Maximal Averages in the Plane", "abstract": "Using variants of the TT* method we give a self-contained proof of the result of Alfonseca, Soria and Vargas on maximal operators on arbitrary directions in $\\rr^2$. We also give a sharp $L^2$ estimate for a maximal function extending a Theorem of Cordoba."}
{"category": "Math", "title": "Probability models characterized by generalized reversed lack of memory property", "abstract": "A binary operator * over real numbers is said to be associative if $(x*y)*z=x*(y*z)$ and is said to be reducible if $x*y=x*z$ or $y*w=z*w$ if and only if $z=y$. The operation is said to have an identity element $\\tilde{e}$ if $x*\\tilde{e}=x$. In this paper a characterization of a subclass of the reversed generalized Pareto distribution (Castillo and Hadi (1995)) in terms of the reversed lack of memory property (Asha and Rejeesh (2007)) is generalized using this binary operation and probability distributions are characterized using the same. This idea is further generalized to the bivariate case."}
{"category": "Math", "title": "Low dimensional discriminant loci and scrolls", "abstract": "Smooth complex polarized varieties $(X,L)$ with a vector subspace $V \\subseteq H^0(X,L)$ spanning $L$ are classified under the assumption that the locus ${\\Cal D}(X,V)$ of singular elements of $|V|$ has codimension equal to $\\dim(X)-i$, $i=3,4,5$, the last case under the additional assumption that $X$ has Picard number one. In fact it is proven that this codimension cannot be $\\dim(X)-4$ while it is $\\dim(X)-3$ if and only if $(X,L)$ is a scroll over a smooth curve. When the codimension is $\\dim(X)-5$ and the Picard number is one only the Pl\\\"ucker embedding of the Grassmannian of lines in $\\Bbb P^4$ or one of its hyperplane sections appear. One of the main ingredients is the computation of the top Chern class of the first jet bundle of scrolls and hyperquadric fibrations. Further consequences of these computations are also provided."}
{"category": "Math", "title": "The Neron model over the Igusa curves", "abstract": "We analyze the geometry of rational p-division points in degenerating families of elliptic curves in characteristic p. We classify the possible Kodaira symbols and determine for the Igusa moduli problem the reduction type of the universal curve. Special attention is paid to characteristic 2 and 3 where wild ramification and stacky phenomena show up."}
{"category": "Math", "title": "On nowhere continuous Costas functions and infinite Golomb rulers", "abstract": "We prove the existence of nowhere continuous bijections that satisfy the Costas property, as well as (countably and uncountably) infinite Golomb rulers. We define and prove the existence of real and rational Costas clouds, namely nowhere continuous Costas injections whose graphs are everywhere dense in a region of the real plane, based on nonlinear solutions of Cauchy's functional equation. We also give 2 constructive examples of a nowhere continuous function, that satisfies a constrained form of the Costas property (over rational or algebraic displacements only, that is), based on the indicator function of a dense subset of the reals."}
{"category": "Math", "title": "Unsolvability of the isomorphism problem for [free abelian]-by-free groups", "abstract": "The isomorphism problem for [free abelian]-by-free groups is unsolvable."}
{"category": "Math", "title": "The Covariant Measure of SLE on the Boundary", "abstract": "We construct a natural measure mu supported on the intersection of a chordal SLE(kappa) curve gamma with the real line R, in the range 4 < kappa < 8. The measure is a function of the SLE path in question. Assuming that boundary measures transform in a ``d-dimensional'' way (where d is the Hausdorff dimension of gamma intersected with R), we show that the measure we construct is (up to multiplicative constant) the unique measure-valued function of the SLE path that satisfies the Domain Markov property."}
{"category": "Math", "title": "New inclusion and coincidence theorems for summing multilinear mappings", "abstract": "In this paper we obtain new inclusion and coincidence theorems for absolutely or multiple summing multilinear mappings. In particular, we derive optimal coincidence theorems of Bohnenblust-Hille type for multilinear forms on K-convex Banach spaces of cotype 2."}
{"category": "Math", "title": "On Okuyama's theorems about Alvis-Curtis duality", "abstract": "The purpose of this paper is to report on the unpublished manuscript [O] by T. Okuyama where are proved some conjectures generalizing to homotopy categories the theorems of [CaRi] and [LS] holding in derived categories. We refer to the latter references and [CaEn]~\\S 4 for a broader introduction to the subject. The main theme is the one of complexes related with the Coxeter complex and the action of parabolic subgroups on them, either for finite groups with BN-pairs or for finite dimensional Hecke algebras. Okuyama's contractions prove a quite efficient tool in a number of situations (see the proof of Solomon-Tits theorem in \\S~6). We often stray away from Okuyama's proofs when it allows simplifications."}
{"category": "Math", "title": "K3 surfaces with non-symplectic involution and compact irreducible G_2-manifolds", "abstract": "We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds were previously obtained in math.DG/0012189 by blowing up curves in Fano threefolds. In this paper, we give further suitable algebraic threefolds using theory of K3 surfaces with non-symplectic involution due to Nikulin. These threefolds are not obtainable from Fano threefolds, as above, and admit matching pairs leading to topologically new examples of compact irreducible G_2-manifolds. `Geography' of the values of Betti numbers b^2,b^3 for the new (and previously known) examples of compact irreducible G_2 manifolds is also discussed."}
{"category": "Math", "title": "Closed 1-Forms in Topology and Geometric Group Theory", "abstract": "In this article we describe relations of the topology of closed 1-forms to the group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CW- complexes and show that many properties of the group theoretic version have analogous statements. In particular we show the relation between Sigma invariants and finiteness properties of certain infinite covering spaces. We also discuss applications of these invariants to the Lusternik- Schnirelmann category of a closed 1-form and to the existence of a non- singular closed 1-form in a given cohomology class on a high-dimensional closed manifold."}
{"category": "Math", "title": "Algebraic Connectivity and Degree Sequences of Trees", "abstract": "We investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. We show that such trees are caterpillars and that the vertex degrees are non-decreasing on every path on non-pendant vertices starting at the characteristic set of the Fiedler vector."}
{"category": "Math", "title": "Zero-class admissibility of observation operators", "abstract": "An admissible observation operator is zero-class admissible if the norm of the output map tends to zero as the time tends to zero. Sufficient and necessary conditions for zero-class admissibility of observation operators are developed and a modified Weiss condition is studied. It is shown that the modified Weiss condition is in general necessary, but not sufficient for zero-class admissibility. For several important classes of C_0-semigroups it is proved that the modified Weiss condition is indeed equivalent to zero-class admissibility. The methods are illustrated by certain PDE examples."}
{"category": "Math", "title": "Infinity-harmonic maps and morphisms", "abstract": "We propose a new notion called \\emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $p\\to \\infty $. Infinity harmoncity appears in many familiar contexts. For example, metric projection onto the orbit of an isometric group action from a tubular neighborhood is infinity harmonic. Unfortunately, infinity-harmonicity is not preserved under composition. Those infinity harmonic maps that always preserve infinity harmonicity under pull back are called infinity harmonic morphisms. We show that infinity harmonic morphisms are precisely horizontally homothetic mas. Many example of infinity-harmonic maps are given, including some very important and well-known classes of maps between Riemannian manifolds."}
{"category": "Math", "title": "Vector Fields and Flows on Differentiable Stacks", "abstract": "This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows on a manifold as well as the author's existing results for orbifolds. It sets the scene for a discussion of Morse Theory on a general proper stack and also paves the way for the categorification of other key aspects of differential geometry such as the tangent bundle and the Lie algebra of vector fields."}
{"category": "Math", "title": "Quantum K-theory of Grassmannians", "abstract": "We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the ordinary (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through 3 general points is irreducible and rational. Our applications include Pieri and Giambelli formulas for the quantum K-theory ring of a Grassmannian, which determine the multiplication in this ring. Our formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of Chaput, Manivel, and Perrin."}
{"category": "Math", "title": "The mapping class group of a punctured surface is generated by three elements", "abstract": "Let $\\Sigma_{g,p}$ be a closed oriented surface of genus $g\\geq 1$ with $p$ punctures. Let $\\rm Mod(\\Sigma_{\\textit{g,p}})$ be the mapping class group of $\\Sigma_{g,p}$. Wajnryb proved in [Wa] that for $p=0, 1$ $\\rm Mod({\\Sigma_{\\textit{g,p}}})$ is generated by two elements. Korkmaz proved in [Ko] that one of these generators can be taken as a Dehn twist. For $p\\geq 2$, We proved that $\\rm Mod(\\Sigma_{\\textit{g,p}})$ is generated by three elements."}
{"category": "Math", "title": "On asymptotic normality of sequential LS-estimates of unstable autoregressive processes", "abstract": "For estimating the unknown parameters in an unstable autoregressive AR(p), the paper proposes sequential least squares estimates with a special stopping time defined by the trace of the observed Fisher information matrix. The limiting distribution of the sequential LSE is shown to be normal for the parameter vector lying both inside the stability region and on some part of its boundary in contrast to the ordinary LSE. The asymptotic normality of the sequential LSE is provided by a new property of the observed Fisher information matrix which holds both inside the stability region of AR(p) process and on the part of its boundary. The asymptotic distribution of the stopping time is derived."}
{"category": "Math", "title": "On theta functions of order four", "abstract": "We prove that the fourth powers of theta functions with even characteristics form a basis of the space of even theta functions of order four on a principally polarized Abelian variety without vanishing theta-null."}
{"category": "Math", "title": "Orbit measures, random matrix theory and interlaced determinantal processes", "abstract": "A connection between representation of compact groups and some invariant ensembles of Hermitian matrices is described. We focus on two types of invariant ensembles which extend the Gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction mulltiplicities. We show that a large class of them are determinantal."}
{"category": "Math", "title": "Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions", "abstract": "In this paper we consider a multi-dimensional damped semiliear wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. We firstly prove the local existence by using the Faedo-Galerkin approximations combined with a contraction mapping theorem. Secondly, the exponential growth of the energy and the $L^p$ norm of the solution is presented."}
{"category": "Math", "title": "On linear resolution of powers of an ideal", "abstract": "In this paper we give a generalization of a result of Herzog, Hibi, and Zheng providing an upper bound for regularity of powers of an ideal. As the main result of the paper, we give a simple criterion in terms of Rees algebra of a given ideal to show that high enough powers of this ideal have linear resolution. We apply the criterion to two important ideals $J,J_{1}$ for which we show that $J^{k},$ and $J_{1}^{k}$ have linear resolution if and only if $k\\neq 2.$ The procedures we include in this work is encoded in computer algebra package CoCoA."}
{"category": "Math", "title": "Power series over generalized Krull domains", "abstract": "We resolve an open problem in commutative algebra and Field Arithmetic, posed by Jarden -- Let R be a generalized Krull domain. Is the ring R[[X]] of formal power series over R a generalized Krull domain? We show that the answer is negative."}
{"category": "Math", "title": "Quantitative unique continuation, logarithmic convexity of Gaussian means and Hardy's uncertainty principle", "abstract": "In this paper we describe some recent works on quantitative unique continuation for elliptic, parabolic and dispersive equations. The elliptic results are joint work with J.Bourgain, while the remainder of the works discussed are joint works with L.Escauriaza, G.Ponce and L.Vega."}
{"category": "Math", "title": "A Strong Law of Large Numbers with Applications to Self-Similar Stable Processes", "abstract": "Let $p \\in (0, \\infty)$ be a constant and let $\\{\\xi_n\\} \\subset L^p(\\Omega, {\\mathcal F}, \\P)$ be a sequence of random variables. For any integers $m, n \\ge 0$, denote $S_{m, n} = \\sum_{k=m}^{m + n} \\xi_k$. It is proved that, if there exist a nondecreasing function $\\varphi: \\R_+\\to \\R_+$ (which satisfies a mild regularity condition) and an appropriately chosen integer $a\\ge 2$ such that $$ \\sum_{n=0}^\\infty \\sup_{k \\ge 0} \\E\\bigg|\\frac{S_{k, a^n}} {\\varphi(a^n)} \\bigg|^p < \\infty,$$ Then $$ \\lim_{n \\to \\infty} \\frac{S_{0, n}} {\\varphi(n)} = 0\\qquad \\hbox{a.s.} $$ This extends Theorem 1 in Levental, Chobanyan and Salehi \\cite{chobanyan-l-s} and can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes."}
{"category": "Math", "title": "The Kobayashi metric in the normal direction and the mapping problem", "abstract": "Estimates of the Kobayashi metric in the normal direction are used to study the mapping problem in several complex variables."}
{"category": "Math", "title": "Multiple polylogarithm values at roots of unity", "abstract": "For any positive integer $N$ let $\\mu_N$ be the group of the $N$th roots of unity. In this note we shall study the $\\Q$-linear relations among values of multiple polylogarithms evaluated at $\\mmu_N$. We show that the standard relations considered by Racinet do not provide all the possible relations in the following cases: (i) level N=4, weight $w=3$ or 4, and (ii) $w=2$, $7<N<50$, and $N$ is a power of 2 or 3, or $N$ has at least two prime factors. We further find some (presumably all) of the missing relations in (i) by using the octahedral symmetry of $\\P^1-(\\{0,\\infty\\}\\cup \\mu_4)$. We also prove some other results when $N=p$ or $N=p^2$ ($p$ prime $\\ge 5$) by using the motivic fundamental group of $\\P^1-(\\{0,\\infty\\}\\cup\\mu_N)$."}
{"category": "Math", "title": "The Structure of Commutative Automorphic Loops", "abstract": "An \\emph{automorphic loop} (or \\emph{A-loop}) is a loop whose inner mappings are automorphisms. Every element of a commutative A-loop generates a group, and $(xy)^{-1} = x^{-1}y^{-1}$ holds. Let $Q$ be a finite commutative A-loop and $p$ a prime. The loop $Q$ has order a power of $p$ if and only if every element of $Q$ has order a power of $p$. The loop $Q$ decomposes as a direct product of a loop of odd order and a loop of order a power of 2. If $Q$ is of odd order, it is solvable. If $A$ is a subloop of $Q$ then $|A|$ divides $|Q|$. If $p$ divides $|Q|$ then $Q$ contains an element of order $p$. If there is a finite simple nonassociative commutative A-loop, it is of exponent 2."}
{"category": "Math", "title": "On a Speculated Relation Between Chv\\'atal-Sankoff Constants of Several Sequences", "abstract": "It is well known that, when normalized by n, the expected length of a longest common subsequence of d sequences of length n over an alphabet of size sigma converges to a constant gamma_{sigma,d}. We disprove a speculation by Steele regarding a possible relation between gamma_{2,d} and gamma_{2,2}. In order to do that we also obtain new lower bounds for gamma_{sigma,d}, when both sigma and d are small integers."}
{"category": "Math", "title": "The Stratified Structure of Spaces of Smooth Orbifold Mappings", "abstract": "We consider four notions of maps between smooth C^r orbifolds O, P with O compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of C^r maps between O and P with the C^r topology carries the structure of a smooth C^\\infty Banach (r finite)/Frechet (r=infty) manifold. For the notion of complete reduced orbifold map, the corresponding set of C^r maps between O and P with the C^r topology carries the structure of a smooth C^\\infty Banach (r finite)/Frechet (r=infty) orbifold. The remaining two notions carry a stratified structure: The C^r orbifold maps between O and P is locally a stratified space with strata modeled on smooth C^\\infty Banach (r finite)/Frechet (r=infty) manifolds while the set of C^r reduced orbifold maps between O and P locally has the structure of a stratified space with strata modeled on smooth C^\\infty Banach (r finite)/Frechet (r=infty) orbifolds. Furthermore, we give the explicit relationship between these notions of orbifold map. Applying our results to the special case of orbifold diffeomorphism groups, we show they inherit the structure of C^\\infty Banach (r finite)/Frechet (r=infty) manifolds. In fact, for r finite they are topological groups, and for r=infty they are convenient Frechet Lie groups."}
{"category": "Math", "title": "A gap principle for dynamics", "abstract": "Let $f_1,...,f_g\\in {\\mathbb C}(z)$ be rational functions, let $\\Phi=(f_1,...,f_g)$ denote their coordinatewise action on $({\\mathbb P}^1)^g$, let $V\\subset ({\\mathbb P}^1)^g$ be a proper subvariety, and let $P=(x_1,...,x_g)\\in ({\\mathbb P}^1)^g({\\mathbb C})$ be a nonpreperiodic point for $\\Phi$. We show that if $V$ does not contain any periodic subvarieties of positive dimension, then the set of $n$ such that $\\Phi^n(P) \\in V({\\mathbb C})$ must be very sparse. In particular, for any $k$ and any sufficiently large $N$, the number of $n \\leq N$ such that $\\Phi^n(P) \\in V({\\mathbb C})$ is less than $\\log^k N$, where $\\log^k$ denotes the $k$-th iterate of the $\\log$ function. This can be interpreted as an analog of the gap principle of Davenport-Roth and Mumford."}
{"category": "Math", "title": "On the Slope of Hyperelliptic Lefschetz Fibrations and the Number of Separating Vanishing Cycles", "abstract": "In this article we find an upper and lower bound for the slope of genus g hyperelliptic Lefschetz fibrations, which is sharp when g = 2, and demonstrate the strong connection, in general, between the slope of hyperelliptic genus g Lefschetz fibrations and the number of separating vanishing cycles. Specifically, we show that the slope is greater than 4-4/g if and only if the fibration contains separating vanishing cycles. We also improve the existing bound on s/n, the ratio of number of separating vanishing cycles to the number of non-separating vanishing cycles, for hyperelliptic Lefschetz fibrations of genus g>1. In particular we show that s<=n for such fibrations when g>5."}
{"category": "Math", "title": "David George Kendall, a biographical account", "abstract": "This biographical account of the life and work of David Kendall includes details of his personal and professional activities. Kendall is probably best known for his work in applied probability, especially queueing theory, and in stochastic analysis and spatial statistics."}
{"category": "Math", "title": "On the Distribution of the Euler Function of Shifted Smooth Numbers", "abstract": "We give asymptotic formulas for some average values of the Euler function on shifted smooth numbers. The result is based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to A. Granville and \\'E. Fouvry & G. Tenenbaum."}
{"category": "Math", "title": "On the Mathematics of the Law of Mass Action", "abstract": "In 1864,Waage and Guldberg formulated the \"law of mass action.\" Since that time, chemists, chemical engineers, physicists and mathematicians have amassed a great deal of knowledge on the topic. In our view, sufficient understanding has been acquired to warrant a formal mathematical consolidation. A major goal of this consolidation is to solidify the mathematical foundations of mass action chemistry -- to provide precise definitions, elucidate what can now be proved, and indicate what is only conjectured. In addition, we believe that the law of mass action is of intrinsic mathematical interest and should be made available in a form that might transcend its application to chemistry alone. We present the law of mass action in the context of a dynamical theory of sets of binomials over the complex numbers."}
{"category": "Math", "title": "Zeros of Systems of ${\\mathfrak p}$-adic Quadratic Forms", "abstract": "It is shown that a system of $r$ quadratic forms over a ${\\mathfrak p}$-adic field has a non-trivial common zero as soon as the number of variables exceeds $4r$, providing that the residue class field has cardinality at least $(2r)^r$."}
{"category": "Math", "title": "Volume entropy of Hilbert Geometries", "abstract": "It is shown that the volume entropy of a Hilbert geometry associated to an $n$-dimensional convex body of class $C^{1,1}$ equals $n-1$. To achieve this result, a new projective invariant of convex bodies, similar to the centro-affine area, is constructed. In the case $n=2$, and without any assumption on the boundary, it is shown that the entropy is bounded above by $\\frac{2}{3-d} \\leq 1$, where $d$ is the Minkowski dimension of the extremal set of $K$. An example of a plane Hilbert geometry with entropy strictly between 0 and 1 is constructed."}
{"category": "Math", "title": "Spectra of Ruelle transfer operators for Axiom A flows (Revised)", "abstract": "For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to these of Dolgopyat (1998) for geodesic flows on compact surfaces (for general potentials)and transitive Anosov flows on compact manifolds with C^1 jointly non-integrable horocycle foliations (for the Sinai-Bowen-Ruelle potential). Here we deal with general potentials. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, closed orbit counting functions, decay of correlations for H\\\"older continuous potentials."}
{"category": "Math", "title": "A robust spectral method for finding lumpings and meta stable states of non-reversible Markov chains", "abstract": "A spectral method for identifying lumping in large Markov chains is presented. Identification of meta stable states is treated as a special case. The method is based on spectral analysis of a self-adjoint matrix that is a function of the original transition matrix. It is demonstrated that the technique is more robust than existing methods when applied to noisy non-reversible Markov chains."}
{"category": "Math", "title": "(G,m)-multiparking functions", "abstract": "The conceptions of $G$-parking functions and $G$-multiparking functions were introduced in [15] and [12] respectively. In this paper, let $G$ be a connected graph with vertex set $\\{1,2,...,n\\}$ and $m\\in V(G)$. We give the definition of $(G,m)$-multiparking function. This definition unifies the conceptions of $G$-parking function and $G$-multiparking function. We construct bijections between the set of $(G,m)$-multiparking functions and the set of $\\mathcal{F}_{G,m}$ of spanning color $m$-forests of $G$. Furthermore we define the $(G,m)$-multiparking complement function, give the reciprocity theorem for $(G,m)$-multiparking function and extend the results [25,12] to $(G,m)$-multiparking function. Finally, we use a combinatorial methods to give a recursion of the generating function of the sum $\\sum\\limits_{i=1}^na_i$ of $G$-parking functions $(a_1,...,a_n)$."}
{"category": "Math", "title": "Compatible Complex Structures on Twistor Spaces", "abstract": "Let (M,g) be a Riemannian 4-manifold. The twistor space Z->M is a CP1-bundle whose total space Z admits a natural metric h. The aim of this article is to study properties of complex structures on (Z,h) which are compatible with the CP1-fibration and the metric h. The results obtained enable us to translate some metric properties on M in terms of complex properties on its twistor space Z."}
{"category": "Math", "title": "Analysis of a class of non linear subdivision schemes and associated multi-resolution transforms", "abstract": "This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multi-resolution transforms. These schemes are defined as a perturbation of a linear subdivision scheme. Assuming a contractivity property, stability and convergence are derived. These results are then applied to various schemes such as uncentered interpolatory linear scheme, WENO scheme [13], Power-P scheme [16] and a non linear scheme using local spherical coordinates [18]."}
{"category": "Math", "title": "$p$-Adic multiresolution analyses", "abstract": "We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating a MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate Haar MRA. We also suggest a method of constructing sets of wavelet functions and prove that any set of wavelet functions generates a $p$-adic wavelet frame."}
{"category": "Math", "title": "Cox rings, semigroups and automorphisms of affine algebraic varieties", "abstract": "We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results on the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Cox ring normalizing the grading. It follows that the automorphism group of a non-degenerate affine toric variety of dimension $\\geq 2$ has infinite dimension. We obtain a wild automorphism of the three-dimensional quadratic cone that rises to Anick's automorphism of the polynomial algebra in four variables."}
{"category": "Math", "title": "Inference for the limiting cluster size distribution of extreme values", "abstract": "Any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound Poisson under appropriate long range dependence conditions. Typically exceedances appear in clusters. The underlying Poisson points represent the cluster positions and the multiplicities correspond to the cluster sizes. In the present paper we introduce estimators of the limiting cluster size probabilities, which are constructed through a recursive algorithm. We derive estimators of the extremal index which plays a key role in determining the intensity of cluster positions. We study the asymptotic properties of the estimators and investigate their finite sample behavior on simulated data."}
{"category": "Math", "title": "A note on zeros of Eisenstein series for genus zero Fuchsian groups", "abstract": "Let $\\Gamma \\subseteq \\text{SL}_2(\\mathbb{R})$ be a genus zero Fuchsian group of the first kind having $\\infty$ as a cusp, and let $E_{2 k}^{\\Gamma}$ be the holomorphic Eisenstein series associated with $\\Gamma$ for the $\\infty$ cusp that does not vanish at $\\infty$ but vanishes at all the other cusps. In the paper \"On zeros of Eisenstein series for genus zero Fuchsian groups\", under assumptions on $\\Gamma$, and on a certain fundamental domain $\\mathcal{F}$, H. Hahn proved that all but at most $c(\\Gamma, \\mathcal{F})$ (a constant) of the zeros of $E_{2 k}^{\\Gamma}$ lie on a certain subset of $\\{z \\in \\mathfrak{H} : j_{\\Gamma}(z) \\in \\mathbb{R}\\}$. In this note, we consider a small generalization of Hahn's result on the domain locating the zeros of $E_{2 k}^{\\Gamma}$. We can prove most of the zeros of $E_{2 k}^{\\Gamma}$ in $\\mathcal{F}$ lie on its lower arcs under the same assumption."}
{"category": "Math", "title": "A strong uniform convergence rate of a kernel conditional quantile estimator under random left-truncation and dependent data", "abstract": "In this paper we study some asymptotic properties of the kernel conditional quantile estimator with randomly left-truncated data which exhibit some kind of dependence. We extend the result obtained by Lemdani, Ould-Sa\\\"id and Poulin [16] in the iid case. The uniform strong convergence rate of the estimator under strong mixing hypothesis is obtained."}
{"category": "Math", "title": "A remark on double cosets", "abstract": "If a soluble group $G$ contains two finitely generated abelian subgroups $A,B$ such that the number of double cosets $AgB$ is finite, then $G$ is shown to be virtually polycyclic."}
{"category": "Math", "title": "Generalised linear mixed model analysis via sequential Monte Carlo sampling", "abstract": "We present a sequential Monte Carlo sampler algorithm for the Bayesian analysis of generalised linear mixed models (GLMMs). These models support a variety of interesting regression-type analyses, but performing inference is often extremely difficult, even when using the Bayesian approach combined with Markov chain Monte Carlo (MCMC). The Sequential Monte Carlo sampler (SMC) is a new and general method for producing samples from posterior distributions. In this article we demonstrate use of the SMC method for performing inference for GLMMs. We demonstrate the effectiveness of the method on both simulated and real data, and find that sequential Monte Carlo is a competitive alternative to the available MCMC techniques."}
{"category": "Math", "title": "Estimating the multivariate extremal index function", "abstract": "The multivariate extremal index function relates the asymptotic distribution of the vector of pointwise maxima of a multivariate stationary sequence to that of the independent sequence from the same stationary distribution. It also measures the degree of clustering of extremes in the multivariate process. In this paper, we construct nonparametric estimators of this function and prove their asymptotic normality under long-range dependence and moment conditions. The results are illustrated by means of a simulation study."}
{"category": "Math", "title": "Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression via model selection", "abstract": "The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper, 2007, for estimating a unknown nonparametric regression. We prove that this procedure is asymptotically efficient for a quadratic risk, i.e. the asymptotic quadratic risk for this procedure coincides with the Pinsker constant which gives a sharp lower bound for the quadratic risk over all possible estimators."}
{"category": "Math", "title": "Analysis of a Population Model Structured by the Cells Molecular Content", "abstract": "We study the mathematical properties of a general model of cell division structured with several internal variables. We begin with a simpler and specific model with two variables, we solve the eigenvalue problem with strong or weak assumptions, and deduce from it the long-time convergence. The main difficulty comes from natural degeneracy of birth terms that we overcome with a regularization technique. We then extend the results to the case with several parameters and recall the link between this simplified model and the one presented in \\cite{CBBP1}; an application to the non-linear problem is also given, leading to robust subpolynomial growth of the total population."}
{"category": "Math", "title": "A group with three real irreducible characters: answering a question of Moreto and Navarro", "abstract": "In this paper, we construct a group with three real irreducible characters whose Sylow 2-subgroup is an iterated central extension of a Suzuki 2-group. This answers a question raised by Moreto and Navarro who asked whether such a group exists."}
{"category": "Math", "title": "Existence theorems in linear chaos", "abstract": "Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. We treat questions of characterizing which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property. In particular, we characterize countable inductive limits of separable Banach spaces that admit a hypercyclic operator, show that there is a non-mixing hypercyclic operator on a separable infinite dimensional complex Fr\\'echet space $X$ if and only if $X$ is non-isomorphic to the space $\\omega$ of all sequences with coordinatewise convergence topology. It is also shown for any $k\\in\\N$, any separable infinite dimensional Fr\\'echet space $X$ non-isomorphic to $\\omega$ admits a mixing uniformly continuous group $\\{T_t\\}_{t\\in C^n}$ of continuous linear operators and that there is no supercyclic strongly continuous operator semigroup $\\{T_t\\}_{t\\geq 0}$ on $\\omega$. We specify a wide class of Fr\\'echet spaces $X$, including all infinite dimensional Banach spaces with separable dual, such that there is a hypercyclic operator $T$ on $X$ for which the dual operator $T'$ is also hypercyclic. An extension of the Salas theorem on hypercyclicity of a perturbation of the identity by adding a backward weighted shift is presented and its various applications are outlined."}
{"category": "Math", "title": "On the classification of fake lens spaces", "abstract": "In the first part of the paper we present a classification of fake lens spaces of dimension >= 5 whose fundamental group is the cyclic group of order N >= 2. The classification uses and extends the results of Wall and others in the case N = 2 and N odd and the results of the authors of the present paper in the case N a power of 2. In the second part we study the suspension map between the simple structure sets of lens spaces of different dimensions. As an application we obtain an inductive geometric description of the torsion invariants of fake lens spaces."}
{"category": "Math", "title": "Short note on the perturbation of operators with dyadic products", "abstract": "In this paper we use abstract vector spaces and their duals without any canonical basis. Some of our results can be extended to infinite dimensional vector spaces too, but here we consider only finite dimensional spaces. We focus on a general perturbation problem. Assume that $B:V\\to V$ is a linear operator, which is perturbated to $B'=B+Q$. We examine the question how the determinant and the inverse change, because of this perturbation. In our approach the operator $Q$ is given as a sum of dyadic products $Q=\\sum_{i=1}^{k}v_{i}\\otimes p_{i}$, where $v_{i}\\in V$ and $p_{i}\\in V^{*}$. In this paper we derive an $m$-th order ($m\\in\\mathbb{N}$) approximation formula for $\\det B'$ and $(B')^{-1}$, which gives the exact result if $m\\geq k$."}
{"category": "Math", "title": "Espaces de fonctions \\`a moyenne fractionnaire int\\'egrable sur les groupes localement compacts", "abstract": "Let $G$ be a locally compact group which is $\\sigma $-compact, endowed with a left Haar measure $\\lambda .$ Denote by $e$ the unit element of $G$, and by $B$ an open relatively compact and symmetric neighbourhood of $e$. For every $(p,q) $ belonging to $[ 1 ; +\\infty ] ^{2}$, we give an equivalent and a priori more manageable definition of the Banach space $L_{(q,p)}^{\\pi}(G),$ defined by R. C. Busby and H. A. Smith in \\cite% {1}. In the case $G$ is a group of homogeneous type, we look at the subspaces $(L^{q},L^{p}) ^{\\alpha}(G)$ of the space $% L_{(q,p)}^{\\pi}(G)$. Theses subspaces are extensions to non abelian groups of the spaces of functions with integrable mean, defined by I. Fofana in \\cite{2}. Finally we show that $L^{\\alpha ,+\\infty}(G)$ is a complex subspace of $(L^{q},L^{p}) ^{\\alpha}(G)$."}
{"category": "Math", "title": "Complexity of links in 3-manifolds", "abstract": "We introduce a natural-valued complexity c(X) for pairs X=(M,L), where M is a closed orientable 3-manifold and L is a link contained in M. The definition employs simple spines, but for well-behaved X's we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveev's recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0,2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0,2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates."}
{"category": "Math", "title": "Quantized mixed tensor space and Schur-Weyl duality", "abstract": "Let $R$ be a commutative ring with one and $q$ an invertible element of $R$. The (specialized) quantum group ${\\mathbf U} = U_q(\\mathfrak{gl}_n)$ over $R$ of the general linear group acts on mixed tensor space $V^{\\otimes r}\\otimes {V^*}^{\\otimes s}$ where $V$ denotes the natural $\\mathbf U$-module $R^n$, $r,s$ are nonnegative integers and $V^*$ is the dual $\\mathbf U$-module to $V$. The image of $\\mathbf U$ in $\\mathrm{End}_R(V^{\\otimes r}\\otimes {V^*}^{\\otimes s})$ is called the rational $q$-Schur algebra $S_{q}(n;r,s)$. We construct a bideterminant basis of $S_{q}(n;r,s)$. There is an action of a $q$-deformation $\\mathfrak{B}_{r,s}^n(q)$ of the walled Brauer algebra on mixed tensor space centralizing the action of $\\mathbf U$. We show that $\\mathrm{End}_{\\mathfrak{B}_{r,s}^n(q)}(V^{\\otimes r}\\otimes {V^*}^{\\otimes s})=S_{q}(n;r,s)$. By \\cite{dipperdotystoll} the image of $\\mathfrak{B}_{r,s}^n(q)$ in $\\mathrm{End}_R(V^{\\otimes r}\\otimes {V^*}^{\\otimes s})$ is $\\mathrm{End}_{\\mathbf U}(V^{\\otimes r}\\otimes {V^*}^{\\otimes s})$. Thus mixed tensor space as $\\mathbf U$-$\\mathfrak{B}_{r,s}^n(q)$-bimodule satisfies Schur-Weyl duality."}
{"category": "Math", "title": "On positively curved 4-manifolds with S^1-symmetry", "abstract": "It is well-known by the work of Hsiang and Kleiner that every closed oriented positively curved 4-dimensional manifold with an effective isometric S^1-action is homeomorphic to S^4 or CP^2. As stated, it is a topological classification. The primary goal of this paper is to show that it is indeed a diffeomorphism classification for such 4-dimensional manifolds. The proof of this diffeomorphism classification also shows an even stronger statement that every positively curved simply connected 4-manifold with an isometric circle action admits another smooth circle action which extends to a 2-dimensional torus action and is equivariantly diffeomorphic to a linear action on S^4 or CP^2. The main strategy is to analyze all possible topological configurations of effective circle actions on simply connected 4-manifolds by using the so-called replacement trick of Pao."}
{"category": "Math", "title": "Minimal Surfaces in the Three-Dimensional Sphere and Minimal Hypersurfaces of Type Number Two", "abstract": "We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson equation. We obtain a classification theorem for bi-umbilical hypersurfaces of type number two. We prove that any minimal hypersurface of type number two with involutive distribution is generated by a minimal surface in the three-dimensional Euclidean space, or in the three dimensional sphere. Thus we prove that the theory of minimal hypersurfaces of type number two with involutive distribution is locally equivalent to the theory of minimal surfaces in the three dimensional Euclidean space or in the three-dimensional sphere."}
{"category": "Math", "title": "Constant mean curvature tori as stationary solutions to the Davey-Stewartson equation", "abstract": "A well known result of Da Rios and Levi-Civita says that a closed planar curve is elastic if and only if it is stationary under the localized induction (or smoke ring) equation, where stationary means that the evolution under the localized induction equation is by rigid motions. We prove an analogous result for surfaces: an immersion of a torus into the conformal 3-sphere has constant mean curvature with respect to a space form subgeometry if and only if it is stationary under the Davey-Stewartson flow."}
{"category": "Math", "title": "Koszulity of splitting algebras associated with cell complexes", "abstract": "We associate to a good cell decomposition of a manifold M a quadratic algebra and show that the Koszulity of the algebra implies a restriction on the Euler characteristic of M. For a two-dimensional manifold M the algebra is Koszul if and only if the Euler characteristic of M is two."}
{"category": "Math", "title": "On spun-normal and twisted squares surfaces", "abstract": "Given a 3 manifold M with torus boundary and an ideal triangulation, Yoshida and Tillmann give different methods to construct surfaces embedded in M from ideal points of the deformation variety. Yoshida builds a surface from twisted squares whereas Tillmann produces a spun-normal surface. We investigate the relation between the generated surfaces and extend a result of Tillmann's (that if the ideal point of the deformation variety corresponds to an ideal point of the character variety then the generated spun-normal surface is detected by the character variety) to the generated twisted squares surfaces."}
{"category": "Math", "title": "Uniqueness of solutions for an elliptic equation modeling MEMS", "abstract": "We study the effect of the parameter $\\lambda$, the dimension $N$, the profile $f$ and the geometry of the domain $\\Omega \\subset\\mathbb{R}^N$, on the question of uniqueness of the solutions to the following elliptic boundary value problem with a singular nonlinearity: $$ 180pt {{array}{ll} -\\Delta u= \\frac{\\lambda f(x)}{(1-u)^2} & \\hbox{in}\\Omega 0<u<1 &\\hbox{in}\\Omega u=0 &\\hbox{on}\\partial \\Omega. {array}. 130pt (S)_{\\lambda, f} $$ This equation has been proposed as a model for a simple electrostatic Micro-Electromechanical System (MEMS) device consisting of a thin dielectric elastic membrane with boundary supported at 0 below a rigid ground plate located at height z = 1."}
{"category": "Math", "title": "On the Minkowski Measure", "abstract": "The Minkowski Question Mark function relates the continued-fraction representation of the real numbers, to their binary expansion. This function is peculiar in many ways; one is that its derivative is 'singular'. One can show by classical techniques that its derivative must vanish on all rationals. Since the Question Mark itself is continuous, one concludes that the derivative must be non-zero on the irrationals, and is thus a discontinuous-everywhere function. This derivative is the subject of this essay. Various results are presented here: First, a simple but formal measure-theoretic construction of the derivative is given, making it clear that it has a very concrete existence as a Lebesgue-Stieltjes measure, and thus is safe to manipulate in various familiar ways. Next, an exact result is given, expressing the measure as an infinite product of piece-wise continuous functions, with each piece being a Mobius transform of the form (ax+b)/(cx+d). This construction is then shown to be the Haar measure of a certain transfer operator. A general proof is given that any transfer operator can be understood to be nothing more nor less than a push-forward on a Banach space; such push-forwards induce an invariant measure, the Haar measure, of which the Minkowski measure can serve as a prototypical example. Some minor notes pertaining to it's relation to the Gauss-Kuzmin-Wirsing operator are made."}
{"category": "Math", "title": "Regularity of the extremal solution in a MEMS model with advection", "abstract": "We consider the regularity of the extremal solution of the nonlinear eigenvalue problem (S)_\\lambda \\qquad {rcr} -\\Delta u + c(x) \\cdot \\nabla u &=& \\frac{\\lambda}{(1-u)^2} \\qquad {in $ \\Omega$}, u &=& 0 \\qquad {on $ \\pOm$}, where $ \\Omega $ is a smooth bounded domain in $ \\IR^N$ and $ c(x)$ is a smooth bounded vector field on $\\bar \\Omega$. We show that, just like in the advection-free model ($c\\equiv 0$), all semi-stable solutions are smooth if (and only if) the dimension $N\\leq 7$. The novelty here comes from the lack of a suitable variational characterization for the semi-stability assumption. We overcome this difficulty by using a general version of Hardy's inequality. In a forthcoming paper \\cite{CG2}, we indicate how this method applies to many other nonlinear eigenvalue problems involving advection (including the Gelfand problem), showing that they all essentially have the same critical dimension as their advection-free counterparts."}
{"category": "Math", "title": "Log canonical thresholds of binomial ideals", "abstract": "We prove that the log canonical thresholds of a large class of binomial ideals, such as complete intersection binomial ideals and the defining ideals of space monomial curves, are computable by linear programming."}
{"category": "Math", "title": "Set theory for category theory", "abstract": "Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number of such \"set-theoretic foundations for category theory,\" and describe their implications for the everyday use of category theory. We assume the reader has some basic knowledge of category theory, but little or no prior experience with formal logic or set theory."}
{"category": "Math", "title": "Expansive actions on uniform spaces and surjunctive maps", "abstract": "We present a uniform version of a result of M. Gromov on the surjunctivity of maps commuting with expansive group actions and discuss several applications. We prove in particular that for any group $\\Gamma$ and any field $\\K$, the space of $\\Gamma$-marked groups $G$ such that the group algebra $\\K[G]$ is stably finite is compact."}
{"category": "Math", "title": "Directly Indecomposables in Semidegenerate Varieties of Connected po-Groupoids", "abstract": "We study varieties with a term-definable poset structure, \"po-groupoids\". It is known that connected posets have the \"strict refinement property\" (SRP). In [arXiv:0808.1860v1 [math.LO]] it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. We obtain such a set for semidegenerate varieties of connected po-groupoids and show its quantifier complexity is bounded in general."}
{"category": "Math", "title": "Motives associated to sums of graphs", "abstract": "The Feynman amplitude associated to a graph is a period of a certain motive. The sum of these motive classes over all connected graphs with no multiple edges or tadpoles and n vertices is defined in the Grothendieck ring of varieties. This sum is shown to lie in the subring generated by the affine line. It follows from work of Belkale and Brosnan that motives of individual graphs do not lie in this subring."}
{"category": "Math", "title": "Accuracy of the Tracy-Widom limit for the largest eigenvalue in white Wishart matrices", "abstract": "Let A be a p-variate real Wishart matrix on n degrees of freedom with identity covariance. The distribution of the largest eigenvalue in A has important applications in multivariate statistics. Consider the asymptotics when p grows in proportion to n, it is known from Johnstone (2001) that after centering and scaling, these distributions approach the orthogonal Tracy-Widom law for real-valued data, which can be numerically evaluated and tabulated in software. Under the same assumption, we show that more carefully chosen centering and scaling constants improve the accuracy of the distributional approximation by the Tracy-Widom limit to second order: O(min(n,p)^{-2/3}). Together with the numerical simulation, it implies that the Tracy-Widom law is an attractive approximation to the distributions of these largest eigenvalues, which is important for using the asymptotic result in practice. We also provide a parallel accuracy result for the smallest eigenvalue of A when n > p."}
{"category": "Math", "title": "Approximation theorems for Banach-valued almost periodic and semi-almost periodic holomorphic functions", "abstract": "The paper studies semi-almost periodic holomorphic functions on a polydisk which have, in a sense, the weakest possible discontinuities on the boundary torus. The basic result used in the proofs is an extension of the classical Bohr approximation theorem for almost periodic holomorphic functions on a strip to the case of Banach-valued almost periodic holomorphic functions."}
{"category": "Math", "title": "On the relative and bi-relative K-theory of rings of finite characteristic", "abstract": "We prove that the relative K-groups associated with a nilpotent extension of Z/p^N Z-algebras and the bi-relative K-groups associated with a Milnor square of Z/p^N Z-algebras are p-primary torsion groups of bounded exponent. We also show that, in general, the cyclotomic trace map extends from Quillen K-theory to Bass completed non-connective algebraic K-theory."}
{"category": "Math", "title": "Stratifying modular representations of finite groups", "abstract": "We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras."}
{"category": "Math", "title": "2-universal Hermitian lattices over imaginary quadratic fields", "abstract": "A positive definite Hermitian lattice is said to be 2-universal if it represents all positive definite binary Hermitian lattices. We find all 2-universal ternary and quaternary Hermitian lattices over imaginary quadratic number fields."}
{"category": "Math", "title": "Enumeration of bilaterally symmetric 3-noncrossing partitions", "abstract": "Schutzenberger's theorem for the ordinary RSK correspondence naturally extends to Chen et. al's correspondence for matchings and partitions. Thus the counting of bilaterally symmetric $k$-noncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous result for 3-noncrossing partitions, we use a different technique to develop a Maple package for 2-dimensional vacillating lattice walk enumeration problems. The package also applies to the hesitating case. As applications, we find several interesting relations for some special bilaterally symmetric partitions."}
{"category": "Math", "title": "Double Schubert polynomials for the classical groups", "abstract": "For each infinite series of the classical Lie groups of type B,C or D, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur's Q- or P-functions defined earlier by Ivanov."}
{"category": "Math", "title": "The Toric Geometry of Triangulated Polygons in Euclidean Space", "abstract": "Speyer and Sturmfels [SpSt] associated Gr\\\"obner toric degenerations $\\mathrm{Gr}_2(\\C^n)^{\\tree}$ of $\\mathrm{Gr}_2(\\C^n)$ to each trivalent tree $\\tree$ with $n$ leaves. These degenerations induce toric degenerations $M_{\\br}^{\\tree}$ of $M_{\\br}$, the space of $n$ ordered, weighted (by $\\br$) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers as stratified symplectic spaces and describe the action of the compact part of the torus as \"bendings of polygons.\" We prove the conjecture of Foth and Hu [FH] that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida [KY]."}
{"category": "Math", "title": "Gorenstein Semigroup Algebras of Weighted Trees", "abstract": "We classify exactly when the toric algebras $\\C[S_{\\tree}(\\br)]$ are Gorenstein. These algebras arise as toric deformations of algebras of invariants of the Cox-Nagata ring of the blow-up of $n-1$ points on $\\mathbb{P}^{n-3}$, or equivalently algebras of the ring of global sections for the Pl\\\"ucker embedding of weight varieties of the Grassmanian $Gr_2(\\C^n)$, and algebras of global sections for embeddings of moduli of weighted points on $\\mathbb{P}^1$. As a corollary, we find exactly when these families of rings are Gorenstein as well."}
{"category": "Math", "title": "Very elementary interpretations of the Euler-Mascheroni constant from counting divisors in intervals", "abstract": "Theorem 1 Let F:N-->R stand for any function which a) $F$ monotonically weakly increases; b) $F$ tends to infinity; and c) such that $q/F(q)$ tends to infinity. Let Z_F(q) equal the number of divisors of q less than sqrt{F(q)} minus the number of divisors of q between sqrt{F(q)} and F(q). Then, on the average, Z_F(q) equals Euler's constant Theorem 2 Fix a in (0,1). Write A for the average number of divisors of n that lie in (0,sqrt{a n}) minus the number of that lie in (sqrt{a n},a n)$. Then A= (sum_{i=1}^{\\lceil {1-a}/a \\rceil} \\frac{1}{i}) - ln(1/a)."}
{"category": "Math", "title": "The Sectional Curvature of the Tangent Bundles with General Natural Lifted Metrics", "abstract": "We study some properties of the tangent bundles with metrics of general natural lifted type. We consider a Riemannian manifold $(M,g)$ and we find the conditions under which the Riemannian manifold $(TM,G)$, where $TM$ is the tangent bundle of $M$ and $G$ is the general natural lifted metric of $g$, has constant sectional curvature."}
{"category": "Math", "title": "On the behaviour of the Atiyah Conjecture under taking subgroups and under taking quotients with finite kernel", "abstract": "We state and prove a condition under which the strong Atiyah Conjecture carries over to subgroups. Moreover, we show that if a group satisfies the (strong) Atiyah Conjecture then any quotient with finite kernel does."}
{"category": "Math", "title": "Cotangent Bundles with General Natural Kahler Structures", "abstract": "We study the conditions under which an almost Hermitian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ is K\\\" ahlerian. First, we obtain the algebraic conditions under which the manifold $(T^*M,G,J)$ is almost Hermitian. Next we get the integrability conditions for the almost complex structure $J$, then the conditions under which the associated 2-form is closed. The manifold $(T^*M,G,J)$ is K\\\" ahlerian iff it is almost Kahlerian and the almost complex structure $J$ is integrable. It follows that the family of Kahlerian structures of above type on $T^*M$ depends on three essential parameters (one is a certain proportionality factor, the other two are parameters involved in the definition of $J$)."}
{"category": "Math", "title": "Postulation of general quartuple fat point schemes in P^3", "abstract": "We study the postulation of a general union $Y$ of double, triple, and quartuple points of $\\mathbb{P}^3$. We prove that $Y$ has the expected postulation in degree $d\\ge 41$, using the Horace differential lemma. We also discuss the cases of low degree with the aid of computer algebra."}
{"category": "Math", "title": "A rigorous lower confidence bound for the expectation of a positive random variable", "abstract": "Given an IID sample from a positive distribution, we provide a method for constructing rigorous finite sample lower confidence bounds for the expectation of the distribution. The method is based on constructing rigorous confidence regions for the cdf of the distribution. We provide some analysis of the asymptotical behavior of the rigorous LCBs. We apply the method to obtain an LCB for a particular, controversial, empirical data set, where the validity of standard methods has been called into question."}
{"category": "Math", "title": "Harmonic almost contact structures via the intrinsic torsion", "abstract": "We go further on the study of harmonicity for almost contact metric structures already initiated by Vergara-Diaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric structures. Additionally, we study the harmonicity of such structures as a map into the quotient bundle of the oriented orthonormal frames by the action of the structural group U(n)x1. Finally, by using a Bochner type formula proved by Bor and Hernandez Lamoneda, we display some examples which give the absolute minimum for the energy."}
{"category": "Math", "title": "Numerical Solution of an Inverse Problem in Size-Structured Population Dynamics", "abstract": "We consider a size-structured model for cell division and address the question of determining the division (birth) rate from the measured stable size distribution of the population. We propose a new regularization technique based on a filtering approach. We prove convergence of the algorithm and validate the theoretical results by implementing numerical simulations, based on classical techniques. We compare the results for direct and inverse problems, for the filtering method and for the quasi-reversibility method proposed in [Perthame-Zubelli]."}
{"category": "Math", "title": "On Alexander Polynomials of Certain (2,5) Torus Curves", "abstract": "In this paper, we compute Alexander polynomials of a torus curve C of type (2, 5), C : f(x, y) = f_2(x, y)^5 + f_5(x, y)^2 = 0, under the assumption that the origin O is the unique inner singularity and f2 = 0 is an irreducible conic. We show that the Alexander polynomial remains the same with that of a generic torus curve as long as C is irreducible."}
{"category": "Math", "title": "Discrete Mechanics and Optimal Control: an Analysis", "abstract": "The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated."}
{"category": "Math", "title": "A dichotomy for Borel functions", "abstract": "The dichotomy discovered by Solecki in \\cite{Sol} states that any Baire class 1 function is either $\\sigma$-continuous or \"includes\" the Pawlikowski function $P$. The aim of this paper is to give an argument which is simpler than the original proof of Solecki and gives a stronger statement: a dichotomy for all Borel functions."}
{"category": "Math", "title": "The telescope conjecture for hereditary rings via Ext-orthogonal pairs", "abstract": "For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exact sequence is constructed. The pairs of finite type are characterized and two consequences for the class of hereditary rings are established: homological epimorphisms and universal localizations coincide, and the telescope conjecture for the derived category holds true. However, we present examples showing that neither of these two statements is true in general for rings of global dimension 2."}
{"category": "Math", "title": "The rough path associated to the multidimensional analytic fbm with any Hurst parameter", "abstract": "In this paper, we consider a complex-valued d-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion. This process has been introduced by the second author of the article, and both its real and imaginary parts, restricted on the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on the analytic fBm can be constructed for any value of the Hurst parameter in (0,1/2). This allows in particular to solve differential equations driven by this process in a neighborhood of 0 of the complex upper half-plane, thanks to a variant of the usual rough path theory due to Gubinelli."}
{"category": "Math", "title": "Hamiltonian inclusions with convex dissipation with a view towards applications", "abstract": "We propose a generalization of hamiltonian mechanics, as a hamiltonian inclusion with convex dissipation function. We obtain a dynamical version of the approach of Mielke to quasistatic rate-independent processes. Then we show that a class of models of dynamical brittle damage can be formulated in this setting."}
{"category": "Math", "title": "A bijection between noncrossing and nonnesting partitions of types A and B", "abstract": "The total number of noncrossing partitions of type $\\Psi$ is the $n$th Catalan number $\\frac{1}{n+1}\\binom{2n}{n}$ when $\\Psi=A_{n-1}$, and the binomial $\\binom{2n}{n}$ when $\\Psi=B_n$, and these numbers coincide with the correspondent number of nonnesting partitions. For type A, there are several bijective proofs of this equality, being the intuitive map that locally converts each crossing to a nesting one of them. In this paper we present a bijection between nonnesting and noncrossing partitions of types A and B that generalizes the type A bijection that locally converts each crossing to a nesting."}
{"category": "Math", "title": "Hodge polynomials and birational types of moduli spaces of coherent systems on elliptic curves", "abstract": "In this paper we consider moduli spaces of coherent systems on an elliptic curve. We compute their Hodge polynomials and determine their birational types in some cases. Moreover we prove that certain moduli spaces of coherent systems are isomorphic. This last result uses the Fourier-Mukai transform of coherent systems introduced by Hern\\'andez Ruiperez and Tejero Prieto."}
{"category": "Math", "title": "On extensions of d.c. functions and convex functions", "abstract": "We show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems, concerning extendability of continuous convex functions from a closed subspace of a normed linear space, complement recent results of J.Borwein, V.Montesinos and J.Vanderwerff."}
{"category": "Math", "title": "Uniqueness results for convex Hamilton-Jacobi equations under $p>1$ growth conditions on data", "abstract": "Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like $o(1+|x|^p)$ at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like $O(1+|x|^p)$ at infinity. This latter case encompasses some equations related to backward stochastic differential equations."}
{"category": "Math", "title": "A note on 3-colorable plane graphs without 5- and 7-cycles", "abstract": "Borodin et al figured out a gap of the paper published at J. Combinatorial Theory Ser. B (Vol.96 (2006) 958--963), and gave a new proof with the similar technique. The purpose of this note is to fix the gap by slightly revising the definition of special faces, and adding a few lines of explanation in the proofs (new added text are all in black font)."}
{"category": "Math", "title": "Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem", "abstract": "The cyclohedron (Bott-Taubes polytope) arises both as the polyhedral realization of the poset of all cyclic bracketings of a circular word and as an essential part of the Fulton-MacPherson compactification of the configuration space of n distinct, labelled points on the circle S^1. The \"polygonal pegs problem\" asks whether every simple, closed curve in the plane or in the higher dimensional space admits an inscribed polygon of a given shape. We develop a new approach to the polygonal pegs problem based on the Fulton-MacPherson (Axelrod-Singer, Kontsevich) compactification of the configuration space of (cyclically) ordered n-element subsets in S^1. Among the results obtained by this method are proofs of Grunbaum's conjecture about affine regular hexagons inscribed in smooth Jordan curves and a new proof of the conjecture of Hadwiger about inscribed parallelograms in smooth, simple, closed curves in the 3-space (originally established by Victor Makeev)."}
{"category": "Math", "title": "Higher-dimensional categories with finite derivation type", "abstract": "We study convergent (terminating and confluent) presentations of n-categories. Using the notion of polygraph (or computad), we introduce the homotopical property of finite derivation type for n-categories, generalizing the one introduced by Squier for word rewriting systems. We characterize this property by using the notion of critical branching. In particular, we define sufficient conditions for an n-category to have finite derivation type. Through examples, we present several techniques based on derivations of 2-categories to study convergent presentations by 3-polygraphs."}
{"category": "Math", "title": "Timelike $B_2$-slant helices in Minkowski space $E_1^4$", "abstract": "We consider a unit speed timelike curve $\\alpha$ in Minkowski 4-space $E_1^4$ and denote the Frenet frame of $\\alpha$ by $\\{T,N,B_1,B_2\\}$. We say that $\\alpha$ is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction $U$ of $E_1^4$. In this work we study those helices where the function $<B_2,U>$ is constant and we give different characterizations of such curves."}
{"category": "Math", "title": "Extensions of Lie Brackets", "abstract": "We provide a framework for extensions of Lie algebroids, including non-abelian extensions and Lie algebroids over different bases. Our approach involves Ehresmann connections, which allows straight generalizations of classical constructions. We exhibit a filtration in cohomology and explain the associated spectral sequence. We also give a description of the groupoid integrating an extension in case a complete connection can be fixed. The problem of integrability is also studied."}
{"category": "Math", "title": "On slant helices in Minkowski space $E_1^3$", "abstract": "We consider a curve $\\alpha=\\alpha(s)$ in Minkowski 3-space $E_1^3$ and denote by $\\{T,N,B}$ the Frenet frame of $\\alpha$. We say that $\\alpha$ is a slant helix if there exists a fixed direction $U$ of $E_1^3$ such that the function $<N(s),U>$ is constant. In this work we give characterizations of slant helices in terms of the curvature and torsion of $\\alpha$."}
{"category": "Math", "title": "Arithmetic groups and the affine E8 Dynkin diagram", "abstract": "Several decades ago, John McKay suggested a correspondence between nodes of the affine E8 Dynkin diagram and certain conjugacy classes in the Monster group. Thanks to Monstrous Moonshine, this correspondence can be recast as an assignment of discrete subgroups of PSL2 to nodes of the affine E8 Dynkin diagram. The purpose of this article is to give an explanation for this latter correspondence using elementary properties of the group PSL2. We also obtain a super analogue of McKay's observation, in which conjugacy classes of the Monster are replaced by conjugacy classes of Conway's group -- the automorphism group of the Leech lattice."}
{"category": "Math", "title": "L^2-Torsion and Bounded Measure Equivalence of Groups", "abstract": "This paper has been withdrawn by the author due to an error in the proof of Proposition 4.8."}
{"category": "Math", "title": "On the K-stability of complete intersections in polarized manifolds", "abstract": "We consider the problem of existence of constant scalar curvature Kaehler metrics on complete intersections of sections of vector bundles. In particular we give general formulas relating the Futaki invariant of such a manifold to the weight of sections defining it and to the Futaki invariant of the ambient manifold. As applications we give a new Mukai-Umemura-Tian like example of Fano 5-fold admitting no Kaehler-Einstein metric and a strong evidence of K-stability of complete intersections on Grassmannians."}
{"category": "Math", "title": "Two counterexamples in rational and interval dynamics", "abstract": "In rational dynamics, we prove the existence of a polynomial that satisfies the Topological Collet-Eckmann condition, but which has a recurrent critical orbit that is not Collet-Eckmann. This shows that the converse of the main theorem in [11] does not hold. In interval dynamics, we show that the Collet-Eckmann property for recurrent critical orbits is not a topological invariant for real polynomials with negative Schwarzian derivative. This contradicts a conjecture of Swiatek [22]."}
{"category": "Math", "title": "Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number", "abstract": "This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number $k$, optimal order (with respect to $h$) error estimate in the broken $H^1$-norm and sub-optimal order estimate in the $L^2$-norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size $h$ is restricted to the preasymptotic regime (i.e., $k^2 h \\gtrsim 1$). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to $k$) in the error bounds. The novelties of the proposed interior penalty discontinuous Galerkin methods include: first, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To the end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in \\cite{cummings00,Cummings_Feng06,hetmaniuk07}."}
{"category": "Math", "title": "A modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations", "abstract": "This paper develops a fully discrete modified characteristic finite element method for a coupled system consisting of the fully nonlinear Monge-Amp\\'ere equation and a transport equation. The system is the Eulerian formulation in the dual space for the B. J. Hoskins' semigeostrophic flow equations, which are widely used in meteorology to model slowly varying flows constrained by rotation and stratification. To overcome the difficulty caused by the strong nonlinearity, we first formulate (at the differential level) a vanishing moment approximation of the semigeostrophic flow equations, a methodology recently proposed by the authors \\cite{Feng1,Feng2}, which involves approximating the fully nonlinear Monge-Amp\\'ere equation by a family of fourth order quasilinear equations. We then construct a fully discrete modified characteristic finite element method for the regularized problem. It is shown that under certain mesh and time stepping constraints, the proposed numerical method converges with an optimal order rate of convergence. In particular, the obtained error bounds show explicit dependence on the regularization parameter $\\vepsi$. Numerical tests are also presented to validate the theoretical results and to gauge the efficiency of the proposed fully discrete modified characteristic finite element method."}
{"category": "Math", "title": "Sumsets and the convex hull", "abstract": "We extend Freiman's inequality on the cardinality of the sumset of a $d$ dimensional set. We consider different sets related by an inclusion of their convex hull, and one of them added possibly several times."}
{"category": "Math", "title": "Plunnecke's inequality for different summands", "abstract": "The aim of this paper is to prove a general version of Pl\\\"unnecke's inequality. Namely, assume that for finite sets $A$, $B_1, ... B_k$ we have information on the size of the sumsets $A+B_{i_1}+... +B_{i_l}$ for all choices of indices $i_1, ... i_l.$ Then we prove the existence of a non-empty subset $X$ of $A$ such that we have `good control' over the size of the sumset $X+B_1+... +B_k$. As an application of this result we generalize an inequality of \\cite{gymr} concerning the submultiplicativity of cardinalities of sumsets."}
{"category": "Math", "title": "The Geometry and Dynamics of Interacting Rigid Bodies and Point Vortices", "abstract": "We derive the equations of motion for a planar rigid body of circular shape moving in a 2D perfect fluid with point vortices using symplectic reduction by stages. After formulating the theory as a mechanical system on a configuration space which is the product of a space of embeddings and the special Euclidian group in two dimensions, we divide out by the particle relabelling symmetry and then by the residual rotational and translational symmetry. The result of the first stage reduction is that the system is described by a non-standard magnetic symplectic form encoding the effects of the fluid, while at the second stage, a careful analysis of the momentum map shows the existence of two equivalent Poisson structures for this problem. For the solid-fluid system, we hence recover the ad hoc Poisson structures calculated by Shashikanth, Marsden, Burdick and Kelly on the one hand, and Borisov, Mamaev, and Ramodanov on the other hand. As a side result, we obtain a convenient expression for the symplectic leaves of the reduced system and we shed further light on the interplay between curvatures and cocycles in the description of the dynamics."}
{"category": "Math", "title": "A corollary of the b-function lemma", "abstract": "Let $X$ be an algebraic variety, $f$ a regular function, $j:U\\subset X$ the complement to the locus of vanishing of $f$, and $M$ a holonomic D-module on $U$. Consider the $D_U[s]$-module $M\\otimes \"f^s\"$. The goal of this note is to describe all $D_X[s]$-submodules $N\\subset j_*(M\\otimes \"f^s\")$ such that $j^*(N)\\simeq M\\otimes \"f^s\"$."}
{"category": "Math", "title": "The fully residually F quotients of F*<x,y>", "abstract": "We describe the fully residually F; or limit groups relative to F; (where F is a free group) that arise from systems of equations in two variables over F that have coefficients in F."}
{"category": "Math", "title": "Characterizing hyperbolic spaces and real trees", "abstract": "Let X be a geodesic metric space. Gromov proved that there exists k>0 such that if every sufficiently large triangle T satisfies the Rips condition with constant k times pr(T), where pr(T) is the perimeter T, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for k. We also show that if all the triangles T in X satisfy the Rips condition with constant k times pr(T), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is hyperbolic if and only if every asymptotic cone of X is a real tree."}
{"category": "Math", "title": "Quivers with relations arising from Koszul algebras of $\\mathfrak g$-invariants", "abstract": "Let $\\mathfrak g$ be a complex simple Lie algebra and let $\\Psi$ be an extremal set of positive roots. One associates with $\\Psi$ an infinite dimensional Koszul algebra $\\bold S_\\Psi^{\\lie g}$ which is a graded subalgebra of the locally finite part of $((\\bold V)^{op}\\tensor S(\\lie g))^{\\lie g}$, where $\\bold V$ is the direct sum of all simple finite dimensional $\\lie g$-modules. We describe the structure of the algebra $\\bold S_\\Psi^{\\lie g}$ explicitly in terms of an infinite quiver with relations for $\\lie g$ of types $A$ and $C$. We also describe several infinite families of quivers and finite dimensional algebras arising from this construction."}
{"category": "Math", "title": "On the regularization of conservative maps", "abstract": "We show that smooth maps are $C^1$-dense among $C^1$ volume preserving maps."}
{"category": "Math", "title": "The Fundamental Group of Balanced Simplicial Complexes and Posets", "abstract": "We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets."}
{"category": "Math", "title": "Kernel algebras and generalized Fourier-Mukai transforms", "abstract": "We introduce and study kernel algebras, i.e., algebras in the category of sheaves on a square of a scheme, where the latter category is equipped with a monoidal structure via a natural convolution operation. We show that many interesting categories, such as D-modules, equivariant sheaves and their twisted versions, arise as categories of modules over kernel algebras. We develop the techniques of constructing derived equivalences between these module categories. As one application we generalize the results of math.AG/9901009 concerning modules over algebras of twisted differential operators on abelian varieties. As another application we recover and generalize the results of Laumon in alg-geom/9603004 concerning an analog of the Fourier transform for derived categories of quasicoherent sheaves on a dual pair of generalized 1-motives."}
{"category": "Math", "title": "Conditional Limit Results for Type I Polar Distributions", "abstract": "Let (S_1,S_2)=(R \\cos(\\Theta), R \\sin (\\Theta)) be a bivariate random vector with associated random radius R which has distribution function $F$ being further independent of the random angle \\Theta. In this paper we investigate the asymptotic behaviour of the conditional survivor probability \\Psi_{\\rho,u}(y):=\\pk{\\rho S_1+ \\sqrt{1- \\rho^2} S_2> y \\lvert S_1> u}, \\rho \\in (-1,1),\\in R when u approaches the upper endpoint of F. On the density function of \\Theta we require a certain local asymptotic behaviour at 0, whereas for F we require that it belongs to the Gumbel max-domain of attraction. The main result of this contribution is an asymptotic expansion of \\Psi_{\\rho,u}, which is then utilised to construct two estimators for the conditional distribution function 1- \\Psi_{\\rho,u}. Further, we allow \\Theta to depend on u."}
{"category": "Math", "title": "Two analytical formulae of the temperature inside a body by using partial lateral and initial data", "abstract": "This paper considerers the problem of computing the value of a solution of the heat equation at a given point inside a bounded domain after the initial time. It is assumed that the initial value of the solution inside the domain (possibly in a part of the domain) is known; the boundary value and the normal derivative on a part of the boundary of the domain over a finite time interval are known. Two analytical formulae for the problem are given. Both formulae make use of a special fundamental solution having a large parameter of the backward heat equation."}
{"category": "Math", "title": "Curvature flow to Nirenberg problem", "abstract": "In this note, we study the curvature flow to Nirenberg problem on $S^2$ with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed non-negative Gaussian curvature $f$ has its positive part, which possesses non-degenerate critical points such that $\\Delta_{S^2} f>0$ at the saddle points."}
{"category": "Math", "title": "The exact distribution of the sample variance from bounded continuous random variables", "abstract": "For a sample of absolutely bounded i.i.d. random variables with a continuous density the cumulative distribution function of the sample variance is represented by a univariate integral over a Fourier series. If the density is a polynomial or a trigonometrical polynomial the coefficients of this series are simple finite terms containing only the error function, the exponential function and powers. In more general cases - e.g. for all beta densities - the coefficients are given by some series expansions. The method is generalized to positive semi-definite quadratic forms of bounded independent but not necessarily identically distributed random variables if the form matrix differs from a diagonal matrix D > 0 only by a matrix of rank 1"}
{"category": "Math", "title": "Fundamental theorem of hyperbolic geometry without the injectivity assumption", "abstract": "Let $\\mathbb{H}^n$ be the $n-$dimensional hyperbolic space. It is well known that, if $f: \\mathbb{H}^n\\to \\mathbb{H}^n$ is a bijection that preserves $r-$dimensional hyperplanes, then $f$ is an isometry. In this paper we make neither injectivity nor $r-$hyperplane preserving assumptions on $f$ and prove the following result: Suppose that $f: \\mathbb{H}^n\\to \\mathbb{H}^n$ is a surjective map and maps an $r-$hyperplane into an $r-$hyperplane, then $f$ is an isometry. The Euclidean version was obtained by A. Chubarev and I. Pinelis in 1999 among other things. Our proof is essentially different from their and the similar problem arising in the spherical case is open."}
{"category": "Math", "title": "Powers of sequences and convergence of ergodic averages", "abstract": "A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\\mathcal{B},\\mu,T)$ and any bounded measurable function $f$, the averages $ \\frac1N \\sum_{n=1}^N f(T^{s_n}x)$ converge in the $L^2$ norm. We construct a sequence $(s_n)$ that is good for the mean ergodic theorem, but the sequence $(s_n^2)$ is not. Furthermore, we show that for any set of bad exponents $B$, there is a sequence $(s_n)$ where $(s_n^k)$ is good for the mean ergodic theorem exactly when $k$ is not in $B$. We then extend this result to multiple ergodic averages. We also prove a similar result for pointwise convergence of single ergodic averages."}
{"category": "Math", "title": "Fibr\\'es de Schwarzenberger et fibr\\'es logarithmiques g\\'en\\'eralis\\'es", "abstract": "We propose a generalization of logarithmic and Schwarzenberger bundles over $\\P^n=\\P^n(\\C)$ when the rank is greater than $n$. The first ones are associated to finite sets of points on $\\P^{n\\vee}$ and the second ones to curves with degree greater than $n$ on $\\P^{n\\vee}$. On the projective plane we show that two logarithmic bundles are isomorphic if and only if they are associated to the same set of points or if the two sets of points belong to a curve of degree equal to the rank of the considered bundles."}
{"category": "Math", "title": "Abelianisation of orthogonal groups and the fundamental group of modular varieties", "abstract": "We study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms. We show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry, in particular the moduli of K3 surfaces. We give applications to modular forms and to computing the fundamental groups of some moduli spaces."}
{"category": "Math", "title": "Towards a Better Understanding of the Semigroup Tree", "abstract": "In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier. These regularites admit two different types of behavior and in this work we investigate which of the two types takes place in particular for well-known classes of semigroups. Also we study the question of what kind of chains appear in the tree and characterize the properties (like being (in)finite) thereof. We conclude with some thoughts that show how this study of the semigroup tree may help in solving the conjecture of Fibonacci-like behavior of the number of semigroups with given genus."}
{"category": "Math", "title": "Drinfel'd doubles and Shapovalov determinants", "abstract": "The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztig's small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument. Key words: Hopf algebra, Nichols algebra, quantum group, representation"}
{"category": "Math", "title": "Large localizations of finite groups", "abstract": "We construct examples of localizations in the category of groups which take the Mathieu group $M_{11}$ to groups of arbitrarily large cardinality which are ``abelian up to finitely many generators''. The paper is part of a broader study on the group theoretic properties which are or are not preserved by localizations."}
{"category": "Math", "title": "On the existence of shortest directed networks", "abstract": "A directed network connecting a set A to a set B is a digraph containing an a-b path for each a in A and b in B. Vertices in the directed network not in A or B are called Steiner points. We show that in a finitely compact metric space in which geodesics exist, any two finite sets A and B are connected by a shortest directed network. We also bound the number of Steiner points by a function of the sizes of A and B. Previously, such an existence result was known only for the Euclidean plane [M. Alfaro, Pacific J. Math. 167 (1995) 201-214]. The main difficulty is that, unlike the undirected case (Steiner minimal trees), the underlying graphs need not be acyclic. Existence in the undirected case was first shown by E. J. Cockayne [Canad. Math. Bull. 10 (1967) 431-450]."}
{"category": "Math", "title": "On finiteness and rigidity of J-holomorphic curves in symplectic three-folds", "abstract": "Given a symplectic three-fold $(M,\\omega)$ we show that for a generic almost complex structure $J$ which is compatible with $\\omega$, there are finitely many $J$-holomorphic curves in $M$ of any genus $g\\geq 0$ representing a homology class $\\beta$ in $\\H_2(M,\\Z)$ with $c_1(M).\\beta=0$, provided that the divisibility of $\\beta$ is at most 4 (i.e. if $\\beta=n\\alpha$ with $\\alpha\\in H_2(M,\\Z)$ and $n\\in \\Z$ then $n\\leq 4$). Moreover, each such curve is embedded and 4-rigid."}
{"category": "Math", "title": "Lelek's problem is not a metric problem", "abstract": "We show that Lelek's problem on the chainability of continua with span zero is not a metric problem: from a non-metric counterexample one can construct a metric one."}
{"category": "Math", "title": "Some Two-Step Procedures for Variable Selection in High-Dimensional Linear Regression", "abstract": "We study the problem of high-dimensional variable selection via some two-step procedures. First we show that given some good initial estimator which is $\\ell_{\\infty}$-consistent but not necessarily variable selection consistent, we can apply the nonnegative Garrote, adaptive Lasso or hard-thresholding procedure to obtain a final estimator that is both estimation and variable selection consistent. Unlike the Lasso, our results do not require the irrepresentable condition which could fail easily even for moderate $p_n$ (Zhao and Yu, 2007) and it also allows $p_n$ to grow almost as fast as $\\exp(n)$ (for hard-thresholding there is no restriction on $p_n$). We also study the conditions under which the Ridge regression can be used as an initial estimator. We show that under a relaxed identifiable condition, the Ridge estimator is $\\ell_{\\infty}$-consistent. Such a condition is usually satisfied when $p_n\\le n$ and does not require the partial orthogonality between relevant and irrelevant covariates which is needed for the univariate regression in (Huang et al., 2008). Our numerical studies show that when using the Lasso or Ridge as initial estimator, the two-step procedures have a higher sparsity recovery rate than the Lasso or adaptive Lasso with univariate regression used in (Huang et al., 2008)."}
{"category": "Math", "title": "Conformally flat tangent bundles with general natural lifted metrics", "abstract": "We study the conditions under which the tangent bundle $(TM,G)$ of an $n$-dimensional Riemannian manifold $(M,g)$ is conformally flat, where $G$ is a general natural lifted metric of $g$. We prove that the base manifold must have constant sectional curvature and we find some expressions for the natural lifted metric $G$, such that the tangent bundle $(TM,G)$ become conformally flat."}
{"category": "Math", "title": "The Holomorphic Sectional Curvature of General Natural K\\\"Ahler Structures on Cotangent Bundles", "abstract": "We study the conditions under which a K\\\"ahlerian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ has constant holomorphic sectional curvature. We obtain that a certain parameter involved in the condition for $(T^*M,G,J)$ to be a K\\\"ahlerian manifold, is expressed as a rational function of the other two, their derivatives, the constant sectional curvature of the base manifold $(M,g)$, and the constant holomorphic sectional curvature of the general natural K\\\"ahlerian structure $(G,J)$."}
{"category": "Math", "title": "Maximal solutions for $-\\Delta u+u^q=0$ in open or finely open sets", "abstract": "We study the existence and uniqueness of new classes of solutions of the superlinear equation $-\\Delta u+u^q=0$ (q>1) in a domain of R^N or in a finely open set for the topology associated to the Bessel capacity C_{2,q'}. Condition of existence or uniqueness of solutions with boundary blow-up are obtained generalizing the results of Dhersin-Le Gall and of Labutin."}
{"category": "Math", "title": "Random Chain Recurrent Sets for Random Dynamical Systems", "abstract": "It is known by the Conley's theorem that the chain recurrent set $CR(\\phi)$ of a deterministic flow $\\phi$ on a compact metric space is the complement of the union of sets $B(A)-A$, where $A$ varies over the collection of attractors and $B(A)$ is the basin of attraction of $A$. It has recently been shown that a similar decomposition result holds for random dynamical systems on noncompact separable complete metric spaces, but under a so-called \\emph{absorbing condition}. In the present paper, the authors introduce a notion of random chain recurrent sets for random dynamical systems, and then prove the random Conley's theorem on noncompact separable complete metric spaces \\emph{without} the absorbing condition."}
{"category": "Math", "title": "Topology of Fatou components for endomorphisms of CP^k: Linking with the Green's Current", "abstract": "Little is known about the global topology of the Fatou set $U(f)$ for holomorphic endomorphisms $f: \\mathbb{CP}^k \\to \\mathbb{CP}^k$, when $k >1$. Classical theory describes $U(f)$ as the complement in $ \\mathbb{CP}^k$ of the support of a dynamically-defined closed positive $(1,1)$ current. Given any closed positive $(1,1)$ current $S$ on $ \\mathbb{CP}^k$, we give a definition of linking number between closed loops in $\\mathbb{CP}^k \\setminus \\supp S$ and the current $S$. It has the property that if $lk(\\gamma,S) \\neq 0$, then $\\gamma$ represents a non-trivial homology element in $H_1(\\mathbb{CP}^k \\setminus \\supp S)$. As an application, we use these linking numbers to establish that many classes of endomorphisms of $\\mathbb{CP}^2$ have Fatou components with infinitely generated first homology. For example, we prove that the Fatou set has infinitely generated first homology for any polynomial endomorphism of $\\mathbb{CP}^2$ for which the restriction to the line at infinity is hyperbolic and has disconnected Julia set. In addition we show that a polynomial skew product of $\\mathbb{CP}^2$ has Fatou set with infinitely generated first homology if some vertical Julia set is disconnected. We then conclude with a section of concrete examples and questions for further study."}
{"category": "Math", "title": "f-categories and Tate motives", "abstract": "Using Beilinson's theory of f-categories, we prove that the triangulated category of Tate motives over a field k is equivalent to the bounded derived category of its heart, provided that k is algebraic over the rationals. This answers a question asked by Levine."}
{"category": "Math", "title": "Ample divisors on moduli spaces of weighted pointed rational curves, with applications to log MMP for $\\bar{M}_{0,n}$", "abstract": "We introduce a new technique for proving positivity of certain divisor classes on $\\bar{M}_{0,n}$ and its weighted variants. Our methods give an unconditional description of the spaces of symmetric weighted pointed rational curves as log canonical models of $\\bar{M}_{0,n}$."}
{"category": "Math", "title": "Finite Chevalley groups and loop groups", "abstract": "Let p, ell be distinct primes and let q be a power of p. Let G be a connected compact Lie group. We show that there exists an integer b such that the mod ell cohomology of the classifying space of a finite Chevalley group G(F_q) is isomorphic to the mod ell cohomology of the classifying space of the loop group LG for q=p^{ab}, a>0."}
{"category": "Math", "title": "On a theorem of Faltings on formal functions", "abstract": "In 1980, Faltings proved, by deep local algebra methods, a local result regarding formal functions which has the following global geometric fact as a consequence. Theorem: Let k be an algebraically closed field (of any characteristic). Let Y be a closed subvariety of a projective irreducible variety X defined over k. Assume that X \\subseteq P^n, dim(X)=d>2 and Y is the intersection of X with r hyperplanes of P^n, with r \\le d-1. Then, every formal rational function on X along Y can be (uniquely) extended to a rational function on X. Due to its importance, the aim of this paper is to provide two elementary global geometric proofs of this theorem."}
{"category": "Math", "title": "Kummer generators and lambda invariants", "abstract": "Let $F_0=\\mathbf Q(\\sqrt{-d})$ be an imaginary quadratic field with $3\\nmid d$ and let $K_0=\\mathbf Q(\\sqrt{3d})$. Let $\\varepsilon_0$ be the fundamental unit of $K_0$ and let $\\lambda$ be the Iwasawa $\\lambda$-invariant for the cyclotomic $\\mathbf Z_3$-extension of $F_0$. The theory of 3-adic $L$-functions gives conditions for $\\lambda\\ge 2$ in terms of $\\epsilon_0$ and the class numbers of $F_0$ and $K_0$. We construct units of $K_1$, the first level of the $\\mathbf Z_3$-extension of $K_0$, that potentially occur as Kummer generators of unramified extensions of $F_1(\\zeta_3)$ and which give an algebraic interpretation of the condition that $\\lambda\\ge 2$. We also discuss similar results on $\\lambda\\ge 2$ that arise from work of Gross-Koblitz."}
{"category": "Math", "title": "Operator splittings and spatial approximations for evolution equations", "abstract": "The convergence of various operator splitting procedures, such as the sequential, the Strang and the weighted splitting, is investigated in the presence of a spatial approximation. To this end a variant of Chernoff's product formula is proved. The methods are applied to abstract partial delay differential equations."}
{"category": "Math", "title": "On the cable expansion formula", "abstract": "In this paper, a generalized version of Morton's formula is proved. Using this formula, one can write down the colored Jones polynomials of cabling of an knot in terms of the colored Jones polynomials of the original knot."}
{"category": "Math", "title": "OPED reconstruction algorithm for limited angle problem", "abstract": "The structure of the reconstruction algorithm OPED permits a natural way to generate additional data, while still preserving the essential feature of the algorithm. This provides a method for image reconstruction for limited angel problems. In stead of completing the set of data, the set of discrete sine transforms of the data is completed. This is achieved by solving systems of linear equations that have, upon choosing appropriate parameters, positive definite coefficient matrices. Numerical examples are presented."}
{"category": "Math", "title": "A strictly stationary, N-tuplewise independent counterexample to the central limit theorem", "abstract": "For an arbitrary integer N that is at least 2, this paper gives a construction of a strictly stationary, N-tuplewise independent sequence of (non-degenerate) bounded random variables such that the Central Limit Theorem fails to hold. The sequence is in part an adaptation of a non-stationary example with similar properties constructed by one of the authors (ARP) in a paper published in 1998."}
{"category": "Math", "title": "On the Number of Rational Iterated Pre-images of the Origin Under Quadratic Dynamical Systems", "abstract": "For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article ``Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems,'' by two of the present authors and five others, it was shown that the number of rational iterated pre-images of the origin is bounded as one varies the morphism in a certain one-dimensional family. Subject to the validity of the Birch and Swinnerton-Dyer conjecture and some other related conjectures for the L-series of a specific abelian variety and using a number of modern tools for locating rational points on high genus curves, we show that the maximum number of rational iterated pre-images is six. We also provide further insight into the geometry of the ``pre-image curves.''"}
{"category": "Math", "title": "Random sampling of long-memory stationary processe", "abstract": "This paper investigates the second order properties of a stationary process after random sampling. While a short memory process gives always rise to a short memory one, we prove that long-memory can disappear when the sampling law has heavy enough tails. We prove that under rather general conditions the existence of the spectral density is preserved by random sampling. We also investigate the effects of deterministic sampling on seasonal long-memory."}
{"category": "Math", "title": "Most actions on regular trees are almost free", "abstract": "Let T be a d-regular tree (d > 2) and A=Aut(T), its automorphism group. Let G be a group generated by n independent Haar-random elements of A. We show that almost surely, every nontrivial element of G has finitely many fixed points on T."}
{"category": "Math", "title": "A simple, fast and stabilized flowing finite volume method for solving general curve evolution equations", "abstract": "A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of evolving family of plane curves. A curve can be evolved in the normal direction by a combination of fourth order terms related to the intrinsic Laplacian of the curvature, second order terms related to the curvature, first order terms related to anisotropy and by a given external velocity field. The evolution is numerically stabilized by an asymptotically uniform tangential redistribution of grid points yielding the first order intrinsic advective terms in the governing system of equations. By using a semi-implicit in time discretization it can be numerically approximated by a solution to linear penta-diagonal systems of equations (in presence of the fourth order terms) or tri-diagonal systems (in the case of the second order terms). Various numerical experiments of plane curve evolutions, including, in particular, nonlinear, anisotropic and regularized backward curvature flows, surface diffusion and Willmore flows, are presented and discussed."}
{"category": "Math", "title": "Chains on suspension spectra", "abstract": "We define and study a homological version of Sullivan's rational de Rham complex for simplicial sets. This new functor can be generalised to simplicial symmetric spectra and in that context it has excellent categorical properties which promise to make a number of interesting applications much more straightforward."}
{"category": "Math", "title": "Values of Noncommutative Polynomials, Lie Skew-Ideals and the Tracial Nullstellensatz", "abstract": "A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for involutions of the first kind and four for involutions of the second kind) and this classification result is used to characterize noncommutative polynomials via their values in these algebras. As an application, we deduce that a polynomial is a sum of commutators and a polynomial identity of $d\\times d$ matrices if and only if all of its values in the algebra of $d\\times d$ matrices have zero trace."}
{"category": "Math", "title": "Hypersurfaces of Constant Curvature in Hyperbolic Space I", "abstract": "We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions."}
{"category": "Math", "title": "Hypersurfaces of Constant Curvature in Hyperbolic Space II", "abstract": "We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their curvature quotients."}
{"category": "Math", "title": "The stable moduli space of flat connections over a surface", "abstract": "We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy type of the infinite symmetric product of M^g, generalizing a well-known fact for the torus. Over a non-orientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori, whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surfaces, and show a close analogy with the Quillen-Lichtenbaum conjectures in algebraic K-theory. The proofs utilize Tyler Lawson's work in deformation K-theory, and rely heavily on Yang-Mills theory and gauge theory."}
{"category": "Math", "title": "A homotopy-theoretic view of Bott-Taubes integrals and knot spaces", "abstract": "We construct cohomology classes in the space of knots by considering a bundle over this space and \"integrating along the fiber\" classes coming from the cohomology of configuration spaces using a Pontrjagin-Thom construction. The bundle we consider is essentially the one considered by Bott and Taubes, who integrated differential forms along the fiber to get knot invariants. By doing this \"integration\" homotopy-theoretically, we are able to produce integral cohomology classes. We then show how this integration is compatible with the homology operations on the space of long knots, as studied by Budney and Cohen. In particular we derive a product formula for evaluations of cohomology classes on homology classes, with respect to connect-sum of knots."}
{"category": "Math", "title": "Spectral Theory of Elliptic Operators in Exterior Domains", "abstract": "We consider various closed (and self-adjoint) extensions of elliptic differential expressions of the type $\\cA=\\sum_{0\\le |\\alpha|,|\\beta|\\le m}(-1)^\\alpha D^\\alpha a_{\\alpha, \\beta}(x)D^\\beta$, $a_{\\alpha, \\beta}(\\cdot)\\in C^{\\infty}({\\overline\\Omega})$, on smooth (bounded or unbounded) domains in $\\bbR^n$ with compact boundary. Using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, we prove various trace ideal properties of powers of resolvent differences of these closed realizations of $\\cA$ and derive estimates on eigenvalues of certain self-adjoint realizations in spectral gaps of the Dirichlet realization. Our results extend classical theorems due to Visik, Povzner, Birman, and Grubb."}
{"category": "Math", "title": "On connectivity of fibers with positive marginals in multiple logistic regression", "abstract": "In this paper we consider exact tests of a multiple logistic regression, where the levels of covariates are equally spaced, via Markov beses. In usual application of multiple logistic regression, the sample size is positive for each combination of levels of the covariates. In this case we do not need a whole Markov basis, which guarantees connectivity of all fibers. We first give an explicit Markov basis for multiple Poisson regression. By the Lawrence lifting of this basis, in the case of bivariate logistic regression, we show a simple subset of the Markov basis which connects all fibers with a positive sample size for each combination of levels of covariates."}
{"category": "Math", "title": "Bounds on the roots of the Steiner polynomial", "abstract": "We consider the Steiner polynomial of a C^2 convex body K in R^n (n \\leq 5). The opposites of the real parts of the roots of the Steiner polynomial are bounded below by the minimum value and above by the maximum value of the principal radii of curvature of the boundary of K."}
{"category": "Math", "title": "Self-mapping degrees of torus bundles and torus semi-bundles", "abstract": "Each closed oriented 3-manifold $M$ is naturally associated with a set of integers $D(M)$, the degrees of all self-maps on $M$. $D(M)$ is determined for each torus bundle and torus semi-bundle $M$. The structure of torus semi-bundle is studied in detail. The paper is a part of a project to determine $D(M)$ for all 3-manifolds in Thurston's picture."}
{"category": "Math", "title": "Self-mapping Degrees of 3-Manifolds", "abstract": "For each closed oriented 3-manifold $M$ in Thurston's picture, the set of degrees of self-maps on $M$ is given."}
{"category": "Math", "title": "Noncommutative resolution, F-blowups and D-modules", "abstract": "We explain the isomorphism between the $G$-Hilbert scheme and the F-blowup from the noncommutative viewpoint after Van den Bergh. In doing this, we immediately and naturally arrive at the notion of $D$-modules. We also find, as a byproduct, a canonical way to construct a noncommutative resolution at least for a few classes of singularities in positive characteristic."}
{"category": "Math", "title": "Test vectors for trilinear forms : the case of two principal series", "abstract": "Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three admissible, irreducible, finite dimensional representations of G, the space of G-invariant linear forms has dimension at most one. When a non zero linear form exists, one wants to find an element of V which is not in its kernel: this is a test vector. Gross and Prasad found explicit test vectors when the three representations are unramified principal series, and when they are all unramified twists of the Steinberg representation. In this paper we decribe explicit test vectors when two of the representations are principal series."}
{"category": "Math", "title": "On systems of Hecke eigenvalues in cohomology of certain subgroups of GL_n(F)", "abstract": "We show how there is a natural action on the cohomology groups attached to certain subgroups of GL_n(F) of the Hecke operators defined as elements in an adelic double coset algebra. Our main result is, that if a system of eigenvalues for Hecke operators occur in the cohomology groups with coefficients in certain modules, then by changing the groups the system also occur in the cohomology groups with coefficients in a 1-dimensional module."}
{"category": "Math", "title": "Holomorphic Functions on Bundles Over Annuli", "abstract": "We consider a family E_m(D,M) of holomorphic bundles constructed as follows: to any given M in GL_n(Z), we associate a \"multiplicative automorphism\" f of (C*)^n. Now let D be a f-invariant Stein Reinhardt domain in (C*)^n. Then E_m(D,M) is defined as the flat bundle over the annulus of modulus m>0, with fiber D, and monodromy f. We show that the function theory on E_m(D,M) depends nontrivially on the parameters m, M and D. Our main result is that E_m(D,M) is Stein if and only if m log(r(M)) <= 2 \\pi^2, where r(M) denotes the max of the spectral radii of M and its inverse. As corollaries, we: -- obtain a classification result for Reinhardt domains in all dimensions; -- establish a similarity between two known counterexamples to a question of J.-P. Serre; -- suggest a potential reformulation of a disproved conjecture of Siu Y.-T."}
{"category": "Math", "title": "Rough Volterra equations 2: convolutional generalized integrals", "abstract": "We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory allowing to handle generalized integrals weighted by an exponential coefficient. The results are applied to the fractional Brownian motion with Hurst coefficient greater than 1/3"}
{"category": "Math", "title": "Representation of solutions of the Gauss hypergeometric equation by the multiple polylogarithms, functional relations of the multiple polylogarithms and relations of the multiple zeta values", "abstract": "In this article, we express solutions of the Gauss hypergeometric equation as a series of the multiple polylogarithms by using iterated integral. This representation is the most simple case of a semisimple representation of solutions of the formal KZ equation. Moreover, combining this representation with the connection relations of solutions of the Gauss hypergeometric equation, we obtain various relations of the multiple polylogarithms of one variable and the multiple zeta values."}
{"category": "Math", "title": "A basic set for the alternating group", "abstract": "This article concerns the $p$-basic set existence problem in the representation theory of finite groups. We show that, for any odd prime $p$, the alternating group $\\A_n$ has a $p$-basic set. More precisely, we prove that the symmetric group $\\sym_n$ has a $p$-basic set with some additional properties, allowing us to deduce a $p$-basic set for $\\A_n$. Our main tool is the generalized perfect isometries introduced by K\\\"ulshammer, Olsson and Robinson. As a consequence we obtain some results on the decomposition number of $\\A_n$."}
{"category": "Math", "title": "Invariant distributions on non-distinguished nilpotent orbits with application to the Gelfand property of (GL(2n,R),Sp(2n,R))", "abstract": "We study invariant distributions on the tangent space to a symmetric space. We prove that an invariant distribution with the property that both its support and the support of its Fourier transform are contained in the set of non-distinguished nilpotent orbits, must vanish. We deduce, using recent developments in the theory of invariant distributions on symmetric spaces that the symmetric pair (GL(2n,R),Sp(2n,R)) is a Gelfand pair. More precisely, we show that for any irreducible smooth admissible Frechet representation $(\\pi,E)$ of GL(2n,R) the space of continuous functionals $Hom_{Sp_{2n}(R)}(E,C)$ is at most one dimensional. Such a result was previously proven for p-adic fields in [HR] for the field of complex numbers in [S]."}
{"category": "Math", "title": "A note on the dynamical zeta function of general toral endomorphisms", "abstract": "It is well-known that the Artin-Mazur dynamical zeta function of a hyperbolic or quasi-hyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. We give an elementary proof of this fact that extends to the case of general toral endomorphisms without change. The result is a closed formula that can be calculated by integer arithmetic only. We also address the functional equation and the relation between the Artin-Mazur and Lefschetz zeta functions."}
{"category": "Math", "title": "Automorphic lifts of prescribed types", "abstract": "We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of $n$-dimensional mod $p$ Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for $n$-dimensional automorphic Galois representations."}
{"category": "Math", "title": "Nonlinear diffusive-dispersive limits for multidimensional conservation laws", "abstract": "We consider a class of multidimensional conservation laws with vanishing nonlinear diffusion and dispersion terms. Under a condition on the relative size of the diffusion and dispersion coefficients, we establish that the diffusive-dispersive solutions are uniformly bounded in a space Lp ($p$ arbitrary large, depending on the nonlinearity of the diffusion) and converge to the classical, entropy solution of the corresponding multidimensional, hyperbolic conservation law. Previous results were restricted to one-dimensional equations and specific spaces Lp. Our proof is based on DiPerna's uniqueness theorem in the class of entropy measure-valued solutions."}
{"category": "Math", "title": "Extremal Bases, Geometrically Separated Domains and Applications", "abstract": "We introduce the notion of extremal basis of tangent vector fields at a boundary point of finite type of a pseudo-convex domain in $\\mathbb{C}^n$. Then we define the class of geometrically separated domains at a boundary point, and give a description of their complex geometry. Examples of such domains are given, for instance, by locally lineally convex domains, domains with locally diagonalizable Levi form, and domains for which the Levi form have comparable eigenvalues at a point. Moreover we show that these domains are localizable. Then we define the notion of \"adapted pluri-subharmonic function\" to these domains, and we give sufficient conditions for his existence. Then we show that all the sharp estimates for the Bergman ans Szeg\\\"o projections are valid in this case. Finally we apply these results to the examples to get global and local sharp estimates, improving, for examlple, a result of Fefferman, Kohn and Machedon on the Szeg\\\"o projection."}
{"category": "Math", "title": "The Orbit Group of a Quandle", "abstract": "We define the notion of the orbit group of a quandle via its connectivity and compute the orbit groups for some basic quandles. We also show that the orbit group counts the number of orbits of certain quandles."}
{"category": "Math", "title": "The Distribution of the Domination Number of Class Cover Catch Digraphs for Non-uniform One-dimensional Data", "abstract": "For two or more classes of points in $\\R^d$ with $d \\ge 1$, the class cover catch digraphs (CCCDs) can be constructed using the relative positions of the points from one class with respect to the points from the other class. The CCCDs were introduced by (Priebe, DeVinney, and Mar-chette, (2001). On the distribution of the domination number of random class catch cover di-graphs. Statistics and Probability Letters, 55:239-246) who investigated the case of two classes, $\\X$ and $\\Y$. They calculated the exact (finite sample) distribution of the domination number of the CCCDs based on $\\X$ points relative to $\\Y$ points both of which were uniformly distri-buted on a bounded interval. We investigate the distribution of the domination number of the CCCDs based on data from non-uniform $\\X$ points on an interval with end points from $\\Y$. Then we extend these calculations for multiple $\\Y$ points on bounded intervals."}
{"category": "Math", "title": "Kahler-Ricci flow on stable Fano manifolds", "abstract": "We study the Kahler-Ricci flow on Fano manifolds. We show that if the curvature is bounded along the flow and if the manifold is K-polystable and asymptotically Chow semistable, then the flow converges exponentially fast to a Kahler-Einstein metric."}
{"category": "Math", "title": "Strong solutions of a class of SDEs with jumps", "abstract": "We study a class of stochastic integral equations with jumps under non-Lipschitz conditions. We use the method of Euler approximations to obtain the existence of the solution and give some sufficient conditions for the strong uniqueness."}
{"category": "Math", "title": "Reidemeister torsion for linear representations and Seifert surgery on knots", "abstract": "We study an invariant of a 3-manifold which consists of Reidemeister torsion for linear representations which pass through a finite group. We show a Dehn surgery formula on this invariant and compute that of a Seifert manifold over $S^2$. As a consequence we obtain a necessary condition for a result of Dehn surgery along a knot to be Seifert fibered, which can be applied even in a case where abelian Reidemeister torsion gives no information."}
{"category": "Math", "title": "Operator extensions of Hua's inequality", "abstract": "We give an extension of Hua's inequality in pre-Hilbert $C^*$-modules without using convexity or the classical Hua's inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert $C^*$-modules, another extension of Hua's inequality is obtained. We also present an operator Hua's inequality, which is equivalent to operator convexity of given continuous real function."}
{"category": "Math", "title": "Extremal metrics on del Pezzo threefolds", "abstract": "We prove the existence of Kahler-Einstein metrics on a nonsingular section of the Grassmannian $\\mathrm{Gr}(2, 5)\\subset\\mathbb{P}^9$ by a linear subspace of codimension 3, and the Fermat hypersurface of degree 6 in $\\mathbb{P}(1,1,1,2,3)$. We also show that a global log canonical threshold of the Mukai--Umemura variety is equal to 1/2."}
{"category": "Math", "title": "On a linear form for Catalan's constant", "abstract": "It is shown how Andrews' multidimensional extension of Watson's transformation between a very-well-poised $_8\\phi_7$-series and a balanced $_4\\phi_3$-series can be used to give a straightforward proof of a conjecture of Zudilin and the second author on the arithmetic behaviour of the coefficients of certain linear forms of 1 and Catalan's constant. This proof is considerably simpler and more stream-lined than the first proof, due to the second author."}
{"category": "Math", "title": "Ramanujan congruences for a class of eta quotients", "abstract": "We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes $\\ell$ for which their coefficients $c(n)$ obey congruences of the form $c(\\ell n + a) \\equiv 0 \\pmod \\ell$. We apply this result to obtain a complete characterization of the congruences of the same form that the sequences $c_N(n)$ satisfy, where $c_N(n)$ is defined by $ \\sum_{n=0}^{\\infty} c_N(n)q^n = \\prod_{n=1}^{\\infty} \\frac{1}{(1-q^n)(1-q^{Nn})}$. This last result answers a question of H.-C. Chan."}
{"category": "Math", "title": "There are no $\\mathcal{C}^5$-Regular Pure $y$-Global Landsberg Surfaces", "abstract": "We show that there are not pure $\\mathcal{C}^5$ regular y-global Landsberg surfaced. The proof is based on the averaged connection associated with the linear Chern's connection and the classification of irreducibles holonomies of torsion-free affine connections. The structure consists on exausting all the possible cases and showing that in dimension 2 Landsberg condition implies Berwald condition."}
{"category": "Math", "title": "The Euler class of planar groups", "abstract": "This is an exposition of the homological classification of actions of surface groups on the plane, in every degree of smoothness."}
{"category": "Math", "title": "Bifurcations in a class of polycycles involving two saddle-nodes on a Mobius band", "abstract": "In this paper we study the bifurcations of a class of polycycles, called lips, occurring in generic three-parameter smooth families of vector fields on a M\\\"obius band. The lips consists of a set of polycycles formed by two saddle-nodes, one attracting and the other repelling, connected by the hyperbolic separatrices of the saddle-nodes and by orbits interior to both nodal sectors. We determine, under certain genericity hypotheses, the maximum number of limits cycles that may bifurcate from a graphic belonging to the lips and we describe its bifurcation diagram."}
{"category": "Math", "title": "Regularity of Dirac-harmonic maps", "abstract": "For any $n$-dimensional compact spin Riemannian manifold $M$ with a given spin structure and a spinor bundle $\\Sigma M$, and any compact Riemannian manifold $N$, we show an $\\epsilon$-regularity theorem for weakly Dirac-harmonic maps . As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when $n = 2$. For $n \\ge 3$, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in $R^n$. If, additions, $\\psi\\in W^{1,p}$ for some $p>2n/3$, then we obtain an energy monotonicity formula and prove a partial regularity theorem for any such a stationary Dirac-harmonic map."}
{"category": "Math", "title": "Global Regularity of the 4D Restricted Euler Equations", "abstract": "We are concerned with the critical threshold phenomena in the Restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for 3D and the 4D restricted Euler equations. It is well known that the 3D RE solutions blow up. Projected on the 3-sphere, the set of initial eigenvalues which give rise to bounded stable solutions is reduced to a single point, which confirms that 3D RE blowup is generic. In contrast, we identify a surprisingly rich set of the initial spectrum on the 4-sphere which yields global smooth solutions; thus, 4D regularity is generic."}
{"category": "Math", "title": "Centroids and the Rapid Decay property in mapping class groups", "abstract": "We study a notion of a Lipschitz, permutation-invariant \"centroid\" for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S)."}
{"category": "Math", "title": "Embeddings of vertex operator algebras associated to orthogonal affine Lie algebras", "abstract": "Let $L_{D_{\\ell}}(-\\ell +{3/2},0)$ (resp. $L_{B_{\\ell}}(-\\ell +{3/2},0)$) be the simple vertex operator algebra associated to affine Lie algebra of type $D_{\\ell}^{(1)}$ (resp. $B_{\\ell}^{(1)}$) with the lowest admissible half-integer level $-\\ell + {3/2}$. We show that $L_{D_{\\ell}}(-\\ell +{3/2},0)$ is a vertex subalgebra of $L_{B_{\\ell}}(-\\ell +{3/2},0)$ with the same conformal vector. For $\\ell =4$, $L_{D_{4}}(-{5/2},0)$ is a vertex subalgebra of three copies of $L_{B_{4}}(-{5/2},0)$ contained in $L_{F_{4}}(-{5/2},0)$, and all five of these vertex operator algebras have the same conformal vector."}
{"category": "Math", "title": "The homology of path spaces and Floer homology with conormal boundary conditions", "abstract": "We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold satisfying non-local conormal boundary conditions. We prove that the homology of this chain complex is isomorphic to the singular homology of the natural path space associated to the boundary conditions."}
{"category": "Math", "title": "Markov bases of binary graph models of K_4-minor free graphs", "abstract": "Markov width of a graph is a graph invariant defined as the maximum degree of a Markov basis element for the corresponding graph model for binary contingency tables. We show that a graph has Markov width at most four if and only if it contains no $K_4$ as a minor, answering a question of Develin and Sullivant. We also present a lower bound of order $\\Omega(n^{2-\\varepsilon})$ on the Markov width of $K_n$."}
{"category": "Math", "title": "Asymptotic Behavior of Stochastic Wave Equations with Critical Exponents on R^3", "abstract": "The existence of a random attractor in H^1(R^3) \\times L^2(R^3) is proved for the damped semilinear stochastic wave equation defined on the entire space R^3. The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method."}
{"category": "Math", "title": "Convolutions of long-tailed and subexponential distributions", "abstract": "Convolutions of long-tailed and subexponential distributions play a major role in the analysis of many stochastic systems. We study these convolutions, proving some important new results through a simple and coherent approach, and showing also that the standard properties of such convolutions follow as easy consequences."}
{"category": "Math", "title": "Floer homology of cotangent bundles and the loop product", "abstract": "We prove that the pair-of-pants product on the Floer homology of the cotangent bundle of a compact manifold M corresponds to the Chas-Sullivan loop product on the singular homology of the loop space of M. We also prove related results concerning the Floer homological interpretation of the Pontrjagin product and of the Serre fibration. The techniques include a Fredholm theory for Cauchy-Riemann operators with jumping Lagrangian boundary conditions of conormal type, and a new cobordism argument replacing the standard gluing technique."}
{"category": "Math", "title": "Complete classification of compact four-manifolds with positive isotropic curvature", "abstract": "In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to $\\mathbb{S}^4,$ or $\\mathbb{R}\\mathbb{P}^4$ or quotients of $\\mathbb{S}^3\\times \\mathbb{R}$ by a cocompact fixed point free subgroup of the isometry group of the standard metric of $\\mathbb{S}^3\\times \\mathbb{R}$, or a connected sum of them."}
{"category": "Math", "title": "On infinitesimal Cherednik algebras of gl_2", "abstract": "We prove that the center of an infinitesimal Cherednik algebra of $\\mf{gl}_2$ is the polynomial algebra of two variables over the field of characteristic 0. In positive characteristic we show that any infinitesimal Cherednik algebra is a finitely generated module over its center."}
{"category": "Math", "title": "Point singularities of 3D stationary Navier-Stokes flows", "abstract": "This article characterizes the singularities of very weak solutions of 3D stationary Navier-Stokes equations in a punctured ball which are sufficiently small in weak $L^3$."}
{"category": "Math", "title": "A note on conditional Akaike information for Poisson regression with random effects", "abstract": "A popular model selection approach for generalized linear mixed-effects models is the Akaike information criterion, or AIC. Among others, \\cite{vaida05} pointed out the distinction between the marginal and conditional inference depending on the focus of research. The conditional AIC was derived for the linear mixed-effects model which was later generalized by \\cite{liang08}. We show that the similar strategy extends to Poisson regression with random effects, where condition AIC can be obtained based on our observations. Simulation studies demonstrate the usage of the criterion."}
{"category": "Math", "title": "The Normal Distribution as a Limit of Generalized Sato-Tate Measures", "abstract": "With suitable order of limits, as p, m, and n all tend to infinity, the distribution of the normalized trace of Frobenius on H^1 of a \"random\" plane curve of degree n over the field with p^m elements, tends to a Gaussian distribution. The same is true of a \"random\" curve of genus g over the field with p^m elements."}
{"category": "Math", "title": "Local tube realizations of CR-manifolds and maximal abelian subalgebras", "abstract": "For every CR-manifold germ (M,a) local tube realizations are characterized by certain abelian subalgebras of the real Lie algebra hol(M,a) of all germs of (real-analytic) infinitesimal transformations. For instance, if M is holomorphically non-degenerate, every such subalgebra is maximal abelian and the classification of all local tube realizations for (M,a) reduces to a purely algebraic problem."}
{"category": "Math", "title": "Tori Embedded in S3 with Dense Asymptotic Lines", "abstract": "In this paper are given examples of tori T^2 embedded in S^3 with all their asymptotic lines dense."}
{"category": "Math", "title": "Normal Subgroups of Profinite Groups of Non-negative Deficiency", "abstract": "We initiate the study of profinite groups of non-negative deficiency. The principal focus of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group $G$ of non-negative deficiency gives rather strong consequences for the structure of $G$. To make this precise we introduce the notion of $p$-deficiency ($p$ a prime) for a profinite group $G$. This concept is more useful in the study of profinite groups then the notion of deficiency. We prove that if the $p$-deficiency of $G$ is positive and $N$ is a finitely generated normal subgroup such that the $p$-Sylow subgroup of $G/N$ is infinite and $p$ divides the order of $N$ then we have $\\cd_p(G)=2$, $\\cd_p(N)=1$ and $\\vcd_p(G/N)=1$ for the cohomological $p$-dimensions; moreover either the $p$-Sylow subgroup of $G/N$ is virtually cyclic or the $p$-Sylow subgroup of $N$ is cyclic. A profinite Poincar\\'e duality group $G$ of dimension 3 at a prime $p$ ($PD^3$-group) has deficiency 0. In this case we show that for $N$ and $p$ as above either $N$ is $PD^1$ at $p$ and $G/N$ is virtually $PD^2$ at $p$ or $N$ is $PD^2$ at $p$ and $G/N$ is virtually $PD^1$ at $p$. In particular if $G$ is pro-$p$ then either $N$ is infinite cyclic and $G/N$ is virtually Demushkin or $N$ is Demushkin and $G/N$ is virtually infinite cyclic. We apply this results to deduce structural information on the profinite completions of ascending HNN-extensions of free groups. We also give some implications of our theory to the congruence kernels of certain arithmetic groups."}
{"category": "Math", "title": "Tilting, deformations and representations of linear groups over Euclidean algebras", "abstract": "We consider the dual space of linear groups over Dynkinian and Euclidean algebras, i.e. finite dimensional algebras derived equivalent to the path algebra of Dynkin or Euclidean quiver. We prove that this space contains an open dense subset isomorphic to the product of dual spaces of full linear groups and, perhaps, one more (explicitly described) space. The proof uses the technique of bimodule categories, deformations and representations of quivers."}
{"category": "Math", "title": "Robust ergodic properties in partially hyperbolic dynamics", "abstract": "We study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting. Earlier work of Bonatti, Viana about existence and finitude of physical measures is extended to the case of local diffeomorphisms. Moreover, we prove that such systems constitute a C^2-open set in which statistical stability is a dense property. In contrast, all mostly contracting systems are shown to be stable under small random perturbations."}
{"category": "Math", "title": "A Remark on Nonlinear Dirac Equations", "abstract": "For a $n$-dimensional spin manifold $M$ with a fixed spin structure and a spinor bundle $\\Sigma M$, we prove an $\\epsilon$-regularity theorem for weak solutions to the nonlinear Dirac equation of cubic nonlinearity. This, in particular, answers a regularity question raised by Chen-Jost-Wang when $n=2$."}
{"category": "Math", "title": "High degree graphs contain large-star factors", "abstract": "We show that any finite simple graph with minimum degree $d$ contains a spanning star forest in which every connected component is of size at least $\\Omega((d/\\log d)^{1/3})$. This settles a problem of J. Kratochvil."}
{"category": "Math", "title": "Continued Fraction Expansions of Matrix Eigenvectors", "abstract": "We examine various properties of the continued fraction expansions of matrix eigenvector slopes of matrices from the SL(2, Z) group. We calculate the average period length, maximum period length, average period sum, maximum period sum and the distributions of 1s 2s and 3s in the periods versus the radius of the Ball within which the matrices are located. We also prove that the periods of continued fraction expansions from the real irrational roots of x2 + px + q = 0 are always palindromes."}
{"category": "Math", "title": "Cohomology rings of certain seven dimensional manifolds", "abstract": "We calculate the cohomology rings of a collection of seven dimensional manifolds supporting an S^3 x S^3-action with one dimensional orbit space. These manifolds are of interest to differential geometers studying non-negative and positive sectional curvature. From this collection, we identify several families of manifolds for which there exist well-known topological invariants distinguishing homeomorphism and diffeomorphism types."}
{"category": "Math", "title": "Integral Deligne Cohomology for Real Varieties", "abstract": "We develop an integral version of Deligne cohomology for smooth proper real varieties. For this purpose the role played by singular cohomology in the complex case has to be replaced by ordinary bigraded G-equivariant cohomology, where G=Gal(C/R). This is the G-equivariant counterpart of singular cohomology. We establish the basic properties of the theory and give a geometric interpretation for the groups in dimension 2 in weights 1 and 2."}
{"category": "Math", "title": "Abstract commensurators of profinite groups", "abstract": "In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group $G$ is a group $Comm(G)$ which depends only on the commensurability class of $G$. We study various properties of $Comm(G)$; in particular, we find two natural ways to turn it into a topological group. We also use $Comm(G)$ to study topological groups which contain $G$ as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we construct a topologically simple group which contains the pro-2 completion of the Grigorchuk group as an open subgroup. On the other hand, we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups. Several celebrated rigidity theorems, like Pink's analogue of Mostow's strong rigidity theorem for simple algebraic groups defined over local fields and the Neukirch-Uchida theorem, can be reformulated as structure theorems for the commensurators of certain profinite groups."}
{"category": "Math", "title": "Global existence and long term behavior of 2d electro-hydrodynamics", "abstract": "We study the equations of a two dimensional incompressible Newtonian fluid coupled with a dispersive parabolic-elliptic system on bounded domains. Global in time weak solutions are shown to exist and converge with a rate to the stationary solution for L^2 initial data. This paper extends and improves on a body of work surrounding the Debye-Huckel system to the hydrodyanamical case."}
{"category": "Math", "title": "The Equitable Basis for sl_2", "abstract": "This article contains an investigation of the equitable basis for the Lie algebra sl_2. Denoting this basis by {x,y,z}, we have [x,y] = 2x + 2y, [y,z] = 2y + 2z, [z, x] = 2z + 2x. One focus of our study is the group of automorphisms G generated by exp(ad x*), exp(ad y*), exp(ad z*), where {x*,y*,z*} is the basis for sl_2 dual to {x,y,z} with respect to the trace form (u,v) = tr(uv). We show that G is isomorphic to the modular group PSL_2(Z). Another focus of our investigation is the lattice L=Zx+Zy+Zz. We prove that the orbit G(x) equals {u in L |(u,u)=2}. We determine the precise relationship between (i) the group G, (ii) the group of automorphisms for sl_2 that preserve L, (iii) the group of automorphisms and antiautomorphisms for sl_2 that preserve L, and (iv) the group of isometries for (,) that preserve L. We obtain analogous results for the lattice L* =Zx*+Zy*+Zz*. Relative to the equitable basis, the matrix of the trace form is a Cartan matrix of hyperbolic type; consequently,we identify the equitable basis with the set of simple roots of the corresponding Kac-Moody Lie algebra g. Then L is the root lattice for g and 1/2L* is the weight lattice, and G(x) coincides with the set of real roots for g. Using L, L*, and G, we give several descriptions of the isotropic roots for g and show that each isotropic root has multiplicity 1. We describe the finite-dimensional sl_2-modules from the point of view of the equitable basis. In the final section, we establish a connection between the Weyl group orbit of the fundamental weights of g and Pythagorean triples."}
{"category": "Math", "title": "Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type", "abstract": "We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras."}
{"category": "Math", "title": "Applied Categories and Functors for Undergraduates", "abstract": "These are lecture notes for a 1-semester undergraduate course (in computer science, mathematics, physics, engineering, chemistry or biology) in applied categorical meta-language. The only necessary background for comprehensive reading of these notes are first-year calculus and linear algebra."}
{"category": "Math", "title": "Arithmetic harmonic analysis on character and quiver varieties", "abstract": "We present a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a Riemann surface of genus g to GL_n(C) with fixed generic semi-simple conjugacy classes at k punctures. Using the character table of GL_n(F_q) we calculate the E-polynomial of these character varieties and confirm that it is as predicted by our main conjecture. Then, using the character table of gl_n(F_q), we calculate the E-polynomial of certain associated comet-shaped quiver varieties, the additive analogues of our character variety, and find that it is the pure part of our conjectured mixed Hodge polynomial. Finally, we observe that the pure part of our conjectured mixed Hodge polynomial also equals certain multiplicities in the tensor product of irreducible representations of GL_n(F_q). This implies a curious connection between the representation theory of GL_n(F_q) and Kac-Moody algebras associated with comet-shaped, typically wild, quivers."}
{"category": "Math", "title": "Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling", "abstract": "This study proposes improved chain-ratio type estimator for estimating population mean using some known values of population parameter(s) of the second auxiliary character. The proposed estimators have been compared with two-phase ratio estimator and some other chain type estimators. The performances of the proposed estimators have been supposed with a numerical illustration."}
{"category": "Math", "title": "The fundamental isomorphism conjecture via non-commutative motives", "abstract": "Given a group, we construct a fundamental additive functor on its orbit category. We prove that any isomorphism conjecture valid for this fundamental isomorphism functor holds for all additive functors, like K-theory, cyclic homology, topological Hochschild homology, etc. Finally, we reduce this fundamental isomorphism conjecture for K-theoretic ones."}
{"category": "Math", "title": "Proof of the the Riemann hypothesis from the density and Lindelof hypotheses via a power sum method", "abstract": "The Riemann hypothesis is equivalent to the $\\varpi$-form of the prime number theorem as $\\varpi(x) =O(x\\sp{1/2} \\log\\sp{2} x)$, where $\\varpi(x) =\\sum\\sb{n\\le x}\\ \\bigl(\\Lambda(n) -1\\big)$ with the sum running through the set of all natural integers. Let ${\\mathsf Z}(s) = -\\tfrac{\\zeta\\sp{\\prime}(s)}{\\zeta(s)} -\\zeta(s)$. We use the classical integral formula for the Heaviside function in the form of ${\\mathsf H}(x) =\\int\\sb{m -i\\infty} \\sp{m +i\\infty} \\tfrac{x\\sp{s}}{s} \\dd s$ where $m >0$, and ${\\mathsf H}(x)$ is 0 when $\\tfrac{1}{2} <x <1$, $\\tfrac{1}{2}$ when $x=1$, and 1 when $x >1$. However, we diverge from the literature by applying Cauchy's residue theorem to the function ${\\mathsf Z}(s) \\cdot \\tfrac{x\\sp{s}} {s}$, rather than $-\\tfrac{\\zeta\\sp{\\prime}(s)} {\\zeta(s)} \\cdot \\tfrac{x\\sp{s}}{s}$, so that we may utilize the formula for $\\tfrac{1}{2}< m <1$, under certain conditions. Starting with the estimate on $\\varpi(x)$ from the trivial zero-free region $\\sigma >1$ of ${\\mathsf Z}(s)$, we use induction to reduce the size of the exponent $\\theta$ in $\\varpi(x) =O(x\\sp{\\theta} \\log\\sp{2} x)$, while we also use induction on $x$ when $\\theta$ is fixed. We prove that the Riemann hypothesis is valid under the assumptions of the explicit strong density hypothesis and the Lindel\\\"of hypothesis recently proven, via a result of the implication on the zero free regions from the remainder terms of the prime number theorem by the power sum method of Tur\\'an."}
{"category": "Math", "title": "Proof of the strong Density Hypothesis", "abstract": "The Riemann hypothesis, conjectured by Bernhard Riemann in 1859, claims that the non-trivial zeros of $\\zeta(s)$ lie on the line $\\Re(s) =1/2$. The density hypothesis is a conjectured estimate $N(\\lambda, T) =O\\bigl(T\\sp{2(1-\\lambda) +\\epsilon} \\bigr)$ for any $\\epsilon >0$, where $N(\\lambda, T)$ is the number of zeros of $\\zeta(s)$ when $\\Re(s) \\ge\\lambda$ and $0 <\\Im(s) \\le T$, with $1/2 \\le \\lambda \\le 1$ and $T >0$. The Riemann-von Mangoldt Theorem confirms this estimate when $\\lambda =1/2$, with $T\\sp{\\epsilon}$ being replaced by $\\log T$. In an attempt to transform Backlund's proof of the Riemann-von Mangoldt Theorem to a proof of the density hypothesis by convexity, we discovered a different approach utilizing an auxiliary function. The crucial point is that this function should be devised to be symmetric with respect to $\\Re(s) =1/2$ and about the size of the Euler Gamma function on the right hand side of the line $\\Re(s) =1/2$. Moreover, it should be analytic and without any zeros in the concerned region. We indeed found such a function, which we call pseudo-Gamma function. With its help, we are able to establish a proof of the density hypothesis. Actually, we give the result explicitly and our result is even stronger than the original density hypothesis, namely it yields $N(\\lambda, T) \\le 8.734 \\log T$ for any $1/2 < \\lambda < 1$ and $T\\ge 2445999554999$."}
{"category": "Math", "title": "Constant Rate Distributions on Partially Ordered Sets", "abstract": "We consider probability distributions with constant rate on partially ordered sets, generalizing distributions in the usual reliability setting that have constant failure rate. In spite of the minimal algebraic structure, there is a surprisingly rich theory, including moment results and results concerning ladder variables and point processes. We concentrate mostly on discrete posets, particularly posets whose graphs are rooted trees. We pose some questions on the existence of constant rate distributions for general discrete posets."}
{"category": "Math", "title": "On Serre's conjecture for mod l Galois representations over totally real fields", "abstract": "In 1987 Serre conjectured that any mod l (\"ell\", not \"1\") two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where l is unramified. The hard work is in formulating an analogue of the \"weight\" part of Serre's conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a \"mod l Langlands philosophy\". Using ideas of Emerton and Vigneras, we formulate a mod l local-global principle for the group D^*, where D is a quaternion algebra over a totally real field, split above l and at 0 or 1 infinite places, and show how it implies the conjecture."}
{"category": "Math", "title": "The Lagrangian Conley Conjecture", "abstract": "We prove a Lagrangian analogue of the Conley conjecture: given a 1-periodic Tonelli Lagrangian with global flow on a closed configuration space, the associated Euler-Lagrange system has infinitely many periodic solutions. More precisely, we show that there exist infinitely many contractible integer periodic solutions with a priori bounded mean action and either infinitely many of them are 1-periodic or they have unbounded period."}
{"category": "Math", "title": "Linear Relations Among Poincare Series via Harmonic Weak Maass Forms", "abstract": "We discuss the problem of the vanishing of Poincar\\'e series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan's mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincar\\'e series and the connection to Ramanujan's mock theta functions."}
{"category": "Math", "title": "Explicit Estimate on Primes between Consecutive Cubes", "abstract": "We give an explicit form of Ingham's Theorem on primes in the short intervals, and show that there is at least one prime between every two consecutive cubes $x\\sp{3}$ and $(x+1)\\sp{3}$ if $\\log\\log x\\ge 15$."}
{"category": "Math", "title": "Constructions of commutative automorphic loops", "abstract": "A loop whose inner mappings are automorphisms is an \\emph{automorphic loop} (or \\emph{A-loop}). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of 2. We initiate the classification of commutative A-loops of small orders and also of order $p^3$, where $p$ is a prime."}
{"category": "Math", "title": "Linear systems in P^3 with low degrees and low multiplicities", "abstract": "We prove that the linear system of hypersurfaces in P^3 of degree d, 14 <= d <= 40, with double, triple and quadruple points in general position are non-special. This solves the cases that have not been completed in a paper by E. Ballico and M.C. Brambilla."}
{"category": "Math", "title": "On the stabilization of persistently excited linear systems", "abstract": "We consider control systems of the type $\\dot x = A x +\\alpha(t)bu$, where $u\\in\\R$, $(A,b)$ is a controllable pair and $\\alpha$ is an unknown time-varying signal with values in $[0,1]$ satisfying a persistent excitation condition i.e., $\\int_t^{t+T}\\al(s)ds\\geq \\mu$ for every $t\\geq 0$, with $0<\\mu\\leq T$ independent on $t$. We prove that such a system is stabilizable with a linear feedback depending only on the pair $(T,\\mu)$ if the eigenvalues of $A$ have non-positive real part. We also show that stabilizability does not hold for arbitrary matrices $A$. Moreover, the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon in dependence of the parameter $\\mu/T$."}
{"category": "Math", "title": "Forgetting of the initial distribution for non-ergodic Hidden Markov Chains", "abstract": "In this paper, the forgetting of the initial distribution for a non-ergodic Hidden Markov Models (HMM) is studied. A new set of conditions is proposed to establish the forgetting property of the filter, which significantly extends all the existing results. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using generic models of non-ergodic HMM and extend all the results known so far."}
{"category": "Math", "title": "On two-sided p-values for non-symmetric distributions", "abstract": "Two-sided statistical tests and p-values are well defined only when the test statistic in question has a symmetric distribution. A new two-sided p-value called conditional p-value $P_C$ is introduced here. It is closely related to the doubled p-value and has an intuitive appeal. Its use is advocated for both continuous and discrete distributions. An important advantage of this p-value is that equivalent 1-sided tests are transformed into $P_C$-equivalent 2-sided tests. It is compared to the widely used doubled and minimum likelihood p-values. Examples include the variance test, the binomial and the Fisher's exact test."}
{"category": "Math", "title": "Counting Quiver Representations over Finite Fields Via Graph Enumeration", "abstract": "Let $\\Gamma$ be a quiver on n vertices $v_1, v_2, ..., v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\\alpha \\in \\N^n$. Hua gave a formula for $A_{\\Gamma}(\\alpha, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\\Gamma$ over the finite field $\\F_q$ with dimension vector $\\alpha$. Kac showed that $A_{\\Gamma}(\\bm{\\alpha}, q)$ is a polynomial in q with integer coefficients. Using Hua's formula, we show that for each non-negative integer s, the s-th derivative of $A_{\\Gamma}(\\alpha,q)$ with respect to q, when evaluated at q = 1, is a polynomial in the variables $g_{ij}$, and we compute the highest degree terms in this polynomial. Our formulas for these coefficients depend on the enumeration of certain families of connected graphs."}
{"category": "Math", "title": "Equivariant Quantizations of Symmetric Algebras", "abstract": "Let g be a Lie bialgebra and let V be a finite-dimensional g-module. We study deformations of the symmetric algebra of V which are equivariant with respect to an action of the quantized enveloping algebra of g. In particular we investigate such quantizations obtained from the quantization of certain Lie bialgebra structures on the semidirect product of g and V. We classify these structure in the important special case, when g is complex, simple, with quasitriangular Lie bialgebra structure and V is a simple g-module. We then introduce more a general notion, co-Poisson module algebras and their quantizations, to further address the problem and show that many known examples of quantized symmetric algebras can be described in this language."}
{"category": "Math", "title": "Homological obstructions to string orientations", "abstract": "We observe that the Poincare duality isomorphism for a string manifold is an isomorphism of modules over the subalgebra A(2) of the modulo 2 Steenrod algebra. In particular, the pattern of the operations Sq^1, Sq^2, and Sq^4 on the cohomology of a string manifold has a symmetry around the middle dimension. We characterize this kind of cohomology operation duality in term of the annihilator of the Thom class of the negative tangent bundle, and in terms of the vanishing of top-degree cohomology operations. We also indicate how the existence of such an operation-preserving duality implies the integrality of certain polynomials in the Pontryagin classes of the manifold."}
{"category": "Math", "title": "Summability of multilinear mappings: Littlewood, Orlicz and beyond", "abstract": "In this paper we prove a plenty of new results concerning summabililty properties of multilinear mappings between Banach spaces, such as an extension of Littlewood's 4/3 Theorem. Among other features, it is shown that every continuous n-linear form on the disc algebra or the Hardy space is (1;2,...,2)-summing, the role of the Littlewood-Orlicz property in the theory is established and the interplay with almost summing multilinear mappings is explored."}
{"category": "Math", "title": "Special Riemannian geometries and the Magic Square of Lie algebras", "abstract": "We investigate nonintegrable Riemannian geometries modelled after certain symmetric spaces related to the Freudenthal-Tits Magic Square. The collection of four such structures found by Nurowski is extended by further eight. A focus is given to those admitting a compatible connection with completely skew torsion."}
{"category": "Math", "title": "The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces", "abstract": "We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define local coordinates and construct foliations on the moduli spaces. We use these constructions to give a new proof of the Diaz' bound on the dimension of complete subvarieties of the moduli spaces. Geometrically, we study the properties of meromorphic differentials with real periods, and their degenerations."}
{"category": "Math", "title": "The approximate fixed point property in Hausdorff topological vector spaces and applications", "abstract": "We establish an approximate fixed point result for self-maps on compact convex subsets of Hausdorff topological vector spaces where continuity is not a necessary condition."}
{"category": "Math", "title": "Towards a Global Springer Theory I: The affine Weyl group action", "abstract": "We propose a generalization of Springer representations to the context of groups over a global function field. The global counterpart of the Grothendieck simultaneous resolution is the parabolic Hitchin fibration. We construct an action of the affine Weyl group on the direct image complex of the parabolic Hitchin fibration. In particular, we get representations of the affine Weyl group on the cohomology of parabolic Hitchin fibers, providing the first step towards a global Springer theory."}
{"category": "Math", "title": "On collisions of Brownian particles", "abstract": "We examine the behavior of $n$ Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence and for the presence of triple collisions among the particles. As an application to the Atlas model for equity markets, we study a special construction of such systems of diffusing particles using Brownian motions with reflection on polyhedral domains."}
{"category": "Math", "title": "Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets", "abstract": "The problem of construction of Barabanov norms for analysis of properties of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. The method of Barabanov norms was the key instrument in disproving the Lagarias-Wang Finiteness Conjecture. The related constructions were essentially based on the study of the geometrical properties of the unit balls of some specific Barabanov norms. In this context the situation when one fails to find among current publications any detailed analysis of the geometrical properties of the unit balls of Barabanov norms looks a bit paradoxical. Partially this is explained by the fact that Barabanov norms are defined nonconstructively, by an implicit procedure. So, even in simplest cases it is very difficult to visualize the shape of their unit balls. The present work may be treated as the first step to make up this deficiency. In the paper two iteration procedure are considered that allow to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets."}
{"category": "Math", "title": "Quantizations of modules of differential operators", "abstract": "Fix a manifold M, and let V be an infinite dimensional Lie algebra of vector fields on M. Assume that V contains a finite dimensional semisimple maximal subalgebra A, the projective or conformal subalgebra. A projective or conformal quantization of a V-module of differential operators on M is a decomposition into irreducible A-modules. We survey recent results on projective quantizations and their applications to cohomology, geometric equivalences and symmetries of differential operator modules, and indecomposable modules."}
{"category": "Math", "title": "On explicit a priori estimates of the joint spectral radius by the generalized Gelfand formula", "abstract": "In various problems of control theory, non-autonomous and multivalued dynamical systems, wavelet theory and other fields of mathematics information about the rate of growth of matrix products with factors taken from some matrix set plays a key role. One of the most prominent quantities characterizing the exponential rate of growth of matrix products is the so-called joint or generalized spectral radius. In the work some explicit a priori estimates for the joint spectral radius with the help of the generalized Gelfand formula are obtained. These estimates are based on the notion of the measure of irreducibility (quasi-controllability) of matrix sets proposed previously by A. Pokrovskii and the author."}
{"category": "Math", "title": "Quasi-K\\\"ahler groups, 3-manifold groups, and formality", "abstract": "In this note, we address the following question: Which 1-formal groups occur as fundamental groups of both quasi-K\\\"ahler manifolds and closed, connected, orientable 3-manifolds. We classify all such groups, at the level of Malcev completions, and compute their coranks. Dropping the assumption on realizability by 3-manifolds, we show that the corank equals the isotropy index of the cup-product map in degree one. Finally, we examine the formality properties of smooth affine surfaces and quasi-homogeneous isolated surface singularities. In the latter case, we describe explicitly the positive-dimensional components of the first characteristic variety for the associated singularity link."}
{"category": "Math", "title": "Duality Theorem for Motives", "abstract": "Using Dold--Puppe category approach to the duality in topology, we prove general duality theorem for the category of motives. As one of the applications of this general result we obtain, in particular, a generalization of Friedlander--Voevodsky's duality to the case of arbitrary base field characteristic."}
{"category": "Math", "title": "Examples of hyperlinear groups without factorization property", "abstract": "In this note we give an example of a group which is locally embeddable into finite groups (in particular it is initially subamenable, sofic and hence hyperlinear) but does not have Kirchberg's factorization property. This group provides also an example of a sofic Kazhdan group which is not residually finite, answering a question of Elek and Szabo. We also give an example of a group which is not initially subamenable but hyperlinear. Finally, we point out a mistake in an assertion of Kirchberg and provide an example of a group which does not have the factorization property and is still a subgroup of a connected finite-dimensional Lie group."}
{"category": "Math", "title": "Phase transition for the Ising model on the Critical Lorentzian triangulation", "abstract": "Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disagreement percolation method and on a variant of Peierls method. The critical temperature is shown to be constant a.s."}
{"category": "Math", "title": "Adjoints of Composition Operators on Hardy Spaces of the Half-Plane", "abstract": "Building on techniques used in the case of the disc, we use a variety of methods to develop formulae for the adjoints of composition operators on Hardy spaces of the upper half-plane. In doing so, we prove a slight extension of a known necessary condition for the boundedness of such operators, and use it to provide a complete classification of the bounded composition operators with rational symbol. We then consider some specific examples, comparing our formulae with each other, and with other easily deduced formulae for simple cases."}
{"category": "Math", "title": "Discrete complex analysis on isoradial graphs", "abstract": "We study discrete complex analysis and potential theory on a large family of planar graphs, the so-called isoradial ones. Along with discrete analogues of several classical results, we prove uniform convergence of discrete harmonic measures, Green's functions and Poisson kernels to their continuous counterparts. Among other applications, the results can be used to establish universality of the critical Ising and other lattice models."}
{"category": "Math", "title": "On the classification of geometric families of 4-dimensional Galois representations", "abstract": "We give a classification theorem for certain four-dimensional families of geometric $\\lambda$-adic Galois representations attached to a pure motive. More precisely, we consider families attached to the cohomology of a smooth projective variety defined over $\\Q$ with coefficients in a quadratic imaginary field, non-selfdual and with four different Hodge-Tate weights. We prove that the image is as large as possible for almost every $\\lambda$ provided that the family is irreducible and not induced from a family of smaller dimension. If we restrict to semistable families an even simpler classification is given. A version of the main result is given for the case where the family is attached to an automorphic form."}
{"category": "Math", "title": "On the combinatorial classification of toric log del Pezzo surfaces", "abstract": "Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is derived in terms of the index l. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for all l<17 is obtained."}
{"category": "Math", "title": "On the existence of k-homogeneous Latin bitrades", "abstract": "Following the earlier work on {homogeneous Latin bitrades by Cavenagh, Donovan, and Dr'apal (2003 and 2004) Bean, Bidkhori, Khosravi, and E. S. Mahmoodian (2005) we prove the following results. All k-homogeneous Latin bitrades of volume km exist, for 1) all odd number k and m is greater than or equal to k, 2) all even number k > 2 and m is greater than or equal to min{k +u,3k/2}, where u is any odd number which divides k, 3) all m is greater than or equal to k, where k is greater than or equal to 3 and is lesser than or equal to 37."}
{"category": "Math", "title": "Rare events, escape rates and quasistationarity: some exact formulae", "abstract": "We present a common framework to study decay and exchanges rates in a wide class of dynamical systems. Several applications, ranging form the metric theory of continuons fractions and the Shannon capacity of contrained systems to the decay rate of metastable states, are given."}
{"category": "Math", "title": "Entire functions in weighted $L_2$ and zero modes of the Pauli operator with non-signdefinite magnetic field", "abstract": "For a real non-signdefinite function $B(z)$, $z\\in \\C$, we investigate the dimension of the space of entire analytical functions square integrable with weight $e^{\\pm 2F}$, where the function $F(z)=F(x_1,x_2)$ satisfies the Poisson equation $\\D F=B$. The answer is known for the function $B$ with constant sign. We discuss some classes of non-signdefinite positively homogeneous functions $B$, where both infinite and zero dimension may occur. In the former case we present a method of constructing entire functions with prescribed behavior at infinity in different directions. The topic is closely related with the question of the dimension of the zero energy subspace (zero modes) for the Pauli operator."}
{"category": "Math", "title": "An alternative construction of B-M and B-T unitals in Desarguesian planes", "abstract": "We present a new construction of non-classical unitals from a classical unital $U$ in $PG(2,q^2)$. The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model $\\Pi$ of $PG(2,q^2)$ with the following three properties: 1. points of $\\Pi$ are those of $PG(2,q^2)$; 2. lines of $\\Pi$ are certain lines and conics of $PG(2,q^2)$; 3. the points in $U$ form a non-classical B-M unital in $\\Pi$. Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree."}
{"category": "Math", "title": "Geometry of Third-Order Ordinary Differential Equations and Its Applications in General Relativity", "abstract": "A PhD thesis written under supervision of Pawel Nurowski and defended at the Faculty of Physics of the University of Warsaw. We adress the problems of local equivalence and geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are described in a uniform manner by the Cartan method of equivalence. This includes conformal, Weyl and metric geometries in three and six dimensions and contact projective geometry in dimension three. Respective connections for these geometries are given and their curvatures are expressed by contact, point or fibre-preserving relative invariants of the ODEs. We construct Cartan coframes which yield the full set of local invariants and solve the local problem of contact and point equivalence of the ODEs. We explicitly describe ODEs admitting at least four-dimensional Lie group of contact or point symmetries and real ODEs fibre-preserving equivalent to II, IV, V, VI, VII and XI Chazy classes."}
{"category": "Math", "title": "A lower bound for the error term in Weyl's law for certain Heisenberg manifolds, II", "abstract": "This article is concerned with estimations from below for the remainder term in Weyl's law for the spectral counting function of certain rational (2l+1)-dimensional Heisenberg manifolds. Concentrating on the case of odd l, it continues the work done in part I which dealt with even l."}
{"category": "Math", "title": "The q-Log-convexity of the Generating Functions of the Squares of Binomial Coefficients", "abstract": "We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence $\\{\\sum_{k=0}^n{n\\choose k}^2q^k\\}_{n\\geq 0}$. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then the principal specialization leads to the q-log-convexity. We also prove that a technical condition of Liu and Wang holds for the squares of the binomial coefficients. Hence we deduce that the linear transformation with respect to the triangular array $\\{{n\\choose k}^2\\}_{0\\leq k\\leq n}$ is log-convexity preserving."}
{"category": "Math", "title": "The Geometry of Filtering", "abstract": "Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space of a principal bundle, and for the operators on the diffeomorphism group arising from stochastic flows. Classical non-linear filtering problems also lead to such conffigurations. A basic tool is the, possibly, non-linear \"semi-connection\" induced by this set up, leading to a canonical decomposition of the operator on the domain space. Topics discussed include: generalised Wietzenbock curvatures arising in the equivariant case, skew -product decompositions of diffusion processes, conditioned processes, classical filtering, decomposition of stochastic flows, and connections determined by stochastic differential equations."}
{"category": "Math", "title": "The gl_2 Bethe algebra associated with a nilpotent element", "abstract": "To any 2x2-matrix K one assigns a commutative subalgebra B^{K}\\subset U(gl_2[t]) called a Bethe algebra. We describe relations between the Bethe algebras, associated with the zero matrix and a nilpotent matrix."}
{"category": "Math", "title": "Rational functions with real multipliers", "abstract": "Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f) belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle."}
{"category": "Math", "title": "The reducts of equality up to primitive positive interdefinability", "abstract": "We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain."}
{"category": "Math", "title": "A generalized portmanteau test of independence between two stationary time series", "abstract": "We propose generalized portmanteau-type test statistics in the frequency domain to test independence between two stationary time series. The test statistics are formed analogous to the one in Chen and Deo (2004, Econometric Theory 20, 382-416), who extended the applicability of portmanteau goodness-of-fit test to the long memory case. Under the null hypothesis of independence, the asymptotic standard normal distributions of the proposed statistics are derived under fairly mild conditions. In particular, each time series is allowed to possess short memory, long memory or anti-persistence. A simulation study shows that the tests have reasonable size and power properties."}
{"category": "Math", "title": "Global uniqueness from partial Cauchy data in two dimensions", "abstract": "We prove for a two dimensional bounded domain that the Cauchy data for the Schroedinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results."}
{"category": "Math", "title": "Upward and Downward Runs on Partially Ordered Sets", "abstract": "We consider Markov chains on partially ordered sets that generalize the success-runs and remaining life chains in reliability theory. We find conditions for recurrence and transience and give simple expressions for the invariant distributions. We study a number of special cases, including rooted trees, uniform posets, and posets associated with positive semigroups."}
{"category": "Math", "title": "Distribution of values of $L$-functions at the edge of the critical strip", "abstract": "We prove several results on the distribution of values of $L$-functions at the edge of the critical strip, by constructing and studying a large class of random Euler products. Among new applications, we study families of symmetric power $L$-functions of holomorphic cusp forms in the level aspect (assuming the automorphy of these $L$-functions) at $s=1$, functions in the Selberg class (in the height aspect), and quadratic twists of a fixed $GL(m)/{\\Bbb Q}$-automorphic cusp form at $s=1$."}
{"category": "Math", "title": "Equivariant asymptotics for Toeplitz operators", "abstract": "In recent years, the Tian-Zelditch asymptotic expansion for the equivariant components of the Szeg\\\"{o} kernel of a polarized complex projective manifold, and its subsequent generalizations in terms of scaling limits, have played an important role in algebraic, symplectic, and differential geometry. A natural question is whether there exist generalizations in which the projector onto the spaces of holomorphic sections can be replaced by the projector onto more general (non-complete) linear series. One case that lends itself to such analysis, and which is natural from the point of view of geometric quantization, is given by the linear series determined by imposing spectral bounds on an invariant self-adjoint Toeplitz operator. In this paper we focus on the asymptotics of the spectral projectors associated to slowly shrinking spectral bands."}
{"category": "Math", "title": "Non-uniform hyperbolicity in complex dynamics", "abstract": "We study rational functions satisfying summability conditions - a family of weak conditions on the expansion along the critical orbits. Assuming their appropriate versions, we derive many nice properties: There exists a unique, ergodic, and non-atomic conformal measure with exponent equal to the Hausdorff dimension of the Julia set. There is an absolutely continuous invariant measure with respect to this conformal measure. The Minkowski dimension of the Julia set is strictly less than 2. Either the Julia set is the whole sphere, or the dynamics is unstable. For such polynomials and Blaschke products the Julia set is conformally removable. The connected components of the boundary of invariant Fatou components are locally connected. Finally, we derive a conformal analogue of Jakobson-Benedicks-Carleson theorem and prove the external continuity of the Hausdorff dimension of Julia sets for almost all points in the Mandelbrot set with respect to the harmonic measure. Some of the results extend to the multimodal maps of an interval."}
{"category": "Math", "title": "Counting interesting elections", "abstract": "We provide an elementary proof of a formula for the number of northeast lattice paths that lie in a certain region of the plane. Equivalently, this formula counts the lattice points inside the Pitman--Stanley polytope of an n-tuple."}
{"category": "Math", "title": "Szego limit theorems on the Sierpinski gasket", "abstract": "We use the existence of localized eigenfunctions of the Laplacian on the Sierpinski gasket to formulate and prove analogues of the strong Szego limit theorem in this fractal setting. Furthermore, we recast some of our results in terms of equally distributed sequences."}
{"category": "Math", "title": "Sur la dualite et la descente d'Iwasawa", "abstract": "We develop a formalism for studying descent and codescent in the context of Iwasawa theory. The main result essentially states that to control descent or codescent amounts to the same. Arithmetic applications are given."}
{"category": "Math", "title": "Central Limit Theorem and recurrence for random walks in bistochastic random environments", "abstract": "We prove the annealed Central Limit Theorem for random walks in bistochastic random environments on $Z^d$ with zero local drift. The proof is based on a \"dynamicist's interpretation\" of the system, and requires a much weaker condition than the customary uniform ellipticity. Moreover, recurrence is derived for $d \\le 2$."}
{"category": "Math", "title": "A note on Artin's constant", "abstract": "We suggest a new approach to Artin's constant that leads to its representation as an infinite sum divided by another infinite sum. The same approach works well for Stephens' constant and higher rank Artin's constants. The main results are theoretical but there are interesting experimental and computational aspects."}
{"category": "Math", "title": "Well-posedness results for triply nonlinear degenerate parabolic equations", "abstract": "We study the well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problem $$ b(u)_t - {\\rm div} \\tilde{\\mathfrak a}(u,\\nabla\\phi(u))+\\psi(u)=f, \\quad u|_{t=0}=u_0 $$ in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities $b,\\phi$ and $\\psi$ are supposed to be continuous non-decreasing, and the nonlinearity $\\tilde{\\mathfrak a}$ falls within the Leray-Lions framework. Some restrictions are imposed on the dependence of $\\tilde{\\mathfrak a}(u,\\nabla\\phi(u))$ on $u$ and also on the set where $\\phi$ degenerates. A model case is $\\tilde{\\mathfrak a}(u,\\nabla\\phi(u)) =\\tilde{\\mathfrak{f}}(b(u),\\psi(u),\\phi(u))+k(u)\\mathfrak{a}_0(\\nabla\\phi(u)),$ with $\\phi$ which is strictly increasing except on a locally finite number of segments, and $\\mathfrak{a}_0$ which is of the Leray-Lions kind. We are interested in existence, uniqueness and stability of entropy solutions. If $b=\\mathrm{Id}$, we obtain a general continuous dependence result on data $u_0,f$ and nonlinearities $b,\\psi,\\phi,\\tilde{\\mathfrak{a}}$. Similar result is shown for the degenerate elliptic problem which corresponds to the case of $b\\equiv 0$ and general non-decreasing surjective $\\psi$. Existence, uniqueness and continuous dependence on data $u_0,f$ are shown when $[b+\\psi](\\R)=\\R$ and $\\phi\\circ [b+\\psi]^{-1}$ is continuous."}
{"category": "Math", "title": "A new construction of the asymptotic algebra associated to the $q$-Schur algebra", "abstract": "We denote by A the ring of Laurent polynomials in the indeterminate v and by K its field of fractions. In this paper, we are interested in representation theory of the \"generic\" q-Schur algebra S_q(n,r) over A. We will associate to every non-degenerate symmetrising trace form \\tau on KS_q(n,r) a subalgebra J_{\\tau} of KS_q(n,r) which is isomorphic to the \"asymptotic\" algebra \\J(n,r)_A defined by J. Du. As a consequence, we give a new criterion for James' conjecture."}
{"category": "Math", "title": "GIT Constructions of Moduli Spaces of Stable Curves and Maps", "abstract": "This largely expository paper first gives an introduction to Hilbert stability and its use in Gieseker's GIT construction of $\\overline{M}_g$. Then I review recent work in this area--variants for unpointed curves that arise in Hassett's log minimal model program, starting with Schubert's moduli space of pseudostable curves, and constructions for weighted pointed stable curves and for pointed stable maps due to Swinarski and to Baldwin and Swinarski respectively. The focus is on the steps at which new ideas are needed. Finally, I list open problems in the area, particularly some arising in the log minimal model program that seem inaccessible to current techniques."}
{"category": "Math", "title": "Embeddings of 3-manifolds in S^4 from the point of view of the 11-tetrahedron census", "abstract": "This is a collection of notes on embedding problems for 3-manifolds. The main question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 manifolds in the census, of which 13400 are orientable. Of the 13400 orientable manifolds, only 149 of them have hyperbolic torsion linking forms and are thus candidates for embedability in the 4-sphere. The majority of this paper is devoted to the embedding problem for these 149 manifolds. At present 41 are known to embed. Among the remaining manifolds, embeddings into homotopy 4-spheres are constructed for 4. 67 manifolds are known to not embed in the 4-sphere. This leaves 37 unresolved cases, of which only 3 are geometric manifolds i.e. having a trivial JSJ-decomposition."}
{"category": "Math", "title": "$L^1$ is complemented in the dual space $L^{\\infty *}$", "abstract": "We show $L^1$ is complemented in the dual space $L^{\\infty *}$ for a finite regular complex measure on a compact Hausdorff space"}
{"category": "Math", "title": "On a problem of Specker about Euclidean representations of finite graphs", "abstract": "Say that a graph $G$ is \\emph{representable in $\\R ^n$} if there is a map $f$ from its vertex set into the Euclidean space $\\R ^n$ such that $\\| f(x) - f(x')\\| = \\| f(y) - f(y')\\|$ iff $\\{x,x'\\}$ and $\\{y, y'\\}$ are both edges or both non-edges in $G$. The purpose of this note is to present the proof of the following result, due to Einhorn and Schoenberg: if $G$ finite is neither complete nor independent, then it is representable in $\\R ^{|G|-2}$. A similar result also holds in the case of finite complete edge-colored graphs."}
{"category": "Math", "title": "2-Vector Spaces and Groupoids", "abstract": "This paper describes a relationship between essentially finite groupoids and 2-vector spaces. In particular, we show to construct 2-vector spaces of Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation--a weak functor--from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail."}
{"category": "Math", "title": "On polytopes associated to factorisations of prime-powers", "abstract": "We study polytopes associated to factorisations of prime powers. These polytopes have explicit descriptions either in terms of their vertices or as intersections of closed halfspaces associated to their facets. We give formulae for their $f-$vectors."}
{"category": "Math", "title": "Plain Varieties", "abstract": "Algebraic varieties which are locally isomorphic to open subsets of affine space will be called {\\em plain}. Plain varieties are smooth and rational. The converse is true for curves and surfaces, and unknown in general. It is shown that plain varieties are stable under blowup in smooth centers."}
{"category": "Math", "title": "A 3-local characterization of M_12 and SL_3(3)", "abstract": "We identify the sporadic simple group $\\mathrm{M}_{12}$ and the simple group $\\mathrm{SL}_3(3)$ from some part of their 3-local structure and give a graph theoretic analogue of the resulting theorem."}
{"category": "Math", "title": "On the linearity of HNN-extensions with abelian base group", "abstract": "We show that an HNN-extension with finitely generated abelian base group is Z-linear if and only if it is residually finite."}
{"category": "Math", "title": "Abstract Hodge decomposition and minimal models for cyclic algebras", "abstract": "We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summation over trees. This minimal model is unique up to homotopy."}
{"category": "Math", "title": "Presentation of right-angled Soergel categories by generators and relations", "abstract": "Soergel bimodule category B is a categorification of the Hecke algebra of a Coxeter system (W,S). We find a presentation of B (as a tensor category) by generators and relations when W is a right-angled Coxeter group."}
{"category": "Math", "title": "W-types in sheaves", "abstract": "We give a concrete description of W-types in categories of sheaves."}
{"category": "Math", "title": "Toward a classification of prime ideals in Pr\\\"ufer domains", "abstract": "The primary purpose of this paper is give a classification scheme for the nonzero primes of a Pr\\\"ufer domain based on five properties. A prime $P$ of a Pr\\\"ufer domain $R$ could be sharp or not sharp, antesharp or not, divisorial or not, branched or unbranched, idempotent or not. Based on these five basic properties, there are six types of maximal ideals and twelve types of nonmaximal (nonzero) primes. Both characterizations and examples are given for each type that exists."}
{"category": "Math", "title": "Graded Brauer Tree Algebras", "abstract": "In this paper we construct non-negative gradings on a basic Brauer tree algebra $A_{\\Gamma}$ corresponding to an arbitrary Brauer tree $\\Gamma$ of type $(m,e)$. We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra $A_S$, whose tree is a star with the exceptional vertex in the middle, to $A_{\\Gamma}$. The grading on $A_S$ comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green's walk around $\\Gamma$ (cf. [\\ref{Zak}]). By computing endomorphism rings of these tilting complexes we get graded algebras. We also compute ${\\rm Out}^K(A_{\\Gamma})$, the group of outer automorphisms that fix isomorphism classes of simple $A_{\\Gamma}$-modules, where $\\Gamma$ is an arbitrary Brauer tree, and we prove that there is unique grading on $A_{\\Gamma}$ up to graded Morita equivalence and rescaling."}
{"category": "Math", "title": "Thom polynomials and Schur functions: towards the singularities $A_i(-)$", "abstract": "We develop algebro-combinatorial tools for computing the Thom polynomials for the Morin singularities $A_i(-)$ ($i\\ge 0$). The main tool is the function $F^{(i)}_r$ defined as a combination of Schur functions with certain numerical specializations of Schur polynomials as their coefficients. We show that the Thom polynomial ${\\cal T}^{A_i}$ for the singularity $A_i$ (any $i$) associated with maps $({\\bf C}^{\\bullet},0) \\to ({\\bf C}^{\\bullet+k},0)$, with any parameter $k\\ge 0$, under the assumption that $\\Sigma^j=\\emptyset$ for all $j\\ge 2$, is given by $F^{(i)}_{k+1}$. Equivalently, this says that \"the 1-part\" of ${\\cal T}^{A_i}$ equals $F^{(i)}_{k+1}$. We investigate 2 examples when ${\\cal T}^{A_i}$ apart from its 1-part consists also of the 2-part being a single Schur function with some multiplicity. Our computations combine the characterization of Thom polynomials via the \"method of restriction equations\" of Rim\\'anyi et al. with the techniques of Schur functions."}
{"category": "Math", "title": "Thom polynomials and Schur funcions: the singularities $A_3(-)$", "abstract": "Combining the \"method of restriction equations\" of Rim\\'anyi et al. with the techniques of symmetric functions, we establish the Schur function expansions of the Thom polynomials for the Morin singularities $A_3: ({\\bf C}^{\\bullet},0)\\to ({\\bf C}^{\\bullet + k},0)$ for any nonnegative integer $k$."}
{"category": "Math", "title": "Block-diagonalisation of matrices and operators", "abstract": "In this short note we deal with a constructive scheme to decompose a continuous family of matrices $A(\\rho)$ asymptotically as $\\rho\\to0$ into blocks corresponding to groups of eigenvalues of the limit matrix A(0). We also discuss the extension of the scheme to matrix families depending upon additional parameters and operators on Hilbert spaces."}
{"category": "Math", "title": "An Indicator Function Limit Theorem in Dynamical Systems", "abstract": "We show by a constructive proof that in all aperiodic dynamical system, for all sequences $(a_n)_{n\\in\\N}\\subset\\R_+$ such that $a_n\\nearrow\\infty$ and $\\frac{a_n}{n}\\to 0$ as $n\\to\\infty$, there exists a set $A\\in\\A$ having the property that the sequence of the distributions of $(\\frac{1}{a_{n}}S_{n}(\\ind_A-\\mu(A)))_{n\\in\\N}$ is dense in the space of all probability measures on $\\R$."}
{"category": "Math", "title": "Amenable actions of amalgamated free products", "abstract": "We prove that the amalgamated free product of two free groups of rank two over a common cyclic subgroup, admits an amenable, faithful, transitive action on an infinite countable set. We also show that any finite index subgroup admits such an action, which applies for example to surface groups and fundamental groups of surface bundles over $\\mathbb{S}^1$."}
{"category": "Math", "title": "The area above the Dyck path of a permutation", "abstract": "In this paper we study a mapping from permutations to Dyck paths. A Dyck path gives rise to a (Young) diagram and we give relationships between statistics on permutations and statistics on their corresponding diagrams. The distribution of the size of this diagram is discussed and a generalisation given of a parity result due to Simion and Schmidt. We propose a filling of the diagram which determines the permutation uniquely. Diagram containment on a restricted class of permutations is shown to be related to the strong Bruhat poset."}
{"category": "Math", "title": "Stability and Total Variation Estimates on General Scalar Balance Laws", "abstract": "Consider the general scalar balance law $\\partial_t u + \\Div f(t, x,u) = F(t,x,u)$ in several space dimensions. The aim of this note is to estimate the dependence of its solutions from the flow $f$ and from the source $F$. To this aim, a bound on the total variation in the space variables of the solution is obtained. This result is then applied to obtain well posedness and stability estimates for a balance law with a non local source."}
{"category": "Math", "title": "On Normalized Ricci Flow and Smooth Structures on Four-Manifolds with $b^+=1$", "abstract": "We find an obstruction to the existence of non-singular solutions to the normalized Ricci flow on four-manifolds with $b^+=1$. By using this obstruction, we study the relationship between the existence or non-existence of non-singular solutions of the normalized Ricci flow and exotic smooth structures on the topological 4-manifolds ${\\mathbb C}{P}^2 # l \\overline{{\\mathbb C}{P}^2}$, where $5 \\leq l \\leq 8$."}
{"category": "Math", "title": "Structures conformes asymptotiquement plates", "abstract": "In the first part of this article we revisit the theory of weighted spinors on conformal manifolds. In the second part we introduce the notions of asymptotically flat Weyl structures and of associated mass, and we prove a conformal version of the positive mass theorem on conformal spin manifolds."}
{"category": "Math", "title": "Parabolic Harnack Inequality and Local Limit Theorem for Percolation Clusters", "abstract": "We consider the random walk on supercritical percolation clusters in the d-dimensional Euclidean lattice. Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function."}
{"category": "Math", "title": "A visible factor of the special L-value", "abstract": "Let~$A$ be a quotient of $J_0(N)$ associated to a newform $f$ such that the special $L$-value of $A$ (at $s=1$) is non-zero. We give a formula for the ratio of the special $L$-value to the real period of $A$ that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is non-trivial in general and is related to certain congruences of $f$ with eigenforms of positive analytic rank. We use the techniques of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime $q$ divides this factor, then $q$ divides either the order of the Shafarevich-Tate group or the order of a component group of $A$. Suppose $p$ is an odd prime such that $p^2$ does not divide $N$, $p$ does not divide the order of the rational torsion subgroup of $A$, and $f$ is congruent modulo a prime ideal over $p$ to an eigenform whose associated abelian variety has positive Mordell-Weil rank. Then we show that $p$ divides the factor mentioned above; in particular, $p$ divides the numerator of the ratio of the special $L$-value to the real period of $A$. Both of these results are as implied by the second part of the Birch and Swinnerton-Dyer conjecture, and thus provide theoretical evidence towards the conjecture."}
{"category": "Math", "title": "On Izumi's theorem on comparison of valuations", "abstract": "We prove that the sequence of MacLane key polynomials constructed in \\cite{Mac1} and \\cite{Sp2} for a valuation extension $(K,\\nu)\\subset (K(x),\\mu)$ is finite, provided that both $\\nu$ and $\\mu$ are divisorial and $\\mu$ is centered over an analytically irreducible local domain $(R,\\frak{m})\\subset K[x]$. As a corollary, we prove Izumi's theorem on comparison of divisorial valuations. %We show that the existence of Izumi constants is equivalent to the finiteness of the sequence of the MacLane key-polynomials. We give explicit bounds for the Izumi constant in terms of the key polynomials of the valuations. We show that this bound can be attained in some cases."}
{"category": "Math", "title": "Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one", "abstract": "Let $E$ be an optimal elliptic curve over $\\Q$ of conductor $N$ having analytic rank one, i.e., such that the $L$-function $L_E(s)$ of $E$ vanishes to order one at $s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing $N$ split and such that the $L$-function of $E$ over $K$ vanishes to order one at $s=1$. Suppose there is another optimal elliptic curve over $\\Q$ of the same conductor $N$ whose Mordell-Weil rank is greater than one and whose associated newform is congruent to the newform associated to $E$ modulo an integer $r$. The theory of visibility then shows that under certain additional hypotheses, $r$ divides the order of the Shafarevich-Tate group of $E$ over $K$. We show that under somewhat similar hypotheses, $r$ divides the order of the Shafarevich-Tate group of $E$ over $K$. We show that under somewhat similar hypotheses, $r$ also divides the Birch and Swinnerton-Dyer {\\em conjectural} order of the Shafarevich-Tate group of $E$ over $K$, which provides new theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture in the analytic rank one case."}
{"category": "Math", "title": "A representation-valued relative Riemann-Hurwitz theorem and the Hurwitz-Hodge bundle", "abstract": "We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interpreted as a representation-ring-valued relative Riemann-Hurwitz formula for families of admissible G-covers."}
{"category": "Math", "title": "Quasisymmetric Schur functions", "abstract": "We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the $t$ parameter from Hall-Littlewood theory."}
{"category": "Math", "title": "Critical points of pairs of varieties of algebras", "abstract": "For a class V of algebras, denote by Conc(V) the class of all semilattices isomorphic to the semilattice Conc(A) of all compact congruences of A, for some A in V. For classes V1 and V2 of algebras, we denote by crit(V1,V2) the smallest cardinality of a semilattice in Conc(V1) which is not in Conc(V2) if it exists, infinity otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties V1 and V2, crit(V1,V2) is either finite, or aleph_n for some natural number n, or infinity. We also find two finitely generated modular lattice varieties V1 and V2 such that crit(V1,V2)=aleph_1, thus answering a question by J. Tuma and F. Wehrung."}
{"category": "Math", "title": "Semiprojectivity of universal C*-algebras generated by algebraic elements", "abstract": "Let $p$ be a polynomial in one variable whose roots either all have multiplicity more than 1 or all have multiplicity exactly 1. It is shown that the universal $C^*$-algebra of a relation $p(x)=0$, $\\|x\\| \\le 1$ is semiprojective. In the case of all roots multiple it is shown that the universal $C^*$-algebra is also residually finite-dimensional. Applications to polynomially compact operators are given."}
{"category": "Math", "title": "The Maximal C*-Algebra of Quotients as an Operator Bimodule", "abstract": "We establish a description of the maximal C*-algebra of quotients of a unital C*-algebra $A$ as a direct limit of spaces of completely bounded bimodule homomorphisms from certain operator submodules of the Haagerup tensor product $A\\otimes_h A$ labelled by the essential closed right ideals of $A$ into $A$. In addition the invariance of the construction of the maximal C*-algebra of quotients under strong Morita equivalence is proved."}
{"category": "Math", "title": "Uniqueness of self-similar solutions to the network flow in a given topological class", "abstract": "In this paper we study the uniqueness of expanding self-similar solutions to the network flow in a fixed topological class. We prove the result via the parabolic Allen-Cahn approximation proved in \\cite{triodginz}. Moreover, we prove that any regular evolution of connected tree-like network (with an initial condition that might be not regular) is unique in a given a topological class."}
{"category": "Math", "title": "Multirate integration of axisymmetric step-flow equations", "abstract": "We present a multirate method that is particularly suited for integrating the systems of Ordinary Differential Equations (ODEs) that arise in step models of surface evolution. The surface of a crystal lattice, that is slightly miscut from a plane of symmetry, consists of a series of terraces separated by steps. Under the assumption of axisymmetry, the step radii satisfy a system of ODEs that reflects the steps' response to step line tension and step-step interactions. Two main problems arise in the numerical solution of these equations. First, the trajectory of the innermost step can become singular, resulting in a divergent step velocity. Second, when a step bunching instability arises, the motion of steps within a bunch becomes very strongly stable, resulting in \"local stiffness\". The multirate method introduced in this paper ensures that small time steps are taken for singular and locally stiff components, while larger time steps are taken for the remaining ones. Special consideration is given to the construction of high order interpolants during run time which ensures fourth order accuracy of scheme for components of the solution sufficiently far away from singular trajectories."}
{"category": "Math", "title": "Asymptotic behavior of a nonlocal parabolic problem in Ohmic heating process", "abstract": "In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem \\[ u_{t}=\\Delta u+\\displaystyle\\frac{\\lambda f(u)}{\\big(\\int_{\\Omega}f(u)dx\\big)^{p}}, x\\in \\Omega, t>0, \\] with homogeneous Dirichlet boundary condition, where $\\lambda>0, p>0$, $f$ is nonincreasing. It is found that: (a) For $0<p\\leq1$, $u(x,t)$ is globally bounded and the unique stationary solution is globally asymptotically stable for any $\\lambda>0$; (b) For $1<p<2$, $u(x,t)$ is globally bounded for any $\\lambda>0$; (c) For $p=2$, if $0<\\lambda<2|\\partial\\Omega|^2$, then $u(x,t)$ is globally bounded, if $\\lambda=2|\\partial\\Omega|^2$, there is no stationary solution and $u(x,t)$ is a global solution and $u(x,t)\\to\\infty$ as $t\\to\\infty$ for all $x\\in\\Omega$, if $\\lambda>2|\\partial\\Omega|^2$, there is no stationary solution and $u(x,t)$ blows up in finite time for all $x\\in\\Omega$; (d) For $p>2$, there exists a $\\lambda^*>0$ such that for $\\lambda>\\lambda^*$, or for $0<\\lambda\\leq\\lambda^*$ and $u_0(x)$ sufficiently large, $u(x,t)$ blows up in finite time. Moreover, some formal asymptotic estimates for the behavior of $u(x,t)$ as it blows up are obtained for $p\\geq2$."}
{"category": "Math", "title": "Compactness of the Complex Green Operator on CR-Manifolds of Hypersurface Type", "abstract": "The purpose of this article is to study compactness of the complex Green operator on CR manifolds of hypersurface type. We introduce (CR-P_q), a potential theoretic condition on $(0,q)$-forms that generalizes Catlin's property (P_q) to CR manifolds of arbitrary codimension. We prove that if an embedded CR-manifold of hypersurface type satisfies (CR-P_q) and (CR-P_{n-1-q}) and is of real dimension at least five, then the complex Green operator is a compact operator on the Sobolev spaces $H^s_{0,q}(M)$, if $1\\leq q \\leq n-2$ and $s\\geq 0$. We use CR-plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle."}
{"category": "Math", "title": "Radial Solutions for Hamiltonian Elliptic Systems with Weights", "abstract": "We prove the existence of infinitely many radial solutions for elliptic systems in Rn with power weights. A key tool for the proof will be a weighted imbedding theorem for fractional-order Sobolev spaces, that could be of independent interest."}
{"category": "Math", "title": "Parametrization of holomorphic Segre preserving maps", "abstract": "In this paper, we explore holomorphic Segre preserving maps. First, we investigate holomorphic Segre preserving maps sending the complexification $\\mathcal{M}$ of a generic real analytic submanifold $M \\subseteq \\C^N$ of finite type at some point $p$ into the complexification $\\mathcal{M}'$ of a generic real analytic submanifold $M' \\subseteq \\C^{N'}$, finitely nondegenerate at some point $p'$. We prove that for a fixed $M$ and $M'$, the germs at $(p,\\bar{p})$ of Segre submersive holomorphic Segre preserving maps sending $(\\M,(p,\\bar{p}))$ into $(\\M',(p', \\bar{p}'))$ can be parametrized by their $r$-jets at $(p,\\bar{p})$, for some fixed $r$ depending only on $M$ and $M'$. (If, in addition, $M$ and $M'$ are both real algebraic, then we prove that any such map must be holomorphic algebraic.) From this parametrization, it follows that the set of germs of holomorphic Segre preserving automorphisms $\\mathcal{H}$ of the complexification $\\mathcal{M}$ of a real analytic submanifold finitely nondegenerate and of finite type at some point $p$, and such that $\\mathcal{H}$ fixes $(p,\\bar{p})$, is an algebraic complex Lie group. We then explore the relationship between this automorphism group and the group of automorphisms of $M$ at $p$."}
{"category": "Math", "title": "Isomorphism and Morita equivalence of graph algebras", "abstract": "For any countable graph $E$, we investigate the relationship between the Leavitt path algebra $L_{\\C}(E)$ and the graph C*-algebra $C^*(E)$. For graphs $E$ and $F$, we examine ring homomorphisms, ring *-homomorphisms, algebra homomorphisms, and algebra *-homomorphisms between $L_{\\C}(E)$ and $L_{\\C}(F)$. We prove that in certain situations isomorphisms between $L_{\\C}(E)$ and $L_{\\C}(F)$ yield *-isomorphisms between the corresponding C*-algebras $C^*(E)$ and $C^*(F)$. Conversely, we show that *-isomorphisms between $C^*(E)$ and $C^*(F)$ produce isomorphisms between $L_{\\C}(E)$ and $L_{\\C}(F)$ in specific cases. The relationship between Leavitt path algebras and graph C*-algebras is also explored in the context of Morita equivalence."}
{"category": "Math", "title": "Geometric properties and related results for holomorphic Segre preserving maps", "abstract": "In this paper, we examine holomorphic Segre preserving maps between the complexifications of real hypersurfaces in $\\mathbb{C}^{n+1}$. In particular, we find several sufficient conditions ensuring that Segre transversality and total Segre nondegeneracy of the maps must hold."}
{"category": "Math", "title": "Multi-variable subordination distributions for free additive convolution", "abstract": "Let k be a positive integer and let D_k denote the space of joint distributions for k-tuples of selfadjoint elements in C*-probability space. The paper studies the concept of \"subordination distribution of \\mu \\boxplus \\nu with respect to \\nu\" for \\mu, \\nu \\in D_k, where \\boxplus is the operation of free additive convolution on D_k. The main tools used in this study are combinatorial properties of R-transforms for joint distributions and a related operator model, with operators acting on the full Fock space Multi-variable subordination turns out to have nice relations to a process of evolution towards \\boxplus-infinite divisibility on D_k that was recently found by Belinschi and Nica (arXiv:0711.3787). Most notably, one gets better insight into a connection which this process was known to have with free Brownian motion."}
{"category": "Math", "title": "Everywhere regularity of certain types of parabolic systems", "abstract": "In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. I show that when potential is a convex function that depends only on the norm of the solution, then bounded weak solutions of these parabolic systems are everywhere Holder continuous and thus everywhere smooth. I also show that the method used to prove this result can be easily adopted to simplify the proof of the result due to Wiegner on everywhere regularity of bounded weak solutions of strongly coupled parabolic systems."}
{"category": "Math", "title": "Notions of Lawvere theory", "abstract": "Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of universal algebra, can be generalized in three ways: replacing Set by another category, working in an enriched setting, and by working with another class of limits than finite products. An important special case involves working with sifted-colimit-preserving monads rather than filtered-colimit-preserving ones."}
{"category": "Math", "title": "The Rational qKZ Equation and Shifted Non-Symmetric Jack Polynomials", "abstract": "We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra $gl_N$. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one."}
{"category": "Math", "title": "The spectra ko and ku are not Thom spectra: an approach using THH", "abstract": "We apply an announced result of Blumberg-Cohen-Schlichtkrull to reprove (under restricted hypotheses) a theorem of Mahowald: the connective real and complex K-theory spectra are not Thom spectra."}
{"category": "Math", "title": "Strong Rational Connectedness of Surfaces", "abstract": "We discuss the strong rational connectedness of smooth rationally connected surfaces. We prove in lots of cases, including the smooth locus of a log del Pezzo surface, the rational connectedness indeed implies the strong rational conectedness. This confirms a conjecture due to Hassett and Tschinkel in \\cite{ht08}."}
{"category": "Math", "title": "New bijective links on planar maps via orientations", "abstract": "This article presents new bijections on planar maps. At first a bijection is established between bipolar orientations on planar maps and specific \"transversal structures\" on triangulations of the 4-gon with no separating 3-cycle, which are called irreducible triangulations. This bijection specializes to a bijection between rooted non-separable maps and rooted irreducible triangulations. This yields in turn a bijection between rooted loopless maps and rooted triangulations, based on the observation that loopless maps and triangulations are decomposed in a similar way into components that are respectively non-separable maps and irreducible triangulations. This gives another bijective proof (after Wormald's construction published in 1980) of the fact that rooted loopless maps with $n$ edges are equinumerous to rooted triangulations with $n$ inner vertices."}
{"category": "Math", "title": "Dissections, orientations, and trees, with applications to optimal mesh encoding and to random sampling", "abstract": "We present a bijection between some quadrangular dissections of an hexagon and unrooted binary trees, with interesting consequences for enumeration, mesh compression and graph sampling. Our bijection yields an efficient uniform random sampler for 3-connected planar graphs, which turns out to be determinant for the quadratic complexity of the current best known uniform random sampler for labelled planar graphs [{\\bf Fusy, Analysis of Algorithms 2005}]. It also provides an encoding for the set $\\mathcal{P}(n)$ of $n$-edge 3-connected planar graphs that matches the entropy bound $\\frac1n\\log_2|\\mathcal{P}(n)|=2+o(1)$ bits per edge (bpe). This solves a theoretical problem recently raised in mesh compression, as these graphs abstract the combinatorial part of meshes with spherical topology. We also achieve the {optimal parametric rate} $\\frac1n\\log_2|\\mathcal{P}(n,i,j)|$ bpe for graphs of $\\mathcal{P}(n)$ with $i$ vertices and $j$ faces, matching in particular the optimal rate for triangulations. Our encoding relies on a linear time algorithm to compute an orientation associated to the minimal Schnyder wood of a 3-connected planar map. This algorithm is of independent interest, and it is for instance a key ingredient in a recent straight line drawing algorithm for 3-connected planar graphs [\\bf Bonichon et al., Graph Drawing 2005]."}
{"category": "Math", "title": "A bijection between noncrossing and nonnesting partitions for classical reflection groups", "abstract": "We present an elementary type preserving bijection between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis."}
{"category": "Math", "title": "An upper bound on the number of zeros of a piecewise polinomial function", "abstract": "A precise tie between a univariate spline's knots and its zeros abundance and dissemination is formulated. As an application, a conjecture formulated by De Concini and Procesi is shown to be true in the special univariate, unimodular case. As a supplement, the same conjecture is shown, through computing a counterexample, to be false when unimodularity hypothesis is dropped."}
{"category": "Math", "title": "Classification of finite-growth general Kac-Moody superalgebras", "abstract": "A contragredient Lie superalgebra is a superalgebra defined by a Cartan matrix. A contragredient Lie superalgebra has finite-growth if the dimensions of the graded components (in the natural grading) depend polynomially on the degree. In this paper we classify finite-growth contragredient Lie superalgebras. Previously, such a classification was known only for the symmetrizable case."}
{"category": "Math", "title": "Alexey Vasilyevich Pogorelov, the mathematician of an incredible power", "abstract": "Life and the mathematical legacy of the great mathematician A.V. Pogorelov."}
{"category": "Math", "title": "Bayesian evidence for finite element model updating", "abstract": "This paper considers the problem of model selection within the context of finite element model updating. Given that a number of FEM updating models, with different updating parameters, can be designed, this paper proposes using the Bayesian evidence statistic to assess the probability of each updating model. This makes it possible then to evaluate the need for alternative updating parameters in the updating of the initial FE model. The model evidences are compared using the Bayes factor, which is the ratio of evidences. The Jeffrey scale is used to determine the differences in the models. The Bayesian evidence is calculated by integrating the likelihood of the data given the model and its parameters over the a priori model parameter space using the new nested sampling algorithm. The nested algorithm samples this likelihood distribution by using a hard likelihood-value constraint on the sampling region while providing the posterior samples of the updating model parameters as a by-product. This method is used to calculate the evidence of a number of plausible finite element models."}
{"category": "Math", "title": "Regular Kac-Moody superalgebras and integrable highest weight modules", "abstract": "We define regular Kac-Moody superalgebras and classify them using integrable modules. We give conditions for irreducible highest weight modules of regular Kac-Moody superalgebras to be integrable. This paper is a major part of the proof for the classification of finite-growth contragredient Lie superalgebras."}
{"category": "Math", "title": "A Collection of Problems on Spectrally Bounded Operators", "abstract": "We discuss several open problems on spectrally bounded operators, some new, some old, adding in a few new insights."}
{"category": "Math", "title": "From random walks to rough paths", "abstract": "Donsker's invariance principle is shown to hold for random walks in rough path topology. As application, we obtain Donsker-type weak limit theorems for stochastic integrals and differential equations."}
{"category": "Math", "title": "On the Geometry of Principal Homogeneous Spaces", "abstract": "Let $B$ be a curve defined over an algebraically closed field $k$ and let $X\\to B$ be an elliptic surface with base curve $B$. We investigate the geometry of everywhere locally trivial principal homogeneous spaces for $X$, i.e. elements of the Tate-Shafarevich group. If $Y$ is such a principal homogeneous space of order $n$, we find strong restrictions on the $\\mathbb{P}^{n-1}$ bundle over $B$ into which $Y$ embeds. Examples for small values of $n$ show that, in at least some cases, these restrictions are sharp. Finally, we determine these bundles in case $k$ has characteristic zero, $B = \\mathbb{P}^1$, and $X$ is generic in a suitable sense."}
{"category": "Math", "title": "Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes", "abstract": "We study asymptotic behavior of maximum likelihood estimator for a time inhomogeneous diffusion process given by a SDE $dX_t=\\alpha b(t)X_t dt + \\sigma(t) dB_t$, $t\\in[0,T)$, with a parameter $\\alpha\\in R$, where $T\\in(0,\\infty]$ and $(B_t)_{t\\in[0,T)}$ is a standard Wiener process. We formulate sufficient conditions under which the MLE of $\\alpha$ normalized by Fisher information converges to the limit distribution of Dickey-Fuller statistics. Next we study a SDE $dY_t=\\alpha b(t)a(Y_t) dt + \\sigma(t) dB_t$, $t\\in[0,T)$, with a perturbed drift satisfying $a(x)=x+O(1+|x|^\\gamma)$ with some $\\gamma\\in[0,1)$. We give again sufficient conditions under which the MLE of $\\alpha$ normalized by Fisher information converges to the limit distribution of Dickey-Fuller statistics."}
{"category": "Math", "title": "Exceptional del Pezzo hypersurfaces", "abstract": "We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in $\\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. As a corollary we obtain the existence of orbifold K\\\"ahler--Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-formed del Pezzo weighted hypersurfaces in $\\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$."}
{"category": "Math", "title": "Snowballs are Quasiballs", "abstract": "We introduce snowballs, which are compact sets in $\\R^3$ homeomorphic to the unit ball. They are 3-dimensional analogs of domains in the plane bounded by snowflake curves. For each snowball $B$ a quasiconformal map $f\\colon \\R^3\\to \\R^3$ is constructed that maps $B$ to the unit ball."}
{"category": "Math", "title": "Contact projective structures and chains", "abstract": "Contact projective structures have been profoundly studied by D.J.F. Fox. He associated to a contact projective structure a canonical projective structure on the same manifold. We interpret Fox' construction in terms of the equivalent parabolic (Cartan) geometries, showing that it is an analog of Fefferman's construction of a conformal structure associated to a CR structure. We show that, on the level of Cartan connections, this Fefferman--type construction is compatible with normality if and only if the initial structure has vanishing contact torsion. This leads to a geometric description of the paths that have to be added to the contact geodesics of a contact projective structure in order to obtain the subordinate projective structure. They are exactly the chains associated to the contact projective structure, which are analogs of the Chern-Moser chains in CR geometry. Finally, we analyze the consequences for the geometry of chains and prove that a chain-preserving contactomorphism must be a morphism of contact projective structures."}
{"category": "Math", "title": "Non-discrete Euclidean Buildings for the Ree and Suzuki groups", "abstract": "We call a non-discrete Euclidean building a Bruhat-Tits space if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of non-discrete Euclidean buildings introduced and studied by Bruhat and Tits. We give the complete classification of Bruhat-Tits spaces whose building at infinity is the fixed point set of a polarity of an ambient building of type B_2, F_4 or G_2 associated with a Ree or Suzuki group endowed with the usual root datum. (In the B_2 and G_2 cases, this fixed point set is a building of rank one; in the F_4 case, it is a generalized octagon whose Weyl group is not crystallographic.) We also show that each of these Bruhat-Tits spaces has a natural embedding in the unique Bruhat-Tits space whose building at infinity is the corresponding ambient building."}
{"category": "Math", "title": "Derangements and Euler's difference table for $C_\\ell\\wr S_n$", "abstract": "Euler's difference table associated to the sequence $\\{n!\\}$ leads naturally to the counting formula for the derangements. In this paper we study Euler's difference table associated to the sequence $\\{\\ell^n n!\\}$ and the generalized derangement problem. For the coefficients appearing in the later table we will give the combinatorial interpretations in terms of two kinds of $k$-successions of the group $C_\\ell\\wr S_n$. In particular for $\\ell=1$ we recover the known results for the symmetric groups while for $\\ell=2$ we obtain the corresponding results for the hyperoctahedral groups."}
{"category": "Math", "title": "Fix-Euler-Mahonian statistics on wreath products", "abstract": "In 1997 Clarke et al. studied a $q$-analogue of Euler's difference table for $n!$ using a key bijection $\\Psi$ on symmetric groups. In this paper we extend their results to the wreath product of a cyclic group with the symmetric group. In particular we obtain a new mahonian statistic \\emph{fmaf} on wreath products. We also show that Foata and Han's two recent transformations on the symmetric groups provide indeed a factorization of $\\Psi$."}
{"category": "Math", "title": "A Graph Bottleneck Inequality", "abstract": "For a weighted directed multigraph, let $f_{ij}$ be the total weight of spanning converging forests that have vertex $i$ in a tree converging to $j$. We prove that $f_{ij} f_{jk} = f_{ik} f_{jj}$ if and only if every directed path from $i$ to $k$ contains $j$ (a graph bottleneck equality). Otherwise, $f_{ij} f_{jk} < f_{ik} f_{jj}$ (a graph bottleneck inequality). In a companion paper (P. Chebotarev, A new family of graph distances, arXiv preprint arXiv:0810.2717}. Submitted), this inequality underlies, by ensuring the triangle inequality, the construction of a new family of graph distances. This stems from the fact that the graph bottleneck inequality is a multiplicative counterpart of the triangle inequality for proximities."}
{"category": "Math", "title": "All bounded type Siegel disks of rational maps are quasi-disks", "abstract": "We prove that every bounded type Siegel disk of a rational map must be a quasi-disk with at least one critical point on its boundary. This verifies Douady-Sullivan conjecture for bounded type Siegel disks."}
{"category": "Math", "title": "Non-orientable surface-plus-one-relation groups", "abstract": "Recently Dicks-Linnell determined the $L^2$-Betti numbers of the orientable surface-plus-one-relation groups, and their arguments involved some results that were obtained topologically by Hempel and Howie. Using algebraic arguments, we now extend all these results of Hempel and Howie to a larger class of two-relator groups, and we then apply the extended results to determine the $L^2$-Betti numbers of the non-orientable surface-plus-one-relation groups."}
{"category": "Math", "title": "Jordan gradings on exceptional simple Lie algebras", "abstract": "Models of all the gradings on the exceptional simple Lie algebras induced by Jordan subgroups of their groups of automorphisms are provided."}
{"category": "Math", "title": "Specific permutation polynomials over finite fields", "abstract": "We present new classes of permutation polynomials over finite fields."}
{"category": "Math", "title": "Second-order elliptic and parabolic equations with $B(\\mathbb R^{2}, VMO)$ coefficients", "abstract": "The solvability in Sobolev spaces $W^{1,2}_p$ is proved for nondivergence form second order parabolic equations for $p>2$ close to 2. The leading coefficients are assumed to be measurable in the time variable and two coordinates of space variables, and almost VMO (vanishing mean oscillation) with respect to the other coordinates. This implies the $W^{2}_p$-solvability for the same $p$ of nondivergence form elliptic equations with leading coefficients measurable in two coordinates and VMO in the others. Under slightly different assumptions, we also obtain the solvability results when $p=2$."}
{"category": "Math", "title": "Krein resolvent formulas for elliptic boundary problems in nonsmooth domains", "abstract": "The paper reports on a recent construction of M-functions and Krein resolvent formulas for general closed extensions of an adjoint pair, and their implementation to boundary value problems for second-order strongly elliptic operators on smooth domains. The results are then extended to domains with $C^{1,1}$ H\\\"older smoothness, by use of a recently developed calculus of pseudodifferential boundary operators with nonsmooth symbols."}
{"category": "Math", "title": "Quiver Presentations for Descent Algebras of Exceptional Type", "abstract": "The descent algebra of a finite Coxeter group $W$ is a basic algebra, and as such it has a presentation as quiver with relations. In recent work, we have developed a combinatorial framework which allows us to systematically compute such a quiver presentation for a Coxeter group of a given type. In this article, we use that framework to determine quiver presentations for the descent algebras of the Coxeter groups of exceptional or non-crystallographic type, i.e., of type $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, $H_4$ or $I_2(m)$."}
{"category": "Math", "title": "Chen-Ruan cohomology of M_{1,n} and \\bar{M}_{1,n}", "abstract": "In this work we compute the Chen--Ruan cohomology and the stringy Chow ring of the moduli spaces of smooth and stable $n$-pointed curves of genus 1. We suggest a definition for an Orbifold Tautological Ring in genus 1, which is both a subring of the Chen--Ruan cohomology and of the stringy Chow ring."}
{"category": "Math", "title": "Rank one perturbations and singular integral operators", "abstract": "We consider rank one perturbations $A_\\alpha=A+\\alpha(\\cdot,\\varphi)\\varphi$ of a self-adjoint operator $A$ with cyclic vector $\\varphi\\in\\mathcal H_{-1}(A)$ on a Hilbert space $\\mathcal H$. The spectral representation of the perturbed operator $A_\\alpha$ is given by a singular integral operator of special form. Such operators exhibit what we call 'rigidity' and are connected with two weight estimates for the Hilbert transform. Also, some results about two weight estimates of Cauchy (Hilbert) transforms are proved. In particular, it is proved that the regularized Cauchy transforms $T_\\varepsilon$ are uniformly (in $\\varepsilon$) bounded operators from $L^2(\\mu)$ to $L^2(\\mu_\\alpha)$, where $\\mu$ and $\\mu_\\alpha$ are the spectral measures of $A$ and $A_\\alpha$, respectively. As an application, a sufficient condition for $A_\\alpha$ to have a pure absolutely continuous spectrum on a closed interval is given in terms of the density of the spectral measure of $A$ with respect to $\\varphi$. Some examples, like Jacobi matrices and Schr\\\"odinger operators with $L^2$ potentials are considered."}
{"category": "Math", "title": "The noncommutative Choquet boundary II: Hyperrigidity", "abstract": "A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space $A\\subseteq \\mathcal B(H)$ and every sequence of unital completely positive linear maps $\\phi_1, \\phi_2,...$ from $\\mathcal B(H)$ to itself, $$ \\lim_{n\\to\\infty}\\|\\phi_n(g)-g\\|=0, \\forall g\\in G \\implies \\lim_{n\\to\\infty}\\|\\phi_n(a)-a\\|=0, \\forall a\\in A. $$ We show that one can determine whether a given set G of generators is hyperrigid by examining the noncommutative Choquet boundary of the operator space spanned by $G\\cup G^*$. We present a variety of concrete applications and discuss prospects for further development."}
{"category": "Math", "title": "Concentration of the spectral measure of large Wishart matrices with dependent entries", "abstract": "We derive concentration inequalities for the spectral measure of large random matrices, allowing for certain forms of dependence. Our main focus is on empirical covariance (Wishart) matrices, but general symmetric random matrices are also considered."}
{"category": "Math", "title": "Phase portraits for quadratic homogeneous polynomial vector fields on S^2", "abstract": "Let X be a homogeneous polynomial vector field of degree 2 on S^2. We show that if X has at least a non--hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on S^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16^{th} Hilbert's problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on S^2 of degree n."}
{"category": "Math", "title": "Noncommutative localization in algebraic $L$-theory", "abstract": "Given a noncommutative (Cohn) localization $A \\to \\sigma^{-1}A$ which is injective and stably flat we obtain a lifting theorem for induced f.g. projective $\\sigma^{-1}A$-module chain complexes and localization exact sequences in algebraic $L$-theory, matching the algebraic $K$-theory localization exact sequence of Neeman and Ranicki."}
{"category": "Math", "title": "Schlesinger transformations for algebraic Painleve VI solutions", "abstract": "Various Schlesinger transformations can be combined with a direct pull-back of a hypergeometric 2x2 system to obtain $RS^2_4$-pullback transformations to isomonodromic 2x2 Fuchsian systems with 4 singularities. The corresponding Painleve VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This paper demonstrates a direct computation of Schlesinger transformations acting on several apparent singular points, and presents an algebraic procedure (via syzygies) of computing algebraic Painleve VI solutions without deriving full RS-pullback transformations."}
{"category": "Math", "title": "Wreath Hecke algebras and centralizer construction for wreath products", "abstract": "Generalizing the centralizer construction of Molev and Olshanski on symmetric groups, we study the structures of the centralizer $\\mZ_{m,n}$ of the wreath product $G_{n-m}$ in the group algebra of $G_n$ for any $n\\geq m$. We establish the connection between $\\mZ_{m,n}$ and a generalization of degenerate affine Hecke algebras introduced in our earlier work."}
{"category": "Math", "title": "LU-decomposition of a noncommutative linear system and Jacobi polynomials", "abstract": "In this paper we obtain the LU-decomposition of a noncommutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\\lieg)^{K}\\to U(\\liek)^{M}\\otimes U(\\liea)$. This LU-decomposition can be transformed into very simple matrix identities, where the entries of the matrices involved belong to a special class of Jacobi polynomials. In particular, each entry of the L part of the original system is expressed in terms of a single ultraspherical Jacobi polynomial. In turns, these matrix identities yield a biorthogonality relation between the ultraspherical Jacobi polynomials."}
{"category": "Math", "title": "A note on quasi-Gorenstein rings", "abstract": "In this paper, after giving a criterion for a Noetherian local ring to be quasi-Gorenstein, we obtain some sufficient conditions for a quasi- Gorenstein ring to be Gorenstein. In the course, we provide a slight generalization of a theorem of Evans and Griffith."}
{"category": "Math", "title": "Yet another look at Harris' ergodic theorem for Markov chains", "abstract": "The aim of this note is to present an elementary proof of a variation of Harris' ergodic theorem of Markov chains. This theorem, dating back to the fifties essentially states that a Markov chain is uniquely ergodic if it admits a ``small'' set which is visited infinitely often. This gives an extension of the ideas of Doeblin to the unbounded state space setting. Often this is established by finding a Lyapunov function with ``small'' level sets. This topic has been studied by many authors (cf. Harris, Hasminskii, Nummelin, Meyn and Tweedie). If the Lyapunov function is strong enough, one has a spectral gap in a weighted supremum norm (cf. Meyn and Tweedie). Traditional proofs of this result rely on the decomposition of the Markov chain into excursions away from the small set and a careful analysis of the exponential tail of the length of these excursions. There have been other variations which have made use of Poisson equations or worked at getting explicit constants. The present proof is very direct, and relies instead on introducing a family of equivalent weighted norms indexed by a parameter $\\beta$ and to make an appropriate choice of this parameter that allows to combine in a very elementary way the two ingredients (existence of a Lyapunov function and irreducibility) that are crucial in obtaining a spectral gap. The original motivation of this proof was the authors' work on spectral gaps in Wasserstein metrics. The proof presented in this note is a version of our reasoning in the total variation setting which we used to guide the calculations in arXiv:math/0602479. While we initially produced it for that purpose, we hope that it will be of interest in its own right."}
{"category": "Math", "title": "Coset bounds for algebraic geometric codes", "abstract": "For a given curve X and divisor class C, we give lower bounds on the degree of a divisor A such that A and A-C belong to specified semigroups of divisors. For suitable choices of the semigroups we obtain (1) lower bounds for the size of a party A that can recover the secret in an algebraic geometric linear secret sharing scheme with adversary threshold C, and (2) lower bounds for the support A of a codeword in a geometric Goppa code with designed minimum support C. Our bounds include and improve both the order bound and the floor bound. The bounds are illustrated for two-point codes on general Hermitian and Suzuki curves."}
{"category": "Math", "title": "Toric moment mappings and Riemannian structures", "abstract": "Coadjoint orbits for the group SO(6) parametrize Riemannian G-reductions in six dimensions, and we use this correspondence to interpret symplectic fibrations between these orbits, and to analyse moment polytopes associated to the standard Hamiltonian torus action on the coadjoint orbits. The theory is then applied to describe so-called intrinsic torsion varieties of Riemannian structures on the Iwasawa manifold."}
{"category": "Math", "title": "Maps on Divisor Class Groups Induced by Ring Homomorphisms of Finite Flat Dimension", "abstract": "Let f: A\\to B be a ring homomorphism between Noetherian normal integral domains. We establish a general criterion for f to induce a homomorphism Cl(f): Cl(A)\\to Cl(B) on divisor class groups. For instance, this criterion applies whenever f has finite flat dimension; this special case generalizes the more classical situations where f is flat or is surjective with kernel generated by an A-regular element. We extend some of Spiroff's work on the kernels of induced maps to this more general setting."}
{"category": "Math", "title": "Limit Theorems for Individual-Based Models in Economics and Finance", "abstract": "There is a widespread recent interest in using ideas from statistical physics to model certain types of problems in economics and finance. The main idea is to derive the macroscopic behavior of the market from the random local interactions between agents. Our purpose is to present a general framework that encompasses a broad range of models, by proving a law of large numbers and a central limit theorem for certain interacting particle systems with very general state spaces. To do this we draw inspiration from some work done in mathematical ecology and mathematical physics. The first result is proved for the system seen as a measure-valued process, while to prove the second one we will need to introduce a chain of embeddings of some abstract Banach and Hilbert spaces of test functions and prove that the fluctuations converge to the solution of a certain generalized Gaussian stochastic differential equation taking values in the dual of one of these spaces."}
{"category": "Math", "title": "Small probability events for two-layer geophysical flows under uncertainty", "abstract": "The stochastics two-layer quasi-geostrophic flow model is an intermediate system between the single-layer two dimensional barotropic flow model and the continuously stratified three dimensional baroclinic flow model. This model is widely used to investigate basic mechanisms in geophysical flows, such as baroclinic effects, the Gulf Stream and subtropical gyres. A large deviation principle for the two-layer quasi-geostrophic flow model under uncertainty is proved. The proof is based on the Laplace principle and a variational approach. This approach does not require the exponential tightness estimates which are needed in other methods for establishing large deviation principles."}
{"category": "Math", "title": "Naturality of Rieffel's Morita equivalence for proper actions", "abstract": "Suppose that a locally compact group $G$ acts freely and properly on the right of a locally compact space $T$. Rieffel proved that if $\\alpha$ is an action of $G$ on a $C^*$-algebra $A$ and there is an equivariant embedding of $C_0(T)$ in $M(A)$, then the action $\\alpha$ of $G$ on $A$ is proper, and the crossed product $A\\rtimes_{\\alpha,r}G$ is Morita equivalent to a generalised fixed-point algebra $\\Fix(A,\\alpha)$ in $M(A)^\\alpha$. We show that the assignment $(A,\\alpha)\\mapsto\\Fix(A,\\alpha)$ extends to a functor $\\Fix$ on a category of $C^*$-dynamical systems in which the isomorphisms are Morita equivalences, and that Rieffel's Morita equivalence implements a natural isomorphism between a crossed-product functor and $\\Fix$. From this, we deduce naturality of Mansfield imprimitivity for crossed products by coactions, improving results of Echterhoff-Kaliszewski-Quigg-Raeburn and Kaliszewski-Quigg Raeburn, and naturality of a Morita equivalence for graph algebras due to Kumjian and Pask."}
{"category": "Math", "title": "Concentration Inequalities and Laws of Large Numbers under Epistemic Irrelevance", "abstract": "This paper presents concentration inequalities and laws of large numbers under weak assumptions of irrelevance, expressed through lower and upper expectations. The results are variants and extensions of De Cooman and Miranda's recent inequalities and laws of large numbers. The proofs indicate connections between concepts of irrelevance for lower/upper expectations and the standard theory of martingales."}
{"category": "Math", "title": "Classes of permutation polynomials based on cyclotomy and an additive analogue", "abstract": "I present a construction of permutation polynomials based on cyclotomy, an additive analogue of this construction, and a generalization of this additive analogue which appears to have no multiplicative analogue. These constructions generalize recent results of J. Marcos from arXiv:0810.2738v1."}
{"category": "Math", "title": "On a theorem of Carlitz", "abstract": "Carlitz proved that, for any prime power q other than 2, the group of all permutations of the finite field F_q is generated by the permutations induced by degree-one polynomials and x^{q-2}. His proof relies on a remarkable polynomial which appears to have been found by magic. We show here that no magic is required: there is a straightforward way to produce a simple polynomial which has the same remarkable properties as the complicated polynomial in Carlitz's proof. We also identify the crucial subtlety which allows such simple polynomials to exist, and discuss some consequences."}
{"category": "Math", "title": "Symplectic spreads and permutation polynomials", "abstract": "Every symplectic spread of PG(3,q), or equivalently every ovoid of Q(4,q), is shown to give rise to a certain family of permutation polynomials of GF(q) and conversely. This leads to an algebraic proof of the existence of the Tits-Luneburg spread of W(2^{2h+1}) and the Ree-Tits spread of W(3^{2h+1}), as well as to a new family of low-degree permutation polynomials over GF(3^{2h+1}). We prove the permutation property of the latter polynomials via an odd characteristic analogue of Dobbertin's approach to uniformly representable permutation polynomials over GF(2^n). These new permutation polynomials were later used by Ding, Wang, and Xiang in arXiv:math/0609586 to produce new skew Hadamard difference sets."}
{"category": "Math", "title": "Algebraic Cycles of a Fixed Degree", "abstract": "In this paper, the homotopy groups of Chow variety $C_{p,d}(P^n)$ of effective $p$-cycles of degree $d$ is proved to be stable in the sense that $p$ or $n$ increases. We also obtain a negative answer to a question by Lawson and Michelsohn on homotopy groups for the space of degree two cycles."}
{"category": "Math", "title": "Everywhere ramified towers of global function fields", "abstract": "We consider a tower of function fields F_0 < F_1 < ... over a finite field such that every place of every F_i ramified in the tower and the sequence genus(F_i)/[F_i:F_0] has a finite limit. We also construct a tower in which every place ramifies and the sequence N_i/[F_i:F_0] has a positive limit, where N_i is the number of degree-one places of F_i. These towers answer questions posed by Stichtenoth."}
{"category": "Math", "title": "Solving polynomial differential equations by transforming them to linear functional-differential equations", "abstract": "We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order Abel differential equation with two nonlinear terms in order to demonstrate in as much detail as possible the computations necessary for a complete solution. We mention in our section on further developments that the basic transformation idea can be generalized to apply to differential equations of any order, to a system of ordinary differential equations without first differentially eliminating the multiple dependent variables, and even to partial differential equations."}
{"category": "Math", "title": "Spectral Theory of the Riemann Zeta-Function: Chapter 6: Appendix", "abstract": "The main aim of this article is to develop, in a fully detailed fashion, a {\\bf unified} theory of the spectral theory of mean values of individual automorphic L-functions which is a natural extension of the fourth moment of the Riemann zeta-function but does not admit any analogous argument and requires a genuinely new method. Thus we first develop a relatively self-contained account of the theory of automorphic representations, especially highlighting the Kirillov model, with which we resolve the problem on the mean value of those L-functions. As another reward, we gain a geometrical understanding of sum formulas involving Kloosterman sums, which is in fact a considerably simplified account of Cogdell-Piatetski-Shapiro's method. Our reasoning is quite explicit in contrast to theirs."}
{"category": "Math", "title": "Engel subalgebras of Leibniz algebras", "abstract": "Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that a left Leibniz algebra, all of whose maximal subalgebras are right ideals, is nilpotent. A primitive Leibniz algebra is shown to split over its minimal ideal and that all the complements to its minimal ideal are conjugate. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements. Cartan subalgebras are shown to have a property analogous to intravariance."}
{"category": "Math", "title": "Elliptic periods and primality proving", "abstract": "We define the ring of elliptic periods modulo an integer $n$ and give an elliptic version of the AKS primality criterion."}
{"category": "Math", "title": "On accuracy of approximation of the spectral radius by the Gelfand formula", "abstract": "The famous Gelfand formula $\\rho(A)= \\limsup_{n\\to\\infty}\\|A^{n}\\|^{1/n}$ for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities $\\|A^{n}\\|^{1/n}$ to $\\rho(A)$. In the paper this deficiency is made up to some extent. By using the Bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities $\\|A^{n}\\|^{1/n}$ to $\\rho(A)$. The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets."}
{"category": "Math", "title": "Six bijections between deco polyominoes and permutations", "abstract": "In this paper we establish six bijections between a particular class of polyominoes, called deco polyominoes, enumerated according to their directed height by n!, and permutations. Each of these bijections allows us to establish different correspondences between classical statistics on deco polyominoes and on permutations."}
{"category": "Math", "title": "On the exhaustive generation of convex permutominoes", "abstract": "A permutomino of size n is a polyomino determined by a pair of permutations of size n+1, such that they differ in each position. In this paper, after recalling some enumerative results about permutominoes, we give a first algorithm for the exhaustive generation of a particular class of permutominoes, the convex permutominoes, proving that its cost is proportional to the number of generated objects."}
{"category": "Math", "title": "Uniqueness of Schrodinger flow via energy inequality", "abstract": "In this short note, we show a uniqueness result of the energy solutions for the Cauchy problem of Schrodinger flow in the whole space $R^n$ provided there is a smooth solution in the energy class."}
{"category": "Math", "title": "Stein's method and stochastic analysis of Rademacher functionals", "abstract": "We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein's method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not require the classical use of exchangeable pairs, we employ a chaos expansion in order to construct an explicit exchangeable pair vector for any random variable which depends on a finite set of Rademacher variables. Among several examples, which include random variables which depend on infinitely many Rademacher variables, we provide three main applications: (i) to CLTs for multilinear forms belonging to a fixed chaos, (ii) to the Gaussian approximation of weighted infinite 2-runs, and (iii) to the computation of explicit bounds in CLTs for multiple integrals over sparse sets. This last application provides an alternate proof (and several refinements) of a recent result by Blei and Janson."}
{"category": "Math", "title": "At infinity of finite-dimensional CAT(0) spaces", "abstract": "We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space $X$ has a non-empty intersection in the visual bordification $ \\bar{X} = X \\cup \\partial X$. Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups."}
{"category": "Math", "title": "Tableaux Combinatorics for the Asymmetric Exclusion Process II", "abstract": "The results of this paper have been subsumed by those of our new paper arXiv:0910.1858"}
{"category": "Math", "title": "On Sums of Indicator Functions in Dynamical Systems", "abstract": "In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every aperiodic dynamical system, for every increasing sequence $(a_n)_{n\\in\\N}\\subset\\R_+$ such that $a_n\\nearrow\\infty$ and $\\frac{a_n}{n}\\to 0$ as $n\\to\\infty$, there exists a measurable set $A$ such that the sequence of the distributions of the partial sums $\\frac{1}{a_n}\\sum_{i=0}^{n-1}(\\ind_A-\\mu(A))\\circ T^i$ is dense in the set of the probability measures on $\\R$. Further, in the ergodic case, we prove that there exists a dense $G_\\delta$ of such sets."}
{"category": "Math", "title": "Properties of weight posets for weight multiplicity free representations", "abstract": "We study weight posets of weight multiplicity free (=wmf) representations $R$ of reductive Lie algebras. Specifically, we are interested in relations between $\\dim R$ and the number of edges in the Hasse diagram of the corresponding weight poset, $# E(R)$. We compute the number of edges and upper covering polynomials for the weight posets of all wmf-representations. We also point out non-trivial isomorphisms between weight posets of different irreducible wmf-representations. Our main results concern wmf-representations associated with periodic gradings or Z-gradings of simple Lie algebras. For Z-gradings, we prove that $0< 2dim R-# E(R) < h$, where $h$ is the Coxeter number of $\\mathfrak g$. For periodic gradings, we prove that $0\\le 2dim R-# E(R)$."}
{"category": "Math", "title": "Non-K\\\"ahler Expanding Ricci Solitons", "abstract": "We produce new examples of non-K\\\"ahler complete expanding gradient Ricci solitons on trivial vector bundles over a product of Einstein manifolds with positive scalar curvature."}
{"category": "Math", "title": "Lie-Rinehart cohomology and integrable connections on modules of rank one", "abstract": "Let $k$ be an algebraically closed field of characteristic 0, let $R$ be a commutative $k$-algebra, and let $M$ be a torsion free $R$-module of rank one with a connection $\\nabla$. We consider the Lie-Rinehart cohomology with values in $End_{R}(M)$ with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on $M$. When $R$ is an isolated singularity of dimension $d\\geq2$, we relate the Lie-Rinehart cohomology to the topological cohomology of the link of the singularity, and when $R$ is a quasi-homogenous hypersurface of dimension two, we give a complete computation of the cohomology."}
{"category": "Math", "title": "Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions", "abstract": "We consider a process $(X_t)_{t\\in[0,T)}$ given by the SDE $dX_t = \\alpha b(t)X_t dt + \\sigma(t) dB_t$, $t\\in[0,T)$, with initial condition $X_0=0$, where $T\\in(0,\\infty]$, $\\alpha\\in R$, $(B_t)_{t\\in[0,T)}$ is a standard Wiener process, $b:[0,T)\\to R\\setminus\\{0\\}$ and $\\sigma:[0,T)\\to(0,\\infty)$ are continuously differentiable functions. Assuming that $b$ and $\\sigma$ satisfy a certain differential equation we derive an explicit formula for the joint Laplace transform of $\\int_0^t\\frac{b(s)^2}{\\sigma(s)^2}(X_s)^2 ds$ and $(X_t)^2$ for all $t\\in[0,T)$. As an application, we study asymptotic behavior of the maximum likelihood estimator of $\\alpha$ for $\\sign(\\alpha-K)=\\sign(K)$, $K\\ne0$, and for $\\alpha=K$, $K\\ne0$. As an example, we examine the so-called $\\alpha$-Wiener bridges given by SDE $dX_t = -\\frac{\\alpha}{T-t}X_t dt + dB_t$, $t\\in[0,T)$, with initial condition $X_0=0$."}
{"category": "Math", "title": "Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups", "abstract": "An integral hyperbolic lattice is called reflective if its automorphism group is generated by reflections, up to finite index. Since 1981, it is known that their number is essentially finite. We show that K3 surfaces over C with reflective Picard lattices can be characterized in terms of compositions of their self-correspondences via moduli of sheaves with primitive isotropic Mukai vector: Their self-correspondences with integral action on the Picard lattice are numerically equivalent to compositions of a finite number of especially simple self-correspondences via moduli of sheaves. This relates two topics: Self-correspondences of K3 surfaces via moduli of sheaves and Arithmetic hyperbolic reflection groups. It also raises several natural unsolved related problems."}
{"category": "Math", "title": "Helicoidal graphs with prescribed mean curvature", "abstract": "We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms."}
{"category": "Math", "title": "Global solutions for two-phase Hele-Shaw bubble for a near-circular initial shape", "abstract": "Using a vortex sheet method we prove global existence of a near circular initial bubble in a Hele-Shaw cell with surface tension and generally finite nonzero viscosity ratio between fluids inside and outside the bubble. The circular shape is shown to be asymptotically stable for all sufficiently smooth small perturbation. The initial condition in this case, while smooth, need not be analytic."}
{"category": "Math", "title": "On the Limiting Shape of Markovian Random Young Tableaux", "abstract": "Let $(X_n)_{n \\ge 0}$ be an irreducible, aperiodic, homogeneous Markov chain, with state space an ordered finite alphabet of size $m$. Using combinatorial constructions and weak invariance principles, we obtain the limiting shape of the associated Young tableau as a multidimensional Brownian functional. Since the length of the top row of the Young tableau is also the length of the longest (weakly) increasing subsequence of $(X_k)_{1\\le k \\le n}$, the corresponding limiting law follows. We relate our results to a conjecture of Kuperberg by showing that, under a cyclic condition, a spectral characterization of the Markov transition matrix delineates precisely when the limiting shape is the spectrum of the traceless GUE. For $m=3$, all cyclic Markov chains have such a limiting shape, a fact previously known for $m=2$. However, this is no longer true for $m \\ge 4$."}
{"category": "Math", "title": "Polyhedral Methods in Numerical Algebraic Geometry", "abstract": "In numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These certificates are the leading terms of a Puiseux series expansion of the curve starting at infinity. The vector of powers of the first term in the series is a tropism. For proper algebraic curves, we relate the computation of tropisms to the calculation of mixed volumes. With this relationship, the computation of tropisms and Puiseux series expansions could be used as a preprocessing stage prior to a more expensive witness set computation. Systems with few monomials have fewer isolated solutions and fewer data are needed to represent their positive dimensional solution sets."}
{"category": "Math", "title": "$p$-Dirac Operators", "abstract": "We introduce non-linear Dirac operators in $\\mathbb{R}^{n}$ associated to the $p$-harmonic equation and we extend to other contexts including spin manifolds and the sphere."}
{"category": "Math", "title": "Mathematical modeling of antigenicity for HIV dynamics", "abstract": "This contribution is devoted to a new model of HIV multiplication motivated by the patent of one of the authors. We take into account the antigenic diversity through what we define \"antigenicity\", whether of the virus or of the adapted lymphocytes. We model the interaction of the immune system and the viral strains by two processes. On the one hand, the presence of a given viral quasi-species generates antigenically adapted lymphocytes. On the other hand, the lymphocytes kill only viruses for which they have been designed. We consider also the mutation and multiplication of the virus. An original infection term is derived. So as to compare our system of differential equations with well-known models, we study some of them and compare their predictions to ours in the reduced case of only one antigenicity. In this particular case, our model does not yield any major qualitative difference. We prove mathematically that, in this case, our model is biologically consistent (positive fields) and has a unique continuous solution for long time evolution. In conclusion, this model improves the ability to simulate more advanced phases of the disease."}
{"category": "Math", "title": "Some local-global non-vanishing results of theta lifts for symplectic-orthogonal dual pairs", "abstract": "Following the approach of B. Roberts, we characterize the non-vanishing of global theta lifts for symplectic-orthogonal dual pairs in terms of its local counterpart. In particular, we replace the temperedness assumption present in Robert's work by a certain weaker assumption, and apply our results to small rank similitude groups. Among our applications is a certain instance of Langlands functorial transfer of a (non-generic) cuspidal automorphic representation of $GSp(4)$ to GL(4)."}
{"category": "Math", "title": "From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices", "abstract": "The famous \\emph{circular law} asserts that if $M_n$ is an $n \\times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the normalized matrix $\\frac{1}{\\sqrt{n}} M_n$ converges almost surely to the uniform distribution on the unit disk $\\{z \\in \\C: |z| \\leq 1 \\}$. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the full circular law was recently established in \\cite{TVcir2}. In this survey we describe some of the key ingredients used in the establishment of the circular law, in particular recent advances in understanding the Littlewood-Offord problem and its inverse."}
{"category": "Math", "title": "Monochromatic boxes in colored grids", "abstract": "A $d$-dimensional grid is a set of the form $R = [a_1] \\times ... \\times [a_d]$. A $d$-dimensional box is a set of the form $\\{b_1,c_1\\} \\times ... \\times \\{b_d,c_d\\}$. When a grid is $c$-colored, must it admit a monochromatic box? If so, we say that $R$ is $c$-guaranteed. This question is a relaxation of one attack on bounding the van der Waerden numbers, and also arises as a natural hypergraph Ramsey problem (viz. the Ramsey numbers of hyperoctahedra). We give conditions on the $a_i$ for $R$ to be $c$-guaranteed that are asymptotically tight, and analyze the set of minimally $c$-guaranteed grids."}
{"category": "Math", "title": "$k^*$-Metrizable Spaces and their Applications", "abstract": "In this paper we introduce and study so-called $k^*$-metrizable spaces forming a new class of generalized metric spaces, and display various applications of such spaces in topological algebra, functional analysis, and measure theory. By definition, a Hausdorff topological space $X$ is $k^*$-metrizable if $X$ is the image of a metrizable space $M$ under a continuous map $f:M\\to X$ having a section $s:X\\to M$ that preserves precompact sets in the sense that the image $s(K)$ of any compact set $K\\subset X$ has compact closure in $X$."}
{"category": "Math", "title": "(Metrically) quarter-stratifiable spaces and their applications in the theory of separately continuous functions", "abstract": "We introduce and study (metrically) quarter-stratifiable spaces and then apply them to generalize Rudin and Kuratowski-Montgomery theorems about the Baire and Borel complexity of separately continuous functions."}
{"category": "Math", "title": "Free topological inverse semigroups as infinite-dimensional manifolds", "abstract": "Let $K$ be a complete quasivariety of topological inverse Clifford semigroups, containing all topological semilattices. It is shown that the free topological inverse semigroup $F(X,K)$ of $X$ in the class $K$ is an $R^\\infty$-manifold if and only if $X$ has no isolated points and $F(X,K)$ is a retract of an $R^\\infty$-manifold. We derive from this that for any retract $X$ of an $R^\\infty$-manifold the free topological inverse semigroup $F(X,K)$ is an $R^\\infty$-manifold if and only if the space $X$ has no isolated points. Also we show that for any homotopically equivalent retracts $X,Y$ of $R^\\infty$-manifolds with no isolated points the free topological inverse semigroups $F(X,K)$ and $F(Y,K)$ are homeomorphic. This allows us to construct non-homeomorphic spaces whose free topological inverse semigroups are homeomorphic."}
{"category": "Math", "title": "Behavior of lacunary series at the natural boundary", "abstract": "We develop a local theory of lacunary Dirichlet series of the form $\\sum\\limits_{k=1}^{\\infty}c_k\\exp(-zg(k)), \\Re(z)>0$ as $z$ approaches the boundary $i\\RR$, under the assumption $g'\\to\\infty$ and further assumptions on $c_k$. These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with $c_k=1$, the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at $z=0$. When sufficient analyticity information on $g$ exists, we obtain Borel summable expansions at points on the boundary, giving exact local description. Borel summability of the expansions provides property-preserving extensions beyond the barrier. The singular behavior has remarkable universality and self-similarity features. If $g(k)=k^b$, $c_k=1$, $b=n$ or $b=(n+1)/n$, $n\\in\\NN$, behavior near the boundary is roughly of the standard form $\\Re(z)^{-b'}Q(x)$ where $Q(x)=1/q$ if $x=p/q\\in\\QQ$ and zero otherwise. The B\\\"otcher map at infinity of polynomial iterations of the form $x_{n+1}=\\lambda P(x_n)$, $|\\lambda|<\\lambda_0(P)$, turns out to have uniformly convergent Fourier expansions in terms of simple lacunary series. For the quadratic map $P(x) =x-x^2$, $\\lambda_0=1$, and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set."}
{"category": "Math", "title": "Oscillator topologies on a paratopological group and related number invariants", "abstract": "We introduce and study oscillator topologies on paratopological groups and define certain related number invariants. As an application we prove that a Hausdorff paratopological group $G$ admits a weaker Hausdorff group topology provided $G$ is 3-oscillating. A paratopological group $G$ is 3-oscillating (resp. 2-oscillating) provided for any neighborhood $U$ of the unity $e$ of $G$ there is a neighborhood $V\\subset G$ of $e$ such that $V^{-1}VV^{-1}\\subset UU^{-1}U$ (resp. $V^{-1}V\\subset UU^{-1}$). The class of 2-oscillating paratopological groups includes all collapsing, all nilpotent paratopological groups, all paratopological groups satisfying a positive law, all paratopological SIN-group and all saturated paratopological groups (the latter means that for any nonempty open set $U\\subset G$ the set $U^{-1}$ has nonempty interior). We prove that each totally bounded paratopological group $G$ is countably cellular; moreover, every cardinal of uncountable cofinality is a precaliber of $G$. Also we give an example of a saturated paratopological group which is not isomorphic to its mirror paratopological group as well as an example of a 2-oscillating paratopological group whose mirror paratopological group is not 2-oscillating."}
{"category": "Math", "title": "Each second countable abelian group is a subgroup of a second countable divisible group", "abstract": "It is shown that each pseudonorm defined on a subgroup $H$ of an abelian group $G$ can be extended to a pseudonorm on $G$ such that the densities of the obtained pseudometrizable topological groups coincide. We derive from this that any Hausdorff $\\omega$-bounded group topology on $H$ can be extended to a Hausdorff $\\omega$-bounded group topology on $G$. In its turn this result implies that each separable metrizable abelian group $H$ is a subgroup of a separable metrizable divisible group $G$. This result essentially relies on the Axiom of Choice and is not true under the Axiom of Determinacy (which contradicts to the Axiom of Choice but implies the Countable Axiom of Choice)."}
{"category": "Math", "title": "The $cl-core$ of an ideal", "abstract": "We expand the notion of core to $cl$-core for Nakayama closures $cl$. In the characteristic $p>0$ setting, when $cl$ is the tight closure, denoted by *, we give some examples of ideals when the core and the *-core differ. We note that *-core$(I)=$ core$(I)$, if $I$ is an ideal in a one-dimensional domain with infinite residue field or if $I$ is an ideal generated by a system of parameters in any Noetherian ring. More generally, we show the same result in a Cohen--Macaulay normal local domain with infinite perfect residue field, if the analytic spread, $\\ell$, is equal to the *-spread and $I$ is $G_{\\ell}$ and weakly-$(\\ell-1)$-residually $S_2$. This last is dependent on our result that generalizes the notion of general minimal reductions to general minimal *-reductions. We also determine that the *-core of a tightly closed ideal in certain one-dimensional semigroup rings is tightly closed and therefore integrally closed."}
{"category": "Math", "title": "Un metodo adaptativo para el modelo Bidominio en electrocardiologia", "abstract": "This paper presents a finite-volume method, together with fully adaptive multi-resolution scheme to obtain spatial adaptation, and a Runge-Kutta-Fehlberg scheme with a local time-varying step to obtain temporal adaptation, to solve numerically the known \"bidominio\" equations that model the electrical activity of the tissue in the myocardium. Two simple models are considered for membrane flows and ionic currents. First we define an approximate solution and we verify its convergence to the corresponding weak solution of the continuum problem, obtaining in this way an alternative demonstration that the continuum problem is well-posed. Next we introduce the multiresolution technique and derive an optimal noise reduction threshold. The efficiency and precision of our method is seen in the reduction of machine time, memory usage, and errors in comparison to other methods. ----- En este trabajo se presenta un metodo de volumenes finitos enriquecido con un esquema de multiresolucion completamente adaptativo para obtener adaptatividad espacial, y un esquema Runge-Kutta-Fehlberg con paso temporal de variacion local para obtener adaptatividad temporal, para resolver numericamente las conocidas ecuaciones \"bidominio\" que modelan la actividad electrica del tejido en el miocardio. Se consideran dos modelos simples para las corrientes de membrana y corrientes ionicas. En primer lugar definimos una solucion aproximada y nos referimos a su convergencia a la correspondiente solucion debil del problema continuo, obteniendo de este modo una demostracion alternativa de que el problema continuo es bien puesto. Luego de introducir la tecnica de multiresolucion, se deriva un umbral optimo para descartar la informacion no significativa, y tanto la eficiencia como la precision de nuestro metodo es vista en terminos de la aceleracion de tiempo de maquina, compresion de memoria computacional y errores en diferentes normas."}
{"category": "Math", "title": "Centralizers in Domains of Finite Gelfand-Kirillov Dimension", "abstract": "We study centralizers of elements in domains. We generalize a result of the author and Small, showing that if $A$ is a finitely generated noetherian domain and $a\\in A$ is not algebraic over the extended centre of $A$, then the centralizer of $a$ has Gelfand-Kirillov dimension at most one less than the Gelfand-Kirillov dimension of $A$. In the case that $A$ is a finitely generated noetherian domain of GK dimension 3 over the complex numbers, we show that the centralizer of an element a $A$ that is not algebraic over the extended centre of $A$ satisfies a polynomial identity."}
{"category": "Math", "title": "Regularity of solutions for the critical $N$-dimensional Burgers' equation", "abstract": "We consider the fractional Burgers' equation on $\\R^N$ with the critical dissipation term. We follow the parabolic De-Giorgi's method of Caffarelli and Vasseur \\cite{Driftdiffusion} and show existence of smooth solutions given any initial datum in $L^2(\\R^N)$."}
{"category": "Math", "title": "On the homotopy type of a cofibred category", "abstract": "In this paper we describe two ways on which cofibred categories give rise to bisimplicial sets. The \"fibred nerve\" is a natural extension of Segal's classical nerve of a category, and it constitutes an alternative simplicial description of the homotopy type of the total category. If the fibration is splitting, then one can construct the \"cleaved nerve\", a smaller variant which emerges from a distinguished closed cleavage. We interpret some classical theorems by Thomason and Quillen in terms of our constructions, and use the fibred and cleaved nerve to establish new results on homotopy and homology of small categories."}
{"category": "Math", "title": "alpha-Wiener bridges: singularity of induced measures and sample path properties", "abstract": "Let us consider the process $(X_t^{(\\alpha)})_{t\\in[0,T)}$ given by the SDE $dX_t^{(\\alpha)} = -\\frac{\\alpha}{T-t}X_t^{(\\alpha)} dt+ dB_t$, $t\\in[0,T)$, where $\\alpha\\in R$, $T\\in(0,\\infty)$, and $(B_t)_{t\\geq 0}$ is a standard Wiener process. In case of $\\alpha>0$ the process $X^{(\\alpha)}$ is known as an $\\alpha$-Wiener bridge, in case of $\\alpha=1$ as the usual Wiener bridge. We prove that for all $\\alpha,\\beta\\in R$, $\\alpha\\ne\\beta$, the probability measures induced by the processes $X^{(\\alpha)}$ and $X^{(\\beta)}$ are singular on C[0,T). Further, we investigate regularity properties of $X_t^{(\\alpha)}$ as $t\\uparrow T$."}
{"category": "Math", "title": "On a quadratic estimate related to the Kato conjecture and boundary value problems", "abstract": "We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms."}
{"category": "Math", "title": "Numerical Range and Quasi-Sectorial Contractions", "abstract": "We apply a method developed by one of the authors, see \\cite{Arl1}, to localize the numerical range of \\textit{quasi-sectorial} contractions semigroups. Our main theorem establishes a relation between the numerical range of quasi-sectorial contraction semigroups $\\{\\exp(- t S)\\}_{t\\ge 0}$, and the maximal {sectorial} generators $S$. We also give a new prove of the rate $O(1/n)$ for the operator-norm Euler formula approximation: $\\exp(- t S)=\\lim\\limits_{n\\to \\infty}(I+tS/n)^{-n}$, $t\\ge 0$, for this class of semigroups."}
{"category": "Math", "title": "Weighted norm inequalities, off-diagonal estimates and elliptic operators", "abstract": "We give an overview of the generalized Calder\\'on-Zygmund theory for \"non-integral\" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\\BMO$ functions. $L^p-L^q$ off-diagonal estimates when $p\\le q$ play an important role and we present them. They are particularly well suited to the semigroups generated by second order elliptic operators and the range of exponents $(p,q)$ rules the $L^p$ theory for many operators constructed from the semigroup and its gradient. Such applications are summarized."}
{"category": "Math", "title": "Occupancy Schemes Associated to Yule Processes", "abstract": "An occupancy problem with an infinite number of bins and a random probability vector for the locations of the balls is considered. The respective sizes of bins are related to the split times of a Yule process. The asymptotic behavior of the landscape of first empty bins, i.e., the set of corresponding indices represented by point processes, is analyzed and convergences in distribution to mixed Poisson processes are established. Additionally, the influence of the random environment, the random probability vector, is analyzed. It is represented by two main components: an i.i.d. sequence and a fixed random variable. Each of these components has a specific impact on the qualitative behavior of the stochastic model. It is shown in particular that for some values of the parameters, some rare events, which are identified, play an important role on average values of the number of empty bins in some regions."}
{"category": "Math", "title": "Milnor fibrations of meromorphic functions", "abstract": "In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g : the local Milnor fibrations given on Milnor tubes over punctured discs around the critical values of f/g, and the Milnor fibration on a sphere."}
{"category": "Math", "title": "Le logo du CNRS est-il convexe ?", "abstract": "In october 2008, CNRS adopts a new logo with a round shape. We study the mathematical representation of this shape, and in particular its convexity."}
{"category": "Math", "title": "Bidual as a weak nonstandard hull", "abstract": "We construct the weak nonstandard hull of a normed linear space X from *X (the nonstandard extension of X) using the weak topology on X. The bidual (i.e. the second dual) X\" is shown to be isometrically isomorphic to the weak nonstandard hull of X. Examples and applications to C*-algebras are given, including a simple proof of the Sherman-Takeda Theorem. As a consequence, the weak nonstandard hull of a C*-algebra is always a von Neumann algebra. Moreover a natural representation of the Arens product is given."}
{"category": "Math", "title": "Probabilistic characterisation of Besov-Lipschitz spaces on metric measure spaces", "abstract": "We give a probabilistic characterisation of the Besov-Lipschitz spaces $Lip(\\alpha,p,q)(X)$ on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of \\cite{Jon,KPP1,KPP2,GHL} which were valid for $\\alpha=\\frac{d_w}{2},p=2$, $q=\\infty,$ where $d_w$ is the walk dimension of the space $X.$"}
{"category": "Math", "title": "On Vanishing and Cofiniteness of Generalized Local Cohomology Modules", "abstract": "In this paper, some results on vanishing and non-vanishing of generalized local cohomology modules are presented and some relations between those modules and, Ext and ordinary local cohomology modules are studied. Also, several cofiniteness propositions for generalized local cohomology modules are established which, among other things, provide an alternative answer to a question in [Y2]."}
{"category": "Math", "title": "A primitive derivation and logarithmic differential forms of Coxeter arrangements", "abstract": "Let $W$ be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case."}
{"category": "Math", "title": "Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves", "abstract": "In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces $L^p(\\Gamma)$ over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht B\\\"ottcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces $L^{p(\\cdot)}(\\Gamma)$ where $p:\\Gamma\\to(1,\\infty)$ satisfies the Dini-Lipschitz condition. One of the main ingredients of the proof is a new sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves."}
{"category": "Math", "title": "Asymptotic normality for deconvolution kernel density estimators from random fields", "abstract": "The paper discusses the estimation of a continuous density function of the target random field $X_{\\bf{i}}$, $\\bf{i}\\in \\mathbb {Z}^N$ which is contaminated by measurement errors. In particular, the observed random field $Y_{\\bf{i}}$, $\\bf{i}\\in \\mathbb {Z}^N$ is such that $Y_{\\bf{i}}=X_{\\bf{i}}+\\epsilon_{\\bf{i}}$, where the random error $\\epsilon_{\\bf{i}}$ is from a known distribution and independent of the target random field. Compared to the existing results, the paper is improved in two directions. First, the random vectors in contrast to univariate random variables are investigated. Second, a random field with a certain spatial interactions instead of i. i. d. random variables is studied. Asymptotic normality of the proposed estimator is established under appropriate conditions."}
{"category": "Math", "title": "On the number of cutpoints of the transient nearest neighbor random walk on the line", "abstract": "We consider transient nearest neighbor random walks on the positive part of the real line. We give criteria for the finiteness of the number of cutpoints and strong cutpoints. Examples and open problems are presented."}
{"category": "Math", "title": "About extension of upper semicontinuous multi-valued maps and applications", "abstract": "We formulate a multi-valued version of the Tietze-Urysohn extension theorem. Precisely, we prove that any upper semicontinuous multi-valued map with nonempty closed convex values defined on a closed subset (resp. closed perfectly normal subset) of a completely normal (resp. of a normal) space $X$ into the unit interval $[0,1]$ can be extended to the whole space $X$. The extension is upper semicontinuous with nonempty closed convex values. We apply this result for the extension of real semicontinuous functions, the characterization of completely normal spaces, the existence of Gale-Mas-Colell and Shafer-Sonnenschein type fixed point theorems and the existence of equilibrium for qualitative games."}
{"category": "Math", "title": "A model for infection on graphs", "abstract": "We address the question of understanding the effect of the underlying network topology on the spread of a virus and the dissemination of information when users are mobile performing independent random walks on a graph. To this end we propose a simple model of infection that enables to study the coincidence time of two random walkers on an arbitrary graph. By studying the coincidence time of a susceptible and an infected individual both moving in the graph we obtain estimates of the infection probability. The main result of this paper is to pinpoint the impact of the network topology on the infection probability. More precisely, we prove that for homogeneous graph including regular graphs and the classical Erdos-Renyi model, the coincidence time is inversely proportional to the number of nodes in the graph. We then study the model on power-law graphs, that exhibit heterogeneous connectivity patterns, and show the existence of a phase transition for the coincidence time depending on the parameter of the power-law of the degree distribution."}
{"category": "Math", "title": "Cluster structures from 2-Calabi-Yau categories with loops", "abstract": "We generalise the notion of cluster structures from the work of Buan-Iyama-Reiten-Scott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Hom-finite 2-Calabi-Yau category, the set of maximal rigid objects satisfies these axioms whenever there are no 2-cycles in the quivers of their endomorphism rings. We apply this result to the cluster category of a tube, and show that this category forms a good model for the combinatorics of a type B cluster algebra."}
{"category": "Math", "title": "Double-critical graphs and complete minors", "abstract": "A connected $k$-chromatic graph $G$ is double-critical if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. The only known double-critical $k$-chromatic graph is the complete $k$-graph $K_k$. The conjecture that there are no other double-critical graphs is a special case of a conjecture from 1966, due to Erd\\H{o}s and Lov\\'asz. The conjecture has been verified for $k \\leq 5$. We prove for $k=6$ and $k=7$ that any non-complete double-critical $k$-chromatic graph is 6-connected and has $K_k$ as a minor."}
{"category": "Math", "title": "Vector bundles on Fano threefolds of genus 7 and Brill-Noether loci", "abstract": "Given a smooth prime Fano threefold $X$ of genus 7 we consider its homologically projectively dual curve $\\Gamma$ and the natural integral functor $\\Phi^{!}:D^b(X) \\to D^b(\\Gamma)$. We prove that, for $d\\geq 6$, $\\Phi^{!}$ gives a birational map from a component of the moduli scheme $M_X(2,1,d)$ of rank 2 stable sheaves on $X$ with $c_1=1$, $c_2=d$ to a generically smooth $2d-9$-dimensional component of the Brill-Noether variety $W^{2d-11}_{d-5,5d-24}$ of stable vector bundles on $\\Gamma$ of rank $d-5$ and degree $5d-24$ with at least $2d-10$ sections. This map turns out to be an isomorphism for $d=6$, and the moduli space $M_X(2,1,6)$ is fine. For general $X$, this moduli space is a smooth irreducible threefold."}
{"category": "Math", "title": "Polyak-Viro formulas for coefficients of the Conway polynomial", "abstract": "We describe the Polyak-Viro arrow diagram formulas for the coefficients of the Conway polynomial. As a consequence, we obtain the Conway polynomial as a state sum over some subsets of the crossings of the knot diagram. It turns out to be a simplification of a special case of Jaeger's state model for the HOMFLY polynomial."}
{"category": "Math", "title": "Torsion bounds for elliptic curves and Drinfeld modules", "abstract": "We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and Pink-Ruetsche, respectively. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules."}
{"category": "Math", "title": "Une classe d'espaces pr\\'ehomog\\`enes de type parabolique faiblement sph\\'eriques", "abstract": "For absolutely simple, finite-dimensional Lie algebras g of rank at least 2, defined over a local field of characteristic 0 and admitting a graduation: g=g(-2)+g(-1)+g(0)+g(1)+g(2) given by an element H such that 2H is simple, we construct parabolic subgroups P of the automorphism group of g which centralize H, having geometric prehomogeneous action on g(1) and g(-1). We study the structure of these prehomogeneous vector spaces. We prove that the Zeta functions associated to the fundamental invariants for the P action on g(1) and g(-1) have meromorphic extensions which satisfy functional equations. We give the explicit calculus of the coefficients of these functional equations and the Bernstein polynomials associated to these fundamental invariants in the archimedian case, by reducing the problem to a similar problem for centralizers of pair of commuting sl(2) Lie algebras. This work is a generalization of well-known results when g(2)=0."}
{"category": "Math", "title": "Uniform growth of groups acting on Cartan-Hadamard spaces", "abstract": "Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2 \\leq K \\leq -1$. There exist a positive constant $C(n,a)$ such that for any finitely generated discrete group $\\Gamma$ acting on $X$, then either $\\Gamma$ is virtually nilpotent or the algebraic entropy $Ent (\\Gamma) \\geq C(n,a)$."}
{"category": "Math", "title": "Linear Dynamical Systems over Finite Rings", "abstract": "The problem of linking the structure of a finite linear dynamical system with its dynamics is well understood when the phase space is a vector space over a finite field. The cycle structure of such a system can be described by the elementary divisors of the linear function, and the problem of determining whether the system is a fixed point system can be answered by computing and factoring the system's characteristic polynomial and minimal polynomial. It has become clear recently that the study of finite linear dynamical systems must be extended to embrace finite rings. The difficulty of dealing with an arbitrary finite commutative ring is that it lacks of unique factorization. In this paper, an efficient algorithm is provided for analyzing the cycle structure of a linear dynamical system over a finite commutative ring. In particular, for a given commutative ring $R$ such that $|R|=q$, where $q$ is a positive integer, the algorithm determines whether a given linear system over $R^n$ is a fixed point system or not in time $O(n^3\\log(n\\log(q)))$."}
{"category": "Math", "title": "Rings without a Gorenstein analogue of the Govorov-Lazard Theorem", "abstract": "It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the Govorov-Lazard Theorem. We show that, in fact, there is a large class of rings without such an analogue. Namely, let R be a commutative local noetherian ring. Then the analogue fails for R if it has a dualizing complex, is henselian, not Gorenstein, and has a finitely generated Gorenstein projective module which is not free. The proof is based on a theory of Gorenstein projective (pre)envelopes. We show, among other things, that the finitely generated Gorenstein projective modules form an enveloping class in mod R if and only if R is Gorenstein or has the property that each finitely generated Gorenstein projective module is free. This is analogous to a recent result on covers by Christensen, Piepmeyer, Striuli, and Takahashi, and their methods are an important input to our work."}
{"category": "Math", "title": "Exponential random graphs as models of overlay networks", "abstract": "In this paper, we give an analytic solution for graphs with n nodes and E edges for which the probability of obtaining a given graph G is specified in terms of the degree sequence of G. We describe how this model naturally appears in the context of load balancing in communication networks, namely Peer-to-Peer overlays. We then analyse the degree distribution of such graphs and show that the degrees are concentrated around their mean value. Finally, we derive asymptotic results on the number of edges crossing a graph cut and use these results $(i)$ to compute the graph expansion and conductance, and $(ii)$ to analyse the graph resilience to random failures."}
{"category": "Math", "title": "Optimal Switching of One-Dimensional Reflected BSDEs, and Associated Multi-Dimensional BSDEs with Oblique Reflection", "abstract": "In this paper, an optimal switching problem is proposed for one-dimensional reflected backward stochastic differential equations (RBSDEs, for short) where the generators, the terminal values and the barriers are all switched with positive costs. The value process is characterized by a system of multi-dimensional RBSDEs with oblique reflection, whose existence and uniqueness are by no means trivial and are therefore carefully examined. Existence is shown using both methods of the Picard iteration and penalization, but under some different conditions. Uniqueness is proved by representation either as the value process to our optimal switching problem for one-dimensional RBSDEs, or as the equilibrium value process to a stochastic differential game of switching and stopping. Finally, the switched RBSDE is interpreted as a real option."}
{"category": "Math", "title": "Inferring sparse Gaussian graphical models with latent structure", "abstract": "Our concern is selecting the concentration matrix's nonzero coefficients for a sparse Gaussian graphical model in a high-dimensional setting. This corresponds to estimating the graph of conditional dependencies between the variables. We describe a novel framework taking into account a latent structure on the concentration matrix. This latent structure is used to drive a penalty matrix and thus to recover a graphical model with a constrained topology. Our method uses an $\\ell_1$ penalized likelihood criterion. Inference of the graph of conditional dependencies between the variates and of the hidden variables is performed simultaneously in an iterative \\textsc{em}-like algorithm. The performances of our method is illustrated on synthetic as well as real data, the latter concerning breast cancer."}
{"category": "Math", "title": "Numerical Measures for Two-Graphs", "abstract": "We study characteristics which might distinguish two-graphs by introducing different numerical measures on the collection of graphs on $n$ vertices. Two conjectures are stated, one using these numerical measures and the other using the deck of a graph, which suggest that there is a finite set of conditions differentiating two-graphs. We verify that, among the four non-trivial non-isomorphic regular two-graphs on 26 vertices, both conjectures hold."}
{"category": "Math", "title": "Spectra of winner-take-all stochastic neural networks", "abstract": "During the recent few years, in response to empirical findings suggesting scale-free self-organisation phenomena emerging in complex nervous systems at a mesoscale level, there has been significant search for suitable models and theoretical explanations in neuroscientific literature, see the recent survey by Bullmore and Sporns (2009). In Piekniewski and Schreiber (2008) we have developed a simple and tractable mathematical model shedding some light on a particular class of the afore-mentioned phenomena, namely on mesoscopic level self-organisation of functional brain networks under fMRI imaging, where we have achieved a high degree of agreement with existing empirical reports. Being addressed to the neuroscientific community, our work Piekniewski and Schreiber (2008) relied on semi-rigorous study of information flow structure in a class of recurrent neural networks exhibiting asymptotic scale-free behaviour and admitting a description in terms of the so-called winner-take-all dynamics. The purpose of the present paper is to define and study these winner-take-all networks with full mathematical rigour in context of their asymptotic spectral properties, well known to be of interest for neuroscientific community. Our main result is a limit theorem for spectra of the spike-flow graphs induced by the winner-take-all dynamics. We provide an explicit characterisation of the limit spectral measure expressed in terms of zeros of Bessel's J-function."}
{"category": "Math", "title": "Kahler-Einstein Structures of General Natural Lifted Type on the Cotangent Bundles", "abstract": "We study the conditions under which the cotangent bundle $T^*M$ of a Riemaannian manifold $(M,g)$, endowed with a K\\\"ahlerian structure $(G,J)$ of general natural lift type (see \\cite{Druta1}), is Einstein. We first obtain a general natural K\\\"ahler-Einstein structure on the cotangent bundle $T^*M$. In this case, a certain parameter, $\\lambda$ involved in the condition for $(T^*M,G,J)$ to be a K\\\"ahlerian manifold, is expressed as a rational function of the other two, the value of the constant sectional curvature, $c$, of the base manifold $(M,g)$ and the constant $\\rho$ involved in the condition for the structure of being Einstein. This expression of $\\lambda$ is just that involved in the condition for the K\\\"ahlerian manifold to have constant holomorphic sectional curvature (see \\cite{Druta2}). In the second case, we obtain a general natural K\\\"ahler-Einstein structure only on $T_0M$, the bundle of nonzero cotangent vectors to $M$. For this structure, $\\lambda$ is expressed as another function of the other two parameters, their derivatives, $c$ and $\\rho$."}
{"category": "Math", "title": "The Necessary Structure of Congruences in Free Semigroups", "abstract": "A characterization of congruences in free semigroups is presented."}
{"category": "Math", "title": "Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type $A_{n-1}$", "abstract": "It is known that the Lawrence-Krammer representation of the Artin group of type $A_{n-1}$ based on the two parameters $t$ and $q$ that was used by Krammer and independently by Bigelow to show the linearity of the braid group on $n$ strands is generically irreducible. Here, we recover this result and show further that for some complex specializations of the parameters the representation is reducible. We give all the values of the parameters for which the representation is reducible as well as the dimensions of the invariant subspaces. We deduce some results of semisimplicity of the Birman-Murakami-Wenzl algebra of type $A_{n-1}$."}
{"category": "Math", "title": "Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations", "abstract": "We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation."}
{"category": "Math", "title": "Closed classes of functions, generalized constraints and clusters", "abstract": "Classes of functions of several variables on arbitrary non-empty domains that are closed under permutation of variables and addition of dummy variables are characterized in terms of generalized constraints, and hereby Hellerstein's Galois theory of functions and generalized constraints is extended to infinite domains. Furthermore, classes of operations on arbitrary non-empty domains that are closed under permutation of variables, addition of dummy variables and composition are characterized in terms of clusters, and a Galois connection is established between operations and clusters."}
{"category": "Math", "title": "Precise estimates for the subelliptic heat kernel on H-type groups", "abstract": "We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type. Specifically, we show that there exist positive constants $C_1$, $C_2$ and a polynomial correction function $Q_t$ on $G$ such that $$C_1 Q_t e^{-\\frac{d^2}{4t}} \\le p_t \\le C_2 Q_t e^{-\\frac{d^2}{4t}}$$ where $p_t$ is the heat kernel, and $d$ the Carnot-Carath\\'eodory distance on $G$. We also obtain similar bounds on the norm of its subelliptic gradient $|\\nabla p_t|$. Along the way, we record explicit formulas for the distance function $d$ and the subriemannian geodesics of H-type groups."}
{"category": "Math", "title": "The critical number of finite abelian groups", "abstract": "Let G be an additive, finite abelian group. The critical number $\\mathsf{cr}(G)$ of $G$ is the smallest positive integer $\\ell$ such that for every subset $S \\subset G \\setminus \\{0\\}$ with $|S| \\ge \\ell$ the following holds: Every element of $G$ can be written as a nonempty sum of distinct elements from $S$. The critical number was first studied by P. Erd\\H{o}s and H. Heilbronn in 1964, and due to the contributions of many authors the value of $\\mathsf {cr}(G)$ is known for all finite abelian groups $G$ except for $G \\cong \\mathbb{Z}/pq\\mathbb{Z}$ where $p,q$ are primes such that $p+\\lfloor2\\sqrt{p-2}\\rfloor+1<q<2p$. We determine that $\\mathsf {cr}(G)=p+q-2$ for such groups."}
{"category": "Math", "title": "Weak Error for stable driven SDEs: expansion of the densities", "abstract": "Consider a multidimensional SDE of the form $X_t = x+\\int_{0}^{t} b(X_{s-})ds+\\int{0}^{t} f(X_{s-})dZ_s$ where $(Z_s)_{s\\ge 0}$ is a symmetric stable process. Under suitable assumptions on the coefficients the unique strong solution of the above equation admits a density w.r.t. the Lebesgue measure and so does its Euler scheme. Using a parametrix approach, we derive an error expansion at order 1 w.r.t. the time step for the difference of these densities."}
{"category": "Math", "title": "A symplectic resolution for the binary tetrahedral group", "abstract": "We describe an explicit symplectic resolution for the quotient singularity arising from the four-dimensional symplectic represenation of the binary tetrahedral group."}
{"category": "Math", "title": "The Combinatorics of Al-Salam-Chihara $q$-Laguerre polynomials", "abstract": "We describe various aspects of the Al-Salam-Chihara $q$-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Grassmann cells."}
{"category": "Math", "title": "On formal codegrees of fusion categories", "abstract": "We prove a general result which implies that the global and Frobenius-Perron dimensions of a fusion category generate Galois invariant ideals in the ring of algebraic integers."}
{"category": "Math", "title": "Algebraic Cuntz-Pimsner rings", "abstract": "From a system consisting of a right non-degenerate ring $R$, a pair of $R$-bimodules $Q$ and $P$ and an $R$-bimodule homomorphism $\\psi:P\\otimes Q\\longrightarrow R$ we construct a $\\Z$-graded ring $\\mathcal{T}_{(P,Q,\\psi)}$ called the Toeplitz ring and (for certain systems) a $\\Z$-graded quotient $\\mathcal{O}_{(P,Q,\\psi)}$ of $\\mathcal{T}_{(P,Q,\\psi)}$ called the Cuntz-Pimsner ring. These rings are the algebraic analogs of the Toeplitz $C^*$-algebra and the Cuntz-Pimsner $C^*$-algebra associated to a $C^*$-correspondence (also called a Hilbert bimodule). This new construction generalizes for example the algebraic crossed product by a single automorphism, corner skew Laurent polynomial ring by a single corner automorphism and Leavitt path algebras. We also describe the structure of the graded ideals of our graded rings in terms of pairs of ideals of the coefficient ring."}
{"category": "Math", "title": "Vanishing viscosity limit for an expanding domain in space", "abstract": "We study the limiting behavior of viscous incompressible flows when the fluid domain is allowed to expand as the viscosity vanishes. We describe precise conditions under which the limiting flow satisfies the full space Euler equations. The argument is based on truncation and on energy estimates, following the structure of the proof of Kato's criterion for the vanishing viscosity limit. This work complements previous work by the authors, see [Kelliher, Comm. Math. Phys. 278 (2008), 753-773] and [arXiv:0801.4935v1]."}
{"category": "Math", "title": "Cohomology of bundles on homological Hopf manifold", "abstract": "We discuss the properties of complex manifolds having rational homology of $S^1 \\times S^{2n-1}$ including those constructed by Hopf, Kodaira and Brieskorn-van de Ven. We extend certain previously known vanishing properties of cohomology of bundles on such manifolds.As an application we consider degeneration of Hodge-deRham spectral sequence in this non Kahler setting."}
{"category": "Math", "title": "Self-Similar Markov Processes on Cantor Set", "abstract": "We define analogues of Brownian motion on the triadic Cantor set by introducing a few natural requirements on the Markov semigroup. We give a detailed description of these symmetric self-similar processes and study their properties such as mixing and moment asymptotics."}
{"category": "Math", "title": "The fractional Galois ideal for arbitrary order of vanishing", "abstract": "We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those $L$-functions of the extension which are non-zero at the special point $s = 0$, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include $L$-functions with a simple zero at $s = 0$, and was shown by the present author to provide class-group annihilators not existing in the Stickelberger ideal. The version presented in this article deals with $L$-functions of arbitrary order of vanishing at $s = 0$, and we give evidence using results of Popescu and Rubin that it is closely related to the Fitting ideal of the class-group, a canonical ideal of annihilators. Finally, we prove an equality involving Stark elements and class-groups originally due to B\\\"uy\\\"ukboduk, but under a slightly different assumption, the advantage being that we need none of the Kolyvagin system machinery used in the original proof."}
{"category": "Math", "title": "Generating varieties for affine Grassmannians", "abstract": "We study the topological group structure (coming from loop multiplication) on an affine Grassmannian. In particular, we study finite-dimensional subvarieties that generate the homology ring. We show that there is a canonical family of generating Schubert varieties, namely those defined by the negative of the coroot associated to the highest root. These not only generate the homology, but generate the affine Grassmannian itself in a precise sense."}
{"category": "Math", "title": "Schrodinger Operators with Purely Discrete Spectrum", "abstract": "We prove $-\\Delta +V$ has purely discrete spectrum if $V\\geq 0$ and, for all $M$, $|\\{x\\mid V(x)<M\\}|<\\infty$ and various extensions."}
{"category": "Math", "title": "Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum", "abstract": "By combining some ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for OPRL in the a.c. spectral region is implied by convergence of $\\frac{1}{n} K_n(x,x)$ for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. We then show that these hypotheses are always valid for ergodic Jacobi matrices with a.c. spectrum and prove that the limit of $\\frac{1}{n} K_n(x,x)$ is $\\rho_\\infty(x)/w(x)$ where $\\rho_\\infty$ is the density of zeros and $w$ is the a.c. weight of the spectral measure."}
{"category": "Math", "title": "A Singular Value Thresholding Algorithm for Matrix Completion", "abstract": "This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and arises in many important applications as in the task of recovering a large matrix from a small subset of its entries (the famous Netflix problem). Off-the-shelf algorithms such as interior point methods are not directly amenable to large problems of this kind with over a million unknown entries. This paper develops a simple first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank. The algorithm is iterative and produces a sequence of matrices (X^k, Y^k) and at each step, mainly performs a soft-thresholding operation on the singular values of the matrix Y^k. There are two remarkable features making this attractive for low-rank matrix completion problems. The first is that the soft-thresholding operation is applied to a sparse matrix; the second is that the rank of the iterates X^k is empirically nondecreasing. Both these facts allow the algorithm to make use of very minimal storage space and keep the computational cost of each iteration low. We provide numerical examples in which 1,000 by 1,000 matrices are recovered in less than a minute on a modest desktop computer. We also demonstrate that our approach is amenable to very large scale problems by recovering matrices of rank about 10 with nearly a billion unknowns from just about 0.4% of their sampled entries. Our methods are connected with linearized Bregman iterations for l1 minimization, and we develop a framework in which one can understand these algorithms in terms of well-known Lagrange multiplier algorithms."}
{"category": "Math", "title": "State estimation for dynamical system described by linear equation with uncertainty", "abstract": "In this paper we investigate a problem of state estimation for the dynamical system described by the linear operator equation with unknown parameters in Hilbert space. We present explicit expressions for linear minimax estimation and error provided that any pair of uncertain parameters belongs to the quadratic bounding set. As an application of the main result we present the solution of minimax estimation problem for the linear descriptor differential equation with constant matrices."}
{"category": "Math", "title": "A comment to: On 3-colorable planar graphs without short cycles", "abstract": "Let G be a graph. It was proved that if G is a planar graph without {4, 6, 7}-cycles and without two 5-cycles sharing exactly one edge, then G 3-colorable. We observed that the proof of this result is not correct."}
{"category": "Math", "title": "Abstract Geometric Algebra. Orthogonal and Symplectic Geometries", "abstract": "Our main interest in this paper is chiefly concerned with the conditions characterizing \\textit{orthogonal and symplectic abstract differential geometries}. A detailed account about the sheaf-theoretic version of the \\textit{symplectic Gram-Schmidt theorem} and of the \\textit{Witt's theorem} is also given."}
{"category": "Math", "title": "Set-membership state estimation framework for uncertain linear differential-algebraic equations", "abstract": "We investigate a state estimation problem for the dynamical system described by uncertain linear operator equation in Hilbert space. The uncertainty is supposed to admit a set-membership description. We present explicit expressions for linear minimax estimation and error provided that any pair of uncertain parameters belongs to the quadratic bounding set. We introduce a new notion of minimax directional observability and index of non-causality for linear noncausal DAEs. Application of these notions to the state estimation problem for linear uncertain noncausal DAEs allows to derive new minimax recursive estimator for both continuous and discrete time. We illustrate the benefits of non-causality of the plant applying our approach to scalar nonlinear set-membership state estimation problem. Numerical example is presented."}
{"category": "Math", "title": "Closed Weingarten hypersurfaces in warped product manifolds", "abstract": "Given a compact Riemannian manifold $M$, we consider a warped product $\\bar M = I \\times_h M$ where $I$ is an open interval in $\\Rr$. We suppose that the mean curvature of the fibers do not change sign. Given a positive differentiable function $\\psi$ in $\\bar M$, we find a closed hypersurface $\\Sigma$ which is solution of an equation of the form $F(B)=\\psi$, where $B$ is the second fundamental form of $\\Sigma$ and $F$ is a function satisfying certain structural properties. As examples, we may exhibit examples of hypersurfaces with prescribed higher order mean curvature."}
{"category": "Math", "title": "Existence of isometric immersions into nilpotent Lie groups", "abstract": "We establish necessary and sufficient conditions for existence of isometric immersions of a simply connected Riemannian manifold into a two-step nilpotent Lie group. This comprises the case of immersions into $H$-type groups."}
{"category": "Math", "title": "The Avrunin-Scott theorem for quantum complete intersections", "abstract": "We prove the Avrunin-Scott theorem for quantum complete intersections; the rank variety of a module is isomorphic to its support variety."}
{"category": "Math", "title": "Existence and nonexistence of traveling waves for a nonlocal monostable equation", "abstract": "We consider the nonlocal analogue of the Fisher-KPP equation. We do not assume that the Borel-measure is absolutely continuous with respect to the Lebesgue measure. We gives a sufficient condition for existence of traveling waves, and a necessary condition for existence of periodic traveling waves."}
{"category": "Math", "title": "Clifford Algebras and Graphs", "abstract": "I show how to associate a Clifford algebra to a graph. I describe the structure of these Clifford graph algebras and provide many examples and pictures. I describe which graphs correspond to isomorphic Clifford algebras and also discuss other related sets of graphs. This construction can be used to build models of representations of simply-laced compact Lie groups."}
{"category": "Math", "title": "Falling Factorials, Generating Functions, and Conjoint Ranking Tables", "abstract": "We investigate the coefficients generated by expressing the falling factorial $(xy)_k$ as a linear combination of falling factorial products $(x)_l (y)_m$ for $l,m =1,...,k$. Algebraic and combinatoric properties of these coefficients are discussed, including recurrence relations, closed-form formulae, relations with Stirling numbers, and a combinatorial characterization in terms of conjoint ranking tables."}
{"category": "Math", "title": "The variance of arithmetic measures associated to closed geodesics on the modular surface", "abstract": "We determine the variance for the fluctuations of the arithmetic measures obtained by collecting all closed geodesics on the modular surface with the same discriminant and ordering them by the latter. This arithmetic variance differs by subtle factors from the variance that one gets when considering individual closed geodesics when ordered by their length. The arithmetic variance is the same one that appears in the fluctuations of measures associated with quantum states on the modular surface."}
{"category": "Math", "title": "On the middle convolution of local systems. With an Appendix by M. Dettweiler and S. Reiter", "abstract": "We study the middle convolution of local systems and realize special linear groups as Galois groups over the rationals. In the Appendix to this paper, written jointly with Stefan Reiter, we prove the existence of a new motivic local system with $G_2$-monodromy."}
{"category": "Math", "title": "Rigid local systems and potential automorphy: The $G_2$-case", "abstract": "We use a certain rigid local system in order to prove the potential automorphy of certain Galois representations with values in $G_2,$ found by N. Katz and the author."}
{"category": "Math", "title": "The rate of convergence of the Walk on Spheres Algorithm", "abstract": "In this paper we examine the rate of convergence of one of the standard algorithms for emulating exit probabilities of Brownian motion, the Walk on Spheres (WoS) algorithm. We obtain the complete characterization of the rate of convergence of WoS in terms of the local geomnetry of a domain."}
{"category": "Math", "title": "Differential Geometry of Curves and Surfaces in Lorentz-Minkowski space", "abstract": "We review part of the classical theory of curves and surfaces in $3$-dimensional Lorentz-Minkowski space. We focus in spacelike surfaces with constant mean curvature pointing the differences and similarities with the Euclidean space."}
{"category": "Math", "title": "Backward Ricci Flow on Locally Homogeneous Three-manifolds", "abstract": "In this paper, we study the backward Ricci flow on locally homogeneous 3-manifolds. We describe the long time behavior and show that, typically and after a proper re-scaling, there is convergence to a sub-Riemannian geometry. A similar behavior was observed by the authors in the case of the cross curvature flow."}
{"category": "Math", "title": "Translation Covers Among Triangular Billiard Surfaces", "abstract": "We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of translation surfaces, which we call the fingerprint of a point, that is preserved under balanced translation covers."}
{"category": "Math", "title": "Selmer varieties for curves with CM Jacobians", "abstract": "We study the Selmer variety associated to a canonical quotient of the $\\Q_p$-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over $\\Q$ whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multi-variable Iwasawa theory is used to prove dimension bounds, which, in turn, lead to a new proof of Diophantine finiteness over $\\Q$ for such curves."}
{"category": "Math", "title": "A Selberg Integral Type Formula for an sl_2 One-Dimensional Space of Conformal Blocks", "abstract": "For distinct complex numbers $z_1,...,z_{2N}$, we give a polynomial $P(y_1,...,y_{2N})$ in the variables $y_1,...,y_{2N}$, which is homogeneous of degree $N$, linear with respect to each variable, $sl_2$-invariant with respect to a natural $sl_2$-action, and is of order $N-1$ at $(y_1,...,y_{2N})=(z_1,...,z_{2N})$. We give also a Selberg integral type formula for the associated one-dimensional space of conformal blocks."}
{"category": "Math", "title": "Lifting to cluster-tilting objects in higher cluster categories", "abstract": "In this note, we consider the $d$-cluster-tilted algebras, the endomorphism algebras of $d$-cluster-tilting objects in $d$-cluster categories. We show that a tilting module over such an algebra lifts to a $d$-cluster-tilting object in this $d$-cluster category."}
{"category": "Math", "title": "Squares in (2^2-1)...(n^2-1) and p-adic valuation", "abstract": "In this paper, we determine all the squares in the sequence $\\{\\prod_{k=2}^n(k^2-1)\\}_{n=2}^\\infty $. From this, one deduces that there are infinitely many squares in this sequence. We also give a formula for the $p$-adic valuation of the terms in this sequence."}
{"category": "Math", "title": "Finite time blow-up for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system", "abstract": "Finite time blow-up is shown to occur for radially symmetric solutions to a critical quasilinear Smoluchowski-Poisson system provided that the mass of the initial condition exceeds an explicit threshold. In the supercritical case, blow-up is shown to take place for any positive mass. The proof relies on a novel identity of virial type."}
{"category": "Math", "title": "Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system", "abstract": "Finite time blow-up is shown to occur for solutions to a one-dimensional quasilinear parabolic-parabolic chemotaxis system as soon as the mean value of the initial condition exceeds some threshold value. The proof combines a novel identity of virial type with the boundedness from below of the Liapunov functional associated to the system, the latter being peculiar to the one-dimensional setting."}
{"category": "Math", "title": "Mean Curvature Flow and Bernstein-Calabi Results for Spacelike Graphs", "abstract": "This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian manifolds."}
{"category": "Math", "title": "On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields", "abstract": "Let $m\\geq -1$ be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations \\[ X^3-mX^2Y-(m+3)XY^2-Y^3=\\lambda \\] where $\\lambda>0$ is a divisor of $m^2+3m+9$ and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazaki's result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors $\\lambda$ of $m^2+3m+9$. As a consequence, we obtain another proof of Okazaki's theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for $m=-1,0,1,2,3,5,12,54,66,1259,2389$."}
{"category": "Math", "title": "Labeled Partitions with Colored Permutations", "abstract": "In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations in the sense that the colors are endowed with a cyclic structure. We use labeled partitions with colored permutations to derive the generating function of the $\\mathrm{fmaj}_k$ indices of colored permutations. The second result is a combinatorial treatment of a relation on the q-derangement numbers with respect to colored permutations which leads to the formula of Chow for signed permutations and the formula of Faliharimalala and Zeng [10] on colored permutations. The third result is an involution on permutations that implies the generating function formula for the signed q-counting of the major indices due to Gessel and Simon. This involution can be extended to signed permutations. In this way, we obtain a combinatorial interpretation of a formula of Adin, Gessel and Roichman."}
{"category": "Math", "title": "Angle-deformations in Coxeter groups", "abstract": "The isomorphism problem for Coxeter groups has been reduced to its 'reflection preserving version' by B. Howlett and the second author. Thus, in order to solve it, it suffices to determine for a given Coxeter system (W,R) all Coxeter generating sets S of W which are contained in R^W, the set of reflections of (W,R). In this paper, we provide a further reduction: it suffices to determine all Coxeter generating sets S in R^W which are sharp-angled with respect to R."}
{"category": "Math", "title": "Existence of traveling wave solutions for a nonlocal bistable equation: an abstract approach", "abstract": "We consider traveling fronts to the nonlocal bistable equation. We do not assume that the Borel-measure is absolutely continuous with respect to the Lebesgue measure. We show that there is a traveling wave solution with monotone profile. In order to prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation."}
{"category": "Math", "title": "Realizations of Countable Groups as Fundamental Groups of Compacta", "abstract": "It is an open question (Pawlikowski) whether every finitely generated group can be realized as a fundamental group of a compact metric space. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of four dimensional Euclidean space. According to theorems of Shelah (see also Pawlikowski) such space can not be locally path connected if the group is not finitely generated. This constructions complements realization of groups in the context of compact Hausdorff spaces, that was studied by Keesling and Rudyak, and Przezdziecki ."}
{"category": "Math", "title": "Moduli of bordered Riemann Surfaces - Complex structure and K\\\"ahler geometry", "abstract": "This paper has been withdrawn by the authors."}
{"category": "Math", "title": "On the sum of superoptimal singular values", "abstract": "We discuss the following extremal problem and its relevance to the sum of the so-called superoptimal singular values of a matrix function: Given an $m\\times n$ matrix function $\\Phi$ on the unit circle $\\mathbb{T}$, when is there a matrix function $\\Psi_{*}$ in the set $A_{k}^{n,m}$ such that \\int_{\\mathbb{T}}{\\rm trace}(\\Phi(\\zeta)\\Psi_{*}(\\zeta))dm(\\zeta)=\\sup_{\\Psi\\in A_{k}^{n,m}}|\\int_{\\mathbb{T}}{\\rm trace}(\\Phi(\\zeta)\\Psi(\\zeta))dm(\\zeta)|? The set $A_{k}^{n,m}$ is defined by A_{k}^{n,m}={\\Psi\\in H_{0}^{1}: \\|\\Psi\\|_{L^{1}}\\leq 1, {\\rm rank}\\Psi(\\zeta)\\leq k{a.e.}\\zeta\\in T}. We introduce Hankel-type operators on spaces of matrix functions and prove that this problem has a solution if and only if the corresponding Hankel-type operator has a maximizing vector. We also characterize the smallest number $k$ for which \\int_{\\mathbb{T}}{\\rm trace}(\\Phi(\\zeta)\\Psi(\\zeta))dm(\\zeta) equals the sum of all the superoptimal singular values of an admissible matrix function $\\Phi$ for some $\\Psi\\in A_{k}^{n,m}$. Moreover, we provide a representation of any such function $\\Psi$ when $\\Phi$ is an admissible very badly approximable unitary-valued $n\\times n$ matrix function."}
{"category": "Math", "title": "Results on the diffusion equation with rough coefficients", "abstract": "We study the behaviour of the solutions of the stationary diffusion equation as a function of a possibly rough ($L^{\\infty}$-) diffusivity. This includes the boundary behaviour of the solution maps, associating to each diffusivity the solution corresponding to some fixed source function, when the diffusivity approaches infinite values in parts of the medium. In $n$-dimensions, $n \\geq 1$, by assuming a weak notion of convergence on the set of diffusivities, we prove the strong sequential continuity of the solution maps. In 1-dimension, we prove a stronger result, i.e., the unique extendability of the map of solution operators, associating to each diffusivity the corresponding solution operator, to a sequentially continuous map in the operator norm on a set containing `diffusivities' assuming infinite values in parts of the medium. In this case, we also give explicit estimates on the convergence behaviour of the map."}
{"category": "Math", "title": "Distinguishing Bing-Whitehead Cantor Sets", "abstract": "Bing-Whitehead Cantor sets were introduced by DeGryse and Osborne in dimension three and greater to produce examples of Cantor sets that were non standard (wild), but still had simply connected complement. In contrast to an earlier example of Kirkor, the construction techniques could be generalized to dimensions bigger than three. These Cantor sets in $S^{3}$ are constructed by using Bing or Whitehead links as stages in defining sequences. Ancel and Starbird, and separately Wright characterized the number of Bing links needed in such constructions so as to produce Cantor sets. However it was unknown whether varying the number of Bing and Whitehead links in the construction would produce non equivalent Cantor sets. Using a generalization of geometric index, and a careful analysis of three dimensional intersection patterns, we prove that Bing-Whitehead Cantor sets are equivalently embedded in $S^3$ if and only if their defining sequences differ by some finite number of Whitehead constructions. As a consequence, there are uncountably many non equivalent such Cantor sets in $S^{3}$ constructed with genus one tori and with simply connected complement."}
{"category": "Math", "title": "Automorphism groups of cyclic codes", "abstract": "In this article we study the automorphism groups of binary cyclic codes. In particular, we provide explicit constructions for codes whose automorphism groups can be described as (a) direct products of two symmetric groups or (b) iterated wreath products of several symmetric groups. Interestingly, some of the codes we consider also arise in the context of regular lattice graphs and permutation decoding."}
{"category": "Math", "title": "Global Well-posedness of Korteweg-de Vries equation in $H^{-3/4}(\\R)$", "abstract": "We prove that the Korteweg-de Vries initial-value problem is globally well-posed in $H^{-3/4}(\\R)$ and the modified Korteweg-de Vries initial-value problem is globally well-posed in $H^{1/4}(\\R)$. The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation at $s=-3/4$ by constructing some special resolution spaces in order to avoid some 'logarithmic divergence' from the high-high interactions. Our local solution has almost the same properties as those for $H^s (s>-3/4)$ solution which enable us to apply the I-method to extend it to a global solution."}
{"category": "Math", "title": "Weighted $\\theta$-Incomplete Pluripotential Theory", "abstract": "Weighted pluripotential theory is a rapidly developing area; and Callaghan \\cite{Callaghan} recently introduced $\\theta$-incomplete polynomials in \\cd for $d>1$. In this paper we combine these two theories by defining weighted $\\theta$-incomplete pluripotential theory. We define weighted $\\theta$-incomplete extremal functions and obtain a Siciak-Zahariuta type equality in terms of $\\theta$-incomplete polynomials. Finally we prove that the extremal functions can be recovered using orthonormal polynomials and we demonstrate a result on strong asymptotics of Bergman functions in the spirit of \\cite{BermanCn}."}
{"category": "Math", "title": "Global Existence for the Vlasov-Poisson System in Bounded Domains", "abstract": "In this paper we prove global existence for solutions of the Vlasov-Poisson system in convex bounded domains with specular boundary conditions and with a prescribed outward electrical field at the boundary."}
{"category": "Math", "title": "On the Existence of Exponentially Decreasing Solutions of the Nonlinear Landau Damping Problem", "abstract": "In this paper we prove the existence of a large class of periodic solutions of the Vlasov-Poisson in one space dimension that decay exponentially as t goes to infinity. The exponential decay is well known for the linearized version of the Landau damping problem and it has been proved in [4] for a class of solutions of the Vlasov-Poisson system that behaves asymptotically as free streaming solutions and are sufficiently flat in the space of velocities. The results in this paper enlarge the class of possible asymptotic limits, replacing the fatness condition in [4] by a stability condition for the linearized problem."}
{"category": "Math", "title": "Induced Modules for Affine Lie Algebras", "abstract": "We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra ${\\mathcal P}$ of an affine Lie algebra ${\\mathfrak G}$, our main result establishes the equivalence between a certain category of ${\\mathcal P}$-induced ${\\mathfrak G}$-modules and the category of weight ${\\mathcal P}$-modules with injective action of the central element of ${\\mathfrak G}$. In particular, the induction functor preserves irreducible modules. If ${\\mathcal P}$ is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra ${\\mathcal P}^{ps}$, ${\\mathcal P}\\subset {\\mathcal P}^{ps}$. The structure of ${\\mathcal P}$-induced modules in this case is fully determined by the structure of ${\\mathcal P}^{ps}$-induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S. K\\\"onig, V. Mazorchuk [Forum Math. 13 (2001), 641-661], B. Cox [Pacific J. Math. 165 (1994), 269-294] and I. Dimitrov, V. Futorny, I. Penkov [Comm. Math. Phys. 250 (2004), 47-63]."}
{"category": "Math", "title": "Toric degenerations of Gelfand-Cetlin systems and potential functions", "abstract": "We define a toric degeneration of an integrable system on a projective manifold, and prove the existence of a toric degeneration of the Gelfand-Cetlin system on the flag manifold of type A. As an application, we calculate the potential function for a Lagrangian torus fiber of the Gelfand-Cetlin system."}
{"category": "Math", "title": "Koszul duality for stratified algebras I. Quasi-hereditary algebras", "abstract": "We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary algebras admitting linear tilting (co)resolutions of standard and costandard modules. We show that such algebras are Koszul, that the class of these algebras is closed with respect to both dualities and that on this class these two dualities commute. All arguments reduce to short computations in the bounded derived category of graded modules."}
{"category": "Math", "title": "Hypocoercivity for kinetic equations with linear relaxation terms", "abstract": "This note is devoted to a simple method for proving hypocoercivity of the solutions of a kinetic equation involving a linear time relaxation operator, i.e. the construction of an adapted Lyapunov functional satisfying a Gronwall-type inequality. The method clearly distinguishes the coercivity at microscopic level, which directly arises from the properties of the relaxation operator, and a spectral gap inequality at the macroscopic level for the spatial density, which is connected to the diffusion limit. It improves on previously known results. Our approach is illustrated by the linear BGK model and a relaxation operator which corresponds at macroscopic level to the linearized fast diffusion."}
{"category": "Math", "title": "Evaluating Azumaya algebras on cubic surfaces", "abstract": "Let X be a cubic surface over a local number field k. Given an Azumaya algebra on X, we describe the local evaluation map X(k) -> Q/Z in two cases, showing a sharp dependence on the geometry of the reduction of X. We show that a suitably generic cubic surface over a number field, whose reduction at some prime is a cone, has no Brauer-Manin obstruction. This extends results of Colliot-Th\\'el\\`ene, Kanevsky and Sansuc."}
{"category": "Math", "title": "Excited Brownian Motions", "abstract": "We study a natural continuous time version of excited random walks, introduced by Norris, Rogers and Williams about twenty years ago. We obtain a necessary and sufficient condition for recurrence and for positive speed. This is analogous to results for excited (or cookie) random walks."}
{"category": "Math", "title": "The classical umbral calculus: Sheffer sequences", "abstract": "Following the approach of Rota and Taylor \\cite{SIAM}, we present an innovative theory of Sheffer sequences in which the main properties are encoded by using umbrae. This syntax allows us noteworthy computational simplifications and conceptual clarifications in many results involving Sheffer sequences. To give an indication of the effectiveness of the theory, we describe applications to the well-known connection constants problem, to Lagrange inversion formula and to solving some recurrence relations."}
{"category": "Math", "title": "Hochschild homology of certain Soergel bimodules", "abstract": "In this paper we compute Hochschild homology of certain Soergel bimodules. Moreover, we describe explicitly the graded bimodule maps between Soergel bimodules. This computations are motivated by the categorifications of the colored HOMFLY-PT polynomial for links via Hochschild homology of Soergel bimodules."}
{"category": "Math", "title": "Notes de lecture de l'article \"Partial sums of the M\\\"obius function\" de Kannan Soundararajan", "abstract": "Kannan Soundararajan recently obtained a new estimate, conditional to the Riemann hypothesis, for the summatory function of the Mobius function. In this expository article we describe his method, with detailed computations."}
{"category": "Math", "title": "On the convergence of the hp-BEM with quasi-uniform meshes for the electric field integral equation on polyhedral surfaces", "abstract": "In this paper the hp-version of the boundary element method is applied to the electric field integral equation on a piecewise plane (open or closed) Lipschitz surface. The underlying meshes are supposed to be quasi-uniform. We use H(div)-conforming discretisations with quadrilateral elements of Raviart-Thomas type and establish quasi-optimal convergence of hp-approximations. Main ingredient of our analysis is a new H^{-1/2}(div)-conforming p-interpolation operator that assumes only H^r \\cap H^{-1/2}(div)-regularity (r > 0) and for which we show quasi-stability with respect to polynomial degrees."}
{"category": "Math", "title": "Artin characters, Hurwitz trees and the lifting problem", "abstract": "We study finite groups of automorphisms of the $p$-adic open disk. In particular, we generalize results of Green, Matignon and Henrio from cyclic groups of order $p$ to arbitrary finite groups. As an application, we produce a counterexample to a question of Chinburg, Guralnick and Harbater, concerning the local lifting problem for generalized quaternion groups."}
{"category": "Math", "title": "On regularization methods of EM-Kaczmarz type", "abstract": "We consider regularization methods of Kaczmarz type in connection with the expectation-maximization (EM) algorithm for solving ill-posed equations. For noisy data, our methods are stabilized extensions of the well established ordered-subsets expectation-maximization iteration (OS-EM). We show monotonicity properties of the methods and present a numerical experiment which indicates that the extended OS-EM methods we propose are much faster than the standard EM algorithm."}
{"category": "Math", "title": "A Quantum Canonical Embedding", "abstract": "This paper presents an English version of a chapter of the L.L. Vaksman book `Quantum Bounded Symmetric Domains', see arXiv:0803.3769 [math.QA]. This chapter deals with a quantum analog of a canonical embedding of a bounded symmetric domain."}
{"category": "Math", "title": "Cocycle and orbit superrigidity for lattices in SL(n,R) acting on homogeneous spaces", "abstract": "We prove cocycle and orbit equivalence superrigidity for lattices in SL(n,R) acting linearly on R^n, as well as acting projectively on certain flag manifolds, including the real projective space. The proof combines operator algebraic techniques with the property (T) in the sense of Zimmer for the action of SL(n,Z) on R^n, n \\geq 4. We also show that the restriction of the associated orbit equivalence relation to a subset of finite Lebesgue measure, provides a II_1 equivalence relation with property (T) and yet fundamental group equal to R_+."}
{"category": "Math", "title": "Singular McKay correspondence for normal surfaces", "abstract": "We define the singular orbifold elliptic genus and $E$-function for all normal surfaces without strictly log-canonical singularities, and prove the analogue of the McKay correspondence in this setting. Our invariants generalize the stringy invariants defined by Willem Veys for this class of singularities. We show that the ability to define these invariants is closely linked to rigidity phenomena associated to the elliptic genus."}
{"category": "Math", "title": "Cluster expansion formulas and perfect matchings", "abstract": "We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\\gamma}$."}
{"category": "Math", "title": "Magnetically-induced buckling of a whirling conducting rod with applications to electrodynamic space tethers", "abstract": "We study the effect of a magnetic field on the behaviour of a slender conducting elastic structure, motivated by stability problems of electrodynamic space tethers. Both statical (buckling) and dynamical (whirling) instability are considered and we also compute post-buckling configurations. The equations used are the geometrically exact Kirchhoff equations. Magnetic buckling of a welded rod is found to be described by a surprisingly degenerate bifurcation, which is unfolded when both transverse anisotropy of the rod and angular velocity are considered. By solving the linearised equations about the (quasi-) stationary solutions we find various secondary instabilities. Our results are relevant for current designs of electrodynamic space tethers and potentially for future applications in nano- and molecular wires."}
{"category": "Math", "title": "Scaling Limits for Width Two Partially Ordered Sets: The Incomparability Window", "abstract": "We study the structure of a uniformly randomly chosen partial order of width 2 on n elements. We show that under the appropriate scaling, the number of incomparable elements converges to the height of a one dimensional Brownian excursion at a uniformly chosen random time in the interval [0,1], which follows the Rayleigh distribution."}
{"category": "Math", "title": "Interlacing and non-orthogonality of spectral polynomials for the Lam\\'e operator", "abstract": "Polynomial solutions to the generalized Lam\\'e equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830's in various contexts including the solution of Laplace equations on an ellipsoid. Recently there has been renewed interest in the distribution of the zeros of Van Vleck polynomials as the degree of the corresponding Stieltjes polynomials increases. In this paper we show that the zeros of Van Vleck polynomials corresponding to Stieltjes polynomials of successive degrees interlace. We also show that the spectral polynomials formed from the Van Vleck zeros are not orthogonal with respect to any weight. This furnishes a counterexample, coming from a second order differential equation, to the converse of the well known theorem that the zeros of orthogonal polynomials interlace."}
{"category": "Math", "title": "(Non)Automaticity of number theoretic functions", "abstract": "Denote by $\\lambda(n)$ Liouville's function concerning the parity of the number of prime divisors of $n$. Using a theorem of Allouche, Mend\\`es France, and Peyri\\`ere and many classical results from the theory of the distribution of prime numbers, we prove that $\\lambda(n)$ is not $k$--automatic for any $k> 2$. This yields that $\\sum_{n=1}^\\infty \\lambda(n) X^n\\in\\mathbb{F}_p[[X]]$ is transcendental over $\\mathbb{F}_p(X)$ for any prime $p>2$. Similar results are proven (or reproven) for many common number--theoretic functions, including $\\phi$, $\\mu$, $\\Omega$, $\\omega$, $\\rho$, and others."}
{"category": "Math", "title": "The vanishing viscosity limit for a dyadic model", "abstract": "A dyadic shell model for the Navier-Stokes equations is studied in the context of turbulence. The model is an infinite nonlinearly coupled system of ODEs. It is proved that the unique fixed point is a global attractor, which converges to the global attractor of the inviscid system as viscosity goes to zero. This implies that the average dissipation rate for the viscous system converges to the anomalous dissipation rate for the inviscid system (which is positive) as viscosity goes to zero. This phenomenon is called the dissipation anomaly predicted by Kolmogorov's theory for the actual Navier-Stokes equations."}
{"category": "Math", "title": "Second maximal subgroups of the finite alternating and symmetric groups", "abstract": "A subgroup of a finite group G is said to be second maximal if it is maximal in every maximal subgroup of G that contains it. A question which has received considerable attention asks: can every positive integer occur as the number of the maximal subgroups that contain a given second maximal subgroup in some finite group G? Various reduction arguments are available except when G is almost simple. Following the classification of the finite simple groups, finite almost simple groups fall into three categories: alternating and symmetric groups, almost simple groups of Lie type, sporadic groups and automorphism groups of sporadic groups. This thesis investigates the finite alternating and symmetric groups, and finds that in such groups, except three well known examples, no second maximal subgroup can be contained in more than 3 maximal subgroups."}
{"category": "Math", "title": "Foundations of a Multi-way Spectral Clustering Framework for Hybrid Linear Modeling", "abstract": "The problem of Hybrid Linear Modeling (HLM) is to model and segment data using a mixture of affine subspaces. Different strategies have been proposed to solve this problem, however, rigorous analysis justifying their performance is missing. This paper suggests the Theoretical Spectral Curvature Clustering (TSCC) algorithm for solving the HLM problem, and provides careful analysis to justify it. The TSCC algorithm is practically a combination of Govindu's multi-way spectral clustering framework (CVPR 2005) and Ng et al.'s spectral clustering algorithm (NIPS 2001). The main result of this paper states that if the given data is sampled from a mixture of distributions concentrated around affine subspaces, then with high sampling probability the TSCC algorithm segments well the different underlying clusters. The goodness of clustering depends on the within-cluster errors, the between-clusters interaction, and a tuning parameter applied by TSCC. The proof also provides new insights for the analysis of Ng et al. (NIPS 2001)."}
{"category": "Math", "title": "A survey on Cox rings", "abstract": "We survey the construction of the Cox ring of an algebraic variety X and study the birational geometry of X when its Cox ring is finitely generated."}
{"category": "Math", "title": "Decomposition of Almost Poisson Structure of Non-Self-Adjoint Dynamical Systems", "abstract": "Non-self-adjoint dynamical systems, e.g., nonholonomic systems, can admit an almost Poisson structure, which is formulated by a kind of Poisson bracket satisfying the usual properties except for the Jacobi identity. A general theory of the almost Poisson structure is investigated based on a decomposition of the bracket into a sum of a Poisson one and an almost Poisson one. The corresponding relation between Poisson structure and symplectic structure is proved, making use of Jacobiizer and symplecticizer. Based on analysis of pseudo-symplectic structure of constraint submanifold of Chaplygin nonholonomic systems, an almost Poisson bracket for the systems is constructed and decomposed into a sum of a canonical Poisson one and an almost Poisson one. Similarly, an almost Poisson structure, which can be decomposed into a sum of canonical one and an almost Lie-Poisson one, is also constructed on an affine space with torsion whose autoparallels are utilized to described the free motion of some non-self-adjoint systems. The decomposition of the almost Poisson bracket directly leads to a decomposition of a dynamical vector field into a sum of usual Hamiltionian vector field and an almost Hamiltonian one, which is useful to simplifying the integration of vector fields."}
{"category": "Math", "title": "Abstract integrals in algebra: coalgebras, Hopf algebras and compact groups", "abstract": "We generalize the results on existence and uniqueness of integrals from compact groups and Hopf algebras in a pure (co)algebraic setting, and find a series of new results on (quasi)-co-Frobenius and semiperfect coalgebras. For a coalgebra $C$, we introduce the generalized space of integrals $\\int_M=\\Hom^C(C,M)$ associated to a right $C$-comodule $M$ and study connections between \"uniqueness of integrals\" $\\dim(\\int_M)\\leq \\dim(M)$ and \"existence of integrals\" $\\dim(\\int_M)\\geq \\dim(M)$ for all $M$ and representation theoretic properties of $C$: (quasi)-co-Frobenius, semiperfect. We show that a coalgebra is co-Frobenius if and only if existence and uniqueness of integrals holds for any finite dimensional $M$. We give the interpretation for $\\int_M$ for the coalgebra of representative functions of a compact group - they will be \"quantum\"-invariant vector integrals. As applications, new proofs of well known characterizations of co-Frobenius coalgebras and Hopf algebras are obtained, as well as the uniqueness of integrals in Hopf algebras. We also give the consequences for the representation theory of infinite dimensional algebras. We give an extensive class of examples which show that the results of the paper are the best possible. These examples are then used to give all the previously unknown connections between the various important classes of coalgebras appearing in literature."}
{"category": "Math", "title": "Circular thin position for knots in the 3-sphere", "abstract": "A regular circle-valued Morse function on the knot complement C(K) = S^3\\K is a function f from C(K) to S^1 which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle decomposition on the knot exterior E(K) = S^3\\N (K), with the property that every regular level surface contains a Seifert surface for the knot. We rearrange the handles in such a way that the regular surfaces are as simple as possible. To make this precise the concept of circular width for E(K) is introduced. When E(K) is endowed with a handle decomposition which realizes the circular width we will say that the knot K is in circular thin position. We use this to show that many knots have more than one non-isotopic incompressible Seifert surface. We also analyze the behavior of the circular width under some knot operations."}
{"category": "Math", "title": "Isometry groups of proper CAT(0)-spaces", "abstract": "Let G be a closed subgroup of the isometry group of a proper CAT(0)-space X. We show that if G is non-elementary and contains a rank-one element then its second bounded cohomology group with coefficients in the regular representation is non-trivial. As a consequence, up to passing to an open subgroup of finite index, either G is a compact extension of a totally disconnected group or G is a compact extension of a simple Lie group of rank one."}
{"category": "Math", "title": "A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups", "abstract": "We define the superclasses for a classical finite unipotent group $U$ of type $B_{n}(q)$, $C_{n}(q)$, or $D_{n}(q)$, and show that, together with the supercharacters defined in a previous paper, they form a supercharacter theory. In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As as consequence, we (re)obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we define the supercharacter table of $U$, and prove various orthogonality relations for supercharacters (similar to the well-known orthogonality relations for irreducible characters)."}
{"category": "Math", "title": "Compact C*-quantum groupoids", "abstract": "We propose a definition of compact quantum groupoids in the setting of C*-algebras, associate to such a quantum groupoid a regular C*-pseudo-multiplicative unitary, and use this unitary to construct a dual Hopf C*-bimodule and to pass to a measurable quantum groupoid in the sense of Enock and Lesieur. Moreover, we discuss examples related to compact and to \\'etale groupoids and study principal compact C*-quantum groupoids."}
{"category": "Math", "title": "Analytic continuation and embeddings in weighted backward shift invariant subspaces", "abstract": "By a famous result, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation a.e. on $\\T$. More can be said if the spectrum of the associated inner function has holes on $\\T$. Then the functions of the invariant subspaces even extend analytically through these holes. We will discuss the situation in weighted backward shift invariant subspaces. The results on analytic continuation will be applied to consider some embeddings of weighted invariant subspaces into their unweighted companions. Such weighted versions of invariant subspaces appear naturally in the context of Toeplitz operators. A connection between the spectrum of the inner function and the approximate point spectrum of the backward shift in the weighted situation is established in the spirit of results by Aleman, Richter and Ross."}
{"category": "Math", "title": "Rank-one isometries of proper CAT(0)-spaces", "abstract": "Let G be a non-elementary group of isometries of a proper CAT(0)-space with limit set L. We survey properties of the action of G on L under the assumption that G contains a rank-one element. Among others, we show that there is a dense orbit for the action of G on the complement of the diagonal in LxL and that pairs of fixed points of rank-one elements are dense in the complement of the diagonal of LxL."}
{"category": "Math", "title": "Symplectic geometry of semisimple orbits", "abstract": "We prove that any coadjoint orbit with real eigenvalues of a complex semisimple Lie group, equipped with the real part of the canonical holomorphic symplectic form, is symplectomorphic to the cotangent bundle of a (partial) flag manifold. Moreover, we generalize this result to hyperbolic orbits in a real semisimple Lie algebra."}
{"category": "Math", "title": "Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains", "abstract": "We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $\\phi (\\Omega)$ parametrized by Lipschitz homeomorphisms $\\phi $ defined on a fixed reference domain $\\Omega$. Given two open sets $\\phi (\\Omega)$, $\\tilde \\phi (\\Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\\|\\tilde \\phi -\\phi \\|_{W^{1,p}(\\Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem."}
{"category": "Math", "title": "Comparaison des param\\`etres de Langlands et des exposants \\`a l'int\\'erieur d'un paquet d'Arthur", "abstract": "In this paper, one proves an idea expressed by Clozel: inside an Arthur's packet, one has the representations in the Langlands' packet inside the Arthur's packet and more tempered representations than these representations"}
{"category": "Math", "title": "Alexey Vasilyevich Pogorelov, the mathematician of an incredible power", "abstract": "Life and the mathematical legacy of the great mathematician A.V. Pogorelov."}
{"category": "Math", "title": "Leaf-wise intersections and Rabinowitz Floer homology", "abstract": "In this article we explain how critical points of a particular perturbation of the Rabinowitz action functional give rise to leaf-wise intersection points in hypersurfaces of restricted contact type. This is used to derive existence and multiplicity results for leaf-wise intersection points in hypersurfaces of restricted contact type in general exact symplectic manifolds. The notion of leaf-wise intersection points was introduced by Moser."}
{"category": "Math", "title": "Contributions to the Geometric and Ergodic Theory of Conservative Flows", "abstract": "We prove the following dichotomy for vector fields in a C1-residual subset of volume-preserving flows: for Lebesgue almost every point all Lyapunov exponents equal to zero or its orbit has a dominated splitting. As a consequence if we have a vector field in this residual that cannot be C1-approximated by a vector field having elliptic periodic orbits, then, there exists a full measure set such that every orbit of this set admits a dominated splitting for the linear Poincare flow. Moreover, we prove that a volume-preserving and C1-stably ergodic flow can be C1-approximated by another volume-preserving flow which is non-uniformly hyperbolic."}
{"category": "Math", "title": "Virtual Crossing Number and the Arrow Polynomial", "abstract": "We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus."}
{"category": "Math", "title": "A family of determinants associated with a square matrix", "abstract": "We associate with a matrix over an arbitrary field an infinite family of matrices whose sizes vary from one to infinity; their entries are traces of powers of the original matrix. We explicitly evaluate the determinants of matrices in our family. The work is motivated by applications to graph spectra."}
{"category": "Math", "title": "The KdV/KP-I limit of the Nonlinear Schrodinger equation", "abstract": "We justify rigorously the convergence of the amplitude of solutions of Nonlinear-Schr\\\"odinger type Equations with non zero limit at infinity to an asymptotic regime governed by the Korteweg-de Vries equation in dimension 1 and the Kadomtsev-Petviashvili I equation in dimensions 2 and more. We get two types of results. In the one-dimensional case, we prove directly by energy bounds that there is no vortex formation for the global solution of the NLS equation in the energy space and deduce from this the convergence towards the unique solution in the energy space of the KdV equation. In arbitrary dimensions, we use an hydrodynamic reformulation of NLS and recast the problem as a singular limit for an hyperbolic system. We thus prove that smooth $H^s$ solutions exist on a time interval independent of the small parameter. We then pass to the limit by a compactness argument and obtain the KdV/KP-I equation."}
{"category": "Math", "title": "Asymptotic Vassiliev Invariants for Vector Fields", "abstract": "We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of $\\R^3$. More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field. As an application, we determine the asymptotic Alexander and Jones polynomials and give a formula for the asymptotic Kontsevich integral."}
{"category": "Math", "title": "The space of volume forms", "abstract": "S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its perturbed equation and prove that the space of volume forms is an infinite dimensional non-positively curved metric space in the sense of Alexandrov."}
{"category": "Math", "title": "Corrigendum to \"Approximation by C^{p}-smooth, Lipschitz functions on Banach spaces\" [J. Math. Anal. Appl., 315 (2006), 599-605]", "abstract": "In this erratum, we recover the results from an earlier paper of the author's which contained a gap. Specifically, we prove that if X is a Banach space with an unconditional basis and admits a C^{p}-smooth, Lipschitz bump function, and Y is a convex subset of X, then any uniformly continuous function f: Y->R can be uniformly approximated by Lipschitz, C^{p}-smooth functions K:X->R. Also, if Z is any Banach space and f:X->Z is L-Lipschitz, then the approximates K:X->Z can be chosen CL-Lipschitz and C^{p}-smooth, for some constant C depending only on X."}
{"category": "Math", "title": "A note on sigular time of mean curvature flow", "abstract": "We show that mean curvature flow of a compact submanifold in a complete Riemannian manifold cannot form singularity at time infinity if the ambient Riemannian manifold has bounded geometry and satisfies certain curvature and volume growth conditions ."}
{"category": "Math", "title": "Infinitesimal adjunction and polar curves", "abstract": "The polar curves of foliations $\\mathcal F$ having a curve $C$ of separatrices generalize the classical polar curves associated to hamiltonian foliations of $C$. As in the classical theory, the equisingularity type ${\\wp}({\\mathcal F})$ of a generic polar curve depends on the analytical type of ${\\mathcal F}$, and hence of $C$. In this paper we find the equisingularity types $\\epsilon (C)$ of $C$, that we call kind singularities, such that ${\\wp}({\\mathcal F})$ is completely determined by $\\epsilon (C)$ for Zariski-general foliations $\\mathcal F$. Our proofs are mainly based on the adjunction properties of the polar curves. The foliation-like framework is necessary, otherwise we do not get the right concept of general foliation in Zariski sense and, as we show by examples, the hamiltonian case can be out of the set of general foliations."}
{"category": "Math", "title": "Quaternionic contact manifolds with a closed fundamental 4-form", "abstract": "We show that the fundamental 4-form on a quaternionic contact manifold of dimension at least eleven is closed if and only if the torsion endomorphism of the Biquard connection vanishes. This condition characterizes quaternionic contact structures which are locally qc homothetic to 3-Sasakian structures."}
{"category": "Math", "title": "Cycle factorizations and one-faced graph embeddings", "abstract": "Consider factorizations into transpositions of an n-cycle in the symmetric group S_n. To every such factorization we assign a monomial in variables w_{ij} that retains the transpositions used, but forgets their order. Summing over all possible factorizations of n-cycles we obtain a polynomial that happens to admit a closed expression. From this expression we deduce a formula for the number of 1-faced embeddings of a given graph."}
{"category": "Math", "title": "On continuous choice of retractions onto nonconvex subsets", "abstract": "For a Banach space $B$ and for a class $\\A$ of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements $A \\in \\A$ can be chosen to depend continuously on $A$, whenever nonconvexity of each $A \\in \\A$ is less than $\\f{1}{2}$. The key geometric argument is that the set of all uniform retractions onto an $\\a-$paraconvex set (in the spirit of E. Michael) is $\\frac{\\a}{1-\\a}-$paraconvex subset in the space of continuous mappings of $B$ into itself. For a Hilbert space $H$ the estimate $\\frac{\\a}{1-\\a}$ can be improved to $\\frac{\\a (1+\\a^{2})}{1-\\a^{2}}$ and the constant $\\f{1}{2}$ can be reduced to the root of the equation $\\a+ \\a^{2}+a^{3}=1$."}
{"category": "Math", "title": "On the homotopy classification of maps", "abstract": "We establish certain conditions which imply that a map $f:X\\to Y$ of topological spaces is null homotopic when the induced integral cohomology homomorphism is trivial; one of them is: $H^*(X)$ and $\\pi_*(Y)$ have no torsion and $H^*(Y)$ is polynomial."}
{"category": "Math", "title": "Structurally damped plate and wave equations with random point force in arbitrary space dimensions", "abstract": "In this paper we consider structurally damped plate and wave equations with point and distributed random forces. In order to treat space dimensions more than one, we work in the setting of $L^q$--spaces with (possibly small) $q\\in(1,2)$. We establish existence, uniqueness and regularity of mild and weak solutions to the stochastic equations employing recent theory for stochastic evolution equations in UMD Banach spaces."}
{"category": "Math", "title": "Approximation by Lipschitz, C^{p} smooth functions on weakly compactly generated Banach spaces", "abstract": "The following result was announced in the earlier version(s) of this paper: On weakly compactly generated Banach spaces which admit a Lipschitz, C^{p} smooth bump function, one can uniformly approximate uniformly continuous, bounded, real-valued functions by Lipschitz, C^{p} smooth functions. Unfortunately, there is a gap in the proof which renders the proof in its present form incorrect"}
{"category": "Math", "title": "General Resolvents for Monotone Operators: Characterization and Extension", "abstract": "Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the resolvents of monotone operators. For example, the proximal mappings in the sense of Moreau are precisely the resolvents of subdifferential operators. More general notions of \"resolvent\", \"proximal mapping\" and \"firmly nonexpansive\" have been studied. One important class, popularized chiefly by Alber and by Kohsaka and Takahashi, is based on the normalized duality mapping. Furthermore, Censor and Lent pioneered the use of the gradient of a well behaved convex functions in a Bregman-distance based framework. It is known that resolvents are firmly nonexpansive, but the converse has been an open problem for the latter framework. In this note, we build on the very recent characterization of maximal monotonicity due to Martinez-Legaz to provide a framework for studying resolvents in which firmly nonexpansive mappings are always resolvents. This framework includes classical resolvents, resolvents based on the normalized duality mapping, resolvents based on Bregman distances, and even resolvents based on (nonsymmetric) rotators. As a by-product of recent work on the proximal average, we obtain a constructive Kirszbraun-Valentine extension result for generalized firmly nonexpansive mappings. Several examples illustrate our results."}
{"category": "Math", "title": "The Radial Masa in a Free Group Factor is Maximal Injective", "abstract": "The radial (or Laplacian) masa in a free group factor is the abelian von Neumann algebra generated by the sum of the generators (of the free group) and their inverses. The main result of this paper is that the radial masa is a maximal injective von Neumann subalgebra of a free group factor. We also investigate tensor products of maximal injective algebras. Given two inclusions $B_i\\subset M_i$ of type $\\mathrm{I}$ von Neumann algebras in finite von Neumann algebras such that each $B_i$ is maximal injective in $M_i$, we show that the tensor product $B_1\\ \\bar{\\otimes}\\ B_2$ is maximal injective in $M_1\\ \\bar{\\otimes}\\ M_2$ provided at least one of the inclusions satisfies the asymptotic orthogonality property we establish for the radial masa. In particular it follows that finite tensor products of generator and radial masas will be maximal injective in the corresponding tensor product of free group factors."}
{"category": "Math", "title": "Metric properties of higher-dimensional Thompson's groups", "abstract": "Higher-dimensional Thompson's groups nV are finitely presented groups described by Brin which generalize dyadic self-maps of the unit interval to dyadic self-maps of n-dimensional unit cubes. We describe some of the metric properties of higher-dimensional Thompson's groups. We give descriptions of elements based upon tree-pair diagrams and give upper and lower bounds for word length in terms of the size of the diagrams. Though these upper and lower bounds are somewhat separated, we show that there are elements realizing the lower bounds and that the fraction of elements which are close to the upper bound converges to 1, showing that the bounds are optimal and that the upper bound is generically achieved."}
{"category": "Math", "title": "Report on \"Geometry and representation theory of tensors for computer science, statistics and other areas.\"", "abstract": "This is a technical report on the proceedings of the workshop held July 21 to July 25, 2008 at the American Institute of Mathematics, Palo Alto, California, organized by Joseph Landsberg, Lek-Heng Lim, Jason Morton, and Jerzy Weyman. We include a list of open problems coming from applications in 4 different areas: signal processing, the Mulmuley-Sohoni approach to P vs. NP, matchgates and holographic algorithms, and entanglement and quantum information theory. We emphasize the interactions between geometry and representation theory and these applied areas."}
{"category": "Math", "title": "Multistage Hypothesis Tests for the Mean of a Normal Distribution", "abstract": "In this paper, we have developed new multistage tests which guarantee prescribed level of power and are more efficient than previous tests in terms of average sampling number and the number of sampling operations. Without truncation, the maximum sampling numbers of our testing plans are absolutely bounded. Based on geometrical arguments, we have derived extremely tight bounds for the operating characteristic function. To reduce the computational complexity for the relevant integrals, we propose adaptive scanning algorithms which are not only useful for present hypothesis testing problem but also for other problem areas."}
{"category": "Math", "title": "Matroid polytopes and their volumes", "abstract": "We express the matroid polytope $P_M$ of a matroid $M$ as a signed Minkowski sum of simplices, and obtain a formula for the volume of $P_M$. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian $Gr_{k,n}$. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of $M$. Our proofs are based on a natural extension of Postnikov's theory of generalized permutohedra."}
{"category": "Math", "title": "Orbit equivalence for Cantor minimal Z^d-systems", "abstract": "We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and Z^d-actions."}
{"category": "Math", "title": "On injectivity of quasiregular mappings", "abstract": "We give sufficient conditions for a planar quasiregular mapping to be injective in terms of the range of the differential matrix."}
{"category": "Math", "title": "Hyperbolic structures on closed spacelike manifolds", "abstract": "In this paper, we study the intrinsic mean curvature flow on certain closed spacelike manifolds, and prove the existence of hyperbolic structures on them."}
{"category": "Math", "title": "Cofiniteness and coassociated primes of local cohomology modules", "abstract": "Let $R$ be a noetherian ring, $\\fa$ an ideal of $R$ such that $\\dim R/\\fa=1$ and $M$ a finite $R$--module. We will study cofiniteness and some other properties of the local cohomology modules $\\lc^{i}_{\\fa}(M)$. For an arbitrary ideal $\\fa$ and an $R$--module $M$ (not necessarily finite), we will characterize $\\fa$--cofinite artinian local cohomology modules. Certain sets of coassociated primes of top local cohomology modules over local rings are characterized."}
{"category": "Math", "title": "Design of Cooperative Processes in a Customer-Supplier Relationship: An Approach Based on Simulation and Decision Theory", "abstract": "Performance improvement in supply chains, taking into account customer demand in the tactical planning process is essential. It is more and more difficult for the customers to ensure a certain level of demand over a medium term horizon as their own customers ask them for personalisation and fast adaptation. It is thus necessary to develop methods and decision support systems to reconcile the order and book processes. In this context, this paper intends firstly to relate decision under uncertainty and the industrial point of view based on the notion of risk management. This serves as a basis for the definition of an approach based on simulation and decision theory that is dedicated to the design of cooperative processes in a customer-supplier relationship. This approach includes the evaluation, in terms of risk, of different cooperative processes using a simulation-dedicated tool. The evaluation process is based on an exploitation of decision theory concepts and methods. The implementation of the approach is illustrated on an academic example typical of the aeronautics supply chain."}
{"category": "Math", "title": "Marginal likelihood for parallel series", "abstract": "Suppose that $k$ series, all having the same autocorrelation function, are observed in parallel at $n$ points in time or space. From a single series of moderate length, the autocorrelation parameter $\\beta$ can be estimated with limited accuracy, so we aim to increase the information by formulating a suitable model for the joint distribution of all series. Three Gaussian models of increasing complexity are considered, two of which assume that the series are independent. This paper studies the rate at which the information for $\\beta$ accumulates as $k$ increases, possibly even beyond $n$. The profile log likelihood for the model with $k(k+1)/2$ covariance parameters behaves anomalously in two respects. On the one hand, it is a log likelihood, so the derivatives satisfy the Bartlett identities. On the other hand, the Fisher information for $\\beta$ increases to a maximum at $k=n/2$, decreasing to zero for $k\\ge n$. In any parametric statistical model, one expects the Fisher information to increase with additional data; decreasing Fisher information is an anomaly demanding an explanation."}
{"category": "Math", "title": "Involutions of Iwahori-Hecke algebras and representations of fixed subalgebras", "abstract": "We establish branching rules between some Iwahori-Hecke algebra of type B and their subalgebras which are defined as fixed subalgebras by involutions including Goldman involution. The Iwahori-Hecke algebra of type D is one of such fixed subalgebras. We also obtain branching rules between those fixed subalgebras and their intersection subalgebra. We determine basic sets of irreducible representations of those fixed subalgebras and their intersection subalgebra by making use of generalized Clifford theory."}
{"category": "Math", "title": "The central limit theorem under random truncation", "abstract": "Under left truncation, data $(X_i,Y_i)$ are observed only when $Y_i\\le X_i$. Usually, the distribution function $F$ of the $X_i$ is the target of interest. In this paper, we study linear functionals $\\int\\varphi \\mathrm{d}F_n$ of the nonparametric maximum likelihood estimator (MLE) of $F$, the Lynden-Bell estimator $F_n$. A useful representation of $\\int \\varphi \\mathrm{d}F_n$ is derived which yields asymptotic normality under optimal moment conditions on the score function $\\varphi$. No continuity assumption on $F$ is required. As a by-product, we obtain the distributional convergence of the Lynden-Bell empirical process on the whole real line."}
{"category": "Math", "title": "Asymptotic normality and consistency of a two-stage generalized least squares estimator in the growth curve model", "abstract": "Let $\\mathbf{Y}=\\mathbf{X}\\bolds{\\Theta}\\mathbf{Z}'+\\bolds{\\mathcal {E}}$ be the growth curve model with $\\bolds{\\mathcal{E}}$ distributed with mean $\\mathbf{0}$ and covariance $\\mathbf{I}_n\\otimes\\bolds{\\Sigma}$, where $\\bolds{\\Theta}$, $\\bolds{\\Sigma}$ are unknown matrices of parameters and $\\mathbf{X}$, $\\mathbf{Z}$ are known matrices. For the estimable parametric transformation of the form $\\bolds {\\gamma}=\\mathbf{C}\\bolds{\\Theta}\\mathbf{D}'$ with given $\\mathbf{C}$ and $\\mathbf{D}$, the two-stage generalized least-squares estimator $\\hat{\\bolds \\gamma}(\\mathbf{Y})$ defined in (7) converges in probability to $\\bolds\\gamma$ as the sample size $n$ tends to infinity and, further, $\\sqrt{n}[\\hat{\\bolds{\\gamma}}(\\mathbf{Y})-\\bolds {\\gamma}]$ converges in distribution to the multivariate normal distribution $\\ma thcal{N}(\\mathbf{0},(\\mathbf{C}\\mathbf{R}^{-1}\\mathbf{C}')\\otimes(\\mat hbf{D}(\\mathbf{Z}'\\bolds{\\Sigma}^{-1}\\mathbf{Z})^{-1}\\mathbf{D}'))$ under the condition that $\\lim_{n\\to\\infty}\\mathbf{X}'\\mathbf{X}/n=\\mathbf{R}$ for some positive definite matrix $\\mathbf{R}$. Moreover, the unbiased and invariant quadratic estimator $\\hat{\\bolds{\\Sigma}}(\\mathbf{Y})$ defined in (6) is also proved to be consistent with the second-order parameter matrix $\\bolds{\\Sigma}$."}
{"category": "Math", "title": "Ball throwing on spheres", "abstract": "Ball throwing on Euclidean spaces has been considered for a while. A suitable renormalization leads to a fractional Brownian motion as limit object. In this paper we investigate ball throwing on spheres. A different behavior is exhibited: we still get a Gaussian limit but which is no longer a fractional Brownian motion. However the limit is locally self-similar when the self-similarity index $H$ is less than 1/2."}
{"category": "Math", "title": "On cubic Berwald spaces", "abstract": "We show that, for Finsler spaces with cubic metric, Landsberg spaces are Berwaldian. Also, for decomposable metrics, we determine specific conditions for a space with cubic metric to be of Berwald type, thus refining the result in [6]."}
{"category": "Math", "title": "Testing for changes in polynomial regression", "abstract": "We consider a nonlinear polynomial regression model in which we wish to test the null hypothesis of structural stability in the regression parameters against the alternative of a break at an unknown time. We derive the extreme value distribution of a maximum-type test statistic which is asymptotically equivalent to the maximally selected likelihood ratio. The resulting test is easy to apply and has good size and power, even in small samples."}
{"category": "Math", "title": "On the parity of generalized partition functions III", "abstract": "Improving on some results of J.-L. Nicolas \\cite {Ndeb}, the elements of the set ${\\cal A}={\\cal A}(1+z+z^3+z^4+z^5)$, for which the partition function $p({\\cal A},n)$ (i.e. the number of partitions of $n$ with parts in ${\\cal A}$) is even for all $n\\geq 6$ are determined. An asymptotic estimate to the counting function of this set is also given."}
{"category": "Math", "title": "Upper bound for the Lempert function of smooth domains", "abstract": "An upper estimate for the Lempert function of any $C^{1+\\epsilon}$-smooth bounded domain in $\\Bbb C^n$ is found in terms of the boundary distance."}
{"category": "Math", "title": "Fibonacci Modules and Multiple Fibonacci Sequences", "abstract": "Double Fibonacci sequences are introduced and they are related to operations with Fibonacci modules. Generalizations and examples are also discussed."}
{"category": "Math", "title": "Solving differential equations", "abstract": "The theme of this paper is to `solve' an absolutely irreducible differential module explicitly in terms of modules of lower dimension and finite extensions of the differential field $K$. Representations of semi-simple Lie algebras and differential Galois theory are the main tools. The results extend the classical work of G. Fano."}
{"category": "Math", "title": "Notes on Calabi-Yau ordinary differential equations", "abstract": "We investigate the structures of Calabi-Yau differential equations and the relations to the arithmetic of the pencils of Calabi-Yau varieties behind the equations. This provides explanations of some observations and computations in a recent paper by Samol and van Straten."}
{"category": "Math", "title": "Fibred torti-rational knots", "abstract": "A torti-rational knot, denoted by K(2a,b|r), is a knot obtained from the 2-bridge link B(2a,b) by applying Dehn twists an arbitrary number of times, r, along one component of B(2a,b). We determine the genus of K(2a,b|r) and solve a question of when K(2a,b|r) is fibred. In most cases, the Alexander polynomials determine the genus and fibredness of these knots. We develop both algebraic and geometric techniques to describe the genus and fibredness by means of continued fraction expansions of b/2a. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same question for the satellite knots of tunnel number one."}
{"category": "Math", "title": "Asymptotically CAT(0) Groups", "abstract": "We study the general theory of asymptotically CAT(0) groups, explaining why such a group has finitely many conjugacy classes of finite subgroups, is $F_\\infty$ and has solvable word problem. We provide techniques to combine asymptotically CAT(0) groups via direct products, amalgams and HNN extensions. The universal cover of the Lie group $PSL(2,\\mathbb{R})$ is shown to be an asymptotically CAT(0) metric space. Therefore, co-compact lattices in $\\widetilde{PSL(2,\\mathbb{R})}$ provide the first examples of asymptotically CAT(0) groups which are neither CAT(0) nor hyperbolic. Another source of examples is shown to be the class of relatively hyperbolic groups."}
{"category": "Math", "title": "Totally Null Surfaces in Neutral Kaehler 4-Manifolds", "abstract": "We study the totally null surfaces of the neutral Kaehler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual ($\\alpha$-planes) or anti-self-dual ($\\beta$-planes) and so we consider $\\alpha$-surfaces and $\\beta$-surfaces. The metric of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the $\\alpha$-planes are integrable and $\\alpha$-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The $\\beta$-surfaces are less known and our interest is mainly in their description. In particular, we classify the $\\beta$-surfaces of the neutral Kaehler metric on $TN$, the tangent bundle to a Riemannian 2-manifold $N$. These include the spaces of oriented geodesics in Euclidean and Lorentz 3-space, for which we show that the $\\beta$-surfaces are affine tangent bundles to curves of constant geodesic curvature on $S^2$ and $H^2$, respectively. In addition, we construct the $\\beta$-surfaces of the space of oriented geodesics of hyperbolic 3-space."}
{"category": "Math", "title": "Isoperimetric functions for subdirect products and Bestvina-Brady groups", "abstract": "In this thesis we investigate the Dehn functions of two different classes of groups: subdirect products, in particular subdirect products of limit groups; and Bestvina-Brady groups. Let D = \\Gamma_1 \\times ... \\times \\Gamma_n be a direct product of n \\geq 3 finitely presented groups and let H be a subgroup of D. Suppose that each \\Gamma_i contains a finite index subgroup \\Gamma_i' \\leq \\Gamma_i such that the commutator subgroup [D', D'] of D' = \\Gamma_1' \\times ... \\times \\Gamma_n' is contained in H. Suppose furthermore that, for each i, the subgroup \\Gamma_i H has finite index in D. We prove that H is finitely presented and satisfies an isoperimetric inequality given in terms of area-radius pairs for the \\Gamma_i and the dimension of (D'/H) \\otimes \\Q. In the case that each \\Gamma_i admits a polynomial-polynomial area-radius pair, it will follow that H satisfies a polynomial isoperimetric inequality. As a corollary we obtain that if K is a subgroup of a direct product of n limit groups and if K is of type FP_m(\\Q), where m = \\max {2, n-1}, then K is finitely presented and satisfies a polynomial isoperimetric inequality. In particular, we obtain that all finitely presented subgroups of a direct product of at most 3 limit groups satisfy a polynomial isoperimetric inequality. We also prove that if B is a finitely presented Bestvina-Brady group, then B admits a quartic isoperimetric function."}
{"category": "Math", "title": "A measure-theoretic approach to the theory of dense hypergraphs", "abstract": "In this paper we develop a measure-theoretic method to treat problems in hypergraph theory. Our central theorem is a correspondence principle between three objects: An increasing hypergraph sequence, a measurable set in an ultraproduct space and a measurable set in a finite dimensional Lebesgue space. Using this correspondence principle we build up the theory of dense hypergraphs from scratch. Along these lines we give new proofs for the Hypergraph Removal Lemma, the Hypergraph Regularity Lemma, the Counting Lemma and the Testability of Hereditary Hypergraph Properties. We prove various new results including a strengthening of the Regularity Lemma and an Inverse Counting Lemma. We also prove the equivalence of various notions for convergence of hypergraphs and we construct limit objects for such sequences. We prove that the limit objects are unique up to a certain family of measure preserving transformations. As our main tool we study the integral and measure theory on the ultraproduct of finite measure spaces which is interesting on its own right."}
{"category": "Math", "title": "Model theoretic forcing in analysis", "abstract": "We present a framework for model theoretic forcing in a non-first-order context, and present some applications of this framework to Banach space theory."}
{"category": "Math", "title": "Discrete approximation of the free Fock space", "abstract": "We prove that the free Fock space ${\\F}(\\R^+;\\C)$, which is very commonly used in Free Probability Theory, is the continuous free product of copies of the space $\\C^2$. We describe an explicit embedding and approximation of this continuous free product structure by means of a discrete-time approximation: the free toy Fock space, a countable free product of copies of $\\C^2$. We show that the basic creation, annihilation and gauge operators of the free Fock space are also limits of elementary operators on the free toy Fock space. When applying these constructions and results to the probabilistic interpretations of these spaces, we recover some discrete approximations of the semi-circular Brownian motion and of the free Poisson process. All these results are also extended to the higher multiplicity case, that is, ${\\F}(\\R^+;\\C^N)$ is the continuous free product of copies of the space $\\C^{N+1}$."}
{"category": "Math", "title": "Data volume and power of multiple tests with small sample size per null", "abstract": "In multiple hypothesis testing, the volume of data, defined as the number of replications per null times the total number of nulls, usually defines the amount of resource required. On the other hand, power is an important measure of performance for multiple testing. Due to practical constraints, the number of replications per null may not be large enough in terms of the difference between false and true nulls. For the case where the population fraction of false nulls is constant, we show that, as the difference between false and true nulls becomes increasingly subtle while the number of replications per null cannot increase fast enough, (1) in order to have enough chance that the data to be collected will yield some trustworthy findings, as measured by a conditional version of the positive false discovery rate (pFDR), the volume of data has to grow at a rate much faster than in the case where the number of replications per null can be large enough, and (2) in order to control the pFDR asymptotically, power has to decay to 0 in a rate highly sensitive to rejection criterion and there is no asymptotically most powerful procedures among those that control the pFDR asymptotically at the same level."}
{"category": "Math", "title": "Strong Convergence towards self-similarity for one-dimensional dissipative Maxwell models", "abstract": "We prove the propagation of regularity, uniformly in time, for the scaled solutions of one-dimensional dissipative Maxwell models. This result together with the weak convergence towards the stationary state proven by Pareschi and Toscani in 2006 implies the strong convergence in Sobolev norms and in the L^1 norm towards it depending on the regularity of the initial data. In the case of the one-dimensional inelastic Boltzmann equation, the result does not depend of the degree of inelasticity. This generalizes a recent result of Carlen, Carrillo and Carvalho (arXiv:0805.1051v1), in which, for weak inelasticity, propagation of regularity for the scaled inelastic Boltzmann equation was found by means of a precise control of the growth of the Fisher information."}
{"category": "Math", "title": "On finite Thurston type orderings of braid groups", "abstract": "We prove that for any finite Thurston-type ordering $<_{T}$ on the braid group\\ $B_{n}$, the restriction to the positive braid monoid $(B_{n}^{+},<_{T})$ is a\\ well-ordered set of order type $\\omega^{\\omega^{n-2}}$. The proof uses a combi\\ natorial description of the ordering $<_{T}$. Our combinatorial description is \\ based on a new normal form for positive braids which we call the $\\C$-normal fo\\ rm. It can be seen as a generalization of Burckel's normal form and Dehornoy's \\ $\\Phi$-normal form (alternating normal form)."}
{"category": "Math", "title": "SU(r,L) is separably unirational", "abstract": "We show that the moduli space of SU_X(r,L) of rank r bundles of fixed determinant L on a smooth projective curve X is separably unirational."}
{"category": "Math", "title": "Projections in several complex variables", "abstract": "This work consists two parts. In the first part, we completely study the heat equation method of Menikoff-Sjostrand and apply it to the Kohn Laplacian defined on a compact orientable connected CR manifold. We then get the full asymptotic expansion of the Szego projection for (0,q) forms when the Levi formis nondegenerate. This generalizes a result of Boutet de Monvel and Sjostrand for (0,0) forms. Our main tool is Fourier integral operators with complex valued phase functions of Melin and Sjostrand. In the second part, we obtain the full asymptotic expansion of the Bergman projection for (0,q) forms when the Levi form is non-degenerate. This also generalizes a result of Boutet de Monvel and Sjostrand for (0,0) forms. We introduce a new operator analogous to the Kohn Laplacian defined on the boundary of a domain and we apply the heat equation method of Menikoff and Sjostrand to this operator. We obtain a description of a new Szego projection up to smoothing operators. Finally, by using the Poisson operator, we get our main result."}
{"category": "Math", "title": "On perturbations of Hilbert spaces and probability algebras with a generic automorphism", "abstract": "We prove that $IHS_A$, the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is $\\aleph_0$-stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set. Similarly, $APr_A$, the theory of atomless probability algebras equipped with a generic automorphism is $\\aleph_0$-stable up to perturbation. However, not allowing perturbation it is not even superstable."}
{"category": "Math", "title": "Stability and stable groups in continuous logic", "abstract": "We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity."}
{"category": "Math", "title": "Hyperoctahedral species", "abstract": "We introduce a new definition for the species of type B, or H-species, analog to the classical species (of type A), but on which we consider the action of the groups Bn of signed permutations. We are interested in algebraic structure on these H-species and give examples of Hopf monoids. The natural way to get a graded vector space from a species, given in this paper in terms of functors, will allow us to deepen our understanding of these species. In particular, the image of the classical species of set compositions under a given functor is isomorphic to the combinatorial Hopf algebra DQSym."}
{"category": "Math", "title": "Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras", "abstract": "We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C*-algebra, an Exel-Laca algebra, and an ultragraph C*-algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C*-algebras and Exel-Laca algebras, and that all simple AF-algebras are either graph C*-algebras or Exel-Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C*-algebra of a row-finite graph with no sinks."}
{"category": "Math", "title": "On the oscillation properties of eigenfunctions of Sturm--Liouville problem with singular coefficients", "abstract": "In the paper we consider singular spectral Sturm--Liouville problem $-(py')'+(q-\\lambda r)y=0$, $(U-1)y^{\\vee}+i(U+1)y^{\\wedge}=0$, where function $p\\in L_{\\infty}[0,1]$ is uniformly positive, generalized functions $q,r\\in W_2^{-1}[0,1]$ are real-valued and unitary matrix $U\\in\\mathbb C^{2\\times 2}$ is diagonal. The main goal is to prove that well-known (for smooth case) facts about number and distribution of zeros of eigenfunctions hold in general case."}
{"category": "Math", "title": "On the implementation of exponential methods for semilinear parabolic equations", "abstract": "The time integration of semilinear parabolic problems by exponential methods of different kinds is considered. A new algorithm for the implementation of these methods is proposed. The algorithm evaluates the operators required by the exponential methods by means of a quadrature formula that converges like $O(e^{-cK/\\ln K})$, with $K$ the number of quadrature nodes. The algorithm allows also the evaluation of the associated scalar mappings and in this case the quadrature converges like $O(e^{-cK})$. The technique is based on the numerical inversion of sectorial Laplace transforms. Several numerical illustrations are provided to test the algorithm."}
{"category": "Math", "title": "Elementary combinatorics of the HOMFLYPT polynomial", "abstract": "We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3."}
{"category": "Math", "title": "Nonexpansive iterations in uniformly convex $W$-hyperbolic spaces", "abstract": "We propose the class of uniformly convex $W$-hyperbolic spaces with monotone modulus of uniform convexity ($UCW$-hyperbolic spaces for short) as an appropriate setting for the study of nonexpansive iterations. $UCW$-hyperbolic spaces are a natural generalization both of uniformly convex normed spaces and CAT(0)-spaces. Furthermore, we apply proof mining techniques to get effective rates of asymptotic regularity for Ishikawa iterations of nonexpansive self-mappings of closed convex subsets in $UCW$-hyperbolic spaces. These effective results are new even for uniformly convex Banach spaces."}
{"category": "Math", "title": "Algebraic properties of edge ideals via combinatorial topology", "abstract": "We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature regarding linearity, Betti numbers, and (sequentially) Cohen-Macaulay properties of edges ideals associated to chordal, complements of chordal, and Ferrers graphs, as well as trees and forests. Our approach unifies (and in many cases strengthens) these results and also provides combinatorial/enumerative interpretations of certain algebraic properties. We apply our setup to obtain new results regarding algebraic properties of edge ideals in the context of local changes to a graph (adding whiskers and ears) as well as bounded vertex degree. These methods also lead to recursive relations among certain generating functions of Betti numbers which we use to establish new formulas for the projective dimension of edge ideals. We use only well-known tools from combinatorial topology along the lines of independence complexes of graphs, (not necessarily pure) vertex decomposability, shellability, etc."}
{"category": "Math", "title": "Counting integral points on universal torsors", "abstract": "Manin's conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of integral points in certain regions on universal torsors. As an application, we prove Manin's conjecture for a singular quartic del Pezzo surface."}
{"category": "Math", "title": "The Donaldson equation", "abstract": "In this short note, we solve a Dirichlet problem for a fully nonlinear elliptic equation. The operator is introduced by S. Donaldson and it is relevant to the geometry of the space of volume forms."}
{"category": "Math", "title": "The cohomology of line bundles of splice-quotient singularities", "abstract": "We provide several results on splice-quotient singularities: a combinatorial expression of the dimension of the first cohomology of all `natural' line bundles, an equivariant Campillo-Delgado-Gusein-Zade type formula about the dimension of relative sections of line bundles (proving that the equivariant, divisorial multi-variable Hilbert series is topological), a combinatorial description of divisors of analytic function-germs, and an expression for the multiplicity of the singularity from its resolution graph. Additional, we establish a new formula for the Seiberg-Witten invariants of any rational homology sphere singularity link."}
{"category": "Math", "title": "Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions", "abstract": "Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized $L^1\\cap L^p\\to L^p$ stability for all $p \\ge 2$ and dimensions $d \\ge 1$ and nonlinear $L^1\\cap H^s\\to L^p\\cap H^s$ stability and $L^2$-asymptotic behavior for $p\\ge 2$ and $d\\ge 3$. The behavior can in general be rather complicated, involving both convective (i.e., wave-like) and diffusive effects."}
{"category": "Math", "title": "The Best Constant, the Nonexistence of Extremal Functions and Related Results for an Improved Hardy-Sobolev Inequality", "abstract": "We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\\mathbb{R}^N$. We also discuss the connection of the related functional spaces and as a result we obtain some Caffarelli - Kohn - Nirenberg inequalities. Our starting point is the existence of a minimizer for the Bliss' inequality and the indirect dependence of the Hardy inequality at the origin."}
{"category": "Math", "title": "Obstructions to the existence of monotone Lagrangian embeddings into cotangent bundles of manifolds fibered over the circle", "abstract": "We extend constructions and results of Damian to get topological obstructions to the existence of closed monotone Lagrangian embeddings into the cotangent bundle of a space which is the total space of a fibration over the circle."}
{"category": "Math", "title": "A Natural Connection on (2,3) Sub-Riemannian Manifolds", "abstract": "We build an analogue for the Levi-Civita connection on Riemannian manifolds for sub-Riemannian manfiolds modeled on the Heisenberg group. We demonstrate some geometric properties of this connection to justify our choice and show that this connection is unique in having these properties."}
{"category": "Math", "title": "From Knothe's transport to Brenier's map and a continuation method for optimal transport", "abstract": "A simple procedure to map two probability measures in $\\mathbb{R}^d$ is the so-called \\emph{Knothe-Rosenblatt rearrangement}, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem."}
{"category": "Math", "title": "On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces with too many lines", "abstract": "We show that the Debarre-de Jong conjecture that the Fano scheme of lines on a smooth hypersurface of degree at most n in n-dimensional projective space must have its expected dimension, and the Beheshti-Starr conjecture that bounds the dimension of the Fano scheme of lines for hypersurfaces of degree at least n in n-dimensional projective space, reduce to determining if the intersection of the top Chern classes of certain vector bundles is nonzero."}
{"category": "Math", "title": "Models and van Kampen theorems for directed homotopy theory", "abstract": "We study topological spaces with a distinguished set of paths, called directed paths. Since these directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, they do assemble into a category, called the fundamental category. We define models of the fundamental category, such as the fundamental bipartite graph, and minimal extremal models which are shown to generalize the fundamental group. In addition, we prove van Kampen theorems for subcategories, retracts, and models of the fundamental category."}
{"category": "Math", "title": "Etude de deux classes de groupes nilpotents de pas deux", "abstract": "The aim of my PhD work is to study the $L^p$-boundedness of operators on two classes of two-step nilpotent Lie groups, using Plancherel formulas and spherical functions as tools. The first class of groups consists of the groups of Heisenberg type, and the second, of the two-step free nilpotent Lie groups (denoted $N_{v,2}$ for $v$ generators). In the latter case, we develop a radial Fourier calculus. Our study has focused on the maximal functions associated with Kor\\'anyi spheres, together with their square functions, and the convolution operator defined with the radial Fourier calculus on the two-step free nilpotent Lie group (radial Fourier multipliers problem). In fact, one chapter of this work is devoted to the proof of $L^p$-inequalities for the maximal spherical function on the two considered classes of groups. Our method is based on interpolation for the same operator family as in the euclidean case, on $L^p$-boundedness for the standard maximal function, and $L^2$-inequalities for square functions. These $L^2$-inequalities are based on Plancherel formula and on the properties of bounded spherical functions for the orthogonal group. On $N_{v,2}$, we construct the bounded spherical functions using representations of the semidirect product of $N_{v,2}$ with the orthogonal group. We also obtain some properties of the Kohn sublaplacian and the radial Plancherel measure. Then we present a first study of the radial Fourier multiplier problem, with the aim of giving our solutions for some technicals difficulties."}
{"category": "Math", "title": "Correlators and Descendants of Subcritical Stein Manifolds", "abstract": "We determine contact homology algebra of a subcritical Stein-fillable contact manifold whose first Chern class vanishes. We also compute the genus-0 one point correlators and gravitational descendants of compactly supported closed forms of their subcritical Stein fillings. This is a step towards determining the full potential function of the filling as defined in \\cite{EliashbergGiventalHofer}. These invariants also give a canonical presentation of the cylindrical contact homology. With respect to this presentation, we determine the degree-2 differential in the Bourgeois--Oancea exact sequence of \\cite{Oancea}. As a further application, we proved that if a K\\\"{a}hler manifold $M^{2n}$ admits a subcritical polarization and $c_1$ vanishes in the subcritical complement, then $M$ is uniruled."}
{"category": "Math", "title": "The Spherical Maximal Function on the Free Two-step Nilpotent Lie Group", "abstract": "We consider here the free two step nilpotent Lie group, provided with the homogeneous Kor\\'anyi norm; we prove the $L^p$-boundedness of the maximal function corresponding to the homogeneous unit sphere, for some $p$."}
{"category": "Math", "title": "On converting a side-pairing to a handle decomposition", "abstract": "We give a method for obtaining a handle decomposition of an $n$-manifold if the manifold is given by isometric side-pairings of a polyhedron in $\\en$, $\\sn$ or $\\hn$. Every cycle of $k$-faces on the polyhedron corresponds to an $(n-k)$-handle of the manifold. Two applications of the method are given. One helps recognize when a noncompact hyperbolic 3-manifold is a complement of a link in $S^3$ (and automatically produces the link diagram), the other shows that a topological $S^4$ described by the author in \\cite{Ivansic3} is diffeomorphic to the standard differentiable $S^4$."}
{"category": "Math", "title": "On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems", "abstract": "We consider the computation of stable approximations to the exact solution $x^\\dag$ of nonlinear ill-posed inverse problems $F(x)=y$ with nonlinear operators $F:X\\to Y$ between two Hilbert spaces $X$ and $Y$ by the Newton type methods $$ x_{k+1}^\\delta=x_0-g_{\\alpha_k} (F'(x_k^\\delta)^*F'(x_k^\\delta)) F'(x_k^\\delta)^* (F(x_k^\\delta)-y^\\delta-F'(x_k^\\delta)(x_k^\\delta-x_0)) $$ in the case that only available data is a noise $y^\\delta$ of $y$ satisfying $\\|y^\\delta-y\\|\\le \\delta$ with a given small noise level $\\delta>0$. We terminate the iteration by the discrepancy principle in which the stopping index $k_\\delta$ is determined as the first integer such that $$ \\|F(x_{k_\\delta}^\\delta)-y^\\delta\\|\\le \\tau \\delta <\\|F(x_k^\\delta)-y^\\delta\\|, \\qquad 0\\le k<k_\\delta $$ with a given number $\\tau>1$. Under certain conditions on $\\{\\alpha_k\\}$, $\\{g_\\alpha\\}$ and $F$, we prove that $x_{k_\\delta}^\\delta$ converges to $x^\\dag$ as $\\delta\\to 0$ and establish various order optimal convergence rate results. It is remarkable that we even can show the order optimality under merely the Lipschitz condition on the Fr\\'{e}chet derivative $F'$ of $F$ if $x_0-x^\\dag$ is smooth enough."}
{"category": "Math", "title": "Planar algebras: a category theoretic point of view", "abstract": "We define Jones's planar algebra as a map of multicategories and constuct a planar algebra starting from a 1-cell in a pivotal strict 2-category. We prove finiteness results for the affine representations of finite depth planar algebras. We also show that the radius of convergence of the dimension of an affine representation of the planar algebra associated to a finite depth subfactor is at least as big as the inverse-square of the modulus."}
{"category": "Math", "title": "The existence results for solutions of indefinite scalar curvature problem", "abstract": "In this paper, we consider the indefinite scalar curvature problem on $R^n$. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on $R^n$ (similarly, on $S^n$) has at least one solution. The key observation in our proof is that we use the bifurcation method to get a large solution and then after establishing the Harnack inequality for solutions near the critical points of the prescribed scalar curvature and taking limit, we find the nontrivial positive solution to the indefinite scalar curvature problem."}
{"category": "Math", "title": "Survey on recent invariants on classical knot theory", "abstract": "The survey we are presenting is over 22 years old but it has still some ideas which where never published (except in Polish). This survey is the base of the third Chapter of my book: KNOTS: From combinatorics of knot diagrams to combinatorial topology based on knots, which is still in preparation (but compare http://arxiv.org/pdf/math.GT/0512630). The purpose of this survey is to present a new combinatorial method of constructing invariants of isotopy classes of tame links. The period of time between the spring of 1984 and the summer of 1985 was full of discoveries which revolutionized the knot theory and will have a deep impact on some other branches of mathematics. It started by the discovery of Jones of the new polynomial invariant of links (in May 1984), and the last big step (which will be described in this survey) has been made by Kauffman in August 1985 when Kauffman applied his method which allowed him to unify almost all previous work. This survey is far from being complete, even if we limit ourselves to the purely combinatorial methods and to the period May, 1984 -- September, 1985."}
{"category": "Math", "title": "Numerical properties of isotrivial fibrations", "abstract": "In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations $\\varphi \\colon X \\lr C$, where $X$ is a smooth, projective surface and $C$ is a curve. In particular we prove that, if $g(C) \\geq 1$ and $X$ is neither ruled nor isomorphic to a quasi-bundle, then $K_X^2 \\leq 8 \\chi(\\mO_X)-2$; this inequality is sharp and if equality holds then $X$ is a minimal surface of general type whose canonical model has precisely two ordinary double points as singularities. Under the further assumption that $K_X$ is ample, we obtain $K_X^2 \\leq 8 \\chi(\\mO_X)-5$ and the inequality is also sharp. This improves previous results of Serrano and Tan."}
{"category": "Math", "title": "Extended obstruction tensors and renormalized volume coefficients", "abstract": "The behavior under conformal change of the renormalized volume coefficients associated to a pseudo-Riemannian metric is investigated. It is shown that they define second order fully nonlinear operators in the conformal factor whose algebraic structure is elucidated via the introduction of \"extended obstruction tensors\". These together with the Schouten tensor constitute building blocks for the coefficients in the ambient metric expansion. The renormalized volume coefficients have recently been considered by Chang-Fang motivated by comparison with the elementary symmetric functions of the eigenvalues of the Schouten tensor."}
{"category": "Math", "title": "Buried Points in Julia Sets", "abstract": "We give an introduction to buried points in Julia sets and a list of questions about buried points, written to encourage aficionados of topology and dynamics to work on these questions."}
{"category": "Math", "title": "Estimating high-dimensional intervention effects from observational data", "abstract": "We assume that we have observational data generated from an unknown underlying directed acyclic graph (DAG) model. A DAG is typically not identifiable from observational data, but it is possible to consistently estimate the equivalence class of a DAG. Moreover, for any given DAG, causal effects can be estimated using intervention calculus. In this paper, we combine these two parts. For each DAG in the estimated equivalence class, we use intervention calculus to estimate the causal effects of the covariates on the response. This yields a collection of estimated causal effects for each covariate. We show that the distinct values in this set can be consistently estimated by an algorithm that uses only local information of the graph. This local approach is computationally fast and feasible in high-dimensional problems. We propose to use summary measures of the set of possible causal effects to determine variable importance. In particular, we use the minimum absolute value of this set, since that is a lower bound on the size of the causal effect. We demonstrate the merits of our methods in a simulation study and on a data set about riboflavin production."}
{"category": "Math", "title": "Fefferman-Stein inequalities for the \\mathbb{Z}_2^d Dunkl maximal operator", "abstract": "In this article, we establish the Fefferman-Stein inequalities for the Dunkl maximal operator associated with a finite reflection group generated by the sign changes. Similar results are also given for a large class of operators related to Dunkl's analysis."}
{"category": "Math", "title": "Modular elliptic directions with complex multiplication (with an application to Gross's elliptic curves)", "abstract": "For every normalized newform f in S_2(Gamma_1(N)) with complex multiplication, we study the modular parametrizations of elliptic curves C from the abelian variety A_f. We apply the results obtained when C is Gross's elliptic curve A(p)."}
{"category": "Math", "title": "Nonnegative Factorization and The Maximum Edge Biclique Problem", "abstract": "Nonnegative Matrix Factorization (NMF) is a data analysis technique which allows compression and interpretation of nonnegative data. NMF became widely studied after the publication of the seminal paper by Lee and Seung (Learning the Parts of Objects by Nonnegative Matrix Factorization, Nature, 1999, vol. 401, pp. 788--791), which introduced an algorithm based on Multiplicative Updates (MU). More recently, another class of methods called Hierarchical Alternating Least Squares (HALS) was introduced that seems to be much more efficient in practice. In this paper, we consider the problem of approximating a not necessarily nonnegative matrix with the product of two nonnegative matrices, which we refer to as Nonnegative Factorization (NF); this is the subproblem that HALS methods implicitly try to solve at each iteration. We prove that NF is NP-hard for any fixed factorization rank, using a reduction to the maximum edge biclique problem. We also generalize the multiplicative updates to NF, which allows us to shed some light on the differences between the MU and HALS algorithms for NMF and give an explanation for the better performance of HALS. Finally, we link stationary points of NF with feasible solutions of the biclique problem to obtain a new type of biclique finding algorithm (based on MU) whose iterations have an algorithmic complexity proportional to the number of edges in the graph, and show that it performs better than comparable existing methods."}
{"category": "Math", "title": "A relaxation scheme for computation of the joint spectral radius of matrix sets", "abstract": "The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets."}
{"category": "Math", "title": "Chern subrings", "abstract": "Let p be an odd prime. We show that for a simply-connected semisimple complex linear algebraic group, if its integral homology has p-torsion, the Chern classes do not generate the Chow ring of its classifying space."}
{"category": "Math", "title": "Orbifold quantum D-modules associated to weighted projective spaces", "abstract": "We construct in an abstract fashion the orbifold quantum cohomology (quantum orbifold cohomology) of weighted projective space, starting from the orbifold quantum differential operator. We obtain the product, grading, and intersection form by making use of the associated self-adjoint D-module and the Birkhoff factorization procedure. The method extends to the more difficult case of Fano hypersurfaces in weighted projective space. However, in contrast to the case of weighted projective space itself or a Fano hypersurface in projective space, a \"small Birkhoff cell\" can appear in the construction; we give an example of this phenomenon."}
{"category": "Math", "title": "Smoothed weighted empirical likelihood ratio confidence intervals for quantiles", "abstract": "Thus far, likelihood-based interval estimates for quantiles have not been studied in the literature on interval censored case 2 data and partly interval censored data, and, in this context, the use of smoothing has not been considered for any type of censored data. This article constructs smoothed weighted empirical likelihood ratio confidence intervals (WELRCI) for quantiles in a unified framework for various types of censored data, including right censored data, doubly censored data, interval censored data and partly interval censored data. The fourth order expansion of the weighted empirical log-likelihood ratio is derived and the theoretical coverage accuracy equation for the proposed WELRCI is established, which generally guarantees at least `first order' accuracy. In particular, for right censored data, we show that the coverage accuracy is at least $O(n^{-1/2})$ and our simulation studies show that in comparison with empirical likelihood-based methods, the smoothing used in WELRCI generally provides a shorter confidence interval with comparable coverage accuracy. For interval censored data, it is interesting to find that with an adjusted rate $n^{-1/3}$, the weighted empirical log-likelihood ratio has an asymptotic distribution completely different from that obtained by the empirical likelihood approach and the resulting WELRCI perform favorably in the available comparison simulation studies."}
{"category": "Math", "title": "Estimates for covering numbers in Schauder's theorem about adjoints of compact operators", "abstract": "Let T:X --> Y be a bounded linear map between Banach spaces X and Y. Let S:Y' --> X' be its adjoint. Let B(X) and B(Y') be the closed unit balls of X and Y' respectively. We obtain apparently new estimates for the covering numbers of the set S(B(Y')). These are expressed in terms of the covering numbers of T(B(X)), or, more generally, in terms of the covering numbers of a \"significant\" subset of T(B(X)). The latter more general estimates are best possible. These estimates follow from our new quantitative version of an abstract compactness result which generalizes classical theorems of Arzela-Ascoli and of Schauder. Analogous estimates also hold for the covering numbers of T(B(X)), in terms of the covering numbers of S(B(Y')) or in terms of a suitable \"significant\" subset of S(B(Y'))."}
{"category": "Math", "title": "Masures affines", "abstract": "We give an abstract definition of affine hovels which generalizes the definition of affine buildings (eventually non simplicial) given by Jacques Tits and includes the hovels built by Stephane Gaussent and the author for some Kac-Moody groups over ultrametric fields. We prove that, in such an affine hovel I, there exist retractions with center a sector germ and that we can add at the infinity of I a pair of twin buildings or two microaffine buildings. For some affine hovels I, we prove that the residue at a point of I has a natural structure of pair of twin buildings and that there exists on I a preorder which induces on each apartment the preorder associated to the Tits cone."}
{"category": "Math", "title": "On the spectrum of the twisted Dolbeault Laplacian on line bundles over K\\\"ahler manifolds", "abstract": "We use Dirac operator techniques to establish a sharp lower bound for the first eigenvalue of the twisted Dolbeault Laplacian on holomorphic line bundles over compact K\\\"ahler manifolds."}
{"category": "Math", "title": "Quasi-energy function for diffeomorphisms with wild separatrices", "abstract": "According to Pixton, there are Morse-Smale diffeomorphisms of the 3-sphere which have no energy function, that is a Lyapunov function whose critical points are all periodic points of the diffeomorphism. We introduce the concept of quasi-energy function for a Morse-Smale diffeomorphism as a Lyapunov function with the least number of critical points and construct a quasi-energy function for any diffeomorphism from some class of Morse-Smale diffeomorphisms on the 3-sphere."}
{"category": "Math", "title": "Adaptive estimation of the conditional intensity of marker-dependent counting processes", "abstract": "We propose in this work an original estimator of the conditional intensity of a marker-dependent counting process, that is, a counting process with covariates. We use model selection methods and provide a non asymptotic bound for the risk of our estimator on a compact set. We show that our estimator reaches automatically a convergence rate over a functional class with a given (unknown) anisotropic regularity. Then, we prove a lower bound which establishes that this rate is optimal. Lastly, we provide a short illustration of the way the estimator works in the context of conditional hazard estimation."}
{"category": "Math", "title": "Symmetry and Time Changed Brownian Motions", "abstract": "In this paper we examine which Brownian Subordination with drift exhibits the symmetry property introduced by Fajardo and Mordecki (2006). We obtain that when the subordination results in a L\\'evy process, a necessary and sufficient condition for the symmetry to hold is that drift must be equal to -1/2."}
{"category": "Math", "title": "Smooth projective toric varieties whose nontrivial nef line bundles are big", "abstract": "For any $n\\geq 3$, we explicitly construct smooth projective toric $n$-folds of Picard number $\\geq 5$, where any nontrivial nef line bundles are big."}
{"category": "Math", "title": "Exponential algebraicity in exponential fields", "abstract": "I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property. A corollary is that there are at most countably many essential counterexamples to Schanuel's conjecture."}
{"category": "Math", "title": "A solution to a problem of Cassels and Diophantine properties of cubic numbers", "abstract": "We prove that almost any pair of real numbers a,b, satisfies the following inhomogeneous uniform version of Littlewood's conjecture: (*) forall x,y in R, liminf_{|n|\\to\\infty} |n|<na - x> <nb - y> = 0, where <-> denotes the distance from the nearest integer. The existence of even a single pair that satisfies (*), solves a problem of Cassels from the 50's. We then prove that if 1,a,b span a totally real number field, then a,b, satisfy (*). It is further shown that if 1,a,b, are linearly dependent over Q, a,b cannot satisfy (*). The results are then applied to give examples of irregular orbit closures of the diagonal groups of a new type."}
{"category": "Math", "title": "Holomorphic shadows in the eyes of model theory", "abstract": "We define a subset of an almost complex manifold (M,J) to be a holomorphic shadow if it is the image of a J-holomorphic map from a compact complex manifold. Notice that a J-holomorphic curve is a holomorphic shadow, and so is a complex subvariety of a compact complex manifold. We show that under some conditions on an almost complex structure J on a manifold M, the holomorphic shadows in the Cartesian products of (M,J) form a Zariski-type structure. Checking this leads to non-trivial geometric questions and results. We then apply the work of Hrushovski and Zilber on Zariski-type structures. We also restate results of Gromov and McDuff on J-holomorphic curves in symplectic geometry in the language of shadows structures."}
{"category": "Math", "title": "On a generalization of Littlewood's conjecture", "abstract": "We present a class of lattices in R^d (d >= 2) which we call GL-lattices and conjecture that any lattice is such. This conjecture is referred to as GLC. Littlewood's conjecture amounts to saying that Z^2 is GL. We then prove existence of GL lattices by first establishing a dimension bound for the set of possible exceptions. Existence of vectors (GL-vectors) in R^d with special Diophantine properties is proved by similar methods. For dimension d >= 3 we give explicit constructions of GL lattices (and in fact a much stronger property). We also show that GLC is implied by a conjecture of G. A. Margulis concerning bounded orbits of the diagonal group. The unifying theme of the methods is to exploit rigidity results in dynamics and derive results in Diophantine approximations or the geometry of numbers."}
{"category": "Math", "title": "Measure theoretical entropy of covers", "abstract": "In this paper we introduce three notions of measure theoretical entropy of a measurable cover U in a measure theoretical dynamical system. Two of them were already introduced in [R] and the new one is defined only in the ergodic case. We then prove that these three notions coincide, thus answering a question posed in [R] and recover a variational inequality (proved in [GW]) and a proof of the classical variational principle based on a comparison between the entropies of covers and partitions."}
{"category": "Math", "title": "Unitarity of SL(2)-conformal blocks in genus zero", "abstract": "This submission has been withdrawn by arXiv administrators due to an unresolved conflict between the authors. This article was submitted without consent of E. Looijenga."}
{"category": "Math", "title": "Discrete Morse theory for totally non-negative flag varieties", "abstract": "In a seminal 1994 paper, Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_{\\geq 0} of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a \"remarkable polyhedral subspace\", and conjectured a decomposition into cells, which was subsequently proven by the first author. Subsequently the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's that (G/P)_{\\geq 0} -- the closure of the top-dimensional cell -- is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_{> 0} is homotopic to a sphere, is new for all G/P other than projective space."}
{"category": "Math", "title": "A formal system for Euclid's Elements", "abstract": "We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning."}
{"category": "Math", "title": "Heegaard genus, cut number, weak p-congruence, and quantum invariants", "abstract": "We use quantum invariants to define a 3-manifold invariant j_p which lies in the non-negative integers. We relate j_p to the Heegard genus, and the cut number. We show that j_$ is an invariant of weak p-congruence."}
{"category": "Math", "title": "What is the optimal shape of a pipe?", "abstract": "We consider an incompressible fluid in a three-dimensional pipe, following the Navier-Stokes system with classical boundary conditions. We are interested in the following question: is there any optimal shape for the criterion \"energy dissipated by the fluid\"? Moreover, is the cylinder the optimal shape? We prove that there exists an optimal shape in a reasonable class of admissible domains, but the cylinder is not optimal. For that purpose, we explicit the first order optimality condition, thanks to adjoint state and we prove that it is impossible that the adjoint state be a solution of this over-determined system when the domain is the cylinder. At last, we show some numerical simulations for that problem."}
{"category": "Math", "title": "Topics in Mitigating Radar Bias", "abstract": "In this paper, we investigate two topics related to mitigating the effect of radar bias in ballistic missile tracking applications. We determine the absolute bias between two radars in polar coordinates when their relative bias is given in rectangular coordinates. Using this result, we then obtain the optimized steady-state filter to handle the random bias."}
{"category": "Math", "title": "Algorithms for translational tiling", "abstract": "In this paper we study algorithms for tiling problems. We show that the conditions $(T1)$ and $(T2)$ of Coven and Meyerowitz, conjectured to be necessary and sufficient for a finite set $A$ to tile the integers, can be checked in time polynomial in ${diam}(A)$. We also give heuristic algorithms to find all non-periodic tilings of a cyclic group $Z_N$. In particular we carry out a full classification of all non-periodic tilings of $Z_{144}$."}
{"category": "Math", "title": "Mathematical Foundations of Consciousness", "abstract": "We employ the Zermelo-Fraenkel Axioms that characterize sets as mathematical primitives. The Anti-foundation Axiom plays a significant role in our development, since among other of its features, its replacement for the Axiom of Foundation in the Zermelo-Fraenkel Axioms motivates Platonic interpretations. These interpretations also depend on such allied notions for sets as pictures, graphs, decorations, labelings and various mappings that we use. A syntax and semantics of operators acting on sets is developed. Such features enable construction of a theory of non-well-founded sets that we use to frame mathematical foundations of consciousness. To do this we introduce a supplementary axiomatic system that characterizes experience and consciousness as primitives. The new axioms proceed through characterization of so- called consciousness operators. The Russell operator plays a central role and is shown to be one example of a consciousness operator. Neural networks supply striking examples of non-well-founded graphs the decorations of which generate associated sets, each with a Platonic aspect. Employing our foundations, we show how the supervening of consciousness on its neural correlates in the brain enables the framing of a theory of consciousness by applying appropriate consciousness operators to the generated sets in question."}
{"category": "Math", "title": "The noncommutative Choquet boundary III: Operator systems in matrix algebras", "abstract": "We classify operator systems $S\\subseteq \\mathcal B(H)$ that act on finite dimensional Hilbert spaces by making use of the noncommutative Choquet boundary. S is said to be {\\em reduced} when its boundary ideal is 0. In the category of operator systems, that property functions as semisimplicity does in the category of complex Banach algebras. We construct explicit examples of reduced operator systems using sequences of \"parameterizing maps\" $\\Gamma_k: \\mathbb C^r\\to \\mathcal B(H_k)$, $k=1,..., N$. We show that every reduced operator system is isomorphic to one of these, and that two sequences give rise to isomorphic operator systems if and only if they are \"unitarily equivalent\" parameterizing sequences. Finally, we construct nonreduced operator systems $S$ that have a given boundary ideal $K$ and a given reduced image in $C^*(S)/K$, and show that these constructed examples exhaust the possibilities."}
{"category": "Math", "title": "On the Chebyshev properties of system of eigenfunctions for Sturm--Liouville problem with singular coefficients", "abstract": "In the paper we consider singular spectral Sturm--Liouville problem $-(py')'+(q-\\lambda r)y=0$, $(U-1)y^{\\vee}+i(U+1)y^{\\wedge}=0$, where function $p\\in L_{\\infty}[0,1]$ is uniformly positive, generalized function $q\\in W_2^{-1}[0,1]$ is real-valued, generalized weight function $r\\in W_2^{-1}[0,1]$ is positive and unitary matrix $U\\in\\mathbb C^{2\\times 2}$ is diagonal. The goal is to prove that well-known (for smooth case) facts about Chebyshev property of eigenfunctions hold in general case."}
{"category": "Math", "title": "Minimal Distortion Morphs Generated by Time-Dependent Vector Fields", "abstract": "A morph between two Riemannian $n$-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian $n$-manifolds and, with respect to these classes, prove the existence of minimal distortion morphs and diffeomorphisms. In particular, we consider the class of time-dependent vector fields (on an open subset $\\Omega$ of $ \\R^{n+1}$ in which the manifolds are embedded) that generate morphs between two manifolds $M$ and $N$ via an evolution equation, define the bending and the morphing distortion energies for these morphs, and prove the existence of minimizers of the corresponding functionals in the set of time-dependent vector fields that generate morphs between $M$ and $N$ and are $L^2$ functions from $[0,1]$ to the Sobolev space $W^{k,2}_0(\\Omega,\\R^{n+1})$."}
{"category": "Math", "title": "On 3-lattices and spherical designs", "abstract": "An integral lattice which is generated by some vectors of norm $q$ is called $q$-lattice. Classification of 3-lattices of dimension at most four is given by Mimura (On 3-lattice, 2006). As a expansion, we give a classification of 3-lattices of dimension at most seven. In addition, we consider the spherical designs from its shells."}
{"category": "Math", "title": "$\\ell$- Volterra Quadratic Stochastic Operators: Lyapunov Functions, Trajectories", "abstract": "We consider $\\ell$-Volterra quadratic stochastic operators defined on $(m-1)$-dimensional simplex, where $\\ell\\in\\{0,1,...,m\\}$. Under some conditions on coefficients of such operators we describe Lyapunov functions and apply them to obtain upper estimates for the set of $\\omega$- limit points of trajectories. We describe a set of fixed points of $\\ell$-Volterra operators."}
{"category": "Math", "title": "Superdiffusivity for a Brownian polymer in a continuous Gaussian environment", "abstract": "This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a Gaussian field $W$ on ${\\mathbb{R}}_+\\times{\\mathbb{R}}$ which is white noise in time and function-valued in space. According to the behavior of the spatial covariance of $W$, we give a lower bound on the power growth (wandering exponent) of the polymer when the time parameter goes to infinity: the polymer is proved to be superdiffusive, with a wandering exponent exceeding any $\\alpha<3/5$."}
{"category": "Math", "title": "Walks with small steps in the quarter plane", "abstract": "Let S be a subset of {-1,0,1}^2 not containing (0,0). We address the enumeration of plane lattice walks with steps in S, that start from (0,0) and always remain in the first quadrant. A priori, there are 2^8 problems of this type, but some are trivial. Some others are equivalent to a model of walks confined to a half-plane: such models can be solved systematically using the kernel method, which leads to algebraic generating functions. We focus on the remaining cases, and show that there are 79 inherently different problems to study. To each of them, we associate a group G of birational transformations. We show that this group is finite in exactly 23 cases. We present a unified way of solving 22 of the 23 models associated with a finite group. For each of them, the generating function is found to be D-finite. The 23rd model, known as Gessel's walks, has recently been proved by Bostan et al. to have an algebraic (and hence D-finite) solution. We conjecture that the remaining 56 models, associated with an infinite group, have a non-D-finite generating function. Our approach allows us to recover and refine some known results, and also to obtain new results. For instance, we prove that walks with N, E, W, S, SW and NE steps have an algebraic generating function."}
{"category": "Math", "title": "Homotopy invariants of Gauss phrases", "abstract": "Equivalence relations can be defined on Gauss phrases using combinatorial moves. In this paper we consider two closely related equivalence relations on Gauss phrases, homotopy and open homotopy. In particular, in each case, we define a new invariant and determine the values that it can attain."}
{"category": "Math", "title": "Moduli of Stable Sheaves on a Smooth Quadric in $\\PP_3$", "abstract": "We prove that the moduli space of stable sheaves of rank 2 with a certain Chern classes on a smooth quadric $Q$ in $\\PP_3$, is isomorphic to $\\PP_3$. Using this identification, we give a new proof that a certain Brill-Noether locus on a non-hyperelliptic curve of genus 4, is isomorphic to the Donagi-Izadi cubic threefold."}
{"category": "Math", "title": "The Atiyah algebroid of the path fibration over a Lie group", "abstract": "Let G be a connected Lie group, LG its loop group, and PG->G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A=T(PG)/LG of this bundle. Given a symmetric bilinear form on the Lie algebra g and the corresponding central extension of Lg, we consider the lifting problem for A, and show how the cohomology class of the Cartan 3-form on G arises as an obstruction. This involves the construction of a 2-form on PG with differential the pull-back of the Cartan form. In the second part of this paper we obtain similar LG-invariant primitives for the higher degree analogues of the Cartan form, and for their G-equivariant extensions."}
{"category": "Math", "title": "Totally geodesic submanifolds in Riemannian symmetric spaces", "abstract": "In the first part of this expository article, the most important constructions and classification results concerning totally geodesic submanifolds in Riemannian symmetric spaces are summarized. In the second part, I describe the results of my classification of the totally geodesic submanifolds in the Riemannian symmetric spaces of rank 2."}
{"category": "Math", "title": "On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation I", "abstract": "The fact that the Korteweg-de-Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation was derived several years ago in the physical literature. In this paper, we provide a rigorous proof of this fact, and compute a precise estimate for the error term. Our proof relies on the integrability of both the equations. In particular, we give a relation between the invariants of the two equations, which, we hope, is of independent interest."}
{"category": "Math", "title": "Wind speed classification using Dirichlet mixtures", "abstract": "Wind energy production is very sensitive to instantaneous wind speed fluctuations. Thus rapid variation of wind speed due to changes in the local meteorological conditions can lead to electrical power variations of the order of the nominal power output. In small grids, as they exist for example on some islands in the French West Indies, such fluctuations can cause instabilities in case of intermediate power shortages. To palliate these difficulties, it is essential to identify and characterize the wind speed distributions. This allows to anticipate the eventuality of power shortage or power surge. Therefore, it is of interest to categorize wind speed fluctuations into distinct classes and to estimate the probability of a distribution to belong to a class. This paper presents a method for classifying wind speed histograms by estimating a finite mixture of Dirichlet distributions. The SAEM algorithm that we use provides a fine distinction between wind speed distribution classes. It's a new nonparametric method for wind speed sequences classification. However, we show that the wind speed distributions in each class correspond to specific Gram- Charlier densities."}
{"category": "Math", "title": "Kshirsagar--Tan independence property of beta matrices and related characterizations", "abstract": "A new independence property of univariate beta distributions, related to the results of Kshirsagar and Tan for beta matrices, is presented. Conversely, a characterization of univariate beta laws through this independence property is proved. A related characterization of a family of $2\\times2$ random matrices including beta matrices is also obtained. The main technical challenge was a problem involving the solution of a related functional equation."}
{"category": "Math", "title": "Central limit theorems for double Poisson integrals", "abstract": "Motivated by second order asymptotic results, we characterize the convergence in law of double integrals, with respect to Poisson random measures, toward a standard Gaussian distribution. Our conditions are expressed in terms of contractions of the kernels. To prove our main results, we use the theory of stable convergence of generalized stochastic integrals developed by Peccati and Taqqu. One of the advantages of our approach is that the conditions are expressed directly in terms of the kernel appearing in the multiple integral and do not make any explicit use of asymptotic dependence properties such as mixing. We illustrate our techniques by an application involving linear and quadratic functionals of generalized Ornstein--Uhlenbeck processes, as well as examples concerning random hazard rates."}
{"category": "Math", "title": "Moebius transformations preserving fixed anharmonic ratio", "abstract": "O. Kobayashi in 2007 proved that differentiable mappings preserving anharmonic ratio are Moebius transformations. We strengthen his result and prove, that the requirement of differentiability and even of injectivity can be omitted."}
{"category": "Math", "title": "Stability of symplectic leaves", "abstract": "We find computable criteria for stability of symplectic leaves of Poisson manifolds. Using Poisson geometry as an inspiration, we also give a general criterion for stability of leaves of Lie algebroids, including singular ones. This not only extends but also provides a new approach (and proofs) to the classical stability results for foliations and group actions."}
{"category": "Math", "title": "Local times of multifractional Brownian sheets", "abstract": "Denote by $H(t)=(H_1(t),...,H_N(t))$ a function in $t\\in{\\mathbb{R}}_+^N$ with values in $(0,1)^N$. Let $\\{B^{H(t)}(t)\\}=\\{B^{H(t)}(t),t\\in{\\mathbb{R}}^N_+\\}$ be an $(N,d)$-multifractional Brownian sheet (mfBs) with Hurst functional $H(t)$. Under some regularity conditions on the function $H(t)$, we prove the existence, joint continuity and the H\\\"{o}lder regularity of the local times of $\\{B^{H(t)}(t)\\}$. We also determine the Hausdorff dimensions of the level sets of $\\{B^{H(t)}(t)\\}$. Our results extend the corresponding results for fractional Brownian sheets and multifractional Brownian motion to multifractional Brownian sheets."}
{"category": "Math", "title": "Homomorphisms of higher categories", "abstract": "We describe a construction that to each algebraically specified notion of higher-dimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associative and unital composition. We give two applications of this construction. The first is to tricategories; and here we do not obtain the trihomomorphisms defined by Gordon, Power and Street, but only something equivalent in a suitable sense. The second is to Batanin's weak omega-categories."}
{"category": "Math", "title": "A Schanuel property for exponentially transcendental powers", "abstract": "We prove the analogue of Schanuel's conjecture for raising to the power of an exponentially transcendental real number. All but countably many real numbers are exponentially transcendental. We also give a more general result for several powers in a context which encompasses the complex case."}
{"category": "Math", "title": "The simplest problem in the collective dynamics of neural networks: Is synchrony stable?", "abstract": "For spiking neural networks we consider the stability problem of global synchrony, arguably the simplest non-trivial collective dynamics in such networks. We find that even this simplest dynamical problem -- local stability of synchrony -- is non-trivial to solve and requires novel methods for its solution. The dynamics in the vicinity of the synchronous state is determined by a multitude of linear operators, in contrast to a single stability matrix in conventional linear stability theory. This unusual property qualitatively depends on network topology and may be neglected for globally coupled homogeneous networks. For generic networks, however, the number of operators increases exponentially with the size of the network. We present methods to treat this multi-operator problem exactly. First, based on the Gershgorin and Perron-Frobenius theorems, we derive bounds on the eigenvalues that provide important information about the synchronization process but are not sufficient to establish the asymptotic stability or instability of the synchronous state. We then present a complete analysis of asymptotic stability for topologically strongly connected networks using simple graph-theoretical considerations. For inhibitory interactions between dissipative (leaky) oscillatory neurons the synchronous state is stable, independent of the parameters and the network connectivity. These results indicate that pulse-like interactions play a profound role in network dynamical systems, and in particular in the dynamics of biological synchronization, unless the coupling is homogeneous and all-to-all."}
{"category": "Math", "title": "Conformal Structure of Minimal Surfaces with Finite Topology", "abstract": "In this paper, we show that a complete embedded minimal surface in $\\Real^3$ with finite topology and one end is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic to that of the helicoid. More precisely, if $g$ is the stereographic projection of the Gauss map, then in a neighborhood of the puncture, $g(p) = \\exp(i\\alpha z(p) + F(p))$, where $\\alpha \\in \\Real$, $z=x_3+ix_3^*$ is a holomorphic coordinate defined in this neighborhood and $F(p)$ is holomorphic in the neighborhood and extends over the puncture with a zero there. This further implies that the end is actually Hausdorff close to a helicoid."}
{"category": "Math", "title": "Higher localized analytic indices and strict deformation quantization", "abstract": "This paper is concerned with the localization of higher analytic indices for Lie groupoids. Let $\\gr$ be a Lie groupoid with Lie algebroid $A\\gr$. Let $\\tau$ be a (periodic) cyclic cocycle over the convolution algebra $\\cg$. We say that $\\tau$ can be localized if there is a correspondence K^0(A^*\\gr)\\stackrel{Ind_{\\tau}}{\\longrightarrow}\\mathbb{C} satisfying $Ind_{\\tau}(a)=< ind D_a,\\tau>$ (Connes pairing). In this case, we call $Ind_{\\tau}$ the higher localized index associated to $\\tau$. In {Ca4} we use the algebra of functions over the tangent groupoid introduced in {Ca2}, which is in fact a strict deformation quantization of the Schwartz algebra $\\sw(A\\gr)$, to prove the following results: \\item Every bounded continuous cyclic cocycle can be localized. \\item If $\\gr$ is {\\'e}tale, every cyclic cocycle can be localized. We will recall this results with the difference that in this paper, a formula for higher localized indices will be given in terms of an asymptotic limit of a pairing at the level of the deformation algebra mentioned above. We will discuss how the higher index formulas of Connes-Moscovici, Gorokhovsky-Lott fit in this unifying setting."}
{"category": "Math", "title": "Skewness Premium with L\\'evy Processes", "abstract": "We study the skewness premium (SK) introduced by Bates (1991) in a general context using L\\'evy Processes. Under a symmetry condition Fajardo and Mordecki (2006) obtain that SK is given by the Bate's $x%$ rule. In this paper we study SK under the absence of that symmetry condition. More exactly, we derive sufficient conditions for SK to be positive, in terms of the characteristic triplet of the L\\'evy Process under the risk neutral measure."}
{"category": "Math", "title": "Supporting degrees of multi-graded local cohomolgoy modules", "abstract": "For a finitely generated graded module $M$ over a positively-graded commutative Noetherian ring $R$, the second author established in 1999 some restrictions, which can be formulated in terms of the Castelnuovo regularity of $M$ or the so-called $a^*$-invariant of $M$, on the supporting degrees of a graded-indecomposable graded-injective direct summand, with associated prime ideal containing the irrelevant ideal of $R$, of any term in the minimal graded-injective resolution of $M$. Earlier, in 1995, T. Marley had established connections between finitely graded local cohomology modules of $M$ and local behaviour of $M$ across $\\Proj(R)$. The purpose of this paper is to present some multi-graded analogues of the above-mentioned work."}
{"category": "Math", "title": "Sharp large deviations for the fractional Ornstein-Uhlenbeck process", "abstract": "We investigate the sharp large deviation properties of the energy and the maximum likelihood estimator for the Ornstein-Uhlenbeck process driven by a fractional Brownian motion with Hurst index greater than one half."}
{"category": "Math", "title": "A version of geometric motivic integration that specializes to p-adic integration via point counting", "abstract": "We give a version of geometric motivic integration that specializes to p-adic integration via point counting. This has been done before for stable sets; we extend this to more general sets. The main problem in doing this is that it requires to take limits, hence the measure will have to take values in a completion of the (localized) Grothendieck ring of varieties. The standard choice is to complete with respect to the dimension filtration; however, since the point counting homomorphism is not continuous with respect to this topology we have to use a stronger one. The first part of the paper is devoted to defining this topology; in the second part we will then see that many of the standard constructions of geometric motivic integration work also in this setting."}
{"category": "Math", "title": "Quadratic functors on pointed categories", "abstract": "We study polynomial functors of degree 2, called quadratic, with values in the category of abelian groups $Ab$, and whose source category is an arbitrary category $\\C$ with null object such that all objects are colimits of copies of a generating object $E$ which is small and regular projective; this includes all pointed algebraic varieties. More specifically, we are interested in such quadratic functors $F$ from $\\C$ to $Ab$ which preserve filtered colimits and suitable coequalizers; one may take reflexive ones if $\\C$ is Mal'cev and Barr exact. A functorial equivalence is established between such functors $F:\\C\\to Ab$ and certain minimal algebraic data which we call quadratic $\\C$-modules: these involve the values on $E$ of the cross-effects of $F$ and certain structure maps generalizing the second Hopf invariant and the Whitehead product. Applying this general result to the case where $E$ is a cogroup these data take a particularly simple form. This application extends results of Baues and Pirashvili obtained for $\\C$ being the category of groups or of modules over some ring; here quadratic $\\C$-modules are equivalent with abelian square groups or quadratic $R$-modules, respectively."}
{"category": "Math", "title": "The law of series", "abstract": "We consider an ergodic process on finitely many states, with positive entropy. Our first main result asserts that the distribution function of the normalized waiting time for the first visit to a small (i.e., over a long block) cylinder set $B$ is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function $1-e^{-t}$. That is, the occurrences of so understood \"rare event\" $B$ along the time axis can appear either with gap sizes of nearly exponential distribution (like in the independent Bernoulli process), or they \"attract\" each-other. Our second main result states that a {\\it typical} ergodic process of positive entropy has the following property: the distribution functions of the normalized hitting times for the majority of cylinders $B$ of lengths $n'$ converge to zero along a \\sq\\ $n'$ whose upper density is 1. The occurrences of such a cylinder $B$ \"strongly attract\", i.e., they appear in \"series\" of many frequent repetitions separated by huge gaps of nearly complete absence. These results, when properly and carefully interpreted, shed some new light, in purely statistical terms, independently from physics, on a century old (and so far rather avoided by serious science) common-sense phenomenon known as {\\it the law of series}, asserting that rare events in reality, once occurred, have a mysterious tendency for untimely repetitions."}
{"category": "Math", "title": "Quasi-isometries between visual hyperbolic spaces", "abstract": "We prove that a PQ-symmetric homeomorphism between two complete metric spaces can be extended to a quasi-isometry between their hyperbolic approximations. This result is used to prove that two visual Gromov hyperbolic spaces are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundaries."}
{"category": "Math", "title": "Logarithmic dimension bounds for the maximal function along a polynomial curve", "abstract": "Let M denote the maximal function along the polynomial curve p(t)=(t,t^2,...,t^d) in R^d: M(f)=sup_{r>0} (1/2r) \\int_{|t|<r} |f(x-p(t))| dt. We show that the L^2-norm of this operator grows at most logarithmically with the parameter d: ||M||_2 < c log d ||f||_2, where c>0 is an absolute constant. The proof depends on the explicit construction of a \"parabolic\" semi-group of operators which is a mixture of stable semi-groups."}
{"category": "Math", "title": "Spontaneous clustering in theoretical and some empirical stationary processes", "abstract": "In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increased frequency separated by longer breaks. Such behavior, contradicting the theoretical \"unbiased behavior\" with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper we prove, using ergodic theory and the notion of category, that clustering (even very strong) is in fact typical for \"rare events\" defined as long cylinder sets in processes generated by a finite partition of an arbitrary (infinite aperiodic) ergodic measure preserving transformation."}
{"category": "Math", "title": "Constructing Stein manifolds after Eliashberg", "abstract": "A unified summary is given of the existence theory of Stein manifolds in all dimensions, based on published and pending literature. Eliashberg's characterization of manifolds admitting Stein structures requires an extra delicate hypothesis in complex dimension 2, which can be eliminated by passing to the topological setting and invoking Freedman theory. The situation is quite similar if one asks which open subsets of a fixed complex manifold can be made Stein by an isotopy. As an application of these theorems, one can construct uncountably many diffeomorphism types of exotic R^4's realized as Stein open subsets of C^2 (i.e. domains of holomorphy). More generally, every domain of holomorphy in C^2 is topologically isotopic to other such domains realizing uncountably many diffeomorphism types. Any tame n-complex in a complex n-manifold can be isotoped to become a nested intersection of Stein open subsets, provided the isotopy is topological when n=2. In the latter case, the Stein neighborhoods are homeomorphic, but frequently realize uncountably many diffeomorphism types. It is also proved that every exhausting Morse function can be subdivided to yield a locally finite handlebody of the same maximal index, both in the context of smooth n-manifolds and for Stein surfaces."}
{"category": "Math", "title": "A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary", "abstract": "In this paper, we prove a priori estimates in Lagrangian coordinates for the equations of motion of an incompressible, inviscid, self-gravitating fluid with free boundary. The estimates show that on a finite time interval we control five derivatives of the fluid velocity and five and a half derivatives of the coordinates of the moving domain."}
{"category": "Math", "title": "Generic bounds for Frobenius closure and tight closure", "abstract": "We use geometric and cohomological methods to show that given a degree bound for membership in ideals of a fixed degree type in the polynomial ring P=k[x_0,..., x_d], one obtains a good generic degree bound for membership in the tight closure of an ideal of that degree type in any standard-graded k-algebra R of dimension d+1. This indicates that the tight closure of an ideal behaves more uniformly than the ideal itself. Moreover, if R is normal, one obtains a generic bound for membership in the Frobenius closure. If d is at most 2, then the bound for ideal membership in P can be computed from the known cases of the Froeberg conjecture and yields explicit generic tight closure bounds."}
{"category": "Math", "title": "Transitive orientations in bull-reducible Berge graphs", "abstract": "A bull is a graph with five vertices $r, y, x, z, s$ and five edges $ry$, $yx$, $yz$, $xz$, $zs$. A graph $G$ is bull-reducible if no vertex of $G$ lies in two bulls. We prove that every bull-reducible Berge graph $G$ that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color exactly the vertices of such a graph."}
{"category": "Math", "title": "Quadratic Binomial APN Functions and Absolutely Irreducible Polynomials", "abstract": "We show that many quadratic binomial functions on a finite field of characteristic 2 are not APN infinitely often. This is of interest in the light of recent discoveries of new families of quadratic binomial APN functions. The proof uses the Weil bound from algebraic geometry."}
{"category": "Math", "title": "Some new examples with almost positive curvature", "abstract": "As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of non-negatively curved manifolds which contain points at which all 2-planes have positive curvature. We show that there are generalisations of the well-known Eschenburg spaces and quotients of $\\sph^7 \\x \\sph^7$ which admit metrics with this property."}
{"category": "Math", "title": "Filiform nilsolitons of dimension 8", "abstract": "A Riemannian manifold (M,g) is said to be Einstein if its Ricci tensor satisfies ric(g) = cg, for some real number c. In the homogeneous case, a problem that is still open is the so called Alekseevskii Conjecture. This conjecture says that any homogeneous Einstein space with negative scalar curvature (i.e. c < 0) is a solvmanifold: a simply connected solvable Lie group endowed with a left invariant Riemannian metric. The aim of this paper is to classify Einstein solvmanifolds of dimension 9 whose nilradicals are 7-step nilpotent Lie algebras of dimension 8."}
{"category": "Math", "title": "The loop cohomology of a space with the polynomial cohomology algebra", "abstract": "Given a simply connected space $X$ with the cohomology $H^*(X;{\\mathbb Z}_2)$ to be polynomial, we calculate the loop cohomology algebra $H^*(\\Omega X;{\\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation $Sq_1$ on $H^*(X;{\\mathbb Z}_2).$ As a consequence we obtain that $H^*(\\Omega X;{\\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is multiplicatively decomposable on $H^{\\ast}(X;{\\mathbb Z}_2).$ The last statement in fact contains a converse of a theorem of A. Borel."}
{"category": "Math", "title": "Units of ring spectra and Thom spectra", "abstract": "We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. We recall (from May, Quinn, and Ray) that a commutative ring spectrum A has a spectrum of units gl(A). To a map of spectra f: b -> bgl(A), we associate a commutative A-algebra Thom spectrum Mf, which admits a commutative A-algebra map to R if and only if b -> bgl(A) -> bgl(R) is null. If A is an associative ring spectrum, then to a map of spaces f: B -> BGL(A) we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B -> BGL(A) -> BGL(R) is null. We also note that BGL(A) classifies the twists of A-theory. We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one another and to the classical theory, we characterize the Thom spectrum functor from the perspective of Morita theory."}
{"category": "Math", "title": "Computing Gorenstein Colength", "abstract": "Given an Artinian local ring $R$, we define its Gorenstein colength $g(R)$ to measure how closely we can approximate $R$ by a Gorenstein Artin local ring. In this paper, we show that $R = T/I$ satisfies the inequality $g(R) \\leq \\lambda(R/\\soc(R))$ in the following two cases: (a) $T$ is a power series ring over a field of characteristic zero and $I$ an ideal that is the power of a system of parameters or (b) $T$ is a 2-dimensional regular local ring with infinite residue field and $I$ is primary to the maximal ideal of $T$. In the first case, we compute $g(R)$ by constructing a Gorenstein Artin local ring mapping onto $R$. We further use this construction to show that an ideal that is the $n$th power of a system of parameters is directly linked to the $(n-1)$st power via Gorenstein ideals. A similar method shows that such ideals are also directly linked to themselves via Gorenstein ideals. Keywords: Gorenstein colength; Gorenstein linkage."}
{"category": "Math", "title": "Alternative algebras with the hyperbolic property", "abstract": "We investigate the structure of an alternative finite dimensional $\\Q$-algebra $\\mathfrak{A}$ subject to the condition that for a $\\Z$-order $\\Gamma \\subset \\mathfrak{A}$, and thus for every $\\Z$-order of $\\mathfrak{A}$, the loop of units of $\\U (\\Gamma)$ does not contain a free abelian subgroup of rank two. In particular, we prove that the radical of such an algebra associates with the whole algebra. We also classify $RA$-loops $L$ for which $\\mathbb{Z}L$ has this property. The classification for group rings is still an open problem."}
{"category": "Math", "title": "Online Coordinate Boosting", "abstract": "We present a new online boosting algorithm for adapting the weights of a boosted classifier, which yields a closer approximation to Freund and Schapire's AdaBoost algorithm than previous online boosting algorithms. We also contribute a new way of deriving the online algorithm that ties together previous online boosting work. We assume that the weak hypotheses were selected beforehand, and only their weights are updated during online boosting. The update rule is derived by minimizing AdaBoost's loss when viewed in an incremental form. The equations show that optimization is computationally expensive. However, a fast online approximation is possible. We compare approximation error to batch AdaBoost on synthetic datasets and generalization error on face datasets and the MNIST dataset."}
{"category": "Math", "title": "Manifolds of semi-negative curvature", "abstract": "The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class on nonpositively curved spaces, and several well-known results, such as existence and uniqueness of best approximations from convex closed sets, or the Bruhat-Tits fixed point theorem, are shown to hold in this setting, without dimension restrictions. Homogeneous spaces G/K of Banach-Lie groups of semi-negative curvature are also studied, explicit estimates on the geodesic distance and sectional curvature are obtained. A characterization of convex homogeneous submanifolds is given in terms of the Banach-Lie algebras. A splitting theorem via convex expansive submanifolds is proven, inducing the corresponding splitting of the Banach-Lie group G. Finally, these notions are used to study the structure of the classical Banach-Lie groups of bounded linear operators acting on a Hilbert space, and the splittings induced by conditional expectations in such setting."}
{"category": "Math", "title": "Simplicity of finitely-aligned k-graph C*-algebras", "abstract": "It is shown that no local periodicity is equivalent to the aperiodicity condition for arbitrary finitely-aligned k-graphs. This allows us to conclude that C*(\\Lambda) is simple if and only if \\Lambda is cofinal and has no local periodicity."}
{"category": "Math", "title": "Hyperbolicity of Semigroup Algebras II", "abstract": "In 1996 Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra %over a field of characteristic zero, contains a ${\\mathbb{Z}}$-order with hyperbolic unit group. In this paper we complete this classification by handling the case in which the semigroup is semi-simple."}
{"category": "Math", "title": "Smoothness and jet schemes", "abstract": "This paper shows some criteria for a scheme of finite type over an algebraically closed field to be non-singular in terms of jet schemes. For the base field of characteristic zero, the scheme is non-singular if and only if one of the truncation morphisms of its jet schemes is flat. For the positive characteristic case, we obtain a similar characterization under the reducedness condition on the scheme. We also obtain by a simple discussion that the scheme is non-singular if and only if one of its jet schemes is non-singular."}
{"category": "Math", "title": "Harmonic forms on principal bundles", "abstract": "We show a relationship between Chern-Simons 1- and 3-forms and harmonic forms on a principal bundle. Doing so requires one to consider an adiabatic limit. For the 3-form case, assume that G is simple and the corresponding Chern-Weil 4-form is exact. Then, the Chern-Simons 3-form on the princpal bundle G-bundle, minus a canonical term from the base, is harmonic in the adiabatic limit."}
{"category": "Math", "title": "Nonreflexive Banach SSD spaces", "abstract": "In this paper, we unify the theory of SSD spaces, part of the theory of strongly representable multifunctions, and the theory of the equivalence of various classes of maximally monotone multifunctions."}
{"category": "Math", "title": "Extra extension properties of equidimensional holomorphic mappings: results and open questions", "abstract": "Holomorphic (nondegenerate) mappings between complex manifolds of the same dimension are of special interest. For example, they appear as coverings of complex manifolds. At the same time they have very strong \"extra\" extension properties in compare with mappings in different dimensions. The aim of this paper is to put together the known results on this subject, give some perspective on the general strategy for future progress, prove some new results and formulate open questions."}
{"category": "Math", "title": "Ad-nilpotent Ideals and Equivalence Relations", "abstract": "In this paper we study ad-nilpotent ideals of a complex simple Lie algebra $\\ccg$ and their connections with affine Weyl groups and nilpotent orbits. We define a left equivalence relation for ad-nilpotent ideals based on their normalizer and generators, and prove that the equivalence relation is compatible with the left cell structure of affine Weyl group of $\\ccg$ and Lusztig's star operator for type $\\tilde A_{n-1}$."}
{"category": "Math", "title": "Ad-nilpotent Ideals of Minimal Dimension", "abstract": "n this paper we study ad-nilpotent ideals of a complex simple Lie algebra $\\ccg$ and their connections with affine Weyl groups and nilpotent orbits. We define a left equivalence relation for ad-nilpotent ideals based on their normalizer and generators, and prove that the equivalence relation is compatible with the left cell structure of affine Weyl group of $\\ccg$ and Lusztig's star operator for type $\\tilde A_{n-1}$."}
{"category": "Math", "title": "On ergodicity of some Markov processes", "abstract": "We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak-$^*$ mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer."}
{"category": "Math", "title": "Some structural results on the non-abelian tensor square of groups", "abstract": "We study the non-abelian tensor square $G\\otimes G$ for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G/G' so that $G\\otimes G$ is isomorphic to the direct product of $\\nabla(G)$ and the non-abelian exterior square $G\\wedge G$. For any group G, we characterize the non-abelian exterior square $G\\wedge G$ in terms of a presentation of G. Finally, we apply our results to some classes of groups, such as the classes of free soluble and free nilpotent groups of finite rank, and some classes of finite p-groups."}
{"category": "Math", "title": "Open maps between shift spaces", "abstract": "Given a code from a shift space to an irreducible sofic shift, any two of the following three conditions -- open, constant-to-one, (right or left) closing -- imply the third. If the range is not sofic, then the same result holds when bi-closingness replaces closingness. Properties of open mappings between shift spaces are investigated in detail. In particular, we show that a closing open (or constant-to-one) extension preserves the structure of a sofic shift."}
{"category": "Math", "title": "Automorphism groups of domains that depend on fewer than the maximal number of parameters", "abstract": "We study domains in complex $n$-space with automorphism group that does not depend on the full $n$ dimensions of the ambient space. A sufficient geometric condition is obtained to guarantee that a domain has such a \"thin\" automorphism group. Examples are provided to illustrate the ideas."}
{"category": "Math", "title": "Strong Influence of a Small Fiber on Shear Stress in Fiber-Reinforced Composites", "abstract": "In stiff fiber-reinforced material, the high shear stress concentration occurs in the narrow region between fibers. With the addition of a small geometric change in cross-section, such as a thin fiber or a overhanging part of fiber, the concentration is significantly increased. This paper presents mathematical analysis to explain the rapidly increased growth of the stress by a small particle in cross-section. To do so, we consider two crucial cases where a thin fiber exists between a pair of fibers, and where one of two fibers has a protruding small lump in cross-section. For each case, the optimal lower and upper bounds on the stress associated with the geometrical factors of fibers is established to explain the strongly increased growth of the stress by a small particle."}
{"category": "Math", "title": "On the existence of open and bi-continuing codes", "abstract": "Given an irreducible sofic shift X, we show that an an irreducible SFT Y of lower entropy is a factor of X if and only if it is a factor of X by an open bi-continuing code. If these equivalent conditions hold and Y is mixing, then any code from a proper subshift of X to Y can be extended to an open bi-continuing code on X. These results are still valid when X is assumed to be only an almost specified shift, i.e., a subshift satisfying an irreducible version of the specification property."}
{"category": "Math", "title": "Unital versions of the higher order peak algebras", "abstract": "We construct unital extensions of the higher order peak algebras defined by Krob and the third author in [Ann. Comb. 9 (2005), 411--430.], and show that they can be obtained as homomorphic images of certain subalgebras of the Mantaci-Reutenauer algebras of type B. This generalizes a result of Bergeron, Nyman and the first author [Trans. AMS 356 (2004), 2781--2824.]."}
{"category": "Math", "title": "Stochastic flows with reflection", "abstract": "Some topological properties of stochastic flow $\\varphi_t(x)$ generated by stochastic differential equation in a ${\\mathbb R}^d_+$ with normal reflection at the boundary are investigated. Sobolev differentiability in initial condition is received. The absolute continuity of the measure-valued process $\\mu\\circ\\varphi_t^{-1}$, where $\\mu\\ll\\lambda^d,$ is studied."}
{"category": "Math", "title": "Gram determinant of planar curves", "abstract": "We investigate the Gram determinant of the bilinear form based on curves in a planar surface, with a focus on the disk with two holes. We prove that the determinant based on $n-1$ curves divides the determinant based on $n$ curves. Motivated by the work on Gram determinants based on curves in a disk and curves in an annulus (Temperley-Lieb algebra of type $A$ and $B$, respectively), we calculate several examples of the Gram determinant based on curves in a disk with two holes and advance conjectures on the complete factorization of Gram determinants."}
{"category": "Math", "title": "Endpoint maximal and smoothing estimates for Schroedinger equations", "abstract": "For $\\alpha >1$ we consider the initial value problem for the dispersive equation $i\\partial_t u +(-\\Delta)^{\\alpha/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\\sup_{t\\in[0,1]}|u(\\cdot,t)|$ with initial values in $L^p$-Sobolev spaces, for $p\\in(2+4/(d+1),\\infty)$. This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp $L^p$ space-time estimates (local in time) for the same range of $p$."}
{"category": "Math", "title": "Generalized improper integral definition for finite limit", "abstract": "A generalization of the definition of a one-dimensional improper integral with a finite limit is presented. The new definition extends the range of valid integrals to include integrals which were previously considered to not be integrable. This definition is shown to be equivalent to the infinite limit definition presented in \"Generalized improper integral definition for infinite limit\" (arXiv:0805.3559) via a particular change of variable of integration. The definition preserves linearity and uniqueness. Integrals which are valid under the conventional definition have the same value under the new definition. Criteria for interchanging the order of integration and differentiation, and for interchanging the order with a second integration, are obtained. Examples are provided."}
{"category": "Math", "title": "A note on lattice-face polytopes and their Ehrhart polynomials", "abstract": "We give a new definition of lattice-face polytopes by removing an unnecessary restriction in the paper \"Ehrhart polynomials of lattice-face polytopes\", and show that with the new definition, the Ehrhart polynomial of a lattice-face polytope still has the property that each coefficient is the normalized volume of a projection of the original polytope. Furthermore, we show that the new family of lattice-face polytopes contains all possible combinatorial types of rational polytopes."}
{"category": "Math", "title": "The Yoneda algebra of a K_2 algebra need not be another K_2 algebra", "abstract": "The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K_2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K_2 algebra would be another K_2 algebra. We show that this is not necessarily the case by constructing a monomial K_2 algebra for which the corresponding Yoneda algebra is not K_2."}
{"category": "Math", "title": "Equidistribution of sparse sequences on nilmanifolds", "abstract": "We study equidistribution properties of nil-orbits $(b^nx)_{n\\in\\N}$ when the parameter $n$ is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show that if $X=G/\\Gamma$ is a nilmanifold, $b\\in G$ is an ergodic nilrotation, and $c\\in \\R\\setminus \\Z$ is positive, then the sequence $(b^{[n^c]}x)_{n\\in\\N}$ is equidistributed in $X$ for every $x\\in X$. This is also the case when $n^c$ is replaced with $a(n)$, where $a(t)$ is a function that belongs to some Hardy field, has polynomial growth, and stays logarithmically away from polynomials, and when it is replaced with a random sequence of integers with sub-exponential growth. Similar results have been established by Boshernitzan when $X$ is the circle."}
{"category": "Math", "title": "A Stochastic Representation for Backward Incompressible Navier-Stokes Equations", "abstract": "By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer's forward formulations in \\cite{Co-Iy}. Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero."}
{"category": "Math", "title": "Computing inclusions of Schur modules", "abstract": "We describe a software package for constructing minimal free resolutions of GL_n(Q)-equivariant graded modules M over Q[x_1, ..., x_n] such that for all i, the ith syzygy module of M is generated in a single degree. We do so by describing some algorithms for manipulating polynomial representations of the general linear group GL_n(Q) following ideas of Olver and Eisenbud-Floystad-Weyman."}
{"category": "Math", "title": "On total dominating sets in graphs", "abstract": "A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if every vertex $v\\in V$ is adjacent to an element of $S$. The domination number of a graph $G$ denoted by $\\gamma(G)$ is the minimum cardinality of a dominating set in $G$. Respectively the total domination number of a graph $G$ denoted by $\\gamma_t(G)$ is the minimum cardinality of a total dominating set in $G$. An upper bound for $\\gamma_t(G)$ which has been achieved by Cockayne and et al. in $\\cite{coc}$ is: for any graph $G$ with no isolated vertex and maximum degree $\\Delta(G)$ and $n$ vertices, $\\gamma_t(G)\\leq n-\\Delta(G)+1$. Here we characterize bipartite graphs and trees which achieve this upper bound. Further we present some another upper and lower bounds for $\\gamma_t(G)$. Also, for circular complete graphs, we determine the value of $\\gamma_t(G)$."}
{"category": "Math", "title": "Parametrized Borsuk-Ulam problem for projective space bundles", "abstract": "Let $\\pi: E \\to B$ be a fiber bundle with fiber having the mod 2 cohomology algebra of a real or a complex projective space and let $\\pi^{'}: E^{'} \\to B$ be vector bundle such that $\\mathbb{Z}_2$ acts fiber preserving and freely on $E$ and $E^{'}-0$, where 0 stands for the zero section of the bundle $\\pi^{'}:E^{'} \\to B$. For a fiber preserving $\\mathbb{Z}_2$-equivariant map $f:E \\to E^{'}$, we estimate the cohomological dimension of the zero set $Z_f = \\{x \\in E | f(x)= 0\\}.$ As an application, we also estimate the cohomological dimension of the $\\mathbb{Z}_2$-coincidence set $A_f=\\{x \\in E | f(x) = f(T(x)) \\}$ of a fiber preserving map $f:E \\to E^{'}$."}
{"category": "Math", "title": "Polynomial Representation of $F_4$ and a New Combinatorial Identity about Twenty-Four", "abstract": "Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module that are nullified by positive root vectors. In this article, we use partial differential equations to find all the singular vectors of the polynomial representation of the simple Lie algebra of type $F_4$ over its basic irreducible module. As applications, we obtain a new combinatorial identity about the number 24 and explicit generators of invariants. Moreover, we show that the number of irreducible submodules contained in the space of homogeneous harmonic polynomials with degree $k\\geq 2$ is $\\geq [|k/3|]+[|(k-2)/3|]+2$."}
{"category": "Math", "title": "Scientists who engage with society perform better academically", "abstract": "Most scientific institutions acknowledge the importance of opening the so-called 'ivory tower' of academic research through popularization, industrial collaboration or teaching. However, little is known about the actual openness of scientific institutions and how their proclaimed priorities translate into concrete measures. This paper gives an idea of some actual practices by studying three key points: the proportion of researchers who are active in wider dissemination, the academic productivity of these scientists, and the institutional recognition of their wider dissemination activities in terms of their careers. We analyze extensive data about the academic production, career recognition and teaching or public/industrial outreach of several thousand of scientists, from many disciplines, from France's Centre National de la Recherche Scientifique. We find that, contrary to what is often suggested, scientists active in wider dissemination are also more active academically. However, their dissemination activities have almost no impact (positive or negative) on their careers."}
{"category": "Math", "title": "Dynamics of automorphisms on projective complex manifolds", "abstract": "We show that the dynamics of automorphisms on all projective complex manifolds X (of dimension 3, or of any dimension but assuming the Good Minimal Model Program or Mori's Program) are canonically built up from the dynamics on just three types of projective complex manifolds: complex tori, weak Calabi-Yau manifolds and rationally connected manifolds. As a by-product, we confirm the conjecture of Guedj for automorphisms on 3-dimensional projective manifolds, and also determine pi_1(X)."}
{"category": "Math", "title": "Low regularity for a quadratic Schr\\\"odinger equation on the circle", "abstract": "In this paper we consider a Schrodinger equation on the circle with a quadratic nonlinearity. Thanks to an explicit computation of the first Picard iterate, we give a precision on the dynamic of the solution, whose existence was proved by C. E. Kenig, G. Ponce and L. Vega. We also show that the equation is well-posed in a space based on Lp norms in frequencies."}
{"category": "Math", "title": "Metric aspects of noncommutative homogeneous spaces", "abstract": "For a closed cocompact subgroup $\\Gamma$ of a locally compact group $G$, given a compact abelian subgroup $K$ of $G$ and a homomorphism $\\rho:\\hat{K}\\to G$ satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations $C^*(\\hat{G}/\\Gamma, \\rho)$ of the homogeneous space $G/\\Gamma$, generalizing Rieffel's construction of quantum Heisenberg manifolds. We show that when $G$ is a Lie group and $G/\\Gamma$ is connected, given any norm on the Lie algebra of $G$, the seminorm on $C^*(\\hat{G}/\\Gamma, \\rho)$ induced by the derivation map of the canonical $G$-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on $\\rho$ continuously, with respect to quantum Gromov-Hausdorff distances."}
{"category": "Math", "title": "Smooth approximation of Lipschitz projections", "abstract": "We show that any Lipschitz projection-valued function p on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions q with Lipschitz constant close to that of p. This answers a question of Rieffel."}
{"category": "Math", "title": "Left-Garside categories, self-distributivity, and braids", "abstract": "In connection with the emerging theory of Garside categories, we develop the notions of a left-Garside category and of a locally left-Garside monoid. In this framework, the connection between the self-distributivity law LD and braids amounts to the result that a certain category associated with LD is a left-Garside category, which projects onto the standard Garside category of braids. This approach leads to a realistic program for establishing the Embedding Conjecture of [Dehornoy, Braids and Self-distributivity, Birkhauser (2000), Chap. IX]."}
{"category": "Math", "title": "On nearly radial marginals of high-dimensional probability measures", "abstract": "We prove that any absolutely continuous probability measure on a high-dimensional linear space has low-dimensional marginals that are approximately spherically-symmetric."}
{"category": "Math", "title": "Counting Descents in Standard Young Tableaux", "abstract": "This paper deals with the distribution of descent number in standard Young tableaux of certain shapes. A simple explicit formula is presented for the number of tableaux of any shape with two rows, with any specified number of descents. For shapes of three rows with one cell in the third row recursive formulas are given, and are solved in certain cases."}
{"category": "Math", "title": "Iteratively algebraic orders", "abstract": "A short proof of a theorem of M.H. Albert, and its application to lattices."}
{"category": "Math", "title": "Le theoreme de periodicite en K-theorie hermitienne", "abstract": "Bott periodicity plays an important role in topological K-theory. The purpose of this paper is to extend the periodicity theorem in a discrete context, where all classical groups are involved and not just the general linear group. The present paper generalizes previous results of the author [K1] and [K2], where 2 was assumed to be invertible in the rings involved. For the proof, two important ideas have to be mentioned : the first one is due to Ranicki [R] who introduced a kind of \"enlarged\" orthogonal group ; the second one is a genuine cup-product between quadratic forms due to Clauwens [C]. As an example of results obtained, we prove that the higher Witt groups of a finite field of characteristic 2 are all isomorphic to Z/2. They generalize in some sense the Dickson and Arf invariants."}
{"category": "Math", "title": "Elliptic equations in divergence form with partially BMO coefficients", "abstract": "The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients $a^{ij}$ are assumed to be measurable in one direction and have small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that $a^{ij}$ have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We also investigate elliptic equations in Sobolev spaces with mixed norms under the same assumptions on the coefficients."}
{"category": "Math", "title": "Averages of central L-values of Hilbert modular forms with an application to subconvexity", "abstract": "We use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change L-functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity problem for these L-functions. We also establish an equidistribution result for the Hecke eigenvalues weighted by these L-values."}
{"category": "Math", "title": "Geometric entropy of geodesic currents on free groups", "abstract": "A \\emph{geodesic current} on a free group $F$ is an $F$-invariant measure on the set $\\partial^2 F$ of pairs of distinct points of $\\partial F$. The space of geodesic currents on $F$ is a natural companion of Culler-Vogtmann's Outer space $cv(F)$ and studying them together yields new information about both spaces as well as about the group $Out(F)$. The main aim of this paper is to introduce and study the notion of {\\it geometric entropy} $h_T(\\mu)$ of a geodesic current $\\mu$ with respect to a point $T$ of $cv(F)$, which can be viewed as a length function on $F$. The geometric entropy is defined as the slowest rate of exponential decay of $\\mu$-measures of bi-infinite cylinders in $F$, as the $T$-length of the word defining such a cylinder goes to infinity. We obtain an explicit formula for $h_{T'}(\\mu_T)$, where $T,T'$ are arbitrary points in $cv(F)$ and where $\\mu_T$ denotes a Patterson-Sullivan current corresponding to $T$. It involves the volume entropy $h(T)$ and the extremal distortion of distances in $T$ with respect to distances in $T'$. It follows that, given $T$ in the projectivized outer space $CV(F)$, $h_{T'}(\\mu_T)$ as function of $T'\\in CV(F)$ achieves a strict global maximum at $T'=T$. We also show that for any $T\\in cv(F)$ and any geodesic current $\\mu$ on $F$, $h_T(\\mu)\\le h(T)$, where the equality is realized when $\\mu=\\mu_T$. For points $T\\in cv(F)$ with simplicial metric (where all edges have length one), we relate the geometric entropy of a current and the measure-theoretic entropy."}
{"category": "Math", "title": "Analytic subordination results in free probability from non-coassociative derivation-comultiplications", "abstract": "We extend Voiculescu's approach to analytic subordination through the coalgebra of the free difference quotient to non-coassociative derivation-comultiplications appearing in free probability theory. We obtain new proofs of Voiculescu's analytic subordination results for freely Markovian triples, and for multiplication of unitaries which are free with amalgamation."}
{"category": "Math", "title": "Ray Class Groups of Quadratic and Cyclotomic Fields", "abstract": "This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups. Also the explicit structure of ray class groups of regular cyclotomic number field is given."}
{"category": "Math", "title": "q-Wakimoto Modules and Integral Formulae of the Quantum Knizhnik-Zamolodchikov Equations", "abstract": "Matrix elements of intertwining operators between $q$-Wakimoto modules associated to the tensor product of representations of $U_q(\\widehat{sl_2})$ with arbitrary spins are studied. It is shown that they coincide with the Tarasov-Varchenko's formulae of the solutions of the qKZ equations. The result generalizes that of the previous paper [Kuroki K., Nakayashiki A., SIGMA 4 (2008), 049, 13 pages, arXiv:0802.1776]."}
{"category": "Math", "title": "Image and Reciprocal Image of a Measure. Compatibility Theorem", "abstract": "It is proposed that to the usual probability theory, three definitions and a new theorem are added, the resulting theory allows one to displace the central role usually given to the notion of conditional probability. When a mapping $\\phi$ is defined between two measurable spaces, to each measure $\\mu$ introduced on the first space, there corresponds an image $\\phi[\\mu]$ on the second space, and, reciprocally, to each measure $\\nu$ defined on the second space the corresponds a reciprocal image $\\phi^{-1}[\\nu]$ on the first space. As the intersection $\\cap$ of two measures is easy to introduce, a relation like $ \\phi[ \\mu \\cap \\phi^{-1} [\\nu] ] = \\phi[\\mu] \\cap \\nu $ makes sense. It is, indeed, a theorem of the theory. This theorem gives mathematical consistency to inferences drawn from physical measurements."}
{"category": "Math", "title": "Statistical Learning Theory: Models, Concepts, and Results", "abstract": "Statistical learning theory provides the theoretical basis for many of today's machine learning algorithms. In this article we attempt to give a gentle, non-technical overview over the key ideas and insights of statistical learning theory. We target at a broad audience, not necessarily machine learning researchers. This paper can serve as a starting point for people who want to get an overview on the field before diving into technical details."}
{"category": "Math", "title": "A Charlier-Parseval approach to Poisson approximation and its applications", "abstract": "A new approach to Poisson approximation is proposed. The basic idea is very simple and based on properties of the Charlier polynomials and the Parseval identity. Such an approach quickly leads to new effective bounds for several Poisson approximation problems. A selected survey on diverse Poisson approximation results is also given."}
{"category": "Math", "title": "Almost $\\cal D$-split sequences and derived equivalences", "abstract": "In this paper, we introduce almost $\\cal D$-split sequences and establish an elementary but somewhat surprising connection between derived equivalences and Auslander-Reiten sequences via BB-tilting modules. In particular, we obtain derived equivalences from Auslander-Reiten sequences (or $n$-almost split sequences), and Auslander-Reiten triangles."}
{"category": "Math", "title": "Derived equivalences and stable equivalences of Morita type, I", "abstract": "For self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this paper, we first show that each derived equivalence $F$ between the derived categories of Artin algebras $A$ and $B$ arises naturally a functor $\\bar{F}$ between their stable module categories, which can be used to compare certain homological dimensions of $A$ with that of $B$; and then we give a sufficient condition for the functor $\\bar{F}$ to be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classic result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras which are both derived-equivalent and stably equivalent of Morita type, thus they share many common invariants."}
{"category": "Math", "title": "Limits of bifractional Brownian noises", "abstract": "Let $B^{H,K}=(B^{H,K}_{t}, t\\geq 0)$ be a bifractional Brownian motion with two parameters $H\\in (0,1)$ and $K\\in(0,1]$. The main result of this paper is that the increment process generated by the bifractional Brownian motion $(B^{H,K}_{h+t} -B^{H,K}_{h}, t\\geq 0)$ converges when $h\\to \\infty$ to $(2^{(1-K)/{2}}B^{HK}_{t}, t\\geq 0)$, where $(B^{HK}_{t}, t\\geq 0)$ is the fractional Brownian motion with Hurst index $HK$. We also study the behavior of the noise associated to the bifractional Brownian motion and limit theorems to $B^{H,K}$."}
{"category": "Math", "title": "On connectedness and indecomposibility of local cohomology modules", "abstract": "Let $I$ denote an ideal of a local Gorenstein ring $(R, \\mathfrak m)$. Then we show that the local cohomology module $H^c_I(R), c = \\height I,$ is indecomposable if and only if $V(I_d)$ is connected in codimension one. Here $I_d$ denotes the intersection of the highest dimensional primary components of $I.$ This is a partial extension of a result shown by Hochster and Huneke in the case $I$ the maximal ideal. Moreover there is an analysis of connectedness properties in relation to various aspects of local cohomology. Among others we show that the endomorphism ring of $H^c_I(R)$ is a local Noetherian ring if $\\dim R/I = 1.$"}
{"category": "Math", "title": "When weak Hopf algebras are Frobenius", "abstract": "We investigate when a weak Hopf algebra H is Frobenius; we show this is not always true, but it is true if the semisimple base algebra A has all its matrix blocks of the same dimension. However, if A is a semisimple algebra not having this property, there is a weak Hopf algebra H with base A which is not Frobenius (and consequently, it is not Frobenius \"over\" A either). We give, moreover, a categorical counterpart of the result that a Hopf algebra is a Frobenius algebra for a noncoassociative generalization of weak Hopf algebra."}
{"category": "Math", "title": "Adaptive variance function estimation in heteroscedastic nonparametric regression", "abstract": "We consider a wavelet thresholding approach to adaptive variance function estimation in heteroscedastic nonparametric regression. A data-driven estimator is constructed by applying wavelet thresholding to the squared first-order differences of the observations. We show that the variance function estimator is nearly optimally adaptive to the smoothness of both the mean and variance functions. The estimator is shown to achieve the optimal adaptive rate of convergence under the pointwise squared error simultaneously over a range of smoothness classes. The estimator is also adaptively within a logarithmic factor of the minimax risk under the global mean integrated squared error over a collection of spatially inhomogeneous function classes. Numerical implementation and simulation results are also discussed."}
{"category": "Math", "title": "Dynamics of vortices for the Complex Ginzburg-Landau equation", "abstract": "We study a complex Ginzburg-Landau equation in the plane, which has the form of a Gross-Pitaevskii equation with some dissipation added. We focus on the regime corresponding to well-prepared unitary vortices and derive their asymptotic motion law."}
{"category": "Math", "title": "On the $U_p$ operator acting on $p$-adic overconvergent modular forms when $X_0(p)$ has genus 1", "abstract": "In this article we will show how to compute $U_p$ acting on spaces of overconvergent $p$-adic modular forms when $X_0(p)$ has genus 1. We first give a construction of Banach bases for spaces of overconvergent $p$-adic modular forms, and then give an algorithm to approximate both the characteristic power series of the $U_p$ operator and eigenvectors of finite slope for $U_p$, and present some explicit examples. We will also relate this to the conjectures of Clay on the slopes of overconvergent modular forms."}
{"category": "Math", "title": "The mutation class of $D_n$ quivers", "abstract": "We give an explicit description of the mutation class of quivers of type D."}
{"category": "Math", "title": "Stability of non-sticking periodic oscillations obtained via the averaging method in discontinuous systems. I. Smooth outside of the discontinuity surfaces systems", "abstract": "In this paper the statement of the second Bogolyubov's theorem on periodic solutions of smooth systems with small parameter is justified for discountinuous systems. It is assumed that the generating solution intersects the discontinuity hyperplanes transversally and that the system is smooth outside of these hyperplanes. Such a situation is natural for mechanical systems with dry friction and without constraints and sticking of oscillations. To illustrate the result we prove stability of the velocity of vibration-induced displacement."}
{"category": "Math", "title": "On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields", "abstract": "We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if W is such a variety, then every piecewise polynomial function on W can be written as suprema of infima of polynomial functions on W. More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of such varieties, which implies the Pierce-Birkhoff Conjecture."}
{"category": "Math", "title": "Robust nonparametric estimation via wavelet median regression", "abstract": "In this paper we develop a nonparametric regression method that is simultaneously adaptive over a wide range of function classes for the regression function and robust over a large collection of error distributions, including those that are heavy-tailed, and may not even possess variances or means. Our approach is to first use local medians to turn the problem of nonparametric regression with unknown noise distribution into a standard Gaussian regression problem and then apply a wavelet block thresholding procedure to construct an estimator of the regression function. It is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. A key technical result in our development is a quantile coupling theorem which gives a tight bound for the quantile coupling between the sample medians and a normal variable. This median coupling inequality may be of independent interest."}
{"category": "Math", "title": "The Toledo invariant on smooth varieties of general type", "abstract": "We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: assuming that the rank of the target Lie group is not greater than two, we prove that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations."}
{"category": "Math", "title": "On the number of inscribed squares of a simple closed curve in the plane", "abstract": "We show that for every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares."}
{"category": "Math", "title": "A SURE Approach for Digital Signal/Image Deconvolution Problems", "abstract": "In this paper, we are interested in the classical problem of restoring data degraded by a convolution and the addition of a white Gaussian noise. The originality of the proposed approach is two-fold. Firstly, we formulate the restoration problem as a nonlinear estimation problem leading to the minimization of a criterion derived from Stein's unbiased quadratic risk estimate. Secondly, the deconvolution procedure is performed using any analysis and synthesis frames that can be overcomplete or not. New theoretical results concerning the calculation of the variance of the Stein's risk estimate are also provided in this work. Simulations carried out on natural images show the good performance of our method w.r.t. conventional wavelet-based restoration methods."}
{"category": "Math", "title": "Analysis of variance, coefficient of determination and $F$-test for local polynomial regression", "abstract": "This paper provides ANOVA inference for nonparametric local polynomial regression (LPR) in analogy with ANOVA tools for the classical linear regression model. A surprisingly simple and exact local ANOVA decomposition is established, and a local R-squared quantity is defined to measure the proportion of local variation explained by fitting LPR. A global ANOVA decomposition is obtained by integrating local counterparts, and a global R-squared and a symmetric projection matrix are defined. We show that the proposed projection matrix is asymptotically idempotent and asymptotically orthogonal to its complement, naturally leading to an $F$-test for testing for no effect. A by-product result is that the asymptotic bias of the ``projected'' response based on local linear regression is of quartic order of the bandwidth. Numerical results illustrate the behaviors of the proposed R-squared and $F$-test. The ANOVA methodology is also extended to varying coefficient models."}
{"category": "Math", "title": "Densities, Laplace Transforms and Analytic Number Theory", "abstract": "Li showed that the Riemann Hypothesis is equivalent to the nonnegativity of a certain sequence of numbers. Bombieri and Lagarias gave an arithmetic formula for the number sequence based on the Guinand-Weil explicit formula and showed that Li's criterion is equivalent to Weil's criterion for the Riemann Hypothesis. We provide a derivation of the explicit formula based on Laplace transforms and present an alternative expression for Li's criterion that invites a probabilistic interpretation."}
{"category": "Math", "title": "Estimation of distributions, moments and quantiles in deconvolution problems", "abstract": "When using the bootstrap in the presence of measurement error, we must first estimate the target distribution function; we cannot directly resample, since we do not have a sample from the target. These and other considerations motivate the development of estimators of distributions, and of related quantities such as moments and quantiles, in errors-in-variables settings. We show that such estimators have curious and unexpected properties. For example, if the distributions of the variable of interest, $W$, say, and of the observation error are both centered at zero, then the rate of convergence of an estimator of the distribution function of $W$ can be slower at the origin than away from the origin. This is an intrinsic characteristic of the problem, not a quirk of particular estimators; the property holds true for optimal estimators."}
{"category": "Math", "title": "High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications", "abstract": "A high order, deterministic direct numerical method is proposed for the nonrelativistic $2D_{\\bf x} \\times 3D_{\\bf v}$ Vlasov-Maxwell system, coupled with Fokker-Planck-Landau type operators. Such a system is devoted to the modelling of electronic transport and energy deposition in the general frame of Inertial Confinement Fusion applications. It describes the kinetics of plasma physics in the nonlocal thermodynamic equilibrium regime. Strong numerical constraints lead us to develop specific methods and approaches for validation, that might be used in other fields where couplings between equations, multiscale physics, and high dimensionality are involved. Parallelisation (MPI communication standard) and fast algorithms such as the multigrid method are employed, that make this direct approach be computationally affordable for simulations of hundreds of picoseconds, when dealing with configurations that present five dimensions in phase space."}
{"category": "Math", "title": "Derived Representation Theory and the Algebraic K-theory of Fields", "abstract": "We describe the construction which takes as input a profinite group, which when applied the the absolute Galois group of a geometric field F agrees in some cases with the algebraic K-theory of F. We prove that it agrees in the case of a topologically finitely generated abelian absolute Galois group, and conjecture that it agrees for all geometric fields. We also discuss properties of the construction, including relationships with the representation theory of infinite discrete groups."}
{"category": "Math", "title": "A survey of Einstein metrics on 4-manifolds", "abstract": "We survey some aspects of the current state of research on Einstein metrics on compact 4-manifolds. A number of open problems are presented and discussed."}
{"category": "Math", "title": "Complex Valued Analytic Torsion for Flat Bundles and for Holomorphic Bundles with (1,1) Connections", "abstract": "The work of Ray and Singer which introduced analytic torsion, a kind of determinant of the Laplacian operator in topological and holomorphic settings, is naturally generalized in both settings. The couplings are extended in a direct way in the topological setting to general flat bundles and in the holomorphic setting to bundles with (1,1) connections, which using the Newlander-Nirenberg Theorem are seen to be the bundles with both holomorphic and anti-holomorphic structures. The resulting natural generalizations of Laplacians are not always self-adjoint and the corresponding generalizations of analytic torsions are thus not always real-valued. The Cheeger-Muller theorem, on equivalence in a topological setting of analytic torsion to classical topological torsion, generalizes to this complex-valued torsion. On the algebraic side the methods introduced include a notion of torsion associated to a complex equipped with both boundary and coboundry maps."}
{"category": "Math", "title": "Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications", "abstract": "In this paper we establish optimal pointwise decay estimates for non-dispersive (compact) radial solutions to non-linear wave equations in 3 dimensions, in the energy supercritical range. As an application, we show for the full energy supercritical range, in the defocusing case, that if the scale invariant Sobolev norm of a radial solution remains bounded in its maximal interval of existence, then the solution must exist for all times and scatter."}
{"category": "Math", "title": "Simplicial complexes and minimal free resolution of monomial algebras", "abstract": "This paper is concerned with the combinatorial description of the graded minimal free resolution of certain monomial algebras which includes toric rings. Concretely, we explicitly describe how the graded minimal free resolution of those algebras is related to the combinatorics of some simplicial complexes. Our description may be interpreted as an algorithmic procedure to partially compute this resolution."}
{"category": "Math", "title": "Concavity properties for free boundary elliptic problems", "abstract": "We prove some concavity properties connected to nonlinear Bernoulli type free boundary problems. In particular, we prove a Brunn-Minkowski inequality and an Urysohn's type inequality for the Bernoulli Constant and we study the behaviour of the free boundary with respect to the given boundary data. Moreover we prove a uniqueness result regarding the interior non-linear Bernoulli problem."}
{"category": "Math", "title": "Codimension one foliations with Bott-Morse singularities II", "abstract": "We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among other things, a good topological understanding of the leaves one may have around a center-type component of the singular set, and also of the topology of its basin. The stability theorem further allows the description of the topology of the boundary of the basin and how the topology of the leaves changes when passing from inside to outside the basin. This is described via fiberwise Milnor-Wallace surgery. A key-point for this is to show that if the boundary of the basin of a center is non-empty, then it contains a saddle; in this case we say that the center and the saddle are paired. We then describe the possible pairings one may have in dimension three and use a construction motivated by the classical saddle-node bifurcation, that we call foliated surgery, that allows the reduction of certain pairings of singularities of a foliation. This is used together with our previous work on the topic to prove an extension for 3-manifolds of Reeb's sphere recognition theorem."}
{"category": "Math", "title": "Gromov-Witten invariants for G/B and Pontryagin product for \\Omega K", "abstract": "We give an explicit formula for (T-equivariant) 3-pointed genus zero Gromov-Witten invariants for G/B. We derive it by finding an explicit formula for the equivariant Pontryagin product on the homology of the based loop group \\Omega K."}
{"category": "Math", "title": "Hom-bialgebras and comodule Hom-algebras", "abstract": "We study Hom-bialgebras and objects admitting coactions by Hom-bialgebras. In particular, we construct a Hom-bialgebra M representing the functor of 2x2-matrices on Hom-associative algebras. Then we construct a Hom-algebra analogue of the affine plane and show that it is a comodule Hom-algebra over M in a suitable sense. It is also shown that the enveloping Hom-associative algebra of a Hom-Lie algebra is naturally a Hom-bialgebra."}
{"category": "Math", "title": "On Systems of Equations over Free Partially Commutative Groups", "abstract": "Version 2: Corrected Section 3.3: instead of lexicographical normal forms we now use a normal form due to V. Diekert and A. Muscholl. Consequent changes made and some misprints corrected. Using an analogue of Makanin-Razborov diagrams, we give an effective description of the solution set of systems of equations over a partially commutative group (right-angled Artin group) $G$. Equivalently, we give a parametrisation of $Hom(H, G)$, where $H$ is a finitely generated group."}
{"category": "Math", "title": "Elementary equivalence of right-angled Coxeter groups and graph products of finite abelian groups", "abstract": "We show that graph products of finite abelian groups are elementarily equivalent if and only if they are $\\exists\\forall$-equivalent if and only if they are isomorphic. In particular, two right-angled Coxeter groups are elementarily equivalent if and only if they are isomorphic."}
{"category": "Math", "title": "Masaki Kashiwara and Algebraic Analysis", "abstract": "This paper is a brief overview of the main contributions of Masaki Kashiwara in the domain of microlocal and algebraic analysis."}
{"category": "Math", "title": "Shcherbina's Theorem for Finely Holomorphic Functions", "abstract": "We prove an analogue of Sadullaev's theorem concerning the size of the set where a maximal totally real manifold can meet a pluripolar set. The manifold has to be of class C-1 only. This readily leads to a version of Shcherbina's theorem for C-1 functions f that are defined in a neighborhood of certain compact sets K in the complex plane. If the graph of f on K is pluripolar, then f satisfies the Cauchy Riemann equations in the closure of the fine interior of K."}
{"category": "Math", "title": "The profile of bubbling solutions of a class of fourth order geometric equations on 4-manifolds", "abstract": "We study a class of fourth order geometric equations defined on a 4-dimensional compact Riemannian manifold which includes the Q-curvature equation. We obtain sharp estimates on the difference near the blow-up points between a bubbling sequence of solutions and the standard bubble."}
{"category": "Math", "title": "Orientability in Yang-Mills Theory over Nonorientable Surfaces", "abstract": "In arXiv:math/0605587, the first two authors have constructed a gauge-equivariant Morse stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface. This space can be identified with the real locus of the space of connections on the pullback of this bundle over the orientable double cover of this nonorientable surface. In this context, the normal bundles to the Morse strata are real vector bundles. We show that these bundles, and their associated homotopy orbit bundles, are orientable for any n when the nonorientable surface is not homeomorphic to the Klein bottle, and for n<4 when the nonorientable surface is the Klein bottle. We also derive similar orientability results when the structure group is SU(n)."}
{"category": "Math", "title": "In search of a Lebesgue density theorem for R^\\infty", "abstract": "We look at a measure, $\\lambda^\\infty$, on the infinite-dimensional space, ${\\mathbb R}^\\infty$, for which we attempt to put forth an analogue of the Lebesgue density theorem. Although this measure allows us to find partial results, for example for continuous functions, we prove that it is impossible to give an analogous theorem in full generality. In particular, we proved that the Lebesgue density of probability density functions on ${\\mathbb R}^\\infty$ is zero almost everywhere."}
{"category": "Math", "title": "Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations", "abstract": "We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the singular fiber are both rational homology manifolds. Also we introduce quasi-semistable degenerations and calculate the limit mixed Hodge structure by constructing the weight spectral sequence. As a corollary we show non-triviality of the space of vanishing cycles of the Lefschetz pencil associated with a tensor product of any two very ample line bundles except for the case of even-dimensional projective space where two has to be replaced by three."}
{"category": "Math", "title": "Asymptotics of multivariate sequences, part III: quadratic points", "abstract": "We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deformations as well as Fourier-Laplace transforms of generalized functions. We apply the results of the theory to specific combinatorial problems, such as Aztec diamond tilings, cube groves, and multi-set permutations."}
{"category": "Math", "title": "Klazar trees and perfect matchings", "abstract": "Martin Klazar computed the total weight of ordered trees under 12 different notions of weight. The last and perhaps most interesting of these weights, w_{12}, led to a recurrence relation and an identity for which he requested combinatorial explanations. Here we provide such explanations. To do so, we introduce the notion of a \"Klazar violator\" vertex in an increasing ordered tree and observe that w_{12} counts what we call Klazar trees--increasing ordered trees with no Klazar violators. A highlight of the paper is a bijection from n-edge increasing ordered trees to perfect matchings of [2n]={1,2,...,2n} that sends Klazar violators to even numbers matched to a larger odd number. We find the distribution of the latter matches and, in particular, establish the one-summation explicit formula sum_{k=1}^{lfloor n/2 rfloor}(2k-1)!!^2 StirlingPartition{n+1}{2k+1} for the number of perfect matchings of [2n] with no even-to-larger-odd matches. The proofs are mostly bijective."}
{"category": "Math", "title": "On the Study of Richard Tom Robert Identity", "abstract": "In order to estimate the average speed of mosquitoes, a simple experiment was designed by Richard (Lu-Hsing Tsai), Tom (Po-Yu Tsai) and Robert (Hung-Ming Tsai). The result of the experiment was posted in the science exhibitions Taichung Taiwan 1993. The average speed of mosquitoes is inferred by the simple relation that is obtained easily. In this paper, we will show how the data generated by computer. Though the rigorous proof is not shown, a sketch proof will be shown in this paper. There are five figures one table and one fortran computer source program in the end of the paper."}
{"category": "Math", "title": "Constructive proof of extended Kapranov theorem", "abstract": "Kapranov Theorem is a well known generalization of Newton-Puiseux theorem for the case of several variables. This theorem is stated mainly in the context of tropical geometry. We present a new, constructive proof, that also characterizes the possible principal terms of points in a hypersurface contained in the algebraic torus $(K^*)^n$."}
{"category": "Math", "title": "Generalized Demailly-Semple jet bundles and holomorphic mappings into complex manifolds", "abstract": "Motivated by the Green-Griffiths conjecture, we study maximal rank holomorphic maps from $\\C^p$ into complex manifolds. When $p>1$ such maps should in principle be more tractable than entire curves. We extend to this setting the jet-bundles techniques introduced by Semple, Green-Griffiths and Demailly. Our main application is the non-existence of maximal rank holomorphic maps from $\\C^2$ into the very general degree $d$ hypersurface in $\\bP^4$, as soon as $d\\geq 93.$"}
{"category": "Math", "title": "Structured vector bundles define differential K-theory", "abstract": "A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring K, elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary K and the Chern-Weil form. This construction provides a simple model of differential K-theory, c.f.Hopkins-Singer (2005), as well as a useful codification of vector bundles with connection."}
{"category": "Math", "title": "Combinatorial models of expanding dynamical systems", "abstract": "We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally defined \"Julia set\" of the generalized dynamical systems depends only on the associated iterated monodromy group. We show then that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules."}
{"category": "Math", "title": "Low eigenvalues and one-dimensional collapse", "abstract": "Our main result is that if a generic convex domain in $\\R^n$ collapses to a domain in $\\R^{n-1}$, then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the domain need not be smooth, merely Lipschitz continuous. To motivate the general case, we first prove the analogous result for triangular and polygonal domains. In so doing, we prove that the first two eigenvalues of triangular domains cannot be polyhomogeneous on the moduli space of triangles without blowing up a certain point. Our results show that the gap generically diverges under one dimensional collapse and is bounded only if the domain is sufficiently close to a rectangle in two dimensions or a cylinder in higher dimensions."}
{"category": "Math", "title": "A general theory of Finite State Backward Stochastic Difference Equations", "abstract": "By analogy with the theory of Backward Stochastic Differential Equations, we define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. This paper considers these processes as constructions in their own right, not as approximations to the continuous case. We establish the existence and uniqueness of solutions under weaker assumptions than are needed in the continuous time setting, and also establish a comparison theorem for these solutions. The conditions of this theorem are shown to approximate those required in the continuous time setting. We also explore the relationship between the driver $F$ and the set of solutions; in particular, we determine under what conditions the driver is uniquely determined by the solution. Applications to the theory of nonlinear expectations are explored, including a representation result."}
{"category": "Math", "title": "Fibrations of simplicial sets", "abstract": "There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen's axioms for a homotopy model category. The combinatorics underlying these fibrations is purely finitary and seems interesting both for its own sake and for its interaction with homotopy types. To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan's Ex functor act on graphs and on nerves of small categories."}
{"category": "Math", "title": "Linear Multistep Numerical Methods for Ordinary Differential Equations", "abstract": "A review of the most popular Linear Multistep (LM) Methods for solving Ordinary Differential Equations numerically is presented. These methods are first derived from first principles, and are discussed in terms of their order, consistency, and various types of stability. Particular varieties of stability that may not be familiar, are briefly defined first. The methods that are included are the Adams-Bashforth Methods, Adams-Moulton Methods, and Backwards Differentiation Formulas. Advantages and disadvantages of these methods are also described. Not much prior knowledge of numerical methods or ordinary differential equations is required, although knowledge of basic topics from calculus is assumed."}
{"category": "Math", "title": "Function Model of the Teichm\\\"uller space of a closed hyperbolic Riemann Surface", "abstract": "We introduce a function model for the Teichm\\\"uller space of a closed hyperbolic Riemann surface. Then we introduce a new metric by using the maximum norm on the function space on the Teichm\\\"uller space. We prove that the identity map from the Teichm\\\"uller space equipped with the usual Teichm\\\"uller metric to the Teichm\\\"uller space equipped with this new metric is uniformly continuous. Furthermore, we also prove that the inverse of the identity, that is, the identity map from the Teichm\\\"uller space equipped with this new metric to the Teichm\\\"uller space equipped with the usual Teichm\\\"uller metric, is continuous. Therefore, the topology induced by the new metric is just the same as the topology induced by the usual Teichm\\\"uller metric on the Teichm\\\"uller space. We give a remark about the pressure metric and the Weil-Petersson metric."}
{"category": "Math", "title": "Generic Uniqueness of Area Minimizing Disks for Extreme Curves", "abstract": "We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. We also show that the same is true for absolutely area minimizing surfaces."}
{"category": "Math", "title": "Maximum Multiplicity of a Root of the Matching Polynomial of a Tree and Minimum Path Cover", "abstract": "We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it."}
{"category": "Math", "title": "Large deviations for Branching Processes in Random Environment", "abstract": "A branching process in random environment $(Z_n, n \\in \\N)$ is a generalization of Galton Watson processes where at each generation the reproduction law is picked randomly. In this paper we give several results which belong to the class of {\\it large deviations}. By contrast to the Galton-Watson case, here random environments and the branching process can conspire to achieve atypical events such as $Z_n \\le e^{cn}$ when $c$ is smaller than the typical geometric growth rate $\\bar L$ and $ Z_n \\ge e^{cn}$ when $c > \\bar L$. One way to obtain such an atypical rate of growth is to have a typical realization of the branching process in an atypical sequence of environments. This gives us a general lower bound for the rate of decrease of their probability. When each individual leaves at least one offspring in the next generation almost surely, we compute the exact rate function of these events and we show that conditionally on the large deviation event, the trajectory $t \\mapsto \\frac1n \\log Z_{[nt]}, t\\in [0,1]$ converges to a deterministic function $f_c :[0,1] \\mapsto \\R_+$ in probability in the sense of the uniform norm. The most interesting case is when $c < \\bar L$ and we authorize individuals to have only one offspring in the next generation. In this situation, conditionally on $Z_n \\le e^{cn}$, the population size stays fixed at 1 until a time $ \\sim n t_c$. After time $n t_c$ an atypical sequence of environments let $Z_n$ grow with the appropriate rate ($\\neq \\bar L$) to reach $c.$ The corresponding map $f_c(t)$ is piecewise linear and is 0 on $[0,t_c]$ and $f_c(t) = c(t-t_c)/(1-t_c)$ on $[t_c,1].$"}
{"category": "Math", "title": "Newton polyhedra of discriminants of projections", "abstract": "For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving mixed fiber polyhedra and Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and generalizes a number of similar known results. The argument is based on a formula for the support function of a mixed fiber body, which also suggests a new proof for existence of mixed fiber bodies."}
{"category": "Math", "title": "Cyclotomic double affine Hecke algebras and affine parabolic category O, I", "abstract": "Using the orbifold KZ connection we construct a functor from an affine parabolic category O of type A to the category O of a cyclotomic rational double affine Hecke algebra. We give several results concerning this functor."}
{"category": "Math", "title": "On Stepwise Control of the Generalized Familywise Error Rate", "abstract": "A classical approach for dealing with the multiple testing problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of at least one false rejection. In many applications, one might be willing to tolerate more than one false rejection provided the number of such cases is controlled, thereby increasing the ability of the procedure to detect false null hypotheses. This suggests replacing control of the FWER by controlling the probability of $k$ or more false rejections, which is called the $k$-FWER. In this article, a unified approach is presented for deriving the $k$-FWER controlling procedures. We first generalize the well-known closure principle in the context of the FWER to the case of controlling the $k$-FWER. Then, we discuss how to derive the $k$-FWER controlling stepwise (stepdown or stepup) procedures based on marginal $p$-values using this principle. We show that, under certain conditions, generalized closed testing procedures can be reduced to stepwise procedures, and any stepwise procedure is equivalent to a generalized closed testing procedure. Finally, we generalize the well-known Hommel procedure in two directions, and show that any generalized Hommel procedure is equivalent to a generalized closed testing procedure with the same critical values."}
{"category": "Math", "title": "On an elliptic system with symmetric potential possessing two global minima", "abstract": "We consider the system {\\Delta}u - W_u (u) = 0, for u: R^2 -> R^2, W: R^2 -> R, where W_u (u) is a smooth potential, symmetric with respect to the u_1, u_2 axes, possessing two global minima a^\\pm := (\\pma,0) and two connections e^\\pm(x_1) connecting the minima. We prove that there exists an equivariant solution u(x_1, x_2) satisfying u(x_1, x_2) -> a^\\pm, as x_1 -> \\pminfiniti, and u(x_1, x_2) -> e^\\pm(x_1), as x_2 -> \\pminfiniti. The problem above was first studied by Alama, Bronsard, and Gui under related hypotheses to the ones introduced in the present paper. At the expense of one extra symmetry assumption, we avoid their considerations with the normalized energy and strengthen their result. We also provide examples for W."}
{"category": "Math", "title": "On concrete models for local operator spaces", "abstract": "In this short note, we propose a concrete analogue of the space $\\cL(H)$ for local operator spaces, the multinormed $C^*$-algebra $\\displaystyle\\prod_{\\alpha} \\cL(H_{\\alpha})$."}
{"category": "Math", "title": "Mean Curvature flow in Higher Co-dimension", "abstract": "We make several improvements on the results of M.-T. Wang in [8] and his joint paper with M.-P. Tsui [7] concerning the long time existence and convergence for solutions of mean curvature flow in higher co-dimension. Both the curvature condition and lower bound of $*\\Omega$ are weakened. New applications are also obtained."}
{"category": "Math", "title": "Contact pair structures and associated metrics", "abstract": "We introduce the notion of contact pair structure and the corresponding associated metrics, in the same spirit of the geometry of almost contact structures. We prove that, with respect to these metrics, the integral curves of the Reeb vector fields are geodesics and that the leaves of the Reeb action are totally geodesic. Mreover, we show that, in the case of a metric contact pair with decomposable endomorphism, the characteristic foliations are orthogonal and their leaves carry induced contact metric structures."}
{"category": "Math", "title": "Jump-Diffusions in Hilbert Spaces: Existence, Stability and Numerics", "abstract": "By means of an original approach, called \"method of the moving frame\", we establish existence, uniqueness and stability results for mild and weak solutions of stochastic partial differential equations (SPDEs) with path dependent coefficients driven by an infinite dimensional Wiener process and a compensated Poisson random measure. Our approach is based on a time-dependent coordinate transform, which reduces a wide class of SPDEs to a class of simpler SDE problems. We try to present the most general results, which we can obtain in our setting, within a self-contained framework to demonstrate our approach in all details. Also several numerical approaches to SPDEs in the spirit of this setting are presented."}
{"category": "Math", "title": "Prescribing curvatures on three dimensional Riemannian manifolds with boundaries", "abstract": "Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\\partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset of $\\bar M$, some of them may appear on $\\partial M$. We describe the asymptotic behavior of the blowup solutions around each blowup point and derive an energy estimate as a consequence."}
{"category": "Math", "title": "On the Calder\\'on-Zygmund lemma for Sobolev functions", "abstract": "We correct an inaccuracy in the original proof"}
{"category": "Math", "title": "Localisation de faisceaux caract\\`eres", "abstract": "Nous obtenons une formule pour les valeurs de la fonction caract\\'eristique d'un faisceau caract\\`ere en fonction de la th\\'eorie des repr\\'esentations de certains groupes finis, li\\'es au groupe de Weyl. Cette formule, qui g\\'en\\'eralise des r\\'esultats ant\\'erieurs de M{\\oe}glin et de Waldspurger, d\\'epend de la connaissance de certains sous-groupes r\\'eductifs admettant un faisceau caract\\`ere cuspidal. Dans un second temps, afin de rendre la formule plus explicite dans le cas d'un groupe quasi-simple, nous d\\'eterminons ces sous-groupes \\`a conjugaison pr\\`es. We obtain a formula for the values of the characteristic function of a character sheaf, in terms of the representation theory of certain finite groups related to the Weyl group. This formula, a generalization of previous results due to M{\\oe}glin and Waldspurger, depends on knowledge of certain reductive subgroups that admit cuspidal character sheaves. For quasi-simple groups, we make the formula truly explicit by determining all such subgroups upto conjugation."}
{"category": "Math", "title": "Addendum to \"Maximal regularity and Hardy spaces\"", "abstract": "We correct an inaccuracy in a previous article [Auscher, Pascal; Bernicot, Fr\\'ed\\'eric; Zhao, Jiman. Maximal regularity and Hardy spaces. Collect. Math. 59 (2008), no. 1, 103-127.]"}
{"category": "Math", "title": "Uniqueness of Morava K-theory", "abstract": "We show that there is an essentially unique S-algebra structure on the Morava K-theory spectrum K(n), while K(n) has uncountably many MU or \\hE{n}-algebra structures. Here \\hE{n} is the K(n)-localized Johnson-Wilson spectrum. To prove this we set up a spectral sequence computing the homotopy groups of the moduli space of A-infinity structures on a spectrum, and use the theory of S-algebra k-invariants for connective S-algebras due to Dugger and Shipley to show that all the uniqueness obstructions are hit by differentials."}
{"category": "Math", "title": "Dehn twists have roots", "abstract": "We construct nontrivial roots of Dehn twists about nonseparating curves."}
{"category": "Math", "title": "Explicit birational geometry of 3-folds of general type, I", "abstract": "Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V):=\\text{dim} H^0(V, 12K_V)>0$ and $P_{m_0}(V)>1$ for some positive integer $m_0\\leq 24$. A direct consequence is the birationality of the pluricanonical map $\\varphi_m$ for all $m\\geq 126$. Besides, the canonical volume $\\text{Vol}(V)$ has a universal lower bound $\\nu(3)\\geq \\frac{1}{63\\cdot 126^2}$."}
{"category": "Math", "title": "A characterization of sub-riemannian spaces as length dilatation structures constructed via coherent projections", "abstract": "We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained from pairs of length dilatation structures, the first being a tempered one and the second obtained via a coherent projection. Thus we get an intrinsic, synthetic, axiomatic description of sub-riemannian geometry, which transforms the classical construction of a Carnot-Caratheodory distance on a regular sub-riemannian manifold into a model for this abstract sub-riemannian geometry."}
{"category": "Math", "title": "On global H\\\"older estimates for optimal transportation", "abstract": "We generalize a well-known result of L. Caffarelli on Lipschitz estimates for optimal transportation $T$ between uniformly log-concave probability measures. Let $T : \\R^d \\to \\R^d$ be an optimal transportation pushing forward $\\mu = e^{-V}dx$ to $\\nu = e^{-W}dx$. Assume that 1) the second differential quotient of $V$ can be estimated from above by a power function, 2) modulus of convexity of $W$ can be estimated from below by $A_q |x|^{1+q}$, $q \\ge 1$. Under these assumptions we show that $T$ is globally H\\\"older with a dimension-free coefficient. In addition, we study optimal transportation $T$ between $\\mu$ and the uniform measure on a bounded convex set $K \\subset \\R^d$. We get estimates for the Lipschitz constant of $T$ in terms of $d$, ${diam(K)}$ and $D V, D^2 V$."}
{"category": "Math", "title": "Explicit birational geometry of 3-folds of general type, II", "abstract": "Let $V$ be a complex nonsingular projective 3-fold of general type. We shall give a detailed classification up to baskets of singularities on a minimal model of $V$. We show that the $m$-canonical map of $V$ is birational for all $m\\geq 73$ and that the canonical volume $\\text{Vol}(V)\\geq {1/2660}$. When $\\chi(\\mathcal{O}_V)\\leq 1$, our result is $\\text{Vol}(V)\\geq {1/420}$, which is optimal. Other effective results are also included in the paper."}
{"category": "Math", "title": "L2-Homogenization of Heat Equations on Tubular Neighborhoods", "abstract": "We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold. We show that, as the tube radius decreases, the semigroup of a suitably rescaled and renormalized generator can be effectively described by a Hamiltonian on the submanifold with a potential that depends on the geometry of the submanifold and of the embedding."}
{"category": "Math", "title": "Smooth Homogenization of Heat Equations on Tubular Neighborhoods", "abstract": "We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold. We show that, as the tube diameter tends to zero, a suitably rescaled and renormalized semigroup converges to a limit semigroup in Sobolev spaces of arbitrarily large Sobolev index."}
{"category": "Math", "title": "Structural properties of semilinear SPDEs driven by cylindrical stable processes", "abstract": "We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise.We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong Feller properties, and study integrability of trajectories. The obtained results can be applied to semilinear stochastic heat equations with Dirichlet boundary conditions and bounded and Lipschitz nonlinearities."}
{"category": "Math", "title": "Kirillov--Reshetikhin crystals for nonexceptional types", "abstract": "We provide combinatorial models for all Kirillov--Reshetikhin crystals of nonexceptional type, which were recently shown to exist. For types D_n^(1), B_n^(1), A_{2n-1}^(2) we rely on a previous construction using the Dynkin diagram automorphism which interchanges nodes 0 and 1. For type C_n^(1) we use a Dynkin diagram folding and for types A_{2n}^(2), D_{n+1}^(2) a similarity construction. We also show that for types C_n^(1) and D_{n+1}^(2) the analog of the Dynkin diagram automorphism exists on the level of crystals."}
{"category": "Math", "title": "L^p Bernstein estimates and approximation by spherical basis functions", "abstract": "The purpose of this paper is to establish L^p error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L^p Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L^p norm of the function itself. An important step in its proof involves measuring the L^p stability of functions in the approximating space in terms of the l^p norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the L^P norm. Finally, we give a new characterization of Besov spaces on the n-sphere in terms of spaces of SBFs."}
{"category": "Math", "title": "Demonstrative and non-demonstrative reasoning by analogy", "abstract": "The paper analizes a set of issues related to analogy and analogical reasoning, namely: 1) The problem of analogy and its duplicity; 2) The role of analogy in demonstrative reasoning; 3) The role of analogy in non-demonstrative reasoning; 4) The limits of analogy; 5) The convergence, particularly in multiple analogical reasoning, of these two apparently distinct aspects and its methodological and philosophical consequences. The paper, using example from number theory, argues for an heuristc conception of analogy."}
{"category": "Math", "title": "On some modular representations of the Borel subgroup of GL_2(Q_p)", "abstract": "Colmez has given a recipe to associate a smooth modular representation Omega(W) of the Borel subgroup of GL_2(Q_p) to a F_p^bar-representation W of Gal(Qp^bar/Qp) by using Fontaine's theory of (phi,Gamma)-modules. We compute Omega(W) explicitly and we prove that if W is irreducible and dim(W)=2, then Omega(W) is the restriction to the Borel subgroup of GL_2(Q_p) of the supersingular representation associated to W in Breuil's correspondence."}
{"category": "Math", "title": "On notions of harmonicity", "abstract": "In this paper, we address the equivalence of the analytic and probabilistic notions of harmonicity in the context of general symmetric Hunt processes on locally compact separable metric spaces. Extensions to general symmetric right processes on Lusin spaces including infinite dimensional spaces are mentioned at the end of this paper."}
{"category": "Math", "title": "On unique extension of time changed reflecting Brownian motions", "abstract": "Let $D$ be an unbounded domain in $\\RR^d$ with $d\\geq 3$. We show that if $D$ contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on $\\overline D$ is transient. Next assume that RBM $X$ on $\\overline D$ is transient and let $Y$ be its time change by Revuz measure ${\\bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable function $m$ on $\\overline D$. We further show that if there is some $r>0$ so that $D\\setminus \\overline {B(0, r)}$ is an unbounded uniform domain, then $Y$ admits one and only one symmetric diffusion that genuinely extends it and admits no killings. In other words, in this case $X$ (or equivalently, $Y$) has a unique Martin boundary point at infinity."}
{"category": "Math", "title": "Ein gesuchter, dennoch bislang unbekannter elementarer Satz", "abstract": "If and only if each point of a set of the phase-space is in the topological hull of a trajectory running through any other point of this set, we call this set a quasiergodic set. But which are these so defined quasiergodic sets in the case of a given continuous dynamical system, which has piecewise differentiable trajectories in a finit-dimensional real phase-space, which is compact? Let its trajectories define a field of normed tangents, which is continuous in almost each point of the phase-space: Then the topological hulls of all trajectories of the phase-space form a partition of it. Thus the elements of this partition are the quasiergodic sets of the given continuous dynamical system. This is the important but rather trivial statement of the elementary theorem 1.1, which we present in this tractatus. We call this theorem elementary, because it is limited to finit-dimensional real phase-spaces. We also find, that there is a linear homogeneous partial differential equation of first order, describing the invariants of the system, which allow us the construction of the quasiergodic sets. Furthermore we show, that any quasiergodic set is a sensitive attractor, if it is neither a closed trajectory or a fixed point."}
{"category": "Math", "title": "Generalizations of the Goulden-Jackson Cluster Method", "abstract": "We give several modifications of the Goulden-Jackson Cluster method for finding generating functions for words avoiding a given set of forbidden words. Our modifications include functions which can take into account various 'weights' on words, including single letter probability distributions, double letter (i.e. pairwise) probability distributions, and triple letter probability distributions. We also describe an alternative, recursive approach to finding such generating functions. We describe Maple implementations of the various modifications. The accompanying Maple package is available at the website for this paper."}
{"category": "Math", "title": "Ensemble Control of Finite Dimensional Time-Varying Linear Systems", "abstract": "In this article, we investigate the problem of simultaneously steering an uncountable family of finite dimensional time-varying linear systems. We call this class of control problems Ensemble Control, a notion coming from the study of spin dynamics in Nuclear Magnetic Resonance (NMR) spectroscopy and imaging (MRI). This subject involves controlling a continuum of parameterized dynamical systems with the same open-loop control input. From a viewpoint of mathematical control theory, this class of problems is challenging because it requires steering a continuum of dynamical systems between points of interest in an infinite dimensional state space by use of the same control function. The existence of such a control raises fundamental questions of ensemble controllability. We derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for ensemble control of time-varying linear systems. We show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics. In addition, we study the problem of optimal ensemble control of harmonic oscillators to demonstrate our main results. We show that the optimal solutions are pertinent to the study of time-frequency limited signals and prolate spheroidal wave functions. A systematic study of ensemble control systems has immediate applications to systems with parameter uncertainties as well as to broad areas of quantum control systems as arising in coherent spectroscopy and quantum information processing. The new mathematical structures appearing in such problems are an excellent motivation for new developments in control theory."}
{"category": "Math", "title": "A non-negative expansion for small Jensen-Shannon Divergences", "abstract": "In this report, we derive a non-negative series expansion for the Jensen-Shannon divergence (JSD) between two probability distributions. This series expansion is shown to be useful for numerical calculations of the JSD, when the probability distributions are nearly equal, and for which, consequently, small numerical errors dominate evaluation."}
{"category": "Math", "title": "A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity", "abstract": "We consider positive solutions of $\\Delta u-\\mu u+Ku^{\\frac{n+2}{n-2}}=0$ on $B_1$ ($n\\ge 5$) where $\\mu $ and $K>0$ are smooth functions on $B_1$. If $K$ is very sub-harmonic at each critical point of $K$ in $B_{2/3}$ and the maximum of $u$ in $\\bar B_{1/3}$ is comparable to its maximum over $\\bar B_1$, then all positive solutions are uniformly bounded on $\\bar B_{1/3}$. As an application, a priori estimate for solutions of equations defined on $\\mathbb S^n$ is derived."}
{"category": "Math", "title": "Iterated bar complexes of E-infinity algebras and homology theories", "abstract": "We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an E-infinity operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E-infinity algebras. We use this effective definition to prove that the n-fold bar complex B^n(A) admits an extension to categories of algebras over E_n-operads. Then we prove that the n-fold bar complex determines the homology theory associated to a category of algebras over E_n-operads. For n infinite, we obtain an isomorphism between the homology of an infinite bar construction and the usual Gamma-homology with trivial coefficients."}
{"category": "Math", "title": "Combinatorial bases of Feigin-Stoyanovsky's type subspaces of higher-level standard $\\tilde{\\gsl}(\\ell+1,\\C)$-modules", "abstract": "Let $\\tilde{\\mathfrak g}$ be an affine Lie algebra of the type $A_\\ell^{(1)}$. We find a combinatorial basis of Feigin-Stoyanovsky's type subspace $W(\\Lambda)$ given in terms of difference and initial conditions. Linear independence of the generating set is proved inductively by using coefficients of intertwining operators. A basis of $L(\\Lambda)$ is obtained as an \"inductive limit\" of the basis of $W(\\Lambda)$."}
{"category": "Math", "title": "A construction for coisotropic subalgebras of Lie bialgebras", "abstract": "Given a Lie bialgebra (g,g*), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g*. We write down families of examples for the case that g is a classical complex simple Lie algebra."}
{"category": "Math", "title": "Some computations about Kazhdan-Lusztig cells in affine Weyl groups of rank 2", "abstract": "In the last section of the paper \"Generalized induction of Kazhdan-Lusztig cells\" and in \"Kazhdan-Lusztig cells in affine Weyl groups of rank 2\" the author described the partition into Kazhdan-Lusztig cells of the affine Weyl groups of rank 2 for all choices of parameters. The proof of these results relies on some explicit computations with GAP. In these notes we give some details of these computations."}
{"category": "Math", "title": "A generalization of the Collatz problem and conjecture", "abstract": "We introduce an infinite set of integer mappings that generalize the well-known Collatz-Ulam mapping and we conjecture that an infinite subset of these mappings feature the remarkable property of the Collatz conjecture, namely that they converge to unity irrespective of which positive integer is chosen initially."}
{"category": "Math", "title": "Action and Index Spectra and Periodic Orbits in Hamiltonian Dynamics", "abstract": "The main theme of this paper is the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension. These generalizations of the Conley conjecture follow from another result proved here asserting that a Hamiltonian diffeomorphism with a symplectically degenerate maximum on a closed rational manifold has infinitely many periodic orbits. We also show that for a broad class of manifolds and/or Hamiltonian diffeomorphisms the minimal action--index gap remains bounded for some infinite sequence of iterations and, as a consequence, whenever a Hamiltonian diffeomorphism has finitely many periodic orbits, the actions and mean indices of these orbits must satisfy a certain relation. Furthermore, for Hamiltonian diffeomorphisms of the n-dimensional complex projective space with exactly n+1 periodic orbits a stronger result holds. Namely, for such a Hamiltonian diffeomorphism, the difference of the action and the mean index on a periodic orbit is independent of the orbit, provided that the symplectic structure on the projective space is normalized to be in the same cohomology class as the first Chern class."}
{"category": "Math", "title": "Derived Smooth Manifolds", "abstract": "We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local $C^\\infty$-rings that is obtained by patching together homotopy zero-sets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection $A\\cap B$ of submanifolds $A,B\\subset X$ exists on the categorical level in our theory, and a cup product formula $$[A]\\smile[B]=[A\\cap B]$$ holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a {\\em categorification} of intersection theory."}
{"category": "Math", "title": "Global Existence Of Smooth Solutions Of A 3D Loglog Energy-Supercritical Wave Equation", "abstract": "We prove global existence of smooth solutions of the 3D loglog energy-supercritical wave equation $\\partial_{tt} u - \\triangle u = -u^{5} \\log^{c} (log(10+u^{2})) $ with $0 < c < {8/225}$ and smooth initial data $(u(0)=u_{0}, \\partial_{t} u(0)=u_{1})$. First we control the $L_{t}^{4} L_{x}^{12}$ norm of the solution on an arbitrary size time interval by an expression depending on the energy and an \\textit{a priori} upper bound of its $L_{t}^{\\infty} \\tilde{H}^{2}(\\mathbb{R}^{3})$ norm, with $\\tilde{H}^{2}(\\mathbb{R}^{3}):=\\dot{H}^{2}(\\mathbb{R}^{3}) \\cap \\dot{H}^{1}(\\mathbb{R}^{3})$. The proof of this long time estimate relies upon the use of some potential decay estimates \\cite{bahger, shatstruwe} and a modification of an argument in \\cite{taolog}. Then we find an \\textit{a posteriori} upper bound of the $L_{t}^{\\infty} \\tilde{H}^{2}(\\mathbb{R}^{3})$ norm of the solution by combining the long time estimate with an induction on time of the Strichartz estimates."}
{"category": "Math", "title": "The Modular number, Congruence number, and Multiplicity One", "abstract": "Let $N$ be a positive integer and let $f$ be a newform of weight 2 on $\\Gamma_0(N)$. In earlier joint work with K. Ribet and W. Stein, we introduced the notions of the modular number and the congruence number of the quotient abelian variety $A_f$ of $J_0(N)$ associated to the newform $f$. These invariants are analogs of the notions of the modular degree and congruence primes respectively associated to elliptic curves. We show that if $p$ is a prime such that every maximal ideal of the Hecke algebra of characteristic $p$ that contains the annihilator ideal of $f$ satisfies multiplicity one, then the modular number and the congruence number have the same $p$-adic valuation."}
{"category": "Math", "title": "A visible factor for analytic rank one", "abstract": "Let $E$ be an optimal elliptic curve of conductor $N$, such that the $L$-function of $E$ vanishes to order one at $s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing $N$ split and such that the $L$-function of $E$ over $K$ also vanishes to order one at $s=1$. In view of the Gross-Zagier theorem, the second part of the Birch and Swinnerton-Dyer conjecture says that the index in $E(K)$ of the subgroup generated by the Heegner point is equal to the product of the Manin constant of $E$, the Tamagawa numbers of $E$, and the square root of the order of the Shafarevich-Tate group of $E$ (over $K$). We extract an integer factor from the index mentioned above and relate this factor to certain congruences of the newform associated to $E$ with eigenforms of analytic rank bigger than one. We use the theory of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime $q$ divides this factor, then $q$ divides the order of the Shafarevich-Tate group or the order of an arithmetic component group of $E$, as predicted by the second part of the Birch and Swinnerton-Dyer conjecture."}
{"category": "Math", "title": "Squareness in the special L-value and special L-values of twists", "abstract": "Let N be a prime and let A be a quotient of J_0(N) over Q associated to a newform such that the special L-value of A (at s=1) is non-zero. Suppose that the algebraic part of the special L-value of A is divisible by an odd prime q such that q does not divide the numerator of (N-1)/12. Then the Birch and Swinnerton-Dyer conjecture predicts that the q-adic valuations of the algebraic part of the special L-value of A and of the order of the Shafarevich-Tate group are both positive even numbers. Under a certain mod q non-vanishing hypothesis on special L-values of twists of A, we show that the q-adic valuations of the algebraic part of the special L-value of A and of the Birch and Swinnerton-Dyer conjectural order of the Shafarevich-Tate group of A are both positive even numbers. We also give a formula for the algebraic part of the special L-value of A over quadratic imaginary fields K in terms of the free abelian group on isomorphism classes of supersingular elliptic curves in characteristic N (equivalently, over conjugacy classes of maximal orders in the definite quaternion algebra over Q ramified at N and infinity) which shows that this algebraic part is a perfect square up to powers of the prime two and of primes dividing the discriminant of K. Finally, for an optimal elliptic curve E, we give a formula for the special L-value of the twist E_D of E by a negative fundamental discriminant -D, which shows that this special L-value is an integer up to a power of 2, under some hypotheses. In view of the second part of the Birch and Swinnerton-Dyer conjecture, this leads us to the surprising conjecture that the square of the order of the torsion subgroup of E_D divides the product of the order of the Shafarevich-Tate group of E_D and the orders of the arithmetic component groups of E_D, under certain mild hypotheses."}
{"category": "Math", "title": "Rational torsion in elliptic curves and the cuspidal subgroup", "abstract": "Let $A$ be an elliptic curve over $\\Q$ of square free conductor $N$. We prove that if $A$ has a rational torsion point of prime order $r$ such that $r$ does not divide $6N$, then $r$ divides the order of the cuspidal subgroup of $J_0(N)$."}
{"category": "Math", "title": "Representation dimension of $m$-replicated algebras", "abstract": "Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field and $A^{(m)}$ the $m$-replicated algebra of $A$. We prove that the representation dimension of $A^{(m)}$ is at most three, and that the dominant dimension of $A^{(m)}$ is at least $m$."}
{"category": "Math", "title": "Symmetric Schroder paths and restricted involutions", "abstract": "Let $A_k$ be the set of permutations in the symmetric group $S_k$ with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns $A_k$. We present a bijection between symmetric Schroder paths of length $2n$ and involutions of length $n+1$ avoiding $\\mathcal{A}_4$. Statistics such as the number of right-to-left maxima and fixed points of the involution correspond to the number of steps in the symmetric Schroder path of a particular type. For each $k> 2$ we determine the generating function for the number of involutions avoiding the subsequences in $A_k$, according to length, first entry and number of fixed points."}
{"category": "Math", "title": "Partial tilting modules over $m$-replicated algebras", "abstract": "Let $A$ be a hereditary algebra over an algebraically closed field $k$ and $A^{(m)}$ be the $m$-replicated algebra of $A$. Given an $A^{(m)}$-module $T$, we denote by $\\delta (T)$ the number of non isomorphic indecomposable summands of $T$. In this paper, we prove that a partial tilting $A^{(m)}$-module $T$ is a tilting $A^{(m)}$-module if and only if $\\delta (T)=\\delta (A^{(m)})$, and that every partial tilting $A^{(m)}$-module has complements. As an application, we deduce that the tilting quiver $\\mathscr{K}_{A^{(m)}}$ of $A^{(m)}$ is connected. Moreover, we investigate the number of complements to almost tilting modules over duplicated algebras."}
{"category": "Math", "title": "Complete bounded holomorphic curves immersed in C^2 with arbitrary genus", "abstract": "In the previous paper, the authors constructed a complete holomorphic immersion of the unit disk D into C^2 whose image is bounded. In this paper, we shall prove existence of complete holomorphic null immersions of Riemann surfaces with arbitrary genus and finite topology, whose image is bounded in C^2. To construct such immersions, we apply the method used by F. J. Lopez to perturb the genus zero example changing its genus. As an analogue the above construction, we also give a new method to construct complete bounded minimal immersions (resp. weakly complete maximal surface) with arbitrary genus and finite topology in Euclidean 3-space (resp. Lorentz-Minkowski 3-spacetime)."}
{"category": "Math", "title": "Near optimal thresholding estimation of a Poisson intensity on the real line", "abstract": "The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $\\tilde{f}_{n,\\gamma}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a possible logarithmic term. Then, minimax properties of $\\tilde{f}_{n,\\gamma}$ on Besov spaces ${\\cal B}^{\\ensuremath \\alpha}_{p,q}$ are established. Under mild assumptions, we prove that $$\\sup_{f\\in B^{\\ensuremath \\alpha}_{p,q}\\cap \\ensuremath \\mathbb {L}_{\\infty}} \\ensuremath \\mathbb {E}(\\ensuremath | | \\tilde{f}_{n,\\gamma}-f| |_2^2)\\leq C(\\frac{\\log n}{n})^{\\frac{\\ensuremath \\alpha}{\\ensuremath \\alpha+{1/2}+({1/2}-\\frac{1}{p})_+}}$$ and the lower bound of the minimax risk for ${\\cal B}^{\\ensuremath \\alpha}_{p,q}\\cap \\ensuremath \\mathbb {L}_{\\infty}$ coincides with the previous upper bound up to the logarithmic term. This new result has two consequences. First, it establishes that the minimax rate of Besov spaces ${\\cal B}^{\\ensuremath \\alpha}_{p,q}$ with $p\\leq 2$ when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. When $p>2$, the rate exponent, which depends on $p$, deteriorates when $p$ increases, which means that the support plays a harmful role in this case. Furthermore, $\\tilde{f}_{n,\\gamma}$ is adaptive minimax up to a logarithmic term."}
{"category": "Math", "title": "Algorithms for Producing and Ordering Lexical and Nonlexical Sequences out of one Element", "abstract": "This paper deals with algorithms for producing and ordering lexical and nonlexical sequences of a given degree. The notion of \"elementary operations\" on positive integral sequences is introduced. Our main theorem answers the question of when two lexical sequences are adjacent."}
{"category": "Math", "title": "Signal Acquisition from Measurements via Non-Linear Models", "abstract": "We consider the problem of reconstruction of a non-linear finite-parametric model $M=M_p(x),$ with $p=(p_1,...,p_r)$ a set of parameters, from a set of measurements $\\mu_j(M)$. In this paper $\\mu_j(M)$ are always the moments $m_j(M)=\\int x^jM_p(x)dx$. This problem is a central one in Signal Processing, Statistics, and in many other applications. We concentrate on a direct (and somewhat \"naive\") approach to the above problem: we simply substitute the model function $M_p(x)$ into the measurements $\\mu_j$ and compute explicitly the resulting \"symbolic\" expressions of $\\mu_j(M_p)$ in terms of the parameters $p$. Equating these \"symbolic\" expressions to the actual measurement results, we produce a system of nonlinear equations on the parameters $p$, which we consequently try to solve. The aim of this paper is to review some recent results (mostly of \\cite{Vet5,Mil1,Mil2,Put1,Vet4,Vet3,Mil3,Mil4,Vet2}) in this direction, stressing the algebraic structure of the arising systems and mathematical tools required for their solutions. In particular, we discuss the relation of the reconstruction problem above with the recent results of \\cite{bfy,bry,chr,pak1,pak2,pak3,pry,ry} on the vanishing problem of generalized polynomial moments and on the Cauchy-type integrals of algebraic functions. The accompanying paper \\cite{Kis1} (this volume) provides a solution method for a wide class of reconstruction problems as above, based on the study of linear differential equations with rational coefficient, which are satisfied by the moment generating function of the problem."}
{"category": "Math", "title": "Morasses and finite support iterations", "abstract": "We introduce a method of constructing a forcing along a simplified $(\\kappa,1)$-morass such that the forcing satisfies the $\\kappa$-chain condition. Alternatively, this may be seen as a method to thin out a larger forcing to get a chain condition. As an application, we construct a ccc forcing that adds an $\\omega_2$-Suslin tree. Related methods are Shelah's historic forcing and Todorcevic's $\\rho$-functions."}
{"category": "Math", "title": "Limiting Distribution of Frobenius Numbers for $n=3$", "abstract": "The purpose of this paper is to give a complete derivation of the limiting distribution of large Frobenius numbers outlined in earlier work of J. Bourgain and Ya. Sinai and fill some gaps formulated there as hypotheses."}
{"category": "Math", "title": "Substitution Tilings and Separated Nets with Similarities to the Integer Lattice", "abstract": "We show that any primitive substitution tiling of the plane creates a separated net which is biLipschitz to the integer lattice. Then we show that if H is a primitive Pisot substitution in an Euclidean space, for every separated net Y, that corresponds to some tiling of the tiling space, there exists a bijection F between Y and the integer lattice that translate every element of Y a bounded distance. As a corollary we get that we have such an F for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner."}
{"category": "Math", "title": "On the representation ring of the polynomial algebra over a perfect field", "abstract": "We consider the tensor product of modules over the polynomial algebra corresponding to the usual tensor product of linear operators. We present a general description of the representation ring in case the ground field k is perfect. It is made explicit in the special cases when k is real closed respectively algebraically closed. Furthermore, we discuss the generalisation of this problem to representations of quivers. In particular the representation ring of quivers of extended Dynkin type A is provided."}
{"category": "Math", "title": "On General Balance Laws with Boundary", "abstract": "This paper is devoted to general balance laws (with a possibly non local source term) with a non-characteristic boundary. Basic well posedness results are obtained, trying to provide sharp estimates. In particular, bounds tend to blow up as the boundary tends to be characteristic. New uniqueness results for the solutions to conservation and/or balance laws with boundary are also provided."}
{"category": "Math", "title": "The classification of complete stable area-stationary surfaces in the Heisenberg group $\\mathbb{H}^1$", "abstract": "We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$."}
{"category": "Math", "title": "Spectral geometry, link complements and surgery diagrams", "abstract": "We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of hyperbolic alternating link complements is expanding if and only if they have bounded volume. We also provide examples of hyperbolic 3-manifolds which require 'complicated' surgery diagrams, thereby proving that a recent theorem of Constantino and Thurston is sharp. Along the way, we find a new upper bound on the bridge number of a knot that is not tangle composite, in terms of the twist number of any diagram of the knot. The proofs rely on a theorem of Lipton and Tarjan on planar graphs, and also the relationship between many different notions of width for knots and 3-manifolds."}
{"category": "Math", "title": "The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution", "abstract": "In 1976 Procesi and Schacher developed an Artin-Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size."}
{"category": "Math", "title": "Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces", "abstract": "We compute the Szeg\\\"o kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in $\\CC^n$ for Grassmannian manifolds of higher ranks. In particular they provide an infinite family of smoothly bounded strictly pseudo-convex domains on complex manifolds for which the log terms in the Fefferman expansion of the Szeg\\\"o kernel vanish and which are not diffeomorphic to the sphere. The analogous results for the Bergman kernel are also obtained."}
{"category": "Math", "title": "Radon, cosine and sine transforms on Grassmannian manifolds", "abstract": "Let $G_{n,r}(\\bbK)$ be the Grassmannian manifold of $k$-dimensional $\\bbK$-subspaces in $\\bbK^n$ where $\\bbK=\\mathbb R, \\mathbb C, \\mathbb H$ is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine transforms, $\\mathcal R_{r^\\prime, r}$, $\\mathcal C_{r^\\prime, r}$ and $\\mathcal S_{r^\\prime, r}$, from the $L^2$ space $L^2(G_{n,r}(\\bbK))$ to the space $L^2(G_{n,r^\\prime}(\\bbK))$, for $r, r^\\prime \\le n-1$. The $L^2$ spaces are decomposed into irreducible representations of $G$ with multiplicity free. We compute the spectral symbols of the transforms under the decomposition. For that purpose we prove two Bernstein-Sato type formulas on general root systems of type BC for the sine and cosine type functions on the compact torus $\\mathbb R^r/{2\\pi Q^\\vee}$ generalizing our recent results for the hyperbolic sine and cosine functions on the non-compact space $\\mathbb R^r$. We find then also a characterization of the images of the transforms. Our results generalize those of Alesker-Bernstein and Grinberg. We prove further that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and we give the unitary structure of the Stein's complementary series in the compact picture."}
{"category": "Math", "title": "Degenerate p-Laplacian operators on H-type groups and applications to Hardy type inequalities", "abstract": "Let $\\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\\mathfrak G=V\\oplus \\mathfrak t$. We define a class of vector fields $X=\\{X_j\\}$ on $\\mathbb G$ depending on a real parameter $k\\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= \\text{div}_X (|\\na_{X} u|^{p-2} \\na_X u)$. For $k=1$ the vector fields $X=\\{X_j\\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$, for $k=2$ and $\\mathbb G$ being the Heisenberg group they are introduced by Greiner \\cite{Greiner-cjm79}. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$."}
{"category": "Math", "title": "Second order structures for sprays and connections on Frechet manifolds", "abstract": "Ambrose, Palais and Singer \\cite{Ambrose} introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath \\cite{Kumar} extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Frechet manifolds. It is proved that the existence of Christoffel and Hessian structures, connections, sprays and dissections are equivalent on those Frechet manifolds which can be considered as projective limits of Banach manifolds. These concepts provide also an alternative way for the study of ordinary differential equations on non-Banach infinite dimensional manifolds. Concrete examples of the structures are provided using direct and flat connections."}
{"category": "Math", "title": "On the hyperbolic automorphisms of the 2-torus and their Markov partitions", "abstract": "In the paper we give an introduction to Anosov diffeomorphisms, ways to represent their chaotic properties and some historical remarks on this subject. A complete classification of hyperbolic linear automorphisms of 2-torus is presented. We introduce a notion of pre-Markov partition for such automorphisms and give their classification and an algorithm for their construction."}
{"category": "Math", "title": "Choice of neighbor order in nearest-neighbor classification", "abstract": "The $k$th-nearest neighbor rule is arguably the simplest and most intuitively appealing nonparametric classification procedure. However, application of this method is inhibited by lack of knowledge about its properties, in particular, about the manner in which it is influenced by the value of $k$; and by the absence of techniques for empirical choice of $k$. In the present paper we detail the way in which the value of $k$ determines the misclassification error. We consider two models, Poisson and Binomial, for the training samples. Under the first model, data are recorded in a Poisson stream and are \"assigned\" to one or other of the two populations in accordance with the prior probabilities. In particular, the total number of data in both training samples is a Poisson-distributed random variable. Under the Binomial model, however, the total number of data in the training samples is fixed, although again each data value is assigned in a random way. Although the values of risk and regret associated with the Poisson and Binomial models are different, they are asymptotically equivalent to first order, and also to the risks associated with kernel-based classifiers that are tailored to the case of two derivatives. These properties motivate new methods for choosing the value of $k$."}
{"category": "Math", "title": "On the structure of some moduli spaces of finite flat group schemes", "abstract": "We consider the moduli space, in the sense of Kisin, of finite flat models of a 2-dimensional representation with values in a finite field of the absolute Galois group of a totally ramified extension of $mathbb{Q}_p$. We determine the connected components of this space and describe its irreducible components. These results prove a modified version of a conjecture of Kisin."}
{"category": "Math", "title": "Automated computation of robust normal forms of planar analytic vector fields", "abstract": "We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighborhood of the saddle point. The normal form is suitable for computations aimed at enclosing the flow close to the saddle, and the time it takes a trajectory to pass it. Several examples illustrate the usefulness of this method."}
{"category": "Math", "title": "On weak generalized stability and (c,d)-pseudostable random variables via functional equations", "abstract": "In this paper we give a first attempt to define and study stable distributions with respect to the weak generalized convolution, focusing our attention on the symmetric weakly stable distribution. As in the case of the classical convolution, characterization of distributions stable in the sense of the weak generalized convolution depends on solving some functional equations in the class of characteristic functions."}
{"category": "Math", "title": "Aggregation of penalized empirical risk minimizers in regression", "abstract": "We give a general result concerning the rates of convergence of penalized empirical risk minimizers (PERM) in the regression model. Then, we consider the problem of agnostic learning of the regression, and give in this context an oracle inequality and a lower bound for PERM over a finite class. These results hold for a general multivariate random design, the only assumption being the compactness of the support of its law (allowing discrete distributions for instance). Then, using these results, we construct adaptive estimators. We consider as examples adaptive estimation over anisotropic Besov spaces or reproductive kernel Hilbert spaces. Finally, we provide an empirical evidence that aggregation leads to more stable estimators than more standard cross-validation or generalized cross-validation methods for the selection of the smoothing parameter, when the number of observation is small."}
{"category": "Math", "title": "Poincar\\'e lemma and global homotopy formulas with sharp anisotropic H\\\"older estimates in q-concave CR manifolds", "abstract": "We prove sharp anisotropic H\\\"older estimates for the local solutions of the tangential Cauchy-Riemann equation in q-concave CR manifolds and we derive the same kind of estimates for global solutions when the manifold is compact."}
{"category": "Math", "title": "Ten colours in quasiperiodic and regular hyperbolic tilings", "abstract": "Colour symmetries with ten colours are presented for different tilings. In many cases, the existence of these colourings were predicted by group theoretical methods. Only in a few cases explicit constructions were known, sometimes using combination of two-colour and five-colour symmetries. Here we present explicit constructions of several of the predicted colourings for the first time, and discuss them in contrast to already known colourings with ten colours."}
{"category": "Math", "title": "Liouville type theorems for conformal Gaussian curvature equation", "abstract": "In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -\\Delta u=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite total curvature solutions. When $K$ is monotone non-decreasing along every ray starting at origin, we can prove a non-existence result too. We use moving plane method and moving sphere method."}
{"category": "Math", "title": "A class of R\\'{e}nyi information estimators for multidimensional densities", "abstract": "A class of estimators of the R\\'{e}nyi and Tsallis entropies of an unknown distribution $f$ in $\\mathbb{R}^m$ is presented. These estimators are based on the $k$th nearest-neighbor distances computed from a sample of $N$ i.i.d. vectors with distribution $f$. We show that entropies of any order $q$, including Shannon's entropy, can be estimated consistently with minimal assumptions on $f$. Moreover, we show that it is straightforward to extend the nearest-neighbor method to estimate the statistical distance between two distributions using one i.i.d. sample from each. (Wit Correction.)"}
{"category": "Math", "title": "Duality between Hyperbolic and de Sitter Geometry", "abstract": "In this paper we describe trigonometry on the de Sitter surface. For that a characterization of geodesics is given, leading to various types of triangles. We define lengths and angles of these. Then, transferring the concept of polar triangles from spherical geometry into the Minkowski space, we relate hyperbolic with de Sitter triangles such that the proof of the hyperbolic law of cosines for angles becomes much clearer and easier than it is traditionally. Furthermore, polar triangles turn out to be a powerful tool for describing de Sitter trigonometry."}
{"category": "Math", "title": "Burnside-Brauer Theorem and Character Products in Table Algebras", "abstract": "In this paper, we first show that the irreducible characters of a quotient table algebra modulo a normal closed subset can be viewed as the irreducible characters of the table algebra itself. Furthermore, we define the character products for table algebras and give a condition in which the products of two characters are characters. Thereafter, as a main result we state and prove the Burnside-Brauer Theorem on finite groups for table algebras."}
{"category": "Math", "title": "Hermitian modular forms congruent to 1 modulo p", "abstract": "For any natural number $\\ell $ and any prime $p\\equiv 1 \\pmod{4}$ not dividing $\\ell $ there is a Hermitian modular form of arbitrary genus $n$ over $L:=\\Q [\\sqrt{-\\ell}]$ that is congruent to 1 modulo $p$ which is a Hermitian theta series of an $O_L$-lattice of rank $p-1$ admitting a fixed point free automorphism of order $p$. It is shown that also for non-free lattices such theta series are modular forms."}
{"category": "Math", "title": "Coactions on Hochschild Homology of Hopf-Galois Extensions and Their Coinvariants", "abstract": "Let A be an H-Galois extension of B. If M is a Hopf bimodule then HH.(A,M), the Hochschild homology of A with coefficients in M, is a right comodule over the coalgebra C:=H/[H,H]. Given an injective left C-comodule V, we denote the cotensor product of M and V by N. Our aim is to investigate the relationship between the cotensor product of HH.(A,M) and V, on the one hand, and HH.(B,N) on the other hand. The roots of this problem can be found in Lorenz's work on the descent of Hochschild homology of centrally Galois extensions, where HH.(A)^G, the subspace of invariant cycles with respect to the action of the Galois group, and HH.(B) are shown to be isomorphic."}
{"category": "Math", "title": "Euler-Mahonian distributions of type $B_n$", "abstract": "Adin, Brenti, and Roichman introduced the pairs of statistics $(\\ndes, \\nmaj)$ and $(\\fdes, \\fmaj)$. They showed that these pairs are equidistributed over the hyperoctahedral group $B_n$, and can be considered \"Euler-Mahonian\" in that they generalize the Carlitz identity. Further, they asked whether there exists a bijective proof of the equidistribution of their statistics. We give such a bijection, along with a new proof of the generalized Carlitz identity."}
{"category": "Math", "title": "The Algorithmic Behaviour of the F5 Algorithm", "abstract": "This paper has been withdrawn due to an error in the proof of the termination of the algorithm. Moreover, some ideas of this paper are a joint work with John Perry and it is inappropriate to mention only me as an author. John Perry was in spite of the statement in the acknowledgment of previous versions not proofreading this paper. Recently we have found a new solution for ensuring F5's termination, which can be found here : arXiv:1006.0318."}
{"category": "Math", "title": "Interface evolution: water waves in 2-D", "abstract": "We study the free boundary evolution between two irrotational, incompressible and inviscid fluids in 2-D without surface tension. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition. The well-posedness of the full water wave problem was first obtained by Wu \\cite{Wu}. The methods introduced in this paper allows us to consider multiple cases: with or without gravity, but also a closed boundary or a periodic boundary with the fluids placed above and below it. It is assumed that the initial interface does not touch itself, being a part of the evolution problem to check that such property prevails for a short time, as well as it does the Rayleigh-Taylor condition, depending conveniently upon the initial data. The addition of the pressure equality to the contour dynamic equations is obtained as a mathematical consequence, and not as a physical assumption, from the mere fact that we are dealing with weak solutions of Euler's equation in the whole space."}
{"category": "Math", "title": "Orthogonally additive holomorphic functions of bounded type over $C(K)$", "abstract": "It is known that all $k$-homogeneous orthogonally additive polynomials $P$ over $C(K)$ are of the form $$ P(x)=\\int_K x^k d\\mu . $$ Thus $x\\mapsto x^k$ factors all orthogonally additive polynomials through some linear form $\\mu$. We show that no such linearization is possible without homogeneity. However, we also show that every orthogonally additive holomorphic functions of bounded type $f$ over $C(K)$ is of the form $$ f(x)=\\int_K h(x) d\\mu $$ for some $\\mu$ and holomorphic $h\\colon C(K) \\to L^1(\\mu)$ of bounded type."}
{"category": "Math", "title": "Convergent and divergent numbers games for certain collections of edge-weighted graphs", "abstract": "The numbers game is a one-player game played on a finite simple graph with certain \"amplitudes\" assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. Here, the edge amplitudes will be negative integers. Combinatorial methods are used to investigate the convergence and divergence of numbers games played on certain such graphs. The results obtained here provide support for results in a companion paper."}
{"category": "Math", "title": "A note on spectral triples and quasidiagonality", "abstract": "Spectral triples (of compact type) are constructed on arbitrary separable quasidiagonal C*-algebras. On the other hand an example of a spectral triple on a non-quasidiagonal algebra is presented."}
{"category": "Math", "title": "The C$^*$-envelope of a semicrossed product and Nest Representations", "abstract": "Let $X$ be compact Hausdorff, and $\\phi: X \\to X$ a continuous surjection. Let $\\mathcal{A}$ be the semicrossed product algebra corresponding to the relation fU = Uf\\circ \\phi$ or to the relation $Uf = f\\circ \\phi U.$ Then the C$^*$-envelope of $\\mathcal{A}$ is the crossed product of a commutative C$^*$-algebra which contains $C(X)$ as a subalgebra, with respect to a homeomorphism which we construct. We also show there are\"sufficiently many\" nest representations."}
{"category": "Math", "title": "On unitary unipotent representations of $p$-adic groups and affine Hecke algebras with unequal parameters", "abstract": "We determine the unitary dual of the geometric graded Hecke algebras with {unequal} parameters which appear in Lusztig's classification of unipotent representations for {exceptional} $p$-adic groups. The largest such algebra is of type $F_4.$ Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent representations with real central character of the $p$-adic groups. In order for this correspondence to be applicable here, we show (following Lusztig's geometric classification, and Barbasch and Moy's original argument) that the set of tempered modules with real central character for a geometric graded Hecke algebra is linearly independent when restricted to the Weyl group."}
{"category": "Math", "title": "The numbers game and Dynkin diagram classification results", "abstract": "The numbers game is a one-player game played on a finite simple graph with certain \"amplitudes\" assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. Combinatorial reasoning is used to show that those connected graphs with negative integer amplitudes for which the numbers game meets a certain finiteness requirement are precisely the Dynkin diagrams associated with the finite-dimensional complex simple Lie algebras. This strengthens a result originally due to the second author. A more general result is obtained when certain real number amplitudes are allowed. The resulting graphs are in families, each family corresponding to a finite irreducible Coxeter group. These results are used to demonstrate that the only generalized Cartan matrices for which there exist finite edge-colored ranked posets enjoying a certain structure property are the Cartan matrices for the finite-dimensional complex semisimple Lie algebras. In this setting, classifications of the finite-dimensional Kac--Moody algebras and of the finite Coxeter and Weyl groups are re-derived."}
{"category": "Math", "title": "Divisors in the moduli spaces of curves", "abstract": "In this mostly expository paper we review several known results about the cohomology of moduli spaces of smooth and stable curves, focusing in particular on low degree cohomology. We also give a new proof of Harer's theorem describing the second cohomology group of the moduli space of smooth n-pointed curves of given genus"}
{"category": "Math", "title": "Median structures on asymptotic cones and homomorphisms into mapping class groups", "abstract": "The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, sending limits of hierarchy paths onto geodesics, and with image a median subspace. One of the applications is that a group with Kazhdan's property (T) can have only finitely many pairwise non-conjugate homomorphisms into a mapping class group. We also give a new proof of the rank conjecture of Brock and Farb (previously proved by Behrstock and Minsky, and independently by Hamenstaedt)."}
{"category": "Math", "title": "The critical dimension for a fourth order elliptic problem with singular nonlinearity", "abstract": "We study the regularity of the extremal solution of the semilinear biharmonic equation $\\bi u=\\f{\\lambda}{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\\subset\\IR^N$, under Dirichlet boundary conditions $u=\\partial_\\nu u=0$ on $\\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \\cite{LY} regarding the identification of a \"pull-in voltage\" $\\la^*>0$ such that a stable classical solution $u_\\la$ with $0<u_\\la<1$ exists for $\\la\\in (0,\\la^*)$, while there is none of any kind when $\\la>\\la^*$. Our main result asserts that the extremal solution $u_{\\lambda^*}$ is regular $(\\sup_B u_{\\lambda^*} <1)$ provided $ N \\le 8$ while $u_{\\lambda^*} $ is singular ($\\sup_B u_{\\lambda^*} =1$) for $N \\ge 17$, in which case $1-C_0|x|^{4/3}\\leq u_{\\lambda^*} (x) \\leq 1-|x|^{4/3}$ on the unit ball, where $ C_0:= <(\\frac{\\lambda^*}{\\bar{\\lambda}}>)^{1/3}$ and $ \\bar{\\lambda}:= \\frac{8 (N-{2/3}) (N- {8/3})}{9}$. The singular character of the extremal solution for the remaining cases (i.e., when $9\\leq N\\leq 16$) requires a computer assisted proof and will not be addressed in this paper."}
{"category": "Math", "title": "Iwahori-Hecke type algebras associated with the Lie superalgebras A(m,n), B(m,n), C(n) and D(m,n)", "abstract": "In this paper we give Iwahori-Hecke type algebras H_q(g) associated with the Lie superalgebras g=A(m,n), B(m,n), C(n) and D(m,n). We classify the irreducible representations of H_q(g) for generic q."}
{"category": "Math", "title": "Geodesic Webs on a Two-Dimensional Manifold and Euler Equations", "abstract": "We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar results are obtained for planar d-webs, d > 4, provided that additional d-4 second-order invariants vanish."}
{"category": "Math", "title": "Ising vectors in the vertex operator algebra $V_{\\Lambda}^+$ associated with the Leech lattice $\\Lambda$", "abstract": "In this article, we study the Ising vectors in the vertex operator algebra $V_\\Lambda^+$ associated with the Leech lattice $\\Lambda$. The main result is a characterization of the Ising vectors in $V_\\Lambda^+$. We show that for any Ising vector $e$ in $V_\\Lambda^+$, there is a sublattice $E\\cong \\sqrt{2}E_8$ of $\\Lambda$ such that $e\\in V_E^+$. Some properties about their corresponding $\\tau$-involutions in the moonshine vertex operator algebra $V^\\natural$ are also discussed. We show that there is no Ising vector of $\\sigma$-type in $V^\\natural$. Moreover, we compute the centralizer $C_{\\aut V^\\natural}(z, \\tau_e)$ for any Ising vector $e\\in V_\\Lambda^+$, where $z$ is a 2B element in $\\aut V^\\natural$ which fixes $V_\\Lambda^+$. Based on this result, we also obtain an explanation for the 1A case of an observation by Glauberman-Norton (2001), which describes some mysterious relations between the centralizer of $z$ and some 2A elements commuting $z$ in the Monster and the Weyl groups of certain sublattices of the root lattice of type $E_8$ ."}
{"category": "Math", "title": "Geodesic Webs and PDE Systems of Euler Equations", "abstract": "We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x_{1},...,x_{n}) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d > n) of hypersurfaces to be hyperplanar webs. These conditions are systems of generalized Euler equations, and for flat connections we give an explicit construction of their solutions."}
{"category": "Math", "title": "Calculation of Coefficients of the Optimal Quadrature Formulas in the $W_2^{m,m-1}(0,1)$ Space", "abstract": "In this paper problem of construction of optimal quadrature formulas in $W_2^{(m,m-1)}(0,1)$ space is considered. Here by using Sobolev's algorithm when $m=1,2$ we find the optimal coefficients of the quadrature formulas of the form $$ \\int\\limits_0^1\\phi(x)dx\\cong \\sum\\limits_{\\beta=0}^NC_{\\beta}\\phi(x_{\\beta}). $$"}
{"category": "Math", "title": "Properties of Discrete Analogue of the Differential Operator $\\frac{d^{2m}}{dx^{2m}}-\\frac{d^{2m-2}}{dx^{2m-2}}$", "abstract": "In the paper properties of the discrete analogue $D_m(h\\beta)$ of the differential operator $\\frac{d^{2m}}{dx^{2m}}-\\frac{d^{2m-2}}{dx^{2m-2}}$ are studied. It is known, that zeros of differential operator $\\frac{d^{2m}}{dx^{2m}}-\\frac{d^{2m-2}}{dx^{2m-2}}$ are functions $e^x$, $e^{-x}$ and $P_{2m-3}(x)$. It is proved that discrete analogue $D_m(h\\beta)$ of this differential operator also have similar properties."}
{"category": "Math", "title": "On the cohomology groups of holomorphic Banach bundles", "abstract": "We consider a compact complex manifold $M$, and introduce the notion of two holomorphic Banach bundles $E,F$ over $M$ being compact perturbations of one another. Given two such bundles we show that if the cohomology groups $H^q(M,E)$ are finite dimensional then so are the cohomology groups $H^q(M,F)$; as well as a more precise result in the same spirit."}
{"category": "Math", "title": "How hot can a heat bath get?", "abstract": "We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for 'extreme' non-equilibrium statistical mechanics. We provide a full picture of the long-time behaviour of such a system, including the existence / non-existence of a non-equilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state. Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is 'too stiff', then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds."}
{"category": "Math", "title": "A note on Talagrand's transportation inequality and logarithmic Sobolev inequality", "abstract": "We give by simple arguments sufficient conditions, so called Lyapunov conditions, for Talagrand's transportation information inequality and for the logarithmic Sobolev inequality. Those sufficient conditions work even in the case where the Bakry-Emery curvature is not lower bounded. Several new examples are provided."}
{"category": "Math", "title": "Projective pairs of profinite groups", "abstract": "We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic extension of a PAC field K. Then M/K is PAC if and only if the corresponding pair of absolute Galois groups (Gal(M),Gal(K)) is projective. Moreover any projective pair can be realized as absolute Galois groups of a PAC extension of a PAC field. Using this characterization we construct new examples of PAC extensions of relatively small fields, e.g., unbounded abelian extensions of the rational numbers."}
{"category": "Math", "title": "Vector product algebras", "abstract": "Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result."}
{"category": "Math", "title": "Age-dependent equations with non-linear diffusion", "abstract": "We consider the well-posedness of models involving age structure and non-linear diffusion. Such problems arise in the study of population dynamics. It is shown how diffusion and age boundary conditions can be treated that depend non-linearly and possibly non-locally on the density itself. The abstract approach is depicted with examples."}
{"category": "Math", "title": "Approximating the marginal likelihood using copula", "abstract": "Model selection is an important activity in modern data analysis and the conventional Bayesian approach to this problem involves calculation of marginal likelihoods for different models, together with diagnostics which examine specific aspects of model fit. Calculating the marginal likelihood is a difficult computational problem. Our article proposes some extensions of the Laplace approximation for this task that are related to copula models and which are easy to apply. Variations which can be used both with and without simulation from the posterior distribution are considered, as well as use of the approximations with bridge sampling and in random effects models with a large number of latent variables. The use of a t-copula to obtain higher accuracy when multivariate dependence is not well captured by a Gaussian copula is also discussed."}
{"category": "Math", "title": "Complexity of PL-manifolds", "abstract": "We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant). Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspherical manifolds (in particular, on manifolds with non-positive sectional curvature). There are finitely many closed hyperbolic manifolds of any given complexity. On the other hand, there are many closed 4-manifolds of complexity zero (manifolds without 3-handles, doubles of 2-handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary fundamental group)."}
{"category": "Math", "title": "Arithmetic Fujita approximation", "abstract": "We prove an arithmetic analogue of Fujita's approximation theorem in Arakelov geometry, conjectured by Moriwaki, by using slope method and measures associated to $\\mathbb R$-filtrations."}
{"category": "Math", "title": "Numerably Contractible Spaces", "abstract": "Numerably contractible spaces play an important role in the theory of homotopy pushouts and pullbacks. The corresponding results imply that a number of well known weak homotopy equivalences are genuine ones if numerably contractible spaces are involved. In this paper we give a first systematic investigation of numerably contractible spaces. We list the elementary properties of the category of these spaces. We then study simplicial objects in this category. In particular, we show that the topological realization functor preserves fibration sequences if the base is path-connected and numerably contractible in each dimension. Consequently, the loop space functor commutes with realization up to homotopy. We give simple conditions which assure that free algebras over a topological operad are numerably contractible."}
{"category": "Math", "title": "A differential Chevalley theorem", "abstract": "We prove a differential analog of a theorem of Chevalley on extending homomorphisms for rings with commuting derivations, generalizing a theorem of Kac. As a corollary, we establish that, under suitable hypotheses, the image of a differential scheme under a finite morphism is a constructible set. We also obtain a new algebraic characterization of differentially closed fields. We show that similar results hold for differentially closed fields that are saturated, in the sense of model theory. In characteristic p > 0, we obtain related results and establish a differential Nullstellensatz. Analogs of these theorems for difference fields are also considered."}
{"category": "Math", "title": "Remarks on missing faces and generalized lower bounds on face numbers", "abstract": "We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension. Sharp analogues of McMullen's generalized lower bounds, and of Barnette's lower bounds, are conjectured for these families of complexes. Some partial results on these conjectures are presented."}
{"category": "Math", "title": "The Frolicher-Nijenhuis Calculus in Synthetic Differential Geometry", "abstract": "Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity."}
{"category": "Math", "title": "Geometric construction of crystal bases for quantum generalized Kac-Moody algebras", "abstract": "We provide a geometric realization of the crystal $B(\\infty)$ for quantum generalized Kac-Moody algebras in terms of the irreducible components of certain Lagrangian subvarieties in the representation spaces of a quiver."}
{"category": "Math", "title": "A problem in the Kourovka notebook concerning the number of conjugacy classes of a finite group", "abstract": "In this paper, we consider Problem 14.44 in the Kourovka notebook, which is a conjecture about the number of conjugacy classes of a finite group. While elementary, this conjecture is still open and appears to elude any straightforward proof, even in the soluble case. However, we do prove that a minimal soluble counterexample must have certain properties, in particular that it must have Fitting height at least 3 and order at least 2000."}
{"category": "Math", "title": "Two-dimensional quantum random walk", "abstract": "We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a compact algebraic variety. We also compute the probability profile within the limit region, which is essentially a negative power of the Gaussian curvature of the same algebraic variety. Our methods are based on analysis of the space-time generating function, following the methods of Pemantle and Wilson (2002)."}
{"category": "Math", "title": "Neighboring ternary cyclotomic coefficients differ by at most one", "abstract": "A cyclotomic polynomial Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd prime factors. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Eli Leher showed in 2007 that neighboring ternary cyclotomic coefficients differ by at most four. We show that, in fact, they differ by at most one. Consequently, the set of coefficients occurring in a ternary cyclotomic polynomial consists of consecutive integers. As an application we reprove in a simpler way a result of Bachman from 2004 on ternary cyclotomic polynomials with an optimally large set of coefficients."}
{"category": "Math", "title": "Hydrodynamics, probability and the geometry of the diffeomorphisms group", "abstract": "We characterize the solution of Navier-Stokes equation as a stochastic geodesic on the diffeomorphisms group, thus generalizing Arnold's description of the Euler flow."}
{"category": "Math", "title": "Alvis-Curtis duality, central characters, and real-valued characters", "abstract": "We prove that Alvis-Curtis duality preserves the Frobenius-Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius-Schur indicator of a regular real-valued character to its central character. We apply these results to compute the Frobenius-Schur indicators of certain real-valued, irreducible, Frobenius-invariant Deligne-Lusztig characters, and the Frobenius-Schur indicators of real-valued regular and semisimple characters of finite unitary groups."}
{"category": "Math", "title": "Representations of cyclic groups acting on complete simplicial fans", "abstract": "Let $sigma$ be a complete simplicial fan in finite dimensional real Euclidean space $V$, and let $G$ be a cyclic subgroup of $GL(V)$ which acts properly on $\\sigma$. We show that the representation of $G$ carried by the cohomology of $X_{sigma}$, the toric variety associated to $sigma$, is a permutation representation."}
{"category": "Math", "title": "Introduction to Graph-Link Theory", "abstract": "The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with the same equivalence relations we get `graph-links'. On one hand graph-links generalise the notion of virtual link, on the other hand they do not feel link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalisation of the Kauffman-Murasugi-Thistlethwaite theorem on `minmal diagrams' for graph-links"}
{"category": "Math", "title": "Boxicity of Circular Arc Graphs", "abstract": "A $k$-dimensional box is the cartesian product $R_1 \\times R_2 \\times ... \\times R_k$ where each $R_i$ is a closed interval on the real line. The {\\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of a collection of $k$-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. Let $G$ be a circular arc graph with maximum degree $\\Delta$. We show that if $\\Delta <\\lfloor \\frac{n(\\alpha-1)}{2\\alpha}\\rfloor$, $\\alpha \\in \\mathbb{N}$, $\\alpha \\geq 2$ then $box(G) \\leq \\alpha$. We also demonstrate a graph with boxicity $> \\alpha$ but with $\\Delta=n\\frac{(\\alpha-1)}{2\\alpha}+\\frac{n}{2\\alpha(\\alpha+1)}+(\\alpha+2)$. So the result cannot be improved substantially when $\\alpha$ is large. Let $r_{inf}$ be minimum number of arcs passing through any point on the circle with respect to some circular arc representation of $G$. We also show that for any circular arc graph $G$, $box(G) \\leq r_{inf} + 1$ and this bound is tight. Given a family of arcs $F$ on the circle, the circular cover number $L(F)$ is the cardinality of the smallest subset $F'$ of $F$ such that the arcs in $F'$ can cover the circle. Maximum circular cover number $L_{max}(G)$ is defined as the maximum value of $L(F)$ obtained over all possible family of arcs $F$ that can represent $G$. We will show that if $G$ is a circular arc graph with $L_{max}(G)> 4$ then $box(G) \\leq 3$."}
{"category": "Math", "title": "The inverse conjecture for the Gowers norm over finite fields via the correspondence principle", "abstract": "The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\\F$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\\|f\\|_{U^d(V)}$ if and only if it correlates with a phase polynomial $\\phi = e_\\F(P)$ of degree at most $d-1$, thus $P: V \\to \\F$ is a polynomial of degree at most $d-1$. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case $\\charac(F) \\geq d$ from an ergodic theory counterpart, which was recently established by Bergelson and the authors. In low characteristic we obtain a partial result, in which the phase polynomial $\\phi$ is allowed to be of some larger degree $C(d)$. The full inverse conjecture remains open in low characteristic; the counterexamples by Lovett-Meshulam-Samorodnitsky or Green-Tao in this setting can be avoided by a slight reformulation of the conjecture."}
{"category": "Math", "title": "Characterization of Line Arrangement for which the Fundamental Group of the Complement is a Direct Product of Free Groups", "abstract": "Kwai Man Fan proved that if the intersection lattice of a line arrangement does not contain a cycle, then the fundamental group of its complement is a direct sum of infinite and cyclic free groups. He also conjectured that the converse is true as well. The main purpose of this paper is to prove this conjecture"}
{"category": "Math", "title": "Uniform approximation on ideals of multilinear mappings", "abstract": "For each ideal of multilinear mappings $\\cal M$ we explicitly construct a corresponding ideal $^{a}{\\cal M}$ such that multilinear forms in $^{a}{\\cal M}$ are exactly those which can be approximated, in the uniform norm, by multilinear forms in ${\\cal M}$. This construction is then applied to finite type, compact, weakly compact and absolutely summing multilinear mappings. It is also proved that the correspondence ${\\cal M} \\mapsto ^{a}{\\cal M}$ is Aron-Berner stability preserving."}
{"category": "Math", "title": "Exponential Squared Integrability for the Discrepancy Function in Two Dimensions", "abstract": "Let A_N be an N-point distribution in the unit square in the Euclidean plane. We consider the Discrepancy function D_N(x) in two dimensions with respect to rectangles with lower left corner anchored at the origin and upper right corner at the point x. This is the difference between the actual number of points of A_N in such a rectangle and the expected number of points - N x_1x_2 - in the rectangle. We prove sharp estimates for the BMO norm and the exponential squared Orlicz norm of D_N(x). For example we show that necessarily ||D_N||_(expL^2) >c(logN)^(1/2) for some aboslute constant c>0. On the other hand we use a digit scrambled version of the van der Corput set to show that this bound is tight in the case N=2^n, for some positive integer n. These results unify the corresponding classical results of Roth and Schmidt in a sharp fashion."}
{"category": "Math", "title": "The Hall algebra of a spherical object", "abstract": "We determine the Hall algebra, in the sense of Toen, of the algebraic triangulated category generated by a spherical object."}
{"category": "Math", "title": "On the Stochastic Rank of Metric Functions", "abstract": "For a class of integral operators with kernels metric functions on manifold we find some necessary and sufficient conditions to have finite rank. The problem we pose has a stochastic nature and boils down to the following alternative question. For a random sample of discrete points, what will be the probability the symmetric matrix of pairwise distances to have full rank? When the metric is an analytic function, the question finds full and satisfactory answer. As an important application, we consider a class of tensor systems of equations formulating the problem of recovering a manifold distribution from its covariance field and solve this problem for representing manifolds such as Euclidean space and unit sphere."}
{"category": "Math", "title": "Topological regluing of rational functions", "abstract": "Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston--Teichm\\\"uller theory. We will discuss a topological theory of regluing, and trace a direction in which a holomorphic theory can develop."}
{"category": "Math", "title": "Well, Papa, can you multiply triplets?", "abstract": "We show that the classical algebra of quaternions is a commutative $\\Z_2\\times\\Z_2\\times\\Z_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible."}
{"category": "Math", "title": "Limit Behaviour of Sequential Empirical Measure Processes", "abstract": "In this paper, we obtain some uniform laws of large numbers and functional central limit theorems for sequential empirical measure processes indexed by classes of product functions satisfying appropriate Vapnik-Chervonenkis properties."}
{"category": "Math", "title": "Homology of framed links embedded in thickened surfaces", "abstract": "We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based on ideas coming from Asaeda, Przytycki and Sikora's categorification of the Kauffman bracket skein module of I-bundles over surfaces. This is accomplished by borrowing ideas from Bar-Natan's Khovanov homology theory for tangles and cobordisms and using embedded surfaces to generate the chain groups, instead of diagrams."}
{"category": "Math", "title": "An example of Brunet-Derrida behavior for a branching-selection particle system on $\\Z$", "abstract": "We consider a branching-selection particle system on $\\Z$ with $N \\geq 1$ particles. During a branching step, each particle is replaced by two new particles, whose positions are shifted from that of the original particle by independently performing two random walk steps according to the distribution $p \\delta_{1} + (1-p) \\delta_{0}$, from the location of the original particle. During the selection step that follows, only the N rightmost particles are kept among the 2N particles obtained at the branching step, to form a new population of $N$ particles. After a large number of iterated branching-selection steps, the displacement of the whole population of $N$ particles is ballistic, with deterministic asymptotic speed $v_{N}(p)$. As $N$ goes to infinity, $v_{N}(p)$ converges to a finite limit $v_{\\infty}(p)$. The main result is that, for every $0<p<1/2$, as $N$ goes to infinity, the order of magnitude of the difference $v_{\\infty}(p)- v_{N}(p)$ is $\\log(N)^{-2}$. This is called Brunet-Derrida behavior in reference to the 1997 paper by E. Brunet and B. Derrida \"Shift in the velocity of a front due to a cutoff\" (see the reference within the paper), where such a behavior is established for a similar branching-selection particle system, using both numerical simulations and heuristic arguments."}
{"category": "Math", "title": "Stability of the vanishing of the $\\overline\\partial_b$-cohomology under small horizontal perturbations of the CR structure in compact abstract q-concave CR manifolds", "abstract": "We consider perturbations of CR structures which preserve the complex tangent bundle. For a compact generic CR manifold its concavity properties and hence the finiteness of some $\\overline\\partial_b$-cohomology groups are also preserved by such perturbations of the CR structure. Here we study the stability of the vanishing of these groups."}
{"category": "Math", "title": "A note on hyperbolic leaves and wild laminations of rational functions", "abstract": "We study the affine orbifold laminations that were constructed by Lyubich and Minsky. An important question left open in their construction is whether these laminations are always locally compact. We show that this is not the case. The counterexample we construct has the property that the regular leaf space contains (many) hyperbolic leaves that intersect the Julia set; whether this can happen is itself a question raised by Lyubich and Minsky."}
{"category": "Math", "title": "Enriched spin curves on stable curves with two components", "abstract": "L. Maino constructed a moduli space for enriched stable curves, by blowing-up the moduli space of Deligne-Mumford stable curves. We introduce enriched spin curves, showing that a parameter space for these objects is obtained by blowing-up the moduli space of spin curves."}
{"category": "Math", "title": "Touchscreen Voting Machines Cause Long Lines and Disenfranchise Voters", "abstract": "Computerized touchscreen \"Direct Recording Electronic\" DRE voting systems have been used by over 1/3 of American voters in recent elections. In many places, insufficient DRE numbers in combination with lengthy ballots and high voter traffic have caused long lines and disenfranchised voters who left without voting. We have applied computer queuing simulation to the voting process and conclude that far more DREs, at great expense, would be needed to keep waiting times low. Alternatively, paper ballot-optical scan systems can be easily and economically scaled to prevent long lines and meet unexpected contingencies."}
{"category": "Math", "title": "CM-stability of blow-ups and canonical metrics", "abstract": "An asymptotic formula for the Tian-Paul CM-line of a flat family blown-up at a flat closed sub-scheme is given. As an application we prove that the blow-up of a polarized manifold along a (relatively) Chow-unstable submanifold admits no (extremal) constant scalar curvature Kahler metrics in classes making the exceptional divisors sufficiently small. Moreover a geometric characterization of relatively Chow-unstable configuration of points in the projective space is given. From this we get new examples of classes admitting no extremal Kahler metric also in the case of the projective plane blown-up at a finite set of points."}
{"category": "Math", "title": "More Results on Regular Ultrafilters in ZFC", "abstract": "We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters. We also list some problems, and furnish applications to topological spaces and to extended logics."}
{"category": "Math", "title": "Fields of Parametrization and Optimal Affine Reparametrization of Rational Curves", "abstract": "In this paper we present three related results on the subject of fields of parametrization. Let C be a rational curve over a field of characteristic zero. Let K be a field finitely generated over Q, such that it is a field of definition of C but not a field of parametrization. It is known that there are quadratic extensions of K that parametrize C. First, we prove that there are infinitely many quadratic extensions of K that are fields of parametrization of C. As a consequence, we prove that the witness variety, that appear in the context of the parametric Weil's descente method, is always a special curve related to algebraic extensions, called hypercircle. It is possible that the witness variety is not a hypercircle for the given extension, but for an alternative one. We use these two facts to present an algorithm to solve the following optimal reparametrization problem. Given a birational parametrization f(t) of a curve C, compute the affine reparametrization at+b such f(at+b) has coefficients over a field as small as possible. The main advantage of this algorithm is that it does not need to compute any rational point on the curve."}
{"category": "Math", "title": "Spectral Bounds for Dirac Operators on Open Manifolds", "abstract": "We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich's estimate for manifolds with positive scalar curvature as well as the author's estimate on surfaces."}
{"category": "Math", "title": "Approximation by Lipschitz, analytic maps on certain Banach spaces", "abstract": "We show that on separable Banach spaces admitting a separating polynomial, any uniformly continuous, bounded, real-valued function can be uniformly approximated by Lipschitz, analytic maps on bounded sets."}
{"category": "Math", "title": "Evidence of Systematic Bias in 2008 Presidential Polling (preliminary report)", "abstract": "Political polls achieve their results by sampling a small number of potential voters rather than the population as a whole. This leads to sampling error which most polling agencies dutifully report. But factors such as nonrepresentative samples, question wording and nonresponse can produce non-sampling errors. While pollsters are aware of such errors, they are difficult to quantify and seldom reported. When a polling agency, whether by intention or not, produces results with non-sampling errors that systematically favor one candidate over another, then that agency's poll is biased. We analyzed polling data for the (on-going) 2008 Presidential race, and though our methods do not allow us to identify which agencies' polls are biased, they do provide significant evidence that some agencies' polls are. We compared polls produced by major television networks with those produced by Gallup and Rasmussen. We found that, taken as a whole, polls produced by the networks were significantly to the left of those produced by Gallup and Rasmussen. We used the available data to provide a tentative ordering of the major television networks' polls from right to left. Our order was: FOX, CNN, NBC (which partners with the Wall Street Journal), ABC (which partners with the Washington Post), CBS (which partners with the New York Times). These results appear to comport well with the informal perceptions of the political leanings of these agencies. Our findings are preliminary, but they make a case for further research into the causes of and remedies for polling bias."}
{"category": "Math", "title": "Q curvature prescription; forbidden functions and the GJMS null space", "abstract": "On an even conformal manifold $(M,c)$, such that the critical GJMS operator has non-trivial kernel, we identify and discuss the role of a finite dimensional vector space $N(Q)$ of functions determined by the conformal structure. Using these we describe an infinite dimensional class of functions that cannot be the Q-curvature $Q^g$ for any $g$ in $c$. If certain functions arise in $N(Q)$ then $Q^g$ cannot be constant for any $g$ in $c$."}
{"category": "Math", "title": "The Pentagram map: a discrete integrable system", "abstract": "The pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons (a twisted polygon is a map from Z into the projective plane that is periodic modulo a projective transformation). We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation."}
{"category": "Math", "title": "Conjugation-free geometric presentations of fundamental groups of arrangements", "abstract": "We introduce the notion of a conjugation-free geometric presentation for a fundamental group of a line arrangement's complement, and we show that the fundamental groups of the following family of arrangements have a conjugation-free geometric presentation: A real arrangement L, whose graph of multiple points is a union of disjoint cycles, has no line with more than two multiple points, and where the multiplicities of the multiple points are arbitrary. We also compute the exact group structure (by means of a semi-direct product of groups) of the arrangement of 6 lines whose graph consists of a cycle of length 3, and all the multiple points have multiplicity 3."}
{"category": "Math", "title": "A rigidity theorem for quaternionic Kaehler structures", "abstract": "We study the moduli space of quaternionic Kaehler structures on a compact manifold of dimension 4n (n>2) from a point of view of Riemannian geometry, not twistor theory. Then we obtain a rigidity theorem for quaternionic Kaehler structures of nonzero scalar curvature by observing the moduli space."}
{"category": "Math", "title": "Jack deformations of Plancherel measures and traceless Gaussian random matrices", "abstract": "We study random partitions $\\lambda=(\\lambda_1,\\lambda_2,...,\\lambda_d)$ of $n$ whose length is not bigger than a fixed number $d$. Suppose a random partition $\\lambda$ is distributed according to the Jack measure, which is a deformation of the Plancherel measure with a positive parameter $\\alpha>0$. We prove that for all $\\alpha>0$, in the limit as $n \\to \\infty$, the joint distribution of scaled $\\lambda_1,..., \\lambda_d$ converges to the joint distribution of some random variables from a traceless Gaussian $\\beta$-ensemble with $\\beta=2/\\alpha$. We also give a short proof of Regev's asymptotic theorem for the sum of $\\beta$-powers of $f^\\lambda$, the number of standard tableaux of shape $\\lambda$."}
{"category": "Math", "title": "Conformally Osserman manifolds", "abstract": "An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every point is Osserman. We prove that a conformally Osserman manifold of dimension $n \\ne 3, 4, 16$ is locally conformally equivalent either to a Euclidean space or to a rank-one symmetric space."}
{"category": "Math", "title": "Simultaneous Binary Collisions for Collinear Four-Body Problem", "abstract": "In this paper, we use canonical transformations to collectively analytically continue the singularities of the simultaneous binary collision solutions for the collinear four- body problem in both the decoupled case and the coupled case. All the solutions are found and more importantly, we describe the relationship between the decoupled solutions and the coupled solutions."}
{"category": "Math", "title": "A remark on the uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities", "abstract": "We consider the uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities. We deduce the uniqueness from the argument in the classical paper by Peletier and Serrin, thereby recovering a part of the uniqueness result of Ouyang and Shi."}
{"category": "Math", "title": "Forcings constructed along morasses", "abstract": "In a previous paper, we introduced a way of constructing a forcing along a simplified gap-1 morass such that the forcing satisfies a chain condition. Now, we generalize this to gap-2 morasses. As an application, we prove that GCH is consistent with the existence of a 0-dimensional Hausdorff topology on $\\omega_3$ which has spread $\\omega_1$."}
{"category": "Math", "title": "On semilinear elliptic equations with global coupling", "abstract": "We consider nonlinear elliptic equations which contains global coupling as a nonlinear term. We classify the existence of all possible positive solutions to this problem."}
{"category": "Math", "title": "Relative Oscillation Theory for Jacobi Matrices", "abstract": "We develop relative oscillation theory for Jacobi matrices which, rather than counting the number of eigenvalues of one single matrix, counts the difference between the number of eigenvalues of two different matrices. This is done by replacing nodes of solutions associated with one matrix by weighted nodes of Wronskians of solutions of two different matrices."}
{"category": "Math", "title": "Demazure crystals of generalized Verma modules and a flagged RSK correspondence", "abstract": "We prove that the Robinson-Schensted-Knuth correspondence is a $\\gl_{\\infty}$-crystal isomorphism between two realizations of the crystal graph of a generalized Verma module with respect to a maximal parabolic subalgebra of $\\gl_{\\infty}$. %This extends the previously known result that %the RSK correspondence is an isomorphism of bicrystals or double %crystals. A flagged version of the RSK correspondence is derived in a natural way by computing a Demazure crystal graph of a generalized Verma module. As an application, we discuss a relation between a Demazure crystal and plane partitions with a bounded condition."}
{"category": "Math", "title": "Lagrangian Floer theory on compact toric manifolds II : Bulk deformations", "abstract": "This is a continuation of part I in the series of the papers on Lagrangian Floer theory on toric manifolds. Using the deformations of Floer cohomology by the ambient cycles, which we call bulk deformations, we find a continuum of non-displaceable Lagrangian fibers on some compact toric manifolds. We also provide a method of finding all fibers with non-vanishing Floer cohomology with bulk deformations in arbitrary compact toric manifolds, which we call bulk-balanced Lagrangian fibers."}
{"category": "Math", "title": "Gibbs posterior for variable selection in high-dimensional classification and data mining", "abstract": "In the popular approach of \"Bayesian variable selection\" (BVS), one uses prior and posterior distributions to select a subset of candidate variables to enter the model. A completely new direction will be considered here to study BVS with a Gibbs posterior originating in statistical mechanics. The Gibbs posterior is constructed from a risk function of practical interest (such as the classification error) and aims at minimizing a risk function without modeling the data probabilistically. This can improve the performance over the usual Bayesian approach, which depends on a probability model which may be misspecified. Conditions will be provided to achieve good risk performance, even in the presence of high dimensionality, when the number of candidate variables \"$K$\" can be much larger than the sample size \"$n$.\" In addition, we develop a convenient Markov chain Monte Carlo algorithm to implement BVS with the Gibbs posterior."}
{"category": "Math", "title": "The Packing Measure of the Range of Super-Brownian Motion", "abstract": "We prove that the total range of Super-Brownian motion with quadratic branching mechanism has an exact packing measure with respect to the gauge function $g(r)=r^4 (\\log \\log1/r)^{-3}$ in super-critical dimensions $d\\geq 5$. More precisely, we prove that the total occupation measure of Super-Brownian motion is equal to the $g$-packing measure restricted to its range, up to a deterministic multiplicative constant that only depends on space dimension $d$."}
{"category": "Math", "title": "G-Structures defined on pseudo-Riemannian manifolds", "abstract": "Concepts and techniques from the theory of G-structures of higher order are applied to the study of certain structures (volume forms, conformal structures, linear connections and projective structures) defined on a pseudo-Riemanniann manifold. Several relationships between the structures involved have been investigated. The operations allowed on G-structures, such as intersection, inclusion, reduction, extension and prolongation, were used for it."}
{"category": "Math", "title": "Locally well generated homotopy categories of complexes", "abstract": "We show that the homotopy category of complexes K(B) over any finitely accessible additive category B is locally well generated. That is, any localizing subcategory L in K(B) which is generated by a set is well generated in the sense of Neeman. We also show that K(B) itself being well generated is equivalent to B being pure semisimple, a concept which naturally generalizes right pure semisimplicity of a ring R for B = Mod-R."}
{"category": "Math", "title": "Liaison invariants and the Hilbert scheme of codimension 2 subschemes in P^{n+2}", "abstract": "In this paper we study the Hilbert scheme, Hilb(P), of equidimensional locally Cohen-Macaulay codimension 2 subschemes, with a special look to surfaces in P^4 and 3-folds in P^5, and the Hilbert scheme stratification H_{c} of constant cohomology. For every (X) in Hilb(P) we define a number \\delta_X in terms of the graded Betti numbers of the homogeneous ideal of X and we prove that 1 + \\delta_X - \\dim_{(X)} H_{c} and 1 + \\delta_X - \\dim T_{c} are CI-biliaison invariants where T_{c} is the tangent space of H_{c} at (X). As a corollary we get a formula for the dimension of any generically smooth component of Hilb(P) in terms of \\delta_X and the CI-biliaison invariant. Both invariants are equal in this case. Recall that, for space curves C, Martin-Deschamps and Perrin have proved the smoothness of the ``morphism'', H_{c} -> E = isomorphism classes of graded artinian modules, given by sending C onto its Rao-module. For surfaces X in P^4 we have two Rao-modules M_i and an induced extension b in Ext^2(M_2,M_1) and a result of Horrocks and Rao saying that a triple D := (M_1,M_2,b) of modules M_i of finite length and an extension b as above determine a surface X up to biliaison. We prove that the corresponding ``morphism'', H_{c} -> V = isomorphism classes of graded artinian modules M_i commuting with b, is smooth, and we get a smoothness criterion for H_{c}. Moreover we get some smoothness results for Hilb(P), valid also for 3-folds, and we give examples of obstructed surfaces and 3-folds. The linkage result we prove in this paper turns out to be useful in determining the structure and dimension of H_{c}, and for proving the main biliaison theorem above."}
{"category": "Math", "title": "Niceness theorems", "abstract": "Many things in mathematics seem lamost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of variables in its many incarnations such as the representing object of the Witt vectors, the direct sum of the rings of representations of the symmetric groups, the free lambda ring on one generator, the homology and cohomology of the classifying space BU, ... . In addition attention is paid to the phenomenon that solutions to universal problems (adjoint functors) tend to pick up extra structure."}
{"category": "Math", "title": "Detecting orbits along subvarieties via the moment map", "abstract": "Let G be a (real or complex) linear reductive algebraic group acting on an affine variety V. Let W be a subvariety. In this work we study how the G-orbits intersect W. We develop a criterion to determine when the intersection can be described as a finite union of orbits of a reductive subgroup. The conditions of the criterion are easily verified in practice and are used to construct continuous families of (non-isomorphic) nilpotent Lie groups which do not admit left-invariant Ricci soliton metrics. Other applications to the left-invariant geometry of Lie groups are also given. The note finishes by applying our techniques to the adjoint representation. The classical result of finiteness of nilpotent orbits is reproven and it is shown that each of these orbits contains a critical point of the norm squared of the moment map."}
{"category": "Math", "title": "Remarks on the non-commutative Khintchine inequalities for $0<p<2$", "abstract": "We show that the validity of the non-commutative Khintchine inequality for some $q$ with $1<q<2$ implies its validity (with another constant) for all $1\\le p<q$. We prove this for the inequality involving the Rademacher functions, but also for more general \"lacunary\" sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the \"Z(2)-sequences\" previously considered by Harcharras. The result appears to be new in that case. It implies that the space $\\ell^n_1$ contains (as an operator space) a large subspace uniformly isomorphic (as an operator space) to $R_k+C_k$ with $k\\sim n^{\\frac12}$. This naturally raises several interesting questions concerning the best possible such $k$. Unfortunately we cannot settle the validity of the non-commutative Khintchine inequality for $0<p<1$ but we can prove several would be corollaries. For instance, given an infinite scalar matrix $[x_{ij}]$, we give a necessary and sufficient condition for $[\\pm x_{ij}]$ to be in the Schatten class $S_p$ for almost all (independent) choices of signs $\\pm 1$. We also characterize the bounded Schur multipliers from $S_2$ to $S_p$. The latter two characterizations extend to $0<p<1$ results already known for $1\\le p\\le2$. In addition, we observe that the hypercontractive inequalities, proved by Carlen and Lieb for the Fermionic case, remain valid for operator space valued functions, and hence the Kahane inequalities are valid in this setting."}
{"category": "Math", "title": "Conditional Limits of W_p scale Mixture Distributions", "abstract": "In this paper we introduce the class of W_p scale mixture random vectors with a particular radial decomposition and a independent splitting property specified by some random variable W_p, and a positive constant p. We derive several conditional limit results assuming that the distribution of the random radius is in the max-domain of attraction of a univariate extreme value distribution and W_p has a certain tail asymptotic behaviour. As an application we obtain the joint asymptotic distribution of concomitants of order statics considering certain bivariate W_p scale miture samples."}
{"category": "Math", "title": "A few things I learnt from Jurgen Moser", "abstract": "A few remarks on integrable dynamical systems inspired by discussions with Jurgen Moser and by his work."}
{"category": "Math", "title": "Simple Compact Quantum Groups I", "abstract": "The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups $B_u(Q)$ for $Q \\in GL(n, {\\mathbb C})$ satisfying $Q \\bar{Q} = \\pm I_n$, $n \\geq 2$; (b) The quantum automorphism groups $A_{aut}(B, \\tau)$ of finite dimensional $C^*$-algebras $B$ endowed with the canonical trace $\\tau$ %endowed with a tracial functional $tr$ when $\\dim(B) \\geq 4$, including the quantum permutation groups $A_{aut}(X_n)$ on $n$ points ($n \\geq 4$); (c) The standard deformations $K_q$ of simple compact Lie groups $K$ and their twists $K_q^u$, as well as Rieffel's deformation $K_J$."}
{"category": "Math", "title": "A Rigidity Condition for Holomorphic Generator in Strongly Convex Domains", "abstract": "In this paper I give a condition, in terms of boundary behaviour, that forces the infinitesimal generator of a one-parameter semigroup of holomorphic maps to vanish."}
{"category": "Math", "title": "Period and index of genus one curves over number fields", "abstract": "The period of a curve is the smallest positive degree of Galois-invariant divisor classes. The index is the smallest positive degree of rational divisors. We construct examples of genus one curves with prescribed period and index over certain number fields."}
{"category": "Math", "title": "Local energy estimate on Kerr black hole backgrounds", "abstract": "We study dispersive properties for the wave equation in the Kerr space-time with small angular momentum. The main result of thispaper is to establish uniform energy bounds and local energy decay for such backgrounds."}
{"category": "Math", "title": "Large gaps between random eigenvalues", "abstract": "We show that in the point process limit of the bulk eigenvalues of $\\beta$-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size $\\lambda$ is given by \\[\\bigl(\\ kappa_{\\beta}+o(1)\\bigr)\\lambda^{\\gamma_{\\beta}}\\exp\\biggl(-{\\bet a}{64}\\lambda^2+\\biggl({\\beta}{8}-{1}{4}\\biggr)\\lambda\\biggr)\\] as $\\lambda\\to\\infty$, where \\[\\gamma_{\\beta}={1}{4}\\biggl({\\beta}{2}+{2}{\\beta}-3\\biggr)\\] and $\\kappa_{\\beta}$ is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157--165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron--Martin--Girsanov transformation in stochastic calculus."}
{"category": "Math", "title": "A note on the eigenvalues of double band matrices", "abstract": "We consider matrices containing two diagonal bands of positive entries. We show that all eigenvalues of such matrices are of the form $r \\zeta$, where $r$ is a nonnegative real number and $\\zeta$ is a $p$th root of unity, where $p$ is the period of the matrix, which is computed from the distance between the bands."}
{"category": "Math", "title": "Cartan--Helgason theorem, Poisson transform, and Furstenberg--Satake compactifications", "abstract": "The connections between the objects mentioned in the title are used to give a short proof of the Cartan--Helgason theorem and a natural construction of the compactifications."}
{"category": "Math", "title": "Riemannian metrics having common geodesics with Berwald metrics", "abstract": "In Theorem 1, we generalize the results of Szabo for Berwald metrics that are not necessary strictly convex: we show that for every Berwald metric F there always exists a Riemannian metric affine equivalent to F. As an application we show (Corollary 3) that every Berwald projectively flat metric is a Minkowski metric; this statement is a \"Berwald\" version of Hilbert's 4th problem. Further, we investigate geodesic equivalence of Berwald metrics. Theorem 2 gives a system of PDE that has a (nontrivial) solution if and only if the given essentially Berwald metric admits a Riemannian metric that is (nontrivially) geodesically equivalent to it. The system of PDE is linear and of Cauchy-Frobenius type, i.e., the derivatives of unknown functions are explicit expressions of the unknown functions. As an application (Corollary 2), we obtain that geodesic equivalence of an essentially Berwald metric and a Riemannian metric is always affine equivalence provided both metrics are complete."}
{"category": "Math", "title": "Very Twisted Stable Maps", "abstract": "Let X be a smooth projective Deligne-Mumford stack over an algebraically closed field k of characteristic 0. In this paper we construct the moduli stack of very twisted stable maps, extending the notion of twisted stable maps by Abramovich and Vistoli to allow for generic stabilizers on the source curves. We also consider the Gromov-Witten theory given by this construction."}
{"category": "Math", "title": "A sufficient condition for intrinsic knotting of bipartite graphs", "abstract": "We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \\geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six vertices in one part. We also show that there is a constant C_n such that a bipartite graph with exactly n \\geq 5 vertices in one part and |E(G)| \\geq 4 |V(G)| + C_n is intrinsically knotted. Finally, we classify bipartite graphs with ten or fewer vertices with respect to intrinsic knotting."}
{"category": "Math", "title": "Summation of Hyperharmonic Series", "abstract": "We shall show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. More exactly, we give summation formula for the general hyperharmonic series."}
{"category": "Math", "title": "About the Non-Integer Property of Hyperharmonic Numbers", "abstract": "It was proven in 1915 by Leopold Theisinger that the $H_n$ harmonic numbers are never integers. In 1996 Conway and Guy have defined the concept of hyperharmonic numbers. The question naturally arises: are there any integer hyperharmonic numbers? The author gives a partial answer to this question and conjectures that the answer is \"no\"."}
{"category": "Math", "title": "On the Harary-Kauffman Conjecture and Turk's Head Knots", "abstract": "The m,n Turk's Head Knot, THK(m,n), is an \"alternating (m,n) torus knot.\" We prove the Harary-Kauffman conjecture for all THK(m,n) except for the case where m \\geq 5 is odd and n \\geq 3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that none of these knots have prime determinant except for THK(m,2) when P_m is a Pell prime."}
{"category": "Math", "title": "Can lattices in SL(n,R) act on the circle?", "abstract": "This expository paper describes the various methods that have yielded partial results on the conjecture that if n > 2, then no lattice in SL(n,R) has a faithful action on the circle (by homeomorphisms). Topics include amenability, Kazhdan's property (T), bounded cohomology, bounded generation, and the Reeb-Thurston Stability Theorem."}
{"category": "Math", "title": "On the Exel crossed product of topological covering maps", "abstract": "For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product $C^*$-algebras $\\cros$ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical imbedding of $C(X)$ into $\\cros$ is a maximal abelian $C^*$-subalgebra of $\\cros$; any nontrivial two sided ideal of $\\cros$ has non-zero intersection with the imbedded copy of $C(X)$; a certain natural representation of $\\cros$ is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product $C^*$-algebras of homeomorphism dynamical systems."}
{"category": "Math", "title": "Product-type non-commutative polynomial states", "abstract": "In math/0702157, arXiv:0712.4185, we investigated monic multivariate non-commutative orthogonal polynomials, their recursions, states of orthogonality, and corresponding continued fraction expansions. In this note, we collect a number of examples, demonstrating what these general results look like for the most important states on non-commutative polynomials, namely for various product states. In particular, we introduce a notion of a product-type state on polynomials, which covers all the non-commutative universal products and excludes some other familiar non-commutative products, and which guarantees a number of nice properties for the corresponding polynomials."}
{"category": "Math", "title": "Feedback Stabilization Methods for the Numerical Solution of Systems of Ordinary Differential Equations", "abstract": "In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations, by means of tools from nonlinear control theory. Lyapunov-based and Small-Gain feedback stabilization methods are exploited and numerous illustrating applications are presented for systems with a globally asymptotically stable equilibrium point. The obtained results can be used for the control of the global discretization error as well."}
{"category": "Math", "title": "The complex Lorentzian Leech lattice and the bimonster (II)", "abstract": "Let $D$ be the incidence graph of the projective plane over $\\FF_3$. The Artin group of the graph $D$ maps onto the bimonster and a complex hyperbolic reflection group $\\Gamma$ acting on 13 dimensional complex hyperbolic space $Y$. The generators of the Artin group are mapped to elements of order 2 (resp. 3) in the bimonster (resp. $\\Gamma$). Let $Y^{\\circ} \\subseteq Y$ be the complement of the union of the mirrors of $\\Gamma$. Daniel Allcock has conjectured that the orbifold fundamental group of $Y^{\\circ}/\\Gamma$ surjects onto bimonster. In this article we study the reflection group $\\Gamma$. Our main result shows that there is homomorphism from the Artin group of $D$ to the orbifold fundamental group of $Y^{\\circ}/\\Gamma$, obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections in $\\Gamma$. This answers a question in Allcock's article \"A monstrous proposal\" and takes a step towards the proof of Allcock's conjecture. The finite group $\\op{PGL}(3, \\FF_3) \\subseteq \\Aut(D)$ acts on $Y$ and fixes a complex hyperbolic line pointwise. We show that the restriction of $\\Gamma$-invariant meromorphic automorphic forms on $Y$ to the complex hyperbolic line fixed by $\\op{PGL}(3, \\FF_3)$ gives meromorphic modular forms of level 13."}
{"category": "Math", "title": "Calculation of the Norm of the Error Functional of Optimal Quadrature Formulas in the Space $W_2^{(2,1)}(0,1)$", "abstract": "In this paper in the space $W_2^{(2,1)}(0,1)$ square of the norm of the error functional of a optimal quadrature formula is calculated."}
{"category": "Math", "title": "Einstein metrics and GIT stability", "abstract": "In this expository article we review the problem of finding Einstein metrics on compact K\\\"ahler manifolds and Sasaki manifolds. In the former half of this article we see that, in the K\\\"ahler case, the problem fits better with the notion of stability in Geometric Invariant Theory if we extend the problem to that of finding extremal K\\\"ahler metrics or constant scalar curvature K\\\"ahler (cscK) metrics. In the latter half of this paper we see that most of ideas in K\\\"ahler geometry extend to Sasaki geometry as transverse K\\\"ahler geometry. We also summarize recent results about the existence of toric Sasaki-Einstein metrics."}
{"category": "Math", "title": "Aleph-zero-categorical groups and their completions", "abstract": "We embed a countably categorical group G into a locally compact group c(G) with a non-trivial topology and study how topological properties of c(G) are connected with the structure of definable subgroups of G."}
{"category": "Math", "title": "Generalization of l1 constraints for high dimensional regression problems", "abstract": "We focus on the high dimensional linear regression $Y\\sim\\mathcal{N}(X\\beta^{*},\\sigma^{2}I_{n})$, where $\\beta^{*}\\in\\mathds{R}^{p}$ is the parameter of interest. In this setting, several estimators such as the LASSO and the Dantzig Selector are known to satisfy interesting properties whenever the vector $\\beta^{*}$ is sparse. Interestingly both of the LASSO and the Dantzig Selector can be seen as orthogonal projections of 0 into $\\mathcal{DC}(s)=\\{\\beta\\in\\mathds{R}^{p},\\|X'(Y-X\\beta)\\|_{\\infty}\\leq s\\}$ - using an $\\ell_{1}$ distance for the Dantzig Selector and $\\ell_{2}$ for the LASSO. For a well chosen $s>0$, this set is actually a confidence region for $\\beta^{*}$. In this paper, we investigate the properties of estimators defined as projections on $\\mathcal{DC}(s)$ using general distances. We prove that the obtained estimators satisfy oracle properties close to the one of the LASSO and Dantzig Selector. On top of that, it turns out that these estimators can be tuned to exploit a different sparsity or/and slightly different estimation objectives."}
{"category": "Math", "title": "Logical tools for handling change in agent-based systems", "abstract": "We give a unified approach to various results and problems of nonclassical logics"}
{"category": "Math", "title": "Defeasible inheritance systems and reactive diagrams", "abstract": "We give an analysis of defeasible inheritance diagrams, also from the perspective of reactive diagrams."}
{"category": "Math", "title": "Galois groups over function fields of positive characteristic", "abstract": "We describe examples motivated by the work of Serre and Abhyankar."}
{"category": "Math", "title": "Adequacy of Link Families", "abstract": "Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating link we defined a new numerical invariant: adequacy number, and computed adequacy polynomial which is the invariant of alternating link families. Adequacy polynomial distinguishes (up to mutation) all families of alternating knots and links whose generating link has at most $n=12$ crossings."}
{"category": "Math", "title": "On central extensions and definably compact groups in o-minimal structures", "abstract": "We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple group and a commutative group, and (ii) the group (G, .) is elementarily equivalent to (G/G^00, .). We also prove results on the internality of finite covers of G in an o-minimal environment, as well as the full compact domination conjecture. These results depend on key theorems about the interpretability of central and finite extensions of definable groups, in the o-minimal context. These methods and others also yield interpretability results for universal covers of arbitrary definable real Lie groups, from which we can deduce the semialgebraicity of finite covers of Lie groups such as SL(2,R)."}
{"category": "Math", "title": "Product formula for Atiyah-Patodi-Singer index classes and higher signatures", "abstract": "We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C*-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures."}
{"category": "Math", "title": "Lineability of the set of bounded linear non-absolutely summing operators", "abstract": "In this note we solve, except for extremely pathological cases, a question posed by Puglisi and Seoane-Sepulveda on the lineability of the set of bounded non-absolutely summing linear operators. We also show how the idea of the proof can be adapted to several related situations."}
{"category": "Math", "title": "Universality of Crystallographic Pinning", "abstract": "We study traveling waves for reaction diffusion equations on the spatially discrete domain $\\Z^2$. The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. Mallet-Paret has shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth. In this paper we show that crystallographic pinning holds in the horizontal and vertical directions for bistable nonlinearities which satisfy a specific computable generic condition. The proof is based on dynamical systems. In particular, it relies on an examination of the heteroclinic chains which occur as singular limits of wave profiles on the boundary of the pinning region."}
{"category": "Math", "title": "Quantum symmetries and quantum isometries of compact metric spaces", "abstract": "We prove that a compact quantum group with faithful Haar state which has a faithful action on a compact space must be a Kac algebra, with bounded antipode and the square of the antipode being identity. The main tool in proving this is the theory of ergodic quantum group action on $C^*$ algebras. Using the above fact, we also formulate a definition of isometric action of a compact quantum group on a compact metric space, generalizing the definition given by Banica for finite metric spaces, and prove for certain special class of metric spaces the existence of the universal object in the category of those compact quantum groups which act isometrically and are `bigger' than the classical isometry group."}
{"category": "Math", "title": "A Criterion for the Viability of Stochastic Semilinear Control Systems via the Quasi-Tangency Condition", "abstract": "In this paper we study a criterion for the viability of stochastic semilinear control systems on a real, separable Hilbert space. The necessary and sufficient conditions are given using the notion of stochastic quasi-tangency. As a consequence, we prove that approximate viability and the viability property coincide for stochastic linear control systems. We obtain Nagumo's stochastic theorem and we present a method allowing to provide explicit criteria for the viability of smooth sets. We analyze the conditions characterizing the viability of the unit ball. The paper generalizes recent results from the deterministic framework."}
{"category": "Math", "title": "H^1 and BMO for certain locally doubling metric measure spaces of finite measure", "abstract": "In a previous paper the authors developed a H^1-BMO theory for unbounded metric measure spaces $(M,\\rho,m)$ of infinite measure that are locally doubling and satisfy two geometric properties, called \"approximate midpoint\" property and \"isoperimetric\" property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies to a class of unbounded, complete Riemannian manifolds of finite measure and to a class of metric measure spaces of the form (R^n,\\rho_\\phi, m_\\phi), where dm_\\phi=e^{-\\phi} dx and \\rho_\\phi$ is the Riemannian metric corresponding to the length element ds^2=(1+ |grad\\phi|)^2(dx_1^2+...+dx_n^2)$. This generalizes previous work of the last two authors for the Gauss space."}
{"category": "Math", "title": "Computing the Newton polygon of the implicit equation", "abstract": "We consider polynomially and rationally parameterized curves, where the polynomials in the parameterization have fixed supports and generic coefficients. We apply sparse (or toric) elimination theory in order to determine the vertex representation of its implicit polygon, i.e. of the implicit equation's Newton polygon. In particular, we consider mixed subdivisions of the input Newton polygons and regular triangulations of point sets defined by Cayley's trick. We distinguish polynomial and rational parameterizations, where the latter may have the same or different denominators; the implicit polygon is shown to have, respectively, up to 4, 5, or 6 vertices."}
{"category": "Math", "title": "Estimates for functions of the Laplacian on manifolds with bounded geometry", "abstract": "In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below and positive injectivity radius. Denote by L the Laplace-Beltrami operator on M. We assume that the kernel associated to the heat semigroup generated by L satisfies a mild decay condition at infinity. We prove that if m is a bounded holomorphic function in a suitable strip of the complex plane, and satisfies Mihlin-Hormander type conditions of appropriate order at infinity, then the operator m(L) extends to an operator of weak type 1. This partially extends a celebrated result of J. Cheeger, M. Gromov and M. Taylor, who proved similar results under much stronger curvature assumptions on M, but without any assumption on the decay of the heat kernel."}
{"category": "Math", "title": "Entire solutions to equivariant elliptic systems with variational structure", "abstract": "In the present paper we consider the system {\\Delta}u - W_u (u) = 0, where u: R^n to R^n, for a class of potentials W: R^n to R that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial entire solutions connecting the global minima of W along certain directions at infinity."}
{"category": "Math", "title": "Stability of invariant measures", "abstract": "We generalize various notions of stability of invariant sets of dynamical systems to invariant measures, by defining a topology on the set of measures. The defined topology is similar, but not topologically equivalent to weak* topology, and it also differs from topologies induced by the Riesz Representation Theorem. It turns out that the constructed topology is a solution of a limit case of a $p$-optimal transport problem, for $p=\\infty$."}
{"category": "Math", "title": "Convergence of a Family of Series", "abstract": "In this article we construct a family of expressions $\\varepsilon(n)$. For each element E(n) from $\\varepsilon(n)$, the convergence of the series $\\sum_{n \\ge n_E}{E(n)}$ can be determined in accordance to the theorems of this article. Some applications are also presented."}
{"category": "Math", "title": "Spectral Connectivity Analysis", "abstract": "Spectral kernel methods are techniques for transforming data into a coordinate system that efficiently reveals the geometric structure - in particular, the \"connectivity\" - of the data. These methods depend on certain tuning parameters. We analyze the dependence of the method on these tuning parameters. We focus on one particular technique - diffusion maps - but our analysis can be used for other methods as well. We identify the population quantities implicitly being estimated, we explain how these methods relate to classical kernel smoothing and we define an appropriate risk function for analyzing the estimators. We also show that, in some cases, fast rates of convergence are possible even in high dimensions."}
{"category": "Math", "title": "Doubling property for biLipschitz homogeneous geodesic surfaces", "abstract": "In this paper we discuss general properties of geodesic surfaces that are locally biLipschitz homogeneous. In particular, we prove that they are locally doubling and that there exists a special doubling measure analogous to the Haar measure for locally compact groups."}
{"category": "Math", "title": "Multi-parameter Quantum Groups and Quantum Shuffles, (I)", "abstract": "In this article, we study the multi-parameter quantum groups defined by generators and relations associated with symmetrizable generalized Cartan matrices, together with their representations in the category $\\mathcal O$. This presentation will be convenient for our later discussions. We present two explicit descriptions here: as a Hopf 2-cocycle deformation, and as the multi-parameter quantum shuffle realization of the positive part."}
{"category": "Math", "title": "Envelopes of holomorphy and holomorphic discs", "abstract": "The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the continuity principle once. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs. A particular case is the following. Given a contact three-manifold with Stein filling, any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc."}
{"category": "Math", "title": "Formulae of Partial Reduction for Linear Systems of First Order Operator Equations", "abstract": "This paper deals with reduction of non-homogeneous linear systems of first order operator equations with constant coefficients. An equivalent reduced system, consisting of higher order linear operator equations having only one variable and first order linear operator equations in two variables, is obtained by using the rational canonical form."}
{"category": "Math", "title": "Differential transcendency in the theory of linear differential systems with constant coefficients", "abstract": "In this paper we consider a reduction of a non-homogeneous linear system of first order operator equations to a totally reduced system. Obtained results are applied to Cauchy problem for linear differential systems with constant coefficients and to the question of differential transcendency."}
{"category": "Math", "title": "Plans D'Experiences D'Information De Kullback-Leibler Minimale", "abstract": "Experimental designs are tools which can dramatically reduce the number of simulations required by time-consuming computer codes. Because we don't know the true relation between the response and inputs, designs should allow one to fit a variety of models and should provide information about all portions of the experimental region. One strategy for selecting the values of the inputs at which to observe the response is to choose these values so they are spread evenly throughout the experimental region, according to \"space-filling designs\". In this article, we suggest a new method based on comparing the empirical distribution of the points in a design to the uniform distribution with the Kullback-Leibler information. The considered approach consists in estimating this difference or, reciprocally, the Shannon entropy. The entropy is estimated by a Monte Carlo method where the density function is replaced by its kernel density estimator or by using the nearest neighbor distances"}
{"category": "Math", "title": "Recovery of Missing Samples in Oversampling Formulas for Band Limited Functions", "abstract": "In a previous paper, the author constructed frames and oversampling formulas for band-limited functions, in the framework of the theory of shift-invariant spaces. In this article we study the problem of recovering missing samples. We find a sufficient condition for the recovery of a finite set of missing samples. The condition is expressed as a linear independence of the components of a vector W over the space of trigonometric polynomials determined by the frequencies of the missing samples. We apply the theory to the derivative sampling of any order and we illustrate our results with a numerical experiment."}
{"category": "Math", "title": "Calabi flow and projective embeddings", "abstract": "Let X be a smooth subvariety of CP^N. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X, which attempts to deform the given embedding into a balanced one. If L->X is an ample line bundle, considering embeddings via H^0(L^k) gives a sequence of balancing flows. We prove that, provided these flows are started at appropriate points, they converge to Calabi flow for as long as it exists. This result is the parabolic analogue of Donaldson's theorem relating balanced embeddings to metrics with constant scalar curvature [JDG 59(3):479-522, 2001]. In our proof we combine Donaldson's techniques with an asymptotic result of Liu-Ma [arXiv:math/0601260v2] which, as we explain, describes the asymptotic behaviour of the derivative of the map FS\\circ Hilb whose fixed points are balanced metrics."}
{"category": "Math", "title": "A strict non-standard inequality .999... < 1", "abstract": "Is .999... equal to 1? Lightstone's decimal expansions yield an infinity of numbers in [0,1] whose expansion starts with an unbounded number of digits \"9\". We present some non-standard thoughts on the ambiguity of an ellipsis, modeling the cognitive concept of generic limit of B. Cornu and D. Tall. A choice of a non-standard hyperinteger H specifies an H-infinite extended decimal string of 9s, corresponding to an infinitesimally diminished hyperreal value. In our model, the student resistance to the unital evaluation of .999... is directed against an unspoken and unacknowledged application of the standard part function, namely the stripping away of a ghost of an infinitesimal, to echo George Berkeley. So long as the number system has not been specified, the students' hunch that .999... can fall infinitesimally short of 1, can be justified in a mathematically rigorous fashion."}
{"category": "Math", "title": "Transference principles and locally symmetric spaces", "abstract": "We explain how the Transference Principles from Diophantine approximation can be interpreted in terms of geometry of the locally symmetric spaces $T_n=SO(n) \\backslash SL(n,R) /SL(n,Z)$ with $n>1$, and how, via this dictionary, they become transparent geometric remarks and can be easily proved. Indeed, a finite family of linear forms is naturally identified to a locally geodesic ray in a space $T_n$ and the way this family is approximated is reflected by the heights at which the ray rises in the cuspidal end. The only difference between the two types of approximation appearing in a Transference Theorem is that the height is measured with respect to different rays in $W$, a Weyl chamber in $T_n$. Thus the Transference Theorem is equivalent to a relation between the Busemann functions of two rays in $W$. This relation is easy to establish on $W$, because restricted to it the two Busemann functions become two linear forms. Since $T_n$ is at finite Hausdorff distance from $W$, the same relation is satisfied up to a bounded perturbation on the whole of $T_n$."}
{"category": "Math", "title": "New asymptotic estimates for spherical designs", "abstract": "Let N(n, t) be the minimal number of points in a spherical t-design on the unit sphere S^n in R^{n+1}. For each n >= 3, we prove a new asymptotic upper bound N(n, t) <= C(n)t^{a_n}, where C(n) is a constant depending only on n, a_3 <= 4, a_4 <= 7, a_5 <= 9, a_6 <= 11, a_7 <= 12, a_8 <= 16, a_9 <= 19, a_10 <= 22, and a_n < n/2*log_2(2n), n > 10."}
{"category": "Math", "title": "On sutured Floer homology and the equivalence of Seifert surfaces", "abstract": "We study the sutured Floer homology invariants of the sutured manifold obtained by cutting a knot complement along a Seifert surface, R. We show that these invariants are finer than the \"top term\" of the knot Floer homology, which they contain. In particular, we use sutured Floer homology to distinguish two non-isotopic minimal genus Seifert surfaces for the knot 8_3. A key ingredient for this technique is finding appropriate Heegaard diagrams for the sutured manifold associated to the complement of a Seifert surface."}
{"category": "Math", "title": "On the $8n^2$-inequality", "abstract": "We give a complete proof of the so called $8n^2$-inequality, a local inequality for the self-intersection of a movable linear system at an isolated centre of a non canonical singularity. The inequality was suggested and several times published by I.Cheltsov but some of his arguments are faulty. We explain the mistake and replace the faulty piece by a correct argument."}
{"category": "Math", "title": "Lieb-Thirring Inequalities for Fourth-Order Operators in Low Dimensions", "abstract": "This paper considers Lieb-Thirring inequalities for higher order differential operators. A result for general fourth-order operators on the half-line is developed, and the trace inequality tr((-Delta)^2 - C^{HR}_{d,2} / (|x|^4) - V(x))^{-\\gamma} < C_\\gamma \\int_{R^d} V(x)_+^{\\gamma + d/4} dx for gamma \\geq 1 - d/4, where C^{HR}_{d,2} is the sharp constant in the Hardy-Rellich inequality and where C_\\gamma > 0 is independent of V, is proved for dimensions d = 1,3. As a corollary of this inequality a Sobolev-type inequality is obtained."}
{"category": "Math", "title": "Good formal structures on flat meromorphic connections, I: Surfaces", "abstract": "We give a criterion under which one can obtain a good decomposition (in the sense of Malgrange) of a formal flat connection on a complex analytic or algebraic variety of arbitrary dimension. The criterion is stated in terms of the spectral behavior of differential operators, and generalizes Robba's construction of the Hukuhara-Levelt-Turrittin decomposition in the one-dimensional case. As an application, we prove the existence of good formal structures for flat meromorphic connections on surfaces after suitable blowing up; this verifies a conjecture of Sabbah, and extends a result of Mochizuki for algebraic connections. Our proof uses a finiteness argument on the valuative tree associated to a point on a surface, in order to verify the numerical criterion."}
{"category": "Math", "title": "Formality of the constructible derived category for spheres: A combinatorial and a geometric approach", "abstract": "We describe the constructible derived category of sheaves on the $n$-sphere, stratified in a point and its complement, as a dg module category of a formal dg algebra. We prove formality by exploring two different methods: As a combinatorial approach, we reformulate the problem in terms of representations of quivers and prove formality for the 2-sphere, for coefficients in a principal ideal domain. We give a suitable generalization of this formality result for the 2-sphere stratified in several points and their complement. As a geometric approach, we give a description of the underlying dg algebra in terms of differential forms, which allows us to prove formality for $n$-spheres, for real or complex coefficients."}
{"category": "Math", "title": "Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function", "abstract": "We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the $C^1$-topology then it has a finite orbit. As a corollary, we show that if such a group has no finite orbit then each of its subgroups contains either a cyclic group of finite index or a nonabelian free subgroup."}
{"category": "Math", "title": "Linear systems over P1xP1 with base points of multiplicity bounded by three", "abstract": "We propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general positions. As an application we obtain a classification of special linear systems on P1xP1 for which the multiplicities do not exceed 3."}
{"category": "Math", "title": "Tori Embedded in R3 with Dense Principal Lines", "abstract": "In this paper are given examples of tori T2 embedded in R3 with all their principal lines dense. These examples are obtained by stereographic projection of deformations of the Clifford torus in S3."}
{"category": "Math", "title": "Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones", "abstract": "We prove that if U\\subset\\R^n is an open domain whose closure \\overline{U} is compact in the path metric, and F is a Lipschitz function on \\partial{U}, then for each \\beta\\in\\R there exists a unique viscosity solution to the \\beta-biased infinity Laplacian equation \\beta |\\nabla u| + \\Delta_\\infty u=0 on U that extends F, where \\Delta_\\infty u= |\\nabla u|^{-2} \\sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}. In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the \\beta-biased \\eps-game as follows. The starting position is x_0 \\in U. At the k^\\text{th} step the two players toss a suitably biased coin (in our key example, player I wins with odds of \\exp(\\beta\\eps) to 1), and the winner chooses x_k with d(x_k,x_{k-1}) < \\eps. The game ends when x_k \\in \\partial{U}, and player II pays the amount F(x_k) to player I. We prove that the value u^{\\eps}(x_0) of this game exists, and that \\|u^\\eps - u\\|_\\infty \\to 0 as \\eps \\to 0, where u is the unique extension of F to \\overline{U} that satisfies comparison with \\beta-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with \\beta-exponential cones if and only if it is a viscosity solution to the \\beta-biased infinity Laplacian equation."}
{"category": "Math", "title": "Convex PBW-type Lyndon Basis and Restricted Two-parameter Quantum Group of Type G_2", "abstract": "We construct finite-dimensional pointed Hopf algebras \\mathfrak u_{r,s}(G_2) (i.e. restricted 2-parameter quantum groups) from the 2-parameter quantum group U_{r,s}(G_2) defined in \\cite{HS}, which turn out to be of Drinfel'd doubles, where a crucial point is to give a detailed combinatorial construction of the convex PBW-type Lyndon basis for type G_2 in 2-parameter quantum version. After furnishing possible commutation relations among quantum root vectors, we show that the restricted quantum groups are ribbon Hopf algebras under certain conditions through determining their left and right integrals. Besides these, we determine all of the Hopf algebra isomorphisms of u_{r,s}(G_2) in terms of the description of the sets of its left (right) skew-primitive elements."}
{"category": "Math", "title": "Scale-invariant groups", "abstract": "Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups G_n that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his conjecture by showing that the following groups (mostly finite-state self-similar groups) are scale-invariant: the lamplighter groups F\\wr\\Z, where F is any finite Abelian group; the solvable Baumslag-Solitar groups BS(1,m); the affine groups A\\ltimes\\Z^d, for any A\\leq GL(\\Z,d). However, the conjecture remains open with some natural stronger notions of scale-invariance for groups and transitive graphs. We construct scale-invariant tilings of certain Cayley graphs of the discrete Heisenberg group, whose existence is not immediate just from the scale-invariance of the group. We also note that torsion-free non-elementary hyperbolic groups are not scale-invariant."}
{"category": "Math", "title": "Cube diagrams and 3-dimensional Reidemeister-like moves for knots", "abstract": "In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology."}
{"category": "Math", "title": "On Volumes of Arithmetic Line Bundles", "abstract": "We show an arithmetic generalization of the recent work of Lazarsfeld-Mustata which uses Okounkov bodies to study linear series of line bundles. As applications, we derive a log-concavity inequality on volumes of arithmetic line bundles and an arithmetic Fujita approximation theorem for big line bundles."}
{"category": "Math", "title": "Periodic Solutions with Singularities in Two Dimensions in the $n$-body Problem", "abstract": "Analytical methods are used to prove the existence of a periodic, symmetric solution with singularities in the planar 4-body problem. A numerical calculation and simulation are used to generate the orbit. The analytical method easily extends to any even number of bodies. Multiple simultaneous binary collisions are a key feature of the orbits generated."}
{"category": "Math", "title": "The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited", "abstract": "In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These homotopy groups have already been calculated by Shimomura. The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura. An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the K(2)-localization of the spectrum TMF of topological modular forms and related spectra. Even more, the Adams-Novikov differentials for L_{K(2)}V(0) can be read off from those for TMF."}
{"category": "Math", "title": "On the mod - p cohomology of Out(F_{2(p-1)}", "abstract": "We study the mod-p cohomology of the group Out(F_n) of outer automorphisms of the free group F_n in the case n=2(p-1) which is the smallest n for which the p-rank of this group is 2. For p=3 we give a complete computation, at least above the virtual cohomological dimension of Out(F_4) (which is 5). More precisley, we calculate the equivariant cohomology of the p-singular part of outer space for p=3. For a general prime p>3 we give a recursive description in terms of the mod-p cohomology of Aut(F_k) for k less or equal to p-1. In this case we use the Out(F_{2(p-1)})-equivariant cohomology of the poset of elementary abelian p-subgroups of Out(F_n)."}
{"category": "Math", "title": "Competitive or Weak Cooperative Stochastic Lotka-Volterra Systems Conditioned to Non-Extinction", "abstract": "We are interested in the long time behavior of a two-type density-dependent biological population conditioned to non-extinction, in both cases of competition or weak cooperation between the two species. This population is described by a stochastic Lotka-Volterra system, obtained as limit of renormalized interacting birth and death processes. The weak cooperation assumption allows the system not to blow up. We study the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned to non extinction. To this aim we generalize in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes. The existence proof of a quasi-stationary distribution is reduced to the one for a $d$-dimensional Kolmogorov diffusion process under a symmetry assumption. The symmetry we need is satisfied under a local balance condition relying the ecological rates. A novelty is the outlined relation between the uniqueness of the quasi-stationary distribution and the ultracontractivity of the killed semi-group. By a comparison between the killing rates for the populations of each type and the one of the global population, we show that the quasi-stationary distribution can be either supported by individuals of one (the strongest one) type or supported by individuals of the two types. We thus highlight two different long time behaviors depending on the parameters of the model: either the model exhibits an intermediary time scale for which only one type (the dominant trait) is surviving, or there is a positive probability to have coexistence of the two species."}
{"category": "Math", "title": "A note on the Poincar'e series of the invariants of ternary forms", "abstract": "Analogue of Springer's formula for the Poincar\\'e series of the algebra invariants of ternary form is found."}
{"category": "Math", "title": "Stability of mean convex cones under mean curvature flow", "abstract": "We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large times. The proof involves the construction of appropriate barriers."}
{"category": "Math", "title": "Asymptotics for the survival probability in a killed branching random walk", "abstract": "Consider a discrete-time one-dimensional supercritical branching random walk. We study the probability that there exists an infinite ray in the branching random walk that always lies above the line of slope $\\gamma-\\epsilon$, where $\\gamma$ denotes the asymptotic speed of the right-most position in the branching random walk. Under mild general assumptions upon the distribution of the branching random walk, we prove that when $\\epsilon\\to 0$, the probability in question decays like $\\exp\\{- {\\beta + o(1)\\over \\epsilon^{1/2}}\\}$, where $\\beta$ is a positive constant depending on the distribution of the branching random walk. In the special case of i.i.d. Bernoulli$(p)$ random variables (with $0<p<{1\\over 2}$) assigned on a rooted binary tree, this answers an open question of Robin Pemantle."}
{"category": "Math", "title": "Quasi-alternating Montesinos links", "abstract": "The aim of this article is to detect new classes of quasi-alternating links. Quasi-alternating links are a natural generalization of alternating links. Their knot Floer and Khovanov homology are particularly easy to compute. Since knot Floer homology detects the genus of a knot as well as whether a knot is fibered, as provided bounds on unknotting number and slice genus, characterization of quasi-alternating links becomes an interesting open problem. We show that there exist classes of non-alternating Montesinos links, which are quasi-alternating."}
{"category": "Math", "title": "Dispersion of volume under the action of isotropic Brownian flows", "abstract": "We study transport properties of isotropic Brownian flows. Under a transience condition for the two-point motion, we show asymptotic normality of the image of a finite measure under the flow and -- under slightly stronger assumptions -- asymptotic normality of the distribution of the volume of the image of a set under the flow. Finally, we show that for a class of isotropic flows, the volume of the image of a nonempty open set (which is a martingale) converges to a random variable which is almost surely strictly positive."}
{"category": "Math", "title": "Biorthonormal Systems in Freud-type Weighted Spaces with Infinitely Many Zeros - An Interpolation Problem", "abstract": "In a Freud-type weighted ($w$) space, introducing another weight ($v$) with infinitely many roots, we give a complete and minimal system with respect to $vw$, by deleting infinitely many elements from the original orthonormal system with respect to $w$. The construction of the conjugate system implies an interpolation problem at infinitely many nodes. Besides the existence, we give some convergence properties of the solution."}
{"category": "Math", "title": "On Unique Additive Representations of Positive Integers and Some Close Problems", "abstract": "Let, for r>=2, (m_r(n)),n>=0, be Moser sequence such that every nonnegative integer is the unique sum of the form s_k+rs_l. In this article we give an explicit decomposition formulas of such form and an unexpectedly simple recursion relation for Moser's numbers. We also study interesting properties of the sequence (rm_r(n-1)+1),n>=1, and its connection with some important problems. In particular, in the case of r=2 this sequence is surprisingly connected with the numbers solving the combinatorial Josephus-Groer problem. We pose also some open questions."}
{"category": "Math", "title": "Constructing elliptic curves over finite fields with prescribed torsion", "abstract": "We present a method for constructing optimized equations for the modular curve X_1(N) using a local search algorithm on a suitably defined graph of birationally equivalent plane curves. We then apply these equations over a finite field F_q to efficiently generate elliptic curves with nontrivial N-torsion by searching for affine points on X_1(N)(F_q), and we give a fast method for generating curves with (or without) a point of order 4N using X_1(2N)."}
{"category": "Math", "title": "On The Negative K-Theory of Schemes in Finite Characteristic", "abstract": "We study the negative $K$-theory of singular varieties over a field of positive characteristic and in particular, prove the vanishing of $K_i(X)$ for $i < -d-2$ for a $k$-variety of dimension $d$."}
{"category": "Math", "title": "Greedy Polyominoes and first-passage times on random Voronoi tilings", "abstract": "Let N be distributed as a Poisson random set on R^d with intensity comparable to the Lebesgue measure. Consider the Voronoi tiling of R^d, (C_v)_{v\\in N}, where C_v is composed by points x in R^d that are closer to v than to any other v' in N. A polyomino P of size n is a connected union (in the usual R^d topological sense) of n tiles, and we denote by Pi_n the collection of all polyominos P of size n containing the origin. Assume that the weight of a Voronoi tile C_v is given by F(C_v), where F is a nonnegative functional on Voronoi tiles. In this paper we investigate the tail behavior of the maximal weight among polyominoes in Pi_n for some functionals F, mainly when F(C_v) is the number of faces of C_v. Next we apply our results to study self-avoiding paths, first-passage percolation models and the stabbing number on the dual graph, named the Delaunay triangulation. As the main application we show that first passage percolation has at most linear variance."}
{"category": "Math", "title": "Representations and characterizations of polynomial functions on chains", "abstract": "We are interested in representations and characterizations of lattice polynomial functions f:L^n -> L, where L is a given bounded distributive lattice. In companion papers [arXiv 0901.4888, arXiv 0808.2619], we investigated certain representations and provided various characterizations of these functions both as solutions of certain functional equations and in terms of necessary and sufficient conditions. In the present paper, we investigate these representations and characterizations in the special case when L is a chain, i.e., a totally ordered lattice. More precisely, we discuss representations of lattice polynomial functions given in terms of standard simplices and we present new axiomatizations of these functions by relaxing some of the conditions given in [arXiv 0901.4888, arXiv 0808.2619] and by considering further conditions, namely comonotonic minitivity and maxitivity."}
{"category": "Math", "title": "Categories. Beginning Course (in Russian)", "abstract": "This is a short textbook on Category Theory for Russian speaking students. It consists of three chapters: Categories and Functors, Representable Functors (including Adjoint Functors and (Co)limits) and Tensor Categories."}
{"category": "Math", "title": "A local optimal diastolic inequality on the two-sphere", "abstract": "Using a ramified cover of the two-sphere by the torus, we prove a local optimal inequality between the diastole and the area on the two-sphere near a singular metric. This singular metric, made of two equilateral triangles glued along their boundary, has been conjectured by E. Calabi to achieve the best ratio area over the square of the length of a shortest closed geodesic. Our diastolic inequality asserts that this conjecture is to some extent locally true."}
{"category": "Math", "title": "A Pentagonal Crystal, the Golden Section, alcove packing and aperiodic tilings", "abstract": "A Lie theoretic interpretation is given to a pattern with five-fold symmetry occurring in aperiodic Penrose tiling based on isosceles triangles with length ratios equal to the Golden Section. Specifically a $B(\\infty)$ crystal based on that of Kashiwara is constructed exhibiting this five-fold symmetry. It is shown that it can be represented as a Kashiwara $B(\\infty)$ crystal in type $A_4$. Similar crystals with $(2n+1)$-fold symmetry are represented as Kashiwara crystals in type $A_{2n}$. The weight diagrams of the latter inspire higher aperiodic tiling. In another approach alcove packing is seen to give aperiodic tiling in type $A_4$. Finally $2m$-fold symmetry is related to type $B_m$."}
{"category": "Math", "title": "Lower bounds for the principal genus of definite binary quadratic forms", "abstract": "We apply Tatuzawa's version of Siegel's theorem to derive two lower bounds on the size of the principal genus of positive definite binary quadratic forms."}
{"category": "Math", "title": "Generalization of the Logarithm Function and of the Exponential Function with Arbitrary Base", "abstract": "The logarithm function and the exponential function are, by nature, base dependent. Thus, in this paper I introduces an arbitrary base in the logarithm and exponential functions, both dependent on $q$, in order to have $\\log_a(x;q)$ and $a_q^x$. Some of the properties of these functions had been analyzed. The logarithm function was applied to the entropy which resulted in the $S_q = k[1 - \\sum_{i = 1}^Wp_i^{q}]/[1 - e^{1 - q}]$ expression."}
{"category": "Math", "title": "Extending Isotopies of Planar Continua", "abstract": "In this paper we solve the following problem in the affirmative: Let $Z$ be a continuum in the plane $\\complex$ and suppose that $h:Z\\times [0,1]\\to\\complex$ is an isotopy starting at the identity. Can $h$ be extended to an isotopy of the plane? We will provide a new characterization of an accessible point in a planar continuum $Z$ and use it to show that an accessible point is preserved during the isotopy. We show next that the isotopy can be extended over hyperbolic crosscuts. The proof makes use of the notion of a metric external ray, which mimics the notion of a conformal external ray, but is easier to control during an isotopy."}
{"category": "Math", "title": "Remarks on Springer's representations", "abstract": "We give an apriori description of a set of irreducible representations of a Weyl group which parametrize the nilpotent orbits in the Lie algebra of a connected reductive group in arbitrary characteristic. We also answer a question of Serre concerning the conjugacy class of a power of a unipotent element in a connected reductive group."}
{"category": "Math", "title": "Sharp Transitions in Making Squares", "abstract": "In many integer factoring algorithms, one produces a sequence of integers (created in a pseudo-random way), and wishes to rapidly determine a subsequence whose product is a square (which we call a square product). In his lecture at the 1994 International Congress of Mathematicians, Pomerance observed that the following problem encapsulates all of the key issues: Select integers a_1, a_2, >... at random from the interval [1,x], until some (non-empty) subsequence has product equal to a square. Find good estimate for the expected stopping time of this process. A good solution to this problem should help one to determine the optimal choice of parameters for one's factoring algorithm, and therefore this is a central question. Pomerance (1994), using an idea of Schroeppel (1985), showed that with probability 1-o(1) the first subsequence whose product equals a square occurs after at least J_0^{1-o(1)} integers have been selected, but no more than J_0, for an appropriate (explicitly determined) J_0=J_0(x). Herein we determine this expected stopping time up to a constant factor, tightening Pomerance's interval to $$[ (\\pi/4)(e^{-\\gamma} - o(1))J_0, (e^{-\\gamma} + o(1)) J_0],$$ where $\\gamma = 0.577...$ is the Euler-Mascheroni constant. We will also confirm the well established belief that, typically, none of the integers in the square product have large prime factors. We believe the upper of the two bounds to be asymptotically sharp."}
{"category": "Math", "title": "On surfaces with p_g=2q-3", "abstract": "We study minimal complex surfaces S of general type with q(S)=q and p_g(S)=2q-3, q>= 5. We give a complete classification in case that S has a fibration onto a curve of genus >=2. For these surfaces K^2=8\\chi. In general we prove that K^2>=7\\chi-1 and that the stronger inequality K^2\\ge 8\\chi holds under extra assumptions (e.g., if the canonical system has no fixed part or the canonical map has even degree). We also describe the Albanese map of S."}
{"category": "Math", "title": "Hom-Algebras and Hom-Coalgebras", "abstract": "The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom-Lie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom-Hopf algebras and prove some fundamental properties and give examples. Finally, we define the concept of Hom-Lie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group."}
{"category": "Math", "title": "Cohomology of graph hypersurfaces associated to certain Feynman graphs", "abstract": "To any Feynman graph (with 2n edges) we can associate a hypersurface X\\subset\\PP^{2n-1}. We study the middle cohomology H^{2n-2}(X) of such hypersurfaces. S. Bloch, H. Esnault, and D. Kreimer (Commun. Math. Phys. 267, 2006) have computed this cohomology for the first series of examples, the wheel with spokes graphs WS_n, n\\geq 3. Using the same technique, we introduce the generalized zigzag graphs and prove that W_5(H^{2n-2}(X))=\\QQ(-2) for all of them (with W_{*} the weight filtration). Next, we study primitively log divergent graphs with small number of edges and the behavior of graph hypersurfaces under the gluing of graphs."}
{"category": "Math", "title": "On perfectly generating projective classes in triangulated categories", "abstract": "We say that a projective class in a triangulated category with coproducts is perfect if the corresponding ideal is closed under coproducts of maps. We study perfect projective classes and the associated phantom and cellular towers. Given a perfect generating projective class, we show that every object is isomorphic to the homotopy colimit of a cellular tower associated to that object. Using this result and the Neeman's Freyd--style representability theorem we give a new proof of Brown Representability Theorem."}
{"category": "Math", "title": "Discrete multivariate distributions", "abstract": "This article brings in two new discrete distributions: multidimensional Binomial distribution and multidimensional Poisson distribution. Those distributions were created in eventology as more correct generalizations of Binomial and Poisson distributions. Accordingly to eventology new laws take into account full distribution of events. Also, in article its characteristics and properties are described"}
{"category": "Math", "title": "Puiseux power series solutions for systems of equations", "abstract": "We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities."}
{"category": "Math", "title": "Eventological theory of decision making for stock markets", "abstract": "The eventological theory of decision-making, the theory of eventfull decision-making is a theory of decision-making based on eventological principles and using results of mathematical eventology; a theoretical basis of the practical eventology. The beginnings of this theory which have arisen from eventfull representation of the reasonable subject and his decisions in the form of eventological distributions (E-distributions) of sets of events and which are based on the eventological H-theorem are offered. The illustrative example of the eventological decision-making by the reasonable subject on his own eventfull behaviour in the financial or share market is considered."}
{"category": "Math", "title": "On the size of minimal unsatisfiable formulas", "abstract": "An unsatisfiable formula is called minimal if it becomes satisfiable whenever any of its clauses are removed. We construct minimal unsatisfiable $k$-SAT formulas with $\\Omega(n^k)$ clauses for $k \\geq 3$, thereby negatively answering a question of Rosenfeld. This should be compared to the result of Lov\\'asz which asserts that a critically 3-chromatic $k$-uniform hypergraph can have at most $\\binom{n}{k-1}$ edges."}
{"category": "Math", "title": "Large deviations for random walks in random environment on a Galton-Watson tree", "abstract": "Consider a random walk in random environment on a supercritical Galton--Watson tree, and let $\\tau_n$ be the hitting time of generation $n$. The paper presents a large deviation principle for $\\tau_n/n$, both in quenched and annealed cases. Then we investigate the subexponential situation, revealing a polynomial regime similar to the one encountered in one dimension. The paper heavily relies on estimates on the tail distribution of the first regeneration time."}
{"category": "Math", "title": "More results on greedy defining sets", "abstract": "The greedy defining sets of graphs were appeared first time in [M. Zaker, Greedy defining sets of graphs, Australas. J. Combin, 2001]. We show that to determine the greedy defining number of bipartite graphs is an NP-complete problem. This result answers affirmatively the problem mentioned in the previous paper. It is also shown that this number for forests can be determined in polynomial time. Then we present a method for obtaining greedy defining sets in Latin squares and using this method, show that any $n\\times n$ Latin square has a GDS of size at most $n^2-(n\\log n)/4$. Finally we present an application of greedy defining sets in designing practical secret sharing schemes."}
{"category": "Math", "title": "Spurious caustics of Dispersion Relation Preserving schemes", "abstract": "A linear dispersive mechanism leading to a burst in the $L_\\infty$ norm of the error in numerical simulation of polychromatic solutions is identified. This local error pile-up corresponds to the existence of spurious caustics, which are allowed by the dispersive nature of the numerical error. From the mathematical point of view, spurious caustics are related to extrema of the numerical group velocity and are physically associated to interactions between rays defined by the characteristic lines of the discrete system. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes."}
{"category": "Math", "title": "On the branch curve of a general projection of a surface to a plane", "abstract": "In this paper we prove that the branch curve of a general projection of a surface to the plane is irreducible, with only nodes and cusps."}
{"category": "Math", "title": "On the moments and distribution of discrete Choquet integrals from continuous distributions", "abstract": "We study the moments and the distribution of the discrete Choquet integral when regarded as a real function of a random sample drawn from a continuous distribution. Since the discrete Choquet integral includes weighted arithmetic means, ordered weighted averaging functions, and lattice polynomial functions as particular cases, our results encompass the corresponding results for these aggregation functions. After detailing the results obtained in [1] in the uniform case, we present results for the standard exponential case, show how approximations of the moments can be obtained for other continuous distributions such as the standard normal, and elaborate on the asymptotic distribution of the Choquet integral. The results presented in this work can be used to improve the interpretation of discrete Choquet integrals when employed as aggregation functions."}
{"category": "Math", "title": "Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations", "abstract": "We investigate mathematically a nonlinear approximation type approach recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to solve high dimensional partial differential equations. We show the link between the approach and the greedy algorithms of approximation theory studied e.g. in [R.A. DeVore and V.N. Temlyakov, Adv. Comput. Math., 1996]. On the prototypical case of the Poisson equation, we show that a variational version of the approach, based on minimization of energies, converges. On the other hand, we show various theoretical and numerical difficulties arising with the non variational version of the approach, consisting of simply solving the first order optimality equations of the problem. Several unsolved issues are indicated in order to motivate further research."}
{"category": "Math", "title": "The geometric correspondence in some special cases", "abstract": "The geometric Langlands correspondence for function fields over finite fields has been proved by Frenkel, Gaitsgory, Vilonen. The aim of this article is to write translation for curves over the complex field and prove the correspondence in some special cases and some new cases, outside the frame of the usual geometric Langlands correspondence."}
{"category": "Math", "title": "Gaussian Correlation Conjecture for Symmetric Convex Sets", "abstract": "Gaussian correlation conjecture states that the Gaussian measure of the intersection of two symmetric convex sets is greater or equal to the product of the measures."}
{"category": "Math", "title": "Trimming and likelihood: Robust location and dispersion estimation in the elliptical model", "abstract": "Robust estimators of location and dispersion are often used in the elliptical model to obtain an uncontaminated and highly representative subsample by trimming the data outside an ellipsoid based in the associated Mahalanobis distance. Here we analyze some one (or $k$)-step Maximum Likelihood Estimators computed on a subsample obtained with such a procedure. We introduce different models which arise naturally from the ways in which the discarded data can be treated, leading to truncated or censored likelihoods, as well as to a likelihood based on an only outliers gross errors model. Results on existence, uniqueness, robustness and asymptotic properties of the proposed estimators are included. A remarkable fact is that the proposed estimators generally keep the breakdown point of the initial (robust) estimators, but they could improve the rate of convergence of the initial estimator because our estimators always converge at rate $n^{1/2}$, independently of the rate of convergence of the initial estimator."}
{"category": "Math", "title": "First Hitting Time of the Boundary of the Weyl Chamber by Radial Dunkl Processes", "abstract": "We provide two equivalent approaches for computing the tail distribution of the first hitting time of the boundary of the Weyl chamber by a radial Dunkl process. The first approach is based on a spectral problem with initial value. The second one expresses the tail distribution by means of the $W$-invariant Dunkl-Hermite polynomials. Illustrative examples are given by the irreducible root systems of types $A$, $B$, $D$. The paper ends with an interest in the case of Brownian motions for which our formulae take determinantal forms."}
{"category": "Math", "title": "Sharp ill-posedness result for the periodic Benjamin-Ono equation", "abstract": "We prove the discontinuity for the weak $ L^2(\\T) $-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in $ H^s(\\T) $ as soon as $ s<0 $ and thus completes exactly the well-posedness result obtained by the author."}
{"category": "Math", "title": "Generalized Bessel function of Type D", "abstract": "We write down the generalized Bessel function associated with the root system of type $D$ by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type $D$."}
{"category": "Math", "title": "Torus invariant divisors", "abstract": "Using the language of polyhedral divisors and divisorial fans we describe invariant divisors on normal varieties X which admit an effective codimension one torus action. In this picture X is given by a divisorial fan on a smooth projective curve Y. Cartier divisors on X can be described by piecewise affine functions h on the divisorial fan S whereas Weil divisors correspond to certain zero and one dimensional faces of it. Furthermore we provide descriptions of the divisor class group and the canonical divisor. Global sections of line bundles O(D_h) will be determined by a subset of a weight polytope associated to h, and global sections of specific line bundles on the underlying curve Y."}
{"category": "Math", "title": "Twisted convolution, pseudo-differential operators and Fourier modulation spaces", "abstract": "We discuss continuity of the twisted convolution on (weighted) Fourier modulation spaces. We use these results to establish continuity results for the twisted convolution on Lebesgue spaces. For example we prove that if $\\omega$ is an appropriate weight and $1\\le p\\le 2$, then $L^p_{(\\omega)}$ is an algebra under the twisted convolution."}
{"category": "Math", "title": "The curve of lines on a prime Fano threefold of genus 8", "abstract": "We show that a general prime Fano threefold X of genus 8 can be reconstructed from the pair $(\\Gamma,L)$, where $\\Gamma$ is its Fano curve of lines and $L=O_{\\Gamma}(1)$ is the theta-characteristic which gives a natural embedding $\\Gamma \\subset \\matbb{P}^5$."}
{"category": "Math", "title": "Rook placements in Young diagrams and permutation enumeration", "abstract": "Given two operators $\\hat D$ and $\\hat E$ subject to the relation $\\hat D\\hat E -q \\hat E \\hat D =p$, and a word $w$ in $M$ and $N$, the rewriting of $w$ in normal form is combinatorially described by rook placements in a Young diagram. We give enumerative results about these rook placements, particularly in the case where $p=(1-q)/q^2$. This case naturally arises in the context of the PASEP, a random process whose partition function and stationary distribution are expressed using two operators $D$ and $E$ subject to the relation $DE-qED=D+E$ (matrix Ansatz). Using the link obtained by Corteel and Williams between the PASEP, permutation tableaux and permutations, we prove a conjecture of Corteel and Rubey about permutation enumeration. This result gives the generating function for permutations of given size with respect to the number of ascents and occurrences of the pattern 13-2, this is also the moments of the $q$-Laguerre orthogonal polynomials."}
{"category": "Math", "title": "Differences of random Cantor sets and lower spectral radii", "abstract": "We investigate the question under which conditions the algebraic difference between two independent random Cantor sets $C_1$ and $C_2$ almost surely contains an interval, and when not. The natural condition is whether the sum $d_1+d_2$ of the Hausdorff dimensions of the sets is smaller (no interval) or larger (an interval) than 1. Palis conjectured that \\emph{generically} it should be true that $d_1+d_2>1$ should imply that $C_1-C_2$ contains an interval. We prove that for 2-adic random Cantor sets generated by a vector of probabilities $(p_0,p_1)$ the interior of the region where the Palis conjecture does not hold is given by those $p_0,p_1$ which satisfy $p_0+p_1>\\sqrt{2}$ and $p_0p_1(1+p_0^2+p_1^2)<1$. We furthermore prove a general result which characterizes the interval/no interval property in terms of the lower spectral radius of a set of $2\\times 2$ matrices."}
{"category": "Math", "title": "Local antithetic sampling with scrambled nets", "abstract": "We consider the problem of computing an approximation to the integral $I=\\int_{[0,1]^d}f(x) dx$. Monte Carlo (MC) sampling typically attains a root mean squared error (RMSE) of $O(n^{-1/2})$ from $n$ independent random function evaluations. By contrast, quasi-Monte Carlo (QMC) sampling using carefully equispaced evaluation points can attain the rate $O(n^{-1+\\varepsilon})$ for any $\\varepsilon>0$ and randomized QMC (RQMC) can attain the RMSE $O(n^{-3/2+\\varepsilon})$, both under mild conditions on $f$. Classical variance reduction methods for MC can be adapted to QMC. Published results combining QMC with importance sampling and with control variates have found worthwhile improvements, but no change in the error rate. This paper extends the classical variance reduction method of antithetic sampling and combines it with RQMC. One such method is shown to bring a modest improvement in the RMSE rate, attaining $O(n^{-3/2-1/d+\\varepsilon})$ for any $\\varepsilon>0$, for smooth enough $f$."}
{"category": "Math", "title": "Asymptotic Behavior of Individual Orbits of Discrete Systems", "abstract": "We consider the asymptotic behavior of bounded solutions of the difference equations of the form $x(n+1)=Bx(n) + y(n)$ in a Banach space $\\X$, where $n=1,2,...$, $B$ is a linear continuous operator in $\\X$, and $(y(n))$ is a sequence in $\\X$ converging to 0 as $n\\to\\infty$. An obtained result with an elementary proof says that if $\\sigma (B) \\cap \\{|z|=1\\} \\subset \\{1\\}$, then every bounded solution $x(n)$ has the property that $\\lim_{n\\to\\infty} (x(n+1)-x(n)) =0$. This result extends a theorem due to Katznelson-Tzafriri. Moreover, the techniques of the proof are furthered to study the individual stability of solutions of the discrete system. A discussion on further extensions is also given."}
{"category": "Math", "title": "Proper isometric actions", "abstract": "We prove the following to results: (1) A subgroup G of the isometry group of a Riemannian manifold M acts properly on M if and only if G is closed in the isometry group of M. (2) The orbits of an isometric action are closed if and only if the action is orbit equivalent to a proper isometric action."}
{"category": "Math", "title": "On $\\omega_3$-chains in P($\\omega_1$) mod finite", "abstract": "We prove that if there exists a simplified $(\\omega_1,2)$-morass, then there is a ccc forcing which adds an $\\omega_3$-chain in P($\\omega_1$) mod finite and a ccc forcing which adds a family of $\\omega_3$-many strongly almost disjoint functions from $\\omega_1$ to $\\omega$. The idea is to use a finite support iteration of countable forcings which is not linear but three-dimensional."}
{"category": "Math", "title": "Initial-boundary value problems for conservation laws with source terms and the Degasperis-Procesi equation", "abstract": "We consider conservation laws with source terms in a bounded domain with Dirichlet boundary conditions. We first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition. Equipped with this formulation, we go on to establish the well-posedness of entropy solutions to the initial-boundary value problem. The proof utilizes the kinetic formulation and the compensated compactness method. Finally, we make use of these results to demonstrate the well-posedness in a class of discontinuous solutions to the initial-boundary value problem for the Degasperis-Procesi shallow water equation, which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term."}
{"category": "Math", "title": "Topological Hochschild homology of Thom spectra and the free loop space", "abstract": "We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f: X->BF, where BF denotes the classifying space for stable spherical fibrations. To do this, we consider symmetric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identifies the topological Hochschild homology as the Thom spectrum of a certain stable bundle over the free loop space L(BX). This leads to explicit calculations of the topological Hochschild homology for a large class of ring spectra, including all of the classical cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p and HZ."}
{"category": "Math", "title": "The evaluation of Tornheim double sums. Part2", "abstract": "We provide an explicit formula for the Tornheim double series T(a,0,c) in terms of an integral involving the Hurwitz zeta function. For integer values of the parameters, a=m, c=n, we show that in the most interesting case of even weight N:=m+n the Tornheim sum T(m,0,n) can be expressed in terms of zeta values and the family of integrals % \\int_0^1 loggamma(q) B_{k}(q) Cl_{j+1} (2 \\pi q) dq, % with k+j = N, where B_{k}(q) is a Bernoulli polynomial and \\Cl_{j+1}(x) is a Clausen function."}
{"category": "Math", "title": "Auslander-Reiten sequences for homotopists and arithmeticians", "abstract": "We introduce Auslander-Reiten sequences for group algebras and give several recent applications. The first part of the paper is devoted to some fundamental problems in Tate cohomology which are motivated by homotopy theory. In the second part of the paper we interpret Auslander-Reiten sequences in the context of Galois theory and connect them to some important arithmetic objects."}
{"category": "Math", "title": "Extremely non-complex C(K) spaces", "abstract": "We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces $X$ such that the norm equality $\\|Id + T^2\\|=1 + \\|T^2\\|$ holds for every bounded linear operator $T:X\\longrightarrow X$. This answers in the positive Question 4.11 of [Kadets, Martin, Meri, Norm equalities for operators, \\emph{Indiana U. Math. J.} \\textbf{56} (2007), 2385--2411]. More concretely, we show that this is the case of some $C(K)$ spaces with few operators constructed in [Koszmider, Banach spaces of continuous functions with few operators, \\emph{Math. Ann.} \\textbf{330} (2004), 151--183] and [Plebanek, A construction of a Banach space $C(K)$ with few operators, \\emph{Topology Appl.} \\textbf{143} (2004), 217--239]. We also construct compact spaces $K_1$ and $K_2$ such that $C(K_1)$ and $C(K_2)$ are extremely non-complex, $C(K_1)$ contains a complemented copy of $C(2^\\omega)$ and $C(K_2)$ contains a (1-complemented) isometric copy of $\\ell_\\infty$."}
{"category": "Math", "title": "Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection", "abstract": "We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche and Zambotti, we use a method based on infinite dimensional equations, approximation by regular equations and convergence of the approximated semi-group. We obtain existence and uniqueness of solution for nonnegative intial conditions, results on the invariant measures, and on the reflection measures."}
{"category": "Math", "title": "Thom spectra that are symmetric spectra", "abstract": "We analyze the functorial and multiplicative properties of the Thom spectrum functor in the setting of symmetric spectra, and we establish the relevant homotopy invariance."}
{"category": "Math", "title": "Higher topological Hochschild homology of Thom spectra", "abstract": "In this paper we analyze the higher topological Hochschild homology of commutative Thom S-algebras. This includes the case of the classical cobordism spectra MO, MSO, MU, etc. We consider the homotopy orbits of the torus action on iterated topological Hochschild homology and we describe the relationship to topological Andre-Quillen homology."}
{"category": "Math", "title": "A simple formula for the Casson-Walker invariant", "abstract": "Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case - the celebrated Casson-Walker invariant of rational homology spheres. This paper is dedicated to a detailed treatment of 2-component links; a general case will be considered in a forthcoming paper. We present simple Gauss diagram formulas for the Casson-Walker invariant. This enables us to understand/separate its dependence on the unframed link and on the framings. We also obtain skein relations for the Casson-Walker invariant under crossing changes, and study its asymptotic behavior when framings tend to infinity. Finally, we present results of extensive computer calculations."}
{"category": "Math", "title": "Moser stability for locally conformally symplectic structures", "abstract": "We formulate and prove the analogue of Moser's stability theorem for locally conformally symplectic structures. As special cases we recover some results previously proved by Banyaga."}
{"category": "Math", "title": "Closeness of convolutions of probability measures", "abstract": "We derive new explicit bounds for the total variation distance between two convolution products of $n$ probability distributions, one of which having identical convolution factors. Approximations by finite signed measures of arbitrary order are considered as well. We are interested in bounds with magic factors, i.e. roughly speaking $n$ also appears in the denominator. Special emphasis is given to the approximation by the $n$-fold convolution of the arithmetic mean of the distributions under consideration. As an application, we consider the multinomial approximation of the generalized multinomial distribution. It turns out that here the order of some bounds given in Roos (2001) and Loh (1992) can significantly be improved. In particular, it follows that a dimension factor can be dropped. Moreover, better accuracy is achieved in the context of symmetric distributions with finite support. In the course of proof, we use a basic Banach algebra technique for measures on a measurable Abelian group. Though this method was already used by Le Cam (1960), our central arguments seem to be new. We also derive new smoothness bounds for convolutions of probability distributions, which might be of independent interest."}
{"category": "Math", "title": "Canonical divisors on T-varieties", "abstract": "Generalising toric geometry we study compact varieties admitting lower dimensional torus actions. In particular we describe divisors on them in terms of convex geometry and give a criterion for their ampleness. These results may be used to study Fano varieties with small torus actions. As a first result we classify log del Pezzo C*-surfaces of Picard number 1 and Gorenstein index less than 4. In further examples we show how classification might work in higher dimensions and we give explicit descriptions of some equivariant smoothings of Fano threefolds."}
{"category": "Math", "title": "The Corona Theorem for the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in $\\mathbb{C}^{n}$", "abstract": "We prove that the multiplier algebra of the Drury-Arveson Hardy space $H_{n}^{2}$ on the unit ball in $\\mathbb{C}^{n}$ has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space $B_{p}^{\\sigma}$ has the \"baby corona property\" for all $\\sigma \\geq 0$ and $1<p<\\infty $. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems."}
{"category": "Math", "title": "On Bochner-Martinelli residue currents and their annihilator ideals", "abstract": "We study the residue current R^f of Bochner-Martinelli type associated with a tuple f=(f_1,...,f_m) of holomorphic germs at the origin in C^n, whose common zero set equals the origin. Our main results are a geometric description of R^f in terms of the Rees valuations associated with the ideal (f) generated by f and a characterization of when the annihilator ideal of R^f equals (f)."}
{"category": "Math", "title": "Hyperkahler SYZ conjecture and semipositive line bundles", "abstract": "Let $M$ be a compact, holomorphic symplectic Kaehler manifold, and $L$ a non-trivial line bundle admitting a metric of semi-positive curvature. We show that some power of $L$ is effective. This result is related to the hyperkaehler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if $L$ is not big."}
{"category": "Math", "title": "Giroux correspondence, confoliations, and symplectic structures on S^1 x M", "abstract": "Let M be a closed oriented 3-manifold such that S^1 x M admits a symplectic structure w. The goal of this paper is to show that M is a fiber bundle over S^1. The basic idea is to use the obvious S^1-action on S^1 x M by rotating the first factor, and one of the key steps is to show that the S^1-action on S^1 x M is actually symplectic with respect to a symplectic form cohomologous to w. We achieve it by crucially using the recent result or its relative version of Giroux about one-to-one correspondence between open book decompositions of M up to positive stabilization and co-oriented contact structures on M up to contact isotopy."}
{"category": "Math", "title": "An asymptotic theory for randomly forced discrete nonlinear heat equations", "abstract": "We study discrete nonlinear parabolic stochastic heat equations of the form, $u_{n+1}(x)-u_n(x)=(\\mathcal {L}u_n)(x)+\\sigma(u_n(x))\\xi_n(x)$, for $n\\in {\\mathbf{Z}}_+$ and $x\\in {\\mathbf{Z}}^d$, where $\\boldsymbol \\xi:=\\{\\xi_n(x)\\}_{n\\ge 0,x\\in {\\mathbf{Z}}^d}$ denotes random forcing and $\\mathcal {L}$ the generator of a random walk on ${\\mathbf{Z}}^d$. Under mild conditions, we prove that the preceding stochastic PDE has a unique solution that grows at most exponentially in time. And that, under natural conditions, it is \"weakly intermittent.\" Along the way, we establish a comparison principle as well as a finite support property."}
{"category": "Math", "title": "Floer homology for 2-torsion instanton invariants", "abstract": "We construct a variant of Floer homology groups and prove a gluing formula for a variant of Donaldson invariants. As a corollary, the variant of Donaldson invariants is non-trivial for connected sums of 4-manifolds which satisfy a condition for Donaldson invariants. We also show a non-existence result of compact, spin 4-manifolds with boundary some homology 3-spheres and with certain intersection forms."}
{"category": "Math", "title": "Expander graphs based on GRH with an application to elliptic curve cryptography", "abstract": "We present a construction of expander graphs obtained from Cayley graphs of narrow ray class groups, whose eigenvalue bounds follow from the Generalized Riemann Hypothesis. Our result implies that the Cayley graph of (Z/qZ)* with respect to small prime generators is an expander. As another application, we show that the graph of small prime degree isogenies between ordinary elliptic curves achieves non-negligible eigenvalue separation, and explain the relationship between the expansion properties of these graphs and the security of the elliptic curve discrete logarithm problem."}
{"category": "Math", "title": "Springer representations on the Khovanov Springer varieties", "abstract": "Springer varieties are studied because their cohomology carries a natural action of the symmetric group $S_n$ and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties $X_n$ as subvarieties of the product of spheres $(S^2)^n$. We show that if $X_n$ is embedded antipodally in $(S^2)^n$ then the natural $S_n$-action on $(S^2)^n$ induces an $S_n$-representation on the image of $H_*(X_n)$. This representation is the Springer representation. Our construction admits an elementary (and geometrically natural) combinatorial description, which we use to prove that the Springer representation on $H_*(X_n)$ is irreducible in each degree. We explicitly identify the Kazhdan-Lusztig basis for the irreducible representation of $S_n$ corresponding to the partition $(n/2,n/2)$."}
{"category": "Math", "title": "On the vanishing of negative K-groups", "abstract": "Let k be an infinite perfect field of positive characteristic p and assume that strong resolution of singularities holds over k. We prove that, if X is a d-dimensional noetherian scheme whose underlying reduced scheme is essentially of finite type over the field k, then the negative K-group K_q(X) vanishes for every q < -d. This partially affirms a conjecture of Weibel."}
{"category": "Math", "title": "Mahlburg's work on Crank Functions returns to Ramanujan's work and inspiration", "abstract": "Mahlburg (2005) brilliantly showed the importance of crank functions in partition congruences that were originally guessed by Dyson (1944). Ramanujan's partition functions are the centre of these works. Not only for the theory on cranks, but for many other researchers' in India Ramanujan's work inspired for their career in mathematics. This is an undergraduate expository article."}
{"category": "Math", "title": "Estimation of missing data by using the filtering process in a time series modeling", "abstract": "This paper proposed a new method to estimate the missing data by using the filtering process. We used datasets without missing data and randomly missing data to evaluate the new method of estimation by using the Box - Jenkins modeling technique to predict monthly average rainfall for site 5504035 Lahar Ikan Mati at Kepala Batas, P. Pinang station in Malaysia. The rainfall data was collected from the $1^{st}$ January 1969 to $31^{st}$ December 1997 in the station. The data used in the development of the model to predict rainfall were represented by an autoregressive integrated moving - average (ARIMA) model. The model for both datasets was ARIMA$(1,0,0)(0,1,1)_s$. The result checked with the Naive test, which is the Thiel's statistic and was found to be equal to $U=0.72086$ for the complete data and $U=0.726352$ for the missing data, which mean they were good models."}
{"category": "Math", "title": "Asymptotics for Kotz Type III Elliptical Distributions", "abstract": "In this paper we derive the tail asymptotics of a Kotz Type III elliptical random vector. As an application of our asymptotic expansion we derive an approximation for the conditional excess distribution. Furthermore, we discuss the asymptotic dependence of Kotz Type III triangular arrays and provide some details on the estimation of conditional excess distribution and survivor function."}
{"category": "Math", "title": "On the Behrens--Fisher problem: A globally convergent algorithm and a finite-sample study of the Wald, LR and LM Tests", "abstract": "In this paper we provide a provably convergent algorithm for the multivariate Gaussian Maximum Likelihood version of the Behrens--Fisher Problem. Our work builds upon a formulation of the log-likelihood function proposed by Buot and Richards \\citeBR. Instead of focusing on the first order optimality conditions, the algorithm aims directly for the maximization of the log-likelihood function itself to achieve a global solution. Convergence proof and complexity estimates are provided for the algorithm. Computational experiments illustrate the applicability of such methods to high-dimensional data. We also discuss how to extend the proposed methodology to a broader class of problems. We establish a systematic algebraic relation between the Wald, Likelihood Ratio and Lagrangian Multiplier Test ($W\\geq \\mathit{LR}\\geq \\mathit{LM}$) in the context of the Behrens--Fisher Problem. Moreover, we use our algorithm to computationally investigate the finite-sample size and power of the Wald, Likelihood Ratio and Lagrange Multiplier Tests, which previously were only available through asymptotic results. The methods developed here are applicable to much higher dimensional settings than the ones available in the literature. This allows us to better capture the role of high dimensionality on the actual size and power of the tests for finite samples."}
{"category": "Math", "title": "Kaehler-Einstein submanifolds of the infinite dimensional projective space", "abstract": "This paper consists of two main results. In the first one we describe all Kaehler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kaehler-Einstein metric with negative scalar curvature which admits a Kaehler immersion into the infinite dimensional complex projective space."}
{"category": "Math", "title": "Probability measures, L\\'{e}vy measures and analyticity in time", "abstract": "We investigate the relation of the semigroup probability density of an infinite activity L\\'{e}vy process to the corresponding L\\'{e}vy density. For subordinators, we provide three methods to compute the former from the latter. The first method is based on approximating compound Poisson distributions, the second method uses convolution integrals of the upper tail integral of the L\\'{e}vy measure and the third method uses the analytic continuation of the L\\'{e}vy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time. Several concrete examples illustrate the three methods and our results."}
{"category": "Math", "title": "An Algorithmic and a geometric characterization of coarsening at random", "abstract": "We show that the class of conditional distributions satisfying the coarsening at random (CAR) property for discrete data has a simple and robust algorithmic description based on randomized uniform multicovers: combinatorial objects generalizing the notion of partition of a set. However, the complexity of a given CAR mechanism can be large: the maximal \"height\" of the needed multicovers can be exponential in the number of points in the sample space. The results stem from a geometric interpretation of the set of CAR distributions as a convex polytope and a characterization of its extreme points. The hierarchy of CAR models defined in this way could be useful in parsimonious statistical modeling of CAR mechanisms, though the results also raise doubts in applied work as to the meaningfulness of the CAR assumption in its full generality."}
{"category": "Math", "title": "$L$-approximation of $B$-splines by trigonometric polynomials", "abstract": "This note is a continuation of our papers [1,2], devoted to $L$-approximation of characteristic function of $(-h, h)$ by trigonometric polynomials. In the paper [1] the sharp values of the best approximation for the special values of $h$ were found. In [2] we gave the complete solution of the problem for arbitrary values of $h$. In general case [2] the situation is more deep and results are not so simple as in [1]. For applications to the problem of optimal constants in the Jackson-type inequalities we need, however, results on $L$-approximation of $B$-splines and linear combinations of $B$-splines. Here we present some simple results about $L$-approximation of $B$-splines as well as give the the proof of its sharpness for the special values of $h$."}
{"category": "Math", "title": "The Brauer Group of a Smooth Orbifold", "abstract": "Let $k$ be a field and $X/k$ be a smooth quasiprojective orbifold. Let $X\\to \\underline{X}$ be its coarse moduli space. In this paper we study the Brauer group of $X$ and compare it with the Brauer group of the smooth locus of $\\underline{X}$."}
{"category": "Math", "title": "Deformation of a smooth Deligne-Mumford stack via differential graded Lie algebra", "abstract": "For a smooth Deligne-Mumford stack over $\\CC$, we define its associated Kodaira-Spencer differential graded Lie algebra and show that the deformation functor of the stack is isomorphic to the deformation functor of the Kodaira-Spencer algebra if the stack is proper over $\\CC$."}
{"category": "Math", "title": "Residual empirical processes for long and short memory time series", "abstract": "This paper studies the residual empirical process of long- and short-memory time series regression models and establishes its uniform expansion under a general framework. The results are applied to the stochastic regression models and unstable autoregressive models. For the long-memory noise, it is shown that the limit distribution of the Kolmogorov-Smirnov test statistic studied in Ho and Hsing [Ann. Statist. 24 (1996) 992-1024] does not hold when the stochastic regression model includes an unknown intercept or when the characteristic polynomial of the unstable autoregressive model has a unit root. To this end, two new statistics are proposed to test for the distribution of the long-memory noises of stochastic regression models and unstable autoregressive models. (With Correction.)"}
{"category": "Math", "title": "On a Lomonaco-Kauffman conjecture", "abstract": "Samuel J. Lomonaco Jr and Louis H. Kauffman conjectured that tame knot theory and knot mosaic theory are equivalent. We give a proof of the Lomonaco-Kauffman conjecture."}
{"category": "Math", "title": "On Virtual Crossing Number Estimates For Virtual Links", "abstract": "We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in $R^{2}$. We tackle this problem by the so-called $\\xi$-polynomial whose leading (lowest) degree naturally estimates the virtual crossing number. Several sufficient conditions for minimality together with infinite series of new examples are given. We also state several open questions about $M$-diagrams, which are minimal according to our sufficient conditions."}
{"category": "Math", "title": "Infinite genus surfaces and irrational polygonal billiards", "abstract": "We prove that the natural invariant surface associated with the billiard game on an irrational polygonal table is homeomorphic to the Loch Ness monster, that is, the only orientable infinite genus topological real surface with exactly one end."}
{"category": "Math", "title": "The Average Number of Block Interchanges Needed to Sort A Permutation and a recent result of Stanley", "abstract": "We use an interesting result of probabilistic flavor concerning the product of two permutations consisting of one cycle each to find an explicit formula for the average number of block interchanges needed to sort a permutation of length $n$."}
{"category": "Math", "title": "On the dimension of the minimal vertex covers semigroup ring of an unmixed bipartite graph", "abstract": "In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the semigroup ring associated to the set of minimal vertex covers of an unmixed bipartite graph. In this paper we relate the dimension of this semigroup ring to the rank of the Boolean lattice associated to the graph."}
{"category": "Math", "title": "Parabolic subgroups of Garside groups II: ribbons", "abstract": "We introduce and investigate the ribbon groupoid associated with a Garside group. Under a technical hypothesis, we prove that this category is a Garside groupoid. We decompose this groupoid into a semi-direct product of two of its parabolic subgroupoids and provide a groupoid presentation. In order to established the latter result, we describe quasi-centralizers in Garside groups. All results hold in the particular case of Artin-Tits groups of spherical type."}
{"category": "Math", "title": "An elementary approach to extreme values theory", "abstract": "This note presents a rather intuitive approach to extreme value theory. This approach was devised mostly for pedagogical reason."}
{"category": "Math", "title": "Images of the Polar maps for Hypersurfaces", "abstract": "For a projective hypersurface $X \\subset \\P^n$, the images of the polar maps of degree $k$ are studied. The cohomology class defined by these maps is calculated and classical results on dual varieties are presented as applications."}
{"category": "Math", "title": "Tilted Euler characteristic densities for Central Limit random fields, with application to \"bubbles\"", "abstract": "Local increases in the mean of a random field are detected (conservatively) by thresholding a field of test statistics at a level $u$ chosen to control the tail probability or $p$-value of its maximum. This $p$-value is approximated by the expected Euler characteristic (EC) of the excursion set of the test statistic field above $u$, denoted $\\mathbb{E}\\varphi(A_u)$. Under isotropy, one can use the expansion $\\mathbb{E}\\varphi(A_u)=\\sum_k\\mathcal{V}_k\\rho_k(u)$, where $\\mathcal{V}_k$ is an intrinsic volume of the parameter space and $\\rho_k$ is an EC density of the field. EC densities are available for a number of processes, mainly those constructed from (multivariate) Gaussian fields via smooth functions. Using saddlepoint methods, we derive an expansion for $\\rho_k(u)$ for fields which are only approximately Gaussian, but for which higher-order cumulants are available. We focus on linear combinations of $n$ independent non-Gaussian fields, whence a Central Limit theorem is in force. The threshold $u$ is allowed to grow with the sample size $n$, in which case our expression has a smaller relative asymptotic error than the Gaussian EC density. Several illustrative examples including an application to \"bubbles\" data accompany the theory."}
{"category": "Math", "title": "On the non-existence of Tensor Products of Algebraic Cycles", "abstract": "Let $\\otimes$ be the map which classifies the tensor product of two line bundles, an extension of this map to the space of all codimension 1 algebraic cycles is constructed. It is proved that this extension cannot exist in codimension greater than 1."}
{"category": "Math", "title": "Compactified Picard stacks over the moduli stack of stable curves with marked points", "abstract": "In this paper we give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of n-pointed stable curves of genus g, for g greater than 2. These stacks are smooth, irreducible and have dimension 4g-3+n, yielding a geometrically meaningful compactification of the degree d universal Picard stack over the moduli stack of smooth curves with marked points."}
{"category": "Math", "title": "Improved Error Bounds for Dirichlet-to-Neumann Absorbing Boundaries", "abstract": "It has long been known how to construct radiation boundary conditions for the time dependent wave equation. Although arguments suggesting that they are accurate have been given, it is only recently that rigorous error bounds have been proved. Previous estimates show that the error caused by these methods behaves like epsilon C exp(gamma t) for any gamma > 0. We improve these results and show that the error behaves like epsilon t^2."}
{"category": "Math", "title": "Continuous trace C*-algebras, gauge groups and rationalization", "abstract": "Let \\zeta be an n-dimensional complex matrix bundle over a compact metric space X and let A_\\zeta denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA_\\zeta, the group of unitaries of A_\\zeta. The answer turns out to be independent of the bundle \\zeta and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X."}
{"category": "Math", "title": "An isospectral deformation on an orbifold quotient of a nilmanifold", "abstract": "We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's Theorem due to DeTurck and Gordon."}
{"category": "Math", "title": "Algebra in superextensions of twinic groups", "abstract": "Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $\\lambda(X)$ consisting of maximal linked systems on $X$. This semigroup contains the semigroup $\\beta(X)$ of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup $\\lambda(X)$ in the semigroup of all self-maps of the power-set of $X$ and using this representation describe the structure of minimal ideal and minimal left ideals of $\\lambda(X)$ for each twinic group $X$. The class of twinic groups includes all amenable groups and all groups with periodic commutators but does not include the free group with two generators."}
{"category": "Math", "title": "Spectral and geometric bounds on 2-orbifold diffeomorphism type", "abstract": "We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collection of 2-orbifolds satisfying lower bounds on sectional curvature and volume, and an upper bound on diameter. An argument converting spectral data to geometric bounds shows that the first result is a consequence of the second."}
{"category": "Math", "title": "Grid graphs and lattice surfaces", "abstract": "First, we apply Thurston's construction of pseudo-Anosov homeomorphisms to grid graphs and obtain translation surfaces whose Veech groups are commensurable to $(m,n,\\infty)$ triangle groups. These surfaces were first discovered by Bouw and M\\\"oller, however our treatment of the surfaces differs. We construct these surfaces by gluing together polygons in two ways. We use these elementary descriptions to compute the Veech groups, resolve primitivity questions, and describe the surfaces algebraically. Second, we show that some $(m,n, \\infty)$ triangle groups can not arise as Veech groups. This generalizes work of Hubert and Schmidt."}
{"category": "Math", "title": "Optimal cross-validation in density estimation with the $L^2$-loss", "abstract": "We analyze the performance of cross-validation (CV) in the density estimation framework with two purposes: (i) risk estimation and (ii) model selection. The main focus is given to the so-called leave-$p$-out CV procedure (Lpo), where $p$ denotes the cardinality of the test set. Closed-form expressions are settled for the Lpo estimator of the risk of projection estimators. These expressions provide a great improvement upon $V$-fold cross-validation in terms of variability and computational complexity. From a theoretical point of view, closed-form expressions also enable to study the Lpo performance in terms of risk estimation. The optimality of leave-one-out (Loo), that is Lpo with $p=1$, is proved among CV procedures used for risk estimation. Two model selection frameworks are also considered: estimation, as opposed to identification. For estimation with finite sample size $n$, optimality is achieved for $p$ large enough [with $p/n=o(1)$] to balance the overfitting resulting from the structure of the model collection. For identification, model selection consistency is settled for Lpo as long as $p/n$ is conveniently related to the rate of convergence of the best estimator in the collection: (i) $p/n\\to1$ as $n\\to+\\infty$ with a parametric rate, and (ii) $p/n=o(1)$ with some nonparametric estimators. These theoretical results are validated by simulation experiments."}
{"category": "Math", "title": "THH of Thom spectra that are E_\\infty ring spectra", "abstract": "We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E_\\infty classifying map X -> BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative S-algebra (E_\\infty ring spectrum) R can be described as an indexed colimit together with a verification that the Lewis-May operadic Thom spectrum functor preserves indexed colimits. We prove a splitting result THH(Mf) \\htp Mf \\sma BX_+ which yields a convenient description of THH(MU). This splitting holds even when the classifying map f: X -> BG is only a homotopy commutative A_\\infty map, provided that the induced multiplication on Mf extends to an E_\\infty ring structure; this permits us to recover Bokstedt's calculation of THH(HZ)."}
{"category": "Math", "title": "Floer homology on the extended moduli space", "abstract": "Starting from a Heegaard splitting of a three-manifold, we use Lagrangian Floer homology to construct a three-manifold invariant, in the form of a relatively Z/8-graded abelian group. Our motivation is to have a well-defined symplectic side of the Atiyah-Floer Conjecture, for arbitrary three-manifolds. The symplectic manifold used in the construction is the extended moduli space of flat SU(2)-connections on the Heegaard surface. An open subset of this moduli space carries a symplectic form, and each of the two handlebodies in the decomposition gives rise to a Lagrangian inside the open set. In order to define their Floer homology, we compactify the open subset by symplectic cutting; the resulting manifold is only semipositive, but we show that one can still develop a version of Floer homology in this setting."}
{"category": "Math", "title": "Convexity and smoothness of Banach spaces with numerical index one", "abstract": "We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (non-complete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index~1). On the other hand, we show that a lush real Banach space is neither strictly convex nor smooth, unless it is one-dimensional. In particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of any lush infinite-dimensional real space contains a copy of $\\ell_1$."}
{"category": "Math", "title": "Classical metric Diophantine approximation revisited: the Khintchine-Groshev theorem", "abstract": "Under the assumption that the approximating function $\\psi$ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of the set of $\\psi$-approximable matrices in $\\R^{mn}$. The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on $\\psi$ is absolutely necessary when $m=n=1$. On the other hand, it is known that monotonicity is not necessary when $n > 2$ (Sprindzuk) or when $n=1$ and $m>1$ (Gallagher). Surprisingly, when $n=2$ the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multi-dimensional analogue of Catlin's Conjecture."}
{"category": "Math", "title": "Sharp Estimates for the $\\bar{\\partial}$-Neumann Problem on Regular Coordinate Domains", "abstract": "This paper treats subelliptic estimates for the $\\bar{\\partial}$-Neumann problem on a class of domains known as regular coordinate domains. Our main result is that the largest subelliptic gain for a regular coordinate domain is bounded below by a purely algebraic number, the inverse of twice the multiplicity of the ideal associated to a given boundary point."}
{"category": "Math", "title": "The Gelfand-Zeitlin integrable system and its action on generic elements of gl(n) and so(n)", "abstract": "In recent work Bertram Kostant and Nolan Wallach ([KW1], [KW2]) have defined an interesting action of a simply connected Lie group $A$ isomorphic to \\mathbb{C}^{{n\\choose 2}} on gl(n) using a completely integrable system derived from Gelfand-Zeitlin theory. In this paper we show that an analogous action of \\mathbb{C}^{d} exists on the complex orthogonal Lie algebra so(n), where d is half the dimension of a regular adjoint orbit in so(n). In [KW1], Kostant and Wallach describe the orbits of $A$ on a certain Zariski open subset of regular semisimple elements in gl(n). We extend these results to the case of so(n). We also make brief mention of the author's results in [Col1], which describe all $A$-orbits of dimension {n\\choose 2} in gl(n)."}
{"category": "Math", "title": "A note on the $\\hat A$-genus for $\\pi_2$-finite manifolds with $S^1$-symmetry", "abstract": "We construct examples of $S^1$-manifolds with finite second homotopy group and non-vanishing $\\hat A$-genus. This is related to the classification of positive quaternionic Kaehler manifolds."}
{"category": "Math", "title": "Injections of mapping class groups", "abstract": "We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of once-punctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multi-twists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary."}
{"category": "Math", "title": "On the global well-posedness of the one-dimensional Schrodinger map flow", "abstract": "We establish the global well-posedness of the initial value problem for the Schrodinger map flow for maps from the real line into Kahler manifolds and for maps from the circle into Riemann surfaces. This partially resolves a conjecture of W.-Y. Ding."}
{"category": "Math", "title": "Boundary behavior of special cohomology classes arising from the Weil representation", "abstract": "In our previous paper [math.NT/0408050], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p,q)$. This correspondence is realized using theta functions associated to explicitly constructed \"special\" Schwartz forms. Furthermore, the theta functions give rise to generating series of certain \"special cycles\" in $X$ with coefficients. In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel-Sere compactification $\\bar{X}$ of $X$. However, for the $\\Q$-split case for signature $(p,p)$, we have to construct and consider a slightly larger compactification, the \"big\" Borel-Serre compactification. The restriction to each face of $\\bar{X}$ is again a theta series as in [math.NT/0408050], now for a smaller orthogonal group and a larger coefficient system. As application we establish the cohomological nonvanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish."}
{"category": "Math", "title": "Automorphisms of hyperelliptic modular curves $X_0(N)$ in positive characteristic", "abstract": "We study the automorphism groups of the reduction $X_0(N) \\times \\bar{\\mathbb{F}}_p$ of a modular curve $X_0(N)$ over primes $ p\\nmid N$."}
{"category": "Math", "title": "Gram's Law Fails a Positive Proportion of the Time", "abstract": "This paper extends the work done by Titchmarsh on Gram's Law (an attempt to locate the zeroes of the zeta-function on the critical line). Herewith it is shown that a positive proportion of Gram intervals violate Gram's Law; and also that a weaker notion of Gram's Law is valid over a positive proportion of intervals."}
{"category": "Math", "title": "The Bing-Borsuk and the Busemann Conjectures", "abstract": "We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every $n$-dimensional homogeneous ANR is a topological $n$-manifold, whereas the Busemann Conjecture asserts that every $n$-dimensional $G$-space is a topological $n$-manifold. The key object in both cases are so-called {\\it generalized manifolds}, i.e. ENR homology manifolds. We look at the history, from the early beginnings to the present day. We also list several open problems and related conjectures."}
{"category": "Math", "title": "Zeta Functions of Projective Toric Hypersurfaces over Finite Fields", "abstract": "I give a formula for the zeta function of a projective toric hypersurface over a finite field and estimate its Newton polygon. As an application this formula allows us to compute the exact number of rational points on the families of Calabi-Yau manifolds in Mirror Symmetry."}
{"category": "Math", "title": "The non-symmetric operad pre-Lie is free", "abstract": "We prove that the pre-Lie operad is a free non-symmetric operad."}
{"category": "Math", "title": "Higher-dimensional forcing", "abstract": "This is an overview about a method of constructing ccc forcings: Suppose first that a continuous, commutative system of complete embeddings between countable forcings indexed along $\\omega_1$ is given. Then its direct limit satisfies ccc by a well-known theorem on finite support iterations. However, this limit has size at most $\\omega_1$. To get larger forcings, we do not consider linear systems but higher-dimensional systems which are indexed along simplified morasses."}
{"category": "Math", "title": "Derivations in algebras of operator-valued functions", "abstract": "In this paper we study derivations in subalgebras of $L_{0}^{wo}(\\nu ;% \\mathcal{L}(X)) $, the algebra of all weak operator measurable funtions $f:S\\to \\mathcal{L}(X) $, where $% \\mathcal{L}(X) $ is the Banach algebra of all bounded linear operators on a Banach space $X$. It is shown, in particular, that all derivations on $L_{0}^{wo}(\\nu ;\\mathcal{L}(X)) $ are inner whenever $X$ is separable and infinite dimensional. This contrasts strongly with the fact that $L_{0}^{wo}(\\nu ;\\mathcal{L}(X)) $ admits non-trivial non-inner derivations whenever $X$ is finite dimensional and the measure $\\nu $ is non-atomic. As an application of our approach, we study derivations in various algebras of measurable operators affiliated with von Neumann algebras."}
{"category": "Math", "title": "Distribution of the Brownian motion on its way to hitting zero", "abstract": "For the one-dimensional Brownian motion $B=(B_t)_{t\\ge 0}$, started at $x>0$, and the first hitting time $\\tau=\\inf\\{t\\ge 0:B_t=0\\}$, we find the probability density of $B_{u\\tau}$ for a $u\\in(0,1)$, i.e. of the Brownian motion on its way to hitting zero."}
{"category": "Math", "title": "Minimal pseudocompact group topologies on free abelian groups", "abstract": "A Hausdorff topological group G is minimal if every continuous isomorphism f: G --> H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence {\\sigma_n : n\\in N} of cardinals such that w(G) = sup {\\sigma_n : n \\in N} and sup {2^{\\sigma_n} : n \\in N} \\leq |G| \\leq 2^{w(G)}, where w(G) is the weight of G. If G is an infinite minimal abelian group, then either |G| = 2^\\sigma for some cardinal \\sigma, or w(G) = min {\\sigma: |G| \\leq 2^\\sigma}; moreover, the equality |G| = 2^{w(G)} holds whenever cf (w(G)) > \\omega. For a cardinal \\kappa, we denote by F_\\kappa the free abelian group with \\kappa many generators. If F_\\kappa admits a pseudocompact group topology, then \\kappa \\geq c, where c is the cardinality of the continuum. We show that the existence of a minimal pseudocompact group topology on F_c is equivalent to the Lusin's Hypothesis 2^{\\omega_1} = c. For \\kappa > c, we prove that F_\\kappa admits a (zero-dimensional) minimal pseudocompact group topology if and only if F_\\kappa has both a minimal group topology and a pseudocompact group topology. If \\kappa > c, then F_\\kappa admits a connected minimal pseudocompact group topology of weight \\sigma if and only if \\kappa = 2^\\sigma. Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology."}
{"category": "Math", "title": "On iterated almost $\\nu$-stable derived equivalences", "abstract": "In a recent paper \\cite{HuXi3}, we introduced a classes of derived equivalences called almost $\\nu$-stable derived equivalences. The most important property is that an almost $\\nu$-stable derived equivalence always induces a stable equivalence of Morita type, which generalizes a well-known result of Rickard: derived-equivalent self-injective algebras are stably equivalent of Morita type. In this paper, we shall consider the compositions of almost $\\nu$-stable derived equivalences and their quasi-inverses, which is called iterated almost $\\nu$-stable derived equivalences. We give a sufficient and necessary condition for a derived equivalence to be an iterated almost $\\nu$-stable derived equivalence, and give an explicit construction of the stable equivalence functor induced by an iterated almost $\\nu$-stable derived equivalence. As a consequence, we get some new sufficient conditions for a derived finite-dimensional algebras to induce a stable equivalence of Morita type."}
{"category": "Math", "title": "Biharmonic curves on \\textit{LP}-Sasakian manifolds", "abstract": "In this paper we give necessary and sufficient conditions for spacelike and timelike curves in a conformally flat, quasi conformally flat and conformally symmetric 4-dimensional \\textit{LP}-Sasakian manifold to be proper biharmonic. Also, we investigate proper biharmonic curves in the Lorentzian sphere $S^{4}_{1}$."}
{"category": "Math", "title": "Uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities", "abstract": "We consider semilinear elliptic equations with double power nonlineaities. The condition to assure the existence of positive solutions is well-known. In the present paper, we remark that the additional condition to assure uniqueness proposed by Ouyang and Shi is unnecessary."}
{"category": "Math", "title": "Identifying quadric bundle structures on complex projective varieties", "abstract": "In this paper we characterize smooth complex projective varieties that admit a quadric bundle structure on some dense open subset in terms of the geometry of certain families of rational curves."}
{"category": "Math", "title": "Boundary Value Problem for an Oblique Paraxial Model of Light Propagation", "abstract": "We study the Schr\\\"odinger equation which comes from the paraxial approximation of the Helmholtz equation in the case where the direction of propagation is tilted with respect to the boundary of the domain. This model has been proposed in (Doumic, Golse, Sentis, CRAS, 2003). Our primary interest here is in the boundary conditions successively in a half-plane, then in a quadrant of R2. The half-plane problem has been used in (Doumic, Duboc, Golse, Sentis, JCP, to appear) to build a numerical method, which has been introduced in the HERA plateform of CEA."}
{"category": "Math", "title": "Existence and uniqueness conditions of positive solutions to semilinear elliptic equations with double power nonlinearities", "abstract": "In this article we find the equivalent conditions to assure the existence and uniqueness of positive solutions to semilinear elliptic equations wih double power nonlinearities. As a bonus, we give a simpler proof of our former result that the uniqueness condition comes from the existence condition."}
{"category": "Math", "title": "Multiple local whittle estimation in stationary systems", "abstract": "Moving from univariate to bivariate jointly dependent long-memory time series introduces a phase parameter $(\\gamma)$, at the frequency of principal interest, zero; for short-memory series $\\gamma=0$ automatically. The latter case has also been stressed under long memory, along with the ``fractional differencing'' case $\\gamma=(\\delta_2-\\delta_1)\\pi /2$, where $\\delta_1$, $\\delta_2$ are the memory parameters of the two series. We develop time domain conditions under which these are and are not relevant, and relate the consequent properties of cross-autocovariances to ones of the (possibly bilateral) moving average representation which, with martingale difference innovations of arbitrary dimension, is used in asymptotic theory for local Whittle parameter estimates depending on a single smoothing number. Incorporating also a regression parameter $(\\beta)$ which, when nonzero, indicates cointegration, the consistency proof of these implicitly defined estimates is nonstandard due to the $\\beta$ estimate converging faster than the others. We also establish joint asymptotic normality of the estimates, and indicate how this outcome can apply in statistical inference on several questions of interest. Issues of implemention are discussed, along with implications of knowing $\\beta$ and of correct or incorrect specification of $\\gamma$, and possible extensions to higher-dimensional systems and nonstationary series."}
{"category": "Math", "title": "On percolation and the bunkbed conjecture", "abstract": "We study a problem on edge percolation on product graphs $G\\times K_2$. Here $G$ is any finite graph and $K_2$ consists of two vertices $\\{0,1\\}$ connected by an edge. Every edge in $G\\times K_2$ is present with probability $p$ independent of other edges. The Bunkbed conjecture states that for all $G$ and $p$ the probability that $(u,0)$ is in the same component as $(v,0)$ is greater than or equal to the probability that $(u,0)$ is in the same component as $(v,1)$ for every pair of vertices $u,v\\in G$. We generalize this conjecture and formulate and prove similar statements for randomly directed graphs. The methods lead to a proof of the original conjecture for special classes of graphs $G$, in particular outerplanar graphs."}
{"category": "Math", "title": "Classification of double power nonlinear functions", "abstract": "In this article we investigate the nature of the functions, including important double power terms which arise naturally in considering typical nonlinear Schroedinger equations."}
{"category": "Math", "title": "Isomorphism of Hilbert modules over stably finite C*-algebras", "abstract": "It is shown that if A is a stably finite C*-algebra and E is a countably generated Hilbert A-module, then E gives rise to a compact element of the Cuntz semigroup if and only if E is algebraically finitely generated and projective. It follows that if E and F are equivalent in the sense of Coward, Elliott and Ivanescu (CEI) and E is algebraically finitely generated and projective, then E and F are isomorphic. In contrast to this, we exhibit two CEI-equivalent Hilbert modules over a stably finite C*-algebra that are not isomorphic."}
{"category": "Math", "title": "Liouville Theorem for Dunkl Polyharmonic Functions", "abstract": "Assume that $f$ is Dunkl polyharmonic in $\\mathbb{R}^n$ (i.e. $(\\Delta_h)^p f=0$ for some integer $p$, where $\\Delta_h$ is the Dunkl Laplacian associated to a root system $R$ and to a multiplicity function $\\kappa$, defined on $R$ and invariant with respect to the finite Coxeter group). Necessary and successful condition that $f$ is a polynomial of degree $\\le s$ for $s\\ge 2p-2$ is proved. As a direct corollary, a Dunkl harmonic function bounded above or below is constant."}
{"category": "Math", "title": "Regular Moebius transformations of the space of quaternions", "abstract": "Let H be the real algebra of quaternions. The notion of regular function of a quaternionic variable recently presented by G. Gentili and D. C. Struppa developed into a quite rich theory. Several properties of regular quaternionic functions are analogous to those of holomorphic functions of one complex variable, although the diversity of the quaternionic setting introduces new phenomena. This paper studies regular quaternionic transformations. We first find a quaternionic analog to the Casorati-Weierstrass theorem and prove that all regular injective functions from H to itself are affine. In particular, the group Aut(H) of biregular functions on H coincides with the group of regular affine transformations. Inspired by the classical quaternionic linear fractional transformations, we define the regular fractional transformations. We then show that each regular injective function from the Alexandroff compactification of H to itself is a regular fractional transformation. Finally, we study regular Moebius transformations, which map the unit ball B onto itself. All regular bijections from B to itself prove to be regular Moebius transformations."}
{"category": "Math", "title": "A Refinement of the Function $g(m)$ on Grimm Conjecture", "abstract": "In this paper, we refine the function $g(x)$ on Grimm's conjecture and improve a result of Erd\\\"{o}s and Selfridge without using Hall's theorem."}
{"category": "Math", "title": "A note on compact K\\\"ahler-Ricci flow with positive bisectional curvature", "abstract": "We show that for any solution to the K\\\"ahler-Ricci flow with positive bisectional curvature on a compact K\\\"ahler manifold $M^n$, the bisectional curvature has a uniform positive lower bound. As a consequence, the solution converges exponentially fast to an K\\\"ahler-Einstein metric with positive bisectional curvature as t tends to the infinity, provided we assume the Futaki-invariant of $M^n$ is zero. This improves a result of D. Phong, J. Song, J. Sturm and B. Weinkove in which they assumed the stronger condition that Mabuchi K-energy is bounded from below."}
{"category": "Math", "title": "A stochastic epidemiological model and a deterministic limit for BitTorrent-like peer-to-peer file-sharing networks", "abstract": "In this paper, we propose a stochastic model for a file-sharing peer-to-peer network which resembles the popular BitTorrent system: large files are split into chunks and a peer can download or swap from another peer only one chunk at a time. We prove that the fluid limits of a scaled Markov model of this system are of the coagulation form, special cases of which are well-known epidemiological (SIR) models. In addition, Lyapunov stability and settling-time results are explored. We derive conditions under which the BitTorrent incentives under consideration result in shorter mean file-acquisition times for peers compared to client-server (single chunk) systems. Finally, a diffusion approximation is given and some open questions are discussed."}
{"category": "Math", "title": "(n-1)-st Koszul homology and the structure of monomial ideals", "abstract": "Koszul homology of monomial ideals provides a description of the structure of such ideals, not only from a homological point of view (free resolutions, Betti numbers, Hilbert series) but also from an algebraic viewpoint. In this paper we show that, in particular, the homology at degree (n-1), with n the number of indeterminates of the ring, plays an important role for this algebraic description in terms of Stanley and irreducible decompositions."}
{"category": "Math", "title": "Discontinuity growth of interval exchange maps", "abstract": "For an interval exchange map, the number of discontinuities of its iterates either exhibits linear growth or is bounded. This dichotomy is used to prove that the group of interval exchanges does not contain distortion elements, giving examples of groups that do not act faithfully via interval exchanges. As a further application of this dichotomy, a classification of centralizers in the group of interval exchanges is given. This classification of centralizers is used to compute the automorphism group of the interval exchange group."}
{"category": "Math", "title": "Partition relations for Hurewicz-type selection hypotheses", "abstract": "We give a general method to reduce Hurewicz-type selection hypotheses into standard ones. The method covers the known results of this kind and gives some new ones. Building on that, we show how to derive Ramsey theoretic characterizations for these selection hypotheses."}
{"category": "Math", "title": "Monomial ideals, almost complete intersections and the Weak Lefschetz Property", "abstract": "Many algebras are expected to have the Weak Lefschetz property though this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field, and on arithmetic properties of the exponent vectors of the monomials."}
{"category": "Math", "title": "Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups", "abstract": "We study algebraic and topological properties of the convolution semigroups of probability measures on a topological groups and show that a compact Clifford topological semigroup $S$ embeds into the convolution semigroup $P(G)$ over some topological group $G$ if and only if $S$ embeds into the semigroup $\\exp(G)$ of compact subsets of $G$ if and only if $S$ is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup $S$ embeds into the functor-semigroup $F(G)$ over a suitable compact topological group $G$ for each weakly normal monadic functor $F$ in the category of compacta such that $F(G)$ contains a $G$-invariant element (which is an analogue of the Haar measure on $G$)."}
{"category": "Math", "title": "Topology of positively curved 8-dimensional manifolds with symmetry", "abstract": "In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank $\\geq 2$ resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S^8, H P^2 or C P^4. And if M is rationally elliptic then M is rationally isomorphic to a rank one symmetric space. For torsion-free manifolds we derive a much stronger classification. We also study the bordism type of 8-dimensional manifolds of positive sectional curvature and symmetry rank $\\geq 2$. As an illustration we apply our results to various families of 8-manifolds."}
{"category": "Math", "title": "A Free Boundary Isoperimetric Problem in the Hyperbolic Space between parallel horospheres", "abstract": "In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in the horospheres is not included). We reduce the problem to the study of rotationally invariant regions and obtain the possible isoperimetric solutions by studying the behaviour of the profile curves of the rotational surfaces with constant mean curvature in the hyperbolic space."}
{"category": "Math", "title": "Introduction to the Minimal Model Program and the existence of flips", "abstract": "The first aim of this note is to give a concise, but complete and self-contained, presentation of the fundamental theorems of Mori theory - the nonvanishing, base point free, rationality and cone theorems - using modern methods of multiplier ideals, Nadel vanishing, and the subadjunction theorem of Kawamata. The second aim is to write up a complete, detailed proof of existence of flips in dimension n assuming the minimal model program with scaling in dimension n-1."}
{"category": "Math", "title": "Improved lower bound on an Euclidean Ramsey problem", "abstract": "It was previously shown that any two-colour colouring of K(C_n) must contain a monochromatic planar K_4 subgraph for n >= N^*, where 6 <= N^* <= N and N is Graham's number. The bound was later improved to 11 <= N^* <= N. In this article, it is improved to 13 <= N^* <= N."}
{"category": "Math", "title": "Subnormal subalgebras of Leibniz algebras", "abstract": "Zassenhaus has proved that if U is a subnormal subalgebra of a finite-dimensional Lie algebra L and V is a finite-dimensional irreducible L-module, then all U-module composition factors of V are isomorphic. Schenkman has proved that if U is a subnormal subalgebra of a finite-dimensional Lie algebra L, then the nilpotent residual of U is an ideal of L. These useful results generalise to Leibniz algebras."}
{"category": "Math", "title": "Divisors on Rational Normal Scrolls", "abstract": "Let $A$ be the homogeneous coordinate ring of a rational normal scroll. The ring $A$ is equal to the quotient of a polynomial ring $S$ by the ideal generated by the two by two minors of a scroll matrix $\\psi$ with two rows and $\\ell$ catalecticant blocks. The class group of $A$ is cyclic, and is infinite provided $\\ell$ is at least two. One generator of the class group is $[J]$, where $J$ is the ideal of $A$ generated by the entries of the first column of $\\psi$. The positive powers of $J$ are well-understood, in the sense that the $n^{\\text{th}}$ ordinary power, the $n^{th}$ symmetric power, and the $n^{th}$ symbolic power all coincide and therefore all three $n^{th}$ powers are resolved by a generalized Eagon-Northcott complex. The inverse of $[J]$ in the class group of $A$ is $[K]$, where $K$ is the ideal generated by the entries of the first row of $\\psi$. We study the positive powers of $[K]$. We obtain a minimal generating set and a Groebner basis for the preimage in $S$ of the symbolic power $K^{(n)}$. We describe a filtration of $K^{(n)}$ in which all of the factors are Cohen-Macaulay $S$-modules resolved by generalized Eagon-Northcott complexes. We use this filtration to describe the modules in a finely graded resolution of $K^{(n)}$ by free $S$-modules. We calculate the regularity of the graded $S$-module $K^{(n)}$ and we show that the symbolic Rees ring of $K$ is Noetherian."}
{"category": "Math", "title": "Cesaro summation and multiplicative functions on a symmetric group", "abstract": "We investigate the summability in sense of Cesaro and its applications to investigation of the mean values of multiplicative functions on permutations."}
{"category": "Math", "title": "The planar algebra of diagonal subfactors", "abstract": "There is a natural construction which associates to a finitely generated, countable, discrete group $G$ and a 3-cocycle $\\omega$ of $G$ an inclusion of II$_1$ factors, the so-called diagonal subfactors (with cocycle). In the case when the cocycle is trivial these subfactors are well studied and their standard invariant (or planar algebra) is known. We give a description of the planar algebra of these subfactors when a cocycle is present. The action of Jones' planar operad involves the 3-cocycle $\\omega$ explicitly and some interesting identities for 3-cocycles appear when naturality of the action is verified."}
{"category": "Math", "title": "Invariants, cohomology, and automorphic forms of higher order", "abstract": "A general structure theorem on higher order invariants is proven. For an arithmetic group, the structure of the corresponding Hecke module is determined. It is shown that the module does not contain any irreducible submodule. This explains the fact that L-functions of higher order forms have no Euler-product. Higher order cohomology is introduced, classical results of Borel are generalized and a higher order version of Borel's conjecture is stated."}
{"category": "Math", "title": "On the Closing Lemma problem for vector fields of bounded type on the torus", "abstract": "We investigate the open Closing Lemma problem for vector fields on the 2-dimensional torus. Under the assumption of bounded type rotation number, the $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points. Namely, given such a $C^r$ vector field $X$, $r\\geq 4$, with a non-trivially recurrent point $p$, there exists a vector field $Y$ arbitrarily near to $X$ in the $C^r$ topology and obtained from $X$ by a twist perturbation, such that $p$ is a periodic point of $Y$. The proof relies on a new result in 1-dimensional dynamics on the non-existence of semi-wandering intervals of smooth maps of the circle."}
{"category": "Math", "title": "Functions holomorphic along holomorphic vector fields", "abstract": "The main result of the paper is the following generalization of Forelli's theorem: Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with the eigenvalues whose ratios are positive reals. Then any function $\\phi$ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary."}
{"category": "Math", "title": "Does a billiard orbit determine its (polygonal) table?", "abstract": "We introduce a new equivalence relation on the set of all polygonal billiards. We say that two billiards (or polygons) are order equivalent if each of the billiards has an orbit whose footpoints are dense in the boundary and the two sequences of footpoints of these orbits have the same combinatorial order. We study this equivalence relation with additional regularity conditions on the orbit."}
{"category": "Math", "title": "Spectrum of large random reversible Markov chains: two examples", "abstract": "We take on a Random Matrix theory viewpoint to study the spectrum of certain reversible Markov chains in random environment. As the number of states tends to infinity, we consider the global behavior of the spectrum, and the local behavior at the edge, including the so called spectral gap. Results are obtained for two simple models with distinct limiting features. The first model is built on the complete graph while the second is a birth-and-death dynamics. Both models give rise to random matrices with non independent entries."}
{"category": "Math", "title": "Isotropic Ornstein-Uhlenbeck flows", "abstract": "Isotropic Brownian flows (IBFs) are a fairly natural class of stochastic flows which has been studied extensively by various authors. Their rich structure allows for explicit calculations in several situations and makes them a natural object to start with if one wants to study more general stochastic flows. Often the intuition gained by understanding the problem in the context of IBFs transfers to more general situations. However, the obvious link between stochastic flows, random dynamical systems and ergodic theory cannot be exploited in its full strength as the IBF does not have an invariant probability measure but rather an infinite one. Isotropic Ornstein-Uhlenbeck flows are in a sense localized IBFs and do have an invariant probability measure. The imposed linear drift destroys the translation invariance of the IBF, but many other important structure properties like the Markov property of the distance process remain valid and allow for explicit calculations in certain situations. The fact that isotropic Ornstein-Uhlenbeck flows have invariant probability measures allows one to apply techniques from random dynamical systems theory. We demonstrate this by applying the results of Ledrappier and Young to calculate the Hausdorff dimension of the statistical equilibrium of an isotropic Ornstein-Uhlenbeck flow."}
{"category": "Math", "title": "Selection of variables and dimension reduction in high-dimensional non-parametric regression", "abstract": "We consider a $l_1$-penalization procedure in the non-parametric Gaussian regression model. In many concrete examples, the dimension $d$ of the input variable $X$ is very large (sometimes depending on the number of observations). Estimation of a $\\beta$-regular regression function $f$ cannot be faster than the slow rate $n^{-2\\beta/(2\\beta+d)}$. Hopefully, in some situations, $f$ depends only on a few numbers of the coordinates of $X$. In this paper, we construct two procedures. The first one selects, with high probability, these coordinates. Then, using this subset selection method, we run a local polynomial estimator (on the set of interesting coordinates) to estimate the regression function at the rate $n^{-2\\beta/(2\\beta+d^*)}$, where $d^*$, the \"real\" dimension of the problem (exact number of variables whom $f$ depends on), has replaced the dimension $d$ of the design. To achieve this result, we used a $l_1$ penalization method in this non-parametric setup."}
{"category": "Math", "title": "On the limit of large girth graph sequences", "abstract": "We prove that any involution-invariant probability measure on the space of trees with maximum degrees at most d arises as the local limit of a convergent large girth graph sequence. This answers a question of Bollobas and Riordan."}
{"category": "Math", "title": "Long-time existence for semi-linear Klein-Gordon equations with quadratic potential", "abstract": "We prove that small smooth solutions of semi-linear Klein-Gordon equations with quadratic potential exist over a longer interval than the one given by local existence theory, for almost every value of mass. We use normal form for the Sobolev energy. The difficulty in comparison with some similar results on the sphere comes from the fact that two successive eigenvalues $\\lambda, \\lambda'$ of $\\sqrt{-\\Delta+|x|^2}$ may be separated by a distance as small as $\\frac{1}{\\sqrt{\\lambda}}$."}
{"category": "Math", "title": "Stein estimation for the drift of Gaussian processes using the Malliavin calculus", "abstract": "We consider the nonparametric functional estimation of the drift of a Gaussian process via minimax and Bayes estimators. In this context, we construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and superharmonic functionals on Gaussian space. Our results are illustrated by numerical simulations and extend the construction of James--Stein type estimators for Gaussian processes by Berger and Wolpert [J. Multivariate Anal. 13 (1983) 401--424]."}
{"category": "Math", "title": "The skew-torsion holonomy theorem and naturally reductive spaces", "abstract": "We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or the list of rank one symmetric spaces). This result was independently proved, by using an algebraic approach, by Paul-Andy Nagy. We apply this theorem to prove that the canonical connection of a compact naturally reductive space is unique, provided the space does not split off, locally, a sphere or a compact Lie group with a bi-invariant metric. From this it follows easily how to obtain the full isometry group of a naturally reductive space. This generalizes known classification results of Onishchick, for normal homogeneous spaces with simple group of isometries, and Shankar, for homogeneous spaces of positive curvature. This also answers a question posed by J. Wolf and Wang-Ziller. Namely, to explain why the presentation group of an isotropy irreducible space, strongly or not, cannot be enlarged (unless for spheres, or for compact simple Lie groups with a bi-invariant metric)."}
{"category": "Math", "title": "On $D$-spaces and Discrete Families of Sets", "abstract": "We prove several reflection theorems on $D$-spaces, which are Hausdorff topological spaces $X$ in which for every open neighbourhood assignment $U$ there is a closed discrete subspace $D$ such that \\[ \\bigcup\\{U(x): x\\in D\\}=X. \\] The upwards reflection theorems are obtained in the presence of a forcing axiom, while most of the downwards reflection results use large cardinal assumptions. The combinatorial content of arguments showing that a given space is a $D$-space, can be formulated using the concept of discrete families. We note the connection between non-reflection arguments involving discrete families and the well known question of the existence of families allowing partial transversals without having a transversal themselves, and use it to give non-trivial instances of the incompactness phenomenon in the context of discretisations."}
{"category": "Math", "title": "A Simpson correspondence in positive characteristic", "abstract": "We define the $p^m$-curvature map on the sheaf of differential operators of level $m$ on a scheme of positive characteristic $p$ as dual to some divided power map on infinitesimal neighborhhods. This leads to the notion of $p^m$-curvature on differential modules of level $m$. We use this construction to recover Kaneda's description of a semi-linear Azumaya splitting of the sheaf of differential operators of level $m$. Then, using a lifting modulo $p^2$ of Frobenius, we are able to define a Frobenius map on differential operators of level $m$ as dual to some divided Frobenius on infinitesimal neighborhhods. We use this map to build a true Azumaya splitting of the completed sheaf of differential operators of level $m$ (up to an automorphism of the center). From this, we derive the fact that Frobenius pull back gives, when restricted to quasi-nilpotent objects, an equivalence between Higgs-modules and differential modules of level $m$. We end by explaining the relation with related work of Ogus-Vologodski and van der Put in level zero as well as Berthelot's Frobenius descent."}
{"category": "Math", "title": "Rate of convergence to self-similarity for Smoluchowski's coagulation equation with constant coefficients", "abstract": "We show that solutions to Smoluchowski's equation with a constant coagulation kernel and an initial datum with some regularity and exponentially decaying tail converge exponentially fast to a self-similar profile. This convergence holds in a weighted Sobolev norm which implies the L\\^2 convergence of derivatives up to a certain order k depending on the regularity of the initial condition. We prove these results through the study of the linearized coagulation equation in self-similar variables, for which we show a spectral gap in a scale of weighted Sobolev spaces. We also take advantage of the fact that the Laplace or Fourier transforms of this equation can be explicitly solved in this case."}
{"category": "Math", "title": "Phi-modules and coefficient spaces", "abstract": "We define and study certain moduli stacks of modules equipped with a Frobenius semi-linear endomorphism. These stacks can be thought of as parametrizing the coefficients of a variable Galois representation and are global variants of the spaces of Kisin-Breuil $\\Phi$-modules used by Kisin in his study of deformation spaces of local Galois representations. We also define a version of a rigid analytic period map for these spaces, we show how their local structure can be described in terms of \"local models\", and we show how Bruhat-Tits buildings can be used to study their special fibers."}
{"category": "Math", "title": "An algorithm to obtain global solutions of the double confluent Heun equation", "abstract": "A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic expansions are used in the computation of those Wronskians. The feasibility of the method is shown in an example, namely, the Schroedinger equation with a quasi-exactly-solvable potential."}
{"category": "Math", "title": "On the centralizer of diffeomorphisms of the half-line", "abstract": "Let f be a smooth diffeomorphism of the half-line fixing only the origin and Z^r its centralizer in the group of C^r diffeomorphisms. According to well-known results of Szekeres and Kopell, Z^1 is a one-parameter group. On the other hand, Sergeraert constructed an f whose centralizer Z^r, $2\\le r\\le \\infty$, reduces to the group generated by f. We show that Z^r can actually be a proper dense and uncountable subgroup of Z^1 and that this phenomenon is not scarce."}
{"category": "Math", "title": "Discrete extrinsic curvatures based on polar polyhedra concept", "abstract": "Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate \"Measurement of circle\". The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygons (polyhedra) which approximate curvilinear convex body. This sequence allows to approximate length of curve, as well as area and volume of the bodies and to obtain error estimates for approximation. In this work it is shown that a sequence of pairs of locally polar polyhedra allows to construct piecewise-affine approximation to scherical Gauss map, to construct convergent point-wise approximations to mean and Gauss curvature, as well as to obtain natural discretizations of bending energies."}
{"category": "Math", "title": "Asymptotic stability of the wave equation on compact surfaces and locally distributed damping - A sharp result", "abstract": "This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping. We study the case where the damping is effective in a well-chosen subset of arbitrarily small measure."}
{"category": "Math", "title": "Conditional stability of unstable viscous shocks", "abstract": "Continuing a line of investigation initiated by Texier and Zumbrun on dynamics of viscous shock and detonation waves, we show that a linearly unstable Lax-type viscous shock solution of a semilinear strictly parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small $L^1\\cap H^2$ perturbatoins, converging time-asymptotically to a translate of the unperturbed wave. That is, for a shock with $p$ unstable eigenvalues, we establish conditional stability on a codimension-$p$ manifold of initial data, with sharp rates of decay in all $L^p$. For $p=0$, we recover the result of unconditional stability obtained by Howard, Mascia, and Zumbrun."}
{"category": "Math", "title": "The Poisson equation on complete manifolds with positive spectrum and applications", "abstract": "In this paper we investigate the existence of a solution to the Poisson equation on complete manifolds with positive spectrum and Ricci curvature bounded from below. We show that if a function $f$ has decay $f=O(r^{-1-\\varepsilon}) $ for some $\\varepsilon >0,$ where $r$ is the distance function to a fixed point, then the Poisson equation $\\Delta u=f$ has a solution $u$ with at most exponential growth. We apply this result on the Poisson equation to study the existence of harmonic maps between complete manifolds and also existence of Hermitian Einstein metrics on holomorphic vector bundles over complete manifolds, thus extending some results of Li-Tam and Ni. Assuming that the manifold is simply connected and of Ricci curvature between two negative constants, we can prove that in fact the Poisson equation has a bounded solution and we apply this result to the Ricci flow on complete surfaces."}
{"category": "Math", "title": "Uniform Stabilization of the wave equation on compact surfaces and locally distributed damping", "abstract": "This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping. We study the case where the damping is effective on the complement of visible umbilical sets."}
{"category": "Math", "title": "Techniques for the Analytic Proof of the Finite Generation of the Canonical Ring", "abstract": "This article is written for the Proceedings of the Conference on Current Developments in Mathematics in Harvard University, November 16-17, 2007. It is an exposition of the analytic proof of the finite generation of the canonical ring for a compact complex algebraic manifold of general type. It lists and discusses the main techniques and explains how they are put together in the proof. Of the various main techniques some special attention is given to (i) the technique of discrepancy subspaces and (ii) the technique of subspaces of minimum additional vanishing."}
{"category": "Math", "title": "Properties of sums of some elementary functions and modeling of transitional and other processes", "abstract": "The article presents mathematical generalization of results which originated as solutions of practical problems, in particular, the modeling of transitional processes in electrical circuits and problems of resource allocation. However, the presented findings have broader meaning and can be used for approximation of transitional and other processes in different areas of science and technology. We present discovered properties of sums of polynomial, power, and exponential functions of one variable. It is shown that for an equation composed of one type of function there is a corresponding equation composed of functions of the other type. The number of real solutions of such equations and the number of characteristic points of certain appropriate corresponding functions are closely related. In particular, we introduce a method similar to Descartes Rule of Signs that allows finding the maximum number of real solutions for the power equation and equation composed of sums of exponential functions. The discovered properties of these functions allow us to improve the adequacy of mathematical models of real phenomena."}
{"category": "Math", "title": "A Generalization of the Circumcenter of a Set", "abstract": "Let (X, d) be a Cat(k) space and P a bounded subset of X . If k > 0 then it is required that the diameter of P be less than Pi/(4 sqrt(k)) . Let u: P to R be a bounded non-negative function from P to R. The existence of a unique point in X called the barycenter of P relative to u is established. When u=1, the barycenter is simply the circumcenter of P. The barycenter has a number of properties including a scaling, continuity and limit property. Under suitable conditions, the barycenter is a fixed point of an isometry or group of isometries. Barycenters are used to show that a complete Cat(k) space X is an absolute retract if k is less than or equal to 0, and an absolute neighborhood retract if X is complete and of curvature less than or equal to k."}
{"category": "Math", "title": "T-Duality and Homological Mirror Symmetry of Toric Varieties", "abstract": "Let $X_\\Sigma$ be a complete toric variety. The coherent-constructible correspondence $\\kappa$ of \\cite{FLTZ} equates $\\Perf_T(X_\\Sigma)$ with a subcategory $Sh_{cc}(M_\\bR;\\LS)$ of constructible sheaves on a vector space $M_\\bR.$ The microlocalization equivalence $\\mu$ of \\cite{NZ,N} relates these sheaves to a subcategory $Fuk(T^*M_\\bR;\\LS)$ of the Fukaya category of the cotangent $T^*M_\\bR$. When $X_\\Si$ is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, $DCoh_T(X_\\Si)\\cong DFuk(T^*M_\\bR;\\LS)$, which is an equivalence of triangulated tensor categories. The nonequivariant coherent-constructible correspondence $\\bar{\\kappa}$ of \\cite{T} embeds $\\Perf(X_\\Si)$ into a subcategory $Sh_c(T_\\bR^\\vee;\\bar{\\Lambda}_\\Si)$ of constructible sheaves on a compact torus $T_\\bR^\\vee$. When $X_\\Si$ is nonsingular, the composition of $\\bar{\\kappa}$ and microlocalization yields a version of homological mirror symmetry, $DCoh(X_\\Sigma)\\hookrightarrow DFuk(T^*T_\\bR;\\bar{\\Lambda}_\\Si)$, which is a full embedding of triangulated tensor categories. When $X_\\Si$ is nonsingular and projective, the composition $\\tau=\\mu\\circ \\kappa$ is compatible with T-duality, in the following sense. An equivariant ample line bundle $\\cL$ has a hermitian metric invariant under the real torus, whose connection defines a family of flat line bundles over the real torus orbits. This data produces a T-dual Lagrangian brane $\\mathbb L$ on the universal cover $T^*M_\\bR$ of the dual real torus fibration. We prove $\\mathbb L\\cong \\tau(\\cL)$ in $Fuk(T^*M_\\bR;\\LS).$ Thus, equivariant homological mirror symmetry is determined by T-duality."}
{"category": "Math", "title": "The number of constant mean curvature isometric immersions of a surface", "abstract": "In classical surface theory there are but few known examples of surfaces admitting nontrivial isometric deformations and fewer still non-simply-connected ones. We consider the isometric deformability question for an immersion x: M \\to R^3 of an oriented non-simply-connected surface with constant mean curvature H. We prove that the space of all isometric immersions of M with constant mean curvature H is, modulo congruences of R^3, either finite or a circle. When it is a circle then, for the immersion x, every cycle in M has vanishing force and, when H is not 0, also vanishing torque. Our work generalizes a rigidity result for minimal surfaces to constant mean curvature surfaces. Moreover, we identify closed vector-valued 1-forms whose periods give the force and torque."}
{"category": "Math", "title": "The Duffin-Schaeffer Conjecture with extra divergence", "abstract": "Given a nonnegative function $\\psi : \\N \\to \\R $, let $W(\\psi)$ denote the set of real numbers $x$ such that $|nx -a| < \\psi(n) $ for infinitely many reduced rationals $a/n (n>0) $. A consequence of our main result is that $W(\\psi)$ is of full Lebesgue measure if there exists an $\\epsilon > 0 $ such that $$ \\textstyle \\sum_{n\\in\\N}(\\frac{\\psi(n)}{n})^{1+\\epsilon}\\varphi (n)=\\infty . $$ The Duffin-Schaeffer Conjecture is the corresponding statement with $\\epsilon = 0$ and represents a fundamental unsolved problem in metric number theory. Another consequence is that $W(\\psi)$ is of full Hausdorff dimension if the above sum with $\\epsilon = 0$ diverges; i.e. the dimension analogue of the Duffin-Schaeffer Conjecture is true."}
{"category": "Math", "title": "Integration of H\\\"older forms and currents in snowflake spaces", "abstract": "For an oriented $n$-dimensional Lipschitz manifold $M$ we give meaning to the integral $\\int_M f dg_1 \\wedge ... \\wedge dg_n$ in case the functions $f, g_1, >..., g_n$ are merely H\\\"older continuous of a certain order by extending the construction of the Riemann-Stieltjes integral to higher dimensions. More generally, we show that for $\\alpha \\in (\\frac{n}{n+1},1]$ the $n$-dimensional locally normal currents in a locally compact metric space $(X,d)$ represent a subspace of the $n$-dimensional currents in $(X,d^\\alpha)$. On the other hand, for $n \\geq 1$ and $\\alpha \\leq \\frac{n}{n+1}$ the latter space consists of the zero functional only."}
{"category": "Math", "title": "Improved Estimation of High-dimensional Ising Models", "abstract": "We consider the problem of jointly estimating the parameters as well as the structure of binary valued Markov Random Fields, in contrast to earlier work that focus on one of the two problems. We formulate the problem as a maximization of $\\ell_1$-regularized surrogate likelihood that allows us to find a sparse solution. Our optimization technique efficiently incorporates the cutting-plane algorithm in order to obtain a tighter outer bound on the marginal polytope, which results in improvement of both parameter estimates and approximation to marginals. On synthetic data, we compare our algorithm on the two estimation tasks to the other existing methods. We analyze the method in the high-dimensional setting, where the number of dimensions $p$ is allowed to grow with the number of observations $n$. The rate of convergence of the estimate is demonstrated to depend explicitly on the sparsity of the underlying graph."}
{"category": "Math", "title": "Metrics of positive scalar curvature and generalised Morse functions, part 1", "abstract": "It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. We show that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse holds in the case of closed simply connected manifolds of dimension at least five."}
{"category": "Math", "title": "A Cauchy-Kowalevsky theorem for overdetermined systems of nonlinear partial differential equations and geometric applications", "abstract": "We consider a Cauchy problem for an overdetermined system of PDEs, and give necessary and sufficient conditions for solvability of this Cauchy problem for all data. As an application, we find all real tube hypersurfaces in complex space whose Levi number is maximal."}
{"category": "Math", "title": "Three spheres inequalities and unique continuation for a three-dimensional Lam\\'e system of elasticity with C^1 coefficients", "abstract": "Assuming that the Lam\\'{e} moduli $\\mu$, $\\lambda$ are $C^{\\tiny{1}}$ and $n\\geq2$, we prove quantitative estimates of a weak sense of strong unique continuation for thesolutions of the n-dimensional Lam\\'{e} system of the form of three spheres inequalities."}
{"category": "Math", "title": "Group-type subfactors and Hadamard matrices", "abstract": "A hyperfinite $II_1$ subfactor may be obtained from a symmetric commuting square via iteration of the basic construction. For certain commuting squares constructed from Hadamard matrices, we describe this subfactor as a group-type inclusion $R^H \\subset R \\rtimes K$, where $H$ and $K$ are finite groups with outer actions on the hyperfinite $II_1$ factor $R$. We find the group of outer automorphisms generated by $H$ and $K$, and use the method of Bisch and Haagerup to determine the principal and dual principal graphs. In some cases a complete classification is obtained by examining the element of $H^3(H \\ast K / Int R)$ associated with the action."}
{"category": "Math", "title": "A Combinatorial Property of Ideals in Free Profinite Monoids", "abstract": "We prove every regular element of a free profinite monoid generates a prime ideal; in particular the minimal ideal is prime. The latter result was first proved by Almeida and Volkov using techniques from symbolic dynamics; our proof is elementary."}
{"category": "Math", "title": "A self-regulating and patch subdivided population", "abstract": "We consider an interacting particle process on a graph which, from a macroscopic point of view, looks like $\\Z^d$ and, at a microscopic level, is a complete graph of degree $N$ (called a patch). There are two birth rates: an inter-patch one $\\lambda$ and an intra-patch one $\\phi$. Once a site is occupied, there is no breeding from outside the patch and the probability $c(i)$ of success of an intra-patch breeding decreases with the size $i$ of the population in the site. We prove the existence of a critical value $\\lambda_{cr}(\\phi, c, N)$ and a critical value $\\phi_{cr}(\\lambda, c, N)$. We consider a sequence of processes generated by the families of control functions $\\{c_i\\}_{i \\in \\N}$ and degrees $\\{N_i\\}_{i \\in \\N}$; we prove, under mild assumptions, the existence of a critical value $i_{cr}$. Roughly speaking we show that, in the limit, these processes behave as the branching random walk on $\\Z^d$ with external birth rate $\\lambda$ and internal birth rate $\\phi$. Some examples of models that can be seen as particular cases are given."}
{"category": "Math", "title": "Cluster fans, stability conditions, and domains of semi-invariants", "abstract": "We show that the cone of finite stability conditions of a quiver Q without oriented cycles has a fan covering given by (the dual of) the cluster fan of Q. Along the way, we give new proofs of Schofield's results on perpendicular categories. We also study domains of semi-invariants of quivers via quiver exceptional sequences. In particular, we recover Igusa-Orr-Todorov-Weyman's theorem on cluster complexes and domains of semi-invariants for Dynkin quivers."}
{"category": "Math", "title": "Optimal sequential multiple hypothesis tests", "abstract": "This work deals with a general problem of testing multiple hypotheses about the distribution of a discrete-time stochastic process. Both the Bayesian and the conditional settings are considered. The structure of optimal sequential tests is characterized."}
{"category": "Math", "title": "Subspaces of 7 x 7 skew-symmetric matrices related to the group G_2", "abstract": "Let $K$ be a field of characteristic different from 2 and let $C$ be an octonion algebra over $K$. We show that there is a seven-dimensional subspace of $7\\times 7$ skew-symmetric matrices over $K$ which is invariant under the automorphism group of $C$. This subspace consists of elements of rank 6 when $C$ is a division algebra, and elements of rank 4 and 6 when $C$ is a split algebra. In the latter case, the automorphism group is the exceptional group $G_2(K)$."}
{"category": "Math", "title": "On Quadratic Fields Generated by Discriminants of Irreducible Trinomials", "abstract": "A. Mukhopadhyay, M. R. Murty and K. Srinivas (http://arxiv.org/abs/0808.0418) have recently studied various arithmetic properties of the discriminant $\\Delta_n(a,b)$ of the trinomial $f_{n,a,b}(t) = t^n + at + b$, where $n \\ge 5$ is a fixed integer. In particular, it is shown that, under the $abc$-conjecture, for every $n \\equiv 1 \\pmod 4$, the quadratic fields $\\Q(\\sqrt{\\Delta_n(a,b)})$ are pairwise distinct for a positive proportion of such discriminants with integers $a$ and $b$ such that $f_{n,a,b}$ is irreducible over $\\Q$ and $|\\Delta_n(a,b)|\\le X$, as $X\\to \\infty$. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result."}
{"category": "Math", "title": "Estimates from below of the Buffon noodle probability for undercooked noodles", "abstract": "Let $\\Cant_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $\\K_n$ of $\\Cant_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $\\K_n$ is essentially the average length of the projections of $\\K_n$, also known as the Favard length of $\\K_n$. A result due to Bateman and Volberg \\cite{BV} shows that a lower estimate for this Favard length is $c \\frac{\\log n}{n}$. We may bend the needle at each stage, giving us what we will call a noodle, and ask whether the uniform lower estimate $c \\frac{\\log n}{n}$ still holds for these so-called Buffon noodle probabilities. If so, we call the sequence of noodles undercooked. We will define a few classes of noodles and prove that they are undercooked. In particular, we are interested in the case when the noodles are circular arcs of radius $r_n$. We will show that if $r_n \\geq 4^{\\frac{n}{5}}$, then the circular arcs are undercooked noodles."}
{"category": "Math", "title": "Vertex operators and sporadic groups", "abstract": "In the 1980's, the work of Frenkel, Lepowsky and Meurman, along with that of Borcherds, culminated in the notion of vertex operator algebra, and an example whose full symmetry group is the largest sporadic simple group: the Monster. Thus it was shown that the vertex operators of mathematical physics play a role in finite group theory. In this article we describe an extension of this phenomenon by introducing the notion of enhanced vertex operator algebra, and constructing examples that realize other sporadic simple groups, including one that is not involved in the Monster."}
{"category": "Math", "title": "Classification theorems for sumsets modulo a prime", "abstract": "Let $\\Z/pZ$ be the finite field of prime order $p$ and $A$ be a subsequence of $\\Z/pZ$. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of $A$ ? (2) When can one represent every element of $\\Z/pZ$ as a sum of some elements of $A$ ? (3) When can one represent every element of $\\Z/pZ$ as a sum of $l$ elements of $A$ ?"}
{"category": "Math", "title": "Squares in sumsets", "abstract": "A finite set $A$ of integers is square-sum-free if there is no subset of $A$ sums up to a square. In 1986, Erd\\H os posed the problem of determining the largest cardinality of a square-sum-free subset of $\\{1, ..., n \\}$. Answering this question, we show that this maximum cardinality is of order $n^{1/3+o(1)}$."}
{"category": "Math", "title": "On two-point configurations in random set", "abstract": "We show that with high probability a random set of size $\\Theta(n^{1-1/k})$ of $\\{1,...,n\\}$ contains two elements $a$ and $a+d^k$, where $d$ is a positive integer. As a consequence, we prove an analogue of S\\'ark\\\"ozy-F\\\"urstenberg's theorem for random set."}
{"category": "Math", "title": "RO(S^1)-graded TR-groups of F_p, Z and \\ell", "abstract": "We give an algorithm for calculating the RO(S^1)-graded TR-groups of F_p, completing the calculation started by the second author. We also calculate the RO(S^1)-graded TR-groups of Z with mod p coefficients and of the Adams summand \\ell of connective complex K-theory with V(1)-coefficients. Some of these calculations are used elsewhere to compute the algebraic K-theory of certain Z-algebras."}
{"category": "Math", "title": "Emile Borel's difficult days in 1941", "abstract": "The German forces occupying Paris arrested Emile Borel and three other members of the Acad\\'emie des Sciences in October 1941 and released them about five weeks later. Why? We examine some relevant German and French archives and other sources and propose some hypotheses. In the process, we review how the Occupation was structured and how it dealt with French higher education and some French mathematicians."}
{"category": "Math", "title": "1-Saturating Sets, Caps and Round Sets in Binary Spaces", "abstract": "We show that, for a positive integer $r$, every minimal 1-saturating set in ${\\rm PG}(r-1,2)$ of size at least ${11/36} 2^r+3$ is either a complete cap or can be obtained from a complete cap $S$ by fixing some $s\\in S$ and replacing every point $s'\\in S\\setminus\\{s\\}$ by the third point on the line through $s$ and $s'$. Stated algebraically: if $G$ is an elementary abelian 2-group and a set $A\\subseteq G\\setminus\\{0\\}$ with $|A|>{11/36} |G|+3$ satisfies $A\\cup 2A=G$ and is minimal subject to this condition, then either $A$ is a maximal sum-free set, or there are a maximal sum-free set $S\\subseteq G$ and an element $s\\in S$ such that $A=\\{s\\}\\cup\\big(s+(S\\setminus\\{s\\})\\big)$. Since, conversely, every set obtained in this way is a minimal 1-saturating set, and the structure of large sum-free sets in an elementary 2-group is known, this provides a complete description of large minimal 1-saturating sets. Our approach is based on characterizing those large sets $A$ in elementary abelian 2-groups such that, for every proper subset $B$ of $A$, the sumset 2B is a proper subset of 2A."}
{"category": "Math", "title": "Asymptotic dimension of proper CAT(0) spaces which are homeomorphic to the plane", "abstract": "In this paper, we investigate a proper CAT(0) space $(X,d)$ which is homeomorphic to $R^2$ and we show that the asymptotic dimension $asdim (X,d)$ is equal to 2."}
{"category": "Math", "title": "From iterated tilted algebras to cluster-tilted algebras", "abstract": "In this paper the relationship between iterated tilted algebras and cluster-tilted algebras and relation-extensions is studied. In the Dynkin case, it is shown that the relationship is very strong and combinatorial."}
{"category": "Math", "title": "Cohomology rings of almost-direct products of free groups", "abstract": "An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg-Mac Lane space."}
{"category": "Math", "title": "Resonance of basis-conjugating automorphism groups", "abstract": "We determine the structure of the first resonance variety of the cohomology ring of the group of automorphisms of a finitely generated free group which act by conjugation on a given basis."}
{"category": "Math", "title": "Schreier rewriting beyond the classical setting", "abstract": "Using actions of free monoids and free associative algebras, we establish some Schreier-type formulas involving the ranks of actions and the ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish the generalization of the Schreier formula to the case of subgroups of infinite index. We also study and apply large modules over free associative algebras in the spirit of the paper Olshanskii, A. Yu.; Osin, D.V., Large groups and their periodic quotients, Proc. Amer. Math. Soc., 136 (2008), 753 - 759."}
{"category": "Math", "title": "On localization properties of Fourier transforms of hyperfunctions", "abstract": "In [Adv. Math. 196 (2005) 310-345] the author introduced a new generalized function space $\\mathcal U(R^k)$ which can be naturally interpreted as the Fourier transform of the space of Sato's hyperfunctions on $R^k$. It was shown that all Gelfand--Shilov spaces $S^{\\prime 0}_\\alpha(R^k)$ ($\\alpha>1$) of analytic functionals are canonically embedded in $\\mathcal U(R^k)$. While the usual definition of support of a generalized function is inapplicable to elements of $S^{\\prime 0}_\\alpha(R^k)$ and $\\mathcal U(R^k)$, their localization properties can be consistently described using the concept of {\\it carrier cone} introduced by Soloviev [Lett. Math. Phys. 33 (1995) 49-59; Comm. Math. Phys. 184 (1997) 579-596]. In this paper, the relation between carrier cones of elements of $S^{\\prime 0}_\\alpha(R^k)$ and $\\mathcal U(R^k)$ is studied. It is proved that an analytic functional $u\\in S^{\\prime 0}_\\alpha(R^k)$ is carried by a cone $K\\subset R^k$ if and only if its canonical image in $\\mathcal U(R^k)$ is carried by $K$."}
{"category": "Math", "title": "A Uniform Estimate for Fourier Restriction to Simple Curves", "abstract": "We prove a uniform Fourier extension-restriction estimate for a certain class of curves in d-dimensional Euclidean space."}
{"category": "Math", "title": "General Hormander and Mikhlin conditions for multipliers of Besov spaces", "abstract": "Here a new condition for the geometry of Banach spaces is introduced and the operator--valued Fourier multiplier theorems in weighted Besov spaces are obtained. Particularly, connections between the geometry of Banach spaces and Hormander-Mikhlin conditions are established. As an application of main results the regularity properties of degenerate elliptic differential operator equations are investigated."}
{"category": "Math", "title": "The orbit structure of the Gelfand-Zeitlin group on n x n matrices", "abstract": "In recent work (\\cite{KW1},\\cite{KW2}), Kostant and Wallach construct an action of a simply connected Lie group $A\\simeq \\mathbb{C}^{{n\\choose 2}}$ on $gl(n)$ using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In \\cite{KW1}, the authors show that $A$-orbits of dimension ${n\\choose 2}$ form Lagrangian submanifolds of regular adjoint orbits in $gl(n)$. They describe the orbit structure of $A$ on a certain Zariski open subset of regular semisimple elements. In this paper, we describe all $A$-orbits of dimension ${n\\choose 2}$ and thus all polarizations of regular adjoint orbits obtained using Gelfand-Zeitlin theory."}
{"category": "Math", "title": "Holomorphic Engel Structures", "abstract": "Recently there has been renewed interest among differential geometers in the theory of Engel structures. We introduce holomorphic analogues of these structures, and pose the problem of classifying projective manifolds admitting them. Besides providing the basic properties of these varieties, we present two series of examples and characterize them by certain positivity conditions on the Engel structure."}
{"category": "Math", "title": "On complex positive definite functions on Z_n vanishing on squares", "abstract": "We generalize the Sarkozy-Furstenberg theorem on squares in difference sets of integers, and show that, given any positive definite function f:Z_N->C with density at least r(N), where r(N)=O((\\log N)^{-c}), there is a perfect square s<=N/2 such that f(s) is non-zero. We do not rely on the usual analysis of the dichotomy of randomness and periodicity of a set and iterative application of the Hardy-Littlewood method. Instead, we find a bound for the van der Corput property of the set of squares."}
{"category": "Math", "title": "Analytic continuation of Dirichlet series with almost periodic coefficients", "abstract": "We prove that an ordinary Dirichlet series with coefficients a(n)=g(n b) has an abscissa of convergence 0 if g is an odd 1-periodic, real-analytic function and b is Diophantine. We also show that if g is odd and has bounded variation and b is of bounded Diophantine type r>1, then the abscissa of convergence is smaller or equal than 1-1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and b is Diophantine, then the ordinary Dirichlet series has an analytic continuation to the entire complex plane."}
{"category": "Math", "title": "Analogue of Newton-Puiseux series for non-holonomic D-modules and factoring", "abstract": "We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to $D$-modules having infinite-dimensional space of solutions (i. e. non-holonomic $D$-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors."}
{"category": "Math", "title": "Non-holonomic Ideals in the Plane and Absolute Factoring", "abstract": "We study {\\it non-holonomic} overideals of a left differential ideal $J\\subset F[\\partial_x, \\partial_y]$ in two variables where $F$ is a differentially closed field of characteristic zero. The main result states that a principal ideal $J=< P>$ generated by an operator $P$ with a separable {\\it symbol} $symb(P)$, which is a homogeneous polynomial in two variables, has a finite number of maximal non-holonomic overideals. This statement is extended to non-holonomic ideals $J$ with a separable symbol. As an application we show that in case of a second-order operator $P$ the ideal $<P>$ has an infinite number of maximal non-holonomic overideals iff $P$ is essentially ordinary. In case of a third-order operator $P$ we give few sufficient conditions on $<P>$ to have a finite number of maximal non-holonomic overideals."}
{"category": "Math", "title": "The Rees-Suschkewitsch Theorem for simple topological semigroups", "abstract": "We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product of a topological group over topological spaces with a continuous sandwich function. We prove that a simple topological semigroup $S$ is a topological paragroup if one of the following conditions is satisfied: (1) $S$ is completely simple and the maximal subgroups of $S$ are topological groups, (2) $S$ contains an idempotent and the square $S\\times S$ is countably compact or pseudocompact, (3) $S$ is sequentially compact or each power of $S$ is countably compact. The last item generalizes an old Wallace's result saying that each simple compact topological semigroup is a topological paragroup."}
{"category": "Math", "title": "Construction of universal Thom-Whitney-a stratifications, their functoriality and Sard-type Theorem for singular varieties", "abstract": "{\\bf Construction.} For a dominating polynomial mapping {$F: K^n\\to K^l$} with an isolated critical value at 0 ($K$ an algebraically closed field of characteristic zero) we construct a closed {\\it bundle} $G_F \\subset T^{*}K^n $. We restrict $ G_F $ over the critical points $Sing(F)$ of $ F$ in $ F^{-1}(0)$ and partition $Sing(F)$ into {\\it 'quasistrata'} of points with the fibers of $G_F$ of constant dimension. It turns out that T-W-a (Thom and Whitney-a) stratifications 'near' $F^{-1}(0)$ exist iff the fibers of bundle $G_F$ are orthogonal to the tangent spaces at the smooth points of the quasistrata (e. g. when $ l=1$). Also, the latter are the orthogonal complements over an irreducible component $ S $ of a quasistratum only if $S $ is {\\bf universal} for the class of {T-W-a} stratifications, meaning that for any $\\{S_j'\\}_j $ in the class, $ \\Sing (F) = \\cup_j S'_j $, there is a component $S' $ of an $ S_j' $ with $S\\cap S'$ being open and dense in both $S $ and $ S' $. {\\bf Results.} We prove that T-W-a stratifications with only universal strata exist iff all fibers of $G_F$ are the orthogonal complements to the respective tangent spaces to the quasistrata, and then the partition of $\\Sing(F)$ by the latter yields the coarsest {\\it universal T-W-a stratification}. The key ingredient is our version of {\\bf Sard-type Theorem for singular spaces} in which a singular point is considered to be noncritical iff nonsingular points nearby are 'uniformly noncritical' (e. g. for a dominating map $ F: X \\to Z $ meaning that the sum of the absolute values of the $l\\times l$ minors of the Jacobian matrix of $ F $, where $ l = \\dim (Z) $, not only does not vanish but, moreover, is separated from zero by a positive constant)."}
{"category": "Math", "title": "Localized linear polynomial operators and quadrature formulas on the sphere", "abstract": "The purpose of this paper is to construct universal, auto--adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper--)sphere $\\SS^q$ ($q\\ge 2$). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well known formulas as Driscoll--Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178."}
{"category": "Math", "title": "Dirichlet Forms on Laakso and Barlow-Evans Fractals of Arbitrary Dimension", "abstract": "In this paper we explore two constructions of the same family of metric measure spaces. The first construction was introduced by Laakso in 2000 where he used it as an example that Poincar\\'e inequalities can hold on spaces of arbitrary Hausdorff dimension. This was proved using minimal generalized upper gradients. Following Cheeger's work these upper gradients can be used to define a Sobolev space. We show that this leads to a Dirichlet form. The second construction was introduced by Barlow and Evans in 2004 as a way of producing exotic spaces along with Markov processes from simpler spaces and processes. We show that for the correct base process in the Barlow Evans construction that this Markov process corresponds to the Dirichlet form derived from the minimal generalized upper gradients."}
{"category": "Math", "title": "Asymptotic behaviour of variation of pure polarized TERP structures", "abstract": "The purpose of this paper is twofold. One is to give a survey of our study on the reductions of harmonic bundles, and the other is to explain a simple application in the study of TERP structure. In particular, we investigate the asymptotic behaviour of the \"new supersymmetric index\" for variation of pure polarized TERP structures."}
{"category": "Math", "title": "The Figure Eight Knot Group is Conjugacy Separable", "abstract": "We prove that torsion free subgroups of PGL(2,C) (abstractly) commensurable with the Euclidean Bianchi groups are conjugacy separable. As a consequence we deduce the result stated in the title."}
{"category": "Math", "title": "On elements of order p^s in the plane Cremona group over a field of characteristic p", "abstract": "We show that the plane Cremona group over a field of characteristic p > 0 does not contain elements of order of power of p larger than 2 and it does not contain elements of order p^2 unless p =2. Also we describe conjugacy classes of elements of order 4."}
{"category": "Math", "title": "Some examples of lifting problems from quotient algebras", "abstract": "We consider three lifting questions: Given a $C\\sp{*}$-algebra $I$, if there is a unital $C\\sp{*}$-algebra $A$ contains $I$ as an ideal, is every unitary from $A/I$ lifted to a unitary in $A$? is every unitary from $A/I$ lifted to an extremal partial isometry? is every extremal partial isometry from $A/I$ lifted to an extremal partial isometry? We show several constructions of $I$ which serve as working examples or counter-examples for above questions."}
{"category": "Math", "title": "Polynomial Representation of E6 and Its Combinatorial and PDE Implications", "abstract": "In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of E6 into a sum of irreducible submodules. It turns out that the cubic polynomial invariant corresponding to the Dicksons' invariant trilinear form is the unique fundamental invariant. Moreover, we obtain a combinatorial identity saying that the dimensions of certain irreducible modules of E6 are correlated by the binomial coefficients of twenty-six. Furthermore, we find all the polynomial solutions for the invariant differential operator corresponding to the Dickson trilinear form in terms of the irreducible submodules."}
{"category": "Math", "title": "Intrinsically triple-linked graphs in RP^3", "abstract": "Flapan--Naimi--Pommersheim showed that every spatial embedding of $K_{10}$, the complete graph on ten vertices, contains a non-split three-component link; that is, $K_{10}$ is intrinsically triple-linked in $\\mathbb{R}^3$. The work of Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known intrinsically triple-linked graphs in $\\mathbb{R}^3$ to include several other families of graphs. In this paper, we will show that while some of these graphs can be embedded 3-linklessly in $\\mathbb{R}P^3$, $K_{10}$ is intrinsically triple-linked in $\\mathbb{R}P^3$."}
{"category": "Math", "title": "Conjectures about distinction and Asai $L$-functions of generic representations of general linear groups over local fields", "abstract": "Let $K/F$ be a quadratic extension of p-adic fields. The Bernstein-Zelevinsky's classification asserts that generic representations are parabolically induced from quasi-square-integrable representations. We show, following a method developed by Cogdell and Piatetski-Shapiro, that the equality of the Rankin-Selberg type Asai $L$-function of generic representations of $GL(n,K)$ and of the Asai $L$-function of the Langlands parameter, is equivalent to the truth of a conjecture about classification of distinguished generic representations in terms of the inducing quasi-square-integrable representations. As the conjecture is true for principal series representations, this gives the expression of the Asai L-function of such representations."}
{"category": "Math", "title": "Spherical designs via Brouwer fixed point theorem", "abstract": "For each N>=c_d*n^{2d*(d+1)/(d+2)} we prove the existence of a spherical n-design on S^d consisting of N points, where c_d is a constant depending only on $d$."}
{"category": "Math", "title": "The Witten genus and vertex algebras", "abstract": "This article is the first report of an ongoing project aimed at finding a geometric interpretation of the Witten genus and other tmf classes. Section 2 reviews the sheaves of chiral differential operators (CDOs) over a complex manifold, including their construction, obstructions and relation with the Witten genus. In section 3, the structure of each sheaf of CDOs is reorganized in terms of modules over the sheaf of holomorphic functions. This invokes the notion of a differential graded vertex algebroid. The construction of sheaves of CDOs is due to Gorbounov, Malikov and Schechtman, and so is the notion of a vertex algebroid; the differential graded version is first introduced here. Section 4 contains the main result, namely the construction of a sheaf of differential graded conformal vertex algebras that provides a fine resolution of a sheaf of CDOs. This `infinite dimensional Dolbeault complex' plays a role for the Witten genus similar to that of the Dolbeault complex for the Todd genus."}
{"category": "Math", "title": "Global Well-posedness for the fourth order nonlinear Schr\\\"{o}dinger equations with small rough data in high demension", "abstract": "For $n\\geq 2$, we establish the smooth effects for the solutions of the linear fourth order Shr\\\"{o}dinger equation in anisotropic Lebesgue spaces with $\\Box_k$-decomposition. Using these estimates, we study the Cauchy problem for the fourth order nonlinear Schr\\\"{o}dinger equations with three order derivatives and obtain the global well posedness for this problem with small data in modulation space $M^{9/2}_{2,1}({\\Real^{n}})$."}
{"category": "Math", "title": "Sharp Decay Estimates and Vanishing Viscosity for Diffusive Hamilton-Jacobi Equations", "abstract": "Sharp temporal decay estimates are established for the gradient and time derivative of solutions to a viscous Hamilton-Jacobi equation as well the associated Hamilton-Jacobi equation. Special care is given to the dependence of the estimates on the viscosity. The initial condition being only continuous and either bounded or non-negative. The main requirement on the Hamiltonians is that it grows superlinearly or sublinearly at infinity, including in particular H(r) = r^p for r non-negatif and p positif and different from 1."}
{"category": "Math", "title": "Multiplier ideal sheaves and the K\\\"ahler-Ricci flow on toric Fano manifolds with large symmetry", "abstract": "The purpose of this paper is to calculate the support of the multiplier ideal sheaves derived from the K\\\"ahler-Ricci flow on certain toric Fano manifolds with large symmetry. The early idea of this paper has already been in Appendix of \\cite{futaki-sano0711}."}
{"category": "Math", "title": "Galois Theory, discriminants and torsion subgroups of elliptic curves", "abstract": "We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve."}
{"category": "Math", "title": "A fully nonlinear problem with free boundary in the plane", "abstract": "We prove that bounded solutions to an overdetermined fully nonlinear free boundary problem in the plane are one dimensional. Our proof relies on maximum principle techniques and convexity arguments."}
{"category": "Math", "title": "Horseshoes in multidimensional scaling and local kernel methods", "abstract": "Classical multidimensional scaling (MDS) is a method for visualizing high-dimensional point clouds by mapping to low-dimensional Euclidean space. This mapping is defined in terms of eigenfunctions of a matrix of interpoint dissimilarities. In this paper we analyze in detail multidimensional scaling applied to a specific dataset: the 2005 United States House of Representatives roll call votes. Certain MDS and kernel projections output ``horseshoes'' that are characteristic of dimensionality reduction techniques. We show that, in general, a latent ordering of the data gives rise to these patterns when one only has local information. That is, when only the interpoint distances for nearby points are known accurately. Our results provide a rigorous set of results and insight into manifold learning in the special case where the manifold is a curve."}
{"category": "Math", "title": "Exact Categories", "abstract": "We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 x 3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel-Quillen embedding theorem in an appendix."}
{"category": "Math", "title": "C*-Algebras Associated with Iterated Function Systems", "abstract": "We review Kajiwara and Watatani's construction of a C*-algebra from an iterated function system (IFS). If the IFS satisfies the finite branch condition or the open set condition, we build an injective homomorphism from Kajiwara-Watatani algebras to the Cuntz algebra, which can be thought as the algebra of the lifted system, and we give the description of its image. Finally, if the IFS admits a left inverse we show that the Kajiwara-Watatani algebra is isomorphic to an Exel's crossed product."}
{"category": "Math", "title": "Compactification minimale et mauvaise reduction", "abstract": "We construct the minimal compactification of some modular Siegel varieties at their bad reduction places. These varieties parametrize principally polarized abelian schemes endowed with a parahoric level structure at a prime number $p$, and with an auxiliary level structure ; such varieties have bad reduction at $p$. We also sketch the arithmetic theory of the associated Siegel modular forms. ----- Nous construisons la compactification minimale de certaines varietes modulaires de Siegel en leurs places de mauvaise reduction. Ces varietes parametrent des schemas abeliens principalement polarises munis d'une structure de niveau parahorique en un nombre premier $p$, et d'une structure de niveau auxilliaire ; elles ont mauvaise reduction en $p$. Nous esquissons egalement une theorie arithmetique des formes modulaires de Siegel associees a ces varietes."}
{"category": "Math", "title": "Matrix valued Brownian motion and a paper by Polya", "abstract": "We give a geometric description of the motion of eigenvalues of a Brownian motion with values in some matrix spaces. In the second part we consider a paper by Polya where he introduced a function close to the Riemann zeta function, which satisfies Riemann hypothesis. We show that each of these two functions can be related to Brownian motion on a symmetric space."}
{"category": "Math", "title": "On the problem of detecting linear dependence for products of abelian varieties and tori", "abstract": "Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a point in G(K) and let L be a finitely generated subgroup of G(K). Suppose that for all but finitely many primes p of K the point (R mod p) belongs to (L mod p). Does it follow that R belongs to L? We answer this question affirmatively in three cases: if L is cyclic; if L is a free left End_K G-submodule of G(K); if L has a set of generators (as a group) which is a basis of a free left End_K G-submodule of G(K). In general we prove that there exists an integer m (depending only on G, K and the rank of L) such that mR belongs to the left End_K G-submodule of G(K) generated by L."}
{"category": "Math", "title": "Robust adaptive importance sampling for normal random vectors", "abstract": "Adaptive Monte Carlo methods are very efficient techniques designed to tune simulation estimators on-line. In this work, we present an alternative to stochastic approximation to tune the optimal change of measure in the context of importance sampling for normal random vectors. Unlike stochastic approximation, which requires very fine tuning in practice, we propose to use sample average approximation and deterministic optimization techniques to devise a robust and fully automatic variance reduction methodology. The same samples are used in the sample optimization of the importance sampling parameter and in the Monte Carlo computation of the expectation of interest with the optimal measure computed in the previous step. We prove that this highly dependent Monte Carlo estimator is convergent and satisfies a central limit theorem with the optimal limiting variance. Numerical experiments confirm the performance of this estimator: in comparison with the crude Monte Carlo method, the computation time needed to achieve a given precision is divided by a factor between 3 and 15."}
{"category": "Math", "title": "On the quadratic normality and the triple curve of three dimensional subvarieties of ${\\mathbb P}^5$", "abstract": "A well-known conjecture asserts that smooth threefolds $X\\subset\\{\\mathbb P}^5$ are quadratically normal with the only exception of the Palatini scroll. As a corollary of a more general statement we obtain the following result, which is related to the previous conjecture: If $X\\subset\\{\\mathbb P}^5$ is not quadratically normal, then its triple curve is reducible. Similar results are also given for higher dimensional varieties."}
{"category": "Math", "title": "Generalized analytic functions on generalized domains", "abstract": "We define the algebra of Colombeau generalized functions on a subset A of the space of d-dimensional generalized points. If the domain A is open, such generalized functions can be identified with pointwise maps from A into the ring of Colombeau generalized numbers. We study analyticity in these algebras, if the domain is an open subset of the complex generalized points. In particular, if the domain is an open ball for the sharp norm, we characterize analyticity and give a unicity theorem involving the values at generalized points."}
{"category": "Math", "title": "Real Subpairs and Frobenius-Schur Indicators of Characters in 2-Blocks", "abstract": "Let B be a real 2-block of a finite group G. Then B has a real defect class. Let g be an element of such a class. A defect couple of B is (D,E), where E is a Sylow 2-subgroup of the extended centralizer C^*(g) of g, and D is the intersection of E with the centralizer C(g). It is known that (D,E) is uniquely determined up to G-conjugacy. We show that (D,E) determines which B-subpairs are real. We also outline how (D,E) influences the vertices of components of the G-permutation module corresponding to the conjugation action of G on its involutions. We apply these methods to enumerate the Frobenius-Schur indicators of the irreducible characters in a real block that has a dihedral defect group. We also determine the vertices of the components of the involution module in such a block."}
{"category": "Math", "title": "Voronoi cells of discrete point sets", "abstract": "It is well known that all cells of the Voronoi diagram of a Delaunay set are polytopes. For a finite point set, all these cells are still polyhedra. So the question arises, if this observation holds for all discrete point sets: Are always all Voronoi cells of an arbitrary, infinite discrete point set polyhedral? In this paper, an answer to this question will be given. It will be shown that all Voronoi cells of a discrete point set are polytopes if and only if every point of the point set is an inner point. Furthermore, the term of a locally finitely generated discrete point set will be introduced and it will be shown that exactly these sets have the property of possessing only polyhedral Voronoi cells."}
{"category": "Math", "title": "An Arakelov Inequality in Characteristic p and Upper Bound of p-Rank Zero Locus", "abstract": "In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus $g\\geq 1$ over characteristic $p$ with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of algebraic curves of $p-$rank zero in a semi-stable family over characteristic $p$ with nontrivial Kodaira-Spencer map in terms of the genus of a general closed fiber, the genus of the base curve and the number of singular fibres. An extension of the above results to smooth families of Abelian varieties over $k$ with $W_2$-lifting assumption is also included."}
{"category": "Math", "title": "Homotopy groups and twisted homology of arrangements", "abstract": "Recent work of M. Yoshinaga shows that in some instances certain higher homotopy groups of arrangements map onto non-resonant homology. This is in contrast to the usual Hurewicz map to untwisted homology, which is always the zero homomorphism in degree greater than one. In this work we examine this dichotomy, generalizing both results."}
{"category": "Math", "title": "A Hilbert C*-module admitting no frames", "abstract": "We show that every infinite-dimensional commutative unital C*-algebra has a Hilbert C*-module admitting no frames. In particular, this shows that Kasparov's stabilization theorem for countably generated Hilbert C*-modules can not be extended to arbitrary Hilbert C*-modules."}
{"category": "Math", "title": "Distributive Lattices, Polyhedra, and Generalized Flow", "abstract": "A D-polyhedron is a polyhedron $P$ such that if $x,y$ are in $P$ then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, D-polyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact every D-polyhedron corresponds to a directed graph with arc-parameters, such that every point in the polyhedron corresponds to a vertex potential on the graph. Alternatively, an edge-based description of the point set can be given. The objects in this model are dual to generalized flows, i.e., dual to flows with gains and losses. These models can be specialized to yield some cases of distributive lattices that have been studied previously. Particular specializations are: lattices of flows of planar digraphs (Khuller, Naor and Klein), of $\\alpha$-orientations of planar graphs (Felsner), of c-orientations (Propp) and of $\\Delta$-bonds of digraphs (Felsner and Knauer). As an additional application we exhibit a distributive lattice structure on generalized flow of breakeven planar digraphs."}
{"category": "Math", "title": "Precise asymptotic of eigenvalues of resonant quasilinear systems", "abstract": "In this work we study the sequence of variational eigenvalues of a system of resonant type involving $p-$ and $q-$laplacians on $\\Omega \\subset \\R^N$, with a coupling term depending on two parameters $\\alpha$ and $\\beta$ satisfying $\\alpha/p + \\beta/q = 1$. We show that the order of growth of the $k^{th}$ eigenvalue depends on $\\alpha+\\beta$, $\\lam_k = O(k^{\\frac{\\alpha+\\beta}{N}})$."}
{"category": "Math", "title": "On distribution of fractional parts of linear forms", "abstract": "We prove a result on linear forms related to Peres-Schlag's theorem on badly approximable numbers with respect to lacunary sequences."}
{"category": "Math", "title": "The space of subgroups of an abelian group", "abstract": "We carry out the Cantor-Bendixson analysis of the space of all subgroups of any countable abelian group and we deduce a complete classification of such spaces up to homeomorphism."}
{"category": "Math", "title": "Maximal Levi Subgroups Acting on the Building of GL_n(F)", "abstract": "In this paper we give a complete invariant of the action of GL_n(F)\\times GL_m(F) acting on the Euclidean building B[GL_n(F)], where F is a non-archimedian field. We then use this invariant to give a natural metric on the resulting quotient space. In the special case of the torus acting on B[GL_2(F)] this gives a method of calculating of any vertex to any fixed apartment."}
{"category": "Math", "title": "On the zero set of G-equivariant maps", "abstract": "Let $G$ be a finite group acting on vector spaces $V$ and $W$ and consider a smooth $G$-equivariant mapping $f:V\\to W$. This paper addresses the question of the zero set near a zero $x$ of $f$ with isotropy subgroup $G$. It is known from results of Bierstone and Field on $G$-transversality theory that the zero set in a neighborhood of $x$ is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near $x$ using only information from the representations $V$ and $W$. We define an index $s(\\Sigma)$ for isotropy subgroups $\\Sigma$ of $G$ which is the difference of the dimension of the fixed point subspace of $\\Sigma$ in $V$ and $W$. Our main result states that if $V$ contains a subspace $G$-isomorphic to $W$, then for every maximal isotropy subgroup $\\Sigma$ satisfying $s(\\Sigma)>s(G)$, the zero set of $f$ near $x$ contains a smooth manifold of zeros with isotropy subgroup $\\Sigma$ of dimension $s(\\Sigma)$. We also present a systematic method to study the zero sets for group representations $V$ and $W$ which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of $G$-reversible equivariant vector fields."}
{"category": "Math", "title": "Invariant Hilbert schemes and Wonderful varieties", "abstract": "The invariant Hilbert schemes considered in \\cite{BC1} were proved to be affine spaces. The proof relied on the classification of strict wonderful varieties. We obtain in the present article a classification-free proof of the affinity of these invariant Hilbert scheme by means of deformation theoretical arguments. As a consequence we recover in a shorter way the classification of strict wonderful varieties. This provides an alternative and new approach to answer Luna's conjecture."}
{"category": "Math", "title": "Kac's conjecture from Nakajima quiver varieties", "abstract": "We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982."}
{"category": "Math", "title": "The quiver of an algebra associated to the Mantaci-Reutenauer descent algebra and the homology of regular semigroups", "abstract": "We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola's results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regular bands appear in the representation theory of finite Coxeter groups via the Solomon-Tits algebra of its Coxeter complex. In particular, we compute the quiver of Hsiao's algebra, which is related to the Mantaci-Reutenauer descent algebra."}
{"category": "Math", "title": "The multivariate signed Bollobas-Riordan polynomial", "abstract": "We generalise the signed Bollobas-Riordan polynomial of S. Chmutov and I. Pak [Moscow Math. J. 7 (2007), no. 3, 409-418] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [J. Combinatorial Theory, Ser. B, 99 (3): 617-638, 2009] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it."}
{"category": "Math", "title": "Potential automorphy for certain Galois representations to GL_2n", "abstract": "Building upon work of Clozel, Harris, Shepherd-Barron, and Taylor, this paper shows that certain Galois representations become automorphic after one makes a suitably large totally-real extension to the base field. The main innovation here is that the result applies to Galois representations to GL_{2n}, where previous work dealt with representations to GSp_n. The main technique is the consideration of the cohomology the Dwork hypersurface, and in particular, of pieces of this cohomology other than the invariants under the natural group action."}
{"category": "Math", "title": "Meromorphic continuation for the zeta function of a Dwork hypersurface", "abstract": "We consider the one-parameter family of hypersurfaces in $\\Pj^5$ with projective equation (X_1^5+X_2^5+X_3^5+X_4^5+X_5^5) = 5\\lambda X_1 X_2... X_5, (writing $\\lambda$ for the parameter), proving that the Galois representations attached to their cohomologies are potentially automorphic, and hence that the zeta function of the family has meromorphic continuation throughout the complex plane."}
{"category": "Math", "title": "A note on the Busemann-Petty problem for bodies of certain invariance", "abstract": "The Busemann-Petty problem asks whether origin symmetric convex bodies in $\\R^n$ with smaller hyperplane sections necessarily have smaller volume. The answer is affirmative if $n\\leq 3$ and negative if $n\\geq 4.$ We consider a class of convex bodies that have a certain invariance property with respect to their ordered k-tuples of coordinates in $\\R^{kn}$ and prove the corresponding problem."}
{"category": "Math", "title": "Perfectness of Kirillov-Reshetikhin crystals for nonexceptional types", "abstract": "For nonexceptional types, we prove a conjecture of Hatayama et al. about the prefectness of Kirillov-Reshetikhin crystals."}
{"category": "Math", "title": "Markov switching negative binomial models: an application to vehicle accident frequencies", "abstract": "In this paper, two-state Markov switching models are proposed to study accident frequencies. These models assume that there are two unobserved states of roadway safety, and that roadway entities (roadway segments) can switch between these states over time. The states are distinct, in the sense that in the different states accident frequencies are generated by separate counting processes (by separate Poisson or negative binomial processes). To demonstrate the applicability of the approach presented herein, two-state Markov switching negative binomial models are estimated using five-year accident frequencies on Indiana interstate highway segments. Bayesian inference methods and Markov Chain Monte Carlo (MCMC) simulations are used for model estimation. The estimated Markov switching models result in a superior statistical fit relative to the standard (single-state) negative binomial model. It is found that the more frequent state is safer and it is correlated with better weather conditions. The less frequent state is found to be less safe and to be correlated with adverse weather conditions."}
{"category": "Math", "title": "On $k$-free-like groups", "abstract": "A $k$-free like group is a $k$-generated group $G$ with a sequence of $k$-element generating sets $Z_n$ such that the girth of $G$ relative to $Z_n$ is unbounded and the Cheeger constant of $G$ relative to $Z_n$ is bounded away from 0. By a recent result of Benjamini-Nachmias-Peres, this implies that the critical bond percolation probability of the Cayley graph of $G$ relative to $Z_n$ tends to $1/(2k-1)$ as $n\\to \\infty$. Answering a question of Benjamini, we construct many non-free groups that are $k$-free like for all sufficiently large $k$."}
{"category": "Math", "title": "A Priori Bounds for the Vorticity of Axis Symmetric Solutions to the Navier-Stokes Equations", "abstract": "We obtain a pointwise, a priori bound for the vorticity of axis symmetric solutions to the 3 dimensional Navier-Stokes equations. The bound is in the form of a reciprocal of a power of the distance to the axis of symmetry. This seems to be the first general pointwise estimate established for the axis symmetric Navier-Stokes equations."}
{"category": "Math", "title": "Stability of Localized Operators", "abstract": "Let $\\ell^p, 1\\le p\\le \\infty$, be the space of all $p$-summable sequences and $C_a$ be the convolution operator associated with a summable sequence $a$. It is known that the $\\ell^p$- stability of the convolution operator $C_a$ for different $1\\le p\\le \\infty$ are equivalent to each other, i.e., if $C_a$ has $\\ell^p$-stability for some $1\\le p\\le \\infty$ then $C_a$ has $\\ell^q$-stability for all $1\\le q\\le \\infty$. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sj\\\"ostrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the $\\ell^p$- stability (or $L^p$-stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized."}
{"category": "Math", "title": "Asymptotics for a free-boundary model in price formation", "abstract": "We study the asymptotics for large time of solutions to a one dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry-Lions on price formation and dynamic equilibria. The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory."}
{"category": "Math", "title": "For objective causal inference, design trumps analysis", "abstract": "For obtaining causal inferences that are objective, and therefore have the best chance of revealing scientific truths, carefully designed and executed randomized experiments are generally considered to be the gold standard. Observational studies, in contrast, are generally fraught with problems that compromise any claim for objectivity of the resulting causal inferences. The thesis here is that observational studies have to be carefully designed to approximate randomized experiments, in particular, without examining any final outcome data. Often a candidate data set will have to be rejected as inadequate because of lack of data on key covariates, or because of lack of overlap in the distributions of key covariates between treatment and control groups, often revealed by careful propensity score analyses. Sometimes the template for the approximating randomized experiment will have to be altered, and the use of principal stratification can be helpful in doing this. These issues are discussed and illustrated using the framework of potential outcomes to define causal effects, which greatly clarifies critical issues."}
{"category": "Math", "title": "Random survival forests", "abstract": "We introduce random survival forests, a random forests method for the analysis of right-censored survival data. New survival splitting rules for growing survival trees are introduced, as is a new missing data algorithm for imputing missing data. A conservation-of-events principle for survival forests is introduced and used to define ensemble mortality, a simple interpretable measure of mortality that can be used as a predicted outcome. Several illustrative examples are given, including a case study of the prognostic implications of body mass for individuals with coronary artery disease. Computations for all examples were implemented using the freely available R-software package, randomSurvivalForest."}
{"category": "Math", "title": "Geometric realizations of curvature models by manifolds with constant scalar curvature", "abstract": "We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and *-scalar curvature."}
{"category": "Math", "title": "Multiply generated dynamical systems and the duality of higher rank graph algebras", "abstract": "We define a semidirect product groupoid of a system of partially defined local homeomorphisms $T=(T_{1},..., T_{r})$. We prove that this construction gives rise to amenable groupoids. The associated algebra is a Cuntz-like algebra. We use this construction for higher rank graph algebras in order to give a topological interpretation for the duality in $E$-theory between $C^{*}(\\Lambda)$ and $C^{*}(\\Lambda^{op})$."}
{"category": "Math", "title": "Cohomology $C_{\\infty}$-algebra and Rational Homotopy Type", "abstract": "In the rational cohomology of a 1-connected space a structure of $C_{\\infty}$-algebra is constructed and it is shown that this object determines the rational homotopy type"}
{"category": "Math", "title": "Open statistical issues in particle physics", "abstract": "Many statistical issues arise in the analysis of Particle Physics experiments. We give a brief introduction to Particle Physics, before describing the techniques used by Particle Physicists for dealing with statistical problems, and also some of the open statistical questions."}
{"category": "Math", "title": "Pseudodifferential operators on manifolds with linearization", "abstract": "We present in this paper the construction of a pseudodifferential calculus on smooth non-compact manifolds associated to a globally defined and coordinate independant complete symbol calculus, that generalizes the standard pseudodifferential calculus on $\\R^n$. We consider the case of manifolds $M$ with linearization in the sense of Bokobza-Haggiag, such that the associated (abstract) exponential map provides global diffeomorphisms of $M$ with $\\R^n$ at any point. Cartan--Hadamard manifolds are special cases of such manifolds. The abstract exponential map encodes a notion of infinity on the manifold that allows, modulo some hypothesis of $S_\\sigma$-bounded geometry, to define the Schwartz space of rapidly decaying functions, globally defined Fourier transformation and classes of symbols with uniform and decaying control over the $x$ variable. Given a linearization on the manifold with some properties of control at infinity, we construct symbol maps and $\\la$-quantization, explicit Moyal star-product on the cotangent bundle, and classes of pseudodifferential operators. We show that these classes are stable under composition, and that the $\\la$-quantization map gives an algebra isomorphism (which depends on the linearization) between symbols and pseudodifferential operators. We study, in our setting, $L^2$-continuity and give some examples. We show in particular that the hyperbolic 2-space $\\HH$ has a $S_1$-bounded geometry, allowing the construction of a global symbol calculus of pseudodifferential operators on $\\S(\\HH)$."}
{"category": "Math", "title": "An upper bound on the exceptional characteristics for Lusztig's character formula", "abstract": "We develop and study a Lefschetz theory in a combinatorial category associated to a root system and derive an upper bound on the exceptional characteristics for Lusztig's formula for the simple rational characters of a reductive algebraic group. Our bound is huge compared to the Coxeter number. It is, however, given by an explicit formula."}
{"category": "Math", "title": "On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane", "abstract": "To each once-punctured-torus bundle, $T_\\phi$, over the circle with pseudo-Anosov monodromy $\\phi$, there are associated two tessellations of the complex plane: one, $\\Delta(\\phi)$, is (the projection from $\\infty$ of) the triangulation of a horosphere at $\\infty$ induced by the canonical decomposition into ideal tetrahedra, and the other, $CW(\\phi)$, is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through $\\infty$. In this paper, we study the relation between $\\Delta(\\phi)$ and $CW(\\phi)$."}
{"category": "Math", "title": "Predictive learning via rule ensembles", "abstract": "General regression and classification models are constructed as linear combinations of simple rules derived from the data. Each rule consists of a conjunction of a small number of simple statements concerning the values of individual input variables. These rule ensembles are shown to produce predictive accuracy comparable to the best methods. However, their principal advantage lies in interpretation. Because of its simple form, each rule is easy to understand, as is its influence on individual predictions, selected subsets of predictions, or globally over the entire space of joint input variable values. Similarly, the degree of relevance of the respective input variables can be assessed globally, locally in different regions of the input space, or at individual prediction points. Techniques are presented for automatically identifying those variables that are involved in interactions with other variables, the strength and degree of those interactions, as well as the identities of the other variables with which they interact. Graphical representations are used to visualize both main and interaction effects."}
{"category": "Math", "title": "Sequential category aggregation and partitioning approaches for multi-way contingency tables based on survey and census data", "abstract": "Large contingency tables arise in many contexts but especially in the collection of survey and census data by government statistical agencies. Because the vast majority of the variables in this context have a large number of categories, agencies and users need a systematic way of constructing tables which are summaries of such contingency tables. We propose such an approach in this paper by finding members of a class of restricted log-linear models which maximize the likelihood of the data and use this to find a parsimonious means of representing the table. In contrast with more standard approaches for model search in hierarchical log-linear models (HLLM), our procedure systematically reduces the number of categories of the variables. Through a series of examples, we illustrate the extent to which it can preserve the interaction structure found with HLLMs and be used as a data simplification procedure prior to HLL modeling. A feature of the procedure is that it can easily be applied to many tables with millions of cells, providing a new way of summarizing large data sets in many disciplines. The focus is on information and description rather than statistical testing. The procedure may treat each variable in the table in different ways, preserving full detail, treating it as fully nominal, or preserving ordinality."}
{"category": "Math", "title": "Energy dissipation and self-similar solutions for an unforced inviscid dyadic model", "abstract": "A shell-type model of an inviscid fluid, previously considered in the literature, is investigated in absence of external force. Energy dissipation of positive solutions is proved and decay of energy like $t^{-2}$ is established. Self-similar decaying positive solutions are introduced and proved to exist and classified. Coalescence and blow-up are obtained as a consequence, in the class of arbitrary sign solutions."}
{"category": "Math", "title": "Measure changes with extinction", "abstract": "We consider a change of measure by a martingale $Z_t$ and clarify that in general $1/Z_t$ is only a supermartingale under the changed measure. We then give a necessary and sufficient condition for the event that the limit of the martingale is zero to coincide with the event that the martingale hits zero in finite time (up to a set of zero probability)."}
{"category": "Math", "title": "A Sharper discrepancy measure for post-election audits", "abstract": "Post-election audits use the discrepancy between machine counts and a hand tally of votes in a random sample of precincts to infer whether error affected the electoral outcome. The maximum relative overstatement of pairwise margins (MRO) quantifies that discrepancy. The electoral outcome a full hand tally shows must agree with the apparent outcome if the MRO is less than 1. This condition is sharper than previous ones when there are more than two candidates or when voters may vote for more than one candidate. For the 2006 U.S. Senate race in Minnesota, a test using MRO gives a $P$-value of 4.05% for the hypothesis that a full hand tally would find a different winner, less than half the value Stark [Ann. Appl. Statist. 2 (2008) 550--581] finds."}
{"category": "Math", "title": "Testing significance of features by lassoed principal components", "abstract": "We consider the problem of testing the significance of features in high-dimensional settings. In particular, we test for differentially-expressed genes in a microarray experiment. We wish to identify genes that are associated with some type of outcome, such as survival time or cancer type. We propose a new procedure, called Lassoed Principal Components (LPC), that builds upon existing methods and can provide a sizable improvement. For instance, in the case of two-class data, a standard (albeit simple) approach might be to compute a two-sample $t$-statistic for each gene. The LPC method involves projecting these conventional gene scores onto the eigenvectors of the gene expression data covariance matrix and then applying an $L_1$ penalty in order to de-noise the resulting projections. We present a theoretical framework under which LPC is the logical choice for identifying significant genes, and we show that LPC can provide a marked reduction in false discovery rates over the conventional methods on both real and simulated data. Moreover, this flexible procedure can be applied to a variety of types of data and can be used to improve many existing methods for the identification of significant features."}
{"category": "Math", "title": "Sufficient enlargements of minimal volume for finite dimensional normed linear spaces", "abstract": "Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\\to X$ such that $P(B_Y)\\subset A$. The main results of the paper: {\\bf (1)} Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. {\\bf (2)} If a finite dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon."}
{"category": "Math", "title": "Skew-symmetric cluster algebras of finite mutation type", "abstract": "In 2003, Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices. Besides cluster algebras of rank 2 and cluster algebras associated with triangulations of surfaces there are exactly 11 exceptional skew-symmetric cluster algebras of finite mutation type. More precisely, 9 of them are associated with root systems $E_6$, $E_7$, $E_8$, $\\widetilde E_6$, $\\widetilde E_7$, $\\widetilde E_8$, $E_6^{(1,1)}$, $E_7^{(1,1)}$, $E_8^{(1,1)}$; two remaining were recently found by Derksen and Owen. We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type, and discuss growth rate of cluster algebras."}
{"category": "Math", "title": "Branching Brownian motion: Almost sure growth along unscaled paths", "abstract": "We give new results on the growth of the number of particles in a dyadic branching Brownian motion which follow within a fixed distance of a path $f:[0,\\infty)\\to \\mathbb{R}$. We show that it is possible to count the number of particles without rescaling the paths. Our results reveal that the number of particles along certain paths can oscillate dramatically. The methods used are entirely probabilistic, taking advantage of the spine technique developed by, amongst others, Lyons et al, Kyprianou, and Hardy & Harris."}
{"category": "Math", "title": "Inference using shape-restricted regression splines", "abstract": "Regression splines are smooth, flexible, and parsimonious nonparametric function estimators. They are known to be sensitive to knot number and placement, but if assumptions such as monotonicity or convexity may be imposed on the regression function, the shape-restricted regression splines are robust to knot choices. Monotone regression splines were introduced by Ramsay [Statist. Sci. 3 (1998) 425--461], but were limited to quadratic and lower order. In this paper an algorithm for the cubic monotone case is proposed, and the method is extended to convex constraints and variants such as increasing-concave. The restricted versions have smaller squared error loss than the unrestricted splines, although they have the same convergence rates. The relatively small degrees of freedom of the model and the insensitivity of the fits to the knot choices allow for practical inference methods; the computational efficiency allows for back-fitting of additive models. Tests of constant versus increasing and linear versus convex regression function, when implemented with shape-restricted regression splines, have higher power than the standard version using ordinary shape-restricted regression."}
{"category": "Math", "title": "Slowdown estimates for ballistic random walk in random environment", "abstract": "For a random walk in an elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying the a ballisticity condition slightly weaker than condition (T'), We consider the probability of linear slowdown. We show an upper bound for this probability which is very close to the lower bound obtained by the \"naive trap\" analysis. As a tool for obtaining the main result, we show an almost local version of the quenched central limit theorem under the same ballisticity condition."}
{"category": "Math", "title": "Bergman polynomials on an Archipelago: Estimates, Zeros and Shape Reconstruction", "abstract": "Growth estimates of complex orthogonal polynomials with respect to the area measure supported by a disjoint union of planar Jordan domains (called, in short, an archipelago) are obtained by a combination of methods of potential theory and rational approximation theory. The study of the asymptotic behavior of the roots of these polynomials reveals a surprisingly rich geometry, which reflects three characteristics: the relative position of an island in the archipelago, the analytic continuation picture of the Schwarz function of every individual boundary and the singular points of the exterior Green function. By way of explicit example, fine asymptotics are obtained for the lemniscate archipelago $|z^m-1|<r^m, 0<r<1,$ which consists of $m$ islands. The asymptotic analysis of the Christoffel functions associated to the same orthogonal polynomials leads to a very accurate reconstruction algorithm of the shape of the archipelago, knowing only finitely many of its power moments. This work naturally complements a 1969 study by H. Widom of Szeg\\H{o} orthogonal polynomials on an archipelago and the more recent asymptotic analysis of Bergman orthogonal polynomials unveiled by the last two authors and their collaborators."}
{"category": "Math", "title": "Hyperellipticity and Klein bottle companionship in systolic geometry", "abstract": "Given a hyperelliptic Klein surface, we construct companion Klein bottles. Bavard's short loops on companion bottles are studied in relation to the surface to improve an inequality of Gromov's in systolic geometry."}
{"category": "Math", "title": "Perturbation of essential spectra of exterior elliptic problems", "abstract": "For a second-order strongly elliptic differential operator on an exterior domain in R^n it is known from works of Birman and Solomiak that a change of the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptotic formula. We show that one can augment, but not diminish, the essential spectrum by imposition of other Neumann-type non-elliptic boundary conditions. - The results are extended to 2m-order operators, where it is shown that for any selfadjoint realization defined by an elliptic normal boundary condition (other than the Dirichlet condition), one can augment the essential spectrum at will by adding a suitable operator to the mapping from free Diriclet data to Neumann data. We here also show an extension of the spectral asymptotics formula for the difference between inverses of elliptic problems. - The proofs rely on Krein-type formulas for differences between inverses, and cutoff techniques, combined with results on singular Green operators and their spectral asymptotics."}
{"category": "Math", "title": "On the structure of left and right F-, SM- and E-quasigroups", "abstract": "It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM- and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given, for example, finite simple F-quasigroup is a simple group or a simple medial quasigroup. It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isomorphic the direct product of a group and a left S-loop (this is an answer to Belousov \"1a\", problem). Any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented."}
{"category": "Math", "title": "Moments, cumulants and diagram formulae for non-linear functionals of random measures", "abstract": "This survey provides a unified discussion of multiple integrals, moments, cumulants and diagram formulae associated with functionals of completely random measures. Our approach is combinatorial, as it is based on the algebraic formalism of partition lattices and M\\\"obius functions. Gaussian and Poisson measures are treated in great detail. We also present several combinatorial interpretations of some recent CLTs involving sequences of random variables belonging to a fixed Wiener chaos."}
{"category": "Math", "title": "Batch means and spectral variance estimators in Markov chain Monte Carlo", "abstract": "Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean-square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality. Finally, we examine the empirical finite-sample properties of spectral variance and batch means estimators and provide recommendations for practitioners."}
{"category": "Math", "title": "The Beilinson-Drinfeld Grassmannian and symplectic knot homology", "abstract": "Seidel-Smith and Manolescu constructed knot homology theories using symplectic fibrations whose total spaces were certain varieties of matrices. These knot homology theories were associated to $SL(n) $ and tensor products of the standard and dual representations. In this paper, we place their geometric setups in a natural, general framework. For any complex reductive group and any sequence of minuscule dominant weights, we construct a fibration of affine varieties over a configuration space. The middle cohomology of these varieties is isomorphic to the space of invariants in the corresponding tensor product of representations. Our construction uses the Beilinson-Drinfeld Grassmannian and the geometric Satake correspondence."}
{"category": "Math", "title": "The Euler adic dynamical system and path counts in the Euler graph", "abstract": "We give a formula for generalized Eulerian numbers, prove monotonicity of sequences of certain ratios of the Eulerian numbers, and apply these results to obtain a new proof that the natural symmetric measure for the Bratteli-Vershik dynamical system based on the Euler graph is the unique fully supported invariant ergodic Borel probability measure. Key ingredients of the proof are a two-dimensional induction argument and a one-to-one correspondence between most paths from two vertices at the same level to another vertex."}
{"category": "Math", "title": "How to compute the length of a geodesic on a Riemannian manifold with small error in arbitrary Sobolev norms", "abstract": "We compute the length of geodesics on a Riemannian manifold by regular polynomial interpolation of the global solution of the eikonal equation related to the line element $ds^2=g_{ij}dx^idx^j$ of the manifold. Our algorithm approximates the length functional in arbitrarily strong Sobolev norms. Error estimates are obtained where the geometric information is used. It is pointed out how the algorithm can be used to get accurate approximation of solutions of parabolic partial differential equations leading obvious applications to finance and physics."}
{"category": "Math", "title": "On free associative algebras linearly graded by finite groups", "abstract": "As an instance of a linear action of a Hopf algebra on a free associative algebra, we consider finite group gradings of a free algebra induced by gradings on the space spanned by the free generators. The homogeneous component corresponding to the identity of the group is a free subalgebra which is graded by the usual degree. We look into its Hilbert series and prove that it is a rational function by giving an explicit formula. As an application, we show that, under suitable conditions, this subalgebra is finitely generated if and only if the grading on the base vector space is trivial."}
{"category": "Math", "title": "Banach $\\widetilde{\\mathbb C}$-algebras", "abstract": "We study Banach $\\widetilde{\\mathbb C}$-algebras, i.e., complete ultra-pseudo-normed algebras over the ring $\\widetilde{\\mathbb C}$ of Colombeau generalized complex numbers. We develop a spectral theory in such algebras. We show by explicit examples that important parts of classical Banach algebra theory do not hold for general Banach $\\widetilde{\\mathbb C}$-algebras and indicate a particular class of Banach $\\widetilde{\\mathbb C}$-algebras that overcomes these limitations to a large extent. We also investigate C*-algebras over $\\widetilde{\\mathbb C}$."}
{"category": "Math", "title": "On a Structure of the Set of Differential Games Values", "abstract": "In this paper the set of value functions of all-possible zero-sum differential games with terminal payoff is characterized. The necessary and sufficient condition for a given function to be a value of some differential game with terminal payoff is obtained."}
{"category": "Math", "title": "Survival of branching random walks in random environment", "abstract": "We study survival of nearest-neighbour branching random walks in random environment (BRWRE) on ${\\mathbb Z}$. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. $2\\times 2$ random matrices."}
{"category": "Math", "title": "Orthogonal bundles over curves in characteristic two", "abstract": "Let X be a smooth projective curve of genus g \\geq 2 defined over a field of characteristic two. We give examples of stable orthogonal bundles with unstable underlying vector bundles and use them to give counterexamples to Behrend's conjecture on the canonical reduction of principal G-bundles for G= SO(n) with n \\geq 7."}
{"category": "Math", "title": "Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls", "abstract": "We show that the open unit ball of the space of operators from a finite dimensional Hilbert space into a separable Hilbert space (we call it \"operator ball\") has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In the appendix we present results of Itai Shafrir about hyperbolic metrics on the operator ball."}
{"category": "Math", "title": "Compositions of projections in Banach spaces and relations between approximation properties", "abstract": "A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the $\\pi$-property in terms of norms of compositions of projections is found."}
{"category": "Math", "title": "The representation category of any compact group is the bimodule category of a II_1 factor", "abstract": "We prove that given any compact group G, there exists a minimal action of G on a II_1 factor M such that the bimodule category of the fixed-point II_1 factor M^G is naturally equivalent with the representation category of G. In particular, all subfactors of M^G with finite Jones index can be described explicitly."}
{"category": "Math", "title": "Combinatorics of Tripartite Boundary Connections for Trees and Dimers", "abstract": "A grove is a spanning forest of a planar graph in which every component tree contains at least one of a special subset of vertices on the outer face called nodes. For the natural probability measure on groves, we compute various connection probabilities for the nodes in a random grove. In particular, for \"tripartite\" pairings of the nodes, the probability can be computed as a Pfaffian in the entries of the Dirichlet-to-Neumann matrix (discrete Hilbert transform) of the graph. These formulas generalize the determinant formulas given by Curtis, Ingerman, and Morrow, and by Fomin, for parallel pairings. These Pfaffian formulas are used to give exact expressions for reconstruction: reconstructing the conductances of a planar graph from boundary measurements. We prove similar theorems for the double-dimer model on bipartite planar graphs."}
{"category": "Math", "title": "On locally extremal functions on connected spaces", "abstract": "We construct an example of a real-valued continuous non-constant function $f$ defined on a connected complete metric space $X$ such that every point of $X$ is a point of local minimum or local maximum for $f$. The space $X$ is connected but fails to be separably connected."}
{"category": "Math", "title": "On normal stability for nonlinear parabolic equations", "abstract": "We show convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which is normally stable."}
{"category": "Math", "title": "Periodic elements of the free idempotent generated semigroup on a biordered set", "abstract": "We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup."}
{"category": "Math", "title": "The large sieve and random walks on left cosets of arithmetic groups", "abstract": "Applying E. Kowalski's recent generalization of the large sieve we prove that certain properties expected to be typical (irreducibility of the characteristic polynomial, absence of squares among the matrix coefficients...) are indeed verified by most (in a very explicit sense) of the elements of GL(n,A) with fixed determinant (where A is an intermediate ring between Z and Q that we specify) or by (special) orthogonal matrices with integral entries and fixed spinor norm."}
{"category": "Math", "title": "A Lax Formalism for the Elliptic Difference Painlev\\'e Equation", "abstract": "A Lax formalism for the elliptic Painlev\\'e equation is presented. The construction is based on the geometry of the curves on ${\\mathbb P}^1\\times{\\mathbb P}^1$ and described in terms of the point configurations."}
{"category": "Math", "title": "Bootstrapped Morawetz Estimates And Resonant Decomposition For Low Regularity Global Solutions Of Cubic NLS On R^{2}", "abstract": "We prove global well-posedness for the L^{2}-critical cubic defocusing nonlinear Schr\\\"odinger equation on R^{2} with data u_{0} \\in H^{s}(R^{2}) for s > {1/3}."}
{"category": "Math", "title": "Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups", "abstract": "We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic."}
{"category": "Math", "title": "Sur la lin\\'earisation des tissus", "abstract": "We give a simple analytic criterion which characterizes linearizable 1-codimensional webs. Then we give an invariant geometrical interpretation of it, in term of projective connection. We explain then how our approach allows to study linearization of more general objects than 1-codimensional webs. By way of illustration, we treat some explicit interesting examples."}
{"category": "Math", "title": "A non-existence theorem for Morse type holomorphic foliations of codimension one transverse to spheres", "abstract": "We prove that a Morse type codimension one holomorphic foliation is not transverse to a sphere in the complex affine space. Also we characterize the variety of contacts of a linear foliation with concentric spheres."}
{"category": "Math", "title": "Surfaces of bounded mean curvature in Riemannian manifolds", "abstract": "Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the inclusion maps converge in $C^0$ to a map from a surface of genus $g$ to $M$. We also show that, on passing to a further subsequence, the distance functions corresponding to pullback metrics converge to a pseudo-metric of fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface $F\\subset M$ together with bounds on the geometry of $M$ give an upper bound on the diameter of $F$. Our proof is modelled on Gromov's compactness theorem for $J$-holomorphic curves."}
{"category": "Math", "title": "Noncommutative Gelfand Duality and Applications I: The Existence of invariant subspaces", "abstract": "Gelfand duality between unital commutative C*-algebras and Compact Hausdorff spaces is extended to all unital C*-algebras, where the dual objects are what we call compact Hausdorff quantum spaces. We apply this result to obtain, a characterization of unitary groups of C*-algebras, and, for arbitrary bounded Hilbert space operators, (i) A spectral theorem cum continuous functional calculus, and (ii) A proof of the general Invariant Subspace Theorem. Also described is a nonabelian generalization of Pontryagin duality of abelian locally compact groups."}
{"category": "Math", "title": "An application of principal stratification to control for institutionalization at follow-up in studies of substance abuse treatment programs", "abstract": "Participants in longitudinal studies on the effects of drug treatment and criminal justice system interventions are at high risk for institutionalization (e.g., spending time in an environment where their freedom to use drugs, commit crimes, or engage in risky behavior may be circumscribed). Methods used for estimating treatment effects in the presence of institutionalization during follow-up can be highly sensitive to assumptions that are unlikely to be met in applications and thus likely to yield misleading inferences. In this paper we consider the use of principal stratification to control for institutionalization at follow-up. Principal stratification has been suggested for similar problems where outcomes are unobservable for samples of study participants because of dropout, death or other forms of censoring. The method identifies principal strata within which causal effects are well defined and potentially estimable. We extend the method of principal stratification to model institutionalization at follow-up and estimate the effect of residential substance abuse treatment versus outpatient services in a large scale study of adolescent substance abuse treatment programs. Additionally, we discuss practical issues in applying the principal stratification model to data. We show via simulation studies that the model can only recover true effects provided the data meet strenuous demands and that there must be caution taken when implementing principal stratification as a technique to control for post-treatment confounders such as institutionalization."}
{"category": "Math", "title": "A General formulation for standardization of rates as a method to control confounding by measured and unmeasured disease risk factors", "abstract": "Standardization, a common approach for controlling confounding in population-studies or data from disease registries, is defined to be a weighted average of stratum specific rates. Typically, discussions on the construction of a particular standardized rate regard the strata as fixed, and focus on the considerations that affect the specification of weights. Each year the data from the SEER cancer registries are analyzed using a weighting procedure referred to as ``direct standardization for age.'' To evaluate the performance of direct standardization, we define a general class of standardization operators. We regard a particular standardized rate to be the output of an operator and a given data set. Based on the functional form of the operators, we define a subclass of standardization operators that controls for confounding by measured risk factors. Using the fundamental disease probability paradigm for inference, we establish the conclusions that can be drawn from year-to-year contrasts of standardized rates produced by these operators in the presence of unmeasured cancer risk factors. These conclusions take the form of falsifying specific assumptions about the conditional probabilities of disease given all the risk factors (both measured and unmeasured), and the conditional probabilities of the unmeasured risk factors given the measured risk factors. We show the one-to-one correspondence between these falsifications and the inferences made from the contrasts of directly standardized rates reported each year in the Annual Report to the Nation on the Status of Cancer."}
{"category": "Math", "title": "Some Probabilistic and Statistical Properties of a Random Coefficient Autoregressive Model", "abstract": "A statistical inference for random coefficient first-order autoregressive model $[RCAR(1)]$ was investigated by P.M. ROBINSON (1978) in which the coefficients varying over individuals. In this paper we attempt to generalize this result to random coefficient autoregressive model of order $p$ $[RCAR(p)]$. The stationarity condition will derived for this model."}
{"category": "Math", "title": "Stochastic integrals and conditional full support", "abstract": "We present conditions that imply the conditional full support (CFS) property, introduced by Guasoni, R\\'asonyi, and Schachermayer [Ann. Appl. Probab., 18 (2008), pp. 491--520], for processes Z := H + K \\cdot W, where W is a Brownian motion, H is a continuous process, and processes H and K are either progressive or independent of W. Moreover, in the latter case under an additional assumption that K is of finite variation, we present conditions under which Z has CFS also when W is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS."}
{"category": "Math", "title": "On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative L\\'{e}vy processes", "abstract": "We consider the classical optimal dividend control problem which was proposed by de Finetti [Trans. XVth Internat. Congress Actuaries 2 (1957) 433--443]. Recently Avram, Palmowski and Pistorius [Ann. Appl. Probab. 17 (2007) 156--180] studied the case when the risk process is modeled by a general spectrally negative L\\'{e}vy process. We draw upon their results and give sufficient conditions under which the optimal strategy is of barrier type, thereby helping to explain the fact that this particular strategy is not optimal in general. As a consequence, we are able to extend considerably the class of processes for which the barrier strategy proves to be optimal."}
{"category": "Math", "title": "First Nonlinear Syzygies of Ideals Associated to Graphs", "abstract": "Consider an ideal $I\\subset K[x_1,..., x_n]$, with $K$ an arbitrary field, generated by monomials of degree two. Assuming that $I$ does not have a linear resolution, we determine the step $s$ of the minimal graded free resolution of $I$ where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree $s+3$, and we compute the corresponding graded Betti number $\\beta_{s,s+3}$. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1."}
{"category": "Math", "title": "Cyclic shifts of the van der Corput set", "abstract": "In [13], K. Roth showed that the expected value of the $L^2$ discrepancy of the cyclic shifts of the $N$ point van der Corput set is bounded by a constant multiple of $\\sqrt{\\log N}$, thus guaranteeing the existence of a shift with asymptotically minimal $L^2$ discrepancy, [11]. In the present paper, we construct a specific example of such a shift."}
{"category": "Math", "title": "Central Limit Theorem and the Bootstrap for U-Statistics of Strongly Mixing Data", "abstract": "The asymptotic normality of U-statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley to prove a new generalized covariance inequality similar to Yoshihara's. It follows from the Hoeffding-decomposition and this inequality that U-statistics of strongly mixing observations converge to a normal limit if the kernel of the U-statistic fulfills some moment and continuity conditions. The validity of the bootstrap for U-statistics has until now only been established in the case of iid data (see Bickel and Freedman). For mixing data, Politis and Romano proposed the circular block bootstrap, which leads to a consistent estimation of the sample mean's distribution. We extend these results to U-statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of U-statistics under absolute regularity and strong mixing. We also calculate a rate of convergence for the bootstrap variance estimator of a U-statistic and give some simulation results."}
{"category": "Math", "title": "Aggregation of autoregressive processes and long memory", "abstract": "We study the aggregation of AR processes and generalized Ornstein-Uhlenbeck (OU) processes. Mixture of spectral densities with random poles are the main tool. In this context, we apply our results for the aggregation of doubly stochastic interactives processes, see Dacunha-Castelle and Fermin (2006). Thus, we study the relationship between aggregation of autoregressive processes and long memory considering complex interaction structures. We precise a very interesting qualitative phenomena: how the long memory creation depends on the poles concentration near to the boundary of stability (measured in the Prokhorov sense). Our results extends the results given by Oppenheim and Viano (2004), and highlight the importance of the angular dispersion measure of poles in the appearance of the long memory."}
{"category": "Math", "title": "A Hypergraph Dictatorship Test with Perfect Completeness", "abstract": "A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan and serves as a key component in their unique games based $\\PCP$ construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all their low degree influences are $o(1).$ Their test makes $q\\geq3$ queries and has amortized query complexity $1+O(\\frac{\\log q}{q})$ but has an inherent loss of perfect completeness. In this paper we give an adaptive hypergraph dictatorship test that achieves both perfect completeness and amortized query complexity $1+O(\\frac{\\log q}{q})$."}
{"category": "Math", "title": "Enumeration of derangements with descents in prescribed positions", "abstract": "We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point $\\lambda$-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, if a permutation $\\pi$ is chosen uniformly among all permutations on $n$ elements, the events that $\\pi$ has descents in a set $S$ of positions, and that $\\pi$ is a derangement, are positively correlated."}
{"category": "Math", "title": "Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph", "abstract": "We investigate the least common multiple of all subdeterminants, lcmd(A x B), of a Kronecker product of matrices, of which one is an integral matrix A with two columns and the other is the incidence matrix of a complete graph with n vertices. We prove that this quantity is the least common multiple of lcmd(A) to the power n-1 and certain binomial functions of the entries of A."}
{"category": "Math", "title": "The \\infty eigenvalue problem and a problem of optimal transportation", "abstract": "The so-called eigenvalues and eigenfunctions of the infinite Laplacian $\\Delta_\\infty$ are defined through an asymptotic study of that of the usual $p$-Laplacian $\\Delta_p$, this brings to a characterization via a non-linear eigenvalue problem for a PDE satisfied in the viscosity sense. In this paper, we obtain an other characterization of the first eigenvalue via a problem of optimal transportation, and recover properties of the first eigenvalue and corresponding positive eigenfunctions."}
{"category": "Math", "title": "An Elementary Proof of Hawkes's Conjecture on Galton-Watson Trees", "abstract": "In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge function for the boundary of supercritical Galton-Watson trees under a certain assumption on the tail at the infinity of the total mass of the branching measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as other other precise results on fractal properties of the boundary of Galton-Watson trees. The goal of this paper is to provide an elementary proof of Hawkes's conjecture under a less restrictive assumption than in T. Watanabe's paper, by use of size-biased Galton-Watson trees introduced by Lyons, Pemantle and Peres in 1995."}
{"category": "Math", "title": "Optimum and equilibrium in a transport problem with queue effects", "abstract": "Consider a distribution of citizens in an urban area in which some services (supermarkets, post offices...) are present. Each citizen, in order to use a service, spends an amount of time which is due both to the travel time to the service and to the queue time waiting in the service. The choice of the service to be used is made by every citizen in order to be served more quickly. Two types of problems can be considered: a global optimization of the total time spent by the citizens of the whole city (we define a global optimum and we study it with techniques from optimal mass transportation) and an individual optimization, in which each citizen chooses the service trying to minimize just his own time expense (we define the concept of equilibrium and we study it with techniques from game theory). In this framework we are also able to exhibit two time-dependent strategies (based on the notions of prudence and memory respectively) which converge to the equilibrium."}
{"category": "Math", "title": "On right-angled Artin groups without surface subgroups", "abstract": "We study the class N of graphs, the right-angled Artin groups defined on which do not contain surface subgroups. We prove that a presumably smaller class N' is closed under amalgamating along complete subgraphs, and also under adding bisimplicial edges. It follows that chordal graphs and chordal bipartite graphs belong to N'."}
{"category": "Math", "title": "Moduli Spaces of Semistable Sheaves on Projective Deligne-Mumford Stacks", "abstract": "We introduce a notion of Gieseker stability for coherent sheaves on tame Deligne-Mumford stacks with projective moduli scheme and some chosen generating sheaf on the stack in the sense of Olsson and Starr \\cite{MR2007396}. We prove that this stability condition is open, and pure dimensional semistable sheaves form a bounded family. We explicitly construct the moduli stack of semistable sheaves as a finite type global quotient, and study the moduli scheme of stable sheaves and its natural compactification in the same spirit as the seminal paper of Simpson \\cite{MR1307297}. With this general machinery we are able to retrieve, as special cases, results of Lieblich \\cite{MR2309155} and Yoshioka \\cite{MR2306170} about moduli of twisted sheaves and parabolic stability introduced by Maruyama-Yokogawa in \\cite{MR1162674}."}
{"category": "Math", "title": "Fibrations and fundamental groups of Kaehler-Weyl manifolds", "abstract": "We extend the Siu--Beauville theorem to a certain class of compact Kaehler--Weyl manifolds, proving that they fiber holomorphically over hyperbolic Riemannian surfaces whenever they satisfy the necessary topological hypotheses. As applications we obtain restrictions on the fundamental groups of such Kaehler--Weyl manifolds, and show that in certain cases they are in fact Kaehler."}
{"category": "Math", "title": "Grothendieck Duality for Deligne-Mumford Stacks", "abstract": "We prove the existence of the dualizing functor for a separated morphism of algebraic stacks with affine diagonal; then we explicitly develop duality for compact Deligne-Mumford stacks focusing in particular on the morphism from a stack to its coarse moduli space and on representable morphisms. We explicitly compute the dualizing complex for a smooth stack over an algebraically closed field and prove that Serre duality holds for smooth compact Deligne-Mumford stacks in its usual form. We prove also that a proper Cohen-Macaulay stack has a dualizing sheaf and it is an invertible sheaf when it is Gorenstein. As an application of this general machinery we compute the dualizing sheaf of a tame nodal curve."}
{"category": "Math", "title": "A d-bar-theoretical proof of Hartogs' extension theorem on (n-1)-complete spaces", "abstract": "Let X be a connected normal complex space of dimension n>=2 which is (n-1)-complete, and let p: M -> X be a resolution of singularities. By use of Takegoshi's generalization of the Grauert-Riemenschneider vanishing theorem, we deduce H^1_{cpt}(M,O)=0, which in turn implies Hartogs' extension theorem on X by the d-bar-technique of Ehrenpreis."}
{"category": "Math", "title": "Quasi-convex sequences in the circle and the 3-adic integers", "abstract": "In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J_3 of 3-adic integers. These sequences are determined by an increasing sequences of integers. For an increasing sequence \\underline{a}=\\{a_n\\} of integers, put g_n=a_{n+1}-a_n. We prove that: (a) the set \\{0\\}\\cup\\{\\pm 3^{-(a_n+1)} : n\\in N\\} is quasi-convex in T if and only if a_0>0 and g_n>1 for every n\\in N; (b) the set \\{0\\}\\cup\\{\\pm 3^{a_n} : n\\in N\\} is quasi-convex in the group J_3 of 3-adic integers if and only if g_n>1 for every n\\in N. Moreover, we solve an open problem of Dikranjan and de Leo by providing a complete characterization of the sequences \\underline{a} such that \\{0\\}\\cup\\{\\pm 2^{-(a_n+1)} : n\\in N\\} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences \\underline{a} such that the set \\{0\\}\\cup\\{\\pm 2^{-(a_n+1)} : n\\in N\\} is quasi-convex in R."}
{"category": "Math", "title": "The Index of Hypoelliptic Operators on Foliated Manifolds", "abstract": "We present an index theorem for certain hypoelliptic differential operators on foliated manifolds. Our proof is a development of Alain Connes tangent groupoid proof of the Atiyah-Singer index theorem. The paper is largely self-contained."}
{"category": "Math", "title": "Homomorphisms of infinitely generated analytic sheaves", "abstract": "We prove that every homomorphism $\\mathcal{O}^E_\\zeta\\to\\mathcal{O}^F_\\zeta$, with $E$ and $F$ Banach spaces and $\\zeta\\in\\mathbb{C}^m$, is induced by a $\\mathop{\\mathrm{Hom}}(E,F)$-valued holomorphic germ, provided that $1\\leq m<\\infty$. A similar structure theorem is obtained for the homomorphisms of type $\\mathcal{O}^E_\\zeta\\to\\mathcal{S}_\\zeta$, where $\\mathcal{S}_\\zeta$ is a stalk of a coherent sheaf of positive $\\mathfrak{m}_\\zeta$-depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheaves which guarantees the sheaf to be equipped with a unique analytic structure in the sense of Lempert-Patyi."}
{"category": "Math", "title": "Differential posets and Smith normal forms", "abstract": "We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice YF studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young's lattice Y and the r-differential Cartesian products Y^r."}
{"category": "Math", "title": "Representations of Multiloop Algebras", "abstract": "We describe the finite-dimensional simple modules of all the (twisted and untwisted) multiloop algebras and classify them up to isomorphism."}
{"category": "Math", "title": "Contractibility and the Hadwiger Conjecture", "abstract": "Consider the following relaxation of the Hadwiger Conjecture: For each $t$ there exists $N_t$ such that every graph with no $K_t$-minor admits a vertex partition into $\\ceil{\\alpha t+\\beta}$ parts, such that each component of the subgraph induced by each part has at most $N_t$ vertices. The Hadwiger Conjecture corresponds to the case $\\alpha=1$, $\\beta=-1$ and $N_t=1$. Kawarabayashi and Mohar [\\emph{J. Combin. Theory Ser. B}, 2007] proved this relaxation with $\\alpha={31/2}$ and $\\beta=0$ (and $N_t$ a huge function of $t$). This paper proves this relaxation with $\\alpha={7/2}$ and $\\beta=-{3/2}$. The main ingredients in the proof are: (1) a list colouring argument due to Kawarabayashi and Mohar, (2) a recent result of Norine and Thomas that says that every sufficiently large $(t+1)$-connected graph contains a $K_t$-minor, and (3) a new sufficient condition for a graph to have a set of edges whose contraction increases the connectivity."}
{"category": "Math", "title": "Crepant resolution conjecture in all genera for type A singularities", "abstract": "We prove an all genera version of the Crepant Resolution Conjecture of Ruan and Bryan-Graber for type A surface singularities. We are based on a method that explicitly computes Hurwitz-Hodge integrals described in an earlier paper and some recent results by Liu-Xu for some intersection numbers on the Deligne-Mumford moduli spaces. We also generalize our results to some three-dimensional orbifolds."}
{"category": "Math", "title": "The p-adic valuation of k-central binomial coefficients", "abstract": "The coefficients c(n,k) defined by (1-k^2x)^(-1/k) = sum c(n,k) x^n reduce to the central binomial coefficients for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study their divisibility properties."}
{"category": "Math", "title": "Brownian moving averages have conditional full support", "abstract": "We prove that any Brownian moving average \\[X_t=\\int_{-\\infty}^t\\bigl(f(s-t)-f(s)\\bigr) dB_s,\\qquad t\\ge0,\\] satisfies the conditional full support condition introduced by Guasoni, R\\'{a}sonyi and Schachermayer [Ann. Appl. Probab. 18 (2008) 491--520]."}
{"category": "Math", "title": "Endpoint for the div-curl lemma in Hardy spaces", "abstract": "We give a div-curl type lemma for the wedge product of closed differential forms on R^n when they have coefficients respectively in a Hardy space and L^infinity or BMO. In this last case, the wedge product belongs to an appropriate Hardy-Orlicz space."}
{"category": "Math", "title": "On the laws of first hitting times of points for one-dimensional symmetric stable L\\'evy processes", "abstract": "Several aspects of the laws of first hitting times of points are investigated for one-dimensional symmetric stable L\\'evy processes. It\\^o's excursion theory plays a key role in this study."}
{"category": "Math", "title": "A connection between extreme value theory and long time approximation of SDE's", "abstract": "We consider a sequence $(\\xi_n)_{n\\ge1}$ of $i.i.d.$ random values living in the domain of attraction of an extreme value distribution. For such sequence, there exists $(a_n)$ and $(b_n)$, with $a_n>0$ and $b_n\\in\\ER$ for every $n\\ge 1$, such that the sequence $(X_n)$ defined by $X_n=(\\max(\\xi_1,...,\\xi_n)-b_n)/a_n$ converges in distribution to a non degenerated distribution. In this paper, we show that $(X_n)$ can be viewed as an Euler scheme with decreasing step of an ergodic Markov process solution to a SDE with jumps and we derive a functional limit theorem for the sequence $(X_n)$ from some methods used in the long time numerical approximation of ergodic SDE's."}
{"category": "Math", "title": "The shifted plactic monoid", "abstract": "We introduce a shifted analog of the plactic monoid of Lascoux and Sch\\\"utzenberger, the \\emph{shifted plactic monoid}. It can be defined in two different ways: via the \\emph{shifted Knuth relations}, or using Haiman's mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur $P$-function; a shifted counterpart of the Lascoux-Sch\\\"utzenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more."}
{"category": "Math", "title": "Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups", "abstract": "We consider stochastic equations in Hilbert spaces with singular drift in the framework of [Da Prato, R\\\"ockner, PTRF 2002]. We prove a Harnack inequality (in the sense of [Wang, PTRF 1997]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure $\\mu$ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure $\\mu$ for non-continuous drifts."}
{"category": "Math", "title": "Functoriality of the BGG Category O", "abstract": "This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category O. We study functorial properties of O across various setups. The first setup is over a skew group ring, involving a finite group $\\Gamma$ acting on a regular triangular algebra $A$. We develop Clifford theory for $A \\rtimes \\Gamma$, and obtain results on block decomposition, complete reducibility, and enough projectives. O is shown to be a highest weight category when $A$ satisfies one of the \"Conditions (S)\"; the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module. Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories O; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup - and which yield information about O."}
{"category": "Math", "title": "On universal estimates for binary renewal processes", "abstract": "A binary renewal process is a stochastic process $\\{X_n\\}$ taking values in $\\{0,1\\}$ where the lengths of the runs of 1's between successive zeros are independent. After observing ${X_0,X_1,...,X_n}$ one would like to predict the future behavior, and the problem of universal estimators is to do so without any prior knowledge of the distribution. We prove a variety of results of this type, including universal estimates for the expected time to renewal as well as estimates for the conditional distribution of the time to renewal. Some of our results require a moment condition on the time to renewal and we show by an explicit construction how some moment condition is necessary."}
{"category": "Math", "title": "Axiomatic framework for the BGG Category O", "abstract": "The main goal of this paper is to show that a wide variety of infinite-dimensional algebras all share a common structure, including a triangular decomposition and a theory of weights. This structure allows us to define and study the BGG Category O, generalizing previous definitions of it. Having presented our axiomatic framework, we present sufficient conditions that guarantee finite length, enough projectives, and a block decomposition into highest weight categories. The framework is strictly more general than the usual theory of O; this is needed to accommodate (quantized or higher rank) infinitesimal Hecke algebras, in addition to semisimple Lie algebras and their quantum groups. We then present numerous examples, two families of which are studied in detail. These are quantum groups defined using not necessarily the root or weight lattices (for these, we study the center and central characters), and infinitesimal Hecke algebras."}
{"category": "Math", "title": "On a length preserving curve flow", "abstract": "In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C-infinity topology as t goes to infinity."}
{"category": "Math", "title": "A characterisation of compact, fragmentable linear orders", "abstract": "We give a characterisation of fragmentable, compact linearly order spaces. In particular, we show that if $K$ is a compact, fragmentable, linearly ordered space then $K$ is a Radon-Nikod\\'{y}m compact. In addition, we obtain some corollaries in topology and renorming theory."}
{"category": "Math", "title": "Growth Estimates for a Class of Subharmonic Functions in a Half Space", "abstract": "A class of subharmonic functions represented by the modified kernels are proved to have the growth estimates u(x) =o(x_{n}^{1-alpha}|x|^{m+alpha})at infinity in the upper half space of Rn, which generalizes the growth properties of analytic functions and harmonic functions."}
{"category": "Math", "title": "A Central Limit Theorem, and related results, for a two-color randomly reinforced urn", "abstract": "We prove a Central Limit Theorem for the sequence of random compositions of a two-color randomly reinforced urn. As a consequence, we are able to show that the distribution of the urn limit composition has no point masses."}
{"category": "Math", "title": "On transfer inequalities in Diophantine approximation, II", "abstract": "We refine Khintchine Transference Principle which relates the measure of simultaneous rational approximation of an $n$ real numbers with the measure of linear independence of these $n$ numbers. Khintchine's inequalities are known to be optimal. However, they may be sharpened by taking into account two further uniform exponents."}
{"category": "Math", "title": "On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice", "abstract": "This paper deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames, over the very lattice. And we also give axiomatizations for the case of a finite MV chain but this time without canonical constants."}
{"category": "Math", "title": "Homology stability for the special linear group of a field and Milnor-Witt K-theory", "abstract": "Let F be a field of characteristic zero and let f(t,n) be the stabilization homomorphism from the n-th integral homology of SL(t,F) to the n-th homology of SL(t+1,F). We prove the following results: For all n, f(t,n) is an isomorphism if t is at least n+1, and is surjective for t=n, confirming a conjecture of C-H. Sah. Furthermore if n is odd, then f(n,n) is an isomorphism and when n is even the kernel of f(n,n) is the (n+1)st power of the fundamental ideal of the Witt Ring of the field.. If n is even, then the cokernel of f(n-1,n) is naturally isomorphic to the n-th Milnor-Witt K-group of F, MWK(n,F) and when n>2 is odd the cokernel of f(n-1,n) is the square of the nth Milnor K-group of F."}
{"category": "Math", "title": "On the entropy and letter frequencies of powerfree words", "abstract": "We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider their empirical distribution obtained by an enumeration of binary cubefree words up to length 80."}
{"category": "Math", "title": "Asymptotic Behaviour for a Class of Subharmonic Functions in a Half Space", "abstract": "A class of subharmonic functions are proved to have the growth estimates $u(x)= o(x_n^{1-\\frac{\\alpha}{p}}|x|^{\\frac{\\gamma}{p}+\\frac{n-1}{q}-n+\\frac{\\alpha}{p}})$ at infinity in the upper half space of ${\\bf R}^{n}$, which generalizes the growth properties of analytic functions and harmonic functions."}
{"category": "Math", "title": "Growth Estimates for a Class of Subharmonic Functions in a Half Plane", "abstract": "A class of subharmonic functions represented by the modified kernels are proved to have the growth estimates $u(z)= o(y^{1-\\alpha}|z|^{m+\\alpha})$ at infinity in the upper half plane ${\\bf C}_{+}$, which generalizes the growth properties of analytic functions and harmonic functions."}
{"category": "Math", "title": "Maximal rationally connected fibrations and movable curves", "abstract": "A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle."}
{"category": "Math", "title": "Morphisms fixing words associated with exchange of three intervals", "abstract": "We consider words coding exchange of three intervals with permutation (3,2,1), here called 3iet words. Recently, a characterization of substitution invariant 3iet words was provided. We study the opposite question: what are the morphisms fixing a 3iet word? We reveal a narrow connection of such morphisms and morphisms fixing Sturmian words using the new notion of amicability."}
{"category": "Math", "title": "Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping", "abstract": "In this paper we show existence of finite energy solutions for the Cauchy problem associated with a semilinear wave equation with interior damping and supercritical source terms. The main contribution consists in dealing with super-supercritical source terms (terms of the order of $|u|^p$ with $p\\geq 5$ in $n=3$ dimensions), an open and highly recognized problem in the literature on nonlinear wave equations."}
{"category": "Math", "title": "On the existence of star products on quotient spaces of linear Hamiltonian torus actions", "abstract": "We discuss BFV deformation quantization of singular symplectic quotient spaces in the special case of linear Hamiltonian torus actions. In particular, we show that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of Arms, Gotay and Jennings for linear Hamiltonian torus actions. It follows that reduced spaces of such actions admit continuous star products."}
{"category": "Math", "title": "Lois pr\\'e-Lie en interaction", "abstract": "D. Calaque, K. Ebrahimi-Fard and D. Manchon have recently defined a Hopf algebra by introducing a new coproduct on a commutative algebra of rooted forests. The space of primitive elements of the graded dual is endowed with a left pre-Lie product defined in terms of insertion of a tree inside another. In this work we prove a ``derivation'' relation between this pre-Lie structure and the left pre-Lie product defined by grafting."}
{"category": "Math", "title": "A survey of combinatorial aspects in the topology of complex hyperplane arrangements", "abstract": "We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras, higher homotopy groups, cohomology rings, twisted homology with rank 1 complex coefficients, and Milnor fibers."}
{"category": "Math", "title": "Regularizations of residue currents", "abstract": "Under assumptions about complete intersection, we prove that Coleff-Herrera type currents satisfy a robust calculus in the sense that natural regularizations of such currents can be multiplied to yield regularizations of the Coleff-Herrera product of the currents."}
{"category": "Math", "title": "Higher order energy decay rates for damped wave equations with variable coefficients", "abstract": "Under appropriate assumptions the energy of wave equations with damping and variable coefficients $c(x)u_{tt}-\\hbox{div}(b(x)\\nabla u)+a(x)u_t =h(x)$ has been shown to decay. Determining the rate of decay for the higher order energies involving the $k$th order spatial and time derivatives has been an open problem with the exception of some sparse results obtained for $k=1,2,3$. We establish estimates that optimally relate the higher order energies with the first order energy by carefully analyzing the effects of linear damping. The results concern weighted (in time) and also pointwise (in time) energy decay estimates. We also obtain $L^\\infty$ estimates for the solution $u$. As an application we compute explicit decay rates for all energies which involve the dimension $n$ and the bounds for the coefficients $a(x)$ and $b(x)$ in the case $c (x)=1$ and $h(x)=0.$"}
{"category": "Math", "title": "An overview of arithmetic motivic integration", "abstract": "This is an attempt at an elementary exposition, with examples, of the theory of motivic integration developed by R. Cluckers and F. Loeser, with the view towards applications in representation theory of p-adic groups."}
{"category": "Math", "title": "P-values for high-dimensional regression", "abstract": "Assigning significance in high-dimensional regression is challenging. Most computationally efficient selection algorithms cannot guard against inclusion of noise variables. Asymptotically valid p-values are not available. An exception is a recent proposal by Wasserman and Roeder (2008) which splits the data into two parts. The number of variables is then reduced to a manageable size using the first split, while classical variable selection techniques can be applied to the remaining variables, using the data from the second split. This yields asymptotic error control under minimal conditions. It involves, however, a one-time random split of the data. Results are sensitive to this arbitrary choice: it amounts to a `p-value lottery' and makes it difficult to reproduce results. Here, we show that inference across multiple random splits can be aggregated, while keeping asymptotic control over the inclusion of noise variables. We show that the resulting p-values can be used for control of both family-wise error (FWER) and false discovery rate (FDR). In addition, the proposed aggregation is shown to improve power while reducing the number of falsely selected variables substantially."}
{"category": "Math", "title": "Variation and Rough Path Properties of Local Times of L\\'evy Processes", "abstract": "In this paper, we will prove that the local time of a L\\'evy process is of finite $p$-variation in the space variable in the classical sense, a.s. for any $p>2$, $t\\geq 0$, if the L\\'evy measure satisfies $\\int_{R\\setminus \\{0\\}}(|y|^{3\\over 2}\\wedge 1)n(dy)<\\infty$, and is a rough path of roughness $p$ a.s. for any $2<p<3$ under a slightly stronger condition for the L\\'evy measure. Then for any function $g$ of finite $q$-variation ($1\\leq q <3$), we establish the integral $\\int_{-\\infty}^{\\infty}g(x)dL_t^x$ as a Young integral when $1\\leq q<2$ and a Lyons' rough path integral when $2\\leq q<3$. We therefore apply these path integrals to extend the Tanaka-Meyer formula for a continuous function $f$ if $\\nabla ^-f$ exists and is of finite $q$-variation when $1\\leq q<3$, for both continuous semi-martingales and a class of L\\'evy processes."}
{"category": "Math", "title": "Master Equation and Perturbative Chern-Simons theory", "abstract": "We extend the Chern-Simons perturbative invariant of Axelrod and Singer to non-acyclic connections. We construct a solution of the quantum master equation on the space of functions on the cohomology of the connection. We prove that this solution is well defined up to master homotopy. We discuss also invariants of links."}
{"category": "Math", "title": "Geometry of canonical self-similar tilings", "abstract": "We give several different geometric characterizations of the situation in which the parallel set $F_\\epsilon$ of a self-similar set $F$ can be described by the inner $\\epsilon$-parallel set $T_{-\\epsilon}$ of the associated canonical tiling $\\mathcal T$, in the sense of \\cite{SST}. For example, $F_\\epsilon=T_{-\\epsilon} \\cup C_\\epsilon$ if and only if the boundary of the convex hull $C$ of $F$ is a subset of $F$, or if the boundary of $E$, the unbounded portion of the complement of $F$, is the boundary of a convex set. In the characterized situation, the tiling allows one to obtain a tube formula for $F$, i.e., an expression for the volume of $F_\\epsilon$ as a function of $\\epsilon$. On the way, we clarify some geometric properties of canonical tilings. Motivated by the search for tube formulas, we give a generalization of the tiling construction which applies to all self-affine sets $F$ having empty interior and satisfying the open set condition. We also characterize the relation between the parallel sets of $F$ and these tilings."}
{"category": "Math", "title": "Test Functions in Constrained Interpolation", "abstract": "We give a set of test functions for the interpolation problem on $H_1^\\infty$ , the constrained interpolation problem studied by Davidson, Paulsen, Raghupathi and Singh. We show that this set of test functions is minimal."}
{"category": "Math", "title": "Splice diagram determining singularity links and universal abelian covers", "abstract": "To a rational homology sphere graph manifold one can associate a weighted tree invariant called splice diagram. In this article we prove a sufficient numerical condition on the splice diagram for a graph manifold to be a singularity link. We also show that if two manifolds have the same splice diagram, then their universal abelian covers are homeomorphic. To prove the last theorem we have to generalize our notions to orbifolds."}
{"category": "Math", "title": "Plumbers' knots", "abstract": "We construct a new type of geometric knot theory, plumbers' knots, and solve the problems of distinguishing and enumerating such knots at a fixed level of complexity. (v2) Minor edits, added theorem 3.18. (v3) Substantial revisions, essentially completely rewritten in places."}
{"category": "Math", "title": "Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space", "abstract": "Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space."}
{"category": "Math", "title": "Complete intersection points on general surfaces in $\\PP^3$", "abstract": "In this paper we consider the existence of complete intersection points of type $(a,b,c)$, on the generic degree $d$ surface of $\\PP^3$. For any choice of $a, b, c$ we resolve the existence question asymptotically, i.e. for all $d \\gg 0$. For small values of $a, b, c$ we resolve the existence problem completely."}
{"category": "Math", "title": "Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds", "abstract": "We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples. The main ingredient of our proof is the effective equidistribution of expanding horospheres on geometrically finite hyperbolic 3-manifolds under the assumption that the critical exponent of its fundamental group exceeds one."}
{"category": "Math", "title": "The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells", "abstract": "Using the notion of Gamma-convergence, we discuss the limiting behavior of the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like $h^\\beta$ with $2<\\beta<4$. We establish that, for the given scaling regime, the limiting theory reduces to the linear pure bending. Two major ingredients of the proofs are: the density of smooth infinitesimal isometries in the space of $W^{2,2}$ first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces."}
{"category": "Math", "title": "Asymptotic analysis of a family of polynomials associated with the inverse error function", "abstract": "We analyze the sequence of polynomials defined by the differential-difference equation $P_{n+1}(x)=P_{n}^{\\prime}(x)+x(n+1)P_{n}(x)$ asymptotically as $n\\to\\infty$. The polynomials $P_{n}(x)$ arise in the computation of higher derivatives of the inverse error function $\\operatorname{inverf}(x)$. We use singularity analysis and discrete versions of the WKB and ray methods and give numerical results showing the accuracy of our formulas."}
{"category": "Math", "title": "On the cohomology of a simple normal crossings divisor", "abstract": "We establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) $D$ in terms of the simplicial cohomologies of the dual complex $\\Delta(D)$ with coefficients in a presheaf of vector spaces. This presheaf consists precisely of the corresponding cohomology data on the components of $D$ and on their intersections. We use this formula to give a Hodge decomposition for SNC divisors and investigate the toric setting. We also conjecture the existence of such a formula for effective non-reduced divisors with SNC support, and show that this would imply the vanishing of the higher simplicial cohomologies of the dual complex associated to a resolution of an isolated rational singularity."}
{"category": "Math", "title": "Solutions of Polynomial Systems Derived from the Steady Cavity Flow Problem", "abstract": "We propose a general algorithm to enumerate all solutions of a zero-dimensional polynomial system with respect to a given cost function. The algorithm is developed and is used to study a polynomial system obtained by discretizing the steady cavity flow problem in two dimensions. The key technique on which our algorithm is based is to solve polynomial optimization problems via sparse semidefinite programming relaxations (SDPR), which has been adopted successfully to solve reaction-diffusion boundary value problems recently. The cost function to be minimized is derived from discretizing the fluid's kinetic energy. The enumeration algorithm's solutions are shown to converge to the minimal kinetic energy solutions for SDPR of increasing order. We demonstrate the algorithm with SDPR of first and second order on polynomial systems for different scenarios of the cavity flow problem and succeed in deriving the $k$ smallest kinetic energy solutions. The question whether these solutions converge to solutions of the steady cavity flow problem is discussed, and we pose a conjecture for the minimal energy solution for increasing Reynolds number."}
{"category": "Math", "title": "The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture", "abstract": "Bennett, Carbery, and Tao formulated an n-linear analogue of the Kakeya conjecture in R^n. They proved the conjecture except for the endpoint case. We prove the endpoint case."}
{"category": "Math", "title": "On the tunnel number and the Morse-Novikov number of knots", "abstract": "We prove that the Morse-Novikov number of a link L in a 3-sphere is less than or equal to twice the tunnel number of L."}
{"category": "Math", "title": "Solving a characteristic Cauchy problem", "abstract": "In this paper we give a meaning to the nonlinear characteristic Cauchy problem for the Wave Equation in base form by replacing it by a family of non-characteristic problems in an appropriate algebra of generalized functions. We prove existence of a solution and we precise how it depends on the choice made. We also check that in the classical case (non-characteristic) our new solution coincides with the classical one."}
{"category": "Math", "title": "Minoration du rang des courbes elliptiques sur les corps de classes de Hilbert", "abstract": "Soit $E/\\BmQ$ une courbe elliptique. Soit $D<0$ un discriminant fondamental suffisamment grand. Si $E(\\bar{\\BmQ})$ contient des points de Heegner de discriminant $D$, ces points engendrent un sous-groupe dont le rang est sup\\'erieur \\`a $\\pabs{D}^{0.0009}$. Ce r\\'esultat est en accord avec la conjecture de Birch et Swinnerton-Dyer. --- Let $E/\\BmQ$ be an elliptic curve. Let $D<0$ be a sufficiently large fundamental discriminant. If $E(\\bar{\\BmQ})$ contains Heegner points of discriminant $D$, these points generate a subgroup of rank greater than $\\pabs{D}^{0.0009}$. This result is in agreement with the conjecture of Birch and Swinnerton-Dyer."}
{"category": "Math", "title": "Oriented bivariant theories, I", "abstract": "In 1981 W. Fulton and R. MacPherson introduced the notion of bivariant theory (BT), which is a sophisticated unification of covariant theories and contravariant theories. This is for the study of singular spaces. In 2001 M. Levine and F. Morel introduced the notion of algebraic cobordism, which is a universal oriented Borel-Moore functor with products (OBMF) of geometric type, in an attempt to understand better V. Voevodsky's (higher) algebraic cobordism. In this paper we introduce a notion of oriented bivariant theory (OBT), a special case of which is nothing but the oriented Borel-Moore functor with products. The present paper is a first one of the series to try to understand Levine-Morel's algebraic cobordism from a bivariant-theoretical viewpoint, and its first step is to introduce OBT as a unification of BT and OBMF."}
{"category": "Math", "title": "Supersymmetric field theories and cohomology", "abstract": "This is the Ph.D. dissertation of the author. The project has been motivated by the conjecture that the Hopkins-Miller tmf spectrum can be described in terms of `spaces' of conformal field theories. In this dissertation, spaces of field theories are constructed as classifying spaces of categories whose objects are certain types of field theories. If such a category has a symmetric monoidal structure and its components form a group, by work of Segal, its classifying space is an infinite loop space and defines a cohomology theory. This has been carried out for two classes of field theories: (i) For each integer n, there is a category SEFT_n whose objects are the Stolz-Teichner (1|1)-dimensional super Euclidean field theories of degree n. It is proved that the classifying space |SEFT_n| represents degree-n K or KO cohomology, depending on the coefficients of the field theories. (ii) For each integer n, there is a category AFT_n whose objects are a kind of (2|1)-dimensional field theories called `annular field theories,' defined using supergeometric versions of circles and annuli only. It is proved that the classifying space |AFT_n| represents the degree-n elliptic cohomology associated with the Tate curve. To the author's knowledge, this is the first time the definitions of low-dimensional supersymmetric field theories are given in full detail."}
{"category": "Math", "title": "Products of longitudinal pseudodifferential operators on flag varieties", "abstract": "Associated to each set $S$ of simple roots for $SL(n,\\mathbb{C})$ is an equivariant fibration $X\\to X_S$ of the space $X$ of complete flags of $\\mathbb{C}^n$. To each such fibration we associate an algebra $J_S$ of operators on $L^2(X)$ which contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. These form a lattice of operator ideals whose common intersection is the compact operators. As a consequence, the product of fibrewise smoothing operators (for instance) along the fibres of two such fibrations, $X\\to X_S$ and $X\\to X_T$, is a compact operator if $S\\cup T$ is the full set of simple roots. The construction uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as `essential orthogonality of subrepresentations'."}
{"category": "Math", "title": "Homological Stability among Moduli Spaces of Holomorphic Curves in Complex Projective Space", "abstract": "The primary goal of this paper is to find a homotopy theoretic approximation to moduli spaces of holomorphic maps Riemann surfaces into complex projective space. There is a similar treatment of a partial compactification of these moduli spaces of consisting of irreducible stable maps in the sense of Gromov-Witten theory. The arguments follow those from a paper of G. Segal on the topology of the space of rational functions."}
{"category": "Math", "title": "Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison", "abstract": "It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the so-called Mokobodski condition. As for uniqueness of solutions, this holds under mild integrability conditions. However, for practical purposes, existence and uniqueness are not enough. In order to further develop these results in Markovian set-ups, one also needs a (simply or doubly) reflected BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and one further needs a suitable comparison theorem. In this paper, we derive such estimates and comparison results. In the last section, applicability of the results is illustrated with a pricing problem in finance."}
{"category": "Math", "title": "Horizontal normal map on the Heisenberg group", "abstract": "We investigate the notion of H-subdifferential and H-normal map of a function on the Heisenberg group, based on its sub-Riemannian structure. In particular, a characterization of the convexity of a function is given via the nonemptiness of the H-subdifferential at every point."}
{"category": "Math", "title": "Geometric Cobordism Categories", "abstract": "In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex structures along with a holomorphic map to a target complex manifold. A general notion of \"geometric structure\" is defined using sheaf theoretic constructions. Our main theorem is the identification of the homotopy type of such cobordism categories in terms of certain Thom spectra. This extends work of Galatius-Madsen-Tillmann-Weiss who identify the homotopy type of cobordism categories of manifolds with fiberwise structures on their tangent bundles. Interpretations of the main theorem are discussed which have relevance to topological field theories, moduli spaces of geometric structures, and h-principles. Applications of the main theorem to various examples of interest in geometry, particularly holomorphic curves, are elaborated upon."}
{"category": "Math", "title": "High-dimensional stochastic optimization with the generalized Dantzig estimator", "abstract": "We propose a generalized version of the Dantzig selector. We show that it satisfies sparsity oracle inequalities in prediction and estimation. We consider then the particular case of high-dimensional linear regression model selection with the Huber loss function. In this case we derive the sup-norm convergence rate and the sign concentration property of the Dantzig estimators under a mutual coherence assumption on the dictionary."}
{"category": "Math", "title": "Spaces of operator-valued functions measurable with respect to the strong operator topology", "abstract": "Let $X$ and $Y$ be Banach spaces and $(\\Omega,\\Sigma,\\mu)$ a finite measure space. In this note we introduce the space $L^p[\\mu;L(X,Y)]$ consisting of all (equivalence classes of) functions $\\Phi:\\Omega \\mapsto L(X,Y)$ such that $\\omega \\mapsto \\Phi(\\omega)x$ is strongly $\\mu$-measurable for all $x\\in X$ and $\\omega \\mapsto \\Phi(\\omega)f(\\omega)$ belongs to $L^1(\\mu;Y)$ for all $f\\in L^{p'}(\\mu;X)$, $1/p+1/p'=1$. We show that functions in $L^p[\\mu;\\L(X,Y)]$ define operator-valued measures with bounded $p$-variation and use these spaces to obtain an isometric characterization of the space of all $L(X,Y)$-valued multipliers acting boundedly from $L^p(\\mu;X)$ into $L^q(\\mu;Y)$, $1\\le q< p<\\infty$."}
{"category": "Math", "title": "On the SU(2,1) representation space of the Brieskorn homology spheres", "abstract": "In this paper, we give a parameterization of the SU(2,1) representation space of the Brieskorn homology spheres using the trace coordinates. As applications, we give an example which shows that the orbifold Toledo invariant in \\cite{krebs} does not distinguish the connected components of the PU(2,1) representation space."}
{"category": "Math", "title": "Configuration of nilpotent groups and isomorphism", "abstract": "The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if $G_1$ and $G_2$ have the same configuration sets and $H_1$ is a normal subgroup of $G_1$ with abelian quotient, then there is a normal subgroup $H_2$ of $G_2$ such that $\\frac{G_1}{H_1}\\cong\\frac{G_2}{H_2}.$ Also configuration of FC-groups and isomorphism is studied."}
{"category": "Math", "title": "General branching processes in discrete time as random trees", "abstract": "The simple Galton--Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of the family tree. This viewpoint has led to new insights and a revival of classical theory. We show how a similar reinterpretation can shed new light on the more interesting forms of branching processes that allow repeated bearings and, thus, overlapping generations. In particular, we use the stable pedigree law to give a transparent description of a size-biased version of general branching processes in discrete time. This allows us to analyze the $x\\log x$ condition for exponential growth of supercritical general processes as well as relation between simple Galton--Watson and more general branching processes."}
{"category": "Math", "title": "Lower Order Terms for the One-Level Density of Elliptic Curve L-Functions", "abstract": "It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve $L$-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one- and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the L-functions ratios conjectures to calculate the 1-level density for the family of even quadratic twists of an elliptic curve L-function for large but finite conductor. This gives a formula for the leading and lower order terms up to an error term that is conjectured to be significantly smaller. The lower order terms explain many of the features of the zero statistics for relatively small conductor and model the very slow convergence to the infinite conductor limit. However, our main observation is that they do not capture the behaviour of zeros in the important region very close to the critical point and so do not explain Miller's discrepancy. This therefore implies that a more accurate model for statistics near to this point needs to be developed."}
{"category": "Math", "title": "Alexander-equivalent Zariski pairs of irreducible sextics", "abstract": "The existence of Alexander-equivalent Zariski pairs dealing with irreducible curves of degree 6 was proved by A. Degtyarev. However, up to now, no explicit example of such a pair was available (only the existence was known). In this paper, we construct the first concrete example."}
{"category": "Math", "title": "Bifurcation of critical periods from Pleshkan's isochrones", "abstract": "Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities $\\mathscr C_3.$ In this paper we prove that if we perturb any of these isochrones inside $\\mathscr C_3,$ then at most two critical periods bifurcate from its period annulus. Moreover we show that, for each $k=0,1,2,$ there are perturbations giving rise to exactly $k$ critical periods. As a byproduct, we obtain a partial result for the analogous problem in the family of quadratic centers $\\mathscr C_2.$ Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in $\\mathscr C_2.$ We prove that if we perturb three of them inside $\\mathscr C_2,$ then at most one critical period bifurcates from its period annulus. In addition, for each $k=0,1,$ we show that there are perturbations giving rise to exactly $k$ critical periods. The quadratic isochronous center that we do not consider displays some peculiarities that are discussed at the end of the paper."}
{"category": "Math", "title": "Coxeter Elements and Root Bases", "abstract": "Let g be a Lie algebra of type A,D,E with fixed Cartan subalgebra h, root system R and Weyl group W. We show that a choice of Coxeter element C gives a root basis for g. Moreover we show that this root basis gives a purely combinatorial construction of g, where root vectors correspond to vertices of a certain quiver $Gammahat$, and show that with respect to this basis the structure constants of the Lie bracket are given by paths in $Gammahat$. This construction is then related to the constructions of Ringel and Peng and Xiao."}
{"category": "Math", "title": "Transformations birationnelles de petit degr\\'e", "abstract": "Since the end of the XIXth century, we know that each birational map of the complex projective plane is the product of a finite number of quadratic birational maps of the projective plane; this motivates our work which essentially deals with these quadratic maps. We establish algebraic properties such as the classification of one parameter groups of quadratic birational maps or the smoothness of the set of quadratic birational maps in the set of rational maps. We prove that a finite number of generic quadratic birational maps generates a free group. We show that if f is a quadratic birational map or an automorphism of the projective plane, the normal subgroup generated by f is the full group of birational maps of the projective plane, which implies that this group is perfect. We study some dynamical properties: following an idea of Guillot, we translate some invariants for foliations in our context, in particular we obtain that if two generic quadratic birational maps are birationally conjugated, then they are conjugated by an automorphism of the projective plane. We are also interested in the presence of \"invariant objects\": curves, foliations, fibrations. Then follows a more experimental part: we draw orbits of quadratic birational maps with real coefficients and sets analogous to Julia sets for polynomials of one variable. We study birational maps of degree 3 and, by considering the different possible configurations of the exceptional curves, we give the \"classification\" of these maps. We can deduce from this that the set of the birational maps of degree 3 exactly is irreducible, in fact rationally connected."}
{"category": "Math", "title": "A continuous semigroup of notions of independence between the classical and the free one", "abstract": "In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non-commutative random variables. These notions are related to the liberation process introduced by D. Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence."}
{"category": "Math", "title": "On the Riesz Basis Property of the Eigen- and Associated Functions of Periodic and Antiperiodic Sturm-Liouville Problems", "abstract": "The paper deals with the Sturm-Liouville operator $$ Ly=-y^{\\prime\\prime}+q(x)y,\\qquad x\\in\\lbrack0,1], $$ generated in the space $L_{2}=L_{2}[0,1]$ by periodic or antiperiodic boundary conditions. Several theorems on Riesz basis property of the root functions of the operator $L$ are proved. One of the main results is the following. \\textsl{Let $q$ belong to Sobolev space $W_{1}^{p}[0,1]$ with some integer $p\\geq0$ and satisfy the conditions $q^{(k)}(0)=q^{(k)}(1)=0$ for $0\\leq k\\leq s-1$, where s}$\\leq p.$ \\textsl{Let the functions $Q$ and $S$ be defined by the equalities $Q(x)=\\int_{0}^{x}q(t) dt, S(x)=Q^{2}(x)$ and let $q_{n}%, Q_{n},S_{n}$ be the Fourier coefficients of $q,Q,S$ with respect to the trigonometric system $\\{e^{2\\pi inx}\\}_{-\\infty}^{\\infty}$. Assume that the sequence $q_{2n}-S_{2n}+2Q_{0}Q_{2n}$ decreases not faster than the powers $n^{-s-2}$. Then the system of eigen and associated functions of the operator $L$ generated by periodic boundary conditions forms a Riesz basis in the space $L_{2}[0,1]$ (provided that the eigenfunctions are normalized) if and only if the condition $$ q_{2n}-S_{2n}+Q_{0}Q_{2n}\\asymp q_{-2n}-S_{-2n}+2Q_{0}Q_{-2n},\\quad n>1, $$ holds."}
{"category": "Math", "title": "F_p-repr\\'esentations semi-stables", "abstract": "Torsion semi-stable representations can be constructed and studied using Breuil modules. In this paper, we define the notion of pylonet and prove that some categories of Breuil modules naturally define pylonets. As a consequence, we are able to define full subcategories of Breuil's categories with very nice properties (in particular, they are abelian). In a second part of this work, we try to make very explicit some abstract constructions coming from the general theory of pylonets (developped earlier in the paper). These explicitations should be very useful to make computations with torsion semi-stable Galois representations."}
{"category": "Math", "title": "Effective algebraic degeneracy", "abstract": "We prove that any nonconstant entire holomorphic curve from the complex line C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary dimension n (at least 2) must be algebraically degenerate provided X is generic if its degree d = deg(X) satisfies the effective lower bound: d larger than or equal to n^{{(n+1)}^{n+5}}."}
{"category": "Math", "title": "A supplement to a theorem of Merker and Porten: a short proof of Hartogs' extension theorem for $(n-1)$-complete complex spaces", "abstract": "We give a short proof for the Hartogs's extension theorem on (n-1)-complete complex spaces."}
{"category": "Math", "title": "Counting closed geodesics in Moduli space", "abstract": "We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R."}
{"category": "Math", "title": "Sturm and Sylvester algorithms revisited via tridiagonal determinantal representations", "abstract": "First, we show that Sturm algorithm and Sylvester algorithm, which compute the number of real roots of a given univariate polynomial, lead to two dual tridiagonal determinantal representations of the polynomial. Next, we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation."}
{"category": "Math", "title": "Mather invariants in groups of piecewise-linear homeomorphisms", "abstract": "We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in [2] and Kassabov and Matucci in [5]. Thanks to the interplay between the techniques, we produce a simplified point of view of conjugacy that allows us to easily recover centralizers and lends itself to generalization."}
{"category": "Math", "title": "Trees of cylinders and canonical splittings", "abstract": "Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders T_c. This tree only depends on the deformation space of T; in particular, it is invariant under automorphisms of G if T is a JSJ splitting. We thus obtain Out(G)-invariant cyclic or abelian JSJ splittings. Furthermore, T_c has very strong compatibility properties (two trees are compatible if they have a common refinement)."}
{"category": "Math", "title": "On Quasiconvexity and Relative Hyperbolic Structures", "abstract": "Let G be a group which is hyperbolic relative to a collection of subgroups A, and it is also hyperbolic relative to a collection of subgroups B. Suppose that the collection A contains B. We characterize, for subgroups of G, when quasicovexity relative to A implies quasiconvexity relative to B. We also show that quasiconvexity relative to B implies quasiconvexity relative to A. Some applications are discussed."}
{"category": "Math", "title": "Scott and Swarup's regular neighbourhood as a tree of cylinders", "abstract": "Let G be a finitely presented group. Scott and Swarup have constructed a canonical splitting of G which encloses all almost invariant sets over virtually polycyclic subgroups of a given length. We give an alternative construction of this regular neighbourhood, by showing that it is the tree of cylinders of a JSJ splitting."}
{"category": "Math", "title": "Flag Paraproducts", "abstract": "We describe the theory of \"flag paraproducts\" and their relationship with the field of differential equations."}
{"category": "Math", "title": "On the Safe Use of Inconsistent Mathematics", "abstract": "A method is presented for using the consistent part of inconsistent axiomatic systems."}
{"category": "Math", "title": "Regularity of smooth curves in biprojective spaces", "abstract": "Maclagan and Smith \\cite{MaclaganSmith} developed a multigraded version of Castelnuovo-Mumford regularity. Based on their definition we will prove in this paper that for a smooth curve $C\\subseteq \\P^a\\times\\P^b$ $(a, b\\geq 2)$ of bidegree $(d_1,d_2)$ with nondegenerate birational projections the ideal sheaf $\\mathcal{I}_{C|\\P^a\\times\\P^b}$ is $(d_2-b+1,d_1-a+1)$-regular. We also give an example showing that in some cases this bound is the best possible."}
{"category": "Math", "title": "Exceptional divisors which are not uniruled belong to the image of the Nash map", "abstract": "We prove that, if X is a variety over an uncountable algebraically closed field k of characteristic zero, then any irreducible exceptional divisor E on a resolution of singularities of X which is not uniruled, belongs to the image of the Nash map, i.e. corresponds to an irreducible component of the space of arcs on X centered in Sing X. This reduces the Nash problem of arcs to understanding which uniruled essential divisors are in the image of the Nash map, more generally, how to determine the uniruled essential divisors from the space of arcs."}
{"category": "Math", "title": "Asymptotic stability of oscillations of two-bodies vibrating screen with one-sided obstacle without clearances", "abstract": "We prove asymptotic stability of periodic oscillations in two-bodies vibrating screen model under assumption that the frequencies w1 and w2 of the generating system (without obstacle and periodic driving) satisfy the assumption w1:w2=1:2. We also assume that the frequency of the external periodic driving equals to w1. These settings correspond to nonlinear resonance which is a well known phenomenon in industrial implementation of the vibrating screen. The justification is performed over nonsmooth analog of the second Bogolyubov's theorem proposed by the author in his previous papers. It is rigorously proven that the periodic oscillations obtained have two frequencies, in contrast with the case when the obtacle is absent."}
{"category": "Math", "title": "Finite Square Lattice Vertex Cover by a Baseline Set Defined With a Minimum Sublattice", "abstract": "Each straight infinite line defined by two vertices of a finite square point lattice contains (covers) these two points and a - possibly empty - subset of points that happen to be collinear to these. This work documents vertex subsets of minimum order such that the sum of the infinite straight lines associated with the edges of their complete subgraph covers the entire set of vertices (nodes). This is an abstraction to the problem of sending a light signal to all stations (receivers) in a square array with a minimum number of stations also equipped with transmitters to redirect the light to other transmitters."}
{"category": "Math", "title": "Points in a triangle forcing small triangles", "abstract": "An old theorem of Alexander Soifer's is the following: Given five points in a triangle of unit area, there must exist some three of them which form a triangle of area 1/4 or less. It is easy to check that this is not true if \"five\" is replaced by \"four\", but can the theorem be improved in any other way? We discuss in this article two different extensions of the original result. First, we allow the value of \"small\", 1/4, to vary. In particular, our main result is to show that given five points in a triangle of unit area, then there must exist some three of them determining a triangle of area 6/25 or less. Second, we put bounds on the minimum number of small triangles determined by n points in a triangle, and make a conjecture about the asymptotic right answer as n tends to infinity."}
{"category": "Math", "title": "A Family of Runge-Kutta Methods with Zero Phase-Lag and Derivatives for the Numerical Solution of the Schr\\\"odinger Equation and Related Problems", "abstract": "We construct a family of two new optimized explicit Runge-Kutta methods with zero phase-lag and derivatives for the numerical solution of the time-independent radial Schr\\\"odinger equation and related ordinary differential equations with oscillating solutions. The numerical results show the superiority of the new technique of nullifying both the phase-lag and its derivatives."}
{"category": "Math", "title": "Two Optimized Symmetric Eight-Step Implicit Methods for Initial-Value Problems with Oscillating Solutions", "abstract": "In this paper we present two optimized eight-step symmetric implicit methods with phase-lag order ten and infinite (phase-fitted). The methods are constructed to solve numerically the radial time-independent Schr\\\"odinger equation with the use of the Woods-Saxon potential. They can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the two new methods to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved."}
{"category": "Math", "title": "A New Methodology for the Development of Numerical Methods for the Numerical Solution of the Schr\\\"odinger Equation", "abstract": "In the present paper we introduce a new methodology for the construction of numerical methods for the approximate solution of the one-dimensional Schr\\\"odinger equation. The new methodology is based on the requirement of vanishing the phase-lag and its derivatives. The efficiency of the new methodology is proved via error analysis and numerical applications."}
{"category": "Math", "title": "Commutative association schemes", "abstract": "Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the \"commutative case,\" has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work."}
{"category": "Math", "title": "High Order Multistep Methods with Improved Phase-Lag Characteristics for the Integration of the Schr\\\"odinger Equation", "abstract": "In this work we introduce a new family of twelve-step linear multistep methods for the integration of the Schr\\\"odinger equation. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error analysis and numerical applications."}
{"category": "Math", "title": "On certain diophantine equations related to triangular and tetrahedral numbers", "abstract": "In this paper we give solutions of certain diophantine equations related to triangular and tetrahedral numbers and propose several problems connected with these numbers. The material of this paper was presented in part at the 11th International Workshop for Young Mathematicians - NUMBER THEORY, Krak\\'{o}w, 14th-20th september 2008."}
{"category": "Math", "title": "High Order Phase Fitted Multistep Integrators for the Schr\\\"odinger Equation with Improved Frequency Tolerance", "abstract": "In this work we introduce a new family of 14-steps linear multistep methods for the integration of the Schr\\\"odinger equation. The new methods are phase fitted but they are designed in order to improve the frequency tolerance. This is achieved by eliminating the first derivatives of the phase lag function at the fitted frequency forcing the phase lag function to be '\\textit{flat}' enough in the neighbor of the fitted frequency. The efficiency of the new family of methods is proved via error analysis and numerical applications."}
{"category": "Math", "title": "Zero Dispersion and Zero Dissipation Implicit Runge-Kutta Methods for the Numerical Solution of Oscillating IVPs", "abstract": "In this paper we present two new methods based on an implicit Runge-Kutta method Gauss which is of algebraic order fourth and has two stages: the first one has zero dispersion and the second one has zero dispersion and zero dissipation. The efficiency of these methods is measured while integrating the radial Schr\\\"odinger equation and other well known initial value problems."}
{"category": "Math", "title": "A Phase-Fitted Runge-Kutta-Nystr\\\"om method for the Numerical Solution of Initial Value Problems with Oscillating Solutions", "abstract": "A new Runge-Kutta-Nystr\\\"om method, with phase-lag of order infinity, for the integration of second-order periodic initial-value problems is developed in this paper. The new method is based on the Dormand and Prince Runge-Kutta-Nystr\\\"om method of algebraic order four\\cite{pa}. Numerical illustrations indicate that the new method is much more efficient than the classical one."}
{"category": "Math", "title": "Counting arithmetic lattices and surfaces", "abstract": "We give estimates on the number $AL_H(x)$ of arithmetic lattices $\\Gamma$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\\log AL_H(x) / x\\log x$ when $x\\to\\infty$. The proofs use several different techniques: geometric (bounding the number of generators of $\\Gamma$ as a function of its covolume), number theoretic (bounding the number of maximal such $\\Gamma$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $\\Gamma$)."}
{"category": "Math", "title": "A mixed problem for a Boussinesq hyperbolic equation with integral condition", "abstract": "A hyperbolic problem wich combines a classical(Dirichlet) and a non-local contraint is considered.The existence and uniqueness of strong solutions are proved,we use a functionnal analysis method based on a priori estimate and on the density of the range of the operator generated by the considered problem."}
{"category": "Math", "title": "A statistical framework for differential privacy", "abstract": "One goal of statistical privacy research is to construct a data release mechanism that protects individual privacy while preserving information content. An example is a {\\em random mechanism} that takes an input database $X$ and outputs a random database $Z$ according to a distribution $Q_n(\\cdot|X)$. {\\em Differential privacy} is a particular privacy requirement developed by computer scientists in which $Q_n(\\cdot |X)$ is required to be insensitive to changes in one data point in $X$. This makes it difficult to infer from $Z$ whether a given individual is in the original database $X$. We consider differential privacy from a statistical perspective. We consider several data release mechanisms that satisfy the differential privacy requirement. We show that it is useful to compare these schemes by computing the rate of convergence of distributions and densities constructed from the released data. We study a general privacy method, called the exponential mechanism, introduced by McSherry and Talwar (2007). We show that the accuracy of this method is intimately linked to the rate at which the probability that the empirical distribution concentrates in a small ball around the true distribution."}
{"category": "Math", "title": "The Cauchy problem for a short-wave equation", "abstract": "We prove an existence and uniqueness of solution for the Cauchy problem of the simplest nonlinear short-wave equation, $u_{tx}=u-3u^{2}$, with periodic boundary condition."}
{"category": "Math", "title": "Mackey-functor structure on the Brauer groups of a finite Galois covering of schemes", "abstract": "Past studies of the Brauer group of a scheme tells us the importance of the interrelationship among Brauer groups of its finite \\'etale coverings. In this paper, we consider these groups simultaneously, and construct an integrated object \"Brauer-Mackey functor\". We realize this as a {\\it cohomological Mackey functor} on the Galois category of finite \\'etale coverings. For any finite \\'etale covering of schemes, we can associate two homomorphisms for Brauer groups, namely the pull-back and the norm map. These homomorphisms make Brauer groups into a bivariant functor ($=$ Mackey functor) on the Galois category. As a corollary, Restricting to a finite Galois covering of schemes, we obtain a cohomological Mackey functor on its Galois group. This is a generalization of the result for rings by Ford. Moreover, applying Bley and Boltje's theorem, we can derive certain isomorphisms for the Brauer groups of intermediate coverings."}
{"category": "Math", "title": "Cohomology of Substitution Tiling Spaces", "abstract": "Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which \"forces its border.\" One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson-Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space."}
{"category": "Math", "title": "Some consequences of the Karpenko-Merkurjev theorem", "abstract": "We use a recent theorem of N. A. Karpenko and A. S. Merkurjev to settle several questions in the theory of essential dimension."}
{"category": "Math", "title": "Conformal Deformation on Manifolds with Boundary", "abstract": "We consider natural conformal invariants arising from the Gauss-Bonnet formulas on manifolds with boundary, and study conformal deformation problems associated to them. The key technique we used is to derive boundary C^2 estimates directly from C^0 estimates for fully nonlinear equations. The main result has appeared in the author's thesis."}
{"category": "Math", "title": "On the classification of toric singularities", "abstract": "For a toric log variety with standard coefficients, we show that the minimal log discrepancy at a closed invariant point bounds the Cartier index of a neighbourhood."}
{"category": "Math", "title": "Yet another solution to the Burnside problem for matrix semigroups", "abstract": "We use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite."}
{"category": "Math", "title": "On a result of Gelfand, Kapranov, and Zelevinsky", "abstract": "In this paper I give new elementary proofs of basic results of Gelfand, Kapranov and Zelevinskywhich express discriminants and resultants in terms of determinants of direct images of Cayley-Koszul complexes of sheaves."}
{"category": "Math", "title": "Projective duality and K-energy asymptotics", "abstract": "Let X be a smooth, linearly normal n dimensional complex projective variety. Assume that the projective dual of X has codimension one with defining polynomial D(X). In this paper the log of the norm of D(X) is expressed as the restriction to the Bergman metrics of an energy functional on X. We show how, for smooth plane curves, this energy functional reduces to the standard action functionals of Kahler geometry."}
{"category": "Math", "title": "Expanding Measures", "abstract": "We prove that any C^{1+} transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller for maps of the interval with negative Schwarzian derivative. We also show how to construct an induced Markov map F such that every expanding probability of the initial transformation lifts to an invariant probability of F. The induced time is bounded at each point by the corresponding first hyperbolic time (the first time the dynamics exhibits hyperbolic behavior). In particular, F may be used to study decay of correlations and others statistical properties of the initial map, relative to any expanding probability."}
{"category": "Math", "title": "Hyperdiscriminant polytopes, Chow polytopes, and Mabuchi energy asymptotics", "abstract": "Let X be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of X restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n-1) and the Chow form of X. As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G if and only if the hyperdiscriminant polytope dominates the Chow polytope for all maximal algebraic tori H of G ."}
{"category": "Math", "title": "Solution of Partial Differential Equations by Method of Hyperholomorphic functions", "abstract": "It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper we generalize this property to finite-dimensional commutative algebras. We prove that if some basis of a subspace of a commutative algebra satisfies a polynomial equation then components of a hyperholomorphic function on the subspace are solutions of the respective partial differential equation."}
{"category": "Math", "title": "Application of graph combinatorics to rational identities of type A", "abstract": "To a word $w$, we associate the rational function $\\Psi_w = \\prod (x_{w_i} - x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize identities linked to Murnaghan-Nakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain)."}
{"category": "Math", "title": "The fundamental category of a stratified space", "abstract": "The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems. When the space is topologically stratified Treumann, based on unpublished ideas of MacPherson, constructed an `exit category' (in the terminology of this paper, the `fundamental category') which classifies constructible sheaves, equivalently stratified etale covers. This paper generalises this construction to homotopically stratified sets, in addition showing that the fundamental category dually classifies constructible cosheaves, equivalently stratified branched covers. The more general setting has several advantages. It allows us to remove a technical `tameness' condition which appears in Treumann's work; to show that the fundamental groupoid can be recovered by inverting all morphisms and, perhaps most importantly, to reduce computations to the two stratum case. This provides an approach to computing the fundamental category in terms of homotopy groups of strata and homotopy links. We apply these techniques to compute the fundamental category of symmetric products of R^2, stratified by collisions. Two appendices explain the close relations respectively between filtered and pre-ordered spaces and between cosheaves and branched covers (technically locally-connected uniquely-complete spreads)."}
{"category": "Math", "title": "On the Structure of Complex Homogeneous Supermanifolds", "abstract": "For a Lie group $G$ and a closed Lie subgroup $H\\subset G$, it is well known that the coset space $G/H$ can be equipped with the structure of a manifold homogeneous under $G$ and that any $G$-homogeneous manifold is isomorphic to one of this kind. An interesting problem is to find an analogue of this result in the case of supermanifolds. In the classical setting, $G$ is a real or a complex Lie group and $G/H$ is a real and, respectively, a complex manifold. Now, if $G$ is a real Lie supergroup and $H\\subset G$ is a closed Lie subsupergroup, there is a natural way to consider $G/H$ as a supermanfold. Furthermore, any $G$-homogeneous real supermanifold can be obtained in this way, see \\cite{Kostant}. The goal of this paper is to give a proof of this result in the complex case."}
{"category": "Math", "title": "Shifted small deviations and Chung LIL for symmetric alpha-stable processes", "abstract": "Consider a symmetric $\\alpha$-stable L\\'evy process with $\\alpha\\in (1,2)$. We study shifted small ball probabilities for these processes in the uniform topology, when the shift function is an arbitrary continuous function which starts at 0. We obtain the exact rate of decrease for these probabilities including constants. Using these small ball estimates, we obtain a functional LIL for $\\alpha$-stable L\\'evy process with attracting functions that are continuous. It occurs that the limit set for the family of renormalized $\\alpha$-stable L\\'evy processes is equal to the set of all continuous functions on $[0,1]$ which start at 0, under certain choice of normalizing functions."}
{"category": "Math", "title": "Soliton dynamics for the nonlinear Schr\\\"odinger equation with magnetic field", "abstract": "The semiclassical limit of a nonlinear focusing Schr\\\"odinger equation in presence of nonconstant electric and magnetic potentials V,A is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation. The concentration curve of the solutions is a parameterization of the solutions of a Newton ODE involving the electric force as well as the magnetic force via the Lorenz law of electrodynamics."}
{"category": "Math", "title": "Period Doubling in Area-Preserving Maps: An Associated One Dimensional Problem", "abstract": "It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of $\\field{R}^2$. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods by J.-P. Eckmann, H. Koch and P. Wittwer (1982 and 1984). As it is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period doubling universality exists to date. We argue that the period doubling renormalization fixed point for area-preserving maps is almost one dimensional, in the sense that it is close to the following Henon-like map: $$H^*(x,u)=(\\phi(x)-u,x-\\phi(\\phi(x)-u)),$$ where $\\phi$ solves $$\\phi(x)={2 \\over \\lambda} \\phi(\\phi(\\lambda x))-x.$$ We then give a ``proof'' of existence of solutions of small analytic perturbations of this one dimensional problem, and describe some of the properties of this solution. The ``proof'' consists of an analytic argument for factorized inverse branches of $\\phi$ together with verification of several inequalities and inclusions of subsets of $\\field{C}$ numerically. Finally, we suggest an analytic approach to the full period doubling problem for area-preserving maps based on its proximity to the one dimensional. In this respect, the paper is an exploration of a possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system."}
{"category": "Math", "title": "The Center of the Nilcoxeter and 0-Hecke Algebras", "abstract": "We compute the dimension of the center of the 0-Hecke algebra $\\mathcal{H}_n$ and of the Nilcoxeter algebra $\\mathcal{NC}_n$ using a calculus of diagrams on the M\\\"{o}bius band. In the case of the Nilcoxeter algebra, this calculus is shown to produce a basis for $Z(\\mathcal{NC}_n)$ and the table of multiplication in this basis is shown to be trivial. We conjecture that a basis for $Z(\\mathcal{H}_n)$ can also be obtained in a specific way from this topological calculus."}
{"category": "Math", "title": "Polynomial operators and local smoothness classes on the unit interval, II", "abstract": "We prove the existence of quadrature formulas exact for integrating high degree polynomials with respect to Jacobi weights based on scattered data on the unit interval. We also obtain a characterization of local Besov spaces using the coefficients of a tight frame expansion."}
{"category": "Math", "title": "Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization", "abstract": "We consider a distributed multi-agent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent combines weighted averages of the received iterates with its own iterate, and adjusts the iterate by using subgradient information (known with stochastic errors) of its own function and by projecting onto the constraint set. The goal of this paper is to explore the effects of stochastic subgradient errors on the convergence of the algorithm. We first consider the behavior of the algorithm in mean, and then the convergence with probability 1 and in mean square. We consider general stochastic errors that have uniformly bounded second moments and obtain bounds on the limiting performance of the algorithm in mean for diminishing and non-diminishing stepsizes. When the means of the errors diminish, we prove that there is mean consensus between the agents and mean convergence to the optimum function value for diminishing stepsizes. When the mean errors diminish sufficiently fast, we strengthen the results to consensus and convergence of the iterates to an optimal solution with probability 1 and in mean square."}
{"category": "Math", "title": "Beltrami forms, affine surfaces and the Schwarz-Christoffel formula: a worked out example of straightening", "abstract": "Consider the straightening phi of a Beltrami form that is constant on a square, with the corresponding ellipses having a vertical major axis, and null outside. A generalized Schwarz-Christoffel formula is used to express the inverse of phi. The formula is found by introducing an affine Riemann surface. This formula is used to draw on a computer the image of the square by phi, and practical aspects are discussed. The resulting shapes are shown for different values of the constant dilatation ratio of the ellipses (=major axis/minor axis). The limit when this ratio tends to infinity is surprising. A model of this limit is proposed, produced by an affine surface uniformization."}
{"category": "Math", "title": "Reconstructing quasimorphisms from associated partial orders and a question of Polterovich", "abstract": "We show that every continuous homogeneous quasimorphism on a finite-dimensional 1-connected simple Lie group arises as the relative growth of any continuous bi-invariant partial order on that group. More generally we show, that an arbitrary homogeneous quasimorphism can be reconstructed as the relative growth of a partial order subject to a certain sandwich condition. This provides a link between invariant orders and bounded cohomology and allows the concrete computation of relative growth for finite dimensional simple Lie groups as well as certain infinite-dimensional Lie groups arising from symplectic geometry."}
{"category": "Math", "title": "Variations of Independence in Boolean Algebras", "abstract": "We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n -independent. The properties of these classes (n-free and omega-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, n Ind, which is the supremum of the cardinalities of n-independent subsets; i_n, the minimum size of a maximal n -independent subset; and i_omega, the minimum size of an omega-independent subset, are introduced and investigated. The values of i_n and i_omega on P(omega)/fin are shown to be independent of ZFC. Ideal-independence is also considered, and it is shown that the cardinal function p <= s_mm for infinite boolean algebras. We also define and consider moderately generated boolean algebras; that is, those boolean algebras that have a generating set consisting of elements that split finitely many elements of the boolean algebra."}
{"category": "Math", "title": "Shape optimization for low Neumann and Steklov eigenvalues", "abstract": "We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature: they are proved using methods of complex analysis. In particular, we present modernized proofs of the classical inequalities due to Szego and Weinstock for the first nonzero Neumann and Steklov eigenvalues. We also extend the inequality for the second nonzero Neumann eigenvalue, obtained recently by Nadirashvili and the authors, to non-homogeneous membranes with log-subharmonic densities. In the homogeneous case, we show that this inequality is strict, which implies that the maximum of the second nonzero Neumann eigenvalue is not attained in the class of simply-connected membranes of a given mass. The same is true for the second nonzero Steklov eigenvalue, as follows from our results on the Hersch-Payne-Schiffer inequalities."}
{"category": "Math", "title": "Geometric Langlands duality and forms of reductive groups", "abstract": "The category of perverse sheaves on the affine Grassmannian of a complex reductive group $G$ gives a canonical geometric construction of the split form of the Langlands dual group $\\check G_\\bZ$ over the integers. Given a field $k$, we give a Tannakian construction of the quasi-split forms of $\\check G_k$, as well as a construction of the gerbe associated to an inner form of $\\check G_k$."}
{"category": "Math", "title": "Predictability on finite horizon for processes with exponential decrease of energy on higher frequencies", "abstract": "The paper presents sufficient conditions of predictability for continuous time processes in deterministic setting. We found that processes with exponential decay on energy for higher frequencies are predictable in some weak sense on some finite time horizon defined by the rate of decay. Moreover, this predictability can be achieved uniformly over classes of processes. Some explicit formulas for predictors are suggested."}
{"category": "Math", "title": "Uniform Asymptotics of the Meixner Polynomials", "abstract": "Using the steepest descent method of Deift-Zhou, we derive locally uniform asymptotic formulas for the Meixner polynomials. These include an asymptotic formula in a neighborhood of the origin, a result which as far as we are aware has not yet been obtained previously. This particular formula involves a special function, which is the uniformly bounded solution to a scalar Riemann-Hilbert problem, and which is asymptotically (as the polynomial degree $n$ tends to infinity) equal to the constant $\"1\"$ except at the origin. Numerical computation by using our formulas, and comparison with earlier results, are also given."}
{"category": "Math", "title": "Maximizing the number of q-colorings", "abstract": "Let P_G(q) denote the number of proper q-colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four-color problem by minimizing P_G(4) over all planar graphs G. Since then, motivated by a variety of applications, much research was done on minimizing or maximizing P_G(q) over various families of graphs. In this paper, we study an old problem of Linial and Wilf, to find the graphs with n vertices and m edges which maximize the number of q-colorings. We provide the first approach which enables one to solve this problem for many nontrivial ranges of parameters. Using our machinery, we show that for each q >= 4 and sufficiently large m < \\kappa_q n^2 where \\kappa_q is approximately 1/(q log q), the extremal graphs are complete bipartite graphs minus the edges of a star, plus isolated vertices. Moreover, for q = 3, we establish the structure of optimal graphs for all large m <= n^2/4, confirming (in a stronger form) a conjecture of Lazebnik from 1989."}
{"category": "Math", "title": "Cohomology theory in 2-categories", "abstract": "Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups $\\mathrm{SCG}$ are being investigated. In this paper, we consider a 2-categorical analogue of an abelian category, in such a way that it contains $\\mathrm{SCG}$ as an example. As a main theorem, we construct a long cohomology 2-exact sequence from any extension of complexes in such a 2-category. Our axiomatic and self-dual definition will enable us to simplify the proofs, by analogy with abelian categories."}
{"category": "Math", "title": "On Boundary Crossing Probabilities for Diffusion Processes", "abstract": "In this paper, we establish a relationship between the asymptotic form of conditional boundary crossing probabilities and first passage time densities for diffusion processes. Namely, we show that, under broad assumptions, the first crossing time density of a general curvilinear boundary by a general time-homogeneous diffusion process has a product-form, the factors being the transition density of the process and the coefficient of the leading term in the asymptotic representation of the non-crossing probability of the boundary by the respective diffusion bridge (as the end-point of the bridge approaches the boundary). Using a similar technique, we also demonstrate that the boundary crossing probability is a Gateaux differentiable function of the boundary and give an explicit representation of its derivative."}
{"category": "Math", "title": "A classification of spherical conjugacy classes in good characteristic", "abstract": "We classify spherical conjugacy classes in a simple algebraic group over an algebraically closed field of good, odd characteristic."}
{"category": "Math", "title": "Universal pointwise selection rule in multivariate function estimation", "abstract": "In this paper, we study the problem of pointwise estimation of a multivariate function. We develop a general pointwise estimation procedure that is based on selection of estimators from a large parameterized collection. An upper bound on the pointwise risk is established and it is shown that the proposed selection procedure specialized for different collections of estimators leads to minimax and adaptive minimax estimators in various settings."}
{"category": "Math", "title": "A Note on Gorenstein Flat Dimension", "abstract": "Unlike the Gorenstein projective and injective dimensions, the majority of results on the Gorenstein flat dimension have been established only over Noetherian (or coherent) rings. Naturally, one would like to generalize these results to any associative ring. In this direction, we show that the Gorenstein flat dimension is a refinement of the classical flat dimension over any ring; and we investigate the relations between the Gorenstein projective dimension and the Gorenstein flat dimension."}
{"category": "Math", "title": "Spherical designs from norm-3 shell of integral lattices", "abstract": "A set of vectors all of which have a constant (non-zero) norm value in an Euclidean lattice is called a shell of the lattice. Venkov classified strongly perfect lattices of minimum 3 (R\\'{e}seaux et \"designs\" sph\\'{e}rique, 2001), whose minimal shell is a spherical 5-design. This note considers the classification of integral lattices whose shells of norm 3 are 5-designs."}
{"category": "Math", "title": "One group of inequalities with altitudes and medians in triangle", "abstract": "In the article we prove some inequalities that contain relations between altitudes and medians in triangle. At least one of these inequalities has not been considered in the literature before and the main theorem has also not been proved elsewhere in that form. Some immediate corollaries have been presented as well."}
{"category": "Math", "title": "A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter", "abstract": "By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we illustrate our results by simulations."}
{"category": "Math", "title": "Simple finite group schemes and their infinitesimal deformations", "abstract": "We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes."}
{"category": "Math", "title": "The \"north pole problem\" and random orthogonal matrices", "abstract": "This paper is motivated by the following observation. Take a 3 x 3 random (Haar distributed) orthogonal matrix $\\Gamma$, and use it to \"rotate\" the north pole, $x_0$ say, on the unit sphere in $R^3$. This then gives a point $u=\\Gamma x_0$ that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform u, giving $v=\\Gamma u=\\Gamma^2 x_0$. Simulations reported in Marzetta et al (2002) suggest that v is more likely to be in the northern hemisphere than in the southern hemisphere, and, morever, that $w=\\Gamma^3 x_0$ has higher probability of being closer to the poles $\\pm x_0$ than the uniformly distributed point u. In this paper we prove these results, in the general setting of dimension $p\\ge 3$, by deriving the exact distributions of the relevant components of u and v. The essential questions answered are the following. Let x be any fixed point on the unit sphere in $R^p$, where $p\\ge 3$. What are the distributions of $U_2=x'\\Gamma^2 x$ and $U_3=x'\\Gamma^3 x$? It is clear by orthogonal invariance that these distribution do not depend on x, so that we can, without loss of generality, take x to be $x_0=(1,0,...,0)'\\in R^p$. Call this the \"north pole\". Then $x_0'\\Gamma^ k x_0$ is the first component of the vector $\\Gamma^k x_0$. We derive stochastic representations for the exact distributions of $U_2$ and $U_3$ in terms of random variables with known distributions."}
{"category": "Math", "title": "On the vanishing of cohomology in triangulated categories", "abstract": "We study the vanishing of cohomology in triangulated categories admitting a central ring action. In particular, we study vanishing gaps and symmetry."}
{"category": "Math", "title": "C-Ideals of Lie Algebras", "abstract": "A subalgebra B of a Lie algebra L is called a c-ideal of L if there is an ideal C of L such that L = B + C and B \\cap C \\leq B_L, where B_L is the largest ideal of $L$ contained in B. This is analogous to the concept of c-normal subgroup, which has been studied by a number of authors. We obtain some properties of c-ideals and use them to give some characterisations of solvable and supersolvable Lie algebras. We also classify those Lie algebras in which every one-dimensional subalgebra is a c-ideal."}
{"category": "Math", "title": "Value-at-Risk Computation by Fourier Inversion with Explicit Error Bounds", "abstract": "The value-at-risk of a delta-gamma approximated derivatives portfolio can be computed by numerical integration of the characteristic function. However, while the choice of parameters in any numerical integration scheme is paramount, in practice it often relies on ad hoc procedures of trial and error. For normal and multivariate $t$-distributed risk factors, we show how to calculate the necessary parameters for one particular integration scheme as a function of the data (the distribution of risk factors, and delta and gamma) \\emph{in order to satisfy a given error tolerance}. This allows for implementation in a fully automated risk management system. We also demonstrate in simulations that the method is significantly faster than the Monte Carlo method, for a given error tolerance."}
{"category": "Math", "title": "Orbitally but not asymptotically stable ground states for the discrete NLS", "abstract": "We consider examples of discrete nonlinear Schroedinger equations in Z admitting ground states which are orbitally but not asymptotically stable in l ^2(Z). The ground states contain internal modes which decouple from the continuous modes. The absence of leaking of energy from discrete to continues modes leads to an almost conservation and perpetual oscillation of the discrete modes. This is quite different from what is known for nonlinear Schroedinger equations in eucliedean spaces. We do not investigate connections with work on quasi periodic solutions as in M.Johansson & S.Aubry and of Bambusi & Vella"}
{"category": "Math", "title": "Coloring plane graphs with independent crossings", "abstract": "We show that every plane graph with maximum face size four whose all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are 5-colorable."}
{"category": "Math", "title": "Cyclic colorings of plane graphs with independent faces", "abstract": "Let G be a plane graph with maximum face size D. If all faces of G with size four or more are vertex disjoint, then G has a cyclic coloring with D+1 colors, i.e., a coloring such that all vertices incident with the same face receive distinct colors."}
{"category": "Math", "title": "Trasferring $L^p$ eigenfunction bounds from $S^{2n+1}$ to $h^n$", "abstract": "By using the notion of contraction of Lie groups, we transfer $L^p-L^2$ estimates for joint spectral projectors from the unit complex sphere $\\sfera$ in ${{\\mathbb{C}}}^{n+1}$ to the reduced Heisenberg group $h^{n}$. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on $h^n$. As a consequence, we prove, in the spirit of Sogge's work, a discrete restriction theorem for the sub-Laplacian $L$ on $h^n$."}
{"category": "Math", "title": "Irreducible complex skew-Berger algebras", "abstract": "Irreducible skew-Berger algebras $\\g\\subset\\gl(n,\\Co)$, i.e. algebras spanned by the images of the linear maps $R:\\odot^2\\Co^n\\to\\g$ satisfying the Bianchi identity, are classified. These Lie algebras can be interpreted as irreducible complex Berger superalgebras contained in $\\gl(0|n,\\Co)$."}
{"category": "Math", "title": "On the integral of $\\log x\\frac{dy}{y}-\\log y\\frac{dx}{x}$ over the A-polynomial curves", "abstract": "In this note, we study the integral of the 1-form $\\log x\\frac{dy}{y}-\\log y\\frac{dx}{x}$ over certain plane curves defined by A-polynomials of knots. It is quite surprising that a Chern-Simons type invariant of 3-manifolds, which can be geometrically computed, may be used to get the exact values of those integrals. The arithmetic nature of these integrals is still unknown at the moment and deserved further investigation."}
{"category": "Math", "title": "Uniformites et Continuity Spaces", "abstract": "A semigroup A is an abelian semigroup with identity 0. A set of positives in A is an ordered down-directed set P containing with every r an element r/2 with r/2 + r/2 = r. A continuity space is an abstract set X equipped with a map d : XxX to A satisfying d(x, x) = 0 and d(x, z) d(x, y) + d(y, z). A quasi-uniform space is an abstract set X equipped with a filterbase of binary relations {U} such that each U contains the diagonal as well as for some V{U}. For each rP, the set } is seen to be a quasi-uniform filterbase on X . Indeed, the down-directedness of P ensures that U(r) is a filterbase of oversets of the diagonal and U(r) contains U(r/2)U(r/2). One obtains a uniform filterbase by symmetrization, i.e. by intersecting the U(r) with the U(s) = {(y, x)|d(y, x) <s}."}
{"category": "Math", "title": "On peeling procedure applied to a Poisson point process", "abstract": "In the focus of our attention is the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a more large class of Poisson point processes."}
{"category": "Math", "title": "Classification of knotted tori in the 2-metastable dimension", "abstract": "This paper is on the classical Knotting Problem: for a given manifold N and a number m describe the set of isotopy classes of embeddings $N\\to S^m$. We study the specific case of knotted tori, i. e. the embeddings $S^p \\times S^q \\to S^m$. The classification of knotted tori up to isotopy in the metastable dimension range $m>p+\\frac{3}{2}q+2$, $p\\le q$, was given by A. Haefliger, E. Zeeman and A. Skopenkov. We consider the dimensions below the metastable range, and give an explicit criterion for the finiteness of this set of isotopy classes in the 2-metastable dimension: Theorem. Assume that $p+\\frac{4}{3}q+2<m<p+\\frac{3}{2}q+2$ and $m>2p+q+2$. Then the set of smooth embeddings $S^p \\times S^q \\to S^m$ up to isotopy is infinite if and only if either $q+1$ or $p+q+1$ is divisible by 4. Our approach to the classification is based on an analogue of the Koschorke exact sequence from the theory of link maps. This sequence involves a new $\\beta$-invariant of knotted tori. The exactness is proved using embedded surgery and the Habegger-Kaiser techniques of studying the complement."}
{"category": "Math", "title": "A semantics for obligations", "abstract": "We analyze a number of properties obligations have or should have."}
{"category": "Math", "title": "A view of mathematics research productivity at U.S. regional public universities", "abstract": "Statistical summaries of certain kinds of mathematics research output are given for a large sample of U.S. regional public universities. These statistical summaries are reported using a variety of metrics that distinguish between single-authored and collaborative work and account for publication length."}
{"category": "Math", "title": "Representations of the general linear groups which are irreducible over subgroups", "abstract": "We classify all triples $(G,V,H)$ such that $SL_n(q)\\leq G\\leq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $\\FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $V\\dar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups."}
{"category": "Math", "title": "A partial analog of integrability theorem for distributions on p-adic spaces and applications", "abstract": "Let X be a smooth real algebraic variety. Let $\\xi$ be a distribution on it. One can define the singular support of $\\xi$ to be the singular support of the $D_X$-module generated by $\\xi$ (some times it is also called the characteristic variety). A powerful property of the singular support is that it is a coisotropic subvariety of $T^*X$. This is the integrability theorem (see [KKS, Mal, Gab]). This theorem turned out to be useful in representation theory of real reductive groups (see e.g. [AG_AMOT, AS, Say]). The aim of this paper is to give an analog of this theorem to the non-Archimedean case. The theory of D-modules is not available to us so we need a different definition of the singular support. We use the notion wave front set from [Hef] and define the singular support to be its Zariski closure. Then we prove that the singular support satisfies some property that we call weakly coisotropic, which is weaker than being coisotropic but is enough for some applications. We also prove some other properties of the singular support that were trivial in the Archimedean case (using the algebraic definition) but not obvious in the non-Archimedean case."}
{"category": "Math", "title": "Quantitative asymptotics of graphical projection pursuit", "abstract": "There is a result of Diaconis and Freedman which says that, in a limiting sense, for large collections of high-dimensional data most one-dimensional projections of the data are approximately Gaussian. This paper gives quantitative versions of that result. For a set of deterministic vectors $\\{x_i\\}_{i=1}^n$ in $\\R^d$ with $n$ and $d$ fixed, let $\\theta\\in\\s^{d-1}$ be a random point of the sphere and let $\\mu_n^\\theta$ denote the random measure which puts mass $\\frac{1}{n}$ at each of the points $\\inprod{x_1}{\\theta},...,\\inprod{x_n}{\\theta}$. For a fixed bounded Lipschitz test function $f$, $Z$ a standard Gaussian random variable and $\\sigma^2$ a suitable constant, an explicit bound is derived for the quantity $\\ds\\P[|\\int f d\\mu_n^\\theta-\\E f(\\sigma Z)|>\\epsilon]$. A bound is also given for $\\ds\\P[d_{BL}(\\mu_n^\\theta, N(0,\\sigma^2))>\\epsilon]$, where $d_{BL}$ denotes the bounded-Lipschitz distance, which yields a lower bound on the waiting time to finding a non-Gaussian projection of the $\\{x_i\\}$ if directions are tried independently and uniformly on $\\s^{d-1}$."}
{"category": "Math", "title": "A minimalist two-level foundation for constructive mathematics", "abstract": "We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin. One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version introduced in the mentioned paper with collections. The other level is given by an extensional set theory which is interpreted in the first one by means of a quotient model. This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the \"proofs-as-programs\" paradigm and acts as a programming language."}
{"category": "Math", "title": "The Julia set of a post-critically finite endomorphism of PC^2", "abstract": "We construct a combinatorial model of the Julia set of the endomorphism $f(z, w)=((1-2z/w)^2, (1-2/w)^2)$ of $PC^2$."}
{"category": "Math", "title": "Concrete Constructions of Real Equiangular Line Sets", "abstract": "We give some concrete constructions of real equiangular line sets. The emphasis here is on {\\em building blocks} for certain angles which are then used to build up larger equiangular line sets. We concentrate on angles greater than or equal to 1/7."}
{"category": "Math", "title": "A Giambelli formula for isotropic Grassmannians", "abstract": "Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in H^*(X,Z) as a polynomial in certain special Schubert classes. We study theta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X. Furthermore, we prove that theta polynomials are special cases of Billey-Haiman Schubert polynomials and use this connection to express the former as positive linear combinations of products of Schur Q-functions and S-polynomials."}
{"category": "Math", "title": "Brunet-Derrida behavior of branching-selection particle systems on the line", "abstract": "We consider a class of branching-selection particle systems on $\\R$ similar to the one considered by E. Brunet and B. Derrida in their 1997 paper \"Shift in the velocity of a front due to a cutoff\". Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size $N$ of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate $(\\log N)^{-2}$. In this paper, we give a rigorous mathematical proof of this fact, for the class of particle systems we consider. The proof makes use of ideas and results by R. Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison of the particle system with a family of $N$ independent branching random walks killed below a linear space-time barrier."}
{"category": "Math", "title": "Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions", "abstract": "In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained."}
{"category": "Math", "title": "Center stable manifolds for quasilinear parabolic pde and conditional stability of nonclassical viscous shock waves", "abstract": "Motivated by the study of conditional stability of traveling waves, we give an elementary $H^2$ center stable manifold construction for quasilinear parabolic PDE, sidestepping apparently delicate regularity issues by the combination of a carefully chosen implicit fixed-point scheme and \"damping-type\" $H^s$ energy estimates of a type familiar from the study of hyperbolic--parabolic and relaxation systems. An important feature of these methods is that they generalize to situations such as the hyperbolic--parabolic or relaxation case for which parabolic-type smoothing estimates are unavailable. As an application, we show conditional stability of Lax- or undercompressive shock waves of general quasilinear parabolic systems of conservation laws by a pointwise stability analysis on the center stable manifold."}
{"category": "Math", "title": "Logarithm laws for unipotent flows, I", "abstract": "We prove analogues of the logarithm laws of Sullivan and Kleinbock-Margulis in the context of unipotent flows. In particular, we obtain results for one-parameter actions on the space of lattices $SL(n, \\R)/SL(n, \\Z)$. The key lemma for our results says the measure of the set of unimodular lattices in $\\R^n$ that does not intersect a `large' volume subset of $\\R^n$ is `small'. This can be considered as a `random' analogue of the classical Minkowski theorem in the geometry of numbers."}
{"category": "Math", "title": "A Nonseparably Connected Metric Space as a Dense Connected Graph", "abstract": "We present a connected metric space that does not contain any nontrivial separable connected subspace. Our space is a dense connected graph of a function from the real line satisfying Cauchy's equation."}
{"category": "Math", "title": "Symplectic Spinors, Holonomy and Maslov Index", "abstract": "In this note it is shown that the Maslov Index for pairs of Lagrangian Paths as introduced by Cappell, Lee and Miller appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinorbundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold $L$, along closed or non-closed paths therein. In especially, the CLM-Index mod 4 determines the holonomy group of this line bundle w.r.t. the Levi-Civita-connection on $L$, hence its vanishing mod 4 is equivalent to the existence of a trivializing parallel section. Moreover, it is shown that the CLM-Index determines parallel transport in that line-bundle along arbitrary paths when compared to the parallel transport w.r.t. to the canonical flat connection of Euclidean space, if the Lagrangian tangent planes at the endpoints either coincide or are orthogonal. This is derived from a result on parallel transport of certain elements of the dual spinorbundle along closed or endpoint-transversal paths."}
{"category": "Math", "title": "On Analytic Perturbations of a Family of Feigenbaum-like Equations", "abstract": "We prove existence of solutions $(\\phi,\\lambda)$ of a family of of Feigenbaum-like equations \\label{family} \\phi(x)={1+\\eps \\over \\lambda} \\phi(\\phi(\\lambda x)) -\\eps x +\\tau(x), where $\\eps$ is a small real number and $\\tau$ is analytic and small on some complex neighborhood of $(-1,1)$ and real-valued on $\\fR$. The family $(\\ref{family})$ appears in the context of period-doubling renormalization for area-preserving maps (cf. \\cite{GK}). Our proof is a development of ideas of H. Epstein (cf \\cite{Eps1}, \\cite{Eps2}, \\cite{Eps3}) adopted to deal with some significant complications that arise from the presence of terms $\\eps x +\\tau(x)$ in the equation $(\\ref{family})$. The method relies on a construction of novel {\\it a-priori} bounds for unimodal functions which turn out to be very tight. We also obtain good bounds on the scaling parameter $\\lambda$. A byproduct of the method is a new proof of the existence of a Feigenbaum-Coullet-Tresser function."}
{"category": "Math", "title": "Hamiltonian Stationary Lagrangian Tori in Kaehler Manifolds", "abstract": "A Hamiltonian stationary Lagrangian submanifold of a Kaehler manifold is a Lagrangian submanifold whose volume is stationary under Hamiltonian variations. We find a sufficient condition on the curvature of a Kaehler manifold of real dimension four that guarantees the existence of a family of small Hamiltonian stationary Lagrangian tori."}
{"category": "Math", "title": "Strong sums of projections in von Neumann factors", "abstract": "This paper presents necessary and sufficient conditions for a positive bounded operator on a separable Hilbert space to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal), with the sum converging in the strong operator topology if the collection is infinite. A similar necessary condition is given when the operator and the projections are taken in a type II von Neumann factor, and the condition is proven to be also sufficient if the operator is \"diagonalizable\". A simpler necessary and sufficient condition is given in the type III factor case."}
{"category": "Math", "title": "An Algorithm for Unconstrained Quadratically Penalized Convex Optimization", "abstract": "A descent algorithm, \"Quasi-Quadratic Minimization with Memory\" (QQMM), is proposed for unconstrained minimization of the sum, $F$, of a non-negative convex function, $V$, and a quadratic form. Such problems come up in regularized estimation in machine learning and statistics. In addition to values of $F$, QQMM requires the (sub)gradient of $V$. Two features of QQMM help keep low the number of evaluations of the objective function it needs. First, QQMM provides good control over stopping the iterative search. This feature makes QQMM well adapted to statistical problems because in such problems the objective function is based on random data and therefore stopping early is sensible. Secondly, QQMM uses a complex method for determining trial minimizers of $F$. After a description of the problem and algorithm a simulation study comparing QQMM to the popular BFGS optimization algorithm is described. The simulation study and other experiments suggest that QQMM is generally substantially faster than BFGS in the problem domain for which it was designed. A QQMM-BFGS hybrid is also generally substantially faster than BFGS but does better than QQMM when QQMM is very slow."}
{"category": "Math", "title": "Consistency of Random Survival Forests", "abstract": "We prove uniform consistency of Random Survival Forests (RSF), a newly introduced forest ensemble learner for analysis of right-censored survival data. Consistency is proven under general splitting rules, bootstrapping, and random selection of variables--that is, under true implementation of the methodology. A key assumption made is that all variables are factors. Although this assumes that the feature space has finite cardinality, in practice the space can be a extremely large--indeed, current computational procedures do not properly deal with this setting. An indirect consequence of this work is the introduction of new computational methodology for dealing with factors with unlimited number of labels."}
{"category": "Math", "title": "Analysis of spectral methods for the homogeneous Boltzmann equation", "abstract": "The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method is modified in order to enforce the posivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the \"spreading\" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound)."}
{"category": "Math", "title": "Lp estimates for non smooth bilinear Littlewood-Paley square functions on R", "abstract": "In this work, some non smooth bilinear analogues of linear Littlewood-Paley square functions on the real line are studied. These bilinear operators are closely related to the bilinear Hilbert transforms and vector valued version of these ones. Mainly we prove boundedness-properties in Lebesgue spaces for them."}
{"category": "Math", "title": "Tropical rational equivalence on R^r", "abstract": "We introduce an improved version of rational equivalence in tropical intersection theory which can be seen as a replacement of chapter 8 of our previous article arXiv:0709.3705v2. Using this new definition, rational equivalence is compatible with push-forwards of cycles. Moreover, we prove that every tropical cycle in R^r is equivalent to a uniquely determined affine cycle, called its degree."}
{"category": "Math", "title": "Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function", "abstract": "We introduce completely monotonic functions of order $r>0$ and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order."}
{"category": "Math", "title": "Semilinear ordinary differential equation coupled with distributed order fractional differential equation", "abstract": "System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of viscoelasticity and in the system identfica- tion theory. Also, the existence and uniqueness of a solution to a general linear fractional differential equation in the space of tempered distributions is given."}
{"category": "Math", "title": "A Forward semi-Lagrangian Method for the Numerical Solution of the Vlasov Equation", "abstract": "This work deals with the numerical solution of the Vlasov equation. This equation gives a kinetic description of the evolution of a plasma, and is coupled with Poisson's equation for the computation of the self-consistent electric field. The coupled model is non linear. A new semi-Lagrangian method, based on forward integration of the characteristics, is developed. The distribution function is updated on an eulerian grid, and the pseudo-particles located on the mesh's nodes follow the characteristics of the equation forward for one time step, and are deposited on the 16 nearest nodes. This is an explicit way of solving the Vlasov equation on a grid of the phase space, which makes it easier to develop high order time schemes than the backward method."}
{"category": "Math", "title": "Subfields of ample fields I. Rational maps and definability", "abstract": "Pop proved that a smooth curve C over an ample field K that has a K-rational point has |K| many K-rational points. We strengthen this result by showing that there are |K| many K-rational points that do not lie in a given proper subfield, even after applying a rational map. As a consequence we gain insight into the structure of existentially definable subsets of ample fields. In particular, we prove that a perfect ample field has no existentially definable proper infinite subfields."}
{"category": "Math", "title": "Automatic Classification of Restricted Lattice Walks", "abstract": "We propose an experimental mathematics approach leading to the computer-driven discovery of various structural properties of general counting functions coming from enumeration of walks."}
{"category": "Math", "title": "Generic Variables in Acyclic Cluster Algebras and Bases in Affine Cluster Algebras", "abstract": "Let $Q$ be a finite quiver without oriented cycles and $\\mathcal A(Q)$ be the coefficient-free cluster algebra with initial seed $(Q,\\textbf u)$. Using the Caldero-Chapoton map, we introduce and investigate a family of generic variables in $\\Z[\\textbf u^{\\pm 1}]$ containing the cluster monomials of $\\mathcal A(Q)$. The aim of these generic variables is to give an explicit new method for constructing $\\Z$-bases in the cluster algebra $\\mathcal A(Q)$. If $Q$ is an affine quiver with minimal imaginary root $\\delta$, we investigate differences between cluster characters associated to indecomposable representations of dimension vector $\\delta$. We define the notion of \\emph{difference property} which gives an explicit description of these differences. We prove in particular that this property holds for quivers of affine type $\\tilde A$. When $Q$ satisfies the difference property, we prove that generic variables span the cluster algebra $\\mathcal A(Q)$. If $\\mathcal A(Q)$ satisfies some gradability condition, we prove that generic variables are linearly independent over $\\mathbb Z$ in $\\mathcal A(Q)$. In particular, this implies that generic variables form a $\\Z$-basis in a cluster algebra associated to an affine quiver of type $\\tilde A$."}
{"category": "Math", "title": "Parabolic foliations on 3-manifolds", "abstract": "In the paper we prove that every closed orientable three-manifold admits a parabolic foliation."}
{"category": "Math", "title": "Classification of Fuchsian systems and their connection problem", "abstract": "We review the Deligne-Simpson problem, a combinatorial structure of middle convolutions and their relation to a Kac-Moody root system discoverd by Crawley-Boevey. We show with examples that middle convolutions transform the Fuchsian systems with a fixed number of accessory parameters into fundamental systems whose spectral type is in a finite set and we give an explicit connection formula for solutions of Fuchsian differential equations without moduli."}
{"category": "Math", "title": "On the second Tate-Shafarevich group of a 1-motive", "abstract": "We prove finiteness results for Tate--Shafarevich groups in degree 2 associated with 1--motives, rely them to Leopoldt's conjecture, and present an example of a semiabelian variety with an infinite Tate--Shafarevich group in degree 2. We also establish an arithmetic duality theorem for 1--motives over number fields which complements earlier results of Harari and Szamuely in this direction."}
{"category": "Math", "title": "Projective metrics and contraction principles for complex cones", "abstract": "In this article, we consider linearly convex complex cones in complex Banach spaces and we define a new projective metric on these cones. Compared to the hyperbolic gauge of Rugh, it has the advantage of being explicit, and easier to estimate. We prove that this metric also satisfies a contraction principle like Birkhoff's theorem for the Hilbert metric. We are thus able to improve existing results on spectral gaps for complex matrices. Finally, we compare the contraction principles for the hyperbolic gauge and our metric on particular cones, including complexification of Birkhoff cones. It appears that the contraction principles for our metric and the hyperbolic gauge occur simultaneously on these cones. However, we get better contraction rates with our metric."}
{"category": "Math", "title": "Random Complexes and l^2-Betti Numbers", "abstract": "Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first $\\ell^2$-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher $\\ell^2$-Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans, and Martin."}
{"category": "Math", "title": "Automorphisms of cotangent bundles of Lie groups", "abstract": "Let G be a Lie group, $T^*G$ its cotangent bundle with its natural Lie group structure obtained by performing a left trivialization of T^*G and endowing the resulting trivial bundle with the semi-direct product, using the coadjoint action of G on the dual space of its Lie algebra. We investigate the group of automorphisms of the Lie algebra of $T^*G$. More precisely, amongst other results, we fully characterize the space of all derivations of the Lie algebra of $T^*G$. As a byproduct, we also characterize some spaces of operators on G amongst which, the space J of bi-invariant tensors on G and prove that if G has a bi-invariant Riemannian or pseudo-Riemannian metric, then J is isomorphic to the space of linear maps from the Lie algebra of G to its dual space which are equivariant with respect to the adjoint and coadjoint actions, as well as that of bi-invariant bilinear forms on G. We discuss some open problems and possible applications."}
{"category": "Math", "title": "Infinite Bar-Joint Frameworks", "abstract": "Some aspects of a mathematical theory of rigidity and flexibility are developed for general infinite frameworks and two main results are obtained. In the first sufficient conditions, of a uniform local nature, are obtained for the existence of a proper flex of an infinite framework. In the second it is shown how continuous paths in the plane may be simulated by infinite Kempe linkages."}
{"category": "Math", "title": "A new geometric approach to problems in birational geometry", "abstract": "A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: given two mildly singular projective varieties with some of the first variety's pseudonormed spaces being isometric to the corresponding ones of the second variety's, can one construct a birational map between them which induces these isometries? In this work a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence."}
{"category": "Math", "title": "Three methods to prove the existence of integral canonical models of Shimura varieties of Hodge type", "abstract": "This is a survey of the three main methods developed in the last 15 years to prove the existence of integral canonical models of Shimura varieties of Hodge type. The only new part is formed by corrections to results of Kisin."}
{"category": "Math", "title": "On the minimal ramification problem for $\\ell$-groups", "abstract": "Let p be a prime number. It is not known if every finite p-group of rank n>1 can be realized as a Galois group over Q with no more than n ramified primes. We prove that this can be done for the family of finite p-groups which contains all the cyclic groups of p-power order, and is closed under direct products, wreath products, and rank preserving homomorphic images. This family contains the Sylow p-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not p. On the other hand, it does not contain all finite p-groups."}
{"category": "Math", "title": "The Graph of the Hypersimplex", "abstract": "The (k,d)-hypersimplex is a (d-1)-dimensional polytope whose vertices are the (0,1)-vectors that sum to k. When k=1, we get a simplex whose graph is the complete graph with d vertices. Here we show how many of the well known graph parameters and attributes of the complete graph extend to a more general case. In particular we obtain explicit formulas in terms of d and k for the number of vertices, vertex degree, number of edges and the diameter. We show that the graphs are vertex transitive, hamilton connected, obtain the clique number and show how the graphs can be decomposed into self-similar subgraphs. The paper concludes with a discussion of the edge expansion rate of the graph of a (k,d)-hypersimplex which we show is at least d/2, and how this graph can be used to generate a random subset of {1,2,3,...,d} with k elements."}
{"category": "Math", "title": "The structure of typical clusters in large sparse random configurations", "abstract": "The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors. Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices. In this work, we use the configuration model as the stochastic counterpart of Smoluchowski's coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to $\\infty$. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors."}
{"category": "Math", "title": "On Relations Between Urbanik and Mehler Semigroups", "abstract": "It is shown that operator-selfdecomposable measures, or more precisely their Urbanik decomposability semigroups, induce generalized Mehler semigroups of bounded linear operators. Moreover, those semigroups can be represented as random integrals of operator valued functions with respect to stochastic L\\'evy processes. Our Banach space setting is in the contrast with the Hilbert spaces on which so far and most often the generalized Mehler semigroups were studied. Furthermore, we give new proofs of the random integral representation."}
{"category": "Math", "title": "Homeomorphisms of the annulus with a transitive lift", "abstract": "Let $f$ be a homeomorphism of the closed annulus $A$ that preserves orientation, boundary components and that has a lift $\\tilde f$ to the infinite strip $\\tilde A$ which is transitive. We show that, if the rotation number of both boundary components of $A$ is strictly positive, then there exists a closed nonempty connected set $\\Gamma\\subset\\tilde A$ such that $\\Gamma\\subset]-\\infty,0]\\times[0,1]$, $\\Gamma$ is unlimited, the projection of $\\Gamma$ to $A$ is dense, $\\Gamma-(1,0)\\subset\\Gamma$ and $\\tilde{f}(\\Gamma)\\subset \\Gamma.$ Also, if $p_1$ is the projection in the first coordinate in $\\tilde A$, then there exists $d>0$ such that, for any $\\tilde z\\in\\Gamma,$ $$\\limsup_{n\\to\\infty}\\frac{p_1(\\tilde f^n(\\tilde z))-p_1(\\tilde z)}{n}<-d.$$ In particular, using a result of Franks, we show that the rotation set of any homeomorphism of the annulus that preserves orientation, boundary components, which has a transitive lift without fixed points in the boundary is an interval with 0 in its interior."}
{"category": "Math", "title": "Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms", "abstract": "Let F be a characteristic zero differential field with an algebraically closed field of constants, E be a no-new-constant extension of F by antiderivatives of F and let y1, ..., yn be antiderivatives of E. The antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if the derivatives of yi in E satisfies certain conditions. We will discuss a new proof for the Kolchin-Ostrowski theorem and generalize this theorem for a tower of extensions by J-I-E antiderivatives and use this generalized version of the theorem to classify the finitely differentially generated subfields of this tower. In the process, we will show that the J-I-E antiderivatives are algebraically independent over the ground differential field. An example of a J-I-E tower is extensions by iterated logarithms. We will discuss the normality of extensions by iterated logarithms and produce an algorithm to compute its finitely differentially generated subfields."}
{"category": "Math", "title": "The discrepancy of a needle on a checkerboard, II", "abstract": "Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. In a previous paper we showed that for any such coloring there are straight line segments, of arbitrarily large length, such that the difference of their white length minus their black length, in absolute value, is at least the square root of their length, up to a multiplicative constant. For the corresponding \"finite\" problem ($N \\times N$ checkerboard) we had proved that we can color it in such a way that the above quantity is at most $C \\sqrt{N \\log N}$, for any placement of the line segment. In this followup we show that it is possible to color the infinite checkerboard with two colors so that for any line segment $I$ the excess of one color over another is bounded above by $C_\\epsilon \\Abs{I}^{\\frac12+\\epsilon}$, for any $\\epsilon>0$. We also prove lower bounds for the discrepancy of circular arcs. Finally, we make some observations regarding the $L^p$ discrepancies for segments and arcs, $p<2$, for which our $L^2$-based methods fail to give any reasonable estimates."}
{"category": "Math", "title": "A lower bound for Garsia's entropy for certain Bernoulli convolutions", "abstract": "Let $\\beta\\in(1,2)$ be a Pisot number and let $H_\\beta$ denote Garsia's entropy for the Bernoulli convolution associated with $\\beta$. Garsia, in 1963 showed that $H_\\beta<1$ for any Pisot $\\beta$. For the Pisot numbers which satisfy $x^m=x^{m-1}+x^{m-2}+...+x+1$ (with $m\\ge2$) Garsia's entropy has been evaluated with high precision by Alexander and Zagier and later improved by Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other numerical values for $H_\\beta$ are known. In the present paper we show that $H_\\beta>0.81$ for all Pisot $\\beta$, and improve this lower bound for certain ranges of $\\beta$. Our method is computational in nature."}
{"category": "Math", "title": "Time Optimal Return of a Dynamic Object", "abstract": "We solve the problem concerning a time optimal return of a particle with a prescribed velocity to the origin by applying a magnitude-bounded force. The equations of controlled motion are derived and explicitly integrated, and the optimal open-loop control and optimal time are analyzed depending on the parameters of the problem. The qualitative behavior of the solution is established, and the solution is compared with other regimes of motion. The results are of interest for control theory and its applications to controlled flight mechanics."}
{"category": "Math", "title": "Period, index and potential sha", "abstract": "In this paper we advance the theory of O'Neil's period-index obstruction map and derive consequences for the arithmetic of genus one curves over global fields. Our first result implies that for every pair of positive integers (P,I) with P dividing I and I dividing P^2, there exists a number field K and a genus one curve C over K with period P and index I. Second, let E be any elliptic curve over a global field K, and let P > 1 be any integer indivisible by the characteristic of K. We construct infinitely many genus one curves C over K with period P, index P^2, and Jacobian E. We deduce strong consequences on the structure of Sharevich-Tate groups under field extension."}
{"category": "Math", "title": "Note on generating all subsets of a finite set with disjoint unions", "abstract": "We call a family G of subsets of [n] a k-generator of (\\mathbb{P}[n]) if every (x \\subset [n]) can be expressed as a union of at most k disjoint sets in (\\mathcal{G}). Frein, Leveque and Sebo conjectured that for any (n \\geq k), such a family must be at least as large as the k-generator obtained by taking a partition of [n] into classes of sizes as equal as possible, and taking the union of the power-sets of the classes. We generalize a theorem of Alon and Frankl \\cite{alon} in order to show that for fixed k, any k-generator of (\\mathbb{P}[n]) must have size at least (k2^{n/k}(1-o(1))), thereby verifying the conjecture asymptotically for multiples of k."}
{"category": "Math", "title": "Information Percolation with Equilibrium Search Dynamics", "abstract": "We solve for the equilibrium dynamics of information sharing in a large population. Each agent is endowed with signals regarding the likely outcome of a random variable of common concern. Individuals choose the effort with which they search for others from whom they can gather additional information. When two agents meet, they share their information. The information gathered is further shared at subsequent meetings, and so on. Equilibria exist in which agents search maximally until they acquire sufficient information precision, and then minimally. A tax whose proceeds are used to subsidize the costs of search improves information sharing and can in some cases increase welfare. On the other hand, endowing agents with public signals reduces information sharing and can in some cases decrease welfare."}
{"category": "Math", "title": "Information Percolation", "abstract": "For a setting in which a large number of asymmetrically informed agents are randomly matched into groups over time, exchanging their information with each other when matched, we provide an explicit solution for the dynamics of the cross-sectional distribution of posterior beliefs. We also show that convergence of the cross-sectional distribution of beliefs to a common posterior is exponential and that the rate of convergence does not depend on the size of the groups of agents that meet. The rate of convergence is merely the mean rate at which an individual agent is matched."}
{"category": "Math", "title": "The word problem and the metric for the Thompson-Stein groups", "abstract": "We consider the Thompson-Stein group F(n_1,...,n_k) for integers n_1,...,n_k and k greater than 1. We highlight several differences between the cases k=1$ and k>1, including the fact that minimal tree-pair diagram representatives of elements may not be unique when k>1. We establish how to find minimal tree-pair diagram representatives of elements of F(n_1,...,n_k), and we prove several theorems describing the equivalence of trees and tree-pair diagrams. We introduce a unique normal form for elements of F(n_1,...,n_k) (with respect to the standard infinite generating set developed by Melanie Stein) which provides a solution to the word problem, and we give sharp upper and lower bounds on the metric with respect to the standard finite generating set, showing that in the case k>1, the metric is not quasi-isometric to the number of leaves or caret in the minimal tree-pair diagram, as is the case when k=1."}
{"category": "Math", "title": "Cremona transformations, surface automorphisms and plane cubics", "abstract": "We give a method for constructing many examples of automorphisms with positive entropy on rational complex surfaces. The general idea is to begin with a quadratic Cremona transformation that fixes a reduced cubic curve and then use the group structure on the cubic to understand when the indeterminacy and exceptional behavior of the transformation may be eliminated by repeated blowing up."}
{"category": "Math", "title": "Thurston type Theorem for sub-hyperbolic rational maps", "abstract": "In 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a new but simpler proof of this result by adapting the argument in the proof of Thurston's Theorem."}
{"category": "Math", "title": "Dynamics of Siegel Rational Maps with Prescribed Combinatorics", "abstract": "We extend Thurston's combinatorial criterion for postcritically finite rational maps to a class of rational maps with bounded type Siegel disks. The combinatorial characterization of this class of Siegel rational maps plays a special role in the study of general Siegel rational maps. As one of the applications, we prove that for any quadratic rational map with a bounded type Siegel disk, the boundary of the Siegel disk is a quasi-circle which passes through one or both of the critical points."}
{"category": "Math", "title": "On the 2d Zakharov system with L^2 Schr\\\"odinger data", "abstract": "We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schroedinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data - a result which is false for the cubic nonlinear Schroedinger equation in dimension two - and it is optimal because Glangetas-Merle's solutions blow up at that time."}
{"category": "Math", "title": "Sets of integers that do not contain long arithmetic progressions", "abstract": "In 1946, Behrend gave a construction of dense finite sets of integers that do not contain 3-term arithmetic progressions. In 1961, Rankin generalized Behrend's construction to sets avoiding k-term arithmetic progressions, and in 2008 Elkin refined Behrend's 3-term construction. In this work, we combine Elkin's refinement and Rankin's generalization. Arithmetic progressions are handled as a special case of polynomial progressions. In 1946, Behrend gave a construction of dense finite sets of integers that do not contain a 3-term arithmetic progression (AP). In 1961, Rankin generalized Behrend's construction to sets avoiding k-term APs. In 2008, Elkin refined Behrend's 3-term construction, and later in 2008, Green & Wolf found a distinct approach (albeit morally similar) that is technically more straightforward. This work combines Elkin's refinement and Rankin's generalization in the Green & Wolf framework. A curious aspect of the construction is that we induct through sets that do not contain a long polynomial progression in order to construct a set without a long AP. The bounds for r_k(N), the largest size of a subset of {1,2,...,N} that does not contain a k element AP, are (where \\log=\\log_2, for sufficiently large N, with n=\\ceiling{\\log k}): r_3(N) > N (\\sqrt{360}/(e \\pi^{3/2})-\\epsilon) \\sqrt[4]{2\\log N} * 4^{-\\sqrt{2 \\log N}}, r_k(N) > CN 2^{-n 2^{(n-1)/2} \\sqrt[n]{\\log N}+\\frac{1}{2n}\\log\\log N}. The improvement over earlier work is in the simplification of the construction, the explicitness of the bound for r_3, and in the \\log\\log term for general k."}
{"category": "Math", "title": "Effective non-vanishing conjectures for projective threefolds", "abstract": "Let X be a smooth projective threefold, and let A be an ample line bundle such that $K_X+A$ is nef. We show that if $K_X$ or $-K_X$ is pseudoeffective, the adjoint bundle $K_X+A$ has global sections. We also give a very short proof of the Beltrametti-Sommese conjecture in dimension three, recently proven by Fukuma: if A is an ample line bundle such that $K_X+2A$ is nef, the adjoint bundle $K_X+2A$ has global sections."}
{"category": "Math", "title": "A Structure Theory for Small Sum Subsets", "abstract": "We develop a new method leading the structure of finite subsets S and T of an abelian group with $|S+T|\\le |S|+|T|$. We show also how to recover the known results in this area in a relatively short space."}
{"category": "Math", "title": "Some remarks on the action of Quantum Isometry Groups", "abstract": "We give a new sufficient condition on a spectral triple to ensure that the quantum group of orientation and volume preserving isometries defined in \\cite{qorient} has a $C^*$-action on the underlying $C^*$ algebra."}
{"category": "Math", "title": "Quantum Isometry Group for Spectral Triples with Real Structure", "abstract": "Given a spectral triple of compact type with a real structure in the sense of [Dabrowski L., J. Geom. Phys. 56 (2006), 86-107] (which is a modification of Connes' original definition to accommodate examples coming from quantum group theory) and references therein, we prove that there is always a universal object in the category of compact quantum group acting by orientation preserving isometries (in the sense of [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572]) and also preserving the real structure of the spectral triple. This gives a natural definition of quantum isometry group in the context of real spectral triples without fixing a choice of 'volume form' as in [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572]."}
{"category": "Math", "title": "Spatial discretization of Cuntz algebras", "abstract": "The (abstract) Cuntz algebra is generated by non-unitary isometries and has therefore no intrinsic finiteness properties. To approximate the elements of the Cuntz algebra by finite-dimensional objects, we thus consider a spatial discretization of this algebra by the finite sections method. For we represent the Cuntz algebra as a (concrete) algebra of operators on a Hilbert space and associate with each operator in this algebra the sequence of its finite sections. The goal of this paper is to examine the structure of the $C^*$-algebra which is generated by all sequences of this form. Our main results are the fractality of a suitable restriction of this sequence algebra and a necessary and sufficient criterion for the stability of sequences in the restricted algebra. These results are employed to study spectral and pseudospectral approximations of elements of the Cuntz algebra."}
{"category": "Math", "title": "Exact and asymptotic $n$-tuple laws at first and last passage", "abstract": "Understanding the space-time features of how a L\\'evy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory of branching processes to name but a few. In \\cite{KD} a new quintuple law was established for a general L\\'evy process at first passage below a fixed level. In this article we use the quintuple law to establish a family of related joint laws, which we call $n$-tuple laws, for L\\'evy processes, L\\'evy processes conditioned to stay positive and positive self-similar Markov processes at both first and last passage over a fixed level. Here the integer $n$ typically ranges from three to seven. Moreover, we look at asymptotic overshoot and undershoot distributions and relate them to overshoot and undershoot distributions of positive self-similar Markov processes issued from the origin. Although the relation between the $n$-tuple laws for L\\'evy processes and positive self-similar Markov processes are straightforward thanks to the Lamperti transformation, by inter-playing the role of a (conditioned) stable processes as both a (conditioned) L\\'evy processes and a positive self-similar Markov processes, we obtain a suite of completely explicit first and last passage identities for so-called Lamperti-stable L\\'evy processes. This leads further to the introduction of a more general family of L\\'evy processes which we call hypergeometric L\\'evy processes, for which similar explicit identities may be considered."}
{"category": "Math", "title": "L^p-summability of Riesz means for the sublaplacian on complex spheres", "abstract": "In this paper we study the L^p-convergence of the Riesz means for the sublaplacian on the sphere S^{2n-1} in the complex n-dimensional space C^n. We show that the Riesz means of order delta of a function f converge to f in L^p(S^{2n-1}) when delta>delta(p):=(2n-1)|1\\2-1\\p|. The index delta(p) improves the one found by Alexopoulos and Lohoue', $2n|1\\2-1\\p|$, and it coincides with the one found by Mauceri and, with different methods, by Mueller in the case of sublaplacian on the Heisenberg group."}
{"category": "Math", "title": "Discontinuity of the Lempert function of the spectral ball", "abstract": "We give some further criteria for continuity or discontinuity of the Lempert funtion of the spectral ball $\\Omega_n$, with respect to one or both of its arguments, in terms of cyclicity the matrices involved."}
{"category": "Math", "title": "Simulating Protein Conformations through Global Optimization", "abstract": "Many researches have been working on the protein folding problem from more than half century. Protein folding is indeed one of the major unsolved problems in science. In this work, we discuss a model for the simulation of protein conformations. This simple model is based on the idea of imposing few geometric requirements on chains of atoms representing the backbone of a protein conformation. The model leads to the formulation of a global optimization problem, whose solutions correspond to conformations satisfying the desired requirements. The global optimization problem is solved by the recently proposed Monkey Search algorithm. The simplicity of the optimization problem and the effectiveness of the used meta-heuristic search allowed the simulation of a large set of high-quality conformations. We show that, even though only few geometric requirements are imposed, some of the simulated conformation results to be similar (in terms of RMSD) to conformations real proteins actually have in nature."}
{"category": "Math", "title": "Decompositions, approximate structure, transference, and the Hahn-Banach theorem", "abstract": "This paper is partly a survey of certain kinds of results and proofs in additive combinatorics, and partly a discussion of how useful the finite-dimensional Hahn-Banach theorem can be. The most interesting single result is probably a simpler proof of a key step in the proof of the Green-Tao theorem, but several other applications of the method are given. A similarly simplified proof of the Green-Tao transference principle was obtained independently (and expressed in a rather different language) by Reingold, Trevisan, Tulsiani and Vadhan."}
{"category": "Math", "title": "Specializations of elliptic surfaces, and divisibility in the Mordell-Weil group", "abstract": "Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic) degree at most $D$ over $k$, such that $\\ell^n Q=P_t$, for some $n\\geq 1$. The bound obtained depends only on $\\ell$, the surface and section in question, $D$, and the degree $[k(t):k]$; that is, it is uniform across all fibres of bounded degree. In special cases, we obtain more specific, in some instances sharp, bounds."}
{"category": "Math", "title": "Koszul cohomology and applications to moduli", "abstract": "We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of new results, including a complete solution to Green's Conjecture for arbitrary hexagonal curves. Finally, we propose several new conjectures on syzygies, including a Prym-Green conjecture for l-roots of trivial bundles as well as a strong Maximal Rank Conjecture for generic curves. To appear in the Proceedings of Clay Mathematical Institute."}
{"category": "Math", "title": "Resolvents of R-Diagonal Operators", "abstract": "We consider the resolvent $(\\lambda-a)^{-1}$ of any $R$-diagonal operator $a$ in a $\\mathrm{II}_1$-factor. Our main theorem gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the $R$-transform of the operator $|\\lambda-c|^2$ where $c$ is Voiculescu's circular operator, and give an asymptotic formula for the negative moments of $|\\lambda-a|^2$ for any $R$-diagonal $a$. We use a mixture of complex analytic and combinatorial techniques, each giving finer information where the other can give only coarse detail. In particular, we introduce {\\em partition structure diagrams}, a new combinatorial structure arising in free probability."}
{"category": "Math", "title": "Stable constant mean curvature hypersurfaces are area minimizing in small L^1 neighborhoods", "abstract": "We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood."}
{"category": "Math", "title": "An explicit formula for the Hilbert symbol of a formal group", "abstract": "Abrashkin established the Bruckner-Vostokov formula for the Hilbert symbol of a formal group under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ($\\varphi, \\Gamma$)-modules and a cohomological interpretation of Abrashkin's technique. To do this, we build ($\\varphi, \\Gamma$)-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the cup-product and the Kummer map."}
{"category": "Math", "title": "The Calabi invariant for some groups of homeomorphisms", "abstract": "We show that the Calabi homomorphism extends to some groups of homeomorphisms on exact symplectic manifolds. The construction is based on the uniqueness of the generating Hamiltonian (proved by Viterbo) of continuous Hamiltonian isotopies (introduced by Mueller and Oh)."}
{"category": "Math", "title": "A refinement of the inequality between arithmetic and geometric means", "abstract": "In this note we present a refinement of the AM-GM inequality, and then we estimate in a special case the typical size of the improvement."}
{"category": "Math", "title": "The asymptotic distribution of the largest prime divisor", "abstract": "We consider the Dickman function $\\Psi (x,y)$ in the limit when $\\ln x/\\ln y\\to \\infty $ and $\\ln \\ln x \\ln y\\to 0$. The asymptotic value is expressed in terms of the ratio of iterated loragithm of $x$ and $ln y$."}
{"category": "Math", "title": "The strong $ABC$ conjecture over function fields (after McQuillan and Yamanoi)", "abstract": "The $abc$ conjecture predicts a highly non trivial upper bound for the height of an algebraic point in terms of its discriminant and its intersection with a fixed divisor of the projective line counted without multiplicity. We describe the two independent proofs of the strong $abc$ conjecture over function fields given by McQuillan and Yamanoi. The first proof relies on tools from differential and algebraic geometry; the second relies on analytic and topological methods. They correspond respectively to the Nevanlinna and the Ahlfors approach to the Nevanlinna Second Main Theorem."}
{"category": "Math", "title": "The structure of minimizers of the frame potential on fusion frames", "abstract": "In this paper we study the fusion frame potential, that is a generalization of the Benedetto-Fickus (vectorial) frame potential to the finite-dimensional fusion frame setting. The structure of local and global minimizers of this potential is studied, when we restrict the frame potential to suitable sets of fusion frames. These minimizers are related to tight fusion frames as in the classical vector frame case. Still, tight fusion frames are not as frequent as tight frames; indeed we show that there are choices of parameters involved in fusion frames for which no tight fusion frame can exist. Thus, we exhibit necessary and sufficient conditions for the existence of tight fusion frames with prescribed parameters, involving the so-called Horn-Klyachko's compatibility inequalities. The second part of the work is devoted to the study of the minimization of the fusion frame potential on a fixed sequence of subspaces, varying the sequence of weights. We related this problem to the index of the Hadamard product by positive matrices and use it to give different characterizations of these minima."}
{"category": "Math", "title": "The irreducible components of Hilb^{4n}(P^3)", "abstract": "It is shown that Hilb^{4n}(P^3)has exactly two irreducible components."}
{"category": "Math", "title": "Tempered fundamental group and metric graph of a Mumford curve", "abstract": "This paper is an attempt to give some general results on the tempered fundamental group of a $p$-adic smooth algebraic varieties (which is a sort of analog of the topologic fundamental group of complex algebraic varieties in the p-adic world). The main result asserts that one can recover the metric structure of the graph of the stable model of a Mumford curve from the tempered fundamental group of this Mumford curve. We will also prove birational invariance, invariance by algebraically closed extensions and a Kuenneth formula for the tempered fundamental group. We will describe the tempered fundamental group of an abelian variety and link the tempered fundamental group of a curve to the tempered fundamental group of its Jacobian variety."}
{"category": "Math", "title": "Homotopy fiber products of homotopy theories", "abstract": "Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is well-behaved with respect to translating it into the setting of more general homotopy theories, given by complete Segal spaces, where we have well-defined homotopy pullbacks."}
{"category": "Math", "title": "Geometric realizations of generalized algebraic curvature operators", "abstract": "We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric torsion free connection."}
{"category": "Math", "title": "On sequence spaces for Fr\\'echet frames", "abstract": "We analyze the construction of a sequence space $\\widetilde{\\Theta}$, resp. a sequence of sequence spaces, in order to have $\\{g_i\\}$ as a $\\widetilde{\\Theta}$-frame or Banach frame for a Banach space $X$, resp. pre-$F$-frame or $F$-frame for a Fr\\'echet space $X_F=\\cap_{s\\in {\\mathbb N}_0} X_s$, where $\\{X_s\\}_{s\\in {\\mathbb N}_0}$ is a sequence of Banach spaces."}
{"category": "Math", "title": "Any flat bundle on a punctured disc has an oper structure", "abstract": "We prove that any flat G-bundle, where G is a complex connected reductive algebraic group, on the punctured disc admits the structure of an oper. This result is important in the local geometric Langlands correspondence proposed in arXiv:math/0508382. Our proof uses certain deformations of the affine Springer fibers which could be of independent interest. As a byproduct, we construct representations of affine Weyl groups on the homology of these deformations generalizing representations constructed by Lusztig."}
{"category": "Math", "title": "Noncommutative Geometry and Quantum Group Symmetries", "abstract": "Quantum groups and quantum homogeneous spaces - developed by several authors since the 80's - provide a large class of examples of algebras which for many reasons we interpret as `coordinate algebras' over noncommutative spaces. This dissertation is an attempt to understand them from the point of view of Connes' noncommutative geometry."}
{"category": "Math", "title": "Dyson's theorem for curves", "abstract": "Let $\\scriptstyle K$ be a number field and $\\scriptstyle X_1$ and $\\scriptstyle X_2$ two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson Theorem for the product $\\scriptstyle X_1 \\times X_2$. If $\\scriptstyle X_i = {\\bb P}_1$ we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on integral points on hyperbolic curves and it will give some insight on effectiveness. This proof is new and avoids the use of Roth and Mordell-Weil theorems, the theory of Linear Forms in Logarithms and the Schmidt subspace theorem."}
{"category": "Math", "title": "Erratum on \"Hadamard spaces with isolated flats\"", "abstract": "The purpose of this erratum is to correct the proof of Theorem A.0.1 in the appendix to our article ``Hadamard spaces with isolated flats'' math.GR/0411232, which was jointly authored by Mohamad Hindawi, Hruska and Kleiner. In that appendix, many of the results of math.GR/0411232 about CAT(0) spaces with isolated flats are extended to a more general setting in which the isolated subspaces are not necessarily flats. However, one step of that extension does not follow from the argument we used the isolated flats setting. We provide a new proof that fills this gap. In addition, we give a more detailed account of several other parts of Theorem A.0.1, which were sketched in math.GR/0411232."}
{"category": "Math", "title": "Analytic subvarieties with many rational points", "abstract": "We give a generalization of the classical Bombieri--Schneider--Lang criterion in transcendence theory. We give a local notion of $LG$--germ, which is similar to the notion of $E$-- function and Gevrey condition, and which generalize (and replace) the condition on derivatives in the theorem quoted above. Let $K\\subset \\Bbb C$ be a number field and $X$ a quasi--projective variety defined over $K$. Let $\\gamma\\colon M\\to X$ be an holomorphic map of finite order from a parabolic Riemann surface to $X$ such that the Zariski closure of the image of it is strictly bigger then one. Suppose that for every $p\\in X(K)\\cap\\gamma(M)$ the formal germ of $M$ near $P$ is an $LG$-- germ, then we prove that $X(K)\\cap\\gamma(M)$ is a finite set. Then we define the notion of conformally parabolic Kh\\\"aler varieties; this generalize the notion of parabolic Riemann surface. We show that on these varieties we can define a value distribution theory. The complementary of a divisor on a compact Kh\\\"aler manifold is conformally parabolic; in particular every quasi projective variety is. Suppose that $A$ is conformally parabolic variety of dimension $m$ over $\\Bbb C$ with Kh\\\"aler form $\\omega$ and $\\gamma\\colon A\\to X$ is an holomorphic map of finite order such that the Zariski closure of the image is strictly bigger then $m$. Suppose that for every $p\\in X(K)\\cap \\gamma (A)$, the image of $A$ is an $LG$--germ. then we prove that there exists a current $T$ on $A$ of bidegree $(1,1)$ such that $\\int_AT\\wedge\\omega^{m-1}$ explicitly bounded and with Lelong number bigger or equal then one on each point in $\\gamma^{-1}(X(K))$. In particular if $A$ is affine $\\gamma^{-1}(X(K))$ is not Zariski dense."}
{"category": "Math", "title": "Reidemeister torsion and analytic torsion of discs", "abstract": "We study the Reidemeister torsion and the analytic torsion of the $m$ dimensional disc in the Euclidean $m$ dimensional space, using the base for the homology defined by Ray and Singer in \\cite{RS}. We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger-M\\\"{u}ller theorem. We use a formula proved by Br\\\"uning and Ma \\cite{BM}, that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary \\cite{Luc}. Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang \\cite{DF}, and we show that the result obtained using this formula is inconsistent with the direct calculation of the analytic torsion."}
{"category": "Math", "title": "Periodic Solutions with Alternating Singularities in the Collinear Four-body Problem", "abstract": "This paper shows the existence of a periodic orbit with singularity in the symmetric collinear four body problem. In each period of the orbit, there is a binary collision (BC) between the inner two bodies and a simultaneous binary collision (SBC) of the two clusters on both sides of the origin. The system is regularized and the existence is proven by using the implicit function theorem and a continuity argument on differential equations of the regularized Hamiltonian."}
{"category": "Math", "title": "Computads and Multitopic Sets", "abstract": "We compare computads with multitopic sets. Both these kinds of structures have n-dimensional objects (called n-cells and n-pasting diagrams, respectively). The computads form a subclass of the more familiar class of omega-categories, while multitopic sets have been devised by Hermida, Makkai and Power as a vehicle for a definition of the concepts of weak omega-category. Our main result states that the category of multitopic sets is equivalent to that of many-to-one computads, a certain full subcategory of the category of all computads."}
{"category": "Math", "title": "Ramification and moduli spaces of finite flat models", "abstract": "We determine the type of the zeta functions and the range of the dimensions of the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This gives a generalization of Raynaud's theorem on the uniqueness of finite flat models in low ramifications."}
{"category": "Math", "title": "Rational Periodic Points for Degree Two Polynomial Morphisms on Projective Space", "abstract": "This article addresses the existence of $\\Q$-rational periodic points for morphisms of projective space. In particular, we construct an infinitely family of morphisms on $\\P^N$ where each component is a degree 2 homogeneous form in $N+1$ variables which has a $\\Q$-periodic point of primitive period $\\frac{(N+1)(N+2)}{2} + \\lfloor \\frac{N-1}{2}\\rfloor$. This result is then used to show that for $N$ large enough there exists morphisms of $\\P^N$ with $\\Q$-rational periodic points with primitive period larger that $c(k)N^k$ for any $k$ and some constant $c(k)$."}
{"category": "Math", "title": "Inverse pressure estimates and the independence of stable dimension for non-invertible maps", "abstract": "We use the inverse pressure concept to estimate the stable dimension for hyperbolic non-invertible maps which are conformal in the stable fibers. The non-invertible case is different than the diffeomorphism case. In particular we show that if the map is open on the respective basic set, then the stable dimension is constant everywhere. We prove also in this setting the Lipschitz continuity of the stable distribution."}
{"category": "Math", "title": "On the group of strong symplectic homeomorphisms", "abstract": "We generalize the \"hamiltonian topology\" on hamiltonian isotopies to an intrinsic \"symplectic topology\" on the space of symplectic isotopies. We use it to define the group $SSympeo(M,\\omega)$ of strong symplectic homeomorphisms, which generalizes the group $Hameo(M,\\omega)$ of hamiltonian homeomorphisms introduced by Oh and Muller. The group $SSympeo(M,\\omega)$ is arcwise connected, is contained in the identity component of $Sympeo(M,\\omega)$; it contains $Hameo(M,\\omega)$ as a normal subgroup and coincides with it when $M$ is simply connected. Finally its commutator subgroup $[SSympeo(M,\\omega),SSympeo(M,\\omega)]$ is contained in $Hameo(M,\\omega)$."}
{"category": "Math", "title": "On Pebbling Graphs by their Blocks", "abstract": "Graph pebbling is a game played on a connected graph G. A player purchases pebbles at a dollar a piece, and hands them to an adversary who distributes them among the vertices of G (called a configuration) and chooses a target vertex r. The player may make a pebbling move by taking two pebbles off of one vertex and moving one pebble to a neighboring vertex. The player wins the game if he can move k pebbles to r. The value of the game (G,k), called the k-pebbling number of G, is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer m of pebbles so that, from every configuration of size m, one can move k pebbles to any target. In this paper, we use the block structure of graphs to investigate pebbling numbers, and we present the exact pebbling number of the graphs whose blocks are complete. We also provide an upper bound for the k-pebbling number of diameter-two graphs, which can be the basis for further investigation into the pebbling numbers of graphs with blocks that have diameter at most two."}
{"category": "Math", "title": "Detecting integral polyhedral functions", "abstract": "We study the class of real-valued functions on convex subsets of R^n which are computed by the maximum of finitely many affine functionals with integer slopes. We prove several results to the effect that this property of a function can be detected by sampling on small subsets of the domain. In so doing, we recover in a unified way some prior results of the first author (some joint with Liang Xiao). We also prove that a function on R^2 is a tropical polynomial if and only if its restriction to each translate of a generic tropical line is a tropical polynomial."}
{"category": "Math", "title": "Voting in agreeable societies", "abstract": "When can a majority of voters find common ground, that is, a position they all agree upon? How does the shape of the political spectrum influence the outcome? When mathematical objects have a social interpretation, the associated theorems have social applications. In this article we give examples of situations where sets model preferences and develop extensions of classical theorems about convex sets, such as Helly's theorem, that can be used in the analysis of voting in \"agreeable\" societies."}
{"category": "Math", "title": "The Goldston-Pintz-Yildirim Sieve and Maximal Gaps", "abstract": "One field of particular interest in Number Theory concerns the gaps between consecutive primes. Within the last few years, very important results have been achieved on how small these gaps can be. The strongest of these results were obtained by Dan Goldston, Janos Pintz and Cem Yalcin Yildirim. The present work begins by generalizing their results so that they can be applied to related problems in a more direct manner. Additionally, we improve the bound for $F_2$ (concerning the maximal gap in a block of three primes) obtained by the authors' first paper with our generalization."}
{"category": "Math", "title": "The Spherical $\\pi$-Operator", "abstract": "In this article, we define the spherical $\\pi$-operator over domains in the $(n-1)$-D unit sphere $S^n$ of $R^n$ and develop new and analogous results on the operator it self and its mapping properties. We introduce the spherical Dirac operator $\\Gamma_\\alpha$ as an $\\alpha$- shift of of $\\Gamma_omega$, where $\\Gamma_omega$ is the negative of the wedge (or Grassmann) product of $\\omega$ with that of the Dirac operator $D_\\omega$. A gegenbauer polynomial is used as a Cauchy kernel for the spherical Dirac operator $\\Gamma_alpha$."}
{"category": "Math", "title": "Unitary Lie Algebras and Lie Tori of Type BC_r, r \\geq 3", "abstract": "A Lie G-torus of type X_r is a Lie algebra with two gradings -- one by an abelian group G and the other by the root lattice of a finite irreducible root system of type X_r. In this paper we construct a centreless Lie G-torus of type BC_r, which we call a unitary Lie G-torus, as it is a special unitary Lie algebra of a nondegenerate G-graded hermitian form of Witt index r over an associative torus with involution. We prove a structure theorem for centreless Lie G-tori of type BC_r, r \\geq 3, that states that any such Lie torus is bi-isomorphic to a unitary Lie G-torus, and we determine necessary and sufficient conditions for two unitary Lie G-tori to be bi-isomorphic. The motivation to investigate Lie G-tori came from the theory of extended affine Lie algebras, which are natural generalizations of the affine and toroidal Lie algebras. Every extended affine Lie algebra possesses an ideal which is a Lie n-torus of type X_r for some irreducible root system X_r, where by an n-torus we mean that the group G is a free abelian group of rank n for some n \\geq 0. The structure theorem above enables us to classify centreless Lie n-tori of type BC_r, r \\geq 3. We show that they are determined by pairs consisting of a quadratic form K on an n-dimensional Z_2-vector space and of an orbit of the orthogonal group of K. We use that result to construct extended affine Lie algebras of type BC_r, r \\geq 3. Our article completes a large project involving many earlier papers and many authors to determine the centreless Lie n-tori of all types."}
{"category": "Math", "title": "On the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4", "abstract": "A complete description of the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4 is given. As a by-product we get a positive relation among right hand Dehn twists in the mapping class group of a closed orientable surface of genus 3."}
{"category": "Math", "title": "Hyperspaces of Closed Limit Sets", "abstract": "We study Michael's lower semifinite topology and Fell's topology on the collection of all closed limit subsets of a topological space. Special attention is given to the subfamily of all maximal limit sets."}
{"category": "Math", "title": "Quotient Spaces Determined by Algebras of Continuous Functions", "abstract": "we prove that if $X$ is a locally compact $\\sigma$-compact space then on its quotient, $\\gamma(X)$ say, determined by the algebra of all real valued bounded continuous functions on $X$, the quotient topology and the completely regular topology defined by this algebra are equal. It follows from this that if $X$ is second countable locally compact then $\\gamma(X)$ is second countable locally compact Hausdorff if and only if it is first countable. The interest in these results originated in papers of R. J. Archbold, and S. Echterhoff and D. P. Williams where the primitive ideal space of a $C^*$-algebra was considered."}
{"category": "Math", "title": "Lagrangian structures for the Stokes, Navier-Stokes and Euler equations", "abstract": "We prove that the Navier-Stokes, the Euler and the Stokes equations admit a Lagrangian structure using the stochastic embedding of Lagrangian systems. These equations coincide with extremals of an explicit stochastic Lagrangian functional, i.e. they are stochastic Lagrangian systems in the sense of [Cresson-Darses, J. Math. Phys. 48, 072703 (2007]"}
{"category": "Math", "title": "On the connection between two quasilinear elliptic problems with source terms of order 0 or 1", "abstract": "We establish a precise connection between two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\\mathbb{R}^{N}.$ The first one, of the form \\[ -\\Delta_{p}u=\\beta(u)| \\nabla u| ^{p}+\\lambda f(x)+\\alpha, \\] involves a source gradient term with natural growth, where $\\beta$ is nonnegative, $\\lambda>0,f(x)\\geqq0$, and $\\alpha$ is a nonnegative measure. The second one, of the form \\[ -\\Delta_{p}v=\\lambda f(x)(1+g(v))^{p-1}+\\mu, \\] presents a source term of order $0, $where $g$ is nondecreasing, and $\\mu$ is a nonnegative measure. Here $\\beta$ and $g$ can present an asymptote. The correlation gives new results of existence, nonexistence, regularity and multiplicity of the solutions for the two problems, without or with measures. New informations on the extremal solutions are given when $g$ is superlinear."}
{"category": "Math", "title": "The first digit frequencies of primes and Riemann zeta zeros tend to uniformity following a size-dependent generalized Benford's law", "abstract": "Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated sci- entists of all ages to search for local and global patterns in this distribution that eventually could shed light into the ultimate nature of primes. In this work we show that a generalization of the well known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in data sets, describes with astonishing precision the statistical distribution of leading digits in the prime numbers sequence. Moreover, a reciprocal version of this pattern also takes place in the sequence of the nontrivial Riemann zeta zeros. We prove that the prime number theorem is, in the last analysis, the responsible of these patterns. Some new relations concerning the prime numbers distribution are also deduced, including a new approximation to the counting function pi(n). Furthermore, some relations concerning the statistical conformance to this generalized Benford's law are derived. Some applications are finally discussed."}
{"category": "Math", "title": "Compressed word problems in HNN-extensions and amalgamated products", "abstract": "It is shown that the compressed word problem for an HNN-extension with base group H and finite associated subgroups is polynomial time Turing-reducible to the compressed word problem for H. An analogous result for amalgamated free products is shown as well."}
{"category": "Math", "title": "Slopes of trigonal fibred surfaces and of higher dimensional fibrations", "abstract": "We give lower bounds for the slope of higher dimensional fibrations over curves under conditions of GIT-semistability of the fibres, using a generalization of a method of Cornalba and Harris. With the same method we establish a sharp lower bound for the slope of trigonal fibrations of even genus and general Maroni invariant; in particular this result proves a conjecture due to Harris and Stankova-Frenkel."}
{"category": "Math", "title": "The Derivation algebra and automorphism group of the generalized Ramond N=2 superconformal algebra", "abstract": "In this paper, we give the definition of the generalized Ramond N=2 superconformal algebras and discuss the derivation algebra and the automorphism group"}
{"category": "Math", "title": "Homology of spaces of regular loops in the sphere", "abstract": "In this paper we compute the singular homology of the space of immersions of the circle into the $n$-sphere. Equipped with Chas-Sullivan's loop product these homology groups are graded commutative algebras, we also compute these algebras. We enrich Morse spectral sequences for fibrations of free loop spaces together with loop products, this offers some new computational tools for string topology."}
{"category": "Math", "title": "The adic realization of the Morse transformation and the extension of its action on the solenoid", "abstract": "We consider the adic realization of the Morse transformation on the additive group of integer dyadic numbers. We discuss the arithmetic properties of that action. Then we extend that action to the action of the group of rational dyadic numbers on the solenoid."}
{"category": "Math", "title": "On mappings of terms determined by hypersubstitutions", "abstract": "The extensions of hypersubstitutions are mappings on the set of all terms. In the present paper we characterize all hypersubstitutions which provide bijections on the set of all terms. The set of all such hypersubstitutions forms a monoid. On the other hand, one can modify each hypersubstitution to any mapping on the set of terms. For this we can consider mappings from the set of all hypersubstitutions into the set of all mappings on the set of all terms."}
{"category": "Math", "title": "Strong Approximation of Empirical Copula Processes by Gaussian Processes", "abstract": "We provide the strong approximation of empirical copula processes by a Gaussian process. In addition we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm."}
{"category": "Math", "title": "Viscoelastic fluids in thin domains: a mathematical proof", "abstract": "The present paper deals with non Newtonian viscoelastic flows of Oldroyd-B tye in thin domains. Such geometries arise for example in the context of lubrication. More precisely, we justify rigorously the asymptotic model obtained heuristically by proving the mathematical convergence of the Navier-Stokes/Oldroyd-B sytem towards the asymptotic model."}
{"category": "Math", "title": "The Banach space -valued BMO, Carleson's condition, and paraproducts", "abstract": "We define a scale of L^q Carleson norms, all of which characterize the membership of a function in BMO. The phenomenon is analogous to the John-Nirenberg inequality, but on the level of Carleson measures. The classical Carleson condition corresponds to the L^2 case in our theory. The result is applied to give a new proof for the L^p-boundedness of paraproducts with a BMO symbol. A novel feature of the argument is that all p are covered at once in a completely interpolation-free manner. This is achieved by using the L^1 Carleson norm, and indicates the usefulness of this notion. Our approach is chosen so that all these results extend in a natural way to the case of X-valued functions, where X is a Banach space with the UMD property."}
{"category": "Math", "title": "On the computation of classical, boolean and free cumulants", "abstract": "This paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for classical, boolean and free cumulants. This is realized by using a suitable polynomial representation of Abel polynomials. The algorithm relies on the classical umbral calculus, a symbolic language introduced by Rota and Taylor in 1994, that is particularly suited to be implemented by using software for symbolic computations. Here we give a MAPLE procedure. Comparisons with existing procedures, especially for conversions between moments and free cumulants, as well as examples of applications to some well-known distributions (classical and free) end the paper."}
{"category": "Math", "title": "Approximate Bayesian computation (ABC) gives exact results under the assumption of model error", "abstract": "Approximate Bayesian computation (ABC) or likelihood-free inference algorithms are used to find approximations to posterior distributions without making explicit use of the likelihood function, depending instead on simulation of sample data sets from the model. In this paper we show that under the assumption of the existence of a uniform additive model error term, ABC algorithms give exact results when sufficient summaries are used. This interpretation allows the approximation made in many previous application papers to be understood, and should guide the choice of metric and tolerance in future work. ABC algorithms can be generalized by replacing the 0-1 cut-off with an acceptance probability that varies with the distance of the simulated data from the observed data. The acceptance density gives the distribution of the error term, enabling the uniform error usually used to be replaced by a general distribution. This generalization can also be applied to approximate Markov chain Monte Carlo algorithms. In light of this work, ABC algorithms can be seen as calibration techniques for implicit stochastic models, inferring parameter values in light of the computer model, data, prior beliefs about the parameter values, and any measurement or model errors."}
{"category": "Math", "title": "Quantum Witten localization and abelianization for qde solutions", "abstract": "We prove a quantum version of the localization formula of Witten that relates invariants of a git quotient with the equivariant invariants of the action. Using the formula we prove a quantum version of an abelianization formula of S. Martin relating invariants of geometric invariant theory quotients by a group and its maximal torus, conjectured by Bertram, Ciocan-Fontanine, and Kim. By similar techniques we prove a quantum Lefschetz principle for holomorphic symplectic reductions. As an application, we give a formula for the fundamental solution to the quantum differential equation (qde) for the moduli space of points on the projective line and for the smoothed moduli space of framed sheaves on the projective plane (a Nakajima quiver variety)."}
{"category": "Math", "title": "Estimates in the Generalized Morrey Spaces for Linear Parabolic Systems", "abstract": "The purpose of this paper is to study the parabolic system $u_t^i-D_\\alpha(a_{ij}^{\\alpha\\beta}D_\\beta u^j)=-div f^i$ in the generalized Morrey Space $L_{\\phi}^{2,\\lambda}$ . We would like to understand the regularity of the solutions of this system. It will be shown that 1: if $a_{ij}^{\\alpha\\beta}\\in C(\\bar{Q_{T}})$ then $Du\\in L_\\phi^{2,\\lambda}$, and 2: if $a_{ij}^{\\alpha\\beta}\\in VMO(Q_{T})$ then $Du\\in L_\\phi^{2,\\lambda}$. Moreover we will be able to obtain estimates on the gradient of the solutions to the system, which will tell us about the regularity of the solutions."}
{"category": "Math", "title": "Distribution of Normalized Zero-Sets of Random Entire Functions", "abstract": "This paper is concerned with the distribution of normalized zero-sets of random entire functions. The normalization of the zero-set is performed in the same way as that of the counting function for an entire function in Nevanlinna theory. The result generalizes the Shiffman and Zelditch theory on the distribution of the zeroes of random holomorphic sections of powers for positive Hermitian holomorphic line bundles from polynomial functions to entire functions. Our result can also be viewed as the analogy of Nevanlinna's First Main Theorem in the theory of the distribution of zero-sets of random entire functions."}
{"category": "Math", "title": "$p-$Ferrer diagram, $p-$linear ideals and arithmetical rank", "abstract": "In this paper we introduce $p-$Ferrer diagram, note that $1-$ Ferrer diagram are the usual Ferrer diagrams or Ferrer board, and corresponds to planar partitions. To any $p-$Ferrer diagram we associate a $p-$Ferrer ideal. We prove that $p-$Ferrer ideal have Castelnuovo mumford regularity $p+1$. We also study Betti numbers, minimal resolutions of $p-$Ferrer ideals. Every $p-$Ferrer ideal is $p-$joined ideals in a sense defined in a fortcoming paper \\cite{m2}, which extends the notion of linearly joined ideals introduced and developped in the papers \\cite{bm2}, \\cite{bm4},\\cite{eghp} and \\cite{m1}. We can observe the connection between the results on this paper about the Poincar\\'e series of a $p-$Ferrer diagram $\\Phi $and the rook problem, which consist to put $k$ rooks in a non attacking position on the $p-$Ferrer diagram $\\Phi $."}
{"category": "Math", "title": "Reducing conjugacy in the full diffeomorphism group of R to conjugacy in the subgroup of orientation-preserving maps", "abstract": "Let $\\Diffeo=\\Diffeo(\\R)$ denote the group of infinitely-differentiable diffeomorphisms of the real line $\\R$, under the operation of composition, and let $\\Diffeo^+$ be the subgroup of diffeomorphisms of degree +1, i.e. orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements $f,g\\in \\Diffeo$ are conjugate in $\\Diffeo$ to associated conjugacy problems in the subgroup $\\Diffeo^+$. The main result concerns the case when $f$ and $g$ have degree -1, and specifies (in an explicit and verifiable way) precisely what must be added to the assumption that their (compositional) squares are conjugate in $\\Diffeo^+$, in order to ensure that $f$ is conjugated to $g$ by an element of $\\Diffeo^+$. The methods involve formal power series, and results of Kopell on centralisers in the diffeomorphism group of a half-open interval."}
{"category": "Math", "title": "Gevrey solutions of irregular hypergeometric systems in two variables", "abstract": "We describe the Gevrey series solutions at singular points of the irregular hypergeometric system (GKZ system) associated with an affine plane monomial curve. We also describe the irregularity complex of such a system with respect to its singular support."}
{"category": "Math", "title": "Gevrey solutions of the irregular hypergeometric system associated with an affine monomial curve", "abstract": "We describe the Gevrey series solutions at singular points of the irregular hypergeometric system (GKZ system) associated with an affine monomial curve. We also describe the irregularity complex of such a system with respect to its singular support. We use restriction and some results in D-module theory to reduce our study to the two dimensional case."}
{"category": "Math", "title": "A combinatorial description of the $U^2=0$ version of Heegaard Floer homology", "abstract": "We show that every 3--manifold admits a Heegaard diagram in which a truncated version of Heegaard Floer homology (when the holomorpic disks pass through the basepoints at most once) can be computed combinatorially."}
{"category": "Math", "title": "Existence and stability of foliations by J-holomorphic spheres", "abstract": "We study the existence and stability of holomorphic foliations in dimension greater than 4 under perturbations of the underlying almost complex structure. An example is given to show that, unlike in dimension 4, J-holomorphic foliations are not stable under large perturbations of almost complex structure."}
{"category": "Math", "title": "Rationally connected foliations on surfaces", "abstract": "In this short note we study foliations with rationally connected leaves on surfaces. Our main result is that on surfaces there exists a polarisation such that the Harder-Narasimhan filtration of the tangent bundle with respect to this polarisation yields the maximal rationally quotient of the surface."}
{"category": "Math", "title": "Toolbox", "abstract": "Contains various tools for preferential and related logics"}
{"category": "Math", "title": "Double bubbles in $S^3$ and $H^3$", "abstract": "We prove the double bubble conjecture in the three-sphere $S^3$ and hyperbolic three-space $H^3$ in the cases where we can apply Hutchings theory: 1) in $S^3$, each enclosed volume and the complement occupy at least 10% of the volume of $S^3$; 2) in $H^3$, the smaller volume is at least 85% that of the larger. A balancing argument and asymptotic analysis reduce the problem in $S^3$ and $H^3$ to some computer checking. The computer analysis has been designed and fully implemented for both spaces."}
{"category": "Math", "title": "Principal gradient schemes have regular reduced closed subschemes", "abstract": "We prove that principal gradient schemes have regular reduced subschemes. We also obtain a regularity criterion for reduced quotient rings."}
{"category": "Math", "title": "Non-unitarisable representations and random forests", "abstract": "We establish a connection between Dixmier's unitarisability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is non-unitarisable if its first L2-Betti number is non-zero or if it is finitely generated with non-trivial cost. Our criterion also applies to torsion groups constructed by D. Osin, thus providing the first examples of non-unitarisable groups not containing a non-Abelian free subgroup."}
{"category": "Math", "title": "Computing Irreducible Decomposition of Monomial Ideals", "abstract": "The paper presents two algorithms for finding irreducible decomposition of monomial ideals. The first one is recursive, derived from staircase structures of monomial ideals. This algorithm has a good performance for highly non-generic monomial ideals. The second one is an incremental algorithm, which computes decompositions of ideals by adding one generator at a time. Our analysis shows that the second algorithm is more efficient than the first one for generic monomial ideals. Furthermore, the time complexity of the second algorithm is at most $O(n^2p\\ell)$ where $n$ is the number of variables, $p$ is the number of minimal generators and $\\ell$ is the number of irreducible components. Another novelty of the second algorithm is that, for generic monomial ideals, the intermediate storage is always bounded by the final output size which may be exponential in the input size."}
{"category": "Math", "title": "ADI finite difference schemes for option pricing in the Heston model with correlation", "abstract": "This paper deals with the numerical solution of the Heston partial differential equation that plays an important role in financial option pricing, Heston (1993, Rev. Finan. Stud. 6). A feature of this time-dependent, two-dimensional convection-diffusion-reaction equation is the presence of a mixed spatial-derivative term, which stems from the correlation between the two underlying stochastic processes for the asset price and its variance. Semi-discretization of the Heston PDE, using finite difference schemes on a non-uniform grid, gives rise to large systems of stiff ordinary differential equations. For the effective numerical solution of these systems, standard implicit time-stepping methods are often not suitable anymore, and tailored time-discretization methods are required. In the present paper, we investigate four splitting schemes of the Alternating Direction Implicit (ADI) type: the Douglas scheme, the Craig & Sneyd scheme, the Modified Craig & Sneyd scheme, and the Hundsdorfer & Verwer scheme - each of which contains a free parameter. ADI schemes were not originally developed to deal with mixed spatial-derivative terms. Accordingly, we first discuss the adaptation of the above four ADI schemes to the Heston equation. Subsequently, we present various numerical examples with realistic data sets from the literature, where we consider European call options as well as down-and-out barrier options. Combined with ample theoretical stability results for ADI schemes that have recently been obtained in In 't Hout & Welfert (2007, Appl. Numer. Math.), we arrive at three ADI schemes that all prove to be very effective in the numerical solution of the Heston PDE with a mixed derivative term."}
{"category": "Math", "title": "A CM construction for curves of genus 2 with p-rank 1", "abstract": "We construct Weil numbers corresponding to genus-2 curves with $p$-rank 1 over the finite field $\\F_{p^2}$ of $p^2$ elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of $\\F_{p^2}$-valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over $\\F_{p^2}$ out of necessity: we show that curves of $p$-rank 1 over $\\F_p$ for large $p$ cannot be efficiently constructed using explicit CM constructions."}
{"category": "Math", "title": "Noncommutative Koszul Algebras from Combinatorial Topology", "abstract": "Associated to any uniform finite layered graph Gamma there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups H_X(n,k), generalizing the usual cohomology groups H^n(X). Along with several other results, our methods give a new and primarily topological proof of a result of Serconek and Wilson and of Piontkovski."}
{"category": "Math", "title": "On counting rings of integers as Galois modules", "abstract": "Let $K$ be a number field and $G$ a finite abelian group. We study the asymptotic behaviour of the number of tamely ramified $G$-extensions of $K$ with ring of integers of fixed realisable class as a Galois module."}
{"category": "Math", "title": "An extension of a logarithmic form of Cramer's ruin theorem to some FARIMA and related processes", "abstract": "Cramer's theorem provides an estimate for the tail probability of the maximum of a random walk with negative drift and increments having a moment generating function finite in a neighborhood of the origin. The class of (g,F)-processes generalizes in a natural way random walks and fractional ARIMA models used in time series analysis. For those (g,F)-processes with negative drift, we obtain a logarithmic estimate of the tail probability of their maximum, under conditions comparable to Cramer's. Furthermore, we exhibit the most likely paths as well as the most likely behavior of the innovations leading to a large maximum."}
{"category": "Math", "title": "Polynomiality of some hook-length statistics", "abstract": "We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions: \\frac{1}{n!} \\sum_{\\lambda \\vdash n} f_{\\lambda}^2 \\sum_{u \\in \\lambda} \\prod_{i=1}^{r}(h_u^2 - i^2) = \\frac{1}{2(r+1)^2} \\binom{2r}{r}\\binom{2r+2}{r+1} \\prod_{j=0}^{r} (n-j), where $f_{\\lambda}$ is the number of standard Young tableaux of shape $\\lambda$ and $h_u$ is the hook length of the square $u$ of the Young diagram of $\\lambda$. We also obtain other similar formulas."}
{"category": "Math", "title": "Estimation and tests for models satisfying linear constraints with unknown parameter", "abstract": "We introduce estimation and test procedures through divergence minimization for models satisfying linear constraints with unknown parameter. Several statistical examples and motivations are given. These procedures extend the empirical likelihood (EL) method and share common features with generalized empirical likelihood (GEL). We treat the problems of existence and characterization of the divergence projections of probability measures on sets of signed finite measures. Our approach allows for a study of the estimates under misspecification. The asymptotic behavior of the proposed estimates are studied using the dual representation of the divergences and the explicit forms of the divergence projections. We discuss the problem of the choice of the divergence under various respects. Also we handle efficiency and robustness properties of minimum divergence estimates. A simulation study shows that the Hellinger divergence enjoys good efficiency and robustness properties."}
{"category": "Math", "title": "Sufficient conditions on observability grammian for synchronization in arrays of coupled time-varying linear systems", "abstract": "Synchronizability of stable, output-coupled, identical, time-varying linear systems is studied. It is shown that if the observability grammian satisfies a persistence of excitation condition, then there exists a bounded, time-varying linear feedback law that yields exponential synchronization for all fixed, asymmetrical interconnections with connected graphs. Also, a weaker condition on the grammian is given for asymptotic synchronization. No assumption is made on the strength of coupling. Moreover, related to the main problem, a particular array of output-coupled systems that is pertinent to much-studied consensus problems is investigated. In this array, the individual systems are integrators with identical, time-varying, symmetric positive semi-definite output matrices. Trajectories of this array are shown to stay bounded using a time-invariant, quadratic Lyapunov function. Also, sufficient conditions on output matrix for synchronization are provided. All of the results in the paper are generated for both continuous time and discrete time."}
{"category": "Math", "title": "Kernel Regression by Mode Calculation of the Conditional Probability Distribution", "abstract": "The most direct way to express arbitrary dependencies in datasets is to estimate the joint distribution and to apply afterwards the argmax-function to obtain the mode of the corresponding conditional distribution. This method is in practice difficult, because it requires a global optimization of a complicated function, the joint distribution by fixed input variables. This article proposes a method for finding global maxima if the joint distribution is modeled by a kernel density estimation. Some experiments show advantages and shortcomings of the resulting regression method in comparison to the standard Nadaraya-Watson regression technique, which approximates the optimum by the expectation value."}
{"category": "Math", "title": "Pivots, Determinants, and Perfect Matchings of Graphs", "abstract": "We give a characterization of the effect of sequences of pivot operations on a graph by relating it to determinants of adjacency matrices. This allows us to deduce that two sequences of pivot operations are equivalent iff they contain the same set S of vertices (modulo two). Moreover, given a set of vertices S, we characterize whether or not such a sequence using precisely the vertices of S exists. We also relate pivots to perfect matchings to obtain a graph-theoretical characterization. Finally, we consider graphs with self-loops to carry over the results to sequences containing both pivots and local complementation operations."}
{"category": "Math", "title": "Finiteness for the k-factor model and chirality varieties", "abstract": "This paper deals with two families of algebraic varieties arising from applications. First, the k-factor model in statistics, consisting of n-times-n covariance matrices of n observed Gaussian variables that are pairwise independent given k hidden Gaussian variables. Second, chirality varieties inspired by applications in chemistry. A point in such a chirality variety records chirality measurements of all k-subsets among an n-set of ligands. Both classes of varieties are given by a parameterisation, while for applications having polynomial equations would be desirable. For instance, such equations could be used to test whether a given point lies in the variety. We prove that in a precise sense, which is different for the two classes of varieties, these equations are finitely characterisable when k is fixed and n grows."}
{"category": "Math", "title": "Multiplicative Aspects of the Halperin-Carlsson Conjecture", "abstract": "We use the multiplicative structure of the Koszul resolution to give short and simple proofs of some known estimates for the total dimension of the cohomology of spaces which admit free torus actions and analogous results for filtered differential modules over polynomial rings. We also point out the possibility of improving these results in the presence of a multiplicative structure on the so-called minimal Hirsch-Brown model for the equivariant cohomology of the space."}
{"category": "Math", "title": "A unified Pietsch domination theorem", "abstract": "In this paper we prove an abstract version of Pietsch's domination theorem which unify a number of known Pietsch-type domination theorems for classes of mappings that generalize the ideal of absolutely p-summing linear operators. A final result shows that Pietsch-type dominations are totally free from algebraic conditions, such as linearity, multilinearity, etc."}
{"category": "Math", "title": "Flips and variation of moduli schemes of sheaves on a surface", "abstract": "Let $H$ be an ample line bundle on a non-singular projective surface $X$, and $M(H)$ the coarse moduli scheme of rank-two $H$-semistable sheaves with fixed Chern classes on $X$. We show that if $H$ changes and passes through walls to get closer to $K_X$, then $M(H)$ undergoes natural flips with respect to canonical divisors. When $X$ is minimal and its Kodaira dimension is positive, this sequence of flips terminates in $M(H_X)$; $H_X$ is an ample line bundle lying so closely to $K_X$ that the canonical divisor of $M(H_X)$ is nef. Remark that so-called Thaddeus-type flips somewhat differ from flips with respect to canonical divisors."}
{"category": "Math", "title": "Conditions for synchronizability in arrays of coupled linear systems", "abstract": "Synchronization control in arrays of identical output-coupled continuous-time linear systems is studied. Sufficiency of new conditions for the existence of a synchronizing feedback law are analyzed. It is shown that for neutrally stable systems that are detectable form their outputs, a linear feedback law exists under which any number of coupled systems synchronize provided that the (directed, weighted) graph describing the interconnection is fixed and connected. An algorithm generating one such feedback law is presented. It is also shown that for critically unstable systems detectability is not sufficient, whereas full-state coupling is, for the existence of a linear feedback law that is synchronizing for all connected coupling configurations."}
{"category": "Math", "title": "The complexity of certain Specht modules for the symmetric group", "abstract": "During the 2004-2005 academic year the VIGRE algebra research group at the University of Georgia computed the complexities of certain Specht modules S^\\lambda for the symmetric group, using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the p-rank of the defect group of its block. The Georgia group conjectured that, generically, the complexity of a Specht module attains this maximal value; that it is smaller precisely when the Young diagram of $\\lambda$ is built out of $p \\times p$ blocks. We prove one direction of this conjecture by showing these Specht modules do indeed have less than maximal complexity. It turns out that this class of partitions, which has not previously appeared in the literature, arises naturally as the solution to a question about the $p$-weight of partitions and branching."}
{"category": "Math", "title": "On the Existence of $U$-Polygons of Class $c\\geq 4$ in Planar Point Sets", "abstract": "For a finite set $U$ of directions in the Euclidean plane, a convex non-degenerate polygon $P$ is called a $U$-polygon if every line parallel to a direction of $U$ that meets a vertex of $P$ also meets another vertex of $P$. We characterize the numbers of edges of $U$-polygons of class $c\\geq4$ with all their vertices in certain subsets of the plane and derive explicit results in the case of cyclotomic model sets."}
{"category": "Math", "title": "Atomic toposes and countable categoricity", "abstract": "We give a model-theoretic characterization of the class of geometric theories classified by an atomic topos having enough points; in particular, we show that every complete geometric theory classified by an atomic topos is countably categorical. Some applications are also discussed."}
{"category": "Math", "title": "A Note on Coincidence Isometries of Modules in Euclidean Space", "abstract": "It is shown that the coincidence isometries of certain modules in Euclidean $n$-space can be decomposed into a product of at most $n$ coincidence reflections defined by their non-zero elements. This generalizes previous results obtained for lattices to situations that are relevant in quasicrystallography."}
{"category": "Math", "title": "On the residual dependence index of elliptical distributions", "abstract": "The residual dependence index of bivariate Gaussian distributions is determined by the correlation coefficient. This tail index is of certain statistical importance when extremes and related rare events of bivariate samples with asymptotic independent components are being modeled. In this paper we calculate the partial residual dependence indices of a multivariate elliptical random vector assuming that the associated random radius is in the Gumbel max-domain of attraction. Furthermore, we discuss the estimation of these indices when the associated random radius possesses a Weibull-tail distribution."}
{"category": "Math", "title": "Note on the X_(1)-Laguerre orthogonal polynomials", "abstract": "This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Laguerre polynomials which are orthogonal in a weighted Hilbert function space on the positive half-line of the real line. These polynomials are generated by a second-order ordinary linear differential equation with a spectral parameter. Some additional information on the Sturm-Liouville form of this equation is given in this note, together with details of the singular differential operators generated in the weighted Hilbert function space. In particular, structured boundary conditions are given to determine the special self-adjoint operator, whose discrete spectrum and associated eigenvectors yield the X_(1)-Laguerre polynomials."}
{"category": "Math", "title": "Two Strange Constructions in the Euclidean Plane", "abstract": "We present two new constructions in the usual euclidean plane. We only deal with 'Grecian Geometry', with this phrase we mean elementary geometry in the two-dimensional space R 2 . We describe and prove two propositions about 'projections'. The proofs need only elementary analytical knowledge."}
{"category": "Math", "title": "Semiuniform semigroups and convolution", "abstract": "Semiuniform semigroups provide a natural setting for the convolution of generalized finite measures on semigroups. A semiuniform semigroup is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the semigroup is contained in an ambit. In the convolution algebras constructed over ambitable semigroups, topological centres have a tractable characterization."}
{"category": "Math", "title": "Entropy inference and the James-Stein estimator, with application to nonlinear gene association networks", "abstract": "We present a procedure for effective estimation of entropy and mutual information from small-sample data, and apply it to the problem of inferring high-dimensional gene association networks. Specifically, we develop a James-Stein-type shrinkage estimator, resulting in a procedure that is highly efficient statistically as well as computationally. Despite its simplicity, we show that it outperforms eight other entropy estimation procedures across a diverse range of sampling scenarios and data-generating models, even in cases of severe undersampling. We illustrate the approach by analyzing E. coli gene expression data and computing an entropy-based gene-association network from gene expression data. A computer program is available that implements the proposed shrinkage estimator."}
{"category": "Math", "title": "Pseudo-parallel Lagrangian submanifolds are semi-parallel", "abstract": "We prove a conjecture formulated by Pablo M. Chacon and Guillermo A. Lobos in [Pseudo-parallel Lagrangian submanifolds in complex space forms, Differential Geom. Appl.] stating that every Lagrangian pseudo-parallel submanifold of a complex space form of dimension at least 3 is semi-parallel."}
{"category": "Math", "title": "Approximately dual frame pairs in Hilbert spaces and applications to Gabor frames", "abstract": "We discuss the concepts of pseudo-dual frames and approximately dual frames, and illuminate their relationship to classical frames. Approximately dual frames are easier to construct than the classical dual frames, and might be tailored to yield almost perfect reconstruction. For approximately dual frames constructed via perturbation theory, we provide a bound on the deviation from perfect reconstruction. An alternative bound is derived for the rich class of Gabor frames, by using the Walnut representation of the frame operator to estimate the deviation from equality in the duality conditions. As illustration of the results, we construct explicit approximate duals of Gabor frames generated by the Gaussian; these approximate duals yield almost perfect reconstruction. Amazingly, the method applies also to certain Gabor frames that are far from being tight."}
{"category": "Math", "title": "Smooth and irreducible multigraded Hilbert schemes", "abstract": "The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is ZZ[x,y], which establishes a conjecture of Haiman and Sturmfels."}
{"category": "Math", "title": "Complete homogeneous symmetric polynomials in Jucys-Murphy elements and the Weingarten function", "abstract": "A connection is made between complete homogeneous symmetric polynomials in Jucys-Murphy elements and the unitary Weingarten function from random matrix theory. In particular we show that $h_r(J_1,...,J_n),$ the complete homogeneous symmetric polynomial of degree $r$ in the JM elements, coincides with the $r$th term in the asymptotic expansion of the Weingarten function. We use this connection to determine precisely which conjugacy classes occur in the class basis resolution of $h_r(J_1,...,J_n),$ and to explicitly determine the coefficients of the classes of minimal height when $r < n.$ These coefficients, which turn out to be products of Catalan numbers, are governed by the Moebius function of the non-crossing partition lattice $NC(n).$"}
{"category": "Math", "title": "Reversibility in the diffeomorphism group of the real line", "abstract": "An element of a group is said to be reversible if it is conjugate to its inverse. We characterise the reversible elements in the group of diffeomorphisms of the real line, and in the subgroup of order preserving diffeomorphisms."}
{"category": "Math", "title": "Random Forests: some methodological insights", "abstract": "This paper examines from an experimental perspective random forests, the increasingly used statistical method for classification and regression problems introduced by Leo Breiman in 2001. It first aims at confirming, known but sparse, advice for using random forests and at proposing some complementary remarks for both standard problems as well as high dimensional ones for which the number of variables hugely exceeds the sample size. But the main contribution of this paper is twofold: to provide some insights about the behavior of the variable importance index based on random forests and in addition, to propose to investigate two classical issues of variable selection. The first one is to find important variables for interpretation and the second one is more restrictive and try to design a good prediction model. The strategy involves a ranking of explanatory variables using the random forests score of importance and a stepwise ascending variable introduction strategy."}
{"category": "Math", "title": "High-dimensional covariance estimation by minimizing $\\ell_1$-penalized log-determinant divergence", "abstract": "Given i.i.d. observations of a random vector $X \\in \\mathbb{R}^p$, we study the problem of estimating both its covariance matrix $\\Sigma^*$, and its inverse covariance or concentration matrix {$\\Theta^* = (\\Sigma^*)^{-1}$.} We estimate $\\Theta^*$ by minimizing an $\\ell_1$-penalized log-determinant Bregman divergence; in the multivariate Gaussian case, this approach corresponds to $\\ell_1$-penalized maximum likelihood, and the structure of $\\Theta^*$ is specified by the graph of an associated Gaussian Markov random field. We analyze the performance of this estimator under high-dimensional scaling, in which the number of nodes in the graph $p$, the number of edges $s$ and the maximum node degree $d$, are allowed to grow as a function of the sample size $n$. In addition to the parameters $(p,s,d)$, our analysis identifies other key quantities covariance matrix $\\Sigma^*$; and (b) the $\\ell_\\infty$ operator norm of the sub-matrix $\\Gamma^*_{S S}$, where $S$ indexes the graph edges, and $\\Gamma^* = (\\Theta^*)^{-1} \\otimes (\\Theta^*)^{-1}$; and (c) a mutual incoherence or irrepresentability measure on the matrix $\\Gamma^*$ and (d) the rate of decay $1/f(n,\\delta)$ on the probabilities $ \\{|\\hat{\\Sigma}^n_{ij}- \\Sigma^*_{ij}| > \\delta \\}$, where $\\hat{\\Sigma}^n$ is the sample covariance based on $n$ samples. Our first result establishes consistency of our estimate $\\hat{\\Theta}$ in the elementwise maximum-norm. This in turn allows us to derive convergence rates in Frobenius and spectral norms, with improvements upon existing results for graphs with maximum node degrees $d = o(\\sqrt{s})$. In our second result, we show that with probability converging to one, the estimate $\\hat{\\Theta}$ correctly specifies the zero pattern of the concentration matrix $\\Theta^*$."}
{"category": "Math", "title": "Zero-state Markov switching count-data models: an empirical assessment", "abstract": "In this study, a two-state Markov switching count-data model is proposed as an alternative to zero-inflated models to account for the preponderance of zeros sometimes observed in transportation count data, such as the number of accidents occurring on a roadway segment over some period of time. For this accident-frequency case, zero-inflated models assume the existence of two states: one of the states is a zero-accident count state, in which accident probabilities are so low that they cannot be statistically distinguished from zero, and the other state is a normal count state, in which counts can be non-negative integers that are generated by some counting process, for example, a Poisson or negative binomial. In contrast to zero-inflated models, Markov switching models allow specific roadway segments to switch between the two states over time. An important advantage of this Markov switching approach is that it allows for the direct statistical estimation of the specific roadway-segment state (i.e., zero or count state) whereas traditional zero-inflated models do not. To demonstrate the applicability of this approach, a two-state Markov switching negative binomial model (estimated with Bayesian inference) and standard zero-inflated negative binomial models are estimated using five-year accident frequencies on Indiana interstate highway segments. It is shown that the Markov switching model is a viable alternative and results in a superior statistical fit relative to the zero-inflated models."}
{"category": "Math", "title": "Markov switching multinomial logit model: an application to accident injury severities", "abstract": "In this study, two-state Markov switching multinomial logit models are proposed for statistical modeling of accident injury severities. These models assume Markov switching in time between two unobserved states of roadway safety. The states are distinct, in the sense that in different states accident severity outcomes are generated by separate multinomial logit processes. To demonstrate the applicability of the approach presented herein, two-state Markov switching multinomial logit models are estimated for severity outcomes of accidents occurring on Indiana roads over a four-year time interval. Bayesian inference methods and Markov Chain Monte Carlo (MCMC) simulations are used for model estimation. The estimated Markov switching models result in a superior statistical fit relative to the standard (single-state) multinomial logit models. It is found that the more frequent state of roadway safety is correlated with better weather conditions. The less frequent state is found to be correlated with adverse weather conditions."}
{"category": "Math", "title": "Traces of high powers of the Frobenius class in the hyperelliptic ensemble", "abstract": "The zeta function of a curve over a finite field may be expressed in terms of the characteristic polynomial of a unitary symplectic matrix, called the Frobenius class of the curve. We compute the expected value of the trace of the n-th power of the Frobenius class for an ensemble of hyperelliptic curves of genus g over a fixed finite field in the limit of large genus, and compare the results to the corresponding averages over the unitary symplectic group USp(2g). We are able to compute the averages for powers n almost up to 4g, finding agreement with the Random Matrix results except for small n and for n=2g. As an application we compute the one-level density of zeros of the zeta function of the curves, including lower-order terms, for test functions whose Fourier transform is supported in (-2,2). The results confirm in part a conjecture of Katz and Sarnak, that to leading order the low-lying zeros for this ensemble have symplectic statistics."}
{"category": "Math", "title": "Isoperimetric problems on time scales with nabla derivatives", "abstract": "We prove a necessary optimality condition for isoperimetric problems under nabla-differentiable curves. As a consequence, the recent results of [M.R. Caputo, A unified view of ostensibly disparate isoperimetric variational problems, Appl. Math. Lett. (2008), doi:10.1016/j.aml.2008.04.004], that put together seemingly dissimilar optimal control problems in economics and physics, are extended to a generic time scale. We end with an illustrative example of application of our main result to a dynamic optimization problem from economics."}
{"category": "Math", "title": "Polynomial Coefficient Enumeration", "abstract": "Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\\alpha\\in K^*$. For instance, if $K=\\ff_q$ then a matrix formula is obtained for the number of coefficients of $f^n$ that are equal to $\\alpha\\in \\ff_q^*$, as a function of $n$. Many additional results are obtained related to such areas as lattice path enumeration and the enumeration of integer points in convex polytopes."}
{"category": "Math", "title": "A generalization of the Strong Castelnuovo Lemma", "abstract": "We consider a set $X$ of distinct points in the $n$-dimensional projective space over an algebraically closed field $k$. Let $A$ denote the coordinate ring of $X$, and let $a_i(X)=\\dim_k [{\\rm Tor}_i^R(A,k)]_{i+1}$. Green's Strong Castelnuovo Lemma (SCL) shows that if the points are in general position, then $a_{n-1}(X)\\neq 0$ if and only if the points are on a rational normal curve. Cavaliere, Rossi and Valla conjectured that if the points are not necessarily in general position the possible extension of the SCL should be the following: $a_{n-1}(X)\\neq 0$ if and only if either the points are on a rational normal curve or in the union of two linear subspaces whose dimensions add up to $n$. In this work we prove the conjecture."}
{"category": "Math", "title": "Weakly infinite dimensional subsets of R^N", "abstract": "The Continuum Hypothesis implies an Erd\\\"os-Sierpi\\'nski like duality between the ideal of first category subsets of $\\reals^{\\naturals}$, and the ideal of countable dimensional subsets of $\\reals^{\\naturals}$. The algebraic sum of a Hurewicz subset - a dimension theoretic analogue of Sierpinski sets and Lusin sets - of $\\reals^{\\naturals}$ with any compactly countable dimensional subset of $\\reals^{\\naturals}$ has first category."}
{"category": "Math", "title": "On a Diophantine problem with two primes and s powers of two", "abstract": "We refine a recent result of Parsell on the values of the form $\\lambda_1p_1 + \\lambda_2p_2 + \\mu_1 2^{m_1} + ...m + \\mu_s 2^{m_s}, $ where $p_1,p_2$ are prime numbers, $m_1,...c, m_s$ are positive integers, $\\lambda_1 / \\lambda_2$ is negative and irrational and $\\lambda_1 / \\mu_1$, $\\lambda_2/\\mu_2 \\in \\Q$."}
{"category": "Math", "title": "Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets", "abstract": "We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected components of the Julia set, then one of $A$ and $B$ surrounds the other. From this, it is shown that each connected component of the Fatou set is either simply or doubly connected. Moreover, we show that the Julia set of such a semigroup is uniformly perfect. An upper estimate of the cardinality of the set of all connected components of the Julia set of such a semigroup is given. By using this, we give a criterion for the Julia set to be connected. Moreover, we show that for any $n\\in \\Bbb{N} \\cup \\{\\aleph_{0}\\} ,$ there exists a finitely generated polynomial semigroup with bounded planar postcritical set such that the cardinality of the set of all connected components of the Julia set is equal to $n.$ Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomials are found and systematically investigated."}
{"category": "Math", "title": "A characteristic subgroup for fusion systems", "abstract": "As a counterpart for the prime 2 to Glauberman's $ZJ$-theorem, Stellmacher proves that any nontrivial 2-group $S$ has a nontrivial characteristic subgroup $W(S)$ with the following property. For any finite $\\Sigma_4$-free group $G$, with $S$ a Sylow 2-subgroup of $G$ and with $O_2(G)$ self-centralizing, the subgroup $W(S)$ is normal in $G$. We generalize Stellmacher's result to fusion systems. A similar construction of $W(S)$ can be done for odd primes and gives rise to a Glauberman functor."}
{"category": "Math", "title": "A $\\mathbb{Z}$-basis for the cluster algebra associated to an affine quiver", "abstract": "The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in \\cite{CK2005} and \\cite{SZ} respectively. In this paper, we will deduce $\\mathbb{Z}$-bases for cluster algebras for affine types $\\widetilde{A}_{n,n},\\widetilde{D}$ and $\\widetilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube."}
{"category": "Math", "title": "Spherical Averaged Endpoint Strichartz Estimates for The Two-dimensional Schrodinger Equations with Inverse Square Potential", "abstract": "The endpoint Strichartz estimates for two-dimensional Schrodinger equations were recovered by averaging the solutions in L^2 in the angular variable by Tao. For Schrodinger equations with defocusing inverse square potential, we proved that the homogeneous endpoint estimates hold under this setting. In particular, the original versions of endpoint estimates hold for radial data."}
{"category": "Math", "title": "Extended Crystal PDE's", "abstract": "In this paper we show that between PDE's and crystallographic groups there is an unforeseen relation. In fact we prove that integral bordism groups of PDE's can be considered extensions of crystallographic subgroups. In this respect we can consider PDE's as {\\em extended crystals}. Then an algebraic-topological obstruction ({\\em crystal obstruction}), characterizing existence of global smooth solutions for smooth boundary value problems, is obtained. Applications of this new theory to the Ricci-flow equation and Navier-Stokes equation are given that solve some well-known fundamental problems. These results, are also extended to singular PDE's, introducing ({\\em extended crystal singular PDE's}). An application to singular MHD-PDE's, is given following some our previous results on such equations, and showing existence of (finite times stable smooth) global solutions crossing critical nuclear energy production zone."}
{"category": "Math", "title": "Predictability in Nonlinear Dynamical Systems with Model Uncertainty", "abstract": "Nonlinear systems with model uncertainty are often described by stochastic differential equations. Some techniques from random dynamical systems are discussed. They are relevant to better understanding of solution processes of stochastic differential equations and thus may shed lights on predictability in nonlinear systems with model uncertainty."}
{"category": "Math", "title": "Stochastic Modeling of Unresolved Scales in Complex Systems", "abstract": "Model uncertainties and simulation uncertainties occur in mathematical modeling of multiscale complex systems, since some mechanisms or scales are not represented (i.e., \"unresolved\") due to lack in our understanding of these mechanisms or limitations in computational power. The impact of these unresolved scales on the resolved scales needs to be parameterized or taken into account. A stochastic scheme is devised to take the effects of unresolved scales into account, in the context of solving nonlinear partial differential equations. An example is presented to demonstrate this strategy."}
{"category": "Math", "title": "Symmetric matrices related to the Mertens function", "abstract": "In this paper we explore a family of congruences over $\\N^\\ast$ from which one builds a sequence of symmetric matrices related to the Mertens function. From the results of numerical experiments, we formulate a conjecture about the growth of the quadratic norm of these matrices, which implies the Riemann hypothesis. This suggests that matrix analysis methods may come to play a more important role in this classical and difficult problem."}
{"category": "Math", "title": "Pseudo-euclidean Jordan algebras", "abstract": "A Jordan algebra J is said to be pseudo-euclidean if J is endowed with an associative non-degenerate symmetric bilinear form B. B is said an associative scalar product on J. First, we provide a description of the pseudo-euclidean Jordan K-algebras in terms of double extensions and generalized double extensions. In particular, we shall use this description to construct all pseudo-euclidean Jordan algebras of dimension less than or equal to 5. And then, from one of these algebras, we shall construct a twelve dimension Lie algebra by the \"TKK\" construction. Second, a description of symplectic pseudo-euclidean Jordan algebras is provided and finally we describe a particular class of these algebras namely the class of symplectic Jordan-Manin Algebras. In addition to these descriptions, this paper demonstrates that these last two classes are identical and provides several information on the structure of pseudo-euclidean Jordan algebras."}
{"category": "Math", "title": "Ergodic seminorms for commuting transformations and applications", "abstract": "Recently, T. Tao gave a finitary proof a convergence theorem for multiple averages with several commuting transformations and soon later, T. Austin gave an ergodic proof of the same result. Although we give here one more proof of the same theorem, this is not the main goal of this paper. Our main concern is to provide some tools for the case of several commuting transformations, similar to the tools successfully used in the case of a single transformation, with the idea that they will be useful in the solution of other problems."}
{"category": "Math", "title": "Parametric estimation and tests through divergences and duality technique", "abstract": "We introduce estimation and test procedures through divergence optimization for discrete or continuous parametric models. This approach is based on a new dual representation for divergences. We treat point estimation and tests for simple and composite hypotheses, extending maximum likelihood technique. An other view at the maximum likelihood approach, for estimation and test, is given. We prove existence and consistency of the proposed estimates. The limit laws of the estimates and test statistics (including the generalized likelihood ratio one) are given both under the null and the alternative hypotheses, and approximation of the power functions is deduced. A new procedure of construction of confidence regions, when the parameter may be a boundary value of the parameter space, is proposed. Also, a solution to the irregularity problem of the generalized likelihood ratio test pertaining to the number of components in a mixture is given, and a new test is proposed, based on $\\chi ^{2}$-divergence on signed finite measures and duality technique."}
{"category": "Math", "title": "A simple intrinsic reduced-observer for geodesic flow", "abstract": "Aghannan and Rouchon proposed a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. The observer is based on the Riemannian structure of the configuration manifold endowed with the kinetic energy metric and is intrinsic. They proved local convergence. When the system is conservative, we propose a globally convergent intrinsic reduced-observer based on the Jacobi metric. For non-conservative systems the observer can be used as a complement to the one of Aghannan and Rouchon. More generally the reduced-observer provides velocity estimation for geodesic flow with position measurements. Thus it can be (formally) used as a fluid flow soft sensor in the case of a perfect incompressible fluid. When the curvature is negative in all planes the geodesic flow is sensitive to initial conditions. Surprisingly this instability yields faster convergence."}
{"category": "Math", "title": "Existence of natural and conformally invariant quantizations of arbitrary symbols", "abstract": "A quantization over a manifold can be seen as a way to construct a differential operator with prescribed principal symbol. The quantization map is moreover required to be a linear bijection. It is known that there is in general no natural quantization procedure. However, considering manifolds endowed with additional structures, such as projective or pseudo-conformal structures, one can seek for quantizations that depend on this additional structure and that are natural if the dependence with respect to the structure is taken into account. The question of existence of such a quantization was addressed in a series of papers in the context of projective geometry, using the framework of Thomas-Whitehead connections (see Bordemann, Hansoul and Fox). Recently, we recovered these existence results, using the theory of Cartan projective connections. In the present work, we show that our method can be adapted to pseudo-conformal geometry to yield the so-called natural and conformally invariant quantization for arbitrary symbols, still outside some critical situations. Our method is general enough to analyze the problem of invariant quantizations in the context of manifolds with irreducible parabolic geometries."}
{"category": "Math", "title": "Deformations of symplectic vortices", "abstract": "We prove a gluing theorem for a symplectic vortex on a compact complex curve and a collection of holomorphic sphere bubbles. Using the theorem we show that the moduli space of regular stable symplectic vortices on a fixed curve with varying markings has the structure of a stratified-smooth topological orbifold. In addition, we show that the moduli space has a non-canonical $C^1$-orbifold structure."}
{"category": "Math", "title": "Coisotropic Submanifolds, Leafwise Fixed Points, and Presymplectic Embeddings", "abstract": "Let $(M,\\omega)$ be a geometrically bounded symplectic manifold, $N\\subseteq M$ a closed, regular (i.e. \"fibering\") coisotropic submanifold, and $\\phi:M\\to M$ a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of $\\phi$ is bounded below by the sum of the $Z_2$-Betti numbers of $N$, provided that the Hofer distance between $\\phi$ and the identity is small enough and the pair $(N,\\phi)$ is non-degenerate. The bound is optimal if there exists a $Z_2$-perfect Morse function on $N$. A version of the Arnol'd-Givental conjecture for coisotropic submanifolds is also discussed. As an application, I prove a presymplectic non-embedding result."}
{"category": "Math", "title": "The Torsion of Homology Groups of M(E,I)-sets", "abstract": "We consider the torsion of homology groups of right pointed sets over a partially commutative monoid M(E,I)"}
{"category": "Math", "title": "Configuration space integrals for embedding spaces and the Haefliger invariant", "abstract": "Let K be the space of long j-knots in R^n. In this paper we introduce a graph complex D and a linear map I from D to the de Rham complex of K via configuration space integral, and prove that (1) when both n>j>=3 are odd, the map I is a cochain map if restricted to graphs with at most one loop component, (2) when n-j>=2 is even, the map I is a cochain map if restricted to tree graphs, and (3) when n-j >=3 is odd, the map I added a correction term produces a (2n-3j-3)-cocycle of K which gives a new formulation of the Haefliger invariant when n=6k, j=4k-1 for some k."}
{"category": "Math", "title": "Distinction of some induced representations", "abstract": "Let $K/F$ be a quadratic extension of $p$-adic fields, $\\sigma$ the nontrivial element of the Galois group of $K$ over $F$, and $\\pi$ a quasi-square-integrable representation of $GL(n,K)$. Denoting by $\\pi^{\\vee}$ the smooth contragredient of $\\pi$, and by $\\pi^{\\sigma}$ the representation $\\pi\\circ \\sigma$, we show that the representation of $GL(2n, K)$ obtained by normalized parabolic induction of the representation $\\pi^\\vee \\otimes \\pi^\\sigma$ is distinguished with respect to $GL(2n,F)$. This is a step towards the classification of distinguished generic representations of general linear groups over $p$-adic fields."}
{"category": "Math", "title": "On an inequality related to the radial growth of subharmonic functions", "abstract": "It is a classical result that every subharmonic function, defined and ${\\mathcal{L}}^p$-integrable for some $p$, $0<p<+\\infty$, on the unit disk $\\mathbb{D}$ of the complex plane ${\\mathbb{C}}$ is for almost all $\\theta$ of the form $o((1-| z|)^{-1/p})$, uniformly as $z\\to e^{i\\theta}$ in any Stolz domain. Recently Pavlovi\\'c gave a related integral inequality for absolute values of harmonic functions, also defined on the unit disk in the complex plane. We generalize Pavlovi\\'c's result to so called quasi-nearly subharmonic functions defined on rather general domains in ${\\mathbb {R}}^n$, $n\\geq 2$."}
{"category": "Math", "title": "On normal Hopf subalgebras of semisimple Hopf algebras", "abstract": "A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka's criterion for normal Hopf subalgebras. At the level of characters, the image of the induction functor from a normal Hopf subalgebra is isomorphic to the image of the restriction functor."}
{"category": "Math", "title": "An explicit d-bar-integration formula for weighted homogeneous varieties II: forms of higher degree", "abstract": "Let Y be a weighted homogeneous (singular) subvariety of C^n. The main objective of this paper is to present a class of explicit integral formulae for solving the d-bar-equation $\\omega=\\dbar\\lambda$ on the regular part of Y, where $\\omega$ is a d-bar-closed (0,q)-form with compact support and degree q>=1. Particular cases of these formulae yield L^p-bounded solution operators for $1<=p<=\\infty$ if Y is a homogeneous and pure dimensional subvariety with an arbitrary singular locus."}
{"category": "Math", "title": "A note on a composition of two random integral mappings $\\J^\\be$ and some examples", "abstract": "A method of random integral representation, that is, a method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note we show that a composition of two random integral mappings $\\J^\\be$ is again a random integral mapping. We illustrate our results on some examples."}
{"category": "Math", "title": "A calculus on L\\'evy exponents and selfdecomposability on Banach spaces", "abstract": "In infinite dimensional Banach spaces there is no complete characterization of the L\\'evy exponents of infinitely divisible probability measures. Here we propose \\emph{a calculus on L\\'evy exponents} that is derived from some random integrals. As a consequence we prove that \\emph{each} selfdecomposable measure can by factorized as another selfdecomposable measure and its background driving measure that is s-selfdecomposable. This complements a result from the paper of Iksanov-Jurek-Schreiber in the Annals of Probability \\textbf{32}, 2004.}"}
{"category": "Math", "title": "Exact properties of Frobenius numbers and fraction of the symmetric semigroups in the weak limit for n=3", "abstract": "We generalize and prove a hypothesis by V. Arnold on the parity of Frobenius number. For the case of symmetric semigroups with three generators of Frobenius numbers we found an exact formula, which in a sense is the sum of two Sylvester's formulaes. We prove that the fraction of the symmetric semigroups is vanishing in the weak limit."}
{"category": "Math", "title": "Limit theorems for p-variations of solutions of SDEs driven by additive non-Gaussian stable Levy noise", "abstract": "In this paper we study the asymptotic properties of the power variations of stochastic processes of the type X=Y+L, where L is an alpha-stable Levy process, and Y a perturbation which satisfies some mild Lipschitz continuity assumptions. We establish local functional limit theorems for the power variation processes of X. In case X is a solution of a stochastic differential equation driven by L, these limit theorems provide estimators of the stability index alpha. They are applicable for instance to model fitting problems for paleo-climatic temperature time series taken from the Greenland ice core."}
{"category": "Math", "title": "On the probabilities of local behaviors in abelian field extensions", "abstract": "For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime p of K, we determine the probability that p splits into r primes in a random G-extension of K that is unramified at p. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev's density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant."}
{"category": "Math", "title": "Trace expansions for elliptic cone operators with stationary domains", "abstract": "We analyze the behavior of the trace of the resolvent of an elliptic cone differential operator as the spectral parameter tends to infinity. The resolvent splits into two components, one associated with the minimal extension of the operator, and another, of finite rank, depending on the particular choice of domain. We give a full asymptotic expansion of the first component and expand the component of finite rank in the case where the domain is stationary. The results make use, and develop further, our previous investigations on the analytic and geometric structure of the resolvent. The analysis of nonstationary domains, considerably more intricate, is pursued elsewhere."}
{"category": "Math", "title": "Commutative Moufang loops and alternative algebras", "abstract": "We compute the orders of free commutative Moufand loops of exponent 3 with $n\\leq 7$ free generators and find embeddings of such loops into a loop of invertible elements of the free commutative alternative algebra with identity $x^3=0$."}
{"category": "Math", "title": "On Ramification Filtrations and p-adic Differential Equations, II: mixed characteristic case", "abstract": "Let K be a complete discretely valued field of mixed characteristic (0, p) with possibly imperfect residue field. We prove a Hasse-Arf theorem for the arithmetic ramification filtrations on G_K, except possibly in the absolutely unramified and non-logarithmic case, or p=2 and logarithmic case. As an application, we obtain a Hasse-Arf theorem for filtrations on finite flat group schemes over O_K."}
{"category": "Math", "title": "The regularizing effects of resetting in a particle system for the Burgers equation", "abstract": "We study the dissipation mechanism of a stochastic particle system for the Burgers equation. The velocity field of the viscous Burgers and Navier-Stokes equations can be expressed as an expected value of a stochastic process based on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3 (2008) 330-345]. In this paper we study a particle system for the viscous Burgers equations using a Monte-Carlo version of the above; we consider N copies of the above stochastic flow, each driven by independent Wiener processes, and replace the expected value with $\\frac{1}{N}$ times the sum over these copies. A similar construction for the Navier-Stokes equations was studied by Mattingly and the first author of this paper [Iyer and Mattingly Nonlinearity 21 (2008) 2537-2553]. Surprisingly, for any finite N, the particle system for the Burgers equations shocks almost surely in finite time. In contrast to the full expected value, the empirical mean $\\frac{1}{N}\\sum_1^N$ does not regularize the system enough to ensure a time global solution. To avoid these shocks, we consider a resetting procedure, which at first sight should have no regularizing effect at all. However, we prove that this procedure prevents the formation of shocks for any $N\\geq2$, and consequently as $N\\to\\infty$ we get convergence to the solution of the viscous Burgers equation on long time intervals."}
{"category": "Math", "title": "Composition of transpositions and equality of ribbon Schur Q-functions", "abstract": "We introduce a new operation on skew diagrams called composition of transpositions, and use it and a Jacobi-Trudi style formula to derive equalities on skew Schur Q-functions whose indexing shifted skew diagram is an ordinary skew diagram. When this skew diagram is a ribbon, we conjecture necessary and sufficient conditions for equality of ribbon Schur Q-functions. Moreover, we determine all relations between ribbon Schur Q-functions; show they supply a Z-basis for skew Schur Q-functions; assert their irreducibility; and show that the non-commutative analogue of ribbon Schur Q-functions is the flag h-vector of Eulerian posets."}
{"category": "Math", "title": "On a solution of one fuzzy logic problem", "abstract": "In this paper defuzzification method of WABL is investigated, its properties are analyzed. The WABL method is applied to some fuzzy models. The package of applied programs is worked out on the base of proposed algorithms. The obtained in the form of visual- interactive graphs results are compared with knows ones."}
{"category": "Math", "title": "Regularity and dimension spectrum of the equivariant spectral triple for the odd dimensional quantum spheres", "abstract": "The odd dimensional quantum sphere $S_q^{2\\ell+1}$ is a homogeneous space for the quantum group $SU_q(\\ell+1)$. A generic equivariant spectral triple for $S_q^{2\\ell+1}$ on its $L_2$ space was constructed by Chakraborty & Pal. We prove regularity for that spectral triple here. We also compute its dimension spectrum and show that it is simple. We give detailed construction of its smooth function algebra and some related algebras that help proving regularity and in the computation of the dimension spectrum. Following the idea of Connes for $SU_q(2)$, we first study another spectral triple for $S_q^{2\\ell+1}$ equivariant under torus group action constructed by Chakraborty & Pal. We then derive the results for the $SU_q(\\ell+1)$-equivariant triple in the $q=0$ case from those for the torus equivariant triple. For the $q\\neq 0$ case, we deduce regularity and dimension spectrum from the $q=0$ case."}
{"category": "Math", "title": "Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations", "abstract": "The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously. First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time."}
{"category": "Math", "title": "Calculation of two Belyi pairs", "abstract": "We calculate two Belyi pairs using the properties of Mulase-Penkava differential. Details are provided including accurate construction of coordinates, variables and equations. The calculation is a part of the work which results in a catalogue arXiv:0710.2658"}
{"category": "Math", "title": "Specializations of multigradings and the arithmetical rank of lattice ideals", "abstract": "In this article we study specializations of multigradings and apply them to the problem of the computation of the arithmetical rank of a lattice ideal $I_{L_{\\mathcal{G}}} \\subset K[x_{1},...,x_{n}]$. The arithmetical rank of $I_{L_{\\mathcal{G}}}$ equals the $\\mathcal{F}$-homogeneous arithmetical rank of $I_{L_{\\mathcal{G}}}$, for an appropriate specialization $\\mathcal{F}$ of $\\mathcal{G}$. To the lattice ideal $I_{L_{\\mathcal{G}}}$ and every specialization $\\mathcal{F}$ of $\\mathcal{G}$ we associate a simplicial complex. We prove that combinatorial invariants of the simplicial complex provide lower bounds for the $\\mathcal{F}$-homogeneous arithmetical rank of $I_{L_{\\mathcal{G}}}$."}
{"category": "Math", "title": "Two-Dimensional Patterns with Distinct Differences -- Constructions, Bounds, and Maximal Anticodes", "abstract": "A two-dimensional grid with dots is called a \\emph{configuration with distinct differences} if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various two-dimensional shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid."}
{"category": "Math", "title": "Binomial generation of the radical of a lattice ideal", "abstract": "Let $I_{L, \\rho}$ be a lattice ideal. We provide a necessary and sufficient criterion under which a set of binomials in $I_{L, \\rho}$ generate the radical of $I_{L, \\rho}$ up to radical. We apply our results to the problem of determining the minimal number of generators of $I_{L, \\rho}$ or of the $rad(I_{L, \\rho})$ up to radical."}
{"category": "Math", "title": "P^r-scrolls arising from Brill-Noether theory and K3-surfaces", "abstract": "In this paper we study examples of P^r-scrolls defined over primitively polarized K3 surfaces S of genus g, which arise from Brill-Noether theory of the general curve in the primitive linear system on S and from classical Lazarsfeld's results in. We show that such scrolls form an open dense subset of a component H of their Hilbert scheme; moreover, we study some properties of H (e.g. smoothness, dimensional computation, etc.) just in terms of the moduli space of such K3's and of the moduli space of semistable torsion-free sheaves of a given Mukai-vector on S. One of the motivation of this analysis is to try to introducing the use of projective geometry and degeneration techniques in order to studying possible limits of semistable vector-bundles of any rank on a general K3 as well as Brill-Noether theory of vector-bundles on suitable degenerations of projective curves. We conclude the paper by discussing some applications to the Hilbert schemes of geometrically ruled surfaces whose base curve has general moduli."}
{"category": "Math", "title": "Riemannian geometric realizations for Ricci tensors of generalized algebraic curvature operators", "abstract": "We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on M defined near P. We construct a torsion free real analytic (resp. C-k) connection D which is defined near P on the tangent bundle of M whose curvature operator is the given operator R at P and so that D has constant scalar curvature. We show that if R is Ricci symmetric, then D can be chosen to be Ricci symmetric; if R has trace free Ricci tensor, then D can be chosen to have trace free Ricci tensor; if R is Ricci alternating, then D can be chosen to be Ricci alternating."}
{"category": "Math", "title": "S-numbers of elementary operators on C*-algebras", "abstract": "We study the s-numbers of elementary operators acting on C*-algebras. The main results are the following: If $\\tau$ is any tensor norm and $a,b\\in B(H)$ are such that the sequences $s(a),s(b)$ of their singular numbers belong to a stable Calkin space $J$ then the sequence of approximation numbers of $a\\otimes_{\\tau} b$ belongs to $J$. If $A$ is a C*-algebra, $J$ is a stable Calkin space, $s$ is an s-number function, and $a_i, b_i \\in A,$ $i=1,...,m$ are such that $s(\\pi(a_i)), s(\\pi(b_i)) \\in J$, $i=1,...,m$ for some faithful representation $\\pi$ of $A$ then $s(\\sum_{i=1}^{m} M_{a_i,b_i})\\in J$. The converse implication holds if and only if the ideal of compact elements of $A$ has finite spectrum. We also prove a quantitative version of a result of Ylinen."}
{"category": "Math", "title": "Application of Multihomogeneous Covariants to the Essential Dimension of Finite Groups", "abstract": "We investigate essential dimension of finite groups over arbitrary fields and give a systematic treatment of multihomogenization, introduced by H.Kraft, G.Schwarz and the author. We generalize the central extension theorem of Buhler and Reichstein and use multihomogenization to substitute and generalize the stack-involved part of the theorem of Karpenko and Merkurjev about the essential dimension of p-groups. One part of this paper is devoted to the study of completely reducible faithful representations. Amongst results concerning faithful representations of minimal dimension there is a computation of the minimal number of irreducible components needed for a faithful representation."}
{"category": "Math", "title": "The Horrocks correspondence for coherent sheaves on projective spaces", "abstract": "We establish an equivalence between the stable category of coherent sheaves (satisfying a mild restriction) on a projective space and the homotopy category of a certain class of minimal complexes of free modules over the exterior algebra Koszul dual to the homogeneous coordinate algebra of the projective space. We also relate these complexes to the Tate resolutions of the respective sheaves. In this way, we extend from vector bundles to coherent sheaves the results of Coand\\u{a} and Trautmann [Trans. AMS 385 (2005)], which interpret in terms of the BGG correspondence the results of Trautmann [Math. Ann. 237 (1978)] about the correspondence of Horrocks [Proc. London. Math. Soc. 14 (1964)], [Asterisque 71-72 (1980)]. We also give direct proofs of the BGG correspondences for graded modules and for coherent sheaves and of the theorem of Eisenbud, Floystad and Schreyer [Trans. AMS 355 (2003)] describing the linear part of the Tate resolution associated to a coherent sheaf. Moreover, we provide an explicit description of the quotient of the Tate resolution by its linear strand corresponding to the module of global sections of the various twists of the sheaf."}
{"category": "Math", "title": "Rescaled Levy-Loewner hulls and random growth", "abstract": "We consider radial Loewner evolution driven by unimodular L\\'evy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process. The process involves two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov $\\rm{HL}(0)$ model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1. Using a different type of compound Poisson process, where the Poisson kernel is replaced by the heat kernel, as driving function, we recover one case of the aforementioned model and $\\rm{SLE}(\\kappa)$ as limits."}
{"category": "Math", "title": "On the localization principle for the automorphisms of pseudoellipsoids", "abstract": "We show that Alexander's extendibility theorem for a local automorphism of the unit ball is valid also for a local automorphism $f$ of a pseudoellipsoid $\\E^n_{(p_1, ..., p_{k})} \\= \\{z \\in \\C^n : \\sum_{j= 1}^{n - k}|z_j|^2 + |z_{n-k+1}|^{2 p_1} + ... + |z_n|^{2 p_{k}} < 1 \\}$, provided that $f$ is defined on a region $\\U \\subset \\E^n_{(p)}$ such that: i) $\\partial \\U \\cap \\partial \\E^n_{(p)}$ contains an open set of strongly pseudoconvex points; ii) $\\U \\cap \\{z_i = 0 \\} \\neq \\emptyset$ for any $n-k +1 \\leq i \\leq n$. By the counterexamples we exhibit, such hypotheses can be considered as optimal."}
{"category": "Math", "title": "On the QALE geometry of Nakajima's metric", "abstract": "We show that on Hilbert scheme of $n$ points on $\\C^2$, the hyperk\\\"ahler metric construsted by H. Nakajima via hyperk\\\"ahler reduction is the Quasi-Asymptotically Locally Euclidean (QALE in short) metric constructed by D. Joyce."}
{"category": "Math", "title": "A cofinite universal space for proper actions for mapping class groups", "abstract": "We prove that the mapping class group $\\Gamma_{g,n}$ for surfaces of negative Euler characteristic has a cofinite universal space $\\E$ for proper actions (the resulting quotient is a finite $CW$-complex). The approach is to construct a truncated Teichmueller space $\\T_{g,n}(\\epsilon)$ by introducing a lower bound for the length of shortest closed geodesics and showing that $\\T_{g,n}(\\epsilon)$ is a $\\Gamma_{g,n}$ equivariant deformation retract of the Teichmueller space $\\T_{g, n}$. The existence of such a cofinite universal space is important in the study of the cohomology of the group $\\gag$. As an application, we note that there are only finitely many conjugacy classes of finite subgroups of $\\Gamma_{g,n}$. Another application is that the rational Novikov conjecture in K-theory holds for $\\Gamma_{g,n}$."}
{"category": "Math", "title": "Tilings defined by affine Weyl groups", "abstract": "Let W be a Weyl group, presented as a crystallographic reflection group on a Euclidean vector space V, and C an open Weyl chamber. In a recent paper, Waldspurger proved that the images (id-w)(C), for Weyl group elements w, are all disjoint, and their union is the closed cone spanned by the positive roots. We show that similarly, if A is the Weyl alcove, the images (id-w)(A), for affine Weyl group elements w, are all disjoint, and their union is V."}
{"category": "Math", "title": "On bundles related to groupoids of matrix subalgebras", "abstract": "This paper has been withdrawn by the author."}
{"category": "Math", "title": "Distinct Difference Configurations: Multihop Paths and Key Predistribution in Sensor Networks", "abstract": "A distinct difference configuration is a set of points in $\\mathbb{Z}^2$ with the property that the vectors (\\emph{difference vectors}) connecting any two of the points are all distinct. Many specific examples of these configurations have been previously studied: the class of distinct difference configurations includes both Costas arrays and sonar sequences, for example. Motivated by an application of these structures in key predistribution for wireless sensor networks, we define the $k$-hop coverage of a distinct difference configuration to be the number of distinct vectors that can be expressed as the sum of $k$ or fewer difference vectors. This is an important parameter when distinct difference configurations are used in the wireless sensor application, as this parameter describes the density of nodes that can be reached by a short secure path in the network. We provide upper and lower bounds for the $k$-hop coverage of a distinct difference configuration with $m$ points, and exploit a connection with $B_{h}$ sequences to construct configurations with maximal $k$-hop coverage. We also construct distinct difference configurations that enable all small vectors to be expressed as the sum of two of the difference vectors of the configuration, an important task for local secure connectivity in the application."}
{"category": "Math", "title": "Integral pinching results for manifolds with boundary", "abstract": "We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the $L^2$-norm of their scalar curvature and the $L^2$-norm of their Ricci tensor are spherical space forms with totally geodesic boundary. Moreover, we prove also that 4-dimensional Riemannian manifolds with umbilic boundary, positive Yamabe invariant and an explicit integral pinching between the total integral of their $(Q,T)$-curvature and the $L^2$-norm of their Weyl curvature are spherical space forms with totally geodesic boundary. As a consequence of our work, we show that a certain conformally invariant operator which plays an important role in Conformal Geometry has a trivial kernel and is non-negative if the Yamabe invariant is positive and verifies a pinching condition together with the total integral of the $(Q,T)$-curvature. As an application of the latter spectral analysis, we show the existence of conformal metrics with constant $Q$-curvature, constant $T$-curvature, and zero mean curvature under the latter assumptions."}
{"category": "Math", "title": "Every braid admits a short sigma-definite representative", "abstract": "A result by Dehornoy (1992) says that every nontrivial braid admits a sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a sigma-definite word representative that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new normal form called the rotating normal form."}
{"category": "Math", "title": "Axiomatizations of quasi-polynomial functions on bounded chains", "abstract": "Two emergent properties in aggregation theory are investigated, namely horizontal maxitivity and comonotonic maxitivity (as well as their dual counterparts) which are commonly defined by means of certain functional equations. We completely describe the function classes axiomatized by each of these properties, up to weak versions of monotonicity in the cases of horizontal maxitivity and minitivity. While studying the classes axiomatized by combinations of these properties, we introduce the concept of quasi-polynomial function which appears as a natural extension of the well-established notion of polynomial function. We give further axiomatizations for this class both in terms of functional equations and natural relaxations of homogeneity and median decomposability. As noteworthy particular cases, we investigate those subclasses of quasi-term functions and quasi-weighted maximum and minimum functions, and provide characterizations accordingly."}
{"category": "Math", "title": "Almost continuous orbit equivalence for non-singular homeomorphisms", "abstract": "Let $X$ and $Y$ be Polish spaces with non-atomic Borel measures $\\mu$ and $\\nu$ of full support. Suppose that $T$ and $S$ are ergodic non-singular homeomorphisms of $(X,\\mu)$ and $(Y,\\nu)$ with continuous Radon-Nikodym derivatives. Suppose that either they are both of type $III_1$ or that they are both of type $III_\\lambda$, $0<\\lambda<1$ and, in the $III_\\lambda$ case, suppose in addition that both `topological asymptotic ranges' (defined in the article) are $\\log\\lambda\\cdot\\Bbb Z$. Then there exist invariant dense $G_\\delta$-subsets $X'\\subset X$ and $Y'\\subset Y$ of full measure and a non-singular homeomorphism $\\phi: X' \\to Y'$ which is an orbit equivalence between $T|_{X'}$ and $S|_{Y'}$, that is $\\phi\\{T^{i}x\\} = \\{S^{i}x\\}$ for all $x \\in X'$. Moreover the Radon-Nikodym derivative $d\\nu\\circ\\phi/d\\mu$ is continuous on $X'$ and, letting $S' = \\phi^{-1}S \\phi$ we have $Tx= {S'}^{n(x)}x$ and $S' = T^{m(x)}x$ where $n$ and $m$ are continuous on $X'$."}
{"category": "Math", "title": "Theta Correspondence for U(1,1) and U(2)", "abstract": "In this paper, we parametrize certain irreducible supercuspidal representations of U(1,1) and U(2) via explicit induction data. The parametrization depends on traceless elements of negative valuation in a quadratic extension of base field. We use the lattice model of the Weil representation to determine which traceless elements are involved in the theta correspondence for reductive dual pair U(1,1) and U(2)."}
{"category": "Math", "title": "Non-oriented solutions of the eikonal equation", "abstract": "We study a new formulation for the eikonal equation |grad u| =1 on a bounded subset of R^2. Instead of a vector field grad u, we consider a field P of orthogonal projections on 1-dimensional subspaces, with div P in L^2. We prove existence and uniqueness for solutions of the equation P div P=0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular neighbourhood of a regular closed curve. The idea of the proof is to apply a generalized method of characteristics introduced in Jabin, Otto, Perthame, \"Line-energy Ginzburg-Landau models: zero-energy states\", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), to a suitable vector field m satisfying P = m \\otimes m. This formulation provides a useful approach to the analysis of stripe patterns. It is specifically suited to systems where the physical properties of the pattern are invariant under rotation over 180 degrees, such as systems of block copolymers or liquid crystals."}
{"category": "Math", "title": "A two-parameter family of complex Hadamard matrices of order 6 induced by hypocycloids", "abstract": "Constructions of Hadamard matrices from smaller blocks is a well-known technique in the theory of real Hadamard matrices: tensoring Hadamard matrices and the classical arrays of Williamson, Ito are all procedures involving smaller order building blocks. We apply a new block-construction for order 6 to obtain a previously unknown 2-dimensional family of complex Hadamard matrices. Our results extend the families D_6(t) and B_6(t) found by various authors recently. As a direct application the existence of a 2-parameter family of MUB-triplets of order 6 is shown."}
{"category": "Math", "title": "Selections, Extensions and Collectionwise Normality", "abstract": "We demonstrate that the classical Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can be reduced only to compact-valued mappings modulo Dowker's extension theorem for such spaces. The idea used to achieve this reduction is also applied to get a simple direct proof of that selection theorem of Michael's. Some other possible applications are demonstrated as well."}
{"category": "Math", "title": "Differential invariants of generic parabolic Monge-Ampere equations", "abstract": "Some new results on geometry of classical parabolic Monge-Amp\\`ere equations (PMA) are presented. PMAs are either \\emph{integrable}, or \\emph{nonintegrable} according to integrability of its characteristic distribution. All integrable PMAs are locally equivalent to the equation $u_{xx}=0$. We study nonintegrable PMAs by associating with each of them a 1-dimensional distribution on the corresponding first order jet manifold, called the \\emph{directing distribution}. According to some property of this distribution, nonintegrable PMAs are subdivided into three classes, one \\emph{generic} and two \\emph{special} ones. Generic PMAs are completely characterized by their directing distributions, and we study canonical models of the latters, \\emph{projective curve bundles} (PCB). A PCB is a 1-dimensional subbundle of the projectivized cotangent bundle of a 4-dimensional manifold. Differential invariants of projective curves composing such a bundle are used to construct a series of contact differential invariants for corresponding PMAs. These give a solution of the equivalence problem for generic PMAs with respect to contact transformations. The introduced invariants measure in an exact manner nonlinearity of PMAs."}
{"category": "Math", "title": "Convergence of multiple ergodic averages along cubes for several commuting transformations", "abstract": "We prove the norm convergence of multiple ergodic averages along cubes for several commuting transformations, and derive corresponding combinatorial results. The method we use relies primarily on the \"magic extension\" established recently by B. ~Host."}
{"category": "Math", "title": "The categorified Diassociative cooperad", "abstract": "Using representations of quivers of type A, we define an anticyclic cooperad in the category of triangulated categories, which is a categorification of the linear dual of the Diassociative operad."}
{"category": "Math", "title": "An observation concerning uniquely ergodic vector fields on 3-manifolds", "abstract": "This paper proves the following: A volume preserving vector field on a compact 3-manifold whose dual 2-form is exact can not generate uniquely ergodic dynamics unless its asymptotic linking number is zero."}
{"category": "Math", "title": "Embedded contact homology and Seiberg-Witten Floer cohomology I", "abstract": "This is the first of five papers that construct an isomorphism between the embedded contact homology and Seiberg-Witten Floer cohomology of a compact 3-manifold with a given contact 1-form. This paper describes what is involved in the construction."}
{"category": "Math", "title": "Number of Irreducible Polynomials and Pairs of Relatively Prime Polynomials in Several Variables over Finite Fields", "abstract": "We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\\em total degree} and the {\\em vector degree}, are considered. We show that the number of irreducibles can be computed recursively by degree and that the number of relatively prime pairs can be expressed in terms of the number of irreducibles. We also obtain asymptotic formulas for the number of irreducibles and the number of relatively prime pairs. The asymptotic formulas for the number of irreducibles generalize and improve several previous results by Carlitz, Cohen and Bodin."}
{"category": "Math", "title": "Preduals of semigroup algebras", "abstract": "For a locally compact group $G$, the measure convolution algebra $M(G)$ carries a natural coproduct. In previous work, we showed that the canonical predual $C_0(G)$ of $M(G)$ is the unique predual which makes both the product and the coproduct on $M(G)$ weak$^*$-continuous. Given a discrete semigroup $S$, the convolution algebra $\\ell^1(S)$ also carries a coproduct. In this paper we examine preduals for $\\ell^1(S)$ making both the product and the coproduct weak$^*$-continuous. Under certain conditions on $S$, we show that $\\ell^1(S)$ has a unique such predual. Such $S$ include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on $\\ell^1(S)$ when $S$ is either $\\mathbb Z_+\\times\\mathbb Z$ or $(\\mathbb N,\\cdot)$."}
{"category": "Math", "title": "Phase transitions in infinitely generated groups, and related problems in additive number theory", "abstract": "Let A be an infinite set of generators for a group G, and let L_A(r) denote the number of elements of G whose word length with respect to A is exactly r. The purpose of this note is to determine all growth functions L_A(r) associated to infinite generating sets for groups, and to describe a phase transition phenomenon associated with infinite generating sets. A list of open problems is also.included."}
{"category": "Math", "title": "Centers of cyclotomic Sergeev superalgebras", "abstract": "We prove that the natural map from the center of the affine Sergeev superalgebra to the even center of any cyclotomic Sergeev superalgebra of odd level is surjective, hence that the even center of a cyclotomic Sergeev superalgebra of odd level consists of symmetric functions in the squares of its polynomial generators."}
{"category": "Math", "title": "Distributive lattices and cohomology", "abstract": "A resolution of the intersection of a finite number of subgroups of an abelian group by means of their sums is constructed, provided the lattice generated by these subgroups is distributive. This is used for detecting singularities of modules over Dedekind rings. A generalized Chinese remainder theorem is derived as a consequence of the above resolution. The Gelfand-Naimark duality between finite closed coverings of compact Hausdorff spaces and the generalized Chinese remainder theorem is clarified."}
{"category": "Math", "title": "On Effective log Iitaka Fibration for 3-folds and 4-folds", "abstract": "We prove the effectiveness of the log Iitaka fibration in Kodaira codimension two for varieties of dimension$\\le 4$. In particular, we finish the proof of effective log Iitaka fibration in dimension two. Also, we show that for the log Iitaka fibration, if the fiber is of dimension two, the denominator of the moduli part is bounded."}
{"category": "Math", "title": "Separation of Relatively Quasiconvex Subgroups", "abstract": "Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF for closed hyperbolic 3-manifolds. The method is to reduce, via combination and filling theorems, the separability of a quasiconvex subgroup of a relatively hyperbolic group G to the separability of a quasiconvex subgroup of a hyperbolic quotient G/N. A result of Agol, Groves, and Manning is then applied."}
{"category": "Math", "title": "Non-singular solutions of normalized Ricci flow on noncompact manifolds of finite volume", "abstract": "The main result of this paper shows that, if $g(t)$ is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold $M$ of finite volume, then the Euler characteristic number $\\chi(M)\\geq0$. Moreover, $\\chi(M)\\neq 0$, there exist a sequence times $t_k\\to\\infty$, a double sequence of points $\\{p_{k,l}\\}_{l=1}^{N}$ and domains $\\{U_{k,l}\\}_{l=1}^{N}$ with $p_{k,l}\\in U_{k,l}$ satisfying the followings: [(i)] $\\dist_{g(t_k)}(p_{k,l_1},p_{k,l_2})\\to\\infty$ as $k\\to\\infty$, for any fixed $l_1\\neq l_2$; [(ii)] for each $l$, $(U_{k,l},g(t_k),p_{k,l})$ converges in the $C_{loc}^\\infty$ sense to a complete negative Einstein manifold $(M_{\\infty,l},g_{\\infty,l},p_{\\infty,l})$ when $k\\to\\infty$; [(iii)] $\\Vol_{g(t_{k})}(M\\backslash\\bigcup_{l=1}^{N}U_{k,l})\\to0$ as $k\\to\\infty$."}
{"category": "Math", "title": "Indecomposable polynomials and their spectrum", "abstract": "We address some questions concerning indecomposable polynomials and their spectrum. How does the spectrum behave via reduction or specialization, or via a more general ring morphism? Are the indecomposability properties equivalent over a field and over its algebraic closure? How many polynomials are decomposable over a finite field?"}
{"category": "Math", "title": "Extending $T^p$ automorphisms over $\\RR^{p+2}$ and realizing DE attractors", "abstract": "In this paper we consider the realization of DE attractors by self-diffeomorphisms of manifolds. For any expanding self-map $\\phi:M\\to M$ of a connected, closed $p$-dimensional manifold $M$, one can always realize a $(p,q)$-type attractor derived from $\\phi$ by a compactly-supported self-diffeomorphsm of $\\RR^{p+q}$, as long as $q\\geq p+1$. Thus lower codimensional realizations are more interesting, related to the knotting problem below the stable range. We show that for any expanding self-map $\\phi$ of a standard smooth $p$-dimensional torus $T^p$, there is compactly-supported self-diffeomorphism of $\\RR^{p+2}$ realizing an attractor derived from $\\phi$. A key ingredient of the construction is to understand automorphisms of $T^p$ which extend over $\\RR^{p+2}$ as a self-diffeomorphism via the standard unknotted embedding $\\imath_p:T^p\\hookrightarrow\\RR^{p+2}$. We show that these automorphisms form a subgroup $E_{\\imath_p}$ of $\\Aut(T^p)$ of index at most $2^p-1$."}
{"category": "Math", "title": "The asymptotic Schottky problem", "abstract": "Let $\\mathcal M_g$ denote the moduli space of compact Riemann surfaces of genus $g$ and let $\\mathcal A_g$ be the space of principally polarized abelian varieties of (complex) dimension $g$. Let $J:\\mathcal M_g\\longrightarrow \\mathcal A_g$ be the map which associates to a Riemann surface its Jacobian. The map $J$ is injective, and the image $J(\\mathcal M_g)$ is contained in a proper subvariety of $\\mathcal A_g$ when $g\\geq 4$. The classical and long-studied Schottky problem is to characterize the Jacobian locus $\\mathcal J_g:=J(\\mathcal M_g)$ in $\\mathcal A_g$. In this paper we adress a large scale version of this problem posed by Farb and called the {\\em coarse Schottky problem}: How does $\\mathcal J_g$ look \"from far away\", or how \"dense\" is $\\mathcal J_g$ in the sense of coarse geometry? The coarse geometry of the Siegel modular variety $\\mathcal A_g$ is encoded in its asymptotic cone $\\textup{Cone}_\\infty(\\mathcal A_g)$, which is a Euclidean simplicial cone of (real) dimension $g$. Our main result asserts that the Jacobian locus $\\mathcal J_g$ is \"asymptotically large\", or \"coarsely dense\" in $\\mathcal A_g$. More precisely, the subset of $\\textup{Cone}_\\infty(\\mathcal A_g)$ determinded by $\\mathcal J_g$ actually coincides with this cone. The proof also shows that the Jacobian locus of hyperelliptic curves is coarsely dense in $\\mathcal A_g$ as well. We also study the boundary points of the Jacobian locus $\\mathcal J_g$ in $\\mathcal A_g$ and in the Baily-Borel and the Borel-Serre compactification. We show that for large genus $g$ the set of boundary points of $\\mathcal J_g$ in these compactifications is \"small\"."}
{"category": "Math", "title": "The Duistermaat-Heckman formula and the cohomology of moduli spaces of polygons", "abstract": "We give a presentation of the cohomology ring of spatial polygon spaces $M(r)$ with fixed side lengths $r \\in \\mathbb R^n_+$. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in $\\mathbb C^n$ by the $U(1)^n$-action by multiplication, where $U(1)^n$ is the torus of diagonal matrices in the unitary group U(n). We prove that the first Chern classes of the $n$ line bundles associated with the fibration $r$-level set $\\rightarrow M(r)$ generate the cohomology ring $H^* (M(r), \\mathbb C).$ By applying the Duistermaat--Heckman Theorem, we then deduce the relations on these generators from the piece-wise polynomial function that describes the volume of $M(r).$ We also give an explicit description of the birational map between $M(r) $ and $M(r')$ when the lengths vectors $r$ and $r'$ are in different chambers of the moment polytope. This wall-crossing analysis is the key step to prove that the Chern classes above are generators of $H^*(M(r))$ (this is well-known when $M(r)$ is toric, and by wall-crossing we prove that it holds also when $M(r)$ is not toric)."}
{"category": "Math", "title": "Solutions of Aronsson equation near isolated points", "abstract": "We show that for non-negative solution of the Aronsson equation an isolated singularity is either removable, or the solution behaves asymptotically like a general cone. This generalizes the asymptotic behavior theory for infinity harmonic functions by Savin, Wang and Yu."}
{"category": "Math", "title": "Invariant theory and the W_{1+\\infty} algebra with negative integral central charge", "abstract": "The vertex algebra W_{1+\\infty,c} with central charge c may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer n\\geq 1, it was conjectured in the physics literature that W_{1+\\infty,-n} should have a minimal strong generating set consisting of n^2+2n elements. Using a free field realization of W_{1+\\infty,-n} due to Kac-Radul, together with a deformed version of Weyl's first and second fundamental theorems of invariant theory for the standard representation of GL_n, we prove this conjecture. A consequence is that the irreducible, highest-weight representations of W_{1+\\infty,-n} are parametrized by a closed subvariety of C^{n^2+2n}."}
{"category": "Math", "title": "Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension", "abstract": "We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution $u(x,t)$, the graph $x\\mapsto T(x)$ of its blow-up points and $\\SS\\subset $ the set of all characteristic points, and show that the $\\SS$ has an empty interior. Finally, given $x_0\\in \\SS$, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons with alternate signs and that $T(x)$ forms a corner of angle $\\frac \\pi 2$ at $x_0$."}
{"category": "Math", "title": "Brownian motion conditioned to stay in a cone", "abstract": "A result of R. Durrett, D. Iglehart and D. Miller states that Brownian meander is Brownian motion conditioned to stay positive for a unit of time, in the sense that it is the weak limit, as $x$ goes to 0, of Brownian motion started at $x>0$ and conditioned to stay positive for a unit of time. We extend this limit theorem to the case of multidimensional Brownian motion conditioned to stay in a smooth convex cone."}
{"category": "Math", "title": "DG-methods for microlocalization", "abstract": "For a complex manifold $X$ the ring of microdifferential operators $\\E_X$ acts on the microlocalization $\\mu hom(F,\\O_X)$, for $F$ in the derived category of sheaves on $X$. Kashiwara, Schapira, Ivorra, Waschkies proved, as a byproduct of their new microlocalization functor for ind-sheaves, $\\mu_X$, that $\\mu hom(F,\\O_X)$ can in fact be defined as an object of the derived category of $\\E_X$-modules: this follows from the fact that $\\mu_X \\O_X$ is concentrated in one degree. In this paper we prove that the tempered microlocalization also is an object of the derived category of $\\E_X$-modules. Since we don't know whether the tempered version of $\\mu_X \\O_X$ is concentrated in one degree, we introduce a method to build suitable resolutions for which the action of $\\E_X$ is realized in the category of complexes. We define a version of the de Rham algebra on the subanalytic site which is quasi-injective and we work in the category of dg-modules over this de Rham algebra instead of the derived category of sheaves."}
{"category": "Math", "title": "Birkhoff normal form and splitting methods for semi linear Hamiltonian PDEs. Part II: Abstract splitting", "abstract": "We consider Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part. We consider abstract splitting methods associated with this decomposition where no discretization in space is made. We prove a normal form result for the corresponding discrete flow under generic non resonance conditions on the frequencies of the linear operator and on the step size. This result implies the conservation of the regularity of the numerical solution associated with the splitting method over arbitrary long time, provided the initial data is small enough. This result holds for numerical schemes controlling the round-off error at each step to avoid possible high frequency energy drift. We apply this results to nonlinear Schr\\\"odinger equations as well as the nonlinear wave equation.}"}
{"category": "Math", "title": "Module categories over pointed Hopf algebras", "abstract": "We develop some techniques to the study of exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups $u_q(\\mathfrak{sl}_2)$."}
{"category": "Math", "title": "On Level-Raising Congruences", "abstract": "A work of Sorensen is rewritten here to include nontrivial types at the infinite places. This extends results of K. Ribet and R. Taylor on level-raising for algebraic modular forms on D^{\\times}, where D is a definite quaternion algebra over a totally real field F. This is done for any automorphic representations \\pi of an arbitrary reductive group G over F which is compact at infinity. It is not assumed that \\pi_\\infty is trivial. If \\lambda is a finite place of \\bar{\\Q}, and w is a place where \\pi_w is unramified and \\pi_w is congruent to the trivial representation mod \\lambda, then under some mild additional assumptions (relaxing requirements on the relation between w and \\ell which appear in previous works) the existence of a \\tilde{\\pi} congruent to \\pi mod \\lambda such that \\tilde{\\pi}_w has more parahoric fixed vectors than \\pi_w, is proven. In the case where G_w has semisimple rank one, results of Clozel, Bellaiche and Graftieaux according to which \\tilde{\\pi}_w is Steinberg, are sharpened. To provide applications of the main theorem two examples over F of rank greater than one are considered. In the first example G is taken to be a unitary group in three variables and a split place w. In the second G is taken to be an inner form of GSp(2). In both cases, precise satisfiable conditions on a split prime w guaranteeing the existence of a \\tilde{\\pi} congruent to \\pi mod \\lambda such that the component \\tilde{\\pi}_w is generic and Iwahori spherical, are obtained. For symplectic G, to conclude that \\tilde{\\pi}_w is generic, computations of R. Schmidt are used. In particular, if \\pi is of Saito-Kurokawa type, it is congruent to a \\tilde{\\pi} which is not of Saito-Kurokawa type."}
{"category": "Math", "title": "Grapham: Graphical Models with Adaptive Random Walk Metropolis Algorithms", "abstract": "Recently developed adaptive Markov chain Monte Carlo (MCMC) methods have been applied successfully to many problems in Bayesian statistics. Grapham is a new open source implementation covering several such methods, with emphasis on graphical models for directed acyclic graphs. The implemented algorithms include the seminal Adaptive Metropolis algorithm adjusting the proposal covariance according to the history of the chain and a Metropolis algorithm adjusting the proposal scale based on the observed acceptance probability. Different variants of the algorithms allow one, for example, to use these two algorithms together, employ delayed rejection and adjust several parameters of the algorithms. The implemented Metropolis-within-Gibbs update allows arbitrary sampling blocks. The software is written in C and uses a simple extension language Lua in configuration."}
{"category": "Math", "title": "On a q-extension of Mehta's eigenvectors of the finite Fourier transform for q a root of unity", "abstract": "It is shown that the continuous q-Hermite polynomials for q a root of unity have simple transformation properties with respect to the classical Fourier transform. This result is then used to construct q-extended eigenvectors of the finite Fourier transform in terms of these polynomials."}
{"category": "Math", "title": "Decomposing locally compact groups into simple pieces", "abstract": "We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple. Two appendices introduce results and examples around the concept of quasi-product."}
{"category": "Math", "title": "Stability of Fractional-Order Systems with Rational Orders", "abstract": "This paper deals with stability of a certain class of fractional order linear and nonlinear systems. The stability is investigated in the time domain and the frequency domain. The general stability conditions and several illustrative examples are presented as well."}
{"category": "Math", "title": "Bilinear Forms on the Dirichlet Space", "abstract": "Let $\\mathcal{D}$ be the classical Dirichlet space, the Hilbert space of holomorphic functions on the disk. Given a holomorphic symbol function $b$ we define the associated Hankel type bilinear form, initially for polynomials f and g, by $T_{b}(f,g):= < fg,b >_{\\mathcal{D}} $, where we are looking at the inner product in the space $\\mathcal{D}$. We let the norm of $T_{b}$ denotes its norm as a bilinear map from $\\mathcal{D}\\times\\mathcal{D}$ to the complex numbers. We say a function $b$ is in the space $\\mathcal{X}$ if the measure $d\\mu_{b}:=| b^{\\prime}(z)| ^{2}dA$ is a Carleson measure for $\\mathcal{D}$ and norm $\\mathcal{X}$ by $$ \\Vert b\\Vert_{\\mathcal{X}}:=| b(0)| +\\Vert | b^{\\prime}(z)| ^{2}dA\\Vert_{CM(\\mathcal{D})}^{1/2}. $$ Our main result is $T_{b}$ is bounded if and only if $b\\in\\mathcal{X}$ and $$ \\Vert T_{b}\\Vert_{\\mathcal{D\\times D}}\\approx\\Vert b\\Vert_{\\mathcal{X}}. $$"}
{"category": "Math", "title": "A non-periodic and two-dimensional example of elliptic homogenization", "abstract": "The focus in this paper is on elliptic homogenization of a certain kind of possibly non-periodic problems. A non-periodic and two-dimensional example is studied, where we numerically illustrate the homogenized matrix."}
{"category": "Math", "title": "Invariant Prolongation of BGG-Operators in Conformal Geometry", "abstract": "BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor connection which yields an invariant prolongation of the first operator. The explicit calculation is presented in the case of conformal Killing forms."}
{"category": "Math", "title": "Parabolic equations with variably partially VMO coefficients", "abstract": "We prove the $W^{1,2}_{p}$-solvability of second order parabolic equations in nondivergence form in the whole space for $p\\in (1,\\infty)$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing mean oscillation (VMO) in the orthogonal directions and the time variable in each small parabolic cylinder with the direction depending on the cylinder. This extends a recent result by Krylov [17] for elliptic equations and removes the restriction that $p>2$."}
{"category": "Math", "title": "The structure of Valdivia compact lines", "abstract": "We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The conjecture asserted that a compact line is Valdivia compact if its weight does not exceed aleph one, every point of uncountable character is isolated from one side and every closed first countable subspace is metrizable. It turns out that the last condition is not sufficient. On the other hand, we show that the conjecture is valid if the closure of the set of points of uncountable character is scattered. This improves an earlier result of the first author."}
{"category": "Math", "title": "Monoidal transformations of singularities in positive characteristic", "abstract": "A sequence of monoidal transformations is defined, in terms of invariants, for a singular hypersurface embedded in a smooth scheme of positive characteristic. Some examples are added to illustrate the improvement of singularities by this procedure."}
{"category": "Math", "title": "Kangaroo points and oblique polynomials in resolution of positive characteristic", "abstract": "Updated version. Includes comments on the advances in the field from the Kyoto workshop on Resolution of Singularities in Positive Characteristic, December 2008. The article surveys the theory of kangaroo points as they appear in the resolution of singularities in positive characteristic. They represent one of the main obstructions for transcribing the characteristic zero proof of resolution to positive characteristic. Kangaroo points are classified through the concept of oblique polynomials. The results of the article are used in Hironaka's recent program towards the resolution of singularities in positive characteristic."}
{"category": "Math", "title": "Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials of Types A and C. Extended Abstract", "abstract": "A breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type C, which are specializations of the corresponding Macdonald polynomials at q=0. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type A, so our work is a first step towards finding such a formula."}
{"category": "Math", "title": "Some weak indivisibility results in ultrahomogeneous metric spaces", "abstract": "We study the validity of a partition property known as weak indivisibility for the integer and the rational Urysohn metric spaces. We also compare weak indivisiblity to another partition property, called age-indivisibility, and provide an example of a countable ultrahomogeneous metric space which may be age-indivisible but not weakly indivisible."}
{"category": "Math", "title": "An Elementary Classification of Symmetric 2-Cocycles", "abstract": "We present a classification of the so-called \"additive symmetric 2-cocycles\" of arbitrary degree and dimension over Z/p, along with a partial result and some conjectures for m-cocycles over Z/p, m > 2. This expands greatly on a result originally due to Lazard and more recently investigated by Ando, Hopkins, and Strickland, which together with their work culminates in a complete classification of 2-cocycles over an arbitrary commutative ring. The ring classifying these polynomials finds application in algebraic topology, including generalizations of formal group laws and of cubical structures."}
{"category": "Math", "title": "Penalized Orthogonal-Components Regression for Large p Small n Data", "abstract": "We propose a penalized orthogonal-components regression (POCRE) for large p small n data. Orthogonal components are sequentially constructed to maximize, upon standardization, their correlation to the response residuals. A new penalization framework, implemented via empirical Bayes thresholding, is presented to effectively identify sparse predictors of each component. POCRE is computationally efficient owing to its sequential construction of leading sparse principal components. In addition, such construction offers other properties such as grouping highly correlated predictors and allowing for collinear or nearly collinear predictors. With multivariate responses, POCRE can construct common components and thus build up latent-variable models for large p small n data."}
{"category": "Math", "title": "Polyharmonic approximation on the sphere", "abstract": "The purpose of this article is to provide new error estimates for a popular type of SBF approximation on the sphere: approximating by linear combinations of Green's functions of polyharmonic differential operators. We show that the $L_p$ approximation order for this kind of approximation is $\\sigma$ for functions having $L_p$ smoothness $\\sigma$ (for $\\sigma$ up to the order of the underlying differential operator, just as in univariate spline theory). This is an improvement over previous error estimates, which penalized the approximation order when measuring error in $L_p$, p>2 and held only in a restrictive setting when measuring error in $L_p$, p<2."}
{"category": "Math", "title": "On a spherical code in the space of spherical harmonics", "abstract": "In this short note we propose a new method for construction new nice arrangement on the sphere $S^d$ using the spaces of spherical harmonic."}
{"category": "Math", "title": "On the discrete logarithm problem", "abstract": "Let $p>2$ be prime and $g$ a primitive root modulo $p$. We present an argument for the fact that discrete logarithms of the numbers in any arithmetic progression are uniformly distributed in $[1,p]$ and raise some questions on the subject."}
{"category": "Math", "title": "Conformal compactification of asymptotically locally hyperbolic metrics", "abstract": "In this paper we study the extent to which conformally compact asymptotically hyperbolic metrics may be characterized intrinsically. Building on the work of the first author, we prove that decay of sectional curvature to -1 and decay of covariant derivatives of curvature outside an appropriate compact set yield H\\\"older regularity for a conformal compactification of the metric. In the Einstein case, we prove that the estimate on the sectional curvature implies the control of all covariant derivatives of the Weyl tensor, permitting us to strengthen our result."}
{"category": "Math", "title": "The Occurrence-in-subtuple problem", "abstract": "As we go along with a bioinformatic analysis we stumbled over a new combinatorial question. Although the problem is a very special one, there are maybe more applications than only this one we have. This text is mainly about the general combinatorial problem, the exact solution and its derivation. An outline of a real problem of this type is in the discussion."}
{"category": "Math", "title": "A congruence problem for polyhedra", "abstract": "It is well known that to determine a triangle up to congruence requires three measurements: three sides, two sides and the included angle, or one side and two angles. We consider various generalizations of this fact to two and three dimensions. In particular we consider the following question: given a convex polyhedron $P$, how many measurements are required to determine $P$ up to congruence? We show that in general the answer is that the number of measurements required is equal to the number of edges of the polyhedron. However, for many polyhedra fewer measurements suffice; in the case of the unit cube we show that nine carefully chosen measurements are enough. We also prove a number of analogous results for planar polygons. In particular we describe a variety of quadrilaterals, including all rhombi and all rectangles, that can be determined up to congruence with only four measurements, and we prove the existence of $n$-gons requiring only $n$ measurements. Finally, we show that one cannot do better: for any sequence of $n$ distinct points in the plane one needs at least $n$ measurements to determine it up to congruence."}
{"category": "Math", "title": "An Analysis of the Effect of Ghost Force Oscillation on Quasicontinuum Error", "abstract": "The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the optimal rate O($h$) in the discrete $\\ell^\\infty$ norm and O($h^{1/p}$) in the $w^{1,p}$ norm for $1 \\leq p < \\infty.$ where $h$ is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to O($h$) at distance O($h|\\log h|$) in the atomistic region and distance O($h$) in the continuum region. E, Ming, and Yang previously gave a counterexample to convergence in the $w^{1,\\infty}$ norm for a harmonic interatomic potential. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\\ell^\\infty$ and $w^{1,p}$ norms."}
{"category": "Math", "title": "The resolvent kernel for PCF self-similar fractals", "abstract": "For the Laplacian $\\Delta$ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, and also with Neumann boundary conditions. That is, we construct a symmetric function $G^{(\\lambda)}$ which solves $(\\lambda \\mathbb{I} - \\Delta)^{-1} f(x) = \\int G^{(\\lambda)}(x,y) f(y) d\\mu(y)$. The method is similar to Kigami's construction of the Green kernel in \\cite[\\S3.5]{Kig01} and is expressed as a sum of scaled and \"translated\" copies of a certain function $\\psi^{(\\lambda)}$ which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket $SG_3$."}
{"category": "Math", "title": "New Permutation Representations of the Braid Group", "abstract": "We give a new infinite family of group homomorphisms from the braid group B_k to the symmetric group S_{mk} for all k and m \\geq 2. Most known permutation representations of braids are included in this family. We prove that the homomorphisms in this family are non-cyclic and transitive. For any divisor l of m, 1\\leq l < m, we prove in particular that if \\frac{m}{l} is odd then there are 1 + \\frac{m}{l} non-conjugate homomorphisms included in our family. We define a certain natural restriction on homomorphisms B_k to S_n, common to all homomorphisms in our family, which we term 'good', and of which there are two types. We prove that all good homomorphisms B_k to S_{mk} of type 1 are included in the infinite family of homomorphisms we gave. For m=3, we prove that all good homomorphisms B_k to S_{3k} of type 2 are also included in this family. Finally, we refute a conjecture made by Matei and Suciu regarding permutation representations of braids and give an updated conjecture."}
{"category": "Math", "title": "On the size of the set A(A+1)", "abstract": "Let $F_p$ be the field of a prime order $p.$ For a subset $A\\subset F_p$ we consider the product set $A(A+1).$ This set is an image of $A\\times A$ under the polynomial mapping $f(x,y)=xy+x:F_p\\times F_p\\to F_p.$ In the present paper we show that if $|A|<p^{1/2},$ then $$ |A(A+1)|\\ge |A|^{106/105+o(1)}.$$ If $|A|>p^{2/3},$ then we prove that $$|A(A+1)|\\gg \\sqrt{p |A|}$$ and show that this is the optimal in general settings bound up to the implied constant. We also estimate the cardinality of $A(A+1)$ when $A$ is a subset of real numbers. We show that in this case one has the Elekes type bound $$ |A(A+1)|\\gg |A|^{5/4}. $$"}
{"category": "Math", "title": "An information-theoretic derivation of min-cut based clustering", "abstract": "Min-cut clustering, based on minimizing one of two heuristic cost-functions proposed by Shi and Malik, has spawned tremendous research, both analytic and algorithmic, in the graph partitioning and image segmentation communities over the last decade. It is however unclear if these heuristics can be derived from a more general principle facilitating generalization to new problem settings. Motivated by an existing graph partitioning framework, we derive relationships between optimizing relevance information, as defined in the Information Bottleneck method, and the regularized cut in a K-partitioned graph. For fast mixing graphs, we show that the cost functions introduced by Shi and Malik can be well approximated as the rate of loss of predictive information about the location of random walkers on the graph. For graphs generated from a stochastic algorithm designed to model community structure, the optimal information theoretic partition and the optimal min-cut partition are shown to be the same with high probability."}
{"category": "Math", "title": "On The Structure and Automorphism Group of Finite Alexander Quandles", "abstract": "We prove that an Alexander quandle of prime order is generated by any pair of distinct elements. Furthermore, we prove for such a quandle that any ordered pair of distinct elements can be sent to any other such pair by an automorphism of the quandle."}
{"category": "Math", "title": "An Analysis of Node-Based Cluster Summation Rules in the Quasicontinuum Method", "abstract": "We investigate two examples of node-based cluster summation rules that have been proposed for the quasicontinuum method: a force-based approach (Knap & Ortiz, J. Mech. Phys. Solids 49, 2001), and an energy-based approach which is a generalization of the non-local quasicontinuum method (Eidel & Stukowski, J. Mech. Phys. Solids, to appear). We show that, even for the case of nearest neighbour interaction in a one-dimensional periodic chain, both of these approaches create large errors when used with graded and, more generally, non-smooth meshes. These errors cannot be removed by increasing the cluster size. We offer some suggestions how the accuracy of (cluster) summation rules may be improved."}
{"category": "Math", "title": "Dynamics of quasiconformal fields", "abstract": "A uniqueness theorem is established for autonomous systems of ODEs, $\\dot{x}=f(x)$, where $f$ is a Sobolev vector field with additional geometric structure, such as delta-monoticity or reduced quasiconformality. Specifically, through every non-critical point of $f$ there passes a unique integral curve."}
{"category": "Math", "title": "Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term", "abstract": "In this paper, we establish the global well-posedness of the Cauchy problem for the Gross-Pitaevskii equation with an rotational angular momentum term in the space $\\Real^2$."}
{"category": "Math", "title": "Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions", "abstract": "In this paper, we establish the global well-posedness of the Cauchy problem for the Gross-Pitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space $\\Real^3$."}
{"category": "Math", "title": "Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schr\\\"odinger Equations in Multi-dimensional Spaces", "abstract": "We study the wellposedness of Cauchy problem for the fourth order nonlinear Schr\\\"odinger equations i\\partial_t u=-\\eps\\Delta u+\\Delta^2 u+P((\\partial_x^\\alpha u)_{\\abs{\\alpha}\\ls 2}, (\\partial_x^\\alpha \\bar{u})_{\\abs{\\alpha}\\ls 2}),\\quad t\\in \\Real, x\\in\\Real^n, where $\\eps\\in\\{-1,0,1\\}$, $n\\gs 2$ denotes the spatial dimension and $P(\\cdot)$ is a polynomial excluding constant and linear terms."}
{"category": "Math", "title": "Well-posedness for one-dimensional derivative nonlinear Schr\\\"odinger equations", "abstract": "In this paper, we investigate the one-dimensional derivative nonlinear Schr\\\"odinger equations of the form $iu_t-u_{xx}+i\\lambda\\abs{u}^k u_x=0$ with non-zero $\\lambda\\in \\Real$ and any real number $k\\gs 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition."}
{"category": "Math", "title": "Lowering topological entropy over subsets", "abstract": "Let $(X, T)$ be a topological dynamical system (TDS), and $h (T, K)$ the topological entropy of a subset $K$ of $X$. $(X, T)$ is {\\it lowerable} if for each $0\\le h\\le h (T, X)$ there is a non-empty compact subset with entropy $h$; is {\\it hereditarily lowerable} if each non-empty compact subset is lowerable; is {\\it hereditarily uniformly lowerable} if for each non-empty compact subset $K$ and each $0\\le h\\le h (T, K)$ there is a non-empty compact subset $K_h\\subseteq K$ with $h (T, K_h)= h$ and $K_h$ has at most one limit point. It is shown that each TDS with finite entropy is lowerable, and that a TDS $(X, T)$ is hereditarily uniformly lowerable if and only if it is asymptotically $h$-expansive."}
{"category": "Math", "title": "A non-smooth continuous unitary representation of a Banach-Lie group", "abstract": "In this note we show that the representation of the additive group of the Hilbert space $L^2([0,1],\\R)$ on $L^2([0,1],\\C)$ given by the multiplication operators $\\pi(f) := e^{if}$ is continuous but its space of smooth vectors is trivial. This example shows that a continuous unitary representation of an infinite dimensional Lie group need not be smooth."}
{"category": "Math", "title": "The conditional convergence of the Dirichlet series of an L-function", "abstract": "The Dirichlet divisor problem is used as a model to give a conjecture concerning the conditional convergence of the Dirichlet series of an L-function."}
{"category": "Math", "title": "On the Webster scalar curvature problem on the CR sphere with a cylindrical-type symmetry", "abstract": "By variational methods, for a kind of Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature, we construct some multi-peak solutions as the parameter is sufficiently small under certain assumptions. We also obtain the asymptotic behaviors of the solutions."}
{"category": "Math", "title": "Vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties", "abstract": "We use the toric degeneration of Bott-Samelson varieties and the description of cohomolgy of line bundles on toric varieties to deduce vanishings results for the cohomology of lines bundles on Bott-Samelson varieties."}
{"category": "Math", "title": "On Self-mapping Degrees of $S^3$-geometry manifolds", "abstract": "In this paper we determined all of the possible self mapping degrees of the manifolds with $S^3$-geometry, which are supposed to be all 3-manifolds with finite fundamental groups. This is a part of a project to determine all possible self mapping degrees of all closed orientable 3-manifold in Thurston's picture."}
{"category": "Math", "title": "Asymptotic behavior of solutions of the fragmentation equation with shattering: An approach via self-similar Markov processes", "abstract": "The subject of this paper is a fragmentation equation with nonconservative solutions, some mass being lost to a dust of zero-mass particles as a consequence of an intensive splitting. Under some assumptions of regular variation on the fragmentation rate, we describe the large time behavior of solutions. Our approach is based on probabilistic tools: the solutions to the fragmentation equation are constructed via nonincreasing self-similar Markov processes that continuously reach 0 in finite time. Our main probabilistic result describes the asymptotic behavior of these processes conditioned on nonextinction and is then used for the solutions to the fragmentation equation. We note that two parameters significantly influence these large time behaviors: the rate of formation of \"nearly-1 relative masses\" (this rate is related to the behavior near 0 of the L\\'evy measure associated with the corresponding self-similar Markov process) and the distribution of large initial particles. Correctly rescaled, the solutions then converge to a nontrivial limit which is related to the quasi-stationary solutions of the equation. Besides, these quasi-stationary solutions, or, equivalently, the quasi-stationary distributions of the self-similar Markov processes, are fully described."}
{"category": "Math", "title": "Openly factorizable spaces and compact extensions of topological semigroups", "abstract": "We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its the Stone-\\v{C}ech compactification $\\beta S$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f:S\\to Y$ to a second countable space $Y$ can be written as the composition $f=g\\circ p$ of an open map $p:X\\to Z$ onto a second countable space $Z$ and a map $g:Z\\to Y$. We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces."}
{"category": "Math", "title": "A Quotient Restriction Theorem for actions of real reductive groups", "abstract": "We prove a version of the Chevalley Restriction Theorem for the action of a real reductive group G on a topological space X which locally embeds into a holomorphic representation. Assuming that there exists an appropriate quotient X//G for the G-action, we introduce a stratification which is defined with respect to orbit types of closed orbits. Our main result is a description of the quotient X//G in terms of quotients by normalizer subgroups associated to the stratification."}
{"category": "Math", "title": "On the Complexity and Volume of Hyperbolic 3-Manifolds", "abstract": "We compare the volume of a hyperbolic 3-manifold $M$ of finite volume and the complexity of its fundamental group."}
{"category": "Math", "title": "Consensus optimization on manifolds", "abstract": "The present paper considers distributed consensus algorithms that involve N agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e. maximizing the consensus) or balance (i.e. minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group SO(n) and the Grassmann manifold Gr(p,n) are treated as original examples. A link is also drawn with the many existing results on the circle."}
{"category": "Math", "title": "Embedding the bicyclic semigroup into countably compact topological semigroups", "abstract": "We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p,q). We prove that each topological semigroup S with pseudocompact square contains no dense copy of C(p,q). On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonov semigroup containing a copy of C(p,q)."}
{"category": "Math", "title": "Random walks in random Dirichlet environment are transient in dimension $d\\ge 3$", "abstract": "We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $\\Z^d$, RWDE are parameterized by a $2d$-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $d\\ge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for $d\\ge 3$."}
{"category": "Math", "title": "Forgetful maps between Deligne-Mostow ball quotients", "abstract": "We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livn\\'e fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also include a retraction of a 3-dimensional ball quotient onto one of its 1-dimensional totally geodesic complex submanifolds."}
{"category": "Math", "title": "Diophantine properties for q-analogues of Dirichlet's beta function at positive integers", "abstract": "small In this paper, we define $q$-analogues of Dirichlet's beta function at positive integers, which can be written as $\\beta_q(s)=\\sum_{k\\geq1}\\sum_{d|k}\\chi(k/d)d^{s-1}q^k$ for $s\\in\\N^*$, where $q$ is a complex number such that $|q|<1$ and $\\chi$ is the non trivial Dirichlet character modulo 4. For odd $s$, these expressions are connected with the automorphic world, in particular with Eisenstein series of level 4. From this, we derive through Nesterenko's work the transcendance of the numbers $\\beta_q(2s+1)$ for $q$ algebraic such that $0<|q|<1$. Our main result concerns the nature of the numbers $\\beta_q(2s)$: we give a lower bound for the dimension of the vector space over $\\Q$ spanned by $1,\\beta_q(2),\\beta_q(4),...,\\beta_q(A)$, where $1/q\\in\\Z\\setminus\\{-1;1\\}$ and $A$ is an even integer. As consequences, for $1/q\\in\\Z\\setminus\\{-1;1\\}$, on the one hand there is an infinity of irrational numbers among $\\beta_q(2),\\beta_q(4),...$, and on the other hand at least one of the numbers $\\beta_q(2),\\beta_q(4),..., \\beta_q(20)$ is irrational."}
{"category": "Math", "title": "Uniform distribution of Galois conjugates and beta-conjugates of a Parry number near the unit circle and dichotomy of Perron numbers", "abstract": "Concentration and equi-distribution, near the unit circle, in Solomyak's set, of the union of the Galois conjugates and the beta-conjugates of a Parry number $\\beta$ are characterized by means of the Erd\\H{o}s-Tur\\'an approach, and its improvements by Mignotte and Amoroso, applied to the analytical function $f_{\\beta}(z) = -1 + \\sum_{i \\geq 1} t_i z^i$ associated with the R\\'enyi $\\beta$-expansion $d_{\\beta}(1)= 0.t_1 t_2 ...$ of unity. Mignotte's discrepancy function requires the knowledge of the factorization of the Parry polynomial of $\\beta$. This one is investigated using theorems of Cassels, Dobrowolski, Pinner and Vaaler, Smyth, Schinzel in terms of cyclotomic, reciprocal non-cyclotomic and non-reciprocal factors. An upper bound of Mignotte's discrepancy function which arises from the beta-conjugates of $\\beta$ which are roots of cyclotomic factors is linked to the Riemann hypothesis, following Amoroso. An equidistribution limit theorem, following Bilu's theorem, is formulated for the concentration phenomenon of conjugates of Parry numbers near the unit circle. Parry numbers are Perron numbers. Open problems on non-Parry Perron numbers are mentioned in the context of the existence of non-unique factorizations of elements of number fields into irreducible Perron numbers (Lind)."}
{"category": "Math", "title": "Extension of sections via adjoint ideals", "abstract": "The paper contains theorems on extending sections of line bundles from divisors to the ambient space, inspired by various results of Siu, Kawamata, and especially Hacon-McKernan and Takayama. Applications are given to basepoint-freeness, to the invariance of analogues of plurigenera and of part of the boundary of the pseudo-effective cone in families, and to a simplified approach towards part of the results used for proving the existence of flips by Hacon-McKernan."}
{"category": "Math", "title": "On Tits' Centre Conjecture for Fixed Point Subcomplexes", "abstract": "We give a short and uniform proof of a special case of Tits' Centre Conjecture using a theorem of J-P. Serre and a result from our earlier work. We consider fixed point subcomplexes $X^H$ of the building $X = X(G)$ of a connected reductive algebraic group $G$, where $H$ is a subgroup of $G$."}
{"category": "Math", "title": "The Eilenberg-Moore category and a Beck-type theorem for a Morita context", "abstract": "The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a {\\em Morita context} comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a {\\em double adjunction} to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the {\\em moritability} of a pair of functors with a common domain) are derived."}
{"category": "Math", "title": "An Elementary Approach to a Model Problem of Lagerstrom", "abstract": "The equation studied is u\"+((n-1)/r)u'+epsilon u u'+ku'^{2}=0, with boundary conditions u(1)=0, u(infinity)=1. This model equation has been studied by many authors since it was introduced in the 1950s by P. A. Lagerstrom. We use an elementary approach to show that there is an infinite series solution which is uniformly convergent on 1<=r<infinity. The first few terms are easily derived, from which one quickly deduces the inner and outer asymptotic expansions, with no matching procedure or a priori assumptions about the nature of the expansion. We also give a short and elementary existence and uniqueness proof which covers all epsilon > 0, k >= 0, and n >= 1."}
{"category": "Math", "title": "Ext-symmetry over quantum complete intersections", "abstract": "We show that symmetry in the vanishing of cohomology holds for graded modules over quantum complete intersections. Moreover, symmetry holds for all modules if the algebra is symmetric."}
{"category": "Math", "title": "Linear stochastic systems: a white noise approach", "abstract": "Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We prove BIBO type stability theorems for these systems, both in the discrete and continuous time cases. We also consider the case of dissipative systems for both discrete and continuous time systems. We further study $\\ell_1$-$\\ell_2$ stability in the discrete time case, and ${\\mathbf L}_2$-${\\mathbf L}_\\infty$ stability in the continuous time case."}
{"category": "Math", "title": "Stable pairs on elliptic K3 surfaces", "abstract": "We study semistable pairs on elliptic K3 surfaces with a section: we construct a family of moduli spaces of pairs, related by wall crossing phenomena, which can be studied to describe the birational correspondence between moduli spaces of sheaves of rank 2 and Hilbert schemes on the surface. In the 4-dimensional case, this can be used to get the isomorphism between the moduli space and the Hilbert scheme described by Friedman."}
{"category": "Math", "title": "Invariants of stationary AF-algebras and torsion subgroup of elliptic curves with complex multiplication", "abstract": "Let G(A) be an AF-algebra given by periodic Bratteli diagram with the incidence matrix A in GL(n, Z). For a given polynomial p(x) in Z[x] we assign to G(A) a finite abelian group Z^n/p(A) Z^n. It is shown that if p(0)=1 or p(0)=-1 and Z[x]/(p(x)) is a principal ideal domain, then Z^n/p(A) Z^n is an invariant of the strong stable isomorphism class of G(A). For n=2 and p(x)=x-1 we conjecture a formula linking values of the invariant and torsion subgroup of elliptic curves with complex multiplication."}
{"category": "Math", "title": "Enlargements of positive sets", "abstract": "In this paper we introduce the notion of enlargement of a positive set in SSD spaces. To a maximally positive set $A$ we associate a family of enlargements $\\E(A)$ and characterize the smallest and biggest element in this family with respect to the inclusion relation. We also emphasize the existence of a bijection between the subfamily of closed enlargements of $\\E(A)$ and the family of so-called representative functions of $A$. We show that the extremal elements of the latter family are two functions recently introduced and studied by Stephen Simons. In this way we extend to SSD spaces some former results given for monotone and maximally monotone sets in Banach spaces."}
{"category": "Math", "title": "A discrete Faa di Bruno's formula", "abstract": "We derive some formulas that rule the behaviour of finite differences under composition of functions with vector values and arguments."}
{"category": "Math", "title": "Principe d'Heisenberg et fonctions positives", "abstract": "Starting from a problem in number theory, the article investigates the properties of the couples of Fourier transforms on the real line, f and g, real and even, f >= 0 out of an interval (-a, a) and f(0) < 0, g >= 0 out of an interval (-b, b) and g(0) < 0 . How small the product ab can be ? There is a strictly positive lower bound, the exact value is not known. The same problem is considered in several dimensions (where it is related to number theory, as the article points out)."}
{"category": "Math", "title": "On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon", "abstract": "The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated, and characterized up to a measure zero set, by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of another conjecture that we call Aubry-Godr\\`eche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the fractality of sum-of-digits functions, play a fundamental role in the description of the singular continous part of the spectrum, combined with some classical results on Riesz products of Peyri\\`ere and M. Queff\\'elec. The dominant scaling of the sequences of approximant measures on a part of the singular component is controlled by certain inequalities in which are involved the class number and the regulator of real quadratic fields."}
{"category": "Math", "title": "A simple presentation of the handlebody group of genus 2", "abstract": "Using a similar algorithm to Hatcher-Thurston's algorithm for finding a presentation of the mapping class group of a surface, Wajnryb succeeded to find a presentation for the handlebody group. This is long and complicated. In this note I simplify Wajnryb's presentation for the handlebody group of genus g = 2."}
{"category": "Math", "title": "Extension of functions with bounded finite differences", "abstract": "We prove that functions defined on a lattice in a finite dimensional torus with bounded finite differences can be smoothly extended to the whole torus, and relate the bounds on the extension's derivatives with bounds on the original function's finite differences."}
{"category": "Math", "title": "Classifications of linear operators preserving elliptic, positive and non-negative polynomials", "abstract": "We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer-Fock dualities, Hankel forms, and convolutions with non-negative measures. We also establish higher-dimensional analogs of these results. In particular, our classification theorems solve the questions raised in [9] originating from entire function theory and the literature pertaining to Hilbert's 17th problem."}
{"category": "Math", "title": "A non-solvable Galois extension of $\\Q$ ramified at 2 only", "abstract": "In this paper, we show the existence of a non-solvable Galois extension of $\\Q$ which is unramified outside 2. The extension $K$ we construct has degree $2251731094732800=2^{19}(3\\cdot 5\\cdot 17\\cdot 257)^2$ and has root discriminant $\\delta_K <2^{{47/8}}=58.68...$, and is totally complex."}
{"category": "Math", "title": "Words with intervening neighbours in infinite Coxeter groups are reduced", "abstract": "Consider a graph with vertex set S. A word in the alphabet S has the intervening neighbours property if any two occurrences of the same letter are separated by all its graph neighbours. For a Coxeter graph, words represent group elements. Speyer recently proved that words with the intervening neighbours property are irreducible if the group is infinite and irreducible. We present a new and shorter proof using the root automaton for recognition of irreducible words."}
{"category": "Math", "title": "$R$-polynomials of finite monoids of Lie type", "abstract": "This paper concerns the combinatorics of the orbit Hecke algebra associated with the orbit of a two sided Weyl group action on the Renner monoid of a finite monoid of Lie type, $M$. It is shown by Putcha in \\cite{Putcha97} that the Kazhdan-Lusztig involution (\\cite{KL79}) can be extended to the orbit Hecke algebra which enables one to define the $R$-polynomials of the intervals contained in a given orbit. Using the $R$-polynomials, we calculate the M\\\"obius function of the Bruhat-Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group."}
{"category": "Math", "title": "Exactly solved models of polyominoes and polygons", "abstract": "This chapter deals with the exact enumeration of certain classes of self-avoiding polygons and polyominoes on the square lattice. We present three general approaches that apply to many classes of polyominoes. The common principle to all of them is a recursive description of the polyominoes which then translates into a functional equation satisfied by the generating function. The first approach applies to classes of polyominoes having a linear recursive structure and results in a rational generating function. The second approach applies to classes of polyominoes having an algebraic recursive structure and results in an algebraic generating function. The third approach, commonly called the Temperley method, is based on the action of adding a new column to the polyominoes. We conclude by discussing some open questions."}
{"category": "Math", "title": "Notes on Algebraic Cycles and Homotopy theory", "abstract": "We show that a conjecture by Lawson holds, that is, the inclusion from the Chow variety $C_{p,d}(P^n)$ of all effective algebraic p-cycles of degree d in n-dimensional projective space to the space of effective algebraic p-cycles is 2d-connected. As a result, the homotopy and homology groups of $C_{p,d}(P^n)$ are calculated up to 2d. We also show an analogous statement for Chow variety $C_{p,d}(\\P^n)$ over algebraically closed fields of arbitrary characteristic and compute their etale homotopy groups up to 2d."}
{"category": "Math", "title": "Translation-finite sets, and weakly compact derivations from $\\lp{1}(\\Z_+)$ to its dual", "abstract": "We characterize those derivations from the convolution algebra $\\ell^1({\\mathbb Z}_+)$ to its dual which are weakly compact. In particular, we provide examples which are weakly compact but not compact. The characterization is combinatorial, in terms of \"translation-finite\" subsets of ${\\mathbb Z}_+$, and we investigate how this notion relates to other notions of \"smallness\" for infinite subsets of ${\\mathbb Z}_+$. In particular, we show that a set of strictly positive Banach density cannot be translation-finite; the proof has a Ramsey-theoretic flavour."}
{"category": "Math", "title": "On the special values of certain Rankin-Selberg L-functions and applications to odd symmetric power L-functions of modular forms", "abstract": "We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and Kasten-Schmidt. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands functoriality one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blasius and Panchishkin on twisted L-values in some cases. We comment on the compatibility of our results with Deligne's conjecture on the critical values of motivic L-functions. These results, as in the above mentioned works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals."}
{"category": "Math", "title": "Alternating Euler sums at the negative integers", "abstract": "We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers."}
{"category": "Math", "title": "Where to place a hole to achieve a maximal escape rate", "abstract": "A natural question of how the survival probability depends upon a position of a hole was seemingly never addressed in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period) which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole."}
{"category": "Math", "title": "Continuous Wavelets on Compact Manifolds", "abstract": "Let $\\bf M$ be a smooth compact oriented Riemannian manifold, and let $\\Delta_{\\bf M}$ be the Laplace-Beltrami operator on ${\\bf M}$. Say $0 \\neq f \\in \\mathcal{S}(\\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 \\Delta_{\\bf M})$. We show that $K_t$ is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator $f(t^2\\Delta)$ on $\\RR^n$. We define continuous ${\\cal S}$-wavelets on ${\\bf M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous ${\\cal S}$-wavelets on ${\\bf M}$ are analogous to continuous wavelets on $\\RR^n$ in $\\mathcal{S}(\\RR^n)$. In particular, we are able to characterize the H$\\ddot{o}$lder continuous functions on ${\\bf M}$ by the size of their continuous ${\\mathcal{S}}-$wavelet transforms, for H$\\ddot{o}$lder exponents strictly between 0 and 1. If $\\bf M$ is the torus $\\TT^2$ or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for $K_t$, one to be used when $t$ is large, and one to be used when $t$ is small."}
{"category": "Math", "title": "Log canonical thresholds on smooth varieties: the Ascending Chain Condition", "abstract": "Building on results of Koll\\'ar, we prove Shokurov's ACC Conjecture for log canonical thresholds on smooth varieties, and more generally, on varieties with quotient singularities."}
{"category": "Math", "title": "Understanding Terrorist Organizations with a Dynamic Model", "abstract": "Terrorist organizations change over time because of processes such as recruitment and training as well as counter-terrorism (CT) measures, but the effects of these processes are typically studied qualitatively and in separation from each other. Seeking a more quantitative and integrated understanding, we constructed a simple dynamic model where equations describe how these processes change an organization's membership. Analysis of the model yields a number of intuitive as well as novel findings. Most importantly it becomes possible to predict whether counter-terrorism measures would be sufficient to defeat the organization. Furthermore, we can prove in general that an organization would collapse if its strength and its pool of foot soldiers decline simultaneously. In contrast, a simultaneous decline in its strength and its pool of leaders is often insufficient and short-termed. These results and other like them demonstrate the great potential of dynamic models for informing terrorism scholarship and counter-terrorism policy making."}
{"category": "Math", "title": "Towards Proving Legendre's Conjecture", "abstract": "Legendre's conjecture states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. We consider the following question : for all integer n>1 and a fixed integer k<=n does there exist a prime number such that kn < p < (k+1)n ? Bertrand-Chebyshev theorem answers this question affirmatively for k=1. A positive answer for k=n would prove Legendre's conjecture. In this paper, we show that one can determine explicitly a number N(k) such that for all n >= N(k), there is at least one prime between kn and (k+1)n. Our proof is based on Erdos's proof of Bertrand-Chebyshev theorem and uses elementary combinatorial techniques without appealing to the prime number theorem."}
{"category": "Math", "title": "Laumon Spaces and the Calogero-Sutherland Integrable System", "abstract": "This paper contains a proof of a conjecture of Braverman concerning Laumon quasiflag spaces. We consider the generating function Z(m), whose coefficients are the integrals of the equivariant Chern polynomial (with variable m) of the tangent bundles of the Laumon spaces. We prove Braverman's conjecture, which states that Z(m) coincides with the eigenfunction of the Calogero-Sutherland hamiltonian, up to a simple factor which we specify. This conjecture was inspired by the work of Nekrasov in the affine \\hat{sl}_n setting, where a similar conjecture is still open."}
{"category": "Math", "title": "Convergence to Weighted Fractional Brownian Sheets", "abstract": "We define weighted fractional Brownian sheets, which are a class of Gaussian random fields with four parameters that include fractional Brownian sheets as special cases, and we give some of their properties. We show that for certain values of the parameters the weighted fractional Brownian sheets are obtained as limits in law of occupation time fluctuations of a stochastic particle model. In contrast with some known approximations of fractional Brownian sheets which use a kernel in a Volterra type integral representation of fractional Brownian motion with respect to ordinary Brownian motion, our approximation does not make use of a kernel."}
{"category": "Math", "title": "Transgression and twisted anomaly cancellation formulas on odd dimensional manifolds", "abstract": "We compute the transgressed forms of some modularly invariant characteristic forms,which are related to the twisted elliptic genera. We study the modularity properties of these secondary characteristic forms and relations among them. We also get some twisted anomaly cancellation formulas on some odd dimensional manifolds."}
{"category": "Math", "title": "On the existence of embeddings into modules of finite homological dimensions", "abstract": "Let R be a commutative Noetherian local ring. We show that R is Gorenstein if and only if every finitely generated R-module can be embedded in a finitely generated R-module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay."}
{"category": "Math", "title": "A Local Relative Trace Formula for F*\\SL(2,F)", "abstract": "In this note, we derive explicitly the local relative trace formula for the symmetric space F*\\SL(2,F) at the level of Lie algebras, where F is a p-adic field of residue characteristic greater than two and F* is the set of invertible elements in F. This is perhaps one of the simplest non-trivial analogs of the trace formula, and also a motivating example for the author's work (in preparation) on the relative trace formula."}
{"category": "Math", "title": "Partial Correlation Estimation by Joint Sparse Regression Models", "abstract": "In this paper, we propose a computationally efficient approach -- space(Sparse PArtial Correlation Estimation)-- for selecting non-zero partial correlations under the high-dimension-low-sample-size setting. This method assumes the overall sparsity of the partial correlation matrix and employs sparse regression techniques for model fitting. We illustrate the performance of space by extensive simulation studies. It is shown that space performs well in both non-zero partial correlation selection and the identification of hub variables, and also outperforms two existing methods. We then apply space to a microarray breast cancer data set and identify a set of hub genes which may provide important insights on genetic regulatory networks. Finally, we prove that, under a set of suitable assumptions, the proposed procedure is asymptotically consistent in terms of model selection and parameter estimation."}
{"category": "Math", "title": "AV-Courant algebroids and generalized CR structures", "abstract": "We construct a generalization of Courant algebroids which are classified by the third cohomology group $H^3(A,V)$, where $A$ is a Lie Algebroid, and $V$ is an $A$-module. We see that both Courant algebroids and $\\mathcal{E}^1(M)$ structures are examples of them. Finally we introduce generalized CR structures on a manifold, which are a generalization of generalized complex structures, and show that every CR structure and contact structure is an example of a generalized CR structure."}
{"category": "Math", "title": "Asymptotically hyperbolic manifolds with polyhomogeneous metric", "abstract": "We analyze the resolvent and define the scattering matrix for asymptotically hyperbolic manifolds with metrics which have a polyhomogeneous expansion near the boundary, and also prove that there is always an essential singularity of the resolvent in this setting. We use this analysis to prove an inverse result for conformally compact odd dimensional Einstein manifolds."}
{"category": "Math", "title": "Second order Poincar\\'e inequalities and CLTs on Wiener space", "abstract": "We prove infinite-dimensional second order Poincar\\'e inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian fields, Stein's method and Malliavin calculus. We provide two applications: (i) to a new \"second order\" characterization of CLTs on a fixed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated fields."}
{"category": "Math", "title": "Large Deviations estimates for some non-local equations I. Fast decaying kernels and explicit bounds", "abstract": "We study large deviations for some non-local parabolic type equations. We show that, under some assumptions on the non-local term, problems defined in a bounded domain converge with an exponential rate to the solution of the problem defined in the whole space. We compute this rate in different examples, with different kernels defining the non-local term, and it turns out that the estimate of convergence depends strongly on the decay at infinity of that kernel."}
{"category": "Math", "title": "A note on the convergence of parametrised non-resonant invariant manifolds", "abstract": "Truncated Taylor series representations of invariant manifolds are abundant in numerical computations. We present an aposteriori method to compute the convergence radii and error estimates of analytic parametrisations of non-resonant local invariant manifolds of a saddle of an analytic vector field, from such a truncated series. This enables us to obtain local enclosures, as well as existence results, for the invariant manifolds."}
{"category": "Math", "title": "Support varieties and the Hochschild cohomology ring modulo nilpotence", "abstract": "This is a survey paper based on my talks at the 41st Symposium on Ring Theory and Representation Theory, held in Shizuoka University, Japan in September 2008, and will appear in the conference proceedings. The paper begins with a brief introduction to the use of Hochschild cohomology in developing the theory of support varieties for a module over an artin algebra, by Snashall and Solberg (Proc. London Math. Soc.(3) 88 (2004), 705-732). The paper then describes the current status of research concerning the structure of the Hochschild cohomology ring modulo nilpotence."}
{"category": "Math", "title": "The 2D Ising model near criticality: a FK percolation analysis", "abstract": "We study the 2d-Ising model defined on finite boxes at temperatures that are below but very close from the critical point. When the temperature approaches the critical point and the size of the box grows fast enough, we establish large deviations estimates on FK-percolation events that concern the phenomenon of phase coexistence."}
{"category": "Math", "title": "Special L-values of t-motives: a conjecture", "abstract": "We propose a conjecture on special values of $ L $-functions in a function field context with positive characteristic coefficients. For $ M $ a uniformizable $ t $-motive with everywhere good reduction we conjecture a relation between the value of the Goss $ L $-function $ L(M^\\vee, s) $ at $ s = 0 $ and the uniformization of the abelian $ t $-module associated with $ M $. When $ M $ is a power of the Carlitz $ t $-motive the conjecture specializes to a theorem of Anderson and Thakur on Carlitz zeta values. Beyond this case we present numerical evidence."}
{"category": "Math", "title": "Invariant elliptic curves as attractors in the projective plane", "abstract": "Let f be a rational self-map of P^2 which leaves invariant an elliptic curve C with strictly negative transverse Lyapunov exponent. We show that C is an attractor, i.e. it possesses a dense orbit and its basin is of strictly positive measure."}
{"category": "Math", "title": "Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles", "abstract": "We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups $G$ such that $G$ is generated by a compact family $\\Gamma $, the planar postcritical set of $G$ is bounded, and $G$ is (semi-) hyperbolic. In one of the classes, we have that for almost every sequence $\\gamma \\in \\Gamma ^{\\Bbb{N}}$, the Julia set $J_{\\gamma}$ of $\\gamma $ is a Jordan curve but not a quasicircle, the unbounded component of the Fatou set $F_{\\gamma}$ of $\\gamma$ is a John domain, and the bounded component of $F_{\\gamma}$ is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups $G$ such that the planar postcritical set of $G$ is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated."}
{"category": "Math", "title": "Birkhoff normal form and splitting methods for semi linear Hamiltonian PDEs. Part I: Finite dimensional discretization", "abstract": "We consider {\\em discretized} Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a finite dimensional Birkhoff normal form result, we show the almost preservation of the {\\em actions} of the numerical solution associated with the splitting method over arbitrary long time, provided the Sobolev norms of the initial data is small enough, and for asymptotically large level of space approximation. This result holds under {\\em generic} non resonance conditions on the frequencies of the linear operator and on the step size. We apply this results to nonlinear Schr\\\"odinger equations as well as the nonlinear wave equation.}"}
{"category": "Math", "title": "Quantales of open groupoids", "abstract": "It is well known that inverse semigroups are closely related to \\'etale groupoids. In particular, it has recently been shown that there is a (non-functorial) equivalence between localic \\'etale groupoids, on one hand, and complete and infinitely distributive inverse semigroups (abstract complete pseudogroups), on the other. This correspondence is mediated by a class of quantales, known as inverse quantal frames, that are obtained from the inverse semigroups by a simple join completion that yields an equivalence of categories. Hence, we can regard abstract complete pseudogroups as being essentially ``the same'' as inverse quantal frames, and in this paper we exploit this fact in order to find a suitable replacement for inverse semigroups in the context of open groupoids that are not necessarily \\'etale. The interest of such a generalization lies in the importance and ubiquity of open groupoids in areas such as operator algebras, differential geometry and topos theory, and we achieve it by means of a class of quantales, called open quantal frames, which generalize inverse quantal frames and whose properties we study in detail. The resulting correspondence between quantales and open groupoids is not a straightforward generalization of the previous results concerning \\'etale groupoids, and it depends heavily on the existence of inverse semigroups of local bisections of the quantales involved."}
{"category": "Math", "title": "A note on Vasiu-Zink windows", "abstract": "We propose a notion of frames and windows that allows an alternative proof of the Vasiu-Zink classification of p-divisible groups over ramified complete regular local rings by their Breuil windows."}
{"category": "Math", "title": "Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras", "abstract": "We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space."}
{"category": "Math", "title": "Notes on Artin-Tate motives", "abstract": "We first study the weight structure on the triangulated category of Artin-Tate motives over a perfect base field k, building on results of Bondarko's. We then study the t-structure on the triangulated category of Artin-Tate motives, when k is algebraic over the rationals, generalizing a result of Levine's. We finally study the interaction of the weight structure and the t-structure. When k is a number field, this will give a useful criterion identifying the weight structure via realizations."}
{"category": "Math", "title": "A note on the subword complexes in Coxeter groups", "abstract": "We prove that the Stanley--Reisner ideal of the Alexander dual of the subword complexes in Coxeter groups has linear quotients with respect to the lexicographical order of the minimal monomial generators. As a consequence, we obtain a shelling order on the facets of the subword complex. We relate some invariants of the subword complexes or of their dual with invariants of the word. For a particular class of subword complexes, we prove that the Stanley--Reisner ring is a complete intersection ring."}
{"category": "Math", "title": "Averaging lemmas with a force term in the transport equation", "abstract": "We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. Two methods are used: local variable changes or stationary phase. These new results are subjected to two non degeneracy assumptions. We characterize the optimal conditions of these assumptions to compare the obtained regularities according to the space and velocity variables. Our results are mainly in $L^2$, and for constant force, in $L^p$ for $1<p \\leq 2$."}
{"category": "Math", "title": "Courant algebroids and Poisson Geometry", "abstract": "Given a manifold M with an action of a quadratic Lie algebra d, such that all stabilizer algebras are co-isotropic in d, we show that the product M\\times d becomes a Courant algebroid over M. If the bilinear form on d is split, the choice of transverse Lagrangian subspaces g_1, g_2 of d defines a bivector field on M, which is Poisson if (d,g_1,g_2) is a Manin triple. In this way, we recover the Poisson structures of Lu-Yakimov, and in particular the Evens-Lu Poisson structures on the variety of Lagrangian Grassmannians and on the de Concini-Procesi compactifications. Various Poisson maps between such examples are interpreted in terms of the behaviour of Lagrangian splittings under Courant morphisms."}
{"category": "Math", "title": "The Zassenhaus variety of a reductive Lie algebra in positive characteristic", "abstract": "Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let $Z$ be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g* with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand-Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G_2."}
{"category": "Math", "title": "Analytic equivalence of normal crossing functions on a real analytic manifold", "abstract": "By Hironaka Desingularization Theorem, any real analytic function has only normal crossing singularities after a suitable modification. We focus on the analytic equivalence of such functions with only normal crossing singularities. We prove that for such functions $C^{\\infty}$ right equivalence implies analytic equivalence. We prove moreover that the cardinality of the set of equivalence classes is zero or countable."}
{"category": "Math", "title": "Trace Forms of Symbol Algebras", "abstract": "Let S be a symbol algebra. The trace form of S is computed and it is shown how this form can be used to determine whether S is a division algebra or not. In addition, the exterior powers of the trace form of S are computed."}
{"category": "Math", "title": "Concentration of the integral norm of idempotents", "abstract": "This is a companion paper of a recent one, entitled {\\sl Integral concentration of idempotent trigonometric polynomials with gaps}. New results of the present work concern $L^1$ concentration, while the above mentioned paper deals with $L^p$-concentration. Our aim here is two-fold. At the first place we try to explain methods and results, and give further straightforward corollaries. On the other hand, we push forward the methods to obtain a better constant for the possible concentration (in $L^1$ norm) of an idempotent on an arbitrary symmetric measurable set of positive measure. We prove a rather high level $\\gamma_1>0.96$, which contradicts strongly the conjecture of Anderson et al. that there is no positive concentration in $L^1$ norm. The same problem is considered on the group $\\mathbb{Z}/q\\mathbb{Z}$, with $q$ say a prime number. There, the property of absolute integral concentration of idempotent polynomials fails, which is in a way a positive answer to the conjecture mentioned above. Our proof uses recent results of B. Green and S. Konyagin on the Littlewood Problem."}
{"category": "Math", "title": "Coagulation, diffusion and the continuous Smoluchowski equation", "abstract": "The Smoluchowski equation is a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers, or by positive reals, these corresponding to the discrete or the continuous form of the equations. In dimension at least 3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a similar method to that used to derive the discrete form of the equations in Hammond and Rezakhanlou [4]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of [4]. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations."}
{"category": "Math", "title": "Quadratic perturbations of quadratic codimension-four centers", "abstract": "We study the stratum in the set of all quadratic differential systems $\\dot{x}=P_2(x,y), \\dot{y}=Q_2(x,y)$ with a center, known as the codimension-four case $Q_4$. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincar\\'e-Pontryagin-Melnikov integral $I$. We show that the orbits of the unperturbed system are elliptic curves, and $I$ is a complete elliptic integral. Then using Picard-Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center."}
{"category": "Math", "title": "On the Kontsevich integral for knotted trivalent graphs", "abstract": "We construct an extension of the Kontsevich integral of knots to knotted trivalent graphs, which commutes with orientation switches, edge deletions, edge unzips, and connected sums. In 1997 Murakami and Ohtsuki [MO] first constructed such an extension, building on Drinfel'd's theory of associators. We construct a step by step definition, using elementary Kontsevich integral methods, to get a one-parameter family of corrections that all yield invariants well behaved under the graph operations above."}
{"category": "Math", "title": "On the graded center of the stable category of a finite $p$-group", "abstract": "We show that for any finite $p$-group $P$ of rank at least 2 and any algebraically closed field $k$ of characteristic $p$ the graded center $Z^*(\\modbar(kP))$ of the stable module category of finite-dimensional $kP$-modules has infinite dimension in each odd degree, and if $p=2$ also in each even degree. In particular, this provides examples of symmetric algebras $A$ for which $Z^0(\\modbar(A))$ is not finite-dimensional, answering a question raised in [10]"}
{"category": "Math", "title": "On (n,d)-perfect rings", "abstract": "In this paper, we introduce the notion of \"(n,d)-perfect rings\" which is in some way a generalization of the notion of \"S-rings\". After we give some basic results of this rings and we survey the relationship between \"A(n) property\" and \"(n,d)-perfect property\". Finally, we investigate the \"(n,d)-perfect property\" in pullback rings."}
{"category": "Math", "title": "A candidate for a solution to Wall's D(2) problem", "abstract": "We show that Wall's D(2) problem, the Realization problem and the Relation Gap problem could all be solved if it could be shown that the deficiency of a certain group is, as intuition would suggest, less than -1. Note the paper has been withdrawn. A presentation of *_p (C_p x C_p)with deficiency -1 is given on p35 of: Cynthia Hog-Angeloni, Beitrage zum (einfachen) homotopietyp zweidimensionaler komplexe zu freein produkten und anderen gruppentheoretischen konstruktionen : PhD thesis, Frankfurt/Main 1988"}
{"category": "Math", "title": "The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes", "abstract": "We prove localization and Zariski-Mayer-Vietoris for higher Grothendieck-Witt groups, alias hermitian $K$-groups, of schemes admitting an ample family of line-bundles. No assumption on the characteristic is needed, and our schemes can be singular. Along the way, we prove additivity, fibration and approximation theorems for the hermitian $K$-theory of exact categories with weak equivalences and duality."}
{"category": "Math", "title": "The theory of disconjugacy for a second order linear differential equation", "abstract": "This is an introduction to the theory of disconjugacy for a second order linear differential equation. We give new proofs of some of basic results and obtain new sufficient conditions for disconjugacy (in particular, on the whole real axis)."}
{"category": "Math", "title": "Transformations of hypergeometric elliptic integrals", "abstract": "The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1/2,1/4,1/4)$, $(1/2,1/3,1/6)$ and $(1/3,1/3,1/3)$. These form a special class of algebraic transformations of Gauss hypergeometric functions, of arbitrary high degree. The Gauss hypergeometric functions can be identified as elliptic integrals on the genus 1 curves $y=x^3-x$ or $y=x^3-1$. Especially interesting are algebraic transformations of the hypergeometric functions into themselves; these transformations come from isogenies of the respective elliptic curves."}
{"category": "Math", "title": "The Ascending Chain Condition for log canonical thresholds on l.c.i. varieties", "abstract": "Shokurov's ACC Conjecture says that the set of all log canonical thresholds on varieties of bounded dimension satisfies the Ascending Chain Condition. This conjecture was proved for log canonical thresholds on smooth varieties in [EM1]. Here we use this result and inversion of adjunction to establish the conjecture for locally complete intersection varieties."}
{"category": "Math", "title": "Liouville type of theorems with weights for the Navier-Stokes equations and the Euler equations", "abstract": "We study Liouville type of theorems for the Navier-Stokes and the Euler equations on $\\Bbb R^N$, $N\\geq 2$. Specifically, we prove that if a weak solution $(v,p)$ satisfies $|v|^2 +|p| \\in L^1 (0,T; L^1(\\Bbb R^N, w_1(x)dx))$ and $\\int_{\\Bbb R^N} p(x,t)w_2 (x)dx \\geq0$ for some weight functions $w_1(x)$ and $w_2 (x)$, then the solution is trivial, namely $v=0$ almost everywhere on $\\Bbb R^N \\times (0, T)$. Similar results hold for the MHD Equations on $\\Bbb R^N$, $N\\geq3$."}
{"category": "Math", "title": "A note on Fibonacci-type polynomials", "abstract": "We opt to study the convergence of maximal real roots of certain Fibonacci-type polynomials given by $G_n=x^kG_{n-1}+G_{n-2}$. The special cases $k=1$ and $k=2$ are found in [4] and [7], respectively."}
{"category": "Math", "title": "Gotzmann lexsegment ideals", "abstract": "In this paper we characterize the componentwise lexsegment ideals which are componentwise linear and the lexsegment ideals generated in one degree which are Gotzmann."}
{"category": "Math", "title": "Convergence rates of posterior distributions for observations without the iid structure", "abstract": "The classical condition on the existence of uniformly exponentially consistent tests for testing the true density against the complement of its arbitrary neighborhood has been widely adopted in study of asymptotics of Bayesian nonparametric procedures. Because we follow a Bayesian approach, it seems to be more natural to explore alternative and appropriate conditions which incorporate the prior distribution. In this paper we supply a new prior-dependent integration condition to establish general posterior convergence rate theorems for observations which may not be independent and identically distributed. The posterior convergence rates for such observations have recently studied by Ghosal and van der Vaart \\cite{ghv1}. We moreover adopt the Hausdorff $\\alpha$-entropy given by Xing and Ranneby \\cite{xir1}\\cite{xi1}, which is also prior-dependent and smaller than the widely used metric entropies. These lead to extensions of several existing theorems. In particular, we establish a posterior convergence rate theorem for general Markov processes and as its application we improve on the currently known posterior rate of convergence for a nonlinear autoregressive model."}
{"category": "Math", "title": "Clifford indices for vector bundles on curves", "abstract": "For smooth projective curves the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step in distinguishing curves of the same genus. In this paper we generalise this to introduce Clifford indices for semistable vector bundles on curves. We study these invariants, giving some basic properties and carrying out some computations for small ranks and for general and some special curves. For curves whose classical Clifford index is two, we compute all values of our new Clifford indices."}
{"category": "Math", "title": "Unconditional bases and strictly convex dual renormings", "abstract": "We present equivalent conditions for a space $X$ with an unconditional basis to admit an equivalent norm with a strictly convex dual norm."}
{"category": "Math", "title": "Bouvier's conjecture", "abstract": "This paper deals with Bouvier's conjecture which sustains that finite-dimensional non-Noetherian Krull domains need not be Jaffard"}
{"category": "Math", "title": "t-Class semigroups of Noetherian domains", "abstract": "The t-class semigroup of an integral domain is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of a Noetherian domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup."}
{"category": "Math", "title": "Gromov-Witten invariants of toric Calabi-Yau threefolds", "abstract": "Based on the large N duality relating topological string theory on Calabi-Yau 3-folds and Chern-Simons theory on 3-manifolds, M. Aganagic, A. Klemm, M. Marino and C. Vafa proposed the topological vertex (hep-th/0305132), an algorithm on computing Gromov-Witten invariants in all genera of any non-singular toric Calabi-Yau 3-fold. In this expository article, we describe the mathematical theory of the topological vertex developed by J. Li, K. Liu, J. Zhou, and the author (math/0408426)."}
{"category": "Math", "title": "Divided power structures and chain complexes", "abstract": "We interpret divided power structures on the homotopy groups of simplicial commutative rings as having a counterpart in divided power structures on chain complexes coming from a non-standard symmetric monoidal structure."}
{"category": "Math", "title": "Fast rotating Bose-Einstein condensates in an asymmetric trap", "abstract": "We investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating Bose-Einstein condensate. This is done in the framework of the 2D Gross-Pitaevskii equation and requires a symplectic reduction of the quadratic form defining the energy. This reduction allows us to simplify the energy on a Bargmann space and study the asymptotics of large rotational velocity. We characterize two regimes of velocity and anisotropy; in the first one where the behaviour is similar to the isotropic case, we construct an upper bound: a hexagonal Abrikosov lattice of vortices, with an inverted parabola profile. The second regime deals with very large velocities, a case in which we prove that the ground state does not display vortices in the bulk, with a 1D limiting problem. In that case, we show that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction but keeps a fixed profile given by a Gaussian in the other direction. The features of this second regime appear as new phenomena."}
{"category": "Math", "title": "A Construction of Biorthogonal Wavelets With a Compact Operator", "abstract": "We present a construction of biorthogonal wavelets using a compact operator which allows to preserve or increase some properties: regularity/vanishing moments, parity, compact supported. We build then a simple algorithm which computes new filters."}
{"category": "Math", "title": "Generalized power method for sparse principal component analysis", "abstract": "In this paper we develop a new approach to sparse principal component analysis (sparse PCA). We propose two single-unit and two block optimization formulations of the sparse PCA problem, aimed at extracting a single sparse dominant principal component of a data matrix, or more components at once, respectively. While the initial formulations involve nonconvex functions, and are therefore computationally intractable, we rewrite them into the form of an optimization program involving maximization of a convex function on a compact set. The dimension of the search space is decreased enormously if the data matrix has many more columns (variables) than rows. We then propose and analyze a simple gradient method suited for the task. It appears that our algorithm has best convergence properties in the case when either the objective function or the feasible set are strongly convex, which is the case with our single-unit formulations and can be enforced in the block case. Finally, we demonstrate numerically on a set of random and gene expression test problems that our approach outperforms existing algorithms both in quality of the obtained solution and in computational speed."}
{"category": "Math", "title": "Counting decomposable multivariate polynomials", "abstract": "A polynomial f (multivariate over a field) is decomposable if f = g(h) with g univariate of degree at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number over a finite field. The relative error in our approximations is exponentially decaying in the input size."}
{"category": "Math", "title": "A Note on Toric Varieties Associated to Moduli Spaces", "abstract": "In this note we give a brief review of the construction of a toric variety $\\mathcal{V}$ coming from a genus $g \\geq 2$ Riemann surface $\\Sigma^g$ equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L. C. Jeffrey in \\cite{JH1}. In \\cite{T1} A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety $DM_g$ -- the so-called Delzant model of moduli space -- for each genus $g.$ We conclude this note with some basic facts about the moment polytopes of the varieties $\\mathcal{V}.$ In particular, we show that the varieties $DM_g$ constructed by Tyurin, and claimed to be smooth, are in fact singular for $g \\geq 3.$"}
{"category": "Math", "title": "The structure of maximal zero-sum free Sequences", "abstract": "Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum zero. It is known that the maximal length of such a zero-sum free sequence is 2n-2, and Gao and Geroldinger conjectured that every zero-sum free sequence of this length contains an element with multiplicity at least n-2. By recent results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify the conjecture for n prime. Now fix a sequence (a_i) of length 2n-2 with maximal multiplicity of elements at most n-3. There are different approeaches to show that (a_i) contains a zero-sum; some work well when (a_i) does contain elements with high multiplicity, others work well when all multiplicities are small. The aim of this article is to initiate a systematic approach to property B via the highest occurring multiplicities. Our main results are the following: denote by m_1 >= m_2 the two maximal multiplicities of (a_i), and suppose that n is sufficiently big and prime. Then (a_i) contains a zero-sum in any of the following cases: when m_2 >= 2/3n, when m_1 > (1-c)n, and when m_2 < cn, for some constant c > 0 not depending on anything."}
{"category": "Math", "title": "Twenty Digits of Some Integrals of the Prime Zeta Function", "abstract": "The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohen's work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods."}
{"category": "Math", "title": "On the stability of the overconvergence under the direct image by a proper smooth morphism", "abstract": "Up to a translation in the language of arithmetic $\\D$-modules, we prove a conjecture of Berthelot on the preservation of the overconvergence under the direct image by a smooth proper morphism of varieties over a perfect field of characteristic $p>0$."}
{"category": "Math", "title": "The falling appart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation", "abstract": "We present a further analysis of the fragmentation at heights of the normalized Brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the Brownian fragmentation when it is about to reduce to dust are described in a limit theorem."}
{"category": "Math", "title": "The Depth of a Hypersubstitution", "abstract": "For given depth of a we derive a formula for the depth of the image of that term under a given hypersubstitution."}
{"category": "Math", "title": "Multi-Hypersubstitutions and Colored Solid Varieties", "abstract": "Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multi-hypersubstitutions and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice."}
{"category": "Math", "title": "On some cohomological properties of almost complex manifolds", "abstract": "We study a special type of almost complex structures, called pure and full and introduced by T.J. Li and W. Zhang, in relation to symplectic structures and Hard Lefschetz condition. We provide sufficient conditions to the existence of the above type of almost complex structures on compact quotients of Lie groups by discrete subgroups. We obtain families of pure and full almost complex structures on compact nilmanifolds and solvmanifolds. Some of these families are parametrized by real 2-forms which are anti-invariant with respect to the almost complex structures."}
{"category": "Math", "title": "Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions", "abstract": "For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u_0*r^{alpha+a-2}*(r+1)^n. In particular, letting a = 2 yields an improvement to the best previous lower bound on L_n (obtained by Hong and Yang) for all but three choices of alpha,r >= 2."}
{"category": "Math", "title": "Some equivariant constructions in noncommutative algebraic geometry", "abstract": "We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup."}
{"category": "Math", "title": "A note on the abelianizations of finite-index subgroups of the mapping class group", "abstract": "For some $g \\geq 3$, let $\\Gamma$ be a finite index subgroup of the mapping class group of a genus $g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $\\Gamma$ should be finite. In this note, we prove two theorems supporting this conjecture. For the first, let $T_x$ denote the Dehn twist about a simple closed curve $x$. For some $n \\geq 1$, we have $T_x^n \\in \\Gamma$. We prove that $T_x^n$ is torsion in the abelianization of $\\Gamma$. Our second result shows that the abelianization of $\\Gamma$ is finite if $\\Gamma$ contains a \"large chunk\" (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves. This generalizes work of Hain and Boggi."}
{"category": "Math", "title": "Hessenberg Pairs of Linear Transformations", "abstract": "Let $\\fld$ denote a field and $V$ denote a nonzero finite-dimensional vector space over $\\fld$. We consider an ordered pair of linear transformations $A: V \\to V$ and $A^*: V \\to V$ that satisfy (i)--(iii) below. Each of $A, A^*$ is diagonalizable on $V$. There exists an ordering $\\lbrace V_i \\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that A^* V_i \\subseteq V_0 + V_1 + ... + V_{i+1} \\qquad \\qquad (0 \\leq i \\leq d), where $V_{-1} = 0$, $V_{d+1}= 0$. There exists an ordering $\\lbrace V^*_i \\rbrace_{i=0}^{\\delta}$ of the eigenspaces of $A^*$ such that A V^*_i \\subseteq V^*_0 + V^*_1 + ... +V^*_{i+1} \\qquad \\qquad (0 \\leq i \\leq \\delta), where $V^*_{-1} = 0$, $V^*_{\\delta+1}= 0$. We call such a pair a {\\it Hessenberg pair} on $V$. In this paper we obtain some characterizations of Hessenberg pairs. We also explain how Hessenberg pairs are related to tridiagonal pairs."}
{"category": "Math", "title": "Trace operators of modulation, alpha modulation and Besov spaces", "abstract": "In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results."}
{"category": "Math", "title": "Some Examples of Dynamics for Gelfand Tsetlin Patterns", "abstract": "We give examples of stochastic processes in the Gelfand Tsetlin cone in which each component evolves independently apart from a blocking and pushing interaction. The processes give couplings to certain conditioned Markov processes, last passage times and asymetric exclusion processes. An example of a cone valued process whose components cannot escape past a wall at the origin is also considered."}
{"category": "Math", "title": "A Wreath Product Approach to Classical Subgroup Theorems", "abstract": "We provide elementary proofs of the Nielsen-Schreier Theorem and the Kurosh Subgroup Theorem via wreath products. Our proofs are diagrammatic in nature and work simultaneously in the abstract and profinite categories. A new proof that open subgroups of quasifree profinite groups are quasifree is also given."}
{"category": "Math", "title": "Homogeneous interpolation on ten points", "abstract": "In this paper we prove that for all pairs $(d,m)$ with $d/m \\geq 174/55$, the linear system of plane curves of degree $d$ with ten general base points of multiplicity $m$ has the expected dimension."}
{"category": "Math", "title": "The central value of the Rankin-Selberg $L$-functions", "abstract": "Let $f$ be a Maass form for $SL(3, \\mathbb{Z})$ which is fixed and $u_j$ be an orthonormal basis of even Maass forms for $SL(2, \\mathbb{Z}),$ we prove an asymptotic formula for the average of the product of the Rankin-Selberg $L$-function of $f$ and $u_j$ and the $L$-function of $u_j$ at the central value 1/2. This implies simultaneous nonvanishing results of these $L$-functions at $1/2.$"}
{"category": "Math", "title": "Bounds for $GL(3)\\times GL(2)$ $L$-functions and GL(3) $L$-functions", "abstract": "In this paper, we will give the subconvexity bounds for self dual GL(3) $L-$functions in the $t$ aspect as well as subconvexity bounds for self dual $GL(3)\\times GL(2)$ $L-$functions in the GL(2) spectral aspect."}
{"category": "Math", "title": "New Presentations of Thompson's Groups and Applications", "abstract": "We find new presentations for the Thompson's groups $F$, the derived group $F^{'}$ and the intermediate group $D$. These presentations have a common ground in that their relators are the same and only the generating sets differ. As an application of these presentations we extract the following consequences: the cost of the group $F^{'}$ is 1 hence the cost cannot decide the (non)amenability question of $F$; the $II_1$ factor $L(F^{'})$ is inner asymptotically abelian and the reduced $C^*$-algebra of $F$ is not residually finite dimensional."}
{"category": "Math", "title": "On the average indices of closed geodesics on positively curved Finsler spheres", "abstract": "In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ for $n\\ge 6$ with reversibility $\\lambda$ and flag curvature $K$ satisfying $(\\frac{\\lambda}{\\lambda+1})^2<K\\le 1$, either there exist infinitely many prime closed geodesics or there exist $[\\frac{n}{2}]-2$ closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist $n-3$ closed geodesics possessing irrational average indices provided the number of closed geodesics is finite."}
{"category": "Math", "title": "Stability of closed characteristics on symmetric compact convex hypersurfaces in $\\R^{2n}$", "abstract": "In this article, let $\\Sigma\\subset\\R^{2n}$ be a compact convex hypersurface which is symmetric with respect to the origin. We prove that if $\\Sg$ carries finitely many geometrically distinct closed characteristics, then at least $n-1$ of them must be non-hyperbolic; if $\\Sg$ carries exactly $n$ geometrically distinct closed characteristics, then at least two of them must be elliptic."}
{"category": "Math", "title": "On the Burns-Epstein invariants of spherical CR 3-manifolds", "abstract": "In this paper we develop a method to compute the Burns-Epstein invariant of a spherical CR homology sphere, up to an integer, from its holonomy representation. As application, we give a formula for the Burns-Epstein invariant, modulo an integer, of a spherical CR structure on a Seifert fibered homology sphere in terms of its holonomy representation."}
{"category": "Math", "title": "On a generalization of P(3,n)", "abstract": "By using techniques of poset representation theory, we present a formula for the number of partitions of a positive integer into three polygonal numbers."}
{"category": "Math", "title": "Stability of closed characteristics on compact hypersurfaces in $\\R^{2n}$ under pinching condition", "abstract": "In this article, let $\\Sigma\\subset\\R^{2n}$ be a compact convex hypersurface which is $(r, R)$-pinched with $\\frac{R}{r}<\\sqrt{{3/2}}$. Then $\\Sg$ carries at least two strictly elliptic closed characteristics; moreover, $\\Sg$ carries at least $2[\\frac{n+2}{4}]$ non-hyperbolic closed characteristics."}
{"category": "Math", "title": "Indirect Cross-validation for Density Estimation", "abstract": "A new method of bandwidth selection for kernel density estimators is proposed. The method, termed indirect cross-validation, or ICV, makes use of so-called selection kernels. Least squares cross-validation (LSCV) is used to select the bandwidth of a selection-kernel estimator, and this bandwidth is appropriately rescaled for use in a Gaussian kernel estimator. The proposed selection kernels are linear combinations of two Gaussian kernels, and need not be unimodal or positive. Theory is developed showing that the relative error of ICV bandwidths can converge to 0 at a rate of $n^{-1/4}$, which is substantially better than the $n^{-1/10}$ rate of LSCV. Interestingly, the selection kernels that are best for purposes of bandwidth selection are very poor if used to actually estimate the density function. This property appears to be part of the larger and well-documented paradox to the effect that \"the harder the estimation problem, the better cross-validation performs.\" The ICV method uniformly outperforms LSCV in a simulation study, a real data example, and a simulated example in which bandwidths are chosen locally."}
{"category": "Math", "title": "Empirical study of indirect cross-validation", "abstract": "In this paper we provide insight into the empirical properties of indirect cross-validation (ICV), a new method of bandwidth selection for kernel density estimators. First, we describe the method and report on the theoretical results used to develop a practical-purpose model for certain ICV parameters. Next, we provide a detailed description of a numerical study which shows that the ICV method usually outperforms least squares cross-validation (LSCV) in finite samples. One of the major advantages of ICV is its increased stability compared to LSCV. Two real data examples show the benefit of using both ICV and a local version of ICV."}
{"category": "Math", "title": "On the total mean curvature of non-rigid surfaces", "abstract": "Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex."}
{"category": "Math", "title": "Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces", "abstract": "The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine. Affine vector fields on homogeneous spaces can be characterized by their Lie brackets with the projections of right invariant vector fields. A linear system on a homogeneous space is a system whose drift part is affine and whose controlled part is invariant. The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which have its own interest. The present proof makes use of geometric control theory arguments."}
{"category": "Math", "title": "Multiplicities of the discrete series", "abstract": "The purpose of this paper is to show that the multiplicities of a discrete series representation relatively to a compact subgroup can be \"computed\" geometrically, in the way predicted by the \"qantization commutes with reduction\" principle of Guillemin-Sternberg."}
{"category": "Math", "title": "Cutoff phenomena for random walks on random regular graphs", "abstract": "The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on $\\G(n,d)$, a random $d$-regular graph on $n$ vertices. It is well known that almost every such graph for $d\\geq 3$ is an expander, and even essentially Ramanujan, implying a mixing-time of $O(\\log n)$. According to a conjecture of Peres, the simple random walk on $\\G(n,d)$ for such $d$ should then exhibit cutoff with high probability. As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is w.h.p. $(6+o(1))\\log_2 n$. In this work we confirm the above conjectures, and establish cutoff in total-variation, its location and its optimal window, both for simple and for non-backtracking random walks on $\\G(n,d)$. Namely, for any fixed $d\\geq3$, the simple random walk on $\\G(n,d)$ w.h.p. has cutoff at $\\frac{d}{d-2}\\log_{d-1} n$ with window order $\\sqrt{\\log n}$. Surprisingly, the non-backtracking random walk on $\\G(n,d)$ w.h.p. has cutoff already at $\\log_{d-1} n$ with constant window order. We further extend these results to $\\G(n,d)$ for any $d=n^{o(1)}$ that grows with $n$ (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers."}
{"category": "Math", "title": "Configurations of infinitely near points", "abstract": "We present a survey of some aspects and new results on configurations, i.e. disjoint unions of constellations of infinitely near points, local and global theory, with some applications and results on generalized Enriques diagrams, singular foliations, and linear systems defined by clusters."}
{"category": "Math", "title": "Potential functions via toric degenerations", "abstract": "This is a short companion paper to arXiv:0810.3470. We construct an integrable system on an open subset of a Fano manifold equipped with a toric degeneration, and compute the potential function for its Lagrangian torus fibers if the central fiber is a toric Fano variety admitting a small resolution."}
{"category": "Math", "title": "Bifurcation and Secondary Bifurcation of Heavy Periodic Hydroelastic Travelling Waves", "abstract": "The paper deals with a problem of interaction between hydrodynamics and mechanics of nonlinear elastic bodies. The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic Cosserat rod, following Antman's elasticity theory, and the fluid beneath is supposed to be in steady 2D irrotational motion under gravity. Assuming that gravity and the density of the undeformed membrane are prescribed, the free parameters of the problem are the speed of the wave and drift velocity of the membrane. The analysis relies upon a conformal formulation of the hydro-elastic problem developed in previous papers; the basic tool for the study of the bifurcation picture is the implicit function theorem, under some non-resonance assumptions. The most interesting part of the final result is the existence of a symmetry-breaking 'third sheet' of solutions, which bifurcates from primary sheets, and is a hydro-elastic analogue of the phenomenon known as 'Wilton ripples' in the surface tension case."}
{"category": "Math", "title": "Brownian Brownian Motion-1", "abstract": "A classical model of Brownian motion consists of a heavy molecule submerged into a gas of light atoms in a closed container. In this work we study a 2D version of this model, where the molecule is a heavy disk of mass M and the gas is represented by just one point particle of mass m = 1, which interacts with the disk and the walls of the container via elastic collisions. Chaotic behavior of the particles is ensured by convex (scattering) walls of the container. We prove that the position and velocity of the disk, in an appropriate time scale, converge, as M tends to infinity, to a Brownian motion (possibly, inhomogeneous); the scaling regime and the structure of the limit process depend on the initial conditions. Our proofs are based on strong hyperbolicity of the underlying dynamics, fast decay of correlations in systems with elastic collisions (billiards), and methods of averaging theory."}
{"category": "Math", "title": "Nearly-optimal estimates for the stability problem in Hardy spaces", "abstract": "We continue the work of \\cite{TLNT}. Let $E$ be a non-Blaschke subset of the unit disc $\\mathbb{D}$ of the complex plane $\\mathbb{C}$. Fixed $1\\leq p\\leq \\infty$, let $H^p(\\mathbb{D})$ be the Hardy space of holomorphic functions in the disk whose boundary value function is in $L^p(\\partial \\mathbb{D})$. Fixed $0<R<1$. For $\\epsilon >0$ define C_p(\\varepsilon, R) = \\sup \\{\\sup_{|z| \\leq R}|g(z)|: g\\in H^p, \\|g\\|_p\\leq 1, |g(\\zeta)| \\leq \\varepsilon \\forall \\zeta\\in E\\}. In this paper we find upper and lower bounds for $C_p(\\epsilon, R)$ when $\\epsilon$ is small for any non-Blaschke set $E$. The bounds are nearly-optimal for many such sets $E$, including sets contained in a compact subset of $\\mathbb{D}$ and sets contained in a finite union of Stolz angles."}
{"category": "Math", "title": "Note on potential theory for functions in Hardy classes", "abstract": "The purpose of this note is to show that the set functions defined in \\cite{trong-tuyen} can be suitably extended to all subsets $E$ of the unit disk $\\mathbb{D}$. In particular we obtain uniform nearly-optimal estimates for the following quantity D_p(E,\\epsilon, R) = \\sup \\{\\sup_{|z| \\leq R}|g(z)|: g\\in H^p, ||g||_{H^p}\\leq 1, (1-|\\zeta |)|g(\\zeta)| \\leq \\epsilon \\forall \\zeta\\in E\\}."}
{"category": "Math", "title": "The Powers Sum of spatial CPD-semigroups and CP-semigroups", "abstract": "We define spatial CPD-semigroup and construct their Powers sum. We construct the Powers sum for general spatial CP-semigroups. In both cases, we show that the product system of that Powers sum is the product of the spatial product systems of its factors. We show that on the domain of intersection, pointwise bounded CPD-semigroups on the one side and Schur CP-semigroups on the other, the constructions coincide. This summarizes all known results about Powers sums and generalizes them considerably."}
{"category": "Math", "title": "$\\omega$-Lie algebras", "abstract": "We study a certain generalization of Lie algebras where the Jacobian of three elements does not vanish but is equal to an expression depending on a skew-symmetric bilinear form."}
{"category": "Math", "title": "Sprouts game on compact surfaces", "abstract": "Sprouts is a two-player topological game, invented in 1967 by Michael Paterson and John Conway. The game starts with p spots drawn on a sheet of paper, and lasts at most 3p-1 moves: the player who makes the last move wins. Sprouts is a very intricate game and the best known manual analysis only achieved to find a winning strategy up to p=7 spots. Recent computer analysis reached up to p=32. The standard game is played on a plane, or equivalently on a sphere. In this article, we generalize and study the game on any compact surface. First, we describe the possible moves on a compact surface, and the way to implement them in a program. Then, we show that we only need to consider a finite number of surfaces to analyze the game with p spots on any compact surface: if we take a surface with a genus greater than some limit genus, then the game on this surface is equivalent to the game on some smaller surface. Finally, with computer calculation, we observe that the winning player on orientable surfaces seems to be always the same one as on a plane, whereas there are significant differences on non-orientable surfaces."}
{"category": "Math", "title": "Extending the Set of Quadratic Exponential Vectors", "abstract": "We extend the square of white noise algebra over the step functions on R to the test function space of bounded square-integrable functions on R^d, and we show that in the Fock representation the exponential vectors exist for all test functions bounded by 1/2."}
{"category": "Math", "title": "A rank inequality for the Tate Conjecture over global function fields", "abstract": "Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective, geometrically-connected variety defined over a global function field, the algebraic rank is less than or equal to the analytic rank. Also discussed is the analogous (open) question for number fields and an easy extension of Lafforgue's theorem to remove the \"finite-order character\" assumption. All results are likely \"known to the experts\", but don't appear to be written down."}
{"category": "Math", "title": "C*-envelopes of tensor algebras for multivariable dynamics", "abstract": "We give a new very concrete description of the C*-envelope of the tensor algebra associated to multivariable dynamical system. In the surjective case, this C*-envelope is described as a crossed product by an endomorphism, and as a groupoid C*-algebra. In the non-surjective case, it is a full corner of a such an algebra. We also show that when the space is compact, then the C*-envelope is simple if and only if the system is minimal."}
{"category": "Math", "title": "Fixation Probability for Competing Selective Sweeps", "abstract": "We consider a biological population in which a beneficial mutation is undergoing a selective sweep when a second beneficial mutation arises at a linked locus and we investigate the probability that both mutations will eventually fix in the population. Previous work has dealt with the case where the second mutation to arise confers a smaller benefit than the first. In that case population size plays almost no role. Here we consider the opposite case and observe that, by contrast, the probability of both mutations fixing can be heavily dependent on population size. Indeed the key parameter is $\\rho N$, the product of the population size and the recombination rate between the two selected loci. If $\\rho N$ is small, the probability that both mutations fix can be reduced through interference to almost zero while for large $\\rho N$ the mutations barely influence one another. The main rigorous result is a method for calculating the fixation probability of a double mutant in the large population limit."}
{"category": "Math", "title": "On Families of (Phi,Gamma)-modules", "abstract": "Berger and Colmez introduced a theory of families of overconvergent \\'etale (Phi,Gamma)-modules associated to families of p-adic Galois representations over p-adic Banach algebras. However, in contrast with the classical theory of (Phi,Gamma)-modules, the functor they obtain is not an equivalence of categories. In this paper, we prove that when the base is an affinoid space, every family of (overconvergent) \\'etale (Phi,Gamma)-modules can locally be converted into a family of p-adic representations in a unique manner, providing the \"local\" equivalence. There is a global mod p obstruction related to the moduli of residual representations."}
{"category": "Math", "title": "Bounds for the annealed return probability on large finite percolation clusters", "abstract": "Bounds for the expected return probability of the delayed random walk on finite clusters of an invariant percolation on transitive unimodular graphs are derived. They are particularly suited for the case of critical Bernoulli percolation and the associated heavy-tailed cluster size distributions. The upper bound relies on the fact that cartesian products of finite graphs with cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond percolation on the homogeneous tree this bound is sharp. The asymptotic type of the expected return probability for large times t in this case is of order of the 3/4'th power of 1/t."}
{"category": "Math", "title": "On Rank Problems for Planar Webs and Projective Structures", "abstract": "We present old and recent results on rank problems and linearizability of geodesic planar webs."}
{"category": "Math", "title": "A simple proof of a theorem of Fukaya and Oh", "abstract": "We study the moduli space of pseudo pointed holomorphic disks with boundaries mapped in the zero section of the cotangent bundle of a manifold. We define perturbations of the equation for which it is possible to describe explicitly all the solutions of the problem in terms of Morse graphs on the manifold. In particular, this proves that the $A_\\infty$ structure of the zero section of the cotangent bundle is equivalent to the Morse $A_\\infty$ structure of the base manifold."}
{"category": "Math", "title": "Homotopy Equivalences induced by Balanced Pairs", "abstract": "We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which the balanced pair of subcategories gives rise to equivalent homotopy categories of complexes. As an application, we prove that for a left-Gorenstein ring, there exists a triangle-equivalence between the homotopy category of its Gorenstein projective modules and the homotopy category of its Gorenstein injective modules, which restricts to a triangle-equivalence between the homotopy category of projective modules and the homotopy category of injective modules. In the case of commutative Gorenstein rings we prove that up to a natural isomorphism our equivalence extends Iyengar-Krause's equivalence."}
{"category": "Math", "title": "Permutations sortable by n-4 passes through a stack", "abstract": "We characterise and enumerate permutations that are sortable by n-4 passes through a stack. We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a polynomial expression."}
{"category": "Math", "title": "Complete convergence of message passing algorithms for some satisfiability problems", "abstract": "In this paper we analyze the performance of Warning Propagation, a popular message passing algorithm. We show that for 3CNF formulas drawn from a certain distribution over random satisfiable 3CNF formulas, commonly referred to as the planted-assignment distribution, running Warning Propagation in the standard way (run message passing until convergence, simplify the formula according to the resulting assignment, and satisfy the remaining subformula, if necessary, using a simple \"off the shelf\" heuristic) results in a satisfying assignment when the clause-variable ratio is a sufficiently large constant."}
{"category": "Math", "title": "Enrichments over symmetric Picard categories", "abstract": "Categorical rings were introduced by Jibladze and Pirashvili in their paper \"Third Mac Lane cohomology via categorical rings\", Journal of Homotopy and related structures, 2, 2007, 187-216. We call those \"2-rings\". In these notes we present basic definitions and results regarding 2-modules. This is work in progress."}
{"category": "Math", "title": "A volume form on the Khovanov invariant", "abstract": "The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over $\\mathbb{Z}$ is recovered."}
{"category": "Math", "title": "Homotopy theory of C*-algebras", "abstract": "In this work we construct from ground up a homotopy theory of C*-algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C*-algebras. The spaces in C*-homotopy theory are certain hybrids of functors represented by C*-algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C*-algebra circle of complex-valued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory, we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C*-homotopy theory. The stable homotopy category of C*-algebras gives rise to invariants such as stable homotopy groups and bigraded cohomology and homology theories. We work out examples related to the emerging subject of noncommutative motives and zeta functions of C*-algebras. In addition, we employ homotopy theory to define a new type of K-theory of C*-algebras."}
{"category": "Math", "title": "Pattern Recognition on Oriented Matroids: Three-Tope Committees", "abstract": "A three-tope committee K* for a simple oriented matroid M is a 3-subset of its maximal covectors such that every positive halfspace of M contains at least two topes from K*. We consider three-tope committees as the vertex sets of triangles in graphs associated with the topes and enumerate them making use of the properties of the poset of convex subsets of the ground set of M."}
{"category": "Math", "title": "Optimal sequential procedures with Bayes decision rules", "abstract": "In this article, a general problem of sequential statistical inference for general discrete-time stochastic processes is considered. The problem is to minimize an average sample number given that Bayesian risk due to incorrect decision does not exceed some given bound. We characterize the form of optimal sequential stopping rules in this problem. In particular, we have a characterization of the form of optimal sequential decision procedures when the Bayesian risk includes both the loss due to incorrect decision and the cost of observations."}
{"category": "Math", "title": "Homogeneous Lagrangian submanifolds of positive Euler characteristic", "abstract": "We fully classify all Lagrangian submanifolds of a complex Grassmannian which are an orbit of a compact group of isometries and have positive Euler characteristic."}
{"category": "Math", "title": "Factor complexity of infinite words associated with non-simple Parry numbers", "abstract": "The factor complexity of the infinite word $\\ubeta$ canonically associated to a non-simple Parry number $\\beta$ is studied. Our approach is based on the notion of special factors introduced by Berstel and Cassaigne. At first, we give a handy method for determining infinite left special branches; this method is applicable to a broad class of infinite words which are fixed points of a primitive substitution. In the second part of the article, we focus on infinite words $\\ubeta$ only. To complete the description of its special factors, we define and study $(a,b)$-maximal left special factors. This enables us to characterize non-simple Parry numbers $\\beta$ for which the word $\\ubeta$ has affine complexity."}
{"category": "Math", "title": "The Inverse of a Nearly Banded Matrix", "abstract": "A quantitative form of the Nullity Theorem is presented, which establishes a linear relation between the singular values of the two submatrices involved in the theorem up to the first order. The theorem is then extended to function spaces and a corresponding form in infinite dimension is discussed."}
{"category": "Math", "title": "Traces on Hecke algebras and families of p-adic modular forms", "abstract": "In this preprint we prove that any finite slope modular form fits into a p-adic family of modular forms which is indexed by the weight. Here, the term p-adic family means that p-adic congruences between weights entail certain p-adic congruences between the corresponding modular forms. We also show that the dimension of the slope subspace of the space of modular forms does not depend on the weight. Both statements are predicted by the Mazur-Gouvea Conjecture, which has been proven by Coleman using methods from rigid analytic geometry. In contrast our proof is based on a comparison of (topological) trace formulas."}
{"category": "Math", "title": "Minimal Faithful Permutation Degrees for Irreducible Coxeter Groups and Binary Polyhedral Groups", "abstract": "In this article we calculate the minimal faithful permutation degree for all of the irreducible Coxeter groups. We also exhibit new examples of finite groups that possess a quotient whose minimal degree is strictly greater than that of the group."}
{"category": "Math", "title": "Generalized Bunce-Deddens algebras", "abstract": "We define a broad class of crossed product C*-algebras of the form C(G)xG, where G is a discrete countable amenable residually finite group, and G is a profinite completion of G. We show that they are unital separable simple nuclear quasidiagonal C*-algebras, or real rank zero, stable rank one, with comparability of projections and with a unique trace."}
{"category": "Math", "title": "Reduced free products of unital AH algebras and Blackadar and Kirchberg's MF algebras", "abstract": "In the paper, we prove that reduced free products of unital AH algebras with respect to given faithful tracial states, in the sense of Voiculescu, are Blackadar and Kirhcberg's MF algebras. We also show that the reduced free products of unital AH algebras with respect to given faithful tracial states, under mild conditions, are not quasidiagonal. Therefore we conclude, for a large class of AH algebras, the Brown-Douglas-Fillmore extension semigroups of the reduced free products of these AH algebras with respect to given faithful tracial states are not groups. Our result is based on Haagerup and Thorbj{\\o}rsen's work on the reduced C$^*$-algebras of free groups."}
{"category": "Math", "title": "Toric ideals generated by circuits", "abstract": "Let I be the toric ideal of a homogeneous normal configuration. We prove that I is generated by circuits if and only if each unbalanced circuit of I has a \"connector\" which is a linear combination of circuits with a square-free term. In particular if each circuit of I with non-square-free terms is balanced, then I is generated by circuits. As a consequence we prove that the toric ideal of a normal edge subring of a multigraph is generated by circuits with a square-free term."}
{"category": "Math", "title": "Some characterizations of affinely full-dimensional factorial designs", "abstract": "A new class of two-level non-regular fractional factorial designs is defined. We call this class an {\\it affinely full-dimensional factorial design}, meaning that design points in the design of this class are not contained in any affine hyperplane in the vector space over $\\mathbb{F}_2$. The property of the indicator function for this class is also clarified. A fractional factorial design in this class has a desirable property that parameters of the main effect model are simultaneously identifiable. We investigate the property of this class from the viewpoint of $D$-optimality. In particular, for the saturated designs, the $D$-optimal design is chosen from this class for the run sizes $r \\equiv 5,6,7$ (mod 8)."}
{"category": "Math", "title": "A general type of twisted anomaly cancellation formulas", "abstract": "For even dimensional manifolds, we prove some twisted anomaly cancellation formulas which generalize some well-known cancellation formulas. For odd dimensional manifolds, we obtain some modularly invariant characteristic forms by the Chern-Simons transgression and we also get some twisted anomaly cancellation formulas."}
{"category": "Math", "title": "Hodge decomposition in the homology of long knots", "abstract": "The paper describes a natural splitting in the rational homology and homotopy of the spaces of long knots. This decomposition presumably arises from the cabling maps in the same way as a natural decomposition in the homology of loop spaces arises from power maps. The generating function for the Euler characteristics of the terms of this splitting is presented. Based on this generating function we show that both the homology and homotopy ranks of the spaces in question grow at least exponentially. Using natural graph-complexes we show that this splitting on the level of the bialgebra of chord diagrams is exactly the splitting defined earlier by Dr. Bar-Natan. In the Appendix we present tables of computer calculations of the Euler characteristics. These computations give a certain optimism that the Vassiliev invariants of order > 20 can distinguish knots from their inverses."}
{"category": "Math", "title": "An Analytic Approach to Stability", "abstract": "The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order $n$ can be made isomorphic by changing o(n^2) edges. Here we show how the recently developed theory of graph limits can be used to give an analytic approach to stability. As an application, we present a new proof of the Erdos-Simonovits Stability Theorem. Also, we investigate various properties of the edit distance. In particular, we show that the combinatorial and fractional versions are within a constant factor from each other, thus answering a question of Goldreich, Krivelevich, Newman, and Rozenberg."}
{"category": "Math", "title": "Face vectors of two-dimensional Buchsbaum complexes", "abstract": "In this paper, we characterize all possible h-vectors of 2-dimensional Buchsbaum simplicial complexes."}
{"category": "Math", "title": "Singular Hermitian-Einstein monopoles on the product of a circle and a Riemann surface", "abstract": "In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bundle. The singularities of this automorphism correspond to the singularities of the singular monopole. We then consider the complex geometry of the moduli space; in particular, we compute dimensions, both from the complex geometric and the gauge theoretic point of view."}
{"category": "Math", "title": "Parity condition for irreducibility of Heegaard splittings", "abstract": "Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist."}
{"category": "Math", "title": "Monotonicity theorems for Laplace Beltrami operator on Riemannian manifolds", "abstract": "For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace-Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds."}
{"category": "Math", "title": "On the Adams-Riemann-Roch theorem in positive characteristic", "abstract": "We give a new proof of the Adams-Riemann-Roch theorem for a smooth projective morphism $X\\to Y$, in the situation where $Y$ is a regular scheme, which is quasi-projective over $\\mF_p$. We also partially answer a question of B. K\\\"ock."}
{"category": "Math", "title": "Representation dimension of extensions of hereditary algebras", "abstract": "We show that if H is a hereditary finite dimensional algebra, M is a finitely generated H-module and B is a semisimple subalgebra of the endomorphism algebra of M, then the representation dimension of the corresponding triangular matrix algebra is less or equal to 3 whenever one of the following conditions hold: i) H is of finite representation type; ii) H is tame and M is a direct sum of regular and preprojective modules; iii) M has no self-extensions"}
{"category": "Math", "title": "Embeddings of k-connected n-manifolds into R^{2n-k-1}", "abstract": "We obtain estimations for isotopy classes of embeddings of closed k-connected n-manifolds into R^{2n-k-1} for n>2k+5 and k\\ge0. This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action of H_{k+1}(N;Z_2) on the set of embeddings. (For k\\ne1 classification results were obtained by algebraic methods without direct construction of embeddings or homology invariants.) The proof involves reduction to classification of embeddings of punctured manifold and uses parametric connected sum of embeddings."}
{"category": "Math", "title": "Une conjecture sur la torsion des classes de Chern des fibr\\'es de Gauss-Manin", "abstract": "Pour tout $t\\in\\mN$ nous d\\'efinissons un certain entier positif $\\N_t$ et nous conjecturons: si $H$ est un fibr\\'e de Gauss-Manin d'une fibration semi-stable alors la $t$-\\`eme classe de Chern de $H$ est annul\\'ee par $\\N_t$. Nous d\\'emontrons diverses cons\\'equences de cette conjecture. For any $t\\in\\mN$, we define a certain positive integer $\\N_t$ and we conjecture: if $H$ is a Gauss-Manin bundle of a semi-stable fibration then the $t$-th Chern class of $H$ is kiled by $\\N_t$. We prove various consequences of this conjecture."}
{"category": "Math", "title": "Open strings, Lagrangian conductors and Floer functor", "abstract": "We introduce a contravariant functor, called Floer functor, from the category of Lagrangian conductors of a symplectic manifold to the homotopy category of bounded chain complexes of open strings in this manifold. The latter two categories are defined for all symplectic manifolds, whereas Floer functor is defined for semipositive manifolds which are either closed or convex at infinity. We then prove that when the first Chern class of the symplectic manifold vanishes, Lagrangian spheres define Lagrangian conductors so that in particular their integral Floer cohomology is well defined. This requires the introduction of singular almost-complex structures given by symplectic field theory."}
{"category": "Math", "title": "Stable ergodicity of dominated systems", "abstract": "We provide a new approach to stable ergodicity of systems with dominated splittings, based on a geometrical analysis of global stable and unstable manifolds of hyperbolic points. Our method suggests that the lack of uniform size of Pesin's local stable and unstable manifolds - a notorious problem in the theory of non-uniform hyperbolicity - is often less severe than it appeas to be."}
{"category": "Math", "title": "The projective McKay correspondence", "abstract": "Kirillov has described a McKay correspondence for finite subgroups of PSL_{2}(C) that associates to each `height' function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on the projective line P^1 and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. The main goal of this paper is to develop an analogous story for the cotangent bundle of P^1. We show that each height function gives rise to a derived equivalence between equivariant sheaves on the cotangent bundle T*P^1 and modules over the preprojective algebra of an affine Dynkin quiver. These different equivalences are related by spherical twists, which take the place of the reflection functors for P^1."}
{"category": "Math", "title": "Representing geometric morphisms using power locale monads", "abstract": "It it shown that geometric morphisms between elementary toposes can be represented as adjunctions between the corresponding categories of locales. These adjunctions are characterised as those that preserve the order enrichment, commute with the double power locale monad and whose right adjoints preserve finite coproduct. They are also characterised as those adjunctions that preserve the order enrichment and commute with both the upper and the lower power locale monads."}
{"category": "Math", "title": "Types are weak omega-groupoids", "abstract": "We define a notion of weak omega-category internal to a model of Martin-L\\\"of type theory, and prove that each type bears a canonical weak omega-category structure obtained from the tower of iterated identity types over that type. We show that the omega-categories arising in this way are in fact omega-groupoids."}
{"category": "Math", "title": "Vortex invariants and toric manifolds", "abstract": "We consider the symplectic vortex equations for a linear Hamiltonian torus action. We show that the associated genus zero moduli space itself is homotopic (in the sense of a homotopy of regular G-moduli problems) to a toric manifold with combinatorial data directly obtained from the original torus action. This allows to view the wall crossing formula of Cieliebak and Salamon for the computation of vortex invariants as a consequence of a generalized Jeffrey-Kirwan localization formula for integrals over symplectic quotients."}
{"category": "Math", "title": "Holomorphic factorization of mappings into SL_n(C)", "abstract": "We solve Gromov's Vaserstein problem. Namely, we show that a null-homotopic holomorphic mapping from a finite dimensional reduced Stein space into SL_n(C) can be factored into a finite product of unipotent matrices with holomorphic entries."}
{"category": "Math", "title": "Splitting Monoidal Stable Model Categories", "abstract": "If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit S forms a commutative ring. An idempotent e of this ring will split the homotopy category. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to the product of C localised at the object eS and C localised at the object (1-e)S. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is."}
{"category": "Math", "title": "Higher Bers maps", "abstract": "The Bers embebbing realizes the Teichm\\\"uller space of a Fuchsian group $G$ as a open, bounded and contractible subset of the complex Banach space of bounded quadratic differentials for $G$. It utilizes the schlicht model of Teichm\\\"uller space, where each point is represented by an injective holomorphic function on the disc, and the map is constructed via the Schwarzian differential operator. In this paper we prove that a certain class of differential operators acting on functions of the disc induce holomorphic mappings of Teichm\\\"uller spaces, and we also obtain a general formula for the differential of the induced mappings at the origin. The main focus of this work, however, is on two particular series of such mappings, dubbed higher Bers maps, because they are induced by so-called higher Schwarzians -- generalizations of the classical Schwarzian operator. For these maps, we prove several further results. The last section contains a discussion of possible applications, open questions and speculations."}
{"category": "Math", "title": "Classifying Rational G-Spectra for Finite G", "abstract": "We give a new proof that for a finite group G, the category of rational G-equivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model."}
{"category": "Math", "title": "Polynomial parametrization of the solutions of Diophantine equations of genus 0", "abstract": "Let f in Z[X,Y,Z] be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients, such that the projective curve given by f=0 has a function field isomorphic to the rational function field Q(t). We show that all integral solutions of the Diophantine equation f=0 (up to those corresponding to some singular points) can be parametrized by a single triple of integer-valued polynomials. In general, it is not possible to parametrize this set of solutions by a single triple of polynomials with integer coefficients."}
{"category": "Math", "title": "Einstein manifolds with nonnegative isotropic curvature are locally symmetric", "abstract": "Let (M,g) be an Einstein manifold of dimension n \\geq 4 with nonnegative isotropic curvature. We show that (M,g) is locally symmetric."}
{"category": "Math", "title": "Uniform Time Average Consistency of Monte Carlo Particle Filters", "abstract": "We prove that bootstrap type Monte Carlo particle filters approximate the optimal nonlinear filter in a time average sense uniformly with respect to the time horizon when the signal is ergodic and the particle system satisfies a tightness property. The latter is satisfied without further assumptions when the signal state space is compact, as well as in the noncompact setting when the signal is geometrically ergodic and the observations satisfy additional regularity assumptions."}
{"category": "Math", "title": "Gross--Schoen Cycles and Dualising Sheaves", "abstract": "The aim of this paper is to study the modified diagonal cycle in the triple product of a curve over a global field defined by Gross and Schoen. Our main result is an identity between the height of this cycle and the self-intersection of the relative dualising sheaf. We have some applications to the following problems in number theory and algebraic geometry."}
{"category": "Math", "title": "Splitting and composition methods in the numerical integration of differential equations", "abstract": "We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications."}
{"category": "Math", "title": "Qualitative Properties of Local Random Invariant Manifolds for SPDEs with Quadratic Nonlinearity", "abstract": "The qualitative properties of local random invariant manifolds for stochastic partial differential equations with quadratic nonlinearities and multiplicative noise is studied by a cut off technique. By a detail estimates on the Perron fixed point equation describing the local random invariant manifold, the structure near a bifurcation is given."}
{"category": "Math", "title": "Weak omega-categories from intensional type theory", "abstract": "We show that for any type in Martin-L\\\"of Intensional Type Theory, the terms of that type and its higher identity types form a weak omega-category in the sense of Leinster. Precisely, we construct a contractible globular operad of definable composition laws, and give an action of this operad on the terms of any type and its identity types."}
{"category": "Math", "title": "Which infinite abelian groups admit an almost maximally almost-periodic group topology?", "abstract": "A topological group G is said to be almost maximally almost-periodic if its von Neumann radical is non-trivial, but finite. In this paper, we prove that every abelian group with an infinite torsion subgroup admits a (Hausdorff) almost maximally almost-periodic group topology. Some open problems are also formulated."}
{"category": "Math", "title": "Buchsteiner loops: associators and constructions", "abstract": "Let $Q$ be a Buchsteiner loop. We describe the associator calculus in three variables, and show that $|Q| \\ge 32$ if $Q$ is not conjugacy closed. We also show that $|Q| \\ge 64$ if there exists $x \\in Q$ such that $x^2$ is not in the nucleus of $Q$. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that is nilpotency class 3 and possesses an abelian inner mapping group."}
{"category": "Math", "title": "Color Visualization of Blaschke Product Mappings", "abstract": "A visualization of Blaschke product mappings can be obtained by treating them as canonical projections of covering Riemann surfaces and finding fundamental domains and covering transformations corresponding to these surfaces. A working tool is the technique of simultaneous continuation we introduced in previous papers. Here, we are refining this technique for some particular types of Blaschke products for which coloring pre-images of annuli centered at the origin allow us to describe the mappings with a high degree of fidelity."}
{"category": "Math", "title": "On the $\\pd$- and $\\barpd$-Operators of a Generalized Complex Structure", "abstract": "In this note, we prove that the $\\pd$- and $\\barpd$-operators introduced by Gualtieri for a generalized complex structure coincide with the $\\bdees$- and $\\bdel$-operators introduced by Alekseev-Xu for Evens-Lu-Weinstein modules of a Lie bialgebroid."}
{"category": "Math", "title": "A proximal method for composite minimization", "abstract": "We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term. Properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem. Preliminary computational results on both convex and nonconvex examples are promising."}
{"category": "Math", "title": "Quandle and hyperbolic volume", "abstract": "We show that the hyperbolic volume of a hyperbolic knot is a quandle cocycle invariant. Further we show that it completely determines invertibility and positive/negative amphicheirality of hyperbolic knots."}
{"category": "Math", "title": "On David type Siegel Disks of the Sine family", "abstract": "In 2008 Petersen posed a list of questions on the application of trans-quasiconformal Siegel surgery developed by Zakeri and himself. In this paper we extend Petersen-Zakeri's idea so that the surgery can be applied to all the premodels which have no \"free critical points\". We explain how the idea is used in solving three of the questions posed by Petersen. To present the details of the idea, we focus on the solution of one of them: we prove that for typical rotation numbers $0< \\theta < 1$, the boundary of the Siegel disk of $f_{\\theta}(z) = e^{2 \\pi i \\theta} \\sin (z)$ is a Jordan curve which passes through exactly two critical points $\\pi/2$ and $-\\pi/2$."}
{"category": "Math", "title": "Mixed volume and an extension of intersection theory of divisors", "abstract": "Let K(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the number of solutions in X of a system of equations f_1 = ... = f_n = 0 where each f_i is a generic function from the space L_i. In counting the solutions, we neglect the solutions x at which all the functions in some space L_i vanish as well as the solutions at which at least one function from some subspace L_i has a pole. The collection K(X) is a commutative semigroup with respect to a natural multiplication. The intersection index [L_1,..., L_n] can be extended to the Grothendieck group of K(X). This gives an extension of the intersection theory of divisors. The extended theory is applicable even to non-complete varieties. We show that this intersection index enjoys all the main properties of the mixed volume of convex bodies. Our paper is inspired by the Bernstein-Kushnirenko theorem from the Newton polytope theory."}
{"category": "Math", "title": "An algorithmic Littlewood-Richardson rule", "abstract": "We introduce a Littlewood-Richardson rule based on an algorithmic deformation of skew Young diagrams and present a bijection with the classical rule. The result is a direct combinatorial interpretation and proof of the geometric rule presented by Coskun. We also present a corollary regarding the Specht modules of the intermediate diagrams."}
{"category": "Math", "title": "Indecomposable Permutations, Hypermaps and Labeled Dyck Paths", "abstract": "Hypermaps were introduced as an algebraic tool for the representation of embeddings of graphs on an orientable surface. Recently a bijection was given between hypermaps and indecomposable permutations; this sheds new light on the subject by connecting a hypermap to a simpler object. In this paper, a bijection between indecomposable permutations and labelled Dyck paths is proposed, from which a few enumerative results concerning hypermaps and maps follow. We obtain for instance an inductive formula for the number of hypermaps with n darts, p vertices and q hyper-edges; the latter is also the number of indecomposable permutations of with p cycles and q left-to-right maxima. The distribution of these parameters among all permutations is also considered."}
{"category": "Math", "title": "Universally catenarian integral domains, strong S-domains and semistar operations", "abstract": "Let $D$ be an integral domain and $\\star$ a semistar operation stable and of finite type on it. In this paper, we are concerned with the study of the semistar (Krull) dimension theory of polynomial rings over $D$. We introduce and investigate the notions of $\\star$-universally catenarian and $\\star$-stably strong S-domains and prove that, every $\\star$-locally finite dimensional Pr\\\"{u}fer $\\star$-multiplication domain is $\\star$-universally catenarian, and this implies $\\star$-stably strong S-domain. We also give new characterizations of $\\star$-quasi-Pr\\\"{u}fer domains introduced recently by Chang and Fontana, in terms of these notions."}
{"category": "Math", "title": "A bijectional attack on the Razumov-Stroganov conjecture", "abstract": "We attempt to prove the Razumov-Stroganov conjecture using a bijectional approach. We have been unsuccessful but we believe the techniques we present can be used to prove the conjecture."}
{"category": "Math", "title": "The existence of thick triangulations -- an \"elementary\" proof", "abstract": "We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \\cite{pe}, since it mainly uses tools of elementary differential topology. The role played by curvatures in this construction is also emphasized."}
{"category": "Math", "title": "Miyaoka-Yau inequality for minimal projective manifolds of general type", "abstract": "In this short note, we prove the Miyaoka-Yau inequality for minimal projective $n$-manifolds of general type by using K\\\"ahler-Ricci flow."}
{"category": "Math", "title": "Restricted involutions and Motzkin paths", "abstract": "We show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions."}
{"category": "Math", "title": "On approximations by shifts of the Gaussian function", "abstract": "The paper study the discrete sets of translations of the Gaussian function that span the spaces L1(R) and L2(R)."}
{"category": "Math", "title": "Operations in Milnor K-theory", "abstract": "We show that operations in Milnor K-theory mod $p$ of a field are spanned by divided power operations. After giving an explicit formula for divided power operations and extending them to some new cases, we determine for all fields $k$ and all prime numbers $p$, all the operations $K^M_i/p \\to K^M_j/p$ commuting with field extensions over the base field $k$. Moreover, the integral case is discussed and we determine the operations $K^M_i/p \\to K^M_j/p$ for smooth schemes over a field."}
{"category": "Math", "title": "Matricially free random variables", "abstract": "We show that the operatorial framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication. The associated notion of independence, called matricial freeness, can be viewed as a generalization of both freeness and monotone independence. At the same time, the sums of matricially free random variables, called random pseudomatrices, are closely related to Gaussian random matrices. The main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial generalizations of semicirle laws."}
{"category": "Math", "title": "Building suitable sets for locally compact groups by means of continuous selections", "abstract": "If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S \\cup {1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory."}
{"category": "Math", "title": "Finite flat models of constant group schemes of rank two", "abstract": "We calculate the number of the isomorphism class of the finite flat models over the ring of integers of an absolutely ramified $p$-adic field of constant group schemes of rank two over finite fields, by counting the rational points of a moduli space of finite flat models."}
{"category": "Math", "title": "On a conjecture on exponential Diophantine equations", "abstract": "We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\\equiv 2 \\pmod 4$, $b\\equiv 3 \\pmod 4$ and $\\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution."}
{"category": "Math", "title": "On near optimal trajectories for a game associated with the \\infty-Laplacian", "abstract": "A two-player stochastic differential game representation has recently been obtained for solutions of the equation -\\Delta_\\infty u=h in a \\calC^2 domain with Dirichlet boundary condition, where h is continuous and takes values in \\RR\\setminus\\{0\\}. Under appropriate assumptions, including smoothness of u, the vanishing \\delta limit law of the state process, when both players play \\delta-optimally, is identified as a diffusion process with coefficients given explicitly in terms of derivatives of the function u."}
{"category": "Math", "title": "Zero-nonzero patterns for nilpotent matrices over finite fields", "abstract": "Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper we initiate an investigation into which zero-nonzero patterns are potentially nilpotent over F, with a special emphasis on the case that F = Z_p is a finite field. As part of this investigation, we develop methods, using the tools of algebraic geometry and commutative algebra, to eliminate zero-nonzero patterns A as being potentially nilpotent over any field F. We then use these techniques to classify all irreducible zero-nonzero patterns of order two and three that are potentially nilpotent over Z_p for each prime p."}
{"category": "Math", "title": "On sums of three squares", "abstract": "We prove that a positive integer not of the form, 4^{k}(8m+7) can be expressible as a sum of three or fewer squares by using some results of Kane and Sun on mixed sums of squares and triangular numbers."}
{"category": "Math", "title": "On the large genus asymptotics of Weil-Petersson volumes", "abstract": "A relatively fast algorithm for evaluating Weil-Petersson volumes of moduli spaces of complex algebraic curves is proposed. On the basis of numerical data, a conjectural large genus asymptotics of the Weil-Petersson volumes is computed. Asymptotic formulas for the intersection numbers involving $\\psi$-classes are conjectured as well. The accuracy of the formulas is high enough to believe that they are exact."}
{"category": "Math", "title": "Metric and arithmetic properties of mediant-Rosen maps", "abstract": "A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry."}
{"category": "Math", "title": "Flow invariants in the classification of Leavitt path algebras", "abstract": "We analyze in the context of Leavitt path algebras some graph operations introduced in the context of symbolic dynamics by Williams, Parry and Sullivan, and Franks. We show that these operations induce Morita equivalence of the corresponding Leavitt path algebras. As a consequence we obtain our two main results: the first gives sufficient conditions for which the Leavitt path algebras in a certain class are Morita equivalent, while the second gives sufficient conditions which yield isomorphisms. We discuss a possible approach to establishing whether or not these conditions are also in fact necessary. In the final section we present many additional operations on graphs which preserve Morita equivalence (resp., isomorphism) of the corresponding Leavitt path algebras."}
{"category": "Math", "title": "Index Theory for Boundary Value Problems via Continuous Fields of C*-algebras", "abstract": "We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semigroupoid $\\cT^-X$ generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field C*_r(\\cT^-X) of C*-algebras over [0,1]. Its fiber in h=0, C*_r(T^-X), can be identified with the symbol algebra for Boutet de Monvel's calculus; for h\\not=0 the fibers are isomorphic to the algebra K of compact operators. We therefore obtain a natural map K_0(C*_r(T^-X))=K_0(C_0(T*X)) -> K_0(K)=Z. Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map."}
{"category": "Math", "title": "Statistical properties of intermittent maps with unbounded derivative", "abstract": "We study the ergodic and statistical properties of a class of maps of the circle and of the interval of Lorenz type which present indifferent fixed points and points with unbounded derivative. These maps have been previously investigated in the physics literature. We prove in particular that correlations decay polynomially, and that suitable Limit Theorems (convergence to Stable Laws or Central Limit Theorem) hold for H\\\"older continuous observables. We moreover show that the return and hitting times are in the limit exponentially distributed."}
{"category": "Math", "title": "Nets in groups, minimum length $g$-adic representations, and minimal additive complements", "abstract": "The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g > 1, the study of h-nets in the additive group of integers with respect to the generating set A_g = {g^i:i=0,1,2,...} requires a knowledge of the word lengths of integers with respect to A_g. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality problems."}
{"category": "Math", "title": "A case study of an Hamilton-Jacobi equation by the Adomian decompositional method", "abstract": "We present a study of the Adomian's Decomposition Method (ADM) applied to the Hamilton-Jacobi equations ut + H (ux) = 0. We recall the well known characteristics methods in the case of this type of equations to justify the existence or not of solutions. This yields that the ADM gives efficient solutions in time only in ]0; T *[, where T * is the critical time of our equation."}
{"category": "Math", "title": "A Minimal Poset Resolution of Stable Ideals", "abstract": "We use the theory of poset resolutions to construct the minimal free resolution of an arbitrary stable monomial ideal in the polynomial ring whose coefficients are from a field. This resolution is recovered by utilizing a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The structure of the poset under consideration is quite rich and in related analysis, we exhibit a regular CW complex which supports a minimal cellular resolution of a stable monomial ideal."}
{"category": "Math", "title": "Degenerate Sklyanin algebras and Generalized Twisted Homogeneous Coordinate rings", "abstract": "In this work, we introduce the point parameter ring B, a generalized twisted homogeneous coordinate ring associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising geometry of these algebras yields an analogue to a result of Artin-Tate-van den Bergh, namely that B is generated in degree one and thus is a factor of the corresponding degenerate Sklyanin algebra."}
{"category": "Math", "title": "Simultaneous Continuation of Infinitely Many Sinks Near a Quadratic Homoclinic Tangency", "abstract": "We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinksnear the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation."}
{"category": "Math", "title": "Unramified cohomology of finite groups of Lie type", "abstract": "We prove vanishing results for unramified stable cohomology of finite groups of Lie type."}
{"category": "Math", "title": "Fast approximation of solutions of SDE's with oblique reflection on an orthant", "abstract": "We consider the discrete \"fast\" penalization scheme for SDE's driven by general semimartingale on orthant $\\mathbb{R}_{+}^{d}$ with oblique reflection."}
{"category": "Math", "title": "The mean curvature of cylindrically bounded submanifolds", "abstract": "We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\\times\\R^{\\ell}$ in a product Riemannian manifold $N^{n-\\ell}\\times\\R^{\\ell}$. It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabion complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature."}
{"category": "Math", "title": "A Frobenius theorem for Cartan geometries, with applications", "abstract": "We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing the configuration of orbits for local Killing fields in a compact real-analytic Cartan geometry, and an open-dense theorem in the smooth case, which says that if there is a dense orbit, then there is an open, dense, locally homogeneus subset. Combining the Frobenius theorem with the embedding theorem of Bader, Frances, and the author gives a representation theorem that relates the fundamental group of the manifold with the automorphism group."}
{"category": "Math", "title": "Giambelli, Pieri, and tableau formulas via raising operators", "abstract": "We give a direct proof of the equivalence between the Giambelli and Pieri type formulas for Hall-Littlewood functions using Young's raising operators, parallel to joint work with Buch and Kresch for the Schubert classes on isotropic Grassmannians. We prove several closely related mirror identities enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert classes. The raising operator approach is applied to obtain tableau formulas for Hall-Littlewood functions, theta polynomials, and related Stanley symmetric functions. Finally, we introduce the notion of a skew element w of the hyperoctahedral group and identify the set of reduced words for w with the set of standard k-tableaux on a skew Young diagram."}
{"category": "Math", "title": "Combinatorial formulas for Le-coordinates in a totally nonnegative Grassmannian", "abstract": "Postnikov constructed a decomposition of a totally nonnegative Grassmannian into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point belongs to and to determine affine coordinates of the point within this cell. This simplifies Postnikov's description of the inverse boundary measurement map and generalizes formulas for the top cell given by Speyer and Williams. In addition, we identify a particular subset of Pluecker coordinates as a totally positive base for the set of non-vanishing Pluecker coordinates for a given positroid cell."}
{"category": "Math", "title": "An almost sure limit theorem for super-Brownian motion", "abstract": "We establish an almost sure scaling limit theorem for super-Brownian motion on $\\mathbb{R}^d$ associated with the semi-linear equation $u_t = {1/2}\\Delta u +\\beta u-\\alpha u^2$, where $\\alpha$ and $\\beta$ are positive constants. In this case, the spectral theoretical assumptions that required in Chen et al (2008) are not satisfied. An example is given to show that the main results also hold for some sub-domains in $\\mathbb{R}^d$."}
{"category": "Math", "title": "Bass Numbers and Semidualizing Complexes", "abstract": "Let R be a commutative local noetherian ring. We prove that the existence of a chain of semidualizing R-complexes of length (d+1) yields a degree-d polynomial lower bound for the Bass numbers of R. We also show how information about certain Bass numbers of R provide restrictions on the lengths of chains of semidualizing R-complexes. To make this article somewhat self-contained, we also include a survey of some of the basic properties of semidualizing modules, semidualizing complexes and derived categories."}
{"category": "Math", "title": "Cluster-tilted algebras of type $D_n$", "abstract": "Let $H$ be a hereditary algebra of Dynkin type $D_n$ over a field $k$ and $\\mathscr{C}_H$ be the cluster category of $H$. Assume that $n\\geq 5$ and that $T$ and $T'$ are tilting objects in $\\mathscr{C}_H$. We prove that the cluster-tilted algebra $\\Gamma=\\mathrm{End}_{\\mathscr{C}_H}(T)^{\\rm op}$ is isomorphic to $\\Gamma'=\\mathrm{End}_{\\mathscr{C}_H}(T')^{\\rm op}$ if and only if $T=\\tau^iT'$ or $T=\\sigma\\tau^jT'$ for some integers $i$ and $j$, where $\\tau$ is the Auslander-Reiten translation and $\\sigma$ is the automorphism of $\\mathscr{C}_H$ defined in section 4."}
{"category": "Math", "title": "Monodromy at infinity of $A$-hypergeometric functions and toric compactifications", "abstract": "We study $A$-hypergeometric functions introduced by Gelfand-Kapranov-Zelevinsky and prove a formula for the eigenvalues of their monodromy automorphisms defined by the analytic continuaions along large loops contained in complex lines parallel to the coordinate axes. A method of toric compactifications will be used to prove our main theorem."}
{"category": "Math", "title": "Global dimensions of endomorphism algebras for generator-cogenerators over $m$-replicated algebras", "abstract": "Let $A$ be a hereditary artin algebra and $A^{(m)}$ be the $m$-replicated algebra of $A$. We investigate the possibilities for the global dimensions of the endomorphism algebras of generator-cogenerators over $A^{(m)}$."}
{"category": "Math", "title": "Black Boxes", "abstract": "We shall deal comprehensively with Black Boxes, the intention being that provably in ZFC we have a sequence of guesses of extra structure on small subsets, where the guesses are pairwise almost disjoint; by this we mean they have quite little interaction, and are far apart but together are dense. We first deal with the simplest case, where the existence comes from winning a game by just writing down the opponent's moves. We show how it helps when instead of orders we have trees with boundedly many levels, having freedom in the last. After this we quite systematically look at existence of black boxes, and make connection to non-saturation of natural ideals and diamonds on them."}
{"category": "Math", "title": "Askey--Wilson polynomials, quadratic harnesses and martingales", "abstract": "We use orthogonality measures of Askey--Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey--Wilson polynomials are orthogonal martingale polynomials for these processes."}
{"category": "Math", "title": "Kazhdan's Property T for Discrete Quantum Groups", "abstract": "We give a simple definition of property T for discrete quantum groups. We prove the basic expected properties: discrete quantum groups with property T are finitely generated and unimodular. Moreover we show that, for \"I.C.C.\" discrete quantum groups, property T is equivalent to Connes' property T for the dual von Neumann algebra. This allows us to give the first example of a property T discrete quantum group which is not a group using the twisting construction."}
{"category": "Math", "title": "Periodicities of T-systems and Y-systems", "abstract": "The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2)."}
{"category": "Math", "title": "Group actions on geodesic Ptolemy spaces", "abstract": "In this paper we study geodesic Ptolemy metric spaces $X$ which allow proper and cocompact isometric actions of crystallographic or, more generally, virtual polycyclic groups. We show that $X$ is equivariantly rough isometric to a Euclidean space."}
{"category": "Math", "title": "Classification de modules aux diff\\'erences filtr\\'es isogradu\\'es", "abstract": "The local analytic classification of irregular linear q-difference equations (Ramis-Sauloy-Zhang) involves the classfication of filtered q-difference modules with a prescribed associated graded module. We prove in a more general setting the existence for this problem of a moduli scheme which is an affine space. ----- La classification analytique locale des equations aux q-differences irregulieres se ramene a la classification de modules aux q-differences filtres a gradue fixe. Nous degageons ici des hypotheses generales qui assurent l'existence d'un schema de modules pour ce probleme, qui soit de plus un espace affine."}
{"category": "Math", "title": "A generic multiplication in quantised Schur algebras", "abstract": "We define a generic multiplication in quantised Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantised Schur algebras, defined by A. A. Beilinson, G. Lusztig and R. MacPherson, a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied by M. Reineke. We also prove that the subalgebra of the new algebra gives a geometric realisation of a positive part of 0-Schur algebras. Consequently, we obtain a multiplicative basis for the positive part of 0-Schur algebras."}
{"category": "Math", "title": "Natural Boundary Conditions in the Calculus of Variations", "abstract": "We prove necessary optimality conditions for problems of the calculus of variations on time scales with a Lagrangian depending on the free end-point."}
{"category": "Math", "title": "Notes on Cohomologies of Ternary Algebras of Associative Type", "abstract": "The aim of this paper is to investigate the cohomologies for ternary algebras of associative type. We study in particular the cases of partially associative ternary algebras and weak totally associative ternary algebras. Also, we consider the Takhtajan's construction, which was used to construct a cohomology of ternary Nambu-Lie algebras using Chevalley-Eilenberg cohomology of Lie algebras, and discuss it in the case of ternary algebras of associative type. One of the main results of this paper states that a deformation cohomology does not exist for partially associative ternary algebras which implies that their operad is not a Koszul operad."}
{"category": "Math", "title": "Zeros of the hypergeometric polynomial F(-n,b;c;z)", "abstract": "Our interest lies in describing the zero behaviour of Gauss hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are arbitrary parameters. In general, this problem has not been solved and even when $b$ and $c$ are both real, the only cases that have been fully analyzed impose additional restrictions on $b$ and $c$. We review recent results that have been proved for the zeros of several classes of hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are real. We show that the number of real zeros of $F(-n,b; c; z)$ for arbitrary real values of the parameters $b$ and $c$, as well as the intervals in which these zeros (if any) lie, can be deduced from corresponding results for Jacobi polynomials."}
{"category": "Math", "title": "Free joinings of C*-dynamical systems", "abstract": "Joinings of C*-dynamical systems are defined in terms of free products of C*-algebras, as an analogue of joinings of classical dynamical systems. We then consider disjointness in this context, in particular for ergodic versus identity systems. Lastly we show how multi-time correlation functions appearing in quantum statistical mechanics naturally fit into this joining framework."}
{"category": "Math", "title": "Holomorphic Extension of Eigenfunctions", "abstract": "Let X be a Riemannian symmetric space of non-compact type. We prove a theorem of holomorphic extension for eigenfunctions of the Laplace-Beltrami operator on X, by techniques from the theory of partial differential equations."}
{"category": "Math", "title": "Convexity of the zeros of some orthogonal polynomials and related functions", "abstract": "We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases."}
{"category": "Math", "title": "Note on the X_(1)-Jacobi orthogonal polynomials", "abstract": "This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Jacobi polynomials which are orthogonal in a weighted Hilbert function space on the the interval (-1,+1) of the real line. These polynomials are generated by a second-order ordinary linear differential equation with a spectral parameter. Some additional information on the Sturm-Liouville form of this equation is given in this note, together with details of the singular differential operators generated in the weighted Hilbert function space. In particular, structured boundary conditions are given to determine the special self-adjoint operator, whose discrete spectrum and associated eigenvectors yield the X_(1)-Jacobi polynomials."}
{"category": "Math", "title": "Zeros of linear combinations of Laguerre polynomials from different sequences", "abstract": "We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely $R_n=L_n^{\\alpha}+aL_{n}^{\\alpha'}$ and $S_n=L_n^{\\alpha}+bL_{n-1}^{\\alpha'}$. Proofs and numerical counterexamples are given in situations where the zeros of $R_n$, and $S_n$, respectively, interlace (or do not in general) with the zeros of $L_k^{\\alpha}$, $L_k^{\\alpha'}$, $k=n$ or $n-1$. The results we prove hold for continuous, as well as integral, shifts of the parameter $\\alpha$."}
{"category": "Math", "title": "Nonhomogeneous parking functions and noncrossing partitions", "abstract": "For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haiman's notion of a parking function symmetric function."}
{"category": "Math", "title": "A Limit Relation for Dunkl-Bessel Functions of Type A and B", "abstract": "We prove a limit relation for the Dunkl-Bessel function of type $B_N$ with multiplicity parameters $k_1$ on the roots $\\pm e_i$ and $k_2$ on $\\pm e_i\\pm e_j$ where $k_1$ tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkl-type Bessel function of type $A_{N-1}$ with multiplicity $k_2$. For certain values of $k_2$ an improved estimate is obtained from a corresponding limit relation for Bessel functions on matrix cones."}
{"category": "Math", "title": "Relations between slices and quotients of the algebraic cobordism spectrum", "abstract": "We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zero-slice then a relative version of Voevodsky's conjecture on the slices of MGL holds true. We outline the picture for K-theory and rational slices."}
{"category": "Math", "title": "About Very Perfect Numbers", "abstract": "In this short paper we prove that the square of an odd prime number cannot be a very perfect number."}
{"category": "Math", "title": "Unipotent elements in small characteristic, III", "abstract": "We give a uniform description of the decomposition of the unipotent variety of a classical group in arbitrary characteristic into pieces (considered in a non-uniform way in the earlier parts of this paper)."}
{"category": "Math", "title": "Eulerian quasisymmetric functions", "abstract": "We introduce a family of quasisymmetric functions called {\\em Eulerian quasisymmetric functions}, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising $q$-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This $q$-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain $q$-analogs, $(q,p)$-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Bj\\\"orner and Welker."}
{"category": "Math", "title": "Cyclic Homologies of Crossed Modules of Algebras", "abstract": "The Hochschild and (cotriple) cyclic homologies of crossed modules of (not-necessarily-unital) associative algebras are investigated. Wodzicki's excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cyclic homology exact sequence."}
{"category": "Math", "title": "On the Jacobson element and generators of the Lie algebra $\\mathfrak{grt}$ in nonzero characteristic", "abstract": "We state a conjecture (due to M. Duflo) analogous to the Kashiwara--Vergne conjecture in the case of a characteristic $p>2$, where the role of the Campbell--Hausdorff series is played by the Jacobson element. We prove a simpler version of this conjecture using Vergne's explicit rational solution of the Kashiwara--Vergne problem. Our result is related to the structure of the Grothendieck--Teichm\\\"{u}ller Lie algebra $\\mathfrak{grt}$ in characteristic $p$: we conjecture existence of a generator of $\\mathfrak{grt}$ in degree $p-1$, and we provide this generator for $p=3$ and $p=5$."}
{"category": "Math", "title": "Separating Solution of a Quadratic Recurrent Equation", "abstract": "In this paper we consider the recurrent equation $$\\Lambda_{p+1}=\\frac1p\\sum_{q=1}^pf\\bigg(\\frac{q}{p+1}\\bigg)\\Lambda_{q}\\Lambda_{p+1-q}$$ for $p\\ge 1$ with $f\\in C[0,1]$ and $\\Lambda_1=y>0$ given. We give conditions on $f$ that guarantee the existence of $y^{(0)}$ such that the sequence $\\Lambda_p$ with $\\Lambda_1=y^{(0)}$ tends to a finite positive limit as $p\\to \\infty$."}
{"category": "Math", "title": "Nef divisors on $\\bar{M}_{0,n}$ from GIT", "abstract": "We introduce and study the GIT CONE of $\\bar{M}_{0,n}$, which is generated by the pullbacks of the natural ample line bundles on the GIT quotients $(\\mathbb P^1)^n//SL(2)$. We give an explicit formula for these line bundles and prove a number of basic results about the GIT cone. As one application, we prove unconditionally that the log canonical models of $\\bar{M}_{0,n}$ with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of Matt Simpson arXiv:0709.4037. (Cf. also a different proof by Fedorchuk and Smyth arXiv:0810.1677)"}
{"category": "Math", "title": "Poset homology of Rees products, and $q$-Eulerian polynomials", "abstract": "The notion of Rees product of posets was introduced by Bj\\\"orner and Welker, where they study connections between poset topology and commutative algebra. Bj\\\"orner and Welker conjectured and Jonsson proved that the dimension of the top homology of the Rees product of the truncated Boolean algebra $B_n \\setminus \\{0\\}$ and the $n$-chain $C_n$ is equal to the number of derangements in the symmetric group $\\mathfrak S_n$. Here we prove a refinement of this result, which involves the Eulerian numbers, and a $q$-analog of both the refinement and the original conjecture, which comes from replacing the Boolean algebra by the lattice of subspaces of the $n$-dimensional vector space over the $q$ element field, and involves the $(\\maj,\\exc)$-$q$-Eulerian polynomials studied in previous papers of the authors. Equivariant versions of the refinement and the original conjecture are also proved, as are type BC versions (in the sense of Coxeter groups) of the original conjecture and its $q$-analog."}
{"category": "Math", "title": "Cyclotomic Units and Class Groups in Z_p extensions of real abelian fields", "abstract": "For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension."}
{"category": "Math", "title": "How long does it take to catch a wild kangaroo?", "abstract": "We develop probabilistic tools for upper and lower bounding the expected time until two independent random walks on $\\ZZ$ intersect each other. This leads to the first sharp analysis of a non-trivial Birthday attack, proving that Pollard's Kangaroo method solves the discrete logarithm problem $g^x=h$ on a cyclic group in expected time $(2+o(1))\\sqrt{b-a}$ for an average $x\\in_{uar}[a,b]$. Our methods also resolve a conjecture of Pollard's, by showing that the same bound holds when step sizes are generalized from powers of 2 to powers of any fixed $n$."}
{"category": "Math", "title": "Increasing and Decreasing Sequences of Length Two in 01-Fillings of Moon Polyominoes", "abstract": "We put recent results on the symmetry of the joint distribution of the numbers of crossings and nestings of two edges over matchings, set partitions and linked partitions, in the larger context of the enumeration of increasing and decreasing chains of length 2 in fillings of moon polyominoes."}
{"category": "Math", "title": "Phi-entropy inequalities for diffusion semigroups", "abstract": "We obtain and study new $\\Phi$-entropy inequalities for diffusion semigroups, with Poincar\\'e or logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear Fokker-Plank type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The $\\Gamma_2$ criterion of D. Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms."}
{"category": "Math", "title": "Affine T-varieties of complexity one and locally nilpotent derivations", "abstract": "Let X=spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus T of dimension n. Let also D be a homogeneous locally nilpotent derivation on the normal affine Z^n-graded domain A, so that D generates a k_+-action on X that is normalized by the T-action. We provide a complete classification of pairs (X,D) in two cases: for toric varieties (n=\\dim X) and in the case where n=\\dim X-1. This generalizes previously known results for surfaces due to Flenner and Zaidenberg. As an application we compute the homogeneous Makar-Limanov invariant of such varieties. In particular we exhibit a family of non-rational varieties with trivial Makar-Limanov invariant."}
{"category": "Math", "title": "Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models", "abstract": "We study the growth rate of a cell population that follows an age-structured PDE with time-periodic coefficients. Our motivation comes from the comparison between experimental tumor growth curves in mice endowed with intact or disrupted circadian clocks, known to exert their influence on the cell division cycle. We compare the growth rate of the model controlled by a time-periodic control on its coefficients with the growth rate of stationary models of the same nature, but with averaged coefficients. We firstly derive a delay differential equation which allows us to prove several inequalities and equalities on the growth rates. We also discuss about the necessity to take into account the structure of the cell division cycle for chronotherapy modeling. Numerical simulations illustrate the results."}
{"category": "Math", "title": "Poisson boundary of the discrete quantum group A_u(F)^", "abstract": "We identify the Poisson boundary of the dual of the universal compact quantum group A_u(F) with a measurable field of ITPFI factors."}
{"category": "Math", "title": "The Vanishing Approach for the Average Continuous Control of Piecewise Deterministic Markov Processes", "abstract": "The main goal of this paper is to derive sufficient conditions for the existence of an optimal control strategy for the long run average continuous control problem of piecewise deterministic Markov processes (PDMP's) taking values in a general Borel space and with compact action space depending on the state variable. In order to do that we apply the so-called vanishing discount approach to obtain a solution to an average cost optimality inequality associated to the long run average cost problem. Our main assumptions are written in terms of some integro-differential inequalities related to the so-called expected growth condition, and geometric convergence of the post-jump location kernel associated to the PDMP."}
{"category": "Math", "title": "Branching of Representations to Symmetric Subgroups", "abstract": "Let $\\gg$ be the Lie algebra of a compact Lie group and let $\\theta$ be any automorphism of $\\gg$. Let $\\gk$ denote the fixed point subalgebra $\\gg^\\theta$. In this paper we present LiE programs that, for any finite dimensional complex representation $\\pi$ of $\\gg$, give the explicit branching $\\pi|_\\gk$ of $\\pi$ on $\\gk$. Cases of special interest include the cases where $\\theta$ has order 2 (corresponding to compact riemannian symmetric spaces $G/K$), where $\\theta$ has order 3 (corresponding to compact nearly--kaehler homogeneous spaces $G/K$), where $\\theta$ has order 5 (which include the fascinating 5--symmetric space $E_8/A_4A_4$), and the cases where $\\gk$ is the centralizer of a toral subalgebra of $\\gg$."}
{"category": "Math", "title": "Duality, a-invariants and canonical modules of rings arising from linear optimization problems", "abstract": "The aim of this paper is to study integer rounding properties of various systems of linear inequalities to gain insight about the algebraic properties of Rees algebras of monomial ideals and monomial subrings. We study the normality and Gorenstein property--as well as the canonical module and the a-invariant--of Rees algebras and subrings arising from systems with the integer rounding property. We relate the algebraic properties of Rees algebras and monomial subrings with integer rounding properties and present a duality theorem."}
{"category": "Math", "title": "Degree bounds for type-A weight rings and Gelfand--Tsetlin semigroups", "abstract": "A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in $\\C^n$ modulo a twisted action of the maximal torus in $\\SL(n,\\C)$. We show that any weight ring in type A is generated by elements of degree strictly less than the Krull dimension, which is at worst $O(n^2)$. On the other hand, we show that the associated semigroup of Gelfand--Tsetlin patterns can have an essential generator of degree exponential in $n$."}
{"category": "Math", "title": "A proof of Lens Rigidity in the category of Analytic Metrics", "abstract": "Consider a compact Riemannian manifold with boundary. Assume all maximally extended geodesics intersect the boundary at both ends. Then to each maximal geodesic segment one can form a triple consisting of the initial and final vectors of the segment and the length of the segment. The collection of all such triples comprises the lens data. In this paper, it is shown that in the category of analytic Riemannian manifolds, the lens data uniquely determine the metric up to isometry. There are no convexity assumptions on the boundary, and conjugate points are allowed, but with some restriction."}
{"category": "Math", "title": "Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations", "abstract": "In this paper, we first study the existence-uniqueness and large deviation estimate of solutions for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then, we apply them to a large class of semilinear stochastic partial differential equations (SPDE) driven by Brownian motions as well as by fractional Brownian motions, and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Lastly, high order SPDEs in a bounded domain of Euclidean space, second order SPDEs on complete Riemannian manifolds, as well as stochastic Navier-Stokes equations are investigated."}
{"category": "Math", "title": "Expansions of the real field by open sets: definability versus interpretability", "abstract": "An open set U of the real numbers R is produced such that the expansion (R,+,x,U) of the real field by U defines a Borel isomorph of (R,+,x,N) but does not define N. It follows that (R,+,x,U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (R,+,x). In particular, there is a Cantor subset K of R such that for every exponentially bounded o-minimal expansion M of (R,+,x), every subset of R definable in (M,K) either has interior or is Hausdorff null."}
{"category": "Math", "title": "Efficiency and influence function of estimators for ARCH models", "abstract": "This paper proposes a closed-form optimal estimator based on the theory of estimating functions for a class of linear ARCH models. The estimating function (EF) estimator has the advantage over the widely used maximum likelihood (ML) and quasi-maximum likelihood (QML) estimators that (i) it can be easily implemented, (ii) it does not depend on a distributional assumption for the innovation, and (iii) it does not require the use of any numerical optimization procedures or the choice of initial values of the conditional variance equation. In the case of normality, the asymptotic distribution of the ML and QML estimators naturally turn out to be identical and, hence, coincides with ours. Moreover, a robustness property of the EF estimator is derived by means of influence function. Simulation results show that the efficiency benefits of our estimator relative to the ML and QML estimators are substantial for some ARCH innovation distributions."}
{"category": "Math", "title": "Testing the equality of error distributions from k independent GARCH models", "abstract": "In this paper we study the problem of testing the null hypothesis that errors from k independent parametrically specified generalized autoregressive conditional heteroskedasticity (GARCH) models have the same distribution versus a general alternative. First we establish the asymptotic validity of a class of linear test statistics derived from the k residual-based empirical distribution functions. A distinctive feature is that the asymptotic distribution of the test statistics involves terms depending on the distributions of errors and the parameters of the models, and weight functions providing the flexibility to choose scores for investigating power performance. A Monte Carlo study assesses the asymptotic performance in terms of empirical size and power of the three-sample test based on the Wilcoxon and Van der Waerden score generating functions in finite samples. The results demonstrate that the two proposed tests have overall reasonable size and their power is particularly high when the assumption of Gaussian errors is violated. As an illustrative example, the tests are applied to daily individual stock returns of the New York Stock Exchange data."}
{"category": "Math", "title": "Total positivity in loop groups I: whirls and curls", "abstract": "This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group GL_n(\\R[t,t^{-1}]), and for the formal loop group GL_n(\\R((t))) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeroes of the Edrei-Thoma theorem classifying totally positive functions (corresponding to our case n=1). We give a solution to the ``factorization problem'' using limits of ratios of minors. This is in a similar spirit to the Berenstein-Fomin-Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi-Yamada's study of discrete Painlev\\'{e} dynamical systems and Berenstein-Kazhdan's study of geometric crystals."}
{"category": "Math", "title": "Withdraw a paper entitled \"On the growth rate of solutions for 2D incompressible Euler equations\"", "abstract": "This paper has been withdrawn by the author due to a crucial definition error of Triebel space."}
{"category": "Math", "title": "Algebraic Entropy and the Action of Mapping Class Groups on Character Varieties", "abstract": "We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov mapping class. Our calculations are compatible with all known calculations of the topological entropy of this action."}
{"category": "Math", "title": "The length of unknotting tunnels", "abstract": "We show there exist tunnel number one hyperbolic 3-manifolds with arbitrarily long unknotting tunnel. This provides a negative answer to an old question of Colin Adams."}
{"category": "Math", "title": "Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes (extended abstract)", "abstract": "We show that the Kronecker coefficients (the Clebsch-Gordan coefficients of the symmetric group) indexed by two two-row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the behavior of the stretching functions attached to the Kronecker coefficients."}
{"category": "Math", "title": "Table of minimum ranks of graphs of order at most 7 and selected optimal matrices", "abstract": "The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\\neq j$) is nonzero whenever $\\{i,j\\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a difficult parameter to compute. However, there are now a number of known reduction techniques and bounds that can be programmed on a computer; we have developed a program using the open-source mathematics software Sage to implement several techniques. We have also established several additional strategies for computation of minimum rank. These techniques have been used to determine the minimum ranks of all graphs of order 7. This paper contains a list of minimum ranks for all graphs of order at most 7. We also present selected optimal matrices."}
{"category": "Math", "title": "Maximums and minimums of overall survival functions with fixed marginal distributions and transmission of technology", "abstract": "Title: An unlikely result Authors: T.M. Other Comments: This paper has been withdrawn Abstract: This paper has been withdrawn by the author due to the fact that some of the results turned out to be known."}
{"category": "Math", "title": "Two-Parameter Heavy-Traffic Limits for Infinite-Server Queues", "abstract": "In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We consider the random variables $Q^e(t,y)$ and $Q^r(t,y)$ representing the number of customers in the system at time $t$ that have elapsed service times less than or equal to time $y$, or residual service times strictly greater than $y$. We also consider $W^r(t,y)$ representing the total amount of work in service time remaining to be done at time $t+y$ for customers in the system at time $t$. The two-parameter stochastic-process limits in the space $D([0,\\infty),D)$ of $D$-valued functions in $D$ draw on, and extend, previous heavy-traffic limits by Glynn and Whitt (1991), where the case of discrete service-time distributions was treated, and Krichagina and Puhalskii (1997), where it was shown that the variability of service times is captured by the Kiefer process with second argument set equal to the service-time c.d.f."}
{"category": "Math", "title": "Distribution and asymptotics under beta random scaling", "abstract": "Let X,Y,B be three independent random variables such that $X$ has the same distribution function as Y B. Assume that B is a Beta random variable with positive parameters a,b and Y has distribution function H. Pakes and Navarro (2007) show under some mild conditions that the distribution function H_{a,b} of X determines H. Based on that result we derive in this paper a recursive formula for calculation of H, if H_{a,b} is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and H_{a,b}, respectively, and the conditional limiting distribution of bivariate elliptical distributions."}
{"category": "Math", "title": "An operator extension of Bohr's inequality", "abstract": "We establish an operator extension of the following generalization of Bohr's inequality, due to M.P. Vasi\\'c and D.J. Ke\\v{c}ki\\'{c}: $$|\\sum_{i=1}^n z_i|^r \\leq (\\sum_{i=1}^n \\alpha_i^{1/(1-r)})^{r-1}\\sum_{i=1}^n \\alpha_i|z_i|^r \\quad (r>1, z_i \\in{\\mathbb C}, \\alpha_i>0, 1 \\leq i \\leq n) .$$ We also present some norm inequalities related to our noncommutative generalization of Bohr's inequality."}
{"category": "Math", "title": "A characterization of Hilbert $C^*$-modules over finite dimensional $C^*$-algebras", "abstract": "We show that the unit ball of a full Hilbert $C^*$-module is sequentially compact in a certain weak topology if and only if the underlying $C^*$-algebra is finite dimensional. This provides an answer to the question posed in J. Chmieli\\'nski et al [Perturbation of the Wigner equation in inner product $C^*$-modules, J. Math. Phys. 49 (2008), no. 3, 033519; arXiv:0801.2726]."}
{"category": "Math", "title": "Boxicity and Cubicity of Asteroidal Triple free graphs", "abstract": "An axis parallel $d$-dimensional box is the Cartesian product $R_1 \\times R_2 \\times ... \\times R_d$ where each $R_i$ is a closed interval on the real line. The {\\it boxicity} of a graph $G$, denoted as $\\boxi(G)$, is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-dimensional boxes. An axis parallel unit cube in $d$-dimensional space or a $d$-cube is defined as the Cartesian product $R_1 \\times R_2 \\times ... \\times R_d$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\\it cubicity} of $G$, denoted as $\\cub(G)$, is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-cubes. Let $S(m)$ denote a star graph on $m+1$ nodes. We define {\\it claw number} of a graph $G$ as the largest positive integer $k$ such that $S(k)$ is an induced subgraph of $G$ and denote it as $\\claw$. Let $G$ be an AT-free graph with chromatic number $\\chi(G)$ and claw number $\\claw$. In this paper we will show that $\\boxi(G) \\leq \\chi(G)$ and this bound is tight. We also show that $\\cub(G) \\leq \\boxi(G)(\\ceil{\\log_2 \\claw} +2)$ $\\leq$ $\\chi(G)(\\ceil{\\log_2 \\claw} +2)$. If $G$ is an AT-free graph having girth at least 5 then $\\boxi(G) \\leq 2$ and therefore $\\cub(G) \\leq 2\\ceil{\\log_2 \\claw} +4$."}
{"category": "Math", "title": "Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization", "abstract": "Let $T$ be an underlying space with a non-atomic measure $\\sigma$ on it (e.g. $T=\\mathbb R^d$ and $\\sigma$ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of $T$, with freely independent values. Such a process (field), $\\omega=\\omega(t)$, $t\\in T$, is given a rigorous meaning through smearing out with test functions on $T$, with $\\int_T \\sigma(dt)f(t)\\omega(t)$ being a (bounded) linear operator in a full Fock space. We define a set $\\mathbf{CP}$ of all continuous polynomials of $\\omega$, and then define a con-commutative $L^2$-space $L^2(\\tau)$ by taking the closure of $\\mathbf{CP}$ in the norm $\\|P\\|_{L^2(\\tau)}:=\\|P\\Omega\\|$, where $\\Omega$ is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between $L^2(\\tau)$ and a (Fock-space-type) Hilbert space $\\mathbb F=\\mathbb R\\oplus\\bigoplus_{n=1}^\\infty L^2(T^n,\\gamma_n)$, with explicitly given measures $\\gamma_n$. We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set $\\mathbf {CP}$ invariant. (Note that, in the general case, the projection of a continuous monomial of oder $n$ onto the $n$-th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions $\\lambda$ and $\\eta\\ge0$ on $T$, such that, in the $\\mathbb F$ space, $\\omega$ has representation $\\omega(t)=\\di_t^\\dag+\\lambda(t)\\di_t^\\dag\\di_t+\\di_t+\\eta(t)\\di_t^\\dag\\di^2_t$, where $\\di_t^\\dag$ and $\\di_t$ are the usual creation and annihilation operators at point $t$."}
{"category": "Math", "title": "Lowering and raising operators for the free Meixner class of orthogonal polynomials", "abstract": "We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line."}
{"category": "Math", "title": "Ergodic properties of linked-twist maps", "abstract": "We study a class of homeomorphisms of surfaces collectively known as linked-twist maps. We introduce an abstract definition which enables us to give a precise characterisation of a property observed by other authors, namely that such maps fall into one of two classes termed co- and counter-twisting. We single out three specific linked-twist maps, one each on the two-torus, in the plane and on the two-sphere and for each prove a theorem concerning its ergodic properties with respect to the invariant Lebesgue measure. For the map on the torus we prove that there is an invariant, zero-measure Cantor set on which the dynamics are topologically conjugate to a full shift on the space of symbol sequences. Such features are commonly known as topological horseshoes. For the map in the plane we prove that there is a set of full measure on which the dynamics are measure-theoretically isomorphic to a full shift on the space of symbol sequences. This is commonly known as the Bernoulli property and verifies, under certain conditions, a conjecture of Wojtkowski's. We introduce the map on the sphere and prove that it too has the Bernoulli property. We conclude with some conjectures, drawn from our experience, concerning how one might extend the results we have for specific linked-twist maps to the abstract linked-twist maps we have defined."}
{"category": "Math", "title": "Gantmakher-Krein theorem for 2-totally nonnegative operators in ideal spaces", "abstract": "The tensor and exterior squares of a completely continuous non-negative linear operator $A$ acting in the ideal space $X(\\Omega)$ are studied. The theorem representing the point spectrum (except, probably, zero) of the tensor square $A \\otimes A$ in the terms of the spectrum of the initial operator $A$ is proved. The existence of the second (according to the module) positive eigenvalue $\\lambda_2$, or a pair of complex adjoint eigenvalues of a completely continuous non-negative operator $A$ is proved under the additional condition, that its exterior square $A\\wedge A$ is also nonnegative."}
{"category": "Math", "title": "On local compactness in quasilinear elliptic problems", "abstract": "One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work \\cite{BN}, Br\\'ezis and Nirenberg introduced the notion of critical level for these sequences in the case of a critical perturbation of the Laplacian homogeneous eigenvalue problem. In this paper, we give a natural and general formula of the critical level for a large class of nonlinear elliptic critical problems. The sharpness of our formula is established by the construction of suitable Palais-Smale sequences which are not relatively compact."}
{"category": "Math", "title": "Extremal functions for the anisotropic Sobolev inequalities", "abstract": "The existence of multiple nonnegative solutions to the anisotropic critical problem - \\sum_{i=1}^{N} \\frac{\\partial}{\\partial x_i} (| \\frac{\\partial u}{\\partial x_i} |^{p_i-2} \\frac{\\partial u}{\\partial x_i}) = |u|^{p^*-2} u {in} \\mathbb{R}^N is proved in suitable anisotropic Sobolev spaces. The solutions correspond to extremal functions of a certain best Sobolev constant. The main tool in our study is an adaptation of the well-known concentration-compactness lemma of P.-L. Lions to anisotropic operators. Futhermore, we show that the set of nontrival solutions $\\calS$ is included in $L^\\infty(\\R^N)$ and is located outside of a ball of radius $\\tau >0$ in $L^{p^*}(\\R^N)$."}
{"category": "Math", "title": "The Goldbach conjecture resulting from global-local cuspidal representations and deformations of Galois representations", "abstract": "In the basic general frame of the Langlands global program, a local p-adic elliptic semimodule corresponding to a local (left) cuspidal form is constructed from its global equivalent covered by p-th roots. In the same context, global and local bilinear deformations of Galois representations inducing the invariance of their respective residue fields are introduced as well as global and local bilinear quantum deformations leaving invariant the orders of the inertia subgroups. More particularly, the inverse quantum deformation of a closed curve responsible for its splitting directly leads to the Goldbach conjecture."}
{"category": "Math", "title": "Th\\'eor\\`emes d'\\'equidistribution pour les syst\\`emes dynamiques d'origine arithm\\'etique", "abstract": "This survey article is about algebraic dynamics. It is mainly concerned by the arithmetic equidistribution theorems featured by dynamical systems. The contents are: - heights - algebraic dynamics, conjectures - equidistribution theorem on the projective line (after Bilu and Baker). It was written for the sesion 2006 of \\'Etats de la recherche, Syst\\`emes dynamiques polynomiaux."}
{"category": "Math", "title": "Lectures on height zeta functions: At the confluence of algebraic geometry, algebraic number theory, and analysis", "abstract": "This is a survey on the theory of height zeta functions, written on the occasion of a French-Japanese winter school, held in Miura (Kanagawa, Japan) in Jan. 2008. It does not presuppose much knowledge in algebraic geometry. The last chapter of the survey explains recent results obtained in collaboration with Yuri Tschinkel concerning asymptotics of volumes of height balls in analytic geometry over local fields, or in adelic spaces."}
{"category": "Math", "title": "Asymptotic Behaviour and Artinian Property of Graded Local Cohomology Modules", "abstract": "In this paper, considering the difference between the finiteness dimension and cohomological dimension for a finitely generated module, we investigate the asymptotic behavior of grades of components of graded local cohomology modules with respect to irrelevant ideal; as long as we study some artinian and tameness property of such modules."}
{"category": "Math", "title": "Symmetric identities for Euler polynomials", "abstract": "In this paper we establish two symmetric identities on sums of products of Euler polynomials."}
{"category": "Math", "title": "The two uniform infinite quadrangulations of the plane have the same law", "abstract": "We prove that the uniform infinite random quadrangulations defined respectively by Chassaing-Durhuus and Krikun have the same distribution."}
{"category": "Math", "title": "Quantum Giambelli formulas for isotropic Grassmannians", "abstract": "Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class in the small quantum cohomology ring of X as a polynomial in certain special Schubert classes, extending the cohomological Giambelli formulas of arXiv:0811.2781."}
{"category": "Math", "title": "Regularity results for stable-like operators", "abstract": "For $\\alpha\\in [1,2)$ we consider operators of the form $$L f(x)=\\int_{R^d} [f(x+h)-f(x)-1_{(|h|\\leq 1)} \\nabla f(x)\\cdot h] \\frac{A(x,h)}{|h|^{d+\\alpha}}$$ and for $\\alpha\\in (0,1)$ we consider the same operator but where the $\\nabla f$ term is omitted. We prove, under appropriate conditions on $A(x,h)$, that the solution $u$ to $L u=f$ will be in $C^{\\alpha+\\beta}$ if $f\\in C^\\beta$."}
{"category": "Math", "title": "Controlling Multiparticle System on the Line, II - Periodic case", "abstract": "As in arXiv: math. 0809.2365 we consider classical system of interacting particles $\\mathcal{P}_1, ..., \\mathcal{P}_n$ on the line with only neighboring particles involved in interaction. On the contrast to arXiv: math. 0809.2365 now {\\it periodic boundary conditions} are imposed onto the system, i.e. $\\mathcal{P}_1$ and $\\mathcal{P}_n$ are considered neighboring. Periodic Toda lattice would be a typical example. We study possibility to control periodic multiparticle systems by means of forces applied to just few of its particles; mainly we study system controlled by single force. The free dynamics of multiparticle systems in periodic and nonperiodic case differ substantially. We see that also the controlled periodic multiparticle system does not mimic its non-periodic counterpart. Main result established is global controllability by means of single controlling force of the multiparticle system with ageneric potential of interaction. We study the nongeneric potentials for which controllability and accessibility properties may lack. Results are formulated and proven in Sections~2,3."}
{"category": "Math", "title": "A numerical study of the self-similar solutions of the Schroedinger Map", "abstract": "We present a numerical study of the self-similar solutions of the Localized Induction Approximation of a vortex filament. These self-similar solutions, which constitute a one-parameter family, develop a singularity at finite time. We study a number of boundary conditions that allow us reproduce the mechanism of singularity formation. Some related questions are also considered."}
{"category": "Math", "title": "Sharp error terms for return time statistics under mixing conditions", "abstract": "We describe the statistics of repetition times of a string of symbols in a stochastic process. Denote by T(A) the time elapsed until the process spells the finite string A and by S(A) the number of consecutive repetitions of A. We prove that, if the length of the string grows unbondedly, (1) the distribution of T(A), when the process starts with A, is well aproximated by a certain mixture of the point measure at the origin and an exponential law, and (2) S(A) is approximately geometrically distributed. We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and allow to get also approximations for all the moments of T(A) and S(A). To obtain (1) we assume that the process is phi-mixing while to obtain (2) we assume the convergence of certain contidional probabilities."}
{"category": "Math", "title": "Mistuning-based Control Design to Improve Closed-Loop Stability of Vehicular Platoons", "abstract": "We consider a decentralized bidirectional control of a platoon of N identical vehicles moving in a straight line. The control objective is for each vehicle to maintain a constant velocity and inter-vehicular separation using only the local information from itself and its two nearest neighbors. Each vehicle is modeled as a double integrator. To aid the analysis, we use continuous approximation to derive a partial differential equation (PDE) approximation of the discrete platoon dynamics. The PDE model is used to explain the progressive loss of closed-loop stability with increasing number of vehicles, and to devise ways to combat this loss of stability. If every vehicle uses the same controller, we show that the least stable closed-loop eigenvalue approaches zero as O(1/N^2) in the limit of a large number (N) of vehicles. We then show how to ameliorate this loss of stability by small amounts of \"mistuning\", i.e., changing the controller gains from their nominal values. We prove that with arbitrary small amounts of mistuning, the asymptotic behavior of the least stable closed loop eigenvalue can be improved to O(1/N) All the conclusions drawn from analysis of the PDE model are corroborated via numerical calculations of the state-space platoon model."}
{"category": "Math", "title": "Lipschitz Characterisation of Polytopal Hilbert Geometries", "abstract": "We prove that the Hilbert Geometry of a convex set is bi-lipschitz equivalent to a normed vector space if and only if the convex is a polytope."}
{"category": "Math", "title": "Density of isoperimetric spectra", "abstract": "We show that the set of k-dimensional isoperimetric exponents of finitely presented groups is dense in the interval [1, \\infty) for k > 1. Hence there is no higher-dimensional analogue of Gromov's gap (1,2) in the isoperimetric spectrum."}
{"category": "Math", "title": "On the holomorphic point of view in the theory of quantum knot invariants", "abstract": "In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which relates the quantum group and the Weyl quantizations of the moduli space of flat SU(2)-connections on the tours. Two results are emphasized: the reconstruction from Weyl quantization of the restriction to the torus of the modular functor, and a description of a basis of the space of quantum observables on the torus in terms of colored curves, which answers a question related to quantum computing."}
{"category": "Math", "title": "Algebraic Methods in Discrete Analogs of the Kakeya Problem", "abstract": "We prove the joints conjecture, showing that for any $N$ lines in ${\\Bbb R}^3$, there are at most $O(N^{{3 \\over 2}})$ points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$ lines in ${\\Bbb R}^3$ so that no $N$ lines lie in the same plane and so that each line intersects a set $P$ of points in at least $N$ points then the cardinality of the set of points is $\\Omega(N^3)$. Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem."}
{"category": "Math", "title": "Graph Minors and Minimum Degree", "abstract": "Let $\\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$. Then $\\mathcal{D}_k$ is closed under taking minors. By the Robertson-Seymour graph minor theorem, $\\mathcal{D}_k$ is characterised by a finite family of minor-minimal forbidden graphs, which we denote by $\\widehat{\\mathcal{D}}_k$. This paper discusses $\\widehat{\\mathcal{D}}_k$ and related topics. We obtain four main results: We prove that every $(k+1)$-regular graph with less than ${4/3}(k+2)$ vertices is in $\\widehat{\\mathcal{D}}_k$, and this bound is best possible. We characterise the graphs in $\\widehat{\\mathcal{D}}_{k+1}$ that can be obtained from a graph in $\\widehat{\\mathcal{D}}_k$ by adding one new vertex. For $k\\leq 3$ every graph in $\\widehat{\\mathcal{D}}_k$ is $(k+1)$-connected, but for large $k$, we exhibit graphs in $\\widehat{\\mathcal{D}}_k$ with connectivity 1. In fact, we construct graphs in $\\mathcal{D}_k$ with arbitrary block structure. We characterise the complete multipartite graphs in $\\widehat{\\mathcal{D}}_k$, and prove analogous characterisations with minimum degree replaced by connectivity, treewidth, or pathwidth."}
{"category": "Math", "title": "Open Orbits and Augmentations of Dynkin Diagrams", "abstract": "Given any representation V of a complex linear reductive Lie group G_0, we show that a larger semi-simple Lie group G with g=g_0 + V + V* + ..., exists precisely when V has a finite number of G_0-orbits. In particular, V admits an open G_0-orbit. Furthermore, this corresponds to an augmentation of the Dynkin diagram of g_0. The representation theory of g should be useful in describing the geometry of manifolds with stable forms as studied by Hitchin."}
{"category": "Math", "title": "Hilbert geometry of polytopes", "abstract": "It is shown that the Hilbert metric on the interior of a convex polytope is bilipschitz to a normed vector space of the same dimension."}
{"category": "Math", "title": "Both necessary and sufficient conditions for Bayesian exponential consistency", "abstract": "The last decade has seen a remarkable development in the theory of asymptotics of Bayesian nonparametric procedures. Exponential consistency has played an important role in this area. It is known that the condition of $f_0$ being in the Kullback-Leibler support of the prior cannot ensure exponential consistency of posteriors. Many authors have obtained additional sufficient conditions for exponential consistency of posteriors, see, for instance, Schwartz (1965), Barron, Schervish and Wasserman (1999), Ghosal, Ghosh and Ramamoorthi (1999), Walker (2004), Xing and Ranneby (2008). However, given the Kullback-Leibler support condition, less is known about both necessary and sufficient conditions. In this paper we give one type of both necessary and sufficient conditions. As a consequence we derive a simple sufficient condition on Bayesian exponential consistency, which is weaker than the previous sufficient conditions."}
{"category": "Math", "title": "Embedding solenoids in foliations", "abstract": "In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H^1(L;R)= 0 and if the foliation is a product foliation in some saturated open neighbourhood U of L, then there exists a foliation F' on M which is C^1-close to F, and F' has an uncountable set of solenoidal minimal sets contained in U that are pair wise non-homeomorphic. If H^1(L;R) is not 0, then it is known that any sufficiently small perturbation of F contains a saturated product neighbourhood. Thus, our result can be thought of as an instability result complementing the stability results of Reeb, Thurston and Langevin and Rosenberg."}
{"category": "Math", "title": "Invariant Measures on Stationary Bratteli Diagrams", "abstract": "We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we explicitly describe all ergodic probability measures invariant with respect to the tail equivalence relation (or the Vershik map). These measures are completely described by the incidence matrix of the diagram. Since such diagrams correspond to substitution dynamical systems, this description gives an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map."}
{"category": "Math", "title": "Chebyshev Knots", "abstract": "A Chebyshev knot is a knot which admits a parametrization of the form $ x(t)=T_a(t); \\ y(t)=T_b(t) ; \\ z(t)= T_c(t + \\phi), $ where $a,b,c$ are pairwise coprime, $T_n(t)$ is the Chebyshev polynomial of degree $n,$ and $\\phi \\in \\RR .$ Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with $\\phi = 0.$ We also show that every knot is a Chebyshev knot."}
{"category": "Math", "title": "Highest weight categories arising from Khovanov's diagram algebra III: category O", "abstract": "We prove that integral blocks of parabolic category O associated to the subalgebra gl(m) x gl(n) of gl(m+n) are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of 2-Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type A."}
{"category": "Math", "title": "Determination of the biquaternion divisors of zero, including the idempotents and nilpotents", "abstract": "The biquaternion (complexified quaternion) algebra contains idempotents (elements whose square remains unchanged) and nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in the literature, but remain obscure through not being gathered together using modern notation and terminology. Explicit formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in the literature, and we rectify this with the present paper. Using several different representations for biquaternions, we present simple formulae for the idempotents, nilpotents and divisors of zero, and we show that the complex components of a biquaternion divisor of zero must have a sum of squares that vanishes, and that this condition is equivalent to two conditions on the inner product of the real and imaginary parts of the biquaternion, and the equality of the norms of the real and imaginary parts. We give numerical examples of nilpotents, idempotents and other divisors of zero. Finally, we conclude with a statement about the composition of the set of biquaternion divisors of zero, and its subsets, the idempotents and the nilpotents."}
{"category": "Math", "title": "Normal forms for almost non-integrable CR structures", "abstract": "We propose two constructions extending the Chern-Moser normal form to non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One of them translates the Chern-Moser normalization into pure intrinsic setting, whereas the other directly extends the (extrinsic) Chern-Moser normal form by allowing non-CR embeddings that are in some sense \"maximally CR\". One of the main differences with the classical integrable case is the presence of the non-integrability tensor at the same order as the Levi form, making impossible a good quadric approximation - a key tool in the Chern-Moser theory. Partial normal forms are obtained for general almost CR structures of any CR codimension, in particular, for almost-complex structures. Applications are given to the equivalence problem and the Lie group structure of the group of all CR-diffeomorphisms."}
{"category": "Math", "title": "A note on the separability index", "abstract": "In discriminating between objects from different classes, the more separable these classes are the less computationally expensive and complex a classifier can be used. One thus seeks a measure that can quickly capture this separability concept between classes whilst having an intuitive interpretation on what it is quantifying. A previously proposed separability measure, the separability index (SI) has been shown to intuitively capture the class separability property very well. This short note highlights the limitations of this measure and proposes a slight variation to it by combining it with another form of separability measure that captures a quantity not covered by the Separability Index."}
{"category": "Math", "title": "A proof of the rooted tree alternative conjecture", "abstract": "Bonato and Tardif conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or infinite. We prove the analogue of their conjecture for rooted trees. We also discuss the original conjecture for locally finite trees and state some new conjectures."}
{"category": "Math", "title": "A Unified Approach of Parameter Estimation", "abstract": "We introduce a new distance and we use it to parameter estimation purposes. We observe how it operates and we use in its place the usual methods of estimation which we call the methods of the new approach. We realize that we obtain a discretization of the continuous case. Moreover, when it is necessary to consider truncated data nothing is changed in computations."}
{"category": "Math", "title": "Algebraic A-hypergeometric Functions", "abstract": "We formulate and prove a combinatorial criterion to decide if an A-hypergeometric system of differential equations has a full set of algebraic solutions or not. This criterion generalises the so-called interlacing criterion in the case of hypergeometric functions of one variable."}
{"category": "Math", "title": "Katz's middle convolution and Yokoyama's extending operation", "abstract": "We give a concrete relation between Katz's middle convolution and Yokoyama's extension and show the equivalence of both algorithms using these operations for the reduction of Fuchsian systems."}
{"category": "Math", "title": "A simplicial model for proper homotopy types", "abstract": "The singular simplicial set Sing(X) of a space X completely captures its weak homotopy type. We introduce a category of_controlled sets_, yielding _simplicial controlled sets_, such that one can functorially produce a singular simplicial controlled set CSing(MaxCtl(X)) from a locally compact X. We then argue that this CSing(MaxCtl(X)) captures the (weak)_proper_ homotopy type of X. Moreover, our techniques strictly generalize the classical simplicial situation: e.g., one obtains, in a unified way, singular homology with compact supports and (Borel-Moore) singular homology with locally finite supports, as well as the corresponding cohomologies."}
{"category": "Math", "title": "On minimal actions of finite simple groups on homology spheres and Euclidean spaces", "abstract": "We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2,p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces."}
{"category": "Math", "title": "Sobolev spaces on multiple cones", "abstract": "The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $\\RR^n$. The analysis interestingly combines use of Poincar\\'e inequalities and of some Hardy type inequalities."}
{"category": "Math", "title": "The elliptic quantum algebra $U_{q,p}(\\hat{sl_N})$ and its vertex operators", "abstract": "We construct a realization of the elliptic quantum algebra $U_{q,p}(\\hat{sl_N})$ for any given level $k$ in terms of free boson fields and their twisted partners. It can be considered as the elliptic deformation of the Wakimoto realization of the quantum affine algebra $U_{q}(\\hat{sl_N})$. We also construct a family of screening currents, which commute with the currents of $U_{q,p}(\\hat{sl_N})$ up to total q-differences. And we give explicit twisted expressions for the type $I$ and the type $II$ vertex operators of $U_{q,p}(\\hat{sl_N})$ by twisting the known results of the type $I$ vertex operators of the quantum affine algebra $U_{q}(\\hat{sl_N})$ and the new results of the type $II$ vertex operators of $U_{q}(\\hat{sl_N})$ we obtained in this paper."}
{"category": "Math", "title": "Smoothness in Relative Geometry", "abstract": "In \\cite{tva}, Bertrand Toen and Michel Vaquie defined a scheme theory for a closed monoidal category $(C,\\otimes,1)$. In this article, we define a notion of smoothness in this relative (and not necesarilly additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists practically in changing homological finiteness conditions into homotopical ones using Dold-Kahn correspondance. To do this, we provide the category $sC$ of simplicial objects in a monoidal category and all the categories $sA-mod$, $sA-alg$ ($a\\in sComm(C)$) with compatible model structures using the work of Rezk in \\cite{r}. We give then a general notions of smoothness in $sComm(C)$. We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and provide some examples of smooth morphisms in $N-alg$, $Comm(Set)$ and Comm(C)."}
{"category": "Math", "title": "Numerical study of a flow of regular planar curves that develop singularities at finite time", "abstract": "In this paper, we will study the following geometric flow, obtained by Goldstein and Petrich while considering the evolution of a vortex patch in the plane under Euler's equations, X_t = -k_s n - (1/2) k^2 T, with s being the arc-length parameter and k the curvature. Perelman and Vega proved that this flow has a one-parameter family of regular solutions that develop a corner-shaped singularity at finite time. We will give a method to reproduce numerically the evolution of those solutions, as well as the formation of the corner, showing several properties associated to them."}
{"category": "Math", "title": "Covering space theory for directed topology", "abstract": "The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a \"local preorder\" encoding control flow. In the case where time does not loop, the \"locally preordered\" state space splits into causally distinct components. The set of such components often gives a computable invariant of machine behavior. In the general case, no such meaningful partition could exist. However, as we show in this note, the locally preordered geometric realization of a precubical set admits a \"locally monotone\" covering from a state space in which time does not loop. Thus we hope to extend geometric techniques in static program analysis to looping processes."}
{"category": "Math", "title": "Espaces critiques pour le syst\\`eme des equations de Navier-Stokes incompressibles", "abstract": "In this work, we exhibit abstract conditions on a functional space E who insure the existence of a global mild solution for small data in E or the existence of a local mild solution in absence of size constraints for a class of semi-linear parabolic equations, which contains the incompressible Navier-Stokes system as a fundamental example. We also give an abstract criterion toward regularity of the obtained solutions. These conditions, given in terms of Littlewood-Paley estimates for products of spectrally localized elements of $E$, are simple to check in all known cases: Lebesgue, Lorents, Besov, Morrey... spaces. These conditions also apply to non-invariant spaces E and we give full details in the case of some 2-microlocal spaces. The following comments did not show on the first version: This article was written around 1998-99 and never published, because at that time, Koch and Tataru announced their result on well-posedness of Navier-stokes equations with initial data in $BMO^{-1}$. We believe though that some results and counterexamples here are of independent interest and we make them available electronically."}
{"category": "Math", "title": "Periodic Geodesics and Geometry of Compact Lorentzian Manifolds with a Killing Vector Field", "abstract": "We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is never vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold $M$ admits a Lorentzian metric with a never vanishing Killing vector field which is timelike somewhere if and only if $M$ admits a smooth circle action without fixed points."}
{"category": "Math", "title": "Independence Complexes of Cylinders Constructed from Square and Hexagonal Grid Graphs", "abstract": "Fendley, Schoutens and van Eerten [Fendley et al., J. Phys. A: Math. Gen., 38 (2005), pp. 315-322] studied the hard square model at negative activity. They found analytical and numerical evidence that the eigenvalues of the transfer matrix with periodic boundary were all roots of unity. They also conjectured that for an m times n square grid, with doubly periodic boundary, the partition function is equal to 1 when m and n are relatively prime. These conjectures were later proven by Jonsson [Jonsson, Electronic J. Combin., 13(1) (2006), R67]. There, it was also noted that the cylindrical case seemed to have interesting properties when the circumference of the cylinder is odd. In particular, when 3 is a divisor of both the circumference and the width of the cylinder minus 1, the partition function is -2. Otherwise, it is equal to 1. In this paper, we investigate the hard square and hard hexagon models at activity -1, with single periodic boundary, i.e, cylindrical identifications, using both topological and combinatorial techniques. We compute the homology groups of the associated independence complex for small sizes and suggest a matching which, we believe, with further analysis could help solve the conjecture."}
{"category": "Math", "title": "Homological mirror symmetry for the genus two curve", "abstract": "Katzarkov has proposed a generalization of Kontsevich's mirror symmetry conjecture, covering some varieties of general type. We prove a version of this conjecture in the simplest example, relating the Fukaya category of a genus two curve to the category of Landau-Ginzburg branes on a certain singular rational surface."}
{"category": "Math", "title": "Representations of the Renner Monoid", "abstract": "We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalizes the Munn-Solomon representation theory of rook monoids to any Renner monoids. The type map and polytope associated with reductive monoids play a crucial role in our work. It turns out that the irreducible representations of certain parabolic subgroups of the Weyl groups determine the complete set of irreducible representations of the Renner monoids. An analogue of the Munn-Solomon formula for calculating the character of the Renner monoids, in terms of the characters of the parabolic subgroups, is shown."}
{"category": "Math", "title": "Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems", "abstract": "We are concerned with viscous profiles (travelling waves and steady solutions) for mixed hyperbolic-parabolic systems in one space variable. For a class of systems including the compressible Navier Stokes equation, these profiles satisfy a singular ordinary differential equation in the form \\label{e:ab} dU / dt = F(U)/ z (U) . Here U takes values in $R^d$ and $F: R^d \\to R^d$ is a regular function. The real valued function $z (U)$ is as well regular, but the equation is singular because $z (U)$ can attain the value 0. We focus on a small enough neighbourhood of a point $\\bar U$ satisfying $F(\\bar U) = \\vec 0$, $z (\\bar U) =0$. From the point of view of the applications to the study of hyperbolic-parabolic systems this means restricting to systems with small total variation. We discuss how to extend the notions of center manifold and of uniformly stable manifold. Also, we give conditions ensuring that if $z (U) \\neq 0$ at $t=0$ then $z (U) \\neq 0$ at every $t$. We provide an example showing that if $z(U)$ becomes zero in finite time then in general the solution $U$ of equation \\eqref{e:ab} is not continuously differentiable."}
{"category": "Math", "title": "Calculus with a Quaternionic Variable", "abstract": "Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses obstacles for the usual manipulations of calculus; but we show in this paper how many of those obstacles can be overcome. The surprising result is that the first order term in the expansion of F(x+delta) is a compact formula involving both F'(x) and [F(x) - F(x*)]/(x-x*). This advance in the differential calculus for quaternionic variables also leads us to some progress in studying integration."}
{"category": "Math", "title": "A new determinantal formula for the classical discriminant", "abstract": "According to several classical results by Bezout, Sylvester, Cayley, and others, the classical discriminant D_n of degree n polynomials may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f. However, all of the determinantal formulae for D_n appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivial in the sense of having the same cokernel as the 1 x 1 matrix (D_n). In this paper, we construct an explicit non-classical formula: the presentation matrix of the open swallowtail first studied by Arnol'd and Givental. We study the properties of this formula, contrasting them with the properties of the classical formulae."}
{"category": "Math", "title": "Homology representations arising from the half cube, II", "abstract": "In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes of the $n$-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least $k$, and we proved that the homology of such a subcomplex is concentrated in degree $k-1$. This homology group supports a natural action of the Coxeter group $W(D_n)$ of type $D$. In this paper, we explicitly determine the characters (over ${\\Bbb C}$) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group $S_n$ by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of $\\sym_n$ agree (over ${\\Bbb C}$) with the representations of $\\sym_n$ on the $(k-2)$-nd homology of the complement of the $k$-equal real hyperplane arrangement."}
{"category": "Math", "title": "Mixed type multiple orthogonal polynomials for two Nikishin systems", "abstract": "We study the logarithmic and ratio asymptotic of linear forms constructed from a Nikishin system which satisfy orthogonality conditions with respect to a system of measures generated from another Nikishin system. This construction combines type I and type II multiple orthogonal polynomials. The logarithmic asymptotic of the linear forms is expressed in terms of the extremal solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotic is described by means of a conformal representation of an appropriate Riemann surface of genus zero onto the extended complex plane."}
{"category": "Math", "title": "A novel changepoint detection algorithm", "abstract": "We propose an algorithm for simultaneously detecting and locating changepoints in a time series, and a framework for predicting the distribution of the next point in the series. The kernel of the algorithm is a system of equations that computes, for each index i, the probability that the last (most recent) change point occurred at i. We evaluate this algorithm by applying it to the change point detection problem and comparing it to the generalized likelihood ratio (GLR) algorithm. We find that our algorithm is as good as GLR, or better, over a wide range of scenarios, and that the advantage increases as the signal-to-noise ratio decreases."}
{"category": "Math", "title": "The solar Julia sets of basic quadratic Cremer polynomials", "abstract": "In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression."}
{"category": "Math", "title": "On the necessity of Reidemeister move 2 for simplifying immersed planar curves", "abstract": "In 2001, Oestlund conjectured that Reidemeister moves 1 and 3 are sufficient to describe a homotopy from any generic immersion from the circle into the plane to the standard embedding of the circle. We show that this conjecture is false."}
{"category": "Math", "title": "The f-vector of the descent polytope", "abstract": "For a positive integer n and a subset S of [n-1], the descent polytope DP_S is the set of points x_1, ..., x_n in the n-dimensional unit cube [0,1]^n such that x_i >= x_{i+1} for i in S and x_i <= x_{i+1} otherwise. First, we express the f-vector of DP_S as a sum over all subsets of [n-1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5, ...}. We derive a generating function for the f-polynomial F_S(t) of DP_S, written as a formal power series in two non-commuting variables with coefficients in Z[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes."}
{"category": "Math", "title": "The structure of thin Lie algebras up to the second diamond", "abstract": "Thin Lie algebras are Lie algebras L, graded over the positive integers, with all homogeneous components of dimension at most two, and satisfying a more stringent but natural narrowness condition modeled on an analogous one for pro-p groups. The two-dimensional homogeneous components of L, which include that of degree one, are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond. Specifically, if L_k is the second diamond of L, then the quotient L/L^k is a graded Lie algebras of maximal class. In characteristic not two, L/L^k is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values. The quotient L/L^k need not be metabelian in characteristic two. We describe here all the possibilities for L/L^k up to isomorphism. In particular, we prove that k+1 equals a power of two."}
{"category": "Math", "title": "A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings", "abstract": "We prove that a Schur function of rectangular shape $(M^n)$ whose variables are specialized to $x_1,x_1^{-1},...,x_n,x_n^{-1}$ factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at $-x_1,...,-x_n$, if $M$ is even, while it factorizes into a product of a symplectic character and an even orthogonal character, both of rectangular shape, if $M$ is odd. It is furthermore shown that the first factorization implies a factorization theorem for rhombus tilings of a hexagon, which has an equivalent formulation in terms of plane partitions. A similar factorization theorem is proven for the sum of two Schur functions of respective rectangular shapes $(M^n)$ and $(M^{n-1})$."}
{"category": "Math", "title": "q-analog of tableau containment", "abstract": "We prove a $q$-analog of the following result due to McKay, Morse and Wilf: the probability that a random standard Young tableau of size $n$ contains a fixed standard Young tableau of shape $\\lambda\\vdash k$ tends to $f^{\\lambda}/k!$ in the large $n$ limit, where $f^{\\lambda}$ is the number of standard Young tableaux of shape $\\lambda$. We also consider the probability that a random pair $(P,Q)$ of standard Young tableaux of the same shape contains a fixed pair $(A,B)$ of standard Young tableaux."}
{"category": "Math", "title": "Masas and Bimodule Decompositions of $\\rm{II}_{1}$ Factors", "abstract": "The measure-multiplicity-invariant for masas in $\\rm{II}_{1}$ factors was introduced in \\cite{MR2261688} to distinguish masas that have the same Puk\\'{a}nszky invariant. In this paper we study the measure class in the measure-multiplicity-invariant. This is equivalent to studying the standard Hilbert space as an associated bimodule. We characterize the type of any masa depending on the left-right-measure using Baire category methods (selection principle of Jankov and von Neumann). We present a second proof of Chifan's result on normalisers and a measure theoretic proof of the equivalence of weak asymptotic homomorphism property (WAHP) and singularity that appeared in \\cite{MR2417416}."}
{"category": "Math", "title": "On the integers of the form $p^2+b^2+2^n$ and $b_1^2+b_2^2+2^{n^2}$", "abstract": "We prove that the sumset {p^2+b^2+2^n: p is prime and b,n\\in N} has positive lower density. We also construct a residue class with odd modulo, which contains no integer of the form p^2+b^2+2^n. And similar results are established for the sumset {b_1^2+b_2^2+2^{n^2}: b_1,b_2,n\\in N}."}
{"category": "Math", "title": "Only rational homology spheres admit $\\Omega(f)$ to be union of DE attractors", "abstract": "If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $\\Omega(f)$ consists of finitely many orientable $(\\pm)$ attractors derived from expanding maps, then $M$ must be a rational homology sphere; moreover all those attractors are of topological dimension $n-2$. Expanding maps are expanding on (co)homologies."}
{"category": "Math", "title": "$L^{p}$ Boundedness of Riesz transform related to Schr\\\"odinger operators on a manifold", "abstract": "We establish various $L^{p}$ estimates for the Schr\\\"odinger operator $-\\Delta+V$ on Riemannian manifolds satisfying the doubling property and a Poincar\\'e inequality, where $\\Delta $ is the Laplace-Beltrami operator and $V$ belongs to a reverse H\\\"{o}lder class. At the end of this paper we apply our result on Lie groups with polynomial growth."}
{"category": "Math", "title": "Some geometrical aspects of control points for toric patches", "abstract": "We use ideas from algebraic geometry and dynamical systems to explain some ways that control points influence the shape of a B\\'ezier curve or patch. In particular, we establish a generalization of Birch's Theorem and use it to deduce sufficient conditions on the control points for a patch to be injective. We also explain a way that the control points influence the shape via degenerations to regular control polytopes. The natural objects of this investigation are irrational patches, which are a generalization of Krasauskas's toric patches, and include B\\'ezier and tensor product patches as important special cases."}
{"category": "Math", "title": "On minimal prime graphs and posets", "abstract": "We show that there are four infinite prime graphs such that every infinite prime graph with no infinite clique embeds one of these graphs. We derive a similar result for infinite prime posets with no infinite chain or no infinite antichain."}
{"category": "Math", "title": "Claw-freeness, 3-homogeneous subsets of a graph and a reconstruction problem", "abstract": "We describe $Forb\\{K_{1,3}, \\overline {K_{1,3}}\\}$, the class of graphs $G$ such that $G$ and its complement $ \\overline{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths or isolated vertices, and of the complements of these graphs. Considering the hypergraph ${\\mathcal H} ^{(3)}(G)$ made of the 3-element subsets of the vertex set of a graph $G$ on which $G$ induces a clique or an independent subset, we deduce from above a description of the Boolean sum $G\\dot{+}G'$ of two graphs $G$ and $G'$ giving the same hypergraph. We indicate the role of this latter description in a reconstruction problem of graphs up to complementation."}
{"category": "Math", "title": "On scattered posets with finite dimension", "abstract": "We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains."}
{"category": "Math", "title": "Artinian and non-artinian local cohomology modules", "abstract": "Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\\fa$ and $\\fb$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\\fa, \\fb$, $\\fa\\cap\\fb$ and $\\fa+ \\fb$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen-Macaulay if there exists an ideal $\\fa$ such that all local cohomology modules of $M$ with respect to $\\fa$ have finite lengths. Also, when $r$ is an integer such that $0\\leq r< \\dim_R(M)$, any maximal element $\\fq$ of the non-empty set of ideals $\\{\\fa$ : $\\H_\\fa^i(M)$ is not artinian for some $i$, $i\\geq r$$\\}$ is a prime ideal and that all Bass numbers of $\\H_\\fq^i(M)$ are finite for all $i\\geq r$."}
{"category": "Math", "title": "Hermitian Symmetric Spaces of Tube Type and Multivariate Meixner-Pollaczek Polynomials", "abstract": "Harmonic analysis on Hermitian symmetric spaces of tube type is a natural framework for introducing multivariate Meixner-Pollaczek polynomials. Their main properties are established in this setting: generating and determinantal formulae, difference equations. As an application we consider the problem of evaluating moments related to a multivariate Barnes type integral involving the Harish-Chandra $c$-function of a symmetric cone."}
{"category": "Math", "title": "Comparison of volumes of convex bodies in real, complex, and quaternionic spaces", "abstract": "The classical Busemann-Petty problem (1956) asks, whether origin-symmetric convex bodies in $\\mathbb {R}^n$ with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if $n\\le 4$ and negative if $n>4$. The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified exposition of this circle of problems in real, complex, and quaternionic $n$-dimensional spaces. All cases are treated simultaneously. In particular, we show that the Busemann-Petty problem in the quaternionic $n$-dimensional space has an affirmative answer if and only if $n =2$. The method relies on the properties of cosine transforms on the unit sphere. Possible generalizations are discussed."}
{"category": "Math", "title": "On the Taylor Coefficients of the Hurwitz Zeta Function", "abstract": "We find a representation for the Maclaurin coefficients of the Hurwitz zeta-function in terms of semi-convergent series involving the Bernoulli polynomials and the Stirling numbers of the first kind. In particular, this gives a representation for the coefficients of the Riemann zeta function. Our main instrument is a certain series transformation formula. A similar result is proved also for the Maclaurin coefficients of the Lerch zeta function."}
{"category": "Math", "title": "On (strongly) Gorenstein (semi)hereditary rings", "abstract": "In this paper, we introduce and study the rings of Gorenstein homological dimensions small or equal than 1, which we call Gorenstein (semi)hereditary rings, specially particular cases of these rings, which we call strongly Gorenstein (semi)hereditary rings."}
{"category": "Math", "title": "Equations resolving a conjecture of Rado on partition regularity", "abstract": "A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k-1)-regular but not k-regular. We prove this conjecture by showing that the equation $\\sum_{i=1}^{k-1} \\frac{2^i}{2^i-1} x_i = (-1 + \\sum_{i=1}^{k-1} \\frac{2^i}{2^i-1}) x_0$ has this property. This conjecture is part of problem E14 in Richard K. Guy's book \"Unsolved problems in number theory\", where it is attributed to Rado's 1933 thesis, \"Studien zur Kombinatorik\"."}
{"category": "Math", "title": "Power operations for Morava E-theory of height 2 at the prime 2", "abstract": "Explicit calculations of the algebraic theory of power operations for a specific Morava E-theory spectrum are given, without detailed proofs."}
{"category": "Math", "title": "A streamlined proof of Goodwillie's n-excisive approximation", "abstract": "We give a shorter proof of Lemma 1.9 from Goodwillie, \"Calculus III\", which is the key step in proving that the construction P_nF gives an n-excisive functor."}
{"category": "Math", "title": "Structural results for free Araki-Woods factors and their continuous cores", "abstract": "We show that for any type ${\\rm III_1}$ free Araki-Woods factor $\\mathcal{M} = \\Gamma(H_\\R, U_t)\"$ associated with an orthogonal representation $(U_t)$ of $\\R$ on a separable real Hilbert space $H_\\R$, the continuous core $M = \\mathcal{M} \\rtimes_\\sigma \\R$ is a semisolid ${\\rm II_\\infty}$ factor, i.e. for any non-zero finite projection $q \\in M$, the ${\\rm II_1}$ factor $qMq$ is semisolid. If the representation $(U_t)$ is moreover assumed to be mixing, then we prove that the core $M$ is solid. As an application, we construct an example of a non-amenable solid ${\\rm II_1}$ factor $N$ with full fundamental group, i.e. $\\mathcal{F}(N) = \\R^*_+$, which is not isomorphic to any interpolated free group factor $L(\\F_t)$, for $1 < t \\leq +\\infty$."}
{"category": "Math", "title": "The Dirichlet problem for the Bellman equation at resonance", "abstract": "We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and positively homogeneous. Examples of such operators include the Hamilon-Jacobi-Bellman operator and the Pucci extremal operators. In the case that the two principal half-eigenvalues are not equal, we show that the measures which achieve the minimum in this formula provide a partial characterization of the solvability of the corresponding Dirichlet problem at resonance."}
{"category": "Math", "title": "Bounded Martin's Maximum with Many Witnesses", "abstract": "We study a strengthening of Bounded Martin's Maximum which asserts that if a \\Sigma_1 fact holds of \\omega_2^V in a stationary set preserving extension then it holds in V for a stationary set of ordinals less than \\omega_2. We show that this principle implies Global Projective Determinacy, and therefore does not hold in the \\mathbb{P}_{max} model for \\mathsf{BMM}, but that the restriction of this principle to forcings which render \\omega_2^V countably cofinal does hold in the \\mathsf{BMM} model, though it is not a consequence of \\mathsf{BMM}."}
{"category": "Math", "title": "Feedback Differential Invariants", "abstract": "The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found."}
{"category": "Math", "title": "Actions of the derived group of a maximal unipotent subgroup on $G$-varieties", "abstract": "Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. We study actions of $U'$ on affine $G$-varieties. First, we consider the algebra of $U'$ invariants on $G/U$. We prove that $k[G/U]^{U'}$ is a polynomial algebra of Krull dimension $2r$, where $r=rk(G)$. A related result is that, for any simple finite-dimensional $G$-module $V$, $V^{U'}$ is a cyclic $U/U'$-module. Second, we study \"symmetries\" of Poincare series for $U'$-invariants on affine conical $G$-varieties. Third, we obtain a classification of simple $G$-modules $V$ with polynomial algebras of $U'$-invariants (for $G$ simple)."}
{"category": "Math", "title": "A Unified Approach to Distance-Two Colouring of Graphs on Surfaces", "abstract": "In this paper we introduce the notion of $\\Sigma$-colouring of a graph $G$: For given subsets $\\Sigma(v)$ of neighbours of $v$, for every $v\\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that appear together in some $\\Sigma(v)$ receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptotic versions of Wegner's and Borodin's Conjecture on the planar version of these two colourings. Using a recent approach of Havet et al., we reduce the problem to edge-colouring of multigraphs, and then use Kahn's result that the list chromatic index is close to the fractional chromatic index. Our results are based on a strong structural lemma for graphs embeddable in a fixed surface, which also implies that the size of a clique in the square of a graph of maximum degree $\\Delta$ embeddable in some fixed surface is at most $\\frac32\\,\\Delta$ plus a constant."}
{"category": "Math", "title": "Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order", "abstract": "The problem of local feedback equivalence for 1-dimensional control systems of the 1-st order is considered. The algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found."}
{"category": "Math", "title": "Automorphism invariance and identities", "abstract": "If an outer (multilinear) commutator identity holds in a large subgroup of a group, then it holds also in a large characteristic subgroup. Similar assertions are valid for algebras and their ideals or subspaces. Varying the meaning of the word \"large\", we obtain many interesting facts. These results cannot be extended to arbitrary (non-multilinear) identities. As an application, we give a sharp estimate for the `virtual derived length' of (virtually solvable)-by-(virtually solvable) groups."}
{"category": "Math", "title": "Semigroup analysis of structured parasite populations", "abstract": "Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral theory of semigroups to establish conditions for the existence of a positive equilibrium solution of our model. Then, we formulate conditions that guarantee that the linearised system is governed by a positive quasicontraction semigroup on the biologically relevant state space. We also show that the governing linear semigroup is eventually compact, hence growth properties of the semigroup are determined by the spectrum of its generator. In the case of a separable fertility function, we deduce a characteristic equation, and investigate the stability of equilibrium solutions in the general case using positive perturbation arguments."}
{"category": "Math", "title": "Quantization of q-Hamiltonian SU(2)-spaces", "abstract": "We explain how to define the quantization of q-Hamiltonian SU(2)-spaces as push-forwards in twisted K-homology, and prove a `quantization commutes with reduction' theorem for this setting. As applications, we show how the Verlinde formulas for flat SU(2) or SO(3) bundles are obtained by localization in twisted K-homology."}
{"category": "Math", "title": "Hierarchical size-structured populations: The linearized semigroup approach", "abstract": "In the present paper we analyze the linear stability of a hierarchical size-structured population model where the vital rates (mortality, fertility and growth rate) depend both on size and a general functional of the population density (\"environment\"). We derive regularity properties of the governing linear semigroup, implying that linear stability is governed by a dominant real eigenvalue of the semigroup generator, which arises as a zero of an associated characteristic function. In the special case where neither the growth rate nor the mortality depend on the environment, we explicitly calculate the characteristic function and use it to formulate simple conditions for the linear stability of population equilibria. In the general case we derive a dissipativity condition for the linear semigroup, thereby characterizing exponential stability of the steady state."}
{"category": "Math", "title": "Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback", "abstract": "In this work we consider a size-structured cannibalism model with the model ingredients (fertility, growth, and mortality rate) depending on size (ranging over an infinite domain) and on a general function of the standing population (environmental feedback). Our focus is on the asymptotic behavior of the system, in particular on the effect of cannibalism on the long-term dynamics. To this end, we formally linearize the system about steady state and establish conditions in terms of the model ingredients which yield uniform exponential stability of the governing linear semigroup. We also show how the point spectrum of the linearized semigroup generator can be characterized in the special case of a separable attack rate and establish a general instability result. Further spectral analysis allows us to give conditions for asynchronous exponential growth of the linear semigroup."}
{"category": "Math", "title": "Decomposition of D-modules over a hyperplane arrangement in the plane", "abstract": "We consider the D-module defined as the push-forward of a rank one linear system on the complement of a central plane hyperplane arrangement, and calculate its decomposition series, using algebraic calculations in the Weyl algebra."}
{"category": "Math", "title": "Singularities of Hinge Structures", "abstract": "Motivated by the hinge structure present in protein chains and other molecular conformations, we study the singularities of certain maps associated to body-and-hinge and panel-and-hinge chains. These are sequentially articulated systems where two consecutive rigid pieces are connected by a hinge, that is, a codimension two axis. The singularities, or critical points, correspond to a dimensional drop in the linear span of the axes, regarded as points on a Grassmann variety in its Pl\\\"{u}cker embedding. These results are valid in arbitrary dimension. The three dimensional case is also relevant in robotics."}
{"category": "Math", "title": "Demystification of Taylor,Laurent coefficients of Lerch,Hurwitz Zeta functions and Dirichlet L-Function at Unity and Zero and their Bounds", "abstract": "Using elementary methods,we obtain simple,explicit expressions and bounds of higher order derivatives of Hurwitz zeta function and consequently those of Dirichlet L-function and also,of Lerch's Zeta function at unity (and at Zero too)and also obtain their interrelations.We also state elementary complete forms of approximate functional equations of Hurwitz zeta function and Dirichlet L-function in the critical strip."}
{"category": "Math", "title": "Extremal Configurations of Hinge Structures", "abstract": "We study body-and-hinge and panel-and-hinge chains in R^d, with two marked points: one on the first body, the other on the last. For a general chain, the squared distance between the marked points gives a Morse-Bott function on a torus configuration space. Maximal configurations, when the distance between the two marked points reaches a global maximum, have particularly simple geometrical characterizations. The three-dimensional case is relevant for applications to robotics and molecular structures."}
{"category": "Math", "title": "Ascending and descending regions of a discrete Morse function", "abstract": "We present an algorithm which produces a decomposition of a regular cellular complex with a discrete Morse function analogous to the Morse-Smale decomposition of a smooth manifold with respect to a smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of the critical cells in a finite number of steps, and that, after a finite number of subdivisions, all the regions are topological discs. The efficiency of the algorithm is discussed and its performance on several examples is demonstrated."}
{"category": "Math", "title": "On leafwise conformal diffeomorphisms", "abstract": "For every diffeomorphism $\\varphi:M\\to N$ between 3--dimensional Riemannian manifolds $M$ and $N$ there are in general locally two 2--dimensional distributions $D_{\\pm}$ such that $\\varphi$ is conformal on both of them. We state necessary and sufficient conditions for a distribution to be one of $D_{\\pm}$. These are algebraic conditions expressed in terms of the self-adjoint and positive definite operator $(\\varphi_{\\ast})^*\\varphi_{\\ast}$. We investigate integrability condition of $D_+$ and $D_-$. We also show that it is possible to choose coordinate systems in which leafwise conformal diffeomorphism is holomorphic on leaves."}
{"category": "Math", "title": "Multiple equivalent matings with the aeroplane polynomial", "abstract": "We produce arbitrarily large equivalence classes of matings with the aeroplane polynomial. These are obtained by a slight generalisation of the technique of proof of a similar result for Wittner captures."}
{"category": "Math", "title": "Constructing knot tunnels using giant steps", "abstract": "This is the first of three papers that refine and extend portions of our earlier preprint, \"Depth of a knot tunnel.\" Together, they rework the entire preprint. H. Goda, M. Scharlemann, and A. Thompson described a general construction of all tunnels of all tunnel number 1 knots using \"tunnel moves\". We apply the theory that we introduced in \"The tree of knot tunnels\" to study this construction. In particular, we use it to calculate the number of distinct minimal sequences of tunnel moves that can produce a given tunnel. As a consequence, we see that for a sparse infinite set of tunnels, the minimal sequence is unique, but generically a tunnel will have many such constructions."}
{"category": "Math", "title": "A lattice in more than two Kac--Moody groups is arithmetic", "abstract": "Let $\\Gamma$ be an irreducible lattice in a product of n infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n is at least 3, then each Kac-Moody groups is in fact a simple algebraic group over a local field and $\\Gamma$ is an arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n is at least 2: either $\\Gamma$ is an S-arithmetic (hence linear) group, or it is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout."}
{"category": "Math", "title": "A simple proof of exponential decay in the two dimensional percolation model", "abstract": "Kesten showed the exponential decay of percolation probability in the subcritical phase for the two-dimensional percolation model. This result implies his celebrated computation that $p_c=0.5$ for bond percolation in the square lattice, and site percolation in the triangular lattice, respectively. In this paper, we present a simpler proof for Kesten's theorem."}
{"category": "Math", "title": "Model-Based Clustering using multi-allelic loci data with loci selection", "abstract": "We propose a Model-Based Clustering (MBC) method combined with loci selection using multi-allelic loci genetic data. The loci selection problem is regarded as a model selection problem and models in competition are compared with the Bayesian Information Criterion (BIC). The resulting procedure selects the subset of clustering loci, the number of clusters, estimates the proportion of each cluster and the allelic frequencies within each cluster. We prove that the selected model converges in probability to the true model under a single realistic assumption as the size of the sample tends to infinity. The proposed method named MixMoGenD (Mixture Model using Genetic Data) was implemented using c++ programming language. Numerical experiments on simulated data sets was conducted to highlight the interest of the proposed loci selection procedure."}
{"category": "Math", "title": "Cabling sequences of tunnels of torus knots", "abstract": "This is the second of three papers that refine and extend portions of our earlier preprint, \"The depth of a knot tunnel.\" Together, they rework the entire preprint. The theory of tunnel number 1 knots that we introduced in \"The tree of knot tunnels\" yields a parameterization in which each tunnel is described uniquely by a finite sequence of rational parameters and a finite sequence of 0's and 1's, that together encode a procedure for constructing the knot and tunnel. In this paper we calculate these invariants for all tunnels of torus knots."}
{"category": "Math", "title": "Optimal sequential testing of two simple hypotheses in presence of control variables", "abstract": "Suppose that at any stage of a statistical experiment a control variable $X$ that affects the distribution of the observed data $Y$ can be used. The distribution of $Y$ depends on some unknown parameter $\\theta$, and we consider the classical problem of testing a simple hypothesis $H_0: \\theta=\\theta_0$ against a simple alternative $H_1: \\theta=\\theta_1$ allowing the data to be controlled by $X$, in the following sequential context. The experiment starts with assigning a value $X_1$ to the control variable and observing $Y_1$ as a response. After some analysis, we choose another value $X_2$ for the control variable, and observe $Y_2$ as a response, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favour of $H_0$ or $H_1$ is to be taken. In this article, our aim is to characterize the structure of optimal sequential procedures, based on this type of data, for testing a simple hypothesis against a simple alternative."}
{"category": "Math", "title": "Tunnel leveling, depth, and bridge numbers", "abstract": "This is the third of three papers that refine and extend portions of our earlier preprint, \"The depth of a knot tunnel.\" Together, they rework the entire preprint. In this paper, we use the theory of tunnel number 1 knots that we introduced in \"The tree of knot tunnels\" to strengthen the Tunnel Leveling Theorem of H. Goda, M. Scharlemann, and A. Thompson. This yields considerable information about bridge numbers of tunnel number 1 knots. In particular, we calculate the minimum bridge number of a knot as a function of the maximum depth invariant d of its tunnels. The growth of this value is on the order of (1+\\sqrt{2})^d. We also find the maximum bridge number as a function of the number of cabling constructions needed to produce the tunnel, showing in particular that the maximum bridge number of a knot produced by n cabling constructions is the (n+2)nd Fibonacci number. Finally, we examine the special case of the \"middle\" tunnels of torus knots."}
{"category": "Math", "title": "A large deviations bound for the Teichmuller flow", "abstract": "Large deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. The method is that of the first author and follows that of L.-S. Young. A corollary of the main results is a large deviation bound for the Teichm\\\"uller flow on the moduli space of abelian differentials, which extends earlier work of J. Athreya."}
{"category": "Math", "title": "Factorization theorems for dominated polynomials", "abstract": "In this note we prove that the factorization theorem for dominated polynomials previously proved by the authors is equivalent to an alternative factorization scheme that uses classical linear techniques and a linearization process. However, this alternative scheme is shown not to be satisfactory until the equivalence is proved."}
{"category": "Math", "title": "Steenrod homotopy", "abstract": "Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homotopy coincides with strong shape. We attempt to simplify foundations of the theory and to clarify and improve some of its major results. Using geometric tools such as Milnor's telescope compactification, comanifolds (=mock bundles) and the Pontryagin-Thom Construction, we obtain new simple proofs of results by Barratt-Milnor; Cathey; Dydak-Segal; Eda-Kawamura; Edwards-Geoghegan; Fox; Geoghegan-Krasinkiewicz; Jussila; Krasinkiewicz-Minc; Mardesic; Mittag-Leffler/Bourbaki; and of three unpublished results by Shchepin. An error in Lisitsa's proof of the \"Hurewicz theorem in Steenrod homotopy\" is corrected. It is shown that over compacta, R.H.Fox's overlayings are same as I.M.James' uniform covering maps. Other results include: - A morphism between inverse sequences of countable (possibly non-abelian) groups that induces isomorphisms on inverse and derived limits is invertible in the pro-category. This implies the \"Whitehead theorem in Steenrod homotopy\", thereby answering two questions of A.Koyama. - If X is an LC_{n-1} compactum, n>0, its n-dimensional Steenrod homotopy classes are representable by maps S^n\\to X, provided that X is simply connected. The assumption of simply-connectedness cannot be dropped by a well-known example of Dydak and Zdravkovska. - A connected compactum is Steenrod connected (=pointed 1-movable) iff every its uniform covering space has countably many uniform connected components."}
{"category": "Math", "title": "Reducibility of Covers of AFT shifts", "abstract": "In this paper we show that the reducibility structure of several covers of sofic shifts is a flow invariant. In addition, we prove that for an irreducible subshift of almost finite type the left Krieger cover and the past set cover are reducible. We provide an example which shows that there are non almost finite type shifts which have reducible left Krieger covers. As an application we show that the Matsumoto algebra of an irreducible, strictly sofic shift of almost finite type is not simple."}
{"category": "Math", "title": "On vertex, edge, and vertex-edge random graphs", "abstract": "We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertex-edge random graphs. Edge random graphs are Erdos-Renyi random graphs, vertex random graphs are generalizations of geometric random graphs, and vertex-edge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in the edges, in the vertices, or in both. We show that vertex-edge random graphs, ostensibly the most general of the three models, can be approximated arbitrarily closely by vertex random graphs, but that the two categories are distinct."}
{"category": "Math", "title": "Problems on Minkowski sums of convex lattice polytopes", "abstract": "This paper was submitted to the Oberwolfach Conference \"Combinatorial Convexity and Algebraic Geometry\", October 1997. Let $M={\\mathbb Z}^r$. For convex lattice polytopes $P,P'$ in ${\\mathbb R}^r$, when is $(M \\cap P)+ (M \\cap P') = M \\cap (P + P')$? Without any additional condition, the equality obviously does not hold. When the pair $(M,P)$ corresponds to a complex projective toric variety $X$ and an ample divisor $D$ on $X$, it is reasonable to assume that $P'$ corresponds to an ample (or, more generally, a nef) divisor $D'$ on the same $X$. Then the question correspons to the surjectivity of the canonical map \\[ H^0(X,{\\mathcal O}_X(D))\\otimes H^0(X,{\\mathcal O}_X(D'))\\to H^0(X,{\\mathcal O}_X(D+D')).\\] When $X$ is nonsingular, the map is hoped to be surjective, but this remains to be an open question after more than ten years. The paper explores various variations on the question in terms of toric geometry."}
{"category": "Math", "title": "Polynomial Representation of $E_7$ and Its Combinatorial and PDE Implications", "abstract": "In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of $E_7$ into a sum of irreducible submodules. Moreover, we obtain a combinatorial identity, saying that the dimensions of certain irreducible modules of $E_7$ are correlated by the binomial coefficients of fifty-five. Furthermore, we prove that two families of irreducible submodules with three integral parameters are solutions of the fundamental invariant differential operator corresponding to Cartan's unique quartic $E_7$ invariant."}
{"category": "Math", "title": "Index theory and partitioning by enlargeable hypersurfaces", "abstract": "In this paper we state and prove a higher index theorem for an odd-dimensional connected spin riemannian manifold $(M,g)$ which is partitioned by an oriented closed hypersurface $N$. This index theorem generalizes a theorem due to N. Higson and J. Roe in the context of Hilbert modules. Then we apply this theorem to prove that if $N$ is area-enlargeable and if there is a smooth map from $M$ into $N$ such that its restriction to $N$ has non-zero degree then the the scalar curvature of $g$ cannot be uniformly positive."}
{"category": "Math", "title": "Periodic orbits and chaos in fast-slow systems with Bogdanov-Takens type fold points", "abstract": "The existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast-slow type having Bogdanov-Takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow-up method. In particular, the blow-up method is effectively used for analyzing the flow near the Bogdanov-Takens type fold point in order to show that a slow manifold near the fold point is extended along the Boutroux's tritronqu\\'{e}e solution of the first Painlev\\'{e} equation in the blow-up space."}
{"category": "Math", "title": "Geography of non-formal symplectic and contact manifolds", "abstract": "Let (m,b) a pair of natural numbers. For m even (resp. m odd and b greater than or equal to 2) we show that if there is an m-dimensional non-formal compact oriented manifold whose first Betti number equals b, there is also a symplectic (resp. contact) manifold with these properties."}
{"category": "Math", "title": "Canonical Bases of Borcherds-Cartan Type", "abstract": "We study the canonical basis for the negative part of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras associated to two different matrices satisfying certain conditions may coincide. We show that the canonical bases coincide provided that the algebras coincide. We also answer partially a question by Lusztig."}
{"category": "Math", "title": "Solymosi's multiplicative energy bound for complex numbers", "abstract": "We extend the recent Solymosi's sum-product estimate for reals to the complex case."}
{"category": "Math", "title": "Strict 2-Groups are Crossed Modules", "abstract": "The 2-categories of strict 2-groups and crossed modules are introduced and their 2-equivalence is made explicit."}
{"category": "Math", "title": "Commutativity and Ideals in Category Crossed Products", "abstract": "In order to simultaneously generalize matrix rings and group graded crossed products, we introduce category crossed products. For such algebras we describe the center and the commutant of the coefficient ring. We also investigate the connection between on the one hand maximal commutativity of the coefficient ring and on the other hand nonemptyness of intersections of the coefficient ring by nonzero twosided ideals."}
{"category": "Math", "title": "Moduli of Bridgeland semistable objects on $\\mathbb{P}^2$", "abstract": "We give another proof of Le Potier's result and some variants on moduli spaces of semistable sheaves on the projective plane, using the Bridgeland stability conditions. As an application we study the wall-crossing phenomena of the Hilbert schemes of points on the projective plane."}
{"category": "Math", "title": "The cluster complex of an hereditary artin algebra", "abstract": "We show that the cluster complex of an arbitrary hereditary artin algebra has the structure of an abstract simplicial polytope. In particular, the cluster-tilting objects form one equivalence class under mutation."}
{"category": "Math", "title": "Infinite sums of additive unstable Adams operations and cobordism", "abstract": "The elements of the ring of bidegree (0,0) additive unstable operations in complex K-theory can be described explicitly as certain infinite sums of Adams operations. Here we show how to make sense of the same expressions for complex cobordism MU, thus identifying the \"Adams subring\" of the corresponding ring of cobordism operations. We prove that the Adams subring is the centre of the ring of bidegree (0,0) additive unstable cobordism operations. For an odd prime p, the analogous result in the p-local split setting is also proved."}
{"category": "Math", "title": "Parallel hierarchical sampling: a practical multiple-chains sampler for Bayesian model selection", "abstract": "This paper introduces the parallel hierarchical sampler (PHS), a Markov chain Monte Carlo algorithm using several chains simultaneously. The connections between PHS and the parallel tempering (PT) algorithm are illustrated, convergence of PHS joint transition kernel is proved and and its practical advantages are emphasized. We illustrate the inferences obtained using PHS, parallel tempering and the Metropolis-Hastings algorithm for three Bayesian model selection problems, namely Gaussian clustering, the selection of covariates for a linear regression model and the selection of the structure of a treed survival model."}
{"category": "Math", "title": "Hilbert scheme of rational cubic curves via stable maps", "abstract": "The space of smooth rational cubic curves in projective space $\\PP^r$ ($r\\ge 3$) is a smooth quasi-projective variety, which gives us an open subset of the corresponding Hilbert scheme, the moduli space of stable maps, or the moduli space of stable sheaves. By taking its closure, we obtain three compactifications $\\bH$, $\\bM$, and $\\bS$ respectively. In this paper, we compare these compactifications. First, we prove that $\\bH$ is the blow-up of $\\bS$ along a smooth subvariety which is the locus of stable sheaves which are planar (i.e. support is contained in a plane). Next we prove that $\\bS$ is obtained from $\\bM$ by three blow-ups followed by three blow-downs and the centers are described explicitly. Using this, we calculate the cohomology of $\\bS$."}
{"category": "Math", "title": "Conjecture de type de Serre et formes compagnons pour GSp_4", "abstract": "We present a Serre-type conjecture on the modularity of four-dimensional symplectic mod p Galois representations. We assume that the Galois representation is irreducible and odd (in the symplectic sense). The modularity condition is formulated using the etale and the algebraic de Rham cohomology of Siegel modular varieties of level prime to p. We concentrate on the case when the Galois representation is ordinary at p and we give a corresponding list of Serre weights. When the representation is moreover tamely ramified at p, we conjecture that all weights of this list are modular, otherwise we describe a subset of weights on the list that should be modular. We propose a construction of de Rham cohomology classes using the dual BGG complex, which should realise some of these weights."}
{"category": "Math", "title": "An almost all result on $q_1 q_2 \\equiv c \\pmod q$", "abstract": "In this paper we consider the congruence equation $q_1 q_2 \\equiv c \\pmod q$ with $a < q_1 \\leq a + q^{1/2+\\epsilon}$ and $b < q_2 \\leq b + q^{1/2+\\epsilon}$ and show that it has solution for almost all $a$ and $b$. Then we apply it to a question of Fujii and Kitaoka as well as generalize it to more variables. At the end, we will present a new way to attack the above congruence equation question through higher moments."}
{"category": "Math", "title": "On global offensive k-alliances in graphs", "abstract": "We investigate the relationship between global offensive $k$-alliances and some characteristic sets of a graph including $r$-dependent sets and $\\tau$-dominating sets. As a consequence of the study, we obtain bounds on the global offensive $k$-alliance number in terms of several parameters of the graph."}
{"category": "Math", "title": "On combinatorial formulas for the characters of Hecke algebras", "abstract": "Hecke algebras are beautiful q-extensions of Coxeter groups. In this paper, we prove several results on their characters, with an emphasis on characters induced from trivial and sign representations of parabolic subalgebras. While most of the results in type A are known, our proofs are of a combinatorial nature, and generalize to (partial) results in types B and C. We also present complete descriptions of such characters for type I."}
{"category": "Math", "title": "Comparison of Data Imputation Techniques and their Impact", "abstract": "Missing and incomplete information in surveys or databases can be imputed using different statistical and soft-computing techniques. This paper comprehensively compares auto-associative neural networks (NN), neuro-fuzzy (NF) systems and the hybrid combinations the above methods with hot-deck imputation. The tests are conducted on an eight category antenatal survey and also under principal component analysis (PCA) conditions. The neural network outperforms the neuro-fuzzy system for all tests by an average of 5.8%, while the hybrid method is on average 15.9% more accurate yet 50% less computationally efficient than the NN or NF systems acting alone. The global impact assessment of the imputed data is performed by several statistical tests. It is found that although the imputed accuracy is high, the global effect of the imputed data causes the PCA inter-relationships between the dataset to become altered. The standard deviation of the imputed dataset is on average 36.7% lower than the actual dataset which may cause an incorrect interpretation of the results."}
{"category": "Math", "title": "Nonuniform Center Bunching and the Genericity of Ergodicity among $C^1$ Partially Hyperbolic Symplectomorphisms", "abstract": "We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns--Wilkinson and Avila--Santamaria--Viana. Combining this new technique with other constructions, we prove that $C^1$-generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms."}
{"category": "Math", "title": "On Coloring of graph fractional powers", "abstract": "\\noindent Let $G$ be a simple graph. For any $k\\in N$, the $k-$power of $G$ is a simple graph $G^k$ with vertex set $V(G)$ and edge set $\\{xy:d_G(x,y)\\leq k\\}$ and the $k-$subdivision of $G$ is a simple graph $G^{\\frac{1}{k}}$, which is constructed by replacing each edge of $G$ with a path of length $k$. So we can introduce the $m-$power of the $n-$subdivision of $G$, as a fractional power of $G$, that is denoted by $G^{\\frac{m}{n}}$. In other words $G^{\\frac{m}{n}}:=(G^{\\frac{1}{n}})^m$. \\noindent In this paper some results about the coloring of $G^{\\frac{m}{n}}$ are presented when $G$ is a simple and connected graph and $\\frac{m}{n}<1$."}
{"category": "Math", "title": "The Haar State on SU_q(N)", "abstract": "Using matrix corepresentations on SL_q(N) and SU_q(N) we derive a Haar state on SU_q(N) which is nearly identical to that on SU_q(2). This allows us to create an orthonormal basis for the vector space of coordinate functions on SU_q(N)."}
{"category": "Math", "title": "Strongly Contracting Geodesics in Outer Space", "abstract": "We study the Lipschitz metric on Outer Space and prove that fully irreducible elements of Out(F_n) act by hyperbolic isometries with axes which are strongly contracting. As a corollary, we prove that the axes of fully irreducible automorphisms in the Cayley graph of Out(F_n) are stable, meaning that a quasi-geodesic with endpoints on the axis stays within a bounded distance from the axis."}
{"category": "Math", "title": "Determinants of perfect complexes and Euler characteristics in relative K_0-groups", "abstract": "We study the K_0 and K_1-groups of exact and triangulated categories of perfect complexes, and we apply the results to show how determinant functors on triangulated categories can be used for the construction of Euler characteristics in relative algebraic K_0-groups."}
{"category": "Math", "title": "Mean Curvature Motion of Graphs with Constant Contact Angle at a Free Boundary", "abstract": "We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\\mathbb{R}^n$, intersecting $\\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary, and derive a continuation criterion based on the second fundamental form. If the initial graph is concave, we show this is preserved, and that the solution exists only for finite time. This corresponds to a symmetric version of mean curvature motion of a network of hypersurfaces with triple junctions, with constant contact angle at the junctions."}
{"category": "Math", "title": "Reversible biholomorphic germs", "abstract": "Let $G$ be a group. We say that an element $f\\in G$ is {\\em reversible in} $G$ if it is conjugate to its inverse, i.e. there exists $g\\in G$ such that $g^{-1}fg=f^{-1}$. We denote the set of reversible elements by $R(G)$. For $f\\in G$, we denote by $R_f(G)$ the set (possibly empty) of {\\em reversers} of $f$, i.e. the set of $g\\in G$ such that $g^{-1}fg=f^{-1}$. We characterise the elements of $R(G)$ and describe each $R_f(G)$, where $G$ is the the group of biholomorphic germs in one complex variable. That is, we determine all solutions to the equation $ f\\circ g\\circ f = g$, in which $f$ and $g$ are holomorphic functions on some neighbourhood of the origin, with $f(0)=g(0)=0$ and $f'(0)\\not=0\\not=g'(0)$."}
{"category": "Math", "title": "On the structure of the category O for W-algebras", "abstract": "W-algebra (of finite type) W is a certain associative algebra associated with a semisimple Lie algebra, say g, and its nilpotent element, say e. The goal of this paper is to study the category O for W introduced by Brundan, Goodwin and Kleshchev. We establish an equivalence of this category with certain category of g-modules. In the case when e is of principal Levi type (this is always so when g is of type A) the category of g-modules in interest is the category of generalized Whittaker modules introduced McDowel and studied by Milicic-Soergel and Backelin."}
{"category": "Math", "title": "K-trivials are NCR", "abstract": "We show that for every K-trivial real X, there is no representation of a continuous probability measure m such that X is 1-random relative to m."}
{"category": "Math", "title": "Poisson structures on the Teichmueller space of hyperbolic surfaces with conical points", "abstract": "In this paper two Poisson structures on the moduli space of hyperbolic surfaces with conical points are compared: the Weil-Petersson one and the \\eta coming from the representation variety. We show that they are multiple of each other, if the angles do not exceed 2\\pi. Moreover, we exhibit an explicit formula for \\eta in terms of hyperbolic lengths of a suitable system of arcs."}
{"category": "Math", "title": "On the classification of certain fusion categories", "abstract": "We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories -- and consequently, semi-simple Hopf algebras -- of dimension $pq^2$, where $p$ and $q$ are distinct primes. This case is especially interesting because it is the simplest class of dimensions where not all integral fusion categories are group-theoretical. Secondly, we classify a certain family of $\\ZZ/3\\ZZ$-graded fusion categories, which are generalizations of the $\\ZZ/2\\ZZ$-graded Tambara-Yamagami categories. Our proofs are based on the recently developed theory of extensions of fusion categories."}
{"category": "Math", "title": "Semiclassical limits of quantized coordinate rings", "abstract": "This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety V consists of the classical coordinate ring O(V) equipped with an associated Poisson structure. Conjectured relationships between primitive ideals of a generic quantized coordinate ring A and symplectic leaves in V (relative to a semiclassical limit Poisson structure on O(V)) are discussed, as are breakdowns in the connections when the symplectic leaves are not algebraic. This prompts replacement of the differential-geometric concept of symplectic leaves with the algebraic concept of symplectic cores, and a reformulated conjecture is proposed: The primitive spectrum of A should be homeomorphic to the space of symplectic cores in V, and to the Poisson-primitive spectrum of O(V). Various examples, including both quantized coordinate rings and enveloping algebras of solvable Lie algebras, are analyzed to support the choice of symplectic cores to replace symplectic leaves."}
{"category": "Math", "title": "Missing Data using Decision Forest and Computational Intelligence", "abstract": "Autoencoder neural network is implemented to estimate the missing data. Genetic algorithm is implemented for network optimization and estimating the missing data. Missing data is treated as Missing At Random mechanism by implementing maximum likelihood algorithm. The network performance is determined by calculating the mean square error of the network prediction. The network is further optimized by implementing Decision Forest. The impact of missing data is then investigated and decision forrests are found to improve the results."}
{"category": "Math", "title": "Program for calculating bounds on the minimum rank of a graph using Sage", "abstract": "The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\\neq j$) is nonzero whenever $\\{i,j\\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a difficult parameter to compute. However, there are now a number of known reduction techniques and bounds that can be programmed on a computer; we have developed a program using the open-source mathematics software Sage to implement several techniques. In this note, we provide the source code for this program."}
{"category": "Math", "title": "Semidefinite geometry of the numerical range", "abstract": "The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semidefinite cone. Both primal and dual sets can also be viewed as convex hulls of explicit algebraic plane curve components. Several numerical examples illustrate this interplay between algebra, geometry and semidefinite programming duality. Finally, these techniques are used to revisit a theorem in statistics on the independence of quadratic forms in a normally distributed vector."}
{"category": "Math", "title": "Simultaneous confidence intervals for the population cell means, for two-by-two factorial data, that utilize uncertain prior information", "abstract": "Consider a two-by-two factorial experiment with more than 1 replicate. Suppose that we have uncertain prior information that the two-factor interaction is zero. We describe new simultaneous frequentist confidence intervals for the 4 population cell means, with simultaneous confidence coefficient 1-alpha, that utilize this prior information in the following sense. These simultaneous confidence intervals define a cube with expected volume that (a) is relatively small when the two-factor interaction is zero and (b) has maximum value that is not too large. Also, these intervals coincide with the standard simultaneous confidence intervals obtained by Tukey's method, with simultaneous confidence coefficient 1-alpha, when the data strongly contradict the prior information that the two-factor interaction is zero. We illustrate the application of these new simultaneous confidence intervals to a real data set."}
{"category": "Math", "title": "Long time behaviour of viscous scalar conservation laws", "abstract": "This paper is concerned with the stability of stationary solutions of the conservation law $\\partial_t u + \\mathrm{div}_y A(y,u) -\\Delta_y u=0$, where the flux $A$ is periodic with respect to its first variable. Essentially two kinds of asymptotic behaviours are studied here: the case when the equation is set on $\\R$, and the case when it is endowed with periodic boundary conditions. In the whole space case, we first prove the existence of viscous stationary shocks - also called standing shocks - which connect two different periodic stationary solutions to one another. We prove that standing shocks are stable in $L^1$, provided the initial disturbance satisfies some appropriate boundedness conditions. We also extend this result to arbitrary initial data, but with some restrictions on the flux $A$. In the periodic case, we prove that periodic stationary solutions are always stable. The proof of this result relies on the derivation of uniform $L^\\infty$ bounds on the solution of the conservation law, and on sub- and super-solution techniques."}
{"category": "Math", "title": "Gonality of a general ACM curve in projective 3-space", "abstract": "Let C be an ACM (projectively normal) nondegenerate smooth curve in projective 3-space, and suppose C is general in its Hilbert scheme - this is irreducible once the postulation is fixed. Answering a question posed by Peskine, we show the gonality of C is d-l, where d is the degree of the curve, and l is the maximum order of a multisecant line of C. Furthermore l=4 except for two series of cases, in which the postulation of C forces every surface of minimum degree containing C to contain a line as well. We compute the value of l in terms of the postulation of C in these exceptional cases. We also show the Clifford index of C is equal to the gonality minus 2."}
{"category": "Math", "title": "Large deviations for intersection local times in critical dimension", "abstract": "Let $(X_t,t\\geq0)$ be a continuous time simple random walk on $\\mathbb{Z}^d$ ($d\\geq3$), and let $l_T(x)$ be the time spent by $(X_t,t\\geq0)$ on the site $x$ up to time $T$. We prove a large deviations principle for the $q$-fold self-intersection local time $I_T=\\sum_{x\\in\\mathbb{Z}^d}l_T(x)^q$ in the critical case $q=\\frac{d}{d-2}$. When $q$ is integer, we obtain similar results for the intersection local times of $q$ independent simple random walks."}
{"category": "Math", "title": "On asymptotic dimension and a property of Nagata", "abstract": "In this note we prove that every metric space $(X, d)$ of asymptotic dimmension at most $n$ is coarsely equivalent to a metric space $(Y, D)$ that satisfies the following property of Nagata: For every $n+2$ points $y_1,..., y_{n+2}$ in $Y$ and for every $x$ in $Y$ there exist two different $i,j$ such that $D(y_i,y_j)\\le D(x,y_i)$. This solves problem 1400 of the book Open problems in Topology II."}
{"category": "Math", "title": "The nonlinear N-membranes evolution problem", "abstract": "The parabolic N-membranes problem for the p-Laplacian and the complete order constraint on the components of the solution is studied in what concerns the approximation, the regularity and the stability of the variational solutions. We extend to the evolutionary case the characterization of the Lagrange multipliers associated with the ordering constraint in terms of the characteristic functions of the coincidence sets. We give continuous dependence results, and study the asymptotic behavior as $t \\to \\infty$ of the solution and the coincidence sets, showing that they converge to their stationary counterparts."}
{"category": "Math", "title": "Polymorphic evolution sequence and evolutionary branching", "abstract": "We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This process, which generalizes the so-called trait substitution sequence, is called polymorphic evolution sequence. Then we introduce a scaling of the size of mutations and we study the polymorphic evolution sequence in the limit of small mutations. From this study in the neighborhood of evolutionary singularities, we obtain a full mathematical justification of a heuristic criterion for the phenomenon of evolutionary branching. To this end we finely analyze the asymptotic behavior of 3-dimensional competitive Lotka-Volterra systems."}
{"category": "Math", "title": "A Novel Method of Solution for the Fluid Loaded Plate", "abstract": "We study the Cauchy problem associated with the equations governing a fluid loaded plate formulated on either the line or the half-line. We show that in both cases the problem can be solved by employing the unified approach to boundary value problems introduced by on of the authors in the late 1990s. The problem on the full line was analysed by Crighton et. al. using a combination of Laplace and Fourier transforms. The new approach avoids the technical difficulty of the a priori assumption that the amplitude of the plate is in $L^1_{dt}(R^+)$ and furthermore yields a simpler solution representation which immediately implies the problem is well-posed. For the problem on the half-line, a similar analysis yields a solution representation, but this formula involves two unknown functions. The main difficulty with the half-line problem is the characterisation of these two functions. By employing the so-called global relation, we show that the two functions can be obtained via the solution of a complex valued integral equation of the convolution type. This equation can be solved in closed form using the Laplace transform. By prescribing the initial data $\\eta_0$ to be in $H^3(R^+)$, we show that the solution depends continuously on the initial data, and hence, the problem is well-posed."}
{"category": "Math", "title": "Homology of graded Hecke algebras", "abstract": "Let H be a graded Hecke algebra with complex deformation parameters and Weyl group W. We show that the Hochschild, cyclic and periodic cyclic homologies of H are all independent of the parameters, and compute them explicitly. We use this to prove that, if the deformation parameters are real, the collection of irreducible tempered H-modules with real central character forms a Q-basis of the representation ring of W. Our method involves a new interpretation of the periodic cyclic homology of finite type algebras, in terms of the cohomology of a sheaf over the underlying complex affine variety."}
{"category": "Math", "title": "Supermartingale Deomposition with General Index Set", "abstract": "We prove results on the existence of Dol\\'{e}ans-Dade measures and of the Doob-Meyer decomposition for supermartingales indexed by a general index set"}
{"category": "Math", "title": "Groups of quasi-invariance and the Pontryagin duality", "abstract": "A Polish group $G$ is called a group of quasi-invariance or a QI-group, if there exist a locally compact group $X$ and a probability measure $\\mu$ on $X$ such that 1) there exists a continuous monomorphism of $G$ to $X$, and 2) for each $g\\in X$ either $g\\in G$ and the shift $\\mu_g$ is equivalent to $\\mu$ or $g\\not\\in G$ and $\\mu_g$ is orthogonal to $\\mu$. It is proved that $G$ is a $\\sigma$-compact subset of $X$. We show that there exists a quotient group $\\mathbb{T}^H_2$ of $\\ell^2$ modulo a discrete subgroup which is a Polish monothetic non locally quasi-convex (and hence nonreflexive) pathwise connected QI-group, and such that the bidual of $\\mathbb{T}^H_2$ is not a QI-group. It is proved also that the bidual group of a QI-group may be not a saturated subgroup of $X$."}
{"category": "Math", "title": "Categorified central extensions, \\'etale Lie 2-groups and Lie's Third Theorem for locally exponential Lie algebras", "abstract": "Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration procedure for Lie algebra cocycles. This paper remedies the obstructions for integrating cocycles and central extensions from Lie algebras to Lie groups by generalising the integrating objects. Those objects obey the maximal coherence that one can expect. Moreover, we show that they are the universal ones for the integration problem. The main application of this result is that a Mackey-complete locally exponential Lie algebra (e.g., a Banach-Lie algebra) integrates to a Lie 2-group in the sense that there is a natural Lie functor from certain Lie 2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphic Lie algebra."}
{"category": "Math", "title": "Monadic approach to Galois descent and cohomology", "abstract": "We describe a simplified categorical approach to Galois descent theory. It is well known that Galois descent is a special case of Grothendieck descent, and that under mild additional conditions the category of Grothendieck descent data coincides with the Eilenberg-Moore category of algebras over a suitable monad. This also suggests using monads directly, and our monadic approach to Galois descent makes no reference to Grothendieck descent theory at all. In order to make Galois descent constructions perfectly clear, we also describe their connections with some other related constructions of categorical algebra, and make various explicit calculations, especially with 1-cocycles and 1-dimensional non-abelian cohomology, usually omitted in the literature."}
{"category": "Math", "title": "ParaSasakian manifolds with a constant paraholomorphic section curvature", "abstract": "In this paper paraSasakian manifolds with a constant paraholomorphic section curvature are considered."}
{"category": "Math", "title": "Linear series on semistable curves", "abstract": "The dimension of spaces of global sections for line bundles on semistable curves parametrized by the compactified Picard scheme is studied. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: the curve has two components; the curve is any semistable curve and the degree is either 0 or 2g-2; the curve is stable, free from separating nodes, and the degree is at most 4. These results are all shown to be sharp. Applications to the Clifford index, to the combinatorial description of hyperelliptic curves, and to plane quintics are given."}
{"category": "Math", "title": "Smooth Frechet Globalizations of Harish-Chandra Modules", "abstract": "An elementary and self-contained approach to smooth Frechet globalizations of Harish-Chandra modules is presented. We provide applications to the theory of Eisenstein series and Automatic Continuity."}
{"category": "Math", "title": "Strong group coalgebras", "abstract": "We introduce strong group coalgebras, as a generalization of strongly graded coalgebras. We give several characterizations, and study two special types of strong group coalgebras, namely cleft group algebras (or crossed coproduct group coalgebras) and smash coproduct group coalgebras."}
{"category": "Math", "title": "Global homotopy formulas on q-concave CR manifolds for large degrees", "abstract": "Using functional analysis and a Friedrichs approximation lemma for first order differential operators, we derive a global homotopy formula in large degrees for the tangential Cauchy-Riemann operator from local homotopy formulas without loss of regularity."}
{"category": "Math", "title": "Sur la pro-p-extension localement cyclotomique maximale d'un corps de nombres", "abstract": "Let p be a prime number and F be a number field. We consider the Galois group G over the cyclotomic Z_p extension of F of the maximal unramified, p-decomposed, pro-p-extension of the cyclotomic Z_p extension of F. The question whether G is free pro-p was already asked by many authors. In this article, we highlight a link between the freeness of G and the Galois descent for some localisation kernels. Then we give explicit criterions to show that G is not a free pro-p-group."}
{"category": "Math", "title": "Sur un crit\\`ere de Baez-Duarte pour l'hypoth\\`ese de Riemann", "abstract": "Baez-Duarte reformulated the Riemann hypothesis as a statement about a Hilbert space distance, involving the integer dilations of the \"fractional part\" function. Under the assumption of the Riemann hypothesis, we improve on the currently known estimate for this distance."}
{"category": "Math", "title": "Estimating limits from Poisson counting data using Dempster--Shafer analysis", "abstract": "We present a Dempster--Shafer (DS) approach to estimating limits from Poisson counting data with nuisance parameters. Dempster--Shafer is a statistical framework that generalizes Bayesian statistics. DS calculus augments traditional probability by allowing mass to be distributed over power sets of the event space. This eliminates the Bayesian dependence on prior distributions while allowing the incorporation of prior information when it is available. We use the Poisson Dempster--Shafer model (DSM) to derive a posterior DSM for the ``Banff upper limits challenge'' three-Poisson model. The results compare favorably with other approaches, demonstrating the utility of the approach. We argue that the reduced dependence on priors afforded by the Dempster--Shafer framework is both practically and theoretically desirable."}
{"category": "Math", "title": "Hopf-Galois extensions and an exact sequence for $H$-Picard groups", "abstract": "Let $H$ be a Hopf algebra, and $A$ an $H$-Galois extension. We investigate $H$-Morita autoequivalences of $A$, introduce the concept of $H$-Picard group, and we establish an exact sequence linking the $H$-Picard group of $A$ and the Picard group of $A^{{\\rm co}H}$."}
{"category": "Math", "title": "On genericity and weight in the free group", "abstract": "I prove that the generic type of the free nonabelian group has infinite weight (strengthening non superstability of the free group)."}
{"category": "Math", "title": "Q-Fano threefolds of large Fano index, I", "abstract": "We study Q-Fano threefolds of large Fano index. In particular, we prove that the maximum of Fano index is attained for the weighted projective space P(3,4,5,7)."}
{"category": "Math", "title": "Multiplicative Noise Removal Using L1 Fidelity on Frame Coefficients", "abstract": "We address the denoising of images contaminated with multiplicative noise, e.g. speckle noise. Classical ways to solve such problems are filtering, statistical (Bayesian) methods, variational methods, and methods that convert the multiplicative noise into additive noise (using a logarithmic function), shrinkage of the coefficients of the log-image data in a wavelet basis or in a frame, and transform back the result using an exponential function. We propose a method composed of several stages: we use the log-image data and apply a reasonable under-optimal hard-thresholding on its curvelet transform; then we apply a variational method where we minimize a specialized criterion composed of an $\\ell^1$ data-fitting to the thresholded coefficients and a Total Variation regularization (TV) term in the image domain; the restored image is an exponential of the obtained minimizer, weighted in a way that the mean of the original image is preserved. Our restored images combine the advantages of shrinkage and variational methods and avoid their main drawbacks. For the minimization stage, we propose a properly adapted fast minimization scheme based on Douglas-Rachford splitting. The existence of a minimizer of our specialized criterion being proven, we demonstrate the convergence of the minimization scheme. The obtained numerical results outperform the main alternative methods."}
{"category": "Math", "title": "Beyond Kirillov-Reshetikhin modules", "abstract": "In this survey, we shall be concerned with the category of finite-dimensional representations of the untwisted quantum affine algebras when the quantum parameter q is not a root of unity. We review the foundational results of the subject, including the Drinfeld presentation, the classification of simple modules and q-characters. We then concentrate on particular families of irreducible representations whose structure has recently been understood: Kirillov-Reshetikhin modules, minimal affinizations and beyond."}
{"category": "Math", "title": "Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment", "abstract": "The objective of the present paper is to establish exponential large deviation inequalities, and to use them to show exponential concentration inequalities for the free energy of a polymer in general random environment, its rate of convergence, and an expression of its limit value in terms of those of some multiplicative cascades."}
{"category": "Math", "title": "On densest packings of equal balls of $\\rb^{n}$ and Marcinkiewicz spaces", "abstract": "We investigate, by \"a la Marcinkiewicz\" techniques applied to the (asymptotic) density function, how dense systems of equal spheres of $\\rb^{n}, n \\geq 1,$ can be partitioned at infinity in order to allow the computation of their density as a true limit and not a limsup. The density of a packing of equal balls is the norm 1 of the characteristic function of the systems of balls in the sense of Marcinkiewicz. Existence Theorems for densest sphere packings and completely saturated sphere packings of maximal density are given new direct proofs."}
{"category": "Math", "title": "On Bialostocki's conjecture for zero-sum sequences", "abstract": "Let $n$ be a positive even integer, and let $a_1,...,a_n$ and $w_1, ..., w_n$ be integers satisfying $\\sum_{k=1}^n a_k\\equiv\\sum_{k=1}^n w_k =0 (mod n)$. A conjecture of Bialostocki states that there is a permutation $\\sigma$ on {1,...,n} such that $\\sum_{k=1}^n w_k a_{\\sigma(k)}=0 (mod n)$. In this paper we confirm the conjecture when $w_1,...,w_n$ form an arithmetic progression with even common difference."}
{"category": "Math", "title": "Existence and uniqueness of solutions of stochastic functional differential equations", "abstract": "We provide sufficient conditions on the coefficients of a stochastic functional differential equation with bounded memory driven by Brownian motion which guarantee existence and uniqueness of a maximal local and global strong solution for each initial condition. Our results extend those of previous works. For local existence and uniqueness, we only require the coefficients to be continuous and to satisfy a one-sided local Lipschitz (or monotonicity) condition. In an appendix we formulate and prove four lemmas which may be of independent interest: three of them can be viewed as rather general stochastic versions of Gronwall's Lemma, the final one - which we call Dereich-Lemma - provides tail bounds for Hoelder norms of stochastic integrals."}
{"category": "Math", "title": "Consistency Spaces", "abstract": "We introduce the concept of a consistency space. The idea of the consistency space is motivated by the question, Given only the collection of sets of sentences which are logically consistent, is it possible to reconstruct their lattice structure?"}
{"category": "Math", "title": "Multifractal analysis of Lyapunov exponent for the backward continued fraction map", "abstract": "In this note we study the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches and a parabolic fixed point. It turns out that, in strong contrast with the hyperbolic case, the domain of the spectrum is unbounded and points of non-differentiability might exist. Moreover, the spectrum is not concave. We establish conditions that ensure the existence of inflection points. We also study the thermodynamic formalism for such maps. We prove that the pressure function is real analytic in a certain interval and then it becomes equal to zero."}
{"category": "Math", "title": "Holomorphic Parabolic Geometries and Calabi-Yau Manifolds", "abstract": "We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact K\\\"ahler manifolds."}
{"category": "Math", "title": "Palindromic random trigonometric polynomials", "abstract": "We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric polynomials has, on average, many real roots. In the case that the coefficients of a real trigonometric polynomial are independently and identically distributed, but with no other assumptions on the distribution, the expected fraction of real zeros is at least one-half. This result is best possible."}
{"category": "Math", "title": "The symbiosis of C*- and W*-algebras", "abstract": "These days it is common for young operator algebraists to know a lot about C*-algebras, or a lot about von Neumann algebras -- but not both. Though a natural consequence of the breadth and depth of each subject, this is unfortunate as the interplay between the two theories has deep historical roots and has led to many beautiful results. We review some of these connections, in the context of amenability, with the hope of convincing (younger) readers that tribalism impedes progress."}
{"category": "Math", "title": "The escaping set of the exponential", "abstract": "We show that the points that converge to infinity under iteration of the exponential map form a connected subset of the complex plane."}
{"category": "Math", "title": "Occupation times via Bessel functions", "abstract": "This study of occupation time densities for continuous-time Markov processes was inspired by the work of E.Nir et al (2006) in the field of Single Molecule FRET spectroscopy. There, a single molecule fluctuates between two or more states, and the experimental observable depends on the state's occupation time distribution. To mathematically describe the observable there was a need to calculate a single state occupation time distribution. In this paper, we consider a Markov process with countably many states. In order to find a one-stete occupation time density, we use a combination of Fourier and Laplace transforms in the way that allows for inversion of the Fourier transform. We derive an explicit expression for an occupation time density in the case of a simple continuous time random walk on Z. Also we examine the spectral measures in Karlin-McGregor diagonalization in an attempt to represent occupation time densities via modified Bessel functions."}
{"category": "Math", "title": "A Modified Coefficient Ideal for Use with the Strict Transform", "abstract": "Two main algorithmic approaches are known for making Hironaka's proof of resolution of singularities in characteristic zero constructive. Their main difference is the use of different notions of transforms during the resolution process and the use of a stratification by the Hilbert-Samuel function in the one using the strict transform. In this article, a (hybrid-type) algorithmic approach is proposed which allows the use of the strict transform without the full impact of the complexity of the stratification by the Hilbert-Samuel function at each step of the desingularization process. This new approach is not intended to always be superior to the implemented one which uses the weak transform, instead it has its strengths precisely at the weak point of the other one and is thus a candidate to be joined with it by an appropriate heuristic."}
{"category": "Math", "title": "Orthogonality and probability: beyond nearest neighbor transitions", "abstract": "In this article, we will explore why Karlin-McGregor method of using orthogonal polynomials in the study of Markov processes was so successful for one dimensional nearest neighbor processes, but failed beyond nearest neighbor transitions. We will proceed by suggesting and testing possible fixtures."}
{"category": "Math", "title": "Two-generator subgroups of the pure braid group", "abstract": "We show that any two elements of the pure braid group either commute or generate a free group, settling a question of Luis Paris. Our proof involves the theory of 3-manifolds and the theory of group actions on trees."}
{"category": "Math", "title": "Partial Reset in Pulse-Coupled Oscillators", "abstract": "Pulse-coupled threshold units serve as paradigmatic models for a wide range of complex systems. When the state variable of a unit crosses a threshold, the unit sends a pulse that is received by other units, thereby mediating the interactions. At the same time, the state variable of the sending unit is reset. Here we study pulse-coupled models with a reset that may be partial only and is mediated by a partial reset function. Such a partial reset characterizes intrinsic physical or biophysical features of a unit (e.g., resistive coupling between dendrite and soma of compartmental neurons) and at the same time makes possible a rigorous mathematical investigation of the collective network dynamics. The partial reset acts as a desynchronization mechanism. For all-to-all pulse-coupled oscillators an increase in the strength of the partial reset causes a sequence of desynchronizing bifurcations from the fully synchronous state via states with large clusters of synchronized units through states with smaller clusters to complete asynchrony. We analytically derive sufficient and necessary conditions for the existence and stability of periodic cluster states on the local dynamics of the oscillators and on the partial reset function to reveal the mechanism underlying the desynchronization transition. We show that the entire sequence may occur due to arbitrarily small changes of the partial reset and is robust against structural perturbations."}
{"category": "Math", "title": "Completeness of bond market driven by L\\'evy process", "abstract": "The completeness problem of the bond market model with the random factors determined by a Wiener process and Poisson random measure is studied. Hedging portfolios use bonds with maturities in a countable, dense subset of a finite time interval. It is shown that under natural assumptions the market is not complete unless the support of the L\\'evy measure consists of a finite number of points. Explicit constructions of contingent claims which can not be replicated are provided."}
{"category": "Math", "title": "Brownian motion on the Sierpinski carpet", "abstract": "We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpinski carpet that is invariant with respect to the local symmetries of the carpet. Consequently for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined."}
{"category": "Math", "title": "Lectures on Moduli Spaces of Elliptic Curves", "abstract": "These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties through the example of elliptic curves. The reason for working with elliptic curves is that most constructions are elementary and explicit in this case. All four approaches to moduli spaces of curves -- complex analytic, topological, algebro-geometric, and number theoretic -- are considered. Topics covered reflect my own biases. Very little, if anything, in these notes is original, except perhaps the selection of topics and the point of view."}
{"category": "Math", "title": "Approximate factor analysis model building via alternating I-divergence minimization", "abstract": "Given a positive definite covariance matrix $\\widehat \\Sigma$, we strive to construct an optimal \\emph{approximate} factor analysis model $HH^\\top +D$, with $H$ having a prescribed number of columns and $D>0$ diagonal. The optimality criterion we minimize is the I-divergence between the corresponding normal laws. Lifting the problem into a properly chosen larger space enables us to derive an alternating minimization algorithm \\`a la Csisz\\'ar-Tusn\\'ady for the construction of the best approximation. The convergence properties of the algorithm are studied, with special attention given to the case where $D$ is singular."}
{"category": "Math", "title": "Krull dimension for limit groups IV: Adjoining roots", "abstract": "This is the fourth and last paper in a sequence on Krull dimension for limit groups, answering a question of Z. Sela. In it we finish the proof, analyzing limit groups obtained from other limit groups by adjoining roots. We generalize our work on Scott complexity and adjoining roots from the previous paper in the sequence to the category of limit groups."}
{"category": "Math", "title": "Stability of the Ricci Yang-Mills flow at Einstein Yang-Mills metrics", "abstract": "Let $P$ be a principal U(1)-bundle over a closed manifold $M$. On $P$, one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to study the stability of the volume-normalized Ricci Yang-Mills flow at Einstein Yang-Mills metrics in dimension two. In certain cases, we show the presence of a center manifold of fixed points, while in others, we show the existence of an asymptotically stable fixed point."}
{"category": "Math", "title": "Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces", "abstract": "We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation \\[\\partial_t u+|\\partial_x|^{1+\\alpha}\\partial_x u+uu_x=0,\\ u(x,0)=u_0(x),\\] is locally well-posed in the Sobolev spaces $H^s$ for $s>1-\\alpha$ if $0\\leq \\alpha \\leq 1$. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru \\cite{IKT} to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and Tzvetkovin \\cite{MST}. Moreover, as a bi-product we prove that if $0<\\alpha \\leq 1$ the corresponding modified equation (with the nonlinearity $\\pm uuu_x$) is locally well-posed in $H^s$ for $s\\geq 1/2-\\alpha/4$."}
{"category": "Math", "title": "Whitehead products in function spaces: Quillen model formulae", "abstract": "We study Whitehead products in the rational homotopy groups of a general component of a function space. For the component of any based map f: X \\to Y, in either the based or free function space, our main results express the Whitehead product directly in terms of the Quillen minimal model of f. These results follow from a purely algebraic development in the setting of chain complexes of derivations of differential graded Lie algebras, which is of interest in its own right. We apply the results to study the Whitehead length of function space components."}
{"category": "Math", "title": "Combinatorial study on the group of parity alternating permutations", "abstract": "The combinatorial theory for the set of parity alternating permutations is expounded. In view of the numbers of ascents and inversions, several enumerative aspects of the set are investigated. In particular, it is shown that signed Eulerian numbers have intimate relationships to the set."}
{"category": "Math", "title": "Time-dependent Sobolev inequality along the Ricci flow", "abstract": "In this article we get a time-dependent Sobolev inequality along the Ricci flow which generalizes the earlier results of Zhang, Ye, Hsu. As an application of the time-dependent Sobolev inequality, we also get a growth of the ratio of bob-collapsing along the Ricci flow."}
{"category": "Math", "title": "A multiplicative formula for structure constants in the cohomology of flag varieties", "abstract": "Let G be a complex semi-simple Lie group and let P,Q be a pair of parabolic subgroups of G such that Q contains P. Consider the flag varieties G/P, G/Q and Q/P. We show that certain structure constants in H^*(G/P) with respect to the Schubert basis can be written as a product of structure constants coming from H^*(G/Q) and H^*(Q/P) in a very natural way. The primary application is to compute Levi-movable structure constants defined by Belkale and Kumar. We also give a generalization of this product formula in the branching Schubert calculus setting."}
{"category": "Math", "title": "Inverse zero-sum problems in finite Abelian p-groups", "abstract": "In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, the method that we use here enables us to show that, if we denote by exp(G) the exponent of the finite Abelian p-group G which is considered, then a zero-sumfree sequence S with maximal possible length in G must contain at least exp(G)-1 elements of maximal order, which improves a previous result of W. Gao and A. Geroldinger."}
{"category": "Math", "title": "Integration over Tropical Plane Curves and Ultradiscretization", "abstract": "In this article we study holomorphic integrals on tropical plane curves in view of ultradiscretization. We prove that the lattice integrals over tropical curves can be obtained as a certain limit of complex integrals over Riemannian surfaces."}
{"category": "Math", "title": "An introduction to exotic 4-manifolds", "abstract": "This article intends to provide an introduction to the construction of small exotic 4-manifolds. Some of the necessary background is covered. An exposition is given of J. Park's construction in arXiv:math.GT/0311395 of an exotic CP^2#7(-CP^2). This article does not intend to present any new results. It was originally a Master's thesis, and its aim is merely to provide a leisurely introduction to exotic 4-manifolds that might be of use to interested graduate students."}
{"category": "Math", "title": "Asymptotic Geometry in the product of Hadamard spaces with rank one isometries", "abstract": "In this article we study asymptotic properties of certain discrete groups $\\Gamma$ acting by isometries on a product $\\XX=\\XX_1\\times \\XX_2$ of locally compact Hadamard spaces. The motivation comes from the fact that Kac-Moody groups over finite fields, which can be seen as generalizations of arithmetic groups over function fields, belong to this class of groups. Hence one may ask whether classical properties of discrete subgroups of higher rank Lie groups as in [MR1437472] and [MR1933790] hold in this context. In the first part of the paper we describe the structure of the geometric limit set of $\\Gamma$ and prove statements analogous to the results of Benoist in [MR1437472]. The second part is concerned with the exponential growth rate $\\delta_\\theta(\\Gamma)$ of orbit points in $\\XX$ with a prescribed so-called \"slope\" $\\theta\\in (0,\\pi/2)$, which appropriately generalizes the critical exponent in higher rank. In analogy to Quint's result in [MR1933790] we show that the homogeneous extension $\\Psi_\\Gamma$ to $\\RR_{\\ge 0}^2$ of $\\delta_\\theta(\\Gamma)$ as a function of $\\theta$ is upper semi-continuous and concave."}
{"category": "Math", "title": "Contraction of a Generalized Metric Structure", "abstract": "In some scientific fields, a scaling is able to modify the topology of an observed object. Our goal in the present work is to introduce a new formalism adapted to the mathematical representation of this kind of phenomenon. To this end, we introduce a new metric structure - the galactic spaces - which depends on an ordered field extension of R. Moreover, some natural transformations of the category of galactic spaces, the contractions, are of particular interest: they generalize usual homotheties, they have a ratio which may be an infinitesimal, they are able to modify the topology and they satisfy a nice composition rule. With the help of nonstandard extensions we can associate to any metric space an infinite family of galactic spaces; lastly, we study some limit properties of this family."}
{"category": "Math", "title": "On the Dimension of Secant Varieties", "abstract": "We generalize Zak's theorems on tangencies and on linear normality as well as Zak's definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, and we classify varieties for which the bound is attained."}
{"category": "Math", "title": "Stochastic Heat Equation with Multiplicative Fractional-Colored Noise", "abstract": "We consider the stochastic heat equation with multiplicative noise $u_t={1/2}\\Delta u+ u \\diamond \\dot{W}$ in $\\bR_{+} \\times \\bR^d$, where $\\diamond$ denotes the Wick product, and the solution is interpreted in the mild sense. The noise $\\dot W$ is fractional in time (with Hurst index $H \\geq 1/2$), and colored in space (with spatial covariance kernel $f$). We prove that if $f$ is the Riesz kernel of order $\\alpha$, or the Bessel kernel of order $\\alpha<d$, then the sufficient condition for the existence of the solution is $d \\leq 2+\\alpha$ (if $H>1/2$), respectively $d<2+\\alpha$ (if $H=1/2$), whereas if $f$ is the heat kernel or the Poisson kernel, then the equation has a solution for any $d$. We give a representation of the $k$-th order moment of the solution, in terms of an exponential moment of the \"convoluted weighted\" intersection local time of $k$ independent $d$-dimensional Brownian motions."}
{"category": "Math", "title": "The Maximum of the Maximum Rectilinear Crossing Numbers of d-regular Graphs of Order n", "abstract": "We extend known results regarding the maximum rectilinear crossing number of the cycle graph (C_n) and the complete graph (K_n) to the class of general d-regular graphs R_{n,d}. We present the generalized star drawings of the d-regular graphs S_{n,d} of order n where n+d= 1 mod 2 and prove that they maximize the maximum rectilinear crossing numbers. A star-like drawing of S_{n,d} for n = d = 0 mod 2 is introduced and we conjecture that this drawing maximizes the maximum rectilinear crossing numbers, too. We offer a simpler proof of two results initially proved by Furry and Kleitman as partial results in the direction of this conjecture."}
{"category": "Math", "title": "Brief communication. An indefinite Sturm theory", "abstract": "Sturm theory for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. Here we propose a Sturm oscillation theorem for indefinite systems of even order and with Dirichlet boundary conditions having strongly indefinite leading term."}
{"category": "Math", "title": "The Lyapunov spectrum is not always concave", "abstract": "We characterize one-dimensional compact repellers having nonconcave Lyapunov spectra. For linear maps with two branches we give an explicit condition that characterizes non-concave Lyapunov spectra."}
{"category": "Math", "title": "Trek separation for Gaussian graphical models", "abstract": "Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases. Our new trek separation criterion generalizes the familiar $d$-separation criterion. Proofs are based on the trek rule, the resulting matrix factorizations and classical theorems of algebraic combinatorics on the expansions of determinants of path polynomials."}
{"category": "Math", "title": "A central limit theorem for random walk in random environment on marked Galton-Watson trees", "abstract": "In this article we focus on a general model of random walk on random marked trees. We prove a recurrence criterion, analogue to the recurrence criterion proved by R. Lyons and Robin Pemantle (1992) in a slightly different model. In the critical case, we obtain a criterion for the positive/null recurrence. Several regimes appear, as proved (in a similar model), by Y. Hu and Z. Shi (2007). We focus on the \"diffusive\" regime and improve their result in this case, by obtaining a functional Central Limit Theorem. Our result is also an extension of a result by Y. Peres and O. Zeitouni (2008), obtained in the setting of biased random walk in Galton-Watson trees."}
{"category": "Math", "title": "Prediction with Restricted Resources and Finite Automata", "abstract": "We obtain an index of the complexity of a random sequence by allowing the role of the measure in classical probability theory to be played by a function we call the generating mechanism. Typically, this generating mechanism will be a finite automata. We generate a set of biased sequences by applying a finite state automata with a specified number, $m$, of states to the set of all binary sequences. Thus we can index the complexity of our random sequence by the number of states of the automata. We detail optimal algorithms to predict sequences generated in this way."}
{"category": "Math", "title": "n-Linear Algebra of type I and its applications", "abstract": "With the advent of computers, one needs algebraic structures that can simultaneously work with bulk data. One such algebraic structure, namely, n-linear algebras of type I are introduced in this book and its applications to n-Markov chains and n-Leontief models are given. These structures can be thought of as the generalization of bilinear algebras and bivector spaces. Several interesting n-linear algebra properties are proved. This book has four chapters. The first chapter just introduces n-group which is essential for the definition of n-vector spaces and n-linear algebras of type I. Chapter two gives the notion of n-vector spaces and several related results which are analogues of the classical linear algebra theorems. In case of n-vector spaces, we can define several types of linear transformations. The notion of n-best approximations can be used for error correction in coding theory. The notion of n-eigen values can be used in deterministic modal superposition principle for undamped structures, which can find its applications in finite element analysis of mechanical structures with uncertain parameters. Further, it is suggested that the concept of n-matrices can be used in real world problems which adopts fuzzy models like Fuzzy Cognitive Maps, Fuzzy Relational Equations and Bidirectional Associative Memories. The applications of these algebraic structures are given in the third chapter. The fourth chapter suggests problems to further a reader's understanding of the subject."}
{"category": "Math", "title": "The reduced HOMFLY-PT homology for the Conway and the Kinoshita-Terasaka knots", "abstract": "In this paper we compute the reduced HOMFLY-PT homologies of the Conway and the Kinoshita-Terasaka knots and show that they are isomorphic."}
{"category": "Math", "title": "Generalized solutions for the Euler-Bernoulli model with distributional forces", "abstract": "We establish existence and uniqueness of generalized solutions to the initial-boundary value problem corresponding to an Euler-Bernoulli beam model from mechanics. The governing partial differential equation is of order four and involves discontinuous, and even distributional coefficients and right-hand side. The general problem is solved by application of functional analytic techniques to obtain estimates for the solutions to regularized problems. Finally, we prove coherence properties and provide a regularity analysis of the generalized solution."}
{"category": "Math", "title": "Phylogenetic distances for neighbour dependent substitution processes", "abstract": "We consider models of nucleotidic substitution processes where the rate of substitution at a given site depends on the state of its neighbours. For a wide class of such nonreversible models, we show how to compute consistent, mathematically exact, estimators of the time elapsed between aligned sequences, for an ancestral sequence and a present one, and also for two present sequences. In both cases, we provide asymptotic confidence intervals, valid for nucleotidic sequences of finite length. We compute explicit formulas for the estimators and for their confidence intervals in the simplest nontrivial case, the Jukes-Cantor model with CpG influence."}
{"category": "Math", "title": "Canonical models of filtered $A_\\infty$-algebras and Morse complexes", "abstract": "The purpose of this paper is two-fold. First we explain the construction of the canonical model of filtered $A_\\infty$-algebras given in the authors' book [FOOO]. The canonical model plays a crucial role in the study of Lagrangian Floer theory on toric manifolds in our recent papers, arXiv:0802.1703 and arXiv:0810.5654. Then using a variation of the arguments used in that construction, we define a natural filtered $A_\\infty$-structure on the Morse complex of a Morse function and its $A_\\infty$ homotopy to the $A_\\infty$-algebras on a Lagrangian submanifold constructed in [FOOO]. The corresponding graphical moduli spaces `summing over trees' involve holomorphic discs connected by the gradient flow lines."}
{"category": "Math", "title": "Multiparameter ergodic averages for two commuting actions of an amenable group", "abstract": "We find limits of some multiple ergodic averages, generalizing a result of Bergelson to the setting of two commuting transformations and actions of amenable groups."}
{"category": "Math", "title": "A Theorem on Analytic Strong Multiplicity One", "abstract": "Let $K$ be an algebraic number field, and $\\pi=\\otimes\\pi_{v}$ an irreducible, automorphic, cuspidal representation of $\\GL_{m}(\\mathbb{A}_{K})$ with analytic conductor $C(\\pi)$. The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant $c$ depending on $\\varepsilon>0, m,$ and $K$ only, such that $\\pi$ can be decided completely by its local components $\\pi_{v}$ with norm $N(v)<c\\cdot C(\\pi)^{2m+\\varepsilon}.$"}
{"category": "Math", "title": "Essential Variables and Separable Sets in Universal Algebra", "abstract": "The study of essential and strongly essential variables in functions defined on finite sets is a part of $k$-valued logic. We extend the main definitions from functions to terms. This allows us to apply concepts and results of Universal Algebra. On the basis of the concept of a separable set of variables in a term we introduce a new notion of complexity of terms, algebras and varieties and give examples."}
{"category": "Math", "title": "\\'Equations aux q-diff\\'erences lin\\'eaires: factorisation, r\\'esolution et th\\'eor\\`emes d'indices", "abstract": "We describe explicit algorithms for factoring q-difference operators and solving q-difference equations. These are well known results, presented in a \"concrete\" form. ----- Nous decrivons des algorithmes explicites pour la factorisation d'operateurs et la resolution d'equations aux q-differences. Il s'agit d'une presentation \"concrete\" de resultats bien connus."}
{"category": "Math", "title": "On the normal holonomy representation of spacelike submanifolds in pseudo-Riemannian space forms", "abstract": "In this paper we study weakly irreducible holonomy representations of the normal connection of a spacelike submanifold in a pseudo-Riemannian space from. We associate screen representations to weakly irreducible normal holonomy groups and classify the screen representations having the Borel-Lichn\\'erowicz property. In particular, we derive a classification of Lorentzian normal holonomy representations."}
{"category": "Math", "title": "Topological Modular Forms of Level 3", "abstract": "We describe and compute the homotopy of spectra of topological modular forms of level 3. We give some computations related to the \"building complex\" associated to level 3 structures at the prime 2. Finally, we note the existence of a number of connective models of the spectrum TMF(Gamma_0(3))."}
{"category": "Math", "title": "Goldie Ranks of Skew Power Series Rings of Automorphic Type", "abstract": "Let A be a semprime, right noetherian ring equipped with an automorphism alpha, and let B := A[[y; alpha]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals."}
{"category": "Math", "title": "Cubic ergodic averages for actions of amenable groups", "abstract": "We unify and extend some previous results about cubic ergodic averages and sets of positive density in products of groups. This provides a joint generalization of earlier work of the author in the case of two commuting actions of an amenable group, and of Q. Chu's recent result on a a finite number of commuting transformations."}
{"category": "Math", "title": "Codimension and pseudometric in co-Heyting algebras", "abstract": "In this paper we introduce a notion of dimension and codimension for every element of a distributive bounded lattice $L$. These notions prove to have a good behavior when $L$ is a co-Heyting algebra. In this case the codimension gives rise to a pseudometric on $L$ which satisfies the ultrametric triangle inequality. We prove that the Hausdorff completion of $L$ with respect to this pseudometric is precisely the projective limit of all its finite dimensional quotients. This completion has some familiar metric properties, such as the convergence of every monotonic sequence in a compact subset. It coincides with the profinite completion of $L$ if and only if it is compact or equivalently if every finite dimensional quotient of $L$ is finite. In this case we say that $L$ is precompact. If $L$ is precompact and Hausdorff, it inherits many of the remarkable properties of its completion, specially those regarding the join/meet irreducible elements. Since every finitely presented co-Heyting algebra is precompact Hausdorff, all the results we prove on the algebraic structure of the latter apply in particular to the former. As an application, we obtain the existence for every positive integers $n,d$ of a term $t_{n,d}$ such that in every co-Heyting algebra generated by an $n$-tuple $a$, $t_{n,d}(a)$ is precisely the maximal element of codimension $d$."}
{"category": "Math", "title": "On Bellissima's construction of the finitely generated free Heyting algebras, and beyond", "abstract": "We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima's representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal sprectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. For example we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators."}
{"category": "Math", "title": "Generalized low-pass filters and multiresolution analyses", "abstract": "We study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group. These notions are based on the structure of generalized multiresolution analyses. We investigate when the Ruelle operator corresponding to such a filter is a pure isometry, and then use that characterization to study the problem of when a collection of closed subspaces, which satisfies all the conditions of a GMRA except the trivial intersection condition, must in fact have a trivial intersection. In this context, we obtain a generalization of a theorem of Bownik and Rzeszotnik."}
{"category": "Math", "title": "Computing p-adic integrals using motivic integration", "abstract": "We use the theory of motivic integration in order to give a geometric explanation of the behavior of some p-adic integrals."}
{"category": "Math", "title": "Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities", "abstract": "The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\\Theta$ which give rise to self-adjoint Laplacians $-\\Delta_{\\Theta, \\Omega}$ in $L^2(\\Omega; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $\\Omega\\subset\\bbR^n$, $n\\in\\bbN$, $n\\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $\\Omega$, following an approach introduced by Filonov for this type of problems."}
{"category": "Math", "title": "Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I", "abstract": "Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type."}
{"category": "Math", "title": "Left eigenvalues of $2\\times 2$ symplectic matrices", "abstract": "We obtain a complete characterization of the $2\\times 2$ symplectic matrices having an infinite number of left eigenvalues. Previously, we give a new proof of a result from Huang and So about the number of eigenvalues of a quaternionic matrix. This is achieved by applying an algorithm for the resolution of equations due to De Leo et al."}
{"category": "Math", "title": "Index formulas on stratified manifolds", "abstract": "Elliptic operators on stratified manifolds with any finite number of strata are considered. Under certain assumptions on the symbols of operators, we obtain index formulas, which express index as a sum of indices of elliptic operators on the strata."}
{"category": "Math", "title": "Inegalite d'Ahlfors en dimension superieure", "abstract": "We proove an Ahlfors' like inequality for the holomorphic curves of a complex compact Kobayashi hyperbolic manifold."}
{"category": "Math", "title": "The Harish-Chandra isomorphism for Clifford algebras", "abstract": "We study the analogue of the Harish-Chandra homomorphism where the universal enveloping algebra is replaced by the Clifford algebra, $Cl(g)$, of a semisimple Lie algebra $g$. Two main goals are achieved. First, we prove that there is a Harish-Chandra type isomorphism between the subalgebra of $g$-invariants in $Cl(g)$ and the Clifford algebra of the Cartan subalgebra of $g$. Second, the Cartan subalgebra is identified, via this isomorphism, with a graded space of the so-called primitive skew-symmetric invariants of $g$. This leads to a distinguished orthogonal basis of the Cartan subalgebra, which turns out to be induced from the Lie algebra Langlands dual to $g$ via the action of its principal three-dimensional subalgebra. This settles a conjecture of Kostant."}
{"category": "Math", "title": "A formalism for the study of Natural Tensors Fields of type (0,2) on Manifolds and Fibrations", "abstract": "In order to study tensor fields of type (0,2) on manifolds and fibrations we introduce the notion of s-spaces. With the help of these objects we generalized the concept of natural tensor without making use of the theory of natural operators and differential invariants."}
{"category": "Math", "title": "Non-crossing linked partitions and multiplication of free random variables", "abstract": "The material gives a new combinatorial proof of the multiplicative property of the S-transform. In particular, several properties of the coefficients of its inverse are connected to non-crossing linked partitions and planar trees."}
{"category": "Math", "title": "Depth and Stanley depth of multigraded modules", "abstract": "We study the behavior of depth and Stanley depth along short exact sequences of multigraded modules and under reduction modulo an element."}
{"category": "Math", "title": "On an optimal quadrature formula in Sobolev space $L_2^{(m)} (0,1)$", "abstract": "In this paper in the space $L_2^{(m)}(0,1)$ the problem of construction of optimal quadrature formulas is considered. Here the quadrature sum consists on values of integrand at nodes and values of first derivative of integrand at the end points of integration interval. The optimal coefficients are found and norm of the error functional is calculated for arbitrary fixed $N$ and for any $m\\geq 2$. It is shown that when $m=2$ and $m=3$ the Euler-Maclaurin quadrature formula is optimal."}
{"category": "Math", "title": "Harmonic functions for a class of integro-differential operators", "abstract": "We consider the operator $\\sL$ defined on $C^2(\\bR^d)$ functions by \\sL f(x)&=&{1/2}\\sum_{i,j=1}^d a_{ij}(x)\\frac{\\partial^2f(x)}{\\partial x_i\\partial x_j}+\\sum_{i=1}^d b_i(x)\\frac{\\partial f(x)}{\\partial x_i} &+&\\int_{\\bR^d\\backslash\\{0\\}}[f(x+h)-f(x)-1_{(|h|\\leq1)}h\\cdot \\grad f(x)]n(x,h)dh. Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on $n(x,h)$, we establish a Harnack inequality for functions that are nonnegative in $\\bR^d$ and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on $n(x,h)$. A regularity theorem for those nonnegative harmonic functions is also proved"}
{"category": "Math", "title": "Energy-supercritical NLS: critical $\\dot H^s$-bounds imply scattering", "abstract": "We consider two classes of defocusing energy-supercritical nonlinear Schr\\\"odinger equations in dimensions $d\\geq 5$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\\in L_t^\\infty \\dot H^{s_c}_x$, then $u$ is global and scatters."}
{"category": "Math", "title": "A limit approach to group homology", "abstract": "In this paper, we consider for any free presentation $G = F/R$ of a group $G$ the coinvariance $H_{0}(G,R_{ab}^{\\otimes n})$ of the $n$-th tensor power of the relation module $R_{ab}$ and show that the homology group $H_{2n}(G,{\\mathbb Z})$ may be identified with the limit of the groups $H_{0}(G,R_{ab}^{\\otimes n})$, where the limit is taken over the category of these presentations of $G$. We also consider the free Lie ring generated by the relation module $R_{ab}$, in order to relate the limit of the groups $\\gamma_{n}R/[\\gamma_{n}R,F]$ to the $n$-torsion subgroup of $H_{2n}(G,{\\mathbb Z})$."}
{"category": "Math", "title": "$L^2$-Betti numbers and non-unitarizable groups without free subgroups", "abstract": "We show that there exist non-unitarizable groups without non-abelian free subgroups. Both torsion and torsion free examples are constructed. As a by-product, we show that there exist finitely generated torsion groups with non-vanishing first $L^2$-Betti numbers. We also relate the well-known problem of whether every hyperbolic group is residually finite to an open question about approximation of $L^2$-Betti numbers."}
{"category": "Math", "title": "Geometrical description of smooth projective symmetric varieties with Picard number one", "abstract": "In a previous paper we have classified the smooth projective symmetric G-varieties with Picard number one (and G semisimple). In this work we give a geometrical description of such varieties. In particular, we determine their group of automorphisms. When this group, Aut(X), acts non-transitively on X, we describe a G-equivariant embedding of the variety X in a homogeneous variety (with respect to a larger group)."}
{"category": "Math", "title": "A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods", "abstract": "We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme."}
{"category": "Math", "title": "Some remarks on the size of tubular neighborhoods in contact topology and fillability", "abstract": "The well-known tubular neighborhood theorem for contact submanifolds states that a small enough neighborhood of such a submanifold N is uniquely determined by the contact structure on N, and the conformal symplectic structure of the normal bundle. In particular, if the submanifold N has trivial normal bundle then its tubular neighborhood will be contactomorphic to a neighborhood of Nx{0} in the model space NxR^{2k}. In this article we make the observation that if (N,\\xi_N) is a 3-dimensional overtwisted submanifold with trivial normal bundle in (M,\\xi), and if its model neighborhood is sufficiently large, then (M,\\xi) does not admit an exact symplectic filling."}
{"category": "Math", "title": "The space of closed subgroups of $R^n$", "abstract": "The Chabauty space of a topological group is the set of its closed subgroups, endowed with a natural topology. As soon as $n>2$, the Chabauty space of $R^n$ has a rather intricate topology and is not a manifold. By an investigation of its local structure, we fit it into a wider, but too wild, class of topological spaces (namely Goresky-MacPherson stratified spaces). Thanks to a localization theorem, this local study also leads to the main result of this article: the Chabauty space of $R^n$ is simply connected for all $n$. Last, we give an alternative proof of the Hubbard-Pourezza Theorem, which describes the Chabauty space of $R^2$."}
{"category": "Math", "title": "Locally definable homotopy", "abstract": "In \"On o-minimal homotopy groups\", o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in V-definable groups -- which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent."}
{"category": "Math", "title": "Vanishing Cycles in Holomorphic Foliations by Curves and Foliated Shells", "abstract": "The purpose of this paper is the study of vanishing cycles in holomorphic foliations by complex curves on compact complex manifolds. The main result consists in showing that a vanishing cycle comes together with a much richer complex geometric object - we call this object a foliated shell."}
{"category": "Math", "title": "Multiplicative formulas in Cohomology of $G/P$ and in quiver representations", "abstract": "Consider a partial flag variety $X$ which is not a grassmaninan. Consider also its cohomology ring ${\\rm H}^*(X,\\ZZ)$ endowed with the base formed by the Poincar\\'e dual classes of the Schubert varieties. In \\cite{Richmond:recursion}, E. Richmond showed that some coefficient structure of the product in ${\\rm H}^*(X,\\ZZ)$ are products of two such coefficients for smaller flag varieties. Consider now a quiver without oriented cycle. If $\\alpha$ and $\\beta$ denote two dimension-vectors, $\\alpha\\circ\\beta$ denotes the number of $\\alpha$-dimensional subrepresentations of a general $\\alpha+\\beta$-dimensional representation. In \\cite{DW:comb}, H. Derksen and J. Weyman expressed some numbers $\\alpha\\circ\\beta$ as products of two smaller such numbers. The aim of this work is to prove two generalisations of the two above results by the same way."}
{"category": "Math", "title": "Geodesic Webs of Hypersurfaces", "abstract": "In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem."}
{"category": "Math", "title": "Factorization property of generalized s-selfdecomposable measures and class $L^f$ distributions$^1$", "abstract": "The method of \\emph{random integral representation}, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note we will find such a representation for generalized s-selfdecomposable and selfdecomposable distributions that have the \\emph{factorization property}. These are the classes $\\mathcal{U}^f_{\\be}$ and $L^f$, respectively"}
{"category": "Math", "title": "Lowest Weights in Cohomology of Variations of Hodge Structure (II)", "abstract": "Let $X$ be an irreducible complex analytic space with $j:U\\into X$ an immersion of a smooth Zariski open subset, and let $\\bV$ be a variation of Hodge structure of weight $n$ over $U$. Assume $X$ is compact K\\\"ahler. Then provided the local monodromy operators at infinity are quasi-unipotent, $IH^k(X, \\bV)$ is known to carry a pure Hodge structure of weight $k+n$, while $H^k(U,\\bV)$ carries a mixed Hodge structure of weight $\\ge k+n$. In this note it is shown that the image of the natural map $IH^k(X,\\bV) \\to H^k(U,\\bV)$ is the lowest weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complement $X-U$ is not a hypersurface."}
{"category": "Math", "title": "Exotic Bialgebras from 9x9 Unitary Braid Matrices", "abstract": "We present the exotic bialgebras that arise from a 9x9 unitary braid matrix."}
{"category": "Math", "title": "Vorticity internal transition layers for the Navier-Stokes equations", "abstract": "We deal with the incompressible Navier-Stokes equations, in two and three dimensions, when some vortex patches are prescribed as initial data i.e. when there is an internal boundary across which the vorticity is discontinuous. We show -thanks to an asymptotic expansion- that there is a sharp but smooth variation of the fluid vorticity into a internal layer moving with the flow of the Euler equations; as long as this later exists and as t * nu is small, where nu is the viscosity coefficient."}
{"category": "Math", "title": "Pairs of mutually annihilating operators", "abstract": "Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations. The classification of (A,B) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636-654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A,B)--> S^{-1}AS, S^{-1}BS)."}
{"category": "Math", "title": "On complex nilpotent Leibniz superalgebras of nilindex n+m", "abstract": "We present the description up to isomorphism of Leibniz superalgebras with characteristic sequence $(n|m_1,...,m_k)$ and nilindex $n+m,$ where $m=m_1+ >...+m_k,$ $n$ and $m$ ($m\\neq 0$) are dimensions of even and odd parts, respectively."}
{"category": "Math", "title": "On the moduli space of quadruples of points in the boundary of complex hyperbolic space", "abstract": "We consider the space $\\mathcal M$ of ordered quadruples of distinct points in the boundary of complex hyperbolic $n$-space, $\\ch{n},$ up to its holomorphic isometry group ${\\rm PU}(n,1).$ One of the important problems in complex hyperbolic geometry is to construct and describe a moduli space for $\\mathcal M$. For $n=2$, this problem was considered by Falbel, Parker, and Platis. The main purpose of this paper is to construct a moduli space for $\\mathcal M $ for any dimension $n \\geq 1$. The major innovation in our paper is the use of the Gram matrix instead of a standard position of points."}
{"category": "Math", "title": "Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space", "abstract": "We consider an entire graph S of a continuous real function over (N-1)-dimensional Euclidean space with N larger than or equal to 3. Let D be a domain in N-dimensional Euclidean space with S as a boundary. Consider in D the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in D. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Amp\\`ere-type equation."}
{"category": "Math", "title": "Explicit Constructions of the non-Abelian $\\mathbf{p^3}$-Extensions Over $\\mathbf{\\QQ}$", "abstract": "Let $p$ be an odd prime. Let $F/k$ be a cyclic extension of degree $p$ and of characteristic different from $p$. The explicit constructions of the non-abelian $p^{3}$-extensions over $k$, are induced by certain elements in ${F(\\mu_{p})}^{*}$. In this paper we let $k=\\QQ$ and present sufficient conditions for these elements to be suitable for the constructions. Polynomials for the non-abelian groups of order 27 over $\\QQ$ are constructed. We describe explicit realizations of those groups with exactly two ramified primes, without consider Scholz conditions."}
{"category": "Math", "title": "A locally finite tree that behaves like an infinite star", "abstract": "We construct a tree T of maximal degree 3 with infinitely many leaves such that whenever finitely many of them are removed, the remaining tree is isomorphic to T. In this sense T resembles an infinite star."}
{"category": "Math", "title": "A local branching heuristic for MINLPs", "abstract": "Local branching is an improvement heuristic, developed within the context of branch-and-bound algorithms for MILPs, which has proved to be very effective in practice. For the binary case, it is based on defining a neighbourhood of the current incumbent solution by allowing only a few binary variables to flip their value, through the addition of a local branching constraint. The neighbourhood is then explored with a branch-and-bound solver. We propose a local branching scheme for (nonconvex) MINLPs which is based on iteratively solving MILPs and NLPs. Preliminary computational experiments show that this approach is able to improve the incumbent solution on the majority of the test instances, requiring only a short CPU time. Moreover, we provide algorithmic ideas for a primal heuristic whose purpose is to find a first feasible solution, based on the same scheme."}
{"category": "Math", "title": "Dunkl Hyperbolic Equations", "abstract": "We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied."}
{"category": "Math", "title": "The Virtually Cyclic Classifying Space of the Heisenberg Group", "abstract": "We are interested in the relationship between the virtual cohomological dimension (or vcd) of a discrete group Gamma and the smallest possible dimension of a model for the classifying space of Gamma relative to its family of virtually cyclic subgroups. In this paper we construct a model for the virtually cyclic classifying space of the Heisenberg group. This model has dimension 3, which equals the vcd of the Heisenberg group. We also prove that there exists no model of dimension less than 3."}
{"category": "Math", "title": "The length of chains in algebraic lattices", "abstract": "We study how the existence in an algebraic lattice $L$ of a chain of a given type is reflected in the join-semilattice $K(L)$ of its compact elements. We show that for every chain $\\alpha$ of size $\\kappa$, there is a set $\\B$ of at most $2^{\\kappa}$ join-semilattices, each one having a least element such that an algebraic lattice $L$ contains no chain of order type $I(\\alpha)$ if and only if the join-semilattice $K(L)$ of its compact elements contains no join-subsemilattice isomorphic to a member of $\\B$. We show that among the join-subsemilattices of $[\\omega]^{<\\omega}$ belonging to $\\B$, one is embeddable in all the others. We conjecture that if $\\alpha$ is countable, there is a finite set $\\B$."}
{"category": "Math", "title": "Local Convergence of the Proximal Point Method for a Special Class of Nonconvex Functions on Hadamard Manifolds", "abstract": "Local convergence analysis of the proximal point method for special class of nonconvex function on Hadamard manifold is presented in this paper. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, is proved that each cluster point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained."}
{"category": "Math", "title": "Lifts of partial characters with respect to a chain of normal subgroups", "abstract": "In this paper, we consider lifts of $\\pi$-partial characters with the property that the irreducible constituents of their restrictions to certain normal subgroups are also lifts. We will show that such a lift must be induced from what we call an inductive pair, and every character induced from an inductive pair is a such a lift. With this condition, we will get a lower bound on the number of such lifts."}
{"category": "Math", "title": "On the Banach-Mazur Type for Normed Spaces", "abstract": "In order to measure qualitative properties we introduce a notion of a type for arbitrary normed spaces which measures the worst possible growth of partial sums of sequences weakly converging to zero. The ideas can be traced back to Banach and Mazur who used this type to compare the so-called linear dimension of classical Banach spaces. As an application we compare the linear dimension and investigate isomorphy of some classical Banach spaces."}
{"category": "Math", "title": "Groups where all the irreducible characters are super-monomial", "abstract": "Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if $G$ is an $M$-group with odd order, then every irreducible character is super monomial. We prove that the conjecture is true if $G$ is an $M$-group of odd order where every irreducible character is a $\\{p \\}$-lift for some prime $p$. We say that a group where irreducible character is super monomial is a super $M$-group. We use our results to find an example of a super $M$-group that has a subgroup that is not a super $M$-group."}
{"category": "Math", "title": "The Large Sieve and Galois Representations", "abstract": "We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis."}
{"category": "Math", "title": "Some index formulae on the moduli space of stable parabolic vector bundles", "abstract": "We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf."}
{"category": "Math", "title": "Asymptotic expansion of planar canard solutions near a non-generic turning point", "abstract": "This paper deals with the asymptotic study of the so-called canard solutions, which arise in the study of real singularly perturbed ODEs. Starting near an attracting branch of the \"slow curve\", those solutions are crossing a turning point before following for a while a repelling branch of the \"slow curve\". Assuming that the turning point is degenerate (or non-generic), we apply a correspondence presented in a recent paper. This application needs the definition of a family of functions $\\phi$ that is studied in a first part. Then, we use the correspondence is used to compute the asymptotic expansion in the powers of the small parameter for the canard solution."}
{"category": "Math", "title": "First, second, and third change of rings theorems for the Gorenstein homological dimensions", "abstract": "Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established, in this paper, for the Gorenstein projective dimension is a generalization of a result established over Noetherian rings and for finitely generated modules. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this paper for the Gorenstein projective dimension is a generalization of a $G$-dimension of a finitely generated module $M$ over a noetherian ring $R$."}
{"category": "Math", "title": "The Ricci Flow for Nilmanifolds", "abstract": "We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.'s for some of these flows and describe their qualitative properties. We also present some explicit solutions for the evolution of soliton metrics under the Ricci flow."}
{"category": "Math", "title": "Steinberg lattice of the general linear group and its modular reduction", "abstract": "We study the Steinberg lattice of the general linear group when reduced modulo a prime different from the defining characteristic."}
{"category": "Math", "title": "Restricted limits of minimal affinizations", "abstract": "We obtain character formulas of minimal affinizations of representations of quantum groups when the underlying simple Lie algebra is orthogonal and the support of the highest weight is contained in the first three nodes of the Dynkin diagram. We also give a framework for extending our techniques to a more general situation. In particular, for the orthogonal algebras and a highest weight supported in at most one spin node, we realize the restricted classical limit of the corresponding minimal affinizations as a quotient of a module given by generators and relations and, furthermore, show that it projects onto the submodule generated by the top weight space of the tensor product of appropriate restricted Kirillov-Reshetikhin modules. We also prove a conjecture of Chari and Pressley regarding the equivalence of certain minimal affinizations in type D4."}
{"category": "Math", "title": "The number of elements in the mutation class of a quiver of type $D_n$", "abstract": "We show that the number of quivers in the mutation class of a quiver of Dynkin type $D_n$ is given by $\\sum_{d|n} \\phi(n/d)\\binom{2d}{d}/(2n)$ for $n \\geq 5$. To obtain this formula, we give a correspondence between the quivers in the mutation class and certain rooted trees."}
{"category": "Math", "title": "'Fair' Partitions of Polygons - an Introduction", "abstract": "We address the question: Given a positive integer $N$, can any 2D convex polygonal region be partitioned into $N$ convex pieces such that all pieces have the same area and same perimeter? The answer to this question is easily `yes' for $N$=2. We prove the answer to be `yes' for $N$=4 and also discuss higher powers of 2."}
{"category": "Math", "title": "How Euler would compute the Euler-Poincar\\'e characteristic of a Lie superalgebra", "abstract": "The Euler-Poincar\\'e characteristic of a finite-dimensional Lie algebra vanishes. If we want to extend this result to Lie superalgebras, we should deal with infinite sums. We observe that a suitable method of summation, which goes back to Euler, allows to do that, to a certain degree. The mathematics behind it is simple, we just glue the pieces of elementary homological algebra, first-year calculus and pedestrian combinatorics together, and present them in a (hopefully) coherent manner."}
{"category": "Math", "title": "Average-Case Perturbations and Smooth Condition Numbers", "abstract": "We define a new condition number adapted to directionally uniform perturbations. The definitions and theorems can be applied to a large class of problems. We show the relation with the classical condition number, and study some interesting examples."}
{"category": "Math", "title": "Feature selection by Higher Criticism thresholding: optimal phase diagram", "abstract": "We consider two-class linear classification in a high-dimensional, low-sample size setting. Only a small fraction of the features are useful, the useful features are unknown to us, and each useful feature contributes weakly to the classification decision -- this setting was called the rare/weak model (RW Model). We select features by thresholding feature $z$-scores. The threshold is set by {\\it higher criticism} (HC). Let $\\pee_i$ denote the $P$-value associated to the $i$-th $z$-score and $\\pee_{(i)}$ denote the $i$-th order statistic of the collection of $P$-values. The HC threshold (HCT) is the order statistic of the $z$-score corresponding to index $i$ maximizing $(i/n - \\pee_{(i)})/\\sqrt{\\pee_{(i)}(1-\\pee_{(i)})}$. The ideal threshold optimizes the classification error. In \\cite{PNAS} we showed that HCT was numerically close to the ideal threshold. We formalize an asymptotic framework for studying the RW model, considering a sequence of problems with increasingly many features and relatively fewer observations. We show that along this sequence, the limiting performance of ideal HCT is essentially just as good as the limiting performance of ideal thresholding. Our results describe two-dimensional {\\it phase space}, a two-dimensional diagram with coordinates quantifying \"rare\" and \"weak\" in the RW model. Phase space can be partitioned into two regions -- one where ideal threshold classification is successful, and one where the features are so weak and so rare that it must fail. Surprisingly, the regions where ideal HCT succeeds and fails make the exact same partition of the phase diagram. Other threshold methods, such as FDR threshold selection, are successful in a substantially smaller region of the phase space than either HCT or Ideal thresholding."}
{"category": "Math", "title": "Mixing time of exponential random graphs", "abstract": "Exponential random graphs are used extensively in the sociology literature. This model seeks to incorporate in random graphs the notion of reciprocity, that is, the larger than expected number of triangles and other small subgraphs. Sampling from these distributions is crucial for parameter estimation hypothesis testing, and more generally for understanding basic features of the network model itself. In practice sampling is typically carried out using Markov chain Monte Carlo, in particular either the Glauber dynamics or the Metropolis-Hasting procedure. In this paper we characterize the high and low temperature regimes of the exponential random graph model. We establish that in the high temperature regime the mixing time of the Glauber dynamics is $\\Theta(n^2 \\log n)$, where $n$ is the number of vertices in the graph; in contrast, we show that in the low temperature regime the mixing is exponentially slow for any local Markov chain. Our results, moreover, give a rigorous basis for criticisms made of such models. In the high temperature regime, where sampling with MCMC is possible, we show that any finite collection of edges are asymptotically independent; thus, the model does not possess the desired reciprocity property, and is not appreciably different from the Erd\\H{o}s-R\\'enyi random graph."}
{"category": "Math", "title": "Branching rules in the ring of superclass functions of unipotent upper-triangular matrices", "abstract": "It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters in this context."}
{"category": "Math", "title": "Solitary Wave Solutions for the Nonlinear Dirac Equations", "abstract": "In this paper we prove the existence and local uniqueness of stationary states for the nonlinear Dirac equation \\[ i \\sum_{j=0}^{3} \\ga^j \\pd_j \\psi - m\\psi + F(\\bar{\\psi}\\psi)\\psi =0 \\] where $ m >0$ and $ F(s) = |s|^{\\theta}$ for $ 1\\leq \\theta < 2.$ More precisely we show that there exists $\\e_0 > 0$ such that for $\\omega \\in(m - \\e_0, m), $ there exists a solution $ \\psi(t,x) = e^{-i\\omega t}\\phi_{\\omega}(x), x_0 = t, x = (x_1, x_2, x_3),$ and the mapping from $ \\omega $ to $ \\phi_{\\omega} $ is continuous. We prove this result by relating the stationary solutions to the ground states of nonlinear Schr\\\"{o}dinger equations."}
{"category": "Math", "title": "Density of commensurators for uniform lattices of right-angled buildings", "abstract": "Let G be the automorphism group of a regular right-angled building X. The \"standard uniform lattice\" \\Gamma_0 in G is a canonical graph product of finite groups, which acts discretely on X with quotient a chamber. We prove that the commensurator of \\Gamma_0 is dense in G. This result was also obtained by Haglund. For our proof, we develop carefully a technique of \"unfoldings\" of complexes of groups. We use unfoldings to construct a sequence of uniform lattices \\Gamma_n in G, each commensurable to \\Gamma_0, and then apply the theory of group actions on complexes of groups to the sequence \\Gamma_n. As further applications of unfoldings, we determine exactly when the group G is nondiscrete, and we prove that G acts strongly transitively on X."}
{"category": "Math", "title": "Badly approximable affine forms and Schmidt games", "abstract": "For any real number \\t, the set of all real numbers x for which there exists a constant c(x) > 0 such that \\inf_{p \\in \\ZZ} |\\t q - x - p| \\geq c(x)/|q| for all q in \\ZZ {0} is an 1/8-winning set."}
{"category": "Math", "title": "On sums and products in C[x]", "abstract": "We show that under the assumption of a 24-term version of Fermat's Last Theorem, there exists an absolute constant c > 0 such that if S is a set of n > n_0 positive integers satisfying |S.S| < n^(1+c), then the sumset S.S satisfies |S+S| >> n^2. In other words, we prove a weak form of the Erdos-Szemeredi sum-product conjecture, conditional on an extension of Fermat's Last Theorem. Unconditionally, we prove this theorem for when S is a set of n monic polynomials. We also prove an analogue of a theorem of Bourgain and Chang for the ring C[x]."}
{"category": "Math", "title": "A characterization of well-founded algebraic lattices", "abstract": "We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice $L$ is well-founded if and only if $K(L)$, the join-semilattice of compact elements of $L$, is well-founded and contains neither $[\\omega]^{<\\omega}$, nor $\\underline\\Omega(\\omega^*)$ as a join-subsemilattice. As an immediate corollary, we get that an algebraic modular lattice $L$ is well-founded if and only if $K(L)$ is well-founded and contains no infinite independent set. If $K(L)$ is a join-subsemilattice of $I_{<\\omega}(Q)$, the set of finitely generated initial segments of a well-founded poset $Q$, then $L$ is well-founded if and only if $K(L)$ is well-quasi-ordered."}
{"category": "Math", "title": "The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties", "abstract": "Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective threefold Y that admits a non-constant map to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of Y: in all relevant cases the minimal model program leads to a fiber space whose fibration factors the moduli map. A much refined affirmative answer to Viehweg's conjecture for families over threefolds follows as a corollary. For families over surfaces, the moduli map can be often be described quite explicitly. Slightly weaker results are obtained for families of varieties with trivial, or more generally semi-ample canonical bundle."}
{"category": "Math", "title": "Homeomorphic Solutions to Reduced Beltrami Equations", "abstract": "We study differential expressions related to linear families of quasiconformal mappings and give a simple and direct proof to a result due to Alessandrini and Nesi arXiv:0707.0727."}
{"category": "Math", "title": "On almost Blow-analytic equivalence", "abstract": "Approximation of real analytic functions by Nash functions is a classical topic in real geometry. In this paper, we focus on the Nash approximation of an analytic desingularization of a Nash function germ obtained by a sequence of blowings-up along smooth analytic centers. We apply the result to prove that Nash function germs that are analytically equivalent after analytic desingularizations are Nash equivalent after Nash desingularizations. Results are based on a precise Euclidean description of a sequence of blowings-up combined with N\\'eron Desingularization."}
{"category": "Math", "title": "An introduction to wonderful varieties with many examples of type F4", "abstract": "We give an introduction to the theory of wonderful G-varieties, with many examples when G is simple of type F4. We present results and open problems about these varieties: on their classification, on their isotropy groups, on morphisms between them, and on their relations with the representation theory of G."}
{"category": "Math", "title": "Combinatorial Formulas for Classical Lie Weight Systems on Arrow Diagrams", "abstract": "In 2002 Haviv gave a way of assigning Lie tensors to directed trivalent graphs. Weight systems on oriented chord idagrams modulo 6T can then be constructed from such tensors. In this paper we give explicit combinatorial formulas of weight systems using Manin triples constrcted from classical Lie algebras. We then compose these oriented weight systems with the averaging map to get weight systems on unoriented chord diagrams and show that they are the same as the ones obtained by Bar-Natan in 1991. In the last section we carry out calculations on certain examples."}
{"category": "Math", "title": "Axisymmetric Rotating Fluid Equations", "abstract": "We investigate the equations of anisotropic axisymmetric incompressible viscous fluids in the exterior of a cylinder of $\\R^3$, rotating around an inhomogeneous vector $B(t, r)$. We prove uniform local existence with respect to the Rossby number in suitable anisotropic Sobolev spaces. We also obtain the propagation of the isotropic Sobolev regularity."}
{"category": "Math", "title": "On Invertibility of Sobolev Mappings", "abstract": "We prove local and global invertibility of Sobolev solutions of certain differential inclusions which prevent the differential matrix from having negative eigenvalues. Our results are new even for quasiregular mappings in two dimensions."}
{"category": "Math", "title": "Extension Phenomena for Holomorphic Geometric Structures", "abstract": "The most commonly encountered types of complex analytic G-structures and Cartan geometries cannot have singularities of complex codimension 2 or more."}
{"category": "Math", "title": "The Virtual Private Network Design Problem with Concave Costs (Oberwolfach abstract)", "abstract": "The symmetric Virtual Private Network Design (VPND) problem is concerned with buying capacity on links (edges) in a communication network such that certain traffic demands can be met. We investigate a natural generalization of VPND where the cost per unit of capacity may decrease if a larger amount of capacity is reserved (economies of scale principle). The growth of the cost of capacity is modelled by a non-decreasing concave function $f$. We call the problem the concave symmetric Virtual Private Network Design (cVPND) problem. After showing that a generalization of the so-called Pyramidal Routing problem and hence also the cVPND have the so-called tree routing property, we study approximation algorithms for cVPND. For general $f$, using known results on the so-called Single Source Buy at Bulk problem by Grandoni and Italiano, we give a randomized 24.92-approximation algorithm."}
{"category": "Math", "title": "Some algebraic properties of hypergraphs", "abstract": "We consider Stanley--Reisner rings $k[x_1,...,x_n]/I(\\mc{H})$ where $I(\\mc{H})$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize upon some known results about chordal graphs and study a weak form of shellability."}
{"category": "Math", "title": "Dynamical Systems and Topological Surgery", "abstract": "In this paper we try to establish a connection between a three-dimensional Lotka--Volterra dynamical system and two-dimensional topological surgery. There are many physical phenomena exhibiting two-dimensional topological surgery through a `hole drilling' process. By our connection, such phenomena may be modelled mathematically by the above dynamical system."}
{"category": "Math", "title": "Nonsmooth Hormander vector fields and their control balls", "abstract": "We prove a ball-box theorem for nonsmooth Hormander vector fields of step s."}
{"category": "Math", "title": "Minimal coexistence configurations for multispecies systems", "abstract": "We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Neumann boundary conditions in dumbbell-like domains. Under suitable non-degeneracy assumptions, we show that, as the competition rate grows indefinitely, the system reaches a state of coexistence of all the species in spatial segregation. Furthermore, the limit configuration is a local minimizer for the associated free energy."}
{"category": "Math", "title": "Lines on Fermat surfaces", "abstract": "We prove that the Neron-Severi groups of several complex Fermat surfaces are generated by lines. Specifically, we obtain these new results for all degrees up to 100 that are relatively prime to 6. The proof uses reduction modulo a supersingular prime. The techniques are developed in detail. They can be applied to other surfaces and varieties as well."}
{"category": "Math", "title": "Dimension of elliptic harmonic measure of Snowspheres", "abstract": "A metric space $\\mathcal{S}$ is called a \\defn{quasisphere} if there is a quasisymmetric homeomorphism $f\\colon S^2\\to \\mathcal{S}$. We consider the elliptic harmonic measure, i.e., the push forward of 2-dimensional Lebesgue measure by $f$. It is shown that for certain self similar quasispheres $\\mathcal{S}$ (snowspheres) the dimension of the elliptic harmonic measure is strictly less than the Hausdorff dimension of $\\mathcal{S}$."}
{"category": "Math", "title": "Kinetic equations with Maxwell boundary conditions", "abstract": "We prove global stability results of {\\sl DiPerna-Lions} renormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann, Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace theorems of the kind previously introduced by the author for the Vlasov equations, new results concerning weak-weak convergence (the renormalized convergence and the biting $L^1$-weak convergence), as well as the Darroz\\`es-Guiraud information in a crucial way."}
{"category": "Math", "title": "Slow escaping points of meromorphic functions", "abstract": "We show that for any transcendental meromorphic function $f$ there is a point $z$ in the Julia set of $f$ such that the iterates $f^n(z)$ escape, that is, tend to $\\infty$, arbitrarily slowly. The proof uses new covering results for analytic functions. We also introduce several slow escaping sets, in each of which $f^n(z)$ tends to $\\infty$ at a bounded rate, and establish the connections between these sets and the Julia set of $f$. To do this, we show that the iterates of $f$ satisfy a strong distortion estimate in all types of escaping Fatou components except one, which we call a plane-filling wandering domain. We give examples to show how varied the structures of these slow escaping sets can be."}
{"category": "Math", "title": "Evaluating the Impact of Missing Data Imputation through the use of the Random Forest Algorithm", "abstract": "This paper presents an impact assessment for the imputation of missing data. The data set used is HIV Seroprevalence data from an antenatal clinic study survey performed in 2001. Data imputation is performed through five methods: Random Forests, Autoassociative Neural Networks with Genetic Algorithms, Autoassociative Neuro-Fuzzy configurations, and two Random Forest and Neural Network based hybrids. Results indicate that Random Forests are superior in imputing missing data in terms both of accuracy and of computation time, with accuracy increases of up to 32% on average for certain variables when compared with autoassociative networks. While the hybrid systems have significant promise, they are hindered by their Neural Network components. The imputed data is used to test for impact in three ways: through statistical analysis, HIV status classification and through probability prediction with Logistic Regression. Results indicate that these methods are fairly immune to imputed data, and that the impact is not highly significant, with linear correlations of 96% between HIV probability prediction and a set of two imputed variables using the logistic regression analysis."}
{"category": "Math", "title": "Rooted trees and symmetric functions: Zhao's homomorphism and the commutative hexagon", "abstract": "Recent work on perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which turns out to be self-dual. Using some homomorphisms defined by the author and W. Zhao, we describe a commutative diagram that relates the aforementioned Hopf algebras to each other and to the Hopf algebras of symmetric functions, noncommutative symmetric functions, and quasi-symmetric functions."}
{"category": "Math", "title": "Non existence of principal values of signed Riesz transforms of non integer dimension", "abstract": "In this paper we prove that, given s> 0, if E is a subset of R^m with positive and bounded s-dimensional Hausdorff measure H^s and the principal values of the s-dimensional signed Riesz transform of H^s|E exist H^s-almost everywhere in E, then s is integer. Other more general variants of this result are also proven."}
{"category": "Math", "title": "Categorical centers and Reshetikhin-Turaev invariants", "abstract": "The center Z(C) of a spherical fusion category C (over an arbitrary commutative ring) is modular. We give an algorithm for computing the Reshetikhin-Turaev invariant defined with Z(C). It is based on Hopf diagrams and an explicit description of the structure of the coend of Z(C)."}
{"category": "Math", "title": "Spectral norm of products of random and deterministic matrices", "abstract": "We study the spectral norm of matrices M that can be factored as M=BA, where A is a random matrix with independent mean zero entries, and B is a fixed matrix. Under the (4+epsilon)-th moment assumption on the entries of A, we show that the spectral norm of such an m by n matrix M is bounded by \\sqrt{m} + \\sqrt{n}, which is sharp. In other words, in regard to the spectral norm, products of random and deterministic matrices behave similarly to random matrices with independent entries. This result along with the previous work of M. Rudelson and the author implies that the smallest singular value of a random m times n matrix with i.i.d. mean zero entries and bounded (4+epsilon)-th moment is bounded below by \\sqrt{m} - \\sqrt{n-1} with high probability."}
{"category": "Math", "title": "On the Dec group of finite abelian Galois extensions over global fields", "abstract": "If K/F is a finite abelian Galois extension of global fields whose Galois group has exponent t, we prove that there exists a short exact sequence that has as a consequence that if t is square free, then Dec(K/F)=Br_{t}(K/F) which we use to show that prime exponent division algebras over Henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras. Finally, we construct a counterexample to the result for higher exponent division algebras."}
{"category": "Math", "title": "On the characterization of algebraically integrable plane foliations", "abstract": "We give a characterization theorem for non-degenerated plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree $r$ of a non-degenerated foliation as above provides the minimum number, $r+1$, of points in the projective plane through which infinitely many algebraic leaves of the foliation go."}
{"category": "Math", "title": "The double of a Hopf monad", "abstract": "The center Z(C) of an autonomous category C is monadic over C (if certain coends exist in C). The notion of Hopf monad naturally arises if one tries to reconstruct the structure of Z(C) in terms of its monad Z: we show that Z is a quasitriangular Hopf monad on C and Z(C) is isomorphic to the braided category Z-C of Z-modules. More generally, let T be a Hopf monad on an autonomous category C. We construct a Hopf monad Z_T on C, the centralizer of T, and a canonical distributive law of T over Z_T. By Beck's theory, this has two consequences. On one hand, D_T=Z_T T is a quasitriangular Hopf monad on C, called the double of T, and Z(T-C)= D_T-C as braided categories. As an illustration, we define the double D(A) of a Hopf algebra A in a braided autonomous category in such a way that the center of the category of A-modules is the braided category of D(A)-modules (generalizing the Drinfeld double). On the other hand, the canonical distributive law also lifts Z_T to a Hopf monad on T-C which gives the coend of T-C. Hence, for T=Z, an explicit description of the Hopf algebra structure of the coend of Z(C) in terms of the structural morphisms of C, which is useful in quantum topology."}
{"category": "Math", "title": "Theorem on best Diophantine approximations for linear forms", "abstract": "We prove a new quantitative result on the degeneracy of the dimension of the subspace spanned by the best Diophantine approximations for a linear form."}
{"category": "Math", "title": "Heat kernel bounds, ancient $\\kappa$ solutions and the Poincar\\'e conjecture", "abstract": "We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3 dimensional ancient $\\kappa$ solutions to the Ricci flow. As an application, using the $W$ entropy associated with the heat kernel, we give a different and shorter proof of Perelman's classification of backward limits of these ancient solutions. The current paper together with \\cite{Z:2} and a different proof of universal noncollapsing due to Chen and Zhu \\cite{ChZ:1} lead to a simplified proof of the Poincar\\'e conjecture without using reduced distance and reduced volume."}
{"category": "Math", "title": "Uniqueness of contact Hamiltonians of topological strictly contact isotopies", "abstract": "We prove that for regular contact forms there exists a bijective correspondence between the $C^0$ limits of sequences of smooth strictly contact isotopies and the limits with respect to the contact distance of their corresponding Hamiltonians."}
{"category": "Math", "title": "The example of a self-similar continuum which is not an attractor of any zipper", "abstract": "The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper."}
{"category": "Math", "title": "Characterization of linear groups whose reduced C*-algebras are simple", "abstract": "The reduced C*-algebra of a countable linear group G is shown to be simple if and only if G has no nontrivial normal amenable subgroups. Moreover, these conditions are shown to be equivalent to the uniqueness of tracial state on the aforementioned C*-algebra."}
{"category": "Math", "title": "Pullback of varieties by finite maps", "abstract": "We study the local geometry of the pullback of a variety via a finite holomorphic map. In particular, we are looking for properties of $V = F^{-1}(W)$ such that if $V$ has the property $A$, then $W$ must have the property $A$. We show that $A$ can be the property of normality or prefactoriality. We also show that $A$ can be the property of smoothness, under extra assumptions."}
{"category": "Math", "title": "Isolated points in the space of left orderings of a group", "abstract": "Let G be a left orderable group and LO(G) the space of all left orderings. We investigate the circumstances under which a left ordering < of G can correspond to an isolated point in LO(G), in particular we extend known results to cover the case of uncountable groups. With minor technical restrictions on the group G, we also find that no dense left ordering is isolated in LO(G), and that the closure of the set of all dense left orderings of G yields a dense G-delta set within a Cantor set of left orderings in LO(G). Lastly, we show that certain conditions on a discrete left ordering of G can guarantee that it is not isolated in LO(G), and we illustrate these ideas using the Dehornoy ordering of the braid groups."}
{"category": "Math", "title": "Universal deformation rings for the symmetric group S_4", "abstract": "Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S_4 denote the symmetric group on 4 letters. We determine the universal deformation ring R(S_4,V) for every kS_4-module V which has stable endomorphism ring k and show that R(S_4,V) is isomorphic to either k, or W[t]/(t^2,2t), or the group ring W[Z/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation."}
{"category": "Math", "title": "Fluctuation theory and exit systems for positive self-similar Markov processes", "abstract": "For a positive self-similar Markov process, X, we construct a local time for the random set, $\\Theta$, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R,H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set $\\Theta$ and the process X sampled on the local time scale. The process (R,H) is described in terms of a ladder process linked to the L\\'{e}vy process associated to X via Lamperti's transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R,H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012-1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying L\\'{e}vy process oscillates."}
{"category": "Math", "title": "Derived Mackey functors", "abstract": "For a finite group $G$, the so-called $G$-Mackey functors form an abelian category $M(G)$ that has many applications in the study of $G$-equivariant stable homotopy. One would expect that the derived category $D(M(G))$ would be similarly important as the \"homological\" counterpart of the $G$-equivariant stable homotopy category. It turns out that this is not so -- $D(M(G))$ is pathological in many respects. We propose and study a replacement for $D(M(G))$, a certain triangulated category $DM(G)$ of \"derived Mackey functors\" that contains $M(G)$ but is different from $D(M(G))$. We show that standard features of the $G$-equivariant stable homotopy category such as the fixed points functors of two types have exact analogs for the category $DM(G)$."}
{"category": "Math", "title": "On Group bijections $\\phi $ with $\\phi(B)=A$ and $\\forall a\\in B, a\\phi(a) \\notin A$", "abstract": "A {\\em Wakeford pairing} from $S$ onto $T$ is a bijection $\\phi : S \\to T$ such that $x\\phi(x)\\notin T,$ for every $x\\in S.$ The number of such pairings will be denoted by $\\mu(S,T)$. Let $A$ and $ B$ be finite subsets of a group $G$ with $1\\notin B$ and $|A|=|B|.$ Also assume that the order of every element of $B$ is $\\ge |B|$. Extending results due to Losonczy and Eliahou-Lecouvey, we show that $\\mu(B,A)\\neq 0.$ Moreover we show that $\\mu(B,A)\\ge \\min \\{\\frac{||B|+1}{3},\\frac{|B|(q-|B|-1)}{2q-|B|-4}\\},$ unless there is $a\\in A$ such that $|Aa^{-1}\\cap B|=|B|-1$ or $Aa^{-1}$ is a progression. In particular, either $\\mu(B,B) \\ge \\min \\{\\frac{||B|+1}{3},\\frac{|B|(q-|B|-1)}{2q-|B|-4}\\},$ or for some $a\\in B,$ $Ba^{-1}$ is a progression."}
{"category": "Math", "title": "The three-dimensional Finite Larmor Radius Approximation", "abstract": "Following Fr\\'enod and Sonnendr\\\"ucker, we consider the finite Larmor radius regime for a plasma submitted to a large magnetic field and take into account both the quasineutrality and the local thermodynamic equilibrium of the electrons. We then rigorously establish the asymptotic gyrokinetic limit of the rescaled and modified Vlasov-Poisson system in a three-dimensional setting with the help of an averaging lemma."}
{"category": "Math", "title": "The speed of a biased random walk on a percolation cluster at high density", "abstract": "We study the speed of a biased random walk on a percolation cluster on $\\Z^d$ in function of the percolation parameter $p$. We obtain a first order expansion of the speed at $p=1$ which proves that percolating slows down the random walk at least in the case where the drift is along a component of the lattice."}
{"category": "Math", "title": "Magmatic \"Quantum-Like\" Systems", "abstract": "Quantum computation has suggested, among others, the consideration of \"non-quantum\" systems which in certain respects may behave \"quantum-like\". Here, what algebraically appears to be the most general possible known setup, namely, of {\\it magmas} is used in order to construct \"quantum-like\" systems. The resulting magmatic composition of systems has as a well known particular case the tensor products."}
{"category": "Math", "title": "Reduced genus-two Gromov-Witten Invariants for complex manifolds", "abstract": "In this article, we construct the reduced genus-two Gromov-Witten invariants for certain almost K\\\"{a}hler manifold $(X, \\omega, J)$ such that $J$ is integrable and satisfies some regularity conditions. In particular, the standard projective space $(\\P^n, \\omega_0, J_0)$ of dimension $n \\le 7$ satisfies these conditions. This invariant counts the number of simple genus-two $J$-holomorphic curves that satisfy appropriate number of constraints."}
{"category": "Math", "title": "On a class of linearizable planar geodesic webs", "abstract": "We present a complete description of a class of linearizable planar geodesic webs which contain a parallelizable 3-subweb."}
{"category": "Math", "title": "Copulas for Markovian dependence", "abstract": "Copulas have been popular to model dependence for multivariate distributions, but have not been used much in modelling temporal dependence of univariate time series. This paper demonstrates some difficulties with using copulas even for Markov processes: some tractable copulas such as mixtures between copulas of complete co- and countermonotonicity and independence (Fr\\'{e}chet copulas) are shown to imply quite a restricted type of Markov process and Archimedean copulas are shown to be incompatible with Markov chains. We also investigate Markov chains that are spreadable or, equivalently, conditionally i.i.d."}
{"category": "Math", "title": "The diffeomorphism group of a Lie foliation", "abstract": "We explicitly compute the diffeomorphism group of several types of linear foliations (with dense leaves) on the torus $T^n$, $n\\geq 2$, namely codimension one foliations, flows, and the so-called non-quadratic foliations. We show in particular that non-quadratic foliations are rigid, in the sense that they do not admit transverse diffeomorphisms other than $\\pm \\id$ and translations. The computation is an application of a general formula that we prove for the diffeomorphism group of any Lie foliation with dense leaves on a compact manifold. Our results generalize those of P. Donato and P. Iglesias for $T^2$, P. Iglesias and G. Lachaud for codimension one foliations on $T^n$, $n\\geq 2$, and B. Herrera for transcendent foliations. The theoretical setting of the paper is that of J. M. Souriau's diffeological spaces."}
{"category": "Math", "title": "A Bernoulli linked-twist map in the plane", "abstract": "We prove that a Lebesgue measure-preserving linked-twist map defined in the plane is metrically isomorphic to a Bernoulli shift (and thus strongly mixing). This is the first such result for an explicitly defined linked-twist map on a manifold other than the two-torus. Our work builds on that of Wojtkowski who established an ergodic partition for this example using an invariant cone-field in the tangent space."}
{"category": "Math", "title": "Note on on Dedekind type DC sums", "abstract": "In this paper we consider Dedekind type DC sums and prove receprocity laws related to DC sums."}
{"category": "Math", "title": "On the comparison of the Dirichlet and Neumann counting functions", "abstract": "The aim of the paper is to show that L. Friedlander's results on the relation between Dirichlet and Neumann counting functions (Arch. Ration. Mech. Anal. 116, 1991) remain valid in abstract setting."}
{"category": "Math", "title": "Complementary Regions of Knot and Link Diagrams", "abstract": "An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. This paper is an investigation into which sequences, either finite or infinite, are universal. We also consider how to minimize the number of odd-sided faces for projections of knots and links with n components."}
{"category": "Math", "title": "Precise subelliptic estimates for a class of special domains", "abstract": "For the $\\bar\\partial$-Neumann problem on a regular coordinate domain $\\Omega\\subset \\C^{n+1}$, we prove $\\epsilon$-subelliptic estimates for an index $\\epsilon$ which is in some cases better than $\\epsilon=\\frac1{2m}$ ($m$ being the {\\it multiplicity}) as it was previously proved by Catlin and Cho in \\cite{CC08}. This also supplies a much simplified proof of the existing literature. Our approach is founded on the method by Catlin in \\cite{C87} which consists in constructing a family of weights $\\{\\phi^\\delta\\}$ whose Levi form is bigger than $\\delta^{-2\\epsilon}$ on the $\\delta$-strip around $\\partial\\Omega$."}
{"category": "Math", "title": "On a C2-nonlinear subdivision scheme avoiding Gibbs oscillations", "abstract": "This paper is devoted to the presentation and the analysis of a new nonlinear subdivision scheme eliminating the Gibbs oscillations close to discontinuities. Its convergence, stability and order of approximation are analyzed. It is proved that this schemes converges towards limit functions of H\\\"older regularity index larger than 1.192. Numerical estimates provide an H\\\"older regularity index of 2.438. Up to our knowledge, this scheme is the first one that achieves simultaneously the control of the Gibbs phenomenon and regularity index larger than 1 for its limit functions."}
{"category": "Math", "title": "A generalization of strongly Gorenstein projective modules", "abstract": "This paper generalize the idea of the authors in J. Pure Appl. Algebra 210 (2007) 437--445. Namely, we define and study a particular case of Gorenstein projective modules. We investigate some change of rings results for this new kind of modules. Examples over not necessarily Noetherian rings are given."}
{"category": "Math", "title": "Applications of the Graph Minor Theorem to algebraic structures I", "abstract": "We use the Graph Minor Theorem to characterize infinite sequences of finite subsets of factorial and commutative semigroups (here semigroups have a unity element), e.g. the multiplicative semigroup of a unique factorization domain."}
{"category": "Math", "title": "Attractor Networks on Complex Flag Manifolds", "abstract": "Robbin and Salamon showed that attractor-repellor networks and Lyapunov maps are equivalent concepts and illustrate this with the example of linear flows on projective spaces. In these examples the fixed points are linearly ordered with respect to the Smale order which makes the attractor-repellor network overly simple. In this paper we provide a class of examples in which the attractor-repellor network and its lattice structure can be explicitly determined even though the Smale order is not total. They are associated with special flows on complex flag manifolds. In the process we show that the Smale order on the set of fixed points can be identified with the well-known Bruhat order. This could also be derived from results of Kazhdan and Lusztig, but we give a new proof using the Lambda-Lemma of Palis. For the convenience of the reader we also introduce the flag manifolds via elementary dynamical systems using only a minimum of Lie theory."}
{"category": "Math", "title": "Doubling rational normal curves", "abstract": "In this paper, we study double structures supported on rational normal curves. After recalling the general construction of double structures supported on a smooth curve described in \\cite{fer}, we specialize it to double structures on rational normal curves. To every double structure we associate a triple of integers $ (2r,g,n) $ where $ r $ is the degree of the support, $ n \\geq r $ is the dimension of the projective space containing the double curve, and $ g $ is the arithmetic genus of the double curve. We compute also some numerical invariants of the constructed curves, and we show that the family of double structures with a given triple $ (2r,g,n) $ is irreducible. Furthermore, we prove that the general double curve in the families associated to $ (2r,r+1,r) $ and $ (2r,1,2r-1) $ is arithmetically Gorenstein. Finally, we prove that the closure of the locus containing double conics of genus $ g \\leq -2 $ form an irreducible component of the corresponding Hilbert scheme, and that the general double conic is a smooth point of that component. Moreover, we prove that the general double conic in $ \\mathbb{P}^3 $ of arbitrary genus is a smooth point of the corresponding Hilbert scheme."}
{"category": "Math", "title": "Point configurations that are asymmetric yet balanced", "abstract": "A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group."}
{"category": "Math", "title": "On Poncelet's maps", "abstract": "Given two ellipses, one surrounding the other one, Poncelet introduced a map $P$ from the exterior one to itself by using the tangent lines to the interior ellipse. This procedure can be extended to any two smooth, nested and convex ovals and we call this type of maps Poncelet's maps. We recall what he proved around 1814 in the dynamical systems language: In the two ellipses case and when the rotation number of P is rational there exists a natural number such that P^n is the identity, or in other words, the Poncelet's map is conjugated to a rational rotation. In this paper we study general Poncelet's maps and give several examples of algebraic ovals where the corresponding Poncelet's map has a rational rotation number and it is not conjugated to a rotation. Finally, we also provide a new proof of Poncelet's result based on dynamical tools."}
{"category": "Math", "title": "$L^p$-improving estimates for averages on polynomial curves", "abstract": "In the combinatorial method proving of $L^p$-improving estimates for averages along curves pioneered by Christ (IMRN, 1998), it is desirable to estimate the average modulus (with respect to some uniform measure on a set) of a polynomial-like function from below using only the value of the function or its derivatives at some prescribed point. In this paper, it is shown that there is always a relatively large set of points (independent of the particular function to be integrated) for which such estimates are possible. Inequalities of this type are then applied to extend the results of Tao and Wright (JAMS, 2003) to obtain endpoint restricted weak-type estimates for averages over curves given by polynomials."}
{"category": "Math", "title": "Finite groups have even more conjugacy classes", "abstract": "In his paper \"Finite groups have many conjugacy classes\" (J. London Math. Soc (2) 46 (1992), 239-249), L. Pyber proved the to date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber's paper."}
{"category": "Math", "title": "On Taking Square Roots without Quadratic Nonresidues over Finite Fields", "abstract": "We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in $\\tilde{O}(\\log^2 q)$ bit operations over finite fields with $q$ elements. As an application, we construct a deterministic primality proving algorithm, which runs in $\\tilde{O}(\\log^3 N)$ for some integers $N$."}
{"category": "Math", "title": "Identities for the Riemann zeta function", "abstract": "We obtain several expansions for $\\zeta(s)$ involving a sequence of polynomials in $s$, denoted in this paper by $\\alpha_k(s)$. These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some series expansions for the zeta function that are known for integer values of $s$. The expansions also give a different approach to the analytic continuation of the Riemann zeta function."}
{"category": "Math", "title": "Deterministic Primality Proving on Proth Numbers", "abstract": "We present an algorithm to decide the primality of Proth numbers, N=2^e t+1, without assuming any unproven hypothesis. The expected running time and the worst case running time of the algorithm are O ((t log t + log N)log N) and O ((t log t + log N) log^2 N) bit operations, respectively."}
{"category": "Math", "title": "Generalized (\\kappa,\\mu)-space forms", "abstract": "Generalized (\\kappa ,\\mu)-space forms are introduced and studied. We examine in depth the contact metric case and present examples for all possible dimensions. We also analyse the trans-Sasakian case."}
{"category": "Math", "title": "The second moment of Dirichlet twists of Hecke $L$-functions", "abstract": "Fix a Hecke cusp form $f$, and consider the $L$-function of $f$ twisted by a primitive Dirichlet character. As we range over all primitive characters of a large modulus $q$, what is the average behavior of the square of the central value of this $L$-function? Stefanicki proved an asymptotic valid only for $q$ having very few prime factors, and we extend this to almost all $q$."}
{"category": "Math", "title": "Additive Bases in Abelian Groups", "abstract": "Let $G$ be a finite, non-trivial abelian group of exponent $m$, and suppose that $B_1, ..., B_k$ are generating subsets of $G$. We prove that if $k>2m \\ln \\log_2 |G|$, then the multiset union $B_1\\cup...\\cup B_k$ forms an additive basis of $G$; that is, for every $g\\in G$ there exist $A_1\\subset B_1, ..., A_k\\subset B_k$ such that $g=\\sum_{i=1}^k\\sum_{a\\in A_i} a$. This generalizes a result of Alon, Linial, and Meshulam on the additive bases conjecture. As another step towards proving the conjecture, in the case where $B_1, ..., B_k$ are finite subsets of a vector space we obtain lower-bound estimates for the number of distinct values, attained by the sums of the form $\\sum_{i=1}^k \\sum_{a\\in A_i} a$, where $A_i$ vary over all subsets of $B_i$ for each $i=1, >..., k$. Finally, we establish a surprising relation between the additive bases conjecture and the problem of covering the vertices of a unit cube by translates of a lattice, and present a reformulation of (the strong form of) the conjecture in terms of coverings."}
{"category": "Math", "title": "Property A and asymptotic dimension", "abstract": "The purpose of this note is to characterize the asymptotic dimension $asdim(X)$ of metric spaces $X$ in terms similar to Property A of Yu: If $(X,d)$ is a metric space and $n\\ge 0$, then the following conditions are equivalent: [a.] $asdim(X,d)\\leq n$, [b.] For each $R,\\epsilon > 0$ there is $S > 0$ and finite non-empty subsets $A_x\\subset B(x,S)\\times N$, $x\\in X$, such that $\\frac{| A_x\\Delta A_y|}{| A_x\\cap A_y|} < \\epsilon$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\\in X$, [c.] For each $R > 0$ there is $S > 0$ and finite non-empty subsets $A_x\\subset B(x,S)\\times N$, $x\\in X$, such that $\\frac{| A_x\\Delta A_y|}{| A_x\\cap A_y|} < \\frac{1}{n+1}$ if $d(x,y) < R$ and the projection of $A_x$ onto $X$ contains at most $n+1$ elements for all $x\\in X$."}
{"category": "Math", "title": "Poynting-Robertson Effect on the Lyapunov Stability of Equilibrium Points in the Generalized Photogravitational Chermnykh's Problem", "abstract": "The Poynting-Robertson(P-R) effect on Lyapunov stability of equilibrium points is being discussed in the the generalized photogravitational Chermnykh's problem when bigger primary is a sours of radiation and smaller primary is an oblate spheriod. We derived the equations of motion, obtained the equilibrium points and examined the linear stability of the equilibrium points for various values of parameter which have been used in the present problem. We have examined the effect of gravitational potential from the belt. The positions of the equilibrium points are different from the position in classical case. We have seen that due to the P-R effect all the equilibrium points are unstable in Lyapunov sense."}
{"category": "Math", "title": "Dunkl Operators and Canonical Invariants of Reflection Groups", "abstract": "Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions."}
{"category": "Math", "title": "Central extensions of current algebras", "abstract": "This is an old paper put here for archeological purposes. We compute the second cohomology of current Lie algebras of the form $L\\otimes A$, where $L$ belongs to some class of Lie algebras which includes classical simple and Zassenhaus algebras, and of some modular semisimple Lie algebras. The results are largely superseded by subsequent papers, though, perhaps, some tricks and observations used here remain of minor interest."}
{"category": "Math", "title": "Nonparametric Estimation of Variance Function for Functional Data", "abstract": "This article investigates nonparametric estimation of variance functions for functional data when the mean function is unknown. We obtain asymptotic results for the kernel estimator based on squared residuals. Similar to the finite dimensional case, our asymptotic result shows the smoothness of the unknown mean function has an effect on the rate of convergence. Our simulaton studies demonstrate that estimator based on residuals performs much better than that based on conditional second moment of the responses."}
{"category": "Math", "title": "Stable isomorphism of dual operator spaces", "abstract": "We prove that two dual operator spaces $X$ and $Y$ are stably isomorphic if and only if there exist completely isometric normal representations $\\phi$ and $\\psi$ of $X$ and $Y$, respectively, and ternary rings of operators $M_1, M_2$ such that $\\phi (X)= [M_2^*\\psi (Y)M_1]^{-w^*}$ and $\\psi (Y)=[M_2\\phi (X)M_1^*].$ We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. We provide examples motivated by CSL algebra theory."}
{"category": "Math", "title": "Conformal Killing graphs with prescribed mean curvature", "abstract": "We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field."}
{"category": "Math", "title": "Compact Kaehler quotients of algebraic varieties and Geometric Invariant Theory", "abstract": "Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kaehler quotient. Additionally, as a byproduct of our discussion we give an example of a complete Kaehlerian non-projective algebraic surface, which may be of independent interest."}
{"category": "Math", "title": "Examples of scalar-flat hypersurfaces in $\\mathbb{R}^{n+1}$", "abstract": "Given a hypersurface $M$ of null scalar curvature in the unit sphere $\\mathbb{S}^n$, $n\\ge 4$, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in $\\Rr^{n+1}$ as a normal graph over a truncated cone generated by $M$. Furthermore, this graph is 1-stable if the cone is strictly 1-stable."}
{"category": "Math", "title": "Higher order Schwarzian derivatives in interval dynamics", "abstract": "We introduce an infinite sequence of higher order Schwarzian derivatives closely related to the theory of monotone matrix functions. We generalize the classical Koebe lemma to maps with positive Schwarzian derivatives up to some order, obtaining control over derivatives of high order. For a large class of multimodal interval maps we show that all inverse branches of first return maps to sufficiently small neighbourhoods of critical values have their higher order Schwarzian derivatives positive up to any given order."}
{"category": "Math", "title": "A lower bound on Seshadri constants of hyperplane bundles on threefolds", "abstract": "We give the lower bound on Seshadri constants for the case of very ample line bundles on threefolds. We consider the situation when the Seshadri constant is strictly less than 2 and give a version of Bauer's theorem \\cite[Theorem 2.1]{B1} for singular surfaces so we can prove the same result for smooth threefolds."}
{"category": "Math", "title": "Generalization of the \"Stark unit\" for abelian L-functions with multiple zeros", "abstract": "For an abelian extension of number fields we show that the Stark conjecture for all Artin L-functions with zero of order r is equivalent to existence of a special element in the rational span of the r-th exterior power of the Galois module of units of the bigger field."}
{"category": "Math", "title": "Sparse graphs: metrics and random models", "abstract": "Recently, Bollob\\'as, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with $n$ vertices and $\\Theta(n)$ edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function $\\ka:[0,1]^2 \\to [0,\\infty)$. To understand these models, we should like to know when different kernels $\\ka$ give rise to `similar' graphs, and, given a real-world network, how `similar' is it to a typical graph $G(n,\\ka)$ derived from a given kernel $\\ka$. The analogous questions for dense graphs, with $\\Theta(n^2)$ edges, are answered by recent results of Borgs, Chayes, Lov\\'asz, S\\'os, Szegedy and Vesztergombi, who showed that several natural metrics on graphs are equivalent, and moreover that any sequence of graphs converges in each metric to a graphon, i.e., a kernel taking values in $[0,1]$. Possible generalizations of these results to graphs with $o(n^2)$ but $\\omega(n)$ edges are discussed in a companion paper [arXiv:0708.1919]; here we focus only on graphs with $\\Theta(n)$ edges, which turn out to be much harder to handle. Many new phenomena occur, and there are a host of plausible metrics to consider; many of these metrics suggest new random graph models, and vice versa."}
{"category": "Math", "title": "On the complexity of Putinar's Positivstellensatz", "abstract": "We prove an upper bound on the degree complexity of Putinar's Positivstellensatz. This bound is much worse than the one obtained previously for Schm\\\"udgen's Positivstellensatz but it depends on the same parameters. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints."}
{"category": "Math", "title": "Vanishing theorems for Dolbeault cohomology of log homogeneous varieties", "abstract": "We consider a complete nonsingular variety $X$ over $\\bC$, having a normal crossing divisor $D$ such that the associated logarithmic tangent bundle is generated by its global sections. We show that $H^i\\big(X, L^{-1} \\otimes \\Omega_X^j(\\log D)\\big) = 0$ for any nef line bundle $L$ on $X$ and all $i < j - c$, where $c$ is an explicit function of $(X,D,L)$. This implies e.g. the vanishing of $H^i(X, L \\otimes \\Omega_X^j)$ for $L$ ample and $i > j$, and gives back a vanishing theorem of Broer when $X$ is a flag variety."}
{"category": "Math", "title": "Extreme lattices and vexillar designs", "abstract": "We define a notion of vexillar design for the flag variety in the spirit of the spherical designs introduced by Delsarte, Goethals and Seidel. For a finite subgroup of the orthogonal group, we explain how conditions on the group have the orbits of any flag under the group action be a design and point out why the minima of a lattice in the sense of the general Hermite constant forming a 4-design implies being extreme. The reasoning proves useful to show the extremality of many new expected examples ($E_8$, $\\La_{24}$, Barnes-Wall lattices, Thompson-Smith lattice for instance) that were out of reach until now."}
{"category": "Math", "title": "Bieri-Neumann-Strebel-Renz invariants and homology jumping loci", "abstract": "We investigate the relationship between the geometric Bieri-Neumann-Strebel-Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds can be expressed in terms of simpler data, for instance, the resonance varieties associated to the cohomology ring. These techniques yield information on the homological finiteness properties of free abelian covers of a given space, and of normal subgroups with abelian quotients of a given group. We illustrate our results in a variety of geometric and topological contexts, such as toric complexes and Artin kernels, as well as K\\\"ahler and quasi-K\\\"ahler manifolds."}
{"category": "Math", "title": "Equivariant Lie-Rinehart cohomology", "abstract": "In this paper, we study Lie-Rinehart cohomology for quotients of singularities by finite groups, and interpret these cohomology groups in terms of integrable connection on modules."}
{"category": "Math", "title": "Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems", "abstract": "A Banach space operator $T\\in B({\\cal X})$ is polaroid if points $\\lambda\\in\\iso\\sigma\\sigma(T)$ are poles of the resolvent of $T$. Let $\\sigma_a(T)$, $\\sigma_w(T)$, $\\sigma_{aw}(T)$, $\\sigma_{SF_+}(T)$ and $\\sigma_{SF_-}(T)$ denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi--Fredholm and lower semi--Fredholm spectrum of $T$. For $A$, $B$ and $C\\in B({\\cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $\\pi_0(M_C)=\\{\\lambda\\in\\iso\\sigma(M_C) 0<\\dim(M_C-\\lambda)^{-1}(0)<\\infty\\}$, $M_0$ satisfies Weyl's theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$ has SVEP at points $\\lambda\\in\\sigma_w(M_0)\\setminus\\sigma_{SF_+}(A)$ and $B$ has SVEP at points $\\mu\\in\\sigma_w(M_0)\\setminus\\sigma_{SF_-}(B)$, or, (ii) both $A$ and $A^*$ have SVEP at points $\\lambda\\in\\sigma_w(M_0)\\setminus\\sigma_{SF_+}(A)$, or, (iii) $A^*$ has SVEP at points $\\lambda\\in\\sigma_w(M_0)\\setminus\\sigma_{SF_+}(A)$ and $B^*$ has SVEP at points $\\mu\\in\\sigma_w(M_0)\\setminus\\sigma_{SF_-}(B)$, then $\\sigma(M_C)\\setminus\\sigma_w(M_C)=\\pi_0(M_C)$. Here the hypothesis that $\\lambda\\in\\pi_0(M_C)$ are poles of the resolvent of $A$ can not be replaced by the hypothesis $\\lambda\\in\\pi_0(A)$ are poles of the resolvent of $A$. For an operator $T\\in B(\\X)$, let $\\pi_0^a(T)=\\{\\lambda:\\lambda\\in\\iso\\sigma_a(T), 0<\\dim(T-\\lambda)^{-1}(0)<\\infty\\}$. We prove that if $A^*$ and $B^*$ have SVEP, $A$ is polaroid on $\\pi_0^a(\\M)$ and $B$ is polaroid on $\\pi_0^a(B)$, then $\\sigma_a(\\M)\\setminus\\sigma_{aw}(\\M)=\\pi_0^a(\\M)$."}
{"category": "Math", "title": "Heat-kernel estimates for random walk among random conductances with heavy tail", "abstract": "We study models of discrete-time, symmetric, $\\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $\\omega_{xy}\\in[0,1]$, with polynomial tail near 0 with exponent $\\gamma>0$. We first prove for all $d\\geq5$ that the return probability shows an anomalous decay (non-Gaussian) that approches (up to sub-polynomial terms) a random constant times $n^{-2}$ when we push the power $\\gamma$ to zero. In contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense, to the standard decay $n^{-d/2}$ for large values of the parameter $\\gamma$."}
{"category": "Math", "title": "Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields", "abstract": "We construct a 'triangulated analogue' of coniveau spectral sequences: the motif of a variety over a countable field is 'decomposed' (in the sense of Postnikov towers) into the twisted (co)motives of its points; this is generalized to arbitrary Voevodsky's motives. To this end we construct a 'Gersten' weight structure for a certain triangulated category of 'comotives': the latter is defined to contain comotives for all projective limits of smooth varieties; the definition of a weight structure was introduced in a preceding paper. The corresponding weight spectral sequences are essentially coniveau one; they are $DM^{eff}_{gm}$-functorial (starting from $E_2$) and can be computed in terms of the homotopy $t$-structure for the category $DM^-_{eff}$ (similarly to the case of smooth varieties). This extends to motives the seminal coniveau spectral sequence computations of Bloch and Ogus. We also obtain that the cohomology of a smooth semi-local scheme is a direct summand of the cohomology of its generic fibre; cohomology of function fields contain twisted cohomology of their residue fields (for all geometric valuations). We also develop further the general theory of weight structures for triangulated categories (independently from the 'motivic' part of the paper). Besides, we develop a certain theory of 'nice' pairings of triangulated categories; this subject seems to be new."}
{"category": "Math", "title": "Brody curves omitting hyperplanes", "abstract": "A Brody curve, a.k.a. normal curve, is a holomorphic map from the complex line to the complex projective space of dimension n, such that the family of its translations is normal. We prove that Brody curves omitting n hyperplanes in general position have growth order at most one, normal type. This generalizes a result of Clunie and Hayman who proved it for n=1."}
{"category": "Math", "title": "Equipartition of energy for the wave equation associated to the Dunkl-Cherednik Laplacian", "abstract": "This paper is concerned with energy properties of the wave equation associated to the Dunkl-Cherednik Laplacian. We establish the conservation of the total energy, the strict equipartition of energy under suitable assumptions and the asymptotic equipartition in the general case."}
{"category": "Math", "title": "Leaf superposition property for integer rectifiable currents", "abstract": "We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation."}
{"category": "Math", "title": "Singular limits for the Riemann problem. General diffusion, relaxation, and boundary conditions", "abstract": "We consider self-similar approximations of nonlinear hyperbolic systems in one space dimension with Riemann initial data and general diffusion matrix. We assume that the matrix of the system is strictly hyperbolic and the diffusion matrix is close to the identity. No genuine nonlinearity assumption is required. We show the existence of a smooth, self-similar solution which has bounded total variation, uniformly in the diffusion parameter. In the zero-diffusion limit, the solutions converge to a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation and the boundary-value problem in a half-space for the same regularizations."}
{"category": "Math", "title": "Stability of quantized time-delay nonlinear systems: A Lyapunov-Krasowskii-functional approach", "abstract": "Lyapunov-Krasowskii functionals are used to design quantized control laws for nonlinear continuous-time systems in the presence of constant delays in the input. The quantized control law is implemented via hysteresis to prevent chattering. Under appropriate conditions, our analysis applies to stabilizable nonlinear systems for any value of the quantization density. The resulting quantized feedback is parametrized with respect to the quantization density. Moreover, the maximal allowable delay tolerated by the system is characterized as a function of the quantization density."}
{"category": "Math", "title": "Haar method, averaged matrix, wave cancellations, and L1 stability for hyperbolic systems", "abstract": "We develop a version of Haar and Holmgren methods which applies to discontinuous solutions of nonlinear hyperbolic systems and allows us to control the L1 distance between two entropy solutions. The main difficulty is to cope with linear hyperbolic systems with discontinuous coefficients. Our main observation is that, while entropy solutions contain compressive shocks only, the averaged matrix associated with two such solutions has compressive or undercompressive shocks, but no rarefaction-shocks -- which are recognized as a source for non-uniqueness and instability. Our Haar-Holmgren-type method rests on the geometry associated with the averaged matrix and takes into account adjoint problems and wave cancellations along generalized characteristics. It extends the method proposed earlier by LeFloch et al. for genuinely nonlinear systems. In the present paper, we cover solutions with small total variation and a class of systems with general flux that need not be genuinely nonlinear and includes for instance fluid dynamics equations. We prove that solutions generated by Glimm or front tracking schemes depend continuously in the L1 norm upon their initial data, by exhibiting an L1 functional controling the distance between two solutions."}
{"category": "Math", "title": "On the NonKoszulity of (2p+1)-ary partially associative Operads", "abstract": "We present a study of quadratic operads for n-ary algebras and their dual for n odd. We will focus on the ternary case (i.e n=3). The aim is to underline the problem of computing the dual operad and the fact that this last is in general defined in the graded differential operad framework. We prove that the operad associated to 3-ary partially associative algebra is not Koszul. Recall that, if n is even, the operad of n-ary partially associative algebras is Koszul."}
{"category": "Math", "title": "Finite energy solutions to the isentropic Euler equations with geometric effects", "abstract": "Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical symmetry when the origin r=0 is included. These partial differential equations are hyperbolic, but fail to be strictly hyperbolic when the fluid mass density vanishes and vacuum is reached. Furthermore, when geometric effects are taken into account, the sup-norm of solutions can not be controlled since there exist no invariant regions. To overcome these difficulties and to establish an existence theory for solutions with arbitrarily large amplitude, we search for solutions with finite mass and total energy. Our strategy of proof takes advantage of the particular structure of the Euler equations, and leads to a versatile framework covering general compressible fluid problems. We establish first higher-integrability estimates for the mass density and the total energy. Next, we use arguments from the theory of compensated compactness and Young measures, extended here to sequences of solutions with finite mass and total energy. The third ingredient of the proof is a characterization of the unbounded support of entropy admissible Young measures. This requires the study of singular products involving measures and principal values."}
{"category": "Math", "title": "Curvature tensor under the complete non-compact Ricci Flow", "abstract": "We prove that for a solution $(M^n,g(t))$, $t\\in[0,T)$, where $T<\\infty$, to the Ricci flow with bounded curvature on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant $C$ on $M^n\\times [0,T)$, the curvature tensor stays uniformly bounded on $M^n\\times [0,T)$. Some other results are also presented."}
{"category": "Math", "title": "Synchronization of discrete-time dynamical networks with time-varying couplings", "abstract": "We study the local complete synchronization of discrete-time dynamical networks with time-varying couplings. Our conditions for the temporal variation of the couplings are rather general and include both variations in the network structure and in the reaction dynamics; the reactions could, for example, be driven by a random dynamical system. A basic tool is the concept of Hajnal diameter which we extend to infinite Jacobian matrix sequences. The Hajnal diameter can be used to verify synchronization and we show that it is equivalent to other quantities which have been extended to time-varying cases, such as the projection radius, projection Lyapunov exponents, and transverse Lyapunov exponents. Furthermore, these results are used to investigate the synchronization problem in coupled map networks with time-varying topologies and possibly directed and weighted edges. In this case, the Hajnal diameter of the infinite coupling matrices can be used to measure the synchronizability of the network process. As we show, the network is capable of synchronizing some chaotic map if and only if there exists an integer T>0 such that for any time interval of length T, there exists a vertex which can access other vertices by directed paths in that time interval."}
{"category": "Math", "title": "Sequential multiple hypothesis testing in presence of control variables", "abstract": "Suppose that at any stage of a statistical experiment a control variable $X$ that affects the distribution of the observed data $Y$ at this stage can be used. The distribution of $Y$ depends on some unknown parameter $\\theta$, and we consider the problem of testing multiple hypotheses $H_1: \\theta=\\theta_1$, $H_2: \\theta=\\theta_2, ...$, $H_k: \\theta=\\theta_k$ allowing the data to be controlled by $X$, in the following sequential context. The experiment starts with assigning a value $X_1$ to the control variable and observing $Y_1$ as a response. After some analysis, another value $X_2$ for the control variable is chosen, and $Y_2$ as a response is observed, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favor of one of the hypotheses $H_1,...$, $H_k$ is to be taken. In this article, our aim is to characterize the structure of optimal sequential testing procedures based on data obtained from an experiment of this type in the case when the observations $Y_1, Y_2,..., Y_n$ are independent, given controls $X_1,X_2,..., X_n$, $n=1,2,...$."}
{"category": "Math", "title": "Characteristic foliation on a hypersurface of general type in a projective symplectic manifold", "abstract": "Given a projective symplectic manifold $M$ and a non-singular hypersurface $X \\subset M$, the symplectic form of $M$ induces a foliation of rank 1 on $X$, called the characteristic foliation. We study the question when the characteristic foliation is algebraic, namely, all the leaves are algebraic curves. Our main result is that the characteristic foliation of $X$ is not algebraic if $X$ is of general type. For the proof, we first establish an \\'etale version of Reeb stability theorem in foliation theory and then combine it with the positivity of the direct image sheaves associated to families of curves."}
{"category": "Math", "title": "Weak isomorphisms between Bernoulli shifts", "abstract": "In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic."}
{"category": "Math", "title": "Commuting birth-and-death processes", "abstract": "We use methods from combinatorics and algebraic statistics to study analogues of birth-and-death processes that have as their state space a finite subset of the $m$-dimensional lattice and for which the $m$ matrices that record the transition probabilities in each of the lattice directions commute pairwise. One reason such processes are of interest is that the transition matrix is straightforward to diagonalize, and hence it is easy to compute $n$ step transition probabilities. The set of commuting birth-and-death processes decomposes as a union of toric varieties, with the main component being the closure of all processes whose nearest neighbor transition probabilities are positive. We exhibit an explicit monomial parametrization for this main component, and we explore the boundary components using primary decomposition."}
{"category": "Math", "title": "$k$-distant crossings and nestings of matchings and partitions", "abstract": "We define and consider k-distant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng (Electronic J. Combinatorics 2006, research paper 33), we show that the joint distribution of k-distant crossings and nestings is symmetric. We also study the numbers of k-distant noncrossing matchings and partitions for small k, which are counted by well-known sequences, as well as the orthogonal polynomials related to k-distant noncrossing matchings and partitions. We extend Chen et al.'s r-crossings and enhanced r-crossings."}
{"category": "Math", "title": "Square function and heat flow estimates on domains", "abstract": "The first purpose of this note is to provide a proof of the usual square function estimate on Lp (?). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on Gaussian bounds on the heat kernel. We also provide a simple proof of a weaker version of the square function estimate, which is enough in most instances involving dispersive PDEs. Moreover, we obtain, by a relatively simple integration by parts, several useful Lp (?; H) bounds for the derivatives of the heat ?ow with values in a given Hilbert space H."}
{"category": "Math", "title": "The Cauchy-Crofton formula and the Whitney arc property for definable sets", "abstract": "We use the Cauchy-Crofton formula to show that every definable cell (bounded by a ball with rational radius) in an O-minimal expansion of a field extension of the real numbers satisfies the Whitney arc property."}
{"category": "Math", "title": "Props in model categories and homotopy invariance of structures", "abstract": "We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object X homotopy equivalent to an algebra A over a cofibrant prop P inherits a P-algebra structure so that X defines a model of A in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of A-infinity algebras."}
{"category": "Math", "title": "On a connectedness theorem of Debarre", "abstract": "Under a slightly stronger hypothesis, one improves a connectedness result of Debarre [D] for a product of two projective spaces in terms of the extension problem of formal-rational functions (see Theorems 1.3 and 1.4 of the introduction)"}
{"category": "Math", "title": "Some remarks about the second Leibniz cohomology group for Lie algebras", "abstract": "We compare by a very elementary approach the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones. Examples are given of coupled cocycles. Some properties are deduced as to Leibniz deformations. We also consider the class of Lie algebras for which the Koszul 3-form is zero, and prove that it contains all quotients of Borel subalgebras, or of their nilradicals, of finite dimensional semisimple Lie algebras. Finally, a list of Kac-Moody types for indecomposable nilpotent Lie algebras of dimension $\\leqslant 7$ is given."}
{"category": "Math", "title": "Geometric Realizations of Hermitian curvature models", "abstract": "We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant *-scalar curvature which satisfies the Kaehler condition at the point in question."}
{"category": "Math", "title": "On $T_{2n-1}^\\perp$ spaces", "abstract": "This paper is devoted to the inequalities for mean values of functions from $T_{2n-1}^\\perp$. The simple proof of the classical Jackson inequality in the case of the second modulus of continuity may be considered as the consequence of our estimates. The problems of the sharp constants in classical Stechkin's inequality are also discussed."}
{"category": "Math", "title": "Nonparametric estimation of a trend based upon sampled continuous processes", "abstract": "Let X be a second order random process indexed by a compact interval [0,T]. Assume that n independent realizations of X are observed on a fixed grid of p time points. Under mild regularity assumptions on the sample paths of X, we show the asymptotic normality of suitable nonparametric estimators of the trend function mu = EX in the space C([0,T]) as n, p go to infinity and, using Gaussian process theory, we derive approximate simultaneous confidence bands for mu."}
{"category": "Math", "title": "Mean field frozen percolation", "abstract": "We define a modification of the Erdos-Renyi random graph process which can be regarded as the mean field frozen percolation process. We describe the behavior of the process using differential equations and investigate their solutions in order to show the self-organized critical and extremum properties of the critical frozen percolation model. We prove two limit theorems about the distribution of the size of the component of a typical frozen vertex."}
{"category": "Math", "title": "On jet bundles and generalized Verma modules", "abstract": "The aim of this paper is to initiate a study of the jet bundles on the grassmannian $X$ over a field of characteristic zero using higher direct images of $G$-linearized sheaves, Lie theoretic methods, enveloping algebra theoretic methods and generalized Verma modules. We calculate the $P$-module of the dual jet bundle $J^l(L)^*$ and prove it equals the $l$'th piece of the canonical filtration for $H^0(X,L)^*$. We use the results obtained to prove the discriminant of any linear system on any grassmannian is irreducible."}
{"category": "Math", "title": "Cone structure of $L^2$-Wasserstein spaces", "abstract": "The purpose of this paper is to understand the geometric structure of the $L^2$-Wasserstein space $\\pp$ over the Euclidean space.For this sake, we focus on its cone structure.One of our main results is that the $L^2$-Wasserstein space over a Polish space has a cone structure if and only if so does the underlying space.In particular, $\\pp$ turns out to have a cone structure.It is also shown that $\\pp$ splits $\\R^d$ isometrically but not $\\R^{d+1}$."}
{"category": "Math", "title": "Note on the Fenchel transform in the Heisenberg group", "abstract": "Given a real-valued function defined on the Heisenberg group, we provide a definition of abstract convexity and Fenchel transform that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the Euclidean case, a convex function can be characterized via its iterated Fenchel transform; the properties of the H-subdifferential play a crucial role."}
{"category": "Math", "title": "Volumes, Traces and Zeta Functions", "abstract": "Let $Q(x)$ be a quadratic form over $\\mathbb{R}^n$. The Epstein zeta function associated to $Q(x)$ is a well known function in number theory. We generalize the construction of the Epstein zeta function to a class of function $\\phi(x)$ defined in $\\mathbb{R}^n$ that we call $A-$homogeneous, where $A$ is a real aquare matrix of order $n$ having each eigenvalue in the left hal space $\\Re\\lambda>0$. Such a class includes all the homogeneous polynomials (positive outside the origin) and all the norms on $\\mathbb{R}^n$ which are smooth outside the origin. As in the classical (i.e. quadratic) case we prove that such zeta functions are obtained from the Mellin transforms of theta function of Jacobi type associated to the $A-$homogeneous function $\\phi(x)$. We prove that the zeta function associated to a $A-$homogeneous function $\\phi(x)$ which is positive and smooth outside the origin is an entire meromorphic function having a unique simple pole at $s=\\alpha$ the trace of the matrix $A$ with residue given by the product of the trace $\\alpha$ and the Lebesgue volume of the unit ball associated to $\\phi(x)$, that is the volume of the set $x\\in\\R^n$ satisfying $\\phi(x)<1$. We also prove that the theta funtion associated to $\\phi(x)$ has an asymptotic expansion near the origin. We find that the coefficients of such expansion depend on the values that the zeta function associated to $\\phi(x)$ assumes at the negative integers."}
{"category": "Math", "title": "Hardy inequalities for weighted Dirac operatos", "abstract": "An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight $r^{-b}$ for functions in $\\R^n$. The exact Hardy constant $c_b=c_b(n)$ is found and generalized minimizers are given. The constant $c_b$ vanishes on a countable set of $b$, which extends the known case $n=2$, $b=0$ which corresponds to the trivial Hardy inequality in $\\R^2$. Analogous inequalities are proved in the case $c_b=0$ under constraints and, with error terms, for a bounded domain."}
{"category": "Math", "title": "Symplectic implosion and non-reductive quotients", "abstract": "The symplectic implosion construction of Guillemin, Jeffrey and Sjamaar associates to a Hamiltonian action of a compact group K on a symplectic manifold X its 'imploded cross section'. When X is a complex projective variety and K acts linearly on X, this construction is closely related to geometric invariant theory (GIT) for the action on X of a maximal unipotent subgroup U of the complexification G of K. The aim of this paper is to generalise symplectic implosion to give a symplectic construction for GIT-like quotients by unipotent radicals U of arbitrary parabolic subgroups P of the complex reductive group G acting linearly on the projective variety X."}
{"category": "Math", "title": "Partial symmetry, reflection monoids and Coxeter groups", "abstract": "This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders."}
{"category": "Math", "title": "Estimates for invariant metrics on $\\Bbb C$-convex domains", "abstract": "Geometric lower and upper estimates are obtained for invariant metrics on $\\Bbb C$-convex domains containing no complex lines."}
{"category": "Math", "title": "Tutte polynomial and G-parking functions", "abstract": "Let $G$ be a connected graph with vertex set $\\{0,1,2,...,n\\}$. We allow $G$ to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of $G$-parking functions. In particular, we give the definition of the bridge vertex of a $G$-parking function and obtain an expression of the Tutte polynomial $T_G(x,y)$ of $G$ in terms of $G$-parking functions. We find the Tutte polynomial enumerates the $G$-parking function by the number of the bridge vertices."}
{"category": "Math", "title": "Sparse recovery under matrix uncertainty", "abstract": "We consider the model {eqnarray*}y=X\\theta^*+\\xi, Z=X+\\Xi,{eqnarray*} where the random vector $y\\in\\mathbb{R}^n$ and the random $n\\times p$ matrix $Z$ are observed, the $n\\times p$ matrix $X$ is unknown, $\\Xi$ is an $n\\times p$ random noise matrix, $\\xi\\in\\mathbb{R}^n$ is a noise independent of $\\Xi$, and $\\theta^*$ is a vector of unknown parameters to be estimated. The matrix uncertainty is in the fact that $X$ is observed with additive error. For dimensions $p$ that can be much larger than the sample size $n$, we consider the estimation of sparse vectors $\\theta^*$. Under matrix uncertainty, the Lasso and Dantzig selector turn out to be extremely unstable in recovering the sparsity pattern (i.e., of the set of nonzero components of $\\theta^*$), even if the noise level is very small. We suggest new estimators called matrix uncertainty selectors (or, shortly, the MU-selectors) which are close to $\\theta^*$ in different norms and in the prediction risk if the restricted eigenvalue assumption on $X$ is satisfied. We also show that under somewhat stronger assumptions, these estimators recover correctly the sparsity pattern."}
{"category": "Math", "title": "Refinements of Lattice paths with flaws", "abstract": "The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four parameters, namely peak, valley, double descent and double ascent. Let ${p}_{n,m,k}$ be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ peaks. First, we derive the reciprocity theorem for the polynomial $P_{n,m}(x)=\\sum\\limits_{k=1}^np_{n,m,k}x^k$. Then we find the Chung-Feller properties for the sum of $p_{n,m,k}$ and $p_{n,m,n-k}$. Finally, we provide a Chung-Feller type theorem for Dyck paths of length $n$ with $k$ double ascents: the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ double ascents is equal to the number of all the Dyck paths that have semi-length $n$, $k$ double ascents and never pass below the x-axis, which is counted by the Narayana number. Let ${v}_{n,m,k}$ (resp. $d_{n,m,k}$) be the number of all the Dyck paths of semi-length $n$ with $m$ flaws and $k$ valleys (resp. double descents). Some similar results are derived."}
{"category": "Math", "title": "A Unification of Two Refinements of Euler's Partition Theorem", "abstract": "We obtain a unification of two refinements of Euler's partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt's insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction."}
{"category": "Math", "title": "An empirical central limit theorem in L^1 for stationary sequences", "abstract": "In this paper, we derive asymptotic results for L^1-Wasserstein distance between the distribution function and the corresponding empirical distribution function of a stationary sequence. Next, we give some applications to dynamical systems and causal linear processes. To prove our main result, we give a Central Limit Theorem for ergodic stationary sequences of random variables with values in L^1. The conditions obtained are expressed in terms of projective-type conditions. The main tools are martingale approximations."}
{"category": "Math", "title": "On a congruence only holding for primes II", "abstract": "We present a primality criterium based on congruences for cyclotomic polynomials, and point out a way to generalize our result in order to obtain a family of similar criteria. No practical use is aimed however."}
{"category": "Math", "title": "Formulas for the Connes-Moscovici Hopf algebra", "abstract": "We give explicit formulas for the coproduct and the antipode in the Connes-Moscovici Hopf algebra $\\mathcal{H}_{\\tmop{CM}}$. To do so, we first restrict ourselves to a sub-Hopf algebra $\\mathcal{H}^1_{\\tmop{CM}}$ containing the nontrivial elements, namely those for which the coproduct and the antipode are nontrivial. There are two ways to obtain explicit formulas. On one hand, the algebra $\\mathcal{H}^1_{\\tmop{CM}}$ is isomorphic to the Fa\\`a di Bruno Hopf algebra of coordinates on the group of identity-tangent diffeomorphism and computations become easy using substitution automorphisms rather than diffeomorphisms. On the other hand, the algebra $\\mathcal{H}^1_{\\tmop{CM}}$ is isomorphic to a sub-Hopf algebra of the classical shuffle Hopf algebra which appears naturally in resummation theory, in the framework of formal and analytic conjugacy of vector fields. Using the very simple structure of the shuffle Hopf algebra, we derive once again explicit formulas for the coproduct and the antipode in $\\mathcal{H}^1_{\\tmop{CM}}$."}
{"category": "Math", "title": "Remarques sur une conjecture de Lang", "abstract": "The aim of this paper is to study a conjecture predicting a lower bound on the canonical height on abelian varieties, formulated by S. Lang and generalized by J. H. Silverman. We give here an asymptotic result on the height of Heegner points on the modular jacobian $J_{0}(N)$, and we derive non-trivial remarks about the conjecture."}
{"category": "Math", "title": "Minoration de la hauteur de Neron-Tate sur les surfaces abeliennes", "abstract": "This paper contains results concerning a conjecture made by Lang and Silverman predicting a lower bound for the canonical height on abelian varieties of dimension 2 over number fields. The method used here is a local height decomposition. We derive as corollaries uniform bounds on the number of torsion points on families of abelian surfaces and on the number of rational points on families of genus 2 curves."}
{"category": "Math", "title": "Differentiability of the arithmetic volume function", "abstract": "We introduce the positive intersection product in Arakelov geometry and prove that the arithmetic volume function is continuously differentiable. As applications, we compute the distribution function of the asymptotic measure of a Hermitian line bundle and several other arithmetic invariants."}
{"category": "Math", "title": "Almost prime values of the order of elliptic curves over finite fields", "abstract": "Let $E$ be an elliptic curve over $\\Q$ without complex multiplication, and which is not isogenous to a curve with non-trivial rational torsion. For each prime $p$ of good reduction, let $|E(\\F_p)|$ be the order of the group of points of the reduced curve over $\\F_p$. We prove in this paper that, under the GRH, there are at least $2.778 C_E^{\\rm twin} x / (\\log{x})^2$ primes $p$ such that $|E(\\F_p)|$ has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng and Murty & Miri. This is also the first result where the dependence on the conjectural constant $C_E^{\\rm twin}$ appearing in the twin prime conjecture for elliptic curves (also known as Koblitz's conjecture) is made explicit. This is achieved by sieving a slightly different sequence than the one used by previous authors. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant appearing in the upper bound for the number of primes $p$ such that $|E(\\F_p)|$ is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH."}
{"category": "Math", "title": "Universality in Complex Wishart ensembles: The 1 cut case", "abstract": "We studied universality of Wishart ensembles whose covariance matrix has 2 distinct eigenvalues and the number of each of these eigenvalue goes to infinity in the asymptotic limit. In this case, the limiting eigenvalue distribution can be supported on 1 or 2 disjoint intervals. In our previous work the case when the support consists of 2 intervals was studied. This paper complements our previous analysis and studied the case when the support consists of a single interval. By using Riemann-Hilbert analysis, we have shown that under proper rescaling of the eigenvalues, the limiting correlation kernel is given by the sine kernel and the Airy kernel in the bulk and the edge of the spectrum respectively. As a consequence, the behavior of the largest eigenvalue in this model is described by the Tracy-Widom distribution."}
{"category": "Math", "title": "Vibration Spectra of the $m$-Tree Fractal", "abstract": "We introduce a family of post-critically finite fractal trees indexed by the number of branches they possess. Then we produce a Laplacian operator on graph approximations to these fractals and use spectral decimation to describe the spectrum of the Laplacian on these trees. Lastly we consider the behavior of the spectrum as the number of branches increases."}
{"category": "Math", "title": "Intriguing sets of partial quadrangles", "abstract": "The point-line geometry known as a \\textit{partial quadrangle} (introduced by Cameron in 1975) has the property that for every point/line non-incident pair $(P,\\ell)$, there is at most one line through $P$ concurrent with $\\ell$. So in particular, the well-studied objects known as \\textit{generalised quadrangles} are each partial quadrangles. An \\textit{intriguing set} of a generalised quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalised quadrangles by Bamberg, Law and Penttila to partial quadrangles, which surprisingly gives insight into the structure of hemisystems and other intriguing sets of generalised quadrangles."}
{"category": "Math", "title": "Dirac operators for coadjoint orbits of compact Lie groups", "abstract": "The coadjoint orbits of compact Lie groups carry many K\\\"ahler structures, which include a Riemannian metric and a complex structure. We provide a fairly explicit formula for the Levi-Civita connection of the Riemannian metric, and we use the complex structure to give a fairly explicit construction of a canonical Dirac operator for the Riemannian metric, in a way that avoids use of the Spin^c groups. Substantial parts of our results apply to compact almost-Hermitian homogeneous spaces, and to other connections besides the Levi-Civita connection. For these other connections we give a criterion that is both necessary and sufficient for their Dirac operator to be formally self-adjoint. We hope to use the detailed results given here to clarify statements in the literature of high-eneregy physics concerning \"Dirac operators\" for matrix algebras that converge to coadjoint orbits. To facilitate this we employ here only global methods -- we never use local coordinate charts, and we use the cross-section modules of vector bundles."}
{"category": "Math", "title": "Field sensitivity to L^p variations of a scatterer", "abstract": "For the problem of diffraction of harmonic scalar waves by a lossless periodic slab scatterer, we analyze field sensitivity with respect to the material coefficients of the slab. The governing equation is the Helmholtz equation, which describes acoustic or electromagnetic fields. The main theorem establishes the variational Frechet derivative of the scattered field measured in the H^1 (root-mean-square-gradient) norm as a function of the material coefficients measured in an L^p (p-power integral) norm, with 2<p<infinity, as long as these coefficients are bounded above and below by positive constants and do not admit resonance. The derivative is Lipschitz continuous. We also establish the variational derivative of the transmitted energy with respect to the material coefficients in L^p."}
{"category": "Math", "title": "Super-sequences in the arc component of a compact connected group", "abstract": "Let G be an abelian topological group. The symbol \\hat{G} denotes the group of all continuous characters \\chi : G --> T endowed with the compact open topology. A subset E of G is said to be qc-dense in G provided that \\chi(E) \\subseteq \\phi([-1/4,1/4]) holds only for the trivial character \\chi \\in \\hat{G}, where \\phi : R --> T = R/Z is the canonical homomorphism. A super-sequence is a non-empty compact Hausdorff space S with at most one non-isolated point (to which S converges). We prove that an infinite compact abelian group G is connected if and only if its arc component G_a contains a super-sequence converging to 0 that is qc-dense in G. This gives as a corollary a recent theorem of Aussenhofer: For a connected locally compact abelian group G, the restriction homomorphism r : \\hat{G} --> \\hat{G}_a defined by r(\\chi) = \\chi\\restriction_{G_a} for \\chi \\in \\hat{G}, is a topological isomorphism. We also show that an infinite compact group G is connected if and only if its arc component G_a contains a super-sequence S converging to the identity e that generates a dense subgroup of G (equivalently, S \\setminus {e} is an infinite suitable set for G in the sense of Hofmann and Morris)."}
{"category": "Math", "title": "Parametric Bing and Krasinkiewicz maps: revisited", "abstract": "Let $M$ be a complete metric $ANR$-space such that for any metric compactum $K$ the function space $C(K,M)$ contains a dense set of Bing (resp., Krasinkiewicz) maps. It is shown that $M$ has the following property: If $f\\colon X\\to Y$ is a perfect surjection between metric spaces, then $C(X,M)$ with the source limitation topology contains a dense $G_\\delta$-subset of maps $g$ such that all restrictions $g|f^{-1}(y)$, $y\\in Y$, are Bing (resp., Krasinkiewicz) maps. We apply the above result to establish some mapping theorems for extensional dimension."}
{"category": "Math", "title": "On the non-quadraticity of values of the q-exponential function and related q-series", "abstract": "We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;\\lambda)=\\sum_{n=0}^\\infty\\frac{z^n}{\\prod_{j=1}^n(q^j-\\lambda)}, \\qquad |q|>1, \\quad \\lambda\\notin q^{\\mathbb Z_{>0}}, $$ that includes as special cases the Tschakaloff function ($\\lambda=0$) and the $q$-exponential function ($\\lambda=1$). In particular, we prove the non-quadraticity of the numbers $F_q(\\alpha;\\lambda)$ for integral $q$, rational $\\lambda$ and $\\alpha\\notin-\\lambda q^{\\mathbb Z_{>0}}$, $\\alpha\\ne0$."}
{"category": "Math", "title": "Jordan *-homomorphisms on $C^*$-algebras", "abstract": "In this paper, we investigate Jordan *-homomorphisms on $C^*$-algebras associated with the following functional inequality $\\|f(\\frac{b-a}{3})+f(\\frac{a-3c}{3})+f(\\frac{3a+3c-b}{3})\\| \\leq \\|f(a)\\|.$ We moreover prove the superstability and the generalized Hyers-Ulam stability of Jordan *-homomorphisms on $C^*$-algebras associated with the following functional equation $$f(\\frac{b-a}{3})+f(\\frac{a-3c}{3})+f(\\frac{3a+3c-b}{3})=f(a).$$"}
{"category": "Math", "title": "Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces", "abstract": "In this paper, we achieve the general solution and the generalized Hyers-Ulam-Rassias stability of the following functional equation $$f(x+ky)+f(x-ky)=k^2f(x+y)+k^2f(x-y)+2(1-k^2)f(x)\\eqno {2 cm}$$for fixed integers $k$ with $k\\neq0,\\pm1$ in the quasi-Banach spaces."}
{"category": "Math", "title": "Stability of homomorphisms and derivations in $C^*$-ternary algebras", "abstract": "In this paper, we investigate homomorphisms between $C^*$-ternary algebras and derivations on $C^*$-ternary algebras, associated with the following functional equation $$f(\\frac{x_2-x_1}{3})+f(\\frac{x_1-3x_3}{3})+f(\\frac{3x_1+3x_3-x_2}{3})=f(x_1).$$ Moreover, we prove the generalized Hyers-Ulam -Rassias stability of homomorphisms in $C^*$-ternary algebras and of derivations on $C^*$-ternary algebras."}
{"category": "Math", "title": "n-Jordan homomorphisms", "abstract": "Let $n\\in \\Bbb N,$ and let $A,B$ be two rings. An additive map $h: A\\to B$ is called n-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $a \\in {A}$. Every Jordan homomorphism is an n-Jordan homomorphism, for all $n\\geq 2,$ but the converse is false, in general. In this paper we investigate the n-Jordan homomorphisms on Banach algebras. Indeed some results related to continuity are given as well."}
{"category": "Math", "title": "From Schoenberg to Pick-Nevanlinna: Toward a complete picture of the variogram class", "abstract": "We show that a large subclass of variograms is closed under products and that some desirable stability properties, such as the product of special compositions, can be obtained within the proposed setting. We introduce new classes of kernels of Schoenberg-L\\'{e}vy type and demonstrate some important properties of rotationally invariant variograms."}
{"category": "Math", "title": "On the chromatic number of random d-regular graphs", "abstract": "In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k-1 or k. If moreover d>(2k-3)log(k-1), then the value k-1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k+1. Our proof applies the small subgraph conditioning method to the number of balanced k-colourings, where a colouring is balanced if the number of vertices of each colour is equal."}
{"category": "Math", "title": "Stability of a mixed type quadratic, cubic and quartic functional equation", "abstract": "In this paper, we obtain the general solution and the generalized Hyers-Ulam Rassias stability of the functional equation $$3(f(x+2y)+f(x-2y))=12(f(x+y)+f(x-y))+4f(3y)-18f(2y)+36f(y)-18f(x).$$"}
{"category": "Math", "title": "Entropy vs volume for pseudo-Anosov maps", "abstract": "We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichm\\\"uller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a family of pseudo-Anosov maps which violates one side of inequalities under unbounded geometry setting, present an explicit bounding constant for a punctured torus, and provide several observations based on experiments."}
{"category": "Math", "title": "Zero entropy invariant measures for some skew product diffeomorphisms", "abstract": "In this paper we study some skew product diffeomorphisms with nonuniformly hyperbolic structure along fibers. We show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers."}
{"category": "Math", "title": "Generalizations of Chung-Feller Theorem", "abstract": "The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this paper, we find the connections between these two Chung-Feller theorems. We focus on the weighted versions of three classes of lattice paths and give the generalizations of the above two theorems. We prove the Chung-Feller theorems of Dyck type for these three classes of lattice paths and the Chung-Feller theorems of Motzkin type for two of these three classes. From the obtained results, we find an interesting fact that many lattice paths have the Chung-Feller properties of both Dyck type and Motzkin type."}
{"category": "Math", "title": "The L_\\infty-deformation complex of diagrams of algebras", "abstract": "The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L_\\infty-algebra which induces a graded Lie bracket on cohomology. As an example, the L_\\infty-algebra structure on the deformation complex of an associative algebra morphism g, with the underlying cochain complex isomorphic to the Gerstenhaber-Schack complex of g, is constructed. Another example is the deformation complex of a Lie algebra morphism. The last example is the diagram describing two mutually inverse morphisms of vector spaces. Its L_\\infty-deformation complex has a nontrivial constant term. Explicit formulas for the L_\\infty-operations in the above examples are given. A typical deformation complex of a diagram of algebras is a fully-fledged L_\\infty-algebra with nontrivial higher operations."}
{"category": "Math", "title": "Hazard Estimation under Generalized Censoring", "abstract": "This paper focuses on the problem of the estimation of the cumulative hazard function of a distribution on a general complete separable metric space when the data points are subject to censoring by an arbitrary adapted random set. A problem involving observability of the estimator proposed in [8] and [9] is resolved and a functional central limit theorem is proven for the revised estimator. Several examples and applications are discussed, and the validity of bootstrap methods is established in each case."}
{"category": "Math", "title": "Recent Developments in Nonregular Fractional Factorial Designs", "abstract": "Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditional analysis focuses on main effects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs."}
{"category": "Math", "title": "Transformations of locally conformally K\\\"ahler manifolds", "abstract": "We consider several transformation groups of a locally conformally K\\\"ahler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperk\\\"ahler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally K\\\"ahler manifold which is neither Weyl-reducible nor locally conformally hyperk\\\"ahler are holomorphic and conformal"}
{"category": "Math", "title": "Asymptotics for the size of the largest component scaled to \"log n\" in inhomogeneous random graphs", "abstract": "We study the inhomogeneous random graphs in the subcritical case. We derive an exact formula for the size of the largest connected component scaled to $\\log n$ where $n$ is the size of the graph. This generalizes the recent result for the \"rank 1 case\". Here we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, plays the crucial role in the subcritical case as well. But now these are the negative solutions which come into play."}
{"category": "Math", "title": "On the classification of degree 1 elliptic threefolds with constant $j$-invariant", "abstract": "We describe the possible Mordell-Weil groups for degree 1 elliptic threefold with rational base and constant $j$-invariant. Moreover, we classify all such elliptic threefolds if the $j$-invariant is nonzero. We can use this classification to describe a class of singular hypersurfaces in $\\Ps(2,3,1,1,1)$ that admit no variation of Hodge structure."}
{"category": "Math", "title": "The number of Hecke eigenvalues of same signs", "abstract": "We give the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues. This improves upon a recent work of Kohnen, Lau & Shparlinski. Also, we study an analogous problem for short intervals."}
{"category": "Math", "title": "Mathematical model for resistance and optimal strategy", "abstract": "We propose a mathematical model for one pattern of charts studied in technical analysis: in a phase of consolidation, the price of a risky asset goes down $\\xi$ times after hitting a resistance level. We construct a mathematical strategy and we calculate the expectation of the wealth for the logaritmic utility function. Via simulations, we compare the strategy with the standard one."}
{"category": "Math", "title": "Strong Gaussian approximations of product-limit and Quantile Processes for Strong mixing and censored data", "abstract": "In this paper, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate $O((\\log n)^{-\\lambda})$ for some $\\lambda>0,$. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process."}
{"category": "Math", "title": "Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions", "abstract": "We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of coideal subalgebras. We reduce the study to KD(n) by proving that the others are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We derive many examples of lattices of intermediate subfactors of the inclusions of depth 2 associated to those Kac algebras, as well as the corresponding principal graphs, which is the original motivation. Along the way, we extend some general results on the Galois correspondence for depth 2 inclusions, and develop some tools and algorithms for the study of twisted group algebras and their lattices of coideal subalgebras. This research was driven by heavy computer exploration, whose tools and methodology we further describe."}
{"category": "Math", "title": "Towards finite generation of the canonical ring without the MMP", "abstract": "This paper is the first of two steps in a project to prove finite generation of the log canonical ring without Mori theory."}
{"category": "Math", "title": "Semiparametric regression estimation using noisy nonlinear non invertible functions of the observations", "abstract": "We investigate a semiparametric regression model where one gets noisy non linear non invertible functions of the observations. We focus on the application to bearings-only tracking. We first investigate the least squares estimator and prove its consistency and asymptotic normality under mild assumptions. We study the semiparametric likelihood process and prove local asymptotic normality of the model. This allows to define the efficient Fisher information as a lower bound for the asymptotic variance of regular estimators, and to prove that the parametric likelihood estimator is regular and asymptotically efficient. Simulations are presented to illustrate our results."}
{"category": "Math", "title": "Deformation of symmetric functions and the rational Steenrod algebra", "abstract": "In 1999, Reg Wood conjectured that the quotient of Q[x_1,...,x_n] by the action of the rational Steenrod algebra is a graded regular representation of the symmetric group S_n. As pointed out by Reg Wood, the analog of this statement is a well known result when the rational Steenrod algebra is replaced by the ring of symmetric functions; actually, much more is known about the structure of the quotient in this case. We introduce a non-commutative q-deformation of the ring of symmetric functions, which specializes at q=1 to the rational Steenrod algebra. We use this formalism to obtain some partial results. Finally, we describe several conjectures based on an extensive computer exploration. In particular, we extend Reg Wood's conjecture to q formal and to any q complex not of the form -a/b, with a in {1,...,n} and b a positive natural number."}
{"category": "Math", "title": "Existence and uniqueness of constant mean curvature spheres in Sol_3", "abstract": "We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H>1/(\\sqrt{3}), there exists a unique (up to left translations) immersed CMC H sphere S_H in Sol_3 (Hopf-type theorem). Moreover, this sphere S_H is embedded, and is therefore the unique (up to left translations) compact embedded CMC H surface in Sol_3 (Alexandrov-type theorem). The uniqueness parts of these results are also obtained for all real numbers H such that there exists a solution of the isoperimetric problem with mean curvature H."}
{"category": "Math", "title": "A geometric approach to Conn's linearization theorem", "abstract": "We give a soft geometric proof of the classical result due to Conn stating that a Poisson structure is linearizable around a singular point (zero) at which the isotropy Lie algebra is compact and semisimple."}
{"category": "Math", "title": "Maximal inequalities for dual Sobolev spaces $W^{-1,p}$ and applications to interpolation", "abstract": "We firstly describe a maximal inequality for dual Sobolev spaces W^{-1,p}. This one corresponds to a \"Sobolev version\" of usual properties of the Hardy-Littlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one seems to be new and we develop arguments in the general framework of Riemannian manifold. Then we present an application to obtain interpolation results for Sobolev spaces."}
{"category": "Math", "title": "Invariants alg\\'ebriques de graphes et reconstruction", "abstract": "We report on results about a study of algebraic graph invariants, based on computer exploration, and motivated by graph-isomorphism and reconstruction problems."}
{"category": "Math", "title": "Algebraic invariants of graphs; a study based on computer exploration", "abstract": "We consider the ring I_n of polynomial invariants over weighted graphs on n vertices. Our primary interest is the use of this ring to define and explore algebraic versions of isomorphism problems of graphs, such as Ulam's reconstruction conjecture. There is a huge body of literature on invariant theory which provides both general results and algorithms. However, there is a combinatorial explosion in the computations involved and, to our knowledge, the ring I_n has only been completely described for n<=4. This led us to study the ring I_n in its own right. We used intensive computer exploration for small n, and developed PerMuVAR, a library for MuPAD, for computing in invariant rings of permutation groups. We present general properties of the ring I_n, as well as results obtained by computer exploration for small n, including the construction of a medium sized generating set for I_5. We address several conjectures suggested by those results (low degree system of parameters, unimodality), for I_n as well as for more general invariant rings. We also show that some particular sets are not generating, disproving a conjecture of Pouzet related to reconstruction, as well as a lemma of Grigoriev on the invariant ring over digraphs. We finally provide a very simple minimal generating set of the field of invariants."}
{"category": "Math", "title": "Normal approximation for coverage models over binomial point processes", "abstract": "We give error bounds which demonstrate optimal rates of convergence in the CLT for the total covered volume and the number of isolated shapes, for germ-grain models with fixed grain radius over a binomial point process of $n$ points in a toroidal spatial region of volume $n$. The proof is based on Stein's method via size-biased couplings."}
{"category": "Math", "title": "Hyperbolic (1,2)-knots in S^3 with crosscap number two and tunnel number one", "abstract": "A knot in S^3 is said to have crosscap number two if it bounds a once-punctured Klein bottle but not a Moebius band. In this paper we give a method of constructing crosscap number two hyperbolic (1,2)-knots with tunnel number one which are neither 2-bridge nor (1,1)-knots. An explicit infinite family of such knots is discussed in detail."}
{"category": "Math", "title": "Binomial coefficients and the ring of p-adic integers", "abstract": "Let k>1 be an integer and let p be a prime. We show that if $p^a\\le k<2p^a$ or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\\binom{n}{k}: n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a complete system of residues modulo any power of p."}
{"category": "Math", "title": "Arithmetical rank of toric ideals associated to graphs", "abstract": "Let $I_{G} \\subset K[x_{1},...,x_{m}]$ be the toric ideal associated to a finite graph $G$. In this paper we study the binomial arithmetical rank and the $G$-homogeneous arithmetical rank of $I_G$ in 2 cases: $G$ is bipartite, $I_G$ is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the $G$-arithmetical rank coincide with the minimal number of generators of $I_G$."}
{"category": "Math", "title": "Patching and local-global principles for homogeneous spaces over function fields of p-adic curves", "abstract": "This is the final version, to appear in Commentarii Mathematici Helvetici."}
{"category": "Math", "title": "The structure of a local embedding and applications to Chern classes of weighted blow-ups", "abstract": "For a proper local embedding between two Deligne--Mumford stacks Y and X, we find, under certain mild conditions, a new (possibly non-separated) Deligne--Mumford stack X', with an etale, surjective and universally closed map to the target X, and whose fiber product with the image of the local embedding is a finite union of stacks with corresponding etale, surjective and universally closed maps to Y. Moreover, a natural set of weights on the substacks of X' allows the construction of a universally closed push-forward, and thus a comparison between the Chow groups of X' and X. We apply the construction above to the computation of the Chern classes of a weighted blow-up along a regular local embedding via deformation to a weighted normal cone and localization. We describe the stack X' in the case when X is the moduli space of stable maps with local embeddings at the boundary. We apply the methods above to find the Chern classes of the stable map spaces."}
{"category": "Math", "title": "Parameter estimation for rough differential equations", "abstract": "We construct the \"expected signature matching\" estimator for differential equations driven by rough paths and we prove its consistency and asymptotic normality. We use it to estimate parameters of a diffusion and a fractional diffusions, that is, a differential equation driven by fractional Brownian motion."}
{"category": "Math", "title": "Orders of Finite Reductive Monoids", "abstract": "We show four formulas for calculating the orders of finite reductive monoids with zero. As applications, these formulas are then used to calculate the orders of finite reductive monoids induced from the $F_q$-split $\\J$-irreducible monoids $\\overline {K^*\\rho(G_0)}$ where $G_0$ is a simple algebraic group over the algebraic closure of $F_q$, and $\\rho: G_0\\to GL(V)$ is the irreducible representation associated with any dominant weight. Finally, we give an explicit formula for the orders of finite symplectic monoids associated with the last fundamental dominant weight of type $C_l$; the connections to $H$-polynomials and Betti numbers are shown."}
{"category": "Math", "title": "Two-parameter Quantum Affine Algebra $U_{r,s}(\\widehat{\\frak {sl}_n})$, Drinfel'd Realization and Quantum Affine Lyndon Basis", "abstract": "We further define two-parameter quantum affine algebra $U_{r,s}(\\widehat{\\frak {sl}_n})$ $(n>2)$ after the work on the finite cases (see [BW1], [BGH1], [HS] & [BH]), which turns out to be a Drinfel'd double. Of importance for the quantum {\\it affine} cases is that we can work out the compatible two-parameter version of the Drinfel'd realization as a quantum affinization of $U_{r,s}(\\frak{sl}_n)$ and establish the Drinfel'd isomorphism Theorem in the two-parameter setting, via developing a new combinatorial approach (quantum calculation) to the quantum {\\it affine} Lyndon basis we present (with an explicit valid algorithm based on the use of Drinfel'd generators)."}
{"category": "Math", "title": "Accurate numerical linear algebra with Bernstein-Vandermonde matrices", "abstract": "The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein-Vandermonde matrix is considered. Bernstein-Vandermonde matrices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials of degree less than or equal to $n$ the Bernstein basis, a widely used basis in Computer Aided Geometric Design, instead of the monomial basis. Our approach is based on the computation of the bidiagonal factorization of a totally positive Bernstein-Vandermonde matrix (or its inverse) by means of Neville elimination. The explicit expressions obtained for the determinants involved in the process makes the algorithm both fast and accurate."}
{"category": "Math", "title": "Accurate computations with Said-Ball-Vandermonde matrices", "abstract": "A generalization of the Vandermonde matrices which arise when the power basis is replaced by the Said-Ball basis is considered. When the nodes are inside the interval (0,1), then those matrices are strictly totally positive. An algorithm for computing the bidiagonal decomposition of those Said-Ball-Vandermonde matrices is presented, which allows to use known algorithms for totally positive matrices represented by their bidiagonal decomposition. The algorithm is shown to be fast and to guarantee high relative accuracy. Some numerical experiments which illustrate the good behaviour of the algorithm are included."}
{"category": "Math", "title": "Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond", "abstract": "We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in $W^{1,2}$, to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions."}
{"category": "Math", "title": "Constant Mean Curvature Hypersurfaces Condensing to Geodesic Segments and Rays in Riemannian Manifolds", "abstract": "We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scalar curvature acts as a `friction term' which permits the usual analytic gluing construction to be carried out."}
{"category": "Math", "title": "Choosing a penalty for model selection in heteroscedastic regression", "abstract": "We consider the problem of choosing between several models in least-squares regression with heteroscedastic data. We prove that any penalization procedure is suboptimal when the penalty is a function of the dimension of the model, at least for some typical heteroscedastic model selection problems. In particular, Mallows' Cp is suboptimal in this framework. On the contrary, optimal model selection is possible with data-driven penalties such as resampling or $V$-fold penalties. Therefore, it is worth estimating the shape of the penalty from data, even at the price of a higher computational cost. Simulation experiments illustrate the existence of a trade-off between statistical accuracy and computational complexity. As a conclusion, we sketch some rules for choosing a penalty in least-squares regression, depending on what is known about possible variations of the noise-level."}
{"category": "Math", "title": "Orientation-reversing involutions of the genus 3 Arnoux-Yoccoz surface and related surfaces", "abstract": "We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square."}
{"category": "Math", "title": "Non-colliding Jacobi processes as limits of Markov chains on Gelfand-Tsetlin graph", "abstract": "We introduce a stochastic dynamics related to the measures that arise in harmonic analysis on the infinite-dimensional unitary group. Our dynamics is obtained as a limit of a sequence of natural Markov chains on Gelfand-Tsetlin graph. We compute finite-dimensional distributions of the limit Markov process, the generator and eigenfunctions of the semigroup related to this process. The limit process can be identified with Doob h-transform of a family of independent diffusions. Space-time correlation functions of the limit process have a determinantal form."}
{"category": "Math", "title": "Operator Algebras with Unique Preduals", "abstract": "We show that every free semigroup algebras has a (strongly) unique Banach space predual. We also provide a new simpler proof that a weak*-closed unital operator operator algebra containing a weak* dense subalgebra of compact operators has a unique Banach space predual."}
{"category": "Math", "title": "On the Geometry of the Second Fundamental Form of Translation Surfaces in E3", "abstract": "In this paper we study the second fundamental form of translation surfaces in E3. We give a non-existence result for polynomial translation surfaces in E3 with vanishing second Gaussian curvature KII. We classify those translation surfaces for which KII and H are proportional. Finally we obtain that there are no II-minimal translation surfaces in the Euclidean 3-space."}
{"category": "Math", "title": "An L^1 Ergodic Theorem for Sparse Random Subsequences", "abstract": "We prove an L^1 subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of universally L^1-good sequences nearly as sparse as the set of squares. In the process, we prove that a certain deterministic condition implies a weak maximal inequality for a sequence of \\ell^1 convolution operators."}
{"category": "Math", "title": "Donkin-Koppinen filtration for general linear supergroup", "abstract": "We consider a generalization of Donkin-Koppinen filtrations for coordinate superalgebras of general linear supergroups. More precisely, if $G=GL(m|n)$ is a general linear supergroup of (super)degree $(m|n)$, then its coordinate superalgebra $K[G]$ is a natural $G\\times G$-supermodule. For every finitely generated ideal $\\Gamma\\subseteq \\Lambda\\times\\Lambda$, the largest subsupermodule $O_{\\Gamma}(K[G])$ of $K[G]$, which has all composition factors of the form $L(\\lambda)\\otimes L(\\mu)$ where $(\\lambda, \\mu)\\in\\Gamma$, has a decreasing filtration $O_{\\Gamma}(K[G])=V_0\\supseteq V_1\\supseteq...$ such that $\\bigcap_{t\\geq 0}V_t=0$ and $V_t/V_{t+1}\\simeq V_-(\\lambda_t)^*\\otimes H_-^0(\\lambda_t)$ for each $t\\geq 0$. Here $H_-^0(\\lambda)$ is a costandard $G$-supermodule, and $V_-(\\lambda)$ is a standard $G$-supermodule, both of highest weight $\\lambda\\in\\Lambda$ (see \\cite{z}). We deduce the existence of such a filtration from more general facts about standard and costandard filtrations in certain highest weight categories which will be proved in Section 4. Until now, analogous results were known only for highest weight categories with finite sets of weights. We believe that the reader will find the results of Section 4 interesting on its own. Finally, we apply our main result to describe invariants of (co)adjoint action of $G$."}
{"category": "Math", "title": "Microlocal analysis and evolution equations", "abstract": "Lecture notes from 2008 CMI/ETH Summer School on Evolution Equations. These notes are an informal introduction to the applications of microlocal methods in the study of linear evolution equations and spectral theory. Calculi of pseudodifferential operators and Fourier integral operators are discussed and axiomatized, but not constructed: the focus is on how to apply these tools."}
{"category": "Math", "title": "Estimating a monotone trend", "abstract": "Motivated by global warming issues, we consider a time se- ries that consists of a nondecreasing trend observed with station- ary fluctuations, nonparametric estimation of the trend under monotonicity assumption is considered. The rescaled isotonic es- timators at an interior point are shown to converge to Chernoff's distribution under minimal conditions on the stationary errors. Since the isotonic estimators suffer from the spiking problem at the end point, two modifications are proposed. The estima- tion errors for both estimators of the boundary point are shown to have interesting limiting distributions. Approximation accu- racies are assessed through simulations. One highlight of our treatment is the proof of the weak convergence results which involve several recent techniques developed in the study of con- ditional central limit questions. These weak convergences can be shown to hold conditionally given the starting values."}
{"category": "Math", "title": "Ultrahigh dimensional variable selection: beyond the linear model", "abstract": "Variable selection in high-dimensional space characterizes many contemporary problems in scientific discovery and decision making. Many frequently-used techniques are based on independence screening; examples include correlation ranking (Fan and Lv, 2008) or feature selection using a two-sample t-test in high-dimensional classification (Tibshirani et al., 2003). Within the context of the linear model, Fan and Lv (2008)showed that this simple correlation ranking possesses a sure independence screening property under certain conditions and that its revision, called iteratively sure independent screening (ISIS), is needed when the features are marginally unrelated but jointly related to the response variable. In this paper, we extend ISIS, without explicit definition of residuals, to a general pseudo-likelihood framework, which includes generalized linear models as a special case. Even in the least-squares setting, the new method improves ISIS by allowing variable deletion in the iterative process. Our technique allows us to select important features in high-dimensional classification where the popularly used two-sample t-method fails. A new technique is introduced to reduce the false discovery rate in the feature screening stage. Several simulated and two real data examples are presented to illustrate the methodology."}
{"category": "Math", "title": "On the intersections of solvable Hall subgroups in finite groups", "abstract": "In the paper we consider the following conjecture: if a finite group $G$ possesses a solvable $\\pi$-Hall subgroup $H$, then there exist elements $x,y,z,t\\in G$ such that the identity $H\\cap H^x\\cap H^y\\cap H^z\\cap H^t=O_\\pi(G)$ holds. The minimal counter example is shown to be an almost simple group of Lie type."}
{"category": "Math", "title": "Halfway Up To the Mathematical Infinity: On the Ontological and Epistemic Sustainability of Georg Cantor's Transfinite Design", "abstract": "Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability and well-ordering -- and implying an ordinal re-creation of the continuum. During the last hundred years, the mainstream set-theoretical research -- all insights and adjustments due to Kurt G\\\"odel's revolutionary insights and discoveries notwithstanding -- has compliantly centered its efforts on \\emph{ad hoc} axiomatizations of Cantor's intuitive transfinite design. We demonstrate here that the ontological and epistemic \\emph{sustainability} of this design has been irremediably compromised by the underlying it peremptory, Reductionist mindset of the XIXth century's ideology of science."}
{"category": "Math", "title": "Time and Space Varying Copulas", "abstract": "In this article we review existing literature on dynamic copulas and then propose an n-copula which varies in time and space. Our approach makes use of stochastic differential equations, and gives rise to a dynamic copula which is able to capture the dependence between multiple Markov diffusion processes. This model is suitable for pricing basket derivatives in finance and may also be applicable to other areas such as bioinformatics and environmental science."}
{"category": "Math", "title": "On Hofmann's bilinear estimate", "abstract": "Using the framework of a previous article joint with Axelsson and McIntosh, we extend to systems two results of S. Hofmann for real symmetric equations and their perturbations going back to a work of B. Dahlberg for Laplace's equation on Lipschitz domains, The first one is a certain bilinear estimate for a class of weak solutions and the second is a criterion which allows to identify the domain of the generator of the semi-group yielding such solutions."}
{"category": "Math", "title": "Upper bounds on Rubinstein distances on configuration spaces and applications", "abstract": "In this paper, we provide upper bounds on several Rubinstein-type distances on the configuration space equipped with the Poisson measure. Our inequalities involve the two well-known gradients, in the sense of Malliavin calculus, which can be defined on this space. Actually, we show that depending on the distance between configurations which is considered, it is one gradient or the other which is the most effective. Some applications to distance estimates between Poisson and other more sophisticated processes are also provided, and an application of our results to tail and isoperimetric estimates completes this work."}
{"category": "Math", "title": "The elliptic threefold y^2=x^3+16s^6+16t^6-32(t^3s^3+t^3+s^3)+16", "abstract": "We present a method to calculate the rank of $E(\\oQ(s,t))$ for the elliptic curve mentioned in the title. This method uses a generalization of a method from Van Geemen and Werner to calculate $h^4(Y)$ for nodal hypersurfaces $Y$."}
{"category": "Math", "title": "Relevement de formes modulaires de Siegel", "abstract": "Dans cette note, nous montrons que certaines formes modulaires de Siegel de caract\\'eristique p et de genre 2 ou 3 se rel\\`event en caract\\'eristique 0. Ce r\\'esultat g\\'en\\'eralise un th\\'eor\\`eme classique obtenu par Katz pour les formes de genre 1. Nous utilisons des r\\'esultats de Shepherd-Barron et de Hulek et Sankaran, ainsi que des th\\'eor\\`emes d'annulation de la cohomologie coh\\'erente d\\^us \\`a Deligne, Illusie et Raynaud et \\`a Esnault et Viehweg. ----- In this note, we show that cuspidal Siegel modular forms of characteristic p and genus 2 or 3 can be lifted to characteristic 0. This result extends a classical theorem proved by Katz for genus 1 modular forms. We use ampleness results due to Shepherd-Barron, Hulek and Sankaran, and vanishing theorems due to Deligne, Illusie, Raynaud, Esnault and Viehweg."}
{"category": "Math", "title": "On the nonexistence of ternary extremal self-dual codes", "abstract": "In this note, we give a new nonexistence result of ternary extremal self-dual codes."}
{"category": "Math", "title": "On the Siegel-Weil Theorem for Loop Groups (I)", "abstract": "We proved a new Siegel-Weil formula for orthogonal and symplectic groups, which will be used later to prove a generalization of Siegel-Weil formula for loop groups."}
{"category": "Math", "title": "Non parametric estimation of the structural expectation of a stochastic increasing function", "abstract": "This article introduces a non parametric warping model for functional data. When the outcome of an experiment is a sample of curves, data can be seen as realizations of a stochastic process, which takes into account the small variations between the different observed curves. The aim of this work is to define a mean pattern which represents the main behaviour of the set of all the realizations. So we define the structural expectation of the underlying stochastic function. Then we provide empirical estimators of this structural expectation and of each individual warping function. Consistency and asymptotic normality for such estimators are proved."}
{"category": "Math", "title": "Estimation of the distribution of random shifts deformation", "abstract": "Consider discrete values of functions shifted by unobserved translation effects, which are independent realizations of a random variable with unknown distribution $\\mu$, modeling the variability in the response of each individual. Our aim is to construct a nonparametric estimator of the density of these random translation deformations using semiparametric preliminary estimates of the shifts. Building on results of Dalalyan et al. (2006), semiparametric estimators are obtained in our discrete framework and their performance studied. From these estimates we construct a nonparametric estimator of the target density. Both rates of convergence and an algorithm to construct the estimator are provided."}
{"category": "Math", "title": "Kernel Inverse Regression for spatial random fields", "abstract": "In this paper, we propose a dimension reduction model for spatially dependent variables. Namely, we investigate an extension of the \\emph{inverse regression} method under strong mixing condition. This method is based on estimation of the matrix of covariance of the expectation of the explanatory given the dependent variable, called the \\emph{inverse regression}. Then, we study, under strong mixing condition, the weak and strong consistency of this estimate, using a kernel estimate of the \\emph{inverse regression}. We provide the asymptotic behaviour of this estimate. A spatial predictor based on this dimension reduction approach is also proposed. This latter appears as an alternative to the spatial non-parametric predictor."}
{"category": "Math", "title": "Plane sextics via dessins d'enfants", "abstract": "We develop a geometric approach to the study of plane sextics with a triple singular point. As an application, we give an explicit geometric description of all irreducible maximal sextics with a type $\\bold E_7$ singular point and compute their fundamental groups. All groups found are finite; one of them is nonabelian."}
{"category": "Math", "title": "A Clebsch-Gordan formula for SL_3 and applications to rationality", "abstract": "If R, S, T are irreducible SL_3-representations, we give an easy and explicit description of a basis of the space of equivariant maps from R tensor S to T. We apply this method to the rationality problem for invariant function fields. In particular, we prove the rationality of the moduli space of plane curves of degree 34. This uses a criterion which ensures the stable rationality of some quotients of Grassmannians by an SL-action."}
{"category": "Math", "title": "A numerical algorithm for zero counting II: Randomization and Condition", "abstract": "This paper was witdrawn by the authors."}
{"category": "Math", "title": "Symmetric quasi-hereditary envelopes", "abstract": "We show how any finite-dimensional algebra can be realized as an idempotent subquotient of some symmetric quasi-hereditary algebra. In the special case of rigid symmetric algebras we show that they can be realized as centralizer subalgebras of symmetric quasi-hereditary algebras. We also show that the infinite-dimensional symmetric quasi-hereditary algebras we construct admit quasi-hereditary structure with respect to two opposite orders, that they have strong exact Borel and $\\Delta$-subalgebras and the corresponding triangular decompositions."}
{"category": "Math", "title": "Evolution by mean curvature in sub-Riemannian geometries: A stochastic approach", "abstract": "We study the phenomenon of evolution by horizontal mean curvature flow in sub-Riemannian geometries. We use a stochastic approach to prove the existence of a generalized evolution in these spaces. In particular we show that the value function of suitable family of stochastic control problems solves in the viscosity sense the level set equation for the evolution by horizontal mean curvature flow."}
{"category": "Math", "title": "Unityped algebras", "abstract": "The paper is essentially a continuation of B.Plotkin, G.Zhitomirski, \"Some logical invariants of algebras and logical relations between algebras\", St.Peterburg Math. J., {19:5}, (2008) 859 -- 879, whose main notion is that of logic-geometrical equivalence of algebras (LG-equivalence of algebras). This equivalence of algebras is more strict than elementary equivalence. In the paper we introduce the notion of unityped algebras and relate it to LG-equivalence. We show that these notions coincide. The idea of the type is one of the central ideas in M odel Theory. The correspondence introduced in the paper stimulates a bunch of problems which connect universal algebraic geometry and Model Theory. The paper consists of five sections: 1. General view 2. Logical noetherianity 3. Unitypeness and isomorphism 4. Logically perfect algebras 5. Some facts from algebraic logic. We provide a new general view on the subject, arising \"on the territory\" of universal algebraic geometry, which yield applications of algebraic logic and universal geometry in Model Theory."}
{"category": "Math", "title": "K\\\"ahlerian Twistor Spinors", "abstract": "On a K\\\"ahler spin manifold K\\\"ahlerian twistor spinors are a natural analogue of twistor spinors on Riemannian spin manifolds. They are defined as sections in the kernel of a first order differential operator adapted to the K\\\"ahler structure, called K\\\"ahlerian twistor (Penrose) operator. We study K\\\"ahlerian twistor spinors and give a complete description of compact K\\\"ahler manifolds of constant scalar curvature admitting such spinors. As in the Riemannian case, the existence of K\\\"ahlerian twistor spinors is related to the lower bound of the spectrum of the Dirac operator."}
{"category": "Math", "title": "Singular Riemannian Foliations: Exceptional Leaves; Tautness", "abstract": "For a singular Riemannian foliation whose leaves are properly embedded, we show in the first part of this article the existence of global tubular neighbourhoods, and we develop a global description of the foliation as stratification by types of leaves. The second part deals with the further restriction to a foliation without horizontal conjugate points, introduced by Lytchak and Thorbergsson, which in the special case of an isometric group action equals the concept of variationally completeness. Therefrom, we deduce a global geometric description -- the focal points of the leaves are exactly the singular points -- as well as a topological one: tautness of the leaves."}
{"category": "Math", "title": "On the Global Behavior of Solutions to a Planar System of Difference Equations", "abstract": "We establish the relation between local stability of equilibria and slopes of critical curves for a specific class of difference equations. We then use this result to give global behavior results for nonnegative solutions of the system of difference equations \\begin{equation*} %\\tag{LGIN} \\begin{array}{rcl} x_{n+1} & = & \\displaystyle \\frac{b_1 x_n}{1+x_n+c_1 y_{n}} +h_1 y_{n+1} & = & \\displaystyle \\frac{b_2 y_n}{1+y_n+c_2 x_{n}} +h_2 \\end{array} \\quad n=0,1,..., \\quad (x_0,y_0) \\in [0,\\infty)\\times [0,\\infty) \\end{equation*} with positive parameters. In particular, we show that the system has between one and three equilibria, and that the number of equilibria determines global behavior as follows: if there is only one equilibrium, then it is globally asymptotically stable. If there are two equilibria, then one is a local attractor and the other one is nonhyperbolic. If there are three equilibria, then they are linearly ordered in the south-east ordering of the plane, and consist of a local attractor, a saddle point, and another local attractor. Finally, we give sufficient conditions for having a unique equilibrium."}
{"category": "Math", "title": "Non-ergodicity of Nose-Hoover dynamics", "abstract": "The numerical integration of the Nose-Hoover dynamics gives a deterministic method that is used to sample the canonical Gibbs measure. The Nose-Hoover dynamics extends the physical Hamiltonian dynamics by the addition of a \"thermostat\" variable, that is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are consistent with non-ergodicity of the dynamics for some model problems. The authors recently proved the non-ergodicity of the Nose-Hoover dynamics for the one-dimensional harmonic oscillator. In this paper, this result is extended to non-harmonic one-dimensional systems. It is also shown for some multidimensional systems that the averaged dynamics for the limit of infinite thermostat \"mass\" have many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity."}
{"category": "Math", "title": "Distances between pairs of vertices and vertical profile in conditioned Galton--Watson trees", "abstract": "We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the second proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson saying that the vertical profile of a randomly labelled conditioned Galton-Watson tree converges in distribution, after suitable normalization, to the density of ISE (Integrated Superbrownian Excursion)."}
{"category": "Math", "title": "On the restricted Verma modules at the critical level", "abstract": "We study the restricted Verma modules of an affine Kac-Moody algebra at the critical level with special emphasis on their Jordan-H\"older multiplicities. The Feigin-Frenkel conjecture gives a formula for these multiplicities that involves the periodic Kazhdan-Lusztig polynomials. We prove this conjecture for all subgeneric blocks and for the case of anti-dominant simple subquotients."}
{"category": "Math", "title": "A syzygetic approach to the smoothability of zero-dimensional schemes", "abstract": "We consider the question of which zero-dimensional schemes deform to a collection of distinct points; equivalently, we ask which Artinian k-algebras deform to a product of fields. We introduce a syzygetic invariant which sheds light on this question for zero-dimensional schemes of regularity two. This invariant imposes obstructions for smoothability in general, and it completely answers the question of smoothability for certain zero-dimensional schemes of low degree. The tools of this paper also lead to other results about Hilbert schemes of points, including a characterization of nonsmoothable zero-dimensional schemes of minimal degree in every embedding dimension d\\geq 4."}
{"category": "Math", "title": "Convex PBW-Type Lyndon Bases and Restricted Two-parameter Quantum Groups of Type B", "abstract": "We construct convex PBW-type Lyndon bases for two-parameter quantum groups U_{r,s}({so}_{2n+1}) with detailed commutation relations. It turns out that under a certain condition, the restricted two-parameter quantum group u_{r,s}({so}_{2n+1}) (r, s are roots of unity) is of Drinfeld double. All of Hopf isomorphisms of u_{r,s}({so}_{2n+1}), as well as u_{r,s}({sl}_n) are determined. Finally, necessary and sufficient conditions for u_{r,s}({so}_{2n+1}) to be a ribbon Hopf algebra are singled out by describing the left and right integrals."}
{"category": "Math", "title": "Arakelov (in)equalities", "abstract": "We discuss several numerical conditions for families of projective varieties or variations of Hodge structures."}
{"category": "Math", "title": "Minimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds", "abstract": "Theorem A. Let $M^n$ denote a closed Riemannian manifold with nonpositive sectional curvature and let $\\tilde M^n$ be the universal cover of $M^n$ with the lifted metric. Suppose that the universal cover $\\tilde M^n$ contains no totally geodesic embedded Euclidean plane $\\mathbb{R}^2$ (i.e., $M^n$ is a visibility manifold). Then Gromov's simplicial volume $\\| M^n \\|$ is non-zero. Consequently, $M^n$ is non-collapsible while keeping Ricci curvature bounded from below. More precisely, if $Ric_g \\ge -(n-1)$, then $vol(M^n, g) \\ge \\frac{1}{(n-1)^n n!} \\| M^n \\| > 0. Theorem B. (Perelman) Let $M^3$ be a closed a-spherical 3-manifold ($K(\\pi, 1)$-space) with the fundamental group $\\Gamma$. Suppose that $\\Gamma$ contains no subgroups isomorphic to $\\mathbb{Z}\\oplus \\mathbb{Z}$. Then $M^3$ is diffeomorphic to a compact quotient of real hyperbolic space $\\mathbb{H}^3$, i.e., $M^3 \\equiv \\mathbb{H}^3/\\Gamma$. Consequently, $MinVol(M^3) \\ge {1/24}\\| M^3 \\| > 0$. Minimal volume and simplicial norm of all other compact 3-manifolds without boundary and {\\it singular} spaces will also be discussed."}
{"category": "Math", "title": "Floor decompositions of tropical curves : the planar case", "abstract": "In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams. We give here detailed proofs in the tropical geometry framework, in the case when the ambient variety is a complex surface, and give some examples of computations using floor diagrams. The focusing on dimension 2 is motivated by the special combinatoric of floor diagrams compared to arbitrary dimension. We treat a general toric surface case in this dimension: the curve is given by an arbitrary lattice polygon and include computation of Welschinger invariants with pairs of conjugate points. See also \\cite{FM} for combinatorial treatment of floor diagrams in the projective case."}
{"category": "Math", "title": "The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings", "abstract": "Given a projective scheme $X$ over a field $k$, an automorphism $\\sigma$ of $X$, and a $\\sigma$-ample invertible sheaf $L$, one may form the twisted homogeneous coordinate ring $B = B(X, L, \\sigma)$, one of the most fundamental constructions in noncommutative projective algebraic geometry. We study the primitive spectrum of $B$, as well as that of other closely related algebras such as skew and skew-Laurent extensions of commutative algebras. Over an algebraically closed, uncountable field $k$ of characteristic zero, we prove that that the primitive ideals of $B$ are characterized by the usual Dixmier-Moeglin conditions whenever the dimension of $X$ is no more than 2."}
{"category": "Math", "title": "Twisted modules for quantum vertex algebras", "abstract": "We study twisted modules for (weak) quantum vertex algebras and we give a conceptual construction of (weak) quantum vertex algebras and their twisted modules. As an application we construct and classify irreducible twisted modules for a certain family of quantum vertex algebras."}
{"category": "Math", "title": "On the singular scheme of codimension one holomorphic foliations in P^3", "abstract": "In this work, we begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion free. In addition, when the codimension of the singular locus is at least two, it is shown that being reduced is equivalent to the reflexivity of the tangent sheaf. Our main results state on one hand, that the tangent sheaf of a codimension one foliation in P^3 is locally free if and only the singular scheme is a curve, and that it splits if and only if that curve is arithmetically Cohen-Macaulay. On the other hand, we discuss when a split foliation in P^3 is determined by its singular scheme."}
{"category": "Math", "title": "A remarkable sequence of integers", "abstract": "A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature."}
{"category": "Math", "title": "Hartogs' extension theorems on Stein spaces", "abstract": "We discuss various known generalizations of the classical Hartogs' extension theorem on Stein spaces with arbitrary singularities and present an analytic proof based on d-bar methods."}
{"category": "Math", "title": "p-Density, exponential sums and Artin-Schreier curves", "abstract": "In this paper we define the $p$-density of a finite subset $D\\subset\\ma{N}^r$, and show that it gives a good lower bound for the $p$-adic valuation of exponential sums over finite fields of characteristic $p$. We also give an application: when $r=1$, the $p$-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in $D$."}
{"category": "Math", "title": "Global Behavior of Solutions to Two Classes of Second Order Rational Difference Equations", "abstract": "For nonnegative real numbers $\\alpha$, $\\beta$, $\\gamma$, $A$, $B$ and $C$ such that $B+C>0$ and $\\alpha+\\beta+\\gamma >0$, the difference equation \\begin{equation*} x_{n+1}=\\displaystyle\\frac{\\alpha +\\beta x_{n}+\\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \\quad n=0,1,2,... %, \\quad x_{-1},x_{0}\\in [0,\\infty) \\end{equation*} has a unique positive equilibrium. A proof is given here for the following statements: \\medskip \\noindent Theorem 1. {\\it For every choice of positive parameters $\\alpha$, $\\beta$, $\\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \\begin{equation*} x_{n+1}=\\displaystyle\\frac{\\alpha +\\beta x_{n}+\\gamma x_{n-1}}{A+B x_{n}+C x_{n-1}}, \\quad n=0,1,2,..., \\quad x_{-1},x_{0}\\in [0,\\infty) \\end{equation*} converge to the positive equilibrium or to a prime period-two solution.} \\medskip \\noindent Theorem 2. {\\it For every choice of positive parameters $\\alpha$, $\\beta$, $\\gamma$, $A$, $B$ and $C$, all solutions to the difference equation \\begin{equation*} x_{n+1}= \\displaystyle\\frac{\\alpha +\\beta x_{n}+\\gamma x_{n-1}}{B x_{n}+C x_{n-1}}, \\quad n=0,1,2,..., \\quad x_{-1},x_{0}\\in (0,\\infty) \\end{equation*} converge to the positive equilibrium or to a prime period-two solution.}"}
{"category": "Math", "title": "Contact deformations of closed 1-forms on Torus bundles over the circle", "abstract": "If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, every closed, nonsingular 1-form deforms linearly into contact forms."}
{"category": "Math", "title": "Global Attractivity of the Equilibrium of a Difference Equation: An Elementary Proof Assisted by Computer Algebra System", "abstract": "Let $p$ and $q$ be arbitrary positive numbers. It is shown that if $q < p$, then all solutions to the difference equation \\tag{E} x_{n+1} = \\frac{p+q x_n}{1+x_{n-1}}, \\quad n=0,1,2,..., \\quad x_{-1}>0, x_0>0 converge to the positive equilibrium $\\overline{x} = {1/2}(q-1 + \\sqrt{(q-1)^2 + 4 p})$. \\medskip The above result, taken together with the 1993 result of Koci\\'c and Ladas for equation (E) with $q \\geq p$, gives global attractivity of the positive equilibrium of (E) for all positive values of the parameters, thus completing the proof of a conjecture of Ladas."}
{"category": "Math", "title": "Hopf Algebras of Graphs", "abstract": "We define graded Hopf algebras with bases labeled by various types of graphs and hypergraphs, provided with natural embeddings into an algebra of polynomials in infinitely many variables. These algebras are graded by the number of edges and can be considered as generalizations of symmetric or quasi-symmetric functions."}
{"category": "Math", "title": "$d$-Koszul algebras, 2-$d$ determined algebras and 2-$d$-Koszul algebras", "abstract": "The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is $d$-Koszul. It is shown that an algebra which has a reduced \\grb basis that is composed of homogeneous elements of degree $d$ is $d$-Koszul if and only if its associated monomial algebra is $d$-Koszul. The class of 2-$d$-determined algebras and the class 2-$d$-Koszul algebras are introduced. In particular, it shown that 2-$d$-determined monomial algebras are 2-$d$-Koszul algebras and the structure of the ideal of relations of such an algebra is completely determined."}
{"category": "Math", "title": "Stability of equilibria for the $\\mathfrak{so}(4)$ free rigid body", "abstract": "The stability for all generic equilibria of the Lie-Poisson dynamics of the $\\mathfrak{so}(4)$ rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of $\\mathfrak{so}(n)$ are equilibrium points for the rigid body dynamics. In the case of $\\mathfrak{so}(4)$ there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit give three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in $\\mathfrak{so}(3)$. In addition to these coordinate type Cartan equilibria there are others that come in curves."}
{"category": "Math", "title": "Function Spaces Related to the Dirichlet Space", "abstract": "We present results about spaces of holomorphic functions associated to the classical Dirichlet space. The spaces we consider have roles similar to the roles of $H^{1}$ and $BMO$ in the Hardy space theory and we emphasize those analogies."}
{"category": "Math", "title": "A uniqueness result on ordinary differential equations with singular coefficients", "abstract": "We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the results."}
{"category": "Math", "title": "Quantum Exchangeable Sequences of Algebras", "abstract": "We extend the notion of quantum exchangeability, introduced by K\\\"ostler and Speicher in arXiv:0807.0677, to sequences (\\rho_1,\\rho_2,...c) of homomorphisms from an algebra C into a noncommutative probability space (A,\\phi), and prove a free de Finetti theorem: an infinite quantum exchangeable sequence (\\rho_1,\\rho_2,...c) is freely independent and identically distributed with respect to a conditional expectation. As a corollary we obtain a free analogue of the Hewitt Savage zero-one law. As in the classical case, the theorem fails for finite sequences. We give a characterization of finite quantum exchangeable sequences, which can be viewed as a noncommutative analogue of sampling without replacement. We then give an approximation to how far a finite quantum exchangeable sequence is from being freely independent with amalgamation."}
{"category": "Math", "title": "The projectively resolving of some classes over a direct product of rings", "abstract": "In this paper, we study the resolving of $\\mathcal{SGP}(-)$ and $\\mathcal{SGF}(-)$, the classes of all strongly Gorenstein projective and flat modules respectively, over a direct product of commutative rings."}
{"category": "Math", "title": "SK1 for graded division algebras", "abstract": "The reduced Whitehead group $\\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that $\\SK$ of a tame valued division algebra over a henselian field coincides with $\\SK$ of its associated graded division algebra. Furthermore, it is shown that $\\SK$ of a graded division algebra is isomorphic to $\\SK$ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes $\\SK$ for generic abelian crossed products."}
{"category": "Math", "title": "Priority Arguments and Epsilon Substitutions", "abstract": "Kreisel has observed that the termination proof for Hilbert's epsilon-substitution method bears a resemblance to the priority arguments used in recursion theory. We make this precise by proving the termination using a framework for priority arguments due to Lerman and Lempp."}
{"category": "Math", "title": "On maximal Subgroups of the multiplicative group of a division algebra", "abstract": "The question of existence of a maximal subgroup in the multiplicative group D* of a division algebra D finite dimensional over its center F is investigated. We prove that if D* has no maximal subgroup, then deg(D) is not a power of 2, F^{*2} is divisible, and for each odd prime p dividing deg(D), there exist noncyclic division algebras of degree p over F."}
{"category": "Math", "title": "Generalized Moonshine I: Genus zero functions", "abstract": "We introduce a notion of Hecke-monicity for functions on certain moduli spaces associated to torsors of finite groups over elliptic curves, and show that it implies strong invariance properties under linear fractional transformations. Specifically, if a weakly Hecke-monic function has algebraic integer coefficients and a pole at infinity, then it is either a holomorphic genus-zero function invariant under a congruence group or of a certain degenerate type. As a special case, we prove the same conclusion for replicable functions of finite order, which were introduced by Conway and Norton in the context of monstrous moonshine. As an application, we introduce a class of Lie algebras with group actions, and show that the characters derived from them are weakly Hecke-monic. When the Lie algebras come from chiral conformal field theory in a certain sense, then the characters form holomorphic genus-zero functions invariant under a congruence group."}
{"category": "Math", "title": "SK1 of Azumaya algebras over hensel pairs", "abstract": "Let A be an Azumaya algebra of constant rank n^2 over a Hensel pair (R,I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK(A) coincides with its reduction SK(A/IA)."}
{"category": "Math", "title": "Kodaira-Iitaka Dimension on a Normal Prime Divisor", "abstract": "This paper was inspired by work by T. Peternell, M. Schneider and A.J. Sommese on the Kodaira dimension of subvarieties. In it I find a relation between the Kodaira-Iitaka dimension of a divisor on a normal variety and that of related divisors on an irreducible normal subvariety of codimension one. The main result may be stated in a simplified form as: For $X$ a complete normal variety, $Y \\sub X$ an irreducible complete normal divisor and $\\sL$ an invertible sheaf on $X$, there exist integers $n_1 > 0, n_2 \\geq 0$ for which $\\kappa(X,\\sL) - 1 \\leq \\kappa(Y,\\sL^{n_1}(-n_2Y)|_Y)$, where, if $Y$ is not a fixed component of large tensor powers of $\\sL$, we may take $n_1 >> n_2$. This has implications for Kodaira-Iitaka dimension on a subvariety of any codimension."}
{"category": "Math", "title": "Fixed points of compositions of earthquakes", "abstract": "Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichm\\\"uller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichm\\\"uller space. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic AdS manifold. The proof uses some estimates from the geometry of those AdS manifolds."}
{"category": "Math", "title": "Iterated destabilizing modifications for vector bundles with connection", "abstract": "Given a vector bundle with integrable connection $(V,\\nabla)$ on a curve, if $V$ is not itself semistable as a vector bundle then we can iterate a construction involving modification by the destabilizing subobject to obtain a Hodge-like filtration $F^p$ which satisfies Griffiths transversality. The associated graded Higgs bundle is the limit of $(V,t\\nabla)$ under the de Rham to Dolbeault degeneration. We get a stratification of the moduli space of connections, with as minimal stratum the space of opers. The strata have fibrations whose fibers are Lagrangian subspaces of the moduli space."}
{"category": "Math", "title": "Coarse dynamics and fixed point property", "abstract": "We investigate the fixed point property of the group actions on a coarse space and its Higson corona. We deduce the coarse version of Brouwer's fixed point theorem."}
{"category": "Math", "title": "A rank-based selection with cardinal payoffs and a cost of choice", "abstract": "A version of the secretary problem is considered. The ranks of items, whose values are independent, identically distributed random variables $X_1,X_2,...,X_n$ from a uniform distribution on $[0; 1]$, are observed sequentially by the grader. He has to select exactly one item, when it appears, and receives a payoff which is a function of the unobserved realization of random variable assigned to the item diminished by some cost. The methods of analysis are based on the existence of an embedded Markov chain and use the technique of backward induction. The result is a generalization of the selection model considered by Bearden(2006). The asymptotic behaviour of the solution is also investigated."}
{"category": "Math", "title": "Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution", "abstract": "Consider a random sample from a bivariate distribution function $F$ in the max-domain of attraction of an extreme-value distribution function $G$. This $G$ is characterized by two extreme-value indices and a spectral measure, the latter determining the tail dependence structure of $F$. A major issue in multivariate extreme-value theory is the estimation of the spectral measure $\\Phi_p$ with respect to the $L_p$ norm. For every $p\\in[1,\\infty]$, a nonparametric maximum empirical likelihood estimator is proposed for $\\Phi_p$. The main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized. Asymptotic normality of the estimators is proved under conditions that allow for tail independence. Moreover, the conditions are easily verifiable as we demonstrate through a number of theoretical examples. A simulation study shows a substantially improved performance of the new estimators. Two case studies illustrate how to implement the methods in practice."}
{"category": "Math", "title": "Krull dimension of types in a class of first-order theories", "abstract": "We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula--plus a few other technical requirements. The theory of vector spaces and the theory fields are examples. We prove the amalgamation property and the existence of a model-companion. We show that the model-companion is strongly minimal. We also prove that the length of any increasing sequence of prime types is bounded, so every formula has finite Krull dimension."}
{"category": "Math", "title": "Linear independence over tropical semirings and beyond", "abstract": "We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples."}
{"category": "Math", "title": "An Effective Lower Bound for Group Complexity of Finite Semigroups and Automata", "abstract": "The question of computing the group complexity of finite semigroups and automata was first posed in K. Krohn and J. Rhodes, \\textit{Complexity of finite semigroups}, Annals of Mathematics (2) \\textbf{88} (1968), 128--160, motivated by the Prime Decomposition Theorem of K. Krohn and J. Rhodes, \\textit{Algebraic theory of machines, {I}: {P}rime decomposition theorem for finite semigroups and machines}, Transactions of the American Mathematical Society \\textbf{116} (1965), 450--464. Here we provide an effective lower bound for group complexity."}
{"category": "Math", "title": "A deconvolution approach to estimation of a common shape in a shifted curves model", "abstract": "This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model. We show that this problem can be transformed into a linear inverse problem, where the density of the random shifts plays the role of a convolution operator. An adaptive estimator of the mean pattern, based on wavelet thresholding is proposed. We study its consistency for the quadratic risk as the number of observed curves tends to infinity, and this estimator is shown to achieve a near-minimax rate of convergence over a large class of Besov balls. This rate depends both on the smoothness of the common shape of the curves and on the decay of the Fourier coefficients of the density of the random shifts. Hence, this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems, which is a new point of view on curve registration and image warping problems. We also provide a new method to estimate the unknown random shifts between curves. Some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature."}
{"category": "Math", "title": "Canonical connection on quasi-Kaehler manifolds with Norden metric", "abstract": "We study the geometry of the canonical connection on a quasi-Kaehler manifold with Norden metric. We consider the cases when the canonical connection has Kaehler curvature tensor and parallel torsion, and derive conditions for an isotropic-Kaehler manifold. We give the relation between the canonical connection, the B-connection, and the connection with totally skew-symmetric torsion on quasi-Kaehler manifolds with Norden metric."}
{"category": "Math", "title": "Quintic surfaces with maximum and other Picard numbers", "abstract": "This paper investigates the Picard numbers of quintic surfaces. We give the first example of a complex quintic surface in IP^3 with maximum Picard number 45. We also investigate its arithmetic and determine the zeta function. Similar techniques are applied to produce quintic surfaces with several other Picard numbers that have not been achieved before."}
{"category": "Math", "title": "Universal convex coverings", "abstract": "In every dimension $d\\ge1$, we establish the existence of a constant $v_d>0$ and of a subset $\\mathcal U_d$ of $\\mathbb R^d$ such that the following holds: $\\mathcal C+\\mathcal U_d=\\mathbb R^d$ for every convex set $\\mathcal C\\subset \\mathbb R^d$ of volume at least $v_d$ and $\\mathcal U_d$ contains at most $\\log(r)^{d-1}r^d$ points at distance at most $r$ from the origin, for every large $r$."}
{"category": "Math", "title": "\\'Equidistribution et diff\\'erentiabilit\\'e", "abstract": "We propose a criterion of equidistribution by the differentiability of certain arithmetic invariants. Combined with the slope method and the asymptotic measures, this criterion gives a new \"conceptual\" proof to equidistribution results originally obtained via the variation principle."}
{"category": "Math", "title": "On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications", "abstract": "We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure cental limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors."}
{"category": "Math", "title": "Efficient covariance estimation for asynchronous noisy high-frequency data", "abstract": "We focus on estimating the integrated covariance of log-price processes in the presence of market microstructure noise. We construct an efficient unbiased estimator for the quadratic covariation of two It\\^{o} processes in the case where high-frequency asynchronous discrete returns under market microstructure noise are observed. This estimator is based on synchronization and multi-scale methods and attains the optimal rate of convergence. A Monte Carlo study analyzes the finite sample size characteristics of our estimator."}
{"category": "Math", "title": "Long-time behavior in scalar conservation laws", "abstract": "We consider the long-time behavior of the entropy solution of a first-order scalar conservation law on a Riemannian manifold. In the case of the Torus, we show that, under a weak property of genuine non-linearity of the flux, the solution converges to its average value in $L^{p}$, $1\\leq p<+\\infty$. We give a partial result in the general case."}
{"category": "Math", "title": "Geometry of the analytic loop group", "abstract": "We introduce and study a notion of analytic loop group with a Riemann-Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity with non trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf-Poisson structure is isomorphic to the semi-classical limit of the center of the quantum affine algebra (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters, are parameterized by certain G-bundles on an elliptic curve."}
{"category": "Math", "title": "Collective Property of Numbers and Its Mathematical Refutation", "abstract": "A number has the \"collective\" property if the number is the greatest lower bound of a bounded, strictly decreasing sequence on the real line. We prove that numbers with the collective property constitute an empty set."}
{"category": "Math", "title": "Harmonics for Deformed Steenrod Operators", "abstract": "We explore in this paper the spaces of common zeros of several deformations of Steenrod operators."}
{"category": "Math", "title": "Simultaneous linearization of holomorphic germs in presence of resonances", "abstract": "Let $f_1, ..., f_m$ be $m\\ge 2$ germs of biholomorphisms of $\\C^n$, fixing the origin, with $(\\d f_1)_O$ diagonalizable and such that $f_1$ commutes with $f_h$ for any $h=2,..., m$. We prove that, under certain arithmetic conditions on the eigenvalues of $(\\d f_1)_O$ and some restrictions on their resonances, $f_1, ..., f_m$ are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under $f_1,..., f_m$."}
{"category": "Math", "title": "On the transition to the normal phase for superconductors surrounded by normal conductors", "abstract": "For a cylindrical superconductor surrounded by a normal material, we discuss transition to the normal phase of stable, locally stable and critical configurations. Associated with those phase transitions, we define critical magnetic fields and we provide a sufficient condition for which those critical fields coincide. In particular, when the conductivity ratio of the superconducting and the normal material is large, we show that the aforementioned critical magnetic fields coincide, thereby proving that the transition to the normal phase is sharp. One key-ingredient in the paper is the analysis of an elliptic boundary value problem involving `transmission' boundary conditions. Another key-ingredient involves a monotonicity result (with respect to the magnetic field strength) of the first eigenvalue of a magnetic Schroedinger operator with discontinuous coefficients."}
{"category": "Math", "title": "Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions", "abstract": "We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier--Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic \"galloping\", or \"pulsating\", solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier--Stokes equations, in the limit as amplitude (hence also heat release) goes to zero. The analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works."}
{"category": "Math", "title": "A Remark on Gelfand Duality for Spectral Triples", "abstract": "We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable \"metric\" category of spectral triples over commutative pre-C*-algebras. We also construct an embedding of a \"quotient\" of the category of spectral triples introduced in arXiv:math/0502583v1 into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension."}
{"category": "Math", "title": "Semi log resolution", "abstract": "The aim of this note is to discuss resolution theorems that are useful in the study of semi log canonical varieties."}
{"category": "Math", "title": "A Spectral Theorem for Imprimitivity C*-bimodules", "abstract": "After recalling in detail some basic definitions on Hilbert C*-bimodules, Morita equivalence and imprimitivity, we discuss a spectral reconstruction theorem for imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras and consider some of its applications in the theory of commutative full C*-categories."}
{"category": "Math", "title": "A Horizontal Categorification of Gelfand Duality", "abstract": "In the setting of C*-categories, we provide a definition of \"spectrum\" of a commutative full C*-category as a one-dimensional unital saturated Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gelfand duality theorem generalizing the usual Gelfand duality between the categories of commutative unital C*-algebras and compact Hausdorff spaces. Although many of the individual ingredients that appear along the way are well-known, the somehow unconventional way we \"glue\" them together seems to shed some new light on the subject."}
{"category": "Math", "title": "Fully discrete Galerkin schemes for the nonlinear and nonlocal Hartree equation", "abstract": "We study the time dependent Hartree equation in the continuum, the semidiscrete, and the fully discrete setting. We prove existence-uniqueness, regularity, and approximation properties for the respective schemes, and set the stage for a controlled numerical computation of delicate nonlinear and nonlocal features of the Hartree dynamics in various physical applications."}
{"category": "Math", "title": "Quotients by finite equivalence relations", "abstract": "This note studies the existence of quotients by finite set theoretic equivalence relations. May 18: Substantial revisions with a new appendix by C. Raicu"}
{"category": "Math", "title": "Bias correction in a multivariate normal regression model with general parameterization", "abstract": "This paper develops a bias correction scheme for a multivariate normal model under a general parameterization. In the model, the mean vector and the covariance matrix share the same parameters. It includes many important regression models available in the literature as special cases, such as (non)linear regression, errors-in-variables models, and so forth. Moreover, heteroscedastic situations may also be studied within our framework. We derive a general expression for the second-order biases of maximum likelihood estimates of the model parameters and show that it is always possible to obtain the second order bias by means of ordinary weighted lest-squares regressions. We enlighten such general expression with an errors-in-variables model and also conduct some simulations in order to verify the performance of the corrected estimates. The simulation results show that the bias correction scheme yields nearly unbiased estimators. We also present an empirical ilustration."}
{"category": "Math", "title": "Stochastic Vs Worst-case Condition Numbers", "abstract": "We compare Stochastic and Worst-case condition numbers and loss of precision for general computational problems. We show an upper bound for the ratio of Worst-case condition number to the Stochastic condition number of order O(sqrt m). We show an upper bound for the difference between the Worst-case loss of precision and the Stochastic loss of precision of order O(ln m). The results hold if the perturbations are measured norm-wise or componentwise."}
{"category": "Math", "title": "On the annealed large deviation rate function for a multi-dimensional random walk in random environment", "abstract": "We derive properties of the rate function in Varadhan's (annealed) large deviation principle for multidimensional, ballistic random walk in random environment, in a certain neighborhood of the zero set of the rate function. Our approach relates the LDP to that of regeneration times and distances. The analysis of the latter is possible due to the i.i.d. structure of regenerations."}
{"category": "Math", "title": "A remark on a conjecture of Hain and Looijenga", "abstract": "After recalling the various tautological algebras of the moduli space of curves and some of its partial compactifications and stating several well-known results and conjectures concerning these algebras, we prove that the natural extension to the case of pointed curves of a 1996 conjecture of Hain and Looijenga is true if and only if two of the stated conjectures are true."}
{"category": "Math", "title": "Exponential decay for solutions to semilinear damped wave equation", "abstract": "This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyaponuv function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in an article of Gazzola and Squassina."}
{"category": "Math", "title": "Time management in a Poisson fishing model", "abstract": "The aim of the paper is to extend the model of \"fishing problem\". The simple formulation is following. The angler goes to fishing. He buys fishing ticket for a fixed time. There are two places for fishing at the lake. The fishes are caught according to renewal processes which are different at both places. The fishes' weights and the inter-arrival times are given by the sequences of i.i.d. random variables with known distribution functions. These distributions are different for the first and second fishing place. The angler's satisfaction measure is given by difference between the utility function dependent on size of the caught fishes and the cost function connected with time. On each place the angler has another utility functions and another cost functions. In this way, the angler's relative opinion about these two places is modeled. For example, on the one place better sort of fish can be caught with bigger probability or one of the places is more comfortable. Obviously our angler wants to have as much satisfaction as possible and additionally he have to leave the lake before the fixed moment. Therefore his goal is to find two optimal stopping times in order to maximize his satisfaction. The first time corresponds to the moment, when he eventually should change the place and the second time, when he should stop fishing. These stopping times have to be less than the fixed time of fishing. The value of the problem and the optimal stopping times are derived."}
{"category": "Math", "title": "Grid Diagrams, Braids, and Contact Geometry", "abstract": "We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots."}
{"category": "Math", "title": "Ultraspherical type generating functions for orthogonal polynomials", "abstract": "We characterize, up to a conjecture, probability distributions of all order finite moments having ultraspherical type generating functions for orthogonal polynomials."}
{"category": "Math", "title": "On the period map for prime Fano threefolds of degree 10", "abstract": "We prove that the deformations of a smooth complex Fano threefold X with Picard number 1, index 1, and degree 10, are unobstructed. The differential of the period map has two-dimensional kernel. We construct two two-dimensional components of the fiber of the period map through X: one is isomorphic to the variety of conics in X, modulo an involution, another is birationally isomorphic to a moduli space of semistable rank-2 torsion-free sheaves on X, modulo an involution. The threefolds corresponding to points of these components are obtained from X via conic and line (birational) transformations."}
{"category": "Math", "title": "Regularized Multivariate Regression for Identifying Master Predictors with Application to Integrative Genomics Study of Breast Cancer", "abstract": "In this paper, we propose a new method remMap -- REgularized Multivariate regression for identifying MAster Predictors -- for fitting multivariate response regression models under the high-dimension-low-sample-size setting. remMap is motivated by investigating the regulatory relationships among different biological molecules based on multiple types of high dimensional genomic data. Particularly, we are interested in studying the influence of DNA copy number alterations on RNA transcript levels. For this purpose, we model the dependence of the RNA expression levels on DNA copy numbers through multivariate linear regressions and utilize proper regularizations to deal with the high dimensionality as well as to incorporate desired network structures. Criteria for selecting the tuning parameters are also discussed. The performance of the proposed method is illustrated through extensive simulation studies. Finally, remMap is applied to a breast cancer study, in which genome wide RNA transcript levels and DNA copy numbers were measured for 172 tumor samples. We identify a tran-hub region in cytoband 17q12-q21, whose amplification influences the RNA expression levels of more than 30 unlinked genes. These findings may lead to a better understanding of breast cancer pathology."}
{"category": "Math", "title": "Monge-Amp\\`ere equations in big cohomology classes", "abstract": "We define non-pluripolar products of closed positive currents on a compact Kaehler manifold. We show that a positive non-pluripolar measure can be written in a unique way as the top degree self-intersection (in the non-pluripolar sense) of a closed positive current in given big cohomology class. The solution is shown to have minimal singularities in the sense of Demailly if the measure is regular enough. These results are combined with a fixed point argument to construct singular Kaehler-Einstein volume forms with minimal singularities on varieties of general type."}
{"category": "Math", "title": "Intersections on tropical moduli spaces", "abstract": "This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a \"boundary\" divisor and we prove general tropical versions of the WDVV resp. topological recursion equations (under some assumptions). As a direct application, we prove that the toric varieties $\\mathbb{P}^1$, $\\mathbb{P}^2$, $\\mathbb{P}^1 \\times \\mathbb{P}^1$ and with Psi-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for $\\mathbb{P}^2$ in Markwig-Rau-2008). Our approach uses tropical intersection theory and can unify and simplify some parts of the existing tropical enumerative geometry (for rational curves)."}
{"category": "Math", "title": "Predictability in Spatially Extended Systems with Model Uncertainty", "abstract": "Macroscopic models for spatially extended systems under random influences are often described by stochastic partial differential equations (SPDEs). Some techniques for understanding solutions of such equations, such as estimating correlations, Liapunov exponents and impact of noises, are discussed. They are relevant for understanding predictability in spatially extended systems with model uncertainty, for example, in physics, geophysics and biological sciences. The presentation is for a wide audience."}
{"category": "Math", "title": "Symplectic forms and cohomology decomposition of almost complex 4-manifolds", "abstract": "For any compact almost complex manifold $(M,J)$, the last two authors defined two subgroups $H_J^+(M)$, $H_J^-(M)$ of the degree 2 real de Rham cohomology group $H^2(M, \\mathbb{R})$ in arXiv:0708.2520. These are the sets of cohomology classes which can be represented by $J$-invariant, respectively, $J$-anti-invariant real $2-$forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of $H^2(M, \\mathbb{R})$. This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given."}
{"category": "Math", "title": "On multivariate Newton-like inequalities", "abstract": "We study multivariate entire functions and polynomials with non-negative coefficients. A class of {\\bf Strongly Log-Concave} entire functions, generalizing {\\it Minkowski} volume polynomials, is introduced: an entire function $f$ in $m$ variables is called {\\bf Strongly Log-Concave} if the function $(\\partial x_1)^{c_1}...(\\partial x_m)^{c_m} f$ is either zero or $\\log((\\partial x_1)^{c_1}...(\\partial x_m)^{c_m} f)$ is concave on $R_{+}^{m}$. We start with yet another point of view (via {\\it propagation}) on the standard univarite (or homogeneous bivariate) {\\bf Newton Inequlities}. We prove analogues of (univariate) {\\bf Newton Inequlities} in the (multivariate) {\\bf Strongly Log-Concave} case. One of the corollaries of our new Newton(like) inequalities is the fact that the support $supp(f)$ of a {\\bf Strongly Log-Concave} entire function $f$ is discretely convex ($D$-convex in our notation). The proofs are based on a natural convex relaxation of the derivatives $Der_{f}(r_1,...,r_m)$ of $f$ at zero and on the lower bounds on $Der_{f}(r_1,...,r_m)$, which generalize the {\\bf Van Der Waerden-Falikman-Egorychev} inequality for the permanent of doubly-stochastic matrices. A few open questions are posed in the final section."}
{"category": "Math", "title": "A New Family of Covariate-Adjusted Response Adaptive Designs and their Asymptotic Properties", "abstract": "It is often important to incorporating covariate information in the design of clinical trials. In literature, there are many designs of using stratification and covariate-adaptive randomization to balance on certain known covariate. Recently Zhang, Hu, Cheung and Chan (2007) have proposed a family of covariate-adjusted response-adaptive (CARA) designs and studied their asymptotic properties. However, these CARA designs often have high variabilities. In this paper, we propose a new family of covariate-adjusted response-adaptive (CARA) designs. We show that the new designs have smaller variabilities and therefore more efficient."}
{"category": "Math", "title": "The Penrose Transform in the Split Signature", "abstract": "A version of the Penrose transform is introduced in the split signature. It relates the cohomological data with supports on the open subsets of the complex 3-projective space and kernel of differential operators on the (real) Grassmannian of 2-planes in the Euclidean 4-space. As an example we derive a cohomological interpretation of the so-called X-ray transform. Furthermore, a cohomological realization of the so-called \"minimal\" representation of SL(4,R) is given. We also present the split Penrose transform in split instanton backgrounds."}
{"category": "Math", "title": "The Gaussian approximation for multi-color generalized Friedman's urn model", "abstract": "The Friedman's urn model is a popular urn model which is widely used in many disciplines. In particular, it is extensively used in treatment allocation schemes in clinical trials. In this paper, we prove that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a multi-color generalized Friedman's urn model with non-homogeneous generating matrices. The Gaussian process is a solution of a stochastic differential equation. This Gaussian approximation together with the properties of the Gaussian process is important for the understanding of the behavior of the urn process and is also useful for statistical inferences. As an application, we obtain the asymptotic properties including the asymptotic normality and the law of the iterated logarithm for a multi-color generalized Friedman's urn model as well as the randomized-play-the-winner rule as a special case."}
{"category": "Math", "title": "Immigrated urn models - asymptotic properties and applications", "abstract": "Urn models have been widely studied and applied in both scientific and social science disciplines. In clinical studies, the adoption of urn models in treatment allocation schemes has been proved to be beneficial to both researchers, by providing more efficient clinical trials, and patients, by increasing the likelihood of receiving the better treatment. In this paper, we propose a new and general class of immigrated urn (IMU) models that incorporates the immigration mechanism into the urn process. Theoretical properties are developed and the advantages of the IMU models are discussed. In general, the IMU models have smaller variabilities than the classical urn models, yielding more powerful statistical inferences in applications. Illustrative examples are presented to demonstrate the wide applicability of the IMU models. The proposed IMU framework, including many popular classical urn models, not only offers a unify perspective for us to comprehend the urn process, but also enables us to generate several novel urn models with desirable properties."}
{"category": "Math", "title": "Multi-color Randomly Reinforced Urn for Adaptive Designs", "abstract": "This paper is withdrawn"}
{"category": "Math", "title": "Finite planar emulators for K_{4,5} - 4K_2 and K_{1,2,2,2} and Fellows' Conjecture", "abstract": "In 1988 Fellows conjectured that if a finite, connected graph admits a finite planar emulator, then it admits a finite planar cover. We construct a finite planar emulator for K_{4,5} - 4K_2. Archdeacon showed that K_{4,5} - 4K_2 does not admit a finite planar cover; thus K_{4,5} - 4K_2 provides a counterexample to Fellows' Conjecture. It is known that Negami's Planar Cover Conjecture is true if and only if K_{1,2,2,2} admits no finite planar cover. We construct a finite planar emulator for K_{1,2,2,2}. The existence of a finite planar cover for K_{1,2,2,2} is still open."}
{"category": "Math", "title": "Minimal Euclidean representations of graphs", "abstract": "A simple graph G is said to be representable in a real vector space of dimension m if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two distinct vertices is one of only two distinct values a or b, with distance a if the vertices are adjacent and distance b otherwise. The Euclidean representation number of G is the smallest dimension in which G is representable. In this note, we bound the Euclidean representation number of a graph using multiplicities of the eigenvalues of the adjacency matrix. We also give an exact formula for the Euclidean representation number using the main angles of the graph."}
{"category": "Math", "title": "Curvature estimates for submanifolds with prescribed Gauss image and mean curvature", "abstract": "We study that the $n-$graphs defining by smooth map $f:\\Om\\subset \\ir{n}\\to \\ir{m}, m\\ge 2,$ in $\\ir{m+n}$ of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates $$\\sup_{D_R(x)}|B|^2\\le\\f{C}{R^2}$$ under the dimension limitations and the Gauss image restrictions. If there is no dimension limitation we obtain $$\\sup_{D_R(x)}|B|^2\\le C R^{-a}\\sup_{D_{2R}(x)}(2-\\De_f)^{-(\\f{3}{2}+\\f{1}{s})}, \\qquad s=\\min(m, n)$$ with $a<1$ under the condition $$\\De_f=[\\text{det}(\\de_{ij}+\\sum_\\a\\pd{f^\\a}{x^i}\\pd{f^\\a}{x^j})]^{\\f{1}{2}}<2.$$ If the image under the Gauss map is contained in a geodesic ball of the radius $\\f{\\sqrt{2}}{4}\\pi$ in $\\grs{n}{m}$ we also derive corresponding estimates."}
{"category": "Math", "title": "A Lieb-Thirring inequality for singular values", "abstract": "Let $A$ and $B$ be positive semidefinite matrices. We investigate the conditions under which the Lieb-Thirring inequality can be extended to singular values. That is, for which values of $p$ does the majorisation $\\sigma(B^p A^p) \\prec_w \\sigma((BA)^p)$ hold, and for which values its reversed inequality $\\sigma(B^p A^p) \\succ_w \\sigma((BA)^p)$."}
{"category": "Math", "title": "Loewner chains on the universal covering space of a Riemann surface", "abstract": "Let R be a hyperbolic Riemann surface with boundary $\\partial R$ and suppose that $\\gamma:[0,T]\\to R\\cup\\partial R$ is a simple curve growing from the boundary of R. By lifting $R_{t}=R\\setminus \\gamma(0,t]$ to the universal covering space of R (which we assume is the upper half-plane $\\mathbb{H}=\\{z\\in\\mathbb{C}:Im[z]>0\\}$) via the covering map $\\pi:\\mathbb{H}\\to R$, we can define a family of simply-connected domains $D_{t}=\\pi^{-1}(R_{t})$. For each $t\\in[0,T]$, suppose that f_{t} is a conformal map of \\mathbb{H} onto D_{t} such that f(z,t)=f_{t}(z) is differentiable almost everywhere in (0,T) with respect to t. In this paper, we will derive a differential equation that describes how f(z,t) evolves in time t. This should be viewed as an extension of the Loewner differential equation to curves on Riemann surfaces with boundary. The motivation of this paper is the desire to extend Schramm's stochastic Loewner evolution (SLE) to multiply-connected domains and Riemann surfaces."}
{"category": "Math", "title": "Optimal three-ball inequalities and quantitative uniqueness for the Stokes system", "abstract": "In this paper we study the local behavior of a solution to the Stokes system with singular coefficients. One of the main results is the bound on the vanishing order of a nontrivial solution to the Stokes system, which is a quantitative version of the strong unique continuation property. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial \\emph{optimal} three-ball inequalities. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution to the Stokes system from those three-ball inequalities."}
{"category": "Math", "title": "Common fixed point theorems for occasionally weakly compatible maps", "abstract": "In this paper, we establish a common fixed point theorem for two pairs of occasionally weakly compatible single and set-valued maps satisfying a strict contractive condition in a metric space. Our result extends many results existing in the literature as those of Aliouche and Popa [15-20]. Also we establish another common fixed point theorem for four owc single and set-valued maps of Gregu% \\v{s} type which generalizes the results of Djoudi and Nisse, Pathak, Cho, Kang and Madharia and we end our work by giving a third theorem which extends the results given by Elamrani & Mehdaoui and Mbarki."}
{"category": "Math", "title": "The Imprimitive Faithful Complex Characters of the Schur Covers of the Symmetric and Alternating Groups", "abstract": "Using combinatorics and character theory, we determine the imprimitive faithful complex characters, i.e., the irreducible faithful complex characters which are induced from proper subgroups, of the Schur covers of the symmetric and alternating groups. Furthermore, for every imprimitive character we establish all its minimal block stabilizers. As a corollary, we also determine the monomial faithful characters of the Schur covers."}
{"category": "Math", "title": "Semiring Properties of Heyting Algebras", "abstract": "The relationship between Heyting algebras (HA) and semirings is explored. A new class of HAs called Symmetric Heyting algebras (SHAs) is proposed, and a necessary condition on SHAs to be consider semirings is given. We define a new mathematical family called Heyting structures, which are similar to semirings, but with Heyting-algebra operators in place of the usual arithmetic operators usually seen in semirings. The impact of the zero-sum free property of semirings on Heyting structures is shown as also the condition under which it is possible to extend one Heyting structure to another. It is also shown that the union of two or more sets forming Heyting structures is again a Heyting structure, if the operators on the new structure are suitably derived from those of the component structures. The analysis also provides a sufficient condition such that the larger Heyting structure satisfying a monotony law implies that the ones forming the union do so as well."}
{"category": "Math", "title": "On Kummer 3-folds", "abstract": "We investigate a generalization of Kummer construction, as introduced in a recent paper by M. Andreatta and J.A. Wisniewski. The aim of this work is to classify 3-dimensional Kummer varieties by computing their Poincare polynomials."}
{"category": "Math", "title": "The duration problem with multiple exchanges", "abstract": "We treat a version of the multiple-choice secretary problem called the multiple-choice duration problem, in which the objective is to maximize the time of possession of relatively best objects. It is shown that, for the $m$--choice duration problem, there exists a sequence (s1,s2,...,sm) of critical numbers such that, whenever there remain k choices yet to be made, then the optimal strategy immediately selects a relatively best object if it appears at or after time $s_k$ ($1\\leq k\\leq m$). We also exhibit an equivalence between the duration problem and the classical best-choice secretary problem. A simple recursive formula is given for calculating the critical numbers when the number of objects tends to infinity. Extensions are made to models involving an acquisition or replacement cost."}
{"category": "Math", "title": "Quasi-complete homogeneous contact manifold associated to a cubic form", "abstract": "Starting from a cubic form, we give a general construction of a quasi-complete homogeneous manifold endowed with a natural contact structure. We show that it can be compactified into a projective contact manifold if and only if the cubic form is the determinant of a simple cubic Jordan algebra."}
{"category": "Math", "title": "Overdetermined 2D Systems Invariant in One Direction and Their Transfer Functions", "abstract": "In this work we develop a theory of Vessels. This object arises in the study of overdetermined 2D systems invariant in one of the variables, which are usually called time invariant. To each overdetermined time invariant 2D systems there is associated a vessel, which is a collection of system operators satisfying certain relations and vise versa. Such an invariance forces the theory of vessels to resemble a constant (classical) 1D case and as a result many notions are naturally redefined and most theorems are reproved in this setting. The notion of transfer function and its connection to the overdetermined 2D time invariant system (and the corresponding vessel) is one of the topics of this work. It is well known that multiplicative structure of a transfer function of a 1D system is closely connected to the decomposition of the state space into invariant subspaces of the state operator and we generalize this result to a wider class of functions. This class (denoted by $\\boldsymbol {\\mathcal I}$) arises as a class of transfer functions, which intertwine solutions of ODEs with spectral parameters. At the end we present solution of factorization problems for finite dimensional case."}
{"category": "Math", "title": "Frameworks, Symmetry and Rigidity", "abstract": "Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in R^d. These form the basis for a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including constrained point-line systems that appear in CAD, body-pin frameworks, hybrid systems of distance constrained objects and infinite bar-joint frameworks. This leads to generalised forms of the Fowler-Guest character formula together with counting rules in terms of counts of symmetry-fixed elements. Necessary conditions for isostaticity are obtained for asymmetric frameworks, both when symmetries are present in subframeworks and when symmetries occur in partition-derived frameworks."}
{"category": "Math", "title": "A non-abelian Stickelberger theorem", "abstract": "Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring Z_(p)[G] that annihilates the p-part of the class group of L."}
{"category": "Math", "title": "Beyond Totally Reflexive Modules and Back", "abstract": "Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the starting point: with a characterization of Gorenstein rings in terms of total acyclicity of complexes."}
{"category": "Math", "title": "Sur les automorphismes reguliers de C^k", "abstract": "We show the uniqueness for the measure of maximal entropy for regular automorphisms of C^k."}
{"category": "Math", "title": "Two-scale semi-lagrangian simulation of a charged particle beam in a periodic focusing channel", "abstract": "This paper is devoted to numerical simulation of a charged particle beam submitted to a strong oscillating electric field. For that, we consider a two-scale numerical approach as follows: we first recall the two-scale model which is obtained by using two-scale convergence techniques; then, we numerically solve this limit model by using a backward semi-lagrangian method and we propose a new mesh of the phase space which allows us to simplify the solution of the Poisson's equation. Finally, we present some numerical results which have been obtained by the new method, and we validate its efficiency through long time simulations."}
{"category": "Math", "title": "Examples of stable embedded minimal spheres without area bounds", "abstract": "The author proves that there is an open non empty set of metrics on any 3-manifold such that there exists a family of stably embedded minimal 2-spheres whose area is unbounded. This generalizes the work of T. Colding and W. Minicozzi who have shown an analogous result for the torus and B. Dean who showed the positive genus case."}
{"category": "Math", "title": "Convergent finite element methods for compressible barotropic Stokes systems", "abstract": "We propose finite element methods for compressible barotropic Stokes systems. We state convergence results for these methods and outline their proofs. The principal tools of the proofs are higher integrability estimates for the discrete density, equations for the discrete effective viscous flux, and renormalized formulations of the numerical method for the density equation."}
{"category": "Math", "title": "The number of orientable small covers over cubes", "abstract": "We count orientable small covers over cubes. We also get estimates for $O_n/R_n$, where $O_n$ is the number of orientable small covers and $R_n$ is the number of all small covers over an $n$-cube up to the Davis-Januszkiewicz equivalence."}
{"category": "Math", "title": "Birational geometry of Fano double spaces of index two", "abstract": "We study birational geometry of Fano varieties, realized as double covers $\\sigma\\colon V\\to {\\mathbb P}^M$, $M\\geq 5$, branched over generic hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only structures of a rationally connected fiber space on $V$ are the pencils-subsystems of the free linear system $|-\\frac12 K_V|$. The groups of birational and biregular self-maps of the variety $V$ coincide."}
{"category": "Math", "title": "Depth Zero Representations of Nonlinear Covers of $p$-adic Groups", "abstract": "We generalize the methods of Moy-Prasad, in order to define and study the genuine depth zero representations of some nonlinear covers of reductive groups over $p$-adic local fields. In particular, we construct all depth zero supercuspidal representations of the metaplectic group $Mp_{2n}$ over a $p$-adic field of odd residue characteristic."}
{"category": "Math", "title": "A Probablistic Origin for a New Class of Bivariate Polynomials", "abstract": "We present here a probabilistic approach to the generation of new polynomials in two discrete variables. This extends our earlier work on the 'classical' orthogonal polynomials in a previously unexplored direction, resulting in the discovery of an exactly soluble eigenvalue problem corresponding to a bivariate Markov chain with a transition kernel formed by a convolution of simple binomial and trinomial distributions. The solution of the relevant eigenfunction problem, giving the spectral resolution of the kernel, leads to what we believe to be a new class of orthogonal polynomials in two discrete variables. Possibilities for the extension of this approach are discussed."}
{"category": "Math", "title": "Scaling limits for symmetric Ito-Levy processes in random medium", "abstract": "We are concerned with scaling limits of the solutions to stochastic differential equations with stationary coefficients driven by Poisson random measures and Brownian motions. We state an annealed convergence theorem, in which the limit exhibits a diffusive or superdiffusive behavior, depending on the integrability properties of the Poisson random measure"}
{"category": "Math", "title": "Convergent Interpolation to Cauchy Integrals over Analytic Arcs", "abstract": "We consider multipoint Pad\\'e approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini-smooth non-vanishing density, then the diagonal multipoint Pad\\'e approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Pad\\'e approximants is deduced from the analysis of underlying non-Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicit granted the parametrization of the arc."}
{"category": "Math", "title": "An introduction to motivic zeta functions of motives", "abstract": "It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes $I$ with rational parameters, expanded at an integral dimension $D= D_0$, are not only periods (Belkale, Brosnan, Bogner, Weinzierl) but actually multiple zeta values (Broadhurst, Kreimer). In order to determine, at least heuristically, whether this is the case in concrete instances, the philosophy of motives - more specifically, the theory of mixed Tate motives - suggests an arithmetic approach (Kontsevich): counting points of algebraic varieties related to $I$ modulo sufficiently many primes $p$ and checking that the number of points varies polynomially in $p$. On the other hand, Kapranov has introduced a new \"zeta function\", the role of which is precisely to \"interpolate\" between zeta functions of reductions modulo different primes $p$. In this survey, we outline this circle of ideas and some of their recent developments."}
{"category": "Math", "title": "Slope filtrations", "abstract": "Many slope filtrations occur in algebraic geometry, asymptotic analysis, ramification theory, p-adic theories, geometry of numbers... These functorial filtrations, which are indexed by rational (or sometimes real) numbers, have a lot of common properties. We propose a unified abstract treatment of slope filtrations, and survey how new ties between different domains have been woven by dint of deep correspondences between different concrete slope filtrations."}
{"category": "Math", "title": "Optimal stopping of a risk process when claims are covered immediately", "abstract": "The optimal stopping problem for the risk process with interests rates and when claims are covered immediately is considered. An insurance company receives premiums and pays out claims which have occured according to a renewal process and which have been recognized by them. The capital of the company is invested at interest rate $\\alpha\\in\\Re^{+}$, the size of claims increase at rate $\\beta\\in\\Re^{+}$ according to inflation process. The immediate payment of claims decreases the company investment by rate $\\alpha_1$. The aim is to find the stopping time which maximizes the capital of the company. The improvement to the known models by taking into account different scheme of claims payment and the possibility of rejection of the request by the insurance company is made. It leads to essentially new risk process and the solution of optimal stopping problem is different."}
{"category": "Math", "title": "Experimental designs for multiple-level responses, with application to a large-scale educational intervention", "abstract": "Educational research often studies subjects that are in naturally clustered groups of classrooms or schools. When designing a randomized experiment to evaluate an intervention directed at teachers, but with effects on teachers and their students, the power or anticipated variance for the treatment effect needs to be examined at both levels. If the treatment is applied to clusters, power is usually reduced. At the same time, a cluster design decreases the probability of contamination, and contamination can also reduce power to detect a treatment effect. Designs that are optimal at one level may be inefficient for estimating the treatment effect at another level. In this paper we study the efficiency of three designs and their ability to detect a treatment effect: randomize schools to treatment, randomize teachers within schools to treatment, and completely randomize teachers to treatment. The three designs are compared for both the teacher and student level within the mixed model framework, and a simulation study is conducted to compare expected treatment variances for the three designs with various levels of correlation within and between clusters. We present a computer program that study designers can use to explore the anticipated variances of treatment effects under proposed experimental designs and settings."}
{"category": "Math", "title": "Foliations on hypersurfaces in holomorphic symplectic manifolds", "abstract": "Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction $\\sigma|_Y$ of the holomorphic symplectic form induces a rank one foliation on Y. We investigate situations where this foliation has compact leaves; in such cases we obtain a space of leaves Y/F which has dimension 2n-2 and admits a holomorphic symplectic form."}
{"category": "Math", "title": "On Galois Correspondence and Non-Commutative Martingales", "abstract": "The subject of this thesis is Galois correspondence for von Neumann algebras and its interplay with non-commutative probability theory. After a brief introduction to representation theory for compact groups, in particular to Peter-Weyl theorem, and to operator algebras, including von Neumann algebras, automorphism groups, crossed products and decomposition theory, we formulate first steps of a non-commutative version of probability theory and introduce non-abelian analogues of stochastic processes and martingales. The central objects are a von Neumann algebra $\\Ma$ and a compact group $\\Gr$ acting on $\\Ma$, for which we give in three consecutive steps, i.e. for inner, spatial and general automorphism groups one-to-one correspondences between subgroups of $\\Gr$ and von Neumann subalgebras of $\\Ma$. Furthermore, we identify non-abelian martingales in our approach and prove for them a convergence theorem."}
{"category": "Math", "title": "Sectorial forms and degenerate differential operators", "abstract": "If $a$ is a densely defined sectorial form in a Hilbert space which is possibly not closable, then we associate in a natural way a holomorphic semigroup generator with $a$. This allows us to remove in several theorems of semigroup theory the assumption that the form is closed or symmetric. Many examples are provided, ranging from complex sectorial differential operators, to Dirichlet-to-Neumann operators and operators with Robin or Wentzell boundary conditions."}
{"category": "Math", "title": "Spaces of algebraic maps from real projective spaces into complex projective spaces", "abstract": "We study the homotopy types of spaces of algebraic (rational) maps from real projective spaces into complex projective spaces. In a previous paper we have shown that the inclusion of the first space into the second one is a homotopy equivalence. In this paper we prove that the homotopy types of the terms of the natural \"degree\" filtration approximate closer and closer the homotopy type of the space of continuous maps and obtain bounds that describe the closeness of the approximation in terms of the degree. Moreover, we compute the homotopy groups of the spaces in low dimensions."}
{"category": "Math", "title": "Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere", "abstract": "Global quantization of pseudo-differential operators on compact Lie groups is introduced relying on the representation theory of the group rather than on expressions in local coordinates. Operators on the 3-dimensional sphere and on group SU(2) are analysed in detail. A new class of globally defined symbols is introduced giving rise to the usual Hormander's classes of operators $\\Psi^m(G)$, $\\Psi^m(S^3)$ and $\\Psi^m(SU(2))$. Properties of the new class and symbolic calculus are analysed. Properties of symbols as well as $L^2$-boundedness and Sobolev $L^2$--boundedness of operators in this global quantization are established on general compact Lie groups."}
{"category": "Math", "title": "The Siegel modular forms of genus 2 with the simplest divisor", "abstract": "We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus two vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a question formulated during the conference \"Black holes, Black Rings and Modular Forms\" (ENS, Paris, August 2007). These modular forms generalize the classical Igusa form and the forms constructed by Gritsenko and Nikulin in 1998."}
{"category": "Math", "title": "Reflected Backward SDEs with General Jumps", "abstract": "In the first part of this paper we give a solution for the one-dimensional reflected backward stochastic differential equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson point process. The reflecting process is right continuous with left limits (rcll for short) whose jumps are arbitrary. We first prove existence and uniqueness of the solution for a specific coefficient in using a method based on a combination of penalization and the Snell envelope theory. To show the result in the general framework we use a fixed point argument in an appropriate space. The second part of the paper is related to BSDEs with two reflecting barriers. Once more we prove the existence and uniqueness of the solution of the BSDE."}
{"category": "Math", "title": "Overdetermined conservative 2D Systems, Invariant in One Direction and a Generalization of Potapov's theorem", "abstract": "This work is a direct continuation of the authors work arXiv:0812.3779v1. A special case of conservative overdetermined time invariant 2D systems is developed and studied. Defining transfer function of such a systems we obtain a class CI of inner functions $S(\\lambda,t_2)$, which are identity for $\\lambda=\\infty$, satisfy certain regularity assumptions and intertwines solutions of ODEs with a spectral parameter $\\lambda$. Using translation model, we prove that every function in the class CI can be realized as a transfer function of a certain vessel. The highlight of this theory is a generalization Potapov's theorem, which gives a very special formula for such a function in the form of multiplication of Blacke-Potapov products, corresponding to the discrete spectrum of certain system operator $A_1(t_2)$ and of multiplicative integral, corresponding to the continuous spectrum of $A_1(t_2)$. This theorem is proved under a slightly more restrictive assumptions, then the development of the whole theory. Namely, we suppose that the derivative of the transfer function is a continuous function of $t_2$ for almost all $\\lambda$. At the last part zero/pole interpolation problem for matrix functions in CI is considered and a realization theorem of such functions appeared in arXiv:0812.3779v1 (theorem 8.1) is reproved. Hermitian case is also analyzed and the corresponding realization theorem is proved."}
{"category": "Math", "title": "Revisiting R\\'ev\\'esz's stochastic approximation method for the estimation of a regression function", "abstract": "In a pioneer work, R\\'ev\\'esz (1973) introduces the stochastic approximation method to build up a recursive kernel estimator of the regression function $x\\mapsto E(Y|X=x)$. However, according to R\\'ev\\'esz (1977), his estimator has two main drawbacks: on the one hand, its convergence rate is smaller than that of the nonrecursive Nadaraya-Watson's kernel regression estimator, and, on the other hand, the required assumptions on the density of the random variable $X$ are stronger than those usually needed in the framework of regression estimation. We first come back on the study of the convergence rate of R\\'ev\\'esz's estimator. An approach in the proofs completely different from that used in R\\'ev\\'esz (1977) allows us to show that R\\'ev\\'esz's recursive estimator may reach the same optimal convergence rate as Nadaraya-Watson's estimator, but the required assumptions on the density of $X$ remain stronger than the usual ones, and this is inherent to the definition of R\\'ev\\'esz's estimator. To overcome this drawback, we introduce the averaging principle of stochastic approximation algorithms to construct the averaged R\\'ev\\'esz's regression estimator, and give its asymptotic behaviour. Our assumptions on the density of $X$ are then usual in the framework of regression estimation. We prove that the averaged R\\'ev\\'esz's regression estimator may reach the same optimal convergence rate as Nadaraya-Watson's estimator. Moreover, we show that, according to the estimation by confidence intervals point of view, it is better to use the averaged R\\'ev\\'esz's estimator rather than Nadaraya-Watson's estimator."}
{"category": "Math", "title": "Braids, Shuffles and Symmetrizers", "abstract": "Multiplicative analogues of the shuffle elements of the braid group rings are introduced; in local representations they give rise to certain graded associative algebras (b-shuffle algebras). For the Hecke and BMW algebras, the (anti)-symmetrizers have simple expressions in terms of the multiplicative shuffles. The (anti)-symmetrizers can be expressed in terms of the highest multiplicative 1-shuffles (for the Hecke and BMW algebras) and in terms of the highest additive 1-shuffles (for the Hecke algebras). The spectra and multiplicities of eigenvalues of the operators of the multiplication by the multiplicative and additive 1-shuffles are examined."}
{"category": "Math", "title": "On the algebraic index for riemannian \\'etale groupoids", "abstract": "In this paper we construct an explicit quasi-isomorphism to study the cyclic cohomology of a deformation quantization over a riemannian \\'etale groupoid. Such a quasi-isomorphism allows us to propose a general algebraic index problem for riemannian \\'etale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3-dim torus."}
{"category": "Math", "title": "Holomorphic Cartan geometries and Calabi--Yau manifolds", "abstract": "We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous d\\'emontrons que les seules vari\\'et\\'es projectives de Calabi--Yau qui poss\\`edent des g\\'eom\\'etrie holomorphes de Cartan sont les vari\\'et\\'es ab\\'eliennes.)"}
{"category": "Math", "title": "On the validity of Chapman-Enskog expansions for shock waves with small strength", "abstract": "We justify a Chapman-Enskog expansion for discontinuous solutions of hyperbolic conservation laws containing shock waves with small strength. Precisely, we establish pointwise uniform estimates for the difference between the traveling waves of a relaxation model and the traveling waves of the corresponding diffusive equations determined by a Chapman-Enskog expansion procedure to first- or second-order."}
{"category": "Math", "title": "Geometric quantization for proper moment maps: the Vergne conjecture", "abstract": "We establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a noncompact symplectic manifold with proper moment map."}
{"category": "Math", "title": "A note on badly approximable affine forms and winning sets", "abstract": "We give a sketch for an alternative proof of a recent result by J. Tseng."}
{"category": "Math", "title": "Stability in the L1 norm via a linearization method for nonlinear hyperbolic systems", "abstract": "We discuss the existence and uniqueness of discontinuous solutions to adjoint problems associated with nonlinear hyperbolic systems of conservation laws. By generalizing the Haar method for Glimm-type approximations to hyperbolic systems, we establish that entropy solutions depend continuously upon their initial data in the natural L1 norm."}
{"category": "Math", "title": "Repetitions in beta-integers", "abstract": "Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this paper, we study the maximal possible repetition of the same motif occurring in beta-integers -- one dimensional models of quasicrystals. We are interested in beta-integers realizing only a finite number of distinct distances between neighboring elements. In such a case, the problem may be reformulated in terms of combinatorics on words as a study of the index of infinite words coding beta-integers. We will solve a particular case for beta being a quadratic non-simple Parry number."}
{"category": "Math", "title": "Diminishing functionals for nonclassical entropy solutions selected by kinetic relations", "abstract": "We consider nonclassical entropy solutions to scalar conservation laws with concave-convex flux functions, whose set of left- and right-hand admissible states across undercompressive shocks is selected by a kinetic function \\phi. We introduce a new definition for the (generalized) strength of classical and nonclassical shocks, allowing us to propose a generalized notion of total variation functional. Relying only upon the natural assumption that the composite function \\phi o \\phi is uniformly contracting, we prove that the generalized total variation of front-tracking approximations is non-increasing in time, and we conclude with the existence of nonclassical solutions to the initial-value problem. We also propose two definitions of generalized interaction potentials which are adapted to handle nonclassical entropy solutions and we investigate their monotonicity properties. In particular, we exhibit an interaction functional which is globally non-increasing along a splitting-merging interaction pattern."}
{"category": "Math", "title": "Cordes characterization for pseudodifferential operators with symbols valued on a noncommutative C*-algebra", "abstract": "Given a separable unital C*-algebra A, let E denote the Banach-space completion of the A-valued Schwartz space on Rn with norm induced by the A-valued inner product $<f,g>=\\int f(x)^*g(x) dx$. The assignment of the pseudodifferential operator B=b(x,D) with A-valued symbol b(x,\\xi) to each smooth function with bounded derivatives b defines an injective mapping O, from the set of all such symbols to the set of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E. It is known that O is surjective if A is commutative. In this paper, we show that, if O is surjective for A, then it is also surjective for the algebra of k-by-k matrices with entries in A."}
{"category": "Math", "title": "Bounds on ternary cyclotomic coefficients", "abstract": "We present a new bound on $A = \\max_n |a_{pqr}(n)|$, where $a_{pqr}(n)$ are the coefficients of a ternary cyclotomic polynomial. We also prove that two consecutive coefficients of such a polynomial differ by at most one."}
{"category": "Math", "title": "Stochastically stable globally coupled maps with bistable thermodynamic limit", "abstract": "We study systems of globally coupled interval maps, where the identical individual maps have two expanding, fractional linear, onto branches, and where the coupling is introduced via a parameter - common to all individual maps - that depends in an analytic way on the mean field of the system. We show: 1) For the range of coupling parameters we consider, finite-size coupled systems always have a unique invariant probability density which is strictly positive and analytic, and all finite-size systems exhibit exponential decay of correlations. 2) For the same range of parameters, the self-consistent Perron-Frobenius operator which captures essential aspects of the corresponding infinite-size system (arising as the limit of the above when the system size tends to infinity), undergoes a supercritical pitchfork bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium."}
{"category": "Math", "title": "On powers of Stirling matrices", "abstract": "The powers of matrices with Stirling number-coefficients are investigated. It is revealed that the elements of these matrices have a number of properties of the ordinary Stirling numbers. Moreover, \"higher order\" Bell, Fubini and Eulerian numbers can be defined. Hence we give a new interpretation for E. T. Bell's iterated exponential integers. In addition, it is worth to note that these numbers appear in combinatorial physics, in the problem of the normal ordering of quantum field theoretical operators."}
{"category": "Math", "title": "Functionally recursive rings of matrices-Two examples", "abstract": "We define the notions of finite-state and functionally recursive matrices and their growth. We also introduce two rings generated by functionally recursive matrices. The first is isomorphic to the 2-generated free ring. The second is a 2-generated monomial ring such that the multiplicative semigroup of monomials in the generators is nil of degree 5 and the ring has Gelfand Kirillov dimension 1 + log(2)/log(a) where a=1/2(1+sqrt(5))."}
{"category": "Math", "title": "Generalization of n-ary Nambu algebras and beyond", "abstract": "The aim of this paper is to introduce $n$-ary Hom-algebra structures generalizing the $n$-ary algebras of Lie type enclosing $n$-ary Nambu algebras, $n$-ary Nambu-Lie algebras, $n$-ary Lie algebras, and $n$-ary algebras of associative type enclosing $n$-ary totally associative and $n$-ary partially associative algebras. Also, we provide a way to construct examples starting from an $n$-ary algebra and an $n$-ary algebras endomorphism. Several examples could be derived using this process."}
{"category": "Math", "title": "Separation theorems for compact Hausdorff foliations", "abstract": "We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the Gromov-Hausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the Gromov-Hausdorff topology."}
{"category": "Math", "title": "On the Supremum of Certain Families of Stochastic Processes", "abstract": "We consider a family of stochastic processes $\\{X_t^\\epsilon, t \\in T\\}$ on a metric space $T$, with a parameter $\\epsilon \\downarrow 0$. We study the conditions under which \\lim_{\\e \\to 0} \\P \\Big(\\sup_{t \\in T} |X_t^\\e| < \\delta \\Big) =1 when one has the \\textit{a priori} estimate on the modulus of continuity and the value at one point. We compare our problem to the celebrated Kolmogorov continuity criteria for stochastic processes, and finally give an application of our main result for stochastic intergrals with respect to compound Poisson random measures with infinite intensity measures."}
{"category": "Math", "title": "Default times, non arbitrage conditions and change of probability measures", "abstract": "In this paper we give a financial justification, based on non arbitrage conditions, of the $(H)$ hypothesis in default time modelling. We also show how the $(H)$ hypothesis is affected by an equivalent change of probability measure. The main technique used here is the theory of progressive enlargements of filtrations."}
{"category": "Math", "title": "Cobweb Posets and KoDAG Digraphs are Representing Natural Join of Relations, their diBigraphs and the Corresponding Adjacency Matrices", "abstract": "Natural join of $di-bigraphs$ that is directed biparted graphs and their corresponding adjacency matrices is defined and then applied to investigate the so called cobweb posets and their $Hasse$ digraphs called $KoDAGs$. $KoDAGs$ are special orderable directed acyclic graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. $KoDAGs$ appear to be distinguished family of $Ferrers$ digraphs which are natural join of a corresponding ordering chain of one direction directed cliques called $di-bicliques$. These digraphs serve to represent faithfully corresponding relations of arbitrary arity so that all relations of arbitrary arity are their subrelations. Being this $chain -way$ complete if compared with kompletne $Kuratowski$ bipartite graphs their $DAG$ denotation is accompanied with the letter $K$ in front of descriptive abbreviation $oDAG$. The way to join bipartite digraphs of binary into $multi-ary$ relations is the natural join operation either on relations or their digraph representatives. This natural join operation is denoted here by $\\os$ symbol deliberately referring to the direct sum $\\oplus$ of adjacency matrices as it becomes the case for disjoint $di-bigraphs$."}
{"category": "Math", "title": "Gr\\\"obner bases for operads", "abstract": "We define a new monoidal category on collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gr\\\"obner bases for operads, and present operadic versions of Bergman's Diamond Lemma and Buchberger's algorithm. This machinery can be applied to study symmetric operads. In particular, we obtain an effective algorithmic version of Hoffbeck's PBW criterion of Koszulness for (symmetric) quadratic operads."}
{"category": "Math", "title": "Fast finite-energy planes in symplectizations and applications", "abstract": "We define the notion of fast finite-energy planes in the symplectization of a closed 3-dimensional energy level $M$ of contact type. We use them to construct special open book decompositions of $M$ when the contact structure is tight and induced by a (non-degenerate) dynamically convex contact form. The obtained open books have disk-like pages that are global surfaces of section for the Hamiltonian dynamics. Let $S \\subset \\R^4$ be the boundary of a smooth, strictly convex, non-degenerate and bounded domain. We show that a necessary and sufficient condition for a closed Hamiltonian orbit $P\\subset S$ to be the boundary of a disk-like global surface of section for the Hamiltonian dynamics is that $P$ is unknotted and has self-linking number -1."}
{"category": "Math", "title": "Non-Hausdorff groupoids", "abstract": "We present examples of non-Hausdorff, etale, essentially principal groupoids for which three results, known to hold in the Hausdorff case, fail. These results are: (A) the subalgebra of continuous functions on the unit space is maximal abelian within the reduced groupoid C*-algebra, (B) every nonzero ideal of the reduced groupoid C*-algebra has a nonzero intersection with the subalgebra of continuous functions on the unit space, and (C) the open support of a normalizer is a bissection."}
{"category": "Math", "title": "Bijections on two variations of noncrossing partitions", "abstract": "We find bijections on 2-distant noncrossing partitions, 12312-avoiding partitions, 3-Motzkin paths, UH-free Schr{\\\"o}der paths and Schr{\\\"o}der paths without peaks at even height. We also give a direct bijection between 2-distant noncrossing partitions and 12312-avoiding partitions."}
{"category": "Math", "title": "K-stability of constant scalar curvature polarization", "abstract": "In this paper, we shall show that a polarized algebraic manifold is K-stable if the polarization class admits a Kaehler metric of constant scalar curvature. This generalizes the results of Chen-Tian, Donaldson and Stoppa. (Parts of the arguments are based on a forthcoming paper \"A stronger concept of K-stability.\" )"}
{"category": "Math", "title": "Fluctuations of the empirical quantiles of independent Brownian motions", "abstract": "We consider $n$ independent, identically distributed one-dimensional Brownian motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth density function $f$. The empirical quantiles, or pointwise order statistics, are denoted by $B_{j:n}(t)$, and we are interested in a sequence of quantiles $Q_n(t) = B_{j(n):n}(t)$, where $j(n)/n \\to \\alpha \\in (0,1)$. This sequence converges in probability in $C[0,\\infty)$ to $q(t)$, the $\\alpha$-quantile of the law of $B_j(t)$. Our main result establishes the convergence in law in $C[0,\\infty)$ of the fluctuation processes $F_n = n^{1/2}(Q_n - q)$. The limit process $F$ is a centered Gaussian process and we derive an explicit formula for its covariance function. We also show that $F$ has many of the same local properties as $B^{1/4}$, the fractional Brownian motion with Hurst parameter $H = 1/4$. For example, it is a quartic variation process, it has H\\\"older continuous paths with any exponent $\\gamma < 1/4$, and (at least locally) it has increments whose correlation is negative and of the same order of magnitude as those of $B^{1/4}$."}
{"category": "Math", "title": "Asymptotics of the Norm of Elliptical Random Vectors", "abstract": "In this paper we consider elliptical random vectors X in R^d,d>1 with stochastic representation A R U where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A is a given matrix. The main result of this paper is an asymptotic expansion of the tail probability of the norm of X derived under the assumption that R has distribution function is in the Gumbel or the Weibull max-domain of attraction."}
{"category": "Math", "title": "Regularity properties of the distance function to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and application in Riemannian geometry", "abstract": "Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka, is due to Li and Nirenberg. Finally, we give applications of our results in Riemannian geometry. Namely, we show that the distance function to the conjugate locus on a Riemannian manifold is locally semiconcave. Then, we show that if a Riemannian manifold is a deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex."}
{"category": "Math", "title": "Invariably Suboptimal - An attempt to improve the voting rules of Treaties of Nice and Lisbon", "abstract": "We investigate the voting rules in the Council of the European Union. It is known that the current system, according to the Treaty of Nice, and the voting system proposed in the Lisbon treaty both strongly deviate from the square root law by Penrose. This is known to be the ideal voting rule under certain assumptions. In 2004 Slomczynski and Zyczkowski designed a voting system, now known as the Jagiellonian Compromise. It satisfies the square root law with very high accuracy. Each member state in this system obtains a voting weight proportional to the square root of the population. Additionally the quota is fixed in such a way that the voting power of each country is also proportional to the square root of the population. In this paper we investigate to which extent a change of the quota in the Treaty of Nice and the Treaty of Lisbon may bring the voting power closer to the ideal square root distribution. Our computations show that even with optimal quota both systems are way off the ideal power distribution."}
{"category": "Math", "title": "Semisimplicity criteria for irreducible Hopf algebras in positive characteristic", "abstract": "We prove that a finite-dimensional irreducible Hopf algebra $H$ in positive characteristic is semisimple, if and only if it is commutative and semisimple, if and only if the restricted Lie algebra $P(H)$ of the primitives is a torus. This generalizes Hochschild's theorem on restricted Lie algebras, and also generalizes Demazure and Gabriel's and Sweedler's results on group schemes, in the special but essential situation with finiteness assumption added."}
{"category": "Math", "title": "A class of nonlinear elliptic boundary value problems", "abstract": "In this paper second order elliptic boundary value problems on bounded domains $\\Omega\\subset\\dR^n$ with boundary conditions on $\\partial\\Omega$ depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space $L^2(\\Omega)\\oplus (L^2(\\partial\\Omega))^m$, which is a natural generalization of known results on $\\lambda$-linear elliptic boundary value problems and $\\lambda$-rational boundary value problems for ordinary second order differential equations."}
{"category": "Math", "title": "Koszul duality for stratified algebras II. Standardly stratified algebras", "abstract": "We give a complete picture of the interaction between Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras imposing the condition that standard filtrations of projective modules are finite, and develop the tilting theory for such algebras. Under the assumption of existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both Ringel and Koszul duals belong to the class, and that these two dualities on this class commute."}
{"category": "Math", "title": "Interpolation Theorems for Self-adjoint Operators", "abstract": "We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator $L$, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where $L$ is a uniformly elliptic operator or a Schr\\\"odinger operator with electro-magnetic potential."}
{"category": "Math", "title": "On the 0-dimensional cusps of the Kahler moduli of a K3 surface", "abstract": "Let S be a projective K3 surface. It is proved that the 0-dimensional cusps of the Kahler moduli of S are in one-to-one correspondence with the twisted Fourier-Mukai partners of S. This leads to a counting formula for the 0-dimensional cusps of the Kahler moduli. Applications to rational maps between K3 surfaces with large Picard numbers are given. When the Picard number of S is 1, the bijective correspondence is calculated explicitly."}
{"category": "Math", "title": "The Enumeration of Vertex Induced Subgraphs with respect to the Number of Components", "abstract": "Inspired by the study of community structure in connection networks, we introduce the graph polynomial $Q(G;x,y)$, the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of $Q(G;x,y)$ using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property. We relate $Q(G;x,y)$ to other known graph invariants and graph polynomials, among them partition functions, the Tutte polynomial, the independence and matching polynomials, and the universal edge elimination polynomial introduced by I. Averbouch, B. Godlin and J.A. Makowsky (2008). We show that $Q(G;x,y)$ is vertex reconstructible in the sense of Kelly and Ulam, discuss its use in computing residual connectedness reliability. Finally we show that the computation of $Q(G;x,y)$ is $\\sharp \\mathbf{P}$-hard, but Fixed Parameter Tractable for graphs of bounded tree-width and clique-width."}
{"category": "Math", "title": "Kan replacement of simplicial manifolds", "abstract": "We establish a functor $Kan$ from local Kan simplicial manifolds to weak Kan simplicial manifolds. It gives a solution to the problem of extending local Lie groupoids to Lie 2-groupoids."}
{"category": "Math", "title": "Backward errors and linearizations for palindromic matrix polynomials", "abstract": "We derive computable expressions of structured backward errors of approximate eigenelements of *-palindromic and *-anti-palindromic matrix polynomials. We also characterize minimal structured perturbations such that approximate eigenelements are exact eigenelements of the perturbed polynomials. We detect structure preserving linearizations which have almost no adverse effect on the structured backward errors of approximate eigenelements of the *-palindromic and *-anti-palindromic polynomials."}
{"category": "Math", "title": "On Borel complexity of the isomorphism problems for graph-related classes of Lie algebras and finite p-groups", "abstract": "We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for a class of finite dimensional $2$-step nilpotent Lie algebras over a field and for a class of finite $p$-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J.D. Hamkins, and R. Miller. A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarity) is also studied."}
{"category": "Math", "title": "Yet Another Poincare's Polyhedron Theorem", "abstract": "Poincar\\'e's Polyhedron Theorem is a widely known valuable tool in constructing manifolds endowed with a prescribed geometric structure. It is one of the few criteria providing discreteness of groups of isometries. This work contains a version of Poincar\\'e's Polyhedron Theorem that is applicable to constructing fibre bundles over surfaces and also suits geometries of nonconstant curvature. Most conditions of the theorem, being as local as possible, are easy to verify in practice."}
{"category": "Math", "title": "Some convergence results on quadratic forms for random fields and application to empirical covariances", "abstract": "Limit theorems are proved for quadratic forms of Gaussian random fields in presence of long memory. We obtain a non central limit theorem under a minimal integrability condition, which allows isotropic and anisotropic models. We apply our limit theorems and those of Ginovian (99) to obtain the asymptotic behavior of the empirical covariances of Gaussian fields, which is a particular example of quadratic forms. We show that it is possible to obtain a Gaussian limit when the spectral density is not in $L^2$. Therefore the dichotomy observed in dimension $d=1$ between central and non central limit theorems cannot be stated so easily due to possible anisotropic strong dependence in $d>1$."}
{"category": "Math", "title": "Schanuel's conjecture and the exceptional set of $\\gamma$-th arithmetic zeta functions", "abstract": "In this work, we study the arithmetic nature of the numbers of the form $n^{\\g}$, for $n \\in \\N$ and $\\g\\in \\C$. We also consider a related conjecture and we show that it is a consequence of the unipresent Schanuel's conjecture."}
{"category": "Math", "title": "Consistency conditions for brane tilings", "abstract": "Given a brane tiling on a torus, we provide a new way to prove and generalise the recent results of Szendroi, Mozgovoy and Reineke regarding the Donaldson-Thomas theory of the moduli space of framed cyclic representations of the associated algebra. Using only a natural cancellation-type consistency condition, we show that the algebras are 3-Calabi-Yau, and calculate Donaldson-Thomas type invariants of the moduli spaces. Two new ingredients to our proofs are a grading of the algebra by the path category of the associated quiver modulo relations, and a way of assigning winding numbers to pairs of paths in the lift of the brane tiling to R^2. These ideas allow us to generalise the above results to all consistent brane tilings on K(pi,1) surfaces. We also prove a converse: no consistent brane tiling on a sphere gives rise to a 3-Calabi-Yau algebra."}
{"category": "Math", "title": "Embedding into manifolds with torsion", "abstract": "We introduce a class of special geometries associated to the choice of a differential graded algebra contained in \\Lambda R^n. We generalize some known embedding results, that effectively characterize the real analytic Riemannian manifolds that can be realized as submanifolds of a Riemannian manifold with special holonomy, to this more general context. In particular, we consider the case of hypersurfaces inside nearly-Kaehler and alpha-Einstein-Sasaki manifolds, proving that the corresponding evolution equations always admit a solution in the real analytic case."}
{"category": "Math", "title": "Arithmetic McKay correspondence", "abstract": "We propose an arithmetic McKay correspondence which relates suitably defined zeta functions of some Deligne-Mumford stacks to the zeta functions of their crepant resolutions. Some examples are discussed."}
{"category": "Math", "title": "Hardy type spaces on certain noncompact manifolds and applications", "abstract": "In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below, positive injectivity radius and spectral gap b. We introduce a sequence X^1(M), X^2(M), ... of new Hardy spaces on M, the sequence Y^1(M/, Y^2(M), ... of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for purely imaginary powers of the Laplace-Beltrami operator and for more general spectral multipliers associated to the Laplace--Beltrami operator L on M. Under the additional condition that the volume of the geodesic balls of radius r is controlled by C r^a e^{2\\sqrt{b} r} for some real number a and for all large r, we prove also an endpoint result for first order Riesz transforms D L^{-1/2}. In particular, these results apply to Riemannian symmetric spaces of the noncompact type."}
{"category": "Math", "title": "KMS States, Entropy and a Variational Principle for Pressure", "abstract": "We want to relate the concepts of entropy and pressure to that of KMS states for $C^*$-Algebras. Several different definitions of entropy are known in our days. The one we describe here is quite natural and extends the usual one for Dynamical Systems in Thermodynamic Formalism Theory. It has the advantage of been very easy to be introduced. It is basically obtained from transfer operators (also called Ruelle operators). Later we introduce the concept of pressure as a min-max principle. Finally, we consider the concept of a KMS state as an equilibrium state for a potential (in the context of $C^*$-Algebras) and we show that there is a relation between KMS states for certain algebras coming from a continuous transformation and equilibrium measures."}
{"category": "Math", "title": "Structure of the curvature tensor on symplectic spinors", "abstract": "We study symplectic manifolds $(M^{2l},\\omega)$ equipped with a symplectic torsion-free affine (also called Fedosov) connection $\\nabla$ and admitting a metaplectic structure. Let $\\mathcal{S}$ be the so called symplectic spinor bundle and let $R^S$ be the curvature tensor field of the symplectic spinor covariant derivative $\\nabla^S$ associated to the Fedosov connection $\\nabla.$ It is known that the space of symplectic spinor valued exterior differential 2-forms, $\\Gamma(M,\\bigwedge^2T^*M\\otimes {\\mathcal{S}}),$ decomposes into three invariant spaces with respect to the structure group, which is the metaplectic group $Mp(2l,\\mathbb{R})$ in this case. For a symplectic spinor field $\\phi \\in \\Gamma(M,\\mathcal{S}),$ we compute explicitly the projections of $R^S\\phi \\in \\Gamma(M,\\bigwedge^2T^*M \\otimes \\mathcal{S})$ onto the three mentioned invariant spaces in terms of the symplectic Ricci and symplectic Weyl curvature tensor fields of the connection $\\nabla.$ Using this decomposition, we derive a complex of first order differential operators provided the Weyl tensor of the Fedosov connection is trivial."}
{"category": "Math", "title": "A Sliding Blocks Estimator for the Extremal Index", "abstract": "In extreme value statistics for stationary sequences, blocks estimators are usually constructed by using disjoint blocks because exceedances over high thresholds of different blocks can be assumed asymptotically independent. In this paper we focus on the estimation of the extremal index which measures the degree of clustering of extremes. We consider disjoint and sliding blocks estimators and compare their asymptotic properties. In particular we show that the sliding blocks estimator is more efficient than the disjoint version and has a smaller asymptotic bias. Moreover we propose a method to reduce its bias when considering sufficiently large block sizes."}
{"category": "Math", "title": "Critical points of solutions of degenerate elliptic equations in the plane", "abstract": "We study the minimizer u of a convex functional in the plane which is not G\\^ateaux-differentiable. Namely, we show that the set of critical points of any C^1-smooth minimizer can not have isolated points. Also, by means of some appropriate approximating scheme and viscosity solutions, we determine an Euler-Lagrange equation that u must satisfy. By applying the same approximating scheme, we can pair u with a function v which may be regarded as the stream function of u in a suitable generalized sense."}
{"category": "Math", "title": "Polars of real singular curves", "abstract": "Polar varieties have in recent years been used by Bank, Giusti, Heintz, Mbakop, and Pardo, and by Safey El Din and Schost, to find efficient procedures for determining points on all real components of a given non-singular algebraic variety. In this note we review the classical notion of polars and polar varieties, as well as the construction of what we here call reciprocal polar varieties. In particular we consider the case of real affine plane curves, and we give conditions for when the polar varieties of singular curves contain points on all real components."}
{"category": "Math", "title": "Generalized Lagrangian mean curvature flow in K\\\"ahler manifolds that are almost Einstein", "abstract": "We introduce the notion of K\\\"ahler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a K\\\"ahler manifold that is almost Einstein, and which in addition has a trivial canonical bundle, we show that the generalized mean curvature vector field of a Lagrangian submanifold is the dual vector field associated to the Lagrangian angle."}
{"category": "Math", "title": "Proving finitely presented groups are large by computer", "abstract": "We present a theoretical algorithm which, given any finite presentation of a group as input, will terminate with answer yes if and only if the group is large. We then implement a practical version of this algorithm using Magma and apply it to a range of presentations. Our main focus is on 2-generator 1-relator presentations where we have a complete picture of largeness if the relator has exponent sum zero in one generator and word length at most 12, as well as when the relator is in the commutator subgroup and has word length at most 18. Indeed all but a tiny number of presentations define large groups. Finally we look at fundamental groups of closed hyperbolic 3-manifolds, where the algorithm readily determines that a quarter of the groups in the Snappea closed census are large."}
{"category": "Math", "title": "Weak convergence of the periodic multiplicative Selmer algorithm", "abstract": "In order to prove weak convergence of the periodic multiplicative Selmer algorithm we ensure that the periodicity matrix is positive and establish a relation between its entries and eigenvalues. Since we can imply that the limit of these relations exist, we arrive at the desired result."}
{"category": "Math", "title": "Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras", "abstract": "We say that a finite dimensional Lie algebra is quasi-reductive if it has a linear form whose stabilizer for the coadjoint representation, modulo the center, is a reductive Lie algebra with a center consisting of semisimple elements. Parabolic subalgebras of a semisimple Lie algebra are not always quasi-reductive (except in types A or C by work of Panyushev). The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso. In this paper, we investigate the quasi-reductivity of biparabolic subalgebras of reductive Lie algebras. Biparabolic (or seaweed) subalgebras are the intersection of two parabolic subalgebras whose sum is the total Lie algebra. As a main result, we complete the classification of quasi-reductive parabolic subalgebras of reductive Lie algebras by considering the exceptional cases."}
{"category": "Math", "title": "A remark about Mahler's conjecture and the maximum value of box splines", "abstract": "In this paper, we recast a special case of Mahler'c conjecture by the maximum value of box splines. This is the case of polytopes with at most $2n+2$ facets. An asymptotic formula for univariate box splines is given. Based on the formula, Mahler's conjecture is proved in this case provided $n$ is big enough."}
{"category": "Math", "title": "Genus-One Stable Maps, Local Equations, and Vakil-Zinger's desingularization", "abstract": "We describe an algebro-geometric approach to Vakil-Zinger's desingularization of the main component of the moduli of genus one stable maps to projective space. The new approach provides complete local structural results for this moduli space as well as for the desingularization of the entire moduli space and should fully extend to higher genera."}
{"category": "Math", "title": "Wythoff polytopes and low-dimensional homology of Mathieu groups", "abstract": "We describe two methods for computing the low-dimensional integral homology of the Mathieu simple groups and use them to make computations such as $H_5(M_{23},\\ZZ)=\\ZZ_7$ and $H_3(M_{24},\\ZZ)=\\ZZ_{12}$. One method works via Sylow subgroups. The other method uses a Wythoff polytope and perturbation techniques to produce an explicit free $\\ZZ M_n$-resolution. Both methods apply in principle to arbitrary finite groups."}
{"category": "Math", "title": "On the extrinsic geometry of contact structures", "abstract": "In the paper we prove, that extrinsic curvature does not impose restrictions on the topology of a contact structure, except the obvious ones."}
{"category": "Math", "title": "Duality and products in algebraic (co)homology theories", "abstract": "The origin and interplay of products and dualities in algebraic (co)homology theories is ascribed to a $\\times_A$-Hopf algebra structure on the relevant universal enveloping algebra. This provides a unified treatment for example of results by Van den Bergh about Hochschild (co)homology and by Huebschmann about Lie-Rinehart (co)homology."}
{"category": "Math", "title": "On varieties whose universal cover is a product of curves", "abstract": "We investigate a necessary condition for a compact complex manifold X of dimension n in order that its universal cover be the Cartesian product $C^n$ of a curve $C = \\PP^1 or \\HH$: the existence of a semispecial tensor $\\omega$. A semispecial tensor is a non zero section $ 0 \\neq \\omega \\in H^0(X, S^n\\Omega^1_X (-K_X) \\otimes \\eta) $), where $\\eta$ is an invertible sheaf of 2-torsion (i.e., $\\eta^2\\cong \\hol_X$). We show that this condition works out nicely, as a sufficient condition, when coupled with some other simple hypothesis, in the case of dimension $n= 2$ or $ n= 3$; but it is not sufficient alone, even in dimension 2. In the case of K\\\"ahler surfaces we use the above results in order to give a characterization of the surfaces whose universal cover is a product of two curves, distinguishing the 6 possible cases."}
{"category": "Math", "title": "Algebraic Surfaces and their Moduli Spaces: Real, Differentiable and Symplectic Structures", "abstract": "The article surveys published and not yet published results about moduli spaces of algebraic surfaces."}
{"category": "Math", "title": "Some Cobweb Posets Digraphs' Elementary Properties and Questions", "abstract": "A digraph that represents reasonably a scheduling problem should be a directed acyclic graph. Here down we shall deal with special kind of graded $DAGs$ named $KoDAGs$. For their definition and first primary properties see $ [1]$, where natural join of directed biparted graphs and their corresponding adjacency matrices is defined and then applied to investigate cobweb posets and their $Hasse$ digraphs called $KoDAGs$. In this report we extend the notion of cobweb poset while delivering some elementary consequences of the description and observations established in $[1]$."}
{"category": "Math", "title": "Metrics of constant scalar curvatures conformal to a Riemannian product with a round sphere", "abstract": "We consider the conformal class of the Riemannian product $g_0 + g$, where $g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect to the square root of the scalar curvature of $g$. This is obtained by studying radial solutions of the equation $\\Delta u -\\lambda u + \\lambda u^p =0$ on $S^m$, and the number of solutions in terms of $\\lambda$."}
{"category": "Math", "title": "Spectral gaps for periodic Schr\\\"odinger operators with hypersurface magnetic wells: Analysis near the bottom", "abstract": "We consider a periodic magnetic Schr\\\"odinger operator $H^h$, depending on the semiclassical parameter $h>0$, on a noncompact Riemannian manifold $M$ such that $H^1(M, {\\mathbb R})=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We suppose that the magnetic field vanishes regularly on a hypersurface $S$. First, we prove upper and lower estimates for the bottom $\\lambda_0(H^h)$ of the spectrum of the operator $H^h$in $L^2(M)$. Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on $S$, we prove the existence of an arbitrary large number of spectral gaps for the operator $H^h$ in the region close to $\\lambda_0(H^h)$, as $h\\to 0$. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem."}
{"category": "Math", "title": "A note on the relation between fixed point and orbit count sequences", "abstract": "The relation between fixed point and orbit count sequences is investigated from the point of view of linear mappings on the space of arithmetic functions. Spectral and asymptotic properties are derived and several quantities are explicitly given in terms of Gaussian binomial coefficients."}
{"category": "Math", "title": "A smooth pseudo-gradient for the Lagrangian action functional", "abstract": "We study the action functional associated to a smooth Lagrangian function on the cotangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H^1 curves, it is a Lyapunov function for some smooth Morse-Smale vector field, under the generic assumption that all the critical points are non-degenerate. This fact is sufficient to associate a Morse complex to the Lagrangian action functional."}
{"category": "Math", "title": "On the stability of $\\phi$-uniform domains", "abstract": "We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $G \\subset {\\mathbb R}^n$. In the sequel, we investigate a class of domains, so called $\\varphi$-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $\\varphi$ from $[0,\\infty)$ to itself. Finally, we discuss a number of stability properties of $\\varphi$-uniform domains. In particular, we show that the class of $\\varphi$-uniform domains is stable in the sense that removal of a geometric sequence of points from a $\\varphi$-uniform domain yields a $\\varphi_1$-uniform domain."}
{"category": "Math", "title": "Gorenstein dimensions in trivial ring extensions", "abstract": "In this paper, we show that the Gorenstein global dimension of trivial ring extensions is often infinite. Also we study the transfer of Gorenstein properties between a ring and its trivial ring extensions. We conclude with an example showing that, in general, the transfer of the notion of Gorenstein projective module does not carry up to pullback constructions."}
{"category": "Math", "title": "Multiplicative $q$-hypergeometric series arising from real quadratic fields", "abstract": "Andrews, Dyson, and Hickerson showed that 2 $q$-hypergeometric series, going back to Ramanujan, are related to real quadratic fields, which explains interesting properties of their Fourier coefficients. There is also an interesting relation of such series to automorphic forms. Here we construct more such examples arising from interesting combinatorial statistics."}
{"category": "Math", "title": "Geometry of splice-quotient singularities", "abstract": "We obtain a new important basic result on splice-quotient singularities in an elegant combinatorial-geometric way: every level of the divisorial filtration of the ring of functions is generated by monomials of the defining coordinate functions. The elegant way is the language of of line bundles based on Okuma's description of the function ring of the universal abelian cover. As an easy application, we obtain a new proof of the End Curve Theorem of Neumann and Wahl."}
{"category": "Math", "title": "Martingale-Coboundary Representation for a Class of Random Fields", "abstract": "A stationary random sequence admits under some assumptions a representation as the sum of two others: one of them is a martingale difference sequence, and another is a so-called coboundary. Such a representation can be used for proving some limit theorems by means of the martingale approximation. A multivariate version of such a decomposition is presented in the paper for a class of random fields generated by several commuting non-invertible probability preserving transformations. In this representation summands of mixed type appear which behave with respect to some groupof directions of the parameter space as reversed multiparameter martingale differences (in the sense of one of several known definitions) while they look as coboundaries relative to the other directions. Applications to limit theorems will be published elsewhere."}
{"category": "Math", "title": "Infinitely many leaf-wise intersections on cotangent bundles", "abstract": "If the homology of the free loop space of a closed manifold B is infinite dimensional then generically there exist infinitely many leaf-wise intersection points for fiber-wise star-shaped hypersurfaces in T*B."}
{"category": "Math", "title": "The R- and L-orders of the Thompson-Higman monoid M_{k,1} and their complexity", "abstract": "We study the monoid generalization M_{k,1} of the Thompson-Higman groups, and we characterize the R- and the L-preorder of M_{k,1}. Although M_{k,1} has only one non-zero J-class and k-1 non-zero D-classes, the R- and the L-preorder are complicated; in particular, <_R is dense (even within an L-class), and <_L is dense (even within an R-class). We study the computational complexity of the R- and the L-preorder. When inputs are given by words over a finite generating set of M_{k,1}, the R- and the L-preorder decision problems are in P. The main result of the paper is that over a \"circuit-like\" generating set, the R-preorder decision problem of M_{k,1} is Pi_2^P-complete, whereas the L-preorder decision problem is coNP-complete. We also prove related results about circuits: For combinational circuits, the surjectiveness problem is Pi_2^P-complete, whereas the injectiveness problem is coNP-complete."}
{"category": "Math", "title": "Reflection equation algebras, coideal subalgebras, and their centres", "abstract": "Reflection equation algebras and related U_q(g)-comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so called `covariantized' algebras, in particular concerning their centres, invariants, and characters. Generalising M. Noumi's construction of quantum symmetric pairs we define a coideal subalgebra B_f of U_q(g) for each character f of a covariantized algebra. The locally finite part F_l(U_q(g)) of U_q(g) with respect to the left adjoint action is a special example of a covariantized algebra. We show that for each character f of F_l(U_q(g)) the centre Z(B_f) canonically contains the representation ring Rep(g) of the semisimple Lie algebra g. We show moreover that for g=sl_n(C) such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of Rep(sl_n(C)) inside U_q(sl_n(C)). As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifold Gr(m,2m) of m-dimensional subspaces in C^{2m}."}
{"category": "Math", "title": "Irregular Wakimoto modules and the Casimir connection", "abstract": "We study some non-highest weight modules over an affine Kac-Moody algebra at non-critical level. Roughly speaking, these modules are non-commutative localizations of some non-highest weight \"vacuum\" modules. Using free field realization, we embed some rings of differential operators in endomorphism rings of our modules. These rings of differential operators act on a localization of the space of coinvariants of any module over the Kac-Moody algebra with respect to a certain level subalgebra. In a particular case this action is identified with the Casimir connection."}
{"category": "Math", "title": "The rectifiable distance in the unitary Fredholm group", "abstract": "Let $U_c(H)={u: u is unitary and u-1 is compact}$ stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_\\infty$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_c(H)$. If $u_0,u_1,u\\in U_c(H)$ and the geodesic $\\beta$ joining $u_0$ and $u_1$ in $U_c(H)$ verifies $d_\\infty(u,\\beta)<\\pi/2$, then the map $f(s)=d_\\infty(u,\\beta(s))$ is convex for $s\\in[0,1]$. In particular the convexity radius of the geodesic balls in $U_c(H)$ is $\\pi/4$. The same convexity property holds in the $p$-Schatten unitary groups $U_p(H)={u: u is unitary and u-1 is in the p-Schatten class}$, for $p$ an even integer, $p\\ge 4$ (in this case, the distance is strictly convex). The same results hold in the unitary group of a $C^*$-algebra with a faithful finite trace. We apply this convexity result to establish the existence of curves of minimal length with given initial conditions, in the unitary orbit of an operator, under the action of the Fredholm group. We characterize self-adjoint operators $A$ such that this orbit is a submanifold (of the affine space $A+K(H)$, where $K(H)$=compact operators)."}
{"category": "Math", "title": "Manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings", "abstract": "We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed."}
{"category": "Math", "title": "On Gromov-Witten theory of root gerbes", "abstract": "This research announcement discusses our results on Gromov-Witten theory of root gerbes. A complete calculation of genus 0 Gromov-Witten theory of $\\mu_{r}$-root gerbes over a smooth base scheme is obtained by a direct analysis of virtual fundamental classes. Our result verifies the genus 0 part of the so-called decomposition conjecture which compares Gromov-Witten theory of \\'etale gerbes with that of the bases. We also verify this conjecture in all genera for toric gerbes over toric Deligne-Mumford stacks."}
{"category": "Math", "title": "Chen's primes and ternary Goldbach problem", "abstract": "We prove that there exists a k_0>0 such that every sufficiently large odd integer n with 3\\mid n can be represented as p_1+p_2+p_3, where p_1,p_2 are Chen's primes and p_3 is a prime with p_3+2 has at most k_0 prime factors."}
{"category": "Math", "title": "Microlocal study of Lefschetz fixed point formulas for higher-dimensional fixed point sets", "abstract": "We introduce new Lagrangian cycles which encode local contributions of Lefschetz numbers of constructible sheaves into geometric objects. We study their functorial properties and apply them to Lefschetz fixed point formulas with higher-dimensional fixed point sets."}
{"category": "Math", "title": "2-Representations and Equivariant 2D Topological Field Theories", "abstract": "Following ideas of G. Moore and G. Segal, we explicitly construct a G-equivariant topological field theory from an arbitrary Frobenius algebra equipped with a twisted action of a finite group G."}
{"category": "Math", "title": "Random complex dynamics and semigroups of holomorphic maps", "abstract": "We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the averaged system disappears, due to the cooperation of the generators. We investigate the iteration and spectral properties of transition operators. We show that under certain conditions, in the limit stage, \"singular functions on the complex plane\" appear. In particular, we consider the functions $T$ which represent the probability of tending to infinity with respect to the random dynamics of polynomials. Under certain conditions these functions $T$ are complex analogues of the devil's staircase and Lebesgue's singular functions. More precisely, we show that these functions $T$ are continuous on the Riemann sphere and vary only on the Julia sets of associated semigroups. Furthermore, by using ergodic theory and potential theory, we investigate the non-differentiability and regularity of these functions. We find many phenomena which can hold in the random complex dynamics and the dynamics of semigroups of rational maps, but cannot hold in the usual iteration dynamics of a single holomorphic map. We carry out a systematic study of these phenomena and their mechanisms."}
{"category": "Math", "title": "Generalized $q$-boson algebras and their integrable modules", "abstract": "We define the generalized $q$-boson algebra $ \\cmdB$ associated to a pair of Nichols algebras and a skew pairing. We study integrable $ \\cmdB$-modules, generalizing results by M. Kashiwara and T. Nakashima on integrable modules over a $q$-boson (Kashiwara) algebra."}
{"category": "Math", "title": "The structure Jacobi operator for hypersurfaces in CP^2 and CH^2", "abstract": "Using the methods of moving frames, we study real hypersurfaces in complex projective space CP^2 and complex hyperbolic space CH^2 whose structure Jacobi operator has various special properties. Our results complement work of several other authors who worked on such hypersurfaces in CP^n and CH^n for n>2."}
{"category": "Math", "title": "Reid's recipe and derived categories", "abstract": "We prove two existing conjectures which describe the geometrical McKay correspondence for a finite abelian G in SL3(C) such that C^3/G has a single isolated singularity. We do it by studying the relation between the derived category mechanics of computing a certain Fourier-Mukai transform and a piece of toric combinatorics known as `Reid's recipe', effectively providing a categorification of the latter."}
{"category": "Math", "title": "T-structure and the Yamabe invariant", "abstract": "The Yamabe invariant is an invariant of a closed smooth manifold, which contains information about possible scalar curvature on it. It is well-known that a product manifold T^m\\times B where T^m$ is the m-dimensional torus, and B is a closed spin manifold of nonzero \\hat{A}-genus has zero Yamabe invariant. We generalize it to various T-structured manifolds, for example T^m-bundles over such B whose transition functions take values in Sp(m,Z) (or Sp(m-1,Z)\\oplus \\pm 1 for odd m)."}
{"category": "Math", "title": "The dihedral group $\\Dh_5$ as group of symplectic automorphisms on K3 surfaces", "abstract": "We prove that if a K3 surface $X$ admits $\\Z/5\\Z$ as group of symplectic automorphisms, then it actually admits $\\Dh_5$ as group of symplectic automorphisms. The orthogonal complement to the $\\Dh_5$-invariants in the second cohomology group of $X$ is a rank 16 lattice, $L$. It is known that $L$ does not depend on $X$: we prove that it is isometric to a lattice recently described by R. L. Griess Jr. and C. H. Lam. We also give an elementary construction of $L$."}
{"category": "Math", "title": "Hopf Hypersurfaces of Small Hopf Principal Curvature in CH^2", "abstract": "Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space CH^2 with any specified value of the Hopf principal curvature less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces in terms of Weierstrass-type data, and also obtain a classification of pseudo-Einstein hypersurfaces in CH^2."}
{"category": "Math", "title": "Normal forms and linearization of holomorphic dilation type semigroups in several variables", "abstract": "In this paper we study commuting families of holomorphic mappings in $\\mathbb{C}^n$ which form abelian semigroups with respect to their real parameter. Linearization models for holomorphic mappings are been used in the spirit of Schr\\\"oder's classical functional equation."}
{"category": "Math", "title": "Maximal residuated lattices with lifting Boolean center", "abstract": "In this paper we define, inspired by ring theory, the class of maximal residuated lattices with lifting Boolean center and prove a structure theorem for them: any maximal residuated lattice with lifting Boolean center is isomorphic to a finite direct product of local residuated lattices."}
{"category": "Math", "title": "Complex Hessian Equation on K\\\"ahler Manifold", "abstract": "In this paper, complex Hessian equation over K\\\"ahler manifold was studied. Under the condition that the underline K\\\"ahler manifold has non-negative holomorphic bisectional curvature, the existence and regularity of the solution was proved."}
{"category": "Math", "title": "Inequalities for mixed $p$-affine surface area", "abstract": "We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed $p$-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of $L_p$ affine surface areas, mixed $p$-affine surface areas and other functionals via these bodies. The surprising new element is that not necessarily convex bodies provide the tool for these interpretations."}
{"category": "Math", "title": "Uniform convergence for complex $[\\mathbf{0,1}]$-martingales", "abstract": "Positive $T$-martingales were developed as a general framework that extends the positive measure-valued martingales and are meant to model intermittent turbulence. We extend their scope by allowing the martingale to take complex values. We focus on martingales constructed on the interval $T=[0,1]$ and replace random measures by random functions. We specify a large class of such martingales for which we provide a general sufficient condition for almost sure uniform convergence to a nontrivial limit. Such a limit yields new examples of naturally generated multifractal processes that may be of use in multifractal signals modeling."}
{"category": "Math", "title": "Convergence of complex multiplicative cascades", "abstract": "The familiar cascade measures are sequences of random positive measures obtained on $[0,1]$ via $b$-adic independent cascades. To generalize them, this paper allows the random weights invoked in the cascades to take real or complex values. This yields sequences of random functions whose possible strong or weak limits are natural candidates for modeling multifractal phenomena. Their asymptotic behavior is investigated, yielding a sufficient condition for almost sure uniform convergence to nontrivial statistically self-similar limits. Is the limit function a monofractal function in multifractal time? General sufficient conditions are given under which such is the case, as well as examples for which no natural time change can be used. In most cases when the sufficient condition for convergence does not hold, we show that either the limit is 0 or the sequence diverges almost surely. In the later case, a functional central limit theorem holds, under some conditions. It provides a natural normalization making the sequence converge in law to a standard Brownian motion in multifractal time."}
{"category": "Math", "title": "Complex Hessian Equations on Compact Kahler Manifold", "abstract": "This paper has been withdrawn by the author due to a crucial error in the proof of Lemma 2.2."}
{"category": "Math", "title": "Shellable Complexes from Multicomplexes", "abstract": "Suppose a group $G$ acts properly on a simplicial complex $\\Gamma$. Let $l$ be the number of $G$-invariant vertices and $p_1, p_2, ... p_m$ be the sizes of the $G$-orbits having size greater than 1. Then $\\Gamma$ must be a subcomplex of $\\Lambda = \\Delta^{l-1}* \\partial \\Delta^{p_1-1}*... * \\partial \\Delta^{p_m-1}$. A result of Novik gives necessary conditions on the face numbers of Cohen-Macaulay subcomplexes of $\\Lambda$. We show that these conditions are also sufficient, and thus provide a complete characterization of the face numbers of these complexes."}
{"category": "Math", "title": "On a multi-point interpolation problem for generalized Schur functions", "abstract": "The nondegenerate Nevanlinna-Pick-Carath\\'eodory-Fejer interpolation problem with finitely many interpolation conditions always has infinitely many solutions in a generalized Schur class $\\cS_\\kappa$ for every $\\kappa\\ge \\kappa_{\\rm min}$ where the integer $\\kappa_{\\rm min}$ equals the number of negative eigenvalues of the Pick matrix associated to the problem and completely determined by interpolation data. A linear fractional description of all $\\cS_{\\kappa_{\\rm min}}$ solutions of the (nondegenerate) problem is well known. In this paper, we present a similar result for an arbitrary $\\kappa\\ge \\kappa_{\\rm min}$."}
{"category": "Math", "title": "On degenerate Hamburger moment problem and extensions of positive semidefinite Hankel block matrices", "abstract": "In this paper we consider two related objects: singular positive semidefinite Hankel block--matrices and associated degenerate truncated matrix Hamburger moment problems. The description of all solutions of a degenerate matrix Hamburger moment problem is given in terms of a linear fractional transformation. The case of interest is the Hamburger moment problem whose Hankel block--matrix admits a positive semidefinite Hankel extension."}
{"category": "Math", "title": "Boundary rigidity for some classes of meromorphic functions", "abstract": "Sufficient boundary asymptotic conditions are established for a generalized Schur function $f$ to be identically equal to a given rational function $g$ unimodular on the unit circle. Similar rigidity statements are presented for generalized Carath\\'eodory and generalized Nevanlinna functions."}
{"category": "Math", "title": "Motivic characteristic classes", "abstract": "Motivic characteristic classes of possibly singular algebraic varieties are homology class versions of motivic characteristics, not classes in the so-called motivic (co)homology. This paper is a survey on them with more emphasis on capturing infinitude finitely and on the motivic nature, in other words, the scissor relation or additivity."}
{"category": "Math", "title": "Quasi-diagonal flows", "abstract": "We introduce two notions for flows on quasi-diagonal C*-algebras, quasi-diagonal and pseudo-diagonal flows; the former being apparently stronger than the latter. We derive basic facts about these flows and give various examples. In addition we extend results of Voiculescu from quasi-diagonal C*-algebras to these flows."}
{"category": "Math", "title": "Pseudo-Anosov braids with small entropy and the magic 3-manifold", "abstract": "We consider a hyperbolic surface bundle over the circle with the smallest known volume among hyperbolic manifolds having 3 cusps, so called \"the magic manifold\". We compute the entropy function on the fiber face of the unit ball with respect to the Thurston norm, determine homology classes whose representatives are genus 0 fiber surfaces, and describe their monodromies by braids. Among such homology classes whose representatives have n punctures, we decide which one realizes the minimal entropy. It turns out that all the braids with smallest known entropy are derived from monodromies for such homology classes."}
{"category": "Math", "title": "Ternary Goldbach's Problem Involving Primes of a Special Type", "abstract": "Let $\\eta$ be a quadratic irrationality. The variant of a ternary problem of Goldbach involving primes such that $a<\\{\\eta p\\}<b$, where $a$ and $b$ are arbitrary numbers of the interval $(0,1)$, solved in this paper."}
{"category": "Math", "title": "Goldbach Conjecture and the least prime number in an arithmetic progression", "abstract": "In this Note, we try to study the relations between the Goldbach Conjecture and the least prime number in an arithmetic progression. We give a new weakened form of the Goldbach Conjecture. We prove that this weakened form and a weakened form of the Chowla Hypothesis imply that every sufficiently large even integer may be written as the sum of two distinct primes. R\\'{e}sum\\'{e} La conjecture de Goldbach et le plus petit nombre premier dans une progression arithm\\'{e}tique Dans ce document, nous essayons d'\\'{e}tudier les relations entre la conjecture de Goldbach et le plus petit nombre premier dans une progression arithm\\'{e}tique. Nous donnons une nouvelle forme faible de la conjecture de Goldbach. Nous prouvons que cette forme affaiblie et une forme affaiblie de l'hypoth\\`{e}se de Chowla impliquent que tout entier pair suffisamment grand peut \\^{e}tre \\'{e}crit comme une somme de deux nombres premiers distincts."}
{"category": "Math", "title": "A cohomological lower bound for the transverse LS category of a foliated manifold", "abstract": "Let $\\mathcal{F}$ be a compact Hausdorff foliation on a compact manifold. Let ${E_2^{>0,\\bullet}}=\\oplus\\{E_2^{p,q}\\colon p>0,q\\geq 0\\}$ be the subalgebra of cohomology classes with positive transverse degree in the $E_2$ term of the spectral sequence of the foliation. We prove that the saturated transverse Lusternik-Schnirelmann category of $\\mathcal{F}$ is bounded below by the length of the cup product in ${E_2^{>0,\\bullet}}$. Other cohomological bounds are discussed."}
{"category": "Math", "title": "Finite-dimensional pointed Hopf algebras with alternating groups are trivial", "abstract": "It is shown that Nichols algebras over alternating groups A_m, m>4, are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to A_m is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups S_m are all infinite-dimensional, except maybe those related to the transpositions considered in [FK], and the class of type (2,3) in S_5. We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra of (X,q) is infinite dimensional, for q an arbitrary cocycle. arXiv:0904.3978 is included here."}
{"category": "Math", "title": "Fomenko-Mischenko Theory, Hessenberg Varieties, and Polarizations", "abstract": "The symmetric algebra g (denoted S(\\g)) over a Lie algebra \\g (frak g) has the structure of a Poisson algebra. Assume \\g is complex semi-simple. Then results of Fomenko- Mischenko (translation of invariants) and A.Tarasev construct a polynomial subalgebra \\cal H = \\bf C[q_1,...,q_b] of S(\\g) which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of \\g. Let G be the adjoint group of \\g and let \\ell = rank \\g. Identify \\g with its dual so that any G-orbit O in \\g has the structure (KKS) of a symplectic manifold and S(\\g) can be identified with the affine algebra of \\g. An element x \\in \\g is strongly regular if \\{(dq_i)_x\\}, i=1,...,b, are linearly independent. Then the set \\g^{sreg} of all strongly regular elements is Zariski open and dense in \\g, and also \\g^{sreg \\subset \\g^{reg} where \\g^{reg} is the set of all regular elements in \\g. A Hessenberg variety is the b-dimensional affine plane in \\g, obtained by translating a Borel subalgebra by a suitable principal nilpotent element. This variety was introduced in [K2]. Defining Hess to be a particular Hessenberg variety, Tarasev has shown that Hess \\subset \\g^sreg. Let R be the set of all regular G-orbits in \\g. Thus if O \\in R, then O is a symplectic manifold of dim 2n where n= b-\\ell. For any O\\in R let O^{sreg} = \\g^{sreg}\\cap O. We show that O^{sreg} is Zariski open and dense in O so that O^{sreg} is again a symplectic manifold of dim 2n. For any O \\in R let Hess (O) = Hess \\cap O. We prove that Hess(O) is a Lagrangian submanifold of O^{sreg} and Hess =\\sqcup_{O \\in R} Hess(O). The main result here shows that there exists, simultaneously over all O \\in R, an explicit polarization (i.e., a \"fibration\" by Lagrangian submanifolds) of O^{sreg} which makes O^{sreg} simulate, in some sense, the cotangent bundle of Hess(O)."}
{"category": "Math", "title": "Character Sheaves of Algebraic Groups Defined over Non-Archimedean Local Fields", "abstract": "This paper concerns character sheaves of connected reductive algebraic groups defined over non-Archimedean local fields and their relation with characters of smooth representations. Although character sheaves were devised with characters of representations of finite groups of Lie type in mind, character sheaves are perfectly well defined for reductive algebraic groups over any algebraically closed field. Nevertheless, the relation between character sheaves of an algebraic group $G$ over an algebraic closure of a field $K$ and characters of representations of $G(K)$ is well understood only when $K$ is a finite field and when $K$ is the field of complex numbers. In this paper we consider the case when $K$ is a non-Archimedean local field and explain how to match certain character sheaves of a connected reductive algebraic group $G$ with virtual representations of $G(K)$. In the final section of the paper we produce examples of character sheaves of general linear groups and matching admissible virtual representations."}
{"category": "Math", "title": "Stable Systolic Category of Manifolds and the Cup-length", "abstract": "It follows from a theorem of Gromov that the stable systolic category of a closed manifold is bounded from below by the rational cup-length of the manifold. In the paper we study the inequality in the opposite direction. In particular, combining our results with Gromov's theorem, we prove the equality of stable systolic category and rational cup-length for simply connected manifolds of dimension less than 8."}
{"category": "Math", "title": "Toy models for D. H. Lehmer's conjecture", "abstract": "In 1947, Lehmer conjectured that the Ramanujan $\\tau$-function $\\tau (m)$ never vanishes for all positive integers $m$, where the $\\tau (m)$ are the Fourier coefficients of the cusp form $\\Delta_{24}$ of weight 12. Lehmer verified the conjecture in 1947 for $m<214928639999$. In 1973, Serre verified up to $m<10^{15}$, and in 1999, Jordan and Kelly for $m<22689242781695999$. The theory of spherical $t$-design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's $\\tau$-function gives the coefficients of a weighted theta series of the $E_{8}$-lattice. It is shown, by Venkov, de la Harpe, and Pache, that $\\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is an 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. In this paper, we consider toy models of Lehmer's conjecture. Namely, we show that the $m$-th Fourier coefficient of the weighted theta series of the $\\mathbb{Z}^2$-lattice and the $A_{2}$-lattice does not vanish, when the shell of norm $m$ of those lattices is not the empty set. In other words, the spherical 5 (resp. 7)-design does not exist among the shells in the $\\mathbb{Z}^2$-lattice (resp. $A_{2}$-lattice)."}
{"category": "Math", "title": "(Disk, Essential surface) pairs of Heegaard splittings that intersect in one point", "abstract": "We consider a Heegaard splitting M=H_1 \\cup_S H_2 of a 3-manifold M having an essential disk D in H_1 and an essential surface F in H_2 with |D \\cap F|=1. (We require that boundary of F is in S when H_2 is a compressionbody with non-empty \"minus\" boundary.) Let F be a genus g surface with n boundary components. From S, we obtain a genus g(S)+2g+n-2 Heegaard splitting M=H'_1 \\cup_S' H'_2 by cutting H_2 along F and attaching F \\times [0,1] to H_1. As an application, by using a theorem due to Casson and Gordon, we give examples of 3-manifolds having two Heegaard splittings of distinct genera where one of the two Heegaard splittings is a strongly irreducible non-minimal genus splitting and it is obtained from the other by the above construction."}
{"category": "Math", "title": "Exponential Sums and Distinct Points on Arcs", "abstract": "Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to this problem was given by Gregory Freiman in 1961, when he proved a lemma which relates the value of an exponential sum with the distribution of summands in semi-circles of the unit circle in the complex plane. Since then, Freiman's Lemma has been extended by several authors. Rather than residue classes, one has considered the situation for finitely many arbitrary points on the unit circle. So far, Lev is the only author who has taken into consideration that the summands may be bounded away from each other, as is the case with residue classes. In this paper we extend Lev's result by lifting a recent result of ours to the case of the points being bounded away from each other."}
{"category": "Math", "title": "Yangians and cohomology rings of Laumon spaces", "abstract": "Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of $GL_n$. We construct the action of the Yangian of $sl_n$ in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of $sl_n[s^{\\pm1},t]$) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analogue of the Gelfand-Tsetlin basis. The affine analogue of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space $M_{n,d}$ of torsion free sheaves on the plane, of rank $n$ and second Chern class $d$, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center $Z$ of the Yangian of $gl_n$ naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on $M_{n,d}$ is the image of a noncommutative power sum in $Z$."}
{"category": "Math", "title": "Characteristic Classes of Lie Algebroid Morphisms", "abstract": "We extend R. Fernandes' construction of secondary characteristic classes of a Lie algebroid to the case of a base-preserving morphism between two Lie algebroids. Like in the case of a Lie algebroid, the simplest characteristic class of our construction coincides with the modular class of the morphism."}
{"category": "Math", "title": "The uncertainty principle lemma under gravity and the discrete spectrum of Schr\\\"odinger operators", "abstract": "The uncertainty principle lemma for the Laplacian on Euclidean spaces shows the borderline-behavior of a potential for the following question : whether the Schr\\\"odinger operator has a finite or infinite number of the discrete pectrum. In this paper, we will give a generalization of this lemma on Euclidean spaces to that on large classes of complete noncompact manifolds. Replacing Euclidean spaces by some specific classes of complete noncompact manifolds, including hyperbolic spaces, we also establish some criterions for the above-type question."}
{"category": "Math", "title": "Hua Loo Keng's Problem Involving Primes of a Special Type", "abstract": "Let $\\eta$ be a quadratic irrationality. The variant of Hua Loo Keng's problem involving primes such that $a<\\{\\eta p^2\\}<b$, where $a$ and $b$ are arbitrary real numbers of the interval $(0,1)$, solved in this paper."}
{"category": "Math", "title": "Sonine Transform Associated to the Dunkl Kernel on the Real Line", "abstract": "We consider the Dunkl intertwining operator $V_\\alpha$ and its dual ${}^tV_\\alpha$, we define and study the Dunkl Sonine operator and its dual on $\\mathbb{R}$. Next, we introduce complex powers of the Dunkl Laplacian $\\Delta_\\alpha$ and establish inversion formulas for the Dunkl Sonine operator $S_{\\alpha,\\beta}$ and its dual ${}^tS_{\\alpha,\\beta}$. Also, we give a Plancherel formula for the operator ${}^tS_{\\alpha,\\beta}$."}
{"category": "Math", "title": "Growth in the minimal injective resolution of a local ring", "abstract": "Let R be a commutative noetherian local ring with residue field k and assume that it is not Gorenstein. In the minimal injective resolution of R, the injective envelope E of the residue field appears as a summand in every degree starting from the depth of R. The number of copies of E in degree i equals the k-vector space dimension of the cohomology module Ext^i(k,R). These dimensions, known as Bass numbers, form an infinite sequence of invariants of R about which little is known. We prove that it is non-decreasing and grows exponentially if R is Golod, a non-trivial fiber product, or Teter, or if it has radical cube zero."}
{"category": "Math", "title": "Existence of sweeping process in Banach spaces under directional prox-regularity", "abstract": "This paper is devoted to weaken \"classical\" assumptions and give new arguments to prove existence of sweeping process (associated to the proximal normal cone of sets). Mainly we define the concept of a \"directional prox-regularity\" and give assumptions about a Banach space to insure the existence of such sweeping process (which permit to generalize the existing results requiring a Hilbertian structure)."}
{"category": "Math", "title": "An elementary proof of the cross theorem in the Reinhardt case", "abstract": "We present an elementary proof of the cross theorem in the case of Reinhardt domains. The results illustrates the well-known interrelations between the holomorphic geometry of a Reinhardt domain and the convex geometry of its logarithmic image."}
{"category": "Math", "title": "Three Crossed Modules", "abstract": "We introduce the notion of 3-crossed module, which extends the notions of 1-crossed module (Whitehead) and 2-crossed module (Conduch\\'e). We show that the category of 3-crossed modules is equivalent to the category of simplicial groups having a Moore complex of length 3. We make explicit the relationship with the cat$^{3}$-groups (Loday) and the 3-hyper-complexes (Cegarra-Carrasco), which also model algebraically homotopy 4-types."}
{"category": "Math", "title": "Delayed Feedback Control near Hopf Bifurcation", "abstract": "The stability of functional differential equations under delayed feedback is investigated near a Hopf bifurcation. Necessary and sufficient conditions are derived for the stability of the equilibrium solution using averaging theory. The results are used to compare delayed versus undelayed feedback, as well as discrete versus distributed delays. Conditions are obtained for which delayed feedback with partial state information can yield stability where undelayed feedback is ineffective. Furthermore, it is shown that if the feedback is stabilizing (respectively, destabilizing), then a discrete delay is locally the most stabilizing (resp., destabilizing) one among delay distributions having the same mean. The result also holds globally if one considers delays that are symmetrically distributed about their mean."}
{"category": "Math", "title": "Algebraic equations and convex bodies", "abstract": "The well-known Bernstein-Kushnirenko theorem from the theory of Newton polyhedra relates algebraic geometry and the theory of mixed volumes. Recently the authors have found a far-reaching generalization of this theorem to generic systems of algebraic equations on any quasi-projective variety. In the present note we review these results and their applications to algebraic geometry and convex geometry."}
{"category": "Math", "title": "A note on fragmentability and weak-G_delta sets", "abstract": "In terms of fragmentability, we describe a new class of Banach spaces which do not contain weak-G_delta open bounded subsets. In particular, none of these spaces is isomorphic to a separable polyhedral space."}
{"category": "Math", "title": "Phase transition approach to detecting singularities of PDEs", "abstract": "We present a mesh refinement algorithm for detecting singularities of time-dependent partial differential equations. The main idea behind the algorithm is to treat the occurrence of singularities of time-dependent partial differential equations as phase transitions. We show how the mesh refinement algorithm can be used to calculate the blow-up rate as we approach the singularity. This calculation can be done in three different ways: i) the direct approach where one monitors the blowing-up quantity as it approaches the singularity and uses the data to calculate the blow-up rate ; ii) the \"phase transition\" approach (\\`a la Wilson) where one treats the singularity as a fixed point of the renormalization flow equation and proceeds to compute the blow-up rate via an analysis in the vicinity of the fixed point and iii) the \"scaling\" approach (\\`a la Widom-Kadanoff) where one postulates the existence of scaling laws for different quantities close to the singularity, computes the associated exponents and then uses them to estimate the blow-up rate. Our algorithm allows a unified presentation of these three approaches. The inviscid Burgers equation and the supercritical Schrodinger equation are used as instructive examples to illustrate the constructions."}
{"category": "Math", "title": "Module Hom-algebras", "abstract": "We study a twisted version of module algebras called module Hom-algebras. It is shown that module algebras deform into module Hom-algebras via endomorphisms. As an example, we construct certain q-deformations of the usual sl(2)-action on the affine plane."}
{"category": "Math", "title": "Stable concordance of knots in 3-manifolds", "abstract": "Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\\times S^2$. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orientable surface with the circle, and stable link concordance for many links of null-homotopic knots in orientable 3-manifolds."}
{"category": "Math", "title": "On q-Runge domains", "abstract": "In $[2]$, Coltoiu gave an example of a domain $D\\subset\\complexes^{6}$ which is 4-complete such that for every ${\\mathcal{F}}\\in Coh(\\complexes^{6})$ the restriction map $H^{3}(\\complexes^{6},{\\mathcal{F}})\\to H^{3}(D,{\\mathcal{F}})$ has a dense image but $D$ is not 4-Runge in $\\complexes^{6}$. Here, we prove that for every integers $n\\geq 4$ and $1\\leq q\\leq n$ there exists a domain $D\\subset \\complexes^{n}$ which is not ($\\tilde{q}-1$)-Runge in $\\complexes^{n}$ but such that for any coherent analytic sheaf ${\\mathcal{F}}$ on $\\complexes^{n}$ the restriction map $H^{p}(\\complexes^{n},{\\mathcal{F}})\\to H^{3}(D,{\\mathcal{F}})$ has a dense image for all $p\\geq \\tilde{q}-2$ if $q$ does not divide $n$, where $\\tilde{q}=n-[\\frac{n}{q}]+1$ and $[\\frac{n}{q}]$ denotes the integral part of $\\frac{n}{q}$."}
{"category": "Math", "title": "Single-Index Model-Assisted Estimation In Survey Sampling", "abstract": "A model-assisted semiparametric method of estimating finite population totals is investigated to improve the precision of survey estimators by incorporating multivariate auxiliary information. The proposed superpopulation model is a single-index model which has proven to be a simple and efficient semiparametric tool in multivariate regression. A class of estimators based on polynomial spline regression is proposed. These estimators are robust against deviation from single-index models. Under standard design conditions, the proposed estimators are asymptotically design-unbiased, consistent and asymptotically normal. An iterative optimization routine is provided that is sufficiently fast for users to analyze large and complex survey data within seconds. The proposed method has been applied to simulated datasets and MU281 dataset, which have provided strong evidence that corroborates with the asymptotic theory."}
{"category": "Math", "title": "Parameter identifiability and redundancy: theoretical considerations", "abstract": "In this paper we outline general considerations on parameter identifiability, and introduce the notion of weak local identifiability and gradient weak local identifiability. These are based on local properties of the likelihood, in particular the rank of the Hessian matrix. We relate these to the notions of parameter identifiability and redundancy previously introduced by Rothenberg (Econometrica 39 (1971) 577-591) and Catchpole and Morgan (Biometrika 84 (1997) 187-196). Within the exponential family parameter irredundancy, local identifiability, gradient weak local identifiability and weak local identifiability are shown to be equivalent. We consider applications to a recently developed class of cancer models of Little and Wright (Math Biosciences 183 (2003) 111-134) and Little et al. (J Theoret Biol 254 (2008) 229-238) that generalize a large number of other recently used quasi-biological cancer models, in particular those of Armitage and Doll (Br J Cancer 8 (1954) 1-12) and the two-mutation model (Moolgavkar and Venzon Math Biosciences 47 (1979) 55-77)."}
{"category": "Math", "title": "Remplissage De L'Espace Euclidien Par Des Complexes Poly\\'Edriques D'Orientation Impos\\'Ee Et De Rotondit\\'E Uniforme", "abstract": "We build polyhedral complexes in Rn that coincide with dyadic grids with different orientations, while keeping uniform lower bounds (depending only on n) on the flatness of the added polyhedrons including their subfaces in all dimensions. After the definitions and first properties of compact Euclidean polyhedrons and complexes, we introduce a tool allowing us to fill with n-dimensionnal polyhedrons a tubular-shaped open set, the boundary of which is a given n - 1-dimensionnal complex. The main result is proven inductively over n by completing our dyadic grids layer after layer, filling the tube surrounding each layer and using the result in the previous dimension to build the missing parts of the tube boundary. A possible application of this result is a way to find solutions to problems of measure minimization over certain topological classes of sets, in arbitrary dimension and codimension."}
{"category": "Math", "title": "Quasiminimality in mixed Tsirelson spaces", "abstract": "We prove quasiminimality of the regular mixed Tsirelson spaces T[(S_n,\\theta_n)_n] with the sequence (\\frac{\\theta_n}{\\theta^n})_n decreasing, where \\theta=\\lim_n \\theta_n^{1/n}, and quasiminimality of all mixed Tsirelson spaces T[(A_n,\\theta_n)_n]. We prove that under certain assumptions on the sequence (\\theta_n)_n the dual spaces are quasiminimal."}
{"category": "Math", "title": "Homotopy groups of ascending unions of infinite-dimensional manifolds", "abstract": "Let M be a topological manifold modelled on topological vector spaces, which is the union of an ascending sequence of such manifolds M_n. We formulate a mild condition ensuring that the k-th homotopy group of M is the direct limit of the k-th homotopy groups of the steps M_n, for each non-negative integer k. This result is useful for Lie theory, because many important examples of infinite-dimensional Lie groups G can be expressed as ascending unions of finite- or infinite-dimensional Lie groups (whose homotopy groups may be easier to access). Information on the k-th homotopy groups of G, for k=0, k=1 and k=2, is needed to understand the Lie group extensions of G with abelian kernels. The above conclusion remains valid if the union of the steps M_n is merely dense in M (under suitable hypotheses). Also, ascending unions can be replaced by (possibly uncountable) directed unions."}
{"category": "Math", "title": "Crucial words for abelian powers", "abstract": "A word is \"crucial\" with respect to a given set of \"prohibited words\" (or simply \"prohibitions\") if it avoids the prohibitions but it cannot be extended to the right by any letter of its alphabet without creating a prohibition. A \"minimal crucial word\" is a crucial word of the shortest length. A word W contains an \"abelian k-th power\" if W has a factor of the form X_1X_2...X_k where X_i is a permutation of X_1 for 2<= i <= k. When k=2 or 3, one deals with \"abelian squares\" and \"abelian cubes\", respectively. In 2004 (arXiv:math/0205217), Evdokimov and Kitaev showed that a minimal crucial word over an n-letter alphabet A_n = {1,2,..., n} avoiding abelian squares has length 4n-7 for n >= 3. In this paper we show that a minimal crucial word over A_n avoiding abelian cubes has length 9n-13 for n >= 5, and it has length 2, 5, 11, and 20 for n=1, 2, 3, and 4, respectively. Moreover, for n >= 4 and k >= 2, we give a construction of length k^2(n-1)-k-1 of a crucial word over A_n avoiding abelian k-th powers. This construction gives the minimal length for k=2 and k=3. For k >= 4 and n >= 5, we provide a lower bound for the length of crucial words over A_n avoiding abelian k-th powers."}
{"category": "Math", "title": "Polynomial processes and their applications to mathematical Finance", "abstract": "We introduce a class of Markov processes, called $m$-polynomial, for which the calculation of (mixed) moments up to order $m$ only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as L\\'evy-driven SDEs with affine vector fields. Thus, many popular models such as exponential L\\'evy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods."}
{"category": "Math", "title": "On the curvature properties of real time-like hypersurfaces of Kaehler manifolds with Norden metric", "abstract": "A type of almost contact hypersurfaces with Norden metric of a Kaehler manifold with Norden metric is considered. The curvature tensor and the special sectional curvatures are characterized. The canonical connection on such manifolds is studied and the form of the corresponding Kaehler curvature tensor is obtained. Some curvature properties of the manifolds belonging to the widest integrable main class of the considered type of hypersurfaces are given."}
{"category": "Math", "title": "Examples of asymptotically conical Ricci-flat K\\\"{a}hler manifolds", "abstract": "The author has proved that a crepant resolution Y of a Ricci-flat K\\\"{a}hler cone X admits a complete Ricci-flat K\\\"{a}hler metric asymptotic to the cone metric in every K\\\"{a}hler class in H^2_c(Y,\\R). These manifolds are generalizations of the Ricci-flat ALE K\\\"{a}hler spaces known by the work of P. Kronheimer, D. Joyce and others. This article considers further the problem of constructing examples. We show that every 3-dimensional Gorenstein toric K\\\"{a}hler cone admits a crepant resolution for which the above theorem applies. This gives infinitely many examples of asymptotically conical Ricci-flat manifolds. Then other examples are given of which are crepant resolutions hypersurface singularities which are known to admit Ricci-flat K\\\"{a}hler cone metrics by the work of C. Boyer, K. Galicki, J. Koll\\'{a}r, and others. Two families of hypersurface examples are given which are distinguished by the condition b_3(Y)=0 or b_3(Y)>0."}
{"category": "Math", "title": "Large deviation principles for non-uniformly hyperbolic rational maps", "abstract": "We show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called \"Topological Collet-Eckmann\". More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that each H{\\\"o}lder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer."}
{"category": "Math", "title": "Introduction into Calculus over Division Ring", "abstract": "Based on twin representations of division ring in an Abelian group I consider $D$\\Hyph vector spaces over division ring. Morphism of $D$\\Hyph vector spaces is linear map of $D$\\Hyph vector spaces. I consider derivative of function $f$ of continuous division ring as linear map the most close to function $f$. I explore expansion of map into Taylor series and method to find solution of differential equation. The norm in $D$\\Hyph vector space allows considering of continuous mapping of $D$\\Hyph vector spaces. Differential of mapping $f$ of $D$\\Hyph vector spaces is defined as linear mapping the most close to map $f$."}
{"category": "Math", "title": "On Osculating Interpolation", "abstract": "The development of high-degree interpolation polynomials which use the values of the function and its subsequent derivatives is reformulated. Also, we present a variant of new formula in barycentric form."}
{"category": "Math", "title": "On the time continuity of entropy solutions", "abstract": "We show that any entropy solution $u$ of a convection diffusion equation $\\partial_t u + \\div F(u)-\\Delta\\phi(u) =b$ in $\\OT$ belongs to $C([0,T),L^1_{Loc}(\\o\\O))$. The proof does not use the uniqueness of the solution."}
{"category": "Math", "title": "One more pathology of C*-algebraic tensor products", "abstract": "We define a collection of tensor product norms for C*-algebras and show that a symmetric tensor product functor on the category of separable C*-algebras need not be associative."}
{"category": "Math", "title": "Matrix differential equations and scalar polynomials satisfying higher order recursions", "abstract": "We show that any scalar differential operator with a family of polyno- mials as its common eigenfunctions leads canonically to a matrix differen- tial operator with the same property. The construction of the correspond- ing family of matrix valued polynomials has been studied in [D1, D2, DV] but the existence of a differential operator having them as common eigen- functions had not been considered This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [GH] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case."}
{"category": "Math", "title": "Density of rational points on diagonal quartic surfaces", "abstract": "Let a,b,c,d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in PP^3 defined by ax^4+by^4+cz^4+dw^4=0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic topology."}
{"category": "Math", "title": "Framed bordism and Lagrangian embeddings of exotic spheres", "abstract": "In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of the standard sphere. The main ingredients of the construction are (1) the fact that the graph of the Hopf fibration embeds the standard sphere, and hence any Lagrangian which embeds in its cotangent bundle, as a displaceable Lagrangian in the product a symplectic vector space of the appropriate dimension with its complex projective space, and (2) a moduli space of solutions to a perturbed Cauchy-Riemann equation introduced by Gromov."}
{"category": "Math", "title": "Derived tame local and two-point algebras", "abstract": "We determine derived representation type of complete finitely generated local and two-point algebras over an algebraically closed field."}
{"category": "Math", "title": "Card deals, lattice paths, abelian words and combinatorial identities", "abstract": "We give combinatorial interpretations of several related identities associated with the names Barrucand, Strehl and Franel, including one for the Apery numbers. The combinatorial constructs employed are derangement-type card deals as introduced in a previous paper on Barrucand's identity, labeled lattice paths and, following a comment of Jeffrey Shallit, abelian words over a 3-letter alphabet."}
{"category": "Math", "title": "Icosahedral Fibres of the Symmetric Cube and Algebraicity", "abstract": "For any number field F, call a cusp form \\pi on GL(2)/F {\\it special icosahedral}, or just s-icosahedral for short, if \\pi is not solvable polyhedral, and for a suitable \"conjugate\" cusp form \\pi' on GL(2)/F, sym^3(\\pi) is isomorphic to sym^3(\\pi'), and the symmetric fifth power L-series of \\pi equals the Rankin-Selberg L-function L(s, sym^2(\\pi') x \\pi) (up to a finite number of Euler factors). Then the point of this Note is to obtain the following result: Let \\pi be s-icosahedral (of trivial central character). Then \\pi_f is algebraic without local components of Steinberg type, \\pi_\\infty is of Galois type, and \\pi_v is tempered everywhere. Moreover, if \\pi' is also of trivial central character, it is s-icosahedral as well, and the field of rationality \\Q(\\pi_f) (of \\pi_f) is K:=\\Q[\\sqrt{5}], with \\pi'_f being the Galois conjugate of \\pi_f under the non-trivial automorphism of K."}
{"category": "Math", "title": "Asymptotic Analysis of Boundary Layer Correctors and Applications", "abstract": "In this paper we extend the ideas presented in Onofrei and Vernescu [\\textit{Asymptotic Analysis, 54, 2007, 103-123}] and introduce suitable second order boundary layer correctors, to study the $H^1$-norm error estimate for the classical problem in homogenization. Previous second order boundary layer results assume either smooth enough coefficients (which is equivalent to assuming smooth enough correctors $\\chi_j,\\chi_{ij}\\in W^{1,\\infty}$), or smooth homogenized solution $u_0$, to obtain an estimate of order $\\displaystyle O(\\epsilon^{\\frac{3}{2}})$. For this we use the periodic unfolding method developed by Cioranescu, Damlamian and Griso [\\textit{C. R. Acad. Sci. Paris, Ser. I 335, 2002, 99-104}]. We prove that in two dimensions, for nonsmooth coefficients and general data, one obtains an estimate of order $\\displaystyle O(\\epsilon^\\frac{3}{2})$. In three dimenssions the same estimate is obtained assuming $\\chi_j,\\chi_{ij}\\in W^{1,p}$, with $p>3$. We also discuss how our results extend, in the case of nonsmooth coefficients, the convergence proof for the finite element multiscale method proposed by T.Hou et al. [\\textit{ J. of Comp. Phys., 134, 1997, 169-189}] and the first order correctoranalysis for the first eigenvalue of a composite media obtained by Vogelius et al.[\\textit{Proc. Royal Soc. Edinburgh, 127A, 1997, 1263-1299}]."}
{"category": "Math", "title": "On the field intersection problem of quartic generic polynomials via formal Tschirnhausen transformation", "abstract": "Let $k$ be a field of characteristic $\\neq 2$. We give an answer to the field intersection problem of quartic generic polynomials over $k$ via formal Tschirnhausen transformation and multi-resolvent polynomials."}
{"category": "Math", "title": "H. Bohr's theorem for bounded symmetric domains", "abstract": "A theorem of Harald Bohr (1914) states that if f is a holomorphic map from the unit disc into itself, then the sum of absolute values of its Taylor expansion is less than 1 for |z|<1/3. The bound 1/3 is optimal. This result has been extended in a suitable sense by Liu Taishun and Wang Jianfei (2007) to the bounded complex symmetric domains of the four classical series, and to polydiscs. The result of Liu and Wang may be generalized to all bounded symmetric domains, with a proof which does not depend on classification."}
{"category": "Math", "title": "Derived categories for algebras with radical square zero", "abstract": "We determine the derived representation types of algebras with radical square zero and give a description of the indecomposable objects in their bounded derived categories."}
{"category": "Math", "title": "Geometry of Non-Archimedean Gromov-Hausdorff distance", "abstract": "In this paper, we study the geometry of non-Archimedean Gromov-Hausdorff metric. This is the first part of our series work, which we try to establish some facts about the counterpart of Gromov-Hausdorff metric in the non-Archimedean spaces. One of the motivation of this work is to find some implied relations between this geometry and number theory via p-adic analysis, so that we can use the former as a tool to study the relating arithmetic aspects."}
{"category": "Math", "title": "Marden theorem and Poncelet-Darboux curves", "abstract": "The Marden theorem of geometry of polynomials and the great Poncelet theorem from projective geometry of conics by their classical beauty occupy very special places. Our main aim is to present a strong and unexpected relationship between the two theorems. We establish a dynamical equivalence between the full Marden theorem and the Poncelet-Darboux theorem. By introducing a class of {\\it isofocal deformations}, we construct morphisms between the Marden curves and the Poncelet-Darboux curves. Then we present effective criterion in terms of pair of polynomials which defines a Poncelet-Darboux curve of degree $n-1$, for complete decomposition of the curve on $(n-1)/2$ conics if $n$ is odd; if $n$ is even, complete decomposition consists of $(n-2)/2$ conics and a line. This is an important question in the study of special, 'tHooft, instanton bundles."}
{"category": "Math", "title": "Koszul duality for monoids and the operad of enriched rooted trees", "abstract": "We introduce here the notion of Koszul duality for monoids in the monoidal category of species with respect to the ordinary product. To each Koszul monoid we associate a class of Koszul algebras in the sense of Priddy, by taking the corresponding analytic functor. The operad $\\mathscr{A}_M$ of rooted trees enriched with a monoid $M$ was introduced by the author many years ago. One special case of that is the operad of ordinary rooted trees, called in the recent literature the permutative non associative operad. We prove here that $\\mathscr{A}_M$ is Koszul if and only if the corresponding monoid $M$ is Koszul. In this way we obtain a wide family of Koszul operads, extending a recent result of Chapoton and Livernet, and providing an interesting link between Koszul duality for associative algebras and Koszul duality for operads."}
{"category": "Math", "title": "Compatible structures on Lie algebroids and Monge-Amp\\`ere operators", "abstract": "We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between such composite structures. We then show that the non-degenerate Monge-Amp\\`ere structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Amp\\`ere operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold."}
{"category": "Math", "title": "Iterative Solution of the Quasicontinuum Equilibrium Equations with Continuation", "abstract": "We give an analysis of a continuation algorithm for the numerical solution of the force-based quasicontinuum equations. The approximate solution of the force-based quasicontinuum equations is computed by an iterative method using an energy-based quasicontinuum approximation as the preconditioner. The analysis presented in this paper is used to determine an efficient strategy for the parameter step size and number of iterations at each parameter value to achieve a solution to a required tolerance. We present computational results for the deformation of a Lennard-Jones chain under tension to demonstrate the necessity of carefully applying continuation to ensure that the computed solution remains in the domain of convergence of the iterative method as the parameter is increased. These results exhibit fracture before the actual load limit if the parameter step size is too large."}
{"category": "Math", "title": "Polynomial birth-death distribution approximation in Wasserstein distance", "abstract": "The polynomial birth-death distribution (abbr. as PBD) on $\\ci=\\{0,1,2, >...\\}$ or $\\ci=\\{0,1,2, ..., m\\}$ for some finite $m$ introduced in Brown & Xia (2001) is the equilibrium distribution of the birth-death process with birth rates $\\{\\alpha_i\\}$ and death rates $\\{\\beta_i\\}$, where $\\a_i\\ge0$ and $\\b_i\\ge0$ are polynomial functions of $i\\in\\ci$. The family includes Poisson, negative binomial, binomial and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein's factors for the PBD approximation with $\\a_i=a$ and $\\b_i=i+bi(i-1)$ in terms of the Wasserstein distance. The paper complements the work of Brown & Xia (2001) and generalizes the work of Barbour & Xia (2006) where Poisson approximation ($b=0$) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions."}
{"category": "Math", "title": "Vandermonde's quintic and multiple decompositions of the number 1318", "abstract": "This note records a curious numerical identity: the number 1318, connected with Vandermonde's cyclotomic quintic, may be decomposed in two distinct ways as a sum of products of pairs of numbers taken from the set \\{$6, 16, 26, 41$\\}, namely $1318 = 6\\cdot41 + 16\\cdot26 + 16\\cdot41 = 6\\cdot16 + 6\\cdot26 + 26\\cdot41$. Based on the existence of radical solutions of certain families of Abelian and generalized Abelian equations, we conjecture the existence of an infinite number of analogous decompositions, involving arbitrarily large sets of numbers."}
{"category": "Math", "title": "On potentially $K_6-C_5$ graphic sequences", "abstract": "For given a graph $H$, a graphic sequence $\\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\\pi$ containing $H$ as a subgraph. In this paper, we characterize the potentially $K_6-C_5$ -graphic sequences."}
{"category": "Math", "title": "Groupoidification Made Easy", "abstract": "Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of 'degroupoidification': a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang-Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field with q elements."}
{"category": "Math", "title": "The optimal assignment problem for a countable state space", "abstract": "Given a square matrix B=(b_{ij}) with real entries, the optimal assignment problem is to find a bijection s between the rows and the columns maximising the sum of the b_{is(i)}. In discrete optimal control and in the theory of discrete event systems, one often encounters the problem of solving the equation Bf=g for a given vector g, where the same symbol B denotes the corresponding max-plus linear operator, (Bf)_i:=max_j (b_{ij}+f_j). The matrix B is said to be strongly regular when there exists a vector g such that the equation Bf=g has a unique solution f. A result of Butkovic and Hevery shows that B is strongly regular if and only if the associated optimal assignment problem has a unique solution. We establish here an extension of this result which applies to max-plus linear operators over a countable state space. The proofs use the theory developed in a previous work in which we characterised the unique solvability of equations involving Moreau conjugacies over an infinite state space, in terms of the minimality of certain coverings of the state space by generalised subdifferentials."}
{"category": "Math", "title": "On the nonexistence of stationary weak solutions to the compressible fluid equations", "abstract": "In this paper we prove that under some integrability conditions for the density and the velocity fields the only stationary weak solutions to the compressible fluid equations on $\\Bbb R^N$ correspond to the zero density. In the case of compressible magnetohydrodynamics equations similar integrability conditions for density, velocity and the magnetic fields lead to the zero density and the zero magnetic field."}
{"category": "Math", "title": "Equivariant classes of matrix matroid varieties", "abstract": "Consider an integer associated with every subset of the set of columns of an $n\\times k$ matrix. The collection of those matrices for which the rank of a union of columns is the predescribed integer for every subset, will be denoted by $X_C$. We study the equivariant cohomology class represented by the Zariski closure $Y_C$ of this set. We show that the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov-Witten invariants of projective spaces. We also show how to calculate these classes and present their basic properties."}
{"category": "Math", "title": "Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model", "abstract": "The Keller-Segel system describes the collective motion of cells that are attracted by a chemical substance and are able to emit it. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled to an elliptic equation for the chemo-attractant concentration. This paper deals with the rate of convergence towards a unique stationary state in self-similar variables, which describes the intermediate asymptotics of the solutions in the original variables. Although it is known that solutions globally exist for any mass less $8\\pi $, a smaller mass condition is needed in our approach for proving an exponential rate of convergence in self-similar variables."}
{"category": "Math", "title": "A Small Sample Correction for Estimating Attributable Risk in Case-Control Studies", "abstract": "The attributable risk, often called the population attributable risk, is in many epidemiological contexts a more relevant measure of exposure-disease association than the excess risk, relative risk, or odds ratio. When estimating attributable risk with case-control data and a rare disease, we present a simple correction to the standard approach making it essentially unbiased, and also less noisy. As with analogous corrections given in Jewell (1986) for other measures of association, the adjustment often won't make a substantial difference unless the sample size is very small or point estimates are desired within fine strata, but we discuss the possible utility for applications."}
{"category": "Math", "title": "Conditional mode regression: Application to functional time series prediction", "abstract": "We consider $\\alpha$-mixing observations and deal with the estimation of the conditional mode of a scalar response variable $Y$ given a random variable $X$ taking values in a semi-metric space. We provide a convergence rate in $L^p$ norm of the estimator. A useful and typical application to functional times series prediction is given."}
{"category": "Math", "title": "On the Bose-Einstein distribution and Bose condensation", "abstract": "For a system of identical Bose particles sitting on integer energy levels, we give sharp estimates for the convergence of the sequence of occupation numbers to the Bose-Einstein distribution and for the Bose condensation effect."}
{"category": "Math", "title": "Linear forms of a given Diophantine type", "abstract": "We prove a result on the existence of linear forms of a given Diophantine type."}
{"category": "Math", "title": "Eventual regularization for the slightly supercritical quasi-geostrophic equation", "abstract": "We prove that weak solutions of a slightly supercritical quasi-geostrophic equation become smooth for large time. We prove it using a De Giorgi type argument using ideas from a recent paper by Caffarelli and Vasseur."}
{"category": "Math", "title": "On the $H$-ring structure of infinite Grassmannians", "abstract": "The $H$-ring structure of certain infinite(-dimensional) Grassmannians is discussed using various algebraic and analytical methods but so that cellular arguments are avoided. These methods allow us to discuss these Grassmannian in greater generality."}
{"category": "Math", "title": "Some additive relations in the Pascal triangle", "abstract": "We derive some, seemingly new, curious additive relations in the Pascal triangle. They arise in summing up the numbers in the triangle along some vertical line up to some place."}
{"category": "Math", "title": "Binomial theorem and exponent for variables commuting as $yx=qxy$", "abstract": "We state analogs of the binomial theorem and the exponential function for variables $x$, $y$ commuting as $yx=qxy$."}
{"category": "Math", "title": "On the symplectic structure of instanton moduli spaces", "abstract": "We study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of SU(2) instantons on both commutative and non-commutative R^4. We identify global Darboux coordinates and quadratic Hamiltonians on classical phase spaces for which these quiver varieties are natural completions. We also show that the group of non-commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non-commutative instantons. This paper should be viewed as a step towards extending known results for Calogero-Moser spaces to the instanton moduli spaces."}
{"category": "Math", "title": "On Sequential Coloring of Graphs and its Defining Sets", "abstract": "In this paper, based on the contributions of Tucker (1983) and Seb{\\H{o}} (1992), we generalize the concept of a sequential coloring of a graph to a framework in which the algorithm may use a coloring rule-base obtained from suitable forcing structures. In this regard, we introduce the {\\it weak} and {\\it strong sequential defining numbers} for such colorings and as the main results, after proving some basic properties, we show that these two parameters are intrinsically different and their spectra are nontrivial. Also, we consider the natural problems related to the complexity of computing such parameters and we show that in a variety of cases these problems are ${\\bf NP}$-complete. We conjecture that this result does not depend on the rule-base for all nontrivial cases."}
{"category": "Math", "title": "Topological conjugacy classes of affine maps", "abstract": "A map $f: \\ff^n \\to \\ff^n$ over a field $\\ff$ is called affine if it is of the form $f(x)=Ax+b$, where the matrix $A \\in \\ff^{n\\times n}$ is called the linear part of affine map and $b \\in \\ff^n$. The affine maps over $\\ff=\\rr$ or $\\cc$ are investigated. We prove that affine maps having fixed points are topologically conjugate if and only if their linear parts are topologically conjugate. If affine maps have no fixed points and $n=1$ or 2, then they are topologically conjugate if and only if their linear parts are either both singular or both non-singular. Thus we obtain classification up to topological conjugacy of affine maps from $\\ff^n$ to $\\ff^n$, where $\\ff=\\rr$ or $\\cc$, $n\\leq 2$."}
{"category": "Math", "title": "Existence of Symplectic Surfaces", "abstract": "In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be embedded if 2n is at least 6. We also analyze the additional conditions under which embedded symplectic representatives exist in dimension 4."}
{"category": "Math", "title": "Sharp bounds for the number of maximal independent sets in trees of fixed diameter", "abstract": "We obtain sharp lower and upper bounds for the number of maximal (under inclusion) independent sets in trees with fixed number of vertices and diameter. All extremal trees are described up to isomorphism."}
{"category": "Math", "title": "Bivariant K-theory via correspondences", "abstract": "We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth K-oriented maps by a class of K-oriented normal maps, which are maps together with a certain factorisation. Our construction does not use any special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories. We formulate necessary and sufficient conditions for certain duality isomorphisms in the geometric bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, both bivariant K-theories agree if there is such a duality isomorphism."}
{"category": "Math", "title": "The Verlinde bundles and the semihomogeneous Wirtinger duality", "abstract": "We determine the splitting type of the Verlinde vector bundles in higher genus in terms of simple semihomogeneous factors. In agreement with strange duality, the simple factors are interchanged by the Fourier-Mukai transform, and their spaces of sections are naturally dual."}
{"category": "Math", "title": "Rational Normal Scrolls and the Defining Equations of Rees Algebras", "abstract": "Consider a height two ideal, $I$, which is minimally generated by $m$ homogeneous forms of degree $d$ in the polynomial ring $R=k[x,y]$. Suppose that one column in the homogeneous presenting matrix $\\f$ of $I$ has entries of degree $n$ and all of the other entries of $\\f$ are linear. We identify an explicit generating set for the ideal $\\Cal A$ which defines the Rees algebra $\\Cal R=R[It]$; so $\\Cal R=S/\\Cal A$ for the polynomial ring $S=R[T_1,...,T_m]$. We resolve $\\Cal R$ as an $S$-module and $I^s$ as an $R$-module, for all powers $s$. The proof uses the homogeneous coordinate ring, $A=S/H$, of a rational normal scroll, with $H\\subseteq \\Cal A$. The ideal $\\Cal AA$ is isomorphic to the $n^{\\text{th}}$ symbolic power of a height one prime ideal $K$ of $A$. The ideal $K^{(n)}$ is generated by monomials. Whenever possible, we study $A/K^{(n)}$ in place of $A/\\Cal AA$ because the generators of $K^{(n)}$ are much less complicated then the generators of $\\Cal AA$. We obtain a filtration of $K^{(n)}$ in which the factors are polynomial rings, hypersurface rings, or modules resolved by generalized Eagon-Northcott complexes. The generators of $I$ parameterize an algebraic curve $\\Cal C$ in projective $m-1$ space. The defining equations of the special fiber ring $\\Cal R/(x,y)\\Cal R$ yield a solution of the implicitization problem for $\\Cal C$."}
{"category": "Math", "title": "The Full Orbifold $K$-theory of Abelian Symplectic Quotients", "abstract": "In their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifold $K$-theory of an orbifold ${\\mathfrak X}$, analogous to the Chen-Ruan orbifold cohomology of ${\\mathfrak X}$ in that it uses the obstruction bundle as a quantum correction to the multiplicative structure. We give an explicit algorithm for the computation of this orbifold invariant in the case when ${\\mathfrak X}$ arises as an abelian symplectic quotient. Our methods are integral $K$-theoretic analogues of those used in the orbifold cohomology case by Goldin, Holm, and Knutson in 2005. We rely on the $K$-theoretic Kirwan surjectivity methods developed by Harada and Landweber. As a worked class of examples, we compute the full orbifold $K$-theory of weighted projective spaces that occur as a symplectic quotient of a complex affine space by a circle. Our computations hold over the integers, and in the particular case of weighted projective spaces, we show that the associated invariant is torsion-free."}
{"category": "Math", "title": "On Primes In Short Intervals", "abstract": "This note discusses the existence of prime numbers in short intervals. An unconditional elementary argument seems to prove the existence of primes in the short intervals [x, x + y], where y >= x^(1/2)(log x)^e, e > 0, and a sufficiently large number x > 0. Further, an extension of Bertrand's postulate to arithmetic progressions will be considered"}
{"category": "Math", "title": "Socle degrees, Resolutions, and Frobenius powers", "abstract": "We first describe a situation in which every graded Betti number in the tail of the resolution of $\\frac RJ$ may be read from the socle degrees of $\\frac RJ$. Then we apply the above result to the ideals $J$ and $J^{[q]}$; and thereby describe a situation in which the graded Betti numbers in the tail of the resolution of $R/J^{[q]}$ are equal to the graded Betti numbers in the tail of a shift of the resolution of $R/J$."}
{"category": "Math", "title": "Decay of mass for nonlinear equation with fractional Laplacian", "abstract": "The large time behavior of nonnegative solutions to the reaction-diffusion equation $\\partial_t u=-(-\\Delta)^{\\alpha/2}u - u^p,$ $(\\alpha\\in(0,2], p>1)$ posed on $\\mathbb{R}^N$ and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for $p>1+{\\alpha}/{N},$ while nonlinear effects win if $p\\leq1+{\\alpha}/{N}.$"}
{"category": "Math", "title": "Blow up of solutions to generalized Keller--Segel model", "abstract": "The existence and nonexistence of global in time solutions is studied for a class of equations generalizing the chemotaxis model of Keller and Segel. These equations involve L\\'evy diffusion operators and general potential type nonlinear terms."}
{"category": "Math", "title": "Poincare duality and Periodicity, II. James Periodicity", "abstract": "Let K be a connected finite complex. This paper studies the problem of whether one can attach a cell to some iterated suspension S^j K so that the resulting space satisfies Poincare duality. When this is possible, we say that S^j K is a spine. We introduce the notion of quadratic self duality and show that if K is quadratically self dual, then S^j K is a spine whenever j is a suitable power of two. The powers of two come from the James periodicity theorem. We briefly explain how our main results, considered up to bordism, give a new interpretation of the four-fold periodicity of the surgery obstruction groups. We therefore obtain a relationship between James periodicity and the four-fold periodicity in L-theory."}
{"category": "Math", "title": "Functional Equations of $L$-Functions for Symmetric Products of the Kloosterman Sheaf", "abstract": "We determine the (arithmetic) local monodromy at 0 and at $\\infty$ of the Kloosterman sheaf using local Fourier transformations and Laumon's stationary phase principle. We then calculate $\\epsilon$-factors for symmetric products of the Kloosterman sheaf. Using Laumon's product formula, we get functional equations of $L$-functions for these symmetric products, and prove a conjecture of Evans on signs of constants of functional equations."}
{"category": "Math", "title": "Classification of modules of the intermediate series over the twisted N=2 superconformal algebra", "abstract": "In this paper, a classification of modules of the intermediate series over the twisted N=2 superconformal algebra is obtained."}
{"category": "Math", "title": "Lie superbialgebra structures on the centerless twisted N=2 superconformal algebra", "abstract": "In this paper, Lie superbialgebra structures on the centerless twisted N=2 superconformal algebra $\\LL$ are considered which are proved to be coboundary triangular."}
{"category": "Math", "title": "Topological N=2 superconformal superbialgebras", "abstract": "Lie superbialgebra structures on the centerless topological N=2 superconformal algebra $\\TT$ are considered, all of which are proved to be coboundary triangular."}
{"category": "Math", "title": "Stability of cubic and quartic functional equations in non-Archimedean spaces", "abstract": "We prove generalized Hyres-Ulam-Rassias stability of the cubic functional equation $f(kx+y)+f(kx-y)=k[f(x+y)+f(x-y)]+2(k^3-k)f(x)$ for all $k\\in \\Bbb N$ and the quartic functional equation $f(kx+y)+f(kx-y)=k^2[f(x+y)+f(x-y)]+2k^2(k^2-1)f(x)-2(k^2-1)f(y)$ for all $k\\in \\Bbb N$ in non-Archimedean normed spaces."}
{"category": "Math", "title": "Nearly generalized Jordan derivations", "abstract": "Let $A$ be an algebra and let $X$ be an $A$-bimodule. A $\\Bbb C-$linear mapping $d:A \\to X$ is called a generalized Jordan derivation if there exists a Jordan derivation (in the usual sense) $\\delta:A \\to X$ such that $d(a^2)=ad(a)+\\delta(a)a$ for all $a \\in A.$ The main purpose of this paper to prove the Hyers-Ulam-Rassias stability and superstability of the generalized Jordan derivations."}
{"category": "Math", "title": "On the stability of generalized mixed type quadratic and quartic functional equation in quasi-Banach spaces", "abstract": "In this paper, we establish the general solution of the functional equation $$f(nx+y)+f(nx-y)=n^2f(x+y)+n^2f(x-y)+2(f(nx)-n^2f(x))-2(n^2-1)f(y)\\eqno {0 cm}$$for fixed integers $n$ with $n\\neq0,\\pm1$ and investigate the generalized Hyers-Ulam-Rassias stability of this equation in quasi-Banach spaces."}
{"category": "Math", "title": "Stability of a functional equation deriving from cubic and quartic functions", "abstract": "In this paper, we obtain the general solution and the generalized Ulam-Hyers stability of the cubic and quartic functional equation &4(f(3x+y)+f(3x-y))=-12(f(x+y)+f(x-y)) &+12(f(2x+y)+f(2x-y))-8f(y)-192f(x)+f(2y)+30f(2x)."}
{"category": "Math", "title": "The stability of a quadratic type functional equation with the fixed point alternative", "abstract": "In this paper, we achieve the general solution and the generalized Hyers-Ulam-Rassias stability for the quadratic type functional equation &f(x+y+2cz)+f(x+y-2cz)+c^2f(2x)+c^2f(2y) &=2[f(x+y)+c^2f(x+z)+c^2f(x-z)+c^2f(y+z)+c^2f(y-z)] {2.6 cm} for fixed integers $c$ with $c\\neq0,\\pm1$, by using the fixed point alternative."}
{"category": "Math", "title": "2-Kac-Moody algebras", "abstract": "We construct a 2-category associated with a Kac-Moody algebra and we study its 2-representations. This generalizes earlier work with Chuang for type A. We relate categorifications relying on K_0 properties and 2-representations."}
{"category": "Math", "title": "On approximate n-ring homomorphisms and n-ring derivations", "abstract": "Let $A,B$ be two rings and let $ X$ be an $ A-$module. An additive map $h: A\\to B$ is called n-ring homomorphism if $h(\\Pi^n_{i=1}a_i)=\\Pi^n_{i=1}h(a_i),$ for all $a_1,a_2, ...,a_n \\in {A}$. An additive map $D: A\\to X$ is called $n$-ring derivation if $$D(\\Pi^n_{i=1}a_i)=D(a_1)a_2... a_n+a_1D(a_2)a_3... a_n+... +a_1a_2... a_{n-1}D(a_n),$$ for all $a_1,a_2, ...,a_n \\in {\\mathcal A}$. In this paper we investigate the Hyers-Ulam-Rassias stability of $n$-ring homomorphisms and n-ring derivations."}
{"category": "Math", "title": "Stability of a functional equation deriving from quartic and additive functions", "abstract": "In this paper, we obtain the general solution and the generalized Hyers-Ulam Rassias stability of the functional equation $$f(2x+y)+f(2x-y)=4(f(x+y)+f(x-y))-{3/7}(f(2y)-2f(y))+2f(2x)-8f(x).$$"}
{"category": "Math", "title": "Extended finite operator calculus as an example of algebraization of analysis", "abstract": "A calculus of sequences started by professor morgan ward constitutes the general scheme for extensions of classical operator calculus of the distinguished gian carlo rota considered by many afterwards and after ward morgan. Because of the historically now established notation we call the wardian calculus of sequences in its afterwards elaborated form a psi calculus. The psi calculus in parts appears to be almost automatic, natural extension of classical operator calculus or equivalently of umbral calculus . This is a review article based on the turn of the centuries author relevant contributions."}
{"category": "Math", "title": "Pseudocompact group topologies with no infinite compact subsets", "abstract": "We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property $\\h$). Every pseudocompact Abelian group $G$ with cardinality $|G|\\leq 2^{2^\\cc}$ satisfies this inequality and therefore admits a pseudocompact group topology with property $\\h$. Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property $\\h$. We also observe that pseudocompact Abelian groups with property $\\h$ contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact."}
{"category": "Math", "title": "Semiclassical analysis of Schr\\\"odinger operators with magnetic wells", "abstract": "We give a survey of some results, mainly obtained by the authors and their collaborators, on spectral properties of the magnetic Schr\\\"odinger operators in the semiclassical limit. We focus our discussion on asymptotic behavior of the individual eigenvalues for operators on closed manifolds and existence of gaps in intervals close to the bottom of the spectrum of periodic operators."}
{"category": "Math", "title": "On L-infinity morphisms of cyclic chains", "abstract": "Recently the first two authors constructed an L-infinity morphism using the S^1-equivariant version of the Poisson Sigma Model (PSM). Its role in deformation quantization was not entirely clear. We give here a \"good\" interpretation and show that the resulting formality statement is equivalent to formality on cyclic chains as conjectured by Tsygan and proved recently by several authors."}
{"category": "Math", "title": "Thresholding-based Iterative Selection Procedures for Model Selection and Shrinkage", "abstract": "This paper discusses a class of thresholding-based iterative selection procedures (TISP) for model selection and shrinkage. People have long before noticed the weakness of the convex $l_1$-constraint (or the soft-thresholding) in wavelets and have designed many different forms of nonconvex penalties to increase model sparsity and accuracy. But for a nonorthogonal regression matrix, there is great difficulty in both investigating the performance in theory and solving the problem in computation. TISP provides a simple and efficient way to tackle this so that we successfully borrow the rich results in the orthogonal design to solve the nonconvex penalized regression for a general design matrix. Our starting point is, however, thresholding rules rather than penalty functions. Indeed, there is a universal connection between them. But a drawback of the latter is its non-unique form, and our approach greatly facilitates the computation and the analysis. In fact, we are able to build the convergence theorem and explore theoretical properties of the selection and estimation via TISP nonasymptotically. More importantly, a novel Hybrid-TISP is proposed based on hard-thresholding and ridge-thresholding. It provides a fusion between the $l_0$-penalty and the $l_2$-penalty, and adaptively achieves the right balance between shrinkage and selection in statistical modeling. In practice, Hybrid-TISP shows superior performance in test-error and is parsimonious."}
{"category": "Math", "title": "Generalized Harish-Chandra descent, Gelfand pairs and an Archimedean analog of Jacquet-Rallis' Theorem", "abstract": "In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field F of characteristic zero. Our main tool is the Luna Slice Theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pairs (GL(n+k,F), GL(n,F) x GL(k,F)) and (GL(n,E), GL(n,F)) are Gelfand pairs for any local field F and its quadratic extension E. In the non-Archimedean case, the first result was proven earlier by Jacquet and Rallis and the second by Flicker. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-Archimedean F the latter is a classical theorem of Gelfand and Kazhdan."}
{"category": "Math", "title": "Special Classes of Set Codes and Their Applications", "abstract": "In this book, the authors introduce the notion of set codes, set bicodes and set n-codes. These are the most generalized notions of semigroup n-codes and group n-codes. Several types of set n-codes are defined. Several examples are given to enable the reader to understand the concept. These new classes of codes will find applications in cryptography, computer networking (where fragmenting of codes is to be carried out) and data storage (where confidentiality is to be maintained). We also describe the error detection and error correction of these codes. The authors feel that these codes would be appropriate to the computer dominated world. This book has three chapters. Chapter One gives basic concepts to make the book a self-contained one. In Chapter Two, the notion of set codes is introduced. The set bicodes and their generalization to set n-codes (n >= 3) is carried out in Chapter Three. This chapter also gives the applications of these codes in the above-mentioned fields."}
{"category": "Math", "title": "Sign conjugacy classes in symmetric groups", "abstract": "A special type of conjugacy classes in symmetric groups is studied and used to answer a question about odd-degree irreducible characters"}
{"category": "Math", "title": "Interlocking of convex polyhedra: towards a geometric theory of fragmented solids", "abstract": "We present structures comprised of identical convex polyhedra which are interlocked geometrically. These sets cannot be disassembled by removing individual polyhedra by translations and/or rotations. The shapes that permit interlocking arrangements include all five platonic solids. Criteria for interlocking based on transformations of the cross-sections of the elements in a 3D reconstruction of a layer from its middle cross-section are formulated. A generalization to higher dimensions is also given. In particular, an interlocking layer of four-dimensional cubes is described."}
{"category": "Math", "title": "On some definite integrals connecting with certain infinite series", "abstract": "We show some definite integrals connecting to infinite series, studied in Ramanujan's paper, titled \"On question 330 of Professor Sanjana\". We present few recursive methods to evaluate these definite integrals in various cases and we generalize this, to evaluate simliar kind of integrals through infinite series."}
{"category": "Math", "title": "SPM Bulletin 26", "abstract": "This festive issue concludes the civilian year 2008 with details on a special issue of Topology and its Applications dedicated to SPM, and with a quite large list of research announcements."}
{"category": "Math", "title": "Non annulation des fonctions L automorphes au point central", "abstract": "The question about modular forms have recently received a lot of attention; concerning the non-vanishing of automorphic L-functions Michel, Kowalski and Vanderkam proved (among others results) that there's positive proportion of non-vanishing of primitives forms at the critical point. This result was proved by these authors in the prime level case; on the othe hand, Iwaniec, Luo and Sarnak showed the same result for the squarefree level case. In order to understand the influence of the arithmetic shape of level forms in their vanishing, this paper studies a generalisation to the primitives forms with prime powers level."}
{"category": "Math", "title": "Knotting corks", "abstract": "It is known that every exotic smooth structure on a simply connected closed 4-manifold is determined by a codimention zero compact contractible Stein submanifold and an involution on its boundary. Such a pair is called a cork. In this paper, we construct infinitely many knotted imbeddings of corks in 4-manifolds such that they induce infinitely many different exotic smooth structures. We also show that we can imbed an arbitrary finite number of corks disjointly into 4-manifolds, so that the corresponding involutions on the boundary of the contractible 4-manifolds give mutually different exotic structures. Furthermore, we construct similar examples for plugs."}
{"category": "Math", "title": "A Note on Distribution Spaces on Manifolds", "abstract": "26 different concrete representations of the space of vector valued distributions on a smooth manifold of dimension n are presented systematically, most of them new. In the particular case of representations as module homomorphisms acting on sections of the dual bundle resp. on n-forms, the continuity of these homomorphisms is already a consequence of their algebraic properties."}
{"category": "Math", "title": "Center Problem for the Group of Rectangular Paths", "abstract": "We solve the center problem for ODEs \\frac{dv}{dx}=\\sum_{i=1}^{\\infty}a_{i}(x) v^{i+1} such that the first integrals of vectors of their coefficients determine rectangular paths in finite dimensional complex vector spaces."}
{"category": "Math", "title": "Perfect colorings of $Z^2$: Nine colors", "abstract": "We list all perfect colorings of $Z^2$ by 9 or less colors. Keywords: perfect colorings, equitable partitions"}
{"category": "Math", "title": "Space Alternating Penalized Kullback Proximal Point Algorithms for Maximizing Likelihood with Nondifferentiable Penalty", "abstract": "The EM algorithm is a widely used methodology for penalized likelihood estimation. Provable monotonicity and convergence are the hallmarks of the EM algorithm and these properties are well established for smooth likelihood and smooth penalty functions. However, many relaxed versions of variable selection penalties are not smooth. The goal of this paper is to introduce a new class of Space Alternating Penalized Kullback Proximal extensions of the EM algorithm for nonsmooth likelihood inference. We show that the cluster points of the new method are stationary points even when on the boundary of the parameter set. Special attention has been paid to the construction of component-wise version of the method in order to ease the implementation for complicated models. Illustration for the problems of model selection for finite mixtures of regression and to sparse image reconstruction is presented."}
{"category": "Math", "title": "A heat trace anomaly on polygons", "abstract": "Let $\\Omega_0$ be a polygon in $\\RR^2$, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that $\\Omega_\\e$ is a family of surfaces with $\\calC^\\infty$ boundary which converges to $\\Omega_0$ smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov \\cite{Fe}, Kac \\cite{K} and McKean-Singer \\cite{MS} recognized that certain heat trace coefficients, in particular the coefficient of $t^0$, are not continuous as $\\e \\searrow 0$. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain $Z$ which models the corner formation. The result applies both for Dirichlet and Neumann conditions. We also include a discussion of what one might expect in higher dimensions."}
{"category": "Math", "title": "Poisson Geometry of Directed Networks in an Annulus", "abstract": "As a generalization of Postnikov's construction (see arXiv: math/0609764), we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson brackets introduced for the space of edge weights in arXiv: 0805.3541 induce a family of Poisson structures on rational-valued matrix functions and on the space of loops in the Grassmannian. In the former case, this family includes, for a particular kind of networks, the Poisson bracket associated with the trigonometric R-matrix."}
{"category": "Math", "title": "Torelli theorem for the Deligne--Hitchin moduli space", "abstract": "Fix integers $g\\geq 3$ and $r\\geq 2$, with $r\\geq 3$ if $g=3$. Given a compact connected Riemann surface $X$ of genus $g$, let $\\MDH(X)$ denote the corresponding $\\text{SL}(r, {\\mathbb C})$ Deligne--Hitchin moduli space. We prove that the complex analytic space $\\MDH(X)$ determines (up to an isomorphism) the unordered pair $\\{X, \\overline{X}\\}$, where $\\overline{X}$ is the Riemann surface defined by the opposite almost complex structure on $X$."}
{"category": "Math", "title": "Concerning the Wave equation on Asymptotically Euclidean Manifolds", "abstract": "We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on $(\\R^d, \\mathfrak{g})$, $d \\geq 3$, when metric $\\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $ x ^{- \\rho}$ with $\\rho>0$. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for $\\rho> 1$ and $d=3$. Also, we establish the Strauss conjecture when the metric is radial with $\\rho>0$ for $d= 3$."}
{"category": "Math", "title": "Multidimensional latent Markov models in a developmental study of inhibitory control and attentional flexibility in early childhood", "abstract": "We demonstrate the use of a multidimensional extension of the latent Markov model to analyse data from studies with correlated binary responses in developmental psychology. In particular, we consider an experiment based on a battery of tests which was administered to pre-school children, at three time periods, in order to measure their inhibitory control and attentional flexibility abilities. Our model represents these abilities by two latent traits which are associated to each state of a latent Markov chain. The conditional distribution of the tests outcomes given the latent process depends on these abilities through a multidimensional two-parameter logistic parameterisation. We outline an EM algorithm to conduct likelihood inference on the model parameters; we also focus on likelihood ratio testing of hypotheses on the dimensionality of the model and on the transition matrices of the latent process. Through the approach based on the proposed model, we find evidence that supports that inhibitory control and attentional flexibility can be conceptualised as distinct constructs. Furthermore, we outline developmental aspects of participants' performance on these abilities based on inspection of the estimated transition matrices."}
{"category": "Math", "title": "On the Geometry of Discrete Exponential Families with Application to Exponential Random Graph Models", "abstract": "There has been an explosion of interest in statistical models for analyzing network data, and considerable interest in the class of exponential random graph (ERG) models, especially in connection with difficulties in computing maximum likelihood estimates. The issues associated with these difficulties relate to the broader structure of discrete exponential families. This paper re-examines the issues in two parts. First we consider the closure of $k$-dimensional exponential families of distribution with discrete base measure and polyhedral convex support $\\mathrm{P}$. We show that the normal fan of $\\mathrm{P}$ is a geometric object that plays a fundamental role in deriving the statistical and geometric properties of the corresponding extended exponential families. We discuss its relevance to maximum likelihood estimation, both from a theoretical and computational standpoint. Second, we apply our results to the analysis of ERG models. In particular, by means of a detailed example, we provide some characterization of the properties of ERG models, and, in particular, of certain behaviors of ERG models known as degeneracy."}
{"category": "Math", "title": "Regularity of Ornstein-Uhlenbeck processes driven by a L{\\'e}vy white noise", "abstract": "The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by L\\'evy white noise \"obtained by subordination of a Gaussian white noise\". Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general \\cadlag modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well."}
{"category": "Math", "title": "On normal embedding of complex algebraic surfaces", "abstract": "We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface."}
{"category": "Math", "title": "Graphs of $C^*$-correspondences and Fell bundles", "abstract": "We define the notion of a $\\Lambda$-system of $C^*$-correspondences associated to a higher-rank graph $\\Lambda$. Roughly speaking, such a system assigns to each vertex of $\\Lambda$ a $C^*$-algebra, and to each path in $\\Lambda$ a $C^*$-correspondence in a way which carries compositions of paths to balanced tensor products of $C^*$-correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of $C^*$-correspondences to associate a $C^*$-algebra to each $\\Lambda$-system. We then construct a Fell bundle over the path groupoid $\\Gg_\\Lambda$ and show that the $C^*$-algebra of the $\\Lambda$-system coincides with the reduced cross-sectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature."}
{"category": "Math", "title": "Weak Solutions of the Stochastic Landau-Lifshitz-Gilbert Equation", "abstract": "The Landau-Lifshitz-Gilbert equation perturbed by a multiplicative space-dependent noise is considered for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a sphere $\\mathbb S^2$. The regularity of weak solutions is also discussed. Some of the regularity results are new even for the deterministic Landau-Lifshitz-Gilbert equation."}
{"category": "Math", "title": "Counting decomposable univariate polynomials", "abstract": "A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number over a finite field. The tame case, where the field characteristic p does not divide the degree n of f, is reasonably well understood, and we obtain exponentially decreasing error bounds. The wild case, where p divides n, is more challenging and our error bounds are weaker."}
{"category": "Math", "title": "CAT(0) and CAT(-1) fillings of hyperbolic manifolds", "abstract": "We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension $d$ for all $d\\geq 4$. These examples result from applying CAT$(0)$/CAT$(-1)$ filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a number of interesting properties, which are established by analyzing their boundaries at infinity by a kind of Morse-theoretic technique, related to but distinct from ordinary and combinatorial Morse theory."}
{"category": "Math", "title": "The degenerate analogue of Ariki's categorification theorem", "abstract": "We explain how to deduce the degenerate analogue of Ariki's categorification theorem over the ground field C as an application of Schur-Weyl duality for higher levels and the Kazhdan-Lusztig conjecture in finite type A. We also discuss some supplementary topics, including Young modules, tensoring with sign, tilting modules and Ringel duality."}
{"category": "Math", "title": "Formality theorems for Hochschild complexes and their applications", "abstract": "We give a popular introduction to formality theorems for Hochschild complexes and their applications. We review some of the recent results and prove that the truncated Hochschild cochain complex of a polynomial algebra is non-formal."}
{"category": "Math", "title": "p-Adic Spherical Coordinates and Their Applications", "abstract": "On the space $\\mathbb Q_p^n$, where $p\\ne 2$ and $p$ does not divide $n$, we construct a p-adic counterpart of spherical coordinates. As applications, a description of homogeneous distributions on $\\mathbb Q_p^n$ and a skew product decomposition of p-adic L\\'evy processes are given."}
{"category": "Math", "title": "Finite closed coverings of compact quantum spaces", "abstract": "We show that a projective space P^\\infty(Z/2) endowed with the Alexandrov topology is a classifying space for finite closed coverings of compact quantum spaces in the sense that any such a covering is functorially equivalent to a sheaf over this projective space. In technical terms, we prove that the category of finitely supported flabby sheaves of algebras is equivalent to the category of algebras with a finite set of ideals that intersect to zero and generate a distributive lattice. In particular, the Gelfand transform allows us to view finite closed coverings of compact Hausdorff spaces as flabby sheaves of commutative C*-algebras over P^\\infty(Z/2)."}
{"category": "Math", "title": "Quasi-alternating links and odd homology: computations and conjectures", "abstract": "We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagrams, where one is quasi-alternating and the other is not. We provide examples of knots and links with $n\\le 12$ crossings which are homologically thin and have no minimal quasi-alternating diagrams. These links are candidates for homologically thin links that are not quasi-alternating. For one of our candidates [JaSa1], knot $11n_{50}$, J. Greene proved that it is not quasi-alternating, so this is the first example of homologically thin knot which is not quasi-alternating [Gr]. Computations were performed by A. Shumakovitch's program \\emph{KhoHo}, the program \\emph{Knotscape}, and our program \\emph{LinKnot}."}
{"category": "Math", "title": "On Shephard Groups with Large Triangles", "abstract": "Shephard groups are common extensions of Artin and Coxeter groups. They appear, for example, in algebraic study of manifolds. An infinite family of Shephard groups which are not Artin or Coxeter groups is considered. Using techniques form small cancellation theory we show that the groups in this family are bi-automatic."}
{"category": "Math", "title": "Obstructions for Deformations of Complexes", "abstract": "We develop two approaches to obstruction theory for deformations of derived isomorphism classes of complexes $Z^\\bullet$ of modules for a profinite group $G$ over a complete local Noetherian ring $A$ of positive residue characteristic $\\ell$."}
{"category": "Math", "title": "Algebraic monodromy and obstructions to formality", "abstract": "Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more general result about iterated group extensions. As an application, we obtain new criteria for formality of spaces, and 1-formality of groups, illustrated by bundle constructions and various examples from low-dimensional topology and singularity theory."}
{"category": "Math", "title": "On diagrammatic bounds of knot volumes and spectral invariants", "abstract": "In recent years, several families of hyperbolic knots have been shown to have both volume and $\\lambda_1$ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized twist number of a diagram can provide two-sided bounds on either the volume or $\\lambda_1$. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on $\\lambda_1$. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded."}
{"category": "Math", "title": "On a theorem of Mestre and Schoof", "abstract": "A well known theorem of Mestre and Schoof implies that the order of an elliptic curve E over a prime field F_q can be uniquely determined by computing the orders of a few points on E and its quadratic twist, provided that q > 229. We extend this result to all finite fields with q > 49, and all prime fields with q > 29."}
{"category": "Math", "title": "Intrinsic Differential Geometry and the Existence of Quasimeromorphic Mappings", "abstract": "We give a new proof of the existence of nontrivial quasimeromorphic mappings on a smooth Riemannian manifold, using solely the intrinsic geometry of the manifold."}
{"category": "Math", "title": "Locally compact abelian groups admitting non-trivial quasi-convex null sequences", "abstract": "In this paper, we show that for every locally compact abelian group G, the following statements are equivalent: (i) G contains no sequence {x_n} such that {0} \\cup {\\pm x_n : n \\in N} is infinite and quasi-convex in G, and x_n --> 0; (ii) one of the subgroups {g \\in G : 2g=0 and {g \\in G : 3g=0} is open in G; (iii) G contains an open compact subgroup of the form Z_2^\\kappa or Z_3^\\kappa for some cardinal \\kappa."}
{"category": "Math", "title": "Commuting elements, simplicial spaces, and filtrations of classifying spaces", "abstract": "Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q>1 and every topological group G, with realizations B(q,G) that filter the classifying space BG. In particular for q=2 this yields a single space B(2,G) assembled from all the n-tuples of commuting elements in G. Homotopy properties of the B(q,G) are considered for finite groups. Cohomology calculations are provided for compact Lie groups. The spaces B(2,G) are described in detail for transitively commutative groups."}
{"category": "Math", "title": "Galois-Type Extensions and Equivariant Projectivity", "abstract": "The theory of general Galois-type extensions is presented, including the interrelations between coalgebra extensions and algebra (co)extensions, properties of corresponding (co)translation maps, and rudiments of entwinings and factorisations. To achieve broad perspective, this theory is placed in the context of far reaching generalisations of the Galois condition to the setting of corings. At the same time, to bring together K-theory and general Galois theory, the equivariant projectivity of extensions is assumed resulting in the centrepiece concept of a principal extension. Motivated by noncommutative geometry, we employ such extensions as replacements of principal bundles. This brings about the notion of a strong connection and yields finitely generated projective associated modules, which play the role of noncommutative vector bundles. Subsequently, the theory of strong connections is developed. It is purported as a basic ingredient in the construction of the Chern character for Galois-type extensions (called the Chern-Galois character)."}
{"category": "Math", "title": "The Atiyah Patodi Singer signature formula for measured foliations", "abstract": "Let $(X_0,\\mathcal{F}_0) $ be a compact manifold with boundary endowed with a foliation $\\mathcal{F}_0$ which is assumed to be measured and transverse to the boundary. We denote by $\\Lambda$ a holonomy invariant transverse measure on $(X_0,\\mathcal{F}_0) $ and by $\\mathcal{R}_0$ the equivalence relation of the foliation. Let $(X,\\mathcal{F})$ be the corresponding manifold with cylindrical end and extended foliation with equivalence relation $\\mathcal{R}$. In the first part of this work we prove a formula for the $L^2$-$\\Lambda$ index of a longitudinal Dirac-type operator $D^{\\mathcal{F}}$ on $X$ in the spirit of Alain Connes' non commutative geometry $ind_{L^2,\\Lambda}(D^{\\mathcal{F},+}) = <\\hat{A}(T\\mathcal{F})Ch(E/S),C_\\Lambda> + 1/2[\\eta_\\Lambda(D^{\\mathcal{F}_\\partial}) - h^+_{\\Lambda} + h^-_{\\Lambda}].$"}
{"category": "Math", "title": "Graded posets zeta matrix formula", "abstract": "The way to arrive at formula of zeta matrix for any graded posets with the finite set of minimal elements is delivered following the first reference. This is being achieved via adjacency and zeta matrix description of bipartite digraphs chains, the representatives of graded posets. The bipartite digraphs elements of such chains amalgamate to form corresponding cover relation graded poset digraphs with corresponding adjacency matrices being amalgamated throughout natural join as special adequate database operation. The colligation of reachability and connectivity with the presented description is made explicit. The special posets encoded via kodags directed acyclic graphs as cobeb posets cover relations digraphs are recognized as an example of differential posets subfamily. As on this night one reminisce anniversary of death of distinguished johann bernoulli the first this sylvester night article is to commemorate this date."}
{"category": "Math", "title": "The hyperbolic Monge-Ampere equation: classical solutions on the whole plane", "abstract": "The Cauchy problem for the hyperbolic Monge-Ampere equation is considered. The equation has the most general form. Coefficients are arbitrary functions depending on two independent variables, unknown function, and first order derivatives. Sufficient conditions on the existence of a (unique) C^3-solution on the whole plain are formulated."}
{"category": "Math", "title": "Convergent sequences in minimal groups", "abstract": "A Hausdorff topological group G is minimal if every continuous isomorphism f : G --> H between G and a Hausdorff topological group H is open. Clearly, every compact Hausdorff group is minimal. It is well known that every infinite compact Hausdorff group contains a non-trivial convergent sequence. We extend this result to minimal abelian groups by proving that every infinite minimal abelian group contains a non-trivial convergent sequence. Furthermore, we show that \"abelian\" is essential and cannot be dropped. Indeed, for every uncountable regular cardinal kappa we construct a Hausdorff group topology T_kappa on the free group F(kappa) with kappa many generators having the following properties: (i) (F(kappa), T_kappa) is a minimal group; (ii) every subset of F(kappa) of size less than kappa is T_kappa-discrete (and thus also T_kappa-closed); (iii) there are no non-trivial proper T_kappa-closed normal subgroups of F(kappa). In particular, all compact subsets of (F(kappa), T_kappa) are finite, and every Hausdorff quotient group of (F(kappa), T_kappa) is minimal (that is, (F(kappa), T_kappa) is totally minimal)."}
{"category": "Math", "title": "Adjustment coefficient for risk processes in some dependent contexts", "abstract": "Following an article by Muller and Pflug, we study the adjustment coefficient of ruin theory in a context of temporal dependency. We provide a consistent estimator of this coefficient, and perform some simulations."}
{"category": "Math", "title": "Hyperbolic--parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation", "abstract": "We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t tends to +infinity. The case where b(t) behaves like (1+t)^{-p} with p<1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and the difference between solutions of the hyperbolic problem and the corresponding solutions of the parabolic problem converges to zero both as t tends to +infinity and as epsilon goes to 0. In this paper we show that these results cannot be true for p>1, but they remain true in the critical case p=1."}
{"category": "Math", "title": "A hive model determination of multiplicity-free Schur function products and skew Schur functions", "abstract": "The hive model is a combinatorial device that may be used to determine Littlewood-Richardson coefficients and study their properties. It represents an alternative to the use of the Littlewood-Richardson rule. Here properties of hives are used to determine all possible multiplicity-free Schur function products and skew Schur function expansions. This confirms the results of Stembridge, Gutschwager, and Thomas and Yong, and sheds light on the combinatorial origin of the conditions for being multiplicity-free, as well as illustrating some of the key features and power of the hive model."}
{"category": "Math", "title": "Typability in partial groupoids", "abstract": "Adapting a claim of M. Kracht, we establish a characterization of the typable partial applicative algebras."}
{"category": "Math", "title": "Additive Separability, Optimization, and Trivial Webs", "abstract": "In this paper we show that two seemingly unrelated problems in economics, the hypothesis of integrability and the hypothesis of additive separability are linked by the absence of curvature of connections on webs naturally associated with each problem."}
{"category": "Math", "title": "On linear fractional transformations associated with generalized J-inner matrix functions", "abstract": "In this paper we study generalized J-inner matrix valued functions which appear as resolvent matrices in various indefinite interpolation problems. Reproducing kernel indefinite inner product spaces associated with a generalized J-inner matrix valued function W are studied and intensively used in the description of the range of the linear fractional transformation associated with W and applied to the Schur class. For a subclass of generalized J-inner matrix valued function W the notion of associated pair is introduced and factorization formulas for W are found."}
{"category": "Math", "title": "Entropy of shifts on topological graph $C^*$-algebras", "abstract": "We give entropy estimates for two canonical non commutative shifts on $C^*$-algebras associated to some topological graphs $E=(E^0,E^1,s,r)$, defined using a basis of the corresponding Hilbert bimodule $H(E)$. We compare their entropies with the growth entropies associated directly to the topological graph. We illustrate with some examples of topological graphs considered by Katsura, where the vertex and the edge spaces are a union of unit circles and more detailed computations can be done."}
{"category": "Math", "title": "Decomposition of tensor products of modular irreducible representations for $SL_3$ (With an Appendix by C.M. Ringel)", "abstract": "We give an algorithm for working out the indecomposable direct summands in a Krull--Schmidt decomposition of a tensor product of two simple modules for G=SL_3 in characteristics 2 and 3. It is shown that there is a finite family of modules such that every such indecomposable summand is expressible as a twisted tensor product of members of that family. Along the way we obtain the submodule structure of various Weyl and tilting modules. Some of the tilting modules that turn up in characteristic 3 are not rigid; these seem to provide the first example of non-rigid tilting modules for algebraic groups. These non-rigid tilting modules lead to examples of non-rigid projective indecomposable modules for Schur algebras, as shown in the Appendix. Higher characteristics (for SL_3) will be considered in a later paper."}
{"category": "Math", "title": "Duality functors for triple vector bundles", "abstract": "We calculate the group of dualization operations for triple vector bundles, showing that it has order 96 and not 72 as given in Mackenzie's original treatment. The group is a nonsplit extension of S4 by the Klein group. Dualization operations are interpreted as functors on appropriate categories and are said to be equal if they are naturally isomorphic. The method set out here will be applied in a subsequent paper to the case of n-fold vector bundles."}
{"category": "Math", "title": "Conditions for certain ruin for the generalised Ornstein-Uhlenbeck process and the structure of the upper and lower bounds", "abstract": "For a bivariate \\Levy process $(\\xi_t,\\eta_t)_{t\\geq 0}$ the generalised Ornstein-Uhlenbeck (GOU) process is defined as \\[V_t:=e^{\\xi_t}(z+\\int_0^t e^{-\\xi_{s-}}\\ud \\eta_s), t\\ge0,\\]where $z\\in\\mathbb{R}.$ We present conditions on the characteristic triplet of $(\\xi,\\eta)$ which ensure certain ruin for the GOU. We present a detailed analysis on the structure of the upper and lower bounds and the sets of values on which the GOU is almost surely increasing, or decreasing. This paper is the sequel to \\cite{BankovskySly08}, which stated conditions for zero probability of ruin, and completes a significant aspect of the study of the GOU."}
{"category": "Math", "title": "Topological Index Theory for Surfaces in 3-Manifolds", "abstract": "The disk complex of a surface in a 3-manifold is used to define its {\\it topological index}. Surfaces with well-defined topological index are shown to generalize well-known classes, such as incompressible, strongly irreducible, and critical surfaces. The main result is that one may always isotope a surface $H$ with topological index $n$ to meet an incompressible surface $F$ so that the sum of the indices of the components of $H \\setminus N(F)$ is at most $n$. This theorem and its corollaries generalize many known results about surfaces in 3-manifolds, and often provides more efficient proofs. The paper concludes with a list of questions and conjectures, including a natural generalization of Hempel's {\\it distance} to surfaces with topological index $\\ge 2$."}
{"category": "Math", "title": "The Hilbert scheme of the diagonal in a product of projective spaces", "abstract": "The diagonal in a product of projective spaces is cut out by the ideal of 2x2-minors of a matrix of unknowns. The multigraded Hilbert scheme which classifies its degenerations has a unique Borel-fixed ideal. This Hilbert scheme is generally reducible, and its main component is a compactification of PGL(d)^n/PGL(d). For n=2 we recover the manifold of complete collineations. For projective lines we obtain a space of trees that is irreducible but singular. All ideals in our Hilbert scheme are radical. We also explore connections to affine buildings and Deligne schemes."}
{"category": "Math", "title": "Graded Specht modules", "abstract": "Recently, the first two authors have defined a Z-grading on group algebras of symmetric groups and more generally on the cyclotomic Hecke algebras of type G(l,1,d). In this paper we explain how to grade Specht modules over these algebras."}
{"category": "Math", "title": "Non-existence and uniqueness results for supercritical semilinear elliptic equations", "abstract": "Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings."}
{"category": "Math", "title": "Flexible Multivariate Density Estimation with Marginal Adaptation", "abstract": "Our article addresses the problem of flexibly estimating a multivariate density while also attempting to estimate its marginals correctly. We do so by proposing two new estimators that try to capture the best features of mixture of normals and copula estimators while avoiding some of their weaknesses. The first estimator we propose is a mixture of normals copula model that is a flexible alternative to parametric copula models such as the normal and t copula. The second is a marginally adapted mixture of normals estimator that improves on the standard mixture of normals by using information contained in univariate estimates of the marginal densities. We show empirically that copula based approaches can behave much better or much worse than estimators based on mixture of normals depending on the properties of the data. We provide fast and reliable implementations of the estimators and illustrate the methodology on simulated and real data."}
{"category": "Math", "title": "Current and density fluctuations for interacting particle systems with anomalous diffusive behavior", "abstract": "We prove density and current fluctuations for two examples of symmetric, interacting particle systems with anomalous diffusive behavior: the zero-range process with long jumps and the zero-range process with degenerated bond disorder. As an application, we obtain subdiffusive behavior of a tagged particle in a simple exclusion process with variable diffusion coefficient."}
{"category": "Math", "title": "Totally Splittable Polytopes", "abstract": "A split of a polytope is a (necessarily regular) subdivision with exactly two maximal cells. A polytope is totally splittable if each triangulation (without additional vertices) is a common refinement of splits. This paper establishes a complete classification of the totally splittable polytopes."}
{"category": "Math", "title": "Wa\\'zewski Topological Principle and V-bounded Solutions of Nonlinear Systems", "abstract": "We use the Wa\\'zewski topological principle to establish a number of new sufficient conditions for the existence of proper (defined on the entire time axis) solutions of essentially nonlinear nonautonomous systems. The systems under consideration are characterized by the monotonicity property with respect to a certain auxiliary guiding function $W(t,x)$ depending on time and phase coordinates. Another auxiliary function $V(t,x)$, which is positively defined in the phase variables $x$ for any $t$, is used to estimate the deviation of the proper solutions from the origin."}
{"category": "Math", "title": "Connected economically metrizable spaces", "abstract": "A topological space is nonseparably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected sequential topological space X is the image of a nonseparably connected complete metric space Eco(X) under a monotone quotient map. The metric d of the space Eco(X) is economical in the sense that for each infinite subspace A of X the cardinality of the set {d(a,b):a,b in A} does not exceed the density of A. The construction of the space Eco(X) determines a functor Eco from the category Top of topological spaces and their continuous maps into the category Metr of metric spaces and their non-expanding maps."}
{"category": "Math", "title": "Order-invariant measures on causal sets", "abstract": "A causal set is a partially ordered set on a countably infinite ground-set such that each element is above finitely many others. A natural extension of a causal set is an enumeration of its elements which respects the order. We bring together two different classes of random processes. In one class, we are given a fixed causal set, and we consider random natural extensions of this causal set: we think of the random enumeration as being generated one point at a time. In the other class of processes, we generate a random causal set, working from the bottom up, adding one new maximal element at each stage. Processes of both types can exhibit a property called order-invariance: if we stop the process after some fixed number of steps, then, conditioned on the structure of the causal set, every possible order of generation of its elements is equally likely. We develop a framework for the study of order-invariance which includes both types of example: order-invariance is then a property of probability measures on a certain space. Our main result is a description of the extremal order-invariant measures."}
{"category": "Math", "title": "Order-invariant measures on fixed causal sets", "abstract": "A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers -- we call such a linear extension a {\\em natural extension}. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of {\\em order-invariance}: if we condition on the set of the bottom $k$ elements of the natural extension, each possible ordering among these $k$ elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set."}
{"category": "Math", "title": "Strange images of profinite groups", "abstract": "We investigate whether a finitely generated profinite group G could have a finitely generated infinite image. A result of Dan Segal shows that this is impossible if G is prosoluble. We prove that such an image does not exist if G is semisimple or nonuniversal. We also investigate the existense of dense normal subgroups in $G$."}
{"category": "Math", "title": "Spatial Epidemics and Local Times for Critical Branching Random Walks in Dimensions 2 and 3", "abstract": "The behavior at criticality of spatial SIR (susceptible/infected/recovered) epidemic models in dimensions two and three is investigated. In these models, finite populations of size N are situated at the vertices of the integer lattice, and infectious contacts are limited to individuals at the same or at neighboring sites. Susceptible individuals, once infected, remain contagious for one unit of time and then recover, after which they are immune to further infection. It is shown that the measure-valued processes associated with these epidemics, suitably scaled, converge, in the large-N limit, either to a standard Dawson-Watanabe process (super-Brownian motion) or to a Dawson-Watanabe process with location-dependent killing, depending on the size of the the initially infected set. A key element of the argument is a proof of Adler's 1993 conjecture that the local time processes associated with branching random walks converge to the local time density process associated with the limiting super-Brownian motion."}
{"category": "Math", "title": "Kostant homology formulas for oscillator modules of Lie superalgebras", "abstract": "We provide a systematic approach to obtain formulas for characters and Kostant ${\\mathfrak u}$-homology groups of the oscillator modules of the finite dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned ${\\mathfrak u}$-homology computation are worked out."}
{"category": "Math", "title": "On the q-Extensions of the Bernoulli and Euler Numbers, Related Identities and Lerch Zeta Function", "abstract": "Recently, $\\lambda$-Bernoulli and $\\lambda$-Euler numbers are studied in [5, 10]. The purpose of this paper is to present a systematic study of some families of the $q$-extensions of the $\\lambda$-Bernoulli and the $\\lambda$-Euler numbers by using the bosonic $p$-adic $q$-integral and the fermionic $p$-adic $q$-integral. The investigation of these $\\lambda$-$q$-Bernoulli and $\\lambda$-$q$-Euler numbers leads to interesting identities related to these objects. The results of the present paper cover earlier results concerning $q$-Bernoulli and $q$-Euler numbers. By using derivative operator to the generating functions of $\\lambda$-$q$-Bernoulli and $\\lambda$-$q$-Euler numbers, we give the $q$-extensions of Lerch zeta function."}
{"category": "Math", "title": "The Orlik-Terao algebra and 2-formality", "abstract": "The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement A in C^n; it is the quotient of an exterior algebra E(V) on |A| generators. Orlik and Terao introduced a commutative analog S(V)/I of the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hilbert series depends only on the intersection lattice L(A). Motivated by topological considerations, Falk and Randell introduced the property of 2-formality; we study the relation between 2-formality and the Orlik-Terao algebra. Our main result is a necessary and sufficient condition for 2-formality in terms of the quadratic component I_2 of the Orlik-Terao ideal I: 2-formality is determined by the tangent space T_p(V(I_2)) at a generic point p."}
{"category": "Math", "title": "Holonomy Lie algebras and the LCS formula for subarrangements of A_n", "abstract": "If X is the complement of a hypersurface in P^n, then Kohno showed that the nilpotent completion of the fundamental group is isomorphic to the nilpotent completion of the holonomy Lie algebra of X. When X is the complement of a hyperplane arrangement A, the ranks phi_k of the lower central series quotients of the fundamental group of X are known for isolated examples, and for two special classes: if X is hypersolvable (in which case the quadratic closure of the cohomology ring is Koszul), or if the holonomy Lie algebra decomposes in degree three as a direct product of local components. In this paper, we use the holonomy Lie algebra to obtain a formula for phi_k when A is a subarrangement of A_n. This extends Kohno's result for braid arrangements, and provides an instance of an LCS formula for arrangements which are not decomposable or hypersolvable."}
{"category": "Math", "title": "On the homology of the space of singular knots", "abstract": "In this paper we introduce various associative products on the homology of the space of knots and singular knots in $S^n$. We prove that these products are related through a desingularization map. We also compute some of these products and prove the nontriviality of the desingularization morphism."}
{"category": "Math", "title": "Representation of small ball probabilities in Hilbert space and lower bound in regression for functional data", "abstract": "Let $S=\\sum_{i=1}^{+\\infty}\\lambda_{i}Z_{i}$ where the $Z_{i}$'s are i.d.d. positive with $\\mathbb{E}\\| Z\\| ^{3}<+\\infty$ and $(\\lambda_{i})_{i\\in\\mathbb{N}}$ a positive nonincreasing sequence such that $\\sum\\lambda_{i}<+\\infty$. We study the small ball probability $\\mathbb{P}(S<\\epsilon) $ when $\\epsilon\\downarrow0$. We start from a result by Lifshits (1997) who computed this probability by means of the Laplace transform of $S$. We prove that $\\mathbb{P}(S<\\cdot) $ belongs to a class of functions introduced by de Haan, well-known in extreme value theory, the class of Gamma-varying functions, for which an exponential-integral representation is available. This approach allows to derive bounds for the rate in nonparametric regression for functional data at a fixed point $x_{0}$ : $\\mathbb{E}(y|X=x_{0}%) $ where $(y_{i},X_{i})_{1\\leq i\\leq n}$ is a sample in $(\\mathbb{R},\\mathcal{F}) $ and $\\mathcal{F}$ is some space of functions. It turns out that, in a general framework, the minimax lower bound for the risk is of order $(\\log n)^{-\\tau}$ for some $\\tau>0$ depending on the regularity of the data and polynomial rates cannot be achieved."}
{"category": "Math", "title": "Spectral distribution and $L^2$-isoperimetric profile of Laplace operators on groups", "abstract": "We give a formula relating the $L^2$-isoperimetric profile to the spectral distribution of the Laplace operator associated to a finitely generated group $\\Gamma$ or a Riemannian manifold with a cocompact, isometric $\\Gamma$-action. As a consequence, we can apply techniques from geometric group theory to estimate the spectral distribution of the Laplace operator in terms of the growth and the F{\\o}lner's function of the group, generalizing previous estimates by Gromov and Shubin. This leads, in particular, to sharp estimates of the spectral distributions for several classes of solvable groups. Furthermore, we prove the asymptotic invariance of the spectral distribution under changes of measures with finite second moment."}
{"category": "Math", "title": "Matrices of unitary moments", "abstract": "We investigate certain matrices composed of mixed, second-order moments of unitaries. The unitaries are taken from C*-algebras with moments taken with respect to traces, or, alternatively, from matrix algebras with the usual trace. These sets are of interest in light of a theorem of E. Kirchberg about Connes' embedding problem."}
{"category": "Math", "title": "Tensor products of irreducible representations of the group G = GL(3,q)", "abstract": "We describe the tensor products of two irreducible linear complex representations of the finite general linear group G = GL(3,q) in terms of induced representations by linear characters of maximal torii and also in terms of Gelfand-Graev representations. Our results include MacDonald's conjectures for G and at the same time they are extensions to G of finite counterparts to classical results on tensor products of holomorphic and anti-holomorphic representations of the group SL(2, R). Moreover they provide an easy way to decompose these tensor products, with the help of Frobenius reciprocity. We also state some conjectures for the general case of GL(n,q)."}
{"category": "Math", "title": "Parabolic Subgroups of Real Direct Limit Groups", "abstract": "Let $G_R$ be a classical real direct limit Lie group and $g_R$ its Lie algebra. The parabolic subalgebras of the complexification $g_C$ were described by the first two authors. In the present paper we extend these results to $g_R$. This also gives a description of the parabolic subgroups of $G_R$. Furthermore, we give a geometric criterion for a parabolic subgroup $P_C$ of $G_C$ to intersect $G_R$ in a parabolic subgroup. This criterion involves the $G_R$-orbit structure of the flag ind-manifold $G_C/P_C$."}
{"category": "Math", "title": "A finiteness theorem for hyperbolic 3-manifolds", "abstract": "We prove that there are only finitely many closed hyperbolic 3-manifolds with injectivity radius and first eigenvalue of the Laplacian bounded below whose fundamental groups can be generated by a given number of elements. An application to arithmetic manifolds is also given."}
{"category": "Math", "title": "On the Second Boundary Value Problem for a Class of Modified-Hessian Equations", "abstract": "In this paper a new class of modified-Hessian equations, closely related to the Optimal Transportation Equation, will be introduced and studied. In particular, the existence of globally smooth, classical solutions of these equations satisfying the second boundary value problem will be proven. This proof follows a standard method of continuity argument, which subsequently requires various a priori estimates to be made on classical solutions. These estimates are modifications and generalise the corresponding estimates of Trudinger and Wang for the Optimal Transportation Equation. Of particular note is the fact that the global C^2 estimate contained in this paper makes no use of duality in regards to the original equation."}
{"category": "Math", "title": "Invisible Parts of Attractors", "abstract": "This paper deals with the attractors of generic dynamical systems. We introduce the notion of epsilon-invisible set, which is an open set in which almost all orbits spend on average a fraction of time no greater than epsilon. For extraordinarily small values of epsilon (say, smaller than 2^{-100}), these are areas of the phase space which an observer virtually never sees when following a generic orbit. We construct an open set in the space of all dynamical systems which have an epsilon-invisible set that includes parts of attractors of size comparable to the entire attractor of the system, for extraordinarily small values of epsilon. The open set consists of C^1 perturbations of a particular skew product over the Smale-Williams solenoid. Thus for all such perturbations, a sizable portion of the attractor is almost never visited by generic orbits and practically never seen by the observer."}
{"category": "Math", "title": "Representations up to homotopy of Lie algebroids", "abstract": "We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid is defined and shown to coincide with Kalkman's BRST model for equivariant cohomology in the case of group actions."}
{"category": "Math", "title": "Approximate Parametrization of Plane Algebraic Curves by Linear Systems of Curves", "abstract": "It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance $\\epsilon>0$ and an $\\epsilon$-irreducible algebraic affine plane curve $\\mathcal C$ of proper degree $d$, we introduce the notion of $\\epsilon$-rationality, and we provide an algorithm to parametrize approximately affine $\\epsilon$-rational plane curves, without exact singularities at infinity, by means of linear systems of $(d-2)$-degree curves. The algorithm outputs a rational parametrization of a rational curve $\\bar{\\mathcal C}$ of degree at most $d$ which has the same points at infinity as $\\mathcal C$. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that $\\bar{\\mathcal C}$ and $\\mathcal C$ are close in practice."}
{"category": "Math", "title": "The Weil algebra and the Van Est isomorphism", "abstract": "This paper belongs to a series devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman's BRST model, here we introduce the Weil algebra $W(A)$ associated to any Lie algebroid $A$. We then show that this Weil algebra is related to the Bott-Shulman-Stasheff complex (computing the cohomology of the classifying space) via a Van Est map and we prove a Van Est isomorphism theorem. As application, we generalize and find a simpler more conceptual proof of the main result of Bursztyn et.al. on the reconstructions of multiplicative forms and of a result of Weinstein-Xu and Crainic on the reconstruction of connection 1-forms. This reveals the relevance of the Weil algebra and Van Est maps to the integration and the pre-quantization of Poisson (and Dirac) manifolds."}
{"category": "Math", "title": "Beta Jacobi processes", "abstract": "We define and study a multidimensional process that generalizes the eigenvalues of matrix Jacobi processes on the one hand and whose stationary distribution is given by the beta Jacobi ensemble on the other hand."}
{"category": "Math", "title": "Wagner Lift of Riemannian metric to Orthogonal Frame Bundle", "abstract": "In the present work we construct a lift of a metric $g$ on a 2-dimensional oriented Riemannian manifold $M$ to a metric $\\hat{g}$ on the total space $P$ of the orthonormal frame bundle of $M$. We call this lift the \\textit {Wagner lift}. Viktor Vladimirovich Wagner (1908 -1981) proposed a technique to extend a metric defined on a non-holonomic distribution to its prolongation via the Lie brackets. We apply the Wagner construction to the specific case when the distribution is the infinitesimal connection in the orthonormal frame bundle which corresponds to a Levi-Civita connection. We find relations between the geometry of the Riemannian manifold $(M,g)$ and of the total space $(P,G)$ of the orthonormal frame bundle endowed with the lifted metric."}
{"category": "Math", "title": "On a new method for controlling exponential processes", "abstract": "Unlike the classical polynomial case there has not been invented up to very recently a tool similar to the Bernstein-Bezier representation which would allow us to control the behavior of the exponential polynomials. The exponential analog to the classical Bernstein polynomials has been introduced in a recent authors' paper which appeared in Constructive Approximations, and this analog retains all basic properties of the classical Bernstein polynomials. The main purpose of the present paper is to contribute in this direction, by proving some important properties of the \"Bernstein exponential operator\" which has been introduced. We also fix our attention upon some special type of exponential polynomials which are particularly important for the further development of theory of representation of Multivariate data."}
{"category": "Math", "title": "Biextensions of 1-motives in Voevodsky's category of motives", "abstract": "Let k be a perfect field. In this paper we prove that biextensions of 1-motives define multilinear morphisms between 1-motives in Voevodsky's triangulated category of effective geometrical motives over k with rational coefficients."}
{"category": "Math", "title": "Invariance of generalized wordlength patterns", "abstract": "The generalized wordlength pattern (GWLP) introduced by Xu and Wu (2001) for an arbitrary fractional factorial design allows one to extend the use of the minimum aberration criterion to such designs. Ai and Zhang (2004) defined the $J$-characteristics of a design and showed that they uniquely determine the design. While both the GWLP and the $J$-characteristics require indexing the levels of each factor by a cyclic group, we see that the definitions carry over with appropriate changes if instead one uses an arbitrary abelian group. This means that the original definitions rest on an arbitrary choice of group structure. We show that the GWLP of a design is independent of this choice, but that the $J$-characteristics are not. We briefly discuss some implications of these results."}
{"category": "Math", "title": "New estimates for the Beurling-Ahlfors operator on differential forms", "abstract": "We establish new $p$-estimates for the norm of the generalized Beurling--Ahlfors transform $\\mathcal{S}$ acting on form-valued functions. Namely, we prove that $\\norm{\\mathcal{S}}_{L^p(\\R^n;\\Lambda)\\to L^p(\\R^n;\\Lambda)}\\leq n(p^{*}-1)$ where $p^*=\\max\\{p, p/(p-1)\\},$ thus extending the recent Nazarov--Volberg estimates to higher dimensions. The even-dimensional case has important implications for quasiconformal mappings. Some promising prospects for further improvement are discussed at the end."}
{"category": "Math", "title": "Compressible flows with a density-dependent viscosity coefficient", "abstract": "We prove the global existence of weak solutions for the 2-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient ($\\lambda=\\lambda(\\rho)$). Initial data and solutions are small in energy-norm with nonnegative densities having arbitrarily large sup-norm. Then, we show that if there is a vacuum domain at the initial time, then the vacuum domain will retain for all time, and vanishes as time goes to infinity. At last, we show that the condition of $\\mu=$constant will induce a singularity of the system at vacuum. Thus, the viscosity coefficient $\\mu$ plays a key role in the Navier-Stokes equations."}
{"category": "Math", "title": "New approach to q-Genocch, Euler numbers and polynomials and their interpolation functions", "abstract": "We give a new construction of q-Genocchi numbers, Euler numbers of higher order, which are different than the q-Genocchi numbers of Cangul-Ozden-Simsek. By using our q-Genoucchi, Euler nimbers of higher order, we can investigate the interesting relationship between w-q-Euler polynomials and w-q-Genocchi polynomials."}
{"category": "Math", "title": "Generic T-adic exponential sums in one variable", "abstract": "The $T$-adic exponential sum associated to a Laurent polynomial in one variable is studied. An explicit arithmetic polygon is proved to be the generic Newton polygon of the $C$-function of the T-adic exponential sum. It gives the generic Newton polygon of $L$-functions of $p$-power order exponential sums."}
{"category": "Math", "title": "Information, Divergence and Risk for Binary Experiments", "abstract": "We unify f-divergences, Bregman divergences, surrogate loss bounds (regret bounds), proper scoring rules, matching losses, cost curves, ROC-curves and information. We do this by systematically studying integral and variational representations of these objects and in so doing identify their primitives which all are related to cost-sensitive binary classification. As well as clarifying relationships between generative and discriminative views of learning, the new machinery leads to tight and more general surrogate loss bounds and generalised Pinsker inequalities relating f-divergences to variational divergence. The new viewpoint illuminates existing algorithms: it provides a new derivation of Support Vector Machines in terms of divergences and relates Maximum Mean Discrepancy to Fisher Linear Discriminants. It also suggests new techniques for estimating f-divergences."}
{"category": "Math", "title": "Convexity properties of generalized moment maps", "abstract": "In this paper, we consider generalized moment maps for Hamiltonian actions on $H$-twisted generalized complex manifolds introduced by Lin and Tolman \\cite{Lin}. The main purpose of this paper is to show convexity and connectedness properties for generalized moment maps. We study Hamiltonian torus actions on compact $H$-twisted generalized complex manifolds and prove that all components of the generalized moment map are Bott-Morse functions. Based on this, we shall show that the generalized moment maps have a convex image and connected fibers. Furthermore, by applying the arguments of Lerman, Meinrenken, Tolman, and Woodward \\cite{Ler2} we extend our results to the case of Hamiltonian actions of general compact Lie groups on $H$-twisted generalized complex orbifolds."}
{"category": "Math", "title": "Riemann-Stieltjes operators and multipliers on $Q_p$ spaces in the unit ball of $C^n$", "abstract": "This paper is devoted to characterizing the Riemann-Stieltjes operators and pointwise multipliers acting on M${\\rm \\ddot{o}}$bius invariant spaces $Q_p$, which unify BMOA and Bloch space in the scale of $p$. The boundedness and compactness of these operators on $Q_p$ spaces are determined by means of an embedding theorem, i.e. $Q_p$ spaces boundedly embedded in the non-isotropic tent type spaces $T_q^\\infty$."}
{"category": "Math", "title": "New inductive constructions of complete caps in $PG(N,q)$, $q$ even", "abstract": "Some new families of small complete caps in $PG(N,q)$, $q$ even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this paper provide an improvement on the currently known upper bounds on the size of the smallest complete cap in $PG(N,q),$ $N\\geq 4,$ for all $q\\geq 2^{3}.$ In particular, substantial improvements are obtained for infinite values of $q$ square, including $ q=2^{2Cm},$ $C\\geq 5,$ $m\\geq 3;$ for $q=2^{Cm},$ $C\\geq 5,$ $m\\geq 9,$ with $C,m$ odd; and for all $q\\leq 2^{18}.$"}
{"category": "Math", "title": "On Cox rings of K3-surfaces", "abstract": "We study Cox rings of K3-surfaces. A first result is that a K3-surface has a finitely generated Cox ring if and only if its effective cone is polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3-surfaces of Picard number two, and explicitly compute the Cox rings of generic K3-surfaces with a non-symplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces."}
{"category": "Math", "title": "A $C^0$-estimate for the parabolic Monge-Amp\\`{e}re equation on complete non-compact K\\\"ahler manifolds", "abstract": "In this article we study the K\\\"ahler Ricci flow, the corresponding parabolic Monge Amp\\`{e}re equation and complete non-compact K\\\"ahler Ricci flat manifolds. In our main result Theorem \\ref{mainthm} we prove that if $(M, g)$ is sufficiently close to being K\\\"ahler Ricci flat in a suitable sense, then the K\\\"ahler Ricci flow \\eqref{KRF} has a long time smooth solution $g(t)$ converging smoothly uniformly on compact sets to a complete K\\\"ahler Ricci flat metric on $M$. The main step is to obtain a uniform $C^0$-estimates for the corresponding parabolic Monge Amp\\`{e}re equation. Our results on this can be viewed as a parabolic version of the main results in \\cite{TY3} on the elliptic Monge Amp\\`{e}re equation."}
{"category": "Math", "title": "On certain categories of modules for twisted affine Lie algebras", "abstract": "We classify integrable irreducible $\\hat{g}[\\sigma]$-modules in categories E and C, where E is proved to contain the well known evaluation modules and C to unify highest weight modules, evaluation modules and their tensor product modules."}
{"category": "Math", "title": "Rational linking and contact geometry", "abstract": "In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a version of Bennequin's inequality for these knots and classify precisely when the Bennequin bound is sharp for fibered knot types. Finally we study rational unknots and show they are weakly Legendrian and transversely simple. This version of the paper corrects the definition of rational self-linking number in the previous and published version of the paper. With this correction all the main results of the paper remain true as originally stated."}
{"category": "Math", "title": "The Atiyah-Patodi-Singer index theorem for Dirac operators over C*-algebras", "abstract": "We prove an Atiyah-Patodi-Singer index theorem for Dirac operators twisted by C*-vector bundles. We use it to derive a general product formula for eta-forms and to define and study new rho-invariants generalizing Lott's higher rho-form. The higher Atiyah-Patodi-Singer index theorem of Leichtnam-Piazza can be recovered by applying the theorem to Dirac operators twisted by the Mishenko-Fomenko bundle associated to the reduced C*-algebra of the fundamental group."}
{"category": "Math", "title": "Invariant manifolds for random and stochastic partial differential equations", "abstract": "Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable and pseudo-unstable manifolds for a class of \\emph{random} partial differential equations and \\emph{stochastic} partial differential equations is shown. Unlike the invariant manifold theory for stochastic \\emph{ordinary} differential equations, random norms are not used. The result is then applied to a nonlinear stochastic partial differential equation with linear multiplicative noise."}
{"category": "Math", "title": "Confirming Two Conjectures of Su and Wang", "abstract": "Two conjectures of Su and Wang (2008) concerning binomial coefficients are proved. For $n\\geq k\\geq 0$ and $b>a>0$, we show that the finite sequence $C_j=\\binom{n+ja}{k+jb}$ is a P\\'{o}lya frequency sequence. For $n\\geq k\\geq 0$ and $a>b>0$, we show that there exists an integer $m\\geq 0$ such that the infinite sequence $\\binom{n+ja}{k+jb}, j=0, 1,...$, is log-concave for $0\\leq j\\leq m$ and log-convex for $j\\geq m$. The proof of the first result exploits the connection between total positivity and planar networks, while that of the second uses a variation-diminishing property of the Laplace transform."}
{"category": "Math", "title": "Deformations of canonical pairs and Fano varieties", "abstract": "This paper is devoted to the study of various aspects of deformations of log pairs, especially in connection to questions related to the invariance of singularities and log plurigenera. In particular, using recent results from the minimal model program, we obtain an extension theorem for adjoint divisors in the spirit of Siu and Kawamata and more recent works of Hacon and McKernan. Our main motivation however comes from the study of deformations of Fano varieties. Our first application regards the behavior of Mori chamber decompositions in families of Fano varieties: we prove that, in the case of mild singularities, such decomposition is rigid under deformation when the dimension is small. We then turn to analyze deformation properties of toric Fano varieties, and prove that every simplicial toric Fano variety with at most terminal singularities is rigid under deformations (and in particular is not smoothable, if singular)."}
{"category": "Math", "title": "Fusion Rings of Loop Group Representations", "abstract": "We compute the fusion rings of positive energy representations of the loop groups of the simple, simply connected Lie groups."}
{"category": "Math", "title": "The tau constant of a metrized graph and its behavior under graph operations", "abstract": "This paper concerns the tau constant, which is an important invariant of a metrized graph, and which has applications to arithmetic properties of curves. We give several formulas for the tau constant, and show how it changes under graph operations including deletion of an edge, contraction of an edge, and union of graphs along one or two points. We show how the tau constant changes when edges of a graph are replaced by arbitrary graphs. We prove Baker and Rumely's lower bound conjecture on the tau constant for several classes of metrized graphs."}
{"category": "Math", "title": "Non-vanishing theorem for log canonical pairs", "abstract": "We obtain a correct generalization of Shokurov's non-vanishing theorem for log canonical pairs. It implies the base point free theorem for log canonical pairs. We also prove the rationality theorem for log canonical pairs. As a corollary, we obtain the cone theorem for log canonical pairs. We do not need Ambro's theory of quasi-log varieties."}
{"category": "Math", "title": "Anti-Pluricanonical Systems On Q-Fano Threefolds", "abstract": "We investigate birationality of the anti-pluricanonical map $\\phi_{-m}$, the rational map defined by the anti-pluricanonical system $|-mK|$, on $\\mathbb{Q}$-Fano threefolds."}
{"category": "Math", "title": "The Self-Linking Number in Annulus and Pants Open Book Decompositions", "abstract": "We find a self-linking number formula for a given null-homologous transverse link in a contact manifold that is compatible with either an annulus or a pair of pants open book decomposition. It extends Bennequin's self-linking formula for a braid in the standard contact $3$-sphere."}
{"category": "Math", "title": "Optimal regularity for the Signorini problem", "abstract": "We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity $C^{1,1/2}$. This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary $C^{1,\\beta}$ hypersurface, $\\beta>1/2$, additionally, our proof covers any linear elliptic operator in divergence form with smooth coefficients. The main ingredients of the proof are a version of Almgren's monotonicity formula and the optimal regularity of global solutions."}
{"category": "Math", "title": "Einstein and conformally flat critical metrics of the volume functional", "abstract": "Let $R$ be a constant. Let $\\mathcal{M}^R_\\gamma$ be the space of smooth metrics $g$ on a given compact manifold $\\Omega^n$ ($n\\ge 3$) with smooth boundary $\\Sigma $ such that $g$ has constant scalar curvature $R$ and $g|_{\\Sigma}$ is a fixed metric $\\gamma$ on $\\Sigma$. Let $V(g)$ be the volume of $g\\in\\mathcal{M}^R_\\gamma$. In this work, we classify all Einstein or conformally flat metrics which are critical points of $V(\\cdot)$ in $\\mathcal{M}^R_\\gamma$."}
{"category": "Math", "title": "Convergence of ray sequences of Pade approximants to 2F1(a,1;c;z), c>a>0", "abstract": "The Pad\\'e table of $\\phantom{}_2F_1(a,1;c;z)$ is normal for $c>a>0$ (cf. \\cite{3}). For $m \\geq n-1$ and $c \\notin {\\zz}^{\\phantom{}^-}$, the denominator polynomial $Q_{mn}(z)$ in the $[m/n]$ Pad\\'e approximant $P_{mn}(z)/Q_{mn}(z)$ for $\\phantom{}_2F_1(a,1;c;z)$ and the remainder term $Q_{mn}(z)\\phantom{}_2F_1(a,1;c;z)-P_{mn}(z)$ were explicitly evaluated by Pad\\'e (cf. \\cite{2}, \\cite{5} or \\cite{7}). We show that for $c>a>0$ and $m\\geq n-1$, the poles of $P_{mn}(z)/Q_{mn}(z)$ lie on the cut $(1,\\infty)$. We deduce that the sequence of approximants $P_{mn}(z)/Q_{mn}(z)$ converges to $\\phantom{}_2F_1(a,1;c;z)$ as $m \\to \\infty$, $ n/m \\to \\rho$ with $0<\\rho \\leq 1$, uniformly on compact subsets of the unit disc $|z|<1$ for $c>a>0$"}
{"category": "Math", "title": "Back to balls in billiards", "abstract": "We consider a billiard in the plane with periodic configuration of convex scatterers. This system is recurrent, in the sense that almost every orbit comes back arbitrarily close to the initial point. In this paper we study the time needed to get back in an r-ball about the initial point, in the phase space and also for the position, in the limit when r->0. We establish the existence of an almost sure convergence rate, and prove a convergence in distribution for the rescaled return times."}
{"category": "Math", "title": "The Borel Conjecture for hyperbolic and CAT(0)-groups", "abstract": "We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space."}
{"category": "Math", "title": "Kashiwara and Zelevinsky involutions in affine type A", "abstract": "We first describe how the Kashiwara involution on crystals of affine type $A$ is encoded by the combinatorics of aperiodic multisegments. This yields a simple relation between this involution and the Zelevinsky involution on the set of simple modules for the affine Hecke algebras. We then give efficient procedures for computing these involutions. Remarkably, these procedures do not use the underlying crystal structure. They also permit to match explicitly the Ginzburg and Ariki parametrizations of the simple modules associated to affine and cyclotomic Hecke algebras, respectively ."}
{"category": "Math", "title": "An Unusual Proof that the Reals are Uncountable", "abstract": "This somewhat unusual proof for the fact that the reals are uncountable, which is adapted from one of Bourbaki's proofs in \"Fonctions d'une variable reelle\", may be of some interest."}
{"category": "Math", "title": "Likelihood Inference in Exponential Families and Directions of Recession", "abstract": "When in a full exponential family the maximum likelihood estimate (MLE) does not exist, the MLE may exist in the Barndorff-Nielsen completion of the family. We propose a practical algorithm for finding the MLE in the completion based on repeated linear programming using the R contributed package rcdd and illustrate it with two generalized linear model examples. When the MLE for the null hypothesis lies in the completion, likelihood ratio tests of model comparison are almost unchanged from the usual case. Only the degrees of freedom need to be adjusted. When the MLE lies in the completion, confidence intervals are changed much more from the usual case. The MLE of the natural parameter can be thought of as having gone to infinity in a certain direction, which we call a generic direction of recession. We propose a new one-sided confidence interval which says how close to infinity the natural parameter may be. This maps to one-sided confidence intervals for mean values showing how close to the boundary of their support they may be."}
{"category": "Math", "title": "A Law of Likelihood for Composite Hypotheses", "abstract": "The law of likelihood underlies a general framework, known as the likelihood paradigm, for representing and interpreting statistical evidence. As stated, the law applies only to simple hypotheses, and there have been reservations about extending the law to composite hypotheses, despite their tremendous relevance in statistical applications. This paper proposes a generalization of the law of likelihood for composite hypotheses. The generalized law is developed in an axiomatic fashion, illustrated with real examples, and examined in an asymptotic analysis. Previous concerns about including composite hypotheses in the likelihood paradigm are discussed in light of the new developments. The generalized law of likelihood is compared with other likelihood-based methods and its practical implications are noted. Lastly, a discussion is given on how to use the generalized law to interpret published results of hypothesis tests as reduced data when the full data are not available."}
{"category": "Math", "title": "On the ranks and border ranks of symmetric tensors", "abstract": "Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent."}
{"category": "Math", "title": "Product Structures for Legendrian Contact Homology", "abstract": "Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products - and, more generally, an A_\\infty structure - on the linearized LCH. We apply the products and A_\\infty structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors, to reinterpret the duality theorem of the fourth author in terms of the cup product, and to recover higher-order linearizations of the LCH."}
{"category": "Math", "title": "Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data", "abstract": "The goal of this paper is to exhibit a critical mass phenomenon occuring in a model for cell self-organization via chemotaxis. The very well known dichotomy arising in the behavior of the macroscopic Keller-Segel system is derived at the kinetic level, being closer to microscopic features. Indeed, under the assumption of spherical symmetry, we prove that solutions with initial data of large mass blow-up in finite time, whereas solutions with initial data of small mass do not. Blow-up is the consequence of a virial identity and the existence part is derived from a comparison argument. Spherical symmetry is crucial within the two approaches. We also briefly investigate the drift-diffusion limit of such a kinetic model. We recover partially at the limit the Keller-Segel criterion for blow-up, thus arguing in favour of a global link between the two models."}
{"category": "Math", "title": "Derived equivalences of Calabi-Yau fibrations", "abstract": "We consider fibrations by abelian surfaces and K3 surfaces over a one dimensional base that are Calabi-Yau and we obtain dual fibrations that are derived equivalent to the original fibration. Finally, we relate the problem to mirror symmetry."}
{"category": "Math", "title": "Abstract Hardy-Sobolev spaces and interpolation", "abstract": "The purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz inequalities."}
{"category": "Math", "title": "On a geometric black hole of a compact manifold", "abstract": "Using a smooth triangulation and a Riemannian metric on a compact, connected, closed manifold M of dimension n we have got that every such M can be represented as a union of a n-dimensional cell and a connected union K of some subsimplexes of the triangulation. A sufficiently small closed neighborhood of K is called a geometric black hole. Any smooth tensor field T (or other structure) can be deformed into a continuous and sectionally smooth tensor field T1 where T1 has a very simple construction out of the black hole."}
{"category": "Math", "title": "A Family of Nonlinear Fourth Order Equations of Gradient Flow Type", "abstract": "Global existence and long-time behavior of solutions to a family of nonlinear fourth order evolution equations on $R^d$ are studied. These equations constitute gradient flows for the perturbed information functionals $F[u] = 1/(2\\alpha) \\int | D (u^\\alpha) |^2 dx + \\lambda/2 \\int |x|^2 u dx$ with respect to the $L^2$-Wasserstein metric. The value of $\\alpha$ ranges from $\\alpha=1/2$, corresponding to a simplified quantum drift diffusion model, to $\\alpha=1$, corresponding to a thin film type equation."}
{"category": "Math", "title": "Actions of Maximal Growth", "abstract": "We study acts and modules of maximal growth over finitely generated free monoids and free associative algebras as well as free groups and free group algebras. The maximality of the growth implies some other specific properties of these acts and modules that makes them close to the free ones; at the same time, we show that being a strong \"infiniteness\" condition, the maximality of the growth can still be combined with various finiteness conditions, which would normally make finitely generated acts finite and finitely generated modules finite-dimensional."}
{"category": "Math", "title": "Lorenz like flows: exponential decay of correlations for the Poincar\\'e map, logarithm law, quantitative recurrence", "abstract": "In this paper we prove that the Poincar\\'e map associated to a Lorenz like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time $\\tau_r(x,x_0)$ is the time needed for the orbit of a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered at $x_0$, with small radius $r$. As the radius of the ball decreases to 0 its asymptotic behavior is a power law whose exponent is related to the local dimension of the SRB measure at $x_0$: for each $x_0$ such that the local dimension $d_{\\mu}(x_0)$ exists, \\lim_{r\\to 0} \\frac{\\log \\tau_r(x,x_0)}{-\\log r} = d_{\\mu}(x_0)-1 holds for $\\mu$ almost each $x$. In a similar way it is possible to consider a quantitative recurrence indicator quantifying the speed of coming back of an orbit to its starting point. Similar results holds for this recurrence indicator."}
{"category": "Math", "title": "Color Visualization of Blaschke Self-Mappings of the Real Projective Plan", "abstract": "The real projective plan $P^2$ can be endowed with a dianalytic structure making it into a non orientable Klein surface. Dianalytic self-mappings of that surface are projections of analytic self-mappings of the Riemann sphere $\\widehat{\\mathbb{C}}$. It is known that the only analytic bijective self-mappings of $\\widehat{\\mathbb{C}}$ are the Moebius transformations. The Blaschke products are obtained by multiplying particular Moebius transformations. They are no longer one-to-one mappings. However, some of these products can be projected on $P^2$ and they become dianalytic self-mappings of $P^2$. More exactly, they represent canonical projections of non orientable branched covering Klein surfaces over $P^2$. This article is devoted to color visualization of such mappings. The working tool is the technique of simultaneous continuation we introduced in previous papers."}
{"category": "Math", "title": "First cohomology groups of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group", "abstract": "We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization $V$ of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Johnson homomorphism does not extend to the automorphism group of a free group as a crossed homomorphism for the rank of the free group is greater than 4."}
{"category": "Math", "title": "A unifying formulation of the Fokker-Planck-Kolmogorov equation for general stochastic hybrid systems (extended version)", "abstract": "This paper has been withdrawn from the arXiv. It is now published by Elsevier in Nonlinear Analysis: Hybrid Systems, see http://dx.doi.org/10.1016/j.nahs.2009.07.008 . A general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and the notion of probability current (in the case of forced jumps). This work unifies all previously known instances of the FPK equation for stochastic hybrid systems, and provides GSHS practitioners with a tool to derive the correct evolution equation for the probability law of the state in any given example."}
{"category": "Math", "title": "On the categorical meaning of Hausdorff and Gromov distances, I", "abstract": "Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every V-category X, provides the powerset of X with a suitable V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras are order-complete. The Gromov construction may be pursued for any endofunctor K of V-Cat. In order to define the Gromov \"distance\" between V-categories X and Y we use V-modules between X and Y, rather than V-category structures on the disjoint union of X and Y. Hence, we first provide a general extension theorem which, for any K, yields a lax extension K to the category V-Mod of V-categories, with V-modules as morphisms."}
{"category": "Math", "title": "A Mahler measure of a K3-hypersurface expressed as a Dirichlet L-series", "abstract": "We present another example of a 3-variable polynomial defining a K3-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet L-series."}
{"category": "Math", "title": "Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence", "abstract": "We construct a mutually catalytic branching process on a countable site space with infinite \"branching rate\". The finite rate mutually catalytic model, in which the rate of branching of one population at a site is proportional to the mass of the other population at that site, was introduced by Dawson and Perkins in [DP98]. We show that our model is the limit for a class of models and in particular for the Dawson-Perkins model as the rate of branching goes to infinity. Our process is characterized as the unique solution to a martingale problem. We also give a characterization of the process as a weak solution of an infinite system of stochastic integral equations driven by a Poisson noise."}
{"category": "Math", "title": "Symmetries of Parabolic Geometries", "abstract": "We generalize the concept of affine locally symmetric spaces for parabolic geometries. We discuss mainly $|1|$--graded geometries and we show some restrictions on their curvature coming from the existence of symmetries. We use the theory of Weyl structures to discuss more interesting $|1|$--graded geometries which can carry a symmetry in a point with nonzero curvature. More concretely, we discuss the number of different symmetries which can exist at the point with nonzero curvature."}
{"category": "Math", "title": "A new Garside structure for braid groups of type $(e,e,r)$", "abstract": "We describe a new presentation for the complex reflection groups of type $(e,e,r)$ and their braid groups. A diagram for this presentation is proposed. The presentation is a monoid presentation which is shown to give rise to a Garside structure. A detailed study of the combinatorics of this structure leads us to describe it as post-classical."}
{"category": "Math", "title": "A short course on multiplier ideals", "abstract": "These notes are the write-up of my 2008 PCMI lectures on multiplier ideals. They aim to give an introduction to the algebro-geometric side of the theory, with an emphasis on its global aspects. The focus is on concrete examples and applications. The lectures take into account a number of recent perspectives, including adjoint ideals and the resulting simplifications in Siu's theorem on plurigenera in the general type case. While the notes refer to my book [PAG] and other sources for some technical points, the conscientious reader should arrive at a reasonable grasp of the machinery after working through these lectures."}
{"category": "Math", "title": "Spaces admitting homogeneous G2-structures", "abstract": "We classify all seven-dimensional spaces which admit a homogeneous cosymplectic G2-structure. The motivation for this classification is that each of these spaces is a possible principal orbit of a parallel Spin(7)-manifold of cohomogeneity one."}
{"category": "Math", "title": "Exponential bounds for minimum contrast estimators", "abstract": "The paper focuses on general properties of parametric minimum contrast estimators. The quality of estimation is measured in terms of the rate function related to the contrast, thus allowing to derive exponential risk bounds invariant with respect to the detailed probabilistic structure of the model. This approach works well for small or moderate samples and covers the case of a misspecified parametric model. Another important feature of the presented bounds is that they may be used in the case when the parametric set is unbounded and non-compact. These bounds do not rely on the entropy or covering numbers and can be easily computed. The most important statistical fact resulting from the exponential bonds is a concentration inequality which claims that minimum contrast estimators concentrate with a large probability on the level set of the rate function. In typical situations, every such set is a root-n neighborhood of the parameter of interest. We also show that the obtained bounds can help for bounding the estimation risk, constructing confidence sets for the underlying parameters. Our general results are illustrated for the case of an i.i.d. sample. We also consider several popular examples including least absolute deviation estimation and the problem of estimating the location of a change point. What we obtain in these examples slightly differs from the usual asymptotic results presented in statistical literature. This difference is due to the unboundness of the parameter set and a possible model misspecification."}
{"category": "Math", "title": "Higher Apery-like numbers arising from special values of the spectral zeta function for the non-commutative harmonic oscillator", "abstract": "A generalization of the Apery-like numbers, which is used to describe the special values $\\zeta_Q(2)$ and $\\zeta_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a recurrence relation for them, which shows a ladder structure among them. Further, we consider the `rational part' of the higher Apery-like numbers. We discuss several kinds of congruence relations among them, which are regarded as an analogue of the ones among Apery numbers."}
{"category": "Math", "title": "The standard filtration on cohomology with compact supports with an appendix on the base change map and the Lefschetz hyperplane theorem", "abstract": "We describe the standard and Leray filtrations on the cohomology groups with compact supports of a quasi projective variety with coefficients in a constructible complex using flags of hyperplane sections on a partial compactification of a related variety. One of the key ingredients of the proof is the Lefschetz hyperplane theorem for perverse sheaves and, in an appendix, we discuss the base change maps for constructible sheaves on algebraic varieties and their role in a proof, due to Beilinson, of the Lefschetz hyperplane theorem."}
{"category": "Math", "title": "The growth of the infinite long-range percolation cluster", "abstract": "We consider long-range percolation on $\\mathbb{Z}^d$, where the probability that two vertices at distance $r$ are connected by an edge is given by $p(r)=1-\\exp[-\\lambda(r)]\\in(0,1)$ and the presence or absence of different edges are independent. Here, $\\lambda(r)$ is a strictly positive, nonincreasing, regularly varying function. We investigate the asymptotic growth of the size of the $k$-ball around the origin, $|\\mathcal{B}_k|$, that is, the number of vertices that are within graph-distance $k$ of the origin, for $k\\to\\infty$, for different $\\lambda(r)$. We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying $\\lambda(r)$ exist, for which, respectively: $\\bullet$ $|\\mathcal{B}_k|^{1/k}\\to\\infty$ almost surely; $\\bullet$ there exist $1<a_1<a_2<\\infty$ such that $\\lim_{k\\to \\infty}\\mathbb{P}(a_1<|\\mathcal{B}_k|^{1/k}<a_2)=1$; $\\bullet$ $|\\mathcal{B}_k|^{1/k}\\to1$ almost surely. This result can be applied to spatial SIR epidemics. In particular, regimes are identified for which the basic reproduction number, $R_0$, which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics."}
{"category": "Math", "title": "Counting conjugacy classes in the unipotent radical of parabolic subgroups of $\\GL_n(q)$", "abstract": "Let $q$ be a power of a prime $p$. Let $P$ be a parabolic subgroup of the general linear group $\\GL_n(q)$ that is the stabilizer of a flag in $\\FF_q^n$ of length at most 5, and let $U = O_p(P)$. In this note we prove that, as a function of $q$, the number $k(U)$ of conjugacy classes of $U$ is a polynomial in $q$ with integer coefficients."}
{"category": "Math", "title": "A survey of Measured Group Theory", "abstract": "The title refers to the area of research which studies infinite groups using measure-theoretic tools, and studies the restrictions that group structure imposes on ergodic theory of their actions. The paper is a survey of recent developments focused on the notion of Measure Equivalence between groups, and Orbit Equivalence between group actions. We discuss known invariants and classification results (rigidity) in both areas."}
{"category": "Math", "title": "Weak$^*$ dentability index of spaces $C([0,\\alpha])$", "abstract": "We compute the weak$^*$-dentability index of the spaces $C(K)$ where $K$ is a countable compact space. Namely ${Dz}(C([0,\\omega^{\\omega^\\alpha}])) = \\omega^{1+\\alpha+1}$, whenever $0\\le\\alpha<\\omega_1$. More generally, ${Dz}(C(K))=\\omega^{1+\\alpha+1}$ if $K$ is a scattered compact whose height $\\eta(K)$ satisfies $\\omega^\\alpha<\\eta(K)\\leq \\omega^{\\alpha+1}$ with an $\\alpha$ countable."}
{"category": "Math", "title": "Optimisation du th\\'eor\\`eme d'Ax-Sen-Tate et application \\`a un calcul de cohomologie galoisienne p-adique", "abstract": "Let p a prime number, Q_p the field of p-adic numbers, K a finite extension of Q_p, \\bar{K} an algebraic closure, and C_p the completion of Q_p, on which the valuation on Q_p extends. In his proof of the Ax-Sen-Tate theorem, Ax shows that if x in C_p satisfies v(sx - x) > A for all s in the absolute Galois group of K G, then there is a y in K such that v(x-y) >= A - C, with the constant C = p/(p-1)^2. Ax questions the optimality of this constant, which we study here. Introducing the extension of K by p^n-th roots of the uniformizer and relying on Tate's and Colmez's works, we find the optimal constant and some more information about elements in C_p satisfying v(sx - x) >= A for all s in G, we compute the first cohomology group of G with coefficients in the ring of integers of \\bar{K}."}
{"category": "Math", "title": "Age and Winning Professional Golf Tournaments", "abstract": "Most professional golfers and analysts think that winning on the PGA Tour peaks when golfers are in their thirties. Rather than relying on educated guesses, we can actually use available statistical data to determine the actual ages at which golfers peak their golf game. We can also test the hypothesis that age affects winning professional golf tournaments. Using data available from the Golf Channel, the PGA Tour, and LPGA Tour, I calculated and provided the mean, the median, and the mode ages at which professional golfers on the PGA, European PGA, Champions, and LPGA Tours had won over a five-year period. More specifically, the ages at which golfers on the PGA, European PGA, Champions Tour, and LPGA Tours peak their wins are 35, 30, 52, and 25, respectively. The regression analyses I conducted seem to support my hypothesis that age affects winning professional golf tournaments."}
{"category": "Math", "title": "Divisor class groups of graded hypersurfaces", "abstract": "We demonstrate how some classical computations of divisor class groups can be obtained using the theory of rational coefficient Weil divisors and related results of Watanabe."}
{"category": "Math", "title": "Multigraded rings, diagonal subalgebras, and rational singularities", "abstract": "We study the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise naturally as homogeneous coordinate rings of blow-ups of projective varieties. As a consequence of some of the results obtained here, it is shown that there exist standard bigraded hypersurfaces whose rings of invariants under torus actions have rational singularities, but are not of F-regular type. Another application is the construction of families of rings with divisor class groups that are finitely generated, but not discrete in the sense of Danilov."}
{"category": "Math", "title": "Bockstein homomorphisms in local cohomology", "abstract": "Let $R$ be a polynomial ring in finitely many variables over the integers, and fix an ideal $I$ of $R$. We prove that for all but finitely prime integers $p$, the Bockstein homomorphisms on local cohomology, $H^k_I(R/pR)\\to H^{k+1}_I(R/pR)$, are zero. This provides strong evidence for Lyubeznik's conjecture which states that the modules $H^k_I(R)$ have a finite number of associated prime ideals."}
{"category": "Math", "title": "Castelnuovo-Mumford regularity of deficiency modules", "abstract": "Let $d \\in \\N$ and let $M$ be a finitely generated graded module of dimension $\\leq d$ over a Noetherian homogeneous ring $R$ with local Artinian base ring $R_0$. Let $\\beg(M)$, $\\gendeg(M)$ and $\\reg(M)$ respectively denote the beginning, the generating degree and the Castelnuovo-Mumford regularity of $M$. If $i \\in \\N_0$ and $n \\in Z$, let $d^i_M(n)$ denote the $R_0$-length of the $n$-th graded component of the $i$-th $R_+$-transform module $D^i_{R_+}(M)$ of $M$ and let $K^i(M)$ denote the $i$-th deficiency module of $M$. Our main result says, that $\\reg(K^i(M))$ is bounded in terms of $\\beg(M)$ and the \"diagonal values\" $d^j_M(-j)$ with $j = 0,..., d-1$. As an application of this we get a number of further bounding results for $\\reg(K^i(M))$."}
{"category": "Math", "title": "Strict p-negative type of a metric space", "abstract": "Doust and Weston introduced a new method called \"enhanced negative type\" for calculating a non trivial lower bound p(T) on the supremal strict p-negative type of any given finite metric tree (T,d). In the context of finite metric trees any such lower bound p(T) > 1 is deemed to be non trivial. In this paper we refine the technique of enhanced negative type and show how it may be applied more generally to any finite metric space (X,d) that is known to have strict p-negative type for some non negative p. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given by Doust and Weston and, moreover, leads in to one of our main results: The supremal p-negative type of a finite metric space cannot be strict. By way of application we are then able to exhibit large classes of finite metric spaces (such as finite isometric subspaces of Hadamard manifolds) that must have strict p-negative type for some p > 1. We also show that if a metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q in [0,p). This generalizes a well known theorem of Schoenberg and leads to a complete classification of the intervals on which a metric space may have strict p-negative type. (Several of the results in this paper hold more generally for semi-metric spaces.)"}
{"category": "Math", "title": "Isomorphism and Symmetries in Random Phylogenetic Trees", "abstract": "The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations."}
{"category": "Math", "title": "Poincare problem for divisors invariant by one-dimensional foliations on smooth algebraic variety", "abstract": "In this paper we consider the question of bounding the degree of an divisor $D$ invariant by a $\\F$ holomorphic foliation, without rational first integral, on smooth algebraic variety $X$ in terms of degree of $\\F$ and some invariants of $D$ and $X$. Particularly, if $\\F$ is a foliation of degree $d$ on $\\mathbb{P}_{\\mathbb{C}}^2$, whose the number of invariants curves is greater that ${k+2\\choose k}$, we show that there exist a number $\\mathcal{M}(d,k)$ such that if $k>\\mathcal{M}(d,k),$ then $\\F$ admits a rational first integral of degree $\\leq k$. Moreover, there exist a number $\\mathscr{G}(d,k)$, such that if $\\F$ has an algebraic solution of degree $k$ and genus smaller than $\\mathscr{G}(d,k)$, then it has a rational first integral of degree $\\leq k$."}
{"category": "Math", "title": "Train track complex of once-punctured torus and 4-punctured sphere", "abstract": "Consider a compact oriented surface $S$ of genus $g \\geq 0$ and $m \\geq 0$ punctured. The train track complex of $S$ which is defined by Hamenst\\\"adt is a 1-complex whose vertices are isotopy classes of complete train tracks on $S$. Hamenst\\\"adt shows that if $3g-3+m \\geq 2$, the mapping class group acts properly discontinuously and cocompactly on the train track complex. We will prove corresponding results for the excluded case, namely when $S$ is a once-punctured torus or a 4-punctured sphere. To work this out, we redefinition of two complexes for these surfaces."}
{"category": "Math", "title": "Efficient estimation of copula-based semiparametric Markov models", "abstract": "This paper considers the efficient estimation of copula-based semiparametric strictly stationary Markov models. These models are characterized by nonparametric invariant (one-dimensional marginal) distributions and parametric bivariate copula functions where the copulas capture temporal dependence and tail dependence of the processes. The Markov processes generated via tail dependent copulas may look highly persistent and are useful for financial and economic applications. We first show that Markov processes generated via Clayton, Gumbel and Student's $t$ copulas and their survival copulas are all geometrically ergodic. We then propose a sieve maximum likelihood estimation (MLE) for the copula parameter, the invariant distribution and the conditional quantiles. We show that the sieve MLEs of any smooth functional is root-$n$ consistent, asymptotically normal and efficient and that their sieve likelihood ratio statistics are asymptotically chi-square distributed. Monte Carlo studies indicate that, even for Markov models generated via tail dependent copulas and fat-tailed marginals, our sieve MLEs perform very well."}
{"category": "Math", "title": "Almost invariant half-spaces of operators on Banach spaces", "abstract": "We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on a Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite codimension in T(Y). We solve this problem in the affirmative for a large class of operators which includes quasinilpotent weighted shift operators on l_p (1 \\le p < \\infty) or c_0."}
{"category": "Math", "title": "Circular edge-colorings of cubic graphs with girth six", "abstract": "We show that the circular chromatic index of a (sub)cubic graph with girth at least six is at most 7/2."}
{"category": "Math", "title": "Discrete Compactness for p-Version of Tetrahedral Edge Elements", "abstract": "We consider the first family of $\\Hcurl$-conforming Ned\\'el\\'ec finite elements on tetrahedral meshes. Spectral approximation ($p$-version) is achieved by keeping the mesh fixed and raising the polynomial degree $p$ uniformly in all mesh cells. We prove that the associated subspaces of discretely weakly divergence free piecewise polynomial vector fields enjoy a long conjectured discrete compactness property as $p\\to\\infty$. This permits us to conclude asymptotic spectral correctness of spectral Galerkin finite element approximations of Maxwell eigenvalue problems."}
{"category": "Math", "title": "Estimators for Long Range Dependence: An Empirical Study", "abstract": "We present the results of a simulation study into the properties of 12 different estimators of the Hurst parameter, $H$, or the fractional integration parameter, $d$, in long memory time series. We compare and contrast their performance on simulated Fractional Gaussian Noises and fractionally integrated series with lengths between 100 and 10,000 data points and $H$ values between 0.55 and 0.90 or $d$ values between 0.05 and 0.40. We apply all 12 estimators to the Campito Mountain data and estimate the accuracy of their estimates using the Beran goodness of fit test for long memory time series. MCS code: 37M10"}
{"category": "Math", "title": "Local Multigrid in H(curl)", "abstract": "We consider H(curl)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence theory for the the so-called local multigrid correction scheme with hybrid smoothing. We establish that its convergence rate is uniform with respect to the number of refinement steps. The proof relies on corresponding results for local multigrid in a H1-context along with local discrete Helmholtz-type decompositions of the edge element space."}
{"category": "Math", "title": "Remarks on Grassmannian Symmetric Spaces", "abstract": "The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for $|1|$--graded parabolic geometries and for almost Grassmannian structures, in particular. As an application of two general constructions with parabolic geometries, we present an example of non--flat Grassmannian symmetric space. Next we observe there is a distinguished torsion--free affine connection preserving the Grassmannian structure so that, with respect to this connection, the Grassmannian symmetric space is an affine symmetric space in the classical sense."}
{"category": "Math", "title": "Hausdorff leaf spaces for codim-1 foliations", "abstract": "The topology of the Hausdorff leaf spaces (HLS) for a codim-1 foliation is the main topic of this paper. At the beginning, the connection between the Hausdorff leaf space and a warped foliations is examined. Next, the author describes the HLS for all basic constructions of foliations such as transversal and tangential gluing, spinning, turbulization, and suspension. Finally, it is shown that the HLS for any codim-1 foliation on a compact Riemannian manifold is isometric to a finite connected metric graph. In addition, the author proves that for any finite connected metric graph G there exists a compact foliated Riemannian manifold (M,F,g) with codim-1 foliation such that the Hausdorff leaf space for F is isometric to G. Finally, the necessary and sufficient condition for warped foliations of codim-1 to converge to HLS(F) is given."}
{"category": "Math", "title": "Multiplicity-free homogeneous operators in the Cowen-Douglas class", "abstract": "In a recent paper, the authors have constructed a large class of operators in the Cowen-Douglas class Cowen-Douglas class of the unit disc $\\mathbb D$ which are {\\em homogeneous} with respect to the action of the group M\\\"{o}b -- the M\\\"{o}bius group consisting of bi-holomorphic automorphisms of the unit disc $\\mathbb{D}$. The {\\em associated representation} for each of these operators is {\\em multiplicity free}. Here we give a different independent construction of all homogeneous operators in the Cowen-Douglas class with multiplicity free associated representation and verify that they are exactly the examples constructed previously."}
{"category": "Math", "title": "Skew Meadows", "abstract": "A skew meadow is a non-commutative ring with an inverse operator satisfying two special equations and in which the inverse of zero is zero. All skew fields and products of skew fields can be viewed as skew meadows. Conversely, we give an embedding of non-trivial skew meadows into products of skew fields, from which a completeness result for the equational logic of skew fields is derived. The relationship between regularity conditions on rings and skew meadows is investigated."}
{"category": "Math", "title": "Strongly p-embedded subgroups", "abstract": "We study finite groups which possess a strongly p-embedded subgroup for some odd prime p. The main results of the paper will be applied in the ongoing project to classify the simple groups of local characteristic p."}
{"category": "Math", "title": "Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations", "abstract": "We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omn\\`es \\cite{DomOmnes}) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and \"hyperbolic\" $L^\\infty$ weak-$\\star$ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, ...). Our results cover the case of non-Lipschitz nonlinearities."}
{"category": "Math", "title": "Zeros of Meixner and Krawtchouk polynomials", "abstract": "We investigate the zeros of a family of hypergeometric polynomials $_2F_1(-n,-x;a;t)$, $n\\in\\nn$ that are known as the Meixner polynomials for certain values of the parameters $a$ and $t$. When $a=-N$, $N\\in\\nn$ and $t=\\frac1{p}$, the polynomials $K_n(x;p,N)=(-N)_n\\phantom{}_2F_1(-n,-x;-N;\\frac1{p})$, $n=0,1,...N$, $0<p<1$ are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polynomials $K_{n}(x;p,a)$, $0<p<1$ and $a>n-1$, the quasi-orthogonal polynomials $K_{n}(x;p,a)$, $k-1<a<k$, $k=1,...,n-1$ and $p>1$ or $p<0$ as well as the non-orthogonal polynomials $K_{n}(x;p,N)$, $0<p<1$ and $n=N+1,N+2,...$. We also show that the polynomials $K_{n}(x;p,a)$, $a\\in \\rr$ are real-rooted when $p\\rightarrow 0$ We use a generalised Sturmian sequence argument and the discrete orthogonality of the Krawtchouk polynomials for certain parameter values to prove that all the zeros of Meixner polynomials are real and positive for parameter ranges where they are no longer orthogonal."}
{"category": "Math", "title": "On the construction and topological invariance of the Pontryagin classes", "abstract": "We use sheaves and algebraic L-theory to construct the rational Pontryagin classes of fiber bundles with fiber R^n. This amounts to an alternative proof of Novikov's theorem on the topological invariance of the rational Pontryagin classes of vector bundles. Transversality arguments and torus tricks are avoided."}
{"category": "Math", "title": "Meadows and the equational specification of division", "abstract": "The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of equational specifications of data types cannot use field theory in applications to number systems based upon rational, real and complex numbers. We study a new axiomatic concept for number systems with division that uses only equations: a meadow is a commutative ring with a total inverse operator satisfying two equations which imply that the inverse of zero is zero. All fields and products of fields can be viewed as meadows. After reviewing alternate axioms for inverse, we start the development of a theory of meadows. We give a general representation theorem for meadows and find, as a corollary, that the conditional equational theory of meadows coincides with the conditional equational theory of zero totalized fields. We also prove representation results for meadows of finite characteristic."}
{"category": "Math", "title": "Introduction to representation theory", "abstract": "These are lecture notes that arose from a representation theory course given by the first author to the remaining six authors in March 2004 within the framework of the Clay Mathematics Institute Research Academy for high school students, and its extended version given by the first author to MIT undergraduate math students in the Fall of 2008. The notes cover a number of standard topics in representation theory of groups, Lie algebras, and quivers, and contain many problems and exercises. They should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra, and may be used for an undergraduate or introductory graduate course in representation theory."}
{"category": "Math", "title": "Simplicity of a vertex operator algebra whose Griess algebra is the Jordan algebra of symmetric matrices", "abstract": "Let $r \\in \\BC$ be a complex number, and $d \\in \\BZ_{\\ge 2}$ a positive integer greater than or equal to 2. Ashihara and Miyamoto introduced a vertex operator algebra $\\Vam$ of central charge $dr$, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size $d$. In this paper, we prove that the vertex operator algebra $\\Vam$ is simple if and only if $r$ is not an integer. Further, in the case that $r$ is an integer (i.e., $\\Vam$ is not simple), we give a generator system of the maximal proper ideal $I_{r}$ of the VOA $\\Vam$ explicitly."}
{"category": "Math", "title": "A simplicial $A_\\infty$-operad acting on $R$-resolutions", "abstract": "We construct a combinatorial model of an A-infinity-operad which acts simplicially on the cobar resolution (not just its total space) of a simplicial set with respect to a ring R."}
{"category": "Math", "title": "On positive solutions of p-Laplacian-type equations", "abstract": "Let $\\Omega$ be a domain in $\\mathbb{R}^d$, $d\\geq 2$, and $1<p<\\infty$. Fix $V\\in L_{\\mathrm{loc}}^\\infty(\\Omega)$. Consider the functional $Q$ and its G\\^{a}teaux derivative $Q^\\prime$ given by $$Q(u):= \\frac{1}{p}\\int_\\Omega. (|\\nabla u|^p+V|u|^p) \\dx, Q^\\prime (u):= -\\nabla\\cdot(|\\nabla u|^{p-2}\\nabla u)+V|u|^{p-2} u.$$ In this paper we discuss a few aspects of relations between functional-analytic properties of the functional $Q$ and properties of positive solutions of the equation $Q^\\prime (u)=0$."}
{"category": "Math", "title": "Good Frames With A Weak Stability", "abstract": "Let K be an abstract elementary class of models. Assume that there are less than the maximal number of models in K_{\\lambda^{+n}} (namely models in K of power \\lambda^{+n}) for all n. We provide conditions on K_\\lambda, that imply the existence of a model in K_{\\lambda^{+n}} for all n. We do this by providing sufficiently strong conditions on K_\\lambda, that they are inherited by a properly chosen subclass of K_{\\lambda^+}."}
{"category": "Math", "title": "On Non-Separating Contact Hypersurfaces in Symplectic 4-Manifolds", "abstract": "We show that certain classes of contact 3-manifolds do not admit non-separating contact type embeddings into any closed symplectic 4-manifolds, e.g. this is the case for all contact manifolds that are (partially) planar or have Giroux torsion. The latter implies that manifolds with Giroux torsion do not admit contact type embeddings into any closed symplectic 4-manifolds. Similarly, there are symplectic 4-manifolds that can admit smoothly embedded non-separating hypersurfaces, but not of contact type: we observe that this is the case for all symplectic ruled surfaces."}
{"category": "Math", "title": "Bari-Markus property for Riesz projections of 1D periodic Dirac operators", "abstract": "The Dirac operators $$ Ly = i 1 & 0 0 & -1 \\frac{dy}{dx} + v(x) y, \\quad y = y_1 y_2, \\quad x\\in[0,\\pi],$$ with $L^2$-potentials $$ v(x) = 0 & P(x) Q(x) & 0, \\quad P,Q \\in L^2 ([0,\\pi]), $$ considered on $[0,\\pi]$ with periodic, antiperiodic or Dirichlet boundary conditions $(bc)$, have discrete spectra, and the Riesz projections $$ S_N = \\frac{1}{2\\pi i} \\int_{|z|= N-{1/2}} (z-L_{bc})^{-1} dz, \\quad P_n = \\frac{1}{2\\pi i} \\int_{|z-n|= {1/4}} (z-L_{bc})^{-1} dz $$ are well--defined for $|n| \\geq N$ if $N $ is sufficiently large. It is proved that $$\\sum_{|n| > N} \\|P_n - P_n^0\\|^2 < \\infty, $$ where $P_n^0, n \\in \\mathbb{Z},$ are the Riesz projections of the free operator. Then, by the Bari--Markus criterion, the spectral Riesz decompositions $$ f = S_N f + \\sum_{|n| >N} P_n f, \\quad \\forall f \\in L^2; $$ converge unconditionally in $L^2.$"}
{"category": "Math", "title": "An algorithm for computing the integral closure", "abstract": "We present an algorithm for computing the integral closure of a reduced ring that is finitely generated over a finite field."}
{"category": "Math", "title": "A Fast Algorithm for Robust Regression with Penalised Trimmed Squares", "abstract": "The presence of groups containing high leverage outliers makes linear regression a difficult problem due to the masking effect. The available high breakdown estimators based on Least Trimmed Squares often do not succeed in detecting masked high leverage outliers in finite samples. An alternative to the LTS estimator, called Penalised Trimmed Squares (PTS) estimator, was introduced by the authors in \\cite{ZiouAv:05,ZiAvPi:07} and it appears to be less sensitive to the masking problem. This estimator is defined by a Quadratic Mixed Integer Programming (QMIP) problem, where in the objective function a penalty cost for each observation is included which serves as an upper bound on the residual error for any feasible regression line. Since the PTS does not require presetting the number of outliers to delete from the data set, it has better efficiency with respect to other estimators. However, due to the high computational complexity of the resulting QMIP problem, exact solutions for moderately large regression problems is infeasible. In this paper we further establish the theoretical properties of the PTS estimator, such as high breakdown and efficiency, and propose an approximate algorithm called Fast-PTS to compute the PTS estimator for large data sets efficiently. Extensive computational experiments on sets of benchmark instances with varying degrees of outlier contamination, indicate that the proposed algorithm performs well in identifying groups of high leverage outliers in reasonable computational time."}
{"category": "Math", "title": "Topological complexity of collision-free motion planning on surfaces", "abstract": "The topological complexity TC(X) is a numerical homotopy invariant of a topological space X which is motivated by robotics and is similar in spirit to the classical Lusternik-Schnirelmann category of X. Given a mechanical system with configuration space X, the invariant TC(X) measures the complexity of all possible motion planning algorithms designed for the system. In this paper, we compute the topological complexity of the configuration space of n distinct ordered points on an orientable surface. Our main tool is a theorem of B. Totaro describing the cohomology of configuration spaces of algebraic varieties."}
{"category": "Math", "title": "A new characterization of Conrad's property for group orderings, with applications", "abstract": "We provide a pure algebraic version of the dynamical characterization of Conrad's property. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof of a theorem first established by Linnell: an orderable group having infinitely many orderings has uncountably many. This proof is achieved by extending to uncountable orderable groups a result about orderings which may be approximated by their conjugates. This last result is illustrated by an example of an exotic ordering on the free group given in the Appendix."}
{"category": "Math", "title": "On the representability of totally unimodular matrices on bidirected graphs", "abstract": "Seymour's famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called $k$-sums starting from network matrices and their transposes and two compact representation matrices $B_{1}, B_{2}$ of a certain ten element matroid. Given that $B_{1}, B_{2}$ are binet matrices we examine the $k$-sums of network and binet matrices. It is shown that the $k$-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for $k=2,3$. A new class of matrices is introduced the so called {\\em tour matrices}, which generalises network, binet and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under $k$-sums, as well as under pivoting and other elementary operations on its rows and columns. Given the constructive proofs of the above results regarding the $k$-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix."}
{"category": "Math", "title": "Non-vanishing complex vector fields and the Euler characteristic", "abstract": "The existence of a nowhere zero real vector field implies a well-known restriction on a compact manifold. But all manifolds admit nowhere zero complex vector fields. The relation between these observations is clarified."}
{"category": "Math", "title": "Asymptotically periodic L^2 minimizers in strongly segregating diblock copolymers", "abstract": "Using the delta correction to the standard free energy \\cite{bc} in the elastic setting with a quadratic foundation term and some parameters, we introduce a one dimension only model for strong segregation in diblock copolymers, whose sharp interface periodic microstructure is consistent with experiment in low temperatures. The Green's function pattern forming nonlocality is the same as in the Ohta-Kawasaki model. Thus we complete the statement in [31,p.349]: ``The detailed analysis of this model will be given elsewhere. Our preliminary results indicate that the new model exhibits periodic minimizers with sharp interfaces.'' We stress that the result is unexpected, as the functional is not well posed, moreover the instabilities in $L^2$ typically occur only along continuous nondifferentiable ``hairs''. We also improve the derivation done by van der Waals and use it and the above to show the existence of a phase transition with Maxwell's equal area rule. However, this model does not predict the universal critical surface tension exponent, conjectured to be 11/9. Actually, the range $(1.2,1.36)$ has been reported in experiments [21,p. 360]. By simply taking a constant kernel, this exponent is 2. This is the experimentally ($ \\pm 0.1$) verified tricritical exponent, found e.g., at the consolute $0.9$ K point in mixtures of ${}^3$He and ${}^4$He. Thus there is a third unseen phase at the phase transition point."}
{"category": "Math", "title": "The fundamental lemma of Jacquet-Rallis in positive characteristics", "abstract": "We prove both the group version and the Lie algebra version of the Fundamental Lemma appearing in a relative trace formula of Jacquet-Rallis in the function field case when the characteristic is greater than the rank of the relevant groups."}
{"category": "Math", "title": "Phantom Probability", "abstract": "Classical probability theory supports probability measures, assigning a fixed positive real value to each event, these measures are far from satisfactory in formulating real-life occurrences. The main innovation of this paper is the introduction of a new probability measure, enabling varying probabilities that are recorded by ring elements to be assigned to events; this measure still provides a Bayesian model, resembling the classical probability model. By introducing two principles for the possible variation of a probability (also known as uncertainty, ambiguity, or imprecise probability), together with the \"correct\" algebraic structure allowing the framing of these principles, we present the foundations for the theory of phantom probability, generalizing classical probability theory in a natural way. This generalization preserves many of the well-known properties, as well as familiar distribution functions, of classical probability theory: moments, covariance, moment generating functions, the law of large numbers, and the central limit theorem are just a few of the instances demonstrating the concept of phantom probability theory."}
{"category": "Math", "title": "What Hilbert spaces can tell us about bounded functions on the bidisk", "abstract": "We discuss various theorems about bounded analytic functions on the bidisk that were proved using operator theory."}
{"category": "Math", "title": "Cyclic vectors of self-adjoint operators in Hilbert space", "abstract": "A criterion and sufficient conditions for a vector to be a cyclic vector for a class of self-adjoint operators are obtained."}
{"category": "Math", "title": "Finitely forcible graphons", "abstract": "We investigate families of graphs and graphons (graph limits) that are defined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities. Generalizing results of Turan, Erdos-Simonovits and Chung-Graham-Wilson, we construct numerous finitely forcible graphons. Most of these fall into two categories: one type has an algebraic structure and the other type has an iterated (fractal-like) structure. We also give some necessary conditions for forcibility, which imply that finitely forcible graphons are \"rare\", and exhibit simple and explicit non-forcible graphons."}
{"category": "Math", "title": "Convergence and divergence of averages along subsequences in certain Orlicz spaces", "abstract": "The classical theorem of Birkhoff states that the $T^N f(x) = (1/N)\\sum_{k=0}^{N-1} f(\\sigma^k x)$ converges almost everywhere for $x\\in X$ and $f\\in L^{1}(X)$, where $\\sigma$ is a measure preserving transformation of a probability measure space $X$. It was shown that there are operators of the form $T^N f(x)=(1/N)\\sum_{k=0}^{N-1}f(\\sigma^{n_k}x)$ for a subsequence $\\{n_k\\}$ of the positive integers that converge in some $L^p$ spaces while diverging in others. The topic of this talk will examine this phenomenon in the class of Orlicz spaces $\\{L{Log}^\\beta L:\\beta>0\\}$."}
{"category": "Math", "title": "Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlev\\'e equations", "abstract": "We consider orthogonal polynomials on the unit circle with respect to a weight which is a quotient of $q$-gamma functions. We show that the Verblunsky coefficients of these polynomials satisfy discrete Painlev\\'e equations, in a Lax form, which correspond to an $A_3^{(1)}$ surface in Sakai's classification."}
{"category": "Math", "title": "Partial Hopf actions, partial invariants and a Morita context", "abstract": "Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to assure the existence of enveloping actions. This allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this article, we explore some properties of the fixed point subalgebra with relations to a partial action of a Hopf algebra. We also construct, for partial actions of finite dimensional Hopf algebras a Morita context relating the fixed point subalgebra and the partial smash product. This is a generalization of a well known result in the theory of Hopf algebras for the case of partial actions. Finally, we study Hopf-Galois extensions and reobtain some classical results in the partial case."}
{"category": "Math", "title": "On Two Results of Mixed Multiplicities", "abstract": "This paper shows that the main result of Trung-Verma in 2007 [TV] only is an immediate consequence of an improvement version of [Theorem 3.4, Vi1] in 2000."}
{"category": "Math", "title": "Midwest cousins of Barnes-Wall lattices", "abstract": "Given a rational lattice and suitable set of linear transformations, we construct a cousin lattice. Sufficient conditions are given for integrality, evenness and unimodularity. When the input is a Barnes-Wall lattice, we get multi-parameter series of cousins. There is a subseries consisting of unimodular lattices which have ranks $2^{d-1}\\pm 2^{d-k-1}$, for odd integers $d\\ge 3$ and integers $k=1,2, ..., \\frac {d-1}2$. Their minimum norms are moderately high: $2^{\\lfloor \\frac d2 \\rfloor -1}$."}
{"category": "Math", "title": "The mathematics of Donald Gordon Higman", "abstract": "This is about the mathematics and life of Donald Gordon Higman, 1928-2006. He did important work in representation theory of groups and algebras and in algebraic combinatorics. Charles C. Sims and Donald Higman discovered and constructed one of the sporadic simple groups."}
{"category": "Math", "title": "A discrete contact model for crowd motion", "abstract": "The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: We first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people; The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here the underlying mathematical framework, and we explain how recent results by J.F. Edmond and L. Thibault on the sweeping process by uniformly prox-regular sets can be adapted to handle this situation in terms of well-posedness. We propose a numerical scheme for this contact dynamics model, based on a prediction-correction algorithm. Numerical illustrations are finally presented and discussed."}
{"category": "Math", "title": "Uniform approximation of homeomorphisms by diffeomorphisms", "abstract": "We prove that a compactly supported homeomorphism of a smooth manifold of dimension greater or equal to 5 can be approximated uniformly by compactly supported diffeomorphisms if and only if it is isotopic to a diffeomorphism. If the given homeomorphism is in addition volume preserving, then it can be approximated uniformly by volume preserving diffeomorphisms."}
{"category": "Math", "title": "Reflexive representability and stable metrics", "abstract": "It is well-known that a topological group can be represented as a group of isometries of a reflexive Banach space if and only if its topology is induced by weakly almost periodic functions (see \\cite{Shtern:CompactSemitopologicalSemigroups}, \\cite{Megrelishvili:OperatorTopologies} and \\cite{Megrelishvili:TopologicalTransformations}). We show that for a metrisable group this is equivalent to the property that its metric is uniformly equivalent to a stable metric in the sense of Krivine and Maurey (see \\cite{Krivine-Maurey:EspacesDeBanachStables}). This result is used to give a partial negative answer to a problem of Megrelishvili."}
{"category": "Math", "title": "Isometry groups of non-positively curved spaces: discrete subgroups", "abstract": "We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterised amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries."}
{"category": "Math", "title": "Nodal Discontinuous Galerkin Methods on Graphics Processors", "abstract": "Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of accuracy without compromising simulation stability. Lately, another property of DG has been growing in importance: The majority of a DG operator is applied in an element-local way, with weak penalty-based element-to-element coupling. The resulting locality in memory access is one of the factors that enables DG to run on off-the-shelf, massively parallel graphics processors (GPUs). In addition, DG's high-order nature lets it require fewer data points per represented wavelength and hence fewer memory accesses, in exchange for higher arithmetic intensity. Both of these factors work significantly in favor of a GPU implementation of DG. Using a single US$400 Nvidia GTX 280 GPU, we accelerate a solver for Maxwell's equations on a general 3D unstructured grid by a factor of 40 to 60 relative to a serial computation on a current-generation CPU. In many cases, our algorithms exhibit full use of the device's available memory bandwidth. Example computations achieve and surpass 200 gigaflops/s of net application-level floating point work. In this article, we describe and derive the techniques used to reach this level of performance. In addition, we present comprehensive data on the accuracy and runtime behavior of the method."}
{"category": "Math", "title": "Tensor extension properties of C(K)-representations and applications to unconditionality", "abstract": "Let K be any compact set. The C^*-algebra C(K) is nuclear and any bounded homomorphism from C(K) into B(H), the algebra of all bounded operators on some Hilbert space H, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept of R-boundedness. Then we apply these results to operators with a uniformly bounded H^\\infty-calculus, as well as to unconditionality on L^p. We show that any unconditional basis on L^p `is' an unconditional basis on L^2 after an appropriate change of density."}
{"category": "Math", "title": "On a Capacity for Modular Spaces", "abstract": "The purpose of this article is to define a capacity on certain topological measure spaces $X$ with respect to certain function spaces $V$ consisting of measurable functions. In this general theory we will not fix the space $V$ but we emphasize that $V$ can be the classical Sobolev space $W^{1,p}(\\Omega)$, the classical Orlicz-Sobolev space $W^{1,\\Phi}(\\Omega)$, the Haj{\\l}asz-Sobolev space $M^{1,p}(\\Omega)$, the Musielak-Orlicz-Sobolev space (or generalized Orlicz-Sobolev space) and many other spaces. Of particular interest is the space $V:=\\tW^{1,p}(\\Omega)$ given as the closure of $W^{1,p}(\\Omega)\\cap C_c(\\overline\\Omega)$ in $W^{1,p}(\\Omega)$. In this case every function $u\\in V$ (a priori defined only on $\\Omega$) has a trace on the boundary $\\partial\\Omega$ which is unique up to a $\\Cap_{p,\\Omega}$-polar set."}
{"category": "Math", "title": "Einstein metrics with anisotropic boundary behaviour", "abstract": "We construct new examples of complete Einstein metrics on balls. At each point of the boundary at infinity, the metric is asymptotic to a homogeneous Einstein metric on a solvable group, which varies with the point at infinity."}
{"category": "Math", "title": "Doset Hibi rings with an application to invariant theory", "abstract": "We define the concept of a doset Hibi ring and a generalized doset Hibi ring which are subrings of a Hibi ring and are normal affine semigrouprings. We apply the theory of (generalized) doset Hibi rings to analyze the rings of absolute orthogonal invariants and absolute special orthogonal invariants and show that these rings are normal and Cohen-Macaulay and has rational singularities if the characteristic of the base field is zero and is F-rational otherwise. We also state criteria of Gorenstein property of these rings."}
{"category": "Math", "title": "Geometry of genus 9 Fano 4-fold", "abstract": "References to the works of Iliev-Ranestad and Kuznetsov added. ----- In a first part we detail the construction, on a general Fano 4-fold of genus 9, of a canonical set of four stable vector bundles of rank 2, and prove that they are rigid. Those results were already known by Iliev-Ranestad and Kuznetsov with different purposes. In a second part we show that its variety of lines is an hyperplane section of P1xP1xP1xP1. Then we compute the Chow ring of a general Fano 4-fold, which appears to have a rich structure in codimension 2. The 4-bundles gives embeddings in a Grassmannian G(2,6), and the link with the order one congruence discovered by E. Mezzeti and P de Poi is done. We will also describe in this part the normalization of the non quadraticaly normal variety they constructed, and also its variety of plane cubics and detail the zak duality in this case."}
{"category": "Math", "title": "Combinatorics and N-Koszul algebras", "abstract": "The numerical Hilbert series combinatorics and the comodule Hilbert series combinatorics are introduced, and some applications are presented, including the MacMahon Master Theorem."}
{"category": "Math", "title": "Taxon Size Distribution in a Time Homogeneous Birth and Death Process", "abstract": "The number of extant individuals within a lineage, as exemplified by counts of species numbers across genera in a higher taxonomic category, is known to be a highly skewed distribution. Because the sublineages (such as genera in a clade) themselves follow a random birth process, deriving the distribution of lineage sizes involves averaging the solutions to a birth and death process over the distribution of time intervals separating the origin of the lineages. In this paper, we show that the resulting distributions can be represented by hypergeometric functions of the second kind. We also provide approximations of these distributions up to the second order, and compare these results to the asymptotic distributions and numerical approximations used in previous studies. For two limiting cases, one with a relatively high rate of lineage origin, one with a low rate, the cumulative probability densities and percentiles are compared to show that the approximations are robust over a wide rane of parameters. It is proposed that the probability density distributions of lineage size may have a number of relevant applications to biological problems such as the coalescence of genetic lineages and in predicting the number of species in living and extinct higher taxa, as these systems are special instances of the underlying process analyzed in this paper."}
{"category": "Math", "title": "Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime", "abstract": "We study the long-time asymptotics of the doubly nonlinear diffusion equation $\\rho_t={div}({|\\nabla\\rho^m|^{p-2}\\nabla\\rho^m})$ in $\\RR^n$, in the range $\\frac{n-p}{n(p-1)}<m>\\frac{n-p+1}{n(p-1)}$ and $1p\\infty$ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an $L^1$-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval $\\frac{2n}{n+1} p\\frac{2n+1}{n+1}$ where a rate of convergence towards self-similarity was still unknown for the $p$-Laplacian equation."}
{"category": "Math", "title": "A new approach to mutual information. II", "abstract": "A new concept of mutual pressure is introduced for potential functions on both continuous and discrete compound spaces via discrete micro-states of permutations, and its relations with the usual pressure and the mutual information are established. This paper is a continuation of the paper of Hiai and Petz in Banach Center Publications, Vol. 78."}
{"category": "Math", "title": "Big rational surfaces", "abstract": "We prove that the Cox ring of a smooth rational surface with big anticanonical class is finitely generated. We classify surfaces of this type that are blow-ups of the plane at distinct points lying on a (possibly reducible) cubic."}
{"category": "Math", "title": "Rigidity versus flexibility for tight confoliations", "abstract": "In \\cite{confol} Y. Eliashberg and W. Thurston gave a definition of tight confoliations. We give an example of a tight confoliation $\\xi$ on $T^3$ violating the Thurston-Bennequin inequalities. This answers a question from \\cite{confol} negatively. Although the tightness of a confoliation does not imply the Thurston-Bennequin inequalities, it is still possible to prove restrictions on homotopy classes of plane fields which contain tight confoliations. The failure of the Thurston-Bennequin inequalities for tight confoliations is due to the presence of overtwisted stars. Overtwisted stars are particular configurations of Legendrian curves which bound a disc with finitely many punctures on the boundary. We prove that the Thurston-Bennequin inequalities hold for tight confoliations without overtwisted stars and that symplectically fillable confoliations do not admit overtwisted stars."}
{"category": "Math", "title": "A CLT for the L^{2} modulus of continuity of Brownian local time", "abstract": "Let $\\{L^{x}_{t} ; (x,t)\\in R^{1}\\times R^{1}_{+}\\}$ denote the local time of Brownian motion and \\[ \\alpha_{t}:=\\int_{-\\infty}^{\\infty} (L^{x}_{t})^{2} dx . \\] Let $\\eta=N(0,1)$ be independent of $\\alpha_{t}$. For each fixed $t$ \\[ {\\int_{-\\infty}^{\\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\\over h^{3/2}} \\stackrel{\\mathcal{L}}{\\to}({64 \\over 3})^{1/2}\\sqrt{\\alpha_{t}} \\eta, \\] as $h\\rar 0$. Equivalently \\[ {\\int_{-\\infty}^{\\infty} (L^{x+1}_{t}- L^{x}_{t})^{2} dx- 4t\\over t^{3/4}} \\stackrel{\\mathcal{L}}{\\to}({64 \\over 3} )^{1/2}\\sqrt{\\alpha_{1}} \\eta, \\] as $t\\rar\\infty$."}
{"category": "Math", "title": "Mathieu's series: inequalities, asymptotics and positive definiteness", "abstract": "Inequalities, asymptotics and, for some specific cases, asymptotical expansions were obtained for generalized Mathieu's series. A connection between inequalities for Mathieu's series and positive definite and completely monotonic functions."}
{"category": "Math", "title": "Computing Inhomogeneous Groebner Bases", "abstract": "In this paper we describe how an idea centered on the concept of self-saturation allows several improvements in the computation of Groebner bases via Buchberger's Algorithm."}
{"category": "Math", "title": "Generalized golden ratios of ternary alphabets", "abstract": "Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in function of the alphabets."}
{"category": "Math", "title": "Positive definite functions in distance geometry", "abstract": "I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a nonnegative linear combination of Gegenbauer polynomials. This fact play a crucial role in Delsarte's method for finding bounds for the density of sphere packings on spheres and Euclidean spaces. One of the most excited applications of Delsarte's method is a solution of the kissing number problem in dimensions 8 and 24. However, 8 and 24 are the only dimensions in which this method gives a precise result. For other dimensions (for instance, three and four) the upper bounds exceed the lower. We have found an extension of the Delsarte method that allows to solve the kissing number problem (as well as the one-sided kissing number problem) in dimensions three and four. In this paper we also will discuss the maximal cardinalities of spherical two-distance sets. Using the so-called polynomial method and Delsarte's method these cardinalities can be determined for all dimensions $n<40$. Recently, were found extensions of Schoenberg's theorem for multivariate positive-definite functions. Using these extensions and semidefinite programming can be improved some upper bounds for spherical codes."}
{"category": "Math", "title": "Constructing subsets of a given packing index in Abelian groups", "abstract": "By definition, the sharp packing index $\\ind_P^\\sharp(A)$ of a subset $A$ of an abelian group $G$ is the smallest cardinal $\\kappa$ such that for any subset $B\\subset G$ of size $|B|\\ge\\kappa$ the family $\\{b+A:b\\in B\\}$ is not disjoint. We prove that an infinite Abelian group $G$ contains a subset $A$ with given index $\\ind_P^\\sharp(A)=\\kappa$ if and only if one of the following conditions holds: (1) $2\\le \\kappa\\le|G|^+$ and $k\\notin \\{3,4\\}$; (2) $\\kappa=3$ and $G$ is not isomorphic to $\\oplus_{i\\in I} \\mathbb{Z}_3$; (3) $\\kappa=4$ and $G$ is not isomorphic to $\\oplus_{i\\in I} \\mathbb{Z}_2$ or to $\\mathbb{Z}_4\\oplus(\\oplus_{i\\in I} \\mathbb{Z}_2)$."}
{"category": "Math", "title": "F-adjunction", "abstract": "In this paper we study singularities defined by the action of Frobenius in characteristic $p > 0$. We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if $X$ is a Gorenstein normal variety then to every normal center of sharp $F$-purity $W \\subseteq X$ such that $X$ is $F$-pure at the generic point of $W$, there exists a canonically defined $\\bQ$-divisor $\\Delta_{W}$ on $W$ satisfying $(K_X)|_W \\sim_{\\bQ} K_{W} + \\Delta_{W}$. Furthermore, the singularities of $X$ near $W$ are \"the same\" as the singularities of $(W, \\Delta_{W})$. As an application, we show that there are finitely many subschemes of a quasi-projective variety that are compatibly split by a given Frobenius splitting. We also reinterpret Fedder's criterion in this context, which has some surprising implications."}
{"category": "Math", "title": "The sphericity of the Phan geometries of type Bn and Cn and the Phan-type theorem of type F4", "abstract": "We adapt and refine methods developed by Abramenko and Devillers--K\\\"ohl--M\\\"uhlherr in order to establish the sphericity of the Phan geometries of type B_n and C_n, and their generalizations. As an application we determine the finiteness length of the unitary form of certain hyperbolic Kac--Moody groups. We also reproduce the finiteness length of the unitary form of the groups Sp_{2n}(GF(q^2)[t,t^{-1}]). Another application is the first published proof of the Phan-type theorem of type F_4. Within the revision of the classification of the finite simple groups this concludes the revision of Phan's theorems and their extension to the non-simply laced diagrams. We also reproduce the Phan-type theorems of types B_n and C_n."}
{"category": "Math", "title": "Bitangential interpolation in generalized Schur classes", "abstract": "Bitangential interpolation problems in the class of matrix valued functions in the generalized Schur class are considered in both the open unit disc and the open right half plane, including problems in which the solutions is not assumed to be holomorphic at the interpolation points. Linear fractional representations of the set of solutions to these problems are presented for invertible and singular Hermitian Pick matrices. These representations make use of a description of the ranges of linear fractional transformations with suitably chosen domains that was developed in a previous paper."}
{"category": "Math", "title": "Modules in resolving subcategories which are free on the punctured spectrum", "abstract": "Let R be a commutative noetherian local ring, and let X be a resolving subcategory of the category of finitely generated R-modules. In this paper, we study modules in X by relating them to modules in X which are free on the punctured spectrum of R. We do this by investigating nonfree loci and establishing an analogue of the notion of a level in a triangulated category which has been introduced by Avramov, Buchweitz, Iyengar and Miller. As an application, we prove a result on the dimension of the nonfree locus of a resolving subcategory having only countably many nonisomorphic indecomposable modules in it, which is a generalization of a theorem of Huneke and Leuschke."}
{"category": "Math", "title": "Non-crossing linked partitions, the partial order << on NC(n), and the S-transform", "abstract": "The paper establishes a connection between two recent combinatorial developments in free probability: the non-crossing linked partitions introduced by Dykema in 2007 to study the S-transform, and the partial order << on NC(n) introduced by Belinschi and Nica in 2008 in order to study relations between free and Boolean probability. More precisely, one has a canonical bijection between NCL(n) (the set of all non-crossing linked partitions of {1, ..., n}) and the set {(p,q) | p,q in NC(n), p<<q}. As a consequence of this bijection, one gets an alternative description of Dykema's formula expressing the moments of a noncommutative random variable a in terms of the coefficients of the reciprocal S-transform 1/S_a. Moreover, due to the Boolean features of <<, this formula can be simplified to a form which resembles the moment-cumulant formula from c-free probability."}
{"category": "Math", "title": "Notes on a minimal set of generators for the radical ideal defining the diagonal locus of $(\\C^2)^n$", "abstract": "We develop several techniques for the study of the radical ideal $I$ defining the diagonal locus of $(\\C^2)^n$. Using these techniques, we give combinatorial construction of generators for $I$ of certain bi-degrees."}
{"category": "Math", "title": "Modified energy for split-step methods applied to the linear Schr\\\"odinger equation", "abstract": "We consider the linear Schr\\\"odinger equation and its discretization by split-step methods where the part corresponding to the Laplace operator is approximated by the midpoint rule. We show that the numerical solution coincides with the exact solution of a modified partial differential equation at each time step. This shows the existence of a modified energy preserved by the numerical scheme. This energy is close to the exact energy if the numerical solution is smooth. As a consequence, we give uniform regularity estimates for the numerical solution over arbitrary long time"}
{"category": "Math", "title": "On simple A-multigraded minimal resolutions", "abstract": "Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$ are the binomials in $I$. We show that for an appropriate choice of bases every $A$-homogeneous minimal free resolution of $R/I$ is simple. We introduce the gcd-complex $D_{gcd}(\\bf b)$ for a degree $\\mathbf{b}\\in \\A$. We show that the homology of $D_{gcd}(\\bf b)$ determines the $i$-Betti numbers of degree $\\bf b$. We discuss the notion of an indispensable complex of $R/I$. We show that the Koszul complex of a complete intersection lattice ideal $I$ is the indispensable resolution of $R/I$ when the $A$-degrees of the elements of the generating $R$-sequence are incomparable."}
{"category": "Math", "title": "Global Existence and Long-Time Asymptotics for Rotating Fluids in a 3D Layer", "abstract": "The Navier-Stokes-Coriolis system is a simple model for rotating fluids, which allows to study the influence of the Coriolis force on the dynamics of three-dimensional flows. In this paper, we consider the NSC system in an infinite three-dimensional layer delimited by two horizontal planes, with periodic boundary conditions in the vertical direction. If the angular velocity parameter is sufficiently large, depending on the initial data, we prove the existence of global, infinite-energy solutions with nonzero circulation number. We also show that these solutions converge toward two-dimensional Lamb-Oseen vortices as time goes to infinity."}
{"category": "Math", "title": "Infinite Hankel Block Matrices, Extremal Problems", "abstract": "In this paper we use the matrix analogue of eigenvalue $\\rho_{min}^{2}$ to formulate and to solve the extremal Nehary problem. When $\\rho_{min}$ is a scalar, our approach coincides with Adamjan-Arov-Krein approach."}
{"category": "Math", "title": "The Stack of Rational Nodal Curves", "abstract": "In this series of three papers we start to investigate the rational Chow ring of the stack consisting of nodal curves of genus 0, in particular we determine completely the rational Chow ring of the substack consisting of curves with at most 3 nodes. In this first paper we construct the stack of rational nodal curves and its stratification by nodes and show that the map from the universal curve to the stack is not representable in the category of schemes."}
{"category": "Math", "title": "Tautological Classes of the Stack of Rational Nodal Curves", "abstract": "This is the second in a series of three papers in which we investigate the rational Chow ring of the stack consisting of nodal curves of genus. Here we define the basic classes: the classes of strata and the Mumford classes."}
{"category": "Math", "title": "On the generalized Scarf complex of lattice ideals", "abstract": "Let $k$ be a field, $ \\mathcal{L}\\subset \\mathbb{Z}^n$ be a lattice such that $\\L\\cap \\mathbb{N}^n=\\{{\\bf 0}\\}$, and $I_\\L\\subset \\Bbbk[x_1,..., x_n]$ the corresponding lattice ideal. We present the generalized Scarf complex of $I_\\L$ and show that it is indispensable in the sense that it is contained in every minimal free resolution of $R/I_\\L$."}
{"category": "Math", "title": "The Chow Ring of the Stack of Rational Curves with at most 3 Nodes", "abstract": "In this paper we explicit the rational Chow ring of the stack consisting of nodal curves of genus 0 with at most 3 nodes: it is a Q-algebra with 10 generators and 11 relations."}
{"category": "Math", "title": "Computing the number of numerical semigroups using generating functions", "abstract": "This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting the number of these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained."}
{"category": "Math", "title": "A classification of homogeneous operators in the Cowen-Douglas class", "abstract": "A complete list of homogeneous operators in the Cowen-Douglas class $B_n(D)$ is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc."}
{"category": "Math", "title": "Asymptotic behaviour of reversible chemical reaction-diffusion equations", "abstract": "We investigate the asymptotic behavior of the a large class of reversible chemical reaction-diffusion equations with the same diffusion. In particular we prove the optimal rate in two cases : when there is no diffusion and in the classical \"two-by-two\" case."}
{"category": "Math", "title": "On the spectrum of $\\bar{X}$-bounded minimal submanifolds", "abstract": "We prove, under a certain boundedness condition at infinity on the $(\\bar{X}^{\\top}, \\bar{X}^{\\bot})$-component of the second fundamental form, the vanishing of the essential spectrum of a complete minimal $\\bar{X}$-bounded and $\\bar{X}$-properly immersed submanifold on a Riemannian manifold endowed with a strongly convex vector field $\\bar{X}$. The same conclusion also holds for any complete minimal $h$-bounded and $h$-properly immersed submanifold that lies in a open set of a Riemannian manifold $\\oM$ supporting a nonnegative strictly convex function $h$. This extends a recent result of Bessa, Jorge and Montenegro on the spectrum of Martin-Morales minimal surfaces. Our proof uses as main tool an extension of Barta's theorem given in \\cite{BM}"}
{"category": "Math", "title": "A topological lens for a measure-preserving system", "abstract": "We introduce a functor which associates to every measure preserving system (X,B,\\mu,T) a topological system (C_2(\\mu),\\tilde{T}) defined on the space of 2-fold couplings of \\mu, called the topological lens of T. We show that often the topological lens \"magnifies\" the basic measure dynamical properties of T in terms of the corresponding topological properties of \\tilde{T}. Some of our main results are as follows: (i) T is weakly mixing iff \\tilde{T} is topologically transitive (iff it is topologically weakly mixing). (ii) T has zero entropy iff \\tilde{T} has zero topological entropy, and T has positive entropy iff \\tilde{T} has infinite topological entropy. (iii) For T a K-system, the topological lens is a P-system (i.e. it is topologically transitive and the set of periodic points is dense; such systems are also called chaotic in the sense of Devaney)."}
{"category": "Math", "title": "Obstructions to Fibering a Manifold", "abstract": "Given a map f: M \\to M of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S^1, these torsion obstructions are identified with the ones due to Farrell. We have changed the exposition according to the comments of the referee and corrected some typos. The paper will appear in Geometriae Dedicata."}
{"category": "Math", "title": "Graph-Chromatic Implicit Relations", "abstract": "A theory about the implication structure in graph coloring is presented. Discovering hidden relations is a crucial activity in every scientific discipline. The development of mathematical models to study and discover such hidden relations is of the most highest interest. The main contribution presented in this work is a model of hidden relations materialized as implicit-edges and implicit-identities in the graph coloring problem, these relations can be interpreted in physical and chemical models as hidden forces, hidden interactions, hidden reactions or hidden variables. Also this theory can be extended to the complete class of NP-complete problems."}
{"category": "Math", "title": "Decentralized Two-Sided Sequential Tests for A Normal Mean", "abstract": "This article is concerned with decentralized sequential testing of a normal mean $\\mu$ with two-sided alternatives. It is assumed that in a single-sensor network system with limited local memory, i.i.d. normal raw observations are observed at the local sensor, and quantized into binary messages that are sent to the fusion center, which makes a final decision between the null hypothesis $H_0: \\mu = 0$ and the alternative hypothesis $H_1: \\mu = \\pm 1.$ We propose a decentralized sequential test using the idea of tandem quantizers (or equivalently, a one-shot feedback). Surprisingly, our proposed test only uses the quantizers of the form $I(X_{n} \\ge \\lambda),$ but it is shown to be asymptotically Bayes. Moreover, by adopting the principle of invariance, we also investigate decentralized invariant tests with the stationary quantizers of the form $I(|X_{n}| > \\lambda),$ and show that $\\lambda = 0.5$ only leads to a suboptimal decentralized invariant sequential test. Numerical simulations are conducted to support our arguments."}
{"category": "Math", "title": "Stability of the Second Order Delay Differential Equations with a Damping Term", "abstract": "For the delay differential equations $$ \\ddot{x}(t) +a(t)\\dot{x}(g(t))+b(t)x(h(t))=0, g(t)\\leq t, h(t)\\leq t, $$ and $$ \\ddot{x}(t) +a(t)\\dot{x}(t)+b(t)x(t)+a_1(t)\\dot{x}(g(t))+b_1(t)x(h(t))=0 $$ explicit exponential stability conditions are obtained."}
{"category": "Math", "title": "Absolute and Delay-Dependent Stability of Equations with a Distributed Delay: a Bridge from Nonlinear Differential to Difference Equations", "abstract": "We study delay-independent stability in nonlinear models with a distributed delay which have a positive equilibrium. Such models frequently occur in population dynamics and other applications. In particular, we construct a relevant difference equation such that its stability implies stability of the equation with a distributed delay and a finite memory. This result is, generally speaking, incorrect for systems with infinite memory. If the relevant difference equation is unstable, we describe the general delay-independent attracting set and also demonstrate that the equation with a distributed delay is stable for small enough delays."}
{"category": "Math", "title": "Gromov-- Witten Invariants of Toric Fibrations", "abstract": "We prove a conjecture of Artur Elezi in a generalized form suggested by Givental. Namely, our main result relates genus-0 Gromov--Witten invariants of a bundle space with such invariants of the base, provided that the fiber is a toric manifold. When the base is the point, a new proof of mirror theorems by A. Givental and by H. Iritani for toric manifolds is obtained."}
{"category": "Math", "title": "Computational birational geometry of minimal rational surfaces", "abstract": "The classification of minimal rational surfaces and the birational links between them by Iskovskikh, Manin and others is a well-known subject in the theory of algebraic surfaces. We explain algorithms that realise links of type II between minimal del Pezzo surfaces, one of the major classes of birational links, and we describe briefly how this fits into a large project to implement the results of Iskovskikh's programme in Magma."}
{"category": "Math", "title": "Seifert cohomology of trees", "abstract": "To every tree we associate a filtered cochain complex. Its cohomology and the corresponding spectral sequence have clear combinatorial description. If a tree is the Dynkin diagram of a simple plane curve singularity, the graded Euler characteristic of this complex coincides with the Alexander polynomial of the link. In this case we also point the relation to the Heegard-Floer homology theory, constructed by P. Ozsvath and Z. Szabo."}
{"category": "Math", "title": "Projecting the Fokker-Planck Equation onto a finite dimensional exponential family", "abstract": "In the present paper we discuss problems concerning evolutions of densities related to Ito diffusions in the framework of the statistical exponential manifold. We develop a rigorous approach to the problem, and we particularize it to the orthogonal projection of the evolution of the density of a diffusion process onto a finite dimensional exponential manifold. It has been shown by D. Brigo (1996) that the projected evolution can always be interpreted as the evolution of the density of a different diffusion process. We give also a compactness result when the dimension of the exponential family increases, as a first step towards a convergence result to be investigated in the future. The infinite dimensional exponential manifold structure introduced by G. Pistone and C. Sempi is used and some examples are given."}
{"category": "Math", "title": "A Better Way to Deal the Cards", "abstract": "This paper considers the effect of riffle shuffling on decks of cards, allowing for some cards to be indistinguishable from other cards. The dual problem of dealing a game with hands, such as bridge or poker, is also considered. The Gilbert-Shannon-Reeds model of card shuffling is used, along with variation distance for measuring how close to uniform a deck has become. The surprising results are that for a deck with only two types of cards (such as red and black), the shuffler can greatly improve the randomness of the deck by insuring that the top and bottom cards are the same before shuffling. And in the case of dealing cards for a game with \"hands\", such as bridge or poker, the normal method of dealing cyclically around the table is very far from optimal. In the case of a well-shuffled bridge deck, changing to another dealing method is as good as doing 3.7 extra shuffles. How the deck is cut in poker affects its randomness as well."}
{"category": "Math", "title": "Mazurkiewicz manifolds and homogeneity", "abstract": "It is proved that no region of a homogeneous locally compact, locally connected metric space can be cut by an $F_\\sigma$-subset of a \"smaller\" dimension. The result applies to different finite or infinite topological dimensions of metrizable spaces."}
{"category": "Math", "title": "Small curvature laminations in hyperbolic 3-manifolds", "abstract": "We show that if $\\mathcal{L}$ is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of $\\mathcal{L}$ are all in the interval $(-\\delta ,\\delta)$ for a fixed $\\delta\\in[0,1)$ and no complimentary region of $\\mathcal{L}$ is an interval bundle over a surface, then each boundary leaf of $\\mathcal{L}$ has a nontrivial fundamental group. We also prove existence of a fixed constant $\\delta_0 > 0$ such that if $\\mathcal{L}$ is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of $\\mathcal{L}$ are all in the interval $(-\\delta_0 ,\\delta_0)$ and no complimentary region of $\\mathcal{L}$ is an interval bundle over a surface, then each boundary leaf of $\\mathcal{L}$ has a noncyclic fundamental group."}
{"category": "Math", "title": "Some field theoretical properties and an application concerning transcendental numbers", "abstract": "For a proper subfield $K$ of $\\QQ$ we show the existence of an algebraic number $\\alpha$ such that no power $\\alpha^n$, $n\\geq 1$, lies in $K$. As an application it is shown that these numbers, multiplied by convenient Gaussian numbers, can be written in the form $P(T)^{Q(T)}$ for some transcendental numbers $T$ where $P$ and $Q$ are arbitrarily prescribed non-constant rational functions over $\\QQ$."}
{"category": "Math", "title": "Counting Bipartite, k-Colored and Directed Acyclic Multi Graphs Through F-nomial coefficients", "abstract": "F-nomial coefficients encompass among others well-known binomial coefficients or Gaussian coefficients that count subsets of finite set and subspaces of finite vector space respectively. Here, the so called F-cobweb tiling sequences N(a) are considered. For such specific sequences a new interpretation with respect to Kwasniewski general combinatorial interpretation of F-nomial coefficients is unearhed. Namely, for tiling sequences F = N(a)$ the F-nomial coefficients are equal to the number of labeled special bipartite multigraphs denoted here as a-multigraphs G(a,n,k). An explicit relation between the number of k-colored a-multigraphs and multi N(a)-nomial coefficients is established. We also prove that the unsigned values of the first row of inversion matrix for N(a) -nomial coefficients considered here are equal to the numbers of directed acyclic a-multigraphs with n nodes."}
{"category": "Math", "title": "Lie bialgebra structures on the Schr\\\"{o}dinger-Virasoro Lie algebra", "abstract": "In this paper we investigate Lie bialgebra structures on the Schr\\\"odinger-Virasoro algebra $\\LL$. Surprisingly, we find out an interesting fact that not all Lie bialgebra structures on the Schr\\\"odinger-Virasoro algebra are triangular coboundary, which is different from the related known results of some Lie algebras related to the Virasoro algebra."}
{"category": "Math", "title": "Combinatorics of double cosets and fundamental domains for the subgroups of the modular group", "abstract": "As noticed by R. Kulkarni, the conjugacy classes of subgroups of the modular group correspond bijectively to bipartite cuboid graphs. We'll explain how to recover the graph corresponding to a subgroup $G$ of $\\mathrm{PSL}_2(\\mathbb{Z})$ from the combinatorics of the right action of $\\mathrm{PSL}_2(\\mathbb{Z})$ on the right cosets $G\\setminus\\mathrm{PSL}_2(\\mathbb{Z})$. This gives a method of constructing nice fundamental domains (which Kulkarni calls \"special polygons\") for the action of $G$ on the upper half plane. For the classical congruence subgroups $\\Gamma_0(N)$, $\\Gamma_1(N)$, $\\Gamma(N)$ etc. the number of operations the method requires is the index times something that grows not faster than a polynomial in $\\log N$. We also give algorithms to locate a given element of the upper half-plane on the fundamental domain and to write a given element of $G$ as a product of independent generators."}
{"category": "Math", "title": "Degeneracy of triality-symmetric morphisms", "abstract": "We define a new symmetry for morphisms of vector bundles, called triality symmetry, and compute Chern class formulas for the degeneracy loci of such morphisms. In an appendix, we show how to canonically associate an octonion algebra bundle to any rank 2 vector bundle."}
{"category": "Math", "title": "Action selectors and Maslov class rigidity", "abstract": "In this paper we detect new restrictions on the Maslov class of displaceable Lagrangian submanifolds of symplectic manifolds which are symplectically aspherical. These restrictions are established using action selectors for Hamiltonian flows. In particular, we construct and utilize a new action selector for the flows of a special class of Hamiltonian functions which arises naturally in the study of Hamiltonian paths which minimize the Hofer length functional."}
{"category": "Math", "title": "A Characterization On Potentially $K_6-C_4$-graphic Sequences", "abstract": "For given a graph $H$, a graphic sequence $\\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\\pi$ containing $H$ as a subgraph. Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ where $H$ is a subgraph of $K_m$. In this paper, we characterize the potentially $K_6-C_4$-graphic sequences. This characterization implies a theorem due to Hu and Lai [7]."}
{"category": "Math", "title": "A Characterization On Potentially $K_{2,5}$-graphic Sequences", "abstract": "For given a graph $H$, a graphic sequence $\\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\\pi$ containing $H$ as a subgraph. Let $K_m-H$ be the graph obtained from $K_m$ by removing the edges set $E(H)$ where $H$ is a subgraph of $K_m$. In this paper, we characterize potentially $K_{2,5}$-graphic sequences. This characterization implies a special case of a theorem due to Yin et al. [26]."}
{"category": "Math", "title": "Semi-infinite TASEP with a Complex Boundary Mechanism", "abstract": "We consider a totally asymmetric exclusion process on the positive half-line. When particles enter in the system according to a Poisson source, Liggett has computed all the limit distributions when the initial distribution has an asymptotic density. In this paper we consider systems for which particles enter at the boundary according to a complex mechanism depending on the current configuration in a finite neighborhood of the origin. For this kind of models, we prove a strong law of large numbers for the number of particles entered in the system at a given time. Our main tool is a new representation of the model as a multi-type particle system with infinitely many particle types."}
{"category": "Math", "title": "Differential Privacy with Compression", "abstract": "This work studies formal utility and privacy guarantees for a simple multiplicative database transformation, where the data are compressed by a random linear or affine transformation, reducing the number of data records substantially, while preserving the number of original input variables. We provide an analysis framework inspired by a recent concept known as differential privacy (Dwork 06). Our goal is to show that, despite the general difficulty of achieving the differential privacy guarantee, it is possible to publish synthetic data that are useful for a number of common statistical learning applications. This includes high dimensional sparse regression (Zhou et al. 07), principal component analysis (PCA), and other statistical measures (Liu et al. 06) based on the covariance of the initial data."}
{"category": "Math", "title": "Two kinds of conditionings for stable L\\'evy processes", "abstract": "Two kinds of conditionings for one-dimensional stable L\\'evy processes are discussed via $ h $-transforms of excursion measures: One is to stay positive, and the other is to avoid the origin."}
{"category": "Math", "title": "Lattice width directions and Minkowski's 3^d-theorem", "abstract": "We show that the number of lattice directions in which a d-dimensional convex body in R^d has minimum width is at most 3^d-1, with equality only for the regular cross-polytope. This is deduced from a sharpened version of the 3^d-theorem due to Hermann Minkowski (22 June 1864--12 January 1909), for which we provide two independent proofs."}
{"category": "Math", "title": "A cautionary tale on the efficiency of some adaptive Monte Carlo schemes", "abstract": "There is a growing interest in the literature for adaptive Markov chain Monte Carlo methods based on sequences of random transition kernels $\\{P_n\\}$ where the kernel $P_n$ is allowed to have an invariant distribution $\\pi_n$ not necessarily equal to the distribution of interest $\\pi$ (target distribution). These algorithms are designed such that as $n\\to\\infty$, $P_n$ converges to $P$, a kernel that has the correct invariant distribution $\\pi$. Typically, $P$ is a kernel with good convergence properties, but one that cannot be directly implemented. It is then expected that the algorithm will inherit the good convergence properties of $P$. The equi-energy sampler of [Ann. Statist. 34 (2006) 1581--1619] is an example of this type of adaptive MCMC. We show in this paper that the asymptotic variance of this type of adaptive MCMC is always at least as large as the asymptotic variance of the Markov chain with transition kernel $P$. We also show by simulation that the difference can be substantial."}
{"category": "Math", "title": "Pseudo-factorials, elliptic functions, and continued fractions", "abstract": "This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a Dixonian and a Weierstrass function, which parametrize the Fermat cubic curve and are relative to a hexagonal lattice. A continued fraction expansion of the ordinary generating function of pseudo-factorials, first discovered empirically, is established here. This article also provides a characterization of the associated orthogonal polynomials, which appear to form a new family of \"elliptic polynomials\", as well as various other properties of pseudo-factorials, including a hexagonal lattice sum expression and elementary congruences."}
{"category": "Math", "title": "The solution infinite horizon noncooperative differential game with nonlinear dynamics without the Hamilton-Jacobi-Bellman equation", "abstract": "For a non-cooperative m-persons differential game, the value functions ofthe various players satisfy a system of Hamilton-Jacobi-Bellman equations.Nashequilibrium solutions in feedback form can be obtained by studying a related system of P.D.E's.A new approach, which is proposed in this paper allows one to construct the feedback optimal control and cost functions J_i(t,x),i=1,...,m directly,i.e.,without any reference to the corresponding Hamilton-Jacobi-Bellman equations."}
{"category": "Math", "title": "Torelli theorem for graphs and tropical curves", "abstract": "Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by contracting all separating edges are 2-isomorphic. In particular, the strong Torelli theorem holds for 3-connected graphs. Next, using the correspondence between compact tropical curves and metric graphs, we prove a tropical Torelli theorem giving necessary and sufficient conditions for two tropical curves to have the same principally polarized tropical Jacobian. Finally we describe some natural posets associated to a graph and prove that they characterize its Delaunay decomposition."}
{"category": "Math", "title": "Mixture of the Riesz distribution with respect to the multivariate Poisson", "abstract": "The aim of this paper is to study the mixture of the Riesz distribution on symmetric matrices with respect to the multivariate Poisson distribution. We show, in particular, that this distribution is related to the modified Bessel function of the first kind. We also study the generated natural exponential family. We determine the domain of the means and the variance function of this family."}
{"category": "Math", "title": "Reduktionssysteme zur Berechnung einer Aufl\\\"osung der orthogonalen freien Quantengruppen $A_o(n)$", "abstract": "F\\\"ur die freien orthogonalen Quantengruppen $A_o(n)$ wird ein vollst\\\"a ndiges Reduktionssystem angegeben und verifiziert. F\\\"ur den Fall $n = 2$ wird ein endlicher Automat angegeben, der s\\\"amtliche der Basiselemente findet. Weiterhin wird eine Basis f\\\"ur die f\\\"ur die Kerne einer freien Aufl\\\"osung von $A_o(n)$ als Bimodul bewiesen. Abschlie{\\ss}end wird mit der nun verifizierten Aufl\\\"osung die Homologie von $A_o(n)$ explizit berechnet."}
{"category": "Math", "title": "Invariants of Lie algebras extended over commutative algebras without unit", "abstract": "We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras."}
{"category": "Math", "title": "Variation of quasiconformal mappings on lines", "abstract": "We obtain improved regularity of homeomorphic solutions of the reduced Beltrami equation, as compared to the standard Beltrami equation. Such an improvement is not possible in terms of Holder or Sobolev regularity; instead, our results concern the generalized variation of restrictions to lines. Specifically, we prove that the restriction to any line segment has finite p-variation for all p>1 but not necessarily for p=1."}
{"category": "Math", "title": "Ergodicity of Mapping Class Group Actions on SU(2)-character varieties", "abstract": "Let S be a compact orientable surface with genus g and n boundary components d_1,...,d_n. Let b = (b_1, ..., b_n) where b_n lies in [-2,2]. Then the mapping class group of S acts on the relative SU(2)-character variety X comprising conjugacy classes of representations f of the fundamental group F of S, where trace f(d_i) = b_i. This action preserves a symplectic structure on the open dense smooth submanifold of X. corresponding to irreducible representations. This subset has full measure and is connected. In this note we use the symplectic geometry of this space to give a new proof that this action is ergodic."}
{"category": "Math", "title": "The Logarithmic Sobolev Inequality in Infinite dimensions for Unbounded Spin Systems on the Lattice with non Quadratic Interactions", "abstract": "We are interested in the Logarithmic Sobolev Inequality for the infinite volume Gibbs measure with no quadratic interactions. We consider unbounded spin systems on the one dimensional Lattice with interactions that go beyond the usual strict convexity and without uniform bound on the second derivative. We assume that the one dimensional single-site measure with boundaries satisfies the Log-Sobolev inequality uniformly on the boundary conditions and we determine conditions under which the Log-Sobolev Inequality can be extended to the infinite volume Gibbs measure."}
{"category": "Math", "title": "Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces", "abstract": "The conjugacy class of a generic unimodular 2 by 2 complex matrix is determined by its trace, which may be an arbitrary complex number. In the nineteenth century, it was known that a generic pair (X,Y) of such pairs is determined up to conjugacy by the triple of traces (tr(X),tr(Y),tr(XY), which may be an arbitary element of C^3. This paper gives an elementary and detailed proof of this fact, which was published by Vogt in 1889. The folk theorem describing the extension of a representation to a representation of the index-two supergroup which is a free product of three groups of order two, is described in detail, and related to hyperbolic geometry. When n > 2, the classification of conjugacy-classes of n-tuples in SL(2,C) is more complicated. We describe it in detail when n= 3. The deformation spaces of hyperbolic structures on some simple surfaces S whose fundamental group is free of rank two or three are computed in trace coordinates. (We only consider the two orientable surfaces whose fundamental group has rank 3.)"}
{"category": "Math", "title": "Sub-Riemannian geometry of parallelizable spheres", "abstract": "The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere $S^3$ originating from different constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising through its right Lie group action over itself, the one inherited from the natural complex structure of the open unit ball in $\\comp^2$ and the geometry that appears when considering the Hopf map as a principal bundle. The main result of this comparison is that in fact those three structures coincide. In the second place, we present two bracket generating distributions for the seven dimensional sphere $S^7$ of step 2 with ranks 6 and 4. These yield to sub-Riemannian structures for $S^7$ that are not present in the literature until now. One of the distributions can be obtained by considering the CR geometry of $S^7$ inherited from the natural complex structure of the open unit ball in $\\comp^4$. The other one originates from the quaternionic analogous of the Hopf map."}
{"category": "Math", "title": "On Freiman's Theorem in Nilpotent Groups", "abstract": "We generalize a result of Tao which extends Freiman's theorem to the Heisenberg group. We extend it to simply connected nilpotent Lie groups of arbitrary step."}
{"category": "Math", "title": "Qualitative Criterion for Interception in a Pursuit/Evasion Game", "abstract": "A qualitative account is given of a differential pursuit/evasion game. A criterion for the existence of an intercept solution is obtained using future cones that contain all attainable trajectories of target or interceptor originating from an initial position. A sufficient and necessary conditon that an opportunity to intercept always exist is that, after some initial time, the future cone of the target be contained within the future cone of the interceptor. The sufficient condition may be regarded as a kind of Nash equillibrium."}
{"category": "Math", "title": "Subproduct systems", "abstract": "The notion of a subproduct system, a generalization of that of a product system, is introduced. We show that there is an essentially 1 to 1 correspondence between cp-semigroups and pairs (X,T) where X is a subproduct system and T is an injective subproduct system representation. A similar statement holds for subproduct systems and units of subproduct systems. This correspondence is used as a framework for developing a dilation theory for cp-semigroups. Results we obtain: (i) a *-automorphic dilation to semigroups of *-endomorphisms over quite general semigroups; (ii) necessary and sufficient conditions for a semigroup of CP maps to have a *-endomorphic dilation; (iii) an analogue of Parrot's example of three contractions with no isometric dilation, that is, an example of three commuting, contractive normal CP maps on B(H) that admit no *-endomorphic dilation (thereby solving an open problem raised by Bhat in 1998). Special attention is given to subproduct systems over the semigroup N, which are used as a framework for studying tuples of operators satisfying homogeneous polynomial relations, and the operator algebras they generate. As applications we obtain a noncommutative (projective) Nullstellansatz, a model for tuples of operators subject to homogeneous polynomial relations, a complete description of all representations of Matsumoto's subshift C*-algebra when the subshift is of finite type, and a classification of certain operator algebras -- including an interesting non-selfadjoint generalization of the noncommutative tori."}
{"category": "Math", "title": "The Golod Shafarevich counter-example without Hilbert series", "abstract": "Let $F$ be an arbitrary field. The Golod-Shafarevich example of a finitely generated nil $F$-algebra which is infinite dimensional -- is revisited. Here we offer a rather elementary treatment of that example, in which induction replaces Hilbert series techniques. This note also contains a detailed exposition of the construction of that example."}
{"category": "Math", "title": "Manifold approximation of set-valued functions", "abstract": "A new continuity for set-valued functions is introduced, and an existence theorem is proved for such continuous set-valued functions."}
{"category": "Math", "title": "A bracket polynomial for graphs. II. Links, Euler circuits and marked graphs", "abstract": "Let $D$ be an oriented classical or virtual link diagram with directed universe $\\vec{U}$. Let $C$ denote a set of directed Euler circuits, one in each connected component of $U$. There is then an associated looped interlacement graph $L(D,C)$ whose construction involves very little geometric information about the way $D$ is drawn in the plane; consequently $L(D,C)$ is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. $L(D,C)$ is determined by three things: the structure of $\\vec{U}$ as a 2-in, 2-out digraph, the distinction between crossings that make a positive contribution to the writhe and those that make a negative contribution, and the relationship between $C$ and the directed circuits in $\\vec{U}$ arising from the link components; this relationship is indicated by marking the vertices where $C$ does not follow the incident link component(s). We introduce a bracket polynomial for arbitrary marked graphs, defined using either a formula involving matrix nullities or a recursion involving the local complement and pivot operations; the marked-graph bracket of $L(D,C)$ is the same as the Kauffman bracket of $D$. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and non-classical virtual links."}
{"category": "Math", "title": "Contextual Epistemic Logic", "abstract": "One of the highlights of recent informal epistemology is its growing theoretical emphasis upon various notions of context. The present paper addresses the connections between knowledge and context within a formal approach. To this end, a \"contextual epistemic logic\", CEL, is proposed, which consists of an extension of standard S5 epistemic modal logic with appropriate reduction axioms to deal with an extra contextual operator. We describe the axiomatics and supply both a Kripkean and a dialogical semantics for CEL. An illustration of how it may fruitfully be applied to informal epistemological matters is provided."}
{"category": "Math", "title": "The maximal operator associated to a non-symmetric Ornstein-Uhlenbeck semigroup", "abstract": "Let (H_t) be the Ornstein-Uhlenbeck semigroup on R^d with covariance matrix I and drift matrix \\lambda(R-I), where \\lambda>0 and R is a skew-adjoint matrix and denote by \\gamma_\\infty the invariant measure for (H_t). Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L^2(\\gamma_\\infty). We prove that if the matrix R generates a one-parameter group of periodic rotations then the maximal operator associated to the semigroup is of weak type 1 with respect to the invariant measure. We also prove that the maximal operator associated to an arbitrary normal Ornstein-Uhlenbeck semigroup is bounded on L^p(\\gamma_\\infty) if and only if 1<p\\le \\infty."}
{"category": "Math", "title": "An inverse problem in number theory and geometric group theory", "abstract": "This paper describes a new link between combinatorial number theory and geometry. The main result states that A is a finite set of relatively prime positive integers if and only if A = (K-K) \\cap N, where K is a compact set of real numbers such that for every real number x there exists y in K with x \\equiv y mod 1. In one direction, given a finite set A of relatively prime positive integers, the proof constructs an appropriate compact set K such that A = (K-K) \\cap N. In the other direction, a strong form of a fundamental theorem in geometric group theory is applied to prove that (K-K)\\cap N is a finite set of relatively prime positive integers if K satisfies the appropriate geometrical conditions. Some related results and open problems are also discussed."}
{"category": "Math", "title": "Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast", "abstract": "We consider divergence-form scalar elliptic equations and vectorial equations for elasticity with rough ($L^\\infty(\\Omega)$, $\\Omega \\subset \\R^d$) coefficients $a(x)$ that, in particular, model media with non-separated scales and high contrast in material properties. We define the flux norm as the $L^2$ norm of the potential part of the fluxes of solutions, which is equivalent to the usual $H^1$-norm. We show that in the flux norm, the error associated with approximating, in a properly defined finite-dimensional space, the set of solutions of the aforementioned PDEs with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard space (e.g., piecewise polynomial). We refer to this property as the {\\it transfer property}. A simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates. This transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations. The proofs of these homogenization results are based on a new class of elliptic inequalities which play the same role in our approach as the div-curl lemma in classical homogenization."}
{"category": "Math", "title": "Arc distance equals level number", "abstract": "A knot K in 1-bridge position with respect to a genus-g Heegaard surface in a 3-manifold can be moved by isotopy through knots in 1-bridge position until it lies in a union of n parallel genus-g surfaces tubed together by n-1 straight tubes, with K intersecting each tube in two arcs connecting the ends. We prove that the minimum n for which this is possible is equal to a Hempel-type distance invariant defined using an arc complex of the two holed genus-g surface"}
{"category": "Math", "title": "Generalised regular variation of arbitrary order", "abstract": "Let $f$ be a measurable, real function defined in a neighbourhood of infinity. The function $f$ is said to be of generalised regular variation if there exist functions $h \\not\\equiv 0$ and $g > 0$ such that $f(xt) - f(t) = h(x) g(t) + o(g(t))$ as $t \\to \\infty$ for all $x \\in (0, \\infty)$. Zooming in on the remainder term $o(g(t))$ leads eventually to a relation of the form $f(xt) - f(t) = h_1(x) g_1(t) + ... + h_n(x) g_n(t) + o(g_n(t))$, each $g_i$ being of smaller order than its predecessor $g_{i-1}$. The function $f$ is said to be generalised regularly varying of order $n$ with rate vector $\\g = (g_1, >..., g_n)'$. Under general assumptions, $\\g$ itself must be regularly varying in the sense that $\\g(xt) = x^{\\B} \\g(t) + o(g_n(t))$ for some upper triangular matrix $\\B \\in \\RR^{n \\times n}$, and the vector of limit functions $\\h = (h_1, >..., h_n)$ is of the form $\\h(x) = \\c \\int_1^x u^\\B u^{-1} \\du$ for some row vector $\\c \\in \\RR^{1 \\times n}$. The usual results in the theory of regular variation such as uniform convergence and Potter bounds continue to hold. An interesting special case arises when all the rate functions $g_i$ are slowly varying, yielding $\\Pi$-variation of order $n$, the canonical case being that $\\B$ is equivalent to a single Jordan block with zero diagonal. The theory is applied to a long list of special functions."}
{"category": "Math", "title": "$L$-complete Hopf algebroids and their comodules", "abstract": "We investigate Hopf algebroids in the category of $L$-complete modules over a commutative Noetherian regular complete local ring. The main examples are provided by the Hopf algebroids associated to Lubin-Tate spectra in the K(n)-local stable homotopy category and we show that these have Landweber filtrations for all finitely generated discrete modules. Along the way we investigate the canonical Hopf algebras associated to Hopf algebroids over fields and introduce a notion of unipotent Hopf algebroid generalising that for Hopf algebras. In two appendices we continue the discussion of the connections with twisted group rings, and expand on a result of Hovey on the non-exactness of coproducts of L-complete modules."}
{"category": "Math", "title": "Scalar Curvature Behavior for Finite Time Singularity of K\\\"ahler-Ricci Flow", "abstract": "In this short paper, we show that K\\\"ahler-Ricci flows over closed manifolds would have scalar curvature blown-up for finite time singularity. Certain control of the blowing-up is achieved with some mild assumption."}
{"category": "Math", "title": "Semi - Riemannian Geometry with Nonholonomic Constraints", "abstract": "In the present article the geometry of semi-Riemannian manifolds with nonholonomic constraints is studied. These manifolds can be considered as analogues to the sub-Riemannian manifolds, where the positively definite metric is substituted by a nondegenerate metric. To study properties of the exponential map the Christoffel symbols and other differential operators were introduced. We study solutions of the Hamiltonian system and their projections into the underlying manifold. The explicit formulae were found for a specific example of a semi-Riemannian manifold with nonholonomic constraints."}
{"category": "Math", "title": "Quantum-sl(2) action on a divided-power quantum plane at even roots of unity", "abstract": "We describe a nonstandard version of the quantum plane, the one in the basis of divided powers at an even root of unity $q=e^{i\\pi/p}$. It can be regarded as an extension of the \"nearly commutative\" algebra $C[X,Y]$ with $X Y =(-1)^p Y X$ by nilpotents. For this quantum plane, we construct a Wess--Zumino-type de Rham complex and find its decomposition into representations of the $2p^3$-dimensional quantum group $U_q sl(2)$ and its Lusztig extension; the quantum group action is also defined on the algebra of quantum differential operators on the quantum plane."}
{"category": "Math", "title": "The tau constant and the edge connectivity of a metrized graph", "abstract": "The tau constant is an important invariant of a metrized graph, and it has applications in arithmetic properties of curves. We show how the tau constant of a metrized graph changes under successive edge contractions and deletions. We discover identities which we call \"contraction\", \"deletion\", and \"contraction-deletion\" identities on a metrized graph. By establishing a lower bound for the tau constant in terms of the edge connectivity, we prove that Baker and Rumely's lower bound conjecture on the tau constant holds for metrized graphs with edge connectivity 5 or more. We show that proving this conjecture for 3-regular graphs is enough to prove it for all graphs."}
{"category": "Math", "title": "The Logarithmic Sobolev Inequality for Gibbs measures on infinite product of Heisenberg groups", "abstract": "We are interested in the $q$ Logarithmic Sobolev inequality for probability measures on the infinite product of Heisenberg groups. We assume that the one site boundary free measure satisfies either a $q$ Log-Sobolev inequality or a U-Bound inequality, and we determine conditions so that the infinite dimensional Gibbs measure satisfies a $q$ Log-Sobolev inequality."}
{"category": "Math", "title": "Models of PA: Standard Systems without Minimal Ultrafilters", "abstract": "We prove that bold N, the standard model of arithmetic, has an uncountable elementary extension N such that there is no ultrafilter on the Boolean Algebra of subsets of bold N represented in N which is minimal (i.e. as in Rudin-Keisler order for partitions represented in N)."}
{"category": "Math", "title": "The Calabi-Yau equation, symplectic forms and almost complex structures", "abstract": "We discuss a conjecture of Donaldson on a version of Yau's Theorem for symplectic forms with compatible almost complex structures and survey some recent progress on this problem. We also speculate on some future possible directions, and use a monotonicity formula for harmonic maps to obtain a new local estimate in the setting of Donaldson's conjecture."}